LMFDB - Embedded newform 2320.2.g.i.1681.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(1681,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.1681"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.16516096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.4
Root			$1.30397i$ of defining polynomial
Character	$\chi$	$=$	2320.1681
Dual form			2320.2.g.i.1681.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.537080i q^{3} +1.00000 q^{5} +1.29966 q^{7} +2.71155 q^{9} -0.537080i q^{11} -5.71155 q^{13} +0.537080i q^{15} +6.91060i q^{17} +4.30266i q^{19} +0.698024i q^{21} -5.01121 q^{23} +1.00000 q^{25} +3.06756i q^{27} +(0.299664 + 5.37682i) q^{29} -8.44438i q^{31} +0.288455 q^{33} +1.29966 q^{35} +7.98476i q^{37} -3.06756i q^{39} +0.698024i q^{41} +10.2166i q^{43} +2.71155 q^{45} -0.537080i q^{47} -5.31087 q^{49} -3.71155 q^{51} +7.11222 q^{53} -0.537080i q^{55} -2.31087 q^{57} +7.71155 q^{59} -5.91390i q^{61} +3.52410 q^{63} -5.71155 q^{65} +9.01121 q^{67} -2.69142i q^{69} +4.28845 q^{71} -6.21258i q^{73} +0.537080i q^{75} -0.698024i q^{77} +10.2166i q^{79} +6.48711 q^{81} +2.70034 q^{83} +6.91060i q^{85} +(-2.88778 + 0.160944i) q^{87} -9.67948i q^{89} -7.42309 q^{91} +4.53531 q^{93} +4.30266i q^{95} +2.07086i q^{97} -1.45632i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{5} - 14 q^{9} - 4 q^{13} + 8 q^{23} + 6 q^{25} - 6 q^{29} + 32 q^{33} - 14 q^{45} + 14 q^{49} + 8 q^{51} + 28 q^{53} + 32 q^{57} + 16 q^{59} - 16 q^{63} - 4 q^{65} + 16 q^{67} + 56 q^{71} + 38 q^{81}+ \cdots - 48 q^{93}+O(q^{100}) $ 6 * q + 6 * q^5 - 14 * q^9 - 4 * q^13 + 8 * q^23 + 6 * q^25 - 6 * q^29 + 32 * q^33 - 14 * q^45 + 14 * q^49 + 8 * q^51 + 28 * q^53 + 32 * q^57 + 16 * q^59 - 16 * q^63 - 4 * q^65 + 16 * q^67 + 56 * q^71 + 38 * q^81 + 24 * q^83 - 32 * q^87 + 16 * q^91 - 48 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			0.537080i			0.310083i	0.987908	+	0.155042i	$0.0495512\pi$
$3$			0.537080i			0.310083i	−0.987908	+	0.155042i	$0.950449\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	1.29966			0.491227			0.245613	−	0.969368i	$-0.421011\pi$
$7$	1.29966			0.491227			0.245613	+	0.969368i	$0.421011\pi$
$8$		0			0
$9$	2.71155			0.903848
$10$		0			0
$11$		−	0.537080i		−	0.161936i	−0.996717	−	0.0809679i	$-0.974199\pi$
$11$		−	0.537080i		−	0.161936i	0.996717	−	0.0809679i	$-0.0258011\pi$
$12$		0			0
$13$	−5.71155			−1.58410			−0.792049	−	0.610458i	$-0.790985\pi$
$13$	−5.71155			−1.58410			−0.792049	+	0.610458i	$0.790985\pi$
$14$		0			0
$15$			0.537080i			0.138673i
$16$		0			0
$17$			6.91060i			1.67607i	0.545619	+	0.838033i	$0.316295\pi$
$17$			6.91060i			1.67607i	−0.545619	+	0.838033i	$0.683705\pi$
$18$		0			0
$19$			4.30266i			0.987098i	0.869718	+	0.493549i	$0.164301\pi$
$19$			4.30266i			0.987098i	−0.869718	+	0.493549i	$0.835699\pi$
$20$		0			0
$21$			0.698024i			0.152321i
$22$		0			0
$23$	−5.01121			−1.04491			−0.522455	−	0.852667i	$-0.674984\pi$
$23$	−5.01121			−1.04491			−0.522455	+	0.852667i	$0.674984\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			3.06756i			0.590352i
$28$		0			0
$29$	0.299664	+	5.37682i	0.0556462	+	0.998451i
$29$	0.299664	+	5.37682i	0.0556462	+	0.998451i
$30$		0			0
$31$		−	8.44438i		−	1.51666i	−0.651874	−	0.758328i	$-0.726017\pi$
$31$		−	8.44438i		−	1.51666i	0.651874	−	0.758328i	$-0.273983\pi$
$32$		0			0
$33$	0.288455			0.0502136
$34$		0			0
$35$	1.29966			0.219683
$36$		0			0
$37$			7.98476i			1.31269i	0.754462	+	0.656343i	$0.227898\pi$
$37$			7.98476i			1.31269i	−0.754462	+	0.656343i	$0.772102\pi$
$38$		0			0
$39$		−	3.06756i		−	0.491202i
$40$		0			0
$41$			0.698024i			0.109013i	0.998513	+	0.0545065i	$0.0173586\pi$
$41$			0.698024i			0.109013i	−0.998513	+	0.0545065i	$0.982641\pi$
$42$		0			0
$43$			10.2166i			1.55801i	0.627017	+	0.779006i	$0.284276\pi$
$43$			10.2166i			1.55801i	−0.627017	+	0.779006i	$0.715724\pi$
$44$		0			0
$45$	2.71155			0.404213
$46$		0			0
$47$		−	0.537080i		−	0.0783412i	−0.999233	−	0.0391706i	$-0.987528\pi$
$47$		−	0.537080i		−	0.0783412i	0.999233	−	0.0391706i	$-0.0124716\pi$
$48$		0			0
$49$	−5.31087			−0.758696
$50$		0			0
$51$	−3.71155			−0.519720
$52$		0			0
$53$	7.11222			0.976938			0.488469	−	0.872581i	$-0.337556\pi$
$53$	7.11222			0.976938			0.488469	+	0.872581i	$0.337556\pi$
$54$		0			0
$55$		−	0.537080i		−	0.0724199i
$56$		0			0
$57$	−2.31087			−0.306083
$58$		0			0
$59$	7.71155			1.00396			0.501979	−	0.864880i	$-0.332606\pi$
$59$	7.71155			1.00396			0.501979	+	0.864880i	$0.332606\pi$
$60$		0			0
$61$		−	5.91390i		−	0.757197i	−0.925561	−	0.378599i	$-0.876406\pi$
$61$		−	5.91390i		−	0.757197i	0.925561	−	0.378599i	$-0.123594\pi$
$62$		0			0
$63$	3.52410			0.443995
$64$		0			0
$65$	−5.71155			−0.708430
$66$		0			0
$67$	9.01121			1.10089			0.550447	−	0.834870i	$-0.314457\pi$
$67$	9.01121			1.10089			0.550447	+	0.834870i	$0.314457\pi$
$68$		0			0
$69$		−	2.69142i		−	0.324009i
$70$		0			0
$71$	4.28845			0.508946			0.254473	−	0.967080i	$-0.418098\pi$
$71$	4.28845			0.508946			0.254473	+	0.967080i	$0.418098\pi$
$72$		0			0
$73$		−	6.21258i		−	0.727127i	−0.931569	−	0.363563i	$-0.881560\pi$
$73$		−	6.21258i		−	0.727127i	0.931569	−	0.363563i	$-0.118440\pi$
$74$		0			0
$75$			0.537080i			0.0620167i
$76$		0			0
$77$		−	0.698024i		−	0.0795472i
$78$		0			0
$79$			10.2166i			1.14945i	0.818346	+	0.574726i	$0.194892\pi$
$79$			10.2166i			1.14945i	−0.818346	+	0.574726i	$0.805108\pi$
$80$		0			0
$81$	6.48711			0.720790
$82$		0			0
$83$	2.70034			0.296400			0.148200	−	0.988957i	$-0.452652\pi$
$83$	2.70034			0.296400			0.148200	+	0.988957i	$0.452652\pi$
$84$		0			0
$85$			6.91060i			0.749560i
$86$		0			0
$87$	−2.88778	+	0.160944i	−0.309603	+	0.0172550i
$88$		0			0
$89$		−	9.67948i		−	1.02602i	−0.858382	−	0.513011i	$-0.828530\pi$
$89$		−	9.67948i		−	1.02602i	0.858382	−	0.513011i	$-0.171470\pi$
$90$		0			0
$91$	−7.42309			−0.778151
$92$		0			0
$93$	4.53531			0.470289
$94$		0			0
$95$			4.30266i			0.441444i
$96$		0			0
$97$			2.07086i			0.210264i	0.994458	+	0.105132i	$0.0335265\pi$
$97$			2.07086i			0.210264i	−0.994458	+	0.105132i	$0.966474\pi$
$98$		0			0
$99$		−	1.45632i		−	0.146365i
$100$		0			0
$101$			3.06756i			0.305233i	0.988285	+	0.152617i	$0.0487700\pi$
$101$			3.06756i			0.305233i	−0.988285	+	0.152617i	$0.951230\pi$
$102$		0			0
$103$	0.187447			0.0184697			0.00923487	−	0.999957i	$-0.497060\pi$
$103$	0.187447			0.0184697			0.00923487	+	0.999957i	$0.497060\pi$
$104$		0			0
$105$			0.698024i			0.0681201i
$106$		0			0
$107$	−12.7228			−1.22996			−0.614978	−	0.788545i	$-0.710835\pi$
$107$	−12.7228			−1.22996			−0.614978	+	0.788545i	$0.710835\pi$
$108$		0			0
$109$	−11.1122			−1.06436			−0.532179	−	0.846632i	$-0.678627\pi$
$109$	−11.1122			−1.06436			−0.532179	+	0.846632i	$0.678627\pi$
$110$		0			0
$111$	−4.28845			−0.407042
$112$		0			0
$113$			9.60202i			0.903282i	0.892200	+	0.451641i	$0.149161\pi$
$113$			9.60202i			0.903282i	−0.892200	+	0.451641i	$0.850839\pi$
$114$		0			0
$115$	−5.01121			−0.467298
$116$		0			0
$117$	−15.4871			−1.43178
$118$		0			0
$119$			8.98146i			0.823329i
$120$		0			0
$121$	10.7115			0.973777
$122$		0			0
$123$	−0.374895			−0.0338031
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	6.07484i		−	0.539055i	−0.962993	−	0.269528i	$-0.913132\pi$
$127$		−	6.07484i		−	0.539055i	0.962993	−	0.269528i	$-0.0868676\pi$
$128$		0			0
$129$	−5.48711			−0.483113
$130$		0			0
$131$			18.1239i			1.58349i	0.610852	+	0.791744i	$0.290827\pi$
$131$			18.1239i			1.58349i	−0.610852	+	0.791744i	$0.709173\pi$
$132$		0			0
$133$			5.59201i			0.484889i
$134$		0			0
$135$			3.06756i			0.264013i
$136$		0			0
$137$		−	13.2006i		−	1.12781i	−0.825841	−	0.563903i	$-0.809299\pi$
$137$		−	13.2006i		−	1.12781i	0.825841	−	0.563903i	$-0.190701\pi$
$138$		0			0
$139$	−14.5993			−1.23830			−0.619149	−	0.785273i	$-0.712522\pi$
$139$	−14.5993			−1.23830			−0.619149	+	0.785273i	$0.712522\pi$
$140$		0			0
$141$	0.288455			0.0242923
$142$		0			0
$143$			3.06756i			0.256522i
$144$		0			0
$145$	0.299664	+	5.37682i	0.0248857	+	0.446521i
$146$		0			0
$147$		−	2.85236i		−	0.235259i
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$	21.7340			1.76868			0.884342	−	0.466839i	$-0.154607\pi$
$151$	21.7340			1.76868			0.884342	+	0.466839i	$0.154607\pi$
$152$		0			0
$153$			18.7384i			1.51491i
$154$		0			0
$155$		−	8.44438i		−	0.678269i
$156$		0			0
$157$			5.29334i			0.422454i	0.977437	+	0.211227i	$0.0677460\pi$
$157$			5.29334i			0.422454i	−0.977437	+	0.211227i	$0.932254\pi$
$158$		0			0
$159$			3.81983i			0.302932i
$160$		0			0
$161$	−6.51289			−0.513288
$162$		0			0
$163$			6.07484i			0.475819i	0.971287	+	0.237909i	$0.0764621\pi$
$163$			6.07484i			0.475819i	−0.971287	+	0.237909i	$0.923538\pi$
$164$		0			0
$165$	0.288455			0.0224562
$166$		0			0
$167$	−5.01121			−0.387779			−0.193890	−	0.981023i	$-0.562110\pi$
$167$	−5.01121			−0.387779			−0.193890	+	0.981023i	$0.562110\pi$
$168$		0			0
$169$	19.6217			1.50937
$170$		0			0
$171$			11.6669i			0.892187i
$172$		0			0
$173$	−8.31087			−0.631864			−0.315932	−	0.948782i	$-0.602317\pi$
$173$	−8.31087			−0.631864			−0.315932	+	0.948782i	$0.602317\pi$
$174$		0			0
$175$	1.29966			0.0982454
$176$		0			0
$177$			4.14172i			0.311311i
$178$		0			0
$179$	−6.59933			−0.493257			−0.246628	−	0.969110i	$-0.579323\pi$
$179$	−6.59933			−0.493257			−0.246628	+	0.969110i	$0.579323\pi$
$180$		0			0
$181$	−3.48711			−0.259195			−0.129597	−	0.991567i	$-0.541369\pi$
$181$	−3.48711			−0.259195			−0.129597	+	0.991567i	$0.541369\pi$
$182$		0			0
$183$	3.17624			0.234794
$184$		0			0
$185$			7.98476i			0.587051i
$186$		0			0
$187$	3.71155			0.271415
$188$		0			0
$189$			3.98679i			0.289997i
$190$		0			0
$191$			2.68540i			0.194309i	0.995269	+	0.0971544i	$0.0309741\pi$
$191$			2.68540i			0.194309i	−0.995269	+	0.0971544i	$0.969026\pi$
$192$		0			0
$193$			4.76228i			0.342796i	0.985202	+	0.171398i	$0.0548285\pi$
$193$			4.76228i			0.342796i	−0.985202	+	0.171398i	$0.945172\pi$
$194$		0			0
$195$		−	3.06756i		−	0.219672i
$196$		0			0
$197$	16.0224			1.14155			0.570775	−	0.821106i	$-0.306643\pi$
$197$	16.0224			1.14155			0.570775	+	0.821106i	$0.306643\pi$
$198$		0			0
$199$	1.68913			0.119739			0.0598695	−	0.998206i	$-0.480932\pi$
$199$	1.68913			0.119739			0.0598695	+	0.998206i	$0.480932\pi$
$200$		0			0
$201$			4.83974i			0.341369i
$202$		0			0
$203$	0.389463	+	6.98806i	0.0273349	+	0.490466i
$204$		0			0
$205$			0.698024i			0.0487521i
$206$		0			0
$207$	−13.5881			−0.944440
$208$		0			0
$209$	2.31087			0.159846
$210$		0			0
$211$			6.99408i			0.481492i	0.970588	+	0.240746i	$0.0773921\pi$
$211$			6.99408i			0.481492i	−0.970588	+	0.240746i	$0.922608\pi$
$212$		0			0
$213$			2.30324i			0.157816i
$214$		0			0
$215$			10.2166i			0.696764i
$216$		0			0
$217$		−	10.9749i		−	0.745022i
$218$		0			0
$219$	3.33665			0.225470
$220$		0			0
$221$		−	39.4702i		−	2.65505i
$222$		0			0
$223$	−2.98879			−0.200144			−0.100072	−	0.994980i	$-0.531907\pi$
$223$	−2.98879			−0.200144			−0.100072	+	0.994980i	$0.531907\pi$
$224$		0			0
$225$	2.71155			0.180770
$226$		0			0
$227$	23.8574			1.58347			0.791735	−	0.610864i	$-0.209178\pi$
$227$	23.8574			1.58347			0.791735	+	0.610864i	$0.209178\pi$
$228$		0			0
$229$			0.321887i			0.0212709i	0.999943	+	0.0106355i	$0.00338543\pi$
$229$			0.321887i			0.0212709i	−0.999943	+	0.0106355i	$0.996615\pi$
$230$		0			0
$231$	0.374895			0.0246663
$232$		0			0
$233$	−16.0224			−1.04966			−0.524832	−	0.851206i	$-0.675872\pi$
$233$	−16.0224			−1.04966			−0.524832	+	0.851206i	$0.675872\pi$
$234$		0			0
$235$		−	0.537080i		−	0.0350352i
$236$		0			0
$237$	−5.48711			−0.356426
$238$		0			0
$239$	−19.7115			−1.27503			−0.637517	−	0.770436i	$-0.720038\pi$
$239$	−19.7115			−1.27503			−0.637517	+	0.770436i	$0.720038\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$			12.6868i			0.813857i
$244$		0			0
$245$	−5.31087			−0.339299
$246$		0			0
$247$		−	24.5748i		−	1.56366i
$248$		0			0
$249$			1.45030i			0.0919088i
$250$		0			0
$251$		−	24.7900i		−	1.56473i	−0.622818	−	0.782366i	$-0.714012\pi$
$251$		−	24.7900i		−	1.56473i	0.622818	−	0.782366i	$-0.285988\pi$
$252$		0			0
$253$			2.69142i			0.169208i
$254$		0			0
$255$	−3.71155			−0.232426
$256$		0			0
$257$	16.8878			1.05343			0.526715	−	0.850042i	$-0.323423\pi$
$257$	16.8878			1.05343			0.526715	+	0.850042i	$0.323423\pi$
$258$		0			0
$259$			10.3775i			0.644827i
$260$		0			0
$261$	0.812553	+	14.5795i	0.0502957	+	0.902448i
$262$		0			0
$263$			7.74635i			0.477661i	0.971061	+	0.238830i	$0.0767640\pi$
$263$			7.74635i			0.477661i	−0.971061	+	0.238830i	$0.923236\pi$
$264$		0			0
$265$	7.11222			0.436900
$266$		0			0
$267$	5.19866			0.318153
$268$		0			0
$269$		−	11.4517i		−	0.698220i	−0.937082	−	0.349110i	$-0.886484\pi$
$269$		−	11.4517i		−	0.698220i	0.937082	−	0.349110i	$-0.113516\pi$
$270$		0			0
$271$		−	25.8642i		−	1.57114i	−0.618774	−	0.785569i	$-0.712370\pi$
$271$		−	25.8642i		−	1.57114i	0.618774	−	0.785569i	$-0.287630\pi$
$272$		0			0
$273$		−	3.98679i		−	0.241292i
$274$		0			0
$275$		−	0.537080i		−	0.0323871i
$276$		0			0
$277$	−8.88778			−0.534015			−0.267008	−	0.963694i	$-0.586035\pi$
$277$	−8.88778			−0.534015			−0.267008	+	0.963694i	$0.586035\pi$
$278$		0			0
$279$		−	22.8973i		−	1.37083i
$280$		0			0
$281$	−22.5353			−1.34434			−0.672172	−	0.740395i	$-0.734638\pi$
$281$	−22.5353			−1.34434			−0.672172	+	0.740395i	$0.734638\pi$
$282$		0			0
$283$	14.4119			0.856697			0.428349	−	0.903614i	$-0.359096\pi$
$283$	14.4119			0.856697			0.428349	+	0.903614i	$0.359096\pi$
$284$		0			0
$285$	−2.31087			−0.136884
$286$		0			0
$287$			0.907196i			0.0535501i
$288$		0			0
$289$	−30.7564			−1.80920
$290$		0			0
$291$	−1.11222			−0.0651993
$292$		0			0
$293$			0.996698i			0.0582277i	0.999576	+	0.0291139i	$0.00926854\pi$
$293$			0.996698i			0.0582277i	−0.999576	+	0.0291139i	$0.990731\pi$
$294$		0			0
$295$	7.71155			0.448984
$296$		0			0
$297$	1.64752			0.0955990
$298$		0			0
$299$	28.6217			1.65524
$300$		0			0
$301$			13.2781i			0.765337i
$302$		0			0
$303$	−1.64752			−0.0946478
$304$		0			0
$305$		−	5.91390i		−	0.338629i
$306$		0			0
$307$		−	27.1053i		−	1.54698i	−0.633807	−	0.773491i	$-0.718509\pi$
$307$		−	27.1053i		−	1.54698i	0.633807	−	0.773491i	$-0.281491\pi$
$308$		0			0
$309$			0.100674i			0.00572716i
$310$		0			0
$311$		−	18.8761i		−	1.07037i	−0.844736	−	0.535184i	$-0.820242\pi$
$311$		−	18.8761i		−	1.07037i	0.844736	−	0.535184i	$-0.179758\pi$
$312$		0			0
$313$	6.28845			0.355444			0.177722	−	0.984081i	$-0.443127\pi$
$313$	6.28845			0.355444			0.177722	+	0.984081i	$0.443127\pi$
$314$		0			0
$315$	3.52410			0.198560
$316$		0			0
$317$		−	13.5225i		−	0.759501i	−0.925089	−	0.379750i	$-0.876010\pi$
$317$		−	13.5225i		−	0.759501i	0.925089	−	0.379750i	$-0.123990\pi$
$318$		0			0
$319$	2.88778	−	0.160944i	0.161685	−	0.00901111i
$320$		0			0
$321$		−	6.83314i		−	0.381389i
$322$		0			0
$323$	−29.7340			−1.65444
$324$		0			0
$325$	−5.71155			−0.316820
$326$		0			0
$327$		−	5.96815i		−	0.330039i
$328$		0			0
$329$		−	0.698024i		−	0.0384833i
$330$		0			0
$331$			20.8695i			1.14709i	0.819173	+	0.573547i	$0.194433\pi$
$331$			20.8695i			1.14709i	−0.819173	+	0.573547i	$0.805567\pi$
$332$		0			0
$333$			21.6510i			1.18647i
$334$		0			0
$335$	9.01121			0.492335
$336$		0			0
$337$			31.1093i			1.69463i	0.531089	+	0.847316i	$0.321783\pi$
$337$			31.1093i			1.69463i	−0.531089	+	0.847316i	$0.678217\pi$
$338$		0			0
$339$	−5.15705			−0.280093
$340$		0			0
$341$	−4.53531			−0.245601
$342$		0			0
$343$	−16.0000			−0.863919
$344$		0			0
$345$		−	2.69142i		−	0.144901i
$346$		0			0
$347$	−32.7676			−1.75906			−0.879528	−	0.475847i	$-0.842142\pi$
$347$	−32.7676			−1.75906			−0.879528	+	0.475847i	$0.842142\pi$
$348$		0			0
$349$	24.0224			1.28589			0.642945	−	0.765912i	$-0.277712\pi$
$349$	24.0224			1.28589			0.642945	+	0.765912i	$0.277712\pi$
$350$		0			0
$351$		−	17.5205i		−	0.935175i
$352$		0			0
$353$	28.3557			1.50922			0.754611	−	0.656172i	$-0.227826\pi$
$353$	28.3557			1.50922			0.754611	+	0.656172i	$0.227826\pi$
$354$		0			0
$355$	4.28845			0.227608
$356$		0			0
$357$	−4.82376			−0.255301
$358$		0			0
$359$		−	30.7039i		−	1.62049i	−0.586090	−	0.810246i	$-0.699334\pi$
$359$		−	30.7039i		−	1.62049i	0.586090	−	0.810246i	$-0.300666\pi$
$360$		0			0
$361$	0.487111			0.0256374
$362$		0			0
$363$			5.75296i			0.301952i
$364$		0			0
$365$		−	6.21258i		−	0.325181i
$366$		0			0
$367$		−	1.93313i		−	0.100908i	−0.998726	−	0.0504542i	$-0.983933\pi$
$367$		−	1.93313i		−	0.100908i	0.998726	−	0.0504542i	$-0.0160669\pi$
$368$		0			0
$369$			1.89272i			0.0985312i
$370$		0			0
$371$	9.24349			0.479898
$372$		0			0
$373$	0.801344			0.0414920			0.0207460	−	0.999785i	$-0.493396\pi$
$373$	0.801344			0.0414920			0.0207460	+	0.999785i	$0.493396\pi$
$374$		0			0
$375$			0.537080i			0.0277347i
$376$		0			0
$377$	−1.71155	−	30.7100i	−0.0881491	−	1.58164i
$378$		0			0
$379$			25.8642i			1.32855i	0.747486	+	0.664277i	$0.231261\pi$
$379$			25.8642i			1.32855i	−0.747486	+	0.664277i	$0.768739\pi$
$380$		0			0
$381$	3.26268			0.167152
$382$		0			0
$383$	20.7676			1.06117			0.530587	−	0.847630i	$-0.321971\pi$
$383$	20.7676			1.06117			0.530587	+	0.847630i	$0.321971\pi$
$384$		0			0
$385$		−	0.698024i		−	0.0355746i
$386$		0			0
$387$			27.7027i			1.40821i
$388$		0			0
$389$		−	2.52446i		−	0.127995i	−0.997950	−	0.0639975i	$-0.979615\pi$
$389$		−	2.52446i		−	0.127995i	0.997950	−	0.0639975i	$-0.0203850\pi$
$390$		0			0
$391$		−	34.6305i		−	1.75134i
$392$		0			0
$393$	−9.73396			−0.491013
$394$		0			0
$395$			10.2166i			0.514051i
$396$		0			0
$397$	13.2211			0.663547			0.331773	−	0.943359i	$-0.392353\pi$
$397$	13.2211			0.663547			0.331773	+	0.943359i	$0.392353\pi$
$398$		0			0
$399$	−3.00336			−0.150356
$400$		0			0
$401$	25.1795			1.25740			0.628701	−	0.777647i	$-0.283587\pi$
$401$	25.1795			1.25740			0.628701	+	0.777647i	$0.283587\pi$
$402$		0			0
$403$			48.2304i			2.40253i
$404$		0			0
$405$	6.48711			0.322347
$406$		0			0
$407$	4.28845			0.212571
$408$		0			0
$409$			28.1855i			1.39368i	0.717225	+	0.696842i	$0.245412\pi$
$409$			28.1855i			1.39368i	−0.717225	+	0.696842i	$0.754588\pi$
$410$		0			0
$411$	7.08980			0.349714
$412$		0			0
$413$	10.0224			0.493171
$414$		0			0
$415$	2.70034			0.132554
$416$		0			0
$417$		−	7.84101i		−	0.383976i
$418$		0			0
$419$	20.9102			1.02153			0.510765	−	0.859720i	$-0.329362\pi$
$419$	20.9102			1.02153			0.510765	+	0.859720i	$0.329362\pi$
$420$		0			0
$421$		−	4.67278i		−	0.227737i	−0.993496	−	0.113869i	$-0.963676\pi$
$421$		−	4.67278i		−	0.227737i	0.993496	−	0.113869i	$-0.0363243\pi$
$422$		0			0
$423$		−	1.45632i		−	0.0708085i
$424$		0			0
$425$			6.91060i			0.335213i
$426$		0			0
$427$		−	7.68608i		−	0.371956i
$428$		0			0
$429$	−1.64752			−0.0795432
$430$		0			0
$431$	−12.2469			−0.589910			−0.294955	−	0.955511i	$-0.595305\pi$
$431$	−12.2469			−0.589910			−0.294955	+	0.955511i	$0.595305\pi$
$432$		0			0
$433$		−	11.8046i		−	0.567292i	−0.958929	−	0.283646i	$-0.908456\pi$
$433$		−	11.8046i		−	0.567292i	0.958929	−	0.283646i	$-0.0915441\pi$
$434$		0			0
$435$	−2.88778	+	0.160944i	−0.138459	+	0.00771666i
$436$		0			0
$437$		−	21.5615i		−	1.03143i
$438$		0			0
$439$	29.5319			1.40948			0.704741	−	0.709464i	$-0.251063\pi$
$439$	29.5319			1.40948			0.704741	+	0.709464i	$0.251063\pi$
$440$		0			0
$441$	−14.4007			−0.685746
$442$		0			0
$443$			3.28275i			0.155968i	0.996955	+	0.0779841i	$0.0248483\pi$
$443$			3.28275i			0.155968i	−0.996955	+	0.0779841i	$0.975152\pi$
$444$		0			0
$445$		−	9.67948i		−	0.458851i
$446$		0			0
$447$			3.22248i			0.152418i
$448$		0			0
$449$		−	28.9378i		−	1.36566i	−0.730578	−	0.682829i	$-0.760749\pi$
$449$		−	28.9378i		−	1.36566i	0.730578	−	0.682829i	$-0.239251\pi$
$450$		0			0
$451$	0.374895			0.0176531
$452$		0			0
$453$			11.6729i			0.548440i
$454$		0			0
$455$	−7.42309			−0.348000
$456$		0			0
$457$	−14.5353			−0.679933			−0.339966	−	0.940438i	$-0.610416\pi$
$457$	−14.5353			−0.679933			−0.339966	+	0.940438i	$0.610416\pi$
$458$		0			0
$459$	−21.1987			−0.989469
$460$		0			0
$461$		−	22.4265i		−	1.04451i	−0.852790	−	0.522254i	$-0.825091\pi$
$461$		−	22.4265i		−	1.04451i	0.852790	−	0.522254i	$-0.174909\pi$
$462$		0			0
$463$	10.0370			0.466458			0.233229	−	0.972422i	$-0.425071\pi$
$463$	10.0370			0.466458			0.233229	+	0.972422i	$0.425071\pi$
$464$		0			0
$465$	4.53531			0.210320
$466$		0			0
$467$		−	4.35691i		−	0.201614i	−0.994906	−	0.100807i	$-0.967858\pi$
$467$		−	4.35691i		−	0.201614i	0.994906	−	0.100807i	$-0.0321424\pi$
$468$		0			0
$469$	11.7115			0.540789
$470$		0			0
$471$	−2.84295			−0.130996
$472$		0			0
$473$	5.48711			0.252298
$474$		0			0
$475$			4.30266i			0.197420i
$476$		0			0
$477$	19.2851			0.883004
$478$		0			0
$479$		−	28.0125i		−	1.27992i	−0.768406	−	0.639962i	$-0.778950\pi$
$479$		−	28.0125i		−	1.27992i	0.768406	−	0.639962i	$-0.221050\pi$
$480$		0			0
$481$		−	45.6053i		−	2.07942i
$482$		0			0
$483$		−	3.49794i		−	0.159162i
$484$		0			0
$485$			2.07086i			0.0940328i
$486$		0			0
$487$	−19.5241			−0.884721			−0.442361	−	0.896837i	$-0.645859\pi$
$487$	−19.5241			−0.884721			−0.442361	+	0.896837i	$0.645859\pi$
$488$		0			0
$489$	−3.26268			−0.147543
$490$		0			0
$491$		−	36.4087i		−	1.64310i	−0.570137	−	0.821550i	$-0.693110\pi$
$491$		−	36.4087i		−	1.64310i	0.570137	−	0.821550i	$-0.306890\pi$
$492$		0			0
$493$	−37.1571	+	2.07086i	−1.67347	+	0.0932668i
$494$		0			0
$495$		−	1.45632i		−	0.0654566i
$496$		0			0
$497$	5.57355			0.250008
$498$		0			0
$499$	−31.4679			−1.40870			−0.704349	−	0.709854i	$-0.748761\pi$
$499$	−31.4679			−1.40870			−0.704349	+	0.709854i	$0.748761\pi$
$500$		0			0
$501$		−	2.69142i		−	0.120244i
$502$		0			0
$503$		−	8.82051i		−	0.393287i	−0.980475	−	0.196644i	$-0.936996\pi$
$503$		−	8.82051i		−	0.393287i	0.980475	−	0.196644i	$-0.0630042\pi$
$504$		0			0
$505$			3.06756i			0.136504i
$506$		0			0
$507$			10.5384i			0.468029i
$508$		0			0
$509$	14.9102			0.660883			0.330442	−	0.943826i	$-0.392802\pi$
$509$	14.9102			0.660883			0.330442	+	0.943826i	$0.392802\pi$
$510$		0			0
$511$		−	8.07426i		−	0.357184i
$512$		0			0
$513$	−13.1987			−0.582735
$514$		0			0
$515$	0.187447			0.00825991
$516$		0			0
$517$	−0.288455			−0.0126862
$518$		0			0
$519$		−	4.46360i		−	0.195930i
$520$		0			0
$521$	0.512889			0.0224701			0.0112350	−	0.999937i	$-0.496424\pi$
$521$	0.512889			0.0224701			0.0112350	+	0.999937i	$0.496424\pi$
$522$		0			0
$523$	−32.8092			−1.43465			−0.717323	−	0.696741i	$-0.754633\pi$
$523$	−32.8092			−1.43465			−0.717323	+	0.696741i	$0.754633\pi$
$524$		0			0
$525$			0.698024i			0.0304643i
$526$		0			0
$527$	58.3557			2.54201
$528$		0			0
$529$	2.11222			0.0918355
$530$		0			0
$531$	20.9102			0.907425
$532$		0			0
$533$		−	3.98679i		−	0.172687i
$534$		0			0
$535$	−12.7228			−0.550053
$536$		0			0
$537$		−	3.54437i		−	0.152951i
$538$		0			0
$539$			2.85236i			0.122860i
$540$		0			0
$541$		−	25.8160i		−	1.10991i	−0.831879	−	0.554957i	$-0.812735\pi$
$541$		−	25.8160i		−	1.10991i	0.831879	−	0.554957i	$-0.187265\pi$
$542$		0			0
$543$		−	1.87286i		−	0.0803720i
$544$		0			0
$545$	−11.1122			−0.475995
$546$		0			0
$547$	−39.3221			−1.68129			−0.840645	−	0.541586i	$-0.817824\pi$
$547$	−39.3221			−1.68129			−0.840645	+	0.541586i	$0.817824\pi$
$548$		0			0
$549$		−	16.0358i		−	0.684392i
$550$		0			0
$551$	−23.1346	+	1.28935i	−0.985569	+	0.0549283i
$552$		0			0
$553$			13.2781i			0.564642i
$554$		0			0
$555$	−4.28845			−0.182035
$556$		0			0
$557$	−30.2469			−1.28160			−0.640800	−	0.767708i	$-0.721397\pi$
$557$	−30.2469			−1.28160			−0.640800	+	0.767708i	$0.721397\pi$
$558$		0			0
$559$		−	58.3524i		−	2.46804i
$560$		0			0
$561$			1.99340i			0.0841613i
$562$		0			0
$563$		−	18.0696i		−	0.761543i	−0.924669	−	0.380772i	$-0.875658\pi$
$563$		−	18.0696i		−	0.761543i	0.924669	−	0.380772i	$-0.124342\pi$
$564$		0			0
$565$			9.60202i			0.403960i
$566$		0			0
$567$	8.43107			0.354071
$568$		0			0
$569$			0.0542492i			0.00227425i	0.999999	+	0.00113712i	$0.000361957\pi$
$569$			0.0542492i			0.00227425i	−0.999999	+	0.00113712i	$0.999638\pi$
$570$		0			0
$571$	24.8238			1.03884			0.519421	−	0.854518i	$-0.326148\pi$
$571$	24.8238			1.03884			0.519421	+	0.854518i	$0.326148\pi$
$572$		0			0
$573$	−1.44227			−0.0602519
$574$		0			0
$575$	−5.01121			−0.208982
$576$		0			0
$577$		−	33.8007i		−	1.40714i	−0.710625	−	0.703571i	$-0.751588\pi$
$577$		−	33.8007i		−	1.40714i	0.710625	−	0.703571i	$-0.248412\pi$
$578$		0			0
$579$	−2.55773			−0.106295
$580$		0			0
$581$	3.50953			0.145600
$582$		0			0
$583$		−	3.81983i		−	0.158201i
$584$		0			0
$585$	−15.4871			−0.640313
$586$		0			0
$587$	16.2323			0.669978			0.334989	−	0.942222i	$-0.391267\pi$
$587$	16.2323			0.669978			0.334989	+	0.942222i	$0.391267\pi$
$588$		0			0
$589$	36.3333			1.49709
$590$		0			0
$591$			8.60532i			0.353976i
$592$		0			0
$593$	9.08980			0.373273			0.186637	−	0.982429i	$-0.440241\pi$
$593$	9.08980			0.373273			0.186637	+	0.982429i	$0.440241\pi$
$594$		0			0
$595$			8.98146i			0.368204i
$596$		0			0
$597$			0.907196i			0.0371291i
$598$		0			0
$599$		−	13.2841i		−	0.542774i	−0.962470	−	0.271387i	$-0.912518\pi$
$599$		−	13.2841i		−	0.542774i	0.962470	−	0.271387i	$-0.0874824\pi$
$600$		0			0
$601$			18.6609i			0.761196i	0.924741	+	0.380598i	$0.124282\pi$
$601$			18.6609i			0.761196i	−0.924741	+	0.380598i	$0.875718\pi$
$602$		0			0
$603$	24.4343			0.995042
$604$		0			0
$605$	10.7115			0.435486
$606$		0			0
$607$			35.3888i			1.43639i	0.695844	+	0.718193i	$0.255030\pi$
$607$			35.3888i			1.43639i	−0.695844	+	0.718193i	$0.744970\pi$
$608$		0			0
$609$	−3.75315	+	0.209173i	−0.152085	+	0.00847610i
$610$		0			0
$611$			3.06756i			0.124100i
$612$		0			0
$613$	−7.77557			−0.314052			−0.157026	−	0.987594i	$-0.550191\pi$
$613$	−7.77557			−0.314052			−0.157026	+	0.987594i	$0.550191\pi$
$614$		0			0
$615$	−0.374895			−0.0151172
$616$		0			0
$617$			1.31859i			0.0530843i	0.999648	+	0.0265421i	$0.00844961\pi$
$617$			1.31859i			0.0530843i	−0.999648	+	0.0265421i	$0.991550\pi$
$618$		0			0
$619$			49.5198i			1.99037i	0.0980212	+	0.995184i	$0.468749\pi$
$619$			49.5198i			1.99037i	−0.0980212	+	0.995184i	$0.531251\pi$
$620$		0			0
$621$		−	15.3722i		−	0.616864i
$622$		0			0
$623$		−	12.5801i		−	0.504010i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			1.24112i			0.0495657i
$628$		0			0
$629$	−55.1795			−2.20015
$630$		0			0
$631$	−10.2244			−0.407028			−0.203514	−	0.979072i	$-0.565236\pi$
$631$	−10.2244			−0.407028			−0.203514	+	0.979072i	$0.565236\pi$
$632$		0			0
$633$	−3.75638			−0.149303
$634$		0			0
$635$		−	6.07484i		−	0.241073i
$636$		0			0
$637$	30.3333			1.20185
$638$		0			0
$639$	11.6283			0.460010
$640$		0			0
$641$		−	31.8848i		−	1.25937i	−0.776849	−	0.629687i	$-0.783183\pi$
$641$		−	31.8848i		−	1.25937i	0.776849	−	0.629687i	$-0.216817\pi$
$642$		0			0
$643$	29.8350			1.17658			0.588288	−	0.808651i	$-0.299802\pi$
$643$	29.8350			1.17658			0.588288	+	0.808651i	$0.299802\pi$
$644$		0			0
$645$	−5.48711			−0.216055
$646$		0			0
$647$	23.9438			0.941329			0.470665	−	0.882312i	$-0.344014\pi$
$647$	23.9438			0.941329			0.470665	+	0.882312i	$0.344014\pi$
$648$		0			0
$649$		−	4.14172i		−	0.162577i
$650$		0			0
$651$	5.89438			0.231019
$652$		0			0
$653$			34.1769i			1.33744i	0.743513	+	0.668722i	$0.233158\pi$
$653$			34.1769i			1.33744i	−0.743513	+	0.668722i	$0.766842\pi$
$654$		0			0
$655$			18.1239i			0.708158i
$656$		0			0
$657$		−	16.8457i		−	0.657213i
$658$		0			0
$659$		−	9.67346i		−	0.376825i	−0.982090	−	0.188412i	$-0.939666\pi$
$659$		−	9.67346i		−	0.376825i	0.982090	−	0.188412i	$-0.0603341\pi$
$660$		0			0
$661$	27.3591			1.06414			0.532072	−	0.846699i	$-0.321413\pi$
$661$	27.3591			1.06414			0.532072	+	0.846699i	$0.321413\pi$
$662$		0			0
$663$	21.1987			0.823288
$664$		0			0
$665$			5.59201i			0.216849i
$666$		0			0
$667$	−1.50168	−	26.9444i	−0.0581453	−	1.04329i
$668$		0			0
$669$		−	1.60522i		−	0.0620614i
$670$		0			0
$671$	−3.17624			−0.122617
$672$		0			0
$673$	32.0673			1.23610			0.618051	−	0.786138i	$-0.287923\pi$
$673$	32.0673			1.23610			0.618051	+	0.786138i	$0.287923\pi$
$674$		0			0
$675$			3.06756i			0.118070i
$676$		0			0
$677$			21.7053i			0.834202i	0.908860	+	0.417101i	$0.136954\pi$
$677$			21.7053i			0.834202i	−0.908860	+	0.417101i	$0.863046\pi$
$678$		0			0
$679$			2.69142i			0.103287i
$680$		0			0
$681$			12.8133i			0.491008i
$682$		0			0
$683$	8.43430			0.322729			0.161365	−	0.986895i	$-0.448410\pi$
$683$	8.43430			0.322729			0.161365	+	0.986895i	$0.448410\pi$
$684$		0			0
$685$		−	13.2006i		−	0.504370i
$686$		0			0
$687$	−0.172879			−0.00659575
$688$		0			0
$689$	−40.6217			−1.54757
$690$		0			0
$691$	12.7373			0.484551			0.242275	−	0.970208i	$-0.422106\pi$
$691$	12.7373			0.484551			0.242275	+	0.970208i	$0.422106\pi$
$692$		0			0
$693$		−	1.89272i		−	0.0718986i
$694$		0			0
$695$	−14.5993			−0.553784
$696$		0			0
$697$	−4.82376			−0.182713
$698$		0			0
$699$		−	8.60532i		−	0.325483i
$700$		0			0
$701$	−31.5319			−1.19095			−0.595473	−	0.803376i	$-0.703035\pi$
$701$	−31.5319			−1.19095			−0.595473	+	0.803376i	$0.703035\pi$
$702$		0			0
$703$	−34.3557			−1.29575
$704$		0			0
$705$	0.288455			0.0108638
$706$		0			0
$707$			3.98679i			0.149939i
$708$		0			0
$709$	−5.71155			−0.214502			−0.107251	−	0.994232i	$-0.534205\pi$
$709$	−5.71155			−0.214502			−0.107251	+	0.994232i	$0.534205\pi$
$710$		0			0
$711$			27.7027i			1.03893i
$712$		0			0
$713$			42.3165i			1.58477i
$714$		0			0
$715$			3.06756i			0.114720i
$716$		0			0
$717$		−	10.5867i		−	0.395367i
$718$		0			0
$719$	−3.33665			−0.124436			−0.0622180	−	0.998063i	$-0.519817\pi$
$719$	−3.33665			−0.124436			−0.0622180	+	0.998063i	$0.519817\pi$
$720$		0			0
$721$	0.243619			0.00907283
$722$		0			0
$723$			1.07416i			0.0399484i
$724$		0			0
$725$	0.299664	+	5.37682i	0.0111292	+	0.199690i
$726$		0			0
$727$			38.3358i			1.42179i	0.703296	+	0.710897i	$0.251711\pi$
$727$			38.3358i			1.42179i	−0.703296	+	0.710897i	$0.748289\pi$
$728$		0			0
$729$	12.6475			0.468427
$730$		0			0
$731$	−70.6026			−2.61133
$732$		0			0
$733$		−	34.7200i		−	1.28241i	−0.767369	−	0.641205i	$-0.778435\pi$
$733$		−	34.7200i		−	1.28241i	0.767369	−	0.641205i	$-0.221565\pi$
$734$		0			0
$735$		−	2.85236i		−	0.105211i
$736$		0			0
$737$		−	4.83974i		−	0.178274i
$738$		0			0
$739$			4.30266i			0.158276i	0.996864	+	0.0791380i	$0.0252168\pi$
$739$			4.30266i			0.158276i	−0.996864	+	0.0791380i	$0.974783\pi$
$740$		0			0
$741$	13.1987			0.484865
$742$		0			0
$743$			23.7159i			0.870051i	0.900418	+	0.435025i	$0.143261\pi$
$743$			23.7159i			0.870051i	−0.900418	+	0.435025i	$0.856739\pi$
$744$		0			0
$745$	6.00000			0.219823
$746$		0			0
$747$	7.32208			0.267901
$748$		0			0
$749$	−16.5353			−0.604187
$750$		0			0
$751$		−	45.1689i		−	1.64824i	−0.566418	−	0.824118i	$-0.691671\pi$
$751$		−	45.1689i		−	1.64824i	0.566418	−	0.824118i	$-0.308329\pi$
$752$		0			0
$753$	13.3142			0.485198
$754$		0			0
$755$	21.7340			0.790980
$756$		0			0
$757$			3.63387i			0.132075i	0.997817	+	0.0660376i	$0.0210357\pi$
$757$			3.63387i			0.132075i	−0.997817	+	0.0660376i	$0.978964\pi$
$758$		0			0
$759$	−1.44551			−0.0524686
$760$		0			0
$761$	30.4197			1.10271			0.551357	−	0.834269i	$-0.314110\pi$
$761$	30.4197			1.10271			0.551357	+	0.834269i	$0.314110\pi$
$762$		0			0
$763$	−14.4421			−0.522841
$764$		0			0
$765$			18.7384i			0.677488i
$766$		0			0
$767$	−44.0448			−1.59037
$768$		0			0
$769$		−	20.7086i		−	0.746771i	−0.927676	−	0.373385i	$-0.878197\pi$
$769$		−	20.7086i		−	0.746771i	0.927676	−	0.373385i	$-0.121803\pi$
$770$		0			0
$771$			9.07009i			0.326651i
$772$		0			0
$773$			32.0828i			1.15394i	0.816766	+	0.576969i	$0.195765\pi$
$773$			32.0828i			1.15394i	−0.816766	+	0.576969i	$0.804235\pi$
$774$		0			0
$775$		−	8.44438i		−	0.303331i
$776$		0			0
$777$	−5.57355			−0.199950
$778$		0			0
$779$	−3.00336			−0.107607
$780$		0			0
$781$		−	2.30324i		−	0.0824165i
$782$		0			0
$783$	−16.4937	+	0.919237i	−0.589437	+	0.0328508i
$784$		0			0
$785$			5.29334i			0.188927i
$786$		0			0
$787$	6.45348			0.230042			0.115021	−	0.993363i	$-0.463306\pi$
$787$	6.45348			0.230042			0.115021	+	0.993363i	$0.463306\pi$
$788$		0			0
$789$	−4.16041			−0.148115
$790$		0			0
$791$			12.4794i			0.443716i
$792$		0			0
$793$			33.7775i			1.19947i
$794$		0			0
$795$			3.81983i			0.135475i
$796$		0			0
$797$		−	36.9225i		−	1.30786i	−0.756554	−	0.653932i	$-0.773118\pi$
$797$		−	36.9225i		−	1.30786i	0.756554	−	0.653932i	$-0.226882\pi$
$798$		0			0
$799$	3.71155			0.131305
$800$		0			0
$801$		−	26.2463i		−	0.927369i
$802$		0			0
$803$	−3.33665			−0.117748
$804$		0			0
$805$	−6.51289			−0.229549
$806$		0			0
$807$	6.15046			0.216506
$808$		0			0
$809$			22.2596i			0.782604i	0.920262	+	0.391302i	$0.127975\pi$
$809$			22.2596i			0.782604i	−0.920262	+	0.391302i	$0.872025\pi$
$810$		0			0
$811$	16.2469			0.570504			0.285252	−	0.958453i	$-0.407923\pi$
$811$	16.2469			0.570504			0.285252	+	0.958453i	$0.407923\pi$
$812$		0			0
$813$	13.8911			0.487184
$814$		0			0
$815$			6.07484i			0.212793i
$816$		0			0
$817$	−43.9584			−1.53791
$818$		0			0
$819$	−20.1280			−0.703331
$820$		0			0
$821$	−0.352476			−0.0123015			−0.00615075	−	0.999981i	$-0.501958\pi$
$821$	−0.352476			−0.0123015			−0.00615075	+	0.999981i	$0.501958\pi$
$822$		0			0
$823$		−	15.0021i		−	0.522939i	−0.965212	−	0.261469i	$-0.915793\pi$
$823$		−	15.0021i		−	0.522939i	0.965212	−	0.261469i	$-0.0842070\pi$
$824$		0			0
$825$	0.288455			0.0100427
$826$		0			0
$827$		−	27.4152i		−	0.953319i	−0.879088	−	0.476659i	$-0.841848\pi$
$827$		−	27.4152i		−	0.953319i	0.879088	−	0.476659i	$-0.158152\pi$
$828$		0			0
$829$			50.9219i			1.76859i	0.466929	+	0.884295i	$0.345360\pi$
$829$			50.9219i			1.76859i	−0.466929	+	0.884295i	$0.654640\pi$
$830$		0			0
$831$		−	4.77345i		−	0.165589i
$832$		0			0
$833$		−	36.7013i		−	1.27163i
$834$		0			0
$835$	−5.01121			−0.173420
$836$		0			0
$837$	25.9036			0.895360
$838$		0			0
$839$		−	37.5371i		−	1.29592i	−0.761673	−	0.647962i	$-0.775622\pi$
$839$		−	37.5371i		−	1.29592i	0.761673	−	0.647962i	$-0.224378\pi$
$840$		0			0
$841$	−28.8204	+	3.22248i	−0.993807	+	0.111120i
$842$		0			0
$843$		−	12.1033i		−	0.416859i
$844$		0			0
$845$	19.6217			0.675009
$846$		0			0
$847$	13.9214			0.478345
$848$		0			0
$849$			7.74033i			0.265648i
$850$		0			0
$851$		−	40.0133i		−	1.37164i
$852$		0			0
$853$		−	0.942449i		−	0.0322688i	−0.999870	−	0.0161344i	$-0.994864\pi$
$853$		−	0.942449i		−	0.0322688i	0.999870	−	0.0161344i	$-0.00513597\pi$
$854$		0			0
$855$			11.6669i			0.398998i
$856$		0			0
$857$	−2.66335			−0.0909783			−0.0454891	−	0.998965i	$-0.514485\pi$
$857$	−2.66335			−0.0909783			−0.0454891	+	0.998965i	$0.514485\pi$
$858$		0			0
$859$			22.9636i			0.783508i	0.920070	+	0.391754i	$0.128132\pi$
$859$			22.9636i			0.783508i	−0.920070	+	0.391754i	$0.871868\pi$
$860$		0			0
$861$	−0.487237			−0.0166050
$862$		0			0
$863$	13.7485			0.468005			0.234003	−	0.972236i	$-0.424818\pi$
$863$	13.7485			0.468005			0.234003	+	0.972236i	$0.424818\pi$
$864$		0			0
$865$	−8.31087			−0.282578
$866$		0			0
$867$		−	16.5186i		−	0.561002i
$868$		0			0
$869$	5.48711			0.186138
$870$		0			0
$871$	−51.4679			−1.74392
$872$		0			0
$873$			5.61523i			0.190047i
$874$		0			0
$875$	1.29966			0.0439367
$876$		0			0
$877$	−42.1313			−1.42267			−0.711336	−	0.702852i	$-0.751910\pi$
$877$	−42.1313			−1.42267			−0.711336	+	0.702852i	$0.751910\pi$
$878$		0			0
$879$	−0.535307			−0.0180555
$880$		0			0
$881$			13.1232i			0.442131i	0.975259	+	0.221065i	$0.0709535\pi$
$881$			13.1232i			0.442131i	−0.975259	+	0.221065i	$0.929047\pi$
$882$		0			0
$883$	49.5465			1.66737			0.833687	−	0.552238i	$-0.186226\pi$
$883$	49.5465			1.66737			0.833687	+	0.552238i	$0.186226\pi$
$884$		0			0
$885$			4.14172i			0.139222i
$886$		0			0
$887$		−	3.43767i		−	0.115426i	−0.998333	−	0.0577129i	$-0.981619\pi$
$887$		−	3.43767i		−	0.115426i	0.998333	−	0.0577129i	$-0.0183808\pi$
$888$		0			0
$889$		−	7.89526i		−	0.264798i
$890$		0			0
$891$		−	3.48410i		−	0.116722i
$892$		0			0
$893$	2.31087			0.0773304
$894$		0			0
$895$	−6.59933			−0.220591
$896$		0			0
$897$			15.3722i			0.513262i
$898$		0			0
$899$	45.4039	−	2.53048i	1.51431	−	0.0843961i
$900$		0			0
$901$			49.1497i			1.63741i
$902$		0			0
$903$	−7.13140			−0.237318
$904$		0			0
$905$	−3.48711			−0.115916
$906$		0			0
$907$		−	3.44971i		−	0.114546i	−0.998359	−	0.0572729i	$-0.981759\pi$
$907$		−	3.44971i		−	0.114546i	0.998359	−	0.0572729i	$-0.0182405\pi$
$908$		0			0
$909$			8.31782i			0.275885i
$910$		0			0
$911$			30.8709i			1.02280i	0.859343	+	0.511399i	$0.170873\pi$
$911$			30.8709i			1.02280i	−0.859343	+	0.511399i	$0.829127\pi$
$912$		0			0
$913$		−	1.45030i		−	0.0479978i
$914$		0			0
$915$	3.17624			0.105003
$916$		0			0
$917$			23.5549i			0.777852i
$918$		0			0
$919$	49.5768			1.63539			0.817694	−	0.575654i	$-0.195252\pi$
$919$	49.5768			1.63539			0.817694	+	0.575654i	$0.195252\pi$
$920$		0			0
$921$	14.5577			0.479693
$922$		0			0
$923$	−24.4937			−0.806220
$924$		0			0
$925$			7.98476i			0.262537i
$926$		0			0
$927$	0.508272			0.0166938
$928$		0			0
$929$	58.0897			1.90586			0.952930	−	0.303190i	$-0.0980516\pi$
$929$	58.0897			1.90586			0.952930	+	0.303190i	$0.0980516\pi$
$930$		0			0
$931$		−	22.8509i		−	0.748908i
$932$		0			0
$933$	10.1380			0.331903
$934$		0			0
$935$	3.71155			0.121381
$936$		0			0
$937$	54.9550			1.79530			0.897651	−	0.440706i	$-0.145272\pi$
$937$	54.9550			1.79530			0.897651	+	0.440706i	$0.145272\pi$
$938$		0			0
$939$			3.37740i			0.110217i
$940$		0			0
$941$	43.6924			1.42433			0.712165	−	0.702012i	$-0.247715\pi$
$941$	43.6924			1.42433			0.712165	+	0.702012i	$0.247715\pi$
$942$		0			0
$943$		−	3.49794i		−	0.113909i
$944$		0			0
$945$			3.98679i			0.129690i
$946$		0			0
$947$		−	13.9279i		−	0.452596i	−0.974058	−	0.226298i	$-0.927338\pi$
$947$		−	13.9279i		−	0.452596i	0.974058	−	0.226298i	$-0.0726623\pi$
$948$		0			0
$949$			35.4834i			1.15184i
$950$		0			0
$951$	7.26268			0.235508
$952$		0			0
$953$	−12.3525			−0.400136			−0.200068	−	0.979782i	$-0.564116\pi$
$953$	−12.3525			−0.400136			−0.200068	+	0.979782i	$0.564116\pi$
$954$		0			0
$955$			2.68540i			0.0868975i
$956$		0			0
$957$	0.0864396	+	1.55097i	0.00279420	+	0.0501358i
$958$		0			0
$959$		−	17.1564i		−	0.554009i
$960$		0			0
$961$	−40.3075			−1.30024
$962$		0			0
$963$	−34.4983			−1.11169
$964$		0			0
$965$			4.76228i			0.153303i
$966$		0			0
$967$			26.5080i			0.852439i	0.904620	+	0.426219i	$0.140155\pi$
$967$			26.5080i			0.852439i	−0.904620	+	0.426219i	$0.859845\pi$
$968$		0			0
$969$		−	15.9695i		−	0.513015i
$970$		0			0
$971$			22.2656i			0.714536i	0.934002	+	0.357268i	$0.116292\pi$
$971$			22.2656i			0.714536i	−0.934002	+	0.357268i	$0.883708\pi$
$972$		0			0
$973$	−18.9742			−0.608286
$974$		0			0
$975$		−	3.06756i		−	0.0982404i
$976$		0			0
$977$	−18.0864			−0.578636			−0.289318	−	0.957233i	$-0.593429\pi$
$977$	−18.0864			−0.578636			−0.289318	+	0.957233i	$0.593429\pi$
$978$		0			0
$979$	−5.19866			−0.166150
$980$		0			0
$981$	−30.1313			−0.962018
$982$		0			0
$983$		−	21.8894i		−	0.698165i	−0.937092	−	0.349082i	$-0.886493\pi$
$983$		−	21.8894i		−	0.698165i	0.937092	−	0.349082i	$-0.113507\pi$
$984$		0			0
$985$	16.0224			0.510517
$986$		0			0
$987$	0.374895			0.0119330
$988$		0			0
$989$		−	51.1973i		−	1.62798i
$990$		0			0
$991$	14.8878			0.472926			0.236463	−	0.971640i	$-0.424012\pi$
$991$	14.8878			0.472926			0.236463	+	0.971640i	$0.424012\pi$
$992$		0			0
$993$	−11.2086			−0.355694
$994$		0			0
$995$	1.68913			0.0535489
$996$		0			0
$997$		−	38.4735i		−	1.21847i	−0.792991	−	0.609234i	$-0.791477\pi$
$997$		−	38.4735i		−	1.21847i	0.792991	−	0.609234i	$-0.208523\pi$
$998$		0			0
$999$	−24.4937			−0.774946

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.g.i.1681.4		6
4.3	odd	2		145.2.c.b.86.2	✓	6
12.11	even	2		1305.2.d.b.811.5		6
20.3	even	4		725.2.d.c.724.4		12
20.7	even	4		725.2.d.c.724.9		12
20.19	odd	2		725.2.c.e.376.5		6
29.28	even	2	inner	2320.2.g.i.1681.3		6
116.75	even	4		4205.2.a.m.1.2		6
116.99	even	4		4205.2.a.m.1.5		6
116.115	odd	2		145.2.c.b.86.5	yes	6
348.347	even	2		1305.2.d.b.811.2		6
580.347	even	4		725.2.d.c.724.3		12
580.463	even	4		725.2.d.c.724.10		12
580.579	odd	2		725.2.c.e.376.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.b.86.2	✓	6	4.3	odd	2
145.2.c.b.86.5	yes	6	116.115	odd	2
725.2.c.e.376.2		6	580.579	odd	2
725.2.c.e.376.5		6	20.19	odd	2
725.2.d.c.724.3		12	580.347	even	4
725.2.d.c.724.4		12	20.3	even	4
725.2.d.c.724.9		12	20.7	even	4
725.2.d.c.724.10		12	580.463	even	4
1305.2.d.b.811.2		6	348.347	even	2
1305.2.d.b.811.5		6	12.11	even	2
2320.2.g.i.1681.3		6	29.28	even	2	inner
2320.2.g.i.1681.4		6	1.1	even	1	trivial
4205.2.a.m.1.2		6	116.75	even	4
4205.2.a.m.1.5		6	116.99	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.537080i q^{3} +1.00000 q^{5} +1.29966 q^{7} +2.71155 q^{9} -0.537080i q^{11} -5.71155 q^{13} +0.537080i q^{15} +6.91060i q^{17} +4.30266i q^{19} +0.698024i q^{21} -5.01121 q^{23} +1.00000 q^{25} +3.06756i q^{27} +(0.299664 + 5.37682i) q^{29} -8.44438i q^{31} +0.288455 q^{33} +1.29966 q^{35} +7.98476i q^{37} -3.06756i q^{39} +0.698024i q^{41} +10.2166i q^{43} +2.71155 q^{45} -0.537080i q^{47} -5.31087 q^{49} -3.71155 q^{51} +7.11222 q^{53} -0.537080i q^{55} -2.31087 q^{57} +7.71155 q^{59} -5.91390i q^{61} +3.52410 q^{63} -5.71155 q^{65} +9.01121 q^{67} -2.69142i q^{69} +4.28845 q^{71} -6.21258i q^{73} +0.537080i q^{75} -0.698024i q^{77} +10.2166i q^{79} +6.48711 q^{81} +2.70034 q^{83} +6.91060i q^{85} +(-2.88778 + 0.160944i) q^{87} -9.67948i q^{89} -7.42309 q^{91} +4.53531 q^{93} +4.30266i q^{95} +2.07086i q^{97} -1.45632i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{5} - 14 q^{9} - 4 q^{13} + 8 q^{23} + 6 q^{25} - 6 q^{29} + 32 q^{33} - 14 q^{45} + 14 q^{49} + 8 q^{51} + 28 q^{53} + 32 q^{57} + 16 q^{59} - 16 q^{63} - 4 q^{65} + 16 q^{67} + 56 q^{71} + 38 q^{81}+ \cdots - 48 q^{93}+O(q^{100}) \) 6 * q + 6 * q^5 - 14 * q^9 - 4 * q^13 + 8 * q^23 + 6 * q^25 - 6 * q^29 + 32 * q^33 - 14 * q^45 + 14 * q^49 + 8 * q^51 + 28 * q^53 + 32 * q^57 + 16 * q^59 - 16 * q^63 - 4 * q^65 + 16 * q^67 + 56 * q^71 + 38 * q^81 + 24 * q^83 - 32 * q^87 + 16 * q^91 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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