LMFDB - Embedded newform 765.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [765,2,Mod(1,765)] 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("765.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,2,-3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 255)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	765.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.11491 q^{2} +2.47283 q^{4} -1.00000 q^{5} +1.64207 q^{7} +1.00000 q^{8} -2.11491 q^{10} +4.58774 q^{11} +4.94567 q^{13} +3.47283 q^{14} -2.83076 q^{16} -1.00000 q^{17} +0.926221 q^{19} -2.47283 q^{20} +9.70265 q^{22} -2.22982 q^{23} +1.00000 q^{25} +10.4596 q^{26} +4.06058 q^{28} -5.53341 q^{29} +3.66152 q^{31} -7.98680 q^{32} -2.11491 q^{34} -1.64207 q^{35} +9.87189 q^{37} +1.95887 q^{38} -1.00000 q^{40} +2.92622 q^{41} -8.45963 q^{43} +11.3447 q^{44} -4.71585 q^{46} -0.587741 q^{47} -4.30359 q^{49} +2.11491 q^{50} +12.2298 q^{52} +3.64207 q^{53} -4.58774 q^{55} +1.64207 q^{56} -11.7026 q^{58} -7.40530 q^{59} -13.4053 q^{61} +7.74378 q^{62} -11.2298 q^{64} -4.94567 q^{65} -13.9736 q^{67} -2.47283 q^{68} -3.47283 q^{70} +5.89134 q^{71} -7.76322 q^{73} +20.8781 q^{74} +2.29039 q^{76} +7.53341 q^{77} -3.28415 q^{79} +2.83076 q^{80} +6.18869 q^{82} -8.45963 q^{83} +1.00000 q^{85} -17.8913 q^{86} +4.58774 q^{88} -4.22982 q^{89} +8.12115 q^{91} -5.51396 q^{92} -1.24302 q^{94} -0.926221 q^{95} +15.8913 q^{97} -9.10170 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} - 3 q^{5} + 4 q^{7} + 3 q^{8} + 2 q^{11} + 4 q^{13} + 5 q^{14} - 4 q^{16} - 3 q^{17} - 2 q^{20} + 11 q^{22} + 6 q^{23} + 3 q^{25} + 6 q^{26} - 5 q^{28} + 6 q^{29} + 2 q^{31} - 4 q^{32} - 4 q^{35}+ \cdots - q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 - 3 * q^5 + 4 * q^7 + 3 * q^8 + 2 * q^11 + 4 * q^13 + 5 * q^14 - 4 * q^16 - 3 * q^17 - 2 * q^20 + 11 * q^22 + 6 * q^23 + 3 * q^25 + 6 * q^26 - 5 * q^28 + 6 * q^29 + 2 * q^31 - 4 * q^32 - 4 * q^35 + 16 * q^37 + 15 * q^38 - 3 * q^40 + 6 * q^41 + 15 * q^44 - 16 * q^46 + 10 * q^47 - 3 * q^49 + 24 * q^52 + 10 * q^53 - 2 * q^55 + 4 * q^56 - 17 * q^58 + 14 * q^59 - 4 * q^61 - 4 * q^62 - 21 * q^64 - 4 * q^65 - 2 * q^67 - 2 * q^68 - 5 * q^70 - 4 * q^71 + 12 * q^73 + 21 * q^74 - 23 * q^76 - 8 * q^79 + 4 * q^80 + 15 * q^82 + 3 * q^85 - 32 * q^86 + 2 * q^88 - 10 * q^91 - 2 * q^92 - 11 * q^94 + 26 * q^97 - q^98

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 $

$2$	2.11491	1.49547	0.747733	−	0.664000i	$-0.231142\pi$
$2$	2.11491	1.49547	0.747733	+	0.664000i	$0.231142\pi$
$3$	0	0
$3$	0	0
$4$	2.47283	1.23642
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.64207	0.620645	0.310323	−	0.950631i	$-0.399563\pi$
$7$	1.64207	0.620645	0.310323	+	0.950631i	$0.399563\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.11491	−0.668792
$11$	4.58774	1.38326	0.691628	−	0.722254i	$-0.256894\pi$
$11$	4.58774	1.38326	0.691628	+	0.722254i	$0.256894\pi$
$12$	0	0
$13$	4.94567	1.37168	0.685841	−	0.727752i	$-0.259435\pi$
$13$	4.94567	1.37168	0.685841	+	0.727752i	$0.259435\pi$
$14$	3.47283	0.928154
$15$	0	0
$16$	−2.83076	−0.707690
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.926221	0.212490	0.106245	−	0.994340i	$-0.466117\pi$
$19$	0.926221	0.212490	0.106245	+	0.994340i	$0.466117\pi$
$20$	−2.47283	−0.552942
$21$	0	0
$22$	9.70265	2.06861
$23$	−2.22982	−0.464949	−0.232474	−	0.972603i	$-0.574682\pi$
$23$	−2.22982	−0.464949	−0.232474	+	0.972603i	$0.574682\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	10.4596	2.05130
$27$	0	0
$28$	4.06058	0.767377
$29$	−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	$0.393351\pi$
$32$	−7.98680	−1.41188
$33$	0	0
$34$	−2.11491	−0.362704
$35$	−1.64207	−0.277561
$36$	0	0
$37$	9.87189	1.62293	0.811464	−	0.584402i	$-0.198671\pi$
$37$	9.87189	1.62293	0.811464	+	0.584402i	$0.198671\pi$
$38$	1.95887	0.317771
$39$	0	0
$40$	−1.00000	−0.158114
$41$	2.92622	0.456999	0.228499	−	0.973544i	$-0.426618\pi$
$41$	2.92622	0.456999	0.228499	+	0.973544i	$0.426618\pi$
$42$	0	0
$43$	−8.45963	−1.29008	−0.645041	−	0.764148i	$-0.723160\pi$
$43$	−8.45963	−1.29008	−0.645041	+	0.764148i	$0.723160\pi$
$44$	11.3447	1.71028
$45$	0	0
$46$	−4.71585	−0.695315
$47$	−0.587741	−0.0857309	−0.0428655	−	0.999081i	$-0.513649\pi$
$47$	−0.587741	−0.0857309	−0.0428655	+	0.999081i	$0.513649\pi$
$48$	0	0
$49$	−4.30359	−0.614799
$50$	2.11491	0.299093
$51$	0	0
$52$	12.2298	1.69597
$53$	3.64207	0.500277	0.250139	−	0.968210i	$-0.419524\pi$
$53$	3.64207	0.500277	0.250139	+	0.968210i	$0.419524\pi$
$54$	0	0
$55$	−4.58774	−0.618611
$56$	1.64207	0.219431
$57$	0	0
$58$	−11.7026	−1.53663
$59$	−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	$0.828409\pi$
$62$	7.74378	0.983461
$63$	0	0
$64$	−11.2298	−1.40373
$65$	−4.94567	−0.613435
$66$	0	0
$67$	−13.9736	−1.70715	−0.853573	−	0.520973i	$-0.825569\pi$
$67$	−13.9736	−1.70715	−0.853573	+	0.520973i	$0.825569\pi$
$68$	−2.47283	−0.299875
$69$	0	0
$70$	−3.47283	−0.415083
$71$	5.89134	0.699173	0.349586	−	0.936904i	$-0.386322\pi$
$71$	5.89134	0.699173	0.349586	+	0.936904i	$0.386322\pi$
$72$	0	0
$73$	−7.76322	−0.908617	−0.454308	−	0.890845i	$-0.650114\pi$
$73$	−7.76322	−0.908617	−0.454308	+	0.890845i	$0.650114\pi$
$74$	20.8781	2.42703
$75$	0	0
$76$	2.29039	0.262726
$77$	7.53341	0.858512
$78$	0	0
$79$	−3.28415	−0.369495	−0.184748	−	0.982786i	$-0.559147\pi$
$79$	−3.28415	−0.369495	−0.184748	+	0.982786i	$0.559147\pi$
$80$	2.83076	0.316489
$81$	0	0
$82$	6.18869	0.683426
$83$	−8.45963	−0.928565	−0.464283	−	0.885687i	$-0.653688\pi$
$83$	−8.45963	−0.928565	−0.464283	+	0.885687i	$0.653688\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−17.8913	−1.92927
$87$	0	0
$88$	4.58774	0.489055
$89$	−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
$92$	−5.51396	−0.574870
$93$	0	0
$94$	−1.24302	−0.128208
$95$	−0.926221	−0.0950283
$96$	0	0
$97$	15.8913	1.61352	0.806760	−	0.590879i	$-0.201219\pi$
$97$	15.8913	1.61352	0.806760	+	0.590879i	$0.201219\pi$
$98$	−9.10170	−0.919411
$99$	0	0
$100$	2.47283	0.247283
$101$	7.89134	0.785217	0.392609	−	0.919706i	$-0.371573\pi$
$101$	7.89134	0.785217	0.392609	+	0.919706i	$0.371573\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	4.94567	0.484963
$105$	0	0
$106$	7.70265	0.748147
$107$	15.0279	1.45280	0.726402	−	0.687270i	$-0.241191\pi$
$107$	15.0279	1.45280	0.726402	+	0.687270i	$0.241191\pi$
$108$	0	0
$109$	−13.0279	−1.24785	−0.623924	−	0.781485i	$-0.714463\pi$
$109$	−13.0279	−1.24785	−0.623924	+	0.781485i	$0.714463\pi$
$110$	−9.70265	−0.925111
$111$	0	0
$112$	−4.64832	−0.439225
$113$	−15.5140	−1.45943	−0.729715	−	0.683751i	$-0.760347\pi$
$113$	−15.5140	−1.45943	−0.729715	+	0.683751i	$0.760347\pi$
$114$	0	0
$115$	2.22982	0.207931
$116$	−13.6832	−1.27045
$117$	0	0
$118$	−15.6615	−1.44176
$119$	−1.64207	−0.150529
$120$	0	0
$121$	10.0474	0.913397
$122$	−28.3510	−2.56678
$123$	0	0
$124$	9.05433	0.813103
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.5529	−1.55756	−0.778782	−	0.627295i	$-0.784162\pi$
$127$	−17.5529	−1.55756	−0.778782	+	0.627295i	$0.784162\pi$
$128$	−7.77643	−0.687346
$129$	0	0
$130$	−10.4596	−0.917370
$131$	9.89134	0.864210	0.432105	−	0.901823i	$-0.357771\pi$
$131$	9.89134	0.864210	0.432105	+	0.901823i	$0.357771\pi$
$132$	0	0
$133$	1.52092	0.131881
$134$	−29.5529	−2.55298
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	19.1685	1.63768	0.818839	−	0.574024i	$-0.194618\pi$
$137$	19.1685	1.63768	0.818839	+	0.574024i	$0.194618\pi$
$138$	0	0
$139$	13.1755	1.11753	0.558765	−	0.829326i	$-0.311275\pi$
$139$	13.1755	1.11753	0.558765	+	0.829326i	$0.311275\pi$
$140$	−4.06058	−0.343181
$141$	0	0
$142$	12.4596	1.04559
$143$	22.6894	1.89739
$144$	0	0
$145$	5.53341	0.459525
$146$	−16.4185	−1.35880
$147$	0	0
$148$	24.4115	2.00662
$149$	−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	$0.717784\pi$
$152$	0.926221	0.0751264
$153$	0	0
$154$	15.9325	1.28387
$155$	−3.66152	−0.294100
$156$	0	0
$157$	−3.17548	−0.253431	−0.126716	−	0.991939i	$-0.540444\pi$
$157$	−3.17548	−0.253431	−0.126716	+	0.991939i	$0.540444\pi$
$158$	−6.94567	−0.552568
$159$	0	0
$160$	7.98680	0.631412
$161$	−3.66152	−0.288568
$162$	0	0
$163$	3.78963	0.296827	0.148413	−	0.988925i	$-0.452583\pi$
$163$	3.78963	0.296827	0.148413	+	0.988925i	$0.452583\pi$
$164$	7.23606	0.565041
$165$	0	0
$166$	−17.8913	−1.38864
$167$	18.6072	1.43987	0.719934	−	0.694043i	$-0.244172\pi$
$167$	18.6072	1.43987	0.719934	+	0.694043i	$0.244172\pi$
$168$	0	0
$169$	11.4596	0.881510
$170$	2.11491	0.162206
$171$	0	0
$172$	−20.9193	−1.59508
$173$	−3.77018	−0.286642	−0.143321	−	0.989676i	$-0.545778\pi$
$173$	−3.77018	−0.286642	−0.143321	+	0.989676i	$0.545778\pi$
$174$	0	0
$175$	1.64207	0.124129
$176$	−12.9868	−0.978917
$177$	0	0
$178$	−8.94567	−0.670506
$179$	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	$0.483368\pi$
$182$	17.1755	1.27313
$183$	0	0
$184$	−2.22982	−0.164384
$185$	−9.87189	−0.725796
$186$	0	0
$187$	−4.58774	−0.335489
$188$	−1.45339	−0.105999
$189$	0	0
$190$	−1.95887	−0.142111
$191$	−18.7283	−1.35514	−0.677568	−	0.735461i	$-0.736966\pi$
$191$	−18.7283	−1.35514	−0.677568	+	0.735461i	$0.736966\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	33.6087	2.41296
$195$	0	0
$196$	−10.6421	−0.760148
$197$	−21.0668	−1.50095	−0.750474	−	0.660900i	$-0.770175\pi$
$197$	−21.0668	−1.50095	−0.750474	+	0.660900i	$0.770175\pi$
$198$	0	0
$199$	9.17548	0.650433	0.325216	−	0.945640i	$-0.394563\pi$
$199$	9.17548	0.650433	0.325216	+	0.945640i	$0.394563\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	16.6894	1.17427
$203$	−9.08627	−0.637731
$204$	0	0
$205$	−2.92622	−0.204376
$206$	16.9193	1.17882
$207$	0	0
$208$	−14.0000	−0.970725
$209$	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	$0.731622\pi$
$212$	9.00624	0.618551
$213$	0	0
$214$	31.7827	2.17262
$215$	8.45963	0.576942
$216$	0	0
$217$	6.01249	0.408154
$218$	−27.5529	−1.86611
$219$	0	0
$220$	−11.3447	−0.764861
$221$	−4.94567	−0.332682
$222$	0	0
$223$	7.74378	0.518562	0.259281	−	0.965802i	$-0.416514\pi$
$223$	7.74378	0.518562	0.259281	+	0.965802i	$0.416514\pi$
$224$	−13.1149	−0.876277
$225$	0	0
$226$	−32.8106	−2.18253
$227$	3.87885	0.257448	0.128724	−	0.991680i	$-0.458912\pi$
$227$	3.87885	0.257448	0.128724	+	0.991680i	$0.458912\pi$
$228$	0	0
$229$	−2.84396	−0.187934	−0.0939672	−	0.995575i	$-0.529955\pi$
$229$	−2.84396	−0.187934	−0.0939672	+	0.995575i	$0.529955\pi$
$230$	4.71585	0.310954
$231$	0	0
$232$	−5.53341	−0.363286
$233$	15.2966	1.00212	0.501058	−	0.865414i	$-0.332944\pi$
$233$	15.2966	1.00212	0.501058	+	0.865414i	$0.332944\pi$
$234$	0	0
$235$	0.587741	0.0383400
$236$	−18.3121	−1.19201
$237$	0	0
$238$	−3.47283	−0.225110
$239$	−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	$0.333454\pi$
$242$	21.2493	1.36595
$243$	0	0
$244$	−33.1491	−2.12215
$245$	4.30359	0.274947
$246$	0	0
$247$	4.58078	0.291468
$248$	3.66152	0.232507
$249$	0	0
$250$	−2.11491	−0.133758
$251$	20.3774	1.28621	0.643104	−	0.765779i	$-0.277646\pi$
$251$	20.3774	1.28621	0.643104	+	0.765779i	$0.277646\pi$
$252$	0	0
$253$	−10.2298	−0.643143
$254$	−37.1227	−2.32928
$255$	0	0
$256$	6.01320	0.375825
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	16.2104	1.00726
$260$	−12.2298	−0.758461
$261$	0	0
$262$	20.9193	1.29240
$263$	15.9108	0.981101	0.490550	−	0.871413i	$-0.336796\pi$
$263$	15.9108	0.981101	0.490550	+	0.871413i	$0.336796\pi$
$264$	0	0
$265$	−3.64207	−0.223731
$266$	3.21661	0.197223
$267$	0	0
$268$	−34.5544	−2.11074
$269$	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	$0.520400\pi$
$272$	2.83076	0.171640
$273$	0	0
$274$	40.5397	2.44909
$275$	4.58774	0.276651
$276$	0	0
$277$	17.0279	1.02311	0.511554	−	0.859251i	$-0.329070\pi$
$277$	17.0279	1.02311	0.511554	+	0.859251i	$0.329070\pi$
$278$	27.8649	1.67123
$279$	0	0
$280$	−1.64207	−0.0981327
$281$	−20.6072	−1.22932	−0.614661	−	0.788791i	$-0.710707\pi$
$281$	−20.6072	−1.22932	−0.614661	+	0.788791i	$0.710707\pi$
$282$	0	0
$283$	16.6700	0.990929	0.495464	−	0.868628i	$-0.334998\pi$
$283$	16.6700	0.990929	0.495464	+	0.868628i	$0.334998\pi$
$284$	14.5683	0.864469
$285$	0	0
$286$	47.9861	2.83748
$287$	4.80507	0.283634
$288$	0	0
$289$	1.00000	0.0588235
$290$	11.7026	0.687203
$291$	0	0
$292$	−19.1972	−1.12343
$293$	−21.2772	−1.24303	−0.621513	−	0.783404i	$-0.713482\pi$
$293$	−21.2772	−1.24303	−0.621513	+	0.783404i	$0.713482\pi$
$294$	0	0
$295$	7.40530	0.431153
$296$	9.87189	0.573792
$297$	0	0
$298$	−16.8634	−0.976871
$299$	−11.0279	−0.637761
$300$	0	0
$301$	−13.8913	−0.800683
$302$	−32.8517	−1.89040
$303$	0	0
$304$	−2.62191	−0.150377
$305$	13.4053	0.767585
$306$	0	0
$307$	22.3510	1.27564	0.637818	−	0.770187i	$-0.279837\pi$
$307$	22.3510	1.27564	0.637818	+	0.770187i	$0.279837\pi$
$308$	18.6289	1.06148
$309$	0	0
$310$	−7.74378	−0.439817
$311$	11.4512	0.649335	0.324668	−	0.945828i	$-0.394748\pi$
$311$	11.4512	0.649335	0.324668	+	0.945828i	$0.394748\pi$
$312$	0	0
$313$	24.4791	1.38364	0.691820	−	0.722070i	$-0.256809\pi$
$313$	24.4791	1.38364	0.691820	+	0.722070i	$0.256809\pi$
$314$	−6.71585	−0.378997
$315$	0	0
$316$	−8.12115	−0.456850
$317$	26.5419	1.49074	0.745370	−	0.666651i	$-0.232273\pi$
$317$	26.5419	1.49074	0.745370	+	0.666651i	$0.232273\pi$
$318$	0	0
$319$	−25.3859	−1.42133
$320$	11.2298	0.627766
$321$	0	0
$322$	−7.74378	−0.431544
$323$	−0.926221	−0.0515363
$324$	0	0
$325$	4.94567	0.274336
$326$	8.01472	0.443894
$327$	0	0
$328$	2.92622	0.161574
$329$	−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	$0.637544\pi$
$332$	−20.9193	−1.14809
$333$	0	0
$334$	39.3525	2.15327
$335$	13.9736	0.763459
$336$	0	0
$337$	−22.5738	−1.22967	−0.614837	−	0.788654i	$-0.710778\pi$
$337$	−22.5738	−1.22967	−0.614837	+	0.788654i	$0.710778\pi$
$338$	24.2361	1.31827
$339$	0	0
$340$	2.47283	0.134108
$341$	16.7981	0.909669
$342$	0	0
$343$	−18.5613	−1.00222
$344$	−8.45963	−0.456113
$345$	0	0
$346$	−7.97359	−0.428663
$347$	−16.3246	−0.876348	−0.438174	−	0.898890i	$-0.644375\pi$
$347$	−16.3246	−0.876348	−0.438174	+	0.898890i	$0.644375\pi$
$348$	0	0
$349$	−26.3315	−1.40949	−0.704747	−	0.709459i	$-0.748939\pi$
$349$	−26.3315	−1.40949	−0.704747	+	0.709459i	$0.748939\pi$
$350$	3.47283	0.185631
$351$	0	0
$352$	−36.6414	−1.95299
$353$	−29.9930	−1.59637	−0.798184	−	0.602413i	$-0.794206\pi$
$353$	−29.9930	−1.59637	−0.798184	+	0.602413i	$0.794206\pi$
$354$	0	0
$355$	−5.89134	−0.312680
$356$	−10.4596	−0.554359
$357$	0	0
$358$	46.7283	2.46967
$359$	−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
$362$	2.97208	0.156209
$363$	0	0
$364$	20.0823	1.05260
$365$	7.76322	0.406346
$366$	0	0
$367$	24.9193	1.30077	0.650387	−	0.759603i	$-0.274607\pi$
$367$	24.9193	1.30077	0.650387	+	0.759603i	$0.274607\pi$
$368$	6.31207	0.329039
$369$	0	0
$370$	−20.8781	−1.08540
$371$	5.98055	0.310495
$372$	0	0
$373$	−2.25622	−0.116823	−0.0584114	−	0.998293i	$-0.518604\pi$
$373$	−2.25622	−0.116823	−0.0584114	+	0.998293i	$0.518604\pi$
$374$	−9.70265	−0.501712
$375$	0	0
$376$	−0.587741	−0.0303105
$377$	−27.3664	−1.40944
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−2.29039	−0.117495
$381$	0	0
$382$	−39.6087	−2.02656
$383$	13.9666	0.713662	0.356831	−	0.934169i	$-0.383857\pi$
$383$	13.9666	0.713662	0.356831	+	0.934169i	$0.383857\pi$
$384$	0	0
$385$	−7.53341	−0.383938
$386$	4.22982	0.215292
$387$	0	0
$388$	39.2966	1.99498
$389$	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
$392$	−4.30359	−0.217364
$393$	0	0
$394$	−44.5544	−2.24462
$395$	3.28415	0.165243
$396$	0	0
$397$	19.4681	0.977076	0.488538	−	0.872543i	$-0.337530\pi$
$397$	19.4681	0.977076	0.488538	+	0.872543i	$0.337530\pi$
$398$	19.4053	0.972700
$399$	0	0
$400$	−2.83076	−0.141538
$401$	−34.8734	−1.74149	−0.870747	−	0.491731i	$-0.836364\pi$
$401$	−34.8734	−1.74149	−0.870747	+	0.491731i	$0.836364\pi$
$402$	0	0
$403$	18.1087	0.902057
$404$	19.5140	0.970856
$405$	0	0
$406$	−19.2166	−0.953704
$407$	45.2897	2.24493
$408$	0	0
$409$	−29.9108	−1.47899	−0.739497	−	0.673160i	$-0.764936\pi$
$409$	−29.9108	−1.47899	−0.739497	+	0.673160i	$0.764936\pi$
$410$	−6.18869	−0.305637
$411$	0	0
$412$	19.7827	0.974622
$413$	−12.1600	−0.598357
$414$	0	0
$415$	8.45963	0.415267
$416$	−39.5000	−1.93665
$417$	0	0
$418$	8.98680	0.439559
$419$	−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	$0.719171\pi$
$422$	−40.8664	−1.98935
$423$	0	0
$424$	3.64207	0.176875
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−22.0125	−1.06526
$428$	37.1616	1.79627
$429$	0	0
$430$	17.8913	0.862797
$431$	1.76322	0.0849315	0.0424658	−	0.999098i	$-0.486479\pi$
$431$	1.76322	0.0849315	0.0424658	+	0.999098i	$0.486479\pi$
$432$	0	0
$433$	21.1057	1.01428	0.507138	−	0.861865i	$-0.330703\pi$
$433$	21.1057	1.01428	0.507138	+	0.861865i	$0.330703\pi$
$434$	12.7159	0.610380
$435$	0	0
$436$	−32.2159	−1.54286
$437$	−2.06530	−0.0987968
$438$	0	0
$439$	−33.7174	−1.60924	−0.804621	−	0.593789i	$-0.797632\pi$
$439$	−33.7174	−1.60924	−0.804621	+	0.593789i	$0.797632\pi$
$440$	−4.58774	−0.218712
$441$	0	0
$442$	−10.4596	−0.497514
$443$	40.7019	1.93381	0.966904	−	0.255142i	$-0.0821223\pi$
$443$	40.7019	1.93381	0.966904	+	0.255142i	$0.0821223\pi$
$444$	0	0
$445$	4.22982	0.200512
$446$	16.3774	0.775491
$447$	0	0
$448$	−18.4402	−0.871217
$449$	−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
$452$	−38.3635	−1.80447
$453$	0	0
$454$	8.20341	0.385005
$455$	−8.12115	−0.380725
$456$	0	0
$457$	29.1491	1.36354	0.681768	−	0.731568i	$-0.261211\pi$
$457$	29.1491	1.36354	0.681768	+	0.731568i	$0.261211\pi$
$458$	−6.01472	−0.281049
$459$	0	0
$460$	5.51396	0.257090
$461$	15.0932	0.702962	0.351481	−	0.936195i	$-0.385678\pi$
$461$	15.0932	0.702962	0.351481	+	0.936195i	$0.385678\pi$
$462$	0	0
$463$	37.6351	1.74905	0.874526	−	0.484979i	$-0.161173\pi$
$463$	37.6351	1.74905	0.874526	+	0.484979i	$0.161173\pi$
$464$	15.6638	0.727172
$465$	0	0
$466$	32.3510	1.49863
$467$	9.72433	0.449988	0.224994	−	0.974360i	$-0.427764\pi$
$467$	9.72433	0.449988	0.224994	+	0.974360i	$0.427764\pi$
$468$	0	0
$469$	−22.9457	−1.05953
$470$	1.24302	0.0573362
$471$	0	0
$472$	−7.40530	−0.340856
$473$	−38.8106	−1.78451
$474$	0	0
$475$	0.926221	0.0424979
$476$	−4.06058	−0.186116
$477$	0	0
$478$	−17.1755	−0.785588
$479$	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
$482$	32.8106	1.49448
$483$	0	0
$484$	24.8455	1.12934
$485$	−15.8913	−0.721588
$486$	0	0
$487$	1.43171	0.0648768	0.0324384	−	0.999474i	$-0.489673\pi$
$487$	1.43171	0.0648768	0.0324384	+	0.999474i	$0.489673\pi$
$488$	−13.4053	−0.606829
$489$	0	0
$490$	9.10170	0.411173
$491$	30.0947	1.35816	0.679078	−	0.734066i	$-0.262380\pi$
$491$	30.0947	1.35816	0.679078	+	0.734066i	$0.262380\pi$
$492$	0	0
$493$	5.53341	0.249212
$494$	9.68793	0.435880
$495$	0	0
$496$	−10.3649	−0.465397
$497$	9.67401	0.433939
$498$	0	0
$499$	3.62263	0.162171	0.0810855	−	0.996707i	$-0.474161\pi$
$499$	3.62263	0.162171	0.0810855	+	0.996707i	$0.474161\pi$
$500$	−2.47283	−0.110588
$501$	0	0
$502$	43.0963	1.92348
$503$	−22.5544	−1.00565	−0.502825	−	0.864388i	$-0.667706\pi$
$503$	−22.5544	−1.00565	−0.502825	+	0.864388i	$0.667706\pi$
$504$	0	0
$505$	−7.89134	−0.351160
$506$	−21.6351	−0.961798
$507$	0	0
$508$	−43.4053	−1.92580
$509$	−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
$512$	28.2702	1.24938
$513$	0	0
$514$	−4.22982	−0.186569
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−2.69641	−0.118588
$518$	34.2834	1.50633
$519$	0	0
$520$	−4.94567	−0.216882
$521$	−26.8734	−1.17735	−0.588673	−	0.808372i	$-0.700349\pi$
$521$	−26.8734	−1.17735	−0.588673	+	0.808372i	$0.700349\pi$
$522$	0	0
$523$	41.7996	1.82777	0.913885	−	0.405973i	$-0.133067\pi$
$523$	41.7996	1.82777	0.913885	+	0.405973i	$0.133067\pi$
$524$	24.4596	1.06852
$525$	0	0
$526$	33.6498	1.46720
$527$	−3.66152	−0.159498
$528$	0	0
$529$	−18.0279	−0.783823
$530$	−7.70265	−0.334582
$531$	0	0
$532$	3.76099	0.163060
$533$	14.4721	0.626857
$534$	0	0
$535$	−15.0279	−0.649714
$536$	−13.9736	−0.603567
$537$	0	0
$538$	10.7306	0.462628
$539$	−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	$0.284784\pi$
$542$	−4.45963	−0.191558
$543$	0	0
$544$	7.98680	0.342431
$545$	13.0279	0.558055
$546$	0	0
$547$	13.8844	0.593653	0.296827	−	0.954931i	$-0.404072\pi$
$547$	13.8844	0.593653	0.296827	+	0.954931i	$0.404072\pi$
$548$	47.4006	2.02485
$549$	0	0
$550$	9.70265	0.413722
$551$	−5.12516	−0.218339
$552$	0	0
$553$	−5.39281	−0.229326
$554$	36.0125	1.53002
$555$	0	0
$556$	32.5808	1.38173
$557$	−0.108664	−0.00460426	−0.00230213	−	0.999997i	$-0.500733\pi$
$557$	−0.108664	−0.00460426	−0.00230213	+	0.999997i	$0.500733\pi$
$558$	0	0
$559$	−41.8385	−1.76958
$560$	4.64832	0.196427
$561$	0	0
$562$	−43.5823	−1.83841
$563$	8.54885	0.360291	0.180145	−	0.983640i	$-0.442343\pi$
$563$	8.54885	0.360291	0.180145	+	0.983640i	$0.442343\pi$
$564$	0	0
$565$	15.5140	0.652677
$566$	35.2555	1.48190
$567$	0	0
$568$	5.89134	0.247195
$569$	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	$0.562053\pi$
$572$	56.1072	2.34596
$573$	0	0
$574$	10.1623	0.424165
$575$	−2.22982	−0.0929897
$576$	0	0
$577$	−21.2313	−0.883872	−0.441936	−	0.897047i	$-0.645708\pi$
$577$	−21.2313	−0.883872	−0.441936	+	0.897047i	$0.645708\pi$
$578$	2.11491	0.0879686
$579$	0	0
$580$	13.6832	0.568164
$581$	−13.8913	−0.576310
$582$	0	0
$583$	16.7089	0.692012
$584$	−7.76322	−0.321245
$585$	0	0
$586$	−44.9993	−1.85890
$587$	14.2617	0.588645	0.294323	−	0.955706i	$-0.404906\pi$
$587$	14.2617	0.588645	0.294323	+	0.955706i	$0.404906\pi$
$588$	0	0
$589$	3.39138	0.139739
$590$	15.6615	0.644775
$591$	0	0
$592$	−27.9450	−1.14853
$593$	42.4138	1.74173	0.870863	−	0.491527i	$-0.163561\pi$
$593$	42.4138	1.74173	0.870863	+	0.491527i	$0.163561\pi$
$594$	0	0
$595$	1.64207	0.0673185
$596$	−19.7174	−0.807655
$597$	0	0
$598$	−23.3230	−0.953750
$599$	−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	$0.437121\pi$
$602$	−29.3789	−1.19739
$603$	0	0
$604$	−38.4115	−1.56294
$605$	−10.0474	−0.408484
$606$	0	0
$607$	−30.7647	−1.24870	−0.624351	−	0.781144i	$-0.714637\pi$
$607$	−30.7647	−1.24870	−0.624351	+	0.781144i	$0.714637\pi$
$608$	−7.39754	−0.300010
$609$	0	0
$610$	28.3510	1.14790
$611$	−2.90677	−0.117595
$612$	0	0
$613$	−0.147558	−0.00595982	−0.00297991	−	0.999996i	$-0.500949\pi$
$613$	−0.147558	−0.00595982	−0.00297991	+	0.999996i	$0.500949\pi$
$614$	47.2702	1.90767
$615$	0	0
$616$	7.53341	0.303530
$617$	−25.6226	−1.03153	−0.515764	−	0.856731i	$-0.672492\pi$
$617$	−25.6226	−1.03153	−0.515764	+	0.856731i	$0.672492\pi$
$618$	0	0
$619$	−18.5544	−0.745763	−0.372882	−	0.927879i	$-0.621630\pi$
$619$	−18.5544	−0.745763	−0.372882	+	0.927879i	$0.621630\pi$
$620$	−9.05433	−0.363631
$621$	0	0
$622$	24.2181	0.971058
$623$	−6.94567	−0.278272
$624$	0	0
$625$	1.00000	0.0400000
$626$	51.7710	2.06918
$627$	0	0
$628$	−7.85244	−0.313347
$629$	−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	$0.189141\pi$
$632$	−3.28415	−0.130636
$633$	0	0
$634$	56.1336	2.22935
$635$	17.5529	0.696564
$636$	0	0
$637$	−21.2841	−0.843309
$638$	−53.6887	−2.12556
$639$	0	0
$640$	7.77643	0.307390
$641$	−13.3300	−0.526503	−0.263252	−	0.964727i	$-0.584795\pi$
$641$	−13.3300	−0.526503	−0.263252	+	0.964727i	$0.584795\pi$
$642$	0	0
$643$	−25.1685	−0.992550	−0.496275	−	0.868165i	$-0.665299\pi$
$643$	−25.1685	−0.992550	−0.496275	+	0.868165i	$0.665299\pi$
$644$	−9.05433	−0.356791
$645$	0	0
$646$	−1.95887	−0.0770708
$647$	−6.40378	−0.251759	−0.125879	−	0.992046i	$-0.540175\pi$
$647$	−6.40378	−0.251759	−0.125879	+	0.992046i	$0.540175\pi$
$648$	0	0
$649$	−33.9736	−1.33358
$650$	10.4596	0.410260
$651$	0	0
$652$	9.37113	0.367002
$653$	48.8884	1.91315	0.956575	−	0.291486i	$-0.0941496\pi$
$653$	48.8884	1.91315	0.956575	+	0.291486i	$0.0941496\pi$
$654$	0	0
$655$	−9.89134	−0.386486
$656$	−8.28343	−0.323414
$657$	0	0
$658$	−2.04113	−0.0795715
$659$	−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	$0.495207\pi$
$662$	−32.2276	−1.25256
$663$	0	0
$664$	−8.45963	−0.328297
$665$	−1.52092	−0.0589789
$666$	0	0
$667$	12.3385	0.477748
$668$	46.0125	1.78028
$669$	0	0
$670$	29.5529	1.14173
$671$	−61.5000	−2.37418
$672$	0	0
$673$	−19.2144	−0.740660	−0.370330	−	0.928900i	$-0.620755\pi$
$673$	−19.2144	−0.740660	−0.370330	+	0.928900i	$0.620755\pi$
$674$	−47.7415	−1.83894
$675$	0	0
$676$	28.3378	1.08991
$677$	−27.2144	−1.04593	−0.522967	−	0.852353i	$-0.675175\pi$
$677$	−27.2144	−1.04593	−0.522967	+	0.852353i	$0.675175\pi$
$678$	0	0
$679$	26.0947	1.00142
$680$	1.00000	0.0383482
$681$	0	0
$682$	35.5264	1.36038
$683$	−23.2702	−0.890410	−0.445205	−	0.895429i	$-0.646869\pi$
$683$	−23.2702	−0.890410	−0.445205	+	0.895429i	$0.646869\pi$
$684$	0	0
$685$	−19.1685	−0.732392
$686$	−39.2555	−1.49878
$687$	0	0
$688$	23.9472	0.912978
$689$	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	$0.309097\pi$
$692$	−9.32304	−0.354409
$693$	0	0
$694$	−34.5249	−1.31055
$695$	−13.1755	−0.499775
$696$	0	0
$697$	−2.92622	−0.110839
$698$	−55.6887	−2.10785
$699$	0	0
$700$	4.06058	0.153475
$701$	23.7702	0.897787	0.448894	−	0.893585i	$-0.351818\pi$
$701$	23.7702	0.897787	0.448894	+	0.893585i	$0.351818\pi$
$702$	0	0
$703$	9.14355	0.344856
$704$	−51.5195	−1.94171
$705$	0	0
$706$	−63.4325	−2.38731
$707$	12.9582	0.487342
$708$	0	0
$709$	−42.2423	−1.58644	−0.793221	−	0.608933i	$-0.791598\pi$
$709$	−42.2423	−1.58644	−0.793221	+	0.608933i	$0.791598\pi$
$710$	−12.4596	−0.467602
$711$	0	0
$712$	−4.22982	−0.158519
$713$	−8.16451	−0.305763
$714$	0	0
$715$	−22.6894	−0.848537
$716$	54.6366	2.04187
$717$	0	0
$718$	−54.9317	−2.05003
$719$	−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
$722$	−38.3689	−1.42794
$723$	0	0
$724$	3.47507	0.129150
$725$	−5.53341	−0.205506
$726$	0	0
$727$	−7.57926	−0.281099	−0.140550	−	0.990074i	$-0.544887\pi$
$727$	−7.57926	−0.281099	−0.140550	+	0.990074i	$0.544887\pi$
$728$	8.12115	0.300990
$729$	0	0
$730$	16.4185	0.607676
$731$	8.45963	0.312891
$732$	0	0
$733$	−33.1102	−1.22295	−0.611476	−	0.791263i	$-0.709424\pi$
$733$	−33.1102	−1.22295	−0.611476	+	0.791263i	$0.709424\pi$
$734$	52.7019	1.94526
$735$	0	0
$736$	17.8091	0.656451
$737$	−64.1072	−2.36142
$738$	0	0
$739$	22.8634	0.841044	0.420522	−	0.907282i	$-0.361847\pi$
$739$	22.8634	0.841044	0.420522	+	0.907282i	$0.361847\pi$
$740$	−24.4115	−0.897386
$741$	0	0
$742$	12.6483	0.464334
$743$	−47.7438	−1.75155	−0.875775	−	0.482720i	$-0.839649\pi$
$743$	−47.7438	−1.75155	−0.875775	+	0.482720i	$0.839649\pi$
$744$	0	0
$745$	7.97359	0.292130
$746$	−4.77170	−0.174704
$747$	0	0
$748$	−11.3447	−0.414804
$749$	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	$0.457089\pi$
$752$	1.66376	0.0606709
$753$	0	0
$754$	−57.8774	−2.10777
$755$	15.5334	0.565319
$756$	0	0
$757$	−2.25622	−0.0820038	−0.0410019	−	0.999159i	$-0.513055\pi$
$757$	−2.25622	−0.0820038	−0.0410019	+	0.999159i	$0.513055\pi$
$758$	59.2174	2.15087
$759$	0	0
$760$	−0.926221	−0.0335976
$761$	−8.02641	−0.290957	−0.145479	−	0.989361i	$-0.546472\pi$
$761$	−8.02641	−0.290957	−0.145479	+	0.989361i	$0.546472\pi$
$762$	0	0
$763$	−21.3928	−0.774472
$764$	−46.3121	−1.67551
$765$	0	0
$766$	29.5381	1.06726
$767$	−36.6241	−1.32242
$768$	0	0
$769$	37.1810	1.34078	0.670391	−	0.742008i	$-0.266127\pi$
$769$	37.1810	1.34078	0.670391	+	0.742008i	$0.266127\pi$
$770$	−15.9325	−0.574166
$771$	0	0
$772$	4.94567	0.177998
$773$	−17.4806	−0.628733	−0.314367	−	0.949302i	$-0.601792\pi$
$773$	−17.4806	−0.628733	−0.314367	+	0.949302i	$0.601792\pi$
$774$	0	0
$775$	3.66152	0.131526
$776$	15.8913	0.570466
$777$	0	0
$778$	8.66000	0.310476
$779$	2.71033	0.0971075
$780$	0	0
$781$	27.0279	0.967135
$782$	4.71585	0.168639
$783$	0	0
$784$	12.1824	0.435087
$785$	3.17548	0.113338
$786$	0	0
$787$	26.7787	0.954556	0.477278	−	0.878752i	$-0.341623\pi$
$787$	26.7787	0.954556	0.477278	+	0.878752i	$0.341623\pi$
$788$	−52.0947	−1.85580
$789$	0	0
$790$	6.94567	0.247116
$791$	−25.4751	−0.905789
$792$	0	0
$793$	−66.2982	−2.35432
$794$	41.1732	1.46118
$795$	0	0
$796$	22.6894	0.804206
$797$	29.5723	1.04750	0.523752	−	0.851871i	$-0.324532\pi$
$797$	29.5723	1.04750	0.523752	+	0.851871i	$0.324532\pi$
$798$	0	0
$799$	0.587741	0.0207928
$800$	−7.98680	−0.282376
$801$	0	0
$802$	−73.7540	−2.60435
$803$	−35.6157	−1.25685
$804$	0	0
$805$	3.66152	0.129052
$806$	38.2982	1.34899
$807$	0	0
$808$	7.89134	0.277616
$809$	−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	$0.408882\pi$
$812$	−22.4688	−0.788501
$813$	0	0
$814$	95.7835	3.35721
$815$	−3.78963	−0.132745
$816$	0	0
$817$	−7.83549	−0.274129
$818$	−63.2585	−2.21178
$819$	0	0
$820$	−7.23606	−0.252694
$821$	30.4596	1.06305	0.531524	−	0.847043i	$-0.321619\pi$
$821$	30.4596	1.06305	0.531524	+	0.847043i	$0.321619\pi$
$822$	0	0
$823$	5.68097	0.198026	0.0990130	−	0.995086i	$-0.468431\pi$
$823$	5.68097	0.198026	0.0990130	+	0.995086i	$0.468431\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−25.7174	−0.894822
$827$	20.3774	0.708591	0.354295	−	0.935134i	$-0.384721\pi$
$827$	20.3774	0.708591	0.354295	+	0.935134i	$0.384721\pi$
$828$	0	0
$829$	−52.9387	−1.83864	−0.919319	−	0.393514i	$-0.871259\pi$
$829$	−52.9387	−1.83864	−0.919319	+	0.393514i	$0.871259\pi$
$830$	17.8913	0.621017
$831$	0	0
$832$	−55.5389	−1.92547
$833$	4.30359	0.149111
$834$	0	0
$835$	−18.6072	−0.643928
$836$	10.5077	0.363417
$837$	0	0
$838$	−19.1344	−0.660985
$839$	−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
$842$	−55.1468	−1.90049
$843$	0	0
$844$	−47.7827	−1.64475
$845$	−11.4596	−0.394223
$846$	0	0
$847$	16.4985	0.566896
$848$	−10.3098	−0.354041
$849$	0	0
$850$	−2.11491	−0.0725407
$851$	−22.0125	−0.754578
$852$	0	0
$853$	11.2144	0.383973	0.191986	−	0.981398i	$-0.438507\pi$
$853$	11.2144	0.383973	0.191986	+	0.981398i	$0.438507\pi$
$854$	−46.5544	−1.59306
$855$	0	0
$856$	15.0279	0.513644
$857$	−30.2812	−1.03439	−0.517193	−	0.855869i	$-0.673023\pi$
$857$	−30.2812	−1.03439	−0.517193	+	0.855869i	$0.673023\pi$
$858$	0	0
$859$	13.2074	0.450631	0.225316	−	0.974286i	$-0.427659\pi$
$859$	13.2074	0.450631	0.225316	+	0.974286i	$0.427659\pi$
$860$	20.9193	0.713341
$861$	0	0
$862$	3.72906	0.127012
$863$	−4.62664	−0.157492	−0.0787462	−	0.996895i	$-0.525092\pi$
$863$	−4.62664	−0.157492	−0.0787462	+	0.996895i	$0.525092\pi$
$864$	0	0
$865$	3.77018	0.128190
$866$	44.6366	1.51681
$867$	0	0
$868$	14.8679	0.504649
$869$	−15.0668	−0.511107
$870$	0	0
$871$	−69.1087	−2.34166
$872$	−13.0279	−0.441181
$873$	0	0
$874$	−4.36792	−0.147747
$875$	−1.64207	−0.0555122
$876$	0	0
$877$	−28.8609	−0.974564	−0.487282	−	0.873245i	$-0.662012\pi$
$877$	−28.8609	−0.974564	−0.487282	+	0.873245i	$0.662012\pi$
$878$	−71.3091	−2.40657
$879$	0	0
$880$	12.9868	0.437785
$881$	45.5195	1.53359	0.766795	−	0.641892i	$-0.221850\pi$
$881$	45.5195	1.53359	0.766795	+	0.641892i	$0.221850\pi$
$882$	0	0
$883$	−4.54189	−0.152847	−0.0764233	−	0.997075i	$-0.524350\pi$
$883$	−4.54189	−0.152847	−0.0764233	+	0.997075i	$0.524350\pi$
$884$	−12.2298	−0.411333
$885$	0	0
$886$	86.0808	2.89194
$887$	5.64903	0.189676	0.0948380	−	0.995493i	$-0.469767\pi$
$887$	5.64903	0.189676	0.0948380	+	0.995493i	$0.469767\pi$
$888$	0	0
$889$	−28.8231	−0.966695
$890$	8.94567	0.299859
$891$	0	0
$892$	19.1491	0.641158
$893$	−0.544378	−0.0182169
$894$	0	0
$895$	−22.0947	−0.738546
$896$	−12.7695	−0.426598
$897$	0	0
$898$	−42.5280	−1.41918
$899$	−20.2607	−0.675732
$900$	0	0
$901$	−3.64207	−0.121335
$902$	28.3921	0.945353
$903$	0	0
$904$	−15.5140	−0.515987
$905$	−1.40530	−0.0467137
$906$	0	0
$907$	20.2882	0.673657	0.336829	−	0.941566i	$-0.390646\pi$
$907$	20.2882	0.673657	0.336829	+	0.941566i	$0.390646\pi$
$908$	9.59175	0.318313
$909$	0	0
$910$	−17.1755	−0.569362
$911$	−4.38433	−0.145259	−0.0726297	−	0.997359i	$-0.523139\pi$
$911$	−4.38433	−0.145259	−0.0726297	+	0.997359i	$0.523139\pi$
$912$	0	0
$913$	−38.8106	−1.28444
$914$	61.6476	2.03912
$915$	0	0
$916$	−7.03265	−0.232365
$917$	16.2423	0.536368
$918$	0	0
$919$	52.8984	1.74496	0.872478	−	0.488653i	$-0.162512\pi$
$919$	52.8984	1.74496	0.872478	+	0.488653i	$0.162512\pi$
$920$	2.22982	0.0735148
$921$	0	0
$922$	31.9208	1.05125
$923$	29.1366	0.959043
$924$	0	0
$925$	9.87189	0.324586
$926$	79.5948	2.61565
$927$	0	0
$928$	44.1942	1.45075
$929$	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
$932$	37.8260	1.23903
$933$	0	0
$934$	20.5661	0.672942
$935$	4.58774	0.150035
$936$	0	0
$937$	−13.7871	−0.450406	−0.225203	−	0.974312i	$-0.572305\pi$
$937$	−13.7871	−0.450406	−0.225203	+	0.974312i	$0.572305\pi$
$938$	−48.5280	−1.58449
$939$	0	0
$940$	1.45339	0.0474043
$941$	43.1616	1.40703	0.703513	−	0.710682i	$-0.251614\pi$
$941$	43.1616	1.40703	0.703513	+	0.710682i	$0.251614\pi$
$942$	0	0
$943$	−6.52493	−0.212481
$944$	20.9626	0.682275
$945$	0	0
$946$	−82.0808	−2.66868
$947$	3.83549	0.124637	0.0623183	−	0.998056i	$-0.480151\pi$
$947$	3.83549	0.124637	0.0623183	+	0.998056i	$0.480151\pi$
$948$	0	0
$949$	−38.3943	−1.24633
$950$	1.95887	0.0635542
$951$	0	0
$952$	−1.64207	−0.0532199
$953$	32.8176	1.06306	0.531532	−	0.847038i	$-0.321616\pi$
$953$	32.8176	1.06306	0.531532	+	0.847038i	$0.321616\pi$
$954$	0	0
$955$	18.7283	0.606035
$956$	−20.0823	−0.649507
$957$	0	0
$958$	27.7827	0.897617
$959$	31.4761	1.01642
$960$	0	0
$961$	−17.5933	−0.567525
$962$	103.256	3.32912
$963$	0	0
$964$	38.3635	1.23560
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	10.2159	0.328521	0.164261	−	0.986417i	$-0.447476\pi$
$967$	10.2159	0.328521	0.164261	+	0.986417i	$0.447476\pi$
$968$	10.0474	0.322935
$969$	0	0
$970$	−33.6087	−1.07911
$971$	45.2966	1.45364	0.726819	−	0.686829i	$-0.240998\pi$
$971$	45.2966	1.45364	0.726819	+	0.686829i	$0.240998\pi$
$972$	0	0
$973$	21.6351	0.693590
$974$	3.02792	0.0970210
$975$	0	0
$976$	37.9472	1.21466
$977$	−38.4596	−1.23043	−0.615216	−	0.788358i	$-0.710931\pi$
$977$	−38.4596	−1.23043	−0.615216	+	0.788358i	$0.710931\pi$
$978$	0	0
$979$	−19.4053	−0.620196
$980$	10.6421	0.339949
$981$	0	0
$982$	63.6476	2.03108
$983$	12.4163	0.396017	0.198009	−	0.980200i	$-0.436553\pi$
$983$	12.4163	0.396017	0.198009	+	0.980200i	$0.436553\pi$
$984$	0	0
$985$	21.0668	0.671245
$986$	11.7026	0.372688
$987$	0	0
$988$	11.3275	0.360376
$989$	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	$0.651362\pi$
$992$	−29.2438	−0.928492
$993$	0	0
$994$	20.4596	0.648940
$995$	−9.17548	−0.290882
$996$	0	0
$997$	−30.6266	−0.969955	−0.484978	−	0.874527i	$-0.661172\pi$
$997$	−30.6266	−0.969955	−0.484978	+	0.874527i	$0.661172\pi$
$998$	7.66152	0.242521
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.a.c.1.1	✓	3	3.2	odd	2
765.2.a.j.1.3		3	1.1	even	1	trivial
1275.2.a.p.1.3		3	15.14	odd	2
1275.2.b.i.1174.1		6	15.2	even	4
1275.2.b.i.1174.6		6	15.8	even	4
3825.2.a.bb.1.1		3	5.4	even	2
4080.2.a.br.1.2		3	12.11	even	2
4335.2.a.s.1.1		3	51.50	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11491 q^{2} +2.47283 q^{4} -1.00000 q^{5} +1.64207 q^{7} +1.00000 q^{8} -2.11491 q^{10} +4.58774 q^{11} +4.94567 q^{13} +3.47283 q^{14} -2.83076 q^{16} -1.00000 q^{17} +0.926221 q^{19} -2.47283 q^{20} +9.70265 q^{22} -2.22982 q^{23} +1.00000 q^{25} +10.4596 q^{26} +4.06058 q^{28} -5.53341 q^{29} +3.66152 q^{31} -7.98680 q^{32} -2.11491 q^{34} -1.64207 q^{35} +9.87189 q^{37} +1.95887 q^{38} -1.00000 q^{40} +2.92622 q^{41} -8.45963 q^{43} +11.3447 q^{44} -4.71585 q^{46} -0.587741 q^{47} -4.30359 q^{49} +2.11491 q^{50} +12.2298 q^{52} +3.64207 q^{53} -4.58774 q^{55} +1.64207 q^{56} -11.7026 q^{58} -7.40530 q^{59} -13.4053 q^{61} +7.74378 q^{62} -11.2298 q^{64} -4.94567 q^{65} -13.9736 q^{67} -2.47283 q^{68} -3.47283 q^{70} +5.89134 q^{71} -7.76322 q^{73} +20.8781 q^{74} +2.29039 q^{76} +7.53341 q^{77} -3.28415 q^{79} +2.83076 q^{80} +6.18869 q^{82} -8.45963 q^{83} +1.00000 q^{85} -17.8913 q^{86} +4.58774 q^{88} -4.22982 q^{89} +8.12115 q^{91} -5.51396 q^{92} -1.24302 q^{94} -0.926221 q^{95} +15.8913 q^{97} -9.10170 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} - 3 q^{5} + 4 q^{7} + 3 q^{8} + 2 q^{11} + 4 q^{13} + 5 q^{14} - 4 q^{16} - 3 q^{17} - 2 q^{20} + 11 q^{22} + 6 q^{23} + 3 q^{25} + 6 q^{26} - 5 q^{28} + 6 q^{29} + 2 q^{31} - 4 q^{32} - 4 q^{35}+ \cdots - q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 - 3 * q^5 + 4 * q^7 + 3 * q^8 + 2 * q^11 + 4 * q^13 + 5 * q^14 - 4 * q^16 - 3 * q^17 - 2 * q^20 + 11 * q^22 + 6 * q^23 + 3 * q^25 + 6 * q^26 - 5 * q^28 + 6 * q^29 + 2 * q^31 - 4 * q^32 - 4 * q^35 + 16 * q^37 + 15 * q^38 - 3 * q^40 + 6 * q^41 + 15 * q^44 - 16 * q^46 + 10 * q^47 - 3 * q^49 + 24 * q^52 + 10 * q^53 - 2 * q^55 + 4 * q^56 - 17 * q^58 + 14 * q^59 - 4 * q^61 - 4 * q^62 - 21 * q^64 - 4 * q^65 - 2 * q^67 - 2 * q^68 - 5 * q^70 - 4 * q^71 + 12 * q^73 + 21 * q^74 - 23 * q^76 - 8 * q^79 + 4 * q^80 + 15 * q^82 + 3 * q^85 - 32 * q^86 + 2 * q^88 - 10 * q^91 - 2 * q^92 - 11 * q^94 + 26 * q^97 - q^98

Introduction

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