LMFDB - Embedded newform 1280.4.d.a.641.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.d (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:	$75.5224448073$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.a.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000i q^{3} +5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+6.00000i q^{3} +5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +16.0000i q^{11} +58.0000i q^{13} -30.0000 q^{15} -70.0000 q^{17} -4.00000i q^{19} -204.000i q^{21} -134.000 q^{23} -25.0000 q^{25} +108.000i q^{27} -242.000i q^{29} -100.000 q^{31} -96.0000 q^{33} -170.000i q^{35} +438.000i q^{37} -348.000 q^{39} +138.000 q^{41} +178.000i q^{43} -45.0000i q^{45} -22.0000 q^{47} +813.000 q^{49} -420.000i q^{51} -162.000i q^{53} -80.0000 q^{55} +24.0000 q^{57} -268.000i q^{59} +250.000i q^{61} +306.000 q^{63} -290.000 q^{65} -422.000i q^{67} -804.000i q^{69} -852.000 q^{71} -306.000 q^{73} -150.000i q^{75} -544.000i q^{77} +456.000 q^{79} -891.000 q^{81} -434.000i q^{83} -350.000i q^{85} +1452.00 q^{87} +726.000 q^{89} -1972.00i q^{91} -600.000i q^{93} +20.0000 q^{95} +1378.00 q^{97} -144.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 68 q^{7} - 18 q^{9}+O(q^{10}) $ 2 * q - 68 * q^7 - 18 * q^9	$ 2 q - 68 q^{7} - 18 q^{9} - 60 q^{15} - 140 q^{17} - 268 q^{23} - 50 q^{25} - 200 q^{31} - 192 q^{33} - 696 q^{39} + 276 q^{41} - 44 q^{47} + 1626 q^{49} - 160 q^{55} + 48 q^{57} + 612 q^{63} - 580 q^{65} - 1704 q^{71} - 612 q^{73} + 912 q^{79} - 1782 q^{81} + 2904 q^{87} + 1452 q^{89} + 40 q^{95} + 2756 q^{97}+O(q^{100}) $ 2 * q - 68 * q^7 - 18 * q^9 - 60 * q^15 - 140 * q^17 - 268 * q^23 - 50 * q^25 - 200 * q^31 - 192 * q^33 - 696 * q^39 + 276 * q^41 - 44 * q^47 + 1626 * q^49 - 160 * q^55 + 48 * q^57 + 612 * q^63 - 580 * q^65 - 1704 * q^71 - 612 * q^73 + 912 * q^79 - 1782 * q^81 + 2904 * q^87 + 1452 * q^89 + 40 * q^95 + 2756 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	6.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	6.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	5.00000i	0.447214i
$5$	5.00000i	0.447214i
$6$	0	0
$7$	−34.0000	−1.83583	−0.917914	−	0.396780i	$-0.870128\pi$
$7$	−34.0000	−1.83583	−0.917914	+	0.396780i	$0.870128\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	16.0000i	0.438562i	0.975662	+	0.219281i	$0.0703711\pi$
$11$	16.0000i	0.438562i	−0.975662	+	0.219281i	$0.929629\pi$
$12$	0	0
$13$	58.0000i	1.23741i	0.785624	+	0.618704i	$0.212342\pi$
$13$	58.0000i	1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$	0	0
$15$	−30.0000	−0.516398
$16$	0	0
$17$	−70.0000	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$17$	−70.0000	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$18$	0	0
$19$	− 4.00000i	− 0.0482980i	−0.999708	−	0.0241490i	$-0.992312\pi$
$19$	− 4.00000i	− 0.0482980i	0.999708	−	0.0241490i	$-0.00768762\pi$
$20$	0	0
$21$	− 204.000i	− 2.11983i
$22$	0	0
$23$	−134.000	−1.21482	−0.607412	−	0.794387i	$-0.707792\pi$
$23$	−134.000	−1.21482	−0.607412	+	0.794387i	$0.707792\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	108.000i	0.769800i
$28$	0	0
$29$	− 242.000i	− 1.54960i	−0.632209	−	0.774798i	$-0.717852\pi$
$29$	− 242.000i	− 1.54960i	0.632209	−	0.774798i	$-0.282148\pi$
$30$	0	0
$31$	−100.000	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$31$	−100.000	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$32$	0	0
$33$	−96.0000	−0.506408
$34$	0	0
$35$	− 170.000i	− 0.821007i
$36$	0	0
$37$	438.000i	1.94613i	0.230534	+	0.973064i	$0.425953\pi$
$37$	438.000i	1.94613i	−0.230534	+	0.973064i	$0.574047\pi$
$38$	0	0
$39$	−348.000	−1.42884
$40$	0	0
$41$	138.000	0.525658	0.262829	−	0.964842i	$-0.415344\pi$
$41$	138.000	0.525658	0.262829	+	0.964842i	$0.415344\pi$
$42$	0	0
$43$	178.000i	0.631273i	0.948880	+	0.315637i	$0.102218\pi$
$43$	178.000i	0.631273i	−0.948880	+	0.315637i	$0.897782\pi$
$44$	0	0
$45$	− 45.0000i	− 0.149071i
$46$	0	0
$47$	−22.0000	−0.0682772	−0.0341386	−	0.999417i	$-0.510869\pi$
$47$	−22.0000	−0.0682772	−0.0341386	+	0.999417i	$0.510869\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	0	0
$51$	− 420.000i	− 1.15317i
$52$	0	0
$53$	− 162.000i	− 0.419857i	−0.977717	−	0.209928i	$-0.932677\pi$
$53$	− 162.000i	− 0.419857i	0.977717	−	0.209928i	$-0.0673231\pi$
$54$	0	0
$55$	−80.0000	−0.196131
$56$	0	0
$57$	24.0000	0.0557698
$58$	0	0
$59$	− 268.000i	− 0.591367i	−0.955286	−	0.295683i	$-0.904453\pi$
$59$	− 268.000i	− 0.591367i	0.955286	−	0.295683i	$-0.0955473\pi$
$60$	0	0
$61$	250.000i	0.524741i	0.964967	+	0.262371i	$0.0845043\pi$
$61$	250.000i	0.524741i	−0.964967	+	0.262371i	$0.915496\pi$
$62$	0	0
$63$	306.000	0.611942
$64$	0	0
$65$	−290.000	−0.553386
$66$	0	0
$67$	− 422.000i	− 0.769485i	−0.923024	−	0.384743i	$-0.874290\pi$
$67$	− 422.000i	− 0.769485i	0.923024	−	0.384743i	$-0.125710\pi$
$68$	0	0
$69$	− 804.000i	− 1.40276i
$70$	0	0
$71$	−852.000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−852.000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−306.000	−0.490611	−0.245305	−	0.969446i	$-0.578888\pi$
$73$	−306.000	−0.490611	−0.245305	+	0.969446i	$0.578888\pi$
$74$	0	0
$75$	− 150.000i	− 0.230940i
$76$	0	0
$77$	− 544.000i	− 0.805124i
$78$	0	0
$79$	456.000	0.649418	0.324709	−	0.945814i	$-0.394734\pi$
$79$	456.000	0.649418	0.324709	+	0.945814i	$0.394734\pi$
$80$	0	0
$81$	−891.000	−1.22222
$82$	0	0
$83$	− 434.000i	− 0.573948i	−0.957938	−	0.286974i	$-0.907351\pi$
$83$	− 434.000i	− 0.573948i	0.957938	−	0.286974i	$-0.0926493\pi$
$84$	0	0
$85$	− 350.000i	− 0.446622i
$86$	0	0
$87$	1452.00	1.78932
$88$	0	0
$89$	726.000	0.864672	0.432336	−	0.901712i	$-0.357689\pi$
$89$	726.000	0.864672	0.432336	+	0.901712i	$0.357689\pi$
$90$	0	0
$91$	− 1972.00i	− 2.27167i
$92$	0	0
$93$	− 600.000i	− 0.669001i
$94$	0	0
$95$	20.0000	0.0215995
$96$	0	0
$97$	1378.00	1.44242	0.721210	−	0.692717i	$-0.243586\pi$
$97$	1378.00	1.44242	0.721210	+	0.692717i	$0.243586\pi$
$98$	0	0
$99$	− 144.000i	− 0.146187i
$100$	0	0
$101$	− 126.000i	− 0.124133i	−0.998072	−	0.0620667i	$-0.980231\pi$
$101$	− 126.000i	− 0.124133i	0.998072	−	0.0620667i	$-0.0197691\pi$
$102$	0	0
$103$	−1262.00	−1.20727	−0.603634	−	0.797262i	$-0.706281\pi$
$103$	−1262.00	−1.20727	−0.603634	+	0.797262i	$0.706281\pi$
$104$	0	0
$105$	1020.00	0.948017
$106$	0	0
$107$	510.000i	0.460781i	0.973098	+	0.230390i	$0.0740003\pi$
$107$	510.000i	0.460781i	−0.973098	+	0.230390i	$0.926000\pi$
$108$	0	0
$109$	26.0000i	0.0228472i	0.999935	+	0.0114236i	$0.00363633\pi$
$109$	26.0000i	0.0228472i	−0.999935	+	0.0114236i	$0.996364\pi$
$110$	0	0
$111$	−2628.00	−2.24720
$112$	0	0
$113$	1242.00	1.03396	0.516980	−	0.855997i	$-0.327056\pi$
$113$	1242.00	1.03396	0.516980	+	0.855997i	$0.327056\pi$
$114$	0	0
$115$	− 670.000i	− 0.543285i
$116$	0	0
$117$	− 522.000i	− 0.412469i
$118$	0	0
$119$	2380.00	1.83340
$120$	0	0
$121$	1075.00	0.807663
$122$	0	0
$123$	828.000i	0.606978i
$124$	0	0
$125$	− 125.000i	− 0.0894427i
$126$	0	0
$127$	978.000	0.683334	0.341667	−	0.939821i	$-0.389008\pi$
$127$	978.000	0.683334	0.341667	+	0.939821i	$0.389008\pi$
$128$	0	0
$129$	−1068.00	−0.728931
$130$	0	0
$131$	912.000i	0.608258i	0.952631	+	0.304129i	$0.0983654\pi$
$131$	912.000i	0.608258i	−0.952631	+	0.304129i	$0.901635\pi$
$132$	0	0
$133$	136.000i	0.0886669i
$134$	0	0
$135$	−540.000	−0.344265
$136$	0	0
$137$	926.000	0.577471	0.288735	−	0.957409i	$-0.406765\pi$
$137$	926.000	0.577471	0.288735	+	0.957409i	$0.406765\pi$
$138$	0	0
$139$	516.000i	0.314867i	0.987530	+	0.157434i	$0.0503220\pi$
$139$	516.000i	0.314867i	−0.987530	+	0.157434i	$0.949678\pi$
$140$	0	0
$141$	− 132.000i	− 0.0788398i
$142$	0	0
$143$	−928.000	−0.542680
$144$	0	0
$145$	1210.00	0.693000
$146$	0	0
$147$	4878.00i	2.73694i
$148$	0	0
$149$	958.000i	0.526728i	0.964697	+	0.263364i	$0.0848320\pi$
$149$	958.000i	0.526728i	−0.964697	+	0.263364i	$0.915168\pi$
$150$	0	0
$151$	332.000	0.178926	0.0894628	−	0.995990i	$-0.471485\pi$
$151$	332.000	0.178926	0.0894628	+	0.995990i	$0.471485\pi$
$152$	0	0
$153$	630.000	0.332892
$154$	0	0
$155$	− 500.000i	− 0.259103i
$156$	0	0
$157$	− 1022.00i	− 0.519519i	−0.965673	−	0.259759i	$-0.916357\pi$
$157$	− 1022.00i	− 0.519519i	0.965673	−	0.259759i	$-0.0836433\pi$
$158$	0	0
$159$	972.000	0.484809
$160$	0	0
$161$	4556.00	2.23021
$162$	0	0
$163$	926.000i	0.444969i	0.974936	+	0.222484i	$0.0714166\pi$
$163$	926.000i	0.444969i	−0.974936	+	0.222484i	$0.928583\pi$
$164$	0	0
$165$	− 480.000i	− 0.226472i
$166$	0	0
$167$	654.000	0.303042	0.151521	−	0.988454i	$-0.451583\pi$
$167$	654.000	0.303042	0.151521	+	0.988454i	$0.451583\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	36.0000i	0.0160993i
$172$	0	0
$173$	− 1294.00i	− 0.568676i	−0.958724	−	0.284338i	$-0.908226\pi$
$173$	− 1294.00i	− 0.568676i	0.958724	−	0.284338i	$-0.0917738\pi$
$174$	0	0
$175$	850.000	0.367165
$176$	0	0
$177$	1608.00	0.682851
$178$	0	0
$179$	2836.00i	1.18420i	0.805863	+	0.592102i	$0.201702\pi$
$179$	2836.00i	1.18420i	−0.805863	+	0.592102i	$0.798298\pi$
$180$	0	0
$181$	− 1742.00i	− 0.715369i	−0.933842	−	0.357685i	$-0.883566\pi$
$181$	− 1742.00i	− 0.715369i	0.933842	−	0.357685i	$-0.116434\pi$
$182$	0	0
$183$	−1500.00	−0.605919
$184$	0	0
$185$	−2190.00	−0.870335
$186$	0	0
$187$	− 1120.00i	− 0.437981i
$188$	0	0
$189$	− 3672.00i	− 1.41322i
$190$	0	0
$191$	−4460.00	−1.68960	−0.844802	−	0.535079i	$-0.820282\pi$
$191$	−4460.00	−1.68960	−0.844802	+	0.535079i	$0.820282\pi$
$192$	0	0
$193$	−3782.00	−1.41054	−0.705270	−	0.708939i	$-0.749174\pi$
$193$	−3782.00	−1.41054	−0.705270	+	0.708939i	$0.749174\pi$
$194$	0	0
$195$	− 1740.00i	− 0.638995i
$196$	0	0
$197$	− 4474.00i	− 1.61807i	−0.587762	−	0.809034i	$-0.699991\pi$
$197$	− 4474.00i	− 1.61807i	0.587762	−	0.809034i	$-0.300009\pi$
$198$	0	0
$199$	3608.00	1.28525	0.642624	−	0.766182i	$-0.277846\pi$
$199$	3608.00	1.28525	0.642624	+	0.766182i	$0.277846\pi$
$200$	0	0
$201$	2532.00	0.888525
$202$	0	0
$203$	8228.00i	2.84479i
$204$	0	0
$205$	690.000i	0.235081i
$206$	0	0
$207$	1206.00	0.404941
$208$	0	0
$209$	64.0000	0.0211817
$210$	0	0
$211$	256.000i	0.0835250i	0.999128	+	0.0417625i	$0.0132973\pi$
$211$	256.000i	0.0835250i	−0.999128	+	0.0417625i	$0.986703\pi$
$212$	0	0
$213$	− 5112.00i	− 1.64445i
$214$	0	0
$215$	−890.000	−0.282314
$216$	0	0
$217$	3400.00	1.06363
$218$	0	0
$219$	− 1836.00i	− 0.566509i
$220$	0	0
$221$	− 4060.00i	− 1.23577i
$222$	0	0
$223$	5158.00	1.54890	0.774451	−	0.632634i	$-0.218026\pi$
$223$	5158.00	1.54890	0.774451	+	0.632634i	$0.218026\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	2226.00i	0.650858i	0.945566	+	0.325429i	$0.105509\pi$
$227$	2226.00i	0.650858i	−0.945566	+	0.325429i	$0.894491\pi$
$228$	0	0
$229$	− 2086.00i	− 0.601951i	−0.953632	−	0.300975i	$-0.902688\pi$
$229$	− 2086.00i	− 0.601951i	0.953632	−	0.300975i	$-0.0973122\pi$
$230$	0	0
$231$	3264.00	0.929677
$232$	0	0
$233$	5718.00	1.60772	0.803860	−	0.594819i	$-0.202776\pi$
$233$	5718.00	1.60772	0.803860	+	0.594819i	$0.202776\pi$
$234$	0	0
$235$	− 110.000i	− 0.0305345i
$236$	0	0
$237$	2736.00i	0.749883i
$238$	0	0
$239$	3624.00	0.980825	0.490412	−	0.871491i	$-0.336846\pi$
$239$	3624.00	0.980825	0.490412	+	0.871491i	$0.336846\pi$
$240$	0	0
$241$	−82.0000	−0.0219174	−0.0109587	−	0.999940i	$-0.503488\pi$
$241$	−82.0000	−0.0219174	−0.0109587	+	0.999940i	$0.503488\pi$
$242$	0	0
$243$	− 2430.00i	− 0.641500i
$244$	0	0
$245$	4065.00i	1.06001i
$246$	0	0
$247$	232.000	0.0597644
$248$	0	0
$249$	2604.00	0.662738
$250$	0	0
$251$	− 5040.00i	− 1.26742i	−0.773571	−	0.633709i	$-0.781532\pi$
$251$	− 5040.00i	− 1.26742i	0.773571	−	0.633709i	$-0.218468\pi$
$252$	0	0
$253$	− 2144.00i	− 0.532775i
$254$	0	0
$255$	2100.00	0.515714
$256$	0	0
$257$	−2310.00	−0.560676	−0.280338	−	0.959901i	$-0.590447\pi$
$257$	−2310.00	−0.560676	−0.280338	+	0.959901i	$0.590447\pi$
$258$	0	0
$259$	− 14892.0i	− 3.57276i
$260$	0	0
$261$	2178.00i	0.516532i
$262$	0	0
$263$	−4110.00	−0.963625	−0.481813	−	0.876274i	$-0.660021\pi$
$263$	−4110.00	−0.963625	−0.481813	+	0.876274i	$0.660021\pi$
$264$	0	0
$265$	810.000	0.187766
$266$	0	0
$267$	4356.00i	0.998438i
$268$	0	0
$269$	746.000i	0.169087i	0.996420	+	0.0845435i	$0.0269432\pi$
$269$	746.000i	0.169087i	−0.996420	+	0.0845435i	$0.973057\pi$
$270$	0	0
$271$	4596.00	1.03021	0.515105	−	0.857127i	$-0.327753\pi$
$271$	4596.00	1.03021	0.515105	+	0.857127i	$0.327753\pi$
$272$	0	0
$273$	11832.0	2.62310
$274$	0	0
$275$	− 400.000i	− 0.0877124i
$276$	0	0
$277$	2206.00i	0.478504i	0.970957	+	0.239252i	$0.0769023\pi$
$277$	2206.00i	0.478504i	−0.970957	+	0.239252i	$0.923098\pi$
$278$	0	0
$279$	900.000	0.193124
$280$	0	0
$281$	−8278.00	−1.75738	−0.878691	−	0.477392i	$-0.841582\pi$
$281$	−8278.00	−1.75738	−0.878691	+	0.477392i	$0.841582\pi$
$282$	0	0
$283$	1178.00i	0.247438i	0.992317	+	0.123719i	$0.0394821\pi$
$283$	1178.00i	0.247438i	−0.992317	+	0.123719i	$0.960518\pi$
$284$	0	0
$285$	120.000i	0.0249410i
$286$	0	0
$287$	−4692.00	−0.965017
$288$	0	0
$289$	−13.0000	−0.00264604
$290$	0	0
$291$	8268.00i	1.66556i
$292$	0	0
$293$	− 106.000i	− 0.0211351i	−0.999944	−	0.0105676i	$-0.996636\pi$
$293$	− 106.000i	− 0.0211351i	0.999944	−	0.0105676i	$-0.00336382\pi$
$294$	0	0
$295$	1340.00	0.264467
$296$	0	0
$297$	−1728.00	−0.337605
$298$	0	0
$299$	− 7772.00i	− 1.50323i
$300$	0	0
$301$	− 6052.00i	− 1.15891i
$302$	0	0
$303$	756.000	0.143337
$304$	0	0
$305$	−1250.00	−0.234671
$306$	0	0
$307$	− 8134.00i	− 1.51216i	−0.654482	−	0.756078i	$-0.727113\pi$
$307$	− 8134.00i	− 1.51216i	0.654482	−	0.756078i	$-0.272887\pi$
$308$	0	0
$309$	− 7572.00i	− 1.39403i
$310$	0	0
$311$	−4396.00	−0.801525	−0.400763	−	0.916182i	$-0.631255\pi$
$311$	−4396.00	−0.801525	−0.400763	+	0.916182i	$0.631255\pi$
$312$	0	0
$313$	−4826.00	−0.871507	−0.435753	−	0.900066i	$-0.643518\pi$
$313$	−4826.00	−0.871507	−0.435753	+	0.900066i	$0.643518\pi$
$314$	0	0
$315$	1530.00i	0.273669i
$316$	0	0
$317$	7026.00i	1.24486i	0.782677	+	0.622428i	$0.213854\pi$
$317$	7026.00i	1.24486i	−0.782677	+	0.622428i	$0.786146\pi$
$318$	0	0
$319$	3872.00	0.679594
$320$	0	0
$321$	−3060.00	−0.532064
$322$	0	0
$323$	280.000i	0.0482341i
$324$	0	0
$325$	− 1450.00i	− 0.247482i
$326$	0	0
$327$	−156.000	−0.0263817
$328$	0	0
$329$	748.000	0.125345
$330$	0	0
$331$	8808.00i	1.46263i	0.682038	+	0.731316i	$0.261094\pi$
$331$	8808.00i	1.46263i	−0.682038	+	0.731316i	$0.738906\pi$
$332$	0	0
$333$	− 3942.00i	− 0.648710i
$334$	0	0
$335$	2110.00	0.344124
$336$	0	0
$337$	5602.00	0.905520	0.452760	−	0.891632i	$-0.350439\pi$
$337$	5602.00	0.905520	0.452760	+	0.891632i	$0.350439\pi$
$338$	0	0
$339$	7452.00i	1.19391i
$340$	0	0
$341$	− 1600.00i	− 0.254090i
$342$	0	0
$343$	−15980.0	−2.51557
$344$	0	0
$345$	4020.00	0.627332
$346$	0	0
$347$	− 6634.00i	− 1.02632i	−0.858294	−	0.513158i	$-0.828475\pi$
$347$	− 6634.00i	− 1.02632i	0.858294	−	0.513158i	$-0.171525\pi$
$348$	0	0
$349$	3198.00i	0.490501i	0.969460	+	0.245251i	$0.0788703\pi$
$349$	3198.00i	0.490501i	−0.969460	+	0.245251i	$0.921130\pi$
$350$	0	0
$351$	−6264.00	−0.952557
$352$	0	0
$353$	−5230.00	−0.788569	−0.394284	−	0.918988i	$-0.629008\pi$
$353$	−5230.00	−0.788569	−0.394284	+	0.918988i	$0.629008\pi$
$354$	0	0
$355$	− 4260.00i	− 0.636894i
$356$	0	0
$357$	14280.0i	2.11702i
$358$	0	0
$359$	−312.000	−0.0458683	−0.0229342	−	0.999737i	$-0.507301\pi$
$359$	−312.000	−0.0458683	−0.0229342	+	0.999737i	$0.507301\pi$
$360$	0	0
$361$	6843.00	0.997667
$362$	0	0
$363$	6450.00i	0.932609i
$364$	0	0
$365$	− 1530.00i	− 0.219408i
$366$	0	0
$367$	−10790.0	−1.53470	−0.767348	−	0.641231i	$-0.778424\pi$
$367$	−10790.0	−1.53470	−0.767348	+	0.641231i	$0.778424\pi$
$368$	0	0
$369$	−1242.00	−0.175219
$370$	0	0
$371$	5508.00i	0.770785i
$372$	0	0
$373$	4190.00i	0.581635i	0.956778	+	0.290818i	$0.0939273\pi$
$373$	4190.00i	0.581635i	−0.956778	+	0.290818i	$0.906073\pi$
$374$	0	0
$375$	750.000	0.103280
$376$	0	0
$377$	14036.0	1.91748
$378$	0	0
$379$	− 6980.00i	− 0.946012i	−0.881059	−	0.473006i	$-0.843169\pi$
$379$	− 6980.00i	− 0.946012i	0.881059	−	0.473006i	$-0.156831\pi$
$380$	0	0
$381$	5868.00i	0.789047i
$382$	0	0
$383$	−13962.0	−1.86273	−0.931364	−	0.364089i	$-0.881380\pi$
$383$	−13962.0	−1.86273	−0.931364	+	0.364089i	$0.881380\pi$
$384$	0	0
$385$	2720.00	0.360062
$386$	0	0
$387$	− 1602.00i	− 0.210424i
$388$	0	0
$389$	− 3810.00i	− 0.496593i	−0.968684	−	0.248296i	$-0.920129\pi$
$389$	− 3810.00i	− 0.496593i	0.968684	−	0.248296i	$-0.0798707\pi$
$390$	0	0
$391$	9380.00	1.21321
$392$	0	0
$393$	−5472.00	−0.702356
$394$	0	0
$395$	2280.00i	0.290428i
$396$	0	0
$397$	− 9158.00i	− 1.15775i	−0.815416	−	0.578875i	$-0.803492\pi$
$397$	− 9158.00i	− 1.15775i	0.815416	−	0.578875i	$-0.196508\pi$
$398$	0	0
$399$	−816.000	−0.102384
$400$	0	0
$401$	4866.00	0.605976	0.302988	−	0.952994i	$-0.402016\pi$
$401$	4866.00	0.605976	0.302988	+	0.952994i	$0.402016\pi$
$402$	0	0
$403$	− 5800.00i	− 0.716920i
$404$	0	0
$405$	− 4455.00i	− 0.546594i
$406$	0	0
$407$	−7008.00	−0.853498
$408$	0	0
$409$	−13486.0	−1.63042	−0.815208	−	0.579169i	$-0.803377\pi$
$409$	−13486.0	−1.63042	−0.815208	+	0.579169i	$0.803377\pi$
$410$	0	0
$411$	5556.00i	0.666806i
$412$	0	0
$413$	9112.00i	1.08565i
$414$	0	0
$415$	2170.00	0.256677
$416$	0	0
$417$	−3096.00	−0.363577
$418$	0	0
$419$	− 5628.00i	− 0.656195i	−0.944644	−	0.328098i	$-0.893593\pi$
$419$	− 5628.00i	− 0.656195i	0.944644	−	0.328098i	$-0.106407\pi$
$420$	0	0
$421$	− 7938.00i	− 0.918942i	−0.888193	−	0.459471i	$-0.848039\pi$
$421$	− 7938.00i	− 0.918942i	0.888193	−	0.459471i	$-0.151961\pi$
$422$	0	0
$423$	198.000	0.0227591
$424$	0	0
$425$	1750.00	0.199735
$426$	0	0
$427$	− 8500.00i	− 0.963334i
$428$	0	0
$429$	− 5568.00i	− 0.626633i
$430$	0	0
$431$	−1916.00	−0.214131	−0.107066	−	0.994252i	$-0.534145\pi$
$431$	−1916.00	−0.214131	−0.107066	+	0.994252i	$0.534145\pi$
$432$	0	0
$433$	−16510.0	−1.83238	−0.916189	−	0.400746i	$-0.868751\pi$
$433$	−16510.0	−1.83238	−0.916189	+	0.400746i	$0.868751\pi$
$434$	0	0
$435$	7260.00i	0.800208i
$436$	0	0
$437$	536.000i	0.0586736i
$438$	0	0
$439$	−1256.00	−0.136550	−0.0682752	−	0.997667i	$-0.521750\pi$
$439$	−1256.00	−0.136550	−0.0682752	+	0.997667i	$0.521750\pi$
$440$	0	0
$441$	−7317.00	−0.790087
$442$	0	0
$443$	− 12222.0i	− 1.31080i	−0.755282	−	0.655400i	$-0.772500\pi$
$443$	− 12222.0i	− 1.31080i	0.755282	−	0.655400i	$-0.227500\pi$
$444$	0	0
$445$	3630.00i	0.386693i
$446$	0	0
$447$	−5748.00	−0.608213
$448$	0	0
$449$	−5946.00	−0.624965	−0.312482	−	0.949924i	$-0.601160\pi$
$449$	−5946.00	−0.624965	−0.312482	+	0.949924i	$0.601160\pi$
$450$	0	0
$451$	2208.00i	0.230534i
$452$	0	0
$453$	1992.00i	0.206606i
$454$	0	0
$455$	9860.00	1.01592
$456$	0	0
$457$	−1258.00	−0.128768	−0.0643838	−	0.997925i	$-0.520508\pi$
$457$	−1258.00	−0.128768	−0.0643838	+	0.997925i	$0.520508\pi$
$458$	0	0
$459$	− 7560.00i	− 0.768781i
$460$	0	0
$461$	16422.0i	1.65911i	0.558426	+	0.829554i	$0.311405\pi$
$461$	16422.0i	1.65911i	−0.558426	+	0.829554i	$0.688595\pi$
$462$	0	0
$463$	−2658.00	−0.266799	−0.133399	−	0.991062i	$-0.542589\pi$
$463$	−2658.00	−0.266799	−0.133399	+	0.991062i	$0.542589\pi$
$464$	0	0
$465$	3000.00	0.299186
$466$	0	0
$467$	− 3686.00i	− 0.365241i	−0.983183	−	0.182621i	$-0.941542\pi$
$467$	− 3686.00i	− 0.365241i	0.983183	−	0.182621i	$-0.0584580\pi$
$468$	0	0
$469$	14348.0i	1.41264i
$470$	0	0
$471$	6132.00	0.599889
$472$	0	0
$473$	−2848.00	−0.276852
$474$	0	0
$475$	100.000i	0.00965961i
$476$	0	0
$477$	1458.00i	0.139952i
$478$	0	0
$479$	−88.0000	−0.00839420	−0.00419710	−	0.999991i	$-0.501336\pi$
$479$	−88.0000	−0.00839420	−0.00419710	+	0.999991i	$0.501336\pi$
$480$	0	0
$481$	−25404.0	−2.40816
$482$	0	0
$483$	27336.0i	2.57522i
$484$	0	0
$485$	6890.00i	0.645070i
$486$	0	0
$487$	−14714.0	−1.36911	−0.684553	−	0.728963i	$-0.740003\pi$
$487$	−14714.0	−1.36911	−0.684553	+	0.728963i	$0.740003\pi$
$488$	0	0
$489$	−5556.00	−0.513806
$490$	0	0
$491$	− 7344.00i	− 0.675010i	−0.941324	−	0.337505i	$-0.890417\pi$
$491$	− 7344.00i	− 0.675010i	0.941324	−	0.337505i	$-0.109583\pi$
$492$	0	0
$493$	16940.0i	1.54754i
$494$	0	0
$495$	720.000	0.0653770
$496$	0	0
$497$	28968.0	2.61447
$498$	0	0
$499$	− 1604.00i	− 0.143898i	−0.997408	−	0.0719488i	$-0.977078\pi$
$499$	− 1604.00i	− 0.143898i	0.997408	−	0.0719488i	$-0.0229218\pi$
$500$	0	0
$501$	3924.00i	0.349923i
$502$	0	0
$503$	14802.0	1.31210	0.656052	−	0.754715i	$-0.272225\pi$
$503$	14802.0	1.31210	0.656052	+	0.754715i	$0.272225\pi$
$504$	0	0
$505$	630.000	0.0555141
$506$	0	0
$507$	− 7002.00i	− 0.613353i
$508$	0	0
$509$	− 22514.0i	− 1.96054i	−0.197660	−	0.980271i	$-0.563334\pi$
$509$	− 22514.0i	− 1.96054i	0.197660	−	0.980271i	$-0.436666\pi$
$510$	0	0
$511$	10404.0	0.900677
$512$	0	0
$513$	432.000	0.0371799
$514$	0	0
$515$	− 6310.00i	− 0.539906i
$516$	0	0
$517$	− 352.000i	− 0.0299438i
$518$	0	0
$519$	7764.00	0.656651
$520$	0	0
$521$	6710.00	0.564243	0.282121	−	0.959379i	$-0.408962\pi$
$521$	6710.00	0.564243	0.282121	+	0.959379i	$0.408962\pi$
$522$	0	0
$523$	7930.00i	0.663011i	0.943453	+	0.331505i	$0.107557\pi$
$523$	7930.00i	0.663011i	−0.943453	+	0.331505i	$0.892443\pi$
$524$	0	0
$525$	5100.00i	0.423966i
$526$	0	0
$527$	7000.00	0.578605
$528$	0	0
$529$	5789.00	0.475795
$530$	0	0
$531$	2412.00i	0.197122i
$532$	0	0
$533$	8004.00i	0.650454i
$534$	0	0
$535$	−2550.00	−0.206068
$536$	0	0
$537$	−17016.0	−1.36740
$538$	0	0
$539$	13008.0i	1.03951i
$540$	0	0
$541$	4918.00i	0.390834i	0.980720	+	0.195417i	$0.0626061\pi$
$541$	4918.00i	0.390834i	−0.980720	+	0.195417i	$0.937394\pi$
$542$	0	0
$543$	10452.0	0.826037
$544$	0	0
$545$	−130.000	−0.0102176
$546$	0	0
$547$	3922.00i	0.306568i	0.988182	+	0.153284i	$0.0489849\pi$
$547$	3922.00i	0.306568i	−0.988182	+	0.153284i	$0.951015\pi$
$548$	0	0
$549$	− 2250.00i	− 0.174914i
$550$	0	0
$551$	−968.000	−0.0748424
$552$	0	0
$553$	−15504.0	−1.19222
$554$	0	0
$555$	− 13140.0i	− 1.00498i
$556$	0	0
$557$	17786.0i	1.35299i	0.736446	+	0.676496i	$0.236503\pi$
$557$	17786.0i	1.35299i	−0.736446	+	0.676496i	$0.763497\pi$
$558$	0	0
$559$	−10324.0	−0.781143
$560$	0	0
$561$	6720.00	0.505737
$562$	0	0
$563$	− 20266.0i	− 1.51707i	−0.651633	−	0.758535i	$-0.725916\pi$
$563$	− 20266.0i	− 1.51707i	0.651633	−	0.758535i	$-0.274084\pi$
$564$	0	0
$565$	6210.00i	0.462401i
$566$	0	0
$567$	30294.0	2.24379
$568$	0	0
$569$	−13358.0	−0.984177	−0.492088	−	0.870545i	$-0.663766\pi$
$569$	−13358.0	−0.984177	−0.492088	+	0.870545i	$0.663766\pi$
$570$	0	0
$571$	16360.0i	1.19903i	0.800364	+	0.599514i	$0.204639\pi$
$571$	16360.0i	1.19903i	−0.800364	+	0.599514i	$0.795361\pi$
$572$	0	0
$573$	− 26760.0i	− 1.95099i
$574$	0	0
$575$	3350.00	0.242965
$576$	0	0
$577$	−15574.0	−1.12366	−0.561832	−	0.827251i	$-0.689903\pi$
$577$	−15574.0	−1.12366	−0.561832	+	0.827251i	$0.689903\pi$
$578$	0	0
$579$	− 22692.0i	− 1.62875i
$580$	0	0
$581$	14756.0i	1.05367i
$582$	0	0
$583$	2592.00	0.184133
$584$	0	0
$585$	2610.00	0.184462
$586$	0	0
$587$	6654.00i	0.467870i	0.972252	+	0.233935i	$0.0751604\pi$
$587$	6654.00i	0.467870i	−0.972252	+	0.233935i	$0.924840\pi$
$588$	0	0
$589$	400.000i	0.0279825i
$590$	0	0
$591$	26844.0	1.86838
$592$	0	0
$593$	−17742.0	−1.22863	−0.614314	−	0.789062i	$-0.710567\pi$
$593$	−17742.0	−1.22863	−0.614314	+	0.789062i	$0.710567\pi$
$594$	0	0
$595$	11900.0i	0.819920i
$596$	0	0
$597$	21648.0i	1.48408i
$598$	0	0
$599$	15840.0	1.08048	0.540238	−	0.841512i	$-0.318334\pi$
$599$	15840.0	1.08048	0.540238	+	0.841512i	$0.318334\pi$
$600$	0	0
$601$	3002.00	0.203751	0.101875	−	0.994797i	$-0.467516\pi$
$601$	3002.00	0.203751	0.101875	+	0.994797i	$0.467516\pi$
$602$	0	0
$603$	3798.00i	0.256495i
$604$	0	0
$605$	5375.00i	0.361198i
$606$	0	0
$607$	23610.0	1.57875	0.789374	−	0.613912i	$-0.210405\pi$
$607$	23610.0	1.57875	0.789374	+	0.613912i	$0.210405\pi$
$608$	0	0
$609$	−49368.0	−3.28488
$610$	0	0
$611$	− 1276.00i	− 0.0844868i
$612$	0	0
$613$	− 23850.0i	− 1.57144i	−0.618583	−	0.785720i	$-0.712293\pi$
$613$	− 23850.0i	− 1.57144i	0.618583	−	0.785720i	$-0.287707\pi$
$614$	0	0
$615$	−4140.00	−0.271449
$616$	0	0
$617$	5334.00	0.348037	0.174018	−	0.984742i	$-0.444325\pi$
$617$	5334.00	0.348037	0.174018	+	0.984742i	$0.444325\pi$
$618$	0	0
$619$	− 2164.00i	− 0.140515i	−0.997529	−	0.0702573i	$-0.977618\pi$
$619$	− 2164.00i	− 0.140515i	0.997529	−	0.0702573i	$-0.0223820\pi$
$620$	0	0
$621$	− 14472.0i	− 0.935171i
$622$	0	0
$623$	−24684.0	−1.58739
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	384.000i	0.0244585i
$628$	0	0
$629$	− 30660.0i	− 1.94355i
$630$	0	0
$631$	−25220.0	−1.59111	−0.795557	−	0.605879i	$-0.792821\pi$
$631$	−25220.0	−1.59111	−0.795557	+	0.605879i	$0.792821\pi$
$632$	0	0
$633$	−1536.00	−0.0964463
$634$	0	0
$635$	4890.00i	0.305596i
$636$	0	0
$637$	47154.0i	2.93298i
$638$	0	0
$639$	7668.00	0.474713
$640$	0	0
$641$	−12306.0	−0.758280	−0.379140	−	0.925339i	$-0.623780\pi$
$641$	−12306.0	−0.758280	−0.379140	+	0.925339i	$0.623780\pi$
$642$	0	0
$643$	27414.0i	1.68134i	0.541547	+	0.840671i	$0.317839\pi$
$643$	27414.0i	1.68134i	−0.541547	+	0.840671i	$0.682161\pi$
$644$	0	0
$645$	− 5340.00i	− 0.325988i
$646$	0	0
$647$	−21834.0	−1.32671	−0.663356	−	0.748304i	$-0.730869\pi$
$647$	−21834.0	−1.32671	−0.663356	+	0.748304i	$0.730869\pi$
$648$	0	0
$649$	4288.00	0.259351
$650$	0	0
$651$	20400.0i	1.22817i
$652$	0	0
$653$	− 23998.0i	− 1.43815i	−0.694931	−	0.719077i	$-0.744565\pi$
$653$	− 23998.0i	− 1.43815i	0.694931	−	0.719077i	$-0.255435\pi$
$654$	0	0
$655$	−4560.00	−0.272021
$656$	0	0
$657$	2754.00	0.163537
$658$	0	0
$659$	32004.0i	1.89180i	0.324452	+	0.945902i	$0.394820\pi$
$659$	32004.0i	1.89180i	−0.324452	+	0.945902i	$0.605180\pi$
$660$	0	0
$661$	8526.00i	0.501699i	0.968026	+	0.250849i	$0.0807099\pi$
$661$	8526.00i	0.501699i	−0.968026	+	0.250849i	$0.919290\pi$
$662$	0	0
$663$	24360.0	1.42694
$664$	0	0
$665$	−680.000	−0.0396530
$666$	0	0
$667$	32428.0i	1.88248i
$668$	0	0
$669$	30948.0i	1.78852i
$670$	0	0
$671$	−4000.00	−0.230132
$672$	0	0
$673$	8178.00	0.468408	0.234204	−	0.972187i	$-0.424752\pi$
$673$	8178.00	0.468408	0.234204	+	0.972187i	$0.424752\pi$
$674$	0	0
$675$	− 2700.00i	− 0.153960i
$676$	0	0
$677$	16646.0i	0.944989i	0.881334	+	0.472495i	$0.156646\pi$
$677$	16646.0i	0.944989i	−0.881334	+	0.472495i	$0.843354\pi$
$678$	0	0
$679$	−46852.0	−2.64803
$680$	0	0
$681$	−13356.0	−0.751546
$682$	0	0
$683$	− 22446.0i	− 1.25750i	−0.777608	−	0.628750i	$-0.783567\pi$
$683$	− 22446.0i	− 1.25750i	0.777608	−	0.628750i	$-0.216433\pi$
$684$	0	0
$685$	4630.00i	0.258253i
$686$	0	0
$687$	12516.0	0.695073
$688$	0	0
$689$	9396.00	0.519534
$690$	0	0
$691$	− 35336.0i	− 1.94536i	−0.232147	−	0.972681i	$-0.574575\pi$
$691$	− 35336.0i	− 1.94536i	0.232147	−	0.972681i	$-0.425425\pi$
$692$	0	0
$693$	4896.00i	0.268375i
$694$	0	0
$695$	−2580.00	−0.140813
$696$	0	0
$697$	−9660.00	−0.524962
$698$	0	0
$699$	34308.0i	1.85643i
$700$	0	0
$701$	3482.00i	0.187608i	0.995591	+	0.0938041i	$0.0299027\pi$
$701$	3482.00i	0.187608i	−0.995591	+	0.0938041i	$0.970097\pi$
$702$	0	0
$703$	1752.00	0.0939942
$704$	0	0
$705$	660.000	0.0352582
$706$	0	0
$707$	4284.00i	0.227887i
$708$	0	0
$709$	19402.0i	1.02773i	0.857872	+	0.513863i	$0.171786\pi$
$709$	19402.0i	1.02773i	−0.857872	+	0.513863i	$0.828214\pi$
$710$	0	0
$711$	−4104.00	−0.216473
$712$	0	0
$713$	13400.0	0.703834
$714$	0	0
$715$	− 4640.00i	− 0.242694i
$716$	0	0
$717$	21744.0i	1.13256i
$718$	0	0
$719$	9896.00	0.513294	0.256647	−	0.966505i	$-0.417382\pi$
$719$	9896.00	0.513294	0.256647	+	0.966505i	$0.417382\pi$
$720$	0	0
$721$	42908.0	2.21633
$722$	0	0
$723$	− 492.000i	− 0.0253080i
$724$	0	0
$725$	6050.00i	0.309919i
$726$	0	0
$727$	494.000	0.0252014	0.0126007	−	0.999921i	$-0.495989\pi$
$727$	494.000	0.0252014	0.0126007	+	0.999921i	$0.495989\pi$
$728$	0	0
$729$	−9477.00	−0.481481
$730$	0	0
$731$	− 12460.0i	− 0.630437i
$732$	0	0
$733$	9282.00i	0.467720i	0.972270	+	0.233860i	$0.0751357\pi$
$733$	9282.00i	0.467720i	−0.972270	+	0.233860i	$0.924864\pi$
$734$	0	0
$735$	−24390.0	−1.22400
$736$	0	0
$737$	6752.00	0.337467
$738$	0	0
$739$	3252.00i	0.161877i	0.996719	+	0.0809383i	$0.0257917\pi$
$739$	3252.00i	0.161877i	−0.996719	+	0.0809383i	$0.974208\pi$
$740$	0	0
$741$	1392.00i	0.0690100i
$742$	0	0
$743$	−4710.00	−0.232561	−0.116281	−	0.993216i	$-0.537097\pi$
$743$	−4710.00	−0.232561	−0.116281	+	0.993216i	$0.537097\pi$
$744$	0	0
$745$	−4790.00	−0.235560
$746$	0	0
$747$	3906.00i	0.191316i
$748$	0	0
$749$	− 17340.0i	− 0.845914i
$750$	0	0
$751$	−25764.0	−1.25185	−0.625927	−	0.779882i	$-0.715279\pi$
$751$	−25764.0	−1.25185	−0.625927	+	0.779882i	$0.715279\pi$
$752$	0	0
$753$	30240.0	1.46349
$754$	0	0
$755$	1660.00i	0.0800180i
$756$	0	0
$757$	30094.0i	1.44489i	0.691426	+	0.722447i	$0.256983\pi$
$757$	30094.0i	1.44489i	−0.691426	+	0.722447i	$0.743017\pi$
$758$	0	0
$759$	12864.0	0.615196
$760$	0	0
$761$	−22362.0	−1.06521	−0.532603	−	0.846365i	$-0.678786\pi$
$761$	−22362.0	−1.06521	−0.532603	+	0.846365i	$0.678786\pi$
$762$	0	0
$763$	− 884.000i	− 0.0419436i
$764$	0	0
$765$	3150.00i	0.148874i
$766$	0	0
$767$	15544.0	0.731762
$768$	0	0
$769$	−30398.0	−1.42546	−0.712731	−	0.701438i	$-0.752542\pi$
$769$	−30398.0	−1.42546	−0.712731	+	0.701438i	$0.752542\pi$
$770$	0	0
$771$	− 13860.0i	− 0.647413i
$772$	0	0
$773$	− 1290.00i	− 0.0600234i	−0.999550	−	0.0300117i	$-0.990446\pi$
$773$	− 1290.00i	− 0.0600234i	0.999550	−	0.0300117i	$-0.00955445\pi$
$774$	0	0
$775$	2500.00	0.115874
$776$	0	0
$777$	89352.0	4.12546
$778$	0	0
$779$	− 552.000i	− 0.0253883i
$780$	0	0
$781$	− 13632.0i	− 0.624573i
$782$	0	0
$783$	26136.0	1.19288
$784$	0	0
$785$	5110.00	0.232336
$786$	0	0
$787$	− 14.0000i	0 0.000634112i	−1.00000		0.000317056i	$-0.999899\pi$
$787$	− 14.0000i	0 0.000634112i	1.00000		0.000317056i	$-0.000100922\pi$
$788$	0	0
$789$	− 24660.0i	− 1.11270i
$790$	0	0
$791$	−42228.0	−1.89817
$792$	0	0
$793$	−14500.0	−0.649319
$794$	0	0
$795$	4860.00i	0.216813i
$796$	0	0
$797$	− 38814.0i	− 1.72505i	−0.506017	−	0.862523i	$-0.668883\pi$
$797$	− 38814.0i	− 1.72505i	0.506017	−	0.862523i	$-0.331117\pi$
$798$	0	0
$799$	1540.00	0.0681868
$800$	0	0
$801$	−6534.00	−0.288224
$802$	0	0
$803$	− 4896.00i	− 0.215163i
$804$	0	0
$805$	22780.0i	0.997378i
$806$	0	0
$807$	−4476.00	−0.195245
$808$	0	0
$809$	−27402.0	−1.19086	−0.595428	−	0.803408i	$-0.703018\pi$
$809$	−27402.0	−1.19086	−0.595428	+	0.803408i	$0.703018\pi$
$810$	0	0
$811$	− 28576.0i	− 1.23729i	−0.785672	−	0.618643i	$-0.787683\pi$
$811$	− 28576.0i	− 1.23729i	0.785672	−	0.618643i	$-0.212317\pi$
$812$	0	0
$813$	27576.0i	1.18958i
$814$	0	0
$815$	−4630.00	−0.198996
$816$	0	0
$817$	712.000	0.0304893
$818$	0	0
$819$	17748.0i	0.757223i
$820$	0	0
$821$	− 31762.0i	− 1.35018i	−0.737733	−	0.675092i	$-0.764104\pi$
$821$	− 31762.0i	− 1.35018i	0.737733	−	0.675092i	$-0.235896\pi$
$822$	0	0
$823$	20506.0	0.868523	0.434261	−	0.900787i	$-0.357009\pi$
$823$	20506.0	0.868523	0.434261	+	0.900787i	$0.357009\pi$
$824$	0	0
$825$	2400.00	0.101282
$826$	0	0
$827$	13014.0i	0.547208i	0.961842	+	0.273604i	$0.0882158\pi$
$827$	13014.0i	0.547208i	−0.961842	+	0.273604i	$0.911784\pi$
$828$	0	0
$829$	− 22790.0i	− 0.954800i	−0.878686	−	0.477400i	$-0.841579\pi$
$829$	− 22790.0i	− 0.954800i	0.878686	−	0.477400i	$-0.158421\pi$
$830$	0	0
$831$	−13236.0	−0.552529
$832$	0	0
$833$	−56910.0	−2.36712
$834$	0	0
$835$	3270.00i	0.135525i
$836$	0	0
$837$	− 10800.0i	− 0.446001i
$838$	0	0
$839$	23696.0	0.975062	0.487531	−	0.873106i	$-0.337898\pi$
$839$	23696.0	0.975062	0.487531	+	0.873106i	$0.337898\pi$
$840$	0	0
$841$	−34175.0	−1.40125
$842$	0	0
$843$	− 49668.0i	− 2.02925i
$844$	0	0
$845$	− 5835.00i	− 0.237550i
$846$	0	0
$847$	−36550.0	−1.48273
$848$	0	0
$849$	−7068.00	−0.285716
$850$	0	0
$851$	− 58692.0i	− 2.36420i
$852$	0	0
$853$	− 5306.00i	− 0.212982i	−0.994314	−	0.106491i	$-0.966038\pi$
$853$	− 5306.00i	− 0.212982i	0.994314	−	0.106491i	$-0.0339616\pi$
$854$	0	0
$855$	−180.000	−0.00719985
$856$	0	0
$857$	21054.0	0.839196	0.419598	−	0.907710i	$-0.362171\pi$
$857$	21054.0	0.839196	0.419598	+	0.907710i	$0.362171\pi$
$858$	0	0
$859$	7364.00i	0.292499i	0.989248	+	0.146249i	$0.0467202\pi$
$859$	7364.00i	0.292499i	−0.989248	+	0.146249i	$0.953280\pi$
$860$	0	0
$861$	− 28152.0i	− 1.11431i
$862$	0	0
$863$	−17226.0	−0.679467	−0.339733	−	0.940522i	$-0.610337\pi$
$863$	−17226.0	−0.679467	−0.339733	+	0.940522i	$0.610337\pi$
$864$	0	0
$865$	6470.00	0.254320
$866$	0	0
$867$	− 78.0000i	− 0.00305539i
$868$	0	0
$869$	7296.00i	0.284810i
$870$	0	0
$871$	24476.0	0.952167
$872$	0	0
$873$	−12402.0	−0.480807
$874$	0	0
$875$	4250.00i	0.164201i
$876$	0	0
$877$	21202.0i	0.816352i	0.912903	+	0.408176i	$0.133835\pi$
$877$	21202.0i	0.816352i	−0.912903	+	0.408176i	$0.866165\pi$
$878$	0	0
$879$	636.000	0.0244047
$880$	0	0
$881$	−29490.0	−1.12774	−0.563872	−	0.825862i	$-0.690689\pi$
$881$	−29490.0	−1.12774	−0.563872	+	0.825862i	$0.690689\pi$
$882$	0	0
$883$	− 2570.00i	− 0.0979472i	−0.998800	−	0.0489736i	$-0.984405\pi$
$883$	− 2570.00i	− 0.0979472i	0.998800	−	0.0489736i	$-0.0155950\pi$
$884$	0	0
$885$	8040.00i	0.305380i
$886$	0	0
$887$	36334.0	1.37540	0.687698	−	0.725997i	$-0.258621\pi$
$887$	36334.0	1.37540	0.687698	+	0.725997i	$0.258621\pi$
$888$	0	0
$889$	−33252.0	−1.25448
$890$	0	0
$891$	− 14256.0i	− 0.536020i
$892$	0	0
$893$	88.0000i	0.00329766i
$894$	0	0
$895$	−14180.0	−0.529592
$896$	0	0
$897$	46632.0	1.73578
$898$	0	0
$899$	24200.0i	0.897792i
$900$	0	0
$901$	11340.0i	0.419301i
$902$	0	0
$903$	36312.0	1.33819
$904$	0	0
$905$	8710.00	0.319923
$906$	0	0
$907$	− 12474.0i	− 0.456662i	−0.973584	−	0.228331i	$-0.926673\pi$
$907$	− 12474.0i	− 0.456662i	0.973584	−	0.228331i	$-0.0733268\pi$
$908$	0	0
$909$	1134.00i	0.0413778i
$910$	0	0
$911$	41132.0	1.49590	0.747949	−	0.663756i	$-0.231039\pi$
$911$	41132.0	1.49590	0.747949	+	0.663756i	$0.231039\pi$
$912$	0	0
$913$	6944.00	0.251712
$914$	0	0
$915$	− 7500.00i	− 0.270975i
$916$	0	0
$917$	− 31008.0i	− 1.11666i
$918$	0	0
$919$	−38416.0	−1.37892	−0.689460	−	0.724324i	$-0.742152\pi$
$919$	−38416.0	−1.37892	−0.689460	+	0.724324i	$0.742152\pi$
$920$	0	0
$921$	48804.0	1.74609
$922$	0	0
$923$	− 49416.0i	− 1.76224i
$924$	0	0
$925$	− 10950.0i	− 0.389226i
$926$	0	0
$927$	11358.0	0.402423
$928$	0	0
$929$	41302.0	1.45864	0.729319	−	0.684174i	$-0.239837\pi$
$929$	41302.0	1.45864	0.729319	+	0.684174i	$0.239837\pi$
$930$	0	0
$931$	− 3252.00i	− 0.114479i
$932$	0	0
$933$	− 26376.0i	− 0.925521i
$934$	0	0
$935$	5600.00	0.195871
$936$	0	0
$937$	26150.0	0.911722	0.455861	−	0.890051i	$-0.349331\pi$
$937$	26150.0	0.911722	0.455861	+	0.890051i	$0.349331\pi$
$938$	0	0
$939$	− 28956.0i	− 1.00633i
$940$	0	0
$941$	35254.0i	1.22130i	0.791899	+	0.610652i	$0.209093\pi$
$941$	35254.0i	1.22130i	−0.791899	+	0.610652i	$0.790907\pi$
$942$	0	0
$943$	−18492.0	−0.638582
$944$	0	0
$945$	18360.0	0.632011
$946$	0	0
$947$	− 18550.0i	− 0.636530i	−0.948002	−	0.318265i	$-0.896900\pi$
$947$	− 18550.0i	− 0.636530i	0.948002	−	0.318265i	$-0.103100\pi$
$948$	0	0
$949$	− 17748.0i	− 0.607086i
$950$	0	0
$951$	−42156.0	−1.43744
$952$	0	0
$953$	−17322.0	−0.588788	−0.294394	−	0.955684i	$-0.595118\pi$
$953$	−17322.0	−0.588788	−0.294394	+	0.955684i	$0.595118\pi$
$954$	0	0
$955$	− 22300.0i	− 0.755614i
$956$	0	0
$957$	23232.0i	0.784727i
$958$	0	0
$959$	−31484.0	−1.06014
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	− 4590.00i	− 0.153594i
$964$	0	0
$965$	− 18910.0i	− 0.630813i
$966$	0	0
$967$	35190.0	1.17025	0.585126	−	0.810942i	$-0.301045\pi$
$967$	35190.0	1.17025	0.585126	+	0.810942i	$0.301045\pi$
$968$	0	0
$969$	−1680.00	−0.0556960
$970$	0	0
$971$	− 40696.0i	− 1.34500i	−0.740096	−	0.672501i	$-0.765220\pi$
$971$	− 40696.0i	− 1.34500i	0.740096	−	0.672501i	$-0.234780\pi$
$972$	0	0
$973$	− 17544.0i	− 0.578042i
$974$	0	0
$975$	8700.00	0.285767
$976$	0	0
$977$	44306.0	1.45084	0.725422	−	0.688304i	$-0.241645\pi$
$977$	44306.0	1.45084	0.725422	+	0.688304i	$0.241645\pi$
$978$	0	0
$979$	11616.0i	0.379212i
$980$	0	0
$981$	− 234.000i	− 0.00761574i
$982$	0	0
$983$	−18798.0	−0.609932	−0.304966	−	0.952363i	$-0.598645\pi$
$983$	−18798.0	−0.609932	−0.304966	+	0.952363i	$0.598645\pi$
$984$	0	0
$985$	22370.0	0.723622
$986$	0	0
$987$	4488.00i	0.144736i
$988$	0	0
$989$	− 23852.0i	− 0.766885i
$990$	0	0
$991$	−2468.00	−0.0791106	−0.0395553	−	0.999217i	$-0.512594\pi$
$991$	−2468.00	−0.0791106	−0.0395553	+	0.999217i	$0.512594\pi$
$992$	0	0
$993$	−52848.0	−1.68890
$994$	0	0
$995$	18040.0i	0.574780i
$996$	0	0
$997$	61086.0i	1.94043i	0.242237	+	0.970217i	$0.422119\pi$
$997$	61086.0i	1.94043i	−0.242237	+	0.970217i	$0.577881\pi$
$998$	0	0
$999$	−47304.0	−1.49813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.a.641.2		2
4.3	odd	2		1280.4.d.p.641.1		2
8.3	odd	2		1280.4.d.p.641.2		2
8.5	even	2	inner	1280.4.d.a.641.1		2
16.3	odd	4		320.4.a.l.1.1		1
16.5	even	4		80.4.a.e.1.1		1
16.11	odd	4		40.4.a.a.1.1	✓	1
16.13	even	4		320.4.a.c.1.1		1
48.5	odd	4		720.4.a.bd.1.1		1
48.11	even	4		360.4.a.h.1.1		1
80.19	odd	4		1600.4.a.j.1.1		1
80.27	even	4		200.4.c.c.49.2		2
80.29	even	4		1600.4.a.br.1.1		1
80.37	odd	4		400.4.c.f.49.1		2
80.43	even	4		200.4.c.c.49.1		2
80.53	odd	4		400.4.c.f.49.2		2
80.59	odd	4		200.4.a.i.1.1		1
80.69	even	4		400.4.a.e.1.1		1
112.27	even	4		1960.4.a.h.1.1		1
240.59	even	4		1800.4.a.bi.1.1		1
240.107	odd	4		1800.4.f.j.649.1		2
240.203	odd	4		1800.4.f.j.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.a.1.1	✓	1	16.11	odd	4
80.4.a.e.1.1		1	16.5	even	4
200.4.a.i.1.1		1	80.59	odd	4
200.4.c.c.49.1		2	80.43	even	4
200.4.c.c.49.2		2	80.27	even	4
320.4.a.c.1.1		1	16.13	even	4
320.4.a.l.1.1		1	16.3	odd	4
360.4.a.h.1.1		1	48.11	even	4
400.4.a.e.1.1		1	80.69	even	4
400.4.c.f.49.1		2	80.37	odd	4
400.4.c.f.49.2		2	80.53	odd	4
720.4.a.bd.1.1		1	48.5	odd	4
1280.4.d.a.641.1		2	8.5	even	2	inner
1280.4.d.a.641.2		2	1.1	even	1	trivial
1280.4.d.p.641.1		2	4.3	odd	2
1280.4.d.p.641.2		2	8.3	odd	2
1600.4.a.j.1.1		1	80.19	odd	4
1600.4.a.br.1.1		1	80.29	even	4
1800.4.a.bi.1.1		1	240.59	even	4
1800.4.f.j.649.1		2	240.107	odd	4
1800.4.f.j.649.2		2	240.203	odd	4
1960.4.a.h.1.1		1	112.27	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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{3} +5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+6.00000i q^{3} +5.00000i q^{5} -34.0000 q^{7} -9.00000 q^{9} +16.0000i q^{11} +58.0000i q^{13} -30.0000 q^{15} -70.0000 q^{17} -4.00000i q^{19} -204.000i q^{21} -134.000 q^{23} -25.0000 q^{25} +108.000i q^{27} -242.000i q^{29} -100.000 q^{31} -96.0000 q^{33} -170.000i q^{35} +438.000i q^{37} -348.000 q^{39} +138.000 q^{41} +178.000i q^{43} -45.0000i q^{45} -22.0000 q^{47} +813.000 q^{49} -420.000i q^{51} -162.000i q^{53} -80.0000 q^{55} +24.0000 q^{57} -268.000i q^{59} +250.000i q^{61} +306.000 q^{63} -290.000 q^{65} -422.000i q^{67} -804.000i q^{69} -852.000 q^{71} -306.000 q^{73} -150.000i q^{75} -544.000i q^{77} +456.000 q^{79} -891.000 q^{81} -434.000i q^{83} -350.000i q^{85} +1452.00 q^{87} +726.000 q^{89} -1972.00i q^{91} -600.000i q^{93} +20.0000 q^{95} +1378.00 q^{97} -144.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 68 q^{7} - 18 q^{9}+O(q^{10}) \) 2 * q - 68 * q^7 - 18 * q^9	\( 2 q - 68 q^{7} - 18 q^{9} - 60 q^{15} - 140 q^{17} - 268 q^{23} - 50 q^{25} - 200 q^{31} - 192 q^{33} - 696 q^{39} + 276 q^{41} - 44 q^{47} + 1626 q^{49} - 160 q^{55} + 48 q^{57} + 612 q^{63} - 580 q^{65} - 1704 q^{71} - 612 q^{73} + 912 q^{79} - 1782 q^{81} + 2904 q^{87} + 1452 q^{89} + 40 q^{95} + 2756 q^{97}+O(q^{100}) \) 2 * q - 68 * q^7 - 18 * q^9 - 60 * q^15 - 140 * q^17 - 268 * q^23 - 50 * q^25 - 200 * q^31 - 192 * q^33 - 696 * q^39 + 276 * q^41 - 44 * q^47 + 1626 * q^49 - 160 * q^55 + 48 * q^57 + 612 * q^63 - 580 * q^65 - 1704 * q^71 - 612 * q^73 + 912 * q^79 - 1782 * q^81 + 2904 * q^87 + 1452 * q^89 + 40 * q^95 + 2756 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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