LMFDB - Embedded newform 8280.2.p.a.1241.4

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			1241.4
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.45

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.83098i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.83098i q^{7} -4.73932 q^{11} +3.06773 q^{13} -1.82684 q^{17} +1.23335i q^{19} +(3.39908 + 3.38323i) q^{23} +1.00000 q^{25} +8.94952i q^{29} +3.44531 q^{31} +3.83098i q^{35} -5.97712i q^{37} -2.73764i q^{41} +3.46415i q^{43} +5.37185i q^{47} -7.67639 q^{49} -5.72617 q^{53} +4.73932 q^{55} -5.59696i q^{59} -9.35346i q^{61} -3.06773 q^{65} +7.99740i q^{67} +13.2445i q^{71} +3.71100 q^{73} +18.1562i q^{77} +11.0381i q^{79} +12.7027 q^{83} +1.82684 q^{85} -14.7806 q^{89} -11.7524i q^{91} -1.23335i q^{95} +1.21398i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	3.83098i		−	1.44797i	−0.689814	−	0.723987i	$-0.742308\pi$
$7$		−	3.83098i		−	1.44797i	0.689814	−	0.723987i	$-0.257692\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.73932			−1.42896			−0.714480	−	0.699656i	$-0.753337\pi$
$11$	−4.73932			−1.42896			−0.714480	+	0.699656i	$0.753337\pi$
$12$		0			0
$13$	3.06773			0.850835			0.425418	−	0.904997i	$-0.360127\pi$
$13$	3.06773			0.850835			0.425418	+	0.904997i	$0.360127\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.82684			−0.443075			−0.221537	−	0.975152i	$-0.571107\pi$
$17$	−1.82684			−0.443075			−0.221537	+	0.975152i	$0.571107\pi$
$18$		0			0
$19$			1.23335i			0.282949i	0.989942	+	0.141475i	$0.0451844\pi$
$19$			1.23335i			0.282949i	−0.989942	+	0.141475i	$0.954816\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.39908	+	3.38323i	0.708757	+	0.705453i
$23$	3.39908	+	3.38323i	0.708757	+	0.705453i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.94952i			1.66188i	0.556360	+	0.830942i	$0.312198\pi$
$29$			8.94952i			1.66188i	−0.556360	+	0.830942i	$0.687802\pi$
$30$		0			0
$31$	3.44531			0.618795			0.309398	−	0.950933i	$-0.399873\pi$
$31$	3.44531			0.618795			0.309398	+	0.950933i	$0.399873\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.83098i			0.647553i
$36$		0			0
$37$		−	5.97712i		−	0.982633i	−0.870981	−	0.491316i	$-0.836516\pi$
$37$		−	5.97712i		−	0.982633i	0.870981	−	0.491316i	$-0.163484\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	2.73764i		−	0.427548i	−0.976883	−	0.213774i	$-0.931424\pi$
$41$		−	2.73764i		−	0.427548i	0.976883	−	0.213774i	$-0.0685756\pi$
$42$		0			0
$43$			3.46415i			0.528278i	0.964485	+	0.264139i	$0.0850878\pi$
$43$			3.46415i			0.528278i	−0.964485	+	0.264139i	$0.914912\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.37185i			0.783565i	0.920058	+	0.391783i	$0.128141\pi$
$47$			5.37185i			0.783565i	−0.920058	+	0.391783i	$0.871859\pi$
$48$		0			0
$49$	−7.67639			−1.09663
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.72617			−0.786550			−0.393275	−	0.919421i	$-0.628658\pi$
$53$	−5.72617			−0.786550			−0.393275	+	0.919421i	$0.628658\pi$
$54$		0			0
$55$	4.73932			0.639050
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.59696i		−	0.728662i	−0.931270	−	0.364331i	$-0.881298\pi$
$59$		−	5.59696i		−	0.728662i	0.931270	−	0.364331i	$-0.118702\pi$
$60$		0			0
$61$		−	9.35346i		−	1.19759i	−0.800903	−	0.598794i	$-0.795647\pi$
$61$		−	9.35346i		−	1.19759i	0.800903	−	0.598794i	$-0.204353\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.06773			−0.380505
$66$		0			0
$67$			7.99740i			0.977038i	0.872553	+	0.488519i	$0.162463\pi$
$67$			7.99740i			0.977038i	−0.872553	+	0.488519i	$0.837537\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.2445i			1.57183i	0.618335	+	0.785915i	$0.287808\pi$
$71$			13.2445i			1.57183i	−0.618335	+	0.785915i	$0.712192\pi$
$72$		0			0
$73$	3.71100			0.434340			0.217170	−	0.976134i	$-0.430317\pi$
$73$	3.71100			0.434340			0.217170	+	0.976134i	$0.430317\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			18.1562i			2.06910i
$78$		0			0
$79$			11.0381i			1.24189i	0.783855	+	0.620944i	$0.213250\pi$
$79$			11.0381i			1.24189i	−0.783855	+	0.620944i	$0.786750\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.7027			1.39431			0.697154	−	0.716922i	$-0.254450\pi$
$83$	12.7027			1.39431			0.697154	+	0.716922i	$0.254450\pi$
$84$		0			0
$85$	1.82684			0.198149
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.7806			−1.56674			−0.783369	−	0.621557i	$-0.786500\pi$
$89$	−14.7806			−1.56674			−0.783369	+	0.621557i	$0.786500\pi$
$90$		0			0
$91$		−	11.7524i		−	1.23199i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	1.23335i		−	0.126539i
$96$		0			0
$97$			1.21398i			0.123261i	0.998099	+	0.0616307i	$0.0196301\pi$
$97$			1.21398i			0.123261i	−0.998099	+	0.0616307i	$0.980370\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	9.66448i		−	0.961652i	−0.876816	−	0.480826i	$-0.840337\pi$
$101$		−	9.66448i		−	0.961652i	0.876816	−	0.480826i	$-0.159663\pi$
$102$		0			0
$103$			8.21748i			0.809693i	0.914385	+	0.404846i	$0.132675\pi$
$103$			8.21748i			0.809693i	−0.914385	+	0.404846i	$0.867325\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.2361			1.76295			0.881477	−	0.472226i	$-0.156550\pi$
$107$	18.2361			1.76295			0.881477	+	0.472226i	$0.156550\pi$
$108$		0			0
$109$			4.92684i			0.471906i	0.971764	+	0.235953i	$0.0758211\pi$
$109$			4.92684i			0.471906i	−0.971764	+	0.235953i	$0.924179\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.21455			−0.584615			−0.292308	−	0.956324i	$-0.594423\pi$
$113$	−6.21455			−0.584615			−0.292308	+	0.956324i	$0.594423\pi$
$114$		0			0
$115$	−3.39908	−	3.38323i	−0.316966	−	0.315488i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			6.99860i			0.641561i
$120$		0			0
$121$	11.4612			1.04192
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−2.00919			−0.178286			−0.0891432	−	0.996019i	$-0.528413\pi$
$127$	−2.00919			−0.178286			−0.0891432	+	0.996019i	$0.528413\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	21.2278i		−	1.85468i	−0.374222	−	0.927339i	$-0.622090\pi$
$131$		−	21.2278i		−	1.85468i	0.374222	−	0.927339i	$-0.377910\pi$
$132$		0			0
$133$	4.72493			0.409703
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.76378			0.748741			0.374370	−	0.927279i	$-0.377859\pi$
$137$	8.76378			0.748741			0.374370	+	0.927279i	$0.377859\pi$
$138$		0			0
$139$	13.9098			1.17982			0.589908	−	0.807470i	$-0.299164\pi$
$139$	13.9098			1.17982			0.589908	+	0.807470i	$0.299164\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.5390			−1.21581
$144$		0			0
$145$		−	8.94952i		−	0.743217i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.2285			1.08372			0.541862	−	0.840467i	$-0.317719\pi$
$149$	13.2285			1.08372			0.541862	+	0.840467i	$0.317719\pi$
$150$		0			0
$151$	14.0438			1.14287			0.571434	−	0.820648i	$-0.306387\pi$
$151$	14.0438			1.14287			0.571434	+	0.820648i	$0.306387\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.44531			−0.276734
$156$		0			0
$157$		−	17.5040i		−	1.39697i	−0.715626	−	0.698484i	$-0.753858\pi$
$157$		−	17.5040i		−	1.39697i	0.715626	−	0.698484i	$-0.246142\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.9611	−	13.0218i	1.02148	−	1.02626i
$162$		0			0
$163$	4.34521			0.340343			0.170172	−	0.985414i	$-0.445568\pi$
$163$	4.34521			0.340343			0.170172	+	0.985414i	$0.445568\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			7.47785i			0.578654i	0.957230	+	0.289327i	$0.0934315\pi$
$167$			7.47785i			0.578654i	−0.957230	+	0.289327i	$0.906569\pi$
$168$		0			0
$169$	−3.58903			−0.276079
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	17.5785i		−	1.33647i	−0.743951	−	0.668234i	$-0.767050\pi$
$173$		−	17.5785i		−	1.33647i	0.743951	−	0.668234i	$-0.232950\pi$
$174$		0			0
$175$		−	3.83098i		−	0.289595i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.1037i		−	0.904671i	−0.891848	−	0.452336i	$-0.850591\pi$
$179$		−	12.1037i		−	0.904671i	0.891848	−	0.452336i	$-0.149409\pi$
$180$		0			0
$181$		−	11.0653i		−	0.822477i	−0.911528	−	0.411239i	$-0.865096\pi$
$181$		−	11.0653i		−	0.822477i	0.911528	−	0.411239i	$-0.134904\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			5.97712i			0.439447i
$186$		0			0
$187$	8.65800			0.633136
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.34765			−0.0975126			−0.0487563	−	0.998811i	$-0.515526\pi$
$191$	−1.34765			−0.0975126			−0.0487563	+	0.998811i	$0.515526\pi$
$192$		0			0
$193$	10.0234			0.721503			0.360752	−	0.932662i	$-0.382520\pi$
$193$	10.0234			0.721503			0.360752	+	0.932662i	$0.382520\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			10.4221i			0.742543i	0.928524	+	0.371272i	$0.121078\pi$
$197$			10.4221i			0.742543i	−0.928524	+	0.371272i	$0.878922\pi$
$198$		0			0
$199$		−	13.6291i		−	0.966138i	−0.875582	−	0.483069i	$-0.839522\pi$
$199$		−	13.6291i		−	0.966138i	0.875582	−	0.483069i	$-0.160478\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	34.2854			2.40636
$204$		0			0
$205$			2.73764i			0.191205i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	5.84523i		−	0.404323i
$210$		0			0
$211$	−14.6795			−1.01058			−0.505291	−	0.862949i	$-0.668615\pi$
$211$	−14.6795			−1.01058			−0.505291	+	0.862949i	$0.668615\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3.46415i		−	0.236253i
$216$		0			0
$217$		−	13.1989i		−	0.895999i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.60427			−0.376984
$222$		0			0
$223$	−11.3948			−0.763051			−0.381525	−	0.924358i	$-0.624601\pi$
$223$	−11.3948			−0.763051			−0.381525	+	0.924358i	$0.624601\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.7684			1.17933			0.589664	−	0.807649i	$-0.299260\pi$
$227$	17.7684			1.17933			0.589664	+	0.807649i	$0.299260\pi$
$228$		0			0
$229$			18.8860i			1.24803i	0.781414	+	0.624013i	$0.214499\pi$
$229$			18.8860i			1.24803i	−0.781414	+	0.624013i	$0.785501\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			25.8639i			1.69440i	0.531275	+	0.847200i	$0.321713\pi$
$233$			25.8639i			1.69440i	−0.531275	+	0.847200i	$0.678287\pi$
$234$		0			0
$235$		−	5.37185i		−	0.350421i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4.06685i			0.263063i	0.991312	+	0.131531i	$0.0419894\pi$
$239$			4.06685i			0.263063i	−0.991312	+	0.131531i	$0.958011\pi$
$240$		0			0
$241$			5.78526i			0.372661i	0.982487	+	0.186331i	$0.0596596\pi$
$241$			5.78526i			0.372661i	−0.982487	+	0.186331i	$0.940340\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.67639			0.490426
$246$		0			0
$247$			3.78358i			0.240743i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.16805			0.263085			0.131543	−	0.991311i	$-0.458007\pi$
$251$	4.16805			0.263085			0.131543	+	0.991311i	$0.458007\pi$
$252$		0			0
$253$	−16.1093	−	16.0342i	−1.01278	−	1.00806i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.18353i			0.385718i	0.981226	+	0.192859i	$0.0617760\pi$
$257$			6.18353i			0.385718i	−0.981226	+	0.192859i	$0.938224\pi$
$258$		0			0
$259$	−22.8982			−1.42283
$260$		0			0
$261$		0			0
$262$		0			0
$263$	25.0748			1.54618			0.773090	−	0.634296i	$-0.218710\pi$
$263$	25.0748			1.54618			0.773090	+	0.634296i	$0.218710\pi$
$264$		0			0
$265$	5.72617			0.351756
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	5.22069i		−	0.318311i	−0.987254	−	0.159155i	$-0.949123\pi$
$269$		−	5.22069i		−	0.318311i	0.987254	−	0.159155i	$-0.0508771\pi$
$270$		0			0
$271$	5.65067			0.343254			0.171627	−	0.985162i	$-0.445098\pi$
$271$	5.65067			0.343254			0.171627	+	0.985162i	$0.445098\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.73932			−0.285792
$276$		0			0
$277$	29.8587			1.79404			0.897018	−	0.441995i	$-0.145729\pi$
$277$	29.8587			1.79404			0.897018	+	0.441995i	$0.145729\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.32277			−0.436840			−0.218420	−	0.975855i	$-0.570090\pi$
$281$	−7.32277			−0.436840			−0.218420	+	0.975855i	$0.570090\pi$
$282$		0			0
$283$		−	25.8629i		−	1.53739i	−0.639617	−	0.768694i	$-0.720907\pi$
$283$		−	25.8629i		−	1.53739i	0.639617	−	0.768694i	$-0.279093\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.4878			−0.619078
$288$		0			0
$289$	−13.6626			−0.803685
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.98640			0.466571			0.233285	−	0.972408i	$-0.425052\pi$
$293$	7.98640			0.466571			0.233285	+	0.972408i	$0.425052\pi$
$294$		0			0
$295$			5.59696i			0.325868i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.4275	+	10.3788i	0.603035	+	0.600224i
$300$		0			0
$301$	13.2711			0.764933
$302$		0			0
$303$		0			0
$304$		0			0
$305$			9.35346i			0.535578i
$306$		0			0
$307$	−12.2443			−0.698821			−0.349411	−	0.936970i	$-0.613618\pi$
$307$	−12.2443			−0.698821			−0.349411	+	0.936970i	$0.613618\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.0397i			1.30646i	0.757158	+	0.653232i	$0.226587\pi$
$311$			23.0397i			1.30646i	−0.757158	+	0.653232i	$0.773413\pi$
$312$		0			0
$313$		−	14.5521i		−	0.822535i	−0.911515	−	0.411267i	$-0.865086\pi$
$313$		−	14.5521i		−	0.822535i	0.911515	−	0.411267i	$-0.134914\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	16.8858i		−	0.948403i	−0.880416	−	0.474202i	$-0.842737\pi$
$317$		−	16.8858i		−	0.948403i	0.880416	−	0.474202i	$-0.157263\pi$
$318$		0			0
$319$		−	42.4146i		−	2.37476i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.25314i		−	0.125368i
$324$		0			0
$325$	3.06773			0.170167
$326$		0			0
$327$		0			0
$328$		0			0
$329$	20.5794			1.13458
$330$		0			0
$331$	28.9328			1.59029			0.795146	−	0.606419i	$-0.207394\pi$
$331$	28.9328			1.59029			0.795146	+	0.606419i	$0.207394\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	7.99740i		−	0.436945i
$336$		0			0
$337$		−	14.2938i		−	0.778635i	−0.921104	−	0.389318i	$-0.872711\pi$
$337$		−	14.2938i		−	0.778635i	0.921104	−	0.389318i	$-0.127289\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.3284			−0.884233
$342$		0			0
$343$			2.59123i			0.139913i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			22.5379i			1.20990i	0.796263	+	0.604950i	$0.206807\pi$
$347$			22.5379i			1.20990i	−0.796263	+	0.604950i	$0.793193\pi$
$348$		0			0
$349$	−9.97175			−0.533776			−0.266888	−	0.963728i	$-0.585995\pi$
$349$	−9.97175			−0.533776			−0.266888	+	0.963728i	$0.585995\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	35.1026i		−	1.86833i	−0.356847	−	0.934163i	$-0.616148\pi$
$353$		−	35.1026i		−	1.86833i	0.356847	−	0.934163i	$-0.383852\pi$
$354$		0			0
$355$		−	13.2445i		−	0.702944i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.95305			0.155856			0.0779281	−	0.996959i	$-0.475170\pi$
$359$	2.95305			0.155856			0.0779281	+	0.996959i	$0.475170\pi$
$360$		0			0
$361$	17.4789			0.919940
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.71100			−0.194243
$366$		0			0
$367$			2.40550i			0.125566i	0.998027	+	0.0627830i	$0.0199976\pi$
$367$			2.40550i			0.125566i	−0.998027	+	0.0627830i	$0.980002\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			21.9368i			1.13890i
$372$		0			0
$373$			21.8320i			1.13042i	0.824948	+	0.565208i	$0.191204\pi$
$373$			21.8320i			1.13042i	−0.824948	+	0.565208i	$0.808796\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			27.4547i			1.41399i
$378$		0			0
$379$		−	10.3519i		−	0.531742i	−0.964009	−	0.265871i	$-0.914340\pi$
$379$		−	10.3519i		−	0.531742i	0.964009	−	0.265871i	$-0.0856595\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.44907			0.176239			0.0881197	−	0.996110i	$-0.471914\pi$
$383$	3.44907			0.176239			0.0881197	+	0.996110i	$0.471914\pi$
$384$		0			0
$385$		−	18.1562i		−	0.925327i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.8657			0.601615			0.300808	−	0.953685i	$-0.402744\pi$
$389$	11.8657			0.601615			0.300808	+	0.953685i	$0.402744\pi$
$390$		0			0
$391$	−6.20959	−	6.18064i	−0.314032	−	0.312568i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	11.0381i		−	0.555389i
$396$		0			0
$397$	20.6918			1.03849			0.519246	−	0.854625i	$-0.326213\pi$
$397$	20.6918			1.03849			0.519246	+	0.854625i	$0.326213\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.5863			−0.678467			−0.339233	−	0.940702i	$-0.610168\pi$
$401$	−13.5863			−0.678467			−0.339233	+	0.940702i	$0.610168\pi$
$402$		0			0
$403$	10.5693			0.526493
$404$		0			0
$405$		0			0
$406$		0			0
$407$			28.3275i			1.40414i
$408$		0			0
$409$	35.9652			1.77836			0.889182	−	0.457553i	$-0.151274\pi$
$409$	35.9652			1.77836			0.889182	+	0.457553i	$0.151274\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−21.4418			−1.05508
$414$		0			0
$415$	−12.7027			−0.623553
$416$		0			0
$417$		0			0
$418$		0			0
$419$	27.0958			1.32372			0.661859	−	0.749629i	$-0.269768\pi$
$419$	27.0958			1.32372			0.661859	+	0.749629i	$0.269768\pi$
$420$		0			0
$421$		−	11.5978i		−	0.565242i	−0.959232	−	0.282621i	$-0.908796\pi$
$421$		−	11.5978i		−	0.565242i	0.959232	−	0.282621i	$-0.0912039\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.82684			−0.0886150
$426$		0			0
$427$	−35.8329			−1.73408
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.0480			1.06202			0.531008	−	0.847367i	$-0.321813\pi$
$431$	22.0480			1.06202			0.531008	+	0.847367i	$0.321813\pi$
$432$		0			0
$433$			21.2405i			1.02075i	0.859951	+	0.510376i	$0.170494\pi$
$433$			21.2405i			1.02075i	−0.859951	+	0.510376i	$0.829506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.17270	+	4.19225i	−0.199608	+	0.200542i
$438$		0			0
$439$	36.9491			1.76348			0.881741	−	0.471733i	$-0.156371\pi$
$439$	36.9491			1.76348			0.881741	+	0.471733i	$0.156371\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			24.9001i			1.18304i	0.806291	+	0.591519i	$0.201472\pi$
$443$			24.9001i			1.18304i	−0.806291	+	0.591519i	$0.798528\pi$
$444$		0			0
$445$	14.7806			0.700666
$446$		0			0
$447$		0			0
$448$		0			0
$449$			33.2810i			1.57063i	0.619098	+	0.785313i	$0.287498\pi$
$449$			33.2810i			1.57063i	−0.619098	+	0.785313i	$0.712502\pi$
$450$		0			0
$451$			12.9746i			0.610948i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			11.7524i			0.550961i
$456$		0			0
$457$		−	4.09708i		−	0.191654i	−0.995398	−	0.0958268i	$-0.969451\pi$
$457$		−	4.09708i		−	0.191654i	0.995398	−	0.0958268i	$-0.0305495\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			2.70774i			0.126112i	0.998010	+	0.0630560i	$0.0200847\pi$
$461$			2.70774i			0.126112i	−0.998010	+	0.0630560i	$0.979915\pi$
$462$		0			0
$463$	−8.51011			−0.395498			−0.197749	−	0.980253i	$-0.563363\pi$
$463$	−8.51011			−0.395498			−0.197749	+	0.980253i	$0.563363\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−38.6679			−1.78934			−0.894668	−	0.446731i	$-0.852588\pi$
$467$	−38.6679			−1.78934			−0.894668	+	0.446731i	$0.852588\pi$
$468$		0			0
$469$	30.6379			1.41472
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16.4177i		−	0.754888i
$474$		0			0
$475$			1.23335i			0.0565899i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.07567			−0.277605			−0.138802	−	0.990320i	$-0.544325\pi$
$479$	−6.07567			−0.277605			−0.138802	+	0.990320i	$0.544325\pi$
$480$		0			0
$481$		−	18.3362i		−	0.836058i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.21398i		−	0.0551242i
$486$		0			0
$487$	−32.4344			−1.46974			−0.734871	−	0.678207i	$-0.762757\pi$
$487$	−32.4344			−1.46974			−0.734871	+	0.678207i	$0.762757\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	39.3958i		−	1.77791i	−0.457998	−	0.888953i	$-0.651433\pi$
$491$		−	39.3958i		−	1.77791i	0.457998	−	0.888953i	$-0.348567\pi$
$492$		0			0
$493$		−	16.3494i		−	0.736339i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	50.7393			2.27597
$498$		0			0
$499$	24.4019			1.09238			0.546189	−	0.837662i	$-0.316078\pi$
$499$	24.4019			1.09238			0.546189	+	0.837662i	$0.316078\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−29.3030			−1.30656			−0.653278	−	0.757118i	$-0.726607\pi$
$503$	−29.3030			−1.30656			−0.653278	+	0.757118i	$0.726607\pi$
$504$		0			0
$505$			9.66448i			0.430064i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	21.8026i		−	0.966383i	−0.875515	−	0.483192i	$-0.839477\pi$
$509$		−	21.8026i		−	0.966383i	0.875515	−	0.483192i	$-0.160523\pi$
$510$		0			0
$511$		−	14.2168i		−	0.628913i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	8.21748i		−	0.362106i
$516$		0			0
$517$		−	25.4589i		−	1.11968i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−39.1115			−1.71351			−0.856754	−	0.515726i	$-0.827522\pi$
$521$	−39.1115			−1.71351			−0.856754	+	0.515726i	$0.827522\pi$
$522$		0			0
$523$			21.2591i			0.929596i	0.885417	+	0.464798i	$0.153873\pi$
$523$			21.2591i			0.929596i	−0.885417	+	0.464798i	$0.846127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.29404			−0.274173
$528$		0			0
$529$	0.107469	+	22.9997i	0.00467256	+	0.999989i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.39834i		−	0.363773i
$534$		0			0
$535$	−18.2361			−0.788417
$536$		0			0
$537$		0			0
$538$		0			0
$539$	36.3809			1.56704
$540$		0			0
$541$	−12.5018			−0.537493			−0.268747	−	0.963211i	$-0.586609\pi$
$541$	−12.5018			−0.537493			−0.268747	+	0.963211i	$0.586609\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	4.92684i		−	0.211043i
$546$		0			0
$547$	12.4642			0.532932			0.266466	−	0.963844i	$-0.414144\pi$
$547$	12.4642			0.532932			0.266466	+	0.963844i	$0.414144\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.0379			−0.470229
$552$		0			0
$553$	42.2868			1.79822
$554$		0			0
$555$		0			0
$556$		0			0
$557$	20.0422			0.849216			0.424608	−	0.905377i	$-0.360412\pi$
$557$	20.0422			0.849216			0.424608	+	0.905377i	$0.360412\pi$
$558$		0			0
$559$			10.6271i			0.449478i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.3227			0.477194			0.238597	−	0.971119i	$-0.423313\pi$
$563$	11.3227			0.477194			0.238597	+	0.971119i	$0.423313\pi$
$564$		0			0
$565$	6.21455			0.261448
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.38306			0.351436			0.175718	−	0.984441i	$-0.443775\pi$
$569$	8.38306			0.351436			0.175718	+	0.984441i	$0.443775\pi$
$570$		0			0
$571$			29.1657i			1.22054i	0.792192	+	0.610272i	$0.208940\pi$
$571$			29.1657i			1.22054i	−0.792192	+	0.610272i	$0.791060\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.39908	+	3.38323i	0.141751	+	0.141091i
$576$		0			0
$577$	5.64306			0.234923			0.117462	−	0.993077i	$-0.462524\pi$
$577$	5.64306			0.234923			0.117462	+	0.993077i	$0.462524\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	48.6639i		−	2.01892i
$582$		0			0
$583$	27.1382			1.12395
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.8090i			0.776331i	0.921590	+	0.388166i	$0.126891\pi$
$587$			18.8090i			0.776331i	−0.921590	+	0.388166i	$0.873109\pi$
$588$		0			0
$589$			4.24926i			0.175088i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	27.7968i		−	1.14148i	−0.821132	−	0.570739i	$-0.806657\pi$
$593$		−	27.7968i		−	1.14148i	0.821132	−	0.570739i	$-0.193343\pi$
$594$		0			0
$595$		−	6.99860i		−	0.286915i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	14.0098i		−	0.572426i	−0.958166	−	0.286213i	$-0.907604\pi$
$599$		−	14.0098i		−	0.572426i	0.958166	−	0.286213i	$-0.0923964\pi$
$600$		0			0
$601$	−9.87168			−0.402674			−0.201337	−	0.979522i	$-0.564529\pi$
$601$	−9.87168			−0.402674			−0.201337	+	0.979522i	$0.564529\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.4612			−0.465963
$606$		0			0
$607$	−4.36491			−0.177166			−0.0885831	−	0.996069i	$-0.528234\pi$
$607$	−4.36491			−0.177166			−0.0885831	+	0.996069i	$0.528234\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.4794i			0.666685i
$612$		0			0
$613$		−	21.5908i		−	0.872044i	−0.899936	−	0.436022i	$-0.856387\pi$
$613$		−	21.5908i		−	0.872044i	0.899936	−	0.436022i	$-0.143613\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.1985			−1.37678			−0.688390	−	0.725341i	$-0.741682\pi$
$617$	−34.1985			−1.37678			−0.688390	+	0.725341i	$0.741682\pi$
$618$		0			0
$619$			17.5680i			0.706120i	0.935601	+	0.353060i	$0.114859\pi$
$619$			17.5680i			0.706120i	−0.935601	+	0.353060i	$0.885141\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			56.6240i			2.26859i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			10.9193i			0.435380i
$630$		0			0
$631$		−	7.51233i		−	0.299061i	−0.988757	−	0.149531i	$-0.952224\pi$
$631$		−	7.51233i		−	0.299061i	0.988757	−	0.149531i	$-0.0477762\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.00919			0.0797321
$636$		0			0
$637$	−23.5491			−0.933049
$638$		0			0
$639$		0			0
$640$		0			0
$641$	48.7137			1.92408			0.962038	−	0.272914i	$-0.0879876\pi$
$641$	48.7137			1.92408			0.962038	+	0.272914i	$0.0879876\pi$
$642$		0			0
$643$			19.3889i			0.764624i	0.924033	+	0.382312i	$0.124872\pi$
$643$			19.3889i			0.764624i	−0.924033	+	0.382312i	$0.875128\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			37.9787i			1.49310i	0.665331	+	0.746549i	$0.268290\pi$
$647$			37.9787i			1.49310i	−0.665331	+	0.746549i	$0.731710\pi$
$648$		0			0
$649$			26.5258i			1.04123i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			17.7483i			0.694544i	0.937764	+	0.347272i	$0.112892\pi$
$653$			17.7483i			0.694544i	−0.937764	+	0.347272i	$0.887108\pi$
$654$		0			0
$655$			21.2278i			0.829437i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	43.2114			1.68328			0.841639	−	0.540040i	$-0.181591\pi$
$659$	43.2114			1.68328			0.841639	+	0.540040i	$0.181591\pi$
$660$		0			0
$661$			1.62903i			0.0633618i	0.999498	+	0.0316809i	$0.0100860\pi$
$661$			1.62903i			0.0633618i	−0.999498	+	0.0316809i	$0.989914\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.72493			−0.183225
$666$		0			0
$667$	−30.2783	+	30.4201i	−1.17238	+	1.17787i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			44.3291i			1.71130i
$672$		0			0
$673$	44.4045			1.71167			0.855834	−	0.517250i	$-0.173044\pi$
$673$	44.4045			1.71167			0.855834	+	0.517250i	$0.173044\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.05117			−0.0788327			−0.0394164	−	0.999223i	$-0.512550\pi$
$677$	−2.05117			−0.0788327			−0.0394164	+	0.999223i	$0.512550\pi$
$678$		0			0
$679$	4.65075			0.178479
$680$		0			0
$681$		0			0
$682$		0			0
$683$			29.1374i			1.11491i	0.830207	+	0.557455i	$0.188222\pi$
$683$			29.1374i			1.11491i	−0.830207	+	0.557455i	$0.811778\pi$
$684$		0			0
$685$	−8.76378			−0.334847
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.5664			−0.669225
$690$		0			0
$691$	−1.68232			−0.0639986			−0.0319993	−	0.999488i	$-0.510187\pi$
$691$	−1.68232			−0.0639986			−0.0319993	+	0.999488i	$0.510187\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−13.9098			−0.527630
$696$		0			0
$697$			5.00124i			0.189436i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−5.33136			−0.201363			−0.100681	−	0.994919i	$-0.532102\pi$
$701$	−5.33136			−0.201363			−0.100681	+	0.994919i	$0.532102\pi$
$702$		0			0
$703$	7.37187			0.278035
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−37.0244			−1.39245
$708$		0			0
$709$			6.63680i			0.249250i	0.992204	+	0.124625i	$0.0397728\pi$
$709$			6.63680i			0.249250i	−0.992204	+	0.124625i	$0.960227\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.7109	+	11.6563i	0.438575	+	0.436531i
$714$		0			0
$715$	14.5390			0.543726
$716$		0			0
$717$		0			0
$718$		0			0
$719$			15.2248i			0.567788i	0.958856	+	0.283894i	$0.0916263\pi$
$719$			15.2248i			0.567788i	−0.958856	+	0.283894i	$0.908374\pi$
$720$		0			0
$721$	31.4810			1.17241
$722$		0			0
$723$		0			0
$724$		0			0
$725$			8.94952i			0.332377i
$726$		0			0
$727$			25.6688i			0.952003i	0.879445	+	0.476001i	$0.157914\pi$
$727$			25.6688i			0.952003i	−0.879445	+	0.476001i	$0.842086\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.32846i		−	0.234067i
$732$		0			0
$733$			1.79515i			0.0663054i	0.999450	+	0.0331527i	$0.0105548\pi$
$733$			1.79515i			0.0663054i	−0.999450	+	0.0331527i	$0.989445\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	37.9023i		−	1.39615i
$738$		0			0
$739$	18.2577			0.671621			0.335810	−	0.941930i	$-0.390990\pi$
$739$	18.2577			0.671621			0.335810	+	0.941930i	$0.390990\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11.5447			0.423536			0.211768	−	0.977320i	$-0.432078\pi$
$743$	11.5447			0.423536			0.211768	+	0.977320i	$0.432078\pi$
$744$		0			0
$745$	−13.2285			−0.484656
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	69.8623i		−	2.55271i
$750$		0			0
$751$			20.3392i			0.742187i	0.928596	+	0.371093i	$0.121017\pi$
$751$			20.3392i			0.742187i	−0.928596	+	0.371093i	$0.878983\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−14.0438			−0.511106
$756$		0			0
$757$		−	21.9739i		−	0.798656i	−0.916808	−	0.399328i	$-0.869243\pi$
$757$		−	21.9739i		−	0.798656i	0.916808	−	0.399328i	$-0.130757\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.16438i			0.223458i	0.993739	+	0.111729i	$0.0356389\pi$
$761$			6.16438i			0.223458i	−0.993739	+	0.111729i	$0.964361\pi$
$762$		0			0
$763$	18.8746			0.683307
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	17.1700i		−	0.619971i
$768$		0			0
$769$		−	23.0308i		−	0.830512i	−0.909704	−	0.415256i	$-0.863692\pi$
$769$		−	23.0308i		−	0.830512i	0.909704	−	0.415256i	$-0.136308\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.1719			0.869404			0.434702	−	0.900574i	$-0.356854\pi$
$773$	24.1719			0.869404			0.434702	+	0.900574i	$0.356854\pi$
$774$		0			0
$775$	3.44531			0.123759
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.37647			0.120974
$780$		0			0
$781$		−	62.7698i		−	2.24608i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			17.5040i			0.624743i
$786$		0			0
$787$		−	23.3110i		−	0.830947i	−0.909605	−	0.415473i	$-0.863616\pi$
$787$		−	23.3110i		−	0.830947i	0.909605	−	0.415473i	$-0.136384\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			23.8078i			0.846507i
$792$		0			0
$793$		−	28.6939i		−	1.01895i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−48.6834			−1.72445			−0.862227	−	0.506523i	$-0.830931\pi$
$797$	−48.6834			−1.72445			−0.862227	+	0.506523i	$0.830931\pi$
$798$		0			0
$799$		−	9.81354i		−	0.347178i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.5876			−0.620654
$804$		0			0
$805$	−12.9611	+	13.0218i	−0.456818	+	0.458958i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	19.6292i		−	0.690125i	−0.938580	−	0.345063i	$-0.887858\pi$
$809$		−	19.6292i		−	0.690125i	0.938580	−	0.345063i	$-0.112142\pi$
$810$		0			0
$811$	26.8726			0.943625			0.471812	−	0.881699i	$-0.343600\pi$
$811$	26.8726			0.943625			0.471812	+	0.881699i	$0.343600\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.34521			−0.152206
$816$		0			0
$817$	−4.27250			−0.149476
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.6760i		−	0.721596i	−0.932644	−	0.360798i	$-0.882504\pi$
$821$		−	20.6760i		−	0.721596i	0.932644	−	0.360798i	$-0.117496\pi$
$822$		0			0
$823$	−26.1344			−0.910989			−0.455495	−	0.890238i	$-0.650538\pi$
$823$	−26.1344			−0.910989			−0.455495	+	0.890238i	$0.650538\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.99963			−0.0695339			−0.0347670	−	0.999395i	$-0.511069\pi$
$827$	−1.99963			−0.0695339			−0.0347670	+	0.999395i	$0.511069\pi$
$828$		0			0
$829$	48.8361			1.69615			0.848073	−	0.529879i	$-0.177763\pi$
$829$	48.8361			1.69615			0.848073	+	0.529879i	$0.177763\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.0236			0.485888
$834$		0			0
$835$		−	7.47785i		−	0.258782i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	26.7135			0.922251			0.461126	−	0.887335i	$-0.347446\pi$
$839$	26.7135			0.922251			0.461126	+	0.887335i	$0.347446\pi$
$840$		0			0
$841$	−51.0938			−1.76186
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.58903			0.123466
$846$		0			0
$847$		−	43.9075i		−	1.50868i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20.2220	−	20.3167i	0.693201	−	0.696448i
$852$		0			0
$853$	−29.8267			−1.02125			−0.510624	−	0.859804i	$-0.670586\pi$
$853$	−29.8267			−1.02125			−0.510624	+	0.859804i	$0.670586\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			48.3247i			1.65074i	0.564592	+	0.825370i	$0.309034\pi$
$857$			48.3247i			1.65074i	−0.564592	+	0.825370i	$0.690966\pi$
$858$		0			0
$859$	−16.2133			−0.553190			−0.276595	−	0.960987i	$-0.589206\pi$
$859$	−16.2133			−0.553190			−0.276595	+	0.960987i	$0.589206\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	20.8023i		−	0.708120i	−0.935223	−	0.354060i	$-0.884801\pi$
$863$		−	20.8023i		−	0.708120i	0.935223	−	0.354060i	$-0.115199\pi$
$864$		0			0
$865$			17.5785i			0.597687i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	52.3133i		−	1.77461i
$870$		0			0
$871$			24.5339i			0.831298i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			3.83098i			0.129511i
$876$		0			0
$877$	−34.3919			−1.16133			−0.580666	−	0.814142i	$-0.697208\pi$
$877$	−34.3919			−1.16133			−0.580666	+	0.814142i	$0.697208\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.35042			−0.146569			−0.0732847	−	0.997311i	$-0.523348\pi$
$881$	−4.35042			−0.146569			−0.0732847	+	0.997311i	$0.523348\pi$
$882$		0			0
$883$	38.5042			1.29577			0.647885	−	0.761738i	$-0.275654\pi$
$883$	38.5042			1.29577			0.647885	+	0.761738i	$0.275654\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	56.2517i		−	1.88875i	−0.328876	−	0.944373i	$-0.606670\pi$
$887$		−	56.2517i		−	1.88875i	0.328876	−	0.944373i	$-0.393330\pi$
$888$		0			0
$889$			7.69715i			0.258154i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.62537			−0.221709
$894$		0			0
$895$			12.1037i			0.404581i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			30.8338i			1.02837i
$900$		0			0
$901$	10.4608			0.348501
$902$		0			0
$903$		0			0
$904$		0			0
$905$			11.0653i			0.367823i
$906$		0			0
$907$			11.9088i			0.395424i	0.980260	+	0.197712i	$0.0633511\pi$
$907$			11.9088i			0.395424i	−0.980260	+	0.197712i	$0.936649\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	19.5796			0.648700			0.324350	−	0.945937i	$-0.394854\pi$
$911$	19.5796			0.648700			0.324350	+	0.945937i	$0.394854\pi$
$912$		0			0
$913$	−60.2024			−1.99241
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−81.3231			−2.68553
$918$		0			0
$919$		−	28.0924i		−	0.926682i	−0.886180	−	0.463341i	$-0.846651\pi$
$919$		−	28.0924i		−	0.926682i	0.886180	−	0.463341i	$-0.153349\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			40.6305i			1.33737i
$924$		0			0
$925$		−	5.97712i		−	0.196527i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			8.62045i			0.282828i	0.989951	+	0.141414i	$0.0451648\pi$
$929$			8.62045i			0.282828i	−0.989951	+	0.141414i	$0.954835\pi$
$930$		0			0
$931$		−	9.46766i		−	0.310290i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.65800			−0.283147
$936$		0			0
$937$		−	12.2226i		−	0.399294i	−0.979868	−	0.199647i	$-0.936020\pi$
$937$		−	12.2226i		−	0.399294i	0.979868	−	0.199647i	$-0.0639796\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	40.6606			1.32550			0.662749	−	0.748842i	$-0.269390\pi$
$941$	40.6606			1.32550			0.662749	+	0.748842i	$0.269390\pi$
$942$		0			0
$943$	9.26208	−	9.30546i	0.301615	−	0.303027i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			41.6798i			1.35441i	0.735793	+	0.677207i	$0.236810\pi$
$947$			41.6798i			1.35441i	−0.735793	+	0.677207i	$0.763190\pi$
$948$		0			0
$949$	11.3844			0.369552
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.2913			−0.398155			−0.199077	−	0.979984i	$-0.563795\pi$
$953$	−12.2913			−0.398155			−0.199077	+	0.979984i	$0.563795\pi$
$954$		0			0
$955$	1.34765			0.0436090
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	33.5739i		−	1.08416i
$960$		0			0
$961$	−19.1299			−0.617092
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.0234			−0.322666
$966$		0			0
$967$	−3.21428			−0.103364			−0.0516821	−	0.998664i	$-0.516458\pi$
$967$	−3.21428			−0.103364			−0.0516821	+	0.998664i	$0.516458\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	58.6779			1.88306			0.941531	−	0.336926i	$-0.109387\pi$
$971$	58.6779			1.88306			0.941531	+	0.336926i	$0.109387\pi$
$972$		0			0
$973$		−	53.2882i		−	1.70834i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.5380			−0.689061			−0.344531	−	0.938775i	$-0.611962\pi$
$977$	−21.5380			−0.689061			−0.344531	+	0.938775i	$0.611962\pi$
$978$		0			0
$979$	70.0499			2.23880
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.29237			−0.200695			−0.100348	−	0.994952i	$-0.531996\pi$
$983$	−6.29237			−0.200695			−0.100348	+	0.994952i	$0.531996\pi$
$984$		0			0
$985$		−	10.4221i		−	0.332075i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.7200	+	11.7749i	−0.372675	+	0.374421i
$990$		0			0
$991$	−13.1503			−0.417733			−0.208866	−	0.977944i	$-0.566977\pi$
$991$	−13.1503			−0.417733			−0.208866	+	0.977944i	$0.566977\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			13.6291i			0.432070i
$996$		0			0
$997$	17.7492			0.562123			0.281062	−	0.959690i	$-0.409313\pi$
$997$	17.7492			0.562123			0.281062	+	0.959690i	$0.409313\pi$
$998$		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.1241.4	✓	48
3.2	odd	2		8280.2.p.b.1241.4	yes	48
23.22	odd	2		8280.2.p.b.1241.45	yes	48
69.68	even	2	inner	8280.2.p.a.1241.45	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.4	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.45	yes	48	69.68	even	2	inner
8280.2.p.b.1241.4	yes	48	3.2	odd	2
8280.2.p.b.1241.45	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.83098i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.83098i q^{7} -4.73932 q^{11} +3.06773 q^{13} -1.82684 q^{17} +1.23335i q^{19} +(3.39908 + 3.38323i) q^{23} +1.00000 q^{25} +8.94952i q^{29} +3.44531 q^{31} +3.83098i q^{35} -5.97712i q^{37} -2.73764i q^{41} +3.46415i q^{43} +5.37185i q^{47} -7.67639 q^{49} -5.72617 q^{53} +4.73932 q^{55} -5.59696i q^{59} -9.35346i q^{61} -3.06773 q^{65} +7.99740i q^{67} +13.2445i q^{71} +3.71100 q^{73} +18.1562i q^{77} +11.0381i q^{79} +12.7027 q^{83} +1.82684 q^{85} -14.7806 q^{89} -11.7524i q^{91} -1.23335i q^{95} +1.21398i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 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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