LMFDB - Embedded newform 128.5.f.b.31.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,5,Mod(31,128)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("128.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.f (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.2313552747$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 $ x^14 - 4x^13 + 15x^12 - 34x^11 + 62x^10 - 312x^9 + 1432x^8 - 4960x^7 + 11456x^6 - 19968x^5 + 31744x^4 - 139264x^3 + 491520x^2 - 1048576*x + 2097152
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{42} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.4
Root			$-2.40693 - 1.48549i$ of defining polynomial
Character	$\chi$	$=$	128.31
Dual form			128.5.f.b.95.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0461995 + 0.0461995i) q^{3} +(8.04297 + 8.04297i) q^{5} -49.8797 q^{7} -80.9957i q^{9} +(84.2573 - 84.2573i) q^{11} +(-19.4838 + 19.4838i) q^{13} +0.743162i q^{15} +437.855 q^{17} +(-349.021 - 349.021i) q^{19} +(-2.30442 - 2.30442i) q^{21} -404.840 q^{23} -495.621i q^{25} +(7.48412 - 7.48412i) q^{27} +(1031.65 - 1031.65i) q^{29} -1506.15i q^{31} +7.78529 q^{33} +(-401.181 - 401.181i) q^{35} +(434.262 + 434.262i) q^{37} -1.80028 q^{39} +696.847i q^{41} +(-917.612 + 917.612i) q^{43} +(651.446 - 651.446i) q^{45} +111.917i q^{47} +86.9810 q^{49} +(20.2287 + 20.2287i) q^{51} +(-1041.19 - 1041.19i) q^{53} +1355.36 q^{55} -32.2492i q^{57} +(1711.60 - 1711.60i) q^{59} +(-3711.24 + 3711.24i) q^{61} +4040.04i q^{63} -313.415 q^{65} +(1854.18 + 1854.18i) q^{67} +(-18.7034 - 18.7034i) q^{69} -1161.89 q^{71} -905.295i q^{73} +(22.8975 - 22.8975i) q^{75} +(-4202.72 + 4202.72i) q^{77} +5869.63i q^{79} -6559.96 q^{81} +(7560.06 + 7560.06i) q^{83} +(3521.65 + 3521.65i) q^{85} +95.3236 q^{87} +6439.80i q^{89} +(971.844 - 971.844i) q^{91} +(69.5835 - 69.5835i) q^{93} -5614.33i q^{95} -413.032 q^{97} +(-6824.48 - 6824.48i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{3} + 2 q^{5} - 4 q^{7} - 94 q^{11} + 2 q^{13} - 4 q^{17} + 706 q^{19} + 164 q^{21} + 1148 q^{23} + 1664 q^{27} - 862 q^{29} - 4 q^{33} - 1340 q^{35} + 1826 q^{37} + 2684 q^{39} - 1694 q^{43}+ \cdots - 49214 q^{99}+O(q^{100}) $ 14 * q + 2 * q^3 + 2 * q^5 - 4 * q^7 - 94 * q^11 + 2 * q^13 - 4 * q^17 + 706 * q^19 + 164 * q^21 + 1148 * q^23 + 1664 * q^27 - 862 * q^29 - 4 * q^33 - 1340 * q^35 + 1826 * q^37 + 2684 * q^39 - 1694 * q^43 - 1410 * q^45 + 682 * q^49 + 3012 * q^51 + 482 * q^53 - 11780 * q^55 + 2786 * q^59 + 3778 * q^61 - 2020 * q^65 - 7998 * q^67 - 9628 * q^69 + 19964 * q^71 - 17570 * q^75 + 9508 * q^77 + 1454 * q^81 + 17282 * q^83 - 9948 * q^85 - 49284 * q^87 + 28036 * q^91 - 8896 * q^93 - 4 * q^97 - 49214 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.0461995	+	0.0461995i	0.00513328	+	0.00513328i	0.709669	−	0.704535i	$-0.248845\pi$
$3$	0.0461995	+	0.0461995i	0.00513328	+	0.00513328i	−0.704535	+	0.709669i	$0.748845\pi$
$4$		0			0
$5$	8.04297	+	8.04297i	0.321719	+	0.321719i	0.849426	−	0.527707i	$-0.176948\pi$
$5$	8.04297	+	8.04297i	0.321719	+	0.321719i	−0.527707	+	0.849426i	$0.676948\pi$
$6$		0			0
$7$	−49.8797			−1.01795			−0.508976	−	0.860781i	$-0.669976\pi$
$7$	−49.8797			−1.01795			−0.508976	+	0.860781i	$0.669976\pi$
$8$		0			0
$9$		−	80.9957i		−	0.999947i
$10$		0			0
$11$	84.2573	−	84.2573i	0.696341	−	0.696341i	−0.267278	−	0.963619i	$-0.586124\pi$
$11$	84.2573	−	84.2573i	0.696341	−	0.696341i	0.963619	+	0.267278i	$0.0861243\pi$
$12$		0			0
$13$	−19.4838	+	19.4838i	−0.115289	+	0.115289i	−0.762398	−	0.647109i	$-0.775978\pi$
$13$	−19.4838	+	19.4838i	−0.115289	+	0.115289i	0.647109	+	0.762398i	$0.275978\pi$
$14$		0			0
$15$			0.743162i			0.00330294i
$16$		0			0
$17$	437.855			1.51507			0.757535	−	0.652795i	$-0.226404\pi$
$17$	437.855			1.51507			0.757535	+	0.652795i	$0.226404\pi$
$18$		0			0
$19$	−349.021	−	349.021i	−0.966817	−	0.966817i	0.0326494	−	0.999467i	$-0.489606\pi$
$19$	−349.021	−	349.021i	−0.966817	−	0.966817i	−0.999467	+	0.0326494i	$0.989606\pi$
$20$		0			0
$21$	−2.30442	−	2.30442i	−0.00522543	−	0.00522543i
$22$		0			0
$23$	−404.840			−0.765293			−0.382647	−	0.923895i	$-0.624987\pi$
$23$	−404.840			−0.765293			−0.382647	+	0.923895i	$0.624987\pi$
$24$		0			0
$25$		−	495.621i		−	0.792994i
$26$		0			0
$27$	7.48412	−	7.48412i	0.0102663	−	0.0102663i
$28$		0			0
$29$	1031.65	−	1031.65i	1.22670	−	1.22670i	0.261490	−	0.965206i	$-0.415786\pi$
$29$	1031.65	−	1031.65i	1.22670	−	1.22670i	0.965206	−	0.261490i	$-0.0842139\pi$
$30$		0			0
$31$		−	1506.15i		−	1.56728i	−0.621217	−	0.783638i	$-0.713362\pi$
$31$		−	1506.15i		−	1.56728i	0.621217	−	0.783638i	$-0.286638\pi$
$32$		0			0
$33$	7.78529			0.00714903
$34$		0			0
$35$	−401.181	−	401.181i	−0.327494	−	0.327494i
$36$		0			0
$37$	434.262	+	434.262i	0.317211	+	0.317211i	0.847695	−	0.530484i	$-0.177990\pi$
$37$	434.262	+	434.262i	0.317211	+	0.317211i	−0.530484	+	0.847695i	$0.677990\pi$
$38$		0			0
$39$	−1.80028			−0.00118362
$40$		0			0
$41$			696.847i			0.414543i	0.978283	+	0.207272i	$0.0664584\pi$
$41$			696.847i			0.414543i	−0.978283	+	0.207272i	$0.933542\pi$
$42$		0			0
$43$	−917.612	+	917.612i	−0.496275	+	0.496275i	−0.910276	−	0.414002i	$-0.864131\pi$
$43$	−917.612	+	917.612i	−0.496275	+	0.496275i	0.414002	+	0.910276i	$0.364131\pi$
$44$		0			0
$45$	651.446	−	651.446i	0.321702	−	0.321702i
$46$		0			0
$47$			111.917i			0.0506641i	0.999679	+	0.0253321i	$0.00806431\pi$
$47$			111.917i			0.0506641i	−0.999679	+	0.0253321i	$0.991936\pi$
$48$		0			0
$49$	86.9810			0.0362270
$50$		0			0
$51$	20.2287	+	20.2287i	0.00777728	+	0.00777728i
$52$		0			0
$53$	−1041.19	−	1041.19i	−0.370663	−	0.370663i	0.497056	−	0.867719i	$-0.334414\pi$
$53$	−1041.19	−	1041.19i	−0.370663	−	0.370663i	−0.867719	+	0.497056i	$0.834414\pi$
$54$		0			0
$55$	1355.36			0.448052
$56$		0			0
$57$		−	32.2492i		−	0.00992589i
$58$		0			0
$59$	1711.60	−	1711.60i	0.491697	−	0.491697i	−0.417144	−	0.908841i	$-0.636969\pi$
$59$	1711.60	−	1711.60i	0.491697	−	0.491697i	0.908841	+	0.417144i	$0.136969\pi$
$60$		0			0
$61$	−3711.24	+	3711.24i	−0.997376	+	0.997376i	−0.999997	−	0.00262076i	$-0.999166\pi$
$61$	−3711.24	+	3711.24i	−0.997376	+	0.997376i	0.00262076	+	0.999997i	$0.499166\pi$
$62$		0			0
$63$			4040.04i			1.01790i
$64$		0			0
$65$	−313.415			−0.0741810
$66$		0			0
$67$	1854.18	+	1854.18i	0.413049	+	0.413049i	0.882799	−	0.469750i	$-0.155656\pi$
$67$	1854.18	+	1854.18i	0.413049	+	0.413049i	−0.469750	+	0.882799i	$0.655656\pi$
$68$		0			0
$69$	−18.7034	−	18.7034i	−0.00392846	−	0.00392846i
$70$		0			0
$71$	−1161.89			−0.230489			−0.115244	−	0.993337i	$-0.536765\pi$
$71$	−1161.89			−0.230489			−0.115244	+	0.993337i	$0.536765\pi$
$72$		0			0
$73$		−	905.295i		−	0.169881i	−0.996386	−	0.0849404i	$-0.972930\pi$
$73$		−	905.295i		−	0.169881i	0.996386	−	0.0849404i	$-0.0270700\pi$
$74$		0			0
$75$	22.8975	−	22.8975i	0.00407066	−	0.00407066i
$76$		0			0
$77$	−4202.72	+	4202.72i	−0.708842	+	0.708842i
$78$		0			0
$79$			5869.63i			0.940495i	0.882535	+	0.470247i	$0.155835\pi$
$79$			5869.63i			0.940495i	−0.882535	+	0.470247i	$0.844165\pi$
$80$		0			0
$81$	−6559.96			−0.999842
$82$		0			0
$83$	7560.06	+	7560.06i	1.09741	+	1.09741i	0.994713	+	0.102698i	$0.0327476\pi$
$83$	7560.06	+	7560.06i	1.09741	+	1.09741i	0.102698	+	0.994713i	$0.467252\pi$
$84$		0			0
$85$	3521.65	+	3521.65i	0.487426	+	0.487426i
$86$		0			0
$87$	95.3236			0.0125939
$88$		0			0
$89$			6439.80i			0.813004i	0.913650	+	0.406502i	$0.133252\pi$
$89$			6439.80i			0.813004i	−0.913650	+	0.406502i	$0.866748\pi$
$90$		0			0
$91$	971.844	−	971.844i	0.117358	−	0.117358i
$92$		0			0
$93$	69.5835	−	69.5835i	0.00804527	−	0.00804527i
$94$		0			0
$95$		−	5614.33i		−	0.622087i
$96$		0			0
$97$	−413.032			−0.0438976			−0.0219488	−	0.999759i	$-0.506987\pi$
$97$	−413.032			−0.0438976			−0.0219488	+	0.999759i	$0.506987\pi$
$98$		0			0
$99$	−6824.48	−	6824.48i	−0.696304	−	0.696304i
$100$		0			0
$101$	−8460.63	−	8460.63i	−0.829392	−	0.829392i	0.158041	−	0.987433i	$-0.449482\pi$
$101$	−8460.63	−	8460.63i	−0.829392	−	0.829392i	−0.987433	+	0.158041i	$0.949482\pi$
$102$		0			0
$103$	17007.9			1.60316			0.801578	−	0.597891i	$-0.203994\pi$
$103$	17007.9			1.60316			0.801578	+	0.597891i	$0.203994\pi$
$104$		0			0
$105$		−	37.0687i		−	0.00336224i
$106$		0			0
$107$	−9368.65	+	9368.65i	−0.818294	+	0.818294i	−0.985861	−	0.167567i	$-0.946409\pi$
$107$	−9368.65	+	9368.65i	−0.818294	+	0.818294i	0.167567	+	0.985861i	$0.446409\pi$
$108$		0			0
$109$	8308.73	−	8308.73i	0.699329	−	0.699329i	−0.264937	−	0.964266i	$-0.585351\pi$
$109$	8308.73	−	8308.73i	0.699329	−	0.699329i	0.964266	+	0.264937i	$0.0853510\pi$
$110$		0			0
$111$			40.1254i			0.00325667i
$112$		0			0
$113$	5814.63			0.455371			0.227685	−	0.973735i	$-0.426884\pi$
$113$	5814.63			0.455371			0.227685	+	0.973735i	$0.426884\pi$
$114$		0			0
$115$	−3256.12	−	3256.12i	−0.246209	−	0.246209i
$116$		0			0
$117$	1578.10	+	1578.10i	0.115283	+	0.115283i
$118$		0			0
$119$	−21840.1			−1.54227
$120$		0			0
$121$			442.428i			0.0302185i
$122$		0			0
$123$	−32.1940	+	32.1940i	−0.00212797	+	0.00212797i
$124$		0			0
$125$	9013.12	−	9013.12i	0.576840	−	0.576840i
$126$		0			0
$127$		−	20367.1i		−	1.26276i	−0.775472	−	0.631382i	$-0.782488\pi$
$127$		−	20367.1i		−	1.26276i	0.775472	−	0.631382i	$-0.217512\pi$
$128$		0			0
$129$	−84.7864			−0.00509503
$130$		0			0
$131$	−4414.52	−	4414.52i	−0.257242	−	0.257242i	0.566690	−	0.823931i	$-0.308224\pi$
$131$	−4414.52	−	4414.52i	−0.257242	−	0.257242i	−0.823931	+	0.566690i	$0.808224\pi$
$132$		0			0
$133$	17409.1	+	17409.1i	0.984174	+	0.984174i
$134$		0			0
$135$	120.389			0.00660571
$136$		0			0
$137$		−	11018.7i		−	0.587067i	−0.955949	−	0.293533i	$-0.905169\pi$
$137$		−	11018.7i		−	0.587067i	0.955949	−	0.293533i	$-0.0948312\pi$
$138$		0			0
$139$	14957.2	−	14957.2i	0.774140	−	0.774140i	−0.204688	−	0.978827i	$-0.565618\pi$
$139$	14957.2	−	14957.2i	0.774140	−	0.774140i	0.978827	+	0.204688i	$0.0656178\pi$
$140$		0			0
$141$	−5.17051	+	5.17051i	−0.000260073	+	0.000260073i
$142$		0			0
$143$			3283.30i			0.160560i
$144$		0			0
$145$	16595.1			0.789302
$146$		0			0
$147$	4.01848	+	4.01848i	0.000185963	+	0.000185963i
$148$		0			0
$149$	−15393.9	−	15393.9i	−0.693388	−	0.693388i	0.269588	−	0.962976i	$-0.413112\pi$
$149$	−15393.9	−	15393.9i	−0.693388	−	0.693388i	−0.962976	+	0.269588i	$0.913112\pi$
$150$		0			0
$151$	−16971.9			−0.744347			−0.372174	−	0.928163i	$-0.621387\pi$
$151$	−16971.9			−0.744347			−0.372174	+	0.928163i	$0.621387\pi$
$152$		0			0
$153$		−	35464.4i		−	1.51499i
$154$		0			0
$155$	12113.9	−	12113.9i	0.504222	−	0.504222i
$156$		0			0
$157$	−1167.73	+	1167.73i	−0.0473744	+	0.0473744i	−0.730397	−	0.683023i	$-0.760665\pi$
$157$	−1167.73	+	1167.73i	−0.0473744	+	0.0473744i	0.683023	+	0.730397i	$0.260665\pi$
$158$		0			0
$159$		−	96.2051i		−	0.00380543i
$160$		0			0
$161$	20193.3			0.779032
$162$		0			0
$163$	−28076.2	−	28076.2i	−1.05673	−	1.05673i	−0.998291	−	0.0584383i	$-0.981388\pi$
$163$	−28076.2	−	28076.2i	−1.05673	−	1.05673i	−0.0584383	−	0.998291i	$-0.518612\pi$
$164$		0			0
$165$	62.6168	+	62.6168i	0.00229998	+	0.00229998i
$166$		0			0
$167$	−2929.82			−0.105053			−0.0525264	−	0.998620i	$-0.516727\pi$
$167$	−2929.82			−0.105053			−0.0525264	+	0.998620i	$0.516727\pi$
$168$		0			0
$169$			27801.8i			0.973417i
$170$		0			0
$171$	−28269.2	+	28269.2i	−0.966767	+	0.966767i
$172$		0			0
$173$	−8560.97	+	8560.97i	−0.286043	+	0.286043i	−0.835513	−	0.549470i	$-0.814829\pi$
$173$	−8560.97	+	8560.97i	−0.286043	+	0.286043i	0.549470	+	0.835513i	$0.314829\pi$
$174$		0			0
$175$			24721.4i			0.807230i
$176$		0			0
$177$	158.150			0.00504804
$178$		0			0
$179$	24420.1	+	24420.1i	0.762153	+	0.762153i	0.976711	−	0.214558i	$-0.0688311\pi$
$179$	24420.1	+	24420.1i	0.762153	+	0.762153i	−0.214558	+	0.976711i	$0.568831\pi$
$180$		0			0
$181$	10946.5	+	10946.5i	0.334133	+	0.334133i	0.854154	−	0.520021i	$-0.174076\pi$
$181$	10946.5	+	10946.5i	0.334133	+	0.334133i	−0.520021	+	0.854154i	$0.674076\pi$
$182$		0			0
$183$	−342.915			−0.0102396
$184$		0			0
$185$			6985.52i			0.204106i
$186$		0			0
$187$	36892.5	−	36892.5i	1.05501	−	1.05501i
$188$		0			0
$189$	−373.306	+	373.306i	−0.0104506	+	0.0104506i
$190$		0			0
$191$		−	22384.9i		−	0.613604i	−0.951773	−	0.306802i	$-0.900741\pi$
$191$		−	22384.9i		−	0.613604i	0.951773	−	0.306802i	$-0.0992590\pi$
$192$		0			0
$193$	−30429.5			−0.816920			−0.408460	−	0.912776i	$-0.633934\pi$
$193$	−30429.5			−0.816920			−0.408460	+	0.912776i	$0.633934\pi$
$194$		0			0
$195$	−14.4796	−	14.4796i	−0.000380792	−	0.000380792i
$196$		0			0
$197$	−19093.9	−	19093.9i	−0.491997	−	0.491997i	0.416938	−	0.908935i	$-0.363103\pi$
$197$	−19093.9	−	19093.9i	−0.491997	−	0.491997i	−0.908935	+	0.416938i	$0.863103\pi$
$198$		0			0
$199$	67963.8			1.71621			0.858107	−	0.513470i	$-0.171640\pi$
$199$	67963.8			1.71621			0.858107	+	0.513470i	$0.171640\pi$
$200$		0			0
$201$			171.324i			0.00424060i
$202$		0			0
$203$	−51458.4	+	51458.4i	−1.24872	+	1.24872i
$204$		0			0
$205$	−5604.72	+	5604.72i	−0.133366	+	0.133366i
$206$		0			0
$207$			32790.3i			0.765253i
$208$		0			0
$209$	−58815.1			−1.34647
$210$		0			0
$211$	55219.8	+	55219.8i	1.24031	+	1.24031i	0.959872	+	0.280438i	$0.0904797\pi$
$211$	55219.8	+	55219.8i	1.24031	+	1.24031i	0.280438	+	0.959872i	$0.409520\pi$
$212$		0			0
$213$	−53.6790	−	53.6790i	−0.00118316	−	0.00118316i
$214$		0			0
$215$	−14760.6			−0.319322
$216$		0			0
$217$			75126.4i			1.59541i
$218$		0			0
$219$	41.8242	−	41.8242i	0.000872046	−	0.000872046i
$220$		0			0
$221$	−8531.07	+	8531.07i	−0.174670	+	0.174670i
$222$		0			0
$223$		−	40417.5i		−	0.812754i	−0.913705	−	0.406377i	$-0.866792\pi$
$223$		−	40417.5i		−	0.812754i	0.913705	−	0.406377i	$-0.133208\pi$
$224$		0			0
$225$	−40143.2			−0.792952
$226$		0			0
$227$	1672.85	+	1672.85i	0.0324643	+	0.0324643i	0.723153	−	0.690688i	$-0.242692\pi$
$227$	1672.85	+	1672.85i	0.0324643	+	0.0324643i	−0.690688	+	0.723153i	$0.742692\pi$
$228$		0			0
$229$	26519.0	+	26519.0i	0.505691	+	0.505691i	0.913201	−	0.407510i	$-0.133603\pi$
$229$	26519.0	+	26519.0i	0.505691	+	0.505691i	−0.407510	+	0.913201i	$0.633603\pi$
$230$		0			0
$231$	−388.328			−0.00727737
$232$		0			0
$233$		−	24163.3i		−	0.445087i	−0.974923	−	0.222543i	$-0.928564\pi$
$233$		−	24163.3i		−	0.445087i	0.974923	−	0.222543i	$-0.0714359\pi$
$234$		0			0
$235$	−900.145	+	900.145i	−0.0162996	+	0.0162996i
$236$		0			0
$237$	−271.174	+	271.174i	−0.00482782	+	0.00482782i
$238$		0			0
$239$			76356.4i			1.33675i	0.743825	+	0.668374i	$0.233010\pi$
$239$			76356.4i			1.33675i	−0.743825	+	0.668374i	$0.766990\pi$
$240$		0			0
$241$	40548.1			0.698130			0.349065	−	0.937099i	$-0.386499\pi$
$241$	40548.1			0.698130			0.349065	+	0.937099i	$0.386499\pi$
$242$		0			0
$243$	−909.281	−	909.281i	−0.0153988	−	0.0153988i
$244$		0			0
$245$	699.586	+	699.586i	0.0116549	+	0.0116549i
$246$		0			0
$247$	13600.5			0.222926
$248$		0			0
$249$			698.542i			0.0112666i
$250$		0			0
$251$	10536.0	−	10536.0i	0.167235	−	0.167235i	−0.618528	−	0.785763i	$-0.712271\pi$
$251$	10536.0	−	10536.0i	0.167235	−	0.167235i	0.785763	+	0.618528i	$0.212271\pi$
$252$		0			0
$253$	−34110.7	+	34110.7i	−0.532905	+	0.532905i
$254$		0			0
$255$			325.397i			0.00500419i
$256$		0			0
$257$	37983.9			0.575086			0.287543	−	0.957768i	$-0.407162\pi$
$257$	37983.9			0.575086			0.287543	+	0.957768i	$0.407162\pi$
$258$		0			0
$259$	−21660.9	−	21660.9i	−0.322906	−	0.322906i
$260$		0			0
$261$	−83559.4	−	83559.4i	−1.22663	−	1.22663i
$262$		0			0
$263$	37545.0			0.542801			0.271400	−	0.962467i	$-0.412513\pi$
$263$	37545.0			0.542801			0.271400	+	0.962467i	$0.412513\pi$
$264$		0			0
$265$		−	16748.6i		−	0.238498i
$266$		0			0
$267$	−297.516	+	297.516i	−0.00417338	+	0.00417338i
$268$		0			0
$269$	−4676.63	+	4676.63i	−0.0646291	+	0.0646291i	−0.738683	−	0.674053i	$-0.764552\pi$
$269$	−4676.63	+	4676.63i	−0.0646291	+	0.0646291i	0.674053	+	0.738683i	$0.264552\pi$
$270$		0			0
$271$			2746.12i			0.0373922i	0.999825	+	0.0186961i	$0.00595149\pi$
$271$			2746.12i			0.0373922i	−0.999825	+	0.0186961i	$0.994049\pi$
$272$		0			0
$273$	89.7975			0.00120487
$274$		0			0
$275$	−41759.7	−	41759.7i	−0.552194	−	0.552194i
$276$		0			0
$277$	33056.5	+	33056.5i	0.430822	+	0.430822i	0.888908	−	0.458086i	$-0.151465\pi$
$277$	33056.5	+	33056.5i	0.430822	+	0.430822i	−0.458086	+	0.888908i	$0.651465\pi$
$278$		0			0
$279$	−121992.			−1.56719
$280$		0			0
$281$		−	80033.0i		−	1.01358i	−0.862071	−	0.506788i	$-0.830833\pi$
$281$		−	80033.0i		−	1.01358i	0.862071	−	0.506788i	$-0.169167\pi$
$282$		0			0
$283$	72284.3	−	72284.3i	0.902549	−	0.902549i	−0.0931068	−	0.995656i	$-0.529680\pi$
$283$	72284.3	−	72284.3i	0.902549	−	0.902549i	0.995656	+	0.0931068i	$0.0296798\pi$
$284$		0			0
$285$	259.379	−	259.379i	0.00319334	−	0.00319334i
$286$		0			0
$287$		−	34758.5i		−	0.421985i
$288$		0			0
$289$	108196.			1.29544
$290$		0			0
$291$	−19.0819	−	19.0819i	−0.000225339	−	0.000225339i
$292$		0			0
$293$	84911.3	+	84911.3i	0.989077	+	0.989077i	0.999941	−	0.0108641i	$-0.00345823\pi$
$293$	84911.3	+	84911.3i	0.989077	+	0.989077i	−0.0108641	+	0.999941i	$0.503458\pi$
$294$		0			0
$295$	27532.6			0.316376
$296$		0			0
$297$		−	1261.18i		−	0.0142977i
$298$		0			0
$299$	7887.81	−	7887.81i	0.0882296	−	0.0882296i
$300$		0			0
$301$	45770.2	−	45770.2i	0.505184	−	0.505184i
$302$		0			0
$303$		−	781.754i		−	0.00851500i
$304$		0			0
$305$	−59698.7			−0.641749
$306$		0			0
$307$	55472.5	+	55472.5i	0.588574	+	0.588574i	0.937245	−	0.348671i	$-0.113367\pi$
$307$	55472.5	+	55472.5i	0.588574	+	0.588574i	−0.348671	+	0.937245i	$0.613367\pi$
$308$		0			0
$309$	785.756	+	785.756i	0.00822944	+	0.00822944i
$310$		0			0
$311$	−127048.			−1.31355			−0.656777	−	0.754084i	$-0.728081\pi$
$311$	−127048.			−1.31355			−0.656777	+	0.754084i	$0.728081\pi$
$312$		0			0
$313$		−	25469.3i		−	0.259974i	−0.991516	−	0.129987i	$-0.958507\pi$
$313$		−	25469.3i		−	0.259974i	0.991516	−	0.129987i	$-0.0414935\pi$
$314$		0			0
$315$	−32493.9	+	32493.9i	−0.327477	+	0.327477i
$316$		0			0
$317$	94218.4	−	94218.4i	0.937599	−	0.937599i	−0.0605656	−	0.998164i	$-0.519290\pi$
$317$	94218.4	−	94218.4i	0.937599	−	0.937599i	0.998164	+	0.0605656i	$0.0192904\pi$
$318$		0			0
$319$		−	173848.i		−	1.70840i
$320$		0			0
$321$	−865.654			−0.00840107
$322$		0			0
$323$	−152821.	−	152821.i	−1.46480	−	1.46480i
$324$		0			0
$325$	9656.57	+	9656.57i	0.0914232	+	0.0914232i
$326$		0			0
$327$	767.719			0.00717971
$328$		0			0
$329$		−	5582.38i		−	0.0515737i
$330$		0			0
$331$	−65141.3	+	65141.3i	−0.594567	+	0.594567i	−0.938862	−	0.344295i	$-0.888118\pi$
$331$	−65141.3	+	65141.3i	−0.594567	+	0.594567i	0.344295	+	0.938862i	$0.388118\pi$
$332$		0			0
$333$	35173.4	−	35173.4i	0.317195	−	0.317195i
$334$		0			0
$335$			29826.2i			0.265771i
$336$		0			0
$337$	135004.			1.18874			0.594369	−	0.804193i	$-0.297402\pi$
$337$	135004.			1.18874			0.594369	+	0.804193i	$0.297402\pi$
$338$		0			0
$339$	268.633	+	268.633i	0.00233754	+	0.00233754i
$340$		0			0
$341$	−126904.	−	126904.i	−1.09136	−	1.09136i
$342$		0			0
$343$	115422.			0.981075
$344$		0			0
$345$		−	300.862i		−	0.00252772i
$346$		0			0
$347$	−10849.2	+	10849.2i	−0.0901025	+	0.0901025i	−0.750721	−	0.660619i	$-0.770294\pi$
$347$	−10849.2	+	10849.2i	−0.0901025	+	0.0901025i	0.660619	+	0.750721i	$0.270294\pi$
$348$		0			0
$349$	6073.15	−	6073.15i	0.0498612	−	0.0498612i	−0.681737	−	0.731598i	$-0.738775\pi$
$349$	6073.15	−	6073.15i	0.0498612	−	0.0498612i	0.731598	+	0.681737i	$0.238775\pi$
$350$		0			0
$351$			291.638i			0.00236717i
$352$		0			0
$353$	−143445.			−1.15116			−0.575579	−	0.817746i	$-0.695223\pi$
$353$	−143445.			−1.15116			−0.575579	+	0.817746i	$0.695223\pi$
$354$		0			0
$355$	−9345.08	−	9345.08i	−0.0741526	−	0.0741526i
$356$		0			0
$357$	−1009.00	−	1009.00i	−0.00791690	−	0.00791690i
$358$		0			0
$359$	124076.			0.962718			0.481359	−	0.876523i	$-0.340143\pi$
$359$	124076.			0.962718			0.481359	+	0.876523i	$0.340143\pi$
$360$		0			0
$361$			113310.i			0.869472i
$362$		0			0
$363$	−20.4400	+	20.4400i	−0.000155120	+	0.000155120i
$364$		0			0
$365$	7281.26	−	7281.26i	0.0546538	−	0.0546538i
$366$		0			0
$367$		−	126240.i		−	0.937268i	−0.883392	−	0.468634i	$-0.844746\pi$
$367$		−	126240.i		−	0.937268i	0.883392	−	0.468634i	$-0.155254\pi$
$368$		0			0
$369$	56441.7			0.414521
$370$		0			0
$371$	51934.3	+	51934.3i	0.377317	+	0.377317i
$372$		0			0
$373$	95513.7	+	95513.7i	0.686512	+	0.686512i	0.961459	−	0.274948i	$-0.0886605\pi$
$373$	95513.7	+	95513.7i	0.686512	+	0.686512i	−0.274948	+	0.961459i	$0.588661\pi$
$374$		0			0
$375$	832.804			0.00592216
$376$		0			0
$377$			40200.9i			0.282848i
$378$		0			0
$379$	31220.0	−	31220.0i	0.217347	−	0.217347i	−0.590032	−	0.807380i	$-0.700885\pi$
$379$	31220.0	−	31220.0i	0.217347	−	0.217347i	0.807380	+	0.590032i	$0.200885\pi$
$380$		0			0
$381$	940.951	−	940.951i	0.00648212	−	0.00648212i
$382$		0			0
$383$			253102.i			1.72543i	0.505689	+	0.862716i	$0.331238\pi$
$383$			253102.i			1.72543i	−0.505689	+	0.862716i	$0.668762\pi$
$384$		0			0
$385$	−67604.7			−0.456095
$386$		0			0
$387$	74322.6	+	74322.6i	0.496248	+	0.496248i
$388$		0			0
$389$	43279.4	+	43279.4i	0.286011	+	0.286011i	0.835500	−	0.549490i	$-0.185178\pi$
$389$	43279.4	+	43279.4i	0.286011	+	0.286011i	−0.549490	+	0.835500i	$0.685178\pi$
$390$		0			0
$391$	−177261.			−1.15947
$392$		0			0
$393$		−	407.898i		−	0.00264099i
$394$		0			0
$395$	−47209.2	+	47209.2i	−0.302575	+	0.302575i
$396$		0			0
$397$	−198533.	+	198533.i	−1.25965	+	1.25965i	−0.308395	+	0.951258i	$0.599792\pi$
$397$	−198533.	+	198533.i	−1.25965	+	1.25965i	−0.951258	+	0.308395i	$0.900208\pi$
$398$		0			0
$399$			1608.58i			0.0101041i
$400$		0			0
$401$	−89330.8			−0.555536			−0.277768	−	0.960648i	$-0.589595\pi$
$401$	−89330.8			−0.555536			−0.277768	+	0.960648i	$0.589595\pi$
$402$		0			0
$403$	29345.5	+	29345.5i	0.180689	+	0.180689i
$404$		0			0
$405$	−52761.6	−	52761.6i	−0.321668	−	0.321668i
$406$		0			0
$407$	73179.5			0.441775
$408$		0			0
$409$			26716.3i			0.159709i	0.996807	+	0.0798545i	$0.0254456\pi$
$409$			26716.3i			0.159709i	−0.996807	+	0.0798545i	$0.974554\pi$
$410$		0			0
$411$	509.057	−	509.057i	0.00301358	−	0.00301358i
$412$		0			0
$413$	−85373.9	+	85373.9i	−0.500524	+	0.500524i
$414$		0			0
$415$			121611.i			0.706115i
$416$		0			0
$417$	1382.03			0.00794775
$418$		0			0
$419$	119011.	+	119011.i	0.677889	+	0.677889i	0.959522	−	0.281633i	$-0.0908761\pi$
$419$	119011.	+	119011.i	0.677889	+	0.677889i	−0.281633	+	0.959522i	$0.590876\pi$
$420$		0			0
$421$	−126967.	−	126967.i	−0.716355	−	0.716355i	0.251502	−	0.967857i	$-0.419075\pi$
$421$	−126967.	−	126967.i	−0.716355	−	0.716355i	−0.967857	+	0.251502i	$0.919075\pi$
$422$		0			0
$423$	9064.80			0.0506614
$424$		0			0
$425$		−	217010.i		−	1.20144i
$426$		0			0
$427$	185115.	−	185115.i	1.01528	−	1.01528i
$428$		0			0
$429$	−151.687	+	151.687i	−0.000824201	+	0.000824201i
$430$		0			0
$431$			82986.1i			0.446736i	0.974734	+	0.223368i	$0.0717051\pi$
$431$			82986.1i			0.446736i	−0.974734	+	0.223368i	$0.928295\pi$
$432$		0			0
$433$	−153228.			−0.817265			−0.408633	−	0.912699i	$-0.633994\pi$
$433$	−153228.			−0.817265			−0.408633	+	0.912699i	$0.633994\pi$
$434$		0			0
$435$	766.685	+	766.685i	0.00405171	+	0.00405171i
$436$		0			0
$437$	141298.	+	141298.i	0.739899	+	0.739899i
$438$		0			0
$439$	48984.2			0.254172			0.127086	−	0.991892i	$-0.459438\pi$
$439$	48984.2			0.254172			0.127086	+	0.991892i	$0.459438\pi$
$440$		0			0
$441$		−	7045.09i		−	0.0362251i
$442$		0			0
$443$	−5464.01	+	5464.01i	−0.0278422	+	0.0278422i	−0.720891	−	0.693049i	$-0.756267\pi$
$443$	−5464.01	+	5464.01i	−0.0278422	+	0.0278422i	0.693049	+	0.720891i	$0.256267\pi$
$444$		0			0
$445$	−51795.1	+	51795.1i	−0.261559	+	0.261559i
$446$		0			0
$447$		−	1422.38i		−	0.00711870i
$448$		0			0
$449$	140068.			0.694776			0.347388	−	0.937721i	$-0.387069\pi$
$449$	140068.			0.694776			0.347388	+	0.937721i	$0.387069\pi$
$450$		0			0
$451$	58714.4	+	58714.4i	0.288664	+	0.288664i
$452$		0			0
$453$	−784.092	−	784.092i	−0.00382094	−	0.00382094i
$454$		0			0
$455$	15633.0			0.0755127
$456$		0			0
$457$			219788.i			1.05238i	0.850368	+	0.526188i	$0.176379\pi$
$457$			219788.i			1.05238i	−0.850368	+	0.526188i	$0.823621\pi$
$458$		0			0
$459$	3276.96	−	3276.96i	0.0155541	−	0.0155541i
$460$		0			0
$461$	−124163.	+	124163.i	−0.584239	+	0.584239i	−0.936065	−	0.351826i	$-0.885561\pi$
$461$	−124163.	+	124163.i	−0.584239	+	0.584239i	0.351826	+	0.936065i	$0.385561\pi$
$462$		0			0
$463$			236566.i			1.10354i	0.833995	+	0.551772i	$0.186048\pi$
$463$			236566.i			1.10354i	−0.833995	+	0.551772i	$0.813952\pi$
$464$		0			0
$465$	1119.32			0.00517663
$466$		0			0
$467$	−73415.5	−	73415.5i	−0.336631	−	0.336631i	0.518467	−	0.855098i	$-0.326503\pi$
$467$	−73415.5	−	73415.5i	−0.336631	−	0.336631i	−0.855098	+	0.518467i	$0.826503\pi$
$468$		0			0
$469$	−92485.8	−	92485.8i	−0.420465	−	0.420465i
$470$		0			0
$471$	−107.897			−0.000486372
$472$		0			0
$473$			154631.i			0.691153i
$474$		0			0
$475$	−172982.	+	172982.i	−0.766681	+	0.766681i
$476$		0			0
$477$	−84332.1	+	84332.1i	−0.370643	+	0.370643i
$478$		0			0
$479$		−	212004.i		−	0.924003i	−0.886879	−	0.462002i	$-0.847131\pi$
$479$		−	212004.i		−	0.924003i	0.886879	−	0.462002i	$-0.152869\pi$
$480$		0			0
$481$	−16922.1			−0.0731417
$482$		0			0
$483$	932.920	+	932.920i	0.00399899	+	0.00399899i
$484$		0			0
$485$	−3322.01	−	3322.01i	−0.0141227	−	0.0141227i
$486$		0			0
$487$	26716.3			0.112647			0.0563234	−	0.998413i	$-0.482062\pi$
$487$	26716.3			0.112647			0.0563234	+	0.998413i	$0.482062\pi$
$488$		0			0
$489$		−	2594.22i		−	0.0108490i
$490$		0			0
$491$	−99544.8	+	99544.8i	−0.412910	+	0.412910i	−0.882751	−	0.469841i	$-0.844311\pi$
$491$	−99544.8	+	99544.8i	−0.412910	+	0.412910i	0.469841	+	0.882751i	$0.344311\pi$
$492$		0			0
$493$	451714.	−	451714.i	1.85853	−	1.85853i
$494$		0			0
$495$		−	109778.i		−	0.448028i
$496$		0			0
$497$	57954.9			0.234627
$498$		0			0
$499$	−189887.	−	189887.i	−0.762597	−	0.762597i	0.214194	−	0.976791i	$-0.431287\pi$
$499$	−189887.	−	189887.i	−0.762597	−	0.762597i	−0.976791	+	0.214194i	$0.931287\pi$
$500$		0			0
$501$	−135.356	−	135.356i	−0.000539265	−	0.000539265i
$502$		0			0
$503$	188872.			0.746502			0.373251	−	0.927730i	$-0.378243\pi$
$503$	188872.			0.746502			0.373251	+	0.927730i	$0.378243\pi$
$504$		0			0
$505$		−	136097.i		−	0.533662i
$506$		0			0
$507$	−1284.43	+	1284.43i	−0.00499682	+	0.00499682i
$508$		0			0
$509$	14343.5	−	14343.5i	0.0553629	−	0.0553629i	−0.678883	−	0.734246i	$-0.737536\pi$
$509$	14343.5	−	14343.5i	0.0553629	−	0.0553629i	0.734246	+	0.678883i	$0.237536\pi$
$510$		0			0
$511$			45155.8i			0.172931i
$512$		0			0
$513$	−5224.23			−0.0198513
$514$		0			0
$515$	136794.	+	136794.i	0.515765	+	0.515765i
$516$		0			0
$517$	9429.82	+	9429.82i	0.0352795	+	0.0352795i
$518$		0			0
$519$	−791.025			−0.00293667
$520$		0			0
$521$		−	65377.4i		−	0.240853i	−0.992722	−	0.120427i	$-0.961574\pi$
$521$		−	65377.4i		−	0.240853i	0.992722	−	0.120427i	$-0.0384262\pi$
$522$		0			0
$523$	143634.	−	143634.i	0.525113	−	0.525113i	−0.393998	−	0.919111i	$-0.628908\pi$
$523$	143634.	−	143634.i	0.525113	−	0.525113i	0.919111	+	0.393998i	$0.128908\pi$
$524$		0			0
$525$	−1142.12	+	1142.12i	−0.00414374	+	0.00414374i
$526$		0			0
$527$		−	659477.i		−	2.37453i
$528$		0			0
$529$	−115946.			−0.414327
$530$		0			0
$531$	−138632.	−	138632.i	−0.491671	−	0.491671i
$532$		0			0
$533$	−13577.2	−	13577.2i	−0.0477921	−	0.0477921i
$534$		0			0
$535$	−150704.			−0.526521
$536$		0			0
$537$			2256.40i			0.00782469i
$538$		0			0
$539$	7328.78	−	7328.78i	0.0252263	−	0.0252263i
$540$		0			0
$541$	275122.	−	275122.i	0.940007	−	0.940007i	−0.0582928	−	0.998300i	$-0.518566\pi$
$541$	275122.	−	275122.i	0.940007	−	0.940007i	0.998300	+	0.0582928i	$0.0185657\pi$
$542$		0			0
$543$			1011.45i			0.00343039i
$544$		0			0
$545$	133654.			0.449975
$546$		0			0
$547$	1032.53	+	1032.53i	0.00345087	+	0.00345087i	0.708830	−	0.705379i	$-0.249223\pi$
$547$	1032.53	+	1032.53i	0.00345087	+	0.00345087i	−0.705379	+	0.708830i	$0.749223\pi$
$548$		0			0
$549$	300594.	+	300594.i	0.997323	+	0.997323i
$550$		0			0
$551$	−720136.			−2.37198
$552$		0			0
$553$		−	292775.i		−	0.957379i
$554$		0			0
$555$	−322.727	+	322.727i	−0.00104773	+	0.00104773i
$556$		0			0
$557$	223266.	−	223266.i	0.719635	−	0.719635i	−0.248895	−	0.968530i	$-0.580067\pi$
$557$	223266.	−	223266.i	0.719635	−	0.719635i	0.968530	+	0.248895i	$0.0800674\pi$
$558$		0			0
$559$		−	35757.1i		−	0.114430i
$560$		0			0
$561$	3408.83			0.0108313
$562$		0			0
$563$	−191551.	−	191551.i	−0.604322	−	0.604322i	0.337134	−	0.941457i	$-0.390542\pi$
$563$	−191551.	−	191551.i	−0.604322	−	0.604322i	−0.941457	+	0.337134i	$0.890542\pi$
$564$		0			0
$565$	46766.9	+	46766.9i	0.146501	+	0.146501i
$566$		0			0
$567$	327209.			1.01779
$568$		0			0
$569$			119746.i			0.369858i	0.982752	+	0.184929i	$0.0592055\pi$
$569$			119746.i			0.369858i	−0.982752	+	0.184929i	$0.940795\pi$
$570$		0			0
$571$	−375516.	+	375516.i	−1.15175	+	1.15175i	−0.165544	+	0.986202i	$0.552938\pi$
$571$	−375516.	+	375516.i	−1.15175	+	1.15175i	−0.986202	+	0.165544i	$0.947062\pi$
$572$		0			0
$573$	1034.17	−	1034.17i	0.00314980	−	0.00314980i
$574$		0			0
$575$			200647.i			0.606873i
$576$		0			0
$577$	185281.			0.556518			0.278259	−	0.960506i	$-0.410243\pi$
$577$	185281.			0.556518			0.278259	+	0.960506i	$0.410243\pi$
$578$		0			0
$579$	−1405.83	−	1405.83i	−0.00419348	−	0.00419348i
$580$		0			0
$581$	−377093.	−	377093.i	−1.11711	−	1.11711i
$582$		0			0
$583$	−175456.			−0.516216
$584$		0			0
$585$			25385.3i			0.0741771i
$586$		0			0
$587$	483071.	−	483071.i	1.40196	−	1.40196i	0.608081	−	0.793875i	$-0.291940\pi$
$587$	483071.	−	483071.i	1.40196	−	1.40196i	0.793875	−	0.608081i	$-0.208060\pi$
$588$		0			0
$589$	−525679.	+	525679.i	−1.51527	+	1.51527i
$590$		0			0
$591$		−	1764.26i		−	0.00505112i
$592$		0			0
$593$	636299.			1.80947			0.904736	−	0.425973i	$-0.140068\pi$
$593$	636299.			1.80947			0.904736	+	0.425973i	$0.140068\pi$
$594$		0			0
$595$	−175659.	−	175659.i	−0.496177	−	0.496177i
$596$		0			0
$597$	3139.90	+	3139.90i	0.00880981	+	0.00880981i
$598$		0			0
$599$	410280.			1.14347			0.571737	−	0.820437i	$-0.306270\pi$
$599$	410280.			1.14347			0.571737	+	0.820437i	$0.306270\pi$
$600$		0			0
$601$		−	531872.i		−	1.47251i	−0.676704	−	0.736255i	$-0.736592\pi$
$601$		−	531872.i		−	1.47251i	0.676704	−	0.736255i	$-0.263408\pi$
$602$		0			0
$603$	150181.	−	150181.i	0.413028	−	0.413028i
$604$		0			0
$605$	−3558.44	+	3558.44i	−0.00972184	+	0.00972184i
$606$		0			0
$607$			505448.i			1.37182i	0.727684	+	0.685912i	$0.240597\pi$
$607$			505448.i			1.37182i	−0.727684	+	0.685912i	$0.759403\pi$
$608$		0			0
$609$	−4754.71			−0.0128200
$610$		0			0
$611$	−2180.57	−	2180.57i	−0.00584099	−	0.00584099i
$612$		0			0
$613$	31334.4	+	31334.4i	0.0833873	+	0.0833873i	0.747570	−	0.664183i	$-0.231220\pi$
$613$	31334.4	+	31334.4i	0.0833873	+	0.0833873i	−0.664183	+	0.747570i	$0.731220\pi$
$614$		0			0
$615$	−517.871			−0.00136921
$616$		0			0
$617$			481833.i			1.26569i	0.774280	+	0.632844i	$0.218112\pi$
$617$			481833.i			1.26569i	−0.774280	+	0.632844i	$0.781888\pi$
$618$		0			0
$619$	−55035.5	+	55035.5i	−0.143635	+	0.143635i	−0.775268	−	0.631632i	$-0.782385\pi$
$619$	−55035.5	+	55035.5i	−0.143635	+	0.143635i	0.631632	+	0.775268i	$0.282385\pi$
$620$		0			0
$621$	−3029.87	+	3029.87i	−0.00785672	+	0.00785672i
$622$		0			0
$623$		−	321215.i		−	0.827599i
$624$		0			0
$625$	−164779.			−0.421834
$626$		0			0
$627$	−2717.23	−	2717.23i	−0.00691180	−	0.00691180i
$628$		0			0
$629$	190144.	+	190144.i	0.480597	+	0.480597i
$630$		0			0
$631$	−188802.			−0.474184			−0.237092	−	0.971487i	$-0.576194\pi$
$631$	−188802.			−0.474184			−0.237092	+	0.971487i	$0.576194\pi$
$632$		0			0
$633$			5102.26i			0.0127337i
$634$		0			0
$635$	163812.	−	163812.i	0.406255	−	0.406255i
$636$		0			0
$637$	−1694.72	+	1694.72i	−0.00417656	+	0.00417656i
$638$		0			0
$639$			94108.5i			0.230477i
$640$		0			0
$641$	−442081.			−1.07593			−0.537967	−	0.842966i	$-0.680807\pi$
$641$	−442081.			−1.07593			−0.537967	+	0.842966i	$0.680807\pi$
$642$		0			0
$643$	246339.	+	246339.i	0.595814	+	0.595814i	0.939196	−	0.343382i	$-0.111573\pi$
$643$	246339.	+	246339.i	0.595814	+	0.595814i	−0.343382	+	0.939196i	$0.611573\pi$
$644$		0			0
$645$	−681.935	−	681.935i	−0.00163917	−	0.00163917i
$646$		0			0
$647$	−54549.4			−0.130311			−0.0651555	−	0.997875i	$-0.520754\pi$
$647$	−54549.4			−0.130311			−0.0651555	+	0.997875i	$0.520754\pi$
$648$		0			0
$649$		−	288429.i		−	0.684778i
$650$		0			0
$651$	−3470.80	+	3470.80i	−0.00818970	+	0.00818970i
$652$		0			0
$653$	−231745.	+	231745.i	−0.543481	+	0.543481i	−0.924548	−	0.381066i	$-0.875557\pi$
$653$	−231745.	+	231745.i	−0.543481	+	0.543481i	0.381066	+	0.924548i	$0.375557\pi$
$654$		0			0
$655$		−	71011.8i		−	0.165519i
$656$		0			0
$657$	−73325.0			−0.169872
$658$		0			0
$659$	479757.	+	479757.i	1.10472	+	1.10472i	0.993833	+	0.110883i	$0.0353678\pi$
$659$	479757.	+	479757.i	1.10472	+	1.10472i	0.110883	+	0.993833i	$0.464632\pi$
$660$		0			0
$661$	−515798.	−	515798.i	−1.18053	−	1.18053i	−0.979607	−	0.200922i	$-0.935606\pi$
$661$	−515798.	−	515798.i	−1.18053	−	1.18053i	−0.200922	−	0.979607i	$-0.564394\pi$
$662$		0			0
$663$	−788.263			−0.00179326
$664$		0			0
$665$			280041.i			0.633254i
$666$		0			0
$667$	−417654.	+	417654.i	−0.938782	+	0.938782i
$668$		0			0
$669$	1867.27	−	1867.27i	0.00417210	−	0.00417210i
$670$		0			0
$671$			625397.i			1.38903i
$672$		0			0
$673$	−505551.			−1.11618			−0.558090	−	0.829780i	$-0.688466\pi$
$673$	−505551.			−1.11618			−0.558090	+	0.829780i	$0.688466\pi$
$674$		0			0
$675$	−3709.29	−	3709.29i	−0.00814111	−	0.00814111i
$676$		0			0
$677$	460825.	+	460825.i	1.00545	+	1.00545i	0.999985	+	0.00546007i	$0.00173800\pi$
$677$	460825.	+	460825.i	1.00545	+	1.00545i	0.00546007	+	0.999985i	$0.498262\pi$
$678$		0			0
$679$	20601.9			0.0446857
$680$		0			0
$681$			154.570i			0.000333297i
$682$		0			0
$683$	−120242.	+	120242.i	−0.257761	+	0.257761i	−0.824143	−	0.566382i	$-0.808343\pi$
$683$	−120242.	+	120242.i	−0.257761	+	0.257761i	0.566382	+	0.824143i	$0.308343\pi$
$684$		0			0
$685$	88622.7	−	88622.7i	0.188870	−	0.188870i
$686$		0			0
$687$			2450.33i			0.00519171i
$688$		0			0
$689$	40572.7			0.0854664
$690$		0			0
$691$	−166965.	−	166965.i	−0.349678	−	0.349678i	0.510311	−	0.859990i	$-0.329530\pi$
$691$	−166965.	−	166965.i	−0.349678	−	0.349678i	−0.859990	+	0.510311i	$0.829530\pi$
$692$		0			0
$693$	340403.	+	340403.i	0.708805	+	0.708805i
$694$		0			0
$695$	240600.			0.498111
$696$		0			0
$697$			305118.i			0.628062i
$698$		0			0
$699$	1116.33	−	1116.33i	0.00228475	−	0.00228475i
$700$		0			0
$701$	−39689.8	+	39689.8i	−0.0807687	+	0.0807687i	−0.746337	−	0.665568i	$-0.768189\pi$
$701$	−39689.8	+	39689.8i	−0.0807687	+	0.0807687i	0.665568	+	0.746337i	$0.268189\pi$
$702$		0			0
$703$		−	303133.i		−	0.613371i
$704$		0			0
$705$	−83.1725			−0.000167341
$706$		0			0
$707$	422013.	+	422013.i	0.844281	+	0.844281i
$708$		0			0
$709$	−250742.	−	250742.i	−0.498809	−	0.498809i	0.412258	−	0.911067i	$-0.364740\pi$
$709$	−250742.	−	250742.i	−0.498809	−	0.498809i	−0.911067	+	0.412258i	$0.864740\pi$
$710$		0			0
$711$	475415.			0.940445
$712$		0			0
$713$			609751.i			1.19943i
$714$		0			0
$715$	−26407.5	+	26407.5i	−0.0516553	+	0.0516553i
$716$		0			0
$717$	−3527.63	+	3527.63i	−0.00686190	+	0.00686190i
$718$		0			0
$719$		−	133454.i		−	0.258152i	−0.991635	−	0.129076i	$-0.958799\pi$
$719$		−	133454.i		−	0.258152i	0.991635	−	0.129076i	$-0.0412011\pi$
$720$		0			0
$721$	−848347.			−1.63194
$722$		0			0
$723$	1873.30	+	1873.30i	0.00358370	+	0.00358370i
$724$		0			0
$725$	−511309.	−	511309.i	−0.972763	−	0.972763i
$726$		0			0
$727$	583370.			1.10376			0.551881	−	0.833923i	$-0.313910\pi$
$727$	583370.			1.10376			0.551881	+	0.833923i	$0.313910\pi$
$728$		0			0
$729$			531273.i			0.999684i
$730$		0			0
$731$	−401781.	+	401781.i	−0.751891	+	0.751891i
$732$		0			0
$733$	−534516.	+	534516.i	−0.994838	+	0.994838i	−0.999987	−	0.00514857i	$-0.998361\pi$
$733$	−534516.	+	534516.i	−0.994838	+	0.994838i	0.00514857	+	0.999987i	$0.498361\pi$
$734$		0			0
$735$			64.6410i			0.000119656i
$736$		0			0
$737$	312456.			0.575247
$738$		0			0
$739$	91929.4	+	91929.4i	0.168332	+	0.168332i	0.786246	−	0.617914i	$-0.212022\pi$
$739$	91929.4	+	91929.4i	0.168332	+	0.168332i	−0.617914	+	0.786246i	$0.712022\pi$
$740$		0			0
$741$	628.336	+	628.336i	0.00114434	+	0.00114434i
$742$		0			0
$743$	83329.2			0.150945			0.0754727	−	0.997148i	$-0.475953\pi$
$743$	83329.2			0.150945			0.0754727	+	0.997148i	$0.475953\pi$
$744$		0			0
$745$		−	247625.i		−	0.446151i
$746$		0			0
$747$	612333.	−	612333.i	1.09735	−	1.09735i
$748$		0			0
$749$	467305.	−	467305.i	0.832985	−	0.832985i
$750$		0			0
$751$			318689.i			0.565051i	0.959260	+	0.282525i	$0.0911722\pi$
$751$			318689.i			0.565051i	−0.959260	+	0.282525i	$0.908828\pi$
$752$		0			0
$753$	973.515			0.00171693
$754$		0			0
$755$	−136504.	−	136504.i	−0.239470	−	0.239470i
$756$		0			0
$757$	428881.	+	428881.i	0.748419	+	0.748419i	0.974182	−	0.225763i	$-0.0724875\pi$
$757$	428881.	+	428881.i	0.748419	+	0.748419i	−0.225763	+	0.974182i	$0.572488\pi$
$758$		0			0
$759$	−3151.80			−0.00547110
$760$		0			0
$761$		−	718933.i		−	1.24142i	−0.784040	−	0.620711i	$-0.786844\pi$
$761$		−	718933.i		−	1.24142i	0.784040	−	0.620711i	$-0.213156\pi$
$762$		0			0
$763$	−414437.	+	414437.i	−0.711884	+	0.711884i
$764$		0			0
$765$	285239.	−	285239.i	0.487401	−	0.487401i
$766$		0			0
$767$			66696.8i			0.113374i
$768$		0			0
$769$	421326.			0.712468			0.356234	−	0.934397i	$-0.384061\pi$
$769$	421326.			0.712468			0.356234	+	0.934397i	$0.384061\pi$
$770$		0			0
$771$	1754.84	+	1754.84i	0.00295208	+	0.00295208i
$772$		0			0
$773$	−455325.	−	455325.i	−0.762013	−	0.762013i	0.214673	−	0.976686i	$-0.431132\pi$
$773$	−455325.	−	455325.i	−0.762013	−	0.762013i	−0.976686	+	0.214673i	$0.931132\pi$
$774$		0			0
$775$	−746482.			−1.24284
$776$		0			0
$777$		−	2001.44i		−	0.00331513i
$778$		0			0
$779$	243214.	−	243214.i	0.400788	−	0.400788i
$780$		0			0
$781$	−97898.1	+	97898.1i	−0.160499	+	0.160499i
$782$		0			0
$783$		−	15442.0i		−	0.0251872i
$784$		0			0
$785$	−18784.0			−0.0304825
$786$		0			0
$787$	−541211.	−	541211.i	−0.873811	−	0.873811i	0.119074	−	0.992885i	$-0.462007\pi$
$787$	−541211.	−	541211.i	−0.873811	−	0.873811i	−0.992885	+	0.119074i	$0.962007\pi$
$788$		0			0
$789$	1734.56	+	1734.56i	0.00278635	+	0.00278635i
$790$		0			0
$791$	−290032.			−0.463546
$792$		0			0
$793$		−	144618.i		−	0.229972i
$794$		0			0
$795$	773.775	−	773.775i	0.00122428	−	0.00122428i
$796$		0			0
$797$	−481291.	+	481291.i	−0.757690	+	0.757690i	−0.975901	−	0.218212i	$-0.929978\pi$
$797$	−481291.	+	481291.i	−0.757690	+	0.757690i	0.218212	+	0.975901i	$0.429978\pi$
$798$		0			0
$799$			49003.4i			0.0767597i
$800$		0			0
$801$	521597.			0.812961
$802$		0			0
$803$	−76277.7	−	76277.7i	−0.118295	−	0.118295i
$804$		0			0
$805$	162414.	+	162414.i	0.250629	+	0.250629i
$806$		0			0
$807$	−432.116			−0.000663518
$808$		0			0
$809$			34438.5i			0.0526195i	0.999654	+	0.0263097i	$0.00837562\pi$
$809$			34438.5i			0.0526195i	−0.999654	+	0.0263097i	$0.991624\pi$
$810$		0			0
$811$	−227480.	+	227480.i	−0.345862	+	0.345862i	−0.858566	−	0.512704i	$-0.828644\pi$
$811$	−227480.	+	227480.i	−0.345862	+	0.345862i	0.512704	+	0.858566i	$0.328644\pi$
$812$		0			0
$813$	−126.869	+	126.869i	−0.000191944	+	0.000191944i
$814$		0			0
$815$		−	451633.i		−	0.679939i
$816$		0			0
$817$	640532.			0.959614
$818$		0			0
$819$	−78715.2	−	78715.2i	−0.117352	−	0.117352i
$820$		0			0
$821$	807677.	+	807677.i	1.19826	+	1.19826i	0.974687	+	0.223573i	$0.0717722\pi$
$821$	807677.	+	807677.i	1.19826	+	1.19826i	0.223573	+	0.974687i	$0.428228\pi$
$822$		0			0
$823$	−703593.			−1.03878			−0.519388	−	0.854539i	$-0.673840\pi$
$823$	−703593.			−1.03878			−0.519388	+	0.854539i	$0.673840\pi$
$824$		0			0
$825$		−	3858.56i		−	0.00566914i
$826$		0			0
$827$	775918.	−	775918.i	1.13450	−	1.13450i	0.145081	−	0.989420i	$-0.453656\pi$
$827$	775918.	−	775918.i	1.13450	−	1.13450i	0.989420	−	0.145081i	$-0.0463444\pi$
$828$		0			0
$829$	804056.	−	804056.i	1.16998	−	1.16998i	0.187763	−	0.982214i	$-0.439876\pi$
$829$	804056.	−	804056.i	1.16998	−	1.16998i	0.982214	−	0.187763i	$-0.0601236\pi$
$830$		0			0
$831$			3054.39i			0.00442305i
$832$		0			0
$833$	38085.1			0.0548864
$834$		0			0
$835$	−23564.4	−	23564.4i	−0.0337974	−	0.0337974i
$836$		0			0
$837$	−11272.2	−	11272.2i	−0.0160901	−	0.0160901i
$838$		0			0
$839$	−248652.			−0.353239			−0.176619	−	0.984279i	$-0.556516\pi$
$839$	−248652.			−0.353239			−0.176619	+	0.984279i	$0.556516\pi$
$840$		0			0
$841$		−	1.42133e6i		−	2.00957i
$842$		0			0
$843$	3697.49	−	3697.49i	0.00520297	−	0.00520297i
$844$		0			0
$845$	−223609.	+	223609.i	−0.313166	+	0.313166i
$846$		0			0
$847$		−	22068.2i		−	0.0307610i
$848$		0			0
$849$	6679.00			0.00926608
$850$		0			0
$851$	−175807.	−	175807.i	−0.242760	−	0.242760i
$852$		0			0
$853$	−779867.	−	779867.i	−1.07182	−	1.07182i	−0.997213	−	0.0746081i	$-0.976229\pi$
$853$	−779867.	−	779867.i	−1.07182	−	1.07182i	−0.0746081	−	0.997213i	$-0.523771\pi$
$854$		0			0
$855$	−454737.			−0.622054
$856$		0			0
$857$		−	481792.i		−	0.655992i	−0.944679	−	0.327996i	$-0.893627\pi$
$857$		−	481792.i		−	0.655992i	0.944679	−	0.327996i	$-0.106373\pi$
$858$		0			0
$859$	947889.	−	947889.i	1.28461	−	1.28461i	0.346594	−	0.938015i	$-0.387338\pi$
$859$	947889.	−	947889.i	1.28461	−	1.28461i	0.938015	−	0.346594i	$-0.112662\pi$
$860$		0			0
$861$	1605.83	−	1605.83i	0.00216617	−	0.00216617i
$862$		0			0
$863$		−	1.00011e6i		−	1.34284i	−0.741077	−	0.671421i	$-0.765684\pi$
$863$		−	1.00011e6i		−	1.34284i	0.741077	−	0.671421i	$-0.234316\pi$
$864$		0			0
$865$	−137711.			−0.184051
$866$		0			0
$867$	4998.61	+	4998.61i	0.00664984	+	0.00664984i
$868$		0			0
$869$	494559.	+	494559.i	0.654905	+	0.654905i
$870$		0			0
$871$	−72252.8			−0.0952398
$872$		0			0
$873$			33453.9i			0.0438953i
$874$		0			0
$875$	−449571.	+	449571.i	−0.587195	+	0.587195i
$876$		0			0
$877$	747906.	−	747906.i	0.972406	−	0.972406i	−0.0272234	−	0.999629i	$-0.508667\pi$
$877$	747906.	−	747906.i	0.972406	−	0.972406i	0.999629	+	0.0272234i	$0.00866655\pi$
$878$		0			0
$879$			7845.72i			0.0101544i
$880$		0			0
$881$	−53961.8			−0.0695240			−0.0347620	−	0.999396i	$-0.511067\pi$
$881$	−53961.8			−0.0695240			−0.0347620	+	0.999396i	$0.511067\pi$
$882$		0			0
$883$	629494.	+	629494.i	0.807366	+	0.807366i	0.984234	−	0.176869i	$-0.0565968\pi$
$883$	629494.	+	629494.i	0.807366	+	0.807366i	−0.176869	+	0.984234i	$0.556597\pi$
$884$		0			0
$885$	1271.99	+	1271.99i	0.00162405	+	0.00162405i
$886$		0			0
$887$	880807.			1.11952			0.559762	−	0.828653i	$-0.310893\pi$
$887$	880807.			1.11952			0.559762	+	0.828653i	$0.310893\pi$
$888$		0			0
$889$			1.01590e6i			1.28543i
$890$		0			0
$891$	−552724.	+	552724.i	−0.696231	+	0.696231i
$892$		0			0
$893$	39061.4	−	39061.4i	0.0489829	−	0.0489829i
$894$		0			0
$895$			392821.i			0.490398i
$896$		0			0
$897$	728.826			0.000905814
$898$		0			0
$899$	−1.55383e6	−	1.55383e6i	−1.92257	−	1.92257i
$900$		0			0
$901$	−455891.	−	455891.i	−0.561580	−	0.561580i
$902$		0			0
$903$	4229.12			0.00518650
$904$		0			0
$905$			176085.i			0.214994i
$906$		0			0
$907$	−662568.	+	662568.i	−0.805408	+	0.805408i	−0.983935	−	0.178527i	$-0.942867\pi$
$907$	−662568.	+	662568.i	−0.805408	+	0.805408i	0.178527	+	0.983935i	$0.442867\pi$
$908$		0			0
$909$	−685275.	+	685275.i	−0.829348	+	0.829348i
$910$		0			0
$911$			817906.i			0.985522i	0.870165	+	0.492761i	$0.164012\pi$
$911$			817906.i			0.985522i	−0.870165	+	0.492761i	$0.835988\pi$
$912$		0			0
$913$	1.27398e6			1.52834
$914$		0			0
$915$	−2758.05	−	2758.05i	−0.00329428	−	0.00329428i
$916$		0			0
$917$	220195.	+	220195.i	0.261860	+	0.261860i
$918$		0			0
$919$	−33891.9			−0.0401296			−0.0200648	−	0.999799i	$-0.506387\pi$
$919$	−33891.9			−0.0401296			−0.0200648	+	0.999799i	$0.506387\pi$
$920$		0			0
$921$			5125.61i			0.00604263i
$922$		0			0
$923$	22638.1	−	22638.1i	0.0265728	−	0.0265728i
$924$		0			0
$925$	215230.	−	215230.i	0.251547	−	0.251547i
$926$		0			0
$927$		−	1.37757e6i		−	1.60307i
$928$		0			0
$929$	−207783.			−0.240757			−0.120378	−	0.992728i	$-0.538411\pi$
$929$	−207783.			−0.240757			−0.120378	+	0.992728i	$0.538411\pi$
$930$		0			0
$931$	−30358.2	−	30358.2i	−0.0350249	−	0.0350249i
$932$		0			0
$933$	−5869.57	−	5869.57i	−0.00674284	−	0.00674284i
$934$		0			0
$935$	593450.			0.678830
$936$		0			0
$937$			1.12361e6i			1.27979i	0.768464	+	0.639893i	$0.221022\pi$
$937$			1.12361e6i			1.27979i	−0.768464	+	0.639893i	$0.778978\pi$
$938$		0			0
$939$	1176.67	−	1176.67i	0.00133452	−	0.00133452i
$940$		0			0
$941$	856765.	−	856765.i	0.967570	−	0.967570i	−0.0319200	−	0.999490i	$-0.510162\pi$
$941$	856765.	−	856765.i	0.967570	−	0.967570i	0.999490	+	0.0319200i	$0.0101622\pi$
$942$		0			0
$943$		−	282112.i		−	0.317247i
$944$		0			0
$945$	−6004.97			−0.00672430
$946$		0			0
$947$	246666.	+	246666.i	0.275048	+	0.275048i	0.831129	−	0.556080i	$-0.187695\pi$
$947$	246666.	+	246666.i	0.275048	+	0.275048i	−0.556080	+	0.831129i	$0.687695\pi$
$948$		0			0
$949$	17638.6	+	17638.6i	0.0195853	+	0.0195853i
$950$		0			0
$951$	8705.68			0.00962591
$952$		0			0
$953$		−	219446.i		−	0.241625i	−0.992675	−	0.120812i	$-0.961450\pi$
$953$		−	219446.i		−	0.241625i	0.992675	−	0.120812i	$-0.0385500\pi$
$954$		0			0
$955$	180041.	−	180041.i	0.197408	−	0.197408i
$956$		0			0
$957$	8031.71	−	8031.71i	0.00876968	−	0.00876968i
$958$		0			0
$959$			549607.i			0.597606i
$960$		0			0
$961$	−1.34498e6			−1.45636
$962$		0			0
$963$	758821.	+	758821.i	0.818251	+	0.818251i
$964$		0			0
$965$	−244743.	−	244743.i	−0.262819	−	0.262819i
$966$		0			0
$967$	650799.			0.695975			0.347988	−	0.937499i	$-0.386865\pi$
$967$	650799.			0.695975			0.347988	+	0.937499i	$0.386865\pi$
$968$		0			0
$969$		−	14120.5i		−	0.0150384i
$970$		0			0
$971$	−790825.	+	790825.i	−0.838768	+	0.838768i	−0.988697	−	0.149929i	$-0.952096\pi$
$971$	−790825.	+	790825.i	−0.838768	+	0.838768i	0.149929	+	0.988697i	$0.452096\pi$
$972$		0			0
$973$	−746058.	+	746058.i	−0.788037	+	0.788037i
$974$		0			0
$975$			892.258i			0.000938602i
$976$		0			0
$977$	−1.40812e6			−1.47520			−0.737600	−	0.675237i	$-0.764041\pi$
$977$	−1.40812e6			−1.47520			−0.737600	+	0.675237i	$0.764041\pi$
$978$		0			0
$979$	542600.	+	542600.i	0.566128	+	0.566128i
$980$		0			0
$981$	−672972.	−	672972.i	−0.699292	−	0.699292i
$982$		0			0
$983$	−208227.			−0.215491			−0.107746	−	0.994179i	$-0.534363\pi$
$983$	−208227.			−0.215491			−0.107746	+	0.994179i	$0.534363\pi$
$984$		0			0
$985$		−	307144.i		−	0.316569i
$986$		0			0
$987$	257.903	−	257.903i	0.000264742	−	0.000264742i
$988$		0			0
$989$	371486.	−	371486.i	0.379796	−	0.379796i
$990$		0			0
$991$			170063.i			0.173166i	0.996245	+	0.0865831i	$0.0275948\pi$
$991$			170063.i			0.173166i	−0.996245	+	0.0865831i	$0.972405\pi$
$992$		0			0
$993$	−6019.00			−0.00610416
$994$		0			0
$995$	546631.	+	546631.i	0.552138	+	0.552138i
$996$		0			0
$997$	917799.	+	917799.i	0.923331	+	0.923331i	0.997263	−	0.0739326i	$-0.0235550\pi$
$997$	917799.	+	917799.i	0.923331	+	0.923331i	−0.0739326	+	0.997263i	$0.523555\pi$
$998$		0			0
$999$	6500.15			0.00651317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.f.b.31.4		14
4.3	odd	2		128.5.f.a.31.4		14
8.3	odd	2		64.5.f.a.15.4		14
8.5	even	2		16.5.f.a.11.7	yes	14
16.3	odd	4	inner	128.5.f.b.95.4		14
16.5	even	4		64.5.f.a.47.4		14
16.11	odd	4		16.5.f.a.3.7	✓	14
16.13	even	4		128.5.f.a.95.4		14
24.5	odd	2		144.5.m.a.91.1		14
24.11	even	2		576.5.m.a.271.5		14
48.5	odd	4		576.5.m.a.559.5		14
48.11	even	4		144.5.m.a.19.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.7	✓	14	16.11	odd	4
16.5.f.a.11.7	yes	14	8.5	even	2
64.5.f.a.15.4		14	8.3	odd	2
64.5.f.a.47.4		14	16.5	even	4
128.5.f.a.31.4		14	4.3	odd	2
128.5.f.a.95.4		14	16.13	even	4
128.5.f.b.31.4		14	1.1	even	1	trivial
128.5.f.b.95.4		14	16.3	odd	4	inner
144.5.m.a.19.1		14	48.11	even	4
144.5.m.a.91.1		14	24.5	odd	2
576.5.m.a.271.5		14	24.11	even	2
576.5.m.a.559.5		14	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.f (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0461995 + 0.0461995i) q^{3} +(8.04297 + 8.04297i) q^{5} -49.8797 q^{7} -80.9957i q^{9} +(84.2573 - 84.2573i) q^{11} +(-19.4838 + 19.4838i) q^{13} +0.743162i q^{15} +437.855 q^{17} +(-349.021 - 349.021i) q^{19} +(-2.30442 - 2.30442i) q^{21} -404.840 q^{23} -495.621i q^{25} +(7.48412 - 7.48412i) q^{27} +(1031.65 - 1031.65i) q^{29} -1506.15i q^{31} +7.78529 q^{33} +(-401.181 - 401.181i) q^{35} +(434.262 + 434.262i) q^{37} -1.80028 q^{39} +696.847i q^{41} +(-917.612 + 917.612i) q^{43} +(651.446 - 651.446i) q^{45} +111.917i q^{47} +86.9810 q^{49} +(20.2287 + 20.2287i) q^{51} +(-1041.19 - 1041.19i) q^{53} +1355.36 q^{55} -32.2492i q^{57} +(1711.60 - 1711.60i) q^{59} +(-3711.24 + 3711.24i) q^{61} +4040.04i q^{63} -313.415 q^{65} +(1854.18 + 1854.18i) q^{67} +(-18.7034 - 18.7034i) q^{69} -1161.89 q^{71} -905.295i q^{73} +(22.8975 - 22.8975i) q^{75} +(-4202.72 + 4202.72i) q^{77} +5869.63i q^{79} -6559.96 q^{81} +(7560.06 + 7560.06i) q^{83} +(3521.65 + 3521.65i) q^{85} +95.3236 q^{87} +6439.80i q^{89} +(971.844 - 971.844i) q^{91} +(69.5835 - 69.5835i) q^{93} -5614.33i q^{95} -413.032 q^{97} +(-6824.48 - 6824.48i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{3} + 2 q^{5} - 4 q^{7} - 94 q^{11} + 2 q^{13} - 4 q^{17} + 706 q^{19} + 164 q^{21} + 1148 q^{23} + 1664 q^{27} - 862 q^{29} - 4 q^{33} - 1340 q^{35} + 1826 q^{37} + 2684 q^{39} - 1694 q^{43}+ \cdots - 49214 q^{99}+O(q^{100}) \) 14 * q + 2 * q^3 + 2 * q^5 - 4 * q^7 - 94 * q^11 + 2 * q^13 - 4 * q^17 + 706 * q^19 + 164 * q^21 + 1148 * q^23 + 1664 * q^27 - 862 * q^29 - 4 * q^33 - 1340 * q^35 + 1826 * q^37 + 2684 * q^39 - 1694 * q^43 - 1410 * q^45 + 682 * q^49 + 3012 * q^51 + 482 * q^53 - 11780 * q^55 + 2786 * q^59 + 3778 * q^61 - 2020 * q^65 - 7998 * q^67 - 9628 * q^69 + 19964 * q^71 - 17570 * q^75 + 9508 * q^77 + 1454 * q^81 + 17282 * q^83 - 9948 * q^85 - 49284 * q^87 + 28036 * q^91 - 8896 * q^93 - 4 * q^97 - 49214 * q^99

Introduction

L-functions

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