LMFDB - Embedded newform 2240.2.e.e.2239.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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, 1, 1]))

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

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

chi := DirichletCharacter("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (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:	$8$
Coefficient field:	8.0.12745506816.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2239.6
Root			$-1.00781 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.e.2239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +(-1.22474 + 1.87083i) q^{5} +2.64575 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.41421i q^{3} +(-1.22474 + 1.87083i) q^{5} +2.64575 q^{7} +1.00000 q^{9} -3.46410i q^{11} -2.44949 q^{13} +(-2.64575 - 1.73205i) q^{15} +4.89898 q^{17} +6.48074 q^{19} +3.74166i q^{21} +(-2.00000 - 4.58258i) q^{25} +5.65685i q^{27} +6.00000 q^{29} +4.89898 q^{33} +(-3.24037 + 4.94975i) q^{35} -9.16515i q^{37} -3.46410i q^{39} -7.48331i q^{41} +5.29150 q^{43} +(-1.22474 + 1.87083i) q^{45} +2.82843i q^{47} +7.00000 q^{49} +6.92820i q^{51} +9.16515i q^{53} +(6.48074 + 4.24264i) q^{55} +9.16515i q^{57} +6.48074 q^{59} -11.2250i q^{61} +2.64575 q^{63} +(3.00000 - 4.58258i) q^{65} -5.29150 q^{67} -9.79796 q^{73} +(6.48074 - 2.82843i) q^{75} -9.16515i q^{77} +6.92820i q^{79} -5.00000 q^{81} +9.89949i q^{83} +(-6.00000 + 9.16515i) q^{85} +8.48528i q^{87} +7.48331i q^{89} -6.48074 q^{91} +(-7.93725 + 12.1244i) q^{95} -14.6969 q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 16 q^{25} + 48 q^{29} + 56 q^{49} + 24 q^{65} - 40 q^{81} - 48 q^{85}+O(q^{100}) $ 8 * q + 8 * q^9 - 16 * q^25 + 48 * q^29 + 56 * q^49 + 24 * q^65 - 40 * q^81 - 48 * q^85

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
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$	−1.22474	+	1.87083i	−0.547723	+	0.836660i
$5$	−1.22474	+	1.87083i	−0.547723	+	0.836660i
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$	−2.44949			−0.679366			−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949			−0.679366			−0.339683	+	0.940540i	$0.610320\pi$
$14$		0			0
$15$	−2.64575	−	1.73205i	−0.683130	−	0.447214i
$16$		0			0
$17$	4.89898			1.18818			0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898			1.18818			0.594089	+	0.804400i	$0.297513\pi$
$18$		0			0
$19$	6.48074			1.48678			0.743392	−	0.668856i	$-0.233216\pi$
$19$	6.48074			1.48678			0.743392	+	0.668856i	$0.233216\pi$
$20$		0			0
$21$			3.74166i			0.816497i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	4.89898			0.852803
$34$		0			0
$35$	−3.24037	+	4.94975i	−0.547723	+	0.836660i
$36$		0			0
$37$		−	9.16515i		−	1.50674i	−0.657596	−	0.753371i	$-0.728427\pi$
$37$		−	9.16515i		−	1.50674i	0.657596	−	0.753371i	$-0.271573\pi$
$38$		0			0
$39$		−	3.46410i		−	0.554700i
$40$		0			0
$41$		−	7.48331i		−	1.16870i	−0.811503	−	0.584349i	$-0.801350\pi$
$41$		−	7.48331i		−	1.16870i	0.811503	−	0.584349i	$-0.198650\pi$
$42$		0			0
$43$	5.29150			0.806947			0.403473	−	0.914991i	$-0.367803\pi$
$43$	5.29150			0.806947			0.403473	+	0.914991i	$0.367803\pi$
$44$		0			0
$45$	−1.22474	+	1.87083i	−0.182574	+	0.278887i
$46$		0			0
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$			6.92820i			0.970143i
$52$		0			0
$53$			9.16515i			1.25893i	0.777029	+	0.629465i	$0.216726\pi$
$53$			9.16515i			1.25893i	−0.777029	+	0.629465i	$0.783274\pi$
$54$		0			0
$55$	6.48074	+	4.24264i	0.873863	+	0.572078i
$56$		0			0
$57$			9.16515i			1.21395i
$58$		0			0
$59$	6.48074			0.843721			0.421860	−	0.906661i	$-0.361377\pi$
$59$	6.48074			0.843721			0.421860	+	0.906661i	$0.361377\pi$
$60$		0			0
$61$		−	11.2250i		−	1.43721i	−0.695418	−	0.718605i	$-0.744781\pi$
$61$		−	11.2250i		−	1.43721i	0.695418	−	0.718605i	$-0.255219\pi$
$62$		0			0
$63$	2.64575			0.333333
$64$		0			0
$65$	3.00000	−	4.58258i	0.372104	−	0.568399i
$66$		0			0
$67$	−5.29150			−0.646460			−0.323230	−	0.946320i	$-0.604769\pi$
$67$	−5.29150			−0.646460			−0.323230	+	0.946320i	$0.604769\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−9.79796			−1.14676			−0.573382	−	0.819288i	$-0.694369\pi$
$73$	−9.79796			−1.14676			−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$	6.48074	−	2.82843i	0.748331	−	0.326599i
$76$		0			0
$77$		−	9.16515i		−	1.04447i
$78$		0			0
$79$			6.92820i			0.779484i	0.920924	+	0.389742i	$0.127436\pi$
$79$			6.92820i			0.779484i	−0.920924	+	0.389742i	$0.872564\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$			9.89949i			1.08661i	0.839535	+	0.543305i	$0.182827\pi$
$83$			9.89949i			1.08661i	−0.839535	+	0.543305i	$0.817173\pi$
$84$		0			0
$85$	−6.00000	+	9.16515i	−0.650791	+	0.994100i
$86$		0			0
$87$			8.48528i			0.909718i
$88$		0			0
$89$			7.48331i			0.793230i	0.917985	+	0.396615i	$0.129815\pi$
$89$			7.48331i			0.793230i	−0.917985	+	0.396615i	$0.870185\pi$
$90$		0			0
$91$	−6.48074			−0.679366
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.93725	+	12.1244i	−0.814345	+	1.24393i
$96$		0			0
$97$	−14.6969			−1.49225			−0.746124	−	0.665807i	$-0.768087\pi$
$97$	−14.6969			−1.49225			−0.746124	+	0.665807i	$0.768087\pi$
$98$		0			0
$99$		−	3.46410i		−	0.348155i
$100$		0			0
$101$		−	3.74166i		−	0.372309i	−0.982521	−	0.186154i	$-0.940398\pi$
$101$		−	3.74166i		−	0.372309i	0.982521	−	0.186154i	$-0.0596025\pi$
$102$		0			0
$103$			16.9706i			1.67216i	0.548608	+	0.836080i	$0.315158\pi$
$103$			16.9706i			1.67216i	−0.548608	+	0.836080i	$0.684842\pi$
$104$		0			0
$105$	−7.00000	−	4.58258i	−0.683130	−	0.447214i
$106$		0			0
$107$	−15.8745			−1.53465			−0.767323	−	0.641260i	$-0.778412\pi$
$107$	−15.8745			−1.53465			−0.767323	+	0.641260i	$0.778412\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	12.9615			1.23025
$112$		0			0
$113$		−	9.16515i		−	0.862185i	−0.902308	−	0.431092i	$-0.858128\pi$
$113$		−	9.16515i		−	0.862185i	0.902308	−	0.431092i	$-0.141872\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.44949			−0.226455
$118$		0			0
$119$	12.9615			1.18818
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	10.5830			0.954237
$124$		0			0
$125$	11.0227	+	1.87083i	0.985901	+	0.167332i
$126$		0			0
$127$	10.5830			0.939090			0.469545	−	0.882909i	$-0.344418\pi$
$127$	10.5830			0.939090			0.469545	+	0.882909i	$0.344418\pi$
$128$		0			0
$129$			7.48331i			0.658869i
$130$		0			0
$131$	19.4422			1.69867			0.849337	−	0.527850i	$-0.177002\pi$
$131$	19.4422			1.69867			0.849337	+	0.527850i	$0.177002\pi$
$132$		0			0
$133$	17.1464			1.48678
$134$		0			0
$135$	−10.5830	−	6.92820i	−0.910840	−	0.596285i
$136$		0			0
$137$			18.3303i			1.56606i	0.621982	+	0.783032i	$0.286328\pi$
$137$			18.3303i			1.56606i	−0.621982	+	0.783032i	$0.713672\pi$
$138$		0			0
$139$	−19.4422			−1.64907			−0.824534	−	0.565813i	$-0.808563\pi$
$139$	−19.4422			−1.64907			−0.824534	+	0.565813i	$0.808563\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$			8.48528i			0.709575i
$144$		0			0
$145$	−7.34847	+	11.2250i	−0.610257	+	0.932183i
$146$		0			0
$147$			9.89949i			0.816497i
$148$		0			0
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$	4.89898			0.396059
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.34847			0.586472			0.293236	−	0.956040i	$-0.405268\pi$
$157$	7.34847			0.586472			0.293236	+	0.956040i	$0.405268\pi$
$158$		0			0
$159$	−12.9615			−1.02791
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.29150			0.414462			0.207231	−	0.978292i	$-0.433555\pi$
$163$	5.29150			0.414462			0.207231	+	0.978292i	$0.433555\pi$
$164$		0			0
$165$	−6.00000	+	9.16515i	−0.467099	+	0.713506i
$166$		0			0
$167$			5.65685i			0.437741i	0.975754	+	0.218870i	$0.0702371\pi$
$167$			5.65685i			0.437741i	−0.975754	+	0.218870i	$0.929763\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$	6.48074			0.495595
$172$		0			0
$173$	−7.34847			−0.558694			−0.279347	−	0.960190i	$-0.590118\pi$
$173$	−7.34847			−0.558694			−0.279347	+	0.960190i	$0.590118\pi$
$174$		0			0
$175$	−5.29150	−	12.1244i	−0.400000	−	0.916515i
$176$		0			0
$177$			9.16515i			0.688895i
$178$		0			0
$179$			24.2487i			1.81243i	0.422813	+	0.906217i	$0.361043\pi$
$179$			24.2487i			1.81243i	−0.422813	+	0.906217i	$0.638957\pi$
$180$		0			0
$181$		−	11.2250i		−	0.834346i	−0.908827	−	0.417173i	$-0.863021\pi$
$181$		−	11.2250i		−	0.834346i	0.908827	−	0.417173i	$-0.136979\pi$
$182$		0			0
$183$	15.8745			1.17348
$184$		0			0
$185$	17.1464	+	11.2250i	1.26063	+	0.825276i
$186$		0			0
$187$		−	16.9706i		−	1.24101i
$188$		0			0
$189$			14.9666i			1.08866i
$190$		0			0
$191$			6.92820i			0.501307i	0.968077	+	0.250654i	$0.0806455\pi$
$191$			6.92820i			0.501307i	−0.968077	+	0.250654i	$0.919354\pi$
$192$		0			0
$193$			9.16515i			0.659722i	0.944030	+	0.329861i	$0.107002\pi$
$193$			9.16515i			0.659722i	−0.944030	+	0.329861i	$0.892998\pi$
$194$		0			0
$195$	6.48074	+	4.24264i	0.464095	+	0.303822i
$196$		0			0
$197$		−	9.16515i		−	0.652990i	−0.945199	−	0.326495i	$-0.894132\pi$
$197$		−	9.16515i		−	0.652990i	0.945199	−	0.326495i	$-0.105868\pi$
$198$		0			0
$199$	12.9615			0.918815			0.459408	−	0.888226i	$-0.348062\pi$
$199$	12.9615			0.918815			0.459408	+	0.888226i	$0.348062\pi$
$200$		0			0
$201$		−	7.48331i		−	0.527832i
$202$		0			0
$203$	15.8745			1.11417
$204$		0			0
$205$	14.0000	+	9.16515i	0.977802	+	0.640122i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	22.4499i		−	1.55290i
$210$		0			0
$211$		−	10.3923i		−	0.715436i	−0.933830	−	0.357718i	$-0.883555\pi$
$211$		−	10.3923i		−	0.715436i	0.933830	−	0.357718i	$-0.116445\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.48074	+	9.89949i	−0.441983	+	0.675140i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	13.8564i		−	0.936329i
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$		−	25.4558i		−	1.70465i	−0.523013	−	0.852325i	$-0.675192\pi$
$223$		−	25.4558i		−	1.70465i	0.523013	−	0.852325i	$-0.324808\pi$
$224$		0			0
$225$	−2.00000	−	4.58258i	−0.133333	−	0.305505i
$226$		0			0
$227$			18.3848i			1.22024i	0.792309	+	0.610120i	$0.208879\pi$
$227$			18.3848i			1.22024i	−0.792309	+	0.610120i	$0.791121\pi$
$228$		0			0
$229$		−	11.2250i		−	0.741767i	−0.928679	−	0.370884i	$-0.879055\pi$
$229$		−	11.2250i		−	0.741767i	0.928679	−	0.370884i	$-0.120945\pi$
$230$		0			0
$231$	12.9615			0.852803
$232$		0			0
$233$		−	18.3303i		−	1.20086i	−0.799678	−	0.600429i	$-0.794996\pi$
$233$		−	18.3303i		−	1.20086i	0.799678	−	0.600429i	$-0.205004\pi$
$234$		0			0
$235$	−5.29150	−	3.46410i	−0.345180	−	0.225973i
$236$		0			0
$237$	−9.79796			−0.636446
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$	−8.57321	+	13.0958i	−0.547723	+	0.836660i
$246$		0			0
$247$	−15.8745			−1.01007
$248$		0			0
$249$	−14.0000			−0.887214
$250$		0			0
$251$	−6.48074			−0.409061			−0.204530	−	0.978860i	$-0.565567\pi$
$251$	−6.48074			−0.409061			−0.204530	+	0.978860i	$0.565567\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−12.9615	−	8.48528i	−0.811679	−	0.531369i
$256$		0			0
$257$	14.6969			0.916770			0.458385	−	0.888754i	$-0.348428\pi$
$257$	14.6969			0.916770			0.458385	+	0.888754i	$0.348428\pi$
$258$		0			0
$259$		−	24.2487i		−	1.50674i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	15.8745			0.978864			0.489432	−	0.872041i	$-0.337204\pi$
$263$	15.8745			0.978864			0.489432	+	0.872041i	$0.337204\pi$
$264$		0			0
$265$	−17.1464	−	11.2250i	−1.05330	−	0.689545i
$266$		0			0
$267$	−10.5830			−0.647669
$268$		0			0
$269$			18.7083i			1.14066i	0.821414	+	0.570332i	$0.193186\pi$
$269$			18.7083i			1.14066i	−0.821414	+	0.570332i	$0.806814\pi$
$270$		0			0
$271$	−25.9230			−1.57471			−0.787354	−	0.616501i	$-0.788550\pi$
$271$	−25.9230			−1.57471			−0.787354	+	0.616501i	$0.788550\pi$
$272$		0			0
$273$		−	9.16515i		−	0.554700i
$274$		0			0
$275$	−15.8745	+	6.92820i	−0.957269	+	0.417786i
$276$		0			0
$277$		−	9.16515i		−	0.550681i	−0.961347	−	0.275340i	$-0.911209\pi$
$277$		−	9.16515i		−	0.550681i	0.961347	−	0.275340i	$-0.0887905\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	21.2132i		−	1.26099i	−0.776192	−	0.630497i	$-0.782851\pi$
$283$		−	21.2132i		−	1.26099i	0.776192	−	0.630497i	$-0.217149\pi$
$284$		0			0
$285$	−17.1464	−	11.2250i	−1.01567	−	0.664910i
$286$		0			0
$287$		−	19.7990i		−	1.16870i
$288$		0			0
$289$	7.00000			0.411765
$290$		0			0
$291$		−	20.7846i		−	1.21842i
$292$		0			0
$293$	−31.8434			−1.86031			−0.930155	−	0.367168i	$-0.880327\pi$
$293$	−31.8434			−1.86031			−0.930155	+	0.367168i	$0.880327\pi$
$294$		0			0
$295$	−7.93725	+	12.1244i	−0.462125	+	0.705907i
$296$		0			0
$297$	19.5959			1.13707
$298$		0			0
$299$		0			0
$300$		0			0
$301$	14.0000			0.806947
$302$		0			0
$303$	5.29150			0.303989
$304$		0			0
$305$	21.0000	+	13.7477i	1.20246	+	0.787193i
$306$		0			0
$307$		−	4.24264i		−	0.242140i	−0.992644	−	0.121070i	$-0.961367\pi$
$307$		−	4.24264i		−	0.242140i	0.992644	−	0.121070i	$-0.0386326\pi$
$308$		0			0
$309$	−24.0000			−1.36531
$310$		0			0
$311$	−12.9615			−0.734978			−0.367489	−	0.930028i	$-0.619782\pi$
$311$	−12.9615			−0.734978			−0.367489	+	0.930028i	$0.619782\pi$
$312$		0			0
$313$	9.79796			0.553813			0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796			0.553813			0.276907	+	0.960897i	$0.410691\pi$
$314$		0			0
$315$	−3.24037	+	4.94975i	−0.182574	+	0.278887i
$316$		0			0
$317$			9.16515i			0.514766i	0.966309	+	0.257383i	$0.0828602\pi$
$317$			9.16515i			0.514766i	−0.966309	+	0.257383i	$0.917140\pi$
$318$		0			0
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$		−	22.4499i		−	1.25303i
$322$		0			0
$323$	31.7490			1.76656
$324$		0			0
$325$	4.89898	+	11.2250i	0.271746	+	0.622649i
$326$		0			0
$327$			2.82843i			0.156412i
$328$		0			0
$329$			7.48331i			0.412568i
$330$		0			0
$331$			24.2487i			1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$			24.2487i			1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$		0			0
$333$		−	9.16515i		−	0.502247i
$334$		0			0
$335$	6.48074	−	9.89949i	0.354081	−	0.540867i
$336$		0			0
$337$		−	9.16515i		−	0.499258i	−0.968342	−	0.249629i	$-0.919691\pi$
$337$		−	9.16515i		−	0.499258i	0.968342	−	0.249629i	$-0.0803086\pi$
$338$		0			0
$339$	12.9615			0.703971
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.8745			−0.852188			−0.426094	−	0.904679i	$-0.640111\pi$
$347$	−15.8745			−0.852188			−0.426094	+	0.904679i	$0.640111\pi$
$348$		0			0
$349$		−	11.2250i		−	0.600859i	−0.953804	−	0.300429i	$-0.902870\pi$
$349$		−	11.2250i		−	0.600859i	0.953804	−	0.300429i	$-0.0971300\pi$
$350$		0			0
$351$		−	13.8564i		−	0.739600i
$352$		0			0
$353$	−4.89898			−0.260746			−0.130373	−	0.991465i	$-0.541618\pi$
$353$	−4.89898			−0.260746			−0.130373	+	0.991465i	$0.541618\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			18.3303i			0.970143i
$358$		0			0
$359$			24.2487i			1.27980i	0.768459	+	0.639899i	$0.221024\pi$
$359$			24.2487i			1.27980i	−0.768459	+	0.639899i	$0.778976\pi$
$360$		0			0
$361$	23.0000			1.21053
$362$		0			0
$363$		−	1.41421i		−	0.0742270i
$364$		0			0
$365$	12.0000	−	18.3303i	0.628109	−	0.959452i
$366$		0			0
$367$		−	8.48528i		−	0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$		−	8.48528i		−	0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$		0			0
$369$		−	7.48331i		−	0.389566i
$370$		0			0
$371$			24.2487i			1.25893i
$372$		0			0
$373$		−	9.16515i		−	0.474554i	−0.971442	−	0.237277i	$-0.923745\pi$
$373$		−	9.16515i		−	0.474554i	0.971442	−	0.237277i	$-0.0762548\pi$
$374$		0			0
$375$	−2.64575	+	15.5885i	−0.136626	+	0.804984i
$376$		0			0
$377$	−14.6969			−0.756931
$378$		0			0
$379$			24.2487i			1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$			24.2487i			1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$		0			0
$381$			14.9666i			0.766764i
$382$		0			0
$383$		−	19.7990i		−	1.01168i	−0.862627	−	0.505841i	$-0.831182\pi$
$383$		−	19.7990i		−	1.01168i	0.862627	−	0.505841i	$-0.168818\pi$
$384$		0			0
$385$	17.1464	+	11.2250i	0.873863	+	0.572078i
$386$		0			0
$387$	5.29150			0.268982
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			27.4955i			1.38696i
$394$		0			0
$395$	−12.9615	−	8.48528i	−0.652163	−	0.426941i
$396$		0			0
$397$	7.34847			0.368809			0.184405	−	0.982850i	$-0.440964\pi$
$397$	7.34847			0.368809			0.184405	+	0.982850i	$0.440964\pi$
$398$		0			0
$399$			24.2487i			1.21395i
$400$		0			0
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	6.12372	−	9.35414i	0.304290	−	0.464811i
$406$		0			0
$407$	−31.7490			−1.57374
$408$		0			0
$409$			22.4499i			1.11008i	0.831824	+	0.555039i	$0.187297\pi$
$409$			22.4499i			1.11008i	−0.831824	+	0.555039i	$0.812703\pi$
$410$		0			0
$411$	−25.9230			−1.27869
$412$		0			0
$413$	17.1464			0.843721
$414$		0			0
$415$	−18.5203	−	12.1244i	−0.909124	−	0.595161i
$416$		0			0
$417$		−	27.4955i		−	1.34646i
$418$		0			0
$419$	6.48074			0.316605			0.158302	−	0.987391i	$-0.449398\pi$
$419$	6.48074			0.316605			0.158302	+	0.987391i	$0.449398\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$			2.82843i			0.137523i
$424$		0			0
$425$	−9.79796	−	22.4499i	−0.475271	−	1.08898i
$426$		0			0
$427$		−	29.6985i		−	1.43721i
$428$		0			0
$429$	−12.0000			−0.579365
$430$		0			0
$431$			3.46410i			0.166860i	0.996514	+	0.0834300i	$0.0265875\pi$
$431$			3.46410i			0.166860i	−0.996514	+	0.0834300i	$0.973413\pi$
$432$		0			0
$433$	−4.89898			−0.235430			−0.117715	−	0.993047i	$-0.537557\pi$
$433$	−4.89898			−0.235430			−0.117715	+	0.993047i	$0.537557\pi$
$434$		0			0
$435$	−15.8745	−	10.3923i	−0.761124	−	0.498273i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−12.9615			−0.618618			−0.309309	−	0.950962i	$-0.600098\pi$
$439$	−12.9615			−0.618618			−0.309309	+	0.950962i	$0.600098\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$		0			0
$443$	15.8745			0.754221			0.377110	−	0.926168i	$-0.376918\pi$
$443$	15.8745			0.754221			0.377110	+	0.926168i	$0.376918\pi$
$444$		0			0
$445$	−14.0000	−	9.16515i	−0.663664	−	0.434470i
$446$		0			0
$447$		−	8.48528i		−	0.401340i
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−25.9230			−1.22066
$452$		0			0
$453$	4.89898			0.230174
$454$		0			0
$455$	7.93725	−	12.1244i	0.372104	−	0.568399i
$456$		0			0
$457$		−	27.4955i		−	1.28618i	−0.765789	−	0.643092i	$-0.777651\pi$
$457$		−	27.4955i		−	1.28618i	0.765789	−	0.643092i	$-0.222349\pi$
$458$		0			0
$459$			27.7128i			1.29352i
$460$		0			0
$461$		−	18.7083i		−	0.871332i	−0.900108	−	0.435666i	$-0.856513\pi$
$461$		−	18.7083i		−	0.871332i	0.900108	−	0.435666i	$-0.143487\pi$
$462$		0			0
$463$	26.4575			1.22958			0.614792	−	0.788689i	$-0.289240\pi$
$463$	26.4575			1.22958			0.614792	+	0.788689i	$0.289240\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	7.07107i		−	0.327210i	−0.986526	−	0.163605i	$-0.947688\pi$
$467$		−	7.07107i		−	0.327210i	0.986526	−	0.163605i	$-0.0523123\pi$
$468$		0			0
$469$	−14.0000			−0.646460
$470$		0			0
$471$			10.3923i			0.478852i
$472$		0			0
$473$		−	18.3303i		−	0.842828i
$474$		0			0
$475$	−12.9615	−	29.6985i	−0.594714	−	1.36266i
$476$		0			0
$477$			9.16515i			0.419643i
$478$		0			0
$479$	−25.9230			−1.18445			−0.592225	−	0.805772i	$-0.701750\pi$
$479$	−25.9230			−1.18445			−0.592225	+	0.805772i	$0.701750\pi$
$480$		0			0
$481$			22.4499i			1.02363i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	18.0000	−	27.4955i	0.817338	−	1.24850i
$486$		0			0
$487$	−21.1660			−0.959123			−0.479562	−	0.877508i	$-0.659204\pi$
$487$	−21.1660			−0.959123			−0.479562	+	0.877508i	$0.659204\pi$
$488$		0			0
$489$			7.48331i			0.338407i
$490$		0			0
$491$		−	10.3923i		−	0.468998i	−0.972116	−	0.234499i	$-0.924655\pi$
$491$		−	10.3923i		−	0.468998i	0.972116	−	0.234499i	$-0.0753450\pi$
$492$		0			0
$493$	29.3939			1.32383
$494$		0			0
$495$	6.48074	+	4.24264i	0.291288	+	0.190693i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	17.3205i		−	0.775372i	−0.921791	−	0.387686i	$-0.873274\pi$
$499$		−	17.3205i		−	0.775372i	0.921791	−	0.387686i	$-0.126726\pi$
$500$		0			0
$501$	−8.00000			−0.357414
$502$		0			0
$503$			5.65685i			0.252227i	0.992016	+	0.126113i	$0.0402503\pi$
$503$			5.65685i			0.252227i	−0.992016	+	0.126113i	$0.959750\pi$
$504$		0			0
$505$	7.00000	+	4.58258i	0.311496	+	0.203922i
$506$		0			0
$507$		−	9.89949i		−	0.439652i
$508$		0			0
$509$		−	3.74166i		−	0.165846i	−0.996556	−	0.0829230i	$-0.973574\pi$
$509$		−	3.74166i		−	0.165846i	0.996556	−	0.0829230i	$-0.0264256\pi$
$510$		0			0
$511$	−25.9230			−1.14676
$512$		0			0
$513$			36.6606i			1.61861i
$514$		0			0
$515$	−31.7490	−	20.7846i	−1.39903	−	0.915879i
$516$		0			0
$517$	9.79796			0.430914
$518$		0			0
$519$		−	10.3923i		−	0.456172i
$520$		0			0
$521$			37.4166i			1.63925i	0.572900	+	0.819625i	$0.305818\pi$
$521$			37.4166i			1.63925i	−0.572900	+	0.819625i	$0.694182\pi$
$522$		0			0
$523$		−	29.6985i		−	1.29862i	−0.760522	−	0.649312i	$-0.775057\pi$
$523$		−	29.6985i		−	1.29862i	0.760522	−	0.649312i	$-0.224943\pi$
$524$		0			0
$525$	17.1464	−	7.48331i	0.748331	−	0.326599i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	6.48074			0.281240
$532$		0			0
$533$			18.3303i			0.793974i
$534$		0			0
$535$	19.4422	−	29.6985i	0.840561	−	1.28398i
$536$		0			0
$537$	−34.2929			−1.47985
$538$		0			0
$539$		−	24.2487i		−	1.04447i
$540$		0			0
$541$	−38.0000			−1.63375			−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000			−1.63375			−0.816874	+	0.576816i	$0.804295\pi$
$542$		0			0
$543$	15.8745			0.681240
$544$		0			0
$545$	−2.44949	+	3.74166i	−0.104925	+	0.160275i
$546$		0			0
$547$	26.4575			1.13124			0.565621	−	0.824665i	$-0.308637\pi$
$547$	26.4575			1.13124			0.565621	+	0.824665i	$0.308637\pi$
$548$		0			0
$549$		−	11.2250i		−	0.479070i
$550$		0			0
$551$	38.8844			1.65653
$552$		0			0
$553$			18.3303i			0.779484i
$554$		0			0
$555$	−15.8745	+	24.2487i	−0.673835	+	1.02930i
$556$		0			0
$557$			27.4955i			1.16502i	0.812824	+	0.582510i	$0.197929\pi$
$557$			27.4955i			1.16502i	−0.812824	+	0.582510i	$0.802071\pi$
$558$		0			0
$559$	−12.9615			−0.548212
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$		−	9.89949i		−	0.417214i	−0.978000	−	0.208607i	$-0.933107\pi$
$563$		−	9.89949i		−	0.417214i	0.978000	−	0.208607i	$-0.0668929\pi$
$564$		0			0
$565$	17.1464	+	11.2250i	0.721356	+	0.472238i
$566$		0			0
$567$	−13.2288			−0.555556
$568$		0			0
$569$	36.0000			1.50920			0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000			1.50920			0.754599	+	0.656186i	$0.227831\pi$
$570$		0			0
$571$		−	24.2487i		−	1.01478i	−0.861717	−	0.507388i	$-0.830611\pi$
$571$		−	24.2487i		−	1.01478i	0.861717	−	0.507388i	$-0.169389\pi$
$572$		0			0
$573$	−9.79796			−0.409316
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−4.89898			−0.203947			−0.101974	−	0.994787i	$-0.532516\pi$
$577$	−4.89898			−0.203947			−0.101974	+	0.994787i	$0.532516\pi$
$578$		0			0
$579$	−12.9615			−0.538661
$580$		0			0
$581$			26.1916i			1.08661i
$582$		0			0
$583$	31.7490			1.31491
$584$		0			0
$585$	3.00000	−	4.58258i	0.124035	−	0.189466i
$586$		0			0
$587$		−	9.89949i		−	0.408596i	−0.978909	−	0.204298i	$-0.934509\pi$
$587$		−	9.89949i		−	0.408596i	0.978909	−	0.204298i	$-0.0654911\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	12.9615			0.533164
$592$		0			0
$593$	−24.4949			−1.00588			−0.502942	−	0.864320i	$-0.667749\pi$
$593$	−24.4949			−1.00588			−0.502942	+	0.864320i	$0.667749\pi$
$594$		0			0
$595$	−15.8745	+	24.2487i	−0.650791	+	0.994100i
$596$		0			0
$597$			18.3303i			0.750209i
$598$		0			0
$599$		−	27.7128i		−	1.13231i	−0.824297	−	0.566157i	$-0.808429\pi$
$599$		−	27.7128i		−	1.13231i	0.824297	−	0.566157i	$-0.191571\pi$
$600$		0			0
$601$		−	22.4499i		−	0.915752i	−0.889016	−	0.457876i	$-0.848610\pi$
$601$		−	22.4499i		−	0.915752i	0.889016	−	0.457876i	$-0.151390\pi$
$602$		0			0
$603$	−5.29150			−0.215487
$604$		0			0
$605$	1.22474	−	1.87083i	0.0497930	−	0.0760600i
$606$		0			0
$607$		−	42.4264i		−	1.72203i	−0.508576	−	0.861017i	$-0.669828\pi$
$607$		−	42.4264i		−	1.72203i	0.508576	−	0.861017i	$-0.330172\pi$
$608$		0			0
$609$			22.4499i			0.909718i
$610$		0			0
$611$		−	6.92820i		−	0.280285i
$612$		0			0
$613$		−	27.4955i		−	1.11053i	−0.831673	−	0.555265i	$-0.812617\pi$
$613$		−	27.4955i		−	1.11053i	0.831673	−	0.555265i	$-0.187383\pi$
$614$		0			0
$615$	−12.9615	+	19.7990i	−0.522657	+	0.798372i
$616$		0			0
$617$		−	27.4955i		−	1.10693i	−0.832874	−	0.553463i	$-0.813306\pi$
$617$		−	27.4955i		−	1.10693i	0.832874	−	0.553463i	$-0.186694\pi$
$618$		0			0
$619$	−45.3652			−1.82338			−0.911690	−	0.410878i	$-0.865222\pi$
$619$	−45.3652			−1.82338			−0.911690	+	0.410878i	$0.865222\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			19.7990i			0.793230i
$624$		0			0
$625$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$626$		0			0
$627$	31.7490			1.26793
$628$		0			0
$629$		−	44.8999i		−	1.79028i
$630$		0			0
$631$			13.8564i			0.551615i	0.961213	+	0.275807i	$0.0889452\pi$
$631$			13.8564i			0.551615i	−0.961213	+	0.275807i	$0.911055\pi$
$632$		0			0
$633$	14.6969			0.584151
$634$		0			0
$635$	−12.9615	+	19.7990i	−0.514361	+	0.785699i
$636$		0			0
$637$	−17.1464			−0.679366
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.0000			−0.947943			−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000			−0.947943			−0.473972	+	0.880540i	$0.657180\pi$
$642$		0			0
$643$		−	4.24264i		−	0.167313i	−0.996495	−	0.0836567i	$-0.973340\pi$
$643$		−	4.24264i		−	0.167313i	0.996495	−	0.0836567i	$-0.0266599\pi$
$644$		0			0
$645$	−14.0000	−	9.16515i	−0.551249	−	0.360877i
$646$		0			0
$647$			39.5980i			1.55676i	0.627795	+	0.778379i	$0.283958\pi$
$647$			39.5980i			1.55676i	−0.627795	+	0.778379i	$0.716042\pi$
$648$		0			0
$649$		−	22.4499i		−	0.881237i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			27.4955i			1.07598i	0.842951	+	0.537990i	$0.180816\pi$
$653$			27.4955i			1.07598i	−0.842951	+	0.537990i	$0.819184\pi$
$654$		0			0
$655$	−23.8118	+	36.3731i	−0.930403	+	1.42121i
$656$		0			0
$657$	−9.79796			−0.382255
$658$		0			0
$659$		−	10.3923i		−	0.404827i	−0.979300	−	0.202413i	$-0.935122\pi$
$659$		−	10.3923i		−	0.404827i	0.979300	−	0.202413i	$-0.0648785\pi$
$660$		0			0
$661$			33.6749i			1.30980i	0.755714	+	0.654901i	$0.227290\pi$
$661$			33.6749i			1.30980i	−0.755714	+	0.654901i	$0.772710\pi$
$662$		0			0
$663$		−	16.9706i		−	0.659082i
$664$		0			0
$665$	−21.0000	+	32.0780i	−0.814345	+	1.24393i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	36.0000			1.39184
$670$		0			0
$671$	−38.8844			−1.50112
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	25.9230	−	11.3137i	0.997775	−	0.435465i
$676$		0			0
$677$	−7.34847			−0.282425			−0.141212	−	0.989979i	$-0.545100\pi$
$677$	−7.34847			−0.282425			−0.141212	+	0.989979i	$0.545100\pi$
$678$		0			0
$679$	−38.8844			−1.49225
$680$		0			0
$681$	−26.0000			−0.996322
$682$		0			0
$683$	47.6235			1.82226			0.911132	−	0.412115i	$-0.135210\pi$
$683$	47.6235			1.82226			0.911132	+	0.412115i	$0.135210\pi$
$684$		0			0
$685$	−34.2929	−	22.4499i	−1.31026	−	0.857768i
$686$		0			0
$687$	15.8745			0.605650
$688$		0			0
$689$		−	22.4499i		−	0.855275i
$690$		0			0
$691$	−6.48074			−0.246539			−0.123269	−	0.992373i	$-0.539338\pi$
$691$	−6.48074			−0.246539			−0.123269	+	0.992373i	$0.539338\pi$
$692$		0			0
$693$		−	9.16515i		−	0.348155i
$694$		0			0
$695$	23.8118	−	36.3731i	0.903232	−	1.37971i
$696$		0			0
$697$		−	36.6606i		−	1.38862i
$698$		0			0
$699$	25.9230			0.980497
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$		−	59.3970i		−	2.24020i
$704$		0			0
$705$	4.89898	−	7.48331i	0.184506	−	0.281838i
$706$		0			0
$707$		−	9.89949i		−	0.372309i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$			6.92820i			0.259828i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−15.8745	−	10.3923i	−0.593673	−	0.388650i
$716$		0			0
$717$	14.6969			0.548867
$718$		0			0
$719$	51.8459			1.93353			0.966763	−	0.255673i	$-0.0822970\pi$
$719$	51.8459			1.93353			0.966763	+	0.255673i	$0.0822970\pi$
$720$		0			0
$721$			44.8999i			1.67216i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.0000	−	27.4955i	−0.445669	−	1.02116i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$	25.9230			0.958795
$732$		0			0
$733$	−26.9444			−0.995214			−0.497607	−	0.867403i	$-0.665788\pi$
$733$	−26.9444			−0.995214			−0.497607	+	0.867403i	$0.665788\pi$
$734$		0			0
$735$	−18.5203	−	12.1244i	−0.683130	−	0.447214i
$736$		0			0
$737$			18.3303i			0.675205i
$738$		0			0
$739$			51.9615i			1.91144i	0.294285	+	0.955718i	$0.404919\pi$
$739$			51.9615i			1.91144i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		−	22.4499i		−	0.824719i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	7.34847	−	11.2250i	0.269227	−	0.411251i
$746$		0			0
$747$			9.89949i			0.362204i
$748$		0			0
$749$	−42.0000			−1.53465
$750$		0			0
$751$		−	31.1769i		−	1.13766i	−0.822455	−	0.568831i	$-0.807396\pi$
$751$		−	31.1769i		−	1.13766i	0.822455	−	0.568831i	$-0.192604\pi$
$752$		0			0
$753$		−	9.16515i		−	0.333997i
$754$		0			0
$755$	6.48074	+	4.24264i	0.235858	+	0.154406i
$756$		0			0
$757$			45.8258i			1.66557i	0.553600	+	0.832783i	$0.313254\pi$
$757$			45.8258i			1.66557i	−0.553600	+	0.832783i	$0.686746\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	37.4166i		−	1.35635i	−0.734901	−	0.678175i	$-0.762771\pi$
$761$		−	37.4166i		−	1.35635i	0.734901	−	0.678175i	$-0.237229\pi$
$762$		0			0
$763$	5.29150			0.191565
$764$		0			0
$765$	−6.00000	+	9.16515i	−0.216930	+	0.331367i
$766$		0			0
$767$	−15.8745			−0.573195
$768$		0			0
$769$		−	44.8999i		−	1.61913i	−0.587029	−	0.809566i	$-0.699703\pi$
$769$		−	44.8999i		−	1.61913i	0.587029	−	0.809566i	$-0.300297\pi$
$770$		0			0
$771$			20.7846i			0.748539i
$772$		0			0
$773$	12.2474			0.440510			0.220255	−	0.975442i	$-0.429311\pi$
$773$	12.2474			0.440510			0.220255	+	0.975442i	$0.429311\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	34.2929			1.23025
$778$		0			0
$779$		−	48.4974i		−	1.73760i
$780$		0			0
$781$		0			0
$782$		0			0
$783$			33.9411i			1.21296i
$784$		0			0
$785$	−9.00000	+	13.7477i	−0.321224	+	0.490677i
$786$		0			0
$787$			38.1838i			1.36110i	0.732700	+	0.680552i	$0.238260\pi$
$787$			38.1838i			1.36110i	−0.732700	+	0.680552i	$0.761740\pi$
$788$		0			0
$789$			22.4499i			0.799239i
$790$		0			0
$791$		−	24.2487i		−	0.862185i
$792$		0			0
$793$			27.4955i			0.976392i
$794$		0			0
$795$	15.8745	−	24.2487i	0.563011	−	0.860013i
$796$		0			0
$797$	−36.7423			−1.30148			−0.650740	−	0.759300i	$-0.725541\pi$
$797$	−36.7423			−1.30148			−0.650740	+	0.759300i	$0.725541\pi$
$798$		0			0
$799$			13.8564i			0.490204i
$800$		0			0
$801$			7.48331i			0.264410i
$802$		0			0
$803$			33.9411i			1.19776i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−26.4575			−0.931349
$808$		0			0
$809$	12.0000			0.421898			0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000			0.421898			0.210949	+	0.977497i	$0.432345\pi$
$810$		0			0
$811$	6.48074			0.227570			0.113785	−	0.993505i	$-0.463703\pi$
$811$	6.48074			0.227570			0.113785	+	0.993505i	$0.463703\pi$
$812$		0			0
$813$		−	36.6606i		−	1.28574i
$814$		0			0
$815$	−6.48074	+	9.89949i	−0.227010	+	0.346764i
$816$		0			0
$817$	34.2929			1.19976
$818$		0			0
$819$	−6.48074			−0.226455
$820$		0			0
$821$	−18.0000			−0.628204			−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000			−0.628204			−0.314102	+	0.949389i	$0.601703\pi$
$822$		0			0
$823$	26.4575			0.922251			0.461125	−	0.887335i	$-0.347446\pi$
$823$	26.4575			0.922251			0.461125	+	0.887335i	$0.347446\pi$
$824$		0			0
$825$	−9.79796	−	22.4499i	−0.341121	−	0.781607i
$826$		0			0
$827$	−47.6235			−1.65603			−0.828016	−	0.560704i	$-0.810530\pi$
$827$	−47.6235			−1.65603			−0.828016	+	0.560704i	$0.810530\pi$
$828$		0			0
$829$			56.1249i			1.94930i	0.223742	+	0.974648i	$0.428173\pi$
$829$			56.1249i			1.94930i	−0.223742	+	0.974648i	$0.571827\pi$
$830$		0			0
$831$	12.9615			0.449629
$832$		0			0
$833$	34.2929			1.18818
$834$		0			0
$835$	−10.5830	−	6.92820i	−0.366240	−	0.239760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.9615			0.447480			0.223740	−	0.974649i	$-0.428173\pi$
$839$	12.9615			0.447480			0.223740	+	0.974649i	$0.428173\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			8.48528i			0.292249i
$844$		0			0
$845$	8.57321	−	13.0958i	0.294928	−	0.450509i
$846$		0			0
$847$	−2.64575			−0.0909091
$848$		0			0
$849$	30.0000			1.02960
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−22.0454			−0.754820			−0.377410	−	0.926046i	$-0.623185\pi$
$853$	−22.0454			−0.754820			−0.377410	+	0.926046i	$0.623185\pi$
$854$		0			0
$855$	−7.93725	+	12.1244i	−0.271448	+	0.414644i
$856$		0			0
$857$	9.79796			0.334692			0.167346	−	0.985898i	$-0.446480\pi$
$857$	9.79796			0.334692			0.167346	+	0.985898i	$0.446480\pi$
$858$		0			0
$859$	−6.48074			−0.221120			−0.110560	−	0.993869i	$-0.535264\pi$
$859$	−6.48074			−0.221120			−0.110560	+	0.993869i	$0.535264\pi$
$860$		0			0
$861$	28.0000			0.954237
$862$		0			0
$863$	15.8745			0.540375			0.270187	−	0.962808i	$-0.412914\pi$
$863$	15.8745			0.540375			0.270187	+	0.962808i	$0.412914\pi$
$864$		0			0
$865$	9.00000	−	13.7477i	0.306009	−	0.467437i
$866$		0			0
$867$			9.89949i			0.336204i
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	12.9615			0.439183
$872$		0			0
$873$	−14.6969			−0.497416
$874$		0			0
$875$	29.1633	+	4.94975i	0.985901	+	0.167332i
$876$		0			0
$877$			9.16515i			0.309485i	0.987955	+	0.154743i	$0.0494548\pi$
$877$			9.16515i			0.309485i	−0.987955	+	0.154743i	$0.950545\pi$
$878$		0			0
$879$		−	45.0333i		−	1.51894i
$880$		0			0
$881$			29.9333i			1.00848i	0.863564	+	0.504239i	$0.168227\pi$
$881$			29.9333i			1.00848i	−0.863564	+	0.504239i	$0.831773\pi$
$882$		0			0
$883$	−5.29150			−0.178073			−0.0890366	−	0.996028i	$-0.528379\pi$
$883$	−5.29150			−0.178073			−0.0890366	+	0.996028i	$0.528379\pi$
$884$		0			0
$885$	−17.1464	−	11.2250i	−0.576371	−	0.377323i
$886$		0			0
$887$		−	39.5980i		−	1.32957i	−0.747035	−	0.664785i	$-0.768523\pi$
$887$		−	39.5980i		−	1.32957i	0.747035	−	0.664785i	$-0.231477\pi$
$888$		0			0
$889$	28.0000			0.939090
$890$		0			0
$891$			17.3205i			0.580259i
$892$		0			0
$893$			18.3303i			0.613400i
$894$		0			0
$895$	−45.3652	−	29.6985i	−1.51639	−	0.992711i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			44.8999i			1.49583i
$902$		0			0
$903$			19.7990i			0.658869i
$904$		0			0
$905$	21.0000	+	13.7477i	0.698064	+	0.456990i
$906$		0			0
$907$	−5.29150			−0.175701			−0.0878507	−	0.996134i	$-0.528000\pi$
$907$	−5.29150			−0.175701			−0.0878507	+	0.996134i	$0.528000\pi$
$908$		0			0
$909$		−	3.74166i		−	0.124103i
$910$		0			0
$911$			24.2487i			0.803396i	0.915772	+	0.401698i	$0.131580\pi$
$911$			24.2487i			0.803396i	−0.915772	+	0.401698i	$0.868420\pi$
$912$		0			0
$913$	34.2929			1.13493
$914$		0			0
$915$	−19.4422	+	29.6985i	−0.642740	+	0.981802i
$916$		0			0
$917$	51.4393			1.69867
$918$		0			0
$919$			55.4256i			1.82832i	0.405351	+	0.914161i	$0.367149\pi$
$919$			55.4256i			1.82832i	−0.405351	+	0.914161i	$0.632851\pi$
$920$		0			0
$921$	6.00000			0.197707
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−42.0000	+	18.3303i	−1.38095	+	0.602697i
$926$		0			0
$927$			16.9706i			0.557386i
$928$		0			0
$929$		−	14.9666i		−	0.491039i	−0.969392	−	0.245520i	$-0.921041\pi$
$929$		−	14.9666i		−	0.491039i	0.969392	−	0.245520i	$-0.0789586\pi$
$930$		0			0
$931$	45.3652			1.48678
$932$		0			0
$933$		−	18.3303i		−	0.600107i
$934$		0			0
$935$	31.7490	+	20.7846i	1.03830	+	0.679729i
$936$		0			0
$937$	19.5959			0.640171			0.320085	−	0.947389i	$-0.396288\pi$
$937$	19.5959			0.640171			0.320085	+	0.947389i	$0.396288\pi$
$938$		0			0
$939$			13.8564i			0.452187i
$940$		0			0
$941$			26.1916i			0.853822i	0.904294	+	0.426911i	$0.140398\pi$
$941$			26.1916i			0.853822i	−0.904294	+	0.426911i	$0.859602\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−28.0000	−	18.3303i	−0.910840	−	0.596285i
$946$		0			0
$947$	15.8745			0.515852			0.257926	−	0.966165i	$-0.416961\pi$
$947$	15.8745			0.515852			0.257926	+	0.966165i	$0.416961\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$		0			0
$951$	−12.9615			−0.420305
$952$		0			0
$953$		−	54.9909i		−	1.78133i	−0.454660	−	0.890665i	$-0.650239\pi$
$953$		−	54.9909i		−	1.78133i	0.454660	−	0.890665i	$-0.349761\pi$
$954$		0			0
$955$	−12.9615	−	8.48528i	−0.419424	−	0.274577i
$956$		0			0
$957$	29.3939			0.950169
$958$		0			0
$959$			48.4974i			1.56606i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−15.8745			−0.511549
$964$		0			0
$965$	−17.1464	−	11.2250i	−0.551963	−	0.361345i
$966$		0			0
$967$	−42.3320			−1.36131			−0.680653	−	0.732606i	$-0.738304\pi$
$967$	−42.3320			−1.36131			−0.680653	+	0.732606i	$0.738304\pi$
$968$		0			0
$969$			44.8999i			1.44239i
$970$		0			0
$971$	−6.48074			−0.207977			−0.103988	−	0.994579i	$-0.533160\pi$
$971$	−6.48074			−0.207977			−0.103988	+	0.994579i	$0.533160\pi$
$972$		0			0
$973$	−51.4393			−1.64907
$974$		0			0
$975$	−15.8745	+	6.92820i	−0.508391	+	0.221880i
$976$		0			0
$977$			36.6606i			1.17288i	0.809994	+	0.586438i	$0.199470\pi$
$977$			36.6606i			1.17288i	−0.809994	+	0.586438i	$0.800530\pi$
$978$		0			0
$979$	25.9230			0.828501
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$			11.3137i			0.360851i	0.983589	+	0.180426i	$0.0577475\pi$
$983$			11.3137i			0.360851i	−0.983589	+	0.180426i	$0.942252\pi$
$984$		0			0
$985$	17.1464	+	11.2250i	0.546331	+	0.357657i
$986$		0			0
$987$	−10.5830			−0.336861
$988$		0			0
$989$		0			0
$990$		0			0
$991$			48.4974i			1.54057i	0.637699	+	0.770286i	$0.279886\pi$
$991$			48.4974i			1.54057i	−0.637699	+	0.770286i	$0.720114\pi$
$992$		0			0
$993$	−34.2929			−1.08825
$994$		0			0
$995$	−15.8745	+	24.2487i	−0.503256	+	0.768736i
$996$		0			0
$997$	−22.0454			−0.698185			−0.349093	−	0.937088i	$-0.613510\pi$
$997$	−22.0454			−0.698185			−0.349093	+	0.937088i	$0.613510\pi$
$998$		0			0
$999$	51.8459			1.64033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.e.2239.6		8
4.3	odd	2	inner	2240.2.e.e.2239.2		8
5.4	even	2	inner	2240.2.e.e.2239.4		8
7.6	odd	2	inner	2240.2.e.e.2239.3		8
8.3	odd	2		560.2.e.d.559.7	yes	8
8.5	even	2		560.2.e.d.559.3	yes	8
20.19	odd	2	inner	2240.2.e.e.2239.8		8
28.27	even	2	inner	2240.2.e.e.2239.7		8
35.34	odd	2	inner	2240.2.e.e.2239.5		8
40.3	even	4		2800.2.k.k.2351.3		8
40.13	odd	4		2800.2.k.k.2351.6		8
40.19	odd	2		560.2.e.d.559.1	✓	8
40.27	even	4		2800.2.k.k.2351.5		8
40.29	even	2		560.2.e.d.559.5	yes	8
40.37	odd	4		2800.2.k.k.2351.4		8
56.13	odd	2		560.2.e.d.559.6	yes	8
56.27	even	2		560.2.e.d.559.2	yes	8
140.139	even	2	inner	2240.2.e.e.2239.1		8
280.13	even	4		2800.2.k.k.2351.2		8
280.27	odd	4		2800.2.k.k.2351.1		8
280.69	odd	2		560.2.e.d.559.4	yes	8
280.83	odd	4		2800.2.k.k.2351.7		8
280.139	even	2		560.2.e.d.559.8	yes	8
280.237	even	4		2800.2.k.k.2351.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.e.d.559.1	✓	8	40.19	odd	2
560.2.e.d.559.2	yes	8	56.27	even	2
560.2.e.d.559.3	yes	8	8.5	even	2
560.2.e.d.559.4	yes	8	280.69	odd	2
560.2.e.d.559.5	yes	8	40.29	even	2
560.2.e.d.559.6	yes	8	56.13	odd	2
560.2.e.d.559.7	yes	8	8.3	odd	2
560.2.e.d.559.8	yes	8	280.139	even	2
2240.2.e.e.2239.1		8	140.139	even	2	inner
2240.2.e.e.2239.2		8	4.3	odd	2	inner
2240.2.e.e.2239.3		8	7.6	odd	2	inner
2240.2.e.e.2239.4		8	5.4	even	2	inner
2240.2.e.e.2239.5		8	35.34	odd	2	inner
2240.2.e.e.2239.6		8	1.1	even	1	trivial
2240.2.e.e.2239.7		8	28.27	even	2	inner
2240.2.e.e.2239.8		8	20.19	odd	2	inner
2800.2.k.k.2351.1		8	280.27	odd	4
2800.2.k.k.2351.2		8	280.13	even	4
2800.2.k.k.2351.3		8	40.3	even	4
2800.2.k.k.2351.4		8	40.37	odd	4
2800.2.k.k.2351.5		8	40.27	even	4
2800.2.k.k.2351.6		8	40.13	odd	4
2800.2.k.k.2351.7		8	280.83	odd	4
2800.2.k.k.2351.8		8	280.237	even	4

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

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +(-1.22474 + 1.87083i) q^{5} +2.64575 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{3} +(-1.22474 + 1.87083i) q^{5} +2.64575 q^{7} +1.00000 q^{9} -3.46410i q^{11} -2.44949 q^{13} +(-2.64575 - 1.73205i) q^{15} +4.89898 q^{17} +6.48074 q^{19} +3.74166i q^{21} +(-2.00000 - 4.58258i) q^{25} +5.65685i q^{27} +6.00000 q^{29} +4.89898 q^{33} +(-3.24037 + 4.94975i) q^{35} -9.16515i q^{37} -3.46410i q^{39} -7.48331i q^{41} +5.29150 q^{43} +(-1.22474 + 1.87083i) q^{45} +2.82843i q^{47} +7.00000 q^{49} +6.92820i q^{51} +9.16515i q^{53} +(6.48074 + 4.24264i) q^{55} +9.16515i q^{57} +6.48074 q^{59} -11.2250i q^{61} +2.64575 q^{63} +(3.00000 - 4.58258i) q^{65} -5.29150 q^{67} -9.79796 q^{73} +(6.48074 - 2.82843i) q^{75} -9.16515i q^{77} +6.92820i q^{79} -5.00000 q^{81} +9.89949i q^{83} +(-6.00000 + 9.16515i) q^{85} +8.48528i q^{87} +7.48331i q^{89} -6.48074 q^{91} +(-7.93725 + 12.1244i) q^{95} -14.6969 q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 16 q^{25} + 48 q^{29} + 56 q^{49} + 24 q^{65} - 40 q^{81} - 48 q^{85}+O(q^{100}) \) 8 * q + 8 * q^9 - 16 * q^25 + 48 * q^29 + 56 * q^49 + 24 * q^65 - 40 * q^81 - 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more