LMFDB - Embedded newform 2496.4.a.cf.1.5

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:	$5$
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			1.5
Root			$1.04635$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +22.0884 q^{5} -0.570636 q^{7} +9.00000 q^{9} +14.5147 q^{11} +13.0000 q^{13} +66.2652 q^{15} +62.8354 q^{17} +106.914 q^{19} -1.71191 q^{21} +175.759 q^{23} +362.898 q^{25} +27.0000 q^{27} -66.4653 q^{29} +53.0145 q^{31} +43.5441 q^{33} -12.6045 q^{35} -61.7208 q^{37} +39.0000 q^{39} -244.872 q^{41} -64.2156 q^{43} +198.796 q^{45} -51.2733 q^{47} -342.674 q^{49} +188.506 q^{51} -495.893 q^{53} +320.606 q^{55} +320.742 q^{57} -322.773 q^{59} -499.239 q^{61} -5.13573 q^{63} +287.149 q^{65} +145.751 q^{67} +527.278 q^{69} +1111.55 q^{71} -592.186 q^{73} +1088.69 q^{75} -8.28261 q^{77} +124.450 q^{79} +81.0000 q^{81} +675.769 q^{83} +1387.93 q^{85} -199.396 q^{87} -411.589 q^{89} -7.41827 q^{91} +159.044 q^{93} +2361.56 q^{95} -1633.14 q^{97} +130.632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	22.0884	1.97565	0.987824	−	0.155578i	$-0.0497239\pi$
$5$	22.0884	1.97565	0.987824	+	0.155578i	$0.0497239\pi$
$6$	0	0
$7$	−0.570636	−0.0308115	−0.0154057	−	0.999881i	$-0.504904\pi$
$7$	−0.570636	−0.0308115	−0.0154057	+	0.999881i	$0.504904\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	14.5147	0.397849	0.198925	−	0.980015i	$-0.436255\pi$
$11$	14.5147	0.397849	0.198925	+	0.980015i	$0.436255\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	66.2652	1.14064
$16$	0	0
$17$	62.8354	0.896460	0.448230	−	0.893918i	$-0.352055\pi$
$17$	62.8354	0.896460	0.448230	+	0.893918i	$0.352055\pi$
$18$	0	0
$19$	106.914	1.29093	0.645467	−	0.763788i	$-0.276663\pi$
$19$	106.914	1.29093	0.645467	+	0.763788i	$0.276663\pi$
$20$	0	0
$21$	−1.71191	−0.0177890
$22$	0	0
$23$	175.759	1.59341	0.796704	−	0.604370i	$-0.206575\pi$
$23$	175.759	1.59341	0.796704	+	0.604370i	$0.206575\pi$
$24$	0	0
$25$	362.898	2.90318
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−66.4653	−0.425597	−0.212798	−	0.977096i	$-0.568258\pi$
$29$	−66.4653	−0.425597	−0.212798	+	0.977096i	$0.568258\pi$
$30$	0	0
$31$	53.0145	0.307151	0.153576	−	0.988137i	$-0.450921\pi$
$31$	53.0145	0.307151	0.153576	+	0.988137i	$0.450921\pi$
$32$	0	0
$33$	43.5441	0.229698
$34$	0	0
$35$	−12.6045	−0.0608726
$36$	0	0
$37$	−61.7208	−0.274239	−0.137119	−	0.990555i	$-0.543784\pi$
$37$	−61.7208	−0.274239	−0.137119	+	0.990555i	$0.543784\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−244.872	−0.932747	−0.466373	−	0.884588i	$-0.654440\pi$
$41$	−244.872	−0.932747	−0.466373	+	0.884588i	$0.654440\pi$
$42$	0	0
$43$	−64.2156	−0.227739	−0.113870	−	0.993496i	$-0.536325\pi$
$43$	−64.2156	−0.227739	−0.113870	+	0.993496i	$0.536325\pi$
$44$	0	0
$45$	198.796	0.658549
$46$	0	0
$47$	−51.2733	−0.159127	−0.0795636	−	0.996830i	$-0.525353\pi$
$47$	−51.2733	−0.159127	−0.0795636	+	0.996830i	$0.525353\pi$
$48$	0	0
$49$	−342.674	−0.999051
$50$	0	0
$51$	188.506	0.517571
$52$	0	0
$53$	−495.893	−1.28521	−0.642605	−	0.766198i	$-0.722146\pi$
$53$	−495.893	−1.28521	−0.642605	+	0.766198i	$0.722146\pi$
$54$	0	0
$55$	320.606	0.786010
$56$	0	0
$57$	320.742	0.745321
$58$	0	0
$59$	−322.773	−0.712229	−0.356115	−	0.934442i	$-0.615899\pi$
$59$	−322.773	−0.712229	−0.356115	+	0.934442i	$0.615899\pi$
$60$	0	0
$61$	−499.239	−1.04788	−0.523942	−	0.851754i	$-0.675539\pi$
$61$	−499.239	−1.04788	−0.523942	+	0.851754i	$0.675539\pi$
$62$	0	0
$63$	−5.13573	−0.0102705
$64$	0	0
$65$	287.149	0.547946
$66$	0	0
$67$	145.751	0.265765	0.132883	−	0.991132i	$-0.457577\pi$
$67$	145.751	0.265765	0.132883	+	0.991132i	$0.457577\pi$
$68$	0	0
$69$	527.278	0.919954
$70$	0	0
$71$	1111.55	1.85799	0.928994	−	0.370094i	$-0.120675\pi$
$71$	1111.55	1.85799	0.928994	+	0.370094i	$0.120675\pi$
$72$	0	0
$73$	−592.186	−0.949454	−0.474727	−	0.880133i	$-0.657453\pi$
$73$	−592.186	−0.949454	−0.474727	+	0.880133i	$0.657453\pi$
$74$	0	0
$75$	1088.69	1.67615
$76$	0	0
$77$	−8.28261	−0.0122583
$78$	0	0
$79$	124.450	0.177238	0.0886188	−	0.996066i	$-0.471755\pi$
$79$	124.450	0.177238	0.0886188	+	0.996066i	$0.471755\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	675.769	0.893678	0.446839	−	0.894614i	$-0.352550\pi$
$83$	675.769	0.893678	0.446839	+	0.894614i	$0.352550\pi$
$84$	0	0
$85$	1387.93	1.77109
$86$	0	0
$87$	−199.396	−0.245718
$88$	0	0
$89$	−411.589	−0.490206	−0.245103	−	0.969497i	$-0.578822\pi$
$89$	−411.589	−0.490206	−0.245103	+	0.969497i	$0.578822\pi$
$90$	0	0
$91$	−7.41827	−0.00854556
$92$	0	0
$93$	159.044	0.177334
$94$	0	0
$95$	2361.56	2.55043
$96$	0	0
$97$	−1633.14	−1.70949	−0.854744	−	0.519050i	$-0.826286\pi$
$97$	−1633.14	−1.70949	−0.854744	+	0.519050i	$0.826286\pi$
$98$	0	0
$99$	130.632	0.132616
$100$	0	0
$101$	1695.34	1.67023	0.835114	−	0.550077i	$-0.185402\pi$
$101$	1695.34	1.67023	0.835114	+	0.550077i	$0.185402\pi$
$102$	0	0
$103$	−1055.03	−1.00927	−0.504637	−	0.863331i	$-0.668374\pi$
$103$	−1055.03	−1.00927	−0.504637	+	0.863331i	$0.668374\pi$
$104$	0	0
$105$	−37.8134	−0.0351448
$106$	0	0
$107$	−1434.41	−1.29598	−0.647990	−	0.761649i	$-0.724390\pi$
$107$	−1434.41	−1.29598	−0.647990	+	0.761649i	$0.724390\pi$
$108$	0	0
$109$	210.166	0.184681	0.0923406	−	0.995727i	$-0.470565\pi$
$109$	210.166	0.184681	0.0923406	+	0.995727i	$0.470565\pi$
$110$	0	0
$111$	−185.163	−0.158332
$112$	0	0
$113$	788.100	0.656090	0.328045	−	0.944662i	$-0.393610\pi$
$113$	788.100	0.656090	0.328045	+	0.944662i	$0.393610\pi$
$114$	0	0
$115$	3882.25	3.14801
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−35.8561	−0.0276212
$120$	0	0
$121$	−1120.32	−0.841716
$122$	0	0
$123$	−734.617	−0.538522
$124$	0	0
$125$	5254.78	3.76002
$126$	0	0
$127$	1668.08	1.16550	0.582750	−	0.812652i	$-0.301977\pi$
$127$	1668.08	1.16550	0.582750	+	0.812652i	$0.301977\pi$
$128$	0	0
$129$	−192.647	−0.131485
$130$	0	0
$131$	−2635.33	−1.75763	−0.878817	−	0.477159i	$-0.841667\pi$
$131$	−2635.33	−1.75763	−0.878817	+	0.477159i	$0.841667\pi$
$132$	0	0
$133$	−61.0090	−0.0397756
$134$	0	0
$135$	596.387	0.380214
$136$	0	0
$137$	−1228.31	−0.765997	−0.382999	−	0.923749i	$-0.625109\pi$
$137$	−1228.31	−0.765997	−0.382999	+	0.923749i	$0.625109\pi$
$138$	0	0
$139$	1363.03	0.831734	0.415867	−	0.909426i	$-0.363478\pi$
$139$	1363.03	0.831734	0.415867	+	0.909426i	$0.363478\pi$
$140$	0	0
$141$	−153.820	−0.0918722
$142$	0	0
$143$	188.691	0.110344
$144$	0	0
$145$	−1468.11	−0.840829
$146$	0	0
$147$	−1028.02	−0.576802
$148$	0	0
$149$	298.498	0.164120	0.0820600	−	0.996627i	$-0.473850\pi$
$149$	298.498	0.164120	0.0820600	+	0.996627i	$0.473850\pi$
$150$	0	0
$151$	−3297.44	−1.77710	−0.888550	−	0.458781i	$-0.848286\pi$
$151$	−3297.44	−1.77710	−0.888550	+	0.458781i	$0.848286\pi$
$152$	0	0
$153$	565.518	0.298820
$154$	0	0
$155$	1171.01	0.606823
$156$	0	0
$157$	−1969.65	−1.00124	−0.500620	−	0.865667i	$-0.666895\pi$
$157$	−1969.65	−1.00124	−0.500620	+	0.865667i	$0.666895\pi$
$158$	0	0
$159$	−1487.68	−0.742016
$160$	0	0
$161$	−100.295	−0.0490952
$162$	0	0
$163$	−2046.76	−0.983525	−0.491762	−	0.870729i	$-0.663647\pi$
$163$	−2046.76	−0.983525	−0.491762	+	0.870729i	$0.663647\pi$
$164$	0	0
$165$	961.819	0.453803
$166$	0	0
$167$	2873.01	1.33126	0.665628	−	0.746283i	$-0.268164\pi$
$167$	2873.01	1.33126	0.665628	+	0.746283i	$0.268164\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	962.226	0.430311
$172$	0	0
$173$	534.246	0.234786	0.117393	−	0.993086i	$-0.462546\pi$
$173$	534.246	0.234786	0.117393	+	0.993086i	$0.462546\pi$
$174$	0	0
$175$	−207.083	−0.0894513
$176$	0	0
$177$	−968.320	−0.411206
$178$	0	0
$179$	3504.15	1.46320	0.731599	−	0.681735i	$-0.238774\pi$
$179$	3504.15	1.46320	0.731599	+	0.681735i	$0.238774\pi$
$180$	0	0
$181$	4027.89	1.65409	0.827047	−	0.562133i	$-0.190019\pi$
$181$	4027.89	1.65409	0.827047	+	0.562133i	$0.190019\pi$
$182$	0	0
$183$	−1497.72	−0.604996
$184$	0	0
$185$	−1363.32	−0.541800
$186$	0	0
$187$	912.036	0.356656
$188$	0	0
$189$	−15.4072	−0.00592967
$190$	0	0
$191$	−1516.47	−0.574493	−0.287247	−	0.957857i	$-0.592740\pi$
$191$	−1516.47	−0.574493	−0.287247	+	0.957857i	$0.592740\pi$
$192$	0	0
$193$	429.841	0.160314	0.0801571	−	0.996782i	$-0.474458\pi$
$193$	429.841	0.160314	0.0801571	+	0.996782i	$0.474458\pi$
$194$	0	0
$195$	861.448	0.316357
$196$	0	0
$197$	−630.616	−0.228069	−0.114034	−	0.993477i	$-0.536377\pi$
$197$	−630.616	−0.228069	−0.114034	+	0.993477i	$0.536377\pi$
$198$	0	0
$199$	−4222.79	−1.50425	−0.752124	−	0.659021i	$-0.770971\pi$
$199$	−4222.79	−1.50425	−0.752124	+	0.659021i	$0.770971\pi$
$200$	0	0
$201$	437.252	0.153440
$202$	0	0
$203$	37.9275	0.0131133
$204$	0	0
$205$	−5408.84	−1.84278
$206$	0	0
$207$	1581.84	0.531136
$208$	0	0
$209$	1551.82	0.513597
$210$	0	0
$211$	1464.08	0.477686	0.238843	−	0.971058i	$-0.423232\pi$
$211$	1464.08	0.477686	0.238843	+	0.971058i	$0.423232\pi$
$212$	0	0
$213$	3334.66	1.07271
$214$	0	0
$215$	−1418.42	−0.449933
$216$	0	0
$217$	−30.2520	−0.00946379
$218$	0	0
$219$	−1776.56	−0.548167
$220$	0	0
$221$	816.860	0.248633
$222$	0	0
$223$	5488.41	1.64812	0.824061	−	0.566501i	$-0.191703\pi$
$223$	5488.41	1.64812	0.824061	+	0.566501i	$0.191703\pi$
$224$	0	0
$225$	3266.08	0.967727
$226$	0	0
$227$	1538.34	0.449793	0.224896	−	0.974383i	$-0.427796\pi$
$227$	1538.34	0.449793	0.224896	+	0.974383i	$0.427796\pi$
$228$	0	0
$229$	−2544.64	−0.734300	−0.367150	−	0.930162i	$-0.619666\pi$
$229$	−2544.64	−0.734300	−0.367150	+	0.930162i	$0.619666\pi$
$230$	0	0
$231$	−24.8478	−0.00707735
$232$	0	0
$233$	588.474	0.165460	0.0827301	−	0.996572i	$-0.473636\pi$
$233$	588.474	0.165460	0.0827301	+	0.996572i	$0.473636\pi$
$234$	0	0
$235$	−1132.55	−0.314379
$236$	0	0
$237$	373.351	0.102328
$238$	0	0
$239$	655.416	0.177386	0.0886932	−	0.996059i	$-0.471731\pi$
$239$	655.416	0.177386	0.0886932	+	0.996059i	$0.471731\pi$
$240$	0	0
$241$	−603.681	−0.161355	−0.0806774	−	0.996740i	$-0.525708\pi$
$241$	−603.681	−0.161355	−0.0806774	+	0.996740i	$0.525708\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−7569.13	−1.97377
$246$	0	0
$247$	1389.88	0.358041
$248$	0	0
$249$	2027.31	0.515965
$250$	0	0
$251$	−6332.16	−1.59236	−0.796180	−	0.605060i	$-0.793149\pi$
$251$	−6332.16	−1.59236	−0.796180	+	0.605060i	$0.793149\pi$
$252$	0	0
$253$	2551.09	0.633936
$254$	0	0
$255$	4163.80	1.02254
$256$	0	0
$257$	2114.90	0.513323	0.256662	−	0.966501i	$-0.417377\pi$
$257$	2114.90	0.513323	0.256662	+	0.966501i	$0.417377\pi$
$258$	0	0
$259$	35.2202	0.00844971
$260$	0	0
$261$	−598.188	−0.141866
$262$	0	0
$263$	−3186.29	−0.747054	−0.373527	−	0.927619i	$-0.621852\pi$
$263$	−3186.29	−0.747054	−0.373527	+	0.927619i	$0.621852\pi$
$264$	0	0
$265$	−10953.5	−2.53912
$266$	0	0
$267$	−1234.77	−0.283020
$268$	0	0
$269$	2056.40	0.466100	0.233050	−	0.972465i	$-0.425129\pi$
$269$	2056.40	0.466100	0.233050	+	0.972465i	$0.425129\pi$
$270$	0	0
$271$	399.938	0.0896476	0.0448238	−	0.998995i	$-0.485727\pi$
$271$	399.938	0.0896476	0.0448238	+	0.998995i	$0.485727\pi$
$272$	0	0
$273$	−22.2548	−0.00493378
$274$	0	0
$275$	5267.35	1.15503
$276$	0	0
$277$	4470.24	0.969643	0.484821	−	0.874613i	$-0.338885\pi$
$277$	4470.24	0.969643	0.484821	+	0.874613i	$0.338885\pi$
$278$	0	0
$279$	477.131	0.102384
$280$	0	0
$281$	−401.796	−0.0852993	−0.0426497	−	0.999090i	$-0.513580\pi$
$281$	−401.796	−0.0852993	−0.0426497	+	0.999090i	$0.513580\pi$
$282$	0	0
$283$	−3914.77	−0.822292	−0.411146	−	0.911569i	$-0.634871\pi$
$283$	−3914.77	−0.822292	−0.411146	+	0.911569i	$0.634871\pi$
$284$	0	0
$285$	7084.68	1.47249
$286$	0	0
$287$	139.733	0.0287393
$288$	0	0
$289$	−964.717	−0.196360
$290$	0	0
$291$	−4899.42	−0.986973
$292$	0	0
$293$	433.936	0.0865216	0.0432608	−	0.999064i	$-0.486225\pi$
$293$	433.936	0.0865216	0.0432608	+	0.999064i	$0.486225\pi$
$294$	0	0
$295$	−7129.55	−1.40711
$296$	0	0
$297$	391.897	0.0765661
$298$	0	0
$299$	2284.87	0.441932
$300$	0	0
$301$	36.6438	0.00701699
$302$	0	0
$303$	5086.03	0.964306
$304$	0	0
$305$	−11027.4	−2.07025
$306$	0	0
$307$	4680.64	0.870157	0.435078	−	0.900393i	$-0.356721\pi$
$307$	4680.64	0.870157	0.435078	+	0.900393i	$0.356721\pi$
$308$	0	0
$309$	−3165.09	−0.582705
$310$	0	0
$311$	670.394	0.122233	0.0611167	−	0.998131i	$-0.480534\pi$
$311$	670.394	0.122233	0.0611167	+	0.998131i	$0.480534\pi$
$312$	0	0
$313$	2198.73	0.397060	0.198530	−	0.980095i	$-0.436383\pi$
$313$	2198.73	0.397060	0.198530	+	0.980095i	$0.436383\pi$
$314$	0	0
$315$	−113.440	−0.0202909
$316$	0	0
$317$	7510.62	1.33072	0.665360	−	0.746523i	$-0.268278\pi$
$317$	7510.62	1.33072	0.665360	+	0.746523i	$0.268278\pi$
$318$	0	0
$319$	−964.724	−0.169323
$320$	0	0
$321$	−4303.24	−0.748235
$322$	0	0
$323$	6717.98	1.15727
$324$	0	0
$325$	4717.67	0.805198
$326$	0	0
$327$	630.498	0.106626
$328$	0	0
$329$	29.2584	0.00490295
$330$	0	0
$331$	8426.86	1.39934	0.699671	−	0.714465i	$-0.253330\pi$
$331$	8426.86	1.39934	0.699671	+	0.714465i	$0.253330\pi$
$332$	0	0
$333$	−555.488	−0.0914130
$334$	0	0
$335$	3219.40	0.525058
$336$	0	0
$337$	−9064.12	−1.46515	−0.732573	−	0.680689i	$-0.761681\pi$
$337$	−9064.12	−1.46515	−0.732573	+	0.680689i	$0.761681\pi$
$338$	0	0
$339$	2364.30	0.378794
$340$	0	0
$341$	769.489	0.122200
$342$	0	0
$343$	391.271	0.0615937
$344$	0	0
$345$	11646.7	1.81751
$346$	0	0
$347$	−4496.43	−0.695622	−0.347811	−	0.937565i	$-0.613075\pi$
$347$	−4496.43	−0.695622	−0.347811	+	0.937565i	$0.613075\pi$
$348$	0	0
$349$	−9397.56	−1.44137	−0.720687	−	0.693260i	$-0.756174\pi$
$349$	−9397.56	−1.44137	−0.720687	+	0.693260i	$0.756174\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−8189.26	−1.23476	−0.617380	−	0.786665i	$-0.711806\pi$
$353$	−8189.26	−1.23476	−0.617380	+	0.786665i	$0.711806\pi$
$354$	0	0
$355$	24552.5	3.67073
$356$	0	0
$357$	−107.568	−0.0159471
$358$	0	0
$359$	4176.24	0.613965	0.306983	−	0.951715i	$-0.400681\pi$
$359$	4176.24	0.613965	0.306983	+	0.951715i	$0.400681\pi$
$360$	0	0
$361$	4571.60	0.666511
$362$	0	0
$363$	−3360.97	−0.485965
$364$	0	0
$365$	−13080.4	−1.87579
$366$	0	0
$367$	5272.91	0.749983	0.374991	−	0.927028i	$-0.377646\pi$
$367$	5272.91	0.749983	0.374991	+	0.927028i	$0.377646\pi$
$368$	0	0
$369$	−2203.85	−0.310916
$370$	0	0
$371$	282.975	0.0395992
$372$	0	0
$373$	−9751.86	−1.35371	−0.676853	−	0.736118i	$-0.736657\pi$
$373$	−9751.86	−1.35371	−0.676853	+	0.736118i	$0.736657\pi$
$374$	0	0
$375$	15764.4	2.17085
$376$	0	0
$377$	−864.049	−0.118039
$378$	0	0
$379$	−12570.3	−1.70368	−0.851839	−	0.523803i	$-0.824513\pi$
$379$	−12570.3	−1.70368	−0.851839	+	0.523803i	$0.824513\pi$
$380$	0	0
$381$	5004.25	0.672901
$382$	0	0
$383$	10564.7	1.40948	0.704741	−	0.709464i	$-0.251063\pi$
$383$	10564.7	1.40948	0.704741	+	0.709464i	$0.251063\pi$
$384$	0	0
$385$	−182.950	−0.0242181
$386$	0	0
$387$	−577.941	−0.0759131
$388$	0	0
$389$	9376.03	1.22207	0.611033	−	0.791605i	$-0.290754\pi$
$389$	9376.03	1.22207	0.611033	+	0.791605i	$0.290754\pi$
$390$	0	0
$391$	11043.9	1.42843
$392$	0	0
$393$	−7906.00	−1.01477
$394$	0	0
$395$	2748.91	0.350159
$396$	0	0
$397$	3449.75	0.436116	0.218058	−	0.975936i	$-0.430028\pi$
$397$	3449.75	0.436116	0.218058	+	0.975936i	$0.430028\pi$
$398$	0	0
$399$	−183.027	−0.0229644
$400$	0	0
$401$	6580.34	0.819467	0.409734	−	0.912205i	$-0.365622\pi$
$401$	6580.34	0.819467	0.409734	+	0.912205i	$0.365622\pi$
$402$	0	0
$403$	689.189	0.0851885
$404$	0	0
$405$	1789.16	0.219516
$406$	0	0
$407$	−895.859	−0.109106
$408$	0	0
$409$	−6394.78	−0.773109	−0.386554	−	0.922267i	$-0.626335\pi$
$409$	−6394.78	−0.773109	−0.386554	+	0.922267i	$0.626335\pi$
$410$	0	0
$411$	−3684.93	−0.442249
$412$	0	0
$413$	184.186	0.0219448
$414$	0	0
$415$	14926.7	1.76559
$416$	0	0
$417$	4089.10	0.480202
$418$	0	0
$419$	8951.92	1.04375	0.521873	−	0.853023i	$-0.325233\pi$
$419$	8951.92	1.04375	0.521873	+	0.853023i	$0.325233\pi$
$420$	0	0
$421$	414.532	0.0479883	0.0239941	−	0.999712i	$-0.492362\pi$
$421$	414.532	0.0479883	0.0239941	+	0.999712i	$0.492362\pi$
$422$	0	0
$423$	−461.460	−0.0530424
$424$	0	0
$425$	22802.8	2.60259
$426$	0	0
$427$	284.884	0.0322869
$428$	0	0
$429$	566.073	0.0637069
$430$	0	0
$431$	945.412	0.105659	0.0528293	−	0.998604i	$-0.483176\pi$
$431$	945.412	0.105659	0.0528293	+	0.998604i	$0.483176\pi$
$432$	0	0
$433$	6691.85	0.742702	0.371351	−	0.928493i	$-0.378895\pi$
$433$	6691.85	0.742702	0.371351	+	0.928493i	$0.378895\pi$
$434$	0	0
$435$	−4404.34	−0.485453
$436$	0	0
$437$	18791.1	2.05698
$438$	0	0
$439$	−799.787	−0.0869516	−0.0434758	−	0.999054i	$-0.513843\pi$
$439$	−799.787	−0.0869516	−0.0434758	+	0.999054i	$0.513843\pi$
$440$	0	0
$441$	−3084.07	−0.333017
$442$	0	0
$443$	4856.67	0.520874	0.260437	−	0.965491i	$-0.416133\pi$
$443$	4856.67	0.520874	0.260437	+	0.965491i	$0.416133\pi$
$444$	0	0
$445$	−9091.34	−0.968474
$446$	0	0
$447$	895.493	0.0947548
$448$	0	0
$449$	−16801.4	−1.76594	−0.882969	−	0.469431i	$-0.844459\pi$
$449$	−16801.4	−1.76594	−0.882969	+	0.469431i	$0.844459\pi$
$450$	0	0
$451$	−3554.24	−0.371093
$452$	0	0
$453$	−9892.32	−1.02601
$454$	0	0
$455$	−163.858	−0.0168830
$456$	0	0
$457$	−3603.14	−0.368814	−0.184407	−	0.982850i	$-0.559036\pi$
$457$	−3603.14	−0.368814	−0.184407	+	0.982850i	$0.559036\pi$
$458$	0	0
$459$	1696.55	0.172524
$460$	0	0
$461$	−7442.82	−0.751946	−0.375973	−	0.926631i	$-0.622691\pi$
$461$	−7442.82	−0.751946	−0.375973	+	0.926631i	$0.622691\pi$
$462$	0	0
$463$	1873.49	0.188053	0.0940265	−	0.995570i	$-0.470026\pi$
$463$	1873.49	0.188053	0.0940265	+	0.995570i	$0.470026\pi$
$464$	0	0
$465$	3513.02	0.350349
$466$	0	0
$467$	8925.09	0.884377	0.442188	−	0.896922i	$-0.354202\pi$
$467$	8925.09	0.884377	0.442188	+	0.896922i	$0.354202\pi$
$468$	0	0
$469$	−83.1706	−0.00818862
$470$	0	0
$471$	−5908.94	−0.578067
$472$	0	0
$473$	−932.070	−0.0906060
$474$	0	0
$475$	38798.8	3.74782
$476$	0	0
$477$	−4463.04	−0.428403
$478$	0	0
$479$	2863.80	0.273174	0.136587	−	0.990628i	$-0.456387\pi$
$479$	2863.80	0.273174	0.136587	+	0.990628i	$0.456387\pi$
$480$	0	0
$481$	−802.371	−0.0760602
$482$	0	0
$483$	−300.884	−0.0283452
$484$	0	0
$485$	−36073.5	−3.37734
$486$	0	0
$487$	−2976.01	−0.276911	−0.138456	−	0.990369i	$-0.544214\pi$
$487$	−2976.01	−0.276911	−0.138456	+	0.990369i	$0.544214\pi$
$488$	0	0
$489$	−6140.28	−0.567838
$490$	0	0
$491$	14753.6	1.35605	0.678027	−	0.735037i	$-0.262835\pi$
$491$	14753.6	1.35605	0.678027	+	0.735037i	$0.262835\pi$
$492$	0	0
$493$	−4176.37	−0.381530
$494$	0	0
$495$	2885.46	0.262003
$496$	0	0
$497$	−634.293	−0.0572474
$498$	0	0
$499$	−16239.3	−1.45685	−0.728426	−	0.685124i	$-0.759748\pi$
$499$	−16239.3	−1.45685	−0.728426	+	0.685124i	$0.759748\pi$
$500$	0	0
$501$	8619.02	0.768601
$502$	0	0
$503$	−1975.94	−0.175155	−0.0875774	−	0.996158i	$-0.527913\pi$
$503$	−1975.94	−0.175155	−0.0875774	+	0.996158i	$0.527913\pi$
$504$	0	0
$505$	37447.4	3.29978
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−18898.0	−1.64565	−0.822827	−	0.568292i	$-0.807605\pi$
$509$	−18898.0	−1.64565	−0.822827	+	0.568292i	$0.807605\pi$
$510$	0	0
$511$	337.923	0.0292541
$512$	0	0
$513$	2886.68	0.248440
$514$	0	0
$515$	−23304.0	−1.99397
$516$	0	0
$517$	−744.216	−0.0633087
$518$	0	0
$519$	1602.74	0.135554
$520$	0	0
$521$	19031.4	1.60035	0.800175	−	0.599767i	$-0.204740\pi$
$521$	19031.4	1.60035	0.800175	+	0.599767i	$0.204740\pi$
$522$	0	0
$523$	20162.4	1.68574	0.842870	−	0.538117i	$-0.180864\pi$
$523$	20162.4	1.68574	0.842870	+	0.538117i	$0.180864\pi$
$524$	0	0
$525$	−621.248	−0.0516447
$526$	0	0
$527$	3331.19	0.275349
$528$	0	0
$529$	18724.4	1.53895
$530$	0	0
$531$	−2904.96	−0.237410
$532$	0	0
$533$	−3183.34	−0.258697
$534$	0	0
$535$	−31683.9	−2.56040
$536$	0	0
$537$	10512.4	0.844778
$538$	0	0
$539$	−4973.81	−0.397472
$540$	0	0
$541$	−1067.93	−0.0848688	−0.0424344	−	0.999099i	$-0.513511\pi$
$541$	−1067.93	−0.0848688	−0.0424344	+	0.999099i	$0.513511\pi$
$542$	0	0
$543$	12083.7	0.954992
$544$	0	0
$545$	4642.23	0.364865
$546$	0	0
$547$	18158.4	1.41938	0.709688	−	0.704516i	$-0.248836\pi$
$547$	18158.4	1.41938	0.709688	+	0.704516i	$0.248836\pi$
$548$	0	0
$549$	−4493.15	−0.349295
$550$	0	0
$551$	−7106.07	−0.549417
$552$	0	0
$553$	−71.0159	−0.00546095
$554$	0	0
$555$	−4089.95	−0.312808
$556$	0	0
$557$	−21903.8	−1.66624	−0.833120	−	0.553092i	$-0.813448\pi$
$557$	−21903.8	−1.66624	−0.833120	+	0.553092i	$0.813448\pi$
$558$	0	0
$559$	−834.803	−0.0631635
$560$	0	0
$561$	2736.11	0.205915
$562$	0	0
$563$	−6633.07	−0.496538	−0.248269	−	0.968691i	$-0.579862\pi$
$563$	−6633.07	−0.496538	−0.248269	+	0.968691i	$0.579862\pi$
$564$	0	0
$565$	17407.9	1.29620
$566$	0	0
$567$	−46.2216	−0.00342350
$568$	0	0
$569$	−14739.5	−1.08596	−0.542981	−	0.839745i	$-0.682704\pi$
$569$	−14739.5	−1.08596	−0.542981	+	0.839745i	$0.682704\pi$
$570$	0	0
$571$	22073.0	1.61774	0.808868	−	0.587990i	$-0.200080\pi$
$571$	22073.0	1.61774	0.808868	+	0.587990i	$0.200080\pi$
$572$	0	0
$573$	−4549.42	−0.331684
$574$	0	0
$575$	63782.7	4.62595
$576$	0	0
$577$	9921.43	0.715831	0.357915	−	0.933754i	$-0.383488\pi$
$577$	9921.43	0.715831	0.357915	+	0.933754i	$0.383488\pi$
$578$	0	0
$579$	1289.52	0.0925574
$580$	0	0
$581$	−385.618	−0.0275355
$582$	0	0
$583$	−7197.73	−0.511320
$584$	0	0
$585$	2584.34	0.182649
$586$	0	0
$587$	−22175.4	−1.55924	−0.779622	−	0.626250i	$-0.784589\pi$
$587$	−22175.4	−1.55924	−0.779622	+	0.626250i	$0.784589\pi$
$588$	0	0
$589$	5668.00	0.396512
$590$	0	0
$591$	−1891.85	−0.131676
$592$	0	0
$593$	−19410.7	−1.34419	−0.672093	−	0.740467i	$-0.734605\pi$
$593$	−19410.7	−1.34419	−0.672093	+	0.740467i	$0.734605\pi$
$594$	0	0
$595$	−792.005	−0.0545698
$596$	0	0
$597$	−12668.4	−0.868478
$598$	0	0
$599$	23121.5	1.57716	0.788582	−	0.614930i	$-0.210816\pi$
$599$	23121.5	1.57716	0.788582	+	0.614930i	$0.210816\pi$
$600$	0	0
$601$	−11376.1	−0.772117	−0.386059	−	0.922474i	$-0.626164\pi$
$601$	−11376.1	−0.772117	−0.386059	+	0.922474i	$0.626164\pi$
$602$	0	0
$603$	1311.76	0.0885884
$604$	0	0
$605$	−24746.2	−1.66293
$606$	0	0
$607$	−10687.2	−0.714628	−0.357314	−	0.933984i	$-0.616307\pi$
$607$	−10687.2	−0.714628	−0.357314	+	0.933984i	$0.616307\pi$
$608$	0	0
$609$	113.783	0.00757094
$610$	0	0
$611$	−666.553	−0.0441340
$612$	0	0
$613$	23589.4	1.55427	0.777135	−	0.629334i	$-0.216672\pi$
$613$	23589.4	1.55427	0.777135	+	0.629334i	$0.216672\pi$
$614$	0	0
$615$	−16226.5	−1.06393
$616$	0	0
$617$	−14195.7	−0.926249	−0.463124	−	0.886293i	$-0.653272\pi$
$617$	−14195.7	−0.926249	−0.463124	+	0.886293i	$0.653272\pi$
$618$	0	0
$619$	11206.6	0.727678	0.363839	−	0.931462i	$-0.381466\pi$
$619$	11206.6	0.727678	0.363839	+	0.931462i	$0.381466\pi$
$620$	0	0
$621$	4745.51	0.306651
$622$	0	0
$623$	234.867	0.0151040
$624$	0	0
$625$	70707.6	4.52529
$626$	0	0
$627$	4655.47	0.296526
$628$	0	0
$629$	−3878.25	−0.245844
$630$	0	0
$631$	2757.97	0.173999	0.0869993	−	0.996208i	$-0.472272\pi$
$631$	2757.97	0.173999	0.0869993	+	0.996208i	$0.472272\pi$
$632$	0	0
$633$	4392.25	0.275792
$634$	0	0
$635$	36845.3	2.30262
$636$	0	0
$637$	−4454.77	−0.277087
$638$	0	0
$639$	10004.0	0.619329
$640$	0	0
$641$	−11974.7	−0.737869	−0.368934	−	0.929455i	$-0.620277\pi$
$641$	−11974.7	−0.737869	−0.368934	+	0.929455i	$0.620277\pi$
$642$	0	0
$643$	−16145.5	−0.990227	−0.495114	−	0.868828i	$-0.664874\pi$
$643$	−16145.5	−0.990227	−0.495114	+	0.868828i	$0.664874\pi$
$644$	0	0
$645$	−4255.26	−0.259769
$646$	0	0
$647$	−21437.9	−1.30264	−0.651322	−	0.758802i	$-0.725785\pi$
$647$	−21437.9	−1.30264	−0.651322	+	0.758802i	$0.725785\pi$
$648$	0	0
$649$	−4684.96	−0.283360
$650$	0	0
$651$	−90.7561	−0.00546392
$652$	0	0
$653$	17735.5	1.06286	0.531428	−	0.847104i	$-0.321656\pi$
$653$	17735.5	1.06286	0.531428	+	0.847104i	$0.321656\pi$
$654$	0	0
$655$	−58210.3	−3.47246
$656$	0	0
$657$	−5329.67	−0.316485
$658$	0	0
$659$	−21518.8	−1.27201	−0.636004	−	0.771686i	$-0.719414\pi$
$659$	−21518.8	−1.27201	−0.636004	+	0.771686i	$0.719414\pi$
$660$	0	0
$661$	30544.1	1.79732	0.898660	−	0.438646i	$-0.144542\pi$
$661$	30544.1	1.79732	0.898660	+	0.438646i	$0.144542\pi$
$662$	0	0
$663$	2450.58	0.143548
$664$	0	0
$665$	−1347.59	−0.0785825
$666$	0	0
$667$	−11681.9	−0.678149
$668$	0	0
$669$	16465.2	0.951543
$670$	0	0
$671$	−7246.29	−0.416900
$672$	0	0
$673$	6938.35	0.397405	0.198703	−	0.980060i	$-0.436327\pi$
$673$	6938.35	0.397405	0.198703	+	0.980060i	$0.436327\pi$
$674$	0	0
$675$	9798.24	0.558718
$676$	0	0
$677$	−8394.43	−0.476550	−0.238275	−	0.971198i	$-0.576582\pi$
$677$	−8394.43	−0.476550	−0.238275	+	0.971198i	$0.576582\pi$
$678$	0	0
$679$	931.930	0.0526718
$680$	0	0
$681$	4615.01	0.259688
$682$	0	0
$683$	−18375.5	−1.02946	−0.514729	−	0.857353i	$-0.672108\pi$
$683$	−18375.5	−1.02946	−0.514729	+	0.857353i	$0.672108\pi$
$684$	0	0
$685$	−27131.4	−1.51334
$686$	0	0
$687$	−7633.93	−0.423948
$688$	0	0
$689$	−6446.61	−0.356453
$690$	0	0
$691$	−11058.8	−0.608824	−0.304412	−	0.952541i	$-0.598460\pi$
$691$	−11058.8	−0.608824	−0.304412	+	0.952541i	$0.598460\pi$
$692$	0	0
$693$	−74.5435	−0.00408611
$694$	0	0
$695$	30107.2	1.64321
$696$	0	0
$697$	−15386.6	−0.836170
$698$	0	0
$699$	1765.42	0.0955284
$700$	0	0
$701$	−7955.43	−0.428634	−0.214317	−	0.976764i	$-0.568753\pi$
$701$	−7955.43	−0.428634	−0.214317	+	0.976764i	$0.568753\pi$
$702$	0	0
$703$	−6598.82	−0.354024
$704$	0	0
$705$	−3397.64	−0.181507
$706$	0	0
$707$	−967.425	−0.0514622
$708$	0	0
$709$	12814.7	0.678794	0.339397	−	0.940643i	$-0.389777\pi$
$709$	12814.7	0.678794	0.339397	+	0.940643i	$0.389777\pi$
$710$	0	0
$711$	1120.05	0.0590792
$712$	0	0
$713$	9317.81	0.489417
$714$	0	0
$715$	4167.88	0.218000
$716$	0	0
$717$	1966.25	0.102414
$718$	0	0
$719$	2221.14	0.115208	0.0576039	−	0.998340i	$-0.481654\pi$
$719$	2221.14	0.115208	0.0576039	+	0.998340i	$0.481654\pi$
$720$	0	0
$721$	602.039	0.0310972
$722$	0	0
$723$	−1811.04	−0.0931582
$724$	0	0
$725$	−24120.1	−1.23558
$726$	0	0
$727$	−17320.1	−0.883587	−0.441794	−	0.897117i	$-0.645658\pi$
$727$	−17320.1	−0.883587	−0.441794	+	0.897117i	$0.645658\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4035.01	−0.204159
$732$	0	0
$733$	28448.3	1.43351	0.716755	−	0.697325i	$-0.245626\pi$
$733$	28448.3	1.43351	0.716755	+	0.697325i	$0.245626\pi$
$734$	0	0
$735$	−22707.4	−1.13956
$736$	0	0
$737$	2115.52	0.105735
$738$	0	0
$739$	−19610.1	−0.976141	−0.488070	−	0.872804i	$-0.662299\pi$
$739$	−19610.1	−0.976141	−0.488070	+	0.872804i	$0.662299\pi$
$740$	0	0
$741$	4169.65	0.206715
$742$	0	0
$743$	−36346.5	−1.79465	−0.897325	−	0.441370i	$-0.854493\pi$
$743$	−36346.5	−1.79465	−0.897325	+	0.441370i	$0.854493\pi$
$744$	0	0
$745$	6593.34	0.324243
$746$	0	0
$747$	6081.92	0.297893
$748$	0	0
$749$	818.528	0.0399311
$750$	0	0
$751$	−30350.0	−1.47468	−0.737342	−	0.675520i	$-0.763919\pi$
$751$	−30350.0	−1.47468	−0.737342	+	0.675520i	$0.763919\pi$
$752$	0	0
$753$	−18996.5	−0.919349
$754$	0	0
$755$	−72835.2	−3.51092
$756$	0	0
$757$	19424.8	0.932635	0.466318	−	0.884617i	$-0.345580\pi$
$757$	19424.8	0.932635	0.466318	+	0.884617i	$0.345580\pi$
$758$	0	0
$759$	7653.28	0.366003
$760$	0	0
$761$	−4640.01	−0.221025	−0.110513	−	0.993875i	$-0.535249\pi$
$761$	−4640.01	−0.221025	−0.110513	+	0.993875i	$0.535249\pi$
$762$	0	0
$763$	−119.928	−0.00569030
$764$	0	0
$765$	12491.4	0.590363
$766$	0	0
$767$	−4196.06	−0.197537
$768$	0	0
$769$	34544.3	1.61990	0.809949	−	0.586501i	$-0.199495\pi$
$769$	34544.3	1.61990	0.809949	+	0.586501i	$0.199495\pi$
$770$	0	0
$771$	6344.71	0.296367
$772$	0	0
$773$	9123.70	0.424523	0.212262	−	0.977213i	$-0.431917\pi$
$773$	9123.70	0.424523	0.212262	+	0.977213i	$0.431917\pi$
$774$	0	0
$775$	19238.9	0.891716
$776$	0	0
$777$	105.660	0.00487844
$778$	0	0
$779$	−26180.3	−1.20411
$780$	0	0
$781$	16133.9	0.739199
$782$	0	0
$783$	−1794.56	−0.0819061
$784$	0	0
$785$	−43506.3	−1.97810
$786$	0	0
$787$	−13224.2	−0.598974	−0.299487	−	0.954100i	$-0.596815\pi$
$787$	−13224.2	−0.598974	−0.299487	+	0.954100i	$0.596815\pi$
$788$	0	0
$789$	−9558.87	−0.431312
$790$	0	0
$791$	−449.719	−0.0202151
$792$	0	0
$793$	−6490.10	−0.290631
$794$	0	0
$795$	−32860.5	−1.46596
$796$	0	0
$797$	14713.4	0.653922	0.326961	−	0.945038i	$-0.393975\pi$
$797$	14713.4	0.653922	0.326961	+	0.945038i	$0.393975\pi$
$798$	0	0
$799$	−3221.78	−0.142651
$800$	0	0
$801$	−3704.30	−0.163402
$802$	0	0
$803$	−8595.39	−0.377740
$804$	0	0
$805$	−2215.35	−0.0969949
$806$	0	0
$807$	6169.20	0.269103
$808$	0	0
$809$	2335.80	0.101511	0.0507555	−	0.998711i	$-0.483837\pi$
$809$	2335.80	0.101511	0.0507555	+	0.998711i	$0.483837\pi$
$810$	0	0
$811$	−26446.9	−1.14510	−0.572550	−	0.819870i	$-0.694046\pi$
$811$	−26446.9	−1.14510	−0.572550	+	0.819870i	$0.694046\pi$
$812$	0	0
$813$	1199.81	0.0517581
$814$	0	0
$815$	−45209.7	−1.94310
$816$	0	0
$817$	−6865.55	−0.293997
$818$	0	0
$819$	−66.7645	−0.00284852
$820$	0	0
$821$	−30682.1	−1.30428	−0.652140	−	0.758098i	$-0.726129\pi$
$821$	−30682.1	−1.30428	−0.652140	+	0.758098i	$0.726129\pi$
$822$	0	0
$823$	10242.8	0.433829	0.216914	−	0.976191i	$-0.430401\pi$
$823$	10242.8	0.433829	0.216914	+	0.976191i	$0.430401\pi$
$824$	0	0
$825$	15802.0	0.666856
$826$	0	0
$827$	35864.5	1.50802	0.754008	−	0.656865i	$-0.228118\pi$
$827$	35864.5	1.50802	0.754008	+	0.656865i	$0.228118\pi$
$828$	0	0
$829$	3633.70	0.152236	0.0761179	−	0.997099i	$-0.475747\pi$
$829$	3633.70	0.152236	0.0761179	+	0.997099i	$0.475747\pi$
$830$	0	0
$831$	13410.7	0.559823
$832$	0	0
$833$	−21532.1	−0.895609
$834$	0	0
$835$	63460.1	2.63009
$836$	0	0
$837$	1431.39	0.0591113
$838$	0	0
$839$	22098.2	0.909312	0.454656	−	0.890667i	$-0.349762\pi$
$839$	22098.2	0.909312	0.454656	+	0.890667i	$0.349762\pi$
$840$	0	0
$841$	−19971.4	−0.818867
$842$	0	0
$843$	−1205.39	−0.0492476
$844$	0	0
$845$	3732.94	0.151973
$846$	0	0
$847$	639.298	0.0259345
$848$	0	0
$849$	−11744.3	−0.474751
$850$	0	0
$851$	−10848.0	−0.436975
$852$	0	0
$853$	−8946.55	−0.359114	−0.179557	−	0.983748i	$-0.557466\pi$
$853$	−8946.55	−0.359114	−0.179557	+	0.983748i	$0.557466\pi$
$854$	0	0
$855$	21254.0	0.850144
$856$	0	0
$857$	−22867.5	−0.911479	−0.455740	−	0.890113i	$-0.650625\pi$
$857$	−22867.5	−0.911479	−0.455740	+	0.890113i	$0.650625\pi$
$858$	0	0
$859$	31281.5	1.24250	0.621252	−	0.783611i	$-0.286625\pi$
$859$	31281.5	1.24250	0.621252	+	0.783611i	$0.286625\pi$
$860$	0	0
$861$	419.199	0.0165926
$862$	0	0
$863$	−23071.5	−0.910040	−0.455020	−	0.890481i	$-0.650368\pi$
$863$	−23071.5	−0.910040	−0.455020	+	0.890481i	$0.650368\pi$
$864$	0	0
$865$	11800.6	0.463854
$866$	0	0
$867$	−2894.15	−0.113369
$868$	0	0
$869$	1806.36	0.0705138
$870$	0	0
$871$	1894.76	0.0737100
$872$	0	0
$873$	−14698.3	−0.569829
$874$	0	0
$875$	−2998.57	−0.115852
$876$	0	0
$877$	26100.7	1.00497	0.502485	−	0.864586i	$-0.332419\pi$
$877$	26100.7	1.00497	0.502485	+	0.864586i	$0.332419\pi$
$878$	0	0
$879$	1301.81	0.0499533
$880$	0	0
$881$	48698.2	1.86230	0.931148	−	0.364642i	$-0.118809\pi$
$881$	48698.2	1.86230	0.931148	+	0.364642i	$0.118809\pi$
$882$	0	0
$883$	870.968	0.0331941	0.0165971	−	0.999862i	$-0.494717\pi$
$883$	870.968	0.0331941	0.0165971	+	0.999862i	$0.494717\pi$
$884$	0	0
$885$	−21388.7	−0.812398
$886$	0	0
$887$	33022.5	1.25004	0.625022	−	0.780608i	$-0.285090\pi$
$887$	33022.5	1.25004	0.625022	+	0.780608i	$0.285090\pi$
$888$	0	0
$889$	−951.869	−0.0359107
$890$	0	0
$891$	1175.69	0.0442055
$892$	0	0
$893$	−5481.83	−0.205423
$894$	0	0
$895$	77401.1	2.89076
$896$	0	0
$897$	6854.62	0.255149
$898$	0	0
$899$	−3523.63	−0.130723
$900$	0	0
$901$	−31159.6	−1.15214
$902$	0	0
$903$	109.931	0.00405126
$904$	0	0
$905$	88969.8	3.26791
$906$	0	0
$907$	42835.0	1.56815	0.784076	−	0.620665i	$-0.213137\pi$
$907$	42835.0	1.56815	0.784076	+	0.620665i	$0.213137\pi$
$908$	0	0
$909$	15258.1	0.556742
$910$	0	0
$911$	5584.98	0.203116	0.101558	−	0.994830i	$-0.467617\pi$
$911$	5584.98	0.203116	0.101558	+	0.994830i	$0.467617\pi$
$912$	0	0
$913$	9808.57	0.355549
$914$	0	0
$915$	−33082.2	−1.19526
$916$	0	0
$917$	1503.82	0.0541553
$918$	0	0
$919$	−50229.1	−1.80294	−0.901472	−	0.432838i	$-0.857512\pi$
$919$	−50229.1	−1.80294	−0.901472	+	0.432838i	$0.857512\pi$
$920$	0	0
$921$	14041.9	0.502385
$922$	0	0
$923$	14450.2	0.515313
$924$	0	0
$925$	−22398.4	−0.796166
$926$	0	0
$927$	−9495.28	−0.336425
$928$	0	0
$929$	−13286.2	−0.469222	−0.234611	−	0.972089i	$-0.575382\pi$
$929$	−13286.2	−0.469222	−0.234611	+	0.972089i	$0.575382\pi$
$930$	0	0
$931$	−36636.7	−1.28971
$932$	0	0
$933$	2011.18	0.0705714
$934$	0	0
$935$	20145.4	0.704626
$936$	0	0
$937$	−1627.36	−0.0567380	−0.0283690	−	0.999598i	$-0.509031\pi$
$937$	−1627.36	−0.0567380	−0.0283690	+	0.999598i	$0.509031\pi$
$938$	0	0
$939$	6596.20	0.229243
$940$	0	0
$941$	44267.2	1.53355	0.766774	−	0.641917i	$-0.221861\pi$
$941$	44267.2	1.53355	0.766774	+	0.641917i	$0.221861\pi$
$942$	0	0
$943$	−43038.6	−1.48625
$944$	0	0
$945$	−340.320	−0.0117149
$946$	0	0
$947$	47433.9	1.62766	0.813830	−	0.581103i	$-0.197379\pi$
$947$	47433.9	1.62766	0.813830	+	0.581103i	$0.197379\pi$
$948$	0	0
$949$	−7698.42	−0.263331
$950$	0	0
$951$	22531.8	0.768292
$952$	0	0
$953$	−38935.8	−1.32346	−0.661728	−	0.749744i	$-0.730177\pi$
$953$	−38935.8	−1.32346	−0.661728	+	0.749744i	$0.730177\pi$
$954$	0	0
$955$	−33496.5	−1.13500
$956$	0	0
$957$	−2894.17	−0.0977589
$958$	0	0
$959$	700.918	0.0236015
$960$	0	0
$961$	−26980.5	−0.905658
$962$	0	0
$963$	−12909.7	−0.431994
$964$	0	0
$965$	9494.50	0.316724
$966$	0	0
$967$	18350.7	0.610257	0.305129	−	0.952311i	$-0.401301\pi$
$967$	18350.7	0.610257	0.305129	+	0.952311i	$0.401301\pi$
$968$	0	0
$969$	20153.9	0.668150
$970$	0	0
$971$	3679.77	0.121616	0.0608082	−	0.998149i	$-0.480632\pi$
$971$	3679.77	0.121616	0.0608082	+	0.998149i	$0.480632\pi$
$972$	0	0
$973$	−777.796	−0.0256269
$974$	0	0
$975$	14153.0	0.464881
$976$	0	0
$977$	−16462.8	−0.539092	−0.269546	−	0.962988i	$-0.586874\pi$
$977$	−16462.8	−0.539092	−0.269546	+	0.962988i	$0.586874\pi$
$978$	0	0
$979$	−5974.08	−0.195028
$980$	0	0
$981$	1891.49	0.0615604
$982$	0	0
$983$	−38116.1	−1.23674	−0.618370	−	0.785887i	$-0.712207\pi$
$983$	−38116.1	−1.23674	−0.618370	+	0.785887i	$0.712207\pi$
$984$	0	0
$985$	−13929.3	−0.450583
$986$	0	0
$987$	87.7753	0.00283072
$988$	0	0
$989$	−11286.5	−0.362882
$990$	0	0
$991$	−8909.11	−0.285577	−0.142789	−	0.989753i	$-0.545607\pi$
$991$	−8909.11	−0.285577	−0.142789	+	0.989753i	$0.545607\pi$
$992$	0	0
$993$	25280.6	0.807910
$994$	0	0
$995$	−93274.7	−2.97186
$996$	0	0
$997$	36786.4	1.16854	0.584271	−	0.811559i	$-0.301381\pi$
$997$	36786.4	1.16854	0.584271	+	0.811559i	$0.301381\pi$
$998$	0	0
$999$	−1666.46	−0.0527773

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.4.a.g.1.1	✓	5	8.5	even	2
1248.4.a.l.1.1	yes	5	8.3	odd	2
2496.4.a.ca.1.5		5	4.3	odd	2
2496.4.a.cf.1.5		5	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +22.0884 q^{5} -0.570636 q^{7} +9.00000 q^{9} +14.5147 q^{11} +13.0000 q^{13} +66.2652 q^{15} +62.8354 q^{17} +106.914 q^{19} -1.71191 q^{21} +175.759 q^{23} +362.898 q^{25} +27.0000 q^{27} -66.4653 q^{29} +53.0145 q^{31} +43.5441 q^{33} -12.6045 q^{35} -61.7208 q^{37} +39.0000 q^{39} -244.872 q^{41} -64.2156 q^{43} +198.796 q^{45} -51.2733 q^{47} -342.674 q^{49} +188.506 q^{51} -495.893 q^{53} +320.606 q^{55} +320.742 q^{57} -322.773 q^{59} -499.239 q^{61} -5.13573 q^{63} +287.149 q^{65} +145.751 q^{67} +527.278 q^{69} +1111.55 q^{71} -592.186 q^{73} +1088.69 q^{75} -8.28261 q^{77} +124.450 q^{79} +81.0000 q^{81} +675.769 q^{83} +1387.93 q^{85} -199.396 q^{87} -411.589 q^{89} -7.41827 q^{91} +159.044 q^{93} +2361.56 q^{95} -1633.14 q^{97} +130.632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} + 10 q^{5} + 14 q^{7} + 45 q^{9} - 22 q^{11} + 65 q^{13} + 30 q^{15} - 34 q^{17} + 90 q^{19} + 42 q^{21} + 96 q^{23} + 107 q^{25} + 135 q^{27} + 54 q^{29} + 378 q^{31} - 66 q^{33} - 84 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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