LMFDB - Embedded newform 6900.2.f.i.6349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6900,2,Mod(6349,6900)] 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("6900.6349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,-4,0,0,0,0,0,0,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.i.6349.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.46410i q^{7} -1.00000 q^{9} -6.19615 q^{11} -6.73205i q^{13} +4.26795i q^{17} +2.73205 q^{19} -4.46410 q^{21} -1.00000i q^{23} +1.00000i q^{27} +3.19615 q^{29} -1.00000 q^{31} +6.19615i q^{33} -9.39230i q^{37} -6.73205 q^{39} -4.26795 q^{41} -3.46410i q^{43} +2.73205i q^{47} -12.9282 q^{49} +4.26795 q^{51} -10.6603i q^{53} -2.73205i q^{57} -3.19615 q^{59} -7.26795 q^{61} +4.46410i q^{63} -5.00000i q^{67} -1.00000 q^{69} +10.1244 q^{71} -14.7321i q^{73} +27.6603i q^{77} -4.00000 q^{79} +1.00000 q^{81} +10.6603i q^{83} -3.19615i q^{87} -4.39230 q^{89} -30.0526 q^{91} +1.00000i q^{93} -4.00000i q^{97} +6.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} - 8 q^{29} - 4 q^{31} - 20 q^{39} - 24 q^{41} - 24 q^{49} + 24 q^{51} + 8 q^{59} - 36 q^{61} - 4 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} + 24 q^{89} - 44 q^{91}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 - 8 * q^29 - 4 * q^31 - 20 * q^39 - 24 * q^41 - 24 * q^49 + 24 * q^51 + 8 * q^59 - 36 * q^61 - 4 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 + 24 * q^89 - 44 * q^91 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$.

$n$	$277$	$1201$	$3451$	$4601$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.46410i	− 1.68727i	−0.536916	−	0.843636i	$-0.680411\pi$
$7$	− 4.46410i	− 1.68727i	0.536916	−	0.843636i	$-0.319589\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.19615	−1.86821	−0.934105	−	0.356998i	$-0.883800\pi$
$11$	−6.19615	−1.86821	−0.934105	+	0.356998i	$0.883800\pi$
$12$	0	0
$13$	− 6.73205i	− 1.86713i	−0.358402	−	0.933567i	$-0.616678\pi$
$13$	− 6.73205i	− 1.86713i	0.358402	−	0.933567i	$-0.383322\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.26795i	1.03513i	0.855644	+	0.517565i	$0.173161\pi$
$17$	4.26795i	1.03513i	−0.855644	+	0.517565i	$0.826839\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	0	0
$21$	−4.46410	−0.974147
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.19615	0.593511	0.296755	−	0.954954i	$-0.404095\pi$
$29$	3.19615	0.593511	0.296755	+	0.954954i	$0.404095\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	6.19615i	1.07861i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.39230i	− 1.54409i	−0.635571	−	0.772043i	$-0.719235\pi$
$37$	− 9.39230i	− 1.54409i	0.635571	−	0.772043i	$-0.280765\pi$
$38$	0	0
$39$	−6.73205	−1.07799
$40$	0	0
$41$	−4.26795	−0.666542	−0.333271	−	0.942831i	$-0.608152\pi$
$41$	−4.26795	−0.666542	−0.333271	+	0.942831i	$0.608152\pi$
$42$	0	0
$43$	− 3.46410i	− 0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$	− 3.46410i	− 0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.73205i	0.398511i	0.979948	+	0.199255i	$0.0638523\pi$
$47$	2.73205i	0.398511i	−0.979948	+	0.199255i	$0.936148\pi$
$48$	0	0
$49$	−12.9282	−1.84689
$50$	0	0
$51$	4.26795	0.597632
$52$	0	0
$53$	− 10.6603i	− 1.46430i	−0.681144	−	0.732149i	$-0.738517\pi$
$53$	− 10.6603i	− 1.46430i	0.681144	−	0.732149i	$-0.261483\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.73205i	− 0.361869i
$58$	0	0
$59$	−3.19615	−0.416104	−0.208052	−	0.978118i	$-0.566712\pi$
$59$	−3.19615	−0.416104	−0.208052	+	0.978118i	$0.566712\pi$
$60$	0	0
$61$	−7.26795	−0.930566	−0.465283	−	0.885162i	$-0.654047\pi$
$61$	−7.26795	−0.930566	−0.465283	+	0.885162i	$0.654047\pi$
$62$	0	0
$63$	4.46410i	0.562424i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.1244	1.20154	0.600770	−	0.799422i	$-0.294861\pi$
$71$	10.1244	1.20154	0.600770	+	0.799422i	$0.294861\pi$
$72$	0	0
$73$	− 14.7321i	− 1.72426i	−0.506690	−	0.862128i	$-0.669131\pi$
$73$	− 14.7321i	− 1.72426i	0.506690	−	0.862128i	$-0.330869\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	27.6603i	3.15218i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.6603i	1.17011i	0.810992	+	0.585057i	$0.198928\pi$
$83$	10.6603i	1.17011i	−0.810992	+	0.585057i	$0.801072\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.19615i	− 0.342664i
$88$	0	0
$89$	−4.39230	−0.465583	−0.232792	−	0.972527i	$-0.574786\pi$
$89$	−4.39230	−0.465583	−0.232792	+	0.972527i	$0.574786\pi$
$90$	0	0
$91$	−30.0526	−3.15036
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	0	0
$99$	6.19615	0.622737
$100$	0	0
$101$	−13.1962	−1.31307	−0.656533	−	0.754297i	$-0.727978\pi$
$101$	−13.1962	−1.31307	−0.656533	+	0.754297i	$0.727978\pi$
$102$	0	0
$103$	3.07180i	0.302673i	0.988482	+	0.151337i	$0.0483577\pi$
$103$	3.07180i	0.302673i	−0.988482	+	0.151337i	$0.951642\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.19615i	0.889026i	0.895772	+	0.444513i	$0.146623\pi$
$107$	9.19615i	0.889026i	−0.895772	+	0.444513i	$0.853377\pi$
$108$	0	0
$109$	13.6603	1.30842	0.654208	−	0.756315i	$-0.273002\pi$
$109$	13.6603	1.30842	0.654208	+	0.756315i	$0.273002\pi$
$110$	0	0
$111$	−9.39230	−0.891478
$112$	0	0
$113$	− 4.80385i	− 0.451908i	−0.974138	−	0.225954i	$-0.927450\pi$
$113$	− 4.80385i	− 0.451908i	0.974138	−	0.225954i	$-0.0725499\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.73205i	0.622378i
$118$	0	0
$119$	19.0526	1.74655
$120$	0	0
$121$	27.3923	2.49021
$122$	0	0
$123$	4.26795i	0.384828i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.1244i	1.16460i	0.812975	+	0.582299i	$0.197847\pi$
$127$	13.1244i	1.16460i	−0.812975	+	0.582299i	$0.802153\pi$
$128$	0	0
$129$	−3.46410	−0.304997
$130$	0	0
$131$	5.85641	0.511677	0.255838	−	0.966720i	$-0.417649\pi$
$131$	5.85641	0.511677	0.255838	+	0.966720i	$0.417649\pi$
$132$	0	0
$133$	− 12.1962i	− 1.05754i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	2.73205	0.230080
$142$	0	0
$143$	41.7128i	3.48820i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.9282i	1.06630i
$148$	0	0
$149$	−16.1962	−1.32684	−0.663420	−	0.748247i	$-0.730896\pi$
$149$	−16.1962	−1.32684	−0.663420	+	0.748247i	$0.730896\pi$
$150$	0	0
$151$	18.3923	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$152$	0	0
$153$	− 4.26795i	− 0.345043i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.3205i	1.78137i	0.454621	+	0.890685i	$0.349775\pi$
$157$	22.3205i	1.78137i	−0.454621	+	0.890685i	$0.650225\pi$
$158$	0	0
$159$	−10.6603	−0.845413
$160$	0	0
$161$	−4.46410	−0.351820
$162$	0	0
$163$	23.8564i	1.86858i	0.356517	+	0.934289i	$0.383964\pi$
$163$	23.8564i	1.86858i	−0.356517	+	0.934289i	$0.616036\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.1962i	1.25330i	0.779302	+	0.626648i	$0.215574\pi$
$167$	16.1962i	1.25330i	−0.779302	+	0.626648i	$0.784426\pi$
$168$	0	0
$169$	−32.3205	−2.48619
$170$	0	0
$171$	−2.73205	−0.208925
$172$	0	0
$173$	− 9.85641i	− 0.749369i	−0.927152	−	0.374684i	$-0.877751\pi$
$173$	− 9.85641i	− 0.749369i	0.927152	−	0.374684i	$-0.122249\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.19615i	0.240238i
$178$	0	0
$179$	−4.53590	−0.339029	−0.169514	−	0.985528i	$-0.554220\pi$
$179$	−4.53590	−0.339029	−0.169514	+	0.985528i	$0.554220\pi$
$180$	0	0
$181$	8.39230	0.623795	0.311898	−	0.950116i	$-0.399035\pi$
$181$	8.39230	0.623795	0.311898	+	0.950116i	$0.399035\pi$
$182$	0	0
$183$	7.26795i	0.537262i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 26.4449i	− 1.93384i
$188$	0	0
$189$	4.46410	0.324716
$190$	0	0
$191$	16.1962	1.17191	0.585956	−	0.810343i	$-0.300719\pi$
$191$	16.1962	1.17191	0.585956	+	0.810343i	$0.300719\pi$
$192$	0	0
$193$	3.60770i	0.259688i	0.991534	+	0.129844i	$0.0414476\pi$
$193$	3.60770i	0.259688i	−0.991534	+	0.129844i	$0.958552\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.535898i	0.0381812i	0.999818	+	0.0190906i	$0.00607709\pi$
$197$	0.535898i	0.0381812i	−0.999818	+	0.0190906i	$0.993923\pi$
$198$	0	0
$199$	11.4641	0.812669	0.406334	−	0.913724i	$-0.366807\pi$
$199$	11.4641	0.812669	0.406334	+	0.913724i	$0.366807\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	− 14.2679i	− 1.00141i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−16.9282	−1.17095
$210$	0	0
$211$	−4.07180	−0.280314	−0.140157	−	0.990129i	$-0.544761\pi$
$211$	−4.07180	−0.280314	−0.140157	+	0.990129i	$0.544761\pi$
$212$	0	0
$213$	− 10.1244i	− 0.693709i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.46410i	0.303043i
$218$	0	0
$219$	−14.7321	−0.995500
$220$	0	0
$221$	28.7321	1.93273
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.07180i	0.602116i	0.953606	+	0.301058i	$0.0973398\pi$
$227$	9.07180i	0.602116i	−0.953606	+	0.301058i	$0.902660\pi$
$228$	0	0
$229$	−10.5359	−0.696232	−0.348116	−	0.937452i	$-0.613178\pi$
$229$	−10.5359	−0.696232	−0.348116	+	0.937452i	$0.613178\pi$
$230$	0	0
$231$	27.6603	1.81991
$232$	0	0
$233$	− 20.7846i	− 1.36165i	−0.732448	−	0.680823i	$-0.761622\pi$
$233$	− 20.7846i	− 1.36165i	0.732448	−	0.680823i	$-0.238378\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	22.1244	1.43111	0.715553	−	0.698559i	$-0.246175\pi$
$239$	22.1244	1.43111	0.715553	+	0.698559i	$0.246175\pi$
$240$	0	0
$241$	16.5885	1.06856	0.534278	−	0.845309i	$-0.320583\pi$
$241$	16.5885	1.06856	0.534278	+	0.845309i	$0.320583\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 18.3923i	− 1.17027i
$248$	0	0
$249$	10.6603	0.675566
$250$	0	0
$251$	−3.85641	−0.243414	−0.121707	−	0.992566i	$-0.538837\pi$
$251$	−3.85641	−0.243414	−0.121707	+	0.992566i	$0.538837\pi$
$252$	0	0
$253$	6.19615i	0.389549i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.1244i	0.943431i	0.881751	+	0.471716i	$0.156365\pi$
$257$	15.1244i	0.943431i	−0.881751	+	0.471716i	$0.843635\pi$
$258$	0	0
$259$	−41.9282	−2.60529
$260$	0	0
$261$	−3.19615	−0.197837
$262$	0	0
$263$	− 28.6603i	− 1.76727i	−0.468179	−	0.883633i	$-0.655090\pi$
$263$	− 28.6603i	− 1.76727i	0.468179	−	0.883633i	$-0.344910\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.39230i	0.268805i
$268$	0	0
$269$	−1.73205	−0.105605	−0.0528025	−	0.998605i	$-0.516815\pi$
$269$	−1.73205	−0.105605	−0.0528025	+	0.998605i	$0.516815\pi$
$270$	0	0
$271$	2.60770	0.158406	0.0792031	−	0.996858i	$-0.474762\pi$
$271$	2.60770	0.158406	0.0792031	+	0.996858i	$0.474762\pi$
$272$	0	0
$273$	30.0526i	1.81886i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.8564i	1.91407i	0.289979	+	0.957033i	$0.406352\pi$
$277$	31.8564i	1.91407i	−0.289979	+	0.957033i	$0.593648\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−29.5167	−1.76082	−0.880408	−	0.474217i	$-0.842731\pi$
$281$	−29.5167	−1.76082	−0.880408	+	0.474217i	$0.842731\pi$
$282$	0	0
$283$	3.14359i	0.186867i	0.995626	+	0.0934336i	$0.0297843\pi$
$283$	3.14359i	0.186867i	−0.995626	+	0.0934336i	$0.970216\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	19.0526i	1.12464i
$288$	0	0
$289$	−1.21539	−0.0714935
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	− 7.33975i	− 0.428793i	−0.976747	−	0.214396i	$-0.931222\pi$
$293$	− 7.33975i	− 0.428793i	0.976747	−	0.214396i	$-0.0687784\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 6.19615i	− 0.359537i
$298$	0	0
$299$	−6.73205	−0.389325
$300$	0	0
$301$	−15.4641	−0.891336
$302$	0	0
$303$	13.1962i	0.758099i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.6603i	0.779632i	0.920893	+	0.389816i	$0.127461\pi$
$307$	13.6603i	0.779632i	−0.920893	+	0.389816i	$0.872539\pi$
$308$	0	0
$309$	3.07180	0.174748
$310$	0	0
$311$	9.07180	0.514414	0.257207	−	0.966356i	$-0.417198\pi$
$311$	9.07180	0.514414	0.257207	+	0.966356i	$0.417198\pi$
$312$	0	0
$313$	− 12.3205i	− 0.696396i	−0.937421	−	0.348198i	$-0.886794\pi$
$313$	− 12.3205i	− 0.696396i	0.937421	−	0.348198i	$-0.113206\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 31.5167i	− 1.77015i	−0.465447	−	0.885076i	$-0.654106\pi$
$317$	− 31.5167i	− 1.77015i	0.465447	−	0.885076i	$-0.345894\pi$
$318$	0	0
$319$	−19.8038	−1.10880
$320$	0	0
$321$	9.19615	0.513279
$322$	0	0
$323$	11.6603i	0.648794i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 13.6603i	− 0.755414i
$328$	0	0
$329$	12.1962	0.672396
$330$	0	0
$331$	21.2487	1.16793	0.583967	−	0.811777i	$-0.301500\pi$
$331$	21.2487	1.16793	0.583967	+	0.811777i	$0.301500\pi$
$332$	0	0
$333$	9.39230i	0.514695i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 28.9282i	− 1.57582i	−0.615791	−	0.787910i	$-0.711163\pi$
$337$	− 28.9282i	− 1.57582i	0.615791	−	0.787910i	$-0.288837\pi$
$338$	0	0
$339$	−4.80385	−0.260909
$340$	0	0
$341$	6.19615	0.335540
$342$	0	0
$343$	26.4641i	1.42893i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.3923i	− 1.52418i	−0.647472	−	0.762089i	$-0.724174\pi$
$347$	− 28.3923i	− 1.52418i	0.647472	−	0.762089i	$-0.275826\pi$
$348$	0	0
$349$	6.46410	0.346015	0.173008	−	0.984920i	$-0.444651\pi$
$349$	6.46410	0.346015	0.173008	+	0.984920i	$0.444651\pi$
$350$	0	0
$351$	6.73205	0.359330
$352$	0	0
$353$	32.5885i	1.73451i	0.497865	+	0.867254i	$0.334117\pi$
$353$	32.5885i	1.73451i	−0.497865	+	0.867254i	$0.665883\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 19.0526i	− 1.00837i
$358$	0	0
$359$	5.26795	0.278032	0.139016	−	0.990290i	$-0.455606\pi$
$359$	5.26795	0.278032	0.139016	+	0.990290i	$0.455606\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	− 27.3923i	− 1.43772i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.5359i	1.22856i	0.789087	+	0.614282i	$0.210554\pi$
$367$	23.5359i	1.22856i	−0.789087	+	0.614282i	$0.789446\pi$
$368$	0	0
$369$	4.26795	0.222181
$370$	0	0
$371$	−47.5885	−2.47067
$372$	0	0
$373$	− 2.53590i	− 0.131304i	−0.997843	−	0.0656519i	$-0.979087\pi$
$373$	− 2.53590i	− 0.131304i	0.997843	−	0.0656519i	$-0.0209127\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 21.5167i	− 1.10816i
$378$	0	0
$379$	17.8564	0.917222	0.458611	−	0.888637i	$-0.348347\pi$
$379$	17.8564	0.917222	0.458611	+	0.888637i	$0.348347\pi$
$380$	0	0
$381$	13.1244	0.672381
$382$	0	0
$383$	− 7.05256i	− 0.360369i	−0.983633	−	0.180184i	$-0.942331\pi$
$383$	− 7.05256i	− 0.360369i	0.983633	−	0.180184i	$-0.0576695\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.46410i	0.176090i
$388$	0	0
$389$	−19.4641	−0.986869	−0.493435	−	0.869783i	$-0.664259\pi$
$389$	−19.4641	−0.986869	−0.493435	+	0.869783i	$0.664259\pi$
$390$	0	0
$391$	4.26795	0.215839
$392$	0	0
$393$	− 5.85641i	− 0.295417i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 32.0000i	− 1.60603i	−0.595956	−	0.803017i	$-0.703227\pi$
$397$	− 32.0000i	− 1.60603i	0.595956	−	0.803017i	$-0.296773\pi$
$398$	0	0
$399$	−12.1962	−0.610571
$400$	0	0
$401$	9.60770	0.479785	0.239893	−	0.970799i	$-0.422888\pi$
$401$	9.60770	0.479785	0.239893	+	0.970799i	$0.422888\pi$
$402$	0	0
$403$	6.73205i	0.335347i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	58.1962i	2.88468i
$408$	0	0
$409$	−30.1769	−1.49215	−0.746076	−	0.665861i	$-0.768065\pi$
$409$	−30.1769	−1.49215	−0.746076	+	0.665861i	$0.768065\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	14.2679i	0.702080i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.00000i	0.342791i
$418$	0	0
$419$	−21.1244	−1.03199	−0.515996	−	0.856591i	$-0.672578\pi$
$419$	−21.1244	−1.03199	−0.515996	+	0.856591i	$0.672578\pi$
$420$	0	0
$421$	−31.5167	−1.53603	−0.768014	−	0.640433i	$-0.778755\pi$
$421$	−31.5167	−1.53603	−0.768014	+	0.640433i	$0.778755\pi$
$422$	0	0
$423$	− 2.73205i	− 0.132837i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.4449i	1.57012i
$428$	0	0
$429$	41.7128	2.01391
$430$	0	0
$431$	24.3923	1.17494	0.587468	−	0.809247i	$-0.300125\pi$
$431$	24.3923	1.17494	0.587468	+	0.809247i	$0.300125\pi$
$432$	0	0
$433$	− 8.60770i	− 0.413659i	−0.978377	−	0.206830i	$-0.933685\pi$
$433$	− 8.60770i	− 0.413659i	0.978377	−	0.206830i	$-0.0663146\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.73205i	− 0.130692i
$438$	0	0
$439$	22.7846	1.08745	0.543725	−	0.839263i	$-0.317013\pi$
$439$	22.7846	1.08745	0.543725	+	0.839263i	$0.317013\pi$
$440$	0	0
$441$	12.9282	0.615629
$442$	0	0
$443$	− 12.1962i	− 0.579457i	−0.957109	−	0.289728i	$-0.906435\pi$
$443$	− 12.1962i	− 0.579457i	0.957109	−	0.289728i	$-0.0935650\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.1962i	0.766052i
$448$	0	0
$449$	−7.73205	−0.364898	−0.182449	−	0.983215i	$-0.558402\pi$
$449$	−7.73205	−0.364898	−0.182449	+	0.983215i	$0.558402\pi$
$450$	0	0
$451$	26.4449	1.24524
$452$	0	0
$453$	− 18.3923i	− 0.864146i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.5359i	0.913851i	0.889505	+	0.456925i	$0.151049\pi$
$457$	19.5359i	0.913851i	−0.889505	+	0.456925i	$0.848951\pi$
$458$	0	0
$459$	−4.26795	−0.199211
$460$	0	0
$461$	5.32051	0.247801	0.123900	−	0.992295i	$-0.460460\pi$
$461$	5.32051	0.247801	0.123900	+	0.992295i	$0.460460\pi$
$462$	0	0
$463$	20.5885i	0.956827i	0.878135	+	0.478413i	$0.158788\pi$
$463$	20.5885i	0.956827i	−0.878135	+	0.478413i	$0.841212\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.4449i	− 1.36255i	−0.732030	−	0.681273i	$-0.761427\pi$
$467$	− 29.4449i	− 1.36255i	0.732030	−	0.681273i	$-0.238573\pi$
$468$	0	0
$469$	−22.3205	−1.03067
$470$	0	0
$471$	22.3205	1.02847
$472$	0	0
$473$	21.4641i	0.986920i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.6603i	0.488100i
$478$	0	0
$479$	−16.9808	−0.775871	−0.387935	−	0.921687i	$-0.626812\pi$
$479$	−16.9808	−0.775871	−0.387935	+	0.921687i	$0.626812\pi$
$480$	0	0
$481$	−63.2295	−2.88302
$482$	0	0
$483$	4.46410i	0.203124i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 0.732051i	− 0.0331724i	−0.999862	−	0.0165862i	$-0.994720\pi$
$487$	− 0.732051i	− 0.0331724i	0.999862	−	0.0165862i	$-0.00527979\pi$
$488$	0	0
$489$	23.8564	1.07882
$490$	0	0
$491$	−19.0526	−0.859830	−0.429915	−	0.902869i	$-0.641456\pi$
$491$	−19.0526	−0.859830	−0.429915	+	0.902869i	$0.641456\pi$
$492$	0	0
$493$	13.6410i	0.614360i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 45.1962i	− 2.02732i
$498$	0	0
$499$	21.5359	0.964079	0.482040	−	0.876149i	$-0.339896\pi$
$499$	21.5359	0.964079	0.482040	+	0.876149i	$0.339896\pi$
$500$	0	0
$501$	16.1962	0.723591
$502$	0	0
$503$	6.80385i	0.303369i	0.988429	+	0.151684i	$0.0484697\pi$
$503$	6.80385i	0.303369i	−0.988429	+	0.151684i	$0.951530\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.3205i	1.43540i
$508$	0	0
$509$	−8.78461	−0.389371	−0.194685	−	0.980866i	$-0.562369\pi$
$509$	−8.78461	−0.389371	−0.194685	+	0.980866i	$0.562369\pi$
$510$	0	0
$511$	−65.7654	−2.90929
$512$	0	0
$513$	2.73205i	0.120623i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.9282i	− 0.744502i
$518$	0	0
$519$	−9.85641	−0.432648
$520$	0	0
$521$	−16.0526	−0.703275	−0.351638	−	0.936136i	$-0.614375\pi$
$521$	−16.0526	−0.703275	−0.351638	+	0.936136i	$0.614375\pi$
$522$	0	0
$523$	− 4.78461i	− 0.209216i	−0.994514	−	0.104608i	$-0.966641\pi$
$523$	− 4.78461i	− 0.209216i	0.994514	−	0.104608i	$-0.0333589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.26795i	− 0.185915i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	3.19615	0.138701
$532$	0	0
$533$	28.7321i	1.24452i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.53590i	0.195738i
$538$	0	0
$539$	80.1051	3.45037
$540$	0	0
$541$	−1.07180	−0.0460801	−0.0230401	−	0.999735i	$-0.507335\pi$
$541$	−1.07180	−0.0460801	−0.0230401	+	0.999735i	$0.507335\pi$
$542$	0	0
$543$	− 8.39230i	− 0.360148i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10.9282i	− 0.467256i	−0.972326	−	0.233628i	$-0.924940\pi$
$547$	− 10.9282i	− 0.467256i	0.972326	−	0.233628i	$-0.0750598\pi$
$548$	0	0
$549$	7.26795	0.310189
$550$	0	0
$551$	8.73205	0.371998
$552$	0	0
$553$	17.8564i	0.759332i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 35.9808i	− 1.52455i	−0.647251	−	0.762277i	$-0.724081\pi$
$557$	− 35.9808i	− 1.52455i	0.647251	−	0.762277i	$-0.275919\pi$
$558$	0	0
$559$	−23.3205	−0.986352
$560$	0	0
$561$	−26.4449	−1.11650
$562$	0	0
$563$	− 15.5885i	− 0.656975i	−0.944508	−	0.328488i	$-0.893461\pi$
$563$	− 15.5885i	− 0.656975i	0.944508	−	0.328488i	$-0.106539\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.46410i	− 0.187475i
$568$	0	0
$569$	−7.46410	−0.312911	−0.156456	−	0.987685i	$-0.550007\pi$
$569$	−7.46410	−0.312911	−0.156456	+	0.987685i	$0.550007\pi$
$570$	0	0
$571$	6.58846	0.275718	0.137859	−	0.990452i	$-0.455978\pi$
$571$	6.58846	0.275718	0.137859	+	0.990452i	$0.455978\pi$
$572$	0	0
$573$	− 16.1962i	− 0.676604i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.7846i	1.03180i	0.856650	+	0.515898i	$0.172542\pi$
$577$	24.7846i	1.03180i	−0.856650	+	0.515898i	$0.827458\pi$
$578$	0	0
$579$	3.60770	0.149931
$580$	0	0
$581$	47.5885	1.97430
$582$	0	0
$583$	66.0526i	2.73562i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.0718i	0.456982i	0.973546	+	0.228491i	$0.0733791\pi$
$587$	11.0718i	0.456982i	−0.973546	+	0.228491i	$0.926621\pi$
$588$	0	0
$589$	−2.73205	−0.112572
$590$	0	0
$591$	0.535898	0.0220439
$592$	0	0
$593$	− 34.7321i	− 1.42627i	−0.701024	−	0.713137i	$-0.747274\pi$
$593$	− 34.7321i	− 1.42627i	0.701024	−	0.713137i	$-0.252726\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.4641i	− 0.469194i
$598$	0	0
$599$	−24.3923	−0.996643	−0.498321	−	0.866992i	$-0.666050\pi$
$599$	−24.3923	−0.996643	−0.498321	+	0.866992i	$0.666050\pi$
$600$	0	0
$601$	−27.3923	−1.11736	−0.558678	−	0.829385i	$-0.688691\pi$
$601$	−27.3923	−1.11736	−0.558678	+	0.829385i	$0.688691\pi$
$602$	0	0
$603$	5.00000i	0.203616i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 22.9808i	− 0.932760i	−0.884584	−	0.466380i	$-0.845558\pi$
$607$	− 22.9808i	− 0.932760i	0.884584	−	0.466380i	$-0.154442\pi$
$608$	0	0
$609$	−14.2679	−0.578166
$610$	0	0
$611$	18.3923	0.744073
$612$	0	0
$613$	13.7128i	0.553855i	0.960891	+	0.276928i	$0.0893162\pi$
$613$	13.7128i	0.553855i	−0.960891	+	0.276928i	$0.910684\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.1244i	1.45431i	0.686472	+	0.727156i	$0.259158\pi$
$617$	36.1244i	1.45431i	−0.686472	+	0.727156i	$0.740842\pi$
$618$	0	0
$619$	2.92820	0.117694	0.0588472	−	0.998267i	$-0.481258\pi$
$619$	2.92820	0.117694	0.0588472	+	0.998267i	$0.481258\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	19.6077i	0.785566i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.9282i	0.676047i
$628$	0	0
$629$	40.0859	1.59833
$630$	0	0
$631$	−44.8372	−1.78494	−0.892470	−	0.451107i	$-0.851029\pi$
$631$	−44.8372	−1.78494	−0.892470	+	0.451107i	$0.851029\pi$
$632$	0	0
$633$	4.07180i	0.161839i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	87.0333i	3.44839i
$638$	0	0
$639$	−10.1244	−0.400513
$640$	0	0
$641$	−28.0526	−1.10801	−0.554005	−	0.832514i	$-0.686901\pi$
$641$	−28.0526	−1.10801	−0.554005	+	0.832514i	$0.686901\pi$
$642$	0	0
$643$	− 8.46410i	− 0.333792i	−0.985975	−	0.166896i	$-0.946626\pi$
$643$	− 8.46410i	− 0.333792i	0.985975	−	0.166896i	$-0.0533743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.2679i	0.914757i	0.889272	+	0.457379i	$0.151212\pi$
$647$	23.2679i	0.914757i	−0.889272	+	0.457379i	$0.848788\pi$
$648$	0	0
$649$	19.8038	0.777369
$650$	0	0
$651$	4.46410	0.174962
$652$	0	0
$653$	39.3731i	1.54079i	0.637569	+	0.770394i	$0.279940\pi$
$653$	39.3731i	1.54079i	−0.637569	+	0.770394i	$0.720060\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	14.7321i	0.574752i
$658$	0	0
$659$	−8.44486	−0.328965	−0.164483	−	0.986380i	$-0.552595\pi$
$659$	−8.44486	−0.328965	−0.164483	+	0.986380i	$0.552595\pi$
$660$	0	0
$661$	35.4641	1.37939	0.689697	−	0.724098i	$-0.257744\pi$
$661$	35.4641	1.37939	0.689697	+	0.724098i	$0.257744\pi$
$662$	0	0
$663$	− 28.7321i	− 1.11586i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.19615i	− 0.123756i
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	45.0333	1.73849
$672$	0	0
$673$	37.9090i	1.46128i	0.682761	+	0.730642i	$0.260779\pi$
$673$	37.9090i	1.46128i	−0.682761	+	0.730642i	$0.739221\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.1962i	1.19897i	0.800388	+	0.599483i	$0.204627\pi$
$677$	31.1962i	1.19897i	−0.800388	+	0.599483i	$0.795373\pi$
$678$	0	0
$679$	−17.8564	−0.685266
$680$	0	0
$681$	9.07180	0.347632
$682$	0	0
$683$	− 10.7321i	− 0.410651i	−0.978694	−	0.205325i	$-0.934175\pi$
$683$	− 10.7321i	− 0.410651i	0.978694	−	0.205325i	$-0.0658252\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.5359i	0.401970i
$688$	0	0
$689$	−71.7654	−2.73404
$690$	0	0
$691$	−7.46410	−0.283948	−0.141974	−	0.989870i	$-0.545345\pi$
$691$	−7.46410	−0.283948	−0.141974	+	0.989870i	$0.545345\pi$
$692$	0	0
$693$	− 27.6603i	− 1.05073i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.2154i	− 0.689957i
$698$	0	0
$699$	−20.7846	−0.786146
$700$	0	0
$701$	12.5885	0.475459	0.237730	−	0.971331i	$-0.423597\pi$
$701$	12.5885	0.475459	0.237730	+	0.971331i	$0.423597\pi$
$702$	0	0
$703$	− 25.6603i	− 0.967795i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	58.9090i	2.21550i
$708$	0	0
$709$	−6.87564	−0.258220	−0.129110	−	0.991630i	$-0.541212\pi$
$709$	−6.87564	−0.258220	−0.129110	+	0.991630i	$0.541212\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	1.00000i	0.0374503i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 22.1244i	− 0.826249i
$718$	0	0
$719$	−17.7321	−0.661294	−0.330647	−	0.943755i	$-0.607267\pi$
$719$	−17.7321	−0.661294	−0.330647	+	0.943755i	$0.607267\pi$
$720$	0	0
$721$	13.7128	0.510692
$722$	0	0
$723$	− 16.5885i	− 0.616931i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 29.0000i	− 1.07555i	−0.843088	−	0.537775i	$-0.819265\pi$
$727$	− 29.0000i	− 1.07555i	0.843088	−	0.537775i	$-0.180735\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	14.7846	0.546829
$732$	0	0
$733$	− 53.6410i	− 1.98128i	−0.136515	−	0.990638i	$-0.543590\pi$
$733$	− 53.6410i	− 1.98128i	0.136515	−	0.990638i	$-0.456410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.9808i	1.14119i
$738$	0	0
$739$	36.3205	1.33607	0.668036	−	0.744129i	$-0.267135\pi$
$739$	36.3205	1.33607	0.668036	+	0.744129i	$0.267135\pi$
$740$	0	0
$741$	−18.3923	−0.675658
$742$	0	0
$743$	9.85641i	0.361596i	0.983520	+	0.180798i	$0.0578681\pi$
$743$	9.85641i	0.361596i	−0.983520	+	0.180798i	$0.942132\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.6603i	− 0.390038i
$748$	0	0
$749$	41.0526	1.50003
$750$	0	0
$751$	−24.8756	−0.907725	−0.453863	−	0.891072i	$-0.649954\pi$
$751$	−24.8756	−0.907725	−0.453863	+	0.891072i	$0.649954\pi$
$752$	0	0
$753$	3.85641i	0.140535i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 50.8564i	− 1.84841i	−0.381900	−	0.924204i	$-0.624730\pi$
$757$	− 50.8564i	− 1.84841i	0.381900	−	0.924204i	$-0.375270\pi$
$758$	0	0
$759$	6.19615	0.224906
$760$	0	0
$761$	−5.19615	−0.188360	−0.0941802	−	0.995555i	$-0.530023\pi$
$761$	−5.19615	−0.188360	−0.0941802	+	0.995555i	$0.530023\pi$
$762$	0	0
$763$	− 60.9808i	− 2.20765i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.5167i	0.776922i
$768$	0	0
$769$	−7.66025	−0.276236	−0.138118	−	0.990416i	$-0.544105\pi$
$769$	−7.66025	−0.276236	−0.138118	+	0.990416i	$0.544105\pi$
$770$	0	0
$771$	15.1244	0.544690
$772$	0	0
$773$	− 45.0333i	− 1.61974i	−0.586612	−	0.809868i	$-0.699539\pi$
$773$	− 45.0333i	− 1.61974i	0.586612	−	0.809868i	$-0.300461\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	41.9282i	1.50417i
$778$	0	0
$779$	−11.6603	−0.417772
$780$	0	0
$781$	−62.7321	−2.24473
$782$	0	0
$783$	3.19615i	0.114221i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 24.6077i	− 0.877170i	−0.898690	−	0.438585i	$-0.855480\pi$
$787$	− 24.6077i	− 0.877170i	0.898690	−	0.438585i	$-0.144520\pi$
$788$	0	0
$789$	−28.6603	−1.02033
$790$	0	0
$791$	−21.4449	−0.762492
$792$	0	0
$793$	48.9282i	1.73749i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 7.73205i	− 0.273883i	−0.990579	−	0.136942i	$-0.956273\pi$
$797$	− 7.73205i	− 0.273883i	0.990579	−	0.136942i	$-0.0437273\pi$
$798$	0	0
$799$	−11.6603	−0.412510
$800$	0	0
$801$	4.39230	0.155194
$802$	0	0
$803$	91.2820i	3.22127i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.73205i	0.0609711i
$808$	0	0
$809$	12.8038	0.450159	0.225080	−	0.974340i	$-0.427736\pi$
$809$	12.8038	0.450159	0.225080	+	0.974340i	$0.427736\pi$
$810$	0	0
$811$	−20.0718	−0.704816	−0.352408	−	0.935846i	$-0.614637\pi$
$811$	−20.0718	−0.704816	−0.352408	+	0.935846i	$0.614637\pi$
$812$	0	0
$813$	− 2.60770i	− 0.0914559i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.46410i	− 0.331107i
$818$	0	0
$819$	30.0526	1.05012
$820$	0	0
$821$	25.8564	0.902395	0.451197	−	0.892424i	$-0.350997\pi$
$821$	25.8564	0.902395	0.451197	+	0.892424i	$0.350997\pi$
$822$	0	0
$823$	− 20.7846i	− 0.724506i	−0.932080	−	0.362253i	$-0.882008\pi$
$823$	− 20.7846i	− 0.724506i	0.932080	−	0.362253i	$-0.117992\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.51666i	− 0.157060i	−0.996912	−	0.0785298i	$-0.974977\pi$
$827$	− 4.51666i	− 0.157060i	0.996912	−	0.0785298i	$-0.0250226\pi$
$828$	0	0
$829$	41.4974	1.44127	0.720633	−	0.693317i	$-0.243852\pi$
$829$	41.4974	1.44127	0.720633	+	0.693317i	$0.243852\pi$
$830$	0	0
$831$	31.8564	1.10509
$832$	0	0
$833$	− 55.1769i	− 1.91177i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−18.7846	−0.647745
$842$	0	0
$843$	29.5167i	1.01661i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 122.282i	− 4.20166i
$848$	0	0
$849$	3.14359	0.107888
$850$	0	0
$851$	−9.39230	−0.321964
$852$	0	0
$853$	− 24.5359i	− 0.840093i	−0.907503	−	0.420047i	$-0.862014\pi$
$853$	− 24.5359i	− 0.840093i	0.907503	−	0.420047i	$-0.137986\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.7128i	0.741696i	0.928694	+	0.370848i	$0.120933\pi$
$857$	21.7128i	0.741696i	−0.928694	+	0.370848i	$0.879067\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	0	0
$861$	19.0526	0.649309
$862$	0	0
$863$	− 51.9615i	− 1.76879i	−0.466738	−	0.884395i	$-0.654571\pi$
$863$	− 51.9615i	− 1.76879i	0.466738	−	0.884395i	$-0.345429\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.21539i	0.0412768i
$868$	0	0
$869$	24.7846	0.840760
$870$	0	0
$871$	−33.6603	−1.14053
$872$	0	0
$873$	4.00000i	0.135379i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0.535898i	0.0180960i	0.999959	+	0.00904800i	$0.00288011\pi$
$877$	0.535898i	0.0180960i	−0.999959	+	0.00904800i	$0.997120\pi$
$878$	0	0
$879$	−7.33975	−0.247563
$880$	0	0
$881$	46.0526	1.55155	0.775775	−	0.631010i	$-0.217359\pi$
$881$	46.0526	1.55155	0.775775	+	0.631010i	$0.217359\pi$
$882$	0	0
$883$	− 28.3397i	− 0.953708i	−0.878982	−	0.476854i	$-0.841777\pi$
$883$	− 28.3397i	− 0.953708i	0.878982	−	0.476854i	$-0.158223\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 17.3205i	− 0.581566i	−0.956789	−	0.290783i	$-0.906084\pi$
$887$	− 17.3205i	− 0.581566i	0.956789	−	0.290783i	$-0.0939157\pi$
$888$	0	0
$889$	58.5885	1.96499
$890$	0	0
$891$	−6.19615	−0.207579
$892$	0	0
$893$	7.46410i	0.249777i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.73205i	0.224777i
$898$	0	0
$899$	−3.19615	−0.106598
$900$	0	0
$901$	45.4974	1.51574
$902$	0	0
$903$	15.4641i	0.514613i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 6.07180i	− 0.201611i	−0.994906	−	0.100805i	$-0.967858\pi$
$907$	− 6.07180i	− 0.201611i	0.994906	−	0.100805i	$-0.0321419\pi$
$908$	0	0
$909$	13.1962	0.437689
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	− 66.0526i	− 2.18602i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 26.1436i	− 0.863338i
$918$	0	0
$919$	9.07180	0.299251	0.149625	−	0.988743i	$-0.452193\pi$
$919$	9.07180	0.299251	0.149625	+	0.988743i	$0.452193\pi$
$920$	0	0
$921$	13.6603	0.450121
$922$	0	0
$923$	− 68.1577i	− 2.24344i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.07180i	− 0.100891i
$928$	0	0
$929$	34.5167	1.13245	0.566227	−	0.824249i	$-0.308402\pi$
$929$	34.5167	1.13245	0.566227	+	0.824249i	$0.308402\pi$
$930$	0	0
$931$	−35.3205	−1.15758
$932$	0	0
$933$	− 9.07180i	− 0.296997i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 18.9282i	− 0.618357i	−0.951004	−	0.309179i	$-0.899946\pi$
$937$	− 18.9282i	− 0.618357i	0.951004	−	0.309179i	$-0.100054\pi$
$938$	0	0
$939$	−12.3205	−0.402065
$940$	0	0
$941$	11.8038	0.384794	0.192397	−	0.981317i	$-0.438374\pi$
$941$	11.8038	0.384794	0.192397	+	0.981317i	$0.438374\pi$
$942$	0	0
$943$	4.26795i	0.138984i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.3205i	1.40773i	0.710335	+	0.703864i	$0.248543\pi$
$947$	43.3205i	1.40773i	−0.710335	+	0.703864i	$0.751457\pi$
$948$	0	0
$949$	−99.1769	−3.21942
$950$	0	0
$951$	−31.5167	−1.02200
$952$	0	0
$953$	29.7128i	0.962492i	0.876585	+	0.481246i	$0.159816\pi$
$953$	29.7128i	0.962492i	−0.876585	+	0.481246i	$0.840184\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.8038i	0.640167i
$958$	0	0
$959$	53.5692	1.72984
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	− 9.19615i	− 0.296342i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41.1244i	1.32247i	0.750179	+	0.661235i	$0.229967\pi$
$967$	41.1244i	1.32247i	−0.750179	+	0.661235i	$0.770033\pi$
$968$	0	0
$969$	11.6603	0.374581
$970$	0	0
$971$	−12.6795	−0.406904	−0.203452	−	0.979085i	$-0.565216\pi$
$971$	−12.6795	−0.406904	−0.203452	+	0.979085i	$0.565216\pi$
$972$	0	0
$973$	31.2487i	1.00179i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34.1244i	− 1.09173i	−0.837872	−	0.545867i	$-0.816200\pi$
$977$	− 34.1244i	− 1.09173i	0.837872	−	0.545867i	$-0.183800\pi$
$978$	0	0
$979$	27.2154	0.869808
$980$	0	0
$981$	−13.6603	−0.436138
$982$	0	0
$983$	0.803848i	0.0256388i	0.999918	+	0.0128194i	$0.00408065\pi$
$983$	0.803848i	0.0256388i	−0.999918	+	0.0128194i	$0.995919\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 12.1962i	− 0.388208i
$988$	0	0
$989$	−3.46410	−0.110152
$990$	0	0
$991$	−26.7128	−0.848560	−0.424280	−	0.905531i	$-0.639473\pi$
$991$	−26.7128	−0.848560	−0.424280	+	0.905531i	$0.639473\pi$
$992$	0	0
$993$	− 21.2487i	− 0.674307i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.5359i	1.47381i	0.675998	+	0.736903i	$0.263713\pi$
$997$	46.5359i	1.47381i	−0.675998	+	0.736903i	$0.736287\pi$
$998$	0	0
$999$	9.39230	0.297159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.i.6349.1		4
5.2	odd	4		1380.2.a.h.1.2	✓	2
5.3	odd	4		6900.2.a.s.1.1		2
5.4	even	2	inner	6900.2.f.i.6349.4		4
15.2	even	4		4140.2.a.o.1.2		2
20.7	even	4		5520.2.a.bp.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.h.1.2	✓	2	5.2	odd	4
4140.2.a.o.1.2		2	15.2	even	4
5520.2.a.bp.1.1		2	20.7	even	4
6900.2.a.s.1.1		2	5.3	odd	4
6900.2.f.i.6349.1		4	1.1	even	1	trivial
6900.2.f.i.6349.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.46410i q^{7} -1.00000 q^{9} -6.19615 q^{11} -6.73205i q^{13} +4.26795i q^{17} +2.73205 q^{19} -4.46410 q^{21} -1.00000i q^{23} +1.00000i q^{27} +3.19615 q^{29} -1.00000 q^{31} +6.19615i q^{33} -9.39230i q^{37} -6.73205 q^{39} -4.26795 q^{41} -3.46410i q^{43} +2.73205i q^{47} -12.9282 q^{49} +4.26795 q^{51} -10.6603i q^{53} -2.73205i q^{57} -3.19615 q^{59} -7.26795 q^{61} +4.46410i q^{63} -5.00000i q^{67} -1.00000 q^{69} +10.1244 q^{71} -14.7321i q^{73} +27.6603i q^{77} -4.00000 q^{79} +1.00000 q^{81} +10.6603i q^{83} -3.19615i q^{87} -4.39230 q^{89} -30.0526 q^{91} +1.00000i q^{93} -4.00000i q^{97} +6.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} - 8 q^{29} - 4 q^{31} - 20 q^{39} - 24 q^{41} - 24 q^{49} + 24 q^{51} + 8 q^{59} - 36 q^{61} - 4 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} + 24 q^{89} - 44 q^{91}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^11 + 4 * q^19 - 4 * q^21 - 8 * q^29 - 4 * q^31 - 20 * q^39 - 24 * q^41 - 24 * q^49 + 24 * q^51 + 8 * q^59 - 36 * q^61 - 4 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 + 24 * q^89 - 44 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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