LMFDB - Embedded newform 1440.5.g.a.271.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,5,Mod(271,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))

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

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

chi := DirichletCharacter("1440.271");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1440.g (of order $2$, degree $1$, 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:	$148.852746841$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 14 x^{14} - 84 x^{13} + 628 x^{12} - 1392 x^{11} + 2016 x^{10} - 18048 x^{9} + \cdots + 4294967296 $ x^16 - 6x^15 + 14x^14 - 84x^13 + 628x^12 - 1392x^11 + 2016x^10 - 18048x^9 + 116992x^8 - 288768x^7 + 516096x^6 - 5701632x^5 + 41156608x^4 - 88080384x^3 + 234881024x^2 - 1610612736*x + 4294967296
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{56}\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.5
Root			$-3.73120 - 1.44159i$ of defining polynomial
Character	$\chi$	$=$	1440.271
Dual form			1440.5.g.a.271.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.1803i q^{5} -2.59084i q^{7} -7.95663 q^{11} +54.6467i q^{13} +28.7782 q^{17} +316.702 q^{19} +846.949i q^{23} -125.000 q^{25} +766.738i q^{29} -1194.00i q^{31} -28.9665 q^{35} -2436.16i q^{37} -2433.43 q^{41} -2338.16 q^{43} -2192.51i q^{47} +2394.29 q^{49} +3041.70i q^{53} +88.9578i q^{55} +455.150 q^{59} +3920.95i q^{61} +610.969 q^{65} +4822.82 q^{67} +874.716i q^{71} -7682.80 q^{73} +20.6144i q^{77} -1475.37i q^{79} +6849.50 q^{83} -321.750i q^{85} -13113.1 q^{89} +141.581 q^{91} -3540.84i q^{95} -8603.40 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 192 q^{11} + 704 q^{19} - 2000 q^{25} + 2208 q^{41} - 5568 q^{43} - 2480 q^{49} + 14016 q^{59} + 18880 q^{67} - 7360 q^{73} + 10560 q^{83} - 6816 q^{89} - 24576 q^{91} + 448 q^{97}+O(q^{100}) $ 16 * q + 192 * q^11 + 704 * q^19 - 2000 * q^25 + 2208 * q^41 - 5568 * q^43 - 2480 * q^49 + 14016 * q^59 + 18880 * q^67 - 7360 * q^73 + 10560 * q^83 - 6816 * q^89 - 24576 * q^91 + 448 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	− 2.59084i	− 0.0528743i	−0.999650	−	0.0264372i	$-0.991584\pi$
$7$	− 2.59084i	− 0.0528743i	0.999650	−	0.0264372i	$-0.00841619\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−7.95663	−0.0657573	−0.0328786	−	0.999459i	$-0.510467\pi$
$11$	−7.95663	−0.0657573	−0.0328786	+	0.999459i	$0.510467\pi$
$12$	0	0
$13$	54.6467i	0.323353i	0.986844	+	0.161677i	$0.0516901\pi$
$13$	54.6467i	0.323353i	−0.986844	+	0.161677i	$0.948310\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	28.7782	0.0995785	0.0497892	−	0.998760i	$-0.484145\pi$
$17$	28.7782	0.0995785	0.0497892	+	0.998760i	$0.484145\pi$
$18$	0	0
$19$	316.702	0.877291	0.438646	−	0.898660i	$-0.355458\pi$
$19$	316.702	0.877291	0.438646	+	0.898660i	$0.355458\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	846.949i	1.60104i	0.599308	+	0.800519i	$0.295443\pi$
$23$	846.949i	1.60104i	−0.599308	+	0.800519i	$0.704557\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	766.738i	0.911698i	0.890057	+	0.455849i	$0.150664\pi$
$29$	766.738i	0.911698i	−0.890057	+	0.455849i	$0.849336\pi$
$30$	0	0
$31$	− 1194.00i	− 1.24245i	−0.783631	−	0.621227i	$-0.786634\pi$
$31$	− 1194.00i	− 1.24245i	0.783631	−	0.621227i	$-0.213366\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−28.9665	−0.0236461
$36$	0	0
$37$	− 2436.16i	− 1.77952i	−0.456432	−	0.889758i	$-0.650873\pi$
$37$	− 2436.16i	− 1.77952i	0.456432	−	0.889758i	$-0.349127\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2433.43	−1.44761	−0.723806	−	0.690004i	$-0.757609\pi$
$41$	−2433.43	−1.44761	−0.723806	+	0.690004i	$0.757609\pi$
$42$	0	0
$43$	−2338.16	−1.26455	−0.632277	−	0.774742i	$-0.717880\pi$
$43$	−2338.16	−1.26455	−0.632277	+	0.774742i	$0.717880\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2192.51i	− 0.992537i	−0.868169	−	0.496269i	$-0.834703\pi$
$47$	− 2192.51i	− 0.992537i	0.868169	−	0.496269i	$-0.165297\pi$
$48$	0	0
$49$	2394.29	0.997204
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3041.70i	1.08284i	0.840752	+	0.541420i	$0.182113\pi$
$53$	3041.70i	1.08284i	−0.840752	+	0.541420i	$0.817887\pi$
$54$	0	0
$55$	88.9578i	0.0294075i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	455.150	0.130753	0.0653764	−	0.997861i	$-0.479175\pi$
$59$	455.150	0.130753	0.0653764	+	0.997861i	$0.479175\pi$
$60$	0	0
$61$	3920.95i	1.05374i	0.849947	+	0.526868i	$0.176634\pi$
$61$	3920.95i	1.05374i	−0.849947	+	0.526868i	$0.823366\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	610.969	0.144608
$66$	0	0
$67$	4822.82	1.07436	0.537182	−	0.843466i	$-0.319489\pi$
$67$	4822.82	1.07436	0.537182	+	0.843466i	$0.319489\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	874.716i	0.173520i	0.996229	+	0.0867602i	$0.0276514\pi$
$71$	874.716i	0.173520i	−0.996229	+	0.0867602i	$0.972349\pi$
$72$	0	0
$73$	−7682.80	−1.44170	−0.720848	−	0.693093i	$-0.756248\pi$
$73$	−7682.80	−1.44170	−0.720848	+	0.693093i	$0.756248\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.6144i	0.00347687i
$78$	0	0
$79$	− 1475.37i	− 0.236400i	−0.992990	−	0.118200i	$-0.962288\pi$
$79$	− 1475.37i	− 0.236400i	0.992990	−	0.118200i	$-0.0377123\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6849.50	0.994266	0.497133	−	0.867674i	$-0.334386\pi$
$83$	6849.50	0.994266	0.497133	+	0.867674i	$0.334386\pi$
$84$	0	0
$85$	− 321.750i	− 0.0445328i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13113.1	−1.65548	−0.827740	−	0.561111i	$-0.810374\pi$
$89$	−13113.1	−1.65548	−0.827740	+	0.561111i	$0.810374\pi$
$90$	0	0
$91$	141.581	0.0170971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 3540.84i	− 0.392337i
$96$	0	0
$97$	−8603.40	−0.914380	−0.457190	−	0.889369i	$-0.651144\pi$
$97$	−8603.40	−0.914380	−0.457190	+	0.889369i	$0.651144\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7164.77i	0.702360i	0.936308	+	0.351180i	$0.114219\pi$
$101$	7164.77i	0.702360i	−0.936308	+	0.351180i	$0.885781\pi$
$102$	0	0
$103$	− 8010.36i	− 0.755053i	−0.925999	−	0.377527i	$-0.876775\pi$
$103$	− 8010.36i	− 0.755053i	0.925999	−	0.377527i	$-0.123225\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2003.50	−0.174994	−0.0874969	−	0.996165i	$-0.527887\pi$
$107$	−2003.50	−0.174994	−0.0874969	+	0.996165i	$0.527887\pi$
$108$	0	0
$109$	20282.4i	1.70713i	0.520985	+	0.853566i	$0.325565\pi$
$109$	20282.4i	1.70713i	−0.520985	+	0.853566i	$0.674435\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	570.501	0.0446786	0.0223393	−	0.999750i	$-0.492889\pi$
$113$	570.501	0.0446786	0.0223393	+	0.999750i	$0.492889\pi$
$114$	0	0
$115$	9469.18	0.716006
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 74.5597i	− 0.00526514i
$120$	0	0
$121$	−14577.7	−0.995676
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	6367.53i	0.394788i	0.980324	+	0.197394i	$0.0632478\pi$
$127$	6367.53i	0.394788i	−0.980324	+	0.197394i	$0.936752\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17348.8	1.01094	0.505472	−	0.862843i	$-0.331319\pi$
$131$	17348.8	1.01094	0.505472	+	0.862843i	$0.331319\pi$
$132$	0	0
$133$	− 820.525i	− 0.0463862i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	425.657	0.0226787	0.0113394	−	0.999936i	$-0.496390\pi$
$137$	425.657	0.0226787	0.0113394	+	0.999936i	$0.496390\pi$
$138$	0	0
$139$	21843.4	1.13055	0.565276	−	0.824902i	$-0.308770\pi$
$139$	21843.4	1.13055	0.565276	+	0.824902i	$0.308770\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 434.804i	− 0.0212628i
$144$	0	0
$145$	8572.39	0.407724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19745.1i	0.889381i	0.895684	+	0.444690i	$0.146686\pi$
$149$	19745.1i	0.889381i	−0.895684	+	0.444690i	$0.853314\pi$
$150$	0	0
$151$	− 28379.1i	− 1.24464i	−0.782761	−	0.622322i	$-0.786189\pi$
$151$	− 28379.1i	− 1.24464i	0.782761	−	0.622322i	$-0.213811\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13349.3	−0.555642
$156$	0	0
$157$	− 4457.00i	− 0.180819i	−0.995905	−	0.0904094i	$-0.971182\pi$
$157$	− 4457.00i	− 0.180819i	0.995905	−	0.0904094i	$-0.0288176\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2194.31	0.0846538
$162$	0	0
$163$	1910.59	0.0719107	0.0359553	−	0.999353i	$-0.488553\pi$
$163$	1910.59	0.0719107	0.0359553	+	0.999353i	$0.488553\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	50764.6i	1.82024i	0.414347	+	0.910119i	$0.364010\pi$
$167$	50764.6i	1.82024i	−0.414347	+	0.910119i	$0.635990\pi$
$168$	0	0
$169$	25574.7	0.895443
$170$	0	0
$171$	0	0
$172$	0	0
$173$	42367.9i	1.41561i	0.706407	+	0.707806i	$0.250315\pi$
$173$	42367.9i	1.41561i	−0.706407	+	0.707806i	$0.749685\pi$
$174$	0	0
$175$	323.855i	0.0105749i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−51902.2	−1.61987	−0.809934	−	0.586521i	$-0.800497\pi$
$179$	−51902.2	−1.61987	−0.809934	+	0.586521i	$0.800497\pi$
$180$	0	0
$181$	8176.63i	0.249584i	0.992183	+	0.124792i	$0.0398264\pi$
$181$	8176.63i	0.249584i	−0.992183	+	0.124792i	$0.960174\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−27237.1	−0.795824
$186$	0	0
$187$	−228.977	−0.00654801
$188$	0	0
$189$	0	0
$190$	0	0
$191$	38144.4i	1.04560i	0.852456	+	0.522799i	$0.175112\pi$
$191$	38144.4i	1.04560i	−0.852456	+	0.522799i	$0.824888\pi$
$192$	0	0
$193$	−35295.2	−0.947547	−0.473774	−	0.880647i	$-0.657109\pi$
$193$	−35295.2	−0.947547	−0.473774	+	0.880647i	$0.657109\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6283.20i	0.161901i	0.996718	+	0.0809503i	$0.0257955\pi$
$197$	6283.20i	0.161901i	−0.996718	+	0.0809503i	$0.974204\pi$
$198$	0	0
$199$	− 20814.6i	− 0.525609i	−0.964849	−	0.262805i	$-0.915353\pi$
$199$	− 20814.6i	− 0.525609i	0.964849	−	0.262805i	$-0.0846474\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1986.50	0.0482054
$204$	0	0
$205$	27206.6i	0.647391i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2519.88	−0.0576883
$210$	0	0
$211$	−21192.8	−0.476017	−0.238009	−	0.971263i	$-0.576495\pi$
$211$	−21192.8	−0.476017	−0.238009	+	0.971263i	$0.576495\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	26141.4i	0.565526i
$216$	0	0
$217$	−3093.46	−0.0656939
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1572.63i	0.0321990i
$222$	0	0
$223$	46813.2i	0.941365i	0.882303	+	0.470683i	$0.155992\pi$
$223$	46813.2i	0.941365i	−0.882303	+	0.470683i	$0.844008\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1993.76	0.0386920	0.0193460	−	0.999813i	$-0.493842\pi$
$227$	1993.76	0.0386920	0.0193460	+	0.999813i	$0.493842\pi$
$228$	0	0
$229$	45591.9i	0.869394i	0.900577	+	0.434697i	$0.143144\pi$
$229$	45591.9i	0.869394i	−0.900577	+	0.434697i	$0.856856\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	82293.2	1.51584	0.757918	−	0.652350i	$-0.226217\pi$
$233$	82293.2	1.51584	0.757918	+	0.652350i	$0.226217\pi$
$234$	0	0
$235$	−24513.1	−0.443876
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 104832.i	− 1.83527i	−0.397426	−	0.917634i	$-0.630097\pi$
$239$	− 104832.i	− 1.83527i	0.397426	−	0.917634i	$-0.369903\pi$
$240$	0	0
$241$	10146.4	0.174693	0.0873467	−	0.996178i	$-0.472161\pi$
$241$	10146.4	0.174693	0.0873467	+	0.996178i	$0.472161\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 26768.9i	− 0.445963i
$246$	0	0
$247$	17306.7i	0.283675i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14884.6	0.236260	0.118130	−	0.992998i	$-0.462310\pi$
$251$	14884.6	0.236260	0.118130	+	0.992998i	$0.462310\pi$
$252$	0	0
$253$	− 6738.86i	− 0.105280i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−63894.6	−0.967382	−0.483691	−	0.875239i	$-0.660704\pi$
$257$	−63894.6	−0.967382	−0.483691	+	0.875239i	$0.660704\pi$
$258$	0	0
$259$	−6311.70	−0.0940907
$260$	0	0
$261$	0	0
$262$	0	0
$263$	46265.3i	0.668874i	0.942418	+	0.334437i	$0.108546\pi$
$263$	46265.3i	0.668874i	−0.942418	+	0.334437i	$0.891454\pi$
$264$	0	0
$265$	34007.2	0.484261
$266$	0	0
$267$	0	0
$268$	0	0
$269$	87276.9i	1.20613i	0.797691	+	0.603066i	$0.206054\pi$
$269$	87276.9i	1.20613i	−0.797691	+	0.603066i	$0.793946\pi$
$270$	0	0
$271$	142380.i	1.93869i	0.245696	+	0.969347i	$0.420984\pi$
$271$	142380.i	1.93869i	−0.245696	+	0.969347i	$0.579016\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	994.579	0.0131515
$276$	0	0
$277$	75663.5i	0.986114i	0.869997	+	0.493057i	$0.164121\pi$
$277$	75663.5i	0.986114i	−0.869997	+	0.493057i	$0.835879\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−43571.0	−0.551804	−0.275902	−	0.961186i	$-0.588977\pi$
$281$	−43571.0	−0.551804	−0.275902	+	0.961186i	$0.588977\pi$
$282$	0	0
$283$	−67132.6	−0.838225	−0.419112	−	0.907934i	$-0.637659\pi$
$283$	−67132.6	−0.838225	−0.419112	+	0.907934i	$0.637659\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6304.64i	0.0765415i
$288$	0	0
$289$	−82692.8	−0.990084
$290$	0	0
$291$	0	0
$292$	0	0
$293$	42456.8i	0.494552i	0.968945	+	0.247276i	$0.0795355\pi$
$293$	42456.8i	0.494552i	−0.968945	+	0.247276i	$0.920465\pi$
$294$	0	0
$295$	− 5088.73i	− 0.0584744i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−46283.0	−0.517701
$300$	0	0
$301$	6057.80i	0.0668624i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	43837.5	0.471245
$306$	0	0
$307$	22580.4	0.239582	0.119791	−	0.992799i	$-0.461777\pi$
$307$	22580.4	0.239582	0.119791	+	0.992799i	$0.461777\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	104096.i	1.07625i	0.842865	+	0.538125i	$0.180867\pi$
$311$	104096.i	1.07625i	−0.842865	+	0.538125i	$0.819133\pi$
$312$	0	0
$313$	−84004.7	−0.857462	−0.428731	−	0.903432i	$-0.641039\pi$
$313$	−84004.7	−0.857462	−0.428731	+	0.903432i	$0.641039\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 52955.1i	− 0.526974i	−0.964663	−	0.263487i	$-0.915127\pi$
$317$	− 52955.1i	− 0.526974i	0.964663	−	0.263487i	$-0.0848726\pi$
$318$	0	0
$319$	− 6100.65i	− 0.0599508i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9114.11	0.0873593
$324$	0	0
$325$	− 6830.84i	− 0.0646706i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5680.46	−0.0524797
$330$	0	0
$331$	61731.5	0.563444	0.281722	−	0.959496i	$-0.409094\pi$
$331$	61731.5	0.563444	0.281722	+	0.959496i	$0.409094\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 53920.8i	− 0.480471i
$336$	0	0
$337$	−15465.0	−0.136173	−0.0680865	−	0.997679i	$-0.521689\pi$
$337$	−15465.0	−0.136173	−0.0680865	+	0.997679i	$0.521689\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9500.20i	0.0817004i
$342$	0	0
$343$	− 12423.8i	− 0.105601i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−74344.2	−0.617431	−0.308715	−	0.951154i	$-0.599899\pi$
$347$	−74344.2	−0.617431	−0.308715	+	0.951154i	$0.599899\pi$
$348$	0	0
$349$	82400.1i	0.676514i	0.941054	+	0.338257i	$0.109837\pi$
$349$	82400.1i	0.676514i	−0.941054	+	0.338257i	$0.890163\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−153677.	−1.23327	−0.616635	−	0.787249i	$-0.711505\pi$
$353$	−153677.	−1.23327	−0.616635	+	0.787249i	$0.711505\pi$
$354$	0	0
$355$	9779.62	0.0776007
$356$	0	0
$357$	0	0
$358$	0	0
$359$	235554.i	1.82768i	0.406072	+	0.913841i	$0.366898\pi$
$359$	235554.i	1.82768i	−0.406072	+	0.913841i	$0.633102\pi$
$360$	0	0
$361$	−30020.8	−0.230360
$362$	0	0
$363$	0	0
$364$	0	0
$365$	85896.3i	0.644746i
$366$	0	0
$367$	112718.i	0.836878i	0.908245	+	0.418439i	$0.137423\pi$
$367$	112718.i	0.836878i	−0.908245	+	0.418439i	$0.862577\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7880.56	0.0572545
$372$	0	0
$373$	− 16080.3i	− 0.115578i	−0.998329	−	0.0577892i	$-0.981595\pi$
$373$	− 16080.3i	− 0.115578i	0.998329	−	0.0577892i	$-0.0184051\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−41899.7	−0.294801
$378$	0	0
$379$	−208815.	−1.45373	−0.726864	−	0.686781i	$-0.759023\pi$
$379$	−208815.	−1.45373	−0.726864	+	0.686781i	$0.759023\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	96057.8i	0.654840i	0.944879	+	0.327420i	$0.106179\pi$
$383$	96057.8i	0.654840i	−0.944879	+	0.327420i	$0.893821\pi$
$384$	0	0
$385$	230.476	0.00155490
$386$	0	0
$387$	0	0
$388$	0	0
$389$	168851.i	1.11584i	0.829893	+	0.557922i	$0.188401\pi$
$389$	168851.i	1.11584i	−0.829893	+	0.557922i	$0.811599\pi$
$390$	0	0
$391$	24373.6i	0.159429i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16495.1	−0.105721
$396$	0	0
$397$	− 189472.i	− 1.20216i	−0.799188	−	0.601081i	$-0.794737\pi$
$397$	− 189472.i	− 1.20216i	0.799188	−	0.601081i	$-0.205263\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−120359.	−0.748494	−0.374247	−	0.927329i	$-0.622099\pi$
$401$	−120359.	−0.748494	−0.374247	+	0.927329i	$0.622099\pi$
$402$	0	0
$403$	65248.1	0.401751
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19383.6i	0.117016i
$408$	0	0
$409$	150488.	0.899615	0.449807	−	0.893126i	$-0.351493\pi$
$409$	150488.	0.899615	0.449807	+	0.893126i	$0.351493\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1179.22i	− 0.00691346i
$414$	0	0
$415$	− 76579.7i	− 0.444649i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	288889.	1.64552	0.822761	−	0.568388i	$-0.192433\pi$
$419$	288889.	1.64552	0.822761	+	0.568388i	$0.192433\pi$
$420$	0	0
$421$	− 182304.i	− 1.02857i	−0.857620	−	0.514284i	$-0.828058\pi$
$421$	− 182304.i	− 1.02857i	0.857620	−	0.514284i	$-0.171942\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3597.27	−0.0199157
$426$	0	0
$427$	10158.6	0.0557155
$428$	0	0
$429$	0	0
$430$	0	0
$431$	100495.i	0.540992i	0.962721	+	0.270496i	$0.0871877\pi$
$431$	100495.i	0.540992i	−0.962721	+	0.270496i	$0.912812\pi$
$432$	0	0
$433$	328152.	1.75025	0.875124	−	0.483898i	$-0.160779\pi$
$433$	328152.	1.75025	0.875124	+	0.483898i	$0.160779\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	268230.i	1.40458i
$438$	0	0
$439$	− 155253.i	− 0.805582i	−0.915292	−	0.402791i	$-0.868040\pi$
$439$	− 155253.i	− 0.805582i	0.915292	−	0.402791i	$-0.131960\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−337851.	−1.72154	−0.860772	−	0.508991i	$-0.830019\pi$
$443$	−337851.	−1.72154	−0.860772	+	0.508991i	$0.830019\pi$
$444$	0	0
$445$	146608.i	0.740353i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	245472.	1.21761	0.608806	−	0.793319i	$-0.291649\pi$
$449$	245472.	1.21761	0.608806	+	0.793319i	$0.291649\pi$
$450$	0	0
$451$	19361.9	0.0951910
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1582.92i	− 0.00764605i
$456$	0	0
$457$	−115007.	−0.550671	−0.275336	−	0.961348i	$-0.588789\pi$
$457$	−115007.	−0.550671	−0.275336	+	0.961348i	$0.588789\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 115456.i	− 0.543270i	−0.962400	−	0.271635i	$-0.912436\pi$
$461$	− 115456.i	− 0.543270i	0.962400	−	0.271635i	$-0.0875644\pi$
$462$	0	0
$463$	308334.i	1.43833i	0.694837	+	0.719167i	$0.255476\pi$
$463$	308334.i	1.43833i	−0.694837	+	0.719167i	$0.744524\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−234153.	−1.07366	−0.536829	−	0.843691i	$-0.680378\pi$
$467$	−234153.	−1.07366	−0.536829	+	0.843691i	$0.680378\pi$
$468$	0	0
$469$	− 12495.2i	− 0.0568063i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18603.9	0.0831536
$474$	0	0
$475$	−39587.8	−0.175458
$476$	0	0
$477$	0	0
$478$	0	0
$479$	175880.i	0.766558i	0.923633	+	0.383279i	$0.125205\pi$
$479$	175880.i	0.766558i	−0.923633	+	0.383279i	$0.874795\pi$
$480$	0	0
$481$	133128.	0.575412
$482$	0	0
$483$	0	0
$484$	0	0
$485$	96188.9i	0.408923i
$486$	0	0
$487$	289085.i	1.21890i	0.792826	+	0.609449i	$0.208609\pi$
$487$	289085.i	1.21890i	−0.792826	+	0.609449i	$0.791391\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	182813.	0.758304	0.379152	−	0.925334i	$-0.376216\pi$
$491$	182813.	0.758304	0.379152	+	0.925334i	$0.376216\pi$
$492$	0	0
$493$	22065.3i	0.0907855i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2266.25	0.00917477
$498$	0	0
$499$	−463712.	−1.86229	−0.931145	−	0.364650i	$-0.881189\pi$
$499$	−463712.	−1.86229	−0.931145	+	0.364650i	$0.881189\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 173588.i	− 0.686094i	−0.939318	−	0.343047i	$-0.888541\pi$
$503$	− 173588.i	− 0.686094i	0.939318	−	0.343047i	$-0.111459\pi$
$504$	0	0
$505$	80104.6	0.314105
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 231066.i	− 0.891867i	−0.895066	−	0.445934i	$-0.852872\pi$
$509$	− 231066.i	− 0.891867i	0.895066	−	0.445934i	$-0.147128\pi$
$510$	0	0
$511$	19904.9i	0.0762287i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−89558.5	−0.337670
$516$	0	0
$517$	17445.0i	0.0652665i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39770.4	−0.146516	−0.0732578	−	0.997313i	$-0.523340\pi$
$521$	−39770.4	−0.146516	−0.0732578	+	0.997313i	$0.523340\pi$
$522$	0	0
$523$	71651.8	0.261953	0.130977	−	0.991385i	$-0.458189\pi$
$523$	71651.8	0.261953	0.130977	+	0.991385i	$0.458189\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 34361.1i	− 0.123722i
$528$	0	0
$529$	−437481.	−1.56332
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 132979.i	− 0.468090i
$534$	0	0
$535$	22399.9i	0.0782596i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−19050.5	−0.0655734
$540$	0	0
$541$	335147.i	1.14509i	0.819873	+	0.572546i	$0.194044\pi$
$541$	335147.i	1.14509i	−0.819873	+	0.572546i	$0.805956\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	226764.	0.763452
$546$	0	0
$547$	174207.	0.582225	0.291112	−	0.956689i	$-0.405975\pi$
$547$	174207.	0.582225	0.291112	+	0.956689i	$0.405975\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	242828.i	0.799825i
$552$	0	0
$553$	−3822.45	−0.0124995
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 179035.i	− 0.577069i	−0.957470	−	0.288534i	$-0.906832\pi$
$557$	− 179035.i	− 0.577069i	0.957470	−	0.288534i	$-0.0931680\pi$
$558$	0	0
$559$	− 127773.i	− 0.408898i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−126413.	−0.398819	−0.199410	−	0.979916i	$-0.563902\pi$
$563$	−126413.	−0.398819	−0.199410	+	0.979916i	$0.563902\pi$
$564$	0	0
$565$	− 6378.40i	− 0.0199809i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	160597.	0.496035	0.248017	−	0.968756i	$-0.420221\pi$
$569$	160597.	0.496035	0.248017	+	0.968756i	$0.420221\pi$
$570$	0	0
$571$	372210.	1.14160	0.570802	−	0.821088i	$-0.306633\pi$
$571$	372210.	1.14160	0.570802	+	0.821088i	$0.306633\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 105869.i	− 0.320208i
$576$	0	0
$577$	328066.	0.985393	0.492696	−	0.870201i	$-0.336011\pi$
$577$	328066.	0.985393	0.492696	+	0.870201i	$0.336011\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 17746.0i	− 0.0525712i
$582$	0	0
$583$	− 24201.7i	− 0.0712047i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	128807.	0.373822	0.186911	−	0.982377i	$-0.440152\pi$
$587$	128807.	0.373822	0.186911	+	0.982377i	$0.440152\pi$
$588$	0	0
$589$	− 378142.i	− 1.08999i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	210780.	0.599405	0.299703	−	0.954033i	$-0.403113\pi$
$593$	210780.	0.599405	0.299703	+	0.954033i	$0.403113\pi$
$594$	0	0
$595$	−833.603	−0.00235464
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 187457.i	− 0.522453i	−0.965278	−	0.261226i	$-0.915873\pi$
$599$	− 187457.i	− 0.522453i	0.965278	−	0.261226i	$-0.0841269\pi$
$600$	0	0
$601$	14463.8	0.0400436	0.0200218	−	0.999800i	$-0.493626\pi$
$601$	14463.8	0.0400436	0.0200218	+	0.999800i	$0.493626\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	162984.i	0.445280i
$606$	0	0
$607$	− 579553.i	− 1.57295i	−0.617620	−	0.786477i	$-0.711903\pi$
$607$	− 579553.i	− 1.57295i	0.617620	−	0.786477i	$-0.288097\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	119814.	0.320940
$612$	0	0
$613$	− 377775.i	− 1.00534i	−0.864479	−	0.502669i	$-0.832352\pi$
$613$	− 377775.i	− 1.00534i	0.864479	−	0.502669i	$-0.167648\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−699981.	−1.83872	−0.919361	−	0.393414i	$-0.871294\pi$
$617$	−699981.	−1.83872	−0.919361	+	0.393414i	$0.871294\pi$
$618$	0	0
$619$	117981.	0.307915	0.153957	−	0.988077i	$-0.450798\pi$
$619$	117981.	0.307915	0.153957	+	0.988077i	$0.450798\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	33973.9i	0.0875324i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 70108.2i	− 0.177202i
$630$	0	0
$631$	749113.i	1.88143i	0.339193	+	0.940717i	$0.389846\pi$
$631$	749113.i	1.88143i	−0.339193	+	0.940717i	$0.610154\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	71191.1	0.176554
$636$	0	0
$637$	130840.i	0.322449i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−190520.	−0.463686	−0.231843	−	0.972753i	$-0.574475\pi$
$641$	−190520.	−0.463686	−0.231843	+	0.972753i	$0.574475\pi$
$642$	0	0
$643$	−272152.	−0.658248	−0.329124	−	0.944287i	$-0.606753\pi$
$643$	−272152.	−0.658248	−0.329124	+	0.944287i	$0.606753\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26959.8i	− 0.0644034i	−0.999481	−	0.0322017i	$-0.989748\pi$
$647$	− 26959.8i	− 0.0644034i	0.999481	−	0.0322017i	$-0.0102519\pi$
$648$	0	0
$649$	−3621.46	−0.00859794
$650$	0	0
$651$	0	0
$652$	0	0
$653$	497282.i	1.16621i	0.812397	+	0.583104i	$0.198162\pi$
$653$	497282.i	1.16621i	−0.812397	+	0.583104i	$0.801838\pi$
$654$	0	0
$655$	− 193965.i	− 0.452108i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	150938.	0.347558	0.173779	−	0.984785i	$-0.444402\pi$
$659$	150938.	0.347558	0.173779	+	0.984785i	$0.444402\pi$
$660$	0	0
$661$	− 535544.i	− 1.22572i	−0.790191	−	0.612861i	$-0.790018\pi$
$661$	− 535544.i	− 1.22572i	0.790191	−	0.612861i	$-0.209982\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9173.75	−0.0207445
$666$	0	0
$667$	−649388.	−1.45966
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 31197.5i	− 0.0692908i
$672$	0	0
$673$	−189265.	−0.417868	−0.208934	−	0.977930i	$-0.566999\pi$
$673$	−189265.	−0.417868	−0.208934	+	0.977930i	$0.566999\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 464037.i	− 1.01245i	−0.862400	−	0.506227i	$-0.831040\pi$
$677$	− 464037.i	− 1.01245i	0.862400	−	0.506227i	$-0.168960\pi$
$678$	0	0
$679$	22290.0i	0.0483472i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42942.9	−0.0920556	−0.0460278	−	0.998940i	$-0.514656\pi$
$683$	−42942.9	−0.0920556	−0.0460278	+	0.998940i	$0.514656\pi$
$684$	0	0
$685$	− 4758.99i	− 0.0101422i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−166219.	−0.350140
$690$	0	0
$691$	695800.	1.45723	0.728615	−	0.684924i	$-0.240164\pi$
$691$	695800.	1.45723	0.728615	+	0.684924i	$0.240164\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 244216.i	− 0.505598i
$696$	0	0
$697$	−70029.8	−0.144151
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 243528.i	− 0.495580i	−0.968814	−	0.247790i	$-0.920296\pi$
$701$	− 243528.i	− 0.495580i	0.968814	−	0.247790i	$-0.0797043\pi$
$702$	0	0
$703$	− 771536.i	− 1.56115i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18562.8	0.0371368
$708$	0	0
$709$	210565.i	0.418883i	0.977821	+	0.209442i	$0.0671646\pi$
$709$	210565.i	0.418883i	−0.977821	+	0.209442i	$0.932835\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.01126e6	1.98922
$714$	0	0
$715$	−4861.25	−0.00950903
$716$	0	0
$717$	0	0
$718$	0	0
$719$	174495.i	0.337539i	0.985656	+	0.168770i	$0.0539794\pi$
$719$	174495.i	0.337539i	−0.985656	+	0.168770i	$0.946021\pi$
$720$	0	0
$721$	−20753.6	−0.0399229
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 95842.3i	− 0.182340i
$726$	0	0
$727$	− 511384.i	− 0.967561i	−0.875189	−	0.483781i	$-0.839263\pi$
$727$	− 511384.i	− 0.967561i	0.875189	−	0.483781i	$-0.160737\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−67288.0	−0.125922
$732$	0	0
$733$	384728.i	0.716055i	0.933711	+	0.358027i	$0.116551\pi$
$733$	384728.i	0.716055i	−0.933711	+	0.358027i	$0.883449\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38373.4	−0.0706473
$738$	0	0
$739$	66307.5	0.121415	0.0607077	−	0.998156i	$-0.480664\pi$
$739$	66307.5	0.121415	0.0607077	+	0.998156i	$0.480664\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 786533.i	− 1.42475i	−0.701798	−	0.712376i	$-0.747619\pi$
$743$	− 786533.i	− 1.42475i	0.701798	−	0.712376i	$-0.252381\pi$
$744$	0	0
$745$	220757.	0.397743
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5190.76i	0.00925268i
$750$	0	0
$751$	− 743780.i	− 1.31876i	−0.751811	−	0.659379i	$-0.770819\pi$
$751$	− 743780.i	− 1.31876i	0.751811	−	0.659379i	$-0.229181\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−317288.	−0.556622
$756$	0	0
$757$	378393.i	0.660315i	0.943926	+	0.330158i	$0.107102\pi$
$757$	378393.i	0.660315i	−0.943926	+	0.330158i	$0.892898\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	487660.	0.842070	0.421035	−	0.907044i	$-0.361667\pi$
$761$	487660.	0.842070	0.421035	+	0.907044i	$0.361667\pi$
$762$	0	0
$763$	52548.6	0.0902634
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24872.5i	0.0422793i
$768$	0	0
$769$	−264950.	−0.448034	−0.224017	−	0.974585i	$-0.571917\pi$
$769$	−264950.	−0.448034	−0.224017	+	0.974585i	$0.571917\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	515794.i	0.863212i	0.902062	+	0.431606i	$0.142053\pi$
$773$	515794.i	0.863212i	−0.902062	+	0.431606i	$0.857947\pi$
$774$	0	0
$775$	149250.i	0.248491i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−770674.	−1.26998
$780$	0	0
$781$	− 6959.79i	− 0.0114102i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−49830.8	−0.0808646
$786$	0	0
$787$	537891.	0.868449	0.434225	−	0.900805i	$-0.357022\pi$
$787$	537891.	0.868449	0.434225	+	0.900805i	$0.357022\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1478.08i	− 0.00236235i
$792$	0	0
$793$	−214267.	−0.340729
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 839237.i	− 1.32120i	−0.750739	−	0.660599i	$-0.770302\pi$
$797$	− 839237.i	− 1.32120i	0.750739	−	0.660599i	$-0.229698\pi$
$798$	0	0
$799$	− 63096.6i	− 0.0988353i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	61129.2	0.0948021
$804$	0	0
$805$	− 24533.1i	− 0.0378583i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−514568.	−0.786223	−0.393111	−	0.919491i	$-0.628601\pi$
$809$	−514568.	−0.786223	−0.393111	+	0.919491i	$0.628601\pi$
$810$	0	0
$811$	−174798.	−0.265763	−0.132881	−	0.991132i	$-0.542423\pi$
$811$	−174798.	−0.265763	−0.132881	+	0.991132i	$0.542423\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 21361.1i	− 0.0321594i
$816$	0	0
$817$	−740500.	−1.10938
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.02886e6i	− 1.52641i	−0.646159	−	0.763203i	$-0.723626\pi$
$821$	− 1.02886e6i	− 1.52641i	0.646159	−	0.763203i	$-0.276374\pi$
$822$	0	0
$823$	54452.4i	0.0803928i	0.999192	+	0.0401964i	$0.0127984\pi$
$823$	54452.4i	0.0803928i	−0.999192	+	0.0401964i	$0.987202\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.20898e6	1.76770	0.883848	−	0.467773i	$-0.154944\pi$
$827$	1.20898e6	1.76770	0.883848	+	0.467773i	$0.154944\pi$
$828$	0	0
$829$	366682.i	0.533556i	0.963758	+	0.266778i	$0.0859591\pi$
$829$	366682.i	0.533556i	−0.963758	+	0.266778i	$0.914041\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	68903.2	0.0993001
$834$	0	0
$835$	567566.	0.814035
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1.27074e6i	− 1.80523i	−0.430444	−	0.902617i	$-0.641643\pi$
$839$	− 1.27074e6i	− 1.80523i	0.430444	−	0.902617i	$-0.358357\pi$
$840$	0	0
$841$	119393.	0.168806
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 285934.i	− 0.400454i
$846$	0	0
$847$	37768.5i	0.0526457i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.06330e6	2.84907
$852$	0	0
$853$	927864.i	1.27522i	0.770358	+	0.637612i	$0.220077\pi$
$853$	927864.i	1.27522i	−0.770358	+	0.637612i	$0.779923\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−155651.	−0.211928	−0.105964	−	0.994370i	$-0.533793\pi$
$857$	−155651.	−0.211928	−0.105964	+	0.994370i	$0.533793\pi$
$858$	0	0
$859$	1.04127e6	1.41116	0.705578	−	0.708632i	$-0.250687\pi$
$859$	1.04127e6	1.41116	0.705578	+	0.708632i	$0.250687\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.42220e6i	− 1.90959i	−0.297264	−	0.954795i	$-0.596074\pi$
$863$	− 1.42220e6i	− 1.90959i	0.297264	−	0.954795i	$-0.403926\pi$
$864$	0	0
$865$	473687.	0.633081
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11739.0i	0.0155450i
$870$	0	0
$871$	263551.i	0.347399i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3620.81	0.00472922
$876$	0	0
$877$	− 577824.i	− 0.751271i	−0.926768	−	0.375635i	$-0.877424\pi$
$877$	− 577824.i	− 0.751271i	0.926768	−	0.375635i	$-0.122576\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	95874.2	0.123524	0.0617618	−	0.998091i	$-0.480328\pi$
$881$	95874.2	0.123524	0.0617618	+	0.998091i	$0.480328\pi$
$882$	0	0
$883$	−1.28768e6	−1.65153	−0.825765	−	0.564014i	$-0.809256\pi$
$883$	−1.28768e6	−1.65153	−0.825765	+	0.564014i	$0.809256\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.17289e6i	− 1.49077i	−0.666633	−	0.745386i	$-0.732265\pi$
$887$	− 1.17289e6i	− 1.49077i	0.666633	−	0.745386i	$-0.267735\pi$
$888$	0	0
$889$	16497.3	0.0208741
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 694374.i	− 0.870744i
$894$	0	0
$895$	580284.i	0.724427i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	915484.	1.13274
$900$	0	0
$901$	87534.6i	0.107828i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	91417.5	0.111617
$906$	0	0
$907$	−410980.	−0.499581	−0.249791	−	0.968300i	$-0.580362\pi$
$907$	−410980.	−0.499581	−0.249791	+	0.968300i	$0.580362\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 67611.8i	− 0.0814678i	−0.999170	−	0.0407339i	$-0.987030\pi$
$911$	− 67611.8i	− 0.0814678i	0.999170	−	0.0407339i	$-0.0129696\pi$
$912$	0	0
$913$	−54498.9	−0.0653802
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 44948.0i	− 0.0534529i
$918$	0	0
$919$	423513.i	0.501459i	0.968057	+	0.250729i	$0.0806705\pi$
$919$	423513.i	0.501459i	−0.968057	+	0.250729i	$0.919330\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−47800.4	−0.0561084
$924$	0	0
$925$	304520.i	0.355903i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−963117.	−1.11596	−0.557979	−	0.829855i	$-0.688423\pi$
$929$	−963117.	−1.11596	−0.557979	+	0.829855i	$0.688423\pi$
$930$	0	0
$931$	758276.	0.874839
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2560.04i	0.00292836i
$936$	0	0
$937$	−1.35035e6	−1.53804	−0.769019	−	0.639226i	$-0.779255\pi$
$937$	−1.35035e6	−1.53804	−0.769019	+	0.639226i	$0.779255\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	651491.i	0.735748i	0.929876	+	0.367874i	$0.119914\pi$
$941$	651491.i	0.735748i	−0.929876	+	0.367874i	$0.880086\pi$
$942$	0	0
$943$	− 2.06099e6i	− 2.31768i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.31859e6	−1.47032	−0.735158	−	0.677896i	$-0.762892\pi$
$947$	−1.31859e6	−1.47032	−0.735158	+	0.677896i	$0.762892\pi$
$948$	0	0
$949$	− 419840.i	− 0.466177i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	76396.5	0.0841178	0.0420589	−	0.999115i	$-0.486608\pi$
$953$	76396.5	0.0841178	0.0420589	+	0.999115i	$0.486608\pi$
$954$	0	0
$955$	426468.	0.467605
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1102.81i	− 0.00119912i
$960$	0	0
$961$	−502111.	−0.543692
$962$	0	0
$963$	0	0
$964$	0	0
$965$	394612.i	0.423756i
$966$	0	0
$967$	462106.i	0.494184i	0.968992	+	0.247092i	$0.0794749\pi$
$967$	462106.i	0.494184i	−0.968992	+	0.247092i	$0.920525\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	901828.	0.956501	0.478250	−	0.878223i	$-0.341271\pi$
$971$	901828.	0.956501	0.478250	+	0.878223i	$0.341271\pi$
$972$	0	0
$973$	− 56592.7i	− 0.0597771i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46500.7	−0.0487159	−0.0243580	−	0.999703i	$-0.507754\pi$
$977$	−46500.7	−0.0487159	−0.0243580	+	0.999703i	$0.507754\pi$
$978$	0	0
$979$	104336.	0.108860
$980$	0	0
$981$	0	0
$982$	0	0
$983$	649309.i	0.671961i	0.941869	+	0.335981i	$0.109068\pi$
$983$	649309.i	0.671961i	−0.941869	+	0.335981i	$0.890932\pi$
$984$	0	0
$985$	70248.3	0.0724042
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.98030e6i	− 2.02460i
$990$	0	0
$991$	194704.i	0.198257i	0.995075	+	0.0991285i	$0.0316055\pi$
$991$	194704.i	0.198257i	−0.995075	+	0.0991285i	$0.968395\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−232715.	−0.235060
$996$	0	0
$997$	− 1.72172e6i	− 1.73210i	−0.499959	−	0.866049i	$-0.666652\pi$
$997$	− 1.72172e6i	− 1.73210i	0.499959	−	0.866049i	$-0.333348\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.5.g.a.271.5		16
3.2	odd	2		160.5.g.a.111.6		16
4.3	odd	2		360.5.g.a.91.2		16
8.3	odd	2	inner	1440.5.g.a.271.12		16
8.5	even	2		360.5.g.a.91.1		16
12.11	even	2		40.5.g.a.11.15	✓	16
15.2	even	4		800.5.e.e.399.16		32
15.8	even	4		800.5.e.e.399.17		32
15.14	odd	2		800.5.g.h.751.12		16
24.5	odd	2		40.5.g.a.11.16	yes	16
24.11	even	2		160.5.g.a.111.5		16
60.23	odd	4		200.5.e.e.99.13		32
60.47	odd	4		200.5.e.e.99.20		32
60.59	even	2		200.5.g.h.51.2		16
120.29	odd	2		200.5.g.h.51.1		16
120.53	even	4		200.5.e.e.99.19		32
120.59	even	2		800.5.g.h.751.11		16
120.77	even	4		200.5.e.e.99.14		32
120.83	odd	4		800.5.e.e.399.15		32
120.107	odd	4		800.5.e.e.399.18		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.15	✓	16	12.11	even	2
40.5.g.a.11.16	yes	16	24.5	odd	2
160.5.g.a.111.5		16	24.11	even	2
160.5.g.a.111.6		16	3.2	odd	2
200.5.e.e.99.13		32	60.23	odd	4
200.5.e.e.99.14		32	120.77	even	4
200.5.e.e.99.19		32	120.53	even	4
200.5.e.e.99.20		32	60.47	odd	4
200.5.g.h.51.1		16	120.29	odd	2
200.5.g.h.51.2		16	60.59	even	2
360.5.g.a.91.1		16	8.5	even	2
360.5.g.a.91.2		16	4.3	odd	2
800.5.e.e.399.15		32	120.83	odd	4
800.5.e.e.399.16		32	15.2	even	4
800.5.e.e.399.17		32	15.8	even	4
800.5.e.e.399.18		32	120.107	odd	4
800.5.g.h.751.11		16	120.59	even	2
800.5.g.h.751.12		16	15.14	odd	2
1440.5.g.a.271.5		16	1.1	even	1	trivial
1440.5.g.a.271.12		16	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1440.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-11.1803i q^{5} -2.59084i q^{7} -7.95663 q^{11} +54.6467i q^{13} +28.7782 q^{17} +316.702 q^{19} +846.949i q^{23} -125.000 q^{25} +766.738i q^{29} -1194.00i q^{31} -28.9665 q^{35} -2436.16i q^{37} -2433.43 q^{41} -2338.16 q^{43} -2192.51i q^{47} +2394.29 q^{49} +3041.70i q^{53} +88.9578i q^{55} +455.150 q^{59} +3920.95i q^{61} +610.969 q^{65} +4822.82 q^{67} +874.716i q^{71} -7682.80 q^{73} +20.6144i q^{77} -1475.37i q^{79} +6849.50 q^{83} -321.750i q^{85} -13113.1 q^{89} +141.581 q^{91} -3540.84i q^{95} -8603.40 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 192 q^{11} + 704 q^{19} - 2000 q^{25} + 2208 q^{41} - 5568 q^{43} - 2480 q^{49} + 14016 q^{59} + 18880 q^{67} - 7360 q^{73} + 10560 q^{83} - 6816 q^{89} - 24576 q^{91} + 448 q^{97}+O(q^{100}) \) 16 * q + 192 * q^11 + 704 * q^19 - 2000 * q^25 + 2208 * q^41 - 5568 * q^43 - 2480 * q^49 + 14016 * q^59 + 18880 * q^67 - 7360 * q^73 + 10560 * q^83 - 6816 * q^89 - 24576 * q^91 + 448 * q^97

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