LMFDB - Embedded newform 1134.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(1,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, 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("1134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.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:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	1134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.37228 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.37228 q^{10} +4.37228 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.37228 q^{17} +5.00000 q^{19} -1.37228 q^{20} -4.37228 q^{22} -7.37228 q^{23} -3.11684 q^{25} -2.00000 q^{26} +1.00000 q^{28} +2.74456 q^{29} +2.00000 q^{31} -1.00000 q^{32} +4.37228 q^{34} -1.37228 q^{35} +2.00000 q^{37} -5.00000 q^{38} +1.37228 q^{40} +10.3723 q^{41} +9.11684 q^{43} +4.37228 q^{44} +7.37228 q^{46} +1.00000 q^{49} +3.11684 q^{50} +2.00000 q^{52} +2.74456 q^{53} -6.00000 q^{55} -1.00000 q^{56} -2.74456 q^{58} +7.11684 q^{59} -14.1168 q^{61} -2.00000 q^{62} +1.00000 q^{64} -2.74456 q^{65} +15.1168 q^{67} -4.37228 q^{68} +1.37228 q^{70} +10.1168 q^{71} -5.11684 q^{73} -2.00000 q^{74} +5.00000 q^{76} +4.37228 q^{77} +12.1168 q^{79} -1.37228 q^{80} -10.3723 q^{82} -5.48913 q^{83} +6.00000 q^{85} -9.11684 q^{86} -4.37228 q^{88} +3.25544 q^{89} +2.00000 q^{91} -7.37228 q^{92} -6.86141 q^{95} +9.11684 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{7} - 2 q^{8} - 3 q^{10} + 3 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 3 q^{17} + 10 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} + 11 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^7 - 2 * q^8 - 3 * q^10 + 3 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 3 * q^17 + 10 * q^19 + 3 * q^20 - 3 * q^22 - 9 * q^23 + 11 * q^25 - 4 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - 2 * q^32 + 3 * q^34 + 3 * q^35 + 4 * q^37 - 10 * q^38 - 3 * q^40 + 15 * q^41 + q^43 + 3 * q^44 + 9 * q^46 + 2 * q^49 - 11 * q^50 + 4 * q^52 - 6 * q^53 - 12 * q^55 - 2 * q^56 + 6 * q^58 - 3 * q^59 - 11 * q^61 - 4 * q^62 + 2 * q^64 + 6 * q^65 + 13 * q^67 - 3 * q^68 - 3 * q^70 + 3 * q^71 + 7 * q^73 - 4 * q^74 + 10 * q^76 + 3 * q^77 + 7 * q^79 + 3 * q^80 - 15 * q^82 + 12 * q^83 + 12 * q^85 - q^86 - 3 * q^88 + 18 * q^89 + 4 * q^91 - 9 * q^92 + 15 * q^95 + q^97 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.37228	−0.613703	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	−1.37228	−0.613703	−0.306851	+	0.951757i	$0.599275\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.37228	0.433953
$11$	4.37228	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$11$	4.37228	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.37228	−1.06043	−0.530217	−	0.847862i	$-0.677890\pi$
$17$	−4.37228	−1.06043	−0.530217	+	0.847862i	$0.677890\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.37228	−0.306851
$21$	0	0
$22$	−4.37228	−0.932174
$23$	−7.37228	−1.53723	−0.768613	−	0.639713i	$-0.779053\pi$
$23$	−7.37228	−1.53723	−0.768613	+	0.639713i	$0.779053\pi$
$24$	0	0
$25$	−3.11684	−0.623369
$26$	−2.00000	−0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	2.74456	0.509652	0.254826	−	0.966987i	$-0.417982\pi$
$29$	2.74456	0.509652	0.254826	+	0.966987i	$0.417982\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.37228	0.749840
$35$	−1.37228	−0.231958
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−5.00000	−0.811107
$39$	0	0
$40$	1.37228	0.216977
$41$	10.3723	1.61988	0.809939	−	0.586514i	$-0.199500\pi$
$41$	10.3723	1.61988	0.809939	+	0.586514i	$0.199500\pi$
$42$	0	0
$43$	9.11684	1.39031	0.695153	−	0.718862i	$-0.255337\pi$
$43$	9.11684	1.39031	0.695153	+	0.718862i	$0.255337\pi$
$44$	4.37228	0.659146
$45$	0	0
$46$	7.37228	1.08698
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.11684	0.440788
$51$	0	0
$52$	2.00000	0.277350
$53$	2.74456	0.376995	0.188497	−	0.982074i	$-0.439638\pi$
$53$	2.74456	0.376995	0.188497	+	0.982074i	$0.439638\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.74456	−0.360379
$59$	7.11684	0.926534	0.463267	−	0.886219i	$-0.346677\pi$
$59$	7.11684	0.926534	0.463267	+	0.886219i	$0.346677\pi$
$60$	0	0
$61$	−14.1168	−1.80748	−0.903738	−	0.428085i	$-0.859188\pi$
$61$	−14.1168	−1.80748	−0.903738	+	0.428085i	$0.859188\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.74456	−0.340421
$66$	0	0
$67$	15.1168	1.84682	0.923408	−	0.383819i	$-0.125391\pi$
$67$	15.1168	1.84682	0.923408	+	0.383819i	$0.125391\pi$
$68$	−4.37228	−0.530217
$69$	0	0
$70$	1.37228	0.164019
$71$	10.1168	1.20065	0.600324	−	0.799757i	$-0.295038\pi$
$71$	10.1168	1.20065	0.600324	+	0.799757i	$0.295038\pi$
$72$	0	0
$73$	−5.11684	−0.598881	−0.299441	−	0.954115i	$-0.596800\pi$
$73$	−5.11684	−0.598881	−0.299441	+	0.954115i	$0.596800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	5.00000	0.573539
$77$	4.37228	0.498268
$78$	0	0
$79$	12.1168	1.36325	0.681626	−	0.731701i	$-0.261273\pi$
$79$	12.1168	1.36325	0.681626	+	0.731701i	$0.261273\pi$
$80$	−1.37228	−0.153426
$81$	0	0
$82$	−10.3723	−1.14543
$83$	−5.48913	−0.602510	−0.301255	−	0.953544i	$-0.597406\pi$
$83$	−5.48913	−0.602510	−0.301255	+	0.953544i	$0.597406\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	−9.11684	−0.983095
$87$	0	0
$88$	−4.37228	−0.466087
$89$	3.25544	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$89$	3.25544	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−7.37228	−0.768613
$93$	0	0
$94$	0	0
$95$	−6.86141	−0.703965
$96$	0	0
$97$	9.11684	0.925675	0.462838	−	0.886443i	$-0.346831\pi$
$97$	9.11684	0.925675	0.462838	+	0.886443i	$0.346831\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−3.11684	−0.311684
$101$	7.37228	0.733569	0.366785	−	0.930306i	$-0.380459\pi$
$101$	7.37228	0.733569	0.366785	+	0.930306i	$0.380459\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−2.74456	−0.266575
$107$	−1.62772	−0.157358	−0.0786788	−	0.996900i	$-0.525070\pi$
$107$	−1.62772	−0.157358	−0.0786788	+	0.996900i	$0.525070\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	1.00000	0.0944911
$113$	1.37228	0.129093	0.0645467	−	0.997915i	$-0.479440\pi$
$113$	1.37228	0.129093	0.0645467	+	0.997915i	$0.479440\pi$
$114$	0	0
$115$	10.1168	0.943401
$116$	2.74456	0.254826
$117$	0	0
$118$	−7.11684	−0.655159
$119$	−4.37228	−0.400806
$120$	0	0
$121$	8.11684	0.737895
$122$	14.1168	1.27808
$123$	0	0
$124$	2.00000	0.179605
$125$	11.1386	0.996266
$126$	0	0
$127$	−14.1168	−1.25267	−0.626334	−	0.779555i	$-0.715445\pi$
$127$	−14.1168	−1.25267	−0.626334	+	0.779555i	$0.715445\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.74456	0.240714
$131$	−7.37228	−0.644119	−0.322060	−	0.946719i	$-0.604375\pi$
$131$	−7.37228	−0.644119	−0.322060	+	0.946719i	$0.604375\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	−15.1168	−1.30590
$135$	0	0
$136$	4.37228	0.374920
$137$	16.3723	1.39878	0.699389	−	0.714741i	$-0.253455\pi$
$137$	16.3723	1.39878	0.699389	+	0.714741i	$0.253455\pi$
$138$	0	0
$139$	−21.2337	−1.80102	−0.900509	−	0.434837i	$-0.856806\pi$
$139$	−21.2337	−1.80102	−0.900509	+	0.434837i	$0.856806\pi$
$140$	−1.37228	−0.115979
$141$	0	0
$142$	−10.1168	−0.848987
$143$	8.74456	0.731257
$144$	0	0
$145$	−3.76631	−0.312775
$146$	5.11684	0.423473
$147$	0	0
$148$	2.00000	0.164399
$149$	14.7446	1.20792	0.603961	−	0.797014i	$-0.293588\pi$
$149$	14.7446	1.20792	0.603961	+	0.797014i	$0.293588\pi$
$150$	0	0
$151$	−8.11684	−0.660539	−0.330270	−	0.943887i	$-0.607140\pi$
$151$	−8.11684	−0.660539	−0.330270	+	0.943887i	$0.607140\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	−4.37228	−0.352328
$155$	−2.74456	−0.220449
$156$	0	0
$157$	−8.11684	−0.647795	−0.323897	−	0.946092i	$-0.604993\pi$
$157$	−8.11684	−0.647795	−0.323897	+	0.946092i	$0.604993\pi$
$158$	−12.1168	−0.963964
$159$	0	0
$160$	1.37228	0.108488
$161$	−7.37228	−0.581017
$162$	0	0
$163$	16.2337	1.27152	0.635760	−	0.771887i	$-0.280687\pi$
$163$	16.2337	1.27152	0.635760	+	0.771887i	$0.280687\pi$
$164$	10.3723	0.809939
$165$	0	0
$166$	5.48913	0.426039
$167$	17.4891	1.35335	0.676675	−	0.736282i	$-0.263420\pi$
$167$	17.4891	1.35335	0.676675	+	0.736282i	$0.263420\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	0	0
$172$	9.11684	0.695153
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−3.11684	−0.235611
$176$	4.37228	0.329573
$177$	0	0
$178$	−3.25544	−0.244005
$179$	14.7446	1.10206	0.551030	−	0.834485i	$-0.314235\pi$
$179$	14.7446	1.10206	0.551030	+	0.834485i	$0.314235\pi$
$180$	0	0
$181$	18.1168	1.34661	0.673307	−	0.739363i	$-0.264873\pi$
$181$	18.1168	1.34661	0.673307	+	0.739363i	$0.264873\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	7.37228	0.543492
$185$	−2.74456	−0.201784
$186$	0	0
$187$	−19.1168	−1.39796
$188$	0	0
$189$	0	0
$190$	6.86141	0.497779
$191$	−1.88316	−0.136260	−0.0681302	−	0.997676i	$-0.521703\pi$
$191$	−1.88316	−0.136260	−0.0681302	+	0.997676i	$0.521703\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	−9.11684	−0.654551
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	3.11684	0.220394
$201$	0	0
$202$	−7.37228	−0.518712
$203$	2.74456	0.192631
$204$	0	0
$205$	−14.2337	−0.994124
$206$	10.0000	0.696733
$207$	0	0
$208$	2.00000	0.138675
$209$	21.8614	1.51219
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	2.74456	0.188497
$213$	0	0
$214$	1.62772	0.111269
$215$	−12.5109	−0.853235
$216$	0	0
$217$	2.00000	0.135769
$218$	−14.0000	−0.948200
$219$	0	0
$220$	−6.00000	−0.404520
$221$	−8.74456	−0.588223
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−1.37228	−0.0912828
$227$	−23.7446	−1.57598	−0.787991	−	0.615687i	$-0.788879\pi$
$227$	−23.7446	−1.57598	−0.787991	+	0.615687i	$0.788879\pi$
$228$	0	0
$229$	−20.1168	−1.32936	−0.664679	−	0.747129i	$-0.731432\pi$
$229$	−20.1168	−1.32936	−0.664679	+	0.747129i	$0.731432\pi$
$230$	−10.1168	−0.667085
$231$	0	0
$232$	−2.74456	−0.180189
$233$	−11.7446	−0.769412	−0.384706	−	0.923039i	$-0.625697\pi$
$233$	−11.7446	−0.769412	−0.384706	+	0.923039i	$0.625697\pi$
$234$	0	0
$235$	0	0
$236$	7.11684	0.463267
$237$	0	0
$238$	4.37228	0.283413
$239$	−18.8614	−1.22004	−0.610021	−	0.792385i	$-0.708839\pi$
$239$	−18.8614	−1.22004	−0.610021	+	0.792385i	$0.708839\pi$
$240$	0	0
$241$	0.883156	0.0568891	0.0284445	−	0.999595i	$-0.490945\pi$
$241$	0.883156	0.0568891	0.0284445	+	0.999595i	$0.490945\pi$
$242$	−8.11684	−0.521770
$243$	0	0
$244$	−14.1168	−0.903738
$245$	−1.37228	−0.0876718
$246$	0	0
$247$	10.0000	0.636285
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−11.1386	−0.704467
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	−32.2337	−2.02651
$254$	14.1168	0.885770
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.8614	1.36368	0.681839	−	0.731503i	$-0.261181\pi$
$257$	21.8614	1.36368	0.681839	+	0.731503i	$0.261181\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	−2.74456	−0.170211
$261$	0	0
$262$	7.37228	0.455461
$263$	13.3723	0.824570	0.412285	−	0.911055i	$-0.364731\pi$
$263$	13.3723	0.824570	0.412285	+	0.911055i	$0.364731\pi$
$264$	0	0
$265$	−3.76631	−0.231363
$266$	−5.00000	−0.306570
$267$	0	0
$268$	15.1168	0.923408
$269$	−7.37228	−0.449496	−0.224748	−	0.974417i	$-0.572156\pi$
$269$	−7.37228	−0.449496	−0.224748	+	0.974417i	$0.572156\pi$
$270$	0	0
$271$	−18.2337	−1.10762	−0.553809	−	0.832644i	$-0.686826\pi$
$271$	−18.2337	−1.10762	−0.553809	+	0.832644i	$0.686826\pi$
$272$	−4.37228	−0.265108
$273$	0	0
$274$	−16.3723	−0.989086
$275$	−13.6277	−0.821782
$276$	0	0
$277$	22.2337	1.33589	0.667946	−	0.744209i	$-0.267174\pi$
$277$	22.2337	1.33589	0.667946	+	0.744209i	$0.267174\pi$
$278$	21.2337	1.27351
$279$	0	0
$280$	1.37228	0.0820095
$281$	10.6277	0.633997	0.316998	−	0.948426i	$-0.397325\pi$
$281$	10.6277	0.633997	0.316998	+	0.948426i	$0.397325\pi$
$282$	0	0
$283$	9.88316	0.587493	0.293746	−	0.955883i	$-0.405098\pi$
$283$	9.88316	0.587493	0.293746	+	0.955883i	$0.405098\pi$
$284$	10.1168	0.600324
$285$	0	0
$286$	−8.74456	−0.517077
$287$	10.3723	0.612256
$288$	0	0
$289$	2.11684	0.124520
$290$	3.76631	0.221165
$291$	0	0
$292$	−5.11684	−0.299441
$293$	−4.62772	−0.270354	−0.135177	−	0.990821i	$-0.543160\pi$
$293$	−4.62772	−0.270354	−0.135177	+	0.990821i	$0.543160\pi$
$294$	0	0
$295$	−9.76631	−0.568617
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−14.7446	−0.854130
$299$	−14.7446	−0.852700
$300$	0	0
$301$	9.11684	0.525486
$302$	8.11684	0.467072
$303$	0	0
$304$	5.00000	0.286770
$305$	19.3723	1.10925
$306$	0	0
$307$	−13.0000	−0.741949	−0.370975	−	0.928643i	$-0.620976\pi$
$307$	−13.0000	−0.741949	−0.370975	+	0.928643i	$0.620976\pi$
$308$	4.37228	0.249134
$309$	0	0
$310$	2.74456	0.155881
$311$	−26.2337	−1.48758	−0.743788	−	0.668416i	$-0.766973\pi$
$311$	−26.2337	−1.48758	−0.743788	+	0.668416i	$0.766973\pi$
$312$	0	0
$313$	−2.88316	−0.162966	−0.0814828	−	0.996675i	$-0.525966\pi$
$313$	−2.88316	−0.162966	−0.0814828	+	0.996675i	$0.525966\pi$
$314$	8.11684	0.458060
$315$	0	0
$316$	12.1168	0.681626
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	−1.37228	−0.0767129
$321$	0	0
$322$	7.37228	0.410841
$323$	−21.8614	−1.21640
$324$	0	0
$325$	−6.23369	−0.345783
$326$	−16.2337	−0.899101
$327$	0	0
$328$	−10.3723	−0.572713
$329$	0	0
$330$	0	0
$331$	−12.2337	−0.672424	−0.336212	−	0.941786i	$-0.609146\pi$
$331$	−12.2337	−0.672424	−0.336212	+	0.941786i	$0.609146\pi$
$332$	−5.48913	−0.301255
$333$	0	0
$334$	−17.4891	−0.956962
$335$	−20.7446	−1.13340
$336$	0	0
$337$	9.11684	0.496626	0.248313	−	0.968680i	$-0.420124\pi$
$337$	9.11684	0.496626	0.248313	+	0.968680i	$0.420124\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	6.00000	0.325396
$341$	8.74456	0.473545
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.11684	−0.491547
$345$	0	0
$346$	6.00000	0.322562
$347$	7.11684	0.382052	0.191026	−	0.981585i	$-0.438818\pi$
$347$	7.11684	0.382052	0.191026	+	0.981585i	$0.438818\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	3.11684	0.166602
$351$	0	0
$352$	−4.37228	−0.233043
$353$	−7.62772	−0.405983	−0.202991	−	0.979181i	$-0.565066\pi$
$353$	−7.62772	−0.405983	−0.202991	+	0.979181i	$0.565066\pi$
$354$	0	0
$355$	−13.8832	−0.736841
$356$	3.25544	0.172538
$357$	0	0
$358$	−14.7446	−0.779274
$359$	−6.86141	−0.362131	−0.181066	−	0.983471i	$-0.557955\pi$
$359$	−6.86141	−0.362131	−0.181066	+	0.983471i	$0.557955\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−18.1168	−0.952200
$363$	0	0
$364$	2.00000	0.104828
$365$	7.02175	0.367535
$366$	0	0
$367$	22.2337	1.16059	0.580295	−	0.814407i	$-0.302937\pi$
$367$	22.2337	1.16059	0.580295	+	0.814407i	$0.302937\pi$
$368$	−7.37228	−0.384307
$369$	0	0
$370$	2.74456	0.142683
$371$	2.74456	0.142491
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	19.1168	0.988508
$375$	0	0
$376$	0	0
$377$	5.48913	0.282704
$378$	0	0
$379$	9.11684	0.468301	0.234150	−	0.972200i	$-0.424769\pi$
$379$	9.11684	0.468301	0.234150	+	0.972200i	$0.424769\pi$
$380$	−6.86141	−0.351983
$381$	0	0
$382$	1.88316	0.0963506
$383$	−21.2554	−1.08610	−0.543051	−	0.839700i	$-0.682731\pi$
$383$	−21.2554	−1.08610	−0.543051	+	0.839700i	$0.682731\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	7.00000	0.356291
$387$	0	0
$388$	9.11684	0.462838
$389$	−34.9783	−1.77347	−0.886734	−	0.462280i	$-0.847031\pi$
$389$	−34.9783	−1.77347	−0.886734	+	0.462280i	$0.847031\pi$
$390$	0	0
$391$	32.2337	1.63013
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−16.6277	−0.836631
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	−3.11684	−0.155842
$401$	−0.255437	−0.0127559	−0.00637797	−	0.999980i	$-0.502030\pi$
$401$	−0.255437	−0.0127559	−0.00637797	+	0.999980i	$0.502030\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	7.37228	0.366785
$405$	0	0
$406$	−2.74456	−0.136210
$407$	8.74456	0.433452
$408$	0	0
$409$	29.3505	1.45129	0.725645	−	0.688069i	$-0.241541\pi$
$409$	29.3505	1.45129	0.725645	+	0.688069i	$0.241541\pi$
$410$	14.2337	0.702952
$411$	0	0
$412$	−10.0000	−0.492665
$413$	7.11684	0.350197
$414$	0	0
$415$	7.53262	0.369762
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−21.8614	−1.06928
$419$	27.6060	1.34864	0.674320	−	0.738439i	$-0.264437\pi$
$419$	27.6060	1.34864	0.674320	+	0.738439i	$0.264437\pi$
$420$	0	0
$421$	−0.233688	−0.0113893	−0.00569463	−	0.999984i	$-0.501813\pi$
$421$	−0.233688	−0.0113893	−0.00569463	+	0.999984i	$0.501813\pi$
$422$	16.0000	0.778868
$423$	0	0
$424$	−2.74456	−0.133288
$425$	13.6277	0.661041
$426$	0	0
$427$	−14.1168	−0.683162
$428$	−1.62772	−0.0786788
$429$	0	0
$430$	12.5109	0.603328
$431$	−29.4891	−1.42044	−0.710221	−	0.703979i	$-0.751405\pi$
$431$	−29.4891	−1.42044	−0.710221	+	0.703979i	$0.751405\pi$
$432$	0	0
$433$	−2.88316	−0.138556	−0.0692778	−	0.997597i	$-0.522069\pi$
$433$	−2.88316	−0.138556	−0.0692778	+	0.997597i	$0.522069\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	14.0000	0.670478
$437$	−36.8614	−1.76332
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	8.74456	0.415936
$443$	−22.8832	−1.08721	−0.543606	−	0.839341i	$-0.682941\pi$
$443$	−22.8832	−1.08721	−0.543606	+	0.839341i	$0.682941\pi$
$444$	0	0
$445$	−4.46738	−0.211774
$446$	4.00000	0.189405
$447$	0	0
$448$	1.00000	0.0472456
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	45.3505	2.13547
$452$	1.37228	0.0645467
$453$	0	0
$454$	23.7446	1.11439
$455$	−2.74456	−0.128667
$456$	0	0
$457$	33.4674	1.56554	0.782769	−	0.622312i	$-0.213807\pi$
$457$	33.4674	1.56554	0.782769	+	0.622312i	$0.213807\pi$
$458$	20.1168	0.939998
$459$	0	0
$460$	10.1168	0.471700
$461$	30.8614	1.43736	0.718680	−	0.695341i	$-0.244747\pi$
$461$	30.8614	1.43736	0.718680	+	0.695341i	$0.244747\pi$
$462$	0	0
$463$	−5.88316	−0.273413	−0.136707	−	0.990612i	$-0.543652\pi$
$463$	−5.88316	−0.273413	−0.136707	+	0.990612i	$0.543652\pi$
$464$	2.74456	0.127413
$465$	0	0
$466$	11.7446	0.544056
$467$	−30.0951	−1.39263	−0.696317	−	0.717734i	$-0.745179\pi$
$467$	−30.0951	−1.39263	−0.696317	+	0.717734i	$0.745179\pi$
$468$	0	0
$469$	15.1168	0.698031
$470$	0	0
$471$	0	0
$472$	−7.11684	−0.327579
$473$	39.8614	1.83283
$474$	0	0
$475$	−15.5842	−0.715053
$476$	−4.37228	−0.200403
$477$	0	0
$478$	18.8614	0.862701
$479$	21.2554	0.971186	0.485593	−	0.874185i	$-0.338604\pi$
$479$	21.2554	0.971186	0.485593	+	0.874185i	$0.338604\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−0.883156	−0.0402267
$483$	0	0
$484$	8.11684	0.368947
$485$	−12.5109	−0.568090
$486$	0	0
$487$	−16.3505	−0.740913	−0.370457	−	0.928850i	$-0.620799\pi$
$487$	−16.3505	−0.740913	−0.370457	+	0.928850i	$0.620799\pi$
$488$	14.1168	0.639040
$489$	0	0
$490$	1.37228	0.0619934
$491$	−19.6277	−0.885787	−0.442893	−	0.896574i	$-0.646048\pi$
$491$	−19.6277	−0.885787	−0.442893	+	0.896574i	$0.646048\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	−10.0000	−0.449921
$495$	0	0
$496$	2.00000	0.0898027
$497$	10.1168	0.453802
$498$	0	0
$499$	0.883156	0.0395355	0.0197677	−	0.999805i	$-0.493707\pi$
$499$	0.883156	0.0395355	0.0197677	+	0.999805i	$0.493707\pi$
$500$	11.1386	0.498133
$501$	0	0
$502$	9.00000	0.401690
$503$	2.23369	0.0995952	0.0497976	−	0.998759i	$-0.484142\pi$
$503$	2.23369	0.0995952	0.0497976	+	0.998759i	$0.484142\pi$
$504$	0	0
$505$	−10.1168	−0.450194
$506$	32.2337	1.43296
$507$	0	0
$508$	−14.1168	−0.626334
$509$	16.9783	0.752548	0.376274	−	0.926508i	$-0.377205\pi$
$509$	16.9783	0.752548	0.376274	+	0.926508i	$0.377205\pi$
$510$	0	0
$511$	−5.11684	−0.226356
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−21.8614	−0.964265
$515$	13.7228	0.604699
$516$	0	0
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	2.74456	0.120357
$521$	3.86141	0.169171	0.0845856	−	0.996416i	$-0.473043\pi$
$521$	3.86141	0.169171	0.0845856	+	0.996416i	$0.473043\pi$
$522$	0	0
$523$	−17.8832	−0.781976	−0.390988	−	0.920396i	$-0.627867\pi$
$523$	−17.8832	−0.781976	−0.390988	+	0.920396i	$0.627867\pi$
$524$	−7.37228	−0.322060
$525$	0	0
$526$	−13.3723	−0.583059
$527$	−8.74456	−0.380919
$528$	0	0
$529$	31.3505	1.36307
$530$	3.76631	0.163598
$531$	0	0
$532$	5.00000	0.216777
$533$	20.7446	0.898547
$534$	0	0
$535$	2.23369	0.0965708
$536$	−15.1168	−0.652948
$537$	0	0
$538$	7.37228	0.317842
$539$	4.37228	0.188327
$540$	0	0
$541$	28.2337	1.21386	0.606931	−	0.794755i	$-0.292401\pi$
$541$	28.2337	1.21386	0.606931	+	0.794755i	$0.292401\pi$
$542$	18.2337	0.783204
$543$	0	0
$544$	4.37228	0.187460
$545$	−19.2119	−0.822949
$546$	0	0
$547$	0.883156	0.0377610	0.0188805	−	0.999822i	$-0.493990\pi$
$547$	0.883156	0.0377610	0.0188805	+	0.999822i	$0.493990\pi$
$548$	16.3723	0.699389
$549$	0	0
$550$	13.6277	0.581088
$551$	13.7228	0.584611
$552$	0	0
$553$	12.1168	0.515261
$554$	−22.2337	−0.944619
$555$	0	0
$556$	−21.2337	−0.900509
$557$	6.51087	0.275875	0.137937	−	0.990441i	$-0.455953\pi$
$557$	6.51087	0.275875	0.137937	+	0.990441i	$0.455953\pi$
$558$	0	0
$559$	18.2337	0.771203
$560$	−1.37228	−0.0579895
$561$	0	0
$562$	−10.6277	−0.448303
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−1.88316	−0.0792250
$566$	−9.88316	−0.415420
$567$	0	0
$568$	−10.1168	−0.424493
$569$	−1.11684	−0.0468205	−0.0234103	−	0.999726i	$-0.507452\pi$
$569$	−1.11684	−0.0468205	−0.0234103	+	0.999726i	$0.507452\pi$
$570$	0	0
$571$	29.3505	1.22828	0.614141	−	0.789197i	$-0.289503\pi$
$571$	29.3505	1.22828	0.614141	+	0.789197i	$0.289503\pi$
$572$	8.74456	0.365629
$573$	0	0
$574$	−10.3723	−0.432931
$575$	22.9783	0.958259
$576$	0	0
$577$	27.1168	1.12889	0.564444	−	0.825471i	$-0.309090\pi$
$577$	27.1168	1.12889	0.564444	+	0.825471i	$0.309090\pi$
$578$	−2.11684	−0.0880491
$579$	0	0
$580$	−3.76631	−0.156388
$581$	−5.48913	−0.227727
$582$	0	0
$583$	12.0000	0.496989
$584$	5.11684	0.211737
$585$	0	0
$586$	4.62772	0.191169
$587$	−8.48913	−0.350384	−0.175192	−	0.984534i	$-0.556055\pi$
$587$	−8.48913	−0.350384	−0.175192	+	0.984534i	$0.556055\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	9.76631	0.402073
$591$	0	0
$592$	2.00000	0.0821995
$593$	3.25544	0.133685	0.0668424	−	0.997764i	$-0.478708\pi$
$593$	3.25544	0.133685	0.0668424	+	0.997764i	$0.478708\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	14.7446	0.603961
$597$	0	0
$598$	14.7446	0.602950
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	6.88316	0.280770	0.140385	−	0.990097i	$-0.455166\pi$
$601$	6.88316	0.280770	0.140385	+	0.990097i	$0.455166\pi$
$602$	−9.11684	−0.371575
$603$	0	0
$604$	−8.11684	−0.330270
$605$	−11.1386	−0.452848
$606$	0	0
$607$	−12.2337	−0.496550	−0.248275	−	0.968690i	$-0.579864\pi$
$607$	−12.2337	−0.496550	−0.248275	+	0.968690i	$0.579864\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	−19.3723	−0.784361
$611$	0	0
$612$	0	0
$613$	−1.76631	−0.0713407	−0.0356703	−	0.999364i	$-0.511357\pi$
$613$	−1.76631	−0.0713407	−0.0356703	+	0.999364i	$0.511357\pi$
$614$	13.0000	0.524637
$615$	0	0
$616$	−4.37228	−0.176164
$617$	−9.86141	−0.397005	−0.198503	−	0.980100i	$-0.563608\pi$
$617$	−9.86141	−0.397005	−0.198503	+	0.980100i	$0.563608\pi$
$618$	0	0
$619$	−23.4674	−0.943233	−0.471617	−	0.881804i	$-0.656329\pi$
$619$	−23.4674	−0.943233	−0.471617	+	0.881804i	$0.656329\pi$
$620$	−2.74456	−0.110224
$621$	0	0
$622$	26.2337	1.05188
$623$	3.25544	0.130426
$624$	0	0
$625$	0.298936	0.0119574
$626$	2.88316	0.115234
$627$	0	0
$628$	−8.11684	−0.323897
$629$	−8.74456	−0.348669
$630$	0	0
$631$	14.3505	0.571286	0.285643	−	0.958336i	$-0.407793\pi$
$631$	14.3505	0.571286	0.285643	+	0.958336i	$0.407793\pi$
$632$	−12.1168	−0.481982
$633$	0	0
$634$	6.00000	0.238290
$635$	19.3723	0.768766
$636$	0	0
$637$	2.00000	0.0792429
$638$	−12.0000	−0.475085
$639$	0	0
$640$	1.37228	0.0542442
$641$	−46.2119	−1.82526	−0.912631	−	0.408785i	$-0.865953\pi$
$641$	−46.2119	−1.82526	−0.912631	+	0.408785i	$0.865953\pi$
$642$	0	0
$643$	−25.3505	−0.999727	−0.499864	−	0.866104i	$-0.666617\pi$
$643$	−25.3505	−0.999727	−0.499864	+	0.866104i	$0.666617\pi$
$644$	−7.37228	−0.290509
$645$	0	0
$646$	21.8614	0.860126
$647$	−17.4891	−0.687568	−0.343784	−	0.939049i	$-0.611709\pi$
$647$	−17.4891	−0.687568	−0.343784	+	0.939049i	$0.611709\pi$
$648$	0	0
$649$	31.1168	1.22144
$650$	6.23369	0.244505
$651$	0	0
$652$	16.2337	0.635760
$653$	−15.2554	−0.596991	−0.298496	−	0.954411i	$-0.596485\pi$
$653$	−15.2554	−0.596991	−0.298496	+	0.954411i	$0.596485\pi$
$654$	0	0
$655$	10.1168	0.395298
$656$	10.3723	0.404970
$657$	0	0
$658$	0	0
$659$	−9.25544	−0.360541	−0.180270	−	0.983617i	$-0.557697\pi$
$659$	−9.25544	−0.360541	−0.180270	+	0.983617i	$0.557697\pi$
$660$	0	0
$661$	9.88316	0.384410	0.192205	−	0.981355i	$-0.438436\pi$
$661$	9.88316	0.384410	0.192205	+	0.981355i	$0.438436\pi$
$662$	12.2337	0.475476
$663$	0	0
$664$	5.48913	0.213019
$665$	−6.86141	−0.266074
$666$	0	0
$667$	−20.2337	−0.783452
$668$	17.4891	0.676675
$669$	0	0
$670$	20.7446	0.801432
$671$	−61.7228	−2.38278
$672$	0	0
$673$	−20.1168	−0.775447	−0.387724	−	0.921776i	$-0.626739\pi$
$673$	−20.1168	−0.775447	−0.387724	+	0.921776i	$0.626739\pi$
$674$	−9.11684	−0.351168
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−34.4674	−1.32469	−0.662344	−	0.749199i	$-0.730438\pi$
$677$	−34.4674	−1.32469	−0.662344	+	0.749199i	$0.730438\pi$
$678$	0	0
$679$	9.11684	0.349872
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−8.74456	−0.334847
$683$	−44.8397	−1.71574	−0.857871	−	0.513865i	$-0.828213\pi$
$683$	−44.8397	−1.71574	−0.857871	+	0.513865i	$0.828213\pi$
$684$	0	0
$685$	−22.4674	−0.858434
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	9.11684	0.347576
$689$	5.48913	0.209119
$690$	0	0
$691$	−5.88316	−0.223806	−0.111903	−	0.993719i	$-0.535695\pi$
$691$	−5.88316	−0.223806	−0.111903	+	0.993719i	$0.535695\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−7.11684	−0.270152
$695$	29.1386	1.10529
$696$	0	0
$697$	−45.3505	−1.71777
$698$	22.0000	0.832712
$699$	0	0
$700$	−3.11684	−0.117806
$701$	−3.76631	−0.142252	−0.0711258	−	0.997467i	$-0.522659\pi$
$701$	−3.76631	−0.142252	−0.0711258	+	0.997467i	$0.522659\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	4.37228	0.164787
$705$	0	0
$706$	7.62772	0.287073
$707$	7.37228	0.277263
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	13.8832	0.521026
$711$	0	0
$712$	−3.25544	−0.122003
$713$	−14.7446	−0.552188
$714$	0	0
$715$	−12.0000	−0.448775
$716$	14.7446	0.551030
$717$	0	0
$718$	6.86141	0.256065
$719$	−8.74456	−0.326117	−0.163059	−	0.986616i	$-0.552136\pi$
$719$	−8.74456	−0.326117	−0.163059	+	0.986616i	$0.552136\pi$
$720$	0	0
$721$	−10.0000	−0.372419
$722$	−6.00000	−0.223297
$723$	0	0
$724$	18.1168	0.673307
$725$	−8.55437	−0.317701
$726$	0	0
$727$	−1.76631	−0.0655089	−0.0327544	−	0.999463i	$-0.510428\pi$
$727$	−1.76631	−0.0655089	−0.0327544	+	0.999463i	$0.510428\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	−7.02175	−0.259887
$731$	−39.8614	−1.47433
$732$	0	0
$733$	−23.8832	−0.882144	−0.441072	−	0.897472i	$-0.645402\pi$
$733$	−23.8832	−0.882144	−0.441072	+	0.897472i	$0.645402\pi$
$734$	−22.2337	−0.820660
$735$	0	0
$736$	7.37228	0.271746
$737$	66.0951	2.43464
$738$	0	0
$739$	9.11684	0.335369	0.167684	−	0.985841i	$-0.446371\pi$
$739$	9.11684	0.335369	0.167684	+	0.985841i	$0.446371\pi$
$740$	−2.74456	−0.100892
$741$	0	0
$742$	−2.74456	−0.100756
$743$	43.7228	1.60403	0.802017	−	0.597301i	$-0.203760\pi$
$743$	43.7228	1.60403	0.802017	+	0.597301i	$0.203760\pi$
$744$	0	0
$745$	−20.2337	−0.741305
$746$	10.0000	0.366126
$747$	0	0
$748$	−19.1168	−0.698981
$749$	−1.62772	−0.0594755
$750$	0	0
$751$	0.116844	0.00426370	0.00213185	−	0.999998i	$-0.499321\pi$
$751$	0.116844	0.00426370	0.00213185	+	0.999998i	$0.499321\pi$
$752$	0	0
$753$	0	0
$754$	−5.48913	−0.199902
$755$	11.1386	0.405375
$756$	0	0
$757$	11.7663	0.427654	0.213827	−	0.976872i	$-0.431407\pi$
$757$	11.7663	0.427654	0.213827	+	0.976872i	$0.431407\pi$
$758$	−9.11684	−0.331139
$759$	0	0
$760$	6.86141	0.248889
$761$	12.5109	0.453519	0.226759	−	0.973951i	$-0.427187\pi$
$761$	12.5109	0.453519	0.226759	+	0.973951i	$0.427187\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	−1.88316	−0.0681302
$765$	0	0
$766$	21.2554	0.767990
$767$	14.2337	0.513949
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	6.00000	0.216225
$771$	0	0
$772$	−7.00000	−0.251936
$773$	−11.1386	−0.400627	−0.200314	−	0.979732i	$-0.564196\pi$
$773$	−11.1386	−0.400627	−0.200314	+	0.979732i	$0.564196\pi$
$774$	0	0
$775$	−6.23369	−0.223921
$776$	−9.11684	−0.327276
$777$	0	0
$778$	34.9783	1.25403
$779$	51.8614	1.85813
$780$	0	0
$781$	44.2337	1.58281
$782$	−32.2337	−1.15267
$783$	0	0
$784$	1.00000	0.0357143
$785$	11.1386	0.397553
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	16.6277	0.591588
$791$	1.37228	0.0487927
$792$	0	0
$793$	−28.2337	−1.00261
$794$	22.0000	0.780751
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−36.8614	−1.30570	−0.652849	−	0.757488i	$-0.726426\pi$
$797$	−36.8614	−1.30570	−0.652849	+	0.757488i	$0.726426\pi$
$798$	0	0
$799$	0	0
$800$	3.11684	0.110197
$801$	0	0
$802$	0.255437	0.00901981
$803$	−22.3723	−0.789501
$804$	0	0
$805$	10.1168	0.356572
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−7.37228	−0.259356
$809$	21.8614	0.768606	0.384303	−	0.923207i	$-0.374442\pi$
$809$	21.8614	0.768606	0.384303	+	0.923207i	$0.374442\pi$
$810$	0	0
$811$	24.8832	0.873766	0.436883	−	0.899518i	$-0.356082\pi$
$811$	24.8832	0.873766	0.436883	+	0.899518i	$0.356082\pi$
$812$	2.74456	0.0963153
$813$	0	0
$814$	−8.74456	−0.306497
$815$	−22.2772	−0.780336
$816$	0	0
$817$	45.5842	1.59479
$818$	−29.3505	−1.02622
$819$	0	0
$820$	−14.2337	−0.497062
$821$	−38.2337	−1.33436	−0.667182	−	0.744894i	$-0.732500\pi$
$821$	−38.2337	−1.33436	−0.667182	+	0.744894i	$0.732500\pi$
$822$	0	0
$823$	22.2337	0.775018	0.387509	−	0.921866i	$-0.373336\pi$
$823$	22.2337	0.775018	0.387509	+	0.921866i	$0.373336\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	−7.11684	−0.247627
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−48.2337	−1.67523	−0.837613	−	0.546265i	$-0.816049\pi$
$829$	−48.2337	−1.67523	−0.837613	+	0.546265i	$0.816049\pi$
$830$	−7.53262	−0.261461
$831$	0	0
$832$	2.00000	0.0693375
$833$	−4.37228	−0.151491
$834$	0	0
$835$	−24.0000	−0.830554
$836$	21.8614	0.756093
$837$	0	0
$838$	−27.6060	−0.953632
$839$	−17.4891	−0.603792	−0.301896	−	0.953341i	$-0.597619\pi$
$839$	−17.4891	−0.603792	−0.301896	+	0.953341i	$0.597619\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0.233688	0.00805342
$843$	0	0
$844$	−16.0000	−0.550743
$845$	12.3505	0.424871
$846$	0	0
$847$	8.11684	0.278898
$848$	2.74456	0.0942487
$849$	0	0
$850$	−13.6277	−0.467427
$851$	−14.7446	−0.505437
$852$	0	0
$853$	−17.8832	−0.612308	−0.306154	−	0.951982i	$-0.599042\pi$
$853$	−17.8832	−0.612308	−0.306154	+	0.951982i	$0.599042\pi$
$854$	14.1168	0.483068
$855$	0	0
$856$	1.62772	0.0556343
$857$	51.9565	1.77480	0.887400	−	0.461000i	$-0.152509\pi$
$857$	51.9565	1.77480	0.887400	+	0.461000i	$0.152509\pi$
$858$	0	0
$859$	51.1168	1.74408	0.872042	−	0.489431i	$-0.162795\pi$
$859$	51.1168	1.74408	0.872042	+	0.489431i	$0.162795\pi$
$860$	−12.5109	−0.426617
$861$	0	0
$862$	29.4891	1.00440
$863$	−18.8614	−0.642050	−0.321025	−	0.947071i	$-0.604027\pi$
$863$	−18.8614	−0.642050	−0.321025	+	0.947071i	$0.604027\pi$
$864$	0	0
$865$	8.23369	0.279954
$866$	2.88316	0.0979736
$867$	0	0
$868$	2.00000	0.0678844
$869$	52.9783	1.79716
$870$	0	0
$871$	30.2337	1.02443
$872$	−14.0000	−0.474100
$873$	0	0
$874$	36.8614	1.24686
$875$	11.1386	0.376553
$876$	0	0
$877$	44.7011	1.50945	0.754724	−	0.656043i	$-0.227771\pi$
$877$	44.7011	1.50945	0.754724	+	0.656043i	$0.227771\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	−6.00000	−0.202260
$881$	14.2337	0.479545	0.239773	−	0.970829i	$-0.422927\pi$
$881$	14.2337	0.479545	0.239773	+	0.970829i	$0.422927\pi$
$882$	0	0
$883$	11.3505	0.381976	0.190988	−	0.981592i	$-0.438831\pi$
$883$	11.3505	0.381976	0.190988	+	0.981592i	$0.438831\pi$
$884$	−8.74456	−0.294111
$885$	0	0
$886$	22.8832	0.768775
$887$	−31.7228	−1.06515	−0.532574	−	0.846383i	$-0.678775\pi$
$887$	−31.7228	−1.06515	−0.532574	+	0.846383i	$0.678775\pi$
$888$	0	0
$889$	−14.1168	−0.473464
$890$	4.46738	0.149747
$891$	0	0
$892$	−4.00000	−0.133930
$893$	0	0
$894$	0	0
$895$	−20.2337	−0.676338
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−33.0000	−1.10122
$899$	5.48913	0.183073
$900$	0	0
$901$	−12.0000	−0.399778
$902$	−45.3505	−1.51001
$903$	0	0
$904$	−1.37228	−0.0456414
$905$	−24.8614	−0.826421
$906$	0	0
$907$	−8.88316	−0.294960	−0.147480	−	0.989065i	$-0.547116\pi$
$907$	−8.88316	−0.294960	−0.147480	+	0.989065i	$0.547116\pi$
$908$	−23.7446	−0.787991
$909$	0	0
$910$	2.74456	0.0909814
$911$	−43.3723	−1.43699	−0.718494	−	0.695533i	$-0.755168\pi$
$911$	−43.3723	−1.43699	−0.718494	+	0.695533i	$0.755168\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	−33.4674	−1.10700
$915$	0	0
$916$	−20.1168	−0.664679
$917$	−7.37228	−0.243454
$918$	0	0
$919$	−29.8832	−0.985754	−0.492877	−	0.870099i	$-0.664055\pi$
$919$	−29.8832	−0.985754	−0.492877	+	0.870099i	$0.664055\pi$
$920$	−10.1168	−0.333542
$921$	0	0
$922$	−30.8614	−1.01637
$923$	20.2337	0.666000
$924$	0	0
$925$	−6.23369	−0.204962
$926$	5.88316	0.193333
$927$	0	0
$928$	−2.74456	−0.0900947
$929$	9.76631	0.320422	0.160211	−	0.987083i	$-0.448782\pi$
$929$	9.76631	0.320422	0.160211	+	0.987083i	$0.448782\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−11.7446	−0.384706
$933$	0	0
$934$	30.0951	0.984742
$935$	26.2337	0.857933
$936$	0	0
$937$	−38.4674	−1.25667	−0.628337	−	0.777941i	$-0.716264\pi$
$937$	−38.4674	−1.25667	−0.628337	+	0.777941i	$0.716264\pi$
$938$	−15.1168	−0.493582
$939$	0	0
$940$	0	0
$941$	1.88316	0.0613891	0.0306946	−	0.999529i	$-0.490228\pi$
$941$	1.88316	0.0613891	0.0306946	+	0.999529i	$0.490228\pi$
$942$	0	0
$943$	−76.4674	−2.49012
$944$	7.11684	0.231634
$945$	0	0
$946$	−39.8614	−1.29601
$947$	16.8832	0.548629	0.274314	−	0.961640i	$-0.411549\pi$
$947$	16.8832	0.548629	0.274314	+	0.961640i	$0.411549\pi$
$948$	0	0
$949$	−10.2337	−0.332200
$950$	15.5842	0.505619
$951$	0	0
$952$	4.37228	0.141706
$953$	10.8832	0.352540	0.176270	−	0.984342i	$-0.443597\pi$
$953$	10.8832	0.352540	0.176270	+	0.984342i	$0.443597\pi$
$954$	0	0
$955$	2.58422	0.0836234
$956$	−18.8614	−0.610021
$957$	0	0
$958$	−21.2554	−0.686732
$959$	16.3723	0.528689
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−4.00000	−0.128965
$963$	0	0
$964$	0.883156	0.0284445
$965$	9.60597	0.309227
$966$	0	0
$967$	48.1168	1.54733	0.773667	−	0.633593i	$-0.218421\pi$
$967$	48.1168	1.54733	0.773667	+	0.633593i	$0.218421\pi$
$968$	−8.11684	−0.260885
$969$	0	0
$970$	12.5109	0.401700
$971$	7.37228	0.236588	0.118294	−	0.992979i	$-0.462258\pi$
$971$	7.37228	0.236588	0.118294	+	0.992979i	$0.462258\pi$
$972$	0	0
$973$	−21.2337	−0.680721
$974$	16.3505	0.523905
$975$	0	0
$976$	−14.1168	−0.451869
$977$	22.8832	0.732097	0.366049	−	0.930596i	$-0.380710\pi$
$977$	22.8832	0.732097	0.366049	+	0.930596i	$0.380710\pi$
$978$	0	0
$979$	14.2337	0.454911
$980$	−1.37228	−0.0438359
$981$	0	0
$982$	19.6277	0.626346
$983$	−50.7446	−1.61850	−0.809250	−	0.587464i	$-0.800126\pi$
$983$	−50.7446	−1.61850	−0.809250	+	0.587464i	$0.800126\pi$
$984$	0	0
$985$	8.23369	0.262347
$986$	12.0000	0.382158
$987$	0	0
$988$	10.0000	0.318142
$989$	−67.2119	−2.13722
$990$	0	0
$991$	−20.4674	−0.650168	−0.325084	−	0.945685i	$-0.605393\pi$
$991$	−20.4674	−0.650168	−0.325084	+	0.945685i	$0.605393\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−10.1168	−0.320887
$995$	13.7228	0.435042
$996$	0	0
$997$	12.1168	0.383744	0.191872	−	0.981420i	$-0.438544\pi$
$997$	12.1168	0.383744	0.191872	+	0.981420i	$0.438544\pi$
$998$	−0.883156	−0.0279558
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.a.k.1.1		2
3.2	odd	2		1134.2.a.n.1.2		2
4.3	odd	2		9072.2.a.bm.1.1		2
7.6	odd	2		7938.2.a.bh.1.2		2
9.2	odd	6		378.2.f.c.253.1		4
9.4	even	3		126.2.f.d.43.2	✓	4
9.5	odd	6		378.2.f.c.127.1		4
9.7	even	3		126.2.f.d.85.2	yes	4
12.11	even	2		9072.2.a.bb.1.2		2
21.20	even	2		7938.2.a.bs.1.1		2
36.7	odd	6		1008.2.r.f.337.1		4
36.11	even	6		3024.2.r.f.1009.1		4
36.23	even	6		3024.2.r.f.2017.1		4
36.31	odd	6		1008.2.r.f.673.1		4
63.2	odd	6		2646.2.h.k.361.2		4
63.4	even	3		882.2.h.m.79.1		4
63.5	even	6		2646.2.e.m.2125.2		4
63.11	odd	6		2646.2.e.n.1549.1		4
63.13	odd	6		882.2.f.k.295.1		4
63.16	even	3		882.2.h.m.67.1		4
63.20	even	6		2646.2.f.j.1765.2		4
63.23	odd	6		2646.2.e.n.2125.1		4
63.25	even	3		882.2.e.l.373.1		4
63.31	odd	6		882.2.h.n.79.2		4
63.32	odd	6		2646.2.h.k.667.2		4
63.34	odd	6		882.2.f.k.589.1		4
63.38	even	6		2646.2.e.m.1549.2		4
63.40	odd	6		882.2.e.k.655.1		4
63.41	even	6		2646.2.f.j.883.2		4
63.47	even	6		2646.2.h.l.361.1		4
63.52	odd	6		882.2.e.k.373.2		4
63.58	even	3		882.2.e.l.655.2		4
63.59	even	6		2646.2.h.l.667.1		4
63.61	odd	6		882.2.h.n.67.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.d.43.2	✓	4	9.4	even	3
126.2.f.d.85.2	yes	4	9.7	even	3
378.2.f.c.127.1		4	9.5	odd	6
378.2.f.c.253.1		4	9.2	odd	6
882.2.e.k.373.2		4	63.52	odd	6
882.2.e.k.655.1		4	63.40	odd	6
882.2.e.l.373.1		4	63.25	even	3
882.2.e.l.655.2		4	63.58	even	3
882.2.f.k.295.1		4	63.13	odd	6
882.2.f.k.589.1		4	63.34	odd	6
882.2.h.m.67.1		4	63.16	even	3
882.2.h.m.79.1		4	63.4	even	3
882.2.h.n.67.2		4	63.61	odd	6
882.2.h.n.79.2		4	63.31	odd	6
1008.2.r.f.337.1		4	36.7	odd	6
1008.2.r.f.673.1		4	36.31	odd	6
1134.2.a.k.1.1		2	1.1	even	1	trivial
1134.2.a.n.1.2		2	3.2	odd	2
2646.2.e.m.1549.2		4	63.38	even	6
2646.2.e.m.2125.2		4	63.5	even	6
2646.2.e.n.1549.1		4	63.11	odd	6
2646.2.e.n.2125.1		4	63.23	odd	6
2646.2.f.j.883.2		4	63.41	even	6
2646.2.f.j.1765.2		4	63.20	even	6
2646.2.h.k.361.2		4	63.2	odd	6
2646.2.h.k.667.2		4	63.32	odd	6
2646.2.h.l.361.1		4	63.47	even	6
2646.2.h.l.667.1		4	63.59	even	6
3024.2.r.f.1009.1		4	36.11	even	6
3024.2.r.f.2017.1		4	36.23	even	6
7938.2.a.bh.1.2		2	7.6	odd	2
7938.2.a.bs.1.1		2	21.20	even	2
9072.2.a.bb.1.2		2	12.11	even	2
9072.2.a.bm.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.37228 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.37228 q^{10} +4.37228 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.37228 q^{17} +5.00000 q^{19} -1.37228 q^{20} -4.37228 q^{22} -7.37228 q^{23} -3.11684 q^{25} -2.00000 q^{26} +1.00000 q^{28} +2.74456 q^{29} +2.00000 q^{31} -1.00000 q^{32} +4.37228 q^{34} -1.37228 q^{35} +2.00000 q^{37} -5.00000 q^{38} +1.37228 q^{40} +10.3723 q^{41} +9.11684 q^{43} +4.37228 q^{44} +7.37228 q^{46} +1.00000 q^{49} +3.11684 q^{50} +2.00000 q^{52} +2.74456 q^{53} -6.00000 q^{55} -1.00000 q^{56} -2.74456 q^{58} +7.11684 q^{59} -14.1168 q^{61} -2.00000 q^{62} +1.00000 q^{64} -2.74456 q^{65} +15.1168 q^{67} -4.37228 q^{68} +1.37228 q^{70} +10.1168 q^{71} -5.11684 q^{73} -2.00000 q^{74} +5.00000 q^{76} +4.37228 q^{77} +12.1168 q^{79} -1.37228 q^{80} -10.3723 q^{82} -5.48913 q^{83} +6.00000 q^{85} -9.11684 q^{86} -4.37228 q^{88} +3.25544 q^{89} +2.00000 q^{91} -7.37228 q^{92} -6.86141 q^{95} +9.11684 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{7} - 2 q^{8} - 3 q^{10} + 3 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 3 q^{17} + 10 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} + 11 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^7 - 2 * q^8 - 3 * q^10 + 3 * q^11 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 3 * q^17 + 10 * q^19 + 3 * q^20 - 3 * q^22 - 9 * q^23 + 11 * q^25 - 4 * q^26 + 2 * q^28 - 6 * q^29 + 4 * q^31 - 2 * q^32 + 3 * q^34 + 3 * q^35 + 4 * q^37 - 10 * q^38 - 3 * q^40 + 15 * q^41 + q^43 + 3 * q^44 + 9 * q^46 + 2 * q^49 - 11 * q^50 + 4 * q^52 - 6 * q^53 - 12 * q^55 - 2 * q^56 + 6 * q^58 - 3 * q^59 - 11 * q^61 - 4 * q^62 + 2 * q^64 + 6 * q^65 + 13 * q^67 - 3 * q^68 - 3 * q^70 + 3 * q^71 + 7 * q^73 - 4 * q^74 + 10 * q^76 + 3 * q^77 + 7 * q^79 + 3 * q^80 - 15 * q^82 + 12 * q^83 + 12 * q^85 - q^86 - 3 * q^88 + 18 * q^89 + 4 * q^91 - 9 * q^92 + 15 * q^95 + q^97 - 2 * q^98

Introduction

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