LMFDB - Embedded newform 1275.2.b.i.1174.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1275,2,Mod(1174,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [6,0,0,-4,0,0,0,0,-6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.1809262577$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3356224.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 16x^{2} + 1 $ x^6 + 8x^4 + 16x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 255)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.6
Root			$2.11491i$ of defining polynomial
Character	$\chi$	$=$	1275.1174
Dual form			1275.2.b.i.1174.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.11491i q^{2} +1.00000i q^{3} -2.47283 q^{4} -2.11491 q^{6} -1.64207i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.58774 q^{11} -2.47283i q^{12} +4.94567i q^{13} +3.47283 q^{14} -2.83076 q^{16} -1.00000i q^{17} -2.11491i q^{18} -0.926221 q^{19} +1.64207 q^{21} -9.70265i q^{22} +2.22982i q^{23} +1.00000 q^{24} -10.4596 q^{26} -1.00000i q^{27} +4.06058i q^{28} -5.53341 q^{29} +3.66152 q^{31} -7.98680i q^{32} -4.58774i q^{33} +2.11491 q^{34} +2.47283 q^{36} -9.87189i q^{37} -1.95887i q^{38} -4.94567 q^{39} -2.92622 q^{41} +3.47283i q^{42} -8.45963i q^{43} +11.3447 q^{44} -4.71585 q^{46} -0.587741i q^{47} -2.83076i q^{48} +4.30359 q^{49} +1.00000 q^{51} -12.2298i q^{52} -3.64207i q^{53} +2.11491 q^{54} -1.64207 q^{56} -0.926221i q^{57} -11.7026i q^{58} -7.40530 q^{59} -13.4053 q^{61} +7.74378i q^{62} +1.64207i q^{63} +11.2298 q^{64} +9.70265 q^{66} +13.9736i q^{67} +2.47283i q^{68} -2.22982 q^{69} -5.89134 q^{71} +1.00000i q^{72} -7.76322i q^{73} +20.8781 q^{74} +2.29039 q^{76} +7.53341i q^{77} -10.4596i q^{78} +3.28415 q^{79} +1.00000 q^{81} -6.18869i q^{82} +8.45963i q^{83} -4.06058 q^{84} +17.8913 q^{86} -5.53341i q^{87} +4.58774i q^{88} -4.22982 q^{89} +8.12115 q^{91} -5.51396i q^{92} +3.66152i q^{93} +1.24302 q^{94} +7.98680 q^{96} -15.8913i q^{97} +9.10170i q^{98} +4.58774 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4} - 6 q^{9} - 4 q^{11} + 10 q^{14} - 8 q^{16} + 8 q^{21} + 6 q^{24} - 12 q^{26} + 12 q^{29} + 4 q^{31} + 4 q^{36} - 8 q^{39} - 12 q^{41} + 30 q^{44} - 32 q^{46} + 6 q^{49} + 6 q^{51} - 8 q^{56}+ \cdots + 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^4 - 6 * q^9 - 4 * q^11 + 10 * q^14 - 8 * q^16 + 8 * q^21 + 6 * q^24 - 12 * q^26 + 12 * q^29 + 4 * q^31 + 4 * q^36 - 8 * q^39 - 12 * q^41 + 30 * q^44 - 32 * q^46 + 6 * q^49 + 6 * q^51 - 8 * q^56 + 28 * q^59 - 8 * q^61 + 42 * q^64 + 22 * q^66 + 12 * q^69 + 8 * q^71 + 42 * q^74 - 46 * q^76 + 16 * q^79 + 6 * q^81 + 10 * q^84 + 64 * q^86 - 20 * q^91 + 22 * q^94 + 8 * q^96 + 4 * 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 $

$2$	2.11491i	1.49547i	0.664000	+	0.747733i	$0.268858\pi$
$2$	2.11491i	1.49547i	−0.664000	+	0.747733i	$0.731142\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.47283	−1.23642
$5$	0	0
$5$	0	0
$6$	−2.11491	−0.863407
$7$	− 1.64207i	− 0.620645i	−0.950631	−	0.310323i	$-0.899563\pi$
$7$	− 1.64207i	− 0.620645i	0.950631	−	0.310323i	$-0.100437\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.58774	−1.38326	−0.691628	−	0.722254i	$-0.743106\pi$
$11$	−4.58774	−1.38326	−0.691628	+	0.722254i	$0.743106\pi$
$12$	− 2.47283i	− 0.713846i
$13$	4.94567i	1.37168i	0.727752	+	0.685841i	$0.240565\pi$
$13$	4.94567i	1.37168i	−0.727752	+	0.685841i	$0.759435\pi$
$14$	3.47283	0.928154
$15$	0	0
$16$	−2.83076	−0.707690
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	− 2.11491i	− 0.498488i
$19$	−0.926221	−0.212490	−0.106245	−	0.994340i	$-0.533883\pi$
$19$	−0.926221	−0.212490	−0.106245	+	0.994340i	$0.533883\pi$
$20$	0	0
$21$	1.64207	0.358330
$22$	− 9.70265i	− 2.06861i
$23$	2.22982i	0.464949i	0.972603	+	0.232474i	$0.0746822\pi$
$23$	2.22982i	0.464949i	−0.972603	+	0.232474i	$0.925318\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−10.4596	−2.05130
$27$	− 1.00000i	− 0.192450i
$28$	4.06058i	0.767377i
$29$	−5.53341	−1.02753	−0.513764	−	0.857931i	$-0.671749\pi$
$29$	−5.53341	−1.02753	−0.513764	+	0.857931i	$0.671749\pi$
$30$	0	0
$31$	3.66152	0.657629	0.328814	−	0.944395i	$-0.393351\pi$
$31$	3.66152	0.657629	0.328814	+	0.944395i	$0.393351\pi$
$32$	− 7.98680i	− 1.41188i
$33$	− 4.58774i	− 0.798623i
$34$	2.11491	0.362704
$35$	0	0
$36$	2.47283	0.412139
$37$	− 9.87189i	− 1.62293i	−0.584402	−	0.811464i	$-0.698671\pi$
$37$	− 9.87189i	− 1.62293i	0.584402	−	0.811464i	$-0.301329\pi$
$38$	− 1.95887i	− 0.317771i
$39$	−4.94567	−0.791941
$40$	0	0
$41$	−2.92622	−0.456999	−0.228499	−	0.973544i	$-0.573382\pi$
$41$	−2.92622	−0.456999	−0.228499	+	0.973544i	$0.573382\pi$
$42$	3.47283i	0.535870i
$43$	− 8.45963i	− 1.29008i	−0.764148	−	0.645041i	$-0.776840\pi$
$43$	− 8.45963i	− 1.29008i	0.764148	−	0.645041i	$-0.223160\pi$
$44$	11.3447	1.71028
$45$	0	0
$46$	−4.71585	−0.695315
$47$	− 0.587741i	− 0.0857309i	−0.999081	−	0.0428655i	$-0.986351\pi$
$47$	− 0.587741i	− 0.0857309i	0.999081	−	0.0428655i	$-0.0136487\pi$
$48$	− 2.83076i	− 0.408585i
$49$	4.30359	0.614799
$50$	0	0
$51$	1.00000	0.140028
$52$	− 12.2298i	− 1.69597i
$53$	− 3.64207i	− 0.500277i	−0.968210	−	0.250139i	$-0.919524\pi$
$53$	− 3.64207i	− 0.500277i	0.968210	−	0.250139i	$-0.0804762\pi$
$54$	2.11491	0.287802
$55$	0	0
$56$	−1.64207	−0.219431
$57$	− 0.926221i	− 0.122681i
$58$	− 11.7026i	− 1.53663i
$59$	−7.40530	−0.964088	−0.482044	−	0.876147i	$-0.660105\pi$
$59$	−7.40530	−0.964088	−0.482044	+	0.876147i	$0.660105\pi$
$60$	0	0
$61$	−13.4053	−1.71637	−0.858186	−	0.513338i	$-0.828409\pi$
$61$	−13.4053	−1.71637	−0.858186	+	0.513338i	$0.828409\pi$
$62$	7.74378i	0.983461i
$63$	1.64207i	0.206882i
$64$	11.2298	1.40373
$65$	0	0
$66$	9.70265	1.19431
$67$	13.9736i	1.70715i	0.520973	+	0.853573i	$0.325569\pi$
$67$	13.9736i	1.70715i	−0.520973	+	0.853573i	$0.674431\pi$
$68$	2.47283i	0.299875i
$69$	−2.22982	−0.268438
$70$	0	0
$71$	−5.89134	−0.699173	−0.349586	−	0.936904i	$-0.613678\pi$
$71$	−5.89134	−0.699173	−0.349586	+	0.936904i	$0.613678\pi$
$72$	1.00000i	0.117851i
$73$	− 7.76322i	− 0.908617i	−0.890845	−	0.454308i	$-0.849886\pi$
$73$	− 7.76322i	− 0.908617i	0.890845	−	0.454308i	$-0.150114\pi$
$74$	20.8781	2.42703
$75$	0	0
$76$	2.29039	0.262726
$77$	7.53341i	0.858512i
$78$	− 10.4596i	− 1.18432i
$79$	3.28415	0.369495	0.184748	−	0.982786i	$-0.440853\pi$
$79$	3.28415	0.369495	0.184748	+	0.982786i	$0.440853\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 6.18869i	− 0.683426i
$83$	8.45963i	0.928565i	0.885687	+	0.464283i	$0.153688\pi$
$83$	8.45963i	0.928565i	−0.885687	+	0.464283i	$0.846312\pi$
$84$	−4.06058	−0.443045
$85$	0	0
$86$	17.8913	1.92927
$87$	− 5.53341i	− 0.593244i
$88$	4.58774i	0.489055i
$89$	−4.22982	−0.448360	−0.224180	−	0.974548i	$-0.571970\pi$
$89$	−4.22982	−0.448360	−0.224180	+	0.974548i	$0.571970\pi$
$90$	0	0
$91$	8.12115	0.851328
$92$	− 5.51396i	− 0.574870i
$93$	3.66152i	0.379682i
$94$	1.24302	0.128208
$95$	0	0
$96$	7.98680	0.815149
$97$	− 15.8913i	− 1.61352i	−0.590879	−	0.806760i	$-0.701219\pi$
$97$	− 15.8913i	− 1.61352i	0.590879	−	0.806760i	$-0.298781\pi$
$98$	9.10170i	0.919411i
$99$	4.58774	0.461085
$100$	0	0
$101$	−7.89134	−0.785217	−0.392609	−	0.919706i	$-0.628427\pi$
$101$	−7.89134	−0.785217	−0.392609	+	0.919706i	$0.628427\pi$
$102$	2.11491i	0.209407i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	4.94567	0.484963
$105$	0	0
$106$	7.70265	0.748147
$107$	15.0279i	1.45280i	0.687270	+	0.726402i	$0.258809\pi$
$107$	15.0279i	1.45280i	−0.687270	+	0.726402i	$0.741191\pi$
$108$	2.47283i	0.237949i
$109$	13.0279	1.24785	0.623924	−	0.781485i	$-0.285537\pi$
$109$	13.0279	1.24785	0.623924	+	0.781485i	$0.285537\pi$
$110$	0	0
$111$	9.87189	0.936998
$112$	4.64832i	0.439225i
$113$	15.5140i	1.45943i	0.683751	+	0.729715i	$0.260347\pi$
$113$	15.5140i	1.45943i	−0.683751	+	0.729715i	$0.739653\pi$
$114$	1.95887	0.183465
$115$	0	0
$116$	13.6832	1.27045
$117$	− 4.94567i	− 0.457227i
$118$	− 15.6615i	− 1.44176i
$119$	−1.64207	−0.150529
$120$	0	0
$121$	10.0474	0.913397
$122$	− 28.3510i	− 2.56678i
$123$	− 2.92622i	− 0.263848i
$124$	−9.05433	−0.813103
$125$	0	0
$126$	−3.47283	−0.309385
$127$	17.5529i	1.55756i	0.627295	+	0.778782i	$0.284162\pi$
$127$	17.5529i	1.55756i	−0.627295	+	0.778782i	$0.715838\pi$
$128$	7.77643i	0.687346i
$129$	8.45963	0.744829
$130$	0	0
$131$	−9.89134	−0.864210	−0.432105	−	0.901823i	$-0.642229\pi$
$131$	−9.89134	−0.864210	−0.432105	+	0.901823i	$0.642229\pi$
$132$	11.3447i	0.987431i
$133$	1.52092i	0.131881i
$134$	−29.5529	−2.55298
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	19.1685i	1.63768i	0.574024	+	0.818839i	$0.305382\pi$
$137$	19.1685i	1.63768i	−0.574024	+	0.818839i	$0.694618\pi$
$138$	− 4.71585i	− 0.401440i
$139$	−13.1755	−1.11753	−0.558765	−	0.829326i	$-0.688725\pi$
$139$	−13.1755	−1.11753	−0.558765	+	0.829326i	$0.688725\pi$
$140$	0	0
$141$	0.587741	0.0494968
$142$	− 12.4596i	− 1.04559i
$143$	− 22.6894i	− 1.89739i
$144$	2.83076	0.235897
$145$	0	0
$146$	16.4185	1.35880
$147$	4.30359i	0.354954i
$148$	24.4115i	2.00662i
$149$	−7.97359	−0.653222	−0.326611	−	0.945159i	$-0.605907\pi$
$149$	−7.97359	−0.653222	−0.326611	+	0.945159i	$0.605907\pi$
$150$	0	0
$151$	−15.5334	−1.26409	−0.632045	−	0.774931i	$-0.717784\pi$
$151$	−15.5334	−1.26409	−0.632045	+	0.774931i	$0.717784\pi$
$152$	0.926221i	0.0751264i
$153$	1.00000i	0.0808452i
$154$	−15.9325	−1.28387
$155$	0	0
$156$	12.2298	0.979169
$157$	3.17548i	0.253431i	0.991939	+	0.126716i	$0.0404435\pi$
$157$	3.17548i	0.253431i	−0.991939	+	0.126716i	$0.959556\pi$
$158$	6.94567i	0.552568i
$159$	3.64207	0.288835
$160$	0	0
$161$	3.66152	0.288568
$162$	2.11491i	0.166163i
$163$	3.78963i	0.296827i	0.988925	+	0.148413i	$0.0474166\pi$
$163$	3.78963i	0.296827i	−0.988925	+	0.148413i	$0.952583\pi$
$164$	7.23606	0.565041
$165$	0	0
$166$	−17.8913	−1.38864
$167$	18.6072i	1.43987i	0.694043	+	0.719934i	$0.255828\pi$
$167$	18.6072i	1.43987i	−0.694043	+	0.719934i	$0.744172\pi$
$168$	− 1.64207i	− 0.126689i
$169$	−11.4596	−0.881510
$170$	0	0
$171$	0.926221	0.0708299
$172$	20.9193i	1.59508i
$173$	3.77018i	0.286642i	0.989676	+	0.143321i	$0.0457781\pi$
$173$	3.77018i	0.286642i	−0.989676	+	0.143321i	$0.954222\pi$
$174$	11.7026	0.887176
$175$	0	0
$176$	12.9868	0.978917
$177$	− 7.40530i	− 0.556616i
$178$	− 8.94567i	− 0.670506i
$179$	22.0947	1.65144	0.825719	−	0.564081i	$-0.190770\pi$
$179$	22.0947	1.65144	0.825719	+	0.564081i	$0.190770\pi$
$180$	0	0
$181$	1.40530	0.104455	0.0522275	−	0.998635i	$-0.483368\pi$
$181$	1.40530	0.104455	0.0522275	+	0.998635i	$0.483368\pi$
$182$	17.1755i	1.27313i
$183$	− 13.4053i	− 0.990948i
$184$	2.22982	0.164384
$185$	0	0
$186$	−7.74378	−0.567801
$187$	4.58774i	0.335489i
$188$	1.45339i	0.105999i
$189$	−1.64207	−0.119443
$190$	0	0
$191$	18.7283	1.35514	0.677568	−	0.735461i	$-0.263034\pi$
$191$	18.7283	1.35514	0.677568	+	0.735461i	$0.263034\pi$
$192$	11.2298i	0.810442i
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	33.6087	2.41296
$195$	0	0
$196$	−10.6421	−0.760148
$197$	− 21.0668i	− 1.50095i	−0.660900	−	0.750474i	$-0.729825\pi$
$197$	− 21.0668i	− 1.50095i	0.660900	−	0.750474i	$-0.270175\pi$
$198$	9.70265i	0.689537i
$199$	−9.17548	−0.650433	−0.325216	−	0.945640i	$-0.605437\pi$
$199$	−9.17548	−0.650433	−0.325216	+	0.945640i	$0.605437\pi$
$200$	0	0
$201$	−13.9736	−0.985621
$202$	− 16.6894i	− 1.17427i
$203$	9.08627i	0.637731i
$204$	−2.47283	−0.173133
$205$	0	0
$206$	−16.9193	−1.17882
$207$	− 2.22982i	− 0.154983i
$208$	− 14.0000i	− 0.970725i
$209$	4.24926	0.293928
$210$	0	0
$211$	−19.3230	−1.33025	−0.665127	−	0.746731i	$-0.731622\pi$
$211$	−19.3230	−1.33025	−0.665127	+	0.746731i	$0.731622\pi$
$212$	9.00624i	0.618551i
$213$	− 5.89134i	− 0.403668i
$214$	−31.7827	−2.17262
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 6.01249i	− 0.408154i
$218$	27.5529i	1.86611i
$219$	7.76322	0.524590
$220$	0	0
$221$	4.94567	0.332682
$222$	20.8781i	1.40125i
$223$	7.74378i	0.518562i	0.965802	+	0.259281i	$0.0834855\pi$
$223$	7.74378i	0.518562i	−0.965802	+	0.259281i	$0.916514\pi$
$224$	−13.1149	−0.876277
$225$	0	0
$226$	−32.8106	−2.18253
$227$	3.87885i	0.257448i	0.991680	+	0.128724i	$0.0410882\pi$
$227$	3.87885i	0.257448i	−0.991680	+	0.128724i	$0.958912\pi$
$228$	2.29039i	0.151685i
$229$	2.84396	0.187934	0.0939672	−	0.995575i	$-0.470045\pi$
$229$	2.84396	0.187934	0.0939672	+	0.995575i	$0.470045\pi$
$230$	0	0
$231$	−7.53341	−0.495662
$232$	5.53341i	0.363286i
$233$	− 15.2966i	− 1.00212i	−0.865414	−	0.501058i	$-0.832944\pi$
$233$	− 15.2966i	− 1.00212i	0.865414	−	0.501058i	$-0.167056\pi$
$234$	10.4596	0.683767
$235$	0	0
$236$	18.3121	1.19201
$237$	3.28415i	0.213328i
$238$	− 3.47283i	− 0.225110i
$239$	−8.12115	−0.525314	−0.262657	−	0.964889i	$-0.584599\pi$
$239$	−8.12115	−0.525314	−0.262657	+	0.964889i	$0.584599\pi$
$240$	0	0
$241$	15.5140	0.999342	0.499671	−	0.866215i	$-0.333454\pi$
$241$	15.5140	0.999342	0.499671	+	0.866215i	$0.333454\pi$
$242$	21.2493i	1.36595i
$243$	1.00000i	0.0641500i
$244$	33.1491	2.12215
$245$	0	0
$246$	6.18869	0.394576
$247$	− 4.58078i	− 0.291468i
$248$	− 3.66152i	− 0.232507i
$249$	−8.45963	−0.536107
$250$	0	0
$251$	−20.3774	−1.28621	−0.643104	−	0.765779i	$-0.722354\pi$
$251$	−20.3774	−1.28621	−0.643104	+	0.765779i	$0.722354\pi$
$252$	− 4.06058i	− 0.255792i
$253$	− 10.2298i	− 0.643143i
$254$	−37.1227	−2.32928
$255$	0	0
$256$	6.01320	0.375825
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	17.8913i	1.11387i
$259$	−16.2104	−1.00726
$260$	0	0
$261$	5.53341	0.342509
$262$	− 20.9193i	− 1.29240i
$263$	− 15.9108i	− 0.981101i	−0.871413	−	0.490550i	$-0.836796\pi$
$263$	− 15.9108i	− 0.981101i	0.871413	−	0.490550i	$-0.163204\pi$
$264$	−4.58774	−0.282356
$265$	0	0
$266$	−3.21661	−0.197223
$267$	− 4.22982i	− 0.258860i
$268$	− 34.5544i	− 2.11074i
$269$	5.07378	0.309354	0.154677	−	0.987965i	$-0.450566\pi$
$269$	5.07378	0.309354	0.154677	+	0.987965i	$0.450566\pi$
$270$	0	0
$271$	−2.10866	−0.128092	−0.0640461	−	0.997947i	$-0.520400\pi$
$271$	−2.10866	−0.128092	−0.0640461	+	0.997947i	$0.520400\pi$
$272$	2.83076i	0.171640i
$273$	8.12115i	0.491514i
$274$	−40.5397	−2.44909
$275$	0	0
$276$	5.51396	0.331902
$277$	− 17.0279i	− 1.02311i	−0.859251	−	0.511554i	$-0.829070\pi$
$277$	− 17.0279i	− 1.02311i	0.859251	−	0.511554i	$-0.170930\pi$
$278$	− 27.8649i	− 1.67123i
$279$	−3.66152	−0.219210
$280$	0	0
$281$	20.6072	1.22932	0.614661	−	0.788791i	$-0.289293\pi$
$281$	20.6072	1.22932	0.614661	+	0.788791i	$0.289293\pi$
$282$	1.24302i	0.0740207i
$283$	16.6700i	0.990929i	0.868628	+	0.495464i	$0.165002\pi$
$283$	16.6700i	0.990929i	−0.868628	+	0.495464i	$0.834998\pi$
$284$	14.5683	0.864469
$285$	0	0
$286$	47.9861	2.83748
$287$	4.80507i	0.283634i
$288$	7.98680i	0.470626i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	15.8913	0.931567
$292$	19.1972i	1.12343i
$293$	21.2772i	1.24303i	0.783404	+	0.621513i	$0.213482\pi$
$293$	21.2772i	1.24303i	−0.783404	+	0.621513i	$0.786518\pi$
$294$	−9.10170	−0.530822
$295$	0	0
$296$	−9.87189	−0.573792
$297$	4.58774i	0.266208i
$298$	− 16.8634i	− 0.976871i
$299$	−11.0279	−0.637761
$300$	0	0
$301$	−13.8913	−0.800683
$302$	− 32.8517i	− 1.89040i
$303$	− 7.89134i	− 0.453345i
$304$	2.62191	0.150377
$305$	0	0
$306$	−2.11491	−0.120901
$307$	− 22.3510i	− 1.27564i	−0.770187	−	0.637818i	$-0.779837\pi$
$307$	− 22.3510i	− 1.27564i	0.770187	−	0.637818i	$-0.220163\pi$
$308$	− 18.6289i	− 1.06148i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−11.4512	−0.649335	−0.324668	−	0.945828i	$-0.605252\pi$
$311$	−11.4512	−0.649335	−0.324668	+	0.945828i	$0.605252\pi$
$312$	4.94567i	0.279993i
$313$	24.4791i	1.38364i	0.722070	+	0.691820i	$0.243191\pi$
$313$	24.4791i	1.38364i	−0.722070	+	0.691820i	$0.756809\pi$
$314$	−6.71585	−0.378997
$315$	0	0
$316$	−8.12115	−0.456850
$317$	26.5419i	1.49074i	0.666651	+	0.745370i	$0.267727\pi$
$317$	26.5419i	1.49074i	−0.666651	+	0.745370i	$0.732273\pi$
$318$	7.70265i	0.431943i
$319$	25.3859	1.42133
$320$	0	0
$321$	−15.0279	−0.838777
$322$	7.74378i	0.431544i
$323$	0.926221i	0.0515363i
$324$	−2.47283	−0.137380
$325$	0	0
$326$	−8.01472	−0.443894
$327$	13.0279i	0.720446i
$328$	2.92622i	0.161574i
$329$	−0.965115	−0.0532085
$330$	0	0
$331$	−15.2383	−0.837572	−0.418786	−	0.908085i	$-0.637544\pi$
$331$	−15.2383	−0.837572	−0.418786	+	0.908085i	$0.637544\pi$
$332$	− 20.9193i	− 1.14809i
$333$	9.87189i	0.540976i
$334$	−39.3525	−2.15327
$335$	0	0
$336$	−4.64832	−0.253586
$337$	22.5738i	1.22967i	0.788654	+	0.614837i	$0.210778\pi$
$337$	22.5738i	1.22967i	−0.788654	+	0.614837i	$0.789222\pi$
$338$	− 24.2361i	− 1.31827i
$339$	−15.5140	−0.842603
$340$	0	0
$341$	−16.7981	−0.909669
$342$	1.95887i	0.105924i
$343$	− 18.5613i	− 1.00222i
$344$	−8.45963	−0.456113
$345$	0	0
$346$	−7.97359	−0.428663
$347$	− 16.3246i	− 0.876348i	−0.898890	−	0.438174i	$-0.855625\pi$
$347$	− 16.3246i	− 0.876348i	0.898890	−	0.438174i	$-0.144375\pi$
$348$	13.6832i	0.733497i
$349$	26.3315	1.40949	0.704747	−	0.709459i	$-0.251061\pi$
$349$	26.3315	1.40949	0.704747	+	0.709459i	$0.251061\pi$
$350$	0	0
$351$	4.94567	0.263980
$352$	36.6414i	1.95299i
$353$	29.9930i	1.59637i	0.602413	+	0.798184i	$0.294206\pi$
$353$	29.9930i	1.59637i	−0.602413	+	0.798184i	$0.705794\pi$
$354$	15.6615	0.832400
$355$	0	0
$356$	10.4596	0.554359
$357$	− 1.64207i	− 0.0869078i
$358$	46.7283i	2.46967i
$359$	−25.9736	−1.37083	−0.685417	−	0.728151i	$-0.740380\pi$
$359$	−25.9736	−1.37083	−0.685417	+	0.728151i	$0.740380\pi$
$360$	0	0
$361$	−18.1421	−0.954848
$362$	2.97208i	0.156209i
$363$	10.0474i	0.527350i
$364$	−20.0823	−1.05260
$365$	0	0
$366$	28.3510	1.48193
$367$	− 24.9193i	− 1.30077i	−0.759603	−	0.650387i	$-0.774607\pi$
$367$	− 24.9193i	− 1.30077i	0.759603	−	0.650387i	$-0.225393\pi$
$368$	− 6.31207i	− 0.329039i
$369$	2.92622	0.152333
$370$	0	0
$371$	−5.98055	−0.310495
$372$	− 9.05433i	− 0.469445i
$373$	− 2.25622i	− 0.116823i	−0.998293	−	0.0584114i	$-0.981396\pi$
$373$	− 2.25622i	− 0.116823i	0.998293	−	0.0584114i	$-0.0186035\pi$
$374$	−9.70265	−0.501712
$375$	0	0
$376$	−0.587741	−0.0303105
$377$	− 27.3664i	− 1.40944i
$378$	− 3.47283i	− 0.178623i
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	−17.5529	−0.899260
$382$	39.6087i	2.02656i
$383$	− 13.9666i	− 0.713662i	−0.934169	−	0.356831i	$-0.883857\pi$
$383$	− 13.9666i	− 0.713662i	0.934169	−	0.356831i	$-0.116143\pi$
$384$	−7.77643	−0.396839
$385$	0	0
$386$	−4.22982	−0.215292
$387$	8.45963i	0.430027i
$388$	39.2966i	1.99498i
$389$	4.09474	0.207612	0.103806	−	0.994598i	$-0.466898\pi$
$389$	4.09474	0.207612	0.103806	+	0.994598i	$0.466898\pi$
$390$	0	0
$391$	2.22982	0.112767
$392$	− 4.30359i	− 0.217364i
$393$	− 9.89134i	− 0.498952i
$394$	44.5544	2.24462
$395$	0	0
$396$	−11.3447	−0.570094
$397$	− 19.4681i	− 0.977076i	−0.872543	−	0.488538i	$-0.837530\pi$
$397$	− 19.4681i	− 0.977076i	0.872543	−	0.488538i	$-0.162470\pi$
$398$	− 19.4053i	− 0.972700i
$399$	−1.52092	−0.0761414
$400$	0	0
$401$	34.8734	1.74149	0.870747	−	0.491731i	$-0.163636\pi$
$401$	34.8734	1.74149	0.870747	+	0.491731i	$0.163636\pi$
$402$	− 29.5529i	− 1.47396i
$403$	18.1087i	0.902057i
$404$	19.5140	0.970856
$405$	0	0
$406$	−19.2166	−0.953704
$407$	45.2897i	2.24493i
$408$	− 1.00000i	− 0.0495074i
$409$	29.9108	1.47899	0.739497	−	0.673160i	$-0.235064\pi$
$409$	29.9108	1.47899	0.739497	+	0.673160i	$0.235064\pi$
$410$	0	0
$411$	−19.1685	−0.945513
$412$	− 19.7827i	− 0.974622i
$413$	12.1600i	0.598357i
$414$	4.71585	0.231772
$415$	0	0
$416$	39.5000	1.93665
$417$	− 13.1755i	− 0.645206i
$418$	8.98680i	0.439559i
$419$	−9.04737	−0.441993	−0.220997	−	0.975275i	$-0.570931\pi$
$419$	−9.04737	−0.441993	−0.220997	+	0.975275i	$0.570931\pi$
$420$	0	0
$421$	−26.0753	−1.27083	−0.635416	−	0.772170i	$-0.719171\pi$
$421$	−26.0753	−1.27083	−0.635416	+	0.772170i	$0.719171\pi$
$422$	− 40.8664i	− 1.98935i
$423$	0.587741i	0.0285770i
$424$	−3.64207	−0.176875
$425$	0	0
$426$	12.4596	0.603671
$427$	22.0125i	1.06526i
$428$	− 37.1616i	− 1.79627i
$429$	22.6894	1.09546
$430$	0	0
$431$	−1.76322	−0.0849315	−0.0424658	−	0.999098i	$-0.513521\pi$
$431$	−1.76322	−0.0849315	−0.0424658	+	0.999098i	$0.513521\pi$
$432$	2.83076i	0.136195i
$433$	21.1057i	1.01428i	0.861865	+	0.507138i	$0.169297\pi$
$433$	21.1057i	1.01428i	−0.861865	+	0.507138i	$0.830703\pi$
$434$	12.7159	0.610380
$435$	0	0
$436$	−32.2159	−1.54286
$437$	− 2.06530i	− 0.0987968i
$438$	16.4185i	0.784506i
$439$	33.7174	1.60924	0.804621	−	0.593789i	$-0.202368\pi$
$439$	33.7174	1.60924	0.804621	+	0.593789i	$0.202368\pi$
$440$	0	0
$441$	−4.30359	−0.204933
$442$	10.4596i	0.497514i
$443$	− 40.7019i	− 1.93381i	−0.255142	−	0.966904i	$-0.582122\pi$
$443$	− 40.7019i	− 1.93381i	0.255142	−	0.966904i	$-0.417878\pi$
$444$	−24.4115	−1.15852
$445$	0	0
$446$	−16.3774	−0.775491
$447$	− 7.97359i	− 0.377138i
$448$	− 18.4402i	− 0.871217i
$449$	−20.1087	−0.948987	−0.474493	−	0.880259i	$-0.657369\pi$
$449$	−20.1087	−0.948987	−0.474493	+	0.880259i	$0.657369\pi$
$450$	0	0
$451$	13.4247	0.632147
$452$	− 38.3635i	− 1.80447i
$453$	− 15.5334i	− 0.729823i
$454$	−8.20341	−0.385005
$455$	0	0
$456$	−0.926221	−0.0433743
$457$	− 29.1491i	− 1.36354i	−0.731568	−	0.681768i	$-0.761211\pi$
$457$	− 29.1491i	− 1.36354i	0.731568	−	0.681768i	$-0.238789\pi$
$458$	6.01472i	0.281049i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−15.0932	−0.702962	−0.351481	−	0.936195i	$-0.614322\pi$
$461$	−15.0932	−0.702962	−0.351481	+	0.936195i	$0.614322\pi$
$462$	− 15.9325i	− 0.741245i
$463$	37.6351i	1.74905i	0.484979	+	0.874526i	$0.338827\pi$
$463$	37.6351i	1.74905i	−0.484979	+	0.874526i	$0.661173\pi$
$464$	15.6638	0.727172
$465$	0	0
$466$	32.3510	1.49863
$467$	9.72433i	0.449988i	0.974360	+	0.224994i	$0.0722363\pi$
$467$	9.72433i	0.449988i	−0.974360	+	0.224994i	$0.927764\pi$
$468$	12.2298i	0.565323i
$469$	22.9457	1.05953
$470$	0	0
$471$	−3.17548	−0.146319
$472$	7.40530i	0.340856i
$473$	38.8106i	1.78451i
$474$	−6.94567	−0.319025
$475$	0	0
$476$	4.06058	0.186116
$477$	3.64207i	0.166759i
$478$	− 17.1755i	− 0.785588i
$479$	13.1366	0.600226	0.300113	−	0.953904i	$-0.402976\pi$
$479$	13.1366	0.600226	0.300113	+	0.953904i	$0.402976\pi$
$480$	0	0
$481$	48.8231	2.22614
$482$	32.8106i	1.49448i
$483$	3.66152i	0.166605i
$484$	−24.8455	−1.12934
$485$	0	0
$486$	−2.11491	−0.0959342
$487$	− 1.43171i	− 0.0648768i	−0.999474	−	0.0324384i	$-0.989673\pi$
$487$	− 1.43171i	− 0.0648768i	0.999474	−	0.0324384i	$-0.0103273\pi$
$488$	13.4053i	0.606829i
$489$	−3.78963	−0.171373
$490$	0	0
$491$	−30.0947	−1.35816	−0.679078	−	0.734066i	$-0.737620\pi$
$491$	−30.0947	−1.35816	−0.679078	+	0.734066i	$0.737620\pi$
$492$	7.23606i	0.326227i
$493$	5.53341i	0.249212i
$494$	9.68793	0.435880
$495$	0	0
$496$	−10.3649	−0.465397
$497$	9.67401i	0.433939i
$498$	− 17.8913i	− 0.801730i
$499$	−3.62263	−0.162171	−0.0810855	−	0.996707i	$-0.525839\pi$
$499$	−3.62263	−0.162171	−0.0810855	+	0.996707i	$0.525839\pi$
$500$	0	0
$501$	−18.6072	−0.831308
$502$	− 43.0963i	− 1.92348i
$503$	22.5544i	1.00565i	0.864388	+	0.502825i	$0.167706\pi$
$503$	22.5544i	1.00565i	−0.864388	+	0.502825i	$0.832294\pi$
$504$	1.64207	0.0731438
$505$	0	0
$506$	21.6351	0.961798
$507$	− 11.4596i	− 0.508940i
$508$	− 43.4053i	− 1.92580i
$509$	−16.7283	−0.741471	−0.370735	−	0.928739i	$-0.620894\pi$
$509$	−16.7283	−0.741471	−0.370735	+	0.928739i	$0.620894\pi$
$510$	0	0
$511$	−12.7478	−0.563929
$512$	28.2702i	1.24938i
$513$	0.926221i	0.0408937i
$514$	4.22982	0.186569
$515$	0	0
$516$	−20.9193	−0.920919
$517$	2.69641i	0.118588i
$518$	− 34.2834i	− 1.50633i
$519$	−3.77018	−0.165493
$520$	0	0
$521$	26.8734	1.17735	0.588673	−	0.808372i	$-0.299651\pi$
$521$	26.8734	1.17735	0.588673	+	0.808372i	$0.299651\pi$
$522$	11.7026i	0.512211i
$523$	41.7996i	1.82777i	0.405973	+	0.913885i	$0.366933\pi$
$523$	41.7996i	1.82777i	−0.405973	+	0.913885i	$0.633067\pi$
$524$	24.4596	1.06852
$525$	0	0
$526$	33.6498	1.46720
$527$	− 3.66152i	− 0.159498i
$528$	12.9868i	0.565178i
$529$	18.0279	0.783823
$530$	0	0
$531$	7.40530	0.321363
$532$	− 3.76099i	− 0.163060i
$533$	− 14.4721i	− 0.626857i
$534$	8.94567	0.387117
$535$	0	0
$536$	13.9736	0.603567
$537$	22.0947i	0.953459i
$538$	10.7306i	0.462628i
$539$	−19.7438	−0.850425
$540$	0	0
$541$	29.1102	1.25154	0.625772	−	0.780006i	$-0.284784\pi$
$541$	29.1102	1.25154	0.625772	+	0.780006i	$0.284784\pi$
$542$	− 4.45963i	− 0.191558i
$543$	1.40530i	0.0603071i
$544$	−7.98680	−0.342431
$545$	0	0
$546$	−17.1755	−0.735043
$547$	− 13.8844i	− 0.593653i	−0.954931	−	0.296827i	$-0.904072\pi$
$547$	− 13.8844i	− 0.593653i	0.954931	−	0.296827i	$-0.0959283\pi$
$548$	− 47.4006i	− 2.02485i
$549$	13.4053	0.572124
$550$	0	0
$551$	5.12516	0.218339
$552$	2.22982i	0.0949072i
$553$	− 5.39281i	− 0.229326i
$554$	36.0125	1.53002
$555$	0	0
$556$	32.5808	1.38173
$557$	− 0.108664i	− 0.00460426i	−0.999997	−	0.00230213i	$-0.999267\pi$
$557$	− 0.108664i	− 0.00460426i	0.999997	−	0.00230213i	$-0.000732791\pi$
$558$	− 7.74378i	− 0.327820i
$559$	41.8385	1.76958
$560$	0	0
$561$	−4.58774	−0.193695
$562$	43.5823i	1.83841i
$563$	− 8.54885i	− 0.360291i	−0.983640	−	0.180145i	$-0.942343\pi$
$563$	− 8.54885i	− 0.360291i	0.983640	−	0.180145i	$-0.0576569\pi$
$564$	−1.45339	−0.0611986
$565$	0	0
$566$	−35.2555	−1.48190
$567$	− 1.64207i	− 0.0689606i
$568$	5.89134i	0.247195i
$569$	21.0668	0.883167	0.441583	−	0.897220i	$-0.354417\pi$
$569$	21.0668	0.883167	0.441583	+	0.897220i	$0.354417\pi$
$570$	0	0
$571$	−9.25774	−0.387424	−0.193712	−	0.981058i	$-0.562053\pi$
$571$	−9.25774	−0.387424	−0.193712	+	0.981058i	$0.562053\pi$
$572$	56.1072i	2.34596i
$573$	18.7283i	0.782388i
$574$	−10.1623	−0.424165
$575$	0	0
$576$	−11.2298	−0.467909
$577$	21.2313i	0.883872i	0.897047	+	0.441936i	$0.145708\pi$
$577$	21.2313i	0.883872i	−0.897047	+	0.441936i	$0.854292\pi$
$578$	− 2.11491i	− 0.0879686i
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	13.8913	0.576310
$582$	33.6087i	1.39313i
$583$	16.7089i	0.692012i
$584$	−7.76322	−0.321245
$585$	0	0
$586$	−44.9993	−1.85890
$587$	14.2617i	0.588645i	0.955706	+	0.294323i	$0.0950940\pi$
$587$	14.2617i	0.588645i	−0.955706	+	0.294323i	$0.904906\pi$
$588$	− 10.6421i	− 0.438872i
$589$	−3.39138	−0.139739
$590$	0	0
$591$	21.0668	0.866573
$592$	27.9450i	1.14853i
$593$	− 42.4138i	− 1.74173i	−0.491527	−	0.870863i	$-0.663561\pi$
$593$	− 42.4138i	− 1.74173i	0.491527	−	0.870863i	$-0.336439\pi$
$594$	−9.70265	−0.398105
$595$	0	0
$596$	19.7174	0.807655
$597$	− 9.17548i	− 0.375528i
$598$	− 23.3230i	− 0.953750i
$599$	−8.93318	−0.365000	−0.182500	−	0.983206i	$-0.558419\pi$
$599$	−8.93318	−0.365000	−0.182500	+	0.983206i	$0.558419\pi$
$600$	0	0
$601$	9.62263	0.392515	0.196258	−	0.980552i	$-0.437121\pi$
$601$	9.62263	0.392515	0.196258	+	0.980552i	$0.437121\pi$
$602$	− 29.3789i	− 1.19739i
$603$	− 13.9736i	− 0.569049i
$604$	38.4115	1.56294
$605$	0	0
$606$	16.6894	0.677962
$607$	30.7647i	1.24870i	0.781144	+	0.624351i	$0.214637\pi$
$607$	30.7647i	1.24870i	−0.781144	+	0.624351i	$0.785363\pi$
$608$	7.39754i	0.300010i
$609$	−9.08627	−0.368194
$610$	0	0
$611$	2.90677	0.117595
$612$	− 2.47283i	− 0.0999584i
$613$	− 0.147558i	− 0.00595982i	−0.999996	−	0.00297991i	$-0.999051\pi$
$613$	− 0.147558i	− 0.00595982i	0.999996	−	0.00297991i	$-0.000948536\pi$
$614$	47.2702	1.90767
$615$	0	0
$616$	7.53341	0.303530
$617$	− 25.6226i	− 1.03153i	−0.856731	−	0.515764i	$-0.827508\pi$
$617$	− 25.6226i	− 1.03153i	0.856731	−	0.515764i	$-0.172492\pi$
$618$	− 16.9193i	− 0.680592i
$619$	18.5544	0.745763	0.372882	−	0.927879i	$-0.378370\pi$
$619$	18.5544	0.745763	0.372882	+	0.927879i	$0.378370\pi$
$620$	0	0
$621$	2.22982	0.0894794
$622$	− 24.2181i	− 0.971058i
$623$	6.94567i	0.278272i
$624$	14.0000	0.560449
$625$	0	0
$626$	−51.7710	−2.06918
$627$	4.24926i	0.169699i
$628$	− 7.85244i	− 0.313347i
$629$	−9.87189	−0.393618
$630$	0	0
$631$	41.6282	1.65719	0.828595	−	0.559848i	$-0.189141\pi$
$631$	41.6282	1.65719	0.828595	+	0.559848i	$0.189141\pi$
$632$	− 3.28415i	− 0.130636i
$633$	− 19.3230i	− 0.768022i
$634$	−56.1336	−2.22935
$635$	0	0
$636$	−9.00624	−0.357121
$637$	21.2841i	0.843309i
$638$	53.6887i	2.12556i
$639$	5.89134	0.233058
$640$	0	0
$641$	13.3300	0.526503	0.263252	−	0.964727i	$-0.415205\pi$
$641$	13.3300	0.526503	0.263252	+	0.964727i	$0.415205\pi$
$642$	− 31.7827i	− 1.25436i
$643$	− 25.1685i	− 0.992550i	−0.868165	−	0.496275i	$-0.834701\pi$
$643$	− 25.1685i	− 0.992550i	0.868165	−	0.496275i	$-0.165299\pi$
$644$	−9.05433	−0.356791
$645$	0	0
$646$	−1.95887	−0.0770708
$647$	− 6.40378i	− 0.251759i	−0.992046	−	0.125879i	$-0.959825\pi$
$647$	− 6.40378i	− 0.251759i	0.992046	−	0.125879i	$-0.0401752\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	33.9736	1.33358
$650$	0	0
$651$	6.01249	0.235648
$652$	− 9.37113i	− 0.367002i
$653$	− 48.8884i	− 1.91315i	−0.291486	−	0.956575i	$-0.594150\pi$
$653$	− 48.8884i	− 1.91315i	0.291486	−	0.956575i	$-0.405850\pi$
$654$	−27.5529	−1.07740
$655$	0	0
$656$	8.28343	0.323414
$657$	7.76322i	0.302872i
$658$	− 2.04113i	− 0.0795715i
$659$	−14.6072	−0.569015	−0.284508	−	0.958674i	$-0.591830\pi$
$659$	−14.6072	−0.569015	−0.284508	+	0.958674i	$0.591830\pi$
$660$	0	0
$661$	0.774193	0.0301126	0.0150563	−	0.999887i	$-0.495207\pi$
$661$	0.774193	0.0301126	0.0150563	+	0.999887i	$0.495207\pi$
$662$	− 32.2276i	− 1.25256i
$663$	4.94567i	0.192074i
$664$	8.45963	0.328297
$665$	0	0
$666$	−20.8781	−0.809011
$667$	− 12.3385i	− 0.477748i
$668$	− 46.0125i	− 1.78028i
$669$	−7.74378	−0.299392
$670$	0	0
$671$	61.5000	2.37418
$672$	− 13.1149i	− 0.505919i
$673$	− 19.2144i	− 0.740660i	−0.928900	−	0.370330i	$-0.879245\pi$
$673$	− 19.2144i	− 0.740660i	0.928900	−	0.370330i	$-0.120755\pi$
$674$	−47.7415	−1.83894
$675$	0	0
$676$	28.3378	1.08991
$677$	− 27.2144i	− 1.04593i	−0.852353	−	0.522967i	$-0.824825\pi$
$677$	− 27.2144i	− 1.04593i	0.852353	−	0.522967i	$-0.175175\pi$
$678$	− 32.8106i	− 1.26008i
$679$	−26.0947	−1.00142
$680$	0	0
$681$	−3.87885	−0.148638
$682$	− 35.5264i	− 1.36038i
$683$	23.2702i	0.890410i	0.895429	+	0.445205i	$0.146869\pi$
$683$	23.2702i	0.890410i	−0.895429	+	0.445205i	$0.853131\pi$
$684$	−2.29039	−0.0875753
$685$	0	0
$686$	39.2555	1.49878
$687$	2.84396i	0.108504i
$688$	23.9472i	0.912978i
$689$	18.0125	0.686221
$690$	0	0
$691$	29.6740	1.12885	0.564426	−	0.825484i	$-0.309097\pi$
$691$	29.6740	1.12885	0.564426	+	0.825484i	$0.309097\pi$
$692$	− 9.32304i	− 0.354409i
$693$	− 7.53341i	− 0.286171i
$694$	34.5249	1.31055
$695$	0	0
$696$	−5.53341	−0.209743
$697$	2.92622i	0.110839i
$698$	55.6887i	2.10785i
$699$	15.2966	0.578572
$700$	0	0
$701$	−23.7702	−0.897787	−0.448894	−	0.893585i	$-0.648182\pi$
$701$	−23.7702	−0.897787	−0.448894	+	0.893585i	$0.648182\pi$
$702$	10.4596i	0.394773i
$703$	9.14355i	0.344856i
$704$	−51.5195	−1.94171
$705$	0	0
$706$	−63.4325	−2.38731
$707$	12.9582i	0.487342i
$708$	18.3121i	0.688210i
$709$	42.2423	1.58644	0.793221	−	0.608933i	$-0.208402\pi$
$709$	42.2423	1.58644	0.793221	+	0.608933i	$0.208402\pi$
$710$	0	0
$711$	−3.28415	−0.123165
$712$	4.22982i	0.158519i
$713$	8.16451i	0.305763i
$714$	3.47283	0.129968
$715$	0	0
$716$	−54.6366	−2.04187
$717$	− 8.12115i	− 0.303290i
$718$	− 54.9317i	− 2.05003i
$719$	−12.1281	−0.452302	−0.226151	−	0.974092i	$-0.572614\pi$
$719$	−12.1281	−0.452302	−0.226151	+	0.974092i	$0.572614\pi$
$720$	0	0
$721$	13.1366	0.489232
$722$	− 38.3689i	− 1.42794i
$723$	15.5140i	0.576970i
$724$	−3.47507	−0.129150
$725$	0	0
$726$	−21.2493	−0.788634
$727$	7.57926i	0.281099i	0.990074	+	0.140550i	$0.0448870\pi$
$727$	7.57926i	0.281099i	−0.990074	+	0.140550i	$0.955113\pi$
$728$	− 8.12115i	− 0.300990i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.45963	−0.312891
$732$	33.1491i	1.22523i
$733$	− 33.1102i	− 1.22295i	−0.791263	−	0.611476i	$-0.790576\pi$
$733$	− 33.1102i	− 1.22295i	0.791263	−	0.611476i	$-0.209424\pi$
$734$	52.7019	1.94526
$735$	0	0
$736$	17.8091	0.656451
$737$	− 64.1072i	− 2.36142i
$738$	6.18869i	0.227809i
$739$	−22.8634	−0.841044	−0.420522	−	0.907282i	$-0.638153\pi$
$739$	−22.8634	−0.841044	−0.420522	+	0.907282i	$0.638153\pi$
$740$	0	0
$741$	4.58078	0.168279
$742$	− 12.6483i	− 0.464334i
$743$	47.7438i	1.75155i	0.482720	+	0.875775i	$0.339649\pi$
$743$	47.7438i	1.75155i	−0.482720	+	0.875775i	$0.660351\pi$
$744$	3.66152	0.134238
$745$	0	0
$746$	4.77170	0.174704
$747$	− 8.45963i	− 0.309522i
$748$	− 11.3447i	− 0.414804i
$749$	24.6770	0.901676
$750$	0	0
$751$	7.36640	0.268804	0.134402	−	0.990927i	$-0.457089\pi$
$751$	7.36640	0.268804	0.134402	+	0.990927i	$0.457089\pi$
$752$	1.66376i	0.0606709i
$753$	− 20.3774i	− 0.742593i
$754$	57.8774	2.10777
$755$	0	0
$756$	4.06058	0.147682
$757$	2.25622i	0.0820038i	0.999159	+	0.0410019i	$0.0130550\pi$
$757$	2.25622i	0.0820038i	−0.999159	+	0.0410019i	$0.986945\pi$
$758$	− 59.2174i	− 2.15087i
$759$	10.2298	0.371319
$760$	0	0
$761$	8.02641	0.290957	0.145479	−	0.989361i	$-0.453528\pi$
$761$	8.02641	0.290957	0.145479	+	0.989361i	$0.453528\pi$
$762$	− 37.1227i	− 1.34481i
$763$	− 21.3928i	− 0.774472i
$764$	−46.3121	−1.67551
$765$	0	0
$766$	29.5381	1.06726
$767$	− 36.6241i	− 1.32242i
$768$	6.01320i	0.216983i
$769$	−37.1810	−1.34078	−0.670391	−	0.742008i	$-0.733873\pi$
$769$	−37.1810	−1.34078	−0.670391	+	0.742008i	$0.733873\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	− 4.94567i	− 0.177998i
$773$	17.4806i	0.628733i	0.949302	+	0.314367i	$0.101792\pi$
$773$	17.4806i	0.628733i	−0.949302	+	0.314367i	$0.898208\pi$
$774$	−17.8913	−0.643091
$775$	0	0
$776$	−15.8913	−0.570466
$777$	− 16.2104i	− 0.581544i
$778$	8.66000i	0.310476i
$779$	2.71033	0.0971075
$780$	0	0
$781$	27.0279	0.967135
$782$	4.71585i	0.168639i
$783$	5.53341i	0.197748i
$784$	−12.1824	−0.435087
$785$	0	0
$786$	20.9193	0.746165
$787$	− 26.7787i	− 0.954556i	−0.878752	−	0.477278i	$-0.841623\pi$
$787$	− 26.7787i	− 0.954556i	0.878752	−	0.477278i	$-0.158377\pi$
$788$	52.0947i	1.85580i
$789$	15.9108	0.566439
$790$	0	0
$791$	25.4751	0.905789
$792$	− 4.58774i	− 0.163018i
$793$	− 66.2982i	− 2.35432i
$794$	41.1732	1.46118
$795$	0	0
$796$	22.6894	0.804206
$797$	29.5723i	1.04750i	0.851871	+	0.523752i	$0.175468\pi$
$797$	29.5723i	1.04750i	−0.851871	+	0.523752i	$0.824532\pi$
$798$	− 3.21661i	− 0.113867i
$799$	−0.587741	−0.0207928
$800$	0	0
$801$	4.22982	0.149453
$802$	73.7540i	2.60435i
$803$	35.6157i	1.25685i
$804$	34.5544	1.21864
$805$	0	0
$806$	−38.2982	−1.34899
$807$	5.07378i	0.178605i
$808$	7.89134i	0.277616i
$809$	−18.0778	−0.635581	−0.317791	−	0.948161i	$-0.602941\pi$
$809$	−18.0778	−0.635581	−0.317791	+	0.948161i	$0.602941\pi$
$810$	0	0
$811$	16.0823	0.564724	0.282362	−	0.959308i	$-0.408882\pi$
$811$	16.0823	0.564724	0.282362	+	0.959308i	$0.408882\pi$
$812$	− 22.4688i	− 0.788501i
$813$	− 2.10866i	− 0.0739541i
$814$	−95.7835	−3.35721
$815$	0	0
$816$	−2.83076	−0.0990964
$817$	7.83549i	0.274129i
$818$	63.2585i	2.21178i
$819$	−8.12115	−0.283776
$820$	0	0
$821$	−30.4596	−1.06305	−0.531524	−	0.847043i	$-0.678381\pi$
$821$	−30.4596	−1.06305	−0.531524	+	0.847043i	$0.678381\pi$
$822$	− 40.5397i	− 1.41398i
$823$	5.68097i	0.198026i	0.995086	+	0.0990130i	$0.0315686\pi$
$823$	5.68097i	0.198026i	−0.995086	+	0.0990130i	$0.968431\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−25.7174	−0.894822
$827$	20.3774i	0.708591i	0.935134	+	0.354295i	$0.115279\pi$
$827$	20.3774i	0.708591i	−0.935134	+	0.354295i	$0.884721\pi$
$828$	5.51396i	0.191623i
$829$	52.9387	1.83864	0.919319	−	0.393514i	$-0.128741\pi$
$829$	52.9387	1.83864	0.919319	+	0.393514i	$0.128741\pi$
$830$	0	0
$831$	17.0279	0.590692
$832$	55.5389i	1.92547i
$833$	− 4.30359i	− 0.149111i
$834$	27.8649	0.964884
$835$	0	0
$836$	−10.5077	−0.363417
$837$	− 3.66152i	− 0.126561i
$838$	− 19.1344i	− 0.660985i
$839$	−9.10019	−0.314173	−0.157087	−	0.987585i	$-0.550210\pi$
$839$	−9.10019	−0.314173	−0.157087	+	0.987585i	$0.550210\pi$
$840$	0	0
$841$	1.61862	0.0558144
$842$	− 55.1468i	− 1.90049i
$843$	20.6072i	0.709749i
$844$	47.7827	1.64475
$845$	0	0
$846$	−1.24302	−0.0427359
$847$	− 16.4985i	− 0.566896i
$848$	10.3098i	0.354041i
$849$	−16.6700	−0.572113
$850$	0	0
$851$	22.0125	0.754578
$852$	14.5683i	0.499102i
$853$	11.2144i	0.383973i	0.981398	+	0.191986i	$0.0614930\pi$
$853$	11.2144i	0.383973i	−0.981398	+	0.191986i	$0.938507\pi$
$854$	−46.5544	−1.59306
$855$	0	0
$856$	15.0279	0.513644
$857$	− 30.2812i	− 1.03439i	−0.855869	−	0.517193i	$-0.826977\pi$
$857$	− 30.2812i	− 1.03439i	0.855869	−	0.517193i	$-0.173023\pi$
$858$	47.9861i	1.63822i
$859$	−13.2074	−0.450631	−0.225316	−	0.974286i	$-0.572341\pi$
$859$	−13.2074	−0.450631	−0.225316	+	0.974286i	$0.572341\pi$
$860$	0	0
$861$	−4.80507	−0.163756
$862$	− 3.72906i	− 0.127012i
$863$	4.62664i	0.157492i	0.996895	+	0.0787462i	$0.0250917\pi$
$863$	4.62664i	0.157492i	−0.996895	+	0.0787462i	$0.974908\pi$
$864$	−7.98680	−0.271716
$865$	0	0
$866$	−44.6366	−1.51681
$867$	− 1.00000i	− 0.0339618i
$868$	14.8679i	0.504649i
$869$	−15.0668	−0.511107
$870$	0	0
$871$	−69.1087	−2.34166
$872$	− 13.0279i	− 0.441181i
$873$	15.8913i	0.537840i
$874$	4.36792	0.147747
$875$	0	0
$876$	−19.1972	−0.648612
$877$	28.8609i	0.974564i	0.873245	+	0.487282i	$0.162012\pi$
$877$	28.8609i	0.974564i	−0.873245	+	0.487282i	$0.837988\pi$
$878$	71.3091i	2.40657i
$879$	−21.2772	−0.717662
$880$	0	0
$881$	−45.5195	−1.53359	−0.766795	−	0.641892i	$-0.778150\pi$
$881$	−45.5195	−1.53359	−0.766795	+	0.641892i	$0.778150\pi$
$882$	− 9.10170i	− 0.306470i
$883$	− 4.54189i	− 0.152847i	−0.997075	−	0.0764233i	$-0.975650\pi$
$883$	− 4.54189i	− 0.152847i	0.997075	−	0.0764233i	$-0.0243500\pi$
$884$	−12.2298	−0.411333
$885$	0	0
$886$	86.0808	2.89194
$887$	5.64903i	0.189676i	0.995493	+	0.0948380i	$0.0302333\pi$
$887$	5.64903i	0.189676i	−0.995493	+	0.0948380i	$0.969767\pi$
$888$	− 9.87189i	− 0.331279i
$889$	28.8231	0.966695
$890$	0	0
$891$	−4.58774	−0.153695
$892$	− 19.1491i	− 0.641158i
$893$	0.544378i	0.0182169i
$894$	16.8634	0.563997
$895$	0	0
$896$	12.7695	0.426598
$897$	− 11.0279i	− 0.368212i
$898$	− 42.5280i	− 1.41918i
$899$	−20.2607	−0.675732
$900$	0	0
$901$	−3.64207	−0.121335
$902$	28.3921i	0.945353i
$903$	− 13.8913i	− 0.462275i
$904$	15.5140	0.515987
$905$	0	0
$906$	32.8517	1.09143
$907$	− 20.2882i	− 0.673657i	−0.941566	−	0.336829i	$-0.890646\pi$
$907$	− 20.2882i	− 0.673657i	0.941566	−	0.336829i	$-0.109354\pi$
$908$	− 9.59175i	− 0.318313i
$909$	7.89134	0.261739
$910$	0	0
$911$	4.38433	0.145259	0.0726297	−	0.997359i	$-0.476861\pi$
$911$	4.38433	0.145259	0.0726297	+	0.997359i	$0.476861\pi$
$912$	2.62191i	0.0868201i
$913$	− 38.8106i	− 1.28444i
$914$	61.6476	2.03912
$915$	0	0
$916$	−7.03265	−0.232365
$917$	16.2423i	0.536368i
$918$	− 2.11491i	− 0.0698024i
$919$	−52.8984	−1.74496	−0.872478	−	0.488653i	$-0.837488\pi$
$919$	−52.8984	−1.74496	−0.872478	+	0.488653i	$0.837488\pi$
$920$	0	0
$921$	22.3510	0.736489
$922$	− 31.9208i	− 1.05125i
$923$	− 29.1366i	− 0.959043i
$924$	18.6289	0.612845
$925$	0	0
$926$	−79.5948	−2.61565
$927$	− 8.00000i	− 0.262754i
$928$	44.1942i	1.45075i
$929$	2.76171	0.0906087	0.0453043	−	0.998973i	$-0.485574\pi$
$929$	2.76171	0.0906087	0.0453043	+	0.998973i	$0.485574\pi$
$930$	0	0
$931$	−3.98608	−0.130638
$932$	37.8260i	1.23903i
$933$	− 11.4512i	− 0.374894i
$934$	−20.5661	−0.672942
$935$	0	0
$936$	−4.94567	−0.161654
$937$	13.7871i	0.450406i	0.974312	+	0.225203i	$0.0723046\pi$
$937$	13.7871i	0.450406i	−0.974312	+	0.225203i	$0.927695\pi$
$938$	48.5280i	1.58449i
$939$	−24.4791	−0.798844
$940$	0	0
$941$	−43.1616	−1.40703	−0.703513	−	0.710682i	$-0.748386\pi$
$941$	−43.1616	−1.40703	−0.703513	+	0.710682i	$0.748386\pi$
$942$	− 6.71585i	− 0.218814i
$943$	− 6.52493i	− 0.212481i
$944$	20.9626	0.682275
$945$	0	0
$946$	−82.0808	−2.66868
$947$	3.83549i	0.124637i	0.998056	+	0.0623183i	$0.0198494\pi$
$947$	3.83549i	0.124637i	−0.998056	+	0.0623183i	$0.980151\pi$
$948$	− 8.12115i	− 0.263763i
$949$	38.3943	1.24633
$950$	0	0
$951$	−26.5419	−0.860680
$952$	1.64207i	0.0532199i
$953$	− 32.8176i	− 1.06306i	−0.847038	−	0.531532i	$-0.821616\pi$
$953$	− 32.8176i	− 1.06306i	0.847038	−	0.531532i	$-0.178384\pi$
$954$	−7.70265	−0.249382
$955$	0	0
$956$	20.0823	0.649507
$957$	25.3859i	0.820608i
$958$	27.7827i	0.897617i
$959$	31.4761	1.01642
$960$	0	0
$961$	−17.5933	−0.567525
$962$	103.256i	3.32912i
$963$	− 15.0279i	− 0.484268i
$964$	−38.3635	−1.23560
$965$	0	0
$966$	−7.74378	−0.249152
$967$	− 10.2159i	− 0.328521i	−0.986417	−	0.164261i	$-0.947476\pi$
$967$	− 10.2159i	− 0.328521i	0.986417	−	0.164261i	$-0.0525238\pi$
$968$	− 10.0474i	− 0.322935i
$969$	−0.926221	−0.0297545
$970$	0	0
$971$	−45.2966	−1.45364	−0.726819	−	0.686829i	$-0.759002\pi$
$971$	−45.2966	−1.45364	−0.726819	+	0.686829i	$0.759002\pi$
$972$	− 2.47283i	− 0.0793162i
$973$	21.6351i	0.693590i
$974$	3.02792	0.0970210
$975$	0	0
$976$	37.9472	1.21466
$977$	− 38.4596i	− 1.23043i	−0.788358	−	0.615216i	$-0.789069\pi$
$977$	− 38.4596i	− 1.23043i	0.788358	−	0.615216i	$-0.210931\pi$
$978$	− 8.01472i	− 0.256283i
$979$	19.4053	0.620196
$980$	0	0
$981$	−13.0279	−0.415950
$982$	− 63.6476i	− 2.03108i
$983$	− 12.4163i	− 0.396017i	−0.980200	−	0.198009i	$-0.936553\pi$
$983$	− 12.4163i	− 0.396017i	0.980200	−	0.198009i	$-0.0634474\pi$
$984$	−2.92622	−0.0932845
$985$	0	0
$986$	−11.7026	−0.372688
$987$	− 0.965115i	− 0.0307199i
$988$	11.3275i	0.360376i
$989$	18.8634	0.599822
$990$	0	0
$991$	−28.8231	−0.915595	−0.457798	−	0.889056i	$-0.651362\pi$
$991$	−28.8231	−0.915595	−0.457798	+	0.889056i	$0.651362\pi$
$992$	− 29.2438i	− 0.928492i
$993$	− 15.2383i	− 0.483573i
$994$	−20.4596	−0.648940
$995$	0	0
$996$	20.9193	0.662852
$997$	30.6266i	0.969955i	0.874527	+	0.484978i	$0.161172\pi$
$997$	30.6266i	0.969955i	−0.874527	+	0.484978i	$0.838828\pi$
$998$	− 7.66152i	− 0.242521i
$999$	−9.87189	−0.312333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.2.b.i.1174.6		6
5.2	odd	4		255.2.a.c.1.1	✓	3
5.3	odd	4		1275.2.a.p.1.3		3
5.4	even	2	inner	1275.2.b.i.1174.1		6
15.2	even	4		765.2.a.j.1.3		3
15.8	even	4		3825.2.a.bb.1.1		3
20.7	even	4		4080.2.a.br.1.2		3
85.67	odd	4		4335.2.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.a.c.1.1	✓	3	5.2	odd	4
765.2.a.j.1.3		3	15.2	even	4
1275.2.a.p.1.3		3	5.3	odd	4
1275.2.b.i.1174.1		6	5.4	even	2	inner
1275.2.b.i.1174.6		6	1.1	even	1	trivial
3825.2.a.bb.1.1		3	15.8	even	4
4080.2.a.br.1.2		3	20.7	even	4
4335.2.a.s.1.1		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.11491i q^{2} +1.00000i q^{3} -2.47283 q^{4} -2.11491 q^{6} -1.64207i q^{7} -1.00000i q^{8} -1.00000 q^{9} -4.58774 q^{11} -2.47283i q^{12} +4.94567i q^{13} +3.47283 q^{14} -2.83076 q^{16} -1.00000i q^{17} -2.11491i q^{18} -0.926221 q^{19} +1.64207 q^{21} -9.70265i q^{22} +2.22982i q^{23} +1.00000 q^{24} -10.4596 q^{26} -1.00000i q^{27} +4.06058i q^{28} -5.53341 q^{29} +3.66152 q^{31} -7.98680i q^{32} -4.58774i q^{33} +2.11491 q^{34} +2.47283 q^{36} -9.87189i q^{37} -1.95887i q^{38} -4.94567 q^{39} -2.92622 q^{41} +3.47283i q^{42} -8.45963i q^{43} +11.3447 q^{44} -4.71585 q^{46} -0.587741i q^{47} -2.83076i q^{48} +4.30359 q^{49} +1.00000 q^{51} -12.2298i q^{52} -3.64207i q^{53} +2.11491 q^{54} -1.64207 q^{56} -0.926221i q^{57} -11.7026i q^{58} -7.40530 q^{59} -13.4053 q^{61} +7.74378i q^{62} +1.64207i q^{63} +11.2298 q^{64} +9.70265 q^{66} +13.9736i q^{67} +2.47283i q^{68} -2.22982 q^{69} -5.89134 q^{71} +1.00000i q^{72} -7.76322i q^{73} +20.8781 q^{74} +2.29039 q^{76} +7.53341i q^{77} -10.4596i q^{78} +3.28415 q^{79} +1.00000 q^{81} -6.18869i q^{82} +8.45963i q^{83} -4.06058 q^{84} +17.8913 q^{86} -5.53341i q^{87} +4.58774i q^{88} -4.22982 q^{89} +8.12115 q^{91} -5.51396i q^{92} +3.66152i q^{93} +1.24302 q^{94} +7.98680 q^{96} -15.8913i q^{97} +9.10170i q^{98} +4.58774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} - 6 q^{9} - 4 q^{11} + 10 q^{14} - 8 q^{16} + 8 q^{21} + 6 q^{24} - 12 q^{26} + 12 q^{29} + 4 q^{31} + 4 q^{36} - 8 q^{39} - 12 q^{41} + 30 q^{44} - 32 q^{46} + 6 q^{49} + 6 q^{51} - 8 q^{56}+ \cdots + 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^4 - 6 * q^9 - 4 * q^11 + 10 * q^14 - 8 * q^16 + 8 * q^21 + 6 * q^24 - 12 * q^26 + 12 * q^29 + 4 * q^31 + 4 * q^36 - 8 * q^39 - 12 * q^41 + 30 * q^44 - 32 * q^46 + 6 * q^49 + 6 * q^51 - 8 * q^56 + 28 * q^59 - 8 * q^61 + 42 * q^64 + 22 * q^66 + 12 * q^69 + 8 * q^71 + 42 * q^74 - 46 * q^76 + 16 * q^79 + 6 * q^81 + 10 * q^84 + 64 * q^86 - 20 * q^91 + 22 * q^94 + 8 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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