LMFDB - Embedded newform 620.3.f.c.309.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [620,3,Mod(309,620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("620.309");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 620 = 2^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	620.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.8937763903$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			309.9
Character	$\chi$	$=$	620.309
Dual form			620.3.f.c.309.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59213 q^{3} +(-2.32008 + 4.42913i) q^{5} -1.60644i q^{7} -6.46514 q^{9} +O(q^{10})$	$q-1.59213 q^{3} +(-2.32008 + 4.42913i) q^{5} -1.60644i q^{7} -6.46514 q^{9} -2.01438i q^{11} -1.88658 q^{13} +(3.69386 - 7.05174i) q^{15} -1.86366 q^{17} -6.62236 q^{19} +2.55765i q^{21} +27.0144 q^{23} +(-14.2344 - 20.5519i) q^{25} +24.6224 q^{27} -45.2074i q^{29} +(-13.7468 + 27.7854i) q^{31} +3.20715i q^{33} +(7.11512 + 3.72706i) q^{35} +40.6209 q^{37} +3.00367 q^{39} -6.90007 q^{41} +47.4345 q^{43} +(14.9996 - 28.6349i) q^{45} -57.3430i q^{47} +46.4194 q^{49} +2.96718 q^{51} -0.0248289 q^{53} +(8.92198 + 4.67354i) q^{55} +10.5436 q^{57} -73.5797 q^{59} -81.6110i q^{61} +10.3858i q^{63} +(4.37701 - 8.35589i) q^{65} -50.0912i q^{67} -43.0104 q^{69} +77.8890 q^{71} +120.168 q^{73} +(22.6630 + 32.7212i) q^{75} -3.23598 q^{77} -106.078i q^{79} +18.9842 q^{81} -105.920 q^{83} +(4.32384 - 8.25439i) q^{85} +71.9758i q^{87} -76.7843i q^{89} +3.03066i q^{91} +(21.8866 - 44.2378i) q^{93} +(15.3644 - 29.3313i) q^{95} -161.362i q^{97} +13.0233i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{5} + 44 q^{9}+O(q^{10}) $ 24 * q - 6 * q^5 + 44 * q^9	$ 24 q - 6 q^{5} + 44 q^{9} + 8 q^{19} - 166 q^{25} - 24 q^{31} + 54 q^{35} - 280 q^{39} - 248 q^{41} - 190 q^{45} - 644 q^{49} - 100 q^{51} + 152 q^{59} - 288 q^{69} - 352 q^{71} - 368 q^{81} + 102 q^{95}+O(q^{100}) $ 24 * q - 6 * q^5 + 44 * q^9 + 8 * q^19 - 166 * q^25 - 24 * q^31 + 54 * q^35 - 280 * q^39 - 248 * q^41 - 190 * q^45 - 644 * q^49 - 100 * q^51 + 152 * q^59 - 288 * q^69 - 352 * q^71 - 368 * q^81 + 102 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/620\mathbb{Z}\right)^\times$.

$n$	$311$	$497$	$561$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.59213			−0.530709			−0.265354	−	0.964151i	$-0.585489\pi$
$3$	−1.59213			−0.530709			−0.265354	+	0.964151i	$0.585489\pi$
$4$		0			0
$5$	−2.32008	+	4.42913i	−0.464016	+	0.885827i
$5$	−2.32008	+	4.42913i	−0.464016	+	0.885827i
$6$		0			0
$7$		−	1.60644i		−	0.229491i	−0.993395	−	0.114745i	$-0.963395\pi$
$7$		−	1.60644i		−	0.229491i	0.993395	−	0.114745i	$-0.0366052\pi$
$8$		0			0
$9$	−6.46514			−0.718348
$10$		0			0
$11$		−	2.01438i		−	0.183126i	−0.995799	−	0.0915630i	$-0.970814\pi$
$11$		−	2.01438i		−	0.183126i	0.995799	−	0.0915630i	$-0.0291863\pi$
$12$		0			0
$13$	−1.88658			−0.145121			−0.0725606	−	0.997364i	$-0.523117\pi$
$13$	−1.88658			−0.145121			−0.0725606	+	0.997364i	$0.523117\pi$
$14$		0			0
$15$	3.69386	−	7.05174i	0.246258	−	0.470116i
$16$		0			0
$17$	−1.86366			−0.109627			−0.0548135	−	0.998497i	$-0.517456\pi$
$17$	−1.86366			−0.109627			−0.0548135	+	0.998497i	$0.517456\pi$
$18$		0			0
$19$	−6.62236			−0.348545			−0.174273	−	0.984697i	$-0.555757\pi$
$19$	−6.62236			−0.348545			−0.174273	+	0.984697i	$0.555757\pi$
$20$		0			0
$21$			2.55765i			0.121793i
$22$		0			0
$23$	27.0144			1.17454			0.587270	−	0.809391i	$-0.300203\pi$
$23$	27.0144			1.17454			0.587270	+	0.809391i	$0.300203\pi$
$24$		0			0
$25$	−14.2344	−	20.5519i	−0.569378	−	0.822076i
$26$		0			0
$27$	24.6224			0.911942
$28$		0			0
$29$		−	45.2074i		−	1.55888i	−0.626480	−	0.779438i	$-0.715505\pi$
$29$		−	45.2074i		−	1.55888i	0.626480	−	0.779438i	$-0.284495\pi$
$30$		0			0
$31$	−13.7468	+	27.7854i	−0.443444	+	0.896302i
$31$	−13.7468	+	27.7854i	−0.443444	+	0.896302i
$32$		0			0
$33$			3.20715i			0.0971865i
$34$		0			0
$35$	7.11512	+	3.72706i	0.203289	+	0.106488i
$36$		0			0
$37$	40.6209			1.09786			0.548932	−	0.835867i	$-0.315035\pi$
$37$	40.6209			1.09786			0.548932	+	0.835867i	$0.315035\pi$
$38$		0			0
$39$	3.00367			0.0770171
$40$		0			0
$41$	−6.90007			−0.168294			−0.0841472	−	0.996453i	$-0.526817\pi$
$41$	−6.90007			−0.168294			−0.0841472	+	0.996453i	$0.526817\pi$
$42$		0			0
$43$	47.4345			1.10313			0.551564	−	0.834133i	$-0.314031\pi$
$43$	47.4345			1.10313			0.551564	+	0.834133i	$0.314031\pi$
$44$		0			0
$45$	14.9996	−	28.6349i	0.333325	−	0.636332i
$46$		0			0
$47$		−	57.3430i		−	1.22006i	−0.792376	−	0.610032i	$-0.791156\pi$
$47$		−	57.3430i		−	1.22006i	0.792376	−	0.610032i	$-0.208844\pi$
$48$		0			0
$49$	46.4194			0.947334
$50$		0			0
$51$	2.96718			0.0581800
$52$		0			0
$53$	−0.0248289			−0.000468470			−0.000234235	−	1.00000i	$-0.500075\pi$
$53$	−0.0248289			−0.000468470			−0.000234235		1.00000i	$0.500075\pi$
$54$		0			0
$55$	8.92198	+	4.67354i	0.162218	+	0.0849734i
$56$		0			0
$57$	10.5436			0.184976
$58$		0			0
$59$	−73.5797			−1.24711			−0.623557	−	0.781778i	$-0.714313\pi$
$59$	−73.5797			−1.24711			−0.623557	+	0.781778i	$0.714313\pi$
$60$		0			0
$61$		−	81.6110i		−	1.33788i	−0.743314	−	0.668942i	$-0.766747\pi$
$61$		−	81.6110i		−	1.33788i	0.743314	−	0.668942i	$-0.233253\pi$
$62$		0			0
$63$			10.3858i			0.164854i
$64$		0			0
$65$	4.37701	−	8.35589i	0.0673386	−	0.128552i
$66$		0			0
$67$		−	50.0912i		−	0.747630i	−0.927503	−	0.373815i	$-0.878049\pi$
$67$		−	50.0912i		−	0.747630i	0.927503	−	0.373815i	$-0.121951\pi$
$68$		0			0
$69$	−43.0104			−0.623339
$70$		0			0
$71$	77.8890			1.09703			0.548514	−	0.836141i	$-0.315194\pi$
$71$	77.8890			1.09703			0.548514	+	0.836141i	$0.315194\pi$
$72$		0			0
$73$	120.168			1.64614			0.823070	−	0.567940i	$-0.192259\pi$
$73$	120.168			1.64614			0.823070	+	0.567940i	$0.192259\pi$
$74$		0			0
$75$	22.6630	+	32.7212i	0.302174	+	0.436283i
$76$		0			0
$77$	−3.23598			−0.0420257
$78$		0			0
$79$		−	106.078i		−	1.34276i	−0.741115	−	0.671378i	$-0.765703\pi$
$79$		−	106.078i		−	1.34276i	0.741115	−	0.671378i	$-0.234297\pi$
$80$		0			0
$81$	18.9842			0.234373
$82$		0			0
$83$	−105.920			−1.27614			−0.638072	−	0.769977i	$-0.720268\pi$
$83$	−105.920			−1.27614			−0.638072	+	0.769977i	$0.720268\pi$
$84$		0			0
$85$	4.32384	−	8.25439i	0.0508687	−	0.0971105i
$86$		0			0
$87$			71.9758i			0.827309i
$88$		0			0
$89$		−	76.7843i		−	0.862745i	−0.902174	−	0.431373i	$-0.858029\pi$
$89$		−	76.7843i		−	0.862745i	0.902174	−	0.431373i	$-0.141971\pi$
$90$		0			0
$91$			3.03066i			0.0333040i
$92$		0			0
$93$	21.8866	−	44.2378i	0.235339	−	0.475675i
$94$		0			0
$95$	15.3644	−	29.3313i	0.161731	−	0.308750i
$96$		0			0
$97$		−	161.362i		−	1.66353i	−0.555130	−	0.831764i	$-0.687332\pi$
$97$		−	161.362i		−	1.66353i	0.555130	−	0.831764i	$-0.312668\pi$
$98$		0			0
$99$			13.0233i			0.131548i
$100$		0			0
$101$	−126.932			−1.25675			−0.628374	−	0.777911i	$-0.716279\pi$
$101$	−126.932			−1.25675			−0.628374	+	0.777911i	$0.716279\pi$
$102$		0			0
$103$			35.2504i			0.342237i	0.985250	+	0.171118i	$0.0547381\pi$
$103$			35.2504i			0.342237i	−0.985250	+	0.171118i	$0.945262\pi$
$104$		0			0
$105$	−11.3282	−	5.93395i	−0.107887	−	0.0565139i
$106$		0			0
$107$			175.315i			1.63846i	0.573465	+	0.819230i	$0.305599\pi$
$107$			175.315i			1.63846i	−0.573465	+	0.819230i	$0.694401\pi$
$108$		0			0
$109$	−15.9398			−0.146237			−0.0731183	−	0.997323i	$-0.523295\pi$
$109$	−15.9398			−0.146237			−0.0731183	+	0.997323i	$0.523295\pi$
$110$		0			0
$111$	−64.6736			−0.582645
$112$		0			0
$113$			56.1741i			0.497116i	0.968617	+	0.248558i	$0.0799566\pi$
$113$			56.1741i			0.497116i	−0.968617	+	0.248558i	$0.920043\pi$
$114$		0			0
$115$	−62.6757	+	119.650i	−0.545006	+	1.04044i
$116$		0			0
$117$	12.1970			0.104248
$118$		0			0
$119$			2.99385i			0.0251584i
$120$		0			0
$121$	116.942			0.966465
$122$		0			0
$123$	10.9858			0.0893153
$124$		0			0
$125$	124.052	−	15.3641i	0.992417	−	0.122913i
$126$		0			0
$127$	−45.5941			−0.359009			−0.179504	−	0.983757i	$-0.557449\pi$
$127$	−45.5941			−0.359009			−0.179504	+	0.983757i	$0.557449\pi$
$128$		0			0
$129$	−75.5217			−0.585440
$130$		0			0
$131$	−44.3911			−0.338863			−0.169432	−	0.985542i	$-0.554193\pi$
$131$	−44.3911			−0.338863			−0.169432	+	0.985542i	$0.554193\pi$
$132$		0			0
$133$			10.6384i			0.0799879i
$134$		0			0
$135$	−57.1261	+	109.056i	−0.423156	+	0.807823i
$136$		0			0
$137$	61.5738			0.449444			0.224722	−	0.974423i	$-0.427853\pi$
$137$	61.5738			0.449444			0.224722	+	0.974423i	$0.427853\pi$
$138$		0			0
$139$		−	117.389i		−	0.844522i	−0.906474	−	0.422261i	$-0.861237\pi$
$139$		−	117.389i		−	0.844522i	0.906474	−	0.422261i	$-0.138763\pi$
$140$		0			0
$141$			91.2973i			0.647499i
$142$		0			0
$143$			3.80029i			0.0265754i
$144$		0			0
$145$	200.230	+	104.885i	1.38089	+	0.723344i
$146$		0			0
$147$	−73.9055			−0.502758
$148$		0			0
$149$	54.8677			0.368240			0.184120	−	0.982904i	$-0.441057\pi$
$149$	54.8677			0.368240			0.184120	+	0.982904i	$0.441057\pi$
$150$		0			0
$151$			232.960i			1.54278i	0.636362	+	0.771391i	$0.280439\pi$
$151$			232.960i			1.54278i	−0.636362	+	0.771391i	$0.719561\pi$
$152$		0			0
$153$	12.0488			0.0787503
$154$		0			0
$155$	−91.1715	−	125.351i	−0.588203	−	0.808713i
$156$		0			0
$157$		−	14.1768i		−	0.0902978i	−0.998980	−	0.0451489i	$-0.985624\pi$
$157$		−	14.1768i		−	0.0902978i	0.998980	−	0.0451489i	$-0.0143762\pi$
$158$		0			0
$159$	0.0395307			0.000248621
$160$		0			0
$161$		−	43.3970i		−	0.269546i
$162$		0			0
$163$			90.5189i			0.555331i	0.960678	+	0.277665i	$0.0895606\pi$
$163$			90.5189i			0.555331i	−0.960678	+	0.277665i	$0.910439\pi$
$164$		0			0
$165$	−14.2049	−	7.44086i	−0.0860904	−	0.0450961i
$166$		0			0
$167$	−185.970			−1.11359			−0.556797	−	0.830649i	$-0.687970\pi$
$167$	−185.970			−1.11359			−0.556797	+	0.830649i	$0.687970\pi$
$168$		0			0
$169$	−165.441			−0.978940
$170$		0			0
$171$	42.8144			0.250377
$172$		0			0
$173$			92.6355i			0.535465i	0.963493	+	0.267733i	$0.0862744\pi$
$173$			92.6355i			0.535465i	−0.963493	+	0.267733i	$0.913726\pi$
$174$		0			0
$175$	−33.0153	+	22.8667i	−0.188659	+	0.130667i
$176$		0			0
$177$	117.148			0.661854
$178$		0			0
$179$			4.90300i			0.0273911i	0.999906	+	0.0136955i	$0.00435956\pi$
$179$			4.90300i			0.0273911i	−0.999906	+	0.0136955i	$0.995640\pi$
$180$		0			0
$181$			41.8556i			0.231246i	0.993293	+	0.115623i	$0.0368865\pi$
$181$			41.8556i			0.231246i	−0.993293	+	0.115623i	$0.963114\pi$
$182$		0			0
$183$			129.935i			0.710027i
$184$		0			0
$185$	−94.2439	+	179.916i	−0.509427	+	0.972516i
$186$		0			0
$187$			3.75412i			0.0200755i
$188$		0			0
$189$		−	39.5544i		−	0.209282i
$190$		0			0
$191$	226.964			1.18829			0.594145	−	0.804358i	$-0.297490\pi$
$191$	226.964			1.18829			0.594145	+	0.804358i	$0.297490\pi$
$192$		0			0
$193$			9.99542i			0.0517898i	0.999665	+	0.0258949i	$0.00824352\pi$
$193$			9.99542i			0.0517898i	−0.999665	+	0.0258949i	$0.991756\pi$
$194$		0			0
$195$	−6.96875	+	13.3036i	−0.0357372	+	0.0682238i
$196$		0			0
$197$	−269.638			−1.36872			−0.684362	−	0.729143i	$-0.739919\pi$
$197$	−269.638			−1.36872			−0.684362	+	0.729143i	$0.739919\pi$
$198$		0			0
$199$		−	206.239i		−	1.03638i	−0.855267	−	0.518188i	$-0.826607\pi$
$199$		−	206.239i		−	1.03638i	0.855267	−	0.518188i	$-0.173393\pi$
$200$		0			0
$201$			79.7515i			0.396774i
$202$		0			0
$203$	−72.6228			−0.357748
$204$		0			0
$205$	16.0087	−	30.5613i	0.0780914	−	0.149080i
$206$		0			0
$207$	−174.652			−0.843729
$208$		0			0
$209$			13.3400i			0.0638276i
$210$		0			0
$211$	116.818			0.553640			0.276820	−	0.960922i	$-0.410719\pi$
$211$	116.818			0.553640			0.276820	+	0.960922i	$0.410719\pi$
$212$		0			0
$213$	−124.009			−0.582202
$214$		0			0
$215$	−110.052	+	210.094i	−0.511869	+	0.977180i
$216$		0			0
$217$	44.6354	+	22.0833i	0.205693	+	0.101766i
$218$		0			0
$219$	−191.323			−0.873621
$220$		0			0
$221$	3.51593			0.0159092
$222$		0			0
$223$	−353.024			−1.58307			−0.791534	−	0.611125i	$-0.790717\pi$
$223$	−353.024			−1.58307			−0.791534	+	0.611125i	$0.790717\pi$
$224$		0			0
$225$	92.0276	+	132.871i	0.409011	+	0.590537i
$226$		0			0
$227$			219.707i			0.967873i	0.875103	+	0.483937i	$0.160793\pi$
$227$			219.707i			0.967873i	−0.875103	+	0.483937i	$0.839207\pi$
$228$		0			0
$229$		−	291.138i		−	1.27135i	−0.771958	−	0.635673i	$-0.780723\pi$
$229$		−	291.138i		−	1.27135i	0.771958	−	0.635673i	$-0.219277\pi$
$230$		0			0
$231$	5.15209			0.0223034
$232$		0			0
$233$		−	318.171i		−	1.36554i	−0.730633	−	0.682770i	$-0.760775\pi$
$233$		−	318.171i		−	1.36554i	0.730633	−	0.682770i	$-0.239225\pi$
$234$		0			0
$235$	253.980	+	133.041i	1.08077	+	0.566130i
$236$		0			0
$237$			168.889i			0.712613i
$238$		0			0
$239$		−	106.228i		−	0.444469i	−0.974993	−	0.222234i	$-0.928665\pi$
$239$		−	106.228i		−	0.444469i	0.974993	−	0.222234i	$-0.0713350\pi$
$240$		0			0
$241$		−	260.338i		−	1.08024i	−0.841588	−	0.540120i	$-0.818379\pi$
$241$		−	260.338i		−	1.08024i	0.841588	−	0.540120i	$-0.181621\pi$
$242$		0			0
$243$	−251.827			−1.03633
$244$		0			0
$245$	−107.697	+	205.598i	−0.439578	+	0.839174i
$246$		0			0
$247$	12.4936			0.0505813
$248$		0			0
$249$	168.638			0.677260
$250$		0			0
$251$			481.362i			1.91778i	0.283784	+	0.958888i	$0.408410\pi$
$251$			481.362i			1.91778i	−0.283784	+	0.958888i	$0.591590\pi$
$252$		0			0
$253$		−	54.4174i		−	0.215089i
$254$		0			0
$255$	−6.88410	+	13.1420i	−0.0269965	+	0.0515374i
$256$		0			0
$257$		−	293.961i		−	1.14382i	−0.820317	−	0.571909i	$-0.806203\pi$
$257$		−	293.961i		−	1.14382i	0.820317	−	0.571909i	$-0.193797\pi$
$258$		0			0
$259$		−	65.2549i		−	0.251950i
$260$		0			0
$261$			292.272i			1.11982i
$262$		0			0
$263$	19.0346			0.0723749			0.0361874	−	0.999345i	$-0.488479\pi$
$263$	19.0346			0.0723749			0.0361874	+	0.999345i	$0.488479\pi$
$264$		0			0
$265$	0.0576051	−	0.109970i	0.000217378	−	0.000414983i
$266$		0			0
$267$			122.250i			0.457866i
$268$		0			0
$269$		−	269.571i		−	1.00212i	−0.865411	−	0.501062i	$-0.832943\pi$
$269$		−	269.571i		−	1.00212i	0.865411	−	0.501062i	$-0.167057\pi$
$270$		0			0
$271$			173.665i			0.640831i	0.947277	+	0.320416i	$0.103823\pi$
$271$			173.665i			0.640831i	−0.947277	+	0.320416i	$0.896177\pi$
$272$		0			0
$273$		−	4.82520i		−	0.0176747i
$274$		0			0
$275$	−41.3994	+	28.6736i	−0.150543	+	0.104268i
$276$		0			0
$277$	92.4335			0.333695			0.166848	−	0.985983i	$-0.446641\pi$
$277$	92.4335			0.333695			0.166848	+	0.985983i	$0.446641\pi$
$278$		0			0
$279$	88.8746	−	179.636i	0.318547	−	0.643857i
$280$		0			0
$281$	−86.6016			−0.308191			−0.154095	−	0.988056i	$-0.549246\pi$
$281$	−86.6016			−0.308191			−0.154095	+	0.988056i	$0.549246\pi$
$282$		0			0
$283$		−	368.011i		−	1.30039i	−0.759767	−	0.650196i	$-0.774687\pi$
$283$		−	368.011i		−	1.30039i	0.759767	−	0.650196i	$-0.225313\pi$
$284$		0			0
$285$	−24.4621	+	46.6991i	−0.0858318	+	0.163857i
$286$		0			0
$287$			11.0845i			0.0386220i
$288$		0			0
$289$	−285.527			−0.987982
$290$		0			0
$291$			256.909i			0.882848i
$292$		0			0
$293$			327.750i			1.11860i	0.828965	+	0.559301i	$0.188930\pi$
$293$			327.750i			1.11860i	−0.828965	+	0.559301i	$0.811070\pi$
$294$		0			0
$295$	170.711	−	325.894i	0.578681	−	1.10473i
$296$		0			0
$297$		−	49.5991i		−	0.167000i
$298$		0			0
$299$	−50.9647			−0.170451
$300$		0			0
$301$		−	76.2005i		−	0.253158i
$302$		0			0
$303$	202.091			0.666967
$304$		0			0
$305$	361.466	+	189.344i	1.18513	+	0.620800i
$306$		0			0
$307$			61.5020i			0.200332i	0.994971	+	0.100166i	$0.0319374\pi$
$307$			61.5020i			0.200332i	−0.994971	+	0.100166i	$0.968063\pi$
$308$		0			0
$309$		−	56.1231i		−	0.181628i
$310$		0			0
$311$	379.929			1.22164			0.610819	−	0.791771i	$-0.290840\pi$
$311$	379.929			1.22164			0.610819	+	0.791771i	$0.290840\pi$
$312$		0			0
$313$	592.072			1.89160			0.945802	−	0.324744i	$-0.105278\pi$
$313$	592.072			1.89160			0.945802	+	0.324744i	$0.105278\pi$
$314$		0			0
$315$	−46.0002	−	24.0960i	−0.146032	−	0.0764951i
$316$		0			0
$317$		−	120.174i		−	0.379098i	−0.981871	−	0.189549i	$-0.939297\pi$
$317$		−	120.174i		−	0.379098i	0.981871	−	0.189549i	$-0.0607025\pi$
$318$		0			0
$319$	−91.0651			−0.285470
$320$		0			0
$321$		−	279.124i		−	0.869545i
$322$		0			0
$323$	12.3418			0.0382099
$324$		0			0
$325$	26.8543	+	38.7727i	0.0826287	+	0.119301i
$326$		0			0
$327$	25.3782			0.0776091
$328$		0			0
$329$	−92.1179			−0.279994
$330$		0			0
$331$		−	174.309i		−	0.526613i	−0.964712	−	0.263306i	$-0.915187\pi$
$331$		−	174.309i		−	0.526613i	0.964712	−	0.263306i	$-0.0848130\pi$
$332$		0			0
$333$	−262.620			−0.788648
$334$		0			0
$335$	221.861	+	116.216i	0.662271	+	0.346913i
$336$		0			0
$337$	286.114			0.849002			0.424501	−	0.905428i	$-0.360450\pi$
$337$	286.114			0.849002			0.424501	+	0.905428i	$0.360450\pi$
$338$		0			0
$339$		−	89.4362i		−	0.263824i
$340$		0			0
$341$	55.9704	+	27.6913i	0.164136	+	0.0812060i
$342$		0			0
$343$		−	153.285i		−	0.446895i
$344$		0			0
$345$	99.7876	−	190.499i	0.289239	−	0.552170i
$346$		0			0
$347$	439.597			1.26685			0.633425	−	0.773804i	$-0.281648\pi$
$347$	439.597			1.26685			0.633425	+	0.773804i	$0.281648\pi$
$348$		0			0
$349$	−434.435			−1.24480			−0.622399	−	0.782700i	$-0.713842\pi$
$349$	−434.435			−1.24480			−0.622399	+	0.782700i	$0.713842\pi$
$350$		0			0
$351$	−46.4521			−0.132342
$352$		0			0
$353$	−39.0955			−0.110752			−0.0553760	−	0.998466i	$-0.517636\pi$
$353$	−39.0955			−0.110752			−0.0553760	+	0.998466i	$0.517636\pi$
$354$		0			0
$355$	−180.709	+	344.981i	−0.509039	+	0.971777i
$356$		0			0
$357$		−	4.76658i		−	0.0133518i
$358$		0			0
$359$	−112.621			−0.313708			−0.156854	−	0.987622i	$-0.550135\pi$
$359$	−112.621			−0.313708			−0.156854	+	0.987622i	$0.550135\pi$
$360$		0			0
$361$	−317.144			−0.878516
$362$		0			0
$363$	−186.187			−0.512911
$364$		0			0
$365$	−278.800	+	532.241i	−0.763836	+	1.45819i
$366$		0			0
$367$	288.059			0.784901			0.392451	−	0.919773i	$-0.371627\pi$
$367$	288.059			0.784901			0.392451	+	0.919773i	$0.371627\pi$
$368$		0			0
$369$	44.6099			0.120894
$370$		0			0
$371$			0.0398860i			0.000107510i
$372$		0			0
$373$		−	276.887i		−	0.742323i	−0.928568	−	0.371162i	$-0.878960\pi$
$373$		−	276.887i		−	0.742323i	0.928568	−	0.371162i	$-0.121040\pi$
$374$		0			0
$375$	−197.507	+	24.4616i	−0.526684	+	0.0652310i
$376$		0			0
$377$			85.2871i			0.226226i
$378$		0			0
$379$	−220.436			−0.581625			−0.290812	−	0.956780i	$-0.593926\pi$
$379$	−220.436			−0.581625			−0.290812	+	0.956780i	$0.593926\pi$
$380$		0			0
$381$	72.5916			0.190529
$382$		0			0
$383$	175.051			0.457053			0.228527	−	0.973538i	$-0.426609\pi$
$383$	175.051			0.457053			0.228527	+	0.973538i	$0.426609\pi$
$384$		0			0
$385$	7.50774	−	14.3326i	0.0195006	−	0.0372275i
$386$		0			0
$387$	−306.670			−0.792430
$388$		0			0
$389$		−	316.942i		−	0.814762i	−0.913258	−	0.407381i	$-0.866442\pi$
$389$		−	316.942i		−	0.814762i	0.913258	−	0.407381i	$-0.133558\pi$
$390$		0			0
$391$	−50.3456			−0.128761
$392$		0			0
$393$	70.6762			0.179838
$394$		0			0
$395$	469.833	+	246.109i	1.18945	+	0.623061i
$396$		0			0
$397$		−	251.079i		−	0.632440i	−0.948686	−	0.316220i	$-0.897586\pi$
$397$		−	251.079i		−	0.632440i	0.948686	−	0.316220i	$-0.102414\pi$
$398$		0			0
$399$		−	16.9377i		−	0.0424503i
$400$		0			0
$401$		−	534.043i		−	1.33178i	−0.746050	−	0.665890i	$-0.768052\pi$
$401$		−	534.043i		−	1.33178i	0.746050	−	0.665890i	$-0.231948\pi$
$402$		0			0
$403$	25.9343	−	52.4192i	0.0643531	−	0.130072i
$404$		0			0
$405$	−44.0449	+	84.0835i	−0.108753	+	0.207614i
$406$		0			0
$407$		−	81.8262i		−	0.201047i
$408$		0			0
$409$		−	651.012i		−	1.59172i	−0.605483	−	0.795859i	$-0.707020\pi$
$409$		−	651.012i		−	1.59172i	0.605483	−	0.795859i	$-0.292980\pi$
$410$		0			0
$411$	−98.0333			−0.238524
$412$		0			0
$413$			118.201i			0.286201i
$414$		0			0
$415$	245.743	−	469.133i	0.592152	−	1.13044i
$416$		0			0
$417$			186.897i			0.448195i
$418$		0			0
$419$	9.97920			0.0238167			0.0119083	−	0.999929i	$-0.496209\pi$
$419$	9.97920			0.0238167			0.0119083	+	0.999929i	$0.496209\pi$
$420$		0			0
$421$	99.2573			0.235765			0.117883	−	0.993028i	$-0.462389\pi$
$421$	99.2573			0.235765			0.117883	+	0.993028i	$0.462389\pi$
$422$		0			0
$423$			370.731i			0.876432i
$424$		0			0
$425$	26.5281	+	38.3017i	0.0624191	+	0.0901217i
$426$		0			0
$427$	−131.103			−0.307032
$428$		0			0
$429$		−	6.05054i		−	0.0141038i
$430$		0			0
$431$	−276.377			−0.641246			−0.320623	−	0.947207i	$-0.603892\pi$
$431$	−276.377			−0.641246			−0.320623	+	0.947207i	$0.603892\pi$
$432$		0			0
$433$	420.217			0.970479			0.485240	−	0.874381i	$-0.338732\pi$
$433$	420.217			0.970479			0.485240	+	0.874381i	$0.338732\pi$
$434$		0			0
$435$	−318.791	−	166.990i	−0.732852	−	0.383885i
$436$		0			0
$437$	−178.899			−0.409380
$438$		0			0
$439$	261.937			0.596669			0.298334	−	0.954461i	$-0.403569\pi$
$439$	261.937			0.596669			0.298334	+	0.954461i	$0.403569\pi$
$440$		0			0
$441$	−300.107			−0.680516
$442$		0			0
$443$		−	673.290i		−	1.51984i	−0.650015	−	0.759921i	$-0.725237\pi$
$443$		−	673.290i		−	1.51984i	0.650015	−	0.759921i	$-0.274763\pi$
$444$		0			0
$445$	340.088	+	178.146i	0.764243	+	0.400328i
$446$		0			0
$447$	−87.3563			−0.195428
$448$		0			0
$449$		−	358.542i		−	0.798534i	−0.916835	−	0.399267i	$-0.869265\pi$
$449$		−	358.542i		−	0.798534i	0.916835	−	0.399267i	$-0.130735\pi$
$450$		0			0
$451$			13.8994i			0.0308191i
$452$		0			0
$453$		−	370.902i		−	0.818768i
$454$		0			0
$455$	−13.4232	−	7.03139i	−0.0295016	−	0.0154536i
$456$		0			0
$457$	525.488			1.14986			0.574932	−	0.818201i	$-0.305028\pi$
$457$	525.488			1.14986			0.574932	+	0.818201i	$0.305028\pi$
$458$		0			0
$459$	−45.8878			−0.0999734
$460$		0			0
$461$			101.046i			0.219189i	0.993976	+	0.109594i	$0.0349552\pi$
$461$			101.046i			0.219189i	−0.993976	+	0.109594i	$0.965045\pi$
$462$		0			0
$463$	31.6061			0.0682638			0.0341319	−	0.999417i	$-0.489133\pi$
$463$	31.6061			0.0682638			0.0341319	+	0.999417i	$0.489133\pi$
$464$		0			0
$465$	145.157	+	199.574i	0.312165	+	0.429191i
$466$		0			0
$467$		−	204.987i		−	0.438944i	−0.975619	−	0.219472i	$-0.929567\pi$
$467$		−	204.987i		−	0.438944i	0.975619	−	0.219472i	$-0.0704334\pi$
$468$		0			0
$469$	−80.4683			−0.171574
$470$		0			0
$471$			22.5712i			0.0479218i
$472$		0			0
$473$		−	95.5513i		−	0.202011i
$474$		0			0
$475$	94.2655	+	136.102i	0.198454	+	0.286531i
$476$		0			0
$477$	0.160522			0.000336525
$478$		0			0
$479$	−44.6366			−0.0931870			−0.0465935	−	0.998914i	$-0.514837\pi$
$479$	−44.6366			−0.0931870			−0.0465935	+	0.998914i	$0.514837\pi$
$480$		0			0
$481$	−76.6345			−0.159323
$482$		0			0
$483$			69.0934i			0.143051i
$484$		0			0
$485$	714.694	+	374.373i	1.47360	+	0.771904i
$486$		0			0
$487$	487.179			1.00037			0.500184	−	0.865919i	$-0.333266\pi$
$487$	487.179			1.00037			0.500184	+	0.865919i	$0.333266\pi$
$488$		0			0
$489$		−	144.117i		−	0.294719i
$490$		0			0
$491$		−	85.9961i		−	0.175145i	−0.996158	−	0.0875724i	$-0.972089\pi$
$491$		−	85.9961i		−	0.175145i	0.996158	−	0.0875724i	$-0.0279109\pi$
$492$		0			0
$493$			84.2511i			0.170895i
$494$		0			0
$495$	−57.6818	−	30.2151i	−0.116529	−	0.0610405i
$496$		0			0
$497$		−	125.124i		−	0.251758i
$498$		0			0
$499$		−	500.280i		−	1.00256i	−0.865284	−	0.501282i	$-0.832862\pi$
$499$		−	500.280i		−	1.00256i	0.865284	−	0.501282i	$-0.167138\pi$
$500$		0			0
$501$	296.088			0.590994
$502$		0			0
$503$		−	149.584i		−	0.297384i	−0.988884	−	0.148692i	$-0.952494\pi$
$503$		−	149.584i		−	0.297384i	0.988884	−	0.148692i	$-0.0475062\pi$
$504$		0			0
$505$	294.492	−	562.197i	0.583152	−	1.11326i
$506$		0			0
$507$	263.403			0.519532
$508$		0			0
$509$			575.053i			1.12977i	0.825170	+	0.564885i	$0.191079\pi$
$509$			575.053i			1.12977i	−0.825170	+	0.564885i	$0.808921\pi$
$510$		0			0
$511$		−	193.043i		−	0.377774i
$512$		0			0
$513$	−163.059			−0.317853
$514$		0			0
$515$	−156.129	−	81.7838i	−0.303163	−	0.158804i
$516$		0			0
$517$	−115.511			−0.223425
$518$		0			0
$519$		−	147.487i		−	0.284176i
$520$		0			0
$521$	−525.572			−1.00878			−0.504388	−	0.863477i	$-0.668282\pi$
$521$	−525.572			−1.00878			−0.504388	+	0.863477i	$0.668282\pi$
$522$		0			0
$523$	788.978			1.50856			0.754281	−	0.656551i	$-0.227985\pi$
$523$	788.978			1.50856			0.754281	+	0.656551i	$0.227985\pi$
$524$		0			0
$525$	52.5646	−	36.4067i	0.100123	−	0.0693461i
$526$		0			0
$527$	25.6192	−	51.7824i	0.0486134	−	0.0982589i
$528$		0			0
$529$	200.779			0.379545
$530$		0			0
$531$	475.703			0.895862
$532$		0			0
$533$	13.0175			0.0244231
$534$		0			0
$535$	−776.494	−	406.746i	−1.45139	−	0.760272i
$536$		0			0
$537$		−	7.80620i		−	0.0145367i
$538$		0			0
$539$		−	93.5065i		−	0.173481i
$540$		0			0
$541$	562.368			1.03950			0.519749	−	0.854319i	$-0.326026\pi$
$541$	562.368			1.03950			0.519749	+	0.854319i	$0.326026\pi$
$542$		0			0
$543$		−	66.6394i		−	0.122724i
$544$		0			0
$545$	36.9816	−	70.5995i	0.0678562	−	0.129540i
$546$		0			0
$547$			389.572i			0.712197i	0.934448	+	0.356098i	$0.115893\pi$
$547$			389.572i			0.712197i	−0.934448	+	0.356098i	$0.884107\pi$
$548$		0			0
$549$			527.626i			0.961067i
$550$		0			0
$551$			299.379i			0.543338i
$552$		0			0
$553$	−170.407			−0.308150
$554$		0			0
$555$	150.048	−	286.448i	0.270357	−	0.516123i
$556$		0			0
$557$	−452.326			−0.812075			−0.406038	−	0.913856i	$-0.633090\pi$
$557$	−452.326			−0.812075			−0.406038	+	0.913856i	$0.633090\pi$
$558$		0			0
$559$	−89.4888			−0.160087
$560$		0			0
$561$		−	5.97704i		−	0.0106543i
$562$		0			0
$563$			613.424i			1.08956i	0.838578	+	0.544781i	$0.183387\pi$
$563$			613.424i			1.08956i	−0.838578	+	0.544781i	$0.816613\pi$
$564$		0			0
$565$	−248.802	−	130.328i	−0.440358	−	0.230670i
$566$		0			0
$567$		−	30.4969i		−	0.0537864i
$568$		0			0
$569$			606.320i			1.06559i	0.846245	+	0.532794i	$0.178858\pi$
$569$			606.320i			1.06559i	−0.846245	+	0.532794i	$0.821142\pi$
$570$		0			0
$571$			506.776i			0.887523i	0.896145	+	0.443762i	$0.146356\pi$
$571$			506.776i			0.887523i	−0.896145	+	0.443762i	$0.853644\pi$
$572$		0			0
$573$	−361.354			−0.630636
$574$		0			0
$575$	−384.535	−	555.198i	−0.668757	−	0.965561i
$576$		0			0
$577$		−	921.880i		−	1.59771i	−0.601522	−	0.798856i	$-0.705439\pi$
$577$		−	921.880i		−	1.59771i	0.601522	−	0.798856i	$-0.294561\pi$
$578$		0			0
$579$		−	15.9140i		−	0.0274853i
$580$		0			0
$581$			170.154i			0.292863i
$582$		0			0
$583$			0.0500150i			8.57890e-5i
$584$		0			0
$585$	−28.2980	+	54.0220i	−0.0483726	+	0.0923453i
$586$		0			0
$587$	−20.5955			−0.0350861			−0.0175430	−	0.999846i	$-0.505584\pi$
$587$	−20.5955			−0.0350861			−0.0175430	+	0.999846i	$0.505584\pi$
$588$		0			0
$589$	91.0359	−	184.005i	0.154560	−	0.312402i
$590$		0			0
$591$	429.298			0.726393
$592$		0			0
$593$		−	27.7290i		−	0.0467606i	−0.999727	−	0.0233803i	$-0.992557\pi$
$593$		−	27.7290i		−	0.0467606i	0.999727	−	0.0233803i	$-0.00744286\pi$
$594$		0			0
$595$	−13.2601	−	6.94597i	−0.0222860	−	0.0116739i
$596$		0			0
$597$			328.358i			0.550014i
$598$		0			0
$599$	120.595			0.201327			0.100664	−	0.994921i	$-0.467903\pi$
$599$	120.595			0.201327			0.100664	+	0.994921i	$0.467903\pi$
$600$		0			0
$601$		−	339.888i		−	0.565538i	−0.959188	−	0.282769i	$-0.908747\pi$
$601$		−	339.888i		−	0.565538i	0.959188	−	0.282769i	$-0.0912530\pi$
$602$		0			0
$603$			323.846i			0.537059i
$604$		0			0
$605$	−271.316	+	517.953i	−0.448456	+	0.856120i
$606$		0			0
$607$			473.516i			0.780092i	0.920795	+	0.390046i	$0.127541\pi$
$607$			473.516i			0.780092i	−0.920795	+	0.390046i	$0.872459\pi$
$608$		0			0
$609$	115.625			0.189860
$610$		0			0
$611$			108.182i			0.177057i
$612$		0			0
$613$	452.885			0.738801			0.369400	−	0.929270i	$-0.379563\pi$
$613$	452.885			0.738801			0.369400	+	0.929270i	$0.379563\pi$
$614$		0			0
$615$	−25.4879	+	48.6575i	−0.0414438	+	0.0791179i
$616$		0			0
$617$			416.979i			0.675817i	0.941179	+	0.337909i	$0.109719\pi$
$617$			416.979i			0.675817i	−0.941179	+	0.337909i	$0.890281\pi$
$618$		0			0
$619$		−	281.767i		−	0.455197i	−0.973755	−	0.227599i	$-0.926913\pi$
$619$		−	281.767i		−	0.455197i	0.973755	−	0.227599i	$-0.0730874\pi$
$620$		0			0
$621$	665.161			1.07111
$622$		0			0
$623$	−123.349			−0.197992
$624$		0			0
$625$	−219.761	+	585.090i	−0.351618	+	0.936143i
$626$		0			0
$627$		−	21.2389i		−	0.0338739i
$628$		0			0
$629$	−75.7035			−0.120355
$630$		0			0
$631$			323.375i			0.512480i	0.966613	+	0.256240i	$0.0824837\pi$
$631$			323.375i			0.512480i	−0.966613	+	0.256240i	$0.917516\pi$
$632$		0			0
$633$	−185.989			−0.293822
$634$		0			0
$635$	105.782	−	201.942i	0.166586	−	0.318020i
$636$		0			0
$637$	−87.5736			−0.137478
$638$		0			0
$639$	−503.563			−0.788048
$640$		0			0
$641$		−	22.6197i		−	0.0352882i	−0.999844	−	0.0176441i	$-0.994383\pi$
$641$		−	22.6197i		−	0.0352882i	0.999844	−	0.0176441i	$-0.00561658\pi$
$642$		0			0
$643$	−461.777			−0.718161			−0.359080	−	0.933307i	$-0.616910\pi$
$643$	−461.777			−0.718161			−0.359080	+	0.933307i	$0.616910\pi$
$644$		0			0
$645$	175.217	−	334.496i	0.271654	−	0.518598i
$646$		0			0
$647$	−689.341			−1.06544			−0.532721	−	0.846291i	$-0.678830\pi$
$647$	−689.341			−1.06544			−0.532721	+	0.846291i	$0.678830\pi$
$648$		0			0
$649$			148.218i			0.228379i
$650$		0			0
$651$	−71.0652	−	35.1594i	−0.109163	−	0.0540082i
$652$		0			0
$653$			572.698i			0.877026i	0.898725	+	0.438513i	$0.144495\pi$
$653$			572.698i			0.877026i	−0.898725	+	0.438513i	$0.855505\pi$
$654$		0			0
$655$	102.991	−	196.614i	0.157238	−	0.300174i
$656$		0			0
$657$	−776.904			−1.18250
$658$		0			0
$659$	228.580			0.346859			0.173429	−	0.984846i	$-0.444515\pi$
$659$	228.580			0.346859			0.173429	+	0.984846i	$0.444515\pi$
$660$		0			0
$661$	−483.184			−0.730990			−0.365495	−	0.930813i	$-0.619100\pi$
$661$	−483.184			−0.730990			−0.365495	+	0.930813i	$0.619100\pi$
$662$		0			0
$663$	−5.59780			−0.00844314
$664$		0			0
$665$	−47.1189	−	24.6819i	−0.0708554	−	0.0371157i
$666$		0			0
$667$		−	1221.25i		−	1.83096i
$668$		0			0
$669$	562.059			0.840148
$670$		0			0
$671$	−164.396			−0.245001
$672$		0			0
$673$	139.495			0.207273			0.103637	−	0.994615i	$-0.466952\pi$
$673$	139.495			0.207273			0.103637	+	0.994615i	$0.466952\pi$
$674$		0			0
$675$	−350.487	−	506.038i	−0.519239	−	0.749686i
$676$		0			0
$677$	−613.264			−0.905856			−0.452928	−	0.891547i	$-0.649620\pi$
$677$	−613.264			−0.905856			−0.452928	+	0.891547i	$0.649620\pi$
$678$		0			0
$679$	−259.218			−0.381764
$680$		0			0
$681$		−	349.802i		−	0.513659i
$682$		0			0
$683$			530.112i			0.776153i	0.921627	+	0.388076i	$0.126860\pi$
$683$			530.112i			0.776153i	−0.921627	+	0.388076i	$0.873140\pi$
$684$		0			0
$685$	−142.856	+	272.719i	−0.208549	+	0.398129i
$686$		0			0
$687$			463.529i			0.674715i
$688$		0			0
$689$	0.0468416			6.79849e−5
$690$		0			0
$691$	−992.386			−1.43616			−0.718079	−	0.695961i	$-0.754979\pi$
$691$	−992.386			−1.43616			−0.718079	+	0.695961i	$0.754979\pi$
$692$		0			0
$693$	20.9211			0.0301891
$694$		0			0
$695$	519.930	+	272.351i	0.748100	+	0.391872i
$696$		0			0
$697$	12.8594			0.0184496
$698$		0			0
$699$			506.568i			0.724704i
$700$		0			0
$701$	−585.290			−0.834936			−0.417468	−	0.908692i	$-0.637082\pi$
$701$	−585.290			−0.834936			−0.417468	+	0.908692i	$0.637082\pi$
$702$		0			0
$703$	−269.006			−0.382655
$704$		0			0
$705$	−404.368	−	211.817i	−0.573572	−	0.300450i
$706$		0			0
$707$			203.907i			0.288412i
$708$		0			0
$709$			971.026i			1.36957i	0.728745	+	0.684786i	$0.240104\pi$
$709$			971.026i			1.36957i	−0.728745	+	0.684786i	$0.759896\pi$
$710$		0			0
$711$			685.807i			0.964567i
$712$		0			0
$713$	−371.361	+	750.606i	−0.520842	+	1.05274i
$714$		0			0
$715$	−16.8320	−	8.81698i	−0.0235412	−	0.0123314i
$716$		0			0
$717$			169.128i			0.235883i
$718$		0			0
$719$			566.411i			0.787776i	0.919158	+	0.393888i	$0.128870\pi$
$719$			566.411i			0.787776i	−0.919158	+	0.393888i	$0.871130\pi$
$720$		0			0
$721$	56.6275			0.0785402
$722$		0			0
$723$			414.490i			0.573292i
$724$		0			0
$725$	−929.098	+	643.502i	−1.28151	+	0.887589i
$726$		0			0
$727$			509.485i			0.700804i	0.936599	+	0.350402i	$0.113955\pi$
$727$			509.485i			0.700804i	−0.936599	+	0.350402i	$0.886045\pi$
$728$		0			0
$729$	230.083			0.315614
$730$		0			0
$731$	−88.4017			−0.120933
$732$		0			0
$733$			771.841i			1.05299i	0.850179	+	0.526494i	$0.176494\pi$
$733$			771.841i			1.05299i	−0.850179	+	0.526494i	$0.823506\pi$
$734$		0			0
$735$	171.467	−	327.337i	0.233288	−	0.445357i
$736$		0			0
$737$	−100.903			−0.136910
$738$		0			0
$739$			276.135i			0.373661i	0.982392	+	0.186830i	$0.0598215\pi$
$739$			276.135i			0.373661i	−0.982392	+	0.186830i	$0.940178\pi$
$740$		0			0
$741$	−19.8913			−0.0268439
$742$		0			0
$743$	−477.767			−0.643024			−0.321512	−	0.946905i	$-0.604191\pi$
$743$	−477.767			−0.643024			−0.321512	+	0.946905i	$0.604191\pi$
$744$		0			0
$745$	−127.298	+	243.016i	−0.170869	+	0.326196i
$746$		0			0
$747$	684.787			0.916716
$748$		0			0
$749$	281.633			0.376012
$750$		0			0
$751$	443.995			0.591205			0.295602	−	0.955311i	$-0.404480\pi$
$751$	443.995			0.591205			0.295602	+	0.955311i	$0.404480\pi$
$752$		0			0
$753$		−	766.389i		−	1.01778i
$754$		0			0
$755$	−1031.81	−	540.486i	−1.36664	−	0.715876i
$756$		0			0
$757$	686.038			0.906259			0.453130	−	0.891445i	$-0.350307\pi$
$757$	686.038			0.906259			0.453130	+	0.891445i	$0.350307\pi$
$758$		0			0
$759$			86.6394i			0.114149i
$760$		0			0
$761$		−	923.358i		−	1.21335i	−0.794951	−	0.606674i	$-0.792503\pi$
$761$		−	923.358i		−	1.21335i	0.794951	−	0.606674i	$-0.207497\pi$
$762$		0			0
$763$			25.6063i			0.0335600i
$764$		0			0
$765$	−27.9542	+	53.3657i	−0.0365414	+	0.0697591i
$766$		0			0
$767$	138.814			0.180983
$768$		0			0
$769$	1113.70			1.44824			0.724120	−	0.689674i	$-0.242246\pi$
$769$	1113.70			1.44824			0.724120	+	0.689674i	$0.242246\pi$
$770$		0			0
$771$			468.023i			0.607034i
$772$		0			0
$773$	−1285.39			−1.66285			−0.831427	−	0.555634i	$-0.812476\pi$
$773$	−1285.39			−1.66285			−0.831427	+	0.555634i	$0.812476\pi$
$774$		0			0
$775$	766.720	−	112.987i	0.989316	−	0.145790i
$776$		0			0
$777$			103.894i			0.133712i
$778$		0			0
$779$	45.6947			0.0586582
$780$		0			0
$781$		−	156.898i		−	0.200894i
$782$		0			0
$783$		−	1113.12i		−	1.42160i
$784$		0			0
$785$	62.7907	+	32.8912i	0.0799882	+	0.0418997i
$786$		0			0
$787$	−590.499			−0.750316			−0.375158	−	0.926961i	$-0.622412\pi$
$787$	−590.499			−0.750316			−0.375158	+	0.926961i	$0.622412\pi$
$788$		0			0
$789$	−30.3055			−0.0384100
$790$		0			0
$791$	90.2401			0.114084
$792$		0			0
$793$			153.965i			0.194155i
$794$		0			0
$795$	−0.0917145	+	0.175087i	−0.000115364	+	0.000220235i
$796$		0			0
$797$	−1170.71			−1.46890			−0.734450	−	0.678663i	$-0.762560\pi$
$797$	−1170.71			−1.46890			−0.734450	+	0.678663i	$0.762560\pi$
$798$		0			0
$799$			106.868i			0.133752i
$800$		0			0
$801$			496.421i			0.619751i
$802$		0			0
$803$		−	242.065i		−	0.301451i
$804$		0			0
$805$	192.211	+	100.684i	0.238771	+	0.125074i
$806$		0			0
$807$			429.191i			0.531836i
$808$		0			0
$809$			1543.31i			1.90767i	0.300327	+	0.953836i	$0.402904\pi$
$809$			1543.31i			1.90767i	−0.300327	+	0.953836i	$0.597096\pi$
$810$		0			0
$811$	1112.40			1.37164			0.685818	−	0.727773i	$-0.259445\pi$
$811$	1112.40			1.37164			0.685818	+	0.727773i	$0.259445\pi$
$812$		0			0
$813$		−	276.497i		−	0.340095i
$814$		0			0
$815$	−400.920	−	210.011i	−0.491927	−	0.257683i
$816$		0			0
$817$	−314.128			−0.384490
$818$		0			0
$819$		−	19.5936i		−	0.0239239i
$820$		0			0
$821$		−	417.348i		−	0.508341i	−0.967159	−	0.254170i	$-0.918198\pi$
$821$		−	417.348i		−	0.508341i	0.967159	−	0.254170i	$-0.0818024\pi$
$822$		0			0
$823$	683.988			0.831091			0.415546	−	0.909572i	$-0.363591\pi$
$823$	683.988			0.831091			0.415546	+	0.909572i	$0.363591\pi$
$824$		0			0
$825$	65.9131	−	45.6520i	0.0798947	−	0.0553358i
$826$		0			0
$827$	905.076			1.09441			0.547204	−	0.836999i	$-0.315692\pi$
$827$	905.076			1.09441			0.547204	+	0.836999i	$0.315692\pi$
$828$		0			0
$829$			1096.06i			1.32214i	0.750323	+	0.661072i	$0.229898\pi$
$829$			1096.06i			1.32214i	−0.750323	+	0.661072i	$0.770102\pi$
$830$		0			0
$831$	−147.166			−0.177095
$832$		0			0
$833$	−86.5098			−0.103853
$834$		0			0
$835$	431.466	−	823.686i	0.516726	−	0.986451i
$836$		0			0
$837$	−338.479	+	684.144i	−0.404395	+	0.817376i
$838$		0			0
$839$	−701.813			−0.836488			−0.418244	−	0.908335i	$-0.637354\pi$
$839$	−701.813			−0.836488			−0.418244	+	0.908335i	$0.637354\pi$
$840$		0			0
$841$	−1202.71			−1.43009
$842$		0			0
$843$	137.881			0.163559
$844$		0			0
$845$	383.836	−	732.759i	0.454244	−	0.867171i
$846$		0			0
$847$		−	187.860i		−	0.221795i
$848$		0			0
$849$			585.919i			0.690129i
$850$		0			0
$851$	1097.35			1.28948
$852$		0			0
$853$			575.751i			0.674972i	0.941331	+	0.337486i	$0.109577\pi$
$853$			575.751i			0.674972i	−0.941331	+	0.337486i	$0.890423\pi$
$854$		0			0
$855$	−99.3330	+	189.631i	−0.116179	+	0.221790i
$856$		0			0
$857$			950.070i			1.10860i	0.832317	+	0.554300i	$0.187014\pi$
$857$			950.070i			1.10860i	−0.832317	+	0.554300i	$0.812986\pi$
$858$		0			0
$859$		−	15.9064i		−	0.0185174i	−0.999957	−	0.00925870i	$-0.997053\pi$
$859$		−	15.9064i		−	0.0185174i	0.999957	−	0.00925870i	$-0.00294718\pi$
$860$		0			0
$861$		−	17.6480i		−	0.0204970i
$862$		0			0
$863$	1171.72			1.35772			0.678862	−	0.734266i	$-0.262474\pi$
$863$	1171.72			1.35772			0.678862	+	0.734266i	$0.262474\pi$
$864$		0			0
$865$	−410.295	−	214.922i	−0.474330	−	0.248465i
$866$		0			0
$867$	454.595			0.524331
$868$		0			0
$869$	−213.681			−0.245894
$870$		0			0
$871$			94.5008i			0.108497i
$872$		0			0
$873$			1043.23i			1.19499i
$874$		0			0
$875$	−24.6815	−	199.282i	−0.0282074	−	0.227751i
$876$		0			0
$877$		−	1211.65i		−	1.38158i	−0.723054	−	0.690792i	$-0.757262\pi$
$877$		−	1211.65i		−	1.38158i	0.723054	−	0.690792i	$-0.242738\pi$
$878$		0			0
$879$		−	521.820i		−	0.593652i
$880$		0			0
$881$		−	1517.57i		−	1.72255i	−0.508136	−	0.861277i	$-0.669665\pi$
$881$		−	1517.57i		−	1.72255i	0.508136	−	0.861277i	$-0.330335\pi$
$882$		0			0
$883$	323.245			0.366076			0.183038	−	0.983106i	$-0.441407\pi$
$883$	323.245			0.366076			0.183038	+	0.983106i	$0.441407\pi$
$884$		0			0
$885$	−271.793	+	518.865i	−0.307111	+	0.586288i
$886$		0			0
$887$		−	1452.08i		−	1.63707i	−0.574455	−	0.818536i	$-0.694786\pi$
$887$		−	1452.08i		−	1.63707i	0.574455	−	0.818536i	$-0.305214\pi$
$888$		0			0
$889$			73.2441i			0.0823893i
$890$		0			0
$891$		−	38.2415i		−	0.0429197i
$892$		0			0
$893$			379.746i			0.425248i
$894$		0			0
$895$	−21.7161	−	11.3754i	−0.0242638	−	0.0127099i
$896$		0			0
$897$	81.1423			0.0904596
$898$		0			0
$899$	1256.10	+	621.455i	1.39722	+	0.691273i
$900$		0			0
$901$	0.0462726			5.13569e−5
$902$		0			0
$903$			121.321i			0.134353i
$904$		0			0
$905$	−185.384	−	97.1084i	−0.204844	−	0.107302i
$906$		0			0
$907$		−	1110.78i		−	1.22467i	−0.790597	−	0.612337i	$-0.790230\pi$
$907$		−	1110.78i		−	1.22467i	0.790597	−	0.612337i	$-0.209770\pi$
$908$		0			0
$909$	820.630			0.902783
$910$		0			0
$911$		−	1576.75i		−	1.73079i	−0.501093	−	0.865393i	$-0.667069\pi$
$911$		−	1576.75i		−	1.73079i	0.501093	−	0.865393i	$-0.332931\pi$
$912$		0			0
$913$			213.364i			0.233695i
$914$		0			0
$915$	−575.499	−	301.460i	−0.628961	−	0.329464i
$916$		0			0
$917$			71.3115i			0.0777661i
$918$		0			0
$919$	1556.21			1.69338			0.846689	−	0.532088i	$-0.178592\pi$
$919$	1556.21			1.69338			0.846689	+	0.532088i	$0.178592\pi$
$920$		0			0
$921$		−	97.9188i		−	0.106318i
$922$		0			0
$923$	−146.943			−0.159202
$924$		0			0
$925$	−578.216	−	834.838i	−0.625099	−	0.902527i
$926$		0			0
$927$		−	227.899i		−	0.245845i
$928$		0			0
$929$			740.277i			0.796853i	0.917200	+	0.398427i	$0.130444\pi$
$929$			740.277i			0.796853i	−0.917200	+	0.398427i	$0.869556\pi$
$930$		0			0
$931$	−307.406			−0.330189
$932$		0			0
$933$	−604.895			−0.648333
$934$		0			0
$935$	−16.6275	−	8.70988i	−0.0177834	−	0.00931538i
$936$		0			0
$937$		−	1038.48i		−	1.10830i	−0.832417	−	0.554149i	$-0.813044\pi$
$937$		−	1038.48i		−	1.10830i	0.832417	−	0.554149i	$-0.186956\pi$
$938$		0			0
$939$	−942.653			−1.00389
$940$		0			0
$941$			146.282i			0.155454i	0.996975	+	0.0777271i	$0.0247663\pi$
$941$			146.282i			0.155454i	−0.996975	+	0.0777271i	$0.975234\pi$
$942$		0			0
$943$	−186.401			−0.197669
$944$		0			0
$945$	175.192	+	91.7694i	0.185388	+	0.0971105i
$946$		0			0
$947$	−185.321			−0.195692			−0.0978462	−	0.995202i	$-0.531195\pi$
$947$	−185.321			−0.195692			−0.0978462	+	0.995202i	$0.531195\pi$
$948$		0			0
$949$	−226.706			−0.238890
$950$		0			0
$951$			191.332i			0.201190i
$952$		0			0
$953$	655.282			0.687599			0.343799	−	0.939043i	$-0.388286\pi$
$953$	655.282			0.687599			0.343799	+	0.939043i	$0.388286\pi$
$954$		0			0
$955$	−526.574	+	1005.25i	−0.551386	+	1.05262i
$956$		0			0
$957$	144.987			0.151502
$958$		0			0
$959$		−	98.9144i		−	0.103143i
$960$		0			0
$961$	−583.053	−	763.917i	−0.606715	−	0.794919i
$962$		0			0
$963$		−	1133.44i		−	1.17698i
$964$		0			0
$965$	−44.2711	−	23.1902i	−0.0458767	−	0.0240313i
$966$		0			0
$967$	−306.500			−0.316960			−0.158480	−	0.987362i	$-0.550659\pi$
$967$	−306.500			−0.316960			−0.158480	+	0.987362i	$0.550659\pi$
$968$		0			0
$969$	−19.6497			−0.0202783
$970$		0			0
$971$	−521.154			−0.536719			−0.268359	−	0.963319i	$-0.586481\pi$
$971$	−521.154			−0.536719			−0.268359	+	0.963319i	$0.586481\pi$
$972$		0			0
$973$	−188.577			−0.193810
$974$		0			0
$975$	−42.7555	−	61.7310i	−0.0438518	−	0.0633139i
$976$		0			0
$977$			957.638i			0.980183i	0.871671	+	0.490091i	$0.163037\pi$
$977$			957.638i			0.980183i	−0.871671	+	0.490091i	$0.836963\pi$
$978$		0			0
$979$	−154.673			−0.157991
$980$		0			0
$981$	103.053			0.105049
$982$		0			0
$983$	1000.24			1.01754			0.508771	−	0.860902i	$-0.330100\pi$
$983$	1000.24			1.01754			0.508771	+	0.860902i	$0.330100\pi$
$984$		0			0
$985$	625.583	−	1194.26i	0.635110	−	1.21245i
$986$		0			0
$987$	146.663			0.148595
$988$		0			0
$989$	1281.42			1.29567
$990$		0			0
$991$			755.163i			0.762021i	0.924571	+	0.381011i	$0.124424\pi$
$991$			755.163i			0.762021i	−0.924571	+	0.381011i	$0.875576\pi$
$992$		0			0
$993$			277.521i			0.279478i
$994$		0			0
$995$	913.459	+	478.491i	0.918050	+	0.480896i
$996$		0			0
$997$			1913.31i			1.91907i	0.281598	+	0.959533i	$0.409136\pi$
$997$			1913.31i			1.91907i	−0.281598	+	0.959533i	$0.590864\pi$
$998$		0			0
$999$	1000.19			1.00119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	620.3.f.c.309.9	✓	24
5.2	odd	4		3100.3.d.h.1301.12		24
5.3	odd	4		3100.3.d.h.1301.13		24
5.4	even	2	inner	620.3.f.c.309.16	yes	24
31.30	odd	2	inner	620.3.f.c.309.15	yes	24
155.92	even	4		3100.3.d.h.1301.11		24
155.123	even	4		3100.3.d.h.1301.14		24
155.154	odd	2	inner	620.3.f.c.309.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
620.3.f.c.309.9	✓	24	1.1	even	1	trivial
620.3.f.c.309.10	yes	24	155.154	odd	2	inner
620.3.f.c.309.15	yes	24	31.30	odd	2	inner
620.3.f.c.309.16	yes	24	5.4	even	2	inner
3100.3.d.h.1301.11		24	155.92	even	4
3100.3.d.h.1301.12		24	5.2	odd	4
3100.3.d.h.1301.13		24	5.3	odd	4
3100.3.d.h.1301.14		24	155.123	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 620 = 2^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	620.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.59213 q^{3} +(-2.32008 + 4.42913i) q^{5} -1.60644i q^{7} -6.46514 q^{9} +O(q^{10})\)	\(q-1.59213 q^{3} +(-2.32008 + 4.42913i) q^{5} -1.60644i q^{7} -6.46514 q^{9} -2.01438i q^{11} -1.88658 q^{13} +(3.69386 - 7.05174i) q^{15} -1.86366 q^{17} -6.62236 q^{19} +2.55765i q^{21} +27.0144 q^{23} +(-14.2344 - 20.5519i) q^{25} +24.6224 q^{27} -45.2074i q^{29} +(-13.7468 + 27.7854i) q^{31} +3.20715i q^{33} +(7.11512 + 3.72706i) q^{35} +40.6209 q^{37} +3.00367 q^{39} -6.90007 q^{41} +47.4345 q^{43} +(14.9996 - 28.6349i) q^{45} -57.3430i q^{47} +46.4194 q^{49} +2.96718 q^{51} -0.0248289 q^{53} +(8.92198 + 4.67354i) q^{55} +10.5436 q^{57} -73.5797 q^{59} -81.6110i q^{61} +10.3858i q^{63} +(4.37701 - 8.35589i) q^{65} -50.0912i q^{67} -43.0104 q^{69} +77.8890 q^{71} +120.168 q^{73} +(22.6630 + 32.7212i) q^{75} -3.23598 q^{77} -106.078i q^{79} +18.9842 q^{81} -105.920 q^{83} +(4.32384 - 8.25439i) q^{85} +71.9758i q^{87} -76.7843i q^{89} +3.03066i q^{91} +(21.8866 - 44.2378i) q^{93} +(15.3644 - 29.3313i) q^{95} -161.362i q^{97} +13.0233i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{5} + 44 q^{9}+O(q^{10}) \) 24 * q - 6 * q^5 + 44 * q^9	\( 24 q - 6 q^{5} + 44 q^{9} + 8 q^{19} - 166 q^{25} - 24 q^{31} + 54 q^{35} - 280 q^{39} - 248 q^{41} - 190 q^{45} - 644 q^{49} - 100 q^{51} + 152 q^{59} - 288 q^{69} - 352 q^{71} - 368 q^{81} + 102 q^{95}+O(q^{100}) \) 24 * q - 6 * q^5 + 44 * q^9 + 8 * q^19 - 166 * q^25 - 24 * q^31 + 54 * q^35 - 280 * q^39 - 248 * q^41 - 190 * q^45 - 644 * q^49 - 100 * q^51 + 152 * q^59 - 288 * q^69 - 352 * q^71 - 368 * q^81 + 102 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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