LMFDB - Embedded newform 256.8.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(1,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("256.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.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:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} + 4820x^{2} - 15296 $ x^6 - 163x^4 + 4820x^2 - 15296
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.81430$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +O(q^{10})$	$q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +5452.20 q^{11} +6289.38 q^{13} -13049.8 q^{15} +34587.3 q^{17} +14595.6 q^{19} -38485.4 q^{21} +24667.5 q^{23} +27169.8 q^{25} +110862. q^{27} -171116. q^{29} +111688. q^{31} -219267. q^{33} +310526. q^{35} -103636. q^{37} -252935. q^{39} -71691.3 q^{41} -328419. q^{43} -184847. q^{45} +119043. q^{47} +92230.3 q^{49} -1.39097e6 q^{51} -1.04011e6 q^{53} +1.76919e6 q^{55} -586982. q^{57} -225984. q^{59} -1.55268e6 q^{61} -545133. q^{63} +2.04085e6 q^{65} -316375. q^{67} -992033. q^{69} -538965. q^{71} +2.68512e6 q^{73} -1.09267e6 q^{75} +5.21754e6 q^{77} +8.22632e6 q^{79} -3.21264e6 q^{81} +5.89510e6 q^{83} +1.12233e7 q^{85} +6.88164e6 q^{87} -437005. q^{89} +6.01868e6 q^{91} -4.49169e6 q^{93} +4.73616e6 q^{95} -7.84322e6 q^{97} -3.10585e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) $ 6 * q + 688 * q^7 + 2918 * q^9	$ 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) $ 6 * q + 688 * q^7 + 2918 * q^9 + 17872 * q^15 + 1452 * q^17 + 1296 * q^23 + 39314 * q^25 - 89280 * q^31 + 53880 * q^33 + 328208 * q^39 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 + 1889896 * q^57 + 5776816 * q^63 + 1416480 * q^65 + 7597104 * q^71 - 2089564 * q^73 + 16015904 * q^79 - 723058 * q^81 + 37453776 * q^87 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

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
$3$	−40.2163	−0.859959	−0.429979	−	0.902839i	$-0.641479\pi$
$3$	−40.2163	−0.859959	−0.429979	+	0.902839i	$0.641479\pi$
$4$	0	0
$5$	324.492	1.16094	0.580468	−	0.814283i	$-0.302870\pi$
$5$	324.492	1.16094	0.580468	+	0.814283i	$0.302870\pi$
$6$	0	0
$7$	956.960	1.05451	0.527255	−	0.849707i	$-0.323221\pi$
$7$	956.960	1.05451	0.527255	+	0.849707i	$0.323221\pi$
$8$	0	0
$9$	−569.651	−0.260471
$10$	0	0
$11$	5452.20	1.23509	0.617544	−	0.786537i	$-0.288128\pi$
$11$	5452.20	1.23509	0.617544	+	0.786537i	$0.288128\pi$
$12$	0	0
$13$	6289.38	0.793973	0.396987	−	0.917824i	$-0.370056\pi$
$13$	6289.38	0.793973	0.396987	+	0.917824i	$0.370056\pi$
$14$	0	0
$15$	−13049.8	−0.998357
$16$	0	0
$17$	34587.3	1.70744	0.853720	−	0.520733i	$-0.174341\pi$
$17$	34587.3	1.70744	0.853720	+	0.520733i	$0.174341\pi$
$18$	0	0
$19$	14595.6	0.488186	0.244093	−	0.969752i	$-0.421510\pi$
$19$	14595.6	0.488186	0.244093	+	0.969752i	$0.421510\pi$
$20$	0	0
$21$	−38485.4	−0.906835
$22$	0	0
$23$	24667.5	0.422743	0.211372	−	0.977406i	$-0.432207\pi$
$23$	24667.5	0.422743	0.211372	+	0.977406i	$0.432207\pi$
$24$	0	0
$25$	27169.8	0.347774
$26$	0	0
$27$	110862.	1.08395
$28$	0	0
$29$	−171116.	−1.30286	−0.651429	−	0.758710i	$-0.725830\pi$
$29$	−171116.	−1.30286	−0.651429	+	0.758710i	$0.725830\pi$
$30$	0	0
$31$	111688.	0.673352	0.336676	−	0.941620i	$-0.390697\pi$
$31$	111688.	0.673352	0.336676	+	0.941620i	$0.390697\pi$
$32$	0	0
$33$	−219267.	−1.06212
$34$	0	0
$35$	310526.	1.22422
$36$	0	0
$37$	−103636.	−0.336360	−0.168180	−	0.985756i	$-0.553789\pi$
$37$	−103636.	−0.336360	−0.168180	+	0.985756i	$0.553789\pi$
$38$	0	0
$39$	−252935.	−0.682784
$40$	0	0
$41$	−71691.3	−0.162451	−0.0812256	−	0.996696i	$-0.525883\pi$
$41$	−71691.3	−0.162451	−0.0812256	+	0.996696i	$0.525883\pi$
$42$	0	0
$43$	−328419.	−0.629925	−0.314962	−	0.949104i	$-0.601992\pi$
$43$	−328419.	−0.629925	−0.314962	+	0.949104i	$0.601992\pi$
$44$	0	0
$45$	−184847.	−0.302391
$46$	0	0
$47$	119043.	0.167248	0.0836241	−	0.996497i	$-0.473350\pi$
$47$	119043.	0.167248	0.0836241	+	0.996497i	$0.473350\pi$
$48$	0	0
$49$	92230.3	0.111992
$50$	0	0
$51$	−1.39097e6	−1.46833
$52$	0	0
$53$	−1.04011e6	−0.959648	−0.479824	−	0.877365i	$-0.659300\pi$
$53$	−1.04011e6	−0.959648	−0.479824	+	0.877365i	$0.659300\pi$
$54$	0	0
$55$	1.76919e6	1.43386
$56$	0	0
$57$	−586982.	−0.419820
$58$	0	0
$59$	−225984.	−0.143250	−0.0716250	−	0.997432i	$-0.522818\pi$
$59$	−225984.	−0.143250	−0.0716250	+	0.997432i	$0.522818\pi$
$60$	0	0
$61$	−1.55268e6	−0.875843	−0.437922	−	0.899013i	$-0.644285\pi$
$61$	−1.55268e6	−0.875843	−0.437922	+	0.899013i	$0.644285\pi$
$62$	0	0
$63$	−545133.	−0.274670
$64$	0	0
$65$	2.04085e6	0.921753
$66$	0	0
$67$	−316375.	−0.128511	−0.0642555	−	0.997933i	$-0.520467\pi$
$67$	−316375.	−0.128511	−0.0642555	+	0.997933i	$0.520467\pi$
$68$	0	0
$69$	−992033.	−0.363542
$70$	0	0
$71$	−538965.	−0.178713	−0.0893566	−	0.996000i	$-0.528481\pi$
$71$	−538965.	−0.178713	−0.0893566	+	0.996000i	$0.528481\pi$
$72$	0	0
$73$	2.68512e6	0.807856	0.403928	−	0.914791i	$-0.367645\pi$
$73$	2.68512e6	0.807856	0.403928	+	0.914791i	$0.367645\pi$
$74$	0	0
$75$	−1.09267e6	−0.299071
$76$	0	0
$77$	5.21754e6	1.30241
$78$	0	0
$79$	8.22632e6	1.87720	0.938600	−	0.345007i	$-0.112123\pi$
$79$	8.22632e6	1.87720	0.938600	+	0.345007i	$0.112123\pi$
$80$	0	0
$81$	−3.21264e6	−0.671684
$82$	0	0
$83$	5.89510e6	1.13167	0.565833	−	0.824520i	$-0.308555\pi$
$83$	5.89510e6	1.13167	0.565833	+	0.824520i	$0.308555\pi$
$84$	0	0
$85$	1.12233e7	1.98223
$86$	0	0
$87$	6.88164e6	1.12040
$88$	0	0
$89$	−437005.	−0.0657085	−0.0328542	−	0.999460i	$-0.510460\pi$
$89$	−437005.	−0.0657085	−0.0328542	+	0.999460i	$0.510460\pi$
$90$	0	0
$91$	6.01868e6	0.837253
$92$	0	0
$93$	−4.49169e6	−0.579055
$94$	0	0
$95$	4.73616e6	0.566753
$96$	0	0
$97$	−7.84322e6	−0.872556	−0.436278	−	0.899812i	$-0.643704\pi$
$97$	−7.84322e6	−0.872556	−0.436278	+	0.899812i	$0.643704\pi$
$98$	0	0
$99$	−3.10585e6	−0.321705
$100$	0	0
$101$	−6.19757e6	−0.598545	−0.299272	−	0.954168i	$-0.596744\pi$
$101$	−6.19757e6	−0.598545	−0.299272	+	0.954168i	$0.596744\pi$
$102$	0	0
$103$	6.59816e6	0.594966	0.297483	−	0.954727i	$-0.403853\pi$
$103$	6.59816e6	0.594966	0.297483	+	0.954727i	$0.403853\pi$
$104$	0	0
$105$	−1.24882e7	−1.05278
$106$	0	0
$107$	512845.	0.0404709	0.0202354	−	0.999795i	$-0.493558\pi$
$107$	512845.	0.0404709	0.0202354	+	0.999795i	$0.493558\pi$
$108$	0	0
$109$	1.95882e7	1.44878	0.724388	−	0.689393i	$-0.242123\pi$
$109$	1.95882e7	1.44878	0.724388	+	0.689393i	$0.242123\pi$
$110$	0	0
$111$	4.16785e6	0.289256
$112$	0	0
$113$	1.88876e7	1.23141	0.615705	−	0.787977i	$-0.288871\pi$
$113$	1.88876e7	1.23141	0.615705	+	0.787977i	$0.288871\pi$
$114$	0	0
$115$	8.00438e6	0.490778
$116$	0	0
$117$	−3.58275e6	−0.206807
$118$	0	0
$119$	3.30987e7	1.80051
$120$	0	0
$121$	1.02394e7	0.525441
$122$	0	0
$123$	2.88316e6	0.139701
$124$	0	0
$125$	−1.65345e7	−0.757193
$126$	0	0
$127$	3.96314e7	1.71683	0.858413	−	0.512959i	$-0.171451\pi$
$127$	3.96314e7	1.71683	0.858413	+	0.512959i	$0.171451\pi$
$128$	0	0
$129$	1.32078e7	0.541709
$130$	0	0
$131$	−3.65337e7	−1.41986	−0.709928	−	0.704274i	$-0.751273\pi$
$131$	−3.65337e7	−1.41986	−0.709928	+	0.704274i	$0.751273\pi$
$132$	0	0
$133$	1.39675e7	0.514797
$134$	0	0
$135$	3.59739e7	1.25840
$136$	0	0
$137$	2.56967e7	0.853799	0.426899	−	0.904299i	$-0.359606\pi$
$137$	2.56967e7	0.853799	0.426899	+	0.904299i	$0.359606\pi$
$138$	0	0
$139$	−5.23001e7	−1.65177	−0.825886	−	0.563836i	$-0.809325\pi$
$139$	−5.23001e7	−1.65177	−0.825886	+	0.563836i	$0.809325\pi$
$140$	0	0
$141$	−4.78747e6	−0.143827
$142$	0	0
$143$	3.42910e7	0.980626
$144$	0	0
$145$	−5.55256e7	−1.51254
$146$	0	0
$147$	−3.70916e6	−0.0963085
$148$	0	0
$149$	−1.80406e7	−0.446785	−0.223392	−	0.974729i	$-0.571713\pi$
$149$	−1.80406e7	−0.446785	−0.223392	+	0.974729i	$0.571713\pi$
$150$	0	0
$151$	−3.87385e7	−0.915637	−0.457818	−	0.889046i	$-0.651369\pi$
$151$	−3.87385e7	−0.915637	−0.457818	+	0.889046i	$0.651369\pi$
$152$	0	0
$153$	−1.97027e7	−0.444739
$154$	0	0
$155$	3.62420e7	0.781719
$156$	0	0
$157$	−5.12341e7	−1.05660	−0.528300	−	0.849058i	$-0.677170\pi$
$157$	−5.12341e7	−1.05660	−0.528300	+	0.849058i	$0.677170\pi$
$158$	0	0
$159$	4.18292e7	0.825258
$160$	0	0
$161$	2.36058e7	0.445787
$162$	0	0
$163$	8.57572e7	1.55101	0.775504	−	0.631343i	$-0.217496\pi$
$163$	8.57572e7	1.55101	0.775504	+	0.631343i	$0.217496\pi$
$164$	0	0
$165$	−7.11504e7	−1.23306
$166$	0	0
$167$	1.05871e8	1.75901	0.879503	−	0.475893i	$-0.157875\pi$
$167$	1.05871e8	1.75901	0.879503	+	0.475893i	$0.157875\pi$
$168$	0	0
$169$	−2.31923e7	−0.369606
$170$	0	0
$171$	−8.31441e6	−0.127158
$172$	0	0
$173$	−1.98148e7	−0.290956	−0.145478	−	0.989361i	$-0.546472\pi$
$173$	−1.98148e7	−0.290956	−0.145478	+	0.989361i	$0.546472\pi$
$174$	0	0
$175$	2.60005e7	0.366731
$176$	0	0
$177$	9.08822e6	0.123189
$178$	0	0
$179$	−2.97800e7	−0.388096	−0.194048	−	0.980992i	$-0.562162\pi$
$179$	−2.97800e7	−0.388096	−0.194048	+	0.980992i	$0.562162\pi$
$180$	0	0
$181$	3.96227e6	0.0496671	0.0248335	−	0.999692i	$-0.492094\pi$
$181$	3.96227e6	0.0496671	0.0248335	+	0.999692i	$0.492094\pi$
$182$	0	0
$183$	6.24429e7	0.753189
$184$	0	0
$185$	−3.36290e7	−0.390493
$186$	0	0
$187$	1.88577e8	2.10884
$188$	0	0
$189$	1.06091e8	1.14304
$190$	0	0
$191$	−4.80105e7	−0.498562	−0.249281	−	0.968431i	$-0.580194\pi$
$191$	−4.80105e7	−0.498562	−0.249281	+	0.968431i	$0.580194\pi$
$192$	0	0
$193$	−4.72502e6	−0.0473100	−0.0236550	−	0.999720i	$-0.507530\pi$
$193$	−4.72502e6	−0.0473100	−0.0236550	+	0.999720i	$0.507530\pi$
$194$	0	0
$195$	−8.20754e7	−0.792669
$196$	0	0
$197$	1.14882e8	1.07058	0.535290	−	0.844668i	$-0.320202\pi$
$197$	1.14882e8	1.07058	0.535290	+	0.844668i	$0.320202\pi$
$198$	0	0
$199$	−1.20933e7	−0.108782	−0.0543911	−	0.998520i	$-0.517322\pi$
$199$	−1.20933e7	−0.108782	−0.0543911	+	0.998520i	$0.517322\pi$
$200$	0	0
$201$	1.27234e7	0.110514
$202$	0	0
$203$	−1.63751e8	−1.37388
$204$	0	0
$205$	−2.32632e7	−0.188596
$206$	0	0
$207$	−1.40518e7	−0.110112
$208$	0	0
$209$	7.95784e7	0.602953
$210$	0	0
$211$	1.95850e8	1.43527	0.717636	−	0.696418i	$-0.245224\pi$
$211$	1.95850e8	1.43527	0.717636	+	0.696418i	$0.245224\pi$
$212$	0	0
$213$	2.16752e7	0.153686
$214$	0	0
$215$	−1.06569e8	−0.731302
$216$	0	0
$217$	1.06881e8	0.710057
$218$	0	0
$219$	−1.07986e8	−0.694723
$220$	0	0
$221$	2.17532e8	1.35566
$222$	0	0
$223$	1.08024e8	0.652311	0.326156	−	0.945316i	$-0.394247\pi$
$223$	1.08024e8	0.652311	0.326156	+	0.945316i	$0.394247\pi$
$224$	0	0
$225$	−1.54773e7	−0.0905851
$226$	0	0
$227$	−1.61144e8	−0.914374	−0.457187	−	0.889371i	$-0.651143\pi$
$227$	−1.61144e8	−0.914374	−0.457187	+	0.889371i	$0.651143\pi$
$228$	0	0
$229$	5.27173e7	0.290088	0.145044	−	0.989425i	$-0.453668\pi$
$229$	5.27173e7	0.290088	0.145044	+	0.989425i	$0.453668\pi$
$230$	0	0
$231$	−2.09830e8	−1.12002
$232$	0	0
$233$	1.79423e8	0.929249	0.464625	−	0.885508i	$-0.346189\pi$
$233$	1.79423e8	0.929249	0.464625	+	0.885508i	$0.346189\pi$
$234$	0	0
$235$	3.86285e7	0.194165
$236$	0	0
$237$	−3.30832e8	−1.61431
$238$	0	0
$239$	−8.42441e7	−0.399160	−0.199580	−	0.979882i	$-0.563958\pi$
$239$	−8.42441e7	−0.399160	−0.199580	+	0.979882i	$0.563958\pi$
$240$	0	0
$241$	−2.12302e8	−0.977000	−0.488500	−	0.872564i	$-0.662456\pi$
$241$	−2.12302e8	−0.977000	−0.488500	+	0.872564i	$0.662456\pi$
$242$	0	0
$243$	−1.13255e8	−0.506333
$244$	0	0
$245$	2.99280e7	0.130016
$246$	0	0
$247$	9.17975e7	0.387607
$248$	0	0
$249$	−2.37079e8	−0.973186
$250$	0	0
$251$	−1.18102e8	−0.471411	−0.235706	−	0.971825i	$-0.575740\pi$
$251$	−1.18102e8	−0.471411	−0.235706	+	0.971825i	$0.575740\pi$
$252$	0	0
$253$	1.34492e8	0.522125
$254$	0	0
$255$	−4.51359e8	−1.70463
$256$	0	0
$257$	1.27463e8	0.468402	0.234201	−	0.972188i	$-0.424753\pi$
$257$	1.27463e8	0.468402	0.234201	+	0.972188i	$0.424753\pi$
$258$	0	0
$259$	−9.91755e7	−0.354695
$260$	0	0
$261$	9.74762e7	0.339357
$262$	0	0
$263$	−4.33125e8	−1.46814	−0.734071	−	0.679073i	$-0.762382\pi$
$263$	−4.33125e8	−1.46814	−0.734071	+	0.679073i	$0.762382\pi$
$264$	0	0
$265$	−3.37506e8	−1.11409
$266$	0	0
$267$	1.75747e7	0.0565066
$268$	0	0
$269$	3.44748e8	1.07986	0.539931	−	0.841709i	$-0.318450\pi$
$269$	3.44748e8	1.07986	0.539931	+	0.841709i	$0.318450\pi$
$270$	0	0
$271$	−4.42513e8	−1.35062	−0.675311	−	0.737533i	$-0.735990\pi$
$271$	−4.42513e8	−1.35062	−0.675311	+	0.737533i	$0.735990\pi$
$272$	0	0
$273$	−2.42049e8	−0.720003
$274$	0	0
$275$	1.48135e8	0.429531
$276$	0	0
$277$	−3.18148e8	−0.899395	−0.449697	−	0.893181i	$-0.648468\pi$
$277$	−3.18148e8	−0.899395	−0.449697	+	0.893181i	$0.648468\pi$
$278$	0	0
$279$	−6.36234e7	−0.175389
$280$	0	0
$281$	−1.28497e8	−0.345478	−0.172739	−	0.984968i	$-0.555262\pi$
$281$	−1.28497e8	−0.345478	−0.172739	+	0.984968i	$0.555262\pi$
$282$	0	0
$283$	3.98970e8	1.04637	0.523187	−	0.852218i	$-0.324743\pi$
$283$	3.98970e8	1.04637	0.523187	+	0.852218i	$0.324743\pi$
$284$	0	0
$285$	−1.90471e8	−0.487384
$286$	0	0
$287$	−6.86057e7	−0.171306
$288$	0	0
$289$	7.85942e8	1.91535
$290$	0	0
$291$	3.15425e8	0.750362
$292$	0	0
$293$	−2.00958e8	−0.466732	−0.233366	−	0.972389i	$-0.574974\pi$
$293$	−2.00958e8	−0.466732	−0.233366	+	0.972389i	$0.574974\pi$
$294$	0	0
$295$	−7.33298e7	−0.166304
$296$	0	0
$297$	6.04444e8	1.33878
$298$	0	0
$299$	1.55143e8	0.335647
$300$	0	0
$301$	−3.14284e8	−0.664262
$302$	0	0
$303$	2.49243e8	0.514724
$304$	0	0
$305$	−5.03830e8	−1.01680
$306$	0	0
$307$	1.58918e7	0.0313465	0.0156733	−	0.999877i	$-0.495011\pi$
$307$	1.58918e7	0.0313465	0.0156733	+	0.999877i	$0.495011\pi$
$308$	0	0
$309$	−2.65353e8	−0.511646
$310$	0	0
$311$	−4.87710e8	−0.919391	−0.459695	−	0.888077i	$-0.652041\pi$
$311$	−4.87710e8	−0.919391	−0.459695	+	0.888077i	$0.652041\pi$
$312$	0	0
$313$	3.24731e8	0.598576	0.299288	−	0.954163i	$-0.403251\pi$
$313$	3.24731e8	0.598576	0.299288	+	0.954163i	$0.403251\pi$
$314$	0	0
$315$	−1.76891e8	−0.318874
$316$	0	0
$317$	−1.06084e9	−1.87043	−0.935213	−	0.354086i	$-0.884792\pi$
$317$	−1.06084e9	−1.87043	−0.935213	+	0.354086i	$0.884792\pi$
$318$	0	0
$319$	−9.32958e8	−1.60914
$320$	0	0
$321$	−2.06247e7	−0.0348033
$322$	0	0
$323$	5.04824e8	0.833548
$324$	0	0
$325$	1.70881e8	0.276123
$326$	0	0
$327$	−7.87763e8	−1.24589
$328$	0	0
$329$	1.13919e8	0.176365
$330$	0	0
$331$	2.88487e8	0.437249	0.218624	−	0.975809i	$-0.429843\pi$
$331$	2.88487e8	0.437249	0.218624	+	0.975809i	$0.429843\pi$
$332$	0	0
$333$	5.90363e7	0.0876121
$334$	0	0
$335$	−1.02661e8	−0.149193
$336$	0	0
$337$	−1.10595e8	−0.157410	−0.0787051	−	0.996898i	$-0.525079\pi$
$337$	−1.10595e8	−0.157410	−0.0787051	+	0.996898i	$0.525079\pi$
$338$	0	0
$339$	−7.59590e8	−1.05896
$340$	0	0
$341$	6.08948e8	0.831649
$342$	0	0
$343$	−6.99837e8	−0.936414
$344$	0	0
$345$	−3.21907e8	−0.422049
$346$	0	0
$347$	−1.10651e9	−1.42168	−0.710841	−	0.703352i	$-0.751686\pi$
$347$	−1.10651e9	−1.42168	−0.710841	+	0.703352i	$0.751686\pi$
$348$	0	0
$349$	1.38337e9	1.74201	0.871003	−	0.491278i	$-0.163470\pi$
$349$	1.38337e9	1.74201	0.871003	+	0.491278i	$0.163470\pi$
$350$	0	0
$351$	6.97254e8	0.860630
$352$	0	0
$353$	−2.47617e8	−0.299618	−0.149809	−	0.988715i	$-0.547866\pi$
$353$	−2.47617e8	−0.299618	−0.149809	+	0.988715i	$0.547866\pi$
$354$	0	0
$355$	−1.74890e8	−0.207475
$356$	0	0
$357$	−1.33111e9	−1.54837
$358$	0	0
$359$	1.38641e9	1.58148	0.790738	−	0.612155i	$-0.209697\pi$
$359$	1.38641e9	1.58148	0.790738	+	0.612155i	$0.209697\pi$
$360$	0	0
$361$	−6.80839e8	−0.761674
$362$	0	0
$363$	−4.11789e8	−0.451858
$364$	0	0
$365$	8.71299e8	0.937869
$366$	0	0
$367$	−7.49367e8	−0.791341	−0.395670	−	0.918393i	$-0.629488\pi$
$367$	−7.49367e8	−0.791341	−0.395670	+	0.918393i	$0.629488\pi$
$368$	0	0
$369$	4.08390e7	0.0423139
$370$	0	0
$371$	−9.95340e8	−1.01196
$372$	0	0
$373$	1.49519e9	1.49181	0.745906	−	0.666051i	$-0.232017\pi$
$373$	1.49519e9	1.49181	0.745906	+	0.666051i	$0.232017\pi$
$374$	0	0
$375$	6.64957e8	0.651155
$376$	0	0
$377$	−1.07621e9	−1.03443
$378$	0	0
$379$	−7.92096e7	−0.0747379	−0.0373689	−	0.999302i	$-0.511898\pi$
$379$	−7.92096e7	−0.0747379	−0.0373689	+	0.999302i	$0.511898\pi$
$380$	0	0
$381$	−1.59383e9	−1.47640
$382$	0	0
$383$	4.80285e8	0.436820	0.218410	−	0.975857i	$-0.429913\pi$
$383$	4.80285e8	0.436820	0.218410	+	0.975857i	$0.429913\pi$
$384$	0	0
$385$	1.69305e9	1.51202
$386$	0	0
$387$	1.87084e8	0.164077
$388$	0	0
$389$	1.07150e9	0.922928	0.461464	−	0.887159i	$-0.347324\pi$
$389$	1.07150e9	0.922928	0.461464	+	0.887159i	$0.347324\pi$
$390$	0	0
$391$	8.53180e8	0.721809
$392$	0	0
$393$	1.46925e9	1.22102
$394$	0	0
$395$	2.66937e9	2.17931
$396$	0	0
$397$	2.03185e9	1.62976	0.814882	−	0.579627i	$-0.196802\pi$
$397$	2.03185e9	1.62976	0.814882	+	0.579627i	$0.196802\pi$
$398$	0	0
$399$	−5.61719e8	−0.442705
$400$	0	0
$401$	−2.57759e9	−1.99622	−0.998111	−	0.0614301i	$-0.980434\pi$
$401$	−2.57759e9	−1.99622	−0.998111	+	0.0614301i	$0.980434\pi$
$402$	0	0
$403$	7.02451e8	0.534624
$404$	0	0
$405$	−1.04248e9	−0.779782
$406$	0	0
$407$	−5.65044e8	−0.415434
$408$	0	0
$409$	−3.30242e8	−0.238672	−0.119336	−	0.992854i	$-0.538077\pi$
$409$	−3.30242e8	−0.238672	−0.119336	+	0.992854i	$0.538077\pi$
$410$	0	0
$411$	−1.03343e9	−0.734232
$412$	0	0
$413$	−2.16257e8	−0.151059
$414$	0	0
$415$	1.91291e9	1.31379
$416$	0	0
$417$	2.10331e9	1.42046
$418$	0	0
$419$	5.80021e7	0.0385207	0.0192604	−	0.999815i	$-0.493869\pi$
$419$	5.80021e7	0.0385207	0.0192604	+	0.999815i	$0.493869\pi$
$420$	0	0
$421$	−1.90609e8	−0.124496	−0.0622480	−	0.998061i	$-0.519827\pi$
$421$	−1.90609e8	−0.124496	−0.0622480	+	0.998061i	$0.519827\pi$
$422$	0	0
$423$	−6.78129e7	−0.0435633
$424$	0	0
$425$	9.39731e8	0.593803
$426$	0	0
$427$	−1.48585e9	−0.923586
$428$	0	0
$429$	−1.37906e9	−0.843298
$430$	0	0
$431$	2.42923e9	1.46150	0.730749	−	0.682646i	$-0.239171\pi$
$431$	2.42923e9	1.46150	0.730749	+	0.682646i	$0.239171\pi$
$432$	0	0
$433$	−2.37902e9	−1.40828	−0.704141	−	0.710060i	$-0.748668\pi$
$433$	−2.37902e9	−1.40828	−0.704141	+	0.710060i	$0.748668\pi$
$434$	0	0
$435$	2.23303e9	1.30072
$436$	0	0
$437$	3.60037e8	0.206378
$438$	0	0
$439$	1.33161e9	0.751194	0.375597	−	0.926783i	$-0.377438\pi$
$439$	1.33161e9	0.751194	0.375597	+	0.926783i	$0.377438\pi$
$440$	0	0
$441$	−5.25390e7	−0.0291707
$442$	0	0
$443$	−5.02643e8	−0.274692	−0.137346	−	0.990523i	$-0.543857\pi$
$443$	−5.02643e8	−0.274692	−0.137346	+	0.990523i	$0.543857\pi$
$444$	0	0
$445$	−1.41805e8	−0.0762834
$446$	0	0
$447$	7.25525e8	0.384217
$448$	0	0
$449$	3.14785e9	1.64116	0.820580	−	0.571531i	$-0.193650\pi$
$449$	3.14785e9	1.64116	0.820580	+	0.571531i	$0.193650\pi$
$450$	0	0
$451$	−3.90876e8	−0.200641
$452$	0	0
$453$	1.55792e9	0.787410
$454$	0	0
$455$	1.95301e9	0.971998
$456$	0	0
$457$	2.68422e9	1.31556	0.657782	−	0.753209i	$-0.271495\pi$
$457$	2.68422e9	1.31556	0.657782	+	0.753209i	$0.271495\pi$
$458$	0	0
$459$	3.83442e9	1.85078
$460$	0	0
$461$	1.30434e9	0.620065	0.310033	−	0.950726i	$-0.399660\pi$
$461$	1.30434e9	0.620065	0.310033	+	0.950726i	$0.399660\pi$
$462$	0	0
$463$	2.86853e9	1.34315	0.671577	−	0.740934i	$-0.265617\pi$
$463$	2.86853e9	1.34315	0.671577	+	0.740934i	$0.265617\pi$
$464$	0	0
$465$	−1.45752e9	−0.672246
$466$	0	0
$467$	5.96519e8	0.271029	0.135514	−	0.990775i	$-0.456731\pi$
$467$	5.96519e8	0.271029	0.135514	+	0.990775i	$0.456731\pi$
$468$	0	0
$469$	−3.02758e8	−0.135516
$470$	0	0
$471$	2.06045e9	0.908632
$472$	0	0
$473$	−1.79061e9	−0.778012
$474$	0	0
$475$	3.96561e8	0.169778
$476$	0	0
$477$	5.92497e8	0.249961
$478$	0	0
$479$	−2.16068e9	−0.898289	−0.449144	−	0.893459i	$-0.648271\pi$
$479$	−2.16068e9	−0.898289	−0.449144	+	0.893459i	$0.648271\pi$
$480$	0	0
$481$	−6.51805e8	−0.267061
$482$	0	0
$483$	−9.49337e8	−0.383359
$484$	0	0
$485$	−2.54506e9	−1.01298
$486$	0	0
$487$	−1.41934e8	−0.0556847	−0.0278424	−	0.999612i	$-0.508864\pi$
$487$	−1.41934e8	−0.0556847	−0.0278424	+	0.999612i	$0.508864\pi$
$488$	0	0
$489$	−3.44884e9	−1.33380
$490$	0	0
$491$	−2.38677e9	−0.909966	−0.454983	−	0.890500i	$-0.650355\pi$
$491$	−2.38677e9	−0.909966	−0.454983	+	0.890500i	$0.650355\pi$
$492$	0	0
$493$	−5.91843e9	−2.22455
$494$	0	0
$495$	−1.00782e9	−0.373479
$496$	0	0
$497$	−5.15769e8	−0.188455
$498$	0	0
$499$	−5.23900e9	−1.88754	−0.943771	−	0.330601i	$-0.892749\pi$
$499$	−5.23900e9	−1.88754	−0.943771	+	0.330601i	$0.892749\pi$
$500$	0	0
$501$	−4.25772e9	−1.51267
$502$	0	0
$503$	3.63292e9	1.27282	0.636411	−	0.771350i	$-0.280418\pi$
$503$	3.63292e9	1.27282	0.636411	+	0.771350i	$0.280418\pi$
$504$	0	0
$505$	−2.01106e9	−0.694872
$506$	0	0
$507$	9.32706e8	0.317846
$508$	0	0
$509$	2.58693e9	0.869505	0.434753	−	0.900550i	$-0.356836\pi$
$509$	2.58693e9	0.869505	0.434753	+	0.900550i	$0.356836\pi$
$510$	0	0
$511$	2.56955e9	0.851892
$512$	0	0
$513$	1.61811e9	0.529171
$514$	0	0
$515$	2.14105e9	0.690718
$516$	0	0
$517$	6.49047e8	0.206566
$518$	0	0
$519$	7.96876e8	0.250210
$520$	0	0
$521$	−1.08542e8	−0.0336253	−0.0168127	−	0.999859i	$-0.505352\pi$
$521$	−1.08542e8	−0.0336253	−0.0168127	+	0.999859i	$0.505352\pi$
$522$	0	0
$523$	−6.10725e9	−1.86676	−0.933382	−	0.358884i	$-0.883157\pi$
$523$	−6.10725e9	−1.86676	−0.933382	+	0.358884i	$0.883157\pi$
$524$	0	0
$525$	−1.04564e9	−0.315374
$526$	0	0
$527$	3.86300e9	1.14971
$528$	0	0
$529$	−2.79634e9	−0.821288
$530$	0	0
$531$	1.28732e8	0.0373125
$532$	0	0
$533$	−4.50894e8	−0.128982
$534$	0	0
$535$	1.66414e8	0.0469841
$536$	0	0
$537$	1.19764e9	0.333747
$538$	0	0
$539$	5.02858e8	0.138320
$540$	0	0
$541$	5.39345e8	0.146445	0.0732227	−	0.997316i	$-0.476672\pi$
$541$	5.39345e8	0.146445	0.0732227	+	0.997316i	$0.476672\pi$
$542$	0	0
$543$	−1.59348e8	−0.0427116
$544$	0	0
$545$	6.35620e9	1.68194
$546$	0	0
$547$	8.82287e7	0.0230491	0.0115246	−	0.999934i	$-0.496332\pi$
$547$	8.82287e7	0.0230491	0.0115246	+	0.999934i	$0.496332\pi$
$548$	0	0
$549$	8.84483e8	0.228132
$550$	0	0
$551$	−2.49754e9	−0.636037
$552$	0	0
$553$	7.87226e9	1.97953
$554$	0	0
$555$	1.35243e9	0.335807
$556$	0	0
$557$	−5.57233e8	−0.136629	−0.0683147	−	0.997664i	$-0.521762\pi$
$557$	−5.57233e8	−0.136629	−0.0683147	+	0.997664i	$0.521762\pi$
$558$	0	0
$559$	−2.06555e9	−0.500143
$560$	0	0
$561$	−7.58386e9	−1.81351
$562$	0	0
$563$	−1.17012e9	−0.276344	−0.138172	−	0.990408i	$-0.544123\pi$
$563$	−1.17012e9	−0.276344	−0.138172	+	0.990408i	$0.544123\pi$
$564$	0	0
$565$	6.12887e9	1.42959
$566$	0	0
$567$	−3.07437e9	−0.708297
$568$	0	0
$569$	2.39181e9	0.544295	0.272147	−	0.962256i	$-0.412266\pi$
$569$	2.39181e9	0.544295	0.272147	+	0.962256i	$0.412266\pi$
$570$	0	0
$571$	3.15823e9	0.709933	0.354966	−	0.934879i	$-0.384492\pi$
$571$	3.15823e9	0.709933	0.354966	+	0.934879i	$0.384492\pi$
$572$	0	0
$573$	1.93080e9	0.428743
$574$	0	0
$575$	6.70211e8	0.147019
$576$	0	0
$577$	4.03435e9	0.874296	0.437148	−	0.899390i	$-0.355989\pi$
$577$	4.03435e9	0.874296	0.437148	+	0.899390i	$0.355989\pi$
$578$	0	0
$579$	1.90023e8	0.0406846
$580$	0	0
$581$	5.64138e9	1.19335
$582$	0	0
$583$	−5.67087e9	−1.18525
$584$	0	0
$585$	−1.16257e9	−0.240090
$586$	0	0
$587$	2.72240e9	0.555544	0.277772	−	0.960647i	$-0.410404\pi$
$587$	2.72240e9	0.555544	0.277772	+	0.960647i	$0.410404\pi$
$588$	0	0
$589$	1.63016e9	0.328721
$590$	0	0
$591$	−4.62012e9	−0.920655
$592$	0	0
$593$	−2.29251e9	−0.451460	−0.225730	−	0.974190i	$-0.572477\pi$
$593$	−2.29251e9	−0.451460	−0.225730	+	0.974190i	$0.572477\pi$
$594$	0	0
$595$	1.07402e10	2.09028
$596$	0	0
$597$	4.86346e8	0.0935482
$598$	0	0
$599$	3.58734e9	0.681991	0.340995	−	0.940065i	$-0.389236\pi$
$599$	3.58734e9	0.681991	0.340995	+	0.940065i	$0.389236\pi$
$600$	0	0
$601$	−8.20369e9	−1.54152	−0.770759	−	0.637127i	$-0.780123\pi$
$601$	−8.20369e9	−1.54152	−0.770759	+	0.637127i	$0.780123\pi$
$602$	0	0
$603$	1.80223e8	0.0334734
$604$	0	0
$605$	3.32259e9	0.610004
$606$	0	0
$607$	−4.60087e9	−0.834986	−0.417493	−	0.908680i	$-0.637091\pi$
$607$	−4.60087e9	−0.834986	−0.417493	+	0.908680i	$0.637091\pi$
$608$	0	0
$609$	6.58546e9	1.18148
$610$	0	0
$611$	7.48707e8	0.132791
$612$	0	0
$613$	−8.55728e9	−1.50046	−0.750229	−	0.661178i	$-0.770057\pi$
$613$	−8.55728e9	−1.50046	−0.750229	+	0.661178i	$0.770057\pi$
$614$	0	0
$615$	9.35561e8	0.162184
$616$	0	0
$617$	2.58089e9	0.442355	0.221178	−	0.975234i	$-0.429010\pi$
$617$	2.58089e9	0.442355	0.221178	+	0.975234i	$0.429010\pi$
$618$	0	0
$619$	−5.26641e9	−0.892478	−0.446239	−	0.894914i	$-0.647237\pi$
$619$	−5.26641e9	−0.892478	−0.446239	+	0.894914i	$0.647237\pi$
$620$	0	0
$621$	2.73469e9	0.458234
$622$	0	0
$623$	−4.18197e8	−0.0692903
$624$	0	0
$625$	−7.48796e9	−1.22683
$626$	0	0
$627$	−3.20035e9	−0.518514
$628$	0	0
$629$	−3.58449e9	−0.574314
$630$	0	0
$631$	−8.32515e9	−1.31914	−0.659568	−	0.751645i	$-0.729261\pi$
$631$	−8.32515e9	−1.31914	−0.659568	+	0.751645i	$0.729261\pi$
$632$	0	0
$633$	−7.87635e9	−1.23428
$634$	0	0
$635$	1.28601e10	1.99313
$636$	0	0
$637$	5.80071e8	0.0889187
$638$	0	0
$639$	3.07022e8	0.0465496
$640$	0	0
$641$	4.26190e9	0.639146	0.319573	−	0.947562i	$-0.396460\pi$
$641$	4.26190e9	0.639146	0.319573	+	0.947562i	$0.396460\pi$
$642$	0	0
$643$	−1.26588e10	−1.87782	−0.938908	−	0.344167i	$-0.888161\pi$
$643$	−1.26588e10	−1.87782	−0.938908	+	0.344167i	$0.888161\pi$
$644$	0	0
$645$	4.28582e9	0.628890
$646$	0	0
$647$	7.38061e9	1.07134	0.535670	−	0.844427i	$-0.320059\pi$
$647$	7.38061e9	1.07134	0.535670	+	0.844427i	$0.320059\pi$
$648$	0	0
$649$	−1.23211e9	−0.176926
$650$	0	0
$651$	−4.29837e9	−0.610620
$652$	0	0
$653$	−3.21579e9	−0.451951	−0.225975	−	0.974133i	$-0.572557\pi$
$653$	−3.21579e9	−0.451951	−0.225975	+	0.974133i	$0.572557\pi$
$654$	0	0
$655$	−1.18549e10	−1.64836
$656$	0	0
$657$	−1.52958e9	−0.210423
$658$	0	0
$659$	5.17004e9	0.703711	0.351856	−	0.936054i	$-0.385551\pi$
$659$	5.17004e9	0.703711	0.351856	+	0.936054i	$0.385551\pi$
$660$	0	0
$661$	1.95604e9	0.263435	0.131717	−	0.991287i	$-0.457951\pi$
$661$	1.95604e9	0.263435	0.131717	+	0.991287i	$0.457951\pi$
$662$	0	0
$663$	−8.74835e9	−1.16581
$664$	0	0
$665$	4.53232e9	0.597647
$666$	0	0
$667$	−4.22099e9	−0.550775
$668$	0	0
$669$	−4.34434e9	−0.560961
$670$	0	0
$671$	−8.46551e9	−1.08174
$672$	0	0
$673$	1.54679e9	0.195605	0.0978024	−	0.995206i	$-0.468819\pi$
$673$	1.54679e9	0.195605	0.0978024	+	0.995206i	$0.468819\pi$
$674$	0	0
$675$	3.01211e9	0.376971
$676$	0	0
$677$	8.55209e9	1.05928	0.529642	−	0.848222i	$-0.322326\pi$
$677$	8.55209e9	1.05928	0.529642	+	0.848222i	$0.322326\pi$
$678$	0	0
$679$	−7.50565e9	−0.920119
$680$	0	0
$681$	6.48061e9	0.786323
$682$	0	0
$683$	−7.26976e9	−0.873067	−0.436534	−	0.899688i	$-0.643794\pi$
$683$	−7.26976e9	−0.873067	−0.436534	+	0.899688i	$0.643794\pi$
$684$	0	0
$685$	8.33837e9	0.991206
$686$	0	0
$687$	−2.12010e9	−0.249463
$688$	0	0
$689$	−6.54162e9	−0.761935
$690$	0	0
$691$	−5.19893e9	−0.599434	−0.299717	−	0.954028i	$-0.596892\pi$
$691$	−5.19893e9	−0.599434	−0.299717	+	0.954028i	$0.596892\pi$
$692$	0	0
$693$	−2.97218e9	−0.339241
$694$	0	0
$695$	−1.69709e10	−1.91760
$696$	0	0
$697$	−2.47961e9	−0.277376
$698$	0	0
$699$	−7.21572e9	−0.799116
$700$	0	0
$701$	−1.35221e10	−1.48262	−0.741310	−	0.671163i	$-0.765795\pi$
$701$	−1.35221e10	−1.48262	−0.741310	+	0.671163i	$0.765795\pi$
$702$	0	0
$703$	−1.51263e9	−0.164206
$704$	0	0
$705$	−1.55349e9	−0.166974
$706$	0	0
$707$	−5.93083e9	−0.631172
$708$	0	0
$709$	−1.05901e10	−1.11594	−0.557969	−	0.829862i	$-0.688419\pi$
$709$	−1.05901e10	−1.11594	−0.557969	+	0.829862i	$0.688419\pi$
$710$	0	0
$711$	−4.68613e9	−0.488957
$712$	0	0
$713$	2.75507e9	0.284655
$714$	0	0
$715$	1.11271e10	1.13845
$716$	0	0
$717$	3.38799e9	0.343261
$718$	0	0
$719$	1.49161e10	1.49659	0.748297	−	0.663363i	$-0.230872\pi$
$719$	1.49161e10	1.49659	0.748297	+	0.663363i	$0.230872\pi$
$720$	0	0
$721$	6.31417e9	0.627398
$722$	0	0
$723$	8.53800e9	0.840180
$724$	0	0
$725$	−4.64919e9	−0.453100
$726$	0	0
$727$	8.90159e9	0.859206	0.429603	−	0.903018i	$-0.358654\pi$
$727$	8.90159e9	0.859206	0.429603	+	0.903018i	$0.358654\pi$
$728$	0	0
$729$	1.15808e10	1.10711
$730$	0	0
$731$	−1.13591e10	−1.07556
$732$	0	0
$733$	−7.99792e9	−0.750090	−0.375045	−	0.927007i	$-0.622373\pi$
$733$	−7.99792e9	−0.750090	−0.375045	+	0.927007i	$0.622373\pi$
$734$	0	0
$735$	−1.20359e9	−0.111808
$736$	0	0
$737$	−1.72494e9	−0.158722
$738$	0	0
$739$	1.03852e10	0.946588	0.473294	−	0.880905i	$-0.343065\pi$
$739$	1.03852e10	0.946588	0.473294	+	0.880905i	$0.343065\pi$
$740$	0	0
$741$	−3.69175e9	−0.333326
$742$	0	0
$743$	3.73477e9	0.334044	0.167022	−	0.985953i	$-0.446585\pi$
$743$	3.73477e9	0.334044	0.167022	+	0.985953i	$0.446585\pi$
$744$	0	0
$745$	−5.85402e9	−0.518689
$746$	0	0
$747$	−3.35815e9	−0.294766
$748$	0	0
$749$	4.90772e8	0.0426770
$750$	0	0
$751$	−1.34330e10	−1.15726	−0.578631	−	0.815589i	$-0.696413\pi$
$751$	−1.34330e10	−1.15726	−0.578631	+	0.815589i	$0.696413\pi$
$752$	0	0
$753$	4.74963e9	0.405394
$754$	0	0
$755$	−1.25703e10	−1.06300
$756$	0	0
$757$	6.78007e9	0.568065	0.284033	−	0.958815i	$-0.408328\pi$
$757$	6.78007e9	0.568065	0.284033	+	0.958815i	$0.408328\pi$
$758$	0	0
$759$	−5.40877e9	−0.449006
$760$	0	0
$761$	−8.01137e9	−0.658962	−0.329481	−	0.944162i	$-0.606874\pi$
$761$	−8.01137e9	−0.658962	−0.329481	+	0.944162i	$0.606874\pi$
$762$	0	0
$763$	1.87451e10	1.52775
$764$	0	0
$765$	−6.39335e9	−0.516314
$766$	0	0
$767$	−1.42130e9	−0.113737
$768$	0	0
$769$	1.46553e10	1.16213	0.581063	−	0.813858i	$-0.302637\pi$
$769$	1.46553e10	1.16213	0.581063	+	0.813858i	$0.302637\pi$
$770$	0	0
$771$	−5.12609e9	−0.402806
$772$	0	0
$773$	−2.33296e10	−1.81668	−0.908340	−	0.418233i	$-0.862650\pi$
$773$	−2.33296e10	−1.81668	−0.908340	+	0.418233i	$0.862650\pi$
$774$	0	0
$775$	3.03456e9	0.234174
$776$	0	0
$777$	3.98847e9	0.305023
$778$	0	0
$779$	−1.04638e9	−0.0793064
$780$	0	0
$781$	−2.93855e9	−0.220726
$782$	0	0
$783$	−1.89703e10	−1.41224
$784$	0	0
$785$	−1.66250e10	−1.22664
$786$	0	0
$787$	−5.27740e8	−0.0385930	−0.0192965	−	0.999814i	$-0.506143\pi$
$787$	−5.27740e8	−0.0385930	−0.0192965	+	0.999814i	$0.506143\pi$
$788$	0	0
$789$	1.74187e10	1.26254
$790$	0	0
$791$	1.80747e10	1.29853
$792$	0	0
$793$	−9.76536e9	−0.695396
$794$	0	0
$795$	1.35732e10	0.958072
$796$	0	0
$797$	2.41514e9	0.168981	0.0844907	−	0.996424i	$-0.473074\pi$
$797$	2.41514e9	0.168981	0.0844907	+	0.996424i	$0.473074\pi$
$798$	0	0
$799$	4.11738e9	0.285566
$800$	0	0
$801$	2.48940e8	0.0171152
$802$	0	0
$803$	1.46398e10	0.997773
$804$	0	0
$805$	7.65988e9	0.517531
$806$	0	0
$807$	−1.38645e10	−0.928636
$808$	0	0
$809$	−1.16364e10	−0.772679	−0.386340	−	0.922357i	$-0.626261\pi$
$809$	−1.16364e10	−0.772679	−0.386340	+	0.922357i	$0.626261\pi$
$810$	0	0
$811$	1.44359e10	0.950320	0.475160	−	0.879899i	$-0.342390\pi$
$811$	1.44359e10	0.950320	0.475160	+	0.879899i	$0.342390\pi$
$812$	0	0
$813$	1.77962e10	1.16148
$814$	0	0
$815$	2.78275e10	1.80062
$816$	0	0
$817$	−4.79348e9	−0.307521
$818$	0	0
$819$	−3.42855e9	−0.218080
$820$	0	0
$821$	−7.63805e9	−0.481706	−0.240853	−	0.970562i	$-0.577427\pi$
$821$	−7.63805e9	−0.481706	−0.240853	+	0.970562i	$0.577427\pi$
$822$	0	0
$823$	2.16446e10	1.35348	0.676738	−	0.736224i	$-0.263393\pi$
$823$	2.16446e10	1.35348	0.676738	+	0.736224i	$0.263393\pi$
$824$	0	0
$825$	−5.95746e9	−0.369379
$826$	0	0
$827$	−1.57823e10	−0.970288	−0.485144	−	0.874434i	$-0.661233\pi$
$827$	−1.57823e10	−0.970288	−0.485144	+	0.874434i	$0.661233\pi$
$828$	0	0
$829$	2.63296e10	1.60510	0.802552	−	0.596582i	$-0.203475\pi$
$829$	2.63296e10	1.60510	0.802552	+	0.596582i	$0.203475\pi$
$830$	0	0
$831$	1.27947e10	0.773442
$832$	0	0
$833$	3.19000e9	0.191220
$834$	0	0
$835$	3.43541e10	2.04210
$836$	0	0
$837$	1.23820e10	0.729882
$838$	0	0
$839$	1.84995e9	0.108142	0.0540710	−	0.998537i	$-0.482780\pi$
$839$	1.84995e9	0.108142	0.0540710	+	0.998537i	$0.482780\pi$
$840$	0	0
$841$	1.20307e10	0.697438
$842$	0	0
$843$	5.16767e9	0.297097
$844$	0	0
$845$	−7.52569e9	−0.429090
$846$	0	0
$847$	9.79866e9	0.554083
$848$	0	0
$849$	−1.60451e10	−0.899839
$850$	0	0
$851$	−2.55643e9	−0.142194
$852$	0	0
$853$	−6.28088e7	−0.00346496	−0.00173248	−	0.999998i	$-0.500551\pi$
$853$	−6.28088e7	−0.00346496	−0.00173248	+	0.999998i	$0.500551\pi$
$854$	0	0
$855$	−2.69796e9	−0.147623
$856$	0	0
$857$	6.25595e9	0.339516	0.169758	−	0.985486i	$-0.445701\pi$
$857$	6.25595e9	0.339516	0.169758	+	0.985486i	$0.445701\pi$
$858$	0	0
$859$	−2.48119e10	−1.33562	−0.667811	−	0.744331i	$-0.732769\pi$
$859$	−2.48119e10	−1.33562	−0.667811	+	0.744331i	$0.732769\pi$
$860$	0	0
$861$	2.75907e9	0.147316
$862$	0	0
$863$	1.28944e10	0.682909	0.341455	−	0.939898i	$-0.389080\pi$
$863$	1.28944e10	0.682909	0.341455	+	0.939898i	$0.389080\pi$
$864$	0	0
$865$	−6.42973e9	−0.337782
$866$	0	0
$867$	−3.16077e10	−1.64712
$868$	0	0
$869$	4.48516e10	2.31851
$870$	0	0
$871$	−1.98980e9	−0.102034
$872$	0	0
$873$	4.46789e9	0.227276
$874$	0	0
$875$	−1.58229e10	−0.798468
$876$	0	0
$877$	−2.75670e10	−1.38004	−0.690018	−	0.723792i	$-0.742397\pi$
$877$	−2.75670e10	−1.38004	−0.690018	+	0.723792i	$0.742397\pi$
$878$	0	0
$879$	8.08177e9	0.401371
$880$	0	0
$881$	2.74343e10	1.35169	0.675847	−	0.737042i	$-0.263778\pi$
$881$	2.74343e10	1.35169	0.675847	+	0.737042i	$0.263778\pi$
$882$	0	0
$883$	−2.09906e10	−1.02604	−0.513018	−	0.858378i	$-0.671473\pi$
$883$	−2.09906e10	−1.02604	−0.513018	+	0.858378i	$0.671473\pi$
$884$	0	0
$885$	2.94905e9	0.143015
$886$	0	0
$887$	2.39134e10	1.15056	0.575280	−	0.817957i	$-0.304893\pi$
$887$	2.39134e10	1.15056	0.575280	+	0.817957i	$0.304893\pi$
$888$	0	0
$889$	3.79257e10	1.81041
$890$	0	0
$891$	−1.75160e10	−0.829588
$892$	0	0
$893$	1.73751e9	0.0816483
$894$	0	0
$895$	−9.66337e9	−0.450555
$896$	0	0
$897$	−6.23927e9	−0.288643
$898$	0	0
$899$	−1.91117e10	−0.877282
$900$	0	0
$901$	−3.59744e10	−1.63854
$902$	0	0
$903$	1.26393e10	0.571238
$904$	0	0
$905$	1.28572e9	0.0576603
$906$	0	0
$907$	3.84425e10	1.71075	0.855373	−	0.518012i	$-0.173328\pi$
$907$	3.84425e10	1.71075	0.855373	+	0.518012i	$0.173328\pi$
$908$	0	0
$909$	3.53045e9	0.155904
$910$	0	0
$911$	3.64507e10	1.59732	0.798660	−	0.601783i	$-0.205543\pi$
$911$	3.64507e10	1.59732	0.798660	+	0.601783i	$0.205543\pi$
$912$	0	0
$913$	3.21413e10	1.39771
$914$	0	0
$915$	2.02622e10	0.874405
$916$	0	0
$917$	−3.49613e10	−1.49725
$918$	0	0
$919$	−2.37343e10	−1.00872	−0.504361	−	0.863493i	$-0.668272\pi$
$919$	−2.37343e10	−1.00872	−0.504361	+	0.863493i	$0.668272\pi$
$920$	0	0
$921$	−6.39111e8	−0.0269567
$922$	0	0
$923$	−3.38976e9	−0.141894
$924$	0	0
$925$	−2.81577e9	−0.116977
$926$	0	0
$927$	−3.75864e9	−0.154972
$928$	0	0
$929$	−4.23952e10	−1.73485	−0.867424	−	0.497570i	$-0.834226\pi$
$929$	−4.23952e10	−1.73485	−0.867424	+	0.497570i	$0.834226\pi$
$930$	0	0
$931$	1.34616e9	0.0546730
$932$	0	0
$933$	1.96139e10	0.790638
$934$	0	0
$935$	6.11916e10	2.44823
$936$	0	0
$937$	−3.70014e10	−1.46937	−0.734683	−	0.678411i	$-0.762669\pi$
$937$	−3.70014e10	−1.46937	−0.734683	+	0.678411i	$0.762669\pi$
$938$	0	0
$939$	−1.30595e10	−0.514750
$940$	0	0
$941$	2.10617e10	0.824005	0.412003	−	0.911183i	$-0.364829\pi$
$941$	2.10617e10	0.824005	0.412003	+	0.911183i	$0.364829\pi$
$942$	0	0
$943$	−1.76844e9	−0.0686752
$944$	0	0
$945$	3.44256e10	1.32700
$946$	0	0
$947$	1.35146e10	0.517104	0.258552	−	0.965997i	$-0.416755\pi$
$947$	1.35146e10	0.517104	0.258552	+	0.965997i	$0.416755\pi$
$948$	0	0
$949$	1.68877e10	0.641416
$950$	0	0
$951$	4.26628e10	1.60849
$952$	0	0
$953$	−4.26831e10	−1.59746	−0.798731	−	0.601688i	$-0.794495\pi$
$953$	−4.26831e10	−1.59746	−0.798731	+	0.601688i	$0.794495\pi$
$954$	0	0
$955$	−1.55790e10	−0.578799
$956$	0	0
$957$	3.75201e10	1.38380
$958$	0	0
$959$	2.45907e10	0.900340
$960$	0	0
$961$	−1.50383e10	−0.546597
$962$	0	0
$963$	−2.92142e8	−0.0105415
$964$	0	0
$965$	−1.53323e9	−0.0549239
$966$	0	0
$967$	−2.32274e10	−0.826053	−0.413026	−	0.910719i	$-0.635528\pi$
$967$	−2.32274e10	−0.826053	−0.413026	+	0.910719i	$0.635528\pi$
$968$	0	0
$969$	−2.03021e10	−0.716817
$970$	0	0
$971$	4.85678e10	1.70248	0.851238	−	0.524780i	$-0.175852\pi$
$971$	4.85678e10	1.70248	0.851238	+	0.524780i	$0.175852\pi$
$972$	0	0
$973$	−5.00491e10	−1.74181
$974$	0	0
$975$	−6.87221e9	−0.237454
$976$	0	0
$977$	1.54549e10	0.530193	0.265096	−	0.964222i	$-0.414596\pi$
$977$	1.54549e10	0.530193	0.265096	+	0.964222i	$0.414596\pi$
$978$	0	0
$979$	−2.38264e9	−0.0811557
$980$	0	0
$981$	−1.11584e10	−0.377364
$982$	0	0
$983$	1.59192e10	0.534546	0.267273	−	0.963621i	$-0.413877\pi$
$983$	1.59192e10	0.534546	0.267273	+	0.963621i	$0.413877\pi$
$984$	0	0
$985$	3.72782e10	1.24288
$986$	0	0
$987$	−4.58142e9	−0.151667
$988$	0	0
$989$	−8.10126e9	−0.266296
$990$	0	0
$991$	1.49883e10	0.489208	0.244604	−	0.969623i	$-0.421342\pi$
$991$	1.49883e10	0.489208	0.244604	+	0.969623i	$0.421342\pi$
$992$	0	0
$993$	−1.16019e10	−0.376016
$994$	0	0
$995$	−3.92416e9	−0.126289
$996$	0	0
$997$	−3.44101e10	−1.09965	−0.549824	−	0.835281i	$-0.685305\pi$
$997$	−3.44101e10	−1.09965	−0.549824	+	0.835281i	$0.685305\pi$
$998$	0	0
$999$	−1.14893e10	−0.364598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.a.r.1.2		6
4.3	odd	2		256.8.a.q.1.5		6
8.3	odd	2		256.8.a.q.1.2		6
8.5	even	2	inner	256.8.a.r.1.5		6
16.3	odd	4		32.8.b.a.17.5		6
16.5	even	4		8.8.b.a.5.1	✓	6
16.11	odd	4		32.8.b.a.17.2		6
16.13	even	4		8.8.b.a.5.2	yes	6
48.5	odd	4		72.8.d.b.37.6		6
48.11	even	4		288.8.d.b.145.2		6
48.29	odd	4		72.8.d.b.37.5		6
48.35	even	4		288.8.d.b.145.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.1	✓	6	16.5	even	4
8.8.b.a.5.2	yes	6	16.13	even	4
32.8.b.a.17.2		6	16.11	odd	4
32.8.b.a.17.5		6	16.3	odd	4
72.8.d.b.37.5		6	48.29	odd	4
72.8.d.b.37.6		6	48.5	odd	4
256.8.a.q.1.2		6	8.3	odd	2
256.8.a.q.1.5		6	4.3	odd	2
256.8.a.r.1.2		6	1.1	even	1	trivial
256.8.a.r.1.5		6	8.5	even	2	inner
288.8.d.b.145.2		6	48.11	even	4
288.8.d.b.145.5		6	48.35	even	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +O(q^{10})\)	\(q-40.2163 q^{3} +324.492 q^{5} +956.960 q^{7} -569.651 q^{9} +5452.20 q^{11} +6289.38 q^{13} -13049.8 q^{15} +34587.3 q^{17} +14595.6 q^{19} -38485.4 q^{21} +24667.5 q^{23} +27169.8 q^{25} +110862. q^{27} -171116. q^{29} +111688. q^{31} -219267. q^{33} +310526. q^{35} -103636. q^{37} -252935. q^{39} -71691.3 q^{41} -328419. q^{43} -184847. q^{45} +119043. q^{47} +92230.3 q^{49} -1.39097e6 q^{51} -1.04011e6 q^{53} +1.76919e6 q^{55} -586982. q^{57} -225984. q^{59} -1.55268e6 q^{61} -545133. q^{63} +2.04085e6 q^{65} -316375. q^{67} -992033. q^{69} -538965. q^{71} +2.68512e6 q^{73} -1.09267e6 q^{75} +5.21754e6 q^{77} +8.22632e6 q^{79} -3.21264e6 q^{81} +5.89510e6 q^{83} +1.12233e7 q^{85} +6.88164e6 q^{87} -437005. q^{89} +6.01868e6 q^{91} -4.49169e6 q^{93} +4.73616e6 q^{95} -7.84322e6 q^{97} -3.10585e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 688 q^{7} + 2918 q^{9}+O(q^{10}) \) 6 * q + 688 * q^7 + 2918 * q^9	\( 6 q + 688 q^{7} + 2918 q^{9} + 17872 q^{15} + 1452 q^{17} + 1296 q^{23} + 39314 q^{25} - 89280 q^{31} + 53880 q^{33} + 328208 q^{39} - 521244 q^{41} + 1566432 q^{47} - 511050 q^{49} + 3270256 q^{55} + 1889896 q^{57} + 5776816 q^{63} + 1416480 q^{65} + 7597104 q^{71} - 2089564 q^{73} + 16015904 q^{79} - 723058 q^{81} + 37453776 q^{87} - 2169084 q^{89} + 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) 6 * q + 688 * q^7 + 2918 * q^9 + 17872 * q^15 + 1452 * q^17 + 1296 * q^23 + 39314 * q^25 - 89280 * q^31 + 53880 * q^33 + 328208 * q^39 - 521244 * q^41 + 1566432 * q^47 - 511050 * q^49 + 3270256 * q^55 + 1889896 * q^57 + 5776816 * q^63 + 1416480 * q^65 + 7597104 * q^71 - 2089564 * q^73 + 16015904 * q^79 - 723058 * q^81 + 37453776 * q^87 - 2169084 * q^89 + 48537936 * q^95 - 1088308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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