LMFDB - Embedded newform 1350.4.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1350.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{7} +8.00000 q^{8} -60.0000 q^{11} +79.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -108.000 q^{17} +11.0000 q^{19} -120.000 q^{22} -132.000 q^{23} +158.000 q^{26} +28.0000 q^{28} -96.0000 q^{29} +20.0000 q^{31} +32.0000 q^{32} -216.000 q^{34} +169.000 q^{37} +22.0000 q^{38} -192.000 q^{41} -488.000 q^{43} -240.000 q^{44} -264.000 q^{46} +204.000 q^{47} -294.000 q^{49} +316.000 q^{52} +360.000 q^{53} +56.0000 q^{56} -192.000 q^{58} -156.000 q^{59} +83.0000 q^{61} +40.0000 q^{62} +64.0000 q^{64} -47.0000 q^{67} -432.000 q^{68} -216.000 q^{71} +511.000 q^{73} +338.000 q^{74} +44.0000 q^{76} -420.000 q^{77} -529.000 q^{79} -384.000 q^{82} -1128.00 q^{83} -976.000 q^{86} -480.000 q^{88} -36.0000 q^{89} +553.000 q^{91} -528.000 q^{92} +408.000 q^{94} -605.000 q^{97} -588.000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	7.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	0	0
$13$	79.0000	1.68544	0.842718	−	0.538356i	$-0.180954\pi$
$13$	79.0000	1.68544	0.842718	+	0.538356i	$0.180954\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−108.000	−1.54081	−0.770407	−	0.637552i	$-0.779947\pi$
$17$	−108.000	−1.54081	−0.770407	+	0.637552i	$0.779947\pi$
$18$	0	0
$19$	11.0000	0.132820	0.0664098	−	0.997792i	$-0.478846\pi$
$19$	11.0000	0.132820	0.0664098	+	0.997792i	$0.478846\pi$
$20$	0	0
$21$	0	0
$22$	−120.000	−1.16291
$23$	−132.000	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$23$	−132.000	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$24$	0	0
$25$	0	0
$26$	158.000	1.19178
$27$	0	0
$28$	28.0000	0.188982
$29$	−96.0000	−0.614716	−0.307358	−	0.951594i	$-0.599445\pi$
$29$	−96.0000	−0.614716	−0.307358	+	0.951594i	$0.599445\pi$
$30$	0	0
$31$	20.0000	0.115874	0.0579372	−	0.998320i	$-0.481548\pi$
$31$	20.0000	0.115874	0.0579372	+	0.998320i	$0.481548\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−216.000	−1.08952
$35$	0	0
$36$	0	0
$37$	169.000	0.750903	0.375452	−	0.926842i	$-0.377488\pi$
$37$	169.000	0.750903	0.375452	+	0.926842i	$0.377488\pi$
$38$	22.0000	0.0939177
$39$	0	0
$40$	0	0
$41$	−192.000	−0.731350	−0.365675	−	0.930743i	$-0.619162\pi$
$41$	−192.000	−0.731350	−0.365675	+	0.930743i	$0.619162\pi$
$42$	0	0
$43$	−488.000	−1.73068	−0.865341	−	0.501184i	$-0.832898\pi$
$43$	−488.000	−1.73068	−0.865341	+	0.501184i	$0.832898\pi$
$44$	−240.000	−0.822304
$45$	0	0
$46$	−264.000	−0.846189
$47$	204.000	0.633116	0.316558	−	0.948573i	$-0.397473\pi$
$47$	204.000	0.633116	0.316558	+	0.948573i	$0.397473\pi$
$48$	0	0
$49$	−294.000	−0.857143
$50$	0	0
$51$	0	0
$52$	316.000	0.842718
$53$	360.000	0.933015	0.466508	−	0.884517i	$-0.345512\pi$
$53$	360.000	0.933015	0.466508	+	0.884517i	$0.345512\pi$
$54$	0	0
$55$	0	0
$56$	56.0000	0.133631
$57$	0	0
$58$	−192.000	−0.434670
$59$	−156.000	−0.344228	−0.172114	−	0.985077i	$-0.555060\pi$
$59$	−156.000	−0.344228	−0.172114	+	0.985077i	$0.555060\pi$
$60$	0	0
$61$	83.0000	0.174214	0.0871071	−	0.996199i	$-0.472238\pi$
$61$	83.0000	0.174214	0.0871071	+	0.996199i	$0.472238\pi$
$62$	40.0000	0.0819356
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−47.0000	−0.0857010	−0.0428505	−	0.999081i	$-0.513644\pi$
$67$	−47.0000	−0.0857010	−0.0428505	+	0.999081i	$0.513644\pi$
$68$	−432.000	−0.770407
$69$	0	0
$70$	0	0
$71$	−216.000	−0.361049	−0.180525	−	0.983570i	$-0.557780\pi$
$71$	−216.000	−0.361049	−0.180525	+	0.983570i	$0.557780\pi$
$72$	0	0
$73$	511.000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	511.000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	338.000	0.530969
$75$	0	0
$76$	44.0000	0.0664098
$77$	−420.000	−0.621603
$78$	0	0
$79$	−529.000	−0.753382	−0.376691	−	0.926339i	$-0.622938\pi$
$79$	−529.000	−0.753382	−0.376691	+	0.926339i	$0.622938\pi$
$80$	0	0
$81$	0	0
$82$	−384.000	−0.517143
$83$	−1128.00	−1.49174	−0.745868	−	0.666094i	$-0.767965\pi$
$83$	−1128.00	−1.49174	−0.745868	+	0.666094i	$0.767965\pi$
$84$	0	0
$85$	0	0
$86$	−976.000	−1.22378
$87$	0	0
$88$	−480.000	−0.581456
$89$	−36.0000	−0.0428763	−0.0214382	−	0.999770i	$-0.506825\pi$
$89$	−36.0000	−0.0428763	−0.0214382	+	0.999770i	$0.506825\pi$
$90$	0	0
$91$	553.000	0.637035
$92$	−528.000	−0.598346
$93$	0	0
$94$	408.000	0.447681
$95$	0	0
$96$	0	0
$97$	−605.000	−0.633283	−0.316641	−	0.948545i	$-0.602555\pi$
$97$	−605.000	−0.633283	−0.316641	+	0.948545i	$0.602555\pi$
$98$	−588.000	−0.606092
$99$	0	0
$100$	0	0
$101$	−1248.00	−1.22951	−0.614756	−	0.788718i	$-0.710745\pi$
$101$	−1248.00	−1.22951	−0.614756	+	0.788718i	$0.710745\pi$
$102$	0	0
$103$	−965.000	−0.923148	−0.461574	−	0.887102i	$-0.652715\pi$
$103$	−965.000	−0.923148	−0.461574	+	0.887102i	$0.652715\pi$
$104$	632.000	0.595891
$105$	0	0
$106$	720.000	0.659741
$107$	1332.00	1.20345	0.601726	−	0.798703i	$-0.294480\pi$
$107$	1332.00	1.20345	0.601726	+	0.798703i	$0.294480\pi$
$108$	0	0
$109$	−1942.00	−1.70651	−0.853256	−	0.521492i	$-0.825376\pi$
$109$	−1942.00	−1.70651	−0.853256	+	0.521492i	$0.825376\pi$
$110$	0	0
$111$	0	0
$112$	112.000	0.0944911
$113$	516.000	0.429568	0.214784	−	0.976662i	$-0.431095\pi$
$113$	516.000	0.429568	0.214784	+	0.976662i	$0.431095\pi$
$114$	0	0
$115$	0	0
$116$	−384.000	−0.307358
$117$	0	0
$118$	−312.000	−0.243406
$119$	−756.000	−0.582373
$120$	0	0
$121$	2269.00	1.70473
$122$	166.000	0.123188
$123$	0	0
$124$	80.0000	0.0579372
$125$	0	0
$126$	0	0
$127$	52.0000	0.0363327	0.0181664	−	0.999835i	$-0.494217\pi$
$127$	52.0000	0.0363327	0.0181664	+	0.999835i	$0.494217\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	−48.0000	−0.0320136	−0.0160068	−	0.999872i	$-0.505095\pi$
$131$	−48.0000	−0.0320136	−0.0160068	+	0.999872i	$0.505095\pi$
$132$	0	0
$133$	77.0000	0.0502011
$134$	−94.0000	−0.0605997
$135$	0	0
$136$	−864.000	−0.544760
$137$	2364.00	1.47423	0.737117	−	0.675765i	$-0.236186\pi$
$137$	2364.00	1.47423	0.737117	+	0.675765i	$0.236186\pi$
$138$	0	0
$139$	173.000	0.105566	0.0527830	−	0.998606i	$-0.483191\pi$
$139$	173.000	0.105566	0.0527830	+	0.998606i	$0.483191\pi$
$140$	0	0
$141$	0	0
$142$	−432.000	−0.255300
$143$	−4740.00	−2.77188
$144$	0	0
$145$	0	0
$146$	1022.00	0.579324
$147$	0	0
$148$	676.000	0.375452
$149$	1608.00	0.884111	0.442055	−	0.896988i	$-0.354249\pi$
$149$	1608.00	0.884111	0.442055	+	0.896988i	$0.354249\pi$
$150$	0	0
$151$	−997.000	−0.537316	−0.268658	−	0.963236i	$-0.586580\pi$
$151$	−997.000	−0.537316	−0.268658	+	0.963236i	$0.586580\pi$
$152$	88.0000	0.0469588
$153$	0	0
$154$	−840.000	−0.439540
$155$	0	0
$156$	0	0
$157$	−614.000	−0.312118	−0.156059	−	0.987748i	$-0.549879\pi$
$157$	−614.000	−0.312118	−0.156059	+	0.987748i	$0.549879\pi$
$158$	−1058.00	−0.532721
$159$	0	0
$160$	0	0
$161$	−924.000	−0.452307
$162$	0	0
$163$	−2693.00	−1.29406	−0.647031	−	0.762464i	$-0.723989\pi$
$163$	−2693.00	−1.29406	−0.647031	+	0.762464i	$0.723989\pi$
$164$	−768.000	−0.365675
$165$	0	0
$166$	−2256.00	−1.05482
$167$	1164.00	0.539359	0.269680	−	0.962950i	$-0.413082\pi$
$167$	1164.00	0.539359	0.269680	+	0.962950i	$0.413082\pi$
$168$	0	0
$169$	4044.00	1.84069
$170$	0	0
$171$	0	0
$172$	−1952.00	−0.865341
$173$	−3648.00	−1.60319	−0.801596	−	0.597866i	$-0.796016\pi$
$173$	−3648.00	−1.60319	−0.801596	+	0.597866i	$0.796016\pi$
$174$	0	0
$175$	0	0
$176$	−960.000	−0.411152
$177$	0	0
$178$	−72.0000	−0.0303181
$179$	−1800.00	−0.751611	−0.375805	−	0.926699i	$-0.622634\pi$
$179$	−1800.00	−0.751611	−0.375805	+	0.926699i	$0.622634\pi$
$180$	0	0
$181$	−547.000	−0.224631	−0.112315	−	0.993673i	$-0.535827\pi$
$181$	−547.000	−0.224631	−0.112315	+	0.993673i	$0.535827\pi$
$182$	1106.00	0.450452
$183$	0	0
$184$	−1056.00	−0.423094
$185$	0	0
$186$	0	0
$187$	6480.00	2.53403
$188$	816.000	0.316558
$189$	0	0
$190$	0	0
$191$	−3156.00	−1.19560	−0.597801	−	0.801644i	$-0.703959\pi$
$191$	−3156.00	−1.19560	−0.597801	+	0.801644i	$0.703959\pi$
$192$	0	0
$193$	−1127.00	−0.420328	−0.210164	−	0.977666i	$-0.567400\pi$
$193$	−1127.00	−0.420328	−0.210164	+	0.977666i	$0.567400\pi$
$194$	−1210.00	−0.447799
$195$	0	0
$196$	−1176.00	−0.428571
$197$	1116.00	0.403613	0.201806	−	0.979425i	$-0.435319\pi$
$197$	1116.00	0.403613	0.201806	+	0.979425i	$0.435319\pi$
$198$	0	0
$199$	−3283.00	−1.16948	−0.584738	−	0.811222i	$-0.698803\pi$
$199$	−3283.00	−1.16948	−0.584738	+	0.811222i	$0.698803\pi$
$200$	0	0
$201$	0	0
$202$	−2496.00	−0.869396
$203$	−672.000	−0.232341
$204$	0	0
$205$	0	0
$206$	−1930.00	−0.652764
$207$	0	0
$208$	1264.00	0.421359
$209$	−660.000	−0.218436
$210$	0	0
$211$	−295.000	−0.0962495	−0.0481247	−	0.998841i	$-0.515324\pi$
$211$	−295.000	−0.0962495	−0.0481247	+	0.998841i	$0.515324\pi$
$212$	1440.00	0.466508
$213$	0	0
$214$	2664.00	0.850969
$215$	0	0
$216$	0	0
$217$	140.000	0.0437964
$218$	−3884.00	−1.20669
$219$	0	0
$220$	0	0
$221$	−8532.00	−2.59694
$222$	0	0
$223$	2644.00	0.793970	0.396985	−	0.917825i	$-0.370056\pi$
$223$	2644.00	0.793970	0.396985	+	0.917825i	$0.370056\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	1032.00	0.303751
$227$	−6024.00	−1.76135	−0.880676	−	0.473719i	$-0.842911\pi$
$227$	−6024.00	−1.76135	−0.880676	+	0.473719i	$0.842911\pi$
$228$	0	0
$229$	−4462.00	−1.28759	−0.643793	−	0.765200i	$-0.722640\pi$
$229$	−4462.00	−1.28759	−0.643793	+	0.765200i	$0.722640\pi$
$230$	0	0
$231$	0	0
$232$	−768.000	−0.217335
$233$	−1008.00	−0.283417	−0.141709	−	0.989908i	$-0.545260\pi$
$233$	−1008.00	−0.283417	−0.141709	+	0.989908i	$0.545260\pi$
$234$	0	0
$235$	0	0
$236$	−624.000	−0.172114
$237$	0	0
$238$	−1512.00	−0.411800
$239$	5064.00	1.37056	0.685278	−	0.728281i	$-0.259681\pi$
$239$	5064.00	1.37056	0.685278	+	0.728281i	$0.259681\pi$
$240$	0	0
$241$	6257.00	1.67240	0.836201	−	0.548423i	$-0.184772\pi$
$241$	6257.00	1.67240	0.836201	+	0.548423i	$0.184772\pi$
$242$	4538.00	1.20543
$243$	0	0
$244$	332.000	0.0871071
$245$	0	0
$246$	0	0
$247$	869.000	0.223859
$248$	160.000	0.0409678
$249$	0	0
$250$	0	0
$251$	2160.00	0.543179	0.271590	−	0.962413i	$-0.412451\pi$
$251$	2160.00	0.543179	0.271590	+	0.962413i	$0.412451\pi$
$252$	0	0
$253$	7920.00	1.96809
$254$	104.000	0.0256911
$255$	0	0
$256$	256.000	0.0625000
$257$	−168.000	−0.0407765	−0.0203882	−	0.999792i	$-0.506490\pi$
$257$	−168.000	−0.0407765	−0.0203882	+	0.999792i	$0.506490\pi$
$258$	0	0
$259$	1183.00	0.283815
$260$	0	0
$261$	0	0
$262$	−96.0000	−0.0226370
$263$	−2424.00	−0.568328	−0.284164	−	0.958776i	$-0.591716\pi$
$263$	−2424.00	−0.568328	−0.284164	+	0.958776i	$0.591716\pi$
$264$	0	0
$265$	0	0
$266$	154.000	0.0354975
$267$	0	0
$268$	−188.000	−0.0428505
$269$	396.000	0.0897567	0.0448783	−	0.998992i	$-0.485710\pi$
$269$	396.000	0.0897567	0.0448783	+	0.998992i	$0.485710\pi$
$270$	0	0
$271$	1811.00	0.405942	0.202971	−	0.979185i	$-0.434940\pi$
$271$	1811.00	0.405942	0.202971	+	0.979185i	$0.434940\pi$
$272$	−1728.00	−0.385204
$273$	0	0
$274$	4728.00	1.04244
$275$	0	0
$276$	0	0
$277$	3022.00	0.655503	0.327752	−	0.944764i	$-0.393709\pi$
$277$	3022.00	0.655503	0.327752	+	0.944764i	$0.393709\pi$
$278$	346.000	0.0746464
$279$	0	0
$280$	0	0
$281$	3072.00	0.652171	0.326086	−	0.945340i	$-0.394270\pi$
$281$	3072.00	0.652171	0.326086	+	0.945340i	$0.394270\pi$
$282$	0	0
$283$	8080.00	1.69719	0.848597	−	0.529039i	$-0.177448\pi$
$283$	8080.00	1.69719	0.848597	+	0.529039i	$0.177448\pi$
$284$	−864.000	−0.180525
$285$	0	0
$286$	−9480.00	−1.96001
$287$	−1344.00	−0.276424
$288$	0	0
$289$	6751.00	1.37411
$290$	0	0
$291$	0	0
$292$	2044.00	0.409644
$293$	−4668.00	−0.930742	−0.465371	−	0.885116i	$-0.654079\pi$
$293$	−4668.00	−0.930742	−0.465371	+	0.885116i	$0.654079\pi$
$294$	0	0
$295$	0	0
$296$	1352.00	0.265484
$297$	0	0
$298$	3216.00	0.625161
$299$	−10428.0	−2.01695
$300$	0	0
$301$	−3416.00	−0.654136
$302$	−1994.00	−0.379940
$303$	0	0
$304$	176.000	0.0332049
$305$	0	0
$306$	0	0
$307$	−6752.00	−1.25523	−0.627617	−	0.778522i	$-0.715970\pi$
$307$	−6752.00	−1.25523	−0.627617	+	0.778522i	$0.715970\pi$
$308$	−1680.00	−0.310802
$309$	0	0
$310$	0	0
$311$	−1812.00	−0.330383	−0.165191	−	0.986262i	$-0.552824\pi$
$311$	−1812.00	−0.330383	−0.165191	+	0.986262i	$0.552824\pi$
$312$	0	0
$313$	−6203.00	−1.12017	−0.560087	−	0.828434i	$-0.689232\pi$
$313$	−6203.00	−1.12017	−0.560087	+	0.828434i	$0.689232\pi$
$314$	−1228.00	−0.220701
$315$	0	0
$316$	−2116.00	−0.376691
$317$	10968.0	1.94329	0.971647	−	0.236436i	$-0.0759793\pi$
$317$	10968.0	1.94329	0.971647	+	0.236436i	$0.0759793\pi$
$318$	0	0
$319$	5760.00	1.01097
$320$	0	0
$321$	0	0
$322$	−1848.00	−0.319829
$323$	−1188.00	−0.204650
$324$	0	0
$325$	0	0
$326$	−5386.00	−0.915040
$327$	0	0
$328$	−1536.00	−0.258571
$329$	1428.00	0.239295
$330$	0	0
$331$	8165.00	1.35586	0.677929	−	0.735127i	$-0.262878\pi$
$331$	8165.00	1.35586	0.677929	+	0.735127i	$0.262878\pi$
$332$	−4512.00	−0.745868
$333$	0	0
$334$	2328.00	0.381385
$335$	0	0
$336$	0	0
$337$	6523.00	1.05439	0.527197	−	0.849743i	$-0.323243\pi$
$337$	6523.00	1.05439	0.527197	+	0.849743i	$0.323243\pi$
$338$	8088.00	1.30157
$339$	0	0
$340$	0	0
$341$	−1200.00	−0.190568
$342$	0	0
$343$	−4459.00	−0.701934
$344$	−3904.00	−0.611888
$345$	0	0
$346$	−7296.00	−1.13363
$347$	6168.00	0.954224	0.477112	−	0.878843i	$-0.341684\pi$
$347$	6168.00	0.954224	0.477112	+	0.878843i	$0.341684\pi$
$348$	0	0
$349$	−6001.00	−0.920419	−0.460209	−	0.887810i	$-0.652226\pi$
$349$	−6001.00	−0.920419	−0.460209	+	0.887810i	$0.652226\pi$
$350$	0	0
$351$	0	0
$352$	−1920.00	−0.290728
$353$	−552.000	−0.0832294	−0.0416147	−	0.999134i	$-0.513250\pi$
$353$	−552.000	−0.0832294	−0.0416147	+	0.999134i	$0.513250\pi$
$354$	0	0
$355$	0	0
$356$	−144.000	−0.0214382
$357$	0	0
$358$	−3600.00	−0.531469
$359$	5004.00	0.735657	0.367829	−	0.929894i	$-0.380101\pi$
$359$	5004.00	0.735657	0.367829	+	0.929894i	$0.380101\pi$
$360$	0	0
$361$	−6738.00	−0.982359
$362$	−1094.00	−0.158838
$363$	0	0
$364$	2212.00	0.318517
$365$	0	0
$366$	0	0
$367$	4291.00	0.610323	0.305161	−	0.952301i	$-0.401290\pi$
$367$	4291.00	0.610323	0.305161	+	0.952301i	$0.401290\pi$
$368$	−2112.00	−0.299173
$369$	0	0
$370$	0	0
$371$	2520.00	0.352647
$372$	0	0
$373$	2833.00	0.393263	0.196632	−	0.980477i	$-0.437000\pi$
$373$	2833.00	0.393263	0.196632	+	0.980477i	$0.437000\pi$
$374$	12960.0	1.79183
$375$	0	0
$376$	1632.00	0.223840
$377$	−7584.00	−1.03606
$378$	0	0
$379$	−5137.00	−0.696227	−0.348113	−	0.937452i	$-0.613178\pi$
$379$	−5137.00	−0.696227	−0.348113	+	0.937452i	$0.613178\pi$
$380$	0	0
$381$	0	0
$382$	−6312.00	−0.845419
$383$	5088.00	0.678811	0.339406	−	0.940640i	$-0.389774\pi$
$383$	5088.00	0.678811	0.339406	+	0.940640i	$0.389774\pi$
$384$	0	0
$385$	0	0
$386$	−2254.00	−0.297217
$387$	0	0
$388$	−2420.00	−0.316641
$389$	−7980.00	−1.04011	−0.520054	−	0.854133i	$-0.674088\pi$
$389$	−7980.00	−1.04011	−0.520054	+	0.854133i	$0.674088\pi$
$390$	0	0
$391$	14256.0	1.84388
$392$	−2352.00	−0.303046
$393$	0	0
$394$	2232.00	0.285397
$395$	0	0
$396$	0	0
$397$	1834.00	0.231853	0.115927	−	0.993258i	$-0.463016\pi$
$397$	1834.00	0.231853	0.115927	+	0.993258i	$0.463016\pi$
$398$	−6566.00	−0.826944
$399$	0	0
$400$	0	0
$401$	1464.00	0.182316	0.0911579	−	0.995836i	$-0.470943\pi$
$401$	1464.00	0.182316	0.0911579	+	0.995836i	$0.470943\pi$
$402$	0	0
$403$	1580.00	0.195299
$404$	−4992.00	−0.614756
$405$	0	0
$406$	−1344.00	−0.164290
$407$	−10140.0	−1.23494
$408$	0	0
$409$	−151.000	−0.0182554	−0.00912771	−	0.999958i	$-0.502905\pi$
$409$	−151.000	−0.0182554	−0.00912771	+	0.999958i	$0.502905\pi$
$410$	0	0
$411$	0	0
$412$	−3860.00	−0.461574
$413$	−1092.00	−0.130106
$414$	0	0
$415$	0	0
$416$	2528.00	0.297946
$417$	0	0
$418$	−1320.00	−0.154458
$419$	11508.0	1.34177	0.670886	−	0.741560i	$-0.265914\pi$
$419$	11508.0	1.34177	0.670886	+	0.741560i	$0.265914\pi$
$420$	0	0
$421$	−6271.00	−0.725962	−0.362981	−	0.931797i	$-0.618241\pi$
$421$	−6271.00	−0.725962	−0.362981	+	0.931797i	$0.618241\pi$
$422$	−590.000	−0.0680587
$423$	0	0
$424$	2880.00	0.329871
$425$	0	0
$426$	0	0
$427$	581.000	0.0658467
$428$	5328.00	0.601726
$429$	0	0
$430$	0	0
$431$	−9468.00	−1.05814	−0.529069	−	0.848579i	$-0.677459\pi$
$431$	−9468.00	−1.05814	−0.529069	+	0.848579i	$0.677459\pi$
$432$	0	0
$433$	−3026.00	−0.335844	−0.167922	−	0.985800i	$-0.553706\pi$
$433$	−3026.00	−0.335844	−0.167922	+	0.985800i	$0.553706\pi$
$434$	280.000	0.0309687
$435$	0	0
$436$	−7768.00	−0.853256
$437$	−1452.00	−0.158944
$438$	0	0
$439$	11180.0	1.21547	0.607736	−	0.794139i	$-0.292078\pi$
$439$	11180.0	1.21547	0.607736	+	0.794139i	$0.292078\pi$
$440$	0	0
$441$	0	0
$442$	−17064.0	−1.83632
$443$	−840.000	−0.0900894	−0.0450447	−	0.998985i	$-0.514343\pi$
$443$	−840.000	−0.0900894	−0.0450447	+	0.998985i	$0.514343\pi$
$444$	0	0
$445$	0	0
$446$	5288.00	0.561422
$447$	0	0
$448$	448.000	0.0472456
$449$	−12780.0	−1.34326	−0.671632	−	0.740885i	$-0.734406\pi$
$449$	−12780.0	−1.34326	−0.671632	+	0.740885i	$0.734406\pi$
$450$	0	0
$451$	11520.0	1.20278
$452$	2064.00	0.214784
$453$	0	0
$454$	−12048.0	−1.24546
$455$	0	0
$456$	0	0
$457$	15658.0	1.60274	0.801368	−	0.598172i	$-0.204106\pi$
$457$	15658.0	1.60274	0.801368	+	0.598172i	$0.204106\pi$
$458$	−8924.00	−0.910461
$459$	0	0
$460$	0	0
$461$	15564.0	1.57242	0.786212	−	0.617956i	$-0.212039\pi$
$461$	15564.0	1.57242	0.786212	+	0.617956i	$0.212039\pi$
$462$	0	0
$463$	4183.00	0.419871	0.209936	−	0.977715i	$-0.432675\pi$
$463$	4183.00	0.419871	0.209936	+	0.977715i	$0.432675\pi$
$464$	−1536.00	−0.153679
$465$	0	0
$466$	−2016.00	−0.200406
$467$	13932.0	1.38051	0.690253	−	0.723568i	$-0.257499\pi$
$467$	13932.0	1.38051	0.690253	+	0.723568i	$0.257499\pi$
$468$	0	0
$469$	−329.000	−0.0323919
$470$	0	0
$471$	0	0
$472$	−1248.00	−0.121703
$473$	29280.0	2.84629
$474$	0	0
$475$	0	0
$476$	−3024.00	−0.291187
$477$	0	0
$478$	10128.0	0.969130
$479$	2712.00	0.258694	0.129347	−	0.991599i	$-0.458712\pi$
$479$	2712.00	0.258694	0.129347	+	0.991599i	$0.458712\pi$
$480$	0	0
$481$	13351.0	1.26560
$482$	12514.0	1.18257
$483$	0	0
$484$	9076.00	0.852367
$485$	0	0
$486$	0	0
$487$	9439.00	0.878279	0.439140	−	0.898419i	$-0.355283\pi$
$487$	9439.00	0.878279	0.439140	+	0.898419i	$0.355283\pi$
$488$	664.000	0.0615940
$489$	0	0
$490$	0	0
$491$	11724.0	1.07759	0.538795	−	0.842437i	$-0.318880\pi$
$491$	11724.0	1.07759	0.538795	+	0.842437i	$0.318880\pi$
$492$	0	0
$493$	10368.0	0.947163
$494$	1738.00	0.158292
$495$	0	0
$496$	320.000	0.0289686
$497$	−1512.00	−0.136464
$498$	0	0
$499$	−11968.0	−1.07367	−0.536835	−	0.843687i	$-0.680380\pi$
$499$	−11968.0	−1.07367	−0.536835	+	0.843687i	$0.680380\pi$
$500$	0	0
$501$	0	0
$502$	4320.00	0.384086
$503$	8892.00	0.788220	0.394110	−	0.919063i	$-0.371053\pi$
$503$	8892.00	0.788220	0.394110	+	0.919063i	$0.371053\pi$
$504$	0	0
$505$	0	0
$506$	15840.0	1.39165
$507$	0	0
$508$	208.000	0.0181664
$509$	6756.00	0.588319	0.294160	−	0.955756i	$-0.404960\pi$
$509$	6756.00	0.588319	0.294160	+	0.955756i	$0.404960\pi$
$510$	0	0
$511$	3577.00	0.309662
$512$	512.000	0.0441942
$513$	0	0
$514$	−336.000	−0.0288333
$515$	0	0
$516$	0	0
$517$	−12240.0	−1.04123
$518$	2366.00	0.200687
$519$	0	0
$520$	0	0
$521$	6228.00	0.523711	0.261856	−	0.965107i	$-0.415666\pi$
$521$	6228.00	0.523711	0.261856	+	0.965107i	$0.415666\pi$
$522$	0	0
$523$	−11639.0	−0.973113	−0.486556	−	0.873649i	$-0.661747\pi$
$523$	−11639.0	−0.973113	−0.486556	+	0.873649i	$0.661747\pi$
$524$	−192.000	−0.0160068
$525$	0	0
$526$	−4848.00	−0.401869
$527$	−2160.00	−0.178541
$528$	0	0
$529$	5257.00	0.432070
$530$	0	0
$531$	0	0
$532$	308.000	0.0251006
$533$	−15168.0	−1.23264
$534$	0	0
$535$	0	0
$536$	−376.000	−0.0302999
$537$	0	0
$538$	792.000	0.0634676
$539$	17640.0	1.40966
$540$	0	0
$541$	17705.0	1.40702	0.703510	−	0.710686i	$-0.251615\pi$
$541$	17705.0	1.40702	0.703510	+	0.710686i	$0.251615\pi$
$542$	3622.00	0.287045
$543$	0	0
$544$	−3456.00	−0.272380
$545$	0	0
$546$	0	0
$547$	−3485.00	−0.272409	−0.136205	−	0.990681i	$-0.543490\pi$
$547$	−3485.00	−0.272409	−0.136205	+	0.990681i	$0.543490\pi$
$548$	9456.00	0.737117
$549$	0	0
$550$	0	0
$551$	−1056.00	−0.0816463
$552$	0	0
$553$	−3703.00	−0.284751
$554$	6044.00	0.463511
$555$	0	0
$556$	692.000	0.0527830
$557$	−19116.0	−1.45417	−0.727083	−	0.686549i	$-0.759125\pi$
$557$	−19116.0	−1.45417	−0.727083	+	0.686549i	$0.759125\pi$
$558$	0	0
$559$	−38552.0	−2.91695
$560$	0	0
$561$	0	0
$562$	6144.00	0.461155
$563$	22368.0	1.67442	0.837210	−	0.546881i	$-0.184185\pi$
$563$	22368.0	1.67442	0.837210	+	0.546881i	$0.184185\pi$
$564$	0	0
$565$	0	0
$566$	16160.0	1.20010
$567$	0	0
$568$	−1728.00	−0.127650
$569$	−8340.00	−0.614466	−0.307233	−	0.951634i	$-0.599403\pi$
$569$	−8340.00	−0.614466	−0.307233	+	0.951634i	$0.599403\pi$
$570$	0	0
$571$	−14677.0	−1.07568	−0.537840	−	0.843047i	$-0.680760\pi$
$571$	−14677.0	−1.07568	−0.537840	+	0.843047i	$0.680760\pi$
$572$	−18960.0	−1.38594
$573$	0	0
$574$	−2688.00	−0.195462
$575$	0	0
$576$	0	0
$577$	10069.0	0.726478	0.363239	−	0.931696i	$-0.381671\pi$
$577$	10069.0	0.726478	0.363239	+	0.931696i	$0.381671\pi$
$578$	13502.0	0.971642
$579$	0	0
$580$	0	0
$581$	−7896.00	−0.563823
$582$	0	0
$583$	−21600.0	−1.53444
$584$	4088.00	0.289662
$585$	0	0
$586$	−9336.00	−0.658134
$587$	1572.00	0.110534	0.0552669	−	0.998472i	$-0.482399\pi$
$587$	1572.00	0.110534	0.0552669	+	0.998472i	$0.482399\pi$
$588$	0	0
$589$	220.000	0.0153904
$590$	0	0
$591$	0	0
$592$	2704.00	0.187726
$593$	1368.00	0.0947336	0.0473668	−	0.998878i	$-0.484917\pi$
$593$	1368.00	0.0947336	0.0473668	+	0.998878i	$0.484917\pi$
$594$	0	0
$595$	0	0
$596$	6432.00	0.442055
$597$	0	0
$598$	−20856.0	−1.42620
$599$	23712.0	1.61744	0.808720	−	0.588194i	$-0.200161\pi$
$599$	23712.0	1.61744	0.808720	+	0.588194i	$0.200161\pi$
$600$	0	0
$601$	−11014.0	−0.747538	−0.373769	−	0.927522i	$-0.621935\pi$
$601$	−11014.0	−0.747538	−0.373769	+	0.927522i	$0.621935\pi$
$602$	−6832.00	−0.462544
$603$	0	0
$604$	−3988.00	−0.268658
$605$	0	0
$606$	0	0
$607$	6415.00	0.428957	0.214478	−	0.976729i	$-0.431195\pi$
$607$	6415.00	0.428957	0.214478	+	0.976729i	$0.431195\pi$
$608$	352.000	0.0234794
$609$	0	0
$610$	0	0
$611$	16116.0	1.06708
$612$	0	0
$613$	−15851.0	−1.04440	−0.522199	−	0.852824i	$-0.674888\pi$
$613$	−15851.0	−1.04440	−0.522199	+	0.852824i	$0.674888\pi$
$614$	−13504.0	−0.887585
$615$	0	0
$616$	−3360.00	−0.219770
$617$	5772.00	0.376616	0.188308	−	0.982110i	$-0.439700\pi$
$617$	5772.00	0.376616	0.188308	+	0.982110i	$0.439700\pi$
$618$	0	0
$619$	−27781.0	−1.80390	−0.901949	−	0.431843i	$-0.857863\pi$
$619$	−27781.0	−1.80390	−0.901949	+	0.431843i	$0.857863\pi$
$620$	0	0
$621$	0	0
$622$	−3624.00	−0.233616
$623$	−252.000	−0.0162057
$624$	0	0
$625$	0	0
$626$	−12406.0	−0.792082
$627$	0	0
$628$	−2456.00	−0.156059
$629$	−18252.0	−1.15700
$630$	0	0
$631$	−29869.0	−1.88442	−0.942208	−	0.335029i	$-0.891254\pi$
$631$	−29869.0	−1.88442	−0.942208	+	0.335029i	$0.891254\pi$
$632$	−4232.00	−0.266361
$633$	0	0
$634$	21936.0	1.37412
$635$	0	0
$636$	0	0
$637$	−23226.0	−1.44466
$638$	11520.0	0.714861
$639$	0	0
$640$	0	0
$641$	−3480.00	−0.214433	−0.107217	−	0.994236i	$-0.534194\pi$
$641$	−3480.00	−0.214433	−0.107217	+	0.994236i	$0.534194\pi$
$642$	0	0
$643$	7432.00	0.455816	0.227908	−	0.973683i	$-0.426812\pi$
$643$	7432.00	0.455816	0.227908	+	0.973683i	$0.426812\pi$
$644$	−3696.00	−0.226153
$645$	0	0
$646$	−2376.00	−0.144710
$647$	−12960.0	−0.787496	−0.393748	−	0.919218i	$-0.628822\pi$
$647$	−12960.0	−0.787496	−0.393748	+	0.919218i	$0.628822\pi$
$648$	0	0
$649$	9360.00	0.566120
$650$	0	0
$651$	0	0
$652$	−10772.0	−0.647031
$653$	23592.0	1.41382	0.706911	−	0.707302i	$-0.250088\pi$
$653$	23592.0	1.41382	0.706911	+	0.707302i	$0.250088\pi$
$654$	0	0
$655$	0	0
$656$	−3072.00	−0.182838
$657$	0	0
$658$	2856.00	0.169207
$659$	−32280.0	−1.90812	−0.954059	−	0.299618i	$-0.903141\pi$
$659$	−32280.0	−1.90812	−0.954059	+	0.299618i	$0.903141\pi$
$660$	0	0
$661$	22619.0	1.33098	0.665490	−	0.746407i	$-0.268223\pi$
$661$	22619.0	1.33098	0.665490	+	0.746407i	$0.268223\pi$
$662$	16330.0	0.958736
$663$	0	0
$664$	−9024.00	−0.527408
$665$	0	0
$666$	0	0
$667$	12672.0	0.735625
$668$	4656.00	0.269680
$669$	0	0
$670$	0	0
$671$	−4980.00	−0.286514
$672$	0	0
$673$	19861.0	1.13757	0.568786	−	0.822486i	$-0.307413\pi$
$673$	19861.0	1.13757	0.568786	+	0.822486i	$0.307413\pi$
$674$	13046.0	0.745568
$675$	0	0
$676$	16176.0	0.920346
$677$	−8292.00	−0.470735	−0.235367	−	0.971906i	$-0.575629\pi$
$677$	−8292.00	−0.470735	−0.235367	+	0.971906i	$0.575629\pi$
$678$	0	0
$679$	−4235.00	−0.239358
$680$	0	0
$681$	0	0
$682$	−2400.00	−0.134752
$683$	19728.0	1.10523	0.552614	−	0.833437i	$-0.313630\pi$
$683$	19728.0	1.10523	0.552614	+	0.833437i	$0.313630\pi$
$684$	0	0
$685$	0	0
$686$	−8918.00	−0.496342
$687$	0	0
$688$	−7808.00	−0.432670
$689$	28440.0	1.57254
$690$	0	0
$691$	27272.0	1.50141	0.750706	−	0.660636i	$-0.229713\pi$
$691$	27272.0	1.50141	0.750706	+	0.660636i	$0.229713\pi$
$692$	−14592.0	−0.801596
$693$	0	0
$694$	12336.0	0.674738
$695$	0	0
$696$	0	0
$697$	20736.0	1.12688
$698$	−12002.0	−0.650834
$699$	0	0
$700$	0	0
$701$	21996.0	1.18513	0.592566	−	0.805522i	$-0.298115\pi$
$701$	21996.0	1.18513	0.592566	+	0.805522i	$0.298115\pi$
$702$	0	0
$703$	1859.00	0.0997347
$704$	−3840.00	−0.205576
$705$	0	0
$706$	−1104.00	−0.0588521
$707$	−8736.00	−0.464712
$708$	0	0
$709$	2009.00	0.106417	0.0532084	−	0.998583i	$-0.483055\pi$
$709$	2009.00	0.106417	0.0532084	+	0.998583i	$0.483055\pi$
$710$	0	0
$711$	0	0
$712$	−288.000	−0.0151591
$713$	−2640.00	−0.138666
$714$	0	0
$715$	0	0
$716$	−7200.00	−0.375805
$717$	0	0
$718$	10008.0	0.520188
$719$	−26280.0	−1.36311	−0.681557	−	0.731765i	$-0.738697\pi$
$719$	−26280.0	−1.36311	−0.681557	+	0.731765i	$0.738697\pi$
$720$	0	0
$721$	−6755.00	−0.348917
$722$	−13476.0	−0.694633
$723$	0	0
$724$	−2188.00	−0.112315
$725$	0	0
$726$	0	0
$727$	−20900.0	−1.06621	−0.533107	−	0.846048i	$-0.678976\pi$
$727$	−20900.0	−1.06621	−0.533107	+	0.846048i	$0.678976\pi$
$728$	4424.00	0.225226
$729$	0	0
$730$	0	0
$731$	52704.0	2.66666
$732$	0	0
$733$	17638.0	0.888778	0.444389	−	0.895834i	$-0.353421\pi$
$733$	17638.0	0.888778	0.444389	+	0.895834i	$0.353421\pi$
$734$	8582.00	0.431563
$735$	0	0
$736$	−4224.00	−0.211547
$737$	2820.00	0.140944
$738$	0	0
$739$	30080.0	1.49731	0.748654	−	0.662961i	$-0.230700\pi$
$739$	30080.0	1.49731	0.748654	+	0.662961i	$0.230700\pi$
$740$	0	0
$741$	0	0
$742$	5040.00	0.249359
$743$	15708.0	0.775600	0.387800	−	0.921744i	$-0.373235\pi$
$743$	15708.0	0.775600	0.387800	+	0.921744i	$0.373235\pi$
$744$	0	0
$745$	0	0
$746$	5666.00	0.278079
$747$	0	0
$748$	25920.0	1.26702
$749$	9324.00	0.454862
$750$	0	0
$751$	20423.0	0.992338	0.496169	−	0.868226i	$-0.334740\pi$
$751$	20423.0	0.992338	0.496169	+	0.868226i	$0.334740\pi$
$752$	3264.00	0.158279
$753$	0	0
$754$	−15168.0	−0.732607
$755$	0	0
$756$	0	0
$757$	4399.00	0.211208	0.105604	−	0.994408i	$-0.466322\pi$
$757$	4399.00	0.211208	0.105604	+	0.994408i	$0.466322\pi$
$758$	−10274.0	−0.492307
$759$	0	0
$760$	0	0
$761$	20652.0	0.983751	0.491875	−	0.870666i	$-0.336312\pi$
$761$	20652.0	0.983751	0.491875	+	0.870666i	$0.336312\pi$
$762$	0	0
$763$	−13594.0	−0.645001
$764$	−12624.0	−0.597801
$765$	0	0
$766$	10176.0	0.479992
$767$	−12324.0	−0.580175
$768$	0	0
$769$	27407.0	1.28520	0.642602	−	0.766200i	$-0.277855\pi$
$769$	27407.0	1.28520	0.642602	+	0.766200i	$0.277855\pi$
$770$	0	0
$771$	0	0
$772$	−4508.00	−0.210164
$773$	5976.00	0.278062	0.139031	−	0.990288i	$-0.455601\pi$
$773$	5976.00	0.278062	0.139031	+	0.990288i	$0.455601\pi$
$774$	0	0
$775$	0	0
$776$	−4840.00	−0.223899
$777$	0	0
$778$	−15960.0	−0.735468
$779$	−2112.00	−0.0971377
$780$	0	0
$781$	12960.0	0.593784
$782$	28512.0	1.30382
$783$	0	0
$784$	−4704.00	−0.214286
$785$	0	0
$786$	0	0
$787$	−24131.0	−1.09298	−0.546491	−	0.837465i	$-0.684037\pi$
$787$	−24131.0	−1.09298	−0.546491	+	0.837465i	$0.684037\pi$
$788$	4464.00	0.201806
$789$	0	0
$790$	0	0
$791$	3612.00	0.162361
$792$	0	0
$793$	6557.00	0.293627
$794$	3668.00	0.163945
$795$	0	0
$796$	−13132.0	−0.584738
$797$	912.000	0.0405329	0.0202664	−	0.999795i	$-0.493549\pi$
$797$	912.000	0.0405329	0.0202664	+	0.999795i	$0.493549\pi$
$798$	0	0
$799$	−22032.0	−0.975515
$800$	0	0
$801$	0	0
$802$	2928.00	0.128917
$803$	−30660.0	−1.34741
$804$	0	0
$805$	0	0
$806$	3160.00	0.138097
$807$	0	0
$808$	−9984.00	−0.434698
$809$	12888.0	0.560096	0.280048	−	0.959986i	$-0.409650\pi$
$809$	12888.0	0.560096	0.280048	+	0.959986i	$0.409650\pi$
$810$	0	0
$811$	−6856.00	−0.296852	−0.148426	−	0.988924i	$-0.547421\pi$
$811$	−6856.00	−0.296852	−0.148426	+	0.988924i	$0.547421\pi$
$812$	−2688.00	−0.116170
$813$	0	0
$814$	−20280.0	−0.873235
$815$	0	0
$816$	0	0
$817$	−5368.00	−0.229868
$818$	−302.000	−0.0129085
$819$	0	0
$820$	0	0
$821$	−636.000	−0.0270360	−0.0135180	−	0.999909i	$-0.504303\pi$
$821$	−636.000	−0.0270360	−0.0135180	+	0.999909i	$0.504303\pi$
$822$	0	0
$823$	−39827.0	−1.68686	−0.843428	−	0.537243i	$-0.819466\pi$
$823$	−39827.0	−1.68686	−0.843428	+	0.537243i	$0.819466\pi$
$824$	−7720.00	−0.326382
$825$	0	0
$826$	−2184.00	−0.0919989
$827$	38124.0	1.60302	0.801512	−	0.597978i	$-0.204029\pi$
$827$	38124.0	1.60302	0.801512	+	0.597978i	$0.204029\pi$
$828$	0	0
$829$	18965.0	0.794550	0.397275	−	0.917700i	$-0.369956\pi$
$829$	18965.0	0.794550	0.397275	+	0.917700i	$0.369956\pi$
$830$	0	0
$831$	0	0
$832$	5056.00	0.210679
$833$	31752.0	1.32070
$834$	0	0
$835$	0	0
$836$	−2640.00	−0.109218
$837$	0	0
$838$	23016.0	0.948776
$839$	−27816.0	−1.14459	−0.572297	−	0.820046i	$-0.693948\pi$
$839$	−27816.0	−1.14459	−0.572297	+	0.820046i	$0.693948\pi$
$840$	0	0
$841$	−15173.0	−0.622125
$842$	−12542.0	−0.513332
$843$	0	0
$844$	−1180.00	−0.0481247
$845$	0	0
$846$	0	0
$847$	15883.0	0.644329
$848$	5760.00	0.233254
$849$	0	0
$850$	0	0
$851$	−22308.0	−0.898600
$852$	0	0
$853$	12337.0	0.495206	0.247603	−	0.968862i	$-0.420357\pi$
$853$	12337.0	0.495206	0.247603	+	0.968862i	$0.420357\pi$
$854$	1162.00	0.0465607
$855$	0	0
$856$	10656.0	0.425484
$857$	−11352.0	−0.452482	−0.226241	−	0.974071i	$-0.572644\pi$
$857$	−11352.0	−0.452482	−0.226241	+	0.974071i	$0.572644\pi$
$858$	0	0
$859$	−2527.00	−0.100373	−0.0501863	−	0.998740i	$-0.515982\pi$
$859$	−2527.00	−0.100373	−0.0501863	+	0.998740i	$0.515982\pi$
$860$	0	0
$861$	0	0
$862$	−18936.0	−0.748217
$863$	−26388.0	−1.04086	−0.520428	−	0.853906i	$-0.674227\pi$
$863$	−26388.0	−1.04086	−0.520428	+	0.853906i	$0.674227\pi$
$864$	0	0
$865$	0	0
$866$	−6052.00	−0.237477
$867$	0	0
$868$	560.000	0.0218982
$869$	31740.0	1.23902
$870$	0	0
$871$	−3713.00	−0.144443
$872$	−15536.0	−0.603343
$873$	0	0
$874$	−2904.00	−0.112390
$875$	0	0
$876$	0	0
$877$	−6383.00	−0.245768	−0.122884	−	0.992421i	$-0.539214\pi$
$877$	−6383.00	−0.245768	−0.122884	+	0.992421i	$0.539214\pi$
$878$	22360.0	0.859469
$879$	0	0
$880$	0	0
$881$	−28908.0	−1.10549	−0.552744	−	0.833351i	$-0.686419\pi$
$881$	−28908.0	−1.10549	−0.552744	+	0.833351i	$0.686419\pi$
$882$	0	0
$883$	−36893.0	−1.40606	−0.703028	−	0.711162i	$-0.748169\pi$
$883$	−36893.0	−1.40606	−0.703028	+	0.711162i	$0.748169\pi$
$884$	−34128.0	−1.29847
$885$	0	0
$886$	−1680.00	−0.0637028
$887$	−45528.0	−1.72343	−0.861714	−	0.507394i	$-0.830609\pi$
$887$	−45528.0	−1.72343	−0.861714	+	0.507394i	$0.830609\pi$
$888$	0	0
$889$	364.000	0.0137325
$890$	0	0
$891$	0	0
$892$	10576.0	0.396985
$893$	2244.00	0.0840903
$894$	0	0
$895$	0	0
$896$	896.000	0.0334077
$897$	0	0
$898$	−25560.0	−0.949831
$899$	−1920.00	−0.0712298
$900$	0	0
$901$	−38880.0	−1.43760
$902$	23040.0	0.850497
$903$	0	0
$904$	4128.00	0.151875
$905$	0	0
$906$	0	0
$907$	31201.0	1.14224	0.571120	−	0.820866i	$-0.306509\pi$
$907$	31201.0	1.14224	0.571120	+	0.820866i	$0.306509\pi$
$908$	−24096.0	−0.880676
$909$	0	0
$910$	0	0
$911$	−23856.0	−0.867601	−0.433801	−	0.901009i	$-0.642828\pi$
$911$	−23856.0	−0.867601	−0.433801	+	0.901009i	$0.642828\pi$
$912$	0	0
$913$	67680.0	2.45332
$914$	31316.0	1.13331
$915$	0	0
$916$	−17848.0	−0.643793
$917$	−336.000	−0.0121000
$918$	0	0
$919$	23492.0	0.843231	0.421616	−	0.906775i	$-0.361463\pi$
$919$	23492.0	0.843231	0.421616	+	0.906775i	$0.361463\pi$
$920$	0	0
$921$	0	0
$922$	31128.0	1.11187
$923$	−17064.0	−0.608525
$924$	0	0
$925$	0	0
$926$	8366.00	0.296894
$927$	0	0
$928$	−3072.00	−0.108667
$929$	15096.0	0.533136	0.266568	−	0.963816i	$-0.414110\pi$
$929$	15096.0	0.533136	0.266568	+	0.963816i	$0.414110\pi$
$930$	0	0
$931$	−3234.00	−0.113845
$932$	−4032.00	−0.141709
$933$	0	0
$934$	27864.0	0.976165
$935$	0	0
$936$	0	0
$937$	14965.0	0.521756	0.260878	−	0.965372i	$-0.415988\pi$
$937$	14965.0	0.521756	0.260878	+	0.965372i	$0.415988\pi$
$938$	−658.000	−0.0229045
$939$	0	0
$940$	0	0
$941$	−19524.0	−0.676370	−0.338185	−	0.941080i	$-0.609813\pi$
$941$	−19524.0	−0.676370	−0.338185	+	0.941080i	$0.609813\pi$
$942$	0	0
$943$	25344.0	0.875201
$944$	−2496.00	−0.0860571
$945$	0	0
$946$	58560.0	2.01263
$947$	11076.0	0.380065	0.190033	−	0.981778i	$-0.439141\pi$
$947$	11076.0	0.380065	0.190033	+	0.981778i	$0.439141\pi$
$948$	0	0
$949$	40369.0	1.38086
$950$	0	0
$951$	0	0
$952$	−6048.00	−0.205900
$953$	−44748.0	−1.52102	−0.760509	−	0.649328i	$-0.775050\pi$
$953$	−44748.0	−1.52102	−0.760509	+	0.649328i	$0.775050\pi$
$954$	0	0
$955$	0	0
$956$	20256.0	0.685278
$957$	0	0
$958$	5424.00	0.182924
$959$	16548.0	0.557208
$960$	0	0
$961$	−29391.0	−0.986573
$962$	26702.0	0.894914
$963$	0	0
$964$	25028.0	0.836201
$965$	0	0
$966$	0	0
$967$	−41519.0	−1.38072	−0.690362	−	0.723464i	$-0.742549\pi$
$967$	−41519.0	−1.38072	−0.690362	+	0.723464i	$0.742549\pi$
$968$	18152.0	0.602714
$969$	0	0
$970$	0	0
$971$	−28404.0	−0.938752	−0.469376	−	0.882999i	$-0.655521\pi$
$971$	−28404.0	−0.938752	−0.469376	+	0.882999i	$0.655521\pi$
$972$	0	0
$973$	1211.00	0.0399002
$974$	18878.0	0.621037
$975$	0	0
$976$	1328.00	0.0435535
$977$	43032.0	1.40913	0.704563	−	0.709642i	$-0.251143\pi$
$977$	43032.0	1.40913	0.704563	+	0.709642i	$0.251143\pi$
$978$	0	0
$979$	2160.00	0.0705147
$980$	0	0
$981$	0	0
$982$	23448.0	0.761971
$983$	18132.0	0.588322	0.294161	−	0.955756i	$-0.404960\pi$
$983$	18132.0	0.588322	0.294161	+	0.955756i	$0.404960\pi$
$984$	0	0
$985$	0	0
$986$	20736.0	0.669745
$987$	0	0
$988$	3476.00	0.111929
$989$	64416.0	2.07109
$990$	0	0
$991$	−44467.0	−1.42537	−0.712685	−	0.701485i	$-0.752521\pi$
$991$	−44467.0	−1.42537	−0.712685	+	0.701485i	$0.752521\pi$
$992$	640.000	0.0204839
$993$	0	0
$994$	−3024.00	−0.0964944
$995$	0	0
$996$	0	0
$997$	−19550.0	−0.621018	−0.310509	−	0.950570i	$-0.600499\pi$
$997$	−19550.0	−0.621018	−0.310509	+	0.950570i	$0.600499\pi$
$998$	−23936.0	−0.759199
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.a.v.1.1		1
3.2	odd	2		1350.4.a.h.1.1		1
5.2	odd	4		1350.4.c.a.649.2		2
5.3	odd	4		1350.4.c.a.649.1		2
5.4	even	2		54.4.a.a.1.1	✓	1
15.2	even	4		1350.4.c.t.649.1		2
15.8	even	4		1350.4.c.t.649.2		2
15.14	odd	2		54.4.a.d.1.1	yes	1
20.19	odd	2		432.4.a.b.1.1		1
40.19	odd	2		1728.4.a.bb.1.1		1
40.29	even	2		1728.4.a.ba.1.1		1
45.4	even	6		162.4.c.h.55.1		2
45.14	odd	6		162.4.c.a.55.1		2
45.29	odd	6		162.4.c.a.109.1		2
45.34	even	6		162.4.c.h.109.1		2
60.59	even	2		432.4.a.m.1.1		1
120.29	odd	2		1728.4.a.e.1.1		1
120.59	even	2		1728.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.a.a.1.1	✓	1	5.4	even	2
54.4.a.d.1.1	yes	1	15.14	odd	2
162.4.c.a.55.1		2	45.14	odd	6
162.4.c.a.109.1		2	45.29	odd	6
162.4.c.h.55.1		2	45.4	even	6
162.4.c.h.109.1		2	45.34	even	6
432.4.a.b.1.1		1	20.19	odd	2
432.4.a.m.1.1		1	60.59	even	2
1350.4.a.h.1.1		1	3.2	odd	2
1350.4.a.v.1.1		1	1.1	even	1	trivial
1350.4.c.a.649.1		2	5.3	odd	4
1350.4.c.a.649.2		2	5.2	odd	4
1350.4.c.t.649.1		2	15.2	even	4
1350.4.c.t.649.2		2	15.8	even	4
1728.4.a.e.1.1		1	120.29	odd	2
1728.4.a.f.1.1		1	120.59	even	2
1728.4.a.ba.1.1		1	40.29	even	2
1728.4.a.bb.1.1		1	40.19	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

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