LMFDB - Embedded newform 1275.2.b.f.1174.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1275,2,Mod(1174,1275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1275.1174");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1275.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:	$10.1809262577$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 255)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.1
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	1275.1174
Dual form			1275.2.b.f.1174.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.30278i q^{2} -1.00000i q^{3} -3.30278 q^{4} -2.30278 q^{6} -3.60555i q^{7} +3.00000i q^{8} -1.00000 q^{9} +5.00000 q^{11} +3.30278i q^{12} -6.60555i q^{13} -8.30278 q^{14} +0.302776 q^{16} -1.00000i q^{17} +2.30278i q^{18} +5.60555 q^{19} -3.60555 q^{21} -11.5139i q^{22} -2.60555i q^{23} +3.00000 q^{24} -15.2111 q^{26} +1.00000i q^{27} +11.9083i q^{28} -0.394449 q^{29} -6.60555 q^{31} +5.30278i q^{32} -5.00000i q^{33} -2.30278 q^{34} +3.30278 q^{36} +6.21110i q^{37} -12.9083i q^{38} -6.60555 q^{39} +9.60555 q^{41} +8.30278i q^{42} +9.21110i q^{43} -16.5139 q^{44} -6.00000 q^{46} -2.21110i q^{47} -0.302776i q^{48} -6.00000 q^{49} -1.00000 q^{51} +21.8167i q^{52} -3.60555i q^{53} +2.30278 q^{54} +10.8167 q^{56} -5.60555i q^{57} +0.908327i q^{58} -4.60555 q^{59} +6.60555 q^{61} +15.2111i q^{62} +3.60555i q^{63} +12.8167 q^{64} -11.5139 q^{66} +12.6056i q^{67} +3.30278i q^{68} -2.60555 q^{69} -1.21110 q^{71} -3.00000i q^{72} +13.0000i q^{73} +14.3028 q^{74} -18.5139 q^{76} -18.0278i q^{77} +15.2111i q^{78} -3.21110 q^{79} +1.00000 q^{81} -22.1194i q^{82} -1.21110i q^{83} +11.9083 q^{84} +21.2111 q^{86} +0.394449i q^{87} +15.0000i q^{88} -8.60555 q^{89} -23.8167 q^{91} +8.60555i q^{92} +6.60555i q^{93} -5.09167 q^{94} +5.30278 q^{96} +3.21110i q^{97} +13.8167i q^{98} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} - 26 q^{14} - 6 q^{16} + 8 q^{19} + 12 q^{24} - 32 q^{26} - 16 q^{29} - 12 q^{31} - 2 q^{34} + 6 q^{36} - 12 q^{39} + 24 q^{41} - 30 q^{44} - 24 q^{46} - 24 q^{49}+ \cdots - 20 q^{99}+O(q^{100}) $ 4 * q - 6 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 - 26 * q^14 - 6 * q^16 + 8 * q^19 + 12 * q^24 - 32 * q^26 - 16 * q^29 - 12 * q^31 - 2 * q^34 + 6 * q^36 - 12 * q^39 + 24 * q^41 - 30 * q^44 - 24 * q^46 - 24 * q^49 - 4 * q^51 + 2 * q^54 - 4 * q^59 + 12 * q^61 + 8 * q^64 - 10 * q^66 + 4 * q^69 + 24 * q^71 + 50 * q^74 - 38 * q^76 + 16 * q^79 + 4 * q^81 + 26 * q^84 + 56 * q^86 - 20 * q^89 - 52 * q^91 - 42 * q^94 + 14 * q^96 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times$.

$n$	$52$	$751$	$851$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 2.30278i	− 1.62831i	−0.580649	−	0.814154i	$-0.697201\pi$
$2$	− 2.30278i	− 1.62831i	0.580649	−	0.814154i	$-0.302799\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−3.30278	−1.65139
$5$	0	0
$5$	0	0
$6$	−2.30278	−0.940104
$7$	− 3.60555i	− 1.36277i	−0.731925	−	0.681385i	$-0.761378\pi$
$7$	− 3.60555i	− 1.36277i	0.731925	−	0.681385i	$-0.238622\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	3.30278i	0.953429i
$13$	− 6.60555i	− 1.83205i	−0.401121	−	0.916025i	$-0.631379\pi$
$13$	− 6.60555i	− 1.83205i	0.401121	−	0.916025i	$-0.368621\pi$
$14$	−8.30278	−2.21901
$15$	0	0
$16$	0.302776	0.0756939
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	2.30278i	0.542769i
$19$	5.60555	1.28600	0.643001	−	0.765865i	$-0.277689\pi$
$19$	5.60555	1.28600	0.643001	+	0.765865i	$0.277689\pi$
$20$	0	0
$21$	−3.60555	−0.786796
$22$	− 11.5139i	− 2.45477i
$23$	− 2.60555i	− 0.543295i	−0.962397	−	0.271647i	$-0.912432\pi$
$23$	− 2.60555i	− 0.543295i	0.962397	−	0.271647i	$-0.0875685\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−15.2111	−2.98314
$27$	1.00000i	0.192450i
$28$	11.9083i	2.25046i
$29$	−0.394449	−0.0732473	−0.0366236	−	0.999329i	$-0.511660\pi$
$29$	−0.394449	−0.0732473	−0.0366236	+	0.999329i	$0.511660\pi$
$30$	0	0
$31$	−6.60555	−1.18639	−0.593196	−	0.805058i	$-0.702134\pi$
$31$	−6.60555	−1.18639	−0.593196	+	0.805058i	$0.702134\pi$
$32$	5.30278i	0.937407i
$33$	− 5.00000i	− 0.870388i
$34$	−2.30278	−0.394923
$35$	0	0
$36$	3.30278	0.550463
$37$	6.21110i	1.02110i	0.859848	+	0.510549i	$0.170558\pi$
$37$	6.21110i	1.02110i	−0.859848	+	0.510549i	$0.829442\pi$
$38$	− 12.9083i	− 2.09401i
$39$	−6.60555	−1.05773
$40$	0	0
$41$	9.60555	1.50014	0.750068	−	0.661361i	$-0.230021\pi$
$41$	9.60555	1.50014	0.750068	+	0.661361i	$0.230021\pi$
$42$	8.30278i	1.28115i
$43$	9.21110i	1.40468i	0.711842	+	0.702340i	$0.247861\pi$
$43$	9.21110i	1.40468i	−0.711842	+	0.702340i	$0.752139\pi$
$44$	−16.5139	−2.48956
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 2.21110i	− 0.322522i	−0.986912	−	0.161261i	$-0.948444\pi$
$47$	− 2.21110i	− 0.322522i	0.986912	−	0.161261i	$-0.0515562\pi$
$48$	− 0.302776i	− 0.0437019i
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	21.8167i	3.02543i
$53$	− 3.60555i	− 0.495261i	−0.968855	−	0.247630i	$-0.920348\pi$
$53$	− 3.60555i	− 0.495261i	0.968855	−	0.247630i	$-0.0796518\pi$
$54$	2.30278	0.313368
$55$	0	0
$56$	10.8167	1.44544
$57$	− 5.60555i	− 0.742473i
$58$	0.908327i	0.119269i
$59$	−4.60555	−0.599592	−0.299796	−	0.954003i	$-0.596919\pi$
$59$	−4.60555	−0.599592	−0.299796	+	0.954003i	$0.596919\pi$
$60$	0	0
$61$	6.60555	0.845754	0.422877	−	0.906187i	$-0.361020\pi$
$61$	6.60555	0.845754	0.422877	+	0.906187i	$0.361020\pi$
$62$	15.2111i	1.93181i
$63$	3.60555i	0.454257i
$64$	12.8167	1.60208
$65$	0	0
$66$	−11.5139	−1.41726
$67$	12.6056i	1.54001i	0.638036	+	0.770007i	$0.279747\pi$
$67$	12.6056i	1.54001i	−0.638036	+	0.770007i	$0.720253\pi$
$68$	3.30278i	0.400520i
$69$	−2.60555	−0.313672
$70$	0	0
$71$	−1.21110	−0.143731	−0.0718657	−	0.997414i	$-0.522895\pi$
$71$	−1.21110	−0.143731	−0.0718657	+	0.997414i	$0.522895\pi$
$72$	− 3.00000i	− 0.353553i
$73$	13.0000i	1.52153i	0.649025	+	0.760767i	$0.275177\pi$
$73$	13.0000i	1.52153i	−0.649025	+	0.760767i	$0.724823\pi$
$74$	14.3028	1.66266
$75$	0	0
$76$	−18.5139	−2.12369
$77$	− 18.0278i	− 2.05445i
$78$	15.2111i	1.72232i
$79$	−3.21110	−0.361277	−0.180639	−	0.983550i	$-0.557816\pi$
$79$	−3.21110	−0.361277	−0.180639	+	0.983550i	$0.557816\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 22.1194i	− 2.44268i
$83$	− 1.21110i	− 0.132936i	−0.997789	−	0.0664679i	$-0.978827\pi$
$83$	− 1.21110i	− 0.132936i	0.997789	−	0.0664679i	$-0.0211730\pi$
$84$	11.9083	1.29930
$85$	0	0
$86$	21.2111	2.28725
$87$	0.394449i	0.0422893i
$88$	15.0000i	1.59901i
$89$	−8.60555	−0.912187	−0.456093	−	0.889932i	$-0.650752\pi$
$89$	−8.60555	−0.912187	−0.456093	+	0.889932i	$0.650752\pi$
$90$	0	0
$91$	−23.8167	−2.49666
$92$	8.60555i	0.897191i
$93$	6.60555i	0.684964i
$94$	−5.09167	−0.525166
$95$	0	0
$96$	5.30278	0.541212
$97$	3.21110i	0.326038i	0.986623	+	0.163019i	$0.0521232\pi$
$97$	3.21110i	0.326038i	−0.986623	+	0.163019i	$0.947877\pi$
$98$	13.8167i	1.39569i
$99$	−5.00000	−0.502519
$100$	0	0
$101$	11.2111	1.11555	0.557773	−	0.829993i	$-0.311656\pi$
$101$	11.2111	1.11555	0.557773	+	0.829993i	$0.311656\pi$
$102$	2.30278i	0.228009i
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	19.8167	1.94318
$105$	0	0
$106$	−8.30278	−0.806437
$107$	9.21110i	0.890471i	0.895414	+	0.445235i	$0.146880\pi$
$107$	9.21110i	0.890471i	−0.895414	+	0.445235i	$0.853120\pi$
$108$	− 3.30278i	− 0.317810i
$109$	−7.21110	−0.690698	−0.345349	−	0.938474i	$-0.612240\pi$
$109$	−7.21110	−0.690698	−0.345349	+	0.938474i	$0.612240\pi$
$110$	0	0
$111$	6.21110	0.589532
$112$	− 1.09167i	− 0.103153i
$113$	− 7.81665i	− 0.735329i	−0.929959	−	0.367664i	$-0.880157\pi$
$113$	− 7.81665i	− 0.735329i	0.929959	−	0.367664i	$-0.119843\pi$
$114$	−12.9083	−1.20898
$115$	0	0
$116$	1.30278	0.120960
$117$	6.60555i	0.610683i
$118$	10.6056i	0.976320i
$119$	−3.60555	−0.330520
$120$	0	0
$121$	14.0000	1.27273
$122$	− 15.2111i	− 1.37715i
$123$	− 9.60555i	− 0.866103i
$124$	21.8167	1.95919
$125$	0	0
$126$	8.30278	0.739670
$127$	− 0.183346i	− 0.0162693i	−0.999967	−	0.00813467i	$-0.997411\pi$
$127$	− 0.183346i	− 0.0162693i	0.999967	−	0.00813467i	$-0.00258937\pi$
$128$	− 18.9083i	− 1.67128i
$129$	9.21110	0.810992
$130$	0	0
$131$	5.21110	0.455296	0.227648	−	0.973743i	$-0.426896\pi$
$131$	5.21110	0.455296	0.227648	+	0.973743i	$0.426896\pi$
$132$	16.5139i	1.43735i
$133$	− 20.2111i	− 1.75252i
$134$	29.0278	2.50762
$135$	0	0
$136$	3.00000	0.257248
$137$	− 10.3944i	− 0.888058i	−0.896012	−	0.444029i	$-0.853549\pi$
$137$	− 10.3944i	− 0.888058i	0.896012	−	0.444029i	$-0.146451\pi$
$138$	6.00000i	0.510754i
$139$	8.42221	0.714362	0.357181	−	0.934035i	$-0.383738\pi$
$139$	8.42221	0.714362	0.357181	+	0.934035i	$0.383738\pi$
$140$	0	0
$141$	−2.21110	−0.186208
$142$	2.78890i	0.234039i
$143$	− 33.0278i	− 2.76192i
$144$	−0.302776	−0.0252313
$145$	0	0
$146$	29.9361	2.47753
$147$	6.00000i	0.494872i
$148$	− 20.5139i	− 1.68623i
$149$	9.39445	0.769623	0.384812	−	0.922995i	$-0.374266\pi$
$149$	9.39445	0.769623	0.384812	+	0.922995i	$0.374266\pi$
$150$	0	0
$151$	11.6056	0.944446	0.472223	−	0.881479i	$-0.343452\pi$
$151$	11.6056	0.944446	0.472223	+	0.881479i	$0.343452\pi$
$152$	16.8167i	1.36401i
$153$	1.00000i	0.0808452i
$154$	−41.5139	−3.34528
$155$	0	0
$156$	21.8167	1.74673
$157$	10.4222i	0.831783i	0.909414	+	0.415891i	$0.136530\pi$
$157$	10.4222i	0.831783i	−0.909414	+	0.415891i	$0.863470\pi$
$158$	7.39445i	0.588271i
$159$	−3.60555	−0.285939
$160$	0	0
$161$	−9.39445	−0.740386
$162$	− 2.30278i	− 0.180923i
$163$	14.8167i	1.16053i	0.814428	+	0.580265i	$0.197051\pi$
$163$	14.8167i	1.16053i	−0.814428	+	0.580265i	$0.802949\pi$
$164$	−31.7250	−2.47730
$165$	0	0
$166$	−2.78890	−0.216460
$167$	− 12.4222i	− 0.961259i	−0.876924	−	0.480630i	$-0.840408\pi$
$167$	− 12.4222i	− 0.961259i	0.876924	−	0.480630i	$-0.159592\pi$
$168$	− 10.8167i	− 0.834523i
$169$	−30.6333	−2.35641
$170$	0	0
$171$	−5.60555	−0.428667
$172$	− 30.4222i	− 2.31967i
$173$	− 3.39445i	− 0.258075i	−0.991640	−	0.129038i	$-0.958811\pi$
$173$	− 3.39445i	− 0.258075i	0.991640	−	0.129038i	$-0.0411888\pi$
$174$	0.908327	0.0688601
$175$	0	0
$176$	1.51388	0.114113
$177$	4.60555i	0.346174i
$178$	19.8167i	1.48532i
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	3.81665	0.283690	0.141845	−	0.989889i	$-0.454697\pi$
$181$	3.81665	0.283690	0.141845	+	0.989889i	$0.454697\pi$
$182$	54.8444i	4.06534i
$183$	− 6.60555i	− 0.488296i
$184$	7.81665	0.576251
$185$	0	0
$186$	15.2111	1.11533
$187$	− 5.00000i	− 0.365636i
$188$	7.30278i	0.532610i
$189$	3.60555	0.262265
$190$	0	0
$191$	11.3944	0.824473	0.412237	−	0.911077i	$-0.364748\pi$
$191$	11.3944	0.824473	0.412237	+	0.911077i	$0.364748\pi$
$192$	− 12.8167i	− 0.924962i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	7.39445	0.530890
$195$	0	0
$196$	19.8167	1.41548
$197$	− 13.2111i	− 0.941252i	−0.882333	−	0.470626i	$-0.844028\pi$
$197$	− 13.2111i	− 0.941252i	0.882333	−	0.470626i	$-0.155972\pi$
$198$	11.5139i	0.818256i
$199$	−4.42221	−0.313482	−0.156741	−	0.987640i	$-0.550099\pi$
$199$	−4.42221	−0.313482	−0.156741	+	0.987640i	$0.550099\pi$
$200$	0	0
$201$	12.6056	0.889127
$202$	− 25.8167i	− 1.81645i
$203$	1.42221i	0.0998192i
$204$	3.30278	0.231241
$205$	0	0
$206$	−27.6333	−1.92530
$207$	2.60555i	0.181098i
$208$	− 2.00000i	− 0.138675i
$209$	28.0278	1.93872
$210$	0	0
$211$	−10.7889	−0.742738	−0.371369	−	0.928485i	$-0.621112\pi$
$211$	−10.7889	−0.742738	−0.371369	+	0.928485i	$0.621112\pi$
$212$	11.9083i	0.817867i
$213$	1.21110i	0.0829834i
$214$	21.2111	1.44996
$215$	0	0
$216$	−3.00000	−0.204124
$217$	23.8167i	1.61678i
$218$	16.6056i	1.12467i
$219$	13.0000	0.878459
$220$	0	0
$221$	−6.60555	−0.444337
$222$	− 14.3028i	− 0.959939i
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	19.1194	1.27747
$225$	0	0
$226$	−18.0000	−1.19734
$227$	3.81665i	0.253320i	0.991946	+	0.126660i	$0.0404257\pi$
$227$	3.81665i	0.253320i	−0.991946	+	0.126660i	$0.959574\pi$
$228$	18.5139i	1.22611i
$229$	−0.211103	−0.0139500	−0.00697502	−	0.999976i	$-0.502220\pi$
$229$	−0.211103	−0.0139500	−0.00697502	+	0.999976i	$0.502220\pi$
$230$	0	0
$231$	−18.0278	−1.18614
$232$	− 1.18335i	− 0.0776905i
$233$	− 10.6056i	− 0.694793i	−0.937718	−	0.347396i	$-0.887066\pi$
$233$	− 10.6056i	− 0.694793i	0.937718	−	0.347396i	$-0.112934\pi$
$234$	15.2111	0.994381
$235$	0	0
$236$	15.2111	0.990158
$237$	3.21110i	0.208584i
$238$	8.30278i	0.538189i
$239$	10.6056	0.686016	0.343008	−	0.939332i	$-0.388554\pi$
$239$	10.6056	0.686016	0.343008	+	0.939332i	$0.388554\pi$
$240$	0	0
$241$	14.6056	0.940826	0.470413	−	0.882446i	$-0.344105\pi$
$241$	14.6056	0.940826	0.470413	+	0.882446i	$0.344105\pi$
$242$	− 32.2389i	− 2.07239i
$243$	− 1.00000i	− 0.0641500i
$244$	−21.8167	−1.39667
$245$	0	0
$246$	−22.1194	−1.41028
$247$	− 37.0278i	− 2.35602i
$248$	− 19.8167i	− 1.25836i
$249$	−1.21110	−0.0767505
$250$	0	0
$251$	0.605551	0.0382221	0.0191110	−	0.999817i	$-0.493916\pi$
$251$	0.605551	0.0382221	0.0191110	+	0.999817i	$0.493916\pi$
$252$	− 11.9083i	− 0.750154i
$253$	− 13.0278i	− 0.819048i
$254$	−0.422205	−0.0264915
$255$	0	0
$256$	−17.9083	−1.11927
$257$	24.4222i	1.52342i	0.647921	+	0.761708i	$0.275639\pi$
$257$	24.4222i	1.52342i	−0.647921	+	0.761708i	$0.724361\pi$
$258$	− 21.2111i	− 1.32055i
$259$	22.3944	1.39152
$260$	0	0
$261$	0.394449	0.0244158
$262$	− 12.0000i	− 0.741362i
$263$	5.42221i	0.334347i	0.985927	+	0.167174i	$0.0534641\pi$
$263$	5.42221i	0.334347i	−0.985927	+	0.167174i	$0.946536\pi$
$264$	15.0000	0.923186
$265$	0	0
$266$	−46.5416	−2.85365
$267$	8.60555i	0.526651i
$268$	− 41.6333i	− 2.54316i
$269$	−22.3944	−1.36541	−0.682707	−	0.730692i	$-0.739197\pi$
$269$	−22.3944	−1.36541	−0.682707	+	0.730692i	$0.739197\pi$
$270$	0	0
$271$	−9.21110	−0.559535	−0.279767	−	0.960068i	$-0.590257\pi$
$271$	−9.21110	−0.559535	−0.279767	+	0.960068i	$0.590257\pi$
$272$	− 0.302776i	− 0.0183585i
$273$	23.8167i	1.44145i
$274$	−23.9361	−1.44603
$275$	0	0
$276$	8.60555	0.517993
$277$	12.7889i	0.768410i	0.923248	+	0.384205i	$0.125524\pi$
$277$	12.7889i	0.768410i	−0.923248	+	0.384205i	$0.874476\pi$
$278$	− 19.3944i	− 1.16320i
$279$	6.60555	0.395464
$280$	0	0
$281$	−32.8444	−1.95933	−0.979667	−	0.200632i	$-0.935700\pi$
$281$	−32.8444	−1.95933	−0.979667	+	0.200632i	$0.935700\pi$
$282$	5.09167i	0.303205i
$283$	− 28.0278i	− 1.66608i	−0.553215	−	0.833039i	$-0.686599\pi$
$283$	− 28.0278i	− 1.66608i	0.553215	−	0.833039i	$-0.313401\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−76.0555	−4.49726
$287$	− 34.6333i	− 2.04434i
$288$	− 5.30278i	− 0.312469i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	3.21110	0.188238
$292$	− 42.9361i	− 2.51264i
$293$	33.2389i	1.94183i	0.239415	+	0.970917i	$0.423045\pi$
$293$	33.2389i	1.94183i	−0.239415	+	0.970917i	$0.576955\pi$
$294$	13.8167	0.805804
$295$	0	0
$296$	−18.6333	−1.08304
$297$	5.00000i	0.290129i
$298$	− 21.6333i	− 1.25318i
$299$	−17.2111	−0.995344
$300$	0	0
$301$	33.2111	1.91426
$302$	− 26.7250i	− 1.53785i
$303$	− 11.2111i	− 0.644061i
$304$	1.69722	0.0973425
$305$	0	0
$306$	2.30278	0.131641
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	59.5416i	3.39270i
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−1.42221	−0.0806459	−0.0403229	−	0.999187i	$-0.512839\pi$
$311$	−1.42221	−0.0806459	−0.0403229	+	0.999187i	$0.512839\pi$
$312$	− 19.8167i	− 1.12190i
$313$	10.2111i	0.577166i	0.957455	+	0.288583i	$0.0931841\pi$
$313$	10.2111i	0.577166i	−0.957455	+	0.288583i	$0.906816\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	10.6056	0.596609
$317$	15.8167i	0.888352i	0.895940	+	0.444176i	$0.146503\pi$
$317$	15.8167i	0.888352i	−0.895940	+	0.444176i	$0.853497\pi$
$318$	8.30278i	0.465597i
$319$	−1.97224	−0.110424
$320$	0	0
$321$	9.21110	0.514114
$322$	21.6333i	1.20558i
$323$	− 5.60555i	− 0.311901i
$324$	−3.30278	−0.183488
$325$	0	0
$326$	34.1194	1.88970
$327$	7.21110i	0.398775i
$328$	28.8167i	1.59113i
$329$	−7.97224	−0.439524
$330$	0	0
$331$	−30.8167	−1.69384	−0.846918	−	0.531723i	$-0.821545\pi$
$331$	−30.8167	−1.69384	−0.846918	+	0.531723i	$0.821545\pi$
$332$	4.00000i	0.219529i
$333$	− 6.21110i	− 0.340366i
$334$	−28.6056	−1.56523
$335$	0	0
$336$	−1.09167	−0.0595556
$337$	− 14.2111i	− 0.774128i	−0.922053	−	0.387064i	$-0.873489\pi$
$337$	− 14.2111i	− 0.774128i	0.922053	−	0.387064i	$-0.126511\pi$
$338$	70.5416i	3.83696i
$339$	−7.81665	−0.424542
$340$	0	0
$341$	−33.0278	−1.78855
$342$	12.9083i	0.698002i
$343$	− 3.60555i	− 0.194681i
$344$	−27.6333	−1.48989
$345$	0	0
$346$	−7.81665	−0.420226
$347$	− 28.2389i	− 1.51594i	−0.652289	−	0.757971i	$-0.726191\pi$
$347$	− 28.2389i	− 1.51594i	0.652289	−	0.757971i	$-0.273809\pi$
$348$	− 1.30278i	− 0.0698361i
$349$	32.6333	1.74682	0.873410	−	0.486985i	$-0.161903\pi$
$349$	32.6333	1.74682	0.873410	+	0.486985i	$0.161903\pi$
$350$	0	0
$351$	6.60555	0.352578
$352$	26.5139i	1.41319i
$353$	− 8.39445i	− 0.446791i	−0.974728	−	0.223396i	$-0.928286\pi$
$353$	− 8.39445i	− 0.446791i	0.974728	−	0.223396i	$-0.0717142\pi$
$354$	10.6056	0.563679
$355$	0	0
$356$	28.4222	1.50637
$357$	3.60555i	0.190826i
$358$	13.8167i	0.730233i
$359$	7.39445	0.390264	0.195132	−	0.980777i	$-0.437486\pi$
$359$	7.39445	0.390264	0.195132	+	0.980777i	$0.437486\pi$
$360$	0	0
$361$	12.4222	0.653800
$362$	− 8.78890i	− 0.461934i
$363$	− 14.0000i	− 0.734809i
$364$	78.6611	4.12296
$365$	0	0
$366$	−15.2111	−0.795097
$367$	2.42221i	0.126438i	0.998000	+	0.0632190i	$0.0201367\pi$
$367$	2.42221i	0.126438i	−0.998000	+	0.0632190i	$0.979863\pi$
$368$	− 0.788897i	− 0.0411241i
$369$	−9.60555	−0.500045
$370$	0	0
$371$	−13.0000	−0.674926
$372$	− 21.8167i	− 1.13114i
$373$	26.4222i	1.36809i	0.729440	+	0.684045i	$0.239781\pi$
$373$	26.4222i	1.36809i	−0.729440	+	0.684045i	$0.760219\pi$
$374$	−11.5139	−0.595368
$375$	0	0
$376$	6.63331	0.342087
$377$	2.60555i	0.134193i
$378$	− 8.30278i	− 0.427049i
$379$	−2.42221	−0.124420	−0.0622102	−	0.998063i	$-0.519815\pi$
$379$	−2.42221	−0.124420	−0.0622102	+	0.998063i	$0.519815\pi$
$380$	0	0
$381$	−0.183346	−0.00939311
$382$	− 26.2389i	− 1.34250i
$383$	− 7.42221i	− 0.379257i	−0.981856	−	0.189628i	$-0.939272\pi$
$383$	− 7.42221i	− 0.379257i	0.981856	−	0.189628i	$-0.0607284\pi$
$384$	−18.9083	−0.964912
$385$	0	0
$386$	13.8167	0.703249
$387$	− 9.21110i	− 0.468227i
$388$	− 10.6056i	− 0.538415i
$389$	5.57779	0.282805	0.141403	−	0.989952i	$-0.454839\pi$
$389$	5.57779	0.282805	0.141403	+	0.989952i	$0.454839\pi$
$390$	0	0
$391$	−2.60555	−0.131768
$392$	− 18.0000i	− 0.909137i
$393$	− 5.21110i	− 0.262865i
$394$	−30.4222	−1.53265
$395$	0	0
$396$	16.5139	0.829854
$397$	− 27.8444i	− 1.39747i	−0.715380	−	0.698735i	$-0.753746\pi$
$397$	− 27.8444i	− 1.39747i	0.715380	−	0.698735i	$-0.246254\pi$
$398$	10.1833i	0.510445i
$399$	−20.2111	−1.01182
$400$	0	0
$401$	1.18335	0.0590935	0.0295467	−	0.999563i	$-0.490594\pi$
$401$	1.18335	0.0590935	0.0295467	+	0.999563i	$0.490594\pi$
$402$	− 29.0278i	− 1.44777i
$403$	43.6333i	2.17353i
$404$	−37.0278	−1.84220
$405$	0	0
$406$	3.27502	0.162536
$407$	31.0555i	1.53936i
$408$	− 3.00000i	− 0.148522i
$409$	−1.00000	−0.0494468	−0.0247234	−	0.999694i	$-0.507871\pi$
$409$	−1.00000	−0.0494468	−0.0247234	+	0.999694i	$0.507871\pi$
$410$	0	0
$411$	−10.3944	−0.512720
$412$	39.6333i	1.95259i
$413$	16.6056i	0.817106i
$414$	6.00000	0.294884
$415$	0	0
$416$	35.0278	1.71738
$417$	− 8.42221i	− 0.412437i
$418$	− 64.5416i	− 3.15683i
$419$	36.6333	1.78965	0.894827	−	0.446413i	$-0.147299\pi$
$419$	36.6333	1.78965	0.894827	+	0.446413i	$0.147299\pi$
$420$	0	0
$421$	−12.2111	−0.595133	−0.297566	−	0.954701i	$-0.596175\pi$
$421$	−12.2111	−0.595133	−0.297566	+	0.954701i	$0.596175\pi$
$422$	24.8444i	1.20941i
$423$	2.21110i	0.107507i
$424$	10.8167	0.525303
$425$	0	0
$426$	2.78890	0.135123
$427$	− 23.8167i	− 1.15257i
$428$	− 30.4222i	− 1.47051i
$429$	−33.0278	−1.59460
$430$	0	0
$431$	29.4222	1.41722	0.708609	−	0.705601i	$-0.249323\pi$
$431$	29.4222	1.41722	0.708609	+	0.705601i	$0.249323\pi$
$432$	0.302776i	0.0145673i
$433$	31.2111i	1.49991i	0.661489	+	0.749955i	$0.269925\pi$
$433$	31.2111i	1.49991i	−0.661489	+	0.749955i	$0.730075\pi$
$434$	54.8444	2.63262
$435$	0	0
$436$	23.8167	1.14061
$437$	− 14.6056i	− 0.698678i
$438$	− 29.9361i	− 1.43040i
$439$	−35.4500	−1.69193	−0.845967	−	0.533235i	$-0.820976\pi$
$439$	−35.4500	−1.69193	−0.845967	+	0.533235i	$0.820976\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	15.2111i	0.723518i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	−20.5139	−0.973546
$445$	0	0
$446$	−59.8722	−2.83503
$447$	− 9.39445i	− 0.444342i
$448$	− 46.2111i	− 2.18327i
$449$	21.6333	1.02094	0.510469	−	0.859896i	$-0.329472\pi$
$449$	21.6333	1.02094	0.510469	+	0.859896i	$0.329472\pi$
$450$	0	0
$451$	48.0278	2.26154
$452$	25.8167i	1.21431i
$453$	− 11.6056i	− 0.545276i
$454$	8.78890	0.412483
$455$	0	0
$456$	16.8167	0.787512
$457$	− 8.60555i	− 0.402551i	−0.979535	−	0.201275i	$-0.935491\pi$
$457$	− 8.60555i	− 0.402551i	0.979535	−	0.201275i	$-0.0645086\pi$
$458$	0.486122i	0.0227150i
$459$	1.00000	0.0466760
$460$	0	0
$461$	−27.0278	−1.25881	−0.629404	−	0.777078i	$-0.716701\pi$
$461$	−27.0278	−1.25881	−0.629404	+	0.777078i	$0.716701\pi$
$462$	41.5139i	1.93140i
$463$	0.788897i	0.0366632i	0.999832	+	0.0183316i	$0.00583545\pi$
$463$	0.788897i	0.0366632i	−0.999832	+	0.0183316i	$0.994165\pi$
$464$	−0.119429	−0.00554437
$465$	0	0
$466$	−24.4222	−1.13134
$467$	− 36.2111i	− 1.67565i	−0.545939	−	0.837825i	$-0.683827\pi$
$467$	− 36.2111i	− 1.67565i	0.545939	−	0.837825i	$-0.316173\pi$
$468$	− 21.8167i	− 1.00848i
$469$	45.4500	2.09868
$470$	0	0
$471$	10.4222	0.480230
$472$	− 13.8167i	− 0.635963i
$473$	46.0555i	2.11763i
$474$	7.39445	0.339638
$475$	0	0
$476$	11.9083	0.545817
$477$	3.60555i	0.165087i
$478$	− 24.4222i	− 1.11705i
$479$	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
$481$	41.0278	1.87070
$482$	− 33.6333i	− 1.53196i
$483$	9.39445i	0.427462i
$484$	−46.2389	−2.10177
$485$	0	0
$486$	−2.30278	−0.104456
$487$	16.8444i	0.763293i	0.924308	+	0.381647i	$0.124643\pi$
$487$	16.8444i	0.763293i	−0.924308	+	0.381647i	$0.875357\pi$
$488$	19.8167i	0.897058i
$489$	14.8167	0.670032
$490$	0	0
$491$	24.4222	1.10216	0.551079	−	0.834453i	$-0.314216\pi$
$491$	24.4222	1.10216	0.551079	+	0.834453i	$0.314216\pi$
$492$	31.7250i	1.43027i
$493$	0.394449i	0.0177651i
$494$	−85.2666	−3.83633
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	4.36669i	0.195873i
$498$	2.78890i	0.124973i
$499$	37.4500	1.67649	0.838245	−	0.545293i	$-0.183582\pi$
$499$	37.4500	1.67649	0.838245	+	0.545293i	$0.183582\pi$
$500$	0	0
$501$	−12.4222	−0.554983
$502$	− 1.39445i	− 0.0622373i
$503$	− 12.7889i	− 0.570229i	−0.958494	−	0.285114i	$-0.907968\pi$
$503$	− 12.7889i	− 0.570229i	0.958494	−	0.285114i	$-0.0920316\pi$
$504$	−10.8167	−0.481812
$505$	0	0
$506$	−30.0000	−1.33366
$507$	30.6333i	1.36047i
$508$	0.605551i	0.0268670i
$509$	−3.81665	−0.169170	−0.0845851	−	0.996416i	$-0.526956\pi$
$509$	−3.81665	−0.169170	−0.0845851	+	0.996416i	$0.526956\pi$
$510$	0	0
$511$	46.8722	2.07350
$512$	3.42221i	0.151242i
$513$	5.60555i	0.247491i
$514$	56.2389	2.48059
$515$	0	0
$516$	−30.4222	−1.33926
$517$	− 11.0555i	− 0.486221i
$518$	− 51.5694i	− 2.26583i
$519$	−3.39445	−0.149000
$520$	0	0
$521$	32.4500	1.42166	0.710829	−	0.703365i	$-0.248320\pi$
$521$	32.4500	1.42166	0.710829	+	0.703365i	$0.248320\pi$
$522$	− 0.908327i	− 0.0397564i
$523$	34.8444i	1.52364i	0.647789	+	0.761820i	$0.275694\pi$
$523$	34.8444i	1.52364i	−0.647789	+	0.761820i	$0.724306\pi$
$524$	−17.2111	−0.751871
$525$	0	0
$526$	12.4861	0.544421
$527$	6.60555i	0.287742i
$528$	− 1.51388i	− 0.0658831i
$529$	16.2111	0.704831
$530$	0	0
$531$	4.60555	0.199864
$532$	66.7527i	2.89410i
$533$	− 63.4500i	− 2.74832i
$534$	19.8167	0.857550
$535$	0	0
$536$	−37.8167	−1.63343
$537$	6.00000i	0.258919i
$538$	51.5694i	2.22331i
$539$	−30.0000	−1.29219
$540$	0	0
$541$	1.39445	0.0599520	0.0299760	−	0.999551i	$-0.490457\pi$
$541$	1.39445	0.0599520	0.0299760	+	0.999551i	$0.490457\pi$
$542$	21.2111i	0.911095i
$543$	− 3.81665i	− 0.163788i
$544$	5.30278	0.227355
$545$	0	0
$546$	54.8444	2.34712
$547$	− 12.8167i	− 0.548001i	−0.961730	−	0.274000i	$-0.911653\pi$
$547$	− 12.8167i	− 0.548001i	0.961730	−	0.274000i	$-0.0883469\pi$
$548$	34.3305i	1.46653i
$549$	−6.60555	−0.281918
$550$	0	0
$551$	−2.21110	−0.0941961
$552$	− 7.81665i	− 0.332699i
$553$	11.5778i	0.492338i
$554$	29.4500	1.25121
$555$	0	0
$556$	−27.8167	−1.17969
$557$	16.7889i	0.711368i	0.934606	+	0.355684i	$0.115752\pi$
$557$	16.7889i	0.711368i	−0.934606	+	0.355684i	$0.884248\pi$
$558$	− 15.2111i	− 0.643937i
$559$	60.8444	2.57344
$560$	0	0
$561$	−5.00000	−0.211100
$562$	75.6333i	3.19040i
$563$	6.21110i	0.261767i	0.991398	+	0.130883i	$0.0417813\pi$
$563$	6.21110i	0.261767i	−0.991398	+	0.130883i	$0.958219\pi$
$564$	7.30278	0.307502
$565$	0	0
$566$	−64.5416	−2.71289
$567$	− 3.60555i	− 0.151419i
$568$	− 3.63331i	− 0.152450i
$569$	−9.21110	−0.386150	−0.193075	−	0.981184i	$-0.561846\pi$
$569$	−9.21110	−0.386150	−0.193075	+	0.981184i	$0.561846\pi$
$570$	0	0
$571$	−25.0278	−1.04738	−0.523690	−	0.851909i	$-0.675445\pi$
$571$	−25.0278	−1.04738	−0.523690	+	0.851909i	$0.675445\pi$
$572$	109.083i	4.56100i
$573$	− 11.3944i	− 0.476010i
$574$	−79.7527	−3.32881
$575$	0	0
$576$	−12.8167	−0.534027
$577$	12.0000i	0.499567i	0.968302	+	0.249783i	$0.0803594\pi$
$577$	12.0000i	0.499567i	−0.968302	+	0.249783i	$0.919641\pi$
$578$	2.30278i	0.0957828i
$579$	6.00000	0.249351
$580$	0	0
$581$	−4.36669	−0.181161
$582$	− 7.39445i	− 0.306510i
$583$	− 18.0278i	− 0.746633i
$584$	−39.0000	−1.61383
$585$	0	0
$586$	76.5416	3.16191
$587$	7.42221i	0.306347i	0.988199	+	0.153174i	$0.0489494\pi$
$587$	7.42221i	0.306347i	−0.988199	+	0.153174i	$0.951051\pi$
$588$	− 19.8167i	− 0.817225i
$589$	−37.0278	−1.52570
$590$	0	0
$591$	−13.2111	−0.543432
$592$	1.88057i	0.0772910i
$593$	− 22.8167i	− 0.936968i	−0.883472	−	0.468484i	$-0.844800\pi$
$593$	− 22.8167i	− 0.936968i	0.883472	−	0.468484i	$-0.155200\pi$
$594$	11.5139	0.472420
$595$	0	0
$596$	−31.0278	−1.27095
$597$	4.42221i	0.180989i
$598$	39.6333i	1.62073i
$599$	−15.2111	−0.621509	−0.310754	−	0.950490i	$-0.600582\pi$
$599$	−15.2111	−0.621509	−0.310754	+	0.950490i	$0.600582\pi$
$600$	0	0
$601$	1.39445	0.0568807	0.0284404	−	0.999595i	$-0.490946\pi$
$601$	1.39445	0.0568807	0.0284404	+	0.999595i	$0.490946\pi$
$602$	− 76.4777i	− 3.11700i
$603$	− 12.6056i	− 0.513338i
$604$	−38.3305	−1.55965
$605$	0	0
$606$	−25.8167	−1.04873
$607$	0.394449i	0.0160102i	0.999968	+	0.00800509i	$0.00254813\pi$
$607$	0.394449i	0.0160102i	−0.999968	+	0.00800509i	$0.997452\pi$
$608$	29.7250i	1.20551i
$609$	1.42221	0.0576307
$610$	0	0
$611$	−14.6056	−0.590877
$612$	− 3.30278i	− 0.133507i
$613$	7.63331i	0.308306i	0.988047	+	0.154153i	$0.0492649\pi$
$613$	7.63331i	0.308306i	−0.988047	+	0.154153i	$0.950735\pi$
$614$	−18.4222	−0.743460
$615$	0	0
$616$	54.0833	2.17908
$617$	− 43.8167i	− 1.76399i	−0.471257	−	0.881996i	$-0.656200\pi$
$617$	− 43.8167i	− 1.76399i	0.471257	−	0.881996i	$-0.343800\pi$
$618$	27.6333i	1.11157i
$619$	7.21110	0.289839	0.144919	−	0.989443i	$-0.453708\pi$
$619$	7.21110	0.289839	0.144919	+	0.989443i	$0.453708\pi$
$620$	0	0
$621$	2.60555	0.104557
$622$	3.27502i	0.131316i
$623$	31.0278i	1.24310i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	23.5139	0.939804
$627$	− 28.0278i	− 1.11932i
$628$	− 34.4222i	− 1.37360i
$629$	6.21110	0.247653
$630$	0	0
$631$	−30.4500	−1.21219	−0.606097	−	0.795391i	$-0.707266\pi$
$631$	−30.4500	−1.21219	−0.606097	+	0.795391i	$0.707266\pi$
$632$	− 9.63331i	− 0.383192i
$633$	10.7889i	0.428820i
$634$	36.4222	1.44651
$635$	0	0
$636$	11.9083	0.472196
$637$	39.6333i	1.57033i
$638$	4.54163i	0.179805i
$639$	1.21110	0.0479105
$640$	0	0
$641$	44.4500	1.75567	0.877834	−	0.478965i	$-0.158988\pi$
$641$	44.4500	1.75567	0.877834	+	0.478965i	$0.158988\pi$
$642$	− 21.2111i	− 0.837135i
$643$	− 15.1833i	− 0.598773i	−0.954132	−	0.299386i	$-0.903218\pi$
$643$	− 15.1833i	− 0.598773i	0.954132	−	0.299386i	$-0.0967819\pi$
$644$	31.0278	1.22266
$645$	0	0
$646$	−12.9083	−0.507871
$647$	37.2111i	1.46292i	0.681885	+	0.731460i	$0.261161\pi$
$647$	37.2111i	1.46292i	−0.681885	+	0.731460i	$0.738839\pi$
$648$	3.00000i	0.117851i
$649$	−23.0278	−0.903919
$650$	0	0
$651$	23.8167	0.933448
$652$	− 48.9361i	− 1.91648i
$653$	− 28.0555i	− 1.09790i	−0.835856	−	0.548949i	$-0.815028\pi$
$653$	− 28.0555i	− 1.09790i	0.835856	−	0.548949i	$-0.184972\pi$
$654$	16.6056	0.649328
$655$	0	0
$656$	2.90833	0.113551
$657$	− 13.0000i	− 0.507178i
$658$	18.3583i	0.715681i
$659$	34.8444	1.35735	0.678673	−	0.734441i	$-0.262555\pi$
$659$	34.8444	1.35735	0.678673	+	0.734441i	$0.262555\pi$
$660$	0	0
$661$	17.4222	0.677645	0.338823	−	0.940850i	$-0.389971\pi$
$661$	17.4222	0.677645	0.338823	+	0.940850i	$0.389971\pi$
$662$	70.9638i	2.75809i
$663$	6.60555i	0.256538i
$664$	3.63331	0.141000
$665$	0	0
$666$	−14.3028	−0.554221
$667$	1.02776i	0.0397949i
$668$	41.0278i	1.58741i
$669$	−26.0000	−1.00522
$670$	0	0
$671$	33.0278	1.27502
$672$	− 19.1194i	− 0.737548i
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	−32.7250	−1.26052
$675$	0	0
$676$	101.175	3.89134
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	18.0000i	0.691286i
$679$	11.5778	0.444315
$680$	0	0
$681$	3.81665	0.146254
$682$	76.0555i	2.91232i
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	18.5139	0.707896
$685$	0	0
$686$	−8.30278	−0.317001
$687$	0.211103i	0.00805406i
$688$	2.78890i	0.106326i
$689$	−23.8167	−0.907342
$690$	0	0
$691$	1.21110	0.0460725	0.0230363	−	0.999735i	$-0.492667\pi$
$691$	1.21110	0.0460725	0.0230363	+	0.999735i	$0.492667\pi$
$692$	11.2111i	0.426182i
$693$	18.0278i	0.684818i
$694$	−65.0278	−2.46842
$695$	0	0
$696$	−1.18335	−0.0448546
$697$	− 9.60555i	− 0.363836i
$698$	− 75.1472i	− 2.84436i
$699$	−10.6056	−0.401139
$700$	0	0
$701$	−13.8167	−0.521848	−0.260924	−	0.965359i	$-0.584027\pi$
$701$	−13.8167	−0.521848	−0.260924	+	0.965359i	$0.584027\pi$
$702$	− 15.2111i	− 0.574106i
$703$	34.8167i	1.31313i
$704$	64.0833	2.41523
$705$	0	0
$706$	−19.3305	−0.727514
$707$	− 40.4222i	− 1.52023i
$708$	− 15.2111i	− 0.571668i
$709$	−36.7889	−1.38164	−0.690818	−	0.723029i	$-0.742749\pi$
$709$	−36.7889	−1.38164	−0.690818	+	0.723029i	$0.742749\pi$
$710$	0	0
$711$	3.21110	0.120426
$712$	− 25.8167i	− 0.967520i
$713$	17.2111i	0.644561i
$714$	8.30278	0.310724
$715$	0	0
$716$	19.8167	0.740583
$717$	− 10.6056i	− 0.396072i
$718$	− 17.0278i	− 0.635470i
$719$	−4.21110	−0.157048	−0.0785238	−	0.996912i	$-0.525021\pi$
$719$	−4.21110	−0.157048	−0.0785238	+	0.996912i	$0.525021\pi$
$720$	0	0
$721$	−43.2666	−1.61133
$722$	− 28.6056i	− 1.06459i
$723$	− 14.6056i	− 0.543186i
$724$	−12.6056	−0.468482
$725$	0	0
$726$	−32.2389	−1.19650
$727$	44.0555i	1.63393i	0.576688	+	0.816964i	$0.304345\pi$
$727$	44.0555i	1.63393i	−0.576688	+	0.816964i	$0.695655\pi$
$728$	− 71.4500i	− 2.64811i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	9.21110	0.340685
$732$	21.8167i	0.806367i
$733$	37.3944i	1.38120i	0.723239	+	0.690598i	$0.242652\pi$
$733$	37.3944i	1.38120i	−0.723239	+	0.690598i	$0.757348\pi$
$734$	5.57779	0.205880
$735$	0	0
$736$	13.8167	0.509289
$737$	63.0278i	2.32166i
$738$	22.1194i	0.814227i
$739$	6.42221	0.236245	0.118122	−	0.992999i	$-0.462313\pi$
$739$	6.42221	0.236245	0.118122	+	0.992999i	$0.462313\pi$
$740$	0	0
$741$	−37.0278	−1.36025
$742$	29.9361i	1.09899i
$743$	− 50.8444i	− 1.86530i	−0.360782	−	0.932650i	$-0.617490\pi$
$743$	− 50.8444i	− 1.86530i	0.360782	−	0.932650i	$-0.382510\pi$
$744$	−19.8167	−0.726514
$745$	0	0
$746$	60.8444	2.22767
$747$	1.21110i	0.0443119i
$748$	16.5139i	0.603807i
$749$	33.2111	1.21351
$750$	0	0
$751$	−53.8722	−1.96582	−0.982912	−	0.184078i	$-0.941070\pi$
$751$	−53.8722	−1.96582	−0.982912	+	0.184078i	$0.941070\pi$
$752$	− 0.669468i	− 0.0244130i
$753$	− 0.605551i	− 0.0220675i
$754$	6.00000	0.218507
$755$	0	0
$756$	−11.9083	−0.433102
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	5.57779i	0.202595i
$759$	−13.0278	−0.472878
$760$	0	0
$761$	−47.4500	−1.72006	−0.860030	−	0.510244i	$-0.829555\pi$
$761$	−47.4500	−1.72006	−0.860030	+	0.510244i	$0.829555\pi$
$762$	0.422205i	0.0152949i
$763$	26.0000i	0.941263i
$764$	−37.6333	−1.36153
$765$	0	0
$766$	−17.0917	−0.617547
$767$	30.4222i	1.09848i
$768$	17.9083i	0.646211i
$769$	19.4222	0.700383	0.350191	−	0.936678i	$-0.386117\pi$
$769$	19.4222	0.700383	0.350191	+	0.936678i	$0.386117\pi$
$770$	0	0
$771$	24.4222	0.879544
$772$	− 19.8167i	− 0.713217i
$773$	44.4500i	1.59875i	0.600830	+	0.799377i	$0.294837\pi$
$773$	44.4500i	1.59875i	−0.600830	+	0.799377i	$0.705163\pi$
$774$	−21.2111	−0.762417
$775$	0	0
$776$	−9.63331	−0.345816
$777$	− 22.3944i	− 0.803396i
$778$	− 12.8444i	− 0.460494i
$779$	53.8444	1.92918
$780$	0	0
$781$	−6.05551	−0.216683
$782$	6.00000i	0.214560i
$783$	− 0.394449i	− 0.0140964i
$784$	−1.81665	−0.0648805
$785$	0	0
$786$	−12.0000	−0.428026
$787$	34.8167i	1.24108i	0.784175	+	0.620540i	$0.213086\pi$
$787$	34.8167i	1.24108i	−0.784175	+	0.620540i	$0.786914\pi$
$788$	43.6333i	1.55437i
$789$	5.42221	0.193036
$790$	0	0
$791$	−28.1833	−1.00208
$792$	− 15.0000i	− 0.533002i
$793$	− 43.6333i	− 1.54946i
$794$	−64.1194	−2.27551
$795$	0	0
$796$	14.6056	0.517680
$797$	54.8167i	1.94171i	0.239677	+	0.970853i	$0.422959\pi$
$797$	54.8167i	1.94171i	−0.239677	+	0.970853i	$0.577041\pi$
$798$	46.5416i	1.64756i
$799$	−2.21110	−0.0782232
$800$	0	0
$801$	8.60555	0.304062
$802$	− 2.72498i	− 0.0962224i
$803$	65.0000i	2.29380i
$804$	−41.6333	−1.46829
$805$	0	0
$806$	100.478	3.53918
$807$	22.3944i	0.788322i
$808$	33.6333i	1.18322i
$809$	−12.4222	−0.436741	−0.218371	−	0.975866i	$-0.570074\pi$
$809$	−12.4222	−0.436741	−0.218371	+	0.975866i	$0.570074\pi$
$810$	0	0
$811$	−23.3944	−0.821490	−0.410745	−	0.911750i	$-0.634731\pi$
$811$	−23.3944	−0.821490	−0.410745	+	0.911750i	$0.634731\pi$
$812$	− 4.69722i	− 0.164840i
$813$	9.21110i	0.323047i
$814$	71.5139	2.50656
$815$	0	0
$816$	−0.302776	−0.0105993
$817$	51.6333i	1.80642i
$818$	2.30278i	0.0805147i
$819$	23.8167	0.832221
$820$	0	0
$821$	−3.21110	−0.112068	−0.0560341	−	0.998429i	$-0.517846\pi$
$821$	−3.21110	−0.112068	−0.0560341	+	0.998429i	$0.517846\pi$
$822$	23.9361i	0.834867i
$823$	− 8.81665i	− 0.307329i	−0.988123	−	0.153665i	$-0.950892\pi$
$823$	− 8.81665i	− 0.307329i	0.988123	−	0.153665i	$-0.0491075\pi$
$824$	36.0000	1.25412
$825$	0	0
$826$	38.2389	1.33050
$827$	− 34.6611i	− 1.20528i	−0.798012	−	0.602642i	$-0.794115\pi$
$827$	− 34.6611i	− 1.20528i	0.798012	−	0.602642i	$-0.205885\pi$
$828$	− 8.60555i	− 0.299064i
$829$	4.21110	0.146258	0.0731288	−	0.997323i	$-0.476702\pi$
$829$	4.21110	0.146258	0.0731288	+	0.997323i	$0.476702\pi$
$830$	0	0
$831$	12.7889	0.443642
$832$	− 84.6611i	− 2.93509i
$833$	6.00000i	0.207888i
$834$	−19.3944	−0.671575
$835$	0	0
$836$	−92.5694	−3.20158
$837$	− 6.60555i	− 0.228321i
$838$	− 84.3583i	− 2.91411i
$839$	33.8444	1.16844	0.584219	−	0.811596i	$-0.301401\pi$
$839$	33.8444	1.16844	0.584219	+	0.811596i	$0.301401\pi$
$840$	0	0
$841$	−28.8444	−0.994635
$842$	28.1194i	0.969060i
$843$	32.8444i	1.13122i
$844$	35.6333	1.22655
$845$	0	0
$846$	5.09167	0.175055
$847$	− 50.4777i	− 1.73443i
$848$	− 1.09167i	− 0.0374882i
$849$	−28.0278	−0.961910
$850$	0	0
$851$	16.1833	0.554758
$852$	− 4.00000i	− 0.137038i
$853$	− 46.8444i	− 1.60392i	−0.597376	−	0.801961i	$-0.703790\pi$
$853$	− 46.8444i	− 1.60392i	0.597376	−	0.801961i	$-0.296210\pi$
$854$	−54.8444	−1.87674
$855$	0	0
$856$	−27.6333	−0.944487
$857$	− 30.0555i	− 1.02668i	−0.858186	−	0.513338i	$-0.828409\pi$
$857$	− 30.0555i	− 1.02668i	0.858186	−	0.513338i	$-0.171591\pi$
$858$	76.0555i	2.59649i
$859$	−41.6056	−1.41956	−0.709782	−	0.704422i	$-0.751206\pi$
$859$	−41.6056	−1.41956	−0.709782	+	0.704422i	$0.751206\pi$
$860$	0	0
$861$	−34.6333	−1.18030
$862$	− 67.7527i	− 2.30767i
$863$	38.6333i	1.31509i	0.753414	+	0.657547i	$0.228406\pi$
$863$	38.6333i	1.31509i	−0.753414	+	0.657547i	$0.771594\pi$
$864$	−5.30278	−0.180404
$865$	0	0
$866$	71.8722	2.44232
$867$	1.00000i	0.0339618i
$868$	− 78.6611i	− 2.66993i
$869$	−16.0555	−0.544646
$870$	0	0
$871$	83.2666	2.82138
$872$	− 21.6333i	− 0.732596i
$873$	− 3.21110i	− 0.108679i
$874$	−33.6333	−1.13766
$875$	0	0
$876$	−42.9361	−1.45068
$877$	6.57779i	0.222116i	0.993814	+	0.111058i	$0.0354240\pi$
$877$	6.57779i	0.222116i	−0.993814	+	0.111058i	$0.964576\pi$
$878$	81.6333i	2.75499i
$879$	33.2389	1.12112
$880$	0	0
$881$	−13.1833	−0.444158	−0.222079	−	0.975029i	$-0.571284\pi$
$881$	−13.1833	−0.444158	−0.222079	+	0.975029i	$0.571284\pi$
$882$	− 13.8167i	− 0.465231i
$883$	28.2389i	0.950313i	0.879901	+	0.475157i	$0.157609\pi$
$883$	28.2389i	0.950313i	−0.879901	+	0.475157i	$0.842391\pi$
$884$	21.8167	0.733773
$885$	0	0
$886$	−27.6333	−0.928359
$887$	− 42.4222i	− 1.42440i	−0.701978	−	0.712199i	$-0.747699\pi$
$887$	− 42.4222i	− 1.42440i	0.701978	−	0.712199i	$-0.252301\pi$
$888$	18.6333i	0.625293i
$889$	−0.661064	−0.0221714
$890$	0	0
$891$	5.00000	0.167506
$892$	85.8722i	2.87521i
$893$	− 12.3944i	− 0.414764i
$894$	−21.6333	−0.723526
$895$	0	0
$896$	−68.1749	−2.27756
$897$	17.2111i	0.574662i
$898$	− 49.8167i	− 1.66240i
$899$	2.60555	0.0869000
$900$	0	0
$901$	−3.60555	−0.120118
$902$	− 110.597i	− 3.68248i
$903$	− 33.2111i	− 1.10520i
$904$	23.4500	0.779934
$905$	0	0
$906$	−26.7250	−0.887878
$907$	− 23.6056i	− 0.783809i	−0.920006	−	0.391905i	$-0.871816\pi$
$907$	− 23.6056i	− 0.783809i	0.920006	−	0.391905i	$-0.128184\pi$
$908$	− 12.6056i	− 0.418330i
$909$	−11.2111	−0.371849
$910$	0	0
$911$	−16.2111	−0.537098	−0.268549	−	0.963266i	$-0.586544\pi$
$911$	−16.2111	−0.537098	−0.268549	+	0.963266i	$0.586544\pi$
$912$	− 1.69722i	− 0.0562007i
$913$	− 6.05551i	− 0.200408i
$914$	−19.8167	−0.655477
$915$	0	0
$916$	0.697224	0.0230369
$917$	− 18.7889i	− 0.620464i
$918$	− 2.30278i	− 0.0760029i
$919$	3.18335	0.105009	0.0525045	−	0.998621i	$-0.483280\pi$
$919$	3.18335	0.105009	0.0525045	+	0.998621i	$0.483280\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	62.2389i	2.04973i
$923$	8.00000i	0.263323i
$924$	59.5416	1.95878
$925$	0	0
$926$	1.81665	0.0596989
$927$	12.0000i	0.394132i
$928$	− 2.09167i	− 0.0686625i
$929$	−40.3944	−1.32530	−0.662649	−	0.748930i	$-0.730568\pi$
$929$	−40.3944	−1.32530	−0.662649	+	0.748930i	$0.730568\pi$
$930$	0	0
$931$	−33.6333	−1.10229
$932$	35.0278i	1.14737i
$933$	1.42221i	0.0465609i
$934$	−83.3860	−2.72847
$935$	0	0
$936$	−19.8167	−0.647728
$937$	5.39445i	0.176229i	0.996110	+	0.0881145i	$0.0280841\pi$
$937$	5.39445i	0.176229i	−0.996110	+	0.0881145i	$0.971916\pi$
$938$	− 104.661i	− 3.41730i
$939$	10.2111	0.333227
$940$	0	0
$941$	25.6333	0.835622	0.417811	−	0.908534i	$-0.362797\pi$
$941$	25.6333	0.835622	0.417811	+	0.908534i	$0.362797\pi$
$942$	− 24.0000i	− 0.781962i
$943$	− 25.0278i	− 0.815016i
$944$	−1.39445	−0.0453854
$945$	0	0
$946$	106.056	3.44816
$947$	− 59.6333i	− 1.93782i	−0.247409	−	0.968911i	$-0.579579\pi$
$947$	− 59.6333i	− 1.93782i	0.247409	−	0.968911i	$-0.420421\pi$
$948$	− 10.6056i	− 0.344452i
$949$	85.8722	2.78753
$950$	0	0
$951$	15.8167	0.512890
$952$	− 10.8167i	− 0.350570i
$953$	− 7.18335i	− 0.232691i	−0.993209	−	0.116346i	$-0.962882\pi$
$953$	− 7.18335i	− 0.232691i	0.993209	−	0.116346i	$-0.0371181\pi$
$954$	8.30278	0.268812
$955$	0	0
$956$	−35.0278	−1.13288
$957$	1.97224i	0.0637536i
$958$	− 24.0000i	− 0.775405i
$959$	−37.4777	−1.21022
$960$	0	0
$961$	12.6333	0.407526
$962$	− 94.4777i	− 3.04608i
$963$	− 9.21110i	− 0.296824i
$964$	−48.2389	−1.55367
$965$	0	0
$966$	21.6333	0.696040
$967$	2.18335i	0.0702117i	0.999384	+	0.0351058i	$0.0111768\pi$
$967$	2.18335i	0.0702117i	−0.999384	+	0.0351058i	$0.988823\pi$
$968$	42.0000i	1.34993i
$969$	−5.60555	−0.180076
$970$	0	0
$971$	45.4500	1.45856	0.729279	−	0.684216i	$-0.239855\pi$
$971$	45.4500	1.45856	0.729279	+	0.684216i	$0.239855\pi$
$972$	3.30278i	0.105937i
$973$	− 30.3667i	− 0.973511i
$974$	38.7889	1.24288
$975$	0	0
$976$	2.00000	0.0640184
$977$	40.0555i	1.28149i	0.767754	+	0.640745i	$0.221374\pi$
$977$	40.0555i	1.28149i	−0.767754	+	0.640745i	$0.778626\pi$
$978$	− 34.1194i	− 1.09102i
$979$	−43.0278	−1.37517
$980$	0	0
$981$	7.21110	0.230233
$982$	− 56.2389i	− 1.79465i
$983$	− 26.6056i	− 0.848585i	−0.905525	−	0.424293i	$-0.860523\pi$
$983$	− 26.6056i	− 0.848585i	0.905525	−	0.424293i	$-0.139477\pi$
$984$	28.8167	0.918641
$985$	0	0
$986$	0.908327	0.0289270
$987$	7.97224i	0.253759i
$988$	122.294i	3.89070i
$989$	24.0000	0.763156
$990$	0	0
$991$	−39.4500	−1.25317	−0.626585	−	0.779353i	$-0.715548\pi$
$991$	−39.4500	−1.25317	−0.626585	+	0.779353i	$0.715548\pi$
$992$	− 35.0278i	− 1.11213i
$993$	30.8167i	0.977937i
$994$	10.0555	0.318941
$995$	0	0
$996$	4.00000	0.126745
$997$	23.0000i	0.728417i	0.931317	+	0.364209i	$0.118661\pi$
$997$	23.0000i	0.728417i	−0.931317	+	0.364209i	$0.881339\pi$
$998$	− 86.2389i	− 2.72984i
$999$	−6.21110	−0.196511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.b.f.1174.1		4
5.2	odd	4		255.2.a.a.1.2	✓	2
5.3	odd	4		1275.2.a.j.1.1		2
5.4	even	2	inner	1275.2.b.f.1174.4		4
15.2	even	4		765.2.a.f.1.1		2
15.8	even	4		3825.2.a.y.1.2		2
20.7	even	4		4080.2.a.bl.1.1		2
85.67	odd	4		4335.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.a.a.1.2	✓	2	5.2	odd	4
765.2.a.f.1.1		2	15.2	even	4
1275.2.a.j.1.1		2	5.3	odd	4
1275.2.b.f.1174.1		4	1.1	even	1	trivial
1275.2.b.f.1174.4		4	5.4	even	2	inner
3825.2.a.y.1.2		2	15.8	even	4
4080.2.a.bl.1.1		2	20.7	even	4
4335.2.a.l.1.2		2	85.67	odd	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:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.30278i q^{2} -1.00000i q^{3} -3.30278 q^{4} -2.30278 q^{6} -3.60555i q^{7} +3.00000i q^{8} -1.00000 q^{9} +5.00000 q^{11} +3.30278i q^{12} -6.60555i q^{13} -8.30278 q^{14} +0.302776 q^{16} -1.00000i q^{17} +2.30278i q^{18} +5.60555 q^{19} -3.60555 q^{21} -11.5139i q^{22} -2.60555i q^{23} +3.00000 q^{24} -15.2111 q^{26} +1.00000i q^{27} +11.9083i q^{28} -0.394449 q^{29} -6.60555 q^{31} +5.30278i q^{32} -5.00000i q^{33} -2.30278 q^{34} +3.30278 q^{36} +6.21110i q^{37} -12.9083i q^{38} -6.60555 q^{39} +9.60555 q^{41} +8.30278i q^{42} +9.21110i q^{43} -16.5139 q^{44} -6.00000 q^{46} -2.21110i q^{47} -0.302776i q^{48} -6.00000 q^{49} -1.00000 q^{51} +21.8167i q^{52} -3.60555i q^{53} +2.30278 q^{54} +10.8167 q^{56} -5.60555i q^{57} +0.908327i q^{58} -4.60555 q^{59} +6.60555 q^{61} +15.2111i q^{62} +3.60555i q^{63} +12.8167 q^{64} -11.5139 q^{66} +12.6056i q^{67} +3.30278i q^{68} -2.60555 q^{69} -1.21110 q^{71} -3.00000i q^{72} +13.0000i q^{73} +14.3028 q^{74} -18.5139 q^{76} -18.0278i q^{77} +15.2111i q^{78} -3.21110 q^{79} +1.00000 q^{81} -22.1194i q^{82} -1.21110i q^{83} +11.9083 q^{84} +21.2111 q^{86} +0.394449i q^{87} +15.0000i q^{88} -8.60555 q^{89} -23.8167 q^{91} +8.60555i q^{92} +6.60555i q^{93} -5.09167 q^{94} +5.30278 q^{96} +3.21110i q^{97} +13.8167i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} - 26 q^{14} - 6 q^{16} + 8 q^{19} + 12 q^{24} - 32 q^{26} - 16 q^{29} - 12 q^{31} - 2 q^{34} + 6 q^{36} - 12 q^{39} + 24 q^{41} - 30 q^{44} - 24 q^{46} - 24 q^{49}+ \cdots - 20 q^{99}+O(q^{100}) \) 4 * q - 6 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 - 26 * q^14 - 6 * q^16 + 8 * q^19 + 12 * q^24 - 32 * q^26 - 16 * q^29 - 12 * q^31 - 2 * q^34 + 6 * q^36 - 12 * q^39 + 24 * q^41 - 30 * q^44 - 24 * q^46 - 24 * q^49 - 4 * q^51 + 2 * q^54 - 4 * q^59 + 12 * q^61 + 8 * q^64 - 10 * q^66 + 4 * q^69 + 24 * q^71 + 50 * q^74 - 38 * q^76 + 16 * q^79 + 4 * q^81 + 26 * q^84 + 56 * q^86 - 20 * q^89 - 52 * q^91 - 42 * q^94 + 14 * q^96 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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