LMFDB - Embedded newform 2352.4.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{137}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 34 $ x^2 - x - 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.35235$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +2.70470 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +2.70470 q^{5} +9.00000 q^{9} -64.1141 q^{11} -70.8188 q^{13} -8.11410 q^{15} +14.7047 q^{17} +159.047 q^{19} +172.570 q^{23} -117.685 q^{25} -27.0000 q^{27} +18.2282 q^{29} +149.409 q^{31} +192.342 q^{33} +41.7718 q^{37} +212.456 q^{39} +50.2081 q^{41} +388.000 q^{43} +24.3423 q^{45} -494.550 q^{47} -44.1141 q^{51} +469.772 q^{53} -173.409 q^{55} -477.141 q^{57} +343.275 q^{59} +72.4966 q^{61} -191.544 q^{65} -293.597 q^{67} -517.711 q^{69} -629.940 q^{71} -696.685 q^{73} +353.054 q^{75} +640.685 q^{79} +81.0000 q^{81} +1072.72 q^{83} +39.7718 q^{85} -54.6846 q^{87} -667.430 q^{89} -448.228 q^{93} +430.174 q^{95} -1307.50 q^{97} -577.027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 18 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 18 * q^5 + 18 * q^9	$ 2 q - 6 q^{3} - 18 q^{5} + 18 q^{9} - 58 q^{11} - 48 q^{13} + 54 q^{15} + 6 q^{17} + 84 q^{19} - 6 q^{23} + 186 q^{25} - 54 q^{27} - 104 q^{29} + 252 q^{31} + 174 q^{33} + 224 q^{37} + 144 q^{39} - 438 q^{41} + 776 q^{43} - 162 q^{45} - 240 q^{47} - 18 q^{51} + 1080 q^{53} - 300 q^{55} - 252 q^{57} + 312 q^{59} + 660 q^{61} - 664 q^{65} + 396 q^{67} + 18 q^{69} - 66 q^{71} - 972 q^{73} - 558 q^{75} + 860 q^{79} + 162 q^{81} + 2520 q^{83} + 220 q^{85} + 312 q^{87} - 1686 q^{89} - 756 q^{93} + 1984 q^{95} - 2100 q^{97} - 522 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 18 * q^5 + 18 * q^9 - 58 * q^11 - 48 * q^13 + 54 * q^15 + 6 * q^17 + 84 * q^19 - 6 * q^23 + 186 * q^25 - 54 * q^27 - 104 * q^29 + 252 * q^31 + 174 * q^33 + 224 * q^37 + 144 * q^39 - 438 * q^41 + 776 * q^43 - 162 * q^45 - 240 * q^47 - 18 * q^51 + 1080 * q^53 - 300 * q^55 - 252 * q^57 + 312 * q^59 + 660 * q^61 - 664 * q^65 + 396 * q^67 + 18 * q^69 - 66 * q^71 - 972 * q^73 - 558 * q^75 + 860 * q^79 + 162 * q^81 + 2520 * q^83 + 220 * q^85 + 312 * q^87 - 1686 * q^89 - 756 * q^93 + 1984 * q^95 - 2100 * q^97 - 522 * q^99

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	2.70470	0.241916	0.120958	−	0.992658i	$-0.461403\pi$
$5$	2.70470	0.241916	0.120958	+	0.992658i	$0.461403\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−64.1141	−1.75738	−0.878688	−	0.477397i	$-0.841580\pi$
$11$	−64.1141	−1.75738	−0.878688	+	0.477397i	$0.841580\pi$
$12$	0	0
$13$	−70.8188	−1.51089	−0.755446	−	0.655211i	$-0.772580\pi$
$13$	−70.8188	−1.51089	−0.755446	+	0.655211i	$0.772580\pi$
$14$	0	0
$15$	−8.11410	−0.139670
$16$	0	0
$17$	14.7047	0.209789	0.104895	−	0.994483i	$-0.466550\pi$
$17$	14.7047	0.209789	0.104895	+	0.994483i	$0.466550\pi$
$18$	0	0
$19$	159.047	1.92041	0.960207	−	0.279288i	$-0.0900983\pi$
$19$	159.047	1.92041	0.960207	+	0.279288i	$0.0900983\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	172.570	1.56450	0.782249	−	0.622966i	$-0.214073\pi$
$23$	172.570	1.56450	0.782249	+	0.622966i	$0.214073\pi$
$24$	0	0
$25$	−117.685	−0.941477
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	18.2282	0.116720	0.0583602	−	0.998296i	$-0.481413\pi$
$29$	18.2282	0.116720	0.0583602	+	0.998296i	$0.481413\pi$
$30$	0	0
$31$	149.409	0.865636	0.432818	−	0.901481i	$-0.357519\pi$
$31$	149.409	0.865636	0.432818	+	0.901481i	$0.357519\pi$
$32$	0	0
$33$	192.342	1.01462
$34$	0	0
$35$	0	0
$36$	0	0
$37$	41.7718	0.185601	0.0928006	−	0.995685i	$-0.470418\pi$
$37$	41.7718	0.185601	0.0928006	+	0.995685i	$0.470418\pi$
$38$	0	0
$39$	212.456	0.872314
$40$	0	0
$41$	50.2081	0.191248	0.0956242	−	0.995418i	$-0.469515\pi$
$41$	50.2081	0.191248	0.0956242	+	0.995418i	$0.469515\pi$
$42$	0	0
$43$	388.000	1.37603	0.688017	−	0.725695i	$-0.258482\pi$
$43$	388.000	1.37603	0.688017	+	0.725695i	$0.258482\pi$
$44$	0	0
$45$	24.3423	0.0806386
$46$	0	0
$47$	−494.550	−1.53484	−0.767421	−	0.641143i	$-0.778460\pi$
$47$	−494.550	−1.53484	−0.767421	+	0.641143i	$0.778460\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−44.1141	−0.121122
$52$	0	0
$53$	469.772	1.21751	0.608756	−	0.793358i	$-0.291669\pi$
$53$	469.772	1.21751	0.608756	+	0.793358i	$0.291669\pi$
$54$	0	0
$55$	−173.409	−0.425137
$56$	0	0
$57$	−477.141	−1.10875
$58$	0	0
$59$	343.275	0.757468	0.378734	−	0.925506i	$-0.376359\pi$
$59$	343.275	0.757468	0.378734	+	0.925506i	$0.376359\pi$
$60$	0	0
$61$	72.4966	0.152168	0.0760839	−	0.997101i	$-0.475758\pi$
$61$	72.4966	0.152168	0.0760839	+	0.997101i	$0.475758\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−191.544	−0.365509
$66$	0	0
$67$	−293.597	−0.535353	−0.267676	−	0.963509i	$-0.586256\pi$
$67$	−293.597	−0.535353	−0.267676	+	0.963509i	$0.586256\pi$
$68$	0	0
$69$	−517.711	−0.903263
$70$	0	0
$71$	−629.940	−1.05296	−0.526479	−	0.850188i	$-0.676488\pi$
$71$	−629.940	−1.05296	−0.526479	+	0.850188i	$0.676488\pi$
$72$	0	0
$73$	−696.685	−1.11700	−0.558498	−	0.829506i	$-0.688622\pi$
$73$	−696.685	−1.11700	−0.558498	+	0.829506i	$0.688622\pi$
$74$	0	0
$75$	353.054	0.543562
$76$	0	0
$77$	0	0
$78$	0	0
$79$	640.685	0.912439	0.456219	−	0.889867i	$-0.349203\pi$
$79$	640.685	0.912439	0.456219	+	0.889867i	$0.349203\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1072.72	1.41864	0.709318	−	0.704888i	$-0.249003\pi$
$83$	1072.72	1.41864	0.709318	+	0.704888i	$0.249003\pi$
$84$	0	0
$85$	39.7718	0.0507513
$86$	0	0
$87$	−54.6846	−0.0673886
$88$	0	0
$89$	−667.430	−0.794914	−0.397457	−	0.917621i	$-0.630107\pi$
$89$	−667.430	−0.794914	−0.397457	+	0.917621i	$0.630107\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−448.228	−0.499775
$94$	0	0
$95$	430.174	0.464579
$96$	0	0
$97$	−1307.50	−1.36863	−0.684314	−	0.729188i	$-0.739898\pi$
$97$	−1307.50	−1.36863	−0.684314	+	0.729188i	$0.739898\pi$
$98$	0	0
$99$	−577.027	−0.585792
$100$	0	0
$101$	−720.342	−0.709671	−0.354835	−	0.934929i	$-0.615463\pi$
$101$	−720.342	−0.709671	−0.354835	+	0.934929i	$0.615463\pi$
$102$	0	0
$103$	−1762.51	−1.68607	−0.843035	−	0.537858i	$-0.819234\pi$
$103$	−1762.51	−1.68607	−0.843035	+	0.537858i	$0.819234\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−139.309	−0.125864	−0.0629322	−	0.998018i	$-0.520045\pi$
$107$	−139.309	−0.125864	−0.0629322	+	0.998018i	$0.520045\pi$
$108$	0	0
$109$	569.544	0.500481	0.250240	−	0.968184i	$-0.419490\pi$
$109$	569.544	0.500481	0.250240	+	0.968184i	$0.419490\pi$
$110$	0	0
$111$	−125.315	−0.107157
$112$	0	0
$113$	−1843.48	−1.53469	−0.767344	−	0.641236i	$-0.778422\pi$
$113$	−1843.48	−1.53469	−0.767344	+	0.641236i	$0.778422\pi$
$114$	0	0
$115$	466.751	0.378476
$116$	0	0
$117$	−637.369	−0.503631
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2779.62	2.08837
$122$	0	0
$123$	−150.624	−0.110417
$124$	0	0
$125$	−656.389	−0.469674
$126$	0	0
$127$	506.510	0.353902	0.176951	−	0.984220i	$-0.443377\pi$
$127$	506.510	0.353902	0.176951	+	0.984220i	$0.443377\pi$
$128$	0	0
$129$	−1164.00	−0.794453
$130$	0	0
$131$	−1054.82	−0.703511	−0.351756	−	0.936092i	$-0.614415\pi$
$131$	−1054.82	−0.703511	−0.351756	+	0.936092i	$0.614415\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−73.0269	−0.0465567
$136$	0	0
$137$	−1690.00	−1.05392	−0.526958	−	0.849891i	$-0.676667\pi$
$137$	−1690.00	−1.05392	−0.526958	+	0.849891i	$0.676667\pi$
$138$	0	0
$139$	1732.30	1.05706	0.528530	−	0.848915i	$-0.322743\pi$
$139$	1732.30	1.05706	0.528530	+	0.848915i	$0.322743\pi$
$140$	0	0
$141$	1483.65	0.886142
$142$	0	0
$143$	4540.48	2.65520
$144$	0	0
$145$	49.3018	0.0282365
$146$	0	0
$147$	0	0
$148$	0	0
$149$	338.121	0.185906	0.0929528	−	0.995671i	$-0.470369\pi$
$149$	338.121	0.185906	0.0929528	+	0.995671i	$0.470369\pi$
$150$	0	0
$151$	−6.17440	−0.00332759	−0.00166379	−	0.999999i	$-0.500530\pi$
$151$	−6.17440	−0.00332759	−0.00166379	+	0.999999i	$0.500530\pi$
$152$	0	0
$153$	132.342	0.0699297
$154$	0	0
$155$	404.108	0.209411
$156$	0	0
$157$	−731.691	−0.371945	−0.185972	−	0.982555i	$-0.559544\pi$
$157$	−731.691	−0.371945	−0.185972	+	0.982555i	$0.559544\pi$
$158$	0	0
$159$	−1409.32	−0.702931
$160$	0	0
$161$	0	0
$162$	0	0
$163$	761.249	0.365801	0.182901	−	0.983131i	$-0.441451\pi$
$163$	761.249	0.365801	0.182901	+	0.983131i	$0.441451\pi$
$164$	0	0
$165$	520.228	0.245453
$166$	0	0
$167$	−2731.03	−1.26547	−0.632736	−	0.774368i	$-0.718068\pi$
$167$	−2731.03	−1.26547	−0.632736	+	0.774368i	$0.718068\pi$
$168$	0	0
$169$	2818.30	1.28280
$170$	0	0
$171$	1431.42	0.640138
$172$	0	0
$173$	3721.16	1.63534	0.817672	−	0.575684i	$-0.195264\pi$
$173$	3721.16	1.63534	0.817672	+	0.575684i	$0.195264\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1029.83	−0.437324
$178$	0	0
$179$	2386.15	0.996366	0.498183	−	0.867072i	$-0.334001\pi$
$179$	2386.15	0.996366	0.498183	+	0.867072i	$0.334001\pi$
$180$	0	0
$181$	−3819.54	−1.56853	−0.784266	−	0.620424i	$-0.786960\pi$
$181$	−3819.54	−1.56853	−0.784266	+	0.620424i	$0.786960\pi$
$182$	0	0
$183$	−217.490	−0.0878541
$184$	0	0
$185$	112.980	0.0448998
$186$	0	0
$187$	−942.779	−0.368678
$188$	0	0
$189$	0	0
$190$	0	0
$191$	214.168	0.0811343	0.0405671	−	0.999177i	$-0.487084\pi$
$191$	214.168	0.0811343	0.0405671	+	0.999177i	$0.487084\pi$
$192$	0	0
$193$	3778.44	1.40921	0.704607	−	0.709598i	$-0.251123\pi$
$193$	3778.44	1.40921	0.704607	+	0.709598i	$0.251123\pi$
$194$	0	0
$195$	574.631	0.211026
$196$	0	0
$197$	3395.01	1.22784	0.613919	−	0.789369i	$-0.289592\pi$
$197$	3395.01	1.22784	0.613919	+	0.789369i	$0.289592\pi$
$198$	0	0
$199$	3926.17	1.39859	0.699294	−	0.714834i	$-0.253498\pi$
$199$	3926.17	1.39859	0.699294	+	0.714834i	$0.253498\pi$
$200$	0	0
$201$	880.792	0.309086
$202$	0	0
$203$	0	0
$204$	0	0
$205$	135.798	0.0462660
$206$	0	0
$207$	1553.13	0.521499
$208$	0	0
$209$	−10197.2	−3.37489
$210$	0	0
$211$	−816.108	−0.266271	−0.133135	−	0.991098i	$-0.542505\pi$
$211$	−816.108	−0.266271	−0.133135	+	0.991098i	$0.542505\pi$
$212$	0	0
$213$	1889.82	0.607926
$214$	0	0
$215$	1049.42	0.332884
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2090.05	0.644898
$220$	0	0
$221$	−1041.37	−0.316969
$222$	0	0
$223$	−630.094	−0.189212	−0.0946059	−	0.995515i	$-0.530159\pi$
$223$	−630.094	−0.189212	−0.0946059	+	0.995515i	$0.530159\pi$
$224$	0	0
$225$	−1059.16	−0.313826
$226$	0	0
$227$	−827.759	−0.242028	−0.121014	−	0.992651i	$-0.538615\pi$
$227$	−827.759	−0.242028	−0.121014	+	0.992651i	$0.538615\pi$
$228$	0	0
$229$	−4280.78	−1.23529	−0.617646	−	0.786456i	$-0.711914\pi$
$229$	−4280.78	−1.23529	−0.617646	+	0.786456i	$0.711914\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	764.846	0.215050	0.107525	−	0.994202i	$-0.465707\pi$
$233$	764.846	0.215050	0.107525	+	0.994202i	$0.465707\pi$
$234$	0	0
$235$	−1337.61	−0.371302
$236$	0	0
$237$	−1922.05	−0.526797
$238$	0	0
$239$	−3467.43	−0.938449	−0.469225	−	0.883079i	$-0.655467\pi$
$239$	−3467.43	−0.938449	−0.469225	+	0.883079i	$0.655467\pi$
$240$	0	0
$241$	−4350.54	−1.16283	−0.581416	−	0.813606i	$-0.697501\pi$
$241$	−4350.54	−1.16283	−0.581416	+	0.813606i	$0.697501\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11263.5	−2.90154
$248$	0	0
$249$	−3218.17	−0.819050
$250$	0	0
$251$	−522.335	−0.131353	−0.0656763	−	0.997841i	$-0.520920\pi$
$251$	−522.335	−0.131353	−0.0656763	+	0.997841i	$0.520920\pi$
$252$	0	0
$253$	−11064.2	−2.74941
$254$	0	0
$255$	−119.315	−0.0293013
$256$	0	0
$257$	−4629.93	−1.12376	−0.561881	−	0.827218i	$-0.689922\pi$
$257$	−4629.93	−1.12376	−0.561881	+	0.827218i	$0.689922\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	164.054	0.0389068
$262$	0	0
$263$	−4357.01	−1.02154	−0.510770	−	0.859717i	$-0.670640\pi$
$263$	−4357.01	−1.02154	−0.510770	+	0.859717i	$0.670640\pi$
$264$	0	0
$265$	1270.59	0.294535
$266$	0	0
$267$	2002.29	0.458944
$268$	0	0
$269$	−4187.23	−0.949070	−0.474535	−	0.880237i	$-0.657384\pi$
$269$	−4187.23	−0.949070	−0.474535	+	0.880237i	$0.657384\pi$
$270$	0	0
$271$	−6508.31	−1.45886	−0.729431	−	0.684054i	$-0.760215\pi$
$271$	−6508.31	−1.45886	−0.729431	+	0.684054i	$0.760215\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7545.24	1.65453
$276$	0	0
$277$	−3212.03	−0.696722	−0.348361	−	0.937360i	$-0.613262\pi$
$277$	−3212.03	−0.696722	−0.348361	+	0.937360i	$0.613262\pi$
$278$	0	0
$279$	1344.68	0.288545
$280$	0	0
$281$	1047.83	0.222449	0.111224	−	0.993795i	$-0.464523\pi$
$281$	1047.83	0.222449	0.111224	+	0.993795i	$0.464523\pi$
$282$	0	0
$283$	6936.63	1.45703	0.728516	−	0.685029i	$-0.240210\pi$
$283$	6936.63	1.45703	0.728516	+	0.685029i	$0.240210\pi$
$284$	0	0
$285$	−1290.52	−0.268225
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4696.77	−0.955989
$290$	0	0
$291$	3922.51	0.790177
$292$	0	0
$293$	823.738	0.164243	0.0821217	−	0.996622i	$-0.473830\pi$
$293$	823.738	0.164243	0.0821217	+	0.996622i	$0.473830\pi$
$294$	0	0
$295$	928.456	0.183243
$296$	0	0
$297$	1731.08	0.338207
$298$	0	0
$299$	−12221.2	−2.36379
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2161.03	0.409729
$304$	0	0
$305$	196.082	0.0368118
$306$	0	0
$307$	−8658.16	−1.60960	−0.804800	−	0.593546i	$-0.797728\pi$
$307$	−8658.16	−1.60960	−0.804800	+	0.593546i	$0.797728\pi$
$308$	0	0
$309$	5287.53	0.973453
$310$	0	0
$311$	−2648.97	−0.482988	−0.241494	−	0.970402i	$-0.577637\pi$
$311$	−2648.97	−0.482988	−0.241494	+	0.970402i	$0.577637\pi$
$312$	0	0
$313$	−27.2208	−0.00491569	−0.00245785	−	0.999997i	$-0.500782\pi$
$313$	−27.2208	−0.00491569	−0.00245785	+	0.999997i	$0.500782\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6919.25	−1.22594	−0.612971	−	0.790105i	$-0.710026\pi$
$317$	−6919.25	−1.22594	−0.612971	+	0.790105i	$0.710026\pi$
$318$	0	0
$319$	−1168.68	−0.205122
$320$	0	0
$321$	417.927	0.0726679
$322$	0	0
$323$	2338.74	0.402882
$324$	0	0
$325$	8334.28	1.42247
$326$	0	0
$327$	−1708.63	−0.288953
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8973.80	−1.49017	−0.745083	−	0.666972i	$-0.767590\pi$
$331$	−8973.80	−1.49017	−0.745083	+	0.666972i	$0.767590\pi$
$332$	0	0
$333$	375.946	0.0618670
$334$	0	0
$335$	−794.093	−0.129510
$336$	0	0
$337$	3825.65	0.618387	0.309194	−	0.950999i	$-0.399941\pi$
$337$	3825.65	0.618387	0.309194	+	0.950999i	$0.399941\pi$
$338$	0	0
$339$	5530.43	0.886052
$340$	0	0
$341$	−9579.25	−1.52125
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1400.25	−0.218513
$346$	0	0
$347$	−8795.51	−1.36071	−0.680357	−	0.732881i	$-0.738175\pi$
$347$	−8795.51	−1.36071	−0.680357	+	0.732881i	$0.738175\pi$
$348$	0	0
$349$	−7841.92	−1.20277	−0.601387	−	0.798958i	$-0.705385\pi$
$349$	−7841.92	−1.20277	−0.601387	+	0.798958i	$0.705385\pi$
$350$	0	0
$351$	1912.11	0.290771
$352$	0	0
$353$	−1885.40	−0.284277	−0.142139	−	0.989847i	$-0.545398\pi$
$353$	−1885.40	−0.284277	−0.142139	+	0.989847i	$0.545398\pi$
$354$	0	0
$355$	−1703.80	−0.254727
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4189.95	0.615981	0.307991	−	0.951389i	$-0.400343\pi$
$359$	4189.95	0.615981	0.307991	+	0.951389i	$0.400343\pi$
$360$	0	0
$361$	18436.9	2.68799
$362$	0	0
$363$	−8338.85	−1.20572
$364$	0	0
$365$	−1884.32	−0.270219
$366$	0	0
$367$	−9444.08	−1.34326	−0.671631	−	0.740886i	$-0.734406\pi$
$367$	−9444.08	−1.34326	−0.671631	+	0.740886i	$0.734406\pi$
$368$	0	0
$369$	451.873	0.0637495
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11745.2	1.63041	0.815203	−	0.579175i	$-0.196625\pi$
$373$	11745.2	1.63041	0.815203	+	0.579175i	$0.196625\pi$
$374$	0	0
$375$	1969.17	0.271166
$376$	0	0
$377$	−1290.90	−0.176352
$378$	0	0
$379$	13493.4	1.82878	0.914390	−	0.404834i	$-0.132671\pi$
$379$	13493.4	1.82878	0.914390	+	0.404834i	$0.132671\pi$
$380$	0	0
$381$	−1519.53	−0.204325
$382$	0	0
$383$	6921.48	0.923423	0.461711	−	0.887030i	$-0.347236\pi$
$383$	6921.48	0.923423	0.461711	+	0.887030i	$0.347236\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3492.00	0.458678
$388$	0	0
$389$	−14049.2	−1.83116	−0.915579	−	0.402138i	$-0.868267\pi$
$389$	−14049.2	−1.83116	−0.915579	+	0.402138i	$0.868267\pi$
$390$	0	0
$391$	2537.60	0.328214
$392$	0	0
$393$	3164.46	0.406172
$394$	0	0
$395$	1732.86	0.220733
$396$	0	0
$397$	−5403.10	−0.683057	−0.341529	−	0.939871i	$-0.610945\pi$
$397$	−5403.10	−0.683057	−0.341529	+	0.939871i	$0.610945\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	322.994	0.0402234	0.0201117	−	0.999798i	$-0.493598\pi$
$401$	322.994	0.0402234	0.0201117	+	0.999798i	$0.493598\pi$
$402$	0	0
$403$	−10581.0	−1.30788
$404$	0	0
$405$	219.081	0.0268795
$406$	0	0
$407$	−2678.16	−0.326171
$408$	0	0
$409$	10375.5	1.25437	0.627185	−	0.778871i	$-0.284207\pi$
$409$	10375.5	1.25437	0.627185	+	0.778871i	$0.284207\pi$
$410$	0	0
$411$	5070.00	0.608478
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2901.40	0.343191
$416$	0	0
$417$	−5196.89	−0.610294
$418$	0	0
$419$	−13078.3	−1.52487	−0.762433	−	0.647068i	$-0.775995\pi$
$419$	−13078.3	−1.52487	−0.762433	+	0.647068i	$0.775995\pi$
$420$	0	0
$421$	−2406.12	−0.278544	−0.139272	−	0.990254i	$-0.544476\pi$
$421$	−2406.12	−0.278544	−0.139272	+	0.990254i	$0.544476\pi$
$422$	0	0
$423$	−4450.95	−0.511614
$424$	0	0
$425$	−1730.52	−0.197512
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13621.5	−1.53298
$430$	0	0
$431$	14524.3	1.62323	0.811613	−	0.584195i	$-0.198590\pi$
$431$	14524.3	1.62323	0.811613	+	0.584195i	$0.198590\pi$
$432$	0	0
$433$	−13632.0	−1.51296	−0.756482	−	0.654014i	$-0.773084\pi$
$433$	−13632.0	−1.51296	−0.756482	+	0.654014i	$0.773084\pi$
$434$	0	0
$435$	−147.905	−0.0163024
$436$	0	0
$437$	27446.8	3.00448
$438$	0	0
$439$	544.026	0.0591457	0.0295728	−	0.999563i	$-0.490585\pi$
$439$	544.026	0.0591457	0.0295728	+	0.999563i	$0.490585\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5856.32	−0.628086	−0.314043	−	0.949409i	$-0.601684\pi$
$443$	−5856.32	−0.628086	−0.314043	+	0.949409i	$0.601684\pi$
$444$	0	0
$445$	−1805.20	−0.192302
$446$	0	0
$447$	−1014.36	−0.107333
$448$	0	0
$449$	−8548.36	−0.898490	−0.449245	−	0.893408i	$-0.648307\pi$
$449$	−8548.36	−0.898490	−0.449245	+	0.893408i	$0.648307\pi$
$450$	0	0
$451$	−3219.05	−0.336095
$452$	0	0
$453$	18.5232	0.00192118
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7222.35	0.739272	0.369636	−	0.929177i	$-0.379482\pi$
$457$	7222.35	0.739272	0.369636	+	0.929177i	$0.379482\pi$
$458$	0	0
$459$	−397.027	−0.0403739
$460$	0	0
$461$	3618.93	0.365619	0.182810	−	0.983148i	$-0.441481\pi$
$461$	3618.93	0.365619	0.182810	+	0.983148i	$0.441481\pi$
$462$	0	0
$463$	−7517.89	−0.754614	−0.377307	−	0.926088i	$-0.623150\pi$
$463$	−7517.89	−0.754614	−0.377307	+	0.926088i	$0.623150\pi$
$464$	0	0
$465$	−1212.32	−0.120903
$466$	0	0
$467$	−3677.56	−0.364405	−0.182202	−	0.983261i	$-0.558323\pi$
$467$	−3677.56	−0.364405	−0.182202	+	0.983261i	$0.558323\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2195.07	0.214742
$472$	0	0
$473$	−24876.3	−2.41821
$474$	0	0
$475$	−18717.4	−1.80803
$476$	0	0
$477$	4227.95	0.405837
$478$	0	0
$479$	13044.6	1.24431	0.622153	−	0.782896i	$-0.286258\pi$
$479$	13044.6	1.24431	0.622153	+	0.782896i	$0.286258\pi$
$480$	0	0
$481$	−2958.23	−0.280423
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3536.40	−0.331092
$486$	0	0
$487$	−9080.46	−0.844917	−0.422459	−	0.906382i	$-0.638833\pi$
$487$	−9080.46	−0.844917	−0.422459	+	0.906382i	$0.638833\pi$
$488$	0	0
$489$	−2283.75	−0.211195
$490$	0	0
$491$	−11300.9	−1.03870	−0.519349	−	0.854562i	$-0.673825\pi$
$491$	−11300.9	−1.03870	−0.519349	+	0.854562i	$0.673825\pi$
$492$	0	0
$493$	268.040	0.0244867
$494$	0	0
$495$	−1560.68	−0.141712
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5993.29	−0.537668	−0.268834	−	0.963187i	$-0.586638\pi$
$499$	−5993.29	−0.537668	−0.268834	+	0.963187i	$0.586638\pi$
$500$	0	0
$501$	8193.10	0.730620
$502$	0	0
$503$	−9330.15	−0.827059	−0.413530	−	0.910491i	$-0.635704\pi$
$503$	−9330.15	−0.827059	−0.413530	+	0.910491i	$0.635704\pi$
$504$	0	0
$505$	−1948.31	−0.171680
$506$	0	0
$507$	−8454.91	−0.740623
$508$	0	0
$509$	−8438.69	−0.734849	−0.367425	−	0.930053i	$-0.619760\pi$
$509$	−8438.69	−0.734849	−0.367425	+	0.930053i	$0.619760\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4294.27	−0.369584
$514$	0	0
$515$	−4767.06	−0.407887
$516$	0	0
$517$	31707.7	2.69729
$518$	0	0
$519$	−11163.5	−0.944167
$520$	0	0
$521$	11099.5	0.933354	0.466677	−	0.884428i	$-0.345451\pi$
$521$	11099.5	0.933354	0.466677	+	0.884428i	$0.345451\pi$
$522$	0	0
$523$	2079.76	0.173884	0.0869422	−	0.996213i	$-0.472290\pi$
$523$	2079.76	0.173884	0.0869422	+	0.996213i	$0.472290\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2197.02	0.181601
$528$	0	0
$529$	17613.6	1.44765
$530$	0	0
$531$	3089.48	0.252489
$532$	0	0
$533$	−3555.68	−0.288956
$534$	0	0
$535$	−376.789	−0.0304486
$536$	0	0
$537$	−7158.46	−0.575252
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18154.1	1.44271	0.721354	−	0.692566i	$-0.243520\pi$
$541$	18154.1	1.44271	0.721354	+	0.692566i	$0.243520\pi$
$542$	0	0
$543$	11458.6	0.905593
$544$	0	0
$545$	1540.44	0.121074
$546$	0	0
$547$	12923.0	1.01015	0.505073	−	0.863077i	$-0.331466\pi$
$547$	12923.0	1.01015	0.505073	+	0.863077i	$0.331466\pi$
$548$	0	0
$549$	652.469	0.0507226
$550$	0	0
$551$	2899.14	0.224152
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−338.941	−0.0259229
$556$	0	0
$557$	17153.3	1.30486	0.652432	−	0.757847i	$-0.273749\pi$
$557$	17153.3	1.30486	0.652432	+	0.757847i	$0.273749\pi$
$558$	0	0
$559$	−27477.7	−2.07904
$560$	0	0
$561$	2828.34	0.212856
$562$	0	0
$563$	−15580.4	−1.16632	−0.583159	−	0.812358i	$-0.698183\pi$
$563$	−15580.4	−1.16632	−0.583159	+	0.812358i	$0.698183\pi$
$564$	0	0
$565$	−4986.05	−0.371265
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5886.08	−0.433668	−0.216834	−	0.976208i	$-0.569573\pi$
$569$	−5886.08	−0.433668	−0.216834	+	0.976208i	$0.569573\pi$
$570$	0	0
$571$	7028.85	0.515146	0.257573	−	0.966259i	$-0.417077\pi$
$571$	7028.85	0.515146	0.257573	+	0.966259i	$0.417077\pi$
$572$	0	0
$573$	−642.504	−0.0468429
$574$	0	0
$575$	−20308.9	−1.47294
$576$	0	0
$577$	−5400.24	−0.389627	−0.194814	−	0.980840i	$-0.562410\pi$
$577$	−5400.24	−0.389627	−0.194814	+	0.980840i	$0.562410\pi$
$578$	0	0
$579$	−11335.3	−0.813610
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−30119.0	−2.13962
$584$	0	0
$585$	−1723.89	−0.121836
$586$	0	0
$587$	49.2876	0.00346562	0.00173281	−	0.999998i	$-0.499448\pi$
$587$	49.2876	0.00346562	0.00173281	+	0.999998i	$0.499448\pi$
$588$	0	0
$589$	23763.1	1.66238
$590$	0	0
$591$	−10185.0	−0.708893
$592$	0	0
$593$	13358.7	0.925088	0.462544	−	0.886596i	$-0.346937\pi$
$593$	13358.7	0.925088	0.462544	+	0.886596i	$0.346937\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11778.5	−0.807475
$598$	0	0
$599$	−2507.44	−0.171037	−0.0855186	−	0.996337i	$-0.527255\pi$
$599$	−2507.44	−0.171037	−0.0855186	+	0.996337i	$0.527255\pi$
$600$	0	0
$601$	−8526.26	−0.578691	−0.289345	−	0.957225i	$-0.593438\pi$
$601$	−8526.26	−0.578691	−0.289345	+	0.957225i	$0.593438\pi$
$602$	0	0
$603$	−2642.38	−0.178451
$604$	0	0
$605$	7518.03	0.505209
$606$	0	0
$607$	8403.41	0.561918	0.280959	−	0.959720i	$-0.409348\pi$
$607$	8403.41	0.561918	0.280959	+	0.959720i	$0.409348\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35023.5	2.31898
$612$	0	0
$613$	6519.91	0.429587	0.214793	−	0.976660i	$-0.431092\pi$
$613$	6519.91	0.429587	0.214793	+	0.976660i	$0.431092\pi$
$614$	0	0
$615$	−407.394	−0.0267117
$616$	0	0
$617$	−3562.51	−0.232449	−0.116225	−	0.993223i	$-0.537079\pi$
$617$	−3562.51	−0.232449	−0.116225	+	0.993223i	$0.537079\pi$
$618$	0	0
$619$	−19199.6	−1.24669	−0.623343	−	0.781949i	$-0.714226\pi$
$619$	−19199.6	−1.24669	−0.623343	+	0.781949i	$0.714226\pi$
$620$	0	0
$621$	−4659.40	−0.301088
$622$	0	0
$623$	0	0
$624$	0	0
$625$	12935.2	0.827855
$626$	0	0
$627$	30591.5	1.94849
$628$	0	0
$629$	614.242	0.0389371
$630$	0	0
$631$	29850.0	1.88322	0.941608	−	0.336712i	$-0.109315\pi$
$631$	29850.0	1.88322	0.941608	+	0.336712i	$0.109315\pi$
$632$	0	0
$633$	2448.32	0.153732
$634$	0	0
$635$	1369.96	0.0856144
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5669.46	−0.350986
$640$	0	0
$641$	−21452.1	−1.32185	−0.660925	−	0.750452i	$-0.729836\pi$
$641$	−21452.1	−1.32185	−0.660925	+	0.750452i	$0.729836\pi$
$642$	0	0
$643$	31670.1	1.94238	0.971188	−	0.238313i	$-0.0765943\pi$
$643$	31670.1	1.94238	0.971188	+	0.238313i	$0.0765943\pi$
$644$	0	0
$645$	−3148.27	−0.192191
$646$	0	0
$647$	−24750.4	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$647$	−24750.4	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$648$	0	0
$649$	−22008.8	−1.33116
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10186.4	0.610449	0.305225	−	0.952280i	$-0.401268\pi$
$653$	10186.4	0.610449	0.305225	+	0.952280i	$0.401268\pi$
$654$	0	0
$655$	−2852.97	−0.170190
$656$	0	0
$657$	−6270.16	−0.372332
$658$	0	0
$659$	−7856.84	−0.464429	−0.232215	−	0.972665i	$-0.574597\pi$
$659$	−7856.84	−0.464429	−0.232215	+	0.972665i	$0.574597\pi$
$660$	0	0
$661$	19490.1	1.14686	0.573432	−	0.819253i	$-0.305612\pi$
$661$	19490.1	1.14686	0.573432	+	0.819253i	$0.305612\pi$
$662$	0	0
$663$	3124.11	0.183002
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3145.65	0.182609
$668$	0	0
$669$	1890.28	0.109241
$670$	0	0
$671$	−4648.05	−0.267416
$672$	0	0
$673$	−4872.89	−0.279102	−0.139551	−	0.990215i	$-0.544566\pi$
$673$	−4872.89	−0.279102	−0.139551	+	0.990215i	$0.544566\pi$
$674$	0	0
$675$	3177.48	0.181187
$676$	0	0
$677$	−24279.8	−1.37836	−0.689179	−	0.724591i	$-0.742029\pi$
$677$	−24279.8	−1.37836	−0.689179	+	0.724591i	$0.742029\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2483.28	0.139735
$682$	0	0
$683$	−12278.3	−0.687871	−0.343936	−	0.938993i	$-0.611760\pi$
$683$	−12278.3	−0.687871	−0.343936	+	0.938993i	$0.611760\pi$
$684$	0	0
$685$	−4570.94	−0.254959
$686$	0	0
$687$	12842.3	0.713196
$688$	0	0
$689$	−33268.7	−1.83953
$690$	0	0
$691$	8367.76	0.460672	0.230336	−	0.973111i	$-0.426017\pi$
$691$	8367.76	0.460672	0.230336	+	0.973111i	$0.426017\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4685.34	0.255720
$696$	0	0
$697$	738.295	0.0401218
$698$	0	0
$699$	−2294.54	−0.124159
$700$	0	0
$701$	−510.228	−0.0274908	−0.0137454	−	0.999906i	$-0.504375\pi$
$701$	−510.228	−0.0274908	−0.0137454	+	0.999906i	$0.504375\pi$
$702$	0	0
$703$	6643.68	0.356431
$704$	0	0
$705$	4012.83	0.214372
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12705.0	0.672985	0.336493	−	0.941686i	$-0.390759\pi$
$709$	12705.0	0.672985	0.336493	+	0.941686i	$0.390759\pi$
$710$	0	0
$711$	5766.16	0.304146
$712$	0	0
$713$	25783.7	1.35429
$714$	0	0
$715$	12280.6	0.642336
$716$	0	0
$717$	10402.3	0.541814
$718$	0	0
$719$	−19865.9	−1.03042	−0.515210	−	0.857064i	$-0.672286\pi$
$719$	−19865.9	−1.03042	−0.515210	+	0.857064i	$0.672286\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	13051.6	0.671362
$724$	0	0
$725$	−2145.18	−0.109890
$726$	0	0
$727$	3822.62	0.195011	0.0975055	−	0.995235i	$-0.468914\pi$
$727$	3822.62	0.195011	0.0975055	+	0.995235i	$0.468914\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	5705.42	0.288677
$732$	0	0
$733$	12366.8	0.623161	0.311580	−	0.950220i	$-0.399142\pi$
$733$	12366.8	0.623161	0.311580	+	0.950220i	$0.399142\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18823.7	0.940816
$738$	0	0
$739$	−10276.9	−0.511557	−0.255779	−	0.966735i	$-0.582332\pi$
$739$	−10276.9	−0.511557	−0.255779	+	0.966735i	$0.582332\pi$
$740$	0	0
$741$	33790.6	1.67521
$742$	0	0
$743$	−27250.9	−1.34554	−0.672771	−	0.739851i	$-0.734896\pi$
$743$	−27250.9	−1.34554	−0.672771	+	0.739851i	$0.734896\pi$
$744$	0	0
$745$	914.515	0.0449735
$746$	0	0
$747$	9654.52	0.472879
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2674.31	−0.129943	−0.0649713	−	0.997887i	$-0.520696\pi$
$751$	−2674.31	−0.129943	−0.0649713	+	0.997887i	$0.520696\pi$
$752$	0	0
$753$	1567.01	0.0758365
$754$	0	0
$755$	−16.6999	−0.000804996 0
$756$	0	0
$757$	−15084.1	−0.724231	−0.362115	−	0.932133i	$-0.617945\pi$
$757$	−15084.1	−0.724231	−0.362115	+	0.932133i	$0.617945\pi$
$758$	0	0
$759$	33192.6	1.58737
$760$	0	0
$761$	24803.1	1.18149	0.590743	−	0.806860i	$-0.298835\pi$
$761$	24803.1	1.18149	0.590743	+	0.806860i	$0.298835\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	357.946	0.0169171
$766$	0	0
$767$	−24310.3	−1.14445
$768$	0	0
$769$	1080.40	0.0506637	0.0253318	−	0.999679i	$-0.491936\pi$
$769$	1080.40	0.0506637	0.0253318	+	0.999679i	$0.491936\pi$
$770$	0	0
$771$	13889.8	0.648804
$772$	0	0
$773$	2971.20	0.138249	0.0691247	−	0.997608i	$-0.477979\pi$
$773$	2971.20	0.138249	0.0691247	+	0.997608i	$0.477979\pi$
$774$	0	0
$775$	−17583.2	−0.814976
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7985.45	0.367276
$780$	0	0
$781$	40388.0	1.85044
$782$	0	0
$783$	−492.161	−0.0224629
$784$	0	0
$785$	−1979.01	−0.0899793
$786$	0	0
$787$	13716.6	0.621274	0.310637	−	0.950529i	$-0.399458\pi$
$787$	13716.6	0.621274	0.310637	+	0.950529i	$0.399458\pi$
$788$	0	0
$789$	13071.0	0.589786
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5134.12	−0.229909
$794$	0	0
$795$	−3811.78	−0.170050
$796$	0	0
$797$	−2618.52	−0.116377	−0.0581887	−	0.998306i	$-0.518532\pi$
$797$	−2618.52	−0.116377	−0.0581887	+	0.998306i	$0.518532\pi$
$798$	0	0
$799$	−7272.22	−0.321993
$800$	0	0
$801$	−6006.87	−0.264971
$802$	0	0
$803$	44667.3	1.96298
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12561.7	0.547946
$808$	0	0
$809$	827.878	0.0359785	0.0179893	−	0.999838i	$-0.494274\pi$
$809$	827.878	0.0359785	0.0179893	+	0.999838i	$0.494274\pi$
$810$	0	0
$811$	11981.9	0.518795	0.259397	−	0.965771i	$-0.416476\pi$
$811$	11981.9	0.518795	0.259397	+	0.965771i	$0.416476\pi$
$812$	0	0
$813$	19524.9	0.842274
$814$	0	0
$815$	2058.95	0.0884931
$816$	0	0
$817$	61710.2	2.64256
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25832.3	−1.09812	−0.549058	−	0.835784i	$-0.685013\pi$
$821$	−25832.3	−1.09812	−0.549058	+	0.835784i	$0.685013\pi$
$822$	0	0
$823$	13384.1	0.566879	0.283439	−	0.958990i	$-0.408525\pi$
$823$	13384.1	0.566879	0.283439	+	0.958990i	$0.408525\pi$
$824$	0	0
$825$	−22635.7	−0.955242
$826$	0	0
$827$	−5108.38	−0.214796	−0.107398	−	0.994216i	$-0.534252\pi$
$827$	−5108.38	−0.214796	−0.107398	+	0.994216i	$0.534252\pi$
$828$	0	0
$829$	34769.2	1.45668	0.728338	−	0.685218i	$-0.240293\pi$
$829$	34769.2	1.45668	0.728338	+	0.685218i	$0.240293\pi$
$830$	0	0
$831$	9636.08	0.402253
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−7386.63	−0.306137
$836$	0	0
$837$	−4034.05	−0.166592
$838$	0	0
$839$	4354.20	0.179170	0.0895851	−	0.995979i	$-0.471446\pi$
$839$	4354.20	0.179170	0.0895851	+	0.995979i	$0.471446\pi$
$840$	0	0
$841$	−24056.7	−0.986376
$842$	0	0
$843$	−3143.48	−0.128431
$844$	0	0
$845$	7622.66	0.310328
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20809.9	−0.841218
$850$	0	0
$851$	7208.58	0.290372
$852$	0	0
$853$	25709.6	1.03198	0.515991	−	0.856594i	$-0.327424\pi$
$853$	25709.6	1.03198	0.515991	+	0.856594i	$0.327424\pi$
$854$	0	0
$855$	3871.57	0.154860
$856$	0	0
$857$	−41466.8	−1.65284	−0.826418	−	0.563057i	$-0.809625\pi$
$857$	−41466.8	−1.65284	−0.826418	+	0.563057i	$0.809625\pi$
$858$	0	0
$859$	27860.3	1.10661	0.553306	−	0.832978i	$-0.313366\pi$
$859$	27860.3	1.10661	0.553306	+	0.832978i	$0.313366\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31885.8	1.25771	0.628857	−	0.777521i	$-0.283523\pi$
$863$	31885.8	1.25771	0.628857	+	0.777521i	$0.283523\pi$
$864$	0	0
$865$	10064.6	0.395616
$866$	0	0
$867$	14090.3	0.551940
$868$	0	0
$869$	−41076.9	−1.60350
$870$	0	0
$871$	20792.2	0.808860
$872$	0	0
$873$	−11767.5	−0.456209
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25529.7	−0.982982	−0.491491	−	0.870883i	$-0.663548\pi$
$877$	−25529.7	−0.982982	−0.491491	+	0.870883i	$0.663548\pi$
$878$	0	0
$879$	−2471.21	−0.0948259
$880$	0	0
$881$	−34416.4	−1.31614	−0.658069	−	0.752958i	$-0.728626\pi$
$881$	−34416.4	−1.31614	−0.658069	+	0.752958i	$0.728626\pi$
$882$	0	0
$883$	−24910.1	−0.949369	−0.474684	−	0.880156i	$-0.657438\pi$
$883$	−24910.1	−0.949369	−0.474684	+	0.880156i	$0.657438\pi$
$884$	0	0
$885$	−2785.37	−0.105796
$886$	0	0
$887$	−14817.5	−0.560906	−0.280453	−	0.959868i	$-0.590485\pi$
$887$	−14817.5	−0.560906	−0.280453	+	0.959868i	$0.590485\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5193.24	−0.195264
$892$	0	0
$893$	−78656.8	−2.94753
$894$	0	0
$895$	6453.83	0.241037
$896$	0	0
$897$	36663.7	1.36473
$898$	0	0
$899$	2723.46	0.101037
$900$	0	0
$901$	6907.85	0.255421
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10330.7	−0.379453
$906$	0	0
$907$	37979.4	1.39039	0.695196	−	0.718820i	$-0.255318\pi$
$907$	37979.4	1.39039	0.695196	+	0.718820i	$0.255318\pi$
$908$	0	0
$909$	−6483.08	−0.236557
$910$	0	0
$911$	30092.3	1.09440	0.547202	−	0.837001i	$-0.315693\pi$
$911$	30092.3	1.09440	0.547202	+	0.837001i	$0.315693\pi$
$912$	0	0
$913$	−68776.8	−2.49308
$914$	0	0
$915$	−588.245	−0.0212533
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−31083.4	−1.11572	−0.557859	−	0.829936i	$-0.688377\pi$
$919$	−31083.4	−1.11572	−0.557859	+	0.829936i	$0.688377\pi$
$920$	0	0
$921$	25974.5	0.929303
$922$	0	0
$923$	44611.6	1.59091
$924$	0	0
$925$	−4915.90	−0.174739
$926$	0	0
$927$	−15862.6	−0.562024
$928$	0	0
$929$	−5772.29	−0.203857	−0.101928	−	0.994792i	$-0.532501\pi$
$929$	−5772.29	−0.203857	−0.101928	+	0.994792i	$0.532501\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	7946.90	0.278853
$934$	0	0
$935$	−2549.93	−0.0891890
$936$	0	0
$937$	−21399.5	−0.746096	−0.373048	−	0.927812i	$-0.621687\pi$
$937$	−21399.5	−0.746096	−0.373048	+	0.927812i	$0.621687\pi$
$938$	0	0
$939$	81.6625	0.00283808
$940$	0	0
$941$	−11275.1	−0.390603	−0.195302	−	0.980743i	$-0.562569\pi$
$941$	−11275.1	−0.390603	−0.195302	+	0.980743i	$0.562569\pi$
$942$	0	0
$943$	8664.44	0.299208
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45888.3	−1.57462	−0.787312	−	0.616554i	$-0.788528\pi$
$947$	−45888.3	−1.57462	−0.787312	+	0.616554i	$0.788528\pi$
$948$	0	0
$949$	49338.4	1.68766
$950$	0	0
$951$	20757.7	0.707798
$952$	0	0
$953$	−35717.5	−1.21406	−0.607032	−	0.794677i	$-0.707640\pi$
$953$	−35717.5	−1.21406	−0.607032	+	0.794677i	$0.707640\pi$
$954$	0	0
$955$	579.260	0.0196277
$956$	0	0
$957$	3506.05	0.118427
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−7467.83	−0.250674
$962$	0	0
$963$	−1253.78	−0.0419548
$964$	0	0
$965$	10219.6	0.340911
$966$	0	0
$967$	6864.36	0.228276	0.114138	−	0.993465i	$-0.463589\pi$
$967$	6864.36	0.228276	0.114138	+	0.993465i	$0.463589\pi$
$968$	0	0
$969$	−7016.22	−0.232604
$970$	0	0
$971$	13333.7	0.440680	0.220340	−	0.975423i	$-0.429283\pi$
$971$	13333.7	0.440680	0.220340	+	0.975423i	$0.429283\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−25002.8	−0.821264
$976$	0	0
$977$	2222.94	0.0727923	0.0363962	−	0.999337i	$-0.488412\pi$
$977$	2222.94	0.0727923	0.0363962	+	0.999337i	$0.488412\pi$
$978$	0	0
$979$	42791.6	1.39696
$980$	0	0
$981$	5125.89	0.166827
$982$	0	0
$983$	−30862.1	−1.00137	−0.500686	−	0.865629i	$-0.666919\pi$
$983$	−30862.1	−1.00137	−0.500686	+	0.865629i	$0.666919\pi$
$984$	0	0
$985$	9182.48	0.297034
$986$	0	0
$987$	0	0
$988$	0	0
$989$	66957.4	2.15280
$990$	0	0
$991$	11593.9	0.371637	0.185819	−	0.982584i	$-0.440506\pi$
$991$	11593.9	0.371637	0.185819	+	0.982584i	$0.440506\pi$
$992$	0	0
$993$	26921.4	0.860347
$994$	0	0
$995$	10619.1	0.338341
$996$	0	0
$997$	−36631.7	−1.16363	−0.581814	−	0.813322i	$-0.697657\pi$
$997$	−36631.7	−1.16363	−0.581814	+	0.813322i	$0.697657\pi$
$998$	0	0
$999$	−1127.84	−0.0357190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bm.1.2		2
4.3	odd	2		1176.4.a.t.1.2	yes	2
7.6	odd	2		2352.4.a.ce.1.1		2
28.27	even	2		1176.4.a.s.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.s.1.1	✓	2	28.27	even	2
1176.4.a.t.1.2	yes	2	4.3	odd	2
2352.4.a.bm.1.2		2	1.1	even	1	trivial
2352.4.a.ce.1.1		2	7.6	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +2.70470 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +2.70470 q^{5} +9.00000 q^{9} -64.1141 q^{11} -70.8188 q^{13} -8.11410 q^{15} +14.7047 q^{17} +159.047 q^{19} +172.570 q^{23} -117.685 q^{25} -27.0000 q^{27} +18.2282 q^{29} +149.409 q^{31} +192.342 q^{33} +41.7718 q^{37} +212.456 q^{39} +50.2081 q^{41} +388.000 q^{43} +24.3423 q^{45} -494.550 q^{47} -44.1141 q^{51} +469.772 q^{53} -173.409 q^{55} -477.141 q^{57} +343.275 q^{59} +72.4966 q^{61} -191.544 q^{65} -293.597 q^{67} -517.711 q^{69} -629.940 q^{71} -696.685 q^{73} +353.054 q^{75} +640.685 q^{79} +81.0000 q^{81} +1072.72 q^{83} +39.7718 q^{85} -54.6846 q^{87} -667.430 q^{89} -448.228 q^{93} +430.174 q^{95} -1307.50 q^{97} -577.027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 18 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 18 * q^5 + 18 * q^9	\( 2 q - 6 q^{3} - 18 q^{5} + 18 q^{9} - 58 q^{11} - 48 q^{13} + 54 q^{15} + 6 q^{17} + 84 q^{19} - 6 q^{23} + 186 q^{25} - 54 q^{27} - 104 q^{29} + 252 q^{31} + 174 q^{33} + 224 q^{37} + 144 q^{39} - 438 q^{41} + 776 q^{43} - 162 q^{45} - 240 q^{47} - 18 q^{51} + 1080 q^{53} - 300 q^{55} - 252 q^{57} + 312 q^{59} + 660 q^{61} - 664 q^{65} + 396 q^{67} + 18 q^{69} - 66 q^{71} - 972 q^{73} - 558 q^{75} + 860 q^{79} + 162 q^{81} + 2520 q^{83} + 220 q^{85} + 312 q^{87} - 1686 q^{89} - 756 q^{93} + 1984 q^{95} - 2100 q^{97} - 522 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 18 * q^5 + 18 * q^9 - 58 * q^11 - 48 * q^13 + 54 * q^15 + 6 * q^17 + 84 * q^19 - 6 * q^23 + 186 * q^25 - 54 * q^27 - 104 * q^29 + 252 * q^31 + 174 * q^33 + 224 * q^37 + 144 * q^39 - 438 * q^41 + 776 * q^43 - 162 * q^45 - 240 * q^47 - 18 * q^51 + 1080 * q^53 - 300 * q^55 - 252 * q^57 + 312 * q^59 + 660 * q^61 - 664 * q^65 + 396 * q^67 + 18 * q^69 - 66 * q^71 - 972 * q^73 - 558 * q^75 + 860 * q^79 + 162 * q^81 + 2520 * q^83 + 220 * q^85 + 312 * q^87 - 1686 * q^89 - 756 * q^93 + 1984 * q^95 - 2100 * q^97 - 522 * q^99

Introduction

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