LMFDB - Embedded newform 800.4.c.c.449.1

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:	$1$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.4.c.c.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{3} -10.0000i q^{7} +2.00000 q^{9} -15.0000 q^{11} +8.00000i q^{13} +21.0000i q^{17} -105.000 q^{19} -50.0000 q^{21} +10.0000i q^{23} -145.000i q^{27} +20.0000 q^{29} -230.000 q^{31} +75.0000i q^{33} +54.0000i q^{37} +40.0000 q^{39} -195.000 q^{41} +300.000i q^{43} -480.000i q^{47} +243.000 q^{49} +105.000 q^{51} +322.000i q^{53} +525.000i q^{57} -560.000 q^{59} -730.000 q^{61} -20.0000i q^{63} +255.000i q^{67} +50.0000 q^{69} -40.0000 q^{71} +317.000i q^{73} +150.000i q^{77} +830.000 q^{79} -671.000 q^{81} -75.0000i q^{83} -100.000i q^{87} +705.000 q^{89} +80.0000 q^{91} +1150.00i q^{93} +1434.00i q^{97} -30.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} - 30 q^{11} - 210 q^{19} - 100 q^{21} + 40 q^{29} - 460 q^{31} + 80 q^{39} - 390 q^{41} + 486 q^{49} + 210 q^{51} - 1120 q^{59} - 1460 q^{61} + 100 q^{69} - 80 q^{71} + 1660 q^{79} - 1342 q^{81}+ \cdots - 60 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 30 * q^11 - 210 * q^19 - 100 * q^21 + 40 * q^29 - 460 * q^31 + 80 * q^39 - 390 * q^41 + 486 * q^49 + 210 * q^51 - 1120 * q^59 - 1460 * q^61 + 100 * q^69 - 80 * q^71 + 1660 * q^79 - 1342 * q^81 + 1410 * q^89 + 160 * q^91 - 60 * q^99

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
$3$	− 5.00000i	− 0.962250i	−0.876652	−	0.481125i	$-0.840228\pi$
$3$	− 5.00000i	− 0.962250i	0.876652	−	0.481125i	$-0.159772\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 10.0000i	− 0.539949i	−0.962867	−	0.269975i	$-0.912985\pi$
$7$	− 10.0000i	− 0.539949i	0.962867	−	0.269975i	$-0.0870153\pi$
$8$	0	0
$9$	2.00000	0.0740741
$10$	0	0
$11$	−15.0000	−0.411152	−0.205576	−	0.978641i	$-0.565907\pi$
$11$	−15.0000	−0.411152	−0.205576	+	0.978641i	$0.565907\pi$
$12$	0	0
$13$	8.00000i	0.170677i	0.996352	+	0.0853385i	$0.0271972\pi$
$13$	8.00000i	0.170677i	−0.996352	+	0.0853385i	$0.972803\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	21.0000i	0.299603i	0.988716	+	0.149801i	$0.0478634\pi$
$17$	21.0000i	0.299603i	−0.988716	+	0.149801i	$0.952137\pi$
$18$	0	0
$19$	−105.000	−1.26782	−0.633912	−	0.773405i	$-0.718552\pi$
$19$	−105.000	−1.26782	−0.633912	+	0.773405i	$0.718552\pi$
$20$	0	0
$21$	−50.0000	−0.519566
$22$	0	0
$23$	10.0000i	0.0906584i	0.998972	+	0.0453292i	$0.0144337\pi$
$23$	10.0000i	0.0906584i	−0.998972	+	0.0453292i	$0.985566\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 145.000i	− 1.03353i
$28$	0	0
$29$	20.0000	0.128066	0.0640329	−	0.997948i	$-0.479604\pi$
$29$	20.0000	0.128066	0.0640329	+	0.997948i	$0.479604\pi$
$30$	0	0
$31$	−230.000	−1.33256	−0.666278	−	0.745704i	$-0.732113\pi$
$31$	−230.000	−1.33256	−0.666278	+	0.745704i	$0.732113\pi$
$32$	0	0
$33$	75.0000i	0.395631i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	54.0000i	0.239934i	0.992778	+	0.119967i	$0.0382788\pi$
$37$	54.0000i	0.239934i	−0.992778	+	0.119967i	$0.961721\pi$
$38$	0	0
$39$	40.0000	0.164234
$40$	0	0
$41$	−195.000	−0.742778	−0.371389	−	0.928477i	$-0.621118\pi$
$41$	−195.000	−0.742778	−0.371389	+	0.928477i	$0.621118\pi$
$42$	0	0
$43$	300.000i	1.06394i	0.846762	+	0.531972i	$0.178549\pi$
$43$	300.000i	1.06394i	−0.846762	+	0.531972i	$0.821451\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 480.000i	− 1.48969i	−0.667240	−	0.744843i	$-0.732525\pi$
$47$	− 480.000i	− 1.48969i	0.667240	−	0.744843i	$-0.267475\pi$
$48$	0	0
$49$	243.000	0.708455
$50$	0	0
$51$	105.000	0.288293
$52$	0	0
$53$	322.000i	0.834530i	0.908785	+	0.417265i	$0.137011\pi$
$53$	322.000i	0.834530i	−0.908785	+	0.417265i	$0.862989\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	525.000i	1.21996i
$58$	0	0
$59$	−560.000	−1.23569	−0.617846	−	0.786299i	$-0.711994\pi$
$59$	−560.000	−1.23569	−0.617846	+	0.786299i	$0.711994\pi$
$60$	0	0
$61$	−730.000	−1.53224	−0.766122	−	0.642695i	$-0.777816\pi$
$61$	−730.000	−1.53224	−0.766122	+	0.642695i	$0.777816\pi$
$62$	0	0
$63$	− 20.0000i	− 0.0399962i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	255.000i	0.464973i	0.972600	+	0.232487i	$0.0746862\pi$
$67$	255.000i	0.464973i	−0.972600	+	0.232487i	$0.925314\pi$
$68$	0	0
$69$	50.0000	0.0872361
$70$	0	0
$71$	−40.0000	−0.0668609	−0.0334305	−	0.999441i	$-0.510643\pi$
$71$	−40.0000	−0.0668609	−0.0334305	+	0.999441i	$0.510643\pi$
$72$	0	0
$73$	317.000i	0.508247i	0.967172	+	0.254124i	$0.0817870\pi$
$73$	317.000i	0.508247i	−0.967172	+	0.254124i	$0.918213\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	150.000i	0.222001i
$78$	0	0
$79$	830.000	1.18205	0.591027	−	0.806652i	$-0.298723\pi$
$79$	830.000	1.18205	0.591027	+	0.806652i	$0.298723\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	− 75.0000i	− 0.0991846i	−0.998770	−	0.0495923i	$-0.984208\pi$
$83$	− 75.0000i	− 0.0991846i	0.998770	−	0.0495923i	$-0.0157922\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 100.000i	− 0.123231i
$88$	0	0
$89$	705.000	0.839661	0.419831	−	0.907602i	$-0.362089\pi$
$89$	705.000	0.839661	0.419831	+	0.907602i	$0.362089\pi$
$90$	0	0
$91$	80.0000	0.0921569
$92$	0	0
$93$	1150.00i	1.28225i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1434.00i	1.50104i	0.660849	+	0.750519i	$0.270196\pi$
$97$	1434.00i	1.50104i	−0.660849	+	0.750519i	$0.729804\pi$
$98$	0	0
$99$	−30.0000	−0.0304557
$100$	0	0
$101$	−1902.00	−1.87382	−0.936911	−	0.349567i	$-0.886329\pi$
$101$	−1902.00	−1.87382	−0.936911	+	0.349567i	$0.886329\pi$
$102$	0	0
$103$	− 1480.00i	− 1.41581i	−0.706306	−	0.707906i	$-0.749640\pi$
$103$	− 1480.00i	− 1.41581i	0.706306	−	0.707906i	$-0.250360\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1945.00i	1.75729i	0.477474	+	0.878646i	$0.341553\pi$
$107$	1945.00i	1.75729i	−0.477474	+	0.878646i	$0.658447\pi$
$108$	0	0
$109$	246.000	0.216170	0.108085	−	0.994142i	$-0.465528\pi$
$109$	246.000	0.216170	0.108085	+	0.994142i	$0.465528\pi$
$110$	0	0
$111$	270.000	0.230876
$112$	0	0
$113$	753.000i	0.626870i	0.949610	+	0.313435i	$0.101480\pi$
$113$	753.000i	0.626870i	−0.949610	+	0.313435i	$0.898520\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.0000i	0.0126427i
$118$	0	0
$119$	210.000	0.161770
$120$	0	0
$121$	−1106.00	−0.830954
$122$	0	0
$123$	975.000i	0.714738i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1490.00i	1.04107i	0.853840	+	0.520536i	$0.174268\pi$
$127$	1490.00i	1.04107i	−0.853840	+	0.520536i	$0.825732\pi$
$128$	0	0
$129$	1500.00	1.02378
$130$	0	0
$131$	780.000	0.520221	0.260110	−	0.965579i	$-0.416241\pi$
$131$	780.000	0.520221	0.260110	+	0.965579i	$0.416241\pi$
$132$	0	0
$133$	1050.00i	0.684561i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2749.00i	1.71433i	0.515044	+	0.857164i	$0.327776\pi$
$137$	2749.00i	1.71433i	−0.515044	+	0.857164i	$0.672224\pi$
$138$	0	0
$139$	735.000	0.448503	0.224251	−	0.974531i	$-0.428006\pi$
$139$	735.000	0.448503	0.224251	+	0.974531i	$0.428006\pi$
$140$	0	0
$141$	−2400.00	−1.43345
$142$	0	0
$143$	− 120.000i	− 0.0701742i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1215.00i	− 0.681711i
$148$	0	0
$149$	−836.000	−0.459650	−0.229825	−	0.973232i	$-0.573815\pi$
$149$	−836.000	−0.459650	−0.229825	+	0.973232i	$0.573815\pi$
$150$	0	0
$151$	1790.00	0.964690	0.482345	−	0.875981i	$-0.339785\pi$
$151$	1790.00	0.964690	0.482345	+	0.875981i	$0.339785\pi$
$152$	0	0
$153$	42.0000i	0.0221928i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1374.00i	− 0.698453i	−0.937038	−	0.349227i	$-0.886444\pi$
$157$	− 1374.00i	− 0.698453i	0.937038	−	0.349227i	$-0.113556\pi$
$158$	0	0
$159$	1610.00	0.803027
$160$	0	0
$161$	100.000	0.0489510
$162$	0	0
$163$	− 1895.00i	− 0.910600i	−0.890338	−	0.455300i	$-0.849532\pi$
$163$	− 1895.00i	− 0.910600i	0.890338	−	0.455300i	$-0.150468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 720.000i	− 0.333624i	−0.985989	−	0.166812i	$-0.946653\pi$
$167$	− 720.000i	− 0.333624i	0.985989	−	0.166812i	$-0.0533474\pi$
$168$	0	0
$169$	2133.00	0.970869
$170$	0	0
$171$	−210.000	−0.0939129
$172$	0	0
$173$	2512.00i	1.10395i	0.833860	+	0.551976i	$0.186126\pi$
$173$	2512.00i	1.10395i	−0.833860	+	0.551976i	$0.813874\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2800.00i	1.18904i
$178$	0	0
$179$	−165.000	−0.0688976	−0.0344488	−	0.999406i	$-0.510968\pi$
$179$	−165.000	−0.0688976	−0.0344488	+	0.999406i	$0.510968\pi$
$180$	0	0
$181$	−3158.00	−1.29686	−0.648432	−	0.761273i	$-0.724575\pi$
$181$	−3158.00	−1.29686	−0.648432	+	0.761273i	$0.724575\pi$
$182$	0	0
$183$	3650.00i	1.47440i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 315.000i	− 0.123182i
$188$	0	0
$189$	−1450.00	−0.558053
$190$	0	0
$191$	−3290.00	−1.24637	−0.623183	−	0.782076i	$-0.714161\pi$
$191$	−3290.00	−1.24637	−0.623183	+	0.782076i	$0.714161\pi$
$192$	0	0
$193$	− 197.000i	− 0.0734734i	−0.999325	−	0.0367367i	$-0.988304\pi$
$193$	− 197.000i	− 0.0734734i	0.999325	−	0.0367367i	$-0.0116963\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1746.00i	− 0.631459i	−0.948849	−	0.315729i	$-0.897751\pi$
$197$	− 1746.00i	− 0.631459i	0.948849	−	0.315729i	$-0.102249\pi$
$198$	0	0
$199$	−4660.00	−1.65999	−0.829997	−	0.557768i	$-0.811658\pi$
$199$	−4660.00	−1.65999	−0.829997	+	0.557768i	$0.811658\pi$
$200$	0	0
$201$	1275.00	0.447421
$202$	0	0
$203$	− 200.000i	− 0.0691490i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	20.0000i	0.00671544i
$208$	0	0
$209$	1575.00	0.521268
$210$	0	0
$211$	−265.000	−0.0864614	−0.0432307	−	0.999065i	$-0.513765\pi$
$211$	−265.000	−0.0864614	−0.0432307	+	0.999065i	$0.513765\pi$
$212$	0	0
$213$	200.000i	0.0643370i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2300.00i	0.719512i
$218$	0	0
$219$	1585.00	0.489061
$220$	0	0
$221$	−168.000	−0.0511353
$222$	0	0
$223$	1060.00i	0.318309i	0.987254	+	0.159154i	$0.0508768\pi$
$223$	1060.00i	0.318309i	−0.987254	+	0.159154i	$0.949123\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4660.00i	− 1.36253i	−0.732035	−	0.681267i	$-0.761429\pi$
$227$	− 4660.00i	− 1.36253i	0.732035	−	0.681267i	$-0.238571\pi$
$228$	0	0
$229$	−1660.00	−0.479021	−0.239511	−	0.970894i	$-0.576987\pi$
$229$	−1660.00	−0.479021	−0.239511	+	0.970894i	$0.576987\pi$
$230$	0	0
$231$	750.000	0.213621
$232$	0	0
$233$	− 3462.00i	− 0.973404i	−0.873568	−	0.486702i	$-0.838200\pi$
$233$	− 3462.00i	− 0.973404i	0.873568	−	0.486702i	$-0.161800\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4150.00i	− 1.13743i
$238$	0	0
$239$	−4020.00	−1.08800	−0.544000	−	0.839085i	$-0.683091\pi$
$239$	−4020.00	−1.08800	−0.544000	+	0.839085i	$0.683091\pi$
$240$	0	0
$241$	3985.00	1.06513	0.532565	−	0.846389i	$-0.321228\pi$
$241$	3985.00	1.06513	0.532565	+	0.846389i	$0.321228\pi$
$242$	0	0
$243$	− 560.000i	− 0.147835i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 840.000i	− 0.216388i
$248$	0	0
$249$	−375.000	−0.0954404
$250$	0	0
$251$	−6625.00	−1.66600	−0.833001	−	0.553272i	$-0.813379\pi$
$251$	−6625.00	−1.66600	−0.833001	+	0.553272i	$0.813379\pi$
$252$	0	0
$253$	− 150.000i	− 0.0372744i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2246.00i	0.545143i	0.962136	+	0.272571i	$0.0878741\pi$
$257$	2246.00i	0.545143i	−0.962136	+	0.272571i	$0.912126\pi$
$258$	0	0
$259$	540.000	0.129552
$260$	0	0
$261$	40.0000	0.00948635
$262$	0	0
$263$	− 3950.00i	− 0.926112i	−0.886329	−	0.463056i	$-0.846753\pi$
$263$	− 3950.00i	− 0.926112i	0.886329	−	0.463056i	$-0.153247\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3525.00i	− 0.807964i
$268$	0	0
$269$	−2656.00	−0.602004	−0.301002	−	0.953623i	$-0.597321\pi$
$269$	−2656.00	−0.602004	−0.301002	+	0.953623i	$0.597321\pi$
$270$	0	0
$271$	−3110.00	−0.697118	−0.348559	−	0.937287i	$-0.613329\pi$
$271$	−3110.00	−0.697118	−0.348559	+	0.937287i	$0.613329\pi$
$272$	0	0
$273$	− 400.000i	− 0.0886780i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6884.00i	− 1.49321i	−0.665267	−	0.746606i	$-0.731682\pi$
$277$	− 6884.00i	− 1.49321i	0.665267	−	0.746606i	$-0.268318\pi$
$278$	0	0
$279$	−460.000	−0.0987078
$280$	0	0
$281$	4630.00	0.982928	0.491464	−	0.870898i	$-0.336462\pi$
$281$	4630.00	0.982928	0.491464	+	0.870898i	$0.336462\pi$
$282$	0	0
$283$	− 215.000i	− 0.0451605i	−0.999745	−	0.0225803i	$-0.992812\pi$
$283$	− 215.000i	− 0.0451605i	0.999745	−	0.0225803i	$-0.00718813\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1950.00i	0.401062i
$288$	0	0
$289$	4472.00	0.910238
$290$	0	0
$291$	7170.00	1.44437
$292$	0	0
$293$	− 1602.00i	− 0.319419i	−0.987164	−	0.159710i	$-0.948944\pi$
$293$	− 1602.00i	− 0.319419i	0.987164	−	0.159710i	$-0.0510558\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2175.00i	0.424937i
$298$	0	0
$299$	−80.0000	−0.0154733
$300$	0	0
$301$	3000.00	0.574475
$302$	0	0
$303$	9510.00i	1.80309i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1885.00i	0.350432i	0.984530	+	0.175216i	$0.0560624\pi$
$307$	1885.00i	0.350432i	−0.984530	+	0.175216i	$0.943938\pi$
$308$	0	0
$309$	−7400.00	−1.36237
$310$	0	0
$311$	9250.00	1.68656	0.843279	−	0.537476i	$-0.180622\pi$
$311$	9250.00	1.68656	0.843279	+	0.537476i	$0.180622\pi$
$312$	0	0
$313$	− 8162.00i	− 1.47394i	−0.675925	−	0.736970i	$-0.736256\pi$
$313$	− 8162.00i	− 1.47394i	0.675925	−	0.736970i	$-0.263744\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6924.00i	− 1.22678i	−0.789779	−	0.613392i	$-0.789805\pi$
$317$	− 6924.00i	− 1.22678i	0.789779	−	0.613392i	$-0.210195\pi$
$318$	0	0
$319$	−300.000	−0.0526545
$320$	0	0
$321$	9725.00	1.69096
$322$	0	0
$323$	− 2205.00i	− 0.379844i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1230.00i	− 0.208010i
$328$	0	0
$329$	−4800.00	−0.804354
$330$	0	0
$331$	−8075.00	−1.34091	−0.670456	−	0.741949i	$-0.733902\pi$
$331$	−8075.00	−1.34091	−0.670456	+	0.741949i	$0.733902\pi$
$332$	0	0
$333$	108.000i	0.0177729i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4741.00i	− 0.766346i	−0.923677	−	0.383173i	$-0.874831\pi$
$337$	− 4741.00i	− 0.766346i	0.923677	−	0.383173i	$-0.125169\pi$
$338$	0	0
$339$	3765.00	0.603206
$340$	0	0
$341$	3450.00	0.547883
$342$	0	0
$343$	− 5860.00i	− 0.922479i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8705.00i	1.34671i	0.739319	+	0.673356i	$0.235148\pi$
$347$	8705.00i	1.34671i	−0.739319	+	0.673356i	$0.764852\pi$
$348$	0	0
$349$	1470.00	0.225465	0.112733	−	0.993625i	$-0.464040\pi$
$349$	1470.00	0.225465	0.112733	+	0.993625i	$0.464040\pi$
$350$	0	0
$351$	1160.00	0.176399
$352$	0	0
$353$	− 1998.00i	− 0.301254i	−0.988591	−	0.150627i	$-0.951871\pi$
$353$	− 1998.00i	− 0.301254i	0.988591	−	0.150627i	$-0.0481293\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1050.00i	− 0.155664i
$358$	0	0
$359$	8190.00	1.20404	0.602022	−	0.798480i	$-0.294362\pi$
$359$	8190.00	1.20404	0.602022	+	0.798480i	$0.294362\pi$
$360$	0	0
$361$	4166.00	0.607377
$362$	0	0
$363$	5530.00i	0.799586i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5340.00i	0.759525i	0.925084	+	0.379763i	$0.123994\pi$
$367$	5340.00i	0.759525i	−0.925084	+	0.379763i	$0.876006\pi$
$368$	0	0
$369$	−390.000	−0.0550206
$370$	0	0
$371$	3220.00	0.450604
$372$	0	0
$373$	− 9378.00i	− 1.30181i	−0.759160	−	0.650904i	$-0.774390\pi$
$373$	− 9378.00i	− 1.30181i	0.759160	−	0.650904i	$-0.225610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	160.000i	0.0218579i
$378$	0	0
$379$	4045.00	0.548226	0.274113	−	0.961697i	$-0.411616\pi$
$379$	4045.00	0.548226	0.274113	+	0.961697i	$0.411616\pi$
$380$	0	0
$381$	7450.00	1.00177
$382$	0	0
$383$	− 8090.00i	− 1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$383$	− 8090.00i	− 1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	600.000i	0.0788106i
$388$	0	0
$389$	−6574.00	−0.856851	−0.428425	−	0.903577i	$-0.640932\pi$
$389$	−6574.00	−0.856851	−0.428425	+	0.903577i	$0.640932\pi$
$390$	0	0
$391$	−210.000	−0.0271615
$392$	0	0
$393$	− 3900.00i	− 0.500583i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7036.00i	− 0.889488i	−0.895658	−	0.444744i	$-0.853295\pi$
$397$	− 7036.00i	− 0.889488i	0.895658	−	0.444744i	$-0.146705\pi$
$398$	0	0
$399$	5250.00	0.658719
$400$	0	0
$401$	8277.00	1.03076	0.515379	−	0.856963i	$-0.327651\pi$
$401$	8277.00	1.03076	0.515379	+	0.856963i	$0.327651\pi$
$402$	0	0
$403$	− 1840.00i	− 0.227437i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 810.000i	− 0.0986492i
$408$	0	0
$409$	2179.00	0.263434	0.131717	−	0.991287i	$-0.457951\pi$
$409$	2179.00	0.263434	0.131717	+	0.991287i	$0.457951\pi$
$410$	0	0
$411$	13745.0	1.64961
$412$	0	0
$413$	5600.00i	0.667211i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3675.00i	− 0.431572i
$418$	0	0
$419$	−12245.0	−1.42770	−0.713851	−	0.700297i	$-0.753051\pi$
$419$	−12245.0	−1.42770	−0.713851	+	0.700297i	$0.753051\pi$
$420$	0	0
$421$	−660.000	−0.0764048	−0.0382024	−	0.999270i	$-0.512163\pi$
$421$	−660.000	−0.0764048	−0.0382024	+	0.999270i	$0.512163\pi$
$422$	0	0
$423$	− 960.000i	− 0.110347i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7300.00i	0.827334i
$428$	0	0
$429$	−600.000	−0.0675251
$430$	0	0
$431$	−7470.00	−0.834843	−0.417421	−	0.908713i	$-0.637066\pi$
$431$	−7470.00	−0.834843	−0.417421	+	0.908713i	$0.637066\pi$
$432$	0	0
$433$	− 1173.00i	− 0.130187i	−0.997879	−	0.0650933i	$-0.979266\pi$
$433$	− 1173.00i	− 0.130187i	0.997879	−	0.0650933i	$-0.0207345\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1050.00i	− 0.114939i
$438$	0	0
$439$	−5660.00	−0.615346	−0.307673	−	0.951492i	$-0.599550\pi$
$439$	−5660.00	−0.615346	−0.307673	+	0.951492i	$0.599550\pi$
$440$	0	0
$441$	486.000	0.0524781
$442$	0	0
$443$	− 1115.00i	− 0.119583i	−0.998211	−	0.0597915i	$-0.980956\pi$
$443$	− 1115.00i	− 0.119583i	0.998211	−	0.0597915i	$-0.0190436\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4180.00i	0.442298i
$448$	0	0
$449$	−4089.00	−0.429781	−0.214891	−	0.976638i	$-0.568940\pi$
$449$	−4089.00	−0.429781	−0.214891	+	0.976638i	$0.568940\pi$
$450$	0	0
$451$	2925.00	0.305394
$452$	0	0
$453$	− 8950.00i	− 0.928273i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4519.00i	− 0.462560i	−0.972887	−	0.231280i	$-0.925709\pi$
$457$	− 4519.00i	− 0.462560i	0.972887	−	0.231280i	$-0.0742913\pi$
$458$	0	0
$459$	3045.00	0.309648
$460$	0	0
$461$	10068.0	1.01717	0.508583	−	0.861013i	$-0.330170\pi$
$461$	10068.0	1.01717	0.508583	+	0.861013i	$0.330170\pi$
$462$	0	0
$463$	− 1460.00i	− 0.146548i	−0.997312	−	0.0732742i	$-0.976655\pi$
$463$	− 1460.00i	− 0.146548i	0.997312	−	0.0732742i	$-0.0233448\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4740.00i	0.469681i	0.972034	+	0.234841i	$0.0754568\pi$
$467$	4740.00i	0.469681i	−0.972034	+	0.234841i	$0.924543\pi$
$468$	0	0
$469$	2550.00	0.251062
$470$	0	0
$471$	−6870.00	−0.672087
$472$	0	0
$473$	− 4500.00i	− 0.437442i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	644.000i	0.0618171i
$478$	0	0
$479$	8610.00	0.821296	0.410648	−	0.911794i	$-0.365302\pi$
$479$	8610.00	0.821296	0.410648	+	0.911794i	$0.365302\pi$
$480$	0	0
$481$	−432.000	−0.0409512
$482$	0	0
$483$	− 500.000i	− 0.0471031i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17710.0i	− 1.64788i	−0.566678	−	0.823939i	$-0.691772\pi$
$487$	− 17710.0i	− 1.64788i	0.566678	−	0.823939i	$-0.308228\pi$
$488$	0	0
$489$	−9475.00	−0.876226
$490$	0	0
$491$	−8660.00	−0.795968	−0.397984	−	0.917392i	$-0.630290\pi$
$491$	−8660.00	−0.795968	−0.397984	+	0.917392i	$0.630290\pi$
$492$	0	0
$493$	420.000i	0.0383689i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	400.000i	0.0361015i
$498$	0	0
$499$	−17300.0	−1.55201	−0.776006	−	0.630725i	$-0.782758\pi$
$499$	−17300.0	−1.55201	−0.776006	+	0.630725i	$0.782758\pi$
$500$	0	0
$501$	−3600.00	−0.321030
$502$	0	0
$503$	− 1860.00i	− 0.164877i	−0.996596	−	0.0824387i	$-0.973729\pi$
$503$	− 1860.00i	− 0.164877i	0.996596	−	0.0824387i	$-0.0262709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 10665.0i	− 0.934219i
$508$	0	0
$509$	5870.00	0.511165	0.255583	−	0.966787i	$-0.417733\pi$
$509$	5870.00	0.511165	0.255583	+	0.966787i	$0.417733\pi$
$510$	0	0
$511$	3170.00	0.274428
$512$	0	0
$513$	15225.0i	1.31033i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7200.00i	0.612487i
$518$	0	0
$519$	12560.0	1.06228
$520$	0	0
$521$	−12187.0	−1.02480	−0.512401	−	0.858746i	$-0.671244\pi$
$521$	−12187.0	−1.02480	−0.512401	+	0.858746i	$0.671244\pi$
$522$	0	0
$523$	5315.00i	0.444376i	0.975004	+	0.222188i	$0.0713199\pi$
$523$	5315.00i	0.444376i	−0.975004	+	0.222188i	$0.928680\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4830.00i	− 0.399237i
$528$	0	0
$529$	12067.0	0.991781
$530$	0	0
$531$	−1120.00	−0.0915327
$532$	0	0
$533$	− 1560.00i	− 0.126775i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	825.000i	0.0662968i
$538$	0	0
$539$	−3645.00	−0.291282
$540$	0	0
$541$	−1672.00	−0.132874	−0.0664371	−	0.997791i	$-0.521163\pi$
$541$	−1672.00	−0.132874	−0.0664371	+	0.997791i	$0.521163\pi$
$542$	0	0
$543$	15790.0i	1.24791i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10595.0i	− 0.828171i	−0.910238	−	0.414085i	$-0.864101\pi$
$547$	− 10595.0i	− 0.828171i	0.910238	−	0.414085i	$-0.135899\pi$
$548$	0	0
$549$	−1460.00	−0.113500
$550$	0	0
$551$	−2100.00	−0.162365
$552$	0	0
$553$	− 8300.00i	− 0.638249i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15264.0i	1.16114i	0.814209	+	0.580571i	$0.197171\pi$
$557$	15264.0i	1.16114i	−0.814209	+	0.580571i	$0.802829\pi$
$558$	0	0
$559$	−2400.00	−0.181591
$560$	0	0
$561$	−1575.00	−0.118532
$562$	0	0
$563$	15400.0i	1.15281i	0.817164	+	0.576406i	$0.195545\pi$
$563$	15400.0i	1.15281i	−0.817164	+	0.576406i	$0.804455\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6710.00i	0.496990i
$568$	0	0
$569$	−14569.0	−1.07340	−0.536700	−	0.843773i	$-0.680329\pi$
$569$	−14569.0	−1.07340	−0.536700	+	0.843773i	$0.680329\pi$
$570$	0	0
$571$	−7780.00	−0.570198	−0.285099	−	0.958498i	$-0.592026\pi$
$571$	−7780.00	−0.570198	−0.285099	+	0.958498i	$0.592026\pi$
$572$	0	0
$573$	16450.0i	1.19932i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19829.0i	− 1.43066i	−0.698786	−	0.715331i	$-0.746276\pi$
$577$	− 19829.0i	− 1.43066i	0.698786	−	0.715331i	$-0.253724\pi$
$578$	0	0
$579$	−985.000	−0.0706998
$580$	0	0
$581$	−750.000	−0.0535546
$582$	0	0
$583$	− 4830.00i	− 0.343119i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	795.000i	0.0558998i	0.999609	+	0.0279499i	$0.00889788\pi$
$587$	795.000i	0.0558998i	−0.999609	+	0.0279499i	$0.991102\pi$
$588$	0	0
$589$	24150.0	1.68945
$590$	0	0
$591$	−8730.00	−0.607621
$592$	0	0
$593$	21457.0i	1.48589i	0.669352	+	0.742946i	$0.266572\pi$
$593$	21457.0i	1.48589i	−0.669352	+	0.742946i	$0.733428\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23300.0i	1.59733i
$598$	0	0
$599$	−26570.0	−1.81239	−0.906194	−	0.422862i	$-0.861026\pi$
$599$	−26570.0	−1.81239	−0.906194	+	0.422862i	$0.861026\pi$
$600$	0	0
$601$	−25245.0	−1.71342	−0.856710	−	0.515799i	$-0.827495\pi$
$601$	−25245.0	−1.71342	−0.856710	+	0.515799i	$0.827495\pi$
$602$	0	0
$603$	510.000i	0.0344425i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 14040.0i	− 0.938824i	−0.882979	−	0.469412i	$-0.844466\pi$
$607$	− 14040.0i	− 0.938824i	0.882979	−	0.469412i	$-0.155534\pi$
$608$	0	0
$609$	−1000.00	−0.0665387
$610$	0	0
$611$	3840.00	0.254255
$612$	0	0
$613$	− 19502.0i	− 1.28496i	−0.766304	−	0.642478i	$-0.777906\pi$
$613$	− 19502.0i	− 1.28496i	0.766304	−	0.642478i	$-0.222094\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6286.00i	0.410154i	0.978746	+	0.205077i	$0.0657444\pi$
$617$	6286.00i	0.410154i	−0.978746	+	0.205077i	$0.934256\pi$
$618$	0	0
$619$	−2420.00	−0.157137	−0.0785687	−	0.996909i	$-0.525035\pi$
$619$	−2420.00	−0.157137	−0.0785687	+	0.996909i	$0.525035\pi$
$620$	0	0
$621$	1450.00	0.0936981
$622$	0	0
$623$	− 7050.00i	− 0.453374i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7875.00i	− 0.501590i
$628$	0	0
$629$	−1134.00	−0.0718848
$630$	0	0
$631$	2290.00	0.144475	0.0722373	−	0.997387i	$-0.476986\pi$
$631$	2290.00	0.144475	0.0722373	+	0.997387i	$0.476986\pi$
$632$	0	0
$633$	1325.00i	0.0831975i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1944.00i	0.120917i
$638$	0	0
$639$	−80.0000	−0.00495266
$640$	0	0
$641$	−15150.0	−0.933524	−0.466762	−	0.884383i	$-0.654580\pi$
$641$	−15150.0	−0.933524	−0.466762	+	0.884383i	$0.654580\pi$
$642$	0	0
$643$	− 7860.00i	− 0.482066i	−0.970517	−	0.241033i	$-0.922514\pi$
$643$	− 7860.00i	− 0.482066i	0.970517	−	0.241033i	$-0.0774861\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2420.00i	− 0.147048i	−0.997293	−	0.0735240i	$-0.976575\pi$
$647$	− 2420.00i	− 0.147048i	0.997293	−	0.0735240i	$-0.0234245\pi$
$648$	0	0
$649$	8400.00	0.508057
$650$	0	0
$651$	11500.0	0.692351
$652$	0	0
$653$	20462.0i	1.22625i	0.789987	+	0.613124i	$0.210087\pi$
$653$	20462.0i	1.22625i	−0.789987	+	0.613124i	$0.789913\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	634.000i	0.0376479i
$658$	0	0
$659$	32205.0	1.90369	0.951843	−	0.306587i	$-0.0991870\pi$
$659$	32205.0	1.90369	0.951843	+	0.306587i	$0.0991870\pi$
$660$	0	0
$661$	−27100.0	−1.59466	−0.797328	−	0.603546i	$-0.793754\pi$
$661$	−27100.0	−1.59466	−0.797328	+	0.603546i	$0.793754\pi$
$662$	0	0
$663$	840.000i	0.0492050i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	200.000i	0.0116102i
$668$	0	0
$669$	5300.00	0.306293
$670$	0	0
$671$	10950.0	0.629985
$672$	0	0
$673$	24182.0i	1.38506i	0.721387	+	0.692532i	$0.243505\pi$
$673$	24182.0i	1.38506i	−0.721387	+	0.692532i	$0.756495\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25704.0i	1.45921i	0.683869	+	0.729605i	$0.260296\pi$
$677$	25704.0i	1.45921i	−0.683869	+	0.729605i	$0.739704\pi$
$678$	0	0
$679$	14340.0	0.810484
$680$	0	0
$681$	−23300.0	−1.31110
$682$	0	0
$683$	− 6525.00i	− 0.365552i	−0.983155	−	0.182776i	$-0.941492\pi$
$683$	− 6525.00i	− 0.365552i	0.983155	−	0.182776i	$-0.0585084\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8300.00i	0.460939i
$688$	0	0
$689$	−2576.00	−0.142435
$690$	0	0
$691$	−28955.0	−1.59407	−0.797033	−	0.603935i	$-0.793599\pi$
$691$	−28955.0	−1.59407	−0.797033	+	0.603935i	$0.793599\pi$
$692$	0	0
$693$	300.000i	0.0164445i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4095.00i	− 0.222538i
$698$	0	0
$699$	−17310.0	−0.936659
$700$	0	0
$701$	9720.00	0.523708	0.261854	−	0.965107i	$-0.415666\pi$
$701$	9720.00	0.523708	0.261854	+	0.965107i	$0.415666\pi$
$702$	0	0
$703$	− 5670.00i	− 0.304194i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19020.0i	1.01177i
$708$	0	0
$709$	25140.0	1.33167	0.665834	−	0.746100i	$-0.268076\pi$
$709$	25140.0	1.33167	0.665834	+	0.746100i	$0.268076\pi$
$710$	0	0
$711$	1660.00	0.0875596
$712$	0	0
$713$	− 2300.00i	− 0.120807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20100.0i	1.04693i
$718$	0	0
$719$	−30030.0	−1.55762	−0.778811	−	0.627259i	$-0.784177\pi$
$719$	−30030.0	−1.55762	−0.778811	+	0.627259i	$0.784177\pi$
$720$	0	0
$721$	−14800.0	−0.764467
$722$	0	0
$723$	− 19925.0i	− 1.02492i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36100.0i	1.84164i	0.389984	+	0.920822i	$0.372481\pi$
$727$	36100.0i	1.84164i	−0.389984	+	0.920822i	$0.627519\pi$
$728$	0	0
$729$	−20917.0	−1.06269
$730$	0	0
$731$	−6300.00	−0.318760
$732$	0	0
$733$	− 5368.00i	− 0.270493i	−0.990812	−	0.135247i	$-0.956817\pi$
$733$	− 5368.00i	− 0.270493i	0.990812	−	0.135247i	$-0.0431827\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3825.00i	− 0.191175i
$738$	0	0
$739$	25540.0	1.27132	0.635659	−	0.771970i	$-0.280728\pi$
$739$	25540.0	1.27132	0.635659	+	0.771970i	$0.280728\pi$
$740$	0	0
$741$	−4200.00	−0.208220
$742$	0	0
$743$	18730.0i	0.924814i	0.886668	+	0.462407i	$0.153014\pi$
$743$	18730.0i	0.924814i	−0.886668	+	0.462407i	$0.846986\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 150.000i	− 0.00734701i
$748$	0	0
$749$	19450.0	0.948849
$750$	0	0
$751$	29940.0	1.45476	0.727381	−	0.686234i	$-0.240737\pi$
$751$	29940.0	1.45476	0.727381	+	0.686234i	$0.240737\pi$
$752$	0	0
$753$	33125.0i	1.60311i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16216.0i	0.778574i	0.921117	+	0.389287i	$0.127279\pi$
$757$	16216.0i	0.778574i	−0.921117	+	0.389287i	$0.872721\pi$
$758$	0	0
$759$	−750.000	−0.0358673
$760$	0	0
$761$	−25843.0	−1.23102	−0.615511	−	0.788128i	$-0.711050\pi$
$761$	−25843.0	−1.23102	−0.615511	+	0.788128i	$0.711050\pi$
$762$	0	0
$763$	− 2460.00i	− 0.116721i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4480.00i	− 0.210904i
$768$	0	0
$769$	27475.0	1.28839	0.644196	−	0.764860i	$-0.277192\pi$
$769$	27475.0	1.28839	0.644196	+	0.764860i	$0.277192\pi$
$770$	0	0
$771$	11230.0	0.524564
$772$	0	0
$773$	3948.00i	0.183699i	0.995773	+	0.0918497i	$0.0292779\pi$
$773$	3948.00i	0.183699i	−0.995773	+	0.0918497i	$0.970722\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2700.00i	− 0.124661i
$778$	0	0
$779$	20475.0	0.941711
$780$	0	0
$781$	600.000	0.0274900
$782$	0	0
$783$	− 2900.00i	− 0.132360i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22100.0i	1.00099i	0.865739	+	0.500496i	$0.166849\pi$
$787$	22100.0i	1.00099i	−0.865739	+	0.500496i	$0.833151\pi$
$788$	0	0
$789$	−19750.0	−0.891152
$790$	0	0
$791$	7530.00	0.338478
$792$	0	0
$793$	− 5840.00i	− 0.261519i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16026.0i	0.712259i	0.934437	+	0.356129i	$0.115904\pi$
$797$	16026.0i	0.712259i	−0.934437	+	0.356129i	$0.884096\pi$
$798$	0	0
$799$	10080.0	0.446314
$800$	0	0
$801$	1410.00	0.0621971
$802$	0	0
$803$	− 4755.00i	− 0.208967i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13280.0i	0.579279i
$808$	0	0
$809$	−35770.0	−1.55452	−0.777260	−	0.629180i	$-0.783391\pi$
$809$	−35770.0	−1.55452	−0.777260	+	0.629180i	$0.783391\pi$
$810$	0	0
$811$	30620.0	1.32579	0.662894	−	0.748714i	$-0.269328\pi$
$811$	30620.0	1.32579	0.662894	+	0.748714i	$0.269328\pi$
$812$	0	0
$813$	15550.0i	0.670802i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 31500.0i	− 1.34889i
$818$	0	0
$819$	160.000	0.00682644
$820$	0	0
$821$	−9150.00	−0.388961	−0.194481	−	0.980906i	$-0.562302\pi$
$821$	−9150.00	−0.388961	−0.194481	+	0.980906i	$0.562302\pi$
$822$	0	0
$823$	28940.0i	1.22574i	0.790183	+	0.612871i	$0.209985\pi$
$823$	28940.0i	1.22574i	−0.790183	+	0.612871i	$0.790015\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 25765.0i	− 1.08336i	−0.840586	−	0.541679i	$-0.817789\pi$
$827$	− 25765.0i	− 1.08336i	0.840586	−	0.541679i	$-0.182211\pi$
$828$	0	0
$829$	41584.0	1.74219	0.871093	−	0.491118i	$-0.163412\pi$
$829$	41584.0	1.74219	0.871093	+	0.491118i	$0.163412\pi$
$830$	0	0
$831$	−34420.0	−1.43684
$832$	0	0
$833$	5103.00i	0.212255i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	33350.0i	1.37723i
$838$	0	0
$839$	20320.0	0.836143	0.418072	−	0.908414i	$-0.362706\pi$
$839$	20320.0	0.836143	0.418072	+	0.908414i	$0.362706\pi$
$840$	0	0
$841$	−23989.0	−0.983599
$842$	0	0
$843$	− 23150.0i	− 0.945822i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11060.0i	0.448673i
$848$	0	0
$849$	−1075.00	−0.0434557
$850$	0	0
$851$	−540.000	−0.0217520
$852$	0	0
$853$	− 38882.0i	− 1.56072i	−0.625330	−	0.780360i	$-0.715036\pi$
$853$	− 38882.0i	− 1.56072i	0.625330	−	0.780360i	$-0.284964\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15269.0i	0.608610i	0.952575	+	0.304305i	$0.0984243\pi$
$857$	15269.0i	0.608610i	−0.952575	+	0.304305i	$0.901576\pi$
$858$	0	0
$859$	19375.0	0.769577	0.384788	−	0.923005i	$-0.374274\pi$
$859$	19375.0	0.769577	0.384788	+	0.923005i	$0.374274\pi$
$860$	0	0
$861$	9750.00	0.385922
$862$	0	0
$863$	− 22900.0i	− 0.903274i	−0.892202	−	0.451637i	$-0.850840\pi$
$863$	− 22900.0i	− 0.903274i	0.892202	−	0.451637i	$-0.149160\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 22360.0i	− 0.875877i
$868$	0	0
$869$	−12450.0	−0.486004
$870$	0	0
$871$	−2040.00	−0.0793602
$872$	0	0
$873$	2868.00i	0.111188i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 23444.0i	− 0.902677i	−0.892353	−	0.451338i	$-0.850947\pi$
$877$	− 23444.0i	− 0.902677i	0.892353	−	0.451338i	$-0.149053\pi$
$878$	0	0
$879$	−8010.00	−0.307361
$880$	0	0
$881$	3750.00	0.143406	0.0717030	−	0.997426i	$-0.477157\pi$
$881$	3750.00	0.143406	0.0717030	+	0.997426i	$0.477157\pi$
$882$	0	0
$883$	37595.0i	1.43281i	0.697684	+	0.716406i	$0.254214\pi$
$883$	37595.0i	1.43281i	−0.697684	+	0.716406i	$0.745786\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11420.0i	0.432295i	0.976361	+	0.216148i	$0.0693493\pi$
$887$	11420.0i	0.432295i	−0.976361	+	0.216148i	$0.930651\pi$
$888$	0	0
$889$	14900.0	0.562126
$890$	0	0
$891$	10065.0	0.378440
$892$	0	0
$893$	50400.0i	1.88866i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	400.000i	0.0148892i
$898$	0	0
$899$	−4600.00	−0.170655
$900$	0	0
$901$	−6762.00	−0.250028
$902$	0	0
$903$	− 15000.0i	− 0.552789i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2620.00i	0.0959158i	0.998849	+	0.0479579i	$0.0152713\pi$
$907$	2620.00i	0.0959158i	−0.998849	+	0.0479579i	$0.984729\pi$
$908$	0	0
$909$	−3804.00	−0.138802
$910$	0	0
$911$	46500.0	1.69112	0.845562	−	0.533877i	$-0.179266\pi$
$911$	46500.0	1.69112	0.845562	+	0.533877i	$0.179266\pi$
$912$	0	0
$913$	1125.00i	0.0407799i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7800.00i	− 0.280893i
$918$	0	0
$919$	−45190.0	−1.62207	−0.811034	−	0.584999i	$-0.801095\pi$
$919$	−45190.0	−1.62207	−0.811034	+	0.584999i	$0.801095\pi$
$920$	0	0
$921$	9425.00	0.337203
$922$	0	0
$923$	− 320.000i	− 0.0114116i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2960.00i	− 0.104875i
$928$	0	0
$929$	−15166.0	−0.535609	−0.267804	−	0.963473i	$-0.586298\pi$
$929$	−15166.0	−0.535609	−0.267804	+	0.963473i	$0.586298\pi$
$930$	0	0
$931$	−25515.0	−0.898196
$932$	0	0
$933$	− 46250.0i	− 1.62289i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 23789.0i	− 0.829405i	−0.909957	−	0.414703i	$-0.863886\pi$
$937$	− 23789.0i	− 0.829405i	0.909957	−	0.414703i	$-0.136114\pi$
$938$	0	0
$939$	−40810.0	−1.41830
$940$	0	0
$941$	47472.0	1.64457	0.822286	−	0.569074i	$-0.192698\pi$
$941$	47472.0	1.64457	0.822286	+	0.569074i	$0.192698\pi$
$942$	0	0
$943$	− 1950.00i	− 0.0673391i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 19760.0i	− 0.678050i	−0.940777	−	0.339025i	$-0.889903\pi$
$947$	− 19760.0i	− 0.678050i	0.940777	−	0.339025i	$-0.110097\pi$
$948$	0	0
$949$	−2536.00	−0.0867461
$950$	0	0
$951$	−34620.0	−1.18047
$952$	0	0
$953$	− 36337.0i	− 1.23512i	−0.786523	−	0.617561i	$-0.788121\pi$
$953$	− 36337.0i	− 1.23512i	0.786523	−	0.617561i	$-0.211879\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1500.00i	0.0506668i
$958$	0	0
$959$	27490.0	0.925650
$960$	0	0
$961$	23109.0	0.775704
$962$	0	0
$963$	3890.00i	0.130170i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 33880.0i	− 1.12669i	−0.826222	−	0.563344i	$-0.809515\pi$
$967$	− 33880.0i	− 1.12669i	0.826222	−	0.563344i	$-0.190485\pi$
$968$	0	0
$969$	−11025.0	−0.365505
$970$	0	0
$971$	−17175.0	−0.567633	−0.283817	−	0.958879i	$-0.591601\pi$
$971$	−17175.0	−0.567633	−0.283817	+	0.958879i	$0.591601\pi$
$972$	0	0
$973$	− 7350.00i	− 0.242169i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13311.0i	− 0.435882i	−0.975962	−	0.217941i	$-0.930066\pi$
$977$	− 13311.0i	− 0.435882i	0.975962	−	0.217941i	$-0.0699340\pi$
$978$	0	0
$979$	−10575.0	−0.345228
$980$	0	0
$981$	492.000	0.0160126
$982$	0	0
$983$	53110.0i	1.72324i	0.507553	+	0.861621i	$0.330550\pi$
$983$	53110.0i	1.72324i	−0.507553	+	0.861621i	$0.669450\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24000.0i	0.773990i
$988$	0	0
$989$	−3000.00	−0.0964555
$990$	0	0
$991$	−8990.00	−0.288170	−0.144085	−	0.989565i	$-0.546024\pi$
$991$	−8990.00	−0.288170	−0.144085	+	0.989565i	$0.546024\pi$
$992$	0	0
$993$	40375.0i	1.29029i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1236.00i	0.0392623i	0.999807	+	0.0196311i	$0.00624919\pi$
$997$	1236.00i	0.0392623i	−0.999807	+	0.0196311i	$0.993751\pi$
$998$	0	0
$999$	7830.00	0.247978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.c.c.449.1		2
4.3	odd	2		800.4.c.d.449.2		2
5.2	odd	4		800.4.a.b.1.1	✓	1
5.3	odd	4		800.4.a.i.1.1	yes	1
5.4	even	2	inner	800.4.c.c.449.2		2
20.3	even	4		800.4.a.c.1.1	yes	1
20.7	even	4		800.4.a.j.1.1	yes	1
20.19	odd	2		800.4.c.d.449.1		2
40.3	even	4		1600.4.a.bp.1.1		1
40.13	odd	4		1600.4.a.l.1.1		1
40.27	even	4		1600.4.a.k.1.1		1
40.37	odd	4		1600.4.a.bq.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.b.1.1	✓	1	5.2	odd	4
800.4.a.c.1.1	yes	1	20.3	even	4
800.4.a.i.1.1	yes	1	5.3	odd	4
800.4.a.j.1.1	yes	1	20.7	even	4
800.4.c.c.449.1		2	1.1	even	1	trivial
800.4.c.c.449.2		2	5.4	even	2	inner
800.4.c.d.449.1		2	20.19	odd	2
800.4.c.d.449.2		2	4.3	odd	2
1600.4.a.k.1.1		1	40.27	even	4
1600.4.a.l.1.1		1	40.13	odd	4
1600.4.a.bp.1.1		1	40.3	even	4
1600.4.a.bq.1.1		1	40.37	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)

\(f(q)\)	\(=\)	\(q-5.00000i q^{3} -10.0000i q^{7} +2.00000 q^{9} -15.0000 q^{11} +8.00000i q^{13} +21.0000i q^{17} -105.000 q^{19} -50.0000 q^{21} +10.0000i q^{23} -145.000i q^{27} +20.0000 q^{29} -230.000 q^{31} +75.0000i q^{33} +54.0000i q^{37} +40.0000 q^{39} -195.000 q^{41} +300.000i q^{43} -480.000i q^{47} +243.000 q^{49} +105.000 q^{51} +322.000i q^{53} +525.000i q^{57} -560.000 q^{59} -730.000 q^{61} -20.0000i q^{63} +255.000i q^{67} +50.0000 q^{69} -40.0000 q^{71} +317.000i q^{73} +150.000i q^{77} +830.000 q^{79} -671.000 q^{81} -75.0000i q^{83} -100.000i q^{87} +705.000 q^{89} +80.0000 q^{91} +1150.00i q^{93} +1434.00i q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 30 q^{11} - 210 q^{19} - 100 q^{21} + 40 q^{29} - 460 q^{31} + 80 q^{39} - 390 q^{41} + 486 q^{49} + 210 q^{51} - 1120 q^{59} - 1460 q^{61} + 100 q^{69} - 80 q^{71} + 1660 q^{79} - 1342 q^{81}+ \cdots - 60 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 30 * q^11 - 210 * q^19 - 100 * q^21 + 40 * q^29 - 460 * q^31 + 80 * q^39 - 390 * q^41 + 486 * q^49 + 210 * q^51 - 1120 * q^59 - 1460 * q^61 + 100 * q^69 - 80 * q^71 + 1660 * q^79 - 1342 * q^81 + 1410 * q^89 + 160 * q^91 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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