LMFDB - Embedded newform 405.4.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 $ x^7 - 2x^6 - 44x^5 + 74x^4 + 479x^3 - 460x^2 - 1200x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-3.04174$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04174 q^{2} +1.25219 q^{4} -5.00000 q^{5} -13.7122 q^{7} -20.5251 q^{8} +O(q^{10})$	\(q+3.04174 q^{2} +1.25219 q^{4} -5.00000 q^{5} -13.7122 q^{7} -20.5251 q^{8} -15.2087 q^{10} +31.8068 q^{11} +58.2432 q^{13} -41.7090 q^{14} -72.4495 q^{16} +109.055 q^{17} +129.695 q^{19} -6.26094 q^{20} +96.7481 q^{22} +79.6684 q^{23} +25.0000 q^{25} +177.161 q^{26} -17.1703 q^{28} +9.03537 q^{29} -33.3809 q^{31} -56.1719 q^{32} +331.717 q^{34} +68.5611 q^{35} -22.1645 q^{37} +394.500 q^{38} +102.625 q^{40} +121.740 q^{41} +10.1417 q^{43} +39.8281 q^{44} +242.331 q^{46} -441.492 q^{47} -154.975 q^{49} +76.0435 q^{50} +72.9314 q^{52} -593.610 q^{53} -159.034 q^{55} +281.445 q^{56} +27.4833 q^{58} +442.460 q^{59} +144.576 q^{61} -101.536 q^{62} +408.736 q^{64} -291.216 q^{65} +862.721 q^{67} +136.557 q^{68} +208.545 q^{70} -818.541 q^{71} +495.052 q^{73} -67.4187 q^{74} +162.403 q^{76} -436.142 q^{77} +1170.53 q^{79} +362.248 q^{80} +370.300 q^{82} -424.043 q^{83} -545.274 q^{85} +30.8485 q^{86} -652.838 q^{88} +1031.37 q^{89} -798.643 q^{91} +99.7598 q^{92} -1342.90 q^{94} -648.477 q^{95} +1598.71 q^{97} -471.394 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8	$ 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8} + 10 q^{10} - 23 q^{11} + 96 q^{13} + 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 180 q^{20} + 311 q^{22} - 96 q^{23} + 175 q^{25} + 358 q^{26} + 337 q^{28} + 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 110 q^{35} + 404 q^{37} - 305 q^{38} + 90 q^{40} + 47 q^{41} + 525 q^{43} - 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 50 q^{50} + 1682 q^{52} - 506 q^{53} + 115 q^{55} + 981 q^{56} + 1183 q^{58} + 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 480 q^{65} + 1093 q^{67} - 2473 q^{68} - 105 q^{70} - 328 q^{71} + 2085 q^{73} + 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} - 1620 q^{80} - 62 q^{82} - 1290 q^{83} + 805 q^{85} + 2569 q^{86} + 2271 q^{88} + 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1395 q^{95} + 1787 q^{97} - 1279 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8 + 10 * q^10 - 23 * q^11 + 96 * q^13 + 21 * q^14 + 324 * q^16 - 161 * q^17 + 279 * q^19 - 180 * q^20 + 311 * q^22 - 96 * q^23 + 175 * q^25 + 358 * q^26 + 337 * q^28 + 296 * q^29 + 244 * q^31 + 314 * q^32 + 125 * q^34 - 110 * q^35 + 404 * q^37 - 305 * q^38 + 90 * q^40 + 47 * q^41 + 525 * q^43 - 55 * q^44 + 717 * q^46 - 164 * q^47 + 1225 * q^49 - 50 * q^50 + 1682 * q^52 - 506 * q^53 + 115 * q^55 + 981 * q^56 + 1183 * q^58 + 85 * q^59 + 828 * q^61 + 786 * q^62 + 2236 * q^64 - 480 * q^65 + 1093 * q^67 - 2473 * q^68 - 105 * q^70 - 328 * q^71 + 2085 * q^73 + 1316 * q^74 + 2789 * q^76 - 24 * q^77 + 2110 * q^79 - 1620 * q^80 - 62 * q^82 - 1290 * q^83 + 805 * q^85 + 2569 * q^86 + 2271 * q^88 + 3048 * q^89 + 3338 * q^91 - 2763 * q^92 - 517 * q^94 - 1395 * q^95 + 1787 * q^97 - 1279 * q^98

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 $

$2$	3.04174	1.07542	0.537709	−	0.843131i	$-0.319290\pi$
$2$	3.04174	1.07542	0.537709	+	0.843131i	$0.319290\pi$
$3$	0	0
$3$	0	0
$4$	1.25219	0.156523
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−13.7122	−0.740390	−0.370195	−	0.928954i	$-0.620709\pi$
$7$	−13.7122	−0.740390	−0.370195	+	0.928954i	$0.620709\pi$
$8$	−20.5251	−0.907090
$9$	0	0
$10$	−15.2087	−0.480941
$11$	31.8068	0.871829	0.435914	−	0.899988i	$-0.356425\pi$
$11$	31.8068	0.871829	0.435914	+	0.899988i	$0.356425\pi$
$12$	0	0
$13$	58.2432	1.24260	0.621298	−	0.783574i	$-0.286606\pi$
$13$	58.2432	1.24260	0.621298	+	0.783574i	$0.286606\pi$
$14$	−41.7090	−0.796229
$15$	0	0
$16$	−72.4495	−1.13202
$17$	109.055	1.55586	0.777932	−	0.628348i	$-0.216269\pi$
$17$	109.055	1.55586	0.777932	+	0.628348i	$0.216269\pi$
$18$	0	0
$19$	129.695	1.56601	0.783004	−	0.622017i	$-0.213687\pi$
$19$	129.695	1.56601	0.783004	+	0.622017i	$0.213687\pi$
$20$	−6.26094	−0.0699994
$21$	0	0
$22$	96.7481	0.937580
$23$	79.6684	0.722261	0.361131	−	0.932515i	$-0.382391\pi$
$23$	79.6684	0.722261	0.361131	+	0.932515i	$0.382391\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	177.161	1.33631
$27$	0	0
$28$	−17.1703	−0.115888
$29$	9.03537	0.0578561	0.0289280	−	0.999581i	$-0.490791\pi$
$29$	9.03537	0.0578561	0.0289280	+	0.999581i	$0.490791\pi$
$30$	0	0
$31$	−33.3809	−0.193400	−0.0966998	−	0.995314i	$-0.530829\pi$
$31$	−33.3809	−0.193400	−0.0966998	+	0.995314i	$0.530829\pi$
$32$	−56.1719	−0.310309
$33$	0	0
$34$	331.717	1.67320
$35$	68.5611	0.331113
$36$	0	0
$37$	−22.1645	−0.0984817	−0.0492408	−	0.998787i	$-0.515680\pi$
$37$	−22.1645	−0.0984817	−0.0492408	+	0.998787i	$0.515680\pi$
$38$	394.500	1.68411
$39$	0	0
$40$	102.625	0.405663
$41$	121.740	0.463720	0.231860	−	0.972749i	$-0.425519\pi$
$41$	121.740	0.463720	0.231860	+	0.972749i	$0.425519\pi$
$42$	0	0
$43$	10.1417	0.0359674	0.0179837	−	0.999838i	$-0.494275\pi$
$43$	10.1417	0.0359674	0.0179837	+	0.999838i	$0.494275\pi$
$44$	39.8281	0.136462
$45$	0	0
$46$	242.331	0.776733
$47$	−441.492	−1.37018	−0.685088	−	0.728460i	$-0.740236\pi$
$47$	−441.492	−1.37018	−0.685088	+	0.728460i	$0.740236\pi$
$48$	0	0
$49$	−154.975	−0.451822
$50$	76.0435	0.215084
$51$	0	0
$52$	72.9314	0.194495
$53$	−593.610	−1.53846	−0.769232	−	0.638969i	$-0.779361\pi$
$53$	−593.610	−1.53846	−0.769232	+	0.638969i	$0.779361\pi$
$54$	0	0
$55$	−159.034	−0.389894
$56$	281.445	0.671601
$57$	0	0
$58$	27.4833	0.0622195
$59$	442.460	0.976328	0.488164	−	0.872752i	$-0.337667\pi$
$59$	442.460	0.976328	0.488164	+	0.872752i	$0.337667\pi$
$60$	0	0
$61$	144.576	0.303460	0.151730	−	0.988422i	$-0.451516\pi$
$61$	144.576	0.303460	0.151730	+	0.988422i	$0.451516\pi$
$62$	−101.536	−0.207985
$63$	0	0
$64$	408.736	0.798312
$65$	−291.216	−0.555706
$66$	0	0
$67$	862.721	1.57311	0.786553	−	0.617523i	$-0.211864\pi$
$67$	862.721	1.57311	0.786553	+	0.617523i	$0.211864\pi$
$68$	136.557	0.243529
$69$	0	0
$70$	208.545	0.356084
$71$	−818.541	−1.36821	−0.684105	−	0.729384i	$-0.739807\pi$
$71$	−818.541	−1.36821	−0.684105	+	0.729384i	$0.739807\pi$
$72$	0	0
$73$	495.052	0.793719	0.396859	−	0.917879i	$-0.370100\pi$
$73$	495.052	0.793719	0.396859	+	0.917879i	$0.370100\pi$
$74$	−67.4187	−0.105909
$75$	0	0
$76$	162.403	0.245117
$77$	−436.142	−0.645494
$78$	0	0
$79$	1170.53	1.66702	0.833510	−	0.552505i	$-0.186328\pi$
$79$	1170.53	1.66702	0.833510	+	0.552505i	$0.186328\pi$
$80$	362.248	0.506256
$81$	0	0
$82$	370.300	0.498693
$83$	−424.043	−0.560781	−0.280390	−	0.959886i	$-0.590464\pi$
$83$	−424.043	−0.560781	−0.280390	+	0.959886i	$0.590464\pi$
$84$	0	0
$85$	−545.274	−0.695804
$86$	30.8485	0.0386799
$87$	0	0
$88$	−652.838	−0.790827
$89$	1031.37	1.22837	0.614183	−	0.789163i	$-0.289486\pi$
$89$	1031.37	1.22837	0.614183	+	0.789163i	$0.289486\pi$
$90$	0	0
$91$	−798.643	−0.920006
$92$	99.7598	0.113051
$93$	0	0
$94$	−1342.90	−1.47351
$95$	−648.477	−0.700340
$96$	0	0
$97$	1598.71	1.67345	0.836725	−	0.547623i	$-0.184467\pi$
$97$	1598.71	1.67345	0.836725	+	0.547623i	$0.184467\pi$
$98$	−471.394	−0.485897
$99$	0	0
$100$	31.3047	0.0313047
$101$	−214.000	−0.210830	−0.105415	−	0.994428i	$-0.533617\pi$
$101$	−214.000	−0.210830	−0.105415	+	0.994428i	$0.533617\pi$
$102$	0	0
$103$	1672.32	1.59979	0.799897	−	0.600138i	$-0.204888\pi$
$103$	1672.32	1.59979	0.799897	+	0.600138i	$0.204888\pi$
$104$	−1195.45	−1.12715
$105$	0	0
$106$	−1805.61	−1.65449
$107$	−600.699	−0.542727	−0.271363	−	0.962477i	$-0.587475\pi$
$107$	−600.699	−0.542727	−0.271363	+	0.962477i	$0.587475\pi$
$108$	0	0
$109$	−771.570	−0.678009	−0.339005	−	0.940785i	$-0.610090\pi$
$109$	−771.570	−0.678009	−0.339005	+	0.940785i	$0.610090\pi$
$110$	−483.740	−0.419299
$111$	0	0
$112$	993.444	0.838140
$113$	−1166.68	−0.971254	−0.485627	−	0.874166i	$-0.661409\pi$
$113$	−1166.68	−0.971254	−0.485627	+	0.874166i	$0.661409\pi$
$114$	0	0
$115$	−398.342	−0.323005
$116$	11.3140	0.00905583
$117$	0	0
$118$	1345.85	1.04996
$119$	−1495.38	−1.15195
$120$	0	0
$121$	−319.327	−0.239915
$122$	439.763	0.326347
$123$	0	0
$124$	−41.7991	−0.0302716
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1630.10	1.13896	0.569479	−	0.822006i	$-0.307145\pi$
$127$	1630.10	1.13896	0.569479	+	0.822006i	$0.307145\pi$
$128$	1692.64	1.16883
$129$	0	0
$130$	−885.803	−0.597616
$131$	258.825	0.172623	0.0863115	−	0.996268i	$-0.472492\pi$
$131$	258.825	0.172623	0.0863115	+	0.996268i	$0.472492\pi$
$132$	0	0
$133$	−1778.41	−1.15946
$134$	2624.17	1.69175
$135$	0	0
$136$	−2238.36	−1.41131
$137$	1052.77	0.656527	0.328264	−	0.944586i	$-0.393537\pi$
$137$	1052.77	0.656527	0.328264	+	0.944586i	$0.393537\pi$
$138$	0	0
$139$	2384.50	1.45504	0.727519	−	0.686088i	$-0.240673\pi$
$139$	2384.50	1.45504	0.727519	+	0.686088i	$0.240673\pi$
$140$	85.8514	0.0518269
$141$	0	0
$142$	−2489.79	−1.47140
$143$	1852.53	1.08333
$144$	0	0
$145$	−45.1769	−0.0258740
$146$	1505.82	0.853579
$147$	0	0
$148$	−27.7541	−0.0154147
$149$	1146.29	0.630253	0.315126	−	0.949050i	$-0.397953\pi$
$149$	1146.29	0.630253	0.315126	+	0.949050i	$0.397953\pi$
$150$	0	0
$151$	−1166.12	−0.628458	−0.314229	−	0.949347i	$-0.601746\pi$
$151$	−1166.12	−0.628458	−0.314229	+	0.949347i	$0.601746\pi$
$152$	−2662.01	−1.42051
$153$	0	0
$154$	−1326.63	−0.694175
$155$	166.904	0.0864909
$156$	0	0
$157$	−3475.62	−1.76678	−0.883390	−	0.468638i	$-0.844745\pi$
$157$	−3475.62	−1.76678	−0.883390	+	0.468638i	$0.844745\pi$
$158$	3560.44	1.79274
$159$	0	0
$160$	280.860	0.138774
$161$	−1092.43	−0.534755
$162$	0	0
$163$	−2223.32	−1.06837	−0.534183	−	0.845369i	$-0.679381\pi$
$163$	−2223.32	−1.06837	−0.534183	+	0.845369i	$0.679381\pi$
$164$	152.441	0.0725831
$165$	0	0
$166$	−1289.83	−0.603074
$167$	−1587.98	−0.735819	−0.367909	−	0.929862i	$-0.619926\pi$
$167$	−1587.98	−0.735819	−0.367909	+	0.929862i	$0.619926\pi$
$168$	0	0
$169$	1195.26	0.544044
$170$	−1658.58	−0.748280
$171$	0	0
$172$	12.6993	0.00562974
$173$	1507.80	0.662634	0.331317	−	0.943520i	$-0.392507\pi$
$173$	1507.80	0.662634	0.331317	+	0.943520i	$0.392507\pi$
$174$	0	0
$175$	−342.806	−0.148078
$176$	−2304.39	−0.986931
$177$	0	0
$178$	3137.15	1.32101
$179$	1572.66	0.656683	0.328341	−	0.944559i	$-0.393510\pi$
$179$	1572.66	0.656683	0.328341	+	0.944559i	$0.393510\pi$
$180$	0	0
$181$	1984.41	0.814918	0.407459	−	0.913223i	$-0.366415\pi$
$181$	1984.41	0.814918	0.407459	+	0.913223i	$0.366415\pi$
$182$	−2429.27	−0.989391
$183$	0	0
$184$	−1635.20	−0.655156
$185$	110.823	0.0440423
$186$	0	0
$187$	3468.69	1.35645
$188$	−552.831	−0.214465
$189$	0	0
$190$	−1972.50	−0.753158
$191$	−2348.35	−0.889639	−0.444819	−	0.895620i	$-0.646732\pi$
$191$	−2348.35	−0.889639	−0.444819	+	0.895620i	$0.646732\pi$
$192$	0	0
$193$	−234.169	−0.0873362	−0.0436681	−	0.999046i	$-0.513904\pi$
$193$	−234.169	−0.0873362	−0.0436681	+	0.999046i	$0.513904\pi$
$194$	4862.87	1.79966
$195$	0	0
$196$	−194.058	−0.0707207
$197$	−717.846	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$197$	−717.846	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$198$	0	0
$199$	1701.45	0.606094	0.303047	−	0.952976i	$-0.401996\pi$
$199$	1701.45	0.606094	0.303047	+	0.952976i	$0.401996\pi$
$200$	−513.127	−0.181418
$201$	0	0
$202$	−650.933	−0.226730
$203$	−123.895	−0.0428361
$204$	0	0
$205$	−608.698	−0.207382
$206$	5086.77	1.72045
$207$	0	0
$208$	−4219.69	−1.40665
$209$	4125.19	1.36529
$210$	0	0
$211$	−4938.66	−1.61133	−0.805667	−	0.592368i	$-0.798193\pi$
$211$	−4938.66	−1.61133	−0.805667	+	0.592368i	$0.798193\pi$
$212$	−743.311	−0.240806
$213$	0	0
$214$	−1827.17	−0.583658
$215$	−50.7086	−0.0160851
$216$	0	0
$217$	457.726	0.143191
$218$	−2346.92	−0.729143
$219$	0	0
$220$	−199.141	−0.0610275
$221$	6351.70	1.93331
$222$	0	0
$223$	−3053.15	−0.916833	−0.458416	−	0.888737i	$-0.651583\pi$
$223$	−3053.15	−0.916833	−0.458416	+	0.888737i	$0.651583\pi$
$224$	770.242	0.229750
$225$	0	0
$226$	−3548.73	−1.04450
$227$	−839.953	−0.245593	−0.122797	−	0.992432i	$-0.539186\pi$
$227$	−839.953	−0.245593	−0.122797	+	0.992432i	$0.539186\pi$
$228$	0	0
$229$	1401.51	0.404429	0.202214	−	0.979341i	$-0.435186\pi$
$229$	1401.51	0.404429	0.202214	+	0.979341i	$0.435186\pi$
$230$	−1211.65	−0.347365
$231$	0	0
$232$	−185.452	−0.0524807
$233$	−2856.99	−0.803295	−0.401647	−	0.915794i	$-0.631562\pi$
$233$	−2856.99	−0.803295	−0.401647	+	0.915794i	$0.631562\pi$
$234$	0	0
$235$	2207.46	0.612761
$236$	554.043	0.152818
$237$	0	0
$238$	−4548.57	−1.23882
$239$	−1709.83	−0.462761	−0.231381	−	0.972863i	$-0.574324\pi$
$239$	−1709.83	−0.462761	−0.231381	+	0.972863i	$0.574324\pi$
$240$	0	0
$241$	−4332.45	−1.15800	−0.579000	−	0.815328i	$-0.696557\pi$
$241$	−4332.45	−1.15800	−0.579000	+	0.815328i	$0.696557\pi$
$242$	−971.309	−0.258009
$243$	0	0
$244$	181.036	0.0474987
$245$	774.875	0.202061
$246$	0	0
$247$	7553.86	1.94591
$248$	685.146	0.175431
$249$	0	0
$250$	−380.218	−0.0961883
$251$	−1724.73	−0.433722	−0.216861	−	0.976202i	$-0.569582\pi$
$251$	−1724.73	−0.433722	−0.216861	+	0.976202i	$0.569582\pi$
$252$	0	0
$253$	2534.00	0.629688
$254$	4958.33	1.22486
$255$	0	0
$256$	1878.70	0.458666
$257$	−5853.25	−1.42068	−0.710342	−	0.703856i	$-0.751460\pi$
$257$	−5853.25	−1.42068	−0.710342	+	0.703856i	$0.751460\pi$
$258$	0	0
$259$	303.925	0.0729149
$260$	−364.657	−0.0869810
$261$	0	0
$262$	787.277	0.185642
$263$	392.360	0.0919923	0.0459962	−	0.998942i	$-0.485354\pi$
$263$	392.360	0.0919923	0.0459962	+	0.998942i	$0.485354\pi$
$264$	0	0
$265$	2968.05	0.688022
$266$	−5409.47	−1.24690
$267$	0	0
$268$	1080.29	0.246228
$269$	−4610.33	−1.04497	−0.522485	−	0.852648i	$-0.674995\pi$
$269$	−4610.33	−1.04497	−0.522485	+	0.852648i	$0.674995\pi$
$270$	0	0
$271$	−1155.72	−0.259058	−0.129529	−	0.991576i	$-0.541347\pi$
$271$	−1155.72	−0.259058	−0.129529	+	0.991576i	$0.541347\pi$
$272$	−7900.97	−1.76128
$273$	0	0
$274$	3202.25	0.706041
$275$	795.170	0.174366
$276$	0	0
$277$	−608.410	−0.131970	−0.0659852	−	0.997821i	$-0.521019\pi$
$277$	−608.410	−0.131970	−0.0659852	+	0.997821i	$0.521019\pi$
$278$	7253.02	1.56477
$279$	0	0
$280$	−1407.22	−0.300349
$281$	2749.24	0.583650	0.291825	−	0.956472i	$-0.405737\pi$
$281$	2749.24	0.583650	0.291825	+	0.956472i	$0.405737\pi$
$282$	0	0
$283$	1021.56	0.214578	0.107289	−	0.994228i	$-0.465783\pi$
$283$	1021.56	0.214578	0.107289	+	0.994228i	$0.465783\pi$
$284$	−1024.97	−0.214157
$285$	0	0
$286$	5634.91	1.16503
$287$	−1669.32	−0.343334
$288$	0	0
$289$	6979.96	1.42071
$290$	−137.416	−0.0278254
$291$	0	0
$292$	619.898	0.124236
$293$	−5232.50	−1.04330	−0.521648	−	0.853161i	$-0.674683\pi$
$293$	−5232.50	−1.04330	−0.521648	+	0.853161i	$0.674683\pi$
$294$	0	0
$295$	−2212.30	−0.436627
$296$	454.929	0.0893317
$297$	0	0
$298$	3486.71	0.677785
$299$	4640.14	0.897479
$300$	0	0
$301$	−139.065	−0.0266299
$302$	−3547.02	−0.675855
$303$	0	0
$304$	−9396.36	−1.77276
$305$	−722.881	−0.135712
$306$	0	0
$307$	−4912.68	−0.913296	−0.456648	−	0.889648i	$-0.650950\pi$
$307$	−4912.68	−0.913296	−0.456648	+	0.889648i	$0.650950\pi$
$308$	−546.132	−0.101035
$309$	0	0
$310$	507.680	0.0930138
$311$	−828.860	−0.151126	−0.0755632	−	0.997141i	$-0.524075\pi$
$311$	−828.860	−0.151126	−0.0755632	+	0.997141i	$0.524075\pi$
$312$	0	0
$313$	9237.21	1.66811	0.834054	−	0.551683i	$-0.186014\pi$
$313$	9237.21	1.66811	0.834054	+	0.551683i	$0.186014\pi$
$314$	−10571.9	−1.90003
$315$	0	0
$316$	1465.72	0.260928
$317$	−9098.41	−1.61204	−0.806021	−	0.591887i	$-0.798383\pi$
$317$	−9098.41	−1.61204	−0.806021	+	0.591887i	$0.798383\pi$
$318$	0	0
$319$	287.386	0.0504406
$320$	−2043.68	−0.357016
$321$	0	0
$322$	−3322.89	−0.575086
$323$	14143.9	2.43649
$324$	0	0
$325$	1456.08	0.248519
$326$	−6762.76	−1.14894
$327$	0	0
$328$	−2498.72	−0.420636
$329$	6053.84	1.01447
$330$	0	0
$331$	5963.45	0.990275	0.495137	−	0.868815i	$-0.335118\pi$
$331$	5963.45	0.990275	0.495137	+	0.868815i	$0.335118\pi$
$332$	−530.982	−0.0877754
$333$	0	0
$334$	−4830.23	−0.791313
$335$	−4313.60	−0.703515
$336$	0	0
$337$	596.854	0.0964768	0.0482384	−	0.998836i	$-0.484639\pi$
$337$	596.854	0.0964768	0.0482384	+	0.998836i	$0.484639\pi$
$338$	3635.69	0.585075
$339$	0	0
$340$	−682.786	−0.108910
$341$	−1061.74	−0.168611
$342$	0	0
$343$	6828.34	1.07492
$344$	−208.160	−0.0326256
$345$	0	0
$346$	4586.32	0.712608
$347$	−4562.10	−0.705782	−0.352891	−	0.935664i	$-0.614801\pi$
$347$	−4562.10	−0.705782	−0.352891	+	0.935664i	$0.614801\pi$
$348$	0	0
$349$	675.267	0.103571	0.0517854	−	0.998658i	$-0.483509\pi$
$349$	675.267	0.103571	0.0517854	+	0.998658i	$0.483509\pi$
$350$	−1042.73	−0.159246
$351$	0	0
$352$	−1786.65	−0.270536
$353$	−3361.83	−0.506890	−0.253445	−	0.967350i	$-0.581564\pi$
$353$	−3361.83	−0.506890	−0.253445	+	0.967350i	$0.581564\pi$
$354$	0	0
$355$	4092.70	0.611882
$356$	1291.46	0.192268
$357$	0	0
$358$	4783.63	0.706208
$359$	7701.61	1.13224	0.566122	−	0.824322i	$-0.308443\pi$
$359$	7701.61	1.13224	0.566122	+	0.824322i	$0.308443\pi$
$360$	0	0
$361$	9961.87	1.45238
$362$	6036.07	0.876378
$363$	0	0
$364$	−1000.05	−0.144003
$365$	−2475.26	−0.354962
$366$	0	0
$367$	5909.50	0.840526	0.420263	−	0.907402i	$-0.361938\pi$
$367$	5909.50	0.840526	0.420263	+	0.907402i	$0.361938\pi$
$368$	−5771.94	−0.817617
$369$	0	0
$370$	337.093	0.0473639
$371$	8139.72	1.13906
$372$	0	0
$373$	−8424.38	−1.16943	−0.584716	−	0.811238i	$-0.698794\pi$
$373$	−8424.38	−1.16943	−0.584716	+	0.811238i	$0.698794\pi$
$374$	10550.8	1.45875
$375$	0	0
$376$	9061.67	1.24287
$377$	526.249	0.0718917
$378$	0	0
$379$	−3510.24	−0.475749	−0.237875	−	0.971296i	$-0.576451\pi$
$379$	−3510.24	−0.475749	−0.237875	+	0.971296i	$0.576451\pi$
$380$	−812.014	−0.109620
$381$	0	0
$382$	−7143.09	−0.956733
$383$	5732.03	0.764733	0.382367	−	0.924011i	$-0.375109\pi$
$383$	5732.03	0.764733	0.382367	+	0.924011i	$0.375109\pi$
$384$	0	0
$385$	2180.71	0.288674
$386$	−712.283	−0.0939229
$387$	0	0
$388$	2001.89	0.261934
$389$	14367.2	1.87261	0.936306	−	0.351184i	$-0.114221\pi$
$389$	14367.2	1.87261	0.936306	+	0.351184i	$0.114221\pi$
$390$	0	0
$391$	8688.23	1.12374
$392$	3180.88	0.409843
$393$	0	0
$394$	−2183.50	−0.279196
$395$	−5852.63	−0.745514
$396$	0	0
$397$	−9014.42	−1.13960	−0.569800	−	0.821784i	$-0.692979\pi$
$397$	−9014.42	−1.13960	−0.569800	+	0.821784i	$0.692979\pi$
$398$	5175.37	0.651804
$399$	0	0
$400$	−1811.24	−0.226405
$401$	10511.1	1.30897	0.654487	−	0.756073i	$-0.272885\pi$
$401$	10511.1	1.30897	0.654487	+	0.756073i	$0.272885\pi$
$402$	0	0
$403$	−1944.21	−0.240317
$404$	−267.969	−0.0329998
$405$	0	0
$406$	−376.857	−0.0460667
$407$	−704.982	−0.0858591
$408$	0	0
$409$	3318.20	0.401160	0.200580	−	0.979677i	$-0.435717\pi$
$409$	3318.20	0.401160	0.200580	+	0.979677i	$0.435717\pi$
$410$	−1851.50	−0.223022
$411$	0	0
$412$	2094.06	0.250405
$413$	−6067.11	−0.722864
$414$	0	0
$415$	2120.22	0.250789
$416$	−3271.63	−0.385589
$417$	0	0
$418$	12547.8	1.46826
$419$	−486.138	−0.0566812	−0.0283406	−	0.999598i	$-0.509022\pi$
$419$	−486.138	−0.0566812	−0.0283406	+	0.999598i	$0.509022\pi$
$420$	0	0
$421$	5051.93	0.584836	0.292418	−	0.956291i	$-0.405540\pi$
$421$	5051.93	0.584836	0.292418	+	0.956291i	$0.405540\pi$
$422$	−15022.1	−1.73286
$423$	0	0
$424$	12183.9	1.39553
$425$	2726.37	0.311173
$426$	0	0
$427$	−1982.46	−0.224679
$428$	−752.188	−0.0849495
$429$	0	0
$430$	−154.242	−0.0172982
$431$	6944.24	0.776084	0.388042	−	0.921642i	$-0.373152\pi$
$431$	6944.24	0.776084	0.388042	+	0.921642i	$0.373152\pi$
$432$	0	0
$433$	−13738.3	−1.52476	−0.762378	−	0.647131i	$-0.775968\pi$
$433$	−13738.3	−1.52476	−0.762378	+	0.647131i	$0.775968\pi$
$434$	1392.28	0.153990
$435$	0	0
$436$	−966.150	−0.106124
$437$	10332.6	1.13107
$438$	0	0
$439$	8481.83	0.922132	0.461066	−	0.887366i	$-0.347467\pi$
$439$	8481.83	0.922132	0.461066	+	0.887366i	$0.347467\pi$
$440$	3264.19	0.353668
$441$	0	0
$442$	19320.2	2.07912
$443$	−558.195	−0.0598660	−0.0299330	−	0.999552i	$-0.509529\pi$
$443$	−558.195	−0.0598660	−0.0299330	+	0.999552i	$0.509529\pi$
$444$	0	0
$445$	−5156.83	−0.549342
$446$	−9286.88	−0.985979
$447$	0	0
$448$	−5604.68	−0.591063
$449$	14775.0	1.55295	0.776476	−	0.630147i	$-0.217005\pi$
$449$	14775.0	1.55295	0.776476	+	0.630147i	$0.217005\pi$
$450$	0	0
$451$	3872.15	0.404285
$452$	−1460.90	−0.152024
$453$	0	0
$454$	−2554.92	−0.264115
$455$	3993.22	0.411439
$456$	0	0
$457$	−14207.9	−1.45431	−0.727154	−	0.686474i	$-0.759158\pi$
$457$	−14207.9	−1.45431	−0.727154	+	0.686474i	$0.759158\pi$
$458$	4263.02	0.434930
$459$	0	0
$460$	−498.799	−0.0505579
$461$	−7702.51	−0.778182	−0.389091	−	0.921199i	$-0.627211\pi$
$461$	−7702.51	−0.778182	−0.389091	+	0.921199i	$0.627211\pi$
$462$	0	0
$463$	7909.07	0.793878	0.396939	−	0.917845i	$-0.370072\pi$
$463$	7909.07	0.793878	0.396939	+	0.917845i	$0.370072\pi$
$464$	−654.608	−0.0654945
$465$	0	0
$466$	−8690.22	−0.863877
$467$	11639.8	1.15337	0.576685	−	0.816967i	$-0.304346\pi$
$467$	11639.8	1.15337	0.576685	+	0.816967i	$0.304346\pi$
$468$	0	0
$469$	−11829.8	−1.16471
$470$	6714.52	0.658974
$471$	0	0
$472$	−9081.53	−0.885618
$473$	322.576	0.0313574
$474$	0	0
$475$	3242.38	0.313202
$476$	−1872.50	−0.180307
$477$	0	0
$478$	−5200.87	−0.497662
$479$	−18695.9	−1.78337	−0.891687	−	0.452653i	$-0.850478\pi$
$479$	−18695.9	−1.78337	−0.891687	+	0.452653i	$0.850478\pi$
$480$	0	0
$481$	−1290.93	−0.122373
$482$	−13178.2	−1.24533
$483$	0	0
$484$	−399.857	−0.0375523
$485$	−7993.56	−0.748390
$486$	0	0
$487$	−2249.54	−0.209315	−0.104658	−	0.994508i	$-0.533375\pi$
$487$	−2249.54	−0.209315	−0.104658	+	0.994508i	$0.533375\pi$
$488$	−2967.44	−0.275266
$489$	0	0
$490$	2356.97	0.217300
$491$	1988.35	0.182756	0.0913778	−	0.995816i	$-0.470873\pi$
$491$	1988.35	0.182756	0.0913778	+	0.995816i	$0.470873\pi$
$492$	0	0
$493$	985.351	0.0900162
$494$	22976.9	2.09267
$495$	0	0
$496$	2418.43	0.218933
$497$	11224.0	1.01301
$498$	0	0
$499$	2413.90	0.216556	0.108278	−	0.994121i	$-0.465466\pi$
$499$	2413.90	0.216556	0.108278	+	0.994121i	$0.465466\pi$
$500$	−156.523	−0.0139999
$501$	0	0
$502$	−5246.19	−0.466432
$503$	−8758.56	−0.776391	−0.388196	−	0.921577i	$-0.626902\pi$
$503$	−8758.56	−0.776391	−0.388196	+	0.921577i	$0.626902\pi$
$504$	0	0
$505$	1070.00	0.0942860
$506$	7707.77	0.677178
$507$	0	0
$508$	2041.19	0.178274
$509$	−11848.0	−1.03174	−0.515869	−	0.856667i	$-0.672531\pi$
$509$	−11848.0	−1.03174	−0.515869	+	0.856667i	$0.672531\pi$
$510$	0	0
$511$	−6788.27	−0.587662
$512$	−7826.64	−0.675570
$513$	0	0
$514$	−17804.1	−1.52783
$515$	−8361.60	−0.715449
$516$	0	0
$517$	−14042.5	−1.19456
$518$	924.460	0.0784140
$519$	0	0
$520$	5977.23	0.504075
$521$	3816.55	0.320933	0.160466	−	0.987041i	$-0.448700\pi$
$521$	3816.55	0.320933	0.160466	+	0.987041i	$0.448700\pi$
$522$	0	0
$523$	−12158.9	−1.01658	−0.508288	−	0.861187i	$-0.669722\pi$
$523$	−12158.9	−1.01658	−0.508288	+	0.861187i	$0.669722\pi$
$524$	324.097	0.0270195
$525$	0	0
$526$	1193.46	0.0989302
$527$	−3640.35	−0.300903
$528$	0	0
$529$	−5819.94	−0.478338
$530$	9028.04	0.739911
$531$	0	0
$532$	−2226.90	−0.181482
$533$	7090.50	0.576217
$534$	0	0
$535$	3003.49	0.242715
$536$	−17707.4	−1.42695
$537$	0	0
$538$	−14023.4	−1.12378
$539$	−4929.26	−0.393911
$540$	0	0
$541$	14919.8	1.18568	0.592840	−	0.805321i	$-0.298007\pi$
$541$	14919.8	1.18568	0.592840	+	0.805321i	$0.298007\pi$
$542$	−3515.39	−0.278596
$543$	0	0
$544$	−6125.82	−0.482799
$545$	3857.85	0.303215
$546$	0	0
$547$	−1027.16	−0.0802889	−0.0401444	−	0.999194i	$-0.512782\pi$
$547$	−1027.16	−0.0802889	−0.0401444	+	0.999194i	$0.512782\pi$
$548$	1318.27	0.102762
$549$	0	0
$550$	2418.70	0.187516
$551$	1171.85	0.0906031
$552$	0	0
$553$	−16050.5	−1.23425
$554$	−1850.63	−0.141923
$555$	0	0
$556$	2985.84	0.227748
$557$	−10590.8	−0.805648	−0.402824	−	0.915277i	$-0.631971\pi$
$557$	−10590.8	−0.805648	−0.402824	+	0.915277i	$0.631971\pi$
$558$	0	0
$559$	590.685	0.0446929
$560$	−4967.22	−0.374827
$561$	0	0
$562$	8362.46	0.627668
$563$	−2192.07	−0.164093	−0.0820467	−	0.996628i	$-0.526146\pi$
$563$	−2192.07	−0.164093	−0.0820467	+	0.996628i	$0.526146\pi$
$564$	0	0
$565$	5833.38	0.434358
$566$	3107.32	0.230761
$567$	0	0
$568$	16800.6	1.24109
$569$	−16569.8	−1.22081	−0.610405	−	0.792089i	$-0.708993\pi$
$569$	−16569.8	−1.22081	−0.610405	+	0.792089i	$0.708993\pi$
$570$	0	0
$571$	7279.33	0.533503	0.266752	−	0.963765i	$-0.414050\pi$
$571$	7279.33	0.533503	0.266752	+	0.963765i	$0.414050\pi$
$572$	2319.71	0.169567
$573$	0	0
$574$	−5077.64	−0.369227
$575$	1991.71	0.144452
$576$	0	0
$577$	−17938.0	−1.29422	−0.647112	−	0.762395i	$-0.724024\pi$
$577$	−17938.0	−1.29422	−0.647112	+	0.762395i	$0.724024\pi$
$578$	21231.2	1.52786
$579$	0	0
$580$	−56.5699	−0.00404989
$581$	5814.58	0.415197
$582$	0	0
$583$	−18880.8	−1.34128
$584$	−10161.0	−0.719974
$585$	0	0
$586$	−15915.9	−1.12198
$587$	11509.7	0.809293	0.404646	−	0.914473i	$-0.367395\pi$
$587$	11509.7	0.809293	0.404646	+	0.914473i	$0.367395\pi$
$588$	0	0
$589$	−4329.35	−0.302865
$590$	−6729.24	−0.469557
$591$	0	0
$592$	1605.81	0.111484
$593$	−9612.80	−0.665684	−0.332842	−	0.942983i	$-0.608008\pi$
$593$	−9612.80	−0.665684	−0.332842	+	0.942983i	$0.608008\pi$
$594$	0	0
$595$	7476.92	0.515166
$596$	1435.37	0.0986493
$597$	0	0
$598$	14114.1	0.965165
$599$	7371.99	0.502857	0.251429	−	0.967876i	$-0.419100\pi$
$599$	7371.99	0.502857	0.251429	+	0.967876i	$0.419100\pi$
$600$	0	0
$601$	−24190.6	−1.64185	−0.820927	−	0.571033i	$-0.806543\pi$
$601$	−24190.6	−1.64185	−0.820927	+	0.571033i	$0.806543\pi$
$602$	−423.001	−0.0286383
$603$	0	0
$604$	−1460.20	−0.0983684
$605$	1596.63	0.107293
$606$	0	0
$607$	−1317.73	−0.0881138	−0.0440569	−	0.999029i	$-0.514028\pi$
$607$	−1317.73	−0.0881138	−0.0440569	+	0.999029i	$0.514028\pi$
$608$	−7285.23	−0.485946
$609$	0	0
$610$	−2198.82	−0.145947
$611$	−25713.9	−1.70257
$612$	0	0
$613$	−4137.52	−0.272615	−0.136307	−	0.990667i	$-0.543523\pi$
$613$	−4137.52	−0.272615	−0.136307	+	0.990667i	$0.543523\pi$
$614$	−14943.1	−0.982174
$615$	0	0
$616$	8951.86	0.585521
$617$	−7791.61	−0.508393	−0.254197	−	0.967153i	$-0.581811\pi$
$617$	−7791.61	−0.508393	−0.254197	+	0.967153i	$0.581811\pi$
$618$	0	0
$619$	21956.4	1.42569	0.712844	−	0.701322i	$-0.247407\pi$
$619$	21956.4	1.42569	0.712844	+	0.701322i	$0.247407\pi$
$620$	208.996	0.0135379
$621$	0	0
$622$	−2521.18	−0.162524
$623$	−14142.3	−0.909471
$624$	0	0
$625$	625.000	0.0400000
$626$	28097.2	1.79391
$627$	0	0
$628$	−4352.13	−0.276543
$629$	−2417.15	−0.153224
$630$	0	0
$631$	−11152.7	−0.703618	−0.351809	−	0.936072i	$-0.614433\pi$
$631$	−11152.7	−0.703618	−0.351809	+	0.936072i	$0.614433\pi$
$632$	−24025.2	−1.51214
$633$	0	0
$634$	−27675.0	−1.73362
$635$	−8150.48	−0.509357
$636$	0	0
$637$	−9026.23	−0.561432
$638$	874.155	0.0542447
$639$	0	0
$640$	−8463.22	−0.522716
$641$	4809.44	0.296352	0.148176	−	0.988961i	$-0.452660\pi$
$641$	4809.44	0.296352	0.148176	+	0.988961i	$0.452660\pi$
$642$	0	0
$643$	−2553.03	−0.156581	−0.0782904	−	0.996931i	$-0.524946\pi$
$643$	−2553.03	−0.156581	−0.0782904	+	0.996931i	$0.524946\pi$
$644$	−1367.93	−0.0837018
$645$	0	0
$646$	43022.1	2.62025
$647$	8446.00	0.513210	0.256605	−	0.966516i	$-0.417396\pi$
$647$	8446.00	0.513210	0.256605	+	0.966516i	$0.417396\pi$
$648$	0	0
$649$	14073.2	0.851191
$650$	4429.01	0.267262
$651$	0	0
$652$	−2784.01	−0.167224
$653$	8094.87	0.485110	0.242555	−	0.970138i	$-0.422015\pi$
$653$	8094.87	0.485110	0.242555	+	0.970138i	$0.422015\pi$
$654$	0	0
$655$	−1294.12	−0.0771994
$656$	−8819.97	−0.524942
$657$	0	0
$658$	18414.2	1.09097
$659$	2175.33	0.128587	0.0642936	−	0.997931i	$-0.479521\pi$
$659$	2175.33	0.128587	0.0642936	+	0.997931i	$0.479521\pi$
$660$	0	0
$661$	−8876.95	−0.522350	−0.261175	−	0.965291i	$-0.584110\pi$
$661$	−8876.95	−0.522350	−0.261175	+	0.965291i	$0.584110\pi$
$662$	18139.3	1.06496
$663$	0	0
$664$	8703.53	0.508679
$665$	8892.06	0.518525
$666$	0	0
$667$	719.834	0.0417872
$668$	−1988.45	−0.115173
$669$	0	0
$670$	−13120.9	−0.756572
$671$	4598.51	0.264565
$672$	0	0
$673$	−9167.92	−0.525108	−0.262554	−	0.964917i	$-0.584565\pi$
$673$	−9167.92	−0.525108	−0.262554	+	0.964917i	$0.584565\pi$
$674$	1815.47	0.103753
$675$	0	0
$676$	1496.70	0.0851557
$677$	−18877.9	−1.07169	−0.535846	−	0.844316i	$-0.680007\pi$
$677$	−18877.9	−1.07169	−0.535846	+	0.844316i	$0.680007\pi$
$678$	0	0
$679$	−21921.9	−1.23901
$680$	11191.8	0.631156
$681$	0	0
$682$	−3229.54	−0.181328
$683$	−27207.5	−1.52426	−0.762128	−	0.647426i	$-0.775845\pi$
$683$	−27207.5	−1.52426	−0.762128	+	0.647426i	$0.775845\pi$
$684$	0	0
$685$	−5263.85	−0.293608
$686$	20770.1	1.15598
$687$	0	0
$688$	−734.762	−0.0407159
$689$	−34573.7	−1.91169
$690$	0	0
$691$	−17407.2	−0.958325	−0.479162	−	0.877726i	$-0.659060\pi$
$691$	−17407.2	−0.958325	−0.479162	+	0.877726i	$0.659060\pi$
$692$	1888.04	0.103718
$693$	0	0
$694$	−13876.7	−0.759011
$695$	−11922.5	−0.650713
$696$	0	0
$697$	13276.3	0.721486
$698$	2053.99	0.111382
$699$	0	0
$700$	−429.257	−0.0231777
$701$	18543.0	0.999086	0.499543	−	0.866289i	$-0.333501\pi$
$701$	18543.0	0.999086	0.499543	+	0.866289i	$0.333501\pi$
$702$	0	0
$703$	−2874.63	−0.154223
$704$	13000.6	0.695991
$705$	0	0
$706$	−10225.8	−0.545118
$707$	2934.42	0.156096
$708$	0	0
$709$	26233.8	1.38961	0.694803	−	0.719200i	$-0.255492\pi$
$709$	26233.8	1.38961	0.694803	+	0.719200i	$0.255492\pi$
$710$	12448.9	0.658029
$711$	0	0
$712$	−21168.9	−1.11424
$713$	−2659.40	−0.139685
$714$	0	0
$715$	−9262.65	−0.484480
$716$	1969.27	0.102786
$717$	0	0
$718$	23426.3	1.21763
$719$	−3043.06	−0.157840	−0.0789199	−	0.996881i	$-0.525147\pi$
$719$	−3043.06	−0.157840	−0.0789199	+	0.996881i	$0.525147\pi$
$720$	0	0
$721$	−22931.2	−1.18447
$722$	30301.4	1.56192
$723$	0	0
$724$	2484.86	0.127554
$725$	225.884	0.0115712
$726$	0	0
$727$	21942.5	1.11940	0.559698	−	0.828696i	$-0.310917\pi$
$727$	21942.5	1.11940	0.559698	+	0.828696i	$0.310917\pi$
$728$	16392.2	0.834528
$729$	0	0
$730$	−7529.10	−0.381732
$731$	1106.00	0.0559603
$732$	0	0
$733$	−26576.9	−1.33921	−0.669604	−	0.742718i	$-0.733536\pi$
$733$	−26576.9	−1.33921	−0.669604	+	0.742718i	$0.733536\pi$
$734$	17975.2	0.903917
$735$	0	0
$736$	−4475.13	−0.224124
$737$	27440.4	1.37148
$738$	0	0
$739$	28787.7	1.43298	0.716489	−	0.697598i	$-0.245748\pi$
$739$	28787.7	1.43298	0.716489	+	0.697598i	$0.245748\pi$
$740$	138.771	0.00689366
$741$	0	0
$742$	24758.9	1.22497
$743$	−4503.02	−0.222341	−0.111171	−	0.993801i	$-0.535460\pi$
$743$	−4503.02	−0.222341	−0.111171	+	0.993801i	$0.535460\pi$
$744$	0	0
$745$	−5731.44	−0.281858
$746$	−25624.8	−1.25763
$747$	0	0
$748$	4343.45	0.212316
$749$	8236.92	0.401830
$750$	0	0
$751$	5705.65	0.277233	0.138617	−	0.990346i	$-0.455734\pi$
$751$	5705.65	0.277233	0.138617	+	0.990346i	$0.455734\pi$
$752$	31985.9	1.55107
$753$	0	0
$754$	1600.71	0.0773136
$755$	5830.58	0.281055
$756$	0	0
$757$	17397.6	0.835305	0.417652	−	0.908607i	$-0.362853\pi$
$757$	17397.6	0.835305	0.417652	+	0.908607i	$0.362853\pi$
$758$	−10677.2	−0.511629
$759$	0	0
$760$	13310.0	0.635271
$761$	−35514.4	−1.69171	−0.845857	−	0.533410i	$-0.820910\pi$
$761$	−35514.4	−1.69171	−0.845857	+	0.533410i	$0.820910\pi$
$762$	0	0
$763$	10579.9	0.501992
$764$	−2940.58	−0.139249
$765$	0	0
$766$	17435.3	0.822408
$767$	25770.3	1.21318
$768$	0	0
$769$	31263.7	1.46605	0.733027	−	0.680199i	$-0.238107\pi$
$769$	31263.7	1.46605	0.733027	+	0.680199i	$0.238107\pi$
$770$	6633.16	0.310445
$771$	0	0
$772$	−293.224	−0.0136702
$773$	−6676.47	−0.310654	−0.155327	−	0.987863i	$-0.549643\pi$
$773$	−6676.47	−0.310654	−0.155327	+	0.987863i	$0.549643\pi$
$774$	0	0
$775$	−834.522	−0.0386799
$776$	−32813.7	−1.51797
$777$	0	0
$778$	43701.3	2.01384
$779$	15789.1	0.726189
$780$	0	0
$781$	−26035.2	−1.19284
$782$	26427.3	1.20849
$783$	0	0
$784$	11227.9	0.511473
$785$	17378.1	0.790128
$786$	0	0
$787$	9433.83	0.427293	0.213646	−	0.976911i	$-0.431466\pi$
$787$	9433.83	0.427293	0.213646	+	0.976911i	$0.431466\pi$
$788$	−898.878	−0.0406361
$789$	0	0
$790$	−17802.2	−0.801739
$791$	15997.7	0.719107
$792$	0	0
$793$	8420.57	0.377078
$794$	−27419.5	−1.22555
$795$	0	0
$796$	2130.54	0.0948679
$797$	−23476.3	−1.04338	−0.521689	−	0.853136i	$-0.674698\pi$
$797$	−23476.3	−1.04338	−0.521689	+	0.853136i	$0.674698\pi$
$798$	0	0
$799$	−48146.9	−2.13181
$800$	−1404.30	−0.0620618
$801$	0	0
$802$	31972.0	1.40769
$803$	15746.0	0.691987
$804$	0	0
$805$	5462.16	0.239150
$806$	−5913.78	−0.258442
$807$	0	0
$808$	4392.38	0.191242
$809$	−33269.8	−1.44586	−0.722932	−	0.690919i	$-0.757206\pi$
$809$	−33269.8	−1.44586	−0.722932	+	0.690919i	$0.757206\pi$
$810$	0	0
$811$	27892.8	1.20771	0.603853	−	0.797096i	$-0.293631\pi$
$811$	27892.8	1.20771	0.603853	+	0.797096i	$0.293631\pi$
$812$	−155.140	−0.00670485
$813$	0	0
$814$	−2144.37	−0.0923345
$815$	11116.6	0.477788
$816$	0	0
$817$	1315.33	0.0563252
$818$	10093.1	0.431415
$819$	0	0
$820$	−762.204	−0.0324601
$821$	−3697.32	−0.157171	−0.0785856	−	0.996907i	$-0.525040\pi$
$821$	−3697.32	−0.157171	−0.0785856	+	0.996907i	$0.525040\pi$
$822$	0	0
$823$	7144.51	0.302602	0.151301	−	0.988488i	$-0.451654\pi$
$823$	7144.51	0.302602	0.151301	+	0.988488i	$0.451654\pi$
$824$	−34324.6	−1.45116
$825$	0	0
$826$	−18454.6	−0.777381
$827$	19866.9	0.835357	0.417679	−	0.908595i	$-0.362844\pi$
$827$	19866.9	0.835357	0.417679	+	0.908595i	$0.362844\pi$
$828$	0	0
$829$	−36736.5	−1.53910	−0.769548	−	0.638589i	$-0.779518\pi$
$829$	−36736.5	−1.53910	−0.769548	+	0.638589i	$0.779518\pi$
$830$	6449.15	0.269703
$831$	0	0
$832$	23806.1	0.991979
$833$	−16900.8	−0.702974
$834$	0	0
$835$	7939.91	0.329068
$836$	5165.52	0.213700
$837$	0	0
$838$	−1478.71	−0.0609560
$839$	30012.3	1.23497	0.617485	−	0.786583i	$-0.288152\pi$
$839$	30012.3	1.23497	0.617485	+	0.786583i	$0.288152\pi$
$840$	0	0
$841$	−24307.4	−0.996653
$842$	15366.6	0.628943
$843$	0	0
$844$	−6184.13	−0.252212
$845$	−5976.32	−0.243304
$846$	0	0
$847$	4378.68	0.177631
$848$	43006.8	1.74158
$849$	0	0
$850$	8292.92	0.334641
$851$	−1765.81	−0.0711295
$852$	0	0
$853$	−5381.02	−0.215994	−0.107997	−	0.994151i	$-0.534444\pi$
$853$	−5381.02	−0.215994	−0.107997	+	0.994151i	$0.534444\pi$
$854$	−6030.13	−0.241624
$855$	0	0
$856$	12329.4	0.492302
$857$	14639.8	0.583530	0.291765	−	0.956490i	$-0.405758\pi$
$857$	14639.8	0.583530	0.291765	+	0.956490i	$0.405758\pi$
$858$	0	0
$859$	29540.0	1.17333	0.586667	−	0.809828i	$-0.300440\pi$
$859$	29540.0	1.17333	0.586667	+	0.809828i	$0.300440\pi$
$860$	−63.4966	−0.00251769
$861$	0	0
$862$	21122.6	0.834615
$863$	3387.63	0.133622	0.0668112	−	0.997766i	$-0.478717\pi$
$863$	3387.63	0.133622	0.0668112	+	0.997766i	$0.478717\pi$
$864$	0	0
$865$	−7538.98	−0.296339
$866$	−41788.3	−1.63975
$867$	0	0
$868$	573.159	0.0224128
$869$	37230.7	1.45336
$870$	0	0
$871$	50247.6	1.95474
$872$	15836.5	0.615015
$873$	0	0
$874$	31429.2	1.21637
$875$	1714.03	0.0662225
$876$	0	0
$877$	−244.104	−0.00939888	−0.00469944	−	0.999989i	$-0.501496\pi$
$877$	−244.104	−0.00939888	−0.00469944	+	0.999989i	$0.501496\pi$
$878$	25799.5	0.991677
$879$	0	0
$880$	11521.9	0.441369
$881$	13910.2	0.531948	0.265974	−	0.963980i	$-0.414307\pi$
$881$	13910.2	0.531948	0.265974	+	0.963980i	$0.414307\pi$
$882$	0	0
$883$	−8805.87	−0.335607	−0.167803	−	0.985820i	$-0.553667\pi$
$883$	−8805.87	−0.335607	−0.167803	+	0.985820i	$0.553667\pi$
$884$	7953.52	0.302608
$885$	0	0
$886$	−1697.89	−0.0643810
$887$	13280.8	0.502735	0.251368	−	0.967892i	$-0.419120\pi$
$887$	13280.8	0.502735	0.251368	+	0.967892i	$0.419120\pi$
$888$	0	0
$889$	−22352.2	−0.843273
$890$	−15685.8	−0.590773
$891$	0	0
$892$	−3823.11	−0.143506
$893$	−57259.5	−2.14571
$894$	0	0
$895$	−7863.30	−0.293677
$896$	−23209.9	−0.865389
$897$	0	0
$898$	44941.7	1.67007
$899$	−301.609	−0.0111893
$900$	0	0
$901$	−64736.1	−2.39364
$902$	11778.1	0.434775
$903$	0	0
$904$	23946.2	0.881015
$905$	−9922.06	−0.364442
$906$	0	0
$907$	−1084.24	−0.0396932	−0.0198466	−	0.999803i	$-0.506318\pi$
$907$	−1084.24	−0.0396932	−0.0198466	+	0.999803i	$0.506318\pi$
$908$	−1051.78	−0.0384411
$909$	0	0
$910$	12146.3	0.442469
$911$	50188.7	1.82528	0.912638	−	0.408769i	$-0.134042\pi$
$911$	50188.7	1.82528	0.912638	+	0.408769i	$0.134042\pi$
$912$	0	0
$913$	−13487.5	−0.488905
$914$	−43216.8	−1.56399
$915$	0	0
$916$	1754.95	0.0633026
$917$	−3549.06	−0.127808
$918$	0	0
$919$	30376.1	1.09033	0.545166	−	0.838328i	$-0.316467\pi$
$919$	30376.1	1.09033	0.545166	+	0.838328i	$0.316467\pi$
$920$	8176.01	0.292995
$921$	0	0
$922$	−23429.1	−0.836871
$923$	−47674.4	−1.70013
$924$	0	0
$925$	−554.113	−0.0196963
$926$	24057.3	0.853750
$927$	0	0
$928$	−507.534	−0.0179533
$929$	−46197.6	−1.63153	−0.815766	−	0.578382i	$-0.803684\pi$
$929$	−46197.6	−1.63153	−0.815766	+	0.578382i	$0.803684\pi$
$930$	0	0
$931$	−20099.5	−0.707557
$932$	−3577.49	−0.125734
$933$	0	0
$934$	35405.1	1.24035
$935$	−17343.4	−0.606621
$936$	0	0
$937$	37004.7	1.29017	0.645085	−	0.764111i	$-0.276822\pi$
$937$	37004.7	1.29017	0.645085	+	0.764111i	$0.276822\pi$
$938$	−35983.2	−1.25255
$939$	0	0
$940$	2764.16	0.0959115
$941$	43814.4	1.51786	0.758932	−	0.651170i	$-0.225722\pi$
$941$	43814.4	1.51786	0.758932	+	0.651170i	$0.225722\pi$
$942$	0	0
$943$	9698.80	0.334927
$944$	−32056.0	−1.10523
$945$	0	0
$946$	981.191	0.0337223
$947$	4609.73	0.158180	0.0790898	−	0.996867i	$-0.474799\pi$
$947$	4609.73	0.158180	0.0790898	+	0.996867i	$0.474799\pi$
$948$	0	0
$949$	28833.4	0.986272
$950$	9862.49	0.336822
$951$	0	0
$952$	30692.9	1.04492
$953$	4281.80	0.145542	0.0727708	−	0.997349i	$-0.476816\pi$
$953$	4281.80	0.145542	0.0727708	+	0.997349i	$0.476816\pi$
$954$	0	0
$955$	11741.8	0.397859
$956$	−2141.03	−0.0724330
$957$	0	0
$958$	−56868.0	−1.91787
$959$	−14435.8	−0.486086
$960$	0	0
$961$	−28676.7	−0.962597
$962$	−3926.68	−0.131602
$963$	0	0
$964$	−5425.05	−0.181254
$965$	1170.85	0.0390579
$966$	0	0
$967$	−36229.8	−1.20483	−0.602416	−	0.798182i	$-0.705795\pi$
$967$	−36229.8	−1.20483	−0.602416	+	0.798182i	$0.705795\pi$
$968$	6554.21	0.217624
$969$	0	0
$970$	−24314.3	−0.804832
$971$	844.928	0.0279249	0.0139624	−	0.999903i	$-0.495555\pi$
$971$	844.928	0.0279249	0.0139624	+	0.999903i	$0.495555\pi$
$972$	0	0
$973$	−32696.7	−1.07730
$974$	−6842.53	−0.225101
$975$	0	0
$976$	−10474.5	−0.343524
$977$	45844.9	1.50124	0.750619	−	0.660735i	$-0.229755\pi$
$977$	45844.9	1.50124	0.750619	+	0.660735i	$0.229755\pi$
$978$	0	0
$979$	32804.5	1.07093
$980$	970.288	0.0316273
$981$	0	0
$982$	6048.05	0.196539
$983$	−40202.0	−1.30442	−0.652210	−	0.758039i	$-0.726158\pi$
$983$	−40202.0	−1.30442	−0.652210	+	0.758039i	$0.726158\pi$
$984$	0	0
$985$	3589.23	0.116104
$986$	2997.18	0.0968050
$987$	0	0
$988$	9458.86	0.304581
$989$	807.974	0.0259778
$990$	0	0
$991$	1797.91	0.0576313	0.0288157	−	0.999585i	$-0.490826\pi$
$991$	1797.91	0.0576313	0.0288157	+	0.999585i	$0.490826\pi$
$992$	1875.07	0.0600136
$993$	0	0
$994$	34140.5	1.08941
$995$	−8507.26	−0.271053
$996$	0	0
$997$	9132.07	0.290086	0.145043	−	0.989425i	$-0.453668\pi$
$997$	9132.07	0.290086	0.145043	+	0.989425i	$0.453668\pi$
$998$	7342.47	0.232888
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.m.1.6		7
3.2	odd	2		405.4.a.n.1.2		7
5.4	even	2		2025.4.a.bb.1.2		7
9.2	odd	6		135.4.e.c.91.6		14
9.4	even	3		45.4.e.c.16.2	✓	14
9.5	odd	6		135.4.e.c.46.6		14
9.7	even	3		45.4.e.c.31.2	yes	14
15.14	odd	2		2025.4.a.ba.1.6		7
45.4	even	6		225.4.e.d.151.6		14
45.7	odd	12		225.4.k.d.49.11		28
45.13	odd	12		225.4.k.d.124.11		28
45.22	odd	12		225.4.k.d.124.4		28
45.34	even	6		225.4.e.d.76.6		14
45.43	odd	12		225.4.k.d.49.4		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.2	✓	14	9.4	even	3
45.4.e.c.31.2	yes	14	9.7	even	3
135.4.e.c.46.6		14	9.5	odd	6
135.4.e.c.91.6		14	9.2	odd	6
225.4.e.d.76.6		14	45.34	even	6
225.4.e.d.151.6		14	45.4	even	6
225.4.k.d.49.4		28	45.43	odd	12
225.4.k.d.49.11		28	45.7	odd	12
225.4.k.d.124.4		28	45.22	odd	12
225.4.k.d.124.11		28	45.13	odd	12
405.4.a.m.1.6		7	1.1	even	1	trivial
405.4.a.n.1.2		7	3.2	odd	2
2025.4.a.ba.1.6		7	15.14	odd	2
2025.4.a.bb.1.2		7	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04174 q^{2} +1.25219 q^{4} -5.00000 q^{5} -13.7122 q^{7} -20.5251 q^{8} +O(q^{10})\)	\(q+3.04174 q^{2} +1.25219 q^{4} -5.00000 q^{5} -13.7122 q^{7} -20.5251 q^{8} -15.2087 q^{10} +31.8068 q^{11} +58.2432 q^{13} -41.7090 q^{14} -72.4495 q^{16} +109.055 q^{17} +129.695 q^{19} -6.26094 q^{20} +96.7481 q^{22} +79.6684 q^{23} +25.0000 q^{25} +177.161 q^{26} -17.1703 q^{28} +9.03537 q^{29} -33.3809 q^{31} -56.1719 q^{32} +331.717 q^{34} +68.5611 q^{35} -22.1645 q^{37} +394.500 q^{38} +102.625 q^{40} +121.740 q^{41} +10.1417 q^{43} +39.8281 q^{44} +242.331 q^{46} -441.492 q^{47} -154.975 q^{49} +76.0435 q^{50} +72.9314 q^{52} -593.610 q^{53} -159.034 q^{55} +281.445 q^{56} +27.4833 q^{58} +442.460 q^{59} +144.576 q^{61} -101.536 q^{62} +408.736 q^{64} -291.216 q^{65} +862.721 q^{67} +136.557 q^{68} +208.545 q^{70} -818.541 q^{71} +495.052 q^{73} -67.4187 q^{74} +162.403 q^{76} -436.142 q^{77} +1170.53 q^{79} +362.248 q^{80} +370.300 q^{82} -424.043 q^{83} -545.274 q^{85} +30.8485 q^{86} -652.838 q^{88} +1031.37 q^{89} -798.643 q^{91} +99.7598 q^{92} -1342.90 q^{94} -648.477 q^{95} +1598.71 q^{97} -471.394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8} + 10 q^{10} - 23 q^{11} + 96 q^{13} + 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 180 q^{20} + 311 q^{22} - 96 q^{23} + 175 q^{25} + 358 q^{26} + 337 q^{28} + 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 110 q^{35} + 404 q^{37} - 305 q^{38} + 90 q^{40} + 47 q^{41} + 525 q^{43} - 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 50 q^{50} + 1682 q^{52} - 506 q^{53} + 115 q^{55} + 981 q^{56} + 1183 q^{58} + 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 480 q^{65} + 1093 q^{67} - 2473 q^{68} - 105 q^{70} - 328 q^{71} + 2085 q^{73} + 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} - 1620 q^{80} - 62 q^{82} - 1290 q^{83} + 805 q^{85} + 2569 q^{86} + 2271 q^{88} + 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1395 q^{95} + 1787 q^{97} - 1279 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8 + 10 * q^10 - 23 * q^11 + 96 * q^13 + 21 * q^14 + 324 * q^16 - 161 * q^17 + 279 * q^19 - 180 * q^20 + 311 * q^22 - 96 * q^23 + 175 * q^25 + 358 * q^26 + 337 * q^28 + 296 * q^29 + 244 * q^31 + 314 * q^32 + 125 * q^34 - 110 * q^35 + 404 * q^37 - 305 * q^38 + 90 * q^40 + 47 * q^41 + 525 * q^43 - 55 * q^44 + 717 * q^46 - 164 * q^47 + 1225 * q^49 - 50 * q^50 + 1682 * q^52 - 506 * q^53 + 115 * q^55 + 981 * q^56 + 1183 * q^58 + 85 * q^59 + 828 * q^61 + 786 * q^62 + 2236 * q^64 - 480 * q^65 + 1093 * q^67 - 2473 * q^68 - 105 * q^70 - 328 * q^71 + 2085 * q^73 + 1316 * q^74 + 2789 * q^76 - 24 * q^77 + 2110 * q^79 - 1620 * q^80 - 62 * q^82 - 1290 * q^83 + 805 * q^85 + 2569 * q^86 + 2271 * q^88 + 3048 * q^89 + 3338 * q^91 - 2763 * q^92 - 517 * q^94 - 1395 * q^95 + 1787 * q^97 - 1279 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more