LMFDB - Embedded newform 2240.2.l.b.1569.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1569,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2240.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.l (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3x^{10} + 11x^{8} + 22x^{6} + 99x^{4} + 243x^{2} + 729 $ x^12 + 3x^10 + 11x^8 + 22x^6 + 99x^4 + 243*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.2
Root			$1.55635 - 0.760115i$ of defining polynomial
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.b.1569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11270 q^{3} +(2.08433 + 0.809661i) q^{5} +1.00000i q^{7} +6.68890 q^{9} +O(q^{10})$	$q-3.11270 q^{3} +(2.08433 + 0.809661i) q^{5} +1.00000i q^{7} +6.68890 q^{9} -2.64844i q^{11} -0.479769 q^{13} +(-6.48791 - 2.52023i) q^{15} +1.49338i q^{17} -2.16867i q^{19} -3.11270i q^{21} -6.33734i q^{23} +(3.68890 + 3.37521i) q^{25} -11.4824 q^{27} -5.25704i q^{29} +6.22540 q^{31} +8.24379i q^{33} +(-0.809661 + 2.08433i) q^{35} -11.3778 q^{37} +1.49338 q^{39} +2.00000 q^{41} -9.98906 q^{43} +(13.9419 + 5.41574i) q^{45} +7.68890i q^{47} -1.00000 q^{49} -4.64844i q^{51} +3.04046 q^{53} +(2.14434 - 5.52023i) q^{55} +6.75041i q^{57} +9.20913i q^{59} -10.8315i q^{61} +6.68890i q^{63} +(-1.00000 - 0.388450i) q^{65} +9.73717 q^{67} +19.7262i q^{69} -0.525015 q^{71} -9.98906i q^{73} +(-11.4824 - 10.5060i) q^{75} +2.64844 q^{77} -13.9442 q^{79} +15.6747 q^{81} +11.3565 q^{83} +(-1.20913 + 3.11270i) q^{85} +16.3636i q^{87} +2.95954 q^{89} -0.479769i q^{91} -19.3778 q^{93} +(1.75589 - 4.52023i) q^{95} -7.97066i q^{97} -17.7151i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{5} + 24 q^{9}+O(q^{10}) $ 12 * q + 4 * q^5 + 24 * q^9	$ 12 q + 4 q^{5} + 24 q^{9} - 20 q^{13} - 12 q^{25} - 24 q^{37} + 24 q^{41} + 48 q^{45} - 12 q^{49} + 8 q^{53} - 12 q^{65} + 4 q^{77} + 20 q^{81} + 56 q^{85} + 64 q^{89} - 120 q^{93}+O(q^{100}) $ 12 * q + 4 * q^5 + 24 * q^9 - 20 * q^13 - 12 * q^25 - 24 * q^37 + 24 * q^41 + 48 * q^45 - 12 * q^49 + 8 * q^53 - 12 * q^65 + 4 * q^77 + 20 * q^81 + 56 * q^85 + 64 * q^89 - 120 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−3.11270			−1.79712			−0.898559	−	0.438853i	$-0.855385\pi$
$3$	−3.11270			−1.79712			−0.898559	+	0.438853i	$0.855385\pi$
$4$		0			0
$5$	2.08433	+	0.809661i	0.932143	+	0.362091i
$5$	2.08433	+	0.809661i	0.932143	+	0.362091i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	6.68890			2.22963
$10$		0			0
$11$		−	2.64844i		−	0.798534i	−0.916835	−	0.399267i	$-0.869265\pi$
$11$		−	2.64844i		−	0.798534i	0.916835	−	0.399267i	$-0.130735\pi$
$12$		0			0
$13$	−0.479769			−0.133064			−0.0665321	−	0.997784i	$-0.521193\pi$
$13$	−0.479769			−0.133064			−0.0665321	+	0.997784i	$0.521193\pi$
$14$		0			0
$15$	−6.48791	−	2.52023i	−1.67517	−	0.650721i
$16$		0			0
$17$			1.49338i			0.362197i	0.983465	+	0.181099i	$0.0579654\pi$
$17$			1.49338i			0.362197i	−0.983465	+	0.181099i	$0.942035\pi$
$18$		0			0
$19$		−	2.16867i		−	0.497527i	−0.968564	−	0.248763i	$-0.919976\pi$
$19$		−	2.16867i		−	0.497527i	0.968564	−	0.248763i	$-0.0800241\pi$
$20$		0			0
$21$		−	3.11270i		−	0.679247i
$22$		0			0
$23$		−	6.33734i		−	1.32143i	−0.750639	−	0.660713i	$-0.770254\pi$
$23$		−	6.33734i		−	1.32143i	0.750639	−	0.660713i	$-0.229746\pi$
$24$		0			0
$25$	3.68890	+	3.37521i	0.737780	+	0.675041i
$26$		0			0
$27$	−11.4824			−2.20980
$28$		0			0
$29$		−	5.25704i		−	0.976207i	−0.872786	−	0.488104i	$-0.837689\pi$
$29$		−	5.25704i		−	0.976207i	0.872786	−	0.488104i	$-0.162311\pi$
$30$		0			0
$31$	6.22540			1.11811			0.559057	−	0.829129i	$-0.311163\pi$
$31$	6.22540			1.11811			0.559057	+	0.829129i	$0.311163\pi$
$32$		0			0
$33$			8.24379i			1.43506i
$34$		0			0
$35$	−0.809661	+	2.08433i	−0.136858	+	0.352317i
$36$		0			0
$37$	−11.3778			−1.87050			−0.935249	−	0.353990i	$-0.884825\pi$
$37$	−11.3778			−1.87050			−0.935249	+	0.353990i	$0.884825\pi$
$38$		0			0
$39$	1.49338			0.239132
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	−9.98906			−1.52332			−0.761658	−	0.647979i	$-0.775615\pi$
$43$	−9.98906			−1.52332			−0.761658	+	0.647979i	$0.775615\pi$
$44$		0			0
$45$	13.9419	+	5.41574i	2.07834	+	0.807331i
$46$		0			0
$47$			7.68890i			1.12154i	0.827971	+	0.560771i	$0.189495\pi$
$47$			7.68890i			1.12154i	−0.827971	+	0.560771i	$0.810505\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	4.64844i		−	0.650912i
$52$		0			0
$53$	3.04046			0.417639			0.208820	−	0.977954i	$-0.433038\pi$
$53$	3.04046			0.417639			0.208820	+	0.977954i	$0.433038\pi$
$54$		0			0
$55$	2.14434	−	5.52023i	0.289142	−	0.744348i
$56$		0			0
$57$			6.75041i			0.894114i
$58$		0			0
$59$			9.20913i			1.19893i	0.800402	+	0.599463i	$0.204619\pi$
$59$			9.20913i			1.19893i	−0.800402	+	0.599463i	$0.795381\pi$
$60$		0			0
$61$		−	10.8315i		−	1.38683i	−0.720539	−	0.693414i	$-0.756106\pi$
$61$		−	10.8315i		−	1.38683i	0.720539	−	0.693414i	$-0.243894\pi$
$62$		0			0
$63$			6.68890i			0.842722i
$64$		0			0
$65$	−1.00000	−	0.388450i	−0.124035	−	0.0481814i
$66$		0			0
$67$	9.73717			1.18958			0.594792	−	0.803879i	$-0.297234\pi$
$67$	9.73717			1.18958			0.594792	+	0.803879i	$0.297234\pi$
$68$		0			0
$69$			19.7262i			2.37476i
$70$		0			0
$71$	−0.525015			−0.0623078			−0.0311539	−	0.999515i	$-0.509918\pi$
$71$	−0.525015			−0.0623078			−0.0311539	+	0.999515i	$0.509918\pi$
$72$		0			0
$73$		−	9.98906i		−	1.16913i	−0.811347	−	0.584565i	$-0.801265\pi$
$73$		−	9.98906i		−	1.16913i	0.811347	−	0.584565i	$-0.198735\pi$
$74$		0			0
$75$	−11.4824	−	10.5060i	−1.32588	−	1.21313i
$76$		0			0
$77$	2.64844			0.301818
$78$		0			0
$79$	−13.9442			−1.56884			−0.784421	−	0.620228i	$-0.787040\pi$
$79$	−13.9442			−1.56884			−0.784421	+	0.620228i	$0.787040\pi$
$80$		0			0
$81$	15.6747			1.74163
$82$		0			0
$83$	11.3565			1.24654			0.623268	−	0.782008i	$-0.285804\pi$
$83$	11.3565			1.24654			0.623268	+	0.782008i	$0.285804\pi$
$84$		0			0
$85$	−1.20913	+	3.11270i	−0.131149	+	0.337620i
$86$		0			0
$87$			16.3636i			1.75436i
$88$		0			0
$89$	2.95954			0.313711			0.156855	−	0.987622i	$-0.449864\pi$
$89$	2.95954			0.313711			0.156855	+	0.987622i	$0.449864\pi$
$90$		0			0
$91$		−	0.479769i		−	0.0502935i
$92$		0			0
$93$	−19.3778			−2.00938
$94$		0			0
$95$	1.75589	−	4.52023i	0.180150	−	0.463766i
$96$		0			0
$97$		−	7.97066i		−	0.809298i	−0.914472	−	0.404649i	$-0.867394\pi$
$97$		−	7.97066i		−	0.809298i	0.914472	−	0.404649i	$-0.132606\pi$
$98$		0			0
$99$		−	17.7151i		−	1.78044i
$100$		0			0
$101$		−	14.0701i		−	1.40003i	−0.714128	−	0.700015i	$-0.753177\pi$
$101$		−	14.0701i		−	1.40003i	0.714128	−	0.700015i	$-0.246823\pi$
$102$		0			0
$103$			3.35156i			0.330239i	0.986274	+	0.165120i	$0.0528010\pi$
$103$			3.35156i			0.330239i	−0.986274	+	0.165120i	$0.947199\pi$
$104$		0			0
$105$	2.52023	−	6.48791i	0.245949	−	0.633155i
$106$		0			0
$107$	−5.70038			−0.551077			−0.275538	−	0.961290i	$-0.588856\pi$
$107$	−5.70038			−0.551077			−0.275538	+	0.961290i	$0.588856\pi$
$108$		0			0
$109$			14.7211i			1.41002i	0.709196	+	0.705012i	$0.249058\pi$
$109$			14.7211i			1.41002i	−0.709196	+	0.705012i	$0.750942\pi$
$110$		0			0
$111$	35.4157			3.36151
$112$		0			0
$113$		−	16.2145i		−	1.52533i	−0.646795	−	0.762664i	$-0.723891\pi$
$113$		−	16.2145i		−	1.52533i	0.646795	−	0.762664i	$-0.276109\pi$
$114$		0			0
$115$	5.13109	−	13.2091i	0.478477	−	1.23176i
$116$		0			0
$117$	−3.20913			−0.296684
$118$		0			0
$119$	−1.49338			−0.136898
$120$		0			0
$121$	3.98578			0.362343
$122$		0			0
$123$	−6.22540			−0.561325
$124$		0			0
$125$	4.95613	+	10.0218i	0.443289	+	0.896379i
$126$		0			0
$127$		−	8.25642i		−	0.732638i	−0.930489	−	0.366319i	$-0.880618\pi$
$127$		−	8.25642i		−	0.732638i	0.930489	−	0.366319i	$-0.119382\pi$
$128$		0			0
$129$	31.0929			2.73758
$130$		0			0
$131$		−	11.2901i		−	0.986416i	−0.869911	−	0.493208i	$-0.835824\pi$
$131$		−	11.2901i		−	0.986416i	0.869911	−	0.493208i	$-0.164176\pi$
$132$		0			0
$133$	2.16867			0.188047
$134$		0			0
$135$	−23.9332	−	9.29688i	−2.05984	−	0.800148i
$136$		0			0
$137$		−	13.5008i		−	1.15345i	−0.816937	−	0.576727i	$-0.804330\pi$
$137$		−	13.5008i		−	1.15345i	0.816937	−	0.576727i	$-0.195670\pi$
$138$		0			0
$139$		−	21.8838i		−	1.85616i	−0.372381	−	0.928080i	$-0.621458\pi$
$139$		−	21.8838i		−	1.85616i	0.372381	−	0.928080i	$-0.378542\pi$
$140$		0			0
$141$		−	23.9332i		−	2.01554i
$142$		0			0
$143$			1.27064i			0.106256i
$144$		0			0
$145$	4.25642	−	10.9574i	0.353476	−	0.909964i
$146$		0			0
$147$	3.11270			0.256731
$148$		0			0
$149$		−	13.5008i		−	1.10603i	−0.833171	−	0.553015i	$-0.813477\pi$
$149$		−	13.5008i		−	1.10603i	0.833171	−	0.553015i	$-0.186523\pi$
$150$		0			0
$151$	17.4559			1.42054			0.710272	−	0.703927i	$-0.248572\pi$
$151$	17.4559			1.42054			0.710272	+	0.703927i	$0.248572\pi$
$152$		0			0
$153$			9.98906i			0.807567i
$154$		0			0
$155$	12.9758	+	5.04046i	1.04224	+	0.404860i
$156$		0			0
$157$	9.91225			0.791084			0.395542	−	0.918448i	$-0.370557\pi$
$157$	9.91225			0.791084			0.395542	+	0.918448i	$0.370557\pi$
$158$		0			0
$159$	−9.46404			−0.750547
$160$		0			0
$161$	6.33734			0.499452
$162$		0			0
$163$	18.9493			1.48423			0.742113	−	0.670275i	$-0.233824\pi$
$163$	18.9493			1.48423			0.742113	+	0.670275i	$0.233824\pi$
$164$		0			0
$165$	−6.67467	+	17.1828i	−0.519623	+	1.33768i
$166$		0			0
$167$		−	16.8103i		−	1.30082i	−0.759584	−	0.650409i	$-0.774597\pi$
$167$		−	16.8103i		−	1.30082i	0.759584	−	0.650409i	$-0.225403\pi$
$168$		0			0
$169$	−12.7698			−0.982294
$170$		0			0
$171$		−	14.5060i		−	1.10930i
$172$		0			0
$173$	−15.5202			−1.17998			−0.589991	−	0.807410i	$-0.700869\pi$
$173$	−15.5202			−1.17998			−0.589991	+	0.807410i	$0.700869\pi$
$174$		0			0
$175$	−3.37521	+	3.68890i	−0.255142	+	0.278855i
$176$		0			0
$177$		−	28.6653i		−	2.15461i
$178$		0			0
$179$		−	8.70312i		−	0.650502i	−0.945628	−	0.325251i	$-0.894551\pi$
$179$		−	8.70312i		−	0.650502i	0.945628	−	0.325251i	$-0.105449\pi$
$180$		0			0
$181$			14.8683i			1.10515i	0.833463	+	0.552575i	$0.186355\pi$
$181$			14.8683i			1.10515i	−0.833463	+	0.552575i	$0.813645\pi$
$182$		0			0
$183$			33.7151i			2.49229i
$184$		0			0
$185$	−23.7151	−	9.21216i	−1.74357	−	0.677291i
$186$		0			0
$187$	3.95512			0.289227
$188$		0			0
$189$		−	11.4824i		−	0.835224i
$190$		0			0
$191$	0.443349			0.0320796			0.0160398	−	0.999871i	$-0.494894\pi$
$191$	0.443349			0.0320796			0.0160398	+	0.999871i	$0.494894\pi$
$192$		0			0
$193$		−	13.2277i		−	0.952151i	−0.879405	−	0.476075i	$-0.842059\pi$
$193$		−	13.2277i		−	0.952151i	0.879405	−	0.476075i	$-0.157941\pi$
$194$		0			0
$195$	3.11270	+	1.20913i	0.222905	+	0.0865876i
$196$		0			0
$197$	7.71514			0.549681			0.274840	−	0.961490i	$-0.411375\pi$
$197$	7.71514			0.549681			0.274840	+	0.961490i	$0.411375\pi$
$198$		0			0
$199$	9.46404			0.670888			0.335444	−	0.942060i	$-0.391114\pi$
$199$	9.46404			0.670888			0.335444	+	0.942060i	$0.391114\pi$
$200$		0			0
$201$	−30.3089			−2.13782
$202$		0			0
$203$	5.25704			0.368972
$204$		0			0
$205$	4.16867	+	1.61932i	0.291152	+	0.113098i
$206$		0			0
$207$		−	42.3898i		−	2.94630i
$208$		0			0
$209$	−5.74358			−0.397292
$210$		0			0
$211$		−	0.392023i		−	0.0269880i	−0.999909	−	0.0134940i	$-0.995705\pi$
$211$		−	0.392023i		−	0.0269880i	0.999909	−	0.0134940i	$-0.00429540\pi$
$212$		0			0
$213$	1.63421			0.111974
$214$		0			0
$215$	−20.8205	−	8.08775i	−1.41995	−	0.551580i
$216$		0			0
$217$			6.22540i			0.422608i
$218$		0			0
$219$			31.0929i			2.10107i
$220$		0			0
$221$		−	0.716477i		−	0.0481955i
$222$		0			0
$223$			19.2422i			1.28855i	0.764793	+	0.644276i	$0.222841\pi$
$223$			19.2422i			1.28855i	−0.764793	+	0.644276i	$0.777159\pi$
$224$		0			0
$225$	24.6747	+	22.5764i	1.64498	+	1.50509i
$226$		0			0
$227$	18.5503			1.23122			0.615612	−	0.788050i	$-0.288909\pi$
$227$	18.5503			1.23122			0.615612	+	0.788050i	$0.288909\pi$
$228$		0			0
$229$		−	5.65611i		−	0.373766i	−0.982382	−	0.186883i	$-0.940161\pi$
$229$		−	5.65611i		−	0.373766i	0.982382	−	0.186883i	$-0.0598385\pi$
$230$		0			0
$231$	−8.24379			−0.542402
$232$		0			0
$233$			6.47729i			0.424341i	0.977233	+	0.212171i	$0.0680532\pi$
$233$			6.47729i			0.424341i	−0.977233	+	0.212171i	$0.931947\pi$
$234$		0			0
$235$	−6.22540	+	16.0262i	−0.406100	+	1.04544i
$236$		0			0
$237$	43.4040			2.81940
$238$		0			0
$239$	−29.9067			−1.93451			−0.967253	−	0.253813i	$-0.918315\pi$
$239$	−29.9067			−1.93451			−0.967253	+	0.253813i	$0.918315\pi$
$240$		0			0
$241$	0.256415			0.0165172			0.00825858	−	0.999966i	$-0.497371\pi$
$241$	0.256415			0.0165172			0.00825858	+	0.999966i	$0.497371\pi$
$242$		0			0
$243$	−14.3432			−0.920120
$244$		0			0
$245$	−2.08433	−	0.809661i	−0.133163	−	0.0517273i
$246$		0			0
$247$			1.04046i			0.0662030i
$248$		0			0
$249$	−35.3493			−2.24017
$250$		0			0
$251$			15.1282i			0.954884i	0.878663	+	0.477442i	$0.158436\pi$
$251$			15.1282i			0.954884i	−0.878663	+	0.477442i	$0.841564\pi$
$252$		0			0
$253$	−16.7840			−1.05520
$254$		0			0
$255$	3.76366	−	9.68890i	0.235689	−	0.606742i
$256$		0			0
$257$			19.2012i			1.19774i	0.800847	+	0.598869i	$0.204383\pi$
$257$			19.2012i			1.19774i	−0.800847	+	0.598869i	$0.795617\pi$
$258$		0			0
$259$		−	11.3778i		−	0.706982i
$260$		0			0
$261$		−	35.1638i		−	2.17658i
$262$		0			0
$263$		−	6.75560i		−	0.416568i	−0.978068	−	0.208284i	$-0.933212\pi$
$263$		−	6.75560i		−	0.416568i	0.978068	−	0.208284i	$-0.0667878\pi$
$264$		0			0
$265$	6.33734	+	2.46174i	0.389300	+	0.151224i
$266$		0			0
$267$	−9.21216			−0.563775
$268$		0			0
$269$			27.3191i			1.66567i	0.553519	+	0.832836i	$0.313284\pi$
$269$			27.3191i			1.66567i	−0.553519	+	0.832836i	$0.686716\pi$
$270$		0			0
$271$	−7.52732			−0.457252			−0.228626	−	0.973514i	$-0.573423\pi$
$271$	−7.52732			−0.457252			−0.228626	+	0.973514i	$0.573423\pi$
$272$		0			0
$273$			1.49338i			0.0903834i
$274$		0			0
$275$	8.93903	−	9.76982i	0.539044	−	0.589142i
$276$		0			0
$277$	−3.55329			−0.213497			−0.106748	−	0.994286i	$-0.534044\pi$
$277$	−3.55329			−0.213497			−0.106748	+	0.994286i	$0.534044\pi$
$278$		0			0
$279$	41.6411			2.49299
$280$		0			0
$281$	−8.90485			−0.531219			−0.265610	−	0.964081i	$-0.585573\pi$
$281$	−8.90485			−0.531219			−0.265610	+	0.964081i	$0.585573\pi$
$282$		0			0
$283$	−9.33810			−0.555092			−0.277546	−	0.960712i	$-0.589521\pi$
$283$	−9.33810			−0.555092			−0.277546	+	0.960712i	$0.589521\pi$
$284$		0			0
$285$	−5.46554	+	14.0701i	−0.323751	+	0.833442i
$286$		0			0
$287$			2.00000i			0.118056i
$288$		0			0
$289$	14.7698			0.868813
$290$		0			0
$291$			24.8103i			1.45440i
$292$		0			0
$293$	21.3163			1.24531			0.622655	−	0.782496i	$-0.286054\pi$
$293$	21.3163			1.24531			0.622655	+	0.782496i	$0.286054\pi$
$294$		0			0
$295$	−7.45627	+	19.1949i	−0.434121	+	1.11757i
$296$		0			0
$297$			30.4105i			1.76460i
$298$		0			0
$299$			3.04046i			0.175834i
$300$		0			0
$301$		−	9.98906i		−	0.575760i
$302$		0			0
$303$			43.7961i			2.51602i
$304$		0			0
$305$	8.76982	−	22.5764i	0.502159	−	1.29272i
$306$		0			0
$307$	−1.81078			−0.103347			−0.0516734	−	0.998664i	$-0.516455\pi$
$307$	−1.81078			−0.103347			−0.0516734	+	0.998664i	$0.516455\pi$
$308$		0			0
$309$		−	10.4324i		−	0.593479i
$310$		0			0
$311$	23.2168			1.31650			0.658251	−	0.752799i	$-0.271297\pi$
$311$	23.2168			1.31650			0.658251	+	0.752799i	$0.271297\pi$
$312$		0			0
$313$		−	7.71878i		−	0.436291i	−0.975916	−	0.218146i	$-0.929999\pi$
$313$		−	7.71878i		−	0.436291i	0.975916	−	0.218146i	$-0.0700008\pi$
$314$		0			0
$315$	−5.41574	+	13.9419i	−0.305142	+	0.785537i
$316$		0			0
$317$	24.3373			1.36692			0.683461	−	0.729987i	$-0.260474\pi$
$317$	24.3373			1.36692			0.683461	+	0.729987i	$0.260474\pi$
$318$		0			0
$319$	−13.9229			−0.779535
$320$		0			0
$321$	17.7436			0.990350
$322$		0			0
$323$	3.23864			0.180203
$324$		0			0
$325$	−1.76982	−	1.61932i	−0.0981720	−	0.0898238i
$326$		0			0
$327$		−	45.8223i		−	2.53398i
$328$		0			0
$329$	−7.68890			−0.423903
$330$		0			0
$331$		−	27.6342i		−	1.51891i	−0.650558	−	0.759457i	$-0.725465\pi$
$331$		−	27.6342i		−	1.51891i	0.650558	−	0.759457i	$-0.274535\pi$
$332$		0			0
$333$	−76.1049			−4.17053
$334$		0			0
$335$	20.2955	+	7.88380i	1.10886	+	0.430738i
$336$		0			0
$337$		−	16.2145i		−	0.883258i	−0.897198	−	0.441629i	$-0.854401\pi$
$337$		−	16.2145i		−	0.883258i	0.897198	−	0.441629i	$-0.145599\pi$
$338$		0			0
$339$			50.4707i			2.74119i
$340$		0			0
$341$		−	16.4876i		−	0.892853i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−15.9716	+	41.1161i	−0.859879	+	2.21361i
$346$		0			0
$347$	5.95227			0.319535			0.159767	−	0.987155i	$-0.448926\pi$
$347$	5.95227			0.319535			0.159767	+	0.987155i	$0.448926\pi$
$348$		0			0
$349$			4.35419i			0.233075i	0.993186	+	0.116537i	$0.0371794\pi$
$349$			4.35419i			0.233075i	−0.993186	+	0.116537i	$0.962821\pi$
$350$		0			0
$351$	5.50892			0.294045
$352$		0			0
$353$			5.78205i			0.307748i	0.988091	+	0.153874i	$0.0491749\pi$
$353$			5.78205i			0.307748i	−0.988091	+	0.153874i	$0.950825\pi$
$354$		0			0
$355$	−1.09431	−	0.425084i	−0.0580798	−	0.0225611i
$356$		0			0
$357$	4.64844			0.246021
$358$		0			0
$359$	−4.54056			−0.239642			−0.119821	−	0.992796i	$-0.538232\pi$
$359$	−4.54056			−0.239642			−0.119821	+	0.992796i	$0.538232\pi$
$360$		0			0
$361$	14.2969			0.752467
$362$		0			0
$363$	−12.4065			−0.651174
$364$		0			0
$365$	8.08775	−	20.8205i	0.423332	−	1.08980i
$366$		0			0
$367$		−	28.0262i		−	1.46296i	−0.681864	−	0.731479i	$-0.738830\pi$
$367$		−	28.0262i		−	1.46296i	0.681864	−	0.731479i	$-0.261170\pi$
$368$		0			0
$369$	13.3778			0.696420
$370$		0			0
$371$			3.04046i			0.157853i
$372$		0			0
$373$	9.01201			0.466624			0.233312	−	0.972402i	$-0.425044\pi$
$373$	9.01201			0.466624			0.233312	+	0.972402i	$0.425044\pi$
$374$		0			0
$375$	−15.4269	−	31.1949i	−0.796643	−	1.61090i
$376$		0			0
$377$			2.52217i			0.129898i
$378$		0			0
$379$		−	21.7151i		−	1.11543i	−0.830032	−	0.557716i	$-0.811678\pi$
$379$		−	21.7151i		−	1.11543i	0.830032	−	0.557716i	$-0.188322\pi$
$380$		0			0
$381$			25.6997i			1.31664i
$382$		0			0
$383$			1.40625i			0.0718559i	0.999354	+	0.0359279i	$0.0114387\pi$
$383$			1.40625i			0.0718559i	−0.999354	+	0.0359279i	$0.988561\pi$
$384$		0			0
$385$	5.52023	+	2.14434i	0.281337	+	0.109285i
$386$		0			0
$387$	−66.8158			−3.39644
$388$		0			0
$389$			36.6359i			1.85751i	0.370688	+	0.928757i	$0.379122\pi$
$389$			36.6359i			1.85751i	−0.370688	+	0.928757i	$0.620878\pi$
$390$		0			0
$391$	9.46404			0.478617
$392$		0			0
$393$			35.1425i			1.77271i
$394$		0			0
$395$	−29.0643	−	11.2901i	−1.46238	−	0.568064i
$396$		0			0
$397$	−14.6132			−0.733414			−0.366707	−	0.930337i	$-0.619515\pi$
$397$	−14.6132			−0.733414			−0.366707	+	0.930337i	$0.619515\pi$
$398$		0			0
$399$	−6.75041			−0.337943
$400$		0			0
$401$	−13.7698			−0.687632			−0.343816	−	0.939037i	$-0.611720\pi$
$401$	−13.7698			−0.687632			−0.343816	+	0.939037i	$0.611720\pi$
$402$		0			0
$403$	−2.98676			−0.148781
$404$		0			0
$405$	32.6713	+	12.6912i	1.62345	+	0.630629i
$406$		0			0
$407$			30.1334i			1.49366i
$408$		0			0
$409$	25.5396			1.26285			0.631427	−	0.775435i	$-0.282470\pi$
$409$	25.5396			1.26285			0.631427	+	0.775435i	$0.282470\pi$
$410$		0			0
$411$			42.0240i			2.07289i
$412$		0			0
$413$	−9.20913			−0.453152
$414$		0			0
$415$	23.6707	+	9.19491i	1.16195	+	0.451360i
$416$		0			0
$417$			68.1177i			3.33574i
$418$		0			0
$419$		−	2.45353i		−	0.119863i	−0.998202	−	0.0599315i	$-0.980912\pi$
$419$		−	2.45353i		−	0.119863i	0.998202	−	0.0599315i	$-0.0190882\pi$
$420$		0			0
$421$			2.01839i			0.0983705i	0.998790	+	0.0491852i	$0.0156625\pi$
$421$			2.01839i			0.0983705i	−0.998790	+	0.0491852i	$0.984338\pi$
$422$		0			0
$423$			51.4303i			2.50063i
$424$		0			0
$425$	−5.04046	+	5.50892i	−0.244498	+	0.267222i
$426$		0			0
$427$	10.8315			0.524172
$428$		0			0
$429$		−	3.95512i		−	0.190955i
$430$		0			0
$431$	−1.47214			−0.0709103			−0.0354551	−	0.999371i	$-0.511288\pi$
$431$	−1.47214			−0.0709103			−0.0354551	+	0.999371i	$0.511288\pi$
$432$		0			0
$433$			2.46174i			0.118304i	0.998249	+	0.0591519i	$0.0188396\pi$
$433$			2.46174i			0.118304i	−0.998249	+	0.0591519i	$0.981160\pi$
$434$		0			0
$435$	−13.2489	+	34.1072i	−0.635238	+	1.63531i
$436$		0			0
$437$	−13.7436			−0.657445
$438$		0			0
$439$	−19.7262			−0.941481			−0.470741	−	0.882272i	$-0.656013\pi$
$439$	−19.7262			−0.941481			−0.470741	+	0.882272i	$0.656013\pi$
$440$		0			0
$441$	−6.68890			−0.318519
$442$		0			0
$443$	12.1777			0.578579			0.289289	−	0.957242i	$-0.406581\pi$
$443$	12.1777			0.578579			0.289289	+	0.957242i	$0.406581\pi$
$444$		0			0
$445$	6.16867	+	2.39622i	0.292423	+	0.113592i
$446$		0			0
$447$			42.0240i			1.98767i
$448$		0			0
$449$	−26.3636			−1.24417			−0.622087	−	0.782948i	$-0.713715\pi$
$449$	−26.3636			−1.24417			−0.622087	+	0.782948i	$0.713715\pi$
$450$		0			0
$451$		−	5.29688i		−	0.249420i
$452$		0			0
$453$	−54.3351			−2.55289
$454$		0			0
$455$	0.388450	−	1.00000i	0.0182108	−	0.0468807i
$456$		0			0
$457$		−	33.5887i		−	1.57121i	−0.618725	−	0.785607i	$-0.712351\pi$
$457$		−	33.5887i		−	1.57121i	0.618725	−	0.785607i	$-0.287649\pi$
$458$		0			0
$459$		−	17.1476i		−	0.800382i
$460$		0			0
$461$		−	4.85796i		−	0.226258i	−0.993580	−	0.113129i	$-0.963913\pi$
$461$		−	4.85796i		−	0.226258i	0.993580	−	0.113129i	$-0.0360873\pi$
$462$		0			0
$463$			1.24440i			0.0578323i	0.999582	+	0.0289162i	$0.00920558\pi$
$463$			1.24440i			0.0578323i	−0.999582	+	0.0289162i	$0.990794\pi$
$464$		0			0
$465$	−40.3898	−	15.6894i	−1.87303	−	0.727580i
$466$		0			0
$467$	25.2794			1.16979			0.584896	−	0.811108i	$-0.301135\pi$
$467$	25.2794			1.16979			0.584896	+	0.811108i	$0.301135\pi$
$468$		0			0
$469$			9.73717i			0.449621i
$470$		0			0
$471$	−30.8539			−1.42167
$472$		0			0
$473$			26.4554i			1.21642i
$474$		0			0
$475$	7.31971	−	8.00000i	0.335851	−	0.367065i
$476$		0			0
$477$	20.3373			0.931183
$478$		0			0
$479$	−11.1489			−0.509405			−0.254703	−	0.967019i	$-0.581978\pi$
$479$	−11.1489			−0.509405			−0.254703	+	0.967019i	$0.581978\pi$
$480$		0			0
$481$	5.45872			0.248896
$482$		0			0
$483$	−19.7262			−0.897574
$484$		0			0
$485$	6.45353	−	16.6135i	0.293040	−	0.754381i
$486$		0			0
$487$		−	16.2564i		−	0.736648i	−0.929697	−	0.368324i	$-0.879932\pi$
$487$		−	16.2564i		−	0.736648i	0.929697	−	0.368324i	$-0.120068\pi$
$488$		0			0
$489$	−58.9836			−2.66733
$490$		0			0
$491$			41.7938i			1.88613i	0.332609	+	0.943065i	$0.392071\pi$
$491$			41.7938i			1.88613i	−0.332609	+	0.943065i	$0.607929\pi$
$492$		0			0
$493$	7.85074			0.353580
$494$		0			0
$495$	14.3432	−	36.9243i	0.644681	−	1.65962i
$496$		0			0
$497$		−	0.525015i		−	0.0235501i
$498$		0			0
$499$			15.6080i			0.698709i	0.936991	+	0.349355i	$0.113599\pi$
$499$			15.6080i			0.698709i	−0.936991	+	0.349355i	$0.886401\pi$
$500$		0			0
$501$			52.3254i			2.33772i
$502$		0			0
$503$			6.39202i			0.285006i	0.989794	+	0.142503i	$0.0455151\pi$
$503$			6.39202i			0.285006i	−0.989794	+	0.142503i	$0.954485\pi$
$504$		0			0
$505$	11.3920	−	29.3268i	0.506938	−	1.30503i
$506$		0			0
$507$	39.7486			1.76530
$508$		0			0
$509$			6.15988i			0.273032i	0.990638	+	0.136516i	$0.0435905\pi$
$509$			6.15988i			0.273032i	−0.990638	+	0.136516i	$0.956410\pi$
$510$		0			0
$511$	9.98906			0.441890
$512$		0			0
$513$			24.9016i			1.09943i
$514$		0			0
$515$	−2.71363	+	6.98578i	−0.119577	+	0.307830i
$516$		0			0
$517$	20.3636			0.895589
$518$		0			0
$519$	48.3098			2.12057
$520$		0			0
$521$	−36.8081			−1.61259			−0.806295	−	0.591513i	$-0.798531\pi$
$521$	−36.8081			−1.61259			−0.806295	+	0.591513i	$0.798531\pi$
$522$		0			0
$523$	12.4065			0.542499			0.271250	−	0.962509i	$-0.412563\pi$
$523$	12.4065			0.542499			0.271250	+	0.962509i	$0.412563\pi$
$524$		0			0
$525$	10.5060	−	11.4824i	0.458520	−	0.501135i
$526$		0			0
$527$			9.29688i			0.404978i
$528$		0			0
$529$	−17.1618			−0.746167
$530$		0			0
$531$			61.5989i			2.67317i
$532$		0			0
$533$	−0.959539			−0.0415622
$534$		0			0
$535$	−11.8815	−	4.61538i	−0.513682	−	0.199540i
$536$		0			0
$537$			27.0902i			1.16903i
$538$		0			0
$539$			2.64844i			0.114076i
$540$		0			0
$541$			11.4824i			0.493668i	0.969058	+	0.246834i	$0.0793903\pi$
$541$			11.4824i			0.493668i	−0.969058	+	0.246834i	$0.920610\pi$
$542$		0			0
$543$		−	46.2804i		−	1.98608i
$544$		0			0
$545$	−11.9191	+	30.6836i	−0.510557	+	1.31434i
$546$		0			0
$547$	−16.4663			−0.704050			−0.352025	−	0.935991i	$-0.614507\pi$
$547$	−16.4663			−0.704050			−0.352025	+	0.935991i	$0.614507\pi$
$548$		0			0
$549$		−	72.4507i		−	3.09212i
$550$		0			0
$551$	−11.4008			−0.485689
$552$		0			0
$553$		−	13.9442i		−	0.592967i
$554$		0			0
$555$	73.8181	+	28.6747i	3.13340	+	1.21717i
$556$		0			0
$557$	−39.7676			−1.68501			−0.842504	−	0.538690i	$-0.818919\pi$
$557$	−39.7676			−1.68501			−0.842504	+	0.538690i	$0.818919\pi$
$558$		0			0
$559$	4.79244			0.202699
$560$		0			0
$561$	−12.3111			−0.519775
$562$		0			0
$563$	−14.0914			−0.593880			−0.296940	−	0.954896i	$-0.595966\pi$
$563$	−14.0914			−0.593880			−0.296940	+	0.954896i	$0.595966\pi$
$564$		0			0
$565$	13.1282	−	33.7963i	0.552308	−	1.42182i
$566$		0			0
$567$			15.6747i			0.658274i
$568$		0			0
$569$	0.622202			0.0260841			0.0130420	−	0.999915i	$-0.495848\pi$
$569$	0.622202			0.0260841			0.0130420	+	0.999915i	$0.495848\pi$
$570$		0			0
$571$		−	32.4707i		−	1.35886i	−0.733741	−	0.679429i	$-0.762228\pi$
$571$		−	32.4707i		−	1.35886i	0.733741	−	0.679429i	$-0.237772\pi$
$572$		0			0
$573$	−1.38001			−0.0576508
$574$		0			0
$575$	21.3898	−	23.3778i	0.892017	−	0.974922i
$576$		0			0
$577$		−	0.858567i		−	0.0357426i	−0.999840	−	0.0178713i	$-0.994311\pi$
$577$		−	0.858567i		−	0.0357426i	0.999840	−	0.0178713i	$-0.00568892\pi$
$578$		0			0
$579$			41.1739i			1.71113i
$580$		0			0
$581$			11.3565i			0.471147i
$582$		0			0
$583$		−	8.05247i		−	0.333499i
$584$		0			0
$585$	−6.68890	−	2.59831i	−0.276552	−	0.107427i
$586$		0			0
$587$	−1.09431			−0.0451669			−0.0225834	−	0.999745i	$-0.507189\pi$
$587$	−1.09431			−0.0451669			−0.0225834	+	0.999745i	$0.507189\pi$
$588$		0			0
$589$		−	13.5008i		−	0.556292i
$590$		0			0
$591$	−24.0149			−0.987841
$592$		0			0
$593$		−	31.7337i		−	1.30315i	−0.758586	−	0.651573i	$-0.774109\pi$
$593$		−	31.7337i		−	1.30315i	0.758586	−	0.651573i	$-0.225891\pi$
$594$		0			0
$595$	−3.11270	−	1.20913i	−0.127608	−	0.0495695i
$596$		0			0
$597$	−29.4587			−1.20567
$598$		0			0
$599$	33.4185			1.36544			0.682722	−	0.730678i	$-0.260796\pi$
$599$	33.4185			1.36544			0.682722	+	0.730678i	$0.260796\pi$
$600$		0			0
$601$	14.9595			0.610212			0.305106	−	0.952318i	$-0.401308\pi$
$601$	14.9595			0.610212			0.305106	+	0.952318i	$0.401308\pi$
$602$		0			0
$603$	65.1310			2.65234
$604$		0			0
$605$	8.30769	+	3.22713i	0.337756	+	0.131201i
$606$		0			0
$607$			5.43248i			0.220498i	0.993904	+	0.110249i	$0.0351648\pi$
$607$			5.43248i			0.220498i	−0.993904	+	0.110249i	$0.964835\pi$
$608$		0			0
$609$	−16.3636			−0.663085
$610$		0			0
$611$		−	3.68890i		−	0.149237i
$612$		0			0
$613$	−18.5938			−0.750995			−0.375497	−	0.926823i	$-0.622528\pi$
$613$	−18.5938			−0.750995			−0.375497	+	0.926823i	$0.622528\pi$
$614$		0			0
$615$	−12.9758	−	5.04046i	−0.523235	−	0.203251i
$616$		0			0
$617$		−	41.6623i		−	1.67726i	−0.544700	−	0.838631i	$-0.683357\pi$
$617$		−	41.6623i		−	1.67726i	0.544700	−	0.838631i	$-0.316643\pi$
$618$		0			0
$619$			34.5585i			1.38902i	0.719482	+	0.694511i	$0.244379\pi$
$619$			34.5585i			1.38902i	−0.719482	+	0.694511i	$0.755621\pi$
$620$		0			0
$621$			72.7681i			2.92008i
$622$		0			0
$623$			2.95954i			0.118571i
$624$		0			0
$625$	2.21595	+	24.9016i	0.0886382	+	0.996064i
$626$		0			0
$627$	17.8781			0.713981
$628$		0			0
$629$		−	16.9914i		−	0.677490i
$630$		0			0
$631$	−33.9223			−1.35043			−0.675213	−	0.737623i	$-0.735948\pi$
$631$	−33.9223			−1.35043			−0.675213	+	0.737623i	$0.735948\pi$
$632$		0			0
$633$			1.22025i			0.0485006i
$634$		0			0
$635$	6.68489	−	17.2091i	0.265282	−	0.682923i
$636$		0			0
$637$	0.479769			0.0190092
$638$		0			0
$639$	−3.51177			−0.138924
$640$		0			0
$641$	4.62220			0.182566			0.0912830	−	0.995825i	$-0.470903\pi$
$641$	4.62220			0.182566			0.0912830	+	0.995825i	$0.470903\pi$
$642$		0			0
$643$	13.3749			0.527454			0.263727	−	0.964597i	$-0.415048\pi$
$643$	13.3749			0.527454			0.263727	+	0.964597i	$0.415048\pi$
$644$		0			0
$645$	64.8081	+	25.1747i	2.55182	+	0.991254i
$646$		0			0
$647$			20.3373i			0.799543i	0.916615	+	0.399772i	$0.130911\pi$
$647$			20.3373i			0.799543i	−0.916615	+	0.399772i	$0.869089\pi$
$648$		0			0
$649$	24.3898			0.957384
$650$		0			0
$651$		−	19.3778i		−	0.759476i
$652$		0			0
$653$	−29.1875			−1.14220			−0.571098	−	0.820882i	$-0.693482\pi$
$653$	−29.1875			−1.14220			−0.571098	+	0.820882i	$0.693482\pi$
$654$		0			0
$655$	9.14111	−	23.5322i	0.357173	−	0.919481i
$656$		0			0
$657$		−	66.8158i		−	2.60673i
$658$		0			0
$659$			15.3231i			0.596904i	0.954425	+	0.298452i	$0.0964702\pi$
$659$			15.3231i			0.596904i	−0.954425	+	0.298452i	$0.903530\pi$
$660$		0			0
$661$		−	19.4088i		−	0.754916i	−0.926027	−	0.377458i	$-0.876798\pi$
$661$		−	19.4088i		−	0.754916i	0.926027	−	0.377458i	$-0.123202\pi$
$662$		0			0
$663$			2.23018i			0.0866130i
$664$		0			0
$665$	4.52023	+	1.75589i	0.175287	+	0.0680903i
$666$		0			0
$667$	−33.3156			−1.28999
$668$		0			0
$669$		−	59.8952i		−	2.31568i
$670$		0			0
$671$	−28.6865			−1.10743
$672$		0			0
$673$			14.5509i			0.560894i	0.959869	+	0.280447i	$0.0904828\pi$
$673$			14.5509i			0.560894i	−0.959869	+	0.280447i	$0.909517\pi$
$674$		0			0
$675$	−42.3575	−	38.7556i	−1.63034	−	1.49170i
$676$		0			0
$677$	−16.1424			−0.620404			−0.310202	−	0.950671i	$-0.600397\pi$
$677$	−16.1424			−0.620404			−0.310202	+	0.950671i	$0.600397\pi$
$678$		0			0
$679$	7.97066			0.305886
$680$		0			0
$681$	−57.7414			−2.21265
$682$		0			0
$683$	19.2012			0.734714			0.367357	−	0.930080i	$-0.380263\pi$
$683$	19.2012			0.734714			0.367357	+	0.930080i	$0.380263\pi$
$684$		0			0
$685$	10.9311	−	28.1402i	0.417655	−	1.07518i
$686$		0			0
$687$			17.6058i			0.671702i
$688$		0			0
$689$	−1.45872			−0.0555728
$690$		0			0
$691$		−	44.5869i		−	1.69617i	−0.529863	−	0.848083i	$-0.677757\pi$
$691$		−	44.5869i		−	1.69617i	0.529863	−	0.848083i	$-0.322243\pi$
$692$		0			0
$693$	17.7151			0.672942
$694$		0			0
$695$	17.7185	−	45.6132i	0.672099	−	1.73021i
$696$		0			0
$697$			2.98676i			0.113131i
$698$		0			0
$699$		−	20.1618i		−	0.762591i
$700$		0			0
$701$		−	9.79759i		−	0.370050i	−0.982734	−	0.185025i	$-0.940763\pi$
$701$		−	9.79759i		−	0.370050i	0.982734	−	0.185025i	$-0.0592366\pi$
$702$		0			0
$703$			24.6747i			0.930623i
$704$		0			0
$705$	19.3778	−	49.8849i	0.729810	−	1.87877i
$706$		0			0
$707$	14.0701			0.529161
$708$		0			0
$709$			28.2219i			1.05990i	0.848030	+	0.529948i	$0.177789\pi$
$709$			28.2219i			1.05990i	−0.848030	+	0.529948i	$0.822211\pi$
$710$		0			0
$711$	−93.2712			−3.49794
$712$		0			0
$713$		−	39.4525i		−	1.47751i
$714$		0			0
$715$	−1.02879	+	2.64844i	−0.0384745	+	0.0990460i
$716$		0			0
$717$	93.0907			3.47654
$718$		0			0
$719$	−19.1800			−0.715292			−0.357646	−	0.933857i	$-0.616421\pi$
$719$	−19.1800			−0.715292			−0.357646	+	0.933857i	$0.616421\pi$
$720$		0			0
$721$	−3.35156			−0.124819
$722$		0			0
$723$	−0.798144			−0.0296833
$724$		0			0
$725$	17.7436	−	19.3927i	0.658980	−	0.720226i
$726$		0			0
$727$			20.4992i			0.760273i	0.924930	+	0.380136i	$0.124123\pi$
$727$			20.4992i			0.760273i	−0.924930	+	0.380136i	$0.875877\pi$
$728$		0			0
$729$	−2.37780			−0.0880666
$730$		0			0
$731$		−	14.9174i		−	0.551741i
$732$		0			0
$733$	24.2474			0.895597			0.447799	−	0.894134i	$-0.352208\pi$
$733$	24.2474			0.895597			0.447799	+	0.894134i	$0.352208\pi$
$734$		0			0
$735$	6.48791	+	2.52023i	0.239310	+	0.0929601i
$736$		0			0
$737$		−	25.7883i		−	0.949924i
$738$		0			0
$739$		−	47.3756i		−	1.74274i	−0.490627	−	0.871370i	$-0.663232\pi$
$739$		−	47.3756i		−	1.74274i	0.490627	−	0.871370i	$-0.336768\pi$
$740$		0			0
$741$		−	3.23864i		−	0.118975i
$742$		0			0
$743$		−	1.66266i		−	0.0609972i	−0.999535	−	0.0304986i	$-0.990290\pi$
$743$		−	1.66266i		−	0.0609972i	0.999535	−	0.0304986i	$-0.00970951\pi$
$744$		0			0
$745$	10.9311	−	28.1402i	0.400484	−	1.03098i
$746$		0			0
$747$	75.9624			2.77932
$748$		0			0
$749$		−	5.70038i		−	0.208288i
$750$		0			0
$751$	3.57220			0.130351			0.0651756	−	0.997874i	$-0.479239\pi$
$751$	3.57220			0.130351			0.0651756	+	0.997874i	$0.479239\pi$
$752$		0			0
$753$		−	47.0896i		−	1.71604i
$754$		0			0
$755$	36.3840	+	14.1334i	1.32415	+	0.514367i
$756$		0			0
$757$	−15.2684			−0.554940			−0.277470	−	0.960734i	$-0.589496\pi$
$757$	−15.2684			−0.554940			−0.277470	+	0.960734i	$0.589496\pi$
$758$		0			0
$759$	52.2437			1.89633
$760$		0			0
$761$	−11.9191			−0.432066			−0.216033	−	0.976386i	$-0.569312\pi$
$761$	−11.9191			−0.432066			−0.216033	+	0.976386i	$0.569312\pi$
$762$		0			0
$763$	−14.7211			−0.532939
$764$		0			0
$765$	−8.08775	+	20.8205i	−0.292413	+	0.752768i
$766$		0			0
$767$		−	4.41826i		−	0.159534i
$768$		0			0
$769$	38.9026			1.40287			0.701433	−	0.712736i	$-0.252544\pi$
$769$	38.9026			1.40287			0.701433	+	0.712736i	$0.252544\pi$
$770$		0			0
$771$		−	59.7676i		−	2.15248i
$772$		0			0
$773$	5.32994			0.191705			0.0958523	−	0.995396i	$-0.469442\pi$
$773$	5.32994			0.191705			0.0958523	+	0.995396i	$0.469442\pi$
$774$		0			0
$775$	22.9649	+	21.0120i	0.824922	+	0.754774i
$776$		0			0
$777$			35.4157i			1.27053i
$778$		0			0
$779$		−	4.33734i		−	0.155401i
$780$		0			0
$781$			1.39047i			0.0497549i
$782$		0			0
$783$			60.3636i			2.15722i
$784$		0			0
$785$	20.6605	+	8.02556i	0.737403	+	0.286445i
$786$		0			0
$787$	24.7757			0.883157			0.441578	−	0.897223i	$-0.354419\pi$
$787$	24.7757			0.883157			0.441578	+	0.897223i	$0.354419\pi$
$788$		0			0
$789$			21.0281i			0.748622i
$790$		0			0
$791$	16.2145			0.576520
$792$		0			0
$793$			5.19661i			0.184537i
$794$		0			0
$795$	−19.7262	−	7.66266i	−0.699617	−	0.271767i
$796$		0			0
$797$	−51.1123			−1.81049			−0.905246	−	0.424888i	$-0.860314\pi$
$797$	−51.1123			−1.81049			−0.905246	+	0.424888i	$0.860314\pi$
$798$		0			0
$799$	−11.4824			−0.406219
$800$		0			0
$801$	19.7961			0.699459
$802$		0			0
$803$	−26.4554			−0.933591
$804$		0			0
$805$	13.2091	+	5.13109i	0.465561	+	0.180847i
$806$		0			0
$807$		−	85.0360i		−	2.99341i
$808$		0			0
$809$	−17.6080			−0.619064			−0.309532	−	0.950889i	$-0.600172\pi$
$809$	−17.6080			−0.619064			−0.309532	+	0.950889i	$0.600172\pi$
$810$		0			0
$811$			1.77886i			0.0624642i	0.999512	+	0.0312321i	$0.00994310\pi$
$811$			1.77886i			0.0624642i	−0.999512	+	0.0312321i	$0.990057\pi$
$812$		0			0
$813$	23.4303			0.821735
$814$		0			0
$815$	39.4967	+	15.3425i	1.38351	+	0.537425i
$816$		0			0
$817$			21.6630i			0.757891i
$818$		0			0
$819$		−	3.20913i		−	0.112136i
$820$		0			0
$821$			48.8348i			1.70435i	0.523259	+	0.852174i	$0.324716\pi$
$821$			48.8348i			1.70435i	−0.523259	+	0.852174i	$0.675284\pi$
$822$		0			0
$823$			2.54128i			0.0885834i	0.999019	+	0.0442917i	$0.0141031\pi$
$823$			2.54128i			0.0885834i	−0.999019	+	0.0442917i	$0.985897\pi$
$824$		0			0
$825$	−27.8245	+	30.4105i	−0.968725	+	1.05876i
$826$		0			0
$827$	−1.15983			−0.0403311			−0.0201655	−	0.999797i	$-0.506419\pi$
$827$	−1.15983			−0.0403311			−0.0201655	+	0.999797i	$0.506419\pi$
$828$		0			0
$829$			39.7699i			1.38126i	0.723206	+	0.690632i	$0.242668\pi$
$829$			39.7699i			1.38126i	−0.723206	+	0.690632i	$0.757332\pi$
$830$		0			0
$831$	11.0603			0.383679
$832$		0			0
$833$		−	1.49338i		−	0.0517425i
$834$		0			0
$835$	13.6106	−	35.0382i	0.471015	−	1.21255i
$836$		0			0
$837$	−71.4827			−2.47081
$838$		0			0
$839$	−28.9384			−0.999064			−0.499532	−	0.866295i	$-0.666495\pi$
$839$	−28.9384			−0.999064			−0.499532	+	0.866295i	$0.666495\pi$
$840$		0			0
$841$	1.36357			0.0470198
$842$		0			0
$843$	27.7181			0.954663
$844$		0			0
$845$	−26.6166	−	10.3392i	−0.915638	−	0.355680i
$846$		0			0
$847$			3.98578i			0.136953i
$848$		0			0
$849$	29.0667			0.997566
$850$		0			0
$851$			72.1049i			2.47173i
$852$		0			0
$853$	−39.0861			−1.33828			−0.669141	−	0.743135i	$-0.733338\pi$
$853$	−39.0861			−1.33828			−0.669141	+	0.743135i	$0.733338\pi$
$854$		0			0
$855$	11.7449	−	30.2354i	0.401669	−	1.03403i
$856$		0			0
$857$			57.6036i			1.96770i	0.178987	+	0.983851i	$0.442718\pi$
$857$			57.6036i			1.96770i	−0.178987	+	0.983851i	$0.557282\pi$
$858$		0			0
$859$		−	10.1687i		−	0.346950i	−0.984838	−	0.173475i	$-0.944500\pi$
$859$		−	10.1687i		−	0.346950i	0.984838	−	0.173475i	$-0.0554997\pi$
$860$		0			0
$861$		−	6.22540i		−	0.212161i
$862$		0			0
$863$		−	28.7840i		−	0.979820i	−0.871773	−	0.489910i	$-0.837030\pi$
$863$		−	28.7840i		−	0.979820i	0.871773	−	0.489910i	$-0.162970\pi$
$864$		0			0
$865$	−32.3493	−	12.5661i	−1.09991	−	0.427261i
$866$		0			0
$867$	−45.9740			−1.56136
$868$		0			0
$869$			36.9303i			1.25277i
$870$		0			0
$871$	−4.67160			−0.158291
$872$		0			0
$873$		−	53.3150i		−	1.80444i
$874$		0			0
$875$	−10.0218	+	4.95613i	−0.338799	+	0.167548i
$876$		0			0
$877$	7.48717			0.252824			0.126412	−	0.991978i	$-0.459654\pi$
$877$	7.48717			0.252824			0.126412	+	0.991978i	$0.459654\pi$
$878$		0			0
$879$	−66.3512			−2.23797
$880$		0			0
$881$	33.3778			1.12453			0.562263	−	0.826958i	$-0.309931\pi$
$881$	33.3778			1.12453			0.562263	+	0.826958i	$0.309931\pi$
$882$		0			0
$883$	−6.75041			−0.227170			−0.113585	−	0.993528i	$-0.536233\pi$
$883$	−6.75041			−0.227170			−0.113585	+	0.993528i	$0.536233\pi$
$884$		0			0
$885$	23.2091	−	59.7480i	0.780166	−	2.00841i
$886$		0			0
$887$			30.5801i			1.02678i	0.858156	+	0.513390i	$0.171610\pi$
$887$			30.5801i			1.02678i	−0.858156	+	0.513390i	$0.828390\pi$
$888$		0			0
$889$	8.25642			0.276911
$890$		0			0
$891$		−	41.5134i		−	1.39075i
$892$		0			0
$893$	16.6747			0.557997
$894$		0			0
$895$	7.04658	−	18.1402i	0.235541	−	0.606361i
$896$		0			0
$897$		−	9.46404i		−	0.315995i
$898$		0			0
$899$		−	32.7271i		−	1.09151i
$900$		0			0
$901$			4.54056i			0.151268i
$902$		0			0
$903$			31.0929i			1.03471i
$904$		0			0
$905$	−12.0382	+	30.9904i	−0.400165	+	1.03016i
$906$		0			0
$907$	−17.7683			−0.589985			−0.294993	−	0.955500i	$-0.595317\pi$
$907$	−17.7683			−0.589985			−0.294993	+	0.955500i	$0.595317\pi$
$908$		0			0
$909$		−	94.1136i		−	3.12155i
$910$		0			0
$911$	29.4422			0.975462			0.487731	−	0.872994i	$-0.337825\pi$
$911$	29.4422			0.975462			0.487731	+	0.872994i	$0.337825\pi$
$912$		0			0
$913$		−	30.0770i		−	0.995402i
$914$		0			0
$915$	−27.2978	+	70.2736i	−0.902438	+	2.32317i
$916$		0			0
$917$	11.2901			0.372830
$918$		0			0
$919$	34.4685			1.13701			0.568506	−	0.822679i	$-0.307522\pi$
$919$	34.4685			1.13701			0.568506	+	0.822679i	$0.307522\pi$
$920$		0			0
$921$	5.63643			0.185727
$922$		0			0
$923$	0.251886			0.00829093
$924$		0			0
$925$	−41.9716	−	38.4024i	−1.38002	−	1.26266i
$926$		0			0
$927$			22.4183i			0.736312i
$928$		0			0
$929$	7.74358			0.254059			0.127029	−	0.991899i	$-0.459456\pi$
$929$	7.74358			0.254059			0.127029	+	0.991899i	$0.459456\pi$
$930$		0			0
$931$			2.16867i			0.0710752i
$932$		0			0
$933$	−72.2668			−2.36591
$934$		0			0
$935$	8.24379	+	3.20231i	0.269601	+	0.104727i
$936$		0			0
$937$			5.02639i			0.164205i	0.996624	+	0.0821025i	$0.0261635\pi$
$937$			5.02639i			0.164205i	−0.996624	+	0.0821025i	$0.973836\pi$
$938$		0			0
$939$			24.0262i			0.784067i
$940$		0			0
$941$			37.8331i			1.23332i	0.787228	+	0.616662i	$0.211516\pi$
$941$			37.8331i			1.23332i	−0.787228	+	0.616662i	$0.788484\pi$
$942$		0			0
$943$		−	12.6747i		−	0.412744i
$944$		0			0
$945$	9.29688	−	23.9332i	0.302427	−	0.778548i
$946$		0			0
$947$	−42.8009			−1.39084			−0.695421	−	0.718602i	$-0.744782\pi$
$947$	−42.8009			−1.39084			−0.695421	+	0.718602i	$0.744782\pi$
$948$		0			0
$949$			4.79244i			0.155569i
$950$		0			0
$951$	−75.7548			−2.45652
$952$		0			0
$953$			32.8118i			1.06288i	0.847096	+	0.531440i	$0.178349\pi$
$953$			32.8118i			1.06288i	−0.847096	+	0.531440i	$0.821651\pi$
$954$		0			0
$955$	0.924087	+	0.358962i	0.0299027	+	0.0116157i
$956$		0			0
$957$	43.3379			1.40092
$958$		0			0
$959$	13.5008			0.435964
$960$		0			0
$961$	7.75560			0.250181
$962$		0			0
$963$	−38.1293			−1.22870
$964$		0			0
$965$	10.7099	−	27.5709i	0.344765	−	0.887540i
$966$		0			0
$967$		−	60.6326i		−	1.94981i	−0.222616	−	0.974906i	$-0.571460\pi$
$967$		−	60.6326i		−	1.94981i	0.222616	−	0.974906i	$-0.428540\pi$
$968$		0			0
$969$	−10.0809			−0.323846
$970$		0			0
$971$			37.8702i			1.21531i	0.794201	+	0.607656i	$0.207890\pi$
$971$			37.8702i			1.21531i	−0.794201	+	0.607656i	$0.792110\pi$
$972$		0			0
$973$	21.8838			0.701563
$974$		0			0
$975$	5.50892	+	5.04046i	0.176427	+	0.161424i
$976$		0			0
$977$		−	8.57734i		−	0.274414i	−0.990542	−	0.137207i	$-0.956188\pi$
$977$		−	8.57734i		−	0.274414i	0.990542	−	0.137207i	$-0.0438125\pi$
$978$		0			0
$979$		−	7.83816i		−	0.250509i
$980$		0			0
$981$			98.4678i			3.14384i
$982$		0			0
$983$			26.6200i			0.849046i	0.905417	+	0.424523i	$0.139558\pi$
$983$			26.6200i			0.849046i	−0.905417	+	0.424523i	$0.860442\pi$
$984$		0			0
$985$	16.0809	+	6.24664i	0.512381	+	0.199035i
$986$		0			0
$987$	23.9332			0.761803
$988$		0			0
$989$			63.3040i			2.01295i
$990$		0			0
$991$	13.5221			0.429543			0.214771	−	0.976664i	$-0.431099\pi$
$991$	13.5221			0.429543			0.214771	+	0.976664i	$0.431099\pi$
$992$		0			0
$993$			86.0170i			2.72967i
$994$		0			0
$995$	19.7262	+	7.66266i	0.625363	+	0.242923i
$996$		0			0
$997$	21.0451			0.666504			0.333252	−	0.942838i	$-0.391854\pi$
$997$	21.0451			0.666504			0.333252	+	0.942838i	$0.391854\pi$
$998$		0			0
$999$	130.645			4.13342

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.b.1569.2	yes	12
4.3	odd	2	inner	2240.2.l.b.1569.12	yes	12
5.4	even	2		2240.2.l.a.1569.12	yes	12
8.3	odd	2		2240.2.l.a.1569.1	✓	12
8.5	even	2		2240.2.l.a.1569.11	yes	12
20.19	odd	2		2240.2.l.a.1569.2	yes	12
40.19	odd	2	inner	2240.2.l.b.1569.11	yes	12
40.29	even	2	inner	2240.2.l.b.1569.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.a.1569.1	✓	12	8.3	odd	2
2240.2.l.a.1569.2	yes	12	20.19	odd	2
2240.2.l.a.1569.11	yes	12	8.5	even	2
2240.2.l.a.1569.12	yes	12	5.4	even	2
2240.2.l.b.1569.1	yes	12	40.29	even	2	inner
2240.2.l.b.1569.2	yes	12	1.1	even	1	trivial
2240.2.l.b.1569.11	yes	12	40.19	odd	2	inner
2240.2.l.b.1569.12	yes	12	4.3	odd	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.11270 q^{3} +(2.08433 + 0.809661i) q^{5} +1.00000i q^{7} +6.68890 q^{9} +O(q^{10})\)	\(q-3.11270 q^{3} +(2.08433 + 0.809661i) q^{5} +1.00000i q^{7} +6.68890 q^{9} -2.64844i q^{11} -0.479769 q^{13} +(-6.48791 - 2.52023i) q^{15} +1.49338i q^{17} -2.16867i q^{19} -3.11270i q^{21} -6.33734i q^{23} +(3.68890 + 3.37521i) q^{25} -11.4824 q^{27} -5.25704i q^{29} +6.22540 q^{31} +8.24379i q^{33} +(-0.809661 + 2.08433i) q^{35} -11.3778 q^{37} +1.49338 q^{39} +2.00000 q^{41} -9.98906 q^{43} +(13.9419 + 5.41574i) q^{45} +7.68890i q^{47} -1.00000 q^{49} -4.64844i q^{51} +3.04046 q^{53} +(2.14434 - 5.52023i) q^{55} +6.75041i q^{57} +9.20913i q^{59} -10.8315i q^{61} +6.68890i q^{63} +(-1.00000 - 0.388450i) q^{65} +9.73717 q^{67} +19.7262i q^{69} -0.525015 q^{71} -9.98906i q^{73} +(-11.4824 - 10.5060i) q^{75} +2.64844 q^{77} -13.9442 q^{79} +15.6747 q^{81} +11.3565 q^{83} +(-1.20913 + 3.11270i) q^{85} +16.3636i q^{87} +2.95954 q^{89} -0.479769i q^{91} -19.3778 q^{93} +(1.75589 - 4.52023i) q^{95} -7.97066i q^{97} -17.7151i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{5} + 24 q^{9}+O(q^{10}) \) 12 * q + 4 * q^5 + 24 * q^9	\( 12 q + 4 q^{5} + 24 q^{9} - 20 q^{13} - 12 q^{25} - 24 q^{37} + 24 q^{41} + 48 q^{45} - 12 q^{49} + 8 q^{53} - 12 q^{65} + 4 q^{77} + 20 q^{81} + 56 q^{85} + 64 q^{89} - 120 q^{93}+O(q^{100}) \) 12 * q + 4 * q^5 + 24 * q^9 - 20 * q^13 - 12 * q^25 - 24 * q^37 + 24 * q^41 + 48 * q^45 - 12 * q^49 + 8 * q^53 - 12 * q^65 + 4 * q^77 + 20 * q^81 + 56 * q^85 + 64 * q^89 - 120 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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