LMFDB - Embedded newform 1280.3.e.l.639.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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([1, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			639.9
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.l.639.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.20423i q^{3} +(1.52118 - 4.76298i) q^{5} +3.29905 q^{7} -8.67551 q^{9} +O(q^{10})$	$q-4.20423i q^{3} +(1.52118 - 4.76298i) q^{5} +3.29905 q^{7} -8.67551 q^{9} +15.0633 q^{11} +7.02352 q^{13} +(-20.0247 - 6.39540i) q^{15} -15.9181i q^{17} -12.8928 q^{19} -13.8699i q^{21} +3.49441 q^{23} +(-20.3720 - 14.4907i) q^{25} -1.36422i q^{27} +13.1653i q^{29} -38.2295i q^{31} -63.3296i q^{33} +(5.01845 - 15.7133i) q^{35} +68.5124 q^{37} -29.5285i q^{39} +57.3992 q^{41} +51.9027i q^{43} +(-13.1970 + 41.3213i) q^{45} -34.0218 q^{47} -38.1163 q^{49} -66.9233 q^{51} +79.9172 q^{53} +(22.9141 - 71.7464i) q^{55} +54.2042i q^{57} -114.933 q^{59} -115.485i q^{61} -28.6209 q^{63} +(10.6841 - 33.4529i) q^{65} -63.3874i q^{67} -14.6913i q^{69} -40.1984i q^{71} -2.16321i q^{73} +(-60.9223 + 85.6485i) q^{75} +49.6946 q^{77} +78.1874i q^{79} -83.8151 q^{81} +61.5753i q^{83} +(-75.8177 - 24.2144i) q^{85} +55.3501 q^{87} +6.04779 q^{89} +23.1709 q^{91} -160.725 q^{93} +(-19.6123 + 61.4082i) q^{95} +135.240i q^{97} -130.682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{9}+O(q^{10}) $ 24 * q - 72 * q^9	$ 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) $ 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

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
$3$		−	4.20423i		−	1.40141i	−0.713452	−	0.700704i	$-0.752869\pi$
$3$		−	4.20423i		−	1.40141i	0.713452	−	0.700704i	$-0.247131\pi$
$4$		0			0
$5$	1.52118	−	4.76298i	0.304237	−	0.952597i
$5$	1.52118	−	4.76298i	0.304237	−	0.952597i
$6$		0			0
$7$	3.29905			0.471292			0.235646	−	0.971839i	$-0.424279\pi$
$7$	3.29905			0.471292			0.235646	+	0.971839i	$0.424279\pi$
$8$		0			0
$9$	−8.67551			−0.963946
$10$		0			0
$11$	15.0633			1.36939			0.684697	−	0.728828i	$-0.259935\pi$
$11$	15.0633			1.36939			0.684697	+	0.728828i	$0.259935\pi$
$12$		0			0
$13$	7.02352			0.540271			0.270136	−	0.962822i	$-0.412931\pi$
$13$	7.02352			0.540271			0.270136	+	0.962822i	$0.412931\pi$
$14$		0			0
$15$	−20.0247	−	6.39540i	−1.33498	−	0.426360i
$16$		0			0
$17$		−	15.9181i		−	0.936359i	−0.883633	−	0.468180i	$-0.844910\pi$
$17$		−	15.9181i		−	0.936359i	0.883633	−	0.468180i	$-0.155090\pi$
$18$		0			0
$19$	−12.8928			−0.678569			−0.339284	−	0.940684i	$-0.610185\pi$
$19$	−12.8928			−0.678569			−0.339284	+	0.940684i	$0.610185\pi$
$20$		0			0
$21$		−	13.8699i		−	0.660473i
$22$		0			0
$23$	3.49441			0.151931			0.0759655	−	0.997110i	$-0.475796\pi$
$23$	3.49441			0.151931			0.0759655	+	0.997110i	$0.475796\pi$
$24$		0			0
$25$	−20.3720	−	14.4907i	−0.814880	−	0.579629i
$26$		0			0
$27$		−	1.36422i		−	0.0505267i
$28$		0			0
$29$			13.1653i			0.453977i	0.973897	+	0.226989i	$0.0728880\pi$
$29$			13.1653i			0.453977i	−0.973897	+	0.226989i	$0.927112\pi$
$30$		0			0
$31$		−	38.2295i		−	1.23321i	−0.787273	−	0.616605i	$-0.788508\pi$
$31$		−	38.2295i		−	1.23321i	0.787273	−	0.616605i	$-0.211492\pi$
$32$		0			0
$33$		−	63.3296i		−	1.91908i
$34$		0			0
$35$	5.01845	−	15.7133i	0.143384	−	0.448951i
$36$		0			0
$37$	68.5124			1.85169			0.925843	−	0.377908i	$-0.123356\pi$
$37$	68.5124			1.85169			0.925843	+	0.377908i	$0.123356\pi$
$38$		0			0
$39$		−	29.5285i		−	0.757140i
$40$		0			0
$41$	57.3992			1.39998			0.699990	−	0.714153i	$-0.253188\pi$
$41$	57.3992			1.39998			0.699990	+	0.714153i	$0.253188\pi$
$42$		0			0
$43$			51.9027i			1.20704i	0.797348	+	0.603520i	$0.206236\pi$
$43$			51.9027i			1.20704i	−0.797348	+	0.603520i	$0.793764\pi$
$44$		0			0
$45$	−13.1970	+	41.3213i	−0.293268	+	0.918251i
$46$		0			0
$47$	−34.0218			−0.723868			−0.361934	−	0.932204i	$-0.617884\pi$
$47$	−34.0218			−0.723868			−0.361934	+	0.932204i	$0.617884\pi$
$48$		0			0
$49$	−38.1163			−0.777883
$50$		0			0
$51$	−66.9233			−1.31222
$52$		0			0
$53$	79.9172			1.50787			0.753936	−	0.656948i	$-0.228153\pi$
$53$	79.9172			1.50787			0.753936	+	0.656948i	$0.228153\pi$
$54$		0			0
$55$	22.9141	−	71.7464i	0.416620	−	1.30448i
$56$		0			0
$57$			54.2042i			0.950952i
$58$		0			0
$59$	−114.933			−1.94802			−0.974012	−	0.226497i	$-0.927273\pi$
$59$	−114.933			−1.94802			−0.974012	+	0.226497i	$0.927273\pi$
$60$		0			0
$61$		−	115.485i		−	1.89320i	−0.322404	−	0.946602i	$-0.604491\pi$
$61$		−	115.485i		−	1.89320i	0.322404	−	0.946602i	$-0.395509\pi$
$62$		0			0
$63$	−28.6209			−0.454300
$64$		0			0
$65$	10.6841	−	33.4529i	0.164370	−	0.514660i
$66$		0			0
$67$		−	63.3874i		−	0.946081i	−0.881041	−	0.473040i	$-0.843156\pi$
$67$		−	63.3874i		−	0.946081i	0.881041	−	0.473040i	$-0.156844\pi$
$68$		0			0
$69$		−	14.6913i		−	0.212917i
$70$		0			0
$71$		−	40.1984i		−	0.566175i	−0.959094	−	0.283088i	$-0.908641\pi$
$71$		−	40.1984i		−	0.566175i	0.959094	−	0.283088i	$-0.0913587\pi$
$72$		0			0
$73$		−	2.16321i		−	0.0296331i	−0.999890	−	0.0148165i	$-0.995284\pi$
$73$		−	2.16321i		−	0.0296331i	0.999890	−	0.0148165i	$-0.00471642\pi$
$74$		0			0
$75$	−60.9223	+	85.6485i	−0.812298	+	1.14198i
$76$		0			0
$77$	49.6946			0.645385
$78$		0			0
$79$			78.1874i			0.989714i	0.868975	+	0.494857i	$0.164780\pi$
$79$			78.1874i			0.989714i	−0.868975	+	0.494857i	$0.835220\pi$
$80$		0			0
$81$	−83.8151			−1.03475
$82$		0			0
$83$			61.5753i			0.741871i	0.928659	+	0.370935i	$0.120963\pi$
$83$			61.5753i			0.741871i	−0.928659	+	0.370935i	$0.879037\pi$
$84$		0			0
$85$	−75.8177	−	24.2144i	−0.891973	−	0.284875i
$86$		0			0
$87$	55.3501			0.636208
$88$		0			0
$89$	6.04779			0.0679527			0.0339764	−	0.999423i	$-0.489183\pi$
$89$	6.04779			0.0679527			0.0339764	+	0.999423i	$0.489183\pi$
$90$		0			0
$91$	23.1709			0.254626
$92$		0			0
$93$	−160.725			−1.72823
$94$		0			0
$95$	−19.6123	+	61.4082i	−0.206445	+	0.646402i
$96$		0			0
$97$			135.240i			1.39422i	0.716962	+	0.697112i	$0.245532\pi$
$97$			135.240i			1.39422i	−0.716962	+	0.697112i	$0.754468\pi$
$98$		0			0
$99$	−130.682			−1.32002
$100$		0			0
$101$			112.667i			1.11552i	0.830002	+	0.557760i	$0.188339\pi$
$101$			112.667i			1.11552i	−0.830002	+	0.557760i	$0.811661\pi$
$102$		0			0
$103$	−38.6668			−0.375406			−0.187703	−	0.982226i	$-0.560104\pi$
$103$	−38.6668			−0.375406			−0.187703	+	0.982226i	$0.560104\pi$
$104$		0			0
$105$	−66.0623	−	21.0987i	−0.629164	−	0.200940i
$106$		0			0
$107$			198.172i			1.85207i	0.377436	+	0.926036i	$0.376806\pi$
$107$			198.172i			1.85207i	−0.377436	+	0.926036i	$0.623194\pi$
$108$		0			0
$109$			48.3119i			0.443229i	0.975134	+	0.221614i	$0.0711326\pi$
$109$			48.3119i			0.443229i	−0.975134	+	0.221614i	$0.928867\pi$
$110$		0			0
$111$		−	288.042i		−	2.59497i
$112$		0			0
$113$		−	75.4754i		−	0.667924i	−0.942587	−	0.333962i	$-0.891614\pi$
$113$		−	75.4754i		−	0.667924i	0.942587	−	0.333962i	$-0.108386\pi$
$114$		0			0
$115$	5.31564	−	16.6438i	0.0462230	−	0.144729i
$116$		0			0
$117$	−60.9327			−0.520792
$118$		0			0
$119$		−	52.5146i		−	0.441299i
$120$		0			0
$121$	105.904			0.875239
$122$		0			0
$123$		−	241.319i		−	1.96194i
$124$		0			0
$125$	−100.009	+	74.9884i	−0.800069	+	0.599908i
$126$		0			0
$127$	−69.4328			−0.546715			−0.273357	−	0.961913i	$-0.588134\pi$
$127$	−69.4328			−0.546715			−0.273357	+	0.961913i	$0.588134\pi$
$128$		0			0
$129$	218.211			1.69156
$130$		0			0
$131$	−145.813			−1.11308			−0.556538	−	0.830822i	$-0.687871\pi$
$131$	−145.813			−1.11308			−0.556538	+	0.830822i	$0.687871\pi$
$132$		0			0
$133$	−42.5340			−0.319804
$134$		0			0
$135$	−6.49776	−	2.07523i	−0.0481316	−	0.0153721i
$136$		0			0
$137$			176.955i			1.29165i	0.763487	+	0.645823i	$0.223486\pi$
$137$			176.955i			1.29165i	−0.763487	+	0.645823i	$0.776514\pi$
$138$		0			0
$139$	−56.3119			−0.405122			−0.202561	−	0.979270i	$-0.564926\pi$
$139$	−56.3119			−0.405122			−0.202561	+	0.979270i	$0.564926\pi$
$140$		0			0
$141$			143.035i			1.01444i
$142$		0			0
$143$	105.798			0.739844
$144$		0			0
$145$	62.7063	+	20.0269i	0.432457	+	0.138117i
$146$		0			0
$147$			160.249i			1.09013i
$148$		0			0
$149$		−	97.1333i		−	0.651901i	−0.945387	−	0.325951i	$-0.894316\pi$
$149$		−	97.1333i		−	0.651901i	0.945387	−	0.325951i	$-0.105684\pi$
$150$		0			0
$151$		−	64.8671i		−	0.429583i	−0.976660	−	0.214792i	$-0.931093\pi$
$151$		−	64.8671i		−	0.429583i	0.976660	−	0.214792i	$-0.0689073\pi$
$152$		0			0
$153$			138.098i			0.902600i
$154$		0			0
$155$	−182.086	−	58.1541i	−1.17475	−	0.375188i
$156$		0			0
$157$	235.678			1.50113			0.750566	−	0.660795i	$-0.229781\pi$
$157$	235.678			1.50113			0.750566	+	0.660795i	$0.229781\pi$
$158$		0			0
$159$		−	335.990i		−	2.11314i
$160$		0			0
$161$	11.5282			0.0716040
$162$		0			0
$163$			130.259i			0.799135i	0.916704	+	0.399568i	$0.130840\pi$
$163$			130.259i			0.799135i	−0.916704	+	0.399568i	$0.869160\pi$
$164$		0			0
$165$	−301.638	−	96.3360i	−1.82811	−	0.583854i
$166$		0			0
$167$	−96.7078			−0.579088			−0.289544	−	0.957165i	$-0.593504\pi$
$167$	−96.7078			−0.579088			−0.289544	+	0.957165i	$0.593504\pi$
$168$		0			0
$169$	−119.670			−0.708107
$170$		0			0
$171$	111.852			0.654103
$172$		0			0
$173$	−134.520			−0.777571			−0.388785	−	0.921328i	$-0.627105\pi$
$173$	−134.520			−0.777571			−0.388785	+	0.921328i	$0.627105\pi$
$174$		0			0
$175$	−67.2082	−	47.8056i	−0.384047	−	0.273175i
$176$		0			0
$177$			483.206i			2.72998i
$178$		0			0
$179$	8.17197			0.0456535			0.0228267	−	0.999739i	$-0.492733\pi$
$179$	8.17197			0.0456535			0.0228267	+	0.999739i	$0.492733\pi$
$180$		0			0
$181$			126.462i			0.698682i	0.936996	+	0.349341i	$0.113595\pi$
$181$			126.462i			0.698682i	−0.936996	+	0.349341i	$0.886405\pi$
$182$		0			0
$183$	−485.527			−2.65315
$184$		0			0
$185$	104.220	−	326.323i	0.563351	−	1.76391i
$186$		0			0
$187$		−	239.780i		−	1.28224i
$188$		0			0
$189$		−	4.50063i		−	0.0238129i
$190$		0			0
$191$		−	40.9687i		−	0.214496i	−0.994232	−	0.107248i	$-0.965796\pi$
$191$		−	40.9687i		−	0.214496i	0.994232	−	0.107248i	$-0.0342039\pi$
$192$		0			0
$193$		−	152.812i		−	0.791772i	−0.918300	−	0.395886i	$-0.870438\pi$
$193$		−	152.812i		−	0.791772i	0.918300	−	0.395886i	$-0.129562\pi$
$194$		0			0
$195$	−140.644	−	44.9182i	−0.721249	−	0.230350i
$196$		0			0
$197$	140.383			0.712605			0.356303	−	0.934371i	$-0.384037\pi$
$197$	140.383			0.712605			0.356303	+	0.934371i	$0.384037\pi$
$198$		0			0
$199$		−	108.191i		−	0.543675i	−0.962343	−	0.271838i	$-0.912369\pi$
$199$		−	108.191i		−	0.543675i	0.962343	−	0.271838i	$-0.0876313\pi$
$200$		0			0
$201$	−266.495			−1.32585
$202$		0			0
$203$			43.4331i			0.213956i
$204$		0			0
$205$	87.3147	−	273.391i	0.425925	−	1.33362i
$206$		0			0
$207$	−30.3158			−0.146453
$208$		0			0
$209$	−194.209			−0.929228
$210$		0			0
$211$	310.419			1.47118			0.735591	−	0.677426i	$-0.236904\pi$
$211$	310.419			1.47118			0.735591	+	0.677426i	$0.236904\pi$
$212$		0			0
$213$	−169.003			−0.793443
$214$		0			0
$215$	247.212	+	78.9536i	1.14982	+	0.367226i
$216$		0			0
$217$		−	126.121i		−	0.581203i
$218$		0			0
$219$	−9.09464			−0.0415280
$220$		0			0
$221$		−	111.801i		−	0.505888i
$222$		0			0
$223$	339.638			1.52304			0.761520	−	0.648141i	$-0.224453\pi$
$223$	339.638			1.52304			0.761520	+	0.648141i	$0.224453\pi$
$224$		0			0
$225$	176.738	+	125.715i	0.785500	+	0.558731i
$226$		0			0
$227$		−	143.631i		−	0.632738i	−0.948636	−	0.316369i	$-0.897536\pi$
$227$		−	143.631i		−	0.632738i	0.948636	−	0.316369i	$-0.102464\pi$
$228$		0			0
$229$			153.318i			0.669512i	0.942305	+	0.334756i	$0.108654\pi$
$229$			153.318i			0.669512i	−0.942305	+	0.334756i	$0.891346\pi$
$230$		0			0
$231$		−	208.927i		−	0.904448i
$232$		0			0
$233$		−	40.0757i		−	0.171999i	−0.996295	−	0.0859994i	$-0.972592\pi$
$233$		−	40.0757i		−	0.171999i	0.996295	−	0.0859994i	$-0.0274083\pi$
$234$		0			0
$235$	−51.7534	+	162.045i	−0.220227	+	0.689555i
$236$		0			0
$237$	328.717			1.38699
$238$		0			0
$239$			246.016i			1.02936i	0.857383	+	0.514678i	$0.172089\pi$
$239$			246.016i			1.02936i	−0.857383	+	0.514678i	$0.827911\pi$
$240$		0			0
$241$	−235.407			−0.976795			−0.488397	−	0.872621i	$-0.662418\pi$
$241$	−235.407			−0.976795			−0.488397	+	0.872621i	$0.662418\pi$
$242$		0			0
$243$			340.100i			1.39959i
$244$		0			0
$245$	−57.9818	+	181.547i	−0.236661	+	0.741009i
$246$		0			0
$247$	−90.5529			−0.366611
$248$		0			0
$249$	258.876			1.03966
$250$		0			0
$251$	−91.4700			−0.364422			−0.182211	−	0.983259i	$-0.558325\pi$
$251$	−91.4700			−0.364422			−0.182211	+	0.983259i	$0.558325\pi$
$252$		0			0
$253$	52.6375			0.208053
$254$		0			0
$255$	−101.803	+	318.755i	−0.399226	+	1.25002i
$256$		0			0
$257$		−	102.175i		−	0.397570i	−0.980043	−	0.198785i	$-0.936300\pi$
$257$		−	102.175i		−	0.397570i	0.980043	−	0.198785i	$-0.0636995\pi$
$258$		0			0
$259$	226.026			0.872686
$260$		0			0
$261$		−	114.216i		−	0.437610i
$262$		0			0
$263$	230.059			0.874748			0.437374	−	0.899280i	$-0.355909\pi$
$263$	230.059			0.874748			0.437374	+	0.899280i	$0.355909\pi$
$264$		0			0
$265$	121.569	−	380.644i	0.458750	−	1.43639i
$266$		0			0
$267$		−	25.4263i		−	0.0952295i
$268$		0			0
$269$		−	312.598i		−	1.16207i	−0.813877	−	0.581037i	$-0.802647\pi$
$269$		−	312.598i		−	1.16207i	0.813877	−	0.581037i	$-0.197353\pi$
$270$		0			0
$271$			115.473i			0.426101i	0.977041	+	0.213050i	$0.0683398\pi$
$271$			115.473i			0.426101i	−0.977041	+	0.213050i	$0.931660\pi$
$272$		0			0
$273$		−	97.4158i		−	0.356834i
$274$		0			0
$275$	−306.870	−	218.279i	−1.11589	−	0.793741i
$276$		0			0
$277$	303.255			1.09478			0.547391	−	0.836877i	$-0.315621\pi$
$277$	303.255			1.09478			0.547391	+	0.836877i	$0.315621\pi$
$278$		0			0
$279$			331.661i			1.18875i
$280$		0			0
$281$	−82.8094			−0.294695			−0.147348	−	0.989085i	$-0.547074\pi$
$281$	−82.8094			−0.294695			−0.147348	+	0.989085i	$0.547074\pi$
$282$		0			0
$283$		−	336.752i		−	1.18994i	−0.803749	−	0.594968i	$-0.797165\pi$
$283$		−	336.752i		−	1.18994i	0.803749	−	0.594968i	$-0.202835\pi$
$284$		0			0
$285$	258.174	+	82.4546i	0.905873	+	0.289314i
$286$		0			0
$287$	189.363			0.659800
$288$		0			0
$289$	35.6137			0.123231
$290$		0			0
$291$	568.579			1.95388
$292$		0			0
$293$	472.607			1.61299			0.806496	−	0.591239i	$-0.201361\pi$
$293$	472.607			1.61299			0.806496	+	0.591239i	$0.201361\pi$
$294$		0			0
$295$	−174.835	+	547.426i	−0.592660	+	1.85568i
$296$		0			0
$297$		−	20.5497i		−	0.0691910i
$298$		0			0
$299$	24.5431			0.0820840
$300$		0			0
$301$			171.230i			0.568869i
$302$		0			0
$303$	473.679			1.56330
$304$		0			0
$305$	−550.055	−	175.675i	−1.80346	−	0.575982i
$306$		0			0
$307$			52.5748i			0.171253i	0.996327	+	0.0856267i	$0.0272892\pi$
$307$			52.5748i			0.171253i	−0.996327	+	0.0856267i	$0.972711\pi$
$308$		0			0
$309$			162.564i			0.526097i
$310$		0			0
$311$		−	277.403i		−	0.891971i	−0.895040	−	0.445985i	$-0.852853\pi$
$311$		−	277.403i		−	0.891971i	0.895040	−	0.445985i	$-0.147147\pi$
$312$		0			0
$313$		−	89.6314i		−	0.286362i	−0.989696	−	0.143181i	$-0.954267\pi$
$313$		−	89.6314i		−	0.286362i	0.989696	−	0.143181i	$-0.0457332\pi$
$314$		0			0
$315$	−43.5377	+	136.321i	−0.138215	+	0.432765i
$316$		0			0
$317$	308.408			0.972897			0.486449	−	0.873709i	$-0.338292\pi$
$317$	308.408			0.972897			0.486449	+	0.873709i	$0.338292\pi$
$318$		0			0
$319$			198.314i			0.621674i
$320$		0			0
$321$	833.158			2.59551
$322$		0			0
$323$			205.229i			0.635384i
$324$		0			0
$325$	−143.083	−	101.776i	−0.440256	−	0.313157i
$326$		0			0
$327$	203.114			0.621145
$328$		0			0
$329$	−112.240			−0.341154
$330$		0			0
$331$	181.666			0.548839			0.274419	−	0.961610i	$-0.411514\pi$
$331$	181.666			0.548839			0.274419	+	0.961610i	$0.411514\pi$
$332$		0			0
$333$	−594.380			−1.78493
$334$		0			0
$335$	−301.913	−	96.4239i	−0.901233	−	0.287832i
$336$		0			0
$337$		−	111.419i		−	0.330621i	−0.986242	−	0.165310i	$-0.947137\pi$
$337$		−	111.419i		−	0.330621i	0.986242	−	0.165310i	$-0.0528625\pi$
$338$		0			0
$339$	−317.316			−0.936034
$340$		0			0
$341$		−	575.864i		−	1.68875i
$342$		0			0
$343$	−287.401			−0.837903
$344$		0			0
$345$	−69.9744	−	22.3482i	−0.202824	−	0.0647773i
$346$		0			0
$347$			518.938i			1.49550i	0.663982	+	0.747749i	$0.268865\pi$
$347$			518.938i			1.49550i	−0.663982	+	0.747749i	$0.731135\pi$
$348$		0			0
$349$			299.596i			0.858441i	0.903200	+	0.429221i	$0.141212\pi$
$349$			299.596i			0.858441i	−0.903200	+	0.429221i	$0.858788\pi$
$350$		0			0
$351$		−	9.58164i		−	0.0272981i
$352$		0			0
$353$		−	594.251i		−	1.68343i	−0.539922	−	0.841715i	$-0.681546\pi$
$353$		−	594.251i		−	1.68343i	0.539922	−	0.841715i	$-0.318454\pi$
$354$		0			0
$355$	−191.464	−	61.1492i	−0.539336	−	0.172251i
$356$		0			0
$357$	−220.783			−0.618440
$358$		0			0
$359$			229.119i			0.638216i	0.947718	+	0.319108i	$0.103383\pi$
$359$			229.119i			0.638216i	−0.947718	+	0.319108i	$0.896617\pi$
$360$		0			0
$361$	−194.776			−0.539545
$362$		0			0
$363$		−	445.244i		−	1.22657i
$364$		0			0
$365$	−10.3033	−	3.29064i	−0.0282283	−	0.00901546i
$366$		0			0
$367$	−90.0376			−0.245334			−0.122667	−	0.992448i	$-0.539145\pi$
$367$	−90.0376			−0.245334			−0.122667	+	0.992448i	$0.539145\pi$
$368$		0			0
$369$	−497.967			−1.34950
$370$		0			0
$371$	263.651			0.710649
$372$		0			0
$373$	423.979			1.13667			0.568336	−	0.822796i	$-0.307587\pi$
$373$	423.979			1.13667			0.568336	+	0.822796i	$0.307587\pi$
$374$		0			0
$375$	315.268	+	420.459i	0.840716	+	1.12122i
$376$		0			0
$377$			92.4671i			0.245271i
$378$		0			0
$379$	410.116			1.08210			0.541051	−	0.840990i	$-0.318027\pi$
$379$	410.116			1.08210			0.541051	+	0.840990i	$0.318027\pi$
$380$		0			0
$381$			291.911i			0.766171i
$382$		0			0
$383$	−741.842			−1.93692			−0.968462	−	0.249163i	$-0.919845\pi$
$383$	−741.842			−1.93692			−0.968462	+	0.249163i	$0.919845\pi$
$384$		0			0
$385$	75.5946	−	236.695i	0.196350	−	0.614791i
$386$		0			0
$387$		−	450.283i		−	1.16352i
$388$		0			0
$389$		−	179.045i		−	0.460270i	−0.973159	−	0.230135i	$-0.926083\pi$
$389$		−	179.045i		−	0.460270i	0.973159	−	0.230135i	$-0.0739168\pi$
$390$		0			0
$391$		−	55.6245i		−	0.142262i
$392$		0			0
$393$			613.030i			1.55987i
$394$		0			0
$395$	372.405	+	118.937i	0.942798	+	0.301107i
$396$		0			0
$397$	−81.9906			−0.206525			−0.103263	−	0.994654i	$-0.532928\pi$
$397$	−81.9906			−0.206525			−0.103263	+	0.994654i	$0.532928\pi$
$398$		0			0
$399$			178.822i			0.448176i
$400$		0			0
$401$	−146.981			−0.366537			−0.183269	−	0.983063i	$-0.558668\pi$
$401$	−146.981			−0.366537			−0.183269	+	0.983063i	$0.558668\pi$
$402$		0			0
$403$		−	268.506i		−	0.666268i
$404$		0			0
$405$	−127.498	+	399.210i	−0.314810	+	0.985703i
$406$		0			0
$407$	1032.02			2.53569
$408$		0			0
$409$	−142.051			−0.347313			−0.173656	−	0.984806i	$-0.555558\pi$
$409$	−142.051			−0.347313			−0.173656	+	0.984806i	$0.555558\pi$
$410$		0			0
$411$	743.961			1.81012
$412$		0			0
$413$	−379.171			−0.918089
$414$		0			0
$415$	293.282	+	93.6673i	0.706704	+	0.225704i
$416$		0			0
$417$			236.748i			0.567741i
$418$		0			0
$419$	−225.166			−0.537389			−0.268694	−	0.963225i	$-0.586592\pi$
$419$	−225.166			−0.537389			−0.268694	+	0.963225i	$0.586592\pi$
$420$		0			0
$421$		−	360.197i		−	0.855575i	−0.903879	−	0.427788i	$-0.859293\pi$
$421$		−	360.197i		−	0.855575i	0.903879	−	0.427788i	$-0.140707\pi$
$422$		0			0
$423$	295.157			0.697770
$424$		0			0
$425$	−230.665	+	324.284i	−0.542741	+	0.763021i
$426$		0			0
$427$		−	380.992i		−	0.892253i
$428$		0			0
$429$		−	444.797i		−	1.03682i
$430$		0			0
$431$		−	54.0689i		−	0.125450i	−0.998031	−	0.0627249i	$-0.980021\pi$
$431$		−	54.0689i		−	0.125450i	0.998031	−	0.0627249i	$-0.0199791\pi$
$432$		0			0
$433$			169.117i			0.390571i	0.980746	+	0.195285i	$0.0625633\pi$
$433$			169.117i			0.390571i	−0.980746	+	0.195285i	$0.937437\pi$
$434$		0			0
$435$	84.1976	−	263.631i	0.193558	−	0.606049i
$436$		0			0
$437$	−45.0528			−0.103096
$438$		0			0
$439$			645.558i			1.47052i	0.677785	+	0.735260i	$0.262940\pi$
$439$			645.558i			1.47052i	−0.677785	+	0.735260i	$0.737060\pi$
$440$		0			0
$441$	330.678			0.749837
$442$		0			0
$443$			478.360i			1.07982i	0.841723	+	0.539910i	$0.181542\pi$
$443$			478.360i			1.07982i	−0.841723	+	0.539910i	$0.818458\pi$
$444$		0			0
$445$	9.19980	−	28.8055i	0.0206737	−	0.0647315i
$446$		0			0
$447$	−408.370			−0.913580
$448$		0			0
$449$	628.735			1.40030			0.700150	−	0.713996i	$-0.253116\pi$
$449$	628.735			1.40030			0.700150	+	0.713996i	$0.253116\pi$
$450$		0			0
$451$	864.623			1.91712
$452$		0			0
$453$	−272.716			−0.602022
$454$		0			0
$455$	35.2472	−	110.363i	0.0774664	−	0.242555i
$456$		0			0
$457$		−	121.818i		−	0.266560i	−0.991078	−	0.133280i	$-0.957449\pi$
$457$		−	121.818i		−	0.266560i	0.991078	−	0.133280i	$-0.0425510\pi$
$458$		0			0
$459$	−21.7158			−0.0473112
$460$		0			0
$461$			152.537i			0.330883i	0.986220	+	0.165441i	$0.0529049\pi$
$461$			152.537i			0.330883i	−0.986220	+	0.165441i	$0.947095\pi$
$462$		0			0
$463$	368.130			0.795097			0.397549	−	0.917581i	$-0.369861\pi$
$463$	368.130			0.795097			0.397549	+	0.917581i	$0.369861\pi$
$464$		0			0
$465$	−244.493	+	765.533i	−0.525791	+	1.64631i
$466$		0			0
$467$		−	741.896i		−	1.58864i	−0.607498	−	0.794321i	$-0.707827\pi$
$467$		−	741.896i		−	1.58864i	0.607498	−	0.794321i	$-0.292173\pi$
$468$		0			0
$469$		−	209.118i		−	0.445881i
$470$		0			0
$471$		−	990.842i		−	2.10370i
$472$		0			0
$473$			781.828i			1.65291i
$474$		0			0
$475$	262.652	+	186.826i	0.552952	+	0.393318i
$476$		0			0
$477$	−693.323			−1.45351
$478$		0			0
$479$			620.600i			1.29562i	0.761803	+	0.647808i	$0.224314\pi$
$479$			620.600i			1.29562i	−0.761803	+	0.647808i	$0.775686\pi$
$480$		0			0
$481$	481.198			1.00041
$482$		0			0
$483$		−	48.4673i		−	0.100346i
$484$		0			0
$485$	644.145	+	205.724i	1.32813	+	0.424174i
$486$		0			0
$487$	−525.738			−1.07954			−0.539772	−	0.841811i	$-0.681490\pi$
$487$	−525.738			−1.07954			−0.539772	+	0.841811i	$0.681490\pi$
$488$		0			0
$489$	547.639			1.11992
$490$		0			0
$491$	695.125			1.41573			0.707867	−	0.706346i	$-0.249658\pi$
$491$	695.125			1.41573			0.707867	+	0.706346i	$0.249658\pi$
$492$		0			0
$493$	209.567			0.425086
$494$		0			0
$495$	−198.791	+	622.437i	−0.401599	+	1.25745i
$496$		0			0
$497$		−	132.617i		−	0.266834i
$498$		0			0
$499$	494.103			0.990186			0.495093	−	0.868840i	$-0.335134\pi$
$499$	494.103			0.990186			0.495093	+	0.868840i	$0.335134\pi$
$500$		0			0
$501$			406.581i			0.811539i
$502$		0			0
$503$	−881.187			−1.75186			−0.875931	−	0.482436i	$-0.839752\pi$
$503$	−881.187			−1.75186			−0.875931	+	0.482436i	$0.839752\pi$
$504$		0			0
$505$	536.633	+	171.388i	1.06264	+	0.339382i
$506$		0			0
$507$			503.120i			0.992347i
$508$		0			0
$509$			676.617i			1.32931i	0.747152	+	0.664653i	$0.231421\pi$
$509$			676.617i			1.32931i	−0.747152	+	0.664653i	$0.768579\pi$
$510$		0			0
$511$		−	7.13654i		−	0.0139658i
$512$		0			0
$513$			17.5886i			0.0342858i
$514$		0			0
$515$	−58.8193	+	184.169i	−0.114212	+	0.357610i
$516$		0			0
$517$	−512.482			−0.991261
$518$		0			0
$519$			565.551i			1.08969i
$520$		0			0
$521$	139.017			0.266828			0.133414	−	0.991060i	$-0.457406\pi$
$521$	139.017			0.266828			0.133414	+	0.991060i	$0.457406\pi$
$522$		0			0
$523$		−	506.388i		−	0.968237i	−0.875002	−	0.484119i	$-0.839140\pi$
$523$		−	506.388i		−	0.968237i	0.875002	−	0.484119i	$-0.160860\pi$
$524$		0			0
$525$	−200.986	+	282.558i	−0.382830	+	0.538207i
$526$		0			0
$527$	−608.542			−1.15473
$528$		0			0
$529$	−516.789			−0.976917
$530$		0			0
$531$	997.106			1.87779
$532$		0			0
$533$	403.144			0.756369
$534$		0			0
$535$	943.888	+	301.455i	1.76428	+	0.563468i
$536$		0			0
$537$		−	34.3568i		−	0.0639792i
$538$		0			0
$539$	−574.158			−1.06523
$540$		0			0
$541$		−	771.126i		−	1.42537i	−0.701484	−	0.712686i	$-0.747479\pi$
$541$		−	771.126i		−	1.42537i	0.701484	−	0.712686i	$-0.252521\pi$
$542$		0			0
$543$	531.673			0.979139
$544$		0			0
$545$	230.109	+	73.4913i	0.422218	+	0.134846i
$546$		0			0
$547$			551.675i			1.00855i	0.863544	+	0.504274i	$0.168240\pi$
$547$			551.675i			1.00855i	−0.863544	+	0.504274i	$0.831760\pi$
$548$		0			0
$549$			1001.90i			1.82495i
$550$		0			0
$551$		−	169.738i		−	0.308055i
$552$		0			0
$553$			257.944i			0.466445i
$554$		0			0
$555$	−1371.94	−	438.164i	−2.47196	−	0.789484i
$556$		0			0
$557$	−971.376			−1.74394			−0.871972	−	0.489556i	$-0.837159\pi$
$557$	−971.376			−1.74394			−0.871972	+	0.489556i	$0.837159\pi$
$558$		0			0
$559$			364.540i			0.652129i
$560$		0			0
$561$	−1008.09			−1.79695
$562$		0			0
$563$		−	721.489i		−	1.28151i	−0.767747	−	0.640754i	$-0.778622\pi$
$563$		−	721.489i		−	1.28151i	0.767747	−	0.640754i	$-0.221378\pi$
$564$		0			0
$565$	−359.488	−	114.812i	−0.636262	−	0.203207i
$566$		0			0
$567$	−276.510			−0.487672
$568$		0			0
$569$	288.369			0.506799			0.253400	−	0.967362i	$-0.418451\pi$
$569$	288.369			0.506799			0.253400	+	0.967362i	$0.418451\pi$
$570$		0			0
$571$	706.199			1.23678			0.618388	−	0.785873i	$-0.287786\pi$
$571$	706.199			1.23678			0.618388	+	0.785873i	$0.287786\pi$
$572$		0			0
$573$	−172.242			−0.300596
$574$		0			0
$575$	−71.1882	−	50.6366i	−0.123806	−	0.0880637i
$576$		0			0
$577$		−	749.954i		−	1.29975i	−0.760042	−	0.649873i	$-0.774822\pi$
$577$		−	749.954i		−	1.29975i	0.760042	−	0.649873i	$-0.225178\pi$
$578$		0			0
$579$	−642.456			−1.10960
$580$		0			0
$581$			203.140i			0.349638i
$582$		0			0
$583$	1203.82			2.06487
$584$		0			0
$585$	−92.6897	+	290.221i	−0.158444	+	0.496105i
$586$		0			0
$587$		−	310.310i		−	0.528637i	−0.964435	−	0.264319i	$-0.914853\pi$
$587$		−	310.310i		−	0.528637i	0.964435	−	0.264319i	$-0.0851471\pi$
$588$		0			0
$589$			492.886i			0.836818i
$590$		0			0
$591$		−	590.203i		−	0.998651i
$592$		0			0
$593$		−	19.1336i		−	0.0322657i	−0.999870	−	0.0161329i	$-0.994865\pi$
$593$		−	19.1336i		−	0.0322657i	0.999870	−	0.0161329i	$-0.00513547\pi$
$594$		0			0
$595$	−250.126	−	79.8843i	−0.420380	−	0.134259i
$596$		0			0
$597$	−454.861			−0.761911
$598$		0			0
$599$		−	704.211i		−	1.17564i	−0.808990	−	0.587822i	$-0.799985\pi$
$599$		−	704.211i		−	1.17564i	0.808990	−	0.587822i	$-0.200015\pi$
$600$		0			0
$601$	284.816			0.473903			0.236952	−	0.971521i	$-0.423852\pi$
$601$	284.816			0.473903			0.236952	+	0.971521i	$0.423852\pi$
$602$		0			0
$603$			549.918i			0.911971i
$604$		0			0
$605$	161.099	−	504.419i	0.266280	−	0.833750i
$606$		0			0
$607$	1114.68			1.83638			0.918191	−	0.396138i	$-0.129650\pi$
$607$	1114.68			1.83638			0.918191	+	0.396138i	$0.129650\pi$
$608$		0			0
$609$	182.602			0.299840
$610$		0			0
$611$	−238.953			−0.391085
$612$		0			0
$613$	−81.0578			−0.132231			−0.0661156	−	0.997812i	$-0.521061\pi$
$613$	−81.0578			−0.132231			−0.0661156	+	0.997812i	$0.521061\pi$
$614$		0			0
$615$	−1149.40	−	367.090i	−1.86894	−	0.596895i
$616$		0			0
$617$		−	205.581i		−	0.333194i	−0.986025	−	0.166597i	$-0.946722\pi$
$617$		−	205.581i		−	0.333194i	0.986025	−	0.166597i	$-0.0532780\pi$
$618$		0			0
$619$	−227.970			−0.368287			−0.184144	−	0.982899i	$-0.558951\pi$
$619$	−227.970			−0.368287			−0.184144	+	0.982899i	$0.558951\pi$
$620$		0			0
$621$		−	4.76715i		−	0.00767658i
$622$		0			0
$623$	19.9519			0.0320256
$624$		0			0
$625$	205.037	+	590.411i	0.328060	+	0.944657i
$626$		0			0
$627$			816.497i			1.30223i
$628$		0			0
$629$		−	1090.59i		−	1.73384i
$630$		0			0
$631$			936.003i			1.48336i	0.670751	+	0.741682i	$0.265972\pi$
$631$			936.003i			1.48336i	−0.670751	+	0.741682i	$0.734028\pi$
$632$		0			0
$633$		−	1305.07i		−	2.06173i
$634$		0			0
$635$	−105.620	+	330.707i	−0.166331	+	0.520799i
$636$		0			0
$637$	−267.711			−0.420268
$638$		0			0
$639$			348.742i			0.545762i
$640$		0			0
$641$	−308.970			−0.482013			−0.241007	−	0.970524i	$-0.577478\pi$
$641$	−308.970			−0.482013			−0.241007	+	0.970524i	$0.577478\pi$
$642$		0			0
$643$			556.493i			0.865463i	0.901523	+	0.432732i	$0.142450\pi$
$643$			556.493i			0.865463i	−0.901523	+	0.432732i	$0.857550\pi$
$644$		0			0
$645$	331.939	−	1039.33i	0.514633	−	1.61137i
$646$		0			0
$647$	−142.746			−0.220628			−0.110314	−	0.993897i	$-0.535186\pi$
$647$	−142.746			−0.220628			−0.110314	+	0.993897i	$0.535186\pi$
$648$		0			0
$649$	−1731.28			−2.66761
$650$		0			0
$651$	−530.241			−0.814502
$652$		0			0
$653$	1034.40			1.58408			0.792040	−	0.610469i	$-0.209019\pi$
$653$	1034.40			1.58408			0.792040	+	0.610469i	$0.209019\pi$
$654$		0			0
$655$	−221.808	+	694.504i	−0.338638	+	1.06031i
$656$		0			0
$657$			18.7670i			0.0285647i
$658$		0			0
$659$	−690.873			−1.04837			−0.524183	−	0.851606i	$-0.675629\pi$
$659$	−690.873			−1.04837			−0.524183	+	0.851606i	$0.675629\pi$
$660$		0			0
$661$			926.204i			1.40122i	0.713546	+	0.700608i	$0.247088\pi$
$661$			926.204i			1.40122i	−0.713546	+	0.700608i	$0.752912\pi$
$662$		0			0
$663$	−470.038			−0.708956
$664$		0			0
$665$	−64.7019	+	202.588i	−0.0972961	+	0.304644i
$666$		0			0
$667$			46.0052i			0.0689733i
$668$		0			0
$669$		−	1427.92i		−	2.13440i
$670$		0			0
$671$		−	1739.60i		−	2.59254i
$672$		0			0
$673$			729.553i			1.08403i	0.840368	+	0.542016i	$0.182339\pi$
$673$			729.553i			1.08403i	−0.840368	+	0.542016i	$0.817661\pi$
$674$		0			0
$675$	−19.7686	+	27.7919i	−0.0292868	+	0.0411732i
$676$		0			0
$677$	−229.836			−0.339492			−0.169746	−	0.985488i	$-0.554295\pi$
$677$	−229.836			−0.339492			−0.169746	+	0.985488i	$0.554295\pi$
$678$		0			0
$679$			446.162i			0.657087i
$680$		0			0
$681$	−603.859			−0.886724
$682$		0			0
$683$		−	76.7080i		−	0.112310i	−0.998422	−	0.0561552i	$-0.982116\pi$
$683$		−	76.7080i		−	0.112310i	0.998422	−	0.0561552i	$-0.0178842\pi$
$684$		0			0
$685$	842.836	+	269.182i	1.23042	+	0.392966i
$686$		0			0
$687$	644.585			0.938260
$688$		0			0
$689$	561.300			0.814660
$690$		0			0
$691$	−1008.82			−1.45995			−0.729974	−	0.683475i	$-0.760468\pi$
$691$	−1008.82			−1.45995			−0.729974	+	0.683475i	$0.760468\pi$
$692$		0			0
$693$	−431.126			−0.622116
$694$		0			0
$695$	−85.6607	+	268.213i	−0.123253	+	0.385917i
$696$		0			0
$697$		−	913.687i		−	1.31088i
$698$		0			0
$699$	−168.487			−0.241041
$700$		0			0
$701$			692.820i			0.988331i	0.869368	+	0.494165i	$0.164526\pi$
$701$			692.820i			0.988331i	−0.869368	+	0.494165i	$0.835474\pi$
$702$		0			0
$703$	−883.317			−1.25650
$704$		0			0
$705$	681.275	+	217.583i	0.966348	+	0.308628i
$706$		0			0
$707$			371.695i			0.525736i
$708$		0			0
$709$			16.2200i			0.0228773i	0.999935	+	0.0114386i	$0.00364111\pi$
$709$			16.2200i			0.0228773i	−0.999935	+	0.0114386i	$0.996359\pi$
$710$		0			0
$711$		−	678.316i		−	0.954030i
$712$		0			0
$713$		−	133.590i		−	0.187363i
$714$		0			0
$715$	160.938	−	503.912i	0.225088	−	0.704773i
$716$		0			0
$717$	1034.31			1.44255
$718$		0			0
$719$		−	1007.80i		−	1.40168i	−0.713321	−	0.700838i	$-0.752810\pi$
$719$		−	1007.80i		−	1.40168i	0.713321	−	0.700838i	$-0.247190\pi$
$720$		0			0
$721$	−127.564			−0.176926
$722$		0			0
$723$			989.706i			1.36889i
$724$		0			0
$725$	190.775	−	268.204i	0.263139	−	0.369937i
$726$		0			0
$727$	58.2441			0.0801157			0.0400579	−	0.999197i	$-0.487246\pi$
$727$	58.2441			0.0801157			0.0400579	+	0.999197i	$0.487246\pi$
$728$		0			0
$729$	675.519			0.926638
$730$		0			0
$731$	826.194			1.13022
$732$		0			0
$733$	1340.52			1.82882			0.914409	−	0.404791i	$-0.132656\pi$
$733$	1340.52			1.82882			0.914409	+	0.404791i	$0.132656\pi$
$734$		0			0
$735$	763.265	+	243.769i	1.03846	+	0.331658i
$736$		0			0
$737$		−	954.826i		−	1.29556i
$738$		0			0
$739$	806.666			1.09156			0.545782	−	0.837927i	$-0.316233\pi$
$739$	806.666			1.09156			0.545782	+	0.837927i	$0.316233\pi$
$740$		0			0
$741$			380.705i			0.513772i
$742$		0			0
$743$	1051.63			1.41538			0.707692	−	0.706521i	$-0.249736\pi$
$743$	1051.63			1.41538			0.707692	+	0.706521i	$0.249736\pi$
$744$		0			0
$745$	−462.644	−	147.758i	−0.620999	−	0.198332i
$746$		0			0
$747$		−	534.197i		−	0.715123i
$748$		0			0
$749$			653.777i			0.872867i
$750$		0			0
$751$		−	392.921i		−	0.523196i	−0.965177	−	0.261598i	$-0.915750\pi$
$751$		−	392.921i		−	0.523196i	0.965177	−	0.261598i	$-0.0842495\pi$
$752$		0			0
$753$			384.561i			0.510705i
$754$		0			0
$755$	−308.961	−	98.6747i	−0.409220	−	0.130695i
$756$		0			0
$757$	−485.104			−0.640825			−0.320412	−	0.947278i	$-0.603822\pi$
$757$	−485.104			−0.640825			−0.320412	+	0.947278i	$0.603822\pi$
$758$		0			0
$759$		−	221.300i		−	0.291568i
$760$		0			0
$761$	−46.6127			−0.0612519			−0.0306260	−	0.999531i	$-0.509750\pi$
$761$	−46.6127			−0.0612519			−0.0306260	+	0.999531i	$0.509750\pi$
$762$		0			0
$763$			159.383i			0.208890i
$764$		0			0
$765$	657.757	+	210.072i	0.859813	+	0.274604i
$766$		0			0
$767$	−807.237			−1.05246
$768$		0			0
$769$	859.032			1.11708			0.558538	−	0.829479i	$-0.311362\pi$
$769$	859.032			1.11708			0.558538	+	0.829479i	$0.311362\pi$
$770$		0			0
$771$	−429.569			−0.557158
$772$		0			0
$773$	1004.82			1.29990			0.649951	−	0.759976i	$-0.274789\pi$
$773$	1004.82			1.29990			0.649951	+	0.759976i	$0.274789\pi$
$774$		0			0
$775$	−553.974	+	778.812i	−0.714805	+	1.00492i
$776$		0			0
$777$		−	950.263i		−	1.22299i
$778$		0			0
$779$	−740.036			−0.949982
$780$		0			0
$781$		−	605.522i		−	0.775317i
$782$		0			0
$783$	17.9604			0.0229380
$784$		0			0
$785$	358.509	−	1122.53i	0.456699	−	1.42997i
$786$		0			0
$787$		−	250.881i		−	0.318781i	−0.987216	−	0.159391i	$-0.949047\pi$
$787$		−	250.881i		−	0.318781i	0.987216	−	0.159391i	$-0.0509529\pi$
$788$		0			0
$789$		−	967.219i		−	1.22588i
$790$		0			0
$791$		−	248.997i		−	0.314787i
$792$		0			0
$793$		−	811.115i		−	1.02284i
$794$		0			0
$795$	−1600.31	−	511.102i	−2.01297	−	0.642896i
$796$		0			0
$797$	−803.365			−1.00799			−0.503993	−	0.863708i	$-0.668136\pi$
$797$	−803.365			−1.00799			−0.503993	+	0.863708i	$0.668136\pi$
$798$		0			0
$799$			541.563i			0.677801i
$800$		0			0
$801$	−52.4677			−0.0655027
$802$		0			0
$803$		−	32.5852i		−	0.0405793i
$804$		0			0
$805$	17.5366	−	54.9088i	0.0217845	−	0.0682097i
$806$		0			0
$807$	−1314.23			−1.62854
$808$		0			0
$809$	254.545			0.314642			0.157321	−	0.987548i	$-0.449714\pi$
$809$	254.545			0.314642			0.157321	+	0.987548i	$0.449714\pi$
$810$		0			0
$811$	1316.92			1.62382			0.811912	−	0.583779i	$-0.198427\pi$
$811$	1316.92			1.62382			0.811912	+	0.583779i	$0.198427\pi$
$812$		0			0
$813$	485.476			0.597141
$814$		0			0
$815$	620.422	+	198.148i	0.761254	+	0.243126i
$816$		0			0
$817$		−	669.172i		−	0.819060i
$818$		0			0
$819$	−201.020			−0.245445
$820$		0			0
$821$			597.540i			0.727820i	0.931434	+	0.363910i	$0.118558\pi$
$821$			597.540i			0.727820i	−0.931434	+	0.363910i	$0.881442\pi$
$822$		0			0
$823$	−1511.87			−1.83702			−0.918510	−	0.395399i	$-0.870606\pi$
$823$	−1511.87			−1.83702			−0.918510	+	0.395399i	$0.870606\pi$
$824$		0			0
$825$	−917.693	+	1290.15i	−1.11236	+	1.56382i
$826$		0			0
$827$		−	750.703i		−	0.907743i	−0.891067	−	0.453871i	$-0.850043\pi$
$827$		−	750.703i		−	0.907743i	0.891067	−	0.453871i	$-0.149957\pi$
$828$		0			0
$829$			451.313i			0.544406i	0.962240	+	0.272203i	$0.0877523\pi$
$829$			451.313i			0.544406i	−0.962240	+	0.272203i	$0.912248\pi$
$830$		0			0
$831$		−	1274.95i		−	1.53424i
$832$		0			0
$833$			606.739i			0.728379i
$834$		0			0
$835$	−147.110	+	460.617i	−0.176180	+	0.551638i
$836$		0			0
$837$	−52.1535			−0.0623101
$838$		0			0
$839$		−	1191.13i		−	1.41970i	−0.704354	−	0.709849i	$-0.748763\pi$
$839$		−	1191.13i		−	1.41970i	0.704354	−	0.709849i	$-0.251237\pi$
$840$		0			0
$841$	667.674			0.793905
$842$		0			0
$843$			348.149i			0.412988i
$844$		0			0
$845$	−182.040	+	569.987i	−0.215432	+	0.674540i
$846$		0			0
$847$	349.382			0.412494
$848$		0			0
$849$	−1415.78			−1.66759
$850$		0			0
$851$	239.411			0.281329
$852$		0			0
$853$	−572.320			−0.670949			−0.335475	−	0.942049i	$-0.608897\pi$
$853$	−572.320			−0.670949			−0.335475	+	0.942049i	$0.608897\pi$
$854$		0			0
$855$	170.147	−	532.748i	0.199002	−	0.623097i
$856$		0			0
$857$		−	621.587i		−	0.725306i	−0.931924	−	0.362653i	$-0.881871\pi$
$857$		−	621.587i		−	0.725306i	0.931924	−	0.362653i	$-0.118129\pi$
$858$		0			0
$859$	−394.083			−0.458770			−0.229385	−	0.973336i	$-0.573671\pi$
$859$	−394.083			−0.458770			−0.229385	+	0.973336i	$0.573671\pi$
$860$		0			0
$861$		−	796.123i		−	0.924649i
$862$		0			0
$863$	−472.620			−0.547648			−0.273824	−	0.961780i	$-0.588289\pi$
$863$	−472.620			−0.547648			−0.273824	+	0.961780i	$0.588289\pi$
$864$		0			0
$865$	−204.629	+	640.715i	−0.236565	+	0.740711i
$866$		0			0
$867$		−	149.728i		−	0.172697i
$868$		0			0
$869$			1177.76i			1.35531i
$870$		0			0
$871$		−	445.203i		−	0.511140i
$872$		0			0
$873$		−	1173.27i		−	1.34396i
$874$		0			0
$875$	−329.933	+	247.390i	−0.377067	+	0.282732i
$876$		0			0
$877$	−407.706			−0.464887			−0.232443	−	0.972610i	$-0.574672\pi$
$877$	−407.706			−0.464887			−0.232443	+	0.972610i	$0.574672\pi$
$878$		0			0
$879$		−	1986.95i		−	2.26046i
$880$		0			0
$881$	−843.968			−0.957966			−0.478983	−	0.877824i	$-0.658994\pi$
$881$	−843.968			−0.957966			−0.478983	+	0.877824i	$0.658994\pi$
$882$		0			0
$883$		−	868.718i		−	0.983826i	−0.870644	−	0.491913i	$-0.836298\pi$
$883$		−	868.718i		−	0.983826i	0.870644	−	0.491913i	$-0.163702\pi$
$884$		0			0
$885$	2301.50	+	735.045i	2.60057	+	0.830559i
$886$		0			0
$887$	−110.857			−0.124979			−0.0624896	−	0.998046i	$-0.519904\pi$
$887$	−110.857			−0.124979			−0.0624896	+	0.998046i	$0.519904\pi$
$888$		0			0
$889$	−229.062			−0.257663
$890$		0			0
$891$	−1262.53			−1.41699
$892$		0			0
$893$	438.637			0.491194
$894$		0			0
$895$	12.4311	−	38.9230i	0.0138895	−	0.0434893i
$896$		0			0
$897$		−	103.185i		−	0.115033i
$898$		0			0
$899$	503.305			0.559849
$900$		0			0
$901$		−	1272.13i		−	1.41191i
$902$		0			0
$903$	719.888			0.797218
$904$		0			0
$905$	602.334	+	192.371i	0.665562	+	0.212565i
$906$		0			0
$907$			198.356i			0.218694i	0.994004	+	0.109347i	$0.0348760\pi$
$907$			198.356i			0.218694i	−0.994004	+	0.109347i	$0.965124\pi$
$908$		0			0
$909$		−	977.448i		−	1.07530i
$910$		0			0
$911$		−	964.205i		−	1.05840i	−0.848496	−	0.529201i	$-0.822492\pi$
$911$		−	964.205i		−	1.05840i	0.848496	−	0.529201i	$-0.177508\pi$
$912$		0			0
$913$			927.529i			1.01591i
$914$		0			0
$915$	−738.575	+	2312.56i	−0.807186	+	2.52738i
$916$		0			0
$917$	−481.043			−0.524584
$918$		0			0
$919$		−	1004.28i		−	1.09279i	−0.837526	−	0.546397i	$-0.815999\pi$
$919$		−	1004.28i		−	1.09279i	0.837526	−	0.546397i	$-0.184001\pi$
$920$		0			0
$921$	221.036			0.239996
$922$		0			0
$923$		−	282.335i		−	0.305888i
$924$		0			0
$925$	−1395.73	−	992.795i	−1.50890	−	1.07329i
$926$		0			0
$927$	335.454			0.361871
$928$		0			0
$929$	39.4532			0.0424685			0.0212342	−	0.999775i	$-0.493240\pi$
$929$	39.4532			0.0424685			0.0212342	+	0.999775i	$0.493240\pi$
$930$		0			0
$931$	491.426			0.527847
$932$		0			0
$933$	−1166.26			−1.25002
$934$		0			0
$935$	−1142.07	−	364.749i	−1.22146	−	0.390106i
$936$		0			0
$937$			727.845i			0.776782i	0.921495	+	0.388391i	$0.126969\pi$
$937$			727.845i			0.776782i	−0.921495	+	0.388391i	$0.873031\pi$
$938$		0			0
$939$	−376.831			−0.401311
$940$		0			0
$941$		−	1077.46i		−	1.14501i	−0.819900	−	0.572506i	$-0.805971\pi$
$941$		−	1077.46i		−	1.14501i	0.819900	−	0.572506i	$-0.194029\pi$
$942$		0			0
$943$	200.577			0.212700
$944$		0			0
$945$	−21.4364	−	6.84628i	−0.0226840	−	0.00724474i
$946$		0			0
$947$			153.952i			0.162568i	0.996691	+	0.0812840i	$0.0259021\pi$
$947$			153.952i			0.162568i	−0.996691	+	0.0812840i	$0.974098\pi$
$948$		0			0
$949$		−	15.1934i		−	0.0160099i
$950$		0			0
$951$		−	1296.62i		−	1.36343i
$952$		0			0
$953$			1524.18i			1.59935i	0.600435	+	0.799673i	$0.294994\pi$
$953$			1524.18i			1.59935i	−0.600435	+	0.799673i	$0.705006\pi$
$954$		0			0
$955$	−195.133	−	62.3209i	−0.204328	−	0.0652575i
$956$		0			0
$957$	833.756			0.871219
$958$		0			0
$959$			583.784i			0.608743i
$960$		0			0
$961$	−500.496			−0.520807
$962$		0			0
$963$		−	1719.24i		−	1.78530i
$964$		0			0
$965$	−727.841	−	232.455i	−0.754239	−	0.240886i
$966$		0			0
$967$	349.260			0.361179			0.180590	−	0.983559i	$-0.442199\pi$
$967$	349.260			0.361179			0.180590	+	0.983559i	$0.442199\pi$
$968$		0			0
$969$	862.829			0.890433
$970$		0			0
$971$	−745.065			−0.767317			−0.383659	−	0.923475i	$-0.625336\pi$
$971$	−745.065			−0.767317			−0.383659	+	0.923475i	$0.625336\pi$
$972$		0			0
$973$	−185.776			−0.190931
$974$		0			0
$975$	−427.889	+	601.554i	−0.438861	+	0.616979i
$976$		0			0
$977$			1205.72i			1.23410i	0.786923	+	0.617051i	$0.211673\pi$
$977$			1205.72i			1.23410i	−0.786923	+	0.617051i	$0.788327\pi$
$978$		0			0
$979$	91.0999			0.0930540
$980$		0			0
$981$		−	419.131i		−	0.427249i
$982$		0			0
$983$	−427.425			−0.434817			−0.217408	−	0.976081i	$-0.569760\pi$
$983$	−427.425			−0.434817			−0.217408	+	0.976081i	$0.569760\pi$
$984$		0			0
$985$	213.549	−	668.643i	0.216801	−	0.678825i
$986$		0			0
$987$			471.880i			0.478096i
$988$		0			0
$989$			181.370i			0.183387i
$990$		0			0
$991$			924.615i			0.933012i	0.884518	+	0.466506i	$0.154487\pi$
$991$			924.615i			0.933012i	−0.884518	+	0.466506i	$0.845513\pi$
$992$		0			0
$993$		−	763.763i		−	0.769148i
$994$		0			0
$995$	−515.313	−	164.579i	−0.517903	−	0.165406i
$996$		0			0
$997$	720.253			0.722420			0.361210	−	0.932484i	$-0.382364\pi$
$997$	720.253			0.722420			0.361210	+	0.932484i	$0.382364\pi$
$998$		0			0
$999$		−	93.4661i		−	0.0935596i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.e.l.639.9		24
4.3	odd	2		1280.3.e.k.639.16		24
5.4	even	2	inner	1280.3.e.l.639.16		24
8.3	odd	2	inner	1280.3.e.l.639.15		24
8.5	even	2		1280.3.e.k.639.10		24
16.3	odd	4		640.3.h.b.639.4	yes	24
16.5	even	4		640.3.h.a.639.4	yes	24
16.11	odd	4		640.3.h.a.639.21	yes	24
16.13	even	4		640.3.h.b.639.21	yes	24
20.19	odd	2		1280.3.e.k.639.9		24
40.19	odd	2	inner	1280.3.e.l.639.10		24
40.29	even	2		1280.3.e.k.639.15		24
80.19	odd	4		640.3.h.b.639.22	yes	24
80.29	even	4		640.3.h.b.639.3	yes	24
80.59	odd	4		640.3.h.a.639.3	✓	24
80.69	even	4		640.3.h.a.639.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.h.a.639.3	✓	24	80.59	odd	4
640.3.h.a.639.4	yes	24	16.5	even	4
640.3.h.a.639.21	yes	24	16.11	odd	4
640.3.h.a.639.22	yes	24	80.69	even	4
640.3.h.b.639.3	yes	24	80.29	even	4
640.3.h.b.639.4	yes	24	16.3	odd	4
640.3.h.b.639.21	yes	24	16.13	even	4
640.3.h.b.639.22	yes	24	80.19	odd	4
1280.3.e.k.639.9		24	20.19	odd	2
1280.3.e.k.639.10		24	8.5	even	2
1280.3.e.k.639.15		24	40.29	even	2
1280.3.e.k.639.16		24	4.3	odd	2
1280.3.e.l.639.9		24	1.1	even	1	trivial
1280.3.e.l.639.10		24	40.19	odd	2	inner
1280.3.e.l.639.15		24	8.3	odd	2	inner
1280.3.e.l.639.16		24	5.4	even	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.20423i q^{3} +(1.52118 - 4.76298i) q^{5} +3.29905 q^{7} -8.67551 q^{9} +O(q^{10})\)	\(q-4.20423i q^{3} +(1.52118 - 4.76298i) q^{5} +3.29905 q^{7} -8.67551 q^{9} +15.0633 q^{11} +7.02352 q^{13} +(-20.0247 - 6.39540i) q^{15} -15.9181i q^{17} -12.8928 q^{19} -13.8699i q^{21} +3.49441 q^{23} +(-20.3720 - 14.4907i) q^{25} -1.36422i q^{27} +13.1653i q^{29} -38.2295i q^{31} -63.3296i q^{33} +(5.01845 - 15.7133i) q^{35} +68.5124 q^{37} -29.5285i q^{39} +57.3992 q^{41} +51.9027i q^{43} +(-13.1970 + 41.3213i) q^{45} -34.0218 q^{47} -38.1163 q^{49} -66.9233 q^{51} +79.9172 q^{53} +(22.9141 - 71.7464i) q^{55} +54.2042i q^{57} -114.933 q^{59} -115.485i q^{61} -28.6209 q^{63} +(10.6841 - 33.4529i) q^{65} -63.3874i q^{67} -14.6913i q^{69} -40.1984i q^{71} -2.16321i q^{73} +(-60.9223 + 85.6485i) q^{75} +49.6946 q^{77} +78.1874i q^{79} -83.8151 q^{81} +61.5753i q^{83} +(-75.8177 - 24.2144i) q^{85} +55.3501 q^{87} +6.04779 q^{89} +23.1709 q^{91} -160.725 q^{93} +(-19.6123 + 61.4082i) q^{95} +135.240i q^{97} -130.682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{9}+O(q^{10}) \) 24 * q - 72 * q^9	\( 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) \) 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more