LMFDB - Embedded newform 8649.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 961)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.28825$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} -1.00000 q^{7} -7.47214 q^{8} +5.85410 q^{10} +4.24264 q^{11} +2.62210 q^{13} +2.61803 q^{14} +9.85410 q^{16} +3.70246 q^{17} +1.00000 q^{19} -10.8541 q^{20} -11.1074 q^{22} -2.62210 q^{23} -6.86474 q^{26} -4.85410 q^{28} -0.540182 q^{29} -10.8541 q^{32} -9.69316 q^{34} +2.23607 q^{35} -4.24264 q^{37} -2.61803 q^{38} +16.7082 q^{40} +1.47214 q^{41} +9.69316 q^{43} +20.5942 q^{44} +6.86474 q^{46} +9.70820 q^{47} -6.00000 q^{49} +12.7279 q^{52} -13.7295 q^{53} -9.48683 q^{55} +7.47214 q^{56} +1.41421 q^{58} -11.9443 q^{59} -13.9358 q^{61} +8.70820 q^{64} -5.86319 q^{65} +6.00000 q^{67} +17.9721 q^{68} -5.85410 q^{70} +1.47214 q^{71} -4.24264 q^{73} +11.1074 q^{74} +4.85410 q^{76} -4.24264 q^{77} +1.62054 q^{79} -22.0344 q^{80} -3.85410 q^{82} -3.16228 q^{83} -8.27895 q^{85} -25.3770 q^{86} -31.7016 q^{88} -15.3500 q^{89} -2.62210 q^{91} -12.7279 q^{92} -25.4164 q^{94} -2.23607 q^{95} -7.00000 q^{97} +15.7082 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 6 q^{4} - 4 q^{7} - 12 q^{8} + 10 q^{10} + 6 q^{14} + 26 q^{16} + 4 q^{19} - 30 q^{20} - 6 q^{28} - 30 q^{32} - 6 q^{38} + 40 q^{40} - 12 q^{41} + 12 q^{47} - 24 q^{49} + 12 q^{56} - 12 q^{59}+ \cdots + 36 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 + 6 * q^4 - 4 * q^7 - 12 * q^8 + 10 * q^10 + 6 * q^14 + 26 * q^16 + 4 * q^19 - 30 * q^20 - 6 * q^28 - 30 * q^32 - 6 * q^38 + 40 * q^40 - 12 * q^41 + 12 * q^47 - 24 * q^49 + 12 * q^56 - 12 * q^59 + 8 * q^64 + 24 * q^67 - 10 * q^70 - 12 * q^71 + 6 * q^76 - 30 * q^80 - 2 * q^82 - 48 * q^94 - 28 * q^97 + 36 * q^98

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 $

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	0	0
$3$	0	0
$4$	4.85410	2.42705
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−7.47214	−2.64180
$9$	0	0
$10$	5.85410	1.85123
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	2.62210	0.727239	0.363619	−	0.931548i	$-0.381541\pi$
$13$	2.62210	0.727239	0.363619	+	0.931548i	$0.381541\pi$
$14$	2.61803	0.699699
$15$	0	0
$16$	9.85410	2.46353
$17$	3.70246	0.897978	0.448989	−	0.893537i	$-0.351784\pi$
$17$	3.70246	0.897978	0.448989	+	0.893537i	$0.351784\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−10.8541	−2.42705
$21$	0	0
$22$	−11.1074	−2.36810
$23$	−2.62210	−0.546745	−0.273372	−	0.961908i	$-0.588139\pi$
$23$	−2.62210	−0.546745	−0.273372	+	0.961908i	$0.588139\pi$
$24$	0	0
$25$	0	0
$26$	−6.86474	−1.34629
$27$	0	0
$28$	−4.85410	−0.917339
$29$	−0.540182	−0.100309	−0.0501546	−	0.998741i	$-0.515971\pi$
$29$	−0.540182	−0.100309	−0.0501546	+	0.998741i	$0.515971\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	−10.8541	−1.91875
$33$	0	0
$34$	−9.69316	−1.66236
$35$	2.23607	0.377964
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	−2.61803	−0.424701
$39$	0	0
$40$	16.7082	2.64180
$41$	1.47214	0.229909	0.114955	−	0.993371i	$-0.463328\pi$
$41$	1.47214	0.229909	0.114955	+	0.993371i	$0.463328\pi$
$42$	0	0
$43$	9.69316	1.47819	0.739097	−	0.673599i	$-0.235253\pi$
$43$	9.69316	1.47819	0.739097	+	0.673599i	$0.235253\pi$
$44$	20.5942	3.10469
$45$	0	0
$46$	6.86474	1.01215
$47$	9.70820	1.41609	0.708044	−	0.706169i	$-0.249578\pi$
$47$	9.70820	1.41609	0.708044	+	0.706169i	$0.249578\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	12.7279	1.76505
$53$	−13.7295	−1.88589	−0.942944	−	0.332951i	$-0.891956\pi$
$53$	−13.7295	−1.88589	−0.942944	+	0.332951i	$0.891956\pi$
$54$	0	0
$55$	−9.48683	−1.27920
$56$	7.47214	0.998506
$57$	0	0
$58$	1.41421	0.185695
$59$	−11.9443	−1.55501	−0.777506	−	0.628876i	$-0.783515\pi$
$59$	−11.9443	−1.55501	−0.777506	+	0.628876i	$0.783515\pi$
$60$	0	0
$61$	−13.9358	−1.78430	−0.892148	−	0.451742i	$-0.850803\pi$
$61$	−13.9358	−1.78430	−0.892148	+	0.451742i	$0.850803\pi$
$62$	0	0
$63$	0	0
$64$	8.70820	1.08853
$65$	−5.86319	−0.727239
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	17.9721	2.17944
$69$	0	0
$70$	−5.85410	−0.699699
$71$	1.47214	0.174710	0.0873552	−	0.996177i	$-0.472158\pi$
$71$	1.47214	0.174710	0.0873552	+	0.996177i	$0.472158\pi$
$72$	0	0
$73$	−4.24264	−0.496564	−0.248282	−	0.968688i	$-0.579866\pi$
$73$	−4.24264	−0.496564	−0.248282	+	0.968688i	$0.579866\pi$
$74$	11.1074	1.29121
$75$	0	0
$76$	4.85410	0.556804
$77$	−4.24264	−0.483494
$78$	0	0
$79$	1.62054	0.182326	0.0911628	−	0.995836i	$-0.470942\pi$
$79$	1.62054	0.182326	0.0911628	+	0.995836i	$0.470942\pi$
$80$	−22.0344	−2.46353
$81$	0	0
$82$	−3.85410	−0.425614
$83$	−3.16228	−0.347105	−0.173553	−	0.984825i	$-0.555525\pi$
$83$	−3.16228	−0.347105	−0.173553	+	0.984825i	$0.555525\pi$
$84$	0	0
$85$	−8.27895	−0.897978
$86$	−25.3770	−2.73648
$87$	0	0
$88$	−31.7016	−3.37940
$89$	−15.3500	−1.62710	−0.813549	−	0.581496i	$-0.802468\pi$
$89$	−15.3500	−1.62710	−0.813549	+	0.581496i	$0.802468\pi$
$90$	0	0
$91$	−2.62210	−0.274870
$92$	−12.7279	−1.32698
$93$	0	0
$94$	−25.4164	−2.62150
$95$	−2.23607	−0.229416
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	15.7082	1.58677
$99$	0	0
$100$	0	0
$101$	−2.23607	−0.222497	−0.111249	−	0.993793i	$-0.535485\pi$
$101$	−2.23607	−0.222497	−0.111249	+	0.993793i	$0.535485\pi$
$102$	0	0
$103$	10.7082	1.05511	0.527555	−	0.849521i	$-0.323109\pi$
$103$	10.7082	1.05511	0.527555	+	0.849521i	$0.323109\pi$
$104$	−19.5927	−1.92122
$105$	0	0
$106$	35.9442	3.49121
$107$	−1.47214	−0.142317	−0.0711584	−	0.997465i	$-0.522670\pi$
$107$	−1.47214	−0.142317	−0.0711584	+	0.997465i	$0.522670\pi$
$108$	0	0
$109$	−1.29180	−0.123732	−0.0618658	−	0.998084i	$-0.519705\pi$
$109$	−1.29180	−0.123732	−0.0618658	+	0.998084i	$0.519705\pi$
$110$	24.8369	2.36810
$111$	0	0
$112$	−9.85410	−0.931125
$113$	9.76393	0.918513	0.459257	−	0.888304i	$-0.348116\pi$
$113$	9.76393	0.918513	0.459257	+	0.888304i	$0.348116\pi$
$114$	0	0
$115$	5.86319	0.546745
$116$	−2.62210	−0.243456
$117$	0	0
$118$	31.2705	2.87868
$119$	−3.70246	−0.339404
$120$	0	0
$121$	7.00000	0.636364
$122$	36.4844	3.30314
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	9.48683	0.841820	0.420910	−	0.907102i	$-0.361711\pi$
$127$	9.48683	0.841820	0.420910	+	0.907102i	$0.361711\pi$
$128$	−1.09017	−0.0963583
$129$	0	0
$130$	15.3500	1.34629
$131$	−7.41641	−0.647975	−0.323987	−	0.946061i	$-0.605024\pi$
$131$	−7.41641	−0.647975	−0.323987	+	0.946061i	$0.605024\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−15.7082	−1.35698
$135$	0	0
$136$	−27.6653	−2.37228
$137$	−2.62210	−0.224021	−0.112010	−	0.993707i	$-0.535729\pi$
$137$	−2.62210	−0.224021	−0.112010	+	0.993707i	$0.535729\pi$
$138$	0	0
$139$	17.9721	1.52437	0.762187	−	0.647356i	$-0.224125\pi$
$139$	17.9721	1.52437	0.762187	+	0.647356i	$0.224125\pi$
$140$	10.8541	0.917339
$141$	0	0
$142$	−3.85410	−0.323429
$143$	11.1246	0.930287
$144$	0	0
$145$	1.20788	0.100309
$146$	11.1074	0.919253
$147$	0	0
$148$	−20.5942	−1.69283
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	−12.5216	−1.01899	−0.509496	−	0.860473i	$-0.670168\pi$
$151$	−12.5216	−1.01899	−0.509496	+	0.860473i	$0.670168\pi$
$152$	−7.47214	−0.606070
$153$	0	0
$154$	11.1074	0.895058
$155$	0	0
$156$	0	0
$157$	8.70820	0.694990	0.347495	−	0.937682i	$-0.387032\pi$
$157$	8.70820	0.694990	0.347495	+	0.937682i	$0.387032\pi$
$158$	−4.24264	−0.337526
$159$	0	0
$160$	24.2705	1.91875
$161$	2.62210	0.206650
$162$	0	0
$163$	14.4164	1.12918	0.564590	−	0.825371i	$-0.309034\pi$
$163$	14.4164	1.12918	0.564590	+	0.825371i	$0.309034\pi$
$164$	7.14590	0.558001
$165$	0	0
$166$	8.27895	0.642571
$167$	−1.08036	−0.0836010	−0.0418005	−	0.999126i	$-0.513309\pi$
$167$	−1.08036	−0.0836010	−0.0418005	+	0.999126i	$0.513309\pi$
$168$	0	0
$169$	−6.12461	−0.471124
$170$	21.6746	1.66236
$171$	0	0
$172$	47.0516	3.58765
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	0	0
$176$	41.8074	3.15135
$177$	0	0
$178$	40.1869	3.01213
$179$	13.2681	0.991705	0.495852	−	0.868407i	$-0.334856\pi$
$179$	13.2681	0.991705	0.495852	+	0.868407i	$0.334856\pi$
$180$	0	0
$181$	−8.69161	−0.646042	−0.323021	−	0.946392i	$-0.604699\pi$
$181$	−8.69161	−0.646042	−0.323021	+	0.946392i	$0.604699\pi$
$182$	6.86474	0.508848
$183$	0	0
$184$	19.5927	1.44439
$185$	9.48683	0.697486
$186$	0	0
$187$	15.7082	1.14870
$188$	47.1246	3.43692
$189$	0	0
$190$	5.85410	0.424701
$191$	−9.76393	−0.706493	−0.353247	−	0.935530i	$-0.614922\pi$
$191$	−9.76393	−0.706493	−0.353247	+	0.935530i	$0.614922\pi$
$192$	0	0
$193$	−20.7082	−1.49061	−0.745305	−	0.666724i	$-0.767696\pi$
$193$	−20.7082	−1.49061	−0.745305	+	0.666724i	$0.767696\pi$
$194$	18.3262	1.31575
$195$	0	0
$196$	−29.1246	−2.08033
$197$	−1.08036	−0.0769727	−0.0384863	−	0.999259i	$-0.512254\pi$
$197$	−1.08036	−0.0769727	−0.0384863	+	0.999259i	$0.512254\pi$
$198$	0	0
$199$	−23.2163	−1.64576	−0.822880	−	0.568215i	$-0.807634\pi$
$199$	−23.2163	−1.64576	−0.822880	+	0.568215i	$0.807634\pi$
$200$	0	0
$201$	0	0
$202$	5.85410	0.411893
$203$	0.540182	0.0379133
$204$	0	0
$205$	−3.29180	−0.229909
$206$	−28.0344	−1.95325
$207$	0	0
$208$	25.8384	1.79157
$209$	4.24264	0.293470
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−66.6443	−4.57715
$213$	0	0
$214$	3.85410	0.263461
$215$	−21.6746	−1.47819
$216$	0	0
$217$	0	0
$218$	3.38197	0.229056
$219$	0	0
$220$	−46.0501	−3.10469
$221$	9.70820	0.653044
$222$	0	0
$223$	−16.1452	−1.08117	−0.540583	−	0.841291i	$-0.681796\pi$
$223$	−16.1452	−1.08117	−0.540583	+	0.841291i	$0.681796\pi$
$224$	10.8541	0.725220
$225$	0	0
$226$	−25.5623	−1.70038
$227$	−2.29180	−0.152112	−0.0760559	−	0.997104i	$-0.524233\pi$
$227$	−2.29180	−0.152112	−0.0760559	+	0.997104i	$0.524233\pi$
$228$	0	0
$229$	−4.24264	−0.280362	−0.140181	−	0.990126i	$-0.544768\pi$
$229$	−4.24264	−0.280362	−0.140181	+	0.990126i	$0.544768\pi$
$230$	−15.3500	−1.01215
$231$	0	0
$232$	4.03631	0.264997
$233$	13.4721	0.882589	0.441294	−	0.897362i	$-0.354519\pi$
$233$	13.4721	0.882589	0.441294	+	0.897362i	$0.354519\pi$
$234$	0	0
$235$	−21.7082	−1.41609
$236$	−57.9787	−3.77409
$237$	0	0
$238$	9.69316	0.628314
$239$	12.1089	0.783262	0.391631	−	0.920122i	$-0.371911\pi$
$239$	12.1089	0.783262	0.391631	+	0.920122i	$0.371911\pi$
$240$	0	0
$241$	23.2163	1.49549	0.747747	−	0.663984i	$-0.231136\pi$
$241$	23.2163	1.49549	0.747747	+	0.663984i	$0.231136\pi$
$242$	−18.3262	−1.17806
$243$	0	0
$244$	−67.6458	−4.33058
$245$	13.4164	0.857143
$246$	0	0
$247$	2.62210	0.166840
$248$	0	0
$249$	0	0
$250$	−29.2705	−1.85123
$251$	11.0286	0.696117	0.348058	−	0.937473i	$-0.386841\pi$
$251$	11.0286	0.696117	0.348058	+	0.937473i	$0.386841\pi$
$252$	0	0
$253$	−11.1246	−0.699398
$254$	−24.8369	−1.55840
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−6.05573	−0.377746	−0.188873	−	0.982002i	$-0.560483\pi$
$257$	−6.05573	−0.377746	−0.188873	+	0.982002i	$0.560483\pi$
$258$	0	0
$259$	4.24264	0.263625
$260$	−28.4605	−1.76505
$261$	0	0
$262$	19.4164	1.19955
$263$	18.5123	1.14152	0.570759	−	0.821118i	$-0.306649\pi$
$263$	18.5123	1.14152	0.570759	+	0.821118i	$0.306649\pi$
$264$	0	0
$265$	30.7000	1.88589
$266$	2.61803	0.160522
$267$	0	0
$268$	29.1246	1.77907
$269$	2.16073	0.131742	0.0658709	−	0.997828i	$-0.479017\pi$
$269$	2.16073	0.131742	0.0658709	+	0.997828i	$0.479017\pi$
$270$	0	0
$271$	18.5911	1.12933	0.564665	−	0.825320i	$-0.309006\pi$
$271$	18.5911	1.12933	0.564665	+	0.825320i	$0.309006\pi$
$272$	36.4844	2.21219
$273$	0	0
$274$	6.86474	0.414714
$275$	0	0
$276$	0	0
$277$	−10.1058	−0.607200	−0.303600	−	0.952800i	$-0.598189\pi$
$277$	−10.1058	−0.607200	−0.303600	+	0.952800i	$0.598189\pi$
$278$	−47.0516	−2.82197
$279$	0	0
$280$	−16.7082	−0.998506
$281$	−13.3607	−0.797031	−0.398516	−	0.917162i	$-0.630475\pi$
$281$	−13.3607	−0.797031	−0.398516	+	0.917162i	$0.630475\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	7.14590	0.424031
$285$	0	0
$286$	−29.1246	−1.72217
$287$	−1.47214	−0.0868974
$288$	0	0
$289$	−3.29180	−0.193635
$290$	−3.16228	−0.185695
$291$	0	0
$292$	−20.5942	−1.20519
$293$	2.29180	0.133888	0.0669441	−	0.997757i	$-0.478675\pi$
$293$	2.29180	0.133888	0.0669441	+	0.997757i	$0.478675\pi$
$294$	0	0
$295$	26.7082	1.55501
$296$	31.7016	1.84262
$297$	0	0
$298$	−35.1246	−2.03471
$299$	−6.87539	−0.397614
$300$	0	0
$301$	−9.69316	−0.558705
$302$	32.7820	1.88639
$303$	0	0
$304$	9.85410	0.565172
$305$	31.1614	1.78430
$306$	0	0
$307$	24.1246	1.37686	0.688432	−	0.725301i	$-0.258299\pi$
$307$	24.1246	1.37686	0.688432	+	0.725301i	$0.258299\pi$
$308$	−20.5942	−1.17346
$309$	0	0
$310$	0	0
$311$	−25.4721	−1.44439	−0.722196	−	0.691688i	$-0.756867\pi$
$311$	−25.4721	−1.44439	−0.722196	+	0.691688i	$0.756867\pi$
$312$	0	0
$313$	11.5200	0.651151	0.325576	−	0.945516i	$-0.394442\pi$
$313$	11.5200	0.651151	0.325576	+	0.945516i	$0.394442\pi$
$314$	−22.7984	−1.28659
$315$	0	0
$316$	7.86629	0.442513
$317$	−26.2361	−1.47356	−0.736782	−	0.676130i	$-0.763656\pi$
$317$	−26.2361	−1.47356	−0.736782	+	0.676130i	$0.763656\pi$
$318$	0	0
$319$	−2.29180	−0.128316
$320$	−19.4721	−1.08853
$321$	0	0
$322$	−6.86474	−0.382557
$323$	3.70246	0.206010
$324$	0	0
$325$	0	0
$326$	−37.7426	−2.09037
$327$	0	0
$328$	−11.0000	−0.607373
$329$	−9.70820	−0.535231
$330$	0	0
$331$	−31.2889	−1.71979	−0.859897	−	0.510467i	$-0.829473\pi$
$331$	−31.2889	−1.71979	−0.859897	+	0.510467i	$0.829473\pi$
$332$	−15.3500	−0.842442
$333$	0	0
$334$	2.82843	0.154765
$335$	−13.4164	−0.733017
$336$	0	0
$337$	15.3500	0.836169	0.418084	−	0.908408i	$-0.362702\pi$
$337$	15.3500	0.836169	0.418084	+	0.908408i	$0.362702\pi$
$338$	16.0344	0.872159
$339$	0	0
$340$	−40.1869	−2.17944
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	−72.4286	−3.90509
$345$	0	0
$346$	47.1246	2.53343
$347$	−21.5958	−1.15932	−0.579661	−	0.814858i	$-0.696815\pi$
$347$	−21.5958	−1.15932	−0.579661	+	0.814858i	$0.696815\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	−46.0501	−2.45448
$353$	9.56564	0.509128	0.254564	−	0.967056i	$-0.418068\pi$
$353$	9.56564	0.509128	0.254564	+	0.967056i	$0.418068\pi$
$354$	0	0
$355$	−3.29180	−0.174710
$356$	−74.5106	−3.94905
$357$	0	0
$358$	−34.7363	−1.83587
$359$	−14.2361	−0.751351	−0.375675	−	0.926751i	$-0.622589\pi$
$359$	−14.2361	−0.751351	−0.375675	+	0.926751i	$0.622589\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	22.7549	1.19597
$363$	0	0
$364$	−12.7279	−0.667124
$365$	9.48683	0.496564
$366$	0	0
$367$	−6.65841	−0.347566	−0.173783	−	0.984784i	$-0.555599\pi$
$367$	−6.65841	−0.347566	−0.173783	+	0.984784i	$0.555599\pi$
$368$	−25.8384	−1.34692
$369$	0	0
$370$	−24.8369	−1.29121
$371$	13.7295	0.712799
$372$	0	0
$373$	−9.29180	−0.481111	−0.240555	−	0.970635i	$-0.577330\pi$
$373$	−9.29180	−0.481111	−0.240555	+	0.970635i	$0.577330\pi$
$374$	−41.1246	−2.12650
$375$	0	0
$376$	−72.5410	−3.74102
$377$	−1.41641	−0.0729487
$378$	0	0
$379$	7.41641	0.380955	0.190478	−	0.981692i	$-0.438996\pi$
$379$	7.41641	0.380955	0.190478	+	0.981692i	$0.438996\pi$
$380$	−10.8541	−0.556804
$381$	0	0
$382$	25.5623	1.30788
$383$	−6.40337	−0.327197	−0.163598	−	0.986527i	$-0.552310\pi$
$383$	−6.40337	−0.327197	−0.163598	+	0.986527i	$0.552310\pi$
$384$	0	0
$385$	9.48683	0.483494
$386$	54.2148	2.75946
$387$	0	0
$388$	−33.9787	−1.72501
$389$	−16.8129	−0.852450	−0.426225	−	0.904617i	$-0.640157\pi$
$389$	−16.8129	−0.852450	−0.426225	+	0.904617i	$0.640157\pi$
$390$	0	0
$391$	−9.70820	−0.490965
$392$	44.8328	2.26440
$393$	0	0
$394$	2.82843	0.142494
$395$	−3.62365	−0.182326
$396$	0	0
$397$	14.7082	0.738184	0.369092	−	0.929393i	$-0.379669\pi$
$397$	14.7082	0.738184	0.369092	+	0.929393i	$0.379669\pi$
$398$	60.7811	3.04668
$399$	0	0
$400$	0	0
$401$	−0.540182	−0.0269754	−0.0134877	−	0.999909i	$-0.504293\pi$
$401$	−0.540182	−0.0269754	−0.0134877	+	0.999909i	$0.504293\pi$
$402$	0	0
$403$	0	0
$404$	−10.8541	−0.540012
$405$	0	0
$406$	−1.41421	−0.0701862
$407$	−18.0000	−0.892227
$408$	0	0
$409$	−30.0810	−1.48741	−0.743706	−	0.668507i	$-0.766934\pi$
$409$	−30.0810	−1.48741	−0.743706	+	0.668507i	$0.766934\pi$
$410$	8.61803	0.425614
$411$	0	0
$412$	51.9787	2.56081
$413$	11.9443	0.587739
$414$	0	0
$415$	7.07107	0.347105
$416$	−28.4605	−1.39539
$417$	0	0
$418$	−11.1074	−0.543280
$419$	−30.5967	−1.49475	−0.747374	−	0.664403i	$-0.768686\pi$
$419$	−30.5967	−1.49475	−0.747374	+	0.664403i	$0.768686\pi$
$420$	0	0
$421$	26.4164	1.28746	0.643728	−	0.765254i	$-0.277387\pi$
$421$	26.4164	1.28746	0.643728	+	0.765254i	$0.277387\pi$
$422$	13.0902	0.637220
$423$	0	0
$424$	102.588	4.98214
$425$	0	0
$426$	0	0
$427$	13.9358	0.674401
$428$	−7.14590	−0.345410
$429$	0	0
$430$	56.7448	2.73648
$431$	35.8885	1.72869	0.864345	−	0.502899i	$-0.167733\pi$
$431$	35.8885	1.72869	0.864345	+	0.502899i	$0.167733\pi$
$432$	0	0
$433$	25.8384	1.24171	0.620857	−	0.783924i	$-0.286785\pi$
$433$	25.8384	1.24171	0.620857	+	0.783924i	$0.286785\pi$
$434$	0	0
$435$	0	0
$436$	−6.27051	−0.300303
$437$	−2.62210	−0.125432
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	70.8869	3.37940
$441$	0	0
$442$	−25.4164	−1.20894
$443$	−6.05573	−0.287716	−0.143858	−	0.989598i	$-0.545951\pi$
$443$	−6.05573	−0.287716	−0.143858	+	0.989598i	$0.545951\pi$
$444$	0	0
$445$	34.3237	1.62710
$446$	42.2688	2.00148
$447$	0	0
$448$	−8.70820	−0.411424
$449$	33.7047	1.59062	0.795311	−	0.606201i	$-0.207307\pi$
$449$	33.7047	1.59062	0.795311	+	0.606201i	$0.207307\pi$
$450$	0	0
$451$	6.24574	0.294101
$452$	47.3951	2.22928
$453$	0	0
$454$	6.00000	0.281594
$455$	5.86319	0.274870
$456$	0	0
$457$	−11.3137	−0.529233	−0.264616	−	0.964354i	$-0.585245\pi$
$457$	−11.3137	−0.529233	−0.264616	+	0.964354i	$0.585245\pi$
$458$	11.1074	0.519014
$459$	0	0
$460$	28.4605	1.32698
$461$	5.16538	0.240576	0.120288	−	0.992739i	$-0.461618\pi$
$461$	5.16538	0.240576	0.120288	+	0.992739i	$0.461618\pi$
$462$	0	0
$463$	3.82998	0.177994	0.0889971	−	0.996032i	$-0.471634\pi$
$463$	3.82998	0.177994	0.0889971	+	0.996032i	$0.471634\pi$
$464$	−5.32300	−0.247114
$465$	0	0
$466$	−35.2705	−1.63387
$467$	15.6525	0.724310	0.362155	−	0.932118i	$-0.382041\pi$
$467$	15.6525	0.724310	0.362155	+	0.932118i	$0.382041\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	56.8328	2.62150
$471$	0	0
$472$	89.2492	4.10803
$473$	41.1246	1.89091
$474$	0	0
$475$	0	0
$476$	−17.9721	−0.823751
$477$	0	0
$478$	−31.7016	−1.45000
$479$	−16.5279	−0.755177	−0.377589	−	0.925973i	$-0.623247\pi$
$479$	−16.5279	−0.755177	−0.377589	+	0.925973i	$0.623247\pi$
$480$	0	0
$481$	−11.1246	−0.507239
$482$	−60.7811	−2.76850
$483$	0	0
$484$	33.9787	1.54449
$485$	15.6525	0.710742
$486$	0	0
$487$	−28.0779	−1.27233	−0.636166	−	0.771552i	$-0.719481\pi$
$487$	−28.0779	−1.27233	−0.636166	+	0.771552i	$0.719481\pi$
$488$	104.130	4.71375
$489$	0	0
$490$	−35.1246	−1.58677
$491$	−15.8902	−0.717115	−0.358557	−	0.933508i	$-0.616731\pi$
$491$	−15.8902	−0.717115	−0.358557	+	0.933508i	$0.616731\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	−6.86474	−0.308859
$495$	0	0
$496$	0	0
$497$	−1.47214	−0.0660343
$498$	0	0
$499$	−18.3848	−0.823016	−0.411508	−	0.911406i	$-0.634998\pi$
$499$	−18.3848	−0.823016	−0.411508	+	0.911406i	$0.634998\pi$
$500$	54.2705	2.42705
$501$	0	0
$502$	−28.8732	−1.28867
$503$	32.8885	1.46643	0.733214	−	0.679998i	$-0.238019\pi$
$503$	32.8885	1.46643	0.733214	+	0.679998i	$0.238019\pi$
$504$	0	0
$505$	5.00000	0.222497
$506$	29.1246	1.29475
$507$	0	0
$508$	46.0501	2.04314
$509$	−7.94510	−0.352160	−0.176080	−	0.984376i	$-0.556342\pi$
$509$	−7.94510	−0.352160	−0.176080	+	0.984376i	$0.556342\pi$
$510$	0	0
$511$	4.24264	0.187683
$512$	40.3050	1.78124
$513$	0	0
$514$	15.8541	0.699294
$515$	−23.9443	−1.05511
$516$	0	0
$517$	41.1884	1.81146
$518$	−11.1074	−0.488030
$519$	0	0
$520$	43.8105	1.92122
$521$	13.4164	0.587784	0.293892	−	0.955839i	$-0.405049\pi$
$521$	13.4164	0.587784	0.293892	+	0.955839i	$0.405049\pi$
$522$	0	0
$523$	−0.618993	−0.0270667	−0.0135333	−	0.999908i	$-0.504308\pi$
$523$	−0.618993	−0.0270667	−0.0135333	+	0.999908i	$0.504308\pi$
$524$	−36.0000	−1.57267
$525$	0	0
$526$	−48.4658	−2.11321
$527$	0	0
$528$	0	0
$529$	−16.1246	−0.701070
$530$	−80.3737	−3.49121
$531$	0	0
$532$	−4.85410	−0.210452
$533$	3.86008	0.167199
$534$	0	0
$535$	3.29180	0.142317
$536$	−44.8328	−1.93648
$537$	0	0
$538$	−5.65685	−0.243884
$539$	−25.4558	−1.09646
$540$	0	0
$541$	25.8328	1.11064	0.555320	−	0.831637i	$-0.312596\pi$
$541$	25.8328	1.11064	0.555320	+	0.831637i	$0.312596\pi$
$542$	−48.6722	−2.09065
$543$	0	0
$544$	−40.1869	−1.72300
$545$	2.88854	0.123732
$546$	0	0
$547$	2.41641	0.103318	0.0516591	−	0.998665i	$-0.483549\pi$
$547$	2.41641	0.103318	0.0516591	+	0.998665i	$0.483549\pi$
$548$	−12.7279	−0.543710
$549$	0	0
$550$	0	0
$551$	−0.540182	−0.0230125
$552$	0	0
$553$	−1.62054	−0.0689126
$554$	26.4574	1.12407
$555$	0	0
$556$	87.2385	3.69974
$557$	−10.6460	−0.451086	−0.225543	−	0.974233i	$-0.572416\pi$
$557$	−10.6460	−0.451086	−0.225543	+	0.974233i	$0.572416\pi$
$558$	0	0
$559$	25.4164	1.07500
$560$	22.0344	0.931125
$561$	0	0
$562$	34.9787	1.47549
$563$	−9.76393	−0.411501	−0.205750	−	0.978605i	$-0.565963\pi$
$563$	−9.76393	−0.411501	−0.205750	+	0.978605i	$0.565963\pi$
$564$	0	0
$565$	−21.8328	−0.918513
$566$	−62.8328	−2.64106
$567$	0	0
$568$	−11.0000	−0.461550
$569$	−11.7264	−0.491595	−0.245798	−	0.969321i	$-0.579050\pi$
$569$	−11.7264	−0.491595	−0.245798	+	0.969321i	$0.579050\pi$
$570$	0	0
$571$	2.62210	0.109731	0.0548657	−	0.998494i	$-0.482527\pi$
$571$	2.62210	0.109731	0.0548657	+	0.998494i	$0.482527\pi$
$572$	54.0000	2.25785
$573$	0	0
$574$	3.85410	0.160867
$575$	0	0
$576$	0	0
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	8.61803	0.358463
$579$	0	0
$580$	5.86319	0.243456
$581$	3.16228	0.131193
$582$	0	0
$583$	−58.2492	−2.41244
$584$	31.7016	1.31182
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−15.7326	−0.649353	−0.324676	−	0.945825i	$-0.605255\pi$
$587$	−15.7326	−0.649353	−0.324676	+	0.945825i	$0.605255\pi$
$588$	0	0
$589$	0	0
$590$	−69.9230	−2.87868
$591$	0	0
$592$	−41.8074	−1.71827
$593$	−5.18034	−0.212731	−0.106366	−	0.994327i	$-0.533921\pi$
$593$	−5.18034	−0.212731	−0.106366	+	0.994327i	$0.533921\pi$
$594$	0	0
$595$	8.27895	0.339404
$596$	65.1246	2.66761
$597$	0	0
$598$	18.0000	0.736075
$599$	−6.05573	−0.247430	−0.123715	−	0.992318i	$-0.539481\pi$
$599$	−6.05573	−0.247430	−0.123715	+	0.992318i	$0.539481\pi$
$600$	0	0
$601$	−18.1784	−0.741514	−0.370757	−	0.928730i	$-0.620902\pi$
$601$	−18.1784	−0.741514	−0.370757	+	0.928730i	$0.620902\pi$
$602$	25.3770	1.03429
$603$	0	0
$604$	−60.7811	−2.47315
$605$	−15.6525	−0.636364
$606$	0	0
$607$	15.1246	0.613889	0.306945	−	0.951727i	$-0.400693\pi$
$607$	15.1246	0.613889	0.306945	+	0.951727i	$0.400693\pi$
$608$	−10.8541	−0.440192
$609$	0	0
$610$	−81.5816	−3.30314
$611$	25.4558	1.02983
$612$	0	0
$613$	−0.588890	−0.0237850	−0.0118925	−	0.999929i	$-0.503786\pi$
$613$	−0.588890	−0.0237850	−0.0118925	+	0.999929i	$0.503786\pi$
$614$	−63.1591	−2.54889
$615$	0	0
$616$	31.7016	1.27729
$617$	6.87539	0.276793	0.138396	−	0.990377i	$-0.455805\pi$
$617$	6.87539	0.276793	0.138396	+	0.990377i	$0.455805\pi$
$618$	0	0
$619$	−25.8384	−1.03853	−0.519267	−	0.854612i	$-0.673795\pi$
$619$	−25.8384	−1.03853	−0.519267	+	0.854612i	$0.673795\pi$
$620$	0	0
$621$	0	0
$622$	66.6869	2.67390
$623$	15.3500	0.614985
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−30.1599	−1.20543
$627$	0	0
$628$	42.2705	1.68678
$629$	−15.7082	−0.626327
$630$	0	0
$631$	23.6290	0.940654	0.470327	−	0.882492i	$-0.344136\pi$
$631$	23.6290	0.940654	0.470327	+	0.882492i	$0.344136\pi$
$632$	−12.1089	−0.481667
$633$	0	0
$634$	68.6869	2.72791
$635$	−21.2132	−0.841820
$636$	0	0
$637$	−15.7326	−0.623347
$638$	6.00000	0.237542
$639$	0	0
$640$	2.43769	0.0963583
$641$	−8.48528	−0.335148	−0.167574	−	0.985859i	$-0.553593\pi$
$641$	−8.48528	−0.335148	−0.167574	+	0.985859i	$0.553593\pi$
$642$	0	0
$643$	4.65530	0.183587	0.0917936	−	0.995778i	$-0.470740\pi$
$643$	4.65530	0.183587	0.0917936	+	0.995778i	$0.470740\pi$
$644$	12.7279	0.501550
$645$	0	0
$646$	−9.69316	−0.381372
$647$	31.7804	1.24942	0.624708	−	0.780858i	$-0.285218\pi$
$647$	31.7804	1.24942	0.624708	+	0.780858i	$0.285218\pi$
$648$	0	0
$649$	−50.6753	−1.98918
$650$	0	0
$651$	0	0
$652$	69.9787	2.74058
$653$	−24.6525	−0.964726	−0.482363	−	0.875971i	$-0.660221\pi$
$653$	−24.6525	−0.964726	−0.482363	+	0.875971i	$0.660221\pi$
$654$	0	0
$655$	16.5836	0.647975
$656$	14.5066	0.566387
$657$	0	0
$658$	25.4164	0.990835
$659$	−24.8197	−0.966837	−0.483418	−	0.875389i	$-0.660605\pi$
$659$	−24.8197	−0.966837	−0.483418	+	0.875389i	$0.660605\pi$
$660$	0	0
$661$	7.83282	0.304661	0.152331	−	0.988330i	$-0.451322\pi$
$661$	7.83282	0.304661	0.152331	+	0.988330i	$0.451322\pi$
$662$	81.9155	3.18374
$663$	0	0
$664$	23.6290	0.916982
$665$	2.23607	0.0867110
$666$	0	0
$667$	1.41641	0.0548435
$668$	−5.24419	−0.202904
$669$	0	0
$670$	35.1246	1.35698
$671$	−59.1246	−2.28248
$672$	0	0
$673$	−39.9805	−1.54114	−0.770568	−	0.637357i	$-0.780027\pi$
$673$	−39.9805	−1.54114	−0.770568	+	0.637357i	$0.780027\pi$
$674$	−40.1869	−1.54794
$675$	0	0
$676$	−29.7295	−1.14344
$677$	−21.2132	−0.815290	−0.407645	−	0.913141i	$-0.633650\pi$
$677$	−21.2132	−0.815290	−0.407645	+	0.913141i	$0.633650\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	61.8614	2.37228
$681$	0	0
$682$	0	0
$683$	41.1803	1.57572	0.787861	−	0.615853i	$-0.211189\pi$
$683$	41.1803	1.57572	0.787861	+	0.615853i	$0.211189\pi$
$684$	0	0
$685$	5.86319	0.224021
$686$	−34.0344	−1.29944
$687$	0	0
$688$	95.5174	3.64157
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−48.4164	−1.84185	−0.920923	−	0.389743i	$-0.872564\pi$
$691$	−48.4164	−1.84185	−0.920923	+	0.389743i	$0.872564\pi$
$692$	−87.3738	−3.32145
$693$	0	0
$694$	56.5384	2.14617
$695$	−40.1869	−1.52437
$696$	0	0
$697$	5.45052	0.206453
$698$	−15.7082	−0.594564
$699$	0	0
$700$	0	0
$701$	3.65248	0.137952	0.0689761	−	0.997618i	$-0.478027\pi$
$701$	3.65248	0.137952	0.0689761	+	0.997618i	$0.478027\pi$
$702$	0	0
$703$	−4.24264	−0.160014
$704$	36.9458	1.39245
$705$	0	0
$706$	−25.0432	−0.942513
$707$	2.23607	0.0840960
$708$	0	0
$709$	41.1884	1.54686	0.773432	−	0.633880i	$-0.218538\pi$
$709$	41.1884	1.54686	0.773432	+	0.633880i	$0.218538\pi$
$710$	8.61803	0.323429
$711$	0	0
$712$	114.697	4.29847
$713$	0	0
$714$	0	0
$715$	−24.8754	−0.930287
$716$	64.4047	2.40692
$717$	0	0
$718$	37.2705	1.39092
$719$	−37.5648	−1.40093	−0.700465	−	0.713687i	$-0.747024\pi$
$719$	−37.5648	−1.40093	−0.700465	+	0.713687i	$0.747024\pi$
$720$	0	0
$721$	−10.7082	−0.398794
$722$	47.1246	1.75380
$723$	0	0
$724$	−42.1900	−1.56798
$725$	0	0
$726$	0	0
$727$	−30.4164	−1.12808	−0.564041	−	0.825747i	$-0.690754\pi$
$727$	−30.4164	−1.12808	−0.564041	+	0.825747i	$0.690754\pi$
$728$	19.5927	0.726152
$729$	0	0
$730$	−24.8369	−0.919253
$731$	35.8885	1.32739
$732$	0	0
$733$	−14.4164	−0.532482	−0.266241	−	0.963906i	$-0.585782\pi$
$733$	−14.4164	−0.532482	−0.266241	+	0.963906i	$0.585782\pi$
$734$	17.4319	0.643424
$735$	0	0
$736$	28.4605	1.04907
$737$	25.4558	0.937678
$738$	0	0
$739$	35.1490	1.29298	0.646489	−	0.762924i	$-0.276237\pi$
$739$	35.1490	1.29298	0.646489	+	0.762924i	$0.276237\pi$
$740$	46.0501	1.69283
$741$	0	0
$742$	−35.9442	−1.31955
$743$	−36.4844	−1.33848	−0.669242	−	0.743045i	$-0.733381\pi$
$743$	−36.4844	−1.33848	−0.669242	+	0.743045i	$0.733381\pi$
$744$	0	0
$745$	−30.0000	−1.09911
$746$	24.3262	0.890647
$747$	0	0
$748$	76.2492	2.78795
$749$	1.47214	0.0537907
$750$	0	0
$751$	−37.5410	−1.36989	−0.684946	−	0.728594i	$-0.740174\pi$
$751$	−37.5410	−1.36989	−0.684946	+	0.728594i	$0.740174\pi$
$752$	95.6656	3.48857
$753$	0	0
$754$	3.70820	0.135045
$755$	27.9991	1.01899
$756$	0	0
$757$	−9.28050	−0.337306	−0.168653	−	0.985676i	$-0.553942\pi$
$757$	−9.28050	−0.337306	−0.168653	+	0.985676i	$0.553942\pi$
$758$	−19.4164	−0.705236
$759$	0	0
$760$	16.7082	0.606070
$761$	40.1869	1.45677	0.728386	−	0.685167i	$-0.240271\pi$
$761$	40.1869	1.45677	0.728386	+	0.685167i	$0.240271\pi$
$762$	0	0
$763$	1.29180	0.0467662
$764$	−47.3951	−1.71470
$765$	0	0
$766$	16.7642	0.605716
$767$	−31.3190	−1.13086
$768$	0	0
$769$	54.1246	1.95178	0.975892	−	0.218255i	$-0.0700365\pi$
$769$	54.1246	1.95178	0.975892	+	0.218255i	$0.0700365\pi$
$770$	−24.8369	−0.895058
$771$	0	0
$772$	−100.520	−3.61778
$773$	−11.1862	−0.402339	−0.201170	−	0.979556i	$-0.564474\pi$
$773$	−11.1862	−0.402339	−0.201170	+	0.979556i	$0.564474\pi$
$774$	0	0
$775$	0	0
$776$	52.3050	1.87764
$777$	0	0
$778$	44.0168	1.57808
$779$	1.47214	0.0527447
$780$	0	0
$781$	6.24574	0.223490
$782$	25.4164	0.908889
$783$	0	0
$784$	−59.1246	−2.11159
$785$	−19.4721	−0.694990
$786$	0	0
$787$	37.5648	1.33904	0.669520	−	0.742794i	$-0.266500\pi$
$787$	37.5648	1.33904	0.669520	+	0.742794i	$0.266500\pi$
$788$	−5.24419	−0.186817
$789$	0	0
$790$	9.48683	0.337526
$791$	−9.76393	−0.347165
$792$	0	0
$793$	−36.5410	−1.29761
$794$	−38.5066	−1.36655
$795$	0	0
$796$	−112.694	−3.99434
$797$	−13.7295	−0.486323	−0.243161	−	0.969986i	$-0.578184\pi$
$797$	−13.7295	−0.486323	−0.243161	+	0.969986i	$0.578184\pi$
$798$	0	0
$799$	35.9442	1.27162
$800$	0	0
$801$	0	0
$802$	1.41421	0.0499376
$803$	−18.0000	−0.635206
$804$	0	0
$805$	−5.86319	−0.206650
$806$	0	0
$807$	0	0
$808$	16.7082	0.587793
$809$	47.5130	1.67047	0.835234	−	0.549895i	$-0.185332\pi$
$809$	47.5130	1.67047	0.835234	+	0.549895i	$0.185332\pi$
$810$	0	0
$811$	−16.5836	−0.582329	−0.291164	−	0.956673i	$-0.594043\pi$
$811$	−16.5836	−0.582329	−0.291164	+	0.956673i	$0.594043\pi$
$812$	2.62210	0.0920175
$813$	0	0
$814$	47.1246	1.65172
$815$	−32.2361	−1.12918
$816$	0	0
$817$	9.69316	0.339121
$818$	78.7532	2.75354
$819$	0	0
$820$	−15.9787	−0.558001
$821$	23.8353	0.831858	0.415929	−	0.909397i	$-0.363457\pi$
$821$	23.8353	0.831858	0.415929	+	0.909397i	$0.363457\pi$
$822$	0	0
$823$	13.5231	0.471387	0.235694	−	0.971827i	$-0.424264\pi$
$823$	13.5231	0.471387	0.235694	+	0.971827i	$0.424264\pi$
$824$	−80.0132	−2.78739
$825$	0	0
$826$	−31.2705	−1.08804
$827$	−18.3547	−0.638255	−0.319127	−	0.947712i	$-0.603390\pi$
$827$	−18.3547	−0.638255	−0.319127	+	0.947712i	$0.603390\pi$
$828$	0	0
$829$	−51.0879	−1.77436	−0.887178	−	0.461427i	$-0.847338\pi$
$829$	−51.0879	−1.77436	−0.887178	+	0.461427i	$0.847338\pi$
$830$	−18.5123	−0.642571
$831$	0	0
$832$	22.8337	0.791618
$833$	−22.2148	−0.769696
$834$	0	0
$835$	2.41577	0.0836010
$836$	20.5942	0.712266
$837$	0	0
$838$	80.1033	2.76712
$839$	19.4164	0.670329	0.335164	−	0.942160i	$-0.391208\pi$
$839$	19.4164	0.670329	0.335164	+	0.942160i	$0.391208\pi$
$840$	0	0
$841$	−28.7082	−0.989938
$842$	−69.1591	−2.38338
$843$	0	0
$844$	−24.2705	−0.835425
$845$	13.6950	0.471124
$846$	0	0
$847$	−7.00000	−0.240523
$848$	−135.292	−4.64593
$849$	0	0
$850$	0	0
$851$	11.1246	0.381347
$852$	0	0
$853$	−32.2918	−1.10565	−0.552825	−	0.833297i	$-0.686450\pi$
$853$	−32.2918	−1.10565	−0.552825	+	0.833297i	$0.686450\pi$
$854$	−36.4844	−1.24847
$855$	0	0
$856$	11.0000	0.375972
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	41.6011	1.41941	0.709705	−	0.704499i	$-0.248828\pi$
$859$	41.6011	1.41941	0.709705	+	0.704499i	$0.248828\pi$
$860$	−105.211	−3.58765
$861$	0	0
$862$	−93.9574	−3.20020
$863$	−31.0826	−1.05806	−0.529032	−	0.848602i	$-0.677445\pi$
$863$	−31.0826	−1.05806	−0.529032	+	0.848602i	$0.677445\pi$
$864$	0	0
$865$	40.2492	1.36851
$866$	−67.6458	−2.29870
$867$	0	0
$868$	0	0
$869$	6.87539	0.233232
$870$	0	0
$871$	15.7326	0.533078
$872$	9.65248	0.326874
$873$	0	0
$874$	6.86474	0.232203
$875$	−11.1803	−0.377964
$876$	0	0
$877$	−28.7082	−0.969407	−0.484704	−	0.874678i	$-0.661073\pi$
$877$	−28.7082	−0.969407	−0.484704	+	0.874678i	$0.661073\pi$
$878$	65.4508	2.20886
$879$	0	0
$880$	−93.4842	−3.15135
$881$	7.24730	0.244168	0.122084	−	0.992520i	$-0.461042\pi$
$881$	7.24730	0.244168	0.122084	+	0.992520i	$0.461042\pi$
$882$	0	0
$883$	33.9411	1.14221	0.571105	−	0.820877i	$-0.306515\pi$
$883$	33.9411	1.14221	0.571105	+	0.820877i	$0.306515\pi$
$884$	47.1246	1.58497
$885$	0	0
$886$	15.8541	0.532629
$887$	−25.3607	−0.851528	−0.425764	−	0.904834i	$-0.639995\pi$
$887$	−25.3607	−0.851528	−0.425764	+	0.904834i	$0.639995\pi$
$888$	0	0
$889$	−9.48683	−0.318178
$890$	−89.8606	−3.01213
$891$	0	0
$892$	−78.3706	−2.62404
$893$	9.70820	0.324873
$894$	0	0
$895$	−29.6684	−0.991705
$896$	1.09017	0.0364200
$897$	0	0
$898$	−88.2400	−2.94461
$899$	0	0
$900$	0	0
$901$	−50.8328	−1.69349
$902$	−16.3516	−0.544448
$903$	0	0
$904$	−72.9574	−2.42653
$905$	19.4350	0.646042
$906$	0	0
$907$	11.5410	0.383213	0.191607	−	0.981472i	$-0.438630\pi$
$907$	11.5410	0.383213	0.191607	+	0.981472i	$0.438630\pi$
$908$	−11.1246	−0.369183
$909$	0	0
$910$	−15.3500	−0.508848
$911$	−26.9188	−0.891859	−0.445929	−	0.895068i	$-0.647127\pi$
$911$	−26.9188	−0.891859	−0.445929	+	0.895068i	$0.647127\pi$
$912$	0	0
$913$	−13.4164	−0.444018
$914$	29.6197	0.979732
$915$	0	0
$916$	−20.5942	−0.680452
$917$	7.41641	0.244911
$918$	0	0
$919$	−42.5410	−1.40330	−0.701649	−	0.712522i	$-0.747553\pi$
$919$	−42.5410	−1.40330	−0.701649	+	0.712522i	$0.747553\pi$
$920$	−43.8105	−1.44439
$921$	0	0
$922$	−13.5231	−0.445361
$923$	3.86008	0.127056
$924$	0	0
$925$	0	0
$926$	−10.0270	−0.329508
$927$	0	0
$928$	5.86319	0.192468
$929$	10.6460	0.349284	0.174642	−	0.984632i	$-0.444123\pi$
$929$	10.6460	0.349284	0.174642	+	0.984632i	$0.444123\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	65.3951	2.14209
$933$	0	0
$934$	−40.9787	−1.34086
$935$	−35.1246	−1.14870
$936$	0	0
$937$	24.2492	0.792188	0.396094	−	0.918210i	$-0.370366\pi$
$937$	24.2492	0.792188	0.396094	+	0.918210i	$0.370366\pi$
$938$	15.7082	0.512891
$939$	0	0
$940$	−105.374	−3.43692
$941$	25.5347	0.832406	0.416203	−	0.909272i	$-0.363361\pi$
$941$	25.5347	0.832406	0.416203	+	0.909272i	$0.363361\pi$
$942$	0	0
$943$	−3.86008	−0.125702
$944$	−117.700	−3.83081
$945$	0	0
$946$	−107.666	−3.50051
$947$	40.1869	1.30590	0.652949	−	0.757402i	$-0.273532\pi$
$947$	40.1869	1.30590	0.652949	+	0.757402i	$0.273532\pi$
$948$	0	0
$949$	−11.1246	−0.361120
$950$	0	0
$951$	0	0
$952$	27.6653	0.896637
$953$	18.6699	0.604778	0.302389	−	0.953185i	$-0.402216\pi$
$953$	18.6699	0.604778	0.302389	+	0.953185i	$0.402216\pi$
$954$	0	0
$955$	21.8328	0.706493
$956$	58.7780	1.90102
$957$	0	0
$958$	43.2705	1.39801
$959$	2.62210	0.0846719
$960$	0	0
$961$	0	0
$962$	29.1246	0.939015
$963$	0	0
$964$	112.694	3.62964
$965$	46.3050	1.49061
$966$	0	0
$967$	−44.0168	−1.41549	−0.707743	−	0.706470i	$-0.750287\pi$
$967$	−44.0168	−1.41549	−0.707743	+	0.706470i	$0.750287\pi$
$968$	−52.3050	−1.68114
$969$	0	0
$970$	−40.9787	−1.31575
$971$	−53.8885	−1.72937	−0.864683	−	0.502318i	$-0.832481\pi$
$971$	−53.8885	−1.72937	−0.864683	+	0.502318i	$0.832481\pi$
$972$	0	0
$973$	−17.9721	−0.576160
$974$	73.5090	2.35538
$975$	0	0
$976$	−137.325	−4.39566
$977$	42.4853	1.35922	0.679612	−	0.733571i	$-0.262148\pi$
$977$	42.4853	1.35922	0.679612	+	0.733571i	$0.262148\pi$
$978$	0	0
$979$	−65.1246	−2.08139
$980$	65.1246	2.08033
$981$	0	0
$982$	41.6011	1.32754
$983$	53.9163	1.71966	0.859832	−	0.510577i	$-0.170568\pi$
$983$	53.9163	1.71966	0.859832	+	0.510577i	$0.170568\pi$
$984$	0	0
$985$	2.41577	0.0769727
$986$	5.23607	0.166750
$987$	0	0
$988$	12.7279	0.404929
$989$	−25.4164	−0.808195
$990$	0	0
$991$	13.9358	0.442685	0.221343	−	0.975196i	$-0.428956\pi$
$991$	13.9358	0.442685	0.221343	+	0.975196i	$0.428956\pi$
$992$	0	0
$993$	0	0
$994$	3.85410	0.122245
$995$	51.9132	1.64576
$996$	0	0
$997$	−58.6656	−1.85796	−0.928980	−	0.370131i	$-0.879313\pi$
$997$	−58.6656	−1.85796	−0.928980	+	0.370131i	$0.879313\pi$
$998$	48.1320	1.52359
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.r.1.2		4
3.2	odd	2		961.2.a.h.1.4	yes	4
31.30	odd	2	inner	8649.2.a.r.1.1		4
93.2	odd	10		961.2.d.j.531.2		8
93.5	odd	6		961.2.c.h.521.3		8
93.8	odd	10		961.2.d.h.374.1		8
93.11	even	30		961.2.g.i.338.2		16
93.14	odd	30		961.2.g.i.816.1		16
93.17	even	30		961.2.g.i.816.2		16
93.20	odd	30		961.2.g.i.338.1		16
93.23	even	10		961.2.d.h.374.2		8
93.26	even	6		961.2.c.h.521.4		8
93.29	even	10		961.2.d.j.531.1		8
93.35	odd	10		961.2.d.h.388.1		8
93.38	odd	30		961.2.g.i.235.2		16
93.41	odd	30		961.2.g.p.844.1		16
93.44	even	30		961.2.g.p.448.1		16
93.47	odd	10		961.2.d.j.628.2		8
93.50	odd	30		961.2.g.p.547.2		16
93.53	even	30		961.2.g.i.732.1		16
93.56	odd	6		961.2.c.h.439.3		8
93.59	odd	30		961.2.g.p.846.1		16
93.65	even	30		961.2.g.p.846.2		16
93.68	even	6		961.2.c.h.439.4		8
93.71	odd	30		961.2.g.i.732.2		16
93.74	even	30		961.2.g.p.547.1		16
93.77	even	10		961.2.d.j.628.1		8
93.80	odd	30		961.2.g.p.448.2		16
93.83	even	30		961.2.g.p.844.2		16
93.86	even	30		961.2.g.i.235.1		16
93.89	even	10		961.2.d.h.388.2		8
93.92	even	2		961.2.a.h.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.h.1.3	✓	4	93.92	even	2
961.2.a.h.1.4	yes	4	3.2	odd	2
961.2.c.h.439.3		8	93.56	odd	6
961.2.c.h.439.4		8	93.68	even	6
961.2.c.h.521.3		8	93.5	odd	6
961.2.c.h.521.4		8	93.26	even	6
961.2.d.h.374.1		8	93.8	odd	10
961.2.d.h.374.2		8	93.23	even	10
961.2.d.h.388.1		8	93.35	odd	10
961.2.d.h.388.2		8	93.89	even	10
961.2.d.j.531.1		8	93.29	even	10
961.2.d.j.531.2		8	93.2	odd	10
961.2.d.j.628.1		8	93.77	even	10
961.2.d.j.628.2		8	93.47	odd	10
961.2.g.i.235.1		16	93.86	even	30
961.2.g.i.235.2		16	93.38	odd	30
961.2.g.i.338.1		16	93.20	odd	30
961.2.g.i.338.2		16	93.11	even	30
961.2.g.i.732.1		16	93.53	even	30
961.2.g.i.732.2		16	93.71	odd	30
961.2.g.i.816.1		16	93.14	odd	30
961.2.g.i.816.2		16	93.17	even	30
961.2.g.p.448.1		16	93.44	even	30
961.2.g.p.448.2		16	93.80	odd	30
961.2.g.p.547.1		16	93.74	even	30
961.2.g.p.547.2		16	93.50	odd	30
961.2.g.p.844.1		16	93.41	odd	30
961.2.g.p.844.2		16	93.83	even	30
961.2.g.p.846.1		16	93.59	odd	30
961.2.g.p.846.2		16	93.65	even	30
8649.2.a.r.1.1		4	31.30	odd	2	inner
8649.2.a.r.1.2		4	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} +4.85410 q^{4} -2.23607 q^{5} -1.00000 q^{7} -7.47214 q^{8} +5.85410 q^{10} +4.24264 q^{11} +2.62210 q^{13} +2.61803 q^{14} +9.85410 q^{16} +3.70246 q^{17} +1.00000 q^{19} -10.8541 q^{20} -11.1074 q^{22} -2.62210 q^{23} -6.86474 q^{26} -4.85410 q^{28} -0.540182 q^{29} -10.8541 q^{32} -9.69316 q^{34} +2.23607 q^{35} -4.24264 q^{37} -2.61803 q^{38} +16.7082 q^{40} +1.47214 q^{41} +9.69316 q^{43} +20.5942 q^{44} +6.86474 q^{46} +9.70820 q^{47} -6.00000 q^{49} +12.7279 q^{52} -13.7295 q^{53} -9.48683 q^{55} +7.47214 q^{56} +1.41421 q^{58} -11.9443 q^{59} -13.9358 q^{61} +8.70820 q^{64} -5.86319 q^{65} +6.00000 q^{67} +17.9721 q^{68} -5.85410 q^{70} +1.47214 q^{71} -4.24264 q^{73} +11.1074 q^{74} +4.85410 q^{76} -4.24264 q^{77} +1.62054 q^{79} -22.0344 q^{80} -3.85410 q^{82} -3.16228 q^{83} -8.27895 q^{85} -25.3770 q^{86} -31.7016 q^{88} -15.3500 q^{89} -2.62210 q^{91} -12.7279 q^{92} -25.4164 q^{94} -2.23607 q^{95} -7.00000 q^{97} +15.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 6 q^{4} - 4 q^{7} - 12 q^{8} + 10 q^{10} + 6 q^{14} + 26 q^{16} + 4 q^{19} - 30 q^{20} - 6 q^{28} - 30 q^{32} - 6 q^{38} + 40 q^{40} - 12 q^{41} + 12 q^{47} - 24 q^{49} + 12 q^{56} - 12 q^{59}+ \cdots + 36 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 + 6 * q^4 - 4 * q^7 - 12 * q^8 + 10 * q^10 + 6 * q^14 + 26 * q^16 + 4 * q^19 - 30 * q^20 - 6 * q^28 - 30 * q^32 - 6 * q^38 + 40 * q^40 - 12 * q^41 + 12 * q^47 - 24 * q^49 + 12 * q^56 - 12 * q^59 + 8 * q^64 + 24 * q^67 - 10 * q^70 - 12 * q^71 + 6 * q^76 - 30 * q^80 - 2 * q^82 - 48 * q^94 - 28 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

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