LMFDB - Embedded newform 2640.2.d.j.529.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(529,2640)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2640.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.d (of order $2$, degree $1$, not 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:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 49x^{6} - 8x^{5} + 72x^{3} + 256x^{2} + 128x + 32 $ x^10 + 49x^6 - 8x^5 + 72x^3 + 256x^2 + 128*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.10
Root			$-0.874489 - 0.874489i$ of defining polynomial
Character	$\chi$	$=$	2640.529
Dual form			2640.2.d.j.529.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(2.05798 - 0.874489i) q^{5} -1.46193i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(2.05798 - 0.874489i) q^{5} -1.46193i q^{7} -1.00000 q^{9} -1.00000 q^{11} +5.53182i q^{13} +(0.874489 + 2.05798i) q^{15} +8.15198i q^{17} -2.94108 q^{19} +1.46193 q^{21} -2.87779i q^{23} +(3.47054 - 3.59936i) q^{25} -1.00000i q^{27} -5.70292 q^{29} -5.49579 q^{31} -1.00000i q^{33} +(-1.27844 - 3.00861i) q^{35} +7.86493i q^{37} -5.53182 q^{39} -0.761840 q^{41} +0.841766i q^{43} +(-2.05798 + 0.874489i) q^{45} +12.1858i q^{47} +4.86277 q^{49} -8.15198 q^{51} -8.00661i q^{53} +(-2.05798 + 0.874489i) q^{55} -2.94108i q^{57} +6.57194 q^{59} -8.37575 q^{61} +1.46193i q^{63} +(4.83752 + 11.3844i) q^{65} +1.44312i q^{67} +2.87779 q^{69} -10.1880 q^{71} +5.07584i q^{73} +(3.59936 + 3.47054i) q^{75} +1.46193i q^{77} +13.8189 q^{79} +1.00000 q^{81} +6.51896i q^{83} +(7.12882 + 16.7766i) q^{85} -5.70292i q^{87} +14.4917 q^{89} +8.08712 q^{91} -5.49579i q^{93} +(-6.05267 + 2.57194i) q^{95} -0.207125i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^5 - 10 * q^9	$ 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 12 q^{19} + 4 q^{21} + 14 q^{25} - 24 q^{29} - 8 q^{31} - 16 q^{35} + 16 q^{39} - 16 q^{41} + 2 q^{45} - 26 q^{49} - 12 q^{51} + 2 q^{55} + 40 q^{59} - 12 q^{61} - 8 q^{69} - 8 q^{71} - 4 q^{75} + 60 q^{79} + 10 q^{81} + 52 q^{85} + 28 q^{89} - 20 q^{95} + 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 12 * q^19 + 4 * q^21 + 14 * q^25 - 24 * q^29 - 8 * q^31 - 16 * q^35 + 16 * q^39 - 16 * q^41 + 2 * q^45 - 26 * q^49 - 12 * q^51 + 2 * q^55 + 40 * q^59 - 12 * q^61 - 8 * q^69 - 8 * q^71 - 4 * q^75 + 60 * q^79 + 10 * q^81 + 52 * q^85 + 28 * q^89 - 20 * q^95 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	2.05798	−	0.874489i	0.920355	−	0.391083i
$5$	2.05798	−	0.874489i	0.920355	−	0.391083i
$6$		0			0
$7$		−	1.46193i		−	0.552556i	−0.961078	−	0.276278i	$-0.910899\pi$
$7$		−	1.46193i		−	0.552556i	0.961078	−	0.276278i	$-0.0891011\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			5.53182i			1.53425i	0.641497	+	0.767126i	$0.278314\pi$
$13$			5.53182i			1.53425i	−0.641497	+	0.767126i	$0.721686\pi$
$14$		0			0
$15$	0.874489	+	2.05798i	0.225792	+	0.531367i
$16$		0			0
$17$			8.15198i			1.97715i	0.150743	+	0.988573i	$0.451834\pi$
$17$			8.15198i			1.97715i	−0.150743	+	0.988573i	$0.548166\pi$
$18$		0			0
$19$	−2.94108			−0.674730			−0.337365	−	0.941374i	$-0.609536\pi$
$19$	−2.94108			−0.674730			−0.337365	+	0.941374i	$0.609536\pi$
$20$		0			0
$21$	1.46193			0.319019
$22$		0			0
$23$		−	2.87779i		−	0.600062i	−0.953930	−	0.300031i	$-0.903003\pi$
$23$		−	2.87779i		−	0.600062i	0.953930	−	0.300031i	$-0.0969970\pi$
$24$		0			0
$25$	3.47054	−	3.59936i	0.694108	−	0.719871i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−5.70292			−1.05901			−0.529503	−	0.848308i	$-0.677621\pi$
$29$	−5.70292			−1.05901			−0.529503	+	0.848308i	$0.677621\pi$
$30$		0			0
$31$	−5.49579			−0.987074			−0.493537	−	0.869725i	$-0.664296\pi$
$31$	−5.49579			−0.987074			−0.493537	+	0.869725i	$0.664296\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−1.27844	−	3.00861i	−0.216096	−	0.508548i
$36$		0			0
$37$			7.86493i			1.29299i	0.762920	+	0.646493i	$0.223765\pi$
$37$			7.86493i			1.29299i	−0.762920	+	0.646493i	$0.776235\pi$
$38$		0			0
$39$	−5.53182			−0.885800
$40$		0			0
$41$	−0.761840			−0.118979			−0.0594897	−	0.998229i	$-0.518947\pi$
$41$	−0.761840			−0.118979			−0.0594897	+	0.998229i	$0.518947\pi$
$42$		0			0
$43$			0.841766i			0.128368i	0.997938	+	0.0641841i	$0.0204445\pi$
$43$			0.841766i			0.128368i	−0.997938	+	0.0641841i	$0.979556\pi$
$44$		0			0
$45$	−2.05798	+	0.874489i	−0.306785	+	0.130361i
$46$		0			0
$47$			12.1858i			1.77749i	0.458403	+	0.888744i	$0.348422\pi$
$47$			12.1858i			1.77749i	−0.458403	+	0.888744i	$0.651578\pi$
$48$		0			0
$49$	4.86277			0.694681
$50$		0			0
$51$	−8.15198			−1.14151
$52$		0			0
$53$		−	8.00661i		−	1.09979i	−0.835233	−	0.549896i	$-0.814667\pi$
$53$		−	8.00661i		−	1.09979i	0.835233	−	0.549896i	$-0.185333\pi$
$54$		0			0
$55$	−2.05798	+	0.874489i	−0.277498	+	0.117916i
$56$		0			0
$57$		−	2.94108i		−	0.389555i
$58$		0			0
$59$	6.57194			0.855594			0.427797	−	0.903875i	$-0.359290\pi$
$59$	6.57194			0.855594			0.427797	+	0.903875i	$0.359290\pi$
$60$		0			0
$61$	−8.37575			−1.07240			−0.536202	−	0.844089i	$-0.680142\pi$
$61$	−8.37575			−1.07240			−0.536202	+	0.844089i	$0.680142\pi$
$62$		0			0
$63$			1.46193i			0.184185i
$64$		0			0
$65$	4.83752	+	11.3844i	0.600020	+	1.41206i
$66$		0			0
$67$			1.44312i			0.176306i	0.996107	+	0.0881528i	$0.0280964\pi$
$67$			1.44312i			0.176306i	−0.996107	+	0.0881528i	$0.971904\pi$
$68$		0			0
$69$	2.87779			0.346446
$70$		0			0
$71$	−10.1880			−1.20909			−0.604547	−	0.796569i	$-0.706646\pi$
$71$	−10.1880			−1.20909			−0.604547	+	0.796569i	$0.706646\pi$
$72$		0			0
$73$			5.07584i			0.594082i	0.954865	+	0.297041i	$0.0959997\pi$
$73$			5.07584i			0.594082i	−0.954865	+	0.297041i	$0.904000\pi$
$74$		0			0
$75$	3.59936	+	3.47054i	0.415618	+	0.400743i
$76$		0			0
$77$			1.46193i			0.166602i
$78$		0			0
$79$	13.8189			1.55474			0.777372	−	0.629041i	$-0.216552\pi$
$79$	13.8189			1.55474			0.777372	+	0.629041i	$0.216552\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			6.51896i			0.715549i	0.933808	+	0.357774i	$0.116464\pi$
$83$			6.51896i			0.715549i	−0.933808	+	0.357774i	$0.883536\pi$
$84$		0			0
$85$	7.12882	+	16.7766i	0.773229	+	1.81968i
$86$		0			0
$87$		−	5.70292i		−	0.611417i
$88$		0			0
$89$	14.4917			1.53612			0.768059	−	0.640379i	$-0.221223\pi$
$89$	14.4917			1.53612			0.768059	+	0.640379i	$0.221223\pi$
$90$		0			0
$91$	8.08712			0.847760
$92$		0			0
$93$		−	5.49579i		−	0.569887i
$94$		0			0
$95$	−6.05267	+	2.57194i	−0.620991	+	0.263875i
$96$		0			0
$97$		−	0.207125i		−	0.0210303i	−0.999945	−	0.0105152i	$-0.996653\pi$
$97$		−	0.207125i		−	0.0210303i	0.999945	−	0.0105152i	$-0.00334714\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$	17.8427			1.77542			0.887708	−	0.460407i	$-0.152297\pi$
$101$	17.8427			1.77542			0.887708	+	0.460407i	$0.152297\pi$
$102$		0			0
$103$			15.0636i			1.48426i	0.670253	+	0.742132i	$0.266185\pi$
$103$			15.0636i			1.48426i	−0.670253	+	0.742132i	$0.733815\pi$
$104$		0			0
$105$	3.00861	−	1.27844i	0.293610	−	0.124763i
$106$		0			0
$107$			6.24410i			0.603640i	0.953365	+	0.301820i	$0.0975942\pi$
$107$			6.24410i			0.603640i	−0.953365	+	0.301820i	$0.902406\pi$
$108$		0			0
$109$	−5.37984			−0.515295			−0.257648	−	0.966239i	$-0.582947\pi$
$109$	−5.37984			−0.515295			−0.257648	+	0.966239i	$0.582947\pi$
$110$		0			0
$111$	−7.86493			−0.746506
$112$		0			0
$113$		−	0.681283i		−	0.0640897i	−0.999486	−	0.0320448i	$-0.989798\pi$
$113$		−	0.681283i		−	0.0640897i	0.999486	−	0.0320448i	$-0.0102019\pi$
$114$		0			0
$115$	−2.51660	−	5.92243i	−0.234674	−	0.552270i
$116$		0			0
$117$		−	5.53182i		−	0.511417i
$118$		0			0
$119$	11.9176			1.09248
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	0.761840i		−	0.0686928i
$124$		0			0
$125$	3.99469	−	10.4423i	0.357296	−	0.933991i
$126$		0			0
$127$		−	16.6521i		−	1.47764i	−0.673904	−	0.738819i	$-0.735384\pi$
$127$		−	16.6521i		−	1.47764i	0.673904	−	0.738819i	$-0.264616\pi$
$128$		0			0
$129$	−0.841766			−0.0741134
$130$		0			0
$131$	−5.44312			−0.475568			−0.237784	−	0.971318i	$-0.576421\pi$
$131$	−5.44312			−0.475568			−0.237784	+	0.971318i	$0.576421\pi$
$132$		0			0
$133$			4.29964i			0.372826i
$134$		0			0
$135$	−0.874489	−	2.05798i	−0.0752640	−	0.177122i
$136$		0			0
$137$			8.12473i			0.694142i	0.937839	+	0.347071i	$0.112824\pi$
$137$			8.12473i			0.694142i	−0.937839	+	0.347071i	$0.887176\pi$
$138$		0			0
$139$	−16.9286			−1.43586			−0.717931	−	0.696114i	$-0.754911\pi$
$139$	−16.9286			−1.43586			−0.717931	+	0.696114i	$0.754911\pi$
$140$		0			0
$141$	−12.1858			−1.02623
$142$		0			0
$143$		−	5.53182i		−	0.462594i
$144$		0			0
$145$	−11.7365	+	4.98714i	−0.974661	+	0.414159i
$146$		0			0
$147$			4.86277i			0.401074i
$148$		0			0
$149$	−13.2009			−1.08146			−0.540729	−	0.841197i	$-0.681851\pi$
$149$	−13.2009			−1.08146			−0.540729	+	0.841197i	$0.681851\pi$
$150$		0			0
$151$	−13.8189			−1.12456			−0.562282	−	0.826946i	$-0.690076\pi$
$151$	−13.8189			−1.12456			−0.562282	+	0.826946i	$0.690076\pi$
$152$		0			0
$153$		−	8.15198i		−	0.659049i
$154$		0			0
$155$	−11.3102	+	4.80601i	−0.908458	+	0.386028i
$156$		0			0
$157$			2.51927i			0.201060i	0.994934	+	0.100530i	$0.0320538\pi$
$157$			2.51927i			0.201060i	−0.994934	+	0.100530i	$0.967946\pi$
$158$		0			0
$159$	8.00661			0.634966
$160$		0			0
$161$	−4.20712			−0.331568
$162$		0			0
$163$		−	7.42590i		−	0.581641i	−0.956778	−	0.290821i	$-0.906072\pi$
$163$		−	7.42590i		−	0.581641i	0.956778	−	0.290821i	$-0.0939283\pi$
$164$		0			0
$165$	−0.874489	−	2.05798i	−0.0680789	−	0.160213i
$166$		0			0
$167$			13.2830i			1.02787i	0.857830	+	0.513933i	$0.171812\pi$
$167$			13.2830i			1.02787i	−0.857830	+	0.513933i	$0.828188\pi$
$168$		0			0
$169$	−17.6010			−1.35393
$170$		0			0
$171$	2.94108			0.224910
$172$		0			0
$173$		−	3.71295i		−	0.282290i	−0.989989	−	0.141145i	$-0.954922\pi$
$173$		−	3.71295i		−	0.282290i	0.989989	−	0.141145i	$-0.0450784\pi$
$174$		0			0
$175$	−5.26199	−	5.07367i	−0.397769	−	0.383534i
$176$		0			0
$177$			6.57194i			0.493977i
$178$		0			0
$179$	−3.63774			−0.271898			−0.135949	−	0.990716i	$-0.543408\pi$
$179$	−3.63774			−0.271898			−0.135949	+	0.990716i	$0.543408\pi$
$180$		0			0
$181$	0.571939			0.0425119			0.0212560	−	0.999774i	$-0.493234\pi$
$181$	0.571939			0.0425119			0.0212560	+	0.999774i	$0.493234\pi$
$182$		0			0
$183$		−	8.37575i		−	0.619153i
$184$		0			0
$185$	6.87779	+	16.1858i	0.505665	+	1.19001i
$186$		0			0
$187$		−	8.15198i		−	0.596132i
$188$		0			0
$189$	−1.46193			−0.106340
$190$		0			0
$191$	1.35411			0.0979802			0.0489901	−	0.998799i	$-0.484400\pi$
$191$	1.35411			0.0979802			0.0489901	+	0.998799i	$0.484400\pi$
$192$		0			0
$193$			4.05420i			0.291828i	0.989297	+	0.145914i	$0.0466122\pi$
$193$			4.05420i			0.291828i	−0.989297	+	0.145914i	$0.953388\pi$
$194$		0			0
$195$	−11.3844	+	4.83752i	−0.815251	+	0.346422i
$196$		0			0
$197$		−	4.28705i		−	0.305440i	−0.988270	−	0.152720i	$-0.951197\pi$
$197$		−	4.28705i		−	0.305440i	0.988270	−	0.152720i	$-0.0488032\pi$
$198$		0			0
$199$	15.2278			1.07947			0.539736	−	0.841835i	$-0.318524\pi$
$199$	15.2278			1.07947			0.539736	+	0.841835i	$0.318524\pi$
$200$		0			0
$201$	−1.44312			−0.101790
$202$		0			0
$203$			8.33725i			0.585160i
$204$		0			0
$205$	−1.56785	+	0.666221i	−0.109503	+	0.0465309i
$206$		0			0
$207$			2.87779i			0.200021i
$208$		0			0
$209$	2.94108			0.203439
$210$		0			0
$211$	11.6883			0.804652			0.402326	−	0.915496i	$-0.368202\pi$
$211$	11.6883			0.804652			0.402326	+	0.915496i	$0.368202\pi$
$212$		0			0
$213$		−	10.1880i		−	0.698071i
$214$		0			0
$215$	0.736115	+	1.73234i	0.0502026	+	0.118144i
$216$		0			0
$217$			8.03445i			0.545414i
$218$		0			0
$219$	−5.07584			−0.342993
$220$		0			0
$221$	−45.0953			−3.03344
$222$		0			0
$223$		−	23.2956i		−	1.55999i	−0.625789	−	0.779993i	$-0.715223\pi$
$223$		−	23.2956i		−	1.55999i	0.625789	−	0.779993i	$-0.284777\pi$
$224$		0			0
$225$	−3.47054	+	3.59936i	−0.231369	+	0.239957i
$226$		0			0
$227$			28.0674i			1.86290i	0.363869	+	0.931450i	$0.381456\pi$
$227$			28.0674i			1.86290i	−0.363869	+	0.931450i	$0.618544\pi$
$228$		0			0
$229$	−23.7041			−1.56641			−0.783207	−	0.621761i	$-0.786417\pi$
$229$	−23.7041			−1.56641			−0.783207	+	0.621761i	$0.786417\pi$
$230$		0			0
$231$	−1.46193			−0.0961877
$232$		0			0
$233$		−	15.7850i		−	1.03411i	−0.855952	−	0.517055i	$-0.827028\pi$
$233$		−	15.7850i		−	1.03411i	0.855952	−	0.517055i	$-0.172972\pi$
$234$		0			0
$235$	10.6564	+	25.0782i	0.695146	+	1.63592i
$236$		0			0
$237$			13.8189i			0.897632i
$238$		0			0
$239$	23.8564			1.54314			0.771572	−	0.636142i	$-0.219471\pi$
$239$	23.8564			1.54314			0.771572	+	0.636142i	$0.219471\pi$
$240$		0			0
$241$	6.60739			0.425619			0.212810	−	0.977094i	$-0.431739\pi$
$241$	6.60739			0.425619			0.212810	+	0.977094i	$0.431739\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	10.0075	−	4.25244i	0.639354	−	0.271678i
$246$		0			0
$247$		−	16.2695i		−	1.03520i
$248$		0			0
$249$	−6.51896			−0.413122
$250$		0			0
$251$	30.7656			1.94191			0.970954	−	0.239265i	$-0.0769065\pi$
$251$	30.7656			1.94191			0.970954	+	0.239265i	$0.0769065\pi$
$252$		0			0
$253$			2.87779i			0.180925i
$254$		0			0
$255$	−16.7766	+	7.12882i	−1.05059	+	0.446424i
$256$		0			0
$257$		−	7.72864i		−	0.482100i	−0.970513	−	0.241050i	$-0.922508\pi$
$257$		−	7.72864i		−	0.482100i	0.970513	−	0.241050i	$-0.0774917\pi$
$258$		0			0
$259$	11.4980			0.714448
$260$		0			0
$261$	5.70292			0.353002
$262$		0			0
$263$		−	5.80507i		−	0.357956i	−0.983853	−	0.178978i	$-0.942721\pi$
$263$		−	5.80507i		−	0.357956i	0.983853	−	0.178978i	$-0.0572791\pi$
$264$		0			0
$265$	−7.00169	−	16.4774i	−0.430110	−	1.01220i
$266$		0			0
$267$			14.4917i			0.886878i
$268$		0			0
$269$	19.6036			1.19525			0.597625	−	0.801776i	$-0.296111\pi$
$269$	19.6036			1.19525			0.597625	+	0.801776i	$0.296111\pi$
$270$		0			0
$271$	−17.0291			−1.03445			−0.517223	−	0.855850i	$-0.673034\pi$
$271$	−17.0291			−1.03445			−0.517223	+	0.855850i	$0.673034\pi$
$272$		0			0
$273$			8.08712i			0.489455i
$274$		0			0
$275$	−3.47054	+	3.59936i	−0.209281	+	0.217049i
$276$		0			0
$277$		−	2.68002i		−	0.161027i	−0.996754	−	0.0805135i	$-0.974344\pi$
$277$		−	2.68002i		−	0.161027i	0.996754	−	0.0805135i	$-0.0256560\pi$
$278$		0			0
$279$	5.49579			0.329025
$280$		0			0
$281$	29.1626			1.73970			0.869849	−	0.493319i	$-0.164216\pi$
$281$	29.1626			1.73970			0.869849	+	0.493319i	$0.164216\pi$
$282$		0			0
$283$			14.3484i			0.852926i	0.904505	+	0.426463i	$0.140241\pi$
$283$			14.3484i			0.852926i	−0.904505	+	0.426463i	$0.859759\pi$
$284$		0			0
$285$	−2.57194	−	6.05267i	−0.152349	−	0.358529i
$286$		0			0
$287$			1.11375i			0.0657428i
$288$		0			0
$289$	−49.4548			−2.90911
$290$		0			0
$291$	0.207125			0.0121419
$292$		0			0
$293$			31.8145i			1.85862i	0.369296	+	0.929312i	$0.379599\pi$
$293$			31.8145i			1.85862i	−0.369296	+	0.929312i	$0.620401\pi$
$294$		0			0
$295$	13.5249	−	5.74709i	0.787450	−	0.334608i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	15.9194			0.920645
$300$		0			0
$301$	1.23060			0.0709306
$302$		0			0
$303$			17.8427i			1.02504i
$304$		0			0
$305$	−17.2371	+	7.32450i	−0.986993	+	0.419400i
$306$		0			0
$307$			16.5383i			0.943893i	0.881627	+	0.471947i	$0.156448\pi$
$307$			16.5383i			0.943893i	−0.881627	+	0.471947i	$0.843552\pi$
$308$		0			0
$309$	−15.0636			−0.856941
$310$		0			0
$311$	0.0357178			0.00202537			0.00101269	−	0.999999i	$-0.499678\pi$
$311$	0.0357178			0.00202537			0.00101269	+	0.999999i	$0.499678\pi$
$312$		0			0
$313$		−	20.7139i		−	1.17082i	−0.810738	−	0.585409i	$-0.800934\pi$
$313$		−	20.7139i		−	1.17082i	0.810738	−	0.585409i	$-0.199066\pi$
$314$		0			0
$315$	1.27844	+	3.00861i	0.0720319	+	0.169516i
$316$		0			0
$317$		−	20.5293i		−	1.15304i	−0.817084	−	0.576519i	$-0.804411\pi$
$317$		−	20.5293i		−	1.15304i	0.817084	−	0.576519i	$-0.195589\pi$
$318$		0			0
$319$	5.70292			0.319302
$320$		0			0
$321$	−6.24410			−0.348512
$322$		0			0
$323$		−	23.9756i		−	1.33404i
$324$		0			0
$325$	19.9110	+	19.1984i	1.10446	+	1.06494i
$326$		0			0
$327$		−	5.37984i		−	0.297506i
$328$		0			0
$329$	17.8148			0.982163
$330$		0			0
$331$	−29.4805			−1.62039			−0.810196	−	0.586159i	$-0.800639\pi$
$331$	−29.4805			−1.62039			−0.810196	+	0.586159i	$0.800639\pi$
$332$		0			0
$333$		−	7.86493i		−	0.430996i
$334$		0			0
$335$	1.26199	+	2.96991i	0.0689501	+	0.162264i
$336$		0			0
$337$			6.65403i			0.362468i	0.983440	+	0.181234i	$0.0580091\pi$
$337$			6.65403i			0.362468i	−0.983440	+	0.181234i	$0.941991\pi$
$338$		0			0
$339$	0.681283			0.0370022
$340$		0			0
$341$	5.49579			0.297614
$342$		0			0
$343$		−	17.3425i		−	0.936407i
$344$		0			0
$345$	5.92243	−	2.51660i	0.318853	−	0.135489i
$346$		0			0
$347$			21.2109i			1.13866i	0.822109	+	0.569331i	$0.192798\pi$
$347$			21.2109i			1.13866i	−0.822109	+	0.569331i	$0.807202\pi$
$348$		0			0
$349$	−7.72959			−0.413756			−0.206878	−	0.978367i	$-0.566330\pi$
$349$	−7.72959			−0.413756			−0.206878	+	0.978367i	$0.566330\pi$
$350$		0			0
$351$	5.53182			0.295267
$352$		0			0
$353$		−	12.9508i		−	0.689301i	−0.938731	−	0.344651i	$-0.887997\pi$
$353$		−	12.9508i		−	0.689301i	0.938731	−	0.344651i	$-0.112003\pi$
$354$		0			0
$355$	−20.9667	+	8.90930i	−1.11280	+	0.472857i
$356$		0			0
$357$			11.9176i			0.630746i
$358$		0			0
$359$	0.178018			0.00939541			0.00469770	−	0.999989i	$-0.498505\pi$
$359$	0.178018			0.00939541			0.00469770	+	0.999989i	$0.498505\pi$
$360$		0			0
$361$	−10.3501			−0.544740
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	4.43876	+	10.4460i	0.232335	+	0.546766i
$366$		0			0
$367$		−	13.6835i		−	0.714275i	−0.934052	−	0.357137i	$-0.883753\pi$
$367$		−	13.6835i		−	0.714275i	0.934052	−	0.357137i	$-0.116247\pi$
$368$		0			0
$369$	0.761840			0.0396598
$370$		0			0
$371$	−11.7051			−0.607697
$372$		0			0
$373$		−	17.1176i		−	0.886314i	−0.896444	−	0.443157i	$-0.853858\pi$
$373$		−	17.1176i		−	0.886314i	0.896444	−	0.443157i	$-0.146142\pi$
$374$		0			0
$375$	10.4423	+	3.99469i	0.539240	+	0.206285i
$376$		0			0
$377$		−	31.5475i		−	1.62478i
$378$		0			0
$379$	−12.4390			−0.638950			−0.319475	−	0.947595i	$-0.603507\pi$
$379$	−12.4390			−0.638950			−0.319475	+	0.947595i	$0.603507\pi$
$380$		0			0
$381$	16.6521			0.853115
$382$		0			0
$383$		−	23.7716i		−	1.21467i	−0.794445	−	0.607336i	$-0.792238\pi$
$383$		−	23.7716i		−	1.21467i	0.794445	−	0.607336i	$-0.207762\pi$
$384$		0			0
$385$	1.27844	+	3.00861i	0.0651553	+	0.153333i
$386$		0			0
$387$		−	0.841766i		−	0.0427894i
$388$		0			0
$389$	−34.0799			−1.72792			−0.863959	−	0.503561i	$-0.832023\pi$
$389$	−34.0799			−1.72792			−0.863959	+	0.503561i	$0.832023\pi$
$390$		0			0
$391$	23.4597			1.18641
$392$		0			0
$393$		−	5.44312i		−	0.274569i
$394$		0			0
$395$	28.4389	−	12.0844i	1.43092	−	0.608035i
$396$		0			0
$397$		−	1.04642i		−	0.0525182i	−0.999655	−	0.0262591i	$-0.991641\pi$
$397$		−	1.04642i		−	0.0525182i	0.999655	−	0.0262591i	$-0.00835950\pi$
$398$		0			0
$399$	−4.29964			−0.215251
$400$		0			0
$401$	27.0476			1.35069			0.675347	−	0.737500i	$-0.263994\pi$
$401$	27.0476			1.35069			0.675347	+	0.737500i	$0.263994\pi$
$402$		0			0
$403$		−	30.4017i		−	1.51442i
$404$		0			0
$405$	2.05798	−	0.874489i	0.102262	−	0.0434537i
$406$		0			0
$407$		−	7.86493i		−	0.389850i
$408$		0			0
$409$	−14.0921			−0.696810			−0.348405	−	0.937344i	$-0.613277\pi$
$409$	−14.0921			−0.696810			−0.348405	+	0.937344i	$0.613277\pi$
$410$		0			0
$411$	−8.12473			−0.400763
$412$		0			0
$413$		−	9.60769i		−	0.472764i
$414$		0			0
$415$	5.70076	+	13.4159i	0.279839	+	0.658559i
$416$		0			0
$417$		−	16.9286i		−	0.828996i
$418$		0			0
$419$	16.3372			0.798127			0.399063	−	0.916923i	$-0.369335\pi$
$419$	16.3372			0.798127			0.399063	+	0.916923i	$0.369335\pi$
$420$		0			0
$421$	27.2056			1.32592			0.662960	−	0.748655i	$-0.269300\pi$
$421$	27.2056			1.32592			0.662960	+	0.748655i	$0.269300\pi$
$422$		0			0
$423$		−	12.1858i		−	0.592496i
$424$		0			0
$425$	29.3419	+	28.2918i	1.42329	+	1.37235i
$426$		0			0
$427$			12.2447i			0.592564i
$428$		0			0
$429$	5.53182			0.267079
$430$		0			0
$431$	36.0094			1.73451			0.867257	−	0.497861i	$-0.165881\pi$
$431$	36.0094			1.73451			0.867257	+	0.497861i	$0.165881\pi$
$432$		0			0
$433$			8.40993i			0.404155i	0.979370	+	0.202078i	$0.0647693\pi$
$433$			8.40993i			0.404155i	−0.979370	+	0.202078i	$0.935231\pi$
$434$		0			0
$435$	−4.98714	−	11.7365i	−0.239115	−	0.562721i
$436$		0			0
$437$			8.46382i			0.404879i
$438$		0			0
$439$	−1.03293			−0.0492989			−0.0246494	−	0.999696i	$-0.507847\pi$
$439$	−1.03293			−0.0492989			−0.0246494	+	0.999696i	$0.507847\pi$
$440$		0			0
$441$	−4.86277			−0.231560
$442$		0			0
$443$		−	4.93636i		−	0.234533i	−0.993100	−	0.117267i	$-0.962587\pi$
$443$		−	4.93636i		−	0.234533i	0.993100	−	0.117267i	$-0.0374132\pi$
$444$		0			0
$445$	29.8236	−	12.6728i	1.41377	−	0.600750i
$446$		0			0
$447$		−	13.2009i		−	0.624380i
$448$		0			0
$449$	−5.25138			−0.247828			−0.123914	−	0.992293i	$-0.539545\pi$
$449$	−5.25138			−0.247828			−0.123914	+	0.992293i	$0.539545\pi$
$450$		0			0
$451$	0.761840			0.0358736
$452$		0			0
$453$		−	13.8189i		−	0.649267i
$454$		0			0
$455$	16.6431	−	7.07209i	0.780241	−	0.331545i
$456$		0			0
$457$		−	6.78059i		−	0.317183i	−0.987344	−	0.158591i	$-0.949305\pi$
$457$		−	6.78059i		−	0.317183i	0.987344	−	0.158591i	$-0.0506953\pi$
$458$		0			0
$459$	8.15198			0.380502
$460$		0			0
$461$	12.2757			0.571735			0.285868	−	0.958269i	$-0.407718\pi$
$461$	12.2757			0.571735			0.285868	+	0.958269i	$0.407718\pi$
$462$		0			0
$463$		−	9.53996i		−	0.443360i	−0.975120	−	0.221680i	$-0.928846\pi$
$463$		−	9.53996i		−	0.443360i	0.975120	−	0.221680i	$-0.0711540\pi$
$464$		0			0
$465$	−4.80601	−	11.3102i	−0.222873	−	0.524499i
$466$		0			0
$467$			20.7684i			0.961047i	0.876982	+	0.480523i	$0.159553\pi$
$467$			20.7684i			0.961047i	−0.876982	+	0.480523i	$0.840447\pi$
$468$		0			0
$469$	2.10974			0.0974187
$470$		0			0
$471$	−2.51927			−0.116082
$472$		0			0
$473$		−	0.841766i		−	0.0387044i
$474$		0			0
$475$	−10.2071	+	10.5860i	−0.468335	+	0.485718i
$476$		0			0
$477$			8.00661i			0.366598i
$478$		0			0
$479$	20.8393			0.952172			0.476086	−	0.879399i	$-0.342055\pi$
$479$	20.8393			0.952172			0.476086	+	0.879399i	$0.342055\pi$
$480$		0			0
$481$	−43.5074			−1.98377
$482$		0			0
$483$		−	4.20712i		−	0.191431i
$484$		0			0
$485$	−0.181128	−	0.426258i	−0.00822461	−	0.0193554i
$486$		0			0
$487$		−	27.9618i		−	1.26707i	−0.773715	−	0.633534i	$-0.781604\pi$
$487$		−	27.9618i		−	1.26707i	0.773715	−	0.633534i	$-0.218396\pi$
$488$		0			0
$489$	7.42590			0.335811
$490$		0			0
$491$	−18.7214			−0.844887			−0.422444	−	0.906389i	$-0.638828\pi$
$491$	−18.7214			−0.844887			−0.422444	+	0.906389i	$0.638828\pi$
$492$		0			0
$493$		−	46.4901i		−	2.09381i
$494$		0			0
$495$	2.05798	−	0.874489i	0.0924992	−	0.0393053i
$496$		0			0
$497$			14.8941i			0.668093i
$498$		0			0
$499$	−39.2701			−1.75797			−0.878987	−	0.476846i	$-0.841780\pi$
$499$	−39.2701			−1.75797			−0.878987	+	0.476846i	$0.841780\pi$
$500$		0			0
$501$	−13.2830			−0.593439
$502$		0			0
$503$			18.3723i			0.819182i	0.912269	+	0.409591i	$0.134329\pi$
$503$			18.3723i			0.819182i	−0.912269	+	0.409591i	$0.865671\pi$
$504$		0			0
$505$	36.7199	−	15.6032i	1.63401	−	0.694335i
$506$		0			0
$507$		−	17.6010i		−	0.781690i
$508$		0			0
$509$	30.0010			1.32977			0.664885	−	0.746946i	$-0.268480\pi$
$509$	30.0010			1.32977			0.664885	+	0.746946i	$0.268480\pi$
$510$		0			0
$511$	7.42050			0.328264
$512$		0			0
$513$			2.94108i			0.129852i
$514$		0			0
$515$	13.1730	+	31.0006i	0.580471	+	1.36605i
$516$		0			0
$517$		−	12.1858i		−	0.535933i
$518$		0			0
$519$	3.71295			0.162980
$520$		0			0
$521$	−13.2902			−0.582252			−0.291126	−	0.956685i	$-0.594030\pi$
$521$	−13.2902			−0.582252			−0.291126	+	0.956685i	$0.594030\pi$
$522$		0			0
$523$		−	10.5841i		−	0.462812i	−0.972857	−	0.231406i	$-0.925667\pi$
$523$		−	10.5841i		−	0.462812i	0.972857	−	0.231406i	$-0.0743325\pi$
$524$		0			0
$525$	5.07367	−	5.26199i	0.221433	−	0.229652i
$526$		0			0
$527$		−	44.8016i		−	1.95159i
$528$		0			0
$529$	14.7183			0.639926
$530$		0			0
$531$	−6.57194			−0.285198
$532$		0			0
$533$		−	4.21436i		−	0.182544i
$534$		0			0
$535$	5.46040	+	12.8502i	0.236074	+	0.555563i
$536$		0			0
$537$		−	3.63774i		−	0.156980i
$538$		0			0
$539$	−4.86277			−0.209454
$540$		0			0
$541$	1.43751			0.0618034			0.0309017	−	0.999522i	$-0.490162\pi$
$541$	1.43751			0.0618034			0.0309017	+	0.999522i	$0.490162\pi$
$542$		0			0
$543$			0.571939i			0.0245443i
$544$		0			0
$545$	−11.0716	+	4.70461i	−0.474255	+	0.201523i
$546$		0			0
$547$			24.2050i			1.03493i	0.855703	+	0.517467i	$0.173125\pi$
$547$			24.2050i			1.03493i	−0.855703	+	0.517467i	$0.826875\pi$
$548$		0			0
$549$	8.37575			0.357468
$550$		0			0
$551$	16.7727			0.714542
$552$		0			0
$553$		−	20.2022i		−	0.859084i
$554$		0			0
$555$	−16.1858	+	6.87779i	−0.687051	+	0.291946i
$556$		0			0
$557$			24.9627i			1.05770i	0.848714	+	0.528852i	$0.177377\pi$
$557$			24.9627i			1.05770i	−0.848714	+	0.528852i	$0.822623\pi$
$558$		0			0
$559$	−4.65650			−0.196949
$560$		0			0
$561$	8.15198			0.344177
$562$		0			0
$563$			17.9295i			0.755639i	0.925879	+	0.377819i	$0.123326\pi$
$563$			17.9295i			0.755639i	−0.925879	+	0.377819i	$0.876674\pi$
$564$		0			0
$565$	−0.595774	−	1.40206i	−0.0250644	−	0.0589853i
$566$		0			0
$567$		−	1.46193i		−	0.0613952i
$568$		0			0
$569$	17.0185			0.713454			0.356727	−	0.934209i	$-0.383893\pi$
$569$	17.0185			0.713454			0.356727	+	0.934209i	$0.383893\pi$
$570$		0			0
$571$	−19.1862			−0.802918			−0.401459	−	0.915877i	$-0.631497\pi$
$571$	−19.1862			−0.802918			−0.401459	+	0.915877i	$0.631497\pi$
$572$		0			0
$573$			1.35411i			0.0565689i
$574$		0			0
$575$	−10.3582	−	9.98750i	−0.431967	−	0.416507i
$576$		0			0
$577$			38.8226i			1.61621i	0.589041	+	0.808103i	$0.299506\pi$
$577$			38.8226i			1.61621i	−0.589041	+	0.808103i	$0.700494\pi$
$578$		0			0
$579$	−4.05420			−0.168487
$580$		0			0
$581$	9.53024			0.395381
$582$		0			0
$583$			8.00661i			0.331600i
$584$		0			0
$585$	−4.83752	−	11.3844i	−0.200007	−	0.470685i
$586$		0			0
$587$			9.51046i			0.392539i	0.980550	+	0.196269i	$0.0628827\pi$
$587$			9.51046i			0.392539i	−0.980550	+	0.196269i	$0.937117\pi$
$588$		0			0
$589$	16.1636			0.666008
$590$		0			0
$591$	4.28705			0.176346
$592$		0			0
$593$			8.42150i			0.345829i	0.984937	+	0.172915i	$0.0553185\pi$
$593$			8.42150i			0.345829i	−0.984937	+	0.172915i	$0.944681\pi$
$594$		0			0
$595$	24.5262	−	10.4218i	1.00547	−	0.427253i
$596$		0			0
$597$			15.2278i			0.623233i
$598$		0			0
$599$	−2.35820			−0.0963536			−0.0481768	−	0.998839i	$-0.515341\pi$
$599$	−2.35820			−0.0963536			−0.0481768	+	0.998839i	$0.515341\pi$
$600$		0			0
$601$	28.7258			1.17175			0.585874	−	0.810402i	$-0.300751\pi$
$601$	28.7258			1.17175			0.585874	+	0.810402i	$0.300751\pi$
$602$		0			0
$603$		−	1.44312i		−	0.0587685i
$604$		0			0
$605$	2.05798	−	0.874489i	0.0836687	−	0.0355530i
$606$		0			0
$607$		−	26.8796i		−	1.09101i	−0.838107	−	0.545506i	$-0.816338\pi$
$607$		−	26.8796i		−	1.09101i	0.838107	−	0.545506i	$-0.183662\pi$
$608$		0			0
$609$	−8.33725			−0.337842
$610$		0			0
$611$	−67.4099			−2.72711
$612$		0			0
$613$		−	37.0774i		−	1.49754i	−0.662828	−	0.748772i	$-0.730644\pi$
$613$		−	37.0774i		−	1.49754i	0.662828	−	0.748772i	$-0.269356\pi$
$614$		0			0
$615$	−0.666221	−	1.56785i	−0.0268646	−	0.0632218i
$616$		0			0
$617$		−	37.8045i		−	1.52195i	−0.648781	−	0.760975i	$-0.724721\pi$
$617$		−	37.8045i		−	1.52195i	0.648781	−	0.760975i	$-0.275279\pi$
$618$		0			0
$619$	36.1427			1.45270			0.726348	−	0.687327i	$-0.241216\pi$
$619$	36.1427			1.45270			0.726348	+	0.687327i	$0.241216\pi$
$620$		0			0
$621$	−2.87779			−0.115482
$622$		0			0
$623$		−	21.1858i		−	0.848792i
$624$		0			0
$625$	−0.910720	−	24.9834i	−0.0364288	−	0.999336i
$626$		0			0
$627$			2.94108i			0.117455i
$628$		0			0
$629$	−64.1148			−2.55642
$630$		0			0
$631$	28.5692			1.13732			0.568660	−	0.822573i	$-0.307462\pi$
$631$	28.5692			1.13732			0.568660	+	0.822573i	$0.307462\pi$
$632$		0			0
$633$			11.6883i			0.464566i
$634$		0			0
$635$	−14.5621	−	34.2697i	−0.577880	−	1.35995i
$636$		0			0
$637$			26.9000i			1.06582i
$638$		0			0
$639$	10.1880			0.403031
$640$		0			0
$641$	22.8677			0.903221			0.451610	−	0.892215i	$-0.350850\pi$
$641$	22.8677			0.903221			0.451610	+	0.892215i	$0.350850\pi$
$642$		0			0
$643$		−	24.1144i		−	0.950979i	−0.879722	−	0.475489i	$-0.842271\pi$
$643$		−	24.1144i		−	0.950979i	0.879722	−	0.475489i	$-0.157729\pi$
$644$		0			0
$645$	−1.73234	+	0.736115i	−0.0682106	+	0.0289845i
$646$		0			0
$647$		−	9.01192i		−	0.354295i	−0.984184	−	0.177148i	$-0.943313\pi$
$647$		−	9.01192i		−	0.354295i	0.984184	−	0.177148i	$-0.0566870\pi$
$648$		0			0
$649$	−6.57194			−0.257971
$650$		0			0
$651$	−8.03445			−0.314895
$652$		0			0
$653$			29.7653i			1.16480i	0.812901	+	0.582402i	$0.197887\pi$
$653$			29.7653i			1.16480i	−0.812901	+	0.582402i	$0.802113\pi$
$654$		0			0
$655$	−11.2018	+	4.75995i	−0.437691	+	0.185987i
$656$		0			0
$657$		−	5.07584i		−	0.198027i
$658$		0			0
$659$	13.2923			0.517795			0.258898	−	0.965905i	$-0.416641\pi$
$659$	13.2923			0.517795			0.258898	+	0.965905i	$0.416641\pi$
$660$		0			0
$661$	−34.4814			−1.34117			−0.670585	−	0.741833i	$-0.733957\pi$
$661$	−34.4814			−1.34117			−0.670585	+	0.741833i	$0.733957\pi$
$662$		0			0
$663$		−	45.0953i		−	1.75136i
$664$		0			0
$665$	3.75999	+	8.84856i	0.145806	+	0.343133i
$666$		0			0
$667$			16.4118i			0.635468i
$668$		0			0
$669$	23.2956			0.900658
$670$		0			0
$671$	8.37575			0.323342
$672$		0			0
$673$		−	12.6453i		−	0.487441i	−0.969846	−	0.243720i	$-0.921632\pi$
$673$		−	12.6453i		−	0.487441i	0.969846	−	0.243720i	$-0.0783679\pi$
$674$		0			0
$675$	−3.59936	−	3.47054i	−0.138539	−	0.133581i
$676$		0			0
$677$		−	39.2780i		−	1.50958i	−0.655968	−	0.754788i	$-0.727740\pi$
$677$		−	39.2780i		−	1.50958i	0.655968	−	0.754788i	$-0.272260\pi$
$678$		0			0
$679$	−0.302801			−0.0116204
$680$		0			0
$681$	−28.0674			−1.07555
$682$		0			0
$683$		−	14.3297i		−	0.548310i	−0.961685	−	0.274155i	$-0.911602\pi$
$683$		−	14.3297i		−	0.548310i	0.961685	−	0.274155i	$-0.0883982\pi$
$684$		0			0
$685$	7.10498	+	16.7205i	0.271467	+	0.638857i
$686$		0			0
$687$		−	23.7041i		−	0.904369i
$688$		0			0
$689$	44.2911			1.68736
$690$		0			0
$691$	9.38389			0.356980			0.178490	−	0.983942i	$-0.442879\pi$
$691$	9.38389			0.356980			0.178490	+	0.983942i	$0.442879\pi$
$692$		0			0
$693$		−	1.46193i		−	0.0555340i
$694$		0			0
$695$	−34.8386	+	14.8038i	−1.32150	+	0.561542i
$696$		0			0
$697$		−	6.21051i		−	0.235240i
$698$		0			0
$699$	15.7850			0.597044
$700$		0			0
$701$	36.4376			1.37623			0.688114	−	0.725602i	$-0.258439\pi$
$701$	36.4376			1.37623			0.688114	+	0.725602i	$0.258439\pi$
$702$		0			0
$703$		−	23.1314i		−	0.872416i
$704$		0			0
$705$	−25.0782	+	10.6564i	−0.944500	+	0.401343i
$706$		0			0
$707$		−	26.0847i		−	0.981017i
$708$		0			0
$709$	34.6133			1.29993			0.649965	−	0.759964i	$-0.274783\pi$
$709$	34.6133			1.29993			0.649965	+	0.759964i	$0.274783\pi$
$710$		0			0
$711$	−13.8189			−0.518248
$712$		0			0
$713$			15.8158i			0.592305i
$714$		0			0
$715$	−4.83752	−	11.3844i	−0.180913	−	0.425751i
$716$		0			0
$717$			23.8564i			0.890935i
$718$		0			0
$719$	−18.1740			−0.677775			−0.338887	−	0.940827i	$-0.610051\pi$
$719$	−18.1740			−0.677775			−0.338887	+	0.940827i	$0.610051\pi$
$720$		0			0
$721$	22.0219			0.820140
$722$		0			0
$723$			6.60739i			0.245731i
$724$		0			0
$725$	−19.7922	+	20.5268i	−0.735064	+	0.762347i
$726$		0			0
$727$			25.6296i			0.950548i	0.879838	+	0.475274i	$0.157651\pi$
$727$			25.6296i			0.950548i	−0.879838	+	0.475274i	$0.842349\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−6.86206			−0.253803
$732$		0			0
$733$			17.7092i			0.654105i	0.945006	+	0.327052i	$0.106055\pi$
$733$			17.7092i			0.654105i	−0.945006	+	0.327052i	$0.893945\pi$
$734$		0			0
$735$	4.25244	+	10.0075i	0.156854	+	0.369131i
$736$		0			0
$737$		−	1.44312i		−	0.0531581i
$738$		0			0
$739$	11.7327			0.431594			0.215797	−	0.976438i	$-0.430765\pi$
$739$	11.7327			0.431594			0.215797	+	0.976438i	$0.430765\pi$
$740$		0			0
$741$	16.2695			0.597676
$742$		0			0
$743$		−	47.7099i		−	1.75031i	−0.483846	−	0.875153i	$-0.660761\pi$
$743$		−	47.7099i		−	1.75031i	0.483846	−	0.875153i	$-0.339239\pi$
$744$		0			0
$745$	−27.1671	+	11.5440i	−0.995325	+	0.422940i
$746$		0			0
$747$		−	6.51896i		−	0.238516i
$748$		0			0
$749$	9.12842			0.333545
$750$		0			0
$751$	−3.10493			−0.113301			−0.0566503	−	0.998394i	$-0.518042\pi$
$751$	−3.10493			−0.113301			−0.0566503	+	0.998394i	$0.518042\pi$
$752$		0			0
$753$			30.7656i			1.12116i
$754$		0			0
$755$	−28.4389	+	12.0844i	−1.03500	+	0.439798i
$756$		0			0
$757$		−	12.5989i		−	0.457914i	−0.973437	−	0.228957i	$-0.926468\pi$
$757$		−	12.5989i		−	0.457914i	0.973437	−	0.228957i	$-0.0735316\pi$
$758$		0			0
$759$	−2.87779			−0.104457
$760$		0			0
$761$	4.96464			0.179968			0.0899840	−	0.995943i	$-0.471318\pi$
$761$	4.96464			0.179968			0.0899840	+	0.995943i	$0.471318\pi$
$762$		0			0
$763$			7.86493i			0.284730i
$764$		0			0
$765$	−7.12882	−	16.7766i	−0.257743	−	0.606559i
$766$		0			0
$767$			36.3548i			1.31270i
$768$		0			0
$769$	33.2623			1.19947			0.599734	−	0.800199i	$-0.295273\pi$
$769$	33.2623			1.19947			0.599734	+	0.800199i	$0.295273\pi$
$770$		0			0
$771$	7.72864			0.278340
$772$		0			0
$773$			8.57639i			0.308471i	0.988034	+	0.154236i	$0.0492915\pi$
$773$			8.57639i			0.308471i	−0.988034	+	0.154236i	$0.950709\pi$
$774$		0			0
$775$	−19.0734	+	19.7813i	−0.685135	+	0.710566i
$776$		0			0
$777$			11.4980i			0.412487i
$778$		0			0
$779$	2.24063			0.0802789
$780$		0			0
$781$	10.1880			0.364556
$782$		0			0
$783$			5.70292i			0.203806i
$784$		0			0
$785$	2.20307	+	5.18460i	0.0786310	+	0.185046i
$786$		0			0
$787$			3.85774i			0.137514i	0.997633	+	0.0687568i	$0.0219032\pi$
$787$			3.85774i			0.137514i	−0.997633	+	0.0687568i	$0.978097\pi$
$788$		0			0
$789$	5.80507			0.206666
$790$		0			0
$791$	−0.995986			−0.0354132
$792$		0			0
$793$		−	46.3331i		−	1.64534i
$794$		0			0
$795$	16.4774	−	7.00169i	0.584394	−	0.248324i
$796$		0			0
$797$		−	48.0706i		−	1.70275i	−0.524560	−	0.851373i	$-0.675770\pi$
$797$		−	48.0706i		−	1.70275i	0.524560	−	0.851373i	$-0.324230\pi$
$798$		0			0
$799$	−99.3388			−3.51435
$800$		0			0
$801$	−14.4917			−0.512039
$802$		0			0
$803$		−	5.07584i		−	0.179122i
$804$		0			0
$805$	−8.65817	+	3.67908i	−0.305160	+	0.129671i
$806$		0			0
$807$			19.6036i			0.690078i
$808$		0			0
$809$	13.2257			0.464989			0.232495	−	0.972598i	$-0.425311\pi$
$809$	13.2257			0.464989			0.232495	+	0.972598i	$0.425311\pi$
$810$		0			0
$811$	2.02478			0.0710998			0.0355499	−	0.999368i	$-0.488682\pi$
$811$	2.02478			0.0710998			0.0355499	+	0.999368i	$0.488682\pi$
$812$		0			0
$813$		−	17.0291i		−	0.597238i
$814$		0			0
$815$	−6.49386	−	15.2823i	−0.227470	−	0.535317i
$816$		0			0
$817$		−	2.47570i		−	0.0866138i
$818$		0			0
$819$	−8.08712			−0.282587
$820$		0			0
$821$	−21.5769			−0.753039			−0.376519	−	0.926409i	$-0.622879\pi$
$821$	−21.5769			−0.753039			−0.376519	+	0.926409i	$0.622879\pi$
$822$		0			0
$823$		−	5.95799i		−	0.207683i	−0.994594	−	0.103841i	$-0.966887\pi$
$823$		−	5.95799i		−	0.207683i	0.994594	−	0.103841i	$-0.0331134\pi$
$824$		0			0
$825$	−3.59936	−	3.47054i	−0.125313	−	0.120829i
$826$		0			0
$827$			53.8102i			1.87116i	0.353110	+	0.935582i	$0.385124\pi$
$827$			53.8102i			1.87116i	−0.353110	+	0.935582i	$0.614876\pi$
$828$		0			0
$829$	10.0918			0.350504			0.175252	−	0.984524i	$-0.443926\pi$
$829$	10.0918			0.350504			0.175252	+	0.984524i	$0.443926\pi$
$830$		0			0
$831$	2.68002			0.0929690
$832$		0			0
$833$			39.6412i			1.37349i
$834$		0			0
$835$	11.6158	+	27.3360i	0.401981	+	0.946002i
$836$		0			0
$837$			5.49579i			0.189962i
$838$		0			0
$839$	7.14443			0.246653			0.123327	−	0.992366i	$-0.460644\pi$
$839$	7.14443			0.246653			0.123327	+	0.992366i	$0.460644\pi$
$840$		0			0
$841$	3.52327			0.121492
$842$		0			0
$843$			29.1626i			1.00441i
$844$		0			0
$845$	−36.2225	+	15.3919i	−1.24609	+	0.529498i
$846$		0			0
$847$		−	1.46193i		−	0.0502324i
$848$		0			0
$849$	−14.3484			−0.492437
$850$		0			0
$851$	22.6337			0.775872
$852$		0			0
$853$			12.2758i			0.420314i	0.977668	+	0.210157i	$0.0673975\pi$
$853$			12.2758i			0.420314i	−0.977668	+	0.210157i	$0.932602\pi$
$854$		0			0
$855$	6.05267	−	2.57194i	0.206997	−	0.0879585i
$856$		0			0
$857$			11.3507i			0.387732i	0.981028	+	0.193866i	$0.0621027\pi$
$857$			11.3507i			0.387732i	−0.981028	+	0.193866i	$0.937897\pi$
$858$		0			0
$859$	−16.9170			−0.577200			−0.288600	−	0.957450i	$-0.593190\pi$
$859$	−16.9170			−0.577200			−0.288600	+	0.957450i	$0.593190\pi$
$860$		0			0
$861$	−1.11375			−0.0379567
$862$		0			0
$863$			3.16421i			0.107711i	0.998549	+	0.0538555i	$0.0171510\pi$
$863$			3.16421i			0.107711i	−0.998549	+	0.0538555i	$0.982849\pi$
$864$		0			0
$865$	−3.24693	−	7.64116i	−0.110399	−	0.259807i
$866$		0			0
$867$		−	49.4548i		−	1.67957i
$868$		0			0
$869$	−13.8189			−0.468773
$870$		0			0
$871$	−7.98310			−0.270497
$872$		0			0
$873$			0.207125i			0.00701011i
$874$		0			0
$875$	−15.2659	−	5.83995i	−0.516083	−	0.197426i
$876$		0			0
$877$		−	9.92790i		−	0.335242i	−0.985852	−	0.167621i	$-0.946392\pi$
$877$		−	9.92790i		−	0.335242i	0.985852	−	0.167621i	$-0.0536084\pi$
$878$		0			0
$879$	−31.8145			−1.07308
$880$		0			0
$881$	4.40175			0.148299			0.0741493	−	0.997247i	$-0.476376\pi$
$881$	4.40175			0.148299			0.0741493	+	0.997247i	$0.476376\pi$
$882$		0			0
$883$		−	20.7286i		−	0.697573i	−0.937202	−	0.348787i	$-0.886594\pi$
$883$		−	20.7286i		−	0.697573i	0.937202	−	0.348787i	$-0.113406\pi$
$884$		0			0
$885$	5.74709	+	13.5249i	0.193186	+	0.454635i
$886$		0			0
$887$		−	29.6923i		−	0.996969i	−0.866899	−	0.498484i	$-0.833890\pi$
$887$		−	29.6923i		−	0.996969i	0.866899	−	0.498484i	$-0.166110\pi$
$888$		0			0
$889$	−24.3442			−0.816478
$890$		0			0
$891$	−1.00000			−0.0335013
$892$		0			0
$893$		−	35.8395i		−	1.19932i
$894$		0			0
$895$	−7.48639	+	3.18117i	−0.250243	+	0.106335i
$896$		0			0
$897$			15.9194i			0.531535i
$898$		0			0
$899$	31.3421			1.04532
$900$		0			0
$901$	65.2697			2.17445
$902$		0			0
$903$			1.23060i			0.0409518i
$904$		0			0
$905$	1.17704	−	0.500155i	0.0391261	−	0.0166257i
$906$		0			0
$907$			3.86493i			0.128333i	0.997939	+	0.0641665i	$0.0204389\pi$
$907$			3.86493i			0.128333i	−0.997939	+	0.0641665i	$0.979561\pi$
$908$		0			0
$909$	−17.8427			−0.591805
$910$		0			0
$911$	−8.63338			−0.286037			−0.143018	−	0.989720i	$-0.545681\pi$
$911$	−8.63338			−0.286037			−0.143018	+	0.989720i	$0.545681\pi$
$912$		0			0
$913$		−	6.51896i		−	0.215746i
$914$		0			0
$915$	−7.32450	−	17.2371i	−0.242140	−	0.569841i
$916$		0			0
$917$			7.95745i			0.262778i
$918$		0			0
$919$	−1.99689			−0.0658713			−0.0329357	−	0.999457i	$-0.510486\pi$
$919$	−1.99689			−0.0658713			−0.0329357	+	0.999457i	$0.510486\pi$
$920$		0			0
$921$	−16.5383			−0.544957
$922$		0			0
$923$		−	56.3583i		−	1.85505i
$924$		0			0
$925$	28.3087	+	27.2956i	0.930784	+	0.897472i
$926$		0			0
$927$		−	15.0636i		−	0.494755i
$928$		0			0
$929$	4.53399			0.148755			0.0743777	−	0.997230i	$-0.476303\pi$
$929$	4.53399			0.148755			0.0743777	+	0.997230i	$0.476303\pi$
$930$		0			0
$931$	−14.3018			−0.468722
$932$		0			0
$933$			0.0357178i			0.00116935i
$934$		0			0
$935$	−7.12882	−	16.7766i	−0.233137	−	0.548653i
$936$		0			0
$937$			12.9662i			0.423586i	0.977315	+	0.211793i	$0.0679303\pi$
$937$			12.9662i			0.423586i	−0.977315	+	0.211793i	$0.932070\pi$
$938$		0			0
$939$	20.7139			0.675972
$940$		0			0
$941$	−28.3147			−0.923033			−0.461516	−	0.887132i	$-0.652694\pi$
$941$	−28.3147			−0.923033			−0.461516	+	0.887132i	$0.652694\pi$
$942$		0			0
$943$			2.19242i			0.0713950i
$944$		0			0
$945$	−3.00861	+	1.27844i	−0.0978702	+	0.0415876i
$946$		0			0
$947$			40.6237i			1.32009i	0.751225	+	0.660046i	$0.229463\pi$
$947$			40.6237i			1.32009i	−0.751225	+	0.660046i	$0.770537\pi$
$948$		0			0
$949$	−28.0786			−0.911470
$950$		0			0
$951$	20.5293			0.665707
$952$		0			0
$953$		−	52.2000i		−	1.69093i	−0.534034	−	0.845463i	$-0.679325\pi$
$953$		−	52.2000i		−	1.69093i	0.534034	−	0.845463i	$-0.320675\pi$
$954$		0			0
$955$	2.78674	−	1.18416i	0.0901766	−	0.0383184i
$956$		0			0
$957$			5.70292i			0.184349i
$958$		0			0
$959$	11.8778			0.383553
$960$		0			0
$961$	−0.796257			−0.0256857
$962$		0			0
$963$		−	6.24410i		−	0.201213i
$964$		0			0
$965$	3.54535	+	8.34345i	0.114129	+	0.268585i
$966$		0			0
$967$			36.8428i			1.18478i	0.805650	+	0.592392i	$0.201816\pi$
$967$			36.8428i			1.18478i	−0.805650	+	0.592392i	$0.798184\pi$
$968$		0			0
$969$	23.9756			0.770208
$970$		0			0
$971$	29.6252			0.950719			0.475360	−	0.879792i	$-0.342318\pi$
$971$	29.6252			0.950719			0.475360	+	0.879792i	$0.342318\pi$
$972$		0			0
$973$			24.7483i			0.793395i
$974$		0			0
$975$	−19.1984	+	19.9110i	−0.614841	+	0.637662i
$976$		0			0
$977$			12.5176i			0.400475i	0.979747	+	0.200238i	$0.0641714\pi$
$977$			12.5176i			0.400475i	−0.979747	+	0.200238i	$0.935829\pi$
$978$		0			0
$979$	−14.4917			−0.463157
$980$		0			0
$981$	5.37984			0.171765
$982$		0			0
$983$		−	0.548105i		−	0.0174818i	−0.999962	−	0.00874091i	$-0.997218\pi$
$983$		−	0.548105i		−	0.0174818i	0.999962	−	0.00874091i	$-0.00278235\pi$
$984$		0			0
$985$	−3.74898	−	8.82265i	−0.119452	−	0.281113i
$986$		0			0
$987$			17.8148i			0.567052i
$988$		0			0
$989$	2.42243			0.0770288
$990$		0			0
$991$	40.8459			1.29751			0.648756	−	0.760997i	$-0.275290\pi$
$991$	40.8459			1.29751			0.648756	+	0.760997i	$0.275290\pi$
$992$		0			0
$993$		−	29.4805i		−	0.935534i
$994$		0			0
$995$	31.3385	−	13.3166i	0.993497	−	0.422163i
$996$		0			0
$997$			21.1758i			0.670644i	0.942104	+	0.335322i	$0.108845\pi$
$997$			21.1758i			0.670644i	−0.942104	+	0.335322i	$0.891155\pi$
$998$		0			0
$999$	7.86493			0.248835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.d.j.529.10		10
4.3	odd	2		1320.2.d.d.529.5	✓	10
5.4	even	2	inner	2640.2.d.j.529.5		10
12.11	even	2		3960.2.d.g.3169.2		10
20.3	even	4		6600.2.a.ca.1.4		5
20.7	even	4		6600.2.a.by.1.2		5
20.19	odd	2		1320.2.d.d.529.10	yes	10
60.59	even	2		3960.2.d.g.3169.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.d.d.529.5	✓	10	4.3	odd	2
1320.2.d.d.529.10	yes	10	20.19	odd	2
2640.2.d.j.529.5		10	5.4	even	2	inner
2640.2.d.j.529.10		10	1.1	even	1	trivial
3960.2.d.g.3169.1		10	60.59	even	2
3960.2.d.g.3169.2		10	12.11	even	2
6600.2.a.by.1.2		5	20.7	even	4
6600.2.a.ca.1.4		5	20.3	even	4

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:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(2.05798 - 0.874489i) q^{5} -1.46193i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(2.05798 - 0.874489i) q^{5} -1.46193i q^{7} -1.00000 q^{9} -1.00000 q^{11} +5.53182i q^{13} +(0.874489 + 2.05798i) q^{15} +8.15198i q^{17} -2.94108 q^{19} +1.46193 q^{21} -2.87779i q^{23} +(3.47054 - 3.59936i) q^{25} -1.00000i q^{27} -5.70292 q^{29} -5.49579 q^{31} -1.00000i q^{33} +(-1.27844 - 3.00861i) q^{35} +7.86493i q^{37} -5.53182 q^{39} -0.761840 q^{41} +0.841766i q^{43} +(-2.05798 + 0.874489i) q^{45} +12.1858i q^{47} +4.86277 q^{49} -8.15198 q^{51} -8.00661i q^{53} +(-2.05798 + 0.874489i) q^{55} -2.94108i q^{57} +6.57194 q^{59} -8.37575 q^{61} +1.46193i q^{63} +(4.83752 + 11.3844i) q^{65} +1.44312i q^{67} +2.87779 q^{69} -10.1880 q^{71} +5.07584i q^{73} +(3.59936 + 3.47054i) q^{75} +1.46193i q^{77} +13.8189 q^{79} +1.00000 q^{81} +6.51896i q^{83} +(7.12882 + 16.7766i) q^{85} -5.70292i q^{87} +14.4917 q^{89} +8.08712 q^{91} -5.49579i q^{93} +(-6.05267 + 2.57194i) q^{95} -0.207125i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^5 - 10 * q^9	\( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 12 q^{19} + 4 q^{21} + 14 q^{25} - 24 q^{29} - 8 q^{31} - 16 q^{35} + 16 q^{39} - 16 q^{41} + 2 q^{45} - 26 q^{49} - 12 q^{51} + 2 q^{55} + 40 q^{59} - 12 q^{61} - 8 q^{69} - 8 q^{71} - 4 q^{75} + 60 q^{79} + 10 q^{81} + 52 q^{85} + 28 q^{89} - 20 q^{95} + 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 + 12 * q^19 + 4 * q^21 + 14 * q^25 - 24 * q^29 - 8 * q^31 - 16 * q^35 + 16 * q^39 - 16 * q^41 + 2 * q^45 - 26 * q^49 - 12 * q^51 + 2 * q^55 + 40 * q^59 - 12 * q^61 - 8 * q^69 - 8 * q^71 - 4 * q^75 + 60 * q^79 + 10 * q^81 + 52 * q^85 + 28 * q^89 - 20 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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