LMFDB - Embedded newform 1024.4.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,4,Mod(1,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1024.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1024.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:	$60.4179558459$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 36x^{8} + 405x^{6} - 1380x^{4} + 420x^{2} - 32 $ x^10 - 36x^8 + 405x^6 - 1380x^4 + 420x^2 - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.94652$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.07024 q^{3} +11.6331 q^{5} +2.67171 q^{7} -25.8546 q^{9} +O(q^{10})$	$q-1.07024 q^{3} +11.6331 q^{5} +2.67171 q^{7} -25.8546 q^{9} +63.9525 q^{11} +50.0586 q^{13} -12.4503 q^{15} +72.4991 q^{17} +27.4961 q^{19} -2.85937 q^{21} -139.462 q^{23} +10.3299 q^{25} +56.5672 q^{27} -93.3995 q^{29} +188.682 q^{31} -68.4447 q^{33} +31.0803 q^{35} -118.886 q^{37} -53.5748 q^{39} +104.629 q^{41} +44.5275 q^{43} -300.770 q^{45} +488.151 q^{47} -335.862 q^{49} -77.5916 q^{51} -211.510 q^{53} +743.968 q^{55} -29.4275 q^{57} +402.624 q^{59} +322.538 q^{61} -69.0758 q^{63} +582.338 q^{65} -196.789 q^{67} +149.258 q^{69} +453.655 q^{71} -259.747 q^{73} -11.0555 q^{75} +170.862 q^{77} -323.190 q^{79} +637.533 q^{81} -797.471 q^{83} +843.392 q^{85} +99.9602 q^{87} -866.853 q^{89} +133.742 q^{91} -201.935 q^{93} +319.866 q^{95} -936.077 q^{97} -1653.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) $ 10 * q + 28 * q^7 + 54 * q^9	$ 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) $ 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * 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
$3$	−1.07024	−0.205968	−0.102984	−	0.994683i	$-0.532839\pi$
$3$	−1.07024	−0.205968	−0.102984	+	0.994683i	$0.532839\pi$
$4$	0	0
$5$	11.6331	1.04050	0.520250	−	0.854014i	$-0.325839\pi$
$5$	11.6331	1.04050	0.520250	+	0.854014i	$0.325839\pi$
$6$	0	0
$7$	2.67171	0.144259	0.0721293	−	0.997395i	$-0.477021\pi$
$7$	2.67171	0.144259	0.0721293	+	0.997395i	$0.477021\pi$
$8$	0	0
$9$	−25.8546	−0.957577
$10$	0	0
$11$	63.9525	1.75295	0.876473	−	0.481451i	$-0.159890\pi$
$11$	63.9525	1.75295	0.876473	+	0.481451i	$0.159890\pi$
$12$	0	0
$13$	50.0586	1.06798	0.533990	−	0.845491i	$-0.320692\pi$
$13$	50.0586	1.06798	0.533990	+	0.845491i	$0.320692\pi$
$14$	0	0
$15$	−12.4503	−0.214310
$16$	0	0
$17$	72.4991	1.03433	0.517165	−	0.855886i	$-0.326987\pi$
$17$	72.4991	1.03433	0.517165	+	0.855886i	$0.326987\pi$
$18$	0	0
$19$	27.4961	0.332002	0.166001	−	0.986126i	$-0.446915\pi$
$19$	27.4961	0.332002	0.166001	+	0.986126i	$0.446915\pi$
$20$	0	0
$21$	−2.85937	−0.0297127
$22$	0	0
$23$	−139.462	−1.26434	−0.632170	−	0.774830i	$-0.717835\pi$
$23$	−139.462	−1.26434	−0.632170	+	0.774830i	$0.717835\pi$
$24$	0	0
$25$	10.3299	0.0826390
$26$	0	0
$27$	56.5672	0.403199
$28$	0	0
$29$	−93.3995	−0.598064	−0.299032	−	0.954243i	$-0.596664\pi$
$29$	−93.3995	−0.598064	−0.299032	+	0.954243i	$0.596664\pi$
$30$	0	0
$31$	188.682	1.09317	0.546584	−	0.837404i	$-0.315928\pi$
$31$	188.682	1.09317	0.546584	+	0.837404i	$0.315928\pi$
$32$	0	0
$33$	−68.4447	−0.361051
$34$	0	0
$35$	31.0803	0.150101
$36$	0	0
$37$	−118.886	−0.528238	−0.264119	−	0.964490i	$-0.585081\pi$
$37$	−118.886	−0.528238	−0.264119	+	0.964490i	$0.585081\pi$
$38$	0	0
$39$	−53.5748	−0.219970
$40$	0	0
$41$	104.629	0.398545	0.199272	−	0.979944i	$-0.436142\pi$
$41$	104.629	0.398545	0.199272	+	0.979944i	$0.436142\pi$
$42$	0	0
$43$	44.5275	0.157916	0.0789579	−	0.996878i	$-0.474841\pi$
$43$	44.5275	0.157916	0.0789579	+	0.996878i	$0.474841\pi$
$44$	0	0
$45$	−300.770	−0.996358
$46$	0	0
$47$	488.151	1.51498	0.757491	−	0.652846i	$-0.226425\pi$
$47$	488.151	1.51498	0.757491	+	0.652846i	$0.226425\pi$
$48$	0	0
$49$	−335.862	−0.979189
$50$	0	0
$51$	−77.5916	−0.213039
$52$	0	0
$53$	−211.510	−0.548173	−0.274087	−	0.961705i	$-0.588376\pi$
$53$	−211.510	−0.548173	−0.274087	+	0.961705i	$0.588376\pi$
$54$	0	0
$55$	743.968	1.82394
$56$	0	0
$57$	−29.4275	−0.0683819
$58$	0	0
$59$	402.624	0.888426	0.444213	−	0.895921i	$-0.353483\pi$
$59$	402.624	0.888426	0.444213	+	0.895921i	$0.353483\pi$
$60$	0	0
$61$	322.538	0.676997	0.338498	−	0.940967i	$-0.390081\pi$
$61$	322.538	0.676997	0.338498	+	0.940967i	$0.390081\pi$
$62$	0	0
$63$	−69.0758	−0.138139
$64$	0	0
$65$	582.338	1.11123
$66$	0	0
$67$	−196.789	−0.358829	−0.179415	−	0.983774i	$-0.557420\pi$
$67$	−196.789	−0.358829	−0.179415	+	0.983774i	$0.557420\pi$
$68$	0	0
$69$	149.258	0.260414
$70$	0	0
$71$	453.655	0.758294	0.379147	−	0.925336i	$-0.376217\pi$
$71$	453.655	0.758294	0.379147	+	0.925336i	$0.376217\pi$
$72$	0	0
$73$	−259.747	−0.416454	−0.208227	−	0.978081i	$-0.566769\pi$
$73$	−259.747	−0.416454	−0.208227	+	0.978081i	$0.566769\pi$
$74$	0	0
$75$	−11.0555	−0.0170210
$76$	0	0
$77$	170.862	0.252877
$78$	0	0
$79$	−323.190	−0.460275	−0.230138	−	0.973158i	$-0.573918\pi$
$79$	−323.190	−0.460275	−0.230138	+	0.973158i	$0.573918\pi$
$80$	0	0
$81$	637.533	0.874531
$82$	0	0
$83$	−797.471	−1.05462	−0.527312	−	0.849672i	$-0.676800\pi$
$83$	−797.471	−1.05462	−0.527312	+	0.849672i	$0.676800\pi$
$84$	0	0
$85$	843.392	1.07622
$86$	0	0
$87$	99.9602	0.123182
$88$	0	0
$89$	−866.853	−1.03243	−0.516215	−	0.856459i	$-0.672659\pi$
$89$	−866.853	−1.03243	−0.516215	+	0.856459i	$0.672659\pi$
$90$	0	0
$91$	133.742	0.154065
$92$	0	0
$93$	−201.935	−0.225158
$94$	0	0
$95$	319.866	0.345448
$96$	0	0
$97$	−936.077	−0.979837	−0.489919	−	0.871768i	$-0.662974\pi$
$97$	−936.077	−0.979837	−0.489919	+	0.871768i	$0.662974\pi$
$98$	0	0
$99$	−1653.47	−1.67858
$100$	0	0
$101$	−2.24639	−0.00221311	−0.00110656	−	0.999999i	$-0.500352\pi$
$101$	−2.24639	−0.00221311	−0.00110656	+	0.999999i	$0.500352\pi$
$102$	0	0
$103$	1388.28	1.32807	0.664036	−	0.747700i	$-0.268842\pi$
$103$	1388.28	1.32807	0.664036	+	0.747700i	$0.268842\pi$
$104$	0	0
$105$	−33.2635	−0.0309160
$106$	0	0
$107$	1161.81	1.04969	0.524845	−	0.851198i	$-0.324123\pi$
$107$	1161.81	1.04969	0.524845	+	0.851198i	$0.324123\pi$
$108$	0	0
$109$	753.489	0.662121	0.331060	−	0.943610i	$-0.392594\pi$
$109$	753.489	0.662121	0.331060	+	0.943610i	$0.392594\pi$
$110$	0	0
$111$	127.237	0.108800
$112$	0	0
$113$	67.2680	0.0560003	0.0280002	−	0.999608i	$-0.491086\pi$
$113$	67.2680	0.0560003	0.0280002	+	0.999608i	$0.491086\pi$
$114$	0	0
$115$	−1622.38	−1.31554
$116$	0	0
$117$	−1294.24	−1.02267
$118$	0	0
$119$	193.696	0.149211
$120$	0	0
$121$	2758.92	2.07282
$122$	0	0
$123$	−111.979	−0.0820876
$124$	0	0
$125$	−1333.97	−0.954514
$126$	0	0
$127$	1903.59	1.33005	0.665026	−	0.746820i	$-0.268421\pi$
$127$	1903.59	1.33005	0.665026	+	0.746820i	$0.268421\pi$
$128$	0	0
$129$	−47.6552	−0.0325257
$130$	0	0
$131$	−1298.86	−0.866271	−0.433136	−	0.901329i	$-0.642593\pi$
$131$	−1298.86	−0.866271	−0.433136	+	0.901329i	$0.642593\pi$
$132$	0	0
$133$	73.4615	0.0478941
$134$	0	0
$135$	658.054	0.419528
$136$	0	0
$137$	477.234	0.297612	0.148806	−	0.988866i	$-0.452457\pi$
$137$	477.234	0.297612	0.148806	+	0.988866i	$0.452457\pi$
$138$	0	0
$139$	2140.97	1.30643	0.653217	−	0.757171i	$-0.273419\pi$
$139$	2140.97	1.30643	0.653217	+	0.757171i	$0.273419\pi$
$140$	0	0
$141$	−522.440	−0.312038
$142$	0	0
$143$	3201.37	1.87211
$144$	0	0
$145$	−1086.53	−0.622285
$146$	0	0
$147$	359.454	0.201682
$148$	0	0
$149$	−530.829	−0.291861	−0.145930	−	0.989295i	$-0.546618\pi$
$149$	−530.829	−0.291861	−0.145930	+	0.989295i	$0.546618\pi$
$150$	0	0
$151$	2997.52	1.61546	0.807730	−	0.589553i	$-0.200696\pi$
$151$	2997.52	1.61546	0.807730	+	0.589553i	$0.200696\pi$
$152$	0	0
$153$	−1874.43	−0.990451
$154$	0	0
$155$	2194.96	1.13744
$156$	0	0
$157$	2134.06	1.08482	0.542409	−	0.840115i	$-0.317512\pi$
$157$	2134.06	1.08482	0.542409	+	0.840115i	$0.317512\pi$
$158$	0	0
$159$	226.368	0.112906
$160$	0	0
$161$	−372.601	−0.182392
$162$	0	0
$163$	2015.52	0.968515	0.484258	−	0.874925i	$-0.339090\pi$
$163$	2015.52	0.968515	0.484258	+	0.874925i	$0.339090\pi$
$164$	0	0
$165$	−796.227	−0.375674
$166$	0	0
$167$	792.415	0.367179	0.183590	−	0.983003i	$-0.441228\pi$
$167$	792.415	0.367179	0.183590	+	0.983003i	$0.441228\pi$
$168$	0	0
$169$	308.861	0.140583
$170$	0	0
$171$	−710.900	−0.317917
$172$	0	0
$173$	−1094.03	−0.480794	−0.240397	−	0.970675i	$-0.577278\pi$
$173$	−1094.03	−0.480794	−0.240397	+	0.970675i	$0.577278\pi$
$174$	0	0
$175$	27.5984	0.0119214
$176$	0	0
$177$	−430.905	−0.182988
$178$	0	0
$179$	602.526	0.251592	0.125796	−	0.992056i	$-0.459852\pi$
$179$	602.526	0.251592	0.125796	+	0.992056i	$0.459852\pi$
$180$	0	0
$181$	3702.50	1.52047	0.760234	−	0.649649i	$-0.225084\pi$
$181$	3702.50	1.52047	0.760234	+	0.649649i	$0.225084\pi$
$182$	0	0
$183$	−345.194	−0.139440
$184$	0	0
$185$	−1383.02	−0.549631
$186$	0	0
$187$	4636.50	1.81312
$188$	0	0
$189$	151.131	0.0581649
$190$	0	0
$191$	3216.39	1.21848	0.609240	−	0.792986i	$-0.291475\pi$
$191$	3216.39	1.21848	0.609240	+	0.792986i	$0.291475\pi$
$192$	0	0
$193$	2852.57	1.06390	0.531950	−	0.846776i	$-0.321459\pi$
$193$	2852.57	1.06390	0.531950	+	0.846776i	$0.321459\pi$
$194$	0	0
$195$	−623.243	−0.228879
$196$	0	0
$197$	2275.50	0.822956	0.411478	−	0.911420i	$-0.365013\pi$
$197$	2275.50	0.822956	0.411478	+	0.911420i	$0.365013\pi$
$198$	0	0
$199$	−747.136	−0.266146	−0.133073	−	0.991106i	$-0.542484\pi$
$199$	−747.136	−0.266146	−0.133073	+	0.991106i	$0.542484\pi$
$200$	0	0
$201$	210.612	0.0739074
$202$	0	0
$203$	−249.536	−0.0862758
$204$	0	0
$205$	1217.17	0.414685
$206$	0	0
$207$	3605.73	1.21070
$208$	0	0
$209$	1758.44	0.581981
$210$	0	0
$211$	−3149.64	−1.02763	−0.513816	−	0.857901i	$-0.671769\pi$
$211$	−3149.64	−1.02763	−0.513816	+	0.857901i	$0.671769\pi$
$212$	0	0
$213$	−485.520	−0.156185
$214$	0	0
$215$	517.995	0.164311
$216$	0	0
$217$	504.102	0.157699
$218$	0	0
$219$	277.993	0.0857763
$220$	0	0
$221$	3629.20	1.10464
$222$	0	0
$223$	−358.053	−0.107520	−0.0537601	−	0.998554i	$-0.517121\pi$
$223$	−358.053	−0.107520	−0.0537601	+	0.998554i	$0.517121\pi$
$224$	0	0
$225$	−267.075	−0.0791332
$226$	0	0
$227$	−4886.67	−1.42881	−0.714404	−	0.699733i	$-0.753302\pi$
$227$	−4886.67	−1.42881	−0.714404	+	0.699733i	$0.753302\pi$
$228$	0	0
$229$	−2022.37	−0.583589	−0.291794	−	0.956481i	$-0.594252\pi$
$229$	−2022.37	−0.583589	−0.291794	+	0.956481i	$0.594252\pi$
$230$	0	0
$231$	−182.864	−0.0520847
$232$	0	0
$233$	−926.479	−0.260496	−0.130248	−	0.991481i	$-0.541577\pi$
$233$	−926.479	−0.260496	−0.130248	+	0.991481i	$0.541577\pi$
$234$	0	0
$235$	5678.73	1.57634
$236$	0	0
$237$	345.892	0.0948021
$238$	0	0
$239$	792.472	0.214480	0.107240	−	0.994233i	$-0.465799\pi$
$239$	792.472	0.214480	0.107240	+	0.994233i	$0.465799\pi$
$240$	0	0
$241$	−1449.01	−0.387299	−0.193650	−	0.981071i	$-0.562033\pi$
$241$	−1449.01	−0.387299	−0.193650	+	0.981071i	$0.562033\pi$
$242$	0	0
$243$	−2209.63	−0.583325
$244$	0	0
$245$	−3907.13	−1.01885
$246$	0	0
$247$	1376.42	0.354572
$248$	0	0
$249$	853.487	0.217219
$250$	0	0
$251$	−5062.94	−1.27319	−0.636594	−	0.771199i	$-0.719657\pi$
$251$	−5062.94	−1.27319	−0.636594	+	0.771199i	$0.719657\pi$
$252$	0	0
$253$	−8918.93	−2.21632
$254$	0	0
$255$	−902.634	−0.221667
$256$	0	0
$257$	−4708.87	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$257$	−4708.87	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$258$	0	0
$259$	−317.629	−0.0762028
$260$	0	0
$261$	2414.81	0.572692
$262$	0	0
$263$	2967.82	0.695830	0.347915	−	0.937526i	$-0.386890\pi$
$263$	2967.82	0.695830	0.347915	+	0.937526i	$0.386890\pi$
$264$	0	0
$265$	−2460.53	−0.570374
$266$	0	0
$267$	927.743	0.212648
$268$	0	0
$269$	−938.519	−0.212723	−0.106362	−	0.994328i	$-0.533920\pi$
$269$	−938.519	−0.212723	−0.106362	+	0.994328i	$0.533920\pi$
$270$	0	0
$271$	−8058.74	−1.80640	−0.903199	−	0.429223i	$-0.858788\pi$
$271$	−8058.74	−1.80640	−0.903199	+	0.429223i	$0.858788\pi$
$272$	0	0
$273$	−143.136	−0.0317326
$274$	0	0
$275$	660.622	0.144862
$276$	0	0
$277$	−682.325	−0.148003	−0.0740017	−	0.997258i	$-0.523577\pi$
$277$	−682.325	−0.148003	−0.0740017	+	0.997258i	$0.523577\pi$
$278$	0	0
$279$	−4878.29	−1.04679
$280$	0	0
$281$	5899.10	1.25235	0.626175	−	0.779682i	$-0.284619\pi$
$281$	5899.10	1.25235	0.626175	+	0.779682i	$0.284619\pi$
$282$	0	0
$283$	−961.519	−0.201966	−0.100983	−	0.994888i	$-0.532199\pi$
$283$	−961.519	−0.201966	−0.100983	+	0.994888i	$0.532199\pi$
$284$	0	0
$285$	−342.334	−0.0711513
$286$	0	0
$287$	279.538	0.0574935
$288$	0	0
$289$	343.118	0.0698388
$290$	0	0
$291$	1001.83	0.201815
$292$	0	0
$293$	−5024.51	−1.00183	−0.500913	−	0.865498i	$-0.667002\pi$
$293$	−5024.51	−1.00183	−0.500913	+	0.865498i	$0.667002\pi$
$294$	0	0
$295$	4683.78	0.924407
$296$	0	0
$297$	3617.62	0.706786
$298$	0	0
$299$	−6981.26	−1.35029
$300$	0	0
$301$	118.964	0.0227807
$302$	0	0
$303$	2.40419	0.000455831 0
$304$	0	0
$305$	3752.13	0.704415
$306$	0	0
$307$	3868.67	0.719207	0.359603	−	0.933105i	$-0.382912\pi$
$307$	3868.67	0.719207	0.359603	+	0.933105i	$0.382912\pi$
$308$	0	0
$309$	−1485.80	−0.273541
$310$	0	0
$311$	5796.70	1.05692	0.528458	−	0.848960i	$-0.322771\pi$
$311$	5796.70	1.05692	0.528458	+	0.848960i	$0.322771\pi$
$312$	0	0
$313$	−8362.62	−1.51017	−0.755085	−	0.655627i	$-0.772404\pi$
$313$	−8362.62	−1.51017	−0.755085	+	0.655627i	$0.772404\pi$
$314$	0	0
$315$	−803.569	−0.143733
$316$	0	0
$317$	487.064	0.0862973	0.0431487	−	0.999069i	$-0.486261\pi$
$317$	487.064	0.0862973	0.0431487	+	0.999069i	$0.486261\pi$
$318$	0	0
$319$	−5973.13	−1.04837
$320$	0	0
$321$	−1243.42	−0.216203
$322$	0	0
$323$	1993.44	0.343400
$324$	0	0
$325$	517.099	0.0882569
$326$	0	0
$327$	−806.416	−0.136376
$328$	0	0
$329$	1304.20	0.218549
$330$	0	0
$331$	−3801.21	−0.631218	−0.315609	−	0.948889i	$-0.602209\pi$
$331$	−3801.21	−0.631218	−0.315609	+	0.948889i	$0.602209\pi$
$332$	0	0
$333$	3073.76	0.505828
$334$	0	0
$335$	−2289.27	−0.373361
$336$	0	0
$337$	1795.31	0.290199	0.145099	−	0.989417i	$-0.453650\pi$
$337$	1795.31	0.290199	0.145099	+	0.989417i	$0.453650\pi$
$338$	0	0
$339$	−71.9931	−0.0115343
$340$	0	0
$341$	12066.7	1.91627
$342$	0	0
$343$	−1813.72	−0.285515
$344$	0	0
$345$	1736.34	0.270960
$346$	0	0
$347$	−2782.23	−0.430426	−0.215213	−	0.976567i	$-0.569045\pi$
$347$	−2782.23	−0.430426	−0.215213	+	0.976567i	$0.569045\pi$
$348$	0	0
$349$	10413.4	1.59718	0.798589	−	0.601876i	$-0.205580\pi$
$349$	10413.4	1.59718	0.798589	+	0.601876i	$0.205580\pi$
$350$	0	0
$351$	2831.68	0.430609
$352$	0	0
$353$	10644.3	1.60493	0.802466	−	0.596698i	$-0.203521\pi$
$353$	10644.3	1.60493	0.802466	+	0.596698i	$0.203521\pi$
$354$	0	0
$355$	5277.43	0.789005
$356$	0	0
$357$	−207.302	−0.0307327
$358$	0	0
$359$	−7459.42	−1.09664	−0.548319	−	0.836269i	$-0.684732\pi$
$359$	−7459.42	−1.09664	−0.548319	+	0.836269i	$0.684732\pi$
$360$	0	0
$361$	−6102.96	−0.889775
$362$	0	0
$363$	−2952.72	−0.426935
$364$	0	0
$365$	−3021.68	−0.433320
$366$	0	0
$367$	6251.35	0.889149	0.444574	−	0.895742i	$-0.353355\pi$
$367$	6251.35	0.889149	0.444574	+	0.895742i	$0.353355\pi$
$368$	0	0
$369$	−2705.14	−0.381637
$370$	0	0
$371$	−565.094	−0.0790787
$372$	0	0
$373$	−12603.3	−1.74953	−0.874763	−	0.484552i	$-0.838983\pi$
$373$	−12603.3	−1.74953	−0.874763	+	0.484552i	$0.838983\pi$
$374$	0	0
$375$	1427.68	0.196600
$376$	0	0
$377$	−4675.45	−0.638721
$378$	0	0
$379$	1674.47	0.226943	0.113472	−	0.993541i	$-0.463803\pi$
$379$	1674.47	0.226943	0.113472	+	0.993541i	$0.463803\pi$
$380$	0	0
$381$	−2037.31	−0.273949
$382$	0	0
$383$	−2880.38	−0.384283	−0.192142	−	0.981367i	$-0.561543\pi$
$383$	−2880.38	−0.384283	−0.192142	+	0.981367i	$0.561543\pi$
$384$	0	0
$385$	1987.66	0.263119
$386$	0	0
$387$	−1151.24	−0.151217
$388$	0	0
$389$	13073.3	1.70397	0.851984	−	0.523567i	$-0.175399\pi$
$389$	13073.3	1.70397	0.851984	+	0.523567i	$0.175399\pi$
$390$	0	0
$391$	−10110.9	−1.30774
$392$	0	0
$393$	1390.09	0.178424
$394$	0	0
$395$	−3759.72	−0.478916
$396$	0	0
$397$	−6021.44	−0.761228	−0.380614	−	0.924734i	$-0.624287\pi$
$397$	−6021.44	−0.761228	−0.380614	+	0.924734i	$0.624287\pi$
$398$	0	0
$399$	−78.6216	−0.00986467
$400$	0	0
$401$	−12722.6	−1.58437	−0.792187	−	0.610278i	$-0.791058\pi$
$401$	−12722.6	−1.58437	−0.792187	+	0.610278i	$0.791058\pi$
$402$	0	0
$403$	9445.14	1.16748
$404$	0	0
$405$	7416.51	0.909949
$406$	0	0
$407$	−7603.08	−0.925972
$408$	0	0
$409$	−232.991	−0.0281678	−0.0140839	−	0.999901i	$-0.504483\pi$
$409$	−232.991	−0.0281678	−0.0140839	+	0.999901i	$0.504483\pi$
$410$	0	0
$411$	−510.756	−0.0612987
$412$	0	0
$413$	1075.69	0.128163
$414$	0	0
$415$	−9277.08	−1.09734
$416$	0	0
$417$	−2291.35	−0.269084
$418$	0	0
$419$	8663.03	1.01006	0.505032	−	0.863101i	$-0.331481\pi$
$419$	8663.03	1.01006	0.505032	+	0.863101i	$0.331481\pi$
$420$	0	0
$421$	−11749.9	−1.36023	−0.680113	−	0.733107i	$-0.738069\pi$
$421$	−11749.9	−1.36023	−0.680113	+	0.733107i	$0.738069\pi$
$422$	0	0
$423$	−12620.9	−1.45071
$424$	0	0
$425$	748.907	0.0854760
$426$	0	0
$427$	861.727	0.0976626
$428$	0	0
$429$	−3426.24	−0.385596
$430$	0	0
$431$	−8737.57	−0.976506	−0.488253	−	0.872702i	$-0.662366\pi$
$431$	−8737.57	−0.976506	−0.488253	+	0.872702i	$0.662366\pi$
$432$	0	0
$433$	11627.5	1.29049	0.645247	−	0.763974i	$-0.276755\pi$
$433$	11627.5	1.29049	0.645247	+	0.763974i	$0.276755\pi$
$434$	0	0
$435$	1162.85	0.128171
$436$	0	0
$437$	−3834.66	−0.419763
$438$	0	0
$439$	−17631.8	−1.91690	−0.958450	−	0.285261i	$-0.907920\pi$
$439$	−17631.8	−1.91690	−0.958450	+	0.285261i	$0.907920\pi$
$440$	0	0
$441$	8683.57	0.937649
$442$	0	0
$443$	−6434.41	−0.690085	−0.345043	−	0.938587i	$-0.612136\pi$
$443$	−6434.41	−0.690085	−0.345043	+	0.938587i	$0.612136\pi$
$444$	0	0
$445$	−10084.2	−1.07424
$446$	0	0
$447$	568.116	0.0601140
$448$	0	0
$449$	−12926.5	−1.35867	−0.679334	−	0.733830i	$-0.737731\pi$
$449$	−12926.5	−1.35867	−0.679334	+	0.733830i	$0.737731\pi$
$450$	0	0
$451$	6691.30	0.698627
$452$	0	0
$453$	−3208.07	−0.332734
$454$	0	0
$455$	1555.84	0.160305
$456$	0	0
$457$	9320.32	0.954018	0.477009	−	0.878898i	$-0.341721\pi$
$457$	9320.32	0.954018	0.477009	+	0.878898i	$0.341721\pi$
$458$	0	0
$459$	4101.07	0.417041
$460$	0	0
$461$	18222.1	1.84098	0.920488	−	0.390771i	$-0.127792\pi$
$461$	18222.1	1.84098	0.920488	+	0.390771i	$0.127792\pi$
$462$	0	0
$463$	−7038.37	−0.706482	−0.353241	−	0.935532i	$-0.614920\pi$
$463$	−7038.37	−0.706482	−0.353241	+	0.935532i	$0.614920\pi$
$464$	0	0
$465$	−2349.14	−0.234277
$466$	0	0
$467$	8487.77	0.841043	0.420522	−	0.907283i	$-0.361847\pi$
$467$	8487.77	0.841043	0.420522	+	0.907283i	$0.361847\pi$
$468$	0	0
$469$	−525.761	−0.0517642
$470$	0	0
$471$	−2283.96	−0.223438
$472$	0	0
$473$	2847.65	0.276818
$474$	0	0
$475$	284.031	0.0274363
$476$	0	0
$477$	5468.51	0.524918
$478$	0	0
$479$	−587.317	−0.0560234	−0.0280117	−	0.999608i	$-0.508918\pi$
$479$	−587.317	−0.0560234	−0.0280117	+	0.999608i	$0.508918\pi$
$480$	0	0
$481$	−5951.28	−0.564148
$482$	0	0
$483$	398.773	0.0375669
$484$	0	0
$485$	−10889.5	−1.01952
$486$	0	0
$487$	8366.45	0.778481	0.389240	−	0.921136i	$-0.372738\pi$
$487$	8366.45	0.778481	0.389240	+	0.921136i	$0.372738\pi$
$488$	0	0
$489$	−2157.10	−0.199483
$490$	0	0
$491$	−2162.76	−0.198786	−0.0993929	−	0.995048i	$-0.531690\pi$
$491$	−2162.76	−0.198786	−0.0993929	+	0.995048i	$0.531690\pi$
$492$	0	0
$493$	−6771.38	−0.618596
$494$	0	0
$495$	−19235.0	−1.74656
$496$	0	0
$497$	1212.03	0.109390
$498$	0	0
$499$	16071.8	1.44183	0.720913	−	0.693026i	$-0.243723\pi$
$499$	16071.8	1.44183	0.720913	+	0.693026i	$0.243723\pi$
$500$	0	0
$501$	−848.077	−0.0756273
$502$	0	0
$503$	−12570.2	−1.11427	−0.557137	−	0.830421i	$-0.688100\pi$
$503$	−12570.2	−1.11427	−0.557137	+	0.830421i	$0.688100\pi$
$504$	0	0
$505$	−26.1326	−0.00230274
$506$	0	0
$507$	−330.556	−0.0289556
$508$	0	0
$509$	−16801.4	−1.46308	−0.731542	−	0.681797i	$-0.761199\pi$
$509$	−16801.4	−1.46308	−0.731542	+	0.681797i	$0.761199\pi$
$510$	0	0
$511$	−693.968	−0.0600770
$512$	0	0
$513$	1555.38	0.133863
$514$	0	0
$515$	16150.1	1.38186
$516$	0	0
$517$	31218.5	2.65568
$518$	0	0
$519$	1170.87	0.0990283
$520$	0	0
$521$	−6612.98	−0.556085	−0.278042	−	0.960569i	$-0.589686\pi$
$521$	−6612.98	−0.556085	−0.278042	+	0.960569i	$0.589686\pi$
$522$	0	0
$523$	−7253.92	−0.606485	−0.303243	−	0.952913i	$-0.598069\pi$
$523$	−7253.92	−0.606485	−0.303243	+	0.952913i	$0.598069\pi$
$524$	0	0
$525$	−29.5370	−0.00245543
$526$	0	0
$527$	13679.3	1.13070
$528$	0	0
$529$	7282.60	0.598553
$530$	0	0
$531$	−10409.7	−0.850737
$532$	0	0
$533$	5237.59	0.425638
$534$	0	0
$535$	13515.5	1.09220
$536$	0	0
$537$	−644.849	−0.0518199
$538$	0	0
$539$	−21479.2	−1.71647
$540$	0	0
$541$	15511.9	1.23273	0.616365	−	0.787461i	$-0.288605\pi$
$541$	15511.9	1.23273	0.616365	+	0.787461i	$0.288605\pi$
$542$	0	0
$543$	−3962.57	−0.313168
$544$	0	0
$545$	8765.44	0.688936
$546$	0	0
$547$	−18510.4	−1.44689	−0.723445	−	0.690382i	$-0.757442\pi$
$547$	−18510.4	−1.44689	−0.723445	+	0.690382i	$0.757442\pi$
$548$	0	0
$549$	−8339.09	−0.648277
$550$	0	0
$551$	−2568.12	−0.198558
$552$	0	0
$553$	−863.469	−0.0663986
$554$	0	0
$555$	1480.17	0.113207
$556$	0	0
$557$	7141.59	0.543266	0.271633	−	0.962401i	$-0.412436\pi$
$557$	7141.59	0.543266	0.271633	+	0.962401i	$0.412436\pi$
$558$	0	0
$559$	2228.98	0.168651
$560$	0	0
$561$	−4962.18	−0.373446
$562$	0	0
$563$	−4594.86	−0.343961	−0.171981	−	0.985100i	$-0.555017\pi$
$563$	−4594.86	−0.343961	−0.171981	+	0.985100i	$0.555017\pi$
$564$	0	0
$565$	782.538	0.0582683
$566$	0	0
$567$	1703.30	0.126159
$568$	0	0
$569$	−2806.05	−0.206741	−0.103371	−	0.994643i	$-0.532963\pi$
$569$	−2806.05	−0.206741	−0.103371	+	0.994643i	$0.532963\pi$
$570$	0	0
$571$	−17025.4	−1.24779	−0.623897	−	0.781506i	$-0.714452\pi$
$571$	−17025.4	−1.24779	−0.623897	+	0.781506i	$0.714452\pi$
$572$	0	0
$573$	−3442.31	−0.250968
$574$	0	0
$575$	−1440.62	−0.104484
$576$	0	0
$577$	7206.84	0.519973	0.259987	−	0.965612i	$-0.416282\pi$
$577$	7206.84	0.519973	0.259987	+	0.965612i	$0.416282\pi$
$578$	0	0
$579$	−3052.95	−0.219130
$580$	0	0
$581$	−2130.61	−0.152138
$582$	0	0
$583$	−13526.6	−0.960918
$584$	0	0
$585$	−15056.1	−1.06409
$586$	0	0
$587$	−14675.2	−1.03188	−0.515938	−	0.856626i	$-0.672557\pi$
$587$	−14675.2	−1.03188	−0.515938	+	0.856626i	$0.672557\pi$
$588$	0	0
$589$	5188.01	0.362934
$590$	0	0
$591$	−2435.33	−0.169503
$592$	0	0
$593$	4758.60	0.329531	0.164766	−	0.986333i	$-0.447313\pi$
$593$	4758.60	0.329531	0.164766	+	0.986333i	$0.447313\pi$
$594$	0	0
$595$	2253.29	0.155254
$596$	0	0
$597$	799.617	0.0548176
$598$	0	0
$599$	14256.4	0.972455	0.486227	−	0.873832i	$-0.338373\pi$
$599$	14256.4	0.972455	0.486227	+	0.873832i	$0.338373\pi$
$600$	0	0
$601$	−10385.2	−0.704862	−0.352431	−	0.935838i	$-0.614645\pi$
$601$	−10385.2	−0.704862	−0.352431	+	0.935838i	$0.614645\pi$
$602$	0	0
$603$	5087.89	0.343606
$604$	0	0
$605$	32094.9	2.15677
$606$	0	0
$607$	−16243.6	−1.08618	−0.543088	−	0.839676i	$-0.682745\pi$
$607$	−16243.6	−1.08618	−0.543088	+	0.839676i	$0.682745\pi$
$608$	0	0
$609$	267.064	0.0177701
$610$	0	0
$611$	24436.1	1.61797
$612$	0	0
$613$	−707.817	−0.0466369	−0.0233185	−	0.999728i	$-0.507423\pi$
$613$	−707.817	−0.0466369	−0.0233185	+	0.999728i	$0.507423\pi$
$614$	0	0
$615$	−1302.66	−0.0854121
$616$	0	0
$617$	11575.9	0.755316	0.377658	−	0.925945i	$-0.376729\pi$
$617$	11575.9	0.755316	0.377658	+	0.925945i	$0.376729\pi$
$618$	0	0
$619$	−25959.4	−1.68562	−0.842808	−	0.538214i	$-0.819099\pi$
$619$	−25959.4	−1.68562	−0.842808	+	0.538214i	$0.819099\pi$
$620$	0	0
$621$	−7888.97	−0.509780
$622$	0	0
$623$	−2315.98	−0.148937
$624$	0	0
$625$	−16809.5	−1.07581
$626$	0	0
$627$	−1881.96	−0.119870
$628$	0	0
$629$	−8619.15	−0.546372
$630$	0	0
$631$	−10224.8	−0.645079	−0.322539	−	0.946556i	$-0.604537\pi$
$631$	−10224.8	−0.645079	−0.322539	+	0.946556i	$0.604537\pi$
$632$	0	0
$633$	3370.88	0.211660
$634$	0	0
$635$	22144.8	1.38392
$636$	0	0
$637$	−16812.8	−1.04576
$638$	0	0
$639$	−11729.0	−0.726125
$640$	0	0
$641$	19804.4	1.22032	0.610162	−	0.792277i	$-0.291104\pi$
$641$	19804.4	1.22032	0.610162	+	0.792277i	$0.291104\pi$
$642$	0	0
$643$	−22175.9	−1.36008	−0.680041	−	0.733174i	$-0.738038\pi$
$643$	−22175.9	−1.36008	−0.680041	+	0.733174i	$0.738038\pi$
$644$	0	0
$645$	−554.380	−0.0338429
$646$	0	0
$647$	9232.26	0.560985	0.280493	−	0.959856i	$-0.409502\pi$
$647$	9232.26	0.560985	0.280493	+	0.959856i	$0.409502\pi$
$648$	0	0
$649$	25748.8	1.55736
$650$	0	0
$651$	−539.511	−0.0324810
$652$	0	0
$653$	−27857.1	−1.66942	−0.834710	−	0.550690i	$-0.814365\pi$
$653$	−27857.1	−1.66942	−0.834710	+	0.550690i	$0.814365\pi$
$654$	0	0
$655$	−15109.8	−0.901355
$656$	0	0
$657$	6715.66	0.398786
$658$	0	0
$659$	5498.55	0.325028	0.162514	−	0.986706i	$-0.448040\pi$
$659$	5498.55	0.325028	0.162514	+	0.986706i	$0.448040\pi$
$660$	0	0
$661$	11469.6	0.674908	0.337454	−	0.941342i	$-0.390434\pi$
$661$	11469.6	0.674908	0.337454	+	0.941342i	$0.390434\pi$
$662$	0	0
$663$	−3884.13	−0.227522
$664$	0	0
$665$	854.587	0.0498338
$666$	0	0
$667$	13025.7	0.756156
$668$	0	0
$669$	383.204	0.0221458
$670$	0	0
$671$	20627.1	1.18674
$672$	0	0
$673$	−28428.2	−1.62827	−0.814135	−	0.580676i	$-0.802788\pi$
$673$	−28428.2	−1.62827	−0.814135	+	0.580676i	$0.802788\pi$
$674$	0	0
$675$	584.333	0.0333200
$676$	0	0
$677$	−23995.5	−1.36222	−0.681110	−	0.732181i	$-0.738502\pi$
$677$	−23995.5	−1.36222	−0.681110	+	0.732181i	$0.738502\pi$
$678$	0	0
$679$	−2500.92	−0.141350
$680$	0	0
$681$	5229.92	0.294289
$682$	0	0
$683$	13506.0	0.756650	0.378325	−	0.925673i	$-0.376500\pi$
$683$	13506.0	0.756650	0.378325	+	0.925673i	$0.376500\pi$
$684$	0	0
$685$	5551.73	0.309665
$686$	0	0
$687$	2164.42	0.120201
$688$	0	0
$689$	−10587.9	−0.585439
$690$	0	0
$691$	−29500.1	−1.62408	−0.812038	−	0.583605i	$-0.801642\pi$
$691$	−29500.1	−1.62408	−0.812038	+	0.583605i	$0.801642\pi$
$692$	0	0
$693$	−4417.57	−0.242150
$694$	0	0
$695$	24906.1	1.35934
$696$	0	0
$697$	7585.52	0.412227
$698$	0	0
$699$	991.557	0.0536540
$700$	0	0
$701$	−33227.5	−1.79028	−0.895139	−	0.445787i	$-0.852924\pi$
$701$	−33227.5	−1.79028	−0.895139	+	0.445787i	$0.852924\pi$
$702$	0	0
$703$	−3268.91	−0.175376
$704$	0	0
$705$	−6077.62	−0.324676
$706$	0	0
$707$	−6.00170	−0.000319261 0
$708$	0	0
$709$	6448.04	0.341553	0.170777	−	0.985310i	$-0.445372\pi$
$709$	6448.04	0.341553	0.170777	+	0.985310i	$0.445372\pi$
$710$	0	0
$711$	8355.95	0.440749
$712$	0	0
$713$	−26313.9	−1.38214
$714$	0	0
$715$	37242.0	1.94793
$716$	0	0
$717$	−848.138	−0.0441761
$718$	0	0
$719$	6494.67	0.336871	0.168436	−	0.985713i	$-0.446128\pi$
$719$	6494.67	0.336871	0.168436	+	0.985713i	$0.446128\pi$
$720$	0	0
$721$	3709.08	0.191586
$722$	0	0
$723$	1550.80	0.0797714
$724$	0	0
$725$	−964.806	−0.0494234
$726$	0	0
$727$	−24866.4	−1.26856	−0.634280	−	0.773103i	$-0.718704\pi$
$727$	−24866.4	−1.26856	−0.634280	+	0.773103i	$0.718704\pi$
$728$	0	0
$729$	−14848.5	−0.754384
$730$	0	0
$731$	3228.20	0.163337
$732$	0	0
$733$	21092.1	1.06283	0.531414	−	0.847112i	$-0.321661\pi$
$733$	21092.1	1.06283	0.531414	+	0.847112i	$0.321661\pi$
$734$	0	0
$735$	4181.58	0.209850
$736$	0	0
$737$	−12585.1	−0.629008
$738$	0	0
$739$	−11952.7	−0.594977	−0.297489	−	0.954725i	$-0.596149\pi$
$739$	−11952.7	−0.594977	−0.297489	+	0.954725i	$0.596149\pi$
$740$	0	0
$741$	−1473.10	−0.0730305
$742$	0	0
$743$	5622.43	0.277614	0.138807	−	0.990319i	$-0.455673\pi$
$743$	5622.43	0.277614	0.138807	+	0.990319i	$0.455673\pi$
$744$	0	0
$745$	−6175.21	−0.303681
$746$	0	0
$747$	20618.3	1.00988
$748$	0	0
$749$	3104.02	0.151427
$750$	0	0
$751$	32314.9	1.57016	0.785079	−	0.619396i	$-0.212622\pi$
$751$	32314.9	1.57016	0.785079	+	0.619396i	$0.212622\pi$
$752$	0	0
$753$	5418.58	0.262236
$754$	0	0
$755$	34870.5	1.68089
$756$	0	0
$757$	−17950.4	−0.861847	−0.430923	−	0.902389i	$-0.641812\pi$
$757$	−17950.4	−0.861847	−0.430923	+	0.902389i	$0.641812\pi$
$758$	0	0
$759$	9545.42	0.456491
$760$	0	0
$761$	13108.2	0.624404	0.312202	−	0.950016i	$-0.398933\pi$
$761$	13108.2	0.624404	0.312202	+	0.950016i	$0.398933\pi$
$762$	0	0
$763$	2013.10	0.0955166
$764$	0	0
$765$	−21805.5	−1.03056
$766$	0	0
$767$	20154.8	0.948822
$768$	0	0
$769$	23661.2	1.10955	0.554776	−	0.832000i	$-0.312804\pi$
$769$	23661.2	1.10955	0.554776	+	0.832000i	$0.312804\pi$
$770$	0	0
$771$	5039.63	0.235406
$772$	0	0
$773$	30222.4	1.40624	0.703120	−	0.711071i	$-0.251790\pi$
$773$	30222.4	1.40624	0.703120	+	0.711071i	$0.251790\pi$
$774$	0	0
$775$	1949.06	0.0903384
$776$	0	0
$777$	339.940	0.0156954
$778$	0	0
$779$	2876.89	0.132318
$780$	0	0
$781$	29012.3	1.32925
$782$	0	0
$783$	−5283.35	−0.241139
$784$	0	0
$785$	24825.8	1.12875
$786$	0	0
$787$	−29544.2	−1.33817	−0.669083	−	0.743188i	$-0.733313\pi$
$787$	−29544.2	−1.33817	−0.669083	+	0.743188i	$0.733313\pi$
$788$	0	0
$789$	−3176.28	−0.143319
$790$	0	0
$791$	179.720	0.00807853
$792$	0	0
$793$	16145.8	0.723020
$794$	0	0
$795$	2633.36	0.117479
$796$	0	0
$797$	9665.91	0.429591	0.214795	−	0.976659i	$-0.431092\pi$
$797$	9665.91	0.429591	0.214795	+	0.976659i	$0.431092\pi$
$798$	0	0
$799$	35390.5	1.56699
$800$	0	0
$801$	22412.1	0.988631
$802$	0	0
$803$	−16611.5	−0.730021
$804$	0	0
$805$	−4334.52	−0.189778
$806$	0	0
$807$	1004.44	0.0438142
$808$	0	0
$809$	−15807.0	−0.686952	−0.343476	−	0.939162i	$-0.611604\pi$
$809$	−15807.0	−0.686952	−0.343476	+	0.939162i	$0.611604\pi$
$810$	0	0
$811$	16295.5	0.705565	0.352783	−	0.935705i	$-0.385236\pi$
$811$	16295.5	0.705565	0.352783	+	0.935705i	$0.385236\pi$
$812$	0	0
$813$	8624.81	0.372061
$814$	0	0
$815$	23446.9	1.00774
$816$	0	0
$817$	1224.33	0.0524284
$818$	0	0
$819$	−3457.84	−0.147529
$820$	0	0
$821$	−436.743	−0.0185657	−0.00928286	−	0.999957i	$-0.502955\pi$
$821$	−436.743	−0.0185657	−0.00928286	+	0.999957i	$0.502955\pi$
$822$	0	0
$823$	−5633.49	−0.238604	−0.119302	−	0.992858i	$-0.538066\pi$
$823$	−5633.49	−0.238604	−0.119302	+	0.992858i	$0.538066\pi$
$824$	0	0
$825$	−707.026	−0.0298369
$826$	0	0
$827$	−22394.7	−0.941645	−0.470822	−	0.882228i	$-0.656043\pi$
$827$	−22394.7	−0.941645	−0.470822	+	0.882228i	$0.656043\pi$
$828$	0	0
$829$	8768.39	0.367357	0.183678	−	0.982986i	$-0.441199\pi$
$829$	8768.39	0.367357	0.183678	+	0.982986i	$0.441199\pi$
$830$	0	0
$831$	730.253	0.0304840
$832$	0	0
$833$	−24349.7	−1.01281
$834$	0	0
$835$	9218.27	0.382050
$836$	0	0
$837$	10673.2	0.440764
$838$	0	0
$839$	30644.3	1.26098	0.630488	−	0.776199i	$-0.282855\pi$
$839$	30644.3	1.26098	0.630488	+	0.776199i	$0.282855\pi$
$840$	0	0
$841$	−15665.5	−0.642319
$842$	0	0
$843$	−6313.47	−0.257945
$844$	0	0
$845$	3593.02	0.146277
$846$	0	0
$847$	7371.03	0.299022
$848$	0	0
$849$	1029.06	0.0415986
$850$	0	0
$851$	16580.1	0.667871
$852$	0	0
$853$	43559.2	1.74846	0.874232	−	0.485508i	$-0.161365\pi$
$853$	43559.2	1.74846	0.874232	+	0.485508i	$0.161365\pi$
$854$	0	0
$855$	−8270.00	−0.330793
$856$	0	0
$857$	−41788.5	−1.66566	−0.832828	−	0.553533i	$-0.813279\pi$
$857$	−41788.5	−1.66566	−0.832828	+	0.553533i	$0.813279\pi$
$858$	0	0
$859$	16849.6	0.669267	0.334633	−	0.942348i	$-0.391388\pi$
$859$	16849.6	0.669267	0.334633	+	0.942348i	$0.391388\pi$
$860$	0	0
$861$	−299.174	−0.0118418
$862$	0	0
$863$	27636.7	1.09011	0.545054	−	0.838401i	$-0.316509\pi$
$863$	27636.7	1.09011	0.545054	+	0.838401i	$0.316509\pi$
$864$	0	0
$865$	−12727.0	−0.500266
$866$	0	0
$867$	−367.220	−0.0143846
$868$	0	0
$869$	−20668.8	−0.806837
$870$	0	0
$871$	−9850.95	−0.383223
$872$	0	0
$873$	24201.9	0.938270
$874$	0	0
$875$	−3563.98	−0.137697
$876$	0	0
$877$	−16855.0	−0.648977	−0.324488	−	0.945890i	$-0.605192\pi$
$877$	−16855.0	−0.648977	−0.324488	+	0.945890i	$0.605192\pi$
$878$	0	0
$879$	5377.45	0.206344
$880$	0	0
$881$	13330.0	0.509759	0.254880	−	0.966973i	$-0.417964\pi$
$881$	13330.0	0.509759	0.254880	+	0.966973i	$0.417964\pi$
$882$	0	0
$883$	−35598.7	−1.35673	−0.678365	−	0.734725i	$-0.737311\pi$
$883$	−35598.7	−1.35673	−0.678365	+	0.734725i	$0.737311\pi$
$884$	0	0
$885$	−5012.78	−0.190399
$886$	0	0
$887$	48821.3	1.84810	0.924048	−	0.382278i	$-0.124860\pi$
$887$	48821.3	1.84810	0.924048	+	0.382278i	$0.124860\pi$
$888$	0	0
$889$	5085.84	0.191871
$890$	0	0
$891$	40771.8	1.53301
$892$	0	0
$893$	13422.2	0.502977
$894$	0	0
$895$	7009.27	0.261781
$896$	0	0
$897$	7471.64	0.278117
$898$	0	0
$899$	−17622.8	−0.653785
$900$	0	0
$901$	−15334.3	−0.566992
$902$	0	0
$903$	−127.321	−0.00469210
$904$	0	0
$905$	43071.7	1.58205
$906$	0	0
$907$	−7523.90	−0.275443	−0.137722	−	0.990471i	$-0.543978\pi$
$907$	−7523.90	−0.275443	−0.137722	+	0.990471i	$0.543978\pi$
$908$	0	0
$909$	58.0796	0.00211923
$910$	0	0
$911$	26016.0	0.946155	0.473077	−	0.881021i	$-0.343143\pi$
$911$	26016.0	0.946155	0.473077	+	0.881021i	$0.343143\pi$
$912$	0	0
$913$	−51000.2	−1.84870
$914$	0	0
$915$	−4015.69	−0.145087
$916$	0	0
$917$	−3470.16	−0.124967
$918$	0	0
$919$	24082.1	0.864413	0.432206	−	0.901775i	$-0.357735\pi$
$919$	24082.1	0.864413	0.432206	+	0.901775i	$0.357735\pi$
$920$	0	0
$921$	−4140.41	−0.148134
$922$	0	0
$923$	22709.3	0.809844
$924$	0	0
$925$	−1228.08	−0.0436530
$926$	0	0
$927$	−35893.5	−1.27173
$928$	0	0
$929$	9324.93	0.329323	0.164661	−	0.986350i	$-0.447347\pi$
$929$	9324.93	0.329323	0.164661	+	0.986350i	$0.447347\pi$
$930$	0	0
$931$	−9234.89	−0.325093
$932$	0	0
$933$	−6203.87	−0.217691
$934$	0	0
$935$	53937.0	1.88656
$936$	0	0
$937$	−15535.2	−0.541636	−0.270818	−	0.962631i	$-0.587294\pi$
$937$	−15535.2	−0.541636	−0.270818	+	0.962631i	$0.587294\pi$
$938$	0	0
$939$	8950.03	0.311047
$940$	0	0
$941$	−48118.3	−1.66696	−0.833482	−	0.552547i	$-0.813656\pi$
$941$	−48118.3	−1.66696	−0.833482	+	0.552547i	$0.813656\pi$
$942$	0	0
$943$	−14591.8	−0.503896
$944$	0	0
$945$	1758.13	0.0605205
$946$	0	0
$947$	−5396.67	−0.185183	−0.0925915	−	0.995704i	$-0.529515\pi$
$947$	−5396.67	−0.185183	−0.0925915	+	0.995704i	$0.529515\pi$
$948$	0	0
$949$	−13002.6	−0.444765
$950$	0	0
$951$	−521.277	−0.0177745
$952$	0	0
$953$	21759.7	0.739629	0.369815	−	0.929106i	$-0.379421\pi$
$953$	21759.7	0.739629	0.369815	+	0.929106i	$0.379421\pi$
$954$	0	0
$955$	37416.7	1.26783
$956$	0	0
$957$	6392.70	0.215932
$958$	0	0
$959$	1275.03	0.0429331
$960$	0	0
$961$	5809.78	0.195018
$962$	0	0
$963$	−30038.2	−1.00516
$964$	0	0
$965$	33184.4	1.10699
$966$	0	0
$967$	−19338.5	−0.643108	−0.321554	−	0.946891i	$-0.604205\pi$
$967$	−19338.5	−0.643108	−0.321554	+	0.946891i	$0.604205\pi$
$968$	0	0
$969$	−2133.47	−0.0707294
$970$	0	0
$971$	−5918.37	−0.195602	−0.0978009	−	0.995206i	$-0.531181\pi$
$971$	−5918.37	−0.195602	−0.0978009	+	0.995206i	$0.531181\pi$
$972$	0	0
$973$	5720.03	0.188464
$974$	0	0
$975$	−553.421	−0.0181781
$976$	0	0
$977$	−17841.9	−0.584249	−0.292125	−	0.956380i	$-0.594362\pi$
$977$	−17841.9	−0.584249	−0.292125	+	0.956380i	$0.594362\pi$
$978$	0	0
$979$	−55437.4	−1.80979
$980$	0	0
$981$	−19481.1	−0.634032
$982$	0	0
$983$	61512.0	1.99586	0.997929	−	0.0643304i	$-0.0204912\pi$
$983$	61512.0	1.99586	0.997929	+	0.0643304i	$0.0204912\pi$
$984$	0	0
$985$	26471.2	0.856286
$986$	0	0
$987$	−1395.81	−0.0450142
$988$	0	0
$989$	−6209.89	−0.199659
$990$	0	0
$991$	1827.73	0.0585870	0.0292935	−	0.999571i	$-0.490674\pi$
$991$	1827.73	0.0585870	0.0292935	+	0.999571i	$0.490674\pi$
$992$	0	0
$993$	4068.21	0.130011
$994$	0	0
$995$	−8691.53	−0.276925
$996$	0	0
$997$	−35379.4	−1.12385	−0.561925	−	0.827188i	$-0.689939\pi$
$997$	−35379.4	−1.12385	−0.561925	+	0.827188i	$0.689939\pi$
$998$	0	0
$999$	−6725.07	−0.212985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.4.a.n.1.5		10
4.3	odd	2		1024.4.a.m.1.6		10
8.3	odd	2		1024.4.a.m.1.5		10
8.5	even	2	inner	1024.4.a.n.1.6		10
16.3	odd	4		1024.4.b.k.513.6		10
16.5	even	4		1024.4.b.j.513.6		10
16.11	odd	4		1024.4.b.k.513.5		10
16.13	even	4		1024.4.b.j.513.5		10
32.3	odd	8		128.4.e.a.33.3		10
32.5	even	8		16.4.e.a.5.3	✓	10
32.11	odd	8		128.4.e.a.97.3		10
32.13	even	8		16.4.e.a.13.3	yes	10
32.19	odd	8		64.4.e.a.17.3		10
32.21	even	8		128.4.e.b.97.3		10
32.27	odd	8		64.4.e.a.49.3		10
32.29	even	8		128.4.e.b.33.3		10
96.5	odd	8		144.4.k.a.37.3		10
96.59	even	8		576.4.k.a.433.2		10
96.77	odd	8		144.4.k.a.109.3		10
96.83	even	8		576.4.k.a.145.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.3	✓	10	32.5	even	8
16.4.e.a.13.3	yes	10	32.13	even	8
64.4.e.a.17.3		10	32.19	odd	8
64.4.e.a.49.3		10	32.27	odd	8
128.4.e.a.33.3		10	32.3	odd	8
128.4.e.a.97.3		10	32.11	odd	8
128.4.e.b.33.3		10	32.29	even	8
128.4.e.b.97.3		10	32.21	even	8
144.4.k.a.37.3		10	96.5	odd	8
144.4.k.a.109.3		10	96.77	odd	8
576.4.k.a.145.2		10	96.83	even	8
576.4.k.a.433.2		10	96.59	even	8
1024.4.a.m.1.5		10	8.3	odd	2
1024.4.a.m.1.6		10	4.3	odd	2
1024.4.a.n.1.5		10	1.1	even	1	trivial
1024.4.a.n.1.6		10	8.5	even	2	inner
1024.4.b.j.513.5		10	16.13	even	4
1024.4.b.j.513.6		10	16.5	even	4
1024.4.b.k.513.5		10	16.11	odd	4
1024.4.b.k.513.6		10	16.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.07024 q^{3} +11.6331 q^{5} +2.67171 q^{7} -25.8546 q^{9} +O(q^{10})\)	\(q-1.07024 q^{3} +11.6331 q^{5} +2.67171 q^{7} -25.8546 q^{9} +63.9525 q^{11} +50.0586 q^{13} -12.4503 q^{15} +72.4991 q^{17} +27.4961 q^{19} -2.85937 q^{21} -139.462 q^{23} +10.3299 q^{25} +56.5672 q^{27} -93.3995 q^{29} +188.682 q^{31} -68.4447 q^{33} +31.0803 q^{35} -118.886 q^{37} -53.5748 q^{39} +104.629 q^{41} +44.5275 q^{43} -300.770 q^{45} +488.151 q^{47} -335.862 q^{49} -77.5916 q^{51} -211.510 q^{53} +743.968 q^{55} -29.4275 q^{57} +402.624 q^{59} +322.538 q^{61} -69.0758 q^{63} +582.338 q^{65} -196.789 q^{67} +149.258 q^{69} +453.655 q^{71} -259.747 q^{73} -11.0555 q^{75} +170.862 q^{77} -323.190 q^{79} +637.533 q^{81} -797.471 q^{83} +843.392 q^{85} +99.9602 q^{87} -866.853 q^{89} +133.742 q^{91} -201.935 q^{93} +319.866 q^{95} -936.077 q^{97} -1653.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 28 q^{7} + 54 q^{9}+O(q^{10}) \) 10 * q + 28 * q^7 + 54 * q^9	\( 10 q + 28 q^{7} + 54 q^{9} + 124 q^{15} + 4 q^{17} + 276 q^{23} + 50 q^{25} + 368 q^{31} - 4 q^{33} + 732 q^{39} + 944 q^{47} - 94 q^{49} + 1380 q^{55} + 108 q^{57} + 2628 q^{63} - 492 q^{65} + 3468 q^{71} - 296 q^{73} + 4416 q^{79} - 482 q^{81} + 6036 q^{87} + 88 q^{89} + 6900 q^{95} - 4 q^{97}+O(q^{100}) \) 10 * q + 28 * q^7 + 54 * q^9 + 124 * q^15 + 4 * q^17 + 276 * q^23 + 50 * q^25 + 368 * q^31 - 4 * q^33 + 732 * q^39 + 944 * q^47 - 94 * q^49 + 1380 * q^55 + 108 * q^57 + 2628 * q^63 - 492 * q^65 + 3468 * q^71 - 296 * q^73 + 4416 * q^79 - 482 * q^81 + 6036 * q^87 + 88 * q^89 + 6900 * q^95 - 4 * q^97

Introduction

L-functions

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Database

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