LMFDB - Embedded newform 1134.2.a.k.1.2

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			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +4.37228 q^{5} +1.00000 q^{7} -1.00000 q^{8} -4.37228 q^{10} -1.37228 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.37228 q^{17} +5.00000 q^{19} +4.37228 q^{20} +1.37228 q^{22} -1.62772 q^{23} +14.1168 q^{25} -2.00000 q^{26} +1.00000 q^{28} -8.74456 q^{29} +2.00000 q^{31} -1.00000 q^{32} -1.37228 q^{34} +4.37228 q^{35} +2.00000 q^{37} -5.00000 q^{38} -4.37228 q^{40} +4.62772 q^{41} -8.11684 q^{43} -1.37228 q^{44} +1.62772 q^{46} +1.00000 q^{49} -14.1168 q^{50} +2.00000 q^{52} -8.74456 q^{53} -6.00000 q^{55} -1.00000 q^{56} +8.74456 q^{58} -10.1168 q^{59} +3.11684 q^{61} -2.00000 q^{62} +1.00000 q^{64} +8.74456 q^{65} -2.11684 q^{67} +1.37228 q^{68} -4.37228 q^{70} -7.11684 q^{71} +12.1168 q^{73} -2.00000 q^{74} +5.00000 q^{76} -1.37228 q^{77} -5.11684 q^{79} +4.37228 q^{80} -4.62772 q^{82} +17.4891 q^{83} +6.00000 q^{85} +8.11684 q^{86} +1.37228 q^{88} +14.7446 q^{89} +2.00000 q^{91} -1.62772 q^{92} +21.8614 q^{95} -8.11684 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.37228	1.95534	0.977672	−	0.210138i	$-0.0673912\pi$
$5$	4.37228	1.95534	0.977672	+	0.210138i	$0.0673912\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−4.37228	−1.38264
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	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
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	4.37228	0.977672
$21$	0	0
$22$	1.37228	0.292571
$23$	−1.62772	−0.339403	−0.169701	−	0.985496i	$-0.554280\pi$
$23$	−1.62772	−0.339403	−0.169701	+	0.985496i	$0.554280\pi$
$24$	0	0
$25$	14.1168	2.82337
$26$	−2.00000	−0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−8.74456	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$30$	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
$34$	−1.37228	−0.235344
$35$	4.37228	0.739050
$36$	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
$40$	−4.37228	−0.691318
$41$	4.62772	0.722728	0.361364	−	0.932425i	$-0.382311\pi$
$41$	4.62772	0.722728	0.361364	+	0.932425i	$0.382311\pi$
$42$	0	0
$43$	−8.11684	−1.23781	−0.618904	−	0.785467i	$-0.712423\pi$
$43$	−8.11684	−1.23781	−0.618904	+	0.785467i	$0.712423\pi$
$44$	−1.37228	−0.206879
$45$	0	0
$46$	1.62772	0.239994
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−14.1168	−1.99642
$51$	0	0
$52$	2.00000	0.277350
$53$	−8.74456	−1.20116	−0.600579	−	0.799565i	$-0.705063\pi$
$53$	−8.74456	−1.20116	−0.600579	+	0.799565i	$0.705063\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	−1.00000	−0.133631
$57$	0	0
$58$	8.74456	1.14822
$59$	−10.1168	−1.31710	−0.658550	−	0.752537i	$-0.728830\pi$
$59$	−10.1168	−1.31710	−0.658550	+	0.752537i	$0.728830\pi$
$60$	0	0
$61$	3.11684	0.399071	0.199535	−	0.979891i	$-0.436057\pi$
$61$	3.11684	0.399071	0.199535	+	0.979891i	$0.436057\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	8.74456	1.08463
$66$	0	0
$67$	−2.11684	−0.258614	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	−2.11684	−0.258614	−0.129307	+	0.991605i	$0.541275\pi$
$68$	1.37228	0.166414
$69$	0	0
$70$	−4.37228	−0.522588
$71$	−7.11684	−0.844614	−0.422307	−	0.906453i	$-0.638780\pi$
$71$	−7.11684	−0.844614	−0.422307	+	0.906453i	$0.638780\pi$
$72$	0	0
$73$	12.1168	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$73$	12.1168	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	5.00000	0.573539
$77$	−1.37228	−0.156386
$78$	0	0
$79$	−5.11684	−0.575690	−0.287845	−	0.957677i	$-0.592939\pi$
$79$	−5.11684	−0.575690	−0.287845	+	0.957677i	$0.592939\pi$
$80$	4.37228	0.488836
$81$	0	0
$82$	−4.62772	−0.511046
$83$	17.4891	1.91968	0.959840	−	0.280546i	$-0.0905157\pi$
$83$	17.4891	1.91968	0.959840	+	0.280546i	$0.0905157\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	8.11684	0.875262
$87$	0	0
$88$	1.37228	0.146286
$89$	14.7446	1.56292	0.781460	−	0.623955i	$-0.214475\pi$
$89$	14.7446	1.56292	0.781460	+	0.623955i	$0.214475\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−1.62772	−0.169701
$93$	0	0
$94$	0	0
$95$	21.8614	2.24293
$96$	0	0
$97$	−8.11684	−0.824141	−0.412070	−	0.911152i	$-0.635194\pi$
$97$	−8.11684	−0.824141	−0.412070	+	0.911152i	$0.635194\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	14.1168	1.41168
$101$	1.62772	0.161964	0.0809820	−	0.996716i	$-0.474194\pi$
$101$	1.62772	0.161964	0.0809820	+	0.996716i	$0.474194\pi$
$102$	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
$106$	8.74456	0.849347
$107$	−7.37228	−0.712705	−0.356353	−	0.934352i	$-0.615980\pi$
$107$	−7.37228	−0.712705	−0.356353	+	0.934352i	$0.615980\pi$
$108$	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
$113$	−4.37228	−0.411310	−0.205655	−	0.978625i	$-0.565932\pi$
$113$	−4.37228	−0.411310	−0.205655	+	0.978625i	$0.565932\pi$
$114$	0	0
$115$	−7.11684	−0.663649
$116$	−8.74456	−0.811912
$117$	0	0
$118$	10.1168	0.931331
$119$	1.37228	0.125797
$120$	0	0
$121$	−9.11684	−0.828804
$122$	−3.11684	−0.282186
$123$	0	0
$124$	2.00000	0.179605
$125$	39.8614	3.56531
$126$	0	0
$127$	3.11684	0.276575	0.138288	−	0.990392i	$-0.455840\pi$
$127$	3.11684	0.276575	0.138288	+	0.990392i	$0.455840\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.74456	−0.766949
$131$	−1.62772	−0.142214	−0.0711072	−	0.997469i	$-0.522653\pi$
$131$	−1.62772	−0.142214	−0.0711072	+	0.997469i	$0.522653\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	2.11684	0.182867
$135$	0	0
$136$	−1.37228	−0.117672
$137$	10.6277	0.907987	0.453994	−	0.891005i	$-0.349999\pi$
$137$	10.6277	0.907987	0.453994	+	0.891005i	$0.349999\pi$
$138$	0	0
$139$	13.2337	1.12247	0.561233	−	0.827658i	$-0.310327\pi$
$139$	13.2337	1.12247	0.561233	+	0.827658i	$0.310327\pi$
$140$	4.37228	0.369525
$141$	0	0
$142$	7.11684	0.597232
$143$	−2.74456	−0.229512
$144$	0	0
$145$	−38.2337	−3.17513
$146$	−12.1168	−1.00280
$147$	0	0
$148$	2.00000	0.164399
$149$	3.25544	0.266696	0.133348	−	0.991069i	$-0.457427\pi$
$149$	3.25544	0.266696	0.133348	+	0.991069i	$0.457427\pi$
$150$	0	0
$151$	9.11684	0.741918	0.370959	−	0.928649i	$-0.379029\pi$
$151$	9.11684	0.741918	0.370959	+	0.928649i	$0.379029\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	1.37228	0.110582
$155$	8.74456	0.702380
$156$	0	0
$157$	9.11684	0.727603	0.363802	−	0.931476i	$-0.381479\pi$
$157$	9.11684	0.727603	0.363802	+	0.931476i	$0.381479\pi$
$158$	5.11684	0.407074
$159$	0	0
$160$	−4.37228	−0.345659
$161$	−1.62772	−0.128282
$162$	0	0
$163$	−18.2337	−1.42817	−0.714086	−	0.700058i	$-0.753158\pi$
$163$	−18.2337	−1.42817	−0.714086	+	0.700058i	$0.753158\pi$
$164$	4.62772	0.361364
$165$	0	0
$166$	−17.4891	−1.35742
$167$	−5.48913	−0.424761	−0.212381	−	0.977187i	$-0.568122\pi$
$167$	−5.48913	−0.424761	−0.212381	+	0.977187i	$0.568122\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	0	0
$172$	−8.11684	−0.618904
$173$	−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
$175$	14.1168	1.06713
$176$	−1.37228	−0.103440
$177$	0	0
$178$	−14.7446	−1.10515
$179$	3.25544	0.243323	0.121661	−	0.992572i	$-0.461178\pi$
$179$	3.25544	0.243323	0.121661	+	0.992572i	$0.461178\pi$
$180$	0	0
$181$	0.883156	0.0656445	0.0328222	−	0.999461i	$-0.489550\pi$
$181$	0.883156	0.0656445	0.0328222	+	0.999461i	$0.489550\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	1.62772	0.119997
$185$	8.74456	0.642913
$186$	0	0
$187$	−1.88316	−0.137710
$188$	0	0
$189$	0	0
$190$	−21.8614	−1.58599
$191$	−19.1168	−1.38325	−0.691623	−	0.722259i	$-0.743104\pi$
$191$	−19.1168	−1.38325	−0.691623	+	0.722259i	$0.743104\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	8.11684	0.582755
$195$	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	$0.615329\pi$
$200$	−14.1168	−0.998212
$201$	0	0
$202$	−1.62772	−0.114526
$203$	−8.74456	−0.613748
$204$	0	0
$205$	20.2337	1.41318
$206$	10.0000	0.696733
$207$	0	0
$208$	2.00000	0.138675
$209$	−6.86141	−0.474613
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−8.74456	−0.600579
$213$	0	0
$214$	7.37228	0.503959
$215$	−35.4891	−2.42034
$216$	0	0
$217$	2.00000	0.135769
$218$	−14.0000	−0.948200
$219$	0	0
$220$	−6.00000	−0.404520
$221$	2.74456	0.184619
$222$	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
$226$	4.37228	0.290840
$227$	−12.2554	−0.813422	−0.406711	−	0.913557i	$-0.633324\pi$
$227$	−12.2554	−0.813422	−0.406711	+	0.913557i	$0.633324\pi$
$228$	0	0
$229$	−2.88316	−0.190524	−0.0952622	−	0.995452i	$-0.530369\pi$
$229$	−2.88316	−0.190524	−0.0952622	+	0.995452i	$0.530369\pi$
$230$	7.11684	0.469271
$231$	0	0
$232$	8.74456	0.574109
$233$	−0.255437	−0.0167343	−0.00836713	−	0.999965i	$-0.502663\pi$
$233$	−0.255437	−0.0167343	−0.00836713	+	0.999965i	$0.502663\pi$
$234$	0	0
$235$	0	0
$236$	−10.1168	−0.658550
$237$	0	0
$238$	−1.37228	−0.0889518
$239$	9.86141	0.637881	0.318941	−	0.947775i	$-0.396673\pi$
$239$	9.86141	0.637881	0.318941	+	0.947775i	$0.396673\pi$
$240$	0	0
$241$	18.1168	1.16701	0.583504	−	0.812110i	$-0.301681\pi$
$241$	18.1168	1.16701	0.583504	+	0.812110i	$0.301681\pi$
$242$	9.11684	0.586053
$243$	0	0
$244$	3.11684	0.199535
$245$	4.37228	0.279335
$246$	0	0
$247$	10.0000	0.636285
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−39.8614	−2.52106
$251$	−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
$253$	2.23369	0.140431
$254$	−3.11684	−0.195568
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.86141	−0.428003	−0.214001	−	0.976833i	$-0.568650\pi$
$257$	−6.86141	−0.428003	−0.214001	+	0.976833i	$0.568650\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	8.74456	0.542315
$261$	0	0
$262$	1.62772	0.100561
$263$	7.62772	0.470345	0.235173	−	0.971954i	$-0.424434\pi$
$263$	7.62772	0.470345	0.235173	+	0.971954i	$0.424434\pi$
$264$	0	0
$265$	−38.2337	−2.34868
$266$	−5.00000	−0.306570
$267$	0	0
$268$	−2.11684	−0.129307
$269$	−1.62772	−0.0992438	−0.0496219	−	0.998768i	$-0.515802\pi$
$269$	−1.62772	−0.0992438	−0.0496219	+	0.998768i	$0.515802\pi$
$270$	0	0
$271$	16.2337	0.986126	0.493063	−	0.869994i	$-0.335877\pi$
$271$	16.2337	0.986126	0.493063	+	0.869994i	$0.335877\pi$
$272$	1.37228	0.0832068
$273$	0	0
$274$	−10.6277	−0.642044
$275$	−19.3723	−1.16819
$276$	0	0
$277$	−12.2337	−0.735051	−0.367526	−	0.930013i	$-0.619795\pi$
$277$	−12.2337	−0.735051	−0.367526	+	0.930013i	$0.619795\pi$
$278$	−13.2337	−0.793704
$279$	0	0
$280$	−4.37228	−0.261294
$281$	16.3723	0.976688	0.488344	−	0.872651i	$-0.337601\pi$
$281$	16.3723	0.976688	0.488344	+	0.872651i	$0.337601\pi$
$282$	0	0
$283$	27.1168	1.61193	0.805965	−	0.591964i	$-0.201647\pi$
$283$	27.1168	1.61193	0.805965	+	0.591964i	$0.201647\pi$
$284$	−7.11684	−0.422307
$285$	0	0
$286$	2.74456	0.162289
$287$	4.62772	0.273166
$288$	0	0
$289$	−15.1168	−0.889226
$290$	38.2337	2.24516
$291$	0	0
$292$	12.1168	0.709085
$293$	−10.3723	−0.605955	−0.302978	−	0.952998i	$-0.597981\pi$
$293$	−10.3723	−0.605955	−0.302978	+	0.952998i	$0.597981\pi$
$294$	0	0
$295$	−44.2337	−2.57538
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−3.25544	−0.188582
$299$	−3.25544	−0.188267
$300$	0	0
$301$	−8.11684	−0.467847
$302$	−9.11684	−0.524615
$303$	0	0
$304$	5.00000	0.286770
$305$	13.6277	0.780321
$306$	0	0
$307$	−13.0000	−0.741949	−0.370975	−	0.928643i	$-0.620976\pi$
$307$	−13.0000	−0.741949	−0.370975	+	0.928643i	$0.620976\pi$
$308$	−1.37228	−0.0781930
$309$	0	0
$310$	−8.74456	−0.496658
$311$	8.23369	0.466890	0.233445	−	0.972370i	$-0.425000\pi$
$311$	8.23369	0.466890	0.233445	+	0.972370i	$0.425000\pi$
$312$	0	0
$313$	−20.1168	−1.13707	−0.568536	−	0.822659i	$-0.692490\pi$
$313$	−20.1168	−1.13707	−0.568536	+	0.822659i	$0.692490\pi$
$314$	−9.11684	−0.514493
$315$	0	0
$316$	−5.11684	−0.287845
$317$	−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
$320$	4.37228	0.244418
$321$	0	0
$322$	1.62772	0.0907092
$323$	6.86141	0.381779
$324$	0	0
$325$	28.2337	1.56612
$326$	18.2337	1.00987
$327$	0	0
$328$	−4.62772	−0.255523
$329$	0	0
$330$	0	0
$331$	22.2337	1.22207	0.611037	−	0.791602i	$-0.290753\pi$
$331$	22.2337	1.22207	0.611037	+	0.791602i	$0.290753\pi$
$332$	17.4891	0.959840
$333$	0	0
$334$	5.48913	0.300352
$335$	−9.25544	−0.505679
$336$	0	0
$337$	−8.11684	−0.442153	−0.221076	−	0.975257i	$-0.570957\pi$
$337$	−8.11684	−0.442153	−0.221076	+	0.975257i	$0.570957\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	6.00000	0.325396
$341$	−2.74456	−0.148626
$342$	0	0
$343$	1.00000	0.0539949
$344$	8.11684	0.437631
$345$	0	0
$346$	6.00000	0.322562
$347$	−10.1168	−0.543101	−0.271550	−	0.962424i	$-0.587536\pi$
$347$	−10.1168	−0.543101	−0.271550	+	0.962424i	$0.587536\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−14.1168	−0.754577
$351$	0	0
$352$	1.37228	0.0731428
$353$	−13.3723	−0.711735	−0.355867	−	0.934536i	$-0.615815\pi$
$353$	−13.3723	−0.711735	−0.355867	+	0.934536i	$0.615815\pi$
$354$	0	0
$355$	−31.1168	−1.65151
$356$	14.7446	0.781460
$357$	0	0
$358$	−3.25544	−0.172055
$359$	21.8614	1.15380	0.576900	−	0.816814i	$-0.304262\pi$
$359$	21.8614	1.15380	0.576900	+	0.816814i	$0.304262\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−0.883156	−0.0464177
$363$	0	0
$364$	2.00000	0.104828
$365$	52.9783	2.77301
$366$	0	0
$367$	−12.2337	−0.638593	−0.319297	−	0.947655i	$-0.603447\pi$
$367$	−12.2337	−0.638593	−0.319297	+	0.947655i	$0.603447\pi$
$368$	−1.62772	−0.0848507
$369$	0	0
$370$	−8.74456	−0.454608
$371$	−8.74456	−0.453995
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	1.88316	0.0973757
$375$	0	0
$376$	0	0
$377$	−17.4891	−0.900736
$378$	0	0
$379$	−8.11684	−0.416934	−0.208467	−	0.978029i	$-0.566847\pi$
$379$	−8.11684	−0.416934	−0.208467	+	0.978029i	$0.566847\pi$
$380$	21.8614	1.12147
$381$	0	0
$382$	19.1168	0.978103
$383$	−32.7446	−1.67317	−0.836584	−	0.547838i	$-0.815451\pi$
$383$	−32.7446	−1.67317	−0.836584	+	0.547838i	$0.815451\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	7.00000	0.356291
$387$	0	0
$388$	−8.11684	−0.412070
$389$	10.9783	0.556619	0.278310	−	0.960491i	$-0.410226\pi$
$389$	10.9783	0.556619	0.278310	+	0.960491i	$0.410226\pi$
$390$	0	0
$391$	−2.23369	−0.112962
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−22.3723	−1.12567
$396$	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
$400$	14.1168	0.705842
$401$	−11.7446	−0.586495	−0.293248	−	0.956036i	$-0.594736\pi$
$401$	−11.7446	−0.586495	−0.293248	+	0.956036i	$0.594736\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	1.62772	0.0809820
$405$	0	0
$406$	8.74456	0.433985
$407$	−2.74456	−0.136043
$408$	0	0
$409$	−22.3505	−1.10516	−0.552581	−	0.833459i	$-0.686357\pi$
$409$	−22.3505	−1.10516	−0.552581	+	0.833459i	$0.686357\pi$
$410$	−20.2337	−0.999271
$411$	0	0
$412$	−10.0000	−0.492665
$413$	−10.1168	−0.497817
$414$	0	0
$415$	76.4674	3.75364
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	6.86141	0.335602
$419$	−12.6060	−0.615842	−0.307921	−	0.951412i	$-0.599633\pi$
$419$	−12.6060	−0.615842	−0.307921	+	0.951412i	$0.599633\pi$
$420$	0	0
$421$	34.2337	1.66845	0.834224	−	0.551426i	$-0.185916\pi$
$421$	34.2337	1.66845	0.834224	+	0.551426i	$0.185916\pi$
$422$	16.0000	0.778868
$423$	0	0
$424$	8.74456	0.424674
$425$	19.3723	0.939694
$426$	0	0
$427$	3.11684	0.150835
$428$	−7.37228	−0.356353
$429$	0	0
$430$	35.4891	1.71144
$431$	−6.51087	−0.313618	−0.156809	−	0.987629i	$-0.550121\pi$
$431$	−6.51087	−0.313618	−0.156809	+	0.987629i	$0.550121\pi$
$432$	0	0
$433$	−20.1168	−0.966754	−0.483377	−	0.875412i	$-0.660590\pi$
$433$	−20.1168	−0.966754	−0.483377	+	0.875412i	$0.660590\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	14.0000	0.670478
$437$	−8.13859	−0.389322
$438$	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
$442$	−2.74456	−0.130546
$443$	−40.1168	−1.90601	−0.953004	−	0.302956i	$-0.902026\pi$
$443$	−40.1168	−1.90601	−0.953004	+	0.302956i	$0.902026\pi$
$444$	0	0
$445$	64.4674	3.05605
$446$	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
$451$	−6.35053	−0.299035
$452$	−4.37228	−0.205655
$453$	0	0
$454$	12.2554	0.575176
$455$	8.74456	0.409951
$456$	0	0
$457$	−35.4674	−1.65909	−0.829547	−	0.558437i	$-0.811401\pi$
$457$	−35.4674	−1.65909	−0.829547	+	0.558437i	$0.811401\pi$
$458$	2.88316	0.134721
$459$	0	0
$460$	−7.11684	−0.331825
$461$	2.13859	0.0996042	0.0498021	−	0.998759i	$-0.484141\pi$
$461$	2.13859	0.0996042	0.0498021	+	0.998759i	$0.484141\pi$
$462$	0	0
$463$	−23.1168	−1.07433	−0.537165	−	0.843477i	$-0.680505\pi$
$463$	−23.1168	−1.07433	−0.537165	+	0.843477i	$0.680505\pi$
$464$	−8.74456	−0.405956
$465$	0	0
$466$	0.255437	0.0118329
$467$	33.0951	1.53146	0.765729	−	0.643163i	$-0.222378\pi$
$467$	33.0951	1.53146	0.765729	+	0.643163i	$0.222378\pi$
$468$	0	0
$469$	−2.11684	−0.0977468
$470$	0	0
$471$	0	0
$472$	10.1168	0.465665
$473$	11.1386	0.512153
$474$	0	0
$475$	70.5842	3.23863
$476$	1.37228	0.0628984
$477$	0	0
$478$	−9.86141	−0.451050
$479$	32.7446	1.49614	0.748069	−	0.663621i	$-0.230981\pi$
$479$	32.7446	1.49614	0.748069	+	0.663621i	$0.230981\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	−18.1168	−0.825200
$483$	0	0
$484$	−9.11684	−0.414402
$485$	−35.4891	−1.61148
$486$	0	0
$487$	35.3505	1.60189	0.800943	−	0.598741i	$-0.204332\pi$
$487$	35.3505	1.60189	0.800943	+	0.598741i	$0.204332\pi$
$488$	−3.11684	−0.141093
$489$	0	0
$490$	−4.37228	−0.197520
$491$	−25.3723	−1.14504	−0.572518	−	0.819892i	$-0.694033\pi$
$491$	−25.3723	−1.14504	−0.572518	+	0.819892i	$0.694033\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	−10.0000	−0.449921
$495$	0	0
$496$	2.00000	0.0898027
$497$	−7.11684	−0.319234
$498$	0	0
$499$	18.1168	0.811021	0.405511	−	0.914090i	$-0.367094\pi$
$499$	18.1168	0.811021	0.405511	+	0.914090i	$0.367094\pi$
$500$	39.8614	1.78266
$501$	0	0
$502$	9.00000	0.401690
$503$	−32.2337	−1.43723	−0.718615	−	0.695409i	$-0.755223\pi$
$503$	−32.2337	−1.43723	−0.718615	+	0.695409i	$0.755223\pi$
$504$	0	0
$505$	7.11684	0.316695
$506$	−2.23369	−0.0992995
$507$	0	0
$508$	3.11684	0.138288
$509$	−28.9783	−1.28444	−0.642219	−	0.766521i	$-0.721986\pi$
$509$	−28.9783	−1.28444	−0.642219	+	0.766521i	$0.721986\pi$
$510$	0	0
$511$	12.1168	0.536018
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.86141	0.302644
$515$	−43.7228	−1.92666
$516$	0	0
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	0	0
$520$	−8.74456	−0.383474
$521$	−24.8614	−1.08920	−0.544599	−	0.838697i	$-0.683318\pi$
$521$	−24.8614	−1.08920	−0.544599	+	0.838697i	$0.683318\pi$
$522$	0	0
$523$	−35.1168	−1.53555	−0.767776	−	0.640718i	$-0.778637\pi$
$523$	−35.1168	−1.53555	−0.767776	+	0.640718i	$0.778637\pi$
$524$	−1.62772	−0.0711072
$525$	0	0
$526$	−7.62772	−0.332584
$527$	2.74456	0.119555
$528$	0	0
$529$	−20.3505	−0.884806
$530$	38.2337	1.66077
$531$	0	0
$532$	5.00000	0.216777
$533$	9.25544	0.400897
$534$	0	0
$535$	−32.2337	−1.39358
$536$	2.11684	0.0914337
$537$	0	0
$538$	1.62772	0.0701759
$539$	−1.37228	−0.0591083
$540$	0	0
$541$	−6.23369	−0.268007	−0.134004	−	0.990981i	$-0.542783\pi$
$541$	−6.23369	−0.268007	−0.134004	+	0.990981i	$0.542783\pi$
$542$	−16.2337	−0.697297
$543$	0	0
$544$	−1.37228	−0.0588361
$545$	61.2119	2.62203
$546$	0	0
$547$	18.1168	0.774620	0.387310	−	0.921949i	$-0.373404\pi$
$547$	18.1168	0.774620	0.387310	+	0.921949i	$0.373404\pi$
$548$	10.6277	0.453994
$549$	0	0
$550$	19.3723	0.826037
$551$	−43.7228	−1.86265
$552$	0	0
$553$	−5.11684	−0.217590
$554$	12.2337	0.519760
$555$	0	0
$556$	13.2337	0.561233
$557$	29.4891	1.24949	0.624747	−	0.780827i	$-0.285202\pi$
$557$	29.4891	1.24949	0.624747	+	0.780827i	$0.285202\pi$
$558$	0	0
$559$	−16.2337	−0.686612
$560$	4.37228	0.184763
$561$	0	0
$562$	−16.3723	−0.690623
$563$	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
$565$	−19.1168	−0.804252
$566$	−27.1168	−1.13981
$567$	0	0
$568$	7.11684	0.298616
$569$	16.1168	0.675653	0.337827	−	0.941208i	$-0.390308\pi$
$569$	16.1168	0.675653	0.337827	+	0.941208i	$0.390308\pi$
$570$	0	0
$571$	−22.3505	−0.935341	−0.467670	−	0.883903i	$-0.654907\pi$
$571$	−22.3505	−0.935341	−0.467670	+	0.883903i	$0.654907\pi$
$572$	−2.74456	−0.114756
$573$	0	0
$574$	−4.62772	−0.193157
$575$	−22.9783	−0.958259
$576$	0	0
$577$	9.88316	0.411441	0.205721	−	0.978611i	$-0.434046\pi$
$577$	9.88316	0.411441	0.205721	+	0.978611i	$0.434046\pi$
$578$	15.1168	0.628778
$579$	0	0
$580$	−38.2337	−1.58757
$581$	17.4891	0.725571
$582$	0	0
$583$	12.0000	0.496989
$584$	−12.1168	−0.501399
$585$	0	0
$586$	10.3723	0.428475
$587$	14.4891	0.598030	0.299015	−	0.954248i	$-0.403342\pi$
$587$	14.4891	0.598030	0.299015	+	0.954248i	$0.403342\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	44.2337	1.82107
$591$	0	0
$592$	2.00000	0.0821995
$593$	14.7446	0.605487	0.302743	−	0.953072i	$-0.402098\pi$
$593$	14.7446	0.605487	0.302743	+	0.953072i	$0.402098\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	3.25544	0.133348
$597$	0	0
$598$	3.25544	0.133125
$599$	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
$601$	24.1168	0.983747	0.491873	−	0.870667i	$-0.336312\pi$
$601$	24.1168	0.983747	0.491873	+	0.870667i	$0.336312\pi$
$602$	8.11684	0.330818
$603$	0	0
$604$	9.11684	0.370959
$605$	−39.8614	−1.62060
$606$	0	0
$607$	22.2337	0.902438	0.451219	−	0.892413i	$-0.350989\pi$
$607$	22.2337	0.902438	0.451219	+	0.892413i	$0.350989\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	−13.6277	−0.551770
$611$	0	0
$612$	0	0
$613$	−36.2337	−1.46346	−0.731732	−	0.681592i	$-0.761288\pi$
$613$	−36.2337	−1.46346	−0.731732	+	0.681592i	$0.761288\pi$
$614$	13.0000	0.524637
$615$	0	0
$616$	1.37228	0.0552908
$617$	18.8614	0.759332	0.379666	−	0.925124i	$-0.376039\pi$
$617$	18.8614	0.759332	0.379666	+	0.925124i	$0.376039\pi$
$618$	0	0
$619$	45.4674	1.82749	0.913744	−	0.406290i	$-0.133178\pi$
$619$	45.4674	1.82749	0.913744	+	0.406290i	$0.133178\pi$
$620$	8.74456	0.351190
$621$	0	0
$622$	−8.23369	−0.330141
$623$	14.7446	0.590728
$624$	0	0
$625$	103.701	4.14804
$626$	20.1168	0.804031
$627$	0	0
$628$	9.11684	0.363802
$629$	2.74456	0.109433
$630$	0	0
$631$	−37.3505	−1.48690	−0.743451	−	0.668791i	$-0.766812\pi$
$631$	−37.3505	−1.48690	−0.743451	+	0.668791i	$0.766812\pi$
$632$	5.11684	0.203537
$633$	0	0
$634$	6.00000	0.238290
$635$	13.6277	0.540800
$636$	0	0
$637$	2.00000	0.0792429
$638$	−12.0000	−0.475085
$639$	0	0
$640$	−4.37228	−0.172830
$641$	34.2119	1.35129	0.675645	−	0.737227i	$-0.263865\pi$
$641$	34.2119	1.35129	0.675645	+	0.737227i	$0.263865\pi$
$642$	0	0
$643$	26.3505	1.03916	0.519582	−	0.854421i	$-0.326088\pi$
$643$	26.3505	1.03916	0.519582	+	0.854421i	$0.326088\pi$
$644$	−1.62772	−0.0641411
$645$	0	0
$646$	−6.86141	−0.269958
$647$	5.48913	0.215800	0.107900	−	0.994162i	$-0.465587\pi$
$647$	5.48913	0.215800	0.107900	+	0.994162i	$0.465587\pi$
$648$	0	0
$649$	13.8832	0.544962
$650$	−28.2337	−1.10742
$651$	0	0
$652$	−18.2337	−0.714086
$653$	−26.7446	−1.04660	−0.523298	−	0.852150i	$-0.675298\pi$
$653$	−26.7446	−1.04660	−0.523298	+	0.852150i	$0.675298\pi$
$654$	0	0
$655$	−7.11684	−0.278078
$656$	4.62772	0.180682
$657$	0	0
$658$	0	0
$659$	−20.7446	−0.808093	−0.404047	−	0.914738i	$-0.632397\pi$
$659$	−20.7446	−0.808093	−0.404047	+	0.914738i	$0.632397\pi$
$660$	0	0
$661$	27.1168	1.05472	0.527361	−	0.849641i	$-0.323181\pi$
$661$	27.1168	1.05472	0.527361	+	0.849641i	$0.323181\pi$
$662$	−22.2337	−0.864137
$663$	0	0
$664$	−17.4891	−0.678710
$665$	21.8614	0.847749
$666$	0	0
$667$	14.2337	0.551131
$668$	−5.48913	−0.212381
$669$	0	0
$670$	9.25544	0.357569
$671$	−4.27719	−0.165119
$672$	0	0
$673$	−2.88316	−0.111137	−0.0555687	−	0.998455i	$-0.517697\pi$
$673$	−2.88316	−0.111137	−0.0555687	+	0.998455i	$0.517697\pi$
$674$	8.11684	0.312649
$675$	0	0
$676$	−9.00000	−0.346154
$677$	34.4674	1.32469	0.662344	−	0.749199i	$-0.269562\pi$
$677$	34.4674	1.32469	0.662344	+	0.749199i	$0.269562\pi$
$678$	0	0
$679$	−8.11684	−0.311496
$680$	−6.00000	−0.230089
$681$	0	0
$682$	2.74456	0.105095
$683$	29.8397	1.14178	0.570891	−	0.821026i	$-0.306598\pi$
$683$	29.8397	1.14178	0.570891	+	0.821026i	$0.306598\pi$
$684$	0	0
$685$	46.4674	1.77543
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−8.11684	−0.309452
$689$	−17.4891	−0.666283
$690$	0	0
$691$	−23.1168	−0.879406	−0.439703	−	0.898143i	$-0.644916\pi$
$691$	−23.1168	−0.879406	−0.439703	+	0.898143i	$0.644916\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	10.1168	0.384030
$695$	57.8614	2.19481
$696$	0	0
$697$	6.35053	0.240544
$698$	22.0000	0.832712
$699$	0	0
$700$	14.1168	0.533567
$701$	−38.2337	−1.44407	−0.722033	−	0.691858i	$-0.756792\pi$
$701$	−38.2337	−1.44407	−0.722033	+	0.691858i	$0.756792\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	−1.37228	−0.0517198
$705$	0	0
$706$	13.3723	0.503272
$707$	1.62772	0.0612167
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	31.1168	1.16779
$711$	0	0
$712$	−14.7446	−0.552576
$713$	−3.25544	−0.121917
$714$	0	0
$715$	−12.0000	−0.448775
$716$	3.25544	0.121661
$717$	0	0
$718$	−21.8614	−0.815860
$719$	2.74456	0.102355	0.0511775	−	0.998690i	$-0.483703\pi$
$719$	2.74456	0.102355	0.0511775	+	0.998690i	$0.483703\pi$
$720$	0	0
$721$	−10.0000	−0.372419
$722$	−6.00000	−0.223297
$723$	0	0
$724$	0.883156	0.0328222
$725$	−123.446	−4.58466
$726$	0	0
$727$	−36.2337	−1.34383	−0.671917	−	0.740627i	$-0.734529\pi$
$727$	−36.2337	−1.34383	−0.671917	+	0.740627i	$0.734529\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	−52.9783	−1.96081
$731$	−11.1386	−0.411976
$732$	0	0
$733$	−41.1168	−1.51869	−0.759343	−	0.650691i	$-0.774479\pi$
$733$	−41.1168	−1.51869	−0.759343	+	0.650691i	$0.774479\pi$
$734$	12.2337	0.451554
$735$	0	0
$736$	1.62772	0.0599985
$737$	2.90491	0.107004
$738$	0	0
$739$	−8.11684	−0.298583	−0.149291	−	0.988793i	$-0.547699\pi$
$739$	−8.11684	−0.298583	−0.149291	+	0.988793i	$0.547699\pi$
$740$	8.74456	0.321457
$741$	0	0
$742$	8.74456	0.321023
$743$	−13.7228	−0.503441	−0.251721	−	0.967800i	$-0.580996\pi$
$743$	−13.7228	−0.503441	−0.251721	+	0.967800i	$0.580996\pi$
$744$	0	0
$745$	14.2337	0.521482
$746$	10.0000	0.366126
$747$	0	0
$748$	−1.88316	−0.0688550
$749$	−7.37228	−0.269377
$750$	0	0
$751$	−17.1168	−0.624603	−0.312301	−	0.949983i	$-0.601100\pi$
$751$	−17.1168	−0.624603	−0.312301	+	0.949983i	$0.601100\pi$
$752$	0	0
$753$	0	0
$754$	17.4891	0.636916
$755$	39.8614	1.45071
$756$	0	0
$757$	46.2337	1.68039	0.840196	−	0.542283i	$-0.182440\pi$
$757$	46.2337	1.68039	0.840196	+	0.542283i	$0.182440\pi$
$758$	8.11684	0.294817
$759$	0	0
$760$	−21.8614	−0.792997
$761$	35.4891	1.28648	0.643240	−	0.765665i	$-0.277590\pi$
$761$	35.4891	1.28648	0.643240	+	0.765665i	$0.277590\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	−19.1168	−0.691623
$765$	0	0
$766$	32.7446	1.18311
$767$	−20.2337	−0.730596
$768$	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
$773$	−39.8614	−1.43372	−0.716858	−	0.697220i	$-0.754420\pi$
$773$	−39.8614	−1.43372	−0.716858	+	0.697220i	$0.754420\pi$
$774$	0	0
$775$	28.2337	1.01418
$776$	8.11684	0.291378
$777$	0	0
$778$	−10.9783	−0.393589
$779$	23.1386	0.829026
$780$	0	0
$781$	9.76631	0.349466
$782$	2.23369	0.0798765
$783$	0	0
$784$	1.00000	0.0357143
$785$	39.8614	1.42271
$786$	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
$790$	22.3723	0.795970
$791$	−4.37228	−0.155460
$792$	0	0
$793$	6.23369	0.221365
$794$	22.0000	0.780751
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−8.13859	−0.288284	−0.144142	−	0.989557i	$-0.546042\pi$
$797$	−8.13859	−0.288284	−0.144142	+	0.989557i	$0.546042\pi$
$798$	0	0
$799$	0	0
$800$	−14.1168	−0.499106
$801$	0	0
$802$	11.7446	0.414715
$803$	−16.6277	−0.586779
$804$	0	0
$805$	−7.11684	−0.250836
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−1.62772	−0.0572629
$809$	−6.86141	−0.241234	−0.120617	−	0.992699i	$-0.538487\pi$
$809$	−6.86141	−0.241234	−0.120617	+	0.992699i	$0.538487\pi$
$810$	0	0
$811$	42.1168	1.47892	0.739461	−	0.673199i	$-0.235080\pi$
$811$	42.1168	1.47892	0.739461	+	0.673199i	$0.235080\pi$
$812$	−8.74456	−0.306874
$813$	0	0
$814$	2.74456	0.0961969
$815$	−79.7228	−2.79257
$816$	0	0
$817$	−40.5842	−1.41986
$818$	22.3505	0.781468
$819$	0	0
$820$	20.2337	0.706591
$821$	−3.76631	−0.131445	−0.0657226	−	0.997838i	$-0.520935\pi$
$821$	−3.76631	−0.131445	−0.0657226	+	0.997838i	$0.520935\pi$
$822$	0	0
$823$	−12.2337	−0.426440	−0.213220	−	0.977004i	$-0.568395\pi$
$823$	−12.2337	−0.426440	−0.213220	+	0.977004i	$0.568395\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	10.1168	0.352010
$827$	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
$829$	−13.7663	−0.478124	−0.239062	−	0.971004i	$-0.576840\pi$
$829$	−13.7663	−0.478124	−0.239062	+	0.971004i	$0.576840\pi$
$830$	−76.4674	−2.65422
$831$	0	0
$832$	2.00000	0.0693375
$833$	1.37228	0.0475467
$834$	0	0
$835$	−24.0000	−0.830554
$836$	−6.86141	−0.237307
$837$	0	0
$838$	12.6060	0.435466
$839$	5.48913	0.189506	0.0947528	−	0.995501i	$-0.469794\pi$
$839$	5.48913	0.189506	0.0947528	+	0.995501i	$0.469794\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	−34.2337	−1.17977
$843$	0	0
$844$	−16.0000	−0.550743
$845$	−39.3505	−1.35370
$846$	0	0
$847$	−9.11684	−0.313258
$848$	−8.74456	−0.300290
$849$	0	0
$850$	−19.3723	−0.664464
$851$	−3.25544	−0.111595
$852$	0	0
$853$	−35.1168	−1.20238	−0.601189	−	0.799107i	$-0.705306\pi$
$853$	−35.1168	−1.20238	−0.601189	+	0.799107i	$0.705306\pi$
$854$	−3.11684	−0.106656
$855$	0	0
$856$	7.37228	0.251979
$857$	−39.9565	−1.36489	−0.682444	−	0.730938i	$-0.739083\pi$
$857$	−39.9565	−1.36489	−0.682444	+	0.730938i	$0.739083\pi$
$858$	0	0
$859$	33.8832	1.15608	0.578039	−	0.816009i	$-0.303818\pi$
$859$	33.8832	1.15608	0.578039	+	0.816009i	$0.303818\pi$
$860$	−35.4891	−1.21017
$861$	0	0
$862$	6.51087	0.221761
$863$	9.86141	0.335686	0.167843	−	0.985814i	$-0.446320\pi$
$863$	9.86141	0.335686	0.167843	+	0.985814i	$0.446320\pi$
$864$	0	0
$865$	−26.2337	−0.891972
$866$	20.1168	0.683598
$867$	0	0
$868$	2.00000	0.0678844
$869$	7.02175	0.238197
$870$	0	0
$871$	−4.23369	−0.143453
$872$	−14.0000	−0.474100
$873$	0	0
$874$	8.13859	0.275292
$875$	39.8614	1.34756
$876$	0	0
$877$	−58.7011	−1.98219	−0.991097	−	0.133141i	$-0.957494\pi$
$877$	−58.7011	−1.98219	−0.991097	+	0.133141i	$0.957494\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−20.2337	−0.681690	−0.340845	−	0.940119i	$-0.610713\pi$
$881$	−20.2337	−0.681690	−0.340845	+	0.940119i	$0.610713\pi$
$882$	0	0
$883$	−40.3505	−1.35790	−0.678952	−	0.734183i	$-0.737565\pi$
$883$	−40.3505	−1.35790	−0.678952	+	0.734183i	$0.737565\pi$
$884$	2.74456	0.0923096
$885$	0	0
$886$	40.1168	1.34775
$887$	25.7228	0.863688	0.431844	−	0.901948i	$-0.357863\pi$
$887$	25.7228	0.863688	0.431844	+	0.901948i	$0.357863\pi$
$888$	0	0
$889$	3.11684	0.104536
$890$	−64.4674	−2.16095
$891$	0	0
$892$	−4.00000	−0.133930
$893$	0	0
$894$	0	0
$895$	14.2337	0.475780
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−33.0000	−1.10122
$899$	−17.4891	−0.583295
$900$	0	0
$901$	−12.0000	−0.399778
$902$	6.35053	0.211450
$903$	0	0
$904$	4.37228	0.145420
$905$	3.86141	0.128357
$906$	0	0
$907$	−26.1168	−0.867196	−0.433598	−	0.901107i	$-0.642756\pi$
$907$	−26.1168	−0.867196	−0.433598	+	0.901107i	$0.642756\pi$
$908$	−12.2554	−0.406711
$909$	0	0
$910$	−8.74456	−0.289879
$911$	−37.6277	−1.24666	−0.623331	−	0.781958i	$-0.714221\pi$
$911$	−37.6277	−1.24666	−0.623331	+	0.781958i	$0.714221\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	35.4674	1.17316
$915$	0	0
$916$	−2.88316	−0.0952622
$917$	−1.62772	−0.0537520
$918$	0	0
$919$	−47.1168	−1.55424	−0.777121	−	0.629352i	$-0.783321\pi$
$919$	−47.1168	−1.55424	−0.777121	+	0.629352i	$0.783321\pi$
$920$	7.11684	0.234635
$921$	0	0
$922$	−2.13859	−0.0704308
$923$	−14.2337	−0.468508
$924$	0	0
$925$	28.2337	0.928318
$926$	23.1168	0.759667
$927$	0	0
$928$	8.74456	0.287054
$929$	44.2337	1.45126	0.725630	−	0.688085i	$-0.241548\pi$
$929$	44.2337	1.45126	0.725630	+	0.688085i	$0.241548\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−0.255437	−0.00836713
$933$	0	0
$934$	−33.0951	−1.08290
$935$	−8.23369	−0.269270
$936$	0	0
$937$	30.4674	0.995326	0.497663	−	0.867371i	$-0.334192\pi$
$937$	30.4674	0.995326	0.497663	+	0.867371i	$0.334192\pi$
$938$	2.11684	0.0691174
$939$	0	0
$940$	0	0
$941$	19.1168	0.623191	0.311596	−	0.950215i	$-0.399137\pi$
$941$	19.1168	0.623191	0.311596	+	0.950215i	$0.399137\pi$
$942$	0	0
$943$	−7.53262	−0.245296
$944$	−10.1168	−0.329275
$945$	0	0
$946$	−11.1386	−0.362147
$947$	34.1168	1.10865	0.554324	−	0.832301i	$-0.312977\pi$
$947$	34.1168	1.10865	0.554324	+	0.832301i	$0.312977\pi$
$948$	0	0
$949$	24.2337	0.786659
$950$	−70.5842	−2.29005
$951$	0	0
$952$	−1.37228	−0.0444759
$953$	28.1168	0.910794	0.455397	−	0.890288i	$-0.349497\pi$
$953$	28.1168	0.910794	0.455397	+	0.890288i	$0.349497\pi$
$954$	0	0
$955$	−83.5842	−2.70472
$956$	9.86141	0.318941
$957$	0	0
$958$	−32.7446	−1.05793
$959$	10.6277	0.343187
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−4.00000	−0.128965
$963$	0	0
$964$	18.1168	0.583504
$965$	−30.6060	−0.985241
$966$	0	0
$967$	30.8832	0.993135	0.496568	−	0.867998i	$-0.334593\pi$
$967$	30.8832	0.993135	0.496568	+	0.867998i	$0.334593\pi$
$968$	9.11684	0.293026
$969$	0	0
$970$	35.4891	1.13949
$971$	1.62772	0.0522360	0.0261180	−	0.999659i	$-0.491685\pi$
$971$	1.62772	0.0522360	0.0261180	+	0.999659i	$0.491685\pi$
$972$	0	0
$973$	13.2337	0.424253
$974$	−35.3505	−1.13270
$975$	0	0
$976$	3.11684	0.0997677
$977$	40.1168	1.28345	0.641726	−	0.766934i	$-0.278219\pi$
$977$	40.1168	1.28345	0.641726	+	0.766934i	$0.278219\pi$
$978$	0	0
$979$	−20.2337	−0.646671
$980$	4.37228	0.139667
$981$	0	0
$982$	25.3723	0.809662
$983$	−39.2554	−1.25205	−0.626027	−	0.779801i	$-0.715320\pi$
$983$	−39.2554	−1.25205	−0.626027	+	0.779801i	$0.715320\pi$
$984$	0	0
$985$	−26.2337	−0.835875
$986$	12.0000	0.382158
$987$	0	0
$988$	10.0000	0.318142
$989$	13.2119	0.420115
$990$	0	0
$991$	48.4674	1.53962	0.769808	−	0.638275i	$-0.220352\pi$
$991$	48.4674	1.53962	0.769808	+	0.638275i	$0.220352\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	7.11684	0.225733
$995$	−43.7228	−1.38611
$996$	0	0
$997$	−5.11684	−0.162052	−0.0810260	−	0.996712i	$-0.525820\pi$
$997$	−5.11684	−0.162052	−0.0810260	+	0.996712i	$0.525820\pi$
$998$	−18.1168	−0.573479
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.a.k.1.2		2
3.2	odd	2		1134.2.a.n.1.1		2
4.3	odd	2		9072.2.a.bm.1.2		2
7.6	odd	2		7938.2.a.bh.1.1		2
9.2	odd	6		378.2.f.c.253.2		4
9.4	even	3		126.2.f.d.43.1	✓	4
9.5	odd	6		378.2.f.c.127.2		4
9.7	even	3		126.2.f.d.85.1	yes	4
12.11	even	2		9072.2.a.bb.1.1		2
21.20	even	2		7938.2.a.bs.1.2		2
36.7	odd	6		1008.2.r.f.337.2		4
36.11	even	6		3024.2.r.f.1009.2		4
36.23	even	6		3024.2.r.f.2017.2		4
36.31	odd	6		1008.2.r.f.673.2		4
63.2	odd	6		2646.2.h.k.361.1		4
63.4	even	3		882.2.h.m.79.2		4
63.5	even	6		2646.2.e.m.2125.1		4
63.11	odd	6		2646.2.e.n.1549.2		4
63.13	odd	6		882.2.f.k.295.2		4
63.16	even	3		882.2.h.m.67.2		4
63.20	even	6		2646.2.f.j.1765.1		4
63.23	odd	6		2646.2.e.n.2125.2		4
63.25	even	3		882.2.e.l.373.2		4
63.31	odd	6		882.2.h.n.79.1		4
63.32	odd	6		2646.2.h.k.667.1		4
63.34	odd	6		882.2.f.k.589.2		4
63.38	even	6		2646.2.e.m.1549.1		4
63.40	odd	6		882.2.e.k.655.2		4
63.41	even	6		2646.2.f.j.883.1		4
63.47	even	6		2646.2.h.l.361.2		4
63.52	odd	6		882.2.e.k.373.1		4
63.58	even	3		882.2.e.l.655.1		4
63.59	even	6		2646.2.h.l.667.2		4
63.61	odd	6		882.2.h.n.67.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.d.43.1	✓	4	9.4	even	3
126.2.f.d.85.1	yes	4	9.7	even	3
378.2.f.c.127.2		4	9.5	odd	6
378.2.f.c.253.2		4	9.2	odd	6
882.2.e.k.373.1		4	63.52	odd	6
882.2.e.k.655.2		4	63.40	odd	6
882.2.e.l.373.2		4	63.25	even	3
882.2.e.l.655.1		4	63.58	even	3
882.2.f.k.295.2		4	63.13	odd	6
882.2.f.k.589.2		4	63.34	odd	6
882.2.h.m.67.2		4	63.16	even	3
882.2.h.m.79.2		4	63.4	even	3
882.2.h.n.67.1		4	63.61	odd	6
882.2.h.n.79.1		4	63.31	odd	6
1008.2.r.f.337.2		4	36.7	odd	6
1008.2.r.f.673.2		4	36.31	odd	6
1134.2.a.k.1.2		2	1.1	even	1	trivial
1134.2.a.n.1.1		2	3.2	odd	2
2646.2.e.m.1549.1		4	63.38	even	6
2646.2.e.m.2125.1		4	63.5	even	6
2646.2.e.n.1549.2		4	63.11	odd	6
2646.2.e.n.2125.2		4	63.23	odd	6
2646.2.f.j.883.1		4	63.41	even	6
2646.2.f.j.1765.1		4	63.20	even	6
2646.2.h.k.361.1		4	63.2	odd	6
2646.2.h.k.667.1		4	63.32	odd	6
2646.2.h.l.361.2		4	63.47	even	6
2646.2.h.l.667.2		4	63.59	even	6
3024.2.r.f.1009.2		4	36.11	even	6
3024.2.r.f.2017.2		4	36.23	even	6
7938.2.a.bh.1.1		2	7.6	odd	2
7938.2.a.bs.1.2		2	21.20	even	2
9072.2.a.bb.1.1		2	12.11	even	2
9072.2.a.bm.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +4.37228 q^{5} +1.00000 q^{7} -1.00000 q^{8} -4.37228 q^{10} -1.37228 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.37228 q^{17} +5.00000 q^{19} +4.37228 q^{20} +1.37228 q^{22} -1.62772 q^{23} +14.1168 q^{25} -2.00000 q^{26} +1.00000 q^{28} -8.74456 q^{29} +2.00000 q^{31} -1.00000 q^{32} -1.37228 q^{34} +4.37228 q^{35} +2.00000 q^{37} -5.00000 q^{38} -4.37228 q^{40} +4.62772 q^{41} -8.11684 q^{43} -1.37228 q^{44} +1.62772 q^{46} +1.00000 q^{49} -14.1168 q^{50} +2.00000 q^{52} -8.74456 q^{53} -6.00000 q^{55} -1.00000 q^{56} +8.74456 q^{58} -10.1168 q^{59} +3.11684 q^{61} -2.00000 q^{62} +1.00000 q^{64} +8.74456 q^{65} -2.11684 q^{67} +1.37228 q^{68} -4.37228 q^{70} -7.11684 q^{71} +12.1168 q^{73} -2.00000 q^{74} +5.00000 q^{76} -1.37228 q^{77} -5.11684 q^{79} +4.37228 q^{80} -4.62772 q^{82} +17.4891 q^{83} +6.00000 q^{85} +8.11684 q^{86} +1.37228 q^{88} +14.7446 q^{89} +2.00000 q^{91} -1.62772 q^{92} +21.8614 q^{95} -8.11684 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{7} - 2 q^{8} - 3 q^{10} + 3 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 3 q^{17} + 10 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} + 11 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 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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