LMFDB - Embedded newform 2240.2.k.g.1791.2

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			1791.2
Root			$1.08568 + 1.64772i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.g.1791.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.29545 q^{3} +1.00000i q^{5} +(2.01507 + 1.71449i) q^{7} +7.85998 q^{9} +O(q^{10})$	$q-3.29545 q^{3} +1.00000i q^{5} +(2.01507 + 1.71449i) q^{7} +7.85998 q^{9} +1.43526i q^{11} -5.60035i q^{13} -3.29545i q^{15} -1.85877i q^{17} +3.04101 q^{19} +(-6.64057 - 5.65002i) q^{21} -0.989130i q^{23} -1.00000 q^{25} -16.0158 q^{27} +5.89054 q^{29} -7.82964 q^{31} -4.72983i q^{33} +(-1.71449 + 2.01507i) q^{35} +3.93243 q^{37} +18.4557i q^{39} -8.37831i q^{41} +0.890743i q^{43} +7.85998i q^{45} -0.347743 q^{47} +(1.12103 + 6.90965i) q^{49} +6.12550i q^{51} -12.1711 q^{53} -1.43526 q^{55} -10.0215 q^{57} -4.32153 q^{59} +13.2238i q^{61} +(15.8384 + 13.4759i) q^{63} +5.60035 q^{65} -15.3113i q^{67} +3.25963i q^{69} -0.574410i q^{71} -13.6466i q^{73} +3.29545 q^{75} +(-2.46075 + 2.89216i) q^{77} -3.17455i q^{79} +29.1994 q^{81} +7.59090 q^{83} +1.85877 q^{85} -19.4120 q^{87} -8.08220i q^{89} +(9.60177 - 11.2851i) q^{91} +25.8022 q^{93} +3.04101i q^{95} +4.66048i q^{97} +11.2811i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.29545			−1.90263			−0.951314	−	0.308223i	$-0.900266\pi$
$3$	−3.29545			−1.90263			−0.951314	+	0.308223i	$0.900266\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	2.01507	+	1.71449i	0.761626	+	0.648017i
$7$	2.01507	+	1.71449i	0.761626	+	0.648017i
$8$		0			0
$9$	7.85998			2.61999
$10$		0			0
$11$			1.43526i			0.432748i	0.976311	+	0.216374i	$0.0694230\pi$
$11$			1.43526i			0.432748i	−0.976311	+	0.216374i	$0.930577\pi$
$12$		0			0
$13$		−	5.60035i		−	1.55326i	−0.629958	−	0.776629i	$-0.716928\pi$
$13$		−	5.60035i		−	1.55326i	0.629958	−	0.776629i	$-0.283072\pi$
$14$		0			0
$15$		−	3.29545i		−	0.850881i
$16$		0			0
$17$		−	1.85877i		−	0.450819i	−0.974264	−	0.225410i	$-0.927628\pi$
$17$		−	1.85877i		−	0.450819i	0.974264	−	0.225410i	$-0.0723720\pi$
$18$		0			0
$19$	3.04101			0.697656			0.348828	−	0.937187i	$-0.386580\pi$
$19$	3.04101			0.697656			0.348828	+	0.937187i	$0.386580\pi$
$20$		0			0
$21$	−6.64057	−	5.65002i	−1.44909	−	1.23294i
$22$		0			0
$23$		−	0.989130i		−	0.206248i	−0.994669	−	0.103124i	$-0.967116\pi$
$23$		−	0.989130i		−	0.206248i	0.994669	−	0.103124i	$-0.0328838\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−16.0158			−3.08225
$28$		0			0
$29$	5.89054			1.09385			0.546923	−	0.837183i	$-0.315799\pi$
$29$	5.89054			1.09385			0.546923	+	0.837183i	$0.315799\pi$
$30$		0			0
$31$	−7.82964			−1.40624			−0.703122	−	0.711069i	$-0.748211\pi$
$31$	−7.82964			−1.40624			−0.703122	+	0.711069i	$0.748211\pi$
$32$		0			0
$33$		−	4.72983i		−	0.823358i
$34$		0			0
$35$	−1.71449	+	2.01507i	−0.289802	+	0.340609i
$36$		0			0
$37$	3.93243			0.646487			0.323244	−	0.946316i	$-0.395227\pi$
$37$	3.93243			0.646487			0.323244	+	0.946316i	$0.395227\pi$
$38$		0			0
$39$			18.4557i			2.95527i
$40$		0			0
$41$		−	8.37831i		−	1.30847i	−0.756290	−	0.654236i	$-0.772990\pi$
$41$		−	8.37831i		−	1.30847i	0.756290	−	0.654236i	$-0.227010\pi$
$42$		0			0
$43$			0.890743i			0.135837i	0.997691	+	0.0679185i	$0.0216358\pi$
$43$			0.890743i			0.135837i	−0.997691	+	0.0679185i	$0.978364\pi$
$44$		0			0
$45$			7.85998i			1.17170i
$46$		0			0
$47$	−0.347743			−0.0507236			−0.0253618	−	0.999678i	$-0.508074\pi$
$47$	−0.347743			−0.0507236			−0.0253618	+	0.999678i	$0.508074\pi$
$48$		0			0
$49$	1.12103	+	6.90965i	0.160147	+	0.987093i
$50$		0			0
$51$			6.12550i			0.857741i
$52$		0			0
$53$	−12.1711			−1.67183			−0.835915	−	0.548859i	$-0.815062\pi$
$53$	−12.1711			−1.67183			−0.835915	+	0.548859i	$0.815062\pi$
$54$		0			0
$55$	−1.43526			−0.193531
$56$		0			0
$57$	−10.0215			−1.32738
$58$		0			0
$59$	−4.32153			−0.562616			−0.281308	−	0.959618i	$-0.590768\pi$
$59$	−4.32153			−0.562616			−0.281308	+	0.959618i	$0.590768\pi$
$60$		0			0
$61$			13.2238i			1.69314i	0.532280	+	0.846568i	$0.321335\pi$
$61$			13.2238i			1.69314i	−0.532280	+	0.846568i	$0.678665\pi$
$62$		0			0
$63$	15.8384	+	13.4759i	1.99545	+	1.69780i
$64$		0			0
$65$	5.60035			0.694639
$66$		0			0
$67$		−	15.3113i		−	1.87057i	−0.353895	−	0.935285i	$-0.615143\pi$
$67$		−	15.3113i		−	1.87057i	0.353895	−	0.935285i	$-0.384857\pi$
$68$		0			0
$69$			3.25963i			0.392413i
$70$		0			0
$71$		−	0.574410i		−	0.0681700i	−0.999419	−	0.0340850i	$-0.989148\pi$
$71$		−	0.574410i		−	0.0681700i	0.999419	−	0.0340850i	$-0.0108517\pi$
$72$		0			0
$73$		−	13.6466i		−	1.59721i	−0.601854	−	0.798606i	$-0.705571\pi$
$73$		−	13.6466i		−	1.59721i	0.601854	−	0.798606i	$-0.294429\pi$
$74$		0			0
$75$	3.29545			0.380526
$76$		0			0
$77$	−2.46075	+	2.89216i	−0.280428	+	0.329592i
$78$		0			0
$79$		−	3.17455i		−	0.357165i	−0.983925	−	0.178582i	$-0.942849\pi$
$79$		−	3.17455i		−	0.357165i	0.983925	−	0.178582i	$-0.0571511\pi$
$80$		0			0
$81$	29.1994			3.24438
$82$		0			0
$83$	7.59090			0.833210			0.416605	−	0.909088i	$-0.363220\pi$
$83$	7.59090			0.833210			0.416605	+	0.909088i	$0.363220\pi$
$84$		0			0
$85$	1.85877			0.201612
$86$		0			0
$87$	−19.4120			−2.08118
$88$		0			0
$89$		−	8.08220i		−	0.856711i	−0.903610	−	0.428356i	$-0.859093\pi$
$89$		−	8.08220i		−	0.856711i	0.903610	−	0.428356i	$-0.140907\pi$
$90$		0			0
$91$	9.60177	−	11.2851i	1.00654	−	1.18300i
$92$		0			0
$93$	25.8022			2.67556
$94$		0			0
$95$			3.04101i			0.312001i
$96$		0			0
$97$			4.66048i			0.473200i	0.971607	+	0.236600i	$0.0760332\pi$
$97$			4.66048i			0.473200i	−0.971607	+	0.236600i	$0.923967\pi$
$98$		0			0
$99$			11.2811i			1.13380i
$100$		0			0
$101$		−	0.482485i		−	0.0480090i	−0.999712	−	0.0240045i	$-0.992358\pi$
$101$		−	0.482485i		−	0.0480090i	0.999712	−	0.0240045i	$-0.00764161\pi$
$102$		0			0
$103$	8.52386			0.839880			0.419940	−	0.907552i	$-0.362051\pi$
$103$	8.52386			0.839880			0.419940	+	0.907552i	$0.362051\pi$
$104$		0			0
$105$	5.65002	−	6.64057i	0.551386	−	0.648053i
$106$		0			0
$107$		−	15.2510i		−	1.47437i	−0.675692	−	0.737184i	$-0.736155\pi$
$107$		−	15.2510i		−	1.47437i	0.675692	−	0.737184i	$-0.263845\pi$
$108$		0			0
$109$	17.8125			1.70613			0.853066	−	0.521804i	$-0.174741\pi$
$109$	17.8125			1.70613			0.853066	+	0.521804i	$0.174741\pi$
$110$		0			0
$111$	−12.9591			−1.23002
$112$		0			0
$113$	−17.7201			−1.66697			−0.833485	−	0.552542i	$-0.813658\pi$
$113$	−17.7201			−1.66697			−0.833485	+	0.552542i	$0.813658\pi$
$114$		0			0
$115$	0.989130			0.0922369
$116$		0			0
$117$		−	44.0187i		−	4.06953i
$118$		0			0
$119$	3.18686	−	3.74556i	0.292139	−	0.343355i
$120$		0			0
$121$	8.94002			0.812729
$122$		0			0
$123$			27.6103i			2.48954i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	6.21856i		−	0.551808i	−0.961185	−	0.275904i	$-0.911023\pi$
$127$		−	6.21856i		−	0.551808i	0.961185	−	0.275904i	$-0.0889771\pi$
$128$		0			0
$129$		−	2.93540i		−	0.258447i
$130$		0			0
$131$	−5.28635			−0.461870			−0.230935	−	0.972969i	$-0.574179\pi$
$131$	−5.28635			−0.461870			−0.230935	+	0.972969i	$0.574179\pi$
$132$		0			0
$133$	6.12786	+	5.21380i	0.531353	+	0.452093i
$134$		0			0
$135$		−	16.0158i		−	1.37842i
$136$		0			0
$137$	7.48277			0.639297			0.319648	−	0.947536i	$-0.396435\pi$
$137$	7.48277			0.639297			0.319648	+	0.947536i	$0.396435\pi$
$138$		0			0
$139$	13.9691			1.18484			0.592421	−	0.805629i	$-0.298172\pi$
$139$	13.9691			1.18484			0.592421	+	0.805629i	$0.298172\pi$
$140$		0			0
$141$	1.14597			0.0965081
$142$		0			0
$143$	8.03797			0.672169
$144$		0			0
$145$			5.89054i			0.489183i
$146$		0			0
$147$	−3.69429	−	22.7704i	−0.304700	−	1.87807i
$148$		0			0
$149$	−0.118251			−0.00968748			−0.00484374	−	0.999988i	$-0.501542\pi$
$149$	−0.118251			−0.00968748			−0.00484374	+	0.999988i	$0.501542\pi$
$150$		0			0
$151$		−	12.2429i		−	0.996315i	−0.867087	−	0.498158i	$-0.834010\pi$
$151$		−	12.2429i		−	0.996315i	0.867087	−	0.498158i	$-0.165990\pi$
$152$		0			0
$153$		−	14.6099i		−	1.18114i
$154$		0			0
$155$		−	7.82964i		−	0.628892i
$156$		0			0
$157$		−	5.60455i		−	0.447292i	−0.974670	−	0.223646i	$-0.928204\pi$
$157$		−	5.60455i		−	0.447292i	0.974670	−	0.223646i	$-0.0717959\pi$
$158$		0			0
$159$	40.1092			3.18087
$160$		0			0
$161$	1.69586	−	1.99317i	0.133652	−	0.157084i
$162$		0			0
$163$		−	11.0820i		−	0.868012i	−0.900910	−	0.434006i	$-0.857100\pi$
$163$		−	11.0820i		−	0.868012i	0.900910	−	0.434006i	$-0.142900\pi$
$164$		0			0
$165$	4.72983			0.368217
$166$		0			0
$167$	−2.22069			−0.171842			−0.0859212	−	0.996302i	$-0.527383\pi$
$167$	−2.22069			−0.171842			−0.0859212	+	0.996302i	$0.527383\pi$
$168$		0			0
$169$	−18.3640			−1.41261
$170$		0			0
$171$	23.9023			1.82786
$172$		0			0
$173$			13.3236i			1.01297i	0.862248	+	0.506486i	$0.169056\pi$
$173$			13.3236i			1.01297i	−0.862248	+	0.506486i	$0.830944\pi$
$174$		0			0
$175$	−2.01507	−	1.71449i	−0.152325	−	0.129603i
$176$		0			0
$177$	14.2414			1.07045
$178$		0			0
$179$		−	25.6905i		−	1.92020i	−0.279658	−	0.960100i	$-0.590221\pi$
$179$		−	25.6905i		−	1.92020i	0.279658	−	0.960100i	$-0.409779\pi$
$180$		0			0
$181$		−	2.87811i		−	0.213928i	−0.994263	−	0.106964i	$-0.965887\pi$
$181$		−	2.87811i		−	0.213928i	0.994263	−	0.106964i	$-0.0341130\pi$
$182$		0			0
$183$		−	43.5784i		−	3.22141i
$184$		0			0
$185$			3.93243i			0.289118i
$186$		0			0
$187$	2.66783			0.195091
$188$		0			0
$189$	−32.2730	−	27.4590i	−2.34752	−	1.99735i
$190$		0			0
$191$			16.9335i			1.22526i	0.790369	+	0.612632i	$0.209889\pi$
$191$			16.9335i			1.22526i	−0.790369	+	0.612632i	$0.790111\pi$
$192$		0			0
$193$	18.4434			1.32759			0.663794	−	0.747915i	$-0.268945\pi$
$193$	18.4434			1.32759			0.663794	+	0.747915i	$0.268945\pi$
$194$		0			0
$195$	−18.4557			−1.32164
$196$		0			0
$197$	−6.64444			−0.473396			−0.236698	−	0.971583i	$-0.576065\pi$
$197$	−6.64444			−0.473396			−0.236698	+	0.971583i	$0.576065\pi$
$198$		0			0
$199$	15.0193			1.06469			0.532345	−	0.846527i	$-0.321311\pi$
$199$	15.0193			1.06469			0.532345	+	0.846527i	$0.321311\pi$
$200$		0			0
$201$			50.4575i			3.55900i
$202$		0			0
$203$	11.8699	+	10.0993i	0.833101	+	0.708831i
$204$		0			0
$205$	8.37831			0.585167
$206$		0			0
$207$		−	7.77455i		−	0.540369i
$208$		0			0
$209$			4.36465i			0.301909i
$210$		0			0
$211$			22.4412i			1.54492i	0.635066	+	0.772458i	$0.280973\pi$
$211$			22.4412i			1.54492i	−0.635066	+	0.772458i	$0.719027\pi$
$212$		0			0
$213$			1.89294i			0.129702i
$214$		0			0
$215$	−0.890743			−0.0607481
$216$		0			0
$217$	−15.7773	−	13.4239i	−1.07103	−	0.911271i
$218$		0			0
$219$			44.9717i			3.03890i
$220$		0			0
$221$	−10.4098			−0.700239
$222$		0			0
$223$	−2.57577			−0.172486			−0.0862432	−	0.996274i	$-0.527486\pi$
$223$	−2.57577			−0.172486			−0.0862432	+	0.996274i	$0.527486\pi$
$224$		0			0
$225$	−7.85998			−0.523999
$226$		0			0
$227$	11.4113			0.757393			0.378696	−	0.925521i	$-0.376372\pi$
$227$	11.4113			0.757393			0.378696	+	0.925521i	$0.376372\pi$
$228$		0			0
$229$		−	20.3747i		−	1.34640i	−0.739462	−	0.673198i	$-0.764920\pi$
$229$		−	20.3747i		−	1.34640i	0.739462	−	0.673198i	$-0.235080\pi$
$230$		0			0
$231$	8.10926	−	9.53095i	0.533550	−	0.627090i
$232$		0			0
$233$	3.46425			0.226950			0.113475	−	0.993541i	$-0.463802\pi$
$233$	3.46425			0.226950			0.113475	+	0.993541i	$0.463802\pi$
$234$		0			0
$235$		−	0.347743i		−	0.0226843i
$236$		0			0
$237$			10.4616i			0.679552i
$238$		0			0
$239$			16.0893i			1.04073i	0.853944	+	0.520364i	$0.174204\pi$
$239$			16.0893i			1.04073i	−0.853944	+	0.520364i	$0.825796\pi$
$240$		0			0
$241$		−	5.45521i		−	0.351401i	−0.984444	−	0.175701i	$-0.943781\pi$
$241$		−	5.45521i		−	0.351401i	0.984444	−	0.175701i	$-0.0562191\pi$
$242$		0			0
$243$	−48.1776			−3.09060
$244$		0			0
$245$	−6.90965	+	1.12103i	−0.441441	+	0.0716199i
$246$		0			0
$247$		−	17.0308i		−	1.08364i
$248$		0			0
$249$	−25.0154			−1.58529
$250$		0			0
$251$	19.1058			1.20595			0.602975	−	0.797760i	$-0.293982\pi$
$251$	19.1058			1.20595			0.602975	+	0.797760i	$0.293982\pi$
$252$		0			0
$253$	1.41966			0.0892533
$254$		0			0
$255$	−6.12550			−0.383593
$256$		0			0
$257$			10.7668i			0.671612i	0.941931	+	0.335806i	$0.109009\pi$
$257$			10.7668i			0.671612i	−0.941931	+	0.335806i	$0.890991\pi$
$258$		0			0
$259$	7.92412	+	6.74212i	0.492381	+	0.418935i
$260$		0			0
$261$	46.2995			2.86587
$262$		0			0
$263$			10.7609i			0.663545i	0.943359	+	0.331773i	$0.107647\pi$
$263$			10.7609i			0.663545i	−0.943359	+	0.331773i	$0.892353\pi$
$264$		0			0
$265$		−	12.1711i		−	0.747665i
$266$		0			0
$267$			26.6345i			1.63000i
$268$		0			0
$269$			17.9573i			1.09488i	0.836846	+	0.547439i	$0.184397\pi$
$269$			17.9573i			1.09488i	−0.836846	+	0.547439i	$0.815603\pi$
$270$		0			0
$271$	−16.5637			−1.00617			−0.503085	−	0.864237i	$-0.667802\pi$
$271$	−16.5637			−1.00617			−0.503085	+	0.864237i	$0.667802\pi$
$272$		0			0
$273$	−31.6421	+	37.1895i	−1.91507	+	2.25081i
$274$		0			0
$275$		−	1.43526i		−	0.0865495i
$276$		0			0
$277$	−7.37156			−0.442914			−0.221457	−	0.975170i	$-0.571081\pi$
$277$	−7.37156			−0.442914			−0.221457	+	0.975170i	$0.571081\pi$
$278$		0			0
$279$	−61.5408			−3.68435
$280$		0			0
$281$	−9.12029			−0.544071			−0.272035	−	0.962287i	$-0.587697\pi$
$281$	−9.12029			−0.544071			−0.272035	+	0.962287i	$0.587697\pi$
$282$		0			0
$283$	27.4579			1.63221			0.816103	−	0.577907i	$-0.196130\pi$
$283$	27.4579			1.63221			0.816103	+	0.577907i	$0.196130\pi$
$284$		0			0
$285$		−	10.0215i		−	0.593623i
$286$		0			0
$287$	14.3646	−	16.8829i	0.847913	−	0.996566i
$288$		0			0
$289$	13.5450			0.796762
$290$		0			0
$291$		−	15.3584i		−	0.900324i
$292$		0			0
$293$			22.5135i			1.31525i	0.753345	+	0.657625i	$0.228439\pi$
$293$			22.5135i			1.31525i	−0.753345	+	0.657625i	$0.771561\pi$
$294$		0			0
$295$		−	4.32153i		−	0.251609i
$296$		0			0
$297$		−	22.9869i		−	1.33384i
$298$		0			0
$299$	−5.53948			−0.320356
$300$		0			0
$301$	−1.52717	+	1.79491i	−0.0880247	+	0.103457i
$302$		0			0
$303$			1.59000i			0.0913433i
$304$		0			0
$305$	−13.2238			−0.757194
$306$		0			0
$307$	−11.3657			−0.648677			−0.324339	−	0.945941i	$-0.605142\pi$
$307$	−11.3657			−0.648677			−0.324339	+	0.945941i	$0.605142\pi$
$308$		0			0
$309$	−28.0899			−1.59798
$310$		0			0
$311$	14.5545			0.825309			0.412654	−	0.910888i	$-0.364602\pi$
$311$	14.5545			0.825309			0.412654	+	0.910888i	$0.364602\pi$
$312$		0			0
$313$			7.75419i			0.438292i	0.975692	+	0.219146i	$0.0703272\pi$
$313$			7.75419i			0.438292i	−0.975692	+	0.219146i	$0.929673\pi$
$314$		0			0
$315$	−13.4759	+	15.8384i	−0.759280	+	0.892395i
$316$		0			0
$317$	9.31447			0.523153			0.261577	−	0.965183i	$-0.415758\pi$
$317$	9.31447			0.523153			0.261577	+	0.965183i	$0.415758\pi$
$318$		0			0
$319$			8.45446i			0.473359i
$320$		0			0
$321$			50.2589i			2.80518i
$322$		0			0
$323$		−	5.65256i		−	0.314517i
$324$		0			0
$325$			5.60035i			0.310652i
$326$		0			0
$327$	−58.7003			−3.24613
$328$		0			0
$329$	−0.700728	−	0.596203i	−0.0386324	−	0.0328698i
$330$		0			0
$331$		−	22.9879i		−	1.26353i	−0.775161	−	0.631764i	$-0.782331\pi$
$331$		−	22.9879i		−	1.26353i	0.775161	−	0.631764i	$-0.217669\pi$
$332$		0			0
$333$	30.9088			1.69379
$334$		0			0
$335$	15.3113			0.836544
$336$		0			0
$337$	6.17653			0.336457			0.168228	−	0.985748i	$-0.446195\pi$
$337$	6.17653			0.336457			0.168228	+	0.985748i	$0.446195\pi$
$338$		0			0
$339$	58.3958			3.17163
$340$		0			0
$341$		−	11.2376i		−	0.608549i
$342$		0			0
$343$	−9.58759	+	15.8454i	−0.517681	+	0.855573i
$344$		0			0
$345$	−3.25963			−0.175493
$346$		0			0
$347$		−	9.47042i		−	0.508399i	−0.967152	−	0.254199i	$-0.918188\pi$
$347$		−	9.47042i		−	0.508399i	0.967152	−	0.254199i	$-0.0818119\pi$
$348$		0			0
$349$		−	22.5004i		−	1.20442i	−0.798339	−	0.602209i	$-0.794288\pi$
$349$		−	22.5004i		−	1.20442i	0.798339	−	0.602209i	$-0.205712\pi$
$350$		0			0
$351$			89.6943i			4.78753i
$352$		0			0
$353$		−	27.2798i		−	1.45196i	−0.687717	−	0.725979i	$-0.741387\pi$
$353$		−	27.2798i		−	1.45196i	0.687717	−	0.725979i	$-0.258613\pi$
$354$		0			0
$355$	0.574410			0.0304865
$356$		0			0
$357$	−10.5021	+	12.3433i	−0.555831	+	0.653278i
$358$		0			0
$359$			15.4879i			0.817421i	0.912664	+	0.408711i	$0.134021\pi$
$359$			15.4879i			0.817421i	−0.912664	+	0.408711i	$0.865979\pi$
$360$		0			0
$361$	−9.75224			−0.513276
$362$		0			0
$363$	−29.4614			−1.54632
$364$		0			0
$365$	13.6466			0.714295
$366$		0			0
$367$	0.442314			0.0230886			0.0115443	−	0.999933i	$-0.496325\pi$
$367$	0.442314			0.0230886			0.0115443	+	0.999933i	$0.496325\pi$
$368$		0			0
$369$		−	65.8534i		−	3.42819i
$370$		0			0
$371$	−24.5256	−	20.8673i	−1.27331	−	1.08337i
$372$		0			0
$373$	−1.61184			−0.0834581			−0.0417291	−	0.999129i	$-0.513287\pi$
$373$	−1.61184			−0.0834581			−0.0417291	+	0.999129i	$0.513287\pi$
$374$		0			0
$375$			3.29545i			0.170176i
$376$		0			0
$377$		−	32.9891i		−	1.69903i
$378$		0			0
$379$		−	14.5583i		−	0.747811i	−0.927467	−	0.373905i	$-0.878018\pi$
$379$		−	14.5583i		−	0.747811i	0.927467	−	0.373905i	$-0.121982\pi$
$380$		0			0
$381$			20.4929i			1.04989i
$382$		0			0
$383$	−5.85765			−0.299312			−0.149656	−	0.988738i	$-0.547817\pi$
$383$	−5.85765			−0.299312			−0.149656	+	0.988738i	$0.547817\pi$
$384$		0			0
$385$	−2.89216	−	2.46075i	−0.147398	−	0.125411i
$386$		0			0
$387$			7.00122i			0.355892i
$388$		0			0
$389$	−0.672103			−0.0340770			−0.0170385	−	0.999855i	$-0.505424\pi$
$389$	−0.672103			−0.0340770			−0.0170385	+	0.999855i	$0.505424\pi$
$390$		0			0
$391$	−1.83857			−0.0929805
$392$		0			0
$393$	17.4209			0.878768
$394$		0			0
$395$	3.17455			0.159729
$396$		0			0
$397$			4.62099i			0.231921i	0.993254	+	0.115960i	$0.0369945\pi$
$397$			4.62099i			0.231921i	−0.993254	+	0.115960i	$0.963005\pi$
$398$		0			0
$399$	−20.1941	−	17.1818i	−1.01097	−	0.860166i
$400$		0			0
$401$	29.8603			1.49115			0.745576	−	0.666421i	$-0.232175\pi$
$401$	29.8603			1.49115			0.745576	+	0.666421i	$0.232175\pi$
$402$		0			0
$403$			43.8487i			2.18426i
$404$		0			0
$405$			29.1994i			1.45093i
$406$		0			0
$407$			5.64406i			0.279766i
$408$		0			0
$409$			9.08811i			0.449378i	0.974431	+	0.224689i	$0.0721366\pi$
$409$			9.08811i			0.449378i	−0.974431	+	0.224689i	$0.927863\pi$
$410$		0			0
$411$	−24.6591			−1.21634
$412$		0			0
$413$	−8.70820	−	7.40924i	−0.428502	−	0.364585i
$414$		0			0
$415$			7.59090i			0.372623i
$416$		0			0
$417$	−46.0344			−2.25431
$418$		0			0
$419$	−27.1192			−1.32486			−0.662430	−	0.749124i	$-0.730475\pi$
$419$	−27.1192			−1.32486			−0.662430	+	0.749124i	$0.730475\pi$
$420$		0			0
$421$	13.7788			0.671537			0.335768	−	0.941945i	$-0.391004\pi$
$421$	13.7788			0.671537			0.335768	+	0.941945i	$0.391004\pi$
$422$		0			0
$423$	−2.73326			−0.132895
$424$		0			0
$425$			1.85877i			0.0901638i
$426$		0			0
$427$	−22.6721	+	26.6469i	−1.09718	+	1.28954i
$428$		0			0
$429$	−26.4887			−1.27889
$430$		0			0
$431$		−	29.3812i		−	1.41524i	−0.706591	−	0.707622i	$-0.749768\pi$
$431$		−	29.3812i		−	1.41524i	0.706591	−	0.707622i	$-0.250232\pi$
$432$		0			0
$433$		−	5.04892i		−	0.242636i	−0.992614	−	0.121318i	$-0.961288\pi$
$433$		−	5.04892i		−	0.242636i	0.992614	−	0.121318i	$-0.0387120\pi$
$434$		0			0
$435$		−	19.4120i		−	0.930733i
$436$		0			0
$437$		−	3.00796i		−	0.143890i
$438$		0			0
$439$	10.2167			0.487617			0.243808	−	0.969823i	$-0.421603\pi$
$439$	10.2167			0.487617			0.243808	+	0.969823i	$0.421603\pi$
$440$		0			0
$441$	8.81127	+	54.3098i	0.419584	+	2.58618i
$442$		0			0
$443$			9.75208i			0.463335i	0.972795	+	0.231668i	$0.0744182\pi$
$443$			9.75208i			0.463335i	−0.972795	+	0.231668i	$0.925582\pi$
$444$		0			0
$445$	8.08220			0.383133
$446$		0			0
$447$	0.389689			0.0184317
$448$		0			0
$449$	0.478289			0.0225719			0.0112859	−	0.999936i	$-0.496407\pi$
$449$	0.478289			0.0225719			0.0112859	+	0.999936i	$0.496407\pi$
$450$		0			0
$451$	12.0251			0.566238
$452$		0			0
$453$			40.3459i			1.89562i
$454$		0			0
$455$	11.2851	+	9.60177i	0.529054	+	0.450138i
$456$		0			0
$457$	−1.01299			−0.0473855			−0.0236928	−	0.999719i	$-0.507542\pi$
$457$	−1.01299			−0.0473855			−0.0236928	+	0.999719i	$0.507542\pi$
$458$		0			0
$459$			29.7698i			1.38954i
$460$		0			0
$461$			3.79232i			0.176626i	0.996093	+	0.0883130i	$0.0281476\pi$
$461$			3.79232i			0.176626i	−0.996093	+	0.0883130i	$0.971852\pi$
$462$		0			0
$463$		−	12.6689i		−	0.588775i	−0.955686	−	0.294387i	$-0.904884\pi$
$463$		−	12.6689i		−	0.588775i	0.955686	−	0.294387i	$-0.0951156\pi$
$464$		0			0
$465$			25.8022i			1.19655i
$466$		0			0
$467$	24.6432			1.14035			0.570176	−	0.821523i	$-0.306875\pi$
$467$	24.6432			1.14035			0.570176	+	0.821523i	$0.306875\pi$
$468$		0			0
$469$	26.2511	−	30.8533i	1.21216	−	1.42467i
$470$		0			0
$471$			18.4695i			0.851030i
$472$		0			0
$473$	−1.27845			−0.0587831
$474$		0			0
$475$	−3.04101			−0.139531
$476$		0			0
$477$	−95.6647			−4.38018
$478$		0			0
$479$	1.87711			0.0857674			0.0428837	−	0.999080i	$-0.486346\pi$
$479$	1.87711			0.0857674			0.0428837	+	0.999080i	$0.486346\pi$
$480$		0			0
$481$		−	22.0230i		−	1.00416i
$482$		0			0
$483$	−5.58861	+	6.56839i	−0.254291	+	0.298872i
$484$		0			0
$485$	−4.66048			−0.211622
$486$		0			0
$487$		−	14.4128i		−	0.653106i	−0.945179	−	0.326553i	$-0.894113\pi$
$487$		−	14.4128i		−	0.653106i	0.945179	−	0.326553i	$-0.105887\pi$
$488$		0			0
$489$			36.5203i			1.65150i
$490$		0			0
$491$		−	20.6556i		−	0.932175i	−0.884739	−	0.466087i	$-0.845663\pi$
$491$		−	20.6556i		−	0.932175i	0.884739	−	0.466087i	$-0.154337\pi$
$492$		0			0
$493$		−	10.9492i		−	0.493126i
$494$		0			0
$495$	−11.2811			−0.507049
$496$		0			0
$497$	0.984822	−	1.15748i	0.0441753	−	0.0519200i
$498$		0			0
$499$		−	36.3726i		−	1.62826i	−0.580681	−	0.814131i	$-0.697214\pi$
$499$		−	36.3726i		−	1.62826i	0.580681	−	0.814131i	$-0.302786\pi$
$500$		0			0
$501$	7.31818			0.326952
$502$		0			0
$503$	9.05110			0.403568			0.201784	−	0.979430i	$-0.435326\pi$
$503$	9.05110			0.403568			0.201784	+	0.979430i	$0.435326\pi$
$504$		0			0
$505$	0.482485			0.0214703
$506$		0			0
$507$	60.5175			2.68768
$508$		0			0
$509$			33.9980i			1.50694i	0.657484	+	0.753468i	$0.271621\pi$
$509$			33.9980i			1.50694i	−0.657484	+	0.753468i	$0.728379\pi$
$510$		0			0
$511$	23.3970	−	27.4989i	1.03502	−	1.21648i
$512$		0			0
$513$	−48.7043			−2.15035
$514$		0			0
$515$			8.52386i			0.375606i
$516$		0			0
$517$		−	0.499103i		−	0.0219505i
$518$		0			0
$519$		−	43.9071i		−	1.92731i
$520$		0			0
$521$			15.5810i			0.682618i	0.939951	+	0.341309i	$0.110870\pi$
$521$			15.5810i			0.682618i	−0.939951	+	0.341309i	$0.889130\pi$
$522$		0			0
$523$	−18.6258			−0.814449			−0.407225	−	0.913328i	$-0.633503\pi$
$523$	−18.6258			−0.814449			−0.407225	+	0.913328i	$0.633503\pi$
$524$		0			0
$525$	6.64057	+	5.65002i	0.289818	+	0.246587i
$526$		0			0
$527$			14.5535i			0.633962i
$528$		0			0
$529$	22.0216			0.957462
$530$		0			0
$531$	−33.9672			−1.47405
$532$		0			0
$533$	−46.9215			−2.03240
$534$		0			0
$535$	15.2510			0.659358
$536$		0			0
$537$			84.6618i			3.65343i
$538$		0			0
$539$	−9.91716	+	1.60897i	−0.427162	+	0.0693033i
$540$		0			0
$541$	−7.29718			−0.313730			−0.156865	−	0.987620i	$-0.550139\pi$
$541$	−7.29718			−0.313730			−0.156865	+	0.987620i	$0.550139\pi$
$542$		0			0
$543$			9.48465i			0.407025i
$544$		0			0
$545$			17.8125i			0.763005i
$546$		0			0
$547$			24.6305i			1.05312i	0.850137	+	0.526561i	$0.176519\pi$
$547$			24.6305i			1.05312i	−0.850137	+	0.526561i	$0.823481\pi$
$548$		0			0
$549$			103.939i			4.43601i
$550$		0			0
$551$	17.9132			0.763128
$552$		0			0
$553$	5.44274	−	6.39695i	0.231449	−	0.272026i
$554$		0			0
$555$		−	12.9591i		−	0.550084i
$556$		0			0
$557$	−46.8742			−1.98612			−0.993062	−	0.117595i	$-0.962482\pi$
$557$	−46.8742			−1.98612			−0.993062	+	0.117595i	$0.962482\pi$
$558$		0			0
$559$	4.98847			0.210990
$560$		0			0
$561$	−8.79169			−0.371185
$562$		0			0
$563$	−29.9499			−1.26224			−0.631119	−	0.775686i	$-0.717404\pi$
$563$	−29.9499			−1.26224			−0.631119	+	0.775686i	$0.717404\pi$
$564$		0			0
$565$		−	17.7201i		−	0.745492i
$566$		0			0
$567$	58.8389	+	50.0621i	2.47100	+	2.10241i
$568$		0			0
$569$	13.1362			0.550700			0.275350	−	0.961344i	$-0.411206\pi$
$569$	13.1362			0.550700			0.275350	+	0.961344i	$0.411206\pi$
$570$		0			0
$571$		−	5.48029i		−	0.229343i	−0.993403	−	0.114672i	$-0.963418\pi$
$571$		−	5.48029i		−	0.229343i	0.993403	−	0.114672i	$-0.0365816\pi$
$572$		0			0
$573$		−	55.8034i		−	2.33122i
$574$		0			0
$575$			0.989130i			0.0412496i
$576$		0			0
$577$		−	8.31369i		−	0.346103i	−0.984913	−	0.173052i	$-0.944637\pi$
$577$		−	8.31369i		−	0.346103i	0.984913	−	0.173052i	$-0.0553628\pi$
$578$		0			0
$579$	−60.7794			−2.52591
$580$		0			0
$581$	15.2962	+	13.0145i	0.634594	+	0.539934i
$582$		0			0
$583$		−	17.4687i		−	0.723480i
$584$		0			0
$585$	44.0187			1.81995
$586$		0			0
$587$	−9.79972			−0.404478			−0.202239	−	0.979336i	$-0.564822\pi$
$587$	−9.79972			−0.404478			−0.202239	+	0.979336i	$0.564822\pi$
$588$		0			0
$589$	−23.8100			−0.981075
$590$		0			0
$591$	21.8964			0.900697
$592$		0			0
$593$			11.6228i			0.477290i	0.971107	+	0.238645i	$0.0767033\pi$
$593$			11.6228i			0.477290i	−0.971107	+	0.238645i	$0.923297\pi$
$594$		0			0
$595$	3.74556	+	3.18686i	0.153553	+	0.130648i
$596$		0			0
$597$	−49.4954			−2.02571
$598$		0			0
$599$		−	16.6809i		−	0.681561i	−0.940143	−	0.340781i	$-0.889309\pi$
$599$		−	16.6809i		−	0.681561i	0.940143	−	0.340781i	$-0.110691\pi$
$600$		0			0
$601$			4.67971i			0.190889i	0.995435	+	0.0954446i	$0.0304273\pi$
$601$			4.67971i			0.190889i	−0.995435	+	0.0954446i	$0.969573\pi$
$602$		0			0
$603$		−	120.346i		−	4.90088i
$604$		0			0
$605$			8.94002i			0.363464i
$606$		0			0
$607$	−15.0519			−0.610939			−0.305469	−	0.952202i	$-0.598813\pi$
$607$	−15.0519			−0.610939			−0.305469	+	0.952202i	$0.598813\pi$
$608$		0			0
$609$	−39.1165	−	33.2817i	−1.58508	−	1.34864i
$610$		0			0
$611$			1.94749i			0.0787868i
$612$		0			0
$613$	−4.84689			−0.195764			−0.0978821	−	0.995198i	$-0.531207\pi$
$613$	−4.84689			−0.195764			−0.0978821	+	0.995198i	$0.531207\pi$
$614$		0			0
$615$	−27.6103			−1.11335
$616$		0			0
$617$	21.4324			0.862834			0.431417	−	0.902153i	$-0.358014\pi$
$617$	21.4324			0.862834			0.431417	+	0.902153i	$0.358014\pi$
$618$		0			0
$619$	−16.4388			−0.660731			−0.330365	−	0.943853i	$-0.607172\pi$
$619$	−16.4388			−0.660731			−0.330365	+	0.943853i	$0.607172\pi$
$620$		0			0
$621$			15.8417i			0.635707i
$622$		0			0
$623$	13.8569	−	16.2862i	0.555164	−	0.652493i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	14.3835i		−	0.574421i
$628$		0			0
$629$		−	7.30950i		−	0.291449i
$630$		0			0
$631$			9.54286i			0.379895i	0.981794	+	0.189948i	$0.0608318\pi$
$631$			9.54286i			0.379895i	−0.981794	+	0.189948i	$0.939168\pi$
$632$		0			0
$633$		−	73.9538i		−	2.93940i
$634$		0			0
$635$	6.21856			0.246776
$636$		0			0
$637$	38.6965	−	6.27816i	1.53321	−	0.248750i
$638$		0			0
$639$		−	4.51486i		−	0.178605i
$640$		0			0
$641$	8.68996			0.343233			0.171616	−	0.985164i	$-0.445101\pi$
$641$	8.68996			0.343233			0.171616	+	0.985164i	$0.445101\pi$
$642$		0			0
$643$	36.6682			1.44605			0.723026	−	0.690820i	$-0.242750\pi$
$643$	36.6682			1.44605			0.723026	+	0.690820i	$0.242750\pi$
$644$		0			0
$645$	2.93540			0.115581
$646$		0			0
$647$	5.26708			0.207070			0.103535	−	0.994626i	$-0.466985\pi$
$647$	5.26708			0.207070			0.103535	+	0.994626i	$0.466985\pi$
$648$		0			0
$649$		−	6.20253i		−	0.243471i
$650$		0			0
$651$	51.9932	+	44.2376i	2.03777	+	1.73381i
$652$		0			0
$653$	−36.0492			−1.41072			−0.705358	−	0.708852i	$-0.749214\pi$
$653$	−36.0492			−1.41072			−0.705358	+	0.708852i	$0.749214\pi$
$654$		0			0
$655$		−	5.28635i		−	0.206555i
$656$		0			0
$657$		−	107.262i		−	4.18469i
$658$		0			0
$659$		−	2.91466i		−	0.113539i	−0.998387	−	0.0567695i	$-0.981920\pi$
$659$		−	2.91466i		−	0.113539i	0.998387	−	0.0567695i	$-0.0180800\pi$
$660$		0			0
$661$		−	26.9858i		−	1.04963i	−0.851217	−	0.524814i	$-0.824135\pi$
$661$		−	26.9858i		−	1.04963i	0.851217	−	0.524814i	$-0.175865\pi$
$662$		0			0
$663$	34.3050			1.33229
$664$		0			0
$665$	−5.21380	+	6.12786i	−0.202182	+	0.237628i
$666$		0			0
$667$		−	5.82651i		−	0.225603i
$668$		0			0
$669$	8.48832			0.328177
$670$		0			0
$671$	−18.9796			−0.732701
$672$		0			0
$673$	0.856509			0.0330160			0.0165080	−	0.999864i	$-0.494745\pi$
$673$	0.856509			0.0330160			0.0165080	+	0.999864i	$0.494745\pi$
$674$		0			0
$675$	16.0158			0.616449
$676$		0			0
$677$		−	25.2852i		−	0.971788i	−0.874018	−	0.485894i	$-0.838494\pi$
$677$		−	25.2852i		−	0.971788i	0.874018	−	0.485894i	$-0.161506\pi$
$678$		0			0
$679$	−7.99036	+	9.39121i	−0.306642	+	0.360402i
$680$		0			0
$681$	−37.6053			−1.44104
$682$		0			0
$683$		−	8.97155i		−	0.343287i	−0.985159	−	0.171643i	$-0.945092\pi$
$683$		−	8.97155i		−	0.343287i	0.985159	−	0.171643i	$-0.0549077\pi$
$684$		0			0
$685$			7.48277i			0.285902i
$686$		0			0
$687$			67.1437i			2.56169i
$688$		0			0
$689$			68.1625i			2.59678i
$690$		0			0
$691$	10.2090			0.388368			0.194184	−	0.980965i	$-0.437794\pi$
$691$	10.2090			0.388368			0.194184	+	0.980965i	$0.437794\pi$
$692$		0			0
$693$	−19.3414	+	22.7323i	−0.734720	+	0.863528i
$694$		0			0
$695$			13.9691i			0.529877i
$696$		0			0
$697$	−15.5734			−0.589884
$698$		0			0
$699$	−11.4162			−0.431802
$700$		0			0
$701$	1.01166			0.0382098			0.0191049	−	0.999817i	$-0.493918\pi$
$701$	1.01166			0.0382098			0.0191049	+	0.999817i	$0.493918\pi$
$702$		0			0
$703$	11.9586			0.451026
$704$		0			0
$705$			1.14597i			0.0431597i
$706$		0			0
$707$	0.827217	−	0.972241i	0.0311107	−	0.0365649i
$708$		0			0
$709$	28.4525			1.06855			0.534277	−	0.845309i	$-0.320584\pi$
$709$	28.4525			1.06855			0.534277	+	0.845309i	$0.320584\pi$
$710$		0			0
$711$		−	24.9519i		−	0.935770i
$712$		0			0
$713$			7.74453i			0.290035i
$714$		0			0
$715$			8.03797i			0.300603i
$716$		0			0
$717$		−	53.0214i		−	1.98012i
$718$		0			0
$719$	40.3885			1.50624			0.753119	−	0.657884i	$-0.228548\pi$
$719$	40.3885			1.50624			0.753119	+	0.657884i	$0.228548\pi$
$720$		0			0
$721$	17.1762	+	14.6141i	0.639674	+	0.544257i
$722$		0			0
$723$			17.9774i			0.668586i
$724$		0			0
$725$	−5.89054			−0.218769
$726$		0			0
$727$	26.6097			0.986900			0.493450	−	0.869774i	$-0.335736\pi$
$727$	26.6097			0.986900			0.493450	+	0.869774i	$0.335736\pi$
$728$		0			0
$729$	71.1687			2.63588
$730$		0			0
$731$	1.65569			0.0612379
$732$		0			0
$733$		−	14.8503i		−	0.548507i	−0.961657	−	0.274254i	$-0.911569\pi$
$733$		−	14.8503i		−	0.548507i	0.961657	−	0.274254i	$-0.0884308\pi$
$734$		0			0
$735$	22.7704	−	3.69429i	0.839899	−	0.136266i
$736$		0			0
$737$	21.9757			0.809485
$738$		0			0
$739$		−	18.3496i		−	0.675002i	−0.941325	−	0.337501i	$-0.890418\pi$
$739$		−	18.3496i		−	0.675002i	0.941325	−	0.337501i	$-0.109582\pi$
$740$		0			0
$741$			56.1240i			2.06177i
$742$		0			0
$743$		−	38.1199i		−	1.39848i	−0.714885	−	0.699242i	$-0.753521\pi$
$743$		−	38.1199i		−	1.39848i	0.714885	−	0.699242i	$-0.246479\pi$
$744$		0			0
$745$		−	0.118251i		−	0.00433237i
$746$		0			0
$747$	59.6644			2.18300
$748$		0			0
$749$	26.1477	−	30.7318i	0.955417	−	1.12292i
$750$		0			0
$751$			0.868435i			0.0316896i	0.999874	+	0.0158448i	$0.00504377\pi$
$751$			0.868435i			0.0316896i	−0.999874	+	0.0158448i	$0.994956\pi$
$752$		0			0
$753$	−62.9624			−2.29448
$754$		0			0
$755$	12.2429			0.445566
$756$		0			0
$757$	48.5225			1.76358			0.881791	−	0.471640i	$-0.156338\pi$
$757$	48.5225			1.76358			0.881791	+	0.471640i	$0.156338\pi$
$758$		0			0
$759$	−4.67842			−0.169816
$760$		0			0
$761$		−	18.4801i		−	0.669902i	−0.942236	−	0.334951i	$-0.891280\pi$
$761$		−	18.4801i		−	0.669902i	0.942236	−	0.334951i	$-0.108720\pi$
$762$		0			0
$763$	35.8935	+	30.5395i	1.29943	+	1.10560i
$764$		0			0
$765$	14.6099			0.528223
$766$		0			0
$767$			24.2021i			0.873888i
$768$		0			0
$769$			23.8197i			0.858960i	0.903076	+	0.429480i	$0.141303\pi$
$769$			23.8197i			0.858960i	−0.903076	+	0.429480i	$0.858697\pi$
$770$		0			0
$771$		−	35.4813i		−	1.27783i
$772$		0			0
$773$			2.14017i			0.0769767i	0.999259	+	0.0384883i	$0.0122542\pi$
$773$			2.14017i			0.0769767i	−0.999259	+	0.0384883i	$0.987746\pi$
$774$		0			0
$775$	7.82964			0.281249
$776$		0			0
$777$	−26.1135	−	22.2183i	−0.936818	−	0.797077i
$778$		0			0
$779$		−	25.4786i		−	0.912864i
$780$		0			0
$781$	0.824429			0.0295004
$782$		0			0
$783$	−94.3419			−3.37150
$784$		0			0
$785$	5.60455			0.200035
$786$		0			0
$787$	−20.9675			−0.747411			−0.373706	−	0.927547i	$-0.621913\pi$
$787$	−20.9675			−0.747411			−0.373706	+	0.927547i	$0.621913\pi$
$788$		0			0
$789$		−	35.4620i		−	1.26248i
$790$		0			0
$791$	−35.7074	−	30.3811i	−1.26961	−	1.08023i
$792$		0			0
$793$	74.0581			2.62988
$794$		0			0
$795$			40.1092i			1.42253i
$796$		0			0
$797$			46.1343i			1.63416i	0.576524	+	0.817080i	$0.304409\pi$
$797$			46.1343i			1.63416i	−0.576524	+	0.817080i	$0.695591\pi$
$798$		0			0
$799$			0.646376i			0.0228672i
$800$		0			0
$801$		−	63.5260i		−	2.24458i
$802$		0			0
$803$	19.5864			0.691190
$804$		0			0
$805$	1.99317	+	1.69586i	0.0702500	+	0.0597711i
$806$		0			0
$807$		−	59.1775i		−	2.08315i
$808$		0			0
$809$	−30.7590			−1.08143			−0.540715	−	0.841206i	$-0.681846\pi$
$809$	−30.7590			−1.08143			−0.540715	+	0.841206i	$0.681846\pi$
$810$		0			0
$811$	−26.1227			−0.917293			−0.458647	−	0.888619i	$-0.651666\pi$
$811$	−26.1227			−0.917293			−0.458647	+	0.888619i	$0.651666\pi$
$812$		0			0
$813$	54.5847			1.91437
$814$		0			0
$815$	11.0820			0.388187
$816$		0			0
$817$			2.70876i			0.0947675i
$818$		0			0
$819$	75.4697	−	88.7008i	2.63713	−	3.09946i
$820$		0			0
$821$	−51.8580			−1.80986			−0.904928	−	0.425565i	$-0.860075\pi$
$821$	−51.8580			−1.80986			−0.904928	+	0.425565i	$0.860075\pi$
$822$		0			0
$823$			32.0866i			1.11847i	0.829010	+	0.559234i	$0.188905\pi$
$823$			32.0866i			1.11847i	−0.829010	+	0.559234i	$0.811095\pi$
$824$		0			0
$825$			4.72983i			0.164672i
$826$		0			0
$827$			19.1319i			0.665282i	0.943054	+	0.332641i	$0.107940\pi$
$827$			19.1319i			0.665282i	−0.943054	+	0.332641i	$0.892060\pi$
$828$		0			0
$829$			14.1288i			0.490714i	0.969433	+	0.245357i	$0.0789052\pi$
$829$			14.1288i			0.490714i	−0.969433	+	0.245357i	$0.921095\pi$
$830$		0			0
$831$	24.2926			0.842701
$832$		0			0
$833$	12.8435	−	2.08374i	0.445000	−	0.0721973i
$834$		0			0
$835$		−	2.22069i		−	0.0768503i
$836$		0			0
$837$	125.398			4.33439
$838$		0			0
$839$	−41.3350			−1.42704			−0.713522	−	0.700633i	$-0.752901\pi$
$839$	−41.3350			−1.42704			−0.713522	+	0.700633i	$0.752901\pi$
$840$		0			0
$841$	5.69845			0.196498
$842$		0			0
$843$	30.0554			1.03516
$844$		0			0
$845$		−	18.3640i		−	0.631740i
$846$		0			0
$847$	18.0148	+	15.3276i	0.618996	+	0.526663i
$848$		0			0
$849$	−90.4862			−3.10548
$850$		0			0
$851$		−	3.88968i		−	0.133337i
$852$		0			0
$853$		−	18.9283i		−	0.648092i	−0.946041	−	0.324046i	$-0.894957\pi$
$853$		−	18.9283i		−	0.648092i	0.946041	−	0.324046i	$-0.105043\pi$
$854$		0			0
$855$			23.9023i			0.817442i
$856$		0			0
$857$			24.1669i			0.825527i	0.910838	+	0.412763i	$0.135436\pi$
$857$			24.1669i			0.825527i	−0.910838	+	0.412763i	$0.864564\pi$
$858$		0			0
$859$	−32.7102			−1.11606			−0.558028	−	0.829822i	$-0.688442\pi$
$859$	−32.7102			−1.11606			−0.558028	+	0.829822i	$0.688442\pi$
$860$		0			0
$861$	−47.3377	+	55.6367i	−1.61326	+	1.89609i
$862$		0			0
$863$			41.8221i			1.42364i	0.702361	+	0.711821i	$0.252129\pi$
$863$			41.8221i			1.42364i	−0.702361	+	0.711821i	$0.747871\pi$
$864$		0			0
$865$	−13.3236			−0.453014
$866$		0			0
$867$	−44.6367			−1.51594
$868$		0			0
$869$	4.55631			0.154562
$870$		0			0
$871$	−85.7486			−2.90548
$872$		0			0
$873$			36.6313i			1.23978i
$874$		0			0
$875$	1.71449	−	2.01507i	0.0579604	−	0.0681219i
$876$		0			0
$877$	18.4511			0.623050			0.311525	−	0.950238i	$-0.399160\pi$
$877$	18.4511			0.623050			0.311525	+	0.950238i	$0.399160\pi$
$878$		0			0
$879$		−	74.1919i		−	2.50243i
$880$		0			0
$881$		−	41.0743i		−	1.38383i	−0.721979	−	0.691915i	$-0.756767\pi$
$881$		−	41.0743i		−	1.38383i	0.721979	−	0.691915i	$-0.243233\pi$
$882$		0			0
$883$			30.8276i			1.03743i	0.854946	+	0.518716i	$0.173590\pi$
$883$			30.8276i			1.03743i	−0.854946	+	0.518716i	$0.826410\pi$
$884$		0			0
$885$			14.2414i			0.478719i
$886$		0			0
$887$	39.7932			1.33612			0.668062	−	0.744105i	$-0.267124\pi$
$887$	39.7932			1.33612			0.668062	+	0.744105i	$0.267124\pi$
$888$		0			0
$889$	10.6617	−	12.5308i	0.357581	−	0.420271i
$890$		0			0
$891$			41.9088i			1.40400i
$892$		0			0
$893$	−1.05749			−0.0353876
$894$		0			0
$895$	25.6905			0.858739
$896$		0			0
$897$	18.2551			0.609519
$898$		0			0
$899$	−46.1208			−1.53821
$900$		0			0
$901$			22.6233i			0.753693i
$902$		0			0
$903$	5.03272	−	5.91504i	0.167478	−	0.196840i
$904$		0			0
$905$	2.87811			0.0956715
$906$		0			0
$907$			21.9225i			0.727925i	0.931414	+	0.363962i	$0.118576\pi$
$907$			21.9225i			0.727925i	−0.931414	+	0.363962i	$0.881424\pi$
$908$		0			0
$909$		−	3.79232i		−	0.125783i
$910$		0			0
$911$		−	16.2898i		−	0.539704i	−0.962902	−	0.269852i	$-0.913025\pi$
$911$		−	16.2898i		−	0.539704i	0.962902	−	0.269852i	$-0.0869749\pi$
$912$		0			0
$913$			10.8949i			0.360570i
$914$		0			0
$915$	43.5784			1.44066
$916$		0			0
$917$	−10.6524	−	9.06340i	−0.351772	−	0.299300i
$918$		0			0
$919$			36.2223i			1.19486i	0.801920	+	0.597432i	$0.203812\pi$
$919$			36.2223i			1.19486i	−0.801920	+	0.597432i	$0.796188\pi$
$920$		0			0
$921$	37.4552			1.23419
$922$		0			0
$923$	−3.21690			−0.105886
$924$		0			0
$925$	−3.93243			−0.129297
$926$		0			0
$927$	66.9974			2.20048
$928$		0			0
$929$		−	14.6208i		−	0.479692i	−0.970811	−	0.239846i	$-0.922903\pi$
$929$		−	14.6208i		−	0.479692i	0.970811	−	0.239846i	$-0.0770970\pi$
$930$		0			0
$931$	3.40907	+	21.0123i	0.111728	+	0.688652i
$932$		0			0
$933$	−47.9635			−1.57026
$934$		0			0
$935$			2.66783i			0.0872473i
$936$		0			0
$937$			5.49926i			0.179653i	0.995957	+	0.0898265i	$0.0286312\pi$
$937$			5.49926i			0.179653i	−0.995957	+	0.0898265i	$0.971369\pi$
$938$		0			0
$939$		−	25.5535i		−	0.833908i
$940$		0			0
$941$			13.1893i			0.429959i	0.976619	+	0.214979i	$0.0689684\pi$
$941$			13.1893i			0.429959i	−0.976619	+	0.214979i	$0.931032\pi$
$942$		0			0
$943$	−8.28724			−0.269870
$944$		0			0
$945$	27.4590	−	32.2730i	0.893242	−	1.04984i
$946$		0			0
$947$			3.02749i			0.0983803i	0.998789	+	0.0491901i	$0.0156640\pi$
$947$			3.02749i			0.0983803i	−0.998789	+	0.0491901i	$0.984336\pi$
$948$		0			0
$949$	−76.4258			−2.48089
$950$		0			0
$951$	−30.6954			−0.995366
$952$		0			0
$953$	−3.56005			−0.115321			−0.0576607	−	0.998336i	$-0.518364\pi$
$953$	−3.56005			−0.115321			−0.0576607	+	0.998336i	$0.518364\pi$
$954$		0			0
$955$	−16.9335			−0.547954
$956$		0			0
$957$		−	27.8613i		−	0.900626i
$958$		0			0
$959$	15.0783	+	12.8292i	0.486905	+	0.414275i
$960$		0			0
$961$	30.3032			0.977523
$962$		0			0
$963$		−	119.873i		−	3.86284i
$964$		0			0
$965$			18.4434i			0.593715i
$966$		0			0
$967$		−	36.3856i		−	1.17008i	−0.811004	−	0.585040i	$-0.801079\pi$
$967$		−	36.3856i		−	1.17008i	0.811004	−	0.585040i	$-0.198921\pi$
$968$		0			0
$969$			18.6277i			0.598408i
$970$		0			0
$971$	−41.0067			−1.31597			−0.657984	−	0.753032i	$-0.728590\pi$
$971$	−41.0067			−1.31597			−0.657984	+	0.753032i	$0.728590\pi$
$972$		0			0
$973$	28.1487	+	23.9499i	0.902406	+	0.767798i
$974$		0			0
$975$		−	18.4557i		−	0.591055i
$976$		0			0
$977$	−48.3681			−1.54743			−0.773716	−	0.633533i	$-0.781604\pi$
$977$	−48.3681			−1.54743			−0.773716	+	0.633533i	$0.781604\pi$
$978$		0			0
$979$	11.6001			0.370740
$980$		0			0
$981$	140.006			4.47005
$982$		0			0
$983$	3.47969			0.110985			0.0554924	−	0.998459i	$-0.482327\pi$
$983$	3.47969			0.110985			0.0554924	+	0.998459i	$0.482327\pi$
$984$		0			0
$985$		−	6.64444i		−	0.211709i
$986$		0			0
$987$	2.30921	+	1.96476i	0.0735030	+	0.0625389i
$988$		0			0
$989$	0.881061			0.0280161
$990$		0			0
$991$		−	38.6592i		−	1.22805i	−0.789286	−	0.614025i	$-0.789549\pi$
$991$		−	38.6592i		−	1.22805i	0.789286	−	0.614025i	$-0.210451\pi$
$992$		0			0
$993$			75.7554i			2.40403i
$994$		0			0
$995$			15.0193i			0.476144i
$996$		0			0
$997$			4.22709i			0.133873i	0.997757	+	0.0669366i	$0.0213225\pi$
$997$			4.22709i			0.133873i	−0.997757	+	0.0669366i	$0.978677\pi$
$998$		0			0
$999$	−62.9811			−1.99263

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.g.1791.2		16
4.3	odd	2		2240.2.k.f.1791.16		16
7.6	odd	2		2240.2.k.f.1791.15		16
8.3	odd	2		1120.2.k.a.671.1	✓	16
8.5	even	2		1120.2.k.b.671.15	yes	16
28.27	even	2	inner	2240.2.k.g.1791.1		16
56.13	odd	2		1120.2.k.a.671.2	yes	16
56.27	even	2		1120.2.k.b.671.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.k.a.671.1	✓	16	8.3	odd	2
1120.2.k.a.671.2	yes	16	56.13	odd	2
1120.2.k.b.671.15	yes	16	8.5	even	2
1120.2.k.b.671.16	yes	16	56.27	even	2
2240.2.k.f.1791.15		16	7.6	odd	2
2240.2.k.f.1791.16		16	4.3	odd	2
2240.2.k.g.1791.1		16	28.27	even	2	inner
2240.2.k.g.1791.2		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-3.29545 q^{3} +1.00000i q^{5} +(2.01507 + 1.71449i) q^{7} +7.85998 q^{9} +O(q^{10})\)	\(q-3.29545 q^{3} +1.00000i q^{5} +(2.01507 + 1.71449i) q^{7} +7.85998 q^{9} +1.43526i q^{11} -5.60035i q^{13} -3.29545i q^{15} -1.85877i q^{17} +3.04101 q^{19} +(-6.64057 - 5.65002i) q^{21} -0.989130i q^{23} -1.00000 q^{25} -16.0158 q^{27} +5.89054 q^{29} -7.82964 q^{31} -4.72983i q^{33} +(-1.71449 + 2.01507i) q^{35} +3.93243 q^{37} +18.4557i q^{39} -8.37831i q^{41} +0.890743i q^{43} +7.85998i q^{45} -0.347743 q^{47} +(1.12103 + 6.90965i) q^{49} +6.12550i q^{51} -12.1711 q^{53} -1.43526 q^{55} -10.0215 q^{57} -4.32153 q^{59} +13.2238i q^{61} +(15.8384 + 13.4759i) q^{63} +5.60035 q^{65} -15.3113i q^{67} +3.25963i q^{69} -0.574410i q^{71} -13.6466i q^{73} +3.29545 q^{75} +(-2.46075 + 2.89216i) q^{77} -3.17455i q^{79} +29.1994 q^{81} +7.59090 q^{83} +1.85877 q^{85} -19.4120 q^{87} -8.08220i q^{89} +(9.60177 - 11.2851i) q^{91} +25.8022 q^{93} +3.04101i q^{95} +4.66048i q^{97} +11.2811i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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