LMFDB - Embedded newform 8280.2.a.bj.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.2597.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.878468$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +0.121532 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.121532 q^{7} -2.87847 q^{11} +5.22829 q^{13} +2.22829 q^{17} +1.22829 q^{19} -1.00000 q^{23} +1.00000 q^{25} -9.34983 q^{29} -2.12153 q^{31} -0.121532 q^{35} +5.59289 q^{37} -8.22829 q^{41} +8.00000 q^{43} +10.4566 q^{47} -6.98523 q^{49} +3.59289 q^{53} +2.87847 q^{55} +0.650174 q^{59} -7.33506 q^{61} -5.22829 q^{65} +5.59289 q^{67} +13.9852 q^{71} +12.9427 q^{73} -0.349826 q^{77} -3.51387 q^{79} +11.1068 q^{83} -2.22829 q^{85} +0.486128 q^{89} +0.635404 q^{91} -1.22829 q^{95} +0.635404 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7	$ 3 q - 3 q^{5} + 2 q^{7} - 7 q^{11} - q^{13} - 10 q^{17} - 13 q^{19} - 3 q^{23} + 3 q^{25} - 13 q^{29} - 8 q^{31} - 2 q^{35} + 5 q^{37} - 8 q^{41} + 24 q^{43} - 2 q^{47} - q^{49} - q^{53} + 7 q^{55} + 17 q^{59} + 13 q^{61} + q^{65} + 5 q^{67} + 22 q^{71} + 12 q^{73} + 14 q^{77} - 4 q^{79} + 15 q^{83} + 10 q^{85} + 8 q^{89} - 3 q^{91} + 13 q^{95} - 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - 7 * q^11 - q^13 - 10 * q^17 - 13 * q^19 - 3 * q^23 + 3 * q^25 - 13 * q^29 - 8 * q^31 - 2 * q^35 + 5 * q^37 - 8 * q^41 + 24 * q^43 - 2 * q^47 - q^49 - q^53 + 7 * q^55 + 17 * q^59 + 13 * q^61 + q^65 + 5 * q^67 + 22 * q^71 + 12 * q^73 + 14 * q^77 - 4 * q^79 + 15 * q^83 + 10 * q^85 + 8 * q^89 - 3 * q^91 + 13 * q^95 - 3 * 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
$7$	0.121532	0.0459347	0.0229674	−	0.999736i	$-0.492689\pi$
$7$	0.121532	0.0459347	0.0229674	+	0.999736i	$0.492689\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.87847	−0.867891	−0.433945	−	0.900939i	$-0.642879\pi$
$11$	−2.87847	−0.867891	−0.433945	+	0.900939i	$0.642879\pi$
$12$	0	0
$13$	5.22829	1.45007	0.725034	−	0.688713i	$-0.241824\pi$
$13$	5.22829	1.45007	0.725034	+	0.688713i	$0.241824\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.22829	0.540441	0.270220	−	0.962799i	$-0.412903\pi$
$17$	2.22829	0.540441	0.270220	+	0.962799i	$0.412903\pi$
$18$	0	0
$19$	1.22829	0.281790	0.140895	−	0.990025i	$-0.455002\pi$
$19$	1.22829	0.281790	0.140895	+	0.990025i	$0.455002\pi$
$20$	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
$29$	−9.34983	−1.73622	−0.868110	−	0.496373i	$-0.834665\pi$
$29$	−9.34983	−1.73622	−0.868110	+	0.496373i	$0.834665\pi$
$30$	0	0
$31$	−2.12153	−0.381038	−0.190519	−	0.981683i	$-0.561017\pi$
$31$	−2.12153	−0.381038	−0.190519	+	0.981683i	$0.561017\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.121532	−0.0205426
$36$	0	0
$37$	5.59289	0.919465	0.459733	−	0.888057i	$-0.347945\pi$
$37$	5.59289	0.919465	0.459733	+	0.888057i	$0.347945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.22829	−1.28504	−0.642522	−	0.766267i	$-0.722112\pi$
$41$	−8.22829	−1.28504	−0.642522	+	0.766267i	$0.722112\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.4566	1.52525	0.762625	−	0.646841i	$-0.223910\pi$
$47$	10.4566	1.52525	0.762625	+	0.646841i	$0.223910\pi$
$48$	0	0
$49$	−6.98523	−0.997890
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.59289	0.493521	0.246761	−	0.969076i	$-0.420634\pi$
$53$	3.59289	0.493521	0.246761	+	0.969076i	$0.420634\pi$
$54$	0	0
$55$	2.87847	0.388133
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.650174	0.0846455	0.0423227	−	0.999104i	$-0.486524\pi$
$59$	0.650174	0.0846455	0.0423227	+	0.999104i	$0.486524\pi$
$60$	0	0
$61$	−7.33506	−0.939158	−0.469579	−	0.882891i	$-0.655594\pi$
$61$	−7.33506	−0.939158	−0.469579	+	0.882891i	$0.655594\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.22829	−0.648490
$66$	0	0
$67$	5.59289	0.683280	0.341640	−	0.939831i	$-0.389018\pi$
$67$	5.59289	0.683280	0.341640	+	0.939831i	$0.389018\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.9852	1.65974	0.829871	−	0.557956i	$-0.188414\pi$
$71$	13.9852	1.65974	0.829871	+	0.557956i	$0.188414\pi$
$72$	0	0
$73$	12.9427	1.51483	0.757415	−	0.652934i	$-0.226462\pi$
$73$	12.9427	1.51483	0.757415	+	0.652934i	$0.226462\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.349826	−0.0398663
$78$	0	0
$79$	−3.51387	−0.395342	−0.197671	−	0.980268i	$-0.563338\pi$
$79$	−3.51387	−0.395342	−0.197671	+	0.980268i	$0.563338\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.1068	1.21913	0.609563	−	0.792738i	$-0.291345\pi$
$83$	11.1068	1.21913	0.609563	+	0.792738i	$0.291345\pi$
$84$	0	0
$85$	−2.22829	−0.241692
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.486128	0.0515294	0.0257647	−	0.999668i	$-0.491798\pi$
$89$	0.486128	0.0515294	0.0257647	+	0.999668i	$0.491798\pi$
$90$	0	0
$91$	0.635404	0.0666085
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.22829	−0.126020
$96$	0	0
$97$	0.635404	0.0645155	0.0322578	−	0.999480i	$-0.489730\pi$
$97$	0.635404	0.0645155	0.0322578	+	0.999480i	$0.489730\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.3498	−1.32836	−0.664179	−	0.747574i	$-0.731219\pi$
$101$	−13.3498	−1.32836	−0.664179	+	0.747574i	$0.731219\pi$
$102$	0	0
$103$	−7.33506	−0.722745	−0.361372	−	0.932422i	$-0.617692\pi$
$103$	−7.33506	−0.722745	−0.361372	+	0.932422i	$0.617692\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.0495	−1.55156	−0.775781	−	0.631003i	$-0.782644\pi$
$107$	−16.0495	−1.55156	−0.775781	+	0.631003i	$0.782644\pi$
$108$	0	0
$109$	1.01477	0.0971973	0.0485987	−	0.998818i	$-0.484524\pi$
$109$	1.01477	0.0971973	0.0485987	+	0.998818i	$0.484524\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.3203	−1.62936	−0.814678	−	0.579914i	$-0.803086\pi$
$113$	−17.3203	−1.62936	−0.814678	+	0.579914i	$0.803086\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.270809	0.0248250
$120$	0	0
$121$	−2.71442	−0.246766
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.75694	0.155903	0.0779514	−	0.996957i	$-0.475162\pi$
$127$	1.75694	0.155903	0.0779514	+	0.996957i	$0.475162\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.2135	1.76606	0.883032	−	0.469313i	$-0.155498\pi$
$131$	20.2135	1.76606	0.883032	+	0.469313i	$0.155498\pi$
$132$	0	0
$133$	0.149277	0.0129439
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.9279	1.53169	0.765844	−	0.643027i	$-0.222322\pi$
$137$	17.9279	1.53169	0.765844	+	0.643027i	$0.222322\pi$
$138$	0	0
$139$	−10.8637	−0.921447	−0.460723	−	0.887544i	$-0.652410\pi$
$139$	−10.8637	−0.921447	−0.460723	+	0.887544i	$0.652410\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.0495	−1.25850
$144$	0	0
$145$	9.34983	0.776461
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.7144	1.28738	0.643688	−	0.765288i	$-0.277404\pi$
$149$	15.7144	1.28738	0.643688	+	0.765288i	$0.277404\pi$
$150$	0	0
$151$	−8.39234	−0.682959	−0.341479	−	0.939889i	$-0.610928\pi$
$151$	−8.39234	−0.682959	−0.341479	+	0.939889i	$0.610928\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.12153	0.170406
$156$	0	0
$157$	−19.5633	−1.56133	−0.780663	−	0.624953i	$-0.785118\pi$
$157$	−19.5633	−1.56133	−0.780663	+	0.624953i	$0.785118\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.121532	−0.00957805
$162$	0	0
$163$	1.01477	0.0794829	0.0397415	−	0.999210i	$-0.487347\pi$
$163$	1.01477	0.0794829	0.0397415	+	0.999210i	$0.487347\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	14.3351	1.10270
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.63540	0.200366	0.100183	−	0.994969i	$-0.468057\pi$
$173$	2.63540	0.200366	0.100183	+	0.994969i	$0.468057\pi$
$174$	0	0
$175$	0.121532	0.00918695
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.4566	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$179$	14.4566	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$180$	0	0
$181$	10.7422	0.798459	0.399229	−	0.916851i	$-0.369278\pi$
$181$	10.7422	0.798459	0.399229	+	0.916851i	$0.369278\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.59289	−0.411197
$186$	0	0
$187$	−6.41407	−0.469043
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.6701	1.64035	0.820176	−	0.572112i	$-0.193876\pi$
$191$	22.6701	1.64035	0.820176	+	0.572112i	$0.193876\pi$
$192$	0	0
$193$	2.24306	0.161459	0.0807296	−	0.996736i	$-0.474275\pi$
$193$	2.24306	0.161459	0.0807296	+	0.996736i	$0.474275\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.57812	0.539919	0.269959	−	0.962872i	$-0.412990\pi$
$197$	7.57812	0.539919	0.269959	+	0.962872i	$0.412990\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	−1.13630	−0.0797528
$204$	0	0
$205$	8.22829	0.574689
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.53560	−0.244563
$210$	0	0
$211$	−3.10676	−0.213878	−0.106939	−	0.994266i	$-0.534105\pi$
$211$	−3.10676	−0.213878	−0.106939	+	0.994266i	$0.534105\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−0.257834	−0.0175029
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.6502	0.783676
$222$	0	0
$223$	18.2135	1.21967	0.609834	−	0.792529i	$-0.291236\pi$
$223$	18.2135	1.21967	0.609834	+	0.792529i	$0.291236\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.9705	−0.927252	−0.463626	−	0.886031i	$-0.653452\pi$
$227$	−13.9705	−0.927252	−0.463626	+	0.886031i	$0.653452\pi$
$228$	0	0
$229$	12.2135	0.807092	0.403546	−	0.914959i	$-0.367777\pi$
$229$	12.2135	0.807092	0.403546	+	0.914959i	$0.367777\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.75694	−0.639198	−0.319599	−	0.947553i	$-0.603548\pi$
$233$	−9.75694	−0.639198	−0.319599	+	0.947553i	$0.603548\pi$
$234$	0	0
$235$	−10.4566	−0.682113
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.62063	−0.557622	−0.278811	−	0.960346i	$-0.589940\pi$
$239$	−8.62063	−0.557622	−0.278811	+	0.960346i	$0.589940\pi$
$240$	0	0
$241$	3.18578	0.205214	0.102607	−	0.994722i	$-0.467282\pi$
$241$	3.18578	0.205214	0.102607	+	0.994722i	$0.467282\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.98523	0.446270
$246$	0	0
$247$	6.42188	0.408614
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.2283	1.59240	0.796198	−	0.605036i	$-0.206841\pi$
$251$	25.2283	1.59240	0.796198	+	0.605036i	$0.206841\pi$
$252$	0	0
$253$	2.87847	0.180968
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6701	0.665583	0.332792	−	0.943000i	$-0.392009\pi$
$257$	10.6701	0.665583	0.332792	+	0.943000i	$0.392009\pi$
$258$	0	0
$259$	0.679714	0.0422354
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.6849	1.15216	0.576080	−	0.817394i	$-0.304582\pi$
$263$	18.6849	1.15216	0.576080	+	0.817394i	$0.304582\pi$
$264$	0	0
$265$	−3.59289	−0.220709
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.8637	1.51597	0.757983	−	0.652274i	$-0.226185\pi$
$269$	24.8637	1.51597	0.757983	+	0.652274i	$0.226185\pi$
$270$	0	0
$271$	0.578119	0.0351183	0.0175591	−	0.999846i	$-0.494410\pi$
$271$	0.578119	0.0351183	0.0175591	+	0.999846i	$0.494410\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.87847	−0.173578
$276$	0	0
$277$	−9.51387	−0.571633	−0.285817	−	0.958284i	$-0.592265\pi$
$277$	−9.51387	−0.571633	−0.285817	+	0.958284i	$0.592265\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.1562	0.784835	0.392418	−	0.919787i	$-0.371639\pi$
$281$	13.1562	0.784835	0.392418	+	0.919787i	$0.371639\pi$
$282$	0	0
$283$	24.5061	1.45673	0.728367	−	0.685187i	$-0.240280\pi$
$283$	24.5061	1.45673	0.728367	+	0.685187i	$0.240280\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−12.0347	−0.707924
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.29254	0.367614	0.183807	−	0.982962i	$-0.441158\pi$
$293$	6.29254	0.367614	0.183807	+	0.982962i	$0.441158\pi$
$294$	0	0
$295$	−0.650174	−0.0378546
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.22829	−0.302360
$300$	0	0
$301$	0.972255	0.0560398
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.33506	0.420004
$306$	0	0
$307$	13.3351	0.761072	0.380536	−	0.924766i	$-0.375740\pi$
$307$	13.3351	0.761072	0.380536	+	0.924766i	$0.375740\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.4270	1.38513	0.692565	−	0.721355i	$-0.256480\pi$
$311$	24.4270	1.38513	0.692565	+	0.721355i	$0.256480\pi$
$312$	0	0
$313$	26.5486	1.50061	0.750307	−	0.661089i	$-0.229906\pi$
$313$	26.5486	1.50061	0.750307	+	0.661089i	$0.229906\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.9852	0.953986	0.476993	−	0.878907i	$-0.341727\pi$
$317$	16.9852	0.953986	0.476993	+	0.878907i	$0.341727\pi$
$318$	0	0
$319$	26.9132	1.50685
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.73700	0.152291
$324$	0	0
$325$	5.22829	0.290014
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.27081	0.0700620
$330$	0	0
$331$	17.8914	0.983403	0.491701	−	0.870764i	$-0.336375\pi$
$331$	17.8914	0.983403	0.491701	+	0.870764i	$0.336375\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.59289	−0.305572
$336$	0	0
$337$	−4.52864	−0.246691	−0.123345	−	0.992364i	$-0.539362\pi$
$337$	−4.52864	−0.246691	−0.123345	+	0.992364i	$0.539362\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.10676	0.330700
$342$	0	0
$343$	−1.69965	−0.0917725
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.0937869	0.00503475	0.00251737	−	0.999997i	$-0.499199\pi$
$347$	0.0937869	0.00503475	0.00251737	+	0.999997i	$0.499199\pi$
$348$	0	0
$349$	8.07902	0.432460	0.216230	−	0.976342i	$-0.430624\pi$
$349$	8.07902	0.432460	0.216230	+	0.976342i	$0.430624\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	−13.9852	−0.742259
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.69965	−0.459150	−0.229575	−	0.973291i	$-0.573734\pi$
$359$	−8.69965	−0.459150	−0.229575	+	0.973291i	$0.573734\pi$
$360$	0	0
$361$	−17.4913	−0.920594
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.9427	−0.677453
$366$	0	0
$367$	31.8064	1.66028	0.830141	−	0.557554i	$-0.188260\pi$
$367$	31.8064	1.66028	0.830141	+	0.557554i	$0.188260\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.436651	0.0226698
$372$	0	0
$373$	−13.4288	−0.695319	−0.347660	−	0.937621i	$-0.613023\pi$
$373$	−13.4288	−0.695319	−0.347660	+	0.937621i	$0.613023\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−48.8836	−2.51764
$378$	0	0
$379$	31.8212	1.63454	0.817272	−	0.576252i	$-0.195485\pi$
$379$	31.8212	1.63454	0.817272	+	0.576252i	$0.195485\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−34.3480	−1.75510	−0.877551	−	0.479483i	$-0.840824\pi$
$383$	−34.3480	−1.75510	−0.877551	+	0.479483i	$0.840824\pi$
$384$	0	0
$385$	0.349826	0.0178288
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.14928	−0.413185	−0.206592	−	0.978427i	$-0.566237\pi$
$389$	−8.14928	−0.413185	−0.206592	+	0.978427i	$0.566237\pi$
$390$	0	0
$391$	−2.22829	−0.112690
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.51387	0.176802
$396$	0	0
$397$	19.0625	0.956717	0.478359	−	0.878165i	$-0.341232\pi$
$397$	19.0625	0.956717	0.478359	+	0.878165i	$0.341232\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.9705	−0.997277	−0.498639	−	0.866810i	$-0.666167\pi$
$401$	−19.9705	−0.997277	−0.498639	+	0.866810i	$0.666167\pi$
$402$	0	0
$403$	−11.0920	−0.552531
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.0990	−0.797996
$408$	0	0
$409$	−10.3351	−0.511036	−0.255518	−	0.966804i	$-0.582246\pi$
$409$	−10.3351	−0.511036	−0.255518	+	0.966804i	$0.582246\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.0790169	0.00388817
$414$	0	0
$415$	−11.1068	−0.545209
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.72740	−0.279802	−0.139901	−	0.990166i	$-0.544678\pi$
$419$	−5.72740	−0.279802	−0.139901	+	0.990166i	$0.544678\pi$
$420$	0	0
$421$	−8.44361	−0.411516	−0.205758	−	0.978603i	$-0.565966\pi$
$421$	−8.44361	−0.411516	−0.205758	+	0.978603i	$0.565966\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.22829	0.108088
$426$	0	0
$427$	−0.891443	−0.0431400
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.3698	−1.70370	−0.851851	−	0.523785i	$-0.824520\pi$
$431$	−35.3698	−1.70370	−0.851851	+	0.523785i	$0.824520\pi$
$432$	0	0
$433$	13.3923	0.643595	0.321797	−	0.946809i	$-0.395713\pi$
$433$	13.3923	0.643595	0.321797	+	0.946809i	$0.395713\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.22829	−0.0587573
$438$	0	0
$439$	−21.1137	−1.00770	−0.503852	−	0.863790i	$-0.668084\pi$
$439$	−21.1137	−1.00770	−0.503852	+	0.863790i	$0.668084\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1692	1.48089	0.740447	−	0.672115i	$-0.234614\pi$
$443$	31.1692	1.48089	0.740447	+	0.672115i	$0.234614\pi$
$444$	0	0
$445$	−0.486128	−0.0230447
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.09199	0.287499	0.143749	−	0.989614i	$-0.454084\pi$
$449$	6.09199	0.287499	0.143749	+	0.989614i	$0.454084\pi$
$450$	0	0
$451$	23.6849	1.11528
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.635404	−0.0297882
$456$	0	0
$457$	22.1345	1.03541	0.517704	−	0.855560i	$-0.326787\pi$
$457$	22.1345	1.03541	0.517704	+	0.855560i	$0.326787\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.5434	0.537630	0.268815	−	0.963192i	$-0.413368\pi$
$461$	11.5434	0.537630	0.268815	+	0.963192i	$0.413368\pi$
$462$	0	0
$463$	30.4270	1.41406	0.707032	−	0.707181i	$-0.250033\pi$
$463$	30.4270	1.41406	0.707032	+	0.707181i	$0.250033\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.3776	−0.480217	−0.240108	−	0.970746i	$-0.577183\pi$
$467$	−10.3776	−0.480217	−0.240108	+	0.970746i	$0.577183\pi$
$468$	0	0
$469$	0.679714	0.0313863
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−23.0277	−1.05882
$474$	0	0
$475$	1.22829	0.0563580
$476$	0	0
$477$	0	0
$478$	0	0
$479$	29.6128	1.35304	0.676522	−	0.736422i	$-0.263486\pi$
$479$	29.6128	1.35304	0.676522	+	0.736422i	$0.263486\pi$
$480$	0	0
$481$	29.2413	1.33329
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.635404	−0.0288522
$486$	0	0
$487$	11.1562	0.505537	0.252769	−	0.967527i	$-0.418659\pi$
$487$	11.1562	0.505537	0.252769	+	0.967527i	$0.418659\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.2630	−1.72679	−0.863393	−	0.504533i	$-0.831665\pi$
$491$	−38.2630	−1.72679	−0.863393	+	0.504533i	$0.831665\pi$
$492$	0	0
$493$	−20.8342	−0.938323
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.69965	0.0762398
$498$	0	0
$499$	6.40711	0.286822	0.143411	−	0.989663i	$-0.454193\pi$
$499$	6.40711	0.286822	0.143411	+	0.989663i	$0.454193\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.0920	0.806682	0.403341	−	0.915050i	$-0.367849\pi$
$503$	18.0920	0.806682	0.403341	+	0.915050i	$0.367849\pi$
$504$	0	0
$505$	13.3498	0.594059
$506$	0	0
$507$	0	0
$508$	0	0
$509$	34.1285	1.51272	0.756359	−	0.654156i	$-0.226976\pi$
$509$	34.1285	1.51272	0.756359	+	0.654156i	$0.226976\pi$
$510$	0	0
$511$	1.57295	0.0695833
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.33506	0.323221
$516$	0	0
$517$	−30.0990	−1.32375
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−35.8854	−1.57217	−0.786085	−	0.618119i	$-0.787895\pi$
$521$	−35.8854	−1.57217	−0.786085	+	0.618119i	$0.787895\pi$
$522$	0	0
$523$	11.7865	0.515387	0.257693	−	0.966227i	$-0.417038\pi$
$523$	11.7865	0.515387	0.257693	+	0.966227i	$0.417038\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.72740	−0.205929
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−43.0199	−1.86340
$534$	0	0
$535$	16.0495	0.693879
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.1068	0.866060
$540$	0	0
$541$	−17.2413	−0.741260	−0.370630	−	0.928781i	$-0.620858\pi$
$541$	−17.2413	−0.741260	−0.370630	+	0.928781i	$0.620858\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.01477	−0.0434680
$546$	0	0
$547$	14.6571	0.626694	0.313347	−	0.949639i	$-0.398550\pi$
$547$	14.6571	0.626694	0.313347	+	0.949639i	$0.398550\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.4843	−0.489249
$552$	0	0
$553$	−0.427048	−0.0181599
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.07902	−0.0880908	−0.0440454	−	0.999030i	$-0.514025\pi$
$557$	−2.07902	−0.0880908	−0.0440454	+	0.999030i	$0.514025\pi$
$558$	0	0
$559$	41.8264	1.76907
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.10676	0.215224	0.107612	−	0.994193i	$-0.465680\pi$
$563$	5.10676	0.215224	0.107612	+	0.994193i	$0.465680\pi$
$564$	0	0
$565$	17.3203	0.728670
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	10.0642	0.421176	0.210588	−	0.977575i	$-0.432462\pi$
$571$	10.0642	0.421176	0.210588	+	0.977575i	$0.432462\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−15.7274	−0.654740	−0.327370	−	0.944896i	$-0.606162\pi$
$577$	−15.7274	−0.654740	−0.327370	+	0.944896i	$0.606162\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.34983	0.0560002
$582$	0	0
$583$	−10.3420	−0.428323
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.7344	0.773250	0.386625	−	0.922237i	$-0.373641\pi$
$587$	18.7344	0.773250	0.386625	+	0.922237i	$0.373641\pi$
$588$	0	0
$589$	−2.60586	−0.107373
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.21532	0.132037	0.0660187	−	0.997818i	$-0.478970\pi$
$593$	3.21532	0.132037	0.0660187	+	0.997818i	$0.478970\pi$
$594$	0	0
$595$	−0.270809	−0.0111021
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.7639	−0.930108	−0.465054	−	0.885282i	$-0.653965\pi$
$599$	−22.7639	−0.930108	−0.465054	+	0.885282i	$0.653965\pi$
$600$	0	0
$601$	4.30552	0.175626	0.0878128	−	0.996137i	$-0.472012\pi$
$601$	4.30552	0.175626	0.0878128	+	0.996137i	$0.472012\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.71442	0.110357
$606$	0	0
$607$	11.0868	0.450000	0.225000	−	0.974359i	$-0.427762\pi$
$607$	11.0868	0.450000	0.225000	+	0.974359i	$0.427762\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	54.6701	2.21172
$612$	0	0
$613$	29.9409	1.20930	0.604651	−	0.796490i	$-0.293313\pi$
$613$	29.9409	1.20930	0.604651	+	0.796490i	$0.293313\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.0130	−1.40957	−0.704785	−	0.709421i	$-0.748956\pi$
$617$	−35.0130	−1.40957	−0.704785	+	0.709421i	$0.748956\pi$
$618$	0	0
$619$	4.98523	0.200373	0.100187	−	0.994969i	$-0.468056\pi$
$619$	4.98523	0.200373	0.100187	+	0.994969i	$0.468056\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.0590800	0.00236699
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.4626	0.496916
$630$	0	0
$631$	−3.64237	−0.145000	−0.0725002	−	0.997368i	$-0.523098\pi$
$631$	−3.64237	−0.145000	−0.0725002	+	0.997368i	$0.523098\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.75694	−0.0697219
$636$	0	0
$637$	−36.5208	−1.44701
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−38.6997	−1.52854	−0.764272	−	0.644894i	$-0.776902\pi$
$641$	−38.6997	−1.52854	−0.764272	+	0.644894i	$0.776902\pi$
$642$	0	0
$643$	−28.8637	−1.13827	−0.569137	−	0.822243i	$-0.692722\pi$
$643$	−28.8637	−1.13827	−0.569137	+	0.822243i	$0.692722\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.8854	0.781777	0.390888	−	0.920438i	$-0.372168\pi$
$647$	19.8854	0.781777	0.390888	+	0.920438i	$0.372168\pi$
$648$	0	0
$649$	−1.87151	−0.0734630
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.1988	1.06437	0.532185	−	0.846628i	$-0.321371\pi$
$653$	27.1988	1.06437	0.532185	+	0.846628i	$0.321371\pi$
$654$	0	0
$655$	−20.2135	−0.789808
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.7274	0.690561	0.345281	−	0.938499i	$-0.387784\pi$
$659$	17.7274	0.690561	0.345281	+	0.938499i	$0.387784\pi$
$660$	0	0
$661$	40.7048	1.58323	0.791617	−	0.611018i	$-0.209240\pi$
$661$	40.7048	1.58323	0.791617	+	0.611018i	$0.209240\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.149277	−0.00578871
$666$	0	0
$667$	9.34983	0.362027
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.1137	0.815086
$672$	0	0
$673$	12.2135	0.470797	0.235398	−	0.971899i	$-0.424361\pi$
$673$	12.2135	0.470797	0.235398	+	0.971899i	$0.424361\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.5061	−0.634380	−0.317190	−	0.948362i	$-0.602739\pi$
$677$	−16.5061	−0.634380	−0.317190	+	0.948362i	$0.602739\pi$
$678$	0	0
$679$	0.0772219	0.00296350
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.1988	1.11726	0.558630	−	0.829417i	$-0.311327\pi$
$683$	29.1988	1.11726	0.558630	+	0.829417i	$0.311327\pi$
$684$	0	0
$685$	−17.9279	−0.684992
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18.7847	0.715639
$690$	0	0
$691$	−15.9150	−0.605434	−0.302717	−	0.953080i	$-0.597894\pi$
$691$	−15.9150	−0.605434	−0.302717	+	0.953080i	$0.597894\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.8637	0.412084
$696$	0	0
$697$	−18.3351	−0.694490
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−34.5981	−1.30675	−0.653375	−	0.757034i	$-0.726648\pi$
$701$	−34.5981	−1.30675	−0.653375	+	0.757034i	$0.726648\pi$
$702$	0	0
$703$	6.86971	0.259096
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.62243	−0.0610177
$708$	0	0
$709$	35.8212	1.34529	0.672646	−	0.739964i	$-0.265158\pi$
$709$	35.8212	1.34529	0.672646	+	0.739964i	$0.265158\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.12153	0.0794520
$714$	0	0
$715$	15.0495	0.562819
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32.5781	1.21496	0.607479	−	0.794335i	$-0.292181\pi$
$719$	32.5781	1.21496	0.607479	+	0.794335i	$0.292181\pi$
$720$	0	0
$721$	−0.891443	−0.0331991
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.34983	−0.347244
$726$	0	0
$727$	−31.9557	−1.18517	−0.592585	−	0.805508i	$-0.701893\pi$
$727$	−31.9557	−1.18517	−0.592585	+	0.805508i	$0.701893\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.8264	0.659331
$732$	0	0
$733$	−2.19359	−0.0810220	−0.0405110	−	0.999179i	$-0.512899\pi$
$733$	−2.19359	−0.0810220	−0.0405110	+	0.999179i	$0.512899\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0990	−0.593013
$738$	0	0
$739$	−51.7473	−1.90356	−0.951778	−	0.306787i	$-0.900746\pi$
$739$	−51.7473	−1.90356	−0.951778	+	0.306787i	$0.900746\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.4436	0.529885	0.264942	−	0.964264i	$-0.414647\pi$
$743$	14.4436	0.529885	0.264942	+	0.964264i	$0.414647\pi$
$744$	0	0
$745$	−15.7144	−0.575732
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.95052	−0.0712706
$750$	0	0
$751$	17.4288	0.635987	0.317994	−	0.948093i	$-0.396991\pi$
$751$	17.4288	0.635987	0.317994	+	0.948093i	$0.396991\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.39234	0.305429
$756$	0	0
$757$	−34.9331	−1.26967	−0.634833	−	0.772650i	$-0.718931\pi$
$757$	−34.9331	−1.26967	−0.634833	+	0.772650i	$0.718931\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.2482	0.842748	0.421374	−	0.906887i	$-0.361548\pi$
$761$	23.2482	0.842748	0.421374	+	0.906887i	$0.361548\pi$
$762$	0	0
$763$	0.123327	0.00446473
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.39930	0.122742
$768$	0	0
$769$	−48.6997	−1.75615	−0.878077	−	0.478519i	$-0.841174\pi$
$769$	−48.6997	−1.75615	−0.878077	+	0.478519i	$0.841174\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.6128	0.561554	0.280777	−	0.959773i	$-0.409408\pi$
$773$	15.6128	0.561554	0.280777	+	0.959773i	$0.409408\pi$
$774$	0	0
$775$	−2.12153	−0.0762077
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.1068	−0.362112
$780$	0	0
$781$	−40.2560	−1.44047
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.5633	0.698246
$786$	0	0
$787$	24.2630	0.864883	0.432441	−	0.901662i	$-0.357652\pi$
$787$	24.2630	0.864883	0.432441	+	0.901662i	$0.357652\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.10497	−0.0748440
$792$	0	0
$793$	−38.3498	−1.36184
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−44.5911	−1.57950	−0.789749	−	0.613430i	$-0.789789\pi$
$797$	−44.5911	−1.57950	−0.789749	+	0.613430i	$0.789789\pi$
$798$	0	0
$799$	23.3003	0.824307
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−37.2552	−1.31471
$804$	0	0
$805$	0.121532	0.00428344
$806$	0	0
$807$	0	0
$808$	0	0
$809$	44.7144	1.57208	0.786038	−	0.618179i	$-0.212129\pi$
$809$	44.7144	1.57208	0.786038	+	0.618179i	$0.212129\pi$
$810$	0	0
$811$	1.19179	0.0418495	0.0209247	−	0.999781i	$-0.493339\pi$
$811$	1.19179	0.0418495	0.0209247	+	0.999781i	$0.493339\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.01477	−0.0355458
$816$	0	0
$817$	9.82635	0.343780
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	26.7551	0.932626	0.466313	−	0.884620i	$-0.345582\pi$
$823$	26.7551	0.932626	0.466313	+	0.884620i	$0.345582\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.2907	0.948992	0.474496	−	0.880258i	$-0.342630\pi$
$827$	27.2907	0.948992	0.474496	+	0.880258i	$0.342630\pi$
$828$	0	0
$829$	34.2925	1.19103	0.595515	−	0.803344i	$-0.296948\pi$
$829$	34.2925	1.19103	0.595515	+	0.803344i	$0.296948\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15.5651	−0.539300
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.6701	1.40409	0.702044	−	0.712133i	$-0.252271\pi$
$839$	40.6701	1.40409	0.702044	+	0.712133i	$0.252271\pi$
$840$	0	0
$841$	58.4192	2.01446
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14.3351	−0.493141
$846$	0	0
$847$	−0.329889	−0.0113351
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−5.59289	−0.191722
$852$	0	0
$853$	−34.6276	−1.18563	−0.592813	−	0.805340i	$-0.701983\pi$
$853$	−34.6276	−1.18563	−0.592813	+	0.805340i	$0.701983\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.4861	−0.904749	−0.452374	−	0.891828i	$-0.649423\pi$
$857$	−26.4861	−0.904749	−0.452374	+	0.891828i	$0.649423\pi$
$858$	0	0
$859$	40.9627	1.39763	0.698814	−	0.715304i	$-0.253712\pi$
$859$	40.9627	1.39763	0.698814	+	0.715304i	$0.253712\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.5712	0.632170	0.316085	−	0.948731i	$-0.397632\pi$
$863$	18.5712	0.632170	0.316085	+	0.948731i	$0.397632\pi$
$864$	0	0
$865$	−2.63540	−0.0896064
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.1146	0.343113
$870$	0	0
$871$	29.2413	0.990803
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.121532	−0.00410853
$876$	0	0
$877$	−28.4913	−0.962083	−0.481041	−	0.876698i	$-0.659741\pi$
$877$	−28.4913	−0.962083	−0.481041	+	0.876698i	$0.659741\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.91318	0.165529	0.0827645	−	0.996569i	$-0.473625\pi$
$881$	4.91318	0.165529	0.0827645	+	0.996569i	$0.473625\pi$
$882$	0	0
$883$	30.1710	1.01534	0.507668	−	0.861553i	$-0.330508\pi$
$883$	30.1710	1.01534	0.507668	+	0.861553i	$0.330508\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25.7865	−0.865825	−0.432913	−	0.901436i	$-0.642514\pi$
$887$	−25.7865	−0.865825	−0.432913	+	0.901436i	$0.642514\pi$
$888$	0	0
$889$	0.213524	0.00716136
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.8438	0.429800
$894$	0	0
$895$	−14.4566	−0.483230
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.8360	0.661566
$900$	0	0
$901$	8.00601	0.266719
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.7422	−0.357082
$906$	0	0
$907$	−44.3776	−1.47353	−0.736767	−	0.676147i	$-0.763648\pi$
$907$	−44.3776	−1.47353	−0.736767	+	0.676147i	$0.763648\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.7968	−1.58358	−0.791789	−	0.610794i	$-0.790850\pi$
$911$	−47.7968	−1.58358	−0.791789	+	0.610794i	$0.790850\pi$
$912$	0	0
$913$	−31.9705	−1.05807
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.45659	0.0811237
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	73.1189	2.40674
$924$	0	0
$925$	5.59289	0.183893
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.0199	−0.788069	−0.394034	−	0.919096i	$-0.628921\pi$
$929$	−24.0199	−0.788069	−0.394034	+	0.919096i	$0.628921\pi$
$930$	0	0
$931$	−8.57991	−0.281195
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.41407	0.209763
$936$	0	0
$937$	0.384533	0.0125621	0.00628107	−	0.999980i	$-0.498001\pi$
$937$	0.384533	0.0125621	0.00628107	+	0.999980i	$0.498001\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.0920	−0.817976	−0.408988	−	0.912540i	$-0.634118\pi$
$941$	−25.0920	−0.817976	−0.408988	+	0.912540i	$0.634118\pi$
$942$	0	0
$943$	8.22829	0.267950
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.7126	−1.45297	−0.726483	−	0.687185i	$-0.758846\pi$
$947$	−44.7126	−1.45297	−0.726483	+	0.687185i	$0.758846\pi$
$948$	0	0
$949$	67.6683	2.19661
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.2500	0.688356	0.344178	−	0.938904i	$-0.388158\pi$
$953$	21.2500	0.688356	0.344178	+	0.938904i	$0.388158\pi$
$954$	0	0
$955$	−22.6701	−0.733588
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.17882	0.0703577
$960$	0	0
$961$	−26.4991	−0.854810
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.24306	−0.0722068
$966$	0	0
$967$	−41.0417	−1.31981	−0.659906	−	0.751349i	$-0.729404\pi$
$967$	−41.0417	−1.31981	−0.659906	+	0.751349i	$0.729404\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.2760	−1.13206	−0.566030	−	0.824385i	$-0.691521\pi$
$971$	−35.2760	−1.13206	−0.566030	+	0.824385i	$0.691521\pi$
$972$	0	0
$973$	−1.32029	−0.0423264
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.79164	0.217284	0.108642	−	0.994081i	$-0.465350\pi$
$977$	6.79164	0.217284	0.108642	+	0.994081i	$0.465350\pi$
$978$	0	0
$979$	−1.39930	−0.0447219
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.5981	−0.625081	−0.312540	−	0.949904i	$-0.601180\pi$
$983$	−19.5981	−0.625081	−0.312540	+	0.949904i	$0.601180\pi$
$984$	0	0
$985$	−7.57812	−0.241459
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−37.2995	−1.18486	−0.592429	−	0.805623i	$-0.701831\pi$
$991$	−37.2995	−1.18486	−0.592429	+	0.805623i	$0.701831\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	−19.7274	−0.624773	−0.312386	−	0.949955i	$-0.601128\pi$
$997$	−19.7274	−0.624773	−0.312386	+	0.949955i	$0.601128\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bj.1.2		3
3.2	odd	2		920.2.a.h.1.2	✓	3
12.11	even	2		1840.2.a.s.1.2		3
15.2	even	4		4600.2.e.p.4049.3		6
15.8	even	4		4600.2.e.p.4049.4		6
15.14	odd	2		4600.2.a.x.1.2		3
24.5	odd	2		7360.2.a.by.1.2		3
24.11	even	2		7360.2.a.cc.1.2		3
60.59	even	2		9200.2.a.ce.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.2	✓	3	3.2	odd	2
1840.2.a.s.1.2		3	12.11	even	2
4600.2.a.x.1.2		3	15.14	odd	2
4600.2.e.p.4049.3		6	15.2	even	4
4600.2.e.p.4049.4		6	15.8	even	4
7360.2.a.by.1.2		3	24.5	odd	2
7360.2.a.cc.1.2		3	24.11	even	2
8280.2.a.bj.1.2		3	1.1	even	1	trivial
9200.2.a.ce.1.2		3	60.59	even	2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.121532 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.121532 q^{7} -2.87847 q^{11} +5.22829 q^{13} +2.22829 q^{17} +1.22829 q^{19} -1.00000 q^{23} +1.00000 q^{25} -9.34983 q^{29} -2.12153 q^{31} -0.121532 q^{35} +5.59289 q^{37} -8.22829 q^{41} +8.00000 q^{43} +10.4566 q^{47} -6.98523 q^{49} +3.59289 q^{53} +2.87847 q^{55} +0.650174 q^{59} -7.33506 q^{61} -5.22829 q^{65} +5.59289 q^{67} +13.9852 q^{71} +12.9427 q^{73} -0.349826 q^{77} -3.51387 q^{79} +11.1068 q^{83} -2.22829 q^{85} +0.486128 q^{89} +0.635404 q^{91} -1.22829 q^{95} +0.635404 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7	\( 3 q - 3 q^{5} + 2 q^{7} - 7 q^{11} - q^{13} - 10 q^{17} - 13 q^{19} - 3 q^{23} + 3 q^{25} - 13 q^{29} - 8 q^{31} - 2 q^{35} + 5 q^{37} - 8 q^{41} + 24 q^{43} - 2 q^{47} - q^{49} - q^{53} + 7 q^{55} + 17 q^{59} + 13 q^{61} + q^{65} + 5 q^{67} + 22 q^{71} + 12 q^{73} + 14 q^{77} - 4 q^{79} + 15 q^{83} + 10 q^{85} + 8 q^{89} - 3 q^{91} + 13 q^{95} - 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - 7 * q^11 - q^13 - 10 * q^17 - 13 * q^19 - 3 * q^23 + 3 * q^25 - 13 * q^29 - 8 * q^31 - 2 * q^35 + 5 * q^37 - 8 * q^41 + 24 * q^43 - 2 * q^47 - q^49 - q^53 + 7 * q^55 + 17 * q^59 + 13 * q^61 + q^65 + 5 * q^67 + 22 * q^71 + 12 * q^73 + 14 * q^77 - 4 * q^79 + 15 * q^83 + 10 * q^85 + 8 * q^89 - 3 * q^91 + 13 * q^95 - 3 * q^97

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