LMFDB - Embedded newform 312.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, 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("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.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:	$18.4085959218$
Analytic rank:	$0$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.84636$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-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	$-0.660509\pi$
$5$	−10.8037	−0.966311	−0.483156	+	0.875535i	$0.660509\pi$
$6$	0	0
$7$	−32.1891	−1.73805	−0.869025	−	0.494769i	$-0.835253\pi$
$7$	−32.1891	−1.73805	−0.869025	+	0.494769i	$0.835253\pi$
$8$	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	$-0.433813\pi$
$23$	45.5417	0.412874	0.206437	+	0.978460i	$0.433813\pi$
$24$	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	$-0.497547\pi$
$29$	2.40702	0.0154128	0.00770642	+	0.999970i	$0.497547\pi$
$30$	0	0
$31$	73.7965	0.427556	0.213778	−	0.976882i	$-0.431423\pi$
$31$	73.7965	0.427556	0.213778	+	0.976882i	$0.431423\pi$
$32$	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	$-0.149502\pi$
$37$	401.383	1.78343	0.891715	+	0.452597i	$0.149502\pi$
$38$	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	$-0.477687\pi$
$47$	45.1373	0.140084	0.0700420	+	0.997544i	$0.477687\pi$
$48$	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	$-0.301947\pi$
$53$	449.761	1.16565	0.582825	+	0.812598i	$0.301947\pi$
$54$	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	$-0.132037\pi$
$61$	872.044	1.83039	0.915195	+	0.403012i	$0.132037\pi$
$62$	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	$-0.491237\pi$
$71$	32.9352	0.0550520	0.0275260	+	0.999621i	$0.491237\pi$
$72$	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	$-0.419644\pi$
$79$	350.766	0.499547	0.249774	+	0.968304i	$0.419644\pi$
$80$	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	$-0.412964\pi$
$101$	548.199	0.540077	0.270039	+	0.962849i	$0.412964\pi$
$102$	0	0
$103$	−917.200	−0.877422	−0.438711	−	0.898628i	$-0.644565\pi$
$103$	−917.200	−0.877422	−0.438711	+	0.898628i	$0.644565\pi$
$104$	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	$-0.607379\pi$
$109$	−753.302	−0.661957	−0.330978	+	0.943638i	$0.607379\pi$
$110$	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	$-0.337588\pi$
$127$	1397.95	0.976758	0.488379	+	0.872632i	$0.337588\pi$
$128$	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	$-0.676086\pi$
$149$	−1911.19	−1.05081	−0.525405	+	0.850852i	$0.676086\pi$
$150$	0	0
$151$	−2007.47	−1.08189	−0.540946	−	0.841057i	$-0.681934\pi$
$151$	−2007.47	−1.08189	−0.540946	+	0.841057i	$0.681934\pi$
$152$	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	$-0.215423\pi$
$157$	3067.26	1.55920	0.779599	+	0.626280i	$0.215423\pi$
$158$	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	$-0.779944\pi$
$167$	−3325.23	−1.54080	−0.770401	+	0.637560i	$0.779944\pi$
$168$	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	$-0.603385\pi$
$173$	−1452.26	−0.638226	−0.319113	+	0.947717i	$0.603385\pi$
$174$	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	$-0.303711\pi$
$181$	2816.51	1.15663	0.578313	+	0.815815i	$0.303711\pi$
$182$	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	$-0.764389\pi$
$191$	−3897.94	−1.47667	−0.738337	+	0.674432i	$0.764389\pi$
$192$	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	$-0.321171\pi$
$197$	2945.95	1.06543	0.532717	+	0.846294i	$0.321171\pi$
$198$	0	0
$199$	−535.794	−0.190862	−0.0954308	−	0.995436i	$-0.530423\pi$
$199$	−535.794	−0.190862	−0.0954308	+	0.995436i	$0.530423\pi$
$200$	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	$-0.383488\pi$
$223$	2383.77	0.715827	0.357913	+	0.933755i	$0.383488\pi$
$224$	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	$-0.739265\pi$
$229$	−4732.77	−1.36572	−0.682861	+	0.730548i	$0.739265\pi$
$230$	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	$-0.282875\pi$
$239$	4658.76	1.26088	0.630440	+	0.776238i	$0.282875\pi$
$240$	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	$-0.107813\pi$
$263$	8045.64	1.88637	0.943186	+	0.332267i	$0.107813\pi$
$264$	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	$-0.582498\pi$
$269$	−2261.41	−0.512568	−0.256284	+	0.966602i	$0.582498\pi$
$270$	0	0
$271$	6001.69	1.34530	0.672651	−	0.739960i	$-0.265156\pi$
$271$	6001.69	1.34530	0.672651	+	0.739960i	$0.265156\pi$
$272$	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	$-0.561148\pi$
$277$	−1760.39	−0.381848	−0.190924	+	0.981605i	$0.561148\pi$
$278$	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	$-0.212415\pi$
$293$	7878.93	1.57096	0.785482	+	0.618885i	$0.212415\pi$
$294$	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	$-0.178001\pi$
$311$	9298.24	1.69535	0.847676	+	0.530513i	$0.178001\pi$
$312$	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	$-0.0658690\pi$
$317$	11047.2	1.95733	0.978666	+	0.205460i	$0.0658690\pi$
$318$	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	$-0.412184\pi$
$349$	3551.95	0.544790	0.272395	+	0.962186i	$0.412184\pi$
$350$	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	$-0.467708\pi$
$359$	1377.77	0.202551	0.101275	+	0.994858i	$0.467708\pi$
$360$	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	$-0.508836\pi$
$367$	−390.300	−0.0555136	−0.0277568	+	0.999615i	$0.508836\pi$
$368$	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	$-0.387745\pi$
$373$	4976.32	0.690788	0.345394	+	0.938458i	$0.387745\pi$
$374$	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	$-0.692853\pi$
$383$	−8536.93	−1.13895	−0.569474	+	0.822010i	$0.692853\pi$
$384$	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	$-0.766101\pi$
$389$	−11385.0	−1.48391	−0.741954	+	0.670451i	$0.766101\pi$
$390$	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	$-0.680922\pi$
$397$	−8515.61	−1.07654	−0.538270	+	0.842773i	$0.680922\pi$
$398$	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	$-0.501359\pi$
$421$	−73.7508	−0.00853775	−0.00426888	+	0.999991i	$0.501359\pi$
$422$	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	$-0.442168\pi$
$431$	3233.48	0.361371	0.180686	+	0.983541i	$0.442168\pi$
$432$	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	$-0.623594\pi$
$439$	−6964.75	−0.757197	−0.378598	+	0.925561i	$0.623594\pi$
$440$	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	$-0.526277\pi$
$461$	−1632.36	−0.164916	−0.0824582	+	0.996595i	$0.526277\pi$
$462$	0	0
$463$	5566.52	0.558743	0.279372	−	0.960183i	$-0.409874\pi$
$463$	5566.52	0.558743	0.279372	+	0.960183i	$0.409874\pi$
$464$	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	$-0.608845\pi$
$479$	−7030.66	−0.670645	−0.335323	+	0.942103i	$0.608845\pi$
$480$	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	$-0.789714\pi$
$487$	−16972.0	−1.57921	−0.789604	+	0.613617i	$0.789714\pi$
$488$	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	$-0.440913\pi$
$503$	4164.12	0.369123	0.184562	+	0.982821i	$0.440913\pi$
$504$	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	$-0.264619\pi$
$509$	15477.5	1.34779	0.673897	+	0.738826i	$0.264619\pi$
$510$	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	$-0.671026\pi$
$541$	−12880.6	−1.02362	−0.511812	+	0.859097i	$0.671026\pi$
$542$	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	$-0.550402\pi$
$557$	−4145.64	−0.315361	−0.157681	+	0.987490i	$0.550402\pi$
$558$	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	$-0.344184\pi$
$599$	13786.3	0.940388	0.470194	+	0.882563i	$0.344184\pi$
$600$	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	$-0.443602\pi$
$607$	5271.71	0.352508	0.176254	+	0.984345i	$0.443602\pi$
$608$	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	$-0.338317\pi$
$613$	14763.7	0.972760	0.486380	+	0.873747i	$0.338317\pi$
$614$	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	$-0.381124\pi$
$631$	11565.8	0.729680	0.364840	+	0.931070i	$0.381124\pi$
$632$	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	$-0.310022\pi$
$647$	18498.7	1.12405	0.562025	+	0.827120i	$0.310022\pi$
$648$	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	$-0.655168\pi$
$653$	−15631.9	−0.936790	−0.468395	+	0.883519i	$0.655168\pi$
$654$	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	$-0.802566\pi$
$661$	−27657.5	−1.62746	−0.813730	+	0.581244i	$0.802566\pi$
$662$	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	$-0.337542\pi$
$677$	17210.1	0.977012	0.488506	+	0.872560i	$0.337542\pi$
$678$	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	$-0.579601\pi$
$701$	−9186.30	−0.494952	−0.247476	+	0.968894i	$0.579601\pi$
$702$	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	$-0.610261\pi$
$709$	−12818.9	−0.679018	−0.339509	+	0.940603i	$0.610261\pi$
$710$	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	$-0.682186\pi$
$719$	−20883.9	−1.08323	−0.541613	+	0.840628i	$0.682186\pi$
$720$	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	$-0.492707\pi$
$727$	898.193	0.0458214	0.0229107	+	0.999738i	$0.492707\pi$
$728$	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	$-0.322452\pi$
$733$	21008.4	1.05861	0.529306	+	0.848431i	$0.322452\pi$
$734$	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	$-0.467620\pi$
$743$	4113.25	0.203096	0.101548	+	0.994831i	$0.467620\pi$
$744$	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	$-0.378697\pi$
$751$	15309.1	0.743857	0.371928	+	0.928261i	$0.378697\pi$
$752$	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	$-0.784752\pi$
$757$	−32489.1	−1.55989	−0.779944	+	0.625850i	$0.784752\pi$
$758$	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	$-0.417047\pi$
$773$	11075.3	0.515332	0.257666	+	0.966234i	$0.417047\pi$
$774$	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	$-0.317294\pi$
$797$	24434.7	1.08597	0.542986	+	0.839742i	$0.317294\pi$
$798$	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	$-0.400128\pi$
$821$	14520.7	0.617268	0.308634	+	0.951181i	$0.400128\pi$
$822$	0	0
$823$	−12198.7	−0.516670	−0.258335	−	0.966055i	$-0.583174\pi$
$823$	−12198.7	−0.516670	−0.258335	+	0.966055i	$0.583174\pi$
$824$	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	$-0.499601\pi$
$829$	59.8986	0.00250949	0.00125474	+	0.999999i	$0.499601\pi$
$830$	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	$-0.457026\pi$
$839$	6541.95	0.269193	0.134597	+	0.990900i	$0.457026\pi$
$840$	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	$-0.794689\pi$
$853$	−39815.6	−1.59819	−0.799097	+	0.601202i	$0.794689\pi$
$854$	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	$-0.272997\pi$
$863$	33171.9	1.30844	0.654220	+	0.756304i	$0.272997\pi$
$864$	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	$-0.467150\pi$
$877$	5351.15	0.206038	0.103019	+	0.994679i	$0.467150\pi$
$878$	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	$-0.456397\pi$
$887$	7214.75	0.273109	0.136554	+	0.990633i	$0.456397\pi$
$888$	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	$-0.758193\pi$
$911$	−39873.8	−1.45014	−0.725070	+	0.688675i	$0.758193\pi$
$912$	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	$-0.734442\pi$
$919$	−37427.3	−1.34343	−0.671715	+	0.740810i	$0.734442\pi$
$920$	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	$-0.732632\pi$
$941$	−38535.5	−1.33499	−0.667493	+	0.744616i	$0.732632\pi$
$942$	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	$-0.377757\pi$
$967$	22532.8	0.749334	0.374667	+	0.927159i	$0.377757\pi$
$968$	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	$-0.180997\pi$
$983$	51940.4	1.68529	0.842646	+	0.538468i	$0.180997\pi$
$984$	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	$-0.210525\pi$
$991$	49237.5	1.57828	0.789142	+	0.614210i	$0.210525\pi$
$992$	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	$-0.494385\pi$
$997$	1110.55	0.0352774	0.0176387	+	0.999844i	$0.494385\pi$
$998$	0	0
$999$	−10837.3	−0.343221

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.g.1.1	✓	3	1.1	even	1	trivial
624.4.a.u.1.1		3	4.3	odd	2
936.4.a.k.1.3		3	3.2	odd	2
1872.4.a.bj.1.3		3	12.11	even	2
2496.4.a.bk.1.3		3	8.3	odd	2
2496.4.a.bo.1.3		3	8.5	even	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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.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

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