LMFDB - Embedded newform 1150.2.b.i.599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,2,Mod(599,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("1150.599");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$9.18279623245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.2
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.2.b.i.599.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} +0.618034i q^{7} +1.00000i q^{8} +0.381966 q^{9} -2.85410 q^{11} -1.61803i q^{12} -7.09017i q^{13} +0.618034 q^{14} +1.00000 q^{16} -6.09017i q^{17} -0.381966i q^{18} -1.85410 q^{19} -1.00000 q^{21} +2.85410i q^{22} +1.00000i q^{23} -1.61803 q^{24} -7.09017 q^{26} +5.47214i q^{27} -0.618034i q^{28} +9.23607 q^{29} +9.09017 q^{31} -1.00000i q^{32} -4.61803i q^{33} -6.09017 q^{34} -0.381966 q^{36} -6.47214i q^{37} +1.85410i q^{38} +11.4721 q^{39} +3.32624 q^{41} +1.00000i q^{42} +2.85410 q^{44} +1.00000 q^{46} +3.70820i q^{47} +1.61803i q^{48} +6.61803 q^{49} +9.85410 q^{51} +7.09017i q^{52} +0.472136i q^{53} +5.47214 q^{54} -0.618034 q^{56} -3.00000i q^{57} -9.23607i q^{58} -1.70820 q^{59} -9.32624 q^{61} -9.09017i q^{62} +0.236068i q^{63} -1.00000 q^{64} -4.61803 q^{66} -14.4721i q^{67} +6.09017i q^{68} -1.61803 q^{69} -4.09017 q^{71} +0.381966i q^{72} +3.23607i q^{73} -6.47214 q^{74} +1.85410 q^{76} -1.76393i q^{77} -11.4721i q^{78} -1.52786 q^{79} -7.70820 q^{81} -3.32624i q^{82} -6.94427i q^{83} +1.00000 q^{84} +14.9443i q^{87} -2.85410i q^{88} +10.4721 q^{89} +4.38197 q^{91} -1.00000i q^{92} +14.7082i q^{93} +3.70820 q^{94} +1.61803 q^{96} -12.3820i q^{97} -6.61803i q^{98} -1.09017 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46}+ \cdots + 18 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 2 * q^6 + 6 * q^9 + 2 * q^11 - 2 * q^14 + 4 * q^16 + 6 * q^19 - 4 * q^21 - 2 * q^24 - 6 * q^26 + 28 * q^29 + 14 * q^31 - 2 * q^34 - 6 * q^36 + 28 * q^39 - 18 * q^41 - 2 * q^44 + 4 * q^46 + 22 * q^49 + 26 * q^51 + 4 * q^54 + 2 * q^56 + 20 * q^59 - 6 * q^61 - 4 * q^64 - 14 * q^66 - 2 * q^69 + 6 * q^71 - 8 * q^74 - 6 * q^76 - 24 * q^79 - 4 * q^81 + 4 * q^84 + 24 * q^89 + 22 * q^91 - 12 * q^94 + 2 * q^96 + 18 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.61803i	0.934172i	0.884212	+	0.467086i	$0.154696\pi$
$3$	1.61803i	0.934172i	−0.884212	+	0.467086i	$0.845304\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.61803	0.660560
$7$	0.618034i	0.233595i	0.993156	+	0.116797i	$0.0372628\pi$
$7$	0.618034i	0.233595i	−0.993156	+	0.116797i	$0.962737\pi$
$8$	1.00000i	0.353553i
$9$	0.381966	0.127322
$10$	0	0
$11$	−2.85410	−0.860544	−0.430272	−	0.902699i	$-0.641582\pi$
$11$	−2.85410	−0.860544	−0.430272	+	0.902699i	$0.641582\pi$
$12$	− 1.61803i	− 0.467086i
$13$	− 7.09017i	− 1.96646i	−0.182372	−	0.983230i	$-0.558377\pi$
$13$	− 7.09017i	− 1.96646i	0.182372	−	0.983230i	$-0.441623\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.00000	0.250000
$17$	− 6.09017i	− 1.47708i	−0.674208	−	0.738542i	$-0.735515\pi$
$17$	− 6.09017i	− 1.47708i	0.674208	−	0.738542i	$-0.264485\pi$
$18$	− 0.381966i	− 0.0900303i
$19$	−1.85410	−0.425360	−0.212680	−	0.977122i	$-0.568219\pi$
$19$	−1.85410	−0.425360	−0.212680	+	0.977122i	$0.568219\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	2.85410i	0.608497i
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	−1.61803	−0.330280
$25$	0	0
$26$	−7.09017	−1.39050
$27$	5.47214i	1.05311i
$28$	− 0.618034i	− 0.116797i
$29$	9.23607	1.71509	0.857547	−	0.514405i	$-0.171987\pi$
$29$	9.23607	1.71509	0.857547	+	0.514405i	$0.171987\pi$
$30$	0	0
$31$	9.09017	1.63264	0.816321	−	0.577598i	$-0.196010\pi$
$31$	9.09017	1.63264	0.816321	+	0.577598i	$0.196010\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.61803i	− 0.803897i
$34$	−6.09017	−1.04446
$35$	0	0
$36$	−0.381966	−0.0636610
$37$	− 6.47214i	− 1.06401i	−0.846740	−	0.532006i	$-0.821438\pi$
$37$	− 6.47214i	− 1.06401i	0.846740	−	0.532006i	$-0.178562\pi$
$38$	1.85410i	0.300775i
$39$	11.4721	1.83701
$40$	0	0
$41$	3.32624	0.519471	0.259736	−	0.965680i	$-0.416365\pi$
$41$	3.32624	0.519471	0.259736	+	0.965680i	$0.416365\pi$
$42$	1.00000i	0.154303i
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	2.85410	0.430272
$45$	0	0
$46$	1.00000	0.147442
$47$	3.70820i	0.540897i	0.962734	+	0.270449i	$0.0871720\pi$
$47$	3.70820i	0.540897i	−0.962734	+	0.270449i	$0.912828\pi$
$48$	1.61803i	0.233543i
$49$	6.61803	0.945433
$50$	0	0
$51$	9.85410	1.37985
$52$	7.09017i	0.983230i
$53$	0.472136i	0.0648529i	0.999474	+	0.0324264i	$0.0103235\pi$
$53$	0.472136i	0.0648529i	−0.999474	+	0.0324264i	$0.989677\pi$
$54$	5.47214	0.744663
$55$	0	0
$56$	−0.618034	−0.0825883
$57$	− 3.00000i	− 0.397360i
$58$	− 9.23607i	− 1.21276i
$59$	−1.70820	−0.222389	−0.111195	−	0.993799i	$-0.535468\pi$
$59$	−1.70820	−0.222389	−0.111195	+	0.993799i	$0.535468\pi$
$60$	0	0
$61$	−9.32624	−1.19410	−0.597051	−	0.802203i	$-0.703661\pi$
$61$	−9.32624	−1.19410	−0.597051	+	0.802203i	$0.703661\pi$
$62$	− 9.09017i	− 1.15445i
$63$	0.236068i	0.0297418i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.61803	−0.568441
$67$	− 14.4721i	− 1.76805i	−0.467437	−	0.884026i	$-0.654823\pi$
$67$	− 14.4721i	− 1.76805i	0.467437	−	0.884026i	$-0.345177\pi$
$68$	6.09017i	0.738542i
$69$	−1.61803	−0.194788
$70$	0	0
$71$	−4.09017	−0.485414	−0.242707	−	0.970100i	$-0.578035\pi$
$71$	−4.09017	−0.485414	−0.242707	+	0.970100i	$0.578035\pi$
$72$	0.381966i	0.0450151i
$73$	3.23607i	0.378753i	0.981905	+	0.189377i	$0.0606467\pi$
$73$	3.23607i	0.378753i	−0.981905	+	0.189377i	$0.939353\pi$
$74$	−6.47214	−0.752371
$75$	0	0
$76$	1.85410	0.212680
$77$	− 1.76393i	− 0.201019i
$78$	− 11.4721i	− 1.29896i
$79$	−1.52786	−0.171898	−0.0859491	−	0.996300i	$-0.527392\pi$
$79$	−1.52786	−0.171898	−0.0859491	+	0.996300i	$0.527392\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	− 3.32624i	− 0.367322i
$83$	− 6.94427i	− 0.762233i	−0.924527	−	0.381116i	$-0.875540\pi$
$83$	− 6.94427i	− 0.762233i	0.924527	−	0.381116i	$-0.124460\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	0	0
$87$	14.9443i	1.60219i
$88$	− 2.85410i	− 0.304248i
$89$	10.4721	1.11004	0.555022	−	0.831836i	$-0.312710\pi$
$89$	10.4721	1.11004	0.555022	+	0.831836i	$0.312710\pi$
$90$	0	0
$91$	4.38197	0.459355
$92$	− 1.00000i	− 0.104257i
$93$	14.7082i	1.52517i
$94$	3.70820	0.382472
$95$	0	0
$96$	1.61803	0.165140
$97$	− 12.3820i	− 1.25720i	−0.777730	−	0.628599i	$-0.783629\pi$
$97$	− 12.3820i	− 1.25720i	0.777730	−	0.628599i	$-0.216371\pi$
$98$	− 6.61803i	− 0.668522i
$99$	−1.09017	−0.109566
$100$	0	0
$101$	−0.291796	−0.0290348	−0.0145174	−	0.999895i	$-0.504621\pi$
$101$	−0.291796	−0.0290348	−0.0145174	+	0.999895i	$0.504621\pi$
$102$	− 9.85410i	− 0.975701i
$103$	16.5623i	1.63193i	0.578100	+	0.815966i	$0.303795\pi$
$103$	16.5623i	1.63193i	−0.578100	+	0.815966i	$0.696205\pi$
$104$	7.09017	0.695248
$105$	0	0
$106$	0.472136	0.0458579
$107$	− 18.1803i	− 1.75756i	−0.477227	−	0.878780i	$-0.658358\pi$
$107$	− 18.1803i	− 1.75756i	0.477227	−	0.878780i	$-0.341642\pi$
$108$	− 5.47214i	− 0.526557i
$109$	11.5623	1.10747	0.553734	−	0.832694i	$-0.313202\pi$
$109$	11.5623	1.10747	0.553734	+	0.832694i	$0.313202\pi$
$110$	0	0
$111$	10.4721	0.993971
$112$	0.618034i	0.0583987i
$113$	1.05573i	0.0993145i	0.998766	+	0.0496573i	$0.0158129\pi$
$113$	1.05573i	0.0993145i	−0.998766	+	0.0496573i	$0.984187\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	−9.23607	−0.857547
$117$	− 2.70820i	− 0.250374i
$118$	1.70820i	0.157253i
$119$	3.76393	0.345039
$120$	0	0
$121$	−2.85410	−0.259464
$122$	9.32624i	0.844358i
$123$	5.38197i	0.485276i
$124$	−9.09017	−0.816321
$125$	0	0
$126$	0.236068	0.0210306
$127$	− 16.1803i	− 1.43577i	−0.696160	−	0.717886i	$-0.745110\pi$
$127$	− 16.1803i	− 1.43577i	0.696160	−	0.717886i	$-0.254890\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	2.94427	0.257242	0.128621	−	0.991694i	$-0.458945\pi$
$131$	2.94427	0.257242	0.128621	+	0.991694i	$0.458945\pi$
$132$	4.61803i	0.401948i
$133$	− 1.14590i	− 0.0993620i
$134$	−14.4721	−1.25020
$135$	0	0
$136$	6.09017	0.522228
$137$	10.3262i	0.882230i	0.897451	+	0.441115i	$0.145417\pi$
$137$	10.3262i	0.882230i	−0.897451	+	0.441115i	$0.854583\pi$
$138$	1.61803i	0.137736i
$139$	−12.7639	−1.08262	−0.541311	−	0.840822i	$-0.682072\pi$
$139$	−12.7639	−1.08262	−0.541311	+	0.840822i	$0.682072\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	4.09017i	0.343239i
$143$	20.2361i	1.69223i
$144$	0.381966	0.0318305
$145$	0	0
$146$	3.23607	0.267819
$147$	10.7082i	0.883198i
$148$	6.47214i	0.532006i
$149$	7.85410	0.643433	0.321717	−	0.946836i	$-0.395740\pi$
$149$	7.85410	0.643433	0.321717	+	0.946836i	$0.395740\pi$
$150$	0	0
$151$	−2.56231	−0.208517	−0.104259	−	0.994550i	$-0.533247\pi$
$151$	−2.56231	−0.208517	−0.104259	+	0.994550i	$0.533247\pi$
$152$	− 1.85410i	− 0.150388i
$153$	− 2.32624i	− 0.188065i
$154$	−1.76393	−0.142142
$155$	0	0
$156$	−11.4721	−0.918506
$157$	3.70820i	0.295947i	0.988991	+	0.147973i	$0.0472750\pi$
$157$	3.70820i	0.295947i	−0.988991	+	0.147973i	$0.952725\pi$
$158$	1.52786i	0.121550i
$159$	−0.763932	−0.0605838
$160$	0	0
$161$	−0.618034	−0.0487079
$162$	7.70820i	0.605614i
$163$	1.38197i	0.108244i	0.998534	+	0.0541220i	$0.0172360\pi$
$163$	1.38197i	0.108244i	−0.998534	+	0.0541220i	$0.982764\pi$
$164$	−3.32624	−0.259736
$165$	0	0
$166$	−6.94427	−0.538980
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	− 1.00000i	− 0.0771517i
$169$	−37.2705	−2.86696
$170$	0	0
$171$	−0.708204	−0.0541577
$172$	0	0
$173$	− 1.43769i	− 0.109306i	−0.998505	−	0.0546529i	$-0.982595\pi$
$173$	− 1.43769i	− 0.109306i	0.998505	−	0.0546529i	$-0.0174052\pi$
$174$	14.9443	1.13292
$175$	0	0
$176$	−2.85410	−0.215136
$177$	− 2.76393i	− 0.207750i
$178$	− 10.4721i	− 0.784920i
$179$	−2.18034	−0.162966	−0.0814831	−	0.996675i	$-0.525966\pi$
$179$	−2.18034	−0.162966	−0.0814831	+	0.996675i	$0.525966\pi$
$180$	0	0
$181$	−12.1459	−0.902797	−0.451399	−	0.892322i	$-0.649075\pi$
$181$	−12.1459	−0.902797	−0.451399	+	0.892322i	$0.649075\pi$
$182$	− 4.38197i	− 0.324813i
$183$	− 15.0902i	− 1.11550i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	14.7082	1.07846
$187$	17.3820i	1.27110i
$188$	− 3.70820i	− 0.270449i
$189$	−3.38197	−0.246002
$190$	0	0
$191$	−13.7082	−0.991891	−0.495945	−	0.868354i	$-0.665178\pi$
$191$	−13.7082	−0.991891	−0.495945	+	0.868354i	$0.665178\pi$
$192$	− 1.61803i	− 0.116772i
$193$	0.763932i	0.0549890i	0.999622	+	0.0274945i	$0.00875288\pi$
$193$	0.763932i	0.0549890i	−0.999622	+	0.0274945i	$0.991247\pi$
$194$	−12.3820	−0.888973
$195$	0	0
$196$	−6.61803	−0.472717
$197$	22.5623i	1.60750i	0.594969	+	0.803749i	$0.297164\pi$
$197$	22.5623i	1.60750i	−0.594969	+	0.803749i	$0.702836\pi$
$198$	1.09017i	0.0774750i
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	23.4164	1.65167
$202$	0.291796i	0.0205307i
$203$	5.70820i	0.400637i
$204$	−9.85410	−0.689925
$205$	0	0
$206$	16.5623	1.15395
$207$	0.381966i	0.0265485i
$208$	− 7.09017i	− 0.491615i
$209$	5.29180	0.366041
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	− 0.472136i	− 0.0324264i
$213$	− 6.61803i	− 0.453460i
$214$	−18.1803	−1.24278
$215$	0	0
$216$	−5.47214	−0.372332
$217$	5.61803i	0.381377i
$218$	− 11.5623i	− 0.783098i
$219$	−5.23607	−0.353821
$220$	0	0
$221$	−43.1803	−2.90462
$222$	− 10.4721i	− 0.702844i
$223$	− 20.9443i	− 1.40253i	−0.712900	−	0.701266i	$-0.752618\pi$
$223$	− 20.9443i	− 1.40253i	0.712900	−	0.701266i	$-0.247382\pi$
$224$	0.618034	0.0412941
$225$	0	0
$226$	1.05573	0.0702260
$227$	18.7639i	1.24541i	0.782458	+	0.622703i	$0.213965\pi$
$227$	18.7639i	1.24541i	−0.782458	+	0.622703i	$0.786035\pi$
$228$	3.00000i	0.198680i
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	2.85410	0.187786
$232$	9.23607i	0.606378i
$233$	6.29180i	0.412189i	0.978532	+	0.206095i	$0.0660755\pi$
$233$	6.29180i	0.412189i	−0.978532	+	0.206095i	$0.933925\pi$
$234$	−2.70820	−0.177041
$235$	0	0
$236$	1.70820	0.111195
$237$	− 2.47214i	− 0.160582i
$238$	− 3.76393i	− 0.243979i
$239$	20.3607	1.31702	0.658511	−	0.752571i	$-0.271186\pi$
$239$	20.3607	1.31702	0.658511	+	0.752571i	$0.271186\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	2.85410i	0.183469i
$243$	3.94427i	0.253025i
$244$	9.32624	0.597051
$245$	0	0
$246$	5.38197	0.343142
$247$	13.1459i	0.836453i
$248$	9.09017i	0.577226i
$249$	11.2361	0.712057
$250$	0	0
$251$	6.14590	0.387926	0.193963	−	0.981009i	$-0.437866\pi$
$251$	6.14590	0.387926	0.193963	+	0.981009i	$0.437866\pi$
$252$	− 0.236068i	− 0.0148709i
$253$	− 2.85410i	− 0.179436i
$254$	−16.1803	−1.01524
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.81966i	0.487777i	0.969803	+	0.243888i	$0.0784231\pi$
$257$	7.81966i	0.487777i	−0.969803	+	0.243888i	$0.921577\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	3.52786	0.218369
$262$	− 2.94427i	− 0.181898i
$263$	20.7426i	1.27905i	0.768772	+	0.639523i	$0.220868\pi$
$263$	20.7426i	1.27905i	−0.768772	+	0.639523i	$0.779132\pi$
$264$	4.61803	0.284220
$265$	0	0
$266$	−1.14590	−0.0702595
$267$	16.9443i	1.03697i
$268$	14.4721i	0.884026i
$269$	14.1803	0.864591	0.432295	−	0.901732i	$-0.357704\pi$
$269$	14.1803	0.864591	0.432295	+	0.901732i	$0.357704\pi$
$270$	0	0
$271$	−30.3262	−1.84219	−0.921094	−	0.389341i	$-0.872703\pi$
$271$	−30.3262	−1.84219	−0.921094	+	0.389341i	$0.872703\pi$
$272$	− 6.09017i	− 0.369271i
$273$	7.09017i	0.429117i
$274$	10.3262	0.623831
$275$	0	0
$276$	1.61803	0.0973942
$277$	− 29.4164i	− 1.76746i	−0.467996	−	0.883730i	$-0.655024\pi$
$277$	− 29.4164i	− 1.76746i	0.467996	−	0.883730i	$-0.344976\pi$
$278$	12.7639i	0.765530i
$279$	3.47214	0.207871
$280$	0	0
$281$	22.7639	1.35798	0.678991	−	0.734146i	$-0.262417\pi$
$281$	22.7639	1.35798	0.678991	+	0.734146i	$0.262417\pi$
$282$	6.00000i	0.357295i
$283$	− 26.9443i	− 1.60167i	−0.598885	−	0.800835i	$-0.704389\pi$
$283$	− 26.9443i	− 1.60167i	0.598885	−	0.800835i	$-0.295611\pi$
$284$	4.09017	0.242707
$285$	0	0
$286$	20.2361	1.19658
$287$	2.05573i	0.121346i
$288$	− 0.381966i	− 0.0225076i
$289$	−20.0902	−1.18177
$290$	0	0
$291$	20.0344	1.17444
$292$	− 3.23607i	− 0.189377i
$293$	− 19.8885i	− 1.16190i	−0.813939	−	0.580951i	$-0.802681\pi$
$293$	− 19.8885i	− 1.16190i	0.813939	−	0.580951i	$-0.197319\pi$
$294$	10.7082	0.624515
$295$	0	0
$296$	6.47214	0.376185
$297$	− 15.6180i	− 0.906250i
$298$	− 7.85410i	− 0.454976i
$299$	7.09017	0.410035
$300$	0	0
$301$	0	0
$302$	2.56231i	0.147444i
$303$	− 0.472136i	− 0.0271235i
$304$	−1.85410	−0.106340
$305$	0	0
$306$	−2.32624	−0.132982
$307$	28.4508i	1.62378i	0.583813	+	0.811888i	$0.301560\pi$
$307$	28.4508i	1.62378i	−0.583813	+	0.811888i	$0.698440\pi$
$308$	1.76393i	0.100509i
$309$	−26.7984	−1.52451
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	11.4721i	0.649482i
$313$	12.7984i	0.723407i	0.932293	+	0.361703i	$0.117805\pi$
$313$	12.7984i	0.723407i	−0.932293	+	0.361703i	$0.882195\pi$
$314$	3.70820	0.209266
$315$	0	0
$316$	1.52786	0.0859491
$317$	11.0902i	0.622886i	0.950265	+	0.311443i	$0.100812\pi$
$317$	11.0902i	0.622886i	−0.950265	+	0.311443i	$0.899188\pi$
$318$	0.763932i	0.0428392i
$319$	−26.3607	−1.47591
$320$	0	0
$321$	29.4164	1.64186
$322$	0.618034i	0.0344417i
$323$	11.2918i	0.628292i
$324$	7.70820	0.428234
$325$	0	0
$326$	1.38197	0.0765400
$327$	18.7082i	1.03457i
$328$	3.32624i	0.183661i
$329$	−2.29180	−0.126351
$330$	0	0
$331$	19.2361	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$331$	19.2361	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$332$	6.94427i	0.381116i
$333$	− 2.47214i	− 0.135472i
$334$	8.00000	0.437741
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	− 13.6738i	− 0.744857i	−0.928061	−	0.372429i	$-0.878525\pi$
$337$	− 13.6738i	− 0.744857i	0.928061	−	0.372429i	$-0.121475\pi$
$338$	37.2705i	2.02725i
$339$	−1.70820	−0.0927769
$340$	0	0
$341$	−25.9443	−1.40496
$342$	0.708204i	0.0382953i
$343$	8.41641i	0.454443i
$344$	0	0
$345$	0	0
$346$	−1.43769	−0.0772909
$347$	6.38197i	0.342602i	0.985219	+	0.171301i	$0.0547970\pi$
$347$	6.38197i	0.342602i	−0.985219	+	0.171301i	$0.945203\pi$
$348$	− 14.9443i	− 0.801097i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	38.7984	2.07090
$352$	2.85410i	0.152124i
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	−2.76393	−0.146901
$355$	0	0
$356$	−10.4721	−0.555022
$357$	6.09017i	0.322326i
$358$	2.18034i	0.115235i
$359$	−26.3607	−1.39126	−0.695632	−	0.718399i	$-0.744875\pi$
$359$	−26.3607	−1.39126	−0.695632	+	0.718399i	$0.744875\pi$
$360$	0	0
$361$	−15.5623	−0.819069
$362$	12.1459i	0.638374i
$363$	− 4.61803i	− 0.242384i
$364$	−4.38197	−0.229677
$365$	0	0
$366$	−15.0902	−0.788776
$367$	− 6.47214i	− 0.337843i	−0.985630	−	0.168921i	$-0.945972\pi$
$367$	− 6.47214i	− 0.337843i	0.985630	−	0.168921i	$-0.0540284\pi$
$368$	1.00000i	0.0521286i
$369$	1.27051	0.0661401
$370$	0	0
$371$	−0.291796	−0.0151493
$372$	− 14.7082i	− 0.762585i
$373$	20.1803i	1.04490i	0.852670	+	0.522449i	$0.174982\pi$
$373$	20.1803i	1.04490i	−0.852670	+	0.522449i	$0.825018\pi$
$374$	17.3820	0.898800
$375$	0	0
$376$	−3.70820	−0.191236
$377$	− 65.4853i	− 3.37266i
$378$	3.38197i	0.173950i
$379$	22.4508	1.15322	0.576611	−	0.817019i	$-0.304375\pi$
$379$	22.4508	1.15322	0.576611	+	0.817019i	$0.304375\pi$
$380$	0	0
$381$	26.1803	1.34126
$382$	13.7082i	0.701373i
$383$	− 17.8885i	− 0.914062i	−0.889451	−	0.457031i	$-0.848913\pi$
$383$	− 17.8885i	− 0.914062i	0.889451	−	0.457031i	$-0.151087\pi$
$384$	−1.61803	−0.0825700
$385$	0	0
$386$	0.763932	0.0388831
$387$	0	0
$388$	12.3820i	0.628599i
$389$	−21.3262	−1.08128	−0.540642	−	0.841253i	$-0.681818\pi$
$389$	−21.3262	−1.08128	−0.540642	+	0.841253i	$0.681818\pi$
$390$	0	0
$391$	6.09017	0.307993
$392$	6.61803i	0.334261i
$393$	4.76393i	0.240309i
$394$	22.5623	1.13667
$395$	0	0
$396$	1.09017	0.0547831
$397$	− 7.32624i	− 0.367693i	−0.982955	−	0.183847i	$-0.941145\pi$
$397$	− 7.32624i	− 0.367693i	0.982955	−	0.183847i	$-0.0588550\pi$
$398$	2.00000i	0.100251i
$399$	1.85410	0.0928212
$400$	0	0
$401$	−1.70820	−0.0853036	−0.0426518	−	0.999090i	$-0.513581\pi$
$401$	−1.70820	−0.0853036	−0.0426518	+	0.999090i	$0.513581\pi$
$402$	− 23.4164i	− 1.16790i
$403$	− 64.4508i	− 3.21053i
$404$	0.291796	0.0145174
$405$	0	0
$406$	5.70820	0.283293
$407$	18.4721i	0.915630i
$408$	9.85410i	0.487851i
$409$	30.2148	1.49402	0.747012	−	0.664810i	$-0.231488\pi$
$409$	30.2148	1.49402	0.747012	+	0.664810i	$0.231488\pi$
$410$	0	0
$411$	−16.7082	−0.824155
$412$	− 16.5623i	− 0.815966i
$413$	− 1.05573i	− 0.0519490i
$414$	0.381966	0.0187726
$415$	0	0
$416$	−7.09017	−0.347624
$417$	− 20.6525i	− 1.01136i
$418$	− 5.29180i	− 0.258830i
$419$	−14.4721	−0.707010	−0.353505	−	0.935433i	$-0.615010\pi$
$419$	−14.4721	−0.707010	−0.353505	+	0.935433i	$0.615010\pi$
$420$	0	0
$421$	13.7426	0.669776	0.334888	−	0.942258i	$-0.391302\pi$
$421$	13.7426	0.669776	0.334888	+	0.942258i	$0.391302\pi$
$422$	− 14.0000i	− 0.681509i
$423$	1.41641i	0.0688681i
$424$	−0.472136	−0.0229289
$425$	0	0
$426$	−6.61803	−0.320645
$427$	− 5.76393i	− 0.278936i
$428$	18.1803i	0.878780i
$429$	−32.7426	−1.58083
$430$	0	0
$431$	3.34752	0.161245	0.0806223	−	0.996745i	$-0.474309\pi$
$431$	3.34752	0.161245	0.0806223	+	0.996745i	$0.474309\pi$
$432$	5.47214i	0.263278i
$433$	8.50658i	0.408800i	0.978887	+	0.204400i	$0.0655243\pi$
$433$	8.50658i	0.408800i	−0.978887	+	0.204400i	$0.934476\pi$
$434$	5.61803	0.269674
$435$	0	0
$436$	−11.5623	−0.553734
$437$	− 1.85410i	− 0.0886937i
$438$	5.23607i	0.250189i
$439$	13.3820	0.638686	0.319343	−	0.947639i	$-0.396538\pi$
$439$	13.3820	0.638686	0.319343	+	0.947639i	$0.396538\pi$
$440$	0	0
$441$	2.52786	0.120374
$442$	43.1803i	2.05388i
$443$	25.0902i	1.19207i	0.802958	+	0.596035i	$0.203258\pi$
$443$	25.0902i	1.19207i	−0.802958	+	0.596035i	$0.796742\pi$
$444$	−10.4721	−0.496986
$445$	0	0
$446$	−20.9443	−0.991740
$447$	12.7082i	0.601077i
$448$	− 0.618034i	− 0.0291994i
$449$	1.56231	0.0737298	0.0368649	−	0.999320i	$-0.488263\pi$
$449$	1.56231	0.0737298	0.0368649	+	0.999320i	$0.488263\pi$
$450$	0	0
$451$	−9.49342	−0.447028
$452$	− 1.05573i	− 0.0496573i
$453$	− 4.14590i	− 0.194791i
$454$	18.7639	0.880635
$455$	0	0
$456$	3.00000	0.140488
$457$	37.7771i	1.76714i	0.468301	+	0.883569i	$0.344866\pi$
$457$	37.7771i	1.76714i	−0.468301	+	0.883569i	$0.655134\pi$
$458$	10.0000i	0.467269i
$459$	33.3262	1.55554
$460$	0	0
$461$	−39.2361	−1.82741	−0.913703	−	0.406383i	$-0.866790\pi$
$461$	−39.2361	−1.82741	−0.913703	+	0.406383i	$0.866790\pi$
$462$	− 2.85410i	− 0.132785i
$463$	2.00000i	0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$	2.00000i	0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$	9.23607	0.428774
$465$	0	0
$466$	6.29180	0.291462
$467$	17.1246i	0.792433i	0.918157	+	0.396216i	$0.129677\pi$
$467$	17.1246i	0.792433i	−0.918157	+	0.396216i	$0.870323\pi$
$468$	2.70820i	0.125187i
$469$	8.94427	0.413008
$470$	0	0
$471$	−6.00000	−0.276465
$472$	− 1.70820i	− 0.0786265i
$473$	0	0
$474$	−2.47214	−0.113549
$475$	0	0
$476$	−3.76393	−0.172520
$477$	0.180340i	0.00825720i
$478$	− 20.3607i	− 0.931276i
$479$	31.8885	1.45702	0.728512	−	0.685033i	$-0.240212\pi$
$479$	31.8885	1.45702	0.728512	+	0.685033i	$0.240212\pi$
$480$	0	0
$481$	−45.8885	−2.09234
$482$	0	0
$483$	− 1.00000i	− 0.0455016i
$484$	2.85410	0.129732
$485$	0	0
$486$	3.94427	0.178916
$487$	19.8197i	0.898115i	0.893503	+	0.449057i	$0.148240\pi$
$487$	19.8197i	0.898115i	−0.893503	+	0.449057i	$0.851760\pi$
$488$	− 9.32624i	− 0.422179i
$489$	−2.23607	−0.101118
$490$	0	0
$491$	6.18034	0.278915	0.139457	−	0.990228i	$-0.455464\pi$
$491$	6.18034	0.278915	0.139457	+	0.990228i	$0.455464\pi$
$492$	− 5.38197i	− 0.242638i
$493$	− 56.2492i	− 2.53334i
$494$	13.1459	0.591462
$495$	0	0
$496$	9.09017	0.408161
$497$	− 2.52786i	− 0.113390i
$498$	− 11.2361i	− 0.503500i
$499$	−12.3607	−0.553340	−0.276670	−	0.960965i	$-0.589231\pi$
$499$	−12.3607	−0.553340	−0.276670	+	0.960965i	$0.589231\pi$
$500$	0	0
$501$	−12.9443	−0.578307
$502$	− 6.14590i	− 0.274305i
$503$	36.3262i	1.61971i	0.586632	+	0.809853i	$0.300453\pi$
$503$	36.3262i	1.61971i	−0.586632	+	0.809853i	$0.699547\pi$
$504$	−0.236068	−0.0105153
$505$	0	0
$506$	−2.85410	−0.126880
$507$	− 60.3050i	− 2.67824i
$508$	16.1803i	0.717886i
$509$	−36.6525	−1.62459	−0.812296	−	0.583245i	$-0.801783\pi$
$509$	−36.6525	−1.62459	−0.812296	+	0.583245i	$0.801783\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	− 10.1459i	− 0.447952i
$514$	7.81966	0.344910
$515$	0	0
$516$	0	0
$517$	− 10.5836i	− 0.465466i
$518$	− 4.00000i	− 0.175750i
$519$	2.32624	0.102111
$520$	0	0
$521$	−15.5279	−0.680288	−0.340144	−	0.940373i	$-0.610476\pi$
$521$	−15.5279	−0.680288	−0.340144	+	0.940373i	$0.610476\pi$
$522$	− 3.52786i	− 0.154410i
$523$	− 26.0000i	− 1.13690i	−0.822718	−	0.568450i	$-0.807543\pi$
$523$	− 26.0000i	− 1.13690i	0.822718	−	0.568450i	$-0.192457\pi$
$524$	−2.94427	−0.128621
$525$	0	0
$526$	20.7426	0.904422
$527$	− 55.3607i	− 2.41155i
$528$	− 4.61803i	− 0.200974i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−0.652476	−0.0283150
$532$	1.14590i	0.0496810i
$533$	− 23.5836i	− 1.02152i
$534$	16.9443	0.733250
$535$	0	0
$536$	14.4721	0.625101
$537$	− 3.52786i	− 0.152239i
$538$	− 14.1803i	− 0.611358i
$539$	−18.8885	−0.813587
$540$	0	0
$541$	22.8328	0.981659	0.490830	−	0.871256i	$-0.336694\pi$
$541$	22.8328	0.981659	0.490830	+	0.871256i	$0.336694\pi$
$542$	30.3262i	1.30262i
$543$	− 19.6525i	− 0.843368i
$544$	−6.09017	−0.261114
$545$	0	0
$546$	7.09017	0.303431
$547$	− 27.9230i	− 1.19390i	−0.802278	−	0.596950i	$-0.796379\pi$
$547$	− 27.9230i	− 1.19390i	0.802278	−	0.596950i	$-0.203621\pi$
$548$	− 10.3262i	− 0.441115i
$549$	−3.56231	−0.152036
$550$	0	0
$551$	−17.1246	−0.729533
$552$	− 1.61803i	− 0.0688681i
$553$	− 0.944272i	− 0.0401545i
$554$	−29.4164	−1.24978
$555$	0	0
$556$	12.7639	0.541311
$557$	22.8328i	0.967457i	0.875218	+	0.483729i	$0.160718\pi$
$557$	22.8328i	0.967457i	−0.875218	+	0.483729i	$0.839282\pi$
$558$	− 3.47214i	− 0.146987i
$559$	0	0
$560$	0	0
$561$	−28.1246	−1.18742
$562$	− 22.7639i	− 0.960239i
$563$	− 13.8885i	− 0.585332i	−0.956215	−	0.292666i	$-0.905458\pi$
$563$	− 13.8885i	− 0.585332i	0.956215	−	0.292666i	$-0.0945425\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	−26.9443	−1.13255
$567$	− 4.76393i	− 0.200066i
$568$	− 4.09017i	− 0.171620i
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	15.9787	0.668688	0.334344	−	0.942451i	$-0.391485\pi$
$571$	15.9787	0.668688	0.334344	+	0.942451i	$0.391485\pi$
$572$	− 20.2361i	− 0.846113i
$573$	− 22.1803i	− 0.926597i
$574$	2.05573	0.0858044
$575$	0	0
$576$	−0.381966	−0.0159153
$577$	3.52786i	0.146867i	0.997300	+	0.0734335i	$0.0233957\pi$
$577$	3.52786i	0.146867i	−0.997300	+	0.0734335i	$0.976604\pi$
$578$	20.0902i	0.835641i
$579$	−1.23607	−0.0513692
$580$	0	0
$581$	4.29180	0.178054
$582$	− 20.0344i	− 0.830454i
$583$	− 1.34752i	− 0.0558087i
$584$	−3.23607	−0.133909
$585$	0	0
$586$	−19.8885	−0.821588
$587$	− 13.6180i	− 0.562076i	−0.959697	−	0.281038i	$-0.909321\pi$
$587$	− 13.6180i	− 0.562076i	0.959697	−	0.281038i	$-0.0906788\pi$
$588$	− 10.7082i	− 0.441599i
$589$	−16.8541	−0.694461
$590$	0	0
$591$	−36.5066	−1.50168
$592$	− 6.47214i	− 0.266003i
$593$	39.2361i	1.61123i	0.592438	+	0.805616i	$0.298166\pi$
$593$	39.2361i	1.61123i	−0.592438	+	0.805616i	$0.701834\pi$
$594$	−15.6180	−0.640816
$595$	0	0
$596$	−7.85410	−0.321717
$597$	− 3.23607i	− 0.132443i
$598$	− 7.09017i	− 0.289939i
$599$	18.3820	0.751067	0.375533	−	0.926809i	$-0.377460\pi$
$599$	18.3820	0.751067	0.375533	+	0.926809i	$0.377460\pi$
$600$	0	0
$601$	−33.2705	−1.35713	−0.678566	−	0.734539i	$-0.737398\pi$
$601$	−33.2705	−1.35713	−0.678566	+	0.734539i	$0.737398\pi$
$602$	0	0
$603$	− 5.52786i	− 0.225112i
$604$	2.56231	0.104259
$605$	0	0
$606$	−0.472136	−0.0191792
$607$	− 26.4721i	− 1.07447i	−0.843432	−	0.537235i	$-0.819469\pi$
$607$	− 26.4721i	− 1.07447i	0.843432	−	0.537235i	$-0.180531\pi$
$608$	1.85410i	0.0751938i
$609$	−9.23607	−0.374264
$610$	0	0
$611$	26.2918	1.06365
$612$	2.32624i	0.0940326i
$613$	19.3050i	0.779720i	0.920874	+	0.389860i	$0.127477\pi$
$613$	19.3050i	0.779720i	−0.920874	+	0.389860i	$0.872523\pi$
$614$	28.4508	1.14818
$615$	0	0
$616$	1.76393	0.0710708
$617$	− 34.0902i	− 1.37242i	−0.727404	−	0.686209i	$-0.759273\pi$
$617$	− 34.0902i	− 1.37242i	0.727404	−	0.686209i	$-0.240727\pi$
$618$	26.7984i	1.07799i
$619$	2.79837	0.112476	0.0562381	−	0.998417i	$-0.482089\pi$
$619$	2.79837	0.112476	0.0562381	+	0.998417i	$0.482089\pi$
$620$	0	0
$621$	−5.47214	−0.219589
$622$	− 4.00000i	− 0.160385i
$623$	6.47214i	0.259301i
$624$	11.4721	0.459253
$625$	0	0
$626$	12.7984	0.511526
$627$	8.56231i	0.341946i
$628$	− 3.70820i	− 0.147973i
$629$	−39.4164	−1.57164
$630$	0	0
$631$	42.0689	1.67474	0.837368	−	0.546640i	$-0.184093\pi$
$631$	42.0689	1.67474	0.837368	+	0.546640i	$0.184093\pi$
$632$	− 1.52786i	− 0.0607752i
$633$	22.6525i	0.900355i
$634$	11.0902	0.440447
$635$	0	0
$636$	0.763932	0.0302919
$637$	− 46.9230i	− 1.85916i
$638$	26.3607i	1.04363i
$639$	−1.56231	−0.0618039
$640$	0	0
$641$	0.360680	0.0142460	0.00712300	−	0.999975i	$-0.497733\pi$
$641$	0.360680	0.0142460	0.00712300	+	0.999975i	$0.497733\pi$
$642$	− 29.4164i	− 1.16097i
$643$	− 8.29180i	− 0.326997i	−0.986544	−	0.163498i	$-0.947722\pi$
$643$	− 8.29180i	− 0.326997i	0.986544	−	0.163498i	$-0.0522778\pi$
$644$	0.618034	0.0243540
$645$	0	0
$646$	11.2918	0.444270
$647$	36.2492i	1.42510i	0.701619	+	0.712552i	$0.252461\pi$
$647$	36.2492i	1.42510i	−0.701619	+	0.712552i	$0.747539\pi$
$648$	− 7.70820i	− 0.302807i
$649$	4.87539	0.191376
$650$	0	0
$651$	−9.09017	−0.356272
$652$	− 1.38197i	− 0.0541220i
$653$	− 8.03444i	− 0.314412i	−0.987566	−	0.157206i	$-0.949751\pi$
$653$	− 8.03444i	− 0.314412i	0.987566	−	0.157206i	$-0.0502487\pi$
$654$	18.7082	0.731549
$655$	0	0
$656$	3.32624	0.129868
$657$	1.23607i	0.0482236i
$658$	2.29180i	0.0893435i
$659$	46.2492	1.80161	0.900807	−	0.434220i	$-0.142976\pi$
$659$	46.2492	1.80161	0.900807	+	0.434220i	$0.142976\pi$
$660$	0	0
$661$	18.6738	0.726325	0.363163	−	0.931726i	$-0.381697\pi$
$661$	18.6738	0.726325	0.363163	+	0.931726i	$0.381697\pi$
$662$	− 19.2361i	− 0.747631i
$663$	− 69.8673i	− 2.71342i
$664$	6.94427	0.269490
$665$	0	0
$666$	−2.47214	−0.0957933
$667$	9.23607i	0.357622i
$668$	− 8.00000i	− 0.309529i
$669$	33.8885	1.31021
$670$	0	0
$671$	26.6180	1.02758
$672$	1.00000i	0.0385758i
$673$	− 10.9443i	− 0.421871i	−0.977500	−	0.210935i	$-0.932349\pi$
$673$	− 10.9443i	− 0.421871i	0.977500	−	0.210935i	$-0.0676510\pi$
$674$	−13.6738	−0.526694
$675$	0	0
$676$	37.2705	1.43348
$677$	50.9443i	1.95795i	0.203986	+	0.978974i	$0.434610\pi$
$677$	50.9443i	1.95795i	−0.203986	+	0.978974i	$0.565390\pi$
$678$	1.70820i	0.0656032i
$679$	7.65248	0.293675
$680$	0	0
$681$	−30.3607	−1.16342
$682$	25.9443i	0.993458i
$683$	− 31.5623i	− 1.20770i	−0.797099	−	0.603849i	$-0.793633\pi$
$683$	− 31.5623i	− 1.20770i	0.797099	−	0.603849i	$-0.206367\pi$
$684$	0.708204	0.0270789
$685$	0	0
$686$	8.41641	0.321340
$687$	− 16.1803i	− 0.617318i
$688$	0	0
$689$	3.34752	0.127531
$690$	0	0
$691$	−29.2361	−1.11219	−0.556096	−	0.831118i	$-0.687701\pi$
$691$	−29.2361	−1.11219	−0.556096	+	0.831118i	$0.687701\pi$
$692$	1.43769i	0.0546529i
$693$	− 0.673762i	− 0.0255941i
$694$	6.38197	0.242256
$695$	0	0
$696$	−14.9443	−0.566461
$697$	− 20.2574i	− 0.767302i
$698$	− 2.00000i	− 0.0757011i
$699$	−10.1803	−0.385056
$700$	0	0
$701$	43.3394	1.63691	0.818453	−	0.574573i	$-0.194832\pi$
$701$	43.3394	1.63691	0.818453	+	0.574573i	$0.194832\pi$
$702$	− 38.7984i	− 1.46435i
$703$	12.0000i	0.452589i
$704$	2.85410	0.107568
$705$	0	0
$706$	24.0000	0.903252
$707$	− 0.180340i	− 0.00678238i
$708$	2.76393i	0.103875i
$709$	26.0902	0.979837	0.489918	−	0.871768i	$-0.337027\pi$
$709$	26.0902	0.979837	0.489918	+	0.871768i	$0.337027\pi$
$710$	0	0
$711$	−0.583592	−0.0218864
$712$	10.4721i	0.392460i
$713$	9.09017i	0.340430i
$714$	6.09017	0.227919
$715$	0	0
$716$	2.18034	0.0814831
$717$	32.9443i	1.23033i
$718$	26.3607i	0.983772i
$719$	−35.2705	−1.31537	−0.657684	−	0.753294i	$-0.728464\pi$
$719$	−35.2705	−1.31537	−0.657684	+	0.753294i	$0.728464\pi$
$720$	0	0
$721$	−10.2361	−0.381211
$722$	15.5623i	0.579169i
$723$	0	0
$724$	12.1459	0.451399
$725$	0	0
$726$	−4.61803	−0.171391
$727$	− 28.2016i	− 1.04594i	−0.852351	−	0.522970i	$-0.824824\pi$
$727$	− 28.2016i	− 1.04594i	0.852351	−	0.522970i	$-0.175176\pi$
$728$	4.38197i	0.162406i
$729$	−29.5066	−1.09284
$730$	0	0
$731$	0	0
$732$	15.0902i	0.557749i
$733$	− 29.4164i	− 1.08652i	−0.839565	−	0.543260i	$-0.817190\pi$
$733$	− 29.4164i	− 1.08652i	0.839565	−	0.543260i	$-0.182810\pi$
$734$	−6.47214	−0.238891
$735$	0	0
$736$	1.00000	0.0368605
$737$	41.3050i	1.52149i
$738$	− 1.27051i	− 0.0467681i
$739$	13.8885	0.510898	0.255449	−	0.966822i	$-0.417777\pi$
$739$	13.8885	0.510898	0.255449	+	0.966822i	$0.417777\pi$
$740$	0	0
$741$	−21.2705	−0.781392
$742$	0.291796i	0.0107122i
$743$	33.6312i	1.23381i	0.787038	+	0.616904i	$0.211613\pi$
$743$	33.6312i	1.23381i	−0.787038	+	0.616904i	$0.788387\pi$
$744$	−14.7082	−0.539229
$745$	0	0
$746$	20.1803	0.738855
$747$	− 2.65248i	− 0.0970490i
$748$	− 17.3820i	− 0.635548i
$749$	11.2361	0.410557
$750$	0	0
$751$	−47.0132	−1.71553	−0.857767	−	0.514038i	$-0.828149\pi$
$751$	−47.0132	−1.71553	−0.857767	+	0.514038i	$0.828149\pi$
$752$	3.70820i	0.135224i
$753$	9.94427i	0.362389i
$754$	−65.4853	−2.38483
$755$	0	0
$756$	3.38197	0.123001
$757$	17.8885i	0.650170i	0.945685	+	0.325085i	$0.105393\pi$
$757$	17.8885i	0.650170i	−0.945685	+	0.325085i	$0.894607\pi$
$758$	− 22.4508i	− 0.815452i
$759$	4.61803	0.167624
$760$	0	0
$761$	46.8673	1.69894	0.849468	−	0.527640i	$-0.176923\pi$
$761$	46.8673	1.69894	0.849468	+	0.527640i	$0.176923\pi$
$762$	− 26.1803i	− 0.948414i
$763$	7.14590i	0.258699i
$764$	13.7082	0.495945
$765$	0	0
$766$	−17.8885	−0.646339
$767$	12.1115i	0.437319i
$768$	1.61803i	0.0583858i
$769$	6.58359	0.237410	0.118705	−	0.992930i	$-0.462126\pi$
$769$	6.58359	0.237410	0.118705	+	0.992930i	$0.462126\pi$
$770$	0	0
$771$	−12.6525	−0.455668
$772$	− 0.763932i	− 0.0274945i
$773$	28.9443i	1.04105i	0.853845	+	0.520527i	$0.174264\pi$
$773$	28.9443i	1.04105i	−0.853845	+	0.520527i	$0.825736\pi$
$774$	0	0
$775$	0	0
$776$	12.3820	0.444487
$777$	6.47214i	0.232187i
$778$	21.3262i	0.764583i
$779$	−6.16718	−0.220962
$780$	0	0
$781$	11.6738	0.417720
$782$	− 6.09017i	− 0.217784i
$783$	50.5410i	1.80619i
$784$	6.61803	0.236358
$785$	0	0
$786$	4.76393	0.169924
$787$	2.87539i	0.102497i	0.998686	+	0.0512483i	$0.0163200\pi$
$787$	2.87539i	0.102497i	−0.998686	+	0.0512483i	$0.983680\pi$
$788$	− 22.5623i	− 0.803749i
$789$	−33.5623	−1.19485
$790$	0	0
$791$	−0.652476	−0.0231994
$792$	− 1.09017i	− 0.0387375i
$793$	66.1246i	2.34815i
$794$	−7.32624	−0.259998
$795$	0	0
$796$	2.00000	0.0708881
$797$	− 13.7082i	− 0.485569i	−0.970080	−	0.242785i	$-0.921939\pi$
$797$	− 13.7082i	− 0.485569i	0.970080	−	0.242785i	$-0.0780609\pi$
$798$	− 1.85410i	− 0.0656345i
$799$	22.5836	0.798950
$800$	0	0
$801$	4.00000	0.141333
$802$	1.70820i	0.0603188i
$803$	− 9.23607i	− 0.325934i
$804$	−23.4164	−0.825833
$805$	0	0
$806$	−64.4508	−2.27018
$807$	22.9443i	0.807677i
$808$	− 0.291796i	− 0.0102653i
$809$	46.7426	1.64338	0.821692	−	0.569932i	$-0.193030\pi$
$809$	46.7426	1.64338	0.821692	+	0.569932i	$0.193030\pi$
$810$	0	0
$811$	21.8197	0.766192	0.383096	−	0.923709i	$-0.374858\pi$
$811$	21.8197	0.766192	0.383096	+	0.923709i	$0.374858\pi$
$812$	− 5.70820i	− 0.200319i
$813$	− 49.0689i	− 1.72092i
$814$	18.4721	0.647448
$815$	0	0
$816$	9.85410	0.344963
$817$	0	0
$818$	− 30.2148i	− 1.05644i
$819$	1.67376	0.0584860
$820$	0	0
$821$	−33.0557	−1.15365	−0.576826	−	0.816867i	$-0.695709\pi$
$821$	−33.0557	−1.15365	−0.576826	+	0.816867i	$0.695709\pi$
$822$	16.7082i	0.582766i
$823$	25.4164i	0.885960i	0.896531	+	0.442980i	$0.146079\pi$
$823$	25.4164i	0.885960i	−0.896531	+	0.442980i	$0.853921\pi$
$824$	−16.5623	−0.576975
$825$	0	0
$826$	−1.05573	−0.0367335
$827$	− 21.7082i	− 0.754868i	−0.926036	−	0.377434i	$-0.876806\pi$
$827$	− 21.7082i	− 0.754868i	0.926036	−	0.377434i	$-0.123194\pi$
$828$	− 0.381966i	− 0.0132742i
$829$	−18.9443	−0.657962	−0.328981	−	0.944337i	$-0.606705\pi$
$829$	−18.9443	−0.657962	−0.328981	+	0.944337i	$0.606705\pi$
$830$	0	0
$831$	47.5967	1.65111
$832$	7.09017i	0.245807i
$833$	− 40.3050i	− 1.39648i
$834$	−20.6525	−0.715137
$835$	0	0
$836$	−5.29180	−0.183021
$837$	49.7426i	1.71936i
$838$	14.4721i	0.499932i
$839$	−33.0132	−1.13974	−0.569870	−	0.821735i	$-0.693007\pi$
$839$	−33.0132	−1.13974	−0.569870	+	0.821735i	$0.693007\pi$
$840$	0	0
$841$	56.3050	1.94155
$842$	− 13.7426i	− 0.473603i
$843$	36.8328i	1.26859i
$844$	−14.0000	−0.481900
$845$	0	0
$846$	1.41641	0.0486971
$847$	− 1.76393i	− 0.0606094i
$848$	0.472136i	0.0162132i
$849$	43.5967	1.49624
$850$	0	0
$851$	6.47214	0.221862
$852$	6.61803i	0.226730i
$853$	− 10.7984i	− 0.369729i	−0.982764	−	0.184865i	$-0.940815\pi$
$853$	− 10.7984i	− 0.369729i	0.982764	−	0.184865i	$-0.0591847\pi$
$854$	−5.76393	−0.197238
$855$	0	0
$856$	18.1803	0.621391
$857$	6.58359i	0.224891i	0.993658	+	0.112446i	$0.0358684\pi$
$857$	6.58359i	0.224891i	−0.993658	+	0.112446i	$0.964132\pi$
$858$	32.7426i	1.11782i
$859$	24.0689	0.821220	0.410610	−	0.911811i	$-0.365316\pi$
$859$	24.0689	0.821220	0.410610	+	0.911811i	$0.365316\pi$
$860$	0	0
$861$	−3.32624	−0.113358
$862$	− 3.34752i	− 0.114017i
$863$	32.7639i	1.11530i	0.830077	+	0.557649i	$0.188296\pi$
$863$	32.7639i	1.11530i	−0.830077	+	0.557649i	$0.811704\pi$
$864$	5.47214	0.186166
$865$	0	0
$866$	8.50658	0.289065
$867$	− 32.5066i	− 1.10398i
$868$	− 5.61803i	− 0.190688i
$869$	4.36068	0.147926
$870$	0	0
$871$	−102.610	−3.47680
$872$	11.5623i	0.391549i
$873$	− 4.72949i	− 0.160069i
$874$	−1.85410	−0.0627159
$875$	0	0
$876$	5.23607	0.176910
$877$	− 18.7426i	− 0.632894i	−0.948610	−	0.316447i	$-0.897510\pi$
$877$	− 18.7426i	− 0.632894i	0.948610	−	0.316447i	$-0.102490\pi$
$878$	− 13.3820i	− 0.451619i
$879$	32.1803	1.08542
$880$	0	0
$881$	8.58359	0.289189	0.144594	−	0.989491i	$-0.453812\pi$
$881$	8.58359	0.289189	0.144594	+	0.989491i	$0.453812\pi$
$882$	− 2.52786i	− 0.0851176i
$883$	15.5623i	0.523713i	0.965107	+	0.261857i	$0.0843348\pi$
$883$	15.5623i	0.523713i	−0.965107	+	0.261857i	$0.915665\pi$
$884$	43.1803	1.45231
$885$	0	0
$886$	25.0902	0.842921
$887$	5.16718i	0.173497i	0.996230	+	0.0867485i	$0.0276477\pi$
$887$	5.16718i	0.173497i	−0.996230	+	0.0867485i	$0.972352\pi$
$888$	10.4721i	0.351422i
$889$	10.0000	0.335389
$890$	0	0
$891$	22.0000	0.737028
$892$	20.9443i	0.701266i
$893$	− 6.87539i	− 0.230076i
$894$	12.7082	0.425026
$895$	0	0
$896$	−0.618034	−0.0206471
$897$	11.4721i	0.383043i
$898$	− 1.56231i	− 0.0521348i
$899$	83.9574	2.80014
$900$	0	0
$901$	2.87539	0.0957931
$902$	9.49342i	0.316096i
$903$	0	0
$904$	−1.05573	−0.0351130
$905$	0	0
$906$	−4.14590	−0.137738
$907$	− 7.12461i	− 0.236569i	−0.992980	−	0.118284i	$-0.962261\pi$
$907$	− 7.12461i	− 0.236569i	0.992980	−	0.118284i	$-0.0377395\pi$
$908$	− 18.7639i	− 0.622703i
$909$	−0.111456	−0.00369677
$910$	0	0
$911$	36.0689	1.19502	0.597508	−	0.801863i	$-0.296158\pi$
$911$	36.0689	1.19502	0.597508	+	0.801863i	$0.296158\pi$
$912$	− 3.00000i	− 0.0993399i
$913$	19.8197i	0.655935i
$914$	37.7771	1.24955
$915$	0	0
$916$	10.0000	0.330409
$917$	1.81966i	0.0600905i
$918$	− 33.3262i	− 1.09993i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−46.0344	−1.51689
$922$	39.2361i	1.29217i
$923$	29.0000i	0.954547i
$924$	−2.85410	−0.0938931
$925$	0	0
$926$	2.00000	0.0657241
$927$	6.32624i	0.207781i
$928$	− 9.23607i	− 0.303189i
$929$	3.52786	0.115745	0.0578727	−	0.998324i	$-0.481568\pi$
$929$	3.52786	0.115745	0.0578727	+	0.998324i	$0.481568\pi$
$930$	0	0
$931$	−12.2705	−0.402150
$932$	− 6.29180i	− 0.206095i
$933$	6.47214i	0.211888i
$934$	17.1246	0.560334
$935$	0	0
$936$	2.70820	0.0885204
$937$	36.7984i	1.20215i	0.799192	+	0.601075i	$0.205261\pi$
$937$	36.7984i	1.20215i	−0.799192	+	0.601075i	$0.794739\pi$
$938$	− 8.94427i	− 0.292041i
$939$	−20.7082	−0.675787
$940$	0	0
$941$	−22.4934	−0.733265	−0.366632	−	0.930366i	$-0.619489\pi$
$941$	−22.4934	−0.733265	−0.366632	+	0.930366i	$0.619489\pi$
$942$	6.00000i	0.195491i
$943$	3.32624i	0.108317i
$944$	−1.70820	−0.0555973
$945$	0	0
$946$	0	0
$947$	54.6869i	1.77709i	0.458793	+	0.888543i	$0.348282\pi$
$947$	54.6869i	1.77709i	−0.458793	+	0.888543i	$0.651718\pi$
$948$	2.47214i	0.0802912i
$949$	22.9443	0.744803
$950$	0	0
$951$	−17.9443	−0.581883
$952$	3.76393i	0.121990i
$953$	3.79837i	0.123041i	0.998106	+	0.0615207i	$0.0195950\pi$
$953$	3.79837i	0.123041i	−0.998106	+	0.0615207i	$0.980405\pi$
$954$	0.180340	0.00583872
$955$	0	0
$956$	−20.3607	−0.658511
$957$	− 42.6525i	− 1.37876i
$958$	− 31.8885i	− 1.03027i
$959$	−6.38197	−0.206084
$960$	0	0
$961$	51.6312	1.66552
$962$	45.8885i	1.47951i
$963$	− 6.94427i	− 0.223776i
$964$	0	0
$965$	0	0
$966$	−1.00000	−0.0321745
$967$	16.5410i	0.531923i	0.963984	+	0.265962i	$0.0856895\pi$
$967$	16.5410i	0.531923i	−0.963984	+	0.265962i	$0.914311\pi$
$968$	− 2.85410i	− 0.0917343i
$969$	−18.2705	−0.586933
$970$	0	0
$971$	34.2705	1.09979	0.549896	−	0.835233i	$-0.314667\pi$
$971$	34.2705	1.09979	0.549896	+	0.835233i	$0.314667\pi$
$972$	− 3.94427i	− 0.126513i
$973$	− 7.88854i	− 0.252895i
$974$	19.8197	0.635063
$975$	0	0
$976$	−9.32624	−0.298526
$977$	23.5623i	0.753825i	0.926249	+	0.376912i	$0.123014\pi$
$977$	23.5623i	0.753825i	−0.926249	+	0.376912i	$0.876986\pi$
$978$	2.23607i	0.0715016i
$979$	−29.8885	−0.955242
$980$	0	0
$981$	4.41641	0.141005
$982$	− 6.18034i	− 0.197223i
$983$	− 14.2705i	− 0.455159i	−0.973760	−	0.227579i	$-0.926919\pi$
$983$	− 14.2705i	− 0.455159i	0.973760	−	0.227579i	$-0.0730811\pi$
$984$	−5.38197	−0.171571
$985$	0	0
$986$	−56.2492	−1.79134
$987$	− 3.70820i	− 0.118033i
$988$	− 13.1459i	− 0.418227i
$989$	0	0
$990$	0	0
$991$	27.5066	0.873775	0.436888	−	0.899516i	$-0.356081\pi$
$991$	27.5066	0.873775	0.436888	+	0.899516i	$0.356081\pi$
$992$	− 9.09017i	− 0.288613i
$993$	31.1246i	0.987710i
$994$	−2.52786	−0.0801790
$995$	0	0
$996$	−11.2361	−0.356028
$997$	− 57.1935i	− 1.81134i	−0.423987	−	0.905668i	$-0.639370\pi$
$997$	− 57.1935i	− 1.81134i	0.423987	−	0.905668i	$-0.360630\pi$
$998$	12.3607i	0.391270i
$999$	35.4164	1.12053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.2.b.i.599.2		4
5.2	odd	4		230.2.a.c.1.2	✓	2
5.3	odd	4		1150.2.a.j.1.1		2
5.4	even	2	inner	1150.2.b.i.599.3		4
15.2	even	4		2070.2.a.u.1.1		2
20.3	even	4		9200.2.a.bu.1.2		2
20.7	even	4		1840.2.a.l.1.1		2
40.27	even	4		7360.2.a.bn.1.2		2
40.37	odd	4		7360.2.a.bh.1.1		2
115.22	even	4		5290.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.c.1.2	✓	2	5.2	odd	4
1150.2.a.j.1.1		2	5.3	odd	4
1150.2.b.i.599.2		4	1.1	even	1	trivial
1150.2.b.i.599.3		4	5.4	even	2	inner
1840.2.a.l.1.1		2	20.7	even	4
2070.2.a.u.1.1		2	15.2	even	4
5290.2.a.o.1.2		2	115.22	even	4
7360.2.a.bh.1.1		2	40.37	odd	4
7360.2.a.bn.1.2		2	40.27	even	4
9200.2.a.bu.1.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.61803i q^{3} -1.00000 q^{4} +1.61803 q^{6} +0.618034i q^{7} +1.00000i q^{8} +0.381966 q^{9} -2.85410 q^{11} -1.61803i q^{12} -7.09017i q^{13} +0.618034 q^{14} +1.00000 q^{16} -6.09017i q^{17} -0.381966i q^{18} -1.85410 q^{19} -1.00000 q^{21} +2.85410i q^{22} +1.00000i q^{23} -1.61803 q^{24} -7.09017 q^{26} +5.47214i q^{27} -0.618034i q^{28} +9.23607 q^{29} +9.09017 q^{31} -1.00000i q^{32} -4.61803i q^{33} -6.09017 q^{34} -0.381966 q^{36} -6.47214i q^{37} +1.85410i q^{38} +11.4721 q^{39} +3.32624 q^{41} +1.00000i q^{42} +2.85410 q^{44} +1.00000 q^{46} +3.70820i q^{47} +1.61803i q^{48} +6.61803 q^{49} +9.85410 q^{51} +7.09017i q^{52} +0.472136i q^{53} +5.47214 q^{54} -0.618034 q^{56} -3.00000i q^{57} -9.23607i q^{58} -1.70820 q^{59} -9.32624 q^{61} -9.09017i q^{62} +0.236068i q^{63} -1.00000 q^{64} -4.61803 q^{66} -14.4721i q^{67} +6.09017i q^{68} -1.61803 q^{69} -4.09017 q^{71} +0.381966i q^{72} +3.23607i q^{73} -6.47214 q^{74} +1.85410 q^{76} -1.76393i q^{77} -11.4721i q^{78} -1.52786 q^{79} -7.70820 q^{81} -3.32624i q^{82} -6.94427i q^{83} +1.00000 q^{84} +14.9443i q^{87} -2.85410i q^{88} +10.4721 q^{89} +4.38197 q^{91} -1.00000i q^{92} +14.7082i q^{93} +3.70820 q^{94} +1.61803 q^{96} -12.3820i q^{97} -6.61803i q^{98} -1.09017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46}+ \cdots + 18 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 2 * q^6 + 6 * q^9 + 2 * q^11 - 2 * q^14 + 4 * q^16 + 6 * q^19 - 4 * q^21 - 2 * q^24 - 6 * q^26 + 28 * q^29 + 14 * q^31 - 2 * q^34 - 6 * q^36 + 28 * q^39 - 18 * q^41 - 2 * q^44 + 4 * q^46 + 22 * q^49 + 26 * q^51 + 4 * q^54 + 2 * q^56 + 20 * q^59 - 6 * q^61 - 4 * q^64 - 14 * q^66 - 2 * q^69 + 6 * q^71 - 8 * q^74 - 6 * q^76 - 24 * q^79 - 4 * q^81 + 4 * q^84 + 24 * q^89 + 22 * q^91 - 12 * q^94 + 2 * q^96 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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