LMFDB - Embedded newform 6900.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.79129 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.79129 q^{7} +1.00000 q^{9} -1.79129 q^{11} +2.20871 q^{13} -4.58258 q^{17} +5.58258 q^{19} -3.79129 q^{21} -1.00000 q^{23} -1.00000 q^{27} -1.58258 q^{29} +1.79129 q^{31} +1.79129 q^{33} +3.00000 q^{37} -2.20871 q^{39} -0.582576 q^{41} +6.58258 q^{43} -1.00000 q^{47} +7.37386 q^{49} +4.58258 q^{51} +4.20871 q^{53} -5.58258 q^{57} -3.79129 q^{59} -1.37386 q^{61} +3.79129 q^{63} +1.20871 q^{67} +1.00000 q^{69} +8.95644 q^{71} +13.0000 q^{73} -6.79129 q^{77} -1.41742 q^{79} +1.00000 q^{81} +1.00000 q^{83} +1.58258 q^{87} -5.00000 q^{89} +8.37386 q^{91} -1.79129 q^{93} -4.62614 q^{97} -1.79129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 3 q^{7} + 2 q^{9} + q^{11} + 9 q^{13} + 2 q^{19} - 3 q^{21} - 2 q^{23} - 2 q^{27} + 6 q^{29} - q^{31} - q^{33} + 6 q^{37} - 9 q^{39} + 8 q^{41} + 4 q^{43} - 2 q^{47} + q^{49} + 13 q^{53} - 2 q^{57} - 3 q^{59} + 11 q^{61} + 3 q^{63} + 7 q^{67} + 2 q^{69} - 5 q^{71} + 26 q^{73} - 9 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} - 6 q^{87} - 10 q^{89} + 3 q^{91} + q^{93} - 23 q^{97} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^7 + 2 * q^9 + q^11 + 9 * q^13 + 2 * q^19 - 3 * q^21 - 2 * q^23 - 2 * q^27 + 6 * q^29 - q^31 - q^33 + 6 * q^37 - 9 * q^39 + 8 * q^41 + 4 * q^43 - 2 * q^47 + q^49 + 13 * q^53 - 2 * q^57 - 3 * q^59 + 11 * q^61 + 3 * q^63 + 7 * q^67 + 2 * q^69 - 5 * q^71 + 26 * q^73 - 9 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 - 6 * q^87 - 10 * q^89 + 3 * q^91 + q^93 - 23 * q^97 + q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.79129	1.43297	0.716486	−	0.697601i	$-0.245749\pi$
$7$	3.79129	1.43297	0.716486	+	0.697601i	$0.245749\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.79129	−0.540094	−0.270047	−	0.962847i	$-0.587039\pi$
$11$	−1.79129	−0.540094	−0.270047	+	0.962847i	$0.587039\pi$
$12$	0	0
$13$	2.20871	0.612587	0.306293	−	0.951937i	$-0.400911\pi$
$13$	2.20871	0.612587	0.306293	+	0.951937i	$0.400911\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.58258	−1.11144	−0.555719	−	0.831370i	$-0.687557\pi$
$17$	−4.58258	−1.11144	−0.555719	+	0.831370i	$0.687557\pi$
$18$	0	0
$19$	5.58258	1.28073	0.640365	−	0.768070i	$-0.278783\pi$
$19$	5.58258	1.28073	0.640365	+	0.768070i	$0.278783\pi$
$20$	0	0
$21$	−3.79129	−0.827327
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.58258	−0.293877	−0.146938	−	0.989146i	$-0.546942\pi$
$29$	−1.58258	−0.293877	−0.146938	+	0.989146i	$0.546942\pi$
$30$	0	0
$31$	1.79129	0.321725	0.160862	−	0.986977i	$-0.448572\pi$
$31$	1.79129	0.321725	0.160862	+	0.986977i	$0.448572\pi$
$32$	0	0
$33$	1.79129	0.311823
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−2.20871	−0.353677
$40$	0	0
$41$	−0.582576	−0.0909830	−0.0454915	−	0.998965i	$-0.514485\pi$
$41$	−0.582576	−0.0909830	−0.0454915	+	0.998965i	$0.514485\pi$
$42$	0	0
$43$	6.58258	1.00383	0.501917	−	0.864916i	$-0.332628\pi$
$43$	6.58258	1.00383	0.501917	+	0.864916i	$0.332628\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	0	0
$49$	7.37386	1.05341
$50$	0	0
$51$	4.58258	0.641689
$52$	0	0
$53$	4.20871	0.578111	0.289056	−	0.957312i	$-0.406659\pi$
$53$	4.20871	0.578111	0.289056	+	0.957312i	$0.406659\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.58258	−0.739430
$58$	0	0
$59$	−3.79129	−0.493584	−0.246792	−	0.969069i	$-0.579376\pi$
$59$	−3.79129	−0.493584	−0.246792	+	0.969069i	$0.579376\pi$
$60$	0	0
$61$	−1.37386	−0.175905	−0.0879526	−	0.996125i	$-0.528032\pi$
$61$	−1.37386	−0.175905	−0.0879526	+	0.996125i	$0.528032\pi$
$62$	0	0
$63$	3.79129	0.477657
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.20871	0.147668	0.0738338	−	0.997271i	$-0.476477\pi$
$67$	1.20871	0.147668	0.0738338	+	0.997271i	$0.476477\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	8.95644	1.06293	0.531467	−	0.847079i	$-0.321641\pi$
$71$	8.95644	1.06293	0.531467	+	0.847079i	$0.321641\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.79129	−0.773939
$78$	0	0
$79$	−1.41742	−0.159473	−0.0797363	−	0.996816i	$-0.525408\pi$
$79$	−1.41742	−0.159473	−0.0797363	+	0.996816i	$0.525408\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.58258	0.169670
$88$	0	0
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	0	0
$91$	8.37386	0.877819
$92$	0	0
$93$	−1.79129	−0.185748
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.62614	−0.469713	−0.234856	−	0.972030i	$-0.575462\pi$
$97$	−4.62614	−0.469713	−0.234856	+	0.972030i	$0.575462\pi$
$98$	0	0
$99$	−1.79129	−0.180031
$100$	0	0
$101$	−3.62614	−0.360814	−0.180407	−	0.983592i	$-0.557741\pi$
$101$	−3.62614	−0.360814	−0.180407	+	0.983592i	$0.557741\pi$
$102$	0	0
$103$	8.41742	0.829393	0.414697	−	0.909960i	$-0.363888\pi$
$103$	8.41742	0.829393	0.414697	+	0.909960i	$0.363888\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.16515	−0.789355	−0.394677	−	0.918820i	$-0.629144\pi$
$107$	−8.16515	−0.789355	−0.394677	+	0.918820i	$0.629144\pi$
$108$	0	0
$109$	9.74773	0.933663	0.466831	−	0.884346i	$-0.345395\pi$
$109$	9.74773	0.933663	0.466831	+	0.884346i	$0.345395\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	11.9564	1.12477	0.562384	−	0.826876i	$-0.309884\pi$
$113$	11.9564	1.12477	0.562384	+	0.826876i	$0.309884\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.20871	0.204196
$118$	0	0
$119$	−17.3739	−1.59266
$120$	0	0
$121$	−7.79129	−0.708299
$122$	0	0
$123$	0.582576	0.0525291
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	0	0
$129$	−6.58258	−0.579563
$130$	0	0
$131$	8.16515	0.713393	0.356696	−	0.934220i	$-0.383903\pi$
$131$	8.16515	0.713393	0.356696	+	0.934220i	$0.383903\pi$
$132$	0	0
$133$	21.1652	1.83525
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−12.9564	−1.09895	−0.549475	−	0.835510i	$-0.685172\pi$
$139$	−12.9564	−1.09895	−0.549475	+	0.835510i	$0.685172\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	0	0
$143$	−3.95644	−0.330854
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.37386	−0.608186
$148$	0	0
$149$	−12.5826	−1.03080	−0.515402	−	0.856948i	$-0.672358\pi$
$149$	−12.5826	−1.03080	−0.515402	+	0.856948i	$0.672358\pi$
$150$	0	0
$151$	−0.582576	−0.0474093	−0.0237047	−	0.999719i	$-0.507546\pi$
$151$	−0.582576	−0.0474093	−0.0237047	+	0.999719i	$0.507546\pi$
$152$	0	0
$153$	−4.58258	−0.370479
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.16515	0.252607	0.126303	−	0.991992i	$-0.459689\pi$
$157$	3.16515	0.252607	0.126303	+	0.991992i	$0.459689\pi$
$158$	0	0
$159$	−4.20871	−0.333773
$160$	0	0
$161$	−3.79129	−0.298795
$162$	0	0
$163$	−11.7477	−0.920153	−0.460077	−	0.887879i	$-0.652178\pi$
$163$	−11.7477	−0.920153	−0.460077	+	0.887879i	$0.652178\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.1216	1.78920	0.894601	−	0.446865i	$-0.147460\pi$
$167$	23.1216	1.78920	0.894601	+	0.446865i	$0.147460\pi$
$168$	0	0
$169$	−8.12159	−0.624738
$170$	0	0
$171$	5.58258	0.426910
$172$	0	0
$173$	0.626136	0.0476043	0.0238021	−	0.999717i	$-0.492423\pi$
$173$	0.626136	0.0476043	0.0238021	+	0.999717i	$0.492423\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.79129	0.284971
$178$	0	0
$179$	8.53901	0.638236	0.319118	−	0.947715i	$-0.396613\pi$
$179$	8.53901	0.638236	0.319118	+	0.947715i	$0.396613\pi$
$180$	0	0
$181$	−16.7477	−1.24485	−0.622424	−	0.782680i	$-0.713852\pi$
$181$	−16.7477	−1.24485	−0.622424	+	0.782680i	$0.713852\pi$
$182$	0	0
$183$	1.37386	0.101559
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.20871	0.600280
$188$	0	0
$189$	−3.79129	−0.275776
$190$	0	0
$191$	−2.16515	−0.156665	−0.0783324	−	0.996927i	$-0.524960\pi$
$191$	−2.16515	−0.156665	−0.0783324	+	0.996927i	$0.524960\pi$
$192$	0	0
$193$	20.1652	1.45152	0.725760	−	0.687948i	$-0.241488\pi$
$193$	20.1652	1.45152	0.725760	+	0.687948i	$0.241488\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.7477	1.47821	0.739107	−	0.673588i	$-0.235248\pi$
$197$	20.7477	1.47821	0.739107	+	0.673588i	$0.235248\pi$
$198$	0	0
$199$	−7.58258	−0.537515	−0.268757	−	0.963208i	$-0.586613\pi$
$199$	−7.58258	−0.537515	−0.268757	+	0.963208i	$0.586613\pi$
$200$	0	0
$201$	−1.20871	−0.0852560
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−6.74773	−0.464533	−0.232266	−	0.972652i	$-0.574614\pi$
$211$	−6.74773	−0.464533	−0.232266	+	0.972652i	$0.574614\pi$
$212$	0	0
$213$	−8.95644	−0.613685
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.79129	0.461023
$218$	0	0
$219$	−13.0000	−0.878459
$220$	0	0
$221$	−10.1216	−0.680852
$222$	0	0
$223$	−1.62614	−0.108894	−0.0544471	−	0.998517i	$-0.517340\pi$
$223$	−1.62614	−0.108894	−0.0544471	+	0.998517i	$0.517340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.582576	−0.0386669	−0.0193335	−	0.999813i	$-0.506154\pi$
$227$	−0.582576	−0.0386669	−0.0193335	+	0.999813i	$0.506154\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	6.79129	0.446834
$232$	0	0
$233$	17.7913	1.16555	0.582773	−	0.812635i	$-0.301968\pi$
$233$	17.7913	1.16555	0.582773	+	0.812635i	$0.301968\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.41742	0.0920716
$238$	0	0
$239$	−27.5826	−1.78417	−0.892084	−	0.451869i	$-0.850757\pi$
$239$	−27.5826	−1.78417	−0.892084	+	0.451869i	$0.850757\pi$
$240$	0	0
$241$	16.3739	1.05473	0.527367	−	0.849638i	$-0.323179\pi$
$241$	16.3739	1.05473	0.527367	+	0.849638i	$0.323179\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.3303	0.784558
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	0	0
$251$	20.2087	1.27556	0.637781	−	0.770218i	$-0.279852\pi$
$251$	20.2087	1.27556	0.637781	+	0.770218i	$0.279852\pi$
$252$	0	0
$253$	1.79129	0.112617
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.20871	0.324911	0.162455	−	0.986716i	$-0.448059\pi$
$257$	5.20871	0.324911	0.162455	+	0.986716i	$0.448059\pi$
$258$	0	0
$259$	11.3739	0.706737
$260$	0	0
$261$	−1.58258	−0.0979590
$262$	0	0
$263$	−3.53901	−0.218225	−0.109113	−	0.994029i	$-0.534801\pi$
$263$	−3.53901	−0.218225	−0.109113	+	0.994029i	$0.534801\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.00000	0.305995
$268$	0	0
$269$	−0.747727	−0.0455897	−0.0227949	−	0.999740i	$-0.507256\pi$
$269$	−0.747727	−0.0455897	−0.0227949	+	0.999740i	$0.507256\pi$
$270$	0	0
$271$	10.1652	0.617489	0.308744	−	0.951145i	$-0.400091\pi$
$271$	10.1652	0.617489	0.308744	+	0.951145i	$0.400091\pi$
$272$	0	0
$273$	−8.37386	−0.506809
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.791288	−0.0475439	−0.0237719	−	0.999717i	$-0.507568\pi$
$277$	−0.791288	−0.0475439	−0.0237719	+	0.999717i	$0.507568\pi$
$278$	0	0
$279$	1.79129	0.107242
$280$	0	0
$281$	12.1652	0.725712	0.362856	−	0.931845i	$-0.381802\pi$
$281$	12.1652	0.725712	0.362856	+	0.931845i	$0.381802\pi$
$282$	0	0
$283$	−2.58258	−0.153518	−0.0767591	−	0.997050i	$-0.524457\pi$
$283$	−2.58258	−0.153518	−0.0767591	+	0.997050i	$0.524457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.20871	−0.130376
$288$	0	0
$289$	4.00000	0.235294
$290$	0	0
$291$	4.62614	0.271189
$292$	0	0
$293$	14.1652	0.827537	0.413768	−	0.910382i	$-0.364212\pi$
$293$	14.1652	0.827537	0.413768	+	0.910382i	$0.364212\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.79129	0.103941
$298$	0	0
$299$	−2.20871	−0.127733
$300$	0	0
$301$	24.9564	1.43847
$302$	0	0
$303$	3.62614	0.208316
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.3303	−1.38860	−0.694302	−	0.719684i	$-0.744287\pi$
$307$	−24.3303	−1.38860	−0.694302	+	0.719684i	$0.744287\pi$
$308$	0	0
$309$	−8.41742	−0.478851
$310$	0	0
$311$	−2.16515	−0.122774	−0.0613872	−	0.998114i	$-0.519552\pi$
$311$	−2.16515	−0.122774	−0.0613872	+	0.998114i	$0.519552\pi$
$312$	0	0
$313$	24.5826	1.38949	0.694745	−	0.719256i	$-0.255517\pi$
$313$	24.5826	1.38949	0.694745	+	0.719256i	$0.255517\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.7913	1.89791	0.948954	−	0.315415i	$-0.102144\pi$
$317$	33.7913	1.89791	0.948954	+	0.315415i	$0.102144\pi$
$318$	0	0
$319$	2.83485	0.158721
$320$	0	0
$321$	8.16515	0.455734
$322$	0	0
$323$	−25.5826	−1.42345
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.74773	−0.539051
$328$	0	0
$329$	−3.79129	−0.209020
$330$	0	0
$331$	−7.53901	−0.414382	−0.207191	−	0.978301i	$-0.566432\pi$
$331$	−7.53901	−0.414382	−0.207191	+	0.978301i	$0.566432\pi$
$332$	0	0
$333$	3.00000	0.164399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.9129	−0.812356	−0.406178	−	0.913794i	$-0.633139\pi$
$337$	−14.9129	−0.812356	−0.406178	+	0.913794i	$0.633139\pi$
$338$	0	0
$339$	−11.9564	−0.649385
$340$	0	0
$341$	−3.20871	−0.173762
$342$	0	0
$343$	1.41742	0.0765337
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−14.5826	−0.780587	−0.390294	−	0.920690i	$-0.627627\pi$
$349$	−14.5826	−0.780587	−0.390294	+	0.920690i	$0.627627\pi$
$350$	0	0
$351$	−2.20871	−0.117892
$352$	0	0
$353$	6.95644	0.370254	0.185127	−	0.982715i	$-0.440730\pi$
$353$	6.95644	0.370254	0.185127	+	0.982715i	$0.440730\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	17.3739	0.919522
$358$	0	0
$359$	5.74773	0.303353	0.151677	−	0.988430i	$-0.451533\pi$
$359$	5.74773	0.303353	0.151677	+	0.988430i	$0.451533\pi$
$360$	0	0
$361$	12.1652	0.640271
$362$	0	0
$363$	7.79129	0.408937
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−26.9129	−1.40484	−0.702420	−	0.711762i	$-0.747897\pi$
$367$	−26.9129	−1.40484	−0.702420	+	0.711762i	$0.747897\pi$
$368$	0	0
$369$	−0.582576	−0.0303277
$370$	0	0
$371$	15.9564	0.828417
$372$	0	0
$373$	12.1652	0.629888	0.314944	−	0.949110i	$-0.398014\pi$
$373$	12.1652	0.629888	0.314944	+	0.949110i	$0.398014\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.49545	−0.180025
$378$	0	0
$379$	19.1652	0.984448	0.492224	−	0.870469i	$-0.336184\pi$
$379$	19.1652	0.984448	0.492224	+	0.870469i	$0.336184\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	0	0
$383$	35.7913	1.82885	0.914425	−	0.404756i	$-0.132644\pi$
$383$	35.7913	1.82885	0.914425	+	0.404756i	$0.132644\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.58258	0.334611
$388$	0	0
$389$	−0.626136	−0.0317464	−0.0158732	−	0.999874i	$-0.505053\pi$
$389$	−0.626136	−0.0317464	−0.0158732	+	0.999874i	$0.505053\pi$
$390$	0	0
$391$	4.58258	0.231751
$392$	0	0
$393$	−8.16515	−0.411877
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.3303	0.518463	0.259232	−	0.965815i	$-0.416531\pi$
$397$	10.3303	0.518463	0.259232	+	0.965815i	$0.416531\pi$
$398$	0	0
$399$	−21.1652	−1.05958
$400$	0	0
$401$	29.3739	1.46686	0.733430	−	0.679765i	$-0.237918\pi$
$401$	29.3739	1.46686	0.733430	+	0.679765i	$0.237918\pi$
$402$	0	0
$403$	3.95644	0.197084
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.37386	−0.266373
$408$	0	0
$409$	32.7477	1.61927	0.809635	−	0.586933i	$-0.199665\pi$
$409$	32.7477	1.61927	0.809635	+	0.586933i	$0.199665\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	−14.3739	−0.707292
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.9564	0.634480
$418$	0	0
$419$	7.12159	0.347912	0.173956	−	0.984753i	$-0.444345\pi$
$419$	7.12159	0.347912	0.173956	+	0.984753i	$0.444345\pi$
$420$	0	0
$421$	9.53901	0.464903	0.232452	−	0.972608i	$-0.425325\pi$
$421$	9.53901	0.464903	0.232452	+	0.972608i	$0.425325\pi$
$422$	0	0
$423$	−1.00000	−0.0486217
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.20871	−0.252067
$428$	0	0
$429$	3.95644	0.191019
$430$	0	0
$431$	−23.7477	−1.14389	−0.571944	−	0.820293i	$-0.693811\pi$
$431$	−23.7477	−1.14389	−0.571944	+	0.820293i	$0.693811\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.58258	−0.267051
$438$	0	0
$439$	−9.33030	−0.445311	−0.222656	−	0.974897i	$-0.571473\pi$
$439$	−9.33030	−0.445311	−0.222656	+	0.974897i	$0.571473\pi$
$440$	0	0
$441$	7.37386	0.351136
$442$	0	0
$443$	27.4955	1.30635	0.653174	−	0.757208i	$-0.273437\pi$
$443$	27.4955	1.30635	0.653174	+	0.757208i	$0.273437\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.5826	0.595135
$448$	0	0
$449$	29.2867	1.38213	0.691063	−	0.722794i	$-0.257143\pi$
$449$	29.2867	1.38213	0.691063	+	0.722794i	$0.257143\pi$
$450$	0	0
$451$	1.04356	0.0491394
$452$	0	0
$453$	0.582576	0.0273718
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	4.58258	0.213896
$460$	0	0
$461$	−27.3739	−1.27493	−0.637464	−	0.770480i	$-0.720017\pi$
$461$	−27.3739	−1.27493	−0.637464	+	0.770480i	$0.720017\pi$
$462$	0	0
$463$	−2.12159	−0.0985987	−0.0492993	−	0.998784i	$-0.515699\pi$
$463$	−2.12159	−0.0985987	−0.0492993	+	0.998784i	$0.515699\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.41742	−0.296963	−0.148481	−	0.988915i	$-0.547439\pi$
$467$	−6.41742	−0.296963	−0.148481	+	0.988915i	$0.547439\pi$
$468$	0	0
$469$	4.58258	0.211604
$470$	0	0
$471$	−3.16515	−0.145842
$472$	0	0
$473$	−11.7913	−0.542164
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.20871	0.192704
$478$	0	0
$479$	−1.58258	−0.0723097	−0.0361549	−	0.999346i	$-0.511511\pi$
$479$	−1.58258	−0.0723097	−0.0361549	+	0.999346i	$0.511511\pi$
$480$	0	0
$481$	6.62614	0.302126
$482$	0	0
$483$	3.79129	0.172510
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.74773	0.396397	0.198199	−	0.980162i	$-0.436491\pi$
$487$	8.74773	0.396397	0.198199	+	0.980162i	$0.436491\pi$
$488$	0	0
$489$	11.7477	0.531251
$490$	0	0
$491$	34.1216	1.53989	0.769943	−	0.638113i	$-0.220285\pi$
$491$	34.1216	1.53989	0.769943	+	0.638113i	$0.220285\pi$
$492$	0	0
$493$	7.25227	0.326626
$494$	0	0
$495$	0	0
$496$	0	0
$497$	33.9564	1.52315
$498$	0	0
$499$	17.8348	0.798397	0.399199	−	0.916864i	$-0.369288\pi$
$499$	17.8348	0.798397	0.399199	+	0.916864i	$0.369288\pi$
$500$	0	0
$501$	−23.1216	−1.03300
$502$	0	0
$503$	12.4955	0.557145	0.278572	−	0.960415i	$-0.410139\pi$
$503$	12.4955	0.557145	0.278572	+	0.960415i	$0.410139\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.12159	0.360693
$508$	0	0
$509$	0.956439	0.0423934	0.0211967	−	0.999775i	$-0.493252\pi$
$509$	0.956439	0.0423934	0.0211967	+	0.999775i	$0.493252\pi$
$510$	0	0
$511$	49.2867	2.18032
$512$	0	0
$513$	−5.58258	−0.246477
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.79129	0.0787807
$518$	0	0
$519$	−0.626136	−0.0274843
$520$	0	0
$521$	−4.74773	−0.208002	−0.104001	−	0.994577i	$-0.533164\pi$
$521$	−4.74773	−0.208002	−0.104001	+	0.994577i	$0.533164\pi$
$522$	0	0
$523$	−7.00000	−0.306089	−0.153044	−	0.988219i	$-0.548908\pi$
$523$	−7.00000	−0.306089	−0.153044	+	0.988219i	$0.548908\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.20871	−0.357577
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.79129	−0.164528
$532$	0	0
$533$	−1.28674	−0.0557350
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8.53901	−0.368486
$538$	0	0
$539$	−13.2087	−0.568940
$540$	0	0
$541$	15.9129	0.684148	0.342074	−	0.939673i	$-0.388871\pi$
$541$	15.9129	0.684148	0.342074	+	0.939673i	$0.388871\pi$
$542$	0	0
$543$	16.7477	0.718714
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.4955	0.448753	0.224377	−	0.974503i	$-0.427965\pi$
$547$	10.4955	0.448753	0.224377	+	0.974503i	$0.427965\pi$
$548$	0	0
$549$	−1.37386	−0.0586351
$550$	0	0
$551$	−8.83485	−0.376377
$552$	0	0
$553$	−5.37386	−0.228520
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.2432	−1.62042	−0.810208	−	0.586143i	$-0.800646\pi$
$557$	−38.2432	−1.62042	−0.810208	+	0.586143i	$0.800646\pi$
$558$	0	0
$559$	14.5390	0.614935
$560$	0	0
$561$	−8.20871	−0.346572
$562$	0	0
$563$	28.3739	1.19582	0.597908	−	0.801565i	$-0.295999\pi$
$563$	28.3739	1.19582	0.597908	+	0.801565i	$0.295999\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.79129	0.159219
$568$	0	0
$569$	25.4174	1.06555	0.532777	−	0.846256i	$-0.321148\pi$
$569$	25.4174	1.06555	0.532777	+	0.846256i	$0.321148\pi$
$570$	0	0
$571$	−24.0780	−1.00763	−0.503817	−	0.863810i	$-0.668071\pi$
$571$	−24.0780	−1.00763	−0.503817	+	0.863810i	$0.668071\pi$
$572$	0	0
$573$	2.16515	0.0904505
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−38.1216	−1.58702	−0.793511	−	0.608556i	$-0.791749\pi$
$577$	−38.1216	−1.58702	−0.793511	+	0.608556i	$0.791749\pi$
$578$	0	0
$579$	−20.1652	−0.838035
$580$	0	0
$581$	3.79129	0.157289
$582$	0	0
$583$	−7.53901	−0.312234
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.9564	0.823690	0.411845	−	0.911254i	$-0.364884\pi$
$587$	19.9564	0.823690	0.411845	+	0.911254i	$0.364884\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	−20.7477	−0.853447
$592$	0	0
$593$	24.4955	1.00591	0.502954	−	0.864313i	$-0.332247\pi$
$593$	24.4955	1.00591	0.502954	+	0.864313i	$0.332247\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.58258	0.310334
$598$	0	0
$599$	−44.5826	−1.82159	−0.910797	−	0.412854i	$-0.864532\pi$
$599$	−44.5826	−1.82159	−0.910797	+	0.412854i	$0.864532\pi$
$600$	0	0
$601$	40.3303	1.64511	0.822554	−	0.568687i	$-0.192549\pi$
$601$	40.3303	1.64511	0.822554	+	0.568687i	$0.192549\pi$
$602$	0	0
$603$	1.20871	0.0492226
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.9564	−1.45943	−0.729713	−	0.683753i	$-0.760347\pi$
$607$	−35.9564	−1.45943	−0.729713	+	0.683753i	$0.760347\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	−2.20871	−0.0893549
$612$	0	0
$613$	−1.53901	−0.0621602	−0.0310801	−	0.999517i	$-0.509895\pi$
$613$	−1.53901	−0.0621602	−0.0310801	+	0.999517i	$0.509895\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.2867	1.01801	0.509003	−	0.860765i	$-0.330014\pi$
$617$	25.2867	1.01801	0.509003	+	0.860765i	$0.330014\pi$
$618$	0	0
$619$	11.7913	0.473932	0.236966	−	0.971518i	$-0.423847\pi$
$619$	11.7913	0.473932	0.236966	+	0.971518i	$0.423847\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−18.9564	−0.759474
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.0000	0.399362
$628$	0	0
$629$	−13.7477	−0.548158
$630$	0	0
$631$	40.3739	1.60726	0.803629	−	0.595131i	$-0.202900\pi$
$631$	40.3739	1.60726	0.803629	+	0.595131i	$0.202900\pi$
$632$	0	0
$633$	6.74773	0.268198
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.2867	0.645304
$638$	0	0
$639$	8.95644	0.354311
$640$	0	0
$641$	−7.04356	−0.278204	−0.139102	−	0.990278i	$-0.544422\pi$
$641$	−7.04356	−0.278204	−0.139102	+	0.990278i	$0.544422\pi$
$642$	0	0
$643$	−1.46099	−0.0576156	−0.0288078	−	0.999585i	$-0.509171\pi$
$643$	−1.46099	−0.0576156	−0.0288078	+	0.999585i	$0.509171\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	6.79129	0.266581
$650$	0	0
$651$	−6.79129	−0.266172
$652$	0	0
$653$	−10.1652	−0.397793	−0.198897	−	0.980020i	$-0.563736\pi$
$653$	−10.1652	−0.397793	−0.198897	+	0.980020i	$0.563736\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.0000	0.507178
$658$	0	0
$659$	−49.4519	−1.92637	−0.963186	−	0.268835i	$-0.913361\pi$
$659$	−49.4519	−1.92637	−0.963186	+	0.268835i	$0.913361\pi$
$660$	0	0
$661$	−12.1216	−0.471475	−0.235738	−	0.971817i	$-0.575751\pi$
$661$	−12.1216	−0.471475	−0.235738	+	0.971817i	$0.575751\pi$
$662$	0	0
$663$	10.1216	0.393090
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.58258	0.0612776
$668$	0	0
$669$	1.62614	0.0628701
$670$	0	0
$671$	2.46099	0.0950053
$672$	0	0
$673$	13.4955	0.520212	0.260106	−	0.965580i	$-0.416243\pi$
$673$	13.4955	0.520212	0.260106	+	0.965580i	$0.416243\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.5390	1.36588	0.682938	−	0.730477i	$-0.260702\pi$
$677$	35.5390	1.36588	0.682938	+	0.730477i	$0.260702\pi$
$678$	0	0
$679$	−17.5390	−0.673086
$680$	0	0
$681$	0.582576	0.0223243
$682$	0	0
$683$	−48.9129	−1.87160	−0.935800	−	0.352532i	$-0.885321\pi$
$683$	−48.9129	−1.87160	−0.935800	+	0.352532i	$0.885321\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.00000	−0.0381524
$688$	0	0
$689$	9.29583	0.354143
$690$	0	0
$691$	14.1652	0.538868	0.269434	−	0.963019i	$-0.413163\pi$
$691$	14.1652	0.538868	0.269434	+	0.963019i	$0.413163\pi$
$692$	0	0
$693$	−6.79129	−0.257980
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.66970	0.101122
$698$	0	0
$699$	−17.7913	−0.672928
$700$	0	0
$701$	12.1652	0.459471	0.229736	−	0.973253i	$-0.426214\pi$
$701$	12.1652	0.459471	0.229736	+	0.973253i	$0.426214\pi$
$702$	0	0
$703$	16.7477	0.631652
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.7477	−0.517036
$708$	0	0
$709$	17.2523	0.647923	0.323961	−	0.946070i	$-0.394985\pi$
$709$	17.2523	0.647923	0.323961	+	0.946070i	$0.394985\pi$
$710$	0	0
$711$	−1.41742	−0.0531576
$712$	0	0
$713$	−1.79129	−0.0670843
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.5826	1.03009
$718$	0	0
$719$	18.5826	0.693013	0.346507	−	0.938048i	$-0.387368\pi$
$719$	18.5826	0.693013	0.346507	+	0.938048i	$0.387368\pi$
$720$	0	0
$721$	31.9129	1.18850
$722$	0	0
$723$	−16.3739	−0.608951
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−30.1652	−1.11570
$732$	0	0
$733$	24.4955	0.904760	0.452380	−	0.891825i	$-0.350575\pi$
$733$	24.4955	0.904760	0.452380	+	0.891825i	$0.350575\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.16515	−0.0797544
$738$	0	0
$739$	18.9564	0.697324	0.348662	−	0.937249i	$-0.386636\pi$
$739$	18.9564	0.697324	0.348662	+	0.937249i	$0.386636\pi$
$740$	0	0
$741$	−12.3303	−0.452965
$742$	0	0
$743$	31.4519	1.15386	0.576929	−	0.816794i	$-0.304251\pi$
$743$	31.4519	1.15386	0.576929	+	0.816794i	$0.304251\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.00000	0.0365881
$748$	0	0
$749$	−30.9564	−1.13112
$750$	0	0
$751$	42.1216	1.53704	0.768519	−	0.639827i	$-0.220994\pi$
$751$	42.1216	1.53704	0.768519	+	0.639827i	$0.220994\pi$
$752$	0	0
$753$	−20.2087	−0.736446
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−54.0345	−1.96392	−0.981958	−	0.189099i	$-0.939443\pi$
$757$	−54.0345	−1.96392	−0.981958	+	0.189099i	$0.939443\pi$
$758$	0	0
$759$	−1.79129	−0.0650196
$760$	0	0
$761$	19.3303	0.700723	0.350361	−	0.936615i	$-0.386059\pi$
$761$	19.3303	0.700723	0.350361	+	0.936615i	$0.386059\pi$
$762$	0	0
$763$	36.9564	1.33791
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.37386	−0.302363
$768$	0	0
$769$	−51.9564	−1.87360	−0.936799	−	0.349869i	$-0.886226\pi$
$769$	−51.9564	−1.87360	−0.936799	+	0.349869i	$0.886226\pi$
$770$	0	0
$771$	−5.20871	−0.187587
$772$	0	0
$773$	−50.3303	−1.81026	−0.905128	−	0.425140i	$-0.860225\pi$
$773$	−50.3303	−1.81026	−0.905128	+	0.425140i	$0.860225\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.3739	−0.408035
$778$	0	0
$779$	−3.25227	−0.116525
$780$	0	0
$781$	−16.0436	−0.574084
$782$	0	0
$783$	1.58258	0.0565566
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−36.4955	−1.30092	−0.650461	−	0.759539i	$-0.725424\pi$
$787$	−36.4955	−1.30092	−0.650461	+	0.759539i	$0.725424\pi$
$788$	0	0
$789$	3.53901	0.125992
$790$	0	0
$791$	45.3303	1.61176
$792$	0	0
$793$	−3.03447	−0.107757
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.5826	−1.01245	−0.506223	−	0.862402i	$-0.668959\pi$
$797$	−28.5826	−1.01245	−0.506223	+	0.862402i	$0.668959\pi$
$798$	0	0
$799$	4.58258	0.162120
$800$	0	0
$801$	−5.00000	−0.176666
$802$	0	0
$803$	−23.2867	−0.821771
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.747727	0.0263212
$808$	0	0
$809$	50.0780	1.76065	0.880325	−	0.474371i	$-0.157325\pi$
$809$	50.0780	1.76065	0.880325	+	0.474371i	$0.157325\pi$
$810$	0	0
$811$	−28.4955	−1.00061	−0.500305	−	0.865849i	$-0.666779\pi$
$811$	−28.4955	−1.00061	−0.500305	+	0.865849i	$0.666779\pi$
$812$	0	0
$813$	−10.1652	−0.356507
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.7477	1.28564
$818$	0	0
$819$	8.37386	0.292606
$820$	0	0
$821$	−22.4955	−0.785097	−0.392548	−	0.919731i	$-0.628406\pi$
$821$	−22.4955	−0.785097	−0.392548	+	0.919731i	$0.628406\pi$
$822$	0	0
$823$	−24.4519	−0.852339	−0.426170	−	0.904643i	$-0.640137\pi$
$823$	−24.4519	−0.852339	−0.426170	+	0.904643i	$0.640137\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.4174	−0.605663	−0.302832	−	0.953044i	$-0.597932\pi$
$827$	−17.4174	−0.605663	−0.302832	+	0.953044i	$0.597932\pi$
$828$	0	0
$829$	−17.0780	−0.593144	−0.296572	−	0.955010i	$-0.595844\pi$
$829$	−17.0780	−0.593144	−0.296572	+	0.955010i	$0.595844\pi$
$830$	0	0
$831$	0.791288	0.0274495
$832$	0	0
$833$	−33.7913	−1.17080
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.79129	−0.0619160
$838$	0	0
$839$	−26.7042	−0.921930	−0.460965	−	0.887418i	$-0.652497\pi$
$839$	−26.7042	−0.921930	−0.460965	+	0.887418i	$0.652497\pi$
$840$	0	0
$841$	−26.4955	−0.913636
$842$	0	0
$843$	−12.1652	−0.418990
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−29.5390	−1.01497
$848$	0	0
$849$	2.58258	0.0886338
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.956439	0.0326713	0.0163357	−	0.999867i	$-0.494800\pi$
$857$	0.956439	0.0326713	0.0163357	+	0.999867i	$0.494800\pi$
$858$	0	0
$859$	0.704166	0.0240258	0.0120129	−	0.999928i	$-0.496176\pi$
$859$	0.704166	0.0240258	0.0120129	+	0.999928i	$0.496176\pi$
$860$	0	0
$861$	2.20871	0.0752727
$862$	0	0
$863$	−27.6606	−0.941578	−0.470789	−	0.882246i	$-0.656031\pi$
$863$	−27.6606	−0.941578	−0.470789	+	0.882246i	$0.656031\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.00000	−0.135847
$868$	0	0
$869$	2.53901	0.0861302
$870$	0	0
$871$	2.66970	0.0904592
$872$	0	0
$873$	−4.62614	−0.156571
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.79129	0.0604875	0.0302437	−	0.999543i	$-0.490372\pi$
$877$	1.79129	0.0604875	0.0302437	+	0.999543i	$0.490372\pi$
$878$	0	0
$879$	−14.1652	−0.477779
$880$	0	0
$881$	−41.8693	−1.41061	−0.705307	−	0.708902i	$-0.749191\pi$
$881$	−41.8693	−1.41061	−0.705307	+	0.708902i	$0.749191\pi$
$882$	0	0
$883$	−29.1652	−0.981485	−0.490743	−	0.871305i	$-0.663275\pi$
$883$	−29.1652	−0.981485	−0.490743	+	0.871305i	$0.663275\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.37386	0.314744	0.157372	−	0.987539i	$-0.449698\pi$
$887$	9.37386	0.314744	0.157372	+	0.987539i	$0.449698\pi$
$888$	0	0
$889$	22.7477	0.762934
$890$	0	0
$891$	−1.79129	−0.0600104
$892$	0	0
$893$	−5.58258	−0.186814
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.20871	0.0737468
$898$	0	0
$899$	−2.83485	−0.0945475
$900$	0	0
$901$	−19.2867	−0.642535
$902$	0	0
$903$	−24.9564	−0.830498
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−52.2432	−1.73471	−0.867353	−	0.497693i	$-0.834181\pi$
$907$	−52.2432	−1.73471	−0.867353	+	0.497693i	$0.834181\pi$
$908$	0	0
$909$	−3.62614	−0.120271
$910$	0	0
$911$	−37.0000	−1.22586	−0.612932	−	0.790135i	$-0.710010\pi$
$911$	−37.0000	−1.22586	−0.612932	+	0.790135i	$0.710010\pi$
$912$	0	0
$913$	−1.79129	−0.0592830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.9564	1.02227
$918$	0	0
$919$	25.9129	0.854787	0.427393	−	0.904066i	$-0.359432\pi$
$919$	25.9129	0.854787	0.427393	+	0.904066i	$0.359432\pi$
$920$	0	0
$921$	24.3303	0.801711
$922$	0	0
$923$	19.7822	0.651139
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.41742	0.276464
$928$	0	0
$929$	−13.2867	−0.435924	−0.217962	−	0.975957i	$-0.569941\pi$
$929$	−13.2867	−0.435924	−0.217962	+	0.975957i	$0.569941\pi$
$930$	0	0
$931$	41.1652	1.34913
$932$	0	0
$933$	2.16515	0.0708839
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.0780	1.21129	0.605643	−	0.795736i	$-0.292916\pi$
$937$	37.0780	1.21129	0.605643	+	0.795736i	$0.292916\pi$
$938$	0	0
$939$	−24.5826	−0.802222
$940$	0	0
$941$	21.8693	0.712919	0.356460	−	0.934311i	$-0.383984\pi$
$941$	21.8693	0.712919	0.356460	+	0.934311i	$0.383984\pi$
$942$	0	0
$943$	0.582576	0.0189713
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.79129	−0.155696	−0.0778480	−	0.996965i	$-0.524805\pi$
$947$	−4.79129	−0.155696	−0.0778480	+	0.996965i	$0.524805\pi$
$948$	0	0
$949$	28.7133	0.932072
$950$	0	0
$951$	−33.7913	−1.09576
$952$	0	0
$953$	−11.0000	−0.356325	−0.178162	−	0.984001i	$-0.557015\pi$
$953$	−11.0000	−0.356325	−0.178162	+	0.984001i	$0.557015\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.83485	−0.0916376
$958$	0	0
$959$	−7.58258	−0.244854
$960$	0	0
$961$	−27.7913	−0.896493
$962$	0	0
$963$	−8.16515	−0.263118
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.3303	−1.20046	−0.600231	−	0.799827i	$-0.704925\pi$
$967$	−37.3303	−1.20046	−0.600231	+	0.799827i	$0.704925\pi$
$968$	0	0
$969$	25.5826	0.821831
$970$	0	0
$971$	−19.9129	−0.639035	−0.319517	−	0.947580i	$-0.603521\pi$
$971$	−19.9129	−0.639035	−0.319517	+	0.947580i	$0.603521\pi$
$972$	0	0
$973$	−49.1216	−1.57477
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.7477	1.23965	0.619825	−	0.784740i	$-0.287204\pi$
$977$	38.7477	1.23965	0.619825	+	0.784740i	$0.287204\pi$
$978$	0	0
$979$	8.95644	0.286249
$980$	0	0
$981$	9.74773	0.311221
$982$	0	0
$983$	36.0345	1.14932	0.574661	−	0.818392i	$-0.305134\pi$
$983$	36.0345	1.14932	0.574661	+	0.818392i	$0.305134\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.79129	0.120678
$988$	0	0
$989$	−6.58258	−0.209314
$990$	0	0
$991$	−41.0780	−1.30489	−0.652443	−	0.757838i	$-0.726256\pi$
$991$	−41.0780	−1.30489	−0.652443	+	0.757838i	$0.726256\pi$
$992$	0	0
$993$	7.53901	0.239243
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−32.6261	−1.03328	−0.516640	−	0.856203i	$-0.672817\pi$
$997$	−32.6261	−1.03328	−0.516640	+	0.856203i	$0.672817\pi$
$998$	0	0
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.a.o.1.2	✓	2
5.2	odd	4		6900.2.f.n.6349.4		4
5.3	odd	4		6900.2.f.n.6349.1		4
5.4	even	2		6900.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6900.2.a.o.1.2	✓	2	1.1	even	1	trivial
6900.2.a.q.1.1	yes	2	5.4	even	2
6900.2.f.n.6349.1		4	5.3	odd	4
6900.2.f.n.6349.4		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.79129 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.79129 q^{7} +1.00000 q^{9} -1.79129 q^{11} +2.20871 q^{13} -4.58258 q^{17} +5.58258 q^{19} -3.79129 q^{21} -1.00000 q^{23} -1.00000 q^{27} -1.58258 q^{29} +1.79129 q^{31} +1.79129 q^{33} +3.00000 q^{37} -2.20871 q^{39} -0.582576 q^{41} +6.58258 q^{43} -1.00000 q^{47} +7.37386 q^{49} +4.58258 q^{51} +4.20871 q^{53} -5.58258 q^{57} -3.79129 q^{59} -1.37386 q^{61} +3.79129 q^{63} +1.20871 q^{67} +1.00000 q^{69} +8.95644 q^{71} +13.0000 q^{73} -6.79129 q^{77} -1.41742 q^{79} +1.00000 q^{81} +1.00000 q^{83} +1.58258 q^{87} -5.00000 q^{89} +8.37386 q^{91} -1.79129 q^{93} -4.62614 q^{97} -1.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 3 q^{7} + 2 q^{9} + q^{11} + 9 q^{13} + 2 q^{19} - 3 q^{21} - 2 q^{23} - 2 q^{27} + 6 q^{29} - q^{31} - q^{33} + 6 q^{37} - 9 q^{39} + 8 q^{41} + 4 q^{43} - 2 q^{47} + q^{49} + 13 q^{53} - 2 q^{57} - 3 q^{59} + 11 q^{61} + 3 q^{63} + 7 q^{67} + 2 q^{69} - 5 q^{71} + 26 q^{73} - 9 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} - 6 q^{87} - 10 q^{89} + 3 q^{91} + q^{93} - 23 q^{97} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^7 + 2 * q^9 + q^11 + 9 * q^13 + 2 * q^19 - 3 * q^21 - 2 * q^23 - 2 * q^27 + 6 * q^29 - q^31 - q^33 + 6 * q^37 - 9 * q^39 + 8 * q^41 + 4 * q^43 - 2 * q^47 + q^49 + 13 * q^53 - 2 * q^57 - 3 * q^59 + 11 * q^61 + 3 * q^63 + 7 * q^67 + 2 * q^69 - 5 * q^71 + 26 * q^73 - 9 * q^77 - 12 * q^79 + 2 * q^81 + 2 * q^83 - 6 * q^87 - 10 * q^89 + 3 * q^91 + q^93 - 23 * q^97 + q^99

Introduction

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