LMFDB - Embedded newform 2240.2.k.f.1791.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1791,2240)]

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([1, 0, 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("2240.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.k (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:	$17.8864900528$
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} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} + \cdots + 5956 $ x^16 - 4x^15 - 6x^14 + 68x^13 - 126x^12 - 148x^11 + 1006x^10 - 1516x^9 - 179x^8 + 3992x^7 - 5596x^6 - 488x^5 + 16080x^4 - 33776x^3 + 37344x^2 - 22336*x + 5956
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 1120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.9
Root			$-3.01485 + 0.0876273i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.f.1791.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.175255 q^{3} -1.00000i q^{5} +(0.684520 + 2.55567i) q^{7} -2.96929 q^{9} +O(q^{10})$	$q-0.175255 q^{3} -1.00000i q^{5} +(0.684520 + 2.55567i) q^{7} -2.96929 q^{9} +0.592758i q^{11} -0.918367i q^{13} +0.175255i q^{15} +4.66066i q^{17} +7.36557 q^{19} +(-0.119965 - 0.447892i) q^{21} -5.99653i q^{23} -1.00000 q^{25} +1.04614 q^{27} -0.178356 q^{29} -5.80152 q^{31} -0.103884i q^{33} +(2.55567 - 0.684520i) q^{35} -5.77996 q^{37} +0.160948i q^{39} +11.8435i q^{41} +3.02292i q^{43} +2.96929i q^{45} -8.37738 q^{47} +(-6.06286 + 3.49881i) q^{49} -0.816802i q^{51} +7.66033 q^{53} +0.592758 q^{55} -1.29085 q^{57} -8.01640 q^{59} +9.66990i q^{61} +(-2.03254 - 7.58850i) q^{63} -0.918367 q^{65} +3.12911i q^{67} +1.05092i q^{69} +7.70266i q^{71} +13.7867i q^{73} +0.175255 q^{75} +(-1.51489 + 0.405755i) q^{77} +2.07898i q^{79} +8.72452 q^{81} +5.59308 q^{83} +4.66066 q^{85} +0.0312576 q^{87} +4.95531i q^{89} +(2.34704 - 0.628641i) q^{91} +1.01674 q^{93} -7.36557i q^{95} +6.76250i q^{97} -1.76007i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^7 + 16 * q^9	$ 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) $ 16 * q - 4 * q^7 + 16 * q^9 - 8 * q^19 - 4 * q^21 - 16 * q^25 + 48 * q^27 - 8 * q^29 - 16 * q^37 - 8 * q^47 - 4 * q^49 - 16 * q^53 + 8 * q^55 + 16 * q^57 - 8 * q^59 - 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 + 64 * q^83 - 16 * q^87 - 64 * q^91 + 48 * 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
$3$	−0.175255			−0.101183			−0.0505916	−	0.998719i	$-0.516111\pi$
$3$	−0.175255			−0.101183			−0.0505916	+	0.998719i	$0.516111\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	0.684520	+	2.55567i	0.258724	+	0.965951i
$7$	0.684520	+	2.55567i	0.258724	+	0.965951i
$8$		0			0
$9$	−2.96929			−0.989762
$10$		0			0
$11$			0.592758i			0.178723i	0.995999	+	0.0893617i	$0.0284827\pi$
$11$			0.592758i			0.178723i	−0.995999	+	0.0893617i	$0.971517\pi$
$12$		0			0
$13$		−	0.918367i		−	0.254709i	−0.991857	−	0.127355i	$-0.959351\pi$
$13$		−	0.918367i		−	0.254709i	0.991857	−	0.127355i	$-0.0406486\pi$
$14$		0			0
$15$			0.175255i			0.0452505i
$16$		0			0
$17$			4.66066i			1.13038i	0.824962	+	0.565188i	$0.191196\pi$
$17$			4.66066i			1.13038i	−0.824962	+	0.565188i	$0.808804\pi$
$18$		0			0
$19$	7.36557			1.68978			0.844889	−	0.534942i	$-0.179666\pi$
$19$	7.36557			1.68978			0.844889	+	0.534942i	$0.179666\pi$
$20$		0			0
$21$	−0.119965	−	0.447892i	−0.0261786	−	0.0977381i
$22$		0			0
$23$		−	5.99653i		−	1.25036i	−0.780479	−	0.625181i	$-0.785025\pi$
$23$		−	5.99653i		−	1.25036i	0.780479	−	0.625181i	$-0.214975\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.04614			0.201331
$28$		0			0
$29$	−0.178356			−0.0331198			−0.0165599	−	0.999863i	$-0.505271\pi$
$29$	−0.178356			−0.0331198			−0.0165599	+	0.999863i	$0.505271\pi$
$30$		0			0
$31$	−5.80152			−1.04198			−0.520992	−	0.853562i	$-0.674438\pi$
$31$	−5.80152			−1.04198			−0.520992	+	0.853562i	$0.674438\pi$
$32$		0			0
$33$		−	0.103884i		−	0.0180838i
$34$		0			0
$35$	2.55567	−	0.684520i	0.431987	−	0.115705i
$36$		0			0
$37$	−5.77996			−0.950220			−0.475110	−	0.879926i	$-0.657592\pi$
$37$	−5.77996			−0.950220			−0.475110	+	0.879926i	$0.657592\pi$
$38$		0			0
$39$			0.160948i			0.0257723i
$40$		0			0
$41$			11.8435i			1.84964i	0.380402	+	0.924821i	$0.375786\pi$
$41$			11.8435i			1.84964i	−0.380402	+	0.924821i	$0.624214\pi$
$42$		0			0
$43$			3.02292i			0.460991i	0.973073	+	0.230495i	$0.0740346\pi$
$43$			3.02292i			0.460991i	−0.973073	+	0.230495i	$0.925965\pi$
$44$		0			0
$45$			2.96929i			0.442635i
$46$		0			0
$47$	−8.37738			−1.22197			−0.610984	−	0.791643i	$-0.709226\pi$
$47$	−8.37738			−1.22197			−0.610984	+	0.791643i	$0.709226\pi$
$48$		0			0
$49$	−6.06286	+	3.49881i	−0.866123	+	0.499830i
$50$		0			0
$51$		−	0.816802i		−	0.114375i
$52$		0			0
$53$	7.66033			1.05223			0.526113	−	0.850414i	$-0.323649\pi$
$53$	7.66033			1.05223			0.526113	+	0.850414i	$0.323649\pi$
$54$		0			0
$55$	0.592758			0.0799275
$56$		0			0
$57$	−1.29085			−0.170977
$58$		0			0
$59$	−8.01640			−1.04365			−0.521823	−	0.853054i	$-0.674748\pi$
$59$	−8.01640			−1.04365			−0.521823	+	0.853054i	$0.674748\pi$
$60$		0			0
$61$			9.66990i			1.23810i	0.785350	+	0.619052i	$0.212483\pi$
$61$			9.66990i			1.23810i	−0.785350	+	0.619052i	$0.787517\pi$
$62$		0			0
$63$	−2.03254	−	7.58850i	−0.256076	−	0.956062i
$64$		0			0
$65$	−0.918367			−0.113909
$66$		0			0
$67$			3.12911i			0.382282i	0.981563	+	0.191141i	$0.0612187\pi$
$67$			3.12911i			0.382282i	−0.981563	+	0.191141i	$0.938781\pi$
$68$		0			0
$69$			1.05092i			0.126516i
$70$		0			0
$71$			7.70266i			0.914137i	0.889431	+	0.457069i	$0.151101\pi$
$71$			7.70266i			0.914137i	−0.889431	+	0.457069i	$0.848899\pi$
$72$		0			0
$73$			13.7867i			1.61361i	0.590815	+	0.806807i	$0.298806\pi$
$73$			13.7867i			1.61361i	−0.590815	+	0.806807i	$0.701194\pi$
$74$		0			0
$75$	0.175255			0.0202367
$76$		0			0
$77$	−1.51489	+	0.405755i	−0.172638	+	0.0462401i
$78$		0			0
$79$			2.07898i			0.233904i	0.993138	+	0.116952i	$0.0373123\pi$
$79$			2.07898i			0.233904i	−0.993138	+	0.116952i	$0.962688\pi$
$80$		0			0
$81$	8.72452			0.969391
$82$		0			0
$83$	5.59308			0.613921			0.306960	−	0.951722i	$-0.400688\pi$
$83$	5.59308			0.613921			0.306960	+	0.951722i	$0.400688\pi$
$84$		0			0
$85$	4.66066			0.505519
$86$		0			0
$87$	0.0312576			0.00335117
$88$		0			0
$89$			4.95531i			0.525262i	0.964896	+	0.262631i	$0.0845902\pi$
$89$			4.95531i			0.525262i	−0.964896	+	0.262631i	$0.915410\pi$
$90$		0			0
$91$	2.34704	−	0.628641i	0.246037	−	0.0658994i
$92$		0			0
$93$	1.01674			0.105431
$94$		0			0
$95$		−	7.36557i		−	0.755692i
$96$		0			0
$97$			6.76250i			0.686628i	0.939221	+	0.343314i	$0.111549\pi$
$97$			6.76250i			0.686628i	−0.939221	+	0.343314i	$0.888451\pi$
$98$		0			0
$99$		−	1.76007i		−	0.176894i
$100$		0			0
$101$		−	10.4782i		−	1.04262i	−0.853368	−	0.521308i	$-0.825444\pi$
$101$		−	10.4782i		−	1.04262i	0.853368	−	0.521308i	$-0.174556\pi$
$102$		0			0
$103$	8.82237			0.869294			0.434647	−	0.900601i	$-0.356873\pi$
$103$	8.82237			0.869294			0.434647	+	0.900601i	$0.356873\pi$
$104$		0			0
$105$	−0.447892	+	0.119965i	−0.0437098	+	0.0117074i
$106$		0			0
$107$		−	7.60897i		−	0.735587i	−0.929907	−	0.367794i	$-0.880113\pi$
$107$		−	7.60897i		−	0.735587i	0.929907	−	0.367794i	$-0.119887\pi$
$108$		0			0
$109$	−15.5149			−1.48606			−0.743029	−	0.669259i	$-0.766612\pi$
$109$	−15.5149			−1.48606			−0.743029	+	0.669259i	$0.766612\pi$
$110$		0			0
$111$	1.01297			0.0961464
$112$		0			0
$113$	−13.7479			−1.29329			−0.646646	−	0.762790i	$-0.723829\pi$
$113$	−13.7479			−1.29329			−0.646646	+	0.762790i	$0.723829\pi$
$114$		0			0
$115$	−5.99653			−0.559179
$116$		0			0
$117$			2.72689i			0.252101i
$118$		0			0
$119$	−11.9111	+	3.19032i	−1.09189	+	0.292456i
$120$		0			0
$121$	10.6486			0.968058
$122$		0			0
$123$		−	2.07563i		−	0.187153i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	2.64272i		−	0.234504i	−0.993102	−	0.117252i	$-0.962592\pi$
$127$		−	2.64272i		−	0.234504i	0.993102	−	0.117252i	$-0.0374085\pi$
$128$		0			0
$129$		−	0.529780i		−	0.0466445i
$130$		0			0
$131$	15.2007			1.32809			0.664044	−	0.747693i	$-0.268839\pi$
$131$	15.2007			1.32809			0.664044	+	0.747693i	$0.268839\pi$
$132$		0			0
$133$	5.04188	+	18.8239i	0.437187	+	1.63224i
$134$		0			0
$135$		−	1.04614i		−	0.0900378i
$136$		0			0
$137$	−12.0409			−1.02873			−0.514363	−	0.857572i	$-0.671972\pi$
$137$	−12.0409			−1.02873			−0.514363	+	0.857572i	$0.671972\pi$
$138$		0			0
$139$	0.262345			0.0222518			0.0111259	−	0.999938i	$-0.496458\pi$
$139$	0.262345			0.0222518			0.0111259	+	0.999938i	$0.496458\pi$
$140$		0			0
$141$	1.46817			0.123643
$142$		0			0
$143$	0.544370			0.0455225
$144$		0			0
$145$			0.178356i			0.0148116i
$146$		0			0
$147$	1.06254	−	0.613183i	0.0876372	−	0.0505745i
$148$		0			0
$149$	−1.80523			−0.147891			−0.0739453	−	0.997262i	$-0.523559\pi$
$149$	−1.80523			−0.147891			−0.0739453	+	0.997262i	$0.523559\pi$
$150$		0			0
$151$			13.3315i			1.08490i	0.840087	+	0.542451i	$0.182504\pi$
$151$			13.3315i			1.08490i	−0.840087	+	0.542451i	$0.817496\pi$
$152$		0			0
$153$		−	13.8388i		−	1.11880i
$154$		0			0
$155$			5.80152i			0.465989i
$156$		0			0
$157$		−	1.82912i		−	0.145980i	−0.997333	−	0.0729900i	$-0.976746\pi$
$157$		−	1.82912i		−	0.145980i	0.997333	−	0.0729900i	$-0.0232541\pi$
$158$		0			0
$159$	−1.34251			−0.106468
$160$		0			0
$161$	15.3251	−	4.10475i	1.20779	−	0.323499i
$162$		0			0
$163$			15.9737i			1.25116i	0.780161	+	0.625579i	$0.215137\pi$
$163$			15.9737i			1.25116i	−0.780161	+	0.625579i	$0.784863\pi$
$164$		0			0
$165$	−0.103884			−0.00808733
$166$		0			0
$167$	16.4935			1.27631			0.638154	−	0.769909i	$-0.279698\pi$
$167$	16.4935			1.27631			0.638154	+	0.769909i	$0.279698\pi$
$168$		0			0
$169$	12.1566			0.935123
$170$		0			0
$171$	−21.8705			−1.67248
$172$		0			0
$173$			9.31174i			0.707958i	0.935253	+	0.353979i	$0.115172\pi$
$173$			9.31174i			0.707958i	−0.935253	+	0.353979i	$0.884828\pi$
$174$		0			0
$175$	−0.684520	−	2.55567i	−0.0517449	−	0.193190i
$176$		0			0
$177$	1.40491			0.105600
$178$		0			0
$179$			8.65612i			0.646988i	0.946230	+	0.323494i	$0.104858\pi$
$179$			8.65612i			0.646988i	−0.946230	+	0.323494i	$0.895142\pi$
$180$		0			0
$181$			17.0374i			1.26638i	0.773996	+	0.633190i	$0.218255\pi$
$181$			17.0374i			1.26638i	−0.773996	+	0.633190i	$0.781745\pi$
$182$		0			0
$183$		−	1.69469i		−	0.125275i
$184$		0			0
$185$			5.77996i			0.424951i
$186$		0			0
$187$	−2.76265			−0.202025
$188$		0			0
$189$	0.716107	+	2.67360i	0.0520891	+	0.194476i
$190$		0			0
$191$			5.63997i			0.408094i	0.978961	+	0.204047i	$0.0654095\pi$
$191$			5.63997i			0.408094i	−0.978961	+	0.204047i	$0.934590\pi$
$192$		0			0
$193$	−16.6423			−1.19794			−0.598971	−	0.800771i	$-0.704424\pi$
$193$	−16.6423			−1.19794			−0.598971	+	0.800771i	$0.704424\pi$
$194$		0			0
$195$	0.160948			0.0115257
$196$		0			0
$197$	−22.4493			−1.59945			−0.799724	−	0.600368i	$-0.795021\pi$
$197$	−22.4493			−1.59945			−0.799724	+	0.600368i	$0.795021\pi$
$198$		0			0
$199$	19.6785			1.39497			0.697484	−	0.716600i	$-0.254303\pi$
$199$	19.6785			1.39497			0.697484	+	0.716600i	$0.254303\pi$
$200$		0			0
$201$		−	0.548391i		−	0.0386805i
$202$		0			0
$203$	−0.122088	−	0.455818i	−0.00856890	−	0.0319921i
$204$		0			0
$205$	11.8435			0.827185
$206$		0			0
$207$			17.8054i			1.23756i
$208$		0			0
$209$			4.36600i			0.302003i
$210$		0			0
$211$		−	28.3685i		−	1.95297i	−0.215592	−	0.976484i	$-0.569168\pi$
$211$		−	28.3685i		−	1.95297i	0.215592	−	0.976484i	$-0.430832\pi$
$212$		0			0
$213$		−	1.34993i		−	0.0924954i
$214$		0			0
$215$	3.02292			0.206161
$216$		0			0
$217$	−3.97126	−	14.8267i	−0.269586	−	1.00651i
$218$		0			0
$219$		−	2.41619i		−	0.163271i
$220$		0			0
$221$	4.28019			0.287917
$222$		0			0
$223$	12.2006			0.817009			0.408505	−	0.912756i	$-0.366050\pi$
$223$	12.2006			0.817009			0.408505	+	0.912756i	$0.366050\pi$
$224$		0			0
$225$	2.96929			0.197952
$226$		0			0
$227$	−10.1178			−0.671545			−0.335772	−	0.941943i	$-0.608997\pi$
$227$	−10.1178			−0.671545			−0.335772	+	0.941943i	$0.608997\pi$
$228$		0			0
$229$			14.3809i			0.950316i	0.879900	+	0.475158i	$0.157609\pi$
$229$			14.3809i			0.950316i	−0.879900	+	0.475158i	$0.842391\pi$
$230$		0			0
$231$	0.265492	−	0.0711105i	0.0174681	−	0.00467872i
$232$		0			0
$233$	−16.0223			−1.04966			−0.524829	−	0.851208i	$-0.675871\pi$
$233$	−16.0223			−1.04966			−0.524829	+	0.851208i	$0.675871\pi$
$234$		0			0
$235$			8.37738i			0.546480i
$236$		0			0
$237$		−	0.364351i		−	0.0236672i
$238$		0			0
$239$		−	13.3532i		−	0.863745i	−0.901935	−	0.431873i	$-0.857853\pi$
$239$		−	13.3532i		−	0.863745i	0.901935	−	0.431873i	$-0.142147\pi$
$240$		0			0
$241$			11.9377i			0.768976i	0.923130	+	0.384488i	$0.125622\pi$
$241$			11.9377i			0.768976i	−0.923130	+	0.384488i	$0.874378\pi$
$242$		0			0
$243$	−4.66744			−0.299417
$244$		0			0
$245$	3.49881	+	6.06286i	0.223531	+	0.387342i
$246$		0			0
$247$		−	6.76429i		−	0.430402i
$248$		0			0
$249$	−0.980213			−0.0621185
$250$		0			0
$251$	6.82359			0.430701			0.215351	−	0.976537i	$-0.430911\pi$
$251$	6.82359			0.430701			0.215351	+	0.976537i	$0.430911\pi$
$252$		0			0
$253$	3.55449			0.223469
$254$		0			0
$255$	−0.816802			−0.0511501
$256$		0			0
$257$		−	10.6984i		−	0.667348i	−0.942689	−	0.333674i	$-0.891712\pi$
$257$		−	10.6984i		−	0.667348i	0.942689	−	0.333674i	$-0.108288\pi$
$258$		0			0
$259$	−3.95650	−	14.7717i	−0.245845	−	0.917866i
$260$		0			0
$261$	0.529589			0.0327807
$262$		0			0
$263$		−	14.3268i		−	0.883428i	−0.897156	−	0.441714i	$-0.854371\pi$
$263$		−	14.3268i		−	0.883428i	0.897156	−	0.441714i	$-0.145629\pi$
$264$		0			0
$265$		−	7.66033i		−	0.470570i
$266$		0			0
$267$		−	0.868441i		−	0.0531478i
$268$		0			0
$269$		−	11.8502i		−	0.722522i	−0.932465	−	0.361261i	$-0.882346\pi$
$269$		−	11.8502i		−	0.722522i	0.932465	−	0.361261i	$-0.117654\pi$
$270$		0			0
$271$	−2.81616			−0.171069			−0.0855347	−	0.996335i	$-0.527260\pi$
$271$	−2.81616			−0.171069			−0.0855347	+	0.996335i	$0.527260\pi$
$272$		0			0
$273$	−0.411329	+	0.110172i	−0.0248948	+	0.00666792i
$274$		0			0
$275$		−	0.592758i		−	0.0357447i
$276$		0			0
$277$	−19.2082			−1.15411			−0.577053	−	0.816707i	$-0.695797\pi$
$277$	−19.2082			−1.15411			−0.577053	+	0.816707i	$0.695797\pi$
$278$		0			0
$279$	17.2264			1.03132
$280$		0			0
$281$	13.7001			0.817279			0.408639	−	0.912696i	$-0.366003\pi$
$281$	13.7001			0.817279			0.408639	+	0.912696i	$0.366003\pi$
$282$		0			0
$283$	−32.3170			−1.92105			−0.960524	−	0.278197i	$-0.910263\pi$
$283$	−32.3170			−1.92105			−0.960524	+	0.278197i	$0.910263\pi$
$284$		0			0
$285$			1.29085i			0.0764633i
$286$		0			0
$287$	−30.2680	+	8.10711i	−1.78666	+	0.478547i
$288$		0			0
$289$	−4.72175			−0.277750
$290$		0			0
$291$		−	1.18516i		−	0.0694752i
$292$		0			0
$293$		−	15.9329i		−	0.930812i	−0.885097	−	0.465406i	$-0.845908\pi$
$293$		−	15.9329i		−	0.930812i	0.885097	−	0.465406i	$-0.154092\pi$
$294$		0			0
$295$			8.01640i			0.466733i
$296$		0			0
$297$			0.620111i			0.0359825i
$298$		0			0
$299$	−5.50701			−0.318479
$300$		0			0
$301$	−7.72557	+	2.06925i	−0.445294	+	0.119269i
$302$		0			0
$303$			1.83635i			0.105495i
$304$		0			0
$305$	9.66990			0.553697
$306$		0			0
$307$	12.4507			0.710599			0.355299	−	0.934753i	$-0.384379\pi$
$307$	12.4507			0.710599			0.355299	+	0.934753i	$0.384379\pi$
$308$		0			0
$309$	−1.54616			−0.0879580
$310$		0			0
$311$	−0.139752			−0.00792461			−0.00396230	−	0.999992i	$-0.501261\pi$
$311$	−0.139752			−0.00792461			−0.00396230	+	0.999992i	$0.501261\pi$
$312$		0			0
$313$		−	21.0649i		−	1.19066i	−0.803481	−	0.595330i	$-0.797021\pi$
$313$		−	21.0649i		−	1.19066i	0.803481	−	0.595330i	$-0.202979\pi$
$314$		0			0
$315$	−7.58850	+	2.03254i	−0.427564	+	0.114520i
$316$		0			0
$317$	−16.7576			−0.941201			−0.470601	−	0.882346i	$-0.655963\pi$
$317$	−16.7576			−0.941201			−0.470601	+	0.882346i	$0.655963\pi$
$318$		0			0
$319$		−	0.105722i		−	0.00591929i
$320$		0			0
$321$			1.33351i			0.0744291i
$322$		0			0
$323$			34.3284i			1.91008i
$324$		0			0
$325$			0.918367i			0.0509418i
$326$		0			0
$327$	2.71906			0.150364
$328$		0			0
$329$	−5.73449	−	21.4098i	−0.316153	−	1.18036i
$330$		0			0
$331$		−	21.6412i		−	1.18951i	−0.803908	−	0.594753i	$-0.797250\pi$
$331$		−	21.6412i		−	1.18951i	0.803908	−	0.594753i	$-0.202750\pi$
$332$		0			0
$333$	17.1624			0.940492
$334$		0			0
$335$	3.12911			0.170962
$336$		0			0
$337$	1.47241			0.0802074			0.0401037	−	0.999196i	$-0.487231\pi$
$337$	1.47241			0.0802074			0.0401037	+	0.999196i	$0.487231\pi$
$338$		0			0
$339$	2.40938			0.130860
$340$		0			0
$341$		−	3.43890i		−	0.186227i
$342$		0			0
$343$	−13.0919	−	13.0997i	−0.706899	−	0.707315i
$344$		0			0
$345$	1.05092			0.0565796
$346$		0			0
$347$		−	29.8364i		−	1.60170i	−0.598863	−	0.800851i	$-0.704381\pi$
$347$		−	29.8364i		−	1.60170i	0.598863	−	0.800851i	$-0.295619\pi$
$348$		0			0
$349$		−	3.61540i		−	0.193528i	−0.995307	−	0.0967639i	$-0.969151\pi$
$349$		−	3.61540i		−	0.193528i	0.995307	−	0.0967639i	$-0.0308492\pi$
$350$		0			0
$351$		−	0.960744i		−	0.0512807i
$352$		0			0
$353$		−	14.0808i		−	0.749444i	−0.927137	−	0.374722i	$-0.877738\pi$
$353$		−	14.0808i		−	0.749444i	0.927137	−	0.374722i	$-0.122262\pi$
$354$		0			0
$355$	7.70266			0.408815
$356$		0			0
$357$	2.08747	−	0.559117i	0.110481	−	0.0295916i
$358$		0			0
$359$			23.5512i			1.24298i	0.783420	+	0.621492i	$0.213473\pi$
$359$			23.5512i			1.24298i	−0.783420	+	0.621492i	$0.786527\pi$
$360$		0			0
$361$	35.2516			1.85535
$362$		0			0
$363$	−1.86622			−0.0979513
$364$		0			0
$365$	13.7867			0.721630
$366$		0			0
$367$	21.6605			1.13067			0.565335	−	0.824861i	$-0.308747\pi$
$367$	21.6605			1.13067			0.565335	+	0.824861i	$0.308747\pi$
$368$		0			0
$369$		−	35.1667i		−	1.83071i
$370$		0			0
$371$	5.24365	+	19.5772i	0.272237	+	1.01640i
$372$		0			0
$373$	−2.20377			−0.114107			−0.0570535	−	0.998371i	$-0.518171\pi$
$373$	−2.20377			−0.114107			−0.0570535	+	0.998371i	$0.518171\pi$
$374$		0			0
$375$		−	0.175255i		−	0.00905011i
$376$		0			0
$377$			0.163796i			0.00843592i
$378$		0			0
$379$			5.42429i			0.278627i	0.990248	+	0.139313i	$0.0444896\pi$
$379$			5.42429i			0.278627i	−0.990248	+	0.139313i	$0.955510\pi$
$380$		0			0
$381$			0.463149i			0.0237278i
$382$		0			0
$383$	−35.7105			−1.82472			−0.912361	−	0.409387i	$-0.865743\pi$
$383$	−35.7105			−1.82472			−0.912361	+	0.409387i	$0.865743\pi$
$384$		0			0
$385$	0.405755	+	1.51489i	0.0206792	+	0.0772061i
$386$		0			0
$387$		−	8.97591i		−	0.456271i
$388$		0			0
$389$	17.9311			0.909143			0.454572	−	0.890710i	$-0.349792\pi$
$389$	17.9311			0.909143			0.454572	+	0.890710i	$0.349792\pi$
$390$		0			0
$391$	27.9478			1.41338
$392$		0			0
$393$	−2.66399			−0.134380
$394$		0			0
$395$	2.07898			0.104605
$396$		0			0
$397$			3.98481i			0.199992i	0.994988	+	0.0999960i	$0.0318830\pi$
$397$			3.98481i			0.199992i	−0.994988	+	0.0999960i	$0.968117\pi$
$398$		0			0
$399$	−0.883613	−	3.29898i	−0.0442360	−	0.165156i
$400$		0			0
$401$	18.2612			0.911920			0.455960	−	0.890000i	$-0.349296\pi$
$401$	18.2612			0.911920			0.455960	+	0.890000i	$0.349296\pi$
$402$		0			0
$403$			5.32792i			0.265403i
$404$		0			0
$405$		−	8.72452i		−	0.433525i
$406$		0			0
$407$		−	3.42612i		−	0.169827i
$408$		0			0
$409$			27.3567i			1.35270i	0.736581	+	0.676350i	$0.236439\pi$
$409$			27.3567i			1.35270i	−0.736581	+	0.676350i	$0.763561\pi$
$410$		0			0
$411$	2.11023			0.104090
$412$		0			0
$413$	−5.48739	−	20.4872i	−0.270017	−	1.00811i
$414$		0			0
$415$		−	5.59308i		−	0.274554i
$416$		0			0
$417$	−0.0459771			−0.00225151
$418$		0			0
$419$	28.9443			1.41402			0.707012	−	0.707202i	$-0.250043\pi$
$419$	28.9443			1.41402			0.707012	+	0.707202i	$0.250043\pi$
$420$		0			0
$421$	37.0861			1.80747			0.903733	−	0.428096i	$-0.140815\pi$
$421$	37.0861			1.80747			0.903733	+	0.428096i	$0.140815\pi$
$422$		0			0
$423$	24.8748			1.20946
$424$		0			0
$425$		−	4.66066i		−	0.226075i
$426$		0			0
$427$	−24.7130	+	6.61924i	−1.19595	+	0.320328i
$428$		0			0
$429$	−0.0954033			−0.00460611
$430$		0			0
$431$		−	9.95359i		−	0.479448i	−0.970841	−	0.239724i	$-0.922943\pi$
$431$		−	9.95359i		−	0.479448i	0.970841	−	0.239724i	$-0.0770569\pi$
$432$		0			0
$433$			2.31840i			0.111415i	0.998447	+	0.0557075i	$0.0177414\pi$
$433$			2.31840i			0.111415i	−0.998447	+	0.0557075i	$0.982259\pi$
$434$		0			0
$435$		−	0.0312576i		−	0.00149869i
$436$		0			0
$437$		−	44.1679i		−	2.11284i
$438$		0			0
$439$	−11.3966			−0.543930			−0.271965	−	0.962307i	$-0.587674\pi$
$439$	−11.3966			−0.543930			−0.271965	+	0.962307i	$0.587674\pi$
$440$		0			0
$441$	18.0024	−	10.3890i	0.857256	−	0.494713i
$442$		0			0
$443$			15.1643i			0.720480i	0.932860	+	0.360240i	$0.117305\pi$
$443$			15.1643i			0.720480i	−0.932860	+	0.360240i	$0.882695\pi$
$444$		0			0
$445$	4.95531			0.234904
$446$		0			0
$447$	0.316375			0.0149640
$448$		0			0
$449$	−9.56741			−0.451514			−0.225757	−	0.974184i	$-0.572486\pi$
$449$	−9.56741			−0.451514			−0.225757	+	0.974184i	$0.572486\pi$
$450$		0			0
$451$	−7.02033			−0.330574
$452$		0			0
$453$		−	2.33641i		−	0.109774i
$454$		0			0
$455$	−0.628641	−	2.34704i	−0.0294711	−	0.110031i
$456$		0			0
$457$	−20.8894			−0.977164			−0.488582	−	0.872518i	$-0.662486\pi$
$457$	−20.8894			−0.977164			−0.488582	+	0.872518i	$0.662486\pi$
$458$		0			0
$459$			4.87572i			0.227579i
$460$		0			0
$461$		−	31.1127i		−	1.44906i	−0.689242	−	0.724531i	$-0.742057\pi$
$461$		−	31.1127i		−	1.44906i	0.689242	−	0.724531i	$-0.257943\pi$
$462$		0			0
$463$			9.86624i			0.458523i	0.973365	+	0.229261i	$0.0736311\pi$
$463$			9.86624i			0.458523i	−0.973365	+	0.229261i	$0.926369\pi$
$464$		0			0
$465$		−	1.01674i		−	0.0471503i
$466$		0			0
$467$	38.4039			1.77712			0.888560	−	0.458761i	$-0.151706\pi$
$467$	38.4039			1.77712			0.888560	+	0.458761i	$0.151706\pi$
$468$		0			0
$469$	−7.99696	+	2.14194i	−0.369265	+	0.0989056i
$470$		0			0
$471$			0.320562i			0.0147707i
$472$		0			0
$473$	−1.79186			−0.0823898
$474$		0			0
$475$	−7.36557			−0.337956
$476$		0			0
$477$	−22.7457			−1.04145
$478$		0			0
$479$	−39.6879			−1.81339			−0.906693	−	0.421790i	$-0.861402\pi$
$479$	−39.6879			−1.81339			−0.906693	+	0.421790i	$0.861402\pi$
$480$		0			0
$481$			5.30813i			0.242030i
$482$		0			0
$483$	−2.68580	+	0.719376i	−0.122208	+	0.0327327i
$484$		0			0
$485$	6.76250			0.307069
$486$		0			0
$487$		−	6.19642i		−	0.280787i	−0.990096	−	0.140393i	$-0.955163\pi$
$487$		−	6.19642i		−	0.280787i	0.990096	−	0.140393i	$-0.0448367\pi$
$488$		0			0
$489$		−	2.79947i		−	0.126596i
$490$		0			0
$491$		−	35.9414i		−	1.62201i	−0.585037	−	0.811007i	$-0.698920\pi$
$491$		−	35.9414i		−	1.62201i	0.585037	−	0.811007i	$-0.301080\pi$
$492$		0			0
$493$		−	0.831255i		−	0.0374378i
$494$		0			0
$495$	−1.76007			−0.0791092
$496$		0			0
$497$	−19.6854	+	5.27263i	−0.883012	+	0.236510i
$498$		0			0
$499$		−	3.20678i		−	0.143555i	−0.997421	−	0.0717776i	$-0.977133\pi$
$499$		−	3.20678i		−	0.143555i	0.997421	−	0.0717776i	$-0.0228672\pi$
$500$		0			0
$501$	−2.89057			−0.129141
$502$		0			0
$503$	35.1483			1.56718			0.783592	−	0.621276i	$-0.213385\pi$
$503$	35.1483			1.56718			0.783592	+	0.621276i	$0.213385\pi$
$504$		0			0
$505$	−10.4782			−0.466272
$506$		0			0
$507$	−2.13050			−0.0946188
$508$		0			0
$509$		−	17.5919i		−	0.779745i	−0.920869	−	0.389873i	$-0.872519\pi$
$509$		−	17.5919i		−	0.779745i	0.920869	−	0.389873i	$-0.127481\pi$
$510$		0			0
$511$	−35.2343	+	9.43729i	−1.55867	+	0.417481i
$512$		0			0
$513$	7.70545			0.340204
$514$		0			0
$515$		−	8.82237i		−	0.388760i
$516$		0			0
$517$		−	4.96577i		−	0.218394i
$518$		0			0
$519$		−	1.63192i		−	0.0716335i
$520$		0			0
$521$			10.8837i			0.476822i	0.971164	+	0.238411i	$0.0766265\pi$
$521$			10.8837i			0.476822i	−0.971164	+	0.238411i	$0.923374\pi$
$522$		0			0
$523$	32.9897			1.44254			0.721269	−	0.692655i	$-0.243559\pi$
$523$	32.9897			1.44254			0.721269	+	0.692655i	$0.243559\pi$
$524$		0			0
$525$	0.119965	+	0.447892i	0.00523572	+	0.0195476i
$526$		0			0
$527$		−	27.0389i		−	1.17783i
$528$		0			0
$529$	−12.9584			−0.563407
$530$		0			0
$531$	23.8030			1.03296
$532$		0			0
$533$	10.8767			0.471121
$534$		0			0
$535$	−7.60897			−0.328965
$536$		0			0
$537$		−	1.51702i		−	0.0654644i
$538$		0			0
$539$	−2.07395	−	3.59381i	−0.0893314	−	0.154797i
$540$		0			0
$541$	32.9719			1.41757			0.708786	−	0.705423i	$-0.249243\pi$
$541$	32.9719			1.41757			0.708786	+	0.705423i	$0.249243\pi$
$542$		0			0
$543$		−	2.98588i		−	0.128137i
$544$		0			0
$545$			15.5149i			0.664586i
$546$		0			0
$547$			41.1285i			1.75853i	0.476337	+	0.879263i	$0.341964\pi$
$547$			41.1285i			1.75853i	−0.476337	+	0.879263i	$0.658036\pi$
$548$		0			0
$549$		−	28.7127i		−	1.22543i
$550$		0			0
$551$	−1.31369			−0.0559651
$552$		0			0
$553$	−5.31319	+	1.42311i	−0.225940	+	0.0605166i
$554$		0			0
$555$		−	1.01297i		−	0.0429980i
$556$		0			0
$557$	10.1818			0.431417			0.215709	−	0.976458i	$-0.430794\pi$
$557$	10.1818			0.431417			0.215709	+	0.976458i	$0.430794\pi$
$558$		0			0
$559$	2.77615			0.117418
$560$		0			0
$561$	0.484166			0.0204415
$562$		0			0
$563$	−1.18055			−0.0497543			−0.0248771	−	0.999691i	$-0.507919\pi$
$563$	−1.18055			−0.0497543			−0.0248771	+	0.999691i	$0.507919\pi$
$564$		0			0
$565$			13.7479i			0.578378i
$566$		0			0
$567$	5.97211	+	22.2970i	0.250805	+	0.936384i
$568$		0			0
$569$	−2.13871			−0.0896595			−0.0448298	−	0.998995i	$-0.514275\pi$
$569$	−2.13871			−0.0896595			−0.0448298	+	0.998995i	$0.514275\pi$
$570$		0			0
$571$			31.4475i			1.31604i	0.753002	+	0.658018i	$0.228605\pi$
$571$			31.4475i			1.31604i	−0.753002	+	0.658018i	$0.771395\pi$
$572$		0			0
$573$		−	0.988430i		−	0.0412923i
$574$		0			0
$575$			5.99653i			0.250073i
$576$		0			0
$577$		−	12.5682i		−	0.523223i	−0.965173	−	0.261611i	$-0.915746\pi$
$577$		−	12.5682i		−	0.523223i	0.965173	−	0.261611i	$-0.0842539\pi$
$578$		0			0
$579$	2.91665			0.121212
$580$		0			0
$581$	3.82858	+	14.2941i	0.158836	+	0.593017i
$582$		0			0
$583$			4.54072i			0.188058i
$584$		0			0
$585$	2.72689			0.112743
$586$		0			0
$587$	9.99523			0.412547			0.206274	−	0.978494i	$-0.433866\pi$
$587$	9.99523			0.412547			0.206274	+	0.978494i	$0.433866\pi$
$588$		0			0
$589$	−42.7315			−1.76072
$590$		0			0
$591$	3.93435			0.161837
$592$		0			0
$593$			8.69470i			0.357048i	0.983936	+	0.178524i	$0.0571323\pi$
$593$			8.69470i			0.357048i	−0.983936	+	0.178524i	$0.942868\pi$
$594$		0			0
$595$	3.19032	+	11.9111i	0.130790	+	0.488307i
$596$		0			0
$597$	−3.44874			−0.141147
$598$		0			0
$599$			14.0357i			0.573482i	0.958008	+	0.286741i	$0.0925720\pi$
$599$			14.0357i			0.573482i	−0.958008	+	0.286741i	$0.907428\pi$
$600$		0			0
$601$		−	6.42097i		−	0.261917i	−0.991388	−	0.130958i	$-0.958195\pi$
$601$		−	6.42097i		−	0.261917i	0.991388	−	0.130958i	$-0.0418054\pi$
$602$		0			0
$603$		−	9.29122i		−	0.378368i
$604$		0			0
$605$		−	10.6486i		−	0.432929i
$606$		0			0
$607$	−8.82832			−0.358331			−0.179165	−	0.983819i	$-0.557340\pi$
$607$	−8.82832			−0.358331			−0.179165	+	0.983819i	$0.557340\pi$
$608$		0			0
$609$	0.0213965	+	0.0798841i	0.000867030	+	0.00323707i
$610$		0			0
$611$			7.69351i			0.311246i
$612$		0			0
$613$	−7.71638			−0.311662			−0.155831	−	0.987784i	$-0.549805\pi$
$613$	−7.71638			−0.311662			−0.155831	+	0.987784i	$0.549805\pi$
$614$		0			0
$615$	−2.07563			−0.0836973
$616$		0			0
$617$	12.7495			0.513274			0.256637	−	0.966508i	$-0.417386\pi$
$617$	12.7495			0.513274			0.256637	+	0.966508i	$0.417386\pi$
$618$		0			0
$619$	−34.9911			−1.40641			−0.703206	−	0.710987i	$-0.748249\pi$
$619$	−34.9911			−1.40641			−0.703206	+	0.710987i	$0.748249\pi$
$620$		0			0
$621$		−	6.27324i		−	0.251736i
$622$		0			0
$623$	−12.6641	+	3.39201i	−0.507378	+	0.135898i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	0.765162i		−	0.0305576i
$628$		0			0
$629$		−	26.9384i		−	1.07411i
$630$		0			0
$631$		−	1.24732i		−	0.0496551i	−0.999692	−	0.0248275i	$-0.992096\pi$
$631$		−	1.24732i		−	0.0496551i	0.999692	−	0.0248275i	$-0.00790366\pi$
$632$		0			0
$633$			4.97171i			0.197608i
$634$		0			0
$635$	−2.64272			−0.104873
$636$		0			0
$637$	3.21319	+	5.56793i	0.127311	+	0.220609i
$638$		0			0
$639$		−	22.8714i		−	0.904778i
$640$		0			0
$641$	23.8413			0.941675			0.470838	−	0.882220i	$-0.343952\pi$
$641$	23.8413			0.941675			0.470838	+	0.882220i	$0.343952\pi$
$642$		0			0
$643$	28.4770			1.12302			0.561512	−	0.827469i	$-0.310220\pi$
$643$	28.4770			1.12302			0.561512	+	0.827469i	$0.310220\pi$
$644$		0			0
$645$	−0.529780			−0.0208601
$646$		0			0
$647$	−29.0144			−1.14067			−0.570337	−	0.821411i	$-0.693187\pi$
$647$	−29.0144			−1.14067			−0.570337	+	0.821411i	$0.693187\pi$
$648$		0			0
$649$		−	4.75179i		−	0.186524i
$650$		0			0
$651$	0.695981	+	2.59845i	0.0272776	+	0.101841i
$652$		0			0
$653$	11.5774			0.453060			0.226530	−	0.974004i	$-0.427262\pi$
$653$	11.5774			0.453060			0.226530	+	0.974004i	$0.427262\pi$
$654$		0			0
$655$		−	15.2007i		−	0.593939i
$656$		0			0
$657$		−	40.9367i		−	1.59709i
$658$		0			0
$659$		−	13.5692i		−	0.528582i	−0.964443	−	0.264291i	$-0.914862\pi$
$659$		−	13.5692i		−	0.528582i	0.964443	−	0.264291i	$-0.0851379\pi$
$660$		0			0
$661$		−	18.7841i		−	0.730618i	−0.930886	−	0.365309i	$-0.880963\pi$
$661$		−	18.7841i		−	0.730618i	0.930886	−	0.365309i	$-0.119037\pi$
$662$		0			0
$663$	−0.750124			−0.0291324
$664$		0			0
$665$	18.8239	−	5.04188i	0.729961	−	0.195516i
$666$		0			0
$667$			1.06952i			0.0414118i
$668$		0			0
$669$	−2.13820			−0.0826677
$670$		0			0
$671$	−5.73191			−0.221278
$672$		0			0
$673$	27.7251			1.06872			0.534361	−	0.845256i	$-0.320552\pi$
$673$	27.7251			1.06872			0.534361	+	0.845256i	$0.320552\pi$
$674$		0			0
$675$	−1.04614			−0.0402661
$676$		0			0
$677$		−	23.0066i		−	0.884214i	−0.896962	−	0.442107i	$-0.854231\pi$
$677$		−	23.0066i		−	0.884214i	0.896962	−	0.442107i	$-0.145769\pi$
$678$		0			0
$679$	−17.2827	+	4.62907i	−0.663249	+	0.177647i
$680$		0			0
$681$	1.77320			0.0679491
$682$		0			0
$683$		−	30.8198i		−	1.17929i	−0.807663	−	0.589644i	$-0.799268\pi$
$683$		−	30.8198i		−	1.17929i	0.807663	−	0.589644i	$-0.200732\pi$
$684$		0			0
$685$			12.0409i			0.460060i
$686$		0			0
$687$		−	2.52032i		−	0.0961561i
$688$		0			0
$689$		−	7.03499i		−	0.268012i
$690$		0			0
$691$	−3.61044			−0.137348			−0.0686738	−	0.997639i	$-0.521877\pi$
$691$	−3.61044			−0.137348			−0.0686738	+	0.997639i	$0.521877\pi$
$692$		0			0
$693$	4.49815	−	1.20480i	0.170871	−	0.0457667i
$694$		0			0
$695$		−	0.262345i		−	0.00995131i
$696$		0			0
$697$	−55.1985			−2.09079
$698$		0			0
$699$	2.80799			0.106208
$700$		0			0
$701$	40.0896			1.51416			0.757081	−	0.653321i	$-0.226625\pi$
$701$	40.0896			1.51416			0.757081	+	0.653321i	$0.226625\pi$
$702$		0			0
$703$	−42.5727			−1.60566
$704$		0			0
$705$		−	1.46817i		−	0.0552947i
$706$		0			0
$707$	26.7787	−	7.17252i	1.00712	−	0.269750i
$708$		0			0
$709$	−39.0714			−1.46736			−0.733679	−	0.679497i	$-0.762198\pi$
$709$	−39.0714			−1.46736			−0.733679	+	0.679497i	$0.762198\pi$
$710$		0			0
$711$		−	6.17309i		−	0.231509i
$712$		0			0
$713$			34.7890i			1.30286i
$714$		0			0
$715$		−	0.544370i		−	0.0203583i
$716$		0			0
$717$			2.34021i			0.0873966i
$718$		0			0
$719$	30.0351			1.12012			0.560061	−	0.828452i	$-0.310778\pi$
$719$	30.0351			1.12012			0.560061	+	0.828452i	$0.310778\pi$
$720$		0			0
$721$	6.03909	+	22.5470i	0.224907	+	0.839695i
$722$		0			0
$723$		−	2.09214i		−	0.0778075i
$724$		0			0
$725$	0.178356			0.00662396
$726$		0			0
$727$	2.90088			0.107588			0.0537938	−	0.998552i	$-0.482869\pi$
$727$	2.90088			0.107588			0.0537938	+	0.998552i	$0.482869\pi$
$728$		0			0
$729$	−25.3556			−0.939095
$730$		0			0
$731$	−14.0888			−0.521093
$732$		0			0
$733$			4.56272i			0.168528i	0.996443	+	0.0842640i	$0.0268539\pi$
$733$			4.56272i			0.168528i	−0.996443	+	0.0842640i	$0.973146\pi$
$734$		0			0
$735$	−0.613183	−	1.06254i	−0.0226176	−	0.0391925i
$736$		0			0
$737$	−1.85481			−0.0683227
$738$		0			0
$739$		−	46.0250i		−	1.69306i	−0.532345	−	0.846528i	$-0.678689\pi$
$739$		−	46.0250i		−	1.69306i	0.532345	−	0.846528i	$-0.321311\pi$
$740$		0			0
$741$			1.18547i			0.0435495i
$742$		0			0
$743$		−	11.5444i		−	0.423523i	−0.977321	−	0.211761i	$-0.932080\pi$
$743$		−	11.5444i		−	0.423523i	0.977321	−	0.211761i	$-0.0679199\pi$
$744$		0			0
$745$			1.80523i			0.0661386i
$746$		0			0
$747$	−16.6075			−0.607635
$748$		0			0
$749$	19.4460	−	5.20850i	0.710541	−	0.190314i
$750$		0			0
$751$			24.1473i			0.881148i	0.897716	+	0.440574i	$0.145225\pi$
$751$			24.1473i			0.881148i	−0.897716	+	0.440574i	$0.854775\pi$
$752$		0			0
$753$	−1.19587			−0.0435798
$754$		0			0
$755$	13.3315			0.485183
$756$		0			0
$757$	−8.41738			−0.305935			−0.152967	−	0.988231i	$-0.548883\pi$
$757$	−8.41738			−0.305935			−0.152967	+	0.988231i	$0.548883\pi$
$758$		0			0
$759$	−0.622941			−0.0226113
$760$		0			0
$761$		−	4.47545i		−	0.162235i	−0.996705	−	0.0811175i	$-0.974151\pi$
$761$		−	4.47545i		−	0.162235i	0.996705	−	0.0811175i	$-0.0258489\pi$
$762$		0			0
$763$	−10.6203	−	39.6509i	−0.384480	−	1.43546i
$764$		0			0
$765$	−13.8388			−0.500344
$766$		0			0
$767$			7.36199i			0.265826i
$768$		0			0
$769$			7.36710i			0.265665i	0.991139	+	0.132832i	$0.0424072\pi$
$769$			7.36710i			0.265665i	−0.991139	+	0.132832i	$0.957593\pi$
$770$		0			0
$771$			1.87494i			0.0675245i
$772$		0			0
$773$		−	28.4840i		−	1.02450i	−0.858837	−	0.512248i	$-0.828813\pi$
$773$		−	28.4840i		−	1.02450i	0.858837	−	0.512248i	$-0.171187\pi$
$774$		0			0
$775$	5.80152			0.208397
$776$		0			0
$777$	0.693395	+	2.58880i	0.0248754	+	0.0928727i
$778$		0			0
$779$			87.2340i			3.12548i
$780$		0			0
$781$	−4.56582			−0.163378
$782$		0			0
$783$	−0.186586			−0.00666803
$784$		0			0
$785$	−1.82912			−0.0652842
$786$		0			0
$787$	−47.0355			−1.67663			−0.838317	−	0.545183i	$-0.816460\pi$
$787$	−47.0355			−1.67663			−0.838317	+	0.545183i	$0.816460\pi$
$788$		0			0
$789$			2.51084i			0.0893881i
$790$		0			0
$791$	−9.41071	−	35.1350i	−0.334606	−	1.24926i
$792$		0			0
$793$	8.88051			0.315356
$794$		0			0
$795$			1.34251i			0.0476138i
$796$		0			0
$797$		−	18.0906i		−	0.640802i	−0.947282	−	0.320401i	$-0.896182\pi$
$797$		−	18.0906i		−	0.640802i	0.947282	−	0.320401i	$-0.103818\pi$
$798$		0			0
$799$		−	39.0441i		−	1.38128i
$800$		0			0
$801$		−	14.7137i		−	0.519885i
$802$		0			0
$803$	−8.17219			−0.288390
$804$		0			0
$805$	−4.10475	−	15.3251i	−0.144673	−	0.540140i
$806$		0			0
$807$			2.07681i			0.0731071i
$808$		0			0
$809$	−0.640394			−0.0225150			−0.0112575	−	0.999937i	$-0.503583\pi$
$809$	−0.640394			−0.0225150			−0.0112575	+	0.999937i	$0.503583\pi$
$810$		0			0
$811$	15.7593			0.553385			0.276692	−	0.960959i	$-0.410762\pi$
$811$	15.7593			0.553385			0.276692	+	0.960959i	$0.410762\pi$
$812$		0			0
$813$	0.493545			0.0173094
$814$		0			0
$815$	15.9737			0.559535
$816$		0			0
$817$			22.2655i			0.778972i
$818$		0			0
$819$	−6.96903	+	1.86661i	−0.243518	+	0.0652248i
$820$		0			0
$821$	26.1931			0.914146			0.457073	−	0.889429i	$-0.348898\pi$
$821$	26.1931			0.914146			0.457073	+	0.889429i	$0.348898\pi$
$822$		0			0
$823$		−	11.0721i		−	0.385949i	−0.981204	−	0.192974i	$-0.938187\pi$
$823$		−	11.0721i		−	0.385949i	0.981204	−	0.192974i	$-0.0618134\pi$
$824$		0			0
$825$			0.103884i			0.00361676i
$826$		0			0
$827$			26.7722i			0.930959i	0.885059	+	0.465479i	$0.154118\pi$
$827$			26.7722i			0.930959i	−0.885059	+	0.465479i	$0.845882\pi$
$828$		0			0
$829$			38.7643i			1.34634i	0.739488	+	0.673170i	$0.235068\pi$
$829$			38.7643i			1.34634i	−0.739488	+	0.673170i	$0.764932\pi$
$830$		0			0
$831$	3.36632			0.116776
$832$		0			0
$833$	−16.3068	−	28.2569i	−0.564996	−	0.979045i
$834$		0			0
$835$		−	16.4935i		−	0.570782i
$836$		0			0
$837$	−6.06923			−0.209783
$838$		0			0
$839$	30.9819			1.06961			0.534807	−	0.844974i	$-0.320384\pi$
$839$	30.9819			1.06961			0.534807	+	0.844974i	$0.320384\pi$
$840$		0			0
$841$	−28.9682			−0.998903
$842$		0			0
$843$	−2.40100			−0.0826949
$844$		0			0
$845$		−	12.1566i		−	0.418200i
$846$		0			0
$847$	7.28921	+	27.2144i	0.250460	+	0.935097i
$848$		0			0
$849$	5.66371			0.194378
$850$		0			0
$851$			34.6597i			1.18812i
$852$		0			0
$853$			41.7085i			1.42807i	0.700109	+	0.714036i	$0.253135\pi$
$853$			41.7085i			1.42807i	−0.700109	+	0.714036i	$0.746865\pi$
$854$		0			0
$855$			21.8705i			0.747955i
$856$		0			0
$857$			21.1855i			0.723684i	0.932239	+	0.361842i	$0.117852\pi$
$857$			21.1855i			0.723684i	−0.932239	+	0.361842i	$0.882148\pi$
$858$		0			0
$859$	31.5673			1.07706			0.538532	−	0.842605i	$-0.318979\pi$
$859$	31.5673			1.07706			0.538532	+	0.842605i	$0.318979\pi$
$860$		0			0
$861$	5.30461	−	1.42081i	0.180781	−	0.0484210i
$862$		0			0
$863$			31.6656i			1.07791i	0.842335	+	0.538954i	$0.181180\pi$
$863$			31.6656i			1.07791i	−0.842335	+	0.538954i	$0.818820\pi$
$864$		0			0
$865$	9.31174			0.316609
$866$		0			0
$867$	0.827507			0.0281036
$868$		0			0
$869$	−1.23233			−0.0418041
$870$		0			0
$871$	2.87367			0.0973706
$872$		0			0
$873$		−	20.0798i		−	0.679598i
$874$		0			0
$875$	−2.55567	+	0.684520i	−0.0863973	+	0.0231410i
$876$		0			0
$877$	−18.4511			−0.623051			−0.311525	−	0.950238i	$-0.600840\pi$
$877$	−18.4511			−0.623051			−0.311525	+	0.950238i	$0.600840\pi$
$878$		0			0
$879$			2.79232i			0.0941826i
$880$		0			0
$881$		−	16.5257i		−	0.556764i	−0.960470	−	0.278382i	$-0.910202\pi$
$881$		−	16.5257i		−	0.556764i	0.960470	−	0.278382i	$-0.0897981\pi$
$882$		0			0
$883$		−	27.3241i		−	0.919529i	−0.888041	−	0.459765i	$-0.847934\pi$
$883$		−	27.3241i		−	0.919529i	0.888041	−	0.459765i	$-0.152066\pi$
$884$		0			0
$885$		−	1.40491i		−	0.0472256i
$886$		0			0
$887$	23.6963			0.795643			0.397822	−	0.917463i	$-0.369766\pi$
$887$	23.6963			0.795643			0.397822	+	0.917463i	$0.369766\pi$
$888$		0			0
$889$	6.75392	−	1.80900i	0.226519	−	0.0606718i
$890$		0			0
$891$			5.17153i			0.173253i
$892$		0			0
$893$	−61.7042			−2.06485
$894$		0			0
$895$	8.65612			0.289342
$896$		0			0
$897$	0.965129			0.0322247
$898$		0			0
$899$	1.03473			0.0345103
$900$		0			0
$901$			35.7022i			1.18941i
$902$		0			0
$903$	1.35394	−	0.362645i	0.0450563	−	0.0120681i
$904$		0			0
$905$	17.0374			0.566343
$906$		0			0
$907$			4.79364i			0.159170i	0.996828	+	0.0795851i	$0.0253595\pi$
$907$			4.79364i			0.159170i	−0.996828	+	0.0795851i	$0.974640\pi$
$908$		0			0
$909$			31.1127i			1.03194i
$910$		0			0
$911$			48.7763i			1.61603i	0.589162	+	0.808015i	$0.299458\pi$
$911$			48.7763i			1.61603i	−0.589162	+	0.808015i	$0.700542\pi$
$912$		0			0
$913$			3.31535i			0.109722i
$914$		0			0
$915$	−1.69469			−0.0560248
$916$		0			0
$917$	10.4052	+	38.8478i	0.343609	+	1.28287i
$918$		0			0
$919$			41.6787i			1.37485i	0.726253	+	0.687427i	$0.241260\pi$
$919$			41.6787i			1.37485i	−0.726253	+	0.687427i	$0.758740\pi$
$920$		0			0
$921$	−2.18204			−0.0719007
$922$		0			0
$923$	7.07386			0.232839
$924$		0			0
$925$	5.77996			0.190044
$926$		0			0
$927$	−26.1961			−0.860394
$928$		0			0
$929$			32.0777i			1.05243i	0.850350	+	0.526217i	$0.176390\pi$
$929$			32.0777i			1.05243i	−0.850350	+	0.526217i	$0.823610\pi$
$930$		0			0
$931$	−44.6565	+	25.7707i	−1.46356	+	0.844602i
$932$		0			0
$933$	0.0244922			0.000801838
$934$		0			0
$935$			2.76265i			0.0903482i
$936$		0			0
$937$			12.9480i			0.422991i	0.977379	+	0.211496i	$0.0678334\pi$
$937$			12.9480i			0.422991i	−0.977379	+	0.211496i	$0.932167\pi$
$938$		0			0
$939$			3.69172i			0.120475i
$940$		0			0
$941$			56.9926i			1.85790i	0.370199	+	0.928952i	$0.379289\pi$
$941$			56.9926i			1.85790i	−0.370199	+	0.928952i	$0.620711\pi$
$942$		0			0
$943$	71.0198			2.31272
$944$		0			0
$945$	2.67360	−	0.716107i	0.0869721	−	0.0232950i
$946$		0			0
$947$		−	35.4821i		−	1.15301i	−0.817092	−	0.576507i	$-0.804415\pi$
$947$		−	35.4821i		−	1.15301i	0.817092	−	0.576507i	$-0.195585\pi$
$948$		0			0
$949$	12.6613			0.411002
$950$		0			0
$951$	2.93685			0.0952338
$952$		0			0
$953$	5.91458			0.191592			0.0957960	−	0.995401i	$-0.469460\pi$
$953$	5.91458			0.191592			0.0957960	+	0.995401i	$0.469460\pi$
$954$		0			0
$955$	5.63997			0.182505
$956$		0			0
$957$			0.0185282i			0.000598933i
$958$		0			0
$959$	−8.24226	−	30.7726i	−0.266157	−	0.993699i
$960$		0			0
$961$	2.65761			0.0857294
$962$		0			0
$963$			22.5932i			0.728056i
$964$		0			0
$965$			16.6423i			0.535736i
$966$		0			0
$967$		−	23.2001i		−	0.746066i	−0.927818	−	0.373033i	$-0.878318\pi$
$967$		−	23.2001i		−	0.746066i	0.927818	−	0.373033i	$-0.121682\pi$
$968$		0			0
$969$		−	6.01621i		−	0.193269i
$970$		0			0
$971$	−14.9461			−0.479642			−0.239821	−	0.970817i	$-0.577089\pi$
$971$	−14.9461			−0.479642			−0.239821	+	0.970817i	$0.577089\pi$
$972$		0			0
$973$	0.179580	+	0.670466i	0.00575708	+	0.0214942i
$974$		0			0
$975$		−	0.160948i		−	0.00515446i
$976$		0			0
$977$	−10.1449			−0.324564			−0.162282	−	0.986744i	$-0.551885\pi$
$977$	−10.1449			−0.324564			−0.162282	+	0.986744i	$0.551885\pi$
$978$		0			0
$979$	−2.93730			−0.0938767
$980$		0			0
$981$	46.0682			1.47084
$982$		0			0
$983$	−25.0960			−0.800438			−0.400219	−	0.916420i	$-0.631066\pi$
$983$	−25.0960			−0.800438			−0.400219	+	0.916420i	$0.631066\pi$
$984$		0			0
$985$			22.4493i			0.715295i
$986$		0			0
$987$	1.00500	+	3.75217i	0.0319894	+	0.119433i
$988$		0			0
$989$	18.1270			0.576405
$990$		0			0
$991$		−	43.0554i		−	1.36770i	−0.729622	−	0.683850i	$-0.760304\pi$
$991$		−	43.0554i		−	1.36770i	0.729622	−	0.683850i	$-0.239696\pi$
$992$		0			0
$993$			3.79271i			0.120358i
$994$		0			0
$995$		−	19.6785i		−	0.623849i
$996$		0			0
$997$			51.2522i			1.62317i	0.584233	+	0.811586i	$0.301396\pi$
$997$			51.2522i			1.62317i	−0.584233	+	0.811586i	$0.698604\pi$
$998$		0			0
$999$	−6.04668			−0.191308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.f.1791.9		16
4.3	odd	2		2240.2.k.g.1791.7		16
7.6	odd	2		2240.2.k.g.1791.8		16
8.3	odd	2		1120.2.k.b.671.10	yes	16
8.5	even	2		1120.2.k.a.671.8	yes	16
28.27	even	2	inner	2240.2.k.f.1791.10		16
56.13	odd	2		1120.2.k.b.671.9	yes	16
56.27	even	2		1120.2.k.a.671.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.k.a.671.7	✓	16	56.27	even	2
1120.2.k.a.671.8	yes	16	8.5	even	2
1120.2.k.b.671.9	yes	16	56.13	odd	2
1120.2.k.b.671.10	yes	16	8.3	odd	2
2240.2.k.f.1791.9		16	1.1	even	1	trivial
2240.2.k.f.1791.10		16	28.27	even	2	inner
2240.2.k.g.1791.7		16	4.3	odd	2
2240.2.k.g.1791.8		16	7.6	odd	2

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.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.175255 q^{3} -1.00000i q^{5} +(0.684520 + 2.55567i) q^{7} -2.96929 q^{9} +O(q^{10})\)	\(q-0.175255 q^{3} -1.00000i q^{5} +(0.684520 + 2.55567i) q^{7} -2.96929 q^{9} +0.592758i q^{11} -0.918367i q^{13} +0.175255i q^{15} +4.66066i q^{17} +7.36557 q^{19} +(-0.119965 - 0.447892i) q^{21} -5.99653i q^{23} -1.00000 q^{25} +1.04614 q^{27} -0.178356 q^{29} -5.80152 q^{31} -0.103884i q^{33} +(2.55567 - 0.684520i) q^{35} -5.77996 q^{37} +0.160948i q^{39} +11.8435i q^{41} +3.02292i q^{43} +2.96929i q^{45} -8.37738 q^{47} +(-6.06286 + 3.49881i) q^{49} -0.816802i q^{51} +7.66033 q^{53} +0.592758 q^{55} -1.29085 q^{57} -8.01640 q^{59} +9.66990i q^{61} +(-2.03254 - 7.58850i) q^{63} -0.918367 q^{65} +3.12911i q^{67} +1.05092i q^{69} +7.70266i q^{71} +13.7867i q^{73} +0.175255 q^{75} +(-1.51489 + 0.405755i) q^{77} +2.07898i q^{79} +8.72452 q^{81} +5.59308 q^{83} +4.66066 q^{85} +0.0312576 q^{87} +4.95531i q^{89} +(2.34704 - 0.628641i) q^{91} +1.01674 q^{93} -7.36557i q^{95} +6.76250i q^{97} -1.76007i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^7 + 16 * q^9	\( 16 q - 4 q^{7} + 16 q^{9} - 8 q^{19} - 4 q^{21} - 16 q^{25} + 48 q^{27} - 8 q^{29} - 16 q^{37} - 8 q^{47} - 4 q^{49} - 16 q^{53} + 8 q^{55} + 16 q^{57} - 8 q^{59} - 60 q^{63} + 8 q^{65} + 8 q^{77} + 40 q^{81} + 64 q^{83} - 16 q^{87} - 64 q^{91} + 48 q^{93}+O(q^{100}) \) 16 * q - 4 * q^7 + 16 * q^9 - 8 * q^19 - 4 * q^21 - 16 * q^25 + 48 * q^27 - 8 * q^29 - 16 * q^37 - 8 * q^47 - 4 * q^49 - 16 * q^53 + 8 * q^55 + 16 * q^57 - 8 * q^59 - 60 * q^63 + 8 * q^65 + 8 * q^77 + 40 * q^81 + 64 * q^83 - 16 * q^87 - 64 * q^91 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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