LMFDB - Embedded newform 2448.4.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,4,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2448.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:	$144.436675694$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 10 $ x^3 - x^2 - 14*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.75985$ of defining polynomial
Character	$\chi$	$=$	2448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.65616 q^{5} +31.5852 q^{7} -7.18910 q^{11} +84.3331 q^{13} -17.0000 q^{17} +37.0838 q^{19} +150.218 q^{23} -66.3832 q^{25} +11.5846 q^{29} +53.2865 q^{31} +241.822 q^{35} -99.2134 q^{37} -118.249 q^{41} +456.016 q^{43} +571.014 q^{47} +654.627 q^{49} -462.867 q^{53} -55.0409 q^{55} +48.0674 q^{59} +59.5236 q^{61} +645.668 q^{65} +740.787 q^{67} -930.437 q^{71} -697.419 q^{73} -227.070 q^{77} -1036.04 q^{79} -22.2043 q^{83} -130.155 q^{85} +369.726 q^{89} +2663.68 q^{91} +283.920 q^{95} +1139.56 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{5} + 8 q^{7} + 34 q^{11} + 36 q^{13} - 51 q^{17} + 142 q^{19} + 110 q^{23} - 193 q^{25} - 90 q^{29} + 148 q^{31} + 416 q^{35} + 110 q^{37} - 720 q^{41} + 146 q^{43} + 500 q^{47} + 379 q^{49}+ \cdots + 402 q^{97}+O(q^{100}) $ 3 * q - 8 * q^5 + 8 * q^7 + 34 * q^11 + 36 * q^13 - 51 * q^17 + 142 * q^19 + 110 * q^23 - 193 * q^25 - 90 * q^29 + 148 * q^31 + 416 * q^35 + 110 * q^37 - 720 * q^41 + 146 * q^43 + 500 * q^47 + 379 * q^49 - 610 * q^53 - 430 * q^55 - 216 * q^59 - 18 * q^61 + 966 * q^65 + 1404 * q^67 - 960 * q^71 - 794 * q^73 - 48 * q^77 + 276 * q^79 - 1552 * q^83 + 136 * q^85 - 1394 * q^89 + 3968 * q^91 - 602 * q^95 + 402 * q^97

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
$5$	7.65616	0.684788	0.342394	−	0.939557i	$-0.388762\pi$
$5$	7.65616	0.684788	0.342394	+	0.939557i	$0.388762\pi$
$6$	0	0
$7$	31.5852	1.70544	0.852721	−	0.522366i	$-0.174951\pi$
$7$	31.5852	1.70544	0.852721	+	0.522366i	$0.174951\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−7.18910	−0.197054	−0.0985271	−	0.995134i	$-0.531413\pi$
$11$	−7.18910	−0.197054	−0.0985271	+	0.995134i	$0.531413\pi$
$12$	0	0
$13$	84.3331	1.79921	0.899607	−	0.436700i	$-0.143853\pi$
$13$	84.3331	1.79921	0.899607	+	0.436700i	$0.143853\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	37.0838	0.447769	0.223885	−	0.974616i	$-0.428126\pi$
$19$	37.0838	0.447769	0.223885	+	0.974616i	$0.428126\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	150.218	1.36185	0.680925	−	0.732353i	$-0.261578\pi$
$23$	150.218	1.36185	0.680925	+	0.732353i	$0.261578\pi$
$24$	0	0
$25$	−66.3832	−0.531066
$26$	0	0
$27$	0	0
$28$	0	0
$29$	11.5846	0.0741792	0.0370896	−	0.999312i	$-0.488191\pi$
$29$	11.5846	0.0741792	0.0370896	+	0.999312i	$0.488191\pi$
$30$	0	0
$31$	53.2865	0.308727	0.154364	−	0.988014i	$-0.450667\pi$
$31$	53.2865	0.308727	0.154364	+	0.988014i	$0.450667\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	241.822	1.16787
$36$	0	0
$37$	−99.2134	−0.440827	−0.220413	−	0.975407i	$-0.570741\pi$
$37$	−99.2134	−0.440827	−0.220413	+	0.975407i	$0.570741\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−118.249	−0.450425	−0.225213	−	0.974310i	$-0.572308\pi$
$41$	−118.249	−0.450425	−0.225213	+	0.974310i	$0.572308\pi$
$42$	0	0
$43$	456.016	1.61725	0.808626	−	0.588323i	$-0.200211\pi$
$43$	456.016	1.61725	0.808626	+	0.588323i	$0.200211\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	571.014	1.77215	0.886073	−	0.463545i	$-0.153423\pi$
$47$	571.014	1.77215	0.886073	+	0.463545i	$0.153423\pi$
$48$	0	0
$49$	654.627	1.90853
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−462.867	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$53$	−462.867	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$54$	0	0
$55$	−55.0409	−0.134940
$56$	0	0
$57$	0	0
$58$	0	0
$59$	48.0674	0.106065	0.0530325	−	0.998593i	$-0.483111\pi$
$59$	48.0674	0.106065	0.0530325	+	0.998593i	$0.483111\pi$
$60$	0	0
$61$	59.5236	0.124938	0.0624689	−	0.998047i	$-0.480103\pi$
$61$	59.5236	0.124938	0.0624689	+	0.998047i	$0.480103\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	645.668	1.23208
$66$	0	0
$67$	740.787	1.35077	0.675384	−	0.737466i	$-0.263978\pi$
$67$	740.787	1.35077	0.675384	+	0.737466i	$0.263978\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−930.437	−1.55525	−0.777623	−	0.628730i	$-0.783575\pi$
$71$	−930.437	−1.55525	−0.777623	+	0.628730i	$0.783575\pi$
$72$	0	0
$73$	−697.419	−1.11817	−0.559087	−	0.829109i	$-0.688848\pi$
$73$	−697.419	−1.11817	−0.559087	+	0.829109i	$0.688848\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−227.070	−0.336065
$78$	0	0
$79$	−1036.04	−1.47549	−0.737747	−	0.675077i	$-0.764110\pi$
$79$	−1036.04	−1.47549	−0.737747	+	0.675077i	$0.764110\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−22.2043	−0.0293643	−0.0146822	−	0.999892i	$-0.504674\pi$
$83$	−22.2043	−0.0293643	−0.0146822	+	0.999892i	$0.504674\pi$
$84$	0	0
$85$	−130.155	−0.166085
$86$	0	0
$87$	0	0
$88$	0	0
$89$	369.726	0.440346	0.220173	−	0.975461i	$-0.429338\pi$
$89$	369.726	0.440346	0.220173	+	0.975461i	$0.429338\pi$
$90$	0	0
$91$	2663.68	3.06846
$92$	0	0
$93$	0	0
$94$	0	0
$95$	283.920	0.306627
$96$	0	0
$97$	1139.56	1.19283	0.596415	−	0.802676i	$-0.296591\pi$
$97$	1139.56	1.19283	0.596415	+	0.802676i	$0.296591\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−703.083	−0.692667	−0.346334	−	0.938111i	$-0.612573\pi$
$101$	−703.083	−0.692667	−0.346334	+	0.938111i	$0.612573\pi$
$102$	0	0
$103$	897.160	0.858250	0.429125	−	0.903245i	$-0.358822\pi$
$103$	897.160	0.858250	0.429125	+	0.903245i	$0.358822\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1901.21	−1.71773	−0.858864	−	0.512203i	$-0.828829\pi$
$107$	−1901.21	−1.71773	−0.858864	+	0.512203i	$0.828829\pi$
$108$	0	0
$109$	584.555	0.513671	0.256836	−	0.966455i	$-0.417320\pi$
$109$	584.555	0.513671	0.256836	+	0.966455i	$0.417320\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	63.4225	0.0527990	0.0263995	−	0.999651i	$-0.491596\pi$
$113$	63.4225	0.0527990	0.0263995	+	0.999651i	$0.491596\pi$
$114$	0	0
$115$	1150.09	0.932578
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−536.949	−0.413631
$120$	0	0
$121$	−1279.32	−0.961170
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1465.26	−1.04845
$126$	0	0
$127$	−175.543	−0.122653	−0.0613266	−	0.998118i	$-0.519533\pi$
$127$	−175.543	−0.122653	−0.0613266	+	0.998118i	$0.519533\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1865.42	1.24414	0.622070	−	0.782961i	$-0.286292\pi$
$131$	1865.42	1.24414	0.622070	+	0.782961i	$0.286292\pi$
$132$	0	0
$133$	1171.30	0.763644
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1057.57	−0.659519	−0.329760	−	0.944065i	$-0.606968\pi$
$137$	−1057.57	−0.659519	−0.329760	+	0.944065i	$0.606968\pi$
$138$	0	0
$139$	−904.833	−0.552136	−0.276068	−	0.961138i	$-0.589032\pi$
$139$	−904.833	−0.552136	−0.276068	+	0.961138i	$0.589032\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−606.279	−0.354543
$144$	0	0
$145$	88.6932	0.0507970
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−809.001	−0.444805	−0.222402	−	0.974955i	$-0.571390\pi$
$149$	−809.001	−0.444805	−0.222402	+	0.974955i	$0.571390\pi$
$150$	0	0
$151$	352.121	0.189769	0.0948847	−	0.995488i	$-0.469752\pi$
$151$	352.121	0.189769	0.0948847	+	0.995488i	$0.469752\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	407.970	0.211413
$156$	0	0
$157$	537.882	0.273424	0.136712	−	0.990611i	$-0.456346\pi$
$157$	537.882	0.273424	0.136712	+	0.990611i	$0.456346\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4744.66	2.32256
$162$	0	0
$163$	−1922.74	−0.923933	−0.461966	−	0.886897i	$-0.652856\pi$
$163$	−1922.74	−0.923933	−0.461966	+	0.886897i	$0.652856\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2971.76	−1.37702	−0.688509	−	0.725228i	$-0.741734\pi$
$167$	−2971.76	−1.37702	−0.688509	+	0.725228i	$0.741734\pi$
$168$	0	0
$169$	4915.07	2.23717
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−988.564	−0.434446	−0.217223	−	0.976122i	$-0.569700\pi$
$173$	−988.564	−0.434446	−0.217223	+	0.976122i	$0.569700\pi$
$174$	0	0
$175$	−2096.73	−0.905702
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1937.65	−0.809089	−0.404545	−	0.914518i	$-0.632570\pi$
$179$	−1937.65	−0.809089	−0.404545	+	0.914518i	$0.632570\pi$
$180$	0	0
$181$	−2180.07	−0.895267	−0.447634	−	0.894217i	$-0.647733\pi$
$181$	−2180.07	−0.895267	−0.447634	+	0.894217i	$0.647733\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−759.594	−0.301873
$186$	0	0
$187$	122.215	0.0477927
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1675.78	0.634845	0.317423	−	0.948284i	$-0.397183\pi$
$191$	1675.78	0.634845	0.317423	+	0.948284i	$0.397183\pi$
$192$	0	0
$193$	−257.961	−0.0962094	−0.0481047	−	0.998842i	$-0.515318\pi$
$193$	−257.961	−0.0962094	−0.0481047	+	0.998842i	$0.515318\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	693.466	0.250799	0.125399	−	0.992106i	$-0.459979\pi$
$197$	693.466	0.250799	0.125399	+	0.992106i	$0.459979\pi$
$198$	0	0
$199$	240.295	0.0855984	0.0427992	−	0.999084i	$-0.486372\pi$
$199$	240.295	0.0855984	0.0427992	+	0.999084i	$0.486372\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	365.901	0.126508
$204$	0	0
$205$	−905.335	−0.308446
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−266.599	−0.0882348
$210$	0	0
$211$	−268.114	−0.0874774	−0.0437387	−	0.999043i	$-0.513927\pi$
$211$	−268.114	−0.0874774	−0.0437387	+	0.999043i	$0.513927\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3491.33	1.10747
$216$	0	0
$217$	1683.07	0.526516
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1433.66	−0.436374
$222$	0	0
$223$	5524.43	1.65894	0.829468	−	0.558554i	$-0.188643\pi$
$223$	5524.43	1.65894	0.829468	+	0.558554i	$0.188643\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−384.400	−0.112394	−0.0561972	−	0.998420i	$-0.517898\pi$
$227$	−384.400	−0.112394	−0.0561972	+	0.998420i	$0.517898\pi$
$228$	0	0
$229$	1395.48	0.402690	0.201345	−	0.979520i	$-0.435469\pi$
$229$	1395.48	0.402690	0.201345	+	0.979520i	$0.435469\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3409.39	−0.958613	−0.479307	−	0.877648i	$-0.659112\pi$
$233$	−3409.39	−0.958613	−0.479307	+	0.877648i	$0.659112\pi$
$234$	0	0
$235$	4371.77	1.21354
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1509.18	−0.408456	−0.204228	−	0.978923i	$-0.565468\pi$
$239$	−1509.18	−0.408456	−0.204228	+	0.978923i	$0.565468\pi$
$240$	0	0
$241$	3406.91	0.910615	0.455307	−	0.890334i	$-0.349529\pi$
$241$	3406.91	0.910615	0.455307	+	0.890334i	$0.349529\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5011.93	1.30694
$246$	0	0
$247$	3127.39	0.805633
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3394.43	0.853605	0.426802	−	0.904345i	$-0.359640\pi$
$251$	3394.43	0.853605	0.426802	+	0.904345i	$0.359640\pi$
$252$	0	0
$253$	−1079.93	−0.268358
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1778.91	−0.431772	−0.215886	−	0.976419i	$-0.569264\pi$
$257$	−1778.91	−0.431772	−0.215886	+	0.976419i	$0.569264\pi$
$258$	0	0
$259$	−3133.68	−0.751804
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4316.88	−1.01213	−0.506065	−	0.862495i	$-0.668900\pi$
$263$	−4316.88	−1.01213	−0.506065	+	0.862495i	$0.668900\pi$
$264$	0	0
$265$	−3543.79	−0.821483
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6546.31	−1.48378	−0.741888	−	0.670524i	$-0.766069\pi$
$269$	−6546.31	−1.48378	−0.741888	+	0.670524i	$0.766069\pi$
$270$	0	0
$271$	−3785.20	−0.848466	−0.424233	−	0.905553i	$-0.639456\pi$
$271$	−3785.20	−0.848466	−0.424233	+	0.905553i	$0.639456\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	477.236	0.104649
$276$	0	0
$277$	−3521.06	−0.763755	−0.381878	−	0.924213i	$-0.624722\pi$
$277$	−3521.06	−0.763755	−0.381878	+	0.924213i	$0.624722\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2922.30	0.620391	0.310195	−	0.950673i	$-0.399606\pi$
$281$	2922.30	0.620391	0.310195	+	0.950673i	$0.399606\pi$
$282$	0	0
$283$	−735.075	−0.154402	−0.0772008	−	0.997016i	$-0.524598\pi$
$283$	−735.075	−0.154402	−0.0772008	+	0.997016i	$0.524598\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3734.93	−0.768175
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8702.92	1.73526	0.867628	−	0.497213i	$-0.165643\pi$
$293$	8702.92	1.73526	0.867628	+	0.497213i	$0.165643\pi$
$294$	0	0
$295$	368.011	0.0726320
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12668.3	2.45026
$300$	0	0
$301$	14403.4	2.75813
$302$	0	0
$303$	0	0
$304$	0	0
$305$	455.722	0.0855559
$306$	0	0
$307$	−2516.95	−0.467916	−0.233958	−	0.972247i	$-0.575168\pi$
$307$	−2516.95	−0.467916	−0.233958	+	0.972247i	$0.575168\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6593.31	1.20216	0.601081	−	0.799188i	$-0.294737\pi$
$311$	6593.31	1.20216	0.601081	+	0.799188i	$0.294737\pi$
$312$	0	0
$313$	4392.99	0.793312	0.396656	−	0.917967i	$-0.370171\pi$
$313$	4392.99	0.793312	0.396656	+	0.917967i	$0.370171\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2601.23	−0.460882	−0.230441	−	0.973086i	$-0.574017\pi$
$317$	−2601.23	−0.460882	−0.230441	+	0.973086i	$0.574017\pi$
$318$	0	0
$319$	−83.2825	−0.0146173
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−630.425	−0.108600
$324$	0	0
$325$	−5598.30	−0.955502
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18035.6	3.02229
$330$	0	0
$331$	4670.49	0.775568	0.387784	−	0.921750i	$-0.373241\pi$
$331$	4670.49	0.775568	0.387784	+	0.921750i	$0.373241\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5671.58	0.924989
$336$	0	0
$337$	1801.67	0.291226	0.145613	−	0.989342i	$-0.453485\pi$
$337$	1801.67	0.291226	0.145613	+	0.989342i	$0.453485\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−383.082	−0.0608360
$342$	0	0
$343$	9842.83	1.54945
$344$	0	0
$345$	0	0
$346$	0	0
$347$	168.340	0.0260431	0.0130216	−	0.999915i	$-0.495855\pi$
$347$	168.340	0.0260431	0.0130216	+	0.999915i	$0.495855\pi$
$348$	0	0
$349$	−4447.85	−0.682200	−0.341100	−	0.940027i	$-0.610799\pi$
$349$	−4447.85	−0.682200	−0.341100	+	0.940027i	$0.610799\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10509.5	1.58460	0.792298	−	0.610135i	$-0.208885\pi$
$353$	10509.5	1.58460	0.792298	+	0.610135i	$0.208885\pi$
$354$	0	0
$355$	−7123.57	−1.06501
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8342.99	1.22654	0.613268	−	0.789875i	$-0.289855\pi$
$359$	8342.99	1.22654	0.613268	+	0.789875i	$0.289855\pi$
$360$	0	0
$361$	−5483.79	−0.799503
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5339.55	−0.765712
$366$	0	0
$367$	−352.402	−0.0501232	−0.0250616	−	0.999686i	$-0.507978\pi$
$367$	−352.402	−0.0501232	−0.0250616	+	0.999686i	$0.507978\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14619.8	−2.04588
$372$	0	0
$373$	12563.2	1.74397	0.871983	−	0.489537i	$-0.162834\pi$
$373$	12563.2	1.74397	0.871983	+	0.489537i	$0.162834\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	976.961	0.133464
$378$	0	0
$379$	1770.57	0.239969	0.119984	−	0.992776i	$-0.461716\pi$
$379$	1770.57	0.239969	0.119984	+	0.992776i	$0.461716\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4330.57	0.577759	0.288880	−	0.957365i	$-0.406717\pi$
$383$	4330.57	0.577759	0.288880	+	0.957365i	$0.406717\pi$
$384$	0	0
$385$	−1738.48	−0.230133
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10295.5	−1.34191	−0.670957	−	0.741496i	$-0.734117\pi$
$389$	−10295.5	−1.34191	−0.670957	+	0.741496i	$0.734117\pi$
$390$	0	0
$391$	−2553.70	−0.330297
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7932.12	−1.01040
$396$	0	0
$397$	93.1792	0.0117797	0.00588983	−	0.999983i	$-0.498125\pi$
$397$	93.1792	0.0117797	0.00588983	+	0.999983i	$0.498125\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13320.9	1.65889	0.829443	−	0.558591i	$-0.188658\pi$
$401$	13320.9	1.65889	0.829443	+	0.558591i	$0.188658\pi$
$402$	0	0
$403$	4493.82	0.555466
$404$	0	0
$405$	0	0
$406$	0	0
$407$	713.255	0.0868667
$408$	0	0
$409$	9272.21	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$409$	9272.21	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1518.22	0.180888
$414$	0	0
$415$	−170.000	−0.0201083
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11325.4	1.32049	0.660244	−	0.751051i	$-0.270453\pi$
$419$	11325.4	1.32049	0.660244	+	0.751051i	$0.270453\pi$
$420$	0	0
$421$	6934.54	0.802776	0.401388	−	0.915908i	$-0.368528\pi$
$421$	6934.54	0.802776	0.401388	+	0.915908i	$0.368528\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1128.52	0.128802
$426$	0	0
$427$	1880.07	0.213074
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2776.55	−0.310306	−0.155153	−	0.987890i	$-0.549587\pi$
$431$	−2776.55	−0.310306	−0.155153	+	0.987890i	$0.549587\pi$
$432$	0	0
$433$	−3252.22	−0.360951	−0.180476	−	0.983579i	$-0.557764\pi$
$433$	−3252.22	−0.360951	−0.180476	+	0.983579i	$0.557764\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5570.65	0.609795
$438$	0	0
$439$	13345.7	1.45093	0.725464	−	0.688260i	$-0.241625\pi$
$439$	13345.7	1.45093	0.725464	+	0.688260i	$0.241625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11639.6	1.24834	0.624169	−	0.781290i	$-0.285438\pi$
$443$	11639.6	1.24834	0.624169	+	0.781290i	$0.285438\pi$
$444$	0	0
$445$	2830.68	0.301544
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6937.72	0.729201	0.364601	−	0.931164i	$-0.381206\pi$
$449$	6937.72	0.729201	0.364601	+	0.931164i	$0.381206\pi$
$450$	0	0
$451$	850.106	0.0887582
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20393.6	2.10124
$456$	0	0
$457$	−10285.8	−1.05284	−0.526422	−	0.850224i	$-0.676467\pi$
$457$	−10285.8	−1.05284	−0.526422	+	0.850224i	$0.676467\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	625.833	0.0632277	0.0316138	−	0.999500i	$-0.489935\pi$
$461$	625.833	0.0632277	0.0316138	+	0.999500i	$0.489935\pi$
$462$	0	0
$463$	6055.97	0.607872	0.303936	−	0.952692i	$-0.401699\pi$
$463$	6055.97	0.607872	0.303936	+	0.952692i	$0.401699\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−815.966	−0.0808531	−0.0404265	−	0.999183i	$-0.512872\pi$
$467$	−815.966	−0.0808531	−0.0404265	+	0.999183i	$0.512872\pi$
$468$	0	0
$469$	23397.9	2.30366
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3278.35	−0.318686
$474$	0	0
$475$	−2461.74	−0.237795
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16219.1	−1.54712	−0.773559	−	0.633724i	$-0.781525\pi$
$479$	−16219.1	−1.54712	−0.773559	+	0.633724i	$0.781525\pi$
$480$	0	0
$481$	−8366.97	−0.793142
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8724.63	0.816835
$486$	0	0
$487$	−2725.13	−0.253568	−0.126784	−	0.991930i	$-0.540465\pi$
$487$	−2725.13	−0.253568	−0.126784	+	0.991930i	$0.540465\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8344.13	−0.766935	−0.383468	−	0.923554i	$-0.625270\pi$
$491$	−8344.13	−0.766935	−0.383468	+	0.923554i	$0.625270\pi$
$492$	0	0
$493$	−196.937	−0.0179911
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−29388.1	−2.65238
$498$	0	0
$499$	−13762.0	−1.23461	−0.617306	−	0.786723i	$-0.711776\pi$
$499$	−13762.0	−1.23461	−0.617306	+	0.786723i	$0.711776\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11909.5	−1.05570	−0.527852	−	0.849336i	$-0.677003\pi$
$503$	−11909.5	−1.05570	−0.527852	+	0.849336i	$0.677003\pi$
$504$	0	0
$505$	−5382.92	−0.474330
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−742.666	−0.0646721	−0.0323360	−	0.999477i	$-0.510295\pi$
$509$	−742.666	−0.0646721	−0.0323360	+	0.999477i	$0.510295\pi$
$510$	0	0
$511$	−22028.1	−1.90698
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6868.80	0.587719
$516$	0	0
$517$	−4105.07	−0.349209
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4815.73	0.404953	0.202477	−	0.979287i	$-0.435101\pi$
$521$	4815.73	0.404953	0.202477	+	0.979287i	$0.435101\pi$
$522$	0	0
$523$	16249.5	1.35858	0.679291	−	0.733869i	$-0.262287\pi$
$523$	16249.5	1.35858	0.679291	+	0.733869i	$0.262287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−905.871	−0.0748773
$528$	0	0
$529$	10398.4	0.854637
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9972.33	−0.810412
$534$	0	0
$535$	−14556.0	−1.17628
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4706.18	−0.376085
$540$	0	0
$541$	−5458.04	−0.433751	−0.216876	−	0.976199i	$-0.569587\pi$
$541$	−5458.04	−0.433751	−0.216876	+	0.976199i	$0.569587\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4475.44	0.351756
$546$	0	0
$547$	−17237.0	−1.34735	−0.673677	−	0.739026i	$-0.735286\pi$
$547$	−17237.0	−1.34735	−0.673677	+	0.739026i	$0.735286\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	429.600	0.0332152
$552$	0	0
$553$	−32723.7	−2.51637
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8452.71	0.643003	0.321502	−	0.946909i	$-0.395812\pi$
$557$	8452.71	0.643003	0.321502	+	0.946909i	$0.395812\pi$
$558$	0	0
$559$	38457.3	2.90978
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8547.32	−0.639834	−0.319917	−	0.947446i	$-0.603655\pi$
$563$	−8547.32	−0.639834	−0.319917	+	0.947446i	$0.603655\pi$
$564$	0	0
$565$	485.573	0.0361561
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19464.8	−1.43411	−0.717055	−	0.697017i	$-0.754510\pi$
$569$	−19464.8	−1.43411	−0.717055	+	0.697017i	$0.754510\pi$
$570$	0	0
$571$	3839.06	0.281366	0.140683	−	0.990055i	$-0.455070\pi$
$571$	3839.06	0.281366	0.140683	+	0.990055i	$0.455070\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9971.94	−0.723232
$576$	0	0
$577$	−18797.7	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$577$	−18797.7	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−701.328	−0.0500791
$582$	0	0
$583$	3327.60	0.236390
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4217.92	0.296580	0.148290	−	0.988944i	$-0.452623\pi$
$587$	4217.92	0.296580	0.148290	+	0.988944i	$0.452623\pi$
$588$	0	0
$589$	1976.07	0.138238
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3011.92	0.208575	0.104287	−	0.994547i	$-0.466744\pi$
$593$	3011.92	0.208575	0.104287	+	0.994547i	$0.466744\pi$
$594$	0	0
$595$	−4110.97	−0.283249
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15137.1	1.03253	0.516266	−	0.856428i	$-0.327322\pi$
$599$	15137.1	1.03253	0.516266	+	0.856428i	$0.327322\pi$
$600$	0	0
$601$	−18980.1	−1.28821	−0.644104	−	0.764938i	$-0.722770\pi$
$601$	−18980.1	−1.28821	−0.644104	+	0.764938i	$0.722770\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9794.65	−0.658197
$606$	0	0
$607$	4593.63	0.307166	0.153583	−	0.988136i	$-0.450919\pi$
$607$	4593.63	0.307166	0.153583	+	0.988136i	$0.450919\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48155.3	3.18847
$612$	0	0
$613$	−25654.3	−1.69032	−0.845160	−	0.534513i	$-0.820495\pi$
$613$	−25654.3	−1.69032	−0.845160	+	0.534513i	$0.820495\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7170.97	0.467897	0.233948	−	0.972249i	$-0.424835\pi$
$617$	7170.97	0.467897	0.233948	+	0.972249i	$0.424835\pi$
$618$	0	0
$619$	13560.6	0.880525	0.440263	−	0.897869i	$-0.354885\pi$
$619$	13560.6	0.880525	0.440263	+	0.897869i	$0.354885\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11677.9	0.750985
$624$	0	0
$625$	−2920.36	−0.186903
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1686.63	0.106916
$630$	0	0
$631$	1414.98	0.0892700	0.0446350	−	0.999003i	$-0.485788\pi$
$631$	1414.98	0.0892700	0.0446350	+	0.999003i	$0.485788\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1343.99	−0.0839914
$636$	0	0
$637$	55206.7	3.43386
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21708.4	1.33764	0.668822	−	0.743422i	$-0.266799\pi$
$641$	21708.4	1.33764	0.668822	+	0.743422i	$0.266799\pi$
$642$	0	0
$643$	−7537.23	−0.462270	−0.231135	−	0.972922i	$-0.574244\pi$
$643$	−7537.23	−0.462270	−0.231135	+	0.972922i	$0.574244\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32667.3	−1.98498	−0.992491	−	0.122316i	$-0.960968\pi$
$647$	−32667.3	−1.98498	−0.992491	+	0.122316i	$0.960968\pi$
$648$	0	0
$649$	−345.561	−0.0209006
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25845.8	1.54889	0.774443	−	0.632644i	$-0.218030\pi$
$653$	25845.8	1.54889	0.774443	+	0.632644i	$0.218030\pi$
$654$	0	0
$655$	14281.9	0.851972
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15741.2	−0.930485	−0.465243	−	0.885183i	$-0.654033\pi$
$659$	−15741.2	−0.930485	−0.465243	+	0.885183i	$0.654033\pi$
$660$	0	0
$661$	23495.2	1.38254	0.691269	−	0.722598i	$-0.257052\pi$
$661$	23495.2	1.38254	0.691269	+	0.722598i	$0.257052\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8967.67	0.522934
$666$	0	0
$667$	1740.21	0.101021
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−427.921	−0.0246195
$672$	0	0
$673$	−7057.34	−0.404221	−0.202110	−	0.979363i	$-0.564780\pi$
$673$	−7057.34	−0.404221	−0.202110	+	0.979363i	$0.564780\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−20756.4	−1.17833	−0.589167	−	0.808011i	$-0.700544\pi$
$677$	−20756.4	−1.17833	−0.589167	+	0.808011i	$0.700544\pi$
$678$	0	0
$679$	35993.2	2.03430
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7013.65	−0.392928	−0.196464	−	0.980511i	$-0.562946\pi$
$683$	−7013.65	−0.392928	−0.196464	+	0.980511i	$0.562946\pi$
$684$	0	0
$685$	−8096.91	−0.451631
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39035.0	−2.15837
$690$	0	0
$691$	4897.47	0.269622	0.134811	−	0.990871i	$-0.456957\pi$
$691$	4897.47	0.269622	0.134811	+	0.990871i	$0.456957\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6927.54	−0.378096
$696$	0	0
$697$	2010.24	0.109244
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17525.5	−0.944264	−0.472132	−	0.881528i	$-0.656515\pi$
$701$	−17525.5	−0.944264	−0.472132	+	0.881528i	$0.656515\pi$
$702$	0	0
$703$	−3679.21	−0.197388
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22207.1	−1.18130
$708$	0	0
$709$	32564.3	1.72493	0.862466	−	0.506115i	$-0.168919\pi$
$709$	32564.3	1.72493	0.862466	+	0.506115i	$0.168919\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8004.58	0.420440
$714$	0	0
$715$	−4641.77	−0.242786
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14498.2	0.752007	0.376004	−	0.926618i	$-0.377298\pi$
$719$	14498.2	0.752007	0.376004	+	0.926618i	$0.377298\pi$
$720$	0	0
$721$	28337.0	1.46370
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−769.020	−0.0393941
$726$	0	0
$727$	25787.6	1.31556	0.657778	−	0.753212i	$-0.271497\pi$
$727$	25787.6	1.31556	0.657778	+	0.753212i	$0.271497\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7752.28	−0.392241
$732$	0	0
$733$	−4177.45	−0.210502	−0.105251	−	0.994446i	$-0.533565\pi$
$733$	−4177.45	−0.210502	−0.105251	+	0.994446i	$0.533565\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5325.59	−0.266175
$738$	0	0
$739$	−14115.5	−0.702636	−0.351318	−	0.936256i	$-0.614266\pi$
$739$	−14115.5	−0.702636	−0.351318	+	0.936256i	$0.614266\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17992.0	0.888376	0.444188	−	0.895934i	$-0.353492\pi$
$743$	17992.0	0.888376	0.444188	+	0.895934i	$0.353492\pi$
$744$	0	0
$745$	−6193.84	−0.304597
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−60050.2	−2.92949
$750$	0	0
$751$	−2055.99	−0.0998989	−0.0499495	−	0.998752i	$-0.515906\pi$
$751$	−2055.99	−0.0998989	−0.0499495	+	0.998752i	$0.515906\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2695.89	0.129952
$756$	0	0
$757$	12132.4	0.582508	0.291254	−	0.956646i	$-0.405927\pi$
$757$	12132.4	0.582508	0.291254	+	0.956646i	$0.405927\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−9319.01	−0.443908	−0.221954	−	0.975057i	$-0.571243\pi$
$761$	−9319.01	−0.443908	−0.221954	+	0.975057i	$0.571243\pi$
$762$	0	0
$763$	18463.3	0.876037
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4053.67	0.190834
$768$	0	0
$769$	38790.8	1.81903	0.909514	−	0.415672i	$-0.136454\pi$
$769$	38790.8	1.81903	0.909514	+	0.415672i	$0.136454\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12857.4	−0.598252	−0.299126	−	0.954214i	$-0.596695\pi$
$773$	−12857.4	−0.598252	−0.299126	+	0.954214i	$0.596695\pi$
$774$	0	0
$775$	−3537.33	−0.163954
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4385.14	−0.201687
$780$	0	0
$781$	6689.00	0.306468
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4118.11	0.187238
$786$	0	0
$787$	26862.0	1.21668	0.608339	−	0.793677i	$-0.291836\pi$
$787$	26862.0	1.21668	0.608339	+	0.793677i	$0.291836\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2003.22	0.0900457
$792$	0	0
$793$	5019.81	0.224790
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12471.5	0.554285	0.277142	−	0.960829i	$-0.410613\pi$
$797$	12471.5	0.554285	0.277142	+	0.960829i	$0.410613\pi$
$798$	0	0
$799$	−9707.23	−0.429809
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5013.82	0.220341
$804$	0	0
$805$	36325.9	1.59046
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24760.8	1.07607	0.538037	−	0.842921i	$-0.319166\pi$
$809$	24760.8	1.07607	0.538037	+	0.842921i	$0.319166\pi$
$810$	0	0
$811$	−11237.3	−0.486556	−0.243278	−	0.969957i	$-0.578223\pi$
$811$	−11237.3	−0.486556	−0.243278	+	0.969957i	$0.578223\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14720.8	−0.632698
$816$	0	0
$817$	16910.8	0.724156
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39976.4	−1.69937	−0.849687	−	0.527287i	$-0.823209\pi$
$821$	−39976.4	−1.69937	−0.849687	+	0.527287i	$0.823209\pi$
$822$	0	0
$823$	−36877.0	−1.56191	−0.780955	−	0.624588i	$-0.785267\pi$
$823$	−36877.0	−1.56191	−0.780955	+	0.624588i	$0.785267\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14311.9	−0.601781	−0.300890	−	0.953659i	$-0.597284\pi$
$827$	−14311.9	−0.601781	−0.300890	+	0.953659i	$0.597284\pi$
$828$	0	0
$829$	−12629.7	−0.529130	−0.264565	−	0.964368i	$-0.585228\pi$
$829$	−12629.7	−0.529130	−0.264565	+	0.964368i	$0.585228\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11128.7	−0.462888
$834$	0	0
$835$	−22752.3	−0.942965
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18683.1	−0.768786	−0.384393	−	0.923170i	$-0.625589\pi$
$839$	−18683.1	−0.768786	−0.384393	+	0.923170i	$0.625589\pi$
$840$	0	0
$841$	−24254.8	−0.994497
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37630.6	1.53199
$846$	0	0
$847$	−40407.5	−1.63922
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14903.6	−0.600340
$852$	0	0
$853$	−35648.7	−1.43094	−0.715468	−	0.698646i	$-0.753786\pi$
$853$	−35648.7	−1.43094	−0.715468	+	0.698646i	$0.753786\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49860.8	1.98741	0.993707	−	0.112010i	$-0.0357290\pi$
$857$	49860.8	1.98741	0.993707	+	0.112010i	$0.0357290\pi$
$858$	0	0
$859$	−21487.0	−0.853466	−0.426733	−	0.904378i	$-0.640336\pi$
$859$	−21487.0	−0.853466	−0.426733	+	0.904378i	$0.640336\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15067.6	−0.594330	−0.297165	−	0.954826i	$-0.596041\pi$
$863$	−15067.6	−0.594330	−0.297165	+	0.954826i	$0.596041\pi$
$864$	0	0
$865$	−7568.60	−0.297503
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7448.23	0.290752
$870$	0	0
$871$	62472.8	2.43032
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−46280.6	−1.78808
$876$	0	0
$877$	−24852.6	−0.956912	−0.478456	−	0.878111i	$-0.658803\pi$
$877$	−24852.6	−0.956912	−0.478456	+	0.878111i	$0.658803\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1840.51	−0.0703842	−0.0351921	−	0.999381i	$-0.511204\pi$
$881$	−1840.51	−0.0703842	−0.0351921	+	0.999381i	$0.511204\pi$
$882$	0	0
$883$	−49803.7	−1.89811	−0.949054	−	0.315115i	$-0.897957\pi$
$883$	−49803.7	−1.89811	−0.949054	+	0.315115i	$0.897957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36314.7	1.37467	0.687333	−	0.726342i	$-0.258781\pi$
$887$	36314.7	1.37467	0.687333	+	0.726342i	$0.258781\pi$
$888$	0	0
$889$	−5544.58	−0.209178
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21175.4	0.793512
$894$	0	0
$895$	−14835.0	−0.554054
$896$	0	0
$897$	0	0
$898$	0	0
$899$	617.300	0.0229011
$900$	0	0
$901$	7868.75	0.290950
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16691.0	−0.613068
$906$	0	0
$907$	−33679.5	−1.23297	−0.616487	−	0.787365i	$-0.711445\pi$
$907$	−33679.5	−1.23297	−0.616487	+	0.787365i	$0.711445\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32437.2	1.17968	0.589841	−	0.807519i	$-0.299190\pi$
$911$	32437.2	1.17968	0.589841	+	0.807519i	$0.299190\pi$
$912$	0	0
$913$	159.629	0.00578636
$914$	0	0
$915$	0	0
$916$	0	0
$917$	58919.7	2.12181
$918$	0	0
$919$	3511.45	0.126042	0.0630208	−	0.998012i	$-0.479927\pi$
$919$	3511.45	0.126042	0.0630208	+	0.998012i	$0.479927\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−78466.6	−2.79822
$924$	0	0
$925$	6586.11	0.234108
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13527.2	−0.477733	−0.238866	−	0.971052i	$-0.576776\pi$
$929$	−13527.2	−0.477733	−0.238866	+	0.971052i	$0.576776\pi$
$930$	0	0
$931$	24276.1	0.854583
$932$	0	0
$933$	0	0
$934$	0	0
$935$	935.695	0.0327278
$936$	0	0
$937$	−8862.80	−0.309002	−0.154501	−	0.987993i	$-0.549377\pi$
$937$	−8862.80	−0.309002	−0.154501	+	0.987993i	$0.549377\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22824.0	−0.790693	−0.395346	−	0.918532i	$-0.629375\pi$
$941$	−22824.0	−0.790693	−0.395346	+	0.918532i	$0.629375\pi$
$942$	0	0
$943$	−17763.1	−0.613412
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26308.4	−0.902754	−0.451377	−	0.892333i	$-0.649067\pi$
$947$	−26308.4	−0.902754	−0.451377	+	0.892333i	$0.649067\pi$
$948$	0	0
$949$	−58815.5	−2.01184
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11947.5	−0.406106	−0.203053	−	0.979168i	$-0.565086\pi$
$953$	−11947.5	−0.406106	−0.203053	+	0.979168i	$0.565086\pi$
$954$	0	0
$955$	12830.1	0.434734
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33403.5	−1.12477
$960$	0	0
$961$	−26951.5	−0.904688
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1974.99	−0.0658830
$966$	0	0
$967$	−21505.3	−0.715163	−0.357581	−	0.933882i	$-0.616399\pi$
$967$	−21505.3	−0.715163	−0.357581	+	0.933882i	$0.616399\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45685.3	1.50990	0.754950	−	0.655783i	$-0.227661\pi$
$971$	45685.3	1.50990	0.754950	+	0.655783i	$0.227661\pi$
$972$	0	0
$973$	−28579.4	−0.941636
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31710.3	−1.03839	−0.519193	−	0.854657i	$-0.673767\pi$
$977$	−31710.3	−1.03839	−0.519193	+	0.854657i	$0.673767\pi$
$978$	0	0
$979$	−2657.99	−0.0867721
$980$	0	0
$981$	0	0
$982$	0	0
$983$	46038.9	1.49381	0.746904	−	0.664932i	$-0.231540\pi$
$983$	46038.9	1.49381	0.746904	+	0.664932i	$0.231540\pi$
$984$	0	0
$985$	5309.28	0.171744
$986$	0	0
$987$	0	0
$988$	0	0
$989$	68501.8	2.20246
$990$	0	0
$991$	−14394.9	−0.461420	−0.230710	−	0.973023i	$-0.574105\pi$
$991$	−14394.9	−0.461420	−0.230710	+	0.973023i	$0.574105\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1839.74	0.0586167
$996$	0	0
$997$	−33473.4	−1.06330	−0.531652	−	0.846963i	$-0.678428\pi$
$997$	−33473.4	−1.06330	−0.531652	+	0.846963i	$0.678428\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.4.a.bd.1.3		3
3.2	odd	2		816.4.a.s.1.1		3
4.3	odd	2		153.4.a.f.1.1		3
12.11	even	2		51.4.a.e.1.3	✓	3
60.59	even	2		1275.4.a.q.1.1		3
84.83	odd	2		2499.4.a.n.1.3		3
204.203	even	2		867.4.a.k.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.e.1.3	✓	3	12.11	even	2
153.4.a.f.1.1		3	4.3	odd	2
816.4.a.s.1.1		3	3.2	odd	2
867.4.a.k.1.3		3	204.203	even	2
1275.4.a.q.1.1		3	60.59	even	2
2448.4.a.bd.1.3		3	1.1	even	1	trivial
2499.4.a.n.1.3		3	84.83	odd	2

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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2448.a (trivial)

\(f(q)\)	\(=\)	\(q+7.65616 q^{5} +31.5852 q^{7} -7.18910 q^{11} +84.3331 q^{13} -17.0000 q^{17} +37.0838 q^{19} +150.218 q^{23} -66.3832 q^{25} +11.5846 q^{29} +53.2865 q^{31} +241.822 q^{35} -99.2134 q^{37} -118.249 q^{41} +456.016 q^{43} +571.014 q^{47} +654.627 q^{49} -462.867 q^{53} -55.0409 q^{55} +48.0674 q^{59} +59.5236 q^{61} +645.668 q^{65} +740.787 q^{67} -930.437 q^{71} -697.419 q^{73} -227.070 q^{77} -1036.04 q^{79} -22.2043 q^{83} -130.155 q^{85} +369.726 q^{89} +2663.68 q^{91} +283.920 q^{95} +1139.56 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} + 8 q^{7} + 34 q^{11} + 36 q^{13} - 51 q^{17} + 142 q^{19} + 110 q^{23} - 193 q^{25} - 90 q^{29} + 148 q^{31} + 416 q^{35} + 110 q^{37} - 720 q^{41} + 146 q^{43} + 500 q^{47} + 379 q^{49}+ \cdots + 402 q^{97}+O(q^{100}) \) 3 * q - 8 * q^5 + 8 * q^7 + 34 * q^11 + 36 * q^13 - 51 * q^17 + 142 * q^19 + 110 * q^23 - 193 * q^25 - 90 * q^29 + 148 * q^31 + 416 * q^35 + 110 * q^37 - 720 * q^41 + 146 * q^43 + 500 * q^47 + 379 * q^49 - 610 * q^53 - 430 * q^55 - 216 * q^59 - 18 * q^61 + 966 * q^65 + 1404 * q^67 - 960 * q^71 - 794 * q^73 - 48 * q^77 + 276 * q^79 - 1552 * q^83 + 136 * q^85 - 1394 * q^89 + 3968 * q^91 - 602 * q^95 + 402 * q^97

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