LMFDB - Embedded newform 1120.2.h.b.111.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(111,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1120.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.94324502638$
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} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 $ x^16 - x^15 - 2x^12 + 6x^11 - 12x^9 + 8x^8 - 24x^7 + 48x^5 - 32x^4 - 128x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.1
Root			$-0.275585 - 1.38710i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.b.111.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.19977i q^{3} +1.00000 q^{5} +(2.59303 + 0.525543i) q^{7} -7.23851 q^{9} +O(q^{10})$	$q-3.19977i q^{3} +1.00000 q^{5} +(2.59303 + 0.525543i) q^{7} -7.23851 q^{9} +3.34588 q^{11} +3.90251 q^{13} -3.19977i q^{15} +2.92992i q^{17} -6.33672i q^{19} +(1.68161 - 8.29709i) q^{21} -3.44642i q^{23} +1.00000 q^{25} +13.5622i q^{27} -2.68130i q^{29} +2.52241 q^{31} -10.7060i q^{33} +(2.59303 + 0.525543i) q^{35} -4.70905i q^{37} -12.4871i q^{39} +5.59232i q^{41} -8.62439 q^{43} -7.23851 q^{45} +0.506742 q^{47} +(6.44761 + 2.72550i) q^{49} +9.37508 q^{51} +11.4136i q^{53} +3.34588 q^{55} -20.2760 q^{57} +0.802275i q^{59} -7.97236 q^{61} +(-18.7697 - 3.80415i) q^{63} +3.90251 q^{65} -6.37503 q^{67} -11.0278 q^{69} -1.11901i q^{71} +5.91619i q^{73} -3.19977i q^{75} +(8.67597 + 1.75840i) q^{77} +10.5412i q^{79} +21.6805 q^{81} -4.29261i q^{83} +2.92992i q^{85} -8.57954 q^{87} +2.00722i q^{89} +(10.1193 + 2.05094i) q^{91} -8.07111i q^{93} -6.33672i q^{95} -6.06903i q^{97} -24.2192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 - 16 * q^9	$ 16 q + 16 q^{5} - 16 q^{9} + 4 q^{11} - 4 q^{21} + 16 q^{25} + 16 q^{31} + 4 q^{43} - 16 q^{45} - 8 q^{49} + 40 q^{51} + 4 q^{55} - 16 q^{57} - 8 q^{61} - 28 q^{63} - 20 q^{67} - 40 q^{69} - 4 q^{77} + 24 q^{81} - 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) $ 16 * q + 16 * q^5 - 16 * q^9 + 4 * q^11 - 4 * q^21 + 16 * q^25 + 16 * q^31 + 4 * q^43 - 16 * q^45 - 8 * q^49 + 40 * q^51 + 4 * q^55 - 16 * q^57 - 8 * q^61 - 28 * q^63 - 20 * q^67 - 40 * q^69 - 4 * q^77 + 24 * q^81 - 72 * q^87 + 32 * q^91 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$		−	3.19977i		−	1.84739i	−0.383133	−	0.923693i	$-0.625155\pi$
$3$		−	3.19977i		−	1.84739i	0.383133	−	0.923693i	$-0.374845\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.59303	+	0.525543i	0.980073	+	0.198637i
$7$	2.59303	+	0.525543i	0.980073	+	0.198637i
$8$		0			0
$9$	−7.23851			−2.41284
$10$		0			0
$11$	3.34588			1.00882			0.504411	−	0.863464i	$-0.331710\pi$
$11$	3.34588			1.00882			0.504411	+	0.863464i	$0.331710\pi$
$12$		0			0
$13$	3.90251			1.08236			0.541181	−	0.840906i	$-0.317977\pi$
$13$	3.90251			1.08236			0.541181	+	0.840906i	$0.317977\pi$
$14$		0			0
$15$		−	3.19977i		−	0.826176i
$16$		0			0
$17$			2.92992i			0.710611i	0.934750	+	0.355306i	$0.115623\pi$
$17$			2.92992i			0.710611i	−0.934750	+	0.355306i	$0.884377\pi$
$18$		0			0
$19$		−	6.33672i		−	1.45374i	−0.686774	−	0.726871i	$-0.740974\pi$
$19$		−	6.33672i		−	1.45374i	0.686774	−	0.726871i	$-0.259026\pi$
$20$		0			0
$21$	1.68161	−	8.29709i	0.366958	−	1.81057i
$22$		0			0
$23$		−	3.44642i		−	0.718629i	−0.933216	−	0.359315i	$-0.883010\pi$
$23$		−	3.44642i		−	0.718629i	0.933216	−	0.359315i	$-0.116990\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			13.5622i			2.61005i
$28$		0			0
$29$		−	2.68130i		−	0.497905i	−0.968516	−	0.248952i	$-0.919914\pi$
$29$		−	2.68130i		−	0.497905i	0.968516	−	0.248952i	$-0.0800863\pi$
$30$		0			0
$31$	2.52241			0.453037			0.226519	−	0.974007i	$-0.427266\pi$
$31$	2.52241			0.453037			0.226519	+	0.974007i	$0.427266\pi$
$32$		0			0
$33$		−	10.7060i		−	1.86368i
$34$		0			0
$35$	2.59303	+	0.525543i	0.438302	+	0.0888330i
$36$		0			0
$37$		−	4.70905i		−	0.774164i	−0.922045	−	0.387082i	$-0.873483\pi$
$37$		−	4.70905i		−	0.774164i	0.922045	−	0.387082i	$-0.126517\pi$
$38$		0			0
$39$		−	12.4871i		−	1.99954i
$40$		0			0
$41$			5.59232i			0.873374i	0.899614	+	0.436687i	$0.143848\pi$
$41$			5.59232i			0.873374i	−0.899614	+	0.436687i	$0.856152\pi$
$42$		0			0
$43$	−8.62439			−1.31521			−0.657604	−	0.753364i	$-0.728430\pi$
$43$	−8.62439			−1.31521			−0.657604	+	0.753364i	$0.728430\pi$
$44$		0			0
$45$	−7.23851			−1.07905
$46$		0			0
$47$	0.506742			0.0739159			0.0369580	−	0.999317i	$-0.488233\pi$
$47$	0.506742			0.0739159			0.0369580	+	0.999317i	$0.488233\pi$
$48$		0			0
$49$	6.44761	+	2.72550i	0.921087	+	0.389357i
$50$		0			0
$51$	9.37508			1.31277
$52$		0			0
$53$			11.4136i			1.56777i	0.620904	+	0.783886i	$0.286765\pi$
$53$			11.4136i			1.56777i	−0.620904	+	0.783886i	$0.713235\pi$
$54$		0			0
$55$	3.34588			0.451159
$56$		0			0
$57$	−20.2760			−2.68562
$58$		0			0
$59$			0.802275i			0.104447i	0.998635	+	0.0522236i	$0.0166309\pi$
$59$			0.802275i			0.104447i	−0.998635	+	0.0522236i	$0.983369\pi$
$60$		0			0
$61$	−7.97236			−1.02076			−0.510378	−	0.859950i	$-0.670494\pi$
$61$	−7.97236			−1.02076			−0.510378	+	0.859950i	$0.670494\pi$
$62$		0			0
$63$	−18.7697	−	3.80415i	−2.36476	−	0.479277i
$64$		0			0
$65$	3.90251			0.484047
$66$		0			0
$67$	−6.37503			−0.778834			−0.389417	−	0.921062i	$-0.627323\pi$
$67$	−6.37503			−0.778834			−0.389417	+	0.921062i	$0.627323\pi$
$68$		0			0
$69$	−11.0278			−1.32759
$70$		0			0
$71$		−	1.11901i		−	0.132802i	−0.997793	−	0.0664011i	$-0.978848\pi$
$71$		−	1.11901i		−	0.132802i	0.997793	−	0.0664011i	$-0.0211517\pi$
$72$		0			0
$73$			5.91619i			0.692438i	0.938154	+	0.346219i	$0.112535\pi$
$73$			5.91619i			0.692438i	−0.938154	+	0.346219i	$0.887465\pi$
$74$		0			0
$75$		−	3.19977i		−	0.369477i
$76$		0			0
$77$	8.67597	+	1.75840i	0.988719	+	0.200389i
$78$		0			0
$79$			10.5412i			1.18598i	0.805211	+	0.592989i	$0.202052\pi$
$79$			10.5412i			1.18598i	−0.805211	+	0.592989i	$0.797948\pi$
$80$		0			0
$81$	21.6805			2.40894
$82$		0			0
$83$		−	4.29261i		−	0.471175i	−0.971853	−	0.235588i	$-0.924299\pi$
$83$		−	4.29261i		−	0.471175i	0.971853	−	0.235588i	$-0.0757014\pi$
$84$		0			0
$85$			2.92992i			0.317795i
$86$		0			0
$87$	−8.57954			−0.919823
$88$		0			0
$89$			2.00722i			0.212764i	0.994325	+	0.106382i	$0.0339267\pi$
$89$			2.00722i			0.212764i	−0.994325	+	0.106382i	$0.966073\pi$
$90$		0			0
$91$	10.1193	+	2.05094i	1.06079	+	0.214997i
$92$		0			0
$93$		−	8.07111i		−	0.836935i
$94$		0			0
$95$		−	6.33672i		−	0.650134i
$96$		0			0
$97$		−	6.06903i		−	0.616217i	−0.951351	−	0.308108i	$-0.900304\pi$
$97$		−	6.06903i		−	0.616217i	0.951351	−	0.308108i	$-0.0996959\pi$
$98$		0			0
$99$	−24.2192			−2.43412
$100$		0			0
$101$	11.2324			1.11767			0.558834	−	0.829279i	$-0.311249\pi$
$101$	11.2324			1.11767			0.558834	+	0.829279i	$0.311249\pi$
$102$		0			0
$103$	−0.403587			−0.0397667			−0.0198833	−	0.999802i	$-0.506329\pi$
$103$	−0.403587			−0.0397667			−0.0198833	+	0.999802i	$0.506329\pi$
$104$		0			0
$105$	1.68161	−	8.29709i	0.164109	−	0.809713i
$106$		0			0
$107$	−17.9070			−1.73114			−0.865568	−	0.500791i	$-0.833043\pi$
$107$	−17.9070			−1.73114			−0.865568	+	0.500791i	$0.833043\pi$
$108$		0			0
$109$		−	14.4062i		−	1.37986i	−0.723874	−	0.689932i	$-0.757641\pi$
$109$		−	14.4062i		−	1.37986i	0.723874	−	0.689932i	$-0.242359\pi$
$110$		0			0
$111$	−15.0679			−1.43018
$112$		0			0
$113$	−6.85909			−0.645249			−0.322625	−	0.946527i	$-0.604565\pi$
$113$	−6.85909			−0.645249			−0.322625	+	0.946527i	$0.604565\pi$
$114$		0			0
$115$		−	3.44642i		−	0.321381i
$116$		0			0
$117$	−28.2484			−2.61156
$118$		0			0
$119$	−1.53980	+	7.59738i	−0.141153	+	0.696451i
$120$		0			0
$121$	0.194920			0.0177200
$122$		0			0
$123$	17.8941			1.61346
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	5.87400i		−	0.521233i	−0.965442	−	0.260617i	$-0.916074\pi$
$127$		−	5.87400i		−	0.521233i	0.965442	−	0.260617i	$-0.0839259\pi$
$128$		0			0
$129$			27.5960i			2.42970i
$130$		0			0
$131$		−	4.47680i		−	0.391139i	−0.980690	−	0.195570i	$-0.937344\pi$
$131$		−	4.47680i		−	0.391139i	0.980690	−	0.195570i	$-0.0626556\pi$
$132$		0			0
$133$	3.33022	−	16.4313i	0.288767	−	1.42477i
$134$		0			0
$135$			13.5622i			1.16725i
$136$		0			0
$137$	7.90390			0.675276			0.337638	−	0.941276i	$-0.390372\pi$
$137$	7.90390			0.675276			0.337638	+	0.941276i	$0.390372\pi$
$138$		0			0
$139$			9.80123i			0.831329i	0.909518	+	0.415665i	$0.136451\pi$
$139$			9.80123i			0.831329i	−0.909518	+	0.415665i	$0.863549\pi$
$140$		0			0
$141$		−	1.62146i		−	0.136551i
$142$		0			0
$143$	13.0573			1.09191
$144$		0			0
$145$		−	2.68130i		−	0.222670i
$146$		0			0
$147$	8.72096	−	20.6308i	0.719292	−	1.70160i
$148$		0			0
$149$			11.5613i			0.947135i	0.880758	+	0.473567i	$0.157034\pi$
$149$			11.5613i			0.947135i	−0.880758	+	0.473567i	$0.842966\pi$
$150$		0			0
$151$		−	0.509248i		−	0.0414420i	−0.999785	−	0.0207210i	$-0.993404\pi$
$151$		−	0.509248i		−	0.0414420i	0.999785	−	0.0207210i	$-0.00659617\pi$
$152$		0			0
$153$		−	21.2083i		−	1.71459i
$154$		0			0
$155$	2.52241			0.202604
$156$		0			0
$157$	9.98993			0.797283			0.398642	−	0.917107i	$-0.369482\pi$
$157$	9.98993			0.797283			0.398642	+	0.917107i	$0.369482\pi$
$158$		0			0
$159$	36.5207			2.89628
$160$		0			0
$161$	1.81124	−	8.93668i	0.142746	−	0.704309i
$162$		0			0
$163$	−19.7736			−1.54879			−0.774393	−	0.632705i	$-0.781944\pi$
$163$	−19.7736			−1.54879			−0.774393	+	0.632705i	$0.781944\pi$
$164$		0			0
$165$		−	10.7060i		−	0.833464i
$166$		0			0
$167$	20.6423			1.59735			0.798673	−	0.601765i	$-0.205536\pi$
$167$	20.6423			1.59735			0.798673	+	0.601765i	$0.205536\pi$
$168$		0			0
$169$	2.22962			0.171509
$170$		0			0
$171$			45.8684i			3.50764i
$172$		0			0
$173$	12.1166			0.921205			0.460602	−	0.887607i	$-0.347633\pi$
$173$	12.1166			0.921205			0.460602	+	0.887607i	$0.347633\pi$
$174$		0			0
$175$	2.59303	+	0.525543i	0.196015	+	0.0397273i
$176$		0			0
$177$	2.56709			0.192954
$178$		0			0
$179$	14.6327			1.09370			0.546851	−	0.837230i	$-0.315827\pi$
$179$	14.6327			1.09370			0.546851	+	0.837230i	$0.315827\pi$
$180$		0			0
$181$	23.1384			1.71986			0.859932	−	0.510408i	$-0.170506\pi$
$181$	23.1384			1.71986			0.859932	+	0.510408i	$0.170506\pi$
$182$		0			0
$183$			25.5097i			1.88573i
$184$		0			0
$185$		−	4.70905i		−	0.346217i
$186$		0			0
$187$			9.80318i			0.716879i
$188$		0			0
$189$	−7.12754	+	35.1673i	−0.518452	+	2.55804i
$190$		0			0
$191$			2.41554i			0.174782i	0.996174	+	0.0873911i	$0.0278530\pi$
$191$			2.41554i			0.174782i	−0.996174	+	0.0873911i	$0.972147\pi$
$192$		0			0
$193$	−7.38549			−0.531619			−0.265810	−	0.964026i	$-0.585639\pi$
$193$	−7.38549			−0.531619			−0.265810	+	0.964026i	$0.585639\pi$
$194$		0			0
$195$		−	12.4871i		−	0.894222i
$196$		0			0
$197$			20.9305i			1.49124i	0.666372	+	0.745619i	$0.267846\pi$
$197$			20.9305i			1.49124i	−0.666372	+	0.745619i	$0.732154\pi$
$198$		0			0
$199$	4.89644			0.347099			0.173550	−	0.984825i	$-0.444476\pi$
$199$	4.89644			0.347099			0.173550	+	0.984825i	$0.444476\pi$
$200$		0			0
$201$			20.3986i			1.43881i
$202$		0			0
$203$	1.40914	−	6.95269i	0.0989021	−	0.487983i
$204$		0			0
$205$			5.59232i			0.390585i
$206$		0			0
$207$			24.9470i			1.73393i
$208$		0			0
$209$		−	21.2019i		−	1.46657i
$210$		0			0
$211$	−2.25447			−0.155204			−0.0776020	−	0.996984i	$-0.524726\pi$
$211$	−2.25447			−0.155204			−0.0776020	+	0.996984i	$0.524726\pi$
$212$		0			0
$213$	−3.58058			−0.245337
$214$		0			0
$215$	−8.62439			−0.588179
$216$		0			0
$217$	6.54067	+	1.32563i	0.444010	+	0.0899898i
$218$		0			0
$219$	18.9304			1.27920
$220$		0			0
$221$			11.4341i			0.769139i
$222$		0			0
$223$	−24.4820			−1.63944			−0.819718	−	0.572768i	$-0.805870\pi$
$223$	−24.4820			−1.63944			−0.819718	+	0.572768i	$0.805870\pi$
$224$		0			0
$225$	−7.23851			−0.482567
$226$		0			0
$227$			5.16178i			0.342599i	0.985219	+	0.171300i	$0.0547966\pi$
$227$			5.16178i			0.342599i	−0.985219	+	0.171300i	$0.945203\pi$
$228$		0			0
$229$	9.48995			0.627114			0.313557	−	0.949569i	$-0.398479\pi$
$229$	9.48995			0.627114			0.313557	+	0.949569i	$0.398479\pi$
$230$		0			0
$231$	5.62648	−	27.7611i	0.370195	−	1.82655i
$232$		0			0
$233$	−2.55217			−0.167198			−0.0835991	−	0.996499i	$-0.526642\pi$
$233$	−2.55217			−0.167198			−0.0835991	+	0.996499i	$0.526642\pi$
$234$		0			0
$235$	0.506742			0.0330562
$236$		0			0
$237$	33.7294			2.19096
$238$		0			0
$239$		−	8.15573i		−	0.527550i	−0.964584	−	0.263775i	$-0.915032\pi$
$239$		−	8.15573i		−	0.527550i	0.964584	−	0.263775i	$-0.0849677\pi$
$240$		0			0
$241$		−	2.66753i		−	0.171831i	−0.996302	−	0.0859155i	$-0.972619\pi$
$241$		−	2.66753i		−	0.171831i	0.996302	−	0.0859155i	$-0.0273815\pi$
$242$		0			0
$243$		−	28.6857i		−	1.84019i
$244$		0			0
$245$	6.44761	+	2.72550i	0.411923	+	0.174126i
$246$		0			0
$247$		−	24.7291i		−	1.57348i
$248$		0			0
$249$	−13.7353			−0.870442
$250$		0			0
$251$		−	8.52244i		−	0.537932i	−0.963150	−	0.268966i	$-0.913318\pi$
$251$		−	8.52244i		−	0.537932i	0.963150	−	0.268966i	$-0.0866819\pi$
$252$		0			0
$253$		−	11.5313i		−	0.724968i
$254$		0			0
$255$	9.37508			0.587090
$256$		0			0
$257$			14.1473i			0.882487i	0.897387	+	0.441244i	$0.145462\pi$
$257$			14.1473i			0.882487i	−0.897387	+	0.441244i	$0.854538\pi$
$258$		0			0
$259$	2.47481	−	12.2107i	0.153777	−	0.758737i
$260$		0			0
$261$			19.4086i			1.20136i
$262$		0			0
$263$			31.1418i			1.92029i	0.279510	+	0.960143i	$0.409828\pi$
$263$			31.1418i			1.92029i	−0.279510	+	0.960143i	$0.590172\pi$
$264$		0			0
$265$			11.4136i			0.701129i
$266$		0			0
$267$	6.42262			0.393058
$268$		0			0
$269$	−1.39567			−0.0850956			−0.0425478	−	0.999094i	$-0.513547\pi$
$269$	−1.39567			−0.0850956			−0.0425478	+	0.999094i	$0.513547\pi$
$270$		0			0
$271$	22.4663			1.36473			0.682366	−	0.731011i	$-0.260951\pi$
$271$	22.4663			1.36473			0.682366	+	0.731011i	$0.260951\pi$
$272$		0			0
$273$	6.56253	−	32.3795i	0.397182	−	1.95970i
$274$		0			0
$275$	3.34588			0.201764
$276$		0			0
$277$		−	12.0828i		−	0.725985i	−0.931792	−	0.362993i	$-0.881755\pi$
$277$		−	12.0828i		−	0.725985i	0.931792	−	0.362993i	$-0.118245\pi$
$278$		0			0
$279$	−18.2584			−1.09310
$280$		0			0
$281$	−5.84893			−0.348918			−0.174459	−	0.984664i	$-0.555818\pi$
$281$	−5.84893			−0.348918			−0.174459	+	0.984664i	$0.555818\pi$
$282$		0			0
$283$		−	10.1804i		−	0.605164i	−0.953123	−	0.302582i	$-0.902151\pi$
$283$		−	10.1804i		−	0.605164i	0.953123	−	0.302582i	$-0.0978486\pi$
$284$		0			0
$285$	−20.2760			−1.20105
$286$		0			0
$287$	−2.93900	+	14.5011i	−0.173484	+	0.855970i
$288$		0			0
$289$	8.41554			0.495032
$290$		0			0
$291$	−19.4195			−1.13839
$292$		0			0
$293$	−30.6832			−1.79253			−0.896266	−	0.443518i	$-0.853730\pi$
$293$	−30.6832			−1.79253			−0.896266	+	0.443518i	$0.853730\pi$
$294$		0			0
$295$			0.802275i			0.0467102i
$296$		0			0
$297$			45.3776i			2.63308i
$298$		0			0
$299$		−	13.4497i		−	0.777817i
$300$		0			0
$301$	−22.3633	−	4.53249i	−1.28900	−	0.261248i
$302$		0			0
$303$		−	35.9412i		−	2.06477i
$304$		0			0
$305$	−7.97236			−0.456496
$306$		0			0
$307$			25.8632i			1.47609i	0.674750	+	0.738046i	$0.264251\pi$
$307$			25.8632i			1.47609i	−0.674750	+	0.738046i	$0.735749\pi$
$308$		0			0
$309$			1.29139i			0.0734644i
$310$		0			0
$311$	−10.8246			−0.613806			−0.306903	−	0.951741i	$-0.599293\pi$
$311$	−10.8246			−0.613806			−0.306903	+	0.951741i	$0.599293\pi$
$312$		0			0
$313$		−	27.8999i		−	1.57699i	−0.615038	−	0.788497i	$-0.710859\pi$
$313$		−	27.8999i		−	1.57699i	0.615038	−	0.788497i	$-0.289141\pi$
$314$		0			0
$315$	−18.7697	−	3.80415i	−1.05755	−	0.214339i
$316$		0			0
$317$			7.49682i			0.421064i	0.977587	+	0.210532i	$0.0675195\pi$
$317$			7.49682i			0.421064i	−0.977587	+	0.210532i	$0.932480\pi$
$318$		0			0
$319$		−	8.97131i		−	0.502297i
$320$		0			0
$321$			57.2983i			3.19808i
$322$		0			0
$323$	18.5661			1.03305
$324$		0			0
$325$	3.90251			0.216473
$326$		0			0
$327$	−46.0965			−2.54914
$328$		0			0
$329$	1.31400	+	0.266315i	0.0724430	+	0.0146824i
$330$		0			0
$331$	−12.3429			−0.678429			−0.339214	−	0.940709i	$-0.610161\pi$
$331$	−12.3429			−0.678429			−0.339214	+	0.940709i	$0.610161\pi$
$332$		0			0
$333$			34.0865i			1.86793i
$334$		0			0
$335$	−6.37503			−0.348305
$336$		0			0
$337$	30.1764			1.64382			0.821908	−	0.569620i	$-0.192910\pi$
$337$	30.1764			1.64382			0.821908	+	0.569620i	$0.192910\pi$
$338$		0			0
$339$			21.9475i			1.19202i
$340$		0			0
$341$	8.43967			0.457034
$342$		0			0
$343$	15.2865	+	10.4558i	0.825392	+	0.564560i
$344$		0			0
$345$	−11.0278			−0.593714
$346$		0			0
$347$	21.3398			1.14558			0.572789	−	0.819703i	$-0.305861\pi$
$347$	21.3398			1.14558			0.572789	+	0.819703i	$0.305861\pi$
$348$		0			0
$349$	−16.3885			−0.877255			−0.438627	−	0.898669i	$-0.644535\pi$
$349$	−16.3885			−0.877255			−0.438627	+	0.898669i	$0.644535\pi$
$350$		0			0
$351$			52.9268i			2.82502i
$352$		0			0
$353$		−	3.84670i		−	0.204739i	−0.994746	−	0.102370i	$-0.967358\pi$
$353$		−	3.84670i		−	0.204739i	0.994746	−	0.102370i	$-0.0326425\pi$
$354$		0			0
$355$		−	1.11901i		−	0.0593910i
$356$		0			0
$357$	24.3099	+	4.92700i	1.28661	+	0.260765i
$358$		0			0
$359$			27.6613i			1.45991i	0.683495	+	0.729955i	$0.260459\pi$
$359$			27.6613i			1.45991i	−0.683495	+	0.729955i	$0.739541\pi$
$360$		0			0
$361$	−21.1540			−1.11337
$362$		0			0
$363$		−	0.623698i		−	0.0327357i
$364$		0			0
$365$			5.91619i			0.309668i
$366$		0			0
$367$	−2.94732			−0.153849			−0.0769244	−	0.997037i	$-0.524510\pi$
$367$	−2.94732			−0.153849			−0.0769244	+	0.997037i	$0.524510\pi$
$368$		0			0
$369$		−	40.4801i		−	2.10731i
$370$		0			0
$371$	−5.99831	+	29.5957i	−0.311417	+	1.53653i
$372$		0			0
$373$			12.6247i			0.653683i	0.945079	+	0.326841i	$0.105984\pi$
$373$			12.6247i			0.653683i	−0.945079	+	0.326841i	$0.894016\pi$
$374$		0			0
$375$		−	3.19977i		−	0.165235i
$376$		0			0
$377$		−	10.4638i		−	0.538914i
$378$		0			0
$379$	20.9882			1.07809			0.539045	−	0.842277i	$-0.318785\pi$
$379$	20.9882			1.07809			0.539045	+	0.842277i	$0.318785\pi$
$380$		0			0
$381$	−18.7954			−0.962919
$382$		0			0
$383$	11.4237			0.583724			0.291862	−	0.956460i	$-0.405725\pi$
$383$	11.4237			0.583724			0.291862	+	0.956460i	$0.405725\pi$
$384$		0			0
$385$	8.67597	+	1.75840i	0.442168	+	0.0896166i
$386$		0			0
$387$	62.4277			3.17338
$388$		0			0
$389$		−	22.7966i		−	1.15583i	−0.816096	−	0.577917i	$-0.803866\pi$
$389$		−	22.7966i		−	1.15583i	0.816096	−	0.577917i	$-0.196134\pi$
$390$		0			0
$391$	10.0978			0.510666
$392$		0			0
$393$	−14.3247			−0.722586
$394$		0			0
$395$			10.5412i			0.530385i
$396$		0			0
$397$	−29.8779			−1.49953			−0.749763	−	0.661706i	$-0.769833\pi$
$397$	−29.8779			−1.49953			−0.749763	+	0.661706i	$0.769833\pi$
$398$		0			0
$399$	−52.5763	−	10.6559i	−2.63211	−	0.533463i
$400$		0			0
$401$	−16.8397			−0.840933			−0.420467	−	0.907308i	$-0.638134\pi$
$401$	−16.8397			−0.840933			−0.420467	+	0.907308i	$0.638134\pi$
$402$		0			0
$403$	9.84372			0.490351
$404$		0			0
$405$	21.6805			1.07731
$406$		0			0
$407$		−	15.7559i		−	0.780993i
$408$		0			0
$409$		−	26.9755i		−	1.33385i	−0.745123	−	0.666927i	$-0.767609\pi$
$409$		−	26.9755i		−	1.33385i	0.745123	−	0.666927i	$-0.232391\pi$
$410$		0			0
$411$		−	25.2906i		−	1.24750i
$412$		0			0
$413$	−0.421630	+	2.08032i	−0.0207470	+	0.102366i
$414$		0			0
$415$		−	4.29261i		−	0.210716i
$416$		0			0
$417$	31.3616			1.53579
$418$		0			0
$419$			15.0474i			0.735114i	0.930001	+	0.367557i	$0.119806\pi$
$419$			15.0474i			0.735114i	−0.930001	+	0.367557i	$0.880194\pi$
$420$		0			0
$421$			27.5257i			1.34152i	0.741674	+	0.670761i	$0.234032\pi$
$421$			27.5257i			1.34152i	−0.741674	+	0.670761i	$0.765968\pi$
$422$		0			0
$423$	−3.66806			−0.178347
$424$		0			0
$425$			2.92992i			0.142122i
$426$		0			0
$427$	−20.6726	−	4.18982i	−1.00042	−	0.202759i
$428$		0			0
$429$		−	41.7805i		−	2.01718i
$430$		0			0
$431$			22.5778i			1.08753i	0.839236	+	0.543767i	$0.183002\pi$
$431$			22.5778i			1.08753i	−0.839236	+	0.543767i	$0.816998\pi$
$432$		0			0
$433$			23.4737i			1.12808i	0.825749	+	0.564038i	$0.190753\pi$
$433$			23.4737i			1.12808i	−0.825749	+	0.564038i	$0.809247\pi$
$434$		0			0
$435$	−8.57954			−0.411357
$436$		0			0
$437$	−21.8390			−1.04470
$438$		0			0
$439$	12.3485			0.589363			0.294681	−	0.955596i	$-0.404786\pi$
$439$	12.3485			0.589363			0.294681	+	0.955596i	$0.404786\pi$
$440$		0			0
$441$	−46.6711	−	19.7285i	−2.22243	−	0.939454i
$442$		0			0
$443$	−24.9234			−1.18415			−0.592074	−	0.805883i	$-0.701691\pi$
$443$	−24.9234			−1.18415			−0.592074	+	0.805883i	$0.701691\pi$
$444$		0			0
$445$			2.00722i			0.0951512i
$446$		0			0
$447$	36.9933			1.74972
$448$		0			0
$449$	33.7360			1.59210			0.796051	−	0.605230i	$-0.206919\pi$
$449$	33.7360			1.59210			0.796051	+	0.605230i	$0.206919\pi$
$450$		0			0
$451$			18.7112i			0.881078i
$452$		0			0
$453$	−1.62947			−0.0765594
$454$		0			0
$455$	10.1193	+	2.05094i	0.474402	+	0.0961495i
$456$		0			0
$457$	−23.0696			−1.07915			−0.539576	−	0.841937i	$-0.681416\pi$
$457$	−23.0696			−1.07915			−0.539576	+	0.841937i	$0.681416\pi$
$458$		0			0
$459$	−39.7363			−1.85473
$460$		0			0
$461$	4.90911			0.228640			0.114320	−	0.993444i	$-0.463531\pi$
$461$	4.90911			0.228640			0.114320	+	0.993444i	$0.463531\pi$
$462$		0			0
$463$		−	30.9829i		−	1.43990i	−0.694027	−	0.719949i	$-0.744165\pi$
$463$		−	30.9829i		−	1.43990i	0.694027	−	0.719949i	$-0.255835\pi$
$464$		0			0
$465$		−	8.07111i		−	0.374289i
$466$		0			0
$467$		−	20.4881i		−	0.948078i	−0.880504	−	0.474039i	$-0.842796\pi$
$467$		−	20.4881i		−	0.948078i	0.880504	−	0.474039i	$-0.157204\pi$
$468$		0			0
$469$	−16.5306	−	3.35035i	−0.763314	−	0.154705i
$470$		0			0
$471$		−	31.9655i		−	1.47289i
$472$		0			0
$473$	−28.8562			−1.32681
$474$		0			0
$475$		−	6.33672i		−	0.290749i
$476$		0			0
$477$		−	82.6171i		−	3.78278i
$478$		0			0
$479$	35.1704			1.60698			0.803489	−	0.595319i	$-0.202974\pi$
$479$	35.1704			1.60698			0.803489	+	0.595319i	$0.202974\pi$
$480$		0			0
$481$		−	18.3772i		−	0.837926i
$482$		0			0
$483$	−28.5953	−	5.79556i	−1.30113	−	0.263707i
$484$		0			0
$485$		−	6.06903i		−	0.275580i
$486$		0			0
$487$		−	7.86977i		−	0.356614i	−0.983975	−	0.178307i	$-0.942938\pi$
$487$		−	7.86977i		−	0.356614i	0.983975	−	0.178307i	$-0.0570619\pi$
$488$		0			0
$489$			63.2708i			2.86121i
$490$		0			0
$491$	−2.88992			−0.130420			−0.0652102	−	0.997872i	$-0.520772\pi$
$491$	−2.88992			−0.130420			−0.0652102	+	0.997872i	$0.520772\pi$
$492$		0			0
$493$	7.85601			0.353817
$494$		0			0
$495$	−24.2192			−1.08857
$496$		0			0
$497$	0.588089	−	2.90163i	0.0263794	−	0.130156i
$498$		0			0
$499$	−16.5134			−0.739240			−0.369620	−	0.929183i	$-0.620512\pi$
$499$	−16.5134			−0.739240			−0.369620	+	0.929183i	$0.620512\pi$
$500$		0			0
$501$		−	66.0504i		−	2.95092i
$502$		0			0
$503$	7.23320			0.322513			0.161256	−	0.986913i	$-0.448445\pi$
$503$	7.23320			0.322513			0.161256	+	0.986913i	$0.448445\pi$
$504$		0			0
$505$	11.2324			0.499837
$506$		0			0
$507$		−	7.13426i		−	0.316843i
$508$		0			0
$509$	26.3986			1.17010			0.585048	−	0.810999i	$-0.301076\pi$
$509$	26.3986			1.17010			0.585048	+	0.810999i	$0.301076\pi$
$510$		0			0
$511$	−3.10921	+	15.3409i	−0.137543	+	0.678640i
$512$		0			0
$513$	85.9401			3.79435
$514$		0			0
$515$	−0.403587			−0.0177842
$516$		0			0
$517$	1.69550			0.0745679
$518$		0			0
$519$		−	38.7702i		−	1.70182i
$520$		0			0
$521$			29.5635i			1.29520i	0.761981	+	0.647599i	$0.224227\pi$
$521$			29.5635i			1.29520i	−0.761981	+	0.647599i	$0.775773\pi$
$522$		0			0
$523$		−	21.6311i		−	0.945862i	−0.881100	−	0.472931i	$-0.843196\pi$
$523$		−	21.6311i		−	0.945862i	0.881100	−	0.472931i	$-0.156804\pi$
$524$		0			0
$525$	1.68161	−	8.29709i	0.0733917	−	0.362115i
$526$		0			0
$527$			7.39046i			0.321933i
$528$		0			0
$529$	11.1222			0.483572
$530$		0			0
$531$		−	5.80727i		−	0.252014i
$532$		0			0
$533$			21.8241i			0.945307i
$534$		0			0
$535$	−17.9070			−0.774188
$536$		0			0
$537$		−	46.8213i		−	2.02049i
$538$		0			0
$539$	21.5729	+	9.11919i	0.929212	+	0.392791i
$540$		0			0
$541$		−	3.88138i		−	0.166874i	−0.996513	−	0.0834368i	$-0.973410\pi$
$541$		−	3.88138i		−	0.166874i	0.996513	−	0.0834368i	$-0.0265897\pi$
$542$		0			0
$543$		−	74.0375i		−	3.17725i
$544$		0			0
$545$		−	14.4062i		−	0.617094i
$546$		0			0
$547$	4.88320			0.208791			0.104395	−	0.994536i	$-0.466709\pi$
$547$	4.88320			0.208791			0.104395	+	0.994536i	$0.466709\pi$
$548$		0			0
$549$	57.7080			2.46292
$550$		0			0
$551$	−16.9906			−0.723826
$552$		0			0
$553$	−5.53985	+	27.3337i	−0.235579	+	1.16235i
$554$		0			0
$555$	−15.0679			−0.639596
$556$		0			0
$557$			44.6377i			1.89136i	0.325099	+	0.945680i	$0.394602\pi$
$557$			44.6377i			1.89136i	−0.325099	+	0.945680i	$0.605398\pi$
$558$		0			0
$559$	−33.6568			−1.42353
$560$		0			0
$561$	31.3679			1.32435
$562$		0			0
$563$			28.5884i			1.20486i	0.798172	+	0.602429i	$0.205800\pi$
$563$			28.5884i			1.20486i	−0.798172	+	0.602429i	$0.794200\pi$
$564$		0			0
$565$	−6.85909			−0.288564
$566$		0			0
$567$	56.2181	+	11.3940i	2.36094	+	0.478504i
$568$		0			0
$569$	−5.12489			−0.214846			−0.107423	−	0.994213i	$-0.534260\pi$
$569$	−5.12489			−0.214846			−0.107423	+	0.994213i	$0.534260\pi$
$570$		0			0
$571$	−36.7465			−1.53779			−0.768897	−	0.639373i	$-0.779194\pi$
$571$	−36.7465			−1.53779			−0.768897	+	0.639373i	$0.779194\pi$
$572$		0			0
$573$	7.72916			0.322890
$574$		0			0
$575$		−	3.44642i		−	0.143726i
$576$		0			0
$577$			2.55301i			0.106283i	0.998587	+	0.0531417i	$0.0169235\pi$
$577$			2.55301i			0.106283i	−0.998587	+	0.0531417i	$0.983076\pi$
$578$		0			0
$579$			23.6319i			0.982106i
$580$		0			0
$581$	2.25595	−	11.1309i	0.0935926	−	0.461786i
$582$		0			0
$583$			38.1884i			1.58160i
$584$		0			0
$585$	−28.2484			−1.16793
$586$		0			0
$587$		−	33.3508i		−	1.37653i	−0.725457	−	0.688267i	$-0.758372\pi$
$587$		−	33.3508i		−	1.37653i	0.725457	−	0.688267i	$-0.241628\pi$
$588$		0			0
$589$		−	15.9838i		−	0.658600i
$590$		0			0
$591$	66.9728			2.75489
$592$		0			0
$593$			10.9350i			0.449048i	0.974469	+	0.224524i	$0.0720827\pi$
$593$			10.9350i			0.449048i	−0.974469	+	0.224524i	$0.927917\pi$
$594$		0			0
$595$	−1.53980	+	7.59738i	−0.0631257	+	0.311462i
$596$		0			0
$597$		−	15.6675i		−	0.641227i
$598$		0			0
$599$			25.8459i			1.05603i	0.849234	+	0.528017i	$0.177064\pi$
$599$			25.8459i			1.05603i	−0.849234	+	0.528017i	$0.822936\pi$
$600$		0			0
$601$		−	6.70199i		−	0.273380i	−0.990614	−	0.136690i	$-0.956354\pi$
$601$		−	6.70199i		−	0.273380i	0.990614	−	0.136690i	$-0.0436464\pi$
$602$		0			0
$603$	46.1457			1.87920
$604$		0			0
$605$	0.194920			0.00792462
$606$		0			0
$607$	−37.4686			−1.52080			−0.760402	−	0.649453i	$-0.774998\pi$
$607$	−37.4686			−1.52080			−0.760402	+	0.649453i	$0.774998\pi$
$608$		0			0
$609$	−22.2470	−	4.50891i	−0.901494	−	0.182710i
$610$		0			0
$611$	1.97757			0.0800038
$612$		0			0
$613$		−	20.0776i		−	0.810928i	−0.914111	−	0.405464i	$-0.867110\pi$
$613$		−	20.0776i		−	0.810928i	0.914111	−	0.405464i	$-0.132890\pi$
$614$		0			0
$615$	17.8941			0.721561
$616$		0			0
$617$	27.8737			1.12215			0.561077	−	0.827764i	$-0.310387\pi$
$617$	27.8737			1.12215			0.561077	+	0.827764i	$0.310387\pi$
$618$		0			0
$619$		−	25.0328i		−	1.00615i	−0.864242	−	0.503076i	$-0.832202\pi$
$619$		−	25.0328i		−	1.00615i	0.864242	−	0.503076i	$-0.167798\pi$
$620$		0			0
$621$	46.7412			1.87566
$622$		0			0
$623$	−1.05488	+	5.20477i	−0.0422628	+	0.208525i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−67.8412			−2.70931
$628$		0			0
$629$	13.7972			0.550129
$630$		0			0
$631$		−	17.2836i		−	0.688048i	−0.938961	−	0.344024i	$-0.888210\pi$
$631$		−	17.2836i		−	0.688048i	0.938961	−	0.344024i	$-0.111790\pi$
$632$		0			0
$633$			7.21378i			0.286722i
$634$		0			0
$635$		−	5.87400i		−	0.233103i
$636$		0			0
$637$	25.1619	+	10.6363i	0.996950	+	0.421425i
$638$		0			0
$639$			8.09997i			0.320430i
$640$		0			0
$641$	−5.40646			−0.213542			−0.106771	−	0.994284i	$-0.534051\pi$
$641$	−5.40646			−0.213542			−0.106771	+	0.994284i	$0.534051\pi$
$642$		0			0
$643$			8.51562i			0.335823i	0.985802	+	0.167912i	$0.0537024\pi$
$643$			8.51562i			0.335823i	−0.985802	+	0.167912i	$0.946298\pi$
$644$		0			0
$645$			27.5960i			1.08659i
$646$		0			0
$647$	−9.29106			−0.365269			−0.182635	−	0.983181i	$-0.558463\pi$
$647$	−9.29106			−0.365269			−0.182635	+	0.983181i	$0.558463\pi$
$648$		0			0
$649$			2.68432i			0.105369i
$650$		0			0
$651$	4.24171	−	20.9286i	0.166246	−	0.820257i
$652$		0			0
$653$		−	13.4994i		−	0.528272i	−0.964486	−	0.264136i	$-0.914913\pi$
$653$		−	13.4994i		−	0.528272i	0.964486	−	0.264136i	$-0.0850867\pi$
$654$		0			0
$655$		−	4.47680i		−	0.174923i
$656$		0			0
$657$		−	42.8244i		−	1.67074i
$658$		0			0
$659$	32.9215			1.28244			0.641219	−	0.767358i	$-0.278429\pi$
$659$	32.9215			1.28244			0.641219	+	0.767358i	$0.278429\pi$
$660$		0			0
$661$	−28.6462			−1.11421			−0.557104	−	0.830443i	$-0.688088\pi$
$661$	−28.6462			−1.11421			−0.557104	+	0.830443i	$0.688088\pi$
$662$		0			0
$663$	36.5864			1.42090
$664$		0			0
$665$	3.33022	−	16.4313i	0.129140	−	0.637179i
$666$		0			0
$667$	−9.24090			−0.357809
$668$		0			0
$669$			78.3367i			3.02867i
$670$		0			0
$671$	−26.6746			−1.02976
$672$		0			0
$673$	4.30988			0.166134			0.0830669	−	0.996544i	$-0.473528\pi$
$673$	4.30988			0.166134			0.0830669	+	0.996544i	$0.473528\pi$
$674$		0			0
$675$			13.5622i			0.522011i
$676$		0			0
$677$	−23.3619			−0.897870			−0.448935	−	0.893564i	$-0.648197\pi$
$677$	−23.3619			−0.897870			−0.448935	+	0.893564i	$0.648197\pi$
$678$		0			0
$679$	3.18954	−	15.7372i	0.122403	−	0.603937i
$680$		0			0
$681$	16.5165			0.632913
$682$		0			0
$683$	−40.0020			−1.53063			−0.765317	−	0.643653i	$-0.777418\pi$
$683$	−40.0020			−1.53063			−0.765317	+	0.643653i	$0.777418\pi$
$684$		0			0
$685$	7.90390			0.301993
$686$		0			0
$687$		−	30.3656i		−	1.15852i
$688$		0			0
$689$			44.5416i			1.69690i
$690$		0			0
$691$			38.8983i			1.47976i	0.672738	+	0.739881i	$0.265118\pi$
$691$			38.8983i			1.47976i	−0.672738	+	0.739881i	$0.734882\pi$
$692$		0			0
$693$	−62.8011	−	12.7282i	−2.38562	−	0.483505i
$694$		0			0
$695$			9.80123i			0.371782i
$696$		0			0
$697$	−16.3851			−0.620629
$698$		0			0
$699$			8.16634i			0.308880i
$700$		0			0
$701$		−	3.06613i		−	0.115806i	−0.998322	−	0.0579031i	$-0.981559\pi$
$701$		−	3.06613i		−	0.115806i	0.998322	−	0.0579031i	$-0.0184415\pi$
$702$		0			0
$703$	−29.8400			−1.12544
$704$		0			0
$705$		−	1.62146i		−	0.0610676i
$706$		0			0
$707$	29.1260	+	5.90313i	1.09540	+	0.222010i
$708$		0			0
$709$		−	38.5857i		−	1.44912i	−0.689214	−	0.724558i	$-0.742044\pi$
$709$		−	38.5857i		−	1.44912i	0.689214	−	0.724558i	$-0.257956\pi$
$710$		0			0
$711$		−	76.3026i		−	2.86157i
$712$		0			0
$713$		−	8.69328i		−	0.325566i
$714$		0			0
$715$	13.0573			0.488317
$716$		0			0
$717$	−26.0964			−0.974589
$718$		0			0
$719$	−31.4806			−1.17403			−0.587014	−	0.809577i	$-0.699697\pi$
$719$	−31.4806			−1.17403			−0.587014	+	0.809577i	$0.699697\pi$
$720$		0			0
$721$	−1.04651	−	0.212103i	−0.0389742	−	0.00789911i
$722$		0			0
$723$	−8.53548			−0.317438
$724$		0			0
$725$		−	2.68130i		−	0.0995810i
$726$		0			0
$727$	1.97285			0.0731689			0.0365844	−	0.999331i	$-0.488352\pi$
$727$	1.97285			0.0731689			0.0365844	+	0.999331i	$0.488352\pi$
$728$		0			0
$729$	−26.7463			−0.990603
$730$		0			0
$731$		−	25.2688i		−	0.934601i
$732$		0			0
$733$	1.04466			0.0385855			0.0192928	−	0.999814i	$-0.493859\pi$
$733$	1.04466			0.0385855			0.0192928	+	0.999814i	$0.493859\pi$
$734$		0			0
$735$	8.72096	−	20.6308i	0.321677	−	0.760980i
$736$		0			0
$737$	−21.3301			−0.785704
$738$		0			0
$739$	33.3529			1.22691			0.613454	−	0.789730i	$-0.289780\pi$
$739$	33.3529			1.22691			0.613454	+	0.789730i	$0.289780\pi$
$740$		0			0
$741$	−79.1275			−2.90682
$742$		0			0
$743$			2.33616i			0.0857054i	0.999081	+	0.0428527i	$0.0136446\pi$
$743$			2.33616i			0.0857054i	−0.999081	+	0.0428527i	$0.986355\pi$
$744$		0			0
$745$			11.5613i			0.423572i
$746$		0			0
$747$			31.0721i			1.13687i
$748$		0			0
$749$	−46.4334	−	9.41091i	−1.69664	−	0.343867i
$750$		0			0
$751$			22.6980i			0.828261i	0.910218	+	0.414130i	$0.135914\pi$
$751$			22.6980i			0.828261i	−0.910218	+	0.414130i	$0.864086\pi$
$752$		0			0
$753$	−27.2698			−0.993768
$754$		0			0
$755$		−	0.509248i		−	0.0185334i
$756$		0			0
$757$		−	0.859830i		−	0.0312510i	−0.999878	−	0.0156255i	$-0.995026\pi$
$757$		−	0.859830i		−	0.0312510i	0.999878	−	0.0156255i	$-0.00497396\pi$
$758$		0			0
$759$	−36.8976			−1.33930
$760$		0			0
$761$		−	9.59745i		−	0.347907i	−0.984754	−	0.173954i	$-0.944346\pi$
$761$		−	9.59745i		−	0.347907i	0.984754	−	0.173954i	$-0.0556543\pi$
$762$		0			0
$763$	7.57108	−	37.3557i	0.274091	−	1.35237i
$764$		0			0
$765$		−	21.2083i		−	0.766787i
$766$		0			0
$767$			3.13089i			0.113050i
$768$		0			0
$769$		−	46.0671i		−	1.66122i	−0.556854	−	0.830610i	$-0.687992\pi$
$769$		−	46.0671i		−	1.66122i	0.556854	−	0.830610i	$-0.312008\pi$
$770$		0			0
$771$	45.2682			1.63029
$772$		0			0
$773$	−7.98484			−0.287195			−0.143597	−	0.989636i	$-0.545867\pi$
$773$	−7.98484			−0.287195			−0.143597	+	0.989636i	$0.545867\pi$
$774$		0			0
$775$	2.52241			0.0906075
$776$		0			0
$777$	−39.0715	−	7.91882i	−1.40168	−	0.284086i
$778$		0			0
$779$	35.4370			1.26966
$780$		0			0
$781$		−	3.74408i		−	0.133974i
$782$		0			0
$783$	36.3644			1.29956
$784$		0			0
$785$	9.98993			0.356556
$786$		0			0
$787$			56.0482i			1.99790i	0.0457828	+	0.998951i	$0.485422\pi$
$787$			56.0482i			1.99790i	−0.0457828	+	0.998951i	$0.514578\pi$
$788$		0			0
$789$	99.6465			3.54751
$790$		0			0
$791$	−17.7858	−	3.60475i	−0.632391	−	0.128170i
$792$		0			0
$793$	−31.1122			−1.10483
$794$		0			0
$795$	36.5207			1.29526
$796$		0			0
$797$	−23.7352			−0.840744			−0.420372	−	0.907352i	$-0.638100\pi$
$797$	−23.7352			−0.840744			−0.420372	+	0.907352i	$0.638100\pi$
$798$		0			0
$799$			1.48472i			0.0525255i
$800$		0			0
$801$		−	14.5292i		−	0.513366i
$802$		0			0
$803$			19.7949i			0.698546i
$804$		0			0
$805$	1.81124	−	8.93668i	0.0638380	−	0.314977i
$806$		0			0
$807$			4.46582i			0.157204i
$808$		0			0
$809$	22.4179			0.788172			0.394086	−	0.919074i	$-0.371061\pi$
$809$	22.4179			0.788172			0.394086	+	0.919074i	$0.371061\pi$
$810$		0			0
$811$		−	17.3731i		−	0.610054i	−0.952344	−	0.305027i	$-0.901335\pi$
$811$		−	17.3731i		−	0.610054i	0.952344	−	0.305027i	$-0.0986655\pi$
$812$		0			0
$813$		−	71.8870i		−	2.52119i
$814$		0			0
$815$	−19.7736			−0.692638
$816$		0			0
$817$			54.6503i			1.91197i
$818$		0			0
$819$	−73.2489	−	14.8457i	−2.55952	−	0.518752i
$820$		0			0
$821$			32.6279i			1.13872i	0.822088	+	0.569361i	$0.192809\pi$
$821$			32.6279i			1.13872i	−0.822088	+	0.569361i	$0.807191\pi$
$822$		0			0
$823$		−	6.06037i		−	0.211251i	−0.994406	−	0.105626i	$-0.966315\pi$
$823$		−	6.06037i		−	0.211251i	0.994406	−	0.105626i	$-0.0336845\pi$
$824$		0			0
$825$		−	10.7060i		−	0.372736i
$826$		0			0
$827$	−21.7214			−0.755328			−0.377664	−	0.925943i	$-0.623273\pi$
$827$	−21.7214			−0.755328			−0.377664	+	0.925943i	$0.623273\pi$
$828$		0			0
$829$	−4.44507			−0.154384			−0.0771919	−	0.997016i	$-0.524595\pi$
$829$	−4.44507			−0.154384			−0.0771919	+	0.997016i	$0.524595\pi$
$830$		0			0
$831$	−38.6622			−1.34118
$832$		0			0
$833$	−7.98550	+	18.8910i	−0.276681	+	0.654535i
$834$		0			0
$835$	20.6423			0.714355
$836$		0			0
$837$			34.2095i			1.18245i
$838$		0			0
$839$	−9.23265			−0.318746			−0.159373	−	0.987218i	$-0.550947\pi$
$839$	−9.23265			−0.318746			−0.159373	+	0.987218i	$0.550947\pi$
$840$		0			0
$841$	21.8106			0.752091
$842$		0			0
$843$			18.7152i			0.644586i
$844$		0			0
$845$	2.22962			0.0767012
$846$		0			0
$847$	0.505433	+	0.102439i	0.0173669	+	0.00351984i
$848$		0			0
$849$	−32.5750			−1.11797
$850$		0			0
$851$	−16.2294			−0.556337
$852$		0			0
$853$	5.33487			0.182662			0.0913312	−	0.995821i	$-0.470888\pi$
$853$	5.33487			0.182662			0.0913312	+	0.995821i	$0.470888\pi$
$854$		0			0
$855$			45.8684i			1.56867i
$856$		0			0
$857$			3.81449i			0.130301i	0.997875	+	0.0651503i	$0.0207527\pi$
$857$			3.81449i			0.130301i	−0.997875	+	0.0651503i	$0.979247\pi$
$858$		0			0
$859$		−	40.9650i		−	1.39771i	−0.715264	−	0.698854i	$-0.753694\pi$
$859$		−	40.9650i		−	1.39771i	0.715264	−	0.698854i	$-0.246306\pi$
$860$		0			0
$861$	46.4000	+	9.40413i	1.58131	+	0.320492i
$862$		0			0
$863$			20.2817i			0.690399i	0.938529	+	0.345199i	$0.112189\pi$
$863$			20.2817i			0.690399i	−0.938529	+	0.345199i	$0.887811\pi$
$864$		0			0
$865$	12.1166			0.411975
$866$		0			0
$867$		−	26.9278i		−	0.914515i
$868$		0			0
$869$			35.2696i			1.19644i
$870$		0			0
$871$	−24.8786			−0.842980
$872$		0			0
$873$			43.9307i			1.48683i
$874$		0			0
$875$	2.59303	+	0.525543i	0.0876604	+	0.0177666i
$876$		0			0
$877$			42.3958i			1.43160i	0.698304	+	0.715802i	$0.253938\pi$
$877$			42.3958i			1.43160i	−0.698304	+	0.715802i	$0.746062\pi$
$878$		0			0
$879$			98.1791i			3.31150i
$880$		0			0
$881$		−	18.8445i		−	0.634887i	−0.948277	−	0.317443i	$-0.897176\pi$
$881$		−	18.8445i		−	0.634887i	0.948277	−	0.317443i	$-0.102824\pi$
$882$		0			0
$883$	−36.2096			−1.21855			−0.609274	−	0.792959i	$-0.708539\pi$
$883$	−36.2096			−1.21855			−0.609274	+	0.792959i	$0.708539\pi$
$884$		0			0
$885$	2.56709			0.0862918
$886$		0			0
$887$	−21.5740			−0.724385			−0.362193	−	0.932103i	$-0.617972\pi$
$887$	−21.5740			−0.724385			−0.362193	+	0.932103i	$0.617972\pi$
$888$		0			0
$889$	3.08704	−	15.2315i	0.103536	−	0.510847i
$890$		0			0
$891$	72.5403			2.43019
$892$		0			0
$893$		−	3.21108i		−	0.107455i
$894$		0			0
$895$	14.6327			0.489118
$896$		0			0
$897$	−43.0360			−1.43693
$898$		0			0
$899$		−	6.76332i		−	0.225570i
$900$		0			0
$901$	−33.4409			−1.11408
$902$		0			0
$903$	−14.5029	+	71.5574i	−0.482627	+	2.38128i
$904$		0			0
$905$	23.1384			0.769147
$906$		0			0
$907$	35.3694			1.17442			0.587210	−	0.809435i	$-0.300226\pi$
$907$	35.3694			1.17442			0.587210	+	0.809435i	$0.300226\pi$
$908$		0			0
$909$	−81.3061			−2.69675
$910$		0			0
$911$			59.3513i			1.96640i	0.182540	+	0.983199i	$0.441568\pi$
$911$			59.3513i			1.96640i	−0.182540	+	0.983199i	$0.558432\pi$
$912$		0			0
$913$		−	14.3626i		−	0.475331i
$914$		0			0
$915$			25.5097i			0.843324i
$916$		0			0
$917$	2.35275	−	11.6085i	0.0776946	−	0.383345i
$918$		0			0
$919$		−	54.6687i		−	1.80335i	−0.432411	−	0.901677i	$-0.642337\pi$
$919$		−	54.6687i		−	1.80335i	0.432411	−	0.901677i	$-0.357663\pi$
$920$		0			0
$921$	82.7563			2.72691
$922$		0			0
$923$		−	4.36696i		−	0.143740i
$924$		0			0
$925$		−	4.70905i		−	0.154833i
$926$		0			0
$927$	2.92137			0.0959504
$928$		0			0
$929$			37.8759i			1.24267i	0.783546	+	0.621334i	$0.213409\pi$
$929$			37.8759i			1.24267i	−0.783546	+	0.621334i	$0.786591\pi$
$930$		0			0
$931$	17.2707	−	40.8567i	0.566025	−	1.33902i
$932$		0			0
$933$			34.6361i			1.13394i
$934$		0			0
$935$			9.80318i			0.320598i
$936$		0			0
$937$			55.9352i			1.82732i	0.406476	+	0.913661i	$0.366757\pi$
$937$			55.9352i			1.82732i	−0.406476	+	0.913661i	$0.633243\pi$
$938$		0			0
$939$	−89.2731			−2.91332
$940$		0			0
$941$	60.3400			1.96703			0.983514	−	0.180831i	$-0.0578787\pi$
$941$	60.3400			1.96703			0.983514	+	0.180831i	$0.0578787\pi$
$942$		0			0
$943$	19.2735			0.627632
$944$		0			0
$945$	−7.12754	+	35.1673i	−0.231859	+	1.14399i
$946$		0			0
$947$	−22.0376			−0.716126			−0.358063	−	0.933697i	$-0.616563\pi$
$947$	−22.0376			−0.716126			−0.358063	+	0.933697i	$0.616563\pi$
$948$		0			0
$949$			23.0880i			0.749469i
$950$		0			0
$951$	23.9881			0.777867
$952$		0			0
$953$	21.2857			0.689511			0.344756	−	0.938693i	$-0.387962\pi$
$953$	21.2857			0.689511			0.344756	+	0.938693i	$0.387962\pi$
$954$		0			0
$955$			2.41554i			0.0781650i
$956$		0			0
$957$	−28.7061			−0.927937
$958$		0			0
$959$	20.4951	+	4.15384i	0.661820	+	0.134134i
$960$		0			0
$961$	−24.6375			−0.794757
$962$		0			0
$963$	129.620			4.17695
$964$		0			0
$965$	−7.38549			−0.237747
$966$		0			0
$967$		−	29.9897i		−	0.964405i	−0.876060	−	0.482203i	$-0.839837\pi$
$967$		−	29.9897i		−	0.964405i	0.876060	−	0.482203i	$-0.160163\pi$
$968$		0			0
$969$		−	59.4072i		−	1.90843i
$970$		0			0
$971$			9.10304i			0.292130i	0.989275	+	0.146065i	$0.0466609\pi$
$971$			9.10304i			0.292130i	−0.989275	+	0.146065i	$0.953339\pi$
$972$		0			0
$973$	−5.15097	+	25.4149i	−0.165132	+	0.814764i
$974$		0			0
$975$		−	12.4871i		−	0.399908i
$976$		0			0
$977$	−38.3437			−1.22672			−0.613362	−	0.789802i	$-0.710183\pi$
$977$	−38.3437			−1.22672			−0.613362	+	0.789802i	$0.710183\pi$
$978$		0			0
$979$			6.71591i			0.214641i
$980$		0			0
$981$			104.279i			3.32938i
$982$		0			0
$983$	−11.4686			−0.365792			−0.182896	−	0.983132i	$-0.558547\pi$
$983$	−11.4686			−0.365792			−0.182896	+	0.983132i	$0.558547\pi$
$984$		0			0
$985$			20.9305i			0.666902i
$986$		0			0
$987$	0.852145	−	4.20448i	0.0271241	−	0.133830i
$988$		0			0
$989$			29.7233i			0.945147i
$990$		0			0
$991$		−	27.8094i		−	0.883395i	−0.897164	−	0.441697i	$-0.854376\pi$
$991$		−	27.8094i		−	0.883395i	0.897164	−	0.441697i	$-0.145624\pi$
$992$		0			0
$993$			39.4945i			1.25332i
$994$		0			0
$995$	4.89644			0.155228
$996$		0			0
$997$	51.3033			1.62479			0.812395	−	0.583107i	$-0.198163\pi$
$997$	51.3033			1.62479			0.812395	+	0.583107i	$0.198163\pi$
$998$		0			0
$999$	63.8653			2.02061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.b.111.1		16
4.3	odd	2		280.2.h.b.251.8	yes	16
7.6	odd	2		1120.2.h.a.111.16		16
8.3	odd	2		1120.2.h.a.111.1		16
8.5	even	2		280.2.h.a.251.7	✓	16
28.27	even	2		280.2.h.a.251.8	yes	16
56.13	odd	2		280.2.h.b.251.7	yes	16
56.27	even	2	inner	1120.2.h.b.111.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.7	✓	16	8.5	even	2
280.2.h.a.251.8	yes	16	28.27	even	2
280.2.h.b.251.7	yes	16	56.13	odd	2
280.2.h.b.251.8	yes	16	4.3	odd	2
1120.2.h.a.111.1		16	8.3	odd	2
1120.2.h.a.111.16		16	7.6	odd	2
1120.2.h.b.111.1		16	1.1	even	1	trivial
1120.2.h.b.111.16		16	56.27	even	2	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.19977i q^{3} +1.00000 q^{5} +(2.59303 + 0.525543i) q^{7} -7.23851 q^{9} +O(q^{10})\)	\(q-3.19977i q^{3} +1.00000 q^{5} +(2.59303 + 0.525543i) q^{7} -7.23851 q^{9} +3.34588 q^{11} +3.90251 q^{13} -3.19977i q^{15} +2.92992i q^{17} -6.33672i q^{19} +(1.68161 - 8.29709i) q^{21} -3.44642i q^{23} +1.00000 q^{25} +13.5622i q^{27} -2.68130i q^{29} +2.52241 q^{31} -10.7060i q^{33} +(2.59303 + 0.525543i) q^{35} -4.70905i q^{37} -12.4871i q^{39} +5.59232i q^{41} -8.62439 q^{43} -7.23851 q^{45} +0.506742 q^{47} +(6.44761 + 2.72550i) q^{49} +9.37508 q^{51} +11.4136i q^{53} +3.34588 q^{55} -20.2760 q^{57} +0.802275i q^{59} -7.97236 q^{61} +(-18.7697 - 3.80415i) q^{63} +3.90251 q^{65} -6.37503 q^{67} -11.0278 q^{69} -1.11901i q^{71} +5.91619i q^{73} -3.19977i q^{75} +(8.67597 + 1.75840i) q^{77} +10.5412i q^{79} +21.6805 q^{81} -4.29261i q^{83} +2.92992i q^{85} -8.57954 q^{87} +2.00722i q^{89} +(10.1193 + 2.05094i) q^{91} -8.07111i q^{93} -6.33672i q^{95} -6.06903i q^{97} -24.2192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 - 16 * q^9	\( 16 q + 16 q^{5} - 16 q^{9} + 4 q^{11} - 4 q^{21} + 16 q^{25} + 16 q^{31} + 4 q^{43} - 16 q^{45} - 8 q^{49} + 40 q^{51} + 4 q^{55} - 16 q^{57} - 8 q^{61} - 28 q^{63} - 20 q^{67} - 40 q^{69} - 4 q^{77} + 24 q^{81} - 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) \) 16 * q + 16 * q^5 - 16 * q^9 + 4 * q^11 - 4 * q^21 + 16 * q^25 + 16 * q^31 + 4 * q^43 - 16 * q^45 - 8 * q^49 + 40 * q^51 + 4 * q^55 - 16 * q^57 - 8 * q^61 - 28 * q^63 - 20 * q^67 - 40 * q^69 - 4 * q^77 + 24 * q^81 - 72 * q^87 + 32 * q^91 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more