LMFDB - Embedded newform 2240.2.k.a.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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1791.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2240.1791
Dual form			2240.2.k.a.1791.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +1.00000i q^{5} +(1.73205 - 2.00000i) q^{7} +3.73205i q^{11} -6.46410i q^{13} -1.73205i q^{15} +0.464102i q^{17} -6.00000 q^{19} +(-3.00000 + 3.46410i) q^{21} +5.46410i q^{23} -1.00000 q^{25} +5.19615 q^{27} +5.92820 q^{29} -6.00000 q^{31} -6.46410i q^{33} +(2.00000 + 1.73205i) q^{35} +2.53590 q^{37} +11.1962i q^{39} +3.46410i q^{41} +2.00000i q^{43} -1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} -0.803848i q^{51} -2.00000 q^{53} -3.73205 q^{55} +10.3923 q^{57} -3.46410 q^{59} -2.53590i q^{61} +6.46410 q^{65} +3.46410i q^{67} -9.46410i q^{69} -0.535898i q^{71} -0.928203i q^{73} +1.73205 q^{75} +(7.46410 + 6.46410i) q^{77} +2.66025i q^{79} -9.00000 q^{81} -8.53590 q^{83} -0.464102 q^{85} -10.2679 q^{87} +9.46410i q^{89} +(-12.9282 - 11.1962i) q^{91} +10.3923 q^{93} -6.00000i q^{95} +7.39230i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{19} - 12 q^{21} - 4 q^{25} - 4 q^{29} - 24 q^{31} + 8 q^{35} + 24 q^{37} - 4 q^{49} - 8 q^{53} - 8 q^{55} + 12 q^{65} + 16 q^{77} - 36 q^{81} - 48 q^{83} + 12 q^{85} - 48 q^{87} - 24 q^{91}+O(q^{100}) $ 4 * q - 24 * q^19 - 12 * q^21 - 4 * q^25 - 4 * q^29 - 24 * q^31 + 8 * q^35 + 24 * q^37 - 4 * q^49 - 8 * q^53 - 8 * q^55 + 12 * q^65 + 16 * q^77 - 36 * q^81 - 48 * q^83 + 12 * q^85 - 48 * q^87 - 24 * q^91

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
$3$	−1.73205			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.73205			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.73205i			1.12526i	0.826710	+	0.562628i	$0.190210\pi$
$11$			3.73205i			1.12526i	−0.826710	+	0.562628i	$0.809790\pi$
$12$		0			0
$13$		−	6.46410i		−	1.79282i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$		−	6.46410i		−	1.79282i	0.443227	−	0.896410i	$-0.353834\pi$
$14$		0			0
$15$		−	1.73205i		−	0.447214i
$16$		0			0
$17$			0.464102i			0.112561i	0.998415	+	0.0562806i	$0.0179241\pi$
$17$			0.464102i			0.112561i	−0.998415	+	0.0562806i	$0.982076\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	−3.00000	+	3.46410i	−0.654654	+	0.755929i
$22$		0			0
$23$			5.46410i			1.13934i	0.821872	+	0.569672i	$0.192930\pi$
$23$			5.46410i			1.13934i	−0.821872	+	0.569672i	$0.807070\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$	5.92820			1.10084			0.550420	−	0.834888i	$-0.314468\pi$
$29$	5.92820			1.10084			0.550420	+	0.834888i	$0.314468\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$		−	6.46410i		−	1.12526i
$34$		0			0
$35$	2.00000	+	1.73205i	0.338062	+	0.292770i
$36$		0			0
$37$	2.53590			0.416899			0.208450	−	0.978033i	$-0.433158\pi$
$37$	2.53590			0.416899			0.208450	+	0.978033i	$0.433158\pi$
$38$		0			0
$39$			11.1962i			1.79282i
$40$		0			0
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.73205			−0.252646			−0.126323	−	0.991989i	$-0.540318\pi$
$47$	−1.73205			−0.252646			−0.126323	+	0.991989i	$0.540318\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$		−	0.803848i		−	0.112561i
$52$		0			0
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$	−3.73205			−0.503230
$56$		0			0
$57$	10.3923			1.37649
$58$		0			0
$59$	−3.46410			−0.450988			−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410			−0.450988			−0.225494	+	0.974245i	$0.572400\pi$
$60$		0			0
$61$		−	2.53590i		−	0.324689i	−0.986734	−	0.162344i	$-0.948094\pi$
$61$		−	2.53590i		−	0.324689i	0.986734	−	0.162344i	$-0.0519055\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.46410			0.801773
$66$		0			0
$67$			3.46410i			0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$			3.46410i			0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		−	9.46410i		−	1.13934i
$70$		0			0
$71$		−	0.535898i		−	0.0635994i	−0.999494	−	0.0317997i	$-0.989876\pi$
$71$		−	0.535898i		−	0.0635994i	0.999494	−	0.0317997i	$-0.0101239\pi$
$72$		0			0
$73$		−	0.928203i		−	0.108638i	−0.998524	−	0.0543190i	$-0.982701\pi$
$73$		−	0.928203i		−	0.108638i	0.998524	−	0.0543190i	$-0.0172988\pi$
$74$		0			0
$75$	1.73205			0.200000
$76$		0			0
$77$	7.46410	+	6.46410i	0.850613	+	0.736653i
$78$		0			0
$79$			2.66025i			0.299302i	0.988739	+	0.149651i	$0.0478150\pi$
$79$			2.66025i			0.299302i	−0.988739	+	0.149651i	$0.952185\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	−8.53590			−0.936937			−0.468468	−	0.883480i	$-0.655194\pi$
$83$	−8.53590			−0.936937			−0.468468	+	0.883480i	$0.655194\pi$
$84$		0			0
$85$	−0.464102			−0.0503389
$86$		0			0
$87$	−10.2679			−1.10084
$88$		0			0
$89$			9.46410i			1.00319i	0.865102	+	0.501596i	$0.167254\pi$
$89$			9.46410i			1.00319i	−0.865102	+	0.501596i	$0.832746\pi$
$90$		0			0
$91$	−12.9282	−	11.1962i	−1.35524	−	1.17368i
$92$		0			0
$93$	10.3923			1.07763
$94$		0			0
$95$		−	6.00000i		−	0.615587i
$96$		0			0
$97$			7.39230i			0.750575i	0.926908	+	0.375287i	$0.122456\pi$
$97$			7.39230i			0.750575i	−0.926908	+	0.375287i	$0.877544\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.53590i			0.849354i	0.905345	+	0.424677i	$0.139612\pi$
$101$			8.53590i			0.849354i	−0.905345	+	0.424677i	$0.860388\pi$
$102$		0			0
$103$	−17.1962			−1.69439			−0.847194	−	0.531284i	$-0.821710\pi$
$103$	−17.1962			−1.69439			−0.847194	+	0.531284i	$0.821710\pi$
$104$		0			0
$105$	−3.46410	−	3.00000i	−0.338062	−	0.292770i
$106$		0			0
$107$			18.3923i			1.77805i	0.457857	+	0.889026i	$0.348617\pi$
$107$			18.3923i			1.77805i	−0.457857	+	0.889026i	$0.651383\pi$
$108$		0			0
$109$	−15.9282			−1.52565			−0.762823	−	0.646608i	$-0.776187\pi$
$109$	−15.9282			−1.52565			−0.762823	+	0.646608i	$0.776187\pi$
$110$		0			0
$111$	−4.39230			−0.416899
$112$		0			0
$113$	−1.46410			−0.137731			−0.0688655	−	0.997626i	$-0.521938\pi$
$113$	−1.46410			−0.137731			−0.0688655	+	0.997626i	$0.521938\pi$
$114$		0			0
$115$	−5.46410			−0.509530
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.928203	+	0.803848i	0.0850883	+	0.0736886i
$120$		0			0
$121$	−2.92820			−0.266200
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			8.53590i			0.757438i	0.925512	+	0.378719i	$0.123635\pi$
$127$			8.53590i			0.757438i	−0.925512	+	0.378719i	$0.876365\pi$
$128$		0			0
$129$		−	3.46410i		−	0.304997i
$130$		0			0
$131$	−9.46410			−0.826882			−0.413441	−	0.910531i	$-0.635673\pi$
$131$	−9.46410			−0.826882			−0.413441	+	0.910531i	$0.635673\pi$
$132$		0			0
$133$	−10.3923	+	12.0000i	−0.901127	+	1.04053i
$134$		0			0
$135$			5.19615i			0.447214i
$136$		0			0
$137$	−0.392305			−0.0335169			−0.0167584	−	0.999860i	$-0.505335\pi$
$137$	−0.392305			−0.0335169			−0.0167584	+	0.999860i	$0.505335\pi$
$138$		0			0
$139$	6.92820			0.587643			0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820			0.587643			0.293821	+	0.955860i	$0.405073\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$		0			0
$143$	24.1244			2.01738
$144$		0			0
$145$			5.92820i			0.492310i
$146$		0			0
$147$	1.73205	+	12.0000i	0.142857	+	0.989743i
$148$		0			0
$149$	−16.9282			−1.38681			−0.693406	−	0.720547i	$-0.743891\pi$
$149$	−16.9282			−1.38681			−0.693406	+	0.720547i	$0.743891\pi$
$150$		0			0
$151$		−	4.80385i		−	0.390932i	−0.980711	−	0.195466i	$-0.937378\pi$
$151$		−	4.80385i		−	0.390932i	0.980711	−	0.195466i	$-0.0626219\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	6.00000i		−	0.481932i
$156$		0			0
$157$			19.8564i			1.58471i	0.610058	+	0.792357i	$0.291146\pi$
$157$			19.8564i			1.58471i	−0.610058	+	0.792357i	$0.708854\pi$
$158$		0			0
$159$	3.46410			0.274721
$160$		0			0
$161$	10.9282	+	9.46410i	0.861263	+	0.745876i
$162$		0			0
$163$			20.7846i			1.62798i	0.580881	+	0.813988i	$0.302708\pi$
$163$			20.7846i			1.62798i	−0.580881	+	0.813988i	$0.697292\pi$
$164$		0			0
$165$	6.46410			0.503230
$166$		0			0
$167$	5.19615			0.402090			0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615			0.402090			0.201045	+	0.979582i	$0.435566\pi$
$168$		0			0
$169$	−28.7846			−2.21420
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	20.3205i		−	1.54494i	−0.635051	−	0.772470i	$-0.719021\pi$
$173$		−	20.3205i		−	1.54494i	0.635051	−	0.772470i	$-0.280979\pi$
$174$		0			0
$175$	−1.73205	+	2.00000i	−0.130931	+	0.151186i
$176$		0			0
$177$	6.00000			0.450988
$178$		0			0
$179$			14.3923i			1.07573i	0.843031	+	0.537866i	$0.180769\pi$
$179$			14.3923i			1.07573i	−0.843031	+	0.537866i	$0.819231\pi$
$180$		0			0
$181$		−	12.9282i		−	0.960946i	−0.877010	−	0.480473i	$-0.840465\pi$
$181$		−	12.9282i		−	0.960946i	0.877010	−	0.480473i	$-0.159535\pi$
$182$		0			0
$183$			4.39230i			0.324689i
$184$		0			0
$185$			2.53590i			0.186443i
$186$		0			0
$187$	−1.73205			−0.126660
$188$		0			0
$189$	9.00000	−	10.3923i	0.654654	−	0.755929i
$190$		0			0
$191$			3.19615i			0.231265i	0.993292	+	0.115633i	$0.0368896\pi$
$191$			3.19615i			0.231265i	−0.993292	+	0.115633i	$0.963110\pi$
$192$		0			0
$193$	−2.53590			−0.182538			−0.0912690	−	0.995826i	$-0.529092\pi$
$193$	−2.53590			−0.182538			−0.0912690	+	0.995826i	$0.529092\pi$
$194$		0			0
$195$	−11.1962			−0.801773
$196$		0			0
$197$	−21.3205			−1.51902			−0.759512	−	0.650494i	$-0.774562\pi$
$197$	−21.3205			−1.51902			−0.759512	+	0.650494i	$0.774562\pi$
$198$		0			0
$199$	−3.46410			−0.245564			−0.122782	−	0.992434i	$-0.539182\pi$
$199$	−3.46410			−0.245564			−0.122782	+	0.992434i	$0.539182\pi$
$200$		0			0
$201$		−	6.00000i		−	0.423207i
$202$		0			0
$203$	10.2679	−	11.8564i	0.720669	−	0.832157i
$204$		0			0
$205$	−3.46410			−0.241943
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	22.3923i		−	1.54891i
$210$		0			0
$211$		−	7.19615i		−	0.495404i	−0.968836	−	0.247702i	$-0.920325\pi$
$211$		−	7.19615i		−	0.495404i	0.968836	−	0.247702i	$-0.0796753\pi$
$212$		0			0
$213$			0.928203i			0.0635994i
$214$		0			0
$215$	−2.00000			−0.136399
$216$		0			0
$217$	−10.3923	+	12.0000i	−0.705476	+	0.814613i
$218$		0			0
$219$			1.60770i			0.108638i
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$	−10.2679			−0.687593			−0.343796	−	0.939044i	$-0.611713\pi$
$223$	−10.2679			−0.687593			−0.343796	+	0.939044i	$0.611713\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.33975			−0.221667			−0.110833	−	0.993839i	$-0.535352\pi$
$227$	−3.33975			−0.221667			−0.110833	+	0.993839i	$0.535352\pi$
$228$		0			0
$229$			15.4641i			1.02190i	0.859611	+	0.510948i	$0.170706\pi$
$229$			15.4641i			1.02190i	−0.859611	+	0.510948i	$0.829294\pi$
$230$		0			0
$231$	−12.9282	−	11.1962i	−0.850613	−	0.736653i
$232$		0			0
$233$	−22.9282			−1.50208			−0.751038	−	0.660259i	$-0.770447\pi$
$233$	−22.9282			−1.50208			−0.751038	+	0.660259i	$0.770447\pi$
$234$		0			0
$235$		−	1.73205i		−	0.112987i
$236$		0			0
$237$		−	4.60770i		−	0.299302i
$238$		0			0
$239$		−	27.9808i		−	1.80993i	−0.425491	−	0.904963i	$-0.639899\pi$
$239$		−	27.9808i		−	1.80993i	0.425491	−	0.904963i	$-0.360101\pi$
$240$		0			0
$241$		−	4.39230i		−	0.282933i	−0.989943	−	0.141467i	$-0.954818\pi$
$241$		−	4.39230i		−	0.282933i	0.989943	−	0.141467i	$-0.0451818\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.92820	−	1.00000i	0.442627	−	0.0638877i
$246$		0			0
$247$			38.7846i			2.46781i
$248$		0			0
$249$	14.7846			0.936937
$250$		0			0
$251$	−1.85641			−0.117175			−0.0585877	−	0.998282i	$-0.518660\pi$
$251$	−1.85641			−0.117175			−0.0585877	+	0.998282i	$0.518660\pi$
$252$		0			0
$253$	−20.3923			−1.28205
$254$		0			0
$255$	0.803848			0.0503389
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	4.39230	−	5.07180i	0.272925	−	0.315146i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.53590i			0.279695i	0.990173	+	0.139848i	$0.0446613\pi$
$263$			4.53590i			0.279695i	−0.990173	+	0.139848i	$0.955339\pi$
$264$		0			0
$265$		−	2.00000i		−	0.122859i
$266$		0			0
$267$		−	16.3923i		−	1.00319i
$268$		0			0
$269$			12.0000i			0.731653i	0.930683	+	0.365826i	$0.119214\pi$
$269$			12.0000i			0.731653i	−0.930683	+	0.365826i	$0.880786\pi$
$270$		0			0
$271$	2.53590			0.154045			0.0770224	−	0.997029i	$-0.475459\pi$
$271$	2.53590			0.154045			0.0770224	+	0.997029i	$0.475459\pi$
$272$		0			0
$273$	22.3923	+	19.3923i	1.35524	+	1.17368i
$274$		0			0
$275$		−	3.73205i		−	0.225051i
$276$		0			0
$277$	24.7846			1.48916			0.744581	−	0.667532i	$-0.232649\pi$
$277$	24.7846			1.48916			0.744581	+	0.667532i	$0.232649\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.92820			0.353647			0.176823	−	0.984243i	$-0.443418\pi$
$281$	5.92820			0.353647			0.176823	+	0.984243i	$0.443418\pi$
$282$		0			0
$283$	12.1244			0.720718			0.360359	−	0.932814i	$-0.382654\pi$
$283$	12.1244			0.720718			0.360359	+	0.932814i	$0.382654\pi$
$284$		0			0
$285$			10.3923i			0.615587i
$286$		0			0
$287$	6.92820	+	6.00000i	0.408959	+	0.354169i
$288$		0			0
$289$	16.7846			0.987330
$290$		0			0
$291$		−	12.8038i		−	0.750575i
$292$		0			0
$293$			14.3205i			0.836613i	0.908306	+	0.418307i	$0.137376\pi$
$293$			14.3205i			0.836613i	−0.908306	+	0.418307i	$0.862624\pi$
$294$		0			0
$295$		−	3.46410i		−	0.201688i
$296$		0			0
$297$			19.3923i			1.12526i
$298$		0			0
$299$	35.3205			2.04264
$300$		0			0
$301$	4.00000	+	3.46410i	0.230556	+	0.199667i
$302$		0			0
$303$		−	14.7846i		−	0.849354i
$304$		0			0
$305$	2.53590			0.145205
$306$		0			0
$307$	1.73205			0.0988534			0.0494267	−	0.998778i	$-0.484261\pi$
$307$	1.73205			0.0988534			0.0494267	+	0.998778i	$0.484261\pi$
$308$		0			0
$309$	29.7846			1.69439
$310$		0			0
$311$	−19.8564			−1.12595			−0.562977	−	0.826473i	$-0.690344\pi$
$311$	−19.8564			−1.12595			−0.562977	+	0.826473i	$0.690344\pi$
$312$		0			0
$313$		−	24.4641i		−	1.38279i	−0.722476	−	0.691396i	$-0.756996\pi$
$313$		−	24.4641i		−	1.38279i	0.722476	−	0.691396i	$-0.243004\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.9282			0.950783			0.475391	−	0.879774i	$-0.342306\pi$
$317$	16.9282			0.950783			0.475391	+	0.879774i	$0.342306\pi$
$318$		0			0
$319$			22.1244i			1.23873i
$320$		0			0
$321$		−	31.8564i		−	1.77805i
$322$		0			0
$323$		−	2.78461i		−	0.154940i
$324$		0			0
$325$			6.46410i			0.358564i
$326$		0			0
$327$	27.5885			1.52565
$328$		0			0
$329$	−3.00000	+	3.46410i	−0.165395	+	0.190982i
$330$		0			0
$331$		−	5.60770i		−	0.308227i	−0.988053	−	0.154113i	$-0.950748\pi$
$331$		−	5.60770i		−	0.308227i	0.988053	−	0.154113i	$-0.0492521\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.46410			−0.189264
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	2.53590			0.137731
$340$		0			0
$341$		−	22.3923i		−	1.21261i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$	9.46410			0.509530
$346$		0			0
$347$		−	14.2487i		−	0.764911i	−0.923974	−	0.382455i	$-0.875079\pi$
$347$		−	14.2487i		−	0.764911i	0.923974	−	0.382455i	$-0.124921\pi$
$348$		0			0
$349$		−	29.3205i		−	1.56949i	−0.619818	−	0.784745i	$-0.712794\pi$
$349$		−	29.3205i		−	1.56949i	0.619818	−	0.784745i	$-0.287206\pi$
$350$		0			0
$351$		−	33.5885i		−	1.79282i
$352$		0			0
$353$			13.3923i			0.712800i	0.934333	+	0.356400i	$0.115996\pi$
$353$			13.3923i			0.712800i	−0.934333	+	0.356400i	$0.884004\pi$
$354$		0			0
$355$	0.535898			0.0284425
$356$		0			0
$357$	−1.60770	−	1.39230i	−0.0850883	−	0.0736886i
$358$		0			0
$359$			9.32051i			0.491918i	0.969280	+	0.245959i	$0.0791028\pi$
$359$			9.32051i			0.491918i	−0.969280	+	0.245959i	$0.920897\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$	5.07180			0.266200
$364$		0			0
$365$	0.928203			0.0485844
$366$		0			0
$367$	36.1244			1.88568			0.942838	−	0.333251i	$-0.108146\pi$
$367$	36.1244			1.88568			0.942838	+	0.333251i	$0.108146\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.46410	+	4.00000i	−0.179847	+	0.207670i
$372$		0			0
$373$	24.3923			1.26299			0.631493	−	0.775382i	$-0.282443\pi$
$373$	24.3923			1.26299			0.631493	+	0.775382i	$0.282443\pi$
$374$		0			0
$375$			1.73205i			0.0894427i
$376$		0			0
$377$		−	38.3205i		−	1.97361i
$378$		0			0
$379$			26.3923i			1.35568i	0.735209	+	0.677841i	$0.237084\pi$
$379$			26.3923i			1.35568i	−0.735209	+	0.677841i	$0.762916\pi$
$380$		0			0
$381$		−	14.7846i		−	0.757438i
$382$		0			0
$383$	20.5359			1.04934			0.524668	−	0.851307i	$-0.324190\pi$
$383$	20.5359			1.04934			0.524668	+	0.851307i	$0.324190\pi$
$384$		0			0
$385$	−6.46410	+	7.46410i	−0.329441	+	0.380406i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6.85641			−0.347634			−0.173817	−	0.984778i	$-0.555610\pi$
$389$	−6.85641			−0.347634			−0.173817	+	0.984778i	$0.555610\pi$
$390$		0			0
$391$	−2.53590			−0.128246
$392$		0			0
$393$	16.3923			0.826882
$394$		0			0
$395$	−2.66025			−0.133852
$396$		0			0
$397$		−	5.53590i		−	0.277839i	−0.990304	−	0.138919i	$-0.955637\pi$
$397$		−	5.53590i		−	0.277839i	0.990304	−	0.138919i	$-0.0443629\pi$
$398$		0			0
$399$	18.0000	−	20.7846i	0.901127	−	1.04053i
$400$		0			0
$401$	−23.9282			−1.19492			−0.597459	−	0.801900i	$-0.703823\pi$
$401$	−23.9282			−1.19492			−0.597459	+	0.801900i	$0.703823\pi$
$402$		0			0
$403$			38.7846i			1.93200i
$404$		0			0
$405$		−	9.00000i		−	0.447214i
$406$		0			0
$407$			9.46410i			0.469118i
$408$		0			0
$409$			31.8564i			1.57520i	0.616188	+	0.787599i	$0.288676\pi$
$409$			31.8564i			1.57520i	−0.616188	+	0.787599i	$0.711324\pi$
$410$		0			0
$411$	0.679492			0.0335169
$412$		0			0
$413$	−6.00000	+	6.92820i	−0.295241	+	0.340915i
$414$		0			0
$415$		−	8.53590i		−	0.419011i
$416$		0			0
$417$	−12.0000			−0.587643
$418$		0			0
$419$	−24.2487			−1.18463			−0.592314	−	0.805708i	$-0.701785\pi$
$419$	−24.2487			−1.18463			−0.592314	+	0.805708i	$0.701785\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	0.464102i		−	0.0225122i
$426$		0			0
$427$	−5.07180	−	4.39230i	−0.245441	−	0.212559i
$428$		0			0
$429$	−41.7846			−2.01738
$430$		0			0
$431$		−	17.5885i		−	0.847206i	−0.905848	−	0.423603i	$-0.860765\pi$
$431$		−	17.5885i		−	0.847206i	0.905848	−	0.423603i	$-0.139235\pi$
$432$		0			0
$433$		−	4.14359i		−	0.199128i	−0.995031	−	0.0995642i	$-0.968255\pi$
$433$		−	4.14359i		−	0.199128i	0.995031	−	0.0995642i	$-0.0317449\pi$
$434$		0			0
$435$		−	10.2679i		−	0.492310i
$436$		0			0
$437$		−	32.7846i		−	1.56830i
$438$		0			0
$439$	15.7128			0.749932			0.374966	−	0.927039i	$-0.377654\pi$
$439$	15.7128			0.749932			0.374966	+	0.927039i	$0.377654\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	26.0000i		−	1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$		−	26.0000i		−	1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$		0			0
$445$	−9.46410			−0.448641
$446$		0			0
$447$	29.3205			1.38681
$448$		0			0
$449$	−1.92820			−0.0909975			−0.0454988	−	0.998964i	$-0.514488\pi$
$449$	−1.92820			−0.0909975			−0.0454988	+	0.998964i	$0.514488\pi$
$450$		0			0
$451$	−12.9282			−0.608765
$452$		0			0
$453$			8.32051i			0.390932i
$454$		0			0
$455$	11.1962	−	12.9282i	0.524884	−	0.606084i
$456$		0			0
$457$	27.4641			1.28472			0.642358	−	0.766405i	$-0.277956\pi$
$457$	27.4641			1.28472			0.642358	+	0.766405i	$0.277956\pi$
$458$		0			0
$459$			2.41154i			0.112561i
$460$		0			0
$461$			27.7128i			1.29071i	0.763881	+	0.645357i	$0.223291\pi$
$461$			27.7128i			1.29071i	−0.763881	+	0.645357i	$0.776709\pi$
$462$		0			0
$463$		−	4.39230i		−	0.204128i	−0.994778	−	0.102064i	$-0.967455\pi$
$463$		−	4.39230i		−	0.204128i	0.994778	−	0.102064i	$-0.0325446\pi$
$464$		0			0
$465$			10.3923i			0.481932i
$466$		0			0
$467$	22.5167			1.04195			0.520973	−	0.853573i	$-0.325569\pi$
$467$	22.5167			1.04195			0.520973	+	0.853573i	$0.325569\pi$
$468$		0			0
$469$	6.92820	+	6.00000i	0.319915	+	0.277054i
$470$		0			0
$471$		−	34.3923i		−	1.58471i
$472$		0			0
$473$	−7.46410			−0.343200
$474$		0			0
$475$	6.00000			0.275299
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−37.1769			−1.69866			−0.849328	−	0.527865i	$-0.822993\pi$
$479$	−37.1769			−1.69866			−0.849328	+	0.527865i	$0.822993\pi$
$480$		0			0
$481$		−	16.3923i		−	0.747425i
$482$		0			0
$483$	−18.9282	−	16.3923i	−0.861263	−	0.745876i
$484$		0			0
$485$	−7.39230			−0.335667
$486$		0			0
$487$		−	28.7846i		−	1.30436i	−0.758066	−	0.652178i	$-0.773856\pi$
$487$		−	28.7846i		−	1.30436i	0.758066	−	0.652178i	$-0.226144\pi$
$488$		0			0
$489$		−	36.0000i		−	1.62798i
$490$		0			0
$491$		−	34.1244i		−	1.54001i	−0.638037	−	0.770005i	$-0.720254\pi$
$491$		−	34.1244i		−	1.54001i	0.638037	−	0.770005i	$-0.279746\pi$
$492$		0			0
$493$			2.75129i			0.123912i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.07180	−	0.928203i	−0.0480767	−	0.0416356i
$498$		0			0
$499$		−	5.58846i		−	0.250174i	−0.992146	−	0.125087i	$-0.960079\pi$
$499$		−	5.58846i		−	0.250174i	0.992146	−	0.125087i	$-0.0399210\pi$
$500$		0			0
$501$	−9.00000			−0.402090
$502$		0			0
$503$	15.5885			0.695055			0.347527	−	0.937670i	$-0.387021\pi$
$503$	15.5885			0.695055			0.347527	+	0.937670i	$0.387021\pi$
$504$		0			0
$505$	−8.53590			−0.379842
$506$		0			0
$507$	49.8564			2.21420
$508$		0			0
$509$			1.85641i			0.0822838i	0.999153	+	0.0411419i	$0.0130996\pi$
$509$			1.85641i			0.0822838i	−0.999153	+	0.0411419i	$0.986900\pi$
$510$		0			0
$511$	−1.85641	−	1.60770i	−0.0821226	−	0.0711202i
$512$		0			0
$513$	−31.1769			−1.37649
$514$		0			0
$515$		−	17.1962i		−	0.757753i
$516$		0			0
$517$		−	6.46410i		−	0.284291i
$518$		0			0
$519$			35.1962i			1.54494i
$520$		0			0
$521$			34.3923i			1.50675i	0.657589	+	0.753377i	$0.271576\pi$
$521$			34.3923i			1.50675i	−0.657589	+	0.753377i	$0.728424\pi$
$522$		0			0
$523$	−24.2487			−1.06032			−0.530161	−	0.847897i	$-0.677869\pi$
$523$	−24.2487			−1.06032			−0.530161	+	0.847897i	$0.677869\pi$
$524$		0			0
$525$	3.00000	−	3.46410i	0.130931	−	0.151186i
$526$		0			0
$527$		−	2.78461i		−	0.121300i
$528$		0			0
$529$	−6.85641			−0.298105
$530$		0			0
$531$		0			0
$532$		0			0
$533$	22.3923			0.969918
$534$		0			0
$535$	−18.3923			−0.795169
$536$		0			0
$537$		−	24.9282i		−	1.07573i
$538$		0			0
$539$	25.8564	−	3.73205i	1.11371	−	0.160751i
$540$		0			0
$541$	33.7846			1.45251			0.726257	−	0.687423i	$-0.241258\pi$
$541$	33.7846			1.45251			0.726257	+	0.687423i	$0.241258\pi$
$542$		0			0
$543$			22.3923i			0.960946i
$544$		0			0
$545$		−	15.9282i		−	0.682289i
$546$		0			0
$547$			14.5359i			0.621510i	0.950490	+	0.310755i	$0.100582\pi$
$547$			14.5359i			0.621510i	−0.950490	+	0.310755i	$0.899418\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−35.5692			−1.51530
$552$		0			0
$553$	5.32051	+	4.60770i	0.226251	+	0.195939i
$554$		0			0
$555$		−	4.39230i		−	0.186443i
$556$		0			0
$557$	−5.85641			−0.248144			−0.124072	−	0.992273i	$-0.539595\pi$
$557$	−5.85641			−0.248144			−0.124072	+	0.992273i	$0.539595\pi$
$558$		0			0
$559$	12.9282			0.546805
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$	43.1769			1.81969			0.909845	−	0.414948i	$-0.136200\pi$
$563$	43.1769			1.81969			0.909845	+	0.414948i	$0.136200\pi$
$564$		0			0
$565$		−	1.46410i		−	0.0615952i
$566$		0			0
$567$	−15.5885	+	18.0000i	−0.654654	+	0.755929i
$568$		0			0
$569$	20.9282			0.877356			0.438678	−	0.898644i	$-0.355447\pi$
$569$	20.9282			0.877356			0.438678	+	0.898644i	$0.355447\pi$
$570$		0			0
$571$		−	41.3205i		−	1.72921i	−0.502453	−	0.864605i	$-0.667569\pi$
$571$		−	41.3205i		−	1.72921i	0.502453	−	0.864605i	$-0.332431\pi$
$572$		0			0
$573$		−	5.53590i		−	0.231265i
$574$		0			0
$575$		−	5.46410i		−	0.227869i
$576$		0			0
$577$		−	1.39230i		−	0.0579624i	−0.999580	−	0.0289812i	$-0.990774\pi$
$577$		−	1.39230i		−	0.0579624i	0.999580	−	0.0289812i	$-0.00922630\pi$
$578$		0			0
$579$	4.39230			0.182538
$580$		0			0
$581$	−14.7846	+	17.0718i	−0.613369	+	0.708257i
$582$		0			0
$583$		−	7.46410i		−	0.309132i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.4641			1.13356			0.566782	−	0.823868i	$-0.308188\pi$
$587$	27.4641			1.13356			0.566782	+	0.823868i	$0.308188\pi$
$588$		0			0
$589$	36.0000			1.48335
$590$		0			0
$591$	36.9282			1.51902
$592$		0			0
$593$			23.5359i			0.966504i	0.875481	+	0.483252i	$0.160544\pi$
$593$			23.5359i			0.966504i	−0.875481	+	0.483252i	$0.839456\pi$
$594$		0			0
$595$	−0.803848	+	0.928203i	−0.0329545	+	0.0380526i
$596$		0			0
$597$	6.00000			0.245564
$598$		0			0
$599$		−	14.1244i		−	0.577106i	−0.957464	−	0.288553i	$-0.906826\pi$
$599$		−	14.1244i		−	0.577106i	0.957464	−	0.288553i	$-0.0931741\pi$
$600$		0			0
$601$			26.7846i			1.09257i	0.837600	+	0.546284i	$0.183958\pi$
$601$			26.7846i			1.09257i	−0.837600	+	0.546284i	$0.816042\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	2.92820i		−	0.119048i
$606$		0			0
$607$	−18.8038			−0.763225			−0.381612	−	0.924322i	$-0.624631\pi$
$607$	−18.8038			−0.763225			−0.381612	+	0.924322i	$0.624631\pi$
$608$		0			0
$609$	−17.7846	+	20.5359i	−0.720669	+	0.832157i
$610$		0			0
$611$			11.1962i			0.452948i
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$	6.00000			0.241943
$616$		0			0
$617$	−9.07180			−0.365217			−0.182608	−	0.983186i	$-0.558454\pi$
$617$	−9.07180			−0.365217			−0.182608	+	0.983186i	$0.558454\pi$
$618$		0			0
$619$	−35.3205			−1.41965			−0.709826	−	0.704378i	$-0.751226\pi$
$619$	−35.3205			−1.41965			−0.709826	+	0.704378i	$0.751226\pi$
$620$		0			0
$621$			28.3923i			1.13934i
$622$		0			0
$623$	18.9282	+	16.3923i	0.758342	+	0.656744i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			38.7846i			1.54891i
$628$		0			0
$629$			1.17691i			0.0469267i
$630$		0			0
$631$			26.9090i			1.07123i	0.844463	+	0.535614i	$0.179920\pi$
$631$			26.9090i			1.07123i	−0.844463	+	0.535614i	$0.820080\pi$
$632$		0			0
$633$			12.4641i			0.495404i
$634$		0			0
$635$	−8.53590			−0.338737
$636$		0			0
$637$	−44.7846	+	6.46410i	−1.77443	+	0.256117i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.92820			−0.194652			−0.0973262	−	0.995253i	$-0.531029\pi$
$641$	−4.92820			−0.194652			−0.0973262	+	0.995253i	$0.531029\pi$
$642$		0			0
$643$	31.0526			1.22459			0.612297	−	0.790628i	$-0.290246\pi$
$643$	31.0526			1.22459			0.612297	+	0.790628i	$0.290246\pi$
$644$		0			0
$645$	3.46410			0.136399
$646$		0			0
$647$	10.3923			0.408564			0.204282	−	0.978912i	$-0.434514\pi$
$647$	10.3923			0.408564			0.204282	+	0.978912i	$0.434514\pi$
$648$		0			0
$649$		−	12.9282i		−	0.507476i
$650$		0			0
$651$	18.0000	−	20.7846i	0.705476	−	0.814613i
$652$		0			0
$653$	−38.3923			−1.50241			−0.751203	−	0.660071i	$-0.770526\pi$
$653$	−38.3923			−1.50241			−0.751203	+	0.660071i	$0.770526\pi$
$654$		0			0
$655$		−	9.46410i		−	0.369793i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	20.8038i		−	0.810403i	−0.914227	−	0.405201i	$-0.867201\pi$
$659$		−	20.8038i		−	0.810403i	0.914227	−	0.405201i	$-0.132799\pi$
$660$		0			0
$661$			15.7128i			0.611158i	0.952167	+	0.305579i	$0.0988499\pi$
$661$			15.7128i			0.611158i	−0.952167	+	0.305579i	$0.901150\pi$
$662$		0			0
$663$	−5.19615			−0.201802
$664$		0			0
$665$	−12.0000	−	10.3923i	−0.465340	−	0.402996i
$666$		0			0
$667$			32.3923i			1.25424i
$668$		0			0
$669$	17.7846			0.687593
$670$		0			0
$671$	9.46410			0.365358
$672$		0			0
$673$	49.1769			1.89563			0.947815	−	0.318820i	$-0.103286\pi$
$673$	49.1769			1.89563			0.947815	+	0.318820i	$0.103286\pi$
$674$		0			0
$675$	−5.19615			−0.200000
$676$		0			0
$677$			4.60770i			0.177088i	0.996072	+	0.0885441i	$0.0282214\pi$
$677$			4.60770i			0.177088i	−0.996072	+	0.0885441i	$0.971779\pi$
$678$		0			0
$679$	14.7846	+	12.8038i	0.567381	+	0.491367i
$680$		0			0
$681$	5.78461			0.221667
$682$		0			0
$683$		−	27.3205i		−	1.04539i	−0.852520	−	0.522695i	$-0.824927\pi$
$683$		−	27.3205i		−	1.04539i	0.852520	−	0.522695i	$-0.175073\pi$
$684$		0			0
$685$		−	0.392305i		−	0.0149892i
$686$		0			0
$687$		−	26.7846i		−	1.02190i
$688$		0			0
$689$			12.9282i			0.492525i
$690$		0			0
$691$	−46.6410			−1.77431			−0.887154	−	0.461474i	$-0.847321\pi$
$691$	−46.6410			−1.77431			−0.887154	+	0.461474i	$0.847321\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.92820i			0.262802i
$696$		0			0
$697$	−1.60770			−0.0608958
$698$		0			0
$699$	39.7128			1.50208
$700$		0			0
$701$	−3.78461			−0.142943			−0.0714714	−	0.997443i	$-0.522769\pi$
$701$	−3.78461			−0.142943			−0.0714714	+	0.997443i	$0.522769\pi$
$702$		0			0
$703$	−15.2154			−0.573859
$704$		0			0
$705$			3.00000i			0.112987i
$706$		0			0
$707$	17.0718	+	14.7846i	0.642051	+	0.556032i
$708$		0			0
$709$	−21.0000			−0.788672			−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000			−0.788672			−0.394336	+	0.918966i	$0.629025\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	32.7846i		−	1.22779i
$714$		0			0
$715$			24.1244i			0.902200i
$716$		0			0
$717$			48.4641i			1.80993i
$718$		0			0
$719$	45.4641			1.69552			0.847762	−	0.530376i	$-0.177949\pi$
$719$	45.4641			1.69552			0.847762	+	0.530376i	$0.177949\pi$
$720$		0			0
$721$	−29.7846	+	34.3923i	−1.10924	+	1.28084i
$722$		0			0
$723$			7.60770i			0.282933i
$724$		0			0
$725$	−5.92820			−0.220168
$726$		0			0
$727$	−34.3923			−1.27554			−0.637770	−	0.770227i	$-0.720143\pi$
$727$	−34.3923			−1.27554			−0.637770	+	0.770227i	$0.720143\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−0.928203			−0.0343308
$732$		0			0
$733$		−	18.4641i		−	0.681987i	−0.940066	−	0.340994i	$-0.889237\pi$
$733$		−	18.4641i		−	0.681987i	0.940066	−	0.340994i	$-0.110763\pi$
$734$		0			0
$735$	−12.0000	+	1.73205i	−0.442627	+	0.0638877i
$736$		0			0
$737$	−12.9282			−0.476216
$738$		0			0
$739$		−	7.73205i		−	0.284428i	−0.989836	−	0.142214i	$-0.954578\pi$
$739$		−	7.73205i		−	0.284428i	0.989836	−	0.142214i	$-0.0454221\pi$
$740$		0			0
$741$		−	67.1769i		−	2.46781i
$742$		0			0
$743$		−	9.60770i		−	0.352472i	−0.984348	−	0.176236i	$-0.943608\pi$
$743$		−	9.60770i		−	0.352472i	0.984348	−	0.176236i	$-0.0563922\pi$
$744$		0			0
$745$		−	16.9282i		−	0.620201i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.7846	+	31.8564i	1.34408	+	1.16401i
$750$		0			0
$751$		−	5.58846i		−	0.203926i	−0.994788	−	0.101963i	$-0.967488\pi$
$751$		−	5.58846i		−	0.203926i	0.994788	−	0.101963i	$-0.0325123\pi$
$752$		0			0
$753$	3.21539			0.117175
$754$		0			0
$755$	4.80385			0.174830
$756$		0			0
$757$	−10.1436			−0.368675			−0.184338	−	0.982863i	$-0.559014\pi$
$757$	−10.1436			−0.368675			−0.184338	+	0.982863i	$0.559014\pi$
$758$		0			0
$759$	35.3205			1.28205
$760$		0			0
$761$			6.24871i			0.226516i	0.993566	+	0.113258i	$0.0361286\pi$
$761$			6.24871i			0.226516i	−0.993566	+	0.113258i	$0.963871\pi$
$762$		0			0
$763$	−27.5885	+	31.8564i	−0.998769	+	1.15328i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			22.3923i			0.808539i
$768$		0			0
$769$			18.0000i			0.649097i	0.945869	+	0.324548i	$0.105212\pi$
$769$			18.0000i			0.649097i	−0.945869	+	0.324548i	$0.894788\pi$
$770$		0			0
$771$		−	10.3923i		−	0.374270i
$772$		0			0
$773$			12.4641i			0.448303i	0.974554	+	0.224151i	$0.0719610\pi$
$773$			12.4641i			0.448303i	−0.974554	+	0.224151i	$0.928039\pi$
$774$		0			0
$775$	6.00000			0.215526
$776$		0			0
$777$	−7.60770	+	8.78461i	−0.272925	+	0.315146i
$778$		0			0
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	2.00000			0.0715656
$782$		0			0
$783$	30.8038			1.10084
$784$		0			0
$785$	−19.8564			−0.708706
$786$		0			0
$787$	−15.3397			−0.546803			−0.273401	−	0.961900i	$-0.588149\pi$
$787$	−15.3397			−0.546803			−0.273401	+	0.961900i	$0.588149\pi$
$788$		0			0
$789$		−	7.85641i		−	0.279695i
$790$		0			0
$791$	−2.53590	+	2.92820i	−0.0901662	+	0.104115i
$792$		0			0
$793$	−16.3923			−0.582108
$794$		0			0
$795$			3.46410i			0.122859i
$796$		0			0
$797$		−	40.1769i		−	1.42314i	−0.702616	−	0.711570i	$-0.747985\pi$
$797$		−	40.1769i		−	1.42314i	0.702616	−	0.711570i	$-0.252015\pi$
$798$		0			0
$799$		−	0.803848i		−	0.0284381i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.46410			0.122245
$804$		0			0
$805$	−9.46410	+	10.9282i	−0.333566	+	0.385169i
$806$		0			0
$807$		−	20.7846i		−	0.731653i
$808$		0			0
$809$	−5.92820			−0.208425			−0.104212	−	0.994555i	$-0.533232\pi$
$809$	−5.92820			−0.208425			−0.104212	+	0.994555i	$0.533232\pi$
$810$		0			0
$811$	−42.9282			−1.50741			−0.753707	−	0.657211i	$-0.771736\pi$
$811$	−42.9282			−1.50741			−0.753707	+	0.657211i	$0.771736\pi$
$812$		0			0
$813$	−4.39230			−0.154045
$814$		0			0
$815$	−20.7846			−0.728053
$816$		0			0
$817$		−	12.0000i		−	0.419827i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.7846			0.481086			0.240543	−	0.970638i	$-0.422674\pi$
$821$	13.7846			0.481086			0.240543	+	0.970638i	$0.422674\pi$
$822$		0			0
$823$		−	24.9282i		−	0.868943i	−0.900686	−	0.434471i	$-0.856935\pi$
$823$		−	24.9282i		−	0.868943i	0.900686	−	0.434471i	$-0.143065\pi$
$824$		0			0
$825$			6.46410i			0.225051i
$826$		0			0
$827$			41.8564i			1.45549i	0.685848	+	0.727745i	$0.259432\pi$
$827$			41.8564i			1.45549i	−0.685848	+	0.727745i	$0.740568\pi$
$828$		0			0
$829$			48.2487i			1.67575i	0.545864	+	0.837874i	$0.316201\pi$
$829$			48.2487i			1.67575i	−0.545864	+	0.837874i	$0.683799\pi$
$830$		0			0
$831$	−42.9282			−1.48916
$832$		0			0
$833$	3.21539	−	0.464102i	0.111407	−	0.0160802i
$834$		0			0
$835$			5.19615i			0.179820i
$836$		0			0
$837$	−31.1769			−1.07763
$838$		0			0
$839$	−16.3923			−0.565925			−0.282963	−	0.959131i	$-0.591317\pi$
$839$	−16.3923			−0.565925			−0.282963	+	0.959131i	$0.591317\pi$
$840$		0			0
$841$	6.14359			0.211848
$842$		0			0
$843$	−10.2679			−0.353647
$844$		0			0
$845$		−	28.7846i		−	0.990221i
$846$		0			0
$847$	−5.07180	+	5.85641i	−0.174269	+	0.201229i
$848$		0			0
$849$	−21.0000			−0.720718
$850$		0			0
$851$			13.8564i			0.474991i
$852$		0			0
$853$		−	38.7846i		−	1.32796i	−0.747750	−	0.663980i	$-0.768866\pi$
$853$		−	38.7846i		−	1.32796i	0.747750	−	0.663980i	$-0.231134\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.28719i			0.0781288i	0.999237	+	0.0390644i	$0.0124378\pi$
$857$			2.28719i			0.0781288i	−0.999237	+	0.0390644i	$0.987562\pi$
$858$		0			0
$859$	15.7128			0.536114			0.268057	−	0.963403i	$-0.413618\pi$
$859$	15.7128			0.536114			0.268057	+	0.963403i	$0.413618\pi$
$860$		0			0
$861$	−12.0000	−	10.3923i	−0.408959	−	0.354169i
$862$		0			0
$863$			12.1436i			0.413373i	0.978407	+	0.206686i	$0.0662680\pi$
$863$			12.1436i			0.413373i	−0.978407	+	0.206686i	$0.933732\pi$
$864$		0			0
$865$	20.3205			0.690918
$866$		0			0
$867$	−29.0718			−0.987330
$868$		0			0
$869$	−9.92820			−0.336791
$870$		0			0
$871$	22.3923			0.758734
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.00000	−	1.73205i	−0.0676123	−	0.0585540i
$876$		0			0
$877$	−2.39230			−0.0807824			−0.0403912	−	0.999184i	$-0.512860\pi$
$877$	−2.39230			−0.0807824			−0.0403912	+	0.999184i	$0.512860\pi$
$878$		0			0
$879$		−	24.8038i		−	0.836613i
$880$		0			0
$881$		−	42.9282i		−	1.44629i	−0.690697	−	0.723144i	$-0.742696\pi$
$881$		−	42.9282i		−	1.44629i	0.690697	−	0.723144i	$-0.257304\pi$
$882$		0			0
$883$		−	44.3923i		−	1.49392i	−0.664869	−	0.746960i	$-0.731513\pi$
$883$		−	44.3923i		−	1.49392i	0.664869	−	0.746960i	$-0.268487\pi$
$884$		0			0
$885$			6.00000i			0.201688i
$886$		0			0
$887$	−15.4641			−0.519234			−0.259617	−	0.965712i	$-0.583596\pi$
$887$	−15.4641			−0.519234			−0.259617	+	0.965712i	$0.583596\pi$
$888$		0			0
$889$	17.0718	+	14.7846i	0.572570	+	0.495860i
$890$		0			0
$891$		−	33.5885i		−	1.12526i
$892$		0			0
$893$	10.3923			0.347765
$894$		0			0
$895$	−14.3923			−0.481082
$896$		0			0
$897$	−61.1769			−2.04264
$898$		0			0
$899$	−35.5692			−1.18630
$900$		0			0
$901$		−	0.928203i		−	0.0309229i
$902$		0			0
$903$	−6.92820	−	6.00000i	−0.230556	−	0.199667i
$904$		0			0
$905$	12.9282			0.429748
$906$		0			0
$907$		−	11.6077i		−	0.385427i	−0.981255	−	0.192714i	$-0.938271\pi$
$907$		−	11.6077i		−	0.385427i	0.981255	−	0.192714i	$-0.0617288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	28.2487i		−	0.935922i	−0.883749	−	0.467961i	$-0.844989\pi$
$911$		−	28.2487i		−	0.935922i	0.883749	−	0.467961i	$-0.155011\pi$
$912$		0			0
$913$		−	31.8564i		−	1.05429i
$914$		0			0
$915$	−4.39230			−0.145205
$916$		0			0
$917$	−16.3923	+	18.9282i	−0.541322	+	0.625064i
$918$		0			0
$919$			37.5885i			1.23993i	0.784630	+	0.619964i	$0.212853\pi$
$919$			37.5885i			1.23993i	−0.784630	+	0.619964i	$0.787147\pi$
$920$		0			0
$921$	−3.00000			−0.0988534
$922$		0			0
$923$	−3.46410			−0.114022
$924$		0			0
$925$	−2.53590			−0.0833798
$926$		0			0
$927$		0			0
$928$		0			0
$929$			42.4974i			1.39430i	0.716928	+	0.697148i	$0.245548\pi$
$929$			42.4974i			1.39430i	−0.716928	+	0.697148i	$0.754452\pi$
$930$		0			0
$931$	6.00000	+	41.5692i	0.196642	+	1.36238i
$932$		0			0
$933$	34.3923			1.12595
$934$		0			0
$935$		−	1.73205i		−	0.0566441i
$936$		0			0
$937$		−	4.60770i		−	0.150527i	−0.997164	−	0.0752634i	$-0.976020\pi$
$937$		−	4.60770i		−	0.150527i	0.997164	−	0.0752634i	$-0.0239798\pi$
$938$		0			0
$939$			42.3731i			1.38279i
$940$		0			0
$941$			51.0333i			1.66364i	0.555046	+	0.831819i	$0.312700\pi$
$941$			51.0333i			1.66364i	−0.555046	+	0.831819i	$0.687300\pi$
$942$		0			0
$943$	−18.9282			−0.616387
$944$		0			0
$945$	10.3923	+	9.00000i	0.338062	+	0.292770i
$946$		0			0
$947$		−	18.7846i		−	0.610418i	−0.952285	−	0.305209i	$-0.901274\pi$
$947$		−	18.7846i		−	0.610418i	0.952285	−	0.305209i	$-0.0987263\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$		0			0
$951$	−29.3205			−0.950783
$952$		0			0
$953$	−13.3205			−0.431494			−0.215747	−	0.976449i	$-0.569219\pi$
$953$	−13.3205			−0.431494			−0.215747	+	0.976449i	$0.569219\pi$
$954$		0			0
$955$	−3.19615			−0.103425
$956$		0			0
$957$		−	38.3205i		−	1.23873i
$958$		0			0
$959$	−0.679492	+	0.784610i	−0.0219419	+	0.0253364i
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2.53590i		−	0.0816335i
$966$		0			0
$967$			31.1769i			1.00258i	0.865279	+	0.501291i	$0.167141\pi$
$967$			31.1769i			1.00258i	−0.865279	+	0.501291i	$0.832859\pi$
$968$		0			0
$969$			4.82309i			0.154940i
$970$		0			0
$971$	26.7846			0.859559			0.429780	−	0.902934i	$-0.358591\pi$
$971$	26.7846			0.859559			0.429780	+	0.902934i	$0.358591\pi$
$972$		0			0
$973$	12.0000	−	13.8564i	0.384702	−	0.444216i
$974$		0			0
$975$		−	11.1962i		−	0.358564i
$976$		0			0
$977$	44.2487			1.41564			0.707821	−	0.706392i	$-0.249678\pi$
$977$	44.2487			1.41564			0.707821	+	0.706392i	$0.249678\pi$
$978$		0			0
$979$	−35.3205			−1.12885
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.58846			0.114454			0.0572270	−	0.998361i	$-0.481774\pi$
$983$	3.58846			0.114454			0.0572270	+	0.998361i	$0.481774\pi$
$984$		0			0
$985$		−	21.3205i		−	0.679328i
$986$		0			0
$987$	5.19615	−	6.00000i	0.165395	−	0.190982i
$988$		0			0
$989$	−10.9282			−0.347497
$990$		0			0
$991$			15.1769i			0.482111i	0.970511	+	0.241055i	$0.0774935\pi$
$991$			15.1769i			0.482111i	−0.970511	+	0.241055i	$0.922506\pi$
$992$		0			0
$993$			9.71281i			0.308227i
$994$		0			0
$995$		−	3.46410i		−	0.109819i
$996$		0			0
$997$		−	12.4641i		−	0.394742i	−0.980329	−	0.197371i	$-0.936760\pi$
$997$		−	12.4641i		−	0.394742i	0.980329	−	0.197371i	$-0.0632404\pi$
$998$		0			0
$999$	13.1769			0.416899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.k.a.1791.2		4
4.3	odd	2		2240.2.k.b.1791.4		4
7.6	odd	2		2240.2.k.b.1791.3		4
8.3	odd	2		140.2.g.b.111.1	yes	4
8.5	even	2		140.2.g.a.111.2	yes	4
24.5	odd	2		1260.2.c.a.811.3		4
24.11	even	2		1260.2.c.b.811.4		4
28.27	even	2	inner	2240.2.k.a.1791.1		4
40.3	even	4		700.2.c.f.699.2		4
40.13	odd	4		700.2.c.b.699.4		4
40.19	odd	2		700.2.g.g.251.4		4
40.27	even	4		700.2.c.c.699.3		4
40.29	even	2		700.2.g.f.251.3		4
40.37	odd	4		700.2.c.e.699.1		4
56.3	even	6		980.2.o.c.411.1		4
56.5	odd	6		980.2.o.d.31.2		4
56.11	odd	6		980.2.o.d.411.1		4
56.13	odd	2		140.2.g.b.111.2	yes	4
56.19	even	6		980.2.o.a.31.2		4
56.27	even	2		140.2.g.a.111.1	✓	4
56.37	even	6		980.2.o.c.31.2		4
56.45	odd	6		980.2.o.b.411.2		4
56.51	odd	6		980.2.o.b.31.2		4
56.53	even	6		980.2.o.a.411.2		4
168.83	odd	2		1260.2.c.a.811.4		4
168.125	even	2		1260.2.c.b.811.3		4
280.13	even	4		700.2.c.c.699.4		4
280.27	odd	4		700.2.c.b.699.3		4
280.69	odd	2		700.2.g.g.251.3		4
280.83	odd	4		700.2.c.e.699.2		4
280.139	even	2		700.2.g.f.251.4		4
280.237	even	4		700.2.c.f.699.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.g.a.111.1	✓	4	56.27	even	2
140.2.g.a.111.2	yes	4	8.5	even	2
140.2.g.b.111.1	yes	4	8.3	odd	2
140.2.g.b.111.2	yes	4	56.13	odd	2
700.2.c.b.699.3		4	280.27	odd	4
700.2.c.b.699.4		4	40.13	odd	4
700.2.c.c.699.3		4	40.27	even	4
700.2.c.c.699.4		4	280.13	even	4
700.2.c.e.699.1		4	40.37	odd	4
700.2.c.e.699.2		4	280.83	odd	4
700.2.c.f.699.1		4	280.237	even	4
700.2.c.f.699.2		4	40.3	even	4
700.2.g.f.251.3		4	40.29	even	2
700.2.g.f.251.4		4	280.139	even	2
700.2.g.g.251.3		4	280.69	odd	2
700.2.g.g.251.4		4	40.19	odd	2
980.2.o.a.31.2		4	56.19	even	6
980.2.o.a.411.2		4	56.53	even	6
980.2.o.b.31.2		4	56.51	odd	6
980.2.o.b.411.2		4	56.45	odd	6
980.2.o.c.31.2		4	56.37	even	6
980.2.o.c.411.1		4	56.3	even	6
980.2.o.d.31.2		4	56.5	odd	6
980.2.o.d.411.1		4	56.11	odd	6
1260.2.c.a.811.3		4	24.5	odd	2
1260.2.c.a.811.4		4	168.83	odd	2
1260.2.c.b.811.3		4	168.125	even	2
1260.2.c.b.811.4		4	24.11	even	2
2240.2.k.a.1791.1		4	28.27	even	2	inner
2240.2.k.a.1791.2		4	1.1	even	1	trivial
2240.2.k.b.1791.3		4	7.6	odd	2
2240.2.k.b.1791.4		4	4.3	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)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +1.00000i q^{5} +(1.73205 - 2.00000i) q^{7} +3.73205i q^{11} -6.46410i q^{13} -1.73205i q^{15} +0.464102i q^{17} -6.00000 q^{19} +(-3.00000 + 3.46410i) q^{21} +5.46410i q^{23} -1.00000 q^{25} +5.19615 q^{27} +5.92820 q^{29} -6.00000 q^{31} -6.46410i q^{33} +(2.00000 + 1.73205i) q^{35} +2.53590 q^{37} +11.1962i q^{39} +3.46410i q^{41} +2.00000i q^{43} -1.73205 q^{47} +(-1.00000 - 6.92820i) q^{49} -0.803848i q^{51} -2.00000 q^{53} -3.73205 q^{55} +10.3923 q^{57} -3.46410 q^{59} -2.53590i q^{61} +6.46410 q^{65} +3.46410i q^{67} -9.46410i q^{69} -0.535898i q^{71} -0.928203i q^{73} +1.73205 q^{75} +(7.46410 + 6.46410i) q^{77} +2.66025i q^{79} -9.00000 q^{81} -8.53590 q^{83} -0.464102 q^{85} -10.2679 q^{87} +9.46410i q^{89} +(-12.9282 - 11.1962i) q^{91} +10.3923 q^{93} -6.00000i q^{95} +7.39230i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{19} - 12 q^{21} - 4 q^{25} - 4 q^{29} - 24 q^{31} + 8 q^{35} + 24 q^{37} - 4 q^{49} - 8 q^{53} - 8 q^{55} + 12 q^{65} + 16 q^{77} - 36 q^{81} - 48 q^{83} + 12 q^{85} - 48 q^{87} - 24 q^{91}+O(q^{100}) \) 4 * q - 24 * q^19 - 12 * q^21 - 4 * q^25 - 4 * q^29 - 24 * q^31 + 8 * q^35 + 24 * q^37 - 4 * q^49 - 8 * q^53 - 8 * q^55 + 12 * q^65 + 16 * q^77 - 36 * q^81 - 48 * q^83 + 12 * q^85 - 48 * q^87 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more