LMFDB - Embedded newform 8280.2.p.a.1241.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("8280.1241");

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.p (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.16
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.08780i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -2.08780i q^{7} -4.49251 q^{11} +0.851963 q^{13} -1.37449 q^{17} -4.41323i q^{19} +(0.597102 - 4.75852i) q^{23} +1.00000 q^{25} -9.76229i q^{29} -8.80487 q^{31} +2.08780i q^{35} +8.90566i q^{37} -10.5573i q^{41} +9.03942i q^{43} -3.43761i q^{47} +2.64109 q^{49} +5.27029 q^{53} +4.49251 q^{55} +1.51213i q^{59} +5.74771i q^{61} -0.851963 q^{65} +3.63363i q^{67} -0.419089i q^{71} +5.12192 q^{73} +9.37946i q^{77} +10.5082i q^{79} -4.58133 q^{83} +1.37449 q^{85} -3.95945 q^{89} -1.77873i q^{91} +4.41323i q^{95} +4.55458i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.08780i		−	0.789114i	−0.918871	−	0.394557i	$-0.870898\pi$
$7$		−	2.08780i		−	0.789114i	0.918871	−	0.394557i	$-0.129102\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.49251			−1.35454			−0.677271	−	0.735734i	$-0.736837\pi$
$11$	−4.49251			−1.35454			−0.677271	+	0.735734i	$0.736837\pi$
$12$		0			0
$13$	0.851963			0.236292			0.118146	−	0.992996i	$-0.462305\pi$
$13$	0.851963			0.236292			0.118146	+	0.992996i	$0.462305\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.37449			−0.333362			−0.166681	−	0.986011i	$-0.553305\pi$
$17$	−1.37449			−0.333362			−0.166681	+	0.986011i	$0.553305\pi$
$18$		0			0
$19$		−	4.41323i		−	1.01246i	−0.862397	−	0.506232i	$-0.831038\pi$
$19$		−	4.41323i		−	1.01246i	0.862397	−	0.506232i	$-0.168962\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.597102	−	4.75852i	0.124504	−	0.992219i
$23$	0.597102	−	4.75852i	0.124504	−	0.992219i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.76229i		−	1.81281i	−0.422409	−	0.906405i	$-0.638815\pi$
$29$		−	9.76229i		−	1.81281i	0.422409	−	0.906405i	$-0.361185\pi$
$30$		0			0
$31$	−8.80487			−1.58140			−0.790701	−	0.612203i	$-0.790284\pi$
$31$	−8.80487			−1.58140			−0.790701	+	0.612203i	$0.790284\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			2.08780i			0.352903i
$36$		0			0
$37$			8.90566i			1.46408i	0.681261	+	0.732041i	$0.261432\pi$
$37$			8.90566i			1.46408i	−0.681261	+	0.732041i	$0.738568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	10.5573i		−	1.64877i	−0.566031	−	0.824384i	$-0.691522\pi$
$41$		−	10.5573i		−	1.64877i	0.566031	−	0.824384i	$-0.308478\pi$
$42$		0			0
$43$			9.03942i			1.37850i	0.724524	+	0.689249i	$0.242060\pi$
$43$			9.03942i			1.37850i	−0.724524	+	0.689249i	$0.757940\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.43761i		−	0.501427i	−0.968061	−	0.250714i	$-0.919335\pi$
$47$		−	3.43761i		−	0.501427i	0.968061	−	0.250714i	$-0.0806653\pi$
$48$		0			0
$49$	2.64109			0.377299
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.27029			0.723929			0.361965	−	0.932192i	$-0.382106\pi$
$53$	5.27029			0.723929			0.361965	+	0.932192i	$0.382106\pi$
$54$		0			0
$55$	4.49251			0.605769
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.51213i			0.196862i	0.995144	+	0.0984311i	$0.0313824\pi$
$59$			1.51213i			0.196862i	−0.995144	+	0.0984311i	$0.968618\pi$
$60$		0			0
$61$			5.74771i			0.735919i	0.929842	+	0.367960i	$0.119944\pi$
$61$			5.74771i			0.735919i	−0.929842	+	0.367960i	$0.880056\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.851963			−0.105673
$66$		0			0
$67$			3.63363i			0.443919i	0.975056	+	0.221959i	$0.0712452\pi$
$67$			3.63363i			0.443919i	−0.975056	+	0.221959i	$0.928755\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	0.419089i		−	0.0497367i	−0.999691	−	0.0248683i	$-0.992083\pi$
$71$		−	0.419089i		−	0.0497367i	0.999691	−	0.0248683i	$-0.00791666\pi$
$72$		0			0
$73$	5.12192			0.599476			0.299738	−	0.954022i	$-0.403101\pi$
$73$	5.12192			0.599476			0.299738	+	0.954022i	$0.403101\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			9.37946i			1.06889i
$78$		0			0
$79$			10.5082i			1.18227i	0.806572	+	0.591135i	$0.201320\pi$
$79$			10.5082i			1.18227i	−0.806572	+	0.591135i	$0.798680\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.58133			−0.502866			−0.251433	−	0.967875i	$-0.580902\pi$
$83$	−4.58133			−0.502866			−0.251433	+	0.967875i	$0.580902\pi$
$84$		0			0
$85$	1.37449			0.149084
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.95945			−0.419701			−0.209851	−	0.977733i	$-0.567298\pi$
$89$	−3.95945			−0.419701			−0.209851	+	0.977733i	$0.567298\pi$
$90$		0			0
$91$		−	1.77873i		−	0.186461i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			4.41323i			0.452788i
$96$		0			0
$97$			4.55458i			0.462448i	0.972901	+	0.231224i	$0.0742730\pi$
$97$			4.55458i			0.462448i	−0.972901	+	0.231224i	$0.925727\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			3.64711i			0.362901i	0.983400	+	0.181450i	$0.0580792\pi$
$101$			3.64711i			0.362901i	−0.983400	+	0.181450i	$0.941921\pi$
$102$		0			0
$103$		−	3.74768i		−	0.369270i	−0.982807	−	0.184635i	$-0.940890\pi$
$103$		−	3.74768i		−	0.369270i	0.982807	−	0.184635i	$-0.0591103\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.1909			−1.46856			−0.734278	−	0.678848i	$-0.762479\pi$
$107$	−15.1909			−1.46856			−0.734278	+	0.678848i	$0.762479\pi$
$108$		0			0
$109$		−	19.9455i		−	1.91044i	−0.295902	−	0.955218i	$-0.595620\pi$
$109$		−	19.9455i		−	1.91044i	0.295902	−	0.955218i	$-0.404380\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.3089			1.34607			0.673035	−	0.739611i	$-0.264990\pi$
$113$	14.3089			1.34607			0.673035	+	0.739611i	$0.264990\pi$
$114$		0			0
$115$	−0.597102	+	4.75852i	−0.0556800	+	0.443734i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.86966i			0.263061i
$120$		0			0
$121$	9.18262			0.834783
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−5.58411			−0.495510			−0.247755	−	0.968823i	$-0.579693\pi$
$127$	−5.58411			−0.495510			−0.247755	+	0.968823i	$0.579693\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			8.50726i			0.743283i	0.928376	+	0.371641i	$0.121205\pi$
$131$			8.50726i			0.743283i	−0.928376	+	0.371641i	$0.878795\pi$
$132$		0			0
$133$	−9.21394			−0.798950
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.4296			−1.40368			−0.701838	−	0.712336i	$-0.747637\pi$
$137$	−16.4296			−1.40368			−0.701838	+	0.712336i	$0.747637\pi$
$138$		0			0
$139$	6.67122			0.565845			0.282923	−	0.959143i	$-0.408696\pi$
$139$	6.67122			0.565845			0.282923	+	0.959143i	$0.408696\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.82745			−0.320067
$144$		0			0
$145$			9.76229i			0.810714i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.80732			0.311907			0.155954	−	0.987764i	$-0.450155\pi$
$149$	3.80732			0.311907			0.155954	+	0.987764i	$0.450155\pi$
$150$		0			0
$151$	−2.18877			−0.178120			−0.0890598	−	0.996026i	$-0.528386\pi$
$151$	−2.18877			−0.178120			−0.0890598	+	0.996026i	$0.528386\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.80487			0.707224
$156$		0			0
$157$		−	5.10875i		−	0.407723i	−0.979000	−	0.203861i	$-0.934651\pi$
$157$		−	5.10875i		−	0.407723i	0.979000	−	0.203861i	$-0.0653492\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.93483	−	1.24663i	−0.782974	−	0.0982481i
$162$		0			0
$163$	7.66851			0.600644			0.300322	−	0.953838i	$-0.402906\pi$
$163$	7.66851			0.600644			0.300322	+	0.953838i	$0.402906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.48620i		−	0.656682i	−0.944559	−	0.328341i	$-0.893511\pi$
$167$		−	8.48620i		−	0.656682i	0.944559	−	0.328341i	$-0.106489\pi$
$168$		0			0
$169$	−12.2742			−0.944166
$170$		0			0
$171$		0			0
$172$		0			0
$173$			22.4636i			1.70788i	0.520374	+	0.853938i	$0.325792\pi$
$173$			22.4636i			1.70788i	−0.520374	+	0.853938i	$0.674208\pi$
$174$		0			0
$175$		−	2.08780i		−	0.157823i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			16.5596i			1.23772i	0.785501	+	0.618860i	$0.212405\pi$
$179$			16.5596i			1.23772i	−0.785501	+	0.618860i	$0.787595\pi$
$180$		0			0
$181$			17.2934i			1.28541i	0.766116	+	0.642703i	$0.222187\pi$
$181$			17.2934i			1.28541i	−0.766116	+	0.642703i	$0.777813\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	8.90566i		−	0.654757i
$186$		0			0
$187$	6.17490			0.451553
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.93023			0.212024			0.106012	−	0.994365i	$-0.466192\pi$
$191$	2.93023			0.212024			0.106012	+	0.994365i	$0.466192\pi$
$192$		0			0
$193$	−1.42951			−0.102899			−0.0514493	−	0.998676i	$-0.516384\pi$
$193$	−1.42951			−0.102899			−0.0514493	+	0.998676i	$0.516384\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	20.5997i		−	1.46766i	−0.679331	−	0.733832i	$-0.737730\pi$
$197$		−	20.5997i		−	1.46766i	0.679331	−	0.733832i	$-0.262270\pi$
$198$		0			0
$199$		−	0.625414i		−	0.0443344i	−0.999754	−	0.0221672i	$-0.992943\pi$
$199$		−	0.625414i		−	0.0443344i	0.999754	−	0.0221672i	$-0.00705662\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−20.3817			−1.43052
$204$		0			0
$205$			10.5573i			0.737351i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			19.8265i			1.37142i
$210$		0			0
$211$	−23.3734			−1.60909			−0.804545	−	0.593891i	$-0.797591\pi$
$211$	−23.3734			−1.60909			−0.804545	+	0.593891i	$0.797591\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	9.03942i		−	0.616483i
$216$		0			0
$217$			18.3828i			1.24791i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.17101			−0.0787708
$222$		0			0
$223$	−16.9123			−1.13253			−0.566266	−	0.824223i	$-0.691612\pi$
$223$	−16.9123			−1.13253			−0.566266	+	0.824223i	$0.691612\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.7439			−1.37682			−0.688409	−	0.725322i	$-0.741691\pi$
$227$	−20.7439			−1.37682			−0.688409	+	0.725322i	$0.741691\pi$
$228$		0			0
$229$			6.03765i			0.398979i	0.979900	+	0.199489i	$0.0639284\pi$
$229$			6.03765i			0.398979i	−0.979900	+	0.199489i	$0.936072\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3.92423i		−	0.257085i	−0.991704	−	0.128542i	$-0.958970\pi$
$233$		−	3.92423i		−	0.257085i	0.991704	−	0.128542i	$-0.0410298\pi$
$234$		0			0
$235$			3.43761i			0.224245i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			8.99020i			0.581528i	0.956795	+	0.290764i	$0.0939094\pi$
$239$			8.99020i			0.581528i	−0.956795	+	0.290764i	$0.906091\pi$
$240$		0			0
$241$			24.0358i			1.54828i	0.633014	+	0.774141i	$0.281818\pi$
$241$			24.0358i			1.54828i	−0.633014	+	0.774141i	$0.718182\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.64109			−0.168733
$246$		0			0
$247$		−	3.75991i		−	0.239237i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.1804			0.768820			0.384410	−	0.923162i	$-0.374405\pi$
$251$	12.1804			0.768820			0.384410	+	0.923162i	$0.374405\pi$
$252$		0			0
$253$	−2.68248	+	21.3777i	−0.168646	+	1.34400i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.9760i			0.747044i	0.927621	+	0.373522i	$0.121850\pi$
$257$			11.9760i			0.747044i	−0.927621	+	0.373522i	$0.878150\pi$
$258$		0			0
$259$	18.5932			1.15533
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.77847			−0.171328			−0.0856639	−	0.996324i	$-0.527301\pi$
$263$	−2.77847			−0.171328			−0.0856639	+	0.996324i	$0.527301\pi$
$264$		0			0
$265$	−5.27029			−0.323751
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	5.13690i		−	0.313203i	−0.987662	−	0.156601i	$-0.949946\pi$
$269$		−	5.13690i		−	0.313203i	0.987662	−	0.156601i	$-0.0500538\pi$
$270$		0			0
$271$	22.9942			1.39680			0.698398	−	0.715709i	$-0.253897\pi$
$271$	22.9942			1.39680			0.698398	+	0.715709i	$0.253897\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.49251			−0.270908
$276$		0			0
$277$	16.3476			0.982232			0.491116	−	0.871094i	$-0.336589\pi$
$277$	16.3476			0.982232			0.491116	+	0.871094i	$0.336589\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.46063			0.206444			0.103222	−	0.994658i	$-0.467085\pi$
$281$	3.46063			0.206444			0.103222	+	0.994658i	$0.467085\pi$
$282$		0			0
$283$		−	7.46696i		−	0.443865i	−0.975062	−	0.221932i	$-0.928764\pi$
$283$		−	7.46696i		−	0.443865i	0.975062	−	0.221932i	$-0.0712365\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−22.0415			−1.30107
$288$		0			0
$289$	−15.1108			−0.888870
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.83580			−0.165669			−0.0828345	−	0.996563i	$-0.526397\pi$
$293$	−2.83580			−0.165669			−0.0828345	+	0.996563i	$0.526397\pi$
$294$		0			0
$295$		−	1.51213i		−	0.0880395i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.508709	−	4.05408i	0.0294194	−	0.234453i
$300$		0			0
$301$	18.8725			1.08779
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.74771i		−	0.329113i
$306$		0			0
$307$	28.4498			1.62372			0.811859	−	0.583854i	$-0.198456\pi$
$307$	28.4498			1.62372			0.811859	+	0.583854i	$0.198456\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			27.5155i			1.56026i	0.625615	+	0.780132i	$0.284848\pi$
$311$			27.5155i			1.56026i	−0.625615	+	0.780132i	$0.715152\pi$
$312$		0			0
$313$		−	5.23849i		−	0.296097i	−0.988980	−	0.148048i	$-0.952701\pi$
$313$		−	5.23849i		−	0.296097i	0.988980	−	0.148048i	$-0.0472991\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.3905i		−	1.36991i	−0.728587	−	0.684953i	$-0.759823\pi$
$317$		−	24.3905i		−	1.36991i	0.728587	−	0.684953i	$-0.240177\pi$
$318$		0			0
$319$			43.8571i			2.45553i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.06593i			0.337517i
$324$		0			0
$325$	0.851963			0.0472584
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.17705			−0.395683
$330$		0			0
$331$	20.6177			1.13325			0.566625	−	0.823976i	$-0.308249\pi$
$331$	20.6177			1.13325			0.566625	+	0.823976i	$0.308249\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	3.63363i		−	0.198526i
$336$		0			0
$337$			20.6627i			1.12557i	0.826604	+	0.562784i	$0.190270\pi$
$337$			20.6627i			1.12557i	−0.826604	+	0.562784i	$0.809730\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	39.5559			2.14207
$342$		0			0
$343$		−	20.1287i		−	1.08685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			32.9159i			1.76702i	0.468412	+	0.883510i	$0.344826\pi$
$347$			32.9159i			1.76702i	−0.468412	+	0.883510i	$0.655174\pi$
$348$		0			0
$349$	−3.57359			−0.191290			−0.0956451	−	0.995416i	$-0.530491\pi$
$349$	−3.57359			−0.191290			−0.0956451	+	0.995416i	$0.530491\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			10.2046i			0.543139i	0.962419	+	0.271569i	$0.0875426\pi$
$353$			10.2046i			0.543139i	−0.962419	+	0.271569i	$0.912457\pi$
$354$		0			0
$355$			0.419089i			0.0222429i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0451			−0.635718			−0.317859	−	0.948138i	$-0.602964\pi$
$359$	−12.0451			−0.635718			−0.317859	+	0.948138i	$0.602964\pi$
$360$		0			0
$361$	−0.476579			−0.0250831
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.12192			−0.268094
$366$		0			0
$367$			4.88734i			0.255117i	0.991831	+	0.127559i	$0.0407141\pi$
$367$			4.88734i			0.255117i	−0.991831	+	0.127559i	$0.959286\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	11.0033i		−	0.571263i
$372$		0			0
$373$			20.3866i			1.05558i	0.849375	+	0.527789i	$0.176979\pi$
$373$			20.3866i			1.05558i	−0.849375	+	0.527789i	$0.823021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.31711i		−	0.428353i
$378$		0			0
$379$		−	22.7336i		−	1.16775i	−0.811844	−	0.583874i	$-0.801536\pi$
$379$		−	22.7336i		−	1.16775i	0.811844	−	0.583874i	$-0.198464\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−29.5590			−1.51040			−0.755198	−	0.655497i	$-0.772459\pi$
$383$	−29.5590			−1.51040			−0.755198	+	0.655497i	$0.772459\pi$
$384$		0			0
$385$		−	9.37946i		−	0.478021i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.27162			−0.0644735			−0.0322367	−	0.999480i	$-0.510263\pi$
$389$	−1.27162			−0.0644735			−0.0322367	+	0.999480i	$0.510263\pi$
$390$		0			0
$391$	−0.820709	+	6.54052i	−0.0415050	+	0.330768i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	10.5082i		−	0.528727i
$396$		0			0
$397$	16.4375			0.824975			0.412488	−	0.910963i	$-0.364660\pi$
$397$	16.4375			0.824975			0.412488	+	0.910963i	$0.364660\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.6527			0.931472			0.465736	−	0.884924i	$-0.345790\pi$
$401$	18.6527			0.931472			0.465736	+	0.884924i	$0.345790\pi$
$402$		0			0
$403$	−7.50142			−0.373673
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	40.0087i		−	1.98316i
$408$		0			0
$409$	−23.8862			−1.18110			−0.590548	−	0.807002i	$-0.701088\pi$
$409$	−23.8862			−1.18110			−0.590548	+	0.807002i	$0.701088\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.15702			0.155347
$414$		0			0
$415$	4.58133			0.224889
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.1390			0.837297			0.418649	−	0.908148i	$-0.362504\pi$
$419$	17.1390			0.837297			0.418649	+	0.908148i	$0.362504\pi$
$420$		0			0
$421$		−	5.67575i		−	0.276619i	−0.990389	−	0.138309i	$-0.955833\pi$
$421$		−	5.67575i		−	0.276619i	0.990389	−	0.138309i	$-0.0441669\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.37449			−0.0666725
$426$		0			0
$427$	12.0001			0.580724
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.0453			−1.44723			−0.723616	−	0.690203i	$-0.757521\pi$
$431$	−30.0453			−1.44723			−0.723616	+	0.690203i	$0.757521\pi$
$432$		0			0
$433$			4.92925i			0.236885i	0.992961	+	0.118442i	$0.0377901\pi$
$433$			4.92925i			0.236885i	−0.992961	+	0.118442i	$0.962210\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0004	−	2.63515i	−1.00459	−	0.126056i
$438$		0			0
$439$	−25.2086			−1.20314			−0.601571	−	0.798819i	$-0.705458\pi$
$439$	−25.2086			−1.20314			−0.601571	+	0.798819i	$0.705458\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			24.1385i			1.14686i	0.819256	+	0.573428i	$0.194387\pi$
$443$			24.1385i			1.14686i	−0.819256	+	0.573428i	$0.805613\pi$
$444$		0			0
$445$	3.95945			0.187696
$446$		0			0
$447$		0			0
$448$		0			0
$449$			17.3504i			0.818817i	0.912351	+	0.409409i	$0.134265\pi$
$449$			17.3504i			0.818817i	−0.912351	+	0.409409i	$0.865735\pi$
$450$		0			0
$451$			47.4286i			2.23332i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1.77873i			0.0833881i
$456$		0			0
$457$		−	20.6139i		−	0.964276i	−0.876095	−	0.482138i	$-0.839860\pi$
$457$		−	20.6139i		−	0.964276i	0.876095	−	0.482138i	$-0.160140\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			28.2324i			1.31491i	0.753492	+	0.657457i	$0.228368\pi$
$461$			28.2324i			1.31491i	−0.753492	+	0.657457i	$0.771632\pi$
$462$		0			0
$463$	8.21421			0.381747			0.190873	−	0.981615i	$-0.438868\pi$
$463$	8.21421			0.381747			0.190873	+	0.981615i	$0.438868\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.70080			−0.124978			−0.0624890	−	0.998046i	$-0.519904\pi$
$467$	−2.70080			−0.124978			−0.0624890	+	0.998046i	$0.519904\pi$
$468$		0			0
$469$	7.58629			0.350302
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	40.6097i		−	1.86723i
$474$		0			0
$475$		−	4.41323i		−	0.202493i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−35.9744			−1.64371			−0.821856	−	0.569696i	$-0.807061\pi$
$479$	−35.9744			−1.64371			−0.821856	+	0.569696i	$0.807061\pi$
$480$		0			0
$481$			7.58729i			0.345951i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	4.55458i		−	0.206813i
$486$		0			0
$487$	−5.05303			−0.228975			−0.114487	−	0.993425i	$-0.536523\pi$
$487$	−5.05303			−0.228975			−0.114487	+	0.993425i	$0.536523\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.9383i		−	0.809545i	−0.914417	−	0.404772i	$-0.867351\pi$
$491$		−	17.9383i		−	0.809545i	0.914417	−	0.404772i	$-0.132649\pi$
$492$		0			0
$493$			13.4181i			0.604323i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.874974			−0.0392479
$498$		0			0
$499$	−16.5145			−0.739291			−0.369646	−	0.929173i	$-0.620521\pi$
$499$	−16.5145			−0.739291			−0.369646	+	0.929173i	$0.620521\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	23.3825			1.04257			0.521286	−	0.853382i	$-0.325452\pi$
$503$	23.3825			1.04257			0.521286	+	0.853382i	$0.325452\pi$
$504$		0			0
$505$		−	3.64711i		−	0.162294i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			13.6454i			0.604822i	0.953178	+	0.302411i	$0.0977915\pi$
$509$			13.6454i			0.604822i	−0.953178	+	0.302411i	$0.902209\pi$
$510$		0			0
$511$		−	10.6935i		−	0.473055i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			3.74768i			0.165142i
$516$		0			0
$517$			15.4435i			0.679204i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.7285			−0.776699			−0.388350	−	0.921512i	$-0.626955\pi$
$521$	−17.7285			−0.776699			−0.388350	+	0.921512i	$0.626955\pi$
$522$		0			0
$523$			29.8984i			1.30737i	0.756768	+	0.653683i	$0.226777\pi$
$523$			29.8984i			1.30737i	−0.756768	+	0.653683i	$0.773223\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.1022			0.527180
$528$		0			0
$529$	−22.2869	−	5.68264i	−0.968997	−	0.247071i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.99440i		−	0.389591i
$534$		0			0
$535$	15.1909			0.656759
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11.8651			−0.511067
$540$		0			0
$541$	−42.1580			−1.81251			−0.906257	−	0.422728i	$-0.861073\pi$
$541$	−42.1580			−1.81251			−0.906257	+	0.422728i	$0.861073\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			19.9455i			0.854373i
$546$		0			0
$547$	−13.7388			−0.587428			−0.293714	−	0.955893i	$-0.594891\pi$
$547$	−13.7388			−0.587428			−0.293714	+	0.955893i	$0.594891\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−43.0832			−1.83541
$552$		0			0
$553$	21.9391			0.932947
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.2966			0.775252			0.387626	−	0.921817i	$-0.373295\pi$
$557$	18.2966			0.775252			0.387626	+	0.921817i	$0.373295\pi$
$558$		0			0
$559$			7.70125i			0.325728i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−38.5815			−1.62601			−0.813007	−	0.582254i	$-0.802171\pi$
$563$	−38.5815			−1.62601			−0.813007	+	0.582254i	$0.802171\pi$
$564$		0			0
$565$	−14.3089			−0.601981
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.2458			−0.764904			−0.382452	−	0.923975i	$-0.624920\pi$
$569$	−18.2458			−0.764904			−0.382452	+	0.923975i	$0.624920\pi$
$570$		0			0
$571$		−	23.0512i		−	0.964665i	−0.875988	−	0.482332i	$-0.839790\pi$
$571$		−	23.0512i		−	0.964665i	0.875988	−	0.482332i	$-0.160210\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.597102	−	4.75852i	0.0249009	−	0.198444i
$576$		0			0
$577$	11.7661			0.489831			0.244915	−	0.969544i	$-0.421240\pi$
$577$	11.7661			0.489831			0.244915	+	0.969544i	$0.421240\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			9.56490i			0.396819i
$582$		0			0
$583$	−23.6768			−0.980593
$584$		0			0
$585$		0			0
$586$		0			0
$587$			3.14807i			0.129935i	0.997887	+	0.0649673i	$0.0206943\pi$
$587$			3.14807i			0.129935i	−0.997887	+	0.0649673i	$0.979306\pi$
$588$		0			0
$589$			38.8579i			1.60111i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.42086i		−	0.263673i	−0.991271	−	0.131837i	$-0.957913\pi$
$593$		−	6.42086i		−	0.263673i	0.991271	−	0.131837i	$-0.0420874\pi$
$594$		0			0
$595$		−	2.86966i		−	0.117644i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	2.33422i		−	0.0953737i	−0.998862	−	0.0476868i	$-0.984815\pi$
$599$		−	2.33422i		−	0.0953737i	0.998862	−	0.0476868i	$-0.0151849\pi$
$600$		0			0
$601$	24.1970			0.987017			0.493509	−	0.869741i	$-0.335714\pi$
$601$	24.1970			0.987017			0.493509	+	0.869741i	$0.335714\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.18262			−0.373326
$606$		0			0
$607$	19.0395			0.772791			0.386395	−	0.922333i	$-0.373720\pi$
$607$	19.0395			0.772791			0.386395	+	0.922333i	$0.373720\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.92872i		−	0.118483i
$612$		0			0
$613$			24.6955i			0.997442i	0.866763	+	0.498721i	$0.166197\pi$
$613$			24.6955i			0.997442i	−0.866763	+	0.498721i	$0.833803\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	33.0940			1.33231			0.666156	−	0.745812i	$-0.267938\pi$
$617$	33.0940			1.33231			0.666156	+	0.745812i	$0.267938\pi$
$618$		0			0
$619$			10.5113i			0.422486i	0.977434	+	0.211243i	$0.0677511\pi$
$619$			10.5113i			0.422486i	−0.977434	+	0.211243i	$0.932249\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			8.26655i			0.331192i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	12.2407i		−	0.488069i
$630$		0			0
$631$		−	9.27395i		−	0.369190i	−0.982815	−	0.184595i	$-0.940903\pi$
$631$		−	9.27395i		−	0.369190i	0.982815	−	0.184595i	$-0.0590974\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.58411			0.221599
$636$		0			0
$637$	2.25011			0.0891527
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.0252			0.395970			0.197985	−	0.980205i	$-0.436560\pi$
$641$	10.0252			0.395970			0.197985	+	0.980205i	$0.436560\pi$
$642$		0			0
$643$		−	17.7062i		−	0.698265i	−0.937073	−	0.349132i	$-0.886476\pi$
$643$		−	17.7062i		−	0.698265i	0.937073	−	0.349132i	$-0.113524\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.3029i		−	0.994758i	−0.867533	−	0.497379i	$-0.834296\pi$
$647$		−	25.3029i		−	0.994758i	0.867533	−	0.497379i	$-0.165704\pi$
$648$		0			0
$649$		−	6.79324i		−	0.266658i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			23.9552i			0.937440i	0.883347	+	0.468720i	$0.155285\pi$
$653$			23.9552i			0.937440i	−0.883347	+	0.468720i	$0.844715\pi$
$654$		0			0
$655$		−	8.50726i		−	0.332406i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.4560			0.835808			0.417904	−	0.908491i	$-0.362765\pi$
$659$	21.4560			0.835808			0.417904	+	0.908491i	$0.362765\pi$
$660$		0			0
$661$			17.7077i			0.688748i	0.938833	+	0.344374i	$0.111909\pi$
$661$			17.7077i			0.688748i	−0.938833	+	0.344374i	$0.888091\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9.21394			0.357301
$666$		0			0
$667$	−46.4540	−	5.82908i	−1.79871	−	0.225703i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	25.8216i		−	0.996834i
$672$		0			0
$673$	−12.9611			−0.499613			−0.249807	−	0.968296i	$-0.580367\pi$
$673$	−12.9611			−0.499613			−0.249807	+	0.968296i	$0.580367\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.48072			−0.0569086			−0.0284543	−	0.999595i	$-0.509059\pi$
$677$	−1.48072			−0.0569086			−0.0284543	+	0.999595i	$0.509059\pi$
$678$		0			0
$679$	9.50906			0.364924
$680$		0			0
$681$		0			0
$682$		0			0
$683$			9.95692i			0.380991i	0.981688	+	0.190495i	$0.0610094\pi$
$683$			9.95692i			0.380991i	−0.981688	+	0.190495i	$0.938991\pi$
$684$		0			0
$685$	16.4296			0.627743
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.49009			0.171059
$690$		0			0
$691$	−22.9105			−0.871556			−0.435778	−	0.900054i	$-0.643527\pi$
$691$	−22.9105			−0.871556			−0.435778	+	0.900054i	$0.643527\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.67122			−0.253054
$696$		0			0
$697$			14.5108i			0.549637i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	27.2064			1.02757			0.513786	−	0.857918i	$-0.328243\pi$
$701$	27.2064			1.02757			0.513786	+	0.857918i	$0.328243\pi$
$702$		0			0
$703$	39.3027			1.48233
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.61443			0.286370
$708$		0			0
$709$			45.0465i			1.69176i	0.533375	+	0.845879i	$0.320923\pi$
$709$			45.0465i			1.69176i	−0.533375	+	0.845879i	$0.679077\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.25740	+	41.8981i	−0.196891	+	1.56910i
$714$		0			0
$715$	3.82745			0.143138
$716$		0			0
$717$		0			0
$718$		0			0
$719$			4.91664i			0.183360i	0.995789	+	0.0916798i	$0.0292236\pi$
$719$			4.91664i			0.183360i	−0.995789	+	0.0916798i	$0.970776\pi$
$720$		0			0
$721$	−7.82441			−0.291396
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	9.76229i		−	0.362562i
$726$		0			0
$727$		−	8.90294i		−	0.330192i	−0.986278	−	0.165096i	$-0.947207\pi$
$727$		−	8.90294i		−	0.330192i	0.986278	−	0.165096i	$-0.0527934\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	12.4246i		−	0.459540i
$732$		0			0
$733$		−	30.6642i		−	1.13261i	−0.824196	−	0.566305i	$-0.808373\pi$
$733$		−	30.6642i		−	1.13261i	0.824196	−	0.566305i	$-0.191627\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	16.3241i		−	0.601306i
$738$		0			0
$739$	−40.0159			−1.47201			−0.736004	−	0.676977i	$-0.763289\pi$
$739$	−40.0159			−1.47201			−0.736004	+	0.676977i	$0.763289\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.52777			−0.202794			−0.101397	−	0.994846i	$-0.532331\pi$
$743$	−5.52777			−0.202794			−0.101397	+	0.994846i	$0.532331\pi$
$744$		0			0
$745$	−3.80732			−0.139489
$746$		0			0
$747$		0			0
$748$		0			0
$749$			31.7155i			1.15886i
$750$		0			0
$751$		−	9.37580i		−	0.342128i	−0.985260	−	0.171064i	$-0.945280\pi$
$751$		−	9.37580i		−	0.342128i	0.985260	−	0.171064i	$-0.0547205\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.18877			0.0796575
$756$		0			0
$757$			28.6243i			1.04037i	0.854055	+	0.520183i	$0.174136\pi$
$757$			28.6243i			1.04037i	−0.854055	+	0.520183i	$0.825864\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	25.5421i		−	0.925902i	−0.886384	−	0.462951i	$-0.846791\pi$
$761$		−	25.5421i		−	0.925902i	0.886384	−	0.462951i	$-0.153209\pi$
$762$		0			0
$763$	−41.6423			−1.50755
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.28828i			0.0465170i
$768$		0			0
$769$			10.8175i			0.390089i	0.980794	+	0.195045i	$0.0624851\pi$
$769$			10.8175i			0.390089i	−0.980794	+	0.195045i	$0.937515\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	42.8094			1.53975			0.769873	−	0.638197i	$-0.220319\pi$
$773$	42.8094			1.53975			0.769873	+	0.638197i	$0.220319\pi$
$774$		0			0
$775$	−8.80487			−0.316280
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−46.5916			−1.66932
$780$		0			0
$781$			1.88276i			0.0673704i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			5.10875i			0.182339i
$786$		0			0
$787$		−	26.6522i		−	0.950050i	−0.879972	−	0.475025i	$-0.842439\pi$
$787$		−	26.6522i		−	0.950050i	0.879972	−	0.475025i	$-0.157561\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	29.8742i		−	1.06220i
$792$		0			0
$793$			4.89684i			0.173892i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	24.3212			0.861503			0.430751	−	0.902471i	$-0.358249\pi$
$797$	24.3212			0.861503			0.430751	+	0.902471i	$0.358249\pi$
$798$		0			0
$799$			4.72496i			0.167157i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−23.0103			−0.812015
$804$		0			0
$805$	9.93483	+	1.24663i	0.350157	+	0.0439379i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	8.77619i		−	0.308555i	−0.988028	−	0.154277i	$-0.950695\pi$
$809$		−	8.77619i		−	0.308555i	0.988028	−	0.154277i	$-0.0493049\pi$
$810$		0			0
$811$	−33.4637			−1.17507			−0.587535	−	0.809199i	$-0.699902\pi$
$811$	−33.4637			−1.17507			−0.587535	+	0.809199i	$0.699902\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.66851			−0.268616
$816$		0			0
$817$	39.8930			1.39568
$818$		0			0
$819$		0			0
$820$		0			0
$821$			24.8222i			0.866301i	0.901322	+	0.433150i	$0.142598\pi$
$821$			24.8222i			0.866301i	−0.901322	+	0.433150i	$0.857402\pi$
$822$		0			0
$823$	−15.9478			−0.555906			−0.277953	−	0.960595i	$-0.589656\pi$
$823$	−15.9478			−0.555906			−0.277953	+	0.960595i	$0.589656\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.16508			−0.144834			−0.0724171	−	0.997374i	$-0.523071\pi$
$827$	−4.16508			−0.144834			−0.0724171	+	0.997374i	$0.523071\pi$
$828$		0			0
$829$	46.5476			1.61666			0.808332	−	0.588727i	$-0.200371\pi$
$829$	46.5476			1.61666			0.808332	+	0.588727i	$0.200371\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.63015			−0.125777
$834$		0			0
$835$			8.48620i			0.293677i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	27.7955			0.959608			0.479804	−	0.877376i	$-0.340708\pi$
$839$	27.7955			0.959608			0.479804	+	0.877376i	$0.340708\pi$
$840$		0			0
$841$	−66.3022			−2.28628
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.2742			0.422244
$846$		0			0
$847$		−	19.1715i		−	0.658739i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	42.3777	+	5.31758i	1.45269	+	0.182284i
$852$		0			0
$853$	49.6822			1.70109			0.850544	−	0.525905i	$-0.176273\pi$
$853$	49.6822			1.70109			0.850544	+	0.525905i	$0.176273\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	15.0813i		−	0.515167i	−0.966256	−	0.257584i	$-0.917074\pi$
$857$		−	15.0813i		−	0.515167i	0.966256	−	0.257584i	$-0.0829262\pi$
$858$		0			0
$859$	32.1563			1.09716			0.548579	−	0.836099i	$-0.315169\pi$
$859$	32.1563			1.09716			0.548579	+	0.836099i	$0.315169\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20.5572i		−	0.699777i	−0.936791	−	0.349888i	$-0.886220\pi$
$863$		−	20.5572i		−	0.699777i	0.936791	−	0.349888i	$-0.113780\pi$
$864$		0			0
$865$		−	22.4636i		−	0.763786i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	47.2084i		−	1.60143i
$870$		0			0
$871$			3.09572i			0.104894i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			2.08780i			0.0705805i
$876$		0			0
$877$	−15.3622			−0.518746			−0.259373	−	0.965777i	$-0.583516\pi$
$877$	−15.3622			−0.518746			−0.259373	+	0.965777i	$0.583516\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−47.4257			−1.59781			−0.798906	−	0.601456i	$-0.794588\pi$
$881$	−47.4257			−1.59781			−0.798906	+	0.601456i	$0.794588\pi$
$882$		0			0
$883$	24.7761			0.833781			0.416890	−	0.908957i	$-0.363120\pi$
$883$	24.7761			0.833781			0.416890	+	0.908957i	$0.363120\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	22.5680i		−	0.757759i	−0.925446	−	0.378879i	$-0.876310\pi$
$887$		−	22.5680i		−	0.757759i	0.925446	−	0.378879i	$-0.123690\pi$
$888$		0			0
$889$			11.6585i			0.391014i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.1710			−0.507677
$894$		0			0
$895$		−	16.5596i		−	0.553525i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			85.9557i			2.86678i
$900$		0			0
$901$	−7.24394			−0.241331
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	17.2934i		−	0.574851i
$906$		0			0
$907$		−	39.3783i		−	1.30754i	−0.756695	−	0.653768i	$-0.773187\pi$
$907$		−	39.3783i		−	1.30754i	0.756695	−	0.653768i	$-0.226813\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.8279			−1.35269			−0.676344	−	0.736586i	$-0.736437\pi$
$911$	−40.8279			−1.35269			−0.676344	+	0.736586i	$0.736437\pi$
$912$		0			0
$913$	20.5816			0.681153
$914$		0			0
$915$		0			0
$916$		0			0
$917$	17.7615			0.586535
$918$		0			0
$919$			9.28320i			0.306225i	0.988209	+	0.153112i	$0.0489296\pi$
$919$			9.28320i			0.306225i	−0.988209	+	0.153112i	$0.951070\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	0.357048i		−	0.0117524i
$924$		0			0
$925$			8.90566i			0.292816i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			57.3240i			1.88074i	0.340153	+	0.940370i	$0.389521\pi$
$929$			57.3240i			1.88074i	−0.340153	+	0.940370i	$0.610479\pi$
$930$		0			0
$931$		−	11.6557i		−	0.382001i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.17490			−0.201941
$936$		0			0
$937$		−	31.7569i		−	1.03745i	−0.854940	−	0.518727i	$-0.826406\pi$
$937$		−	31.7569i		−	1.03745i	0.854940	−	0.518727i	$-0.173594\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.7036			0.642318			0.321159	−	0.947025i	$-0.395928\pi$
$941$	19.7036			0.642318			0.321159	+	0.947025i	$0.395928\pi$
$942$		0			0
$943$	−50.2369	−	6.30376i	−1.63594	−	0.205279i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	43.4661i		−	1.41246i	−0.707983	−	0.706229i	$-0.750395\pi$
$947$		−	43.4661i		−	1.41246i	0.707983	−	0.706229i	$-0.249605\pi$
$948$		0			0
$949$	4.36369			0.141651
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.0029			−1.10146			−0.550732	−	0.834682i	$-0.685651\pi$
$953$	−34.0029			−1.10146			−0.550732	+	0.834682i	$0.685651\pi$
$954$		0			0
$955$	−2.93023			−0.0948199
$956$		0			0
$957$		0			0
$958$		0			0
$959$			34.3017i			1.10766i
$960$		0			0
$961$	46.5257			1.50083
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.42951			0.0460176
$966$		0			0
$967$	58.5190			1.88184			0.940922	−	0.338622i	$-0.109961\pi$
$967$	58.5190			1.88184			0.940922	+	0.338622i	$0.109961\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−31.8295			−1.02146			−0.510728	−	0.859742i	$-0.670624\pi$
$971$	−31.8295			−1.02146			−0.510728	+	0.859742i	$0.670624\pi$
$972$		0			0
$973$		−	13.9282i		−	0.446517i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−43.9685			−1.40668			−0.703339	−	0.710855i	$-0.748308\pi$
$977$	−43.9685			−1.40668			−0.703339	+	0.710855i	$0.748308\pi$
$978$		0			0
$979$	17.7879			0.568503
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.7453			−0.916833			−0.458417	−	0.888737i	$-0.651583\pi$
$983$	−28.7453			−0.916833			−0.458417	+	0.888737i	$0.651583\pi$
$984$		0			0
$985$			20.5997i			0.656360i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43.0142	+	5.39745i	1.36777	+	0.171629i
$990$		0			0
$991$	21.3588			0.678484			0.339242	−	0.940699i	$-0.389830\pi$
$991$	21.3588			0.678484			0.339242	+	0.940699i	$0.389830\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			0.625414i			0.0198270i
$996$		0			0
$997$	−32.4428			−1.02747			−0.513737	−	0.857948i	$-0.671739\pi$
$997$	−32.4428			−1.02747			−0.513737	+	0.857948i	$0.671739\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.16	✓	48
3.2	odd	2		8280.2.p.b.1241.16	yes	48
23.22	odd	2		8280.2.p.b.1241.33	yes	48
69.68	even	2	inner	8280.2.p.a.1241.33	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.16	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.33	yes	48	69.68	even	2	inner
8280.2.p.b.1241.16	yes	48	3.2	odd	2
8280.2.p.b.1241.33	yes	48	23.22	odd	2

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.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.08780i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -2.08780i q^{7} -4.49251 q^{11} +0.851963 q^{13} -1.37449 q^{17} -4.41323i q^{19} +(0.597102 - 4.75852i) q^{23} +1.00000 q^{25} -9.76229i q^{29} -8.80487 q^{31} +2.08780i q^{35} +8.90566i q^{37} -10.5573i q^{41} +9.03942i q^{43} -3.43761i q^{47} +2.64109 q^{49} +5.27029 q^{53} +4.49251 q^{55} +1.51213i q^{59} +5.74771i q^{61} -0.851963 q^{65} +3.63363i q^{67} -0.419089i q^{71} +5.12192 q^{73} +9.37946i q^{77} +10.5082i q^{79} -4.58133 q^{83} +1.37449 q^{85} -3.95945 q^{89} -1.77873i q^{91} +4.41323i q^{95} +4.55458i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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