LMFDB - Embedded newform 405.5.d.a.404.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,5,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$41.8648350490$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.6
Character	$\chi$	$=$	405.404
Dual form			405.5.d.a.404.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-6.24699 q^{2} +23.0249 q^{4} +(13.1570 + 21.2578i) q^{5} +11.6987i q^{7} -43.8846 q^{8} +(-82.1915 - 132.797i) q^{10} +163.351i q^{11} +55.8508i q^{13} -73.0819i q^{14} -94.2517 q^{16} +256.666 q^{17} -618.990 q^{19} +(302.938 + 489.459i) q^{20} -1020.45i q^{22} -659.248 q^{23} +(-278.789 + 559.376i) q^{25} -348.899i q^{26} +269.362i q^{28} +141.622i q^{29} -600.047 q^{31} +1290.94 q^{32} -1603.39 q^{34} +(-248.689 + 153.920i) q^{35} -751.406i q^{37} +3866.82 q^{38} +(-577.389 - 932.891i) q^{40} +2355.44i q^{41} -1986.04i q^{43} +3761.15i q^{44} +4118.32 q^{46} +867.508 q^{47} +2264.14 q^{49} +(1741.59 - 3494.42i) q^{50} +1285.96i q^{52} +3196.69 q^{53} +(-3472.49 + 2149.21i) q^{55} -513.394i q^{56} -884.714i q^{58} +2760.61i q^{59} -96.8767 q^{61} +3748.49 q^{62} -6556.49 q^{64} +(-1187.27 + 734.827i) q^{65} +4338.35i q^{67} +5909.72 q^{68} +(1553.56 - 961.535i) q^{70} -8119.50i q^{71} -2097.20i q^{73} +4694.03i q^{74} -14252.2 q^{76} -1911.00 q^{77} +1859.92 q^{79} +(-1240.07 - 2003.58i) q^{80} -14714.4i q^{82} -12163.2 q^{83} +(3376.95 + 5456.16i) q^{85} +12406.8i q^{86} -7168.61i q^{88} +10695.7i q^{89} -653.383 q^{91} -15179.1 q^{92} -5419.31 q^{94} +(-8144.03 - 13158.4i) q^{95} -6495.02i q^{97} -14144.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) $ 44 * q + 324 * q^4 + 28 * q^10 + 2116 * q^16 - 8 * q^19 + 296 * q^25 + 2224 * q^31 + 872 * q^34 + 1700 * q^40 - 5668 * q^46 - 10792 * q^49 - 3072 * q^55 - 5564 * q^61 + 8348 * q^64 - 9564 * q^70 + 3552 * q^76 - 15800 * q^79 - 13532 * q^85 + 1560 * q^91 + 37652 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−6.24699			−1.56175			−0.780874	−	0.624688i	$-0.785226\pi$
$2$	−6.24699			−1.56175			−0.780874	+	0.624688i	$0.785226\pi$
$3$		0			0
$3$		0			0
$4$	23.0249			1.43906
$5$	13.1570	+	21.2578i	0.526279	+	0.850312i
$5$	13.1570	+	21.2578i	0.526279	+	0.850312i
$6$		0			0
$7$			11.6987i			0.238749i	0.992849	+	0.119375i	$0.0380890\pi$
$7$			11.6987i			0.238749i	−0.992849	+	0.119375i	$0.961911\pi$
$8$	−43.8846			−0.685697
$9$		0			0
$10$	−82.1915	−	132.797i	−0.821915	−	1.32797i
$11$			163.351i			1.35001i	0.737813	+	0.675005i	$0.235859\pi$
$11$			163.351i			1.35001i	−0.737813	+	0.675005i	$0.764141\pi$
$12$		0			0
$13$			55.8508i			0.330478i	0.986254	+	0.165239i	$0.0528396\pi$
$13$			55.8508i			0.330478i	−0.986254	+	0.165239i	$0.947160\pi$
$14$		−	73.0819i		−	0.372867i
$15$		0			0
$16$	−94.2517			−0.368171
$17$	256.666			0.888118			0.444059	−	0.895998i	$-0.353538\pi$
$17$	256.666			0.888118			0.444059	+	0.895998i	$0.353538\pi$
$18$		0			0
$19$	−618.990			−1.71465			−0.857327	−	0.514773i	$-0.827876\pi$
$19$	−618.990			−1.71465			−0.857327	+	0.514773i	$0.827876\pi$
$20$	302.938	+	489.459i	0.757345	+	1.22365i
$21$		0			0
$22$		−	1020.45i		−	2.10838i
$23$	−659.248			−1.24622			−0.623108	−	0.782136i	$-0.714130\pi$
$23$	−659.248			−1.24622			−0.623108	+	0.782136i	$0.714130\pi$
$24$		0			0
$25$	−278.789	+	559.376i	−0.446062	+	0.895002i
$26$		−	348.899i		−	0.516124i
$27$		0			0
$28$			269.362i			0.343574i
$29$			141.622i			0.168398i	0.996449	+	0.0841988i	$0.0268331\pi$
$29$			141.622i			0.168398i	−0.996449	+	0.0841988i	$0.973167\pi$
$30$		0			0
$31$	−600.047			−0.624398			−0.312199	−	0.950017i	$-0.601066\pi$
$31$	−600.047			−0.624398			−0.312199	+	0.950017i	$0.601066\pi$
$32$	1290.94			1.26069
$33$		0			0
$34$	−1603.39			−1.38702
$35$	−248.689	+	153.920i	−0.203012	+	0.125649i
$36$		0			0
$37$		−	751.406i		−	0.548872i	−0.961605	−	0.274436i	$-0.911509\pi$
$37$		−	751.406i		−	0.548872i	0.961605	−	0.274436i	$-0.0884912\pi$
$38$	3866.82			2.67786
$39$		0			0
$40$	−577.389	−	932.891i	−0.360868	−	0.583057i
$41$			2355.44i			1.40121i	0.713548	+	0.700606i	$0.247087\pi$
$41$			2355.44i			1.40121i	−0.713548	+	0.700606i	$0.752913\pi$
$42$		0			0
$43$		−	1986.04i		−	1.07412i	−0.843545	−	0.537058i	$-0.819536\pi$
$43$		−	1986.04i		−	1.07412i	0.843545	−	0.537058i	$-0.180464\pi$
$44$			3761.15i			1.94274i
$45$		0			0
$46$	4118.32			1.94628
$47$	867.508			0.392715			0.196358	−	0.980532i	$-0.437089\pi$
$47$	867.508			0.392715			0.196358	+	0.980532i	$0.437089\pi$
$48$		0			0
$49$	2264.14			0.942999
$50$	1741.59	−	3494.42i	0.696636	−	1.39777i
$51$		0			0
$52$			1285.96i			0.475577i
$53$	3196.69			1.13802			0.569009	−	0.822332i	$-0.307327\pi$
$53$	3196.69			1.13802			0.569009	+	0.822332i	$0.307327\pi$
$54$		0			0
$55$	−3472.49	+	2149.21i	−1.14793	+	0.710481i
$56$		−	513.394i		−	0.163710i
$57$		0			0
$58$		−	884.714i		−	0.262995i
$59$			2760.61i			0.793052i	0.918024	+	0.396526i	$0.129784\pi$
$59$			2760.61i			0.793052i	−0.918024	+	0.396526i	$0.870216\pi$
$60$		0			0
$61$	−96.8767			−0.0260351			−0.0130176	−	0.999915i	$-0.504144\pi$
$61$	−96.8767			−0.0260351			−0.0130176	+	0.999915i	$0.504144\pi$
$62$	3748.49			0.975153
$63$		0			0
$64$	−6556.49			−1.60071
$65$	−1187.27	+	734.827i	−0.281010	+	0.173924i
$66$		0			0
$67$			4338.35i			0.966441i	0.875499	+	0.483221i	$0.160533\pi$
$67$			4338.35i			0.966441i	−0.875499	+	0.483221i	$0.839467\pi$
$68$	5909.72			1.27805
$69$		0			0
$70$	1553.56	−	961.535i	0.317053	−	0.196232i
$71$		−	8119.50i		−	1.61069i	−0.592805	−	0.805346i	$-0.701979\pi$
$71$		−	8119.50i		−	1.61069i	0.592805	−	0.805346i	$-0.298021\pi$
$72$		0			0
$73$		−	2097.20i		−	0.393545i	−0.980449	−	0.196772i	$-0.936954\pi$
$73$		−	2097.20i		−	0.393545i	0.980449	−	0.196772i	$-0.0630460\pi$
$74$			4694.03i			0.857200i
$75$		0			0
$76$	−14252.2			−2.46748
$77$	−1911.00			−0.322314
$78$		0			0
$79$	1859.92			0.298016			0.149008	−	0.988836i	$-0.452392\pi$
$79$	1859.92			0.298016			0.149008	+	0.988836i	$0.452392\pi$
$80$	−1240.07	−	2003.58i	−0.193760	−	0.313060i
$81$		0			0
$82$		−	14714.4i		−	2.18834i
$83$	−12163.2			−1.76560			−0.882799	−	0.469752i	$-0.844343\pi$
$83$	−12163.2			−1.76560			−0.882799	+	0.469752i	$0.844343\pi$
$84$		0			0
$85$	3376.95	+	5456.16i	0.467397	+	0.755177i
$86$			12406.8i			1.67750i
$87$		0			0
$88$		−	7168.61i		−	0.925699i
$89$			10695.7i			1.35030i	0.737681	+	0.675150i	$0.235921\pi$
$89$			10695.7i			1.35030i	−0.737681	+	0.675150i	$0.764079\pi$
$90$		0			0
$91$	−653.383			−0.0789015
$92$	−15179.1			−1.79338
$93$		0			0
$94$	−5419.31			−0.613322
$95$	−8144.03	−	13158.4i	−0.902385	−	1.45799i
$96$		0			0
$97$		−	6495.02i		−	0.690299i	−0.938548	−	0.345149i	$-0.887828\pi$
$97$		−	6495.02i		−	0.690299i	0.938548	−	0.345149i	$-0.112172\pi$
$98$	−14144.1			−1.47273
$99$		0			0
$100$	−6419.09	+	12879.6i	−0.641909	+	1.28796i
$101$		−	14101.4i		−	1.38236i	−0.722684	−	0.691179i	$-0.757092\pi$
$101$		−	14101.4i		−	1.38236i	0.722684	−	0.691179i	$-0.242908\pi$
$102$		0			0
$103$		−	19533.7i		−	1.84124i	−0.390458	−	0.920621i	$-0.627683\pi$
$103$		−	19533.7i		−	1.84124i	0.390458	−	0.920621i	$-0.372317\pi$
$104$		−	2450.99i		−	0.226608i
$105$		0			0
$106$	−19969.7			−1.77730
$107$	2000.10			0.174697			0.0873484	−	0.996178i	$-0.472161\pi$
$107$	2000.10			0.174697			0.0873484	+	0.996178i	$0.472161\pi$
$108$		0			0
$109$	−612.947			−0.0515905			−0.0257952	−	0.999667i	$-0.508212\pi$
$109$	−612.947			−0.0515905			−0.0257952	+	0.999667i	$0.508212\pi$
$110$	21692.6	−	13426.1i	1.79278	−	1.10959i
$111$		0			0
$112$		−	1102.62i		−	0.0879006i
$113$	−10229.1			−0.801092			−0.400546	−	0.916277i	$-0.631180\pi$
$113$	−10229.1			−0.801092			−0.400546	+	0.916277i	$0.631180\pi$
$114$		0			0
$115$	−8673.71	−	14014.2i	−0.655857	−	1.05967i
$116$			3260.84i			0.242334i
$117$		0			0
$118$		−	17245.5i		−	1.23855i
$119$			3002.67i			0.212038i
$120$		0			0
$121$	−12042.6			−0.822527
$122$	605.188			0.0406603
$123$		0			0
$124$	−13816.0			−0.898545
$125$	−15559.1	+	1433.26i	−0.995784	+	0.0917287i
$126$		0			0
$127$			12148.6i			0.753217i	0.926373	+	0.376608i	$0.122910\pi$
$127$			12148.6i			0.753217i	−0.926373	+	0.376608i	$0.877090\pi$
$128$	20303.2			1.23921
$129$		0			0
$130$	7416.84	−	4590.46i	0.438866	−	0.271625i
$131$		−	21697.7i		−	1.26436i	−0.774822	−	0.632179i	$-0.782161\pi$
$131$		−	21697.7i		−	1.26436i	0.774822	−	0.632179i	$-0.217839\pi$
$132$		0			0
$133$		−	7241.39i		−	0.409373i
$134$		−	27101.7i		−	1.50934i
$135$		0			0
$136$	−11263.7			−0.608980
$137$	−25304.6			−1.34822			−0.674108	−	0.738633i	$-0.735472\pi$
$137$	−25304.6			−1.34822			−0.674108	+	0.738633i	$0.735472\pi$
$138$		0			0
$139$	−18431.3			−0.953950			−0.476975	−	0.878917i	$-0.658267\pi$
$139$	−18431.3			−0.953950			−0.476975	+	0.878917i	$0.658267\pi$
$140$	−5726.05	+	3543.99i	−0.292145	+	0.180816i
$141$		0			0
$142$			50722.5i			2.51550i
$143$	−9123.29			−0.446149
$144$		0			0
$145$	−3010.58	+	1863.32i	−0.143191	+	0.0886240i
$146$			13101.2i			0.614618i
$147$		0			0
$148$		−	17301.1i		−	0.789859i
$149$			5448.09i			0.245398i	0.992444	+	0.122699i	$0.0391550\pi$
$149$			5448.09i			0.245398i	−0.992444	+	0.122699i	$0.960845\pi$
$150$		0			0
$151$	11836.4			0.519119			0.259559	−	0.965727i	$-0.416423\pi$
$151$	11836.4			0.519119			0.259559	+	0.965727i	$0.416423\pi$
$152$	27164.1			1.17573
$153$		0			0
$154$	11938.0			0.503374
$155$	−7894.79	−	12755.7i	−0.328607	−	0.530933i
$156$		0			0
$157$			13841.4i			0.561539i	0.959775	+	0.280769i	$0.0905896\pi$
$157$			13841.4i			0.561539i	−0.959775	+	0.280769i	$0.909410\pi$
$158$	−11618.9			−0.465427
$159$		0			0
$160$	16984.9	+	27442.6i	0.663473	+	1.07198i
$161$		−	7712.36i		−	0.297533i
$162$		0			0
$163$			6615.52i			0.248994i	0.992220	+	0.124497i	$0.0397317\pi$
$163$			6615.52i			0.248994i	−0.992220	+	0.124497i	$0.960268\pi$
$164$			54233.7i			2.01642i
$165$		0			0
$166$	75983.4			2.75742
$167$	−11135.4			−0.399277			−0.199638	−	0.979870i	$-0.563977\pi$
$167$	−11135.4			−0.399277			−0.199638	+	0.979870i	$0.563977\pi$
$168$		0			0
$169$	25441.7			0.890784
$170$	−21095.8	−	34084.6i	−0.729957	−	1.17940i
$171$		0			0
$172$		−	45728.4i		−	1.54571i
$173$	17766.7			0.593627			0.296813	−	0.954936i	$-0.404076\pi$
$173$	17766.7			0.593627			0.296813	+	0.954936i	$0.404076\pi$
$174$		0			0
$175$	−6543.99	−	3261.47i	−0.213681	−	0.106497i
$176$		−	15396.1i		−	0.497034i
$177$		0			0
$178$		−	66816.1i		−	2.10883i
$179$			26305.1i			0.820983i	0.911864	+	0.410492i	$0.134643\pi$
$179$			26305.1i			0.820983i	−0.911864	+	0.410492i	$0.865357\pi$
$180$		0			0
$181$	−34338.0			−1.04814			−0.524069	−	0.851676i	$-0.675587\pi$
$181$	−34338.0			−1.04814			−0.524069	+	0.851676i	$0.675587\pi$
$182$	4081.68			0.123224
$183$		0			0
$184$	28930.9			0.854527
$185$	15973.2	−	9886.23i	0.466713	−	0.288860i
$186$		0			0
$187$			41926.7i			1.19897i
$188$	19974.3			0.565140
$189$		0			0
$190$	50875.7	+	82200.2i	1.40930	+	2.27701i
$191$			32934.0i			0.902771i	0.892329	+	0.451385i	$0.149070\pi$
$191$			32934.0i			0.902771i	−0.892329	+	0.451385i	$0.850930\pi$
$192$		0			0
$193$		−	26434.1i		−	0.709658i	−0.934931	−	0.354829i	$-0.884539\pi$
$193$		−	26434.1i		−	0.709658i	0.934931	−	0.354829i	$-0.115461\pi$
$194$			40574.3i			1.07807i
$195$		0			0
$196$	52131.6			1.35703
$197$	3784.70			0.0975212			0.0487606	−	0.998810i	$-0.484473\pi$
$197$	3784.70			0.0975212			0.0487606	+	0.998810i	$0.484473\pi$
$198$		0			0
$199$	26809.9			0.677002			0.338501	−	0.940966i	$-0.390080\pi$
$199$	26809.9			0.677002			0.338501	+	0.940966i	$0.390080\pi$
$200$	12234.5	−	24548.0i	0.305863	−	0.613701i
$201$		0			0
$202$			88091.6i			2.15890i
$203$	−1656.80			−0.0402048
$204$		0			0
$205$	−50071.4	+	30990.4i	−1.19147	+	0.737428i
$206$			122027.i			2.87556i
$207$		0			0
$208$		−	5264.03i		−	0.121672i
$209$		−	101113.i		−	2.31480i
$210$		0			0
$211$	12127.6			0.272402			0.136201	−	0.990681i	$-0.456511\pi$
$211$	12127.6			0.272402			0.136201	+	0.990681i	$0.456511\pi$
$212$	73603.5			1.63767
$213$		0			0
$214$	−12494.6			−0.272832
$215$	42218.8	−	26130.3i	0.913334	−	0.565284i
$216$		0			0
$217$		−	7019.78i		−	0.149075i
$218$	3829.07			0.0805714
$219$		0			0
$220$	−79953.8	+	49485.3i	−1.65194	+	1.02242i
$221$			14335.0i			0.293503i
$222$		0			0
$223$		−	83242.7i		−	1.67393i	−0.547258	−	0.836964i	$-0.684328\pi$
$223$		−	83242.7i		−	1.67393i	0.547258	−	0.836964i	$-0.315672\pi$
$224$			15102.4i			0.300988i
$225$		0			0
$226$	63901.4			1.25110
$227$	−14390.6			−0.279271			−0.139636	−	0.990203i	$-0.544593\pi$
$227$	−14390.6			−0.279271			−0.139636	+	0.990203i	$0.544593\pi$
$228$		0			0
$229$	−47890.6			−0.913228			−0.456614	−	0.889665i	$-0.650938\pi$
$229$	−47890.6			−0.913228			−0.456614	+	0.889665i	$0.650938\pi$
$230$	54184.6	+	87546.4i	1.02428	+	1.65494i
$231$		0			0
$232$		−	6215.05i		−	0.115470i
$233$	−34930.9			−0.643425			−0.321713	−	0.946837i	$-0.604259\pi$
$233$	−34930.9			−0.643425			−0.321713	+	0.946837i	$0.604259\pi$
$234$		0			0
$235$	11413.8	+	18441.3i	0.206678	+	0.333930i
$236$			63562.9i			1.14125i
$237$		0			0
$238$		−	18757.6i		−	0.331149i
$239$		−	2819.28i		−	0.0493563i	−0.999695	−	0.0246782i	$-0.992144\pi$
$239$		−	2819.28i		−	0.0493563i	0.999695	−	0.0246782i	$-0.00785610\pi$
$240$		0			0
$241$	−9159.68			−0.157705			−0.0788526	−	0.996886i	$-0.525126\pi$
$241$	−9159.68			−0.157705			−0.0788526	+	0.996886i	$0.525126\pi$
$242$	75230.2			1.28458
$243$		0			0
$244$	−2230.58			−0.0374660
$245$	29789.2	+	48130.6i	0.496280	+	0.801843i
$246$		0			0
$247$		−	34571.1i		−	0.566655i
$248$	26332.8			0.428148
$249$		0			0
$250$	97197.7	−	8953.57i	1.55516	−	0.143257i
$251$			8151.50i			0.129387i	0.997905	+	0.0646934i	$0.0206069\pi$
$251$			8151.50i			0.129387i	−0.997905	+	0.0646934i	$0.979393\pi$
$252$		0			0
$253$		−	107689.i		−	1.68240i
$254$		−	75892.4i		−	1.17634i
$255$		0			0
$256$	−21930.4			−0.334631
$257$	−14535.7			−0.220074			−0.110037	−	0.993927i	$-0.535097\pi$
$257$	−14535.7			−0.220074			−0.110037	+	0.993927i	$0.535097\pi$
$258$		0			0
$259$	8790.50			0.131043
$260$	−27336.7	+	16919.3i	−0.404389	+	0.250286i
$261$		0			0
$262$			135545.i			1.97461i
$263$	−9131.28			−0.132014			−0.0660070	−	0.997819i	$-0.521026\pi$
$263$	−9131.28			−0.132014			−0.0660070	+	0.997819i	$0.521026\pi$
$264$		0			0
$265$	42058.7	+	67954.6i	0.598914	+	0.967670i
$266$			45236.9i			0.639337i
$267$		0			0
$268$			99890.3i			1.39076i
$269$		−	121211.i		−	1.67508i	−0.546373	−	0.837542i	$-0.683992\pi$
$269$		−	121211.i		−	1.67508i	0.546373	−	0.837542i	$-0.316008\pi$
$270$		0			0
$271$	−77369.8			−1.05350			−0.526748	−	0.850021i	$-0.676589\pi$
$271$	−77369.8			−1.05350			−0.526748	+	0.850021i	$0.676589\pi$
$272$	−24191.2			−0.326979
$273$		0			0
$274$	158078.			2.10557
$275$	−91374.8	−	45540.5i	−1.20826	−	0.602188i
$276$		0			0
$277$		−	48724.7i		−	0.635023i	−0.948254	−	0.317512i	$-0.897153\pi$
$277$		−	48724.7i		−	0.635023i	0.948254	−	0.317512i	$-0.102847\pi$
$278$	115140.			1.48983
$279$		0			0
$280$	10913.6	−	6754.71i	0.139205	−	0.0861570i
$281$			85327.5i			1.08063i	0.841463	+	0.540314i	$0.181695\pi$
$281$			85327.5i			1.08063i	−0.841463	+	0.540314i	$0.818305\pi$
$282$		0			0
$283$		−	23888.3i		−	0.298272i	−0.988817	−	0.149136i	$-0.952351\pi$
$283$		−	23888.3i		−	0.298272i	0.988817	−	0.149136i	$-0.0476493\pi$
$284$		−	186951.i		−	2.31788i
$285$		0			0
$286$	56993.2			0.696772
$287$	−27555.6			−0.334539
$288$		0			0
$289$	−17643.5			−0.211247
$290$	18807.1	−	11640.2i	0.223628	−	0.138408i
$291$		0			0
$292$		−	48287.9i		−	0.566334i
$293$	131326.			1.52973			0.764865	−	0.644190i	$-0.222806\pi$
$293$	131326.			1.52973			0.764865	+	0.644190i	$0.222806\pi$
$294$		0			0
$295$	−58684.6	+	36321.3i	−0.674342	+	0.417366i
$296$			32975.2i			0.376360i
$297$		0			0
$298$		−	34034.2i		−	0.383251i
$299$		−	36819.5i		−	0.411847i
$300$		0			0
$301$	23234.1			0.256445
$302$	−73942.1			−0.810733
$303$		0			0
$304$	58340.8			0.631285
$305$	−1274.60	−	2059.39i	−0.0137017	−	0.0221380i
$306$		0			0
$307$		−	53610.8i		−	0.568821i	−0.958703	−	0.284411i	$-0.908202\pi$
$307$		−	53610.8i		−	0.568821i	0.958703	−	0.284411i	$-0.0917979\pi$
$308$	−44000.6			−0.463829
$309$		0			0
$310$	49318.7	+	79684.6i	0.513202	+	0.829184i
$311$			114280.i			1.18155i	0.806838	+	0.590773i	$0.201177\pi$
$311$			114280.i			1.18155i	−0.806838	+	0.590773i	$0.798823\pi$
$312$		0			0
$313$			65154.6i			0.665053i	0.943094	+	0.332526i	$0.107901\pi$
$313$			65154.6i			0.665053i	−0.943094	+	0.332526i	$0.892099\pi$
$314$		−	86466.9i		−	0.876982i
$315$		0			0
$316$	42824.5			0.428863
$317$	51085.7			0.508371			0.254186	−	0.967155i	$-0.418193\pi$
$317$	51085.7			0.508371			0.254186	+	0.967155i	$0.418193\pi$
$318$		0			0
$319$	−23134.2			−0.227338
$320$	−86263.5	−	139377.i	−0.842417	−	1.36110i
$321$		0			0
$322$			48179.1i			0.464672i
$323$	−158874.			−1.52281
$324$		0			0
$325$	−31241.6	−	15570.6i	−0.295779	−	0.147414i
$326$		−	41327.1i		−	0.388866i
$327$		0			0
$328$		−	103367.i		−	0.960807i
$329$			10148.7i			0.0937605i
$330$		0			0
$331$	−147542.			−1.34666			−0.673331	−	0.739341i	$-0.735137\pi$
$331$	−147542.			−1.34666			−0.673331	+	0.739341i	$0.735137\pi$
$332$	−280057.			−2.54080
$333$		0			0
$334$	69562.9			0.623570
$335$	−92223.9	+	57079.6i	−0.821777	+	0.508617i
$336$		0			0
$337$			118156.i			1.04039i	0.854048	+	0.520194i	$0.174141\pi$
$337$			118156.i			1.04039i	−0.854048	+	0.520194i	$0.825859\pi$
$338$	−158934.			−1.39118
$339$		0			0
$340$	77753.9	+	125628.i	0.672612	+	1.08674i
$341$		−	98018.3i		−	0.842944i
$342$		0			0
$343$			54576.2i			0.463890i
$344$			87156.6i			0.736518i
$345$		0			0
$346$	−110988.			−0.927095
$347$	−106626.			−0.885530			−0.442765	−	0.896638i	$-0.646002\pi$
$347$	−106626.			−0.885530			−0.442765	+	0.896638i	$0.646002\pi$
$348$		0			0
$349$	−36958.8			−0.303436			−0.151718	−	0.988424i	$-0.548481\pi$
$349$	−36958.8			−0.303436			−0.151718	+	0.988424i	$0.548481\pi$
$350$	40880.3	+	20374.4i	0.333716	+	0.166322i
$351$		0			0
$352$			210877.i			1.70194i
$353$	−21920.9			−0.175918			−0.0879589	−	0.996124i	$-0.528034\pi$
$353$	−21920.9			−0.175918			−0.0879589	+	0.996124i	$0.528034\pi$
$354$		0			0
$355$	172603.	−	106828.i	1.36959	−	0.847673i
$356$			246268.i			1.94316i
$357$		0			0
$358$		−	164328.i		−	1.28217i
$359$			118640.i			0.920536i	0.887780	+	0.460268i	$0.152247\pi$
$359$			118640.i			0.920536i	−0.887780	+	0.460268i	$0.847753\pi$
$360$		0			0
$361$	252827.			1.94003
$362$	214509.			1.63693
$363$		0			0
$364$	−15044.1			−0.113544
$365$	44581.9	−	27592.8i	0.334636	−	0.207114i
$366$		0			0
$367$			59087.4i			0.438695i	0.975647	+	0.219347i	$0.0703928\pi$
$367$			59087.4i			0.438695i	−0.975647	+	0.219347i	$0.929607\pi$
$368$	62135.3			0.458820
$369$		0			0
$370$	−99784.8	+	61759.2i	−0.728888	+	0.451126i
$371$			37397.2i			0.271701i
$372$		0			0
$373$		−	174789.i		−	1.25631i	−0.778089	−	0.628155i	$-0.783811\pi$
$373$		−	174789.i		−	1.25631i	0.778089	−	0.628155i	$-0.216189\pi$
$374$		−	261916.i		−	1.87249i
$375$		0			0
$376$	−38070.3			−0.269284
$377$	−7909.72			−0.0556517
$378$		0			0
$379$	−113098.			−0.787366			−0.393683	−	0.919246i	$-0.628799\pi$
$379$	−113098.			−0.787366			−0.393683	+	0.919246i	$0.628799\pi$
$380$	−187516.	−	302970.i	−1.29858	−	2.09813i
$381$		0			0
$382$		−	205738.i		−	1.40990i
$383$	−176723.			−1.20475			−0.602374	−	0.798214i	$-0.705778\pi$
$383$	−176723.			−1.20475			−0.602374	+	0.798214i	$0.705778\pi$
$384$		0			0
$385$	−25143.0	−	40623.7i	−0.169627	−	0.274068i
$386$			165133.i			1.10831i
$387$		0			0
$388$		−	149547.i		−	0.993379i
$389$		−	158727.i		−	1.04894i	−0.851428	−	0.524471i	$-0.824263\pi$
$389$		−	158727.i		−	1.04894i	0.851428	−	0.524471i	$-0.175737\pi$
$390$		0			0
$391$	−169207.			−1.10679
$392$	−99361.0			−0.646612
$393$		0			0
$394$	−23643.0			−0.152304
$395$	24470.9	+	39537.8i	0.156840	+	0.253407i
$396$		0			0
$397$		−	17831.6i		−	0.113138i	−0.998399	−	0.0565692i	$-0.981984\pi$
$397$		−	17831.6i		−	0.113138i	0.998399	−	0.0565692i	$-0.0180161\pi$
$398$	−167481.			−1.05731
$399$		0			0
$400$	26276.3	−	52722.2i	0.164227	−	0.329514i
$401$			13313.0i			0.0827919i	0.999143	+	0.0413960i	$0.0131805\pi$
$401$			13313.0i			0.0827919i	−0.999143	+	0.0413960i	$0.986819\pi$
$402$		0			0
$403$		−	33513.1i		−	0.206350i
$404$		−	324684.i		−	1.98929i
$405$		0			0
$406$	10350.0			0.0627898
$407$	122743.			0.740983
$408$		0			0
$409$	−303954.			−1.81703			−0.908515	−	0.417853i	$-0.862783\pi$
$409$	−303954.			−1.81703			−0.908515	+	0.417853i	$0.862783\pi$
$410$	312796.	−	193597.i	1.86077	−	1.15168i
$411$		0			0
$412$		−	449763.i		−	2.64965i
$413$	−32295.7			−0.189341
$414$		0			0
$415$	−160031.	−	258563.i	−0.929196	−	1.50131i
$416$			72100.2i			0.416630i
$417$		0			0
$418$			631651.i			3.61513i
$419$			68885.5i			0.392374i	0.980567	+	0.196187i	$0.0628559\pi$
$419$			68885.5i			0.392374i	−0.980567	+	0.196187i	$0.937144\pi$
$420$		0			0
$421$	−65430.2			−0.369159			−0.184580	−	0.982818i	$-0.559092\pi$
$421$	−65430.2			−0.369159			−0.184580	+	0.982818i	$0.559092\pi$
$422$	−75761.1			−0.425423
$423$		0			0
$424$	−140286.			−0.780336
$425$	−71555.6	+	143573.i	−0.396155	+	0.794867i
$426$		0			0
$427$		−	1133.33i		−	0.00621587i
$428$	46052.2			0.251399
$429$		0			0
$430$	−263741.	+	163236.i	−1.42640	+	0.882831i
$431$			95087.7i			0.511882i	0.966692	+	0.255941i	$0.0823852\pi$
$431$			95087.7i			0.511882i	−0.966692	+	0.255941i	$0.917615\pi$
$432$		0			0
$433$		−	71258.3i		−	0.380067i	−0.981778	−	0.190033i	$-0.939140\pi$
$433$		−	71258.3i		−	0.380067i	0.981778	−	0.190033i	$-0.0608596\pi$
$434$			43852.5i			0.232817i
$435$		0			0
$436$	−14113.0			−0.0742417
$437$	408068.			2.13683
$438$		0			0
$439$	−4316.93			−0.0223999			−0.0111999	−	0.999937i	$-0.503565\pi$
$439$	−4316.93			−0.0223999			−0.0111999	+	0.999937i	$0.503565\pi$
$440$	152389.	−	94317.1i	0.787133	−	0.487175i
$441$		0			0
$442$		−	89550.7i		−	0.458379i
$443$	206204.			1.05073			0.525364	−	0.850878i	$-0.323929\pi$
$443$	206204.			1.05073			0.525364	+	0.850878i	$0.323929\pi$
$444$		0			0
$445$	−227368.	+	140723.i	−1.14818	+	0.710634i
$446$			520017.i			2.61425i
$447$		0			0
$448$		−	76702.6i		−	0.382168i
$449$			65145.6i			0.323141i	0.986861	+	0.161571i	$0.0516560\pi$
$449$			65145.6i			0.323141i	−0.986861	+	0.161571i	$0.948344\pi$
$450$		0			0
$451$	−384763.			−1.89165
$452$	−235525.			−1.15282
$453$		0			0
$454$	89897.8			0.436151
$455$	−8596.54	−	13889.5i	−0.0415241	−	0.0670909i
$456$		0			0
$457$			123244.i			0.590109i	0.955480	+	0.295054i	$0.0953377\pi$
$457$			123244.i			0.590109i	−0.955480	+	0.295054i	$0.904662\pi$
$458$	299172.			1.42623
$459$		0			0
$460$	−199711.	−	322675.i	−0.943816	−	1.52493i
$461$		−	62622.4i		−	0.294665i	−0.989087	−	0.147332i	$-0.952931\pi$
$461$		−	62622.4i		−	0.294665i	0.989087	−	0.147332i	$-0.0470687\pi$
$462$		0			0
$463$		−	357984.i		−	1.66994i	−0.550294	−	0.834971i	$-0.685484\pi$
$463$		−	357984.i		−	1.66994i	0.550294	−	0.834971i	$-0.314516\pi$
$464$		−	13348.2i		−	0.0619991i
$465$		0			0
$466$	218213.			1.00487
$467$	246564.			1.13057			0.565283	−	0.824897i	$-0.308767\pi$
$467$	246564.			1.13057			0.565283	+	0.824897i	$0.308767\pi$
$468$		0			0
$469$	−50753.2			−0.230737
$470$	−71301.7	−	115203.i	−0.322778	−	0.521515i
$471$		0			0
$472$		−	121149.i		−	0.543794i
$473$	324422.			1.45007
$474$		0			0
$475$	172567.	−	346248.i	0.764841	−	1.53462i
$476$			69136.1i			0.305134i
$477$		0			0
$478$			17612.0i			0.0770822i
$479$			319646.i			1.39315i	0.717484	+	0.696575i	$0.245293\pi$
$479$			319646.i			1.39315i	−0.717484	+	0.696575i	$0.754707\pi$
$480$		0			0
$481$	41966.6			0.181390
$482$	57220.4			0.246296
$483$		0			0
$484$	−277280.			−1.18366
$485$	138070.	−	85454.7i	0.586969	−	0.363289i
$486$		0			0
$487$			443400.i			1.86955i	0.355235	+	0.934777i	$0.384401\pi$
$487$			443400.i			1.86955i	−0.355235	+	0.934777i	$0.615599\pi$
$488$	4251.40			0.0178522
$489$		0			0
$490$	−186093.	−	300672.i	−0.775064	−	1.25228i
$491$		−	320334.i		−	1.32874i	−0.747404	−	0.664370i	$-0.768700\pi$
$491$		−	320334.i		−	1.32874i	0.747404	−	0.664370i	$-0.231300\pi$
$492$		0			0
$493$			36349.7i			0.149557i
$494$			215965.i			0.884973i
$495$		0			0
$496$	56555.4			0.229885
$497$	94987.8			0.384552
$498$		0			0
$499$	−168276.			−0.675805			−0.337902	−	0.941181i	$-0.609717\pi$
$499$	−168276.			−0.675805			−0.337902	+	0.941181i	$0.609717\pi$
$500$	−358248.	+	33000.7i	−1.43299	+	0.132003i
$501$		0			0
$502$		−	50922.3i		−	0.202070i
$503$	273094.			1.07938			0.539692	−	0.841863i	$-0.318541\pi$
$503$	273094.			1.07938			0.539692	+	0.841863i	$0.318541\pi$
$504$		0			0
$505$	299766.	−	185532.i	1.17544	−	0.727505i
$506$			672733.i			2.62749i
$507$		0			0
$508$			279721.i			1.08392i
$509$			398255.i			1.53718i	0.639739	+	0.768592i	$0.279042\pi$
$509$			398255.i			1.53718i	−0.639739	+	0.768592i	$0.720958\pi$
$510$		0			0
$511$	24534.6			0.0939586
$512$	−187853.			−0.716602
$513$		0			0
$514$	90804.2			0.343700
$515$	415244.	−	257005.i	1.56563	−	0.969006i
$516$		0			0
$517$			141708.i			0.530169i
$518$	−54914.2			−0.204656
$519$		0			0
$520$	52102.7	−	32247.6i	0.192688	−	0.119259i
$521$			384980.i			1.41828i	0.705067	+	0.709141i	$0.250917\pi$
$521$			384980.i			1.41828i	−0.705067	+	0.709141i	$0.749083\pi$
$522$		0			0
$523$			279373.i			1.02137i	0.859769	+	0.510683i	$0.170608\pi$
$523$			279373.i			1.02137i	−0.859769	+	0.510683i	$0.829392\pi$
$524$		−	499587.i		−	1.81948i
$525$		0			0
$526$	57043.0			0.206173
$527$	−154012.			−0.554539
$528$		0			0
$529$	154767.			0.553055
$530$	−262741.	−	424512.i	−0.935353	−	1.51126i
$531$		0			0
$532$		−	166732.i		−	0.589111i
$533$	−131553.			−0.463070
$534$		0			0
$535$	26315.3	+	42517.8i	0.0919391	+	0.148547i
$536$		−	190387.i		−	0.662686i
$537$		0			0
$538$			757203.i			2.61606i
$539$			369850.i			1.27306i
$540$		0			0
$541$	276109.			0.943377			0.471689	−	0.881765i	$-0.343645\pi$
$541$	276109.			0.943377			0.471689	+	0.881765i	$0.343645\pi$
$542$	483329.			1.64530
$543$		0			0
$544$	331342.			1.11964
$545$	−8064.52	−	13029.9i	−0.0271510	−	0.0438680i
$546$		0			0
$547$		−	313727.i		−	1.04852i	−0.851558	−	0.524261i	$-0.824342\pi$
$547$		−	313727.i		−	1.04852i	0.851558	−	0.524261i	$-0.175658\pi$
$548$	−582638.			−1.94016
$549$		0			0
$550$	570818.	+	284491.i	1.88700	+	0.940466i
$551$		−	87662.8i		−	0.288743i
$552$		0			0
$553$			21758.7i			0.0711513i
$554$			304383.i			0.991746i
$555$		0			0
$556$	−424379.			−1.37279
$557$	37442.2			0.120684			0.0603422	−	0.998178i	$-0.480781\pi$
$557$	37442.2			0.120684			0.0603422	+	0.998178i	$0.480781\pi$
$558$		0			0
$559$	110922.			0.354972
$560$	23439.4	−	14507.2i	0.0747429	−	0.0462602i
$561$		0			0
$562$		−	533041.i		−	1.68767i
$563$	458509.			1.44654			0.723270	−	0.690565i	$-0.242638\pi$
$563$	458509.			1.44654			0.723270	+	0.690565i	$0.242638\pi$
$564$		0			0
$565$	−134585.	−	217449.i	−0.421598	−	0.681179i
$566$			149230.i			0.465826i
$567$		0			0
$568$			356321.i			1.10445i
$569$			99217.5i			0.306453i	0.988191	+	0.153226i	$0.0489664\pi$
$569$			99217.5i			0.306453i	−0.988191	+	0.153226i	$0.951034\pi$
$570$		0			0
$571$	456342.			1.39965			0.699823	−	0.714317i	$-0.253262\pi$
$571$	456342.			1.39965			0.699823	+	0.714317i	$0.253262\pi$
$572$	−210063.			−0.642034
$573$		0			0
$574$	172140.			0.522465
$575$	183791.	−	368768.i	0.555889	−	1.11537i
$576$		0			0
$577$		−	319678.i		−	0.960198i	−0.877214	−	0.480099i	$-0.840601\pi$
$577$		−	319678.i		−	0.960198i	0.877214	−	0.480099i	$-0.159399\pi$
$578$	110219.			0.329914
$579$		0			0
$580$	−69318.4	+	42902.8i	−0.206059	+	0.127535i
$581$		−	142294.i		−	0.421535i
$582$		0			0
$583$			522183.i			1.53633i
$584$			92034.9i			0.269853i
$585$		0			0
$586$	−820392.			−2.38905
$587$	−134772.			−0.391133			−0.195567	−	0.980690i	$-0.562655\pi$
$587$	−134772.			−0.391133			−0.195567	+	0.980690i	$0.562655\pi$
$588$		0			0
$589$	371423.			1.07063
$590$	366602.	−	226899.i	1.05315	−	0.651821i
$591$		0			0
$592$			70821.3i			0.202079i
$593$	−135712.			−0.385929			−0.192965	−	0.981206i	$-0.561810\pi$
$593$	−135712.			−0.385929			−0.192965	+	0.981206i	$0.561810\pi$
$594$		0			0
$595$	−63830.1	+	39506.0i	−0.180298	+	0.111591i
$596$			125442.i			0.353142i
$597$		0			0
$598$			230011.i			0.643201i
$599$		−	114997.i		−	0.320504i	−0.987076	−	0.160252i	$-0.948769\pi$
$599$		−	114997.i		−	0.320504i	0.987076	−	0.160252i	$-0.0512307\pi$
$600$		0			0
$601$	59940.3			0.165947			0.0829736	−	0.996552i	$-0.473558\pi$
$601$	59940.3			0.165947			0.0829736	+	0.996552i	$0.473558\pi$
$602$	−145143.			−0.400502
$603$		0			0
$604$	272533.			0.747042
$605$	−158444.	−	256000.i	−0.432879	−	0.699405i
$606$		0			0
$607$			150638.i			0.408844i	0.978883	+	0.204422i	$0.0655314\pi$
$607$			150638.i			0.408844i	−0.978883	+	0.204422i	$0.934469\pi$
$608$	−799081.			−2.16164
$609$		0			0
$610$	7962.44	+	12865.0i	0.0213986	+	0.0345740i
$611$			48451.0i			0.129784i
$612$		0			0
$613$			404962.i			1.07769i	0.842405	+	0.538845i	$0.181139\pi$
$613$			404962.i			1.07769i	−0.842405	+	0.538845i	$0.818861\pi$
$614$			334906.i			0.888355i
$615$		0			0
$616$	83863.6			0.221010
$617$	−638746.			−1.67787			−0.838934	−	0.544233i	$-0.816821\pi$
$617$	−638746.			−1.67787			−0.838934	+	0.544233i	$0.816821\pi$
$618$		0			0
$619$	−466770.			−1.21821			−0.609104	−	0.793090i	$-0.708471\pi$
$619$	−466770.			−1.21821			−0.609104	+	0.793090i	$0.708471\pi$
$620$	−181777.	−	293698.i	−0.472885	−	0.764044i
$621$		0			0
$622$		−	713908.i		−	1.84528i
$623$	−125126.			−0.322383
$624$		0			0
$625$	−235179.	−	311895.i	−0.602058	−	0.798452i
$626$		−	407020.i		−	1.03864i
$627$		0			0
$628$			318696.i			0.808087i
$629$		−	192861.i		−	0.487463i
$630$		0			0
$631$	−392613.			−0.986066			−0.493033	−	0.870011i	$-0.664112\pi$
$631$	−392613.			−0.986066			−0.493033	+	0.870011i	$0.664112\pi$
$632$	−81621.9			−0.204349
$633$		0			0
$634$	−319132.			−0.793948
$635$	−258253.	+	159839.i	−0.640469	+	0.396402i
$636$		0			0
$637$			126454.i			0.311640i
$638$	144519.			0.355045
$639$		0			0
$640$	267129.	+	431603.i	0.652171	+	1.05372i
$641$			347045.i			0.844635i	0.906448	+	0.422318i	$0.138783\pi$
$641$			347045.i			0.844635i	−0.906448	+	0.422318i	$0.861217\pi$
$642$		0			0
$643$			78210.4i			0.189166i	0.995517	+	0.0945828i	$0.0301517\pi$
$643$			78210.4i			0.189166i	−0.995517	+	0.0945828i	$0.969848\pi$
$644$		−	177577.i		−	0.428168i
$645$		0			0
$646$	992483.			2.37825
$647$	429610.			1.02628			0.513139	−	0.858305i	$-0.328482\pi$
$647$	429610.			1.02628			0.513139	+	0.858305i	$0.328482\pi$
$648$		0			0
$649$	−450950.			−1.07063
$650$	195166.	+	97269.2i	0.461932	+	0.230223i
$651$		0			0
$652$			152322.i			0.358316i
$653$	−10510.8			−0.0246496			−0.0123248	−	0.999924i	$-0.503923\pi$
$653$	−10510.8			−0.0246496			−0.0123248	+	0.999924i	$0.503923\pi$
$654$		0			0
$655$	461245.	−	285475.i	1.07510	−	0.665405i
$656$		−	222004.i		−	0.515885i
$657$		0			0
$658$		−	63399.1i		−	0.146430i
$659$			297455.i			0.684936i	0.939530	+	0.342468i	$0.111263\pi$
$659$			297455.i			0.684936i	−0.939530	+	0.342468i	$0.888737\pi$
$660$		0			0
$661$	48003.7			0.109868			0.0549341	−	0.998490i	$-0.482505\pi$
$661$	48003.7			0.109868			0.0549341	+	0.998490i	$0.482505\pi$
$662$	921691.			2.10315
$663$		0			0
$664$	533778.			1.21067
$665$	153936.	−	95274.7i	0.348094	−	0.215444i
$666$		0			0
$667$		−	93364.3i		−	0.209860i
$668$	−256392.			−0.574582
$669$		0			0
$670$	576122.	−	356576.i	1.28341	−	0.794332i
$671$		−	15824.9i		−	0.0351477i
$672$		0			0
$673$		−	533475.i		−	1.17783i	−0.808194	−	0.588917i	$-0.799555\pi$
$673$		−	533475.i		−	1.17783i	0.808194	−	0.588917i	$-0.200445\pi$
$674$		−	738119.i		−	1.62483i
$675$		0			0
$676$	585793.			1.28189
$677$	−571239.			−1.24635			−0.623175	−	0.782082i	$-0.714158\pi$
$677$	−571239.			−1.24635			−0.623175	+	0.782082i	$0.714158\pi$
$678$		0			0
$679$	75983.4			0.164808
$680$	−148196.	−	239441.i	−0.320493	−	0.517823i
$681$		0			0
$682$			612320.i			1.31647i
$683$	−89045.3			−0.190884			−0.0954420	−	0.995435i	$-0.530426\pi$
$683$	−89045.3			−0.190884			−0.0954420	+	0.995435i	$0.530426\pi$
$684$		0			0
$685$	−332932.	−	537921.i	−0.709537	−	1.14640i
$686$		−	340937.i		−	0.724479i
$687$		0			0
$688$			187188.i			0.395458i
$689$			178538.i			0.376090i
$690$		0			0
$691$	−454814.			−0.952528			−0.476264	−	0.879302i	$-0.658009\pi$
$691$	−454814.			−0.952528			−0.476264	+	0.879302i	$0.658009\pi$
$692$	409076.			0.854263
$693$		0			0
$694$	666090.			1.38297
$695$	−242500.	−	391809.i	−0.502044	−	0.811156i
$696$		0			0
$697$			604561.i			1.24444i
$698$	230882.			0.473891
$699$		0			0
$700$	−150675.	−	75095.1i	−0.307500	−	0.153255i
$701$		−	77719.5i		−	0.158159i	−0.996868	−	0.0790795i	$-0.974802\pi$
$701$		−	77719.5i		−	0.158159i	0.996868	−	0.0790795i	$-0.0251981\pi$
$702$		0			0
$703$			465113.i			0.941126i
$704$		−	1.07101e6i		−	2.16097i
$705$		0			0
$706$	136940.			0.274739
$707$	164969.			0.330037
$708$		0			0
$709$	242676.			0.482763			0.241381	−	0.970430i	$-0.422400\pi$
$709$	242676.			0.482763			0.241381	+	0.970430i	$0.422400\pi$
$710$	−1.07825e6	+	667354.i	−2.13896	+	1.32385i
$711$		0			0
$712$		−	469378.i		−	0.925897i
$713$	395580.			0.778135
$714$		0			0
$715$	−120035.	−	193941.i	−0.234799	−	0.379366i
$716$			605674.i			1.18144i
$717$		0			0
$718$		−	741141.i		−	1.43765i
$719$		−	513112.i		−	0.992555i	−0.868164	−	0.496278i	$-0.834700\pi$
$719$		−	513112.i		−	0.992555i	0.868164	−	0.496278i	$-0.165300\pi$
$720$		0			0
$721$	228520.			0.439595
$722$	−1.57941e6			−3.02985
$723$		0			0
$724$	−790631.			−1.50833
$725$	−79220.2	−	39482.7i	−0.150716	−	0.0751157i
$726$		0			0
$727$			130306.i			0.246545i	0.992373	+	0.123273i	$0.0393390\pi$
$727$			130306.i			0.246545i	−0.992373	+	0.123273i	$0.960661\pi$
$728$	28673.5			0.0541025
$729$		0			0
$730$	−278503.	+	172372.i	−0.522617	+	0.323460i
$731$		−	509749.i		−	0.953941i
$732$		0			0
$733$			831621.i			1.54781i	0.633302	+	0.773904i	$0.281699\pi$
$733$			831621.i			1.54781i	−0.633302	+	0.773904i	$0.718301\pi$
$734$		−	369118.i		−	0.685131i
$735$		0			0
$736$	−851053.			−1.57109
$737$	−708676.			−1.30471
$738$		0			0
$739$	304925.			0.558346			0.279173	−	0.960241i	$-0.409940\pi$
$739$	304925.			0.558346			0.279173	+	0.960241i	$0.409940\pi$
$740$	367783.	−	227630.i	0.671627	−	0.415686i
$741$		0			0
$742$		−	233620.i		−	0.424329i
$743$	−594643.			−1.07716			−0.538578	−	0.842576i	$-0.681038\pi$
$743$	−594643.			−1.07716			−0.538578	+	0.842576i	$0.681038\pi$
$744$		0			0
$745$	−115814.	+	71680.3i	−0.208665	+	0.129148i
$746$			1.09191e6i			1.96204i
$747$		0			0
$748$			965359.i			1.72538i
$749$			23398.7i			0.0417088i
$750$		0			0
$751$	−935776.			−1.65917			−0.829587	−	0.558378i	$-0.811424\pi$
$751$	−935776.			−1.65917			−0.829587	+	0.558378i	$0.811424\pi$
$752$	−81764.1			−0.144586
$753$		0			0
$754$	49412.0			0.0869140
$755$	155731.	+	251616.i	0.273201	+	0.441413i
$756$		0			0
$757$			555428.i			0.969251i	0.874722	+	0.484625i	$0.161044\pi$
$757$			555428.i			0.969251i	−0.874722	+	0.484625i	$0.838956\pi$
$758$	706523.			1.22967
$759$		0			0
$760$	357398.	+	577450.i	0.618763	+	0.999740i
$761$			1.02739e6i			1.77405i	0.461723	+	0.887024i	$0.347231\pi$
$761$			1.02739e6i			1.77405i	−0.461723	+	0.887024i	$0.652769\pi$
$762$		0			0
$763$		−	7170.69i		−	0.0123172i
$764$			758302.i			1.29914i
$765$		0			0
$766$	1.10399e6			1.88151
$767$	−154182.			−0.262086
$768$		0			0
$769$	1.01892e6			1.72301			0.861504	−	0.507750i	$-0.169523\pi$
$769$	1.01892e6			1.72301			0.861504	+	0.507750i	$0.169523\pi$
$770$	157068.	+	253776.i	0.264915	+	0.428025i
$771$		0			0
$772$		−	608642.i		−	1.02124i
$773$	−1.11045e6			−1.85841			−0.929203	−	0.369569i	$-0.879505\pi$
$773$	−1.11045e6			−1.85841			−0.929203	+	0.369569i	$0.879505\pi$
$774$		0			0
$775$	167286.	−	335652.i	0.278520	−	0.558838i
$776$			285032.i			0.473336i
$777$		0			0
$778$			991566.i			1.63818i
$779$		−	1.45799e6i		−	2.40259i
$780$		0			0
$781$	1.32633e6			2.17445
$782$	1.05703e6			1.72852
$783$		0			0
$784$	−213399.			−0.347185
$785$	−294237.	+	182110.i	−0.477483	+	0.295526i
$786$		0			0
$787$		−	178592.i		−	0.288345i	−0.989553	−	0.144172i	$-0.953948\pi$
$787$		−	178592.i		−	0.288345i	0.989553	−	0.144172i	$-0.0460520\pi$
$788$	87142.4			0.140339
$789$		0			0
$790$	−152870.	−	246993.i	−0.244944	−	0.395758i
$791$		−	119668.i		−	0.191260i
$792$		0			0
$793$		−	5410.64i		−	0.00860403i
$794$			111394.i			0.176694i
$795$		0			0
$796$	617297.			0.974244
$797$	−273296.			−0.430246			−0.215123	−	0.976587i	$-0.569015\pi$
$797$	−273296.			−0.430246			−0.215123	+	0.976587i	$0.569015\pi$
$798$		0			0
$799$	222660.			0.348777
$800$	−359900.	+	722124.i	−0.562344	+	1.12832i
$801$		0			0
$802$		−	83166.4i		−	0.129300i
$803$	342580.			0.531289
$804$		0			0
$805$	163948.	−	101471.i	0.252996	−	0.156585i
$806$			209356.i			0.322267i
$807$		0			0
$808$			618836.i			0.947879i
$809$		−	92441.7i		−	0.141244i	−0.997503	−	0.0706222i	$-0.977502\pi$
$809$		−	92441.7i		−	0.141244i	0.997503	−	0.0706222i	$-0.0224985\pi$
$810$		0			0
$811$	496286.			0.754554			0.377277	−	0.926101i	$-0.376861\pi$
$811$	496286.			0.754554			0.377277	+	0.926101i	$0.376861\pi$
$812$	−38147.7			−0.0578571
$813$		0			0
$814$	−766776.			−1.15723
$815$	−140631.	+	87040.1i	−0.211723	+	0.131040i
$816$		0			0
$817$			1.22934e6i			1.84174i
$818$	1.89880e6			2.83774
$819$		0			0
$820$	−1.15289e6	+	713552.i	−1.71459	+	1.06120i
$821$		−	626564.i		−	0.929564i	−0.885425	−	0.464782i	$-0.846133\pi$
$821$		−	626564.i		−	0.929564i	0.885425	−	0.464782i	$-0.153867\pi$
$822$		0			0
$823$		−	1.25079e6i		−	1.84665i	−0.384021	−	0.923325i	$-0.625461\pi$
$823$		−	1.25079e6i		−	1.84665i	0.384021	−	0.923325i	$-0.374539\pi$
$824$			857231.i			1.26253i
$825$		0			0
$826$	201751.			0.295702
$827$	399216.			0.583709			0.291855	−	0.956463i	$-0.405728\pi$
$827$	399216.			0.583709			0.291855	+	0.956463i	$0.405728\pi$
$828$		0			0
$829$	537391.			0.781954			0.390977	−	0.920400i	$-0.372137\pi$
$829$	537391.			0.781954			0.390977	+	0.920400i	$0.372137\pi$
$830$	999711.	+	1.61524e6i	1.45117	+	2.34467i
$831$		0			0
$832$		−	366185.i		−	0.528998i
$833$	581128.			0.837494
$834$		0			0
$835$	−146508.	−	236715.i	−0.210131	−	0.339510i
$836$		−	2.32811e6i		−	3.33113i
$837$		0			0
$838$		−	430327.i		−	0.612789i
$839$			823663.i			1.17011i	0.810995	+	0.585053i	$0.198926\pi$
$839$			823663.i			1.17011i	−0.810995	+	0.585053i	$0.801074\pi$
$840$		0			0
$841$	687224.			0.971642
$842$	408742.			0.576534
$843$		0			0
$844$	279237.			0.392002
$845$	334735.	+	540834.i	0.468801	+	0.757445i
$846$		0			0
$847$		−	140883.i		−	0.196378i
$848$	−301294.			−0.418985
$849$		0			0
$850$	447007.	−	896899.i	0.618695	−	1.24138i
$851$			495363.i			0.684014i
$852$		0			0
$853$			968538.i			1.33112i	0.746343	+	0.665562i	$0.231808\pi$
$853$			968538.i			1.33112i	−0.746343	+	0.665562i	$0.768192\pi$
$854$			7079.93i			0.00970763i
$855$		0			0
$856$	−87773.8			−0.119789
$857$	460057.			0.626398			0.313199	−	0.949688i	$-0.398599\pi$
$857$	460057.			0.626398			0.313199	+	0.949688i	$0.398599\pi$
$858$		0			0
$859$	778471.			1.05501			0.527504	−	0.849552i	$-0.323128\pi$
$859$	778471.			1.05501			0.527504	+	0.849552i	$0.323128\pi$
$860$	972086.	−	601647.i	1.31434	−	0.813476i
$861$		0			0
$862$		−	594012.i		−	0.799430i
$863$	−665897.			−0.894098			−0.447049	−	0.894509i	$-0.647525\pi$
$863$	−665897.			−0.894098			−0.447049	+	0.894509i	$0.647525\pi$
$864$		0			0
$865$	233755.	+	377680.i	0.312413	+	0.504768i
$866$			445150.i			0.593568i
$867$		0			0
$868$		−	161630.i		−	0.214527i
$869$			303820.i			0.402325i
$870$		0			0
$871$	−242301.			−0.319388
$872$	26898.9			0.0353755
$873$		0			0
$874$	−2.54920e6			−3.33719
$875$	−16767.3	−	182022.i	−0.0219002	−	0.237743i
$876$		0			0
$877$			1.30845e6i			1.70122i	0.525800	+	0.850608i	$0.323766\pi$
$877$			1.30845e6i			1.70122i	−0.525800	+	0.850608i	$0.676234\pi$
$878$	26967.8			0.0349830
$879$		0			0
$880$	327288.	−	202566.i	0.422634	−	0.261578i
$881$		−	521436.i		−	0.671814i	−0.941895	−	0.335907i	$-0.890957\pi$
$881$		−	521436.i		−	0.671814i	0.941895	−	0.335907i	$-0.109043\pi$
$882$		0			0
$883$		−	1.26259e6i		−	1.61935i	−0.586882	−	0.809673i	$-0.699645\pi$
$883$		−	1.26259e6i		−	1.61935i	0.586882	−	0.809673i	$-0.300355\pi$
$884$			330062.i			0.422368i
$885$		0			0
$886$	−1.28816e6			−1.64097
$887$	669030.			0.850351			0.425176	−	0.905111i	$-0.360212\pi$
$887$	669030.			0.850351			0.425176	+	0.905111i	$0.360212\pi$
$888$		0			0
$889$	−142124.			−0.179830
$890$	1.42036e6	−	879097.i	1.79316	−	1.10983i
$891$		0			0
$892$		−	1.91666e6i		−	2.40888i
$893$	−536978.			−0.673370
$894$		0			0
$895$	−559189.	+	346096.i	−0.698092	+	0.432066i
$896$			237522.i			0.295861i
$897$		0			0
$898$		−	406964.i		−	0.504665i
$899$		−	84980.0i		−	0.105147i
$900$		0			0
$901$	820482.			1.01069
$902$	2.40361e6			2.95428
$903$		0			0
$904$	448902.			0.549307
$905$	−451784.	−	729951.i	−0.551612	−	0.891244i
$906$		0			0
$907$			1.07224e6i			1.30340i	0.758479	+	0.651698i	$0.225943\pi$
$907$			1.07224e6i			1.30340i	−0.758479	+	0.651698i	$0.774057\pi$
$908$	−331342.			−0.401887
$909$		0			0
$910$	53702.5	+	86767.5i	0.0648503	+	0.104779i
$911$			471637.i			0.568292i	0.958781	+	0.284146i	$0.0917100\pi$
$911$			471637.i			0.568292i	−0.958781	+	0.284146i	$0.908290\pi$
$912$		0			0
$913$		−	1.98687e6i		−	2.38357i
$914$		−	769902.i		−	0.921601i
$915$		0			0
$916$	−1.10268e6			−1.31419
$917$	253835.			0.301865
$918$		0			0
$919$	729318.			0.863547			0.431774	−	0.901982i	$-0.357888\pi$
$919$	729318.			0.863547			0.431774	+	0.901982i	$0.357888\pi$
$920$	380642.	+	615007.i	0.449719	+	0.726615i
$921$		0			0
$922$			391202.i			0.460192i
$923$	453481.			0.532298
$924$		0			0
$925$	420319.	+	209483.i	0.491242	+	0.244831i
$926$			2.23632e6i			2.60803i
$927$		0			0
$928$			182827.i			0.212297i
$929$		−	63798.9i		−	0.0739234i	−0.999317	−	0.0369617i	$-0.988232\pi$
$929$		−	63798.9i		−	0.0739234i	0.999317	−	0.0369617i	$-0.0117680\pi$
$930$		0			0
$931$	−1.40148e6			−1.61692
$932$	−804282.			−0.925926
$933$		0			0
$934$	−1.54028e6			−1.76566
$935$	−891270.	+	551628.i	−1.01950	+	0.630991i
$936$		0			0
$937$			1.06986e6i			1.21856i	0.792953	+	0.609282i	$0.208542\pi$
$937$			1.06986e6i			1.21856i	−0.792953	+	0.609282i	$0.791458\pi$
$938$	317055.			0.360354
$939$		0			0
$940$	262801.	+	424610.i	0.297421	+	0.480545i
$941$		−	1.53752e6i		−	1.73637i	−0.496241	−	0.868185i	$-0.665287\pi$
$941$		−	1.53752e6i		−	1.73637i	0.496241	−	0.868185i	$-0.334713\pi$
$942$		0			0
$943$		−	1.55282e6i		−	1.74621i
$944$		−	260193.i		−	0.291978i
$945$		0			0
$946$	−2.02666e6			−2.26464
$947$	248742.			0.277363			0.138682	−	0.990337i	$-0.455714\pi$
$947$	248742.			0.277363			0.138682	+	0.990337i	$0.455714\pi$
$948$		0			0
$949$	117130.			0.130058
$950$	−1.07803e6	+	2.16301e6i	−1.19449	+	2.39669i
$951$		0			0
$952$		−	131771.i		−	0.145394i
$953$	−399593.			−0.439979			−0.219990	−	0.975502i	$-0.570602\pi$
$953$	−399593.			−0.439979			−0.219990	+	0.975502i	$0.570602\pi$
$954$		0			0
$955$	−700104.	+	433311.i	−0.767637	+	0.475109i
$956$		−	64913.8i		−	0.0710266i
$957$		0			0
$958$		−	1.99682e6i		−	2.17575i
$959$		−	296032.i		−	0.321886i
$960$		0			0
$961$	−563465.			−0.610127
$962$	−262165.			−0.283286
$963$		0			0
$964$	−210901.			−0.226947
$965$	561930.	−	347792.i	0.603431	−	0.373478i
$966$		0			0
$967$			62894.7i			0.0672607i	0.999434	+	0.0336303i	$0.0107069\pi$
$967$			62894.7i			0.0672607i	−0.999434	+	0.0336303i	$0.989293\pi$
$968$	528486.			0.564005
$969$		0			0
$970$	−862521.	+	533835.i	−0.916698	+	0.567366i
$971$		−	625318.i		−	0.663227i	−0.943415	−	0.331613i	$-0.892407\pi$
$971$		−	625318.i		−	0.663227i	0.943415	−	0.331613i	$-0.107593\pi$
$972$		0			0
$973$		−	215622.i		−	0.227755i
$974$		−	2.76992e6i		−	2.91977i
$975$		0			0
$976$	9130.79			0.00958537
$977$	−1.00672e6			−1.05468			−0.527339	−	0.849655i	$-0.676810\pi$
$977$	−1.00672e6			−1.05468			−0.527339	+	0.849655i	$0.676810\pi$
$978$		0			0
$979$	−1.74716e6			−1.82292
$980$	685894.	+	1.10820e6i	0.714176	+	1.15390i
$981$		0			0
$982$			2.00112e6i			2.07516i
$983$	−486877.			−0.503863			−0.251932	−	0.967745i	$-0.581066\pi$
$983$	−486877.			−0.503863			−0.251932	+	0.967745i	$0.581066\pi$
$984$		0			0
$985$	49795.1	+	80454.4i	0.0513233	+	0.0829234i
$986$		−	227076.i		−	0.233570i
$987$		0			0
$988$		−	795996.i		−	0.815449i
$989$			1.30929e6i			1.33858i
$990$		0			0
$991$	−645585.			−0.657364			−0.328682	−	0.944441i	$-0.606604\pi$
$991$	−645585.			−0.657364			−0.328682	+	0.944441i	$0.606604\pi$
$992$	−774627.			−0.787171
$993$		0			0
$994$	−593388.			−0.600573
$995$	352737.	+	569920.i	0.356291	+	0.575663i
$996$		0			0
$997$		−	1.13791e6i		−	1.14476i	−0.819987	−	0.572382i	$-0.806019\pi$
$997$		−	1.13791e6i		−	1.14476i	0.819987	−	0.572382i	$-0.193981\pi$
$998$	1.05122e6			1.05544
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.5.d.a.404.6		44
3.2	odd	2	inner	405.5.d.a.404.40		44
5.4	even	2	inner	405.5.d.a.404.39		44
9.2	odd	6		135.5.h.a.44.3		44
9.4	even	3		135.5.h.a.89.20		44
9.5	odd	6		45.5.h.a.29.3	yes	44
9.7	even	3		45.5.h.a.14.20	yes	44
15.14	odd	2	inner	405.5.d.a.404.5		44
45.4	even	6		135.5.h.a.89.3		44
45.14	odd	6		45.5.h.a.29.20	yes	44
45.29	odd	6		135.5.h.a.44.20		44
45.34	even	6		45.5.h.a.14.3	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.h.a.14.3	✓	44	45.34	even	6
45.5.h.a.14.20	yes	44	9.7	even	3
45.5.h.a.29.3	yes	44	9.5	odd	6
45.5.h.a.29.20	yes	44	45.14	odd	6
135.5.h.a.44.3		44	9.2	odd	6
135.5.h.a.44.20		44	45.29	odd	6
135.5.h.a.89.3		44	45.4	even	6
135.5.h.a.89.20		44	9.4	even	3
405.5.d.a.404.5		44	15.14	odd	2	inner
405.5.d.a.404.6		44	1.1	even	1	trivial
405.5.d.a.404.39		44	5.4	even	2	inner
405.5.d.a.404.40		44	3.2	odd	2	inner

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.24699 q^{2} +23.0249 q^{4} +(13.1570 + 21.2578i) q^{5} +11.6987i q^{7} -43.8846 q^{8} +(-82.1915 - 132.797i) q^{10} +163.351i q^{11} +55.8508i q^{13} -73.0819i q^{14} -94.2517 q^{16} +256.666 q^{17} -618.990 q^{19} +(302.938 + 489.459i) q^{20} -1020.45i q^{22} -659.248 q^{23} +(-278.789 + 559.376i) q^{25} -348.899i q^{26} +269.362i q^{28} +141.622i q^{29} -600.047 q^{31} +1290.94 q^{32} -1603.39 q^{34} +(-248.689 + 153.920i) q^{35} -751.406i q^{37} +3866.82 q^{38} +(-577.389 - 932.891i) q^{40} +2355.44i q^{41} -1986.04i q^{43} +3761.15i q^{44} +4118.32 q^{46} +867.508 q^{47} +2264.14 q^{49} +(1741.59 - 3494.42i) q^{50} +1285.96i q^{52} +3196.69 q^{53} +(-3472.49 + 2149.21i) q^{55} -513.394i q^{56} -884.714i q^{58} +2760.61i q^{59} -96.8767 q^{61} +3748.49 q^{62} -6556.49 q^{64} +(-1187.27 + 734.827i) q^{65} +4338.35i q^{67} +5909.72 q^{68} +(1553.56 - 961.535i) q^{70} -8119.50i q^{71} -2097.20i q^{73} +4694.03i q^{74} -14252.2 q^{76} -1911.00 q^{77} +1859.92 q^{79} +(-1240.07 - 2003.58i) q^{80} -14714.4i q^{82} -12163.2 q^{83} +(3376.95 + 5456.16i) q^{85} +12406.8i q^{86} -7168.61i q^{88} +10695.7i q^{89} -653.383 q^{91} -15179.1 q^{92} -5419.31 q^{94} +(-8144.03 - 13158.4i) q^{95} -6495.02i q^{97} -14144.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) 44 * q + 324 * q^4 + 28 * q^10 + 2116 * q^16 - 8 * q^19 + 296 * q^25 + 2224 * q^31 + 872 * q^34 + 1700 * q^40 - 5668 * q^46 - 10792 * q^49 - 3072 * q^55 - 5564 * q^61 + 8348 * q^64 - 9564 * q^70 + 3552 * q^76 - 15800 * q^79 - 13532 * q^85 + 1560 * q^91 + 37652 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more