LMFDB - Embedded newform 64.22.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	64.a (trivial)

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:	yes
Analytic conductor:	$178.865500344$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5201320x^{3} - 466399708x^{2} + 4990572086304x - 1473608896916400 $ x^5 - 2x^4 - 5201320x^3 - 466399708x^2 + 4990572086304x - 1473608896916400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{48}\cdot 3^{2}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2112.98$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-161942. q^{3} -9.10999e6 q^{5} +4.58736e8 q^{7} +1.57648e10 q^{9} +O(q^{10})$	$q-161942. q^{3} -9.10999e6 q^{5} +4.58736e8 q^{7} +1.57648e10 q^{9} -1.21568e11 q^{11} -2.06102e11 q^{13} +1.47529e12 q^{15} +6.23844e11 q^{17} +1.95292e13 q^{19} -7.42887e13 q^{21} +1.83895e14 q^{23} -3.93845e14 q^{25} -8.59019e14 q^{27} -3.78214e15 q^{29} -6.71258e15 q^{31} +1.96869e16 q^{33} -4.17908e15 q^{35} -1.47214e16 q^{37} +3.33766e16 q^{39} +1.63913e17 q^{41} +9.69596e16 q^{43} -1.43617e17 q^{45} +1.37998e17 q^{47} -3.48107e17 q^{49} -1.01026e17 q^{51} +9.54452e17 q^{53} +1.10748e18 q^{55} -3.16260e18 q^{57} +5.91287e18 q^{59} +8.61362e18 q^{61} +7.23190e18 q^{63} +1.87759e18 q^{65} -1.37765e18 q^{67} -2.97803e19 q^{69} +4.70038e19 q^{71} +2.45764e19 q^{73} +6.37801e19 q^{75} -5.57676e19 q^{77} +6.92981e19 q^{79} -2.57946e19 q^{81} +7.55315e19 q^{83} -5.68321e18 q^{85} +6.12488e20 q^{87} +2.43657e20 q^{89} -9.45466e19 q^{91} +1.08705e21 q^{93} -1.77911e20 q^{95} -1.23815e21 q^{97} -1.91650e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 23144 q^{3} - 18311174 q^{5} + 63978640 q^{7} + 2839988161 q^{9}+O(q^{10}) $ 5 * q - 23144 * q^3 - 18311174 * q^5 + 63978640 * q^7 + 2839988161 * q^9	$ 5 q - 23144 q^{3} - 18311174 q^{5} + 63978640 q^{7} + 2839988161 q^{9} - 25629588280 q^{11} - 26739996110 q^{13} + 850898706352 q^{15} - 88104593910 q^{17} - 13998239618440 q^{19} - 11868255565952 q^{21} + 191593435978416 q^{23} - 41142227484149 q^{25} - 436694294703248 q^{27} - 229446229587518 q^{29} - 30\!\cdots\!60 q^{31}+ \cdots - 45\!\cdots\!40 q^{99}+O(q^{100}) $ 5 * q - 23144 * q^3 - 18311174 * q^5 + 63978640 * q^7 + 2839988161 * q^9 - 25629588280 * q^11 - 26739996110 * q^13 + 850898706352 * q^15 - 88104593910 * q^17 - 13998239618440 * q^19 - 11868255565952 * q^21 + 191593435978416 * q^23 - 41142227484149 * q^25 - 436694294703248 * q^27 - 229446229587518 * q^29 - 3073873710228160 * q^31 + 5433145544904640 * q^33 + 8500692544521248 * q^35 + 11689624248645370 * q^37 + 5755218754185200 * q^39 + 73561781960295890 * q^41 - 59173471060274168 * q^43 + 110307182017064626 * q^45 + 189587882278171232 * q^47 + 198072470828678205 * q^49 - 60381606670165200 * q^51 + 448473144842566250 * q^53 - 276218022333741296 * q^55 - 419697547444918720 * q^57 + 580240954920275560 * q^59 - 60707231290899774 * q^61 + 7329049347459246928 * q^63 - 4215936008336579644 * q^65 - 20784270726401480552 * q^67 - 13151945925537436032 * q^69 + 48791036649307600400 * q^71 - 7687983545663740350 * q^73 - 63701791920635021848 * q^75 - 14991304956050303360 * q^77 + 150036082528549569440 * q^79 + 6961005137522997901 * q^81 - 320036481947166592392 * q^83 + 55351169878564568052 * q^85 + 845870037277169219440 * q^87 + 157308716989748010706 * q^89 - 1517228856976775217760 * q^91 + 285928010618243909120 * q^93 + 2644313286399100711792 * q^95 - 165820666835121495910 * q^97 - 4544456840208531002840 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−161942.	−1.58338	−0.791692	−	0.610921i	$-0.790799\pi$
$3$	−161942.	−1.58338	−0.791692	+	0.610921i	$0.790799\pi$
$4$	0	0
$5$	−9.10999e6	−0.417189	−0.208594	−	0.978002i	$-0.566889\pi$
$5$	−9.10999e6	−0.417189	−0.208594	+	0.978002i	$0.566889\pi$
$6$	0	0
$7$	4.58736e8	0.613810	0.306905	−	0.951740i	$-0.400707\pi$
$7$	4.58736e8	0.613810	0.306905	+	0.951740i	$0.400707\pi$
$8$	0	0
$9$	1.57648e10	1.50710
$10$	0	0
$11$	−1.21568e11	−1.41317	−0.706587	−	0.707627i	$-0.749766\pi$
$11$	−1.21568e11	−1.41317	−0.706587	+	0.707627i	$0.749766\pi$
$12$	0	0
$13$	−2.06102e11	−0.414646	−0.207323	−	0.978273i	$-0.566475\pi$
$13$	−2.06102e11	−0.414646	−0.207323	+	0.978273i	$0.566475\pi$
$14$	0	0
$15$	1.47529e12	0.660570
$16$	0	0
$17$	6.23844e11	0.0750520	0.0375260	−	0.999296i	$-0.488052\pi$
$17$	6.23844e11	0.0750520	0.0375260	+	0.999296i	$0.488052\pi$
$18$	0	0
$19$	1.95292e13	0.730755	0.365377	−	0.930859i	$-0.380940\pi$
$19$	1.95292e13	0.730755	0.365377	+	0.930859i	$0.380940\pi$
$20$	0	0
$21$	−7.42887e13	−0.971896
$22$	0	0
$23$	1.83895e14	0.925608	0.462804	−	0.886461i	$-0.346843\pi$
$23$	1.83895e14	0.925608	0.462804	+	0.886461i	$0.346843\pi$
$24$	0	0
$25$	−3.93845e14	−0.825953
$26$	0	0
$27$	−8.59019e14	−0.802940
$28$	0	0
$29$	−3.78214e15	−1.66940	−0.834698	−	0.550709i	$-0.814358\pi$
$29$	−3.78214e15	−1.66940	−0.834698	+	0.550709i	$0.814358\pi$
$30$	0	0
$31$	−6.71258e15	−1.47093	−0.735465	−	0.677563i	$-0.763036\pi$
$31$	−6.71258e15	−1.47093	−0.735465	+	0.677563i	$0.763036\pi$
$32$	0	0
$33$	1.96869e16	2.23760
$34$	0	0
$35$	−4.17908e15	−0.256075
$36$	0	0
$37$	−1.47214e16	−0.503304	−0.251652	−	0.967818i	$-0.580974\pi$
$37$	−1.47214e16	−0.503304	−0.251652	+	0.967818i	$0.580974\pi$
$38$	0	0
$39$	3.33766e16	0.656544
$40$	0	0
$41$	1.63913e17	1.90715	0.953573	−	0.301162i	$-0.0973745\pi$
$41$	1.63913e17	1.90715	0.953573	+	0.301162i	$0.0973745\pi$
$42$	0	0
$43$	9.69596e16	0.684182	0.342091	−	0.939667i	$-0.388865\pi$
$43$	9.69596e16	0.684182	0.342091	+	0.939667i	$0.388865\pi$
$44$	0	0
$45$	−1.43617e17	−0.628747
$46$	0	0
$47$	1.37998e17	0.382689	0.191344	−	0.981523i	$-0.438715\pi$
$47$	1.37998e17	0.382689	0.191344	+	0.981523i	$0.438715\pi$
$48$	0	0
$49$	−3.48107e17	−0.623238
$50$	0	0
$51$	−1.01026e17	−0.118836
$52$	0	0
$53$	9.54452e17	0.749648	0.374824	−	0.927096i	$-0.377703\pi$
$53$	9.54452e17	0.749648	0.374824	+	0.927096i	$0.377703\pi$
$54$	0	0
$55$	1.10748e18	0.589560
$56$	0	0
$57$	−3.16260e18	−1.15707
$58$	0	0
$59$	5.91287e18	1.50609	0.753047	−	0.657966i	$-0.228583\pi$
$59$	5.91287e18	1.50609	0.753047	+	0.657966i	$0.228583\pi$
$60$	0	0
$61$	8.61362e18	1.54605	0.773023	−	0.634378i	$-0.218744\pi$
$61$	8.61362e18	1.54605	0.773023	+	0.634378i	$0.218744\pi$
$62$	0	0
$63$	7.23190e18	0.925075
$64$	0	0
$65$	1.87759e18	0.172986
$66$	0	0
$67$	−1.37765e18	−0.0923323	−0.0461662	−	0.998934i	$-0.514700\pi$
$67$	−1.37765e18	−0.0923323	−0.0461662	+	0.998934i	$0.514700\pi$
$68$	0	0
$69$	−2.97803e19	−1.46559
$70$	0	0
$71$	4.70038e19	1.71364	0.856822	−	0.515613i	$-0.172436\pi$
$71$	4.70038e19	1.71364	0.856822	+	0.515613i	$0.172436\pi$
$72$	0	0
$73$	2.45764e19	0.669312	0.334656	−	0.942340i	$-0.391380\pi$
$73$	2.45764e19	0.669312	0.334656	+	0.942340i	$0.391380\pi$
$74$	0	0
$75$	6.37801e19	1.30780
$76$	0	0
$77$	−5.57676e19	−0.867419
$78$	0	0
$79$	6.92981e19	0.823449	0.411725	−	0.911308i	$-0.364927\pi$
$79$	6.92981e19	0.823449	0.411725	+	0.911308i	$0.364927\pi$
$80$	0	0
$81$	−2.57946e19	−0.235742
$82$	0	0
$83$	7.55315e19	0.534329	0.267164	−	0.963651i	$-0.413913\pi$
$83$	7.55315e19	0.534329	0.267164	+	0.963651i	$0.413913\pi$
$84$	0	0
$85$	−5.68321e18	−0.0313108
$86$	0	0
$87$	6.12488e20	2.64329
$88$	0	0
$89$	2.43657e20	0.828292	0.414146	−	0.910211i	$-0.364080\pi$
$89$	2.43657e20	0.828292	0.414146	+	0.910211i	$0.364080\pi$
$90$	0	0
$91$	−9.45466e19	−0.254514
$92$	0	0
$93$	1.08705e21	2.32905
$94$	0	0
$95$	−1.77911e20	−0.304863
$96$	0	0
$97$	−1.23815e21	−1.70479	−0.852397	−	0.522896i	$-0.824852\pi$
$97$	−1.23815e21	−1.70479	−0.852397	+	0.522896i	$0.824852\pi$
$98$	0	0
$99$	−1.91650e21	−2.12980
$100$	0	0
$101$	−8.95080e20	−0.806283	−0.403141	−	0.915138i	$-0.632082\pi$
$101$	−8.95080e20	−0.806283	−0.403141	+	0.915138i	$0.632082\pi$
$102$	0	0
$103$	5.58691e20	0.409620	0.204810	−	0.978802i	$-0.434342\pi$
$103$	5.58691e20	0.409620	0.204810	+	0.978802i	$0.434342\pi$
$104$	0	0
$105$	6.76769e20	0.405464
$106$	0	0
$107$	−3.61355e21	−1.77584	−0.887921	−	0.459997i	$-0.847851\pi$
$107$	−3.61355e21	−1.77584	−0.887921	+	0.459997i	$0.847851\pi$
$108$	0	0
$109$	−2.53077e21	−1.02394	−0.511969	−	0.859004i	$-0.671084\pi$
$109$	−2.53077e21	−1.02394	−0.511969	+	0.859004i	$0.671084\pi$
$110$	0	0
$111$	2.38401e21	0.796923
$112$	0	0
$113$	1.41603e21	0.392418	0.196209	−	0.980562i	$-0.437137\pi$
$113$	1.41603e21	0.392418	0.196209	+	0.980562i	$0.437137\pi$
$114$	0	0
$115$	−1.67528e21	−0.386153
$116$	0	0
$117$	−3.24917e21	−0.624915
$118$	0	0
$119$	2.86180e20	0.0460676
$120$	0	0
$121$	7.37848e21	0.997058
$122$	0	0
$123$	−2.65445e22	−3.01974
$124$	0	0
$125$	7.93191e21	0.761767
$126$	0	0
$127$	5.77088e21	0.469141	0.234570	−	0.972099i	$-0.424632\pi$
$127$	5.77088e21	0.469141	0.234570	+	0.972099i	$0.424632\pi$
$128$	0	0
$129$	−1.57018e22	−1.08332
$130$	0	0
$131$	−4.45950e21	−0.261781	−0.130890	−	0.991397i	$-0.541784\pi$
$131$	−4.45950e21	−0.261781	−0.130890	+	0.991397i	$0.541784\pi$
$132$	0	0
$133$	8.95876e21	0.448544
$134$	0	0
$135$	7.82565e21	0.334978
$136$	0	0
$137$	−3.85764e22	−1.41500	−0.707498	−	0.706715i	$-0.750176\pi$
$137$	−3.85764e22	−1.41500	−0.707498	+	0.706715i	$0.750176\pi$
$138$	0	0
$139$	2.87325e22	0.905144	0.452572	−	0.891728i	$-0.350507\pi$
$139$	2.87325e22	0.905144	0.452572	+	0.891728i	$0.350507\pi$
$140$	0	0
$141$	−2.23477e22	−0.605943
$142$	0	0
$143$	2.50554e22	0.585967
$144$	0	0
$145$	3.44553e22	0.696453
$146$	0	0
$147$	5.63731e22	0.986824
$148$	0	0
$149$	5.14968e22	0.782213	0.391106	−	0.920346i	$-0.372092\pi$
$149$	5.14968e22	0.782213	0.391106	+	0.920346i	$0.372092\pi$
$150$	0	0
$151$	9.85091e22	1.30082	0.650412	−	0.759582i	$-0.274596\pi$
$151$	9.85091e22	1.30082	0.650412	+	0.759582i	$0.274596\pi$
$152$	0	0
$153$	9.83479e21	0.113111
$154$	0	0
$155$	6.11515e22	0.613655
$156$	0	0
$157$	−2.05156e23	−1.79944	−0.899721	−	0.436464i	$-0.856230\pi$
$157$	−2.05156e23	−1.79944	−0.899721	+	0.436464i	$0.856230\pi$
$158$	0	0
$159$	−1.54566e23	−1.18698
$160$	0	0
$161$	8.43592e22	0.568147
$162$	0	0
$163$	2.30220e23	1.36199	0.680993	−	0.732289i	$-0.261548\pi$
$163$	2.30220e23	1.36199	0.680993	+	0.732289i	$0.261548\pi$
$164$	0	0
$165$	−1.79348e23	−0.933500
$166$	0	0
$167$	−1.17994e23	−0.541174	−0.270587	−	0.962696i	$-0.587218\pi$
$167$	−1.17994e23	−0.541174	−0.270587	+	0.962696i	$0.587218\pi$
$168$	0	0
$169$	−2.04586e23	−0.828069
$170$	0	0
$171$	3.07875e23	1.10132
$172$	0	0
$173$	−3.50956e22	−0.111114	−0.0555570	−	0.998456i	$-0.517693\pi$
$173$	−3.50956e22	−0.111114	−0.0555570	+	0.998456i	$0.517693\pi$
$174$	0	0
$175$	−1.80671e23	−0.506978
$176$	0	0
$177$	−9.57542e23	−2.38473
$178$	0	0
$179$	−3.90222e23	−0.863683	−0.431842	−	0.901949i	$-0.642136\pi$
$179$	−3.90222e23	−0.863683	−0.431842	+	0.901949i	$0.642136\pi$
$180$	0	0
$181$	4.30461e23	0.847831	0.423915	−	0.905702i	$-0.360655\pi$
$181$	4.30461e23	0.847831	0.423915	+	0.905702i	$0.360655\pi$
$182$	0	0
$183$	−1.39491e24	−2.44798
$184$	0	0
$185$	1.34112e23	0.209973
$186$	0	0
$187$	−7.58393e22	−0.106061
$188$	0	0
$189$	−3.94063e23	−0.492852
$190$	0	0
$191$	−3.08410e23	−0.345365	−0.172682	−	0.984978i	$-0.555243\pi$
$191$	−3.08410e23	−0.345365	−0.172682	+	0.984978i	$0.555243\pi$
$192$	0	0
$193$	−2.67062e23	−0.268077	−0.134039	−	0.990976i	$-0.542795\pi$
$193$	−2.67062e23	−0.268077	−0.134039	+	0.990976i	$0.542795\pi$
$194$	0	0
$195$	−3.04060e23	−0.273903
$196$	0	0
$197$	−1.37335e24	−1.11144	−0.555718	−	0.831371i	$-0.687557\pi$
$197$	−1.37335e24	−1.11144	−0.555718	+	0.831371i	$0.687557\pi$
$198$	0	0
$199$	−9.53826e22	−0.0694244	−0.0347122	−	0.999397i	$-0.511051\pi$
$199$	−9.53826e22	−0.0694244	−0.0347122	+	0.999397i	$0.511051\pi$
$200$	0	0
$201$	2.23100e23	0.146198
$202$	0	0
$203$	−1.73501e24	−1.02469
$204$	0	0
$205$	−1.49325e24	−0.795640
$206$	0	0
$207$	2.89907e24	1.39499
$208$	0	0
$209$	−2.37412e24	−1.03268
$210$	0	0
$211$	3.34901e24	1.31811	0.659053	−	0.752096i	$-0.270957\pi$
$211$	3.34901e24	1.31811	0.659053	+	0.752096i	$0.270957\pi$
$212$	0	0
$213$	−7.61189e24	−2.71335
$214$	0	0
$215$	−8.83300e23	−0.285433
$216$	0	0
$217$	−3.07931e24	−0.902871
$218$	0	0
$219$	−3.97996e24	−1.05978
$220$	0	0
$221$	−1.28576e23	−0.0311200
$222$	0	0
$223$	7.76557e24	1.70991	0.854953	−	0.518706i	$-0.173586\pi$
$223$	7.76557e24	1.70991	0.854953	+	0.518706i	$0.173586\pi$
$224$	0	0
$225$	−6.20891e24	−1.24480
$226$	0	0
$227$	−3.73550e24	−0.682460	−0.341230	−	0.939980i	$-0.610844\pi$
$227$	−3.73550e24	−0.682460	−0.341230	+	0.939980i	$0.610844\pi$
$228$	0	0
$229$	7.02225e24	1.17005	0.585024	−	0.811016i	$-0.301085\pi$
$229$	7.02225e24	1.17005	0.585024	+	0.811016i	$0.301085\pi$
$230$	0	0
$231$	9.03111e24	1.37346
$232$	0	0
$233$	−7.73354e24	−1.07434	−0.537169	−	0.843474i	$-0.680506\pi$
$233$	−7.73354e24	−1.07434	−0.537169	+	0.843474i	$0.680506\pi$
$234$	0	0
$235$	−1.25716e24	−0.159654
$236$	0	0
$237$	−1.12223e25	−1.30384
$238$	0	0
$239$	8.34188e24	0.887332	0.443666	−	0.896192i	$-0.353678\pi$
$239$	8.34188e24	0.887332	0.443666	+	0.896192i	$0.353678\pi$
$240$	0	0
$241$	−2.74514e24	−0.267539	−0.133769	−	0.991013i	$-0.542708\pi$
$241$	−2.74514e24	−0.267539	−0.133769	+	0.991013i	$0.542708\pi$
$242$	0	0
$243$	1.31629e25	1.17621
$244$	0	0
$245$	3.17125e24	0.260008
$246$	0	0
$247$	−4.02501e24	−0.303005
$248$	0	0
$249$	−1.22317e25	−0.846047
$250$	0	0
$251$	−6.00967e24	−0.382188	−0.191094	−	0.981572i	$-0.561203\pi$
$251$	−6.00967e24	−0.382188	−0.191094	+	0.981572i	$0.561203\pi$
$252$	0	0
$253$	−2.23557e25	−1.30804
$254$	0	0
$255$	9.20350e23	0.0495771
$256$	0	0
$257$	9.16372e24	0.454751	0.227376	−	0.973807i	$-0.426986\pi$
$257$	9.16372e24	0.454751	0.227376	+	0.973807i	$0.426986\pi$
$258$	0	0
$259$	−6.75323e24	−0.308933
$260$	0	0
$261$	−5.96249e25	−2.51595
$262$	0	0
$263$	−3.35289e25	−1.30582	−0.652912	−	0.757434i	$-0.726453\pi$
$263$	−3.35289e25	−1.30582	−0.652912	+	0.757434i	$0.726453\pi$
$264$	0	0
$265$	−8.69505e24	−0.312745
$266$	0	0
$267$	−3.94582e25	−1.31150
$268$	0	0
$269$	8.96874e24	0.275634	0.137817	−	0.990458i	$-0.455991\pi$
$269$	8.96874e24	0.275634	0.137817	+	0.990458i	$0.455991\pi$
$270$	0	0
$271$	−4.64216e25	−1.31990	−0.659952	−	0.751307i	$-0.729424\pi$
$271$	−4.64216e25	−1.31990	−0.659952	+	0.751307i	$0.729424\pi$
$272$	0	0
$273$	1.53111e25	0.402993
$274$	0	0
$275$	4.78789e25	1.16722
$276$	0	0
$277$	6.22302e25	1.40593	0.702965	−	0.711224i	$-0.251859\pi$
$277$	6.22302e25	1.40593	0.702965	+	0.711224i	$0.251859\pi$
$278$	0	0
$279$	−1.05823e26	−2.21684
$280$	0	0
$281$	−5.96646e25	−1.15958	−0.579790	−	0.814766i	$-0.696865\pi$
$281$	−5.96646e25	−1.15958	−0.579790	+	0.814766i	$0.696865\pi$
$282$	0	0
$283$	−4.76596e25	−0.859790	−0.429895	−	0.902879i	$-0.641449\pi$
$283$	−4.76596e25	−0.859790	−0.429895	+	0.902879i	$0.641449\pi$
$284$	0	0
$285$	2.88112e25	0.482715
$286$	0	0
$287$	7.51930e25	1.17062
$288$	0	0
$289$	−6.87028e25	−0.994367
$290$	0	0
$291$	2.00509e26	2.69934
$292$	0	0
$293$	−3.88523e25	−0.486751	−0.243375	−	0.969932i	$-0.578255\pi$
$293$	−3.88523e25	−0.486751	−0.243375	+	0.969932i	$0.578255\pi$
$294$	0	0
$295$	−5.38662e25	−0.628326
$296$	0	0
$297$	1.04429e26	1.13469
$298$	0	0
$299$	−3.79011e25	−0.383800
$300$	0	0
$301$	4.44789e25	0.419958
$302$	0	0
$303$	1.44951e26	1.27665
$304$	0	0
$305$	−7.84699e25	−0.644993
$306$	0	0
$307$	1.18817e26	0.911857	0.455928	−	0.890017i	$-0.349307\pi$
$307$	1.18817e26	0.911857	0.455928	+	0.890017i	$0.349307\pi$
$308$	0	0
$309$	−9.04755e25	−0.648585
$310$	0	0
$311$	−2.70804e26	−1.81414	−0.907070	−	0.420980i	$-0.861686\pi$
$311$	−2.70804e26	−1.81414	−0.907070	+	0.420980i	$0.861686\pi$
$312$	0	0
$313$	1.68974e26	1.05829	0.529145	−	0.848532i	$-0.322513\pi$
$313$	1.68974e26	1.05829	0.529145	+	0.848532i	$0.322513\pi$
$314$	0	0
$315$	−6.58825e25	−0.385931
$316$	0	0
$317$	2.44249e26	1.33879	0.669393	−	0.742909i	$-0.266554\pi$
$317$	2.44249e26	1.33879	0.669393	+	0.742909i	$0.266554\pi$
$318$	0	0
$319$	4.59787e26	2.35914
$320$	0	0
$321$	5.85185e26	2.81184
$322$	0	0
$323$	1.21832e25	0.0548446
$324$	0	0
$325$	8.11724e25	0.342478
$326$	0	0
$327$	4.09837e26	1.62129
$328$	0	0
$329$	6.33048e25	0.234898
$330$	0	0
$331$	−9.58024e24	−0.0333567	−0.0166783	−	0.999861i	$-0.505309\pi$
$331$	−9.58024e24	−0.0333567	−0.0166783	+	0.999861i	$0.505309\pi$
$332$	0	0
$333$	−2.32080e26	−0.758531
$334$	0	0
$335$	1.25504e25	0.0385200
$336$	0	0
$337$	−3.03627e26	−0.875440	−0.437720	−	0.899111i	$-0.644214\pi$
$337$	−3.03627e26	−0.875440	−0.437720	+	0.899111i	$0.644214\pi$
$338$	0	0
$339$	−2.29315e26	−0.621349
$340$	0	0
$341$	8.16034e26	2.07868
$342$	0	0
$343$	−4.15915e26	−0.996359
$344$	0	0
$345$	2.71298e26	0.611429
$346$	0	0
$347$	−6.10802e26	−1.29551	−0.647755	−	0.761848i	$-0.724292\pi$
$347$	−6.10802e26	−1.29551	−0.647755	+	0.761848i	$0.724292\pi$
$348$	0	0
$349$	4.95153e26	0.988719	0.494359	−	0.869258i	$-0.335403\pi$
$349$	4.95153e26	0.988719	0.494359	+	0.869258i	$0.335403\pi$
$350$	0	0
$351$	1.77046e26	0.332936
$352$	0	0
$353$	−5.38234e26	−0.953535	−0.476768	−	0.879029i	$-0.658192\pi$
$353$	−5.38234e26	−0.953535	−0.476768	+	0.879029i	$0.658192\pi$
$354$	0	0
$355$	−4.28204e26	−0.714913
$356$	0	0
$357$	−4.63445e25	−0.0729427
$358$	0	0
$359$	1.87205e26	0.277860	0.138930	−	0.990302i	$-0.455634\pi$
$359$	1.87205e26	0.277860	0.138930	+	0.990302i	$0.455634\pi$
$360$	0	0
$361$	−3.32820e26	−0.465997
$362$	0	0
$363$	−1.19489e27	−1.57873
$364$	0	0
$365$	−2.23891e26	−0.279230
$366$	0	0
$367$	2.57966e26	0.303786	0.151893	−	0.988397i	$-0.451463\pi$
$367$	2.57966e26	0.303786	0.151893	+	0.988397i	$0.451463\pi$
$368$	0	0
$369$	2.58407e27	2.87427
$370$	0	0
$371$	4.37842e26	0.460141
$372$	0	0
$373$	1.90971e27	1.89682	0.948408	−	0.317052i	$-0.102693\pi$
$373$	1.90971e27	1.89682	0.948408	+	0.317052i	$0.102693\pi$
$374$	0	0
$375$	−1.28451e27	−1.20617
$376$	0	0
$377$	7.79509e26	0.692208
$378$	0	0
$379$	6.97534e26	0.585941	0.292970	−	0.956122i	$-0.405356\pi$
$379$	6.97534e26	0.585941	0.292970	+	0.956122i	$0.405356\pi$
$380$	0	0
$381$	−9.34548e26	−0.742830
$382$	0	0
$383$	−8.43562e26	−0.634644	−0.317322	−	0.948318i	$-0.602784\pi$
$383$	−8.43562e26	−0.634644	−0.317322	+	0.948318i	$0.602784\pi$
$384$	0	0
$385$	5.08042e26	0.361878
$386$	0	0
$387$	1.52855e27	1.03113
$388$	0	0
$389$	−2.29088e27	−1.46397	−0.731985	−	0.681320i	$-0.761406\pi$
$389$	−2.29088e27	−1.46397	−0.731985	+	0.681320i	$0.761406\pi$
$390$	0	0
$391$	1.14722e26	0.0694687
$392$	0	0
$393$	7.22180e26	0.414499
$394$	0	0
$395$	−6.31305e26	−0.343534
$396$	0	0
$397$	1.15431e27	0.595692	0.297846	−	0.954614i	$-0.403732\pi$
$397$	1.15431e27	0.595692	0.297846	+	0.954614i	$0.403732\pi$
$398$	0	0
$399$	−1.45080e27	−0.710218
$400$	0	0
$401$	−3.36185e27	−1.56157	−0.780786	−	0.624798i	$-0.785181\pi$
$401$	−3.36185e27	−1.56157	−0.780786	+	0.624798i	$0.785181\pi$
$402$	0	0
$403$	1.38348e27	0.609915
$404$	0	0
$405$	2.34989e26	0.0983489
$406$	0	0
$407$	1.78965e27	0.711256
$408$	0	0
$409$	−1.32581e27	−0.500480	−0.250240	−	0.968184i	$-0.580509\pi$
$409$	−1.32581e27	−0.500480	−0.250240	+	0.968184i	$0.580509\pi$
$410$	0	0
$411$	6.24713e27	2.24048
$412$	0	0
$413$	2.71245e27	0.924456
$414$	0	0
$415$	−6.88091e26	−0.222916
$416$	0	0
$417$	−4.65299e27	−1.43319
$418$	0	0
$419$	2.11638e27	0.619935	0.309967	−	0.950747i	$-0.399682\pi$
$419$	2.11638e27	0.619935	0.309967	+	0.950747i	$0.399682\pi$
$420$	0	0
$421$	−3.85871e27	−1.07518	−0.537589	−	0.843207i	$-0.680665\pi$
$421$	−3.85871e27	−1.07518	−0.537589	+	0.843207i	$0.680665\pi$
$422$	0	0
$423$	2.17552e27	0.576752
$424$	0	0
$425$	−2.45698e26	−0.0619894
$426$	0	0
$427$	3.95138e27	0.948978
$428$	0	0
$429$	−4.05752e27	−0.927810
$430$	0	0
$431$	6.54536e26	0.142535	0.0712676	−	0.997457i	$-0.477296\pi$
$431$	6.54536e26	0.142535	0.0712676	+	0.997457i	$0.477296\pi$
$432$	0	0
$433$	2.35082e27	0.487636	0.243818	−	0.969821i	$-0.421600\pi$
$433$	2.35082e27	0.487636	0.243818	+	0.969821i	$0.421600\pi$
$434$	0	0
$435$	−5.57976e27	−1.10275
$436$	0	0
$437$	3.59132e27	0.676393
$438$	0	0
$439$	7.54841e26	0.135512	0.0677561	−	0.997702i	$-0.478416\pi$
$439$	7.54841e26	0.135512	0.0677561	+	0.997702i	$0.478416\pi$
$440$	0	0
$441$	−5.48785e27	−0.939284
$442$	0	0
$443$	1.70820e27	0.278804	0.139402	−	0.990236i	$-0.455482\pi$
$443$	1.70820e27	0.278804	0.139402	+	0.990236i	$0.455482\pi$
$444$	0	0
$445$	−2.21971e27	−0.345554
$446$	0	0
$447$	−8.33950e27	−1.23854
$448$	0	0
$449$	2.53187e27	0.358803	0.179401	−	0.983776i	$-0.442584\pi$
$449$	2.53187e27	0.358803	0.179401	+	0.983776i	$0.442584\pi$
$450$	0	0
$451$	−1.99266e28	−2.69513
$452$	0	0
$453$	−1.59528e28	−2.05970
$454$	0	0
$455$	8.61319e26	0.106180
$456$	0	0
$457$	3.24150e27	0.381615	0.190808	−	0.981627i	$-0.438889\pi$
$457$	3.24150e27	0.381615	0.190808	+	0.981627i	$0.438889\pi$
$458$	0	0
$459$	−5.35893e26	−0.0602622
$460$	0	0
$461$	−1.49606e28	−1.60728	−0.803638	−	0.595118i	$-0.797105\pi$
$461$	−1.49606e28	−1.60728	−0.803638	+	0.595118i	$0.797105\pi$
$462$	0	0
$463$	−1.35427e28	−1.39029	−0.695145	−	0.718869i	$-0.744660\pi$
$463$	−1.35427e28	−1.39029	−0.695145	+	0.718869i	$0.744660\pi$
$464$	0	0
$465$	−9.90300e27	−0.971652
$466$	0	0
$467$	−9.88753e27	−0.927386	−0.463693	−	0.885996i	$-0.653476\pi$
$467$	−9.88753e27	−0.927386	−0.463693	+	0.885996i	$0.653476\pi$
$468$	0	0
$469$	−6.31979e26	−0.0566745
$470$	0	0
$471$	3.32234e28	2.84921
$472$	0	0
$473$	−1.17872e28	−0.966868
$474$	0	0
$475$	−7.69149e27	−0.603570
$476$	0	0
$477$	1.50468e28	1.12980
$478$	0	0
$479$	2.53582e27	0.182220	0.0911099	−	0.995841i	$-0.470959\pi$
$479$	2.53582e27	0.182220	0.0911099	+	0.995841i	$0.470959\pi$
$480$	0	0
$481$	3.03411e27	0.208693
$482$	0	0
$483$	−1.36613e28	−0.899595
$484$	0	0
$485$	1.12796e28	0.711221
$486$	0	0
$487$	−8.98165e27	−0.542378	−0.271189	−	0.962526i	$-0.587417\pi$
$487$	−8.98165e27	−0.542378	−0.271189	+	0.962526i	$0.587417\pi$
$488$	0	0
$489$	−3.72823e28	−2.15655
$490$	0	0
$491$	9.38470e27	0.520074	0.260037	−	0.965599i	$-0.416265\pi$
$491$	9.38470e27	0.520074	0.260037	+	0.965599i	$0.416265\pi$
$492$	0	0
$493$	−2.35947e27	−0.125291
$494$	0	0
$495$	1.74593e28	0.888528
$496$	0	0
$497$	2.15624e28	1.05185
$498$	0	0
$499$	8.46478e27	0.395876	0.197938	−	0.980214i	$-0.436575\pi$
$499$	8.46478e27	0.395876	0.197938	+	0.980214i	$0.436575\pi$
$500$	0	0
$501$	1.91082e28	0.856886
$502$	0	0
$503$	−1.27009e28	−0.546222	−0.273111	−	0.961982i	$-0.588053\pi$
$503$	−1.27009e28	−0.546222	−0.273111	+	0.961982i	$0.588053\pi$
$504$	0	0
$505$	8.15416e27	0.336372
$506$	0	0
$507$	3.31311e28	1.31115
$508$	0	0
$509$	−4.62853e28	−1.75754	−0.878772	−	0.477241i	$-0.841637\pi$
$509$	−4.62853e28	−1.75754	−0.878772	+	0.477241i	$0.841637\pi$
$510$	0	0
$511$	1.12741e28	0.410830
$512$	0	0
$513$	−1.67759e28	−0.586752
$514$	0	0
$515$	−5.08967e27	−0.170889
$516$	0	0
$517$	−1.67761e28	−0.540806
$518$	0	0
$519$	5.68345e27	0.175936
$520$	0	0
$521$	4.62335e28	1.37455	0.687276	−	0.726396i	$-0.258806\pi$
$521$	4.62335e28	1.37455	0.687276	+	0.726396i	$0.258806\pi$
$522$	0	0
$523$	−2.68491e28	−0.766764	−0.383382	−	0.923590i	$-0.625241\pi$
$523$	−2.68491e28	−0.766764	−0.383382	+	0.923590i	$0.625241\pi$
$524$	0	0
$525$	2.92582e28	0.802741
$526$	0	0
$527$	−4.18760e27	−0.110396
$528$	0	0
$529$	−5.65430e27	−0.143250
$530$	0	0
$531$	9.32155e28	2.26984
$532$	0	0
$533$	−3.37829e28	−0.790791
$534$	0	0
$535$	3.29194e28	0.740861
$536$	0	0
$537$	6.31932e28	1.36754
$538$	0	0
$539$	4.23186e28	0.880743
$540$	0	0
$541$	1.32552e28	0.265347	0.132673	−	0.991160i	$-0.457644\pi$
$541$	1.32552e28	0.265347	0.132673	+	0.991160i	$0.457644\pi$
$542$	0	0
$543$	−6.97097e28	−1.34244
$544$	0	0
$545$	2.30553e28	0.427176
$546$	0	0
$547$	6.04300e28	1.07742	0.538711	−	0.842491i	$-0.318912\pi$
$547$	6.04300e28	1.07742	0.538711	+	0.842491i	$0.318912\pi$
$548$	0	0
$549$	1.35792e29	2.33005
$550$	0	0
$551$	−7.38623e28	−1.21992
$552$	0	0
$553$	3.17896e28	0.505441
$554$	0	0
$555$	−2.17183e28	−0.332467
$556$	0	0
$557$	1.04128e29	1.53493	0.767463	−	0.641094i	$-0.221519\pi$
$557$	1.04128e29	1.53493	0.767463	+	0.641094i	$0.221519\pi$
$558$	0	0
$559$	−1.99836e28	−0.283694
$560$	0	0
$561$	1.22816e28	0.167936
$562$	0	0
$563$	−1.16007e29	−1.52808	−0.764040	−	0.645169i	$-0.776787\pi$
$563$	−1.16007e29	−1.52808	−0.764040	+	0.645169i	$0.776787\pi$
$564$	0	0
$565$	−1.29000e28	−0.163713
$566$	0	0
$567$	−1.18329e28	−0.144701
$568$	0	0
$569$	−8.71343e28	−1.02686	−0.513429	−	0.858132i	$-0.671625\pi$
$569$	−8.71343e28	−1.02686	−0.513429	+	0.858132i	$0.671625\pi$
$570$	0	0
$571$	−3.44513e28	−0.391315	−0.195657	−	0.980672i	$-0.562684\pi$
$571$	−3.44513e28	−0.391315	−0.195657	+	0.980672i	$0.562684\pi$
$572$	0	0
$573$	4.99445e28	0.546845
$574$	0	0
$575$	−7.24261e28	−0.764509
$576$	0	0
$577$	4.91815e28	0.500560	0.250280	−	0.968174i	$-0.419477\pi$
$577$	4.91815e28	0.500560	0.250280	+	0.968174i	$0.419477\pi$
$578$	0	0
$579$	4.32485e28	0.424469
$580$	0	0
$581$	3.46491e28	0.327976
$582$	0	0
$583$	−1.16031e29	−1.05938
$584$	0	0
$585$	2.95999e28	0.260708
$586$	0	0
$587$	−1.75632e29	−1.49246	−0.746230	−	0.665689i	$-0.768138\pi$
$587$	−1.75632e29	−1.49246	−0.746230	+	0.665689i	$0.768138\pi$
$588$	0	0
$589$	−1.31091e29	−1.07489
$590$	0	0
$591$	2.22402e29	1.75983
$592$	0	0
$593$	1.47676e28	0.112781	0.0563906	−	0.998409i	$-0.482041\pi$
$593$	1.47676e28	0.112781	0.0563906	+	0.998409i	$0.482041\pi$
$594$	0	0
$595$	−2.60709e27	−0.0192189
$596$	0	0
$597$	1.54464e28	0.109925
$598$	0	0
$599$	−1.42495e29	−0.979077	−0.489539	−	0.871982i	$-0.662835\pi$
$599$	−1.42495e29	−0.979077	−0.489539	+	0.871982i	$0.662835\pi$
$600$	0	0
$601$	1.49306e28	0.0990592	0.0495296	−	0.998773i	$-0.484228\pi$
$601$	1.49306e28	0.0990592	0.0495296	+	0.998773i	$0.484228\pi$
$602$	0	0
$603$	−2.17184e28	−0.139154
$604$	0	0
$605$	−6.72178e28	−0.415962
$606$	0	0
$607$	2.91646e29	1.74331	0.871656	−	0.490118i	$-0.163046\pi$
$607$	2.91646e29	1.74331	0.871656	+	0.490118i	$0.163046\pi$
$608$	0	0
$609$	2.80970e29	1.62248
$610$	0	0
$611$	−2.84417e28	−0.158680
$612$	0	0
$613$	2.08133e29	1.12203	0.561017	−	0.827804i	$-0.310410\pi$
$613$	2.08133e29	1.12203	0.561017	+	0.827804i	$0.310410\pi$
$614$	0	0
$615$	2.41820e29	1.25980
$616$	0	0
$617$	2.42845e29	1.22274	0.611371	−	0.791344i	$-0.290618\pi$
$617$	2.42845e29	1.22274	0.611371	+	0.791344i	$0.290618\pi$
$618$	0	0
$619$	−3.14806e29	−1.53212	−0.766058	−	0.642771i	$-0.777785\pi$
$619$	−3.14806e29	−1.53212	−0.766058	+	0.642771i	$0.777785\pi$
$620$	0	0
$621$	−1.57969e29	−0.743208
$622$	0	0
$623$	1.11774e29	0.508413
$624$	0	0
$625$	1.15541e29	0.508153
$626$	0	0
$627$	3.84470e29	1.63513
$628$	0	0
$629$	−9.18384e27	−0.0377740
$630$	0	0
$631$	−4.38970e29	−1.74633	−0.873165	−	0.487425i	$-0.837936\pi$
$631$	−4.38970e29	−1.74633	−0.873165	+	0.487425i	$0.837936\pi$
$632$	0	0
$633$	−5.42345e29	−2.08707
$634$	0	0
$635$	−5.25727e28	−0.195720
$636$	0	0
$637$	7.17456e28	0.258423
$638$	0	0
$639$	7.41007e29	2.58264
$640$	0	0
$641$	−2.94919e29	−0.994704	−0.497352	−	0.867549i	$-0.665694\pi$
$641$	−2.94919e29	−0.994704	−0.497352	+	0.867549i	$0.665694\pi$
$642$	0	0
$643$	5.27914e28	0.172325	0.0861624	−	0.996281i	$-0.472540\pi$
$643$	5.27914e28	0.172325	0.0861624	+	0.996281i	$0.472540\pi$
$644$	0	0
$645$	1.43043e29	0.451950
$646$	0	0
$647$	3.85044e29	1.17765	0.588824	−	0.808262i	$-0.299591\pi$
$647$	3.85044e29	1.17765	0.588824	+	0.808262i	$0.299591\pi$
$648$	0	0
$649$	−7.18815e29	−2.12837
$650$	0	0
$651$	4.98669e29	1.42959
$652$	0	0
$653$	2.95217e29	0.819506	0.409753	−	0.912196i	$-0.365615\pi$
$653$	2.95217e29	0.819506	0.409753	+	0.912196i	$0.365615\pi$
$654$	0	0
$655$	4.06260e28	0.109212
$656$	0	0
$657$	3.87444e29	1.00872
$658$	0	0
$659$	−7.45934e27	−0.0188106	−0.00940532	−	0.999956i	$-0.502994\pi$
$659$	−7.45934e27	−0.0188106	−0.00940532	+	0.999956i	$0.502994\pi$
$660$	0	0
$661$	−4.16154e29	−1.01657	−0.508287	−	0.861188i	$-0.669721\pi$
$661$	−4.16154e29	−1.01657	−0.508287	+	0.861188i	$0.669721\pi$
$662$	0	0
$663$	2.08218e28	0.0492749
$664$	0	0
$665$	−8.16141e28	−0.187128
$666$	0	0
$667$	−6.95517e29	−1.54521
$668$	0	0
$669$	−1.25757e30	−2.70744
$670$	0	0
$671$	−1.04714e30	−2.18483
$672$	0	0
$673$	−5.91758e29	−1.19670	−0.598351	−	0.801234i	$-0.704177\pi$
$673$	−5.91758e29	−1.19670	−0.598351	+	0.801234i	$0.704177\pi$
$674$	0	0
$675$	3.38320e29	0.663191
$676$	0	0
$677$	9.03724e28	0.171733	0.0858667	−	0.996307i	$-0.472634\pi$
$677$	9.03724e28	0.171733	0.0858667	+	0.996307i	$0.472634\pi$
$678$	0	0
$679$	−5.67987e29	−1.04642
$680$	0	0
$681$	6.04934e29	1.08060
$682$	0	0
$683$	3.24307e29	0.561744	0.280872	−	0.959745i	$-0.409376\pi$
$683$	3.24307e29	0.561744	0.280872	+	0.959745i	$0.409376\pi$
$684$	0	0
$685$	3.51430e29	0.590321
$686$	0	0
$687$	−1.13720e30	−1.85263
$688$	0	0
$689$	−1.96715e29	−0.310839
$690$	0	0
$691$	−7.54055e29	−1.15580	−0.577901	−	0.816107i	$-0.696128\pi$
$691$	−7.54055e29	−1.15580	−0.577901	+	0.816107i	$0.696128\pi$
$692$	0	0
$693$	−8.79167e29	−1.30729
$694$	0	0
$695$	−2.61753e29	−0.377616
$696$	0	0
$697$	1.02256e29	0.143135
$698$	0	0
$699$	1.25238e30	1.70109
$700$	0	0
$701$	−3.20065e29	−0.421890	−0.210945	−	0.977498i	$-0.567654\pi$
$701$	−3.20065e29	−0.421890	−0.210945	+	0.977498i	$0.567654\pi$
$702$	0	0
$703$	−2.87497e29	−0.367792
$704$	0	0
$705$	2.03587e29	0.252793
$706$	0	0
$707$	−4.10606e29	−0.494904
$708$	0	0
$709$	−2.29896e29	−0.268996	−0.134498	−	0.990914i	$-0.542942\pi$
$709$	−2.29896e29	−0.268996	−0.134498	+	0.990914i	$0.542942\pi$
$710$	0	0
$711$	1.09247e30	1.24102
$712$	0	0
$713$	−1.23441e30	−1.36150
$714$	0	0
$715$	−2.28254e29	−0.244459
$716$	0	0
$717$	−1.35090e30	−1.40499
$718$	0	0
$719$	−6.12699e29	−0.618862	−0.309431	−	0.950922i	$-0.600139\pi$
$719$	−6.12699e29	−0.618862	−0.309431	+	0.950922i	$0.600139\pi$
$720$	0	0
$721$	2.56292e29	0.251428
$722$	0	0
$723$	4.44554e29	0.423616
$724$	0	0
$725$	1.48958e30	1.37884
$726$	0	0
$727$	−2.54304e29	−0.228687	−0.114344	−	0.993441i	$-0.536477\pi$
$727$	−2.54304e29	−0.228687	−0.114344	+	0.993441i	$0.536477\pi$
$728$	0	0
$729$	−1.86180e30	−1.62665
$730$	0	0
$731$	6.04876e28	0.0513492
$732$	0	0
$733$	1.16587e30	0.961742	0.480871	−	0.876791i	$-0.340321\pi$
$733$	1.16587e30	0.961742	0.480871	+	0.876791i	$0.340321\pi$
$734$	0	0
$735$	−5.13558e29	−0.411692
$736$	0	0
$737$	1.67478e29	0.130482
$738$	0	0
$739$	−3.56632e29	−0.270056	−0.135028	−	0.990842i	$-0.543112\pi$
$739$	−3.56632e29	−0.270056	−0.135028	+	0.990842i	$0.543112\pi$
$740$	0	0
$741$	6.51819e29	0.479773
$742$	0	0
$743$	−1.33985e29	−0.0958681	−0.0479341	−	0.998851i	$-0.515264\pi$
$743$	−1.33985e29	−0.0958681	−0.0479341	+	0.998851i	$0.515264\pi$
$744$	0	0
$745$	−4.69136e29	−0.326330
$746$	0	0
$747$	1.19074e30	0.805289
$748$	0	0
$749$	−1.65767e30	−1.09003
$750$	0	0
$751$	1.79448e30	1.14741	0.573706	−	0.819061i	$-0.305505\pi$
$751$	1.79448e30	1.14741	0.573706	+	0.819061i	$0.305505\pi$
$752$	0	0
$753$	9.73218e29	0.605150
$754$	0	0
$755$	−8.97417e29	−0.542689
$756$	0	0
$757$	2.78719e30	1.63931	0.819653	−	0.572860i	$-0.194166\pi$
$757$	2.78719e30	1.63931	0.819653	+	0.572860i	$0.194166\pi$
$758$	0	0
$759$	3.62032e30	2.07114
$760$	0	0
$761$	7.47749e29	0.416118	0.208059	−	0.978116i	$-0.433285\pi$
$761$	7.47749e29	0.416118	0.208059	+	0.978116i	$0.433285\pi$
$762$	0	0
$763$	−1.16095e30	−0.628504
$764$	0	0
$765$	−8.95948e28	−0.0471887
$766$	0	0
$767$	−1.21866e30	−0.624497
$768$	0	0
$769$	2.51748e30	1.25528	0.627639	−	0.778504i	$-0.284021\pi$
$769$	2.51748e30	1.25528	0.627639	+	0.778504i	$0.284021\pi$
$770$	0	0
$771$	−1.48399e30	−0.720045
$772$	0	0
$773$	1.04829e29	0.0494989	0.0247495	−	0.999694i	$-0.492121\pi$
$773$	1.04829e29	0.0494989	0.0247495	+	0.999694i	$0.492121\pi$
$774$	0	0
$775$	2.64372e30	1.21492
$776$	0	0
$777$	1.09363e30	0.489159
$778$	0	0
$779$	3.20110e30	1.39366
$780$	0	0
$781$	−5.71415e30	−2.42167
$782$	0	0
$783$	3.24893e30	1.34042
$784$	0	0
$785$	1.86897e30	0.750708
$786$	0	0
$787$	−7.98670e29	−0.312344	−0.156172	−	0.987730i	$-0.549915\pi$
$787$	−7.98670e29	−0.312344	−0.156172	+	0.987730i	$0.549915\pi$
$788$	0	0
$789$	5.42974e30	2.06762
$790$	0	0
$791$	6.49585e29	0.240870
$792$	0	0
$793$	−1.77529e30	−0.641062
$794$	0	0
$795$	1.40809e30	0.495195
$796$	0	0
$797$	−2.38344e30	−0.816379	−0.408190	−	0.912897i	$-0.633840\pi$
$797$	−2.38344e30	−0.816379	−0.408190	+	0.912897i	$0.633840\pi$
$798$	0	0
$799$	8.60893e28	0.0287216
$800$	0	0
$801$	3.84121e30	1.24832
$802$	0	0
$803$	−2.98770e30	−0.945854
$804$	0	0
$805$	−7.68511e29	−0.237025
$806$	0	0
$807$	−1.45242e30	−0.436434
$808$	0	0
$809$	1.54246e29	0.0451601	0.0225801	−	0.999745i	$-0.492812\pi$
$809$	1.54246e29	0.0451601	0.0225801	+	0.999745i	$0.492812\pi$
$810$	0	0
$811$	−5.31290e30	−1.51570	−0.757850	−	0.652429i	$-0.773750\pi$
$811$	−5.31290e30	−1.51570	−0.757850	+	0.652429i	$0.773750\pi$
$812$	0	0
$813$	7.51760e30	2.08992
$814$	0	0
$815$	−2.09730e30	−0.568206
$816$	0	0
$817$	1.89354e30	0.499970
$818$	0	0
$819$	−1.49051e30	−0.383579
$820$	0	0
$821$	2.53829e30	0.636705	0.318352	−	0.947972i	$-0.396870\pi$
$821$	2.53829e30	0.636705	0.318352	+	0.947972i	$0.396870\pi$
$822$	0	0
$823$	−1.09701e30	−0.268233	−0.134117	−	0.990966i	$-0.542820\pi$
$823$	−1.09701e30	−0.268233	−0.134117	+	0.990966i	$0.542820\pi$
$824$	0	0
$825$	−7.75360e30	−1.84815
$826$	0	0
$827$	−1.29685e30	−0.301357	−0.150678	−	0.988583i	$-0.548146\pi$
$827$	−1.29685e30	−0.301357	−0.150678	+	0.988583i	$0.548146\pi$
$828$	0	0
$829$	4.74295e30	1.07455	0.537274	−	0.843408i	$-0.319454\pi$
$829$	4.74295e30	1.07455	0.537274	+	0.843408i	$0.319454\pi$
$830$	0	0
$831$	−1.00777e31	−2.22613
$832$	0	0
$833$	−2.17164e29	−0.0467752
$834$	0	0
$835$	1.07492e30	0.225772
$836$	0	0
$837$	5.76623e30	1.18107
$838$	0	0
$839$	−5.41057e30	−1.08079	−0.540397	−	0.841410i	$-0.681726\pi$
$839$	−5.41057e30	−1.08079	−0.540397	+	0.841410i	$0.681726\pi$
$840$	0	0
$841$	9.17178e30	1.78688
$842$	0	0
$843$	9.66220e30	1.83606
$844$	0	0
$845$	1.86378e30	0.345461
$846$	0	0
$847$	3.38478e30	0.612004
$848$	0	0
$849$	7.71808e30	1.36138
$850$	0	0
$851$	−2.70718e30	−0.465862
$852$	0	0
$853$	3.54864e30	0.595796	0.297898	−	0.954598i	$-0.403714\pi$
$853$	3.54864e30	0.595796	0.297898	+	0.954598i	$0.403714\pi$
$854$	0	0
$855$	−2.80473e30	−0.459460
$856$	0	0
$857$	4.54932e29	0.0727190	0.0363595	−	0.999339i	$-0.488424\pi$
$857$	4.54932e29	0.0727190	0.0363595	+	0.999339i	$0.488424\pi$
$858$	0	0
$859$	−6.46394e30	−1.00825	−0.504126	−	0.863630i	$-0.668185\pi$
$859$	−6.46394e30	−1.00825	−0.504126	+	0.863630i	$0.668185\pi$
$860$	0	0
$861$	−1.21769e31	−1.85355
$862$	0	0
$863$	−1.70017e30	−0.252569	−0.126284	−	0.991994i	$-0.540305\pi$
$863$	−1.70017e30	−0.252569	−0.126284	+	0.991994i	$0.540305\pi$
$864$	0	0
$865$	3.19720e29	0.0463555
$866$	0	0
$867$	1.11259e31	1.57446
$868$	0	0
$869$	−8.42442e30	−1.16368
$870$	0	0
$871$	2.83937e29	0.0382853
$872$	0	0
$873$	−1.95193e31	−2.56930
$874$	0	0
$875$	3.63865e30	0.467580
$876$	0	0
$877$	−1.34635e31	−1.68912	−0.844562	−	0.535458i	$-0.820139\pi$
$877$	−1.34635e31	−1.68912	−0.844562	+	0.535458i	$0.820139\pi$
$878$	0	0
$879$	6.29182e30	0.770713
$880$	0	0
$881$	8.51820e30	1.01883	0.509414	−	0.860522i	$-0.329862\pi$
$881$	8.51820e30	1.01883	0.509414	+	0.860522i	$0.329862\pi$
$882$	0	0
$883$	4.37339e30	0.510776	0.255388	−	0.966839i	$-0.417797\pi$
$883$	4.37339e30	0.510776	0.255388	+	0.966839i	$0.417797\pi$
$884$	0	0
$885$	8.72319e30	0.994881
$886$	0	0
$887$	−5.84217e30	−0.650693	−0.325347	−	0.945595i	$-0.605481\pi$
$887$	−5.84217e30	−0.650693	−0.325347	+	0.945595i	$0.605481\pi$
$888$	0	0
$889$	2.64731e30	0.287963
$890$	0	0
$891$	3.13580e30	0.333144
$892$	0	0
$893$	2.69499e30	0.279652
$894$	0	0
$895$	3.55491e30	0.360319
$896$	0	0
$897$	6.13778e30	0.607702
$898$	0	0
$899$	2.53880e31	2.45556
$900$	0	0
$901$	5.95429e29	0.0562626
$902$	0	0
$903$	−7.20299e30	−0.664954
$904$	0	0
$905$	−3.92150e30	−0.353705
$906$	0	0
$907$	−2.10560e31	−1.85566	−0.927831	−	0.373002i	$-0.878328\pi$
$907$	−2.10560e31	−1.85566	−0.927831	+	0.373002i	$0.878328\pi$
$908$	0	0
$909$	−1.41108e31	−1.21515
$910$	0	0
$911$	1.47793e31	1.24369	0.621844	−	0.783141i	$-0.286384\pi$
$911$	1.47793e31	1.24369	0.621844	+	0.783141i	$0.286384\pi$
$912$	0	0
$913$	−9.18220e30	−0.755099
$914$	0	0
$915$	1.27076e31	1.02127
$916$	0	0
$917$	−2.04574e30	−0.160684
$918$	0	0
$919$	1.28613e31	0.987350	0.493675	−	0.869647i	$-0.335653\pi$
$919$	1.28613e31	0.987350	0.493675	+	0.869647i	$0.335653\pi$
$920$	0	0
$921$	−1.92415e31	−1.44382
$922$	0	0
$923$	−9.68759e30	−0.710556
$924$	0	0
$925$	5.79795e30	0.415706
$926$	0	0
$927$	8.80767e30	0.617339
$928$	0	0
$929$	8.99460e30	0.616335	0.308168	−	0.951332i	$-0.400284\pi$
$929$	8.99460e30	0.616335	0.308168	+	0.951332i	$0.400284\pi$
$930$	0	0
$931$	−6.79825e30	−0.455434
$932$	0	0
$933$	4.38545e31	2.87248
$934$	0	0
$935$	6.90895e29	0.0442476
$936$	0	0
$937$	−2.94052e31	−1.84145	−0.920723	−	0.390218i	$-0.872400\pi$
$937$	−2.94052e31	−1.84145	−0.920723	+	0.390218i	$0.872400\pi$
$938$	0	0
$939$	−2.73640e31	−1.67568
$940$	0	0
$941$	−2.20191e31	−1.31859	−0.659293	−	0.751886i	$-0.729144\pi$
$941$	−2.20191e31	−1.31859	−0.659293	+	0.751886i	$0.729144\pi$
$942$	0	0
$943$	3.01428e31	1.76527
$944$	0	0
$945$	3.58991e30	0.205612
$946$	0	0
$947$	5.55079e30	0.310942	0.155471	−	0.987840i	$-0.450310\pi$
$947$	5.55079e30	0.310942	0.155471	+	0.987840i	$0.450310\pi$
$948$	0	0
$949$	−5.06526e30	−0.277528
$950$	0	0
$951$	−3.95542e31	−2.11981
$952$	0	0
$953$	−2.96303e31	−1.55332	−0.776660	−	0.629920i	$-0.783088\pi$
$953$	−2.96303e31	−1.55332	−0.776660	+	0.629920i	$0.783088\pi$
$954$	0	0
$955$	2.80961e30	0.144082
$956$	0	0
$957$	−7.44588e31	−3.73543
$958$	0	0
$959$	−1.76964e31	−0.868539
$960$	0	0
$961$	2.42333e31	1.16363
$962$	0	0
$963$	−5.69670e31	−2.67638
$964$	0	0
$965$	2.43293e30	0.111839
$966$	0	0
$967$	3.95378e31	1.77842	0.889211	−	0.457497i	$-0.151254\pi$
$967$	3.95378e31	1.77842	0.889211	+	0.457497i	$0.151254\pi$
$968$	0	0
$969$	−1.97297e30	−0.0868400
$970$	0	0
$971$	1.89658e31	0.816899	0.408450	−	0.912781i	$-0.366070\pi$
$971$	1.89658e31	0.816899	0.408450	+	0.912781i	$0.366070\pi$
$972$	0	0
$973$	1.31806e31	0.555586
$974$	0	0
$975$	−1.31452e31	−0.542275
$976$	0	0
$977$	1.51182e30	0.0610391	0.0305196	−	0.999534i	$-0.490284\pi$
$977$	1.51182e30	0.0610391	0.0305196	+	0.999534i	$0.490284\pi$
$978$	0	0
$979$	−2.96208e31	−1.17052
$980$	0	0
$981$	−3.98971e31	−1.54318
$982$	0	0
$983$	−3.50124e31	−1.32559	−0.662796	−	0.748800i	$-0.730630\pi$
$983$	−3.50124e31	−1.32559	−0.662796	+	0.748800i	$0.730630\pi$
$984$	0	0
$985$	1.25112e31	0.463679
$986$	0	0
$987$	−1.02517e31	−0.371934
$988$	0	0
$989$	1.78304e31	0.633285
$990$	0	0
$991$	1.25402e31	0.436046	0.218023	−	0.975944i	$-0.430039\pi$
$991$	1.25402e31	0.436046	0.218023	+	0.975944i	$0.430039\pi$
$992$	0	0
$993$	1.55144e30	0.0528164
$994$	0	0
$995$	8.68935e29	0.0289631
$996$	0	0
$997$	−5.10693e30	−0.166671	−0.0833357	−	0.996522i	$-0.526557\pi$
$997$	−5.10693e30	−0.166671	−0.0833357	+	0.996522i	$0.526557\pi$
$998$	0	0
$999$	1.26459e31	0.404123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.a.o.1.1		5
4.3	odd	2		64.22.a.p.1.5		5
8.3	odd	2		32.22.a.c.1.1	✓	5
8.5	even	2		32.22.a.d.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.22.a.c.1.1	✓	5	8.3	odd	2
32.22.a.d.1.5	yes	5	8.5	even	2
64.22.a.o.1.1		5	1.1	even	1	trivial
64.22.a.p.1.5		5	4.3	odd	2

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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q-161942. q^{3} -9.10999e6 q^{5} +4.58736e8 q^{7} +1.57648e10 q^{9} +O(q^{10})\)	\(q-161942. q^{3} -9.10999e6 q^{5} +4.58736e8 q^{7} +1.57648e10 q^{9} -1.21568e11 q^{11} -2.06102e11 q^{13} +1.47529e12 q^{15} +6.23844e11 q^{17} +1.95292e13 q^{19} -7.42887e13 q^{21} +1.83895e14 q^{23} -3.93845e14 q^{25} -8.59019e14 q^{27} -3.78214e15 q^{29} -6.71258e15 q^{31} +1.96869e16 q^{33} -4.17908e15 q^{35} -1.47214e16 q^{37} +3.33766e16 q^{39} +1.63913e17 q^{41} +9.69596e16 q^{43} -1.43617e17 q^{45} +1.37998e17 q^{47} -3.48107e17 q^{49} -1.01026e17 q^{51} +9.54452e17 q^{53} +1.10748e18 q^{55} -3.16260e18 q^{57} +5.91287e18 q^{59} +8.61362e18 q^{61} +7.23190e18 q^{63} +1.87759e18 q^{65} -1.37765e18 q^{67} -2.97803e19 q^{69} +4.70038e19 q^{71} +2.45764e19 q^{73} +6.37801e19 q^{75} -5.57676e19 q^{77} +6.92981e19 q^{79} -2.57946e19 q^{81} +7.55315e19 q^{83} -5.68321e18 q^{85} +6.12488e20 q^{87} +2.43657e20 q^{89} -9.45466e19 q^{91} +1.08705e21 q^{93} -1.77911e20 q^{95} -1.23815e21 q^{97} -1.91650e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 23144 q^{3} - 18311174 q^{5} + 63978640 q^{7} + 2839988161 q^{9}+O(q^{10}) \) 5 * q - 23144 * q^3 - 18311174 * q^5 + 63978640 * q^7 + 2839988161 * q^9	\( 5 q - 23144 q^{3} - 18311174 q^{5} + 63978640 q^{7} + 2839988161 q^{9} - 25629588280 q^{11} - 26739996110 q^{13} + 850898706352 q^{15} - 88104593910 q^{17} - 13998239618440 q^{19} - 11868255565952 q^{21} + 191593435978416 q^{23} - 41142227484149 q^{25} - 436694294703248 q^{27} - 229446229587518 q^{29} - 30\!\cdots\!60 q^{31}+ \cdots - 45\!\cdots\!40 q^{99}+O(q^{100}) \) 5 * q - 23144 * q^3 - 18311174 * q^5 + 63978640 * q^7 + 2839988161 * q^9 - 25629588280 * q^11 - 26739996110 * q^13 + 850898706352 * q^15 - 88104593910 * q^17 - 13998239618440 * q^19 - 11868255565952 * q^21 + 191593435978416 * q^23 - 41142227484149 * q^25 - 436694294703248 * q^27 - 229446229587518 * q^29 - 3073873710228160 * q^31 + 5433145544904640 * q^33 + 8500692544521248 * q^35 + 11689624248645370 * q^37 + 5755218754185200 * q^39 + 73561781960295890 * q^41 - 59173471060274168 * q^43 + 110307182017064626 * q^45 + 189587882278171232 * q^47 + 198072470828678205 * q^49 - 60381606670165200 * q^51 + 448473144842566250 * q^53 - 276218022333741296 * q^55 - 419697547444918720 * q^57 + 580240954920275560 * q^59 - 60707231290899774 * q^61 + 7329049347459246928 * q^63 - 4215936008336579644 * q^65 - 20784270726401480552 * q^67 - 13151945925537436032 * q^69 + 48791036649307600400 * q^71 - 7687983545663740350 * q^73 - 63701791920635021848 * q^75 - 14991304956050303360 * q^77 + 150036082528549569440 * q^79 + 6961005137522997901 * q^81 - 320036481947166592392 * q^83 + 55351169878564568052 * q^85 + 845870037277169219440 * q^87 + 157308716989748010706 * q^89 - 1517228856976775217760 * q^91 + 285928010618243909120 * q^93 + 2644313286399100711792 * q^95 - 165820666835121495910 * q^97 - 4544456840208531002840 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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