LMFDB - Embedded newform 920.2.e.a.369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(369,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("920.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.e (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:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.a.369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -3.00000i q^{7} -1.00000 q^{9} +6.00000i q^{13} +(2.00000 + 4.00000i) q^{15} -7.00000i q^{17} +4.00000 q^{19} +6.00000 q^{21} +1.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +4.00000i q^{27} +9.00000 q^{29} -3.00000 q^{31} +(-3.00000 - 6.00000i) q^{35} -7.00000i q^{37} -12.0000 q^{39} +9.00000 q^{41} +4.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -2.00000i q^{47} -2.00000 q^{49} +14.0000 q^{51} +7.00000i q^{53} +8.00000i q^{57} -9.00000 q^{59} -2.00000 q^{61} +3.00000i q^{63} +(6.00000 + 12.0000i) q^{65} +13.0000i q^{67} -2.00000 q^{69} -13.0000 q^{71} +4.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +2.00000 q^{79} -11.0000 q^{81} +11.0000i q^{83} +(-7.00000 - 14.0000i) q^{85} +18.0000i q^{87} +10.0000 q^{89} +18.0000 q^{91} -6.00000i q^{93} +(8.00000 - 4.00000i) q^{95} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9} + 4 q^{15} + 8 q^{19} + 12 q^{21} + 6 q^{25} + 18 q^{29} - 6 q^{31} - 6 q^{35} - 24 q^{39} + 18 q^{41} - 4 q^{45} - 4 q^{49} + 28 q^{51} - 18 q^{59} - 4 q^{61} + 12 q^{65} - 4 q^{69}+ \cdots + 16 q^{95}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 + 4 * q^15 + 8 * q^19 + 12 * q^21 + 6 * q^25 + 18 * q^29 - 6 * q^31 - 6 * q^35 - 24 * q^39 + 18 * q^41 - 4 * q^45 - 4 * q^49 + 28 * q^51 - 18 * q^59 - 4 * q^61 + 12 * q^65 - 4 * q^69 - 26 * q^71 + 16 * q^75 + 4 * q^79 - 22 * q^81 - 14 * q^85 + 20 * q^89 + 36 * q^91 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/920\mathbb{Z}\right)^\times$.

$n$	$231$	$281$	$461$	$737$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	2.00000	+	4.00000i	0.516398	+	1.03280i
$16$		0			0
$17$		−	7.00000i		−	1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$		−	7.00000i		−	1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$	−3.00000			−0.538816			−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000			−0.538816			−0.269408	+	0.963026i	$0.586828\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	−	6.00000i	−0.507093	−	1.01419i
$36$		0			0
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$		0			0
$39$	−12.0000			−1.92154
$40$		0			0
$41$	9.00000			1.40556			0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000			1.40556			0.702782	+	0.711405i	$0.251941\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	−2.00000	+	1.00000i	−0.298142	+	0.149071i
$46$		0			0
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$	14.0000			1.96039
$52$		0			0
$53$			7.00000i			0.961524i	0.876851	+	0.480762i	$0.159640\pi$
$53$			7.00000i			0.961524i	−0.876851	+	0.480762i	$0.840360\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			8.00000i			1.05963i
$58$		0			0
$59$	−9.00000			−1.17170			−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000			−1.17170			−0.585850	+	0.810419i	$0.699239\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$			3.00000i			0.377964i
$64$		0			0
$65$	6.00000	+	12.0000i	0.744208	+	1.48842i
$66$		0			0
$67$			13.0000i			1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$			13.0000i			1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$		0			0
$69$	−2.00000			−0.240772
$70$		0			0
$71$	−13.0000			−1.54282			−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000			−1.54282			−0.771408	+	0.636341i	$0.780447\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	8.00000	+	6.00000i	0.923760	+	0.692820i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.00000			0.225018			0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000			0.225018			0.112509	+	0.993651i	$0.464111\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$			11.0000i			1.20741i	0.797209	+	0.603703i	$0.206309\pi$
$83$			11.0000i			1.20741i	−0.797209	+	0.603703i	$0.793691\pi$
$84$		0			0
$85$	−7.00000	−	14.0000i	−0.759257	−	1.51851i
$86$		0			0
$87$			18.0000i			1.92980i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	18.0000			1.88691
$92$		0			0
$93$		−	6.00000i		−	0.622171i
$94$		0			0
$95$	8.00000	−	4.00000i	0.820783	−	0.410391i
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.0000			−1.29355			−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000			−1.29355			−0.646774	+	0.762682i	$0.723882\pi$
$102$		0			0
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$		0			0
$105$	12.0000	−	6.00000i	1.17108	−	0.585540i
$106$		0			0
$107$		−	15.0000i		−	1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$		−	15.0000i		−	1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	14.0000			1.32882
$112$		0			0
$113$		−	5.00000i		−	0.470360i	−0.971952	−	0.235180i	$-0.924432\pi$
$113$		−	5.00000i		−	0.470360i	0.971952	−	0.235180i	$-0.0755680\pi$
$114$		0			0
$115$	1.00000	+	2.00000i	0.0932505	+	0.186501i
$116$		0			0
$117$		−	6.00000i		−	0.554700i
$118$		0			0
$119$	−21.0000			−1.92507
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$			18.0000i			1.62301i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		−	12.0000i		−	1.04053i
$134$		0			0
$135$	4.00000	+	8.00000i	0.344265	+	0.688530i
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	−11.0000			−0.933008			−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000			−0.933008			−0.466504	+	0.884519i	$0.654487\pi$
$140$		0			0
$141$	4.00000			0.336861
$142$		0			0
$143$		0			0
$144$		0			0
$145$	18.0000	−	9.00000i	1.49482	−	0.747409i
$146$		0			0
$147$		−	4.00000i		−	0.329914i
$148$		0			0
$149$	−20.0000			−1.63846			−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000			−1.63846			−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$			7.00000i			0.565916i
$154$		0			0
$155$	−6.00000	+	3.00000i	−0.481932	+	0.240966i
$156$		0			0
$157$		−	13.0000i		−	1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$		−	13.0000i		−	1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$		0			0
$159$	−14.0000			−1.11027
$160$		0			0
$161$	3.00000			0.236433
$162$		0			0
$163$		−	16.0000i		−	1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$		−	16.0000i		−	1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$			18.0000i			1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$			18.0000i			1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$		0			0
$175$	−12.0000	−	9.00000i	−0.907115	−	0.680336i
$176$		0			0
$177$		−	18.0000i		−	1.35296i
$178$		0			0
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		−	4.00000i		−	0.295689i
$184$		0			0
$185$	−7.00000	−	14.0000i	−0.514650	−	1.02930i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	12.0000			0.872872
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	−24.0000	+	12.0000i	−1.71868	+	0.859338i
$196$		0			0
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$	−18.0000			−1.27599			−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000			−1.27599			−0.637993	+	0.770042i	$0.720235\pi$
$200$		0			0
$201$	−26.0000			−1.83390
$202$		0			0
$203$		−	27.0000i		−	1.89503i
$204$		0			0
$205$	18.0000	−	9.00000i	1.25717	−	0.628587i
$206$		0			0
$207$		−	1.00000i		−	0.0695048i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−11.0000			−0.757271			−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000			−0.757271			−0.378636	+	0.925546i	$0.623607\pi$
$212$		0			0
$213$		−	26.0000i		−	1.78149i
$214$		0			0
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$			9.00000i			0.610960i
$218$		0			0
$219$	−8.00000			−0.540590
$220$		0			0
$221$	42.0000			2.82523
$222$		0			0
$223$			2.00000i			0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$			2.00000i			0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$			16.0000i			1.06196i	0.847385	+	0.530979i	$0.178176\pi$
$227$			16.0000i			1.06196i	−0.847385	+	0.530979i	$0.821824\pi$
$228$		0			0
$229$	8.00000			0.528655			0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000			0.528655			0.264327	+	0.964433i	$0.414850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.00000i		−	0.131024i	−0.997852	−	0.0655122i	$-0.979132\pi$
$233$		−	2.00000i		−	0.131024i	0.997852	−	0.0655122i	$-0.0208681\pi$
$234$		0			0
$235$	−2.00000	−	4.00000i	−0.130466	−	0.260931i
$236$		0			0
$237$			4.00000i			0.259828i
$238$		0			0
$239$	15.0000			0.970269			0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000			0.970269			0.485135	+	0.874439i	$0.338771\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$	−4.00000	+	2.00000i	−0.255551	+	0.127775i
$246$		0			0
$247$			24.0000i			1.52708i
$248$		0			0
$249$	−22.0000			−1.39419
$250$		0			0
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	28.0000	−	14.0000i	1.75343	−	0.876714i
$256$		0			0
$257$			26.0000i			1.62184i	0.585160	+	0.810918i	$0.301032\pi$
$257$			26.0000i			1.62184i	−0.585160	+	0.810918i	$0.698968\pi$
$258$		0			0
$259$	−21.0000			−1.30488
$260$		0			0
$261$	−9.00000			−0.557086
$262$		0			0
$263$			9.00000i			0.554964i	0.960731	+	0.277482i	$0.0894999\pi$
$263$			9.00000i			0.554964i	−0.960731	+	0.277482i	$0.910500\pi$
$264$		0			0
$265$	7.00000	+	14.0000i	0.430007	+	0.860013i
$266$		0			0
$267$			20.0000i			1.22398i
$268$		0			0
$269$	−21.0000			−1.28039			−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000			−1.28039			−0.640196	+	0.768211i	$0.721147\pi$
$270$		0			0
$271$	−25.0000			−1.51864			−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000			−1.51864			−0.759321	+	0.650716i	$0.774469\pi$
$272$		0			0
$273$			36.0000i			2.17882i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	3.00000			0.179605
$280$		0			0
$281$	20.0000			1.19310			0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000			1.19310			0.596550	+	0.802576i	$0.296538\pi$
$282$		0			0
$283$		−	19.0000i		−	1.12943i	−0.825285	−	0.564716i	$-0.808986\pi$
$283$		−	19.0000i		−	1.12943i	0.825285	−	0.564716i	$-0.191014\pi$
$284$		0			0
$285$	8.00000	+	16.0000i	0.473879	+	0.947758i
$286$		0			0
$287$		−	27.0000i		−	1.59376i
$288$		0			0
$289$	−32.0000			−1.88235
$290$		0			0
$291$	−4.00000			−0.234484
$292$		0			0
$293$		−	3.00000i		−	0.175262i	−0.996153	−	0.0876309i	$-0.972070\pi$
$293$		−	3.00000i		−	0.175262i	0.996153	−	0.0876309i	$-0.0279296\pi$
$294$		0			0
$295$	−18.0000	+	9.00000i	−1.04800	+	0.524000i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$		−	26.0000i		−	1.49366i
$304$		0			0
$305$	−4.00000	+	2.00000i	−0.229039	+	0.114520i
$306$		0			0
$307$		−	10.0000i		−	0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$		−	10.0000i		−	0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$		0			0
$309$	32.0000			1.82042
$310$		0			0
$311$	28.0000			1.58773			0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000			1.58773			0.793867	+	0.608091i	$0.208065\pi$
$312$		0			0
$313$			1.00000i			0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$			1.00000i			0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$		0			0
$315$	3.00000	+	6.00000i	0.169031	+	0.338062i
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	30.0000			1.67444
$322$		0			0
$323$		−	28.0000i		−	1.55796i
$324$		0			0
$325$	24.0000	+	18.0000i	1.33128	+	0.998460i
$326$		0			0
$327$			4.00000i			0.221201i
$328$		0			0
$329$	−6.00000			−0.330791
$330$		0			0
$331$	1.00000			0.0549650			0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000			0.0549650			0.0274825	+	0.999622i	$0.491251\pi$
$332$		0			0
$333$			7.00000i			0.383598i
$334$		0			0
$335$	13.0000	+	26.0000i	0.710266	+	1.42053i
$336$		0			0
$337$			30.0000i			1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$			30.0000i			1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$		0			0
$345$	−4.00000	+	2.00000i	−0.215353	+	0.107676i
$346$		0			0
$347$		−	28.0000i		−	1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$		−	28.0000i		−	1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$		0			0
$349$	−27.0000			−1.44528			−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000			−1.44528			−0.722638	+	0.691226i	$0.757071\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	−26.0000	+	13.0000i	−1.37994	+	0.689968i
$356$		0			0
$357$		−	42.0000i		−	2.22288i
$358$		0			0
$359$	−10.0000			−0.527780			−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000			−0.527780			−0.263890	+	0.964553i	$0.585006\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	22.0000i		−	1.15470i
$364$		0			0
$365$	4.00000	+	8.00000i	0.209370	+	0.418739i
$366$		0			0
$367$		−	1.00000i		−	0.0521996i	−0.999659	−	0.0260998i	$-0.991691\pi$
$367$		−	1.00000i		−	0.0521996i	0.999659	−	0.0260998i	$-0.00830876\pi$
$368$		0			0
$369$	−9.00000			−0.468521
$370$		0			0
$371$	21.0000			1.09027
$372$		0			0
$373$			10.0000i			0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$			10.0000i			0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$		0			0
$375$	22.0000	+	4.00000i	1.13608	+	0.206559i
$376$		0			0
$377$			54.0000i			2.78114i
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$	16.0000			0.819705
$382$		0			0
$383$			9.00000i			0.459879i	0.973205	+	0.229939i	$0.0738528\pi$
$383$			9.00000i			0.459879i	−0.973205	+	0.229939i	$0.926147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	−14.0000			−0.709828			−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000			−0.709828			−0.354914	+	0.934899i	$0.615490\pi$
$390$		0			0
$391$	7.00000			0.354005
$392$		0			0
$393$			16.0000i			0.807093i
$394$		0			0
$395$	4.00000	−	2.00000i	0.201262	−	0.100631i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$		0			0
$399$	24.0000			1.20150
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$		−	18.0000i		−	0.896644i
$404$		0			0
$405$	−22.0000	+	11.0000i	−1.09319	+	0.546594i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−21.0000			−1.03838			−0.519192	−	0.854658i	$-0.673767\pi$
$409$	−21.0000			−1.03838			−0.519192	+	0.854658i	$0.673767\pi$
$410$		0			0
$411$	−4.00000			−0.197305
$412$		0			0
$413$			27.0000i			1.32858i
$414$		0			0
$415$	11.0000	+	22.0000i	0.539969	+	1.07994i
$416$		0			0
$417$		−	22.0000i		−	1.07734i
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$			2.00000i			0.0972433i
$424$		0			0
$425$	−28.0000	−	21.0000i	−1.35820	−	1.01865i
$426$		0			0
$427$			6.00000i			0.290360i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$		0			0
$433$		−	25.0000i		−	1.20142i	−0.799466	−	0.600712i	$-0.794884\pi$
$433$		−	25.0000i		−	1.20142i	0.799466	−	0.600712i	$-0.205116\pi$
$434$		0			0
$435$	18.0000	+	36.0000i	0.863034	+	1.72607i
$436$		0			0
$437$			4.00000i			0.191346i
$438$		0			0
$439$	4.00000			0.190910			0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000			0.190910			0.0954548	+	0.995434i	$0.469569\pi$
$440$		0			0
$441$	2.00000			0.0952381
$442$		0			0
$443$			28.0000i			1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$			28.0000i			1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$		0			0
$445$	20.0000	−	10.0000i	0.948091	−	0.474045i
$446$		0			0
$447$		−	40.0000i		−	1.89194i
$448$		0			0
$449$	33.0000			1.55737			0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000			1.55737			0.778683	+	0.627417i	$0.215888\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	36.0000	−	18.0000i	1.68771	−	0.843853i
$456$		0			0
$457$		−	13.0000i		−	0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$457$		−	13.0000i		−	0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$458$		0			0
$459$	28.0000			1.30693
$460$		0			0
$461$	−10.0000			−0.465746			−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000			−0.465746			−0.232873	+	0.972507i	$0.574813\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$	−6.00000	−	12.0000i	−0.278243	−	0.556487i
$466$		0			0
$467$			29.0000i			1.34196i	0.741475	+	0.670980i	$0.234126\pi$
$467$			29.0000i			1.34196i	−0.741475	+	0.670980i	$0.765874\pi$
$468$		0			0
$469$	39.0000			1.80085
$470$		0			0
$471$	26.0000			1.19802
$472$		0			0
$473$		0			0
$474$		0			0
$475$	12.0000	−	16.0000i	0.550598	−	0.734130i
$476$		0			0
$477$		−	7.00000i		−	0.320508i
$478$		0			0
$479$	−4.00000			−0.182765			−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000			−0.182765			−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$	42.0000			1.91504
$482$		0			0
$483$			6.00000i			0.273009i
$484$		0			0
$485$	2.00000	+	4.00000i	0.0908153	+	0.181631i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$	32.0000			1.44709
$490$		0			0
$491$	−29.0000			−1.30875			−0.654376	−	0.756169i	$-0.727069\pi$
$491$	−29.0000			−1.30875			−0.654376	+	0.756169i	$0.727069\pi$
$492$		0			0
$493$		−	63.0000i		−	2.83738i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			39.0000i			1.74939i
$498$		0			0
$499$	25.0000			1.11915			0.559577	−	0.828778i	$-0.310964\pi$
$499$	25.0000			1.11915			0.559577	+	0.828778i	$0.310964\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	11.0000i		−	0.490466i	−0.969464	−	0.245233i	$-0.921136\pi$
$503$		−	11.0000i		−	0.490466i	0.969464	−	0.245233i	$-0.0788644\pi$
$504$		0			0
$505$	−26.0000	+	13.0000i	−1.15698	+	0.578492i
$506$		0			0
$507$		−	46.0000i		−	2.04293i
$508$		0			0
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$			16.0000i			0.706417i
$514$		0			0
$515$	−16.0000	−	32.0000i	−0.705044	−	1.41009i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−36.0000			−1.58022
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$			28.0000i			1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$			28.0000i			1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$		0			0
$525$	18.0000	−	24.0000i	0.785584	−	1.04745i
$526$		0			0
$527$			21.0000i			0.914774i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	9.00000			0.390567
$532$		0			0
$533$			54.0000i			2.33900i
$534$		0			0
$535$	−15.0000	−	30.0000i	−0.648507	−	1.29701i
$536$		0			0
$537$			40.0000i			1.72613i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		−	20.0000i		−	0.858282i
$544$		0			0
$545$	4.00000	−	2.00000i	0.171341	−	0.0856706i
$546$		0			0
$547$			4.00000i			0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$			4.00000i			0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$		0			0
$549$	2.00000			0.0853579
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$		−	6.00000i		−	0.255146i
$554$		0			0
$555$	28.0000	−	14.0000i	1.18853	−	0.594267i
$556$		0			0
$557$		−	15.0000i		−	0.635570i	−0.948163	−	0.317785i	$-0.897061\pi$
$557$		−	15.0000i		−	0.635570i	0.948163	−	0.317785i	$-0.102939\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$			17.0000i			0.716465i	0.933632	+	0.358232i	$0.116620\pi$
$563$			17.0000i			0.716465i	−0.933632	+	0.358232i	$0.883380\pi$
$564$		0			0
$565$	−5.00000	−	10.0000i	−0.210352	−	0.420703i
$566$		0			0
$567$			33.0000i			1.38587i
$568$		0			0
$569$	20.0000			0.838444			0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000			0.838444			0.419222	+	0.907884i	$0.362303\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$			32.0000i			1.33682i
$574$		0			0
$575$	4.00000	+	3.00000i	0.166812	+	0.125109i
$576$		0			0
$577$			22.0000i			0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$			22.0000i			0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$		0			0
$579$	−32.0000			−1.32987
$580$		0			0
$581$	33.0000			1.36907
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−6.00000	−	12.0000i	−0.248069	−	0.496139i
$586$		0			0
$587$			36.0000i			1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$			36.0000i			1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	16.0000			0.658152
$592$		0			0
$593$			42.0000i			1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$			42.0000i			1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$		0			0
$595$	−42.0000	+	21.0000i	−1.72183	+	0.860916i
$596$		0			0
$597$		−	36.0000i		−	1.47338i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	7.00000			0.285536			0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000			0.285536			0.142768	+	0.989756i	$0.454400\pi$
$602$		0			0
$603$		−	13.0000i		−	0.529401i
$604$		0			0
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$			30.0000i			1.21766i	0.793300	+	0.608831i	$0.208361\pi$
$607$			30.0000i			1.21766i	−0.793300	+	0.608831i	$0.791639\pi$
$608$		0			0
$609$	54.0000			2.18819
$610$		0			0
$611$	12.0000			0.485468
$612$		0			0
$613$			22.0000i			0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$			22.0000i			0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$		0			0
$615$	18.0000	+	36.0000i	0.725830	+	1.45166i
$616$		0			0
$617$			3.00000i			0.120775i	0.998175	+	0.0603877i	$0.0192337\pi$
$617$			3.00000i			0.120775i	−0.998175	+	0.0603877i	$0.980766\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$	−4.00000			−0.160514
$622$		0			0
$623$		−	30.0000i		−	1.20192i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−49.0000			−1.95376
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		−	22.0000i		−	0.874421i
$634$		0			0
$635$	−8.00000	−	16.0000i	−0.317470	−	0.634941i
$636$		0			0
$637$		−	12.0000i		−	0.475457i
$638$		0			0
$639$	13.0000			0.514272
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		−	5.00000i		−	0.197181i	−0.995128	−	0.0985904i	$-0.968567\pi$
$643$		−	5.00000i		−	0.197181i	0.995128	−	0.0985904i	$-0.0314334\pi$
$644$		0			0
$645$	−16.0000	+	8.00000i	−0.629999	+	0.315000i
$646$		0			0
$647$			2.00000i			0.0786281i	0.999227	+	0.0393141i	$0.0125173\pi$
$647$			2.00000i			0.0786281i	−0.999227	+	0.0393141i	$0.987483\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−18.0000			−0.705476
$652$		0			0
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		0			0
$655$	16.0000	−	8.00000i	0.625172	−	0.312586i
$656$		0			0
$657$		−	4.00000i		−	0.156055i
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−32.0000			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$662$		0			0
$663$			84.0000i			3.26229i
$664$		0			0
$665$	−12.0000	−	24.0000i	−0.465340	−	0.930680i
$666$		0			0
$667$			9.00000i			0.348481i
$668$		0			0
$669$	−4.00000			−0.154649
$670$		0			0
$671$		0			0
$672$		0			0
$673$			8.00000i			0.308377i	0.988041	+	0.154189i	$0.0492764\pi$
$673$			8.00000i			0.308377i	−0.988041	+	0.154189i	$0.950724\pi$
$674$		0			0
$675$	16.0000	+	12.0000i	0.615840	+	0.461880i
$676$		0			0
$677$			3.00000i			0.115299i	0.998337	+	0.0576497i	$0.0183606\pi$
$677$			3.00000i			0.115299i	−0.998337	+	0.0576497i	$0.981639\pi$
$678$		0			0
$679$	6.00000			0.230259
$680$		0			0
$681$	−32.0000			−1.22624
$682$		0			0
$683$		−	18.0000i		−	0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$		−	18.0000i		−	0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$		0			0
$685$	2.00000	+	4.00000i	0.0764161	+	0.152832i
$686$		0			0
$687$			16.0000i			0.610438i
$688$		0			0
$689$	−42.0000			−1.60007
$690$		0			0
$691$	−28.0000			−1.06517			−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000			−1.06517			−0.532585	+	0.846376i	$0.678779\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−22.0000	+	11.0000i	−0.834508	+	0.417254i
$696$		0			0
$697$		−	63.0000i		−	2.38630i
$698$		0			0
$699$	4.00000			0.151294
$700$		0			0
$701$	20.0000			0.755390			0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000			0.755390			0.377695	+	0.925930i	$0.376717\pi$
$702$		0			0
$703$		−	28.0000i		−	1.05604i
$704$		0			0
$705$	8.00000	−	4.00000i	0.301297	−	0.150649i
$706$		0			0
$707$			39.0000i			1.46675i
$708$		0			0
$709$	48.0000			1.80268			0.901339	−	0.433114i	$-0.142585\pi$
$709$	48.0000			1.80268			0.901339	+	0.433114i	$0.142585\pi$
$710$		0			0
$711$	−2.00000			−0.0750059
$712$		0			0
$713$		−	3.00000i		−	0.112351i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			30.0000i			1.12037i
$718$		0			0
$719$	27.0000			1.00693			0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000			1.00693			0.503465	+	0.864016i	$0.332058\pi$
$720$		0			0
$721$	−48.0000			−1.78761
$722$		0			0
$723$		−	44.0000i		−	1.63638i
$724$		0			0
$725$	27.0000	−	36.0000i	1.00275	−	1.33701i
$726$		0			0
$727$		−	1.00000i		−	0.0370879i	−0.999828	−	0.0185440i	$-0.994097\pi$
$727$		−	1.00000i		−	0.0370879i	0.999828	−	0.0185440i	$-0.00590307\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	28.0000			1.03562
$732$		0			0
$733$			31.0000i			1.14501i	0.819901	+	0.572506i	$0.194029\pi$
$733$			31.0000i			1.14501i	−0.819901	+	0.572506i	$0.805971\pi$
$734$		0			0
$735$	−4.00000	−	8.00000i	−0.147542	−	0.295084i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	19.0000			0.698926			0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000			0.698926			0.349463	+	0.936950i	$0.386364\pi$
$740$		0			0
$741$	−48.0000			−1.76332
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$	−40.0000	+	20.0000i	−1.46549	+	0.732743i
$746$		0			0
$747$		−	11.0000i		−	0.402469i
$748$		0			0
$749$	−45.0000			−1.64426
$750$		0			0
$751$	−26.0000			−0.948753			−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000			−0.948753			−0.474377	+	0.880322i	$0.657327\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	13.0000i		−	0.472493i	−0.971693	−	0.236247i	$-0.924083\pi$
$757$		−	13.0000i		−	0.472493i	0.971693	−	0.236247i	$-0.0759173\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−15.0000			−0.543750			−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000			−0.543750			−0.271875	+	0.962333i	$0.587644\pi$
$762$		0			0
$763$		−	6.00000i		−	0.217215i
$764$		0			0
$765$	7.00000	+	14.0000i	0.253086	+	0.506171i
$766$		0			0
$767$		−	54.0000i		−	1.94983i
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$	−52.0000			−1.87273
$772$		0			0
$773$		−	26.0000i		−	0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$		−	26.0000i		−	0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$		0			0
$775$	−9.00000	+	12.0000i	−0.323290	+	0.431053i
$776$		0			0
$777$		−	42.0000i		−	1.50674i
$778$		0			0
$779$	36.0000			1.28983
$780$		0			0
$781$		0			0
$782$		0			0
$783$			36.0000i			1.28654i
$784$		0			0
$785$	−13.0000	−	26.0000i	−0.463990	−	0.927980i
$786$		0			0
$787$		−	7.00000i		−	0.249523i	−0.992187	−	0.124762i	$-0.960183\pi$
$787$		−	7.00000i		−	0.249523i	0.992187	−	0.124762i	$-0.0398166\pi$
$788$		0			0
$789$	−18.0000			−0.640817
$790$		0			0
$791$	−15.0000			−0.533339
$792$		0			0
$793$		−	12.0000i		−	0.426132i
$794$		0			0
$795$	−28.0000	+	14.0000i	−0.993058	+	0.496529i
$796$		0			0
$797$			39.0000i			1.38145i	0.723117	+	0.690725i	$0.242709\pi$
$797$			39.0000i			1.38145i	−0.723117	+	0.690725i	$0.757291\pi$
$798$		0			0
$799$	−14.0000			−0.495284
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$		0			0
$804$		0			0
$805$	6.00000	−	3.00000i	0.211472	−	0.105736i
$806$		0			0
$807$		−	42.0000i		−	1.47847i
$808$		0			0
$809$	33.0000			1.16022			0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000			1.16022			0.580109	+	0.814539i	$0.303010\pi$
$810$		0			0
$811$	5.00000			0.175574			0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000			0.175574			0.0877869	+	0.996139i	$0.472021\pi$
$812$		0			0
$813$		−	50.0000i		−	1.75358i
$814$		0			0
$815$	−16.0000	−	32.0000i	−0.560456	−	1.12091i
$816$		0			0
$817$			16.0000i			0.559769i
$818$		0			0
$819$	−18.0000			−0.628971
$820$		0			0
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$			46.0000i			1.60346i	0.597687	+	0.801730i	$0.296087\pi$
$823$			46.0000i			1.60346i	−0.597687	+	0.801730i	$0.703913\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25.0000i		−	0.869335i	−0.900591	−	0.434668i	$-0.856866\pi$
$827$		−	25.0000i		−	0.869335i	0.900591	−	0.434668i	$-0.143134\pi$
$828$		0			0
$829$	47.0000			1.63238			0.816189	−	0.577785i	$-0.196083\pi$
$829$	47.0000			1.63238			0.816189	+	0.577785i	$0.196083\pi$
$830$		0			0
$831$	−4.00000			−0.138758
$832$		0			0
$833$			14.0000i			0.485071i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	12.0000i		−	0.414781i
$838$		0			0
$839$	26.0000			0.897620			0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000			0.897620			0.448810	+	0.893627i	$0.351848\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$			40.0000i			1.37767i
$844$		0			0
$845$	−46.0000	+	23.0000i	−1.58245	+	0.791224i
$846$		0			0
$847$			33.0000i			1.13389i
$848$		0			0
$849$	38.0000			1.30416
$850$		0			0
$851$	7.00000			0.239957
$852$		0			0
$853$		−	12.0000i		−	0.410872i	−0.978671	−	0.205436i	$-0.934139\pi$
$853$		−	12.0000i		−	0.410872i	0.978671	−	0.205436i	$-0.0658613\pi$
$854$		0			0
$855$	−8.00000	+	4.00000i	−0.273594	+	0.136797i
$856$		0			0
$857$		−	32.0000i		−	1.09310i	−0.837427	−	0.546550i	$-0.815941\pi$
$857$		−	32.0000i		−	1.09310i	0.837427	−	0.546550i	$-0.184059\pi$
$858$		0			0
$859$	15.0000			0.511793			0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000			0.511793			0.255897	+	0.966704i	$0.417629\pi$
$860$		0			0
$861$	54.0000			1.84032
$862$		0			0
$863$		−	30.0000i		−	1.02121i	−0.859815	−	0.510606i	$-0.829421\pi$
$863$		−	30.0000i		−	1.02121i	0.859815	−	0.510606i	$-0.170579\pi$
$864$		0			0
$865$	18.0000	+	36.0000i	0.612018	+	1.22404i
$866$		0			0
$867$		−	64.0000i		−	2.17355i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−78.0000			−2.64293
$872$		0			0
$873$		−	2.00000i		−	0.0676897i
$874$		0			0
$875$	−33.0000	−	6.00000i	−1.11560	−	0.202837i
$876$		0			0
$877$		−	40.0000i		−	1.35070i	−0.737496	−	0.675352i	$-0.763992\pi$
$877$		−	40.0000i		−	1.35070i	0.737496	−	0.675352i	$-0.236008\pi$
$878$		0			0
$879$	6.00000			0.202375
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$			40.0000i			1.34611i	0.739594	+	0.673054i	$0.235018\pi$
$883$			40.0000i			1.34611i	−0.739594	+	0.673054i	$0.764982\pi$
$884$		0			0
$885$	−18.0000	−	36.0000i	−0.605063	−	1.21013i
$886$		0			0
$887$			22.0000i			0.738688i	0.929293	+	0.369344i	$0.120418\pi$
$887$			22.0000i			0.738688i	−0.929293	+	0.369344i	$0.879582\pi$
$888$		0			0
$889$	−24.0000			−0.804934
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.00000i		−	0.267710i
$894$		0			0
$895$	40.0000	−	20.0000i	1.33705	−	0.668526i
$896$		0			0
$897$		−	12.0000i		−	0.400668i
$898$		0			0
$899$	−27.0000			−0.900500
$900$		0			0
$901$	49.0000			1.63243
$902$		0			0
$903$			24.0000i			0.798670i
$904$		0			0
$905$	−20.0000	+	10.0000i	−0.664822	+	0.332411i
$906$		0			0
$907$			41.0000i			1.36138i	0.732570	+	0.680691i	$0.238320\pi$
$907$			41.0000i			1.36138i	−0.732570	+	0.680691i	$0.761680\pi$
$908$		0			0
$909$	13.0000			0.431183
$910$		0			0
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−4.00000	−	8.00000i	−0.132236	−	0.264472i
$916$		0			0
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$	−56.0000			−1.84727			−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000			−1.84727			−0.923635	+	0.383274i	$0.874797\pi$
$920$		0			0
$921$	20.0000			0.659022
$922$		0			0
$923$		−	78.0000i		−	2.56740i
$924$		0			0
$925$	−28.0000	−	21.0000i	−0.920634	−	0.690476i
$926$		0			0
$927$			16.0000i			0.525509i
$928$		0			0
$929$	−27.0000			−0.885841			−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000			−0.885841			−0.442921	+	0.896561i	$0.646058\pi$
$930$		0			0
$931$	−8.00000			−0.262189
$932$		0			0
$933$			56.0000i			1.83336i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			26.0000i			0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$			26.0000i			0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$	−2.00000			−0.0652675
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$			9.00000i			0.293080i
$944$		0			0
$945$	24.0000	−	12.0000i	0.780720	−	0.390360i
$946$		0			0
$947$		−	42.0000i		−	1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$		−	42.0000i		−	1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$	24.0000			0.778253
$952$		0			0
$953$		−	42.0000i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$		−	42.0000i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$		0			0
$955$	32.0000	−	16.0000i	1.03550	−	0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.00000			0.193750
$960$		0			0
$961$	−22.0000			−0.709677
$962$		0			0
$963$			15.0000i			0.483368i
$964$		0			0
$965$	16.0000	+	32.0000i	0.515058	+	1.03012i
$966$		0			0
$967$		−	34.0000i		−	1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$		−	34.0000i		−	1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$		0			0
$969$	56.0000			1.79898
$970$		0			0
$971$	−50.0000			−1.60458			−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000			−1.60458			−0.802288	+	0.596937i	$0.796384\pi$
$972$		0			0
$973$			33.0000i			1.05793i
$974$		0			0
$975$	−36.0000	+	48.0000i	−1.15292	+	1.53723i
$976$		0			0
$977$			3.00000i			0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$			3.00000i			0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−2.00000			−0.0638551
$982$		0			0
$983$			15.0000i			0.478426i	0.970967	+	0.239213i	$0.0768894\pi$
$983$			15.0000i			0.478426i	−0.970967	+	0.239213i	$0.923111\pi$
$984$		0			0
$985$	−8.00000	−	16.0000i	−0.254901	−	0.509802i
$986$		0			0
$987$		−	12.0000i		−	0.381964i
$988$		0			0
$989$	−4.00000			−0.127193
$990$		0			0
$991$	5.00000			0.158830			0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000			0.158830			0.0794151	+	0.996842i	$0.474695\pi$
$992$		0			0
$993$			2.00000i			0.0634681i
$994$		0			0
$995$	−36.0000	+	18.0000i	−1.14128	+	0.570638i
$996$		0			0
$997$		−	20.0000i		−	0.633406i	−0.948525	−	0.316703i	$-0.897424\pi$
$997$		−	20.0000i		−	0.633406i	0.948525	−	0.316703i	$-0.102576\pi$
$998$		0			0
$999$	28.0000			0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.a.369.2	yes	2
4.3	odd	2		1840.2.e.a.369.1		2
5.2	odd	4		4600.2.a.o.1.1		1
5.3	odd	4		4600.2.a.b.1.1		1
5.4	even	2	inner	920.2.e.a.369.1	✓	2
20.3	even	4		9200.2.a.bi.1.1		1
20.7	even	4		9200.2.a.d.1.1		1
20.19	odd	2		1840.2.e.a.369.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.a.369.1	✓	2	5.4	even	2	inner
920.2.e.a.369.2	yes	2	1.1	even	1	trivial
1840.2.e.a.369.1		2	4.3	odd	2
1840.2.e.a.369.2		2	20.19	odd	2
4600.2.a.b.1.1		1	5.3	odd	4
4600.2.a.o.1.1		1	5.2	odd	4
9200.2.a.d.1.1		1	20.7	even	4
9200.2.a.bi.1.1		1	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -3.00000i q^{7} -1.00000 q^{9} +6.00000i q^{13} +(2.00000 + 4.00000i) q^{15} -7.00000i q^{17} +4.00000 q^{19} +6.00000 q^{21} +1.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +4.00000i q^{27} +9.00000 q^{29} -3.00000 q^{31} +(-3.00000 - 6.00000i) q^{35} -7.00000i q^{37} -12.0000 q^{39} +9.00000 q^{41} +4.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -2.00000i q^{47} -2.00000 q^{49} +14.0000 q^{51} +7.00000i q^{53} +8.00000i q^{57} -9.00000 q^{59} -2.00000 q^{61} +3.00000i q^{63} +(6.00000 + 12.0000i) q^{65} +13.0000i q^{67} -2.00000 q^{69} -13.0000 q^{71} +4.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +2.00000 q^{79} -11.0000 q^{81} +11.0000i q^{83} +(-7.00000 - 14.0000i) q^{85} +18.0000i q^{87} +10.0000 q^{89} +18.0000 q^{91} -6.00000i q^{93} +(8.00000 - 4.00000i) q^{95} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9} + 4 q^{15} + 8 q^{19} + 12 q^{21} + 6 q^{25} + 18 q^{29} - 6 q^{31} - 6 q^{35} - 24 q^{39} + 18 q^{41} - 4 q^{45} - 4 q^{49} + 28 q^{51} - 18 q^{59} - 4 q^{61} + 12 q^{65} - 4 q^{69}+ \cdots + 16 q^{95}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 + 4 * q^15 + 8 * q^19 + 12 * q^21 + 6 * q^25 + 18 * q^29 - 6 * q^31 - 6 * q^35 - 24 * q^39 + 18 * q^41 - 4 * q^45 - 4 * q^49 + 28 * q^51 - 18 * q^59 - 4 * q^61 + 12 * q^65 - 4 * q^69 - 26 * q^71 + 16 * q^75 + 4 * q^79 - 22 * q^81 - 14 * q^85 + 20 * q^89 + 36 * q^91 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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