LMFDB - Embedded newform 2304.4.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

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:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.3923 q^{5} -3.46410 q^{7} +O(q^{10})$	$q+10.3923 q^{5} -3.46410 q^{7} +55.4256 q^{13} +90.0000 q^{17} +116.000 q^{19} +103.923 q^{23} -17.0000 q^{25} -259.808 q^{29} +301.377 q^{31} -36.0000 q^{35} -34.6410 q^{37} -54.0000 q^{41} +20.0000 q^{43} +394.908 q^{47} -331.000 q^{49} +488.438 q^{53} -324.000 q^{59} -575.041 q^{61} +576.000 q^{65} -116.000 q^{67} -1101.58 q^{71} -1106.00 q^{73} +148.956 q^{79} +1152.00 q^{83} +935.307 q^{85} +918.000 q^{89} -192.000 q^{91} +1205.51 q^{95} +190.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 180 q^{17} + 232 q^{19} - 34 q^{25} - 72 q^{35} - 108 q^{41} + 40 q^{43} - 662 q^{49} - 648 q^{59} + 1152 q^{65} - 232 q^{67} - 2212 q^{73} + 2304 q^{83} + 1836 q^{89} - 384 q^{91} + 380 q^{97}+O(q^{100}) $ 2 * q + 180 * q^17 + 232 * q^19 - 34 * q^25 - 72 * q^35 - 108 * q^41 + 40 * q^43 - 662 * q^49 - 648 * q^59 + 1152 * q^65 - 232 * q^67 - 2212 * q^73 + 2304 * q^83 + 1836 * q^89 - 384 * q^91 + 380 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	10.3923	0.929516	0.464758	−	0.885438i	$-0.346141\pi$
$5$	10.3923	0.929516	0.464758	+	0.885438i	$0.346141\pi$
$6$	0	0
$7$	−3.46410	−0.187044	−0.0935220	−	0.995617i	$-0.529813\pi$
$7$	−3.46410	−0.187044	−0.0935220	+	0.995617i	$0.529813\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	55.4256	1.18248	0.591242	−	0.806494i	$-0.298638\pi$
$13$	55.4256	1.18248	0.591242	+	0.806494i	$0.298638\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	90.0000	1.28401	0.642006	−	0.766700i	$-0.278102\pi$
$17$	90.0000	1.28401	0.642006	+	0.766700i	$0.278102\pi$
$18$	0	0
$19$	116.000	1.40064	0.700322	−	0.713827i	$-0.253040\pi$
$19$	116.000	1.40064	0.700322	+	0.713827i	$0.253040\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	103.923	0.942150	0.471075	−	0.882093i	$-0.343866\pi$
$23$	103.923	0.942150	0.471075	+	0.882093i	$0.343866\pi$
$24$	0	0
$25$	−17.0000	−0.136000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−259.808	−1.66362	−0.831811	−	0.555058i	$-0.812696\pi$
$29$	−259.808	−1.66362	−0.831811	+	0.555058i	$0.812696\pi$
$30$	0	0
$31$	301.377	1.74609	0.873046	−	0.487637i	$-0.162141\pi$
$31$	301.377	1.74609	0.873046	+	0.487637i	$0.162141\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−36.0000	−0.173860
$36$	0	0
$37$	−34.6410	−0.153918	−0.0769588	−	0.997034i	$-0.524521\pi$
$37$	−34.6410	−0.153918	−0.0769588	+	0.997034i	$0.524521\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−54.0000	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$41$	−54.0000	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$42$	0	0
$43$	20.0000	0.0709296	0.0354648	−	0.999371i	$-0.488709\pi$
$43$	20.0000	0.0709296	0.0354648	+	0.999371i	$0.488709\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	394.908	1.22560	0.612800	−	0.790238i	$-0.290043\pi$
$47$	394.908	1.22560	0.612800	+	0.790238i	$0.290043\pi$
$48$	0	0
$49$	−331.000	−0.965015
$50$	0	0
$51$	0	0
$52$	0	0
$53$	488.438	1.26589	0.632945	−	0.774197i	$-0.281846\pi$
$53$	488.438	1.26589	0.632945	+	0.774197i	$0.281846\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−324.000	−0.714936	−0.357468	−	0.933925i	$-0.616360\pi$
$59$	−324.000	−0.714936	−0.357468	+	0.933925i	$0.616360\pi$
$60$	0	0
$61$	−575.041	−1.20699	−0.603495	−	0.797366i	$-0.706226\pi$
$61$	−575.041	−1.20699	−0.603495	+	0.797366i	$0.706226\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	576.000	1.09914
$66$	0	0
$67$	−116.000	−0.211517	−0.105759	−	0.994392i	$-0.533727\pi$
$67$	−116.000	−0.211517	−0.105759	+	0.994392i	$0.533727\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1101.58	−1.84132	−0.920662	−	0.390361i	$-0.872350\pi$
$71$	−1101.58	−1.84132	−0.920662	+	0.390361i	$0.872350\pi$
$72$	0	0
$73$	−1106.00	−1.77325	−0.886627	−	0.462486i	$-0.846958\pi$
$73$	−1106.00	−1.77325	−0.886627	+	0.462486i	$0.846958\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	148.956	0.212138	0.106069	−	0.994359i	$-0.466174\pi$
$79$	148.956	0.212138	0.106069	+	0.994359i	$0.466174\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1152.00	1.52348	0.761738	−	0.647886i	$-0.224347\pi$
$83$	1152.00	1.52348	0.761738	+	0.647886i	$0.224347\pi$
$84$	0	0
$85$	935.307	1.19351
$86$	0	0
$87$	0	0
$88$	0	0
$89$	918.000	1.09335	0.546673	−	0.837346i	$-0.315894\pi$
$89$	918.000	1.09335	0.546673	+	0.837346i	$0.315894\pi$
$90$	0	0
$91$	−192.000	−0.221177
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1205.51	1.30192
$96$	0	0
$97$	190.000	0.198882	0.0994411	−	0.995043i	$-0.468295\pi$
$97$	190.000	0.198882	0.0994411	+	0.995043i	$0.468295\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.3923	−0.0102383	−0.00511917	−	0.999987i	$-0.501629\pi$
$101$	−10.3923	−0.0102383	−0.00511917	+	0.999987i	$0.501629\pi$
$102$	0	0
$103$	793.279	0.758875	0.379438	−	0.925217i	$-0.376118\pi$
$103$	793.279	0.758875	0.379438	+	0.925217i	$0.376118\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	252.000	0.227680	0.113840	−	0.993499i	$-0.463685\pi$
$107$	252.000	0.227680	0.113840	+	0.993499i	$0.463685\pi$
$108$	0	0
$109$	−457.261	−0.401814	−0.200907	−	0.979610i	$-0.564389\pi$
$109$	−457.261	−0.401814	−0.200907	+	0.979610i	$0.564389\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2214.00	1.84315	0.921573	−	0.388204i	$-0.126904\pi$
$113$	2214.00	1.84315	0.921573	+	0.388204i	$0.126904\pi$
$114$	0	0
$115$	1080.00	0.875744
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−311.769	−0.240167
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1475.71	−1.05593
$126$	0	0
$127$	696.284	0.486498	0.243249	−	0.969964i	$-0.421787\pi$
$127$	696.284	0.486498	0.243249	+	0.969964i	$0.421787\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2268.00	1.51264	0.756321	−	0.654201i	$-0.226995\pi$
$131$	2268.00	1.51264	0.756321	+	0.654201i	$0.226995\pi$
$132$	0	0
$133$	−401.836	−0.261982
$134$	0	0
$135$	0	0
$136$	0	0
$137$	522.000	0.325529	0.162764	−	0.986665i	$-0.447959\pi$
$137$	522.000	0.325529	0.162764	+	0.986665i	$0.447959\pi$
$138$	0	0
$139$	676.000	0.412501	0.206250	−	0.978499i	$-0.433874\pi$
$139$	676.000	0.412501	0.206250	+	0.978499i	$0.433874\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2700.00	−1.54636
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1465.31	0.805660	0.402830	−	0.915275i	$-0.368027\pi$
$149$	1465.31	0.805660	0.402830	+	0.915275i	$0.368027\pi$
$150$	0	0
$151$	−2386.77	−1.28631	−0.643153	−	0.765738i	$-0.722374\pi$
$151$	−2386.77	−1.28631	−0.643153	+	0.765738i	$0.722374\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3132.00	1.62302
$156$	0	0
$157$	2016.11	1.02486	0.512430	−	0.858729i	$-0.328746\pi$
$157$	2016.11	1.02486	0.512430	+	0.858729i	$0.328746\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−360.000	−0.176223
$162$	0	0
$163$	−388.000	−0.186445	−0.0932224	−	0.995645i	$-0.529717\pi$
$163$	−388.000	−0.186445	−0.0932224	+	0.995645i	$0.529717\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2203.17	1.02088	0.510438	−	0.859915i	$-0.329483\pi$
$167$	2203.17	1.02088	0.510438	+	0.859915i	$0.329483\pi$
$168$	0	0
$169$	875.000	0.398270
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−197.454	−0.0867753	−0.0433877	−	0.999058i	$-0.513815\pi$
$173$	−197.454	−0.0867753	−0.0433877	+	0.999058i	$0.513815\pi$
$174$	0	0
$175$	58.8897	0.0254380
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2844.00	−1.18754	−0.593772	−	0.804633i	$-0.702362\pi$
$179$	−2844.00	−1.18754	−0.593772	+	0.804633i	$0.702362\pi$
$180$	0	0
$181$	96.9948	0.0398319	0.0199159	−	0.999802i	$-0.493660\pi$
$181$	96.9948	0.0398319	0.0199159	+	0.999802i	$0.493660\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−360.000	−0.143069
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3200.83	−1.21259	−0.606293	−	0.795241i	$-0.707344\pi$
$191$	−3200.83	−1.21259	−0.606293	+	0.795241i	$0.707344\pi$
$192$	0	0
$193$	−1342.00	−0.500514	−0.250257	−	0.968179i	$-0.580515\pi$
$193$	−1342.00	−0.500514	−0.250257	+	0.968179i	$0.580515\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−966.484	−0.349539	−0.174769	−	0.984609i	$-0.555918\pi$
$197$	−966.484	−0.349539	−0.174769	+	0.984609i	$0.555918\pi$
$198$	0	0
$199$	−1784.01	−0.635504	−0.317752	−	0.948174i	$-0.602928\pi$
$199$	−1784.01	−0.635504	−0.317752	+	0.948174i	$0.602928\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	900.000	0.311171
$204$	0	0
$205$	−561.184	−0.191194
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2764.00	0.901809	0.450904	−	0.892572i	$-0.351102\pi$
$211$	2764.00	0.901809	0.450904	+	0.892572i	$0.351102\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	207.846	0.0659302
$216$	0	0
$217$	−1044.00	−0.326596
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4988.31	1.51832
$222$	0	0
$223$	−4292.02	−1.28886	−0.644428	−	0.764665i	$-0.722905\pi$
$223$	−4292.02	−1.28886	−0.644428	+	0.764665i	$0.722905\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5688.00	1.66311	0.831555	−	0.555443i	$-0.187451\pi$
$227$	5688.00	1.66311	0.831555	+	0.555443i	$0.187451\pi$
$228$	0	0
$229$	−5570.28	−1.60740	−0.803699	−	0.595036i	$-0.797138\pi$
$229$	−5570.28	−1.60740	−0.803699	+	0.595036i	$0.797138\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2718.00	0.764215	0.382108	−	0.924118i	$-0.375198\pi$
$233$	2718.00	0.764215	0.382108	+	0.924118i	$0.375198\pi$
$234$	0	0
$235$	4104.00	1.13921
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3574.95	−0.967550	−0.483775	−	0.875192i	$-0.660735\pi$
$239$	−3574.95	−0.967550	−0.483775	+	0.875192i	$0.660735\pi$
$240$	0	0
$241$	4490.00	1.20011	0.600055	−	0.799959i	$-0.295146\pi$
$241$	4490.00	1.20011	0.600055	+	0.799959i	$0.295146\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3439.85	−0.896996
$246$	0	0
$247$	6429.37	1.65624
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4608.00	1.15878	0.579391	−	0.815050i	$-0.303290\pi$
$251$	4608.00	1.15878	0.579391	+	0.815050i	$0.303290\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4626.00	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−4626.00	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	120.000	0.0287893
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1995.32	−0.467821	−0.233910	−	0.972258i	$-0.575152\pi$
$263$	−1995.32	−0.467821	−0.233910	+	0.972258i	$0.575152\pi$
$264$	0	0
$265$	5076.00	1.17666
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3148.87	0.713717	0.356859	−	0.934158i	$-0.383848\pi$
$269$	3148.87	0.713717	0.356859	+	0.934158i	$0.383848\pi$
$270$	0	0
$271$	−5345.11	−1.19813	−0.599063	−	0.800702i	$-0.704460\pi$
$271$	−5345.11	−1.19813	−0.599063	+	0.800702i	$0.704460\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6526.37	1.41564	0.707818	−	0.706394i	$-0.249679\pi$
$277$	6526.37	1.41564	0.707818	+	0.706394i	$0.249679\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1170.00	−0.248386	−0.124193	−	0.992258i	$-0.539634\pi$
$281$	−1170.00	−0.248386	−0.124193	+	0.992258i	$0.539634\pi$
$282$	0	0
$283$	5740.00	1.20568	0.602840	−	0.797862i	$-0.294036\pi$
$283$	5740.00	1.20568	0.602840	+	0.797862i	$0.294036\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	187.061	0.0384735
$288$	0	0
$289$	3187.00	0.648687
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7991.68	−1.59344	−0.796722	−	0.604346i	$-0.793434\pi$
$293$	−7991.68	−1.59344	−0.796722	+	0.604346i	$0.793434\pi$
$294$	0	0
$295$	−3367.11	−0.664544
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5760.00	1.11408
$300$	0	0
$301$	−69.2820	−0.0132669
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5976.00	−1.12192
$306$	0	0
$307$	5452.00	1.01356	0.506779	−	0.862076i	$-0.330836\pi$
$307$	5452.00	1.01356	0.506779	+	0.862076i	$0.330836\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2203.17	0.401705	0.200852	−	0.979622i	$-0.435629\pi$
$311$	2203.17	0.401705	0.200852	+	0.979622i	$0.435629\pi$
$312$	0	0
$313$	1034.00	0.186726	0.0933628	−	0.995632i	$-0.470238\pi$
$313$	1034.00	0.186726	0.0933628	+	0.995632i	$0.470238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2650.04	0.469530	0.234765	−	0.972052i	$-0.424568\pi$
$317$	2650.04	0.469530	0.234765	+	0.972052i	$0.424568\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10440.0	1.79844
$324$	0	0
$325$	−942.236	−0.160818
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1368.00	−0.229241
$330$	0	0
$331$	4132.00	0.686149	0.343074	−	0.939308i	$-0.388532\pi$
$331$	4132.00	0.686149	0.343074	+	0.939308i	$0.388532\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1205.51	−0.196609
$336$	0	0
$337$	−458.000	−0.0740322	−0.0370161	−	0.999315i	$-0.511785\pi$
$337$	−458.000	−0.0740322	−0.0370161	+	0.999315i	$0.511785\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2334.80	0.367544
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11016.0	1.70424	0.852118	−	0.523350i	$-0.175318\pi$
$347$	11016.0	1.70424	0.852118	+	0.523350i	$0.175318\pi$
$348$	0	0
$349$	2528.79	0.387860	0.193930	−	0.981015i	$-0.437876\pi$
$349$	2528.79	0.387860	0.193930	+	0.981015i	$0.437876\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5562.00	−0.838627	−0.419314	−	0.907841i	$-0.637729\pi$
$353$	−5562.00	−0.838627	−0.419314	+	0.907841i	$0.637729\pi$
$354$	0	0
$355$	−11448.0	−1.71154
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8875.03	1.30475	0.652376	−	0.757895i	$-0.273772\pi$
$359$	8875.03	1.30475	0.652376	+	0.757895i	$0.273772\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11493.9	−1.64827
$366$	0	0
$367$	12799.9	1.82056	0.910282	−	0.413989i	$-0.135865\pi$
$367$	12799.9	1.82056	0.910282	+	0.413989i	$0.135865\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1692.00	−0.236777
$372$	0	0
$373$	−4981.38	−0.691491	−0.345745	−	0.938328i	$-0.612374\pi$
$373$	−4981.38	−0.691491	−0.345745	+	0.938328i	$0.612374\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14400.0	−1.96721
$378$	0	0
$379$	−9892.00	−1.34068	−0.670340	−	0.742054i	$-0.733852\pi$
$379$	−9892.00	−1.34068	−0.670340	+	0.742054i	$0.733852\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8771.11	1.17019	0.585095	−	0.810965i	$-0.301057\pi$
$383$	8771.11	1.17019	0.585095	+	0.810965i	$0.301057\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9903.87	1.29086	0.645432	−	0.763818i	$-0.276677\pi$
$389$	9903.87	1.29086	0.645432	+	0.763818i	$0.276677\pi$
$390$	0	0
$391$	9353.07	1.20973
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1548.00	0.197186
$396$	0	0
$397$	−103.923	−0.0131379	−0.00656895	−	0.999978i	$-0.502091\pi$
$397$	−103.923	−0.0131379	−0.00656895	+	0.999978i	$0.502091\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1062.00	−0.132254	−0.0661269	−	0.997811i	$-0.521064\pi$
$401$	−1062.00	−0.132254	−0.0661269	+	0.997811i	$0.521064\pi$
$402$	0	0
$403$	16704.0	2.06473
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8614.00	1.04141	0.520703	−	0.853738i	$-0.325670\pi$
$409$	8614.00	1.04141	0.520703	+	0.853738i	$0.325670\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1122.37	0.133724
$414$	0	0
$415$	11971.9	1.41609
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10440.0	1.21725	0.608625	−	0.793458i	$-0.291722\pi$
$419$	10440.0	1.21725	0.608625	+	0.793458i	$0.291722\pi$
$420$	0	0
$421$	900.666	0.104266	0.0521328	−	0.998640i	$-0.483398\pi$
$421$	900.666	0.104266	0.0521328	+	0.998640i	$0.483398\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1530.00	−0.174626
$426$	0	0
$427$	1992.00	0.225760
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−394.908	−0.0441346	−0.0220673	−	0.999756i	$-0.507025\pi$
$431$	−394.908	−0.0441346	−0.0220673	+	0.999756i	$0.507025\pi$
$432$	0	0
$433$	12958.0	1.43816	0.719078	−	0.694929i	$-0.244564\pi$
$433$	12958.0	1.43816	0.719078	+	0.694929i	$0.244564\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12055.1	1.31962
$438$	0	0
$439$	−11441.9	−1.24395	−0.621974	−	0.783038i	$-0.713669\pi$
$439$	−11441.9	−1.24395	−0.621974	+	0.783038i	$0.713669\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1800.00	0.193049	0.0965244	−	0.995331i	$-0.469227\pi$
$443$	1800.00	0.193049	0.0965244	+	0.995331i	$0.469227\pi$
$444$	0	0
$445$	9540.14	1.01628
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13626.0	1.43218	0.716092	−	0.698006i	$-0.245929\pi$
$449$	13626.0	1.43218	0.716092	+	0.698006i	$0.245929\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1995.32	−0.205587
$456$	0	0
$457$	−12602.0	−1.28993	−0.644964	−	0.764213i	$-0.723127\pi$
$457$	−12602.0	−1.28993	−0.644964	+	0.764213i	$0.723127\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1839.44	0.185838	0.0929188	−	0.995674i	$-0.470380\pi$
$461$	1839.44	0.185838	0.0929188	+	0.995674i	$0.470380\pi$
$462$	0	0
$463$	−11012.4	−1.10538	−0.552688	−	0.833389i	$-0.686398\pi$
$463$	−11012.4	−1.10538	−0.552688	+	0.833389i	$0.686398\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9144.00	0.906068	0.453034	−	0.891493i	$-0.350342\pi$
$467$	9144.00	0.906068	0.453034	+	0.891493i	$0.350342\pi$
$468$	0	0
$469$	401.836	0.0395630
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1972.00	−0.190488
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6173.03	−0.588837	−0.294418	−	0.955677i	$-0.595126\pi$
$479$	−6173.03	−0.588837	−0.294418	+	0.955677i	$0.595126\pi$
$480$	0	0
$481$	−1920.00	−0.182005
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1974.54	0.184864
$486$	0	0
$487$	−3204.29	−0.298153	−0.149076	−	0.988826i	$-0.547630\pi$
$487$	−3204.29	−0.298153	−0.149076	+	0.988826i	$0.547630\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−396.000	−0.0363976	−0.0181988	−	0.999834i	$-0.505793\pi$
$491$	−396.000	−0.0363976	−0.0181988	+	0.999834i	$0.505793\pi$
$492$	0	0
$493$	−23382.7	−2.13611
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3816.00	0.344408
$498$	0	0
$499$	12436.0	1.11565	0.557827	−	0.829957i	$-0.311635\pi$
$499$	12436.0	1.11565	0.557827	+	0.829957i	$0.311635\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16482.2	−1.46104	−0.730522	−	0.682890i	$-0.760723\pi$
$503$	−16482.2	−1.46104	−0.730522	+	0.682890i	$0.760723\pi$
$504$	0	0
$505$	−108.000	−0.00951671
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9155.62	−0.797280	−0.398640	−	0.917107i	$-0.630518\pi$
$509$	−9155.62	−0.797280	−0.398640	+	0.917107i	$0.630518\pi$
$510$	0	0
$511$	3831.30	0.331676
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8244.00	0.705386
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7650.00	0.643287	0.321644	−	0.946861i	$-0.395765\pi$
$521$	7650.00	0.643287	0.321644	+	0.946861i	$0.395765\pi$
$522$	0	0
$523$	18332.0	1.53270	0.766350	−	0.642423i	$-0.222071\pi$
$523$	18332.0	1.53270	0.766350	+	0.642423i	$0.222071\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27123.9	2.24200
$528$	0	0
$529$	−1367.00	−0.112353
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2992.98	−0.243228
$534$	0	0
$535$	2618.86	0.211632
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16863.2	−1.34012	−0.670062	−	0.742305i	$-0.733733\pi$
$541$	−16863.2	−1.34012	−0.670062	+	0.742305i	$0.733733\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4752.00	−0.373492
$546$	0	0
$547$	−1684.00	−0.131632	−0.0658159	−	0.997832i	$-0.520965\pi$
$547$	−1684.00	−0.131632	−0.0658159	+	0.997832i	$0.520965\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−30137.7	−2.33014
$552$	0	0
$553$	−516.000	−0.0396791
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2275.91	−0.173130	−0.0865652	−	0.996246i	$-0.527589\pi$
$557$	−2275.91	−0.173130	−0.0865652	+	0.996246i	$0.527589\pi$
$558$	0	0
$559$	1108.51	0.0838731
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7992.00	−0.598264	−0.299132	−	0.954212i	$-0.596697\pi$
$563$	−7992.00	−0.598264	−0.299132	+	0.954212i	$0.596697\pi$
$564$	0	0
$565$	23008.6	1.71323
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5526.00	−0.407139	−0.203569	−	0.979061i	$-0.565254\pi$
$569$	−5526.00	−0.407139	−0.203569	+	0.979061i	$0.565254\pi$
$570$	0	0
$571$	13420.0	0.983554	0.491777	−	0.870721i	$-0.336347\pi$
$571$	13420.0	0.983554	0.491777	+	0.870721i	$0.336347\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1766.69	−0.128132
$576$	0	0
$577$	−10178.0	−0.734343	−0.367171	−	0.930153i	$-0.619674\pi$
$577$	−10178.0	−0.734343	−0.367171	+	0.930153i	$0.619674\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3990.65	−0.284957
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18684.0	−1.31375	−0.656875	−	0.754000i	$-0.728122\pi$
$587$	−18684.0	−1.31375	−0.656875	+	0.754000i	$0.728122\pi$
$588$	0	0
$589$	34959.7	2.44565
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5094.00	0.352758	0.176379	−	0.984322i	$-0.443562\pi$
$593$	5094.00	0.352758	0.176379	+	0.984322i	$0.443562\pi$
$594$	0	0
$595$	−3240.00	−0.223239
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19433.6	1.32560	0.662801	−	0.748795i	$-0.269367\pi$
$599$	19433.6	1.32560	0.662801	+	0.748795i	$0.269367\pi$
$600$	0	0
$601$	−27722.0	−1.88154	−0.940769	−	0.339049i	$-0.889895\pi$
$601$	−27722.0	−1.88154	−0.940769	+	0.339049i	$0.889895\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13832.2	−0.929516
$606$	0	0
$607$	−26684.0	−1.78430	−0.892149	−	0.451741i	$-0.850803\pi$
$607$	−26684.0	−1.78430	−0.892149	+	0.451741i	$0.850803\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21888.0	1.44925
$612$	0	0
$613$	16911.7	1.11429	0.557144	−	0.830416i	$-0.311897\pi$
$613$	16911.7	1.11429	0.557144	+	0.830416i	$0.311897\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17694.0	1.15451	0.577256	−	0.816563i	$-0.304124\pi$
$617$	17694.0	1.15451	0.577256	+	0.816563i	$0.304124\pi$
$618$	0	0
$619$	−13652.0	−0.886462	−0.443231	−	0.896407i	$-0.646168\pi$
$619$	−13652.0	−0.886462	−0.443231	+	0.896407i	$0.646168\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3180.05	−0.204504
$624$	0	0
$625$	−13211.0	−0.845504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3117.69	−0.197632
$630$	0	0
$631$	−9162.55	−0.578059	−0.289030	−	0.957320i	$-0.593333\pi$
$631$	−9162.55	−0.578059	−0.289030	+	0.957320i	$0.593333\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7236.00	0.452208
$636$	0	0
$637$	−18345.9	−1.14112
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5202.00	0.320541	0.160270	−	0.987073i	$-0.448763\pi$
$641$	5202.00	0.320541	0.160270	+	0.987073i	$0.448763\pi$
$642$	0	0
$643$	−15892.0	−0.974680	−0.487340	−	0.873212i	$-0.662033\pi$
$643$	−15892.0	−0.974680	−0.487340	+	0.873212i	$0.662033\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−478.046	−0.0290478	−0.0145239	−	0.999895i	$-0.504623\pi$
$647$	−478.046	−0.0290478	−0.0145239	+	0.999895i	$0.504623\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24660.9	1.47788	0.738941	−	0.673770i	$-0.235326\pi$
$653$	24660.9	1.47788	0.738941	+	0.673770i	$0.235326\pi$
$654$	0	0
$655$	23569.7	1.40602
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28260.0	−1.67049	−0.835245	−	0.549878i	$-0.814674\pi$
$659$	−28260.0	−1.67049	−0.835245	+	0.549878i	$0.814674\pi$
$660$	0	0
$661$	−25863.0	−1.52187	−0.760933	−	0.648830i	$-0.775258\pi$
$661$	−25863.0	−1.52187	−0.760933	+	0.648830i	$0.775258\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4176.00	−0.243516
$666$	0	0
$667$	−27000.0	−1.56738
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	190.000	0.0108826	0.00544128	−	0.999985i	$-0.498268\pi$
$673$	190.000	0.0108826	0.00544128	+	0.999985i	$0.498268\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4998.70	0.283775	0.141887	−	0.989883i	$-0.454683\pi$
$677$	4998.70	0.283775	0.141887	+	0.989883i	$0.454683\pi$
$678$	0	0
$679$	−658.179	−0.0371997
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8064.00	0.451772	0.225886	−	0.974154i	$-0.427472\pi$
$683$	8064.00	0.451772	0.225886	+	0.974154i	$0.427472\pi$
$684$	0	0
$685$	5424.78	0.302584
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27072.0	1.49690
$690$	0	0
$691$	19244.0	1.05944	0.529722	−	0.848171i	$-0.322296\pi$
$691$	19244.0	1.05944	0.529722	+	0.848171i	$0.322296\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7025.20	0.383426
$696$	0	0
$697$	−4860.00	−0.264111
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6204.21	−0.334279	−0.167140	−	0.985933i	$-0.553453\pi$
$701$	−6204.21	−0.334279	−0.167140	+	0.985933i	$0.553453\pi$
$702$	0	0
$703$	−4018.36	−0.215584
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.0000	0.00191502
$708$	0	0
$709$	−15020.3	−0.795629	−0.397814	−	0.917466i	$-0.630231\pi$
$709$	−15020.3	−0.795629	−0.397814	+	0.917466i	$0.630231\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31320.0	1.64508
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30740.4	−1.59447	−0.797236	−	0.603668i	$-0.793705\pi$
$719$	−30740.4	−1.59447	−0.797236	+	0.603668i	$0.793705\pi$
$720$	0	0
$721$	−2748.00	−0.141943
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4416.73	0.226253
$726$	0	0
$727$	−12127.8	−0.618701	−0.309351	−	0.950948i	$-0.600112\pi$
$727$	−12127.8	−0.618701	−0.309351	+	0.950948i	$0.600112\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1800.00	0.0910744
$732$	0	0
$733$	12387.6	0.624212	0.312106	−	0.950047i	$-0.398966\pi$
$733$	12387.6	0.624212	0.312106	+	0.950047i	$0.398966\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−11180.0	−0.556513	−0.278256	−	0.960507i	$-0.589756\pi$
$739$	−11180.0	−0.556513	−0.278256	+	0.960507i	$0.589756\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35500.1	1.75286	0.876429	−	0.481532i	$-0.159919\pi$
$743$	35500.1	1.75286	0.876429	+	0.481532i	$0.159919\pi$
$744$	0	0
$745$	15228.0	0.748873
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−872.954	−0.0425862
$750$	0	0
$751$	−37970.0	−1.84493	−0.922467	−	0.386076i	$-0.873830\pi$
$751$	−37970.0	−1.84493	−0.922467	+	0.386076i	$0.873830\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−24804.0	−1.19564
$756$	0	0
$757$	−39047.4	−1.87477	−0.937385	−	0.348296i	$-0.886760\pi$
$757$	−39047.4	−1.87477	−0.937385	+	0.348296i	$0.886760\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12222.0	−0.582191	−0.291095	−	0.956694i	$-0.594020\pi$
$761$	−12222.0	−0.582191	−0.291095	+	0.956694i	$0.594020\pi$
$762$	0	0
$763$	1584.00	0.0751568
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17957.9	−0.845401
$768$	0	0
$769$	−34030.0	−1.59578	−0.797889	−	0.602804i	$-0.794050\pi$
$769$	−34030.0	−1.59578	−0.797889	+	0.602804i	$0.794050\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4873.99	−0.226786	−0.113393	−	0.993550i	$-0.536172\pi$
$773$	−4873.99	−0.226786	−0.113393	+	0.993550i	$0.536172\pi$
$774$	0	0
$775$	−5123.41	−0.237469
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6264.00	−0.288102
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20952.0	0.952623
$786$	0	0
$787$	−30988.0	−1.40356	−0.701781	−	0.712393i	$-0.747611\pi$
$787$	−30988.0	−1.40356	−0.701781	+	0.712393i	$0.747611\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7669.52	−0.344749
$792$	0	0
$793$	−31872.0	−1.42725
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7160.30	0.318232	0.159116	−	0.987260i	$-0.449136\pi$
$797$	7160.30	0.318232	0.159116	+	0.987260i	$0.449136\pi$
$798$	0	0
$799$	35541.7	1.57369
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3741.23	−0.163803
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37530.0	1.63101	0.815503	−	0.578752i	$-0.196460\pi$
$809$	37530.0	1.63101	0.815503	+	0.578752i	$0.196460\pi$
$810$	0	0
$811$	−10852.0	−0.469871	−0.234935	−	0.972011i	$-0.575488\pi$
$811$	−10852.0	−0.469871	−0.234935	+	0.972011i	$0.575488\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4032.21	−0.173303
$816$	0	0
$817$	2320.00	0.0993470
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31353.6	−1.33282	−0.666411	−	0.745584i	$-0.732171\pi$
$821$	−31353.6	−1.33282	−0.666411	+	0.745584i	$0.732171\pi$
$822$	0	0
$823$	32947.1	1.39546	0.697729	−	0.716361i	$-0.254194\pi$
$823$	32947.1	1.39546	0.697729	+	0.716361i	$0.254194\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10044.0	0.422327	0.211163	−	0.977451i	$-0.432275\pi$
$827$	10044.0	0.422327	0.211163	+	0.977451i	$0.432275\pi$
$828$	0	0
$829$	9796.48	0.410429	0.205215	−	0.978717i	$-0.434211\pi$
$829$	9796.48	0.410429	0.205215	+	0.978717i	$0.434211\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−29790.0	−1.23909
$834$	0	0
$835$	22896.0	0.948921
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21054.8	0.866380	0.433190	−	0.901303i	$-0.357388\pi$
$839$	21054.8	0.866380	0.433190	+	0.901303i	$0.357388\pi$
$840$	0	0
$841$	43111.0	1.76764
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9093.27	0.370199
$846$	0	0
$847$	4610.72	0.187044
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3600.00	−0.145013
$852$	0	0
$853$	40703.2	1.63382	0.816911	−	0.576763i	$-0.195684\pi$
$853$	40703.2	1.63382	0.816911	+	0.576763i	$0.195684\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18342.0	−0.731098	−0.365549	−	0.930792i	$-0.619119\pi$
$857$	−18342.0	−0.731098	−0.365549	+	0.930792i	$0.619119\pi$
$858$	0	0
$859$	26324.0	1.04559	0.522796	−	0.852458i	$-0.324889\pi$
$859$	26324.0	1.04559	0.522796	+	0.852458i	$0.324889\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12761.8	0.503378	0.251689	−	0.967808i	$-0.419014\pi$
$863$	12761.8	0.503378	0.251689	+	0.967808i	$0.419014\pi$
$864$	0	0
$865$	−2052.00	−0.0806591
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−6429.37	−0.250116
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5112.00	0.197505
$876$	0	0
$877$	−2459.51	−0.0946999	−0.0473500	−	0.998878i	$-0.515078\pi$
$877$	−2459.51	−0.0946999	−0.0473500	+	0.998878i	$0.515078\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37314.0	−1.42695	−0.713474	−	0.700682i	$-0.752879\pi$
$881$	−37314.0	−1.42695	−0.713474	+	0.700682i	$0.752879\pi$
$882$	0	0
$883$	−18244.0	−0.695311	−0.347655	−	0.937622i	$-0.613022\pi$
$883$	−18244.0	−0.695311	−0.347655	+	0.937622i	$0.613022\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17957.9	−0.679783	−0.339891	−	0.940465i	$-0.610390\pi$
$887$	−17957.9	−0.679783	−0.339891	+	0.940465i	$0.610390\pi$
$888$	0	0
$889$	−2412.00	−0.0909965
$890$	0	0
$891$	0	0
$892$	0	0
$893$	45809.3	1.71663
$894$	0	0
$895$	−29555.7	−1.10384
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−78300.0	−2.90484
$900$	0	0
$901$	43959.4	1.62542
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1008.00	0.0370244
$906$	0	0
$907$	16388.0	0.599950	0.299975	−	0.953947i	$-0.403022\pi$
$907$	16388.0	0.599950	0.299975	+	0.953947i	$0.403022\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25107.8	−0.913127	−0.456564	−	0.889691i	$-0.650920\pi$
$911$	−25107.8	−0.913127	−0.456564	+	0.889691i	$0.650920\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7856.58	−0.282930
$918$	0	0
$919$	27155.1	0.974716	0.487358	−	0.873202i	$-0.337961\pi$
$919$	27155.1	0.974716	0.487358	+	0.873202i	$0.337961\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−61056.0	−2.17734
$924$	0	0
$925$	588.897	0.0209328
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48006.0	−1.69540	−0.847700	−	0.530477i	$-0.822013\pi$
$929$	−48006.0	−1.69540	−0.847700	+	0.530477i	$0.822013\pi$
$930$	0	0
$931$	−38396.0	−1.35164
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7894.00	−0.275225	−0.137612	−	0.990486i	$-0.543943\pi$
$937$	−7894.00	−0.275225	−0.137612	+	0.990486i	$0.543943\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2670.82	0.0925253	0.0462627	−	0.998929i	$-0.485269\pi$
$941$	2670.82	0.0925253	0.0462627	+	0.998929i	$0.485269\pi$
$942$	0	0
$943$	−5611.84	−0.193793
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22356.0	0.767130	0.383565	−	0.923514i	$-0.374696\pi$
$947$	22356.0	0.767130	0.383565	+	0.923514i	$0.374696\pi$
$948$	0	0
$949$	−61300.7	−2.09685
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14958.0	−0.508434	−0.254217	−	0.967147i	$-0.581818\pi$
$953$	−14958.0	−0.508434	−0.254217	+	0.967147i	$0.581818\pi$
$954$	0	0
$955$	−33264.0	−1.12712
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1808.26	−0.0608882
$960$	0	0
$961$	61037.0	2.04884
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13946.5	−0.465236
$966$	0	0
$967$	17476.4	0.581182	0.290591	−	0.956847i	$-0.406148\pi$
$967$	17476.4	0.581182	0.290591	+	0.956847i	$0.406148\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−48528.0	−1.60385	−0.801925	−	0.597425i	$-0.796190\pi$
$971$	−48528.0	−1.60385	−0.801925	+	0.597425i	$0.796190\pi$
$972$	0	0
$973$	−2341.73	−0.0771557
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39978.0	1.30912	0.654560	−	0.756010i	$-0.272854\pi$
$977$	39978.0	1.30912	0.654560	+	0.756010i	$0.272854\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17957.9	−0.582674	−0.291337	−	0.956621i	$-0.594100\pi$
$983$	−17957.9	−0.582674	−0.291337	+	0.956621i	$0.594100\pi$
$984$	0	0
$985$	−10044.0	−0.324902
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2078.46	0.0668263
$990$	0	0
$991$	−1666.23	−0.0534103	−0.0267052	−	0.999643i	$-0.508502\pi$
$991$	−1666.23	−0.0534103	−0.0267052	+	0.999643i	$0.508502\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18540.0	−0.590711
$996$	0	0
$997$	41465.3	1.31717	0.658585	−	0.752506i	$-0.271155\pi$
$997$	41465.3	1.31717	0.658585	+	0.752506i	$0.271155\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.a.bg.1.2		2
3.2	odd	2		768.4.a.g.1.1		2
4.3	odd	2		2304.4.a.be.1.2		2
8.3	odd	2	inner	2304.4.a.bg.1.1		2
8.5	even	2		2304.4.a.be.1.1		2
12.11	even	2		768.4.a.n.1.1		2
16.3	odd	4		576.4.d.g.289.1		4
16.5	even	4		576.4.d.g.289.4		4
16.11	odd	4		576.4.d.g.289.3		4
16.13	even	4		576.4.d.g.289.2		4
24.5	odd	2		768.4.a.n.1.2		2
24.11	even	2		768.4.a.g.1.2		2
48.5	odd	4		192.4.d.a.97.3	yes	4
48.11	even	4		192.4.d.a.97.1	✓	4
48.29	odd	4		192.4.d.a.97.2	yes	4
48.35	even	4		192.4.d.a.97.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.d.a.97.1	✓	4	48.11	even	4
192.4.d.a.97.2	yes	4	48.29	odd	4
192.4.d.a.97.3	yes	4	48.5	odd	4
192.4.d.a.97.4	yes	4	48.35	even	4
576.4.d.g.289.1		4	16.3	odd	4
576.4.d.g.289.2		4	16.13	even	4
576.4.d.g.289.3		4	16.11	odd	4
576.4.d.g.289.4		4	16.5	even	4
768.4.a.g.1.1		2	3.2	odd	2
768.4.a.g.1.2		2	24.11	even	2
768.4.a.n.1.1		2	12.11	even	2
768.4.a.n.1.2		2	24.5	odd	2
2304.4.a.be.1.1		2	8.5	even	2
2304.4.a.be.1.2		2	4.3	odd	2
2304.4.a.bg.1.1		2	8.3	odd	2	inner
2304.4.a.bg.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

\(f(q)\)	\(=\)	\(q+10.3923 q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q+10.3923 q^{5} -3.46410 q^{7} +55.4256 q^{13} +90.0000 q^{17} +116.000 q^{19} +103.923 q^{23} -17.0000 q^{25} -259.808 q^{29} +301.377 q^{31} -36.0000 q^{35} -34.6410 q^{37} -54.0000 q^{41} +20.0000 q^{43} +394.908 q^{47} -331.000 q^{49} +488.438 q^{53} -324.000 q^{59} -575.041 q^{61} +576.000 q^{65} -116.000 q^{67} -1101.58 q^{71} -1106.00 q^{73} +148.956 q^{79} +1152.00 q^{83} +935.307 q^{85} +918.000 q^{89} -192.000 q^{91} +1205.51 q^{95} +190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 180 q^{17} + 232 q^{19} - 34 q^{25} - 72 q^{35} - 108 q^{41} + 40 q^{43} - 662 q^{49} - 648 q^{59} + 1152 q^{65} - 232 q^{67} - 2212 q^{73} + 2304 q^{83} + 1836 q^{89} - 384 q^{91} + 380 q^{97}+O(q^{100}) \) 2 * q + 180 * q^17 + 232 * q^19 - 34 * q^25 - 72 * q^35 - 108 * q^41 + 40 * q^43 - 662 * q^49 - 648 * q^59 + 1152 * q^65 - 232 * q^67 - 2212 * q^73 + 2304 * q^83 + 1836 * q^89 - 384 * q^91 + 380 * q^97

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