LMFDB - Embedded newform 1170.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.65685 q^{11} -1.00000 q^{13} +2.82843 q^{14} +1.00000 q^{16} -0.828427 q^{17} +2.82843 q^{19} -1.00000 q^{20} +5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.82843 q^{28} +8.82843 q^{29} +4.00000 q^{31} -1.00000 q^{32} +0.828427 q^{34} +2.82843 q^{35} -11.6569 q^{37} -2.82843 q^{38} +1.00000 q^{40} +7.65685 q^{41} +9.65685 q^{43} -5.65685 q^{44} -8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -13.3137 q^{53} +5.65685 q^{55} +2.82843 q^{56} -8.82843 q^{58} +2.34315 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +5.65685 q^{67} -0.828427 q^{68} -2.82843 q^{70} +5.65685 q^{71} -14.4853 q^{73} +11.6569 q^{74} +2.82843 q^{76} +16.0000 q^{77} +2.34315 q^{79} -1.00000 q^{80} -7.65685 q^{82} -6.34315 q^{83} +0.828427 q^{85} -9.65685 q^{86} +5.65685 q^{88} +15.6569 q^{89} +2.82843 q^{91} +8.48528 q^{92} -8.00000 q^{94} -2.82843 q^{95} -3.17157 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{25} + 2 q^{26} + 12 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} - 12 q^{37} + 2 q^{40} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 4 q^{53} - 12 q^{58} + 16 q^{59} + 12 q^{61} - 8 q^{62} + 2 q^{64} + 2 q^{65} + 4 q^{68} - 12 q^{73} + 12 q^{74} + 32 q^{77} + 16 q^{79} - 2 q^{80} - 4 q^{82} - 24 q^{83} - 4 q^{85} - 8 q^{86} + 20 q^{89} - 16 q^{94} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^25 + 2 * q^26 + 12 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 12 * q^37 + 2 * q^40 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 4 * q^53 - 12 * q^58 + 16 * q^59 + 12 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 4 * q^68 - 12 * q^73 + 12 * q^74 + 32 * q^77 + 16 * q^79 - 2 * q^80 - 4 * q^82 - 24 * q^83 - 4 * q^85 - 8 * q^86 + 20 * q^89 - 16 * q^94 - 12 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.82843	0.755929
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	5.65685	1.20605
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−2.82843	−0.534522
$29$	8.82843	1.63940	0.819699	−	0.572795i	$-0.194141\pi$
$29$	8.82843	1.63940	0.819699	+	0.572795i	$0.194141\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.828427	0.142074
$35$	2.82843	0.478091
$36$	0	0
$37$	−11.6569	−1.91638	−0.958188	−	0.286141i	$-0.907627\pi$
$37$	−11.6569	−1.91638	−0.958188	+	0.286141i	$0.907627\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	1.00000	0.158114
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	−5.65685	−0.852803
$45$	0	0
$46$	−8.48528	−1.25109
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	2.82843	0.377964
$57$	0	0
$58$	−8.82843	−1.15923
$59$	2.34315	0.305052	0.152526	−	0.988299i	$-0.451259\pi$
$59$	2.34315	0.305052	0.152526	+	0.988299i	$0.451259\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	−0.828427	−0.100462
$69$	0	0
$70$	−2.82843	−0.338062
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−14.4853	−1.69537	−0.847687	−	0.530497i	$-0.822005\pi$
$73$	−14.4853	−1.69537	−0.847687	+	0.530497i	$0.822005\pi$
$74$	11.6569	1.35508
$75$	0	0
$76$	2.82843	0.324443
$77$	16.0000	1.82337
$78$	0	0
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−7.65685	−0.845558
$83$	−6.34315	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$83$	−6.34315	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$84$	0	0
$85$	0.828427	0.0898555
$86$	−9.65685	−1.04133
$87$	0	0
$88$	5.65685	0.603023
$89$	15.6569	1.65962	0.829812	−	0.558044i	$-0.188448\pi$
$89$	15.6569	1.65962	0.829812	+	0.558044i	$0.188448\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	8.48528	0.884652
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−3.17157	−0.322024	−0.161012	−	0.986952i	$-0.551476\pi$
$97$	−3.17157	−0.322024	−0.161012	+	0.986952i	$0.551476\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−16.1421	−1.60620	−0.803101	−	0.595843i	$-0.796818\pi$
$101$	−16.1421	−1.60620	−0.803101	+	0.595843i	$0.796818\pi$
$102$	0	0
$103$	−1.65685	−0.163255	−0.0816274	−	0.996663i	$-0.526012\pi$
$103$	−1.65685	−0.163255	−0.0816274	+	0.996663i	$0.526012\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	13.3137	1.29314
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	8.82843	0.845610	0.422805	−	0.906221i	$-0.361046\pi$
$109$	8.82843	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$110$	−5.65685	−0.539360
$111$	0	0
$112$	−2.82843	−0.267261
$113$	−6.48528	−0.610084	−0.305042	−	0.952339i	$-0.598670\pi$
$113$	−6.48528	−0.610084	−0.305042	+	0.952339i	$0.598670\pi$
$114$	0	0
$115$	−8.48528	−0.791257
$116$	8.82843	0.819699
$117$	0	0
$118$	−2.34315	−0.215704
$119$	2.34315	0.214796
$120$	0	0
$121$	21.0000	1.90909
$122$	−6.00000	−0.543214
$123$	0	0
$124$	4.00000	0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.65685	0.856907	0.428454	−	0.903564i	$-0.359059\pi$
$127$	9.65685	0.856907	0.428454	+	0.903564i	$0.359059\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	6.14214	0.536641	0.268320	−	0.963330i	$-0.413531\pi$
$131$	6.14214	0.536641	0.268320	+	0.963330i	$0.413531\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−5.65685	−0.488678
$135$	0	0
$136$	0.828427	0.0710370
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	0	0
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	2.82843	0.239046
$141$	0	0
$142$	−5.65685	−0.474713
$143$	5.65685	0.473050
$144$	0	0
$145$	−8.82843	−0.733161
$146$	14.4853	1.19881
$147$	0	0
$148$	−11.6569	−0.958188
$149$	−3.65685	−0.299581	−0.149791	−	0.988718i	$-0.547860\pi$
$149$	−3.65685	−0.299581	−0.149791	+	0.988718i	$0.547860\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	−16.0000	−1.28932
$155$	−4.00000	−0.321288
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	−2.34315	−0.186411
$159$	0	0
$160$	1.00000	0.0790569
$161$	−24.0000	−1.89146
$162$	0	0
$163$	11.3137	0.886158	0.443079	−	0.896483i	$-0.353886\pi$
$163$	11.3137	0.886158	0.443079	+	0.896483i	$0.353886\pi$
$164$	7.65685	0.597900
$165$	0	0
$166$	6.34315	0.492324
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.828427	−0.0635375
$171$	0	0
$172$	9.65685	0.736328
$173$	9.31371	0.708108	0.354054	−	0.935225i	$-0.384803\pi$
$173$	9.31371	0.708108	0.354054	+	0.935225i	$0.384803\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	−5.65685	−0.426401
$177$	0	0
$178$	−15.6569	−1.17353
$179$	−7.51472	−0.561676	−0.280838	−	0.959755i	$-0.590612\pi$
$179$	−7.51472	−0.561676	−0.280838	+	0.959755i	$0.590612\pi$
$180$	0	0
$181$	−7.65685	−0.569129	−0.284565	−	0.958657i	$-0.591849\pi$
$181$	−7.65685	−0.569129	−0.284565	+	0.958657i	$0.591849\pi$
$182$	−2.82843	−0.209657
$183$	0	0
$184$	−8.48528	−0.625543
$185$	11.6569	0.857029
$186$	0	0
$187$	4.68629	0.342696
$188$	8.00000	0.583460
$189$	0	0
$190$	2.82843	0.205196
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	0	0
$193$	2.48528	0.178894	0.0894472	−	0.995992i	$-0.471490\pi$
$193$	2.48528	0.178894	0.0894472	+	0.995992i	$0.471490\pi$
$194$	3.17157	0.227706
$195$	0	0
$196$	1.00000	0.0714286
$197$	13.3137	0.948562	0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137	0.948562	0.474281	+	0.880373i	$0.342708\pi$
$198$	0	0
$199$	10.3431	0.733206	0.366603	−	0.930377i	$-0.380521\pi$
$199$	10.3431	0.733206	0.366603	+	0.930377i	$0.380521\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	16.1421	1.13576
$203$	−24.9706	−1.75259
$204$	0	0
$205$	−7.65685	−0.534778
$206$	1.65685	0.115439
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−16.0000	−1.10674
$210$	0	0
$211$	0.686292	0.0472463	0.0236231	−	0.999721i	$-0.492480\pi$
$211$	0.686292	0.0472463	0.0236231	+	0.999721i	$0.492480\pi$
$212$	−13.3137	−0.914389
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−9.65685	−0.658592
$216$	0	0
$217$	−11.3137	−0.768025
$218$	−8.82843	−0.597937
$219$	0	0
$220$	5.65685	0.381385
$221$	0.828427	0.0557260
$222$	0	0
$223$	10.8284	0.725125	0.362563	−	0.931959i	$-0.381902\pi$
$223$	10.8284	0.725125	0.362563	+	0.931959i	$0.381902\pi$
$224$	2.82843	0.188982
$225$	0	0
$226$	6.48528	0.431394
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	4.14214	0.273720	0.136860	−	0.990590i	$-0.456299\pi$
$229$	4.14214	0.273720	0.136860	+	0.990590i	$0.456299\pi$
$230$	8.48528	0.559503
$231$	0	0
$232$	−8.82843	−0.579615
$233$	−5.51472	−0.361281	−0.180641	−	0.983549i	$-0.557817\pi$
$233$	−5.51472	−0.361281	−0.180641	+	0.983549i	$0.557817\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	2.34315	0.152526
$237$	0	0
$238$	−2.34315	−0.151884
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	5.31371	0.342286	0.171143	−	0.985246i	$-0.445254\pi$
$241$	5.31371	0.342286	0.171143	+	0.985246i	$0.445254\pi$
$242$	−21.0000	−1.34993
$243$	0	0
$244$	6.00000	0.384111
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−2.82843	−0.179969
$248$	−4.00000	−0.254000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−10.8284	−0.683484	−0.341742	−	0.939794i	$-0.611017\pi$
$251$	−10.8284	−0.683484	−0.341742	+	0.939794i	$0.611017\pi$
$252$	0	0
$253$	−48.0000	−3.01773
$254$	−9.65685	−0.605925
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.82843	0.301189	0.150595	−	0.988596i	$-0.451881\pi$
$257$	4.82843	0.301189	0.150595	+	0.988596i	$0.451881\pi$
$258$	0	0
$259$	32.9706	2.04869
$260$	1.00000	0.0620174
$261$	0	0
$262$	−6.14214	−0.379462
$263$	−16.4853	−1.01653	−0.508263	−	0.861202i	$-0.669712\pi$
$263$	−16.4853	−1.01653	−0.508263	+	0.861202i	$0.669712\pi$
$264$	0	0
$265$	13.3137	0.817855
$266$	8.00000	0.490511
$267$	0	0
$268$	5.65685	0.345547
$269$	14.4853	0.883183	0.441592	−	0.897216i	$-0.354414\pi$
$269$	14.4853	0.883183	0.441592	+	0.897216i	$0.354414\pi$
$270$	0	0
$271$	7.31371	0.444276	0.222138	−	0.975015i	$-0.428696\pi$
$271$	7.31371	0.444276	0.222138	+	0.975015i	$0.428696\pi$
$272$	−0.828427	−0.0502308
$273$	0	0
$274$	−17.3137	−1.04596
$275$	−5.65685	−0.341121
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	−6.34315	−0.380437
$279$	0	0
$280$	−2.82843	−0.169031
$281$	−8.34315	−0.497710	−0.248855	−	0.968541i	$-0.580054\pi$
$281$	−8.34315	−0.497710	−0.248855	+	0.968541i	$0.580054\pi$
$282$	0	0
$283$	−17.6569	−1.04959	−0.524796	−	0.851228i	$-0.675858\pi$
$283$	−17.6569	−1.04959	−0.524796	+	0.851228i	$0.675858\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	−5.65685	−0.334497
$287$	−21.6569	−1.27836
$288$	0	0
$289$	−16.3137	−0.959630
$290$	8.82843	0.518423
$291$	0	0
$292$	−14.4853	−0.847687
$293$	16.6274	0.971384	0.485692	−	0.874130i	$-0.338568\pi$
$293$	16.6274	0.971384	0.485692	+	0.874130i	$0.338568\pi$
$294$	0	0
$295$	−2.34315	−0.136423
$296$	11.6569	0.677541
$297$	0	0
$298$	3.65685	0.211836
$299$	−8.48528	−0.490716
$300$	0	0
$301$	−27.3137	−1.57434
$302$	−12.0000	−0.690522
$303$	0	0
$304$	2.82843	0.162221
$305$	−6.00000	−0.343559
$306$	0	0
$307$	−21.6569	−1.23602	−0.618011	−	0.786169i	$-0.712061\pi$
$307$	−21.6569	−1.23602	−0.618011	+	0.786169i	$0.712061\pi$
$308$	16.0000	0.911685
$309$	0	0
$310$	4.00000	0.227185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−30.9706	−1.75056	−0.875280	−	0.483617i	$-0.839323\pi$
$313$	−30.9706	−1.75056	−0.875280	+	0.483617i	$0.839323\pi$
$314$	−5.31371	−0.299870
$315$	0	0
$316$	2.34315	0.131812
$317$	−25.3137	−1.42176	−0.710880	−	0.703314i	$-0.751703\pi$
$317$	−25.3137	−1.42176	−0.710880	+	0.703314i	$0.751703\pi$
$318$	0	0
$319$	−49.9411	−2.79617
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	24.0000	1.33747
$323$	−2.34315	−0.130376
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−11.3137	−0.626608
$327$	0	0
$328$	−7.65685	−0.422779
$329$	−22.6274	−1.24749
$330$	0	0
$331$	−8.48528	−0.466393	−0.233197	−	0.972430i	$-0.574919\pi$
$331$	−8.48528	−0.466393	−0.233197	+	0.972430i	$0.574919\pi$
$332$	−6.34315	−0.348125
$333$	0	0
$334$	8.97056	0.490847
$335$	−5.65685	−0.309067
$336$	0	0
$337$	10.9706	0.597605	0.298802	−	0.954315i	$-0.403413\pi$
$337$	10.9706	0.597605	0.298802	+	0.954315i	$0.403413\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	0.828427	0.0449278
$341$	−22.6274	−1.22534
$342$	0	0
$343$	16.9706	0.916324
$344$	−9.65685	−0.520663
$345$	0	0
$346$	−9.31371	−0.500708
$347$	9.65685	0.518407	0.259204	−	0.965823i	$-0.416540\pi$
$347$	9.65685	0.518407	0.259204	+	0.965823i	$0.416540\pi$
$348$	0	0
$349$	12.1421	0.649954	0.324977	−	0.945722i	$-0.394644\pi$
$349$	12.1421	0.649954	0.324977	+	0.945722i	$0.394644\pi$
$350$	2.82843	0.151186
$351$	0	0
$352$	5.65685	0.301511
$353$	−5.31371	−0.282820	−0.141410	−	0.989951i	$-0.545164\pi$
$353$	−5.31371	−0.282820	−0.141410	+	0.989951i	$0.545164\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	15.6569	0.829812
$357$	0	0
$358$	7.51472	0.397165
$359$	28.2843	1.49279	0.746393	−	0.665505i	$-0.231784\pi$
$359$	28.2843	1.49279	0.746393	+	0.665505i	$0.231784\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	7.65685	0.402435
$363$	0	0
$364$	2.82843	0.148250
$365$	14.4853	0.758194
$366$	0	0
$367$	25.6569	1.33928	0.669638	−	0.742687i	$-0.266449\pi$
$367$	25.6569	1.33928	0.669638	+	0.742687i	$0.266449\pi$
$368$	8.48528	0.442326
$369$	0	0
$370$	−11.6569	−0.606011
$371$	37.6569	1.95505
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	−4.68629	−0.242322
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−8.82843	−0.454687
$378$	0	0
$379$	−7.51472	−0.386005	−0.193003	−	0.981198i	$-0.561823\pi$
$379$	−7.51472	−0.386005	−0.193003	+	0.981198i	$0.561823\pi$
$380$	−2.82843	−0.145095
$381$	0	0
$382$	11.3137	0.578860
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	−2.48528	−0.126497
$387$	0	0
$388$	−3.17157	−0.161012
$389$	6.48528	0.328817	0.164408	−	0.986392i	$-0.447429\pi$
$389$	6.48528	0.328817	0.164408	+	0.986392i	$0.447429\pi$
$390$	0	0
$391$	−7.02944	−0.355494
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−13.3137	−0.670735
$395$	−2.34315	−0.117896
$396$	0	0
$397$	30.2843	1.51992	0.759962	−	0.649968i	$-0.225218\pi$
$397$	30.2843	1.51992	0.759962	+	0.649968i	$0.225218\pi$
$398$	−10.3431	−0.518455
$399$	0	0
$400$	1.00000	0.0500000
$401$	26.9706	1.34685	0.673423	−	0.739258i	$-0.264823\pi$
$401$	26.9706	1.34685	0.673423	+	0.739258i	$0.264823\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	−16.1421	−0.803101
$405$	0	0
$406$	24.9706	1.23927
$407$	65.9411	3.26858
$408$	0	0
$409$	−3.65685	−0.180820	−0.0904099	−	0.995905i	$-0.528818\pi$
$409$	−3.65685	−0.180820	−0.0904099	+	0.995905i	$0.528818\pi$
$410$	7.65685	0.378145
$411$	0	0
$412$	−1.65685	−0.0816274
$413$	−6.62742	−0.326114
$414$	0	0
$415$	6.34315	0.311373
$416$	1.00000	0.0490290
$417$	0	0
$418$	16.0000	0.782586
$419$	10.8284	0.529003	0.264502	−	0.964385i	$-0.414793\pi$
$419$	10.8284	0.529003	0.264502	+	0.964385i	$0.414793\pi$
$420$	0	0
$421$	−24.1421	−1.17662	−0.588308	−	0.808637i	$-0.700206\pi$
$421$	−24.1421	−1.17662	−0.588308	+	0.808637i	$0.700206\pi$
$422$	−0.686292	−0.0334081
$423$	0	0
$424$	13.3137	0.646571
$425$	−0.828427	−0.0401846
$426$	0	0
$427$	−16.9706	−0.821263
$428$	4.00000	0.193347
$429$	0	0
$430$	9.65685	0.465695
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−22.9706	−1.10389	−0.551947	−	0.833879i	$-0.686115\pi$
$433$	−22.9706	−1.10389	−0.551947	+	0.833879i	$0.686115\pi$
$434$	11.3137	0.543075
$435$	0	0
$436$	8.82843	0.422805
$437$	24.0000	1.14808
$438$	0	0
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	−5.65685	−0.269680
$441$	0	0
$442$	−0.828427	−0.0394043
$443$	−30.3431	−1.44165	−0.720823	−	0.693119i	$-0.756236\pi$
$443$	−30.3431	−1.44165	−0.720823	+	0.693119i	$0.756236\pi$
$444$	0	0
$445$	−15.6569	−0.742206
$446$	−10.8284	−0.512741
$447$	0	0
$448$	−2.82843	−0.133631
$449$	−26.2843	−1.24043	−0.620216	−	0.784431i	$-0.712955\pi$
$449$	−26.2843	−1.24043	−0.620216	+	0.784431i	$0.712955\pi$
$450$	0	0
$451$	−43.3137	−2.03956
$452$	−6.48528	−0.305042
$453$	0	0
$454$	4.00000	0.187729
$455$	−2.82843	−0.132599
$456$	0	0
$457$	20.8284	0.974313	0.487156	−	0.873315i	$-0.338034\pi$
$457$	20.8284	0.974313	0.487156	+	0.873315i	$0.338034\pi$
$458$	−4.14214	−0.193549
$459$	0	0
$460$	−8.48528	−0.395628
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−3.79899	−0.176554	−0.0882770	−	0.996096i	$-0.528136\pi$
$463$	−3.79899	−0.176554	−0.0882770	+	0.996096i	$0.528136\pi$
$464$	8.82843	0.409849
$465$	0	0
$466$	5.51472	0.255464
$467$	−7.31371	−0.338438	−0.169219	−	0.985578i	$-0.554125\pi$
$467$	−7.31371	−0.338438	−0.169219	+	0.985578i	$0.554125\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	8.00000	0.369012
$471$	0	0
$472$	−2.34315	−0.107852
$473$	−54.6274	−2.51177
$474$	0	0
$475$	2.82843	0.129777
$476$	2.34315	0.107398
$477$	0	0
$478$	16.0000	0.731823
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	−5.31371	−0.242033
$483$	0	0
$484$	21.0000	0.954545
$485$	3.17157	0.144014
$486$	0	0
$487$	16.4853	0.747019	0.373510	−	0.927626i	$-0.378154\pi$
$487$	16.4853	0.747019	0.373510	+	0.927626i	$0.378154\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	1.00000	0.0451754
$491$	38.1421	1.72133	0.860665	−	0.509171i	$-0.170048\pi$
$491$	38.1421	1.72133	0.860665	+	0.509171i	$0.170048\pi$
$492$	0	0
$493$	−7.31371	−0.329393
$494$	2.82843	0.127257
$495$	0	0
$496$	4.00000	0.179605
$497$	−16.0000	−0.717698
$498$	0	0
$499$	0.485281	0.0217242	0.0108621	−	0.999941i	$-0.496542\pi$
$499$	0.485281	0.0217242	0.0108621	+	0.999941i	$0.496542\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	10.8284	0.483296
$503$	23.5147	1.04847	0.524235	−	0.851574i	$-0.324351\pi$
$503$	23.5147	1.04847	0.524235	+	0.851574i	$0.324351\pi$
$504$	0	0
$505$	16.1421	0.718316
$506$	48.0000	2.13386
$507$	0	0
$508$	9.65685	0.428454
$509$	37.3137	1.65390	0.826951	−	0.562275i	$-0.190074\pi$
$509$	37.3137	1.65390	0.826951	+	0.562275i	$0.190074\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−4.82843	−0.212973
$515$	1.65685	0.0730097
$516$	0	0
$517$	−45.2548	−1.99031
$518$	−32.9706	−1.44864
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	−26.9706	−1.18160	−0.590801	−	0.806817i	$-0.701188\pi$
$521$	−26.9706	−1.18160	−0.590801	+	0.806817i	$0.701188\pi$
$522$	0	0
$523$	−10.6274	−0.464704	−0.232352	−	0.972632i	$-0.574642\pi$
$523$	−10.6274	−0.464704	−0.232352	+	0.972632i	$0.574642\pi$
$524$	6.14214	0.268320
$525$	0	0
$526$	16.4853	0.718792
$527$	−3.31371	−0.144347
$528$	0	0
$529$	49.0000	2.13043
$530$	−13.3137	−0.578311
$531$	0	0
$532$	−8.00000	−0.346844
$533$	−7.65685	−0.331655
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−5.65685	−0.244339
$537$	0	0
$538$	−14.4853	−0.624505
$539$	−5.65685	−0.243658
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	−7.31371	−0.314151
$543$	0	0
$544$	0.828427	0.0355185
$545$	−8.82843	−0.378168
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	17.3137	0.739605
$549$	0	0
$550$	5.65685	0.241209
$551$	24.9706	1.06378
$552$	0	0
$553$	−6.62742	−0.281826
$554$	−26.0000	−1.10463
$555$	0	0
$556$	6.34315	0.269009
$557$	−10.6863	−0.452793	−0.226396	−	0.974035i	$-0.572694\pi$
$557$	−10.6863	−0.452793	−0.226396	+	0.974035i	$0.572694\pi$
$558$	0	0
$559$	−9.65685	−0.408441
$560$	2.82843	0.119523
$561$	0	0
$562$	8.34315	0.351934
$563$	30.3431	1.27881	0.639406	−	0.768870i	$-0.279181\pi$
$563$	30.3431	1.27881	0.639406	+	0.768870i	$0.279181\pi$
$564$	0	0
$565$	6.48528	0.272838
$566$	17.6569	0.742173
$567$	0	0
$568$	−5.65685	−0.237356
$569$	−31.6569	−1.32712	−0.663562	−	0.748121i	$-0.730956\pi$
$569$	−31.6569	−1.32712	−0.663562	+	0.748121i	$0.730956\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	5.65685	0.236525
$573$	0	0
$574$	21.6569	0.903940
$575$	8.48528	0.353861
$576$	0	0
$577$	−23.4558	−0.976480	−0.488240	−	0.872710i	$-0.662361\pi$
$577$	−23.4558	−0.976480	−0.488240	+	0.872710i	$0.662361\pi$
$578$	16.3137	0.678561
$579$	0	0
$580$	−8.82843	−0.366580
$581$	17.9411	0.744323
$582$	0	0
$583$	75.3137	3.11918
$584$	14.4853	0.599405
$585$	0	0
$586$	−16.6274	−0.686872
$587$	−2.62742	−0.108445	−0.0542226	−	0.998529i	$-0.517268\pi$
$587$	−2.62742	−0.108445	−0.0542226	+	0.998529i	$0.517268\pi$
$588$	0	0
$589$	11.3137	0.466173
$590$	2.34315	0.0964658
$591$	0	0
$592$	−11.6569	−0.479094
$593$	0.343146	0.0140913	0.00704565	−	0.999975i	$-0.497757\pi$
$593$	0.343146	0.0140913	0.00704565	+	0.999975i	$0.497757\pi$
$594$	0	0
$595$	−2.34315	−0.0960596
$596$	−3.65685	−0.149791
$597$	0	0
$598$	8.48528	0.346989
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	29.3137	1.19573	0.597866	−	0.801596i	$-0.296016\pi$
$601$	29.3137	1.19573	0.597866	+	0.801596i	$0.296016\pi$
$602$	27.3137	1.11322
$603$	0	0
$604$	12.0000	0.488273
$605$	−21.0000	−0.853771
$606$	0	0
$607$	28.9706	1.17588	0.587939	−	0.808905i	$-0.299939\pi$
$607$	28.9706	1.17588	0.587939	+	0.808905i	$0.299939\pi$
$608$	−2.82843	−0.114708
$609$	0	0
$610$	6.00000	0.242933
$611$	−8.00000	−0.323645
$612$	0	0
$613$	22.2843	0.900053	0.450027	−	0.893015i	$-0.351414\pi$
$613$	22.2843	0.900053	0.450027	+	0.893015i	$0.351414\pi$
$614$	21.6569	0.874000
$615$	0	0
$616$	−16.0000	−0.644658
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	−34.8284	−1.39987	−0.699936	−	0.714205i	$-0.746788\pi$
$619$	−34.8284	−1.39987	−0.699936	+	0.714205i	$0.746788\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	−24.0000	−0.962312
$623$	−44.2843	−1.77421
$624$	0	0
$625$	1.00000	0.0400000
$626$	30.9706	1.23783
$627$	0	0
$628$	5.31371	0.212040
$629$	9.65685	0.385044
$630$	0	0
$631$	−33.6569	−1.33986	−0.669929	−	0.742425i	$-0.733676\pi$
$631$	−33.6569	−1.33986	−0.669929	+	0.742425i	$0.733676\pi$
$632$	−2.34315	−0.0932053
$633$	0	0
$634$	25.3137	1.00534
$635$	−9.65685	−0.383221
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	49.9411	1.97719
$639$	0	0
$640$	1.00000	0.0395285
$641$	4.62742	0.182772	0.0913860	−	0.995816i	$-0.470870\pi$
$641$	4.62742	0.182772	0.0913860	+	0.995816i	$0.470870\pi$
$642$	0	0
$643$	39.5980	1.56159	0.780796	−	0.624786i	$-0.214814\pi$
$643$	39.5980	1.56159	0.780796	+	0.624786i	$0.214814\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	2.34315	0.0921898
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	0	0
$649$	−13.2548	−0.520298
$650$	1.00000	0.0392232
$651$	0	0
$652$	11.3137	0.443079
$653$	42.2843	1.65471	0.827356	−	0.561678i	$-0.189844\pi$
$653$	42.2843	1.65471	0.827356	+	0.561678i	$0.189844\pi$
$654$	0	0
$655$	−6.14214	−0.239993
$656$	7.65685	0.298950
$657$	0	0
$658$	22.6274	0.882109
$659$	7.51472	0.292732	0.146366	−	0.989231i	$-0.453242\pi$
$659$	7.51472	0.292732	0.146366	+	0.989231i	$0.453242\pi$
$660$	0	0
$661$	−8.14214	−0.316692	−0.158346	−	0.987384i	$-0.550616\pi$
$661$	−8.14214	−0.316692	−0.158346	+	0.987384i	$0.550616\pi$
$662$	8.48528	0.329790
$663$	0	0
$664$	6.34315	0.246162
$665$	8.00000	0.310227
$666$	0	0
$667$	74.9117	2.90059
$668$	−8.97056	−0.347081
$669$	0	0
$670$	5.65685	0.218543
$671$	−33.9411	−1.31028
$672$	0	0
$673$	32.6274	1.25769	0.628847	−	0.777529i	$-0.283527\pi$
$673$	32.6274	1.25769	0.628847	+	0.777529i	$0.283527\pi$
$674$	−10.9706	−0.422570
$675$	0	0
$676$	1.00000	0.0384615
$677$	−12.3431	−0.474386	−0.237193	−	0.971463i	$-0.576227\pi$
$677$	−12.3431	−0.474386	−0.237193	+	0.971463i	$0.576227\pi$
$678$	0	0
$679$	8.97056	0.344259
$680$	−0.828427	−0.0317687
$681$	0	0
$682$	22.6274	0.866449
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	0	0
$685$	−17.3137	−0.661523
$686$	−16.9706	−0.647939
$687$	0	0
$688$	9.65685	0.368164
$689$	13.3137	0.507212
$690$	0	0
$691$	−27.7990	−1.05752	−0.528762	−	0.848770i	$-0.677343\pi$
$691$	−27.7990	−1.05752	−0.528762	+	0.848770i	$0.677343\pi$
$692$	9.31371	0.354054
$693$	0	0
$694$	−9.65685	−0.366569
$695$	−6.34315	−0.240609
$696$	0	0
$697$	−6.34315	−0.240264
$698$	−12.1421	−0.459587
$699$	0	0
$700$	−2.82843	−0.106904
$701$	−0.142136	−0.00536839	−0.00268419	−	0.999996i	$-0.500854\pi$
$701$	−0.142136	−0.00536839	−0.00268419	+	0.999996i	$0.500854\pi$
$702$	0	0
$703$	−32.9706	−1.24351
$704$	−5.65685	−0.213201
$705$	0	0
$706$	5.31371	0.199984
$707$	45.6569	1.71710
$708$	0	0
$709$	−7.17157	−0.269334	−0.134667	−	0.990891i	$-0.542996\pi$
$709$	−7.17157	−0.269334	−0.134667	+	0.990891i	$0.542996\pi$
$710$	5.65685	0.212298
$711$	0	0
$712$	−15.6569	−0.586765
$713$	33.9411	1.27111
$714$	0	0
$715$	−5.65685	−0.211554
$716$	−7.51472	−0.280838
$717$	0	0
$718$	−28.2843	−1.05556
$719$	29.6569	1.10601	0.553007	−	0.833177i	$-0.313480\pi$
$719$	29.6569	1.10601	0.553007	+	0.833177i	$0.313480\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	11.0000	0.409378
$723$	0	0
$724$	−7.65685	−0.284565
$725$	8.82843	0.327880
$726$	0	0
$727$	−45.9411	−1.70386	−0.851931	−	0.523654i	$-0.824568\pi$
$727$	−45.9411	−1.70386	−0.851931	+	0.523654i	$0.824568\pi$
$728$	−2.82843	−0.104828
$729$	0	0
$730$	−14.4853	−0.536124
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−0.343146	−0.0126744	−0.00633719	−	0.999980i	$-0.502017\pi$
$733$	−0.343146	−0.0126744	−0.00633719	+	0.999980i	$0.502017\pi$
$734$	−25.6569	−0.947012
$735$	0	0
$736$	−8.48528	−0.312772
$737$	−32.0000	−1.17874
$738$	0	0
$739$	−14.1421	−0.520227	−0.260113	−	0.965578i	$-0.583760\pi$
$739$	−14.1421	−0.520227	−0.260113	+	0.965578i	$0.583760\pi$
$740$	11.6569	0.428514
$741$	0	0
$742$	−37.6569	−1.38243
$743$	36.2843	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$743$	36.2843	1.33114	0.665570	+	0.746335i	$0.268188\pi$
$744$	0	0
$745$	3.65685	0.133977
$746$	2.68629	0.0983521
$747$	0	0
$748$	4.68629	0.171348
$749$	−11.3137	−0.413394
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	8.82843	0.321512
$755$	−12.0000	−0.436725
$756$	0	0
$757$	19.9411	0.724773	0.362386	−	0.932028i	$-0.381962\pi$
$757$	19.9411	0.724773	0.362386	+	0.932028i	$0.381962\pi$
$758$	7.51472	0.272947
$759$	0	0
$760$	2.82843	0.102598
$761$	−27.6569	−1.00256	−0.501280	−	0.865285i	$-0.667137\pi$
$761$	−27.6569	−1.00256	−0.501280	+	0.865285i	$0.667137\pi$
$762$	0	0
$763$	−24.9706	−0.903995
$764$	−11.3137	−0.409316
$765$	0	0
$766$	−29.6569	−1.07155
$767$	−2.34315	−0.0846061
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	16.0000	0.576600
$771$	0	0
$772$	2.48528	0.0894472
$773$	53.3137	1.91756	0.958780	−	0.284148i	$-0.0917107\pi$
$773$	53.3137	1.91756	0.958780	+	0.284148i	$0.0917107\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	3.17157	0.113853
$777$	0	0
$778$	−6.48528	−0.232509
$779$	21.6569	0.775937
$780$	0	0
$781$	−32.0000	−1.14505
$782$	7.02944	0.251372
$783$	0	0
$784$	1.00000	0.0357143
$785$	−5.31371	−0.189654
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	13.3137	0.474281
$789$	0	0
$790$	2.34315	0.0833654
$791$	18.3431	0.652207
$792$	0	0
$793$	−6.00000	−0.213066
$794$	−30.2843	−1.07475
$795$	0	0
$796$	10.3431	0.366603
$797$	−16.6274	−0.588973	−0.294487	−	0.955656i	$-0.595149\pi$
$797$	−16.6274	−0.588973	−0.294487	+	0.955656i	$0.595149\pi$
$798$	0	0
$799$	−6.62742	−0.234461
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−26.9706	−0.952364
$803$	81.9411	2.89164
$804$	0	0
$805$	24.0000	0.845889
$806$	4.00000	0.140894
$807$	0	0
$808$	16.1421	0.567878
$809$	−13.3137	−0.468085	−0.234043	−	0.972226i	$-0.575196\pi$
$809$	−13.3137	−0.468085	−0.234043	+	0.972226i	$0.575196\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	−24.9706	−0.876295
$813$	0	0
$814$	−65.9411	−2.31124
$815$	−11.3137	−0.396302
$816$	0	0
$817$	27.3137	0.955586
$818$	3.65685	0.127859
$819$	0	0
$820$	−7.65685	−0.267389
$821$	−34.2843	−1.19653	−0.598265	−	0.801299i	$-0.704143\pi$
$821$	−34.2843	−1.19653	−0.598265	+	0.801299i	$0.704143\pi$
$822$	0	0
$823$	−52.9706	−1.84644	−0.923219	−	0.384275i	$-0.874452\pi$
$823$	−52.9706	−1.84644	−0.923219	+	0.384275i	$0.874452\pi$
$824$	1.65685	0.0577193
$825$	0	0
$826$	6.62742	0.230597
$827$	−9.65685	−0.335802	−0.167901	−	0.985804i	$-0.553699\pi$
$827$	−9.65685	−0.335802	−0.167901	+	0.985804i	$0.553699\pi$
$828$	0	0
$829$	−53.3137	−1.85166	−0.925831	−	0.377938i	$-0.876633\pi$
$829$	−53.3137	−1.85166	−0.925831	+	0.377938i	$0.876633\pi$
$830$	−6.34315	−0.220174
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	−0.828427	−0.0287033
$834$	0	0
$835$	8.97056	0.310439
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−10.8284	−0.374062
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	24.1421	0.831993
$843$	0	0
$844$	0.686292	0.0236231
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−59.3970	−2.04090
$848$	−13.3137	−0.457195
$849$	0	0
$850$	0.828427	0.0284148
$851$	−98.9117	−3.39065
$852$	0	0
$853$	38.2843	1.31083	0.655414	−	0.755270i	$-0.272494\pi$
$853$	38.2843	1.31083	0.655414	+	0.755270i	$0.272494\pi$
$854$	16.9706	0.580721
$855$	0	0
$856$	−4.00000	−0.136717
$857$	20.8284	0.711486	0.355743	−	0.934584i	$-0.384228\pi$
$857$	20.8284	0.711486	0.355743	+	0.934584i	$0.384228\pi$
$858$	0	0
$859$	37.9411	1.29453	0.647267	−	0.762263i	$-0.275912\pi$
$859$	37.9411	1.29453	0.647267	+	0.762263i	$0.275912\pi$
$860$	−9.65685	−0.329296
$861$	0	0
$862$	−16.0000	−0.544962
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	−9.31371	−0.316676
$866$	22.9706	0.780571
$867$	0	0
$868$	−11.3137	−0.384012
$869$	−13.2548	−0.449639
$870$	0	0
$871$	−5.65685	−0.191675
$872$	−8.82843	−0.298968
$873$	0	0
$874$	−24.0000	−0.811812
$875$	2.82843	0.0956183
$876$	0	0
$877$	−51.2548	−1.73075	−0.865376	−	0.501122i	$-0.832921\pi$
$877$	−51.2548	−1.73075	−0.865376	+	0.501122i	$0.832921\pi$
$878$	−22.6274	−0.763638
$879$	0	0
$880$	5.65685	0.190693
$881$	10.2843	0.346486	0.173243	−	0.984879i	$-0.444575\pi$
$881$	10.2843	0.346486	0.173243	+	0.984879i	$0.444575\pi$
$882$	0	0
$883$	31.3137	1.05379	0.526895	−	0.849930i	$-0.323356\pi$
$883$	31.3137	1.05379	0.526895	+	0.849930i	$0.323356\pi$
$884$	0.828427	0.0278630
$885$	0	0
$886$	30.3431	1.01940
$887$	40.4853	1.35936	0.679681	−	0.733508i	$-0.262118\pi$
$887$	40.4853	1.35936	0.679681	+	0.733508i	$0.262118\pi$
$888$	0	0
$889$	−27.3137	−0.916072
$890$	15.6569	0.524819
$891$	0	0
$892$	10.8284	0.362563
$893$	22.6274	0.757198
$894$	0	0
$895$	7.51472	0.251189
$896$	2.82843	0.0944911
$897$	0	0
$898$	26.2843	0.877117
$899$	35.3137	1.17778
$900$	0	0
$901$	11.0294	0.367444
$902$	43.3137	1.44219
$903$	0	0
$904$	6.48528	0.215697
$905$	7.65685	0.254522
$906$	0	0
$907$	8.28427	0.275075	0.137537	−	0.990497i	$-0.456081\pi$
$907$	8.28427	0.275075	0.137537	+	0.990497i	$0.456081\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	2.82843	0.0937614
$911$	−24.9706	−0.827312	−0.413656	−	0.910433i	$-0.635748\pi$
$911$	−24.9706	−0.827312	−0.413656	+	0.910433i	$0.635748\pi$
$912$	0	0
$913$	35.8823	1.18753
$914$	−20.8284	−0.688943
$915$	0	0
$916$	4.14214	0.136860
$917$	−17.3726	−0.573693
$918$	0	0
$919$	41.9411	1.38351	0.691755	−	0.722132i	$-0.256838\pi$
$919$	41.9411	1.38351	0.691755	+	0.722132i	$0.256838\pi$
$920$	8.48528	0.279751
$921$	0	0
$922$	14.0000	0.461065
$923$	−5.65685	−0.186198
$924$	0	0
$925$	−11.6569	−0.383275
$926$	3.79899	0.124843
$927$	0	0
$928$	−8.82843	−0.289807
$929$	33.5980	1.10231	0.551157	−	0.834402i	$-0.314187\pi$
$929$	33.5980	1.10231	0.551157	+	0.834402i	$0.314187\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	−5.51472	−0.180641
$933$	0	0
$934$	7.31371	0.239312
$935$	−4.68629	−0.153258
$936$	0	0
$937$	16.6274	0.543194	0.271597	−	0.962411i	$-0.412448\pi$
$937$	16.6274	0.543194	0.271597	+	0.962411i	$0.412448\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−54.9706	−1.79199	−0.895995	−	0.444065i	$-0.853536\pi$
$941$	−54.9706	−1.79199	−0.895995	+	0.444065i	$0.853536\pi$
$942$	0	0
$943$	64.9706	2.11573
$944$	2.34315	0.0762629
$945$	0	0
$946$	54.6274	1.77609
$947$	−30.3431	−0.986020	−0.493010	−	0.870024i	$-0.664103\pi$
$947$	−30.3431	−0.986020	−0.493010	+	0.870024i	$0.664103\pi$
$948$	0	0
$949$	14.4853	0.470212
$950$	−2.82843	−0.0917663
$951$	0	0
$952$	−2.34315	−0.0759418
$953$	27.8579	0.902405	0.451202	−	0.892422i	$-0.350995\pi$
$953$	27.8579	0.902405	0.451202	+	0.892422i	$0.350995\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	−16.0000	−0.517477
$957$	0	0
$958$	−11.3137	−0.365529
$959$	−48.9706	−1.58134
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−11.6569	−0.375832
$963$	0	0
$964$	5.31371	0.171143
$965$	−2.48528	−0.0800040
$966$	0	0
$967$	−7.51472	−0.241657	−0.120829	−	0.992673i	$-0.538555\pi$
$967$	−7.51472	−0.241657	−0.120829	+	0.992673i	$0.538555\pi$
$968$	−21.0000	−0.674966
$969$	0	0
$970$	−3.17157	−0.101833
$971$	−15.5147	−0.497891	−0.248946	−	0.968517i	$-0.580084\pi$
$971$	−15.5147	−0.497891	−0.248946	+	0.968517i	$0.580084\pi$
$972$	0	0
$973$	−17.9411	−0.575166
$974$	−16.4853	−0.528222
$975$	0	0
$976$	6.00000	0.192055
$977$	8.34315	0.266921	0.133460	−	0.991054i	$-0.457391\pi$
$977$	8.34315	0.266921	0.133460	+	0.991054i	$0.457391\pi$
$978$	0	0
$979$	−88.5685	−2.83066
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−38.1421	−1.21716
$983$	2.34315	0.0747347	0.0373674	−	0.999302i	$-0.488103\pi$
$983$	2.34315	0.0747347	0.0373674	+	0.999302i	$0.488103\pi$
$984$	0	0
$985$	−13.3137	−0.424210
$986$	7.31371	0.232916
$987$	0	0
$988$	−2.82843	−0.0899843
$989$	81.9411	2.60558
$990$	0	0
$991$	−42.9117	−1.36313	−0.681567	−	0.731755i	$-0.738701\pi$
$991$	−42.9117	−1.36313	−0.681567	+	0.731755i	$0.738701\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	16.0000	0.507489
$995$	−10.3431	−0.327900
$996$	0	0
$997$	61.3137	1.94182	0.970912	−	0.239435i	$-0.0769623\pi$
$997$	61.3137	1.94182	0.970912	+	0.239435i	$0.0769623\pi$
$998$	−0.485281	−0.0153613
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.o.1.1		2
3.2	odd	2		390.2.a.h.1.1	✓	2
4.3	odd	2		9360.2.a.ch.1.2		2
5.2	odd	4		5850.2.e.bk.5149.1		4
5.3	odd	4		5850.2.e.bk.5149.4		4
5.4	even	2		5850.2.a.cl.1.2		2
12.11	even	2		3120.2.a.bc.1.2		2
15.2	even	4		1950.2.e.o.1249.3		4
15.8	even	4		1950.2.e.o.1249.2		4
15.14	odd	2		1950.2.a.bd.1.2		2
39.5	even	4		5070.2.b.q.1351.1		4
39.8	even	4		5070.2.b.q.1351.4		4
39.38	odd	2		5070.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.1	✓	2	3.2	odd	2
1170.2.a.o.1.1		2	1.1	even	1	trivial
1950.2.a.bd.1.2		2	15.14	odd	2
1950.2.e.o.1249.2		4	15.8	even	4
1950.2.e.o.1249.3		4	15.2	even	4
3120.2.a.bc.1.2		2	12.11	even	2
5070.2.a.bc.1.2		2	39.38	odd	2
5070.2.b.q.1351.1		4	39.5	even	4
5070.2.b.q.1351.4		4	39.8	even	4
5850.2.a.cl.1.2		2	5.4	even	2
5850.2.e.bk.5149.1		4	5.2	odd	4
5850.2.e.bk.5149.4		4	5.3	odd	4
9360.2.a.ch.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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.82843 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.65685 q^{11} -1.00000 q^{13} +2.82843 q^{14} +1.00000 q^{16} -0.828427 q^{17} +2.82843 q^{19} -1.00000 q^{20} +5.65685 q^{22} +8.48528 q^{23} +1.00000 q^{25} +1.00000 q^{26} -2.82843 q^{28} +8.82843 q^{29} +4.00000 q^{31} -1.00000 q^{32} +0.828427 q^{34} +2.82843 q^{35} -11.6569 q^{37} -2.82843 q^{38} +1.00000 q^{40} +7.65685 q^{41} +9.65685 q^{43} -5.65685 q^{44} -8.48528 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -13.3137 q^{53} +5.65685 q^{55} +2.82843 q^{56} -8.82843 q^{58} +2.34315 q^{59} +6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +5.65685 q^{67} -0.828427 q^{68} -2.82843 q^{70} +5.65685 q^{71} -14.4853 q^{73} +11.6569 q^{74} +2.82843 q^{76} +16.0000 q^{77} +2.34315 q^{79} -1.00000 q^{80} -7.65685 q^{82} -6.34315 q^{83} +0.828427 q^{85} -9.65685 q^{86} +5.65685 q^{88} +15.6569 q^{89} +2.82843 q^{91} +8.48528 q^{92} -8.00000 q^{94} -2.82843 q^{95} -3.17157 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 2 q^{13} + 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{25} + 2 q^{26} + 12 q^{29} + 8 q^{31} - 2 q^{32} - 4 q^{34} - 12 q^{37} + 2 q^{40} + 4 q^{41} + 8 q^{43} + 16 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 4 q^{53} - 12 q^{58} + 16 q^{59} + 12 q^{61} - 8 q^{62} + 2 q^{64} + 2 q^{65} + 4 q^{68} - 12 q^{73} + 12 q^{74} + 32 q^{77} + 16 q^{79} - 2 q^{80} - 4 q^{82} - 24 q^{83} - 4 q^{85} - 8 q^{86} + 20 q^{89} - 16 q^{94} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 2 * q^13 + 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^25 + 2 * q^26 + 12 * q^29 + 8 * q^31 - 2 * q^32 - 4 * q^34 - 12 * q^37 + 2 * q^40 + 4 * q^41 + 8 * q^43 + 16 * q^47 + 2 * q^49 - 2 * q^50 - 2 * q^52 - 4 * q^53 - 12 * q^58 + 16 * q^59 + 12 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 4 * q^68 - 12 * q^73 + 12 * q^74 + 32 * q^77 + 16 * q^79 - 2 * q^80 - 4 * q^82 - 24 * q^83 - 4 * q^85 - 8 * q^86 + 20 * q^89 - 16 * q^94 - 12 * q^97 - 2 * q^98

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