LMFDB - Embedded newform 920.2.e.c.369.4

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024 $ x^16 - 2x^15 + 2x^14 + 6x^13 + 100x^12 - 196x^11 + 210x^10 + 702x^9 + 1572x^8 - 1594x^7 + 2464x^6 + 9568x^5 + 15457x^4 + 4336x^3 + 128x^2 - 512*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.4
Root			$-1.13508 - 1.13508i$ of defining polynomial
Character	$\chi$	$=$	920.369
Dual form			920.2.e.c.369.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51561i q^{3} +(-1.92655 - 1.13508i) q^{5} -4.64022i q^{7} -3.32827 q^{9} +O(q^{10})$	$q-2.51561i q^{3} +(-1.92655 - 1.13508i) q^{5} -4.64022i q^{7} -3.32827 q^{9} -1.64944 q^{11} -1.91631i q^{13} +(-2.85542 + 4.84643i) q^{15} +0.969195i q^{17} +4.91039 q^{19} -11.6730 q^{21} -1.00000i q^{23} +(2.42317 + 4.37359i) q^{25} +0.825798i q^{27} -7.48480 q^{29} +7.77655 q^{31} +4.14935i q^{33} +(-5.26704 + 8.93961i) q^{35} +7.63254i q^{37} -4.82068 q^{39} +5.79719 q^{41} +3.77230i q^{43} +(6.41207 + 3.77787i) q^{45} -2.22605i q^{47} -14.5317 q^{49} +2.43811 q^{51} -11.2963i q^{53} +(3.17773 + 1.87226i) q^{55} -12.3526i q^{57} -10.1233 q^{59} +6.06140 q^{61} +15.4439i q^{63} +(-2.17517 + 3.69186i) q^{65} +12.0618i q^{67} -2.51561 q^{69} +3.01159 q^{71} -16.6439i q^{73} +(11.0022 - 6.09574i) q^{75} +7.65379i q^{77} +0.163787 q^{79} -7.90743 q^{81} +5.66822i q^{83} +(1.10012 - 1.86720i) q^{85} +18.8288i q^{87} +3.90271 q^{89} -8.89211 q^{91} -19.5627i q^{93} +(-9.46010 - 5.57371i) q^{95} +1.79123i q^{97} +5.48979 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) $ 16 * q - 2 * q^5 - 22 * q^9	$ 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Display 100 coefficients 10 coefficients

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
$3$		−	2.51561i		−	1.45239i	−0.687491	−	0.726193i	$-0.741288\pi$
$3$		−	2.51561i		−	1.45239i	0.687491	−	0.726193i	$-0.258712\pi$
$4$		0			0
$5$	−1.92655	−	1.13508i	−0.861578	−	0.507625i
$5$	−1.92655	−	1.13508i	−0.861578	−	0.507625i
$6$		0			0
$7$		−	4.64022i		−	1.75384i	−0.480636	−	0.876920i	$-0.659594\pi$
$7$		−	4.64022i		−	1.75384i	0.480636	−	0.876920i	$-0.340406\pi$
$8$		0			0
$9$	−3.32827			−1.10942
$10$		0			0
$11$	−1.64944			−0.497326			−0.248663	−	0.968590i	$-0.579991\pi$
$11$	−1.64944			−0.497326			−0.248663	+	0.968590i	$0.579991\pi$
$12$		0			0
$13$		−	1.91631i		−	0.531489i	−0.964043	−	0.265745i	$-0.914382\pi$
$13$		−	1.91631i		−	0.531489i	0.964043	−	0.265745i	$-0.0856178\pi$
$14$		0			0
$15$	−2.85542	+	4.84643i	−0.737267	+	1.25134i
$16$		0			0
$17$			0.969195i			0.235064i	0.993069	+	0.117532i	$0.0374983\pi$
$17$			0.969195i			0.235064i	−0.993069	+	0.117532i	$0.962502\pi$
$18$		0			0
$19$	4.91039			1.12652			0.563261	−	0.826279i	$-0.309547\pi$
$19$	4.91039			1.12652			0.563261	+	0.826279i	$0.309547\pi$
$20$		0			0
$21$	−11.6730			−2.54725
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	2.42317	+	4.37359i	0.484634	+	0.874717i
$26$		0			0
$27$			0.825798i			0.158925i
$28$		0			0
$29$	−7.48480			−1.38989			−0.694946	−	0.719062i	$-0.744572\pi$
$29$	−7.48480			−1.38989			−0.694946	+	0.719062i	$0.744572\pi$
$30$		0			0
$31$	7.77655			1.39671			0.698355	−	0.715751i	$-0.253916\pi$
$31$	7.77655			1.39671			0.698355	+	0.715751i	$0.253916\pi$
$32$		0			0
$33$			4.14935i			0.722309i
$34$		0			0
$35$	−5.26704	+	8.93961i	−0.890293	+	1.51107i
$36$		0			0
$37$			7.63254i			1.25478i	0.778704	+	0.627391i	$0.215877\pi$
$37$			7.63254i			1.25478i	−0.778704	+	0.627391i	$0.784123\pi$
$38$		0			0
$39$	−4.82068			−0.771927
$40$		0			0
$41$	5.79719			0.905368			0.452684	−	0.891671i	$-0.350466\pi$
$41$	5.79719			0.905368			0.452684	+	0.891671i	$0.350466\pi$
$42$		0			0
$43$			3.77230i			0.575270i	0.957740	+	0.287635i	$0.0928691\pi$
$43$			3.77230i			0.575270i	−0.957740	+	0.287635i	$0.907131\pi$
$44$		0			0
$45$	6.41207	+	3.77787i	0.955855	+	0.563171i
$46$		0			0
$47$		−	2.22605i		−	0.324702i	−0.986733	−	0.162351i	$-0.948092\pi$
$47$		−	2.22605i		−	0.324702i	0.986733	−	0.162351i	$-0.0519077\pi$
$48$		0			0
$49$	−14.5317			−2.07595
$50$		0			0
$51$	2.43811			0.341404
$52$		0			0
$53$		−	11.2963i		−	1.55166i	−0.630941	−	0.775831i	$-0.717331\pi$
$53$		−	11.2963i		−	1.55166i	0.630941	−	0.775831i	$-0.282669\pi$
$54$		0			0
$55$	3.17773	+	1.87226i	0.428485	+	0.252455i
$56$		0			0
$57$		−	12.3526i		−	1.63614i
$58$		0			0
$59$	−10.1233			−1.31794			−0.658968	−	0.752171i	$-0.729007\pi$
$59$	−10.1233			−1.31794			−0.658968	+	0.752171i	$0.729007\pi$
$60$		0			0
$61$	6.06140			0.776083			0.388042	−	0.921642i	$-0.373152\pi$
$61$	6.06140			0.776083			0.388042	+	0.921642i	$0.373152\pi$
$62$		0			0
$63$			15.4439i			1.94575i
$64$		0			0
$65$	−2.17517	+	3.69186i	−0.269797	+	0.457919i
$66$		0			0
$67$			12.0618i			1.47358i	0.676119	+	0.736792i	$0.263660\pi$
$67$			12.0618i			1.47358i	−0.676119	+	0.736792i	$0.736340\pi$
$68$		0			0
$69$	−2.51561			−0.302843
$70$		0			0
$71$	3.01159			0.357411			0.178705	−	0.983903i	$-0.442809\pi$
$71$	3.01159			0.357411			0.178705	+	0.983903i	$0.442809\pi$
$72$		0			0
$73$		−	16.6439i		−	1.94802i	−0.226495	−	0.974012i	$-0.572727\pi$
$73$		−	16.6439i		−	1.94802i	0.226495	−	0.974012i	$-0.427273\pi$
$74$		0			0
$75$	11.0022	−	6.09574i	1.27043	−	0.703875i
$76$		0			0
$77$			7.65379i			0.872230i
$78$		0			0
$79$	0.163787			0.0184275			0.00921377	−	0.999958i	$-0.497067\pi$
$79$	0.163787			0.0184275			0.00921377	+	0.999958i	$0.497067\pi$
$80$		0			0
$81$	−7.90743			−0.878603
$82$		0			0
$83$			5.66822i			0.622168i	0.950382	+	0.311084i	$0.100692\pi$
$83$			5.66822i			0.622168i	−0.950382	+	0.311084i	$0.899308\pi$
$84$		0			0
$85$	1.10012	−	1.86720i	0.119325	−	0.202526i
$86$		0			0
$87$			18.8288i			2.01866i
$88$		0			0
$89$	3.90271			0.413687			0.206843	−	0.978374i	$-0.433681\pi$
$89$	3.90271			0.413687			0.206843	+	0.978374i	$0.433681\pi$
$90$		0			0
$91$	−8.89211			−0.932147
$92$		0			0
$93$		−	19.5627i		−	2.02856i
$94$		0			0
$95$	−9.46010	−	5.57371i	−0.970586	−	0.571850i
$96$		0			0
$97$			1.79123i			0.181872i	0.995857	+	0.0909362i	$0.0289859\pi$
$97$			1.79123i			0.181872i	−0.995857	+	0.0909362i	$0.971014\pi$
$98$		0			0
$99$	5.48979			0.551745
$100$		0			0
$101$	−7.68775			−0.764960			−0.382480	−	0.923964i	$-0.624930\pi$
$101$	−7.68775			−0.764960			−0.382480	+	0.923964i	$0.624930\pi$
$102$		0			0
$103$			2.30773i			0.227387i	0.993516	+	0.113694i	$0.0362682\pi$
$103$			2.30773i			0.227387i	−0.993516	+	0.113694i	$0.963732\pi$
$104$		0			0
$105$	22.4885	+	13.2498i	2.19466	+	1.29305i
$106$		0			0
$107$		−	10.7032i		−	1.03472i	−0.855768	−	0.517359i	$-0.826915\pi$
$107$		−	10.7032i		−	1.03472i	0.855768	−	0.517359i	$-0.173085\pi$
$108$		0			0
$109$	−7.94739			−0.761222			−0.380611	−	0.924735i	$-0.624286\pi$
$109$	−7.94739			−0.761222			−0.380611	+	0.924735i	$0.624286\pi$
$110$		0			0
$111$	19.2005			1.82243
$112$		0			0
$113$			14.5761i			1.37120i	0.727977	+	0.685602i	$0.240461\pi$
$113$			14.5761i			1.37120i	−0.727977	+	0.685602i	$0.759539\pi$
$114$		0			0
$115$	−1.13508	+	1.92655i	−0.105847	+	0.179651i
$116$		0			0
$117$			6.37800i			0.589646i
$118$		0			0
$119$	4.49728			0.412265
$120$		0			0
$121$	−8.27934			−0.752667
$122$		0			0
$123$		−	14.5834i		−	1.31494i
$124$		0			0
$125$	0.296037	−	11.1764i	0.0264784	−	0.999649i
$126$		0			0
$127$		−	21.8047i		−	1.93485i	−0.253158	−	0.967425i	$-0.581469\pi$
$127$		−	21.8047i		−	1.93485i	0.253158	−	0.967425i	$-0.418531\pi$
$128$		0			0
$129$	9.48962			0.835514
$130$		0			0
$131$	−3.66922			−0.320581			−0.160290	−	0.987070i	$-0.551243\pi$
$131$	−3.66922			−0.320581			−0.160290	+	0.987070i	$0.551243\pi$
$132$		0			0
$133$		−	22.7853i		−	1.97574i
$134$		0			0
$135$	0.937350	−	1.59094i	0.0806742	−	0.136926i
$136$		0			0
$137$			3.42421i			0.292550i	0.989244	+	0.146275i	$0.0467284\pi$
$137$			3.42421i			0.292550i	−0.989244	+	0.146275i	$0.953272\pi$
$138$		0			0
$139$	12.2838			1.04190			0.520950	−	0.853587i	$-0.325578\pi$
$139$	12.2838			1.04190			0.520950	+	0.853587i	$0.325578\pi$
$140$		0			0
$141$	−5.59985			−0.471593
$142$		0			0
$143$			3.16085i			0.264323i
$144$		0			0
$145$	14.4198	+	8.49588i	1.19750	+	0.705544i
$146$		0			0
$147$			36.5560i			3.01509i
$148$		0			0
$149$	−1.42737			−0.116935			−0.0584675	−	0.998289i	$-0.518621\pi$
$149$	−1.42737			−0.116935			−0.0584675	+	0.998289i	$0.518621\pi$
$150$		0			0
$151$	16.9521			1.37954			0.689772	−	0.724027i	$-0.257711\pi$
$151$	16.9521			1.37954			0.689772	+	0.724027i	$0.257711\pi$
$152$		0			0
$153$		−	3.22574i		−	0.260786i
$154$		0			0
$155$	−14.9819	−	8.82704i	−1.20338	−	0.709005i
$156$		0			0
$157$		−	19.6335i		−	1.56693i	−0.621438	−	0.783464i	$-0.713451\pi$
$157$		−	19.6335i		−	1.56693i	0.621438	−	0.783464i	$-0.286549\pi$
$158$		0			0
$159$	−28.4169			−2.25361
$160$		0			0
$161$	−4.64022			−0.365701
$162$		0			0
$163$		−	13.4311i		−	1.05200i	−0.850483	−	0.526002i	$-0.823690\pi$
$163$		−	13.4311i		−	1.05200i	0.850483	−	0.526002i	$-0.176310\pi$
$164$		0			0
$165$	4.70986	−	7.99392i	0.366662	−	0.622326i
$166$		0			0
$167$		−	2.14523i		−	0.166003i	−0.996549	−	0.0830013i	$-0.973549\pi$
$167$		−	2.14523i		−	0.166003i	0.996549	−	0.0830013i	$-0.0264506\pi$
$168$		0			0
$169$	9.32775			0.717519
$170$		0			0
$171$	−16.3431			−1.24979
$172$		0			0
$173$		−	2.04953i		−	0.155823i	−0.996960	−	0.0779115i	$-0.975175\pi$
$173$		−	2.04953i		−	0.155823i	0.996960	−	0.0779115i	$-0.0248252\pi$
$174$		0			0
$175$	20.2944	−	11.2440i	1.53411	−	0.849970i
$176$		0			0
$177$			25.4661i			1.91415i
$178$		0			0
$179$	−18.8683			−1.41028			−0.705140	−	0.709068i	$-0.749116\pi$
$179$	−18.8683			−1.41028			−0.705140	+	0.709068i	$0.749116\pi$
$180$		0			0
$181$	−4.49843			−0.334366			−0.167183	−	0.985926i	$-0.553467\pi$
$181$	−4.49843			−0.334366			−0.167183	+	0.985926i	$0.553467\pi$
$182$		0			0
$183$		−	15.2481i		−	1.12717i
$184$		0			0
$185$	8.66358	−	14.7045i	0.636959	−	1.08109i
$186$		0			0
$187$		−	1.59863i		−	0.116904i
$188$		0			0
$189$	3.83189			0.278729
$190$		0			0
$191$	20.0622			1.45165			0.725823	−	0.687881i	$-0.241459\pi$
$191$	20.0622			1.45165			0.725823	+	0.687881i	$0.241459\pi$
$192$		0			0
$193$			4.77197i			0.343494i	0.985141	+	0.171747i	$0.0549412\pi$
$193$			4.77197i			0.343494i	−0.985141	+	0.171747i	$0.945059\pi$
$194$		0			0
$195$	9.28727	+	5.47188i	0.665075	+	0.391849i
$196$		0			0
$197$		−	15.7807i		−	1.12433i	−0.827026	−	0.562164i	$-0.809969\pi$
$197$		−	15.7807i		−	1.12433i	0.827026	−	0.562164i	$-0.190031\pi$
$198$		0			0
$199$	−12.0195			−0.852039			−0.426020	−	0.904714i	$-0.640085\pi$
$199$	−12.0195			−0.852039			−0.426020	+	0.904714i	$0.640085\pi$
$200$		0			0
$201$	30.3427			2.14021
$202$		0			0
$203$			34.7312i			2.43765i
$204$		0			0
$205$	−11.1686	−	6.58029i	−0.780046	−	0.459588i
$206$		0			0
$207$			3.32827i			0.231331i
$208$		0			0
$209$	−8.09942			−0.560248
$210$		0			0
$211$	16.0791			1.10693			0.553467	−	0.832871i	$-0.313304\pi$
$211$	16.0791			1.10693			0.553467	+	0.832871i	$0.313304\pi$
$212$		0			0
$213$		−	7.57598i		−	0.519098i
$214$		0			0
$215$	4.28188	−	7.26752i	0.292022	−	0.495640i
$216$		0			0
$217$		−	36.0850i		−	2.44961i
$218$		0			0
$219$	−41.8696			−2.82928
$220$		0			0
$221$	1.85728			0.124934
$222$		0			0
$223$		−	18.2248i		−	1.22042i	−0.792239	−	0.610211i	$-0.791085\pi$
$223$		−	18.2248i		−	1.22042i	0.792239	−	0.610211i	$-0.208915\pi$
$224$		0			0
$225$	−8.06496	−	14.5565i	−0.537664	−	0.970432i
$226$		0			0
$227$		−	10.5475i		−	0.700065i	−0.936737	−	0.350033i	$-0.886170\pi$
$227$		−	10.5475i		−	0.700065i	0.936737	−	0.350033i	$-0.113830\pi$
$228$		0			0
$229$	−22.0719			−1.45855			−0.729276	−	0.684219i	$-0.760143\pi$
$229$	−22.0719			−1.45855			−0.729276	+	0.684219i	$0.760143\pi$
$230$		0			0
$231$	19.2539			1.26681
$232$		0			0
$233$			11.9562i			0.783277i	0.920119	+	0.391638i	$0.128092\pi$
$233$			11.9562i			0.783277i	−0.920119	+	0.391638i	$0.871908\pi$
$234$		0			0
$235$	−2.52675	+	4.28858i	−0.164827	+	0.279756i
$236$		0			0
$237$		−	0.412025i		−	0.0267639i
$238$		0			0
$239$	0.163761			0.0105928			0.00529640	−	0.999986i	$-0.498314\pi$
$239$	0.163761			0.0105928			0.00529640	+	0.999986i	$0.498314\pi$
$240$		0			0
$241$	−6.69099			−0.431004			−0.215502	−	0.976503i	$-0.569139\pi$
$241$	−6.69099			−0.431004			−0.215502	+	0.976503i	$0.569139\pi$
$242$		0			0
$243$			22.3694i			1.43500i
$244$		0			0
$245$	27.9960	+	16.4947i	1.78860	+	1.05381i
$246$		0			0
$247$		−	9.40984i		−	0.598734i
$248$		0			0
$249$	14.2590			0.903628
$250$		0			0
$251$	6.90434			0.435798			0.217899	−	0.975971i	$-0.430080\pi$
$251$	6.90434			0.435798			0.217899	+	0.975971i	$0.430080\pi$
$252$		0			0
$253$			1.64944i			0.103700i
$254$		0			0
$255$	−4.69714	−	2.76746i	−0.294146	−	0.173305i
$256$		0			0
$257$			25.3210i			1.57948i	0.613440	+	0.789742i	$0.289785\pi$
$257$			25.3210i			1.57948i	−0.613440	+	0.789742i	$0.710215\pi$
$258$		0			0
$259$	35.4167			2.20069
$260$		0			0
$261$	24.9114			1.54198
$262$		0			0
$263$		−	9.56021i		−	0.589508i	−0.955573	−	0.294754i	$-0.904762\pi$
$263$		−	9.56021i		−	0.589508i	0.955573	−	0.294754i	$-0.0952377\pi$
$264$		0			0
$265$	−12.8222	+	21.7628i	−0.787662	+	1.33688i
$266$		0			0
$267$		−	9.81768i		−	0.600832i
$268$		0			0
$269$	28.7061			1.75024			0.875120	−	0.483905i	$-0.160782\pi$
$269$	28.7061			1.75024			0.875120	+	0.483905i	$0.160782\pi$
$270$		0			0
$271$	3.19440			0.194046			0.0970229	−	0.995282i	$-0.469068\pi$
$271$	3.19440			0.194046			0.0970229	+	0.995282i	$0.469068\pi$
$272$		0			0
$273$			22.3691i			1.35384i
$274$		0			0
$275$	−3.99688	−	7.21398i	−0.241021	−	0.435020i
$276$		0			0
$277$			17.2490i			1.03639i	0.855261	+	0.518197i	$0.173397\pi$
$277$			17.2490i			1.03639i	−0.855261	+	0.518197i	$0.826603\pi$
$278$		0			0
$279$	−25.8825			−1.54954
$280$		0			0
$281$	12.1604			0.725430			0.362715	−	0.931900i	$-0.381850\pi$
$281$	12.1604			0.725430			0.362715	+	0.931900i	$0.381850\pi$
$282$		0			0
$283$			11.0741i			0.658287i	0.944280	+	0.329144i	$0.106760\pi$
$283$			11.0741i			0.658287i	−0.944280	+	0.329144i	$0.893240\pi$
$284$		0			0
$285$	−14.0212	+	23.7979i	−0.830547	+	1.40967i
$286$		0			0
$287$		−	26.9002i		−	1.58787i
$288$		0			0
$289$	16.0607			0.944745
$290$		0			0
$291$	4.50604			0.264149
$292$		0			0
$293$			21.1811i			1.23741i	0.785623	+	0.618706i	$0.212343\pi$
$293$			21.1811i			1.23741i	−0.785623	+	0.618706i	$0.787657\pi$
$294$		0			0
$295$	19.5029	+	11.4908i	1.13551	+	0.669017i
$296$		0			0
$297$		−	1.36211i		−	0.0790374i
$298$		0			0
$299$	−1.91631			−0.110823
$300$		0			0
$301$	17.5043			1.00893
$302$		0			0
$303$			19.3393i			1.11102i
$304$		0			0
$305$	−11.6776	−	6.88020i	−0.668656	−	0.393959i
$306$		0			0
$307$		−	20.2792i		−	1.15739i	−0.815543	−	0.578697i	$-0.803562\pi$
$307$		−	20.2792i		−	1.15739i	0.815543	−	0.578697i	$-0.196438\pi$
$308$		0			0
$309$	5.80534			0.330254
$310$		0			0
$311$	34.4696			1.95459			0.977295	−	0.211882i	$-0.0679592\pi$
$311$	34.4696			1.95459			0.977295	+	0.211882i	$0.0679592\pi$
$312$		0			0
$313$			16.4447i			0.929510i	0.885439	+	0.464755i	$0.153858\pi$
$313$			16.4447i			0.929510i	−0.885439	+	0.464755i	$0.846142\pi$
$314$		0			0
$315$	17.5301	−	29.7534i	0.987712	−	1.67642i
$316$		0			0
$317$		−	1.18172i		−	0.0663720i	−0.999449	−	0.0331860i	$-0.989435\pi$
$317$		−	1.18172i		−	0.0663720i	0.999449	−	0.0331860i	$-0.0105654\pi$
$318$		0			0
$319$	12.3458			0.691230
$320$		0			0
$321$	−26.9251			−1.50281
$322$		0			0
$323$			4.75913i			0.264805i
$324$		0			0
$325$	8.38115	−	4.64355i	0.464903	−	0.257578i
$326$		0			0
$327$			19.9925i			1.10559i
$328$		0			0
$329$	−10.3294			−0.569476
$330$		0			0
$331$	−33.6576			−1.84999			−0.924996	−	0.379977i	$-0.875932\pi$
$331$	−33.6576			−1.84999			−0.924996	+	0.379977i	$0.875932\pi$
$332$		0			0
$333$		−	25.4032i		−	1.39208i
$334$		0			0
$335$	13.6912	−	23.2376i	0.748028	−	1.26961i
$336$		0			0
$337$		−	5.29846i		−	0.288625i	−0.989532	−	0.144313i	$-0.953903\pi$
$337$		−	5.29846i		−	0.288625i	0.989532	−	0.144313i	$-0.0460971\pi$
$338$		0			0
$339$	36.6677			1.99152
$340$		0			0
$341$	−12.8270			−0.694620
$342$		0			0
$343$			34.9487i			1.88705i
$344$		0			0
$345$	4.84643	+	2.85542i	0.260923	+	0.153731i
$346$		0			0
$347$		−	33.6842i		−	1.80826i	−0.427256	−	0.904131i	$-0.640520\pi$
$347$		−	33.6842i		−	1.80826i	0.427256	−	0.904131i	$-0.359480\pi$
$348$		0			0
$349$	−19.2892			−1.03253			−0.516264	−	0.856430i	$-0.672678\pi$
$349$	−19.2892			−1.03253			−0.516264	+	0.856430i	$0.672678\pi$
$350$		0			0
$351$	1.58249			0.0844668
$352$		0			0
$353$			6.05334i			0.322187i	0.986939	+	0.161093i	$0.0515020\pi$
$353$			6.05334i			0.322187i	−0.986939	+	0.161093i	$0.948498\pi$
$354$		0			0
$355$	−5.80198	−	3.41841i	−0.307937	−	0.181431i
$356$		0			0
$357$		−	11.3134i		−	0.598768i
$358$		0			0
$359$	−15.9564			−0.842149			−0.421074	−	0.907026i	$-0.638347\pi$
$359$	−15.9564			−0.842149			−0.421074	+	0.907026i	$0.638347\pi$
$360$		0			0
$361$	5.11195			0.269050
$362$		0			0
$363$			20.8275i			1.09316i
$364$		0			0
$365$	−18.8923	+	32.0653i	−0.988866	+	1.67838i
$366$		0			0
$367$			13.7786i			0.719237i	0.933099	+	0.359619i	$0.117093\pi$
$367$			13.7786i			0.719237i	−0.933099	+	0.359619i	$0.882907\pi$
$368$		0			0
$369$	−19.2946			−1.00444
$370$		0			0
$371$	−52.4172			−2.72137
$372$		0			0
$373$		−	13.0667i		−	0.676566i	−0.941044	−	0.338283i	$-0.890154\pi$
$373$		−	13.0667i		−	0.676566i	0.941044	−	0.338283i	$-0.109846\pi$
$374$		0			0
$375$	−28.1155	−	0.744713i	−1.45188	−	0.0384568i
$376$		0			0
$377$			14.3432i			0.738713i
$378$		0			0
$379$	36.1458			1.85668			0.928342	−	0.371727i	$-0.121234\pi$
$379$	36.1458			1.85668			0.928342	+	0.371727i	$0.121234\pi$
$380$		0			0
$381$	−54.8519			−2.81015
$382$		0			0
$383$			20.4289i			1.04387i	0.852987	+	0.521933i	$0.174789\pi$
$383$			20.4289i			1.04387i	−0.852987	+	0.521933i	$0.825211\pi$
$384$		0			0
$385$	8.68769	−	14.7454i	0.442766	−	0.751495i
$386$		0			0
$387$		−	12.5552i		−	0.638218i
$388$		0			0
$389$	−37.3561			−1.89403			−0.947015	−	0.321189i	$-0.895917\pi$
$389$	−37.3561			−1.89403			−0.947015	+	0.321189i	$0.895917\pi$
$390$		0			0
$391$	0.969195			0.0490143
$392$		0			0
$393$			9.23030i			0.465607i
$394$		0			0
$395$	−0.315544	−	0.185913i	−0.0158768	−	0.00935428i
$396$		0			0
$397$			9.73956i			0.488814i	0.969673	+	0.244407i	$0.0785934\pi$
$397$			9.73956i			0.488814i	−0.969673	+	0.244407i	$0.921407\pi$
$398$		0			0
$399$	−57.3189			−2.86953
$400$		0			0
$401$	−26.8802			−1.34233			−0.671166	−	0.741307i	$-0.734206\pi$
$401$	−26.8802			−1.34233			−0.671166	+	0.741307i	$0.734206\pi$
$402$		0			0
$403$		−	14.9023i		−	0.742336i
$404$		0			0
$405$	15.2340	+	8.97560i	0.756985	+	0.446001i
$406$		0			0
$407$		−	12.5895i		−	0.624036i
$408$		0			0
$409$	−23.8307			−1.17835			−0.589175	−	0.808005i	$-0.700547\pi$
$409$	−23.8307			−1.17835			−0.589175	+	0.808005i	$0.700547\pi$
$410$		0			0
$411$	8.61395			0.424895
$412$		0			0
$413$			46.9742i			2.31145i
$414$		0			0
$415$	6.43391	−	10.9201i	0.315828	−	0.536047i
$416$		0			0
$417$		−	30.9012i		−	1.51324i
$418$		0			0
$419$	−16.8463			−0.822995			−0.411497	−	0.911411i	$-0.634994\pi$
$419$	−16.8463			−0.822995			−0.411497	+	0.911411i	$0.634994\pi$
$420$		0			0
$421$	10.4397			0.508798			0.254399	−	0.967099i	$-0.418122\pi$
$421$	10.4397			0.508798			0.254399	+	0.967099i	$0.418122\pi$
$422$		0			0
$423$			7.40888i			0.360232i
$424$		0			0
$425$	−4.23886	+	2.34852i	−0.205615	+	0.113920i
$426$		0			0
$427$		−	28.1263i		−	1.36113i
$428$		0			0
$429$	7.95145			0.383899
$430$		0			0
$431$	15.3189			0.737886			0.368943	−	0.929452i	$-0.379720\pi$
$431$	15.3189			0.737886			0.368943	+	0.929452i	$0.379720\pi$
$432$		0			0
$433$		−	17.5759i		−	0.844642i	−0.906446	−	0.422321i	$-0.861216\pi$
$433$		−	17.5759i		−	0.844642i	0.906446	−	0.422321i	$-0.138784\pi$
$434$		0			0
$435$	21.3723	−	36.2746i	1.02472	−	1.73923i
$436$		0			0
$437$		−	4.91039i		−	0.234896i
$438$		0			0
$439$	−26.7104			−1.27482			−0.637408	−	0.770527i	$-0.719993\pi$
$439$	−26.7104			−1.27482			−0.637408	+	0.770527i	$0.719993\pi$
$440$		0			0
$441$	48.3654			2.30311
$442$		0			0
$443$		−	14.9540i		−	0.710487i	−0.934774	−	0.355243i	$-0.884398\pi$
$443$		−	14.9540i		−	0.710487i	0.934774	−	0.355243i	$-0.115602\pi$
$444$		0			0
$445$	−7.51876	−	4.42990i	−0.356423	−	0.209998i
$446$		0			0
$447$			3.59071i			0.169835i
$448$		0			0
$449$	6.46847			0.305266			0.152633	−	0.988283i	$-0.451225\pi$
$449$	6.46847			0.305266			0.152633	+	0.988283i	$0.451225\pi$
$450$		0			0
$451$	−9.56213			−0.450263
$452$		0			0
$453$		−	42.6448i		−	2.00363i
$454$		0			0
$455$	17.1311	+	10.0933i	0.803117	+	0.473181i
$456$		0			0
$457$			41.1286i			1.92391i	0.273202	+	0.961957i	$0.411917\pi$
$457$			41.1286i			1.92391i	−0.273202	+	0.961957i	$0.588083\pi$
$458$		0			0
$459$	−0.800359			−0.0373576
$460$		0			0
$461$	12.9053			0.601060			0.300530	−	0.953772i	$-0.402836\pi$
$461$	12.9053			0.601060			0.300530	+	0.953772i	$0.402836\pi$
$462$		0			0
$463$			6.74237i			0.313344i	0.987651	+	0.156672i	$0.0500766\pi$
$463$			6.74237i			0.313344i	−0.987651	+	0.156672i	$0.949923\pi$
$464$		0			0
$465$	−22.2054	+	37.6885i	−1.02975	+	1.74776i
$466$		0			0
$467$			9.30704i			0.430678i	0.976539	+	0.215339i	$0.0690857\pi$
$467$			9.30704i			0.430678i	−0.976539	+	0.215339i	$0.930914\pi$
$468$		0			0
$469$	55.9695			2.58443
$470$		0			0
$471$	−49.3902			−2.27578
$472$		0			0
$473$		−	6.22220i		−	0.286097i
$474$		0			0
$475$	11.8987	+	21.4760i	0.545950	+	0.985387i
$476$		0			0
$477$			37.5970i			1.72145i
$478$		0			0
$479$	−31.6673			−1.44691			−0.723457	−	0.690370i	$-0.757448\pi$
$479$	−31.6673			−1.44691			−0.723457	+	0.690370i	$0.757448\pi$
$480$		0			0
$481$	14.6263			0.666903
$482$		0			0
$483$			11.6730i			0.531139i
$484$		0			0
$485$	2.03320	−	3.45090i	0.0923229	−	0.156697i
$486$		0			0
$487$		−	16.7582i		−	0.759385i	−0.925113	−	0.379693i	$-0.876030\pi$
$487$		−	16.7582i		−	0.759385i	0.925113	−	0.379693i	$-0.123970\pi$
$488$		0			0
$489$	−33.7873			−1.52792
$490$		0			0
$491$	24.5811			1.10933			0.554666	−	0.832073i	$-0.312846\pi$
$491$	24.5811			1.10933			0.554666	+	0.832073i	$0.312846\pi$
$492$		0			0
$493$		−	7.25423i		−	0.326714i
$494$		0			0
$495$	−10.5763	−	6.23138i	−0.475372	−	0.280080i
$496$		0			0
$497$		−	13.9745i		−	0.626841i
$498$		0			0
$499$	−2.25368			−0.100888			−0.0504442	−	0.998727i	$-0.516064\pi$
$499$	−2.25368			−0.100888			−0.0504442	+	0.998727i	$0.516064\pi$
$500$		0			0
$501$	−5.39654			−0.241100
$502$		0			0
$503$		−	10.8452i		−	0.483566i	−0.970330	−	0.241783i	$-0.922268\pi$
$503$		−	10.8452i		−	0.483566i	0.970330	−	0.241783i	$-0.0777322\pi$
$504$		0			0
$505$	14.8108	+	8.72624i	0.659073	+	0.388313i
$506$		0			0
$507$		−	23.4649i		−	1.04211i
$508$		0			0
$509$	−19.6490			−0.870926			−0.435463	−	0.900207i	$-0.643415\pi$
$509$	−19.6490			−0.870926			−0.435463	+	0.900207i	$0.643415\pi$
$510$		0			0
$511$	−77.2316			−3.41652
$512$		0			0
$513$			4.05499i			0.179032i
$514$		0			0
$515$	2.61947	−	4.44595i	0.115427	−	0.195912i
$516$		0			0
$517$			3.67174i			0.161483i
$518$		0			0
$519$	−5.15581			−0.226315
$520$		0			0
$521$	19.9000			0.871836			0.435918	−	0.899986i	$-0.356424\pi$
$521$	19.9000			0.871836			0.435918	+	0.899986i	$0.356424\pi$
$522$		0			0
$523$			20.4617i			0.894726i	0.894352	+	0.447363i	$0.147637\pi$
$523$			20.4617i			0.894726i	−0.894352	+	0.447363i	$0.852363\pi$
$524$		0			0
$525$	−28.2856	−	51.0527i	−1.23448	−	2.22812i
$526$		0			0
$527$			7.53700i			0.328317i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	33.6930			1.46215
$532$		0			0
$533$		−	11.1092i		−	0.481194i
$534$		0			0
$535$	−12.1490	+	20.6202i	−0.525249	+	0.891491i
$536$		0			0
$537$			47.4651i			2.04827i
$538$		0			0
$539$	23.9692			1.03243
$540$		0			0
$541$	−4.86543			−0.209181			−0.104591	−	0.994515i	$-0.533353\pi$
$541$	−4.86543			−0.209181			−0.104591	+	0.994515i	$0.533353\pi$
$542$		0			0
$543$			11.3163i			0.485628i
$544$		0			0
$545$	15.3110	+	9.02096i	0.655852	+	0.386415i
$546$		0			0
$547$			33.3411i			1.42556i	0.701386	+	0.712781i	$0.252565\pi$
$547$			33.3411i			1.42556i	−0.701386	+	0.712781i	$0.747435\pi$
$548$		0			0
$549$	−20.1740			−0.861005
$550$		0			0
$551$	−36.7533			−1.56574
$552$		0			0
$553$		−	0.760011i		−	0.0323189i
$554$		0			0
$555$	−36.9906	−	21.7941i	−1.57016	−	0.925110i
$556$		0			0
$557$		−	19.3928i		−	0.821697i	−0.911704	−	0.410849i	$-0.865232\pi$
$557$		−	19.3928i		−	0.821697i	0.911704	−	0.410849i	$-0.134768\pi$
$558$		0			0
$559$	7.22890			0.305750
$560$		0			0
$561$	−4.02153			−0.169789
$562$		0			0
$563$			7.49245i			0.315769i	0.987458	+	0.157884i	$0.0504674\pi$
$563$			7.49245i			0.315769i	−0.987458	+	0.157884i	$0.949533\pi$
$564$		0			0
$565$	16.5451	−	28.0815i	0.696057	−	1.18140i
$566$		0			0
$567$			36.6922i			1.54093i
$568$		0			0
$569$	3.26375			0.136823			0.0684117	−	0.997657i	$-0.478207\pi$
$569$	3.26375			0.136823			0.0684117	+	0.997657i	$0.478207\pi$
$570$		0			0
$571$	14.9738			0.626636			0.313318	−	0.949648i	$-0.398559\pi$
$571$	14.9738			0.626636			0.313318	+	0.949648i	$0.398559\pi$
$572$		0			0
$573$		−	50.4685i		−	2.10835i
$574$		0			0
$575$	4.37359	−	2.42317i	0.182391	−	0.101053i
$576$		0			0
$577$		−	21.5111i		−	0.895520i	−0.894154	−	0.447760i	$-0.852222\pi$
$577$		−	21.5111i		−	0.895520i	0.894154	−	0.447760i	$-0.147778\pi$
$578$		0			0
$579$	12.0044			0.498886
$580$		0			0
$581$	26.3018			1.09118
$582$		0			0
$583$			18.6326i			0.771681i
$584$		0			0
$585$	7.23957	−	12.2875i	0.299319	−	0.508026i
$586$		0			0
$587$			1.76395i			0.0728059i	0.999337	+	0.0364030i	$0.0115900\pi$
$587$			1.76395i			0.0728059i	−0.999337	+	0.0364030i	$0.988410\pi$
$588$		0			0
$589$	38.1859			1.57342
$590$		0			0
$591$	−39.6980			−1.63296
$592$		0			0
$593$			8.68465i			0.356636i	0.983973	+	0.178318i	$0.0570656\pi$
$593$			8.68465i			0.356636i	−0.983973	+	0.178318i	$0.942934\pi$
$594$		0			0
$595$	−8.66423	−	5.10479i	−0.355199	−	0.209276i
$596$		0			0
$597$			30.2363i			1.23749i
$598$		0			0
$599$	23.3988			0.956049			0.478024	−	0.878347i	$-0.341353\pi$
$599$	23.3988			0.956049			0.478024	+	0.878347i	$0.341353\pi$
$600$		0			0
$601$	21.0536			0.858794			0.429397	−	0.903116i	$-0.358726\pi$
$601$	21.0536			0.858794			0.429397	+	0.903116i	$0.358726\pi$
$602$		0			0
$603$		−	40.1450i		−	1.63483i
$604$		0			0
$605$	15.9505	+	9.39774i	0.648481	+	0.382072i
$606$		0			0
$607$		−	35.0489i		−	1.42259i	−0.702892	−	0.711296i	$-0.748108\pi$
$607$		−	35.0489i		−	1.42259i	0.702892	−	0.711296i	$-0.251892\pi$
$608$		0			0
$609$	87.3699			3.54041
$610$		0			0
$611$	−4.26580			−0.172576
$612$		0			0
$613$			27.5922i			1.11444i	0.830365	+	0.557219i	$0.188132\pi$
$613$			27.5922i			1.11444i	−0.830365	+	0.557219i	$0.811868\pi$
$614$		0			0
$615$	−16.5534	+	28.0957i	−0.667498	+	1.13293i
$616$		0			0
$617$		−	8.22209i		−	0.331009i	−0.986209	−	0.165504i	$-0.947075\pi$
$617$		−	8.22209i		−	0.331009i	0.986209	−	0.165504i	$-0.0529252\pi$
$618$		0			0
$619$	19.5042			0.783940			0.391970	−	0.919978i	$-0.371794\pi$
$619$	19.5042			0.783940			0.391970	+	0.919978i	$0.371794\pi$
$620$		0			0
$621$	0.825798			0.0331381
$622$		0			0
$623$		−	18.1095i		−	0.725540i
$624$		0			0
$625$	−13.2565	+	21.1959i	−0.530260	+	0.847835i
$626$		0			0
$627$			20.3749i			0.813696i
$628$		0			0
$629$	−7.39742			−0.294955
$630$		0			0
$631$	−7.59833			−0.302485			−0.151242	−	0.988497i	$-0.548327\pi$
$631$	−7.59833			−0.302485			−0.151242	+	0.988497i	$0.548327\pi$
$632$		0			0
$633$		−	40.4488i		−	1.60770i
$634$		0			0
$635$	−24.7501	+	42.0077i	−0.982178	+	1.66702i
$636$		0			0
$637$			27.8472i			1.10335i
$638$		0			0
$639$	−10.0234			−0.396520
$640$		0			0
$641$	−2.02749			−0.0800809			−0.0400405	−	0.999198i	$-0.512749\pi$
$641$	−2.02749			−0.0800809			−0.0400405	+	0.999198i	$0.512749\pi$
$642$		0			0
$643$		−	21.7200i		−	0.856554i	−0.903647	−	0.428277i	$-0.859121\pi$
$643$		−	21.7200i		−	0.856554i	0.903647	−	0.428277i	$-0.140879\pi$
$644$		0			0
$645$	−18.2822	−	10.7715i	−0.719861	−	0.424128i
$646$		0			0
$647$		−	12.3675i		−	0.486216i	−0.969999	−	0.243108i	$-0.921833\pi$
$647$		−	12.3675i		−	0.486216i	0.969999	−	0.243108i	$-0.0781670\pi$
$648$		0			0
$649$	16.6978			0.655444
$650$		0			0
$651$	−90.7755			−3.55777
$652$		0			0
$653$			24.8363i			0.971921i	0.873981	+	0.485960i	$0.161530\pi$
$653$			24.8363i			0.971921i	−0.873981	+	0.485960i	$0.838470\pi$
$654$		0			0
$655$	7.06892	+	4.16487i	0.276205	+	0.162735i
$656$		0			0
$657$			55.3955i			2.16118i
$658$		0			0
$659$	29.4867			1.14864			0.574319	−	0.818631i	$-0.305267\pi$
$659$	29.4867			1.14864			0.574319	+	0.818631i	$0.305267\pi$
$660$		0			0
$661$	42.3911			1.64882			0.824411	−	0.565992i	$-0.191507\pi$
$661$	42.3911			1.64882			0.824411	+	0.565992i	$0.191507\pi$
$662$		0			0
$663$		−	4.67218i		−	0.181452i
$664$		0			0
$665$	−25.8633	+	43.8970i	−1.00293	+	1.70225i
$666$		0			0
$667$			7.48480i			0.289813i
$668$		0			0
$669$	−45.8464			−1.77252
$670$		0			0
$671$	−9.99794			−0.385966
$672$		0			0
$673$		−	20.5867i		−	0.793559i	−0.917914	−	0.396779i	$-0.870128\pi$
$673$		−	20.5867i		−	0.793559i	0.917914	−	0.396779i	$-0.129872\pi$
$674$		0			0
$675$	−3.61170	+	2.00105i	−0.139014	+	0.0770203i
$676$		0			0
$677$			12.8416i			0.493545i	0.969073	+	0.246772i	$0.0793700\pi$
$677$			12.8416i			0.493545i	−0.969073	+	0.246772i	$0.920630\pi$
$678$		0			0
$679$	8.31173			0.318975
$680$		0			0
$681$	−26.5335			−1.01676
$682$		0			0
$683$		−	47.1531i		−	1.80426i	−0.431460	−	0.902132i	$-0.642001\pi$
$683$		−	47.1531i		−	1.80426i	0.431460	−	0.902132i	$-0.357999\pi$
$684$		0			0
$685$	3.88676	−	6.59690i	0.148506	−	0.252054i
$686$		0			0
$687$			55.5242i			2.11838i
$688$		0			0
$689$	−21.6472			−0.824691
$690$		0			0
$691$	29.9278			1.13851			0.569253	−	0.822163i	$-0.307233\pi$
$691$	29.9278			1.13851			0.569253	+	0.822163i	$0.307233\pi$
$692$		0			0
$693$		−	25.4739i		−	0.967673i
$694$		0			0
$695$	−23.6653	−	13.9432i	−0.897678	−	0.528894i
$696$		0			0
$697$			5.61860i			0.212820i
$698$		0			0
$699$	30.0771			1.13762
$700$		0			0
$701$	15.7689			0.595585			0.297793	−	0.954631i	$-0.403750\pi$
$701$	15.7689			0.595585			0.297793	+	0.954631i	$0.403750\pi$
$702$		0			0
$703$			37.4788i			1.41354i
$704$		0			0
$705$	10.7884	+	6.35630i	0.406314	+	0.239392i
$706$		0			0
$707$			35.6729i			1.34162i
$708$		0			0
$709$	0.221825			0.00833080			0.00416540	−	0.999991i	$-0.498674\pi$
$709$	0.221825			0.00833080			0.00416540	+	0.999991i	$0.498674\pi$
$710$		0			0
$711$	−0.545129			−0.0204439
$712$		0			0
$713$		−	7.77655i		−	0.291234i
$714$		0			0
$715$	3.58783	−	6.08952i	0.134177	−	0.227735i
$716$		0			0
$717$		−	0.411958i		−	0.0153848i
$718$		0			0
$719$	43.2546			1.61312			0.806562	−	0.591149i	$-0.201325\pi$
$719$	43.2546			1.61312			0.806562	+	0.591149i	$0.201325\pi$
$720$		0			0
$721$	10.7084			0.398801
$722$		0			0
$723$			16.8319i			0.625984i
$724$		0			0
$725$	−18.1369	−	32.7354i	−0.673589	−	1.21576i
$726$		0			0
$727$			8.90379i			0.330223i	0.986275	+	0.165112i	$0.0527985\pi$
$727$			8.90379i			0.330223i	−0.986275	+	0.165112i	$0.947202\pi$
$728$		0			0
$729$	32.5502			1.20556
$730$		0			0
$731$	−3.65609			−0.135226
$732$		0			0
$733$			14.0655i			0.519521i	0.965673	+	0.259761i	$0.0836436\pi$
$733$			14.0655i			0.519521i	−0.965673	+	0.259761i	$0.916356\pi$
$734$		0			0
$735$	41.4941	−	70.4268i	1.53053	−	2.59773i
$736$		0			0
$737$		−	19.8953i		−	0.732852i
$738$		0			0
$739$	37.4096			1.37614			0.688068	−	0.725646i	$-0.258459\pi$
$739$	37.4096			1.37614			0.688068	+	0.725646i	$0.258459\pi$
$740$		0			0
$741$	−23.6714			−0.869592
$742$		0			0
$743$			7.48682i			0.274665i	0.990525	+	0.137332i	$0.0438528\pi$
$743$			7.48682i			0.274665i	−0.990525	+	0.137332i	$0.956147\pi$
$744$		0			0
$745$	2.74990	+	1.62019i	0.100749	+	0.0593591i
$746$		0			0
$747$		−	18.8654i		−	0.690248i
$748$		0			0
$749$	−49.6653			−1.81473
$750$		0			0
$751$	29.0524			1.06014			0.530069	−	0.847955i	$-0.322166\pi$
$751$	29.0524			1.06014			0.530069	+	0.847955i	$0.322166\pi$
$752$		0			0
$753$		−	17.3686i		−	0.632947i
$754$		0			0
$755$	−32.6591	−	19.2421i	−1.18858	−	0.700291i
$756$		0			0
$757$		−	52.0910i		−	1.89328i	−0.322295	−	0.946639i	$-0.604454\pi$
$757$		−	52.0910i		−	1.89328i	0.322295	−	0.946639i	$-0.395546\pi$
$758$		0			0
$759$	4.14935			0.150612
$760$		0			0
$761$	21.8693			0.792762			0.396381	−	0.918086i	$-0.370266\pi$
$761$	21.8693			0.792762			0.396381	+	0.918086i	$0.370266\pi$
$762$		0			0
$763$			36.8777i			1.33506i
$764$		0			0
$765$	−3.66149	+	6.21455i	−0.132381	+	0.224687i
$766$		0			0
$767$			19.3993i			0.700469i
$768$		0			0
$769$	−40.9646			−1.47722			−0.738611	−	0.674132i	$-0.764518\pi$
$769$	−40.9646			−1.47722			−0.738611	+	0.674132i	$0.764518\pi$
$770$		0			0
$771$	63.6977			2.29402
$772$		0			0
$773$			31.3674i			1.12821i	0.825704	+	0.564103i	$0.190778\pi$
$773$			31.3674i			1.12821i	−0.825704	+	0.564103i	$0.809222\pi$
$774$		0			0
$775$	18.8439	+	34.0114i	0.676893	+	1.22173i
$776$		0			0
$777$		−	89.0945i		−	3.19625i
$778$		0			0
$779$	28.4665			1.01992
$780$		0			0
$781$	−4.96746			−0.177750
$782$		0			0
$783$		−	6.18093i		−	0.220888i
$784$		0			0
$785$	−22.2857	+	37.8249i	−0.795411	+	1.35003i
$786$		0			0
$787$			17.8694i			0.636977i	0.947927	+	0.318489i	$0.103175\pi$
$787$			17.8694i			0.636977i	−0.947927	+	0.318489i	$0.896825\pi$
$788$		0			0
$789$	−24.0497			−0.856192
$790$		0			0
$791$	67.6364			2.40487
$792$		0			0
$793$		−	11.6155i		−	0.412480i
$794$		0			0
$795$	54.7466	+	32.2556i	1.94166	+	1.14399i
$796$		0			0
$797$			22.0293i			0.780319i	0.920747	+	0.390159i	$0.127580\pi$
$797$			22.0293i			0.780319i	−0.920747	+	0.390159i	$0.872420\pi$
$798$		0			0
$799$	2.15747			0.0763259
$800$		0			0
$801$	−12.9893			−0.458953
$802$		0			0
$803$			27.4532i			0.968803i
$804$		0			0
$805$	8.93961	+	5.26704i	0.315080	+	0.185639i
$806$		0			0
$807$		−	72.2132i		−	2.54202i
$808$		0			0
$809$	21.0548			0.740246			0.370123	−	0.928983i	$-0.379316\pi$
$809$	21.0548			0.740246			0.370123	+	0.928983i	$0.379316\pi$
$810$		0			0
$811$	−34.0175			−1.19451			−0.597257	−	0.802050i	$-0.703743\pi$
$811$	−34.0175			−1.19451			−0.597257	+	0.802050i	$0.703743\pi$
$812$		0			0
$813$		−	8.03584i		−	0.281829i
$814$		0			0
$815$	−15.2454	+	25.8756i	−0.534024	+	0.906384i
$816$		0			0
$817$			18.5235i			0.648054i
$818$		0			0
$819$	29.5954			1.03415
$820$		0			0
$821$	−2.94320			−0.102718			−0.0513591	−	0.998680i	$-0.516355\pi$
$821$	−2.94320			−0.102718			−0.0513591	+	0.998680i	$0.516355\pi$
$822$		0			0
$823$		−	41.4173i		−	1.44372i	−0.692041	−	0.721859i	$-0.743288\pi$
$823$		−	41.4173i		−	1.44372i	0.692041	−	0.721859i	$-0.256712\pi$
$824$		0			0
$825$	−18.1475	+	10.0546i	−0.631816	+	0.350055i
$826$		0			0
$827$			23.4450i			0.815262i	0.913147	+	0.407631i	$0.133645\pi$
$827$			23.4450i			0.815262i	−0.913147	+	0.407631i	$0.866355\pi$
$828$		0			0
$829$	5.16155			0.179268			0.0896340	−	0.995975i	$-0.471430\pi$
$829$	5.16155			0.179268			0.0896340	+	0.995975i	$0.471430\pi$
$830$		0			0
$831$	43.3918			1.50524
$832$		0			0
$833$		−	14.0840i		−	0.487983i
$834$		0			0
$835$	−2.43501	+	4.13288i	−0.0842670	+	0.143024i
$836$		0			0
$837$			6.42186i			0.221972i
$838$		0			0
$839$	−7.55411			−0.260797			−0.130398	−	0.991462i	$-0.541626\pi$
$839$	−7.55411			−0.260797			−0.130398	+	0.991462i	$0.541626\pi$
$840$		0			0
$841$	27.0222			0.931801
$842$		0			0
$843$		−	30.5908i		−	1.05360i
$844$		0			0
$845$	−17.9704	−	10.5878i	−0.618199	−	0.364231i
$846$		0			0
$847$			38.4180i			1.32006i
$848$		0			0
$849$	27.8581			0.956087
$850$		0			0
$851$	7.63254			0.261640
$852$		0			0
$853$		−	19.3528i		−	0.662628i	−0.943521	−	0.331314i	$-0.892508\pi$
$853$		−	19.3528i		−	0.662628i	0.943521	−	0.331314i	$-0.107492\pi$
$854$		0			0
$855$	31.4858	+	18.5508i	1.07679	+	0.634424i
$856$		0			0
$857$		−	19.6628i		−	0.671667i	−0.941921	−	0.335833i	$-0.890982\pi$
$857$		−	19.6628i		−	0.671667i	0.941921	−	0.335833i	$-0.109018\pi$
$858$		0			0
$859$	22.1341			0.755205			0.377602	−	0.925968i	$-0.376749\pi$
$859$	22.1341			0.755205			0.377602	+	0.925968i	$0.376749\pi$
$860$		0			0
$861$	−67.6704			−2.30620
$862$		0			0
$863$		−	57.4689i		−	1.95626i	−0.207986	−	0.978132i	$-0.566691\pi$
$863$		−	57.4689i		−	1.95626i	0.207986	−	0.978132i	$-0.433309\pi$
$864$		0			0
$865$	−2.32639	+	3.94852i	−0.0790996	+	0.134254i
$866$		0			0
$867$		−	40.4023i		−	1.37213i
$868$		0			0
$869$	−0.270158			−0.00916449
$870$		0			0
$871$	23.1142			0.783194
$872$		0			0
$873$		−	5.96171i		−	0.201773i
$874$		0			0
$875$	−51.8611	−	1.37368i	−1.75323	−	0.0464388i
$876$		0			0
$877$			14.5774i			0.492243i	0.969239	+	0.246121i	$0.0791562\pi$
$877$			14.5774i			0.492243i	−0.969239	+	0.246121i	$0.920844\pi$
$878$		0			0
$879$	53.2832			1.79720
$880$		0			0
$881$	30.1713			1.01650			0.508248	−	0.861211i	$-0.330293\pi$
$881$	30.1713			1.01650			0.508248	+	0.861211i	$0.330293\pi$
$882$		0			0
$883$		−	8.92294i		−	0.300281i	−0.988665	−	0.150140i	$-0.952027\pi$
$883$		−	8.92294i		−	0.300281i	0.988665	−	0.150140i	$-0.0479726\pi$
$884$		0			0
$885$	28.9062	−	49.0617i	0.971671	−	1.64919i
$886$		0			0
$887$			31.3040i			1.05109i	0.850767	+	0.525543i	$0.176138\pi$
$887$			31.3040i			1.05109i	−0.850767	+	0.525543i	$0.823862\pi$
$888$		0			0
$889$	−101.178			−3.39342
$890$		0			0
$891$	13.0429			0.436952
$892$		0			0
$893$		−	10.9308i		−	0.365784i
$894$		0			0
$895$	36.3506	+	21.4171i	1.21507	+	0.715893i
$896$		0			0
$897$			4.82068i			0.160958i
$898$		0			0
$899$	−58.2060			−1.94128
$900$		0			0
$901$	10.9483			0.364740
$902$		0			0
$903$		−	44.0340i		−	1.46536i
$904$		0			0
$905$	8.66645	+	5.10610i	0.288082	+	0.169733i
$906$		0			0
$907$		−	55.2883i		−	1.83582i	−0.396793	−	0.917908i	$-0.629877\pi$
$907$		−	55.2883i		−	1.83582i	0.396793	−	0.917908i	$-0.370123\pi$
$908$		0			0
$909$	25.5869			0.848664
$910$		0			0
$911$	−30.0030			−0.994044			−0.497022	−	0.867738i	$-0.665573\pi$
$911$	−30.0030			−0.994044			−0.497022	+	0.867738i	$0.665573\pi$
$912$		0			0
$913$		−	9.34941i		−	0.309420i
$914$		0			0
$915$	−17.3079	+	29.3762i	−0.572180	+	0.971146i
$916$		0			0
$917$			17.0260i			0.562247i
$918$		0			0
$919$	−5.98146			−0.197310			−0.0986551	−	0.995122i	$-0.531454\pi$
$919$	−5.98146			−0.197310			−0.0986551	+	0.995122i	$0.531454\pi$
$920$		0			0
$921$	−51.0144			−1.68098
$922$		0			0
$923$		−	5.77115i		−	0.189960i
$924$		0			0
$925$	−33.3816	+	18.4949i	−1.09758	+	0.608110i
$926$		0			0
$927$		−	7.68075i		−	0.252269i
$928$		0			0
$929$	59.4966			1.95202			0.976009	−	0.217728i	$-0.0698647\pi$
$929$	59.4966			1.95202			0.976009	+	0.217728i	$0.0698647\pi$
$930$		0			0
$931$	−71.3563			−2.33861
$932$		0			0
$933$		−	86.7119i		−	2.83882i
$934$		0			0
$935$	−1.81458	+	3.07984i	−0.0593432	+	0.100722i
$936$		0			0
$937$		−	59.7229i		−	1.95106i	−0.219863	−	0.975531i	$-0.570561\pi$
$937$		−	59.7229i		−	1.95106i	0.219863	−	0.975531i	$-0.429439\pi$
$938$		0			0
$939$	41.3684			1.35001
$940$		0			0
$941$	41.6385			1.35738			0.678688	−	0.734426i	$-0.262549\pi$
$941$	41.6385			1.35738			0.678688	+	0.734426i	$0.262549\pi$
$942$		0			0
$943$		−	5.79719i		−	0.188782i
$944$		0			0
$945$	−7.38231	−	4.34951i	−0.240147	−	0.141490i
$946$		0			0
$947$		−	23.0566i		−	0.749239i	−0.927179	−	0.374620i	$-0.877773\pi$
$947$		−	23.0566i		−	0.749239i	0.927179	−	0.374620i	$-0.122227\pi$
$948$		0			0
$949$	−31.8950			−1.03535
$950$		0			0
$951$	−2.97274			−0.0963978
$952$		0			0
$953$		−	45.5945i		−	1.47695i	−0.674280	−	0.738476i	$-0.735546\pi$
$953$		−	45.5945i		−	1.47695i	0.674280	−	0.738476i	$-0.264454\pi$
$954$		0			0
$955$	−38.6507	−	22.7722i	−1.25071	−	0.736892i
$956$		0			0
$957$		−	31.0571i		−	1.00393i
$958$		0			0
$959$	15.8891			0.513085
$960$		0			0
$961$	29.4748			0.950800
$962$		0			0
$963$			35.6232i			1.14794i
$964$		0			0
$965$	5.41659	−	9.19343i	0.174366	−	0.295947i
$966$		0			0
$967$		−	47.5979i		−	1.53064i	−0.643647	−	0.765322i	$-0.722580\pi$
$967$		−	47.5979i		−	1.53064i	0.643647	−	0.765322i	$-0.277420\pi$
$968$		0			0
$969$	11.9721			0.384599
$970$		0			0
$971$	20.0384			0.643064			0.321532	−	0.946899i	$-0.395802\pi$
$971$	20.0384			0.643064			0.321532	+	0.946899i	$0.395802\pi$
$972$		0			0
$973$		−	56.9996i		−	1.82732i
$974$		0			0
$975$	−11.6813	−	21.0837i	−0.374102	−	0.675218i
$976$		0			0
$977$			1.86869i			0.0597846i	0.999553	+	0.0298923i	$0.00951644\pi$
$977$			1.86869i			0.0597846i	−0.999553	+	0.0298923i	$0.990484\pi$
$978$		0			0
$979$	−6.43730			−0.205737
$980$		0			0
$981$	26.4511			0.844518
$982$		0			0
$983$			30.3038i			0.966542i	0.875471	+	0.483271i	$0.160551\pi$
$983$			30.3038i			0.966542i	−0.875471	+	0.483271i	$0.839449\pi$
$984$		0			0
$985$	−17.9124	+	30.4022i	−0.570737	+	0.968696i
$986$		0			0
$987$			25.9846i			0.827098i
$988$		0			0
$989$	3.77230			0.119952
$990$		0			0
$991$	−30.1424			−0.957505			−0.478752	−	0.877950i	$-0.658911\pi$
$991$	−30.1424			−0.957505			−0.478752	+	0.877950i	$0.658911\pi$
$992$		0			0
$993$			84.6694i			2.68690i
$994$		0			0
$995$	23.1561	+	13.6431i	0.734099	+	0.432516i
$996$		0			0
$997$			13.9205i			0.440865i	0.975402	+	0.220433i	$0.0707469\pi$
$997$			13.9205i			0.440865i	−0.975402	+	0.220433i	$0.929253\pi$
$998$		0			0
$999$	−6.30294			−0.199416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.e.c.369.4	✓	16
4.3	odd	2		1840.2.e.h.369.13		16
5.2	odd	4		4600.2.a.bk.1.2		8
5.3	odd	4		4600.2.a.bj.1.7		8
5.4	even	2	inner	920.2.e.c.369.13	yes	16
20.3	even	4		9200.2.a.de.1.2		8
20.7	even	4		9200.2.a.dd.1.7		8
20.19	odd	2		1840.2.e.h.369.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.4	✓	16	1.1	even	1	trivial
920.2.e.c.369.13	yes	16	5.4	even	2	inner
1840.2.e.h.369.4		16	20.19	odd	2
1840.2.e.h.369.13		16	4.3	odd	2
4600.2.a.bj.1.7		8	5.3	odd	4
4600.2.a.bk.1.2		8	5.2	odd	4
9200.2.a.dd.1.7		8	20.7	even	4
9200.2.a.de.1.2		8	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)

\(f(q)\)	\(=\)	\(q-2.51561i q^{3} +(-1.92655 - 1.13508i) q^{5} -4.64022i q^{7} -3.32827 q^{9} +O(q^{10})\)	\(q-2.51561i q^{3} +(-1.92655 - 1.13508i) q^{5} -4.64022i q^{7} -3.32827 q^{9} -1.64944 q^{11} -1.91631i q^{13} +(-2.85542 + 4.84643i) q^{15} +0.969195i q^{17} +4.91039 q^{19} -11.6730 q^{21} -1.00000i q^{23} +(2.42317 + 4.37359i) q^{25} +0.825798i q^{27} -7.48480 q^{29} +7.77655 q^{31} +4.14935i q^{33} +(-5.26704 + 8.93961i) q^{35} +7.63254i q^{37} -4.82068 q^{39} +5.79719 q^{41} +3.77230i q^{43} +(6.41207 + 3.77787i) q^{45} -2.22605i q^{47} -14.5317 q^{49} +2.43811 q^{51} -11.2963i q^{53} +(3.17773 + 1.87226i) q^{55} -12.3526i q^{57} -10.1233 q^{59} +6.06140 q^{61} +15.4439i q^{63} +(-2.17517 + 3.69186i) q^{65} +12.0618i q^{67} -2.51561 q^{69} +3.01159 q^{71} -16.6439i q^{73} +(11.0022 - 6.09574i) q^{75} +7.65379i q^{77} +0.163787 q^{79} -7.90743 q^{81} +5.66822i q^{83} +(1.10012 - 1.86720i) q^{85} +18.8288i q^{87} +3.90271 q^{89} -8.89211 q^{91} -19.5627i q^{93} +(-9.46010 - 5.57371i) q^{95} +1.79123i q^{97} +5.48979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) 16 * q - 2 * q^5 - 22 * q^9	\( 16 q - 2 q^{5} - 22 q^{9} + 14 q^{11} + 6 q^{15} - 22 q^{19} + 12 q^{25} - 44 q^{29} + 18 q^{31} + 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} - 38 q^{51} + 30 q^{55} - 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} + 30 q^{71} + 56 q^{75} + 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} - 70 q^{91} + 38 q^{95} - 122 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 22 * q^9 + 14 * q^11 + 6 * q^15 - 22 * q^19 + 12 * q^25 - 44 * q^29 + 18 * q^31 + 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 - 38 * q^51 + 30 * q^55 - 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 + 30 * q^71 + 56 * q^75 + 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 - 70 * q^91 + 38 * q^95 - 122 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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