LMFDB - Embedded newform 800.4.c.l.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.c (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:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.l.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.47214i q^{3} +31.3050i q^{7} +7.00000 q^{9} +O(q^{10})$	$q+4.47214i q^{3} +31.3050i q^{7} +7.00000 q^{9} -8.94427 q^{11} -62.0000i q^{13} +46.0000i q^{17} -107.331 q^{19} -140.000 q^{21} +192.302i q^{23} +152.053i q^{27} +90.0000 q^{29} -152.053 q^{31} -40.0000i q^{33} +214.000i q^{37} +277.272 q^{39} -10.0000 q^{41} -67.0820i q^{43} -398.020i q^{47} -637.000 q^{49} -205.718 q^{51} -678.000i q^{53} -480.000i q^{57} +411.437 q^{59} +250.000 q^{61} +219.135i q^{63} -49.1935i q^{67} -860.000 q^{69} -366.715 q^{71} +522.000i q^{73} -280.000i q^{77} -876.539 q^{79} -491.000 q^{81} +380.132i q^{83} +402.492i q^{87} -970.000 q^{89} +1940.91 q^{91} -680.000i q^{93} +934.000i q^{97} -62.6099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{9}+O(q^{10}) $ 4 * q + 28 * q^9	$ 4 q + 28 q^{9} - 560 q^{21} + 360 q^{29} - 40 q^{41} - 2548 q^{49} + 1000 q^{61} - 3440 q^{69} - 1964 q^{81} - 3880 q^{89}+O(q^{100}) $ 4 * q + 28 * q^9 - 560 * q^21 + 360 * q^29 - 40 * q^41 - 2548 * q^49 + 1000 * q^61 - 3440 * q^69 - 1964 * q^81 - 3880 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	4.47214i	0.860663i	0.902671	+	0.430331i	$0.141603\pi$
$3$	4.47214i	0.860663i	−0.902671	+	0.430331i	$0.858397\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	31.3050i	1.69031i	0.534522	+	0.845154i	$0.320491\pi$
$7$	31.3050i	1.69031i	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	7.00000	0.259259
$10$	0	0
$11$	−8.94427	−0.245164	−0.122582	−	0.992458i	$-0.539117\pi$
$11$	−8.94427	−0.245164	−0.122582	+	0.992458i	$0.539117\pi$
$12$	0	0
$13$	− 62.0000i	− 1.32275i	−0.750057	−	0.661373i	$-0.769974\pi$
$13$	− 62.0000i	− 1.32275i	0.750057	−	0.661373i	$-0.230026\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	46.0000i	0.656273i	0.944630	+	0.328136i	$0.106421\pi$
$17$	46.0000i	0.656273i	−0.944630	+	0.328136i	$0.893579\pi$
$18$	0	0
$19$	−107.331	−1.29597	−0.647986	−	0.761652i	$-0.724389\pi$
$19$	−107.331	−1.29597	−0.647986	+	0.761652i	$0.724389\pi$
$20$	0	0
$21$	−140.000	−1.45479
$22$	0	0
$23$	192.302i	1.74338i	0.490059	+	0.871689i	$0.336975\pi$
$23$	192.302i	1.74338i	−0.490059	+	0.871689i	$0.663025\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	152.053i	1.08380i
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−152.053	−0.880950	−0.440475	−	0.897765i	$-0.645190\pi$
$31$	−152.053	−0.880950	−0.440475	+	0.897765i	$0.645190\pi$
$32$	0	0
$33$	− 40.0000i	− 0.211003i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	214.000i	0.950848i	0.879757	+	0.475424i	$0.157705\pi$
$37$	214.000i	0.950848i	−0.879757	+	0.475424i	$0.842295\pi$
$38$	0	0
$39$	277.272	1.13844
$40$	0	0
$41$	−10.0000	−0.0380912	−0.0190456	−	0.999819i	$-0.506063\pi$
$41$	−10.0000	−0.0380912	−0.0190456	+	0.999819i	$0.506063\pi$
$42$	0	0
$43$	− 67.0820i	− 0.237905i	−0.992900	−	0.118953i	$-0.962046\pi$
$43$	− 67.0820i	− 0.237905i	0.992900	−	0.118953i	$-0.0379536\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 398.020i	− 1.23526i	−0.786469	−	0.617630i	$-0.788093\pi$
$47$	− 398.020i	− 1.23526i	0.786469	−	0.617630i	$-0.211907\pi$
$48$	0	0
$49$	−637.000	−1.85714
$50$	0	0
$51$	−205.718	−0.564830
$52$	0	0
$53$	− 678.000i	− 1.75718i	−0.477578	−	0.878589i	$-0.658485\pi$
$53$	− 678.000i	− 1.75718i	0.477578	−	0.878589i	$-0.341515\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 480.000i	− 1.11540i
$58$	0	0
$59$	411.437	0.907872	0.453936	−	0.891034i	$-0.350019\pi$
$59$	411.437	0.907872	0.453936	+	0.891034i	$0.350019\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	219.135i	0.438228i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 49.1935i	− 0.0897006i	−0.998994	−	0.0448503i	$-0.985719\pi$
$67$	− 49.1935i	− 0.0897006i	0.998994	−	0.0448503i	$-0.0142811\pi$
$68$	0	0
$69$	−860.000	−1.50046
$70$	0	0
$71$	−366.715	−0.612973	−0.306486	−	0.951875i	$-0.599153\pi$
$71$	−366.715	−0.612973	−0.306486	+	0.951875i	$0.599153\pi$
$72$	0	0
$73$	522.000i	0.836924i	0.908234	+	0.418462i	$0.137431\pi$
$73$	522.000i	0.836924i	−0.908234	+	0.418462i	$0.862569\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 280.000i	− 0.414402i
$78$	0	0
$79$	−876.539	−1.24833	−0.624166	−	0.781291i	$-0.714561\pi$
$79$	−876.539	−1.24833	−0.624166	+	0.781291i	$0.714561\pi$
$80$	0	0
$81$	−491.000	−0.673525
$82$	0	0
$83$	380.132i	0.502709i	0.967895	+	0.251355i	$0.0808760\pi$
$83$	380.132i	0.502709i	−0.967895	+	0.251355i	$0.919124\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	402.492i	0.495997i
$88$	0	0
$89$	−970.000	−1.15528	−0.577639	−	0.816292i	$-0.696026\pi$
$89$	−970.000	−1.15528	−0.577639	+	0.816292i	$0.696026\pi$
$90$	0	0
$91$	1940.91	2.23585
$92$	0	0
$93$	− 680.000i	− 0.758201i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	934.000i	0.977663i	0.872378	+	0.488832i	$0.162577\pi$
$97$	934.000i	0.977663i	−0.872378	+	0.488832i	$0.837423\pi$
$98$	0	0
$99$	−62.6099	−0.0635609
$100$	0	0
$101$	−602.000	−0.593082	−0.296541	−	0.955020i	$-0.595833\pi$
$101$	−602.000	−0.593082	−0.296541	+	0.955020i	$0.595833\pi$
$102$	0	0
$103$	− 1829.10i	− 1.74978i	−0.484325	−	0.874888i	$-0.660935\pi$
$103$	− 1829.10i	− 1.74978i	0.484325	−	0.874888i	$-0.339065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1525.00i	1.37782i	0.724845	+	0.688912i	$0.241911\pi$
$107$	1525.00i	1.37782i	−0.724845	+	0.688912i	$0.758089\pi$
$108$	0	0
$109$	−2154.00	−1.89281	−0.946403	−	0.322989i	$-0.895312\pi$
$109$	−2154.00	−1.89281	−0.946403	+	0.322989i	$0.895312\pi$
$110$	0	0
$111$	−957.037	−0.818360
$112$	0	0
$113$	− 2182.00i	− 1.81651i	−0.418420	−	0.908254i	$-0.637416\pi$
$113$	− 2182.00i	− 1.81651i	0.418420	−	0.908254i	$-0.362584\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 434.000i	− 0.342934i
$118$	0	0
$119$	−1440.03	−1.10930
$120$	0	0
$121$	−1251.00	−0.939895
$122$	0	0
$123$	− 44.7214i	− 0.0327837i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1310.34i	− 0.915539i	−0.889071	−	0.457770i	$-0.848648\pi$
$127$	− 1310.34i	− 0.915539i	0.889071	−	0.457770i	$-0.151352\pi$
$128$	0	0
$129$	300.000	0.204756
$130$	0	0
$131$	−205.718	−0.137204	−0.0686019	−	0.997644i	$-0.521854\pi$
$131$	−205.718	−0.137204	−0.0686019	+	0.997644i	$0.521854\pi$
$132$	0	0
$133$	− 3360.00i	− 2.19059i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2094.00i	1.30586i	0.757419	+	0.652929i	$0.226460\pi$
$137$	2094.00i	1.30586i	−0.757419	+	0.652929i	$0.773540\pi$
$138$	0	0
$139$	1377.42	0.840511	0.420256	−	0.907406i	$-0.361940\pi$
$139$	1377.42	0.840511	0.420256	+	0.907406i	$0.361940\pi$
$140$	0	0
$141$	1780.00	1.06314
$142$	0	0
$143$	554.545i	0.324289i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2848.75i	− 1.59837i
$148$	0	0
$149$	334.000	0.183640	0.0918200	−	0.995776i	$-0.470732\pi$
$149$	334.000	0.183640	0.0918200	+	0.995776i	$0.470732\pi$
$150$	0	0
$151$	3139.44	1.69195	0.845973	−	0.533225i	$-0.179020\pi$
$151$	3139.44	1.69195	0.845973	+	0.533225i	$0.179020\pi$
$152$	0	0
$153$	322.000i	0.170145i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 834.000i	− 0.423952i	−0.977275	−	0.211976i	$-0.932010\pi$
$157$	− 834.000i	− 0.423952i	0.977275	−	0.211976i	$-0.0679898\pi$
$158$	0	0
$159$	3032.11	1.51234
$160$	0	0
$161$	−6020.00	−2.94685
$162$	0	0
$163$	− 3090.25i	− 1.48495i	−0.669874	−	0.742475i	$-0.733652\pi$
$163$	− 3090.25i	− 1.48495i	0.669874	−	0.742475i	$-0.266348\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.47214i	− 0.00207224i	−0.999999	−	0.00103612i	$-0.999670\pi$
$167$	− 4.47214i	− 0.00207224i	0.999999	−	0.00103612i	$-0.000329807\pi$
$168$	0	0
$169$	−1647.00	−0.749659
$170$	0	0
$171$	−751.319	−0.335993
$172$	0	0
$173$	− 1838.00i	− 0.807749i	−0.914814	−	0.403874i	$-0.867663\pi$
$173$	− 1838.00i	− 0.807749i	0.914814	−	0.403874i	$-0.132337\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1840.00i	0.781372i
$178$	0	0
$179$	−1842.52	−0.769365	−0.384683	−	0.923049i	$-0.625689\pi$
$179$	−1842.52	−0.769365	−0.384683	+	0.923049i	$0.625689\pi$
$180$	0	0
$181$	1862.00	0.764648	0.382324	−	0.924028i	$-0.375124\pi$
$181$	1862.00	0.764648	0.382324	+	0.924028i	$0.375124\pi$
$182$	0	0
$183$	1118.03i	0.451625i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 411.437i	− 0.160894i
$188$	0	0
$189$	−4760.00	−1.83195
$190$	0	0
$191$	−2066.13	−0.782721	−0.391360	−	0.920237i	$-0.627995\pi$
$191$	−2066.13	−0.782721	−0.391360	+	0.920237i	$0.627995\pi$
$192$	0	0
$193$	3378.00i	1.25986i	0.776650	+	0.629932i	$0.216917\pi$
$193$	3378.00i	1.25986i	−0.776650	+	0.629932i	$0.783083\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 66.0000i	− 0.0238696i	−0.999929	−	0.0119348i	$-0.996201\pi$
$197$	− 66.0000i	− 0.0238696i	0.999929	−	0.0119348i	$-0.00379905\pi$
$198$	0	0
$199$	1216.42	0.433316	0.216658	−	0.976248i	$-0.430484\pi$
$199$	1216.42	0.433316	0.216658	+	0.976248i	$0.430484\pi$
$200$	0	0
$201$	220.000	0.0772020
$202$	0	0
$203$	2817.45i	0.974118i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1346.11i	0.451987i
$208$	0	0
$209$	960.000	0.317725
$210$	0	0
$211$	−5286.06	−1.72468	−0.862341	−	0.506329i	$-0.831002\pi$
$211$	−5286.06	−1.72468	−0.862341	+	0.506329i	$0.831002\pi$
$212$	0	0
$213$	− 1640.00i	− 0.527563i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 4760.00i	− 1.48908i
$218$	0	0
$219$	−2334.45	−0.720310
$220$	0	0
$221$	2852.00	0.868083
$222$	0	0
$223$	2965.03i	0.890371i	0.895438	+	0.445186i	$0.146862\pi$
$223$	2965.03i	0.890371i	−0.895438	+	0.445186i	$0.853138\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4369.28i	1.27753i	0.769402	+	0.638765i	$0.220554\pi$
$227$	4369.28i	1.27753i	−0.769402	+	0.638765i	$0.779446\pi$
$228$	0	0
$229$	3250.00	0.937843	0.468921	−	0.883240i	$-0.344643\pi$
$229$	3250.00	0.937843	0.468921	+	0.883240i	$0.344643\pi$
$230$	0	0
$231$	1252.20	0.356661
$232$	0	0
$233$	3298.00i	0.927293i	0.886020	+	0.463646i	$0.153459\pi$
$233$	3298.00i	0.927293i	−0.886020	+	0.463646i	$0.846541\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3920.00i	− 1.07439i
$238$	0	0
$239$	554.545	0.150086	0.0750429	−	0.997180i	$-0.476091\pi$
$239$	554.545	0.150086	0.0750429	+	0.997180i	$0.476091\pi$
$240$	0	0
$241$	5150.00	1.37652	0.688259	−	0.725465i	$-0.258375\pi$
$241$	5150.00	1.37652	0.688259	+	0.725465i	$0.258375\pi$
$242$	0	0
$243$	1909.60i	0.504119i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6654.54i	1.71424i
$248$	0	0
$249$	−1700.00	−0.432663
$250$	0	0
$251$	1386.36	0.348631	0.174316	−	0.984690i	$-0.444229\pi$
$251$	1386.36	0.348631	0.174316	+	0.984690i	$0.444229\pi$
$252$	0	0
$253$	− 1720.00i	− 0.427413i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4166.00i	1.01116i	0.862780	+	0.505580i	$0.168721\pi$
$257$	4166.00i	1.01116i	−0.862780	+	0.505580i	$0.831279\pi$
$258$	0	0
$259$	−6699.26	−1.60723
$260$	0	0
$261$	630.000	0.149410
$262$	0	0
$263$	961.509i	0.225434i	0.993627	+	0.112717i	$0.0359554\pi$
$263$	961.509i	0.225434i	−0.993627	+	0.112717i	$0.964045\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4337.97i	− 0.994305i
$268$	0	0
$269$	1494.00	0.338627	0.169314	−	0.985562i	$-0.445845\pi$
$269$	1494.00	0.338627	0.169314	+	0.985562i	$0.445845\pi$
$270$	0	0
$271$	5017.74	1.12474	0.562372	−	0.826884i	$-0.309889\pi$
$271$	5017.74	1.12474	0.562372	+	0.826884i	$0.309889\pi$
$272$	0	0
$273$	8680.00i	1.92431i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1006.00i	0.218212i	0.994030	+	0.109106i	$0.0347988\pi$
$277$	1006.00i	0.218212i	−0.994030	+	0.109106i	$0.965201\pi$
$278$	0	0
$279$	−1064.37	−0.228395
$280$	0	0
$281$	−3210.00	−0.681468	−0.340734	−	0.940160i	$-0.610676\pi$
$281$	−3210.00	−0.681468	−0.340734	+	0.940160i	$0.610676\pi$
$282$	0	0
$283$	3635.85i	0.763705i	0.924223	+	0.381853i	$0.124714\pi$
$283$	3635.85i	0.763705i	−0.924223	+	0.381853i	$0.875286\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 313.050i	− 0.0643858i
$288$	0	0
$289$	2797.00	0.569306
$290$	0	0
$291$	−4176.97	−0.841439
$292$	0	0
$293$	− 3622.00i	− 0.722183i	−0.932530	−	0.361091i	$-0.882404\pi$
$293$	− 3622.00i	− 0.722183i	0.932530	−	0.361091i	$-0.117596\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1360.00i	− 0.265708i
$298$	0	0
$299$	11922.7	2.30605
$300$	0	0
$301$	2100.00	0.402133
$302$	0	0
$303$	− 2692.23i	− 0.510443i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2088.49i	− 0.388261i	−0.980976	−	0.194131i	$-0.937811\pi$
$307$	− 2088.49i	− 0.388261i	0.980976	−	0.194131i	$-0.0621886\pi$
$308$	0	0
$309$	8180.00	1.50597
$310$	0	0
$311$	−8899.55	−1.62266	−0.811330	−	0.584589i	$-0.801256\pi$
$311$	−8899.55	−1.62266	−0.811330	+	0.584589i	$0.801256\pi$
$312$	0	0
$313$	8778.00i	1.58518i	0.609754	+	0.792591i	$0.291268\pi$
$313$	8778.00i	1.58518i	−0.609754	+	0.792591i	$0.708732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5046.00i	0.894043i	0.894523	+	0.447021i	$0.147515\pi$
$317$	5046.00i	0.894043i	−0.894523	+	0.447021i	$0.852485\pi$
$318$	0	0
$319$	−804.984	−0.141287
$320$	0	0
$321$	−6820.00	−1.18584
$322$	0	0
$323$	− 4937.24i	− 0.850512i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9632.98i	− 1.62907i
$328$	0	0
$329$	12460.0	2.08797
$330$	0	0
$331$	−313.050	−0.0519842	−0.0259921	−	0.999662i	$-0.508274\pi$
$331$	−313.050	−0.0519842	−0.0259921	+	0.999662i	$0.508274\pi$
$332$	0	0
$333$	1498.00i	0.246516i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2574.00i	0.416067i	0.978122	+	0.208034i	$0.0667064\pi$
$337$	2574.00i	0.416067i	−0.978122	+	0.208034i	$0.933294\pi$
$338$	0	0
$339$	9758.20	1.56340
$340$	0	0
$341$	1360.00	0.215977
$342$	0	0
$343$	− 9203.66i	− 1.44884i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2643.03i	− 0.408892i	−0.978878	−	0.204446i	$-0.934461\pi$
$347$	− 2643.03i	− 0.408892i	0.978878	−	0.204446i	$-0.0655392\pi$
$348$	0	0
$349$	10170.0	1.55985	0.779925	−	0.625873i	$-0.215257\pi$
$349$	10170.0	1.55985	0.779925	+	0.625873i	$0.215257\pi$
$350$	0	0
$351$	9427.26	1.43359
$352$	0	0
$353$	− 318.000i	− 0.0479474i	−0.999713	−	0.0239737i	$-0.992368\pi$
$353$	− 318.000i	− 0.0479474i	0.999713	−	0.0239737i	$-0.00763180\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6440.00i	− 0.954737i
$358$	0	0
$359$	−12378.9	−1.81987	−0.909933	−	0.414755i	$-0.863867\pi$
$359$	−12378.9	−1.81987	−0.909933	+	0.414755i	$0.863867\pi$
$360$	0	0
$361$	4661.00	0.679545
$362$	0	0
$363$	− 5594.64i	− 0.808933i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3072.36i	0.436991i	0.975838	+	0.218496i	$0.0701149\pi$
$367$	3072.36i	0.436991i	−0.975838	+	0.218496i	$0.929885\pi$
$368$	0	0
$369$	−70.0000	−0.00987549
$370$	0	0
$371$	21224.8	2.97017
$372$	0	0
$373$	− 3278.00i	− 0.455036i	−0.973774	−	0.227518i	$-0.926939\pi$
$373$	− 3278.00i	− 0.455036i	0.973774	−	0.227518i	$-0.0730610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5580.00i	− 0.762293i
$378$	0	0
$379$	−5116.12	−0.693397	−0.346699	−	0.937977i	$-0.612697\pi$
$379$	−5116.12	−0.693397	−0.346699	+	0.937977i	$0.612697\pi$
$380$	0	0
$381$	5860.00	0.787971
$382$	0	0
$383$	− 1149.34i	− 0.153338i	−0.997057	−	0.0766690i	$-0.975572\pi$
$383$	− 1149.34i	− 0.153338i	0.997057	−	0.0766690i	$-0.0244285\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 469.574i	− 0.0616791i
$388$	0	0
$389$	−834.000	−0.108703	−0.0543515	−	0.998522i	$-0.517309\pi$
$389$	−834.000	−0.108703	−0.0543515	+	0.998522i	$0.517309\pi$
$390$	0	0
$391$	−8845.88	−1.14413
$392$	0	0
$393$	− 920.000i	− 0.118086i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8734.00i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	8734.00i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	15026.4	1.88536
$400$	0	0
$401$	242.000	0.0301369	0.0150685	−	0.999886i	$-0.495203\pi$
$401$	242.000	0.0301369	0.0150685	+	0.999886i	$0.495203\pi$
$402$	0	0
$403$	9427.26i	1.16527i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1914.07i	− 0.233113i
$408$	0	0
$409$	6514.00	0.787522	0.393761	−	0.919213i	$-0.371174\pi$
$409$	6514.00	0.787522	0.393761	+	0.919213i	$0.371174\pi$
$410$	0	0
$411$	−9364.65	−1.12390
$412$	0	0
$413$	12880.0i	1.53458i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6160.00i	0.723397i
$418$	0	0
$419$	−16081.8	−1.87505	−0.937527	−	0.347913i	$-0.886890\pi$
$419$	−16081.8	−1.87505	−0.937527	+	0.347913i	$0.886890\pi$
$420$	0	0
$421$	7250.00	0.839295	0.419648	−	0.907687i	$-0.362154\pi$
$421$	7250.00	0.839295	0.419648	+	0.907687i	$0.362154\pi$
$422$	0	0
$423$	− 2786.14i	− 0.320252i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7826.24i	0.886975i
$428$	0	0
$429$	−2480.00	−0.279104
$430$	0	0
$431$	4981.96	0.556781	0.278390	−	0.960468i	$-0.410199\pi$
$431$	4981.96	0.556781	0.278390	+	0.960468i	$0.410199\pi$
$432$	0	0
$433$	11482.0i	1.27434i	0.770723	+	0.637171i	$0.219895\pi$
$433$	11482.0i	1.27434i	−0.770723	+	0.637171i	$0.780105\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20640.0i	− 2.25937i
$438$	0	0
$439$	−3792.37	−0.412301	−0.206150	−	0.978520i	$-0.566094\pi$
$439$	−3792.37	−0.412301	−0.206150	+	0.978520i	$0.566094\pi$
$440$	0	0
$441$	−4459.00	−0.481481
$442$	0	0
$443$	− 746.847i	− 0.0800988i	−0.999198	−	0.0400494i	$-0.987248\pi$
$443$	− 746.847i	− 0.0800988i	0.999198	−	0.0400494i	$-0.0127515\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1493.69i	0.158052i
$448$	0	0
$449$	1306.00	0.137269	0.0686347	−	0.997642i	$-0.478136\pi$
$449$	1306.00	0.137269	0.0686347	+	0.997642i	$0.478136\pi$
$450$	0	0
$451$	89.4427	0.00933857
$452$	0	0
$453$	14040.0i	1.45620i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9526.00i	0.975071i	0.873103	+	0.487536i	$0.162104\pi$
$457$	9526.00i	0.975071i	−0.873103	+	0.487536i	$0.837896\pi$
$458$	0	0
$459$	−6994.42	−0.711267
$460$	0	0
$461$	1518.00	0.153363	0.0766815	−	0.997056i	$-0.475568\pi$
$461$	1518.00	0.153363	0.0766815	+	0.997056i	$0.475568\pi$
$462$	0	0
$463$	17293.7i	1.73587i	0.496676	+	0.867936i	$0.334554\pi$
$463$	17293.7i	1.73587i	−0.496676	+	0.867936i	$0.665446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16980.7i	1.68260i	0.540570	+	0.841299i	$0.318208\pi$
$467$	16980.7i	1.68260i	−0.540570	+	0.841299i	$0.681792\pi$
$468$	0	0
$469$	1540.00	0.151622
$470$	0	0
$471$	3729.76	0.364880
$472$	0	0
$473$	600.000i	0.0583256i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4746.00i	− 0.455565i
$478$	0	0
$479$	−3810.26	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$479$	−3810.26	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$480$	0	0
$481$	13268.0	1.25773
$482$	0	0
$483$	− 26922.3i	− 2.53624i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1310.34i	− 0.121924i	−0.998140	−	0.0609620i	$-0.980583\pi$
$487$	− 1310.34i	− 0.121924i	0.998140	−	0.0609620i	$-0.0194168\pi$
$488$	0	0
$489$	13820.0	1.27804
$490$	0	0
$491$	2960.55	0.272114	0.136057	−	0.990701i	$-0.456557\pi$
$491$	2960.55	0.272114	0.136057	+	0.990701i	$0.456557\pi$
$492$	0	0
$493$	4140.00i	0.378207i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11480.0i	− 1.03611i
$498$	0	0
$499$	19319.6	1.73320	0.866598	−	0.499006i	$-0.166301\pi$
$499$	19319.6	1.73320	0.866598	+	0.499006i	$0.166301\pi$
$500$	0	0
$501$	20.0000	0.00178350
$502$	0	0
$503$	3072.36i	0.272345i	0.990685	+	0.136173i	$0.0434802\pi$
$503$	3072.36i	0.272345i	−0.990685	+	0.136173i	$0.956520\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7365.61i	− 0.645203i
$508$	0	0
$509$	−18550.0	−1.61535	−0.807676	−	0.589626i	$-0.799275\pi$
$509$	−18550.0	−1.61535	−0.807676	+	0.589626i	$0.799275\pi$
$510$	0	0
$511$	−16341.2	−1.41466
$512$	0	0
$513$	− 16320.0i	− 1.40457i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3560.00i	0.302841i
$518$	0	0
$519$	8219.79	0.695200
$520$	0	0
$521$	−2102.00	−0.176757	−0.0883784	−	0.996087i	$-0.528168\pi$
$521$	−2102.00	−0.176757	−0.0883784	+	0.996087i	$0.528168\pi$
$522$	0	0
$523$	17696.2i	1.47955i	0.672856	+	0.739773i	$0.265067\pi$
$523$	17696.2i	1.47955i	−0.672856	+	0.739773i	$0.734933\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6994.42i	− 0.578144i
$528$	0	0
$529$	−24813.0	−2.03937
$530$	0	0
$531$	2880.06	0.235374
$532$	0	0
$533$	620.000i	0.0503850i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8240.00i	− 0.662164i
$538$	0	0
$539$	5697.50	0.455304
$540$	0	0
$541$	−9922.00	−0.788503	−0.394251	−	0.919003i	$-0.628996\pi$
$541$	−9922.00	−0.788503	−0.394251	+	0.919003i	$0.628996\pi$
$542$	0	0
$543$	8327.12i	0.658105i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3716.34i	− 0.290493i	−0.989396	−	0.145246i	$-0.953603\pi$
$547$	− 3716.34i	− 0.290493i	0.989396	−	0.145246i	$-0.0463975\pi$
$548$	0	0
$549$	1750.00	0.136044
$550$	0	0
$551$	−9659.81	−0.746864
$552$	0	0
$553$	− 27440.0i	− 2.11007i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15094.0i	1.14821i	0.818781	+	0.574105i	$0.194650\pi$
$557$	15094.0i	1.14821i	−0.818781	+	0.574105i	$0.805350\pi$
$558$	0	0
$559$	−4159.09	−0.314688
$560$	0	0
$561$	1840.00	0.138476
$562$	0	0
$563$	5657.25i	0.423490i	0.977325	+	0.211745i	$0.0679146\pi$
$563$	5657.25i	0.423490i	−0.977325	+	0.211745i	$0.932085\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 15370.7i	− 1.13847i
$568$	0	0
$569$	5906.00	0.435136	0.217568	−	0.976045i	$-0.430188\pi$
$569$	5906.00	0.435136	0.217568	+	0.976045i	$0.430188\pi$
$570$	0	0
$571$	−4892.52	−0.358573	−0.179287	−	0.983797i	$-0.557379\pi$
$571$	−4892.52	−0.358573	−0.179287	+	0.983797i	$0.557379\pi$
$572$	0	0
$573$	− 9240.00i	− 0.673659i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13286.0i	0.958585i	0.877655	+	0.479292i	$0.159107\pi$
$577$	13286.0i	0.958585i	−0.877655	+	0.479292i	$0.840893\pi$
$578$	0	0
$579$	−15106.9	−1.08432
$580$	0	0
$581$	−11900.0	−0.849734
$582$	0	0
$583$	6064.22i	0.430796i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9029.24i	− 0.634884i	−0.948278	−	0.317442i	$-0.897176\pi$
$587$	− 9029.24i	− 0.634884i	0.948278	−	0.317442i	$-0.102824\pi$
$588$	0	0
$589$	16320.0	1.14169
$590$	0	0
$591$	295.161	0.0205437
$592$	0	0
$593$	11442.0i	0.792355i	0.918174	+	0.396178i	$0.129664\pi$
$593$	11442.0i	0.792355i	−0.918174	+	0.396178i	$0.870336\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5440.00i	0.372939i
$598$	0	0
$599$	14149.8	0.965187	0.482593	−	0.875845i	$-0.339695\pi$
$599$	14149.8	0.965187	0.482593	+	0.875845i	$0.339695\pi$
$600$	0	0
$601$	3110.00	0.211081	0.105540	−	0.994415i	$-0.466343\pi$
$601$	3110.00	0.211081	0.105540	+	0.994415i	$0.466343\pi$
$602$	0	0
$603$	− 344.354i	− 0.0232557i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11193.8i	0.748502i	0.927327	+	0.374251i	$0.122100\pi$
$607$	11193.8i	0.748502i	−0.927327	+	0.374251i	$0.877900\pi$
$608$	0	0
$609$	−12600.0	−0.838387
$610$	0	0
$611$	−24677.2	−1.63394
$612$	0	0
$613$	− 5342.00i	− 0.351976i	−0.984392	−	0.175988i	$-0.943688\pi$
$613$	− 5342.00i	− 0.351976i	0.984392	−	0.175988i	$-0.0563120\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19714.0i	− 1.28631i	−0.765734	−	0.643157i	$-0.777624\pi$
$617$	− 19714.0i	− 1.28631i	0.765734	−	0.643157i	$-0.222376\pi$
$618$	0	0
$619$	13166.0	0.854903	0.427451	−	0.904038i	$-0.359411\pi$
$619$	13166.0	0.854903	0.427451	+	0.904038i	$0.359411\pi$
$620$	0	0
$621$	−29240.0	−1.88947
$622$	0	0
$623$	− 30365.8i	− 1.95278i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4293.25i	0.273454i
$628$	0	0
$629$	−9844.00	−0.624016
$630$	0	0
$631$	12262.6	0.773639	0.386820	−	0.922155i	$-0.373574\pi$
$631$	12262.6	0.773639	0.386820	+	0.922155i	$0.373574\pi$
$632$	0	0
$633$	− 23640.0i	− 1.48437i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	39494.0i	2.45653i
$638$	0	0
$639$	−2567.01	−0.158919
$640$	0	0
$641$	−2690.00	−0.165754	−0.0828772	−	0.996560i	$-0.526411\pi$
$641$	−2690.00	−0.165754	−0.0828772	+	0.996560i	$0.526411\pi$
$642$	0	0
$643$	12240.2i	0.750712i	0.926881	+	0.375356i	$0.122480\pi$
$643$	12240.2i	0.750712i	−0.926881	+	0.375356i	$0.877520\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17973.5i	1.09214i	0.837741	+	0.546068i	$0.183876\pi$
$647$	17973.5i	1.09214i	−0.837741	+	0.546068i	$0.816124\pi$
$648$	0	0
$649$	−3680.00	−0.222577
$650$	0	0
$651$	21287.4	1.28159
$652$	0	0
$653$	− 3478.00i	− 0.208430i	−0.994555	−	0.104215i	$-0.966767\pi$
$653$	− 3478.00i	− 0.208430i	0.994555	−	0.104215i	$-0.0332330\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3654.00i	0.216980i
$658$	0	0
$659$	10572.1	0.624934	0.312467	−	0.949929i	$-0.398845\pi$
$659$	10572.1	0.624934	0.312467	+	0.949929i	$0.398845\pi$
$660$	0	0
$661$	−110.000	−0.00647277	−0.00323639	−	0.999995i	$-0.501030\pi$
$661$	−110.000	−0.00647277	−0.00323639	+	0.999995i	$0.501030\pi$
$662$	0	0
$663$	12754.5i	0.747127i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17307.2i	1.00470i
$668$	0	0
$669$	−13260.0	−0.766310
$670$	0	0
$671$	−2236.07	−0.128647
$672$	0	0
$673$	− 14278.0i	− 0.817796i	−0.912580	−	0.408898i	$-0.865913\pi$
$673$	− 14278.0i	− 0.817796i	0.912580	−	0.408898i	$-0.134087\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18386.0i	− 1.04377i	−0.853016	−	0.521884i	$-0.825229\pi$
$677$	− 18386.0i	− 1.04377i	0.853016	−	0.521884i	$-0.174771\pi$
$678$	0	0
$679$	−29238.8	−1.65255
$680$	0	0
$681$	−19540.0	−1.09952
$682$	0	0
$683$	15317.1i	0.858113i	0.903278	+	0.429057i	$0.141154\pi$
$683$	15317.1i	0.858113i	−0.903278	+	0.429057i	$0.858846\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14534.4i	0.807167i
$688$	0	0
$689$	−42036.0	−2.32430
$690$	0	0
$691$	9507.76	0.523433	0.261717	−	0.965145i	$-0.415711\pi$
$691$	9507.76	0.523433	0.261717	+	0.965145i	$0.415711\pi$
$692$	0	0
$693$	− 1960.00i	− 0.107438i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 460.000i	− 0.0249982i
$698$	0	0
$699$	−14749.1	−0.798086
$700$	0	0
$701$	−15830.0	−0.852911	−0.426456	−	0.904508i	$-0.640238\pi$
$701$	−15830.0	−0.852911	−0.426456	+	0.904508i	$0.640238\pi$
$702$	0	0
$703$	− 22968.9i	− 1.23227i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18845.6i	− 1.00249i
$708$	0	0
$709$	20050.0	1.06205	0.531025	−	0.847356i	$-0.321807\pi$
$709$	20050.0	1.06205	0.531025	+	0.847356i	$0.321807\pi$
$710$	0	0
$711$	−6135.77	−0.323642
$712$	0	0
$713$	− 29240.0i	− 1.53583i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2480.00i	0.129173i
$718$	0	0
$719$	−21126.4	−1.09580	−0.547900	−	0.836544i	$-0.684573\pi$
$719$	−21126.4	−1.09580	−0.547900	+	0.836544i	$0.684573\pi$
$720$	0	0
$721$	57260.0	2.95766
$722$	0	0
$723$	23031.5i	1.18472i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11336.9i	0.578351i	0.957276	+	0.289175i	$0.0933811\pi$
$727$	11336.9i	0.578351i	−0.957276	+	0.289175i	$0.906619\pi$
$728$	0	0
$729$	−21797.0	−1.10740
$730$	0	0
$731$	3085.77	0.156131
$732$	0	0
$733$	− 17198.0i	− 0.866607i	−0.901248	−	0.433303i	$-0.857348\pi$
$733$	− 17198.0i	− 0.866607i	0.901248	−	0.433303i	$-0.142652\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	440.000i	0.0219913i
$738$	0	0
$739$	−4597.36	−0.228845	−0.114423	−	0.993432i	$-0.536502\pi$
$739$	−4597.36	−0.228845	−0.114423	+	0.993432i	$0.536502\pi$
$740$	0	0
$741$	−29760.0	−1.47539
$742$	0	0
$743$	− 2419.43i	− 0.119462i	−0.998215	−	0.0597309i	$-0.980976\pi$
$743$	− 2419.43i	− 0.119462i	0.998215	−	0.0597309i	$-0.0190243\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2660.92i	0.130332i
$748$	0	0
$749$	−47740.0	−2.32895
$750$	0	0
$751$	7432.69	0.361149	0.180574	−	0.983561i	$-0.442204\pi$
$751$	7432.69	0.361149	0.180574	+	0.983561i	$0.442204\pi$
$752$	0	0
$753$	6200.00i	0.300054i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11474.0i	− 0.550898i	−0.961316	−	0.275449i	$-0.911174\pi$
$757$	− 11474.0i	− 0.550898i	0.961316	−	0.275449i	$-0.0888265\pi$
$758$	0	0
$759$	7692.07	0.367858
$760$	0	0
$761$	31802.0	1.51488	0.757439	−	0.652906i	$-0.226450\pi$
$761$	31802.0	1.51488	0.757439	+	0.652906i	$0.226450\pi$
$762$	0	0
$763$	− 67430.9i	− 3.19942i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 25509.1i	− 1.20089i
$768$	0	0
$769$	5310.00	0.249003	0.124502	−	0.992219i	$-0.460267\pi$
$769$	5310.00	0.249003	0.124502	+	0.992219i	$0.460267\pi$
$770$	0	0
$771$	−18630.9	−0.870267
$772$	0	0
$773$	37938.0i	1.76525i	0.470082	+	0.882623i	$0.344224\pi$
$773$	37938.0i	1.76525i	−0.470082	+	0.882623i	$0.655776\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 29960.0i	− 1.38328i
$778$	0	0
$779$	1073.31	0.0493651
$780$	0	0
$781$	3280.00	0.150279
$782$	0	0
$783$	13684.7i	0.624588i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 37633.0i	− 1.70454i	−0.523103	−	0.852270i	$-0.675226\pi$
$787$	− 37633.0i	− 1.70454i	0.523103	−	0.852270i	$-0.324774\pi$
$788$	0	0
$789$	−4300.00	−0.194023
$790$	0	0
$791$	68307.4	3.07046
$792$	0	0
$793$	− 15500.0i	− 0.694100i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17526.0i	0.778924i	0.921042	+	0.389462i	$0.127339\pi$
$797$	17526.0i	0.778924i	−0.921042	+	0.389462i	$0.872661\pi$
$798$	0	0
$799$	18308.9	0.810667
$800$	0	0
$801$	−6790.00	−0.299517
$802$	0	0
$803$	− 4668.91i	− 0.205183i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6681.37i	0.291444i
$808$	0	0
$809$	−8970.00	−0.389825	−0.194912	−	0.980821i	$-0.562442\pi$
$809$	−8970.00	−0.389825	−0.194912	+	0.980821i	$0.562442\pi$
$810$	0	0
$811$	−3550.88	−0.153746	−0.0768731	−	0.997041i	$-0.524494\pi$
$811$	−3550.88	−0.153746	−0.0768731	+	0.997041i	$0.524494\pi$
$812$	0	0
$813$	22440.0i	0.968026i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7200.00i	0.308318i
$818$	0	0
$819$	13586.3	0.579665
$820$	0	0
$821$	−15550.0	−0.661022	−0.330511	−	0.943802i	$-0.607221\pi$
$821$	−15550.0	−0.661022	−0.330511	+	0.943802i	$0.607221\pi$
$822$	0	0
$823$	− 26712.1i	− 1.13138i	−0.824619	−	0.565689i	$-0.808610\pi$
$823$	− 26712.1i	− 1.13138i	0.824619	−	0.565689i	$-0.191390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	863.122i	0.0362923i	0.999835	+	0.0181461i	$0.00577641\pi$
$827$	863.122i	0.0362923i	−0.999835	+	0.0181461i	$0.994224\pi$
$828$	0	0
$829$	−19066.0	−0.798781	−0.399391	−	0.916781i	$-0.630778\pi$
$829$	−19066.0	−0.798781	−0.399391	+	0.916781i	$0.630778\pi$
$830$	0	0
$831$	−4498.97	−0.187807
$832$	0	0
$833$	− 29302.0i	− 1.21879i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 23120.0i	− 0.954772i
$838$	0	0
$839$	47744.5	1.96463	0.982315	−	0.187238i	$-0.0599534\pi$
$839$	47744.5	1.96463	0.982315	+	0.187238i	$0.0599534\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	− 14355.6i	− 0.586514i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 39162.5i	− 1.58871i
$848$	0	0
$849$	−16260.0	−0.657293
$850$	0	0
$851$	−41152.6	−1.65769
$852$	0	0
$853$	− 14462.0i	− 0.580503i	−0.956950	−	0.290252i	$-0.906261\pi$
$853$	− 14462.0i	− 0.580503i	0.956950	−	0.290252i	$-0.0937390\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 29346.0i	− 1.16971i	−0.811138	−	0.584854i	$-0.801152\pi$
$857$	− 29346.0i	− 1.16971i	0.811138	−	0.584854i	$-0.198848\pi$
$858$	0	0
$859$	−22807.9	−0.905932	−0.452966	−	0.891528i	$-0.649634\pi$
$859$	−22807.9	−0.905932	−0.452966	+	0.891528i	$0.649634\pi$
$860$	0	0
$861$	1400.00	0.0554145
$862$	0	0
$863$	24753.3i	0.976375i	0.872739	+	0.488187i	$0.162342\pi$
$863$	24753.3i	0.976375i	−0.872739	+	0.488187i	$0.837658\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12508.6i	0.489981i
$868$	0	0
$869$	7840.00	0.306046
$870$	0	0
$871$	−3050.00	−0.118651
$872$	0	0
$873$	6538.00i	0.253468i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32126.0i	1.23696i	0.785799	+	0.618482i	$0.212252\pi$
$877$	32126.0i	1.23696i	−0.785799	+	0.618482i	$0.787748\pi$
$878$	0	0
$879$	16198.1	0.621556
$880$	0	0
$881$	−33570.0	−1.28377	−0.641885	−	0.766801i	$-0.721848\pi$
$881$	−33570.0	−1.28377	−0.641885	+	0.766801i	$0.721848\pi$
$882$	0	0
$883$	− 6435.40i	− 0.245265i	−0.992452	−	0.122632i	$-0.960866\pi$
$883$	− 6435.40i	− 0.245265i	0.992452	−	0.122632i	$-0.0391336\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	46827.7i	1.77263i	0.463084	+	0.886314i	$0.346743\pi$
$887$	46827.7i	1.77263i	−0.463084	+	0.886314i	$0.653257\pi$
$888$	0	0
$889$	41020.0	1.54754
$890$	0	0
$891$	4391.64	0.165124
$892$	0	0
$893$	42720.0i	1.60086i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	53320.0i	1.98473i
$898$	0	0
$899$	−13684.7	−0.507688
$900$	0	0
$901$	31188.0	1.15319
$902$	0	0
$903$	9391.49i	0.346101i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 11980.9i	− 0.438608i	−0.975657	−	0.219304i	$-0.929621\pi$
$907$	− 11980.9i	− 0.438608i	0.975657	−	0.219304i	$-0.0703787\pi$
$908$	0	0
$909$	−4214.00	−0.153762
$910$	0	0
$911$	−24194.3	−0.879903	−0.439951	−	0.898022i	$-0.645004\pi$
$911$	−24194.3	−0.879903	−0.439951	+	0.898022i	$0.645004\pi$
$912$	0	0
$913$	− 3400.00i	− 0.123246i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6440.00i	− 0.231917i
$918$	0	0
$919$	37512.3	1.34648	0.673240	−	0.739424i	$-0.264902\pi$
$919$	37512.3	1.34648	0.673240	+	0.739424i	$0.264902\pi$
$920$	0	0
$921$	9340.00	0.334162
$922$	0	0
$923$	22736.3i	0.810808i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12803.7i	− 0.453646i
$928$	0	0
$929$	21994.0	0.776749	0.388374	−	0.921502i	$-0.373037\pi$
$929$	21994.0	0.776749	0.388374	+	0.921502i	$0.373037\pi$
$930$	0	0
$931$	68370.0	2.40681
$932$	0	0
$933$	− 39800.0i	− 1.39656i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16286.0i	0.567813i	0.958852	+	0.283906i	$0.0916305\pi$
$937$	16286.0i	0.567813i	−0.958852	+	0.283906i	$0.908370\pi$
$938$	0	0
$939$	−39256.4	−1.36431
$940$	0	0
$941$	24302.0	0.841894	0.420947	−	0.907085i	$-0.361698\pi$
$941$	24302.0	0.841894	0.420947	+	0.907085i	$0.361698\pi$
$942$	0	0
$943$	− 1923.02i	− 0.0664073i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 19869.7i	− 0.681815i	−0.940097	−	0.340907i	$-0.889266\pi$
$947$	− 19869.7i	− 0.681815i	0.940097	−	0.340907i	$-0.110734\pi$
$948$	0	0
$949$	32364.0	1.10704
$950$	0	0
$951$	−22566.4	−0.769470
$952$	0	0
$953$	− 22422.0i	− 0.762140i	−0.924546	−	0.381070i	$-0.875556\pi$
$953$	− 22422.0i	− 0.762140i	0.924546	−	0.381070i	$-0.124444\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3600.00i	− 0.121600i
$958$	0	0
$959$	−65552.6	−2.20730
$960$	0	0
$961$	−6671.00	−0.223927
$962$	0	0
$963$	10675.0i	0.357214i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43777.7i	− 1.45584i	−0.685662	−	0.727920i	$-0.740487\pi$
$967$	− 43777.7i	− 1.45584i	0.685662	−	0.727920i	$-0.259513\pi$
$968$	0	0
$969$	22080.0	0.732004
$970$	0	0
$971$	−25714.8	−0.849873	−0.424936	−	0.905223i	$-0.639704\pi$
$971$	−25714.8	−0.849873	−0.424936	+	0.905223i	$0.639704\pi$
$972$	0	0
$973$	43120.0i	1.42072i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 28986.0i	− 0.949175i	−0.880208	−	0.474588i	$-0.842597\pi$
$977$	− 28986.0i	− 0.949175i	0.880208	−	0.474588i	$-0.157403\pi$
$978$	0	0
$979$	8675.94	0.283232
$980$	0	0
$981$	−15078.0	−0.490727
$982$	0	0
$983$	32123.4i	1.04229i	0.853467	+	0.521147i	$0.174496\pi$
$983$	32123.4i	1.04229i	−0.853467	+	0.521147i	$0.825504\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	55722.8i	1.79704i
$988$	0	0
$989$	12900.0	0.414758
$990$	0	0
$991$	−11994.3	−0.384471	−0.192235	−	0.981349i	$-0.561574\pi$
$991$	−11994.3	−0.384471	−0.192235	+	0.981349i	$0.561574\pi$
$992$	0	0
$993$	− 1400.00i	− 0.0447408i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	406.000i	0.0128968i	0.999979	+	0.00644842i	$0.00205261\pi$
$997$	406.000i	0.0128968i	−0.999979	+	0.00644842i	$0.997947\pi$
$998$	0	0
$999$	−32539.3	−1.03053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.l.449.3		4
4.3	odd	2	inner	800.4.c.l.449.2		4
5.2	odd	4		160.4.a.d.1.2	yes	2
5.3	odd	4		800.4.a.o.1.1		2
5.4	even	2	inner	800.4.c.l.449.1		4
15.2	even	4		1440.4.a.bb.1.1		2
20.3	even	4		800.4.a.o.1.2		2
20.7	even	4		160.4.a.d.1.1	✓	2
20.19	odd	2	inner	800.4.c.l.449.4		4
40.3	even	4		1600.4.a.cg.1.1		2
40.13	odd	4		1600.4.a.cg.1.2		2
40.27	even	4		320.4.a.q.1.2		2
40.37	odd	4		320.4.a.q.1.1		2
60.47	odd	4		1440.4.a.bb.1.2		2
80.27	even	4		1280.4.d.v.641.3		4
80.37	odd	4		1280.4.d.v.641.1		4
80.67	even	4		1280.4.d.v.641.2		4
80.77	odd	4		1280.4.d.v.641.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.d.1.1	✓	2	20.7	even	4
160.4.a.d.1.2	yes	2	5.2	odd	4
320.4.a.q.1.1		2	40.37	odd	4
320.4.a.q.1.2		2	40.27	even	4
800.4.a.o.1.1		2	5.3	odd	4
800.4.a.o.1.2		2	20.3	even	4
800.4.c.l.449.1		4	5.4	even	2	inner
800.4.c.l.449.2		4	4.3	odd	2	inner
800.4.c.l.449.3		4	1.1	even	1	trivial
800.4.c.l.449.4		4	20.19	odd	2	inner
1280.4.d.v.641.1		4	80.37	odd	4
1280.4.d.v.641.2		4	80.67	even	4
1280.4.d.v.641.3		4	80.27	even	4
1280.4.d.v.641.4		4	80.77	odd	4
1440.4.a.bb.1.1		2	15.2	even	4
1440.4.a.bb.1.2		2	60.47	odd	4
1600.4.a.cg.1.1		2	40.3	even	4
1600.4.a.cg.1.2		2	40.13	odd	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.47214i q^{3} +31.3050i q^{7} +7.00000 q^{9} +O(q^{10})\)	\(q+4.47214i q^{3} +31.3050i q^{7} +7.00000 q^{9} -8.94427 q^{11} -62.0000i q^{13} +46.0000i q^{17} -107.331 q^{19} -140.000 q^{21} +192.302i q^{23} +152.053i q^{27} +90.0000 q^{29} -152.053 q^{31} -40.0000i q^{33} +214.000i q^{37} +277.272 q^{39} -10.0000 q^{41} -67.0820i q^{43} -398.020i q^{47} -637.000 q^{49} -205.718 q^{51} -678.000i q^{53} -480.000i q^{57} +411.437 q^{59} +250.000 q^{61} +219.135i q^{63} -49.1935i q^{67} -860.000 q^{69} -366.715 q^{71} +522.000i q^{73} -280.000i q^{77} -876.539 q^{79} -491.000 q^{81} +380.132i q^{83} +402.492i q^{87} -970.000 q^{89} +1940.91 q^{91} -680.000i q^{93} +934.000i q^{97} -62.6099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{9}+O(q^{10}) \) 4 * q + 28 * q^9	\( 4 q + 28 q^{9} - 560 q^{21} + 360 q^{29} - 40 q^{41} - 2548 q^{49} + 1000 q^{61} - 3440 q^{69} - 1964 q^{81} - 3880 q^{89}+O(q^{100}) \) 4 * q + 28 * q^9 - 560 * q^21 + 360 * q^29 - 40 * q^41 - 2548 * q^49 + 1000 * q^61 - 3440 * q^69 - 1964 * q^81 - 3880 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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