LMFDB - Embedded newform 8280.2.a.bw.1.5

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			1.5
Root			$4.09437$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -0.627340 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -0.627340 q^{7} -1.35025 q^{11} +2.31878 q^{13} +5.74602 q^{17} -8.04218 q^{19} +1.00000 q^{23} +1.00000 q^{25} -6.21116 q^{29} -7.91514 q^{31} -0.627340 q^{35} +10.0588 q^{37} -2.48776 q^{41} +4.42390 q^{43} -4.68883 q^{47} -6.60645 q^{49} +2.21712 q^{53} -1.35025 q^{55} -0.999511 q^{59} -9.30722 q^{61} +2.31878 q^{65} -11.0106 q^{67} -9.42438 q^{71} +9.73349 q^{73} +0.847065 q^{77} +3.26504 q^{79} +1.04553 q^{83} +5.74602 q^{85} +2.84494 q^{89} -1.45466 q^{91} -8.04218 q^{95} +7.81347 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.627340	−0.237112	−0.118556	−	0.992947i	$-0.537827\pi$
$7$	−0.627340	−0.237112	−0.118556	+	0.992947i	$0.537827\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.35025	−0.407115	−0.203558	−	0.979063i	$-0.565250\pi$
$11$	−1.35025	−0.407115	−0.203558	+	0.979063i	$0.565250\pi$
$12$	0	0
$13$	2.31878	0.643114	0.321557	−	0.946890i	$-0.395794\pi$
$13$	2.31878	0.643114	0.321557	+	0.946890i	$0.395794\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.74602	1.39362	0.696808	−	0.717258i	$-0.254603\pi$
$17$	5.74602	1.39362	0.696808	+	0.717258i	$0.254603\pi$
$18$	0	0
$19$	−8.04218	−1.84500	−0.922501	−	0.385994i	$-0.873859\pi$
$19$	−8.04218	−1.84500	−0.922501	+	0.385994i	$0.873859\pi$
$20$	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
$29$	−6.21116	−1.15338	−0.576692	−	0.816962i	$-0.695657\pi$
$29$	−6.21116	−1.15338	−0.576692	+	0.816962i	$0.695657\pi$
$30$	0	0
$31$	−7.91514	−1.42160	−0.710800	−	0.703394i	$-0.751667\pi$
$31$	−7.91514	−1.42160	−0.710800	+	0.703394i	$0.751667\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.627340	−0.106040
$36$	0	0
$37$	10.0588	1.65366	0.826832	−	0.562449i	$-0.190141\pi$
$37$	10.0588	1.65366	0.826832	+	0.562449i	$0.190141\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.48776	−0.388523	−0.194261	−	0.980950i	$-0.562231\pi$
$41$	−2.48776	−0.388523	−0.194261	+	0.980950i	$0.562231\pi$
$42$	0	0
$43$	4.42390	0.674638	0.337319	−	0.941390i	$-0.390480\pi$
$43$	4.42390	0.674638	0.337319	+	0.941390i	$0.390480\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.68883	−0.683936	−0.341968	−	0.939712i	$-0.611093\pi$
$47$	−4.68883	−0.683936	−0.341968	+	0.939712i	$0.611093\pi$
$48$	0	0
$49$	−6.60645	−0.943778
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.21712	0.304544	0.152272	−	0.988339i	$-0.451341\pi$
$53$	2.21712	0.304544	0.152272	+	0.988339i	$0.451341\pi$
$54$	0	0
$55$	−1.35025	−0.182068
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.999511	−0.130125	−0.0650626	−	0.997881i	$-0.520725\pi$
$59$	−0.999511	−0.130125	−0.0650626	+	0.997881i	$0.520725\pi$
$60$	0	0
$61$	−9.30722	−1.19167	−0.595833	−	0.803108i	$-0.703178\pi$
$61$	−9.30722	−1.19167	−0.595833	+	0.803108i	$0.703178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.31878	0.287609
$66$	0	0
$67$	−11.0106	−1.34516	−0.672578	−	0.740026i	$-0.734813\pi$
$67$	−11.0106	−1.34516	−0.672578	+	0.740026i	$0.734813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.42438	−1.11847	−0.559234	−	0.829010i	$-0.688905\pi$
$71$	−9.42438	−1.11847	−0.559234	+	0.829010i	$0.688905\pi$
$72$	0	0
$73$	9.73349	1.13922	0.569609	−	0.821916i	$-0.307095\pi$
$73$	9.73349	1.13922	0.569609	+	0.821916i	$0.307095\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.847065	0.0965320
$78$	0	0
$79$	3.26504	0.367345	0.183673	−	0.982987i	$-0.441201\pi$
$79$	3.26504	0.367345	0.183673	+	0.982987i	$0.441201\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.04553	0.114761	0.0573807	−	0.998352i	$-0.481725\pi$
$83$	1.04553	0.114761	0.0573807	+	0.998352i	$0.481725\pi$
$84$	0	0
$85$	5.74602	0.623244
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.84494	0.301563	0.150782	−	0.988567i	$-0.451821\pi$
$89$	2.84494	0.301563	0.150782	+	0.988567i	$0.451821\pi$
$90$	0	0
$91$	−1.45466	−0.152490
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.04218	−0.825110
$96$	0	0
$97$	7.81347	0.793338	0.396669	−	0.917962i	$-0.370166\pi$
$97$	7.81347	0.793338	0.396669	+	0.917962i	$0.370166\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.5145	−1.14573	−0.572866	−	0.819649i	$-0.694168\pi$
$101$	−11.5145	−1.14573	−0.572866	+	0.819649i	$0.694168\pi$
$102$	0	0
$103$	6.21129	0.612017	0.306008	−	0.952029i	$-0.401006\pi$
$103$	6.21129	0.612017	0.306008	+	0.952029i	$0.401006\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.34725	−0.710286	−0.355143	−	0.934812i	$-0.615568\pi$
$107$	−7.34725	−0.710286	−0.355143	+	0.934812i	$0.615568\pi$
$108$	0	0
$109$	6.76495	0.647965	0.323983	−	0.946063i	$-0.394978\pi$
$109$	6.76495	0.647965	0.323983	+	0.946063i	$0.394978\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.55728	−0.334641	−0.167320	−	0.985903i	$-0.553511\pi$
$113$	−3.55728	−0.334641	−0.167320	+	0.985903i	$0.553511\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.60471	−0.330443
$120$	0	0
$121$	−9.17683	−0.834257
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.1787	−0.903210	−0.451605	−	0.892218i	$-0.649148\pi$
$127$	−10.1787	−0.903210	−0.451605	+	0.892218i	$0.649148\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.5369	−1.53220	−0.766102	−	0.642719i	$-0.777806\pi$
$131$	−17.5369	−1.53220	−0.766102	+	0.642719i	$0.777806\pi$
$132$	0	0
$133$	5.04518	0.437472
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.923710	0.0789178	0.0394589	−	0.999221i	$-0.487437\pi$
$137$	0.923710	0.0789178	0.0394589	+	0.999221i	$0.487437\pi$
$138$	0	0
$139$	12.3527	1.04775	0.523873	−	0.851796i	$-0.324487\pi$
$139$	12.3527	1.04775	0.523873	+	0.851796i	$0.324487\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.13093	−0.261822
$144$	0	0
$145$	−6.21116	−0.515809
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.56285	−0.128034	−0.0640169	−	0.997949i	$-0.520391\pi$
$149$	−1.56285	−0.128034	−0.0640169	+	0.997949i	$0.520391\pi$
$150$	0	0
$151$	16.1330	1.31288	0.656442	−	0.754377i	$-0.272061\pi$
$151$	16.1330	1.31288	0.656442	+	0.754377i	$0.272061\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.91514	−0.635759
$156$	0	0
$157$	−19.2721	−1.53808	−0.769039	−	0.639201i	$-0.779265\pi$
$157$	−19.2721	−1.53808	−0.769039	+	0.639201i	$0.779265\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.627340	−0.0494413
$162$	0	0
$163$	6.84160	0.535875	0.267938	−	0.963436i	$-0.413658\pi$
$163$	6.84160	0.535875	0.267938	+	0.963436i	$0.413658\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.1235	1.47982	0.739912	−	0.672704i	$-0.234867\pi$
$167$	19.1235	1.47982	0.739912	+	0.672704i	$0.234867\pi$
$168$	0	0
$169$	−7.62325	−0.586404
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.9946	−0.835905	−0.417952	−	0.908469i	$-0.637252\pi$
$173$	−10.9946	−0.835905	−0.417952	+	0.908469i	$0.637252\pi$
$174$	0	0
$175$	−0.627340	−0.0474224
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.188747	0.0141076	0.00705379	−	0.999975i	$-0.497755\pi$
$179$	0.188747	0.0141076	0.00705379	+	0.999975i	$0.497755\pi$
$180$	0	0
$181$	13.6547	1.01495	0.507474	−	0.861667i	$-0.330579\pi$
$181$	13.6547	1.01495	0.507474	+	0.861667i	$0.330579\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.0588	0.739541
$186$	0	0
$187$	−7.75856	−0.567362
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.33019	0.675109	0.337555	−	0.941306i	$-0.390400\pi$
$191$	9.33019	0.675109	0.337555	+	0.941306i	$0.390400\pi$
$192$	0	0
$193$	−15.9196	−1.14592	−0.572959	−	0.819584i	$-0.694204\pi$
$193$	−15.9196	−1.14592	−0.572959	+	0.819584i	$0.694204\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.14012	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$197$	−4.14012	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$198$	0	0
$199$	−11.2974	−0.800852	−0.400426	−	0.916329i	$-0.631138\pi$
$199$	−11.2974	−0.800852	−0.400426	+	0.916329i	$0.631138\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.89651	0.273481
$204$	0	0
$205$	−2.48776	−0.173753
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.8589	0.751129
$210$	0	0
$211$	−5.37625	−0.370116	−0.185058	−	0.982728i	$-0.559247\pi$
$211$	−5.37625	−0.370116	−0.185058	+	0.982728i	$0.559247\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.42390	0.301707
$216$	0	0
$217$	4.96548	0.337079
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.3238	0.896254
$222$	0	0
$223$	0.171237	0.0114669	0.00573344	−	0.999984i	$-0.498175\pi$
$223$	0.171237	0.0114669	0.00573344	+	0.999984i	$0.498175\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.1906	0.875492	0.437746	−	0.899099i	$-0.355777\pi$
$227$	13.1906	0.875492	0.437746	+	0.899099i	$0.355777\pi$
$228$	0	0
$229$	7.06100	0.466604	0.233302	−	0.972404i	$-0.425047\pi$
$229$	7.06100	0.466604	0.233302	+	0.972404i	$0.425047\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.5359	−1.01779	−0.508895	−	0.860829i	$-0.669946\pi$
$233$	−15.5359	−1.01779	−0.508895	+	0.860829i	$0.669946\pi$
$234$	0	0
$235$	−4.68883	−0.305866
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.6546	−1.27135	−0.635675	−	0.771957i	$-0.719278\pi$
$239$	−19.6546	−1.27135	−0.635675	+	0.771957i	$0.719278\pi$
$240$	0	0
$241$	9.43078	0.607490	0.303745	−	0.952753i	$-0.401763\pi$
$241$	9.43078	0.607490	0.303745	+	0.952753i	$0.401763\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.60645	−0.422070
$246$	0	0
$247$	−18.6481	−1.18655
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.9799	−1.26112	−0.630561	−	0.776139i	$-0.717175\pi$
$251$	−19.9799	−1.26112	−0.630561	+	0.776139i	$0.717175\pi$
$252$	0	0
$253$	−1.35025	−0.0848894
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.6706	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$257$	−19.6706	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$258$	0	0
$259$	−6.31031	−0.392104
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.8941	0.856748	0.428374	−	0.903602i	$-0.359087\pi$
$263$	13.8941	0.856748	0.428374	+	0.903602i	$0.359087\pi$
$264$	0	0
$265$	2.21712	0.136196
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−24.5448	−1.49652	−0.748262	−	0.663404i	$-0.769111\pi$
$269$	−24.5448	−1.49652	−0.748262	+	0.663404i	$0.769111\pi$
$270$	0	0
$271$	1.96182	0.119172	0.0595859	−	0.998223i	$-0.481022\pi$
$271$	1.96182	0.119172	0.0595859	+	0.998223i	$0.481022\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.35025	−0.0814231
$276$	0	0
$277$	1.86591	0.112112	0.0560560	−	0.998428i	$-0.482147\pi$
$277$	1.86591	0.112112	0.0560560	+	0.998428i	$0.482147\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.0904	−1.13884	−0.569420	−	0.822046i	$-0.692832\pi$
$281$	−19.0904	−1.13884	−0.569420	+	0.822046i	$0.692832\pi$
$282$	0	0
$283$	−12.4149	−0.737992	−0.368996	−	0.929431i	$-0.620298\pi$
$283$	−12.4149	−0.737992	−0.368996	+	0.929431i	$0.620298\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.56067	0.0921235
$288$	0	0
$289$	16.0168	0.942165
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.6677	−1.49952	−0.749762	−	0.661708i	$-0.769832\pi$
$293$	−25.6677	−1.49952	−0.749762	+	0.661708i	$0.769832\pi$
$294$	0	0
$295$	−0.999511	−0.0581938
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.31878	0.134099
$300$	0	0
$301$	−2.77528	−0.159965
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.30722	−0.532930
$306$	0	0
$307$	−28.4044	−1.62112	−0.810562	−	0.585653i	$-0.800838\pi$
$307$	−28.4044	−1.62112	−0.810562	+	0.585653i	$0.800838\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.2898	0.753596	0.376798	−	0.926296i	$-0.377025\pi$
$311$	13.2898	0.753596	0.376798	+	0.926296i	$0.377025\pi$
$312$	0	0
$313$	16.2524	0.918638	0.459319	−	0.888271i	$-0.348094\pi$
$313$	16.2524	0.918638	0.459319	+	0.888271i	$0.348094\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.3091	−1.36533	−0.682667	−	0.730730i	$-0.739180\pi$
$317$	−24.3091	−1.36533	−0.682667	+	0.730730i	$0.739180\pi$
$318$	0	0
$319$	8.38661	0.469560
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−46.2106	−2.57122
$324$	0	0
$325$	2.31878	0.128623
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.94149	0.162170
$330$	0	0
$331$	14.2774	0.784757	0.392379	−	0.919804i	$-0.371652\pi$
$331$	14.2774	0.784757	0.392379	+	0.919804i	$0.371652\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.0106	−0.601572
$336$	0	0
$337$	−7.07082	−0.385172	−0.192586	−	0.981280i	$-0.561687\pi$
$337$	−7.07082	−0.385172	−0.192586	+	0.981280i	$0.561687\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.6874	0.578756
$342$	0	0
$343$	8.53586	0.460893
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.50299	−0.295416	−0.147708	−	0.989031i	$-0.547190\pi$
$347$	−5.50299	−0.295416	−0.147708	+	0.989031i	$0.547190\pi$
$348$	0	0
$349$	−2.21784	−0.118718	−0.0593592	−	0.998237i	$-0.518906\pi$
$349$	−2.21784	−0.118718	−0.0593592	+	0.998237i	$0.518906\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.8763	0.951457	0.475729	−	0.879592i	$-0.342184\pi$
$353$	17.8763	0.951457	0.475729	+	0.879592i	$0.342184\pi$
$354$	0	0
$355$	−9.42438	−0.500194
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0365	−1.05748	−0.528742	−	0.848783i	$-0.677336\pi$
$359$	−20.0365	−1.05748	−0.528742	+	0.848783i	$0.677336\pi$
$360$	0	0
$361$	45.6767	2.40403
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.73349	0.509474
$366$	0	0
$367$	−4.03743	−0.210752	−0.105376	−	0.994432i	$-0.533605\pi$
$367$	−4.03743	−0.210752	−0.105376	+	0.994432i	$0.533605\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.39089	−0.0722112
$372$	0	0
$373$	21.4729	1.11182	0.555912	−	0.831241i	$-0.312369\pi$
$373$	21.4729	1.11182	0.555912	+	0.831241i	$0.312369\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.4023	−0.741757
$378$	0	0
$379$	6.60661	0.339359	0.169679	−	0.985499i	$-0.445727\pi$
$379$	6.60661	0.339359	0.169679	+	0.985499i	$0.445727\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.1012	−1.12932	−0.564660	−	0.825323i	$-0.690993\pi$
$383$	−22.1012	−1.12932	−0.564660	+	0.825323i	$0.690993\pi$
$384$	0	0
$385$	0.847065	0.0431704
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.49684	0.177297	0.0886485	−	0.996063i	$-0.471745\pi$
$389$	3.49684	0.177297	0.0886485	+	0.996063i	$0.471745\pi$
$390$	0	0
$391$	5.74602	0.290589
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.26504	0.164282
$396$	0	0
$397$	−13.3465	−0.669844	−0.334922	−	0.942246i	$-0.608710\pi$
$397$	−13.3465	−0.669844	−0.334922	+	0.942246i	$0.608710\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.1440	−1.30557	−0.652783	−	0.757545i	$-0.726399\pi$
$401$	−26.1440	−1.30557	−0.652783	+	0.757545i	$0.726399\pi$
$402$	0	0
$403$	−18.3535	−0.914252
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.5819	−0.673232
$408$	0	0
$409$	26.0639	1.28877	0.644387	−	0.764699i	$-0.277112\pi$
$409$	26.0639	1.28877	0.644387	+	0.764699i	$0.277112\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.627033	0.0308543
$414$	0	0
$415$	1.04553	0.0513229
$416$	0	0
$417$	0	0
$418$	0	0
$419$	22.1694	1.08305	0.541523	−	0.840686i	$-0.317848\pi$
$419$	22.1694	1.08305	0.541523	+	0.840686i	$0.317848\pi$
$420$	0	0
$421$	−1.79226	−0.0873495	−0.0436748	−	0.999046i	$-0.513907\pi$
$421$	−1.79226	−0.0873495	−0.0436748	+	0.999046i	$0.513907\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.74602	0.278723
$426$	0	0
$427$	5.83879	0.282559
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.95997	0.0944085	0.0472042	−	0.998885i	$-0.484969\pi$
$431$	1.95997	0.0944085	0.0472042	+	0.998885i	$0.484969\pi$
$432$	0	0
$433$	−6.48430	−0.311616	−0.155808	−	0.987787i	$-0.549798\pi$
$433$	−6.48430	−0.311616	−0.155808	+	0.987787i	$0.549798\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.04218	−0.384710
$438$	0	0
$439$	11.8824	0.567118	0.283559	−	0.958955i	$-0.408485\pi$
$439$	11.8824	0.567118	0.283559	+	0.958955i	$0.408485\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.4288	0.590508	0.295254	−	0.955419i	$-0.404596\pi$
$443$	12.4288	0.590508	0.295254	+	0.955419i	$0.404596\pi$
$444$	0	0
$445$	2.84494	0.134863
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−29.1634	−1.37631	−0.688154	−	0.725565i	$-0.741579\pi$
$449$	−29.1634	−1.37631	−0.688154	+	0.725565i	$0.741579\pi$
$450$	0	0
$451$	3.35910	0.158174
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.45466	−0.0681957
$456$	0	0
$457$	29.0382	1.35835	0.679175	−	0.733977i	$-0.262338\pi$
$457$	29.0382	1.35835	0.679175	+	0.733977i	$0.262338\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	41.8601	1.94962	0.974810	−	0.223037i	$-0.0715971\pi$
$461$	41.8601	1.94962	0.974810	+	0.223037i	$0.0715971\pi$
$462$	0	0
$463$	18.6791	0.868093	0.434046	−	0.900890i	$-0.357085\pi$
$463$	18.6791	0.868093	0.434046	+	0.900890i	$0.357085\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.6118	0.722426	0.361213	−	0.932483i	$-0.382363\pi$
$467$	15.6118	0.722426	0.361213	+	0.932483i	$0.382363\pi$
$468$	0	0
$469$	6.90737	0.318953
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.97336	−0.274655
$474$	0	0
$475$	−8.04218	−0.369001
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−38.7467	−1.77038	−0.885191	−	0.465228i	$-0.845972\pi$
$479$	−38.7467	−1.77038	−0.885191	+	0.465228i	$0.845972\pi$
$480$	0	0
$481$	23.3243	1.06350
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.81347	0.354792
$486$	0	0
$487$	−12.7078	−0.575844	−0.287922	−	0.957654i	$-0.592964\pi$
$487$	−12.7078	−0.575844	−0.287922	+	0.957654i	$0.592964\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−22.7702	−1.02761	−0.513803	−	0.857908i	$-0.671764\pi$
$491$	−22.7702	−1.02761	−0.513803	+	0.857908i	$0.671764\pi$
$492$	0	0
$493$	−35.6895	−1.60737
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.91229	0.265202
$498$	0	0
$499$	1.63295	0.0731007	0.0365503	−	0.999332i	$-0.488363\pi$
$499$	1.63295	0.0731007	0.0365503	+	0.999332i	$0.488363\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.34609	0.372134	0.186067	−	0.982537i	$-0.440426\pi$
$503$	8.34609	0.372134	0.186067	+	0.982537i	$0.440426\pi$
$504$	0	0
$505$	−11.5145	−0.512387
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.87561	−0.127459	−0.0637295	−	0.997967i	$-0.520299\pi$
$509$	−2.87561	−0.127459	−0.0637295	+	0.997967i	$0.520299\pi$
$510$	0	0
$511$	−6.10620	−0.270122
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.21129	0.273702
$516$	0	0
$517$	6.33109	0.278441
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.69979	−0.205902	−0.102951	−	0.994686i	$-0.532828\pi$
$521$	−4.69979	−0.205902	−0.102951	+	0.994686i	$0.532828\pi$
$522$	0	0
$523$	−2.02492	−0.0885436	−0.0442718	−	0.999020i	$-0.514097\pi$
$523$	−2.02492	−0.0885436	−0.0442718	+	0.999020i	$0.514097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.4806	−1.98117
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.76857	−0.249865
$534$	0	0
$535$	−7.34725	−0.317650
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.92034	0.384226
$540$	0	0
$541$	−26.3432	−1.13258	−0.566292	−	0.824205i	$-0.691622\pi$
$541$	−26.3432	−1.13258	−0.566292	+	0.824205i	$0.691622\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.76495	0.289779
$546$	0	0
$547$	43.8919	1.87668	0.938342	−	0.345708i	$-0.112361\pi$
$547$	43.8919	1.87668	0.938342	+	0.345708i	$0.112361\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	49.9512	2.12799
$552$	0	0
$553$	−2.04829	−0.0871020
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.91728	−0.250723	−0.125362	−	0.992111i	$-0.540009\pi$
$557$	−5.91728	−0.250723	−0.125362	+	0.992111i	$0.540009\pi$
$558$	0	0
$559$	10.2580	0.433869
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.9175	−1.13444	−0.567218	−	0.823568i	$-0.691980\pi$
$563$	−26.9175	−1.13444	−0.567218	+	0.823568i	$0.691980\pi$
$564$	0	0
$565$	−3.55728	−0.149656
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.1385	−0.508871	−0.254435	−	0.967090i	$-0.581890\pi$
$569$	−12.1385	−0.508871	−0.254435	+	0.967090i	$0.581890\pi$
$570$	0	0
$571$	−42.2886	−1.76972	−0.884861	−	0.465855i	$-0.845747\pi$
$571$	−42.2886	−1.76972	−0.884861	+	0.465855i	$0.845747\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−3.35667	−0.139740	−0.0698701	−	0.997556i	$-0.522258\pi$
$577$	−3.35667	−0.139740	−0.0698701	+	0.997556i	$0.522258\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.655900	−0.0272113
$582$	0	0
$583$	−2.99366	−0.123985
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.7175	−1.22657	−0.613286	−	0.789861i	$-0.710153\pi$
$587$	−29.7175	−1.22657	−0.613286	+	0.789861i	$0.710153\pi$
$588$	0	0
$589$	63.6550	2.62286
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.9653	1.35372	0.676861	−	0.736110i	$-0.263340\pi$
$593$	32.9653	1.35372	0.676861	+	0.736110i	$0.263340\pi$
$594$	0	0
$595$	−3.60471	−0.147779
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.98074	−0.203508	−0.101754	−	0.994810i	$-0.532445\pi$
$599$	−4.98074	−0.203508	−0.101754	+	0.994810i	$0.532445\pi$
$600$	0	0
$601$	29.5795	1.20657	0.603286	−	0.797525i	$-0.293858\pi$
$601$	29.5795	1.20657	0.603286	+	0.797525i	$0.293858\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.17683	−0.373091
$606$	0	0
$607$	−8.66850	−0.351843	−0.175922	−	0.984404i	$-0.556291\pi$
$607$	−8.66850	−0.351843	−0.175922	+	0.984404i	$0.556291\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.8724	−0.439849
$612$	0	0
$613$	5.21499	0.210631	0.105316	−	0.994439i	$-0.466415\pi$
$613$	5.21499	0.210631	0.105316	+	0.994439i	$0.466415\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.3664	−0.980956	−0.490478	−	0.871454i	$-0.663178\pi$
$617$	−24.3664	−0.980956	−0.490478	+	0.871454i	$0.663178\pi$
$618$	0	0
$619$	−20.1629	−0.810417	−0.405209	−	0.914224i	$-0.632801\pi$
$619$	−20.1629	−0.810417	−0.405209	+	0.914224i	$0.632801\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.78474	−0.0715043
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	57.7984	2.30457
$630$	0	0
$631$	−44.6410	−1.77713	−0.888564	−	0.458752i	$-0.848297\pi$
$631$	−44.6410	−1.77713	−0.888564	+	0.458752i	$0.848297\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.1787	−0.403928
$636$	0	0
$637$	−15.3189	−0.606957
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.4769	−0.966778	−0.483389	−	0.875406i	$-0.660594\pi$
$641$	−24.4769	−0.966778	−0.483389	+	0.875406i	$0.660594\pi$
$642$	0	0
$643$	−16.8712	−0.665333	−0.332667	−	0.943044i	$-0.607948\pi$
$643$	−16.8712	−0.665333	−0.332667	+	0.943044i	$0.607948\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.3206	0.563000	0.281500	−	0.959561i	$-0.409168\pi$
$647$	14.3206	0.563000	0.281500	+	0.959561i	$0.409168\pi$
$648$	0	0
$649$	1.34959	0.0529760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.2996	−0.481320	−0.240660	−	0.970609i	$-0.577364\pi$
$653$	−12.2996	−0.481320	−0.240660	+	0.970609i	$0.577364\pi$
$654$	0	0
$655$	−17.5369	−0.685222
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.80189	−0.0701916	−0.0350958	−	0.999384i	$-0.511174\pi$
$659$	−1.80189	−0.0701916	−0.0350958	+	0.999384i	$0.511174\pi$
$660$	0	0
$661$	−29.1017	−1.13192	−0.565962	−	0.824431i	$-0.691495\pi$
$661$	−29.1017	−1.13192	−0.565962	+	0.824431i	$0.691495\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.04518	0.195644
$666$	0	0
$667$	−6.21116	−0.240497
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.5671	0.485146
$672$	0	0
$673$	34.1899	1.31792	0.658962	−	0.752176i	$-0.270996\pi$
$673$	34.1899	1.31792	0.658962	+	0.752176i	$0.270996\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.93105	0.266382	0.133191	−	0.991090i	$-0.457478\pi$
$677$	6.93105	0.266382	0.133191	+	0.991090i	$0.457478\pi$
$678$	0	0
$679$	−4.90170	−0.188110
$680$	0	0
$681$	0	0
$682$	0	0
$683$	25.2446	0.965957	0.482978	−	0.875632i	$-0.339555\pi$
$683$	25.2446	0.965957	0.482978	+	0.875632i	$0.339555\pi$
$684$	0	0
$685$	0.923710	0.0352931
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.14101	0.195857
$690$	0	0
$691$	9.81842	0.373510	0.186755	−	0.982406i	$-0.440203\pi$
$691$	9.81842	0.373510	0.186755	+	0.982406i	$0.440203\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.3527	0.468566
$696$	0	0
$697$	−14.2947	−0.541452
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.5157	1.00149	0.500743	−	0.865596i	$-0.333060\pi$
$701$	26.5157	1.00149	0.500743	+	0.865596i	$0.333060\pi$
$702$	0	0
$703$	−80.8951	−3.05101
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.22348	0.271667
$708$	0	0
$709$	−15.0779	−0.566264	−0.283132	−	0.959081i	$-0.591373\pi$
$709$	−15.0779	−0.566264	−0.283132	+	0.959081i	$0.591373\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.91514	−0.296424
$714$	0	0
$715$	−3.13093	−0.117090
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.37151	−0.349499	−0.174749	−	0.984613i	$-0.555911\pi$
$719$	−9.37151	−0.349499	−0.174749	+	0.984613i	$0.555911\pi$
$720$	0	0
$721$	−3.89659	−0.145117
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.21116	−0.230677
$726$	0	0
$727$	−16.6604	−0.617900	−0.308950	−	0.951078i	$-0.599978\pi$
$727$	−16.6604	−0.617900	−0.308950	+	0.951078i	$0.599978\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	25.4198	0.940186
$732$	0	0
$733$	10.2290	0.377817	0.188908	−	0.981995i	$-0.439505\pi$
$733$	10.2290	0.377817	0.188908	+	0.981995i	$0.439505\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.8670	0.547634
$738$	0	0
$739$	21.9737	0.808316	0.404158	−	0.914689i	$-0.367565\pi$
$739$	21.9737	0.808316	0.404158	+	0.914689i	$0.367565\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.0976	0.553877	0.276938	−	0.960888i	$-0.410680\pi$
$743$	15.0976	0.553877	0.276938	+	0.960888i	$0.410680\pi$
$744$	0	0
$745$	−1.56285	−0.0572584
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.60922	0.168417
$750$	0	0
$751$	−19.1668	−0.699407	−0.349704	−	0.936860i	$-0.613718\pi$
$751$	−19.1668	−0.699407	−0.349704	+	0.936860i	$0.613718\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.1330	0.587139
$756$	0	0
$757$	−2.54988	−0.0926771	−0.0463385	−	0.998926i	$-0.514755\pi$
$757$	−2.54988	−0.0926771	−0.0463385	+	0.998926i	$0.514755\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−55.0184	−1.99442	−0.997208	−	0.0746748i	$-0.976208\pi$
$761$	−55.0184	−1.99442	−0.997208	+	0.0746748i	$0.976208\pi$
$762$	0	0
$763$	−4.24392	−0.153640
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.31765	−0.0836854
$768$	0	0
$769$	30.3425	1.09418	0.547089	−	0.837074i	$-0.315736\pi$
$769$	30.3425	1.09418	0.547089	+	0.837074i	$0.315736\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.8692	1.32609	0.663047	−	0.748578i	$-0.269263\pi$
$773$	36.8692	1.32609	0.663047	+	0.748578i	$0.269263\pi$
$774$	0	0
$775$	−7.91514	−0.284320
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.0070	0.716826
$780$	0	0
$781$	12.7253	0.455346
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.2721	−0.687850
$786$	0	0
$787$	−33.2534	−1.18536	−0.592678	−	0.805439i	$-0.701929\pi$
$787$	−33.2534	−1.18536	−0.592678	+	0.805439i	$0.701929\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.23162	0.0793473
$792$	0	0
$793$	−21.5814	−0.766378
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.8839	0.916854	0.458427	−	0.888732i	$-0.348413\pi$
$797$	25.8839	0.916854	0.458427	+	0.888732i	$0.348413\pi$
$798$	0	0
$799$	−26.9421	−0.953144
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−13.1426	−0.463793
$804$	0	0
$805$	−0.627340	−0.0221108
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−35.1660	−1.23637	−0.618186	−	0.786032i	$-0.712132\pi$
$809$	−35.1660	−1.23637	−0.618186	+	0.786032i	$0.712132\pi$
$810$	0	0
$811$	47.7014	1.67502	0.837512	−	0.546419i	$-0.184009\pi$
$811$	47.7014	1.67502	0.837512	+	0.546419i	$0.184009\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.84160	0.239651
$816$	0	0
$817$	−35.5778	−1.24471
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−44.1313	−1.54019	−0.770097	−	0.637927i	$-0.779792\pi$
$821$	−44.1313	−1.54019	−0.770097	+	0.637927i	$0.779792\pi$
$822$	0	0
$823$	38.9463	1.35758	0.678791	−	0.734331i	$-0.262504\pi$
$823$	38.9463	1.35758	0.678791	+	0.734331i	$0.262504\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.4861	−1.02533	−0.512666	−	0.858588i	$-0.671342\pi$
$827$	−29.4861	−1.02533	−0.512666	+	0.858588i	$0.671342\pi$
$828$	0	0
$829$	18.3170	0.636176	0.318088	−	0.948061i	$-0.396959\pi$
$829$	18.3170	0.636176	0.318088	+	0.948061i	$0.396959\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−37.9608	−1.31526
$834$	0	0
$835$	19.1235	0.661797
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.4677	−1.08638	−0.543192	−	0.839608i	$-0.682785\pi$
$839$	−31.4677	−1.08638	−0.543192	+	0.839608i	$0.682785\pi$
$840$	0	0
$841$	9.57849	0.330293
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.62325	−0.262248
$846$	0	0
$847$	5.75699	0.197812
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.0588	0.344813
$852$	0	0
$853$	27.6363	0.946250	0.473125	−	0.880995i	$-0.343126\pi$
$853$	27.6363	0.946250	0.473125	+	0.880995i	$0.343126\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.6011	0.567081	0.283541	−	0.958960i	$-0.408491\pi$
$857$	16.6011	0.567081	0.283541	+	0.958960i	$0.408491\pi$
$858$	0	0
$859$	16.0596	0.547947	0.273973	−	0.961737i	$-0.411662\pi$
$859$	16.0596	0.547947	0.273973	+	0.961737i	$0.411662\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	49.7050	1.69198	0.845990	−	0.533199i	$-0.179010\pi$
$863$	49.7050	1.69198	0.845990	+	0.533199i	$0.179010\pi$
$864$	0	0
$865$	−10.9946	−0.373828
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.40861	−0.149552
$870$	0	0
$871$	−25.5311	−0.865089
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.627340	−0.0212079
$876$	0	0
$877$	−26.7142	−0.902075	−0.451037	−	0.892505i	$-0.648946\pi$
$877$	−26.7142	−0.902075	−0.451037	+	0.892505i	$0.648946\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−0.0231374	−0.000779520 0	−0.000389760	−	1.00000i	$-0.500124\pi$
$881$	−0.0231374	−0.000779520 0	−0.000389760		1.00000i	$0.500124\pi$
$882$	0	0
$883$	30.1060	1.01315	0.506574	−	0.862196i	$-0.330912\pi$
$883$	30.1060	1.01315	0.506574	+	0.862196i	$0.330912\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.48357	−0.284851	−0.142425	−	0.989806i	$-0.545490\pi$
$887$	−8.48357	−0.284851	−0.142425	+	0.989806i	$0.545490\pi$
$888$	0	0
$889$	6.38548	0.214162
$890$	0	0
$891$	0	0
$892$	0	0
$893$	37.7084	1.26186
$894$	0	0
$895$	0.188747	0.00630910
$896$	0	0
$897$	0	0
$898$	0	0
$899$	49.1622	1.63965
$900$	0	0
$901$	12.7396	0.424418
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.6547	0.453898
$906$	0	0
$907$	−17.0547	−0.566292	−0.283146	−	0.959077i	$-0.591378\pi$
$907$	−17.0547	−0.566292	−0.283146	+	0.959077i	$0.591378\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.2085	−1.19964	−0.599820	−	0.800135i	$-0.704761\pi$
$911$	−36.2085	−1.19964	−0.599820	+	0.800135i	$0.704761\pi$
$912$	0	0
$913$	−1.41172	−0.0467212
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.0016	0.363304
$918$	0	0
$919$	−29.7478	−0.981290	−0.490645	−	0.871359i	$-0.663239\pi$
$919$	−29.7478	−0.981290	−0.490645	+	0.871359i	$0.663239\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.8531	−0.719303
$924$	0	0
$925$	10.0588	0.330733
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.0832	0.626098	0.313049	−	0.949737i	$-0.398650\pi$
$929$	19.0832	0.626098	0.313049	+	0.949737i	$0.398650\pi$
$930$	0	0
$931$	53.1302	1.74127
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.75856	−0.253732
$936$	0	0
$937$	1.64285	0.0536694	0.0268347	−	0.999640i	$-0.491457\pi$
$937$	1.64285	0.0536694	0.0268347	+	0.999640i	$0.491457\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−35.4789	−1.15658	−0.578289	−	0.815832i	$-0.696280\pi$
$941$	−35.4789	−1.15658	−0.578289	+	0.815832i	$0.696280\pi$
$942$	0	0
$943$	−2.48776	−0.0810126
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.50185	−0.113795	−0.0568975	−	0.998380i	$-0.518121\pi$
$947$	−3.50185	−0.113795	−0.0568975	+	0.998380i	$0.518121\pi$
$948$	0	0
$949$	22.5698	0.732648
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.3043	−0.690114	−0.345057	−	0.938582i	$-0.612140\pi$
$953$	−21.3043	−0.690114	−0.345057	+	0.938582i	$0.612140\pi$
$954$	0	0
$955$	9.33019	0.301918
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.579480	−0.0187124
$960$	0	0
$961$	31.6494	1.02095
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.9196	−0.512470
$966$	0	0
$967$	−27.6522	−0.889234	−0.444617	−	0.895721i	$-0.646660\pi$
$967$	−27.6522	−0.889234	−0.444617	+	0.895721i	$0.646660\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	55.1373	1.76944	0.884721	−	0.466121i	$-0.154349\pi$
$971$	55.1373	1.76944	0.884721	+	0.466121i	$0.154349\pi$
$972$	0	0
$973$	−7.74937	−0.248433
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.73362	−0.0554635	−0.0277317	−	0.999615i	$-0.508828\pi$
$977$	−1.73362	−0.0554635	−0.0277317	+	0.999615i	$0.508828\pi$
$978$	0	0
$979$	−3.84138	−0.122771
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.0475	0.703205	0.351602	−	0.936149i	$-0.385637\pi$
$983$	22.0475	0.703205	0.351602	+	0.936149i	$0.385637\pi$
$984$	0	0
$985$	−4.14012	−0.131915
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.42390	0.140672
$990$	0	0
$991$	−17.4001	−0.552732	−0.276366	−	0.961052i	$-0.589130\pi$
$991$	−17.4001	−0.552732	−0.276366	+	0.961052i	$0.589130\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.2974	−0.358152
$996$	0	0
$997$	−9.32388	−0.295290	−0.147645	−	0.989040i	$-0.547169\pi$
$997$	−9.32388	−0.295290	−0.147645	+	0.989040i	$0.547169\pi$
$998$	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.1.5	yes	7
3.2	odd	2		8280.2.a.bv.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bv.1.5	✓	7	3.2	odd	2
8280.2.a.bw.1.5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.627340 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -0.627340 q^{7} -1.35025 q^{11} +2.31878 q^{13} +5.74602 q^{17} -8.04218 q^{19} +1.00000 q^{23} +1.00000 q^{25} -6.21116 q^{29} -7.91514 q^{31} -0.627340 q^{35} +10.0588 q^{37} -2.48776 q^{41} +4.42390 q^{43} -4.68883 q^{47} -6.60645 q^{49} +2.21712 q^{53} -1.35025 q^{55} -0.999511 q^{59} -9.30722 q^{61} +2.31878 q^{65} -11.0106 q^{67} -9.42438 q^{71} +9.73349 q^{73} +0.847065 q^{77} +3.26504 q^{79} +1.04553 q^{83} +5.74602 q^{85} +2.84494 q^{89} -1.45466 q^{91} -8.04218 q^{95} +7.81347 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} - 6 q^{7}+O(q^{10}) \) 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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