LMFDB - Embedded newform 2496.4.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, 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("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 20x^{3} - 33x^{2} + 17x + 40 $ x^5 - 20x^3 - 33x^2 + 17*x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.81539$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -14.7452 q^{5} -9.44613 q^{7} +9.00000 q^{9} +55.6334 q^{11} +13.0000 q^{13} +44.2356 q^{15} -120.234 q^{17} -136.902 q^{19} +28.3384 q^{21} +62.1708 q^{23} +92.4212 q^{25} -27.0000 q^{27} +171.656 q^{29} +68.3802 q^{31} -166.900 q^{33} +139.285 q^{35} +165.907 q^{37} -39.0000 q^{39} -437.690 q^{41} +191.295 q^{43} -132.707 q^{45} +271.577 q^{47} -253.771 q^{49} +360.703 q^{51} +155.725 q^{53} -820.327 q^{55} +410.706 q^{57} +582.688 q^{59} +261.654 q^{61} -85.0152 q^{63} -191.688 q^{65} +197.665 q^{67} -186.513 q^{69} +69.8308 q^{71} +815.835 q^{73} -277.264 q^{75} -525.521 q^{77} -396.735 q^{79} +81.0000 q^{81} -938.295 q^{83} +1772.88 q^{85} -514.967 q^{87} +572.142 q^{89} -122.800 q^{91} -205.141 q^{93} +2018.65 q^{95} -145.028 q^{97} +500.701 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} + 10 q^{5} - 14 q^{7} + 45 q^{9} + 22 q^{11} + 65 q^{13} - 30 q^{15} - 34 q^{17} - 90 q^{19} + 42 q^{21} - 96 q^{23} + 107 q^{25} - 135 q^{27} + 54 q^{29} - 378 q^{31} - 66 q^{33} + 84 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 + 10 * q^5 - 14 * q^7 + 45 * q^9 + 22 * q^11 + 65 * q^13 - 30 * q^15 - 34 * q^17 - 90 * q^19 + 42 * q^21 - 96 * q^23 + 107 * q^25 - 135 * q^27 + 54 * q^29 - 378 * q^31 - 66 * q^33 + 84 * q^35 + 318 * q^37 - 195 * q^39 - 458 * q^41 + 104 * q^43 + 90 * q^45 - 474 * q^47 + 25 * q^49 + 102 * q^51 + 450 * q^53 - 1116 * q^55 + 270 * q^57 + 822 * q^59 + 546 * q^61 - 126 * q^63 + 130 * q^65 - 130 * q^67 + 288 * q^69 - 1646 * q^71 - 166 * q^73 - 321 * q^75 + 900 * q^77 - 196 * q^79 + 405 * q^81 - 118 * q^83 + 3340 * q^85 - 162 * q^87 - 186 * q^89 - 182 * q^91 + 1134 * q^93 - 564 * q^95 - 2582 * q^97 + 198 * q^99

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
$5$	−14.7452	−1.31885	−0.659426	−	0.751770i	$-0.729201\pi$
$5$	−14.7452	−1.31885	−0.659426	+	0.751770i	$0.729201\pi$
$6$	0	0
$7$	−9.44613	−0.510043	−0.255022	−	0.966935i	$-0.582083\pi$
$7$	−9.44613	−0.510043	−0.255022	+	0.966935i	$0.582083\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	55.6334	1.52492	0.762460	−	0.647036i	$-0.223992\pi$
$11$	55.6334	1.52492	0.762460	+	0.647036i	$0.223992\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	44.2356	0.761439
$16$	0	0
$17$	−120.234	−1.71536	−0.857679	−	0.514186i	$-0.828094\pi$
$17$	−120.234	−1.71536	−0.857679	+	0.514186i	$0.828094\pi$
$18$	0	0
$19$	−136.902	−1.65303	−0.826513	−	0.562917i	$-0.809679\pi$
$19$	−136.902	−1.65303	−0.826513	+	0.562917i	$0.809679\pi$
$20$	0	0
$21$	28.3384	0.294474
$22$	0	0
$23$	62.1708	0.563631	0.281816	−	0.959469i	$-0.409063\pi$
$23$	62.1708	0.563631	0.281816	+	0.959469i	$0.409063\pi$
$24$	0	0
$25$	92.4212	0.739369
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	171.656	1.09916	0.549580	−	0.835441i	$-0.314788\pi$
$29$	171.656	1.09916	0.549580	+	0.835441i	$0.314788\pi$
$30$	0	0
$31$	68.3802	0.396176	0.198088	−	0.980184i	$-0.436527\pi$
$31$	68.3802	0.396176	0.198088	+	0.980184i	$0.436527\pi$
$32$	0	0
$33$	−166.900	−0.880413
$34$	0	0
$35$	139.285	0.672671
$36$	0	0
$37$	165.907	0.737161	0.368580	−	0.929596i	$-0.379844\pi$
$37$	165.907	0.737161	0.368580	+	0.929596i	$0.379844\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−437.690	−1.66721	−0.833605	−	0.552360i	$-0.813727\pi$
$41$	−437.690	−1.66721	−0.833605	+	0.552360i	$0.813727\pi$
$42$	0	0
$43$	191.295	0.678424	0.339212	−	0.940710i	$-0.389840\pi$
$43$	191.295	0.678424	0.339212	+	0.940710i	$0.389840\pi$
$44$	0	0
$45$	−132.707	−0.439617
$46$	0	0
$47$	271.577	0.842843	0.421421	−	0.906865i	$-0.361531\pi$
$47$	271.577	0.842843	0.421421	+	0.906865i	$0.361531\pi$
$48$	0	0
$49$	−253.771	−0.739856
$50$	0	0
$51$	360.703	0.990362
$52$	0	0
$53$	155.725	0.403593	0.201796	−	0.979428i	$-0.435322\pi$
$53$	155.725	0.403593	0.201796	+	0.979428i	$0.435322\pi$
$54$	0	0
$55$	−820.327	−2.01114
$56$	0	0
$57$	410.706	0.954375
$58$	0	0
$59$	582.688	1.28575	0.642877	−	0.765969i	$-0.277741\pi$
$59$	582.688	1.28575	0.642877	+	0.765969i	$0.277741\pi$
$60$	0	0
$61$	261.654	0.549202	0.274601	−	0.961558i	$-0.411454\pi$
$61$	261.654	0.549202	0.274601	+	0.961558i	$0.411454\pi$
$62$	0	0
$63$	−85.0152	−0.170014
$64$	0	0
$65$	−191.688	−0.365784
$66$	0	0
$67$	197.665	0.360427	0.180213	−	0.983628i	$-0.442321\pi$
$67$	197.665	0.360427	0.180213	+	0.983628i	$0.442321\pi$
$68$	0	0
$69$	−186.513	−0.325413
$70$	0	0
$71$	69.8308	0.116724	0.0583619	−	0.998295i	$-0.481412\pi$
$71$	69.8308	0.116724	0.0583619	+	0.998295i	$0.481412\pi$
$72$	0	0
$73$	815.835	1.30803	0.654016	−	0.756481i	$-0.273083\pi$
$73$	815.835	1.30803	0.654016	+	0.756481i	$0.273083\pi$
$74$	0	0
$75$	−277.264	−0.426875
$76$	0	0
$77$	−525.521	−0.777775
$78$	0	0
$79$	−396.735	−0.565015	−0.282507	−	0.959265i	$-0.591166\pi$
$79$	−396.735	−0.565015	−0.282507	+	0.959265i	$0.591166\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−938.295	−1.24086	−0.620429	−	0.784262i	$-0.713042\pi$
$83$	−938.295	−1.24086	−0.620429	+	0.784262i	$0.713042\pi$
$84$	0	0
$85$	1772.88	2.26230
$86$	0	0
$87$	−514.967	−0.634600
$88$	0	0
$89$	572.142	0.681427	0.340713	−	0.940167i	$-0.389331\pi$
$89$	572.142	0.681427	0.340713	+	0.940167i	$0.389331\pi$
$90$	0	0
$91$	−122.800	−0.141461
$92$	0	0
$93$	−205.141	−0.228732
$94$	0	0
$95$	2018.65	2.18010
$96$	0	0
$97$	−145.028	−0.151808	−0.0759041	−	0.997115i	$-0.524184\pi$
$97$	−145.028	−0.151808	−0.0759041	+	0.997115i	$0.524184\pi$
$98$	0	0
$99$	500.701	0.508307
$100$	0	0
$101$	1332.99	1.31324	0.656621	−	0.754221i	$-0.271985\pi$
$101$	1332.99	1.31324	0.656621	+	0.754221i	$0.271985\pi$
$102$	0	0
$103$	−427.202	−0.408675	−0.204337	−	0.978901i	$-0.565504\pi$
$103$	−427.202	−0.408675	−0.204337	+	0.978901i	$0.565504\pi$
$104$	0	0
$105$	−417.856	−0.388367
$106$	0	0
$107$	434.184	0.392281	0.196141	−	0.980576i	$-0.437159\pi$
$107$	434.184	0.392281	0.196141	+	0.980576i	$0.437159\pi$
$108$	0	0
$109$	1070.80	0.940951	0.470476	−	0.882413i	$-0.344082\pi$
$109$	1070.80	0.940951	0.470476	+	0.882413i	$0.344082\pi$
$110$	0	0
$111$	−497.721	−0.425600
$112$	0	0
$113$	−488.658	−0.406806	−0.203403	−	0.979095i	$-0.565200\pi$
$113$	−488.658	−0.406806	−0.203403	+	0.979095i	$0.565200\pi$
$114$	0	0
$115$	−916.722	−0.743346
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	1135.75	0.874906
$120$	0	0
$121$	1764.08	1.32538
$122$	0	0
$123$	1313.07	0.962565
$124$	0	0
$125$	480.382	0.343733
$126$	0	0
$127$	1774.22	1.23966	0.619828	−	0.784738i	$-0.287202\pi$
$127$	1774.22	1.23966	0.619828	+	0.784738i	$0.287202\pi$
$128$	0	0
$129$	−573.885	−0.391688
$130$	0	0
$131$	814.519	0.543243	0.271621	−	0.962404i	$-0.412440\pi$
$131$	814.519	0.543243	0.271621	+	0.962404i	$0.412440\pi$
$132$	0	0
$133$	1293.20	0.843115
$134$	0	0
$135$	398.121	0.253813
$136$	0	0
$137$	1328.55	0.828511	0.414256	−	0.910161i	$-0.364042\pi$
$137$	1328.55	0.828511	0.414256	+	0.910161i	$0.364042\pi$
$138$	0	0
$139$	−735.419	−0.448759	−0.224379	−	0.974502i	$-0.572035\pi$
$139$	−735.419	−0.448759	−0.224379	+	0.974502i	$0.572035\pi$
$140$	0	0
$141$	−814.732	−0.486616
$142$	0	0
$143$	723.235	0.422937
$144$	0	0
$145$	−2531.10	−1.44963
$146$	0	0
$147$	761.312	0.427156
$148$	0	0
$149$	−2360.83	−1.29803	−0.649015	−	0.760775i	$-0.724819\pi$
$149$	−2360.83	−1.29803	−0.649015	+	0.760775i	$0.724819\pi$
$150$	0	0
$151$	−2725.54	−1.46889	−0.734443	−	0.678671i	$-0.762556\pi$
$151$	−2725.54	−1.46889	−0.734443	+	0.678671i	$0.762556\pi$
$152$	0	0
$153$	−1082.11	−0.571786
$154$	0	0
$155$	−1008.28	−0.522497
$156$	0	0
$157$	410.252	0.208546	0.104273	−	0.994549i	$-0.466748\pi$
$157$	410.252	0.208546	0.104273	+	0.994549i	$0.466748\pi$
$158$	0	0
$159$	−467.174	−0.233014
$160$	0	0
$161$	−587.274	−0.287476
$162$	0	0
$163$	1268.90	0.609743	0.304871	−	0.952394i	$-0.401387\pi$
$163$	1268.90	0.609743	0.304871	+	0.952394i	$0.401387\pi$
$164$	0	0
$165$	2460.98	1.16113
$166$	0	0
$167$	−3541.26	−1.64090	−0.820451	−	0.571717i	$-0.806278\pi$
$167$	−3541.26	−1.64090	−0.820451	+	0.571717i	$0.806278\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1232.12	−0.551009
$172$	0	0
$173$	−670.272	−0.294566	−0.147283	−	0.989094i	$-0.547053\pi$
$173$	−670.272	−0.294566	−0.147283	+	0.989094i	$0.547053\pi$
$174$	0	0
$175$	−873.023	−0.377110
$176$	0	0
$177$	−1748.06	−0.742331
$178$	0	0
$179$	−4416.42	−1.84413	−0.922063	−	0.387040i	$-0.873498\pi$
$179$	−4416.42	−1.84413	−0.922063	+	0.387040i	$0.873498\pi$
$180$	0	0
$181$	−1633.78	−0.670929	−0.335464	−	0.942053i	$-0.608893\pi$
$181$	−1633.78	−0.670929	−0.335464	+	0.942053i	$0.608893\pi$
$182$	0	0
$183$	−784.962	−0.317082
$184$	0	0
$185$	−2446.33	−0.972206
$186$	0	0
$187$	−6689.04	−2.61578
$188$	0	0
$189$	255.046	0.0981579
$190$	0	0
$191$	−3961.29	−1.50067	−0.750337	−	0.661055i	$-0.770109\pi$
$191$	−3961.29	−1.50067	−0.750337	+	0.661055i	$0.770109\pi$
$192$	0	0
$193$	−2196.08	−0.819052	−0.409526	−	0.912298i	$-0.634306\pi$
$193$	−2196.08	−0.819052	−0.409526	+	0.912298i	$0.634306\pi$
$194$	0	0
$195$	575.063	0.211185
$196$	0	0
$197$	3511.12	1.26983	0.634916	−	0.772581i	$-0.281035\pi$
$197$	3511.12	1.26983	0.634916	+	0.772581i	$0.281035\pi$
$198$	0	0
$199$	4813.03	1.71450	0.857252	−	0.514896i	$-0.172170\pi$
$199$	4813.03	1.71450	0.857252	+	0.514896i	$0.172170\pi$
$200$	0	0
$201$	−592.994	−0.208092
$202$	0	0
$203$	−1621.48	−0.560619
$204$	0	0
$205$	6453.83	2.19880
$206$	0	0
$207$	559.538	0.187877
$208$	0	0
$209$	−7616.34	−2.52073
$210$	0	0
$211$	319.742	0.104322	0.0521610	−	0.998639i	$-0.483389\pi$
$211$	319.742	0.104322	0.0521610	+	0.998639i	$0.483389\pi$
$212$	0	0
$213$	−209.493	−0.0673906
$214$	0	0
$215$	−2820.69	−0.894741
$216$	0	0
$217$	−645.928	−0.202067
$218$	0	0
$219$	−2447.51	−0.755192
$220$	0	0
$221$	−1563.04	−0.475754
$222$	0	0
$223$	−3261.73	−0.979470	−0.489735	−	0.871871i	$-0.662906\pi$
$223$	−3261.73	−0.979470	−0.489735	+	0.871871i	$0.662906\pi$
$224$	0	0
$225$	831.791	0.246456
$226$	0	0
$227$	−1639.14	−0.479268	−0.239634	−	0.970863i	$-0.577027\pi$
$227$	−1639.14	−0.479268	−0.239634	+	0.970863i	$0.577027\pi$
$228$	0	0
$229$	−5198.91	−1.50023	−0.750117	−	0.661305i	$-0.770003\pi$
$229$	−5198.91	−1.50023	−0.750117	+	0.661305i	$0.770003\pi$
$230$	0	0
$231$	1576.56	0.449049
$232$	0	0
$233$	−5894.74	−1.65741	−0.828707	−	0.559683i	$-0.810923\pi$
$233$	−5894.74	−1.65741	−0.828707	+	0.559683i	$0.810923\pi$
$234$	0	0
$235$	−4004.46	−1.11158
$236$	0	0
$237$	1190.20	0.326211
$238$	0	0
$239$	−3293.00	−0.891239	−0.445620	−	0.895222i	$-0.647017\pi$
$239$	−3293.00	−0.891239	−0.445620	+	0.895222i	$0.647017\pi$
$240$	0	0
$241$	138.548	0.0370317	0.0185159	−	0.999829i	$-0.494106\pi$
$241$	138.548	0.0370317	0.0185159	+	0.999829i	$0.494106\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3741.90	0.975760
$246$	0	0
$247$	−1779.73	−0.458467
$248$	0	0
$249$	2814.89	0.716410
$250$	0	0
$251$	−2812.92	−0.707371	−0.353685	−	0.935364i	$-0.615072\pi$
$251$	−2812.92	−0.707371	−0.353685	+	0.935364i	$0.615072\pi$
$252$	0	0
$253$	3458.78	0.859492
$254$	0	0
$255$	−5318.63	−1.30614
$256$	0	0
$257$	4450.11	1.08012	0.540059	−	0.841627i	$-0.318402\pi$
$257$	4450.11	1.08012	0.540059	+	0.841627i	$0.318402\pi$
$258$	0	0
$259$	−1567.18	−0.375984
$260$	0	0
$261$	1544.90	0.366387
$262$	0	0
$263$	−1789.33	−0.419523	−0.209762	−	0.977753i	$-0.567269\pi$
$263$	−1789.33	−0.419523	−0.209762	+	0.977753i	$0.567269\pi$
$264$	0	0
$265$	−2296.19	−0.532279
$266$	0	0
$267$	−1716.43	−0.393422
$268$	0	0
$269$	7912.65	1.79347	0.896733	−	0.442571i	$-0.145934\pi$
$269$	7912.65	1.79347	0.896733	+	0.442571i	$0.145934\pi$
$270$	0	0
$271$	4234.45	0.949168	0.474584	−	0.880210i	$-0.342599\pi$
$271$	4234.45	0.949168	0.474584	+	0.880210i	$0.342599\pi$
$272$	0	0
$273$	368.399	0.0816723
$274$	0	0
$275$	5141.71	1.12748
$276$	0	0
$277$	6294.69	1.36538	0.682692	−	0.730706i	$-0.260809\pi$
$277$	6294.69	1.36538	0.682692	+	0.730706i	$0.260809\pi$
$278$	0	0
$279$	615.422	0.132059
$280$	0	0
$281$	5274.13	1.11967	0.559837	−	0.828603i	$-0.310864\pi$
$281$	5274.13	1.11967	0.559837	+	0.828603i	$0.310864\pi$
$282$	0	0
$283$	−2871.45	−0.603145	−0.301573	−	0.953443i	$-0.597512\pi$
$283$	−2871.45	−0.603145	−0.301573	+	0.953443i	$0.597512\pi$
$284$	0	0
$285$	−6055.95	−1.25868
$286$	0	0
$287$	4134.48	0.850350
$288$	0	0
$289$	9543.26	1.94245
$290$	0	0
$291$	435.085	0.0876465
$292$	0	0
$293$	−2216.66	−0.441975	−0.220988	−	0.975277i	$-0.570928\pi$
$293$	−2216.66	−0.441975	−0.220988	+	0.975277i	$0.570928\pi$
$294$	0	0
$295$	−8591.85	−1.69572
$296$	0	0
$297$	−1502.10	−0.293471
$298$	0	0
$299$	808.221	0.156323
$300$	0	0
$301$	−1807.00	−0.346026
$302$	0	0
$303$	−3998.97	−0.758200
$304$	0	0
$305$	−3858.14	−0.724317
$306$	0	0
$307$	−6630.04	−1.23256	−0.616281	−	0.787526i	$-0.711362\pi$
$307$	−6630.04	−1.23256	−0.616281	+	0.787526i	$0.711362\pi$
$308$	0	0
$309$	1281.61	0.235949
$310$	0	0
$311$	−6417.22	−1.17005	−0.585027	−	0.811014i	$-0.698916\pi$
$311$	−6417.22	−1.17005	−0.585027	+	0.811014i	$0.698916\pi$
$312$	0	0
$313$	−10030.5	−1.81137	−0.905685	−	0.423952i	$-0.860643\pi$
$313$	−10030.5	−1.81137	−0.905685	+	0.423952i	$0.860643\pi$
$314$	0	0
$315$	1253.57	0.224224
$316$	0	0
$317$	−6080.92	−1.07741	−0.538704	−	0.842495i	$-0.681086\pi$
$317$	−6080.92	−1.07741	−0.538704	+	0.842495i	$0.681086\pi$
$318$	0	0
$319$	9549.79	1.67613
$320$	0	0
$321$	−1302.55	−0.226484
$322$	0	0
$323$	16460.3	2.83553
$324$	0	0
$325$	1201.48	0.205064
$326$	0	0
$327$	−3212.39	−0.543259
$328$	0	0
$329$	−2565.35	−0.429886
$330$	0	0
$331$	4169.50	0.692376	0.346188	−	0.938165i	$-0.387476\pi$
$331$	4169.50	0.692376	0.346188	+	0.938165i	$0.387476\pi$
$332$	0	0
$333$	1493.16	0.245720
$334$	0	0
$335$	−2914.61	−0.475349
$336$	0	0
$337$	8455.48	1.36676	0.683382	−	0.730061i	$-0.260508\pi$
$337$	8455.48	1.36676	0.683382	+	0.730061i	$0.260508\pi$
$338$	0	0
$339$	1465.97	0.234869
$340$	0	0
$341$	3804.23	0.604136
$342$	0	0
$343$	5637.17	0.887402
$344$	0	0
$345$	2750.17	0.429171
$346$	0	0
$347$	−10221.0	−1.58125	−0.790623	−	0.612304i	$-0.790243\pi$
$347$	−10221.0	−1.58125	−0.790623	+	0.612304i	$0.790243\pi$
$348$	0	0
$349$	6210.05	0.952483	0.476241	−	0.879315i	$-0.341999\pi$
$349$	6210.05	0.952483	0.476241	+	0.879315i	$0.341999\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−2938.60	−0.443076	−0.221538	−	0.975152i	$-0.571108\pi$
$353$	−2938.60	−0.443076	−0.221538	+	0.975152i	$0.571108\pi$
$354$	0	0
$355$	−1029.67	−0.153941
$356$	0	0
$357$	−3407.24	−0.505127
$358$	0	0
$359$	−9684.69	−1.42378	−0.711892	−	0.702289i	$-0.752162\pi$
$359$	−9684.69	−1.42378	−0.711892	+	0.702289i	$0.752162\pi$
$360$	0	0
$361$	11883.2	1.73250
$362$	0	0
$363$	−5292.24	−0.765208
$364$	0	0
$365$	−12029.7	−1.72510
$366$	0	0
$367$	2061.20	0.293171	0.146585	−	0.989198i	$-0.453172\pi$
$367$	2061.20	0.293171	0.146585	+	0.989198i	$0.453172\pi$
$368$	0	0
$369$	−3939.21	−0.555737
$370$	0	0
$371$	−1470.99	−0.205850
$372$	0	0
$373$	−8658.96	−1.20200	−0.600998	−	0.799251i	$-0.705230\pi$
$373$	−8658.96	−1.20200	−0.600998	+	0.799251i	$0.705230\pi$
$374$	0	0
$375$	−1441.15	−0.198454
$376$	0	0
$377$	2231.52	0.304852
$378$	0	0
$379$	−4101.61	−0.555898	−0.277949	−	0.960596i	$-0.589655\pi$
$379$	−4101.61	−0.555898	−0.277949	+	0.960596i	$0.589655\pi$
$380$	0	0
$381$	−5322.65	−0.715716
$382$	0	0
$383$	−8906.22	−1.18822	−0.594108	−	0.804386i	$-0.702495\pi$
$383$	−8906.22	−1.18822	−0.594108	+	0.804386i	$0.702495\pi$
$384$	0	0
$385$	7748.92	1.02577
$386$	0	0
$387$	1721.66	0.226141
$388$	0	0
$389$	−10659.9	−1.38941	−0.694703	−	0.719296i	$-0.744464\pi$
$389$	−10659.9	−1.38941	−0.694703	+	0.719296i	$0.744464\pi$
$390$	0	0
$391$	−7475.06	−0.966829
$392$	0	0
$393$	−2443.56	−0.313641
$394$	0	0
$395$	5849.94	0.745170
$396$	0	0
$397$	10652.9	1.34673	0.673365	−	0.739310i	$-0.264848\pi$
$397$	10652.9	1.34673	0.673365	+	0.739310i	$0.264848\pi$
$398$	0	0
$399$	−3879.59	−0.486773
$400$	0	0
$401$	−10685.9	−1.33074	−0.665369	−	0.746514i	$-0.731726\pi$
$401$	−10685.9	−1.33074	−0.665369	+	0.746514i	$0.731726\pi$
$402$	0	0
$403$	888.942	0.109879
$404$	0	0
$405$	−1194.36	−0.146539
$406$	0	0
$407$	9229.98	1.12411
$408$	0	0
$409$	13408.9	1.62110	0.810549	−	0.585670i	$-0.199169\pi$
$409$	13408.9	1.62110	0.810549	+	0.585670i	$0.199169\pi$
$410$	0	0
$411$	−3985.66	−0.478341
$412$	0	0
$413$	−5504.15	−0.655790
$414$	0	0
$415$	13835.4	1.63651
$416$	0	0
$417$	2206.26	0.259091
$418$	0	0
$419$	−12789.8	−1.49123	−0.745615	−	0.666377i	$-0.767844\pi$
$419$	−12789.8	−1.49123	−0.745615	+	0.666377i	$0.767844\pi$
$420$	0	0
$421$	−3851.75	−0.445898	−0.222949	−	0.974830i	$-0.571568\pi$
$421$	−3851.75	−0.445898	−0.222949	+	0.974830i	$0.571568\pi$
$422$	0	0
$423$	2444.20	0.280948
$424$	0	0
$425$	−11112.2	−1.26828
$426$	0	0
$427$	−2471.62	−0.280117
$428$	0	0
$429$	−2169.70	−0.244183
$430$	0	0
$431$	−16659.8	−1.86189	−0.930946	−	0.365157i	$-0.881015\pi$
$431$	−16659.8	−1.86189	−0.930946	+	0.365157i	$0.881015\pi$
$432$	0	0
$433$	−16062.9	−1.78275	−0.891377	−	0.453262i	$-0.850260\pi$
$433$	−16062.9	−1.78275	−0.891377	+	0.453262i	$0.850260\pi$
$434$	0	0
$435$	7593.29	0.836943
$436$	0	0
$437$	−8511.32	−0.931697
$438$	0	0
$439$	620.202	0.0674274	0.0337137	−	0.999432i	$-0.489267\pi$
$439$	620.202	0.0674274	0.0337137	+	0.999432i	$0.489267\pi$
$440$	0	0
$441$	−2283.94	−0.246619
$442$	0	0
$443$	4046.82	0.434019	0.217009	−	0.976170i	$-0.430370\pi$
$443$	4046.82	0.434019	0.217009	+	0.976170i	$0.430370\pi$
$444$	0	0
$445$	−8436.36	−0.898701
$446$	0	0
$447$	7082.49	0.749419
$448$	0	0
$449$	10650.6	1.11945	0.559727	−	0.828677i	$-0.310906\pi$
$449$	10650.6	1.11945	0.559727	+	0.828677i	$0.310906\pi$
$450$	0	0
$451$	−24350.2	−2.54236
$452$	0	0
$453$	8176.63	0.848061
$454$	0	0
$455$	1810.71	0.186565
$456$	0	0
$457$	−950.953	−0.0973385	−0.0486692	−	0.998815i	$-0.515498\pi$
$457$	−950.953	−0.0973385	−0.0486692	+	0.998815i	$0.515498\pi$
$458$	0	0
$459$	3246.32	0.330121
$460$	0	0
$461$	−3844.16	−0.388374	−0.194187	−	0.980965i	$-0.562207\pi$
$461$	−3844.16	−0.388374	−0.194187	+	0.980965i	$0.562207\pi$
$462$	0	0
$463$	4153.25	0.416886	0.208443	−	0.978035i	$-0.433160\pi$
$463$	4153.25	0.416886	0.208443	+	0.978035i	$0.433160\pi$
$464$	0	0
$465$	3024.84	0.301664
$466$	0	0
$467$	8205.41	0.813064	0.406532	−	0.913636i	$-0.366738\pi$
$467$	8205.41	0.813064	0.406532	+	0.913636i	$0.366738\pi$
$468$	0	0
$469$	−1867.17	−0.183833
$470$	0	0
$471$	−1230.76	−0.120404
$472$	0	0
$473$	10642.4	1.03454
$474$	0	0
$475$	−12652.7	−1.22220
$476$	0	0
$477$	1401.52	0.134531
$478$	0	0
$479$	−9472.01	−0.903522	−0.451761	−	0.892139i	$-0.649204\pi$
$479$	−9472.01	−0.903522	−0.451761	+	0.892139i	$0.649204\pi$
$480$	0	0
$481$	2156.79	0.204452
$482$	0	0
$483$	1761.82	0.165975
$484$	0	0
$485$	2138.47	0.200212
$486$	0	0
$487$	13022.3	1.21169	0.605847	−	0.795581i	$-0.292834\pi$
$487$	13022.3	1.21169	0.605847	+	0.795581i	$0.292834\pi$
$488$	0	0
$489$	−3806.71	−0.352035
$490$	0	0
$491$	12947.4	1.19003	0.595017	−	0.803713i	$-0.297145\pi$
$491$	12947.4	1.19003	0.595017	+	0.803713i	$0.297145\pi$
$492$	0	0
$493$	−20638.9	−1.88545
$494$	0	0
$495$	−7382.94	−0.670381
$496$	0	0
$497$	−659.631	−0.0595342
$498$	0	0
$499$	9313.74	0.835551	0.417776	−	0.908550i	$-0.362810\pi$
$499$	9313.74	0.835551	0.417776	+	0.908550i	$0.362810\pi$
$500$	0	0
$501$	10623.8	0.947375
$502$	0	0
$503$	10365.4	0.918828	0.459414	−	0.888222i	$-0.348059\pi$
$503$	10365.4	0.918828	0.459414	+	0.888222i	$0.348059\pi$
$504$	0	0
$505$	−19655.2	−1.73197
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−12031.0	−1.04767	−0.523835	−	0.851820i	$-0.675499\pi$
$509$	−12031.0	−1.04767	−0.523835	+	0.851820i	$0.675499\pi$
$510$	0	0
$511$	−7706.49	−0.667153
$512$	0	0
$513$	3696.36	0.318125
$514$	0	0
$515$	6299.19	0.538981
$516$	0	0
$517$	15108.8	1.28527
$518$	0	0
$519$	2010.82	0.170068
$520$	0	0
$521$	18554.7	1.56026	0.780131	−	0.625616i	$-0.215152\pi$
$521$	18554.7	1.56026	0.780131	+	0.625616i	$0.215152\pi$
$522$	0	0
$523$	1315.45	0.109982	0.0549912	−	0.998487i	$-0.482487\pi$
$523$	1315.45	0.109982	0.0549912	+	0.998487i	$0.482487\pi$
$524$	0	0
$525$	2619.07	0.217725
$526$	0	0
$527$	−8221.64	−0.679583
$528$	0	0
$529$	−8301.79	−0.682320
$530$	0	0
$531$	5244.19	0.428585
$532$	0	0
$533$	−5689.97	−0.462401
$534$	0	0
$535$	−6402.13	−0.517361
$536$	0	0
$537$	13249.3	1.06471
$538$	0	0
$539$	−14118.1	−1.12822
$540$	0	0
$541$	7840.34	0.623073	0.311537	−	0.950234i	$-0.399156\pi$
$541$	7840.34	0.623073	0.311537	+	0.950234i	$0.399156\pi$
$542$	0	0
$543$	4901.35	0.387361
$544$	0	0
$545$	−15789.1	−1.24098
$546$	0	0
$547$	4796.69	0.374939	0.187469	−	0.982270i	$-0.439971\pi$
$547$	4796.69	0.374939	0.187469	+	0.982270i	$0.439971\pi$
$548$	0	0
$549$	2354.89	0.183067
$550$	0	0
$551$	−23500.0	−1.81694
$552$	0	0
$553$	3747.61	0.288182
$554$	0	0
$555$	7339.00	0.561303
$556$	0	0
$557$	15030.4	1.14337	0.571686	−	0.820473i	$-0.306290\pi$
$557$	15030.4	1.14337	0.571686	+	0.820473i	$0.306290\pi$
$558$	0	0
$559$	2486.84	0.188161
$560$	0	0
$561$	20067.1	1.51022
$562$	0	0
$563$	−5085.88	−0.380718	−0.190359	−	0.981715i	$-0.560965\pi$
$563$	−5085.88	−0.380718	−0.190359	+	0.981715i	$0.560965\pi$
$564$	0	0
$565$	7205.36	0.536516
$566$	0	0
$567$	−765.137	−0.0566715
$568$	0	0
$569$	−12964.0	−0.955151	−0.477576	−	0.878591i	$-0.658484\pi$
$569$	−12964.0	−0.955151	−0.477576	+	0.878591i	$0.658484\pi$
$570$	0	0
$571$	−10862.3	−0.796098	−0.398049	−	0.917364i	$-0.630313\pi$
$571$	−10862.3	−0.796098	−0.398049	+	0.917364i	$0.630313\pi$
$572$	0	0
$573$	11883.9	0.866415
$574$	0	0
$575$	5745.90	0.416732
$576$	0	0
$577$	−4502.26	−0.324838	−0.162419	−	0.986722i	$-0.551930\pi$
$577$	−4502.26	−0.324838	−0.162419	+	0.986722i	$0.551930\pi$
$578$	0	0
$579$	6588.23	0.472880
$580$	0	0
$581$	8863.26	0.632892
$582$	0	0
$583$	8663.49	0.615446
$584$	0	0
$585$	−1725.19	−0.121928
$586$	0	0
$587$	−5910.43	−0.415587	−0.207794	−	0.978173i	$-0.566628\pi$
$587$	−5910.43	−0.415587	−0.207794	+	0.978173i	$0.566628\pi$
$588$	0	0
$589$	−9361.39	−0.654889
$590$	0	0
$591$	−10533.4	−0.733138
$592$	0	0
$593$	23680.8	1.63989	0.819945	−	0.572443i	$-0.194004\pi$
$593$	23680.8	1.63989	0.819945	+	0.572443i	$0.194004\pi$
$594$	0	0
$595$	−16746.8	−1.15387
$596$	0	0
$597$	−14439.1	−0.989870
$598$	0	0
$599$	16091.1	1.09760	0.548802	−	0.835952i	$-0.315084\pi$
$599$	16091.1	1.09760	0.548802	+	0.835952i	$0.315084\pi$
$600$	0	0
$601$	−6691.76	−0.454180	−0.227090	−	0.973874i	$-0.572921\pi$
$601$	−6691.76	−0.454180	−0.227090	+	0.973874i	$0.572921\pi$
$602$	0	0
$603$	1778.98	0.120142
$604$	0	0
$605$	−26011.7	−1.74798
$606$	0	0
$607$	8087.54	0.540796	0.270398	−	0.962749i	$-0.412845\pi$
$607$	8087.54	0.540796	0.270398	+	0.962749i	$0.412845\pi$
$608$	0	0
$609$	4864.44	0.323674
$610$	0	0
$611$	3530.50	0.233763
$612$	0	0
$613$	−20105.7	−1.32474	−0.662368	−	0.749179i	$-0.730448\pi$
$613$	−20105.7	−1.32474	−0.662368	+	0.749179i	$0.730448\pi$
$614$	0	0
$615$	−19361.5	−1.26948
$616$	0	0
$617$	24167.0	1.57687	0.788434	−	0.615119i	$-0.210892\pi$
$617$	24167.0	1.57687	0.788434	+	0.615119i	$0.210892\pi$
$618$	0	0
$619$	−1170.34	−0.0759932	−0.0379966	−	0.999278i	$-0.512098\pi$
$619$	−1170.34	−0.0759932	−0.0379966	+	0.999278i	$0.512098\pi$
$620$	0	0
$621$	−1678.61	−0.108471
$622$	0	0
$623$	−5404.53	−0.347557
$624$	0	0
$625$	−18636.0	−1.19270
$626$	0	0
$627$	22849.0	1.45535
$628$	0	0
$629$	−19947.7	−1.26449
$630$	0	0
$631$	−8603.15	−0.542767	−0.271384	−	0.962471i	$-0.587481\pi$
$631$	−8603.15	−0.542767	−0.271384	+	0.962471i	$0.587481\pi$
$632$	0	0
$633$	−959.225	−0.0602303
$634$	0	0
$635$	−26161.2	−1.63492
$636$	0	0
$637$	−3299.02	−0.205199
$638$	0	0
$639$	628.478	0.0389080
$640$	0	0
$641$	−4145.99	−0.255470	−0.127735	−	0.991808i	$-0.540771\pi$
$641$	−4145.99	−0.255470	−0.127735	+	0.991808i	$0.540771\pi$
$642$	0	0
$643$	−10397.1	−0.637667	−0.318833	−	0.947811i	$-0.603291\pi$
$643$	−10397.1	−0.637667	−0.318833	+	0.947811i	$0.603291\pi$
$644$	0	0
$645$	8462.06	0.516579
$646$	0	0
$647$	24039.8	1.46074	0.730371	−	0.683050i	$-0.239347\pi$
$647$	24039.8	1.46074	0.730371	+	0.683050i	$0.239347\pi$
$648$	0	0
$649$	32416.9	1.96067
$650$	0	0
$651$	1937.79	0.116663
$652$	0	0
$653$	5877.28	0.352214	0.176107	−	0.984371i	$-0.443650\pi$
$653$	5877.28	0.352214	0.176107	+	0.984371i	$0.443650\pi$
$654$	0	0
$655$	−12010.2	−0.716457
$656$	0	0
$657$	7342.52	0.436011
$658$	0	0
$659$	−24314.2	−1.43725	−0.718625	−	0.695398i	$-0.755228\pi$
$659$	−24314.2	−1.43725	−0.718625	+	0.695398i	$0.755228\pi$
$660$	0	0
$661$	−29089.2	−1.71171	−0.855854	−	0.517217i	$-0.826968\pi$
$661$	−29089.2	−1.71171	−0.855854	+	0.517217i	$0.826968\pi$
$662$	0	0
$663$	4689.13	0.274677
$664$	0	0
$665$	−19068.4	−1.11194
$666$	0	0
$667$	10672.0	0.619521
$668$	0	0
$669$	9785.20	0.565497
$670$	0	0
$671$	14556.7	0.837490
$672$	0	0
$673$	−3261.01	−0.186779	−0.0933897	−	0.995630i	$-0.529770\pi$
$673$	−3261.01	−0.186779	−0.0933897	+	0.995630i	$0.529770\pi$
$674$	0	0
$675$	−2495.37	−0.142292
$676$	0	0
$677$	15283.6	0.867645	0.433822	−	0.900998i	$-0.357165\pi$
$677$	15283.6	0.867645	0.433822	+	0.900998i	$0.357165\pi$
$678$	0	0
$679$	1369.96	0.0774287
$680$	0	0
$681$	4917.43	0.276705
$682$	0	0
$683$	−25901.5	−1.45109	−0.725544	−	0.688176i	$-0.758412\pi$
$683$	−25901.5	−1.45109	−0.725544	+	0.688176i	$0.758412\pi$
$684$	0	0
$685$	−19589.8	−1.09268
$686$	0	0
$687$	15596.7	0.866161
$688$	0	0
$689$	2024.42	0.111936
$690$	0	0
$691$	−33260.9	−1.83112	−0.915560	−	0.402181i	$-0.868252\pi$
$691$	−33260.9	−1.83112	−0.915560	+	0.402181i	$0.868252\pi$
$692$	0	0
$693$	−4729.69	−0.259258
$694$	0	0
$695$	10843.9	0.591846
$696$	0	0
$697$	52625.3	2.85986
$698$	0	0
$699$	17684.2	0.956908
$700$	0	0
$701$	−12913.7	−0.695783	−0.347891	−	0.937535i	$-0.613102\pi$
$701$	−12913.7	−0.695783	−0.347891	+	0.937535i	$0.613102\pi$
$702$	0	0
$703$	−22713.0	−1.21855
$704$	0	0
$705$	12013.4	0.641774
$706$	0	0
$707$	−12591.6	−0.669810
$708$	0	0
$709$	−4763.90	−0.252344	−0.126172	−	0.992008i	$-0.540269\pi$
$709$	−4763.90	−0.252344	−0.126172	+	0.992008i	$0.540269\pi$
$710$	0	0
$711$	−3570.61	−0.188338
$712$	0	0
$713$	4251.25	0.223297
$714$	0	0
$715$	−10664.2	−0.557791
$716$	0	0
$717$	9878.99	0.514557
$718$	0	0
$719$	−19464.3	−1.00959	−0.504795	−	0.863239i	$-0.668432\pi$
$719$	−19464.3	−1.00959	−0.504795	+	0.863239i	$0.668432\pi$
$720$	0	0
$721$	4035.41	0.208442
$722$	0	0
$723$	−415.643	−0.0213803
$724$	0	0
$725$	15864.6	0.812685
$726$	0	0
$727$	10674.2	0.544546	0.272273	−	0.962220i	$-0.412225\pi$
$727$	10674.2	0.544546	0.272273	+	0.962220i	$0.412225\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−23000.2	−1.16374
$732$	0	0
$733$	18606.8	0.937597	0.468798	−	0.883305i	$-0.344687\pi$
$733$	18606.8	0.937597	0.468798	+	0.883305i	$0.344687\pi$
$734$	0	0
$735$	−11225.7	−0.563355
$736$	0	0
$737$	10996.8	0.549622
$738$	0	0
$739$	5357.18	0.266667	0.133334	−	0.991071i	$-0.457432\pi$
$739$	5357.18	0.266667	0.133334	+	0.991071i	$0.457432\pi$
$740$	0	0
$741$	5339.18	0.264696
$742$	0	0
$743$	−25371.2	−1.25273	−0.626364	−	0.779530i	$-0.715458\pi$
$743$	−25371.2	−1.25273	−0.626364	+	0.779530i	$0.715458\pi$
$744$	0	0
$745$	34810.9	1.71191
$746$	0	0
$747$	−8444.66	−0.413620
$748$	0	0
$749$	−4101.36	−0.200080
$750$	0	0
$751$	9775.61	0.474989	0.237495	−	0.971389i	$-0.423674\pi$
$751$	9775.61	0.474989	0.237495	+	0.971389i	$0.423674\pi$
$752$	0	0
$753$	8438.77	0.408401
$754$	0	0
$755$	40188.7	1.93724
$756$	0	0
$757$	22496.9	1.08014	0.540068	−	0.841621i	$-0.318398\pi$
$757$	22496.9	1.08014	0.540068	+	0.841621i	$0.318398\pi$
$758$	0	0
$759$	−10376.3	−0.496228
$760$	0	0
$761$	−40282.4	−1.91884	−0.959419	−	0.281986i	$-0.909007\pi$
$761$	−40282.4	−1.91884	−0.959419	+	0.281986i	$0.909007\pi$
$762$	0	0
$763$	−10114.9	−0.479926
$764$	0	0
$765$	15955.9	0.754100
$766$	0	0
$767$	7574.94	0.356604
$768$	0	0
$769$	15223.3	0.713872	0.356936	−	0.934129i	$-0.383821\pi$
$769$	15223.3	0.713872	0.356936	+	0.934129i	$0.383821\pi$
$770$	0	0
$771$	−13350.3	−0.623607
$772$	0	0
$773$	−1830.17	−0.0851574	−0.0425787	−	0.999093i	$-0.513557\pi$
$773$	−1830.17	−0.0851574	−0.0425787	+	0.999093i	$0.513557\pi$
$774$	0	0
$775$	6319.78	0.292920
$776$	0	0
$777$	4701.54	0.217074
$778$	0	0
$779$	59920.6	2.75594
$780$	0	0
$781$	3884.93	0.177995
$782$	0	0
$783$	−4634.70	−0.211533
$784$	0	0
$785$	−6049.25	−0.275041
$786$	0	0
$787$	−10219.7	−0.462889	−0.231445	−	0.972848i	$-0.574345\pi$
$787$	−10219.7	−0.462889	−0.231445	+	0.972848i	$0.574345\pi$
$788$	0	0
$789$	5367.98	0.242212
$790$	0	0
$791$	4615.92	0.207488
$792$	0	0
$793$	3401.50	0.152321
$794$	0	0
$795$	6888.57	0.307311
$796$	0	0
$797$	−7383.96	−0.328172	−0.164086	−	0.986446i	$-0.552468\pi$
$797$	−7383.96	−0.328172	−0.164086	+	0.986446i	$0.552468\pi$
$798$	0	0
$799$	−32652.9	−1.44578
$800$	0	0
$801$	5149.28	0.227142
$802$	0	0
$803$	45387.7	1.99464
$804$	0	0
$805$	8659.48	0.379139
$806$	0	0
$807$	−23737.9	−1.03546
$808$	0	0
$809$	−16125.8	−0.700807	−0.350403	−	0.936599i	$-0.613956\pi$
$809$	−16125.8	−0.700807	−0.350403	+	0.936599i	$0.613956\pi$
$810$	0	0
$811$	−38170.9	−1.65273	−0.826363	−	0.563138i	$-0.809594\pi$
$811$	−38170.9	−1.65273	−0.826363	+	0.563138i	$0.809594\pi$
$812$	0	0
$813$	−12703.4	−0.548002
$814$	0	0
$815$	−18710.2	−0.804160
$816$	0	0
$817$	−26188.7	−1.12145
$818$	0	0
$819$	−1105.20	−0.0471535
$820$	0	0
$821$	45756.8	1.94509	0.972547	−	0.232705i	$-0.0747577\pi$
$821$	45756.8	1.94509	0.972547	+	0.232705i	$0.0747577\pi$
$822$	0	0
$823$	35523.2	1.50457	0.752285	−	0.658837i	$-0.228951\pi$
$823$	35523.2	1.50457	0.752285	+	0.658837i	$0.228951\pi$
$824$	0	0
$825$	−15425.1	−0.650950
$826$	0	0
$827$	29187.4	1.22726	0.613630	−	0.789594i	$-0.289709\pi$
$827$	29187.4	1.22726	0.613630	+	0.789594i	$0.289709\pi$
$828$	0	0
$829$	−13111.1	−0.549297	−0.274648	−	0.961545i	$-0.588561\pi$
$829$	−13111.1	−0.549297	−0.274648	+	0.961545i	$0.588561\pi$
$830$	0	0
$831$	−18884.1	−0.788305
$832$	0	0
$833$	30511.9	1.26912
$834$	0	0
$835$	52216.6	2.16411
$836$	0	0
$837$	−1846.26	−0.0762440
$838$	0	0
$839$	19083.7	0.785269	0.392635	−	0.919695i	$-0.371564\pi$
$839$	19083.7	0.785269	0.392635	+	0.919695i	$0.371564\pi$
$840$	0	0
$841$	5076.62	0.208152
$842$	0	0
$843$	−15822.4	−0.646444
$844$	0	0
$845$	−2491.94	−0.101450
$846$	0	0
$847$	−16663.7	−0.676001
$848$	0	0
$849$	8614.36	0.348226
$850$	0	0
$851$	10314.6	0.415487
$852$	0	0
$853$	−14986.8	−0.601570	−0.300785	−	0.953692i	$-0.597249\pi$
$853$	−14986.8	−0.601570	−0.300785	+	0.953692i	$0.597249\pi$
$854$	0	0
$855$	18167.9	0.726699
$856$	0	0
$857$	−32888.7	−1.31092	−0.655459	−	0.755231i	$-0.727525\pi$
$857$	−32888.7	−1.31092	−0.655459	+	0.755231i	$0.727525\pi$
$858$	0	0
$859$	7238.15	0.287500	0.143750	−	0.989614i	$-0.454084\pi$
$859$	7238.15	0.287500	0.143750	+	0.989614i	$0.454084\pi$
$860$	0	0
$861$	−12403.4	−0.490950
$862$	0	0
$863$	−33335.2	−1.31488	−0.657442	−	0.753505i	$-0.728361\pi$
$863$	−33335.2	−1.31488	−0.657442	+	0.753505i	$0.728361\pi$
$864$	0	0
$865$	9883.30	0.388488
$866$	0	0
$867$	−28629.8	−1.12147
$868$	0	0
$869$	−22071.7	−0.861602
$870$	0	0
$871$	2569.64	0.0999644
$872$	0	0
$873$	−1305.25	−0.0506027
$874$	0	0
$875$	−4537.75	−0.175319
$876$	0	0
$877$	39278.6	1.51236	0.756182	−	0.654361i	$-0.227062\pi$
$877$	39278.6	1.51236	0.756182	+	0.654361i	$0.227062\pi$
$878$	0	0
$879$	6649.98	0.255175
$880$	0	0
$881$	8324.62	0.318347	0.159173	−	0.987251i	$-0.449117\pi$
$881$	8324.62	0.318347	0.159173	+	0.987251i	$0.449117\pi$
$882$	0	0
$883$	8434.60	0.321457	0.160729	−	0.986999i	$-0.448616\pi$
$883$	8434.60	0.321457	0.160729	+	0.986999i	$0.448616\pi$
$884$	0	0
$885$	25775.6	0.979024
$886$	0	0
$887$	20767.8	0.786148	0.393074	−	0.919507i	$-0.371412\pi$
$887$	20767.8	0.786148	0.393074	+	0.919507i	$0.371412\pi$
$888$	0	0
$889$	−16759.5	−0.632278
$890$	0	0
$891$	4506.31	0.169436
$892$	0	0
$893$	−37179.5	−1.39324
$894$	0	0
$895$	65121.0	2.43213
$896$	0	0
$897$	−2424.66	−0.0902532
$898$	0	0
$899$	11737.8	0.435460
$900$	0	0
$901$	−18723.4	−0.692306
$902$	0	0
$903$	5421.00	0.199778
$904$	0	0
$905$	24090.5	0.884856
$906$	0	0
$907$	−4837.14	−0.177083	−0.0885416	−	0.996072i	$-0.528221\pi$
$907$	−4837.14	−0.177083	−0.0885416	+	0.996072i	$0.528221\pi$
$908$	0	0
$909$	11996.9	0.437747
$910$	0	0
$911$	−17048.3	−0.620016	−0.310008	−	0.950734i	$-0.600332\pi$
$911$	−17048.3	−0.620016	−0.310008	+	0.950734i	$0.600332\pi$
$912$	0	0
$913$	−52200.6	−1.89221
$914$	0	0
$915$	11574.4	0.418184
$916$	0	0
$917$	−7694.05	−0.277077
$918$	0	0
$919$	36802.1	1.32099	0.660494	−	0.750831i	$-0.270347\pi$
$919$	36802.1	1.32099	0.660494	+	0.750831i	$0.270347\pi$
$920$	0	0
$921$	19890.1	0.711620
$922$	0	0
$923$	907.801	0.0323734
$924$	0	0
$925$	15333.3	0.545034
$926$	0	0
$927$	−3844.82	−0.136225
$928$	0	0
$929$	−45335.7	−1.60109	−0.800547	−	0.599270i	$-0.795457\pi$
$929$	−45335.7	−1.60109	−0.800547	+	0.599270i	$0.795457\pi$
$930$	0	0
$931$	34741.7	1.22300
$932$	0	0
$933$	19251.6	0.675531
$934$	0	0
$935$	98631.3	3.44983
$936$	0	0
$937$	−4469.74	−0.155838	−0.0779188	−	0.996960i	$-0.524828\pi$
$937$	−4469.74	−0.155838	−0.0779188	+	0.996960i	$0.524828\pi$
$938$	0	0
$939$	30091.6	1.04579
$940$	0	0
$941$	−21175.8	−0.733593	−0.366796	−	0.930301i	$-0.619545\pi$
$941$	−21175.8	−0.733593	−0.366796	+	0.930301i	$0.619545\pi$
$942$	0	0
$943$	−27211.5	−0.939692
$944$	0	0
$945$	−3760.70	−0.129456
$946$	0	0
$947$	41219.5	1.41442	0.707209	−	0.707004i	$-0.249954\pi$
$947$	41219.5	1.41442	0.707209	+	0.707004i	$0.249954\pi$
$948$	0	0
$949$	10605.9	0.362783
$950$	0	0
$951$	18242.8	0.622042
$952$	0	0
$953$	−15109.0	−0.513565	−0.256783	−	0.966469i	$-0.582662\pi$
$953$	−15109.0	−0.513565	−0.256783	+	0.966469i	$0.582662\pi$
$954$	0	0
$955$	58410.0	1.97917
$956$	0	0
$957$	−28649.4	−0.967714
$958$	0	0
$959$	−12549.7	−0.422577
$960$	0	0
$961$	−25115.2	−0.843045
$962$	0	0
$963$	3907.65	0.130760
$964$	0	0
$965$	32381.6	1.08021
$966$	0	0
$967$	27743.9	0.922631	0.461315	−	0.887236i	$-0.347378\pi$
$967$	27743.9	0.922631	0.461315	+	0.887236i	$0.347378\pi$
$968$	0	0
$969$	−49380.9	−1.63709
$970$	0	0
$971$	36606.5	1.20984	0.604922	−	0.796284i	$-0.293204\pi$
$971$	36606.5	1.20984	0.604922	+	0.796284i	$0.293204\pi$
$972$	0	0
$973$	6946.87	0.228886
$974$	0	0
$975$	−3604.43	−0.118394
$976$	0	0
$977$	−26268.0	−0.860171	−0.430086	−	0.902788i	$-0.641517\pi$
$977$	−26268.0	−0.860171	−0.430086	+	0.902788i	$0.641517\pi$
$978$	0	0
$979$	31830.3	1.03912
$980$	0	0
$981$	9637.17	0.313650
$982$	0	0
$983$	−24696.3	−0.801311	−0.400656	−	0.916229i	$-0.631218\pi$
$983$	−24696.3	−0.801311	−0.400656	+	0.916229i	$0.631218\pi$
$984$	0	0
$985$	−51772.2	−1.67472
$986$	0	0
$987$	7696.06	0.248195
$988$	0	0
$989$	11893.0	0.382381
$990$	0	0
$991$	52765.4	1.69137	0.845685	−	0.533683i	$-0.179193\pi$
$991$	52765.4	1.69137	0.845685	+	0.533683i	$0.179193\pi$
$992$	0	0
$993$	−12508.5	−0.399744
$994$	0	0
$995$	−70969.1	−2.26118
$996$	0	0
$997$	41868.0	1.32996	0.664981	−	0.746860i	$-0.268440\pi$
$997$	41868.0	1.32996	0.664981	+	0.746860i	$0.268440\pi$
$998$	0	0
$999$	−4479.49	−0.141867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.ca.1.1		5
4.3	odd	2		2496.4.a.cf.1.1		5
8.3	odd	2		1248.4.a.g.1.5	✓	5
8.5	even	2		1248.4.a.l.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.4.a.g.1.5	✓	5	8.3	odd	2
1248.4.a.l.1.5	yes	5	8.5	even	2
2496.4.a.ca.1.1		5	1.1	even	1	trivial
2496.4.a.cf.1.1		5	4.3	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -14.7452 q^{5} -9.44613 q^{7} +9.00000 q^{9} +55.6334 q^{11} +13.0000 q^{13} +44.2356 q^{15} -120.234 q^{17} -136.902 q^{19} +28.3384 q^{21} +62.1708 q^{23} +92.4212 q^{25} -27.0000 q^{27} +171.656 q^{29} +68.3802 q^{31} -166.900 q^{33} +139.285 q^{35} +165.907 q^{37} -39.0000 q^{39} -437.690 q^{41} +191.295 q^{43} -132.707 q^{45} +271.577 q^{47} -253.771 q^{49} +360.703 q^{51} +155.725 q^{53} -820.327 q^{55} +410.706 q^{57} +582.688 q^{59} +261.654 q^{61} -85.0152 q^{63} -191.688 q^{65} +197.665 q^{67} -186.513 q^{69} +69.8308 q^{71} +815.835 q^{73} -277.264 q^{75} -525.521 q^{77} -396.735 q^{79} +81.0000 q^{81} -938.295 q^{83} +1772.88 q^{85} -514.967 q^{87} +572.142 q^{89} -122.800 q^{91} -205.141 q^{93} +2018.65 q^{95} -145.028 q^{97} +500.701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} + 10 q^{5} - 14 q^{7} + 45 q^{9} + 22 q^{11} + 65 q^{13} - 30 q^{15} - 34 q^{17} - 90 q^{19} + 42 q^{21} - 96 q^{23} + 107 q^{25} - 135 q^{27} + 54 q^{29} - 378 q^{31} - 66 q^{33} + 84 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 + 10 * q^5 - 14 * q^7 + 45 * q^9 + 22 * q^11 + 65 * q^13 - 30 * q^15 - 34 * q^17 - 90 * q^19 + 42 * q^21 - 96 * q^23 + 107 * q^25 - 135 * q^27 + 54 * q^29 - 378 * q^31 - 66 * q^33 + 84 * q^35 + 318 * q^37 - 195 * q^39 - 458 * q^41 + 104 * q^43 + 90 * q^45 - 474 * q^47 + 25 * q^49 + 102 * q^51 + 450 * q^53 - 1116 * q^55 + 270 * q^57 + 822 * q^59 + 546 * q^61 - 126 * q^63 + 130 * q^65 - 130 * q^67 + 288 * q^69 - 1646 * q^71 - 166 * q^73 - 321 * q^75 + 900 * q^77 - 196 * q^79 + 405 * q^81 - 118 * q^83 + 3340 * q^85 - 162 * q^87 - 186 * q^89 - 182 * q^91 + 1134 * q^93 - 564 * q^95 - 2582 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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