LMFDB - Embedded newform 1710.4.f.a.341.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1710,4,Mod(341,1710)] 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("1710.341"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1710, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1710.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$100.893266110$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.21
Character	$\chi$	$=$	1710.341
Dual form			1710.4.f.a.341.22

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} +4.15148 q^{7} -8.00000 q^{8} +10.0000i q^{10} -13.6396i q^{11} +65.3955i q^{13} -8.30297 q^{14} +16.0000 q^{16} -44.6878i q^{17} +(29.3300 - 77.4516i) q^{19} -20.0000i q^{20} +27.2792i q^{22} +76.5757i q^{23} -25.0000 q^{25} -130.791i q^{26} +16.6059 q^{28} -175.756 q^{29} -85.0245i q^{31} -32.0000 q^{32} +89.3756i q^{34} -20.7574i q^{35} +367.851i q^{37} +(-58.6600 + 154.903i) q^{38} +40.0000i q^{40} +29.6257 q^{41} +217.300 q^{43} -54.5584i q^{44} -153.151i q^{46} -613.034i q^{47} -325.765 q^{49} +50.0000 q^{50} +261.582i q^{52} +706.146 q^{53} -68.1980 q^{55} -33.2119 q^{56} +351.513 q^{58} -115.406 q^{59} -602.179 q^{61} +170.049i q^{62} +64.0000 q^{64} +326.978 q^{65} -514.633i q^{67} -178.751i q^{68} +41.5148i q^{70} -233.009 q^{71} -42.8481 q^{73} -735.702i q^{74} +(117.320 - 309.806i) q^{76} -56.6245i q^{77} -41.1864i q^{79} -80.0000i q^{80} -59.2514 q^{82} -1308.54i q^{83} -223.439 q^{85} -434.599 q^{86} +109.117i q^{88} +600.186 q^{89} +271.488i q^{91} +306.303i q^{92} +1226.07i q^{94} +(-387.258 - 146.650i) q^{95} -554.623i q^{97} +651.530 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) $ 40 * q - 80 * q^2 + 160 * q^4 - 56 * q^7 - 320 * q^8 + 112 * q^14 + 640 * q^16 - 76 * q^19 - 1000 * q^25 - 224 * q^28 - 120 * q^29 - 1280 * q^32 + 152 * q^38 - 312 * q^41 + 56 * q^43 + 2112 * q^49 + 2000 * q^50 + 288 * q^53 + 448 * q^56 + 240 * q^58 + 912 * q^59 - 848 * q^61 + 2560 * q^64 + 240 * q^65 - 1248 * q^71 + 512 * q^73 - 304 * q^76 + 624 * q^82 - 112 * q^86 - 552 * q^89 - 60 * q^95 - 4224 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−2.00000			−0.707107
$2$	−2.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	4.00000			0.500000
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$	4.15148			0.224159			0.112080	−	0.993699i	$-0.464249\pi$
$7$	4.15148			0.224159			0.112080	+	0.993699i	$0.464249\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$			10.0000i			0.316228i
$11$		−	13.6396i		−	0.373863i	−0.982373	−	0.186931i	$-0.940146\pi$
$11$		−	13.6396i		−	0.373863i	0.982373	−	0.186931i	$-0.0598542\pi$
$12$		0			0
$13$			65.3955i			1.39519i	0.716493	+	0.697594i	$0.245746\pi$
$13$			65.3955i			1.39519i	−0.716493	+	0.697594i	$0.754254\pi$
$14$	−8.30297			−0.158504
$15$		0			0
$16$	16.0000			0.250000
$17$		−	44.6878i		−	0.637552i	−0.947830	−	0.318776i	$-0.896728\pi$
$17$		−	44.6878i		−	0.637552i	0.947830	−	0.318776i	$-0.103272\pi$
$18$		0			0
$19$	29.3300	−	77.4516i	0.354146	−	0.935190i
$19$	29.3300	−	77.4516i	0.354146	−	0.935190i
$20$		−	20.0000i		−	0.223607i
$21$		0			0
$22$			27.2792i			0.264361i
$23$			76.5757i			0.694224i	0.937824	+	0.347112i	$0.112838\pi$
$23$			76.5757i			0.694224i	−0.937824	+	0.347112i	$0.887162\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		−	130.791i		−	0.986547i
$27$		0			0
$28$	16.6059			0.112080
$29$	−175.756			−1.12542			−0.562710	−	0.826655i	$-0.690241\pi$
$29$	−175.756			−1.12542			−0.562710	+	0.826655i	$0.690241\pi$
$30$		0			0
$31$		−	85.0245i		−	0.492608i	−0.969193	−	0.246304i	$-0.920784\pi$
$31$		−	85.0245i		−	0.492608i	0.969193	−	0.246304i	$-0.0792162\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$			89.3756i			0.450817i
$35$		−	20.7574i		−	0.100247i
$36$		0			0
$37$			367.851i			1.63444i	0.576325	+	0.817220i	$0.304486\pi$
$37$			367.851i			1.63444i	−0.576325	+	0.817220i	$0.695514\pi$
$38$	−58.6600	+	154.903i	−0.250419	+	0.661279i
$39$		0			0
$40$			40.0000i			0.158114i
$41$	29.6257			0.112848			0.0564239	−	0.998407i	$-0.482030\pi$
$41$	29.6257			0.112848			0.0564239	+	0.998407i	$0.482030\pi$
$42$		0			0
$43$	217.300			0.770648			0.385324	−	0.922781i	$-0.374090\pi$
$43$	217.300			0.770648			0.385324	+	0.922781i	$0.374090\pi$
$44$		−	54.5584i		−	0.186931i
$45$		0			0
$46$		−	153.151i		−	0.490890i
$47$		−	613.034i		−	1.90256i	−0.308331	−	0.951279i	$-0.599770\pi$
$47$		−	613.034i		−	1.90256i	0.308331	−	0.951279i	$-0.400230\pi$
$48$		0			0
$49$	−325.765			−0.949753
$50$	50.0000			0.141421
$51$		0			0
$52$			261.582i			0.697594i
$53$	706.146			1.83013			0.915063	−	0.403311i	$-0.132141\pi$
$53$	706.146			1.83013			0.915063	+	0.403311i	$0.132141\pi$
$54$		0			0
$55$	−68.1980			−0.167197
$56$	−33.2119			−0.0792522
$57$		0			0
$58$	351.513			0.795792
$59$	−115.406			−0.254655			−0.127327	−	0.991861i	$-0.540640\pi$
$59$	−115.406			−0.254655			−0.127327	+	0.991861i	$0.540640\pi$
$60$		0			0
$61$	−602.179			−1.26395			−0.631977	−	0.774987i	$-0.717756\pi$
$61$	−602.179			−1.26395			−0.631977	+	0.774987i	$0.717756\pi$
$62$			170.049i			0.348327i
$63$		0			0
$64$	64.0000			0.125000
$65$	326.978			0.623947
$66$		0			0
$67$		−	514.633i		−	0.938394i	−0.883093	−	0.469197i	$-0.844543\pi$
$67$		−	514.633i		−	0.938394i	0.883093	−	0.469197i	$-0.155457\pi$
$68$		−	178.751i		−	0.318776i
$69$		0			0
$70$			41.5148i			0.0708853i
$71$	−233.009			−0.389480			−0.194740	−	0.980855i	$-0.562386\pi$
$71$	−233.009			−0.389480			−0.194740	+	0.980855i	$0.562386\pi$
$72$		0			0
$73$	−42.8481			−0.0686985			−0.0343492	−	0.999410i	$-0.510936\pi$
$73$	−42.8481			−0.0686985			−0.0343492	+	0.999410i	$0.510936\pi$
$74$		−	735.702i		−	1.15572i
$75$		0			0
$76$	117.320	−	309.806i	0.177073	−	0.467595i
$77$		−	56.6245i		−	0.0838047i
$78$		0			0
$79$		−	41.1864i		−	0.0586560i	−0.999570	−	0.0293280i	$-0.990663\pi$
$79$		−	41.1864i		−	0.0586560i	0.999570	−	0.0293280i	$-0.00933674\pi$
$80$		−	80.0000i		−	0.111803i
$81$		0			0
$82$	−59.2514			−0.0797954
$83$		−	1308.54i		−	1.73049i	−0.501347	−	0.865246i	$-0.667162\pi$
$83$		−	1308.54i		−	1.73049i	0.501347	−	0.865246i	$-0.332838\pi$
$84$		0			0
$85$	−223.439			−0.285122
$86$	−434.599			−0.544931
$87$		0			0
$88$			109.117i			0.132180i
$89$	600.186			0.714827			0.357414	−	0.933946i	$-0.383659\pi$
$89$	600.186			0.714827			0.357414	+	0.933946i	$0.383659\pi$
$90$		0			0
$91$			271.488i			0.312744i
$92$			306.303i			0.347112i
$93$		0			0
$94$			1226.07i			1.34531i
$95$	−387.258	−	146.650i	−0.418230	−	0.158379i
$96$		0			0
$97$		−	554.623i		−	0.580551i	−0.956943	−	0.290275i	$-0.906253\pi$
$97$		−	554.623i		−	0.580551i	0.956943	−	0.290275i	$-0.0937469\pi$
$98$	651.530			0.671577
$99$		0			0
$100$	−100.000			−0.100000
$101$			963.620i			0.949345i	0.880163	+	0.474672i	$0.157433\pi$
$101$			963.620i			0.949345i	−0.880163	+	0.474672i	$0.842567\pi$
$102$		0			0
$103$			1781.63i			1.70437i	0.523245	+	0.852183i	$0.324721\pi$
$103$			1781.63i			1.70437i	−0.523245	+	0.852183i	$0.675279\pi$
$104$		−	523.164i		−	0.493274i
$105$		0			0
$106$	−1412.29			−1.29409
$107$	−1171.80			−1.05871			−0.529357	−	0.848399i	$-0.677567\pi$
$107$	−1171.80			−1.05871			−0.529357	+	0.848399i	$0.677567\pi$
$108$		0			0
$109$			603.780i			0.530565i	0.964171	+	0.265283i	$0.0854653\pi$
$109$			603.780i			0.530565i	−0.964171	+	0.265283i	$0.914535\pi$
$110$	136.396			0.118226
$111$		0			0
$112$	66.4237			0.0560398
$113$	−739.988			−0.616038			−0.308019	−	0.951380i	$-0.599666\pi$
$113$	−739.988			−0.616038			−0.308019	+	0.951380i	$0.599666\pi$
$114$		0			0
$115$	382.879			0.310466
$116$	−703.026			−0.562710
$117$		0			0
$118$	230.813			0.180068
$119$		−	185.521i		−	0.142913i
$120$		0			0
$121$	1144.96			0.860227
$122$	1204.36			0.893750
$123$		0			0
$124$		−	340.098i		−	0.246304i
$125$			125.000i			0.0894427i
$126$		0			0
$127$			1276.15i			0.891652i	0.895120	+	0.445826i	$0.147090\pi$
$127$			1276.15i			0.891652i	−0.895120	+	0.445826i	$0.852910\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	−653.955			−0.441197
$131$		−	1084.70i		−	0.723442i	−0.932286	−	0.361721i	$-0.882189\pi$
$131$		−	1084.70i		−	0.723442i	0.932286	−	0.361721i	$-0.117811\pi$
$132$		0			0
$133$	121.763	−	321.539i	0.0793849	−	0.209631i
$134$			1029.27i			0.663545i
$135$		0			0
$136$			357.503i			0.225409i
$137$		−	2300.40i		−	1.43457i	−0.696779	−	0.717286i	$-0.745384\pi$
$137$		−	2300.40i		−	1.43457i	0.696779	−	0.717286i	$-0.254616\pi$
$138$		0			0
$139$	1476.92			0.901227			0.450614	−	0.892719i	$-0.351205\pi$
$139$	1476.92			0.901227			0.450614	+	0.892719i	$0.351205\pi$
$140$		−	83.0297i		−	0.0501235i
$141$		0			0
$142$	466.018			0.275404
$143$	891.968			0.521609
$144$		0			0
$145$			878.782i			0.503303i
$146$	85.6962			0.0485772
$147$		0			0
$148$			1471.40i			0.817220i
$149$		−	586.724i		−	0.322593i	−0.986906	−	0.161296i	$-0.948432\pi$
$149$		−	586.724i		−	0.322593i	0.986906	−	0.161296i	$-0.0515675\pi$
$150$		0			0
$151$		−	1640.73i		−	0.884246i	−0.896954	−	0.442123i	$-0.854226\pi$
$151$		−	1640.73i		−	0.884246i	0.896954	−	0.442123i	$-0.145774\pi$
$152$	−234.640	+	619.613i	−0.125209	+	0.330640i
$153$		0			0
$154$			113.249i			0.0592589i
$155$	−425.123			−0.220301
$156$		0			0
$157$	−3695.13			−1.87837			−0.939183	−	0.343416i	$-0.888416\pi$
$157$	−3695.13			−1.87837			−0.939183	+	0.343416i	$0.888416\pi$
$158$			82.3727i			0.0414761i
$159$		0			0
$160$			160.000i			0.0790569i
$161$			317.903i			0.155617i
$162$		0			0
$163$	−1012.89			−0.486724			−0.243362	−	0.969936i	$-0.578250\pi$
$163$	−1012.89			−0.486724			−0.243362	+	0.969936i	$0.578250\pi$
$164$	118.503			0.0564239
$165$		0			0
$166$			2617.08i			1.22364i
$167$	425.197			0.197023			0.0985113	−	0.995136i	$-0.468592\pi$
$167$	425.197			0.197023			0.0985113	+	0.995136i	$0.468592\pi$
$168$		0			0
$169$	−2079.57			−0.946551
$170$	446.878			0.201612
$171$		0			0
$172$	869.198			0.385324
$173$	435.146			0.191234			0.0956171	−	0.995418i	$-0.469518\pi$
$173$	435.146			0.191234			0.0956171	+	0.995418i	$0.469518\pi$
$174$		0			0
$175$	−103.787			−0.0448318
$176$		−	218.233i		−	0.0934657i
$177$		0			0
$178$	−1200.37			−0.505459
$179$	−1268.07			−0.529496			−0.264748	−	0.964318i	$-0.585289\pi$
$179$	−1268.07			−0.529496			−0.264748	+	0.964318i	$0.585289\pi$
$180$		0			0
$181$			1298.55i			0.533262i	0.963799	+	0.266631i	$0.0859105\pi$
$181$			1298.55i			0.533262i	−0.963799	+	0.266631i	$0.914089\pi$
$182$		−	542.977i		−	0.221144i
$183$		0			0
$184$		−	612.606i		−	0.245445i
$185$	1839.25			0.730944
$186$		0			0
$187$	−609.523			−0.238357
$188$		−	2452.14i		−	0.951279i
$189$		0			0
$190$	774.516	+	293.300i	0.295733	+	0.111991i
$191$		−	170.790i		−	0.0647014i	−0.999477	−	0.0323507i	$-0.989701\pi$
$191$		−	170.790i		−	0.0647014i	0.999477	−	0.0323507i	$-0.0102993\pi$
$192$		0			0
$193$		−	503.406i		−	0.187751i	−0.995584	−	0.0938755i	$-0.970074\pi$
$193$		−	503.406i		−	0.187751i	0.995584	−	0.0938755i	$-0.0299256\pi$
$194$			1109.25i			0.410511i
$195$		0			0
$196$	−1303.06			−0.474876
$197$		−	1133.50i		−	0.409941i	−0.978768	−	0.204970i	$-0.934290\pi$
$197$		−	1133.50i		−	0.409941i	0.978768	−	0.204970i	$-0.0657098\pi$
$198$		0			0
$199$	−758.201			−0.270087			−0.135044	−	0.990840i	$-0.543117\pi$
$199$	−758.201			−0.270087			−0.135044	+	0.990840i	$0.543117\pi$
$200$	200.000			0.0707107
$201$		0			0
$202$		−	1927.24i		−	0.671288i
$203$	−729.650			−0.252273
$204$		0			0
$205$		−	148.129i		−	0.0504670i
$206$		−	3563.27i		−	1.20517i
$207$		0			0
$208$			1046.33i			0.348797i
$209$	−1056.41	−	400.049i	−0.349633	−	0.132402i
$210$		0			0
$211$			1242.75i			0.405472i	0.979233	+	0.202736i	$0.0649833\pi$
$211$			1242.75i			0.405472i	−0.979233	+	0.202736i	$0.935017\pi$
$212$	2824.59			0.915063
$213$		0			0
$214$	2343.60			0.748624
$215$		−	1086.50i		−	0.344644i
$216$		0			0
$217$		−	352.978i		−	0.110423i
$218$		−	1207.56i		−	0.375166i
$219$		0			0
$220$	−272.792			−0.0835983
$221$	2922.38			0.889505
$222$		0			0
$223$		−	4924.02i		−	1.47864i	−0.673354	−	0.739320i	$-0.735147\pi$
$223$		−	4924.02i		−	1.47864i	0.673354	−	0.739320i	$-0.264853\pi$
$224$	−132.847			−0.0396261
$225$		0			0
$226$	1479.98			0.435604
$227$	1903.39			0.556531			0.278265	−	0.960504i	$-0.410241\pi$
$227$	1903.39			0.556531			0.278265	+	0.960504i	$0.410241\pi$
$228$		0			0
$229$	−4308.36			−1.24325			−0.621625	−	0.783315i	$-0.713527\pi$
$229$	−4308.36			−1.24325			−0.621625	+	0.783315i	$0.713527\pi$
$230$	−765.757			−0.219533
$231$		0			0
$232$	1406.05			0.397896
$233$		−	6736.17i		−	1.89400i	−0.321239	−	0.946998i	$-0.604099\pi$
$233$		−	6736.17i		−	1.89400i	0.321239	−	0.946998i	$-0.395901\pi$
$234$		0			0
$235$	−3065.17			−0.850850
$236$	−461.626			−0.127327
$237$		0			0
$238$			371.041i			0.101055i
$239$		−	5564.76i		−	1.50608i	−0.657972	−	0.753042i	$-0.728585\pi$
$239$		−	5564.76i		−	1.50608i	0.657972	−	0.753042i	$-0.271415\pi$
$240$		0			0
$241$			5906.43i			1.57870i	0.613944	+	0.789350i	$0.289582\pi$
$241$			5906.43i			1.57870i	−0.613944	+	0.789350i	$0.710418\pi$
$242$	−2289.92			−0.608272
$243$		0			0
$244$	−2408.72			−0.631977
$245$			1628.83i			0.424742i
$246$		0			0
$247$	5064.99	+	1918.05i	1.30477	+	0.494100i
$248$			680.196i			0.174163i
$249$		0			0
$250$		−	250.000i		−	0.0632456i
$251$			2484.79i			0.624856i	0.949941	+	0.312428i	$0.101142\pi$
$251$			2484.79i			0.624856i	−0.949941	+	0.312428i	$0.898858\pi$
$252$		0			0
$253$	1044.46			0.259544
$254$		−	2552.30i		−	0.630493i
$255$		0			0
$256$	256.000			0.0625000
$257$	−5495.16			−1.33377			−0.666884	−	0.745161i	$-0.732372\pi$
$257$	−5495.16			−1.33377			−0.666884	+	0.745161i	$0.732372\pi$
$258$		0			0
$259$			1527.13i			0.366375i
$260$	1307.91			0.311974
$261$		0			0
$262$			2169.40i			0.511550i
$263$			5987.43i			1.40381i	0.712273	+	0.701903i	$0.247666\pi$
$263$			5987.43i			1.40381i	−0.712273	+	0.701903i	$0.752334\pi$
$264$		0			0
$265$		−	3530.73i		−	0.818457i
$266$	−243.526	+	643.078i	−0.0561336	+	0.148232i
$267$		0			0
$268$		−	2058.53i		−	0.469197i
$269$	−7707.70			−1.74701			−0.873507	−	0.486812i	$-0.838160\pi$
$269$	−7707.70			−1.74701			−0.873507	+	0.486812i	$0.838160\pi$
$270$		0			0
$271$	−6368.52			−1.42753			−0.713764	−	0.700387i	$-0.753011\pi$
$271$	−6368.52			−1.42753			−0.713764	+	0.700387i	$0.753011\pi$
$272$		−	715.005i		−	0.159388i
$273$		0			0
$274$			4600.80i			1.01440i
$275$			340.990i			0.0747726i
$276$		0			0
$277$	−4375.82			−0.949160			−0.474580	−	0.880212i	$-0.657400\pi$
$277$	−4375.82			−0.949160			−0.474580	+	0.880212i	$0.657400\pi$
$278$	−2953.84			−0.637264
$279$		0			0
$280$			166.059i			0.0354427i
$281$	−2039.75			−0.433029			−0.216515	−	0.976279i	$-0.569469\pi$
$281$	−2039.75			−0.433029			−0.216515	+	0.976279i	$0.569469\pi$
$282$		0			0
$283$	−3952.88			−0.830299			−0.415149	−	0.909753i	$-0.636271\pi$
$283$	−3952.88			−0.830299			−0.415149	+	0.909753i	$0.636271\pi$
$284$	−932.036			−0.194740
$285$		0			0
$286$	−1783.94			−0.368833
$287$	122.991			0.0252958
$288$		0			0
$289$	2916.00			0.593527
$290$		−	1757.56i		−	0.355889i
$291$		0			0
$292$	−171.392			−0.0343492
$293$	−8906.70			−1.77589			−0.887944	−	0.459952i	$-0.847866\pi$
$293$	−8906.70			−1.77589			−0.887944	+	0.459952i	$0.847866\pi$
$294$		0			0
$295$			577.032i			0.113885i
$296$		−	2942.81i		−	0.577862i
$297$		0			0
$298$			1173.45i			0.228108i
$299$	−5007.71			−0.968573
$300$		0			0
$301$	902.116			0.172748
$302$			3281.47i			0.625256i
$303$		0			0
$304$	469.280	−	1239.23i	0.0885364	−	0.233798i
$305$			3010.90i			0.565257i
$306$		0			0
$307$		−	7611.80i		−	1.41508i	−0.706676	−	0.707538i	$-0.749806\pi$
$307$		−	7611.80i		−	1.41508i	0.706676	−	0.707538i	$-0.250194\pi$
$308$		−	226.498i		−	0.0419024i
$309$		0			0
$310$	850.245			0.155776
$311$		−	5645.48i		−	1.02934i	−0.857388	−	0.514671i	$-0.827914\pi$
$311$		−	5645.48i		−	1.02934i	0.857388	−	0.514671i	$-0.172086\pi$
$312$		0			0
$313$	1932.47			0.348977			0.174488	−	0.984659i	$-0.444173\pi$
$313$	1932.47			0.348977			0.174488	+	0.984659i	$0.444173\pi$
$314$	7390.26			1.32821
$315$		0			0
$316$		−	164.745i		−	0.0293280i
$317$	5098.52			0.903348			0.451674	−	0.892183i	$-0.350827\pi$
$317$	5098.52			0.903348			0.451674	+	0.892183i	$0.350827\pi$
$318$		0			0
$319$			2397.25i			0.420752i
$320$		−	320.000i		−	0.0559017i
$321$		0			0
$322$		−	635.806i		−	0.110038i
$323$	−3461.14	−	1310.69i	−0.596233	−	0.225786i
$324$		0			0
$325$		−	1634.89i		−	0.279038i
$326$	2025.79			0.344166
$327$		0			0
$328$	−237.006			−0.0398977
$329$		−	2545.00i		−	0.426476i
$330$		0			0
$331$		−	4627.19i		−	0.768378i	−0.923254	−	0.384189i	$-0.874481\pi$
$331$		−	4627.19i		−	0.768378i	0.923254	−	0.384189i	$-0.125519\pi$
$332$		−	5234.16i		−	0.865246i
$333$		0			0
$334$	−850.395			−0.139316
$335$	−2573.16			−0.419663
$336$		0			0
$337$		−	4092.12i		−	0.661459i	−0.943726	−	0.330730i	$-0.892705\pi$
$337$		−	4092.12i		−	0.661459i	0.943726	−	0.330730i	$-0.107295\pi$
$338$	4159.15			0.669313
$339$		0			0
$340$	−893.756			−0.142561
$341$	−1159.70			−0.184168
$342$		0			0
$343$	−2776.37			−0.437055
$344$	−1738.40			−0.272465
$345$		0			0
$346$	−870.291			−0.135223
$347$		−	11294.3i		−	1.74728i	−0.486569	−	0.873642i	$-0.661752\pi$
$347$		−	11294.3i		−	1.74728i	0.486569	−	0.873642i	$-0.338248\pi$
$348$		0			0
$349$	−2963.59			−0.454548			−0.227274	−	0.973831i	$-0.572981\pi$
$349$	−2963.59			−0.454548			−0.227274	+	0.973831i	$0.572981\pi$
$350$	207.574			0.0317009
$351$		0			0
$352$			436.467i			0.0660902i
$353$		−	4672.70i		−	0.704541i	−0.935898	−	0.352270i	$-0.885410\pi$
$353$		−	4672.70i		−	0.704541i	0.935898	−	0.352270i	$-0.114590\pi$
$354$		0			0
$355$			1165.04i			0.174181i
$356$	2400.75			0.357414
$357$		0			0
$358$	2536.13			0.374410
$359$		−	3891.41i		−	0.572091i	−0.958216	−	0.286045i	$-0.907659\pi$
$359$		−	3891.41i		−	0.572091i	0.958216	−	0.286045i	$-0.0923408\pi$
$360$		0			0
$361$	−5138.50	−	4543.31i	−0.749162	−	0.662387i
$362$		−	2597.10i		−	0.377073i
$363$		0			0
$364$			1085.95i			0.156372i
$365$			214.240i			0.0307229i
$366$		0			0
$367$	−6600.96			−0.938876			−0.469438	−	0.882966i	$-0.655543\pi$
$367$	−6600.96			−0.938876			−0.469438	+	0.882966i	$0.655543\pi$
$368$			1225.21i			0.173556i
$369$		0			0
$370$	−3678.51			−0.516856
$371$	2931.56			0.410239
$372$		0			0
$373$		−	3309.44i		−	0.459400i	−0.973261	−	0.229700i	$-0.926225\pi$
$373$		−	3309.44i		−	0.459400i	0.973261	−	0.229700i	$-0.0737746\pi$
$374$	1219.05			0.168544
$375$		0			0
$376$			4904.27i			0.672656i
$377$		−	11493.7i		−	1.57017i
$378$		0			0
$379$		−	4421.21i		−	0.599215i	−0.954063	−	0.299608i	$-0.903144\pi$
$379$		−	4421.21i		−	0.599215i	0.954063	−	0.299608i	$-0.0968557\pi$
$380$	−1549.03	−	586.600i	−0.209115	−	0.0791893i
$381$		0			0
$382$			341.581i			0.0457508i
$383$	−13442.0			−1.79335			−0.896675	−	0.442689i	$-0.854025\pi$
$383$	−13442.0			−1.79335			−0.896675	+	0.442689i	$0.854025\pi$
$384$		0			0
$385$	−283.123			−0.0374786
$386$			1006.81i			0.132760i
$387$		0			0
$388$		−	2218.49i		−	0.290275i
$389$			13460.2i			1.75440i	0.480126	+	0.877199i	$0.340591\pi$
$389$			13460.2i			1.75440i	−0.480126	+	0.877199i	$0.659409\pi$
$390$		0			0
$391$	3422.00			0.442604
$392$	2606.12			0.335788
$393$		0			0
$394$			2267.00i			0.289872i
$395$	−205.932			−0.0262318
$396$		0			0
$397$	1872.73			0.236749			0.118375	−	0.992969i	$-0.462232\pi$
$397$	1872.73			0.236749			0.118375	+	0.992969i	$0.462232\pi$
$398$	1516.40			0.190981
$399$		0			0
$400$	−400.000			−0.0500000
$401$	−10551.8			−1.31404			−0.657021	−	0.753873i	$-0.728184\pi$
$401$	−10551.8			−1.31404			−0.657021	+	0.753873i	$0.728184\pi$
$402$		0			0
$403$	5560.22			0.687282
$404$			3854.48i			0.474672i
$405$		0			0
$406$	1459.30			0.178384
$407$	5017.34			0.611057
$408$		0			0
$409$		−	8447.18i		−	1.02124i	−0.859807	−	0.510619i	$-0.829416\pi$
$409$		−	8447.18i		−	1.02124i	0.859807	−	0.510619i	$-0.170584\pi$
$410$			296.257i			0.0356856i
$411$		0			0
$412$			7126.54i			0.852183i
$413$	−479.108			−0.0570832
$414$		0			0
$415$	−6542.70			−0.773900
$416$		−	2092.66i		−	0.246637i
$417$		0			0
$418$	2112.82	+	800.099i	0.247228	+	0.0936223i
$419$			5198.00i			0.606060i	0.952981	+	0.303030i	$0.0979982\pi$
$419$			5198.00i			0.606060i	−0.952981	+	0.303030i	$0.902002\pi$
$420$		0			0
$421$		−	10164.6i		−	1.17671i	−0.808603	−	0.588354i	$-0.799776\pi$
$421$		−	10164.6i		−	1.17671i	0.808603	−	0.588354i	$-0.200224\pi$
$422$		−	2485.50i		−	0.286712i
$423$		0			0
$424$	−5649.17			−0.647047
$425$			1117.20i			0.127510i
$426$		0			0
$427$	−2499.94			−0.283327
$428$	−4687.21			−0.529357
$429$		0			0
$430$			2173.00i			0.243700i
$431$	−14568.7			−1.62819			−0.814097	−	0.580729i	$-0.802768\pi$
$431$	−14568.7			−1.62819			−0.814097	+	0.580729i	$0.802768\pi$
$432$		0			0
$433$		−	11453.2i		−	1.27115i	−0.772039	−	0.635575i	$-0.780763\pi$
$433$		−	11453.2i		−	1.27115i	0.772039	−	0.635575i	$-0.219237\pi$
$434$			705.956i			0.0780806i
$435$		0			0
$436$			2415.12i			0.265283i
$437$	5930.91	+	2245.97i	0.649231	+	0.245856i
$438$		0			0
$439$			17377.6i			1.88926i	0.328133	+	0.944632i	$0.393581\pi$
$439$			17377.6i			1.88926i	−0.328133	+	0.944632i	$0.606419\pi$
$440$	545.584			0.0591129
$441$		0			0
$442$	−5844.76			−0.628975
$443$		−	12355.4i		−	1.32510i	−0.749016	−	0.662552i	$-0.769473\pi$
$443$		−	12355.4i		−	1.32510i	0.749016	−	0.662552i	$-0.230527\pi$
$444$		0			0
$445$		−	3000.93i		−	0.319680i
$446$			9848.04i			1.04556i
$447$		0			0
$448$	265.695			0.0280199
$449$	6284.98			0.660594			0.330297	−	0.943877i	$-0.392851\pi$
$449$	6284.98			0.660594			0.330297	+	0.943877i	$0.392851\pi$
$450$		0			0
$451$		−	404.082i		−	0.0421896i
$452$	−2959.95			−0.308019
$453$		0			0
$454$	−3806.78			−0.393527
$455$	1357.44			0.139863
$456$		0			0
$457$	13350.6			1.36655			0.683275	−	0.730161i	$-0.260555\pi$
$457$	13350.6			1.36655			0.683275	+	0.730161i	$0.260555\pi$
$458$	8616.72			0.879111
$459$		0			0
$460$	1531.51			0.155233
$461$		−	6439.57i		−	0.650587i	−0.945613	−	0.325294i	$-0.894537\pi$
$461$		−	6439.57i		−	0.650587i	0.945613	−	0.325294i	$-0.105463\pi$
$462$		0			0
$463$	−5189.74			−0.520923			−0.260462	−	0.965484i	$-0.583875\pi$
$463$	−5189.74			−0.520923			−0.260462	+	0.965484i	$0.583875\pi$
$464$	−2812.10			−0.281355
$465$		0			0
$466$			13472.3i			1.33926i
$467$			10235.6i			1.01423i	0.861877	+	0.507117i	$0.169289\pi$
$467$			10235.6i			1.01423i	−0.861877	+	0.507117i	$0.830711\pi$
$468$		0			0
$469$		−	2136.49i		−	0.210350i
$470$	6130.34			0.601642
$471$		0			0
$472$	923.251			0.0900341
$473$		−	2963.88i		−	0.288117i
$474$		0			0
$475$	−733.250	+	1936.29i	−0.0708291	+	0.187038i
$476$		−	742.083i		−	0.0714565i
$477$		0			0
$478$			11129.5i			1.06496i
$479$			19901.6i			1.89839i	0.314696	+	0.949193i	$0.398098\pi$
$479$			19901.6i			1.89839i	−0.314696	+	0.949193i	$0.601902\pi$
$480$		0			0
$481$	−24055.8			−2.28035
$482$		−	11812.9i		−	1.11631i
$483$		0			0
$484$	4579.85			0.430113
$485$	−2773.11			−0.259630
$486$		0			0
$487$		−	15569.3i		−	1.44869i	−0.689439	−	0.724344i	$-0.742143\pi$
$487$		−	15569.3i		−	1.44869i	0.689439	−	0.724344i	$-0.257857\pi$
$488$	4817.43			0.446875
$489$		0			0
$490$		−	3257.65i		−	0.300338i
$491$			798.662i			0.0734076i	0.999326	+	0.0367038i	$0.0116858\pi$
$491$			798.662i			0.0734076i	−0.999326	+	0.0367038i	$0.988314\pi$
$492$		0			0
$493$			7854.17i			0.717513i
$494$	−10130.0	−	3836.10i	−0.922610	−	0.349381i
$495$		0			0
$496$		−	1360.39i		−	0.123152i
$497$	−967.333			−0.0873055
$498$		0			0
$499$	−17364.6			−1.55781			−0.778906	−	0.627141i	$-0.784225\pi$
$499$	−17364.6			−1.55781			−0.778906	+	0.627141i	$0.784225\pi$
$500$			500.000i			0.0447214i
$501$		0			0
$502$		−	4969.59i		−	0.441840i
$503$			14649.6i			1.29860i	0.760534	+	0.649298i	$0.224937\pi$
$503$			14649.6i			1.29860i	−0.760534	+	0.649298i	$0.775063\pi$
$504$		0			0
$505$	4818.10			0.424560
$506$	−2088.92			−0.183526
$507$		0			0
$508$			5104.59i			0.445826i
$509$	−11409.3			−0.993529			−0.496764	−	0.867885i	$-0.665479\pi$
$509$	−11409.3			−0.993529			−0.496764	+	0.867885i	$0.665479\pi$
$510$		0			0
$511$	−177.883			−0.0153994
$512$	−512.000			−0.0441942
$513$		0			0
$514$	10990.3			0.943117
$515$	8908.17			0.762215
$516$		0			0
$517$	−8361.54			−0.711296
$518$		−	3054.25i		−	0.259066i
$519$		0			0
$520$	−2615.82			−0.220599
$521$	10248.7			0.861812			0.430906	−	0.902397i	$-0.358194\pi$
$521$	10248.7			0.861812			0.430906	+	0.902397i	$0.358194\pi$
$522$		0			0
$523$			9591.08i			0.801890i	0.916102	+	0.400945i	$0.131318\pi$
$523$			9591.08i			0.801890i	−0.916102	+	0.400945i	$0.868682\pi$
$524$		−	4338.81i		−	0.361721i
$525$		0			0
$526$		−	11974.9i		−	0.992641i
$527$	−3799.56			−0.314064
$528$		0			0
$529$	6303.16			0.518053
$530$			7061.46i			0.578737i
$531$		0			0
$532$	487.052	−	1286.16i	0.0396925	−	0.104816i
$533$			1937.39i			0.157444i
$534$		0			0
$535$			5859.01i			0.473471i
$536$			4117.06i			0.331772i
$537$		0			0
$538$	15415.4			1.23533
$539$			4443.30i			0.355077i
$540$		0			0
$541$	1750.62			0.139122			0.0695609	−	0.997578i	$-0.477840\pi$
$541$	1750.62			0.139122			0.0695609	+	0.997578i	$0.477840\pi$
$542$	12737.0			1.00941
$543$		0			0
$544$			1430.01i			0.112704i
$545$	3018.90			0.237276
$546$		0			0
$547$		−	17617.8i		−	1.37712i	−0.725181	−	0.688558i	$-0.758244\pi$
$547$		−	17617.8i		−	1.37712i	0.725181	−	0.688558i	$-0.241756\pi$
$548$		−	9201.60i		−	0.717286i
$549$		0			0
$550$		−	681.980i		−	0.0528722i
$551$	−5154.94	+	13612.6i	−0.398562	+	1.05248i
$552$		0			0
$553$		−	170.984i		−	0.0131483i
$554$	8751.64			0.671158
$555$		0			0
$556$	5907.67			0.450614
$557$			13846.9i			1.05334i	0.850070	+	0.526670i	$0.176560\pi$
$557$			13846.9i			1.05334i	−0.850070	+	0.526670i	$0.823440\pi$
$558$		0			0
$559$			14210.4i			1.07520i
$560$		−	332.119i		−	0.0250617i
$561$		0			0
$562$	4079.50			0.306198
$563$	−14800.9			−1.10797			−0.553983	−	0.832528i	$-0.686893\pi$
$563$	−14800.9			−1.10797			−0.553983	+	0.832528i	$0.686893\pi$
$564$		0			0
$565$			3699.94i			0.275500i
$566$	7905.77			0.587110
$567$		0			0
$568$	1864.07			0.137702
$569$	−6786.69			−0.500023			−0.250011	−	0.968243i	$-0.580434\pi$
$569$	−6786.69			−0.500023			−0.250011	+	0.968243i	$0.580434\pi$
$570$		0			0
$571$	16430.6			1.20420			0.602102	−	0.798419i	$-0.294330\pi$
$571$	16430.6			1.20420			0.602102	+	0.798419i	$0.294330\pi$
$572$	3567.87			0.260805
$573$		0			0
$574$	−245.981			−0.0178869
$575$		−	1914.39i		−	0.138845i
$576$		0			0
$577$	12651.6			0.912813			0.456407	−	0.889771i	$-0.349136\pi$
$577$	12651.6			0.912813			0.456407	+	0.889771i	$0.349136\pi$
$578$	−5832.00			−0.419687
$579$		0			0
$580$			3515.13i			0.251651i
$581$		−	5432.38i		−	0.387906i
$582$		0			0
$583$		−	9631.55i		−	0.684216i
$584$	342.785			0.0242886
$585$		0			0
$586$	17813.4			1.25574
$587$			11.0824i			0.000779248i	1.00000		0.000389624i	$0.000124021\pi$
$587$			11.0824i			0.000779248i	−1.00000		0.000389624i	$0.999876\pi$
$588$		0			0
$589$	−6585.29	−	2493.77i	−0.460683	−	0.174455i
$590$		−	1154.06i		−	0.0805289i
$591$		0			0
$592$			5885.61i			0.408610i
$593$		−	10519.8i		−	0.728493i	−0.931302	−	0.364247i	$-0.881326\pi$
$593$		−	10519.8i		−	0.728493i	0.931302	−	0.364247i	$-0.118674\pi$
$594$		0			0
$595$	−927.604			−0.0639127
$596$		−	2346.90i		−	0.161296i
$597$		0			0
$598$	10015.4			0.684885
$599$	−6103.55			−0.416335			−0.208167	−	0.978093i	$-0.566750\pi$
$599$	−6103.55			−0.416335			−0.208167	+	0.978093i	$0.566750\pi$
$600$		0			0
$601$		−	27303.0i		−	1.85310i	−0.376170	−	0.926551i	$-0.622759\pi$
$601$		−	27303.0i		−	1.85310i	0.376170	−	0.926551i	$-0.377241\pi$
$602$	−1804.23			−0.122151
$603$		0			0
$604$		−	6562.94i		−	0.442123i
$605$		−	5724.81i		−	0.384705i
$606$		0			0
$607$		−	3520.94i		−	0.235437i	−0.993047	−	0.117719i	$-0.962442\pi$
$607$		−	3520.94i		−	0.235437i	0.993047	−	0.117719i	$-0.0375581\pi$
$608$	−938.560	+	2478.45i	−0.0626047	+	0.165320i
$609$		0			0
$610$		−	6021.79i		−	0.399697i
$611$	40089.7			2.65443
$612$		0			0
$613$	17496.4			1.15281			0.576405	−	0.817164i	$-0.304455\pi$
$613$	17496.4			1.15281			0.576405	+	0.817164i	$0.304455\pi$
$614$			15223.6i			1.00061i
$615$		0			0
$616$			452.996i			0.0296295i
$617$		−	6227.73i		−	0.406352i	−0.979142	−	0.203176i	$-0.934874\pi$
$617$		−	6227.73i		−	0.406352i	0.979142	−	0.203176i	$-0.0651263\pi$
$618$		0			0
$619$	−26012.6			−1.68907			−0.844534	−	0.535502i	$-0.820122\pi$
$619$	−26012.6			−1.68907			−0.844534	+	0.535502i	$0.820122\pi$
$620$	−1700.49			−0.110151
$621$		0			0
$622$			11291.0i			0.727855i
$623$	2491.66			0.160235
$624$		0			0
$625$	625.000			0.0400000
$626$	−3864.94			−0.246764
$627$		0			0
$628$	−14780.5			−0.939183
$629$	16438.5			1.04204
$630$		0			0
$631$	4445.40			0.280457			0.140229	−	0.990119i	$-0.455216\pi$
$631$	4445.40			0.280457			0.140229	+	0.990119i	$0.455216\pi$
$632$			329.491i			0.0207380i
$633$		0			0
$634$	−10197.0			−0.638763
$635$	6380.74			0.398759
$636$		0			0
$637$		−	21303.6i		−	1.32508i
$638$		−	4794.49i		−	0.297517i
$639$		0			0
$640$			640.000i			0.0395285i
$641$	−18974.6			−1.16919			−0.584594	−	0.811326i	$-0.698746\pi$
$641$	−18974.6			−1.16919			−0.584594	+	0.811326i	$0.698746\pi$
$642$		0			0
$643$	26283.2			1.61199			0.805994	−	0.591924i	$-0.201631\pi$
$643$	26283.2			1.61199			0.805994	+	0.591924i	$0.201631\pi$
$644$			1271.61i			0.0778083i
$645$		0			0
$646$	6922.29	+	2621.39i	0.421600	+	0.159655i
$647$			14181.6i			0.861724i	0.902418	+	0.430862i	$0.141790\pi$
$647$			14181.6i			0.861724i	−0.902418	+	0.430862i	$0.858210\pi$
$648$		0			0
$649$			1574.10i			0.0952060i
$650$			3269.78i			0.197309i
$651$		0			0
$652$	−4051.58			−0.243362
$653$			28895.3i			1.73164i	0.500356	+	0.865820i	$0.333203\pi$
$653$			28895.3i			1.73164i	−0.500356	+	0.865820i	$0.666797\pi$
$654$		0			0
$655$	−5423.51			−0.323533
$656$	474.011			0.0282119
$657$		0			0
$658$			5090.00i			0.301564i
$659$	−12577.3			−0.743463			−0.371731	−	0.928340i	$-0.621236\pi$
$659$	−12577.3			−0.743463			−0.371731	+	0.928340i	$0.621236\pi$
$660$		0			0
$661$		−	7017.06i		−	0.412907i	−0.978456	−	0.206454i	$-0.933808\pi$
$661$		−	7017.06i		−	0.412907i	0.978456	−	0.206454i	$-0.0661923\pi$
$662$			9254.38i			0.543326i
$663$		0			0
$664$			10468.3i			0.611822i
$665$	−1607.70	−	608.815i	−0.0937500	−	0.0355020i
$666$		0			0
$667$		−	13458.7i		−	0.781293i
$668$	1700.79			0.0985113
$669$		0			0
$670$	5146.33			0.296746
$671$			8213.48i			0.472545i
$672$		0			0
$673$			16806.6i			0.962625i	0.876549	+	0.481313i	$0.159840\pi$
$673$			16806.6i			0.962625i	−0.876549	+	0.481313i	$0.840160\pi$
$674$			8184.23i			0.467722i
$675$		0			0
$676$	−8318.29			−0.473276
$677$	23241.2			1.31939			0.659697	−	0.751531i	$-0.270684\pi$
$677$	23241.2			1.31939			0.659697	+	0.751531i	$0.270684\pi$
$678$		0			0
$679$		−	2302.51i		−	0.130136i
$680$	1787.51			0.100806
$681$		0			0
$682$	2319.40			0.130226
$683$	16838.9			0.943373			0.471686	−	0.881766i	$-0.343645\pi$
$683$	16838.9			0.943373			0.471686	+	0.881766i	$0.343645\pi$
$684$		0			0
$685$	−11502.0			−0.641560
$686$	5552.74			0.309044
$687$		0			0
$688$	3476.79			0.192662
$689$			46178.8i			2.55337i
$690$		0			0
$691$	29238.0			1.60964			0.804822	−	0.593516i	$-0.202261\pi$
$691$	29238.0			1.60964			0.804822	+	0.593516i	$0.202261\pi$
$692$	1740.58			0.0956171
$693$		0			0
$694$			22588.5i			1.23552i
$695$		−	7384.59i		−	0.403041i
$696$		0			0
$697$		−	1323.91i		−	0.0719463i
$698$	5927.17			0.321414
$699$		0			0
$700$	−415.148			−0.0224159
$701$		−	8103.20i		−	0.436596i	−0.975882	−	0.218298i	$-0.929950\pi$
$701$		−	8103.20i		−	0.436596i	0.975882	−	0.218298i	$-0.0700505\pi$
$702$		0			0
$703$	28490.6	+	10789.1i	1.52851	+	0.578830i
$704$		−	872.934i		−	0.0467329i
$705$		0			0
$706$			9345.40i			0.498185i
$707$			4000.45i			0.212804i
$708$		0			0
$709$	3316.19			0.175659			0.0878295	−	0.996136i	$-0.472007\pi$
$709$	3316.19			0.175659			0.0878295	+	0.996136i	$0.472007\pi$
$710$		−	2330.09i		−	0.123164i
$711$		0			0
$712$	−4801.49			−0.252730
$713$	6510.82			0.341980
$714$		0			0
$715$		−	4459.84i		−	0.233271i
$716$	−5072.27			−0.264748
$717$		0			0
$718$			7782.81i			0.404529i
$719$		−	8484.76i		−	0.440095i	−0.975489	−	0.220047i	$-0.929379\pi$
$719$		−	8484.76i		−	0.440095i	0.975489	−	0.220047i	$-0.0706212\pi$
$720$		0			0
$721$			7396.42i			0.382049i
$722$	10277.0	+	9086.62i	0.529737	+	0.468378i
$723$		0			0
$724$			5194.20i			0.266631i
$725$	4393.91			0.225084
$726$		0			0
$727$	15912.5			0.811776			0.405888	−	0.913923i	$-0.366962\pi$
$727$	15912.5			0.811776			0.405888	+	0.913923i	$0.366962\pi$
$728$		−	2171.91i		−	0.110572i
$729$		0			0
$730$		−	428.481i		−	0.0217244i
$731$		−	9710.64i		−	0.491328i
$732$		0			0
$733$	26040.8			1.31219			0.656097	−	0.754677i	$-0.272206\pi$
$733$	26040.8			1.31219			0.656097	+	0.754677i	$0.272206\pi$
$734$	13201.9			0.663885
$735$		0			0
$736$		−	2450.42i		−	0.122723i
$737$	−7019.38			−0.350831
$738$		0			0
$739$	27765.9			1.38212			0.691059	−	0.722798i	$-0.257144\pi$
$739$	27765.9			1.38212			0.691059	+	0.722798i	$0.257144\pi$
$740$	7357.02			0.365472
$741$		0			0
$742$	−5863.11			−0.290083
$743$	−1321.31			−0.0652413			−0.0326207	−	0.999468i	$-0.510385\pi$
$743$	−1321.31			−0.0652413			−0.0326207	+	0.999468i	$0.510385\pi$
$744$		0			0
$745$	−2933.62			−0.144268
$746$			6618.88i			0.324845i
$747$		0			0
$748$	−2438.09			−0.119179
$749$	−4864.72			−0.237320
$750$		0			0
$751$		−	855.394i		−	0.0415629i	−0.999784	−	0.0207815i	$-0.993385\pi$
$751$		−	855.394i		−	0.0415629i	0.999784	−	0.0207815i	$-0.00661542\pi$
$752$		−	9808.55i		−	0.475640i
$753$		0			0
$754$			22987.4i			1.11028i
$755$	−8203.67			−0.395447
$756$		0			0
$757$	−19685.7			−0.945163			−0.472581	−	0.881287i	$-0.656678\pi$
$757$	−19685.7			−0.945163			−0.472581	+	0.881287i	$0.656678\pi$
$758$			8842.43i			0.423709i
$759$		0			0
$760$	3098.06	+	1173.20i	0.147867	+	0.0559953i
$761$		−	2346.44i		−	0.111772i	−0.998437	−	0.0558859i	$-0.982202\pi$
$761$		−	2346.44i		−	0.111772i	0.998437	−	0.0558859i	$-0.0177983\pi$
$762$		0			0
$763$			2506.58i			0.118931i
$764$		−	683.162i		−	0.0323507i
$765$		0			0
$766$	26884.0			1.26809
$767$		−	7547.06i		−	0.355291i
$768$		0			0
$769$	−2983.65			−0.139913			−0.0699565	−	0.997550i	$-0.522286\pi$
$769$	−2983.65			−0.139913			−0.0699565	+	0.997550i	$0.522286\pi$
$770$	566.245			0.0265014
$771$		0			0
$772$		−	2013.62i		−	0.0938755i
$773$	−673.298			−0.0313284			−0.0156642	−	0.999877i	$-0.504986\pi$
$773$	−673.298			−0.0313284			−0.0156642	+	0.999877i	$0.504986\pi$
$774$		0			0
$775$			2125.61i			0.0985217i
$776$			4436.98i			0.205256i
$777$		0			0
$778$		−	26920.5i		−	1.24055i
$779$	868.922	−	2294.56i	0.0399645	−	0.105534i
$780$		0			0
$781$			3178.15i			0.145612i
$782$	−6844.00			−0.312968
$783$		0			0
$784$	−5212.24			−0.237438
$785$			18475.7i			0.840031i
$786$		0			0
$787$			9634.69i			0.436391i	0.975905	+	0.218195i	$0.0700170\pi$
$787$			9634.69i			0.436391i	−0.975905	+	0.218195i	$0.929983\pi$
$788$		−	4533.99i		−	0.204970i
$789$		0			0
$790$	411.864			0.0185487
$791$	−3072.05			−0.138090
$792$		0			0
$793$		−	39379.8i		−	1.76345i
$794$	−3745.45			−0.167407
$795$		0			0
$796$	−3032.80			−0.135044
$797$	12203.1			0.542354			0.271177	−	0.962529i	$-0.412587\pi$
$797$	12203.1			0.542354			0.271177	+	0.962529i	$0.412587\pi$
$798$		0			0
$799$	−27395.2			−1.21298
$800$	800.000			0.0353553
$801$		0			0
$802$	21103.6			0.929167
$803$			584.430i			0.0256838i
$804$		0			0
$805$	1589.51			0.0695938
$806$	−11120.4			−0.485981
$807$		0			0
$808$		−	7708.96i		−	0.335644i
$809$			12209.4i			0.530604i	0.964165	+	0.265302i	$0.0854716\pi$
$809$			12209.4i			0.530604i	−0.964165	+	0.265302i	$0.914528\pi$
$810$		0			0
$811$			30281.1i			1.31112i	0.755145	+	0.655558i	$0.227566\pi$
$811$			30281.1i			1.31112i	−0.755145	+	0.655558i	$0.772434\pi$
$812$	−2918.60			−0.126136
$813$		0			0
$814$	−10034.7			−0.432082
$815$			5064.47i			0.217670i
$816$		0			0
$817$	6373.40	−	16830.2i	0.272922	−	0.720703i
$818$			16894.4i			0.722124i
$819$		0			0
$820$		−	592.514i		−	0.0252335i
$821$		−	13979.7i		−	0.594270i	−0.954836	−	0.297135i	$-0.903969\pi$
$821$		−	13979.7i		−	0.594270i	0.954836	−	0.297135i	$-0.0960311\pi$
$822$		0			0
$823$	29479.0			1.24857			0.624286	−	0.781196i	$-0.285390\pi$
$823$	29479.0			1.24857			0.624286	+	0.781196i	$0.285390\pi$
$824$		−	14253.1i		−	0.602584i
$825$		0			0
$826$	958.216			0.0403639
$827$	5311.72			0.223345			0.111673	−	0.993745i	$-0.464379\pi$
$827$	5311.72			0.223345			0.111673	+	0.993745i	$0.464379\pi$
$828$		0			0
$829$			18971.2i			0.794807i	0.917644	+	0.397404i	$0.130089\pi$
$829$			18971.2i			0.794807i	−0.917644	+	0.397404i	$0.869911\pi$
$830$	13085.4			0.547230
$831$		0			0
$832$			4185.31i			0.174399i
$833$			14557.7i			0.605517i
$834$		0			0
$835$		−	2125.99i		−	0.0881112i
$836$	−4225.63	−	1600.20i	−0.174816	−	0.0662009i
$837$		0			0
$838$		−	10396.0i		−	0.428549i
$839$	17833.1			0.733811			0.366905	−	0.930258i	$-0.380417\pi$
$839$	17833.1			0.733811			0.366905	+	0.930258i	$0.380417\pi$
$840$		0			0
$841$	6501.34			0.266568
$842$			20329.3i			0.832059i
$843$		0			0
$844$			4971.01i			0.202736i
$845$			10397.9i			0.423311i
$846$		0			0
$847$	4753.29			0.192828
$848$	11298.3			0.457531
$849$		0			0
$850$		−	2234.39i		−	0.0901635i
$851$	−28168.5			−1.13467
$852$		0			0
$853$	−38798.0			−1.55735			−0.778674	−	0.627429i	$-0.784107\pi$
$853$	−38798.0			−1.55735			−0.778674	+	0.627429i	$0.784107\pi$
$854$	4999.88			0.200342
$855$		0			0
$856$	9374.42			0.374312
$857$	43650.6			1.73988			0.869939	−	0.493159i	$-0.164158\pi$
$857$	43650.6			1.73988			0.869939	+	0.493159i	$0.164158\pi$
$858$		0			0
$859$	−9559.58			−0.379707			−0.189854	−	0.981812i	$-0.560801\pi$
$859$	−9559.58			−0.379707			−0.189854	+	0.981812i	$0.560801\pi$
$860$		−	4345.99i		−	0.172322i
$861$		0			0
$862$	29137.5			1.15131
$863$	4318.16			0.170327			0.0851633	−	0.996367i	$-0.472859\pi$
$863$	4318.16			0.170327			0.0851633	+	0.996367i	$0.472859\pi$
$864$		0			0
$865$		−	2175.73i		−	0.0855225i
$866$			22906.5i			0.898838i
$867$		0			0
$868$		−	1411.91i		−	0.0552113i
$869$	−561.765			−0.0219293
$870$		0			0
$871$	33654.7			1.30924
$872$		−	4830.24i		−	0.187583i
$873$		0			0
$874$	−11861.8	−	4491.93i	−0.459076	−	0.173847i
$875$			518.935i			0.0200494i
$876$		0			0
$877$			45060.7i			1.73500i	0.497440	+	0.867498i	$0.334273\pi$
$877$			45060.7i			1.73500i	−0.497440	+	0.867498i	$0.665727\pi$
$878$		−	34755.2i		−	1.33591i
$879$		0			0
$880$	−1091.17			−0.0417991
$881$			2580.52i			0.0986832i	0.998782	+	0.0493416i	$0.0157123\pi$
$881$			2580.52i			0.0986832i	−0.998782	+	0.0493416i	$0.984288\pi$
$882$		0			0
$883$	30115.7			1.14776			0.573880	−	0.818939i	$-0.305438\pi$
$883$	30115.7			1.14776			0.573880	+	0.818939i	$0.305438\pi$
$884$	11689.5			0.444753
$885$		0			0
$886$			24710.7i			0.936990i
$887$	40935.9			1.54960			0.774799	−	0.632208i	$-0.217851\pi$
$887$	40935.9			1.54960			0.774799	+	0.632208i	$0.217851\pi$
$888$		0			0
$889$			5297.91i			0.199872i
$890$			6001.86i			0.226048i
$891$		0			0
$892$		−	19696.1i		−	0.739320i
$893$	−47480.5	−	17980.3i	−1.77925	−	0.673782i
$894$		0			0
$895$			6340.33i			0.236798i
$896$	−531.390			−0.0198130
$897$		0			0
$898$	−12570.0			−0.467110
$899$			14943.6i			0.554391i
$900$		0			0
$901$		−	31556.1i		−	1.16680i
$902$			808.165i			0.0298325i
$903$		0			0
$904$	5919.91			0.217802
$905$	6492.75			0.238482
$906$		0			0
$907$			14928.9i			0.546534i	0.961938	+	0.273267i	$0.0881043\pi$
$907$			14928.9i			0.546534i	−0.961938	+	0.273267i	$0.911896\pi$
$908$	7613.56			0.278265
$909$		0			0
$910$	−2714.88			−0.0988984
$911$	21444.8			0.779909			0.389955	−	0.920834i	$-0.372491\pi$
$911$	21444.8			0.779909			0.389955	+	0.920834i	$0.372491\pi$
$912$		0			0
$913$	−17847.9			−0.646967
$914$	−26701.1			−0.966296
$915$		0			0
$916$	−17233.4			−0.621625
$917$		−	4503.12i		−	0.162166i
$918$		0			0
$919$	−19268.9			−0.691647			−0.345824	−	0.938300i	$-0.612400\pi$
$919$	−19268.9			−0.691647			−0.345824	+	0.938300i	$0.612400\pi$
$920$	−3063.03			−0.109766
$921$		0			0
$922$			12879.1i			0.460035i
$923$		−	15237.7i		−	0.543398i
$924$		0			0
$925$		−	9196.27i		−	0.326888i
$926$	10379.5			0.368348
$927$		0			0
$928$	5624.21			0.198948
$929$			48551.4i			1.71466i	0.514765	+	0.857331i	$0.327879\pi$
$929$			48551.4i			1.71466i	−0.514765	+	0.857331i	$0.672121\pi$
$930$		0			0
$931$	−9554.69	+	25231.0i	−0.336351	+	0.888200i
$932$		−	26944.7i		−	0.946998i
$933$		0			0
$934$		−	20471.2i		−	0.717172i
$935$			3047.62i			0.106597i
$936$		0			0
$937$	3206.26			0.111786			0.0558932	−	0.998437i	$-0.482199\pi$
$937$	3206.26			0.111786			0.0558932	+	0.998437i	$0.482199\pi$
$938$			4272.98i			0.148740i
$939$		0			0
$940$	−12260.7			−0.425425
$941$	−27414.2			−0.949711			−0.474856	−	0.880064i	$-0.657500\pi$
$941$	−27414.2			−0.949711			−0.474856	+	0.880064i	$0.657500\pi$
$942$		0			0
$943$			2268.61i			0.0783416i
$944$	−1846.50			−0.0636637
$945$		0			0
$946$			5927.75i			0.203729i
$947$			22829.9i			0.783392i	0.920095	+	0.391696i	$0.128111\pi$
$947$			22829.9i			0.783392i	−0.920095	+	0.391696i	$0.871889\pi$
$948$		0			0
$949$		−	2802.07i		−	0.0958473i
$950$	1466.50	−	3872.58i	0.0500837	−	0.132256i
$951$		0			0
$952$			1484.17i			0.0505274i
$953$	−31171.4			−1.05954			−0.529769	−	0.848142i	$-0.677721\pi$
$953$	−31171.4			−1.05954			−0.529769	+	0.848142i	$0.677721\pi$
$954$		0			0
$955$	−853.952			−0.0289353
$956$		−	22259.0i		−	0.753042i
$957$		0			0
$958$		−	39803.2i		−	1.34236i
$959$		−	9550.07i		−	0.321572i
$960$		0			0
$961$	22561.8			0.757337
$962$	48111.6			1.61245
$963$		0			0
$964$			23625.7i			0.789350i
$965$	−2517.03			−0.0839648
$966$		0			0
$967$	−29863.0			−0.993101			−0.496551	−	0.868008i	$-0.665400\pi$
$967$	−29863.0			−0.993101			−0.496551	+	0.868008i	$0.665400\pi$
$968$	−9159.69			−0.304136
$969$		0			0
$970$	5546.23			0.183586
$971$	17583.6			0.581138			0.290569	−	0.956854i	$-0.406155\pi$
$971$	17583.6			0.581138			0.290569	+	0.956854i	$0.406155\pi$
$972$		0			0
$973$	6131.40			0.202018
$974$			31138.5i			1.02438i
$975$		0			0
$976$	−9634.87			−0.315988
$977$	−41266.1			−1.35130			−0.675649	−	0.737223i	$-0.736137\pi$
$977$	−41266.1			−1.35130			−0.675649	+	0.737223i	$0.736137\pi$
$978$		0			0
$979$		−	8186.30i		−	0.267247i
$980$			6515.30i			0.212371i
$981$		0			0
$982$		−	1597.32i		−	0.0519070i
$983$	−32351.4			−1.04969			−0.524847	−	0.851197i	$-0.675877\pi$
$983$	−32351.4			−1.04969			−0.524847	+	0.851197i	$0.675877\pi$
$984$		0			0
$985$	−5667.49			−0.183331
$986$		−	15708.3i		−	0.507359i
$987$		0			0
$988$	20259.9	+	7672.20i	0.652383	+	0.247050i
$989$			16639.9i			0.535002i
$990$		0			0
$991$		−	31689.1i		−	1.01578i	−0.861422	−	0.507890i	$-0.830426\pi$
$991$		−	31689.1i		−	1.01578i	0.861422	−	0.507890i	$-0.169574\pi$
$992$			2720.79i			0.0870817i
$993$		0			0
$994$	1934.67			0.0617343
$995$			3791.00i			0.120787i
$996$		0			0
$997$	−27589.3			−0.876391			−0.438196	−	0.898880i	$-0.644382\pi$
$997$	−27589.3			−0.876391			−0.438196	+	0.898880i	$0.644382\pi$
$998$	34729.3			1.10154
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.f.a.341.21	✓	40
3.2	odd	2		1710.4.f.b.341.22	yes	40
19.18	odd	2		1710.4.f.b.341.21	yes	40
57.56	even	2	inner	1710.4.f.a.341.22	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.4.f.a.341.21	✓	40	1.1	even	1	trivial
1710.4.f.a.341.22	yes	40	57.56	even	2	inner
1710.4.f.b.341.21	yes	40	19.18	odd	2
1710.4.f.b.341.22	yes	40	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} +4.15148 q^{7} -8.00000 q^{8} +10.0000i q^{10} -13.6396i q^{11} +65.3955i q^{13} -8.30297 q^{14} +16.0000 q^{16} -44.6878i q^{17} +(29.3300 - 77.4516i) q^{19} -20.0000i q^{20} +27.2792i q^{22} +76.5757i q^{23} -25.0000 q^{25} -130.791i q^{26} +16.6059 q^{28} -175.756 q^{29} -85.0245i q^{31} -32.0000 q^{32} +89.3756i q^{34} -20.7574i q^{35} +367.851i q^{37} +(-58.6600 + 154.903i) q^{38} +40.0000i q^{40} +29.6257 q^{41} +217.300 q^{43} -54.5584i q^{44} -153.151i q^{46} -613.034i q^{47} -325.765 q^{49} +50.0000 q^{50} +261.582i q^{52} +706.146 q^{53} -68.1980 q^{55} -33.2119 q^{56} +351.513 q^{58} -115.406 q^{59} -602.179 q^{61} +170.049i q^{62} +64.0000 q^{64} +326.978 q^{65} -514.633i q^{67} -178.751i q^{68} +41.5148i q^{70} -233.009 q^{71} -42.8481 q^{73} -735.702i q^{74} +(117.320 - 309.806i) q^{76} -56.6245i q^{77} -41.1864i q^{79} -80.0000i q^{80} -59.2514 q^{82} -1308.54i q^{83} -223.439 q^{85} -434.599 q^{86} +109.117i q^{88} +600.186 q^{89} +271.488i q^{91} +306.303i q^{92} +1226.07i q^{94} +(-387.258 - 146.650i) q^{95} -554.623i q^{97} +651.530 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) \) 40 * q - 80 * q^2 + 160 * q^4 - 56 * q^7 - 320 * q^8 + 112 * q^14 + 640 * q^16 - 76 * q^19 - 1000 * q^25 - 224 * q^28 - 120 * q^29 - 1280 * q^32 + 152 * q^38 - 312 * q^41 + 56 * q^43 + 2112 * q^49 + 2000 * q^50 + 288 * q^53 + 448 * q^56 + 240 * q^58 + 912 * q^59 - 848 * q^61 + 2560 * q^64 + 240 * q^65 - 1248 * q^71 + 512 * q^73 - 304 * q^76 + 624 * q^82 - 112 * q^86 - 552 * q^89 - 60 * q^95 - 4224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more