LMFDB - Embedded newform 2400.4.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{21}, \sqrt{141})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 61x^{2} + 154x + 28 $ x^4 - x^3 - 61x^2 + 154x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.60753$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -3.13836 q^{7} +9.00000 q^{9} -54.7891 q^{11} -66.0294 q^{13} -47.6991 q^{17} -139.149 q^{19} +9.41507 q^{21} +100.000 q^{23} -27.0000 q^{27} -75.4151 q^{29} -212.067 q^{31} +164.367 q^{33} -417.245 q^{37} +198.088 q^{39} +493.384 q^{41} -314.767 q^{43} -436.490 q^{47} -333.151 q^{49} +143.097 q^{51} -123.960 q^{53} +417.447 q^{57} +32.7121 q^{59} +573.321 q^{61} -28.2452 q^{63} +297.044 q^{67} -300.000 q^{69} +242.041 q^{71} +811.202 q^{73} +171.948 q^{77} +212.067 q^{79} +81.0000 q^{81} -369.660 q^{83} +226.245 q^{87} -357.321 q^{89} +207.224 q^{91} +636.200 q^{93} +638.217 q^{97} -493.102 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 60 q^{7} + 36 q^{9} - 180 q^{21} + 400 q^{23} - 108 q^{27} - 84 q^{29} + 1248 q^{41} + 192 q^{43} - 440 q^{47} + 844 q^{49} + 552 q^{61} + 540 q^{63} - 408 q^{67} - 1200 q^{69} + 324 q^{81}+ \cdots + 312 q^{89}+O(q^{100}) $ 4 * q - 12 * q^3 + 60 * q^7 + 36 * q^9 - 180 * q^21 + 400 * q^23 - 108 * q^27 - 84 * q^29 + 1248 * q^41 + 192 * q^43 - 440 * q^47 + 844 * q^49 + 552 * q^61 + 540 * q^63 - 408 * q^67 - 1200 * q^69 + 324 * q^81 - 608 * q^83 + 252 * q^87 + 312 * q^89

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.13836	−0.169455	−0.0847277	−	0.996404i	$-0.527002\pi$
$7$	−3.13836	−0.169455	−0.0847277	+	0.996404i	$0.527002\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−54.7891	−1.50178	−0.750888	−	0.660429i	$-0.770374\pi$
$11$	−54.7891	−1.50178	−0.750888	+	0.660429i	$0.770374\pi$
$12$	0	0
$13$	−66.0294	−1.40871	−0.704357	−	0.709846i	$-0.748764\pi$
$13$	−66.0294	−1.40871	−0.704357	+	0.709846i	$0.748764\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−47.6991	−0.680514	−0.340257	−	0.940332i	$-0.610514\pi$
$17$	−47.6991	−0.680514	−0.340257	+	0.940332i	$0.610514\pi$
$18$	0	0
$19$	−139.149	−1.68015	−0.840077	−	0.542466i	$-0.817491\pi$
$19$	−139.149	−1.68015	−0.840077	+	0.542466i	$0.817491\pi$
$20$	0	0
$21$	9.41507	0.0978351
$22$	0	0
$23$	100.000	0.906584	0.453292	−	0.891362i	$-0.350249\pi$
$23$	100.000	0.906584	0.453292	+	0.891362i	$0.350249\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−75.4151	−0.482904	−0.241452	−	0.970413i	$-0.577624\pi$
$29$	−75.4151	−0.482904	−0.241452	+	0.970413i	$0.577624\pi$
$30$	0	0
$31$	−212.067	−1.22865	−0.614327	−	0.789052i	$-0.710572\pi$
$31$	−212.067	−1.22865	−0.614327	+	0.789052i	$0.710572\pi$
$32$	0	0
$33$	164.367	0.867051
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−417.245	−1.85391	−0.926955	−	0.375174i	$-0.877583\pi$
$37$	−417.245	−1.85391	−0.926955	+	0.375174i	$0.877583\pi$
$38$	0	0
$39$	198.088	0.813321
$40$	0	0
$41$	493.384	1.87936	0.939678	−	0.342061i	$-0.111125\pi$
$41$	493.384	1.87936	0.939678	+	0.342061i	$0.111125\pi$
$42$	0	0
$43$	−314.767	−1.11631	−0.558157	−	0.829735i	$-0.688491\pi$
$43$	−314.767	−1.11631	−0.558157	+	0.829735i	$0.688491\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−436.490	−1.35465	−0.677326	−	0.735683i	$-0.736861\pi$
$47$	−436.490	−1.35465	−0.677326	+	0.735683i	$0.736861\pi$
$48$	0	0
$49$	−333.151	−0.971285
$50$	0	0
$51$	143.097	0.392895
$52$	0	0
$53$	−123.960	−0.321268	−0.160634	−	0.987014i	$-0.551354\pi$
$53$	−123.960	−0.321268	−0.160634	+	0.987014i	$0.551354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	417.447	0.970038
$58$	0	0
$59$	32.7121	0.0721822	0.0360911	−	0.999349i	$-0.488509\pi$
$59$	32.7121	0.0721822	0.0360911	+	0.999349i	$0.488509\pi$
$60$	0	0
$61$	573.321	1.20338	0.601690	−	0.798730i	$-0.294494\pi$
$61$	573.321	1.20338	0.601690	+	0.798730i	$0.294494\pi$
$62$	0	0
$63$	−28.2452	−0.0564851
$64$	0	0
$65$	0	0
$66$	0	0
$67$	297.044	0.541637	0.270819	−	0.962630i	$-0.412706\pi$
$67$	297.044	0.541637	0.270819	+	0.962630i	$0.412706\pi$
$68$	0	0
$69$	−300.000	−0.523417
$70$	0	0
$71$	242.041	0.404577	0.202288	−	0.979326i	$-0.435162\pi$
$71$	242.041	0.404577	0.202288	+	0.979326i	$0.435162\pi$
$72$	0	0
$73$	811.202	1.30060	0.650301	−	0.759676i	$-0.274643\pi$
$73$	811.202	1.30060	0.650301	+	0.759676i	$0.274643\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	171.948	0.254484
$78$	0	0
$79$	212.067	0.302017	0.151009	−	0.988532i	$-0.451748\pi$
$79$	212.067	0.302017	0.151009	+	0.988532i	$0.451748\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−369.660	−0.488861	−0.244431	−	0.969667i	$-0.578601\pi$
$83$	−369.660	−0.488861	−0.244431	+	0.969667i	$0.578601\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	226.245	0.278805
$88$	0	0
$89$	−357.321	−0.425572	−0.212786	−	0.977099i	$-0.568254\pi$
$89$	−357.321	−0.425572	−0.212786	+	0.977099i	$0.568254\pi$
$90$	0	0
$91$	207.224	0.238714
$92$	0	0
$93$	636.200	0.709364
$94$	0	0
$95$	0	0
$96$	0	0
$97$	638.217	0.668053	0.334027	−	0.942564i	$-0.391592\pi$
$97$	638.217	0.668053	0.334027	+	0.942564i	$0.391592\pi$
$98$	0	0
$99$	−493.102	−0.500592
$100$	0	0
$101$	815.717	0.803632	0.401816	−	0.915720i	$-0.368379\pi$
$101$	815.717	0.803632	0.401816	+	0.915720i	$0.368379\pi$
$102$	0	0
$103$	298.522	0.285575	0.142788	−	0.989753i	$-0.454393\pi$
$103$	298.522	0.285575	0.142788	+	0.989753i	$0.454393\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1145.66	−1.03509	−0.517547	−	0.855654i	$-0.673155\pi$
$107$	−1145.66	−1.03509	−0.517547	+	0.855654i	$0.673155\pi$
$108$	0	0
$109$	−595.471	−0.523264	−0.261632	−	0.965168i	$-0.584261\pi$
$109$	−595.471	−0.523264	−0.261632	+	0.965168i	$0.584261\pi$
$110$	0	0
$111$	1251.73	1.07035
$112$	0	0
$113$	−1671.31	−1.39136	−0.695681	−	0.718351i	$-0.744897\pi$
$113$	−1671.31	−1.39136	−0.695681	+	0.718351i	$0.744897\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−594.265	−0.469571
$118$	0	0
$119$	149.697	0.115317
$120$	0	0
$121$	1670.85	1.25533
$122$	0	0
$123$	−1480.15	−1.08505
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1856.67	−1.29727	−0.648634	−	0.761100i	$-0.724659\pi$
$127$	−1856.67	−1.29727	−0.648634	+	0.761100i	$0.724659\pi$
$128$	0	0
$129$	944.301	0.644505
$130$	0	0
$131$	409.636	0.273207	0.136603	−	0.990626i	$-0.456381\pi$
$131$	409.636	0.273207	0.136603	+	0.990626i	$0.456381\pi$
$132$	0	0
$133$	436.699	0.284711
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2650.20	−1.65271	−0.826357	−	0.563147i	$-0.809590\pi$
$137$	−2650.20	−1.65271	−0.826357	+	0.563147i	$0.809590\pi$
$138$	0	0
$139$	−452.090	−0.275869	−0.137934	−	0.990441i	$-0.544046\pi$
$139$	−452.090	−0.275869	−0.137934	+	0.990441i	$0.544046\pi$
$140$	0	0
$141$	1309.47	0.782109
$142$	0	0
$143$	3617.70	2.11557
$144$	0	0
$145$	0	0
$146$	0	0
$147$	999.452	0.560772
$148$	0	0
$149$	−184.900	−0.101662	−0.0508308	−	0.998707i	$-0.516187\pi$
$149$	−184.900	−0.101662	−0.0508308	+	0.998707i	$0.516187\pi$
$150$	0	0
$151$	−2552.70	−1.37573	−0.687866	−	0.725838i	$-0.741452\pi$
$151$	−2552.70	−1.37573	−0.687866	+	0.725838i	$0.741452\pi$
$152$	0	0
$153$	−429.292	−0.226838
$154$	0	0
$155$	0	0
$156$	0	0
$157$	894.870	0.454894	0.227447	−	0.973790i	$-0.426962\pi$
$157$	894.870	0.454894	0.227447	+	0.973790i	$0.426962\pi$
$158$	0	0
$159$	371.880	0.185484
$160$	0	0
$161$	−313.836	−0.153626
$162$	0	0
$163$	−1805.72	−0.867700	−0.433850	−	0.900985i	$-0.642845\pi$
$163$	−1805.72	−0.867700	−0.433850	+	0.900985i	$0.642845\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3132.60	−1.45155	−0.725773	−	0.687935i	$-0.758518\pi$
$167$	−3132.60	−1.45155	−0.725773	+	0.687935i	$0.758518\pi$
$168$	0	0
$169$	2162.89	0.984473
$170$	0	0
$171$	−1252.34	−0.560052
$172$	0	0
$173$	−450.274	−0.197882	−0.0989412	−	0.995093i	$-0.531546\pi$
$173$	−450.274	−0.197882	−0.0989412	+	0.995093i	$0.531546\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−98.1362	−0.0416744
$178$	0	0
$179$	2803.93	1.17081	0.585407	−	0.810740i	$-0.300935\pi$
$179$	2803.93	1.17081	0.585407	+	0.810740i	$0.300935\pi$
$180$	0	0
$181$	−1988.07	−0.816422	−0.408211	−	0.912888i	$-0.633847\pi$
$181$	−1988.07	−0.816422	−0.408211	+	0.912888i	$0.633847\pi$
$182$	0	0
$183$	−1719.96	−0.694772
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2613.39	1.02198
$188$	0	0
$189$	84.7356	0.0326117
$190$	0	0
$191$	−2826.96	−1.07095	−0.535475	−	0.844551i	$-0.679867\pi$
$191$	−2826.96	−1.07095	−0.535475	+	0.844551i	$0.679867\pi$
$192$	0	0
$193$	−213.103	−0.0794793	−0.0397397	−	0.999210i	$-0.512653\pi$
$193$	−213.103	−0.0794793	−0.0397397	+	0.999210i	$0.512653\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2411.73	0.872227	0.436113	−	0.899892i	$-0.356355\pi$
$197$	2411.73	0.872227	0.436113	+	0.899892i	$0.356355\pi$
$198$	0	0
$199$	−2287.77	−0.814954	−0.407477	−	0.913216i	$-0.633591\pi$
$199$	−2287.77	−0.814954	−0.407477	+	0.913216i	$0.633591\pi$
$200$	0	0
$201$	−891.132	−0.312714
$202$	0	0
$203$	236.679	0.0818307
$204$	0	0
$205$	0	0
$206$	0	0
$207$	900.000	0.302195
$208$	0	0
$209$	7623.85	2.52322
$210$	0	0
$211$	5405.54	1.76366	0.881831	−	0.471566i	$-0.156311\pi$
$211$	5405.54	1.76366	0.881831	+	0.471566i	$0.156311\pi$
$212$	0	0
$213$	−726.122	−0.233582
$214$	0	0
$215$	0	0
$216$	0	0
$217$	665.541	0.208202
$218$	0	0
$219$	−2433.61	−0.750903
$220$	0	0
$221$	3149.55	0.958649
$222$	0	0
$223$	773.366	0.232235	0.116117	−	0.993235i	$-0.462955\pi$
$223$	773.366	0.232235	0.116117	+	0.993235i	$0.462955\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4088.90	−1.19555	−0.597776	−	0.801663i	$-0.703949\pi$
$227$	−4088.90	−1.19555	−0.597776	+	0.801663i	$0.703949\pi$
$228$	0	0
$229$	5589.06	1.61282	0.806409	−	0.591359i	$-0.201408\pi$
$229$	5589.06	1.61282	0.806409	+	0.591359i	$0.201408\pi$
$230$	0	0
$231$	−515.844	−0.146926
$232$	0	0
$233$	2387.12	0.671181	0.335591	−	0.942008i	$-0.391064\pi$
$233$	2387.12	0.671181	0.335591	+	0.942008i	$0.391064\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−636.200	−0.174370
$238$	0	0
$239$	893.861	0.241921	0.120960	−	0.992657i	$-0.461403\pi$
$239$	893.861	0.241921	0.120960	+	0.992657i	$0.461403\pi$
$240$	0	0
$241$	6396.26	1.70962	0.854812	−	0.518937i	$-0.173672\pi$
$241$	6396.26	1.70962	0.854812	+	0.518937i	$0.173672\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9187.92	2.36686
$248$	0	0
$249$	1108.98	0.282244
$250$	0	0
$251$	−5349.54	−1.34526	−0.672629	−	0.739980i	$-0.734835\pi$
$251$	−5349.54	−1.34526	−0.672629	+	0.739980i	$0.734835\pi$
$252$	0	0
$253$	−5478.91	−1.36149
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3918.97	0.951200	0.475600	−	0.879662i	$-0.342231\pi$
$257$	3918.97	0.951200	0.475600	+	0.879662i	$0.342231\pi$
$258$	0	0
$259$	1309.46	0.314155
$260$	0	0
$261$	−678.736	−0.160968
$262$	0	0
$263$	267.622	0.0627463	0.0313732	−	0.999508i	$-0.490012\pi$
$263$	267.622	0.0627463	0.0313732	+	0.999508i	$0.490012\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1071.96	0.245704
$268$	0	0
$269$	265.957	0.0602814	0.0301407	−	0.999546i	$-0.490404\pi$
$269$	265.957	0.0602814	0.0301407	+	0.999546i	$0.490404\pi$
$270$	0	0
$271$	4658.72	1.04427	0.522135	−	0.852863i	$-0.325136\pi$
$271$	4658.72	1.04427	0.522135	+	0.852863i	$0.325136\pi$
$272$	0	0
$273$	−621.672	−0.137822
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2565.06	0.556388	0.278194	−	0.960525i	$-0.410264\pi$
$277$	2565.06	0.556388	0.278194	+	0.960525i	$0.410264\pi$
$278$	0	0
$279$	−1908.60	−0.409551
$280$	0	0
$281$	3984.09	0.845803	0.422902	−	0.906176i	$-0.361012\pi$
$281$	3984.09	0.845803	0.422902	+	0.906176i	$0.361012\pi$
$282$	0	0
$283$	−1081.27	−0.227119	−0.113560	−	0.993531i	$-0.536225\pi$
$283$	−1081.27	−0.227119	−0.113560	+	0.993531i	$0.536225\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1548.41	−0.318467
$288$	0	0
$289$	−2637.79	−0.536900
$290$	0	0
$291$	−1914.65	−0.385701
$292$	0	0
$293$	8318.01	1.65851	0.829254	−	0.558871i	$-0.188765\pi$
$293$	8318.01	1.65851	0.829254	+	0.558871i	$0.188765\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1479.31	0.289017
$298$	0	0
$299$	−6602.94	−1.27712
$300$	0	0
$301$	987.852	0.189166
$302$	0	0
$303$	−2447.15	−0.463977
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6033.76	−1.12171	−0.560855	−	0.827914i	$-0.689527\pi$
$307$	−6033.76	−1.12171	−0.560855	+	0.827914i	$0.689527\pi$
$308$	0	0
$309$	−895.566	−0.164877
$310$	0	0
$311$	5191.10	0.946497	0.473248	−	0.880929i	$-0.343081\pi$
$311$	5191.10	0.946497	0.473248	+	0.880929i	$0.343081\pi$
$312$	0	0
$313$	469.150	0.0847218	0.0423609	−	0.999102i	$-0.486512\pi$
$313$	469.150	0.0847218	0.0423609	+	0.999102i	$0.486512\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−461.225	−0.0817192	−0.0408596	−	0.999165i	$-0.513010\pi$
$317$	−461.225	−0.0817192	−0.0408596	+	0.999165i	$0.513010\pi$
$318$	0	0
$319$	4131.93	0.725215
$320$	0	0
$321$	3436.98	0.597612
$322$	0	0
$323$	6637.28	1.14337
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1786.41	0.302107
$328$	0	0
$329$	1369.86	0.229553
$330$	0	0
$331$	3917.49	0.650529	0.325264	−	0.945623i	$-0.394547\pi$
$331$	3917.49	0.650529	0.325264	+	0.945623i	$0.394547\pi$
$332$	0	0
$333$	−3755.20	−0.617970
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10961.6	−1.77186	−0.885928	−	0.463824i	$-0.846477\pi$
$337$	−10961.6	−1.77186	−0.885928	+	0.463824i	$0.846477\pi$
$338$	0	0
$339$	5013.94	0.803304
$340$	0	0
$341$	11618.9	1.84516
$342$	0	0
$343$	2122.00	0.334045
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2905.89	−0.449556	−0.224778	−	0.974410i	$-0.572166\pi$
$347$	−2905.89	−0.449556	−0.224778	+	0.974410i	$0.572166\pi$
$348$	0	0
$349$	4016.26	0.616005	0.308002	−	0.951386i	$-0.400340\pi$
$349$	4016.26	0.616005	0.308002	+	0.951386i	$0.400340\pi$
$350$	0	0
$351$	1782.80	0.271107
$352$	0	0
$353$	−4238.54	−0.639078	−0.319539	−	0.947573i	$-0.603528\pi$
$353$	−4238.54	−0.639078	−0.319539	+	0.947573i	$0.603528\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−449.091	−0.0665782
$358$	0	0
$359$	10817.5	1.59033	0.795163	−	0.606395i	$-0.207385\pi$
$359$	10817.5	1.59033	0.795163	+	0.606395i	$0.207385\pi$
$360$	0	0
$361$	12503.4	1.82292
$362$	0	0
$363$	−5012.55	−0.724767
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8143.58	1.15829	0.579144	−	0.815226i	$-0.303387\pi$
$367$	8143.58	1.15829	0.579144	+	0.815226i	$0.303387\pi$
$368$	0	0
$369$	4440.45	0.626452
$370$	0	0
$371$	389.031	0.0544406
$372$	0	0
$373$	2097.52	0.291168	0.145584	−	0.989346i	$-0.453494\pi$
$373$	2097.52	0.291168	0.145584	+	0.989346i	$0.453494\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4979.62	0.680274
$378$	0	0
$379$	4319.09	0.585374	0.292687	−	0.956208i	$-0.405451\pi$
$379$	4319.09	0.585374	0.292687	+	0.956208i	$0.405451\pi$
$380$	0	0
$381$	5570.02	0.748978
$382$	0	0
$383$	2301.09	0.306998	0.153499	−	0.988149i	$-0.450946\pi$
$383$	2301.09	0.306998	0.153499	+	0.988149i	$0.450946\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2832.90	−0.372105
$388$	0	0
$389$	−7648.49	−0.996900	−0.498450	−	0.866918i	$-0.666097\pi$
$389$	−7648.49	−0.996900	−0.498450	+	0.866918i	$0.666097\pi$
$390$	0	0
$391$	−4769.91	−0.616944
$392$	0	0
$393$	−1228.91	−0.157736
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11038.0	−1.39542	−0.697708	−	0.716382i	$-0.745797\pi$
$397$	−11038.0	−1.39542	−0.697708	+	0.716382i	$0.745797\pi$
$398$	0	0
$399$	−1310.10	−0.164378
$400$	0	0
$401$	5971.47	0.743644	0.371822	−	0.928304i	$-0.378733\pi$
$401$	5971.47	0.743644	0.371822	+	0.928304i	$0.378733\pi$
$402$	0	0
$403$	14002.6	1.73082
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22860.5	2.78416
$408$	0	0
$409$	−4103.70	−0.496124	−0.248062	−	0.968744i	$-0.579794\pi$
$409$	−4103.70	−0.496124	−0.248062	+	0.968744i	$0.579794\pi$
$410$	0	0
$411$	7950.60	0.954195
$412$	0	0
$413$	−102.662	−0.0122317
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1356.27	0.159273
$418$	0	0
$419$	−4624.74	−0.539221	−0.269610	−	0.962969i	$-0.586895\pi$
$419$	−4624.74	−0.539221	−0.269610	+	0.962969i	$0.586895\pi$
$420$	0	0
$421$	7026.34	0.813403	0.406702	−	0.913561i	$-0.366679\pi$
$421$	7026.34	0.813403	0.406702	+	0.913561i	$0.366679\pi$
$422$	0	0
$423$	−3928.41	−0.451551
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1799.28	−0.203919
$428$	0	0
$429$	−10853.1	−1.22143
$430$	0	0
$431$	−997.789	−0.111512	−0.0557562	−	0.998444i	$-0.517757\pi$
$431$	−997.789	−0.111512	−0.0557562	+	0.998444i	$0.517757\pi$
$432$	0	0
$433$	285.270	0.0316609	0.0158305	−	0.999875i	$-0.494961\pi$
$433$	285.270	0.0316609	0.0158305	+	0.999875i	$0.494961\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−13914.9	−1.52320
$438$	0	0
$439$	−15424.9	−1.67698	−0.838488	−	0.544920i	$-0.816560\pi$
$439$	−15424.9	−1.67698	−0.838488	+	0.544920i	$0.816560\pi$
$440$	0	0
$441$	−2998.36	−0.323762
$442$	0	0
$443$	−17093.8	−1.83330	−0.916649	−	0.399692i	$-0.869117\pi$
$443$	−17093.8	−1.83330	−0.916649	+	0.399692i	$0.869117\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	554.700	0.0586944
$448$	0	0
$449$	−680.578	−0.0715333	−0.0357667	−	0.999360i	$-0.511387\pi$
$449$	−680.578	−0.0715333	−0.0357667	+	0.999360i	$0.511387\pi$
$450$	0	0
$451$	−27032.1	−2.82237
$452$	0	0
$453$	7658.09	0.794279
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11091.2	1.13528	0.567642	−	0.823275i	$-0.307856\pi$
$457$	11091.2	1.13528	0.567642	+	0.823275i	$0.307856\pi$
$458$	0	0
$459$	1287.88	0.130965
$460$	0	0
$461$	14732.4	1.48840	0.744202	−	0.667955i	$-0.232830\pi$
$461$	14732.4	1.48840	0.744202	+	0.667955i	$0.232830\pi$
$462$	0	0
$463$	10881.4	1.09223	0.546114	−	0.837711i	$-0.316107\pi$
$463$	10881.4	1.09223	0.546114	+	0.837711i	$0.316107\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17667.2	−1.75063	−0.875313	−	0.483556i	$-0.839345\pi$
$467$	−17667.2	−1.75063	−0.875313	+	0.483556i	$0.839345\pi$
$468$	0	0
$469$	−932.230	−0.0917833
$470$	0	0
$471$	−2684.61	−0.262633
$472$	0	0
$473$	17245.8	1.67646
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1115.64	−0.107089
$478$	0	0
$479$	−17668.7	−1.68540	−0.842698	−	0.538386i	$-0.819034\pi$
$479$	−17668.7	−1.68540	−0.842698	+	0.538386i	$0.819034\pi$
$480$	0	0
$481$	27550.4	2.61163
$482$	0	0
$483$	941.507	0.0886958
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12173.3	1.13270	0.566351	−	0.824164i	$-0.308355\pi$
$487$	12173.3	1.13270	0.566351	+	0.824164i	$0.308355\pi$
$488$	0	0
$489$	5417.17	0.500967
$490$	0	0
$491$	−1749.88	−0.160837	−0.0804186	−	0.996761i	$-0.525626\pi$
$491$	−1749.88	−0.160837	−0.0804186	+	0.996761i	$0.525626\pi$
$492$	0	0
$493$	3597.23	0.328623
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−759.610	−0.0685577
$498$	0	0
$499$	−6887.58	−0.617897	−0.308948	−	0.951079i	$-0.599977\pi$
$499$	−6887.58	−0.617897	−0.308948	+	0.951079i	$0.599977\pi$
$500$	0	0
$501$	9397.81	0.838050
$502$	0	0
$503$	−19138.6	−1.69652	−0.848258	−	0.529582i	$-0.822349\pi$
$503$	−19138.6	−1.69652	−0.848258	+	0.529582i	$0.822349\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6488.66	−0.568386
$508$	0	0
$509$	4935.83	0.429817	0.214909	−	0.976634i	$-0.431055\pi$
$509$	4935.83	0.429817	0.214909	+	0.976634i	$0.431055\pi$
$510$	0	0
$511$	−2545.84	−0.220394
$512$	0	0
$513$	3757.02	0.323346
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23914.9	2.03439
$518$	0	0
$519$	1350.82	0.114247
$520$	0	0
$521$	−5552.52	−0.466911	−0.233455	−	0.972368i	$-0.575003\pi$
$521$	−5552.52	−0.466911	−0.233455	+	0.972368i	$0.575003\pi$
$522$	0	0
$523$	15455.8	1.29222	0.646112	−	0.763243i	$-0.276394\pi$
$523$	15455.8	1.29222	0.646112	+	0.763243i	$0.276394\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10115.4	0.836117
$528$	0	0
$529$	−2167.00	−0.178105
$530$	0	0
$531$	294.409	0.0240607
$532$	0	0
$533$	−32577.8	−2.64747
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−8411.79	−0.675970
$538$	0	0
$539$	18253.0	1.45865
$540$	0	0
$541$	−4196.68	−0.333511	−0.166755	−	0.985998i	$-0.553329\pi$
$541$	−4196.68	−0.333511	−0.166755	+	0.985998i	$0.553329\pi$
$542$	0	0
$543$	5964.22	0.471361
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15213.5	−1.18918	−0.594592	−	0.804028i	$-0.702686\pi$
$547$	−15213.5	−1.18918	−0.594592	+	0.804028i	$0.702686\pi$
$548$	0	0
$549$	5159.89	0.401127
$550$	0	0
$551$	10493.9	0.811354
$552$	0	0
$553$	−665.541	−0.0511784
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−638.705	−0.0485867	−0.0242933	−	0.999705i	$-0.507734\pi$
$557$	−638.705	−0.0485867	−0.0242933	+	0.999705i	$0.507734\pi$
$558$	0	0
$559$	20783.9	1.57257
$560$	0	0
$561$	−7840.18	−0.590041
$562$	0	0
$563$	−8809.59	−0.659467	−0.329733	−	0.944074i	$-0.606959\pi$
$563$	−8809.59	−0.659467	−0.329733	+	0.944074i	$0.606959\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−254.207	−0.0188284
$568$	0	0
$569$	−3028.50	−0.223131	−0.111565	−	0.993757i	$-0.535586\pi$
$569$	−3028.50	−0.223131	−0.111565	+	0.993757i	$0.535586\pi$
$570$	0	0
$571$	12574.9	0.921619	0.460810	−	0.887499i	$-0.347559\pi$
$571$	12574.9	0.921619	0.460810	+	0.887499i	$0.347559\pi$
$572$	0	0
$573$	8480.87	0.618313
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8139.23	−0.587245	−0.293623	−	0.955921i	$-0.594861\pi$
$577$	−8139.23	−0.587245	−0.293623	+	0.955921i	$0.594861\pi$
$578$	0	0
$579$	639.310	0.0458874
$580$	0	0
$581$	1160.13	0.0828402
$582$	0	0
$583$	6791.66	0.482473
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2158.65	−0.151783	−0.0758917	−	0.997116i	$-0.524180\pi$
$587$	−2158.65	−0.151783	−0.0758917	+	0.997116i	$0.524180\pi$
$588$	0	0
$589$	29508.8	2.06433
$590$	0	0
$591$	−7235.19	−0.503580
$592$	0	0
$593$	−20751.2	−1.43701	−0.718507	−	0.695520i	$-0.755174\pi$
$593$	−20751.2	−1.43701	−0.718507	+	0.695520i	$0.755174\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6863.31	0.470514
$598$	0	0
$599$	−12392.6	−0.845319	−0.422659	−	0.906289i	$-0.638903\pi$
$599$	−12392.6	−0.845319	−0.422659	+	0.906289i	$0.638903\pi$
$600$	0	0
$601$	−18145.3	−1.23155	−0.615775	−	0.787922i	$-0.711157\pi$
$601$	−18145.3	−1.23155	−0.615775	+	0.787922i	$0.711157\pi$
$602$	0	0
$603$	2673.39	0.180546
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4054.01	0.271083	0.135541	−	0.990772i	$-0.456723\pi$
$607$	4054.01	0.271083	0.135541	+	0.990772i	$0.456723\pi$
$608$	0	0
$609$	−710.038	−0.0472450
$610$	0	0
$611$	28821.2	1.90832
$612$	0	0
$613$	11118.1	0.732551	0.366276	−	0.930506i	$-0.380633\pi$
$613$	11118.1	0.732551	0.366276	+	0.930506i	$0.380633\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3632.66	0.237027	0.118513	−	0.992952i	$-0.462187\pi$
$617$	3632.66	0.237027	0.118513	+	0.992952i	$0.462187\pi$
$618$	0	0
$619$	5849.26	0.379809	0.189904	−	0.981803i	$-0.439182\pi$
$619$	5849.26	0.379809	0.189904	+	0.981803i	$0.439182\pi$
$620$	0	0
$621$	−2700.00	−0.174472
$622$	0	0
$623$	1121.40	0.0721154
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−22871.5	−1.45678
$628$	0	0
$629$	19902.2	1.26161
$630$	0	0
$631$	23408.1	1.47680	0.738400	−	0.674363i	$-0.235582\pi$
$631$	23408.1	1.47680	0.738400	+	0.674363i	$0.235582\pi$
$632$	0	0
$633$	−16216.6	−1.01825
$634$	0	0
$635$	0	0
$636$	0	0
$637$	21997.8	1.36826
$638$	0	0
$639$	2178.37	0.134859
$640$	0	0
$641$	−8461.56	−0.521391	−0.260695	−	0.965421i	$-0.583952\pi$
$641$	−8461.56	−0.521391	−0.260695	+	0.965421i	$0.583952\pi$
$642$	0	0
$643$	−3968.61	−0.243401	−0.121700	−	0.992567i	$-0.538835\pi$
$643$	−3968.61	−0.243401	−0.121700	+	0.992567i	$0.538835\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12865.9	0.781780	0.390890	−	0.920437i	$-0.372167\pi$
$647$	12865.9	0.781780	0.390890	+	0.920437i	$0.372167\pi$
$648$	0	0
$649$	−1792.27	−0.108402
$650$	0	0
$651$	−1996.62	−0.120205
$652$	0	0
$653$	17208.1	1.03125	0.515626	−	0.856814i	$-0.327560\pi$
$653$	17208.1	1.03125	0.515626	+	0.856814i	$0.327560\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7300.82	0.433534
$658$	0	0
$659$	24399.7	1.44230	0.721152	−	0.692777i	$-0.243613\pi$
$659$	24399.7	1.44230	0.721152	+	0.692777i	$0.243613\pi$
$660$	0	0
$661$	−21423.6	−1.26064	−0.630318	−	0.776337i	$-0.717076\pi$
$661$	−21423.6	−1.26064	−0.630318	+	0.776337i	$0.717076\pi$
$662$	0	0
$663$	−9448.64	−0.553476
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7541.51	−0.437794
$668$	0	0
$669$	−2320.10	−0.134081
$670$	0	0
$671$	−31411.7	−1.80721
$672$	0	0
$673$	25564.1	1.46423	0.732114	−	0.681182i	$-0.238534\pi$
$673$	25564.1	1.46423	0.732114	+	0.681182i	$0.238534\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6060.39	−0.344047	−0.172023	−	0.985093i	$-0.555030\pi$
$677$	−6060.39	−0.344047	−0.172023	+	0.985093i	$0.555030\pi$
$678$	0	0
$679$	−2002.95	−0.113205
$680$	0	0
$681$	12266.7	0.690252
$682$	0	0
$683$	−25313.5	−1.41815	−0.709073	−	0.705135i	$-0.750886\pi$
$683$	−25313.5	−1.41815	−0.709073	+	0.705135i	$0.750886\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16767.2	−0.931160
$688$	0	0
$689$	8185.01	0.452575
$690$	0	0
$691$	−17510.4	−0.964004	−0.482002	−	0.876170i	$-0.660090\pi$
$691$	−17510.4	−0.964004	−0.482002	+	0.876170i	$0.660090\pi$
$692$	0	0
$693$	1547.53	0.0848280
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23534.0	−1.27893
$698$	0	0
$699$	−7161.35	−0.387507
$700$	0	0
$701$	11479.3	0.618500	0.309250	−	0.950981i	$-0.399922\pi$
$701$	11479.3	0.618500	0.309250	+	0.950981i	$0.399922\pi$
$702$	0	0
$703$	58059.2	3.11485
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2560.01	−0.136180
$708$	0	0
$709$	−5759.09	−0.305060	−0.152530	−	0.988299i	$-0.548742\pi$
$709$	−5759.09	−0.305060	−0.152530	+	0.988299i	$0.548742\pi$
$710$	0	0
$711$	1908.60	0.100672
$712$	0	0
$713$	−21206.7	−1.11388
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2681.58	−0.139673
$718$	0	0
$719$	10059.1	0.521753	0.260876	−	0.965372i	$-0.415989\pi$
$719$	10059.1	0.521753	0.260876	+	0.965372i	$0.415989\pi$
$720$	0	0
$721$	−936.868	−0.0483922
$722$	0	0
$723$	−19188.8	−0.987052
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6488.65	0.331019	0.165509	−	0.986208i	$-0.447073\pi$
$727$	6488.65	0.331019	0.165509	+	0.986208i	$0.447073\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	15014.1	0.759668
$732$	0	0
$733$	−24085.8	−1.21368	−0.606842	−	0.794823i	$-0.707564\pi$
$733$	−24085.8	−1.21368	−0.606842	+	0.794823i	$0.707564\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16274.8	−0.813418
$738$	0	0
$739$	−35834.3	−1.78375	−0.891873	−	0.452287i	$-0.850608\pi$
$739$	−35834.3	−1.78375	−0.891873	+	0.452287i	$0.850608\pi$
$740$	0	0
$741$	−27563.8	−1.36651
$742$	0	0
$743$	−28929.8	−1.42844	−0.714219	−	0.699922i	$-0.753218\pi$
$743$	−28929.8	−1.42844	−0.714219	+	0.699922i	$0.753218\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3326.94	−0.162954
$748$	0	0
$749$	3595.49	0.175402
$750$	0	0
$751$	11630.6	0.565121	0.282561	−	0.959249i	$-0.408816\pi$
$751$	11630.6	0.565121	0.282561	+	0.959249i	$0.408816\pi$
$752$	0	0
$753$	16048.6	0.776685
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−39403.9	−1.89189	−0.945945	−	0.324327i	$-0.894862\pi$
$757$	−39403.9	−1.89189	−0.945945	+	0.324327i	$0.894862\pi$
$758$	0	0
$759$	16436.7	0.786055
$760$	0	0
$761$	−32189.6	−1.53334	−0.766670	−	0.642042i	$-0.778088\pi$
$761$	−32189.6	−1.53334	−0.766670	+	0.642042i	$0.778088\pi$
$762$	0	0
$763$	1868.80	0.0886699
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2159.96	−0.101684
$768$	0	0
$769$	−14438.3	−0.677061	−0.338530	−	0.940955i	$-0.609930\pi$
$769$	−14438.3	−0.677061	−0.338530	+	0.940955i	$0.609930\pi$
$770$	0	0
$771$	−11756.9	−0.549176
$772$	0	0
$773$	7958.15	0.370291	0.185145	−	0.982711i	$-0.440724\pi$
$773$	7958.15	0.370291	0.185145	+	0.982711i	$0.440724\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3928.39	−0.181377
$778$	0	0
$779$	−68653.8	−3.15761
$780$	0	0
$781$	−13261.2	−0.607584
$782$	0	0
$783$	2036.21	0.0929350
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14650.9	−0.663595	−0.331798	−	0.943351i	$-0.607655\pi$
$787$	−14650.9	−0.663595	−0.331798	+	0.943351i	$0.607655\pi$
$788$	0	0
$789$	−802.866	−0.0362266
$790$	0	0
$791$	5245.18	0.235774
$792$	0	0
$793$	−37856.0	−1.69522
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36677.7	1.63010	0.815050	−	0.579390i	$-0.196709\pi$
$797$	36677.7	1.63010	0.815050	+	0.579390i	$0.196709\pi$
$798$	0	0
$799$	20820.2	0.921860
$800$	0	0
$801$	−3215.89	−0.141857
$802$	0	0
$803$	−44445.1	−1.95322
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−797.872	−0.0348035
$808$	0	0
$809$	−11455.7	−0.497850	−0.248925	−	0.968523i	$-0.580077\pi$
$809$	−11455.7	−0.497850	−0.248925	+	0.968523i	$0.580077\pi$
$810$	0	0
$811$	−7414.38	−0.321028	−0.160514	−	0.987034i	$-0.551315\pi$
$811$	−7414.38	−0.321028	−0.160514	+	0.987034i	$0.551315\pi$
$812$	0	0
$813$	−13976.2	−0.602910
$814$	0	0
$815$	0	0
$816$	0	0
$817$	43799.5	1.87558
$818$	0	0
$819$	1865.02	0.0795713
$820$	0	0
$821$	−20416.0	−0.867872	−0.433936	−	0.900944i	$-0.642876\pi$
$821$	−20416.0	−0.867872	−0.433936	+	0.900944i	$0.642876\pi$
$822$	0	0
$823$	12681.8	0.537132	0.268566	−	0.963261i	$-0.413450\pi$
$823$	12681.8	0.537132	0.268566	+	0.963261i	$0.413450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12540.1	0.527284	0.263642	−	0.964621i	$-0.415076\pi$
$827$	12540.1	0.527284	0.263642	+	0.964621i	$0.415076\pi$
$828$	0	0
$829$	6163.70	0.258232	0.129116	−	0.991630i	$-0.458786\pi$
$829$	6163.70	0.258232	0.129116	+	0.991630i	$0.458786\pi$
$830$	0	0
$831$	−7695.18	−0.321231
$832$	0	0
$833$	15891.0	0.660973
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5725.80	0.236455
$838$	0	0
$839$	−9169.30	−0.377306	−0.188653	−	0.982044i	$-0.560412\pi$
$839$	−9169.30	−0.377306	−0.188653	+	0.982044i	$0.560412\pi$
$840$	0	0
$841$	−18701.6	−0.766803
$842$	0	0
$843$	−11952.3	−0.488325
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5243.72	−0.212723
$848$	0	0
$849$	3243.81	0.131127
$850$	0	0
$851$	−41724.5	−1.68073
$852$	0	0
$853$	−20597.6	−0.826787	−0.413394	−	0.910552i	$-0.635657\pi$
$853$	−20597.6	−0.826787	−0.413394	+	0.910552i	$0.635657\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19022.4	0.758217	0.379109	−	0.925352i	$-0.376231\pi$
$857$	19022.4	0.758217	0.379109	+	0.925352i	$0.376231\pi$
$858$	0	0
$859$	32860.9	1.30524	0.652618	−	0.757687i	$-0.273670\pi$
$859$	32860.9	1.30524	0.652618	+	0.757687i	$0.273670\pi$
$860$	0	0
$861$	4645.24	0.183867
$862$	0	0
$863$	2423.62	0.0955978	0.0477989	−	0.998857i	$-0.484779\pi$
$863$	2423.62	0.0955978	0.0477989	+	0.998857i	$0.484779\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7913.38	0.309980
$868$	0	0
$869$	−11618.9	−0.453562
$870$	0	0
$871$	−19613.6	−0.763011
$872$	0	0
$873$	5743.96	0.222684
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23961.9	−0.922619	−0.461309	−	0.887239i	$-0.652620\pi$
$877$	−23961.9	−0.922619	−0.461309	+	0.887239i	$0.652620\pi$
$878$	0	0
$879$	−24954.0	−0.957541
$880$	0	0
$881$	−38508.8	−1.47264	−0.736318	−	0.676635i	$-0.763437\pi$
$881$	−38508.8	−1.47264	−0.736318	+	0.676635i	$0.763437\pi$
$882$	0	0
$883$	50877.5	1.93903	0.969516	−	0.245029i	$-0.0787975\pi$
$883$	50877.5	1.93903	0.969516	+	0.245029i	$0.0787975\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6563.69	−0.248464	−0.124232	−	0.992253i	$-0.539647\pi$
$887$	−6563.69	−0.248464	−0.124232	+	0.992253i	$0.539647\pi$
$888$	0	0
$889$	5826.90	0.219829
$890$	0	0
$891$	−4437.92	−0.166864
$892$	0	0
$893$	60737.2	2.27603
$894$	0	0
$895$	0	0
$896$	0	0
$897$	19808.8	0.737344
$898$	0	0
$899$	15993.0	0.593322
$900$	0	0
$901$	5912.79	0.218628
$902$	0	0
$903$	−2963.56	−0.109215
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14135.0	0.517468	0.258734	−	0.965949i	$-0.416695\pi$
$907$	14135.0	0.517468	0.258734	+	0.965949i	$0.416695\pi$
$908$	0	0
$909$	7341.45	0.267877
$910$	0	0
$911$	3241.58	0.117891	0.0589453	−	0.998261i	$-0.481226\pi$
$911$	3241.58	0.117891	0.0589453	+	0.998261i	$0.481226\pi$
$912$	0	0
$913$	20253.4	0.734161
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1285.59	−0.0462964
$918$	0	0
$919$	23267.0	0.835154	0.417577	−	0.908642i	$-0.362879\pi$
$919$	23267.0	0.835154	0.417577	+	0.908642i	$0.362879\pi$
$920$	0	0
$921$	18101.3	0.647619
$922$	0	0
$923$	−15981.8	−0.569933
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2686.70	0.0951917
$928$	0	0
$929$	−31142.1	−1.09983	−0.549913	−	0.835222i	$-0.685339\pi$
$929$	−31142.1	−1.09983	−0.549913	+	0.835222i	$0.685339\pi$
$930$	0	0
$931$	46357.5	1.63191
$932$	0	0
$933$	−15573.3	−0.546460
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15438.1	−0.538252	−0.269126	−	0.963105i	$-0.586735\pi$
$937$	−15438.1	−0.538252	−0.269126	+	0.963105i	$0.586735\pi$
$938$	0	0
$939$	−1407.45	−0.0489142
$940$	0	0
$941$	30968.1	1.07283	0.536414	−	0.843955i	$-0.319778\pi$
$941$	30968.1	1.07283	0.536414	+	0.843955i	$0.319778\pi$
$942$	0	0
$943$	49338.4	1.70379
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33855.5	−1.16173	−0.580865	−	0.814000i	$-0.697285\pi$
$947$	−33855.5	−1.16173	−0.580865	+	0.814000i	$0.697285\pi$
$948$	0	0
$949$	−53563.2	−1.83218
$950$	0	0
$951$	1383.68	0.0471806
$952$	0	0
$953$	24702.3	0.839649	0.419825	−	0.907605i	$-0.362092\pi$
$953$	24702.3	0.839649	0.419825	+	0.907605i	$0.362092\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12395.8	−0.418703
$958$	0	0
$959$	8317.27	0.280061
$960$	0	0
$961$	15181.2	0.509591
$962$	0	0
$963$	−10310.9	−0.345032
$964$	0	0
$965$	0	0
$966$	0	0
$967$	49483.3	1.64558	0.822789	−	0.568346i	$-0.192417\pi$
$967$	49483.3	1.64558	0.822789	+	0.568346i	$0.192417\pi$
$968$	0	0
$969$	−19911.8	−0.660125
$970$	0	0
$971$	−8369.30	−0.276605	−0.138303	−	0.990390i	$-0.544165\pi$
$971$	−8369.30	−0.276605	−0.138303	+	0.990390i	$0.544165\pi$
$972$	0	0
$973$	1418.82	0.0467474
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16742.7	0.548257	0.274129	−	0.961693i	$-0.411611\pi$
$977$	16742.7	0.548257	0.274129	+	0.961693i	$0.411611\pi$
$978$	0	0
$979$	19577.3	0.639114
$980$	0	0
$981$	−5359.24	−0.174421
$982$	0	0
$983$	24181.7	0.784615	0.392307	−	0.919834i	$-0.371677\pi$
$983$	24181.7	0.784615	0.392307	+	0.919834i	$0.371677\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4109.59	−0.132533
$988$	0	0
$989$	−31476.7	−1.01203
$990$	0	0
$991$	−17268.6	−0.553536	−0.276768	−	0.960937i	$-0.589263\pi$
$991$	−17268.6	−0.553536	−0.276768	+	0.960937i	$0.589263\pi$
$992$	0	0
$993$	−11752.5	−0.375583
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9396.27	−0.298478	−0.149239	−	0.988801i	$-0.547682\pi$
$997$	−9396.27	−0.298478	−0.149239	+	0.988801i	$0.547682\pi$
$998$	0	0
$999$	11265.6	0.356785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bz.1.1		4
4.3	odd	2		2400.4.a.ca.1.4		4
5.2	odd	4		480.4.f.f.289.5	yes	8
5.3	odd	4		480.4.f.f.289.2	yes	8
5.4	even	2		2400.4.a.ca.1.3		4
15.2	even	4		1440.4.f.m.289.7		8
15.8	even	4		1440.4.f.m.289.6		8
20.3	even	4		480.4.f.f.289.6	yes	8
20.7	even	4		480.4.f.f.289.1	✓	8
20.19	odd	2	inner	2400.4.a.bz.1.2		4
40.3	even	4		960.4.f.u.769.3		8
40.13	odd	4		960.4.f.u.769.7		8
40.27	even	4		960.4.f.u.769.8		8
40.37	odd	4		960.4.f.u.769.4		8
60.23	odd	4		1440.4.f.m.289.5		8
60.47	odd	4		1440.4.f.m.289.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.f.289.1	✓	8	20.7	even	4
480.4.f.f.289.2	yes	8	5.3	odd	4
480.4.f.f.289.5	yes	8	5.2	odd	4
480.4.f.f.289.6	yes	8	20.3	even	4
960.4.f.u.769.3		8	40.3	even	4
960.4.f.u.769.4		8	40.37	odd	4
960.4.f.u.769.7		8	40.13	odd	4
960.4.f.u.769.8		8	40.27	even	4
1440.4.f.m.289.5		8	60.23	odd	4
1440.4.f.m.289.6		8	15.8	even	4
1440.4.f.m.289.7		8	15.2	even	4
1440.4.f.m.289.8		8	60.47	odd	4
2400.4.a.bz.1.1		4	1.1	even	1	trivial
2400.4.a.bz.1.2		4	20.19	odd	2	inner
2400.4.a.ca.1.3		4	5.4	even	2
2400.4.a.ca.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -3.13836 q^{7} +9.00000 q^{9} -54.7891 q^{11} -66.0294 q^{13} -47.6991 q^{17} -139.149 q^{19} +9.41507 q^{21} +100.000 q^{23} -27.0000 q^{27} -75.4151 q^{29} -212.067 q^{31} +164.367 q^{33} -417.245 q^{37} +198.088 q^{39} +493.384 q^{41} -314.767 q^{43} -436.490 q^{47} -333.151 q^{49} +143.097 q^{51} -123.960 q^{53} +417.447 q^{57} +32.7121 q^{59} +573.321 q^{61} -28.2452 q^{63} +297.044 q^{67} -300.000 q^{69} +242.041 q^{71} +811.202 q^{73} +171.948 q^{77} +212.067 q^{79} +81.0000 q^{81} -369.660 q^{83} +226.245 q^{87} -357.321 q^{89} +207.224 q^{91} +636.200 q^{93} +638.217 q^{97} -493.102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 60 q^{7} + 36 q^{9} - 180 q^{21} + 400 q^{23} - 108 q^{27} - 84 q^{29} + 1248 q^{41} + 192 q^{43} - 440 q^{47} + 844 q^{49} + 552 q^{61} + 540 q^{63} - 408 q^{67} - 1200 q^{69} + 324 q^{81}+ \cdots + 312 q^{89}+O(q^{100}) \) 4 * q - 12 * q^3 + 60 * q^7 + 36 * q^9 - 180 * q^21 + 400 * q^23 - 108 * q^27 - 84 * q^29 + 1248 * q^41 + 192 * q^43 - 440 * q^47 + 844 * q^49 + 552 * q^61 + 540 * q^63 - 408 * q^67 - 1200 * q^69 + 324 * q^81 - 608 * q^83 + 252 * q^87 + 312 * q^89

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