LMFDB - Embedded newform 8280.2.p.a.1241.6

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			1241.6
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.43

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.21237i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.21237i q^{7} -4.83602 q^{11} -1.11582 q^{13} +5.43837 q^{17} +1.24380i q^{19} +(1.90128 - 4.40285i) q^{23} +1.00000 q^{25} +2.73619i q^{29} +8.10862 q^{31} +3.21237i q^{35} -7.85549i q^{37} +0.525989i q^{41} -1.68919i q^{43} +2.16253i q^{47} -3.31929 q^{49} -4.59274 q^{53} +4.83602 q^{55} -4.37598i q^{59} +11.3309i q^{61} +1.11582 q^{65} -10.8460i q^{67} -1.30097i q^{71} +15.0201 q^{73} +15.5351i q^{77} -2.06914i q^{79} +2.56550 q^{83} -5.43837 q^{85} +4.37085 q^{89} +3.58444i q^{91} -1.24380i q^{95} -14.8708i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	3.21237i		−	1.21416i	−0.794641	−	0.607080i	$-0.792341\pi$
$7$		−	3.21237i		−	1.21416i	0.794641	−	0.607080i	$-0.207659\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.83602			−1.45812			−0.729058	−	0.684452i	$-0.760041\pi$
$11$	−4.83602			−1.45812			−0.729058	+	0.684452i	$0.760041\pi$
$12$		0			0
$13$	−1.11582			−0.309474			−0.154737	−	0.987956i	$-0.549453\pi$
$13$	−1.11582			−0.309474			−0.154737	+	0.987956i	$0.549453\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.43837			1.31900			0.659499	−	0.751705i	$-0.270768\pi$
$17$	5.43837			1.31900			0.659499	+	0.751705i	$0.270768\pi$
$18$		0			0
$19$			1.24380i			0.285347i	0.989770	+	0.142673i	$0.0455698\pi$
$19$			1.24380i			0.285347i	−0.989770	+	0.142673i	$0.954430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.90128	−	4.40285i	0.396445	−	0.918058i
$23$	1.90128	−	4.40285i	0.396445	−	0.918058i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.73619i			0.508097i	0.967191	+	0.254049i	$0.0817624\pi$
$29$			2.73619i			0.508097i	−0.967191	+	0.254049i	$0.918238\pi$
$30$		0			0
$31$	8.10862			1.45635			0.728176	−	0.685390i	$-0.240368\pi$
$31$	8.10862			1.45635			0.728176	+	0.685390i	$0.240368\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.21237i			0.542989i
$36$		0			0
$37$		−	7.85549i		−	1.29143i	−0.763577	−	0.645717i	$-0.776558\pi$
$37$		−	7.85549i		−	1.29143i	0.763577	−	0.645717i	$-0.223442\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.525989i			0.0821457i	0.999156	+	0.0410729i	$0.0130776\pi$
$41$			0.525989i			0.0821457i	−0.999156	+	0.0410729i	$0.986922\pi$
$42$		0			0
$43$		−	1.68919i		−	0.257599i	−0.991671	−	0.128799i	$-0.958888\pi$
$43$		−	1.68919i		−	0.257599i	0.991671	−	0.128799i	$-0.0411123\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.16253i			0.315437i	0.987484	+	0.157718i	$0.0504139\pi$
$47$			2.16253i			0.315437i	−0.987484	+	0.157718i	$0.949586\pi$
$48$		0			0
$49$	−3.31929			−0.474185
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.59274			−0.630861			−0.315430	−	0.948949i	$-0.602149\pi$
$53$	−4.59274			−0.630861			−0.315430	+	0.948949i	$0.602149\pi$
$54$		0			0
$55$	4.83602			0.652089
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	4.37598i		−	0.569704i	−0.958572	−	0.284852i	$-0.908056\pi$
$59$		−	4.37598i		−	0.569704i	0.958572	−	0.284852i	$-0.0919444\pi$
$60$		0			0
$61$			11.3309i			1.45078i	0.688340	+	0.725389i	$0.258340\pi$
$61$			11.3309i			1.45078i	−0.688340	+	0.725389i	$0.741660\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.11582			0.138401
$66$		0			0
$67$		−	10.8460i		−	1.32504i	−0.749043	−	0.662522i	$-0.769486\pi$
$67$		−	10.8460i		−	1.32504i	0.749043	−	0.662522i	$-0.230514\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.30097i		−	0.154397i	−0.997016	−	0.0771986i	$-0.975402\pi$
$71$		−	1.30097i		−	0.154397i	0.997016	−	0.0771986i	$-0.0245976\pi$
$72$		0			0
$73$	15.0201			1.75796			0.878982	−	0.476855i	$-0.158223\pi$
$73$	15.0201			1.75796			0.878982	+	0.476855i	$0.158223\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			15.5351i			1.77039i
$78$		0			0
$79$		−	2.06914i		−	0.232796i	−0.993203	−	0.116398i	$-0.962865\pi$
$79$		−	2.06914i		−	0.232796i	0.993203	−	0.116398i	$-0.0371348\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.56550			0.281600			0.140800	−	0.990038i	$-0.455033\pi$
$83$	2.56550			0.281600			0.140800	+	0.990038i	$0.455033\pi$
$84$		0			0
$85$	−5.43837			−0.589874
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.37085			0.463309			0.231655	−	0.972798i	$-0.425586\pi$
$89$	4.37085			0.463309			0.231655	+	0.972798i	$0.425586\pi$
$90$		0			0
$91$			3.58444i			0.375751i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1.24380i		−	0.127611i
$96$		0			0
$97$		−	14.8708i		−	1.50990i	−0.655784	−	0.754948i	$-0.727662\pi$
$97$		−	14.8708i		−	1.50990i	0.655784	−	0.754948i	$-0.272338\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			13.0142i			1.29496i	0.762081	+	0.647482i	$0.224178\pi$
$101$			13.0142i			1.29496i	−0.762081	+	0.647482i	$0.775822\pi$
$102$		0			0
$103$		−	17.3667i		−	1.71119i	−0.517643	−	0.855597i	$-0.673190\pi$
$103$		−	17.3667i		−	1.71119i	0.517643	−	0.855597i	$-0.326810\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.9397			−1.34760			−0.673801	−	0.738913i	$-0.735339\pi$
$107$	−13.9397			−1.34760			−0.673801	+	0.738913i	$0.735339\pi$
$108$		0			0
$109$			1.39900i			0.134000i	0.997753	+	0.0670001i	$0.0213428\pi$
$109$			1.39900i			0.134000i	−0.997753	+	0.0670001i	$0.978657\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.11577			−0.857539			−0.428770	−	0.903414i	$-0.641053\pi$
$113$	−9.11577			−0.857539			−0.428770	+	0.903414i	$0.641053\pi$
$114$		0			0
$115$	−1.90128	+	4.40285i	−0.177296	+	0.410568i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	17.4700i		−	1.60148i
$120$		0			0
$121$	12.3871			1.12610
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	4.34954			0.385959			0.192979	−	0.981203i	$-0.438185\pi$
$127$	4.34954			0.385959			0.192979	+	0.981203i	$0.438185\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			0.324570i			0.0283578i	0.999899	+	0.0141789i	$0.00451343\pi$
$131$			0.324570i			0.0283578i	−0.999899	+	0.0141789i	$0.995487\pi$
$132$		0			0
$133$	3.99553			0.346457
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−17.3874			−1.48551			−0.742753	−	0.669566i	$-0.766480\pi$
$137$	−17.3874			−1.48551			−0.742753	+	0.669566i	$0.766480\pi$
$138$		0			0
$139$	−5.76237			−0.488758			−0.244379	−	0.969680i	$-0.578584\pi$
$139$	−5.76237			−0.488758			−0.244379	+	0.969680i	$0.578584\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.39615			0.451249
$144$		0			0
$145$		−	2.73619i		−	0.227228i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.974277			−0.0798159			−0.0399079	−	0.999203i	$-0.512706\pi$
$149$	−0.974277			−0.0798159			−0.0399079	+	0.999203i	$0.512706\pi$
$150$		0			0
$151$	−20.8384			−1.69581			−0.847904	−	0.530150i	$-0.822136\pi$
$151$	−20.8384			−1.69581			−0.847904	+	0.530150i	$0.822136\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.10862			−0.651300
$156$		0			0
$157$		−	6.93487i		−	0.553463i	−0.960947	−	0.276732i	$-0.910749\pi$
$157$		−	6.93487i		−	0.553463i	0.960947	−	0.276732i	$-0.0892513\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.1436	−	6.10762i	−1.11467	−	0.481348i
$162$		0			0
$163$	−11.0112			−0.862463			−0.431232	−	0.902241i	$-0.641921\pi$
$163$	−11.0112			−0.862463			−0.431232	+	0.902241i	$0.641921\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.28964i			0.254560i	0.991867	+	0.127280i	$0.0406247\pi$
$167$			3.28964i			0.254560i	−0.991867	+	0.127280i	$0.959375\pi$
$168$		0			0
$169$	−11.7549			−0.904226
$170$		0			0
$171$		0			0
$172$		0			0
$173$			7.18346i			0.546149i	0.961993	+	0.273074i	$0.0880405\pi$
$173$			7.18346i			0.546149i	−0.961993	+	0.273074i	$0.911960\pi$
$174$		0			0
$175$		−	3.21237i		−	0.242832i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2.24921i		−	0.168114i	−0.996461	−	0.0840568i	$-0.973212\pi$
$179$		−	2.24921i		−	0.168114i	0.996461	−	0.0840568i	$-0.0267877\pi$
$180$		0			0
$181$		−	17.2585i		−	1.28281i	−0.767201	−	0.641406i	$-0.778351\pi$
$181$		−	17.2585i		−	1.28281i	0.767201	−	0.641406i	$-0.221649\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			7.85549i			0.577547i
$186$		0			0
$187$	−26.3001			−1.92325
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.07184			0.294628			0.147314	−	0.989090i	$-0.452937\pi$
$191$	4.07184			0.294628			0.147314	+	0.989090i	$0.452937\pi$
$192$		0			0
$193$	−22.5453			−1.62284			−0.811421	−	0.584462i	$-0.801306\pi$
$193$	−22.5453			−1.62284			−0.811421	+	0.584462i	$0.801306\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.53153i		−	0.465352i	−0.972554	−	0.232676i	$-0.925252\pi$
$197$		−	6.53153i		−	0.465352i	0.972554	−	0.232676i	$-0.0747482\pi$
$198$		0			0
$199$			4.71710i			0.334387i	0.985924	+	0.167193i	$0.0534704\pi$
$199$			4.71710i			0.334387i	−0.985924	+	0.167193i	$0.946530\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.78964			0.616912
$204$		0			0
$205$		−	0.525989i		−	0.0367367i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	6.01503i		−	0.416068i
$210$		0			0
$211$	−16.5835			−1.14165			−0.570826	−	0.821071i	$-0.693377\pi$
$211$	−16.5835			−1.14165			−0.570826	+	0.821071i	$0.693377\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			1.68919i			0.115202i
$216$		0			0
$217$		−	26.0479i		−	1.76824i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.06827			−0.408196
$222$		0			0
$223$	2.44478			0.163714			0.0818571	−	0.996644i	$-0.473915\pi$
$223$	2.44478			0.163714			0.0818571	+	0.996644i	$0.473915\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.3407			1.01820			0.509101	−	0.860707i	$-0.329978\pi$
$227$	15.3407			1.01820			0.509101	+	0.860707i	$0.329978\pi$
$228$		0			0
$229$			2.97821i			0.196806i	0.995147	+	0.0984028i	$0.0313734\pi$
$229$			2.97821i			0.196806i	−0.995147	+	0.0984028i	$0.968627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.8091i		−	0.839153i	−0.907720	−	0.419577i	$-0.862179\pi$
$233$		−	12.8091i		−	0.839153i	0.907720	−	0.419577i	$-0.137821\pi$
$234$		0			0
$235$		−	2.16253i		−	0.141068i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.5684i			1.71857i	0.511499	+	0.859284i	$0.329090\pi$
$239$			26.5684i			1.71857i	−0.511499	+	0.859284i	$0.670910\pi$
$240$		0			0
$241$			23.7504i			1.52990i	0.644089	+	0.764950i	$0.277237\pi$
$241$			23.7504i			1.52990i	−0.644089	+	0.764950i	$0.722763\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.31929			0.212062
$246$		0			0
$247$		−	1.38786i		−	0.0883074i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.84925			−0.306082			−0.153041	−	0.988220i	$-0.548907\pi$
$251$	−4.84925			−0.306082			−0.153041	+	0.988220i	$0.548907\pi$
$252$		0			0
$253$	−9.19465	+	21.2923i	−0.578063	+	1.33863i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	3.00021i		−	0.187148i	−0.995612	−	0.0935741i	$-0.970171\pi$
$257$		−	3.00021i		−	0.187148i	0.995612	−	0.0935741i	$-0.0298292\pi$
$258$		0			0
$259$	−25.2347			−1.56801
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.84409			0.175374			0.0876869	−	0.996148i	$-0.472052\pi$
$263$	2.84409			0.175374			0.0876869	+	0.996148i	$0.472052\pi$
$264$		0			0
$265$	4.59274			0.282129
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	16.7889i		−	1.02364i	−0.859094	−	0.511818i	$-0.828972\pi$
$269$		−	16.7889i		−	1.02364i	0.859094	−	0.511818i	$-0.171028\pi$
$270$		0			0
$271$	29.5354			1.79415			0.897075	−	0.441879i	$-0.145688\pi$
$271$	29.5354			1.79415			0.897075	+	0.441879i	$0.145688\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.83602			−0.291623
$276$		0			0
$277$	−9.51051			−0.571431			−0.285715	−	0.958314i	$-0.592231\pi$
$277$	−9.51051			−0.571431			−0.285715	+	0.958314i	$0.592231\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.89277			−0.470843			−0.235422	−	0.971893i	$-0.575647\pi$
$281$	−7.89277			−0.470843			−0.235422	+	0.971893i	$0.575647\pi$
$282$		0			0
$283$		−	8.11451i		−	0.482358i	−0.970481	−	0.241179i	$-0.922466\pi$
$283$		−	8.11451i		−	0.482358i	0.970481	−	0.241179i	$-0.0775340\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.68967			0.0997381
$288$		0			0
$289$	12.5759			0.739757
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−23.8407			−1.39279			−0.696395	−	0.717658i	$-0.745214\pi$
$293$	−23.8407			−1.39279			−0.696395	+	0.717658i	$0.745214\pi$
$294$		0			0
$295$			4.37598i			0.254779i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.12150	+	4.91281i	−0.122689	+	0.284115i
$300$		0			0
$301$	−5.42629			−0.312766
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	11.3309i		−	0.648807i
$306$		0			0
$307$	−16.2951			−0.930010			−0.465005	−	0.885308i	$-0.653948\pi$
$307$	−16.2951			−0.930010			−0.465005	+	0.885308i	$0.653948\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			0.161476i			0.00915646i	0.999990	+	0.00457823i	$0.00145730\pi$
$311$			0.161476i			0.00915646i	−0.999990	+	0.00457823i	$0.998543\pi$
$312$		0			0
$313$		−	6.29953i		−	0.356071i	−0.984024	−	0.178035i	$-0.943026\pi$
$313$		−	6.29953i		−	0.356071i	0.984024	−	0.178035i	$-0.0569741\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.7178i			1.27596i	0.770054	+	0.637978i	$0.220229\pi$
$317$			22.7178i			1.27596i	−0.770054	+	0.637978i	$0.779771\pi$
$318$		0			0
$319$		−	13.2323i		−	0.740865i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.76423i			0.376372i
$324$		0			0
$325$	−1.11582			−0.0618948
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.94683			0.382991
$330$		0			0
$331$	−31.6012			−1.73696			−0.868478	−	0.495727i	$-0.834902\pi$
$331$	−31.6012			−1.73696			−0.868478	+	0.495727i	$0.834902\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			10.8460i			0.592578i
$336$		0			0
$337$		−	11.8892i		−	0.647649i	−0.946117	−	0.323824i	$-0.895031\pi$
$337$		−	11.8892i		−	0.647649i	0.946117	−	0.323824i	$-0.104969\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−39.2135			−2.12353
$342$		0			0
$343$		−	11.8238i		−	0.638424i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.5186i		−	0.994133i	−0.867713	−	0.497066i	$-0.834411\pi$
$347$		−	18.5186i		−	0.994133i	0.867713	−	0.497066i	$-0.165589\pi$
$348$		0			0
$349$	32.3503			1.73167			0.865837	−	0.500326i	$-0.166787\pi$
$349$	32.3503			1.73167			0.865837	+	0.500326i	$0.166787\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			34.5476i			1.83878i	0.393344	+	0.919391i	$0.371318\pi$
$353$			34.5476i			1.83878i	−0.393344	+	0.919391i	$0.628682\pi$
$354$		0			0
$355$			1.30097i			0.0690486i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.69664			−0.142323			−0.0711617	−	0.997465i	$-0.522671\pi$
$359$	−2.69664			−0.142323			−0.0711617	+	0.997465i	$0.522671\pi$
$360$		0			0
$361$	17.4530			0.918577
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−15.0201			−0.786185
$366$		0			0
$367$		−	13.0713i		−	0.682318i	−0.940006	−	0.341159i	$-0.889180\pi$
$367$		−	13.0713i		−	0.682318i	0.940006	−	0.341159i	$-0.110820\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			14.7535i			0.765966i
$372$		0			0
$373$			23.8822i			1.23657i	0.785953	+	0.618286i	$0.212173\pi$
$373$			23.8822i			1.23657i	−0.785953	+	0.618286i	$0.787827\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.05311i		−	0.157243i
$378$		0			0
$379$		−	32.1187i		−	1.64983i	−0.565260	−	0.824913i	$-0.691224\pi$
$379$		−	32.1187i		−	1.64983i	0.565260	−	0.824913i	$-0.308776\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.2045			0.521426			0.260713	−	0.965416i	$-0.416042\pi$
$383$	10.2045			0.521426			0.260713	+	0.965416i	$0.416042\pi$
$384$		0			0
$385$		−	15.5351i		−	0.791740i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−13.0110			−0.659683			−0.329842	−	0.944036i	$-0.606995\pi$
$389$	−13.0110			−0.659683			−0.329842	+	0.944036i	$0.606995\pi$
$390$		0			0
$391$	10.3399	−	23.9443i	0.522910	−	1.21092i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			2.06914i			0.104110i
$396$		0			0
$397$	−10.9332			−0.548722			−0.274361	−	0.961627i	$-0.588466\pi$
$397$	−10.9332			−0.548722			−0.274361	+	0.961627i	$0.588466\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.1100			1.05418			0.527092	−	0.849808i	$-0.323282\pi$
$401$	21.1100			1.05418			0.527092	+	0.849808i	$0.323282\pi$
$402$		0			0
$403$	−9.04780			−0.450703
$404$		0			0
$405$		0			0
$406$		0			0
$407$			37.9893i			1.88306i
$408$		0			0
$409$	18.5170			0.915606			0.457803	−	0.889054i	$-0.348636\pi$
$409$	18.5170			0.915606			0.457803	+	0.889054i	$0.348636\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.0572			−0.691712
$414$		0			0
$415$	−2.56550			−0.125935
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.69147			0.131487			0.0657435	−	0.997837i	$-0.479058\pi$
$419$	2.69147			0.131487			0.0657435	+	0.997837i	$0.479058\pi$
$420$		0			0
$421$		−	9.45660i		−	0.460887i	−0.973086	−	0.230443i	$-0.925982\pi$
$421$		−	9.45660i		−	0.460887i	0.973086	−	0.230443i	$-0.0740176\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.43837			0.263800
$426$		0			0
$427$	36.3991			1.76148
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.22794			−0.444494			−0.222247	−	0.974990i	$-0.571339\pi$
$431$	−9.22794			−0.444494			−0.222247	+	0.974990i	$0.571339\pi$
$432$		0			0
$433$		−	11.2620i		−	0.541215i	−0.962690	−	0.270608i	$-0.912775\pi$
$433$		−	11.2620i		−	0.541215i	0.962690	−	0.270608i	$-0.0872246\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.47626	+	2.36481i	0.261965	+	0.113124i
$438$		0			0
$439$	−20.3769			−0.972536			−0.486268	−	0.873810i	$-0.661642\pi$
$439$	−20.3769			−0.972536			−0.486268	+	0.873810i	$0.661642\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	29.9591i		−	1.42340i	−0.702484	−	0.711700i	$-0.747926\pi$
$443$		−	29.9591i		−	1.42340i	0.702484	−	0.711700i	$-0.252074\pi$
$444$		0			0
$445$	−4.37085			−0.207198
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.27279i		−	0.0600668i	−0.999549	−	0.0300334i	$-0.990439\pi$
$449$		−	1.27279i		−	0.0600668i	0.999549	−	0.0300334i	$-0.00956137\pi$
$450$		0			0
$451$		−	2.54370i		−	0.119778i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.58444i		−	0.168041i
$456$		0			0
$457$			10.6392i			0.497679i	0.968545	+	0.248839i	$0.0800492\pi$
$457$			10.6392i			0.497679i	−0.968545	+	0.248839i	$0.919951\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			33.9848i			1.58283i	0.611279	+	0.791415i	$0.290655\pi$
$461$			33.9848i			1.58283i	−0.611279	+	0.791415i	$0.709345\pi$
$462$		0			0
$463$	−16.8644			−0.783756			−0.391878	−	0.920017i	$-0.628175\pi$
$463$	−16.8644			−0.783756			−0.391878	+	0.920017i	$0.628175\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	35.2379			1.63062			0.815308	−	0.579028i	$-0.196568\pi$
$467$	35.2379			1.63062			0.815308	+	0.579028i	$0.196568\pi$
$468$		0			0
$469$	−34.8412			−1.60882
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.16895i			0.375609i
$474$		0			0
$475$			1.24380i			0.0570694i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.5360			−1.53230			−0.766150	−	0.642662i	$-0.777830\pi$
$479$	−33.5360			−1.53230			−0.766150	+	0.642662i	$0.777830\pi$
$480$		0			0
$481$			8.76535i			0.399665i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			14.8708i			0.675246i
$486$		0			0
$487$	17.1485			0.777073			0.388537	−	0.921433i	$-0.372981\pi$
$487$	17.1485			0.777073			0.388537	+	0.921433i	$0.372981\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.97838i			0.134413i	0.997739	+	0.0672063i	$0.0214086\pi$
$491$			2.97838i			0.134413i	−0.997739	+	0.0672063i	$0.978591\pi$
$492$		0			0
$493$			14.8804i			0.670180i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.17921			−0.187463
$498$		0			0
$499$	6.69102			0.299531			0.149766	−	0.988722i	$-0.452148\pi$
$499$	6.69102			0.299531			0.149766	+	0.988722i	$0.452148\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13.2709			−0.591719			−0.295860	−	0.955231i	$-0.595606\pi$
$503$	−13.2709			−0.591719			−0.295860	+	0.955231i	$0.595606\pi$
$504$		0			0
$505$		−	13.0142i		−	0.579126i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	37.9942i		−	1.68406i	−0.539430	−	0.842031i	$-0.681360\pi$
$509$		−	37.9942i		−	1.68406i	0.539430	−	0.842031i	$-0.318640\pi$
$510$		0			0
$511$		−	48.2499i		−	2.13445i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			17.3667i			0.765269i
$516$		0			0
$517$		−	10.4580i		−	0.459943i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.0141			−1.35875			−0.679376	−	0.733790i	$-0.737749\pi$
$521$	−31.0141			−1.35875			−0.679376	+	0.733790i	$0.737749\pi$
$522$		0			0
$523$		−	28.3208i		−	1.23838i	−0.785241	−	0.619191i	$-0.787461\pi$
$523$		−	28.3208i		−	1.23838i	0.785241	−	0.619191i	$-0.212539\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	44.0977			1.92093
$528$		0			0
$529$	−15.7702	−	16.7422i	−0.685663	−	0.727920i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.586912i		−	0.0254220i
$534$		0			0
$535$	13.9397			0.602666
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.0522			0.691416
$540$		0			0
$541$	−19.7954			−0.851070			−0.425535	−	0.904942i	$-0.639914\pi$
$541$	−19.7954			−0.851070			−0.425535	+	0.904942i	$0.639914\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1.39900i		−	0.0599267i
$546$		0			0
$547$	42.1167			1.80078			0.900390	−	0.435083i	$-0.143281\pi$
$547$	42.1167			1.80078			0.900390	+	0.435083i	$0.143281\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.40326			−0.144984
$552$		0			0
$553$	−6.64682			−0.282652
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.0742			−0.638716			−0.319358	−	0.947634i	$-0.603467\pi$
$557$	−15.0742			−0.638716			−0.319358	+	0.947634i	$0.603467\pi$
$558$		0			0
$559$			1.88484i			0.0797201i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.2425			0.515958			0.257979	−	0.966151i	$-0.416943\pi$
$563$	12.2425			0.515958			0.257979	+	0.966151i	$0.416943\pi$
$564$		0			0
$565$	9.11577			0.383503
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.2128			−1.14082			−0.570411	−	0.821360i	$-0.693216\pi$
$569$	−27.2128			−1.14082			−0.570411	+	0.821360i	$0.693216\pi$
$570$		0			0
$571$			34.3473i			1.43739i	0.695326	+	0.718695i	$0.255260\pi$
$571$			34.3473i			1.43739i	−0.695326	+	0.718695i	$0.744740\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.90128	−	4.40285i	0.0792890	−	0.183612i
$576$		0			0
$577$	16.2526			0.676603			0.338301	−	0.941038i	$-0.390148\pi$
$577$	16.2526			0.676603			0.338301	+	0.941038i	$0.390148\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	8.24132i		−	0.341908i
$582$		0			0
$583$	22.2106			0.919868
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	26.8886i		−	1.10981i	−0.831914	−	0.554905i	$-0.812755\pi$
$587$		−	26.8886i		−	1.10981i	0.831914	−	0.554905i	$-0.187245\pi$
$588$		0			0
$589$			10.0855i			0.415565i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	25.3923i		−	1.04274i	−0.853331	−	0.521369i	$-0.825422\pi$
$593$		−	25.3923i		−	1.04274i	0.853331	−	0.521369i	$-0.174578\pi$
$594$		0			0
$595$			17.4700i			0.716201i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	15.6984i		−	0.641420i	−0.947177	−	0.320710i	$-0.896078\pi$
$599$		−	15.6984i		−	0.641420i	0.947177	−	0.320710i	$-0.103922\pi$
$600$		0			0
$601$	−21.6788			−0.884295			−0.442147	−	0.896942i	$-0.645783\pi$
$601$	−21.6788			−0.884295			−0.442147	+	0.896942i	$0.645783\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.3871			−0.503607
$606$		0			0
$607$	12.4328			0.504630			0.252315	−	0.967645i	$-0.418808\pi$
$607$	12.4328			0.504630			0.252315	+	0.967645i	$0.418808\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	2.41300i		−	0.0976195i
$612$		0			0
$613$			7.44801i			0.300822i	0.988624	+	0.150411i	$0.0480598\pi$
$613$			7.44801i			0.300822i	−0.988624	+	0.150411i	$0.951940\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.02795			0.0816422			0.0408211	−	0.999166i	$-0.487003\pi$
$617$	2.02795			0.0816422			0.0408211	+	0.999166i	$0.487003\pi$
$618$		0			0
$619$		−	25.4405i		−	1.02254i	−0.859421	−	0.511269i	$-0.829175\pi$
$619$		−	25.4405i		−	1.02254i	0.859421	−	0.511269i	$-0.170825\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	14.0408i		−	0.562532i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	42.7211i		−	1.70340i
$630$		0			0
$631$		−	32.0627i		−	1.27639i	−0.769873	−	0.638197i	$-0.779681\pi$
$631$		−	32.0627i		−	1.27639i	0.769873	−	0.638197i	$-0.220319\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.34954			−0.172606
$636$		0			0
$637$	3.70375			0.146748
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−25.9445			−1.02475			−0.512374	−	0.858762i	$-0.671234\pi$
$641$	−25.9445			−1.02475			−0.512374	+	0.858762i	$0.671234\pi$
$642$		0			0
$643$		−	49.4598i		−	1.95050i	−0.221097	−	0.975252i	$-0.570964\pi$
$643$		−	49.4598i		−	1.95050i	0.221097	−	0.975252i	$-0.429036\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			34.8579i			1.37041i	0.728353	+	0.685203i	$0.240286\pi$
$647$			34.8579i			1.37041i	−0.728353	+	0.685203i	$0.759714\pi$
$648$		0			0
$649$			21.1623i			0.830694i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			28.7991i			1.12700i	0.826118	+	0.563498i	$0.190545\pi$
$653$			28.7991i			1.12700i	−0.826118	+	0.563498i	$0.809455\pi$
$654$		0			0
$655$		−	0.324570i		−	0.0126820i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−29.6751			−1.15598			−0.577989	−	0.816045i	$-0.696162\pi$
$659$	−29.6751			−1.15598			−0.577989	+	0.816045i	$0.696162\pi$
$660$		0			0
$661$			0.509408i			0.0198137i	0.999951	+	0.00990685i	$0.00315350\pi$
$661$			0.509408i			0.0198137i	−0.999951	+	0.00990685i	$0.996847\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.99553			−0.154940
$666$		0			0
$667$	12.0470	+	5.20227i	0.466463	+	0.201433i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	54.7966i		−	2.11540i
$672$		0			0
$673$	−38.8513			−1.49761			−0.748803	−	0.662792i	$-0.769371\pi$
$673$	−38.8513			−1.49761			−0.748803	+	0.662792i	$0.769371\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.0278			−0.616000			−0.308000	−	0.951386i	$-0.599660\pi$
$677$	−16.0278			−0.616000			−0.308000	+	0.951386i	$0.599660\pi$
$678$		0			0
$679$	−47.7703			−1.83326
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	35.5504i		−	1.36030i	−0.733073	−	0.680150i	$-0.761915\pi$
$683$		−	35.5504i		−	1.36030i	0.733073	−	0.680150i	$-0.238085\pi$
$684$		0			0
$685$	17.3874			0.664338
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.12469			0.195235
$690$		0			0
$691$	23.3256			0.887349			0.443674	−	0.896188i	$-0.353675\pi$
$691$	23.3256			0.887349			0.443674	+	0.896188i	$0.353675\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.76237			0.218579
$696$		0			0
$697$			2.86052i			0.108350i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.56254			0.0590163			0.0295082	−	0.999565i	$-0.490606\pi$
$701$	1.56254			0.0590163			0.0295082	+	0.999565i	$0.490606\pi$
$702$		0			0
$703$	9.77064			0.368507
$704$		0			0
$705$		0			0
$706$		0			0
$707$	41.8065			1.57229
$708$		0			0
$709$		−	11.1067i		−	0.417122i	−0.978009	−	0.208561i	$-0.933122\pi$
$709$		−	11.1067i		−	0.417122i	0.978009	−	0.208561i	$-0.0668779\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15.4168	−	35.7011i	0.577364	−	1.33702i
$714$		0			0
$715$	−5.39615			−0.201805
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	22.3821i		−	0.834711i	−0.908743	−	0.417356i	$-0.862957\pi$
$719$		−	22.3821i		−	0.834711i	0.908743	−	0.417356i	$-0.137043\pi$
$720$		0			0
$721$	−55.7882			−2.07766
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2.73619i			0.101619i
$726$		0			0
$727$			26.2750i			0.974486i	0.873267	+	0.487243i	$0.161997\pi$
$727$			26.2750i			0.974486i	−0.873267	+	0.487243i	$0.838003\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	9.18643i		−	0.339772i
$732$		0			0
$733$			14.9939i			0.553814i	0.960897	+	0.276907i	$0.0893094\pi$
$733$			14.9939i			0.553814i	−0.960897	+	0.276907i	$0.910691\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			52.4512i			1.93207i
$738$		0			0
$739$	−31.7230			−1.16695			−0.583475	−	0.812131i	$-0.698307\pi$
$739$	−31.7230			−1.16695			−0.583475	+	0.812131i	$0.698307\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−39.4237			−1.44632			−0.723158	−	0.690683i	$-0.757310\pi$
$743$	−39.4237			−1.44632			−0.723158	+	0.690683i	$0.757310\pi$
$744$		0			0
$745$	0.974277			0.0356947
$746$		0			0
$747$		0			0
$748$		0			0
$749$			44.7794i			1.63620i
$750$		0			0
$751$		−	21.0333i		−	0.767515i	−0.923434	−	0.383757i	$-0.874630\pi$
$751$		−	21.0333i		−	0.767515i	0.923434	−	0.383757i	$-0.125370\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20.8384			0.758388
$756$		0			0
$757$			12.5497i			0.456128i	0.973646	+	0.228064i	$0.0732395\pi$
$757$			12.5497i			0.456128i	−0.973646	+	0.228064i	$0.926760\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			22.4657i			0.814382i	0.913343	+	0.407191i	$0.133492\pi$
$761$			22.4657i			0.814382i	−0.913343	+	0.407191i	$0.866508\pi$
$762$		0			0
$763$	4.49411			0.162698
$764$		0			0
$765$		0			0
$766$		0			0
$767$			4.88282i			0.176309i
$768$		0			0
$769$			1.65714i			0.0597579i	0.999554	+	0.0298790i	$0.00951219\pi$
$769$			1.65714i			0.0597579i	−0.999554	+	0.0298790i	$0.990488\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−16.8888			−0.607448			−0.303724	−	0.952760i	$-0.598230\pi$
$773$	−16.8888			−0.607448			−0.303724	+	0.952760i	$0.598230\pi$
$774$		0			0
$775$	8.10862			0.291270
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.654224			−0.0234400
$780$		0			0
$781$			6.29154i			0.225129i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			6.93487i			0.247516i
$786$		0			0
$787$			18.5226i			0.660258i	0.943936	+	0.330129i	$0.107092\pi$
$787$			18.5226i			0.660258i	−0.943936	+	0.330129i	$0.892908\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			29.2832i			1.04119i
$792$		0			0
$793$		−	12.6433i		−	0.448978i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.4880			−1.07994			−0.539970	−	0.841684i	$-0.681564\pi$
$797$	−30.4880			−1.07994			−0.539970	+	0.841684i	$0.681564\pi$
$798$		0			0
$799$			11.7606i			0.416061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−72.6373			−2.56331
$804$		0			0
$805$	14.1436	+	6.10762i	0.498496	+	0.215265i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	41.9163i		−	1.47370i	−0.676057	−	0.736849i	$-0.736313\pi$
$809$		−	41.9163i		−	1.47370i	0.676057	−	0.736849i	$-0.263687\pi$
$810$		0			0
$811$	16.8281			0.590915			0.295458	−	0.955356i	$-0.404528\pi$
$811$	16.8281			0.590915			0.295458	+	0.955356i	$0.404528\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.0112			0.385705
$816$		0			0
$817$	2.10101			0.0735050
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	19.6346i		−	0.685251i	−0.939472	−	0.342625i	$-0.888684\pi$
$821$		−	19.6346i		−	0.685251i	0.939472	−	0.342625i	$-0.111316\pi$
$822$		0			0
$823$	8.65387			0.301655			0.150827	−	0.988560i	$-0.451806\pi$
$823$	8.65387			0.301655			0.150827	+	0.988560i	$0.451806\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	20.7939			0.723074			0.361537	−	0.932358i	$-0.382252\pi$
$827$	20.7939			0.723074			0.361537	+	0.932358i	$0.382252\pi$
$828$		0			0
$829$	15.2404			0.529322			0.264661	−	0.964341i	$-0.414740\pi$
$829$	15.2404			0.529322			0.264661	+	0.964341i	$0.414740\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.0515			−0.625449
$834$		0			0
$835$		−	3.28964i		−	0.113843i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.65184			−0.126075			−0.0630377	−	0.998011i	$-0.520079\pi$
$839$	−3.65184			−0.126075			−0.0630377	+	0.998011i	$0.520079\pi$
$840$		0			0
$841$	21.5133			0.741837
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.7549			0.404382
$846$		0			0
$847$		−	39.7919i		−	1.36727i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−34.5866	−	14.9355i	−1.18561	−	0.511983i
$852$		0			0
$853$	24.8505			0.850866			0.425433	−	0.904990i	$-0.360122\pi$
$853$	24.8505			0.850866			0.425433	+	0.904990i	$0.360122\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			40.2119i			1.37361i	0.726841	+	0.686806i	$0.240988\pi$
$857$			40.2119i			1.37361i	−0.726841	+	0.686806i	$0.759012\pi$
$858$		0			0
$859$	16.0985			0.549272			0.274636	−	0.961548i	$-0.411443\pi$
$859$	16.0985			0.549272			0.274636	+	0.961548i	$0.411443\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	14.8436i		−	0.505282i	−0.967560	−	0.252641i	$-0.918701\pi$
$863$		−	14.8436i		−	0.505282i	0.967560	−	0.252641i	$-0.0812991\pi$
$864$		0			0
$865$		−	7.18346i		−	0.244245i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			10.0064i			0.339443i
$870$		0			0
$871$			12.1022i			0.410067i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			3.21237i			0.108598i
$876$		0			0
$877$	−6.24768			−0.210969			−0.105485	−	0.994421i	$-0.533639\pi$
$877$	−6.24768			−0.210969			−0.105485	+	0.994421i	$0.533639\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.81466			−0.128519			−0.0642596	−	0.997933i	$-0.520469\pi$
$881$	−3.81466			−0.128519			−0.0642596	+	0.997933i	$0.520469\pi$
$882$		0			0
$883$	−28.2988			−0.952329			−0.476165	−	0.879356i	$-0.657973\pi$
$883$	−28.2988			−0.952329			−0.476165	+	0.879356i	$0.657973\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			11.4467i			0.384343i	0.981361	+	0.192171i	$0.0615529\pi$
$887$			11.4467i			0.384343i	−0.981361	+	0.192171i	$0.938447\pi$
$888$		0			0
$889$		−	13.9723i		−	0.468616i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.68975			−0.0900089
$894$		0			0
$895$			2.24921i			0.0751827i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			22.1867i			0.739969i
$900$		0			0
$901$	−24.9770			−0.832104
$902$		0			0
$903$		0			0
$904$		0			0
$905$			17.2585i			0.573691i
$906$		0			0
$907$			14.6379i			0.486042i	0.970021	+	0.243021i	$0.0781384\pi$
$907$			14.6379i			0.486042i	−0.970021	+	0.243021i	$0.921862\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	35.9197			1.19007			0.595037	−	0.803698i	$-0.297137\pi$
$911$	35.9197			1.19007			0.595037	+	0.803698i	$0.297137\pi$
$912$		0			0
$913$	−12.4068			−0.410605
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.04264			0.0344309
$918$		0			0
$919$			24.6942i			0.814588i	0.913297	+	0.407294i	$0.133527\pi$
$919$			24.6942i			0.814588i	−0.913297	+	0.407294i	$0.866473\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.45166i			0.0477820i
$924$		0			0
$925$		−	7.85549i		−	0.258287i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	11.5670i		−	0.379502i	−0.981832	−	0.189751i	$-0.939232\pi$
$929$		−	11.5670i		−	0.379502i	0.981832	−	0.189751i	$-0.0607681\pi$
$930$		0			0
$931$		−	4.12853i		−	0.135307i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	26.3001			0.860104
$936$		0			0
$937$			26.0335i			0.850478i	0.905081	+	0.425239i	$0.139810\pi$
$937$			26.0335i			0.850478i	−0.905081	+	0.425239i	$0.860190\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	47.8927			1.56126			0.780629	−	0.624995i	$-0.214899\pi$
$941$	47.8927			1.56126			0.780629	+	0.624995i	$0.214899\pi$
$942$		0			0
$943$	2.31585	+	1.00006i	0.0754146	+	0.0325663i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	50.9658i		−	1.65617i	−0.560605	−	0.828083i	$-0.689431\pi$
$947$		−	50.9658i		−	1.65617i	0.560605	−	0.828083i	$-0.310569\pi$
$948$		0			0
$949$	−16.7597			−0.544044
$950$		0			0
$951$		0			0
$952$		0			0
$953$	12.3264			0.399290			0.199645	−	0.979868i	$-0.436021\pi$
$953$	12.3264			0.399290			0.199645	+	0.979868i	$0.436021\pi$
$954$		0			0
$955$	−4.07184			−0.131762
$956$		0			0
$957$		0			0
$958$		0			0
$959$			55.8547i			1.80364i
$960$		0			0
$961$	34.7498			1.12096
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.5453			0.725757
$966$		0			0
$967$	−4.92092			−0.158246			−0.0791230	−	0.996865i	$-0.525212\pi$
$967$	−4.92092			−0.158246			−0.0791230	+	0.996865i	$0.525212\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.42340			0.141954			0.0709768	−	0.997478i	$-0.477388\pi$
$971$	4.42340			0.141954			0.0709768	+	0.997478i	$0.477388\pi$
$972$		0			0
$973$			18.5109i			0.593431i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	53.8741			1.72358			0.861792	−	0.507261i	$-0.169342\pi$
$977$	53.8741			1.72358			0.861792	+	0.507261i	$0.169342\pi$
$978$		0			0
$979$	−21.1375			−0.675558
$980$		0			0
$981$		0			0
$982$		0			0
$983$	45.9370			1.46516			0.732581	−	0.680680i	$-0.238316\pi$
$983$	45.9370			1.46516			0.732581	+	0.680680i	$0.238316\pi$
$984$		0			0
$985$			6.53153i			0.208112i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.43725	−	3.21163i	−0.236491	−	0.102124i
$990$		0			0
$991$	−49.2611			−1.56483			−0.782415	−	0.622757i	$-0.786013\pi$
$991$	−49.2611			−1.56483			−0.782415	+	0.622757i	$0.786013\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4.71710i		−	0.149542i
$996$		0			0
$997$	56.3537			1.78474			0.892369	−	0.451306i	$-0.149042\pi$
$997$	56.3537			1.78474			0.892369	+	0.451306i	$0.149042\pi$
$998$		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.1241.6	✓	48
3.2	odd	2		8280.2.p.b.1241.6	yes	48
23.22	odd	2		8280.2.p.b.1241.43	yes	48
69.68	even	2	inner	8280.2.p.a.1241.43	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.6	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.43	yes	48	69.68	even	2	inner
8280.2.p.b.1241.6	yes	48	3.2	odd	2
8280.2.p.b.1241.43	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.21237i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.21237i q^{7} -4.83602 q^{11} -1.11582 q^{13} +5.43837 q^{17} +1.24380i q^{19} +(1.90128 - 4.40285i) q^{23} +1.00000 q^{25} +2.73619i q^{29} +8.10862 q^{31} +3.21237i q^{35} -7.85549i q^{37} +0.525989i q^{41} -1.68919i q^{43} +2.16253i q^{47} -3.31929 q^{49} -4.59274 q^{53} +4.83602 q^{55} -4.37598i q^{59} +11.3309i q^{61} +1.11582 q^{65} -10.8460i q^{67} -1.30097i q^{71} +15.0201 q^{73} +15.5351i q^{77} -2.06914i q^{79} +2.56550 q^{83} -5.43837 q^{85} +4.37085 q^{89} +3.58444i q^{91} -1.24380i q^{95} -14.8708i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 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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