LMFDB - Embedded newform 2312.4.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.02132 q^{3} -2.91509 q^{5} +14.5048 q^{7} +22.2990 q^{9} +O(q^{10})$	$q-7.02132 q^{3} -2.91509 q^{5} +14.5048 q^{7} +22.2990 q^{9} +11.3355 q^{11} +48.9293 q^{13} +20.4678 q^{15} -71.5192 q^{19} -101.843 q^{21} -37.9001 q^{23} -116.502 q^{25} +33.0076 q^{27} -76.9346 q^{29} +65.2099 q^{31} -79.5905 q^{33} -42.2828 q^{35} -78.6810 q^{37} -343.548 q^{39} +358.424 q^{41} -82.9594 q^{43} -65.0035 q^{45} -11.8038 q^{47} -132.611 q^{49} -565.154 q^{53} -33.0441 q^{55} +502.159 q^{57} -335.131 q^{59} +344.467 q^{61} +323.442 q^{63} -142.633 q^{65} +634.043 q^{67} +266.109 q^{69} -58.2230 q^{71} -229.839 q^{73} +818.000 q^{75} +164.420 q^{77} -327.446 q^{79} -833.828 q^{81} +1429.67 q^{83} +540.183 q^{87} +144.618 q^{89} +709.709 q^{91} -457.859 q^{93} +208.485 q^{95} -426.162 q^{97} +252.771 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9}+O(q^{10}) $ 24 * q + 88 * q^9	$ 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69} - 5248 q^{77} - 3560 q^{81} + 696 q^{83} + 1176 q^{87} - 5504 q^{89} + 616 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

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
$3$	−7.02132	−1.35125	−0.675627	−	0.737244i	$-0.736127\pi$
$3$	−7.02132	−1.35125	−0.675627	+	0.737244i	$0.736127\pi$
$4$	0	0
$5$	−2.91509	−0.260734	−0.130367	−	0.991466i	$-0.541615\pi$
$5$	−2.91509	−0.260734	−0.130367	+	0.991466i	$0.541615\pi$
$6$	0	0
$7$	14.5048	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$7$	14.5048	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$8$	0	0
$9$	22.2990	0.825887
$10$	0	0
$11$	11.3355	0.310709	0.155354	−	0.987859i	$-0.450348\pi$
$11$	11.3355	0.310709	0.155354	+	0.987859i	$0.450348\pi$
$12$	0	0
$13$	48.9293	1.04389	0.521944	−	0.852980i	$-0.325207\pi$
$13$	48.9293	1.04389	0.521944	+	0.852980i	$0.325207\pi$
$14$	0	0
$15$	20.4678	0.352317
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−71.5192	−0.863559	−0.431780	−	0.901979i	$-0.642114\pi$
$19$	−71.5192	−0.863559	−0.431780	+	0.901979i	$0.642114\pi$
$20$	0	0
$21$	−101.843	−1.05828
$22$	0	0
$23$	−37.9001	−0.343596	−0.171798	−	0.985132i	$-0.554958\pi$
$23$	−37.9001	−0.343596	−0.171798	+	0.985132i	$0.554958\pi$
$24$	0	0
$25$	−116.502	−0.932018
$26$	0	0
$27$	33.0076	0.235271
$28$	0	0
$29$	−76.9346	−0.492634	−0.246317	−	0.969189i	$-0.579220\pi$
$29$	−76.9346	−0.492634	−0.246317	+	0.969189i	$0.579220\pi$
$30$	0	0
$31$	65.2099	0.377808	0.188904	−	0.981996i	$-0.439507\pi$
$31$	65.2099	0.377808	0.188904	+	0.981996i	$0.439507\pi$
$32$	0	0
$33$	−79.5905	−0.419846
$34$	0	0
$35$	−42.2828	−0.204203
$36$	0	0
$37$	−78.6810	−0.349597	−0.174798	−	0.984604i	$-0.555927\pi$
$37$	−78.6810	−0.349597	−0.174798	+	0.984604i	$0.555927\pi$
$38$	0	0
$39$	−343.548	−1.41056
$40$	0	0
$41$	358.424	1.36528	0.682640	−	0.730755i	$-0.260832\pi$
$41$	358.424	1.36528	0.682640	+	0.730755i	$0.260832\pi$
$42$	0	0
$43$	−82.9594	−0.294214	−0.147107	−	0.989121i	$-0.546996\pi$
$43$	−82.9594	−0.294214	−0.147107	+	0.989121i	$0.546996\pi$
$44$	0	0
$45$	−65.0035	−0.215337
$46$	0	0
$47$	−11.8038	−0.0366332	−0.0183166	−	0.999832i	$-0.505831\pi$
$47$	−11.8038	−0.0366332	−0.0183166	+	0.999832i	$0.505831\pi$
$48$	0	0
$49$	−132.611	−0.386620
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−565.154	−1.46471	−0.732357	−	0.680921i	$-0.761580\pi$
$53$	−565.154	−1.46471	−0.732357	+	0.680921i	$0.761580\pi$
$54$	0	0
$55$	−33.0441	−0.0810121
$56$	0	0
$57$	502.159	1.16689
$58$	0	0
$59$	−335.131	−0.739497	−0.369749	−	0.929132i	$-0.620556\pi$
$59$	−335.131	−0.739497	−0.369749	+	0.929132i	$0.620556\pi$
$60$	0	0
$61$	344.467	0.723023	0.361512	−	0.932368i	$-0.382261\pi$
$61$	344.467	0.723023	0.361512	+	0.932368i	$0.382261\pi$
$62$	0	0
$63$	323.442	0.646823
$64$	0	0
$65$	−142.633	−0.272177
$66$	0	0
$67$	634.043	1.15613	0.578065	−	0.815991i	$-0.303808\pi$
$67$	634.043	1.15613	0.578065	+	0.815991i	$0.303808\pi$
$68$	0	0
$69$	266.109	0.464286
$70$	0	0
$71$	−58.2230	−0.0973211	−0.0486606	−	0.998815i	$-0.515495\pi$
$71$	−58.2230	−0.0973211	−0.0486606	+	0.998815i	$0.515495\pi$
$72$	0	0
$73$	−229.839	−0.368501	−0.184251	−	0.982879i	$-0.558986\pi$
$73$	−229.839	−0.368501	−0.184251	+	0.982879i	$0.558986\pi$
$74$	0	0
$75$	818.000	1.25939
$76$	0	0
$77$	164.420	0.243342
$78$	0	0
$79$	−327.446	−0.466336	−0.233168	−	0.972437i	$-0.574909\pi$
$79$	−327.446	−0.466336	−0.233168	+	0.972437i	$0.574909\pi$
$80$	0	0
$81$	−833.828	−1.14380
$82$	0	0
$83$	1429.67	1.89068	0.945340	−	0.326086i	$-0.105730\pi$
$83$	1429.67	1.89068	0.945340	+	0.326086i	$0.105730\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	540.183	0.665674
$88$	0	0
$89$	144.618	0.172242	0.0861208	−	0.996285i	$-0.472553\pi$
$89$	144.618	0.172242	0.0861208	+	0.996285i	$0.472553\pi$
$90$	0	0
$91$	709.709	0.817558
$92$	0	0
$93$	−457.859	−0.510514
$94$	0	0
$95$	208.485	0.225159
$96$	0	0
$97$	−426.162	−0.446085	−0.223042	−	0.974809i	$-0.571599\pi$
$97$	−426.162	−0.446085	−0.223042	+	0.974809i	$0.571599\pi$
$98$	0	0
$99$	252.771	0.256610
$100$	0	0
$101$	1157.50	1.14036	0.570178	−	0.821522i	$-0.306874\pi$
$101$	1157.50	1.14036	0.570178	+	0.821522i	$0.306874\pi$
$102$	0	0
$103$	526.282	0.503457	0.251729	−	0.967798i	$-0.419001\pi$
$103$	526.282	0.503457	0.251729	+	0.967798i	$0.419001\pi$
$104$	0	0
$105$	296.881	0.275930
$106$	0	0
$107$	1548.61	1.39915	0.699577	−	0.714557i	$-0.253372\pi$
$107$	1548.61	1.39915	0.699577	+	0.714557i	$0.253372\pi$
$108$	0	0
$109$	2208.11	1.94035	0.970176	−	0.242402i	$-0.0779354\pi$
$109$	2208.11	1.94035	0.970176	+	0.242402i	$0.0779354\pi$
$110$	0	0
$111$	552.445	0.472394
$112$	0	0
$113$	794.502	0.661420	0.330710	−	0.943732i	$-0.392712\pi$
$113$	794.502	0.661420	0.330710	+	0.943732i	$0.392712\pi$
$114$	0	0
$115$	110.482	0.0895871
$116$	0	0
$117$	1091.07	0.862133
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1202.51	−0.903460
$122$	0	0
$123$	−2516.61	−1.84484
$124$	0	0
$125$	704.001	0.503742
$126$	0	0
$127$	−2218.14	−1.54983	−0.774914	−	0.632067i	$-0.782207\pi$
$127$	−2218.14	−1.54983	−0.774914	+	0.632067i	$0.782207\pi$
$128$	0	0
$129$	582.485	0.397557
$130$	0	0
$131$	−2086.61	−1.39167	−0.695833	−	0.718203i	$-0.744965\pi$
$131$	−2086.61	−1.39167	−0.695833	+	0.718203i	$0.744965\pi$
$132$	0	0
$133$	−1037.37	−0.676327
$134$	0	0
$135$	−96.2201	−0.0613430
$136$	0	0
$137$	−1724.58	−1.07548	−0.537740	−	0.843111i	$-0.680722\pi$
$137$	−1724.58	−1.07548	−0.537740	+	0.843111i	$0.680722\pi$
$138$	0	0
$139$	2242.48	1.36838	0.684189	−	0.729305i	$-0.260156\pi$
$139$	2242.48	1.36838	0.684189	+	0.729305i	$0.260156\pi$
$140$	0	0
$141$	82.8782	0.0495007
$142$	0	0
$143$	554.640	0.324345
$144$	0	0
$145$	224.271	0.128446
$146$	0	0
$147$	931.103	0.522422
$148$	0	0
$149$	−3338.27	−1.83545	−0.917724	−	0.397219i	$-0.869975\pi$
$149$	−3338.27	−1.83545	−0.917724	+	0.397219i	$0.869975\pi$
$150$	0	0
$151$	1568.80	0.845479	0.422739	−	0.906251i	$-0.361069\pi$
$151$	1568.80	0.845479	0.422739	+	0.906251i	$0.361069\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−190.093	−0.0985071
$156$	0	0
$157$	1311.84	0.666856	0.333428	−	0.942776i	$-0.391795\pi$
$157$	1311.84	0.666856	0.333428	+	0.942776i	$0.391795\pi$
$158$	0	0
$159$	3968.12	1.97920
$160$	0	0
$161$	−549.733	−0.269100
$162$	0	0
$163$	−3047.12	−1.46422	−0.732112	−	0.681184i	$-0.761465\pi$
$163$	−3047.12	−1.46422	−0.732112	+	0.681184i	$0.761465\pi$
$164$	0	0
$165$	232.013	0.109468
$166$	0	0
$167$	1453.27	0.673397	0.336699	−	0.941612i	$-0.390690\pi$
$167$	1453.27	0.673397	0.336699	+	0.941612i	$0.390690\pi$
$168$	0	0
$169$	197.073	0.0897009
$170$	0	0
$171$	−1594.80	−0.713202
$172$	0	0
$173$	213.846	0.0939792	0.0469896	−	0.998895i	$-0.485037\pi$
$173$	213.846	0.0939792	0.0469896	+	0.998895i	$0.485037\pi$
$174$	0	0
$175$	−1689.84	−0.729943
$176$	0	0
$177$	2353.06	0.999249
$178$	0	0
$179$	−3363.38	−1.40442	−0.702208	−	0.711972i	$-0.747802\pi$
$179$	−3363.38	−1.40442	−0.702208	+	0.711972i	$0.747802\pi$
$180$	0	0
$181$	−4374.80	−1.79656	−0.898278	−	0.439428i	$-0.855181\pi$
$181$	−4374.80	−1.79656	−0.898278	+	0.439428i	$0.855181\pi$
$182$	0	0
$183$	−2418.61	−0.976988
$184$	0	0
$185$	229.362	0.0911516
$186$	0	0
$187$	0	0
$188$	0	0
$189$	478.768	0.184261
$190$	0	0
$191$	−3689.66	−1.39777	−0.698887	−	0.715232i	$-0.746321\pi$
$191$	−3689.66	−1.39777	−0.698887	+	0.715232i	$0.746321\pi$
$192$	0	0
$193$	11.6847	0.00435795	0.00217897	−	0.999998i	$-0.499306\pi$
$193$	11.6847	0.00435795	0.00217897	+	0.999998i	$0.499306\pi$
$194$	0	0
$195$	1001.47	0.367780
$196$	0	0
$197$	2163.13	0.782319	0.391159	−	0.920323i	$-0.372074\pi$
$197$	2163.13	0.782319	0.391159	+	0.920323i	$0.372074\pi$
$198$	0	0
$199$	4722.24	1.68216	0.841082	−	0.540908i	$-0.181919\pi$
$199$	4722.24	1.68216	0.841082	+	0.540908i	$0.181919\pi$
$200$	0	0
$201$	−4451.82	−1.56222
$202$	0	0
$203$	−1115.92	−0.385824
$204$	0	0
$205$	−1044.84	−0.355974
$206$	0	0
$207$	−845.132	−0.283772
$208$	0	0
$209$	−810.708	−0.268315
$210$	0	0
$211$	2072.32	0.676134	0.338067	−	0.941122i	$-0.390227\pi$
$211$	2072.32	0.676134	0.338067	+	0.941122i	$0.390227\pi$
$212$	0	0
$213$	408.803	0.131506
$214$	0	0
$215$	241.834	0.0767114
$216$	0	0
$217$	945.856	0.295893
$218$	0	0
$219$	1613.77	0.497939
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2028.02	0.608996	0.304498	−	0.952513i	$-0.401511\pi$
$223$	2028.02	0.608996	0.304498	+	0.952513i	$0.401511\pi$
$224$	0	0
$225$	−2597.88	−0.769742
$226$	0	0
$227$	486.751	0.142321	0.0711604	−	0.997465i	$-0.477330\pi$
$227$	486.751	0.142321	0.0711604	+	0.997465i	$0.477330\pi$
$228$	0	0
$229$	−6207.06	−1.79115	−0.895576	−	0.444908i	$-0.853236\pi$
$229$	−6207.06	−1.79115	−0.895576	+	0.444908i	$0.853236\pi$
$230$	0	0
$231$	−1154.44	−0.328817
$232$	0	0
$233$	−1941.81	−0.545974	−0.272987	−	0.962018i	$-0.588012\pi$
$233$	−1941.81	−0.545974	−0.272987	+	0.962018i	$0.588012\pi$
$234$	0	0
$235$	34.4091	0.00955150
$236$	0	0
$237$	2299.10	0.630138
$238$	0	0
$239$	1054.29	0.285340	0.142670	−	0.989770i	$-0.454431\pi$
$239$	1054.29	0.285340	0.142670	+	0.989770i	$0.454431\pi$
$240$	0	0
$241$	−5463.91	−1.46042	−0.730210	−	0.683223i	$-0.760578\pi$
$241$	−5463.91	−1.46042	−0.730210	+	0.683223i	$0.760578\pi$
$242$	0	0
$243$	4963.37	1.31029
$244$	0	0
$245$	386.572	0.100805
$246$	0	0
$247$	−3499.38	−0.901458
$248$	0	0
$249$	−10038.2	−2.55479
$250$	0	0
$251$	5648.57	1.42046	0.710229	−	0.703971i	$-0.248592\pi$
$251$	5648.57	1.42046	0.710229	+	0.703971i	$0.248592\pi$
$252$	0	0
$253$	−429.618	−0.106758
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1074.70	0.260849	0.130424	−	0.991458i	$-0.458366\pi$
$257$	1074.70	0.260849	0.130424	+	0.991458i	$0.458366\pi$
$258$	0	0
$259$	−1141.25	−0.273799
$260$	0	0
$261$	−1715.56	−0.406860
$262$	0	0
$263$	4932.16	1.15639	0.578194	−	0.815899i	$-0.303758\pi$
$263$	4932.16	1.15639	0.578194	+	0.815899i	$0.303758\pi$
$264$	0	0
$265$	1647.47	0.381900
$266$	0	0
$267$	−1015.41	−0.232742
$268$	0	0
$269$	−1629.96	−0.369444	−0.184722	−	0.982791i	$-0.559139\pi$
$269$	−1629.96	−0.369444	−0.184722	+	0.982791i	$0.559139\pi$
$270$	0	0
$271$	−8113.14	−1.81859	−0.909296	−	0.416150i	$-0.863379\pi$
$271$	−8113.14	−1.81859	−0.909296	+	0.416150i	$0.863379\pi$
$272$	0	0
$273$	−4983.10	−1.10473
$274$	0	0
$275$	−1320.62	−0.289586
$276$	0	0
$277$	4208.51	0.912869	0.456434	−	0.889757i	$-0.349126\pi$
$277$	4208.51	0.912869	0.456434	+	0.889757i	$0.349126\pi$
$278$	0	0
$279$	1454.11	0.312026
$280$	0	0
$281$	−3145.69	−0.667816	−0.333908	−	0.942606i	$-0.608367\pi$
$281$	−3145.69	−0.667816	−0.333908	+	0.942606i	$0.608367\pi$
$282$	0	0
$283$	−3195.05	−0.671116	−0.335558	−	0.942020i	$-0.608925\pi$
$283$	−3195.05	−0.671116	−0.335558	+	0.942020i	$0.608925\pi$
$284$	0	0
$285$	−1463.84	−0.304247
$286$	0	0
$287$	5198.87	1.06927
$288$	0	0
$289$	0	0
$290$	0	0
$291$	2992.22	0.602774
$292$	0	0
$293$	−5625.09	−1.12158	−0.560788	−	0.827960i	$-0.689502\pi$
$293$	−5625.09	−1.12158	−0.560788	+	0.827960i	$0.689502\pi$
$294$	0	0
$295$	976.937	0.192812
$296$	0	0
$297$	374.159	0.0731006
$298$	0	0
$299$	−1854.42	−0.358676
$300$	0	0
$301$	−1203.31	−0.230424
$302$	0	0
$303$	−8127.20	−1.54091
$304$	0	0
$305$	−1004.15	−0.188516
$306$	0	0
$307$	−3528.70	−0.656005	−0.328002	−	0.944677i	$-0.606375\pi$
$307$	−3528.70	−0.656005	−0.328002	+	0.944677i	$0.606375\pi$
$308$	0	0
$309$	−3695.20	−0.680299
$310$	0	0
$311$	8276.39	1.50904	0.754519	−	0.656278i	$-0.227870\pi$
$311$	8276.39	1.50904	0.754519	+	0.656278i	$0.227870\pi$
$312$	0	0
$313$	−7789.26	−1.40663	−0.703315	−	0.710878i	$-0.748298\pi$
$313$	−7789.26	−1.40663	−0.703315	+	0.710878i	$0.748298\pi$
$314$	0	0
$315$	−942.862	−0.168648
$316$	0	0
$317$	−1931.09	−0.342148	−0.171074	−	0.985258i	$-0.554724\pi$
$317$	−1931.09	−0.342148	−0.171074	+	0.985258i	$0.554724\pi$
$318$	0	0
$319$	−872.095	−0.153066
$320$	0	0
$321$	−10873.3	−1.89061
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5700.37	−0.972922
$326$	0	0
$327$	−15503.8	−2.62191
$328$	0	0
$329$	−171.212	−0.0286906
$330$	0	0
$331$	6335.48	1.05205	0.526026	−	0.850468i	$-0.323681\pi$
$331$	6335.48	1.05205	0.526026	+	0.850468i	$0.323681\pi$
$332$	0	0
$333$	−1754.50	−0.288727
$334$	0	0
$335$	−1848.29	−0.301442
$336$	0	0
$337$	−7014.80	−1.13389	−0.566945	−	0.823756i	$-0.691875\pi$
$337$	−7014.80	−1.13389	−0.566945	+	0.823756i	$0.691875\pi$
$338$	0	0
$339$	−5578.45	−0.893746
$340$	0	0
$341$	739.189	0.117388
$342$	0	0
$343$	−6898.64	−1.08598
$344$	0	0
$345$	−775.731	−0.121055
$346$	0	0
$347$	−1183.86	−0.183150	−0.0915749	−	0.995798i	$-0.529190\pi$
$347$	−1183.86	−0.183150	−0.0915749	+	0.995798i	$0.529190\pi$
$348$	0	0
$349$	2805.34	0.430276	0.215138	−	0.976584i	$-0.430980\pi$
$349$	2805.34	0.430276	0.215138	+	0.976584i	$0.430980\pi$
$350$	0	0
$351$	1615.04	0.245596
$352$	0	0
$353$	158.865	0.0239533	0.0119766	−	0.999928i	$-0.496188\pi$
$353$	158.865	0.0239533	0.0119766	+	0.999928i	$0.496188\pi$
$354$	0	0
$355$	169.725	0.0253749
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8918.70	1.31117	0.655586	−	0.755120i	$-0.272422\pi$
$359$	8918.70	1.31117	0.655586	+	0.755120i	$0.272422\pi$
$360$	0	0
$361$	−1744.01	−0.254266
$362$	0	0
$363$	8443.18	1.22080
$364$	0	0
$365$	670.001	0.0960807
$366$	0	0
$367$	−7723.34	−1.09851	−0.549257	−	0.835653i	$-0.685089\pi$
$367$	−7723.34	−1.09851	−0.549257	+	0.835653i	$0.685089\pi$
$368$	0	0
$369$	7992.48	1.12757
$370$	0	0
$371$	−8197.44	−1.14714
$372$	0	0
$373$	−6385.58	−0.886415	−0.443208	−	0.896419i	$-0.646160\pi$
$373$	−6385.58	−0.886415	−0.443208	+	0.896419i	$0.646160\pi$
$374$	0	0
$375$	−4943.02	−0.680683
$376$	0	0
$377$	−3764.35	−0.514255
$378$	0	0
$379$	4231.50	0.573503	0.286752	−	0.958005i	$-0.407425\pi$
$379$	4231.50	0.573503	0.286752	+	0.958005i	$0.407425\pi$
$380$	0	0
$381$	15574.3	2.09421
$382$	0	0
$383$	−304.246	−0.0405907	−0.0202953	−	0.999794i	$-0.506461\pi$
$383$	−304.246	−0.0405907	−0.0202953	+	0.999794i	$0.506461\pi$
$384$	0	0
$385$	−479.298	−0.0634476
$386$	0	0
$387$	−1849.91	−0.242987
$388$	0	0
$389$	−4317.86	−0.562787	−0.281394	−	0.959592i	$-0.590797\pi$
$389$	−4317.86	−0.562787	−0.281394	+	0.959592i	$0.590797\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	14650.8	1.88050
$394$	0	0
$395$	954.534	0.121589
$396$	0	0
$397$	−3980.92	−0.503266	−0.251633	−	0.967823i	$-0.580968\pi$
$397$	−3980.92	−0.503266	−0.251633	+	0.967823i	$0.580968\pi$
$398$	0	0
$399$	7283.72	0.913889
$400$	0	0
$401$	−12741.8	−1.58677	−0.793384	−	0.608722i	$-0.791682\pi$
$401$	−12741.8	−1.58677	−0.793384	+	0.608722i	$0.791682\pi$
$402$	0	0
$403$	3190.67	0.394389
$404$	0	0
$405$	2430.69	0.298226
$406$	0	0
$407$	−891.892	−0.108623
$408$	0	0
$409$	−13744.6	−1.66168	−0.830839	−	0.556514i	$-0.812139\pi$
$409$	−13744.6	−1.66168	−0.830839	+	0.556514i	$0.812139\pi$
$410$	0	0
$411$	12108.8	1.45325
$412$	0	0
$413$	−4861.01	−0.579164
$414$	0	0
$415$	−4167.61	−0.492964
$416$	0	0
$417$	−15745.2	−1.84903
$418$	0	0
$419$	−5266.40	−0.614035	−0.307017	−	0.951704i	$-0.599331\pi$
$419$	−5266.40	−0.614035	−0.307017	+	0.951704i	$0.599331\pi$
$420$	0	0
$421$	−3995.20	−0.462504	−0.231252	−	0.972894i	$-0.574282\pi$
$421$	−3995.20	−0.462504	−0.231252	+	0.972894i	$0.574282\pi$
$422$	0	0
$423$	−263.212	−0.0302549
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4996.42	0.566262
$428$	0	0
$429$	−3894.30	−0.438272
$430$	0	0
$431$	1424.78	0.159232	0.0796162	−	0.996826i	$-0.474631\pi$
$431$	1424.78	0.159232	0.0796162	+	0.996826i	$0.474631\pi$
$432$	0	0
$433$	−6295.67	−0.698731	−0.349365	−	0.936987i	$-0.613603\pi$
$433$	−6295.67	−0.698731	−0.349365	+	0.936987i	$0.613603\pi$
$434$	0	0
$435$	−1574.68	−0.173564
$436$	0	0
$437$	2710.58	0.296716
$438$	0	0
$439$	−10902.2	−1.18527	−0.592636	−	0.805470i	$-0.701913\pi$
$439$	−10902.2	−1.18527	−0.592636	+	0.805470i	$0.701913\pi$
$440$	0	0
$441$	−2957.08	−0.319305
$442$	0	0
$443$	6459.15	0.692739	0.346370	−	0.938098i	$-0.387414\pi$
$443$	6459.15	0.692739	0.346370	+	0.938098i	$0.387414\pi$
$444$	0	0
$445$	−421.575	−0.0449092
$446$	0	0
$447$	23439.1	2.48016
$448$	0	0
$449$	−2260.33	−0.237576	−0.118788	−	0.992920i	$-0.537901\pi$
$449$	−2260.33	−0.237576	−0.118788	+	0.992920i	$0.537901\pi$
$450$	0	0
$451$	4062.93	0.424204
$452$	0	0
$453$	−11015.1	−1.14246
$454$	0	0
$455$	−2068.87	−0.213165
$456$	0	0
$457$	11621.6	1.18957	0.594787	−	0.803883i	$-0.297236\pi$
$457$	11621.6	1.18957	0.594787	+	0.803883i	$0.297236\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	81.5123	0.00823516	0.00411758	−	0.999992i	$-0.498689\pi$
$461$	81.5123	0.00823516	0.00411758	+	0.999992i	$0.498689\pi$
$462$	0	0
$463$	1412.38	0.141768	0.0708841	−	0.997485i	$-0.477418\pi$
$463$	1412.38	0.141768	0.0708841	+	0.997485i	$0.477418\pi$
$464$	0	0
$465$	1334.70	0.133108
$466$	0	0
$467$	−6195.58	−0.613913	−0.306956	−	0.951724i	$-0.599311\pi$
$467$	−6195.58	−0.613913	−0.306956	+	0.951724i	$0.599311\pi$
$468$	0	0
$469$	9196.67	0.905464
$470$	0	0
$471$	−9210.87	−0.901092
$472$	0	0
$473$	−940.390	−0.0914147
$474$	0	0
$475$	8332.14	0.804853
$476$	0	0
$477$	−12602.3	−1.20969
$478$	0	0
$479$	−9129.62	−0.870863	−0.435431	−	0.900222i	$-0.643404\pi$
$479$	−9129.62	−0.870863	−0.435431	+	0.900222i	$0.643404\pi$
$480$	0	0
$481$	−3849.80	−0.364940
$482$	0	0
$483$	3859.85	0.363622
$484$	0	0
$485$	1242.30	0.116309
$486$	0	0
$487$	4352.88	0.405026	0.202513	−	0.979280i	$-0.435089\pi$
$487$	4352.88	0.405026	0.202513	+	0.979280i	$0.435089\pi$
$488$	0	0
$489$	21394.8	1.97854
$490$	0	0
$491$	352.887	0.0324350	0.0162175	−	0.999868i	$-0.494838\pi$
$491$	352.887	0.0324350	0.0162175	+	0.999868i	$0.494838\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−736.849	−0.0669069
$496$	0	0
$497$	−844.513	−0.0762205
$498$	0	0
$499$	5178.38	0.464561	0.232281	−	0.972649i	$-0.425381\pi$
$499$	5178.38	0.464561	0.232281	+	0.972649i	$0.425381\pi$
$500$	0	0
$501$	−10203.9	−0.909931
$502$	0	0
$503$	3775.18	0.334646	0.167323	−	0.985902i	$-0.446488\pi$
$503$	3775.18	0.334646	0.167323	+	0.985902i	$0.446488\pi$
$504$	0	0
$505$	−3374.23	−0.297329
$506$	0	0
$507$	−1383.71	−0.121209
$508$	0	0
$509$	−19837.4	−1.72746	−0.863729	−	0.503957i	$-0.831877\pi$
$509$	−19837.4	−1.72746	−0.863729	+	0.503957i	$0.831877\pi$
$510$	0	0
$511$	−3333.77	−0.288605
$512$	0	0
$513$	−2360.67	−0.203170
$514$	0	0
$515$	−1534.16	−0.131268
$516$	0	0
$517$	−133.802	−0.0113822
$518$	0	0
$519$	−1501.48	−0.126990
$520$	0	0
$521$	−5001.22	−0.420551	−0.210276	−	0.977642i	$-0.567436\pi$
$521$	−5001.22	−0.420551	−0.210276	+	0.977642i	$0.567436\pi$
$522$	0	0
$523$	−16056.9	−1.34248	−0.671242	−	0.741238i	$-0.734239\pi$
$523$	−16056.9	−1.34248	−0.671242	+	0.741238i	$0.734239\pi$
$524$	0	0
$525$	11864.9	0.986338
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−10730.6	−0.881942
$530$	0	0
$531$	−7473.07	−0.610741
$532$	0	0
$533$	17537.4	1.42520
$534$	0	0
$535$	−4514.33	−0.364806
$536$	0	0
$537$	23615.3	1.89772
$538$	0	0
$539$	−1503.21	−0.120126
$540$	0	0
$541$	14799.8	1.17614	0.588071	−	0.808809i	$-0.299887\pi$
$541$	14799.8	1.17614	0.588071	+	0.808809i	$0.299887\pi$
$542$	0	0
$543$	30716.9	2.42760
$544$	0	0
$545$	−6436.83	−0.505915
$546$	0	0
$547$	−4725.62	−0.369383	−0.184692	−	0.982797i	$-0.559129\pi$
$547$	−4725.62	−0.369383	−0.184692	+	0.982797i	$0.559129\pi$
$548$	0	0
$549$	7681.24	0.597136
$550$	0	0
$551$	5502.30	0.425419
$552$	0	0
$553$	−4749.53	−0.365227
$554$	0	0
$555$	−1610.43	−0.123169
$556$	0	0
$557$	−9858.63	−0.749952	−0.374976	−	0.927034i	$-0.622349\pi$
$557$	−9858.63	−0.749952	−0.374976	+	0.927034i	$0.622349\pi$
$558$	0	0
$559$	−4059.14	−0.307126
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12799.9	0.958171	0.479085	−	0.877768i	$-0.340968\pi$
$563$	12799.9	0.958171	0.479085	+	0.877768i	$0.340968\pi$
$564$	0	0
$565$	−2316.04	−0.172454
$566$	0	0
$567$	−12094.5	−0.895806
$568$	0	0
$569$	−22641.1	−1.66812	−0.834062	−	0.551671i	$-0.813991\pi$
$569$	−22641.1	−1.66812	−0.834062	+	0.551671i	$0.813991\pi$
$570$	0	0
$571$	−13700.4	−1.00411	−0.502054	−	0.864836i	$-0.667422\pi$
$571$	−13700.4	−1.00411	−0.502054	+	0.864836i	$0.667422\pi$
$572$	0	0
$573$	25906.3	1.88875
$574$	0	0
$575$	4415.45	0.320238
$576$	0	0
$577$	6449.44	0.465327	0.232663	−	0.972557i	$-0.425256\pi$
$577$	6449.44	0.465327	0.232663	+	0.972557i	$0.425256\pi$
$578$	0	0
$579$	−82.0421	−0.00588869
$580$	0	0
$581$	20737.1	1.48075
$582$	0	0
$583$	−6406.32	−0.455099
$584$	0	0
$585$	−3180.57	−0.224787
$586$	0	0
$587$	6818.67	0.479449	0.239725	−	0.970841i	$-0.422943\pi$
$587$	6818.67	0.479449	0.239725	+	0.970841i	$0.422943\pi$
$588$	0	0
$589$	−4663.75	−0.326259
$590$	0	0
$591$	−15188.0	−1.05711
$592$	0	0
$593$	8159.21	0.565023	0.282511	−	0.959264i	$-0.408832\pi$
$593$	8159.21	0.565023	0.282511	+	0.959264i	$0.408832\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−33156.4	−2.27303
$598$	0	0
$599$	2928.45	0.199755	0.0998774	−	0.995000i	$-0.468155\pi$
$599$	2928.45	0.199755	0.0998774	+	0.995000i	$0.468155\pi$
$600$	0	0
$601$	12103.4	0.821476	0.410738	−	0.911753i	$-0.365271\pi$
$601$	12103.4	0.821476	0.410738	+	0.911753i	$0.365271\pi$
$602$	0	0
$603$	14138.5	0.954832
$604$	0	0
$605$	3505.41	0.235562
$606$	0	0
$607$	−24827.8	−1.66018	−0.830091	−	0.557628i	$-0.811711\pi$
$607$	−24827.8	−1.66018	−0.830091	+	0.557628i	$0.811711\pi$
$608$	0	0
$609$	7835.24	0.521346
$610$	0	0
$611$	−577.551	−0.0382409
$612$	0	0
$613$	22505.6	1.48286	0.741431	−	0.671029i	$-0.234148\pi$
$613$	22505.6	1.48286	0.741431	+	0.671029i	$0.234148\pi$
$614$	0	0
$615$	7336.15	0.481011
$616$	0	0
$617$	−23598.7	−1.53979	−0.769894	−	0.638171i	$-0.779691\pi$
$617$	−23598.7	−1.53979	−0.769894	+	0.638171i	$0.779691\pi$
$618$	0	0
$619$	8769.75	0.569444	0.284722	−	0.958610i	$-0.408099\pi$
$619$	8769.75	0.569444	0.284722	+	0.958610i	$0.408099\pi$
$620$	0	0
$621$	−1250.99	−0.0808382
$622$	0	0
$623$	2097.66	0.134897
$624$	0	0
$625$	12510.6	0.800676
$626$	0	0
$627$	5692.24	0.362562
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−16385.2	−1.03373	−0.516866	−	0.856066i	$-0.672901\pi$
$631$	−16385.2	−1.03373	−0.516866	+	0.856066i	$0.672901\pi$
$632$	0	0
$633$	−14550.4	−0.913628
$634$	0	0
$635$	6466.08	0.404092
$636$	0	0
$637$	−6488.55	−0.403588
$638$	0	0
$639$	−1298.31	−0.0803763
$640$	0	0
$641$	6403.51	0.394577	0.197288	−	0.980346i	$-0.436786\pi$
$641$	6403.51	0.394577	0.197288	+	0.980346i	$0.436786\pi$
$642$	0	0
$643$	−28512.8	−1.74873	−0.874366	−	0.485268i	$-0.838722\pi$
$643$	−28512.8	−1.74873	−0.874366	+	0.485268i	$0.838722\pi$
$644$	0	0
$645$	−1698.00	−0.103657
$646$	0	0
$647$	21561.7	1.31016	0.655082	−	0.755558i	$-0.272634\pi$
$647$	21561.7	1.31016	0.655082	+	0.755558i	$0.272634\pi$
$648$	0	0
$649$	−3798.89	−0.229768
$650$	0	0
$651$	−6641.16	−0.399827
$652$	0	0
$653$	24695.5	1.47996	0.739978	−	0.672631i	$-0.234836\pi$
$653$	24695.5	1.47996	0.739978	+	0.672631i	$0.234836\pi$
$654$	0	0
$655$	6082.67	0.362854
$656$	0	0
$657$	−5125.16	−0.304341
$658$	0	0
$659$	−16978.4	−1.00362	−0.501810	−	0.864978i	$-0.667332\pi$
$659$	−16978.4	−1.00362	−0.501810	+	0.864978i	$0.667332\pi$
$660$	0	0
$661$	7950.85	0.467855	0.233928	−	0.972254i	$-0.424842\pi$
$661$	7950.85	0.467855	0.233928	+	0.972254i	$0.424842\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3024.03	0.176341
$666$	0	0
$667$	2915.83	0.169267
$668$	0	0
$669$	−14239.4	−0.822908
$670$	0	0
$671$	3904.71	0.224650
$672$	0	0
$673$	−28452.0	−1.62963	−0.814816	−	0.579719i	$-0.803162\pi$
$673$	−28452.0	−1.62963	−0.814816	+	0.579719i	$0.803162\pi$
$674$	0	0
$675$	−3845.46	−0.219277
$676$	0	0
$677$	32155.3	1.82545	0.912725	−	0.408575i	$-0.133974\pi$
$677$	32155.3	1.82545	0.912725	+	0.408575i	$0.133974\pi$
$678$	0	0
$679$	−6181.40	−0.349367
$680$	0	0
$681$	−3417.64	−0.192312
$682$	0	0
$683$	−5041.97	−0.282468	−0.141234	−	0.989976i	$-0.545107\pi$
$683$	−5041.97	−0.282468	−0.141234	+	0.989976i	$0.545107\pi$
$684$	0	0
$685$	5027.30	0.280414
$686$	0	0
$687$	43581.7	2.42030
$688$	0	0
$689$	−27652.5	−1.52900
$690$	0	0
$691$	18241.4	1.00425	0.502124	−	0.864796i	$-0.332552\pi$
$691$	18241.4	1.00425	0.502124	+	0.864796i	$0.332552\pi$
$692$	0	0
$693$	3666.39	0.200973
$694$	0	0
$695$	−6537.02	−0.356782
$696$	0	0
$697$	0	0
$698$	0	0
$699$	13634.0	0.737750
$700$	0	0
$701$	22391.5	1.20644	0.603219	−	0.797575i	$-0.293884\pi$
$701$	22391.5	1.20644	0.603219	+	0.797575i	$0.293884\pi$
$702$	0	0
$703$	5627.20	0.301897
$704$	0	0
$705$	−241.597	−0.0129065
$706$	0	0
$707$	16789.4	0.893110
$708$	0	0
$709$	2919.37	0.154639	0.0773195	−	0.997006i	$-0.475364\pi$
$709$	2919.37	0.154639	0.0773195	+	0.997006i	$0.475364\pi$
$710$	0	0
$711$	−7301.70	−0.385141
$712$	0	0
$713$	−2471.46	−0.129813
$714$	0	0
$715$	−1616.82	−0.0845676
$716$	0	0
$717$	−7402.50	−0.385567
$718$	0	0
$719$	−12516.3	−0.649206	−0.324603	−	0.945850i	$-0.605231\pi$
$719$	−12516.3	−0.649206	−0.324603	+	0.945850i	$0.605231\pi$
$720$	0	0
$721$	7633.62	0.394301
$722$	0	0
$723$	38363.8	1.97340
$724$	0	0
$725$	8963.05	0.459144
$726$	0	0
$727$	−14169.7	−0.722869	−0.361435	−	0.932397i	$-0.617713\pi$
$727$	−14169.7	−0.722869	−0.361435	+	0.932397i	$0.617713\pi$
$728$	0	0
$729$	−12336.1	−0.626737
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−12782.3	−0.644098	−0.322049	−	0.946723i	$-0.604372\pi$
$733$	−12782.3	−0.644098	−0.322049	+	0.946723i	$0.604372\pi$
$734$	0	0
$735$	−2714.25	−0.136213
$736$	0	0
$737$	7187.22	0.359219
$738$	0	0
$739$	21707.5	1.08055	0.540273	−	0.841489i	$-0.318321\pi$
$739$	21707.5	1.08055	0.540273	+	0.841489i	$0.318321\pi$
$740$	0	0
$741$	24570.3	1.21810
$742$	0	0
$743$	−30938.8	−1.52764	−0.763819	−	0.645431i	$-0.776678\pi$
$743$	−30938.8	−1.52764	−0.763819	+	0.645431i	$0.776678\pi$
$744$	0	0
$745$	9731.36	0.478563
$746$	0	0
$747$	31880.1	1.56149
$748$	0	0
$749$	22462.2	1.09580
$750$	0	0
$751$	25559.8	1.24193	0.620966	−	0.783838i	$-0.286741\pi$
$751$	25559.8	1.24193	0.620966	+	0.783838i	$0.286741\pi$
$752$	0	0
$753$	−39660.4	−1.91940
$754$	0	0
$755$	−4573.20	−0.220445
$756$	0	0
$757$	6684.36	0.320934	0.160467	−	0.987041i	$-0.448700\pi$
$757$	6684.36	0.320934	0.160467	+	0.987041i	$0.448700\pi$
$758$	0	0
$759$	3016.49	0.144258
$760$	0	0
$761$	22539.6	1.07366	0.536832	−	0.843689i	$-0.319621\pi$
$761$	22539.6	1.07366	0.536832	+	0.843689i	$0.319621\pi$
$762$	0	0
$763$	32028.2	1.51966
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16397.7	−0.771952
$768$	0	0
$769$	4473.13	0.209760	0.104880	−	0.994485i	$-0.466554\pi$
$769$	4473.13	0.209760	0.104880	+	0.994485i	$0.466554\pi$
$770$	0	0
$771$	−7545.83	−0.352473
$772$	0	0
$773$	−16438.6	−0.764885	−0.382442	−	0.923979i	$-0.624917\pi$
$773$	−16438.6	−0.764885	−0.382442	+	0.923979i	$0.624917\pi$
$774$	0	0
$775$	−7597.09	−0.352123
$776$	0	0
$777$	8013.10	0.369972
$778$	0	0
$779$	−25634.2	−1.17900
$780$	0	0
$781$	−659.989	−0.0302385
$782$	0	0
$783$	−2539.42	−0.115902
$784$	0	0
$785$	−3824.14	−0.173872
$786$	0	0
$787$	6384.35	0.289171	0.144585	−	0.989492i	$-0.453815\pi$
$787$	6384.35	0.289171	0.144585	+	0.989492i	$0.453815\pi$
$788$	0	0
$789$	−34630.3	−1.56257
$790$	0	0
$791$	11524.1	0.518014
$792$	0	0
$793$	16854.5	0.754755
$794$	0	0
$795$	−11567.4	−0.516044
$796$	0	0
$797$	25998.5	1.15548	0.577738	−	0.816223i	$-0.303936\pi$
$797$	25998.5	1.15548	0.577738	+	0.816223i	$0.303936\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	3224.84	0.142252
$802$	0	0
$803$	−2605.35	−0.114497
$804$	0	0
$805$	1602.52	0.0701633
$806$	0	0
$807$	11444.5	0.499213
$808$	0	0
$809$	7851.84	0.341231	0.170616	−	0.985338i	$-0.445424\pi$
$809$	7851.84	0.341231	0.170616	+	0.985338i	$0.445424\pi$
$810$	0	0
$811$	−35084.7	−1.51910	−0.759550	−	0.650449i	$-0.774581\pi$
$811$	−35084.7	−1.51910	−0.759550	+	0.650449i	$0.774581\pi$
$812$	0	0
$813$	56965.0	2.45738
$814$	0	0
$815$	8882.62	0.381772
$816$	0	0
$817$	5933.19	0.254071
$818$	0	0
$819$	15825.8	0.675210
$820$	0	0
$821$	34385.9	1.46172	0.730861	−	0.682526i	$-0.239119\pi$
$821$	34385.9	1.46172	0.730861	+	0.682526i	$0.239119\pi$
$822$	0	0
$823$	−47093.8	−1.99464	−0.997318	−	0.0731849i	$-0.976684\pi$
$823$	−47093.8	−1.99464	−0.997318	+	0.0731849i	$0.976684\pi$
$824$	0	0
$825$	9272.47	0.391304
$826$	0	0
$827$	35062.9	1.47431	0.737157	−	0.675722i	$-0.236168\pi$
$827$	35062.9	1.47431	0.737157	+	0.675722i	$0.236168\pi$
$828$	0	0
$829$	−7193.75	−0.301387	−0.150693	−	0.988581i	$-0.548151\pi$
$829$	−7193.75	−0.301387	−0.150693	+	0.988581i	$0.548151\pi$
$830$	0	0
$831$	−29549.3	−1.23352
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4236.41	−0.175577
$836$	0	0
$837$	2152.42	0.0888871
$838$	0	0
$839$	2465.50	0.101452	0.0507262	−	0.998713i	$-0.483846\pi$
$839$	2465.50	0.101452	0.0507262	+	0.998713i	$0.483846\pi$
$840$	0	0
$841$	−18470.1	−0.757311
$842$	0	0
$843$	22086.9	0.902389
$844$	0	0
$845$	−574.485	−0.0233880
$846$	0	0
$847$	−17442.1	−0.707577
$848$	0	0
$849$	22433.4	0.906848
$850$	0	0
$851$	2982.02	0.120120
$852$	0	0
$853$	−5880.76	−0.236053	−0.118027	−	0.993010i	$-0.537657\pi$
$853$	−5880.76	−0.236053	−0.118027	+	0.993010i	$0.537657\pi$
$854$	0	0
$855$	4648.99	0.185956
$856$	0	0
$857$	−11582.5	−0.461669	−0.230834	−	0.972993i	$-0.574146\pi$
$857$	−11582.5	−0.461669	−0.230834	+	0.972993i	$0.574146\pi$
$858$	0	0
$859$	−9277.33	−0.368496	−0.184248	−	0.982880i	$-0.558985\pi$
$859$	−9277.33	−0.368496	−0.184248	+	0.982880i	$0.558985\pi$
$860$	0	0
$861$	−36502.9	−1.44485
$862$	0	0
$863$	35817.4	1.41279	0.706395	−	0.707818i	$-0.250320\pi$
$863$	35817.4	1.41279	0.706395	+	0.707818i	$0.250320\pi$
$864$	0	0
$865$	−623.380	−0.0245035
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3711.77	−0.144894
$870$	0	0
$871$	31023.3	1.20687
$872$	0	0
$873$	−9502.97	−0.368415
$874$	0	0
$875$	10211.4	0.394524
$876$	0	0
$877$	12766.9	0.491573	0.245786	−	0.969324i	$-0.420954\pi$
$877$	12766.9	0.491573	0.245786	+	0.969324i	$0.420954\pi$
$878$	0	0
$879$	39495.6	1.51553
$880$	0	0
$881$	23963.3	0.916395	0.458197	−	0.888851i	$-0.348495\pi$
$881$	23963.3	0.916395	0.458197	+	0.888851i	$0.348495\pi$
$882$	0	0
$883$	13968.4	0.532360	0.266180	−	0.963923i	$-0.414238\pi$
$883$	13968.4	0.532360	0.266180	+	0.963923i	$0.414238\pi$
$884$	0	0
$885$	−6859.39	−0.260538
$886$	0	0
$887$	−21628.1	−0.818717	−0.409359	−	0.912374i	$-0.634247\pi$
$887$	−21628.1	−0.818717	−0.409359	+	0.912374i	$0.634247\pi$
$888$	0	0
$889$	−32173.7	−1.21380
$890$	0	0
$891$	−9451.89	−0.355388
$892$	0	0
$893$	844.197	0.0316349
$894$	0	0
$895$	9804.54	0.366178
$896$	0	0
$897$	13020.5	0.484662
$898$	0	0
$899$	−5016.89	−0.186121
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8448.82	0.311361
$904$	0	0
$905$	12752.9	0.468422
$906$	0	0
$907$	−30736.6	−1.12524	−0.562620	−	0.826716i	$-0.690207\pi$
$907$	−30736.6	−1.12524	−0.562620	+	0.826716i	$0.690207\pi$
$908$	0	0
$909$	25811.1	0.941804
$910$	0	0
$911$	−31162.6	−1.13333	−0.566664	−	0.823949i	$-0.691766\pi$
$911$	−31162.6	−1.13333	−0.566664	+	0.823949i	$0.691766\pi$
$912$	0	0
$913$	16206.1	0.587450
$914$	0	0
$915$	7050.47	0.254734
$916$	0	0
$917$	−30265.9	−1.08993
$918$	0	0
$919$	41773.3	1.49943	0.749714	−	0.661762i	$-0.230191\pi$
$919$	41773.3	1.49943	0.749714	+	0.661762i	$0.230191\pi$
$920$	0	0
$921$	24776.1	0.886429
$922$	0	0
$923$	−2848.81	−0.101592
$924$	0	0
$925$	9166.51	0.325830
$926$	0	0
$927$	11735.5	0.415799
$928$	0	0
$929$	21716.3	0.766943	0.383472	−	0.923553i	$-0.374728\pi$
$929$	21716.3	0.766943	0.383472	+	0.923553i	$0.374728\pi$
$930$	0	0
$931$	9484.21	0.333869
$932$	0	0
$933$	−58111.2	−2.03909
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6375.27	0.222274	0.111137	−	0.993805i	$-0.464551\pi$
$937$	6375.27	0.222274	0.111137	+	0.993805i	$0.464551\pi$
$938$	0	0
$939$	54690.9	1.90071
$940$	0	0
$941$	−43310.2	−1.50040	−0.750198	−	0.661213i	$-0.770042\pi$
$941$	−43310.2	−1.50040	−0.750198	+	0.661213i	$0.770042\pi$
$942$	0	0
$943$	−13584.3	−0.469105
$944$	0	0
$945$	−1395.65	−0.0480430
$946$	0	0
$947$	8467.96	0.290572	0.145286	−	0.989390i	$-0.453590\pi$
$947$	8467.96	0.290572	0.145286	+	0.989390i	$0.453590\pi$
$948$	0	0
$949$	−11245.8	−0.384674
$950$	0	0
$951$	13558.8	0.462329
$952$	0	0
$953$	−51683.4	−1.75676	−0.878378	−	0.477966i	$-0.841374\pi$
$953$	−51683.4	−1.75676	−0.878378	+	0.477966i	$0.841374\pi$
$954$	0	0
$955$	10755.7	0.364447
$956$	0	0
$957$	6123.26	0.206831
$958$	0	0
$959$	−25014.7	−0.842300
$960$	0	0
$961$	−25538.7	−0.857261
$962$	0	0
$963$	34532.3	1.15554
$964$	0	0
$965$	−34.0620	−0.00113626
$966$	0	0
$967$	−40211.0	−1.33723	−0.668613	−	0.743611i	$-0.733112\pi$
$967$	−40211.0	−1.33723	−0.668613	+	0.743611i	$0.733112\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11348.1	−0.375053	−0.187526	−	0.982260i	$-0.560047\pi$
$971$	−11348.1	−0.375053	−0.187526	+	0.982260i	$0.560047\pi$
$972$	0	0
$973$	32526.7	1.07169
$974$	0	0
$975$	40024.1	1.31466
$976$	0	0
$977$	−51696.4	−1.69285	−0.846425	−	0.532508i	$-0.821250\pi$
$977$	−51696.4	−1.69285	−0.846425	+	0.532508i	$0.821250\pi$
$978$	0	0
$979$	1639.33	0.0535169
$980$	0	0
$981$	49238.5	1.60251
$982$	0	0
$983$	24337.9	0.789684	0.394842	−	0.918749i	$-0.370799\pi$
$983$	24337.9	0.789684	0.394842	+	0.918749i	$0.370799\pi$
$984$	0	0
$985$	−6305.73	−0.203977
$986$	0	0
$987$	1202.13	0.0387683
$988$	0	0
$989$	3144.17	0.101091
$990$	0	0
$991$	30296.6	0.971142	0.485571	−	0.874197i	$-0.338612\pi$
$991$	30296.6	0.971142	0.485571	+	0.874197i	$0.338612\pi$
$992$	0	0
$993$	−44483.4	−1.42159
$994$	0	0
$995$	−13765.8	−0.438597
$996$	0	0
$997$	17949.0	0.570161	0.285081	−	0.958504i	$-0.407980\pi$
$997$	17949.0	0.570161	0.285081	+	0.958504i	$0.407980\pi$
$998$	0	0
$999$	−2597.07	−0.0822499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.r.1.3		24
17.3	odd	16		136.4.n.a.9.1	✓	24
17.6	odd	16		136.4.n.a.121.1	yes	24
17.16	even	2	inner	2312.4.a.r.1.22		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.n.a.9.1	✓	24	17.3	odd	16
136.4.n.a.121.1	yes	24	17.6	odd	16
2312.4.a.r.1.3		24	1.1	even	1	trivial
2312.4.a.r.1.22		24	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-7.02132 q^{3} -2.91509 q^{5} +14.5048 q^{7} +22.2990 q^{9} +O(q^{10})\)	\(q-7.02132 q^{3} -2.91509 q^{5} +14.5048 q^{7} +22.2990 q^{9} +11.3355 q^{11} +48.9293 q^{13} +20.4678 q^{15} -71.5192 q^{19} -101.843 q^{21} -37.9001 q^{23} -116.502 q^{25} +33.0076 q^{27} -76.9346 q^{29} +65.2099 q^{31} -79.5905 q^{33} -42.2828 q^{35} -78.6810 q^{37} -343.548 q^{39} +358.424 q^{41} -82.9594 q^{43} -65.0035 q^{45} -11.8038 q^{47} -132.611 q^{49} -565.154 q^{53} -33.0441 q^{55} +502.159 q^{57} -335.131 q^{59} +344.467 q^{61} +323.442 q^{63} -142.633 q^{65} +634.043 q^{67} +266.109 q^{69} -58.2230 q^{71} -229.839 q^{73} +818.000 q^{75} +164.420 q^{77} -327.446 q^{79} -833.828 q^{81} +1429.67 q^{83} +540.183 q^{87} +144.618 q^{89} +709.709 q^{91} -457.859 q^{93} +208.485 q^{95} -426.162 q^{97} +252.771 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 24 * q + 88 * q^9	\( 24 q + 88 q^{9} - 168 q^{13} - 120 q^{15} + 88 q^{19} - 64 q^{21} + 144 q^{25} - 520 q^{33} + 512 q^{35} - 616 q^{43} - 984 q^{47} + 272 q^{49} - 1640 q^{53} - 2296 q^{55} + 1304 q^{59} - 1960 q^{67} - 2408 q^{69} - 5248 q^{77} - 3560 q^{81} + 696 q^{83} + 1176 q^{87} - 5504 q^{89} + 616 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 168 * q^13 - 120 * q^15 + 88 * q^19 - 64 * q^21 + 144 * q^25 - 520 * q^33 + 512 * q^35 - 616 * q^43 - 984 * q^47 + 272 * q^49 - 1640 * q^53 - 2296 * q^55 + 1304 * q^59 - 1960 * q^67 - 2408 * q^69 - 5248 * q^77 - 3560 * q^81 + 696 * q^83 + 1176 * q^87 - 5504 * q^89 + 616 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more