LMFDB - Embedded newform 2496.4.a.g.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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-3.00000 q^{3} +16.0000 q^{5} -28.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} +16.0000 q^{5} -28.0000 q^{7} +9.00000 q^{9} +34.0000 q^{11} +13.0000 q^{13} -48.0000 q^{15} +138.000 q^{17} +108.000 q^{19} +84.0000 q^{21} +52.0000 q^{23} +131.000 q^{25} -27.0000 q^{27} +190.000 q^{29} +176.000 q^{31} -102.000 q^{33} -448.000 q^{35} -342.000 q^{37} -39.0000 q^{39} +240.000 q^{41} -140.000 q^{43} +144.000 q^{45} -454.000 q^{47} +441.000 q^{49} -414.000 q^{51} -198.000 q^{53} +544.000 q^{55} -324.000 q^{57} -154.000 q^{59} -34.0000 q^{61} -252.000 q^{63} +208.000 q^{65} -656.000 q^{67} -156.000 q^{69} -550.000 q^{71} +614.000 q^{73} -393.000 q^{75} -952.000 q^{77} -8.00000 q^{79} +81.0000 q^{81} +762.000 q^{83} +2208.00 q^{85} -570.000 q^{87} -444.000 q^{89} -364.000 q^{91} -528.000 q^{93} +1728.00 q^{95} +1022.00 q^{97} +306.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	16.0000	1.43108	0.715542	−	0.698570i	$-0.246180\pi$
$5$	16.0000	1.43108	0.715542	+	0.698570i	$0.246180\pi$
$6$	0	0
$7$	−28.0000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−28.0000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	34.0000	0.931944	0.465972	−	0.884799i	$-0.345705\pi$
$11$	34.0000	0.931944	0.465972	+	0.884799i	$0.345705\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−48.0000	−0.826236
$16$	0	0
$17$	138.000	1.96882	0.984409	−	0.175893i	$-0.0562813\pi$
$17$	138.000	1.96882	0.984409	+	0.175893i	$0.0562813\pi$
$18$	0	0
$19$	108.000	1.30405	0.652024	−	0.758199i	$-0.273920\pi$
$19$	108.000	1.30405	0.652024	+	0.758199i	$0.273920\pi$
$20$	0	0
$21$	84.0000	0.872872
$22$	0	0
$23$	52.0000	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$23$	52.0000	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$24$	0	0
$25$	131.000	1.04800
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	190.000	1.21662	0.608312	−	0.793698i	$-0.291847\pi$
$29$	190.000	1.21662	0.608312	+	0.793698i	$0.291847\pi$
$30$	0	0
$31$	176.000	1.01969	0.509847	−	0.860265i	$-0.329702\pi$
$31$	176.000	1.01969	0.509847	+	0.860265i	$0.329702\pi$
$32$	0	0
$33$	−102.000	−0.538058
$34$	0	0
$35$	−448.000	−2.16359
$36$	0	0
$37$	−342.000	−1.51958	−0.759790	−	0.650169i	$-0.774698\pi$
$37$	−342.000	−1.51958	−0.759790	+	0.650169i	$0.774698\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	240.000	0.914188	0.457094	−	0.889418i	$-0.348890\pi$
$41$	240.000	0.914188	0.457094	+	0.889418i	$0.348890\pi$
$42$	0	0
$43$	−140.000	−0.496507	−0.248253	−	0.968695i	$-0.579857\pi$
$43$	−140.000	−0.496507	−0.248253	+	0.968695i	$0.579857\pi$
$44$	0	0
$45$	144.000	0.477028
$46$	0	0
$47$	−454.000	−1.40899	−0.704497	−	0.709707i	$-0.748827\pi$
$47$	−454.000	−1.40899	−0.704497	+	0.709707i	$0.748827\pi$
$48$	0	0
$49$	441.000	1.28571
$50$	0	0
$51$	−414.000	−1.13670
$52$	0	0
$53$	−198.000	−0.513158	−0.256579	−	0.966523i	$-0.582595\pi$
$53$	−198.000	−0.513158	−0.256579	+	0.966523i	$0.582595\pi$
$54$	0	0
$55$	544.000	1.33369
$56$	0	0
$57$	−324.000	−0.752892
$58$	0	0
$59$	−154.000	−0.339815	−0.169908	−	0.985460i	$-0.554347\pi$
$59$	−154.000	−0.339815	−0.169908	+	0.985460i	$0.554347\pi$
$60$	0	0
$61$	−34.0000	−0.0713648	−0.0356824	−	0.999363i	$-0.511360\pi$
$61$	−34.0000	−0.0713648	−0.0356824	+	0.999363i	$0.511360\pi$
$62$	0	0
$63$	−252.000	−0.503953
$64$	0	0
$65$	208.000	0.396911
$66$	0	0
$67$	−656.000	−1.19617	−0.598083	−	0.801434i	$-0.704071\pi$
$67$	−656.000	−1.19617	−0.598083	+	0.801434i	$0.704071\pi$
$68$	0	0
$69$	−156.000	−0.272177
$70$	0	0
$71$	−550.000	−0.919338	−0.459669	−	0.888090i	$-0.652032\pi$
$71$	−550.000	−0.919338	−0.459669	+	0.888090i	$0.652032\pi$
$72$	0	0
$73$	614.000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$73$	614.000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$74$	0	0
$75$	−393.000	−0.605063
$76$	0	0
$77$	−952.000	−1.40897
$78$	0	0
$79$	−8.00000	−0.0113933	−0.00569665	−	0.999984i	$-0.501813\pi$
$79$	−8.00000	−0.0113933	−0.00569665	+	0.999984i	$0.501813\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	762.000	1.00772	0.503858	−	0.863787i	$-0.331914\pi$
$83$	762.000	1.00772	0.503858	+	0.863787i	$0.331914\pi$
$84$	0	0
$85$	2208.00	2.81754
$86$	0	0
$87$	−570.000	−0.702419
$88$	0	0
$89$	−444.000	−0.528808	−0.264404	−	0.964412i	$-0.585175\pi$
$89$	−444.000	−0.528808	−0.264404	+	0.964412i	$0.585175\pi$
$90$	0	0
$91$	−364.000	−0.419314
$92$	0	0
$93$	−528.000	−0.588721
$94$	0	0
$95$	1728.00	1.86620
$96$	0	0
$97$	1022.00	1.06978	0.534889	−	0.844923i	$-0.320354\pi$
$97$	1022.00	1.06978	0.534889	+	0.844923i	$0.320354\pi$
$98$	0	0
$99$	306.000	0.310648
$100$	0	0
$101$	1190.00	1.17237	0.586185	−	0.810177i	$-0.300629\pi$
$101$	1190.00	1.17237	0.586185	+	0.810177i	$0.300629\pi$
$102$	0	0
$103$	224.000	0.214285	0.107143	−	0.994244i	$-0.465830\pi$
$103$	224.000	0.214285	0.107143	+	0.994244i	$0.465830\pi$
$104$	0	0
$105$	1344.00	1.24915
$106$	0	0
$107$	−640.000	−0.578235	−0.289117	−	0.957294i	$-0.593362\pi$
$107$	−640.000	−0.578235	−0.289117	+	0.957294i	$0.593362\pi$
$108$	0	0
$109$	1934.00	1.69948	0.849741	−	0.527200i	$-0.176758\pi$
$109$	1934.00	1.69948	0.849741	+	0.527200i	$0.176758\pi$
$110$	0	0
$111$	1026.00	0.877330
$112$	0	0
$113$	−418.000	−0.347983	−0.173992	−	0.984747i	$-0.555667\pi$
$113$	−418.000	−0.347983	−0.173992	+	0.984747i	$0.555667\pi$
$114$	0	0
$115$	832.000	0.674647
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−3864.00	−2.97657
$120$	0	0
$121$	−175.000	−0.131480
$122$	0	0
$123$	−720.000	−0.527807
$124$	0	0
$125$	96.0000	0.0686920
$126$	0	0
$127$	1040.00	0.726654	0.363327	−	0.931662i	$-0.381641\pi$
$127$	1040.00	0.726654	0.363327	+	0.931662i	$0.381641\pi$
$128$	0	0
$129$	420.000	0.286658
$130$	0	0
$131$	−568.000	−0.378827	−0.189414	−	0.981897i	$-0.560659\pi$
$131$	−568.000	−0.378827	−0.189414	+	0.981897i	$0.560659\pi$
$132$	0	0
$133$	−3024.00	−1.97153
$134$	0	0
$135$	−432.000	−0.275412
$136$	0	0
$137$	528.000	0.329271	0.164635	−	0.986355i	$-0.447355\pi$
$137$	528.000	0.329271	0.164635	+	0.986355i	$0.447355\pi$
$138$	0	0
$139$	−1556.00	−0.949483	−0.474742	−	0.880125i	$-0.657459\pi$
$139$	−1556.00	−0.949483	−0.474742	+	0.880125i	$0.657459\pi$
$140$	0	0
$141$	1362.00	0.813483
$142$	0	0
$143$	442.000	0.258475
$144$	0	0
$145$	3040.00	1.74109
$146$	0	0
$147$	−1323.00	−0.742307
$148$	0	0
$149$	−1524.00	−0.837926	−0.418963	−	0.908003i	$-0.637606\pi$
$149$	−1524.00	−0.837926	−0.418963	+	0.908003i	$0.637606\pi$
$150$	0	0
$151$	3024.00	1.62973	0.814866	−	0.579649i	$-0.196810\pi$
$151$	3024.00	1.62973	0.814866	+	0.579649i	$0.196810\pi$
$152$	0	0
$153$	1242.00	0.656273
$154$	0	0
$155$	2816.00	1.45927
$156$	0	0
$157$	−2198.00	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−2198.00	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	594.000	0.296272
$160$	0	0
$161$	−1456.00	−0.712726
$162$	0	0
$163$	−268.000	−0.128781	−0.0643907	−	0.997925i	$-0.520510\pi$
$163$	−268.000	−0.128781	−0.0643907	+	0.997925i	$0.520510\pi$
$164$	0	0
$165$	−1632.00	−0.770006
$166$	0	0
$167$	−702.000	−0.325284	−0.162642	−	0.986685i	$-0.552002\pi$
$167$	−702.000	−0.325284	−0.162642	+	0.986685i	$0.552002\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	972.000	0.434682
$172$	0	0
$173$	−2066.00	−0.907948	−0.453974	−	0.891015i	$-0.649994\pi$
$173$	−2066.00	−0.907948	−0.453974	+	0.891015i	$0.649994\pi$
$174$	0	0
$175$	−3668.00	−1.58443
$176$	0	0
$177$	462.000	0.196192
$178$	0	0
$179$	−276.000	−0.115247	−0.0576235	−	0.998338i	$-0.518352\pi$
$179$	−276.000	−0.115247	−0.0576235	+	0.998338i	$0.518352\pi$
$180$	0	0
$181$	3474.00	1.42663	0.713316	−	0.700843i	$-0.247192\pi$
$181$	3474.00	1.42663	0.713316	+	0.700843i	$0.247192\pi$
$182$	0	0
$183$	102.000	0.0412025
$184$	0	0
$185$	−5472.00	−2.17465
$186$	0	0
$187$	4692.00	1.83483
$188$	0	0
$189$	756.000	0.290957
$190$	0	0
$191$	3920.00	1.48503	0.742516	−	0.669828i	$-0.233632\pi$
$191$	3920.00	1.48503	0.742516	+	0.669828i	$0.233632\pi$
$192$	0	0
$193$	2186.00	0.815294	0.407647	−	0.913140i	$-0.366349\pi$
$193$	2186.00	0.815294	0.407647	+	0.913140i	$0.366349\pi$
$194$	0	0
$195$	−624.000	−0.229157
$196$	0	0
$197$	1368.00	0.494751	0.247376	−	0.968920i	$-0.420432\pi$
$197$	1368.00	0.494751	0.247376	+	0.968920i	$0.420432\pi$
$198$	0	0
$199$	1072.00	0.381870	0.190935	−	0.981603i	$-0.438848\pi$
$199$	1072.00	0.381870	0.190935	+	0.981603i	$0.438848\pi$
$200$	0	0
$201$	1968.00	0.690607
$202$	0	0
$203$	−5320.00	−1.83936
$204$	0	0
$205$	3840.00	1.30828
$206$	0	0
$207$	468.000	0.157141
$208$	0	0
$209$	3672.00	1.21530
$210$	0	0
$211$	5444.00	1.77621	0.888105	−	0.459640i	$-0.152022\pi$
$211$	5444.00	1.77621	0.888105	+	0.459640i	$0.152022\pi$
$212$	0	0
$213$	1650.00	0.530780
$214$	0	0
$215$	−2240.00	−0.710543
$216$	0	0
$217$	−4928.00	−1.54163
$218$	0	0
$219$	−1842.00	−0.568360
$220$	0	0
$221$	1794.00	0.546052
$222$	0	0
$223$	−96.0000	−0.0288280	−0.0144140	−	0.999896i	$-0.504588\pi$
$223$	−96.0000	−0.0288280	−0.0144140	+	0.999896i	$0.504588\pi$
$224$	0	0
$225$	1179.00	0.349333
$226$	0	0
$227$	198.000	0.0578930	0.0289465	−	0.999581i	$-0.490785\pi$
$227$	198.000	0.0578930	0.0289465	+	0.999581i	$0.490785\pi$
$228$	0	0
$229$	−5922.00	−1.70889	−0.854447	−	0.519538i	$-0.826104\pi$
$229$	−5922.00	−1.70889	−0.854447	+	0.519538i	$0.826104\pi$
$230$	0	0
$231$	2856.00	0.813468
$232$	0	0
$233$	−5114.00	−1.43789	−0.718947	−	0.695065i	$-0.755376\pi$
$233$	−5114.00	−1.43789	−0.718947	+	0.695065i	$0.755376\pi$
$234$	0	0
$235$	−7264.00	−2.01639
$236$	0	0
$237$	24.0000	0.00657792
$238$	0	0
$239$	5226.00	1.41440	0.707200	−	0.707013i	$-0.249958\pi$
$239$	5226.00	1.41440	0.707200	+	0.707013i	$0.249958\pi$
$240$	0	0
$241$	−762.000	−0.203671	−0.101836	−	0.994801i	$-0.532472\pi$
$241$	−762.000	−0.203671	−0.101836	+	0.994801i	$0.532472\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	7056.00	1.83996
$246$	0	0
$247$	1404.00	0.361678
$248$	0	0
$249$	−2286.00	−0.581805
$250$	0	0
$251$	3240.00	0.814769	0.407384	−	0.913257i	$-0.366441\pi$
$251$	3240.00	0.814769	0.407384	+	0.913257i	$0.366441\pi$
$252$	0	0
$253$	1768.00	0.439341
$254$	0	0
$255$	−6624.00	−1.62671
$256$	0	0
$257$	−1386.00	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$257$	−1386.00	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$258$	0	0
$259$	9576.00	2.29739
$260$	0	0
$261$	1710.00	0.405542
$262$	0	0
$263$	−3300.00	−0.773714	−0.386857	−	0.922140i	$-0.626439\pi$
$263$	−3300.00	−0.773714	−0.386857	+	0.922140i	$0.626439\pi$
$264$	0	0
$265$	−3168.00	−0.734372
$266$	0	0
$267$	1332.00	0.305307
$268$	0	0
$269$	−4290.00	−0.972364	−0.486182	−	0.873858i	$-0.661611\pi$
$269$	−4290.00	−0.972364	−0.486182	+	0.873858i	$0.661611\pi$
$270$	0	0
$271$	−2452.00	−0.549625	−0.274813	−	0.961498i	$-0.588616\pi$
$271$	−2452.00	−0.549625	−0.274813	+	0.961498i	$0.588616\pi$
$272$	0	0
$273$	1092.00	0.242091
$274$	0	0
$275$	4454.00	0.976677
$276$	0	0
$277$	42.0000	0.00911024	0.00455512	−	0.999990i	$-0.498550\pi$
$277$	42.0000	0.00911024	0.00455512	+	0.999990i	$0.498550\pi$
$278$	0	0
$279$	1584.00	0.339898
$280$	0	0
$281$	−2288.00	−0.485732	−0.242866	−	0.970060i	$-0.578088\pi$
$281$	−2288.00	−0.485732	−0.242866	+	0.970060i	$0.578088\pi$
$282$	0	0
$283$	1156.00	0.242816	0.121408	−	0.992603i	$-0.461259\pi$
$283$	1156.00	0.242816	0.121408	+	0.992603i	$0.461259\pi$
$284$	0	0
$285$	−5184.00	−1.07745
$286$	0	0
$287$	−6720.00	−1.38212
$288$	0	0
$289$	14131.0	2.87625
$290$	0	0
$291$	−3066.00	−0.617636
$292$	0	0
$293$	8684.00	1.73148	0.865742	−	0.500491i	$-0.166847\pi$
$293$	8684.00	1.73148	0.865742	+	0.500491i	$0.166847\pi$
$294$	0	0
$295$	−2464.00	−0.486304
$296$	0	0
$297$	−918.000	−0.179353
$298$	0	0
$299$	676.000	0.130749
$300$	0	0
$301$	3920.00	0.750648
$302$	0	0
$303$	−3570.00	−0.676868
$304$	0	0
$305$	−544.000	−0.102129
$306$	0	0
$307$	−7552.00	−1.40396	−0.701979	−	0.712197i	$-0.747700\pi$
$307$	−7552.00	−1.40396	−0.701979	+	0.712197i	$0.747700\pi$
$308$	0	0
$309$	−672.000	−0.123718
$310$	0	0
$311$	−2652.00	−0.483541	−0.241770	−	0.970334i	$-0.577728\pi$
$311$	−2652.00	−0.483541	−0.241770	+	0.970334i	$0.577728\pi$
$312$	0	0
$313$	−4426.00	−0.799273	−0.399636	−	0.916674i	$-0.630864\pi$
$313$	−4426.00	−0.799273	−0.399636	+	0.916674i	$0.630864\pi$
$314$	0	0
$315$	−4032.00	−0.721198
$316$	0	0
$317$	4944.00	0.875971	0.437985	−	0.898982i	$-0.355692\pi$
$317$	4944.00	0.875971	0.437985	+	0.898982i	$0.355692\pi$
$318$	0	0
$319$	6460.00	1.13383
$320$	0	0
$321$	1920.00	0.333844
$322$	0	0
$323$	14904.0	2.56743
$324$	0	0
$325$	1703.00	0.290663
$326$	0	0
$327$	−5802.00	−0.981197
$328$	0	0
$329$	12712.0	2.13020
$330$	0	0
$331$	−6088.00	−1.01096	−0.505478	−	0.862839i	$-0.668684\pi$
$331$	−6088.00	−1.01096	−0.505478	+	0.862839i	$0.668684\pi$
$332$	0	0
$333$	−3078.00	−0.506527
$334$	0	0
$335$	−10496.0	−1.71181
$336$	0	0
$337$	6638.00	1.07298	0.536491	−	0.843906i	$-0.319750\pi$
$337$	6638.00	1.07298	0.536491	+	0.843906i	$0.319750\pi$
$338$	0	0
$339$	1254.00	0.200908
$340$	0	0
$341$	5984.00	0.950298
$342$	0	0
$343$	−2744.00	−0.431959
$344$	0	0
$345$	−2496.00	−0.389508
$346$	0	0
$347$	−2292.00	−0.354585	−0.177293	−	0.984158i	$-0.556734\pi$
$347$	−2292.00	−0.354585	−0.177293	+	0.984158i	$0.556734\pi$
$348$	0	0
$349$	9866.00	1.51322	0.756612	−	0.653865i	$-0.226853\pi$
$349$	9866.00	1.51322	0.756612	+	0.653865i	$0.226853\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	2368.00	0.357042	0.178521	−	0.983936i	$-0.442869\pi$
$353$	2368.00	0.357042	0.178521	+	0.983936i	$0.442869\pi$
$354$	0	0
$355$	−8800.00	−1.31565
$356$	0	0
$357$	11592.0	1.71853
$358$	0	0
$359$	−5070.00	−0.745360	−0.372680	−	0.927960i	$-0.621561\pi$
$359$	−5070.00	−0.745360	−0.372680	+	0.927960i	$0.621561\pi$
$360$	0	0
$361$	4805.00	0.700539
$362$	0	0
$363$	525.000	0.0759101
$364$	0	0
$365$	9824.00	1.40880
$366$	0	0
$367$	8584.00	1.22093	0.610465	−	0.792043i	$-0.290983\pi$
$367$	8584.00	1.22093	0.610465	+	0.792043i	$0.290983\pi$
$368$	0	0
$369$	2160.00	0.304729
$370$	0	0
$371$	5544.00	0.775822
$372$	0	0
$373$	−4994.00	−0.693243	−0.346621	−	0.938005i	$-0.612671\pi$
$373$	−4994.00	−0.693243	−0.346621	+	0.938005i	$0.612671\pi$
$374$	0	0
$375$	−288.000	−0.0396593
$376$	0	0
$377$	2470.00	0.337431
$378$	0	0
$379$	1300.00	0.176191	0.0880957	−	0.996112i	$-0.471922\pi$
$379$	1300.00	0.176191	0.0880957	+	0.996112i	$0.471922\pi$
$380$	0	0
$381$	−3120.00	−0.419534
$382$	0	0
$383$	4590.00	0.612371	0.306185	−	0.951972i	$-0.400947\pi$
$383$	4590.00	0.612371	0.306185	+	0.951972i	$0.400947\pi$
$384$	0	0
$385$	−15232.0	−2.01635
$386$	0	0
$387$	−1260.00	−0.165502
$388$	0	0
$389$	−3510.00	−0.457491	−0.228746	−	0.973486i	$-0.573462\pi$
$389$	−3510.00	−0.457491	−0.228746	+	0.973486i	$0.573462\pi$
$390$	0	0
$391$	7176.00	0.928148
$392$	0	0
$393$	1704.00	0.218716
$394$	0	0
$395$	−128.000	−0.0163048
$396$	0	0
$397$	−6230.00	−0.787594	−0.393797	−	0.919197i	$-0.628839\pi$
$397$	−6230.00	−0.787594	−0.393797	+	0.919197i	$0.628839\pi$
$398$	0	0
$399$	9072.00	1.13827
$400$	0	0
$401$	−7500.00	−0.933995	−0.466998	−	0.884259i	$-0.654664\pi$
$401$	−7500.00	−0.933995	−0.466998	+	0.884259i	$0.654664\pi$
$402$	0	0
$403$	2288.00	0.282812
$404$	0	0
$405$	1296.00	0.159009
$406$	0	0
$407$	−11628.0	−1.41616
$408$	0	0
$409$	8254.00	0.997883	0.498941	−	0.866636i	$-0.333722\pi$
$409$	8254.00	0.997883	0.498941	+	0.866636i	$0.333722\pi$
$410$	0	0
$411$	−1584.00	−0.190105
$412$	0	0
$413$	4312.00	0.513752
$414$	0	0
$415$	12192.0	1.44212
$416$	0	0
$417$	4668.00	0.548185
$418$	0	0
$419$	−14808.0	−1.72653	−0.863267	−	0.504747i	$-0.831586\pi$
$419$	−14808.0	−1.72653	−0.863267	+	0.504747i	$0.831586\pi$
$420$	0	0
$421$	−10354.0	−1.19863	−0.599315	−	0.800513i	$-0.704560\pi$
$421$	−10354.0	−1.19863	−0.599315	+	0.800513i	$0.704560\pi$
$422$	0	0
$423$	−4086.00	−0.469665
$424$	0	0
$425$	18078.0	2.06332
$426$	0	0
$427$	952.000	0.107893
$428$	0	0
$429$	−1326.00	−0.149230
$430$	0	0
$431$	15486.0	1.73071	0.865353	−	0.501163i	$-0.167094\pi$
$431$	15486.0	1.73071	0.865353	+	0.501163i	$0.167094\pi$
$432$	0	0
$433$	−2018.00	−0.223970	−0.111985	−	0.993710i	$-0.535721\pi$
$433$	−2018.00	−0.223970	−0.111985	+	0.993710i	$0.535721\pi$
$434$	0	0
$435$	−9120.00	−1.00522
$436$	0	0
$437$	5616.00	0.614759
$438$	0	0
$439$	−8792.00	−0.955853	−0.477926	−	0.878400i	$-0.658611\pi$
$439$	−8792.00	−0.955853	−0.477926	+	0.878400i	$0.658611\pi$
$440$	0	0
$441$	3969.00	0.428571
$442$	0	0
$443$	−2760.00	−0.296008	−0.148004	−	0.988987i	$-0.547285\pi$
$443$	−2760.00	−0.296008	−0.148004	+	0.988987i	$0.547285\pi$
$444$	0	0
$445$	−7104.00	−0.756768
$446$	0	0
$447$	4572.00	0.483777
$448$	0	0
$449$	9532.00	1.00188	0.500939	−	0.865483i	$-0.332988\pi$
$449$	9532.00	1.00188	0.500939	+	0.865483i	$0.332988\pi$
$450$	0	0
$451$	8160.00	0.851972
$452$	0	0
$453$	−9072.00	−0.940927
$454$	0	0
$455$	−5824.00	−0.600073
$456$	0	0
$457$	12862.0	1.31654	0.658270	−	0.752782i	$-0.271288\pi$
$457$	12862.0	1.31654	0.658270	+	0.752782i	$0.271288\pi$
$458$	0	0
$459$	−3726.00	−0.378899
$460$	0	0
$461$	6744.00	0.681344	0.340672	−	0.940182i	$-0.389346\pi$
$461$	6744.00	0.681344	0.340672	+	0.940182i	$0.389346\pi$
$462$	0	0
$463$	9572.00	0.960796	0.480398	−	0.877051i	$-0.340492\pi$
$463$	9572.00	0.960796	0.480398	+	0.877051i	$0.340492\pi$
$464$	0	0
$465$	−8448.00	−0.842509
$466$	0	0
$467$	9104.00	0.902105	0.451052	−	0.892498i	$-0.351049\pi$
$467$	9104.00	0.902105	0.451052	+	0.892498i	$0.351049\pi$
$468$	0	0
$469$	18368.0	1.80843
$470$	0	0
$471$	6594.00	0.645086
$472$	0	0
$473$	−4760.00	−0.462717
$474$	0	0
$475$	14148.0	1.36664
$476$	0	0
$477$	−1782.00	−0.171053
$478$	0	0
$479$	18870.0	1.79998	0.899992	−	0.435906i	$-0.143572\pi$
$479$	18870.0	1.79998	0.899992	+	0.435906i	$0.143572\pi$
$480$	0	0
$481$	−4446.00	−0.421456
$482$	0	0
$483$	4368.00	0.411493
$484$	0	0
$485$	16352.0	1.53094
$486$	0	0
$487$	1744.00	0.162276	0.0811378	−	0.996703i	$-0.474145\pi$
$487$	1744.00	0.162276	0.0811378	+	0.996703i	$0.474145\pi$
$488$	0	0
$489$	804.000	0.0743520
$490$	0	0
$491$	13360.0	1.22796	0.613980	−	0.789322i	$-0.289568\pi$
$491$	13360.0	1.22796	0.613980	+	0.789322i	$0.289568\pi$
$492$	0	0
$493$	26220.0	2.39531
$494$	0	0
$495$	4896.00	0.444563
$496$	0	0
$497$	15400.0	1.38991
$498$	0	0
$499$	17368.0	1.55811	0.779057	−	0.626954i	$-0.215698\pi$
$499$	17368.0	1.55811	0.779057	+	0.626954i	$0.215698\pi$
$500$	0	0
$501$	2106.00	0.187803
$502$	0	0
$503$	5828.00	0.516616	0.258308	−	0.966063i	$-0.416835\pi$
$503$	5828.00	0.516616	0.258308	+	0.966063i	$0.416835\pi$
$504$	0	0
$505$	19040.0	1.67776
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−10744.0	−0.935598	−0.467799	−	0.883835i	$-0.654953\pi$
$509$	−10744.0	−0.935598	−0.467799	+	0.883835i	$0.654953\pi$
$510$	0	0
$511$	−17192.0	−1.48832
$512$	0	0
$513$	−2916.00	−0.250964
$514$	0	0
$515$	3584.00	0.306660
$516$	0	0
$517$	−15436.0	−1.31310
$518$	0	0
$519$	6198.00	0.524204
$520$	0	0
$521$	−12234.0	−1.02875	−0.514377	−	0.857564i	$-0.671977\pi$
$521$	−12234.0	−1.02875	−0.514377	+	0.857564i	$0.671977\pi$
$522$	0	0
$523$	1812.00	0.151498	0.0757488	−	0.997127i	$-0.475865\pi$
$523$	1812.00	0.151498	0.0757488	+	0.997127i	$0.475865\pi$
$524$	0	0
$525$	11004.0	0.914769
$526$	0	0
$527$	24288.0	2.00759
$528$	0	0
$529$	−9463.00	−0.777760
$530$	0	0
$531$	−1386.00	−0.113272
$532$	0	0
$533$	3120.00	0.253550
$534$	0	0
$535$	−10240.0	−0.827502
$536$	0	0
$537$	828.000	0.0665379
$538$	0	0
$539$	14994.0	1.19821
$540$	0	0
$541$	−6098.00	−0.484609	−0.242305	−	0.970200i	$-0.577903\pi$
$541$	−6098.00	−0.484609	−0.242305	+	0.970200i	$0.577903\pi$
$542$	0	0
$543$	−10422.0	−0.823666
$544$	0	0
$545$	30944.0	2.43210
$546$	0	0
$547$	−18332.0	−1.43294	−0.716471	−	0.697616i	$-0.754244\pi$
$547$	−18332.0	−1.43294	−0.716471	+	0.697616i	$0.754244\pi$
$548$	0	0
$549$	−306.000	−0.0237883
$550$	0	0
$551$	20520.0	1.58654
$552$	0	0
$553$	224.000	0.0172250
$554$	0	0
$555$	16416.0	1.25553
$556$	0	0
$557$	20004.0	1.52172	0.760859	−	0.648917i	$-0.224778\pi$
$557$	20004.0	1.52172	0.760859	+	0.648917i	$0.224778\pi$
$558$	0	0
$559$	−1820.00	−0.137706
$560$	0	0
$561$	−14076.0	−1.05934
$562$	0	0
$563$	10988.0	0.822538	0.411269	−	0.911514i	$-0.365086\pi$
$563$	10988.0	0.822538	0.411269	+	0.911514i	$0.365086\pi$
$564$	0	0
$565$	−6688.00	−0.497993
$566$	0	0
$567$	−2268.00	−0.167984
$568$	0	0
$569$	11062.0	0.815014	0.407507	−	0.913202i	$-0.366398\pi$
$569$	11062.0	0.815014	0.407507	+	0.913202i	$0.366398\pi$
$570$	0	0
$571$	−708.000	−0.0518895	−0.0259447	−	0.999663i	$-0.508259\pi$
$571$	−708.000	−0.0518895	−0.0259447	+	0.999663i	$0.508259\pi$
$572$	0	0
$573$	−11760.0	−0.857384
$574$	0	0
$575$	6812.00	0.494052
$576$	0	0
$577$	−2094.00	−0.151082	−0.0755410	−	0.997143i	$-0.524068\pi$
$577$	−2094.00	−0.151082	−0.0755410	+	0.997143i	$0.524068\pi$
$578$	0	0
$579$	−6558.00	−0.470710
$580$	0	0
$581$	−21336.0	−1.52352
$582$	0	0
$583$	−6732.00	−0.478235
$584$	0	0
$585$	1872.00	0.132304
$586$	0	0
$587$	−17854.0	−1.25539	−0.627695	−	0.778460i	$-0.716001\pi$
$587$	−17854.0	−1.25539	−0.627695	+	0.778460i	$0.716001\pi$
$588$	0	0
$589$	19008.0	1.32973
$590$	0	0
$591$	−4104.00	−0.285645
$592$	0	0
$593$	23948.0	1.65839	0.829196	−	0.558958i	$-0.188799\pi$
$593$	23948.0	1.65839	0.829196	+	0.558958i	$0.188799\pi$
$594$	0	0
$595$	−61824.0	−4.25973
$596$	0	0
$597$	−3216.00	−0.220473
$598$	0	0
$599$	18068.0	1.23245	0.616226	−	0.787570i	$-0.288661\pi$
$599$	18068.0	1.23245	0.616226	+	0.787570i	$0.288661\pi$
$600$	0	0
$601$	19942.0	1.35350	0.676748	−	0.736215i	$-0.263389\pi$
$601$	19942.0	1.35350	0.676748	+	0.736215i	$0.263389\pi$
$602$	0	0
$603$	−5904.00	−0.398722
$604$	0	0
$605$	−2800.00	−0.188159
$606$	0	0
$607$	−26376.0	−1.76370	−0.881852	−	0.471526i	$-0.843704\pi$
$607$	−26376.0	−1.76370	−0.881852	+	0.471526i	$0.843704\pi$
$608$	0	0
$609$	15960.0	1.06196
$610$	0	0
$611$	−5902.00	−0.390785
$612$	0	0
$613$	19426.0	1.27995	0.639975	−	0.768396i	$-0.278945\pi$
$613$	19426.0	1.27995	0.639975	+	0.768396i	$0.278945\pi$
$614$	0	0
$615$	−11520.0	−0.755335
$616$	0	0
$617$	−8024.00	−0.523556	−0.261778	−	0.965128i	$-0.584309\pi$
$617$	−8024.00	−0.523556	−0.261778	+	0.965128i	$0.584309\pi$
$618$	0	0
$619$	−20648.0	−1.34073	−0.670366	−	0.742031i	$-0.733863\pi$
$619$	−20648.0	−1.34073	−0.670366	+	0.742031i	$0.733863\pi$
$620$	0	0
$621$	−1404.00	−0.0907256
$622$	0	0
$623$	12432.0	0.799482
$624$	0	0
$625$	−14839.0	−0.949696
$626$	0	0
$627$	−11016.0	−0.701653
$628$	0	0
$629$	−47196.0	−2.99178
$630$	0	0
$631$	−12280.0	−0.774737	−0.387369	−	0.921925i	$-0.626616\pi$
$631$	−12280.0	−0.774737	−0.387369	+	0.921925i	$0.626616\pi$
$632$	0	0
$633$	−16332.0	−1.02550
$634$	0	0
$635$	16640.0	1.03990
$636$	0	0
$637$	5733.00	0.356593
$638$	0	0
$639$	−4950.00	−0.306446
$640$	0	0
$641$	−15878.0	−0.978383	−0.489191	−	0.872176i	$-0.662708\pi$
$641$	−15878.0	−0.978383	−0.489191	+	0.872176i	$0.662708\pi$
$642$	0	0
$643$	−21520.0	−1.31985	−0.659927	−	0.751330i	$-0.729413\pi$
$643$	−21520.0	−1.31985	−0.659927	+	0.751330i	$0.729413\pi$
$644$	0	0
$645$	6720.00	0.410232
$646$	0	0
$647$	−7312.00	−0.444304	−0.222152	−	0.975012i	$-0.571308\pi$
$647$	−7312.00	−0.444304	−0.222152	+	0.975012i	$0.571308\pi$
$648$	0	0
$649$	−5236.00	−0.316689
$650$	0	0
$651$	14784.0	0.890062
$652$	0	0
$653$	−3090.00	−0.185178	−0.0925889	−	0.995704i	$-0.529514\pi$
$653$	−3090.00	−0.185178	−0.0925889	+	0.995704i	$0.529514\pi$
$654$	0	0
$655$	−9088.00	−0.542134
$656$	0	0
$657$	5526.00	0.328143
$658$	0	0
$659$	−13428.0	−0.793749	−0.396875	−	0.917873i	$-0.629905\pi$
$659$	−13428.0	−0.793749	−0.396875	+	0.917873i	$0.629905\pi$
$660$	0	0
$661$	−22598.0	−1.32974	−0.664872	−	0.746958i	$-0.731514\pi$
$661$	−22598.0	−1.32974	−0.664872	+	0.746958i	$0.731514\pi$
$662$	0	0
$663$	−5382.00	−0.315263
$664$	0	0
$665$	−48384.0	−2.82143
$666$	0	0
$667$	9880.00	0.573546
$668$	0	0
$669$	288.000	0.0166438
$670$	0	0
$671$	−1156.00	−0.0665080
$672$	0	0
$673$	6178.00	0.353855	0.176927	−	0.984224i	$-0.443384\pi$
$673$	6178.00	0.353855	0.176927	+	0.984224i	$0.443384\pi$
$674$	0	0
$675$	−3537.00	−0.201688
$676$	0	0
$677$	−22398.0	−1.27153	−0.635764	−	0.771883i	$-0.719315\pi$
$677$	−22398.0	−1.27153	−0.635764	+	0.771883i	$0.719315\pi$
$678$	0	0
$679$	−28616.0	−1.61735
$680$	0	0
$681$	−594.000	−0.0334246
$682$	0	0
$683$	11410.0	0.639226	0.319613	−	0.947548i	$-0.396447\pi$
$683$	11410.0	0.639226	0.319613	+	0.947548i	$0.396447\pi$
$684$	0	0
$685$	8448.00	0.471214
$686$	0	0
$687$	17766.0	0.986631
$688$	0	0
$689$	−2574.00	−0.142325
$690$	0	0
$691$	32488.0	1.78857	0.894285	−	0.447498i	$-0.147685\pi$
$691$	32488.0	1.78857	0.894285	+	0.447498i	$0.147685\pi$
$692$	0	0
$693$	−8568.00	−0.469656
$694$	0	0
$695$	−24896.0	−1.35879
$696$	0	0
$697$	33120.0	1.79987
$698$	0	0
$699$	15342.0	0.830168
$700$	0	0
$701$	−5094.00	−0.274462	−0.137231	−	0.990539i	$-0.543820\pi$
$701$	−5094.00	−0.274462	−0.137231	+	0.990539i	$0.543820\pi$
$702$	0	0
$703$	−36936.0	−1.98160
$704$	0	0
$705$	21792.0	1.16416
$706$	0	0
$707$	−33320.0	−1.77246
$708$	0	0
$709$	−25418.0	−1.34639	−0.673197	−	0.739463i	$-0.735079\pi$
$709$	−25418.0	−1.34639	−0.673197	+	0.739463i	$0.735079\pi$
$710$	0	0
$711$	−72.0000	−0.00379777
$712$	0	0
$713$	9152.00	0.480708
$714$	0	0
$715$	7072.00	0.369899
$716$	0	0
$717$	−15678.0	−0.816605
$718$	0	0
$719$	20428.0	1.05958	0.529788	−	0.848130i	$-0.322271\pi$
$719$	20428.0	1.05958	0.529788	+	0.848130i	$0.322271\pi$
$720$	0	0
$721$	−6272.00	−0.323969
$722$	0	0
$723$	2286.00	0.117590
$724$	0	0
$725$	24890.0	1.27502
$726$	0	0
$727$	38336.0	1.95571	0.977857	−	0.209276i	$-0.0671107\pi$
$727$	38336.0	1.95571	0.977857	+	0.209276i	$0.0671107\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−19320.0	−0.977532
$732$	0	0
$733$	166.000	0.00836473	0.00418237	−	0.999991i	$-0.498669\pi$
$733$	166.000	0.00836473	0.00418237	+	0.999991i	$0.498669\pi$
$734$	0	0
$735$	−21168.0	−1.06230
$736$	0	0
$737$	−22304.0	−1.11476
$738$	0	0
$739$	−25248.0	−1.25678	−0.628392	−	0.777897i	$-0.716286\pi$
$739$	−25248.0	−1.25678	−0.628392	+	0.777897i	$0.716286\pi$
$740$	0	0
$741$	−4212.00	−0.208815
$742$	0	0
$743$	4442.00	0.219329	0.109664	−	0.993969i	$-0.465022\pi$
$743$	4442.00	0.219329	0.109664	+	0.993969i	$0.465022\pi$
$744$	0	0
$745$	−24384.0	−1.19914
$746$	0	0
$747$	6858.00	0.335905
$748$	0	0
$749$	17920.0	0.874209
$750$	0	0
$751$	19848.0	0.964399	0.482200	−	0.876061i	$-0.339838\pi$
$751$	19848.0	0.964399	0.482200	+	0.876061i	$0.339838\pi$
$752$	0	0
$753$	−9720.00	−0.470407
$754$	0	0
$755$	48384.0	2.33228
$756$	0	0
$757$	29166.0	1.40034	0.700169	−	0.713977i	$-0.253108\pi$
$757$	29166.0	1.40034	0.700169	+	0.713977i	$0.253108\pi$
$758$	0	0
$759$	−5304.00	−0.253653
$760$	0	0
$761$	−6240.00	−0.297240	−0.148620	−	0.988894i	$-0.547483\pi$
$761$	−6240.00	−0.297240	−0.148620	+	0.988894i	$0.547483\pi$
$762$	0	0
$763$	−54152.0	−2.56938
$764$	0	0
$765$	19872.0	0.939181
$766$	0	0
$767$	−2002.00	−0.0942478
$768$	0	0
$769$	−39750.0	−1.86401	−0.932004	−	0.362449i	$-0.881941\pi$
$769$	−39750.0	−1.86401	−0.932004	+	0.362449i	$0.881941\pi$
$770$	0	0
$771$	4158.00	0.194224
$772$	0	0
$773$	−9764.00	−0.454317	−0.227158	−	0.973858i	$-0.572943\pi$
$773$	−9764.00	−0.454317	−0.227158	+	0.973858i	$0.572943\pi$
$774$	0	0
$775$	23056.0	1.06864
$776$	0	0
$777$	−28728.0	−1.32640
$778$	0	0
$779$	25920.0	1.19214
$780$	0	0
$781$	−18700.0	−0.856772
$782$	0	0
$783$	−5130.00	−0.234140
$784$	0	0
$785$	−35168.0	−1.59898
$786$	0	0
$787$	−36016.0	−1.63130	−0.815649	−	0.578547i	$-0.803620\pi$
$787$	−36016.0	−1.63130	−0.815649	+	0.578547i	$0.803620\pi$
$788$	0	0
$789$	9900.00	0.446704
$790$	0	0
$791$	11704.0	0.526102
$792$	0	0
$793$	−442.000	−0.0197930
$794$	0	0
$795$	9504.00	0.423990
$796$	0	0
$797$	22290.0	0.990655	0.495328	−	0.868706i	$-0.335048\pi$
$797$	22290.0	0.990655	0.495328	+	0.868706i	$0.335048\pi$
$798$	0	0
$799$	−62652.0	−2.77405
$800$	0	0
$801$	−3996.00	−0.176269
$802$	0	0
$803$	20876.0	0.917432
$804$	0	0
$805$	−23296.0	−1.01997
$806$	0	0
$807$	12870.0	0.561395
$808$	0	0
$809$	−25578.0	−1.11159	−0.555794	−	0.831320i	$-0.687586\pi$
$809$	−25578.0	−1.11159	−0.555794	+	0.831320i	$0.687586\pi$
$810$	0	0
$811$	29900.0	1.29461	0.647306	−	0.762230i	$-0.275895\pi$
$811$	29900.0	1.29461	0.647306	+	0.762230i	$0.275895\pi$
$812$	0	0
$813$	7356.00	0.317326
$814$	0	0
$815$	−4288.00	−0.184297
$816$	0	0
$817$	−15120.0	−0.647469
$818$	0	0
$819$	−3276.00	−0.139771
$820$	0	0
$821$	−16412.0	−0.697665	−0.348832	−	0.937185i	$-0.613422\pi$
$821$	−16412.0	−0.697665	−0.348832	+	0.937185i	$0.613422\pi$
$822$	0	0
$823$	−18552.0	−0.785762	−0.392881	−	0.919589i	$-0.628522\pi$
$823$	−18552.0	−0.785762	−0.392881	+	0.919589i	$0.628522\pi$
$824$	0	0
$825$	−13362.0	−0.563885
$826$	0	0
$827$	−28662.0	−1.20517	−0.602585	−	0.798055i	$-0.705863\pi$
$827$	−28662.0	−1.20517	−0.602585	+	0.798055i	$0.705863\pi$
$828$	0	0
$829$	3686.00	0.154427	0.0772136	−	0.997015i	$-0.475398\pi$
$829$	3686.00	0.154427	0.0772136	+	0.997015i	$0.475398\pi$
$830$	0	0
$831$	−126.000	−0.00525980
$832$	0	0
$833$	60858.0	2.53134
$834$	0	0
$835$	−11232.0	−0.465508
$836$	0	0
$837$	−4752.00	−0.196240
$838$	0	0
$839$	−13370.0	−0.550159	−0.275080	−	0.961421i	$-0.588704\pi$
$839$	−13370.0	−0.550159	−0.275080	+	0.961421i	$0.588704\pi$
$840$	0	0
$841$	11711.0	0.480175
$842$	0	0
$843$	6864.00	0.280437
$844$	0	0
$845$	2704.00	0.110083
$846$	0	0
$847$	4900.00	0.198779
$848$	0	0
$849$	−3468.00	−0.140190
$850$	0	0
$851$	−17784.0	−0.716366
$852$	0	0
$853$	−11398.0	−0.457515	−0.228757	−	0.973483i	$-0.573466\pi$
$853$	−11398.0	−0.457515	−0.228757	+	0.973483i	$0.573466\pi$
$854$	0	0
$855$	15552.0	0.622067
$856$	0	0
$857$	7990.00	0.318475	0.159238	−	0.987240i	$-0.449096\pi$
$857$	7990.00	0.318475	0.159238	+	0.987240i	$0.449096\pi$
$858$	0	0
$859$	7652.00	0.303938	0.151969	−	0.988385i	$-0.451439\pi$
$859$	7652.00	0.303938	0.151969	+	0.988385i	$0.451439\pi$
$860$	0	0
$861$	20160.0	0.797969
$862$	0	0
$863$	1022.00	0.0403120	0.0201560	−	0.999797i	$-0.493584\pi$
$863$	1022.00	0.0403120	0.0201560	+	0.999797i	$0.493584\pi$
$864$	0	0
$865$	−33056.0	−1.29935
$866$	0	0
$867$	−42393.0	−1.66060
$868$	0	0
$869$	−272.000	−0.0106179
$870$	0	0
$871$	−8528.00	−0.331757
$872$	0	0
$873$	9198.00	0.356592
$874$	0	0
$875$	−2688.00	−0.103853
$876$	0	0
$877$	15546.0	0.598576	0.299288	−	0.954163i	$-0.403251\pi$
$877$	15546.0	0.598576	0.299288	+	0.954163i	$0.403251\pi$
$878$	0	0
$879$	−26052.0	−0.999673
$880$	0	0
$881$	11310.0	0.432513	0.216256	−	0.976337i	$-0.430615\pi$
$881$	11310.0	0.432513	0.216256	+	0.976337i	$0.430615\pi$
$882$	0	0
$883$	17260.0	0.657809	0.328904	−	0.944363i	$-0.393321\pi$
$883$	17260.0	0.657809	0.328904	+	0.944363i	$0.393321\pi$
$884$	0	0
$885$	7392.00	0.280768
$886$	0	0
$887$	−832.000	−0.0314947	−0.0157474	−	0.999876i	$-0.505013\pi$
$887$	−832.000	−0.0314947	−0.0157474	+	0.999876i	$0.505013\pi$
$888$	0	0
$889$	−29120.0	−1.09860
$890$	0	0
$891$	2754.00	0.103549
$892$	0	0
$893$	−49032.0	−1.83739
$894$	0	0
$895$	−4416.00	−0.164928
$896$	0	0
$897$	−2028.00	−0.0754882
$898$	0	0
$899$	33440.0	1.24059
$900$	0	0
$901$	−27324.0	−1.01032
$902$	0	0
$903$	−11760.0	−0.433387
$904$	0	0
$905$	55584.0	2.04163
$906$	0	0
$907$	31740.0	1.16197	0.580986	−	0.813913i	$-0.302667\pi$
$907$	31740.0	1.16197	0.580986	+	0.813913i	$0.302667\pi$
$908$	0	0
$909$	10710.0	0.390790
$910$	0	0
$911$	23568.0	0.857127	0.428563	−	0.903512i	$-0.359020\pi$
$911$	23568.0	0.857127	0.428563	+	0.903512i	$0.359020\pi$
$912$	0	0
$913$	25908.0	0.939134
$914$	0	0
$915$	1632.00	0.0589642
$916$	0	0
$917$	15904.0	0.572733
$918$	0	0
$919$	18864.0	0.677112	0.338556	−	0.940946i	$-0.390062\pi$
$919$	18864.0	0.677112	0.338556	+	0.940946i	$0.390062\pi$
$920$	0	0
$921$	22656.0	0.810576
$922$	0	0
$923$	−7150.00	−0.254978
$924$	0	0
$925$	−44802.0	−1.59252
$926$	0	0
$927$	2016.00	0.0714284
$928$	0	0
$929$	−19536.0	−0.689941	−0.344971	−	0.938613i	$-0.612111\pi$
$929$	−19536.0	−0.689941	−0.344971	+	0.938613i	$0.612111\pi$
$930$	0	0
$931$	47628.0	1.67663
$932$	0	0
$933$	7956.00	0.279172
$934$	0	0
$935$	75072.0	2.62579
$936$	0	0
$937$	18174.0	0.633638	0.316819	−	0.948486i	$-0.397385\pi$
$937$	18174.0	0.633638	0.316819	+	0.948486i	$0.397385\pi$
$938$	0	0
$939$	13278.0	0.461460
$940$	0	0
$941$	−51172.0	−1.77275	−0.886376	−	0.462966i	$-0.846785\pi$
$941$	−51172.0	−1.77275	−0.886376	+	0.462966i	$0.846785\pi$
$942$	0	0
$943$	12480.0	0.430970
$944$	0	0
$945$	12096.0	0.416384
$946$	0	0
$947$	3726.00	0.127855	0.0639275	−	0.997955i	$-0.479637\pi$
$947$	3726.00	0.127855	0.0639275	+	0.997955i	$0.479637\pi$
$948$	0	0
$949$	7982.00	0.273031
$950$	0	0
$951$	−14832.0	−0.505742
$952$	0	0
$953$	−40498.0	−1.37656	−0.688279	−	0.725447i	$-0.741633\pi$
$953$	−40498.0	−1.37656	−0.688279	+	0.725447i	$0.741633\pi$
$954$	0	0
$955$	62720.0	2.12521
$956$	0	0
$957$	−19380.0	−0.654615
$958$	0	0
$959$	−14784.0	−0.497810
$960$	0	0
$961$	1185.00	0.0397771
$962$	0	0
$963$	−5760.00	−0.192745
$964$	0	0
$965$	34976.0	1.16675
$966$	0	0
$967$	−28568.0	−0.950036	−0.475018	−	0.879976i	$-0.657558\pi$
$967$	−28568.0	−0.950036	−0.475018	+	0.879976i	$0.657558\pi$
$968$	0	0
$969$	−44712.0	−1.48231
$970$	0	0
$971$	−8676.00	−0.286742	−0.143371	−	0.989669i	$-0.545794\pi$
$971$	−8676.00	−0.286742	−0.143371	+	0.989669i	$0.545794\pi$
$972$	0	0
$973$	43568.0	1.43548
$974$	0	0
$975$	−5109.00	−0.167814
$976$	0	0
$977$	−2796.00	−0.0915578	−0.0457789	−	0.998952i	$-0.514577\pi$
$977$	−2796.00	−0.0915578	−0.0457789	+	0.998952i	$0.514577\pi$
$978$	0	0
$979$	−15096.0	−0.492819
$980$	0	0
$981$	17406.0	0.566494
$982$	0	0
$983$	406.000	0.0131733	0.00658667	−	0.999978i	$-0.497903\pi$
$983$	406.000	0.0131733	0.00658667	+	0.999978i	$0.497903\pi$
$984$	0	0
$985$	21888.0	0.708030
$986$	0	0
$987$	−38136.0	−1.22987
$988$	0	0
$989$	−7280.00	−0.234065
$990$	0	0
$991$	23232.0	0.744691	0.372346	−	0.928094i	$-0.378554\pi$
$991$	23232.0	0.744691	0.372346	+	0.928094i	$0.378554\pi$
$992$	0	0
$993$	18264.0	0.583676
$994$	0	0
$995$	17152.0	0.546487
$996$	0	0
$997$	−6110.00	−0.194088	−0.0970440	−	0.995280i	$-0.530939\pi$
$997$	−6110.00	−0.194088	−0.0970440	+	0.995280i	$0.530939\pi$
$998$	0	0
$999$	9234.00	0.292443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.g.1.1		1
4.3	odd	2		2496.4.a.q.1.1		1
8.3	odd	2		78.4.a.a.1.1	✓	1
8.5	even	2		624.4.a.f.1.1		1
24.5	odd	2		1872.4.a.o.1.1		1
24.11	even	2		234.4.a.k.1.1		1
40.19	odd	2		1950.4.a.o.1.1		1
104.51	odd	2		1014.4.a.i.1.1		1
104.83	even	4		1014.4.b.a.337.2		2
104.99	even	4		1014.4.b.a.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.a.1.1	✓	1	8.3	odd	2
234.4.a.k.1.1		1	24.11	even	2
624.4.a.f.1.1		1	8.5	even	2
1014.4.a.i.1.1		1	104.51	odd	2
1014.4.b.a.337.1		2	104.99	even	4
1014.4.b.a.337.2		2	104.83	even	4
1872.4.a.o.1.1		1	24.5	odd	2
1950.4.a.o.1.1		1	40.19	odd	2
2496.4.a.g.1.1		1	1.1	even	1	trivial
2496.4.a.q.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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