LMFDB - Embedded newform 2352.4.a.cn.1.1

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.391168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 40x^{2} + 382 $ x^4 - 40*x^2 + 382
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.92368$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -13.3405 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -13.3405 q^{5} +9.00000 q^{9} +1.42531 q^{11} +38.7658 q^{13} +40.0215 q^{15} +27.3562 q^{17} +65.4495 q^{19} -2.54332 q^{23} +52.9686 q^{25} -27.0000 q^{27} -63.7544 q^{29} -51.9548 q^{31} -4.27592 q^{33} -335.494 q^{37} -116.297 q^{39} +447.541 q^{41} +170.339 q^{43} -120.064 q^{45} +116.813 q^{47} -82.0685 q^{51} +86.3042 q^{53} -19.0143 q^{55} -196.349 q^{57} -380.892 q^{59} -199.624 q^{61} -517.155 q^{65} -951.939 q^{67} +7.62996 q^{69} -830.527 q^{71} -332.753 q^{73} -158.906 q^{75} -755.996 q^{79} +81.0000 q^{81} +15.2261 q^{83} -364.945 q^{85} +191.263 q^{87} +1554.11 q^{89} +155.864 q^{93} -873.128 q^{95} +101.205 q^{97} +12.8278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9	$ 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9} - 40 q^{11} + 48 q^{13} - 24 q^{15} + 72 q^{17} - 32 q^{19} - 8 q^{23} + 164 q^{25} - 108 q^{27} + 144 q^{29} - 48 q^{31} + 120 q^{33} + 48 q^{37} - 144 q^{39} + 72 q^{41} - 512 q^{43} + 72 q^{45} - 160 q^{47} - 216 q^{51} + 536 q^{53} + 336 q^{55} + 96 q^{57} - 240 q^{59} + 896 q^{61} - 136 q^{65} - 1088 q^{67} + 24 q^{69} - 1288 q^{71} + 1488 q^{73} - 492 q^{75} - 416 q^{79} + 324 q^{81} + 112 q^{83} - 1512 q^{85} - 432 q^{87} + 3160 q^{89} + 144 q^{93} + 240 q^{95} + 2384 q^{97} - 360 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9 - 40 * q^11 + 48 * q^13 - 24 * q^15 + 72 * q^17 - 32 * q^19 - 8 * q^23 + 164 * q^25 - 108 * q^27 + 144 * q^29 - 48 * q^31 + 120 * q^33 + 48 * q^37 - 144 * q^39 + 72 * q^41 - 512 * q^43 + 72 * q^45 - 160 * q^47 - 216 * q^51 + 536 * q^53 + 336 * q^55 + 96 * q^57 - 240 * q^59 + 896 * q^61 - 136 * q^65 - 1088 * q^67 + 24 * q^69 - 1288 * q^71 + 1488 * q^73 - 492 * q^75 - 416 * q^79 + 324 * q^81 + 112 * q^83 - 1512 * q^85 - 432 * q^87 + 3160 * q^89 + 144 * q^93 + 240 * q^95 + 2384 * q^97 - 360 * 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
$5$	−13.3405	−1.19321	−0.596605	−	0.802535i	$-0.703484\pi$
$5$	−13.3405	−1.19321	−0.596605	+	0.802535i	$0.703484\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	1.42531	0.0390678	0.0195339	−	0.999809i	$-0.493782\pi$
$11$	1.42531	0.0390678	0.0195339	+	0.999809i	$0.493782\pi$
$12$	0	0
$13$	38.7658	0.827054	0.413527	−	0.910492i	$-0.364297\pi$
$13$	38.7658	0.827054	0.413527	+	0.910492i	$0.364297\pi$
$14$	0	0
$15$	40.0215	0.688900
$16$	0	0
$17$	27.3562	0.390285	0.195143	−	0.980775i	$-0.437483\pi$
$17$	27.3562	0.390285	0.195143	+	0.980775i	$0.437483\pi$
$18$	0	0
$19$	65.4495	0.790271	0.395135	−	0.918623i	$-0.370698\pi$
$19$	65.4495	0.790271	0.395135	+	0.918623i	$0.370698\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.54332	−0.0230573	−0.0115287	−	0.999934i	$-0.503670\pi$
$23$	−2.54332	−0.0230573	−0.0115287	+	0.999934i	$0.503670\pi$
$24$	0	0
$25$	52.9686	0.423749
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−63.7544	−0.408238	−0.204119	−	0.978946i	$-0.565433\pi$
$29$	−63.7544	−0.408238	−0.204119	+	0.978946i	$0.565433\pi$
$30$	0	0
$31$	−51.9548	−0.301011	−0.150506	−	0.988609i	$-0.548090\pi$
$31$	−51.9548	−0.301011	−0.150506	+	0.988609i	$0.548090\pi$
$32$	0	0
$33$	−4.27592	−0.0225558
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−335.494	−1.49067	−0.745337	−	0.666688i	$-0.767711\pi$
$37$	−335.494	−1.49067	−0.745337	+	0.666688i	$0.767711\pi$
$38$	0	0
$39$	−116.297	−0.477500
$40$	0	0
$41$	447.541	1.70473	0.852367	−	0.522944i	$-0.175166\pi$
$41$	447.541	1.70473	0.852367	+	0.522944i	$0.175166\pi$
$42$	0	0
$43$	170.339	0.604103	0.302052	−	0.953292i	$-0.402328\pi$
$43$	170.339	0.604103	0.302052	+	0.953292i	$0.402328\pi$
$44$	0	0
$45$	−120.064	−0.397737
$46$	0	0
$47$	116.813	0.362531	0.181266	−	0.983434i	$-0.441981\pi$
$47$	116.813	0.362531	0.181266	+	0.983434i	$0.441981\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−82.0685	−0.225331
$52$	0	0
$53$	86.3042	0.223675	0.111838	−	0.993726i	$-0.464326\pi$
$53$	86.3042	0.223675	0.111838	+	0.993726i	$0.464326\pi$
$54$	0	0
$55$	−19.0143	−0.0466161
$56$	0	0
$57$	−196.349	−0.456263
$58$	0	0
$59$	−380.892	−0.840473	−0.420236	−	0.907415i	$-0.638053\pi$
$59$	−380.892	−0.840473	−0.420236	+	0.907415i	$0.638053\pi$
$60$	0	0
$61$	−199.624	−0.419004	−0.209502	−	0.977808i	$-0.567184\pi$
$61$	−199.624	−0.419004	−0.209502	+	0.977808i	$0.567184\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−517.155	−0.986848
$66$	0	0
$67$	−951.939	−1.73579	−0.867894	−	0.496749i	$-0.834527\pi$
$67$	−951.939	−1.73579	−0.867894	+	0.496749i	$0.834527\pi$
$68$	0	0
$69$	7.62996	0.0133122
$70$	0	0
$71$	−830.527	−1.38825	−0.694123	−	0.719857i	$-0.744207\pi$
$71$	−830.527	−1.38825	−0.694123	+	0.719857i	$0.744207\pi$
$72$	0	0
$73$	−332.753	−0.533504	−0.266752	−	0.963765i	$-0.585950\pi$
$73$	−332.753	−0.533504	−0.266752	+	0.963765i	$0.585950\pi$
$74$	0	0
$75$	−158.906	−0.244652
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−755.996	−1.07666	−0.538330	−	0.842734i	$-0.680945\pi$
$79$	−755.996	−1.07666	−0.538330	+	0.842734i	$0.680945\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	15.2261	0.0201360	0.0100680	−	0.999949i	$-0.496795\pi$
$83$	15.2261	0.0201360	0.0100680	+	0.999949i	$0.496795\pi$
$84$	0	0
$85$	−364.945	−0.465692
$86$	0	0
$87$	191.263	0.235696
$88$	0	0
$89$	1554.11	1.85096	0.925481	−	0.378795i	$-0.123661\pi$
$89$	1554.11	1.85096	0.925481	+	0.378795i	$0.123661\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	155.864	0.173789
$94$	0	0
$95$	−873.128	−0.942959
$96$	0	0
$97$	101.205	0.105937	0.0529683	−	0.998596i	$-0.483132\pi$
$97$	101.205	0.105937	0.0529683	+	0.998596i	$0.483132\pi$
$98$	0	0
$99$	12.8278	0.0130226
$100$	0	0
$101$	1439.24	1.41792	0.708961	−	0.705248i	$-0.249164\pi$
$101$	1439.24	1.41792	0.708961	+	0.705248i	$0.249164\pi$
$102$	0	0
$103$	−1605.32	−1.53569	−0.767847	−	0.640633i	$-0.778672\pi$
$103$	−1605.32	−1.53569	−0.767847	+	0.640633i	$0.778672\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	308.070	0.278339	0.139169	−	0.990269i	$-0.455557\pi$
$107$	308.070	0.278339	0.139169	+	0.990269i	$0.455557\pi$
$108$	0	0
$109$	708.959	0.622991	0.311495	−	0.950248i	$-0.399170\pi$
$109$	708.959	0.622991	0.311495	+	0.950248i	$0.399170\pi$
$110$	0	0
$111$	1006.48	0.860641
$112$	0	0
$113$	1107.48	0.921975	0.460988	−	0.887407i	$-0.347495\pi$
$113$	1107.48	0.921975	0.460988	+	0.887407i	$0.347495\pi$
$114$	0	0
$115$	33.9291	0.0275122
$116$	0	0
$117$	348.892	0.275685
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1328.97	−0.998474
$122$	0	0
$123$	−1342.62	−0.984229
$124$	0	0
$125$	960.934	0.687588
$126$	0	0
$127$	2330.27	1.62817	0.814087	−	0.580743i	$-0.197238\pi$
$127$	2330.27	1.62817	0.814087	+	0.580743i	$0.197238\pi$
$128$	0	0
$129$	−511.017	−0.348779
$130$	0	0
$131$	−2494.68	−1.66383	−0.831913	−	0.554907i	$-0.812754\pi$
$131$	−2494.68	−1.66383	−0.831913	+	0.554907i	$0.812754\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	360.193	0.229633
$136$	0	0
$137$	260.269	0.162309	0.0811543	−	0.996702i	$-0.474139\pi$
$137$	260.269	0.162309	0.0811543	+	0.996702i	$0.474139\pi$
$138$	0	0
$139$	−244.245	−0.149040	−0.0745202	−	0.997220i	$-0.523743\pi$
$139$	−244.245	−0.149040	−0.0745202	+	0.997220i	$0.523743\pi$
$140$	0	0
$141$	−350.440	−0.209308
$142$	0	0
$143$	55.2531	0.0323112
$144$	0	0
$145$	850.514	0.487113
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1326.35	−0.729255	−0.364628	−	0.931153i	$-0.618804\pi$
$149$	−1326.35	−0.729255	−0.364628	+	0.931153i	$0.618804\pi$
$150$	0	0
$151$	−406.949	−0.219318	−0.109659	−	0.993969i	$-0.534976\pi$
$151$	−406.949	−0.219318	−0.109659	+	0.993969i	$0.534976\pi$
$152$	0	0
$153$	246.206	0.130095
$154$	0	0
$155$	693.102	0.359170
$156$	0	0
$157$	3135.93	1.59410	0.797052	−	0.603911i	$-0.206392\pi$
$157$	3135.93	1.59410	0.797052	+	0.603911i	$0.206392\pi$
$158$	0	0
$159$	−258.913	−0.129139
$160$	0	0
$161$	0	0
$162$	0	0
$163$	130.980	0.0629394	0.0314697	−	0.999505i	$-0.489981\pi$
$163$	130.980	0.0629394	0.0314697	+	0.999505i	$0.489981\pi$
$164$	0	0
$165$	57.0428	0.0269138
$166$	0	0
$167$	1787.80	0.828407	0.414204	−	0.910184i	$-0.364060\pi$
$167$	1787.80	0.828407	0.414204	+	0.910184i	$0.364060\pi$
$168$	0	0
$169$	−694.213	−0.315982
$170$	0	0
$171$	589.046	0.263424
$172$	0	0
$173$	−609.165	−0.267711	−0.133855	−	0.991001i	$-0.542736\pi$
$173$	−609.165	−0.267711	−0.133855	+	0.991001i	$0.542736\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1142.68	0.485247
$178$	0	0
$179$	−1488.76	−0.621649	−0.310825	−	0.950467i	$-0.600605\pi$
$179$	−1488.76	−0.621649	−0.310825	+	0.950467i	$0.600605\pi$
$180$	0	0
$181$	452.765	0.185932	0.0929661	−	0.995669i	$-0.470365\pi$
$181$	452.765	0.185932	0.0929661	+	0.995669i	$0.470365\pi$
$182$	0	0
$183$	598.872	0.241912
$184$	0	0
$185$	4475.66	1.77869
$186$	0	0
$187$	38.9909	0.0152476
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2723.03	1.03158	0.515789	−	0.856716i	$-0.327499\pi$
$191$	2723.03	1.03158	0.515789	+	0.856716i	$0.327499\pi$
$192$	0	0
$193$	2577.69	0.961380	0.480690	−	0.876891i	$-0.340386\pi$
$193$	2577.69	0.961380	0.480690	+	0.876891i	$0.340386\pi$
$194$	0	0
$195$	1551.46	0.569757
$196$	0	0
$197$	−1433.02	−0.518266	−0.259133	−	0.965842i	$-0.583437\pi$
$197$	−1433.02	−0.518266	−0.259133	+	0.965842i	$0.583437\pi$
$198$	0	0
$199$	−1771.08	−0.630897	−0.315448	−	0.948943i	$-0.602155\pi$
$199$	−1771.08	−0.630897	−0.315448	+	0.948943i	$0.602155\pi$
$200$	0	0
$201$	2855.82	1.00216
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5970.41	−2.03411
$206$	0	0
$207$	−22.8899	−0.00768578
$208$	0	0
$209$	93.2856	0.0308742
$210$	0	0
$211$	5366.25	1.75084	0.875421	−	0.483362i	$-0.160584\pi$
$211$	5366.25	1.75084	0.875421	+	0.483362i	$0.160584\pi$
$212$	0	0
$213$	2491.58	0.801504
$214$	0	0
$215$	−2272.40	−0.720822
$216$	0	0
$217$	0	0
$218$	0	0
$219$	998.259	0.308019
$220$	0	0
$221$	1060.48	0.322787
$222$	0	0
$223$	−2827.33	−0.849023	−0.424511	−	0.905423i	$-0.639554\pi$
$223$	−2827.33	−0.849023	−0.424511	+	0.905423i	$0.639554\pi$
$224$	0	0
$225$	476.718	0.141250
$226$	0	0
$227$	3565.64	1.04255	0.521277	−	0.853387i	$-0.325456\pi$
$227$	3565.64	1.04255	0.521277	+	0.853387i	$0.325456\pi$
$228$	0	0
$229$	−4743.28	−1.36876	−0.684378	−	0.729128i	$-0.739926\pi$
$229$	−4743.28	−1.36876	−0.684378	+	0.729128i	$0.739926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3471.07	0.975955	0.487978	−	0.872856i	$-0.337735\pi$
$233$	3471.07	0.975955	0.487978	+	0.872856i	$0.337735\pi$
$234$	0	0
$235$	−1558.35	−0.432576
$236$	0	0
$237$	2267.99	0.621610
$238$	0	0
$239$	6089.40	1.64808	0.824039	−	0.566533i	$-0.191716\pi$
$239$	6089.40	1.64808	0.824039	+	0.566533i	$0.191716\pi$
$240$	0	0
$241$	4936.25	1.31938	0.659692	−	0.751536i	$-0.270687\pi$
$241$	4936.25	1.31938	0.659692	+	0.751536i	$0.270687\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2537.20	0.653596
$248$	0	0
$249$	−45.6784	−0.0116255
$250$	0	0
$251$	−1263.24	−0.317668	−0.158834	−	0.987305i	$-0.550774\pi$
$251$	−1263.24	−0.317668	−0.158834	+	0.987305i	$0.550774\pi$
$252$	0	0
$253$	−3.62501	−0.000900800 0
$254$	0	0
$255$	1094.83	0.268867
$256$	0	0
$257$	6152.93	1.49342	0.746711	−	0.665149i	$-0.231632\pi$
$257$	6152.93	1.49342	0.746711	+	0.665149i	$0.231632\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−573.789	−0.136079
$262$	0	0
$263$	−4184.54	−0.981102	−0.490551	−	0.871412i	$-0.663204\pi$
$263$	−4184.54	−0.981102	−0.490551	+	0.871412i	$0.663204\pi$
$264$	0	0
$265$	−1151.34	−0.266892
$266$	0	0
$267$	−4662.34	−1.06865
$268$	0	0
$269$	718.978	0.162962	0.0814812	−	0.996675i	$-0.474035\pi$
$269$	718.978	0.162962	0.0814812	+	0.996675i	$0.474035\pi$
$270$	0	0
$271$	1294.42	0.290150	0.145075	−	0.989421i	$-0.453658\pi$
$271$	1294.42	0.290150	0.145075	+	0.989421i	$0.453658\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	75.4965	0.0165549
$276$	0	0
$277$	−2320.92	−0.503431	−0.251716	−	0.967801i	$-0.580995\pi$
$277$	−2320.92	−0.503431	−0.251716	+	0.967801i	$0.580995\pi$
$278$	0	0
$279$	−467.593	−0.100337
$280$	0	0
$281$	6565.23	1.39377	0.696884	−	0.717184i	$-0.254569\pi$
$281$	6565.23	1.39377	0.696884	+	0.717184i	$0.254569\pi$
$282$	0	0
$283$	3728.49	0.783165	0.391583	−	0.920143i	$-0.371928\pi$
$283$	3728.49	0.783165	0.391583	+	0.920143i	$0.371928\pi$
$284$	0	0
$285$	2619.39	0.544418
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4164.64	−0.847678
$290$	0	0
$291$	−303.616	−0.0611626
$292$	0	0
$293$	269.734	0.0537817	0.0268909	−	0.999638i	$-0.491439\pi$
$293$	269.734	0.0537817	0.0268909	+	0.999638i	$0.491439\pi$
$294$	0	0
$295$	5081.28	1.00286
$296$	0	0
$297$	−38.4833	−0.00751860
$298$	0	0
$299$	−98.5938	−0.0190697
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4317.73	−0.818638
$304$	0	0
$305$	2663.08	0.499959
$306$	0	0
$307$	−3151.93	−0.585961	−0.292980	−	0.956118i	$-0.594647\pi$
$307$	−3151.93	−0.585961	−0.292980	+	0.956118i	$0.594647\pi$
$308$	0	0
$309$	4815.95	0.886633
$310$	0	0
$311$	8137.23	1.48366	0.741832	−	0.670585i	$-0.233957\pi$
$311$	8137.23	1.48366	0.741832	+	0.670585i	$0.233957\pi$
$312$	0	0
$313$	3975.26	0.717875	0.358938	−	0.933362i	$-0.383139\pi$
$313$	3975.26	0.717875	0.358938	+	0.933362i	$0.383139\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6079.93	1.07723	0.538617	−	0.842551i	$-0.318947\pi$
$317$	6079.93	1.07723	0.538617	+	0.842551i	$0.318947\pi$
$318$	0	0
$319$	−90.8695	−0.0159489
$320$	0	0
$321$	−924.210	−0.160699
$322$	0	0
$323$	1790.45	0.308431
$324$	0	0
$325$	2053.37	0.350463
$326$	0	0
$327$	−2126.88	−0.359684
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7579.85	−1.25869	−0.629345	−	0.777126i	$-0.716677\pi$
$331$	−7579.85	−1.25869	−0.629345	+	0.777126i	$0.716677\pi$
$332$	0	0
$333$	−3019.45	−0.496891
$334$	0	0
$335$	12699.3	2.07116
$336$	0	0
$337$	10921.5	1.76537	0.882685	−	0.469965i	$-0.155733\pi$
$337$	10921.5	1.76537	0.882685	+	0.469965i	$0.155733\pi$
$338$	0	0
$339$	−3322.45	−0.532303
$340$	0	0
$341$	−74.0515	−0.0117599
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−101.787	−0.0158842
$346$	0	0
$347$	−8410.73	−1.30119	−0.650593	−	0.759426i	$-0.725480\pi$
$347$	−8410.73	−1.30119	−0.650593	+	0.759426i	$0.725480\pi$
$348$	0	0
$349$	12309.3	1.88797	0.943985	−	0.329988i	$-0.107045\pi$
$349$	12309.3	1.88797	0.943985	+	0.329988i	$0.107045\pi$
$350$	0	0
$351$	−1046.68	−0.159167
$352$	0	0
$353$	−4136.83	−0.623742	−0.311871	−	0.950124i	$-0.600956\pi$
$353$	−4136.83	−0.623742	−0.311871	+	0.950124i	$0.600956\pi$
$354$	0	0
$355$	11079.6	1.65647
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3539.56	−0.520365	−0.260182	−	0.965559i	$-0.583783\pi$
$359$	−3539.56	−0.520365	−0.260182	+	0.965559i	$0.583783\pi$
$360$	0	0
$361$	−2575.36	−0.375472
$362$	0	0
$363$	3986.91	0.576469
$364$	0	0
$365$	4439.09	0.636582
$366$	0	0
$367$	−7873.94	−1.11994	−0.559968	−	0.828514i	$-0.689186\pi$
$367$	−7873.94	−1.11994	−0.559968	+	0.828514i	$0.689186\pi$
$368$	0	0
$369$	4027.87	0.568245
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7059.21	−0.979925	−0.489963	−	0.871743i	$-0.662990\pi$
$373$	−7059.21	−0.979925	−0.489963	+	0.871743i	$0.662990\pi$
$374$	0	0
$375$	−2882.80	−0.396979
$376$	0	0
$377$	−2471.49	−0.337634
$378$	0	0
$379$	7629.57	1.03405	0.517025	−	0.855971i	$-0.327040\pi$
$379$	7629.57	1.03405	0.517025	+	0.855971i	$0.327040\pi$
$380$	0	0
$381$	−6990.81	−0.940027
$382$	0	0
$383$	−533.655	−0.0711971	−0.0355985	−	0.999366i	$-0.511334\pi$
$383$	−533.655	−0.0711971	−0.0355985	+	0.999366i	$0.511334\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1533.05	0.201368
$388$	0	0
$389$	−5716.76	−0.745119	−0.372559	−	0.928008i	$-0.621520\pi$
$389$	−5716.76	−0.745119	−0.372559	+	0.928008i	$0.621520\pi$
$390$	0	0
$391$	−69.5755	−0.00899894
$392$	0	0
$393$	7484.04	0.960610
$394$	0	0
$395$	10085.4	1.28468
$396$	0	0
$397$	−7873.77	−0.995398	−0.497699	−	0.867350i	$-0.665822\pi$
$397$	−7873.77	−0.995398	−0.497699	+	0.867350i	$0.665822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	619.386	0.0771338	0.0385669	−	0.999256i	$-0.487721\pi$
$401$	619.386	0.0771338	0.0385669	+	0.999256i	$0.487721\pi$
$402$	0	0
$403$	−2014.07	−0.248953
$404$	0	0
$405$	−1080.58	−0.132579
$406$	0	0
$407$	−478.182	−0.0582374
$408$	0	0
$409$	−135.639	−0.0163983	−0.00819916	−	0.999966i	$-0.502610\pi$
$409$	−135.639	−0.0163983	−0.00819916	+	0.999966i	$0.502610\pi$
$410$	0	0
$411$	−780.807	−0.0937089
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−203.124	−0.0240264
$416$	0	0
$417$	732.736	0.0860486
$418$	0	0
$419$	−4903.63	−0.571738	−0.285869	−	0.958269i	$-0.592282\pi$
$419$	−4903.63	−0.571738	−0.285869	+	0.958269i	$0.592282\pi$
$420$	0	0
$421$	−3294.67	−0.381407	−0.190704	−	0.981648i	$-0.561077\pi$
$421$	−3294.67	−0.381407	−0.190704	+	0.981648i	$0.561077\pi$
$422$	0	0
$423$	1051.32	0.120844
$424$	0	0
$425$	1449.02	0.165383
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−165.759	−0.0186549
$430$	0	0
$431$	9838.69	1.09957	0.549783	−	0.835308i	$-0.314710\pi$
$431$	9838.69	1.09957	0.549783	+	0.835308i	$0.314710\pi$
$432$	0	0
$433$	8862.06	0.983564	0.491782	−	0.870718i	$-0.336346\pi$
$433$	8862.06	0.983564	0.491782	+	0.870718i	$0.336346\pi$
$434$	0	0
$435$	−2551.54	−0.281235
$436$	0	0
$437$	−166.459	−0.0182215
$438$	0	0
$439$	14282.7	1.55280	0.776399	−	0.630242i	$-0.217044\pi$
$439$	14282.7	1.55280	0.776399	+	0.630242i	$0.217044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9833.36	−1.05462	−0.527310	−	0.849673i	$-0.676799\pi$
$443$	−9833.36	−1.05462	−0.527310	+	0.849673i	$0.676799\pi$
$444$	0	0
$445$	−20732.6	−2.20858
$446$	0	0
$447$	3979.06	0.421036
$448$	0	0
$449$	7326.49	0.770063	0.385032	−	0.922903i	$-0.374190\pi$
$449$	7326.49	0.770063	0.385032	+	0.922903i	$0.374190\pi$
$450$	0	0
$451$	637.882	0.0666002
$452$	0	0
$453$	1220.85	0.126624
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1533.54	−0.156971	−0.0784856	−	0.996915i	$-0.525008\pi$
$457$	−1533.54	−0.156971	−0.0784856	+	0.996915i	$0.525008\pi$
$458$	0	0
$459$	−738.617	−0.0751104
$460$	0	0
$461$	−1528.92	−0.154466	−0.0772330	−	0.997013i	$-0.524609\pi$
$461$	−1528.92	−0.154466	−0.0772330	+	0.997013i	$0.524609\pi$
$462$	0	0
$463$	8497.20	0.852912	0.426456	−	0.904508i	$-0.359762\pi$
$463$	8497.20	0.852912	0.426456	+	0.904508i	$0.359762\pi$
$464$	0	0
$465$	−2079.31	−0.207367
$466$	0	0
$467$	8349.59	0.827351	0.413676	−	0.910424i	$-0.364245\pi$
$467$	8349.59	0.827351	0.413676	+	0.910424i	$0.364245\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−9407.78	−0.920356
$472$	0	0
$473$	242.785	0.0236010
$474$	0	0
$475$	3466.77	0.334877
$476$	0	0
$477$	776.738	0.0745585
$478$	0	0
$479$	9408.01	0.897417	0.448709	−	0.893678i	$-0.351884\pi$
$479$	9408.01	0.897417	0.448709	+	0.893678i	$0.351884\pi$
$480$	0	0
$481$	−13005.7	−1.23287
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1350.13	−0.126405
$486$	0	0
$487$	9034.53	0.840644	0.420322	−	0.907375i	$-0.361917\pi$
$487$	9034.53	0.840644	0.420322	+	0.907375i	$0.361917\pi$
$488$	0	0
$489$	−392.939	−0.0363381
$490$	0	0
$491$	−3232.81	−0.297138	−0.148569	−	0.988902i	$-0.547467\pi$
$491$	−3232.81	−0.297138	−0.148569	+	0.988902i	$0.547467\pi$
$492$	0	0
$493$	−1744.08	−0.159329
$494$	0	0
$495$	−171.129	−0.0155387
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9718.05	0.871823	0.435912	−	0.899989i	$-0.356426\pi$
$499$	9718.05	0.871823	0.435912	+	0.899989i	$0.356426\pi$
$500$	0	0
$501$	−5363.40	−0.478281
$502$	0	0
$503$	3056.84	0.270970	0.135485	−	0.990779i	$-0.456741\pi$
$503$	3056.84	0.270970	0.135485	+	0.990779i	$0.456741\pi$
$504$	0	0
$505$	−19200.2	−1.69188
$506$	0	0
$507$	2082.64	0.182432
$508$	0	0
$509$	−8692.26	−0.756931	−0.378465	−	0.925615i	$-0.623548\pi$
$509$	−8692.26	−0.756931	−0.378465	+	0.925615i	$0.623548\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1767.14	−0.152088
$514$	0	0
$515$	21415.7	1.83240
$516$	0	0
$517$	166.495	0.0141633
$518$	0	0
$519$	1827.49	0.154563
$520$	0	0
$521$	15864.8	1.33407	0.667036	−	0.745025i	$-0.267563\pi$
$521$	15864.8	1.33407	0.667036	+	0.745025i	$0.267563\pi$
$522$	0	0
$523$	23437.5	1.95956	0.979779	−	0.200085i	$-0.0641218\pi$
$523$	23437.5	1.95956	0.979779	+	0.200085i	$0.0641218\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1421.28	−0.117480
$528$	0	0
$529$	−12160.5	−0.999468
$530$	0	0
$531$	−3428.03	−0.280158
$532$	0	0
$533$	17349.3	1.40991
$534$	0	0
$535$	−4109.80	−0.332116
$536$	0	0
$537$	4466.28	0.358909
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7285.87	0.579009	0.289505	−	0.957177i	$-0.406509\pi$
$541$	7285.87	0.579009	0.289505	+	0.957177i	$0.406509\pi$
$542$	0	0
$543$	−1358.29	−0.107348
$544$	0	0
$545$	−9457.87	−0.743359
$546$	0	0
$547$	13089.8	1.02318	0.511589	−	0.859230i	$-0.329057\pi$
$547$	13089.8	1.02318	0.511589	+	0.859230i	$0.329057\pi$
$548$	0	0
$549$	−1796.62	−0.139668
$550$	0	0
$551$	−4172.69	−0.322618
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13427.0	−1.02693
$556$	0	0
$557$	−11167.2	−0.849495	−0.424747	−	0.905312i	$-0.639637\pi$
$557$	−11167.2	−0.849495	−0.424747	+	0.905312i	$0.639637\pi$
$558$	0	0
$559$	6603.32	0.499626
$560$	0	0
$561$	−116.973	−0.00880320
$562$	0	0
$563$	−839.230	−0.0628230	−0.0314115	−	0.999507i	$-0.510000\pi$
$563$	−839.230	−0.0628230	−0.0314115	+	0.999507i	$0.510000\pi$
$564$	0	0
$565$	−14774.4	−1.10011
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5207.13	0.383645	0.191823	−	0.981430i	$-0.438560\pi$
$569$	5207.13	0.383645	0.191823	+	0.981430i	$0.438560\pi$
$570$	0	0
$571$	−19877.8	−1.45685	−0.728425	−	0.685125i	$-0.759747\pi$
$571$	−19877.8	−1.45685	−0.728425	+	0.685125i	$0.759747\pi$
$572$	0	0
$573$	−8169.09	−0.595582
$574$	0	0
$575$	−134.716	−0.00977052
$576$	0	0
$577$	2014.01	0.145311	0.0726553	−	0.997357i	$-0.476853\pi$
$577$	2014.01	0.145311	0.0726553	+	0.997357i	$0.476853\pi$
$578$	0	0
$579$	−7733.08	−0.555053
$580$	0	0
$581$	0	0
$582$	0	0
$583$	123.010	0.00873851
$584$	0	0
$585$	−4654.39	−0.328949
$586$	0	0
$587$	−12502.2	−0.879085	−0.439542	−	0.898222i	$-0.644859\pi$
$587$	−12502.2	−0.879085	−0.439542	+	0.898222i	$0.644859\pi$
$588$	0	0
$589$	−3400.41	−0.237881
$590$	0	0
$591$	4299.06	0.299221
$592$	0	0
$593$	3322.15	0.230058	0.115029	−	0.993362i	$-0.463304\pi$
$593$	3322.15	0.230058	0.115029	+	0.993362i	$0.463304\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5313.24	0.364248
$598$	0	0
$599$	13057.6	0.890683	0.445342	−	0.895361i	$-0.353082\pi$
$599$	13057.6	0.890683	0.445342	+	0.895361i	$0.353082\pi$
$600$	0	0
$601$	15364.1	1.04278	0.521392	−	0.853317i	$-0.325413\pi$
$601$	15364.1	1.04278	0.521392	+	0.853317i	$0.325413\pi$
$602$	0	0
$603$	−8567.45	−0.578596
$604$	0	0
$605$	17729.1	1.19139
$606$	0	0
$607$	17502.3	1.17034	0.585170	−	0.810911i	$-0.301028\pi$
$607$	17502.3	1.17034	0.585170	+	0.810911i	$0.301028\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4528.36	0.299833
$612$	0	0
$613$	−8668.35	−0.571144	−0.285572	−	0.958357i	$-0.592184\pi$
$613$	−8668.35	−0.571144	−0.285572	+	0.958357i	$0.592184\pi$
$614$	0	0
$615$	17911.2	1.17439
$616$	0	0
$617$	13110.8	0.855462	0.427731	−	0.903906i	$-0.359313\pi$
$617$	13110.8	0.855462	0.427731	+	0.903906i	$0.359313\pi$
$618$	0	0
$619$	24444.2	1.58723	0.793615	−	0.608421i	$-0.208197\pi$
$619$	24444.2	1.58723	0.793615	+	0.608421i	$0.208197\pi$
$620$	0	0
$621$	68.6696	0.00443739
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19440.4	−1.24419
$626$	0	0
$627$	−279.857	−0.0178252
$628$	0	0
$629$	−9177.84	−0.581788
$630$	0	0
$631$	−18458.3	−1.16452	−0.582262	−	0.813002i	$-0.697832\pi$
$631$	−18458.3	−1.16452	−0.582262	+	0.813002i	$0.697832\pi$
$632$	0	0
$633$	−16098.7	−1.01085
$634$	0	0
$635$	−31087.0	−1.94275
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7474.74	−0.462748
$640$	0	0
$641$	8862.99	0.546126	0.273063	−	0.961996i	$-0.411963\pi$
$641$	8862.99	0.546126	0.273063	+	0.961996i	$0.411963\pi$
$642$	0	0
$643$	27011.4	1.65665	0.828324	−	0.560249i	$-0.189294\pi$
$643$	27011.4	1.65665	0.828324	+	0.560249i	$0.189294\pi$
$644$	0	0
$645$	6817.21	0.416167
$646$	0	0
$647$	28930.3	1.75791	0.878954	−	0.476906i	$-0.158242\pi$
$647$	28930.3	1.75791	0.878954	+	0.476906i	$0.158242\pi$
$648$	0	0
$649$	−542.888	−0.0328354
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23824.8	1.42778	0.713888	−	0.700260i	$-0.246933\pi$
$653$	23824.8	1.42778	0.713888	+	0.700260i	$0.246933\pi$
$654$	0	0
$655$	33280.2	1.98529
$656$	0	0
$657$	−2994.78	−0.177835
$658$	0	0
$659$	−19306.5	−1.14124	−0.570618	−	0.821216i	$-0.693296\pi$
$659$	−19306.5	−1.14124	−0.570618	+	0.821216i	$0.693296\pi$
$660$	0	0
$661$	−6414.99	−0.377480	−0.188740	−	0.982027i	$-0.560440\pi$
$661$	−6414.99	−0.377480	−0.188740	+	0.982027i	$0.560440\pi$
$662$	0	0
$663$	−3181.45	−0.186361
$664$	0	0
$665$	0	0
$666$	0	0
$667$	162.148	0.00941287
$668$	0	0
$669$	8481.99	0.490184
$670$	0	0
$671$	−284.525	−0.0163696
$672$	0	0
$673$	−24090.1	−1.37980	−0.689899	−	0.723906i	$-0.742345\pi$
$673$	−24090.1	−1.37980	−0.689899	+	0.723906i	$0.742345\pi$
$674$	0	0
$675$	−1430.15	−0.0815505
$676$	0	0
$677$	−18477.5	−1.04897	−0.524483	−	0.851421i	$-0.675741\pi$
$677$	−18477.5	−1.04897	−0.524483	+	0.851421i	$0.675741\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−10696.9	−0.601919
$682$	0	0
$683$	12404.8	0.694961	0.347480	−	0.937687i	$-0.387037\pi$
$683$	12404.8	0.694961	0.347480	+	0.937687i	$0.387037\pi$
$684$	0	0
$685$	−3472.11	−0.193668
$686$	0	0
$687$	14229.9	0.790251
$688$	0	0
$689$	3345.65	0.184992
$690$	0	0
$691$	−3690.71	−0.203186	−0.101593	−	0.994826i	$-0.532394\pi$
$691$	−3690.71	−0.203186	−0.101593	+	0.994826i	$0.532394\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3258.35	0.177837
$696$	0	0
$697$	12243.0	0.665332
$698$	0	0
$699$	−10413.2	−0.563468
$700$	0	0
$701$	30121.5	1.62293	0.811465	−	0.584401i	$-0.198671\pi$
$701$	30121.5	1.62293	0.811465	+	0.584401i	$0.198671\pi$
$702$	0	0
$703$	−21957.9	−1.17804
$704$	0	0
$705$	4675.04	0.249748
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36821.1	1.95042	0.975209	−	0.221287i	$-0.0710259\pi$
$709$	36821.1	1.95042	0.975209	+	0.221287i	$0.0710259\pi$
$710$	0	0
$711$	−6803.96	−0.358887
$712$	0	0
$713$	132.138	0.00694052
$714$	0	0
$715$	−737.104	−0.0385540
$716$	0	0
$717$	−18268.2	−0.951518
$718$	0	0
$719$	−29581.9	−1.53438	−0.767188	−	0.641422i	$-0.778345\pi$
$719$	−29581.9	−1.53438	−0.767188	+	0.641422i	$0.778345\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14808.7	−0.761747
$724$	0	0
$725$	−3376.98	−0.172990
$726$	0	0
$727$	−22956.0	−1.17110	−0.585551	−	0.810636i	$-0.699122\pi$
$727$	−22956.0	−1.17110	−0.585551	+	0.810636i	$0.699122\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	4659.82	0.235773
$732$	0	0
$733$	−10773.6	−0.542880	−0.271440	−	0.962455i	$-0.587500\pi$
$733$	−10773.6	−0.542880	−0.271440	+	0.962455i	$0.587500\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1356.80	−0.0678135
$738$	0	0
$739$	−2440.70	−0.121492	−0.0607459	−	0.998153i	$-0.519348\pi$
$739$	−2440.70	−0.121492	−0.0607459	+	0.998153i	$0.519348\pi$
$740$	0	0
$741$	−7611.61	−0.377354
$742$	0	0
$743$	−13502.7	−0.666712	−0.333356	−	0.942801i	$-0.608181\pi$
$743$	−13502.7	−0.666712	−0.333356	+	0.942801i	$0.608181\pi$
$744$	0	0
$745$	17694.2	0.870154
$746$	0	0
$747$	137.035	0.00671199
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22874.0	1.11143	0.555716	−	0.831373i	$-0.312444\pi$
$751$	22874.0	1.11143	0.555716	+	0.831373i	$0.312444\pi$
$752$	0	0
$753$	3789.71	0.183406
$754$	0	0
$755$	5428.90	0.261693
$756$	0	0
$757$	−3770.12	−0.181014	−0.0905069	−	0.995896i	$-0.528849\pi$
$757$	−3770.12	−0.181014	−0.0905069	+	0.995896i	$0.528849\pi$
$758$	0	0
$759$	10.8750	0.000520077 0
$760$	0	0
$761$	−23825.4	−1.13491	−0.567456	−	0.823404i	$-0.692072\pi$
$761$	−23825.4	−1.13491	−0.567456	+	0.823404i	$0.692072\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3284.50	−0.155231
$766$	0	0
$767$	−14765.6	−0.695116
$768$	0	0
$769$	15661.3	0.734408	0.367204	−	0.930140i	$-0.380315\pi$
$769$	15661.3	0.734408	0.367204	+	0.930140i	$0.380315\pi$
$770$	0	0
$771$	−18458.8	−0.862227
$772$	0	0
$773$	−5265.53	−0.245004	−0.122502	−	0.992468i	$-0.539092\pi$
$773$	−5265.53	−0.245004	−0.122502	+	0.992468i	$0.539092\pi$
$774$	0	0
$775$	−2751.97	−0.127553
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29291.3	1.34720
$780$	0	0
$781$	−1183.75	−0.0542357
$782$	0	0
$783$	1721.37	0.0785653
$784$	0	0
$785$	−41834.8	−1.90210
$786$	0	0
$787$	−5288.51	−0.239536	−0.119768	−	0.992802i	$-0.538215\pi$
$787$	−5288.51	−0.239536	−0.119768	+	0.992802i	$0.538215\pi$
$788$	0	0
$789$	12553.6	0.566440
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7738.58	−0.346539
$794$	0	0
$795$	3454.02	0.154090
$796$	0	0
$797$	−19536.6	−0.868285	−0.434143	−	0.900844i	$-0.642949\pi$
$797$	−19536.6	−0.868285	−0.434143	+	0.900844i	$0.642949\pi$
$798$	0	0
$799$	3195.57	0.141491
$800$	0	0
$801$	13987.0	0.616987
$802$	0	0
$803$	−474.275	−0.0208428
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2156.93	−0.0940863
$808$	0	0
$809$	37594.9	1.63383	0.816913	−	0.576761i	$-0.195684\pi$
$809$	37594.9	1.63383	0.816913	+	0.576761i	$0.195684\pi$
$810$	0	0
$811$	−32059.6	−1.38812	−0.694060	−	0.719917i	$-0.744180\pi$
$811$	−32059.6	−1.38812	−0.694060	+	0.719917i	$0.744180\pi$
$812$	0	0
$813$	−3883.27	−0.167518
$814$	0	0
$815$	−1747.33	−0.0750998
$816$	0	0
$817$	11148.6	0.477405
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37886.9	1.61055	0.805276	−	0.592900i	$-0.202017\pi$
$821$	37886.9	1.61055	0.805276	+	0.592900i	$0.202017\pi$
$822$	0	0
$823$	−33341.7	−1.41217	−0.706087	−	0.708125i	$-0.749541\pi$
$823$	−33341.7	−1.41217	−0.706087	+	0.708125i	$0.749541\pi$
$824$	0	0
$825$	−226.490	−0.00955800
$826$	0	0
$827$	−15589.5	−0.655501	−0.327750	−	0.944764i	$-0.606290\pi$
$827$	−15589.5	−0.655501	−0.327750	+	0.944764i	$0.606290\pi$
$828$	0	0
$829$	−34075.0	−1.42759	−0.713796	−	0.700353i	$-0.753026\pi$
$829$	−34075.0	−1.42759	−0.713796	+	0.700353i	$0.753026\pi$
$830$	0	0
$831$	6962.75	0.290656
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−23850.1	−0.988464
$836$	0	0
$837$	1402.78	0.0579297
$838$	0	0
$839$	−29579.3	−1.21715	−0.608577	−	0.793495i	$-0.708259\pi$
$839$	−29579.3	−1.21715	−0.608577	+	0.793495i	$0.708259\pi$
$840$	0	0
$841$	−20324.4	−0.833342
$842$	0	0
$843$	−19695.7	−0.804692
$844$	0	0
$845$	9261.14	0.377033
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−11185.5	−0.452161
$850$	0	0
$851$	853.269	0.0343710
$852$	0	0
$853$	22610.0	0.907563	0.453782	−	0.891113i	$-0.350075\pi$
$853$	22610.0	0.907563	0.453782	+	0.891113i	$0.350075\pi$
$854$	0	0
$855$	−7858.16	−0.314320
$856$	0	0
$857$	23598.3	0.940610	0.470305	−	0.882504i	$-0.344144\pi$
$857$	23598.3	0.940610	0.470305	+	0.882504i	$0.344144\pi$
$858$	0	0
$859$	21212.4	0.842558	0.421279	−	0.906931i	$-0.361581\pi$
$859$	21212.4	0.842558	0.421279	+	0.906931i	$0.361581\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7004.76	0.276298	0.138149	−	0.990411i	$-0.455885\pi$
$863$	7004.76	0.276298	0.138149	+	0.990411i	$0.455885\pi$
$864$	0	0
$865$	8126.55	0.319435
$866$	0	0
$867$	12493.9	0.489407
$868$	0	0
$869$	−1077.53	−0.0420628
$870$	0	0
$871$	−36902.7	−1.43559
$872$	0	0
$873$	910.849	0.0353122
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−47506.2	−1.82916	−0.914578	−	0.404411i	$-0.867477\pi$
$877$	−47506.2	−1.82916	−0.914578	+	0.404411i	$0.867477\pi$
$878$	0	0
$879$	−809.203	−0.0310509
$880$	0	0
$881$	3901.92	0.149216	0.0746079	−	0.997213i	$-0.476229\pi$
$881$	3901.92	0.149216	0.0746079	+	0.997213i	$0.476229\pi$
$882$	0	0
$883$	18715.1	0.713266	0.356633	−	0.934245i	$-0.383925\pi$
$883$	18715.1	0.713266	0.356633	+	0.934245i	$0.383925\pi$
$884$	0	0
$885$	−15243.9	−0.579002
$886$	0	0
$887$	−2727.69	−0.103255	−0.0516274	−	0.998666i	$-0.516441\pi$
$887$	−2727.69	−0.103255	−0.0516274	+	0.998666i	$0.516441\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	115.450	0.00434087
$892$	0	0
$893$	7645.37	0.286498
$894$	0	0
$895$	19860.8	0.741758
$896$	0	0
$897$	295.781	0.0110099
$898$	0	0
$899$	3312.34	0.122884
$900$	0	0
$901$	2360.95	0.0872972
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6040.10	−0.221856
$906$	0	0
$907$	−17941.6	−0.656825	−0.328412	−	0.944534i	$-0.606514\pi$
$907$	−17941.6	−0.656825	−0.328412	+	0.944534i	$0.606514\pi$
$908$	0	0
$909$	12953.2	0.472641
$910$	0	0
$911$	35439.6	1.28887	0.644437	−	0.764657i	$-0.277092\pi$
$911$	35439.6	1.28887	0.644437	+	0.764657i	$0.277092\pi$
$912$	0	0
$913$	21.7019	0.000786669 0
$914$	0	0
$915$	−7989.24	−0.288652
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17329.3	0.622023	0.311012	−	0.950406i	$-0.399332\pi$
$919$	17329.3	0.622023	0.311012	+	0.950406i	$0.399332\pi$
$920$	0	0
$921$	9455.78	0.338305
$922$	0	0
$923$	−32196.0	−1.14815
$924$	0	0
$925$	−17770.7	−0.631672
$926$	0	0
$927$	−14447.8	−0.511898
$928$	0	0
$929$	18347.2	0.647959	0.323979	−	0.946064i	$-0.394979\pi$
$929$	18347.2	0.647959	0.323979	+	0.946064i	$0.394979\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24411.7	−0.856594
$934$	0	0
$935$	−520.158	−0.0181936
$936$	0	0
$937$	11020.9	0.384245	0.192123	−	0.981371i	$-0.438463\pi$
$937$	11020.9	0.384245	0.192123	+	0.981371i	$0.438463\pi$
$938$	0	0
$939$	−11925.8	−0.414465
$940$	0	0
$941$	−35859.7	−1.24229	−0.621144	−	0.783697i	$-0.713332\pi$
$941$	−35859.7	−1.24229	−0.621144	+	0.783697i	$0.713332\pi$
$942$	0	0
$943$	−1138.24	−0.0393066
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41681.1	1.43026	0.715129	−	0.698992i	$-0.246368\pi$
$947$	41681.1	1.43026	0.715129	+	0.698992i	$0.246368\pi$
$948$	0	0
$949$	−12899.4	−0.441237
$950$	0	0
$951$	−18239.8	−0.621941
$952$	0	0
$953$	−51773.4	−1.75982	−0.879908	−	0.475143i	$-0.842396\pi$
$953$	−51773.4	−1.75982	−0.879908	+	0.475143i	$0.842396\pi$
$954$	0	0
$955$	−36326.5	−1.23089
$956$	0	0
$957$	272.608	0.00920813
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27091.7	−0.909392
$962$	0	0
$963$	2772.63	0.0927796
$964$	0	0
$965$	−34387.7	−1.14713
$966$	0	0
$967$	−55592.6	−1.84874	−0.924372	−	0.381491i	$-0.875411\pi$
$967$	−55592.6	−1.84874	−0.924372	+	0.381491i	$0.875411\pi$
$968$	0	0
$969$	−5371.34	−0.178073
$970$	0	0
$971$	22136.6	0.731613	0.365807	−	0.930691i	$-0.380793\pi$
$971$	22136.6	0.731613	0.365807	+	0.930691i	$0.380793\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6160.11	−0.202340
$976$	0	0
$977$	40501.1	1.32625	0.663125	−	0.748509i	$-0.269230\pi$
$977$	40501.1	1.32625	0.663125	+	0.748509i	$0.269230\pi$
$978$	0	0
$979$	2215.09	0.0723130
$980$	0	0
$981$	6380.64	0.207664
$982$	0	0
$983$	−11785.9	−0.382414	−0.191207	−	0.981550i	$-0.561240\pi$
$983$	−11785.9	−0.382414	−0.191207	+	0.981550i	$0.561240\pi$
$984$	0	0
$985$	19117.2	0.618400
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−433.226	−0.0139290
$990$	0	0
$991$	8009.47	0.256740	0.128370	−	0.991726i	$-0.459026\pi$
$991$	8009.47	0.256740	0.128370	+	0.991726i	$0.459026\pi$
$992$	0	0
$993$	22739.6	0.726705
$994$	0	0
$995$	23627.1	0.752792
$996$	0	0
$997$	−22894.7	−0.727265	−0.363633	−	0.931542i	$-0.618464\pi$
$997$	−22894.7	−0.727265	−0.363633	+	0.931542i	$0.618464\pi$
$998$	0	0
$999$	9058.35	0.286880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cn.1.1		4
4.3	odd	2		1176.4.a.be.1.1	yes	4
7.6	odd	2		2352.4.a.co.1.4		4
28.27	even	2		1176.4.a.z.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.z.1.4	✓	4	28.27	even	2
1176.4.a.be.1.1	yes	4	4.3	odd	2
2352.4.a.cn.1.1		4	1.1	even	1	trivial
2352.4.a.co.1.4		4	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -13.3405 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -13.3405 q^{5} +9.00000 q^{9} +1.42531 q^{11} +38.7658 q^{13} +40.0215 q^{15} +27.3562 q^{17} +65.4495 q^{19} -2.54332 q^{23} +52.9686 q^{25} -27.0000 q^{27} -63.7544 q^{29} -51.9548 q^{31} -4.27592 q^{33} -335.494 q^{37} -116.297 q^{39} +447.541 q^{41} +170.339 q^{43} -120.064 q^{45} +116.813 q^{47} -82.0685 q^{51} +86.3042 q^{53} -19.0143 q^{55} -196.349 q^{57} -380.892 q^{59} -199.624 q^{61} -517.155 q^{65} -951.939 q^{67} +7.62996 q^{69} -830.527 q^{71} -332.753 q^{73} -158.906 q^{75} -755.996 q^{79} +81.0000 q^{81} +15.2261 q^{83} -364.945 q^{85} +191.263 q^{87} +1554.11 q^{89} +155.864 q^{93} -873.128 q^{95} +101.205 q^{97} +12.8278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9	\( 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9} - 40 q^{11} + 48 q^{13} - 24 q^{15} + 72 q^{17} - 32 q^{19} - 8 q^{23} + 164 q^{25} - 108 q^{27} + 144 q^{29} - 48 q^{31} + 120 q^{33} + 48 q^{37} - 144 q^{39} + 72 q^{41} - 512 q^{43} + 72 q^{45} - 160 q^{47} - 216 q^{51} + 536 q^{53} + 336 q^{55} + 96 q^{57} - 240 q^{59} + 896 q^{61} - 136 q^{65} - 1088 q^{67} + 24 q^{69} - 1288 q^{71} + 1488 q^{73} - 492 q^{75} - 416 q^{79} + 324 q^{81} + 112 q^{83} - 1512 q^{85} - 432 q^{87} + 3160 q^{89} + 144 q^{93} + 240 q^{95} + 2384 q^{97} - 360 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9 - 40 * q^11 + 48 * q^13 - 24 * q^15 + 72 * q^17 - 32 * q^19 - 8 * q^23 + 164 * q^25 - 108 * q^27 + 144 * q^29 - 48 * q^31 + 120 * q^33 + 48 * q^37 - 144 * q^39 + 72 * q^41 - 512 * q^43 + 72 * q^45 - 160 * q^47 - 216 * q^51 + 536 * q^53 + 336 * q^55 + 96 * q^57 - 240 * q^59 + 896 * q^61 - 136 * q^65 - 1088 * q^67 + 24 * q^69 - 1288 * q^71 + 1488 * q^73 - 492 * q^75 - 416 * q^79 + 324 * q^81 + 112 * q^83 - 1512 * q^85 - 432 * q^87 + 3160 * q^89 + 144 * q^93 + 240 * q^95 + 2384 * q^97 - 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more