LMFDB - Embedded newform 2496.4.a.bj.1.2

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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +13.1231 q^{5} -15.6155 q^{7} +9.00000 q^{9} -55.7235 q^{11} -13.0000 q^{13} +39.3693 q^{15} +23.4773 q^{17} -25.6458 q^{19} -46.8466 q^{21} +189.477 q^{23} +47.2159 q^{25} +27.0000 q^{27} +236.739 q^{29} -47.0625 q^{31} -167.170 q^{33} -204.924 q^{35} +154.648 q^{37} -39.0000 q^{39} -34.6610 q^{41} -398.833 q^{43} +118.108 q^{45} -582.773 q^{47} -99.1553 q^{49} +70.4318 q^{51} +361.015 q^{53} -731.265 q^{55} -76.9375 q^{57} +396.985 q^{59} +211.322 q^{61} -140.540 q^{63} -170.600 q^{65} +85.8277 q^{67} +568.432 q^{69} +651.602 q^{71} +927.329 q^{73} +141.648 q^{75} +870.152 q^{77} +1117.98 q^{79} +81.0000 q^{81} +391.629 q^{83} +308.095 q^{85} +710.216 q^{87} -745.960 q^{89} +203.002 q^{91} -141.187 q^{93} -336.553 q^{95} -173.231 q^{97} -501.511 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 18 q^{5} + 10 q^{7} + 18 q^{9} + 4 q^{11} - 26 q^{13} + 54 q^{15} - 52 q^{17} - 142 q^{19} + 30 q^{21} + 280 q^{23} - 54 q^{25} + 54 q^{27} + 424 q^{29} + 178 q^{31} + 12 q^{33} - 80 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 18 * q^5 + 10 * q^7 + 18 * q^9 + 4 * q^11 - 26 * q^13 + 54 * q^15 - 52 * q^17 - 142 * q^19 + 30 * q^21 + 280 * q^23 - 54 * q^25 + 54 * q^27 + 424 * q^29 + 178 * q^31 + 12 * q^33 - 80 * q^35 - 136 * q^37 - 78 * q^39 - 226 * q^41 - 72 * q^43 + 162 * q^45 - 176 * q^47 + 214 * q^49 - 156 * q^51 + 788 * q^53 - 440 * q^55 - 426 * q^57 + 728 * q^59 + 736 * q^61 + 90 * q^63 - 234 * q^65 + 1054 * q^67 + 840 * q^69 + 660 * q^71 + 24 * q^73 - 162 * q^75 + 2400 * q^77 - 40 * q^79 + 162 * q^81 + 1344 * q^83 - 60 * q^85 + 1272 * q^87 + 314 * q^89 - 130 * q^91 + 534 * q^93 - 904 * q^95 - 264 * q^97 + 36 * 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
$5$	13.1231	1.17377	0.586883	−	0.809672i	$-0.300355\pi$
$5$	13.1231	1.17377	0.586883	+	0.809672i	$0.300355\pi$
$6$	0	0
$7$	−15.6155	−0.843159	−0.421580	−	0.906791i	$-0.638524\pi$
$7$	−15.6155	−0.843159	−0.421580	+	0.906791i	$0.638524\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−55.7235	−1.52739	−0.763694	−	0.645579i	$-0.776616\pi$
$11$	−55.7235	−1.52739	−0.763694	+	0.645579i	$0.776616\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	39.3693	0.677674
$16$	0	0
$17$	23.4773	0.334946	0.167473	−	0.985877i	$-0.446439\pi$
$17$	23.4773	0.334946	0.167473	+	0.985877i	$0.446439\pi$
$18$	0	0
$19$	−25.6458	−0.309661	−0.154830	−	0.987941i	$-0.549483\pi$
$19$	−25.6458	−0.309661	−0.154830	+	0.987941i	$0.549483\pi$
$20$	0	0
$21$	−46.8466	−0.486798
$22$	0	0
$23$	189.477	1.71777	0.858886	−	0.512167i	$-0.171157\pi$
$23$	189.477	1.71777	0.858886	+	0.512167i	$0.171157\pi$
$24$	0	0
$25$	47.2159	0.377727
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	236.739	1.51591	0.757953	−	0.652309i	$-0.226200\pi$
$29$	236.739	1.51591	0.757953	+	0.652309i	$0.226200\pi$
$30$	0	0
$31$	−47.0625	−0.272667	−0.136333	−	0.990663i	$-0.543532\pi$
$31$	−47.0625	−0.272667	−0.136333	+	0.990663i	$0.543532\pi$
$32$	0	0
$33$	−167.170	−0.881838
$34$	0	0
$35$	−204.924	−0.989672
$36$	0	0
$37$	154.648	0.687133	0.343567	−	0.939128i	$-0.388365\pi$
$37$	154.648	0.687133	0.343567	+	0.939128i	$0.388365\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−34.6610	−0.132028	−0.0660139	−	0.997819i	$-0.521028\pi$
$41$	−34.6610	−0.132028	−0.0660139	+	0.997819i	$0.521028\pi$
$42$	0	0
$43$	−398.833	−1.41445	−0.707227	−	0.706987i	$-0.750054\pi$
$43$	−398.833	−1.41445	−0.707227	+	0.706987i	$0.750054\pi$
$44$	0	0
$45$	118.108	0.391255
$46$	0	0
$47$	−582.773	−1.80864	−0.904321	−	0.426854i	$-0.859622\pi$
$47$	−582.773	−1.80864	−0.904321	+	0.426854i	$0.859622\pi$
$48$	0	0
$49$	−99.1553	−0.289082
$50$	0	0
$51$	70.4318	0.193381
$52$	0	0
$53$	361.015	0.935646	0.467823	−	0.883822i	$-0.345038\pi$
$53$	361.015	0.935646	0.467823	+	0.883822i	$0.345038\pi$
$54$	0	0
$55$	−731.265	−1.79280
$56$	0	0
$57$	−76.9375	−0.178783
$58$	0	0
$59$	396.985	0.875983	0.437992	−	0.898979i	$-0.355690\pi$
$59$	396.985	0.875983	0.437992	+	0.898979i	$0.355690\pi$
$60$	0	0
$61$	211.322	0.443558	0.221779	−	0.975097i	$-0.428814\pi$
$61$	211.322	0.443558	0.221779	+	0.975097i	$0.428814\pi$
$62$	0	0
$63$	−140.540	−0.281053
$64$	0	0
$65$	−170.600	−0.325544
$66$	0	0
$67$	85.8277	0.156500	0.0782502	−	0.996934i	$-0.475067\pi$
$67$	85.8277	0.156500	0.0782502	+	0.996934i	$0.475067\pi$
$68$	0	0
$69$	568.432	0.991756
$70$	0	0
$71$	651.602	1.08917	0.544584	−	0.838706i	$-0.316687\pi$
$71$	651.602	1.08917	0.544584	+	0.838706i	$0.316687\pi$
$72$	0	0
$73$	927.329	1.48679	0.743395	−	0.668852i	$-0.233214\pi$
$73$	927.329	1.48679	0.743395	+	0.668852i	$0.233214\pi$
$74$	0	0
$75$	141.648	0.218081
$76$	0	0
$77$	870.152	1.28783
$78$	0	0
$79$	1117.98	1.59218	0.796090	−	0.605178i	$-0.206898\pi$
$79$	1117.98	1.59218	0.796090	+	0.605178i	$0.206898\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	391.629	0.517914	0.258957	−	0.965889i	$-0.416621\pi$
$83$	391.629	0.517914	0.258957	+	0.965889i	$0.416621\pi$
$84$	0	0
$85$	308.095	0.393148
$86$	0	0
$87$	710.216	0.875208
$88$	0	0
$89$	−745.960	−0.888445	−0.444223	−	0.895916i	$-0.646520\pi$
$89$	−745.960	−0.888445	−0.444223	+	0.895916i	$0.646520\pi$
$90$	0	0
$91$	203.002	0.233850
$92$	0	0
$93$	−141.187	−0.157424
$94$	0	0
$95$	−336.553	−0.363470
$96$	0	0
$97$	−173.231	−0.181329	−0.0906647	−	0.995881i	$-0.528899\pi$
$97$	−173.231	−0.181329	−0.0906647	+	0.995881i	$0.528899\pi$
$98$	0	0
$99$	−501.511	−0.509129
$100$	0	0
$101$	1263.45	1.24473	0.622367	−	0.782726i	$-0.286171\pi$
$101$	1263.45	1.24473	0.622367	+	0.782726i	$0.286171\pi$
$102$	0	0
$103$	−874.401	−0.836479	−0.418240	−	0.908337i	$-0.637353\pi$
$103$	−874.401	−0.836479	−0.418240	+	0.908337i	$0.637353\pi$
$104$	0	0
$105$	−614.773	−0.571387
$106$	0	0
$107$	1591.89	1.43826	0.719128	−	0.694877i	$-0.244541\pi$
$107$	1591.89	1.43826	0.719128	+	0.694877i	$0.244541\pi$
$108$	0	0
$109$	−371.788	−0.326705	−0.163352	−	0.986568i	$-0.552231\pi$
$109$	−371.788	−0.326705	−0.163352	+	0.986568i	$0.552231\pi$
$110$	0	0
$111$	463.943	0.396716
$112$	0	0
$113$	1173.98	0.977335	0.488667	−	0.872470i	$-0.337483\pi$
$113$	1173.98	0.977335	0.488667	+	0.872470i	$0.337483\pi$
$114$	0	0
$115$	2486.53	2.01626
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−366.610	−0.282412
$120$	0	0
$121$	1774.11	1.33291
$122$	0	0
$123$	−103.983	−0.0762263
$124$	0	0
$125$	−1020.77	−0.730403
$126$	0	0
$127$	−739.360	−0.516595	−0.258298	−	0.966065i	$-0.583161\pi$
$127$	−739.360	−0.516595	−0.258298	+	0.966065i	$0.583161\pi$
$128$	0	0
$129$	−1196.50	−0.816635
$130$	0	0
$131$	2131.58	1.42166	0.710829	−	0.703365i	$-0.248320\pi$
$131$	2131.58	1.42166	0.710829	+	0.703365i	$0.248320\pi$
$132$	0	0
$133$	400.473	0.261094
$134$	0	0
$135$	354.324	0.225891
$136$	0	0
$137$	−256.934	−0.160229	−0.0801144	−	0.996786i	$-0.525529\pi$
$137$	−256.934	−0.160229	−0.0801144	+	0.996786i	$0.525529\pi$
$138$	0	0
$139$	437.443	0.266931	0.133466	−	0.991053i	$-0.457389\pi$
$139$	437.443	0.266931	0.133466	+	0.991053i	$0.457389\pi$
$140$	0	0
$141$	−1748.32	−1.04422
$142$	0	0
$143$	724.405	0.423621
$144$	0	0
$145$	3106.75	1.77932
$146$	0	0
$147$	−297.466	−0.166902
$148$	0	0
$149$	3406.65	1.87304	0.936522	−	0.350609i	$-0.114025\pi$
$149$	3406.65	1.87304	0.936522	+	0.350609i	$0.114025\pi$
$150$	0	0
$151$	−3275.77	−1.76542	−0.882710	−	0.469918i	$-0.844283\pi$
$151$	−3275.77	−1.76542	−0.882710	+	0.469918i	$0.844283\pi$
$152$	0	0
$153$	211.295	0.111649
$154$	0	0
$155$	−617.606	−0.320047
$156$	0	0
$157$	−1074.34	−0.546128	−0.273064	−	0.961996i	$-0.588037\pi$
$157$	−1074.34	−0.546128	−0.273064	+	0.961996i	$0.588037\pi$
$158$	0	0
$159$	1083.05	0.540196
$160$	0	0
$161$	−2958.79	−1.44835
$162$	0	0
$163$	2031.49	0.976186	0.488093	−	0.872792i	$-0.337693\pi$
$163$	2031.49	0.976186	0.488093	+	0.872792i	$0.337693\pi$
$164$	0	0
$165$	−2193.80	−1.03507
$166$	0	0
$167$	−299.697	−0.138870	−0.0694349	−	0.997586i	$-0.522120\pi$
$167$	−299.697	−0.138870	−0.0694349	+	0.997586i	$0.522120\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−230.813	−0.103220
$172$	0	0
$173$	−2371.16	−1.04206	−0.521028	−	0.853540i	$-0.674451\pi$
$173$	−2371.16	−1.04206	−0.521028	+	0.853540i	$0.674451\pi$
$174$	0	0
$175$	−737.301	−0.318484
$176$	0	0
$177$	1190.95	0.505749
$178$	0	0
$179$	−249.613	−0.104229	−0.0521145	−	0.998641i	$-0.516596\pi$
$179$	−249.613	−0.104229	−0.0521145	+	0.998641i	$0.516596\pi$
$180$	0	0
$181$	3151.45	1.29417	0.647087	−	0.762416i	$-0.275987\pi$
$181$	3151.45	1.29417	0.647087	+	0.762416i	$0.275987\pi$
$182$	0	0
$183$	633.966	0.256088
$184$	0	0
$185$	2029.46	0.806534
$186$	0	0
$187$	−1308.24	−0.511592
$188$	0	0
$189$	−421.619	−0.162266
$190$	0	0
$191$	4739.45	1.79547	0.897735	−	0.440535i	$-0.145211\pi$
$191$	4739.45	1.79547	0.897735	+	0.440535i	$0.145211\pi$
$192$	0	0
$193$	636.818	0.237509	0.118754	−	0.992924i	$-0.462110\pi$
$193$	636.818	0.237509	0.118754	+	0.992924i	$0.462110\pi$
$194$	0	0
$195$	−511.801	−0.187953
$196$	0	0
$197$	3001.13	1.08539	0.542695	−	0.839930i	$-0.317404\pi$
$197$	3001.13	1.08539	0.542695	+	0.839930i	$0.317404\pi$
$198$	0	0
$199$	648.958	0.231173	0.115587	−	0.993297i	$-0.463125\pi$
$199$	648.958	0.231173	0.115587	+	0.993297i	$0.463125\pi$
$200$	0	0
$201$	257.483	0.0903555
$202$	0	0
$203$	−3696.80	−1.27815
$204$	0	0
$205$	−454.860	−0.154970
$206$	0	0
$207$	1705.30	0.572590
$208$	0	0
$209$	1429.08	0.472972
$210$	0	0
$211$	−240.114	−0.0783418	−0.0391709	−	0.999233i	$-0.512472\pi$
$211$	−240.114	−0.0783418	−0.0391709	+	0.999233i	$0.512472\pi$
$212$	0	0
$213$	1954.81	0.628832
$214$	0	0
$215$	−5233.93	−1.66024
$216$	0	0
$217$	734.906	0.229902
$218$	0	0
$219$	2781.99	0.858399
$220$	0	0
$221$	−305.204	−0.0928972
$222$	0	0
$223$	1104.37	0.331633	0.165816	−	0.986157i	$-0.446974\pi$
$223$	1104.37	0.331633	0.165816	+	0.986157i	$0.446974\pi$
$224$	0	0
$225$	424.943	0.125909
$226$	0	0
$227$	497.034	0.145327	0.0726636	−	0.997357i	$-0.476850\pi$
$227$	497.034	0.145327	0.0726636	+	0.997357i	$0.476850\pi$
$228$	0	0
$229$	−2879.05	−0.830800	−0.415400	−	0.909639i	$-0.636358\pi$
$229$	−2879.05	−0.830800	−0.415400	+	0.909639i	$0.636358\pi$
$230$	0	0
$231$	2610.45	0.743529
$232$	0	0
$233$	−3846.91	−1.08163	−0.540814	−	0.841142i	$-0.681884\pi$
$233$	−3846.91	−1.08163	−0.540814	+	0.841142i	$0.681884\pi$
$234$	0	0
$235$	−7647.79	−2.12292
$236$	0	0
$237$	3353.93	0.919246
$238$	0	0
$239$	−2903.25	−0.785755	−0.392877	−	0.919591i	$-0.628520\pi$
$239$	−2903.25	−0.785755	−0.392877	+	0.919591i	$0.628520\pi$
$240$	0	0
$241$	−4801.06	−1.28325	−0.641626	−	0.767018i	$-0.721740\pi$
$241$	−4801.06	−1.28325	−0.641626	+	0.767018i	$0.721740\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1301.23	−0.339315
$246$	0	0
$247$	333.396	0.0858845
$248$	0	0
$249$	1174.89	0.299018
$250$	0	0
$251$	4768.20	1.19907	0.599534	−	0.800350i	$-0.295353\pi$
$251$	4768.20	1.19907	0.599534	+	0.800350i	$0.295353\pi$
$252$	0	0
$253$	−10558.3	−2.62370
$254$	0	0
$255$	924.284	0.226984
$256$	0	0
$257$	6002.18	1.45683	0.728415	−	0.685136i	$-0.240257\pi$
$257$	6002.18	1.45683	0.728415	+	0.685136i	$0.240257\pi$
$258$	0	0
$259$	−2414.91	−0.579363
$260$	0	0
$261$	2130.65	0.505302
$262$	0	0
$263$	3268.33	0.766289	0.383145	−	0.923688i	$-0.374841\pi$
$263$	3268.33	0.766289	0.383145	+	0.923688i	$0.374841\pi$
$264$	0	0
$265$	4737.64	1.09823
$266$	0	0
$267$	−2237.88	−0.512944
$268$	0	0
$269$	8690.14	1.96969	0.984846	−	0.173431i	$-0.0554854\pi$
$269$	8690.14	1.96969	0.984846	+	0.173431i	$0.0554854\pi$
$270$	0	0
$271$	3239.17	0.726073	0.363036	−	0.931775i	$-0.381740\pi$
$271$	3239.17	0.726073	0.363036	+	0.931775i	$0.381740\pi$
$272$	0	0
$273$	609.006	0.135014
$274$	0	0
$275$	−2631.03	−0.576936
$276$	0	0
$277$	220.091	0.0477400	0.0238700	−	0.999715i	$-0.492401\pi$
$277$	220.091	0.0477400	0.0238700	+	0.999715i	$0.492401\pi$
$278$	0	0
$279$	−423.562	−0.0908889
$280$	0	0
$281$	587.233	0.124667	0.0623334	−	0.998055i	$-0.480146\pi$
$281$	587.233	0.124667	0.0623334	+	0.998055i	$0.480146\pi$
$282$	0	0
$283$	−8274.09	−1.73796	−0.868981	−	0.494845i	$-0.835225\pi$
$283$	−8274.09	−1.73796	−0.868981	+	0.494845i	$0.835225\pi$
$284$	0	0
$285$	−1009.66	−0.209849
$286$	0	0
$287$	541.250	0.111320
$288$	0	0
$289$	−4361.82	−0.887812
$290$	0	0
$291$	−519.693	−0.104691
$292$	0	0
$293$	5382.07	1.07312	0.536559	−	0.843863i	$-0.319724\pi$
$293$	5382.07	1.07312	0.536559	+	0.843863i	$0.319724\pi$
$294$	0	0
$295$	5209.67	1.02820
$296$	0	0
$297$	−1504.53	−0.293946
$298$	0	0
$299$	−2463.20	−0.476424
$300$	0	0
$301$	6227.99	1.19261
$302$	0	0
$303$	3790.35	0.718647
$304$	0	0
$305$	2773.20	0.520633
$306$	0	0
$307$	−3095.92	−0.575549	−0.287774	−	0.957698i	$-0.592915\pi$
$307$	−3095.92	−0.575549	−0.287774	+	0.957698i	$0.592915\pi$
$308$	0	0
$309$	−2623.20	−0.482941
$310$	0	0
$311$	−3474.58	−0.633521	−0.316761	−	0.948506i	$-0.602595\pi$
$311$	−3474.58	−0.633521	−0.316761	+	0.948506i	$0.602595\pi$
$312$	0	0
$313$	−3523.92	−0.636370	−0.318185	−	0.948029i	$-0.603073\pi$
$313$	−3523.92	−0.636370	−0.318185	+	0.948029i	$0.603073\pi$
$314$	0	0
$315$	−1844.32	−0.329891
$316$	0	0
$317$	323.396	0.0572988	0.0286494	−	0.999590i	$-0.490879\pi$
$317$	323.396	0.0572988	0.0286494	+	0.999590i	$0.490879\pi$
$318$	0	0
$319$	−13191.9	−2.31537
$320$	0	0
$321$	4775.66	0.830378
$322$	0	0
$323$	−602.094	−0.103720
$324$	0	0
$325$	−613.807	−0.104763
$326$	0	0
$327$	−1115.36	−0.188623
$328$	0	0
$329$	9100.30	1.52497
$330$	0	0
$331$	−1057.74	−0.175646	−0.0878228	−	0.996136i	$-0.527991\pi$
$331$	−1057.74	−0.175646	−0.0878228	+	0.996136i	$0.527991\pi$
$332$	0	0
$333$	1391.83	0.229044
$334$	0	0
$335$	1126.33	0.183695
$336$	0	0
$337$	11538.4	1.86510	0.932550	−	0.361041i	$-0.117578\pi$
$337$	11538.4	1.86510	0.932550	+	0.361041i	$0.117578\pi$
$338$	0	0
$339$	3521.94	0.564265
$340$	0	0
$341$	2622.49	0.416468
$342$	0	0
$343$	6904.49	1.08690
$344$	0	0
$345$	7459.59	1.16409
$346$	0	0
$347$	5211.75	0.806286	0.403143	−	0.915137i	$-0.367918\pi$
$347$	5211.75	0.806286	0.403143	+	0.915137i	$0.367918\pi$
$348$	0	0
$349$	2369.81	0.363476	0.181738	−	0.983347i	$-0.441828\pi$
$349$	2369.81	0.363476	0.181738	+	0.983347i	$0.441828\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−8997.40	−1.35661	−0.678305	−	0.734780i	$-0.737285\pi$
$353$	−8997.40	−1.35661	−0.678305	+	0.734780i	$0.737285\pi$
$354$	0	0
$355$	8551.05	1.27843
$356$	0	0
$357$	−1099.83	−0.163051
$358$	0	0
$359$	−10863.4	−1.59707	−0.798537	−	0.601946i	$-0.794392\pi$
$359$	−10863.4	−1.59707	−0.798537	+	0.601946i	$0.794392\pi$
$360$	0	0
$361$	−6201.29	−0.904110
$362$	0	0
$363$	5322.32	0.769557
$364$	0	0
$365$	12169.4	1.74514
$366$	0	0
$367$	13552.2	1.92757	0.963785	−	0.266682i	$-0.0859273\pi$
$367$	13552.2	1.92757	0.963785	+	0.266682i	$0.0859273\pi$
$368$	0	0
$369$	−311.949	−0.0440092
$370$	0	0
$371$	−5637.44	−0.788899
$372$	0	0
$373$	−2255.64	−0.313118	−0.156559	−	0.987669i	$-0.550040\pi$
$373$	−2255.64	−0.313118	−0.156559	+	0.987669i	$0.550040\pi$
$374$	0	0
$375$	−3062.31	−0.421698
$376$	0	0
$377$	−3077.60	−0.420437
$378$	0	0
$379$	−7698.92	−1.04345	−0.521724	−	0.853114i	$-0.674711\pi$
$379$	−7698.92	−1.04345	−0.521724	+	0.853114i	$0.674711\pi$
$380$	0	0
$381$	−2218.08	−0.298256
$382$	0	0
$383$	−211.235	−0.0281818	−0.0140909	−	0.999901i	$-0.504485\pi$
$383$	−211.235	−0.0281818	−0.0140909	+	0.999901i	$0.504485\pi$
$384$	0	0
$385$	11419.1	1.51161
$386$	0	0
$387$	−3589.50	−0.471485
$388$	0	0
$389$	−3515.25	−0.458175	−0.229088	−	0.973406i	$-0.573574\pi$
$389$	−3515.25	−0.458175	−0.229088	+	0.973406i	$0.573574\pi$
$390$	0	0
$391$	4448.41	0.575360
$392$	0	0
$393$	6394.75	0.820795
$394$	0	0
$395$	14671.3	1.86885
$396$	0	0
$397$	10968.8	1.38667	0.693337	−	0.720613i	$-0.256140\pi$
$397$	10968.8	1.38667	0.693337	+	0.720613i	$0.256140\pi$
$398$	0	0
$399$	1201.42	0.150742
$400$	0	0
$401$	6508.24	0.810488	0.405244	−	0.914208i	$-0.367187\pi$
$401$	6508.24	0.810488	0.405244	+	0.914208i	$0.367187\pi$
$402$	0	0
$403$	611.812	0.0756242
$404$	0	0
$405$	1062.97	0.130418
$406$	0	0
$407$	−8617.51	−1.04952
$408$	0	0
$409$	9363.46	1.13201	0.566006	−	0.824401i	$-0.308488\pi$
$409$	9363.46	1.13201	0.566006	+	0.824401i	$0.308488\pi$
$410$	0	0
$411$	−770.801	−0.0925081
$412$	0	0
$413$	−6199.13	−0.738594
$414$	0	0
$415$	5139.39	0.607910
$416$	0	0
$417$	1312.33	0.154113
$418$	0	0
$419$	−12057.8	−1.40588	−0.702940	−	0.711249i	$-0.748130\pi$
$419$	−12057.8	−1.40588	−0.702940	+	0.711249i	$0.748130\pi$
$420$	0	0
$421$	15937.1	1.84496	0.922479	−	0.386047i	$-0.126160\pi$
$421$	15937.1	1.84496	0.922479	+	0.386047i	$0.126160\pi$
$422$	0	0
$423$	−5244.95	−0.602880
$424$	0	0
$425$	1108.50	0.126518
$426$	0	0
$427$	−3299.90	−0.373990
$428$	0	0
$429$	2173.22	0.244578
$430$	0	0
$431$	−6792.37	−0.759111	−0.379556	−	0.925169i	$-0.623923\pi$
$431$	−6792.37	−0.759111	−0.379556	+	0.925169i	$0.623923\pi$
$432$	0	0
$433$	−3338.00	−0.370471	−0.185235	−	0.982694i	$-0.559305\pi$
$433$	−3338.00	−0.370471	−0.185235	+	0.982694i	$0.559305\pi$
$434$	0	0
$435$	9320.24	1.02729
$436$	0	0
$437$	−4859.30	−0.531927
$438$	0	0
$439$	−13313.1	−1.44738	−0.723688	−	0.690127i	$-0.757555\pi$
$439$	−13313.1	−1.44738	−0.723688	+	0.690127i	$0.757555\pi$
$440$	0	0
$441$	−892.398	−0.0963608
$442$	0	0
$443$	5175.32	0.555050	0.277525	−	0.960718i	$-0.410486\pi$
$443$	5175.32	0.555050	0.277525	+	0.960718i	$0.410486\pi$
$444$	0	0
$445$	−9789.31	−1.04283
$446$	0	0
$447$	10219.9	1.08140
$448$	0	0
$449$	−4090.57	−0.429947	−0.214973	−	0.976620i	$-0.568966\pi$
$449$	−4090.57	−0.429947	−0.214973	+	0.976620i	$0.568966\pi$
$450$	0	0
$451$	1931.43	0.201658
$452$	0	0
$453$	−9827.31	−1.01927
$454$	0	0
$455$	2664.01	0.274486
$456$	0	0
$457$	−4300.78	−0.440224	−0.220112	−	0.975475i	$-0.570642\pi$
$457$	−4300.78	−0.440224	−0.220112	+	0.975475i	$0.570642\pi$
$458$	0	0
$459$	633.886	0.0644603
$460$	0	0
$461$	2578.07	0.260461	0.130231	−	0.991484i	$-0.458428\pi$
$461$	2578.07	0.260461	0.130231	+	0.991484i	$0.458428\pi$
$462$	0	0
$463$	4529.58	0.454660	0.227330	−	0.973818i	$-0.427000\pi$
$463$	4529.58	0.454660	0.227330	+	0.973818i	$0.427000\pi$
$464$	0	0
$465$	−1852.82	−0.184779
$466$	0	0
$467$	4761.89	0.471850	0.235925	−	0.971771i	$-0.424188\pi$
$467$	4761.89	0.471850	0.235925	+	0.971771i	$0.424188\pi$
$468$	0	0
$469$	−1340.24	−0.131955
$470$	0	0
$471$	−3223.03	−0.315307
$472$	0	0
$473$	22224.4	2.16042
$474$	0	0
$475$	−1210.89	−0.116967
$476$	0	0
$477$	3249.14	0.311882
$478$	0	0
$479$	−11621.4	−1.10855	−0.554275	−	0.832333i	$-0.687005\pi$
$479$	−11621.4	−1.10855	−0.554275	+	0.832333i	$0.687005\pi$
$480$	0	0
$481$	−2010.42	−0.190576
$482$	0	0
$483$	−8876.36	−0.836208
$484$	0	0
$485$	−2273.33	−0.212838
$486$	0	0
$487$	−7836.62	−0.729181	−0.364590	−	0.931168i	$-0.618791\pi$
$487$	−7836.62	−0.729181	−0.364590	+	0.931168i	$0.618791\pi$
$488$	0	0
$489$	6094.46	0.563601
$490$	0	0
$491$	−14163.9	−1.30185	−0.650925	−	0.759142i	$-0.725619\pi$
$491$	−14163.9	−1.30185	−0.650925	+	0.759142i	$0.725619\pi$
$492$	0	0
$493$	5557.98	0.507746
$494$	0	0
$495$	−6581.39	−0.597599
$496$	0	0
$497$	−10175.1	−0.918342
$498$	0	0
$499$	−811.953	−0.0728417	−0.0364208	−	0.999337i	$-0.511596\pi$
$499$	−811.953	−0.0728417	−0.0364208	+	0.999337i	$0.511596\pi$
$500$	0	0
$501$	−899.091	−0.0801765
$502$	0	0
$503$	−9718.41	−0.861476	−0.430738	−	0.902477i	$-0.641747\pi$
$503$	−9718.41	−0.861476	−0.430738	+	0.902477i	$0.641747\pi$
$504$	0	0
$505$	16580.4	1.46103
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	10546.5	0.918403	0.459201	−	0.888332i	$-0.348136\pi$
$509$	10546.5	0.918403	0.459201	+	0.888332i	$0.348136\pi$
$510$	0	0
$511$	−14480.7	−1.25360
$512$	0	0
$513$	−692.438	−0.0595943
$514$	0	0
$515$	−11474.9	−0.981831
$516$	0	0
$517$	32474.1	2.76250
$518$	0	0
$519$	−7113.47	−0.601631
$520$	0	0
$521$	1157.10	0.0973005	0.0486503	−	0.998816i	$-0.484508\pi$
$521$	1157.10	0.0973005	0.0486503	+	0.998816i	$0.484508\pi$
$522$	0	0
$523$	19193.0	1.60468	0.802342	−	0.596865i	$-0.203587\pi$
$523$	19193.0	1.60468	0.802342	+	0.596865i	$0.203587\pi$
$524$	0	0
$525$	−2211.90	−0.183877
$526$	0	0
$527$	−1104.90	−0.0913285
$528$	0	0
$529$	23734.6	1.95074
$530$	0	0
$531$	3572.86	0.291994
$532$	0	0
$533$	450.593	0.0366179
$534$	0	0
$535$	20890.5	1.68818
$536$	0	0
$537$	−748.840	−0.0601766
$538$	0	0
$539$	5525.28	0.441541
$540$	0	0
$541$	7809.48	0.620620	0.310310	−	0.950635i	$-0.399567\pi$
$541$	7809.48	0.620620	0.310310	+	0.950635i	$0.399567\pi$
$542$	0	0
$543$	9454.35	0.747192
$544$	0	0
$545$	−4879.01	−0.383475
$546$	0	0
$547$	4979.12	0.389199	0.194599	−	0.980883i	$-0.437659\pi$
$547$	4979.12	0.389199	0.194599	+	0.980883i	$0.437659\pi$
$548$	0	0
$549$	1901.90	0.147853
$550$	0	0
$551$	−6071.36	−0.469417
$552$	0	0
$553$	−17457.8	−1.34246
$554$	0	0
$555$	6088.37	0.465652
$556$	0	0
$557$	−24508.2	−1.86435	−0.932176	−	0.362005i	$-0.882092\pi$
$557$	−24508.2	−1.86435	−0.932176	+	0.362005i	$0.882092\pi$
$558$	0	0
$559$	5184.83	0.392299
$560$	0	0
$561$	−3924.71	−0.295368
$562$	0	0
$563$	14957.3	1.11967	0.559836	−	0.828604i	$-0.310864\pi$
$563$	14957.3	1.11967	0.559836	+	0.828604i	$0.310864\pi$
$564$	0	0
$565$	15406.3	1.14716
$566$	0	0
$567$	−1264.86	−0.0936844
$568$	0	0
$569$	5756.83	0.424146	0.212073	−	0.977254i	$-0.431979\pi$
$569$	5756.83	0.424146	0.212073	+	0.977254i	$0.431979\pi$
$570$	0	0
$571$	−7292.96	−0.534502	−0.267251	−	0.963627i	$-0.586115\pi$
$571$	−7292.96	−0.534502	−0.267251	+	0.963627i	$0.586115\pi$
$572$	0	0
$573$	14218.4	1.03662
$574$	0	0
$575$	8946.34	0.648849
$576$	0	0
$577$	20749.7	1.49709	0.748544	−	0.663085i	$-0.230753\pi$
$577$	20749.7	1.49709	0.748544	+	0.663085i	$0.230753\pi$
$578$	0	0
$579$	1910.46	0.137126
$580$	0	0
$581$	−6115.49	−0.436684
$582$	0	0
$583$	−20117.0	−1.42909
$584$	0	0
$585$	−1535.40	−0.108515
$586$	0	0
$587$	2046.37	0.143889	0.0719444	−	0.997409i	$-0.477080\pi$
$587$	2046.37	0.143889	0.0719444	+	0.997409i	$0.477080\pi$
$588$	0	0
$589$	1206.96	0.0844343
$590$	0	0
$591$	9003.39	0.626650
$592$	0	0
$593$	12664.1	0.876988	0.438494	−	0.898734i	$-0.355512\pi$
$593$	12664.1	0.876988	0.438494	+	0.898734i	$0.355512\pi$
$594$	0	0
$595$	−4811.06	−0.331486
$596$	0	0
$597$	1946.87	0.133468
$598$	0	0
$599$	−5788.48	−0.394843	−0.197421	−	0.980319i	$-0.563257\pi$
$599$	−5788.48	−0.394843	−0.197421	+	0.980319i	$0.563257\pi$
$600$	0	0
$601$	−19622.2	−1.33179	−0.665897	−	0.746044i	$-0.731951\pi$
$601$	−19622.2	−1.33179	−0.665897	+	0.746044i	$0.731951\pi$
$602$	0	0
$603$	772.449	0.0521668
$604$	0	0
$605$	23281.8	1.56453
$606$	0	0
$607$	11386.5	0.761389	0.380695	−	0.924701i	$-0.375685\pi$
$607$	11386.5	0.761389	0.380695	+	0.924701i	$0.375685\pi$
$608$	0	0
$609$	−11090.4	−0.737940
$610$	0	0
$611$	7576.04	0.501627
$612$	0	0
$613$	16046.0	1.05725	0.528623	−	0.848857i	$-0.322709\pi$
$613$	16046.0	1.05725	0.528623	+	0.848857i	$0.322709\pi$
$614$	0	0
$615$	−1364.58	−0.0894718
$616$	0	0
$617$	−25943.8	−1.69280	−0.846402	−	0.532545i	$-0.821236\pi$
$617$	−25943.8	−1.69280	−0.846402	+	0.532545i	$0.821236\pi$
$618$	0	0
$619$	22044.1	1.43139	0.715693	−	0.698415i	$-0.246111\pi$
$619$	22044.1	1.43139	0.715693	+	0.698415i	$0.246111\pi$
$620$	0	0
$621$	5115.89	0.330585
$622$	0	0
$623$	11648.6	0.749101
$624$	0	0
$625$	−19297.6	−1.23505
$626$	0	0
$627$	4287.23	0.273071
$628$	0	0
$629$	3630.71	0.230152
$630$	0	0
$631$	6948.74	0.438392	0.219196	−	0.975681i	$-0.429657\pi$
$631$	6948.74	0.438392	0.219196	+	0.975681i	$0.429657\pi$
$632$	0	0
$633$	−720.341	−0.0452306
$634$	0	0
$635$	−9702.70	−0.606362
$636$	0	0
$637$	1289.02	0.0801770
$638$	0	0
$639$	5864.42	0.363056
$640$	0	0
$641$	−17916.9	−1.10402	−0.552009	−	0.833838i	$-0.686139\pi$
$641$	−17916.9	−1.10402	−0.552009	+	0.833838i	$0.686139\pi$
$642$	0	0
$643$	10248.9	0.628581	0.314290	−	0.949327i	$-0.398233\pi$
$643$	10248.9	0.628581	0.314290	+	0.949327i	$0.398233\pi$
$644$	0	0
$645$	−15701.8	−0.958539
$646$	0	0
$647$	13034.7	0.792035	0.396018	−	0.918243i	$-0.370392\pi$
$647$	13034.7	0.792035	0.396018	+	0.918243i	$0.370392\pi$
$648$	0	0
$649$	−22121.4	−1.33797
$650$	0	0
$651$	2204.72	0.132734
$652$	0	0
$653$	−5517.54	−0.330656	−0.165328	−	0.986239i	$-0.552868\pi$
$653$	−5517.54	−0.330656	−0.165328	+	0.986239i	$0.552868\pi$
$654$	0	0
$655$	27973.0	1.66869
$656$	0	0
$657$	8345.97	0.495597
$658$	0	0
$659$	−31919.9	−1.88683	−0.943416	−	0.331610i	$-0.892408\pi$
$659$	−31919.9	−1.88683	−0.943416	+	0.331610i	$0.892408\pi$
$660$	0	0
$661$	3517.33	0.206972	0.103486	−	0.994631i	$-0.467000\pi$
$661$	3517.33	0.206972	0.103486	+	0.994631i	$0.467000\pi$
$662$	0	0
$663$	−915.613	−0.0536342
$664$	0	0
$665$	5255.45	0.306463
$666$	0	0
$667$	44856.6	2.60398
$668$	0	0
$669$	3313.11	0.191468
$670$	0	0
$671$	−11775.6	−0.677484
$672$	0	0
$673$	−33399.0	−1.91298	−0.956490	−	0.291766i	$-0.905757\pi$
$673$	−33399.0	−1.91298	−0.956490	+	0.291766i	$0.905757\pi$
$674$	0	0
$675$	1274.83	0.0726936
$676$	0	0
$677$	15843.8	0.899450	0.449725	−	0.893167i	$-0.351522\pi$
$677$	15843.8	0.899450	0.449725	+	0.893167i	$0.351522\pi$
$678$	0	0
$679$	2705.09	0.152890
$680$	0	0
$681$	1491.10	0.0839047
$682$	0	0
$683$	−10145.6	−0.568392	−0.284196	−	0.958766i	$-0.591727\pi$
$683$	−10145.6	−0.568392	−0.284196	+	0.958766i	$0.591727\pi$
$684$	0	0
$685$	−3371.77	−0.188071
$686$	0	0
$687$	−8637.16	−0.479663
$688$	0	0
$689$	−4693.20	−0.259502
$690$	0	0
$691$	18650.9	1.02679	0.513396	−	0.858152i	$-0.328387\pi$
$691$	18650.9	1.02679	0.513396	+	0.858152i	$0.328387\pi$
$692$	0	0
$693$	7831.36	0.429277
$694$	0	0
$695$	5740.61	0.313315
$696$	0	0
$697$	−813.745	−0.0442221
$698$	0	0
$699$	−11540.7	−0.624478
$700$	0	0
$701$	−25182.9	−1.35684	−0.678420	−	0.734675i	$-0.737335\pi$
$701$	−25182.9	−1.35684	−0.678420	+	0.734675i	$0.737335\pi$
$702$	0	0
$703$	−3966.07	−0.212778
$704$	0	0
$705$	−22943.4	−1.22567
$706$	0	0
$707$	−19729.4	−1.04951
$708$	0	0
$709$	−16697.6	−0.884473	−0.442236	−	0.896899i	$-0.645815\pi$
$709$	−16697.6	−0.884473	−0.442236	+	0.896899i	$0.645815\pi$
$710$	0	0
$711$	10061.8	0.530727
$712$	0	0
$713$	−8917.27	−0.468379
$714$	0	0
$715$	9506.45	0.497232
$716$	0	0
$717$	−8709.74	−0.453656
$718$	0	0
$719$	−22657.4	−1.17521	−0.587606	−	0.809147i	$-0.699930\pi$
$719$	−22657.4	−1.17521	−0.587606	+	0.809147i	$0.699930\pi$
$720$	0	0
$721$	13654.2	0.705285
$722$	0	0
$723$	−14403.2	−0.740886
$724$	0	0
$725$	11177.8	0.572599
$726$	0	0
$727$	32052.8	1.63518	0.817588	−	0.575803i	$-0.195310\pi$
$727$	32052.8	1.63518	0.817588	+	0.575803i	$0.195310\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−9363.52	−0.473765
$732$	0	0
$733$	−22728.1	−1.14527	−0.572634	−	0.819811i	$-0.694078\pi$
$733$	−22728.1	−1.14527	−0.572634	+	0.819811i	$0.694078\pi$
$734$	0	0
$735$	−3903.68	−0.195904
$736$	0	0
$737$	−4782.62	−0.239037
$738$	0	0
$739$	3741.58	0.186247	0.0931234	−	0.995655i	$-0.470315\pi$
$739$	3741.58	0.186247	0.0931234	+	0.995655i	$0.470315\pi$
$740$	0	0
$741$	1000.19	0.0495854
$742$	0	0
$743$	23932.4	1.18169	0.590844	−	0.806786i	$-0.298795\pi$
$743$	23932.4	1.18169	0.590844	+	0.806786i	$0.298795\pi$
$744$	0	0
$745$	44705.8	2.19852
$746$	0	0
$747$	3524.66	0.172638
$748$	0	0
$749$	−24858.1	−1.21268
$750$	0	0
$751$	15777.6	0.766623	0.383311	−	0.923619i	$-0.374784\pi$
$751$	15777.6	0.766623	0.383311	+	0.923619i	$0.374784\pi$
$752$	0	0
$753$	14304.6	0.692282
$754$	0	0
$755$	−42988.3	−2.07219
$756$	0	0
$757$	−32474.4	−1.55919	−0.779593	−	0.626287i	$-0.784574\pi$
$757$	−32474.4	−1.55919	−0.779593	+	0.626287i	$0.784574\pi$
$758$	0	0
$759$	−31675.0	−1.51480
$760$	0	0
$761$	24471.4	1.16569	0.582844	−	0.812584i	$-0.301940\pi$
$761$	24471.4	1.16569	0.582844	+	0.812584i	$0.301940\pi$
$762$	0	0
$763$	5805.66	0.275464
$764$	0	0
$765$	2772.85	0.131049
$766$	0	0
$767$	−5160.80	−0.242954
$768$	0	0
$769$	5185.33	0.243157	0.121578	−	0.992582i	$-0.461204\pi$
$769$	5185.33	0.243157	0.121578	+	0.992582i	$0.461204\pi$
$770$	0	0
$771$	18006.5	0.841102
$772$	0	0
$773$	−6218.64	−0.289352	−0.144676	−	0.989479i	$-0.546214\pi$
$773$	−6218.64	−0.289352	−0.144676	+	0.989479i	$0.546214\pi$
$774$	0	0
$775$	−2222.10	−0.102994
$776$	0	0
$777$	−7244.72	−0.334495
$778$	0	0
$779$	888.910	0.0408838
$780$	0	0
$781$	−36309.5	−1.66358
$782$	0	0
$783$	6391.94	0.291736
$784$	0	0
$785$	−14098.7	−0.641026
$786$	0	0
$787$	−15933.8	−0.721702	−0.360851	−	0.932623i	$-0.617514\pi$
$787$	−15933.8	−0.721702	−0.360851	+	0.932623i	$0.617514\pi$
$788$	0	0
$789$	9805.00	0.442417
$790$	0	0
$791$	−18332.3	−0.824049
$792$	0	0
$793$	−2747.19	−0.123021
$794$	0	0
$795$	14212.9	0.634063
$796$	0	0
$797$	825.379	0.0366831	0.0183415	−	0.999832i	$-0.494161\pi$
$797$	825.379	0.0366831	0.0183415	+	0.999832i	$0.494161\pi$
$798$	0	0
$799$	−13681.9	−0.605796
$800$	0	0
$801$	−6713.64	−0.296148
$802$	0	0
$803$	−51674.0	−2.27090
$804$	0	0
$805$	−38828.5	−1.70003
$806$	0	0
$807$	26070.4	1.13720
$808$	0	0
$809$	23981.8	1.04222	0.521110	−	0.853490i	$-0.325518\pi$
$809$	23981.8	1.04222	0.521110	+	0.853490i	$0.325518\pi$
$810$	0	0
$811$	−1359.77	−0.0588753	−0.0294376	−	0.999567i	$-0.509372\pi$
$811$	−1359.77	−0.0588753	−0.0294376	+	0.999567i	$0.509372\pi$
$812$	0	0
$813$	9717.52	0.419198
$814$	0	0
$815$	26659.4	1.14581
$816$	0	0
$817$	10228.4	0.438001
$818$	0	0
$819$	1827.02	0.0779501
$820$	0	0
$821$	29276.9	1.24455	0.622273	−	0.782801i	$-0.286209\pi$
$821$	29276.9	1.24455	0.622273	+	0.782801i	$0.286209\pi$
$822$	0	0
$823$	3971.15	0.168196	0.0840982	−	0.996457i	$-0.473199\pi$
$823$	3971.15	0.168196	0.0840982	+	0.996457i	$0.473199\pi$
$824$	0	0
$825$	−7893.10	−0.333094
$826$	0	0
$827$	−18920.5	−0.795564	−0.397782	−	0.917480i	$-0.630220\pi$
$827$	−18920.5	−0.795564	−0.397782	+	0.917480i	$0.630220\pi$
$828$	0	0
$829$	−29902.3	−1.25277	−0.626387	−	0.779512i	$-0.715467\pi$
$829$	−29902.3	−1.25277	−0.626387	+	0.779512i	$0.715467\pi$
$830$	0	0
$831$	660.273	0.0275627
$832$	0	0
$833$	−2327.90	−0.0968269
$834$	0	0
$835$	−3932.95	−0.163001
$836$	0	0
$837$	−1270.69	−0.0524748
$838$	0	0
$839$	−41125.0	−1.69224	−0.846121	−	0.532991i	$-0.821068\pi$
$839$	−41125.0	−1.69224	−0.846121	+	0.532991i	$0.821068\pi$
$840$	0	0
$841$	31656.2	1.29797
$842$	0	0
$843$	1761.70	0.0719765
$844$	0	0
$845$	2217.80	0.0902897
$846$	0	0
$847$	−27703.6	−1.12386
$848$	0	0
$849$	−24822.3	−1.00341
$850$	0	0
$851$	29302.2	1.18034
$852$	0	0
$853$	−2252.79	−0.0904266	−0.0452133	−	0.998977i	$-0.514397\pi$
$853$	−2252.79	−0.0904266	−0.0452133	+	0.998977i	$0.514397\pi$
$854$	0	0
$855$	−3028.98	−0.121157
$856$	0	0
$857$	16608.6	0.662005	0.331002	−	0.943630i	$-0.392613\pi$
$857$	16608.6	0.662005	0.331002	+	0.943630i	$0.392613\pi$
$858$	0	0
$859$	−14112.0	−0.560529	−0.280265	−	0.959923i	$-0.590422\pi$
$859$	−14112.0	−0.560529	−0.280265	+	0.959923i	$0.590422\pi$
$860$	0	0
$861$	1623.75	0.0642709
$862$	0	0
$863$	−40154.6	−1.58387	−0.791934	−	0.610607i	$-0.790925\pi$
$863$	−40154.6	−1.58387	−0.791934	+	0.610607i	$0.790925\pi$
$864$	0	0
$865$	−31116.9	−1.22313
$866$	0	0
$867$	−13085.5	−0.512578
$868$	0	0
$869$	−62297.6	−2.43188
$870$	0	0
$871$	−1115.76	−0.0434054
$872$	0	0
$873$	−1559.08	−0.0604431
$874$	0	0
$875$	15939.8	0.615846
$876$	0	0
$877$	22495.6	0.866161	0.433080	−	0.901355i	$-0.357427\pi$
$877$	22495.6	0.866161	0.433080	+	0.901355i	$0.357427\pi$
$878$	0	0
$879$	16146.2	0.619565
$880$	0	0
$881$	−42296.3	−1.61748	−0.808740	−	0.588166i	$-0.799850\pi$
$881$	−42296.3	−1.61748	−0.808740	+	0.588166i	$0.799850\pi$
$882$	0	0
$883$	−37440.9	−1.42694	−0.713469	−	0.700687i	$-0.752877\pi$
$883$	−37440.9	−1.42694	−0.713469	+	0.700687i	$0.752877\pi$
$884$	0	0
$885$	15629.0	0.593631
$886$	0	0
$887$	−3372.82	−0.127675	−0.0638377	−	0.997960i	$-0.520334\pi$
$887$	−3372.82	−0.127675	−0.0638377	+	0.997960i	$0.520334\pi$
$888$	0	0
$889$	11545.5	0.435572
$890$	0	0
$891$	−4513.60	−0.169710
$892$	0	0
$893$	14945.7	0.560066
$894$	0	0
$895$	−3275.70	−0.122340
$896$	0	0
$897$	−7389.61	−0.275064
$898$	0	0
$899$	−11141.5	−0.413337
$900$	0	0
$901$	8475.65	0.313390
$902$	0	0
$903$	18684.0	0.688554
$904$	0	0
$905$	41356.8	1.51906
$906$	0	0
$907$	30130.3	1.10304	0.551522	−	0.834160i	$-0.314047\pi$
$907$	30130.3	1.10304	0.551522	+	0.834160i	$0.314047\pi$
$908$	0	0
$909$	11371.1	0.414911
$910$	0	0
$911$	26089.9	0.948844	0.474422	−	0.880297i	$-0.342657\pi$
$911$	26089.9	0.948844	0.474422	+	0.880297i	$0.342657\pi$
$912$	0	0
$913$	−21822.9	−0.791055
$914$	0	0
$915$	8319.60	0.300588
$916$	0	0
$917$	−33285.8	−1.19868
$918$	0	0
$919$	5142.55	0.184589	0.0922944	−	0.995732i	$-0.470580\pi$
$919$	5142.55	0.184589	0.0922944	+	0.995732i	$0.470580\pi$
$920$	0	0
$921$	−9287.76	−0.332293
$922$	0	0
$923$	−8470.83	−0.302081
$924$	0	0
$925$	7301.83	0.259549
$926$	0	0
$927$	−7869.61	−0.278826
$928$	0	0
$929$	23311.6	0.823282	0.411641	−	0.911346i	$-0.364956\pi$
$929$	23311.6	0.823282	0.411641	+	0.911346i	$0.364956\pi$
$930$	0	0
$931$	2542.92	0.0895176
$932$	0	0
$933$	−10423.7	−0.365764
$934$	0	0
$935$	−17168.1	−0.600489
$936$	0	0
$937$	−17214.9	−0.600199	−0.300099	−	0.953908i	$-0.597020\pi$
$937$	−17214.9	−0.600199	−0.300099	+	0.953908i	$0.597020\pi$
$938$	0	0
$939$	−10571.8	−0.367409
$940$	0	0
$941$	27871.2	0.965541	0.482770	−	0.875747i	$-0.339631\pi$
$941$	27871.2	0.965541	0.482770	+	0.875747i	$0.339631\pi$
$942$	0	0
$943$	−6567.47	−0.226793
$944$	0	0
$945$	−5532.95	−0.190462
$946$	0	0
$947$	−14956.7	−0.513227	−0.256614	−	0.966514i	$-0.582607\pi$
$947$	−14956.7	−0.513227	−0.256614	+	0.966514i	$0.582607\pi$
$948$	0	0
$949$	−12055.3	−0.412361
$950$	0	0
$951$	970.188	0.0330815
$952$	0	0
$953$	−2056.14	−0.0698898	−0.0349449	−	0.999389i	$-0.511126\pi$
$953$	−2056.14	−0.0698898	−0.0349449	+	0.999389i	$0.511126\pi$
$954$	0	0
$955$	62196.4	2.10746
$956$	0	0
$957$	−39575.7	−1.33678
$958$	0	0
$959$	4012.16	0.135098
$960$	0	0
$961$	−27576.1	−0.925653
$962$	0	0
$963$	14327.0	0.479419
$964$	0	0
$965$	8357.03	0.278780
$966$	0	0
$967$	−25282.1	−0.840764	−0.420382	−	0.907347i	$-0.638104\pi$
$967$	−25282.1	−0.840764	−0.420382	+	0.907347i	$0.638104\pi$
$968$	0	0
$969$	−1806.28	−0.0598825
$970$	0	0
$971$	22919.6	0.757493	0.378747	−	0.925500i	$-0.376355\pi$
$971$	22919.6	0.757493	0.378747	+	0.925500i	$0.376355\pi$
$972$	0	0
$973$	−6830.91	−0.225066
$974$	0	0
$975$	−1841.42	−0.0604848
$976$	0	0
$977$	42964.8	1.40693	0.703463	−	0.710732i	$-0.251636\pi$
$977$	42964.8	1.40693	0.703463	+	0.710732i	$0.251636\pi$
$978$	0	0
$979$	41567.5	1.35700
$980$	0	0
$981$	−3346.09	−0.108902
$982$	0	0
$983$	53757.9	1.74426	0.872132	−	0.489271i	$-0.162737\pi$
$983$	53757.9	1.74426	0.872132	+	0.489271i	$0.162737\pi$
$984$	0	0
$985$	39384.2	1.27399
$986$	0	0
$987$	27300.9	0.880443
$988$	0	0
$989$	−75569.8	−2.42971
$990$	0	0
$991$	28021.7	0.898224	0.449112	−	0.893475i	$-0.351740\pi$
$991$	28021.7	0.898224	0.449112	+	0.893475i	$0.351740\pi$
$992$	0	0
$993$	−3173.22	−0.101409
$994$	0	0
$995$	8516.35	0.271343
$996$	0	0
$997$	−7057.01	−0.224170	−0.112085	−	0.993699i	$-0.535753\pi$
$997$	−7057.01	−0.224170	−0.112085	+	0.993699i	$0.535753\pi$
$998$	0	0
$999$	4175.49	0.132239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bj.1.2		2
4.3	odd	2		2496.4.a.ba.1.2		2
8.3	odd	2		312.4.a.d.1.1	✓	2
8.5	even	2		624.4.a.j.1.1		2
24.5	odd	2		1872.4.a.bi.1.2		2
24.11	even	2		936.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.d.1.1	✓	2	8.3	odd	2
624.4.a.j.1.1		2	8.5	even	2
936.4.a.j.1.2		2	24.11	even	2
1872.4.a.bi.1.2		2	24.5	odd	2
2496.4.a.ba.1.2		2	4.3	odd	2
2496.4.a.bj.1.2		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +13.1231 q^{5} -15.6155 q^{7} +9.00000 q^{9} -55.7235 q^{11} -13.0000 q^{13} +39.3693 q^{15} +23.4773 q^{17} -25.6458 q^{19} -46.8466 q^{21} +189.477 q^{23} +47.2159 q^{25} +27.0000 q^{27} +236.739 q^{29} -47.0625 q^{31} -167.170 q^{33} -204.924 q^{35} +154.648 q^{37} -39.0000 q^{39} -34.6610 q^{41} -398.833 q^{43} +118.108 q^{45} -582.773 q^{47} -99.1553 q^{49} +70.4318 q^{51} +361.015 q^{53} -731.265 q^{55} -76.9375 q^{57} +396.985 q^{59} +211.322 q^{61} -140.540 q^{63} -170.600 q^{65} +85.8277 q^{67} +568.432 q^{69} +651.602 q^{71} +927.329 q^{73} +141.648 q^{75} +870.152 q^{77} +1117.98 q^{79} +81.0000 q^{81} +391.629 q^{83} +308.095 q^{85} +710.216 q^{87} -745.960 q^{89} +203.002 q^{91} -141.187 q^{93} -336.553 q^{95} -173.231 q^{97} -501.511 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 18 q^{5} + 10 q^{7} + 18 q^{9} + 4 q^{11} - 26 q^{13} + 54 q^{15} - 52 q^{17} - 142 q^{19} + 30 q^{21} + 280 q^{23} - 54 q^{25} + 54 q^{27} + 424 q^{29} + 178 q^{31} + 12 q^{33} - 80 q^{35}+ \cdots + 36 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 18 * q^5 + 10 * q^7 + 18 * q^9 + 4 * q^11 - 26 * q^13 + 54 * q^15 - 52 * q^17 - 142 * q^19 + 30 * q^21 + 280 * q^23 - 54 * q^25 + 54 * q^27 + 424 * q^29 + 178 * q^31 + 12 * q^33 - 80 * q^35 - 136 * q^37 - 78 * q^39 - 226 * q^41 - 72 * q^43 + 162 * q^45 - 176 * q^47 + 214 * q^49 - 156 * q^51 + 788 * q^53 - 440 * q^55 - 426 * q^57 + 728 * q^59 + 736 * q^61 + 90 * q^63 - 234 * q^65 + 1054 * q^67 + 840 * q^69 + 660 * q^71 + 24 * q^73 - 162 * q^75 + 2400 * q^77 - 40 * q^79 + 162 * q^81 + 1344 * q^83 - 60 * q^85 + 1272 * q^87 + 314 * q^89 - 130 * q^91 + 534 * q^93 - 904 * q^95 - 264 * q^97 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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