LMFDB - Embedded newform 450.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$450 = 2 \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	450.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:	$26.5508595026$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$ -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} -26.0000 q^{7} +8.00000 q^{8} +28.0000 q^{11} -12.0000 q^{13} -52.0000 q^{14} +16.0000 q^{16} -64.0000 q^{17} -60.0000 q^{19} +56.0000 q^{22} -58.0000 q^{23} -24.0000 q^{26} -104.000 q^{28} -90.0000 q^{29} -128.000 q^{31} +32.0000 q^{32} -128.000 q^{34} -236.000 q^{37} -120.000 q^{38} -242.000 q^{41} -362.000 q^{43} +112.000 q^{44} -116.000 q^{46} +226.000 q^{47} +333.000 q^{49} -48.0000 q^{52} -108.000 q^{53} -208.000 q^{56} -180.000 q^{58} +20.0000 q^{59} +542.000 q^{61} -256.000 q^{62} +64.0000 q^{64} +434.000 q^{67} -256.000 q^{68} +1128.00 q^{71} -632.000 q^{73} -472.000 q^{74} -240.000 q^{76} -728.000 q^{77} -720.000 q^{79} -484.000 q^{82} -478.000 q^{83} -724.000 q^{86} +224.000 q^{88} +490.000 q^{89} +312.000 q^{91} -232.000 q^{92} +452.000 q^{94} -1456.00 q^{97} +666.000 q^{98} +O(q^{100})

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
$7$	−26.0000	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$7$	−26.0000	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	−12.0000	−0.256015	−0.128008	−	0.991773i	$-0.540858\pi$
$13$	−12.0000	−0.256015	−0.128008	+	0.991773i	$0.540858\pi$
$14$	−52.0000	−0.992685
$15$	0	0
$16$	16.0000	0.250000
$17$	−64.0000	−0.913075	−0.456538	−	0.889704i	$-0.650911\pi$
$17$	−64.0000	−0.913075	−0.456538	+	0.889704i	$0.650911\pi$
$18$	0	0
$19$	−60.0000	−0.724471	−0.362235	−	0.932087i	$-0.617986\pi$
$19$	−60.0000	−0.724471	−0.362235	+	0.932087i	$0.617986\pi$
$20$	0	0
$21$	0	0
$22$	56.0000	0.542693
$23$	−58.0000	−0.525819	−0.262909	−	0.964821i	$-0.584682\pi$
$23$	−58.0000	−0.525819	−0.262909	+	0.964821i	$0.584682\pi$
$24$	0	0
$25$	0	0
$26$	−24.0000	−0.181030
$27$	0	0
$28$	−104.000	−0.701934
$29$	−90.0000	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$30$	0	0
$31$	−128.000	−0.741596	−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000	−0.741596	−0.370798	+	0.928714i	$0.620916\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−128.000	−0.645642
$35$	0	0
$36$	0	0
$37$	−236.000	−1.04860	−0.524299	−	0.851534i	$-0.675673\pi$
$37$	−236.000	−1.04860	−0.524299	+	0.851534i	$0.675673\pi$
$38$	−120.000	−0.512278
$39$	0	0
$40$	0	0
$41$	−242.000	−0.921806	−0.460903	−	0.887450i	$-0.652474\pi$
$41$	−242.000	−0.921806	−0.460903	+	0.887450i	$0.652474\pi$
$42$	0	0
$43$	−362.000	−1.28383	−0.641913	−	0.766778i	$-0.721859\pi$
$43$	−362.000	−1.28383	−0.641913	+	0.766778i	$0.721859\pi$
$44$	112.000	0.383742
$45$	0	0
$46$	−116.000	−0.371810
$47$	226.000	0.701393	0.350697	−	0.936489i	$-0.385945\pi$
$47$	226.000	0.701393	0.350697	+	0.936489i	$0.385945\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	0	0
$51$	0	0
$52$	−48.0000	−0.128008
$53$	−108.000	−0.279905	−0.139952	−	0.990158i	$-0.544695\pi$
$53$	−108.000	−0.279905	−0.139952	+	0.990158i	$0.544695\pi$
$54$	0	0
$55$	0	0
$56$	−208.000	−0.496342
$57$	0	0
$58$	−180.000	−0.407503
$59$	20.0000	0.0441318	0.0220659	−	0.999757i	$-0.492976\pi$
$59$	20.0000	0.0441318	0.0220659	+	0.999757i	$0.492976\pi$
$60$	0	0
$61$	542.000	1.13764	0.568820	−	0.822462i	$-0.307400\pi$
$61$	542.000	1.13764	0.568820	+	0.822462i	$0.307400\pi$
$62$	−256.000	−0.524388
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	434.000	0.791366	0.395683	−	0.918387i	$-0.370508\pi$
$67$	434.000	0.791366	0.395683	+	0.918387i	$0.370508\pi$
$68$	−256.000	−0.456538
$69$	0	0
$70$	0	0
$71$	1128.00	1.88548	0.942739	−	0.333531i	$-0.108240\pi$
$71$	1128.00	1.88548	0.942739	+	0.333531i	$0.108240\pi$
$72$	0	0
$73$	−632.000	−1.01329	−0.506644	−	0.862155i	$-0.669114\pi$
$73$	−632.000	−1.01329	−0.506644	+	0.862155i	$0.669114\pi$
$74$	−472.000	−0.741471
$75$	0	0
$76$	−240.000	−0.362235
$77$	−728.000	−1.07745
$78$	0	0
$79$	−720.000	−1.02540	−0.512698	−	0.858569i	$-0.671354\pi$
$79$	−720.000	−1.02540	−0.512698	+	0.858569i	$0.671354\pi$
$80$	0	0
$81$	0	0
$82$	−484.000	−0.651815
$83$	−478.000	−0.632136	−0.316068	−	0.948736i	$-0.602363\pi$
$83$	−478.000	−0.632136	−0.316068	+	0.948736i	$0.602363\pi$
$84$	0	0
$85$	0	0
$86$	−724.000	−0.907801
$87$	0	0
$88$	224.000	0.271346
$89$	490.000	0.583594	0.291797	−	0.956480i	$-0.405747\pi$
$89$	490.000	0.583594	0.291797	+	0.956480i	$0.405747\pi$
$90$	0	0
$91$	312.000	0.359412
$92$	−232.000	−0.262909
$93$	0	0
$94$	452.000	0.495960
$95$	0	0
$96$	0	0
$97$	−1456.00	−1.52407	−0.762033	−	0.647538i	$-0.775799\pi$
$97$	−1456.00	−1.52407	−0.762033	+	0.647538i	$0.775799\pi$
$98$	666.000	0.686491
$99$	0	0
$100$	0	0
$101$	578.000	0.569437	0.284719	−	0.958611i	$-0.408100\pi$
$101$	578.000	0.569437	0.284719	+	0.958611i	$0.408100\pi$
$102$	0	0
$103$	−1462.00	−1.39859	−0.699297	−	0.714831i	$-0.746503\pi$
$103$	−1462.00	−1.39859	−0.699297	+	0.714831i	$0.746503\pi$
$104$	−96.0000	−0.0905151
$105$	0	0
$106$	−216.000	−0.197922
$107$	966.000	0.872773	0.436387	−	0.899759i	$-0.356258\pi$
$107$	966.000	0.872773	0.436387	+	0.899759i	$0.356258\pi$
$108$	0	0
$109$	370.000	0.325134	0.162567	−	0.986698i	$-0.448023\pi$
$109$	370.000	0.325134	0.162567	+	0.986698i	$0.448023\pi$
$110$	0	0
$111$	0	0
$112$	−416.000	−0.350967
$113$	−528.000	−0.439558	−0.219779	−	0.975550i	$-0.570534\pi$
$113$	−528.000	−0.439558	−0.219779	+	0.975550i	$0.570534\pi$
$114$	0	0
$115$	0	0
$116$	−360.000	−0.288148
$117$	0	0
$118$	40.0000	0.0312059
$119$	1664.00	1.28184
$120$	0	0
$121$	−547.000	−0.410969
$122$	1084.00	0.804432
$123$	0	0
$124$	−512.000	−0.370798
$125$	0	0
$126$	0	0
$127$	1534.00	1.07181	0.535907	−	0.844277i	$-0.319970\pi$
$127$	1534.00	1.07181	0.535907	+	0.844277i	$0.319970\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	−12.0000	−0.00800340	−0.00400170	−	0.999992i	$-0.501274\pi$
$131$	−12.0000	−0.00800340	−0.00400170	+	0.999992i	$0.501274\pi$
$132$	0	0
$133$	1560.00	1.01706
$134$	868.000	0.559580
$135$	0	0
$136$	−512.000	−0.322821
$137$	−1224.00	−0.763309	−0.381655	−	0.924305i	$-0.624646\pi$
$137$	−1224.00	−0.763309	−0.381655	+	0.924305i	$0.624646\pi$
$138$	0	0
$139$	3100.00	1.89164	0.945822	−	0.324685i	$-0.105258\pi$
$139$	3100.00	1.89164	0.945822	+	0.324685i	$0.105258\pi$
$140$	0	0
$141$	0	0
$142$	2256.00	1.33323
$143$	−336.000	−0.196488
$144$	0	0
$145$	0	0
$146$	−1264.00	−0.716503
$147$	0	0
$148$	−944.000	−0.524299
$149$	−250.000	−0.137455	−0.0687275	−	0.997635i	$-0.521894\pi$
$149$	−250.000	−0.137455	−0.0687275	+	0.997635i	$0.521894\pi$
$150$	0	0
$151$	2152.00	1.15978	0.579892	−	0.814694i	$-0.303095\pi$
$151$	2152.00	1.15978	0.579892	+	0.814694i	$0.303095\pi$
$152$	−480.000	−0.256139
$153$	0	0
$154$	−1456.00	−0.761869
$155$	0	0
$156$	0	0
$157$	524.000	0.266368	0.133184	−	0.991091i	$-0.457480\pi$
$157$	524.000	0.266368	0.133184	+	0.991091i	$0.457480\pi$
$158$	−1440.00	−0.725065
$159$	0	0
$160$	0	0
$161$	1508.00	0.738180
$162$	0	0
$163$	3518.00	1.69050	0.845249	−	0.534373i	$-0.179452\pi$
$163$	3518.00	1.69050	0.845249	+	0.534373i	$0.179452\pi$
$164$	−968.000	−0.460903
$165$	0	0
$166$	−956.000	−0.446988
$167$	−534.000	−0.247438	−0.123719	−	0.992317i	$-0.539482\pi$
$167$	−534.000	−0.247438	−0.123719	+	0.992317i	$0.539482\pi$
$168$	0	0
$169$	−2053.00	−0.934456
$170$	0	0
$171$	0	0
$172$	−1448.00	−0.641913
$173$	4252.00	1.86863	0.934317	−	0.356444i	$-0.116011\pi$
$173$	4252.00	1.86863	0.934317	+	0.356444i	$0.116011\pi$
$174$	0	0
$175$	0	0
$176$	448.000	0.191871
$177$	0	0
$178$	980.000	0.412664
$179$	−2500.00	−1.04390	−0.521952	−	0.852975i	$-0.674796\pi$
$179$	−2500.00	−1.04390	−0.521952	+	0.852975i	$0.674796\pi$
$180$	0	0
$181$	−2578.00	−1.05868	−0.529340	−	0.848410i	$-0.677561\pi$
$181$	−2578.00	−1.05868	−0.529340	+	0.848410i	$0.677561\pi$
$182$	624.000	0.254143
$183$	0	0
$184$	−464.000	−0.185905
$185$	0	0
$186$	0	0
$187$	−1792.00	−0.700770
$188$	904.000	0.350697
$189$	0	0
$190$	0	0
$191$	768.000	0.290945	0.145473	−	0.989362i	$-0.453530\pi$
$191$	768.000	0.290945	0.145473	+	0.989362i	$0.453530\pi$
$192$	0	0
$193$	2608.00	0.972684	0.486342	−	0.873769i	$-0.338331\pi$
$193$	2608.00	0.972684	0.486342	+	0.873769i	$0.338331\pi$
$194$	−2912.00	−1.07768
$195$	0	0
$196$	1332.00	0.485423
$197$	5116.00	1.85025	0.925127	−	0.379659i	$-0.123959\pi$
$197$	5116.00	1.85025	0.925127	+	0.379659i	$0.123959\pi$
$198$	0	0
$199$	−3480.00	−1.23965	−0.619826	−	0.784739i	$-0.712797\pi$
$199$	−3480.00	−1.23965	−0.619826	+	0.784739i	$0.712797\pi$
$200$	0	0
$201$	0	0
$202$	1156.00	0.402653
$203$	2340.00	0.809043
$204$	0	0
$205$	0	0
$206$	−2924.00	−0.988955
$207$	0	0
$208$	−192.000	−0.0640039
$209$	−1680.00	−0.556019
$210$	0	0
$211$	3132.00	1.02188	0.510938	−	0.859618i	$-0.329298\pi$
$211$	3132.00	1.02188	0.510938	+	0.859618i	$0.329298\pi$
$212$	−432.000	−0.139952
$213$	0	0
$214$	1932.00	0.617144
$215$	0	0
$216$	0	0
$217$	3328.00	1.04110
$218$	740.000	0.229904
$219$	0	0
$220$	0	0
$221$	768.000	0.233761
$222$	0	0
$223$	−62.0000	−0.0186181	−0.00930903	−	0.999957i	$-0.502963\pi$
$223$	−62.0000	−0.0186181	−0.00930903	+	0.999957i	$0.502963\pi$
$224$	−832.000	−0.248171
$225$	0	0
$226$	−1056.00	−0.310814
$227$	−5314.00	−1.55376	−0.776878	−	0.629651i	$-0.783198\pi$
$227$	−5314.00	−1.55376	−0.776878	+	0.629651i	$0.783198\pi$
$228$	0	0
$229$	−190.000	−0.0548277	−0.0274139	−	0.999624i	$-0.508727\pi$
$229$	−190.000	−0.0548277	−0.0274139	+	0.999624i	$0.508727\pi$
$230$	0	0
$231$	0	0
$232$	−720.000	−0.203751
$233$	−2408.00	−0.677053	−0.338526	−	0.940957i	$-0.609928\pi$
$233$	−2408.00	−0.677053	−0.338526	+	0.940957i	$0.609928\pi$
$234$	0	0
$235$	0	0
$236$	80.0000	0.0220659
$237$	0	0
$238$	3328.00	0.906396
$239$	5680.00	1.53727	0.768637	−	0.639685i	$-0.220935\pi$
$239$	5680.00	1.53727	0.768637	+	0.639685i	$0.220935\pi$
$240$	0	0
$241$	−278.000	−0.0743052	−0.0371526	−	0.999310i	$-0.511829\pi$
$241$	−278.000	−0.0743052	−0.0371526	+	0.999310i	$0.511829\pi$
$242$	−1094.00	−0.290599
$243$	0	0
$244$	2168.00	0.568820
$245$	0	0
$246$	0	0
$247$	720.000	0.185476
$248$	−1024.00	−0.262194
$249$	0	0
$250$	0	0
$251$	−3252.00	−0.817787	−0.408893	−	0.912582i	$-0.634085\pi$
$251$	−3252.00	−0.817787	−0.408893	+	0.912582i	$0.634085\pi$
$252$	0	0
$253$	−1624.00	−0.403557
$254$	3068.00	0.757888
$255$	0	0
$256$	256.000	0.0625000
$257$	1536.00	0.372813	0.186407	−	0.982473i	$-0.440316\pi$
$257$	1536.00	0.372813	0.186407	+	0.982473i	$0.440316\pi$
$258$	0	0
$259$	6136.00	1.47209
$260$	0	0
$261$	0	0
$262$	−24.0000	−0.00565926
$263$	−4858.00	−1.13900	−0.569500	−	0.821991i	$-0.692863\pi$
$263$	−4858.00	−1.13900	−0.569500	+	0.821991i	$0.692863\pi$
$264$	0	0
$265$	0	0
$266$	3120.00	0.719171
$267$	0	0
$268$	1736.00	0.395683
$269$	−2610.00	−0.591578	−0.295789	−	0.955253i	$-0.595583\pi$
$269$	−2610.00	−0.591578	−0.295789	+	0.955253i	$0.595583\pi$
$270$	0	0
$271$	−5168.00	−1.15843	−0.579213	−	0.815176i	$-0.696640\pi$
$271$	−5168.00	−1.15843	−0.579213	+	0.815176i	$0.696640\pi$
$272$	−1024.00	−0.228269
$273$	0	0
$274$	−2448.00	−0.539741
$275$	0	0
$276$	0	0
$277$	1924.00	0.417336	0.208668	−	0.977987i	$-0.433087\pi$
$277$	1924.00	0.417336	0.208668	+	0.977987i	$0.433087\pi$
$278$	6200.00	1.33759
$279$	0	0
$280$	0	0
$281$	−3042.00	−0.645803	−0.322901	−	0.946433i	$-0.604658\pi$
$281$	−3042.00	−0.645803	−0.322901	+	0.946433i	$0.604658\pi$
$282$	0	0
$283$	1718.00	0.360864	0.180432	−	0.983587i	$-0.442250\pi$
$283$	1718.00	0.360864	0.180432	+	0.983587i	$0.442250\pi$
$284$	4512.00	0.942739
$285$	0	0
$286$	−672.000	−0.138938
$287$	6292.00	1.29409
$288$	0	0
$289$	−817.000	−0.166294
$290$	0	0
$291$	0	0
$292$	−2528.00	−0.506644
$293$	2292.00	0.456997	0.228498	−	0.973544i	$-0.426618\pi$
$293$	2292.00	0.456997	0.228498	+	0.973544i	$0.426618\pi$
$294$	0	0
$295$	0	0
$296$	−1888.00	−0.370736
$297$	0	0
$298$	−500.000	−0.0971954
$299$	696.000	0.134618
$300$	0	0
$301$	9412.00	1.80232
$302$	4304.00	0.820091
$303$	0	0
$304$	−960.000	−0.181118
$305$	0	0
$306$	0	0
$307$	−5406.00	−1.00501	−0.502503	−	0.864576i	$-0.667587\pi$
$307$	−5406.00	−1.00501	−0.502503	+	0.864576i	$0.667587\pi$
$308$	−2912.00	−0.538723
$309$	0	0
$310$	0	0
$311$	5688.00	1.03710	0.518548	−	0.855048i	$-0.326473\pi$
$311$	5688.00	1.03710	0.518548	+	0.855048i	$0.326473\pi$
$312$	0	0
$313$	−7352.00	−1.32767	−0.663833	−	0.747881i	$-0.731072\pi$
$313$	−7352.00	−1.32767	−0.663833	+	0.747881i	$0.731072\pi$
$314$	1048.00	0.188351
$315$	0	0
$316$	−2880.00	−0.512698
$317$	−3484.00	−0.617290	−0.308645	−	0.951177i	$-0.599876\pi$
$317$	−3484.00	−0.617290	−0.308645	+	0.951177i	$0.599876\pi$
$318$	0	0
$319$	−2520.00	−0.442298
$320$	0	0
$321$	0	0
$322$	3016.00	0.521972
$323$	3840.00	0.661496
$324$	0	0
$325$	0	0
$326$	7036.00	1.19536
$327$	0	0
$328$	−1936.00	−0.325908
$329$	−5876.00	−0.984664
$330$	0	0
$331$	−7868.00	−1.30654	−0.653269	−	0.757125i	$-0.726603\pi$
$331$	−7868.00	−1.30654	−0.653269	+	0.757125i	$0.726603\pi$
$332$	−1912.00	−0.316068
$333$	0	0
$334$	−1068.00	−0.174965
$335$	0	0
$336$	0	0
$337$	−656.000	−0.106037	−0.0530187	−	0.998594i	$-0.516884\pi$
$337$	−656.000	−0.106037	−0.0530187	+	0.998594i	$0.516884\pi$
$338$	−4106.00	−0.660760
$339$	0	0
$340$	0	0
$341$	−3584.00	−0.569163
$342$	0	0
$343$	260.000	0.0409291
$344$	−2896.00	−0.453901
$345$	0	0
$346$	8504.00	1.32132
$347$	−5754.00	−0.890176	−0.445088	−	0.895487i	$-0.646828\pi$
$347$	−5754.00	−0.890176	−0.445088	+	0.895487i	$0.646828\pi$
$348$	0	0
$349$	−3110.00	−0.477004	−0.238502	−	0.971142i	$-0.576656\pi$
$349$	−3110.00	−0.477004	−0.238502	+	0.971142i	$0.576656\pi$
$350$	0	0
$351$	0	0
$352$	896.000	0.135673
$353$	−7808.00	−1.17727	−0.588637	−	0.808397i	$-0.700335\pi$
$353$	−7808.00	−1.17727	−0.588637	+	0.808397i	$0.700335\pi$
$354$	0	0
$355$	0	0
$356$	1960.00	0.291797
$357$	0	0
$358$	−5000.00	−0.738151
$359$	9240.00	1.35841	0.679204	−	0.733949i	$-0.262325\pi$
$359$	9240.00	1.35841	0.679204	+	0.733949i	$0.262325\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	−5156.00	−0.748600
$363$	0	0
$364$	1248.00	0.179706
$365$	0	0
$366$	0	0
$367$	3214.00	0.457137	0.228569	−	0.973528i	$-0.426595\pi$
$367$	3214.00	0.457137	0.228569	+	0.973528i	$0.426595\pi$
$368$	−928.000	−0.131455
$369$	0	0
$370$	0	0
$371$	2808.00	0.392949
$372$	0	0
$373$	348.000	0.0483077	0.0241538	−	0.999708i	$-0.492311\pi$
$373$	348.000	0.0483077	0.0241538	+	0.999708i	$0.492311\pi$
$374$	−3584.00	−0.495519
$375$	0	0
$376$	1808.00	0.247980
$377$	1080.00	0.147541
$378$	0	0
$379$	4940.00	0.669527	0.334764	−	0.942302i	$-0.391344\pi$
$379$	4940.00	0.669527	0.334764	+	0.942302i	$0.391344\pi$
$380$	0	0
$381$	0	0
$382$	1536.00	0.205729
$383$	6142.00	0.819430	0.409715	−	0.912214i	$-0.365628\pi$
$383$	6142.00	0.819430	0.409715	+	0.912214i	$0.365628\pi$
$384$	0	0
$385$	0	0
$386$	5216.00	0.687791
$387$	0	0
$388$	−5824.00	−0.762033
$389$	−3050.00	−0.397535	−0.198768	−	0.980047i	$-0.563694\pi$
$389$	−3050.00	−0.397535	−0.198768	+	0.980047i	$0.563694\pi$
$390$	0	0
$391$	3712.00	0.480112
$392$	2664.00	0.343246
$393$	0	0
$394$	10232.0	1.30833
$395$	0	0
$396$	0	0
$397$	−5396.00	−0.682160	−0.341080	−	0.940034i	$-0.610793\pi$
$397$	−5396.00	−0.682160	−0.341080	+	0.940034i	$0.610793\pi$
$398$	−6960.00	−0.876566
$399$	0	0
$400$	0	0
$401$	−14482.0	−1.80348	−0.901741	−	0.432276i	$-0.857711\pi$
$401$	−14482.0	−1.80348	−0.901741	+	0.432276i	$0.857711\pi$
$402$	0	0
$403$	1536.00	0.189860
$404$	2312.00	0.284719
$405$	0	0
$406$	4680.00	0.572080
$407$	−6608.00	−0.804782
$408$	0	0
$409$	−1090.00	−0.131778	−0.0658888	−	0.997827i	$-0.520988\pi$
$409$	−1090.00	−0.131778	−0.0658888	+	0.997827i	$0.520988\pi$
$410$	0	0
$411$	0	0
$412$	−5848.00	−0.699297
$413$	−520.000	−0.0619553
$414$	0	0
$415$	0	0
$416$	−384.000	−0.0452576
$417$	0	0
$418$	−3360.00	−0.393165
$419$	7180.00	0.837150	0.418575	−	0.908182i	$-0.362530\pi$
$419$	7180.00	0.837150	0.418575	+	0.908182i	$0.362530\pi$
$420$	0	0
$421$	−8138.00	−0.942095	−0.471047	−	0.882108i	$-0.656124\pi$
$421$	−8138.00	−0.942095	−0.471047	+	0.882108i	$0.656124\pi$
$422$	6264.00	0.722575
$423$	0	0
$424$	−864.000	−0.0989612
$425$	0	0
$426$	0	0
$427$	−14092.0	−1.59710
$428$	3864.00	0.436387
$429$	0	0
$430$	0	0
$431$	208.000	0.0232460	0.0116230	−	0.999932i	$-0.496300\pi$
$431$	208.000	0.0232460	0.0116230	+	0.999932i	$0.496300\pi$
$432$	0	0
$433$	−12992.0	−1.44193	−0.720965	−	0.692971i	$-0.756301\pi$
$433$	−12992.0	−1.44193	−0.720965	+	0.692971i	$0.756301\pi$
$434$	6656.00	0.736171
$435$	0	0
$436$	1480.00	0.162567
$437$	3480.00	0.380940
$438$	0	0
$439$	1080.00	0.117416	0.0587080	−	0.998275i	$-0.481302\pi$
$439$	1080.00	0.117416	0.0587080	+	0.998275i	$0.481302\pi$
$440$	0	0
$441$	0	0
$442$	1536.00	0.165294
$443$	−9078.00	−0.973609	−0.486805	−	0.873511i	$-0.661838\pi$
$443$	−9078.00	−0.973609	−0.486805	+	0.873511i	$0.661838\pi$
$444$	0	0
$445$	0	0
$446$	−124.000	−0.0131650
$447$	0	0
$448$	−1664.00	−0.175484
$449$	−14310.0	−1.50408	−0.752039	−	0.659119i	$-0.770929\pi$
$449$	−14310.0	−1.50408	−0.752039	+	0.659119i	$0.770929\pi$
$450$	0	0
$451$	−6776.00	−0.707471
$452$	−2112.00	−0.219779
$453$	0	0
$454$	−10628.0	−1.09867
$455$	0	0
$456$	0	0
$457$	2344.00	0.239929	0.119965	−	0.992778i	$-0.461722\pi$
$457$	2344.00	0.239929	0.119965	+	0.992778i	$0.461722\pi$
$458$	−380.000	−0.0387691
$459$	0	0
$460$	0	0
$461$	−11382.0	−1.14992	−0.574959	−	0.818182i	$-0.694982\pi$
$461$	−11382.0	−1.14992	−0.574959	+	0.818182i	$0.694982\pi$
$462$	0	0
$463$	−16062.0	−1.61223	−0.806117	−	0.591756i	$-0.798435\pi$
$463$	−16062.0	−1.61223	−0.806117	+	0.591756i	$0.798435\pi$
$464$	−1440.00	−0.144074
$465$	0	0
$466$	−4816.00	−0.478749
$467$	17166.0	1.70096	0.850479	−	0.526008i	$-0.176312\pi$
$467$	17166.0	1.70096	0.850479	+	0.526008i	$0.176312\pi$
$468$	0	0
$469$	−11284.0	−1.11097
$470$	0	0
$471$	0	0
$472$	160.000	0.0156030
$473$	−10136.0	−0.985315
$474$	0	0
$475$	0	0
$476$	6656.00	0.640919
$477$	0	0
$478$	11360.0	1.08702
$479$	−7520.00	−0.717323	−0.358661	−	0.933468i	$-0.616767\pi$
$479$	−7520.00	−0.717323	−0.358661	+	0.933468i	$0.616767\pi$
$480$	0	0
$481$	2832.00	0.268458
$482$	−556.000	−0.0525417
$483$	0	0
$484$	−2188.00	−0.205485
$485$	0	0
$486$	0	0
$487$	11814.0	1.09927	0.549634	−	0.835406i	$-0.314767\pi$
$487$	11814.0	1.09927	0.549634	+	0.835406i	$0.314767\pi$
$488$	4336.00	0.402216
$489$	0	0
$490$	0	0
$491$	−14052.0	−1.29156	−0.645782	−	0.763522i	$-0.723468\pi$
$491$	−14052.0	−1.29156	−0.645782	+	0.763522i	$0.723468\pi$
$492$	0	0
$493$	5760.00	0.526202
$494$	1440.00	0.131151
$495$	0	0
$496$	−2048.00	−0.185399
$497$	−29328.0	−2.64696
$498$	0	0
$499$	7620.00	0.683603	0.341802	−	0.939772i	$-0.388963\pi$
$499$	7620.00	0.683603	0.341802	+	0.939772i	$0.388963\pi$
$500$	0	0
$501$	0	0
$502$	−6504.00	−0.578262
$503$	−1818.00	−0.161154	−0.0805772	−	0.996748i	$-0.525676\pi$
$503$	−1818.00	−0.161154	−0.0805772	+	0.996748i	$0.525676\pi$
$504$	0	0
$505$	0	0
$506$	−3248.00	−0.285358
$507$	0	0
$508$	6136.00	0.535907
$509$	−17850.0	−1.55440	−0.777198	−	0.629256i	$-0.783360\pi$
$509$	−17850.0	−1.55440	−0.777198	+	0.629256i	$0.783360\pi$
$510$	0	0
$511$	16432.0	1.42252
$512$	512.000	0.0441942
$513$	0	0
$514$	3072.00	0.263619
$515$	0	0
$516$	0	0
$517$	6328.00	0.538308
$518$	12272.0	1.04093
$519$	0	0
$520$	0	0
$521$	19238.0	1.61772	0.808860	−	0.588001i	$-0.200085\pi$
$521$	19238.0	1.61772	0.808860	+	0.588001i	$0.200085\pi$
$522$	0	0
$523$	6278.00	0.524891	0.262445	−	0.964947i	$-0.415471\pi$
$523$	6278.00	0.524891	0.262445	+	0.964947i	$0.415471\pi$
$524$	−48.0000	−0.00400170
$525$	0	0
$526$	−9716.00	−0.805395
$527$	8192.00	0.677133
$528$	0	0
$529$	−8803.00	−0.723514
$530$	0	0
$531$	0	0
$532$	6240.00	0.508531
$533$	2904.00	0.235997
$534$	0	0
$535$	0	0
$536$	3472.00	0.279790
$537$	0	0
$538$	−5220.00	−0.418309
$539$	9324.00	0.745108
$540$	0	0
$541$	−9818.00	−0.780238	−0.390119	−	0.920764i	$-0.627566\pi$
$541$	−9818.00	−0.780238	−0.390119	+	0.920764i	$0.627566\pi$
$542$	−10336.0	−0.819131
$543$	0	0
$544$	−2048.00	−0.161410
$545$	0	0
$546$	0	0
$547$	12514.0	0.978172	0.489086	−	0.872236i	$-0.337330\pi$
$547$	12514.0	0.978172	0.489086	+	0.872236i	$0.337330\pi$
$548$	−4896.00	−0.381655
$549$	0	0
$550$	0	0
$551$	5400.00	0.417509
$552$	0	0
$553$	18720.0	1.43952
$554$	3848.00	0.295101
$555$	0	0
$556$	12400.0	0.945822
$557$	10596.0	0.806045	0.403022	−	0.915190i	$-0.367960\pi$
$557$	10596.0	0.806045	0.403022	+	0.915190i	$0.367960\pi$
$558$	0	0
$559$	4344.00	0.328679
$560$	0	0
$561$	0	0
$562$	−6084.00	−0.456651
$563$	14002.0	1.04816	0.524080	−	0.851669i	$-0.324409\pi$
$563$	14002.0	1.04816	0.524080	+	0.851669i	$0.324409\pi$
$564$	0	0
$565$	0	0
$566$	3436.00	0.255169
$567$	0	0
$568$	9024.00	0.666617
$569$	7330.00	0.540052	0.270026	−	0.962853i	$-0.412968\pi$
$569$	7330.00	0.540052	0.270026	+	0.962853i	$0.412968\pi$
$570$	0	0
$571$	5812.00	0.425963	0.212981	−	0.977056i	$-0.431683\pi$
$571$	5812.00	0.425963	0.212981	+	0.977056i	$0.431683\pi$
$572$	−1344.00	−0.0982438
$573$	0	0
$574$	12584.0	0.915063
$575$	0	0
$576$	0	0
$577$	−16736.0	−1.20750	−0.603751	−	0.797173i	$-0.706328\pi$
$577$	−16736.0	−1.20750	−0.603751	+	0.797173i	$0.706328\pi$
$578$	−1634.00	−0.117587
$579$	0	0
$580$	0	0
$581$	12428.0	0.887436
$582$	0	0
$583$	−3024.00	−0.214822
$584$	−5056.00	−0.358251
$585$	0	0
$586$	4584.00	0.323146
$587$	−7434.00	−0.522716	−0.261358	−	0.965242i	$-0.584170\pi$
$587$	−7434.00	−0.522716	−0.261358	+	0.965242i	$0.584170\pi$
$588$	0	0
$589$	7680.00	0.537265
$590$	0	0
$591$	0	0
$592$	−3776.00	−0.262150
$593$	25872.0	1.79163	0.895814	−	0.444429i	$-0.146593\pi$
$593$	25872.0	1.79163	0.895814	+	0.444429i	$0.146593\pi$
$594$	0	0
$595$	0	0
$596$	−1000.00	−0.0687275
$597$	0	0
$598$	1392.00	0.0951892
$599$	3720.00	0.253748	0.126874	−	0.991919i	$-0.459506\pi$
$599$	3720.00	0.253748	0.126874	+	0.991919i	$0.459506\pi$
$600$	0	0
$601$	−12958.0	−0.879481	−0.439740	−	0.898125i	$-0.644930\pi$
$601$	−12958.0	−0.879481	−0.439740	+	0.898125i	$0.644930\pi$
$602$	18824.0	1.27443
$603$	0	0
$604$	8608.00	0.579892
$605$	0	0
$606$	0	0
$607$	7214.00	0.482384	0.241192	−	0.970477i	$-0.422462\pi$
$607$	7214.00	0.482384	0.241192	+	0.970477i	$0.422462\pi$
$608$	−1920.00	−0.128070
$609$	0	0
$610$	0	0
$611$	−2712.00	−0.179568
$612$	0	0
$613$	4828.00	0.318109	0.159055	−	0.987270i	$-0.449155\pi$
$613$	4828.00	0.318109	0.159055	+	0.987270i	$0.449155\pi$
$614$	−10812.0	−0.710646
$615$	0	0
$616$	−5824.00	−0.380934
$617$	27656.0	1.80452	0.902260	−	0.431193i	$-0.141907\pi$
$617$	27656.0	1.80452	0.902260	+	0.431193i	$0.141907\pi$
$618$	0	0
$619$	−21220.0	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$619$	−21220.0	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$620$	0	0
$621$	0	0
$622$	11376.0	0.733338
$623$	−12740.0	−0.819289
$624$	0	0
$625$	0	0
$626$	−14704.0	−0.938802
$627$	0	0
$628$	2096.00	0.133184
$629$	15104.0	0.957450
$630$	0	0
$631$	17672.0	1.11491	0.557457	−	0.830206i	$-0.311777\pi$
$631$	17672.0	1.11491	0.557457	+	0.830206i	$0.311777\pi$
$632$	−5760.00	−0.362532
$633$	0	0
$634$	−6968.00	−0.436490
$635$	0	0
$636$	0	0
$637$	−3996.00	−0.248551
$638$	−5040.00	−0.312752
$639$	0	0
$640$	0	0
$641$	−7322.00	−0.451173	−0.225586	−	0.974223i	$-0.572430\pi$
$641$	−7322.00	−0.451173	−0.225586	+	0.974223i	$0.572430\pi$
$642$	0	0
$643$	8238.00	0.505249	0.252624	−	0.967564i	$-0.418706\pi$
$643$	8238.00	0.505249	0.252624	+	0.967564i	$0.418706\pi$
$644$	6032.00	0.369090
$645$	0	0
$646$	7680.00	0.467749
$647$	6426.00	0.390467	0.195233	−	0.980757i	$-0.437454\pi$
$647$	6426.00	0.390467	0.195233	+	0.980757i	$0.437454\pi$
$648$	0	0
$649$	560.000	0.0338705
$650$	0	0
$651$	0	0
$652$	14072.0	0.845249
$653$	−5908.00	−0.354055	−0.177027	−	0.984206i	$-0.556648\pi$
$653$	−5908.00	−0.354055	−0.177027	+	0.984206i	$0.556648\pi$
$654$	0	0
$655$	0	0
$656$	−3872.00	−0.230452
$657$	0	0
$658$	−11752.0	−0.696262
$659$	26780.0	1.58301	0.791503	−	0.611166i	$-0.209299\pi$
$659$	26780.0	1.58301	0.791503	+	0.611166i	$0.209299\pi$
$660$	0	0
$661$	−24538.0	−1.44390	−0.721950	−	0.691945i	$-0.756754\pi$
$661$	−24538.0	−1.44390	−0.721950	+	0.691945i	$0.756754\pi$
$662$	−15736.0	−0.923863
$663$	0	0
$664$	−3824.00	−0.223494
$665$	0	0
$666$	0	0
$667$	5220.00	0.303027
$668$	−2136.00	−0.123719
$669$	0	0
$670$	0	0
$671$	15176.0	0.873119
$672$	0	0
$673$	28848.0	1.65232	0.826158	−	0.563439i	$-0.190522\pi$
$673$	28848.0	1.65232	0.826158	+	0.563439i	$0.190522\pi$
$674$	−1312.00	−0.0749798
$675$	0	0
$676$	−8212.00	−0.467228
$677$	−26884.0	−1.52620	−0.763099	−	0.646282i	$-0.776323\pi$
$677$	−26884.0	−1.52620	−0.763099	+	0.646282i	$0.776323\pi$
$678$	0	0
$679$	37856.0	2.13959
$680$	0	0
$681$	0	0
$682$	−7168.00	−0.402459
$683$	14282.0	0.800125	0.400063	−	0.916488i	$-0.368988\pi$
$683$	14282.0	0.800125	0.400063	+	0.916488i	$0.368988\pi$
$684$	0	0
$685$	0	0
$686$	520.000	0.0289412
$687$	0	0
$688$	−5792.00	−0.320956
$689$	1296.00	0.0716599
$690$	0	0
$691$	−3428.00	−0.188723	−0.0943613	−	0.995538i	$-0.530081\pi$
$691$	−3428.00	−0.188723	−0.0943613	+	0.995538i	$0.530081\pi$
$692$	17008.0	0.934317
$693$	0	0
$694$	−11508.0	−0.629449
$695$	0	0
$696$	0	0
$697$	15488.0	0.841678
$698$	−6220.00	−0.337293
$699$	0	0
$700$	0	0
$701$	−26942.0	−1.45162	−0.725810	−	0.687895i	$-0.758535\pi$
$701$	−26942.0	−1.45162	−0.725810	+	0.687895i	$0.758535\pi$
$702$	0	0
$703$	14160.0	0.759679
$704$	1792.00	0.0959354
$705$	0	0
$706$	−15616.0	−0.832459
$707$	−15028.0	−0.799415
$708$	0	0
$709$	−1950.00	−0.103292	−0.0516458	−	0.998665i	$-0.516447\pi$
$709$	−1950.00	−0.103292	−0.0516458	+	0.998665i	$0.516447\pi$
$710$	0	0
$711$	0	0
$712$	3920.00	0.206332
$713$	7424.00	0.389945
$714$	0	0
$715$	0	0
$716$	−10000.0	−0.521952
$717$	0	0
$718$	18480.0	0.960540
$719$	−12080.0	−0.626576	−0.313288	−	0.949658i	$-0.601430\pi$
$719$	−12080.0	−0.626576	−0.313288	+	0.949658i	$0.601430\pi$
$720$	0	0
$721$	38012.0	1.96344
$722$	−6518.00	−0.335976
$723$	0	0
$724$	−10312.0	−0.529340
$725$	0	0
$726$	0	0
$727$	−17226.0	−0.878785	−0.439393	−	0.898295i	$-0.644806\pi$
$727$	−17226.0	−0.878785	−0.439393	+	0.898295i	$0.644806\pi$
$728$	2496.00	0.127071
$729$	0	0
$730$	0	0
$731$	23168.0	1.17223
$732$	0	0
$733$	788.000	0.0397073	0.0198536	−	0.999803i	$-0.493680\pi$
$733$	788.000	0.0397073	0.0198536	+	0.999803i	$0.493680\pi$
$734$	6428.00	0.323245
$735$	0	0
$736$	−1856.00	−0.0929525
$737$	12152.0	0.607360
$738$	0	0
$739$	−2060.00	−0.102542	−0.0512709	−	0.998685i	$-0.516327\pi$
$739$	−2060.00	−0.102542	−0.0512709	+	0.998685i	$0.516327\pi$
$740$	0	0
$741$	0	0
$742$	5616.00	0.277857
$743$	−3258.00	−0.160867	−0.0804337	−	0.996760i	$-0.525631\pi$
$743$	−3258.00	−0.160867	−0.0804337	+	0.996760i	$0.525631\pi$
$744$	0	0
$745$	0	0
$746$	696.000	0.0341587
$747$	0	0
$748$	−7168.00	−0.350385
$749$	−25116.0	−1.22526
$750$	0	0
$751$	−4528.00	−0.220012	−0.110006	−	0.993931i	$-0.535087\pi$
$751$	−4528.00	−0.220012	−0.110006	+	0.993931i	$0.535087\pi$
$752$	3616.00	0.175348
$753$	0	0
$754$	2160.00	0.104327
$755$	0	0
$756$	0	0
$757$	−18236.0	−0.875560	−0.437780	−	0.899082i	$-0.644235\pi$
$757$	−18236.0	−0.875560	−0.437780	+	0.899082i	$0.644235\pi$
$758$	9880.00	0.473427
$759$	0	0
$760$	0	0
$761$	18678.0	0.889720	0.444860	−	0.895600i	$-0.353253\pi$
$761$	18678.0	0.889720	0.444860	+	0.895600i	$0.353253\pi$
$762$	0	0
$763$	−9620.00	−0.456445
$764$	3072.00	0.145473
$765$	0	0
$766$	12284.0	0.579424
$767$	−240.000	−0.0112984
$768$	0	0
$769$	27390.0	1.28441	0.642203	−	0.766534i	$-0.278020\pi$
$769$	27390.0	1.28441	0.642203	+	0.766534i	$0.278020\pi$
$770$	0	0
$771$	0	0
$772$	10432.0	0.486342
$773$	9252.00	0.430493	0.215247	−	0.976560i	$-0.430944\pi$
$773$	9252.00	0.430493	0.215247	+	0.976560i	$0.430944\pi$
$774$	0	0
$775$	0	0
$776$	−11648.0	−0.538839
$777$	0	0
$778$	−6100.00	−0.281100
$779$	14520.0	0.667822
$780$	0	0
$781$	31584.0	1.44707
$782$	7424.00	0.339491
$783$	0	0
$784$	5328.00	0.242711
$785$	0	0
$786$	0	0
$787$	−5726.00	−0.259352	−0.129676	−	0.991556i	$-0.541394\pi$
$787$	−5726.00	−0.259352	−0.129676	+	0.991556i	$0.541394\pi$
$788$	20464.0	0.925127
$789$	0	0
$790$	0	0
$791$	13728.0	0.617082
$792$	0	0
$793$	−6504.00	−0.291253
$794$	−10792.0	−0.482360
$795$	0	0
$796$	−13920.0	−0.619826
$797$	27236.0	1.21048	0.605238	−	0.796045i	$-0.293078\pi$
$797$	27236.0	1.21048	0.605238	+	0.796045i	$0.293078\pi$
$798$	0	0
$799$	−14464.0	−0.640425
$800$	0	0
$801$	0	0
$802$	−28964.0	−1.27525
$803$	−17696.0	−0.777682
$804$	0	0
$805$	0	0
$806$	3072.00	0.134251
$807$	0	0
$808$	4624.00	0.201326
$809$	−10950.0	−0.475873	−0.237937	−	0.971281i	$-0.576471\pi$
$809$	−10950.0	−0.475873	−0.237937	+	0.971281i	$0.576471\pi$
$810$	0	0
$811$	−8828.00	−0.382236	−0.191118	−	0.981567i	$-0.561211\pi$
$811$	−8828.00	−0.382236	−0.191118	+	0.981567i	$0.561211\pi$
$812$	9360.00	0.404522
$813$	0	0
$814$	−13216.0	−0.569067
$815$	0	0
$816$	0	0
$817$	21720.0	0.930094
$818$	−2180.00	−0.0931808
$819$	0	0
$820$	0	0
$821$	16058.0	0.682616	0.341308	−	0.939951i	$-0.389130\pi$
$821$	16058.0	0.682616	0.341308	+	0.939951i	$0.389130\pi$
$822$	0	0
$823$	−41862.0	−1.77305	−0.886523	−	0.462684i	$-0.846887\pi$
$823$	−41862.0	−1.77305	−0.886523	+	0.462684i	$0.846887\pi$
$824$	−11696.0	−0.494478
$825$	0	0
$826$	−1040.00	−0.0438090
$827$	−12154.0	−0.511047	−0.255524	−	0.966803i	$-0.582248\pi$
$827$	−12154.0	−0.511047	−0.255524	+	0.966803i	$0.582248\pi$
$828$	0	0
$829$	−15390.0	−0.644773	−0.322386	−	0.946608i	$-0.604485\pi$
$829$	−15390.0	−0.644773	−0.322386	+	0.946608i	$0.604485\pi$
$830$	0	0
$831$	0	0
$832$	−768.000	−0.0320019
$833$	−21312.0	−0.886455
$834$	0	0
$835$	0	0
$836$	−6720.00	−0.278010
$837$	0	0
$838$	14360.0	0.591955
$839$	4280.00	0.176117	0.0880584	−	0.996115i	$-0.471934\pi$
$839$	4280.00	0.176117	0.0880584	+	0.996115i	$0.471934\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	−16276.0	−0.666162
$843$	0	0
$844$	12528.0	0.510938
$845$	0	0
$846$	0	0
$847$	14222.0	0.576947
$848$	−1728.00	−0.0699761
$849$	0	0
$850$	0	0
$851$	13688.0	0.551373
$852$	0	0
$853$	−14452.0	−0.580102	−0.290051	−	0.957011i	$-0.593672\pi$
$853$	−14452.0	−0.580102	−0.290051	+	0.957011i	$0.593672\pi$
$854$	−28184.0	−1.12932
$855$	0	0
$856$	7728.00	0.308572
$857$	−22584.0	−0.900181	−0.450090	−	0.892983i	$-0.648608\pi$
$857$	−22584.0	−0.900181	−0.450090	+	0.892983i	$0.648608\pi$
$858$	0	0
$859$	−26740.0	−1.06212	−0.531058	−	0.847336i	$-0.678205\pi$
$859$	−26740.0	−1.06212	−0.531058	+	0.847336i	$0.678205\pi$
$860$	0	0
$861$	0	0
$862$	416.000	0.0164374
$863$	−498.000	−0.0196432	−0.00982162	−	0.999952i	$-0.503126\pi$
$863$	−498.000	−0.0196432	−0.00982162	+	0.999952i	$0.503126\pi$
$864$	0	0
$865$	0	0
$866$	−25984.0	−1.01960
$867$	0	0
$868$	13312.0	0.520552
$869$	−20160.0	−0.786975
$870$	0	0
$871$	−5208.00	−0.202602
$872$	2960.00	0.114952
$873$	0	0
$874$	6960.00	0.269366
$875$	0	0
$876$	0	0
$877$	13244.0	0.509941	0.254970	−	0.966949i	$-0.417934\pi$
$877$	13244.0	0.509941	0.254970	+	0.966949i	$0.417934\pi$
$878$	2160.00	0.0830256
$879$	0	0
$880$	0	0
$881$	−40842.0	−1.56186	−0.780932	−	0.624616i	$-0.785255\pi$
$881$	−40842.0	−1.56186	−0.780932	+	0.624616i	$0.785255\pi$
$882$	0	0
$883$	12078.0	0.460314	0.230157	−	0.973154i	$-0.426076\pi$
$883$	12078.0	0.460314	0.230157	+	0.973154i	$0.426076\pi$
$884$	3072.00	0.116881
$885$	0	0
$886$	−18156.0	−0.688446
$887$	−18294.0	−0.692506	−0.346253	−	0.938141i	$-0.612546\pi$
$887$	−18294.0	−0.692506	−0.346253	+	0.938141i	$0.612546\pi$
$888$	0	0
$889$	−39884.0	−1.50469
$890$	0	0
$891$	0	0
$892$	−248.000	−0.00930903
$893$	−13560.0	−0.508139
$894$	0	0
$895$	0	0
$896$	−3328.00	−0.124086
$897$	0	0
$898$	−28620.0	−1.06354
$899$	11520.0	0.427379
$900$	0	0
$901$	6912.00	0.255574
$902$	−13552.0	−0.500257
$903$	0	0
$904$	−4224.00	−0.155407
$905$	0	0
$906$	0	0
$907$	−22566.0	−0.826121	−0.413060	−	0.910704i	$-0.635540\pi$
$907$	−22566.0	−0.826121	−0.413060	+	0.910704i	$0.635540\pi$
$908$	−21256.0	−0.776878
$909$	0	0
$910$	0	0
$911$	6768.00	0.246140	0.123070	−	0.992398i	$-0.460726\pi$
$911$	6768.00	0.246140	0.123070	+	0.992398i	$0.460726\pi$
$912$	0	0
$913$	−13384.0	−0.485154
$914$	4688.00	0.169656
$915$	0	0
$916$	−760.000	−0.0274139
$917$	312.000	0.0112357
$918$	0	0
$919$	22200.0	0.796856	0.398428	−	0.917200i	$-0.369556\pi$
$919$	22200.0	0.796856	0.398428	+	0.917200i	$0.369556\pi$
$920$	0	0
$921$	0	0
$922$	−22764.0	−0.813115
$923$	−13536.0	−0.482712
$924$	0	0
$925$	0	0
$926$	−32124.0	−1.14002
$927$	0	0
$928$	−2880.00	−0.101876
$929$	6330.00	0.223553	0.111776	−	0.993733i	$-0.464346\pi$
$929$	6330.00	0.223553	0.111776	+	0.993733i	$0.464346\pi$
$930$	0	0
$931$	−19980.0	−0.703349
$932$	−9632.00	−0.338526
$933$	0	0
$934$	34332.0	1.20276
$935$	0	0
$936$	0	0
$937$	19544.0	0.681403	0.340702	−	0.940172i	$-0.389335\pi$
$937$	19544.0	0.681403	0.340702	+	0.940172i	$0.389335\pi$
$938$	−22568.0	−0.785577
$939$	0	0
$940$	0	0
$941$	9898.00	0.342896	0.171448	−	0.985193i	$-0.445155\pi$
$941$	9898.00	0.342896	0.171448	+	0.985193i	$0.445155\pi$
$942$	0	0
$943$	14036.0	0.484703
$944$	320.000	0.0110330
$945$	0	0
$946$	−20272.0	−0.696723
$947$	41406.0	1.42082	0.710409	−	0.703789i	$-0.248510\pi$
$947$	41406.0	1.42082	0.710409	+	0.703789i	$0.248510\pi$
$948$	0	0
$949$	7584.00	0.259417
$950$	0	0
$951$	0	0
$952$	13312.0	0.453198
$953$	25432.0	0.864453	0.432226	−	0.901765i	$-0.357728\pi$
$953$	25432.0	0.864453	0.432226	+	0.901765i	$0.357728\pi$
$954$	0	0
$955$	0	0
$956$	22720.0	0.768637
$957$	0	0
$958$	−15040.0	−0.507224
$959$	31824.0	1.07159
$960$	0	0
$961$	−13407.0	−0.450035
$962$	5664.00	0.189828
$963$	0	0
$964$	−1112.00	−0.0371526
$965$	0	0
$966$	0	0
$967$	−12106.0	−0.402588	−0.201294	−	0.979531i	$-0.564515\pi$
$967$	−12106.0	−0.402588	−0.201294	+	0.979531i	$0.564515\pi$
$968$	−4376.00	−0.145300
$969$	0	0
$970$	0	0
$971$	−7812.00	−0.258186	−0.129093	−	0.991632i	$-0.541207\pi$
$971$	−7812.00	−0.258186	−0.129093	+	0.991632i	$0.541207\pi$
$972$	0	0
$973$	−80600.0	−2.65562
$974$	23628.0	0.777300
$975$	0	0
$976$	8672.00	0.284410
$977$	12576.0	0.411814	0.205907	−	0.978572i	$-0.433986\pi$
$977$	12576.0	0.411814	0.205907	+	0.978572i	$0.433986\pi$
$978$	0	0
$979$	13720.0	0.447899
$980$	0	0
$981$	0	0
$982$	−28104.0	−0.913274
$983$	4342.00	0.140883	0.0704417	−	0.997516i	$-0.477559\pi$
$983$	4342.00	0.140883	0.0704417	+	0.997516i	$0.477559\pi$
$984$	0	0
$985$	0	0
$986$	11520.0	0.372081
$987$	0	0
$988$	2880.00	0.0927379
$989$	20996.0	0.675060
$990$	0	0
$991$	26272.0	0.842137	0.421068	−	0.907029i	$-0.361655\pi$
$991$	26272.0	0.842137	0.421068	+	0.907029i	$0.361655\pi$
$992$	−4096.00	−0.131097
$993$	0	0
$994$	−58656.0	−1.87169
$995$	0	0
$996$	0	0
$997$	−44796.0	−1.42297	−0.711486	−	0.702700i	$-0.751978\pi$
$997$	−44796.0	−1.42297	−0.711486	+	0.702700i	$0.751978\pi$
$998$	15240.0	0.483381
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.k.1.1		1
3.2	odd	2		50.4.a.b.1.1		1
5.2	odd	4		90.4.c.b.19.2		2
5.3	odd	4		90.4.c.b.19.1		2
5.4	even	2		450.4.a.j.1.1		1
12.11	even	2		400.4.a.n.1.1		1
15.2	even	4		10.4.b.a.9.1	✓	2
15.8	even	4		10.4.b.a.9.2	yes	2
15.14	odd	2		50.4.a.d.1.1		1
20.3	even	4		720.4.f.f.289.2		2
20.7	even	4		720.4.f.f.289.1		2
21.20	even	2		2450.4.a.o.1.1		1
24.5	odd	2		1600.4.a.bh.1.1		1
24.11	even	2		1600.4.a.t.1.1		1
60.23	odd	4		80.4.c.a.49.2		2
60.47	odd	4		80.4.c.a.49.1		2
60.59	even	2		400.4.a.h.1.1		1
105.62	odd	4		490.4.c.b.99.1		2
105.83	odd	4		490.4.c.b.99.2		2
105.104	even	2		2450.4.a.bb.1.1		1
120.29	odd	2		1600.4.a.u.1.1		1
120.53	even	4		320.4.c.d.129.2		2
120.59	even	2		1600.4.a.bg.1.1		1
120.77	even	4		320.4.c.d.129.1		2
120.83	odd	4		320.4.c.c.129.1		2
120.107	odd	4		320.4.c.c.129.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.b.a.9.1	✓	2	15.2	even	4
10.4.b.a.9.2	yes	2	15.8	even	4
50.4.a.b.1.1		1	3.2	odd	2
50.4.a.d.1.1		1	15.14	odd	2
80.4.c.a.49.1		2	60.47	odd	4
80.4.c.a.49.2		2	60.23	odd	4
90.4.c.b.19.1		2	5.3	odd	4
90.4.c.b.19.2		2	5.2	odd	4
320.4.c.c.129.1		2	120.83	odd	4
320.4.c.c.129.2		2	120.107	odd	4
320.4.c.d.129.1		2	120.77	even	4
320.4.c.d.129.2		2	120.53	even	4
400.4.a.h.1.1		1	60.59	even	2
400.4.a.n.1.1		1	12.11	even	2
450.4.a.j.1.1		1	5.4	even	2
450.4.a.k.1.1		1	1.1	even	1	trivial
490.4.c.b.99.1		2	105.62	odd	4
490.4.c.b.99.2		2	105.83	odd	4
720.4.f.f.289.1		2	20.7	even	4
720.4.f.f.289.2		2	20.3	even	4
1600.4.a.t.1.1		1	24.11	even	2
1600.4.a.u.1.1		1	120.29	odd	2
1600.4.a.bg.1.1		1	120.59	even	2
1600.4.a.bh.1.1		1	24.5	odd	2
2450.4.a.o.1.1		1	21.20	even	2
2450.4.a.bb.1.1		1	105.104	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	450.4.a.k.1.1

Level	$450$
Weight	$4$
Character	450.1
Self dual	yes
Analytic conductor	$26.551$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$