LMFDB - Embedded newform 2496.4.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$147.268767374$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.36248.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 54x - 90 $ x^3 - x^2 - 54*x - 90
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.84636$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +10.8037 q^{5} +32.1891 q^{7} +9.00000 q^{9} -30.2219 q^{11} -13.0000 q^{13} -32.4111 q^{15} +34.0000 q^{17} +41.8621 q^{19} -96.5674 q^{21} -45.5417 q^{23} -8.28038 q^{25} -27.0000 q^{27} -2.40702 q^{29} -73.7965 q^{31} +90.6658 q^{33} +347.761 q^{35} -401.383 q^{37} +39.0000 q^{39} -353.619 q^{41} -329.274 q^{43} +97.2332 q^{45} -45.1373 q^{47} +693.139 q^{49} -102.000 q^{51} -449.761 q^{53} -326.508 q^{55} -125.586 q^{57} -351.640 q^{59} -872.044 q^{61} +289.702 q^{63} -140.448 q^{65} -177.463 q^{67} +136.625 q^{69} -32.9352 q^{71} +777.906 q^{73} +24.8411 q^{75} -972.818 q^{77} -350.766 q^{79} +81.0000 q^{81} -421.087 q^{83} +367.325 q^{85} +7.22106 q^{87} -1365.76 q^{89} -418.459 q^{91} +221.389 q^{93} +452.265 q^{95} +468.206 q^{97} -271.997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 16 q^{5} + 22 q^{7} + 27 q^{9} - 20 q^{11} - 39 q^{13} + 48 q^{15} + 102 q^{17} - 38 q^{19} - 66 q^{21} - 32 q^{23} + 161 q^{25} - 81 q^{27} - 350 q^{29} - 50 q^{31} + 60 q^{33} + 232 q^{35}+ \cdots - 180 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 16 * q^5 + 22 * q^7 + 27 * q^9 - 20 * q^11 - 39 * q^13 + 48 * q^15 + 102 * q^17 - 38 * q^19 - 66 * q^21 - 32 * q^23 + 161 * q^25 - 81 * q^27 - 350 * q^29 - 50 * q^31 + 60 * q^33 + 232 * q^35 - 542 * q^37 + 117 * q^39 + 500 * q^41 + 420 * q^43 - 144 * q^45 - 324 * q^47 + 1119 * q^49 - 306 * q^51 - 538 * q^53 - 896 * q^55 + 114 * q^57 + 112 * q^59 - 1206 * q^61 + 198 * q^63 + 208 * q^65 + 950 * q^67 + 96 * q^69 + 1008 * q^71 + 1834 * q^73 - 483 * q^75 + 792 * q^77 + 272 * q^79 + 243 * q^81 - 1640 * q^83 - 544 * q^85 + 1050 * q^87 - 1576 * q^89 - 286 * q^91 + 150 * q^93 + 2760 * q^95 + 402 * q^97 - 180 * 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
$5$	10.8037	0.966311	0.483156	−	0.875535i	$-0.339491\pi$
$5$	10.8037	0.966311	0.483156	+	0.875535i	$0.339491\pi$
$6$	0	0
$7$	32.1891	1.73805	0.869025	−	0.494769i	$-0.164747\pi$
$7$	32.1891	1.73805	0.869025	+	0.494769i	$0.164747\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−30.2219	−0.828387	−0.414194	−	0.910189i	$-0.635936\pi$
$11$	−30.2219	−0.828387	−0.414194	+	0.910189i	$0.635936\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	−32.4111	−0.557900
$16$	0	0
$17$	34.0000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	34.0000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	41.8621	0.505465	0.252732	−	0.967536i	$-0.418671\pi$
$19$	41.8621	0.505465	0.252732	+	0.967536i	$0.418671\pi$
$20$	0	0
$21$	−96.5674	−1.00346
$22$	0	0
$23$	−45.5417	−0.412874	−0.206437	−	0.978460i	$-0.566187\pi$
$23$	−45.5417	−0.412874	−0.206437	+	0.978460i	$0.566187\pi$
$24$	0	0
$25$	−8.28038	−0.0662430
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−2.40702	−0.0154128	−0.00770642	−	0.999970i	$-0.502453\pi$
$29$	−2.40702	−0.0154128	−0.00770642	+	0.999970i	$0.502453\pi$
$30$	0	0
$31$	−73.7965	−0.427556	−0.213778	−	0.976882i	$-0.568577\pi$
$31$	−73.7965	−0.427556	−0.213778	+	0.976882i	$0.568577\pi$
$32$	0	0
$33$	90.6658	0.478269
$34$	0	0
$35$	347.761	1.67950
$36$	0	0
$37$	−401.383	−1.78343	−0.891715	−	0.452597i	$-0.850498\pi$
$37$	−401.383	−1.78343	−0.891715	+	0.452597i	$0.850498\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−353.619	−1.34698	−0.673489	−	0.739197i	$-0.735205\pi$
$41$	−353.619	−1.34698	−0.673489	+	0.739197i	$0.735205\pi$
$42$	0	0
$43$	−329.274	−1.16776	−0.583882	−	0.811839i	$-0.698467\pi$
$43$	−329.274	−1.16776	−0.583882	+	0.811839i	$0.698467\pi$
$44$	0	0
$45$	97.2332	0.322104
$46$	0	0
$47$	−45.1373	−0.140084	−0.0700420	−	0.997544i	$-0.522313\pi$
$47$	−45.1373	−0.140084	−0.0700420	+	0.997544i	$0.522313\pi$
$48$	0	0
$49$	693.139	2.02081
$50$	0	0
$51$	−102.000	−0.280056
$52$	0	0
$53$	−449.761	−1.16565	−0.582825	−	0.812598i	$-0.698053\pi$
$53$	−449.761	−1.16565	−0.582825	+	0.812598i	$0.698053\pi$
$54$	0	0
$55$	−326.508	−0.800479
$56$	0	0
$57$	−125.586	−0.291830
$58$	0	0
$59$	−351.640	−0.775927	−0.387963	−	0.921675i	$-0.626821\pi$
$59$	−351.640	−0.775927	−0.387963	+	0.921675i	$0.626821\pi$
$60$	0	0
$61$	−872.044	−1.83039	−0.915195	−	0.403012i	$-0.867963\pi$
$61$	−872.044	−1.83039	−0.915195	+	0.403012i	$0.867963\pi$
$62$	0	0
$63$	289.702	0.579350
$64$	0	0
$65$	−140.448	−0.268006
$66$	0	0
$67$	−177.463	−0.323591	−0.161795	−	0.986824i	$-0.551728\pi$
$67$	−177.463	−0.323591	−0.161795	+	0.986824i	$0.551728\pi$
$68$	0	0
$69$	136.625	0.238373
$70$	0	0
$71$	−32.9352	−0.0550520	−0.0275260	−	0.999621i	$-0.508763\pi$
$71$	−32.9352	−0.0550520	−0.0275260	+	0.999621i	$0.508763\pi$
$72$	0	0
$73$	777.906	1.24722	0.623609	−	0.781736i	$-0.285666\pi$
$73$	777.906	1.24722	0.623609	+	0.781736i	$0.285666\pi$
$74$	0	0
$75$	24.8411	0.0382454
$76$	0	0
$77$	−972.818	−1.43978
$78$	0	0
$79$	−350.766	−0.499547	−0.249774	−	0.968304i	$-0.580356\pi$
$79$	−350.766	−0.499547	−0.249774	+	0.968304i	$0.580356\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−421.087	−0.556871	−0.278436	−	0.960455i	$-0.589816\pi$
$83$	−421.087	−0.556871	−0.278436	+	0.960455i	$0.589816\pi$
$84$	0	0
$85$	367.325	0.468730
$86$	0	0
$87$	7.22106	0.00889861
$88$	0	0
$89$	−1365.76	−1.62663	−0.813315	−	0.581823i	$-0.802340\pi$
$89$	−1365.76	−1.62663	−0.813315	+	0.581823i	$0.802340\pi$
$90$	0	0
$91$	−418.459	−0.482048
$92$	0	0
$93$	221.389	0.246850
$94$	0	0
$95$	452.265	0.488436
$96$	0	0
$97$	468.206	0.490094	0.245047	−	0.969511i	$-0.421197\pi$
$97$	468.206	0.490094	0.245047	+	0.969511i	$0.421197\pi$
$98$	0	0
$99$	−271.997	−0.276129
$100$	0	0
$101$	−548.199	−0.540077	−0.270039	−	0.962849i	$-0.587036\pi$
$101$	−548.199	−0.540077	−0.270039	+	0.962849i	$0.587036\pi$
$102$	0	0
$103$	917.200	0.877422	0.438711	−	0.898628i	$-0.355435\pi$
$103$	917.200	0.877422	0.438711	+	0.898628i	$0.355435\pi$
$104$	0	0
$105$	−1043.28	−0.969657
$106$	0	0
$107$	1279.54	1.15605	0.578025	−	0.816019i	$-0.303823\pi$
$107$	1279.54	1.15605	0.578025	+	0.816019i	$0.303823\pi$
$108$	0	0
$109$	753.302	0.661957	0.330978	−	0.943638i	$-0.392621\pi$
$109$	753.302	0.661957	0.330978	+	0.943638i	$0.392621\pi$
$110$	0	0
$111$	1204.15	1.02966
$112$	0	0
$113$	−1195.29	−0.995074	−0.497537	−	0.867443i	$-0.665762\pi$
$113$	−1195.29	−0.995074	−0.497537	+	0.867443i	$0.665762\pi$
$114$	0	0
$115$	−492.019	−0.398965
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	1094.43	0.843078
$120$	0	0
$121$	−417.634	−0.313775
$122$	0	0
$123$	1060.86	0.777678
$124$	0	0
$125$	−1439.92	−1.03032
$126$	0	0
$127$	−1397.95	−0.976758	−0.488379	−	0.872632i	$-0.662412\pi$
$127$	−1397.95	−0.976758	−0.488379	+	0.872632i	$0.662412\pi$
$128$	0	0
$129$	987.822	0.674208
$130$	0	0
$131$	−1692.92	−1.12909	−0.564546	−	0.825401i	$-0.690949\pi$
$131$	−1692.92	−1.12909	−0.564546	+	0.825401i	$0.690949\pi$
$132$	0	0
$133$	1347.51	0.878523
$134$	0	0
$135$	−291.700	−0.185967
$136$	0	0
$137$	137.034	0.0854567	0.0427283	−	0.999087i	$-0.486395\pi$
$137$	137.034	0.0854567	0.0427283	+	0.999087i	$0.486395\pi$
$138$	0	0
$139$	2586.53	1.57832	0.789161	−	0.614187i	$-0.210516\pi$
$139$	2586.53	1.57832	0.789161	+	0.614187i	$0.210516\pi$
$140$	0	0
$141$	135.412	0.0808775
$142$	0	0
$143$	392.885	0.229753
$144$	0	0
$145$	−26.0047	−0.0148936
$146$	0	0
$147$	−2079.42	−1.16672
$148$	0	0
$149$	1911.19	1.05081	0.525405	−	0.850852i	$-0.323914\pi$
$149$	1911.19	1.05081	0.525405	+	0.850852i	$0.323914\pi$
$150$	0	0
$151$	2007.47	1.08189	0.540946	−	0.841057i	$-0.318066\pi$
$151$	2007.47	1.08189	0.540946	+	0.841057i	$0.318066\pi$
$152$	0	0
$153$	306.000	0.161690
$154$	0	0
$155$	−797.274	−0.413152
$156$	0	0
$157$	−3067.26	−1.55920	−0.779599	−	0.626280i	$-0.784577\pi$
$157$	−3067.26	−1.55920	−0.779599	+	0.626280i	$0.784577\pi$
$158$	0	0
$159$	1349.28	0.672988
$160$	0	0
$161$	−1465.95	−0.717596
$162$	0	0
$163$	1188.86	0.571281	0.285640	−	0.958337i	$-0.407794\pi$
$163$	1188.86	0.571281	0.285640	+	0.958337i	$0.407794\pi$
$164$	0	0
$165$	979.525	0.462157
$166$	0	0
$167$	3325.23	1.54080	0.770401	−	0.637560i	$-0.220056\pi$
$167$	3325.23	1.54080	0.770401	+	0.637560i	$0.220056\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	376.759	0.168488
$172$	0	0
$173$	1452.26	0.638226	0.319113	−	0.947717i	$-0.396615\pi$
$173$	1452.26	0.638226	0.319113	+	0.947717i	$0.396615\pi$
$174$	0	0
$175$	−266.538	−0.115134
$176$	0	0
$177$	1054.92	0.447982
$178$	0	0
$179$	2462.51	1.02825	0.514125	−	0.857715i	$-0.328117\pi$
$179$	2462.51	1.02825	0.514125	+	0.857715i	$0.328117\pi$
$180$	0	0
$181$	−2816.51	−1.15663	−0.578313	−	0.815815i	$-0.696289\pi$
$181$	−2816.51	−1.15663	−0.578313	+	0.815815i	$0.696289\pi$
$182$	0	0
$183$	2616.13	1.05678
$184$	0	0
$185$	−4336.41	−1.72335
$186$	0	0
$187$	−1027.55	−0.401827
$188$	0	0
$189$	−869.106	−0.334488
$190$	0	0
$191$	3897.94	1.47667	0.738337	−	0.674432i	$-0.235611\pi$
$191$	3897.94	1.47667	0.738337	+	0.674432i	$0.235611\pi$
$192$	0	0
$193$	3906.14	1.45684	0.728420	−	0.685131i	$-0.240255\pi$
$193$	3906.14	1.45684	0.728420	+	0.685131i	$0.240255\pi$
$194$	0	0
$195$	421.344	0.154734
$196$	0	0
$197$	−2945.95	−1.06543	−0.532717	−	0.846294i	$-0.678829\pi$
$197$	−2945.95	−1.06543	−0.532717	+	0.846294i	$0.678829\pi$
$198$	0	0
$199$	535.794	0.190862	0.0954308	−	0.995436i	$-0.469577\pi$
$199$	535.794	0.190862	0.0954308	+	0.995436i	$0.469577\pi$
$200$	0	0
$201$	532.390	0.186825
$202$	0	0
$203$	−77.4799	−0.0267883
$204$	0	0
$205$	−3820.39	−1.30160
$206$	0	0
$207$	−409.876	−0.137625
$208$	0	0
$209$	−1265.15	−0.418720
$210$	0	0
$211$	3688.34	1.20339	0.601696	−	0.798725i	$-0.294492\pi$
$211$	3688.34	1.20339	0.601696	+	0.798725i	$0.294492\pi$
$212$	0	0
$213$	98.8057	0.0317843
$214$	0	0
$215$	−3557.37	−1.12842
$216$	0	0
$217$	−2375.44	−0.743114
$218$	0	0
$219$	−2333.72	−0.720082
$220$	0	0
$221$	−442.000	−0.134535
$222$	0	0
$223$	−2383.77	−0.715827	−0.357913	−	0.933755i	$-0.616512\pi$
$223$	−2383.77	−0.715827	−0.357913	+	0.933755i	$0.616512\pi$
$224$	0	0
$225$	−74.5234	−0.0220810
$226$	0	0
$227$	5654.49	1.65331	0.826656	−	0.562708i	$-0.190240\pi$
$227$	5654.49	1.65331	0.826656	+	0.562708i	$0.190240\pi$
$228$	0	0
$229$	4732.77	1.36572	0.682861	−	0.730548i	$-0.260735\pi$
$229$	4732.77	1.36572	0.682861	+	0.730548i	$0.260735\pi$
$230$	0	0
$231$	2918.45	0.831256
$232$	0	0
$233$	−1284.93	−0.361281	−0.180640	−	0.983549i	$-0.557817\pi$
$233$	−1284.93	−0.361281	−0.180640	+	0.983549i	$0.557817\pi$
$234$	0	0
$235$	−487.649	−0.135365
$236$	0	0
$237$	1052.30	0.288414
$238$	0	0
$239$	−4658.76	−1.26088	−0.630440	−	0.776238i	$-0.717125\pi$
$239$	−4658.76	−1.26088	−0.630440	+	0.776238i	$0.717125\pi$
$240$	0	0
$241$	2167.09	0.579230	0.289615	−	0.957143i	$-0.406473\pi$
$241$	2167.09	0.579230	0.289615	+	0.957143i	$0.406473\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	7488.46	1.95274
$246$	0	0
$247$	−544.208	−0.140191
$248$	0	0
$249$	1263.26	0.321510
$250$	0	0
$251$	5219.24	1.31249	0.656246	−	0.754547i	$-0.272143\pi$
$251$	5219.24	1.31249	0.656246	+	0.754547i	$0.272143\pi$
$252$	0	0
$253$	1376.36	0.342020
$254$	0	0
$255$	−1101.98	−0.270621
$256$	0	0
$257$	399.657	0.0970036	0.0485018	−	0.998823i	$-0.484555\pi$
$257$	399.657	0.0970036	0.0485018	+	0.998823i	$0.484555\pi$
$258$	0	0
$259$	−12920.2	−3.09969
$260$	0	0
$261$	−21.6632	−0.00513762
$262$	0	0
$263$	−8045.64	−1.88637	−0.943186	−	0.332267i	$-0.892187\pi$
$263$	−8045.64	−1.88637	−0.943186	+	0.332267i	$0.892187\pi$
$264$	0	0
$265$	−4859.08	−1.12638
$266$	0	0
$267$	4097.28	0.939136
$268$	0	0
$269$	2261.41	0.512568	0.256284	−	0.966602i	$-0.417502\pi$
$269$	2261.41	0.512568	0.256284	+	0.966602i	$0.417502\pi$
$270$	0	0
$271$	−6001.69	−1.34530	−0.672651	−	0.739960i	$-0.734844\pi$
$271$	−6001.69	−1.34530	−0.672651	+	0.739960i	$0.734844\pi$
$272$	0	0
$273$	1255.38	0.278311
$274$	0	0
$275$	250.249	0.0548749
$276$	0	0
$277$	1760.39	0.381848	0.190924	−	0.981605i	$-0.438852\pi$
$277$	1760.39	0.381848	0.190924	+	0.981605i	$0.438852\pi$
$278$	0	0
$279$	−664.168	−0.142519
$280$	0	0
$281$	−6327.06	−1.34321	−0.671603	−	0.740911i	$-0.734394\pi$
$281$	−6327.06	−1.34321	−0.671603	+	0.740911i	$0.734394\pi$
$282$	0	0
$283$	6197.80	1.30184	0.650921	−	0.759146i	$-0.274383\pi$
$283$	6197.80	1.30184	0.650921	+	0.759146i	$0.274383\pi$
$284$	0	0
$285$	−1356.80	−0.281999
$286$	0	0
$287$	−11382.7	−2.34111
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	−1404.62	−0.282956
$292$	0	0
$293$	−7878.93	−1.57096	−0.785482	−	0.618885i	$-0.787585\pi$
$293$	−7878.93	−1.57096	−0.785482	+	0.618885i	$0.787585\pi$
$294$	0	0
$295$	−3799.01	−0.749787
$296$	0	0
$297$	815.992	0.159423
$298$	0	0
$299$	592.043	0.114511
$300$	0	0
$301$	−10599.0	−2.02963
$302$	0	0
$303$	1644.60	0.311814
$304$	0	0
$305$	−9421.29	−1.76873
$306$	0	0
$307$	−3503.65	−0.651348	−0.325674	−	0.945482i	$-0.605591\pi$
$307$	−3503.65	−0.651348	−0.325674	+	0.945482i	$0.605591\pi$
$308$	0	0
$309$	−2751.60	−0.506580
$310$	0	0
$311$	−9298.24	−1.69535	−0.847676	−	0.530513i	$-0.821999\pi$
$311$	−9298.24	−1.69535	−0.847676	+	0.530513i	$0.821999\pi$
$312$	0	0
$313$	7036.43	1.27068	0.635339	−	0.772233i	$-0.280860\pi$
$313$	7036.43	1.27068	0.635339	+	0.772233i	$0.280860\pi$
$314$	0	0
$315$	3129.85	0.559832
$316$	0	0
$317$	−11047.2	−1.95733	−0.978666	−	0.205460i	$-0.934131\pi$
$317$	−11047.2	−1.95733	−0.978666	+	0.205460i	$0.934131\pi$
$318$	0	0
$319$	72.7448	0.0127678
$320$	0	0
$321$	−3838.61	−0.667446
$322$	0	0
$323$	1423.31	0.245186
$324$	0	0
$325$	107.645	0.0183725
$326$	0	0
$327$	−2259.91	−0.382181
$328$	0	0
$329$	−1452.93	−0.243473
$330$	0	0
$331$	−5075.45	−0.842816	−0.421408	−	0.906871i	$-0.638464\pi$
$331$	−5075.45	−0.842816	−0.421408	+	0.906871i	$0.638464\pi$
$332$	0	0
$333$	−3612.45	−0.594477
$334$	0	0
$335$	−1917.26	−0.312689
$336$	0	0
$337$	−6777.58	−1.09554	−0.547772	−	0.836628i	$-0.684524\pi$
$337$	−6777.58	−1.09554	−0.547772	+	0.836628i	$0.684524\pi$
$338$	0	0
$339$	3585.87	0.574506
$340$	0	0
$341$	2230.27	0.354182
$342$	0	0
$343$	11270.7	1.77423
$344$	0	0
$345$	1476.06	0.230343
$346$	0	0
$347$	−8725.29	−1.34985	−0.674925	−	0.737886i	$-0.735824\pi$
$347$	−8725.29	−1.34985	−0.674925	+	0.737886i	$0.735824\pi$
$348$	0	0
$349$	−3551.95	−0.544790	−0.272395	−	0.962186i	$-0.587816\pi$
$349$	−3551.95	−0.544790	−0.272395	+	0.962186i	$0.587816\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	9440.63	1.42344	0.711719	−	0.702464i	$-0.247917\pi$
$353$	9440.63	1.42344	0.711719	+	0.702464i	$0.247917\pi$
$354$	0	0
$355$	−355.822	−0.0531974
$356$	0	0
$357$	−3283.29	−0.486751
$358$	0	0
$359$	−1377.77	−0.202551	−0.101275	−	0.994858i	$-0.532292\pi$
$359$	−1377.77	−0.202551	−0.101275	+	0.994858i	$0.532292\pi$
$360$	0	0
$361$	−5106.56	−0.744505
$362$	0	0
$363$	1252.90	0.181158
$364$	0	0
$365$	8404.25	1.20520
$366$	0	0
$367$	390.300	0.0555136	0.0277568	−	0.999615i	$-0.491164\pi$
$367$	390.300	0.0555136	0.0277568	+	0.999615i	$0.491164\pi$
$368$	0	0
$369$	−3182.58	−0.448993
$370$	0	0
$371$	−14477.4	−2.02596
$372$	0	0
$373$	−4976.32	−0.690788	−0.345394	−	0.938458i	$-0.612255\pi$
$373$	−4976.32	−0.690788	−0.345394	+	0.938458i	$0.612255\pi$
$374$	0	0
$375$	4319.76	0.594857
$376$	0	0
$377$	31.2913	0.00427476
$378$	0	0
$379$	6020.58	0.815980	0.407990	−	0.912986i	$-0.366230\pi$
$379$	6020.58	0.815980	0.407990	+	0.912986i	$0.366230\pi$
$380$	0	0
$381$	4193.86	0.563931
$382$	0	0
$383$	8536.93	1.13895	0.569474	−	0.822010i	$-0.307147\pi$
$383$	8536.93	1.13895	0.569474	+	0.822010i	$0.307147\pi$
$384$	0	0
$385$	−10510.0	−1.39127
$386$	0	0
$387$	−2963.47	−0.389254
$388$	0	0
$389$	11385.0	1.48391	0.741954	−	0.670451i	$-0.233899\pi$
$389$	11385.0	1.48391	0.741954	+	0.670451i	$0.233899\pi$
$390$	0	0
$391$	−1548.42	−0.200273
$392$	0	0
$393$	5078.76	0.651882
$394$	0	0
$395$	−3789.56	−0.482718
$396$	0	0
$397$	8515.61	1.07654	0.538270	−	0.842773i	$-0.319078\pi$
$397$	8515.61	1.07654	0.538270	+	0.842773i	$0.319078\pi$
$398$	0	0
$399$	−4042.52	−0.507215
$400$	0	0
$401$	−2276.15	−0.283456	−0.141728	−	0.989906i	$-0.545266\pi$
$401$	−2276.15	−0.283456	−0.141728	+	0.989906i	$0.545266\pi$
$402$	0	0
$403$	959.354	0.118583
$404$	0	0
$405$	875.099	0.107368
$406$	0	0
$407$	12130.6	1.47737
$408$	0	0
$409$	−797.754	−0.0964460	−0.0482230	−	0.998837i	$-0.515356\pi$
$409$	−797.754	−0.0964460	−0.0482230	+	0.998837i	$0.515356\pi$
$410$	0	0
$411$	−411.101	−0.0493384
$412$	0	0
$413$	−11319.0	−1.34860
$414$	0	0
$415$	−4549.29	−0.538111
$416$	0	0
$417$	−7759.59	−0.911244
$418$	0	0
$419$	14933.0	1.74111	0.870557	−	0.492067i	$-0.163759\pi$
$419$	14933.0	1.74111	0.870557	+	0.492067i	$0.163759\pi$
$420$	0	0
$421$	73.7508	0.00853775	0.00426888	−	0.999991i	$-0.498641\pi$
$421$	73.7508	0.00853775	0.00426888	+	0.999991i	$0.498641\pi$
$422$	0	0
$423$	−406.235	−0.0466947
$424$	0	0
$425$	−281.533	−0.0321326
$426$	0	0
$427$	−28070.3	−3.18131
$428$	0	0
$429$	−1178.66	−0.132648
$430$	0	0
$431$	−3233.48	−0.361371	−0.180686	−	0.983541i	$-0.557832\pi$
$431$	−3233.48	−0.361371	−0.180686	+	0.983541i	$0.557832\pi$
$432$	0	0
$433$	−12812.8	−1.42204	−0.711019	−	0.703172i	$-0.751766\pi$
$433$	−12812.8	−1.42204	−0.711019	+	0.703172i	$0.751766\pi$
$434$	0	0
$435$	78.0141	0.00859883
$436$	0	0
$437$	−1906.47	−0.208693
$438$	0	0
$439$	6964.75	0.757197	0.378598	−	0.925561i	$-0.376406\pi$
$439$	6964.75	0.757197	0.378598	+	0.925561i	$0.376406\pi$
$440$	0	0
$441$	6238.25	0.673605
$442$	0	0
$443$	−1254.97	−0.134594	−0.0672971	−	0.997733i	$-0.521438\pi$
$443$	−1254.97	−0.134594	−0.0672971	+	0.997733i	$0.521438\pi$
$444$	0	0
$445$	−14755.2	−1.57183
$446$	0	0
$447$	−5733.57	−0.606686
$448$	0	0
$449$	−9871.71	−1.03758	−0.518791	−	0.854901i	$-0.673618\pi$
$449$	−9871.71	−1.03758	−0.518791	+	0.854901i	$0.673618\pi$
$450$	0	0
$451$	10687.1	1.11582
$452$	0	0
$453$	−6022.41	−0.624630
$454$	0	0
$455$	−4520.89	−0.465808
$456$	0	0
$457$	−12859.1	−1.31625	−0.658123	−	0.752911i	$-0.728649\pi$
$457$	−12859.1	−1.31625	−0.658123	+	0.752911i	$0.728649\pi$
$458$	0	0
$459$	−918.000	−0.0933520
$460$	0	0
$461$	1632.36	0.164916	0.0824582	−	0.996595i	$-0.473723\pi$
$461$	1632.36	0.164916	0.0824582	+	0.996595i	$0.473723\pi$
$462$	0	0
$463$	−5566.52	−0.558743	−0.279372	−	0.960183i	$-0.590126\pi$
$463$	−5566.52	−0.558743	−0.279372	+	0.960183i	$0.590126\pi$
$464$	0	0
$465$	2391.82	0.238534
$466$	0	0
$467$	2849.53	0.282357	0.141178	−	0.989984i	$-0.454911\pi$
$467$	2849.53	0.282357	0.141178	+	0.989984i	$0.454911\pi$
$468$	0	0
$469$	−5712.38	−0.562417
$470$	0	0
$471$	9201.78	0.900203
$472$	0	0
$473$	9951.30	0.967360
$474$	0	0
$475$	−346.634	−0.0334835
$476$	0	0
$477$	−4047.85	−0.388550
$478$	0	0
$479$	7030.66	0.670645	0.335323	−	0.942103i	$-0.391155\pi$
$479$	7030.66	0.670645	0.335323	+	0.942103i	$0.391155\pi$
$480$	0	0
$481$	5217.98	0.494635
$482$	0	0
$483$	4397.84	0.414304
$484$	0	0
$485$	5058.35	0.473583
$486$	0	0
$487$	16972.0	1.57921	0.789604	−	0.613617i	$-0.210286\pi$
$487$	16972.0	1.57921	0.789604	+	0.613617i	$0.210286\pi$
$488$	0	0
$489$	−3566.58	−0.329829
$490$	0	0
$491$	9969.09	0.916291	0.458145	−	0.888877i	$-0.348514\pi$
$491$	9969.09	0.916291	0.458145	+	0.888877i	$0.348514\pi$
$492$	0	0
$493$	−81.8387	−0.00747633
$494$	0	0
$495$	−2938.57	−0.266826
$496$	0	0
$497$	−1060.16	−0.0956831
$498$	0	0
$499$	−10962.9	−0.983500	−0.491750	−	0.870736i	$-0.663643\pi$
$499$	−10962.9	−0.983500	−0.491750	+	0.870736i	$0.663643\pi$
$500$	0	0
$501$	−9975.68	−0.889582
$502$	0	0
$503$	−4164.12	−0.369123	−0.184562	−	0.982821i	$-0.559087\pi$
$503$	−4164.12	−0.369123	−0.184562	+	0.982821i	$0.559087\pi$
$504$	0	0
$505$	−5922.57	−0.521883
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−15477.5	−1.34779	−0.673897	−	0.738826i	$-0.735381\pi$
$509$	−15477.5	−1.34779	−0.673897	+	0.738826i	$0.735381\pi$
$510$	0	0
$511$	25040.1	2.16773
$512$	0	0
$513$	−1130.28	−0.0972767
$514$	0	0
$515$	9909.14	0.847862
$516$	0	0
$517$	1364.14	0.116044
$518$	0	0
$519$	−4356.78	−0.368480
$520$	0	0
$521$	8442.92	0.709964	0.354982	−	0.934873i	$-0.384487\pi$
$521$	8442.92	0.709964	0.354982	+	0.934873i	$0.384487\pi$
$522$	0	0
$523$	9748.73	0.815071	0.407535	−	0.913189i	$-0.366388\pi$
$523$	9748.73	0.815071	0.407535	+	0.913189i	$0.366388\pi$
$524$	0	0
$525$	799.614	0.0664724
$526$	0	0
$527$	−2509.08	−0.207395
$528$	0	0
$529$	−10093.0	−0.829535
$530$	0	0
$531$	−3164.76	−0.258642
$532$	0	0
$533$	4597.05	0.373584
$534$	0	0
$535$	13823.7	1.11710
$536$	0	0
$537$	−7387.54	−0.593660
$538$	0	0
$539$	−20948.0	−1.67402
$540$	0	0
$541$	12880.6	1.02362	0.511812	−	0.859097i	$-0.328974\pi$
$541$	12880.6	1.02362	0.511812	+	0.859097i	$0.328974\pi$
$542$	0	0
$543$	8449.52	0.667778
$544$	0	0
$545$	8138.44	0.639656
$546$	0	0
$547$	4314.68	0.337262	0.168631	−	0.985679i	$-0.446065\pi$
$547$	4314.68	0.337262	0.168631	+	0.985679i	$0.446065\pi$
$548$	0	0
$549$	−7848.39	−0.610130
$550$	0	0
$551$	−100.763	−0.00779065
$552$	0	0
$553$	−11290.8	−0.868238
$554$	0	0
$555$	13009.2	0.994976
$556$	0	0
$557$	4145.64	0.315361	0.157681	−	0.987490i	$-0.449598\pi$
$557$	4145.64	0.315361	0.157681	+	0.987490i	$0.449598\pi$
$558$	0	0
$559$	4280.56	0.323879
$560$	0	0
$561$	3082.64	0.231995
$562$	0	0
$563$	−8390.08	−0.628063	−0.314032	−	0.949413i	$-0.601680\pi$
$563$	−8390.08	−0.628063	−0.314032	+	0.949413i	$0.601680\pi$
$564$	0	0
$565$	−12913.5	−0.961551
$566$	0	0
$567$	2607.32	0.193117
$568$	0	0
$569$	7739.75	0.570241	0.285121	−	0.958492i	$-0.407966\pi$
$569$	7739.75	0.570241	0.285121	+	0.958492i	$0.407966\pi$
$570$	0	0
$571$	2853.05	0.209101	0.104550	−	0.994520i	$-0.466660\pi$
$571$	2853.05	0.209101	0.104550	+	0.994520i	$0.466660\pi$
$572$	0	0
$573$	−11693.8	−0.852558
$574$	0	0
$575$	377.103	0.0273500
$576$	0	0
$577$	14334.9	1.03426	0.517131	−	0.855906i	$-0.327000\pi$
$577$	14334.9	1.03426	0.517131	+	0.855906i	$0.327000\pi$
$578$	0	0
$579$	−11718.4	−0.841107
$580$	0	0
$581$	−13554.4	−0.967870
$582$	0	0
$583$	13592.7	0.965609
$584$	0	0
$585$	−1264.03	−0.0893355
$586$	0	0
$587$	−20453.1	−1.43814	−0.719071	−	0.694937i	$-0.755432\pi$
$587$	−20453.1	−1.43814	−0.719071	+	0.694937i	$0.755432\pi$
$588$	0	0
$589$	−3089.28	−0.216115
$590$	0	0
$591$	8837.85	0.615128
$592$	0	0
$593$	−15117.6	−1.04689	−0.523443	−	0.852061i	$-0.675353\pi$
$593$	−15117.6	−1.04689	−0.523443	+	0.852061i	$0.675353\pi$
$594$	0	0
$595$	11823.9	0.814675
$596$	0	0
$597$	−1607.38	−0.110194
$598$	0	0
$599$	−13786.3	−0.940388	−0.470194	−	0.882563i	$-0.655816\pi$
$599$	−13786.3	−0.940388	−0.470194	+	0.882563i	$0.655816\pi$
$600$	0	0
$601$	28653.1	1.94474	0.972368	−	0.233454i	$-0.0750028\pi$
$601$	28653.1	1.94474	0.972368	+	0.233454i	$0.0750028\pi$
$602$	0	0
$603$	−1597.17	−0.107864
$604$	0	0
$605$	−4511.99	−0.303204
$606$	0	0
$607$	−5271.71	−0.352508	−0.176254	−	0.984345i	$-0.556398\pi$
$607$	−5271.71	−0.352508	−0.176254	+	0.984345i	$0.556398\pi$
$608$	0	0
$609$	232.440	0.0154662
$610$	0	0
$611$	586.785	0.0388523
$612$	0	0
$613$	−14763.7	−0.972760	−0.486380	−	0.873747i	$-0.661683\pi$
$613$	−14763.7	−0.972760	−0.486380	+	0.873747i	$0.661683\pi$
$614$	0	0
$615$	11461.2	0.751479
$616$	0	0
$617$	16934.0	1.10492	0.552461	−	0.833539i	$-0.313689\pi$
$617$	16934.0	1.10492	0.552461	+	0.833539i	$0.313689\pi$
$618$	0	0
$619$	19432.6	1.26181	0.630906	−	0.775859i	$-0.282683\pi$
$619$	19432.6	1.26181	0.630906	+	0.775859i	$0.282683\pi$
$620$	0	0
$621$	1229.63	0.0794577
$622$	0	0
$623$	−43962.6	−2.82716
$624$	0	0
$625$	−14521.4	−0.929369
$626$	0	0
$627$	3795.46	0.241748
$628$	0	0
$629$	−13647.0	−0.865091
$630$	0	0
$631$	−11565.8	−0.729680	−0.364840	−	0.931070i	$-0.618876\pi$
$631$	−11565.8	−0.729680	−0.364840	+	0.931070i	$0.618876\pi$
$632$	0	0
$633$	−11065.0	−0.694779
$634$	0	0
$635$	−15103.0	−0.943852
$636$	0	0
$637$	−9010.81	−0.560473
$638$	0	0
$639$	−296.417	−0.0183507
$640$	0	0
$641$	−17372.8	−1.07049	−0.535244	−	0.844698i	$-0.679780\pi$
$641$	−17372.8	−1.07049	−0.535244	+	0.844698i	$0.679780\pi$
$642$	0	0
$643$	26204.3	1.60715	0.803574	−	0.595205i	$-0.202929\pi$
$643$	26204.3	1.60715	0.803574	+	0.595205i	$0.202929\pi$
$644$	0	0
$645$	10672.1	0.651495
$646$	0	0
$647$	−18498.7	−1.12405	−0.562025	−	0.827120i	$-0.689978\pi$
$647$	−18498.7	−1.12405	−0.562025	+	0.827120i	$0.689978\pi$
$648$	0	0
$649$	10627.3	0.642768
$650$	0	0
$651$	7126.33	0.429037
$652$	0	0
$653$	15631.9	0.936790	0.468395	−	0.883519i	$-0.344832\pi$
$653$	15631.9	0.936790	0.468395	+	0.883519i	$0.344832\pi$
$654$	0	0
$655$	−18289.8	−1.09105
$656$	0	0
$657$	7001.15	0.415740
$658$	0	0
$659$	2057.87	0.121644	0.0608218	−	0.998149i	$-0.480628\pi$
$659$	2057.87	0.121644	0.0608218	+	0.998149i	$0.480628\pi$
$660$	0	0
$661$	27657.5	1.62746	0.813730	−	0.581244i	$-0.197434\pi$
$661$	27657.5	1.62746	0.813730	+	0.581244i	$0.197434\pi$
$662$	0	0
$663$	1326.00	0.0776736
$664$	0	0
$665$	14558.0	0.848926
$666$	0	0
$667$	109.620	0.00636357
$668$	0	0
$669$	7151.32	0.413283
$670$	0	0
$671$	26354.9	1.51627
$672$	0	0
$673$	18122.1	1.03797	0.518987	−	0.854782i	$-0.326309\pi$
$673$	18122.1	1.03797	0.518987	+	0.854782i	$0.326309\pi$
$674$	0	0
$675$	223.570	0.0127485
$676$	0	0
$677$	−17210.1	−0.977012	−0.488506	−	0.872560i	$-0.662458\pi$
$677$	−17210.1	−0.977012	−0.488506	+	0.872560i	$0.662458\pi$
$678$	0	0
$679$	15071.1	0.851808
$680$	0	0
$681$	−16963.5	−0.954540
$682$	0	0
$683$	−15770.7	−0.883525	−0.441762	−	0.897132i	$-0.645647\pi$
$683$	−15770.7	−0.883525	−0.441762	+	0.897132i	$0.645647\pi$
$684$	0	0
$685$	1480.47	0.0825777
$686$	0	0
$687$	−14198.3	−0.788500
$688$	0	0
$689$	5846.89	0.323293
$690$	0	0
$691$	28419.8	1.56460	0.782301	−	0.622901i	$-0.214046\pi$
$691$	28419.8	1.56460	0.782301	+	0.622901i	$0.214046\pi$
$692$	0	0
$693$	−8755.36	−0.479926
$694$	0	0
$695$	27944.1	1.52515
$696$	0	0
$697$	−12023.1	−0.653380
$698$	0	0
$699$	3854.78	0.208586
$700$	0	0
$701$	9186.30	0.494952	0.247476	−	0.968894i	$-0.420399\pi$
$701$	9186.30	0.494952	0.247476	+	0.968894i	$0.420399\pi$
$702$	0	0
$703$	−16802.7	−0.901461
$704$	0	0
$705$	1462.95	0.0781528
$706$	0	0
$707$	−17646.0	−0.938681
$708$	0	0
$709$	12818.9	0.679018	0.339509	−	0.940603i	$-0.389739\pi$
$709$	12818.9	0.679018	0.339509	+	0.940603i	$0.389739\pi$
$710$	0	0
$711$	−3156.89	−0.166516
$712$	0	0
$713$	3360.82	0.176527
$714$	0	0
$715$	4244.61	0.222013
$716$	0	0
$717$	13976.3	0.727969
$718$	0	0
$719$	20883.9	1.08323	0.541613	−	0.840628i	$-0.317814\pi$
$719$	20883.9	1.08323	0.541613	+	0.840628i	$0.317814\pi$
$720$	0	0
$721$	29523.9	1.52500
$722$	0	0
$723$	−6501.26	−0.334418
$724$	0	0
$725$	19.9311	0.00102099
$726$	0	0
$727$	−898.193	−0.0458214	−0.0229107	−	0.999738i	$-0.507293\pi$
$727$	−898.193	−0.0458214	−0.0229107	+	0.999738i	$0.507293\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−11195.3	−0.566448
$732$	0	0
$733$	−21008.4	−1.05861	−0.529306	−	0.848431i	$-0.677548\pi$
$733$	−21008.4	−1.05861	−0.529306	+	0.848431i	$0.677548\pi$
$734$	0	0
$735$	−22465.4	−1.12741
$736$	0	0
$737$	5363.28	0.268058
$738$	0	0
$739$	−17326.9	−0.862490	−0.431245	−	0.902235i	$-0.641925\pi$
$739$	−17326.9	−0.862490	−0.431245	+	0.902235i	$0.641925\pi$
$740$	0	0
$741$	1632.62	0.0809391
$742$	0	0
$743$	−4113.25	−0.203096	−0.101548	−	0.994831i	$-0.532380\pi$
$743$	−4113.25	−0.203096	−0.101548	+	0.994831i	$0.532380\pi$
$744$	0	0
$745$	20647.9	1.01541
$746$	0	0
$747$	−3789.79	−0.185624
$748$	0	0
$749$	41187.1	2.00927
$750$	0	0
$751$	−15309.1	−0.743857	−0.371928	−	0.928261i	$-0.621303\pi$
$751$	−15309.1	−0.743857	−0.371928	+	0.928261i	$0.621303\pi$
$752$	0	0
$753$	−15657.7	−0.757768
$754$	0	0
$755$	21688.1	1.04544
$756$	0	0
$757$	32489.1	1.55989	0.779944	−	0.625850i	$-0.215248\pi$
$757$	32489.1	1.55989	0.779944	+	0.625850i	$0.215248\pi$
$758$	0	0
$759$	−4129.08	−0.197465
$760$	0	0
$761$	23935.8	1.14017	0.570087	−	0.821585i	$-0.306910\pi$
$761$	23935.8	1.14017	0.570087	+	0.821585i	$0.306910\pi$
$762$	0	0
$763$	24248.1	1.15051
$764$	0	0
$765$	3305.93	0.156243
$766$	0	0
$767$	4571.33	0.215203
$768$	0	0
$769$	7627.82	0.357693	0.178847	−	0.983877i	$-0.442763\pi$
$769$	7627.82	0.357693	0.178847	+	0.983877i	$0.442763\pi$
$770$	0	0
$771$	−1198.97	−0.0560051
$772$	0	0
$773$	−11075.3	−0.515332	−0.257666	−	0.966234i	$-0.582953\pi$
$773$	−11075.3	−0.515332	−0.257666	+	0.966234i	$0.582953\pi$
$774$	0	0
$775$	611.063	0.0283226
$776$	0	0
$777$	38760.5	1.78961
$778$	0	0
$779$	−14803.3	−0.680850
$780$	0	0
$781$	995.367	0.0456044
$782$	0	0
$783$	64.9896	0.00296620
$784$	0	0
$785$	−33137.7	−1.50667
$786$	0	0
$787$	−12170.8	−0.551260	−0.275630	−	0.961264i	$-0.588887\pi$
$787$	−12170.8	−0.551260	−0.275630	+	0.961264i	$0.588887\pi$
$788$	0	0
$789$	24136.9	1.08910
$790$	0	0
$791$	−38475.3	−1.72949
$792$	0	0
$793$	11336.6	0.507659
$794$	0	0
$795$	14577.2	0.650316
$796$	0	0
$797$	−24434.7	−1.08597	−0.542986	−	0.839742i	$-0.682706\pi$
$797$	−24434.7	−1.08597	−0.542986	+	0.839742i	$0.682706\pi$
$798$	0	0
$799$	−1534.67	−0.0679507
$800$	0	0
$801$	−12291.8	−0.542210
$802$	0	0
$803$	−23509.8	−1.03318
$804$	0	0
$805$	−15837.6	−0.693421
$806$	0	0
$807$	−6784.23	−0.295931
$808$	0	0
$809$	12303.4	0.534690	0.267345	−	0.963601i	$-0.413854\pi$
$809$	12303.4	0.534690	0.267345	+	0.963601i	$0.413854\pi$
$810$	0	0
$811$	−12491.2	−0.540845	−0.270422	−	0.962742i	$-0.587163\pi$
$811$	−12491.2	−0.540845	−0.270422	+	0.962742i	$0.587163\pi$
$812$	0	0
$813$	18005.1	0.776710
$814$	0	0
$815$	12844.1	0.552035
$816$	0	0
$817$	−13784.1	−0.590263
$818$	0	0
$819$	−3766.13	−0.160683
$820$	0	0
$821$	−14520.7	−0.617268	−0.308634	−	0.951181i	$-0.599872\pi$
$821$	−14520.7	−0.617268	−0.308634	+	0.951181i	$0.599872\pi$
$822$	0	0
$823$	12198.7	0.516670	0.258335	−	0.966055i	$-0.416826\pi$
$823$	12198.7	0.516670	0.258335	+	0.966055i	$0.416826\pi$
$824$	0	0
$825$	−750.747	−0.0316820
$826$	0	0
$827$	−33507.8	−1.40892	−0.704462	−	0.709742i	$-0.748812\pi$
$827$	−33507.8	−1.40892	−0.704462	+	0.709742i	$0.748812\pi$
$828$	0	0
$829$	−59.8986	−0.00250949	−0.00125474	−	0.999999i	$-0.500399\pi$
$829$	−59.8986	−0.00250949	−0.00125474	+	0.999999i	$0.500399\pi$
$830$	0	0
$831$	−5281.18	−0.220460
$832$	0	0
$833$	23566.7	0.980239
$834$	0	0
$835$	35924.7	1.48889
$836$	0	0
$837$	1992.51	0.0822832
$838$	0	0
$839$	−6541.95	−0.269193	−0.134597	−	0.990900i	$-0.542974\pi$
$839$	−6541.95	−0.269193	−0.134597	+	0.990900i	$0.542974\pi$
$840$	0	0
$841$	−24383.2	−0.999762
$842$	0	0
$843$	18981.2	0.775500
$844$	0	0
$845$	1825.82	0.0743316
$846$	0	0
$847$	−13443.3	−0.545356
$848$	0	0
$849$	−18593.4	−0.751618
$850$	0	0
$851$	18279.7	0.736333
$852$	0	0
$853$	39815.6	1.59819	0.799097	−	0.601202i	$-0.205311\pi$
$853$	39815.6	1.59819	0.799097	+	0.601202i	$0.205311\pi$
$854$	0	0
$855$	4070.39	0.162812
$856$	0	0
$857$	−22194.9	−0.884671	−0.442335	−	0.896850i	$-0.645850\pi$
$857$	−22194.9	−0.884671	−0.442335	+	0.896850i	$0.645850\pi$
$858$	0	0
$859$	−31640.5	−1.25676	−0.628381	−	0.777905i	$-0.716282\pi$
$859$	−31640.5	−1.25676	−0.628381	+	0.777905i	$0.716282\pi$
$860$	0	0
$861$	34148.1	1.35164
$862$	0	0
$863$	−33171.9	−1.30844	−0.654220	−	0.756304i	$-0.727003\pi$
$863$	−33171.9	−1.30844	−0.654220	+	0.756304i	$0.727003\pi$
$864$	0	0
$865$	15689.7	0.616725
$866$	0	0
$867$	11271.0	0.441503
$868$	0	0
$869$	10600.8	0.413818
$870$	0	0
$871$	2307.02	0.0897479
$872$	0	0
$873$	4213.86	0.163365
$874$	0	0
$875$	−46349.7	−1.79075
$876$	0	0
$877$	−5351.15	−0.206038	−0.103019	−	0.994679i	$-0.532850\pi$
$877$	−5351.15	−0.206038	−0.103019	+	0.994679i	$0.532850\pi$
$878$	0	0
$879$	23636.8	0.906996
$880$	0	0
$881$	−2731.65	−0.104462	−0.0522312	−	0.998635i	$-0.516633\pi$
$881$	−2731.65	−0.104462	−0.0522312	+	0.998635i	$0.516633\pi$
$882$	0	0
$883$	6111.50	0.232920	0.116460	−	0.993195i	$-0.462845\pi$
$883$	6111.50	0.232920	0.116460	+	0.993195i	$0.462845\pi$
$884$	0	0
$885$	11397.0	0.432890
$886$	0	0
$887$	−7214.75	−0.273109	−0.136554	−	0.990633i	$-0.543603\pi$
$887$	−7214.75	−0.273109	−0.136554	+	0.990633i	$0.543603\pi$
$888$	0	0
$889$	−44998.9	−1.69765
$890$	0	0
$891$	−2447.98	−0.0920430
$892$	0	0
$893$	−1889.54	−0.0708075
$894$	0	0
$895$	26604.2	0.993609
$896$	0	0
$897$	−1776.13	−0.0661128
$898$	0	0
$899$	177.630	0.00658986
$900$	0	0
$901$	−15291.9	−0.565423
$902$	0	0
$903$	31797.1	1.17181
$904$	0	0
$905$	−30428.7	−1.11766
$906$	0	0
$907$	27365.0	1.00181	0.500905	−	0.865502i	$-0.333001\pi$
$907$	27365.0	1.00181	0.500905	+	0.865502i	$0.333001\pi$
$908$	0	0
$909$	−4933.79	−0.180026
$910$	0	0
$911$	39873.8	1.45014	0.725070	−	0.688675i	$-0.241807\pi$
$911$	39873.8	1.45014	0.725070	+	0.688675i	$0.241807\pi$
$912$	0	0
$913$	12726.1	0.461305
$914$	0	0
$915$	28263.9	1.02117
$916$	0	0
$917$	−54493.6	−1.96242
$918$	0	0
$919$	37427.3	1.34343	0.671715	−	0.740810i	$-0.265558\pi$
$919$	37427.3	1.34343	0.671715	+	0.740810i	$0.265558\pi$
$920$	0	0
$921$	10510.9	0.376056
$922$	0	0
$923$	428.158	0.0152687
$924$	0	0
$925$	3323.60	0.118140
$926$	0	0
$927$	8254.80	0.292474
$928$	0	0
$929$	−50172.4	−1.77191	−0.885954	−	0.463774i	$-0.846495\pi$
$929$	−50172.4	−1.77191	−0.885954	+	0.463774i	$0.846495\pi$
$930$	0	0
$931$	29016.3	1.02145
$932$	0	0
$933$	27894.7	0.978813
$934$	0	0
$935$	−11101.3	−0.388290
$936$	0	0
$937$	23584.4	0.822270	0.411135	−	0.911574i	$-0.365132\pi$
$937$	23584.4	0.822270	0.411135	+	0.911574i	$0.365132\pi$
$938$	0	0
$939$	−21109.3	−0.733627
$940$	0	0
$941$	38535.5	1.33499	0.667493	−	0.744616i	$-0.267368\pi$
$941$	38535.5	1.33499	0.667493	+	0.744616i	$0.267368\pi$
$942$	0	0
$943$	16104.4	0.556132
$944$	0	0
$945$	−9389.55	−0.323219
$946$	0	0
$947$	−35237.3	−1.20914	−0.604572	−	0.796550i	$-0.706656\pi$
$947$	−35237.3	−1.20914	−0.604572	+	0.796550i	$0.706656\pi$
$948$	0	0
$949$	−10112.8	−0.345916
$950$	0	0
$951$	33141.7	1.13007
$952$	0	0
$953$	14173.4	0.481765	0.240883	−	0.970554i	$-0.422563\pi$
$953$	14173.4	0.481765	0.240883	+	0.970554i	$0.422563\pi$
$954$	0	0
$955$	42112.1	1.42693
$956$	0	0
$957$	−218.235	−0.00737150
$958$	0	0
$959$	4410.99	0.148528
$960$	0	0
$961$	−24345.1	−0.817196
$962$	0	0
$963$	11515.8	0.385350
$964$	0	0
$965$	42200.7	1.40776
$966$	0	0
$967$	−22532.8	−0.749334	−0.374667	−	0.927159i	$-0.622243\pi$
$967$	−22532.8	−0.749334	−0.374667	+	0.927159i	$0.622243\pi$
$968$	0	0
$969$	−4269.94	−0.141558
$970$	0	0
$971$	21753.8	0.718962	0.359481	−	0.933152i	$-0.382954\pi$
$971$	21753.8	0.718962	0.359481	+	0.933152i	$0.382954\pi$
$972$	0	0
$973$	83258.1	2.74320
$974$	0	0
$975$	−322.935	−0.0106074
$976$	0	0
$977$	19179.7	0.628059	0.314030	−	0.949413i	$-0.398321\pi$
$977$	19179.7	0.628059	0.314030	+	0.949413i	$0.398321\pi$
$978$	0	0
$979$	41275.9	1.34748
$980$	0	0
$981$	6779.72	0.220652
$982$	0	0
$983$	−51940.4	−1.68529	−0.842646	−	0.538468i	$-0.819003\pi$
$983$	−51940.4	−1.68529	−0.842646	+	0.538468i	$0.819003\pi$
$984$	0	0
$985$	−31827.1	−1.02954
$986$	0	0
$987$	4358.79	0.140569
$988$	0	0
$989$	14995.7	0.482139
$990$	0	0
$991$	−49237.5	−1.57828	−0.789142	−	0.614210i	$-0.789475\pi$
$991$	−49237.5	−1.57828	−0.789142	+	0.614210i	$0.789475\pi$
$992$	0	0
$993$	15226.4	0.486600
$994$	0	0
$995$	5788.55	0.184432
$996$	0	0
$997$	−1110.55	−0.0352774	−0.0176387	−	0.999844i	$-0.505615\pi$
$997$	−1110.55	−0.0352774	−0.0176387	+	0.999844i	$0.505615\pi$
$998$	0	0
$999$	10837.3	0.343221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bk.1.3		3
4.3	odd	2		2496.4.a.bo.1.3		3
8.3	odd	2		312.4.a.g.1.1	✓	3
8.5	even	2		624.4.a.u.1.1		3
24.5	odd	2		1872.4.a.bj.1.3		3
24.11	even	2		936.4.a.k.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.g.1.1	✓	3	8.3	odd	2
624.4.a.u.1.1		3	8.5	even	2
936.4.a.k.1.3		3	24.11	even	2
1872.4.a.bj.1.3		3	24.5	odd	2
2496.4.a.bk.1.3		3	1.1	even	1	trivial
2496.4.a.bo.1.3		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +10.8037 q^{5} +32.1891 q^{7} +9.00000 q^{9} -30.2219 q^{11} -13.0000 q^{13} -32.4111 q^{15} +34.0000 q^{17} +41.8621 q^{19} -96.5674 q^{21} -45.5417 q^{23} -8.28038 q^{25} -27.0000 q^{27} -2.40702 q^{29} -73.7965 q^{31} +90.6658 q^{33} +347.761 q^{35} -401.383 q^{37} +39.0000 q^{39} -353.619 q^{41} -329.274 q^{43} +97.2332 q^{45} -45.1373 q^{47} +693.139 q^{49} -102.000 q^{51} -449.761 q^{53} -326.508 q^{55} -125.586 q^{57} -351.640 q^{59} -872.044 q^{61} +289.702 q^{63} -140.448 q^{65} -177.463 q^{67} +136.625 q^{69} -32.9352 q^{71} +777.906 q^{73} +24.8411 q^{75} -972.818 q^{77} -350.766 q^{79} +81.0000 q^{81} -421.087 q^{83} +367.325 q^{85} +7.22106 q^{87} -1365.76 q^{89} -418.459 q^{91} +221.389 q^{93} +452.265 q^{95} +468.206 q^{97} -271.997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 16 q^{5} + 22 q^{7} + 27 q^{9} - 20 q^{11} - 39 q^{13} + 48 q^{15} + 102 q^{17} - 38 q^{19} - 66 q^{21} - 32 q^{23} + 161 q^{25} - 81 q^{27} - 350 q^{29} - 50 q^{31} + 60 q^{33} + 232 q^{35}+ \cdots - 180 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 16 * q^5 + 22 * q^7 + 27 * q^9 - 20 * q^11 - 39 * q^13 + 48 * q^15 + 102 * q^17 - 38 * q^19 - 66 * q^21 - 32 * q^23 + 161 * q^25 - 81 * q^27 - 350 * q^29 - 50 * q^31 + 60 * q^33 + 232 * q^35 - 542 * q^37 + 117 * q^39 + 500 * q^41 + 420 * q^43 - 144 * q^45 - 324 * q^47 + 1119 * q^49 - 306 * q^51 - 538 * q^53 - 896 * q^55 + 114 * q^57 + 112 * q^59 - 1206 * q^61 + 198 * q^63 + 208 * q^65 + 950 * q^67 + 96 * q^69 + 1008 * q^71 + 1834 * q^73 - 483 * q^75 + 792 * q^77 + 272 * q^79 + 243 * q^81 - 1640 * q^83 - 544 * q^85 + 1050 * q^87 - 1576 * q^89 - 286 * q^91 + 150 * q^93 + 2760 * q^95 + 402 * q^97 - 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more