LMFDB - Embedded newform 6900.2.f.b.6349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(6349,6900)]

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, 1, 0]))

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

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

chi := DirichletCharacter("6900.6349");

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.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.b.6349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.00000i q^{13} -7.00000i q^{17} +6.00000 q^{19} -3.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} +9.00000 q^{29} +9.00000 q^{31} +2.00000i q^{33} -7.00000i q^{37} +2.00000 q^{39} +5.00000 q^{41} +8.00000i q^{47} -2.00000 q^{49} -7.00000 q^{51} +11.0000i q^{53} -6.00000i q^{57} -9.00000 q^{59} +3.00000i q^{63} -3.00000i q^{67} -1.00000 q^{69} +3.00000 q^{71} +6.00000i q^{73} +6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -5.00000i q^{83} -9.00000i q^{87} +6.00000 q^{91} -9.00000i q^{93} -10.0000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 4 q^{11} + 12 q^{19} - 6 q^{21} + 18 q^{29} + 18 q^{31} + 4 q^{39} + 10 q^{41} - 4 q^{49} - 14 q^{51} - 18 q^{59} - 2 q^{69} + 6 q^{71} + 16 q^{79} + 2 q^{81} + 12 q^{91} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^11 + 12 * q^19 - 6 * q^21 + 18 * q^29 + 18 * q^31 + 4 * q^39 + 10 * q^41 - 4 * q^49 - 14 * q^51 - 18 * q^59 - 2 * q^69 + 6 * q^71 + 16 * q^79 + 2 * q^81 + 12 * q^91 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	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
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−7.00000	−0.980196
$52$	0	0
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6.00000i	− 0.794719i
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	3.00000i	0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.00000i	0.683763i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.00000i	− 0.548821i	−0.961613	−	0.274411i	$-0.911517\pi$
$83$	− 5.00000i	− 0.548821i	0.961613	−	0.274411i	$-0.0884828\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 9.00000i	− 0.964901i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	− 9.00000i	− 0.933257i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.00000i	0.483368i	0.970355	+	0.241684i	$0.0776998\pi$
$107$	5.00000i	0.483368i	−0.970355	+	0.241684i	$0.922300\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	0	0
$113$	− 21.0000i	− 1.97551i	−0.156001	−	0.987757i	$-0.549860\pi$
$113$	− 21.0000i	− 1.97551i	0.156001	−	0.987757i	$-0.450140\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−21.0000	−1.92507
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 5.00000i	− 0.450835i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	− 18.0000i	− 1.56080i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.0000i	− 1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$	− 14.0000i	− 1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	7.00000i	0.565916i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.00000i	− 0.399043i	−0.979893	−	0.199522i	$-0.936061\pi$
$157$	− 5.00000i	− 0.399043i	0.979893	−	0.199522i	$-0.0639388\pi$
$158$	0	0
$159$	11.0000	0.872357
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	− 4.00000i	− 0.304114i	−0.988372	−	0.152057i	$-0.951410\pi$
$173$	− 4.00000i	− 0.304114i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.00000i	0.676481i
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.0000i	1.02378i
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	24.0000i	1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$	24.0000i	1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	− 27.0000i	− 1.89503i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	0	0
$213$	− 3.00000i	− 0.205557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 27.0000i	− 1.83288i
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	14.0000	0.941742
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	− 16.0000i	− 1.04819i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	− 16.0000i	− 1.04819i	0.851658	−	0.524097i	$-0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	−5.00000	−0.316862
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	0	0
$259$	−21.0000	−1.30488
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	− 11.0000i	− 0.678289i	−0.940734	−	0.339145i	$-0.889862\pi$
$263$	− 11.0000i	− 0.678289i	0.940734	−	0.339145i	$-0.110138\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	31.0000	1.88312	0.941558	−	0.336851i	$-0.109362\pi$
$271$	31.0000	1.88312	0.941558	+	0.336851i	$0.109362\pi$
$272$	0	0
$273$	− 6.00000i	− 0.363137i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	0	0
$279$	−9.00000	−0.538816
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	5.00000i	0.297219i	0.988896	+	0.148610i	$0.0474798\pi$
$283$	5.00000i	0.297219i	−0.988896	+	0.148610i	$0.952520\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 15.0000i	− 0.885422i
$288$	0	0
$289$	−32.0000	−1.88235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.00000i	− 0.116052i
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 3.00000i	− 0.172345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 30.0000i	− 1.71219i	−0.516818	−	0.856095i	$-0.672884\pi$
$307$	− 30.0000i	− 1.71219i	0.516818	−	0.856095i	$-0.327116\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	29.0000i	1.63918i	0.572953	+	0.819588i	$0.305798\pi$
$313$	29.0000i	1.63918i	−0.572953	+	0.819588i	$0.694202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	− 42.0000i	− 2.33694i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.0000i	0.553001i
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	21.0000	1.15426	0.577132	−	0.816651i	$-0.304172\pi$
$331$	21.0000	1.15426	0.577132	+	0.816651i	$0.304172\pi$
$332$	0	0
$333$	7.00000i	0.383598i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 26.0000i	− 1.41631i	−0.706057	−	0.708155i	$-0.749528\pi$
$337$	− 26.0000i	− 1.41631i	0.706057	−	0.708155i	$-0.250472\pi$
$338$	0	0
$339$	−21.0000	−1.14056
$340$	0	0
$341$	−18.0000	−0.974755
$342$	0	0
$343$	− 15.0000i	− 0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−31.0000	−1.65939	−0.829696	−	0.558216i	$-0.811486\pi$
$349$	−31.0000	−1.65939	−0.829696	+	0.558216i	$0.811486\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	28.0000i	1.49029i	0.666903	+	0.745145i	$0.267620\pi$
$353$	28.0000i	1.49029i	−0.666903	+	0.745145i	$0.732380\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21.0000i	1.11144i
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.00000i	0.365397i	0.983169	+	0.182699i	$0.0584832\pi$
$367$	7.00000i	0.365397i	−0.983169	+	0.182699i	$0.941517\pi$
$368$	0	0
$369$	−5.00000	−0.260290
$370$	0	0
$371$	33.0000	1.71327
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0000i	0.927047i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	− 27.0000i	− 1.37964i	−0.723983	−	0.689818i	$-0.757691\pi$
$383$	− 27.0000i	− 1.37964i	0.723983	−	0.689818i	$-0.242309\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	0	0
$393$	12.0000i	0.605320i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 28.0000i	− 1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$	− 28.0000i	− 1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	18.0000i	0.896644i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.0000i	0.693954i
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	27.0000i	1.32858i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5.00000i	− 0.244851i
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	− 8.00000i	− 0.388973i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	− 1.00000i	− 0.0480569i	−0.999711	−	0.0240285i	$-0.992351\pi$
$433$	− 1.00000i	− 0.0480569i	0.999711	−	0.0240285i	$-0.00764923\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.00000i	− 0.189194i
$448$	0	0
$449$	17.0000	0.802280	0.401140	−	0.916017i	$-0.368614\pi$
$449$	17.0000	0.802280	0.401140	+	0.916017i	$0.368614\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	4.00000i	0.187936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.00000i	0.327446i	0.986506	+	0.163723i	$0.0523504\pi$
$457$	7.00000i	0.327446i	−0.986506	+	0.163723i	$0.947650\pi$
$458$	0	0
$459$	7.00000	0.326732
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.0000i	0.971764i	0.874024	+	0.485882i	$0.161502\pi$
$467$	21.0000i	0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	−5.00000	−0.230388
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 11.0000i	− 0.503655i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	0	0
$489$	14.0000	0.633102
$490$	0	0
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	0	0
$493$	− 63.0000i	− 2.83738i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 9.00000i	− 0.403705i
$498$	0	0
$499$	−15.0000	−0.671492	−0.335746	−	0.941953i	$-0.608988\pi$
$499$	−15.0000	−0.671492	−0.335746	+	0.941953i	$0.608988\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	0	0
$503$	33.0000i	1.47140i	0.677309	+	0.735699i	$0.263146\pi$
$503$	33.0000i	1.47140i	−0.677309	+	0.735699i	$0.736854\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	18.0000	0.796273
$512$	0	0
$513$	6.00000i	0.264906i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 63.0000i	− 2.74432i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	9.00000	0.390567
$532$	0	0
$533$	10.0000i	0.433148i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000i	0.690451i
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	− 12.0000i	− 0.514969i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.0000i	− 1.02617i	−0.858339	−	0.513083i	$-0.828503\pi$
$547$	− 24.0000i	− 1.02617i	0.858339	−	0.513083i	$-0.171497\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	54.0000	2.30048
$552$	0	0
$553$	− 24.0000i	− 1.02058i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11.0000i	− 0.466085i	−0.972467	−	0.233042i	$-0.925132\pi$
$557$	− 11.0000i	− 0.466085i	0.972467	−	0.233042i	$-0.0748681\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	14.0000	0.591080
$562$	0	0
$563$	5.00000i	0.210725i	0.994434	+	0.105362i	$0.0336003\pi$
$563$	5.00000i	0.210725i	−0.994434	+	0.105362i	$0.966400\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.00000i	− 0.125988i
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	− 4.00000i	− 0.167102i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	0	0
$579$	24.0000	0.997406
$580$	0	0
$581$	−15.0000	−0.622305
$582$	0	0
$583$	− 22.0000i	− 0.911147i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.00000i	− 0.0825488i	−0.999148	−	0.0412744i	$-0.986858\pi$
$587$	− 2.00000i	− 0.0825488i	0.999148	−	0.0412744i	$-0.0131418\pi$
$588$	0	0
$589$	54.0000	2.22503
$590$	0	0
$591$	14.0000	0.575883
$592$	0	0
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.0000i	0.572982i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	3.00000i	0.122169i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	−27.0000	−1.09410
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.00000i	− 0.362326i	−0.983453	−	0.181163i	$-0.942014\pi$
$617$	− 9.00000i	− 0.362326i	0.983453	−	0.181163i	$-0.0579862\pi$
$618$	0	0
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.0000i	0.479234i
$628$	0	0
$629$	−49.0000	−1.95376
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	0	0
$633$	23.0000i	0.914168i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	−3.00000	−0.118678
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	23.0000i	0.907031i	0.891248	+	0.453516i	$0.149830\pi$
$643$	23.0000i	0.907031i	−0.891248	+	0.453516i	$0.850170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.0000i	0.393141i	0.980490	+	0.196570i	$0.0629804\pi$
$647$	10.0000i	0.393141i	−0.980490	+	0.196570i	$0.937020\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	−27.0000	−1.05821
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	0	0
$663$	− 14.0000i	− 0.543715i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 16.0000i	− 0.616755i	−0.951264	−	0.308377i	$-0.900214\pi$
$673$	− 16.0000i	− 0.616755i	0.951264	−	0.308377i	$-0.0997859\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 21.0000i	− 0.807096i	−0.914959	−	0.403548i	$-0.867777\pi$
$677$	− 21.0000i	− 0.807096i	0.914959	−	0.403548i	$-0.132223\pi$
$678$	0	0
$679$	−30.0000	−1.15129
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16.0000i	− 0.610438i
$688$	0	0
$689$	−22.0000	−0.838133
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	0	0
$693$	− 6.00000i	− 0.227921i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 35.0000i	− 1.32572i
$698$	0	0
$699$	−16.0000	−0.605176
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	− 42.0000i	− 1.58406i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.00000i	− 0.338480i
$708$	0	0
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	− 9.00000i	− 0.337053i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.0000i	0.784259i
$718$	0	0
$719$	43.0000	1.60363	0.801815	−	0.597573i	$-0.203868\pi$
$719$	43.0000	1.60363	0.801815	+	0.597573i	$0.203868\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	0	0
$723$	− 26.0000i	− 0.966950i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.0000i	0.853023i	0.904482	+	0.426511i	$0.140258\pi$
$727$	23.0000i	0.853023i	−0.904482	+	0.426511i	$0.859742\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	47.0000i	1.73598i	0.496578	+	0.867992i	$0.334590\pi$
$733$	47.0000i	1.73598i	−0.496578	+	0.867992i	$0.665410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.00000i	0.221013i
$738$	0	0
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	32.0000i	1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$	32.0000i	1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.00000i	0.182940i
$748$	0	0
$749$	15.0000	0.548088
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	0	0
$753$	30.0000i	1.09326i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.00000i	− 0.0363456i	−0.999835	−	0.0181728i	$-0.994215\pi$
$757$	− 1.00000i	− 0.0363456i	0.999835	−	0.0181728i	$-0.00578490\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	−39.0000	−1.41375	−0.706874	−	0.707339i	$-0.749895\pi$
$761$	−39.0000	−1.41375	−0.706874	+	0.707339i	$0.749895\pi$
$762$	0	0
$763$	30.0000i	1.08607i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 18.0000i	− 0.649942i
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−8.00000	−0.288113
$772$	0	0
$773$	54.0000i	1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$	54.0000i	1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	21.0000i	0.753371i
$778$	0	0
$779$	30.0000	1.07486
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	9.00000i	0.321634i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31.0000i	− 1.10503i	−0.833503	−	0.552515i	$-0.813668\pi$
$787$	− 31.0000i	− 1.10503i	0.833503	−	0.552515i	$-0.186332\pi$
$788$	0	0
$789$	−11.0000	−0.391610
$790$	0	0
$791$	−63.0000	−2.24002
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.0000i	0.389640i	0.980839	+	0.194820i	$0.0624123\pi$
$797$	11.0000i	0.389640i	−0.980839	+	0.194820i	$0.937588\pi$
$798$	0	0
$799$	56.0000	1.98114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.00000i	− 0.105605i
$808$	0	0
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	− 31.0000i	− 1.08722i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−6.00000	−0.209657
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	− 52.0000i	− 1.81261i	−0.422628	−	0.906303i	$-0.638892\pi$
$823$	− 52.0000i	− 1.81261i	0.422628	−	0.906303i	$-0.361108\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.0000i	1.07798i	0.842314	+	0.538988i	$0.181193\pi$
$827$	31.0000i	1.07798i	−0.842314	+	0.538988i	$0.818807\pi$
$828$	0	0
$829$	−49.0000	−1.70184	−0.850920	−	0.525295i	$-0.823955\pi$
$829$	−49.0000	−1.70184	−0.850920	+	0.525295i	$0.823955\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	14.0000i	0.485071i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.00000i	0.311086i
$838$	0	0
$839$	−2.00000	−0.0690477	−0.0345238	−	0.999404i	$-0.510991\pi$
$839$	−2.00000	−0.0690477	−0.0345238	+	0.999404i	$0.510991\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 10.0000i	− 0.344418i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.0000i	0.721569i
$848$	0	0
$849$	5.00000	0.171600
$850$	0	0
$851$	−7.00000	−0.239957
$852$	0	0
$853$	30.0000i	1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$	30.0000i	1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	43.0000	1.46714	0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000	1.46714	0.733571	+	0.679613i	$0.237852\pi$
$860$	0	0
$861$	−15.0000	−0.511199
$862$	0	0
$863$	− 6.00000i	− 0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$	− 6.00000i	− 0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	32.0000i	1.08678i
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	10.0000i	0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 34.0000i	− 1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	− 34.0000i	− 1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$	0	0
$879$	9.00000	0.303562
$880$	0	0
$881$	−16.0000	−0.539054	−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000	−0.539054	−0.269527	+	0.962993i	$0.586867\pi$
$882$	0	0
$883$	− 12.0000i	− 0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$	− 12.0000i	− 0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.0000i	1.27592i	0.770072	+	0.637958i	$0.220220\pi$
$887$	38.0000i	1.27592i	−0.770072	+	0.637958i	$0.779780\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.00000i	− 0.0667781i
$898$	0	0
$899$	81.0000	2.70150
$900$	0	0
$901$	77.0000	2.56524
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	37.0000i	1.22856i	0.789086	+	0.614282i	$0.210554\pi$
$907$	37.0000i	1.22856i	−0.789086	+	0.614282i	$0.789446\pi$
$908$	0	0
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	10.0000i	0.330952i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.0000i	1.18882i
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	0	0
$923$	6.00000i	0.197492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	57.0000	1.87011	0.935055	−	0.354504i	$-0.115350\pi$
$929$	57.0000	1.87011	0.935055	+	0.354504i	$0.115350\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	46.0000i	1.50275i	0.659873	+	0.751377i	$0.270610\pi$
$937$	46.0000i	1.50275i	−0.659873	+	0.751377i	$0.729390\pi$
$938$	0	0
$939$	29.0000	0.946379
$940$	0	0
$941$	−8.00000	−0.260793	−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000	−0.260793	−0.130396	+	0.991462i	$0.541625\pi$
$942$	0	0
$943$	− 5.00000i	− 0.162822i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	18.0000i	0.581857i
$958$	0	0
$959$	−42.0000	−1.35625
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	− 5.00000i	− 0.161123i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	0	0
$969$	−42.0000	−1.34923
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	− 15.0000i	− 0.480878i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 37.0000i	− 1.18373i	−0.806035	−	0.591867i	$-0.798391\pi$
$977$	− 37.0000i	− 1.18373i	0.806035	−	0.591867i	$-0.201609\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	− 41.0000i	− 1.30770i	−0.756626	−	0.653848i	$-0.773153\pi$
$983$	− 41.0000i	− 1.30770i	0.756626	−	0.653848i	$-0.226847\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−59.0000	−1.87420	−0.937098	−	0.349065i	$-0.886499\pi$
$991$	−59.0000	−1.87420	−0.937098	+	0.349065i	$0.886499\pi$
$992$	0	0
$993$	− 21.0000i	− 0.666415i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.0000i	0.506725i	0.967371	+	0.253363i	$0.0815366\pi$
$997$	16.0000i	0.506725i	−0.967371	+	0.253363i	$0.918463\pi$
$998$	0	0
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.b.6349.1		2
5.2	odd	4		6900.2.a.c.1.1		1
5.3	odd	4		1380.2.a.e.1.1	✓	1
5.4	even	2	inner	6900.2.f.b.6349.2		2
15.8	even	4		4140.2.a.b.1.1		1
20.3	even	4		5520.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.e.1.1	✓	1	5.3	odd	4
4140.2.a.b.1.1		1	15.8	even	4
5520.2.a.m.1.1		1	20.3	even	4
6900.2.a.c.1.1		1	5.2	odd	4
6900.2.f.b.6349.1		2	1.1	even	1	trivial
6900.2.f.b.6349.2		2	5.4	even	2	inner

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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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.00000i q^{13} -7.00000i q^{17} +6.00000 q^{19} -3.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} +9.00000 q^{29} +9.00000 q^{31} +2.00000i q^{33} -7.00000i q^{37} +2.00000 q^{39} +5.00000 q^{41} +8.00000i q^{47} -2.00000 q^{49} -7.00000 q^{51} +11.0000i q^{53} -6.00000i q^{57} -9.00000 q^{59} +3.00000i q^{63} -3.00000i q^{67} -1.00000 q^{69} +3.00000 q^{71} +6.00000i q^{73} +6.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} -5.00000i q^{83} -9.00000i q^{87} +6.00000 q^{91} -9.00000i q^{93} -10.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 4 q^{11} + 12 q^{19} - 6 q^{21} + 18 q^{29} + 18 q^{31} + 4 q^{39} + 10 q^{41} - 4 q^{49} - 14 q^{51} - 18 q^{59} - 2 q^{69} + 6 q^{71} + 16 q^{79} + 2 q^{81} + 12 q^{91} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^11 + 12 * q^19 - 6 * q^21 + 18 * q^29 + 18 * q^31 + 4 * q^39 + 10 * q^41 - 4 * q^49 - 14 * q^51 - 18 * q^59 - 2 * q^69 + 6 * q^71 + 16 * q^79 + 2 * q^81 + 12 * q^91 + 4 * q^99

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