LMFDB - Embedded newform 2352.4.a.ck.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$138.772492334$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.145408.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} + 142 $ x^4 - 24*x^2 + 142
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.25358$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -6.95987 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -6.95987 q^{5} +9.00000 q^{9} -43.9023 q^{11} -83.5699 q^{13} +20.8796 q^{15} +10.4354 q^{17} -4.27642 q^{19} -160.077 q^{23} -76.5601 q^{25} -27.0000 q^{27} +9.93138 q^{29} +133.715 q^{31} +131.707 q^{33} -357.606 q^{37} +250.710 q^{39} -127.174 q^{41} -343.250 q^{43} -62.6389 q^{45} -77.4628 q^{47} -31.3063 q^{51} -460.812 q^{53} +305.555 q^{55} +12.8293 q^{57} -272.735 q^{59} -51.3874 q^{61} +581.636 q^{65} +327.081 q^{67} +480.231 q^{69} +571.877 q^{71} +206.313 q^{73} +229.680 q^{75} -923.918 q^{79} +81.0000 q^{81} +1105.73 q^{83} -72.6293 q^{85} -29.7941 q^{87} -1532.57 q^{89} -401.146 q^{93} +29.7634 q^{95} +97.6059 q^{97} -395.121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 152 q^{17} + 224 q^{19} + 8 q^{23} - 28 q^{25} - 108 q^{27} - 144 q^{29} + 400 q^{31} - 120 q^{33} - 304 q^{37} + 144 q^{39} + 152 q^{41} - 160 q^{43} - 72 q^{45} + 544 q^{47} - 456 q^{51} - 1320 q^{53} + 16 q^{55} - 672 q^{57} + 1040 q^{59} - 896 q^{61} - 648 q^{65} + 416 q^{67} - 24 q^{69} - 248 q^{71} + 752 q^{73} + 84 q^{75} - 864 q^{79} + 324 q^{81} + 1456 q^{83} - 1608 q^{85} + 432 q^{87} - 2936 q^{89} - 1200 q^{93} + 80 q^{95} - 144 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 + 152 * q^17 + 224 * q^19 + 8 * q^23 - 28 * q^25 - 108 * q^27 - 144 * q^29 + 400 * q^31 - 120 * q^33 - 304 * q^37 + 144 * q^39 + 152 * q^41 - 160 * q^43 - 72 * q^45 + 544 * q^47 - 456 * q^51 - 1320 * q^53 + 16 * q^55 - 672 * q^57 + 1040 * q^59 - 896 * q^61 - 648 * q^65 + 416 * q^67 - 24 * q^69 - 248 * q^71 + 752 * q^73 + 84 * q^75 - 864 * q^79 + 324 * q^81 + 1456 * q^83 - 1608 * q^85 + 432 * q^87 - 2936 * q^89 - 1200 * q^93 + 80 * q^95 - 144 * 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
$5$	−6.95987	−0.622510	−0.311255	−	0.950326i	$-0.600749\pi$
$5$	−6.95987	−0.622510	−0.311255	+	0.950326i	$0.600749\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−43.9023	−1.20337	−0.601684	−	0.798734i	$-0.705503\pi$
$11$	−43.9023	−1.20337	−0.601684	+	0.798734i	$0.705503\pi$
$12$	0	0
$13$	−83.5699	−1.78293	−0.891466	−	0.453088i	$-0.850322\pi$
$13$	−83.5699	−1.78293	−0.891466	+	0.453088i	$0.850322\pi$
$14$	0	0
$15$	20.8796	0.359406
$16$	0	0
$17$	10.4354	0.148880	0.0744401	−	0.997225i	$-0.476283\pi$
$17$	10.4354	0.148880	0.0744401	+	0.997225i	$0.476283\pi$
$18$	0	0
$19$	−4.27642	−0.0516357	−0.0258179	−	0.999667i	$-0.508219\pi$
$19$	−4.27642	−0.0516357	−0.0258179	+	0.999667i	$0.508219\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−160.077	−1.45123	−0.725616	−	0.688100i	$-0.758445\pi$
$23$	−160.077	−1.45123	−0.725616	+	0.688100i	$0.758445\pi$
$24$	0	0
$25$	−76.5601	−0.612481
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	9.93138	0.0635935	0.0317967	−	0.999494i	$-0.489877\pi$
$29$	9.93138	0.0635935	0.0317967	+	0.999494i	$0.489877\pi$
$30$	0	0
$31$	133.715	0.774710	0.387355	−	0.921931i	$-0.373389\pi$
$31$	133.715	0.774710	0.387355	+	0.921931i	$0.373389\pi$
$32$	0	0
$33$	131.707	0.694765
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−357.606	−1.58892	−0.794460	−	0.607316i	$-0.792246\pi$
$37$	−357.606	−1.58892	−0.794460	+	0.607316i	$0.792246\pi$
$38$	0	0
$39$	250.710	1.02938
$40$	0	0
$41$	−127.174	−0.484420	−0.242210	−	0.970224i	$-0.577872\pi$
$41$	−127.174	−0.484420	−0.242210	+	0.970224i	$0.577872\pi$
$42$	0	0
$43$	−343.250	−1.21733	−0.608664	−	0.793428i	$-0.708294\pi$
$43$	−343.250	−1.21733	−0.608664	+	0.793428i	$0.708294\pi$
$44$	0	0
$45$	−62.6389	−0.207503
$46$	0	0
$47$	−77.4628	−0.240406	−0.120203	−	0.992749i	$-0.538355\pi$
$47$	−77.4628	−0.240406	−0.120203	+	0.992749i	$0.538355\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−31.3063	−0.0859560
$52$	0	0
$53$	−460.812	−1.19429	−0.597145	−	0.802133i	$-0.703698\pi$
$53$	−460.812	−1.19429	−0.597145	+	0.802133i	$0.703698\pi$
$54$	0	0
$55$	305.555	0.749109
$56$	0	0
$57$	12.8293	0.0298119
$58$	0	0
$59$	−272.735	−0.601816	−0.300908	−	0.953653i	$-0.597290\pi$
$59$	−272.735	−0.601816	−0.300908	+	0.953653i	$0.597290\pi$
$60$	0	0
$61$	−51.3874	−0.107860	−0.0539302	−	0.998545i	$-0.517175\pi$
$61$	−51.3874	−0.107860	−0.0539302	+	0.998545i	$0.517175\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	581.636	1.10989
$66$	0	0
$67$	327.081	0.596407	0.298204	−	0.954502i	$-0.403613\pi$
$67$	327.081	0.596407	0.298204	+	0.954502i	$0.403613\pi$
$68$	0	0
$69$	480.231	0.837869
$70$	0	0
$71$	571.877	0.955905	0.477953	−	0.878386i	$-0.341379\pi$
$71$	571.877	0.955905	0.477953	+	0.878386i	$0.341379\pi$
$72$	0	0
$73$	206.313	0.330783	0.165391	−	0.986228i	$-0.447111\pi$
$73$	206.313	0.330783	0.165391	+	0.986228i	$0.447111\pi$
$74$	0	0
$75$	229.680	0.353616
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−923.918	−1.31581	−0.657904	−	0.753102i	$-0.728557\pi$
$79$	−923.918	−1.31581	−0.657904	+	0.753102i	$0.728557\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1105.73	1.46229	0.731143	−	0.682224i	$-0.238987\pi$
$83$	1105.73	1.46229	0.731143	+	0.682224i	$0.238987\pi$
$84$	0	0
$85$	−72.6293	−0.0926794
$86$	0	0
$87$	−29.7941	−0.0367157
$88$	0	0
$89$	−1532.57	−1.82531	−0.912655	−	0.408731i	$-0.865971\pi$
$89$	−1532.57	−1.82531	−0.912655	+	0.408731i	$0.865971\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−401.146	−0.447279
$94$	0	0
$95$	29.7634	0.0321438
$96$	0	0
$97$	97.6059	0.102169	0.0510844	−	0.998694i	$-0.483732\pi$
$97$	97.6059	0.102169	0.0510844	+	0.998694i	$0.483732\pi$
$98$	0	0
$99$	−395.121	−0.401123
$100$	0	0
$101$	−1326.39	−1.30674	−0.653368	−	0.757040i	$-0.726645\pi$
$101$	−1326.39	−1.30674	−0.653368	+	0.757040i	$0.726645\pi$
$102$	0	0
$103$	780.354	0.746511	0.373255	−	0.927729i	$-0.378242\pi$
$103$	780.354	0.746511	0.373255	+	0.927729i	$0.378242\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	436.263	0.394160	0.197080	−	0.980387i	$-0.436854\pi$
$107$	436.263	0.394160	0.197080	+	0.980387i	$0.436854\pi$
$108$	0	0
$109$	−2092.32	−1.83861	−0.919304	−	0.393547i	$-0.871248\pi$
$109$	−2092.32	−1.83861	−0.919304	+	0.393547i	$0.871248\pi$
$110$	0	0
$111$	1072.82	0.917363
$112$	0	0
$113$	129.323	0.107661	0.0538303	−	0.998550i	$-0.482857\pi$
$113$	129.323	0.107661	0.0538303	+	0.998550i	$0.482857\pi$
$114$	0	0
$115$	1114.12	0.903407
$116$	0	0
$117$	−752.129	−0.594311
$118$	0	0
$119$	0	0
$120$	0	0
$121$	596.415	0.448096
$122$	0	0
$123$	381.522	0.279680
$124$	0	0
$125$	1402.83	1.00379
$126$	0	0
$127$	−392.568	−0.274289	−0.137145	−	0.990551i	$-0.543793\pi$
$127$	−392.568	−0.274289	−0.137145	+	0.990551i	$0.543793\pi$
$128$	0	0
$129$	1029.75	0.702824
$130$	0	0
$131$	952.085	0.634993	0.317496	−	0.948259i	$-0.397158\pi$
$131$	952.085	0.634993	0.317496	+	0.948259i	$0.397158\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	187.917	0.119802
$136$	0	0
$137$	−1850.66	−1.15411	−0.577053	−	0.816706i	$-0.695798\pi$
$137$	−1850.66	−1.15411	−0.577053	+	0.816706i	$0.695798\pi$
$138$	0	0
$139$	−1874.96	−1.14412	−0.572059	−	0.820213i	$-0.693855\pi$
$139$	−1874.96	−1.14412	−0.572059	+	0.820213i	$0.693855\pi$
$140$	0	0
$141$	232.388	0.138799
$142$	0	0
$143$	3668.91	2.14552
$144$	0	0
$145$	−69.1211	−0.0395876
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−956.162	−0.525717	−0.262859	−	0.964834i	$-0.584665\pi$
$149$	−956.162	−0.525717	−0.262859	+	0.964834i	$0.584665\pi$
$150$	0	0
$151$	512.503	0.276205	0.138102	−	0.990418i	$-0.455900\pi$
$151$	512.503	0.276205	0.138102	+	0.990418i	$0.455900\pi$
$152$	0	0
$153$	93.9189	0.0496267
$154$	0	0
$155$	−930.643	−0.482265
$156$	0	0
$157$	573.762	0.291663	0.145832	−	0.989309i	$-0.453414\pi$
$157$	573.762	0.291663	0.145832	+	0.989309i	$0.453414\pi$
$158$	0	0
$159$	1382.44	0.689524
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3889.78	−1.86915	−0.934573	−	0.355771i	$-0.884218\pi$
$163$	−3889.78	−1.86915	−0.934573	+	0.355771i	$0.884218\pi$
$164$	0	0
$165$	−916.664	−0.432498
$166$	0	0
$167$	−167.287	−0.0775154	−0.0387577	−	0.999249i	$-0.512340\pi$
$167$	−167.287	−0.0775154	−0.0387577	+	0.999249i	$0.512340\pi$
$168$	0	0
$169$	4786.92	2.17885
$170$	0	0
$171$	−38.4878	−0.0172119
$172$	0	0
$173$	−2559.23	−1.12471	−0.562355	−	0.826896i	$-0.690104\pi$
$173$	−2559.23	−1.12471	−0.562355	+	0.826896i	$0.690104\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	818.206	0.347458
$178$	0	0
$179$	1214.50	0.507130	0.253565	−	0.967318i	$-0.418397\pi$
$179$	1214.50	0.507130	0.253565	+	0.967318i	$0.418397\pi$
$180$	0	0
$181$	−2555.03	−1.04925	−0.524625	−	0.851334i	$-0.675794\pi$
$181$	−2555.03	−1.04925	−0.524625	+	0.851334i	$0.675794\pi$
$182$	0	0
$183$	154.162	0.0622732
$184$	0	0
$185$	2488.89	0.989119
$186$	0	0
$187$	−458.140	−0.179158
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3256.32	−1.23361	−0.616804	−	0.787117i	$-0.711573\pi$
$191$	−3256.32	−1.23361	−0.616804	+	0.787117i	$0.711573\pi$
$192$	0	0
$193$	96.0749	0.0358323	0.0179161	−	0.999839i	$-0.494297\pi$
$193$	96.0749	0.0358323	0.0179161	+	0.999839i	$0.494297\pi$
$194$	0	0
$195$	−1744.91	−0.640797
$196$	0	0
$197$	1335.35	0.482941	0.241471	−	0.970408i	$-0.422370\pi$
$197$	1335.35	0.482941	0.241471	+	0.970408i	$0.422370\pi$
$198$	0	0
$199$	2907.06	1.03556	0.517778	−	0.855515i	$-0.326759\pi$
$199$	2907.06	1.03556	0.517778	+	0.855515i	$0.326759\pi$
$200$	0	0
$201$	−981.242	−0.344336
$202$	0	0
$203$	0	0
$204$	0	0
$205$	885.115	0.301557
$206$	0	0
$207$	−1440.69	−0.483744
$208$	0	0
$209$	187.745	0.0621368
$210$	0	0
$211$	−721.998	−0.235566	−0.117783	−	0.993039i	$-0.537579\pi$
$211$	−721.998	−0.235566	−0.117783	+	0.993039i	$0.537579\pi$
$212$	0	0
$213$	−1715.63	−0.551892
$214$	0	0
$215$	2388.98	0.757799
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−618.940	−0.190978
$220$	0	0
$221$	−872.088	−0.265443
$222$	0	0
$223$	5906.68	1.77373	0.886863	−	0.462033i	$-0.152880\pi$
$223$	5906.68	1.77373	0.886863	+	0.462033i	$0.152880\pi$
$224$	0	0
$225$	−689.041	−0.204160
$226$	0	0
$227$	4877.87	1.42624	0.713118	−	0.701044i	$-0.247283\pi$
$227$	4877.87	1.42624	0.713118	+	0.701044i	$0.247283\pi$
$228$	0	0
$229$	−4893.35	−1.41206	−0.706030	−	0.708182i	$-0.749516\pi$
$229$	−4893.35	−1.41206	−0.706030	+	0.708182i	$0.749516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3155.44	0.887208	0.443604	−	0.896223i	$-0.353700\pi$
$233$	3155.44	0.887208	0.443604	+	0.896223i	$0.353700\pi$
$234$	0	0
$235$	539.131	0.149655
$236$	0	0
$237$	2771.75	0.759682
$238$	0	0
$239$	4165.43	1.12736	0.563680	−	0.825993i	$-0.309385\pi$
$239$	4165.43	1.12736	0.563680	+	0.825993i	$0.309385\pi$
$240$	0	0
$241$	−6240.57	−1.66801	−0.834006	−	0.551756i	$-0.813958\pi$
$241$	−6240.57	−1.66801	−0.834006	+	0.551756i	$0.813958\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	357.380	0.0920630
$248$	0	0
$249$	−3317.19	−0.844252
$250$	0	0
$251$	387.084	0.0973407	0.0486703	−	0.998815i	$-0.484502\pi$
$251$	387.084	0.0973407	0.0486703	+	0.998815i	$0.484502\pi$
$252$	0	0
$253$	7027.75	1.74637
$254$	0	0
$255$	217.888	0.0535085
$256$	0	0
$257$	4498.89	1.09196	0.545979	−	0.837799i	$-0.316158\pi$
$257$	4498.89	1.09196	0.545979	+	0.837799i	$0.316158\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	89.3824	0.0211978
$262$	0	0
$263$	6791.57	1.59234	0.796171	−	0.605071i	$-0.206855\pi$
$263$	6791.57	1.59234	0.796171	+	0.605071i	$0.206855\pi$
$264$	0	0
$265$	3207.19	0.743458
$266$	0	0
$267$	4597.72	1.05384
$268$	0	0
$269$	8537.49	1.93509	0.967546	−	0.252694i	$-0.0813167\pi$
$269$	8537.49	1.93509	0.967546	+	0.252694i	$0.0813167\pi$
$270$	0	0
$271$	−4743.77	−1.06333	−0.531667	−	0.846953i	$-0.678434\pi$
$271$	−4743.77	−1.06333	−0.531667	+	0.846953i	$0.678434\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3361.17	0.737040
$276$	0	0
$277$	−1004.32	−0.217847	−0.108924	−	0.994050i	$-0.534740\pi$
$277$	−1004.32	−0.217847	−0.108924	+	0.994050i	$0.534740\pi$
$278$	0	0
$279$	1203.44	0.258237
$280$	0	0
$281$	4563.78	0.968869	0.484435	−	0.874828i	$-0.339025\pi$
$281$	4563.78	0.968869	0.484435	+	0.874828i	$0.339025\pi$
$282$	0	0
$283$	5558.25	1.16750	0.583752	−	0.811932i	$-0.301584\pi$
$283$	5558.25	1.16750	0.583752	+	0.811932i	$0.301584\pi$
$284$	0	0
$285$	−89.2901	−0.0185582
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4804.10	−0.977835
$290$	0	0
$291$	−292.818	−0.0589872
$292$	0	0
$293$	2903.23	0.578868	0.289434	−	0.957198i	$-0.406533\pi$
$293$	2903.23	0.578868	0.289434	+	0.957198i	$0.406533\pi$
$294$	0	0
$295$	1898.20	0.374636
$296$	0	0
$297$	1185.36	0.231588
$298$	0	0
$299$	13377.6	2.58745
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3979.16	0.754444
$304$	0	0
$305$	357.650	0.0671442
$306$	0	0
$307$	7974.96	1.48259	0.741295	−	0.671179i	$-0.234212\pi$
$307$	7974.96	1.48259	0.741295	+	0.671179i	$0.234212\pi$
$308$	0	0
$309$	−2341.06	−0.430998
$310$	0	0
$311$	−4744.75	−0.865114	−0.432557	−	0.901607i	$-0.642389\pi$
$311$	−4744.75	−0.865114	−0.432557	+	0.901607i	$0.642389\pi$
$312$	0	0
$313$	9872.97	1.78292	0.891459	−	0.453102i	$-0.149683\pi$
$313$	9872.97	1.78292	0.891459	+	0.453102i	$0.149683\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−311.027	−0.0551073	−0.0275537	−	0.999620i	$-0.508772\pi$
$317$	−311.027	−0.0551073	−0.0275537	+	0.999620i	$0.508772\pi$
$318$	0	0
$319$	−436.011	−0.0765263
$320$	0	0
$321$	−1308.79	−0.227568
$322$	0	0
$323$	−44.6263	−0.00768754
$324$	0	0
$325$	6398.12	1.09201
$326$	0	0
$327$	6276.97	1.06152
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7451.30	−1.23734	−0.618671	−	0.785650i	$-0.712329\pi$
$331$	−7451.30	−1.23734	−0.618671	+	0.785650i	$0.712329\pi$
$332$	0	0
$333$	−3218.45	−0.529640
$334$	0	0
$335$	−2276.44	−0.371269
$336$	0	0
$337$	4112.84	0.664808	0.332404	−	0.943137i	$-0.392140\pi$
$337$	4112.84	0.664808	0.332404	+	0.943137i	$0.392140\pi$
$338$	0	0
$339$	−387.968	−0.0621578
$340$	0	0
$341$	−5870.42	−0.932262
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3342.35	−0.521582
$346$	0	0
$347$	−11459.5	−1.77285	−0.886423	−	0.462876i	$-0.846817\pi$
$347$	−11459.5	−1.77285	−0.886423	+	0.462876i	$0.846817\pi$
$348$	0	0
$349$	−1293.03	−0.198322	−0.0991610	−	0.995071i	$-0.531616\pi$
$349$	−1293.03	−0.198322	−0.0991610	+	0.995071i	$0.531616\pi$
$350$	0	0
$351$	2256.39	0.343125
$352$	0	0
$353$	7265.01	1.09540	0.547702	−	0.836674i	$-0.315503\pi$
$353$	7265.01	1.09540	0.547702	+	0.836674i	$0.315503\pi$
$354$	0	0
$355$	−3980.19	−0.595061
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1381.80	0.203143	0.101572	−	0.994828i	$-0.467613\pi$
$359$	1381.80	0.203143	0.101572	+	0.994828i	$0.467613\pi$
$360$	0	0
$361$	−6840.71	−0.997334
$362$	0	0
$363$	−1789.25	−0.258708
$364$	0	0
$365$	−1435.91	−0.205916
$366$	0	0
$367$	−4502.44	−0.640397	−0.320198	−	0.947351i	$-0.603750\pi$
$367$	−4502.44	−0.640397	−0.320198	+	0.947351i	$0.603750\pi$
$368$	0	0
$369$	−1144.57	−0.161473
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11140.1	1.54642	0.773208	−	0.634152i	$-0.218651\pi$
$373$	11140.1	1.54642	0.773208	+	0.634152i	$0.218651\pi$
$374$	0	0
$375$	−4208.50	−0.579536
$376$	0	0
$377$	−829.964	−0.113383
$378$	0	0
$379$	13060.7	1.77015	0.885073	−	0.465453i	$-0.154108\pi$
$379$	13060.7	1.77015	0.885073	+	0.465453i	$0.154108\pi$
$380$	0	0
$381$	1177.70	0.158361
$382$	0	0
$383$	−11248.6	−1.50072	−0.750362	−	0.661027i	$-0.770121\pi$
$383$	−11248.6	−1.50072	−0.750362	+	0.661027i	$0.770121\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3089.25	−0.405776
$388$	0	0
$389$	−11873.8	−1.54763	−0.773813	−	0.633414i	$-0.781653\pi$
$389$	−11873.8	−1.54763	−0.773813	+	0.633414i	$0.781653\pi$
$390$	0	0
$391$	−1670.47	−0.216060
$392$	0	0
$393$	−2856.26	−0.366613
$394$	0	0
$395$	6430.35	0.819104
$396$	0	0
$397$	13921.3	1.75992	0.879960	−	0.475047i	$-0.157569\pi$
$397$	13921.3	1.75992	0.879960	+	0.475047i	$0.157569\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12830.2	−1.59778	−0.798891	−	0.601475i	$-0.794580\pi$
$401$	−12830.2	−1.59778	−0.798891	+	0.601475i	$0.794580\pi$
$402$	0	0
$403$	−11174.6	−1.38126
$404$	0	0
$405$	−563.750	−0.0691678
$406$	0	0
$407$	15699.7	1.91206
$408$	0	0
$409$	−5606.76	−0.677840	−0.338920	−	0.940815i	$-0.610062\pi$
$409$	−5606.76	−0.677840	−0.338920	+	0.940815i	$0.610062\pi$
$410$	0	0
$411$	5551.98	0.666324
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7695.75	−0.910288
$416$	0	0
$417$	5624.89	0.660556
$418$	0	0
$419$	−77.6276	−0.00905097	−0.00452549	−	0.999990i	$-0.501441\pi$
$419$	−77.6276	−0.00905097	−0.00452549	+	0.999990i	$0.501441\pi$
$420$	0	0
$421$	9742.15	1.12780	0.563900	−	0.825843i	$-0.309301\pi$
$421$	9742.15	1.12780	0.563900	+	0.825843i	$0.309301\pi$
$422$	0	0
$423$	−697.165	−0.0801355
$424$	0	0
$425$	−798.938	−0.0911863
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11006.7	−1.23872
$430$	0	0
$431$	1306.54	0.146018	0.0730092	−	0.997331i	$-0.476740\pi$
$431$	1306.54	0.146018	0.0730092	+	0.997331i	$0.476740\pi$
$432$	0	0
$433$	−4039.33	−0.448309	−0.224154	−	0.974554i	$-0.571962\pi$
$433$	−4039.33	−0.448309	−0.224154	+	0.974554i	$0.571962\pi$
$434$	0	0
$435$	207.363	0.0228559
$436$	0	0
$437$	684.557	0.0749355
$438$	0	0
$439$	58.3195	0.00634040	0.00317020	−	0.999995i	$-0.498991\pi$
$439$	58.3195	0.00634040	0.00317020	+	0.999995i	$0.498991\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16369.5	1.75561	0.877806	−	0.479016i	$-0.159006\pi$
$443$	16369.5	1.75561	0.877806	+	0.479016i	$0.159006\pi$
$444$	0	0
$445$	10666.5	1.13627
$446$	0	0
$447$	2868.49	0.303523
$448$	0	0
$449$	1584.33	0.166523	0.0832617	−	0.996528i	$-0.473466\pi$
$449$	1584.33	0.166523	0.0832617	+	0.996528i	$0.473466\pi$
$450$	0	0
$451$	5583.23	0.582936
$452$	0	0
$453$	−1537.51	−0.159467
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8530.39	−0.873162	−0.436581	−	0.899665i	$-0.643811\pi$
$457$	−8530.39	−0.873162	−0.436581	+	0.899665i	$0.643811\pi$
$458$	0	0
$459$	−281.757	−0.0286520
$460$	0	0
$461$	2593.77	0.262048	0.131024	−	0.991379i	$-0.458174\pi$
$461$	2593.77	0.262048	0.131024	+	0.991379i	$0.458174\pi$
$462$	0	0
$463$	−15296.9	−1.53543	−0.767717	−	0.640789i	$-0.778607\pi$
$463$	−15296.9	−1.53543	−0.767717	+	0.640789i	$0.778607\pi$
$464$	0	0
$465$	2791.93	0.278436
$466$	0	0
$467$	−13942.1	−1.38151	−0.690754	−	0.723090i	$-0.742721\pi$
$467$	−13942.1	−1.38151	−0.690754	+	0.723090i	$0.742721\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1721.28	−0.168392
$472$	0	0
$473$	15069.5	1.46489
$474$	0	0
$475$	327.404	0.0316259
$476$	0	0
$477$	−4147.31	−0.398097
$478$	0	0
$479$	1431.14	0.136515	0.0682574	−	0.997668i	$-0.478256\pi$
$479$	1431.14	0.136515	0.0682574	+	0.997668i	$0.478256\pi$
$480$	0	0
$481$	29885.1	2.83294
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−679.325	−0.0636012
$486$	0	0
$487$	−7664.42	−0.713158	−0.356579	−	0.934265i	$-0.616057\pi$
$487$	−7664.42	−0.713158	−0.356579	+	0.934265i	$0.616057\pi$
$488$	0	0
$489$	11669.3	1.07915
$490$	0	0
$491$	14761.3	1.35676	0.678380	−	0.734711i	$-0.262682\pi$
$491$	14761.3	1.35676	0.678380	+	0.734711i	$0.262682\pi$
$492$	0	0
$493$	103.638	0.00946781
$494$	0	0
$495$	2749.99	0.249703
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11606.1	−1.04121	−0.520604	−	0.853798i	$-0.674293\pi$
$499$	−11606.1	−1.04121	−0.520604	+	0.853798i	$0.674293\pi$
$500$	0	0
$501$	501.862	0.0447535
$502$	0	0
$503$	−1650.23	−0.146283	−0.0731414	−	0.997322i	$-0.523302\pi$
$503$	−1650.23	−0.146283	−0.0731414	+	0.997322i	$0.523302\pi$
$504$	0	0
$505$	9231.48	0.813456
$506$	0	0
$507$	−14360.8	−1.25796
$508$	0	0
$509$	−3728.12	−0.324649	−0.162324	−	0.986737i	$-0.551899\pi$
$509$	−3728.12	−0.324649	−0.162324	+	0.986737i	$0.551899\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	115.463	0.00993730
$514$	0	0
$515$	−5431.17	−0.464710
$516$	0	0
$517$	3400.80	0.289298
$518$	0	0
$519$	7677.69	0.649351
$520$	0	0
$521$	15803.9	1.32895	0.664473	−	0.747312i	$-0.268656\pi$
$521$	15803.9	1.32895	0.664473	+	0.747312i	$0.268656\pi$
$522$	0	0
$523$	−9235.54	−0.772164	−0.386082	−	0.922464i	$-0.626172\pi$
$523$	−9235.54	−0.772164	−0.386082	+	0.922464i	$0.626172\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1395.38	0.115339
$528$	0	0
$529$	13457.6	1.10607
$530$	0	0
$531$	−2454.62	−0.200605
$532$	0	0
$533$	10627.9	0.863688
$534$	0	0
$535$	−3036.34	−0.245369
$536$	0	0
$537$	−3643.51	−0.292792
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8399.16	−0.667483	−0.333741	−	0.942665i	$-0.608311\pi$
$541$	−8399.16	−0.667483	−0.333741	+	0.942665i	$0.608311\pi$
$542$	0	0
$543$	7665.10	0.605784
$544$	0	0
$545$	14562.3	1.14455
$546$	0	0
$547$	−18402.5	−1.43846	−0.719228	−	0.694774i	$-0.755504\pi$
$547$	−18402.5	−1.43846	−0.719228	+	0.694774i	$0.755504\pi$
$548$	0	0
$549$	−462.487	−0.0359535
$550$	0	0
$551$	−42.4708	−0.00328370
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7466.68	−0.571068
$556$	0	0
$557$	7803.37	0.593607	0.296804	−	0.954938i	$-0.404079\pi$
$557$	7803.37	0.593607	0.296804	+	0.954938i	$0.404079\pi$
$558$	0	0
$559$	28685.3	2.17041
$560$	0	0
$561$	1374.42	0.103437
$562$	0	0
$563$	708.972	0.0530721	0.0265361	−	0.999648i	$-0.491552\pi$
$563$	708.972	0.0530721	0.0265361	+	0.999648i	$0.491552\pi$
$564$	0	0
$565$	−900.069	−0.0670198
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15799.2	1.16403	0.582017	−	0.813176i	$-0.302264\pi$
$569$	15799.2	1.16403	0.582017	+	0.813176i	$0.302264\pi$
$570$	0	0
$571$	−22721.9	−1.66529	−0.832647	−	0.553804i	$-0.813176\pi$
$571$	−22721.9	−1.66529	−0.832647	+	0.553804i	$0.813176\pi$
$572$	0	0
$573$	9768.97	0.712224
$574$	0	0
$575$	12255.5	0.888852
$576$	0	0
$577$	19216.2	1.38645	0.693226	−	0.720721i	$-0.256189\pi$
$577$	19216.2	1.38645	0.693226	+	0.720721i	$0.256189\pi$
$578$	0	0
$579$	−288.225	−0.0206878
$580$	0	0
$581$	0	0
$582$	0	0
$583$	20230.7	1.43717
$584$	0	0
$585$	5234.72	0.369964
$586$	0	0
$587$	−19262.1	−1.35440	−0.677201	−	0.735798i	$-0.736807\pi$
$587$	−19262.1	−1.35440	−0.677201	+	0.735798i	$0.736807\pi$
$588$	0	0
$589$	−571.824	−0.0400027
$590$	0	0
$591$	−4006.04	−0.278826
$592$	0	0
$593$	−11871.8	−0.822122	−0.411061	−	0.911608i	$-0.634842\pi$
$593$	−11871.8	−0.822122	−0.411061	+	0.911608i	$0.634842\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8721.17	−0.597879
$598$	0	0
$599$	−17021.1	−1.16104	−0.580520	−	0.814246i	$-0.697151\pi$
$599$	−17021.1	−1.16104	−0.580520	+	0.814246i	$0.697151\pi$
$600$	0	0
$601$	−5518.23	−0.374531	−0.187266	−	0.982309i	$-0.559963\pi$
$601$	−5518.23	−0.374531	−0.187266	+	0.982309i	$0.559963\pi$
$602$	0	0
$603$	2943.73	0.198802
$604$	0	0
$605$	−4150.97	−0.278944
$606$	0	0
$607$	−20388.4	−1.36333	−0.681663	−	0.731667i	$-0.738743\pi$
$607$	−20388.4	−1.36333	−0.681663	+	0.731667i	$0.738743\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6473.55	0.428628
$612$	0	0
$613$	−14237.0	−0.938053	−0.469026	−	0.883184i	$-0.655395\pi$
$613$	−14237.0	−0.938053	−0.469026	+	0.883184i	$0.655395\pi$
$614$	0	0
$615$	−2655.34	−0.174104
$616$	0	0
$617$	−20383.4	−1.32999	−0.664997	−	0.746846i	$-0.731567\pi$
$617$	−20383.4	−1.32999	−0.664997	+	0.746846i	$0.731567\pi$
$618$	0	0
$619$	15997.5	1.03876	0.519380	−	0.854543i	$-0.326163\pi$
$619$	15997.5	1.03876	0.519380	+	0.854543i	$0.326163\pi$
$620$	0	0
$621$	4322.08	0.279290
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−193.527	−0.0123857
$626$	0	0
$627$	−563.235	−0.0358747
$628$	0	0
$629$	−3731.77	−0.236559
$630$	0	0
$631$	−4857.90	−0.306481	−0.153241	−	0.988189i	$-0.548971\pi$
$631$	−4857.90	−0.306481	−0.153241	+	0.988189i	$0.548971\pi$
$632$	0	0
$633$	2165.99	0.136004
$634$	0	0
$635$	2732.22	0.170748
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5146.89	0.318635
$640$	0	0
$641$	24563.9	1.51360	0.756798	−	0.653649i	$-0.226763\pi$
$641$	24563.9	1.51360	0.756798	+	0.653649i	$0.226763\pi$
$642$	0	0
$643$	−6010.79	−0.368650	−0.184325	−	0.982865i	$-0.559010\pi$
$643$	−6010.79	−0.368650	−0.184325	+	0.982865i	$0.559010\pi$
$644$	0	0
$645$	−7166.93	−0.437515
$646$	0	0
$647$	16927.4	1.02857	0.514285	−	0.857619i	$-0.328057\pi$
$647$	16927.4	1.02857	0.514285	+	0.857619i	$0.328057\pi$
$648$	0	0
$649$	11973.7	0.724206
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7702.96	0.461623	0.230812	−	0.972998i	$-0.425862\pi$
$653$	7702.96	0.461623	0.230812	+	0.972998i	$0.425862\pi$
$654$	0	0
$655$	−6626.39	−0.395290
$656$	0	0
$657$	1856.82	0.110261
$658$	0	0
$659$	5899.80	0.348746	0.174373	−	0.984680i	$-0.444210\pi$
$659$	5899.80	0.348746	0.174373	+	0.984680i	$0.444210\pi$
$660$	0	0
$661$	−25442.5	−1.49713	−0.748563	−	0.663064i	$-0.769256\pi$
$661$	−25442.5	−1.49713	−0.748563	+	0.663064i	$0.769256\pi$
$662$	0	0
$663$	2616.26	0.153254
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1589.78	−0.0922889
$668$	0	0
$669$	−17720.1	−1.02406
$670$	0	0
$671$	2256.03	0.129796
$672$	0	0
$673$	−18424.7	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$673$	−18424.7	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$674$	0	0
$675$	2067.12	0.117872
$676$	0	0
$677$	−1189.55	−0.0675307	−0.0337654	−	0.999430i	$-0.510750\pi$
$677$	−1189.55	−0.0675307	−0.0337654	+	0.999430i	$0.510750\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14633.6	−0.823437
$682$	0	0
$683$	19938.4	1.11702	0.558509	−	0.829499i	$-0.311374\pi$
$683$	19938.4	1.11702	0.558509	+	0.829499i	$0.311374\pi$
$684$	0	0
$685$	12880.4	0.718443
$686$	0	0
$687$	14680.1	0.815253
$688$	0	0
$689$	38510.0	2.12934
$690$	0	0
$691$	−5625.32	−0.309692	−0.154846	−	0.987939i	$-0.549488\pi$
$691$	−5625.32	−0.309692	−0.154846	+	0.987939i	$0.549488\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13049.5	0.712225
$696$	0	0
$697$	−1327.11	−0.0721206
$698$	0	0
$699$	−9466.31	−0.512230
$700$	0	0
$701$	−29961.0	−1.61428	−0.807140	−	0.590360i	$-0.798986\pi$
$701$	−29961.0	−1.61428	−0.807140	+	0.590360i	$0.798986\pi$
$702$	0	0
$703$	1529.27	0.0820451
$704$	0	0
$705$	−1617.39	−0.0864036
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14177.2	0.750968	0.375484	−	0.926829i	$-0.377476\pi$
$709$	14177.2	0.750968	0.375484	+	0.926829i	$0.377476\pi$
$710$	0	0
$711$	−8315.26	−0.438603
$712$	0	0
$713$	−21404.8	−1.12428
$714$	0	0
$715$	−25535.2	−1.33561
$716$	0	0
$717$	−12496.3	−0.650882
$718$	0	0
$719$	−425.151	−0.0220521	−0.0110260	−	0.999939i	$-0.503510\pi$
$719$	−425.151	−0.0220521	−0.0110260	+	0.999939i	$0.503510\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18721.7	0.963027
$724$	0	0
$725$	−760.348	−0.0389498
$726$	0	0
$727$	−12678.5	−0.646793	−0.323397	−	0.946264i	$-0.604825\pi$
$727$	−12678.5	−0.646793	−0.323397	+	0.946264i	$0.604825\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3581.96	−0.181236
$732$	0	0
$733$	7973.30	0.401774	0.200887	−	0.979614i	$-0.435618\pi$
$733$	7973.30	0.401774	0.200887	+	0.979614i	$0.435618\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14359.6	−0.717698
$738$	0	0
$739$	−12208.1	−0.607687	−0.303843	−	0.952722i	$-0.598270\pi$
$739$	−12208.1	−0.607687	−0.303843	+	0.952722i	$0.598270\pi$
$740$	0	0
$741$	−1072.14	−0.0531526
$742$	0	0
$743$	−13351.2	−0.659232	−0.329616	−	0.944115i	$-0.606919\pi$
$743$	−13351.2	−0.659232	−0.329616	+	0.944115i	$0.606919\pi$
$744$	0	0
$745$	6654.77	0.327264
$746$	0	0
$747$	9951.58	0.487429
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27971.9	−1.35913	−0.679567	−	0.733614i	$-0.737832\pi$
$751$	−27971.9	−1.35913	−0.679567	+	0.733614i	$0.737832\pi$
$752$	0	0
$753$	−1161.25	−0.0561997
$754$	0	0
$755$	−3566.96	−0.171940
$756$	0	0
$757$	20382.6	0.978623	0.489312	−	0.872109i	$-0.337248\pi$
$757$	20382.6	0.978623	0.489312	+	0.872109i	$0.337248\pi$
$758$	0	0
$759$	−21083.2	−1.00827
$760$	0	0
$761$	−24172.0	−1.15143	−0.575713	−	0.817652i	$-0.695275\pi$
$761$	−24172.0	−1.15143	−0.575713	+	0.817652i	$0.695275\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−653.664	−0.0308931
$766$	0	0
$767$	22792.5	1.07300
$768$	0	0
$769$	1320.89	0.0619409	0.0309705	−	0.999520i	$-0.490140\pi$
$769$	1320.89	0.0619409	0.0309705	+	0.999520i	$0.490140\pi$
$770$	0	0
$771$	−13496.7	−0.630443
$772$	0	0
$773$	28790.1	1.33960	0.669798	−	0.742544i	$-0.266381\pi$
$773$	28790.1	1.33960	0.669798	+	0.742544i	$0.266381\pi$
$774$	0	0
$775$	−10237.3	−0.474495
$776$	0	0
$777$	0	0
$778$	0	0
$779$	543.850	0.0250134
$780$	0	0
$781$	−25106.7	−1.15031
$782$	0	0
$783$	−268.147	−0.0122386
$784$	0	0
$785$	−3993.31	−0.181563
$786$	0	0
$787$	−38138.2	−1.72742	−0.863710	−	0.503989i	$-0.831865\pi$
$787$	−38138.2	−1.72742	−0.863710	+	0.503989i	$0.831865\pi$
$788$	0	0
$789$	−20374.7	−0.919339
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4294.44	0.192308
$794$	0	0
$795$	−9621.58	−0.429235
$796$	0	0
$797$	−29562.7	−1.31388	−0.656941	−	0.753942i	$-0.728150\pi$
$797$	−29562.7	−1.31388	−0.656941	+	0.753942i	$0.728150\pi$
$798$	0	0
$799$	−808.357	−0.0357918
$800$	0	0
$801$	−13793.2	−0.608437
$802$	0	0
$803$	−9057.63	−0.398054
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25612.5	−1.11723
$808$	0	0
$809$	28800.4	1.25163	0.625814	−	0.779972i	$-0.284767\pi$
$809$	28800.4	1.25163	0.625814	+	0.779972i	$0.284767\pi$
$810$	0	0
$811$	23960.3	1.03743	0.518717	−	0.854946i	$-0.326410\pi$
$811$	23960.3	1.03743	0.518717	+	0.854946i	$0.326410\pi$
$812$	0	0
$813$	14231.3	0.613917
$814$	0	0
$815$	27072.4	1.16356
$816$	0	0
$817$	1467.88	0.0628576
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7480.81	0.318005	0.159003	−	0.987278i	$-0.449172\pi$
$821$	7480.81	0.318005	0.159003	+	0.987278i	$0.449172\pi$
$822$	0	0
$823$	37456.0	1.58643	0.793217	−	0.608939i	$-0.208405\pi$
$823$	37456.0	1.58643	0.793217	+	0.608939i	$0.208405\pi$
$824$	0	0
$825$	−10083.5	−0.425531
$826$	0	0
$827$	−18855.9	−0.792847	−0.396423	−	0.918068i	$-0.629749\pi$
$827$	−18855.9	−0.792847	−0.396423	+	0.918068i	$0.629749\pi$
$828$	0	0
$829$	−34027.1	−1.42559	−0.712793	−	0.701375i	$-0.752570\pi$
$829$	−34027.1	−1.42559	−0.712793	+	0.701375i	$0.752570\pi$
$830$	0	0
$831$	3012.95	0.125774
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1164.30	0.0482541
$836$	0	0
$837$	−3610.32	−0.149093
$838$	0	0
$839$	9279.70	0.381848	0.190924	−	0.981605i	$-0.438852\pi$
$839$	9279.70	0.381848	0.190924	+	0.981605i	$0.438852\pi$
$840$	0	0
$841$	−24290.4	−0.995956
$842$	0	0
$843$	−13691.3	−0.559377
$844$	0	0
$845$	−33316.4	−1.35635
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−16674.7	−0.674059
$850$	0	0
$851$	57244.4	2.30589
$852$	0	0
$853$	−29478.9	−1.18328	−0.591641	−	0.806201i	$-0.701520\pi$
$853$	−29478.9	−1.18328	−0.591641	+	0.806201i	$0.701520\pi$
$854$	0	0
$855$	267.870	0.0107146
$856$	0	0
$857$	−18935.8	−0.754765	−0.377383	−	0.926057i	$-0.623176\pi$
$857$	−18935.8	−0.754765	−0.377383	+	0.926057i	$0.623176\pi$
$858$	0	0
$859$	−11678.8	−0.463883	−0.231941	−	0.972730i	$-0.574508\pi$
$859$	−11678.8	−0.463883	−0.231941	+	0.972730i	$0.574508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14625.7	0.576899	0.288450	−	0.957495i	$-0.406860\pi$
$863$	14625.7	0.576899	0.288450	+	0.957495i	$0.406860\pi$
$864$	0	0
$865$	17811.9	0.700143
$866$	0	0
$867$	14412.3	0.564553
$868$	0	0
$869$	40562.1	1.58340
$870$	0	0
$871$	−27334.1	−1.06335
$872$	0	0
$873$	878.454	0.0340563
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26195.9	1.00863	0.504317	−	0.863519i	$-0.331744\pi$
$877$	26195.9	1.00863	0.504317	+	0.863519i	$0.331744\pi$
$878$	0	0
$879$	−8709.68	−0.334210
$880$	0	0
$881$	17065.6	0.652616	0.326308	−	0.945264i	$-0.394195\pi$
$881$	17065.6	0.652616	0.326308	+	0.945264i	$0.394195\pi$
$882$	0	0
$883$	20428.3	0.778557	0.389279	−	0.921120i	$-0.372724\pi$
$883$	20428.3	0.778557	0.389279	+	0.921120i	$0.372724\pi$
$884$	0	0
$885$	−5694.61	−0.216296
$886$	0	0
$887$	−3679.59	−0.139288	−0.0696441	−	0.997572i	$-0.522186\pi$
$887$	−3679.59	−0.139288	−0.0696441	+	0.997572i	$0.522186\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3556.09	−0.133708
$892$	0	0
$893$	331.264	0.0124136
$894$	0	0
$895$	−8452.80	−0.315694
$896$	0	0
$897$	−40132.8	−1.49386
$898$	0	0
$899$	1327.98	0.0492665
$900$	0	0
$901$	−4808.77	−0.177806
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17782.7	0.653168
$906$	0	0
$907$	25374.5	0.928938	0.464469	−	0.885589i	$-0.346245\pi$
$907$	25374.5	0.928938	0.464469	+	0.885589i	$0.346245\pi$
$908$	0	0
$909$	−11937.5	−0.435579
$910$	0	0
$911$	4890.79	0.177869	0.0889347	−	0.996037i	$-0.471654\pi$
$911$	4890.79	0.177869	0.0889347	+	0.996037i	$0.471654\pi$
$912$	0	0
$913$	−48544.2	−1.75967
$914$	0	0
$915$	−1072.95	−0.0387657
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16590.3	−0.595501	−0.297750	−	0.954644i	$-0.596236\pi$
$919$	−16590.3	−0.595501	−0.297750	+	0.954644i	$0.596236\pi$
$920$	0	0
$921$	−23924.9	−0.855974
$922$	0	0
$923$	−47791.7	−1.70431
$924$	0	0
$925$	27378.4	0.973184
$926$	0	0
$927$	7023.19	0.248837
$928$	0	0
$929$	51991.9	1.83617	0.918083	−	0.396388i	$-0.129737\pi$
$929$	51991.9	1.83617	0.918083	+	0.396388i	$0.129737\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	14234.3	0.499474
$934$	0	0
$935$	3188.60	0.111528
$936$	0	0
$937$	26392.7	0.920182	0.460091	−	0.887872i	$-0.347817\pi$
$937$	26392.7	0.920182	0.460091	+	0.887872i	$0.347817\pi$
$938$	0	0
$939$	−29618.9	−1.02937
$940$	0	0
$941$	−40403.8	−1.39971	−0.699854	−	0.714286i	$-0.746752\pi$
$941$	−40403.8	−1.39971	−0.699854	+	0.714286i	$0.746752\pi$
$942$	0	0
$943$	20357.6	0.703006
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18557.3	0.636782	0.318391	−	0.947959i	$-0.396858\pi$
$947$	18557.3	0.636782	0.318391	+	0.947959i	$0.396858\pi$
$948$	0	0
$949$	−17241.6	−0.589763
$950$	0	0
$951$	933.081	0.0318162
$952$	0	0
$953$	−30432.6	−1.03443	−0.517213	−	0.855857i	$-0.673031\pi$
$953$	−30432.6	−1.03443	−0.517213	+	0.855857i	$0.673031\pi$
$954$	0	0
$955$	22663.6	0.767934
$956$	0	0
$957$	1308.03	0.0441825
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−11911.2	−0.399824
$962$	0	0
$963$	3926.37	0.131387
$964$	0	0
$965$	−668.670	−0.0223059
$966$	0	0
$967$	−21280.5	−0.707688	−0.353844	−	0.935304i	$-0.615126\pi$
$967$	−21280.5	−0.707688	−0.353844	+	0.935304i	$0.615126\pi$
$968$	0	0
$969$	133.879	0.00443840
$970$	0	0
$971$	10383.6	0.343179	0.171589	−	0.985169i	$-0.445110\pi$
$971$	10383.6	0.343179	0.171589	+	0.985169i	$0.445110\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−19194.4	−0.630473
$976$	0	0
$977$	−50807.3	−1.66374	−0.831868	−	0.554973i	$-0.812728\pi$
$977$	−50807.3	−1.66374	−0.831868	+	0.554973i	$0.812728\pi$
$978$	0	0
$979$	67283.6	2.19652
$980$	0	0
$981$	−18830.9	−0.612870
$982$	0	0
$983$	21417.1	0.694913	0.347456	−	0.937696i	$-0.387045\pi$
$983$	21417.1	0.694913	0.347456	+	0.937696i	$0.387045\pi$
$984$	0	0
$985$	−9293.84	−0.300636
$986$	0	0
$987$	0	0
$988$	0	0
$989$	54946.3	1.76663
$990$	0	0
$991$	−3420.14	−0.109631	−0.0548155	−	0.998497i	$-0.517457\pi$
$991$	−3420.14	−0.109631	−0.0548155	+	0.998497i	$0.517457\pi$
$992$	0	0
$993$	22353.9	0.714380
$994$	0	0
$995$	−20232.7	−0.644644
$996$	0	0
$997$	−45658.1	−1.45036	−0.725180	−	0.688560i	$-0.758243\pi$
$997$	−45658.1	−1.45036	−0.725180	+	0.688560i	$0.758243\pi$
$998$	0	0
$999$	9655.36	0.305788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.ck.1.2		4
4.3	odd	2		1176.4.a.bc.1.2	yes	4
7.6	odd	2		2352.4.a.cr.1.3		4
28.27	even	2		1176.4.a.bb.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.bb.1.3	✓	4	28.27	even	2
1176.4.a.bc.1.2	yes	4	4.3	odd	2
2352.4.a.ck.1.2		4	1.1	even	1	trivial
2352.4.a.cr.1.3		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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -6.95987 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -6.95987 q^{5} +9.00000 q^{9} -43.9023 q^{11} -83.5699 q^{13} +20.8796 q^{15} +10.4354 q^{17} -4.27642 q^{19} -160.077 q^{23} -76.5601 q^{25} -27.0000 q^{27} +9.93138 q^{29} +133.715 q^{31} +131.707 q^{33} -357.606 q^{37} +250.710 q^{39} -127.174 q^{41} -343.250 q^{43} -62.6389 q^{45} -77.4628 q^{47} -31.3063 q^{51} -460.812 q^{53} +305.555 q^{55} +12.8293 q^{57} -272.735 q^{59} -51.3874 q^{61} +581.636 q^{65} +327.081 q^{67} +480.231 q^{69} +571.877 q^{71} +206.313 q^{73} +229.680 q^{75} -923.918 q^{79} +81.0000 q^{81} +1105.73 q^{83} -72.6293 q^{85} -29.7941 q^{87} -1532.57 q^{89} -401.146 q^{93} +29.7634 q^{95} +97.6059 q^{97} -395.121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 - 8 * q^5 + 36 * q^9	\( 4 q - 12 q^{3} - 8 q^{5} + 36 q^{9} + 40 q^{11} - 48 q^{13} + 24 q^{15} + 152 q^{17} + 224 q^{19} + 8 q^{23} - 28 q^{25} - 108 q^{27} - 144 q^{29} + 400 q^{31} - 120 q^{33} - 304 q^{37} + 144 q^{39} + 152 q^{41} - 160 q^{43} - 72 q^{45} + 544 q^{47} - 456 q^{51} - 1320 q^{53} + 16 q^{55} - 672 q^{57} + 1040 q^{59} - 896 q^{61} - 648 q^{65} + 416 q^{67} - 24 q^{69} - 248 q^{71} + 752 q^{73} + 84 q^{75} - 864 q^{79} + 324 q^{81} + 1456 q^{83} - 1608 q^{85} + 432 q^{87} - 2936 q^{89} - 1200 q^{93} + 80 q^{95} - 144 q^{97} + 360 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 - 8 * q^5 + 36 * q^9 + 40 * q^11 - 48 * q^13 + 24 * q^15 + 152 * q^17 + 224 * q^19 + 8 * q^23 - 28 * q^25 - 108 * q^27 - 144 * q^29 + 400 * q^31 - 120 * q^33 - 304 * q^37 + 144 * q^39 + 152 * q^41 - 160 * q^43 - 72 * q^45 + 544 * q^47 - 456 * q^51 - 1320 * q^53 + 16 * q^55 - 672 * q^57 + 1040 * q^59 - 896 * q^61 - 648 * q^65 + 416 * q^67 - 24 * q^69 - 248 * q^71 + 752 * q^73 + 84 * q^75 - 864 * q^79 + 324 * q^81 + 1456 * q^83 - 1608 * q^85 + 432 * q^87 - 2936 * q^89 - 1200 * q^93 + 80 * q^95 - 144 * q^97 + 360 * q^99

Introduction

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