LMFDB - Embedded newform 2496.4.a.bg.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -1.46410 q^{5} -8.39230 q^{7} +9.00000 q^{9} -34.7846 q^{11} +13.0000 q^{13} -4.39230 q^{15} -108.067 q^{17} -143.244 q^{19} -25.1769 q^{21} -128.708 q^{23} -122.856 q^{25} +27.0000 q^{27} +18.8616 q^{29} -78.5359 q^{31} -104.354 q^{33} +12.2872 q^{35} +327.072 q^{37} +39.0000 q^{39} +327.587 q^{41} +336.918 q^{43} -13.1769 q^{45} +99.2820 q^{47} -272.569 q^{49} -324.200 q^{51} +686.554 q^{53} +50.9282 q^{55} -429.731 q^{57} +242.420 q^{59} +644.851 q^{61} -75.5307 q^{63} -19.0333 q^{65} +871.643 q^{67} -386.123 q^{69} +100.221 q^{71} +604.600 q^{73} -368.569 q^{75} +291.923 q^{77} +1070.39 q^{79} +81.0000 q^{81} -741.672 q^{83} +158.221 q^{85} +56.5847 q^{87} -501.577 q^{89} -109.100 q^{91} -235.608 q^{93} +209.723 q^{95} -1569.71 q^{97} -313.061 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 4 q^{5} + 4 q^{7} + 18 q^{9} - 28 q^{11} + 26 q^{13} + 12 q^{15} - 36 q^{17} - 44 q^{19} + 12 q^{21} - 8 q^{23} - 218 q^{25} + 54 q^{27} + 204 q^{29} - 164 q^{31} - 84 q^{33} + 80 q^{35}+ \cdots - 252 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 4 * q^5 + 4 * q^7 + 18 * q^9 - 28 * q^11 + 26 * q^13 + 12 * q^15 - 36 * q^17 - 44 * q^19 + 12 * q^21 - 8 * q^23 - 218 * q^25 + 54 * q^27 + 204 * q^29 - 164 * q^31 - 84 * q^33 + 80 * q^35 + 668 * q^37 + 78 * q^39 - 100 * q^41 + 272 * q^43 + 36 * q^45 + 60 * q^47 - 462 * q^49 - 108 * q^51 + 708 * q^53 + 88 * q^55 - 132 * q^57 + 180 * q^59 + 1068 * q^61 + 36 * q^63 + 52 * q^65 + 420 * q^67 - 24 * q^69 + 436 * q^71 - 412 * q^73 - 654 * q^75 + 376 * q^77 + 672 * q^79 + 162 * q^81 + 124 * q^83 + 552 * q^85 + 612 * q^87 + 140 * q^89 + 52 * q^91 - 492 * q^93 + 752 * q^95 - 188 * q^97 - 252 * 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
$5$	−1.46410	−0.130953	−0.0654766	−	0.997854i	$-0.520857\pi$
$5$	−1.46410	−0.130953	−0.0654766	+	0.997854i	$0.520857\pi$
$6$	0	0
$7$	−8.39230	−0.453142	−0.226571	−	0.973995i	$-0.572752\pi$
$7$	−8.39230	−0.453142	−0.226571	+	0.973995i	$0.572752\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−34.7846	−0.953450	−0.476725	−	0.879052i	$-0.658176\pi$
$11$	−34.7846	−0.953450	−0.476725	+	0.879052i	$0.658176\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−4.39230	−0.0756059
$16$	0	0
$17$	−108.067	−1.54177	−0.770883	−	0.636977i	$-0.780185\pi$
$17$	−108.067	−1.54177	−0.770883	+	0.636977i	$0.780185\pi$
$18$	0	0
$19$	−143.244	−1.72960	−0.864798	−	0.502120i	$-0.832554\pi$
$19$	−143.244	−1.72960	−0.864798	+	0.502120i	$0.832554\pi$
$20$	0	0
$21$	−25.1769	−0.261622
$22$	0	0
$23$	−128.708	−1.16684	−0.583422	−	0.812169i	$-0.698287\pi$
$23$	−128.708	−1.16684	−0.583422	+	0.812169i	$0.698287\pi$
$24$	0	0
$25$	−122.856	−0.982851
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	18.8616	0.120776	0.0603880	−	0.998175i	$-0.480766\pi$
$29$	18.8616	0.120776	0.0603880	+	0.998175i	$0.480766\pi$
$30$	0	0
$31$	−78.5359	−0.455015	−0.227507	−	0.973776i	$-0.573058\pi$
$31$	−78.5359	−0.455015	−0.227507	+	0.973776i	$0.573058\pi$
$32$	0	0
$33$	−104.354	−0.550475
$34$	0	0
$35$	12.2872	0.0593404
$36$	0	0
$37$	327.072	1.45325	0.726625	−	0.687034i	$-0.241088\pi$
$37$	327.072	1.45325	0.726625	+	0.687034i	$0.241088\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	327.587	1.24782	0.623909	−	0.781497i	$-0.285544\pi$
$41$	327.587	1.24782	0.623909	+	0.781497i	$0.285544\pi$
$42$	0	0
$43$	336.918	1.19487	0.597436	−	0.801917i	$-0.296186\pi$
$43$	336.918	1.19487	0.597436	+	0.801917i	$0.296186\pi$
$44$	0	0
$45$	−13.1769	−0.0436511
$46$	0	0
$47$	99.2820	0.308123	0.154061	−	0.988061i	$-0.450765\pi$
$47$	99.2820	0.308123	0.154061	+	0.988061i	$0.450765\pi$
$48$	0	0
$49$	−272.569	−0.794662
$50$	0	0
$51$	−324.200	−0.890139
$52$	0	0
$53$	686.554	1.77935	0.889674	−	0.456597i	$-0.150932\pi$
$53$	686.554	1.77935	0.889674	+	0.456597i	$0.150932\pi$
$54$	0	0
$55$	50.9282	0.124857
$56$	0	0
$57$	−429.731	−0.998583
$58$	0	0
$59$	242.420	0.534923	0.267462	−	0.963569i	$-0.413815\pi$
$59$	242.420	0.534923	0.267462	+	0.963569i	$0.413815\pi$
$60$	0	0
$61$	644.851	1.35352	0.676760	−	0.736204i	$-0.263383\pi$
$61$	644.851	1.35352	0.676760	+	0.736204i	$0.263383\pi$
$62$	0	0
$63$	−75.5307	−0.151047
$64$	0	0
$65$	−19.0333	−0.0363199
$66$	0	0
$67$	871.643	1.58938	0.794688	−	0.607018i	$-0.207634\pi$
$67$	871.643	1.58938	0.794688	+	0.607018i	$0.207634\pi$
$68$	0	0
$69$	−386.123	−0.673677
$70$	0	0
$71$	100.221	0.167521	0.0837605	−	0.996486i	$-0.473307\pi$
$71$	100.221	0.167521	0.0837605	+	0.996486i	$0.473307\pi$
$72$	0	0
$73$	604.600	0.969357	0.484678	−	0.874692i	$-0.338937\pi$
$73$	604.600	0.969357	0.484678	+	0.874692i	$0.338937\pi$
$74$	0	0
$75$	−368.569	−0.567449
$76$	0	0
$77$	291.923	0.432048
$78$	0	0
$79$	1070.39	1.52441	0.762204	−	0.647337i	$-0.224117\pi$
$79$	1070.39	1.52441	0.762204	+	0.647337i	$0.224117\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−741.672	−0.980832	−0.490416	−	0.871489i	$-0.663155\pi$
$83$	−741.672	−0.980832	−0.490416	+	0.871489i	$0.663155\pi$
$84$	0	0
$85$	158.221	0.201899
$86$	0	0
$87$	56.5847	0.0697301
$88$	0	0
$89$	−501.577	−0.597382	−0.298691	−	0.954350i	$-0.596550\pi$
$89$	−501.577	−0.597382	−0.298691	+	0.954350i	$0.596550\pi$
$90$	0	0
$91$	−109.100	−0.125679
$92$	0	0
$93$	−235.608	−0.262703
$94$	0	0
$95$	209.723	0.226496
$96$	0	0
$97$	−1569.71	−1.64309	−0.821544	−	0.570144i	$-0.806887\pi$
$97$	−1569.71	−1.64309	−0.821544	+	0.570144i	$0.806887\pi$
$98$	0	0
$99$	−313.061	−0.317817
$100$	0	0
$101$	−1639.34	−1.61505	−0.807526	−	0.589832i	$-0.799194\pi$
$101$	−1639.34	−1.61505	−0.807526	+	0.589832i	$0.799194\pi$
$102$	0	0
$103$	−830.333	−0.794322	−0.397161	−	0.917749i	$-0.630005\pi$
$103$	−830.333	−0.794322	−0.397161	+	0.917749i	$0.630005\pi$
$104$	0	0
$105$	36.8616	0.0342602
$106$	0	0
$107$	−1323.67	−1.19593	−0.597963	−	0.801523i	$-0.704023\pi$
$107$	−1323.67	−1.19593	−0.597963	+	0.801523i	$0.704023\pi$
$108$	0	0
$109$	426.441	0.374731	0.187365	−	0.982290i	$-0.440005\pi$
$109$	426.441	0.374731	0.187365	+	0.982290i	$0.440005\pi$
$110$	0	0
$111$	981.215	0.839035
$112$	0	0
$113$	1548.66	1.28925	0.644625	−	0.764499i	$-0.277013\pi$
$113$	1548.66	1.28925	0.644625	+	0.764499i	$0.277013\pi$
$114$	0	0
$115$	188.441	0.152802
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	906.928	0.698638
$120$	0	0
$121$	−121.031	−0.0909323
$122$	0	0
$123$	982.761	0.720428
$124$	0	0
$125$	362.887	0.259661
$126$	0	0
$127$	1456.63	1.01776	0.508878	−	0.860839i	$-0.330060\pi$
$127$	1456.63	1.01776	0.508878	+	0.860839i	$0.330060\pi$
$128$	0	0
$129$	1010.75	0.689860
$130$	0	0
$131$	1822.88	1.21577	0.607885	−	0.794025i	$-0.292018\pi$
$131$	1822.88	1.21577	0.607885	+	0.794025i	$0.292018\pi$
$132$	0	0
$133$	1202.14	0.783752
$134$	0	0
$135$	−39.5307	−0.0252020
$136$	0	0
$137$	609.926	0.380361	0.190181	−	0.981749i	$-0.439093\pi$
$137$	609.926	0.380361	0.190181	+	0.981749i	$0.439093\pi$
$138$	0	0
$139$	−548.687	−0.334813	−0.167407	−	0.985888i	$-0.553539\pi$
$139$	−548.687	−0.334813	−0.167407	+	0.985888i	$0.553539\pi$
$140$	0	0
$141$	297.846	0.177895
$142$	0	0
$143$	−452.200	−0.264440
$144$	0	0
$145$	−27.6152	−0.0158160
$146$	0	0
$147$	−817.708	−0.458799
$148$	0	0
$149$	−1863.89	−1.02480	−0.512402	−	0.858746i	$-0.671244\pi$
$149$	−1863.89	−1.02480	−0.512402	+	0.858746i	$0.671244\pi$
$150$	0	0
$151$	−194.418	−0.104778	−0.0523890	−	0.998627i	$-0.516684\pi$
$151$	−194.418	−0.104778	−0.0523890	+	0.998627i	$0.516684\pi$
$152$	0	0
$153$	−972.600	−0.513922
$154$	0	0
$155$	114.985	0.0595857
$156$	0	0
$157$	215.939	0.109769	0.0548847	−	0.998493i	$-0.482521\pi$
$157$	215.939	0.109769	0.0548847	+	0.998493i	$0.482521\pi$
$158$	0	0
$159$	2059.66	1.02731
$160$	0	0
$161$	1080.15	0.528746
$162$	0	0
$163$	−1331.28	−0.639717	−0.319858	−	0.947465i	$-0.603635\pi$
$163$	−1331.28	−0.639717	−0.319858	+	0.947465i	$0.603635\pi$
$164$	0	0
$165$	152.785	0.0720865
$166$	0	0
$167$	3189.68	1.47799	0.738997	−	0.673709i	$-0.235300\pi$
$167$	3189.68	1.47799	0.738997	+	0.673709i	$0.235300\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1289.19	−0.576532
$172$	0	0
$173$	3407.54	1.49752	0.748759	−	0.662842i	$-0.230650\pi$
$173$	3407.54	1.49752	0.748759	+	0.662842i	$0.230650\pi$
$174$	0	0
$175$	1031.05	0.445371
$176$	0	0
$177$	727.261	0.308838
$178$	0	0
$179$	−1020.46	−0.426106	−0.213053	−	0.977041i	$-0.568341\pi$
$179$	−1020.46	−0.426106	−0.213053	+	0.977041i	$0.568341\pi$
$180$	0	0
$181$	3458.60	1.42031	0.710155	−	0.704046i	$-0.248625\pi$
$181$	3458.60	1.42031	0.710155	+	0.704046i	$0.248625\pi$
$182$	0	0
$183$	1934.55	0.781455
$184$	0	0
$185$	−478.866	−0.190308
$186$	0	0
$187$	3759.06	1.47000
$188$	0	0
$189$	−226.592	−0.0872072
$190$	0	0
$191$	−3014.88	−1.14214	−0.571071	−	0.820901i	$-0.693472\pi$
$191$	−3014.88	−1.14214	−0.571071	+	0.820901i	$0.693472\pi$
$192$	0	0
$193$	539.795	0.201323	0.100661	−	0.994921i	$-0.467904\pi$
$193$	539.795	0.201323	0.100661	+	0.994921i	$0.467904\pi$
$194$	0	0
$195$	−57.1000	−0.0209693
$196$	0	0
$197$	−3630.70	−1.31308	−0.656541	−	0.754291i	$-0.727981\pi$
$197$	−3630.70	−1.31308	−0.656541	+	0.754291i	$0.727981\pi$
$198$	0	0
$199$	−3846.40	−1.37017	−0.685086	−	0.728462i	$-0.740236\pi$
$199$	−3846.40	−1.37017	−0.685086	+	0.728462i	$0.740236\pi$
$200$	0	0
$201$	2614.93	0.917627
$202$	0	0
$203$	−158.292	−0.0547287
$204$	0	0
$205$	−479.621	−0.163406
$206$	0	0
$207$	−1158.37	−0.388948
$208$	0	0
$209$	4982.67	1.64908
$210$	0	0
$211$	−993.169	−0.324041	−0.162020	−	0.986787i	$-0.551801\pi$
$211$	−993.169	−0.324041	−0.162020	+	0.986787i	$0.551801\pi$
$212$	0	0
$213$	300.662	0.0967183
$214$	0	0
$215$	−493.282	−0.156472
$216$	0	0
$217$	659.097	0.206186
$218$	0	0
$219$	1813.80	0.559658
$220$	0	0
$221$	−1404.87	−0.427609
$222$	0	0
$223$	3813.70	1.14522	0.572610	−	0.819828i	$-0.305931\pi$
$223$	3813.70	1.14522	0.572610	+	0.819828i	$0.305931\pi$
$224$	0	0
$225$	−1105.71	−0.327617
$226$	0	0
$227$	3002.36	0.877859	0.438929	−	0.898522i	$-0.355358\pi$
$227$	3002.36	0.877859	0.438929	+	0.898522i	$0.355358\pi$
$228$	0	0
$229$	3848.06	1.11042	0.555212	−	0.831709i	$-0.312637\pi$
$229$	3848.06	1.11042	0.555212	+	0.831709i	$0.312637\pi$
$230$	0	0
$231$	875.769	0.249443
$232$	0	0
$233$	2014.59	0.566438	0.283219	−	0.959055i	$-0.408598\pi$
$233$	2014.59	0.566438	0.283219	+	0.959055i	$0.408598\pi$
$234$	0	0
$235$	−145.359	−0.0403497
$236$	0	0
$237$	3211.17	0.880117
$238$	0	0
$239$	−5499.39	−1.48839	−0.744196	−	0.667961i	$-0.767167\pi$
$239$	−5499.39	−1.48839	−0.744196	+	0.667961i	$0.767167\pi$
$240$	0	0
$241$	197.872	0.0528883	0.0264441	−	0.999650i	$-0.491582\pi$
$241$	197.872	0.0528883	0.0264441	+	0.999650i	$0.491582\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	399.069	0.104064
$246$	0	0
$247$	−1862.17	−0.479704
$248$	0	0
$249$	−2225.01	−0.566283
$250$	0	0
$251$	1708.87	0.429733	0.214867	−	0.976643i	$-0.431068\pi$
$251$	1708.87	0.429733	0.214867	+	0.976643i	$0.431068\pi$
$252$	0	0
$253$	4477.05	1.11253
$254$	0	0
$255$	474.662	0.116567
$256$	0	0
$257$	5410.23	1.31316	0.656578	−	0.754258i	$-0.272003\pi$
$257$	5410.23	1.31316	0.656578	+	0.754258i	$0.272003\pi$
$258$	0	0
$259$	−2744.89	−0.658529
$260$	0	0
$261$	169.754	0.0402587
$262$	0	0
$263$	−3232.63	−0.757917	−0.378959	−	0.925414i	$-0.623718\pi$
$263$	−3232.63	−0.757917	−0.378959	+	0.925414i	$0.623718\pi$
$264$	0	0
$265$	−1005.18	−0.233011
$266$	0	0
$267$	−1504.73	−0.344899
$268$	0	0
$269$	6804.99	1.54241	0.771204	−	0.636588i	$-0.219655\pi$
$269$	6804.99	1.54241	0.771204	+	0.636588i	$0.219655\pi$
$270$	0	0
$271$	−7650.61	−1.71491	−0.857456	−	0.514557i	$-0.827956\pi$
$271$	−7650.61	−1.71491	−0.857456	+	0.514557i	$0.827956\pi$
$272$	0	0
$273$	−327.300	−0.0725608
$274$	0	0
$275$	4273.51	0.937100
$276$	0	0
$277$	−5246.78	−1.13808	−0.569040	−	0.822310i	$-0.692685\pi$
$277$	−5246.78	−1.13808	−0.569040	+	0.822310i	$0.692685\pi$
$278$	0	0
$279$	−706.823	−0.151672
$280$	0	0
$281$	3794.60	0.805576	0.402788	−	0.915293i	$-0.368041\pi$
$281$	3794.60	0.805576	0.402788	+	0.915293i	$0.368041\pi$
$282$	0	0
$283$	−471.574	−0.0990536	−0.0495268	−	0.998773i	$-0.515771\pi$
$283$	−471.574	−0.0990536	−0.0495268	+	0.998773i	$0.515771\pi$
$284$	0	0
$285$	629.169	0.130768
$286$	0	0
$287$	−2749.21	−0.565438
$288$	0	0
$289$	6765.40	1.37704
$290$	0	0
$291$	−4709.12	−0.948638
$292$	0	0
$293$	−30.5867	−0.00609862	−0.00304931	−	0.999995i	$-0.500971\pi$
$293$	−30.5867	−0.00609862	−0.00304931	+	0.999995i	$0.500971\pi$
$294$	0	0
$295$	−354.928	−0.0700499
$296$	0	0
$297$	−939.184	−0.183492
$298$	0	0
$299$	−1673.20	−0.323624
$300$	0	0
$301$	−2827.52	−0.541447
$302$	0	0
$303$	−4918.02	−0.932451
$304$	0	0
$305$	−944.128	−0.177248
$306$	0	0
$307$	7700.81	1.43162	0.715811	−	0.698294i	$-0.246057\pi$
$307$	7700.81	1.43162	0.715811	+	0.698294i	$0.246057\pi$
$308$	0	0
$309$	−2491.00	−0.458602
$310$	0	0
$311$	−2903.37	−0.529374	−0.264687	−	0.964334i	$-0.585269\pi$
$311$	−2903.37	−0.529374	−0.264687	+	0.964334i	$0.585269\pi$
$312$	0	0
$313$	−5645.42	−1.01948	−0.509742	−	0.860328i	$-0.670259\pi$
$313$	−5645.42	−1.01948	−0.509742	+	0.860328i	$0.670259\pi$
$314$	0	0
$315$	110.585	0.0197801
$316$	0	0
$317$	−1066.77	−0.189008	−0.0945041	−	0.995524i	$-0.530127\pi$
$317$	−1066.77	−0.189008	−0.0945041	+	0.995524i	$0.530127\pi$
$318$	0	0
$319$	−656.092	−0.115154
$320$	0	0
$321$	−3971.01	−0.690469
$322$	0	0
$323$	15479.9	2.66663
$324$	0	0
$325$	−1597.13	−0.272594
$326$	0	0
$327$	1279.32	0.216351
$328$	0	0
$329$	−833.205	−0.139623
$330$	0	0
$331$	−2205.86	−0.366300	−0.183150	−	0.983085i	$-0.558629\pi$
$331$	−2205.86	−0.366300	−0.183150	+	0.983085i	$0.558629\pi$
$332$	0	0
$333$	2943.65	0.484417
$334$	0	0
$335$	−1276.17	−0.208134
$336$	0	0
$337$	−439.466	−0.0710363	−0.0355182	−	0.999369i	$-0.511308\pi$
$337$	−439.466	−0.0710363	−0.0355182	+	0.999369i	$0.511308\pi$
$338$	0	0
$339$	4645.97	0.744349
$340$	0	0
$341$	2731.84	0.433834
$342$	0	0
$343$	5166.04	0.813237
$344$	0	0
$345$	565.323	0.0882202
$346$	0	0
$347$	−5767.97	−0.892336	−0.446168	−	0.894949i	$-0.647212\pi$
$347$	−5767.97	−0.892336	−0.446168	+	0.894949i	$0.647212\pi$
$348$	0	0
$349$	2582.12	0.396039	0.198020	−	0.980198i	$-0.436549\pi$
$349$	2582.12	0.396039	0.198020	+	0.980198i	$0.436549\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	8065.93	1.21616	0.608082	−	0.793874i	$-0.291939\pi$
$353$	8065.93	1.21616	0.608082	+	0.793874i	$0.291939\pi$
$354$	0	0
$355$	−146.733	−0.0219374
$356$	0	0
$357$	2720.78	0.403359
$358$	0	0
$359$	315.815	0.0464292	0.0232146	−	0.999731i	$-0.492610\pi$
$359$	315.815	0.0464292	0.0232146	+	0.999731i	$0.492610\pi$
$360$	0	0
$361$	13659.7	1.99150
$362$	0	0
$363$	−363.093	−0.0524998
$364$	0	0
$365$	−885.196	−0.126940
$366$	0	0
$367$	5183.06	0.737203	0.368601	−	0.929588i	$-0.379837\pi$
$367$	5183.06	0.737203	0.368601	+	0.929588i	$0.379837\pi$
$368$	0	0
$369$	2948.28	0.415939
$370$	0	0
$371$	−5761.77	−0.806297
$372$	0	0
$373$	1691.00	0.234737	0.117368	−	0.993088i	$-0.462554\pi$
$373$	1691.00	0.234737	0.117368	+	0.993088i	$0.462554\pi$
$374$	0	0
$375$	1088.66	0.149915
$376$	0	0
$377$	245.200	0.0334972
$378$	0	0
$379$	−4562.78	−0.618402	−0.309201	−	0.950997i	$-0.600062\pi$
$379$	−4562.78	−0.618402	−0.309201	+	0.950997i	$0.600062\pi$
$380$	0	0
$381$	4369.89	0.587602
$382$	0	0
$383$	−4896.69	−0.653287	−0.326644	−	0.945148i	$-0.605918\pi$
$383$	−4896.69	−0.653287	−0.326644	+	0.945148i	$0.605918\pi$
$384$	0	0
$385$	−427.405	−0.0565781
$386$	0	0
$387$	3032.26	0.398291
$388$	0	0
$389$	−1611.91	−0.210096	−0.105048	−	0.994467i	$-0.533500\pi$
$389$	−1611.91	−0.210096	−0.105048	+	0.994467i	$0.533500\pi$
$390$	0	0
$391$	13909.0	1.79900
$392$	0	0
$393$	5468.64	0.701925
$394$	0	0
$395$	−1567.16	−0.199626
$396$	0	0
$397$	−1744.95	−0.220595	−0.110298	−	0.993899i	$-0.535180\pi$
$397$	−1744.95	−0.220595	−0.110298	+	0.993899i	$0.535180\pi$
$398$	0	0
$399$	3606.43	0.452500
$400$	0	0
$401$	−283.494	−0.0353042	−0.0176521	−	0.999844i	$-0.505619\pi$
$401$	−283.494	−0.0353042	−0.0176521	+	0.999844i	$0.505619\pi$
$402$	0	0
$403$	−1020.97	−0.126198
$404$	0	0
$405$	−118.592	−0.0145504
$406$	0	0
$407$	−11377.1	−1.38560
$408$	0	0
$409$	8619.44	1.04206	0.521032	−	0.853537i	$-0.325547\pi$
$409$	8619.44	1.04206	0.521032	+	0.853537i	$0.325547\pi$
$410$	0	0
$411$	1829.78	0.219602
$412$	0	0
$413$	−2034.47	−0.242396
$414$	0	0
$415$	1085.88	0.128443
$416$	0	0
$417$	−1646.06	−0.193304
$418$	0	0
$419$	−10146.0	−1.18297	−0.591483	−	0.806318i	$-0.701457\pi$
$419$	−10146.0	−1.18297	−0.591483	+	0.806318i	$0.701457\pi$
$420$	0	0
$421$	5435.09	0.629193	0.314597	−	0.949225i	$-0.398131\pi$
$421$	5435.09	0.629193	0.314597	+	0.949225i	$0.398131\pi$
$422$	0	0
$423$	893.538	0.102708
$424$	0	0
$425$	13276.7	1.51533
$426$	0	0
$427$	−5411.79	−0.613337
$428$	0	0
$429$	−1356.60	−0.152674
$430$	0	0
$431$	−4851.97	−0.542253	−0.271127	−	0.962544i	$-0.587396\pi$
$431$	−4851.97	−0.542253	−0.271127	+	0.962544i	$0.587396\pi$
$432$	0	0
$433$	−14096.0	−1.56446	−0.782230	−	0.622990i	$-0.785918\pi$
$433$	−14096.0	−1.56446	−0.782230	+	0.622990i	$0.785918\pi$
$434$	0	0
$435$	−82.8457	−0.00913138
$436$	0	0
$437$	18436.5	2.01817
$438$	0	0
$439$	6431.04	0.699173	0.349586	−	0.936904i	$-0.386322\pi$
$439$	6431.04	0.699173	0.349586	+	0.936904i	$0.386322\pi$
$440$	0	0
$441$	−2453.12	−0.264887
$442$	0	0
$443$	−13281.3	−1.42441	−0.712205	−	0.701972i	$-0.752303\pi$
$443$	−13281.3	−1.42441	−0.712205	+	0.701972i	$0.752303\pi$
$444$	0	0
$445$	734.359	0.0782292
$446$	0	0
$447$	−5591.67	−0.591671
$448$	0	0
$449$	2227.04	0.234077	0.117038	−	0.993127i	$-0.462660\pi$
$449$	2227.04	0.234077	0.117038	+	0.993127i	$0.462660\pi$
$450$	0	0
$451$	−11395.0	−1.18973
$452$	0	0
$453$	−583.253	−0.0604936
$454$	0	0
$455$	159.733	0.0164581
$456$	0	0
$457$	14683.2	1.50295	0.751477	−	0.659759i	$-0.229342\pi$
$457$	14683.2	1.50295	0.751477	+	0.659759i	$0.229342\pi$
$458$	0	0
$459$	−2917.80	−0.296713
$460$	0	0
$461$	−3622.75	−0.366005	−0.183003	−	0.983112i	$-0.558582\pi$
$461$	−3622.75	−0.366005	−0.183003	+	0.983112i	$0.558582\pi$
$462$	0	0
$463$	15517.0	1.55753	0.778766	−	0.627314i	$-0.215846\pi$
$463$	15517.0	1.55753	0.778766	+	0.627314i	$0.215846\pi$
$464$	0	0
$465$	344.954	0.0344018
$466$	0	0
$467$	−14315.1	−1.41847	−0.709235	−	0.704973i	$-0.750959\pi$
$467$	−14315.1	−1.41847	−0.709235	+	0.704973i	$0.750959\pi$
$468$	0	0
$469$	−7315.10	−0.720213
$470$	0	0
$471$	647.817	0.0633754
$472$	0	0
$473$	−11719.6	−1.13925
$474$	0	0
$475$	17598.4	1.69994
$476$	0	0
$477$	6178.98	0.593116
$478$	0	0
$479$	−6601.85	−0.629742	−0.314871	−	0.949135i	$-0.601961\pi$
$479$	−6601.85	−0.629742	−0.314871	+	0.949135i	$0.601961\pi$
$480$	0	0
$481$	4251.93	0.403059
$482$	0	0
$483$	3240.46	0.305271
$484$	0	0
$485$	2298.21	0.215168
$486$	0	0
$487$	−616.770	−0.0573892	−0.0286946	−	0.999588i	$-0.509135\pi$
$487$	−616.770	−0.0573892	−0.0286946	+	0.999588i	$0.509135\pi$
$488$	0	0
$489$	−3993.84	−0.369341
$490$	0	0
$491$	−1565.75	−0.143913	−0.0719567	−	0.997408i	$-0.522924\pi$
$491$	−1565.75	−0.143913	−0.0719567	+	0.997408i	$0.522924\pi$
$492$	0	0
$493$	−2038.31	−0.186208
$494$	0	0
$495$	458.354	0.0416191
$496$	0	0
$497$	−841.081	−0.0759108
$498$	0	0
$499$	−2449.31	−0.219732	−0.109866	−	0.993946i	$-0.535042\pi$
$499$	−2449.31	−0.219732	−0.109866	+	0.993946i	$0.535042\pi$
$500$	0	0
$501$	9569.04	0.853320
$502$	0	0
$503$	−7106.77	−0.629970	−0.314985	−	0.949097i	$-0.602000\pi$
$503$	−7106.77	−0.629970	−0.314985	+	0.949097i	$0.602000\pi$
$504$	0	0
$505$	2400.16	0.211496
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−7657.22	−0.666798	−0.333399	−	0.942786i	$-0.608196\pi$
$509$	−7657.22	−0.666798	−0.333399	+	0.942786i	$0.608196\pi$
$510$	0	0
$511$	−5073.99	−0.439256
$512$	0	0
$513$	−3867.58	−0.332861
$514$	0	0
$515$	1215.69	0.104019
$516$	0	0
$517$	−3453.49	−0.293780
$518$	0	0
$519$	10222.6	0.864593
$520$	0	0
$521$	3616.82	0.304138	0.152069	−	0.988370i	$-0.451406\pi$
$521$	3616.82	0.304138	0.152069	+	0.988370i	$0.451406\pi$
$522$	0	0
$523$	−14089.9	−1.17803	−0.589014	−	0.808123i	$-0.700484\pi$
$523$	−14089.9	−1.17803	−0.589014	+	0.808123i	$0.700484\pi$
$524$	0	0
$525$	3093.15	0.257135
$526$	0	0
$527$	8487.11	0.701526
$528$	0	0
$529$	4398.66	0.361524
$530$	0	0
$531$	2181.78	0.178308
$532$	0	0
$533$	4258.63	0.346082
$534$	0	0
$535$	1937.99	0.156610
$536$	0	0
$537$	−3061.39	−0.246012
$538$	0	0
$539$	9481.21	0.757671
$540$	0	0
$541$	16786.5	1.33403	0.667013	−	0.745046i	$-0.267572\pi$
$541$	16786.5	1.33403	0.667013	+	0.745046i	$0.267572\pi$
$542$	0	0
$543$	10375.8	0.820016
$544$	0	0
$545$	−624.353	−0.0490722
$546$	0	0
$547$	−19055.0	−1.48946	−0.744729	−	0.667368i	$-0.767421\pi$
$547$	−19055.0	−1.48946	−0.744729	+	0.667368i	$0.767421\pi$
$548$	0	0
$549$	5803.66	0.451173
$550$	0	0
$551$	−2701.80	−0.208894
$552$	0	0
$553$	−8983.04	−0.690773
$554$	0	0
$555$	−1436.60	−0.109874
$556$	0	0
$557$	13217.4	1.00546	0.502729	−	0.864444i	$-0.332329\pi$
$557$	13217.4	1.00546	0.502729	+	0.864444i	$0.332329\pi$
$558$	0	0
$559$	4379.93	0.331398
$560$	0	0
$561$	11277.2	0.848703
$562$	0	0
$563$	24946.0	1.86741	0.933703	−	0.358048i	$-0.116558\pi$
$563$	24946.0	1.86741	0.933703	+	0.358048i	$0.116558\pi$
$564$	0	0
$565$	−2267.39	−0.168832
$566$	0	0
$567$	−679.777	−0.0503491
$568$	0	0
$569$	−15102.5	−1.11270	−0.556352	−	0.830947i	$-0.687799\pi$
$569$	−15102.5	−1.11270	−0.556352	+	0.830947i	$0.687799\pi$
$570$	0	0
$571$	16747.0	1.22739	0.613697	−	0.789542i	$-0.289682\pi$
$571$	16747.0	1.22739	0.613697	+	0.789542i	$0.289682\pi$
$572$	0	0
$573$	−9044.64	−0.659416
$574$	0	0
$575$	15812.6	1.14683
$576$	0	0
$577$	19788.9	1.42777	0.713885	−	0.700263i	$-0.246934\pi$
$577$	19788.9	1.42777	0.713885	+	0.700263i	$0.246934\pi$
$578$	0	0
$579$	1619.38	0.116234
$580$	0	0
$581$	6224.33	0.444456
$582$	0	0
$583$	−23881.5	−1.69652
$584$	0	0
$585$	−171.300	−0.0121066
$586$	0	0
$587$	−23658.3	−1.66351	−0.831756	−	0.555141i	$-0.812664\pi$
$587$	−23658.3	−1.66351	−0.831756	+	0.555141i	$0.812664\pi$
$588$	0	0
$589$	11249.8	0.786992
$590$	0	0
$591$	−10892.1	−0.758108
$592$	0	0
$593$	−17388.5	−1.20415	−0.602074	−	0.798441i	$-0.705659\pi$
$593$	−17388.5	−1.20415	−0.602074	+	0.798441i	$0.705659\pi$
$594$	0	0
$595$	−1327.84	−0.0914890
$596$	0	0
$597$	−11539.2	−0.791070
$598$	0	0
$599$	9053.40	0.617549	0.308774	−	0.951135i	$-0.400081\pi$
$599$	9053.40	0.617549	0.308774	+	0.951135i	$0.400081\pi$
$600$	0	0
$601$	−13531.6	−0.918414	−0.459207	−	0.888329i	$-0.651866\pi$
$601$	−13531.6	−0.918414	−0.459207	+	0.888329i	$0.651866\pi$
$602$	0	0
$603$	7844.79	0.529792
$604$	0	0
$605$	177.202	0.0119079
$606$	0	0
$607$	2841.08	0.189977	0.0949884	−	0.995478i	$-0.469719\pi$
$607$	2841.08	0.189977	0.0949884	+	0.995478i	$0.469719\pi$
$608$	0	0
$609$	−474.876	−0.0315976
$610$	0	0
$611$	1290.67	0.0854579
$612$	0	0
$613$	23429.1	1.54371	0.771854	−	0.635800i	$-0.219330\pi$
$613$	23429.1	1.54371	0.771854	+	0.635800i	$0.219330\pi$
$614$	0	0
$615$	−1438.86	−0.0943423
$616$	0	0
$617$	−17303.8	−1.12905	−0.564526	−	0.825415i	$-0.690941\pi$
$617$	−17303.8	−1.12905	−0.564526	+	0.825415i	$0.690941\pi$
$618$	0	0
$619$	2669.38	0.173330	0.0866652	−	0.996237i	$-0.472379\pi$
$619$	2669.38	0.173330	0.0866652	+	0.996237i	$0.472379\pi$
$620$	0	0
$621$	−3475.11	−0.224559
$622$	0	0
$623$	4209.39	0.270699
$624$	0	0
$625$	14825.7	0.948848
$626$	0	0
$627$	14948.0	0.952099
$628$	0	0
$629$	−35345.6	−2.24057
$630$	0	0
$631$	12671.5	0.799439	0.399720	−	0.916637i	$-0.369107\pi$
$631$	12671.5	0.799439	0.399720	+	0.916637i	$0.369107\pi$
$632$	0	0
$633$	−2979.51	−0.187085
$634$	0	0
$635$	−2132.66	−0.133278
$636$	0	0
$637$	−3543.40	−0.220400
$638$	0	0
$639$	901.985	0.0558403
$640$	0	0
$641$	6148.11	0.378839	0.189420	−	0.981896i	$-0.439339\pi$
$641$	6148.11	0.378839	0.189420	+	0.981896i	$0.439339\pi$
$642$	0	0
$643$	21495.1	1.31833	0.659164	−	0.752000i	$-0.270910\pi$
$643$	21495.1	1.31833	0.659164	+	0.752000i	$0.270910\pi$
$644$	0	0
$645$	−1479.85	−0.0903394
$646$	0	0
$647$	20055.9	1.21867	0.609335	−	0.792913i	$-0.291436\pi$
$647$	20055.9	1.21867	0.609335	+	0.792913i	$0.291436\pi$
$648$	0	0
$649$	−8432.50	−0.510023
$650$	0	0
$651$	1977.29	0.119042
$652$	0	0
$653$	−13667.9	−0.819091	−0.409546	−	0.912290i	$-0.634313\pi$
$653$	−13667.9	−0.819091	−0.409546	+	0.912290i	$0.634313\pi$
$654$	0	0
$655$	−2668.88	−0.159209
$656$	0	0
$657$	5441.40	0.323119
$658$	0	0
$659$	−14754.7	−0.872173	−0.436087	−	0.899905i	$-0.643636\pi$
$659$	−14754.7	−0.872173	−0.436087	+	0.899905i	$0.643636\pi$
$660$	0	0
$661$	27222.0	1.60184	0.800918	−	0.598774i	$-0.204345\pi$
$661$	27222.0	1.60184	0.800918	+	0.598774i	$0.204345\pi$
$662$	0	0
$663$	−4214.60	−0.246880
$664$	0	0
$665$	−1760.06	−0.102635
$666$	0	0
$667$	−2427.63	−0.140927
$668$	0	0
$669$	11441.1	0.661193
$670$	0	0
$671$	−22430.9	−1.29051
$672$	0	0
$673$	9635.48	0.551888	0.275944	−	0.961174i	$-0.411010\pi$
$673$	9635.48	0.551888	0.275944	+	0.961174i	$0.411010\pi$
$674$	0	0
$675$	−3317.12	−0.189150
$676$	0	0
$677$	16897.8	0.959286	0.479643	−	0.877464i	$-0.340766\pi$
$677$	16897.8	0.959286	0.479643	+	0.877464i	$0.340766\pi$
$678$	0	0
$679$	13173.5	0.744552
$680$	0	0
$681$	9007.09	0.506832
$682$	0	0
$683$	30252.7	1.69486	0.847429	−	0.530908i	$-0.178149\pi$
$683$	30252.7	1.69486	0.847429	+	0.530908i	$0.178149\pi$
$684$	0	0
$685$	−892.993	−0.0498095
$686$	0	0
$687$	11544.2	0.641103
$688$	0	0
$689$	8925.20	0.493502
$690$	0	0
$691$	9471.02	0.521410	0.260705	−	0.965418i	$-0.416045\pi$
$691$	9471.02	0.521410	0.260705	+	0.965418i	$0.416045\pi$
$692$	0	0
$693$	2627.31	0.144016
$694$	0	0
$695$	803.334	0.0438449
$696$	0	0
$697$	−35401.2	−1.92384
$698$	0	0
$699$	6043.77	0.327033
$700$	0	0
$701$	21733.2	1.17097	0.585485	−	0.810683i	$-0.300904\pi$
$701$	21733.2	1.17097	0.585485	+	0.810683i	$0.300904\pi$
$702$	0	0
$703$	−46850.9	−2.51354
$704$	0	0
$705$	−436.077	−0.0232959
$706$	0	0
$707$	13757.8	0.731848
$708$	0	0
$709$	−14647.9	−0.775901	−0.387950	−	0.921680i	$-0.626817\pi$
$709$	−14647.9	−0.775901	−0.387950	+	0.921680i	$0.626817\pi$
$710$	0	0
$711$	9633.51	0.508136
$712$	0	0
$713$	10108.2	0.530931
$714$	0	0
$715$	662.067	0.0346292
$716$	0	0
$717$	−16498.2	−0.859324
$718$	0	0
$719$	−15330.7	−0.795187	−0.397594	−	0.917562i	$-0.630155\pi$
$719$	−15330.7	−0.795187	−0.397594	+	0.917562i	$0.630155\pi$
$720$	0	0
$721$	6968.41	0.359941
$722$	0	0
$723$	593.617	0.0305351
$724$	0	0
$725$	−2317.26	−0.118705
$726$	0	0
$727$	23367.4	1.19209	0.596045	−	0.802951i	$-0.296738\pi$
$727$	23367.4	1.19209	0.596045	+	0.802951i	$0.296738\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−36409.6	−1.84221
$732$	0	0
$733$	18192.0	0.916694	0.458347	−	0.888773i	$-0.348442\pi$
$733$	18192.0	0.916694	0.458347	+	0.888773i	$0.348442\pi$
$734$	0	0
$735$	1197.21	0.0600812
$736$	0	0
$737$	−30319.8	−1.51539
$738$	0	0
$739$	19762.7	0.983741	0.491870	−	0.870668i	$-0.336313\pi$
$739$	19762.7	0.983741	0.491870	+	0.870668i	$0.336313\pi$
$740$	0	0
$741$	−5586.50	−0.276957
$742$	0	0
$743$	−13400.8	−0.661677	−0.330839	−	0.943687i	$-0.607332\pi$
$743$	−13400.8	−0.661677	−0.330839	+	0.943687i	$0.607332\pi$
$744$	0	0
$745$	2728.92	0.134201
$746$	0	0
$747$	−6675.04	−0.326944
$748$	0	0
$749$	11108.7	0.541924
$750$	0	0
$751$	−9368.29	−0.455198	−0.227599	−	0.973755i	$-0.573088\pi$
$751$	−9368.29	−0.455198	−0.227599	+	0.973755i	$0.573088\pi$
$752$	0	0
$753$	5126.62	0.248107
$754$	0	0
$755$	284.647	0.0137210
$756$	0	0
$757$	−5851.52	−0.280947	−0.140474	−	0.990084i	$-0.544863\pi$
$757$	−5851.52	−0.280947	−0.140474	+	0.990084i	$0.544863\pi$
$758$	0	0
$759$	13431.1	0.642318
$760$	0	0
$761$	28716.6	1.36791	0.683953	−	0.729526i	$-0.260260\pi$
$761$	28716.6	1.36791	0.683953	+	0.729526i	$0.260260\pi$
$762$	0	0
$763$	−3578.82	−0.169806
$764$	0	0
$765$	1423.98	0.0672997
$766$	0	0
$767$	3151.47	0.148361
$768$	0	0
$769$	38963.7	1.82713	0.913567	−	0.406687i	$-0.133316\pi$
$769$	38963.7	1.82713	0.913567	+	0.406687i	$0.133316\pi$
$770$	0	0
$771$	16230.7	0.758151
$772$	0	0
$773$	7592.57	0.353280	0.176640	−	0.984275i	$-0.443477\pi$
$773$	7592.57	0.353280	0.176640	+	0.984275i	$0.443477\pi$
$774$	0	0
$775$	9648.64	0.447212
$776$	0	0
$777$	−8234.66	−0.380202
$778$	0	0
$779$	−46924.7	−2.15822
$780$	0	0
$781$	−3486.13	−0.159723
$782$	0	0
$783$	509.262	0.0232434
$784$	0	0
$785$	−316.156	−0.0143747
$786$	0	0
$787$	35488.2	1.60739	0.803696	−	0.595041i	$-0.202864\pi$
$787$	35488.2	1.60739	0.803696	+	0.595041i	$0.202864\pi$
$788$	0	0
$789$	−9697.88	−0.437584
$790$	0	0
$791$	−12996.8	−0.584213
$792$	0	0
$793$	8383.07	0.375399
$794$	0	0
$795$	−3015.55	−0.134529
$796$	0	0
$797$	43362.4	1.92720	0.963599	−	0.267352i	$-0.0861486\pi$
$797$	43362.4	1.92720	0.963599	+	0.267352i	$0.0861486\pi$
$798$	0	0
$799$	−10729.1	−0.475053
$800$	0	0
$801$	−4514.19	−0.199127
$802$	0	0
$803$	−21030.8	−0.924234
$804$	0	0
$805$	−1581.46	−0.0692410
$806$	0	0
$807$	20415.0	0.890510
$808$	0	0
$809$	−42504.0	−1.84717	−0.923586	−	0.383392i	$-0.874756\pi$
$809$	−42504.0	−1.84717	−0.923586	+	0.383392i	$0.874756\pi$
$810$	0	0
$811$	28029.2	1.21361	0.606805	−	0.794851i	$-0.292451\pi$
$811$	28029.2	1.21361	0.606805	+	0.794851i	$0.292451\pi$
$812$	0	0
$813$	−22951.8	−0.990105
$814$	0	0
$815$	1949.13	0.0837730
$816$	0	0
$817$	−48261.3	−2.06665
$818$	0	0
$819$	−981.900	−0.0418930
$820$	0	0
$821$	−21499.0	−0.913911	−0.456955	−	0.889490i	$-0.651060\pi$
$821$	−21499.0	−0.913911	−0.456955	+	0.889490i	$0.651060\pi$
$822$	0	0
$823$	−29687.3	−1.25739	−0.628697	−	0.777650i	$-0.716411\pi$
$823$	−29687.3	−1.25739	−0.628697	+	0.777650i	$0.716411\pi$
$824$	0	0
$825$	12820.5	0.541035
$826$	0	0
$827$	−18571.7	−0.780897	−0.390448	−	0.920625i	$-0.627680\pi$
$827$	−18571.7	−0.780897	−0.390448	+	0.920625i	$0.627680\pi$
$828$	0	0
$829$	−2576.14	−0.107929	−0.0539644	−	0.998543i	$-0.517186\pi$
$829$	−2576.14	−0.107929	−0.0539644	+	0.998543i	$0.517186\pi$
$830$	0	0
$831$	−15740.3	−0.657071
$832$	0	0
$833$	29455.6	1.22518
$834$	0	0
$835$	−4670.02	−0.193548
$836$	0	0
$837$	−2120.47	−0.0875677
$838$	0	0
$839$	13089.6	0.538621	0.269311	−	0.963053i	$-0.413204\pi$
$839$	13089.6	0.538621	0.269311	+	0.963053i	$0.413204\pi$
$840$	0	0
$841$	−24033.2	−0.985413
$842$	0	0
$843$	11383.8	0.465100
$844$	0	0
$845$	−247.433	−0.0100733
$846$	0	0
$847$	1015.73	0.0412052
$848$	0	0
$849$	−1414.72	−0.0571886
$850$	0	0
$851$	−42096.6	−1.69572
$852$	0	0
$853$	−37066.5	−1.48785	−0.743923	−	0.668265i	$-0.767037\pi$
$853$	−37066.5	−1.48785	−0.743923	+	0.668265i	$0.767037\pi$
$854$	0	0
$855$	1887.51	0.0754987
$856$	0	0
$857$	20389.3	0.812701	0.406350	−	0.913717i	$-0.366801\pi$
$857$	20389.3	0.812701	0.406350	+	0.913717i	$0.366801\pi$
$858$	0	0
$859$	−32050.7	−1.27306	−0.636529	−	0.771253i	$-0.719630\pi$
$859$	−32050.7	−1.27306	−0.636529	+	0.771253i	$0.719630\pi$
$860$	0	0
$861$	−8247.63	−0.326456
$862$	0	0
$863$	23477.1	0.926037	0.463018	−	0.886349i	$-0.346766\pi$
$863$	23477.1	0.926037	0.463018	+	0.886349i	$0.346766\pi$
$864$	0	0
$865$	−4988.99	−0.196105
$866$	0	0
$867$	20296.2	0.795035
$868$	0	0
$869$	−37233.1	−1.45345
$870$	0	0
$871$	11331.4	0.440814
$872$	0	0
$873$	−14127.4	−0.547696
$874$	0	0
$875$	−3045.46	−0.117663
$876$	0	0
$877$	−44692.4	−1.72082	−0.860408	−	0.509606i	$-0.829791\pi$
$877$	−44692.4	−1.72082	−0.860408	+	0.509606i	$0.829791\pi$
$878$	0	0
$879$	−91.7601	−0.00352104
$880$	0	0
$881$	8892.46	0.340062	0.170031	−	0.985439i	$-0.445613\pi$
$881$	8892.46	0.340062	0.170031	+	0.985439i	$0.445613\pi$
$882$	0	0
$883$	40356.1	1.53804	0.769022	−	0.639223i	$-0.220744\pi$
$883$	40356.1	1.53804	0.769022	+	0.639223i	$0.220744\pi$
$884$	0	0
$885$	−1064.78	−0.0404433
$886$	0	0
$887$	−30987.8	−1.17302	−0.586511	−	0.809941i	$-0.699499\pi$
$887$	−30987.8	−1.17302	−0.586511	+	0.809941i	$0.699499\pi$
$888$	0	0
$889$	−12224.5	−0.461188
$890$	0	0
$891$	−2817.55	−0.105939
$892$	0	0
$893$	−14221.5	−0.532928
$894$	0	0
$895$	1494.06	0.0557999
$896$	0	0
$897$	−5019.60	−0.186845
$898$	0	0
$899$	−1481.31	−0.0549549
$900$	0	0
$901$	−74193.6	−2.74334
$902$	0	0
$903$	−8482.55	−0.312604
$904$	0	0
$905$	−5063.75	−0.185994
$906$	0	0
$907$	11584.8	0.424110	0.212055	−	0.977258i	$-0.431984\pi$
$907$	11584.8	0.424110	0.212055	+	0.977258i	$0.431984\pi$
$908$	0	0
$909$	−14754.0	−0.538351
$910$	0	0
$911$	19161.5	0.696868	0.348434	−	0.937333i	$-0.386713\pi$
$911$	19161.5	0.696868	0.348434	+	0.937333i	$0.386713\pi$
$912$	0	0
$913$	25798.8	0.935174
$914$	0	0
$915$	−2832.38	−0.102334
$916$	0	0
$917$	−15298.2	−0.550916
$918$	0	0
$919$	39089.3	1.40309	0.701544	−	0.712626i	$-0.252494\pi$
$919$	39089.3	1.40309	0.701544	+	0.712626i	$0.252494\pi$
$920$	0	0
$921$	23102.4	0.826548
$922$	0	0
$923$	1302.87	0.0464620
$924$	0	0
$925$	−40182.9	−1.42833
$926$	0	0
$927$	−7473.00	−0.264774
$928$	0	0
$929$	258.811	0.00914027	0.00457014	−	0.999990i	$-0.498545\pi$
$929$	258.811	0.00914027	0.00457014	+	0.999990i	$0.498545\pi$
$930$	0	0
$931$	39043.8	1.37445
$932$	0	0
$933$	−8710.12	−0.305634
$934$	0	0
$935$	−5503.64	−0.192501
$936$	0	0
$937$	−29811.6	−1.03938	−0.519692	−	0.854354i	$-0.673953\pi$
$937$	−29811.6	−1.03938	−0.519692	+	0.854354i	$0.673953\pi$
$938$	0	0
$939$	−16936.3	−0.588599
$940$	0	0
$941$	39979.7	1.38502	0.692509	−	0.721410i	$-0.256506\pi$
$941$	39979.7	1.38502	0.692509	+	0.721410i	$0.256506\pi$
$942$	0	0
$943$	−42163.0	−1.45601
$944$	0	0
$945$	331.754	0.0114201
$946$	0	0
$947$	−8295.11	−0.284641	−0.142320	−	0.989821i	$-0.545456\pi$
$947$	−8295.11	−0.284641	−0.142320	+	0.989821i	$0.545456\pi$
$948$	0	0
$949$	7859.80	0.268851
$950$	0	0
$951$	−3200.30	−0.109124
$952$	0	0
$953$	19633.5	0.667359	0.333679	−	0.942687i	$-0.391710\pi$
$953$	19633.5	0.667359	0.333679	+	0.942687i	$0.391710\pi$
$954$	0	0
$955$	4414.09	0.149567
$956$	0	0
$957$	−1968.28	−0.0664841
$958$	0	0
$959$	−5118.68	−0.172358
$960$	0	0
$961$	−23623.1	−0.792961
$962$	0	0
$963$	−11913.0	−0.398642
$964$	0	0
$965$	−790.314	−0.0263638
$966$	0	0
$967$	11936.8	0.396961	0.198480	−	0.980105i	$-0.436399\pi$
$967$	11936.8	0.396961	0.198480	+	0.980105i	$0.436399\pi$
$968$	0	0
$969$	46439.6	1.53958
$970$	0	0
$971$	3489.36	0.115323	0.0576616	−	0.998336i	$-0.481636\pi$
$971$	3489.36	0.115323	0.0576616	+	0.998336i	$0.481636\pi$
$972$	0	0
$973$	4604.75	0.151718
$974$	0	0
$975$	−4791.40	−0.157382
$976$	0	0
$977$	28945.7	0.947856	0.473928	−	0.880564i	$-0.342836\pi$
$977$	28945.7	0.947856	0.473928	+	0.880564i	$0.342836\pi$
$978$	0	0
$979$	17447.2	0.569574
$980$	0	0
$981$	3837.97	0.124910
$982$	0	0
$983$	41249.7	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$983$	41249.7	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$984$	0	0
$985$	5315.72	0.171952
$986$	0	0
$987$	−2499.62	−0.0806116
$988$	0	0
$989$	−43363.9	−1.39423
$990$	0	0
$991$	30447.5	0.975981	0.487991	−	0.872849i	$-0.337730\pi$
$991$	30447.5	0.975981	0.487991	+	0.872849i	$0.337730\pi$
$992$	0	0
$993$	−6617.59	−0.211483
$994$	0	0
$995$	5631.53	0.179429
$996$	0	0
$997$	31771.1	1.00923	0.504615	−	0.863345i	$-0.331634\pi$
$997$	31771.1	1.00923	0.504615	+	0.863345i	$0.331634\pi$
$998$	0	0
$999$	8830.94	0.279678

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bg.1.1		2
4.3	odd	2		2496.4.a.z.1.1		2
8.3	odd	2		624.4.a.o.1.2		2
8.5	even	2		312.4.a.a.1.2	✓	2
24.5	odd	2		936.4.a.g.1.1		2
24.11	even	2		1872.4.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.a.1.2	✓	2	8.5	even	2
624.4.a.o.1.2		2	8.3	odd	2
936.4.a.g.1.1		2	24.5	odd	2
1872.4.a.be.1.1		2	24.11	even	2
2496.4.a.z.1.1		2	4.3	odd	2
2496.4.a.bg.1.1		2	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -1.46410 q^{5} -8.39230 q^{7} +9.00000 q^{9} -34.7846 q^{11} +13.0000 q^{13} -4.39230 q^{15} -108.067 q^{17} -143.244 q^{19} -25.1769 q^{21} -128.708 q^{23} -122.856 q^{25} +27.0000 q^{27} +18.8616 q^{29} -78.5359 q^{31} -104.354 q^{33} +12.2872 q^{35} +327.072 q^{37} +39.0000 q^{39} +327.587 q^{41} +336.918 q^{43} -13.1769 q^{45} +99.2820 q^{47} -272.569 q^{49} -324.200 q^{51} +686.554 q^{53} +50.9282 q^{55} -429.731 q^{57} +242.420 q^{59} +644.851 q^{61} -75.5307 q^{63} -19.0333 q^{65} +871.643 q^{67} -386.123 q^{69} +100.221 q^{71} +604.600 q^{73} -368.569 q^{75} +291.923 q^{77} +1070.39 q^{79} +81.0000 q^{81} -741.672 q^{83} +158.221 q^{85} +56.5847 q^{87} -501.577 q^{89} -109.100 q^{91} -235.608 q^{93} +209.723 q^{95} -1569.71 q^{97} -313.061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 4 q^{5} + 4 q^{7} + 18 q^{9} - 28 q^{11} + 26 q^{13} + 12 q^{15} - 36 q^{17} - 44 q^{19} + 12 q^{21} - 8 q^{23} - 218 q^{25} + 54 q^{27} + 204 q^{29} - 164 q^{31} - 84 q^{33} + 80 q^{35}+ \cdots - 252 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 4 * q^5 + 4 * q^7 + 18 * q^9 - 28 * q^11 + 26 * q^13 + 12 * q^15 - 36 * q^17 - 44 * q^19 + 12 * q^21 - 8 * q^23 - 218 * q^25 + 54 * q^27 + 204 * q^29 - 164 * q^31 - 84 * q^33 + 80 * q^35 + 668 * q^37 + 78 * q^39 - 100 * q^41 + 272 * q^43 + 36 * q^45 + 60 * q^47 - 462 * q^49 - 108 * q^51 + 708 * q^53 + 88 * q^55 - 132 * q^57 + 180 * q^59 + 1068 * q^61 + 36 * q^63 + 52 * q^65 + 420 * q^67 - 24 * q^69 + 436 * q^71 - 412 * q^73 - 654 * q^75 + 376 * q^77 + 672 * q^79 + 162 * q^81 + 124 * q^83 + 552 * q^85 + 612 * q^87 + 140 * q^89 + 52 * q^91 - 492 * q^93 + 752 * q^95 - 188 * q^97 - 252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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