LMFDB - Embedded newform 8512.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.85410 q^{9} +5.09017 q^{11} -2.23607 q^{13} +0.381966 q^{15} -5.38197 q^{17} +1.00000 q^{19} +0.381966 q^{21} +4.70820 q^{23} -4.00000 q^{25} -2.23607 q^{27} +5.85410 q^{29} -4.61803 q^{31} +1.94427 q^{33} +1.00000 q^{35} -6.70820 q^{37} -0.854102 q^{39} -11.0902 q^{41} +0.472136 q^{43} -2.85410 q^{45} -8.70820 q^{47} +1.00000 q^{49} -2.05573 q^{51} -1.32624 q^{53} +5.09017 q^{55} +0.381966 q^{57} -2.70820 q^{59} +3.47214 q^{61} -2.85410 q^{63} -2.23607 q^{65} -9.09017 q^{67} +1.79837 q^{69} +11.1803 q^{71} -12.0902 q^{73} -1.52786 q^{75} +5.09017 q^{77} +10.9443 q^{79} +7.70820 q^{81} -12.3820 q^{83} -5.38197 q^{85} +2.23607 q^{87} +10.1803 q^{89} -2.23607 q^{91} -1.76393 q^{93} +1.00000 q^{95} -4.70820 q^{97} -14.5279 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - q^{11} + 3 q^{15} - 13 q^{17} + 2 q^{19} + 3 q^{21} - 4 q^{23} - 8 q^{25} + 5 q^{29} - 7 q^{31} - 14 q^{33} + 2 q^{35} + 5 q^{39} - 11 q^{41} - 8 q^{43} + q^{45}+ \cdots - 38 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + q^9 - q^11 + 3 * q^15 - 13 * q^17 + 2 * q^19 + 3 * q^21 - 4 * q^23 - 8 * q^25 + 5 * q^29 - 7 * q^31 - 14 * q^33 + 2 * q^35 + 5 * q^39 - 11 * q^41 - 8 * q^43 + q^45 - 4 * q^47 + 2 * q^49 - 22 * q^51 + 13 * q^53 - q^55 + 3 * q^57 + 8 * q^59 - 2 * q^61 + q^63 - 7 * q^67 - 21 * q^69 - 13 * q^73 - 12 * q^75 - q^77 + 4 * q^79 + 2 * q^81 - 27 * q^83 - 13 * q^85 - 2 * q^89 - 8 * q^93 + 2 * q^95 + 4 * q^97 - 38 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	5.09017	1.53474	0.767372	−	0.641202i	$-0.221564\pi$
$11$	5.09017	1.53474	0.767372	+	0.641202i	$0.221564\pi$
$12$	0	0
$13$	−2.23607	−0.620174	−0.310087	−	0.950708i	$-0.600358\pi$
$13$	−2.23607	−0.620174	−0.310087	+	0.950708i	$0.600358\pi$
$14$	0	0
$15$	0.381966	0.0986232
$16$	0	0
$17$	−5.38197	−1.30532	−0.652659	−	0.757652i	$-0.726347\pi$
$17$	−5.38197	−1.30532	−0.652659	+	0.757652i	$0.726347\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.381966	0.0833518
$22$	0	0
$23$	4.70820	0.981728	0.490864	−	0.871236i	$-0.336681\pi$
$23$	4.70820	0.981728	0.490864	+	0.871236i	$0.336681\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	5.85410	1.08708	0.543540	−	0.839383i	$-0.317084\pi$
$29$	5.85410	1.08708	0.543540	+	0.839383i	$0.317084\pi$
$30$	0	0
$31$	−4.61803	−0.829423	−0.414712	−	0.909953i	$-0.636118\pi$
$31$	−4.61803	−0.829423	−0.414712	+	0.909953i	$0.636118\pi$
$32$	0	0
$33$	1.94427	0.338454
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.70820	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
$37$	−6.70820	−1.10282	−0.551411	+	0.834234i	$0.685910\pi$
$38$	0	0
$39$	−0.854102	−0.136766
$40$	0	0
$41$	−11.0902	−1.73199	−0.865997	−	0.500050i	$-0.833315\pi$
$41$	−11.0902	−1.73199	−0.865997	+	0.500050i	$0.833315\pi$
$42$	0	0
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	0	0
$45$	−2.85410	−0.425464
$46$	0	0
$47$	−8.70820	−1.27022	−0.635111	−	0.772421i	$-0.719046\pi$
$47$	−8.70820	−1.27022	−0.635111	+	0.772421i	$0.719046\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.05573	−0.287860
$52$	0	0
$53$	−1.32624	−0.182173	−0.0910864	−	0.995843i	$-0.529034\pi$
$53$	−1.32624	−0.182173	−0.0910864	+	0.995843i	$0.529034\pi$
$54$	0	0
$55$	5.09017	0.686358
$56$	0	0
$57$	0.381966	0.0505926
$58$	0	0
$59$	−2.70820	−0.352578	−0.176289	−	0.984338i	$-0.556409\pi$
$59$	−2.70820	−0.352578	−0.176289	+	0.984338i	$0.556409\pi$
$60$	0	0
$61$	3.47214	0.444561	0.222281	−	0.974983i	$-0.428650\pi$
$61$	3.47214	0.444561	0.222281	+	0.974983i	$0.428650\pi$
$62$	0	0
$63$	−2.85410	−0.359583
$64$	0	0
$65$	−2.23607	−0.277350
$66$	0	0
$67$	−9.09017	−1.11054	−0.555271	−	0.831670i	$-0.687385\pi$
$67$	−9.09017	−1.11054	−0.555271	+	0.831670i	$0.687385\pi$
$68$	0	0
$69$	1.79837	0.216499
$70$	0	0
$71$	11.1803	1.32686	0.663431	−	0.748237i	$-0.269100\pi$
$71$	11.1803	1.32686	0.663431	+	0.748237i	$0.269100\pi$
$72$	0	0
$73$	−12.0902	−1.41505	−0.707524	−	0.706690i	$-0.750188\pi$
$73$	−12.0902	−1.41505	−0.707524	+	0.706690i	$0.750188\pi$
$74$	0	0
$75$	−1.52786	−0.176423
$76$	0	0
$77$	5.09017	0.580079
$78$	0	0
$79$	10.9443	1.23133	0.615663	−	0.788009i	$-0.288888\pi$
$79$	10.9443	1.23133	0.615663	+	0.788009i	$0.288888\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	−12.3820	−1.35910	−0.679549	−	0.733630i	$-0.737824\pi$
$83$	−12.3820	−1.35910	−0.679549	+	0.733630i	$0.737824\pi$
$84$	0	0
$85$	−5.38197	−0.583756
$86$	0	0
$87$	2.23607	0.239732
$88$	0	0
$89$	10.1803	1.07911	0.539557	−	0.841949i	$-0.318592\pi$
$89$	10.1803	1.07911	0.539557	+	0.841949i	$0.318592\pi$
$90$	0	0
$91$	−2.23607	−0.234404
$92$	0	0
$93$	−1.76393	−0.182911
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−4.70820	−0.478046	−0.239023	−	0.971014i	$-0.576827\pi$
$97$	−4.70820	−0.478046	−0.239023	+	0.971014i	$0.576827\pi$
$98$	0	0
$99$	−14.5279	−1.46011
$100$	0	0
$101$	3.76393	0.374525	0.187263	−	0.982310i	$-0.440038\pi$
$101$	3.76393	0.374525	0.187263	+	0.982310i	$0.440038\pi$
$102$	0	0
$103$	−12.2361	−1.20566	−0.602828	−	0.797871i	$-0.705959\pi$
$103$	−12.2361	−1.20566	−0.602828	+	0.797871i	$0.705959\pi$
$104$	0	0
$105$	0.381966	0.0372761
$106$	0	0
$107$	−8.23607	−0.796211	−0.398105	−	0.917340i	$-0.630332\pi$
$107$	−8.23607	−0.796211	−0.398105	+	0.917340i	$0.630332\pi$
$108$	0	0
$109$	16.8885	1.61763	0.808815	−	0.588064i	$-0.200110\pi$
$109$	16.8885	1.61763	0.808815	+	0.588064i	$0.200110\pi$
$110$	0	0
$111$	−2.56231	−0.243203
$112$	0	0
$113$	−12.3820	−1.16480	−0.582399	−	0.812903i	$-0.697886\pi$
$113$	−12.3820	−1.16480	−0.582399	+	0.812903i	$0.697886\pi$
$114$	0	0
$115$	4.70820	0.439042
$116$	0	0
$117$	6.38197	0.590013
$118$	0	0
$119$	−5.38197	−0.493364
$120$	0	0
$121$	14.9098	1.35544
$122$	0	0
$123$	−4.23607	−0.381953
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	10.7639	0.955145	0.477572	−	0.878592i	$-0.341517\pi$
$127$	10.7639	0.955145	0.477572	+	0.878592i	$0.341517\pi$
$128$	0	0
$129$	0.180340	0.0158780
$130$	0	0
$131$	6.61803	0.578220	0.289110	−	0.957296i	$-0.406641\pi$
$131$	6.61803	0.578220	0.289110	+	0.957296i	$0.406641\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−2.23607	−0.192450
$136$	0	0
$137$	12.4721	1.06557	0.532783	−	0.846252i	$-0.321146\pi$
$137$	12.4721	1.06557	0.532783	+	0.846252i	$0.321146\pi$
$138$	0	0
$139$	−17.1803	−1.45722	−0.728609	−	0.684930i	$-0.759833\pi$
$139$	−17.1803	−1.45722	−0.728609	+	0.684930i	$0.759833\pi$
$140$	0	0
$141$	−3.32624	−0.280120
$142$	0	0
$143$	−11.3820	−0.951808
$144$	0	0
$145$	5.85410	0.486157
$146$	0	0
$147$	0.381966	0.0315040
$148$	0	0
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	0	0
$151$	−6.61803	−0.538568	−0.269284	−	0.963061i	$-0.586787\pi$
$151$	−6.61803	−0.538568	−0.269284	+	0.963061i	$0.586787\pi$
$152$	0	0
$153$	15.3607	1.24184
$154$	0	0
$155$	−4.61803	−0.370929
$156$	0	0
$157$	−11.3820	−0.908380	−0.454190	−	0.890905i	$-0.650071\pi$
$157$	−11.3820	−0.908380	−0.454190	+	0.890905i	$0.650071\pi$
$158$	0	0
$159$	−0.506578	−0.0401742
$160$	0	0
$161$	4.70820	0.371058
$162$	0	0
$163$	−13.7984	−1.08077	−0.540386	−	0.841417i	$-0.681722\pi$
$163$	−13.7984	−1.08077	−0.540386	+	0.841417i	$0.681722\pi$
$164$	0	0
$165$	1.94427	0.151361
$166$	0	0
$167$	−7.47214	−0.578211	−0.289106	−	0.957297i	$-0.593358\pi$
$167$	−7.47214	−0.578211	−0.289106	+	0.957297i	$0.593358\pi$
$168$	0	0
$169$	−8.00000	−0.615385
$170$	0	0
$171$	−2.85410	−0.218259
$172$	0	0
$173$	23.7082	1.80250	0.901251	−	0.433298i	$-0.142650\pi$
$173$	23.7082	1.80250	0.901251	+	0.433298i	$0.142650\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−1.03444	−0.0777534
$178$	0	0
$179$	12.6180	0.943116	0.471558	−	0.881835i	$-0.343692\pi$
$179$	12.6180	0.943116	0.471558	+	0.881835i	$0.343692\pi$
$180$	0	0
$181$	−24.5066	−1.82156	−0.910780	−	0.412892i	$-0.864519\pi$
$181$	−24.5066	−1.82156	−0.910780	+	0.412892i	$0.864519\pi$
$182$	0	0
$183$	1.32624	0.0980383
$184$	0	0
$185$	−6.70820	−0.493197
$186$	0	0
$187$	−27.3951	−2.00333
$188$	0	0
$189$	−2.23607	−0.162650
$190$	0	0
$191$	−21.0344	−1.52200	−0.760999	−	0.648753i	$-0.775291\pi$
$191$	−21.0344	−1.52200	−0.760999	+	0.648753i	$0.775291\pi$
$192$	0	0
$193$	−21.5623	−1.55209	−0.776044	−	0.630678i	$-0.782777\pi$
$193$	−21.5623	−1.55209	−0.776044	+	0.630678i	$0.782777\pi$
$194$	0	0
$195$	−0.854102	−0.0611635
$196$	0	0
$197$	15.7984	1.12559	0.562794	−	0.826597i	$-0.309727\pi$
$197$	15.7984	1.12559	0.562794	+	0.826597i	$0.309727\pi$
$198$	0	0
$199$	−2.23607	−0.158511	−0.0792553	−	0.996854i	$-0.525254\pi$
$199$	−2.23607	−0.158511	−0.0792553	+	0.996854i	$0.525254\pi$
$200$	0	0
$201$	−3.47214	−0.244906
$202$	0	0
$203$	5.85410	0.410877
$204$	0	0
$205$	−11.0902	−0.774571
$206$	0	0
$207$	−13.4377	−0.933984
$208$	0	0
$209$	5.09017	0.352094
$210$	0	0
$211$	19.9787	1.37539	0.687696	−	0.725999i	$-0.258622\pi$
$211$	19.9787	1.37539	0.687696	+	0.725999i	$0.258622\pi$
$212$	0	0
$213$	4.27051	0.292611
$214$	0	0
$215$	0.472136	0.0321994
$216$	0	0
$217$	−4.61803	−0.313493
$218$	0	0
$219$	−4.61803	−0.312058
$220$	0	0
$221$	12.0344	0.809524
$222$	0	0
$223$	−11.9443	−0.799848	−0.399924	−	0.916548i	$-0.630963\pi$
$223$	−11.9443	−0.799848	−0.399924	+	0.916548i	$0.630963\pi$
$224$	0	0
$225$	11.4164	0.761094
$226$	0	0
$227$	−4.90983	−0.325877	−0.162938	−	0.986636i	$-0.552097\pi$
$227$	−4.90983	−0.325877	−0.162938	+	0.986636i	$0.552097\pi$
$228$	0	0
$229$	3.52786	0.233128	0.116564	−	0.993183i	$-0.462812\pi$
$229$	3.52786	0.233128	0.116564	+	0.993183i	$0.462812\pi$
$230$	0	0
$231$	1.94427	0.127924
$232$	0	0
$233$	8.32624	0.545470	0.272735	−	0.962089i	$-0.412072\pi$
$233$	8.32624	0.545470	0.272735	+	0.962089i	$0.412072\pi$
$234$	0	0
$235$	−8.70820	−0.568061
$236$	0	0
$237$	4.18034	0.271542
$238$	0	0
$239$	22.1246	1.43112	0.715561	−	0.698550i	$-0.246171\pi$
$239$	22.1246	1.43112	0.715561	+	0.698550i	$0.246171\pi$
$240$	0	0
$241$	10.2918	0.662953	0.331476	−	0.943463i	$-0.392453\pi$
$241$	10.2918	0.662953	0.331476	+	0.943463i	$0.392453\pi$
$242$	0	0
$243$	9.65248	0.619207
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.23607	−0.142278
$248$	0	0
$249$	−4.72949	−0.299719
$250$	0	0
$251$	16.3262	1.03050	0.515251	−	0.857039i	$-0.327699\pi$
$251$	16.3262	1.03050	0.515251	+	0.857039i	$0.327699\pi$
$252$	0	0
$253$	23.9656	1.50670
$254$	0	0
$255$	−2.05573	−0.128735
$256$	0	0
$257$	−20.2705	−1.26444	−0.632220	−	0.774789i	$-0.717856\pi$
$257$	−20.2705	−1.26444	−0.632220	+	0.774789i	$0.717856\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	−16.7082	−1.03421
$262$	0	0
$263$	−5.79837	−0.357543	−0.178772	−	0.983891i	$-0.557212\pi$
$263$	−5.79837	−0.357543	−0.178772	+	0.983891i	$0.557212\pi$
$264$	0	0
$265$	−1.32624	−0.0814701
$266$	0	0
$267$	3.88854	0.237975
$268$	0	0
$269$	19.7984	1.20713	0.603564	−	0.797314i	$-0.293747\pi$
$269$	19.7984	1.20713	0.603564	+	0.797314i	$0.293747\pi$
$270$	0	0
$271$	−19.5623	−1.18833	−0.594163	−	0.804345i	$-0.702516\pi$
$271$	−19.5623	−1.18833	−0.594163	+	0.804345i	$0.702516\pi$
$272$	0	0
$273$	−0.854102	−0.0516926
$274$	0	0
$275$	−20.3607	−1.22780
$276$	0	0
$277$	−22.6525	−1.36106	−0.680528	−	0.732722i	$-0.738249\pi$
$277$	−22.6525	−1.36106	−0.680528	+	0.732722i	$0.738249\pi$
$278$	0	0
$279$	13.1803	0.789086
$280$	0	0
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	0	0
$283$	16.3820	0.973807	0.486903	−	0.873456i	$-0.338126\pi$
$283$	16.3820	0.973807	0.486903	+	0.873456i	$0.338126\pi$
$284$	0	0
$285$	0.381966	0.0226257
$286$	0	0
$287$	−11.0902	−0.654632
$288$	0	0
$289$	11.9656	0.703856
$290$	0	0
$291$	−1.79837	−0.105423
$292$	0	0
$293$	10.5279	0.615044	0.307522	−	0.951541i	$-0.400500\pi$
$293$	10.5279	0.615044	0.307522	+	0.951541i	$0.400500\pi$
$294$	0	0
$295$	−2.70820	−0.157678
$296$	0	0
$297$	−11.3820	−0.660449
$298$	0	0
$299$	−10.5279	−0.608842
$300$	0	0
$301$	0.472136	0.0272135
$302$	0	0
$303$	1.43769	0.0825934
$304$	0	0
$305$	3.47214	0.198814
$306$	0	0
$307$	19.7426	1.12677	0.563386	−	0.826194i	$-0.309498\pi$
$307$	19.7426	1.12677	0.563386	+	0.826194i	$0.309498\pi$
$308$	0	0
$309$	−4.67376	−0.265881
$310$	0	0
$311$	12.3262	0.698957	0.349478	−	0.936944i	$-0.386359\pi$
$311$	12.3262	0.698957	0.349478	+	0.936944i	$0.386359\pi$
$312$	0	0
$313$	−8.18034	−0.462380	−0.231190	−	0.972909i	$-0.574262\pi$
$313$	−8.18034	−0.462380	−0.231190	+	0.972909i	$0.574262\pi$
$314$	0	0
$315$	−2.85410	−0.160810
$316$	0	0
$317$	4.47214	0.251180	0.125590	−	0.992082i	$-0.459918\pi$
$317$	4.47214	0.251180	0.125590	+	0.992082i	$0.459918\pi$
$318$	0	0
$319$	29.7984	1.66839
$320$	0	0
$321$	−3.14590	−0.175587
$322$	0	0
$323$	−5.38197	−0.299461
$324$	0	0
$325$	8.94427	0.496139
$326$	0	0
$327$	6.45085	0.356733
$328$	0	0
$329$	−8.70820	−0.480099
$330$	0	0
$331$	34.2705	1.88368	0.941839	−	0.336065i	$-0.109096\pi$
$331$	34.2705	1.88368	0.941839	+	0.336065i	$0.109096\pi$
$332$	0	0
$333$	19.1459	1.04919
$334$	0	0
$335$	−9.09017	−0.496649
$336$	0	0
$337$	6.32624	0.344612	0.172306	−	0.985043i	$-0.444878\pi$
$337$	6.32624	0.344612	0.172306	+	0.985043i	$0.444878\pi$
$338$	0	0
$339$	−4.72949	−0.256871
$340$	0	0
$341$	−23.5066	−1.27295
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.79837	0.0968212
$346$	0	0
$347$	−8.56231	−0.459649	−0.229824	−	0.973232i	$-0.573815\pi$
$347$	−8.56231	−0.459649	−0.229824	+	0.973232i	$0.573815\pi$
$348$	0	0
$349$	−30.2705	−1.62034	−0.810172	−	0.586193i	$-0.800626\pi$
$349$	−30.2705	−1.62034	−0.810172	+	0.586193i	$0.800626\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−19.3820	−1.03160	−0.515799	−	0.856710i	$-0.672505\pi$
$353$	−19.3820	−1.03160	−0.515799	+	0.856710i	$0.672505\pi$
$354$	0	0
$355$	11.1803	0.593391
$356$	0	0
$357$	−2.05573	−0.108801
$358$	0	0
$359$	−23.5623	−1.24357	−0.621785	−	0.783188i	$-0.713592\pi$
$359$	−23.5623	−1.24357	−0.621785	+	0.783188i	$0.713592\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.69505	0.298913
$364$	0	0
$365$	−12.0902	−0.632828
$366$	0	0
$367$	−22.8885	−1.19477	−0.597386	−	0.801954i	$-0.703794\pi$
$367$	−22.8885	−1.19477	−0.597386	+	0.801954i	$0.703794\pi$
$368$	0	0
$369$	31.6525	1.64776
$370$	0	0
$371$	−1.32624	−0.0688548
$372$	0	0
$373$	−21.5066	−1.11357	−0.556784	−	0.830657i	$-0.687965\pi$
$373$	−21.5066	−1.11357	−0.556784	+	0.830657i	$0.687965\pi$
$374$	0	0
$375$	−3.43769	−0.177522
$376$	0	0
$377$	−13.0902	−0.674178
$378$	0	0
$379$	−15.4721	−0.794750	−0.397375	−	0.917656i	$-0.630079\pi$
$379$	−15.4721	−0.794750	−0.397375	+	0.917656i	$0.630079\pi$
$380$	0	0
$381$	4.11146	0.210636
$382$	0	0
$383$	−11.7639	−0.601109	−0.300554	−	0.953765i	$-0.597172\pi$
$383$	−11.7639	−0.601109	−0.300554	+	0.953765i	$0.597172\pi$
$384$	0	0
$385$	5.09017	0.259419
$386$	0	0
$387$	−1.34752	−0.0684985
$388$	0	0
$389$	−3.38197	−0.171472	−0.0857362	−	0.996318i	$-0.527324\pi$
$389$	−3.38197	−0.171472	−0.0857362	+	0.996318i	$0.527324\pi$
$390$	0	0
$391$	−25.3394	−1.28147
$392$	0	0
$393$	2.52786	0.127514
$394$	0	0
$395$	10.9443	0.550666
$396$	0	0
$397$	10.7639	0.540226	0.270113	−	0.962829i	$-0.412939\pi$
$397$	10.7639	0.540226	0.270113	+	0.962829i	$0.412939\pi$
$398$	0	0
$399$	0.381966	0.0191222
$400$	0	0
$401$	15.7426	0.786150	0.393075	−	0.919506i	$-0.371411\pi$
$401$	15.7426	0.786150	0.393075	+	0.919506i	$0.371411\pi$
$402$	0	0
$403$	10.3262	0.514387
$404$	0	0
$405$	7.70820	0.383024
$406$	0	0
$407$	−34.1459	−1.69255
$408$	0	0
$409$	32.2705	1.59567	0.797837	−	0.602873i	$-0.205978\pi$
$409$	32.2705	1.59567	0.797837	+	0.602873i	$0.205978\pi$
$410$	0	0
$411$	4.76393	0.234987
$412$	0	0
$413$	−2.70820	−0.133262
$414$	0	0
$415$	−12.3820	−0.607807
$416$	0	0
$417$	−6.56231	−0.321358
$418$	0	0
$419$	−4.70820	−0.230011	−0.115005	−	0.993365i	$-0.536689\pi$
$419$	−4.70820	−0.230011	−0.115005	+	0.993365i	$0.536689\pi$
$420$	0	0
$421$	−13.1803	−0.642370	−0.321185	−	0.947016i	$-0.604081\pi$
$421$	−13.1803	−0.642370	−0.321185	+	0.947016i	$0.604081\pi$
$422$	0	0
$423$	24.8541	1.20845
$424$	0	0
$425$	21.5279	1.04425
$426$	0	0
$427$	3.47214	0.168028
$428$	0	0
$429$	−4.34752	−0.209900
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	22.5279	1.08262	0.541310	−	0.840823i	$-0.317929\pi$
$433$	22.5279	1.08262	0.541310	+	0.840823i	$0.317929\pi$
$434$	0	0
$435$	2.23607	0.107211
$436$	0	0
$437$	4.70820	0.225224
$438$	0	0
$439$	−16.3607	−0.780853	−0.390426	−	0.920634i	$-0.627672\pi$
$439$	−16.3607	−0.780853	−0.390426	+	0.920634i	$0.627672\pi$
$440$	0	0
$441$	−2.85410	−0.135910
$442$	0	0
$443$	−10.2016	−0.484694	−0.242347	−	0.970190i	$-0.577917\pi$
$443$	−10.2016	−0.484694	−0.242347	+	0.970190i	$0.577917\pi$
$444$	0	0
$445$	10.1803	0.482594
$446$	0	0
$447$	−2.67376	−0.126465
$448$	0	0
$449$	−3.61803	−0.170746	−0.0853728	−	0.996349i	$-0.527208\pi$
$449$	−3.61803	−0.170746	−0.0853728	+	0.996349i	$0.527208\pi$
$450$	0	0
$451$	−56.4508	−2.65817
$452$	0	0
$453$	−2.52786	−0.118769
$454$	0	0
$455$	−2.23607	−0.104828
$456$	0	0
$457$	−35.2148	−1.64728	−0.823639	−	0.567114i	$-0.808060\pi$
$457$	−35.2148	−1.64728	−0.823639	+	0.567114i	$0.808060\pi$
$458$	0	0
$459$	12.0344	0.561720
$460$	0	0
$461$	−5.03444	−0.234477	−0.117239	−	0.993104i	$-0.537404\pi$
$461$	−5.03444	−0.234477	−0.117239	+	0.993104i	$0.537404\pi$
$462$	0	0
$463$	30.7771	1.43033	0.715166	−	0.698954i	$-0.246351\pi$
$463$	30.7771	1.43033	0.715166	+	0.698954i	$0.246351\pi$
$464$	0	0
$465$	−1.76393	−0.0818004
$466$	0	0
$467$	−13.1459	−0.608320	−0.304160	−	0.952621i	$-0.598376\pi$
$467$	−13.1459	−0.608320	−0.304160	+	0.952621i	$0.598376\pi$
$468$	0	0
$469$	−9.09017	−0.419745
$470$	0	0
$471$	−4.34752	−0.200323
$472$	0	0
$473$	2.40325	0.110502
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	3.78522	0.173313
$478$	0	0
$479$	30.9787	1.41545	0.707727	−	0.706486i	$-0.249721\pi$
$479$	30.9787	1.41545	0.707727	+	0.706486i	$0.249721\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	1.79837	0.0818288
$484$	0	0
$485$	−4.70820	−0.213789
$486$	0	0
$487$	−7.29180	−0.330423	−0.165211	−	0.986258i	$-0.552831\pi$
$487$	−7.29180	−0.330423	−0.165211	+	0.986258i	$0.552831\pi$
$488$	0	0
$489$	−5.27051	−0.238341
$490$	0	0
$491$	−40.9443	−1.84779	−0.923895	−	0.382647i	$-0.875013\pi$
$491$	−40.9443	−1.84779	−0.923895	+	0.382647i	$0.875013\pi$
$492$	0	0
$493$	−31.5066	−1.41898
$494$	0	0
$495$	−14.5279	−0.652979
$496$	0	0
$497$	11.1803	0.501507
$498$	0	0
$499$	−18.0902	−0.809827	−0.404914	−	0.914355i	$-0.632698\pi$
$499$	−18.0902	−0.809827	−0.404914	+	0.914355i	$0.632698\pi$
$500$	0	0
$501$	−2.85410	−0.127512
$502$	0	0
$503$	−16.9443	−0.755508	−0.377754	−	0.925906i	$-0.623303\pi$
$503$	−16.9443	−0.755508	−0.377754	+	0.925906i	$0.623303\pi$
$504$	0	0
$505$	3.76393	0.167493
$506$	0	0
$507$	−3.05573	−0.135710
$508$	0	0
$509$	−6.58359	−0.291813	−0.145906	−	0.989298i	$-0.546610\pi$
$509$	−6.58359	−0.291813	−0.145906	+	0.989298i	$0.546610\pi$
$510$	0	0
$511$	−12.0902	−0.534838
$512$	0	0
$513$	−2.23607	−0.0987248
$514$	0	0
$515$	−12.2361	−0.539186
$516$	0	0
$517$	−44.3262	−1.94947
$518$	0	0
$519$	9.05573	0.397502
$520$	0	0
$521$	−28.8885	−1.26563	−0.632815	−	0.774303i	$-0.718101\pi$
$521$	−28.8885	−1.26563	−0.632815	+	0.774303i	$0.718101\pi$
$522$	0	0
$523$	22.4164	0.980201	0.490101	−	0.871666i	$-0.336960\pi$
$523$	22.4164	0.980201	0.490101	+	0.871666i	$0.336960\pi$
$524$	0	0
$525$	−1.52786	−0.0666815
$526$	0	0
$527$	24.8541	1.08266
$528$	0	0
$529$	−0.832816	−0.0362094
$530$	0	0
$531$	7.72949	0.335431
$532$	0	0
$533$	24.7984	1.07414
$534$	0	0
$535$	−8.23607	−0.356076
$536$	0	0
$537$	4.81966	0.207984
$538$	0	0
$539$	5.09017	0.219249
$540$	0	0
$541$	37.4164	1.60866	0.804329	−	0.594185i	$-0.202525\pi$
$541$	37.4164	1.60866	0.804329	+	0.594185i	$0.202525\pi$
$542$	0	0
$543$	−9.36068	−0.401705
$544$	0	0
$545$	16.8885	0.723426
$546$	0	0
$547$	−8.38197	−0.358387	−0.179193	−	0.983814i	$-0.557349\pi$
$547$	−8.38197	−0.358387	−0.179193	+	0.983814i	$0.557349\pi$
$548$	0	0
$549$	−9.90983	−0.422941
$550$	0	0
$551$	5.85410	0.249393
$552$	0	0
$553$	10.9443	0.465398
$554$	0	0
$555$	−2.56231	−0.108764
$556$	0	0
$557$	−14.0344	−0.594658	−0.297329	−	0.954775i	$-0.596096\pi$
$557$	−14.0344	−0.594658	−0.297329	+	0.954775i	$0.596096\pi$
$558$	0	0
$559$	−1.05573	−0.0446525
$560$	0	0
$561$	−10.4640	−0.441791
$562$	0	0
$563$	29.5410	1.24501	0.622503	−	0.782618i	$-0.286116\pi$
$563$	29.5410	1.24501	0.622503	+	0.782618i	$0.286116\pi$
$564$	0	0
$565$	−12.3820	−0.520913
$566$	0	0
$567$	7.70820	0.323714
$568$	0	0
$569$	−26.7082	−1.11967	−0.559833	−	0.828605i	$-0.689135\pi$
$569$	−26.7082	−1.11967	−0.559833	+	0.828605i	$0.689135\pi$
$570$	0	0
$571$	−11.9443	−0.499852	−0.249926	−	0.968265i	$-0.580406\pi$
$571$	−11.9443	−0.499852	−0.249926	+	0.968265i	$0.580406\pi$
$572$	0	0
$573$	−8.03444	−0.335644
$574$	0	0
$575$	−18.8328	−0.785383
$576$	0	0
$577$	25.3262	1.05435	0.527173	−	0.849758i	$-0.323252\pi$
$577$	25.3262	1.05435	0.527173	+	0.849758i	$0.323252\pi$
$578$	0	0
$579$	−8.23607	−0.342279
$580$	0	0
$581$	−12.3820	−0.513691
$582$	0	0
$583$	−6.75078	−0.279589
$584$	0	0
$585$	6.38197	0.263862
$586$	0	0
$587$	−10.3607	−0.427631	−0.213816	−	0.976874i	$-0.568589\pi$
$587$	−10.3607	−0.427631	−0.213816	+	0.976874i	$0.568589\pi$
$588$	0	0
$589$	−4.61803	−0.190283
$590$	0	0
$591$	6.03444	0.248224
$592$	0	0
$593$	40.7082	1.67169	0.835843	−	0.548969i	$-0.184979\pi$
$593$	40.7082	1.67169	0.835843	+	0.548969i	$0.184979\pi$
$594$	0	0
$595$	−5.38197	−0.220639
$596$	0	0
$597$	−0.854102	−0.0349561
$598$	0	0
$599$	46.8541	1.91441	0.957203	−	0.289416i	$-0.0934612\pi$
$599$	46.8541	1.91441	0.957203	+	0.289416i	$0.0934612\pi$
$600$	0	0
$601$	−27.9787	−1.14128	−0.570638	−	0.821202i	$-0.693304\pi$
$601$	−27.9787	−1.14128	−0.570638	+	0.821202i	$0.693304\pi$
$602$	0	0
$603$	25.9443	1.05653
$604$	0	0
$605$	14.9098	0.606171
$606$	0	0
$607$	12.5967	0.511286	0.255643	−	0.966771i	$-0.417713\pi$
$607$	12.5967	0.511286	0.255643	+	0.966771i	$0.417713\pi$
$608$	0	0
$609$	2.23607	0.0906100
$610$	0	0
$611$	19.4721	0.787758
$612$	0	0
$613$	27.0344	1.09191	0.545955	−	0.837814i	$-0.316167\pi$
$613$	27.0344	1.09191	0.545955	+	0.837814i	$0.316167\pi$
$614$	0	0
$615$	−4.23607	−0.170815
$616$	0	0
$617$	−20.1459	−0.811043	−0.405522	−	0.914085i	$-0.632910\pi$
$617$	−20.1459	−0.811043	−0.405522	+	0.914085i	$0.632910\pi$
$618$	0	0
$619$	22.2016	0.892359	0.446179	−	0.894944i	$-0.352784\pi$
$619$	22.2016	0.892359	0.446179	+	0.894944i	$0.352784\pi$
$620$	0	0
$621$	−10.5279	−0.422469
$622$	0	0
$623$	10.1803	0.407867
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	1.94427	0.0776467
$628$	0	0
$629$	36.1033	1.43953
$630$	0	0
$631$	−34.8885	−1.38889	−0.694445	−	0.719545i	$-0.744350\pi$
$631$	−34.8885	−1.38889	−0.694445	+	0.719545i	$0.744350\pi$
$632$	0	0
$633$	7.63119	0.303313
$634$	0	0
$635$	10.7639	0.427154
$636$	0	0
$637$	−2.23607	−0.0885962
$638$	0	0
$639$	−31.9098	−1.26233
$640$	0	0
$641$	8.61803	0.340392	0.170196	−	0.985410i	$-0.445560\pi$
$641$	8.61803	0.340392	0.170196	+	0.985410i	$0.445560\pi$
$642$	0	0
$643$	−1.47214	−0.0580554	−0.0290277	−	0.999579i	$-0.509241\pi$
$643$	−1.47214	−0.0580554	−0.0290277	+	0.999579i	$0.509241\pi$
$644$	0	0
$645$	0.180340	0.00710088
$646$	0	0
$647$	3.65248	0.143594	0.0717968	−	0.997419i	$-0.477127\pi$
$647$	3.65248	0.143594	0.0717968	+	0.997419i	$0.477127\pi$
$648$	0	0
$649$	−13.7852	−0.541117
$650$	0	0
$651$	−1.76393	−0.0691339
$652$	0	0
$653$	−18.1246	−0.709271	−0.354635	−	0.935005i	$-0.615395\pi$
$653$	−18.1246	−0.709271	−0.354635	+	0.935005i	$0.615395\pi$
$654$	0	0
$655$	6.61803	0.258588
$656$	0	0
$657$	34.5066	1.34623
$658$	0	0
$659$	−10.9098	−0.424987	−0.212493	−	0.977163i	$-0.568158\pi$
$659$	−10.9098	−0.424987	−0.212493	+	0.977163i	$0.568158\pi$
$660$	0	0
$661$	−3.52786	−0.137218	−0.0686090	−	0.997644i	$-0.521856\pi$
$661$	−3.52786	−0.137218	−0.0686090	+	0.997644i	$0.521856\pi$
$662$	0	0
$663$	4.59675	0.178523
$664$	0	0
$665$	1.00000	0.0387783
$666$	0	0
$667$	27.5623	1.06722
$668$	0	0
$669$	−4.56231	−0.176389
$670$	0	0
$671$	17.6738	0.682288
$672$	0	0
$673$	−6.49342	−0.250303	−0.125152	−	0.992138i	$-0.539942\pi$
$673$	−6.49342	−0.250303	−0.125152	+	0.992138i	$0.539942\pi$
$674$	0	0
$675$	8.94427	0.344265
$676$	0	0
$677$	−16.9230	−0.650403	−0.325202	−	0.945645i	$-0.605432\pi$
$677$	−16.9230	−0.650403	−0.325202	+	0.945645i	$0.605432\pi$
$678$	0	0
$679$	−4.70820	−0.180684
$680$	0	0
$681$	−1.87539	−0.0718650
$682$	0	0
$683$	−20.7082	−0.792377	−0.396189	−	0.918169i	$-0.629667\pi$
$683$	−20.7082	−0.792377	−0.396189	+	0.918169i	$0.629667\pi$
$684$	0	0
$685$	12.4721	0.476536
$686$	0	0
$687$	1.34752	0.0514113
$688$	0	0
$689$	2.96556	0.112979
$690$	0	0
$691$	−1.12461	−0.0427822	−0.0213911	−	0.999771i	$-0.506810\pi$
$691$	−1.12461	−0.0427822	−0.0213911	+	0.999771i	$0.506810\pi$
$692$	0	0
$693$	−14.5279	−0.551868
$694$	0	0
$695$	−17.1803	−0.651687
$696$	0	0
$697$	59.6869	2.26080
$698$	0	0
$699$	3.18034	0.120292
$700$	0	0
$701$	−2.47214	−0.0933713	−0.0466856	−	0.998910i	$-0.514866\pi$
$701$	−2.47214	−0.0933713	−0.0466856	+	0.998910i	$0.514866\pi$
$702$	0	0
$703$	−6.70820	−0.253005
$704$	0	0
$705$	−3.32624	−0.125273
$706$	0	0
$707$	3.76393	0.141557
$708$	0	0
$709$	9.29180	0.348961	0.174480	−	0.984661i	$-0.444175\pi$
$709$	9.29180	0.348961	0.174480	+	0.984661i	$0.444175\pi$
$710$	0	0
$711$	−31.2361	−1.17144
$712$	0	0
$713$	−21.7426	−0.814268
$714$	0	0
$715$	−11.3820	−0.425661
$716$	0	0
$717$	8.45085	0.315603
$718$	0	0
$719$	−16.3607	−0.610150	−0.305075	−	0.952328i	$-0.598682\pi$
$719$	−16.3607	−0.610150	−0.305075	+	0.952328i	$0.598682\pi$
$720$	0	0
$721$	−12.2361	−0.455695
$722$	0	0
$723$	3.93112	0.146200
$724$	0	0
$725$	−23.4164	−0.869664
$726$	0	0
$727$	−27.4721	−1.01889	−0.509443	−	0.860505i	$-0.670148\pi$
$727$	−27.4721	−1.01889	−0.509443	+	0.860505i	$0.670148\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−2.54102	−0.0939830
$732$	0	0
$733$	14.8328	0.547863	0.273931	−	0.961749i	$-0.411676\pi$
$733$	14.8328	0.547863	0.273931	+	0.961749i	$0.411676\pi$
$734$	0	0
$735$	0.381966	0.0140890
$736$	0	0
$737$	−46.2705	−1.70440
$738$	0	0
$739$	19.1803	0.705560	0.352780	−	0.935706i	$-0.385236\pi$
$739$	19.1803	0.705560	0.352780	+	0.935706i	$0.385236\pi$
$740$	0	0
$741$	−0.854102	−0.0313762
$742$	0	0
$743$	−39.6525	−1.45471	−0.727354	−	0.686262i	$-0.759250\pi$
$743$	−39.6525	−1.45471	−0.727354	+	0.686262i	$0.759250\pi$
$744$	0	0
$745$	−7.00000	−0.256460
$746$	0	0
$747$	35.3394	1.29300
$748$	0	0
$749$	−8.23607	−0.300939
$750$	0	0
$751$	−35.6180	−1.29972	−0.649860	−	0.760054i	$-0.725173\pi$
$751$	−35.6180	−1.29972	−0.649860	+	0.760054i	$0.725173\pi$
$752$	0	0
$753$	6.23607	0.227255
$754$	0	0
$755$	−6.61803	−0.240855
$756$	0	0
$757$	24.8328	0.902564	0.451282	−	0.892381i	$-0.350967\pi$
$757$	24.8328	0.902564	0.451282	+	0.892381i	$0.350967\pi$
$758$	0	0
$759$	9.15403	0.332270
$760$	0	0
$761$	11.9443	0.432980	0.216490	−	0.976285i	$-0.430539\pi$
$761$	11.9443	0.432980	0.216490	+	0.976285i	$0.430539\pi$
$762$	0	0
$763$	16.8885	0.611406
$764$	0	0
$765$	15.3607	0.555367
$766$	0	0
$767$	6.05573	0.218660
$768$	0	0
$769$	16.3607	0.589981	0.294991	−	0.955500i	$-0.404683\pi$
$769$	16.3607	0.589981	0.294991	+	0.955500i	$0.404683\pi$
$770$	0	0
$771$	−7.74265	−0.278845
$772$	0	0
$773$	2.87539	0.103421	0.0517103	−	0.998662i	$-0.483533\pi$
$773$	2.87539	0.103421	0.0517103	+	0.998662i	$0.483533\pi$
$774$	0	0
$775$	18.4721	0.663539
$776$	0	0
$777$	−2.56231	−0.0919222
$778$	0	0
$779$	−11.0902	−0.397347
$780$	0	0
$781$	56.9098	2.03639
$782$	0	0
$783$	−13.0902	−0.467805
$784$	0	0
$785$	−11.3820	−0.406240
$786$	0	0
$787$	−44.3607	−1.58129	−0.790644	−	0.612276i	$-0.790254\pi$
$787$	−44.3607	−1.58129	−0.790644	+	0.612276i	$0.790254\pi$
$788$	0	0
$789$	−2.21478	−0.0788483
$790$	0	0
$791$	−12.3820	−0.440252
$792$	0	0
$793$	−7.76393	−0.275705
$794$	0	0
$795$	−0.506578	−0.0179665
$796$	0	0
$797$	53.2705	1.88694	0.943469	−	0.331460i	$-0.107541\pi$
$797$	53.2705	1.88694	0.943469	+	0.331460i	$0.107541\pi$
$798$	0	0
$799$	46.8673	1.65804
$800$	0	0
$801$	−29.0557	−1.02663
$802$	0	0
$803$	−61.5410	−2.17174
$804$	0	0
$805$	4.70820	0.165942
$806$	0	0
$807$	7.56231	0.266206
$808$	0	0
$809$	43.0689	1.51422	0.757111	−	0.653287i	$-0.226610\pi$
$809$	43.0689	1.51422	0.757111	+	0.653287i	$0.226610\pi$
$810$	0	0
$811$	27.8885	0.979299	0.489650	−	0.871919i	$-0.337125\pi$
$811$	27.8885	0.979299	0.489650	+	0.871919i	$0.337125\pi$
$812$	0	0
$813$	−7.47214	−0.262059
$814$	0	0
$815$	−13.7984	−0.483336
$816$	0	0
$817$	0.472136	0.0165179
$818$	0	0
$819$	6.38197	0.223004
$820$	0	0
$821$	3.41641	0.119233	0.0596167	−	0.998221i	$-0.481012\pi$
$821$	3.41641	0.119233	0.0596167	+	0.998221i	$0.481012\pi$
$822$	0	0
$823$	−28.7639	−1.00265	−0.501324	−	0.865260i	$-0.667153\pi$
$823$	−28.7639	−1.00265	−0.501324	+	0.865260i	$0.667153\pi$
$824$	0	0
$825$	−7.77709	−0.270763
$826$	0	0
$827$	−47.1803	−1.64062	−0.820311	−	0.571918i	$-0.806199\pi$
$827$	−47.1803	−1.64062	−0.820311	+	0.571918i	$0.806199\pi$
$828$	0	0
$829$	−42.2492	−1.46738	−0.733688	−	0.679486i	$-0.762203\pi$
$829$	−42.2492	−1.46738	−0.733688	+	0.679486i	$0.762203\pi$
$830$	0	0
$831$	−8.65248	−0.300151
$832$	0	0
$833$	−5.38197	−0.186474
$834$	0	0
$835$	−7.47214	−0.258584
$836$	0	0
$837$	10.3262	0.356927
$838$	0	0
$839$	8.36068	0.288643	0.144321	−	0.989531i	$-0.453900\pi$
$839$	8.36068	0.288643	0.144321	+	0.989531i	$0.453900\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	0	0
$843$	−4.20163	−0.144712
$844$	0	0
$845$	−8.00000	−0.275208
$846$	0	0
$847$	14.9098	0.512308
$848$	0	0
$849$	6.25735	0.214752
$850$	0	0
$851$	−31.5836	−1.08267
$852$	0	0
$853$	13.2148	0.452466	0.226233	−	0.974073i	$-0.427359\pi$
$853$	13.2148	0.452466	0.226233	+	0.974073i	$0.427359\pi$
$854$	0	0
$855$	−2.85410	−0.0976082
$856$	0	0
$857$	26.6180	0.909255	0.454627	−	0.890682i	$-0.349772\pi$
$857$	26.6180	0.909255	0.454627	+	0.890682i	$0.349772\pi$
$858$	0	0
$859$	14.9787	0.511067	0.255534	−	0.966800i	$-0.417749\pi$
$859$	14.9787	0.511067	0.255534	+	0.966800i	$0.417749\pi$
$860$	0	0
$861$	−4.23607	−0.144365
$862$	0	0
$863$	−7.03444	−0.239455	−0.119728	−	0.992807i	$-0.538202\pi$
$863$	−7.03444	−0.239455	−0.119728	+	0.992807i	$0.538202\pi$
$864$	0	0
$865$	23.7082	0.806103
$866$	0	0
$867$	4.57044	0.155220
$868$	0	0
$869$	55.7082	1.88977
$870$	0	0
$871$	20.3262	0.688728
$872$	0	0
$873$	13.4377	0.454797
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	24.2918	0.820276	0.410138	−	0.912024i	$-0.365481\pi$
$877$	24.2918	0.820276	0.410138	+	0.912024i	$0.365481\pi$
$878$	0	0
$879$	4.02129	0.135635
$880$	0	0
$881$	−21.5066	−0.724575	−0.362288	−	0.932066i	$-0.618004\pi$
$881$	−21.5066	−0.724575	−0.362288	+	0.932066i	$0.618004\pi$
$882$	0	0
$883$	−2.12461	−0.0714989	−0.0357494	−	0.999361i	$-0.511382\pi$
$883$	−2.12461	−0.0714989	−0.0357494	+	0.999361i	$0.511382\pi$
$884$	0	0
$885$	−1.03444	−0.0347724
$886$	0	0
$887$	−2.16718	−0.0727669	−0.0363835	−	0.999338i	$-0.511584\pi$
$887$	−2.16718	−0.0727669	−0.0363835	+	0.999338i	$0.511584\pi$
$888$	0	0
$889$	10.7639	0.361011
$890$	0	0
$891$	39.2361	1.31446
$892$	0	0
$893$	−8.70820	−0.291409
$894$	0	0
$895$	12.6180	0.421774
$896$	0	0
$897$	−4.02129	−0.134267
$898$	0	0
$899$	−27.0344	−0.901649
$900$	0	0
$901$	7.13777	0.237794
$902$	0	0
$903$	0.180340	0.00600134
$904$	0	0
$905$	−24.5066	−0.814626
$906$	0	0
$907$	28.3050	0.939850	0.469925	−	0.882706i	$-0.344281\pi$
$907$	28.3050	0.939850	0.469925	+	0.882706i	$0.344281\pi$
$908$	0	0
$909$	−10.7426	−0.356311
$910$	0	0
$911$	−8.47214	−0.280694	−0.140347	−	0.990102i	$-0.544822\pi$
$911$	−8.47214	−0.280694	−0.140347	+	0.990102i	$0.544822\pi$
$912$	0	0
$913$	−63.0263	−2.08587
$914$	0	0
$915$	1.32624	0.0438441
$916$	0	0
$917$	6.61803	0.218547
$918$	0	0
$919$	−13.3607	−0.440728	−0.220364	−	0.975418i	$-0.570725\pi$
$919$	−13.3607	−0.440728	−0.220364	+	0.975418i	$0.570725\pi$
$920$	0	0
$921$	7.54102	0.248485
$922$	0	0
$923$	−25.0000	−0.822885
$924$	0	0
$925$	26.8328	0.882258
$926$	0	0
$927$	34.9230	1.14702
$928$	0	0
$929$	22.2705	0.730672	0.365336	−	0.930876i	$-0.380954\pi$
$929$	22.2705	0.730672	0.365336	+	0.930876i	$0.380954\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	4.70820	0.154140
$934$	0	0
$935$	−27.3951	−0.895916
$936$	0	0
$937$	−23.8541	−0.779280	−0.389640	−	0.920967i	$-0.627400\pi$
$937$	−23.8541	−0.779280	−0.389640	+	0.920967i	$0.627400\pi$
$938$	0	0
$939$	−3.12461	−0.101968
$940$	0	0
$941$	−12.1115	−0.394822	−0.197411	−	0.980321i	$-0.563253\pi$
$941$	−12.1115	−0.394822	−0.197411	+	0.980321i	$0.563253\pi$
$942$	0	0
$943$	−52.2148	−1.70035
$944$	0	0
$945$	−2.23607	−0.0727393
$946$	0	0
$947$	3.03444	0.0986061	0.0493031	−	0.998784i	$-0.484300\pi$
$947$	3.03444	0.0986061	0.0493031	+	0.998784i	$0.484300\pi$
$948$	0	0
$949$	27.0344	0.877575
$950$	0	0
$951$	1.70820	0.0553923
$952$	0	0
$953$	−38.2148	−1.23790	−0.618949	−	0.785431i	$-0.712441\pi$
$953$	−38.2148	−1.23790	−0.618949	+	0.785431i	$0.712441\pi$
$954$	0	0
$955$	−21.0344	−0.680659
$956$	0	0
$957$	11.3820	0.367927
$958$	0	0
$959$	12.4721	0.402746
$960$	0	0
$961$	−9.67376	−0.312057
$962$	0	0
$963$	23.5066	0.757489
$964$	0	0
$965$	−21.5623	−0.694115
$966$	0	0
$967$	47.0902	1.51432	0.757159	−	0.653231i	$-0.226587\pi$
$967$	47.0902	1.51432	0.757159	+	0.653231i	$0.226587\pi$
$968$	0	0
$969$	−2.05573	−0.0660395
$970$	0	0
$971$	−26.2361	−0.841955	−0.420978	−	0.907071i	$-0.638313\pi$
$971$	−26.2361	−0.841955	−0.420978	+	0.907071i	$0.638313\pi$
$972$	0	0
$973$	−17.1803	−0.550776
$974$	0	0
$975$	3.41641	0.109413
$976$	0	0
$977$	16.7639	0.536326	0.268163	−	0.963374i	$-0.413583\pi$
$977$	16.7639	0.536326	0.268163	+	0.963374i	$0.413583\pi$
$978$	0	0
$979$	51.8197	1.65616
$980$	0	0
$981$	−48.2016	−1.53896
$982$	0	0
$983$	22.8197	0.727834	0.363917	−	0.931431i	$-0.381439\pi$
$983$	22.8197	0.727834	0.363917	+	0.931431i	$0.381439\pi$
$984$	0	0
$985$	15.7984	0.503378
$986$	0	0
$987$	−3.32624	−0.105875
$988$	0	0
$989$	2.22291	0.0706845
$990$	0	0
$991$	29.1246	0.925174	0.462587	−	0.886574i	$-0.346921\pi$
$991$	29.1246	0.925174	0.462587	+	0.886574i	$0.346921\pi$
$992$	0	0
$993$	13.0902	0.415404
$994$	0	0
$995$	−2.23607	−0.0708881
$996$	0	0
$997$	57.7984	1.83049	0.915246	−	0.402894i	$-0.131996\pi$
$997$	57.7984	1.83049	0.915246	+	0.402894i	$0.131996\pi$
$998$	0	0
$999$	15.0000	0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bg.1.1		2
4.3	odd	2		8512.2.a.k.1.2		2
8.3	odd	2		2128.2.a.m.1.1		2
8.5	even	2		532.2.a.b.1.2	✓	2
24.5	odd	2		4788.2.a.l.1.2		2
56.13	odd	2		3724.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.b.1.2	✓	2	8.5	even	2
2128.2.a.m.1.1		2	8.3	odd	2
3724.2.a.g.1.1		2	56.13	odd	2
4788.2.a.l.1.2		2	24.5	odd	2
8512.2.a.k.1.2		2	4.3	odd	2
8512.2.a.bg.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.85410 q^{9} +5.09017 q^{11} -2.23607 q^{13} +0.381966 q^{15} -5.38197 q^{17} +1.00000 q^{19} +0.381966 q^{21} +4.70820 q^{23} -4.00000 q^{25} -2.23607 q^{27} +5.85410 q^{29} -4.61803 q^{31} +1.94427 q^{33} +1.00000 q^{35} -6.70820 q^{37} -0.854102 q^{39} -11.0902 q^{41} +0.472136 q^{43} -2.85410 q^{45} -8.70820 q^{47} +1.00000 q^{49} -2.05573 q^{51} -1.32624 q^{53} +5.09017 q^{55} +0.381966 q^{57} -2.70820 q^{59} +3.47214 q^{61} -2.85410 q^{63} -2.23607 q^{65} -9.09017 q^{67} +1.79837 q^{69} +11.1803 q^{71} -12.0902 q^{73} -1.52786 q^{75} +5.09017 q^{77} +10.9443 q^{79} +7.70820 q^{81} -12.3820 q^{83} -5.38197 q^{85} +2.23607 q^{87} +10.1803 q^{89} -2.23607 q^{91} -1.76393 q^{93} +1.00000 q^{95} -4.70820 q^{97} -14.5279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - q^{11} + 3 q^{15} - 13 q^{17} + 2 q^{19} + 3 q^{21} - 4 q^{23} - 8 q^{25} + 5 q^{29} - 7 q^{31} - 14 q^{33} + 2 q^{35} + 5 q^{39} - 11 q^{41} - 8 q^{43} + q^{45}+ \cdots - 38 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + q^9 - q^11 + 3 * q^15 - 13 * q^17 + 2 * q^19 + 3 * q^21 - 4 * q^23 - 8 * q^25 + 5 * q^29 - 7 * q^31 - 14 * q^33 + 2 * q^35 + 5 * q^39 - 11 * q^41 - 8 * q^43 + q^45 - 4 * q^47 + 2 * q^49 - 22 * q^51 + 13 * q^53 - q^55 + 3 * q^57 + 8 * q^59 - 2 * q^61 + q^63 - 7 * q^67 - 21 * q^69 - 13 * q^73 - 12 * q^75 - q^77 + 4 * q^79 + 2 * q^81 - 27 * q^83 - 13 * q^85 - 2 * q^89 - 8 * q^93 + 2 * q^95 + 4 * q^97 - 38 * q^99

Introduction

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