LMFDB - Embedded newform 2352.4.a.co.1.4

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:	$4$
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			1.4
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	$-0.296516\pi$
$5$	13.3405	1.19321	0.596605	+	0.802535i	$0.296516\pi$
$6$	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	$-0.635703\pi$
$13$	−38.7658	−0.827054	−0.413527	+	0.910492i	$0.635703\pi$
$14$	0	0
$15$	40.0215	0.688900
$16$	0	0
$17$	−27.3562	−0.390285	−0.195143	−	0.980775i	$-0.562517\pi$
$17$	−27.3562	−0.390285	−0.195143	+	0.980775i	$0.562517\pi$
$18$	0	0
$19$	−65.4495	−0.790271	−0.395135	−	0.918623i	$-0.629302\pi$
$19$	−65.4495	−0.790271	−0.395135	+	0.918623i	$0.629302\pi$
$20$	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	$-0.451910\pi$
$31$	51.9548	0.301011	0.150506	+	0.988609i	$0.451910\pi$
$32$	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	$-0.824834\pi$
$41$	−447.541	−1.70473	−0.852367	+	0.522944i	$0.824834\pi$
$42$	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	$-0.558019\pi$
$47$	−116.813	−0.362531	−0.181266	+	0.983434i	$0.558019\pi$
$48$	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	$-0.361947\pi$
$59$	380.892	0.840473	0.420236	+	0.907415i	$0.361947\pi$
$60$	0	0
$61$	199.624	0.419004	0.209502	−	0.977808i	$-0.432816\pi$
$61$	199.624	0.419004	0.209502	+	0.977808i	$0.432816\pi$
$62$	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	$-0.414050\pi$
$73$	332.753	0.533504	0.266752	+	0.963765i	$0.414050\pi$
$74$	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	$-0.503205\pi$
$83$	−15.2261	−0.0201360	−0.0100680	+	0.999949i	$0.503205\pi$
$84$	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	$-0.876339\pi$
$89$	−1554.11	−1.85096	−0.925481	+	0.378795i	$0.876339\pi$
$90$	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	$-0.516868\pi$
$97$	−101.205	−0.105937	−0.0529683	+	0.998596i	$0.516868\pi$
$98$	0	0
$99$	12.8278	0.0130226
$100$	0	0
$101$	−1439.24	−1.41792	−0.708961	−	0.705248i	$-0.750836\pi$
$101$	−1439.24	−1.41792	−0.708961	+	0.705248i	$0.750836\pi$
$102$	0	0
$103$	1605.32	1.53569	0.767847	−	0.640633i	$-0.221328\pi$
$103$	1605.32	1.53569	0.767847	+	0.640633i	$0.221328\pi$
$104$	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	$-0.187246\pi$
$131$	2494.68	1.66383	0.831913	+	0.554907i	$0.187246\pi$
$132$	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	$-0.476257\pi$
$139$	244.245	0.149040	0.0745202	+	0.997220i	$0.476257\pi$
$140$	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	$-0.793608\pi$
$157$	−3135.93	−1.59410	−0.797052	+	0.603911i	$0.793608\pi$
$158$	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	$-0.635940\pi$
$167$	−1787.80	−0.828407	−0.414204	+	0.910184i	$0.635940\pi$
$168$	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	$-0.457264\pi$
$173$	609.165	0.267711	0.133855	+	0.991001i	$0.457264\pi$
$174$	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	$-0.529635\pi$
$181$	−452.765	−0.185932	−0.0929661	+	0.995669i	$0.529635\pi$
$182$	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	$-0.397845\pi$
$199$	1771.08	0.630897	0.315448	+	0.948943i	$0.397845\pi$
$200$	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	$-0.360446\pi$
$223$	2827.33	0.849023	0.424511	+	0.905423i	$0.360446\pi$
$224$	0	0
$225$	476.718	0.141250
$226$	0	0
$227$	−3565.64	−1.04255	−0.521277	−	0.853387i	$-0.674544\pi$
$227$	−3565.64	−1.04255	−0.521277	+	0.853387i	$0.674544\pi$
$228$	0	0
$229$	4743.28	1.36876	0.684378	−	0.729128i	$-0.260074\pi$
$229$	4743.28	1.36876	0.684378	+	0.729128i	$0.260074\pi$
$230$	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	$-0.729313\pi$
$241$	−4936.25	−1.31938	−0.659692	+	0.751536i	$0.729313\pi$
$242$	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	$-0.449226\pi$
$251$	1263.24	0.317668	0.158834	+	0.987305i	$0.449226\pi$
$252$	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	$-0.768368\pi$
$257$	−6152.93	−1.49342	−0.746711	+	0.665149i	$0.768368\pi$
$258$	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	$-0.525965\pi$
$269$	−718.978	−0.162962	−0.0814812	+	0.996675i	$0.525965\pi$
$270$	0	0
$271$	−1294.42	−0.290150	−0.145075	−	0.989421i	$-0.546342\pi$
$271$	−1294.42	−0.290150	−0.145075	+	0.989421i	$0.546342\pi$
$272$	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	$-0.628072\pi$
$283$	−3728.49	−0.783165	−0.391583	+	0.920143i	$0.628072\pi$
$284$	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	$-0.508561\pi$
$293$	−269.734	−0.0537817	−0.0268909	+	0.999638i	$0.508561\pi$
$294$	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	$-0.405353\pi$
$307$	3151.93	0.585961	0.292980	+	0.956118i	$0.405353\pi$
$308$	0	0
$309$	4815.95	0.886633
$310$	0	0
$311$	−8137.23	−1.48366	−0.741832	−	0.670585i	$-0.766043\pi$
$311$	−8137.23	−1.48366	−0.741832	+	0.670585i	$0.766043\pi$
$312$	0	0
$313$	−3975.26	−0.717875	−0.358938	−	0.933362i	$-0.616861\pi$
$313$	−3975.26	−0.717875	−0.358938	+	0.933362i	$0.616861\pi$
$314$	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	$-0.892955\pi$
$349$	−12309.3	−1.88797	−0.943985	+	0.329988i	$0.892955\pi$
$350$	0	0
$351$	−1046.68	−0.159167
$352$	0	0
$353$	4136.83	0.623742	0.311871	−	0.950124i	$-0.399044\pi$
$353$	4136.83	0.623742	0.311871	+	0.950124i	$0.399044\pi$
$354$	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	$-0.310814\pi$
$367$	7873.94	1.11994	0.559968	+	0.828514i	$0.310814\pi$
$368$	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	$-0.488666\pi$
$383$	533.655	0.0711971	0.0355985	+	0.999366i	$0.488666\pi$
$384$	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	$-0.334178\pi$
$397$	7873.77	0.995398	0.497699	+	0.867350i	$0.334178\pi$
$398$	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	$-0.497390\pi$
$409$	135.639	0.0163983	0.00819916	+	0.999966i	$0.497390\pi$
$410$	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	$-0.407718\pi$
$419$	4903.63	0.571738	0.285869	+	0.958269i	$0.407718\pi$
$420$	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	$-0.663654\pi$
$433$	−8862.06	−0.983564	−0.491782	+	0.870718i	$0.663654\pi$
$434$	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	$-0.782956\pi$
$439$	−14282.7	−1.55280	−0.776399	+	0.630242i	$0.782956\pi$
$440$	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	$-0.475391\pi$
$461$	1528.92	0.154466	0.0772330	+	0.997013i	$0.475391\pi$
$462$	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	$-0.635755\pi$
$467$	−8349.59	−0.827351	−0.413676	+	0.910424i	$0.635755\pi$
$468$	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	$-0.648116\pi$
$479$	−9408.01	−0.897417	−0.448709	+	0.893678i	$0.648116\pi$
$480$	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	$-0.543259\pi$
$503$	−3056.84	−0.270970	−0.135485	+	0.990779i	$0.543259\pi$
$504$	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	$-0.376452\pi$
$509$	8692.26	0.756931	0.378465	+	0.925615i	$0.376452\pi$
$510$	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	$-0.732437\pi$
$521$	−15864.8	−1.33407	−0.667036	+	0.745025i	$0.732437\pi$
$522$	0	0
$523$	−23437.5	−1.95956	−0.979779	−	0.200085i	$-0.935878\pi$
$523$	−23437.5	−1.95956	−0.979779	+	0.200085i	$0.935878\pi$
$524$	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	$-0.490000\pi$
$563$	839.230	0.0628230	0.0314115	+	0.999507i	$0.490000\pi$
$564$	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	$-0.523147\pi$
$577$	−2014.01	−0.145311	−0.0726553	+	0.997357i	$0.523147\pi$
$578$	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	$-0.355141\pi$
$587$	12502.2	0.879085	0.439542	+	0.898222i	$0.355141\pi$
$588$	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	$-0.536696\pi$
$593$	−3322.15	−0.230058	−0.115029	+	0.993362i	$0.536696\pi$
$594$	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	$-0.674587\pi$
$601$	−15364.1	−1.04278	−0.521392	+	0.853317i	$0.674587\pi$
$602$	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	$-0.698972\pi$
$607$	−17502.3	−1.17034	−0.585170	+	0.810911i	$0.698972\pi$
$608$	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	$-0.791803\pi$
$619$	−24444.2	−1.58723	−0.793615	+	0.608421i	$0.791803\pi$
$620$	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	$-0.810706\pi$
$643$	−27011.4	−1.65665	−0.828324	+	0.560249i	$0.810706\pi$
$644$	0	0
$645$	6817.21	0.416167
$646$	0	0
$647$	−28930.3	−1.75791	−0.878954	−	0.476906i	$-0.841758\pi$
$647$	−28930.3	−1.75791	−0.878954	+	0.476906i	$0.841758\pi$
$648$	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	$-0.439560\pi$
$661$	6414.99	0.377480	0.188740	+	0.982027i	$0.439560\pi$
$662$	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	$-0.324259\pi$
$677$	18477.5	1.04897	0.524483	+	0.851421i	$0.324259\pi$
$678$	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	$-0.467606\pi$
$691$	3690.71	0.203186	0.101593	+	0.994826i	$0.467606\pi$
$692$	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	$-0.221655\pi$
$719$	29581.9	1.53438	0.767188	+	0.641422i	$0.221655\pi$
$720$	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	$-0.300878\pi$
$727$	22956.0	1.17110	0.585551	+	0.810636i	$0.300878\pi$
$728$	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	$-0.412500\pi$
$733$	10773.6	0.542880	0.271440	+	0.962455i	$0.412500\pi$
$734$	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	$-0.307928\pi$
$761$	23825.4	1.13491	0.567456	+	0.823404i	$0.307928\pi$
$762$	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	$-0.619685\pi$
$769$	−15661.3	−0.734408	−0.367204	+	0.930140i	$0.619685\pi$
$770$	0	0
$771$	−18458.8	−0.862227
$772$	0	0
$773$	5265.53	0.245004	0.122502	−	0.992468i	$-0.460908\pi$
$773$	5265.53	0.245004	0.122502	+	0.992468i	$0.460908\pi$
$774$	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	$-0.461785\pi$
$787$	5288.51	0.239536	0.119768	+	0.992802i	$0.461785\pi$
$788$	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	$-0.357051\pi$
$797$	19536.6	0.868285	0.434143	+	0.900844i	$0.357051\pi$
$798$	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	$-0.255820\pi$
$811$	32059.6	1.38812	0.694060	+	0.719917i	$0.255820\pi$
$812$	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	$-0.246974\pi$
$829$	34075.0	1.42759	0.713796	+	0.700353i	$0.246974\pi$
$830$	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	$-0.291741\pi$
$839$	29579.3	1.21715	0.608577	+	0.793495i	$0.291741\pi$
$840$	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	$-0.649925\pi$
$853$	−22610.0	−0.907563	−0.453782	+	0.891113i	$0.649925\pi$
$854$	0	0
$855$	−7858.16	−0.314320
$856$	0	0
$857$	−23598.3	−0.940610	−0.470305	−	0.882504i	$-0.655856\pi$
$857$	−23598.3	−0.940610	−0.470305	+	0.882504i	$0.655856\pi$
$858$	0	0
$859$	−21212.4	−0.842558	−0.421279	−	0.906931i	$-0.638419\pi$
$859$	−21212.4	−0.842558	−0.421279	+	0.906931i	$0.638419\pi$
$860$	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	$-0.523771\pi$
$881$	−3901.92	−0.149216	−0.0746079	+	0.997213i	$0.523771\pi$
$882$	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	$-0.483559\pi$
$887$	2727.69	0.103255	0.0516274	+	0.998666i	$0.483559\pi$
$888$	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	$-0.605021\pi$
$929$	−18347.2	−0.647959	−0.323979	+	0.946064i	$0.605021\pi$
$930$	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	$-0.561537\pi$
$937$	−11020.9	−0.384245	−0.192123	+	0.981371i	$0.561537\pi$
$938$	0	0
$939$	−11925.8	−0.414465
$940$	0	0
$941$	35859.7	1.24229	0.621144	−	0.783697i	$-0.286668\pi$
$941$	35859.7	1.24229	0.621144	+	0.783697i	$0.286668\pi$
$942$	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	$-0.619207\pi$
$971$	−22136.6	−0.731613	−0.365807	+	0.930691i	$0.619207\pi$
$972$	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	$-0.438760\pi$
$983$	11785.9	0.382414	0.191207	+	0.981550i	$0.438760\pi$
$984$	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	$-0.381536\pi$
$997$	22894.7	0.727265	0.363633	+	0.931542i	$0.381536\pi$
$998$	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.co.1.4		4
4.3	odd	2		1176.4.a.z.1.4	✓	4
7.6	odd	2		2352.4.a.cn.1.1		4
28.27	even	2		1176.4.a.be.1.1	yes	4

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

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

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