LMFDB - Embedded newform 1170.2.a.p.1.2

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(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.60555 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.60555 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} -4.60555 q^{14} +1.00000 q^{16} +2.60555 q^{17} -0.605551 q^{19} -1.00000 q^{20} +2.60555 q^{23} +1.00000 q^{25} -1.00000 q^{26} +4.60555 q^{28} -2.60555 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.60555 q^{34} -4.60555 q^{35} -3.21110 q^{37} +0.605551 q^{38} +1.00000 q^{40} +11.2111 q^{41} -9.21110 q^{43} -2.60555 q^{46} -5.21110 q^{47} +14.2111 q^{49} -1.00000 q^{50} +1.00000 q^{52} -4.60555 q^{56} +2.60555 q^{58} -5.21110 q^{59} +7.21110 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +7.21110 q^{67} +2.60555 q^{68} +4.60555 q^{70} +12.0000 q^{71} -0.605551 q^{73} +3.21110 q^{74} -0.605551 q^{76} +13.2111 q^{79} -1.00000 q^{80} -11.2111 q^{82} -2.60555 q^{85} +9.21110 q^{86} +11.2111 q^{89} +4.60555 q^{91} +2.60555 q^{92} +5.21110 q^{94} +0.605551 q^{95} -0.605551 q^{97} -14.2111 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 2 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} + 8 q^{37} - 6 q^{38} + 2 q^{40} + 8 q^{41} - 4 q^{43} + 2 q^{46} + 4 q^{47} + 14 q^{49} - 2 q^{50} + 2 q^{52} - 2 q^{56} - 2 q^{58} + 4 q^{59} - 4 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{68} + 2 q^{70} + 24 q^{71} + 6 q^{73} - 8 q^{74} + 6 q^{76} + 12 q^{79} - 2 q^{80} - 8 q^{82} + 2 q^{85} + 4 q^{86} + 8 q^{89} + 2 q^{91} - 2 q^{92} - 4 q^{94} - 6 q^{95} + 6 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^23 + 2 * q^25 - 2 * q^26 + 2 * q^28 + 2 * q^29 + 4 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 + 8 * q^37 - 6 * q^38 + 2 * q^40 + 8 * q^41 - 4 * q^43 + 2 * q^46 + 4 * q^47 + 14 * q^49 - 2 * q^50 + 2 * q^52 - 2 * q^56 - 2 * q^58 + 4 * q^59 - 4 * q^62 + 2 * q^64 - 2 * q^65 - 2 * q^68 + 2 * q^70 + 24 * q^71 + 6 * q^73 - 8 * q^74 + 6 * q^76 + 12 * q^79 - 2 * q^80 - 8 * q^82 + 2 * q^85 + 4 * q^86 + 8 * q^89 + 2 * q^91 - 2 * q^92 - 4 * q^94 - 6 * q^95 + 6 * q^97 - 14 * 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
$7$	4.60555	1.74073	0.870367	−	0.492403i	$-0.163881\pi$
$7$	4.60555	1.74073	0.870367	+	0.492403i	$0.163881\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−4.60555	−1.23089
$15$	0	0
$16$	1.00000	0.250000
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	0	0
$19$	−0.605551	−0.138923	−0.0694615	−	0.997585i	$-0.522128\pi$
$19$	−0.605551	−0.138923	−0.0694615	+	0.997585i	$0.522128\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	2.60555	0.543295	0.271647	−	0.962397i	$-0.412432\pi$
$23$	2.60555	0.543295	0.271647	+	0.962397i	$0.412432\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	4.60555	0.870367
$29$	−2.60555	−0.483839	−0.241919	−	0.970296i	$-0.577777\pi$
$29$	−2.60555	−0.483839	−0.241919	+	0.970296i	$0.577777\pi$
$30$	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
$34$	−2.60555	−0.446848
$35$	−4.60555	−0.778480
$36$	0	0
$37$	−3.21110	−0.527902	−0.263951	−	0.964536i	$-0.585026\pi$
$37$	−3.21110	−0.527902	−0.263951	+	0.964536i	$0.585026\pi$
$38$	0.605551	0.0982334
$39$	0	0
$40$	1.00000	0.158114
$41$	11.2111	1.75088	0.875440	−	0.483327i	$-0.160572\pi$
$41$	11.2111	1.75088	0.875440	+	0.483327i	$0.160572\pi$
$42$	0	0
$43$	−9.21110	−1.40468	−0.702340	−	0.711842i	$-0.747861\pi$
$43$	−9.21110	−1.40468	−0.702340	+	0.711842i	$0.747861\pi$
$44$	0	0
$45$	0	0
$46$	−2.60555	−0.384168
$47$	−5.21110	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$47$	−5.21110	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$48$	0	0
$49$	14.2111	2.03016
$50$	−1.00000	−0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	−4.60555	−0.615443
$57$	0	0
$58$	2.60555	0.342126
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	7.21110	0.880976	0.440488	−	0.897758i	$-0.354805\pi$
$67$	7.21110	0.880976	0.440488	+	0.897758i	$0.354805\pi$
$68$	2.60555	0.315970
$69$	0	0
$70$	4.60555	0.550469
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−0.605551	−0.0708744	−0.0354372	−	0.999372i	$-0.511282\pi$
$73$	−0.605551	−0.0708744	−0.0354372	+	0.999372i	$0.511282\pi$
$74$	3.21110	0.373283
$75$	0	0
$76$	−0.605551	−0.0694615
$77$	0	0
$78$	0	0
$79$	13.2111	1.48637	0.743183	−	0.669089i	$-0.233315\pi$
$79$	13.2111	1.48637	0.743183	+	0.669089i	$0.233315\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−11.2111	−1.23806
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2.60555	−0.282612
$86$	9.21110	0.993259
$87$	0	0
$88$	0	0
$89$	11.2111	1.18837	0.594187	−	0.804327i	$-0.297474\pi$
$89$	11.2111	1.18837	0.594187	+	0.804327i	$0.297474\pi$
$90$	0	0
$91$	4.60555	0.482793
$92$	2.60555	0.271647
$93$	0	0
$94$	5.21110	0.537484
$95$	0.605551	0.0621282
$96$	0	0
$97$	−0.605551	−0.0614844	−0.0307422	−	0.999527i	$-0.509787\pi$
$97$	−0.605551	−0.0614844	−0.0307422	+	0.999527i	$0.509787\pi$
$98$	−14.2111	−1.43554
$99$	0	0
$100$	1.00000	0.100000
$101$	−7.81665	−0.777786	−0.388893	−	0.921283i	$-0.627142\pi$
$101$	−7.81665	−0.777786	−0.388893	+	0.921283i	$0.627142\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	11.3944	1.09139	0.545695	−	0.837984i	$-0.316266\pi$
$109$	11.3944	1.09139	0.545695	+	0.837984i	$0.316266\pi$
$110$	0	0
$111$	0	0
$112$	4.60555	0.435184
$113$	−14.6056	−1.37397	−0.686987	−	0.726669i	$-0.741067\pi$
$113$	−14.6056	−1.37397	−0.686987	+	0.726669i	$0.741067\pi$
$114$	0	0
$115$	−2.60555	−0.242969
$116$	−2.60555	−0.241919
$117$	0	0
$118$	5.21110	0.479721
$119$	12.0000	1.10004
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−7.21110	−0.652863
$123$	0	0
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.2111	1.17230	0.586148	−	0.810204i	$-0.300644\pi$
$127$	13.2111	1.17230	0.586148	+	0.810204i	$0.300644\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.00000	0.0877058
$131$	9.39445	0.820797	0.410398	−	0.911906i	$-0.365390\pi$
$131$	9.39445	0.820797	0.410398	+	0.911906i	$0.365390\pi$
$132$	0	0
$133$	−2.78890	−0.241828
$134$	−7.21110	−0.622944
$135$	0	0
$136$	−2.60555	−0.223424
$137$	21.6333	1.84826	0.924129	−	0.382080i	$-0.124792\pi$
$137$	21.6333	1.84826	0.924129	+	0.382080i	$0.124792\pi$
$138$	0	0
$139$	−14.4222	−1.22328	−0.611638	−	0.791138i	$-0.709489\pi$
$139$	−14.4222	−1.22328	−0.611638	+	0.791138i	$0.709489\pi$
$140$	−4.60555	−0.389240
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	2.60555	0.216379
$146$	0.605551	0.0501158
$147$	0	0
$148$	−3.21110	−0.263951
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−20.4222	−1.66194	−0.830968	−	0.556321i	$-0.812213\pi$
$151$	−20.4222	−1.66194	−0.830968	+	0.556321i	$0.812213\pi$
$152$	0.605551	0.0491167
$153$	0	0
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−3.21110	−0.256274	−0.128137	−	0.991756i	$-0.540900\pi$
$157$	−3.21110	−0.256274	−0.128137	+	0.991756i	$0.540900\pi$
$158$	−13.2111	−1.05102
$159$	0	0
$160$	1.00000	0.0790569
$161$	12.0000	0.945732
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	11.2111	0.875440
$165$	0	0
$166$	0	0
$167$	−10.4222	−0.806494	−0.403247	−	0.915091i	$-0.632119\pi$
$167$	−10.4222	−0.806494	−0.403247	+	0.915091i	$0.632119\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.60555	0.199837
$171$	0	0
$172$	−9.21110	−0.702340
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	4.60555	0.348147
$176$	0	0
$177$	0	0
$178$	−11.2111	−0.840308
$179$	25.0278	1.87066	0.935331	−	0.353773i	$-0.115102\pi$
$179$	25.0278	1.87066	0.935331	+	0.353773i	$0.115102\pi$
$180$	0	0
$181$	−4.78890	−0.355956	−0.177978	−	0.984034i	$-0.556956\pi$
$181$	−4.78890	−0.355956	−0.177978	+	0.984034i	$0.556956\pi$
$182$	−4.60555	−0.341386
$183$	0	0
$184$	−2.60555	−0.192084
$185$	3.21110	0.236085
$186$	0	0
$187$	0	0
$188$	−5.21110	−0.380059
$189$	0	0
$190$	−0.605551	−0.0439313
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−7.39445	−0.532264	−0.266132	−	0.963937i	$-0.585746\pi$
$193$	−7.39445	−0.532264	−0.266132	+	0.963937i	$0.585746\pi$
$194$	0.605551	0.0434760
$195$	0	0
$196$	14.2111	1.01508
$197$	−4.42221	−0.315069	−0.157535	−	0.987513i	$-0.550355\pi$
$197$	−4.42221	−0.315069	−0.157535	+	0.987513i	$0.550355\pi$
$198$	0	0
$199$	−9.21110	−0.652958	−0.326479	−	0.945204i	$-0.605862\pi$
$199$	−9.21110	−0.652958	−0.326479	+	0.945204i	$0.605862\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	7.81665	0.549978
$203$	−12.0000	−0.842235
$204$	0	0
$205$	−11.2111	−0.783017
$206$	4.00000	0.278693
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−10.7889	−0.742738	−0.371369	−	0.928485i	$-0.621112\pi$
$211$	−10.7889	−0.742738	−0.371369	+	0.928485i	$0.621112\pi$
$212$	0	0
$213$	0	0
$214$	−12.0000	−0.820303
$215$	9.21110	0.628192
$216$	0	0
$217$	9.21110	0.625290
$218$	−11.3944	−0.771729
$219$	0	0
$220$	0	0
$221$	2.60555	0.175268
$222$	0	0
$223$	−24.6056	−1.64771	−0.823855	−	0.566801i	$-0.808181\pi$
$223$	−24.6056	−1.64771	−0.823855	+	0.566801i	$0.808181\pi$
$224$	−4.60555	−0.307721
$225$	0	0
$226$	14.6056	0.971547
$227$	22.4222	1.48821	0.744107	−	0.668060i	$-0.232875\pi$
$227$	22.4222	1.48821	0.744107	+	0.668060i	$0.232875\pi$
$228$	0	0
$229$	21.8167	1.44169	0.720843	−	0.693099i	$-0.243755\pi$
$229$	21.8167	1.44169	0.720843	+	0.693099i	$0.243755\pi$
$230$	2.60555	0.171805
$231$	0	0
$232$	2.60555	0.171063
$233$	−19.8167	−1.29823	−0.649116	−	0.760689i	$-0.724861\pi$
$233$	−19.8167	−1.29823	−0.649116	+	0.760689i	$0.724861\pi$
$234$	0	0
$235$	5.21110	0.339935
$236$	−5.21110	−0.339214
$237$	0	0
$238$	−12.0000	−0.777844
$239$	10.4222	0.674156	0.337078	−	0.941477i	$-0.390561\pi$
$239$	10.4222	0.674156	0.337078	+	0.941477i	$0.390561\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	7.21110	0.461644
$245$	−14.2111	−0.907914
$246$	0	0
$247$	−0.605551	−0.0385303
$248$	−2.00000	−0.127000
$249$	0	0
$250$	1.00000	0.0632456
$251$	2.60555	0.164461	0.0822305	−	0.996613i	$-0.473796\pi$
$251$	2.60555	0.164461	0.0822305	+	0.996613i	$0.473796\pi$
$252$	0	0
$253$	0	0
$254$	−13.2111	−0.828938
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.0278	−1.56119	−0.780594	−	0.625038i	$-0.785083\pi$
$257$	−25.0278	−1.56119	−0.780594	+	0.625038i	$0.785083\pi$
$258$	0	0
$259$	−14.7889	−0.918937
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	−9.39445	−0.580391
$263$	−25.0278	−1.54328	−0.771639	−	0.636061i	$-0.780563\pi$
$263$	−25.0278	−1.54328	−0.771639	+	0.636061i	$0.780563\pi$
$264$	0	0
$265$	0	0
$266$	2.78890	0.170998
$267$	0	0
$268$	7.21110	0.440488
$269$	−31.8167	−1.93990	−0.969948	−	0.243313i	$-0.921766\pi$
$269$	−31.8167	−1.93990	−0.969948	+	0.243313i	$0.921766\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	2.60555	0.157985
$273$	0	0
$274$	−21.6333	−1.30692
$275$	0	0
$276$	0	0
$277$	−16.7889	−1.00875	−0.504374	−	0.863486i	$-0.668277\pi$
$277$	−16.7889	−1.00875	−0.504374	+	0.863486i	$0.668277\pi$
$278$	14.4222	0.864986
$279$	0	0
$280$	4.60555	0.275234
$281$	−11.2111	−0.668798	−0.334399	−	0.942432i	$-0.608533\pi$
$281$	−11.2111	−0.668798	−0.334399	+	0.942432i	$0.608533\pi$
$282$	0	0
$283$	13.2111	0.785319	0.392659	−	0.919684i	$-0.371555\pi$
$283$	13.2111	0.785319	0.392659	+	0.919684i	$0.371555\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	51.6333	3.04782
$288$	0	0
$289$	−10.2111	−0.600653
$290$	−2.60555	−0.153003
$291$	0	0
$292$	−0.605551	−0.0354372
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	5.21110	0.303402
$296$	3.21110	0.186642
$297$	0	0
$298$	−6.00000	−0.347571
$299$	2.60555	0.150683
$300$	0	0
$301$	−42.4222	−2.44518
$302$	20.4222	1.17517
$303$	0	0
$304$	−0.605551	−0.0347307
$305$	−7.21110	−0.412907
$306$	0	0
$307$	19.2111	1.09644	0.548218	−	0.836336i	$-0.315306\pi$
$307$	19.2111	1.09644	0.548218	+	0.836336i	$0.315306\pi$
$308$	0	0
$309$	0	0
$310$	2.00000	0.113592
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−3.21110	−0.181502	−0.0907511	−	0.995874i	$-0.528927\pi$
$313$	−3.21110	−0.181502	−0.0907511	+	0.995874i	$0.528927\pi$
$314$	3.21110	0.181213
$315$	0	0
$316$	13.2111	0.743183
$317$	−16.4222	−0.922363	−0.461181	−	0.887306i	$-0.652574\pi$
$317$	−16.4222	−0.922363	−0.461181	+	0.887306i	$0.652574\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−1.57779	−0.0877909
$324$	0	0
$325$	1.00000	0.0554700
$326$	−2.00000	−0.110770
$327$	0	0
$328$	−11.2111	−0.619030
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−12.6056	−0.692864	−0.346432	−	0.938075i	$-0.612607\pi$
$331$	−12.6056	−0.692864	−0.346432	+	0.938075i	$0.612607\pi$
$332$	0	0
$333$	0	0
$334$	10.4222	0.570278
$335$	−7.21110	−0.393985
$336$	0	0
$337$	−30.8444	−1.68020	−0.840101	−	0.542430i	$-0.817504\pi$
$337$	−30.8444	−1.68020	−0.840101	+	0.542430i	$0.817504\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−2.60555	−0.141306
$341$	0	0
$342$	0	0
$343$	33.2111	1.79323
$344$	9.21110	0.496629
$345$	0	0
$346$	12.0000	0.645124
$347$	−29.2111	−1.56813	−0.784067	−	0.620676i	$-0.786858\pi$
$347$	−29.2111	−1.56813	−0.784067	+	0.620676i	$0.786858\pi$
$348$	0	0
$349$	9.81665	0.525473	0.262737	−	0.964868i	$-0.415375\pi$
$349$	9.81665	0.525473	0.262737	+	0.964868i	$0.415375\pi$
$350$	−4.60555	−0.246177
$351$	0	0
$352$	0	0
$353$	21.6333	1.15142	0.575712	−	0.817652i	$-0.304725\pi$
$353$	21.6333	1.15142	0.575712	+	0.817652i	$0.304725\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	11.2111	0.594187
$357$	0	0
$358$	−25.0278	−1.32276
$359$	1.57779	0.0832728	0.0416364	−	0.999133i	$-0.486743\pi$
$359$	1.57779	0.0832728	0.0416364	+	0.999133i	$0.486743\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	4.78890	0.251699
$363$	0	0
$364$	4.60555	0.241396
$365$	0.605551	0.0316960
$366$	0	0
$367$	−21.2111	−1.10721	−0.553605	−	0.832779i	$-0.686748\pi$
$367$	−21.2111	−1.10721	−0.553605	+	0.832779i	$0.686748\pi$
$368$	2.60555	0.135824
$369$	0	0
$370$	−3.21110	−0.166937
$371$	0	0
$372$	0	0
$373$	−15.2111	−0.787601	−0.393801	−	0.919196i	$-0.628840\pi$
$373$	−15.2111	−0.787601	−0.393801	+	0.919196i	$0.628840\pi$
$374$	0	0
$375$	0	0
$376$	5.21110	0.268742
$377$	−2.60555	−0.134193
$378$	0	0
$379$	−19.3944	−0.996226	−0.498113	−	0.867112i	$-0.665974\pi$
$379$	−19.3944	−0.996226	−0.498113	+	0.867112i	$0.665974\pi$
$380$	0.605551	0.0310641
$381$	0	0
$382$	−12.0000	−0.613973
$383$	10.4222	0.532550	0.266275	−	0.963897i	$-0.414207\pi$
$383$	10.4222	0.532550	0.266275	+	0.963897i	$0.414207\pi$
$384$	0	0
$385$	0	0
$386$	7.39445	0.376368
$387$	0	0
$388$	−0.605551	−0.0307422
$389$	−19.8167	−1.00474	−0.502372	−	0.864652i	$-0.667539\pi$
$389$	−19.8167	−1.00474	−0.502372	+	0.864652i	$0.667539\pi$
$390$	0	0
$391$	6.78890	0.343329
$392$	−14.2111	−0.717769
$393$	0	0
$394$	4.42221	0.222787
$395$	−13.2111	−0.664723
$396$	0	0
$397$	20.7889	1.04336	0.521682	−	0.853140i	$-0.325305\pi$
$397$	20.7889	1.04336	0.521682	+	0.853140i	$0.325305\pi$
$398$	9.21110	0.461711
$399$	0	0
$400$	1.00000	0.0500000
$401$	9.63331	0.481064	0.240532	−	0.970641i	$-0.422678\pi$
$401$	9.63331	0.481064	0.240532	+	0.970641i	$0.422678\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	−7.81665	−0.388893
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	29.6333	1.46527	0.732636	−	0.680620i	$-0.238290\pi$
$409$	29.6333	1.46527	0.732636	+	0.680620i	$0.238290\pi$
$410$	11.2111	0.553677
$411$	0	0
$412$	−4.00000	−0.197066
$413$	−24.0000	−1.18096
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	0	0
$419$	−35.4500	−1.73184	−0.865922	−	0.500179i	$-0.833267\pi$
$419$	−35.4500	−1.73184	−0.865922	+	0.500179i	$0.833267\pi$
$420$	0	0
$421$	27.0278	1.31725	0.658626	−	0.752470i	$-0.271138\pi$
$421$	27.0278	1.31725	0.658626	+	0.752470i	$0.271138\pi$
$422$	10.7889	0.525195
$423$	0	0
$424$	0	0
$425$	2.60555	0.126388
$426$	0	0
$427$	33.2111	1.60720
$428$	12.0000	0.580042
$429$	0	0
$430$	−9.21110	−0.444199
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	34.8444	1.67452	0.837258	−	0.546808i	$-0.184157\pi$
$433$	34.8444	1.67452	0.837258	+	0.546808i	$0.184157\pi$
$434$	−9.21110	−0.442147
$435$	0	0
$436$	11.3944	0.545695
$437$	−1.57779	−0.0754762
$438$	0	0
$439$	−24.8444	−1.18576	−0.592880	−	0.805291i	$-0.702009\pi$
$439$	−24.8444	−1.18576	−0.592880	+	0.805291i	$0.702009\pi$
$440$	0	0
$441$	0	0
$442$	−2.60555	−0.123933
$443$	−3.63331	−0.172624	−0.0863118	−	0.996268i	$-0.527508\pi$
$443$	−3.63331	−0.172624	−0.0863118	+	0.996268i	$0.527508\pi$
$444$	0	0
$445$	−11.2111	−0.531457
$446$	24.6056	1.16511
$447$	0	0
$448$	4.60555	0.217592
$449$	−33.6333	−1.58725	−0.793627	−	0.608405i	$-0.791810\pi$
$449$	−33.6333	−1.58725	−0.793627	+	0.608405i	$0.791810\pi$
$450$	0	0
$451$	0	0
$452$	−14.6056	−0.686987
$453$	0	0
$454$	−22.4222	−1.05233
$455$	−4.60555	−0.215912
$456$	0	0
$457$	−24.6056	−1.15100	−0.575500	−	0.817802i	$-0.695192\pi$
$457$	−24.6056	−1.15100	−0.575500	+	0.817802i	$0.695192\pi$
$458$	−21.8167	−1.01943
$459$	0	0
$460$	−2.60555	−0.121484
$461$	−11.2111	−0.522153	−0.261077	−	0.965318i	$-0.584078\pi$
$461$	−11.2111	−0.522153	−0.261077	+	0.965318i	$0.584078\pi$
$462$	0	0
$463$	33.8167	1.57159	0.785797	−	0.618485i	$-0.212253\pi$
$463$	33.8167	1.57159	0.785797	+	0.618485i	$0.212253\pi$
$464$	−2.60555	−0.120960
$465$	0	0
$466$	19.8167	0.917989
$467$	34.4222	1.59287	0.796435	−	0.604724i	$-0.206717\pi$
$467$	34.4222	1.59287	0.796435	+	0.604724i	$0.206717\pi$
$468$	0	0
$469$	33.2111	1.53355
$470$	−5.21110	−0.240370
$471$	0	0
$472$	5.21110	0.239860
$473$	0	0
$474$	0	0
$475$	−0.605551	−0.0277846
$476$	12.0000	0.550019
$477$	0	0
$478$	−10.4222	−0.476701
$479$	10.4222	0.476203	0.238101	−	0.971240i	$-0.423475\pi$
$479$	10.4222	0.476203	0.238101	+	0.971240i	$0.423475\pi$
$480$	0	0
$481$	−3.21110	−0.146414
$482$	10.0000	0.455488
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0.605551	0.0274967
$486$	0	0
$487$	39.0278	1.76852	0.884258	−	0.466998i	$-0.154665\pi$
$487$	39.0278	1.76852	0.884258	+	0.466998i	$0.154665\pi$
$488$	−7.21110	−0.326431
$489$	0	0
$490$	14.2111	0.641992
$491$	18.2389	0.823108	0.411554	−	0.911385i	$-0.364986\pi$
$491$	18.2389	0.823108	0.411554	+	0.911385i	$0.364986\pi$
$492$	0	0
$493$	−6.78890	−0.305757
$494$	0.605551	0.0272450
$495$	0	0
$496$	2.00000	0.0898027
$497$	55.2666	2.47905
$498$	0	0
$499$	6.18335	0.276805	0.138402	−	0.990376i	$-0.455803\pi$
$499$	6.18335	0.276805	0.138402	+	0.990376i	$0.455803\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−2.60555	−0.116291
$503$	30.2389	1.34828	0.674142	−	0.738602i	$-0.264514\pi$
$503$	30.2389	1.34828	0.674142	+	0.738602i	$0.264514\pi$
$504$	0	0
$505$	7.81665	0.347837
$506$	0	0
$507$	0	0
$508$	13.2111	0.586148
$509$	33.6333	1.49077	0.745385	−	0.666634i	$-0.232266\pi$
$509$	33.6333	1.49077	0.745385	+	0.666634i	$0.232266\pi$
$510$	0	0
$511$	−2.78890	−0.123374
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	25.0278	1.10393
$515$	4.00000	0.176261
$516$	0	0
$517$	0	0
$518$	14.7889	0.649787
$519$	0	0
$520$	1.00000	0.0438529
$521$	6.78890	0.297427	0.148713	−	0.988880i	$-0.452487\pi$
$521$	6.78890	0.297427	0.148713	+	0.988880i	$0.452487\pi$
$522$	0	0
$523$	9.57779	0.418808	0.209404	−	0.977829i	$-0.432848\pi$
$523$	9.57779	0.418808	0.209404	+	0.977829i	$0.432848\pi$
$524$	9.39445	0.410398
$525$	0	0
$526$	25.0278	1.09126
$527$	5.21110	0.226999
$528$	0	0
$529$	−16.2111	−0.704831
$530$	0	0
$531$	0	0
$532$	−2.78890	−0.120914
$533$	11.2111	0.485607
$534$	0	0
$535$	−12.0000	−0.518805
$536$	−7.21110	−0.311472
$537$	0	0
$538$	31.8167	1.37171
$539$	0	0
$540$	0	0
$541$	27.0278	1.16201	0.581007	−	0.813899i	$-0.302659\pi$
$541$	27.0278	1.16201	0.581007	+	0.813899i	$0.302659\pi$
$542$	−2.00000	−0.0859074
$543$	0	0
$544$	−2.60555	−0.111712
$545$	−11.3944	−0.488085
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	21.6333	0.924129
$549$	0	0
$550$	0	0
$551$	1.57779	0.0672163
$552$	0	0
$553$	60.8444	2.58737
$554$	16.7889	0.713292
$555$	0	0
$556$	−14.4222	−0.611638
$557$	−38.8444	−1.64589	−0.822945	−	0.568121i	$-0.807671\pi$
$557$	−38.8444	−1.64589	−0.822945	+	0.568121i	$0.807671\pi$
$558$	0	0
$559$	−9.21110	−0.389588
$560$	−4.60555	−0.194620
$561$	0	0
$562$	11.2111	0.472912
$563$	−27.6333	−1.16461	−0.582303	−	0.812972i	$-0.697848\pi$
$563$	−27.6333	−1.16461	−0.582303	+	0.812972i	$0.697848\pi$
$564$	0	0
$565$	14.6056	0.614460
$566$	−13.2111	−0.555304
$567$	0	0
$568$	−12.0000	−0.503509
$569$	41.2111	1.72766	0.863830	−	0.503784i	$-0.168059\pi$
$569$	41.2111	1.72766	0.863830	+	0.503784i	$0.168059\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	−51.6333	−2.15513
$575$	2.60555	0.108659
$576$	0	0
$577$	6.18335	0.257416	0.128708	−	0.991683i	$-0.458917\pi$
$577$	6.18335	0.257416	0.128708	+	0.991683i	$0.458917\pi$
$578$	10.2111	0.424726
$579$	0	0
$580$	2.60555	0.108190
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0.605551	0.0250579
$585$	0	0
$586$	30.0000	1.23929
$587$	−22.4222	−0.925463	−0.462732	−	0.886498i	$-0.653131\pi$
$587$	−22.4222	−0.925463	−0.462732	+	0.886498i	$0.653131\pi$
$588$	0	0
$589$	−1.21110	−0.0499026
$590$	−5.21110	−0.214538
$591$	0	0
$592$	−3.21110	−0.131976
$593$	−35.2111	−1.44595	−0.722973	−	0.690876i	$-0.757225\pi$
$593$	−35.2111	−1.44595	−0.722973	+	0.690876i	$0.757225\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	6.00000	0.245770
$597$	0	0
$598$	−2.60555	−0.106549
$599$	−20.8444	−0.851680	−0.425840	−	0.904799i	$-0.640021\pi$
$599$	−20.8444	−0.851680	−0.425840	+	0.904799i	$0.640021\pi$
$600$	0	0
$601$	−25.6333	−1.04560	−0.522802	−	0.852454i	$-0.675113\pi$
$601$	−25.6333	−1.04560	−0.522802	+	0.852454i	$0.675113\pi$
$602$	42.4222	1.72900
$603$	0	0
$604$	−20.4222	−0.830968
$605$	11.0000	0.447214
$606$	0	0
$607$	26.7889	1.08733	0.543664	−	0.839303i	$-0.317037\pi$
$607$	26.7889	1.08733	0.543664	+	0.839303i	$0.317037\pi$
$608$	0.605551	0.0245583
$609$	0	0
$610$	7.21110	0.291969
$611$	−5.21110	−0.210819
$612$	0	0
$613$	−37.6333	−1.51999	−0.759997	−	0.649926i	$-0.774800\pi$
$613$	−37.6333	−1.51999	−0.759997	+	0.649926i	$0.774800\pi$
$614$	−19.2111	−0.775297
$615$	0	0
$616$	0	0
$617$	−38.8444	−1.56382	−0.781909	−	0.623393i	$-0.785754\pi$
$617$	−38.8444	−1.56382	−0.781909	+	0.623393i	$0.785754\pi$
$618$	0	0
$619$	4.60555	0.185113	0.0925564	−	0.995707i	$-0.470496\pi$
$619$	4.60555	0.185113	0.0925564	+	0.995707i	$0.470496\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	−12.0000	−0.481156
$623$	51.6333	2.06864
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.21110	0.128341
$627$	0	0
$628$	−3.21110	−0.128137
$629$	−8.36669	−0.333602
$630$	0	0
$631$	8.78890	0.349880	0.174940	−	0.984579i	$-0.444027\pi$
$631$	8.78890	0.349880	0.174940	+	0.984579i	$0.444027\pi$
$632$	−13.2111	−0.525509
$633$	0	0
$634$	16.4222	0.652209
$635$	−13.2111	−0.524267
$636$	0	0
$637$	14.2111	0.563064
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	−24.0555	−0.948657	−0.474328	−	0.880348i	$-0.657309\pi$
$643$	−24.0555	−0.948657	−0.474328	+	0.880348i	$0.657309\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	1.57779	0.0620775
$647$	−1.02776	−0.0404053	−0.0202026	−	0.999796i	$-0.506431\pi$
$647$	−1.02776	−0.0404053	−0.0202026	+	0.999796i	$0.506431\pi$
$648$	0	0
$649$	0	0
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	2.00000	0.0783260
$653$	17.2111	0.673522	0.336761	−	0.941590i	$-0.390668\pi$
$653$	17.2111	0.673522	0.336761	+	0.941590i	$0.390668\pi$
$654$	0	0
$655$	−9.39445	−0.367071
$656$	11.2111	0.437720
$657$	0	0
$658$	24.0000	0.935617
$659$	7.81665	0.304494	0.152247	−	0.988343i	$-0.451349\pi$
$659$	7.81665	0.304494	0.152247	+	0.988343i	$0.451349\pi$
$660$	0	0
$661$	15.0278	0.584512	0.292256	−	0.956340i	$-0.405594\pi$
$661$	15.0278	0.584512	0.292256	+	0.956340i	$0.405594\pi$
$662$	12.6056	0.489929
$663$	0	0
$664$	0	0
$665$	2.78890	0.108149
$666$	0	0
$667$	−6.78890	−0.262867
$668$	−10.4222	−0.403247
$669$	0	0
$670$	7.21110	0.278589
$671$	0	0
$672$	0	0
$673$	−4.78890	−0.184598	−0.0922992	−	0.995731i	$-0.529422\pi$
$673$	−4.78890	−0.184598	−0.0922992	+	0.995731i	$0.529422\pi$
$674$	30.8444	1.18808
$675$	0	0
$676$	1.00000	0.0384615
$677$	5.21110	0.200279	0.100139	−	0.994973i	$-0.468071\pi$
$677$	5.21110	0.200279	0.100139	+	0.994973i	$0.468071\pi$
$678$	0	0
$679$	−2.78890	−0.107028
$680$	2.60555	0.0999183
$681$	0	0
$682$	0	0
$683$	−34.4222	−1.31713	−0.658565	−	0.752524i	$-0.728836\pi$
$683$	−34.4222	−1.31713	−0.658565	+	0.752524i	$0.728836\pi$
$684$	0	0
$685$	−21.6333	−0.826566
$686$	−33.2111	−1.26801
$687$	0	0
$688$	−9.21110	−0.351170
$689$	0	0
$690$	0	0
$691$	−2.18335	−0.0830584	−0.0415292	−	0.999137i	$-0.513223\pi$
$691$	−2.18335	−0.0830584	−0.0415292	+	0.999137i	$0.513223\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	29.2111	1.10884
$695$	14.4222	0.547065
$696$	0	0
$697$	29.2111	1.10645
$698$	−9.81665	−0.371566
$699$	0	0
$700$	4.60555	0.174073
$701$	−9.39445	−0.354823	−0.177412	−	0.984137i	$-0.556772\pi$
$701$	−9.39445	−0.354823	−0.177412	+	0.984137i	$0.556772\pi$
$702$	0	0
$703$	1.94449	0.0733377
$704$	0	0
$705$	0	0
$706$	−21.6333	−0.814180
$707$	−36.0000	−1.35392
$708$	0	0
$709$	−33.4500	−1.25624	−0.628120	−	0.778117i	$-0.716175\pi$
$709$	−33.4500	−1.25624	−0.628120	+	0.778117i	$0.716175\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	−11.2111	−0.420154
$713$	5.21110	0.195157
$714$	0	0
$715$	0	0
$716$	25.0278	0.935331
$717$	0	0
$718$	−1.57779	−0.0588828
$719$	−6.78890	−0.253183	−0.126592	−	0.991955i	$-0.540404\pi$
$719$	−6.78890	−0.253183	−0.126592	+	0.991955i	$0.540404\pi$
$720$	0	0
$721$	−18.4222	−0.686079
$722$	18.6333	0.693460
$723$	0	0
$724$	−4.78890	−0.177978
$725$	−2.60555	−0.0967677
$726$	0	0
$727$	−0.366692	−0.0135999	−0.00679993	−	0.999977i	$-0.502165\pi$
$727$	−0.366692	−0.0135999	−0.00679993	+	0.999977i	$0.502165\pi$
$728$	−4.60555	−0.170693
$729$	0	0
$730$	−0.605551	−0.0224125
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−1.63331	−0.0603276	−0.0301638	−	0.999545i	$-0.509603\pi$
$733$	−1.63331	−0.0603276	−0.0301638	+	0.999545i	$0.509603\pi$
$734$	21.2111	0.782916
$735$	0	0
$736$	−2.60555	−0.0960419
$737$	0	0
$738$	0	0
$739$	−19.3944	−0.713436	−0.356718	−	0.934212i	$-0.616104\pi$
$739$	−19.3944	−0.713436	−0.356718	+	0.934212i	$0.616104\pi$
$740$	3.21110	0.118042
$741$	0	0
$742$	0	0
$743$	−26.0555	−0.955884	−0.477942	−	0.878391i	$-0.658617\pi$
$743$	−26.0555	−0.955884	−0.477942	+	0.878391i	$0.658617\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	15.2111	0.556918
$747$	0	0
$748$	0	0
$749$	55.2666	2.01940
$750$	0	0
$751$	51.2666	1.87075	0.935373	−	0.353664i	$-0.115064\pi$
$751$	51.2666	1.87075	0.935373	+	0.353664i	$0.115064\pi$
$752$	−5.21110	−0.190029
$753$	0	0
$754$	2.60555	0.0948886
$755$	20.4222	0.743240
$756$	0	0
$757$	22.8444	0.830294	0.415147	−	0.909754i	$-0.363730\pi$
$757$	22.8444	0.830294	0.415147	+	0.909754i	$0.363730\pi$
$758$	19.3944	0.704438
$759$	0	0
$760$	−0.605551	−0.0219657
$761$	−33.6333	−1.21921	−0.609603	−	0.792707i	$-0.708671\pi$
$761$	−33.6333	−1.21921	−0.609603	+	0.792707i	$0.708671\pi$
$762$	0	0
$763$	52.4777	1.89982
$764$	12.0000	0.434145
$765$	0	0
$766$	−10.4222	−0.376570
$767$	−5.21110	−0.188162
$768$	0	0
$769$	22.8444	0.823791	0.411895	−	0.911231i	$-0.364867\pi$
$769$	22.8444	0.823791	0.411895	+	0.911231i	$0.364867\pi$
$770$	0	0
$771$	0	0
$772$	−7.39445	−0.266132
$773$	−16.4222	−0.590666	−0.295333	−	0.955394i	$-0.595431\pi$
$773$	−16.4222	−0.590666	−0.295333	+	0.955394i	$0.595431\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0.605551	0.0217380
$777$	0	0
$778$	19.8167	0.710461
$779$	−6.78890	−0.243237
$780$	0	0
$781$	0	0
$782$	−6.78890	−0.242770
$783$	0	0
$784$	14.2111	0.507539
$785$	3.21110	0.114609
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	−4.42221	−0.157535
$789$	0	0
$790$	13.2111	0.470030
$791$	−67.2666	−2.39173
$792$	0	0
$793$	7.21110	0.256074
$794$	−20.7889	−0.737770
$795$	0	0
$796$	−9.21110	−0.326479
$797$	−13.5778	−0.480950	−0.240475	−	0.970655i	$-0.577303\pi$
$797$	−13.5778	−0.480950	−0.240475	+	0.970655i	$0.577303\pi$
$798$	0	0
$799$	−13.5778	−0.480348
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−9.63331	−0.340164
$803$	0	0
$804$	0	0
$805$	−12.0000	−0.422944
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	7.81665	0.274989
$809$	22.4222	0.788323	0.394161	−	0.919041i	$-0.371035\pi$
$809$	22.4222	0.788323	0.394161	+	0.919041i	$0.371035\pi$
$810$	0	0
$811$	12.9722	0.455517	0.227759	−	0.973718i	$-0.426860\pi$
$811$	12.9722	0.455517	0.227759	+	0.973718i	$0.426860\pi$
$812$	−12.0000	−0.421117
$813$	0	0
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	5.57779	0.195142
$818$	−29.6333	−1.03610
$819$	0	0
$820$	−11.2111	−0.391509
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	37.0278	1.28294
$834$	0	0
$835$	10.4222	0.360675
$836$	0	0
$837$	0	0
$838$	35.4500	1.22460
$839$	13.5778	0.468758	0.234379	−	0.972145i	$-0.424694\pi$
$839$	13.5778	0.468758	0.234379	+	0.972145i	$0.424694\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	−27.0278	−0.931438
$843$	0	0
$844$	−10.7889	−0.371369
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−50.6611	−1.74073
$848$	0	0
$849$	0	0
$850$	−2.60555	−0.0893697
$851$	−8.36669	−0.286807
$852$	0	0
$853$	−48.0555	−1.64539	−0.822695	−	0.568483i	$-0.807530\pi$
$853$	−48.0555	−1.64539	−0.822695	+	0.568483i	$0.807530\pi$
$854$	−33.2111	−1.13646
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−35.4500	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$857$	−35.4500	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$858$	0	0
$859$	47.6333	1.62523	0.812614	−	0.582803i	$-0.198044\pi$
$859$	47.6333	1.62523	0.812614	+	0.582803i	$0.198044\pi$
$860$	9.21110	0.314096
$861$	0	0
$862$	−24.0000	−0.817443
$863$	13.5778	0.462194	0.231097	−	0.972931i	$-0.425769\pi$
$863$	13.5778	0.462194	0.231097	+	0.972931i	$0.425769\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	−34.8444	−1.18406
$867$	0	0
$868$	9.21110	0.312645
$869$	0	0
$870$	0	0
$871$	7.21110	0.244339
$872$	−11.3944	−0.385865
$873$	0	0
$874$	1.57779	0.0533697
$875$	−4.60555	−0.155696
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	24.8444	0.838458
$879$	0	0
$880$	0	0
$881$	−41.2111	−1.38844	−0.694219	−	0.719764i	$-0.744250\pi$
$881$	−41.2111	−1.38844	−0.694219	+	0.719764i	$0.744250\pi$
$882$	0	0
$883$	−14.4222	−0.485346	−0.242673	−	0.970108i	$-0.578024\pi$
$883$	−14.4222	−0.485346	−0.242673	+	0.970108i	$0.578024\pi$
$884$	2.60555	0.0876342
$885$	0	0
$886$	3.63331	0.122063
$887$	2.60555	0.0874858	0.0437429	−	0.999043i	$-0.486072\pi$
$887$	2.60555	0.0874858	0.0437429	+	0.999043i	$0.486072\pi$
$888$	0	0
$889$	60.8444	2.04066
$890$	11.2111	0.375797
$891$	0	0
$892$	−24.6056	−0.823855
$893$	3.15559	0.105598
$894$	0	0
$895$	−25.0278	−0.836586
$896$	−4.60555	−0.153861
$897$	0	0
$898$	33.6333	1.12236
$899$	−5.21110	−0.173800
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	14.6056	0.485773
$905$	4.78890	0.159188
$906$	0	0
$907$	−55.6333	−1.84727	−0.923637	−	0.383269i	$-0.874798\pi$
$907$	−55.6333	−1.84727	−0.923637	+	0.383269i	$0.874798\pi$
$908$	22.4222	0.744107
$909$	0	0
$910$	4.60555	0.152673
$911$	27.6333	0.915532	0.457766	−	0.889073i	$-0.348650\pi$
$911$	27.6333	0.915532	0.457766	+	0.889073i	$0.348650\pi$
$912$	0	0
$913$	0	0
$914$	24.6056	0.813879
$915$	0	0
$916$	21.8167	0.720843
$917$	43.2666	1.42879
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	2.60555	0.0859025
$921$	0	0
$922$	11.2111	0.369218
$923$	12.0000	0.394985
$924$	0	0
$925$	−3.21110	−0.105580
$926$	−33.8167	−1.11128
$927$	0	0
$928$	2.60555	0.0855314
$929$	−32.0555	−1.05171	−0.525854	−	0.850575i	$-0.676254\pi$
$929$	−32.0555	−1.05171	−0.525854	+	0.850575i	$0.676254\pi$
$930$	0	0
$931$	−8.60555	−0.282036
$932$	−19.8167	−0.649116
$933$	0	0
$934$	−34.4222	−1.12633
$935$	0	0
$936$	0	0
$937$	29.6333	0.968078	0.484039	−	0.875046i	$-0.339169\pi$
$937$	29.6333	0.968078	0.484039	+	0.875046i	$0.339169\pi$
$938$	−33.2111	−1.08438
$939$	0	0
$940$	5.21110	0.169967
$941$	21.6333	0.705226	0.352613	−	0.935769i	$-0.385293\pi$
$941$	21.6333	0.705226	0.352613	+	0.935769i	$0.385293\pi$
$942$	0	0
$943$	29.2111	0.951244
$944$	−5.21110	−0.169607
$945$	0	0
$946$	0	0
$947$	−13.5778	−0.441219	−0.220610	−	0.975362i	$-0.570805\pi$
$947$	−13.5778	−0.441219	−0.220610	+	0.975362i	$0.570805\pi$
$948$	0	0
$949$	−0.605551	−0.0196570
$950$	0.605551	0.0196467
$951$	0	0
$952$	−12.0000	−0.388922
$953$	−9.39445	−0.304316	−0.152158	−	0.988356i	$-0.548622\pi$
$953$	−9.39445	−0.304316	−0.152158	+	0.988356i	$0.548622\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	10.4222	0.337078
$957$	0	0
$958$	−10.4222	−0.336726
$959$	99.6333	3.21733
$960$	0	0
$961$	−27.0000	−0.870968
$962$	3.21110	0.103530
$963$	0	0
$964$	−10.0000	−0.322078
$965$	7.39445	0.238036
$966$	0	0
$967$	−19.3944	−0.623683	−0.311842	−	0.950134i	$-0.600946\pi$
$967$	−19.3944	−0.623683	−0.311842	+	0.950134i	$0.600946\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−0.605551	−0.0194431
$971$	26.6056	0.853813	0.426906	−	0.904296i	$-0.359603\pi$
$971$	26.6056	0.853813	0.426906	+	0.904296i	$0.359603\pi$
$972$	0	0
$973$	−66.4222	−2.12940
$974$	−39.0278	−1.25053
$975$	0	0
$976$	7.21110	0.230822
$977$	−14.3667	−0.459631	−0.229816	−	0.973234i	$-0.573812\pi$
$977$	−14.3667	−0.459631	−0.229816	+	0.973234i	$0.573812\pi$
$978$	0	0
$979$	0	0
$980$	−14.2111	−0.453957
$981$	0	0
$982$	−18.2389	−0.582025
$983$	15.6333	0.498625	0.249313	−	0.968423i	$-0.419795\pi$
$983$	15.6333	0.498625	0.249313	+	0.968423i	$0.419795\pi$
$984$	0	0
$985$	4.42221	0.140903
$986$	6.78890	0.216203
$987$	0	0
$988$	−0.605551	−0.0192652
$989$	−24.0000	−0.763156
$990$	0	0
$991$	11.6333	0.369544	0.184772	−	0.982781i	$-0.440845\pi$
$991$	11.6333	0.369544	0.184772	+	0.982781i	$0.440845\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−55.2666	−1.75295
$995$	9.21110	0.292012
$996$	0	0
$997$	19.2111	0.608422	0.304211	−	0.952605i	$-0.401607\pi$
$997$	19.2111	0.608422	0.304211	+	0.952605i	$0.401607\pi$
$998$	−6.18335	−0.195730
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.p.1.2	✓	2
3.2	odd	2		1170.2.a.q.1.2	yes	2
4.3	odd	2		9360.2.a.cf.1.1		2
5.2	odd	4		5850.2.e.bl.5149.2		4
5.3	odd	4		5850.2.e.bl.5149.3		4
5.4	even	2		5850.2.a.ck.1.1		2
12.11	even	2		9360.2.a.cn.1.1		2
15.2	even	4		5850.2.e.bj.5149.4		4
15.8	even	4		5850.2.e.bj.5149.1		4
15.14	odd	2		5850.2.a.ce.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.a.p.1.2	✓	2	1.1	even	1	trivial
1170.2.a.q.1.2	yes	2	3.2	odd	2
5850.2.a.ce.1.1		2	15.14	odd	2
5850.2.a.ck.1.1		2	5.4	even	2
5850.2.e.bj.5149.1		4	15.8	even	4
5850.2.e.bj.5149.4		4	15.2	even	4
5850.2.e.bl.5149.2		4	5.2	odd	4
5850.2.e.bl.5149.3		4	5.3	odd	4
9360.2.a.cf.1.1		2	4.3	odd	2
9360.2.a.cn.1.1		2	12.11	even	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.60555 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +4.60555 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{13} -4.60555 q^{14} +1.00000 q^{16} +2.60555 q^{17} -0.605551 q^{19} -1.00000 q^{20} +2.60555 q^{23} +1.00000 q^{25} -1.00000 q^{26} +4.60555 q^{28} -2.60555 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.60555 q^{34} -4.60555 q^{35} -3.21110 q^{37} +0.605551 q^{38} +1.00000 q^{40} +11.2111 q^{41} -9.21110 q^{43} -2.60555 q^{46} -5.21110 q^{47} +14.2111 q^{49} -1.00000 q^{50} +1.00000 q^{52} -4.60555 q^{56} +2.60555 q^{58} -5.21110 q^{59} +7.21110 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +7.21110 q^{67} +2.60555 q^{68} +4.60555 q^{70} +12.0000 q^{71} -0.605551 q^{73} +3.21110 q^{74} -0.605551 q^{76} +13.2111 q^{79} -1.00000 q^{80} -11.2111 q^{82} -2.60555 q^{85} +9.21110 q^{86} +11.2111 q^{89} +4.60555 q^{91} +2.60555 q^{92} +5.21110 q^{94} +0.605551 q^{95} -0.605551 q^{97} -14.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} + 2 q^{29} + 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} + 8 q^{37} - 6 q^{38} + 2 q^{40} + 8 q^{41} - 4 q^{43} + 2 q^{46} + 4 q^{47} + 14 q^{49} - 2 q^{50} + 2 q^{52} - 2 q^{56} - 2 q^{58} + 4 q^{59} - 4 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{68} + 2 q^{70} + 24 q^{71} + 6 q^{73} - 8 q^{74} + 6 q^{76} + 12 q^{79} - 2 q^{80} - 8 q^{82} + 2 q^{85} + 4 q^{86} + 8 q^{89} + 2 q^{91} - 2 q^{92} - 4 q^{94} - 6 q^{95} + 6 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 6 * q^19 - 2 * q^20 - 2 * q^23 + 2 * q^25 - 2 * q^26 + 2 * q^28 + 2 * q^29 + 4 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 + 8 * q^37 - 6 * q^38 + 2 * q^40 + 8 * q^41 - 4 * q^43 + 2 * q^46 + 4 * q^47 + 14 * q^49 - 2 * q^50 + 2 * q^52 - 2 * q^56 - 2 * q^58 + 4 * q^59 - 4 * q^62 + 2 * q^64 - 2 * q^65 - 2 * q^68 + 2 * q^70 + 24 * q^71 + 6 * q^73 - 8 * q^74 + 6 * q^76 + 12 * q^79 - 2 * q^80 - 8 * q^82 + 2 * q^85 + 4 * q^86 + 8 * q^89 + 2 * q^91 - 2 * q^92 - 4 * q^94 - 6 * q^95 + 6 * q^97 - 14 * q^98

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