LMFDB - Embedded newform 2368.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.48389.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 10 $ x^4 - 2x^3 - 8x^2 + 3*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.26726$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26726 q^{3} +2.21602 q^{5} +3.07555 q^{7} +2.14048 q^{9} -1.14048 q^{11} -2.80828 q^{13} -5.02431 q^{15} +2.00000 q^{17} -3.89752 q^{19} -6.97307 q^{21} -7.62376 q^{23} -0.0892395 q^{25} +1.94876 q^{27} +3.34281 q^{29} -10.3671 q^{31} +2.58576 q^{33} +6.81548 q^{35} -1.00000 q^{37} +6.36711 q^{39} +2.54822 q^{41} -8.43205 q^{43} +4.74335 q^{45} -7.17802 q^{47} +2.45898 q^{49} -4.53452 q^{51} -5.61007 q^{53} -2.52733 q^{55} +8.83671 q^{57} +6.17848 q^{59} +10.6481 q^{61} +6.58314 q^{63} -6.22322 q^{65} +0.318501 q^{67} +17.2851 q^{69} +6.42555 q^{71} +16.0046 q^{73} +0.202329 q^{75} -3.50759 q^{77} -8.69281 q^{79} -10.8398 q^{81} +9.76116 q^{83} +4.43205 q^{85} -7.57902 q^{87} -8.25357 q^{89} -8.63700 q^{91} +23.5050 q^{93} -8.63700 q^{95} -13.2201 q^{97} -2.44117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 5 q^{5} - q^{7} + 8 q^{9} - 4 q^{11} - 5 q^{13} + 8 q^{17} - 2 q^{19} - q^{21} - 9 q^{23} + 7 q^{25} + q^{27} - 7 q^{29} - q^{31} + 3 q^{33} + 12 q^{35} - 4 q^{37} - 15 q^{39} + 2 q^{41}+ \cdots - 44 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 5 * q^5 - q^7 + 8 * q^9 - 4 * q^11 - 5 * q^13 + 8 * q^17 - 2 * q^19 - q^21 - 9 * q^23 + 7 * q^25 + q^27 - 7 * q^29 - q^31 + 3 * q^33 + 12 * q^35 - 4 * q^37 - 15 * q^39 + 2 * q^41 - 6 * q^43 - 29 * q^47 + 9 * q^49 - 4 * q^51 + 5 * q^53 - 5 * q^55 - 32 * q^57 + 10 * q^59 + q^61 - 28 * q^63 - 2 * q^65 + q^67 + 27 * q^69 - 17 * q^71 + 8 * q^73 - 19 * q^75 + 27 * q^77 + 15 * q^79 - 8 * q^81 - 15 * q^83 - 10 * q^85 - 17 * q^87 - 20 * q^89 - 34 * q^91 + 6 * q^93 - 34 * q^95 + 2 * q^97 - 44 * q^99

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
$3$	−2.26726	−1.30900	−0.654502	−	0.756060i	$-0.727122\pi$
$3$	−2.26726	−1.30900	−0.654502	+	0.756060i	$0.727122\pi$
$4$	0	0
$5$	2.21602	0.991036	0.495518	−	0.868598i	$-0.334978\pi$
$5$	2.21602	0.991036	0.495518	+	0.868598i	$0.334978\pi$
$6$	0	0
$7$	3.07555	1.16245	0.581223	−	0.813744i	$-0.302574\pi$
$7$	3.07555	1.16245	0.581223	+	0.813744i	$0.302574\pi$
$8$	0	0
$9$	2.14048	0.713493
$10$	0	0
$11$	−1.14048	−0.343867	−0.171934	−	0.985109i	$-0.555001\pi$
$11$	−1.14048	−0.343867	−0.171934	+	0.985109i	$0.555001\pi$
$12$	0	0
$13$	−2.80828	−0.778878	−0.389439	−	0.921052i	$-0.627331\pi$
$13$	−2.80828	−0.778878	−0.389439	+	0.921052i	$0.627331\pi$
$14$	0	0
$15$	−5.02431	−1.29727
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−3.89752	−0.894153	−0.447077	−	0.894496i	$-0.647535\pi$
$19$	−3.89752	−0.894153	−0.447077	+	0.894496i	$0.647535\pi$
$20$	0	0
$21$	−6.97307	−1.52165
$22$	0	0
$23$	−7.62376	−1.58966	−0.794832	−	0.606829i	$-0.792441\pi$
$23$	−7.62376	−1.58966	−0.794832	+	0.606829i	$0.792441\pi$
$24$	0	0
$25$	−0.0892395	−0.0178479
$26$	0	0
$27$	1.94876	0.375039
$28$	0	0
$29$	3.34281	0.620744	0.310372	−	0.950615i	$-0.399546\pi$
$29$	3.34281	0.620744	0.310372	+	0.950615i	$0.399546\pi$
$30$	0	0
$31$	−10.3671	−1.86199	−0.930994	−	0.365034i	$-0.881057\pi$
$31$	−10.3671	−1.86199	−0.930994	+	0.365034i	$0.881057\pi$
$32$	0	0
$33$	2.58576	0.450124
$34$	0	0
$35$	6.81548	1.15203
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	6.36711	1.01955
$40$	0	0
$41$	2.54822	0.397965	0.198982	−	0.980003i	$-0.436236\pi$
$41$	2.54822	0.397965	0.198982	+	0.980003i	$0.436236\pi$
$42$	0	0
$43$	−8.43205	−1.28588	−0.642938	−	0.765919i	$-0.722285\pi$
$43$	−8.43205	−1.28588	−0.642938	+	0.765919i	$0.722285\pi$
$44$	0	0
$45$	4.74335	0.707097
$46$	0	0
$47$	−7.17802	−1.04702	−0.523511	−	0.852019i	$-0.675378\pi$
$47$	−7.17802	−1.04702	−0.523511	+	0.852019i	$0.675378\pi$
$48$	0	0
$49$	2.45898	0.351283
$50$	0	0
$51$	−4.53452	−0.634960
$52$	0	0
$53$	−5.61007	−0.770602	−0.385301	−	0.922791i	$-0.625902\pi$
$53$	−5.61007	−0.770602	−0.385301	+	0.922791i	$0.625902\pi$
$54$	0	0
$55$	−2.52733	−0.340785
$56$	0	0
$57$	8.83671	1.17045
$58$	0	0
$59$	6.17848	0.804369	0.402185	−	0.915559i	$-0.368251\pi$
$59$	6.17848	0.804369	0.402185	+	0.915559i	$0.368251\pi$
$60$	0	0
$61$	10.6481	1.36335	0.681673	−	0.731657i	$-0.261253\pi$
$61$	10.6481	1.36335	0.681673	+	0.731657i	$0.261253\pi$
$62$	0	0
$63$	6.58314	0.829397
$64$	0	0
$65$	−6.22322	−0.771896
$66$	0	0
$67$	0.318501	0.0389111	0.0194555	−	0.999811i	$-0.493807\pi$
$67$	0.318501	0.0389111	0.0194555	+	0.999811i	$0.493807\pi$
$68$	0	0
$69$	17.2851	2.08088
$70$	0	0
$71$	6.42555	0.762573	0.381286	−	0.924457i	$-0.375481\pi$
$71$	6.42555	0.762573	0.381286	+	0.924457i	$0.375481\pi$
$72$	0	0
$73$	16.0046	1.87319	0.936597	−	0.350409i	$-0.113957\pi$
$73$	16.0046	1.87319	0.936597	+	0.350409i	$0.113957\pi$
$74$	0	0
$75$	0.202329	0.0233630
$76$	0	0
$77$	−3.50759	−0.399727
$78$	0	0
$79$	−8.69281	−0.978018	−0.489009	−	0.872279i	$-0.662641\pi$
$79$	−8.69281	−0.978018	−0.489009	+	0.872279i	$0.662641\pi$
$80$	0	0
$81$	−10.8398	−1.20442
$82$	0	0
$83$	9.76116	1.07143	0.535713	−	0.844400i	$-0.320043\pi$
$83$	9.76116	1.07143	0.535713	+	0.844400i	$0.320043\pi$
$84$	0	0
$85$	4.43205	0.480723
$86$	0	0
$87$	−7.57902	−0.812556
$88$	0	0
$89$	−8.25357	−0.874876	−0.437438	−	0.899248i	$-0.644114\pi$
$89$	−8.25357	−0.874876	−0.437438	+	0.899248i	$0.644114\pi$
$90$	0	0
$91$	−8.63700	−0.905404
$92$	0	0
$93$	23.5050	2.43735
$94$	0	0
$95$	−8.63700	−0.886138
$96$	0	0
$97$	−13.2201	−1.34230	−0.671151	−	0.741321i	$-0.734200\pi$
$97$	−13.2201	−1.34230	−0.671151	+	0.741321i	$0.734200\pi$
$98$	0	0
$99$	−2.44117	−0.245347
$100$	0	0
$101$	−16.3231	−1.62421	−0.812103	−	0.583514i	$-0.801677\pi$
$101$	−16.3231	−1.62421	−0.812103	+	0.583514i	$0.801677\pi$
$102$	0	0
$103$	9.93507	0.978931	0.489466	−	0.872023i	$-0.337192\pi$
$103$	9.93507	0.978931	0.489466	+	0.872023i	$0.337192\pi$
$104$	0	0
$105$	−15.4525	−1.50801
$106$	0	0
$107$	−5.34281	−0.516509	−0.258254	−	0.966077i	$-0.583147\pi$
$107$	−5.34281	−0.516509	−0.258254	+	0.966077i	$0.583147\pi$
$108$	0	0
$109$	−5.79505	−0.555065	−0.277532	−	0.960716i	$-0.589517\pi$
$109$	−5.79505	−0.555065	−0.277532	+	0.960716i	$0.589517\pi$
$110$	0	0
$111$	2.26726	0.215199
$112$	0	0
$113$	11.0690	1.04129	0.520644	−	0.853774i	$-0.325692\pi$
$113$	11.0690	1.04129	0.520644	+	0.853774i	$0.325692\pi$
$114$	0	0
$115$	−16.8944	−1.57541
$116$	0	0
$117$	−6.01107	−0.555724
$118$	0	0
$119$	6.15109	0.563870
$120$	0	0
$121$	−9.69931	−0.881755
$122$	0	0
$123$	−5.77748	−0.520938
$124$	0	0
$125$	−11.2779	−1.00872
$126$	0	0
$127$	−2.43854	−0.216386	−0.108193	−	0.994130i	$-0.534506\pi$
$127$	−2.43854	−0.216386	−0.108193	+	0.994130i	$0.534506\pi$
$128$	0	0
$129$	19.1177	1.68322
$130$	0	0
$131$	−12.3296	−1.07724	−0.538620	−	0.842549i	$-0.681054\pi$
$131$	−12.3296	−1.07724	−0.538620	+	0.842549i	$0.681054\pi$
$132$	0	0
$133$	−11.9870	−1.03941
$134$	0	0
$135$	4.31850	0.371677
$136$	0	0
$137$	−13.5516	−1.15779	−0.578897	−	0.815401i	$-0.696517\pi$
$137$	−13.5516	−1.15779	−0.578897	+	0.815401i	$0.696517\pi$
$138$	0	0
$139$	6.20690	0.526463	0.263231	−	0.964733i	$-0.415212\pi$
$139$	6.20690	0.526463	0.263231	+	0.964733i	$0.415212\pi$
$140$	0	0
$141$	16.2745	1.37056
$142$	0	0
$143$	3.20279	0.267830
$144$	0	0
$145$	7.40774	0.615179
$146$	0	0
$147$	−5.57515	−0.459831
$148$	0	0
$149$	−2.26006	−0.185152	−0.0925758	−	0.995706i	$-0.529510\pi$
$149$	−2.26006	−0.185152	−0.0925758	+	0.995706i	$0.529510\pi$
$150$	0	0
$151$	17.5011	1.42422	0.712110	−	0.702068i	$-0.247740\pi$
$151$	17.5011	1.42422	0.712110	+	0.702068i	$0.247740\pi$
$152$	0	0
$153$	4.28096	0.346095
$154$	0	0
$155$	−22.9738	−1.84530
$156$	0	0
$157$	−11.9661	−0.955000	−0.477500	−	0.878632i	$-0.658457\pi$
$157$	−11.9661	−0.955000	−0.477500	+	0.878632i	$0.658457\pi$
$158$	0	0
$159$	12.7195	1.00872
$160$	0	0
$161$	−23.4472	−1.84790
$162$	0	0
$163$	−1.18452	−0.0927787	−0.0463893	−	0.998923i	$-0.514771\pi$
$163$	−1.18452	−0.0927787	−0.0463893	+	0.998923i	$0.514771\pi$
$164$	0	0
$165$	5.73011	0.446089
$166$	0	0
$167$	−20.2160	−1.56436	−0.782181	−	0.623051i	$-0.785893\pi$
$167$	−20.2160	−1.56436	−0.782181	+	0.623051i	$0.785893\pi$
$168$	0	0
$169$	−5.11355	−0.393350
$170$	0	0
$171$	−8.34256	−0.637972
$172$	0	0
$173$	−22.6791	−1.72426	−0.862131	−	0.506686i	$-0.830870\pi$
$173$	−22.6791	−1.72426	−0.862131	+	0.506686i	$0.830870\pi$
$174$	0	0
$175$	−0.274460	−0.0207472
$176$	0	0
$177$	−14.0082	−1.05292
$178$	0	0
$179$	14.5831	1.08999	0.544997	−	0.838438i	$-0.316531\pi$
$179$	14.5831	1.08999	0.544997	+	0.838438i	$0.316531\pi$
$180$	0	0
$181$	−24.7277	−1.83800	−0.918999	−	0.394260i	$-0.871001\pi$
$181$	−24.7277	−1.83800	−0.918999	+	0.394260i	$0.871001\pi$
$182$	0	0
$183$	−24.1420	−1.78463
$184$	0	0
$185$	−2.21602	−0.162925
$186$	0	0
$187$	−2.28096	−0.166800
$188$	0	0
$189$	5.99350	0.435963
$190$	0	0
$191$	−14.7241	−1.06540	−0.532698	−	0.846305i	$-0.678822\pi$
$191$	−14.7241	−1.06540	−0.532698	+	0.846305i	$0.678822\pi$
$192$	0	0
$193$	18.1997	1.31004	0.655022	−	0.755610i	$-0.272659\pi$
$193$	18.1997	1.31004	0.655022	+	0.755610i	$0.272659\pi$
$194$	0	0
$195$	14.1097	1.01041
$196$	0	0
$197$	−8.52803	−0.607597	−0.303798	−	0.952736i	$-0.598255\pi$
$197$	−8.52803	−0.607597	−0.303798	+	0.952736i	$0.598255\pi$
$198$	0	0
$199$	22.7130	1.61008	0.805041	−	0.593219i	$-0.202143\pi$
$199$	22.7130	1.61008	0.805041	+	0.593219i	$0.202143\pi$
$200$	0	0
$201$	−0.722125	−0.0509348
$202$	0	0
$203$	10.2810	0.721582
$204$	0	0
$205$	5.64691	0.394397
$206$	0	0
$207$	−16.3185	−1.13421
$208$	0	0
$209$	4.44504	0.307470
$210$	0	0
$211$	6.22252	0.428376	0.214188	−	0.976792i	$-0.431289\pi$
$211$	6.22252	0.428376	0.214188	+	0.976792i	$0.431289\pi$
$212$	0	0
$213$	−14.5684	−0.998211
$214$	0	0
$215$	−18.6856	−1.27435
$216$	0	0
$217$	−31.8845	−2.16446
$218$	0	0
$219$	−36.2866	−2.45202
$220$	0	0
$221$	−5.61657	−0.377811
$222$	0	0
$223$	−12.6040	−0.844028	−0.422014	−	0.906589i	$-0.638677\pi$
$223$	−12.6040	−0.844028	−0.422014	+	0.906589i	$0.638677\pi$
$224$	0	0
$225$	−0.191015	−0.0127344
$226$	0	0
$227$	26.3652	1.74992	0.874960	−	0.484196i	$-0.160888\pi$
$227$	26.3652	1.74992	0.874960	+	0.484196i	$0.160888\pi$
$228$	0	0
$229$	12.2957	0.812522	0.406261	−	0.913757i	$-0.366832\pi$
$229$	12.2957	0.812522	0.406261	+	0.913757i	$0.366832\pi$
$230$	0	0
$231$	7.95263	0.523245
$232$	0	0
$233$	14.1439	0.926597	0.463299	−	0.886202i	$-0.346666\pi$
$233$	14.1439	0.926597	0.463299	+	0.886202i	$0.346666\pi$
$234$	0	0
$235$	−15.9067	−1.03764
$236$	0	0
$237$	19.7089	1.28023
$238$	0	0
$239$	13.7171	0.887287	0.443643	−	0.896203i	$-0.353686\pi$
$239$	13.7171	0.887287	0.443643	+	0.896203i	$0.353686\pi$
$240$	0	0
$241$	28.9127	1.86243	0.931216	−	0.364469i	$-0.118749\pi$
$241$	28.9127	1.86243	0.931216	+	0.364469i	$0.118749\pi$
$242$	0	0
$243$	18.7304	1.20155
$244$	0	0
$245$	5.44916	0.348134
$246$	0	0
$247$	10.9453	0.696436
$248$	0	0
$249$	−22.1311	−1.40250
$250$	0	0
$251$	−21.3712	−1.34894	−0.674470	−	0.738302i	$-0.735628\pi$
$251$	−21.3712	−1.34894	−0.674470	+	0.738302i	$0.735628\pi$
$252$	0	0
$253$	8.69474	0.546633
$254$	0	0
$255$	−10.0486	−0.629269
$256$	0	0
$257$	3.64395	0.227304	0.113652	−	0.993521i	$-0.463745\pi$
$257$	3.64395	0.227304	0.113652	+	0.993521i	$0.463745\pi$
$258$	0	0
$259$	−3.07555	−0.191105
$260$	0	0
$261$	7.15521	0.442896
$262$	0	0
$263$	−18.2062	−1.12264	−0.561321	−	0.827598i	$-0.689707\pi$
$263$	−18.2062	−1.12264	−0.561321	+	0.827598i	$0.689707\pi$
$264$	0	0
$265$	−12.4320	−0.763695
$266$	0	0
$267$	18.7130	1.14522
$268$	0	0
$269$	−8.04861	−0.490733	−0.245366	−	0.969430i	$-0.578908\pi$
$269$	−8.04861	−0.490733	−0.245366	+	0.969430i	$0.578908\pi$
$270$	0	0
$271$	−13.8159	−0.839258	−0.419629	−	0.907696i	$-0.637840\pi$
$271$	−13.8159	−0.839258	−0.419629	+	0.907696i	$0.637840\pi$
$272$	0	0
$273$	19.5823	1.18518
$274$	0	0
$275$	0.101776	0.00613731
$276$	0	0
$277$	2.21602	0.133148	0.0665740	−	0.997781i	$-0.478793\pi$
$277$	2.21602	0.133148	0.0665740	+	0.997781i	$0.478793\pi$
$278$	0	0
$279$	−22.1906	−1.32852
$280$	0	0
$281$	−2.10248	−0.125423	−0.0627116	−	0.998032i	$-0.519975\pi$
$281$	−2.10248	−0.125423	−0.0627116	+	0.998032i	$0.519975\pi$
$282$	0	0
$283$	−6.26796	−0.372592	−0.186296	−	0.982494i	$-0.559648\pi$
$283$	−6.26796	−0.372592	−0.186296	+	0.982494i	$0.559648\pi$
$284$	0	0
$285$	19.5823	1.15996
$286$	0	0
$287$	7.83716	0.462613
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	29.9735	1.75708
$292$	0	0
$293$	10.3166	0.602701	0.301350	−	0.953513i	$-0.402563\pi$
$293$	10.3166	0.602701	0.301350	+	0.953513i	$0.402563\pi$
$294$	0	0
$295$	13.6917	0.797159
$296$	0	0
$297$	−2.22252	−0.128964
$298$	0	0
$299$	21.4097	1.23815
$300$	0	0
$301$	−25.9331	−1.49476
$302$	0	0
$303$	37.0087	2.12609
$304$	0	0
$305$	23.5964	1.35112
$306$	0	0
$307$	−11.1040	−0.633737	−0.316868	−	0.948469i	$-0.602631\pi$
$307$	−11.1040	−0.633737	−0.316868	+	0.948469i	$0.602631\pi$
$308$	0	0
$309$	−22.5254	−1.28143
$310$	0	0
$311$	10.3094	0.584591	0.292296	−	0.956328i	$-0.405581\pi$
$311$	10.3094	0.584591	0.292296	+	0.956328i	$0.405581\pi$
$312$	0	0
$313$	2.95218	0.166867	0.0834334	−	0.996513i	$-0.473411\pi$
$313$	2.95218	0.166867	0.0834334	+	0.996513i	$0.473411\pi$
$314$	0	0
$315$	14.5884	0.821963
$316$	0	0
$317$	14.1867	0.796805	0.398403	−	0.917211i	$-0.369565\pi$
$317$	14.1867	0.796805	0.398403	+	0.917211i	$0.369565\pi$
$318$	0	0
$319$	−3.81240	−0.213453
$320$	0	0
$321$	12.1135	0.676112
$322$	0	0
$323$	−7.79505	−0.433728
$324$	0	0
$325$	0.250610	0.0139013
$326$	0	0
$327$	13.1389	0.726582
$328$	0	0
$329$	−22.0763	−1.21711
$330$	0	0
$331$	26.4533	1.45400	0.727002	−	0.686636i	$-0.240913\pi$
$331$	26.4533	1.45400	0.727002	+	0.686636i	$0.240913\pi$
$332$	0	0
$333$	−2.14048	−0.117297
$334$	0	0
$335$	0.705806	0.0385623
$336$	0	0
$337$	2.57857	0.140463	0.0702317	−	0.997531i	$-0.477626\pi$
$337$	2.57857	0.140463	0.0702317	+	0.997531i	$0.477626\pi$
$338$	0	0
$339$	−25.0964	−1.36305
$340$	0	0
$341$	11.8235	0.640277
$342$	0	0
$343$	−13.9661	−0.754099
$344$	0	0
$345$	38.3041	2.06222
$346$	0	0
$347$	10.2810	0.551911	0.275955	−	0.961170i	$-0.411006\pi$
$347$	10.2810	0.551911	0.275955	+	0.961170i	$0.411006\pi$
$348$	0	0
$349$	23.9187	1.28034	0.640171	−	0.768233i	$-0.278864\pi$
$349$	23.9187	1.28034	0.640171	+	0.768233i	$0.278864\pi$
$350$	0	0
$351$	−5.47267	−0.292110
$352$	0	0
$353$	21.9596	1.16879	0.584396	−	0.811468i	$-0.301331\pi$
$353$	21.9596	1.16879	0.584396	+	0.811468i	$0.301331\pi$
$354$	0	0
$355$	14.2392	0.755737
$356$	0	0
$357$	−13.9461	−0.738108
$358$	0	0
$359$	−2.64350	−0.139518	−0.0697592	−	0.997564i	$-0.522223\pi$
$359$	−2.64350	−0.139518	−0.0697592	+	0.997564i	$0.522223\pi$
$360$	0	0
$361$	−3.80932	−0.200490
$362$	0	0
$363$	21.9909	1.15422
$364$	0	0
$365$	35.4665	1.85640
$366$	0	0
$367$	−13.9187	−0.726553	−0.363276	−	0.931681i	$-0.618342\pi$
$367$	−13.9187	−0.726553	−0.363276	+	0.931681i	$0.618342\pi$
$368$	0	0
$369$	5.45441	0.283945
$370$	0	0
$371$	−17.2540	−0.895784
$372$	0	0
$373$	15.2136	0.787733	0.393866	−	0.919168i	$-0.371137\pi$
$373$	15.2136	0.787733	0.393866	+	0.919168i	$0.371137\pi$
$374$	0	0
$375$	25.5699	1.32042
$376$	0	0
$377$	−9.38755	−0.483483
$378$	0	0
$379$	−1.17082	−0.0601412	−0.0300706	−	0.999548i	$-0.509573\pi$
$379$	−1.17082	−0.0601412	−0.0300706	+	0.999548i	$0.509573\pi$
$380$	0	0
$381$	5.52882	0.283250
$382$	0	0
$383$	−17.6310	−0.900900	−0.450450	−	0.892802i	$-0.648736\pi$
$383$	−17.6310	−0.900900	−0.450450	+	0.892802i	$0.648736\pi$
$384$	0	0
$385$	−7.77291	−0.396144
$386$	0	0
$387$	−18.0486	−0.917463
$388$	0	0
$389$	−27.6176	−1.40027	−0.700134	−	0.714012i	$-0.746876\pi$
$389$	−27.6176	−1.40027	−0.700134	+	0.714012i	$0.746876\pi$
$390$	0	0
$391$	−15.2475	−0.771101
$392$	0	0
$393$	27.9544	1.41011
$394$	0	0
$395$	−19.2635	−0.969251
$396$	0	0
$397$	5.85541	0.293874	0.146937	−	0.989146i	$-0.453058\pi$
$397$	5.85541	0.293874	0.146937	+	0.989146i	$0.453058\pi$
$398$	0	0
$399$	27.1777	1.36059
$400$	0	0
$401$	−0.485911	−0.0242653	−0.0121326	−	0.999926i	$-0.503862\pi$
$401$	−0.485911	−0.0242653	−0.0121326	+	0.999926i	$0.503862\pi$
$402$	0	0
$403$	29.1138	1.45026
$404$	0	0
$405$	−24.0212	−1.19362
$406$	0	0
$407$	1.14048	0.0565314
$408$	0	0
$409$	27.0417	1.33712	0.668562	−	0.743656i	$-0.266910\pi$
$409$	27.0417	1.33712	0.668562	+	0.743656i	$0.266910\pi$
$410$	0	0
$411$	30.7251	1.51556
$412$	0	0
$413$	19.0022	0.935037
$414$	0	0
$415$	21.6310	1.06182
$416$	0	0
$417$	−14.0727	−0.689142
$418$	0	0
$419$	−20.5704	−1.00493	−0.502464	−	0.864598i	$-0.667573\pi$
$419$	−20.5704	−1.00493	−0.502464	+	0.864598i	$0.667573\pi$
$420$	0	0
$421$	11.6594	0.568244	0.284122	−	0.958788i	$-0.408298\pi$
$421$	11.6594	0.568244	0.284122	+	0.958788i	$0.408298\pi$
$422$	0	0
$423$	−15.3644	−0.747043
$424$	0	0
$425$	−0.178479	−0.00865751
$426$	0	0
$427$	32.7486	1.58482
$428$	0	0
$429$	−7.26156	−0.350591
$430$	0	0
$431$	13.0690	0.629514	0.314757	−	0.949172i	$-0.398077\pi$
$431$	13.0690	0.629514	0.314757	+	0.949172i	$0.398077\pi$
$432$	0	0
$433$	−26.6454	−1.28050	−0.640249	−	0.768167i	$-0.721169\pi$
$433$	−26.6454	−1.28050	−0.640249	+	0.768167i	$0.721169\pi$
$434$	0	0
$435$	−16.7953	−0.805273
$436$	0	0
$437$	29.7138	1.42140
$438$	0	0
$439$	15.5964	0.744374	0.372187	−	0.928158i	$-0.378608\pi$
$439$	15.5964	0.744374	0.372187	+	0.928158i	$0.378608\pi$
$440$	0	0
$441$	5.26339	0.250638
$442$	0	0
$443$	−30.6546	−1.45644	−0.728221	−	0.685342i	$-0.759653\pi$
$443$	−30.6546	−1.45644	−0.728221	+	0.685342i	$0.759653\pi$
$444$	0	0
$445$	−18.2901	−0.867034
$446$	0	0
$447$	5.12416	0.242364
$448$	0	0
$449$	18.8641	0.890252	0.445126	−	0.895468i	$-0.353159\pi$
$449$	18.8641	0.890252	0.445126	+	0.895468i	$0.353159\pi$
$450$	0	0
$451$	−2.90619	−0.136847
$452$	0	0
$453$	−39.6796	−1.86431
$454$	0	0
$455$	−19.1398	−0.897288
$456$	0	0
$457$	−29.8571	−1.39666	−0.698329	−	0.715777i	$-0.746073\pi$
$457$	−29.8571	−1.39666	−0.698329	+	0.715777i	$0.746073\pi$
$458$	0	0
$459$	3.89752	0.181921
$460$	0	0
$461$	−30.1390	−1.40371	−0.701857	−	0.712318i	$-0.747645\pi$
$461$	−30.1390	−1.40371	−0.701857	+	0.712318i	$0.747645\pi$
$462$	0	0
$463$	−20.1857	−0.938108	−0.469054	−	0.883169i	$-0.655405\pi$
$463$	−20.1857	−0.938108	−0.469054	+	0.883169i	$0.655405\pi$
$464$	0	0
$465$	52.0876	2.41550
$466$	0	0
$467$	−31.2201	−1.44470	−0.722348	−	0.691530i	$-0.756937\pi$
$467$	−31.2201	−1.44470	−0.722348	+	0.691530i	$0.756937\pi$
$468$	0	0
$469$	0.979564	0.0452321
$470$	0	0
$471$	27.1303	1.25010
$472$	0	0
$473$	9.61657	0.442170
$474$	0	0
$475$	0.347813	0.0159588
$476$	0	0
$477$	−12.0082	−0.549819
$478$	0	0
$479$	36.7900	1.68098	0.840490	−	0.541827i	$-0.182267\pi$
$479$	36.7900	1.68098	0.840490	+	0.541827i	$0.182267\pi$
$480$	0	0
$481$	2.80828	0.128047
$482$	0	0
$483$	53.1610	2.41891
$484$	0	0
$485$	−29.2961	−1.33027
$486$	0	0
$487$	17.9461	0.813217	0.406609	−	0.913602i	$-0.366711\pi$
$487$	17.9461	0.813217	0.406609	+	0.913602i	$0.366711\pi$
$488$	0	0
$489$	2.68562	0.121448
$490$	0	0
$491$	29.3428	1.32422	0.662111	−	0.749406i	$-0.269661\pi$
$491$	29.3428	1.32422	0.662111	+	0.749406i	$0.269661\pi$
$492$	0	0
$493$	6.68562	0.301105
$494$	0	0
$495$	−5.40969	−0.243147
$496$	0	0
$497$	19.7621	0.886450
$498$	0	0
$499$	6.17848	0.276587	0.138293	−	0.990391i	$-0.455838\pi$
$499$	6.17848	0.276587	0.138293	+	0.990391i	$0.455838\pi$
$500$	0	0
$501$	45.8350	2.04776
$502$	0	0
$503$	6.14917	0.274178	0.137089	−	0.990559i	$-0.456225\pi$
$503$	6.14917	0.274178	0.137089	+	0.990559i	$0.456225\pi$
$504$	0	0
$505$	−36.1723	−1.60965
$506$	0	0
$507$	11.5938	0.514897
$508$	0	0
$509$	−27.6183	−1.22416	−0.612080	−	0.790796i	$-0.709667\pi$
$509$	−27.6183	−1.22416	−0.612080	+	0.790796i	$0.709667\pi$
$510$	0	0
$511$	49.2228	2.17749
$512$	0	0
$513$	−7.59534	−0.335343
$514$	0	0
$515$	22.0163	0.970156
$516$	0	0
$517$	8.18638	0.360037
$518$	0	0
$519$	51.4195	2.25707
$520$	0	0
$521$	−10.1720	−0.445643	−0.222821	−	0.974859i	$-0.571527\pi$
$521$	−10.1720	−0.445643	−0.222821	+	0.974859i	$0.571527\pi$
$522$	0	0
$523$	−9.41161	−0.411541	−0.205771	−	0.978600i	$-0.565970\pi$
$523$	−9.41161	−0.411541	−0.205771	+	0.978600i	$0.565970\pi$
$524$	0	0
$525$	0.622273	0.0271582
$526$	0	0
$527$	−20.7342	−0.903197
$528$	0	0
$529$	35.1218	1.52703
$530$	0	0
$531$	13.2249	0.573912
$532$	0	0
$533$	−7.15612	−0.309966
$534$	0	0
$535$	−11.8398	−0.511879
$536$	0	0
$537$	−33.0638	−1.42681
$538$	0	0
$539$	−2.80441	−0.120795
$540$	0	0
$541$	2.04371	0.0878659	0.0439329	−	0.999034i	$-0.486011\pi$
$541$	2.04371	0.0878659	0.0439329	+	0.999034i	$0.486011\pi$
$542$	0	0
$543$	56.0643	2.40595
$544$	0	0
$545$	−12.8420	−0.550089
$546$	0	0
$547$	−22.9271	−0.980292	−0.490146	−	0.871640i	$-0.663057\pi$
$547$	−22.9271	−0.980292	−0.490146	+	0.871640i	$0.663057\pi$
$548$	0	0
$549$	22.7920	0.972737
$550$	0	0
$551$	−13.0287	−0.555040
$552$	0	0
$553$	−26.7351	−1.13689
$554$	0	0
$555$	5.02431	0.213270
$556$	0	0
$557$	33.6633	1.42636	0.713179	−	0.700982i	$-0.247255\pi$
$557$	33.6633	1.42636	0.713179	+	0.700982i	$0.247255\pi$
$558$	0	0
$559$	23.6796	1.00154
$560$	0	0
$561$	5.17153	0.218342
$562$	0	0
$563$	10.0486	0.423499	0.211749	−	0.977324i	$-0.432084\pi$
$563$	10.0486	0.423499	0.211749	+	0.977324i	$0.432084\pi$
$564$	0	0
$565$	24.5293	1.03195
$566$	0	0
$567$	−33.3383	−1.40008
$568$	0	0
$569$	33.9947	1.42513	0.712567	−	0.701604i	$-0.247532\pi$
$569$	33.9947	1.42513	0.712567	+	0.701604i	$0.247532\pi$
$570$	0	0
$571$	6.71755	0.281121	0.140560	−	0.990072i	$-0.455110\pi$
$571$	6.71755	0.281121	0.140560	+	0.990072i	$0.455110\pi$
$572$	0	0
$573$	33.3833	1.39461
$574$	0	0
$575$	0.680341	0.0283722
$576$	0	0
$577$	−32.5701	−1.35591	−0.677956	−	0.735102i	$-0.737134\pi$
$577$	−32.5701	−1.35591	−0.677956	+	0.735102i	$0.737134\pi$
$578$	0	0
$579$	−41.2635	−1.71485
$580$	0	0
$581$	30.0209	1.24548
$582$	0	0
$583$	6.39816	0.264985
$584$	0	0
$585$	−13.3207	−0.550742
$586$	0	0
$587$	−30.6948	−1.26691	−0.633454	−	0.773780i	$-0.718363\pi$
$587$	−30.6948	−1.26691	−0.633454	+	0.773780i	$0.718363\pi$
$588$	0	0
$589$	40.4061	1.66490
$590$	0	0
$591$	19.3353	0.795347
$592$	0	0
$593$	−44.7258	−1.83667	−0.918334	−	0.395805i	$-0.870465\pi$
$593$	−44.7258	−1.83667	−0.918334	+	0.395805i	$0.870465\pi$
$594$	0	0
$595$	13.6310	0.558815
$596$	0	0
$597$	−51.4963	−2.10760
$598$	0	0
$599$	4.23268	0.172942	0.0864712	−	0.996254i	$-0.472441\pi$
$599$	4.23268	0.172942	0.0864712	+	0.996254i	$0.472441\pi$
$600$	0	0
$601$	7.01028	0.285955	0.142978	−	0.989726i	$-0.454332\pi$
$601$	7.01028	0.285955	0.142978	+	0.989726i	$0.454332\pi$
$602$	0	0
$603$	0.681744	0.0277628
$604$	0	0
$605$	−21.4939	−0.873851
$606$	0	0
$607$	23.4188	0.950540	0.475270	−	0.879840i	$-0.342350\pi$
$607$	23.4188	0.950540	0.475270	+	0.879840i	$0.342350\pi$
$608$	0	0
$609$	−23.3096	−0.944554
$610$	0	0
$611$	20.1579	0.815502
$612$	0	0
$613$	7.78855	0.314577	0.157288	−	0.987553i	$-0.449725\pi$
$613$	7.78855	0.314577	0.157288	+	0.987553i	$0.449725\pi$
$614$	0	0
$615$	−12.8030	−0.516268
$616$	0	0
$617$	−25.4388	−1.02413	−0.512064	−	0.858948i	$-0.671119\pi$
$617$	−25.4388	−1.02413	−0.512064	+	0.858948i	$0.671119\pi$
$618$	0	0
$619$	21.0948	0.847873	0.423937	−	0.905692i	$-0.360648\pi$
$619$	21.0948	0.847873	0.423937	+	0.905692i	$0.360648\pi$
$620$	0	0
$621$	−14.8569	−0.596187
$622$	0	0
$623$	−25.3842	−1.01700
$624$	0	0
$625$	−24.5458	−0.981834
$626$	0	0
$627$	−10.0781	−0.402479
$628$	0	0
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	3.60937	0.143687	0.0718433	−	0.997416i	$-0.477112\pi$
$631$	3.60937	0.143687	0.0718433	+	0.997416i	$0.477112\pi$
$632$	0	0
$633$	−14.1081	−0.560746
$634$	0	0
$635$	−5.40387	−0.214446
$636$	0	0
$637$	−6.90551	−0.273606
$638$	0	0
$639$	13.7538	0.544090
$640$	0	0
$641$	−36.8109	−1.45394	−0.726972	−	0.686667i	$-0.759073\pi$
$641$	−36.8109	−1.45394	−0.726972	+	0.686667i	$0.759073\pi$
$642$	0	0
$643$	28.3296	1.11721	0.558605	−	0.829434i	$-0.311337\pi$
$643$	28.3296	1.11721	0.558605	+	0.829434i	$0.311337\pi$
$644$	0	0
$645$	42.3652	1.66813
$646$	0	0
$647$	−0.581191	−0.0228490	−0.0114245	−	0.999935i	$-0.503637\pi$
$647$	−0.581191	−0.0228490	−0.0114245	+	0.999935i	$0.503637\pi$
$648$	0	0
$649$	−7.04642	−0.276596
$650$	0	0
$651$	72.2906	2.83329
$652$	0	0
$653$	3.71276	0.145291	0.0726457	−	0.997358i	$-0.476856\pi$
$653$	3.71276	0.145291	0.0726457	+	0.997358i	$0.476856\pi$
$654$	0	0
$655$	−27.3226	−1.06758
$656$	0	0
$657$	34.2574	1.33651
$658$	0	0
$659$	−15.5269	−0.604841	−0.302420	−	0.953175i	$-0.597795\pi$
$659$	−15.5269	−0.604841	−0.302420	+	0.953175i	$0.597795\pi$
$660$	0	0
$661$	−17.2760	−0.671957	−0.335978	−	0.941870i	$-0.609067\pi$
$661$	−17.2760	−0.671957	−0.335978	+	0.941870i	$0.609067\pi$
$662$	0	0
$663$	12.7342	0.494556
$664$	0	0
$665$	−26.5635	−1.03009
$666$	0	0
$667$	−25.4848	−0.986774
$668$	0	0
$669$	28.5766	1.10484
$670$	0	0
$671$	−12.1439	−0.468810
$672$	0	0
$673$	−8.68491	−0.334779	−0.167389	−	0.985891i	$-0.553534\pi$
$673$	−8.68491	−0.334779	−0.167389	+	0.985891i	$0.553534\pi$
$674$	0	0
$675$	−0.173907	−0.00669367
$676$	0	0
$677$	−35.7279	−1.37313	−0.686567	−	0.727067i	$-0.740883\pi$
$677$	−35.7279	−1.37313	−0.686567	+	0.727067i	$0.740883\pi$
$678$	0	0
$679$	−40.6591	−1.56035
$680$	0	0
$681$	−59.7768	−2.29065
$682$	0	0
$683$	8.80932	0.337079	0.168540	−	0.985695i	$-0.446095\pi$
$683$	8.80932	0.337079	0.168540	+	0.985695i	$0.446095\pi$
$684$	0	0
$685$	−30.0307	−1.14742
$686$	0	0
$687$	−27.8775	−1.06359
$688$	0	0
$689$	15.7547	0.600205
$690$	0	0
$691$	−6.36904	−0.242290	−0.121145	−	0.992635i	$-0.538657\pi$
$691$	−6.36904	−0.242290	−0.121145	+	0.992635i	$0.538657\pi$
$692$	0	0
$693$	−7.50793	−0.285203
$694$	0	0
$695$	13.7546	0.521743
$696$	0	0
$697$	5.09644	0.193041
$698$	0	0
$699$	−32.0679	−1.21292
$700$	0	0
$701$	−11.6199	−0.438877	−0.219439	−	0.975626i	$-0.570423\pi$
$701$	−11.6199	−0.438877	−0.219439	+	0.975626i	$0.570423\pi$
$702$	0	0
$703$	3.89752	0.146998
$704$	0	0
$705$	36.0646	1.35827
$706$	0	0
$707$	−50.2024	−1.88805
$708$	0	0
$709$	20.2372	0.760026	0.380013	−	0.924981i	$-0.375920\pi$
$709$	20.2372	0.760026	0.380013	+	0.924981i	$0.375920\pi$
$710$	0	0
$711$	−18.6068	−0.697809
$712$	0	0
$713$	79.0364	2.95994
$714$	0	0
$715$	7.09745	0.265430
$716$	0	0
$717$	−31.1003	−1.16146
$718$	0	0
$719$	−10.8432	−0.404383	−0.202192	−	0.979346i	$-0.564806\pi$
$719$	−10.8432	−0.404383	−0.202192	+	0.979346i	$0.564806\pi$
$720$	0	0
$721$	30.5557	1.13796
$722$	0	0
$723$	−65.5527	−2.43793
$724$	0	0
$725$	−0.298311	−0.0110790
$726$	0	0
$727$	−35.3481	−1.31099	−0.655494	−	0.755200i	$-0.727540\pi$
$727$	−35.3481	−1.31099	−0.655494	+	0.755200i	$0.727540\pi$
$728$	0	0
$729$	−9.94727	−0.368417
$730$	0	0
$731$	−16.8641	−0.623741
$732$	0	0
$733$	−27.3253	−1.00928	−0.504641	−	0.863329i	$-0.668375\pi$
$733$	−27.3253	−1.00928	−0.504641	+	0.863329i	$0.668375\pi$
$734$	0	0
$735$	−12.3547	−0.455709
$736$	0	0
$737$	−0.363243	−0.0133802
$738$	0	0
$739$	27.8208	1.02341	0.511703	−	0.859162i	$-0.329015\pi$
$739$	27.8208	1.02341	0.511703	+	0.859162i	$0.329015\pi$
$740$	0	0
$741$	−24.8160	−0.911638
$742$	0	0
$743$	−37.9335	−1.39164	−0.695822	−	0.718214i	$-0.744960\pi$
$743$	−37.9335	−1.39164	−0.695822	+	0.718214i	$0.744960\pi$
$744$	0	0
$745$	−5.00836	−0.183492
$746$	0	0
$747$	20.8936	0.764455
$748$	0	0
$749$	−16.4320	−0.600414
$750$	0	0
$751$	47.0299	1.71615	0.858073	−	0.513528i	$-0.171662\pi$
$751$	47.0299	1.71615	0.858073	+	0.513528i	$0.171662\pi$
$752$	0	0
$753$	48.4542	1.76577
$754$	0	0
$755$	38.7828	1.41145
$756$	0	0
$757$	−13.0041	−0.472643	−0.236321	−	0.971675i	$-0.575942\pi$
$757$	−13.0041	−0.472643	−0.236321	+	0.971675i	$0.575942\pi$
$758$	0	0
$759$	−19.7132	−0.715546
$760$	0	0
$761$	48.7562	1.76741	0.883705	−	0.468045i	$-0.155041\pi$
$761$	48.7562	1.76741	0.883705	+	0.468045i	$0.155041\pi$
$762$	0	0
$763$	−17.8229	−0.645233
$764$	0	0
$765$	9.48670	0.342992
$766$	0	0
$767$	−17.3509	−0.626505
$768$	0	0
$769$	25.5641	0.921865	0.460933	−	0.887435i	$-0.347515\pi$
$769$	25.5641	0.921865	0.460933	+	0.887435i	$0.347515\pi$
$770$	0	0
$771$	−8.26180	−0.297541
$772$	0	0
$773$	−10.6026	−0.381350	−0.190675	−	0.981653i	$-0.561068\pi$
$773$	−10.6026	−0.381350	−0.190675	+	0.981653i	$0.561068\pi$
$774$	0	0
$775$	0.925156	0.0332326
$776$	0	0
$777$	6.97307	0.250157
$778$	0	0
$779$	−9.93174	−0.355842
$780$	0	0
$781$	−7.32820	−0.262224
$782$	0	0
$783$	6.51433	0.232803
$784$	0	0
$785$	−26.5172	−0.946439
$786$	0	0
$787$	40.9813	1.46083	0.730413	−	0.683006i	$-0.239328\pi$
$787$	40.9813	1.46083	0.730413	+	0.683006i	$0.239328\pi$
$788$	0	0
$789$	41.2782	1.46954
$790$	0	0
$791$	34.0434	1.21044
$792$	0	0
$793$	−29.9028	−1.06188
$794$	0	0
$795$	28.1867	0.999680
$796$	0	0
$797$	45.0984	1.59747	0.798733	−	0.601685i	$-0.205504\pi$
$797$	45.0984	1.59747	0.798733	+	0.601685i	$0.205504\pi$
$798$	0	0
$799$	−14.3560	−0.507880
$800$	0	0
$801$	−17.6666	−0.624218
$802$	0	0
$803$	−18.2529	−0.644130
$804$	0	0
$805$	−51.9596	−1.83134
$806$	0	0
$807$	18.2483	0.642371
$808$	0	0
$809$	13.4342	0.472323	0.236161	−	0.971714i	$-0.424111\pi$
$809$	13.4342	0.472323	0.236161	+	0.971714i	$0.424111\pi$
$810$	0	0
$811$	33.6020	1.17993	0.589963	−	0.807431i	$-0.299142\pi$
$811$	33.6020	1.17993	0.589963	+	0.807431i	$0.299142\pi$
$812$	0	0
$813$	31.3244	1.09859
$814$	0	0
$815$	−2.62492	−0.0919470
$816$	0	0
$817$	32.8641	1.14977
$818$	0	0
$819$	−18.4873	−0.645999
$820$	0	0
$821$	7.15680	0.249774	0.124887	−	0.992171i	$-0.460143\pi$
$821$	7.15680	0.249774	0.124887	+	0.992171i	$0.460143\pi$
$822$	0	0
$823$	17.7282	0.617966	0.308983	−	0.951068i	$-0.400011\pi$
$823$	17.7282	0.617966	0.308983	+	0.951068i	$0.400011\pi$
$824$	0	0
$825$	−0.230752	−0.00803376
$826$	0	0
$827$	30.4129	1.05756	0.528780	−	0.848759i	$-0.322650\pi$
$827$	30.4129	1.05756	0.528780	+	0.848759i	$0.322650\pi$
$828$	0	0
$829$	32.3315	1.12292	0.561460	−	0.827504i	$-0.310240\pi$
$829$	32.3315	1.12292	0.561460	+	0.827504i	$0.310240\pi$
$830$	0	0
$831$	−5.02431	−0.174291
$832$	0	0
$833$	4.91796	0.170397
$834$	0	0
$835$	−44.7992	−1.55034
$836$	0	0
$837$	−20.2030	−0.698319
$838$	0	0
$839$	3.00604	0.103780	0.0518900	−	0.998653i	$-0.483475\pi$
$839$	3.00604	0.103780	0.0518900	+	0.998653i	$0.483475\pi$
$840$	0	0
$841$	−17.8256	−0.614677
$842$	0	0
$843$	4.76687	0.164180
$844$	0	0
$845$	−11.3317	−0.389824
$846$	0	0
$847$	−29.8307	−1.02499
$848$	0	0
$849$	14.2111	0.487724
$850$	0	0
$851$	7.62376	0.261339
$852$	0	0
$853$	−32.2920	−1.10566	−0.552829	−	0.833295i	$-0.686452\pi$
$853$	−32.2920	−1.10566	−0.552829	+	0.833295i	$0.686452\pi$
$854$	0	0
$855$	−18.4873	−0.632253
$856$	0	0
$857$	42.8093	1.46234	0.731169	−	0.682196i	$-0.238975\pi$
$857$	42.8093	1.46234	0.731169	+	0.682196i	$0.238975\pi$
$858$	0	0
$859$	20.5567	0.701384	0.350692	−	0.936491i	$-0.385946\pi$
$859$	20.5567	0.701384	0.350692	+	0.936491i	$0.385946\pi$
$860$	0	0
$861$	−17.7689	−0.605563
$862$	0	0
$863$	−30.2353	−1.02922	−0.514611	−	0.857424i	$-0.672064\pi$
$863$	−30.2353	−1.02922	−0.514611	+	0.857424i	$0.672064\pi$
$864$	0	0
$865$	−50.2575	−1.70880
$866$	0	0
$867$	29.4744	1.00100
$868$	0	0
$869$	9.91396	0.336308
$870$	0	0
$871$	−0.894441	−0.0303070
$872$	0	0
$873$	−28.2974	−0.957723
$874$	0	0
$875$	−34.6856	−1.17259
$876$	0	0
$877$	−31.7830	−1.07323	−0.536617	−	0.843826i	$-0.680298\pi$
$877$	−31.7830	−1.07323	−0.536617	+	0.843826i	$0.680298\pi$
$878$	0	0
$879$	−23.3904	−0.788938
$880$	0	0
$881$	−9.27159	−0.312368	−0.156184	−	0.987728i	$-0.549919\pi$
$881$	−9.27159	−0.312368	−0.156184	+	0.987728i	$0.549919\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	−31.0426	−1.04348
$886$	0	0
$887$	30.8250	1.03500	0.517500	−	0.855683i	$-0.326863\pi$
$887$	30.8250	1.03500	0.517500	+	0.855683i	$0.326863\pi$
$888$	0	0
$889$	−7.49985	−0.251537
$890$	0	0
$891$	12.3625	0.414161
$892$	0	0
$893$	27.9765	0.936198
$894$	0	0
$895$	32.3166	1.08022
$896$	0	0
$897$	−48.5414	−1.62075
$898$	0	0
$899$	−34.6553	−1.15582
$900$	0	0
$901$	−11.2201	−0.373797
$902$	0	0
$903$	58.7972	1.95665
$904$	0	0
$905$	−54.7972	−1.82152
$906$	0	0
$907$	55.9288	1.85709	0.928543	−	0.371225i	$-0.121062\pi$
$907$	55.9288	1.85709	0.928543	+	0.371225i	$0.121062\pi$
$908$	0	0
$909$	−34.9392	−1.15886
$910$	0	0
$911$	18.4663	0.611815	0.305907	−	0.952061i	$-0.401040\pi$
$911$	18.4663	0.611815	0.305907	+	0.952061i	$0.401040\pi$
$912$	0	0
$913$	−11.1324	−0.368428
$914$	0	0
$915$	−53.4992	−1.76863
$916$	0	0
$917$	−37.9202	−1.25223
$918$	0	0
$919$	5.12383	0.169019	0.0845097	−	0.996423i	$-0.473068\pi$
$919$	5.12383	0.169019	0.0845097	+	0.996423i	$0.473068\pi$
$920$	0	0
$921$	25.1756	0.829564
$922$	0	0
$923$	−18.0448	−0.593951
$924$	0	0
$925$	0.0892395	0.00293418
$926$	0	0
$927$	21.2658	0.698460
$928$	0	0
$929$	−0.612450	−0.0200938	−0.0100469	−	0.999950i	$-0.503198\pi$
$929$	−0.612450	−0.0200938	−0.0100469	+	0.999950i	$0.503198\pi$
$930$	0	0
$931$	−9.58393	−0.314101
$932$	0	0
$933$	−23.3741	−0.765233
$934$	0	0
$935$	−5.05465	−0.165305
$936$	0	0
$937$	−3.83662	−0.125337	−0.0626684	−	0.998034i	$-0.519961\pi$
$937$	−3.83662	−0.125337	−0.0626684	+	0.998034i	$0.519961\pi$
$938$	0	0
$939$	−6.69336	−0.218429
$940$	0	0
$941$	−3.05465	−0.0995789	−0.0497894	−	0.998760i	$-0.515855\pi$
$941$	−3.05465	−0.0995789	−0.0497894	+	0.998760i	$0.515855\pi$
$942$	0	0
$943$	−19.4270	−0.632631
$944$	0	0
$945$	13.2817	0.432055
$946$	0	0
$947$	−38.9604	−1.26604	−0.633022	−	0.774134i	$-0.718186\pi$
$947$	−38.9604	−1.26604	−0.633022	+	0.774134i	$0.718186\pi$
$948$	0	0
$949$	−44.9454	−1.45899
$950$	0	0
$951$	−32.1650	−1.04302
$952$	0	0
$953$	42.2529	1.36871	0.684353	−	0.729151i	$-0.260085\pi$
$953$	42.2529	1.36871	0.684353	+	0.729151i	$0.260085\pi$
$954$	0	0
$955$	−32.6289	−1.05585
$956$	0	0
$957$	8.64371	0.279411
$958$	0	0
$959$	−41.6787	−1.34587
$960$	0	0
$961$	76.4771	2.46700
$962$	0	0
$963$	−11.4362	−0.368525
$964$	0	0
$965$	40.3310	1.29830
$966$	0	0
$967$	36.2122	1.16451	0.582253	−	0.813008i	$-0.302171\pi$
$967$	36.2122	1.16451	0.582253	+	0.813008i	$0.302171\pi$
$968$	0	0
$969$	17.6734	0.567752
$970$	0	0
$971$	−11.9615	−0.383864	−0.191932	−	0.981408i	$-0.561475\pi$
$971$	−11.9615	−0.383864	−0.191932	+	0.981408i	$0.561475\pi$
$972$	0	0
$973$	19.0896	0.611985
$974$	0	0
$975$	−0.568198	−0.0181969
$976$	0	0
$977$	−38.3440	−1.22673	−0.613366	−	0.789799i	$-0.710185\pi$
$977$	−38.3440	−1.22673	−0.613366	+	0.789799i	$0.710185\pi$
$978$	0	0
$979$	9.41301	0.300841
$980$	0	0
$981$	−12.4042	−0.396035
$982$	0	0
$983$	31.1742	0.994302	0.497151	−	0.867664i	$-0.334380\pi$
$983$	31.1742	0.994302	0.497151	+	0.867664i	$0.334380\pi$
$984$	0	0
$985$	−18.8983	−0.602150
$986$	0	0
$987$	50.0528	1.59320
$988$	0	0
$989$	64.2839	2.04411
$990$	0	0
$991$	−39.2981	−1.24834	−0.624172	−	0.781287i	$-0.714564\pi$
$991$	−39.2981	−1.24834	−0.624172	+	0.781287i	$0.714564\pi$
$992$	0	0
$993$	−59.9765	−1.90330
$994$	0	0
$995$	50.3326	1.59565
$996$	0	0
$997$	−34.3022	−1.08636	−0.543180	−	0.839616i	$-0.682780\pi$
$997$	−34.3022	−1.08636	−0.543180	+	0.839616i	$0.682780\pi$
$998$	0	0
$999$	−1.94876	−0.0616561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bg.1.2		4
4.3	odd	2		2368.2.a.bh.1.3		4
8.3	odd	2		592.2.a.j.1.2		4
8.5	even	2		296.2.a.d.1.3	✓	4
24.5	odd	2		2664.2.a.r.1.4		4
24.11	even	2		5328.2.a.bp.1.4		4
40.29	even	2		7400.2.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.d.1.3	✓	4	8.5	even	2
592.2.a.j.1.2		4	8.3	odd	2
2368.2.a.bg.1.2		4	1.1	even	1	trivial
2368.2.a.bh.1.3		4	4.3	odd	2
2664.2.a.r.1.4		4	24.5	odd	2
5328.2.a.bp.1.4		4	24.11	even	2
7400.2.a.n.1.2		4	40.29	even	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26726 q^{3} +2.21602 q^{5} +3.07555 q^{7} +2.14048 q^{9} -1.14048 q^{11} -2.80828 q^{13} -5.02431 q^{15} +2.00000 q^{17} -3.89752 q^{19} -6.97307 q^{21} -7.62376 q^{23} -0.0892395 q^{25} +1.94876 q^{27} +3.34281 q^{29} -10.3671 q^{31} +2.58576 q^{33} +6.81548 q^{35} -1.00000 q^{37} +6.36711 q^{39} +2.54822 q^{41} -8.43205 q^{43} +4.74335 q^{45} -7.17802 q^{47} +2.45898 q^{49} -4.53452 q^{51} -5.61007 q^{53} -2.52733 q^{55} +8.83671 q^{57} +6.17848 q^{59} +10.6481 q^{61} +6.58314 q^{63} -6.22322 q^{65} +0.318501 q^{67} +17.2851 q^{69} +6.42555 q^{71} +16.0046 q^{73} +0.202329 q^{75} -3.50759 q^{77} -8.69281 q^{79} -10.8398 q^{81} +9.76116 q^{83} +4.43205 q^{85} -7.57902 q^{87} -8.25357 q^{89} -8.63700 q^{91} +23.5050 q^{93} -8.63700 q^{95} -13.2201 q^{97} -2.44117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 5 q^{5} - q^{7} + 8 q^{9} - 4 q^{11} - 5 q^{13} + 8 q^{17} - 2 q^{19} - q^{21} - 9 q^{23} + 7 q^{25} + q^{27} - 7 q^{29} - q^{31} + 3 q^{33} + 12 q^{35} - 4 q^{37} - 15 q^{39} + 2 q^{41}+ \cdots - 44 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 5 * q^5 - q^7 + 8 * q^9 - 4 * q^11 - 5 * q^13 + 8 * q^17 - 2 * q^19 - q^21 - 9 * q^23 + 7 * q^25 + q^27 - 7 * q^29 - q^31 + 3 * q^33 + 12 * q^35 - 4 * q^37 - 15 * q^39 + 2 * q^41 - 6 * q^43 - 29 * q^47 + 9 * q^49 - 4 * q^51 + 5 * q^53 - 5 * q^55 - 32 * q^57 + 10 * q^59 + q^61 - 28 * q^63 - 2 * q^65 + q^67 + 27 * q^69 - 17 * q^71 + 8 * q^73 - 19 * q^75 + 27 * q^77 + 15 * q^79 - 8 * q^81 - 15 * q^83 - 10 * q^85 - 17 * q^87 - 20 * q^89 - 34 * q^91 + 6 * q^93 - 34 * q^95 + 2 * q^97 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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