LMFDB - Embedded newform 8280.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 $ x^7 - 3x^6 - 18x^5 + 46x^4 + 60x^3 - 76x^2 - 51x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.70204$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -5.18676 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -5.18676 q^{7} +5.53051 q^{11} +1.78900 q^{13} -2.22405 q^{17} -8.22035 q^{19} +1.00000 q^{23} +1.00000 q^{25} +7.06032 q^{29} -2.06612 q^{31} -5.18676 q^{35} -5.02631 q^{37} +11.4917 q^{41} -8.62968 q^{43} +1.37239 q^{47} +19.9025 q^{49} -4.46906 q^{53} +5.53051 q^{55} +11.1486 q^{59} +7.81573 q^{61} +1.78900 q^{65} -1.96903 q^{67} -8.51894 q^{71} -10.7482 q^{73} -28.6854 q^{77} -14.0361 q^{79} +6.83698 q^{83} -2.22405 q^{85} -15.5115 q^{89} -9.27911 q^{91} -8.22035 q^{95} -4.19194 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q + 7 * q^5 - 6 * q^7	$ 7 q + 7 q^{5} - 6 q^{7} - 2 q^{11} - 6 q^{13} - 4 q^{17} - 8 q^{19} + 7 q^{23} + 7 q^{25} + 2 q^{29} - 8 q^{31} - 6 q^{35} - 16 q^{37} + 2 q^{41} - 10 q^{43} - 8 q^{47} + 19 q^{49} - 10 q^{53} - 2 q^{55} - 24 q^{59} + 8 q^{61} - 6 q^{65} - 20 q^{67} - 8 q^{71} + 2 q^{73} - 12 q^{77} - 2 q^{79} - 22 q^{83} - 4 q^{85} - 16 q^{89} - 20 q^{91} - 8 q^{95} + 4 q^{97}+O(q^{100}) $ 7 * q + 7 * q^5 - 6 * q^7 - 2 * q^11 - 6 * q^13 - 4 * q^17 - 8 * q^19 + 7 * q^23 + 7 * q^25 + 2 * q^29 - 8 * q^31 - 6 * q^35 - 16 * q^37 + 2 * q^41 - 10 * q^43 - 8 * q^47 + 19 * q^49 - 10 * q^53 - 2 * q^55 - 24 * q^59 + 8 * q^61 - 6 * q^65 - 20 * q^67 - 8 * q^71 + 2 * q^73 - 12 * q^77 - 2 * q^79 - 22 * q^83 - 4 * q^85 - 16 * q^89 - 20 * q^91 - 8 * q^95 + 4 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−5.18676	−1.96041	−0.980205	−	0.197983i	$-0.936561\pi$
$7$	−5.18676	−1.96041	−0.980205	+	0.197983i	$0.936561\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.53051	1.66751	0.833756	−	0.552132i	$-0.186186\pi$
$11$	5.53051	1.66751	0.833756	+	0.552132i	$0.186186\pi$
$12$	0	0
$13$	1.78900	0.496179	0.248090	−	0.968737i	$-0.420197\pi$
$13$	1.78900	0.496179	0.248090	+	0.968737i	$0.420197\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.22405	−0.539412	−0.269706	−	0.962943i	$-0.586926\pi$
$17$	−2.22405	−0.539412	−0.269706	+	0.962943i	$0.586926\pi$
$18$	0	0
$19$	−8.22035	−1.88588	−0.942939	−	0.332967i	$-0.891950\pi$
$19$	−8.22035	−1.88588	−0.942939	+	0.332967i	$0.891950\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.06032	1.31107	0.655534	−	0.755166i	$-0.272444\pi$
$29$	7.06032	1.31107	0.655534	+	0.755166i	$0.272444\pi$
$30$	0	0
$31$	−2.06612	−0.371086	−0.185543	−	0.982636i	$-0.559404\pi$
$31$	−2.06612	−0.371086	−0.185543	+	0.982636i	$0.559404\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.18676	−0.876722
$36$	0	0
$37$	−5.02631	−0.826320	−0.413160	−	0.910658i	$-0.635575\pi$
$37$	−5.02631	−0.826320	−0.413160	+	0.910658i	$0.635575\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.4917	1.79470	0.897348	−	0.441323i	$-0.145491\pi$
$41$	11.4917	1.79470	0.897348	+	0.441323i	$0.145491\pi$
$42$	0	0
$43$	−8.62968	−1.31601	−0.658007	−	0.753012i	$-0.728600\pi$
$43$	−8.62968	−1.31601	−0.658007	+	0.753012i	$0.728600\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.37239	0.200183	0.100092	−	0.994978i	$-0.468086\pi$
$47$	1.37239	0.200183	0.100092	+	0.994978i	$0.468086\pi$
$48$	0	0
$49$	19.9025	2.84321
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.46906	−0.613873	−0.306936	−	0.951730i	$-0.599304\pi$
$53$	−4.46906	−0.613873	−0.306936	+	0.951730i	$0.599304\pi$
$54$	0	0
$55$	5.53051	0.745734
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.1486	1.45143	0.725713	−	0.687997i	$-0.241510\pi$
$59$	11.1486	1.45143	0.725713	+	0.687997i	$0.241510\pi$
$60$	0	0
$61$	7.81573	1.00070	0.500351	−	0.865823i	$-0.333204\pi$
$61$	7.81573	1.00070	0.500351	+	0.865823i	$0.333204\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.78900	0.221898
$66$	0	0
$67$	−1.96903	−0.240555	−0.120277	−	0.992740i	$-0.538378\pi$
$67$	−1.96903	−0.240555	−0.120277	+	0.992740i	$0.538378\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.51894	−1.01101	−0.505506	−	0.862823i	$-0.668694\pi$
$71$	−8.51894	−1.01101	−0.505506	+	0.862823i	$0.668694\pi$
$72$	0	0
$73$	−10.7482	−1.25799	−0.628994	−	0.777410i	$-0.716533\pi$
$73$	−10.7482	−1.25799	−0.628994	+	0.777410i	$0.716533\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−28.6854	−3.26901
$78$	0	0
$79$	−14.0361	−1.57918	−0.789591	−	0.613633i	$-0.789707\pi$
$79$	−14.0361	−1.57918	−0.789591	+	0.613633i	$0.789707\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.83698	0.750456	0.375228	−	0.926933i	$-0.377564\pi$
$83$	6.83698	0.750456	0.375228	+	0.926933i	$0.377564\pi$
$84$	0	0
$85$	−2.22405	−0.241232
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.5115	−1.64421	−0.822105	−	0.569335i	$-0.807201\pi$
$89$	−15.5115	−1.64421	−0.822105	+	0.569335i	$0.807201\pi$
$90$	0	0
$91$	−9.27911	−0.972715
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.22035	−0.843390
$96$	0	0
$97$	−4.19194	−0.425627	−0.212814	−	0.977093i	$-0.568263\pi$
$97$	−4.19194	−0.425627	−0.212814	+	0.977093i	$0.568263\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.10435	0.209390	0.104695	−	0.994504i	$-0.466613\pi$
$101$	2.10435	0.209390	0.104695	+	0.994504i	$0.466613\pi$
$102$	0	0
$103$	8.51052	0.838566	0.419283	−	0.907856i	$-0.362281\pi$
$103$	8.51052	0.838566	0.419283	+	0.907856i	$0.362281\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.56936	0.828431	0.414216	−	0.910179i	$-0.364056\pi$
$107$	8.56936	0.828431	0.414216	+	0.910179i	$0.364056\pi$
$108$	0	0
$109$	−20.0678	−1.92214	−0.961071	−	0.276300i	$-0.910892\pi$
$109$	−20.0678	−1.92214	−0.961071	+	0.276300i	$0.910892\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.1800	−1.05173	−0.525864	−	0.850569i	$-0.676258\pi$
$113$	−11.1800	−1.05173	−0.525864	+	0.850569i	$0.676258\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.5356	1.05747
$120$	0	0
$121$	19.5866	1.78060
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.7755	−1.39985	−0.699926	−	0.714216i	$-0.746784\pi$
$127$	−15.7755	−1.39985	−0.699926	+	0.714216i	$0.746784\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.23941	−0.545140	−0.272570	−	0.962136i	$-0.587874\pi$
$131$	−6.23941	−0.545140	−0.272570	+	0.962136i	$0.587874\pi$
$132$	0	0
$133$	42.6370	3.69709
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.63201	0.224867	0.112434	−	0.993659i	$-0.464135\pi$
$137$	2.63201	0.224867	0.112434	+	0.993659i	$0.464135\pi$
$138$	0	0
$139$	1.02125	0.0866211	0.0433106	−	0.999062i	$-0.486210\pi$
$139$	1.02125	0.0866211	0.0433106	+	0.999062i	$0.486210\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.89409	0.827385
$144$	0	0
$145$	7.06032	0.586327
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.6707	1.69341	0.846705	−	0.532062i	$-0.178583\pi$
$149$	20.6707	1.69341	0.846705	+	0.532062i	$0.178583\pi$
$150$	0	0
$151$	7.02315	0.571536	0.285768	−	0.958299i	$-0.407751\pi$
$151$	7.02315	0.571536	0.285768	+	0.958299i	$0.407751\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.06612	−0.165955
$156$	0	0
$157$	−16.5529	−1.32107	−0.660535	−	0.750796i	$-0.729670\pi$
$157$	−16.5529	−1.32107	−0.660535	+	0.750796i	$0.729670\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.18676	−0.408774
$162$	0	0
$163$	11.2497	0.881144	0.440572	−	0.897717i	$-0.354776\pi$
$163$	11.2497	0.881144	0.440572	+	0.897717i	$0.354776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.7556	−1.37397	−0.686984	−	0.726673i	$-0.741066\pi$
$167$	−17.7556	−1.37397	−0.686984	+	0.726673i	$0.741066\pi$
$168$	0	0
$169$	−9.79948	−0.753806
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0126	0.761246	0.380623	−	0.924730i	$-0.375710\pi$
$173$	10.0126	0.761246	0.380623	+	0.924730i	$0.375710\pi$
$174$	0	0
$175$	−5.18676	−0.392082
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.4041	−1.15135	−0.575677	−	0.817677i	$-0.695262\pi$
$179$	−15.4041	−1.15135	−0.575677	+	0.817677i	$0.695262\pi$
$180$	0	0
$181$	9.47996	0.704640	0.352320	−	0.935880i	$-0.385393\pi$
$181$	9.47996	0.704640	0.352320	+	0.935880i	$0.385393\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.02631	−0.369542
$186$	0	0
$187$	−12.3001	−0.899476
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.8994	−0.861014	−0.430507	−	0.902587i	$-0.641665\pi$
$191$	−11.8994	−0.861014	−0.430507	+	0.902587i	$0.641665\pi$
$192$	0	0
$193$	24.6195	1.77215	0.886075	−	0.463543i	$-0.153422\pi$
$193$	24.6195	1.77215	0.886075	+	0.463543i	$0.153422\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.98652	0.141534	0.0707670	−	0.997493i	$-0.477455\pi$
$197$	1.98652	0.141534	0.0707670	+	0.997493i	$0.477455\pi$
$198$	0	0
$199$	4.07108	0.288591	0.144296	−	0.989535i	$-0.453908\pi$
$199$	4.07108	0.288591	0.144296	+	0.989535i	$0.453908\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−36.6202	−2.57023
$204$	0	0
$205$	11.4917	0.802613
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−45.4627	−3.14472
$210$	0	0
$211$	2.29266	0.157834	0.0789168	−	0.996881i	$-0.474854\pi$
$211$	2.29266	0.157834	0.0789168	+	0.996881i	$0.474854\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.62968	−0.588539
$216$	0	0
$217$	10.7165	0.727481
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.97883	−0.267645
$222$	0	0
$223$	−8.88382	−0.594905	−0.297452	−	0.954737i	$-0.596137\pi$
$223$	−8.88382	−0.594905	−0.297452	+	0.954737i	$0.596137\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.5779	−1.76404	−0.882018	−	0.471216i	$-0.843815\pi$
$227$	−26.5779	−1.76404	−0.882018	+	0.471216i	$0.843815\pi$
$228$	0	0
$229$	−15.7078	−1.03800	−0.518999	−	0.854775i	$-0.673695\pi$
$229$	−15.7078	−1.03800	−0.518999	+	0.854775i	$0.673695\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.0578	1.31403	0.657016	−	0.753876i	$-0.271818\pi$
$233$	20.0578	1.31403	0.657016	+	0.753876i	$0.271818\pi$
$234$	0	0
$235$	1.37239	0.0895247
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.0908704	0.00587792	0.00293896	−	0.999996i	$-0.499064\pi$
$239$	0.0908704	0.00587792	0.00293896	+	0.999996i	$0.499064\pi$
$240$	0	0
$241$	−13.6392	−0.878575	−0.439288	−	0.898346i	$-0.644769\pi$
$241$	−13.6392	−0.878575	−0.439288	+	0.898346i	$0.644769\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	19.9025	1.27152
$246$	0	0
$247$	−14.7062	−0.935733
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.63107	−0.355430	−0.177715	−	0.984082i	$-0.556870\pi$
$251$	−5.63107	−0.355430	−0.177715	+	0.984082i	$0.556870\pi$
$252$	0	0
$253$	5.53051	0.347700
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.8908	−0.741727	−0.370863	−	0.928687i	$-0.620938\pi$
$257$	−11.8908	−0.741727	−0.370863	+	0.928687i	$0.620938\pi$
$258$	0	0
$259$	26.0703	1.61993
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.2162	−0.629956	−0.314978	−	0.949099i	$-0.601997\pi$
$263$	−10.2162	−0.629956	−0.314978	+	0.949099i	$0.601997\pi$
$264$	0	0
$265$	−4.46906	−0.274532
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0.258728	0.0157749	0.00788746	−	0.999969i	$-0.497489\pi$
$269$	0.258728	0.0157749	0.00788746	+	0.999969i	$0.497489\pi$
$270$	0	0
$271$	−5.19036	−0.315292	−0.157646	−	0.987496i	$-0.550391\pi$
$271$	−5.19036	−0.315292	−0.157646	+	0.987496i	$0.550391\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.53051	0.333503
$276$	0	0
$277$	−19.9592	−1.19923	−0.599615	−	0.800288i	$-0.704680\pi$
$277$	−19.9592	−1.19923	−0.599615	+	0.800288i	$0.704680\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.2250	−0.788938	−0.394469	−	0.918909i	$-0.629071\pi$
$281$	−13.2250	−0.788938	−0.394469	+	0.918909i	$0.629071\pi$
$282$	0	0
$283$	16.2598	0.966544	0.483272	−	0.875470i	$-0.339448\pi$
$283$	16.2598	0.966544	0.483272	+	0.875470i	$0.339448\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−59.6045	−3.51834
$288$	0	0
$289$	−12.0536	−0.709035
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.28858	−0.367383	−0.183691	−	0.982984i	$-0.558805\pi$
$293$	−6.28858	−0.367383	−0.183691	+	0.982984i	$0.558805\pi$
$294$	0	0
$295$	11.1486	0.649098
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.78900	0.103461
$300$	0	0
$301$	44.7601	2.57993
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.81573	0.447527
$306$	0	0
$307$	23.6636	1.35055	0.675275	−	0.737566i	$-0.264025\pi$
$307$	23.6636	1.35055	0.675275	+	0.737566i	$0.264025\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.7175	−0.777848	−0.388924	−	0.921270i	$-0.627153\pi$
$311$	−13.7175	−0.777848	−0.388924	+	0.921270i	$0.627153\pi$
$312$	0	0
$313$	−21.1372	−1.19475	−0.597373	−	0.801964i	$-0.703789\pi$
$313$	−21.1372	−1.19475	−0.597373	+	0.801964i	$0.703789\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9895	0.954227	0.477113	−	0.878842i	$-0.341683\pi$
$317$	16.9895	0.954227	0.477113	+	0.878842i	$0.341683\pi$
$318$	0	0
$319$	39.0472	2.18622
$320$	0	0
$321$	0	0
$322$	0	0
$323$	18.2825	1.01726
$324$	0	0
$325$	1.78900	0.0992359
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.11825	−0.392442
$330$	0	0
$331$	−9.57525	−0.526303	−0.263152	−	0.964754i	$-0.584762\pi$
$331$	−9.57525	−0.526303	−0.263152	+	0.964754i	$0.584762\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.96903	−0.107579
$336$	0	0
$337$	−18.9193	−1.03060	−0.515299	−	0.857011i	$-0.672319\pi$
$337$	−18.9193	−1.03060	−0.515299	+	0.857011i	$0.672319\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.4267	−0.618791
$342$	0	0
$343$	−66.9220	−3.61345
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.5628	1.15755	0.578777	−	0.815486i	$-0.303530\pi$
$347$	21.5628	1.15755	0.578777	+	0.815486i	$0.303530\pi$
$348$	0	0
$349$	1.04297	0.0558291	0.0279145	−	0.999610i	$-0.491113\pi$
$349$	1.04297	0.0558291	0.0279145	+	0.999610i	$0.491113\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.75247	−0.412622	−0.206311	−	0.978486i	$-0.566146\pi$
$353$	−7.75247	−0.412622	−0.206311	+	0.978486i	$0.566146\pi$
$354$	0	0
$355$	−8.51894	−0.452138
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.2843	−1.75668	−0.878340	−	0.478037i	$-0.841349\pi$
$359$	−33.2843	−1.75668	−0.878340	+	0.478037i	$0.841349\pi$
$360$	0	0
$361$	48.5741	2.55653
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.7482	−0.562589
$366$	0	0
$367$	−32.6495	−1.70429	−0.852146	−	0.523305i	$-0.824699\pi$
$367$	−32.6495	−1.70429	−0.852146	+	0.523305i	$0.824699\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	23.1799	1.20344
$372$	0	0
$373$	−4.46944	−0.231419	−0.115710	−	0.993283i	$-0.536914\pi$
$373$	−4.46944	−0.231419	−0.115710	+	0.993283i	$0.536914\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.6309	0.650525
$378$	0	0
$379$	5.91194	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$379$	5.91194	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.80511	0.194432	0.0972161	−	0.995263i	$-0.469006\pi$
$383$	3.80511	0.194432	0.0972161	+	0.995263i	$0.469006\pi$
$384$	0	0
$385$	−28.6854	−1.46195
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11.4995	0.583046	0.291523	−	0.956564i	$-0.405838\pi$
$389$	11.4995	0.583046	0.291523	+	0.956564i	$0.405838\pi$
$390$	0	0
$391$	−2.22405	−0.112475
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.0361	−0.706232
$396$	0	0
$397$	−4.15253	−0.208409	−0.104205	−	0.994556i	$-0.533230\pi$
$397$	−4.15253	−0.208409	−0.104205	+	0.994556i	$0.533230\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−36.5628	−1.82586	−0.912930	−	0.408116i	$-0.866186\pi$
$401$	−36.5628	−1.82586	−0.912930	+	0.408116i	$0.866186\pi$
$402$	0	0
$403$	−3.69629	−0.184125
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.7981	−1.37790
$408$	0	0
$409$	23.9221	1.18287	0.591436	−	0.806352i	$-0.298561\pi$
$409$	23.9221	1.18287	0.591436	+	0.806352i	$0.298561\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−57.8252	−2.84539
$414$	0	0
$415$	6.83698	0.335614
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.34692	−0.212361	−0.106180	−	0.994347i	$-0.533862\pi$
$419$	−4.34692	−0.212361	−0.106180	+	0.994347i	$0.533862\pi$
$420$	0	0
$421$	5.98865	0.291869	0.145935	−	0.989294i	$-0.453381\pi$
$421$	5.98865	0.291869	0.145935	+	0.989294i	$0.453381\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.22405	−0.107882
$426$	0	0
$427$	−40.5383	−1.96179
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.2601	1.02406	0.512030	−	0.858967i	$-0.328894\pi$
$431$	21.2601	1.02406	0.512030	+	0.858967i	$0.328894\pi$
$432$	0	0
$433$	−17.9361	−0.861955	−0.430978	−	0.902363i	$-0.641831\pi$
$433$	−17.9361	−0.861955	−0.430978	+	0.902363i	$0.641831\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.22035	−0.393233
$438$	0	0
$439$	−40.7767	−1.94616	−0.973082	−	0.230459i	$-0.925977\pi$
$439$	−40.7767	−1.94616	−0.973082	+	0.230459i	$0.925977\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.7585	−1.12880	−0.564401	−	0.825501i	$-0.690893\pi$
$443$	−23.7585	−1.12880	−0.564401	+	0.825501i	$0.690893\pi$
$444$	0	0
$445$	−15.5115	−0.735313
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.21479	0.293294	0.146647	−	0.989189i	$-0.453152\pi$
$449$	6.21479	0.293294	0.146647	+	0.989189i	$0.453152\pi$
$450$	0	0
$451$	63.5548	2.99268
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.27911	−0.435012
$456$	0	0
$457$	2.53970	0.118802	0.0594010	−	0.998234i	$-0.481081\pi$
$457$	2.53970	0.118802	0.0594010	+	0.998234i	$0.481081\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.8931	−0.693641	−0.346820	−	0.937932i	$-0.612739\pi$
$461$	−14.8931	−0.693641	−0.346820	+	0.937932i	$0.612739\pi$
$462$	0	0
$463$	−28.1509	−1.30828	−0.654142	−	0.756372i	$-0.726970\pi$
$463$	−28.1509	−1.30828	−0.654142	+	0.756372i	$0.726970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.3752	−1.17423	−0.587113	−	0.809505i	$-0.699736\pi$
$467$	−25.3752	−1.17423	−0.587113	+	0.809505i	$0.699736\pi$
$468$	0	0
$469$	10.2129	0.471586
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−47.7266	−2.19447
$474$	0	0
$475$	−8.22035	−0.377175
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.03440	0.367101	0.183551	−	0.983010i	$-0.441241\pi$
$479$	8.03440	0.367101	0.183551	+	0.983010i	$0.441241\pi$
$480$	0	0
$481$	−8.99207	−0.410003
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.19194	−0.190346
$486$	0	0
$487$	−16.1617	−0.732356	−0.366178	−	0.930545i	$-0.619334\pi$
$487$	−16.1617	−0.732356	−0.366178	+	0.930545i	$0.619334\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19.5159	−0.880740	−0.440370	−	0.897816i	$-0.645153\pi$
$491$	−19.5159	−0.880740	−0.440370	+	0.897816i	$0.645153\pi$
$492$	0	0
$493$	−15.7025	−0.707205
$494$	0	0
$495$	0	0
$496$	0	0
$497$	44.1857	1.98200
$498$	0	0
$499$	16.2002	0.725222	0.362611	−	0.931941i	$-0.381885\pi$
$499$	16.2002	0.725222	0.362611	+	0.931941i	$0.381885\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.7478	1.14804	0.574019	−	0.818842i	$-0.305384\pi$
$503$	25.7478	1.14804	0.574019	+	0.818842i	$0.305384\pi$
$504$	0	0
$505$	2.10435	0.0936422
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.6907	−1.31602	−0.658009	−	0.753010i	$-0.728601\pi$
$509$	−29.6907	−1.31602	−0.658009	+	0.753010i	$0.728601\pi$
$510$	0	0
$511$	55.7486	2.46617
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.51052	0.375018
$516$	0	0
$517$	7.59001	0.333808
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.5883	1.69058	0.845291	−	0.534305i	$-0.179427\pi$
$521$	38.5883	1.69058	0.845291	+	0.534305i	$0.179427\pi$
$522$	0	0
$523$	−29.6118	−1.29483	−0.647416	−	0.762137i	$-0.724150\pi$
$523$	−29.6118	−1.29483	−0.647416	+	0.762137i	$0.724150\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.59515	0.200168
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	20.5586	0.890491
$534$	0	0
$535$	8.56936	0.370486
$536$	0	0
$537$	0	0
$538$	0	0
$539$	110.071	4.74109
$540$	0	0
$541$	7.81422	0.335959	0.167980	−	0.985790i	$-0.446276\pi$
$541$	7.81422	0.335959	0.167980	+	0.985790i	$0.446276\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−20.0678	−0.859608
$546$	0	0
$547$	34.1207	1.45890	0.729448	−	0.684036i	$-0.239777\pi$
$547$	34.1207	1.45890	0.729448	+	0.684036i	$0.239777\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−58.0383	−2.47251
$552$	0	0
$553$	72.8018	3.09585
$554$	0	0
$555$	0	0
$556$	0	0
$557$	37.2090	1.57660	0.788299	−	0.615293i	$-0.210962\pi$
$557$	37.2090	1.57660	0.788299	+	0.615293i	$0.210962\pi$
$558$	0	0
$559$	−15.4385	−0.652979
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.8233	1.29905	0.649523	−	0.760342i	$-0.274969\pi$
$563$	30.8233	1.29905	0.649523	+	0.760342i	$0.274969\pi$
$564$	0	0
$565$	−11.1800	−0.470347
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.5281	0.441361	0.220681	−	0.975346i	$-0.429172\pi$
$569$	10.5281	0.441361	0.220681	+	0.975346i	$0.429172\pi$
$570$	0	0
$571$	−34.7410	−1.45387	−0.726933	−	0.686709i	$-0.759055\pi$
$571$	−34.7410	−1.45387	−0.726933	+	0.686709i	$0.759055\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−12.9327	−0.538397	−0.269199	−	0.963085i	$-0.586759\pi$
$577$	−12.9327	−0.538397	−0.269199	+	0.963085i	$0.586759\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−35.4618	−1.47120
$582$	0	0
$583$	−24.7162	−1.02364
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.4522	1.00925	0.504626	−	0.863338i	$-0.331630\pi$
$587$	24.4522	1.00925	0.504626	+	0.863338i	$0.331630\pi$
$588$	0	0
$589$	16.9842	0.699823
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.5575	0.803130	0.401565	−	0.915831i	$-0.368466\pi$
$593$	19.5575	0.803130	0.401565	+	0.915831i	$0.368466\pi$
$594$	0	0
$595$	11.5356	0.472914
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.5425	−1.20708	−0.603538	−	0.797334i	$-0.706243\pi$
$599$	−29.5425	−1.20708	−0.603538	+	0.797334i	$0.706243\pi$
$600$	0	0
$601$	−22.9919	−0.937861	−0.468930	−	0.883235i	$-0.655361\pi$
$601$	−22.9919	−0.937861	−0.468930	+	0.883235i	$0.655361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.5866	0.796308
$606$	0	0
$607$	−47.5502	−1.93000	−0.965001	−	0.262247i	$-0.915536\pi$
$607$	−47.5502	−1.93000	−0.965001	+	0.262247i	$0.915536\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.45520	0.0993268
$612$	0	0
$613$	−18.6728	−0.754189	−0.377094	−	0.926175i	$-0.623077\pi$
$613$	−18.6728	−0.754189	−0.377094	+	0.926175i	$0.623077\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.222177	−0.00894452	−0.00447226	−	0.999990i	$-0.501424\pi$
$617$	−0.222177	−0.00894452	−0.00447226	+	0.999990i	$0.501424\pi$
$618$	0	0
$619$	27.3634	1.09983	0.549914	−	0.835221i	$-0.314660\pi$
$619$	27.3634	1.09983	0.549914	+	0.835221i	$0.314660\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	80.4542	3.22333
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.1788	0.445727
$630$	0	0
$631$	−28.0642	−1.11722	−0.558608	−	0.829432i	$-0.688664\pi$
$631$	−28.0642	−1.11722	−0.558608	+	0.829432i	$0.688664\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15.7755	−0.626033
$636$	0	0
$637$	35.6055	1.41074
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.31916	0.289089	0.144545	−	0.989498i	$-0.453828\pi$
$641$	7.31916	0.289089	0.144545	+	0.989498i	$0.453828\pi$
$642$	0	0
$643$	−23.0989	−0.910932	−0.455466	−	0.890253i	$-0.650527\pi$
$643$	−23.0989	−0.910932	−0.455466	+	0.890253i	$0.650527\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.19129	0.322033	0.161016	−	0.986952i	$-0.448523\pi$
$647$	8.19129	0.322033	0.161016	+	0.986952i	$0.448523\pi$
$648$	0	0
$649$	61.6576	2.42027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.8688	−1.09059	−0.545296	−	0.838244i	$-0.683583\pi$
$653$	−27.8688	−1.09059	−0.545296	+	0.838244i	$0.683583\pi$
$654$	0	0
$655$	−6.23941	−0.243794
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.9507	1.12776	0.563880	−	0.825856i	$-0.309308\pi$
$659$	28.9507	1.12776	0.563880	+	0.825856i	$0.309308\pi$
$660$	0	0
$661$	26.7973	1.04229	0.521147	−	0.853467i	$-0.325504\pi$
$661$	26.7973	1.04229	0.521147	+	0.853467i	$0.325504\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	42.6370	1.65339
$666$	0	0
$667$	7.06032	0.273377
$668$	0	0
$669$	0	0
$670$	0	0
$671$	43.2250	1.66868
$672$	0	0
$673$	29.5218	1.13798	0.568991	−	0.822344i	$-0.307334\pi$
$673$	29.5218	1.13798	0.568991	+	0.822344i	$0.307334\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.4897	0.518450	0.259225	−	0.965817i	$-0.416533\pi$
$677$	13.4897	0.518450	0.259225	+	0.965817i	$0.416533\pi$
$678$	0	0
$679$	21.7426	0.834404
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.7128	−0.945609	−0.472805	−	0.881167i	$-0.656758\pi$
$683$	−24.7128	−0.945609	−0.472805	+	0.881167i	$0.656758\pi$
$684$	0	0
$685$	2.63201	0.100564
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.99515	−0.304591
$690$	0	0
$691$	19.1923	0.730108	0.365054	−	0.930986i	$-0.381051\pi$
$691$	19.1923	0.730108	0.365054	+	0.930986i	$0.381051\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.02125	0.0387381
$696$	0	0
$697$	−25.5580	−0.968080
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0.866603	0.0327311	0.0163656	−	0.999866i	$-0.494790\pi$
$701$	0.866603	0.0327311	0.0163656	+	0.999866i	$0.494790\pi$
$702$	0	0
$703$	41.3180	1.55834
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.9147	−0.410491
$708$	0	0
$709$	−36.5381	−1.37222	−0.686108	−	0.727499i	$-0.740682\pi$
$709$	−36.5381	−1.37222	−0.686108	+	0.727499i	$0.740682\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.06612	−0.0773768
$714$	0	0
$715$	9.89409	0.370018
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.2845	−1.50236	−0.751179	−	0.660099i	$-0.770514\pi$
$719$	−40.2845	−1.50236	−0.751179	+	0.660099i	$0.770514\pi$
$720$	0	0
$721$	−44.1420	−1.64393
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.06032	0.262214
$726$	0	0
$727$	40.4482	1.50014	0.750071	−	0.661358i	$-0.230019\pi$
$727$	40.4482	1.50014	0.750071	+	0.661358i	$0.230019\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.1929	0.709873
$732$	0	0
$733$	−15.1569	−0.559831	−0.279916	−	0.960025i	$-0.590306\pi$
$733$	−15.1569	−0.559831	−0.279916	+	0.960025i	$0.590306\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.8897	−0.401128
$738$	0	0
$739$	−1.69715	−0.0624307	−0.0312154	−	0.999513i	$-0.509938\pi$
$739$	−1.69715	−0.0624307	−0.0312154	+	0.999513i	$0.509938\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.8438	0.691314	0.345657	−	0.938361i	$-0.387656\pi$
$743$	18.8438	0.691314	0.345657	+	0.938361i	$0.387656\pi$
$744$	0	0
$745$	20.6707	0.757316
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−44.4472	−1.62407
$750$	0	0
$751$	6.98309	0.254816	0.127408	−	0.991850i	$-0.459334\pi$
$751$	6.98309	0.254816	0.127408	+	0.991850i	$0.459334\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.02315	0.255598
$756$	0	0
$757$	−42.1359	−1.53146	−0.765728	−	0.643164i	$-0.777621\pi$
$757$	−42.1359	−1.53146	−0.765728	+	0.643164i	$0.777621\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.6359	1.50930	0.754649	−	0.656128i	$-0.227807\pi$
$761$	41.6359	1.50930	0.754649	+	0.656128i	$0.227807\pi$
$762$	0	0
$763$	104.087	3.76819
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.9449	0.720168
$768$	0	0
$769$	−7.24030	−0.261092	−0.130546	−	0.991442i	$-0.541673\pi$
$769$	−7.24030	−0.261092	−0.130546	+	0.991442i	$0.541673\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23.8204	0.856759	0.428379	−	0.903599i	$-0.359085\pi$
$773$	23.8204	0.856759	0.428379	+	0.903599i	$0.359085\pi$
$774$	0	0
$775$	−2.06612	−0.0742172
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−94.4655	−3.38458
$780$	0	0
$781$	−47.1141	−1.68588
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.5529	−0.590800
$786$	0	0
$787$	24.5172	0.873945	0.436972	−	0.899475i	$-0.356051\pi$
$787$	24.5172	0.873945	0.436972	+	0.899475i	$0.356051\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	57.9881	2.06182
$792$	0	0
$793$	13.9823	0.496527
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0419	0.780765	0.390382	−	0.920653i	$-0.372343\pi$
$797$	22.0419	0.780765	0.390382	+	0.920653i	$0.372343\pi$
$798$	0	0
$799$	−3.05226	−0.107981
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−59.4433	−2.09771
$804$	0	0
$805$	−5.18676	−0.182809
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.44998	0.297085	0.148543	−	0.988906i	$-0.452542\pi$
$809$	8.44998	0.297085	0.148543	+	0.988906i	$0.452542\pi$
$810$	0	0
$811$	−7.40731	−0.260106	−0.130053	−	0.991507i	$-0.541515\pi$
$811$	−7.40731	−0.260106	−0.130053	+	0.991507i	$0.541515\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.2497	0.394060
$816$	0	0
$817$	70.9390	2.48184
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.7103	0.443594	0.221797	−	0.975093i	$-0.428808\pi$
$821$	12.7103	0.443594	0.221797	+	0.975093i	$0.428808\pi$
$822$	0	0
$823$	−1.23216	−0.0429505	−0.0214753	−	0.999769i	$-0.506836\pi$
$823$	−1.23216	−0.0429505	−0.0214753	+	0.999769i	$0.506836\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.6839	1.20608	0.603039	−	0.797712i	$-0.293956\pi$
$827$	34.6839	1.20608	0.603039	+	0.797712i	$0.293956\pi$
$828$	0	0
$829$	−2.83700	−0.0985330	−0.0492665	−	0.998786i	$-0.515688\pi$
$829$	−2.83700	−0.0985330	−0.0492665	+	0.998786i	$0.515688\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−44.2641	−1.53366
$834$	0	0
$835$	−17.7556	−0.614457
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.353131	0.0121914	0.00609571	−	0.999981i	$-0.498060\pi$
$839$	0.353131	0.0121914	0.00609571	+	0.999981i	$0.498060\pi$
$840$	0	0
$841$	20.8481	0.718899
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.79948	−0.337112
$846$	0	0
$847$	−101.591	−3.49071
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.02631	−0.172300
$852$	0	0
$853$	−30.1917	−1.03374	−0.516872	−	0.856063i	$-0.672904\pi$
$853$	−30.1917	−1.03374	−0.516872	+	0.856063i	$0.672904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37.6661	−1.28665	−0.643326	−	0.765593i	$-0.722446\pi$
$857$	−37.6661	−1.28665	−0.643326	+	0.765593i	$0.722446\pi$
$858$	0	0
$859$	36.2701	1.23752	0.618760	−	0.785580i	$-0.287635\pi$
$859$	36.2701	1.23752	0.618760	+	0.785580i	$0.287635\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.17475	−0.244231	−0.122116	−	0.992516i	$-0.538968\pi$
$863$	−7.17475	−0.244231	−0.122116	+	0.992516i	$0.538968\pi$
$864$	0	0
$865$	10.0126	0.340439
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−77.6267	−2.63331
$870$	0	0
$871$	−3.52259	−0.119358
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.18676	−0.175344
$876$	0	0
$877$	−20.5238	−0.693039	−0.346519	−	0.938043i	$-0.612637\pi$
$877$	−20.5238	−0.693039	−0.346519	+	0.938043i	$0.612637\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−53.5933	−1.80561	−0.902803	−	0.430055i	$-0.858494\pi$
$881$	−53.5933	−1.80561	−0.902803	+	0.430055i	$0.858494\pi$
$882$	0	0
$883$	7.63602	0.256973	0.128486	−	0.991711i	$-0.458988\pi$
$883$	7.63602	0.256973	0.128486	+	0.991711i	$0.458988\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.54307	−0.286848	−0.143424	−	0.989661i	$-0.545811\pi$
$887$	−8.54307	−0.286848	−0.143424	+	0.989661i	$0.545811\pi$
$888$	0	0
$889$	81.8238	2.74428
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.2815	−0.377521
$894$	0	0
$895$	−15.4041	−0.514901
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.5875	−0.486519
$900$	0	0
$901$	9.93942	0.331130
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.47996	0.315125
$906$	0	0
$907$	−19.5010	−0.647519	−0.323759	−	0.946139i	$-0.604947\pi$
$907$	−19.5010	−0.647519	−0.323759	+	0.946139i	$0.604947\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.46746	−0.0817505	−0.0408753	−	0.999164i	$-0.513015\pi$
$911$	−2.46746	−0.0817505	−0.0408753	+	0.999164i	$0.513015\pi$
$912$	0	0
$913$	37.8120	1.25139
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.3623	1.06870
$918$	0	0
$919$	44.6343	1.47235	0.736175	−	0.676792i	$-0.236630\pi$
$919$	44.6343	1.47235	0.736175	+	0.676792i	$0.236630\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−15.2404	−0.501643
$924$	0	0
$925$	−5.02631	−0.165264
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.85129	−0.0607389	−0.0303694	−	0.999539i	$-0.509668\pi$
$929$	−1.85129	−0.0607389	−0.0303694	+	0.999539i	$0.509668\pi$
$930$	0	0
$931$	−163.605	−5.36195
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.3001	−0.402258
$936$	0	0
$937$	17.3837	0.567900	0.283950	−	0.958839i	$-0.408355\pi$
$937$	17.3837	0.567900	0.283950	+	0.958839i	$0.408355\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.20116	−0.136954	−0.0684769	−	0.997653i	$-0.521814\pi$
$941$	−4.20116	−0.136954	−0.0684769	+	0.997653i	$0.521814\pi$
$942$	0	0
$943$	11.4917	0.374220
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.27990	−0.171574	−0.0857869	−	0.996314i	$-0.527340\pi$
$947$	−5.27990	−0.171574	−0.0857869	+	0.996314i	$0.527340\pi$
$948$	0	0
$949$	−19.2286	−0.624187
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−41.3188	−1.33845	−0.669223	−	0.743061i	$-0.733373\pi$
$953$	−41.3188	−1.33845	−0.669223	+	0.743061i	$0.733373\pi$
$954$	0	0
$955$	−11.8994	−0.385057
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.6516	−0.440832
$960$	0	0
$961$	−26.7312	−0.862295
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.6195	0.792529
$966$	0	0
$967$	−56.6523	−1.82181	−0.910907	−	0.412612i	$-0.864617\pi$
$967$	−56.6523	−1.82181	−0.910907	+	0.412612i	$0.864617\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.40721	0.301892	0.150946	−	0.988542i	$-0.451768\pi$
$971$	9.40721	0.301892	0.150946	+	0.988542i	$0.451768\pi$
$972$	0	0
$973$	−5.29697	−0.169813
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.7170	1.87852	0.939262	−	0.343200i	$-0.111511\pi$
$977$	58.7170	1.87852	0.939262	+	0.343200i	$0.111511\pi$
$978$	0	0
$979$	−85.7863	−2.74174
$980$	0	0
$981$	0	0
$982$	0	0
$983$	37.9057	1.20900	0.604502	−	0.796603i	$-0.293372\pi$
$983$	37.9057	1.20900	0.604502	+	0.796603i	$0.293372\pi$
$984$	0	0
$985$	1.98652	0.0632959
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.62968	−0.274408
$990$	0	0
$991$	−36.0724	−1.14588	−0.572938	−	0.819599i	$-0.694197\pi$
$991$	−36.0724	−1.14588	−0.572938	+	0.819599i	$0.694197\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.07108	0.129062
$996$	0	0
$997$	−4.37723	−0.138628	−0.0693142	−	0.997595i	$-0.522081\pi$
$997$	−4.37723	−0.138628	−0.0693142	+	0.997595i	$0.522081\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bw.1.1	yes	7
3.2	odd	2		8280.2.a.bv.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bv.1.1	✓	7	3.2	odd	2
8280.2.a.bw.1.1	yes	7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -5.18676 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -5.18676 q^{7} +5.53051 q^{11} +1.78900 q^{13} -2.22405 q^{17} -8.22035 q^{19} +1.00000 q^{23} +1.00000 q^{25} +7.06032 q^{29} -2.06612 q^{31} -5.18676 q^{35} -5.02631 q^{37} +11.4917 q^{41} -8.62968 q^{43} +1.37239 q^{47} +19.9025 q^{49} -4.46906 q^{53} +5.53051 q^{55} +11.1486 q^{59} +7.81573 q^{61} +1.78900 q^{65} -1.96903 q^{67} -8.51894 q^{71} -10.7482 q^{73} -28.6854 q^{77} -14.0361 q^{79} +6.83698 q^{83} -2.22405 q^{85} -15.5115 q^{89} -9.27911 q^{91} -8.22035 q^{95} -4.19194 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q + 7 * q^5 - 6 * q^7	\( 7 q + 7 q^{5} - 6 q^{7} - 2 q^{11} - 6 q^{13} - 4 q^{17} - 8 q^{19} + 7 q^{23} + 7 q^{25} + 2 q^{29} - 8 q^{31} - 6 q^{35} - 16 q^{37} + 2 q^{41} - 10 q^{43} - 8 q^{47} + 19 q^{49} - 10 q^{53} - 2 q^{55} - 24 q^{59} + 8 q^{61} - 6 q^{65} - 20 q^{67} - 8 q^{71} + 2 q^{73} - 12 q^{77} - 2 q^{79} - 22 q^{83} - 4 q^{85} - 16 q^{89} - 20 q^{91} - 8 q^{95} + 4 q^{97}+O(q^{100}) \) 7 * q + 7 * q^5 - 6 * q^7 - 2 * q^11 - 6 * q^13 - 4 * q^17 - 8 * q^19 + 7 * q^23 + 7 * q^25 + 2 * q^29 - 8 * q^31 - 6 * q^35 - 16 * q^37 + 2 * q^41 - 10 * q^43 - 8 * q^47 + 19 * q^49 - 10 * q^53 - 2 * q^55 - 24 * q^59 + 8 * q^61 - 6 * q^65 - 20 * q^67 - 8 * q^71 + 2 * q^73 - 12 * q^77 - 2 * q^79 - 22 * q^83 - 4 * q^85 - 16 * q^89 - 20 * q^91 - 8 * q^95 + 4 * q^97

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