LMFDB - Embedded newform 240.3.e.b.31.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,3,Mod(31,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("240.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	240.e (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:	$6.53952634465$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.3
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	240.31
Dual form			240.3.e.b.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -2.23607 q^{5} +4.28187i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -2.23607 q^{5} +4.28187i q^{7} -3.00000 q^{9} +4.28187i q^{11} -9.41641 q^{13} -3.87298i q^{15} -25.4164 q^{17} +29.3483i q^{19} -7.41641 q^{21} -8.56373i q^{23} +5.00000 q^{25} -5.19615i q^{27} -28.2492 q^{29} +1.63553i q^{31} -7.41641 q^{33} -9.57454i q^{35} -26.5836 q^{37} -16.3097i q^{39} +42.0000 q^{41} +68.8959i q^{43} +6.70820 q^{45} -17.1275i q^{47} +30.6656 q^{49} -44.0225i q^{51} -7.75078 q^{53} -9.57454i q^{55} -50.8328 q^{57} +45.8511i q^{59} +112.833 q^{61} -12.8456i q^{63} +21.0557 q^{65} +74.1886i q^{67} +14.8328 q^{69} -101.515i q^{71} +102.498 q^{73} +8.66025i q^{75} -18.3344 q^{77} -15.4919i q^{79} +9.00000 q^{81} -129.614i q^{83} +56.8328 q^{85} -48.9291i q^{87} -39.1672 q^{89} -40.3198i q^{91} -2.83282 q^{93} -65.6249i q^{95} -32.3344 q^{97} -12.8456i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 16 q^{13} - 48 q^{17} + 24 q^{21} + 20 q^{25} + 48 q^{29} + 24 q^{33} - 160 q^{37} + 168 q^{41} - 92 q^{49} - 192 q^{53} - 96 q^{57} + 344 q^{61} + 120 q^{65} - 48 q^{69} + 88 q^{73} - 288 q^{77} + 36 q^{81} + 120 q^{85} - 264 q^{89} + 96 q^{93} - 344 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 16 * q^13 - 48 * q^17 + 24 * q^21 + 20 * q^25 + 48 * q^29 + 24 * q^33 - 160 * q^37 + 168 * q^41 - 92 * q^49 - 192 * q^53 - 96 * q^57 + 344 * q^61 + 120 * q^65 - 48 * q^69 + 88 * q^73 - 288 * q^77 + 36 * q^81 + 120 * q^85 - 264 * q^89 + 96 * q^93 - 344 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	4.28187i	0.611695i	0.952081	+	0.305848i	$0.0989398\pi$
$7$	4.28187i	0.611695i	−0.952081	+	0.305848i	$0.901060\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	4.28187i	0.389260i	0.980877	+	0.194630i	$0.0623507\pi$
$11$	4.28187i	0.389260i	−0.980877	+	0.194630i	$0.937649\pi$
$12$	0	0
$13$	−9.41641	−0.724339	−0.362170	−	0.932112i	$-0.617964\pi$
$13$	−9.41641	−0.724339	−0.362170	+	0.932112i	$0.617964\pi$
$14$	0	0
$15$	− 3.87298i	− 0.258199i
$16$	0	0
$17$	−25.4164	−1.49508	−0.747541	−	0.664215i	$-0.768766\pi$
$17$	−25.4164	−1.49508	−0.747541	+	0.664215i	$0.768766\pi$
$18$	0	0
$19$	29.3483i	1.54465i	0.635228	+	0.772325i	$0.280906\pi$
$19$	29.3483i	1.54465i	−0.635228	+	0.772325i	$0.719094\pi$
$20$	0	0
$21$	−7.41641	−0.353162
$22$	0	0
$23$	− 8.56373i	− 0.372336i	−0.982518	−	0.186168i	$-0.940393\pi$
$23$	− 8.56373i	− 0.372336i	0.982518	−	0.186168i	$-0.0596069\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−28.2492	−0.974111	−0.487056	−	0.873371i	$-0.661929\pi$
$29$	−28.2492	−0.974111	−0.487056	+	0.873371i	$0.661929\pi$
$30$	0	0
$31$	1.63553i	0.0527589i	0.999652	+	0.0263795i	$0.00839782\pi$
$31$	1.63553i	0.0527589i	−0.999652	+	0.0263795i	$0.991602\pi$
$32$	0	0
$33$	−7.41641	−0.224740
$34$	0	0
$35$	− 9.57454i	− 0.273558i
$36$	0	0
$37$	−26.5836	−0.718475	−0.359238	−	0.933246i	$-0.616963\pi$
$37$	−26.5836	−0.718475	−0.359238	+	0.933246i	$0.616963\pi$
$38$	0	0
$39$	− 16.3097i	− 0.418197i
$40$	0	0
$41$	42.0000	1.02439	0.512195	−	0.858869i	$-0.328832\pi$
$41$	42.0000	1.02439	0.512195	+	0.858869i	$0.328832\pi$
$42$	0	0
$43$	68.8959i	1.60223i	0.598510	+	0.801116i	$0.295760\pi$
$43$	68.8959i	1.60223i	−0.598510	+	0.801116i	$0.704240\pi$
$44$	0	0
$45$	6.70820	0.149071
$46$	0	0
$47$	− 17.1275i	− 0.364414i	−0.983260	−	0.182207i	$-0.941676\pi$
$47$	− 17.1275i	− 0.364414i	0.983260	−	0.182207i	$-0.0583241\pi$
$48$	0	0
$49$	30.6656	0.625829
$50$	0	0
$51$	− 44.0225i	− 0.863186i
$52$	0	0
$53$	−7.75078	−0.146241	−0.0731205	−	0.997323i	$-0.523296\pi$
$53$	−7.75078	−0.146241	−0.0731205	+	0.997323i	$0.523296\pi$
$54$	0	0
$55$	− 9.57454i	− 0.174083i
$56$	0	0
$57$	−50.8328	−0.891804
$58$	0	0
$59$	45.8511i	0.777137i	0.921420	+	0.388569i	$0.127030\pi$
$59$	45.8511i	0.777137i	−0.921420	+	0.388569i	$0.872970\pi$
$60$	0	0
$61$	112.833	1.84972	0.924859	−	0.380310i	$-0.124183\pi$
$61$	112.833	1.84972	0.924859	+	0.380310i	$0.124183\pi$
$62$	0	0
$63$	− 12.8456i	− 0.203898i
$64$	0	0
$65$	21.0557	0.323934
$66$	0	0
$67$	74.1886i	1.10729i	0.832752	+	0.553646i	$0.186764\pi$
$67$	74.1886i	1.10729i	−0.832752	+	0.553646i	$0.813236\pi$
$68$	0	0
$69$	14.8328	0.214968
$70$	0	0
$71$	− 101.515i	− 1.42979i	−0.699230	−	0.714897i	$-0.746474\pi$
$71$	− 101.515i	− 1.42979i	0.699230	−	0.714897i	$-0.253526\pi$
$72$	0	0
$73$	102.498	1.40409	0.702044	−	0.712133i	$-0.252271\pi$
$73$	102.498	1.40409	0.702044	+	0.712133i	$0.252271\pi$
$74$	0	0
$75$	8.66025i	0.115470i
$76$	0	0
$77$	−18.3344	−0.238109
$78$	0	0
$79$	− 15.4919i	− 0.196100i	−0.995181	−	0.0980502i	$-0.968739\pi$
$79$	− 15.4919i	− 0.196100i	0.995181	−	0.0980502i	$-0.0312606\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 129.614i	− 1.56162i	−0.624770	−	0.780809i	$-0.714807\pi$
$83$	− 129.614i	− 1.56162i	0.624770	−	0.780809i	$-0.285193\pi$
$84$	0	0
$85$	56.8328	0.668621
$86$	0	0
$87$	− 48.9291i	− 0.562403i
$88$	0	0
$89$	−39.1672	−0.440081	−0.220040	−	0.975491i	$-0.570619\pi$
$89$	−39.1672	−0.440081	−0.220040	+	0.975491i	$0.570619\pi$
$90$	0	0
$91$	− 40.3198i	− 0.443075i
$92$	0	0
$93$	−2.83282	−0.0304604
$94$	0	0
$95$	− 65.6249i	− 0.690788i
$96$	0	0
$97$	−32.3344	−0.333344	−0.166672	−	0.986012i	$-0.553302\pi$
$97$	−32.3344	−0.333344	−0.166672	+	0.986012i	$0.553302\pi$
$98$	0	0
$99$	− 12.8456i	− 0.129753i
$100$	0	0
$101$	193.416	1.91501	0.957507	−	0.288410i	$-0.0931266\pi$
$101$	193.416	1.91501	0.957507	+	0.288410i	$0.0931266\pi$
$102$	0	0
$103$	39.3090i	0.381641i	0.981625	+	0.190820i	$0.0611148\pi$
$103$	39.3090i	0.381641i	−0.981625	+	0.190820i	$0.938885\pi$
$104$	0	0
$105$	16.5836	0.157939
$106$	0	0
$107$	154.056i	1.43978i	0.694090	+	0.719888i	$0.255807\pi$
$107$	154.056i	1.43978i	−0.694090	+	0.719888i	$0.744193\pi$
$108$	0	0
$109$	−100.833	−0.925072	−0.462536	−	0.886601i	$-0.653060\pi$
$109$	−100.833	−0.925072	−0.462536	+	0.886601i	$0.653060\pi$
$110$	0	0
$111$	− 46.0441i	− 0.414812i
$112$	0	0
$113$	−117.915	−1.04349	−0.521747	−	0.853100i	$-0.674720\pi$
$113$	−117.915	−1.04349	−0.521747	+	0.853100i	$0.674720\pi$
$114$	0	0
$115$	19.1491i	0.166514i
$116$	0	0
$117$	28.2492	0.241446
$118$	0	0
$119$	− 108.830i	− 0.914535i
$120$	0	0
$121$	102.666	0.848476
$122$	0	0
$123$	72.7461i	0.591432i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	70.2928i	0.553487i	0.960944	+	0.276743i	$0.0892552\pi$
$127$	70.2928i	0.553487i	−0.960944	+	0.276743i	$0.910745\pi$
$128$	0	0
$129$	−119.331	−0.925049
$130$	0	0
$131$	122.925i	0.938356i	0.883104	+	0.469178i	$0.155450\pi$
$131$	122.925i	0.938356i	−0.883104	+	0.469178i	$0.844550\pi$
$132$	0	0
$133$	−125.666	−0.944854
$134$	0	0
$135$	11.6190i	0.0860663i
$136$	0	0
$137$	−141.915	−1.03587	−0.517937	−	0.855419i	$-0.673300\pi$
$137$	−141.915	−1.03587	−0.517937	+	0.855419i	$0.673300\pi$
$138$	0	0
$139$	175.704i	1.26406i	0.774945	+	0.632029i	$0.217777\pi$
$139$	175.704i	1.26406i	−0.774945	+	0.632029i	$0.782223\pi$
$140$	0	0
$141$	29.6656	0.210395
$142$	0	0
$143$	− 40.3198i	− 0.281957i
$144$	0	0
$145$	63.1672	0.435636
$146$	0	0
$147$	53.1144i	0.361323i
$148$	0	0
$149$	45.9149	0.308153	0.154077	−	0.988059i	$-0.450760\pi$
$149$	45.9149	0.308153	0.154077	+	0.988059i	$0.450760\pi$
$150$	0	0
$151$	− 162.620i	− 1.07695i	−0.842641	−	0.538476i	$-0.819000\pi$
$151$	− 162.620i	− 1.07695i	0.842641	−	0.538476i	$-0.181000\pi$
$152$	0	0
$153$	76.2492	0.498361
$154$	0	0
$155$	− 3.65715i	− 0.0235945i
$156$	0	0
$157$	−202.413	−1.28926	−0.644628	−	0.764496i	$-0.722988\pi$
$157$	−202.413	−1.28926	−0.644628	+	0.764496i	$0.722988\pi$
$158$	0	0
$159$	− 13.4247i	− 0.0844323i
$160$	0	0
$161$	36.6687	0.227756
$162$	0	0
$163$	− 74.1886i	− 0.455145i	−0.973761	−	0.227572i	$-0.926921\pi$
$163$	− 74.1886i	− 0.455145i	0.973761	−	0.227572i	$-0.0730788\pi$
$164$	0	0
$165$	16.5836	0.100507
$166$	0	0
$167$	267.792i	1.60355i	0.597629	+	0.801773i	$0.296110\pi$
$167$	267.792i	1.60355i	−0.597629	+	0.801773i	$0.703890\pi$
$168$	0	0
$169$	−80.3313	−0.475333
$170$	0	0
$171$	− 88.0450i	− 0.514883i
$172$	0	0
$173$	−187.082	−1.08140	−0.540700	−	0.841216i	$-0.681840\pi$
$173$	−187.082	−1.08140	−0.540700	+	0.841216i	$0.681840\pi$
$174$	0	0
$175$	21.4093i	0.122339i
$176$	0	0
$177$	−79.4164	−0.448680
$178$	0	0
$179$	119.176i	0.665790i	0.942964	+	0.332895i	$0.108025\pi$
$179$	119.176i	0.665790i	−0.942964	+	0.332895i	$0.891975\pi$
$180$	0	0
$181$	−249.161	−1.37658	−0.688290	−	0.725436i	$-0.741638\pi$
$181$	−249.161	−1.37658	−0.688290	+	0.725436i	$0.741638\pi$
$182$	0	0
$183$	195.432i	1.06794i
$184$	0	0
$185$	59.4427	0.321312
$186$	0	0
$187$	− 108.830i	− 0.581977i
$188$	0	0
$189$	22.2492	0.117721
$190$	0	0
$191$	316.676i	1.65799i	0.559257	+	0.828994i	$0.311086\pi$
$191$	316.676i	1.65799i	−0.559257	+	0.828994i	$0.688914\pi$
$192$	0	0
$193$	107.167	0.555270	0.277635	−	0.960687i	$-0.410449\pi$
$193$	107.167	0.555270	0.277635	+	0.960687i	$0.410449\pi$
$194$	0	0
$195$	36.4696i	0.187024i
$196$	0	0
$197$	28.9180	0.146792	0.0733958	−	0.997303i	$-0.476616\pi$
$197$	28.9180	0.146792	0.0733958	+	0.997303i	$0.476616\pi$
$198$	0	0
$199$	− 317.539i	− 1.59567i	−0.602873	−	0.797837i	$-0.705978\pi$
$199$	− 317.539i	− 1.59567i	0.602873	−	0.797837i	$-0.294022\pi$
$200$	0	0
$201$	−128.498	−0.639296
$202$	0	0
$203$	− 120.959i	− 0.595859i
$204$	0	0
$205$	−93.9149	−0.458121
$206$	0	0
$207$	25.6912i	0.124112i
$208$	0	0
$209$	−125.666	−0.601271
$210$	0	0
$211$	− 378.735i	− 1.79495i	−0.441065	−	0.897475i	$-0.645399\pi$
$211$	− 378.735i	− 1.79495i	0.441065	−	0.897475i	$-0.354601\pi$
$212$	0	0
$213$	175.830	0.825492
$214$	0	0
$215$	− 154.056i	− 0.716540i
$216$	0	0
$217$	−7.00311	−0.0322724
$218$	0	0
$219$	177.533i	0.810651i
$220$	0	0
$221$	239.331	1.08295
$222$	0	0
$223$	326.250i	1.46301i	0.681838	+	0.731503i	$0.261181\pi$
$223$	326.250i	1.46301i	−0.681838	+	0.731503i	$0.738819\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 282.512i	− 1.24455i	−0.782800	−	0.622273i	$-0.786209\pi$
$227$	− 282.512i	− 1.24455i	0.782800	−	0.622273i	$-0.213791\pi$
$228$	0	0
$229$	−150.997	−0.659375	−0.329688	−	0.944090i	$-0.606943\pi$
$229$	−150.997	−0.659375	−0.329688	+	0.944090i	$0.606943\pi$
$230$	0	0
$231$	− 31.7561i	− 0.137472i
$232$	0	0
$233$	−115.751	−0.496784	−0.248392	−	0.968660i	$-0.579902\pi$
$233$	−115.751	−0.496784	−0.248392	+	0.968660i	$0.579902\pi$
$234$	0	0
$235$	38.2982i	0.162971i
$236$	0	0
$237$	26.8328	0.113219
$238$	0	0
$239$	204.280i	0.854728i	0.904080	+	0.427364i	$0.140558\pi$
$239$	204.280i	0.854728i	−0.904080	+	0.427364i	$0.859442\pi$
$240$	0	0
$241$	86.3282	0.358208	0.179104	−	0.983830i	$-0.442680\pi$
$241$	86.3282	0.358208	0.179104	+	0.983830i	$0.442680\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−68.5704	−0.279879
$246$	0	0
$247$	− 276.356i	− 1.11885i
$248$	0	0
$249$	224.498	0.901600
$250$	0	0
$251$	154.681i	0.616258i	0.951345	+	0.308129i	$0.0997028\pi$
$251$	154.681i	0.616258i	−0.951345	+	0.308129i	$0.900297\pi$
$252$	0	0
$253$	36.6687	0.144936
$254$	0	0
$255$	98.4373i	0.386029i
$256$	0	0
$257$	473.745	1.84336	0.921682	−	0.387946i	$-0.126815\pi$
$257$	473.745	1.84336	0.921682	+	0.387946i	$0.126815\pi$
$258$	0	0
$259$	− 113.827i	− 0.439488i
$260$	0	0
$261$	84.7477	0.324704
$262$	0	0
$263$	− 319.175i	− 1.21359i	−0.794858	−	0.606796i	$-0.792455\pi$
$263$	− 319.175i	− 1.21359i	0.794858	−	0.606796i	$-0.207545\pi$
$264$	0	0
$265$	17.3313	0.0654010
$266$	0	0
$267$	− 67.8396i	− 0.254081i
$268$	0	0
$269$	456.079	1.69546	0.847730	−	0.530427i	$-0.177969\pi$
$269$	456.079	1.69546	0.847730	+	0.530427i	$0.177969\pi$
$270$	0	0
$271$	144.720i	0.534022i	0.963693	+	0.267011i	$0.0860361\pi$
$271$	144.720i	0.534022i	−0.963693	+	0.267011i	$0.913964\pi$
$272$	0	0
$273$	69.8359	0.255809
$274$	0	0
$275$	21.4093i	0.0778521i
$276$	0	0
$277$	36.4195	0.131478	0.0657392	−	0.997837i	$-0.479059\pi$
$277$	36.4195	0.131478	0.0657392	+	0.997837i	$0.479059\pi$
$278$	0	0
$279$	− 4.90658i	− 0.0175863i
$280$	0	0
$281$	−87.1672	−0.310204	−0.155102	−	0.987899i	$-0.549571\pi$
$281$	−87.1672	−0.310204	−0.155102	+	0.987899i	$0.549571\pi$
$282$	0	0
$283$	− 238.921i	− 0.844244i	−0.906539	−	0.422122i	$-0.861285\pi$
$283$	− 238.921i	− 0.844244i	0.906539	−	0.422122i	$-0.138715\pi$
$284$	0	0
$285$	113.666	0.398827
$286$	0	0
$287$	179.838i	0.626614i
$288$	0	0
$289$	356.994	1.23527
$290$	0	0
$291$	− 56.0048i	− 0.192456i
$292$	0	0
$293$	218.243	0.744857	0.372428	−	0.928061i	$-0.378525\pi$
$293$	218.243	0.744857	0.372428	+	0.928061i	$0.378525\pi$
$294$	0	0
$295$	− 102.526i	− 0.347546i
$296$	0	0
$297$	22.2492	0.0749132
$298$	0	0
$299$	80.6396i	0.269698i
$300$	0	0
$301$	−295.003	−0.980077
$302$	0	0
$303$	335.007i	1.10563i
$304$	0	0
$305$	−252.302	−0.827219
$306$	0	0
$307$	− 462.645i	− 1.50699i	−0.657455	−	0.753494i	$-0.728367\pi$
$307$	− 462.645i	− 1.50699i	0.657455	−	0.753494i	$-0.271633\pi$
$308$	0	0
$309$	−68.0851	−0.220340
$310$	0	0
$311$	573.405i	1.84375i	0.387491	+	0.921874i	$0.373342\pi$
$311$	573.405i	1.84375i	−0.387491	+	0.921874i	$0.626658\pi$
$312$	0	0
$313$	262.498	0.838653	0.419327	−	0.907835i	$-0.362266\pi$
$313$	262.498	0.838653	0.419327	+	0.907835i	$0.362266\pi$
$314$	0	0
$315$	28.7236i	0.0911861i
$316$	0	0
$317$	21.9149	0.0691320	0.0345660	−	0.999402i	$-0.488995\pi$
$317$	21.9149	0.0691320	0.0345660	+	0.999402i	$0.488995\pi$
$318$	0	0
$319$	− 120.959i	− 0.379183i
$320$	0	0
$321$	−266.833	−0.831255
$322$	0	0
$323$	− 745.929i	− 2.30938i
$324$	0	0
$325$	−47.0820	−0.144868
$326$	0	0
$327$	− 174.648i	− 0.534090i
$328$	0	0
$329$	73.3375	0.222910
$330$	0	0
$331$	− 57.8333i	− 0.174723i	−0.996177	−	0.0873615i	$-0.972156\pi$
$331$	− 57.8333i	− 0.174723i	0.996177	−	0.0873615i	$-0.0278435\pi$
$332$	0	0
$333$	79.7508	0.239492
$334$	0	0
$335$	− 165.891i	− 0.495196i
$336$	0	0
$337$	157.502	0.467364	0.233682	−	0.972313i	$-0.424923\pi$
$337$	157.502	0.467364	0.233682	+	0.972313i	$0.424923\pi$
$338$	0	0
$339$	− 204.235i	− 0.602462i
$340$	0	0
$341$	−7.00311	−0.0205370
$342$	0	0
$343$	341.117i	0.994512i
$344$	0	0
$345$	−33.1672	−0.0961368
$346$	0	0
$347$	248.257i	0.715438i	0.933829	+	0.357719i	$0.116445\pi$
$347$	248.257i	0.715438i	−0.933829	+	0.357719i	$0.883555\pi$
$348$	0	0
$349$	53.5016	0.153300	0.0766498	−	0.997058i	$-0.475578\pi$
$349$	53.5016	0.153300	0.0766498	+	0.997058i	$0.475578\pi$
$350$	0	0
$351$	48.9291i	0.139399i
$352$	0	0
$353$	−289.416	−0.819877	−0.409938	−	0.912113i	$-0.634450\pi$
$353$	−289.416	−0.819877	−0.409938	+	0.912113i	$0.634450\pi$
$354$	0	0
$355$	226.995i	0.639423i
$356$	0	0
$357$	188.498	0.528007
$358$	0	0
$359$	− 381.437i	− 1.06250i	−0.847215	−	0.531250i	$-0.821723\pi$
$359$	− 381.437i	− 1.06250i	0.847215	−	0.531250i	$-0.178277\pi$
$360$	0	0
$361$	−500.325	−1.38594
$362$	0	0
$363$	177.822i	0.489868i
$364$	0	0
$365$	−229.193	−0.627927
$366$	0	0
$367$	− 145.640i	− 0.396839i	−0.980117	−	0.198419i	$-0.936419\pi$
$367$	− 145.640i	− 0.396839i	0.980117	−	0.198419i	$-0.0635808\pi$
$368$	0	0
$369$	−126.000	−0.341463
$370$	0	0
$371$	− 33.1878i	− 0.0894549i
$372$	0	0
$373$	−184.079	−0.493509	−0.246755	−	0.969078i	$-0.579364\pi$
$373$	−184.079	−0.493509	−0.246755	+	0.969078i	$0.579364\pi$
$374$	0	0
$375$	− 19.3649i	− 0.0516398i
$376$	0	0
$377$	266.006	0.705587
$378$	0	0
$379$	297.913i	0.786049i	0.919528	+	0.393025i	$0.128571\pi$
$379$	297.913i	0.786049i	−0.919528	+	0.393025i	$0.871429\pi$
$380$	0	0
$381$	−121.751	−0.319556
$382$	0	0
$383$	415.874i	1.08583i	0.839786	+	0.542917i	$0.182680\pi$
$383$	415.874i	1.08583i	−0.839786	+	0.542917i	$0.817320\pi$
$384$	0	0
$385$	40.9969	0.106485
$386$	0	0
$387$	− 206.688i	− 0.534077i
$388$	0	0
$389$	−95.4102	−0.245270	−0.122635	−	0.992452i	$-0.539135\pi$
$389$	−95.4102	−0.245270	−0.122635	+	0.992452i	$0.539135\pi$
$390$	0	0
$391$	217.659i	0.556673i
$392$	0	0
$393$	−212.912	−0.541760
$394$	0	0
$395$	34.6410i	0.0876988i
$396$	0	0
$397$	553.745	1.39482	0.697411	−	0.716671i	$-0.254335\pi$
$397$	553.745	1.39482	0.697411	+	0.716671i	$0.254335\pi$
$398$	0	0
$399$	− 217.659i	− 0.545512i
$400$	0	0
$401$	−111.167	−0.277225	−0.138612	−	0.990347i	$-0.544264\pi$
$401$	−111.167	−0.277225	−0.138612	+	0.990347i	$0.544264\pi$
$402$	0	0
$403$	− 15.4008i	− 0.0382154i
$404$	0	0
$405$	−20.1246	−0.0496904
$406$	0	0
$407$	− 113.827i	− 0.279674i
$408$	0	0
$409$	237.331	0.580272	0.290136	−	0.956985i	$-0.406299\pi$
$409$	237.331	0.580272	0.290136	+	0.956985i	$0.406299\pi$
$410$	0	0
$411$	− 245.804i	− 0.598063i
$412$	0	0
$413$	−196.328	−0.475371
$414$	0	0
$415$	289.826i	0.698377i
$416$	0	0
$417$	−304.328	−0.729804
$418$	0	0
$419$	519.173i	1.23908i	0.784967	+	0.619538i	$0.212680\pi$
$419$	519.173i	1.23908i	−0.784967	+	0.619538i	$0.787320\pi$
$420$	0	0
$421$	−797.659	−1.89468	−0.947339	−	0.320233i	$-0.896239\pi$
$421$	−797.659	−1.89468	−0.947339	+	0.320233i	$0.896239\pi$
$422$	0	0
$423$	51.3824i	0.121471i
$424$	0	0
$425$	−127.082	−0.299017
$426$	0	0
$427$	483.135i	1.13146i
$428$	0	0
$429$	69.8359	0.162788
$430$	0	0
$431$	626.037i	1.45252i	0.687419	+	0.726261i	$0.258744\pi$
$431$	626.037i	1.45252i	−0.687419	+	0.726261i	$0.741256\pi$
$432$	0	0
$433$	63.4953	0.146641	0.0733203	−	0.997308i	$-0.476640\pi$
$433$	63.4953	0.146641	0.0733203	+	0.997308i	$0.476640\pi$
$434$	0	0
$435$	109.409i	0.251514i
$436$	0	0
$437$	251.331	0.575129
$438$	0	0
$439$	56.7662i	0.129308i	0.997908	+	0.0646540i	$0.0205944\pi$
$439$	56.7662i	0.129308i	−0.997908	+	0.0646540i	$0.979406\pi$
$440$	0	0
$441$	−91.9969	−0.208610
$442$	0	0
$443$	− 803.285i	− 1.81329i	−0.421899	−	0.906643i	$-0.638636\pi$
$443$	− 803.285i	− 1.81329i	0.421899	−	0.906643i	$-0.361364\pi$
$444$	0	0
$445$	87.5805	0.196810
$446$	0	0
$447$	79.5269i	0.177912i
$448$	0	0
$449$	−702.827	−1.56532	−0.782658	−	0.622452i	$-0.786136\pi$
$449$	−702.827	−1.56532	−0.782658	+	0.622452i	$0.786136\pi$
$450$	0	0
$451$	179.838i	0.398755i
$452$	0	0
$453$	281.666	0.621778
$454$	0	0
$455$	90.1578i	0.198149i
$456$	0	0
$457$	−228.663	−0.500356	−0.250178	−	0.968200i	$-0.580489\pi$
$457$	−228.663	−0.500356	−0.250178	+	0.968200i	$0.580489\pi$
$458$	0	0
$459$	132.068i	0.287729i
$460$	0	0
$461$	−742.073	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$461$	−742.073	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$462$	0	0
$463$	654.466i	1.41353i	0.707447	+	0.706767i	$0.249847\pi$
$463$	654.466i	1.41353i	−0.707447	+	0.706767i	$0.750153\pi$
$464$	0	0
$465$	6.33437	0.0136223
$466$	0	0
$467$	− 401.155i	− 0.859004i	−0.903066	−	0.429502i	$-0.858689\pi$
$467$	− 401.155i	− 0.859004i	0.903066	−	0.429502i	$-0.141311\pi$
$468$	0	0
$469$	−317.666	−0.677325
$470$	0	0
$471$	− 350.590i	− 0.744353i
$472$	0	0
$473$	−295.003	−0.623685
$474$	0	0
$475$	146.742i	0.308930i
$476$	0	0
$477$	23.2523	0.0487470
$478$	0	0
$479$	− 884.198i	− 1.84593i	−0.384889	−	0.922963i	$-0.625760\pi$
$479$	− 884.198i	− 1.84593i	0.384889	−	0.922963i	$-0.374240\pi$
$480$	0	0
$481$	250.322	0.520420
$482$	0	0
$483$	63.5121i	0.131495i
$484$	0	0
$485$	72.3018	0.149076
$486$	0	0
$487$	608.671i	1.24984i	0.780690	+	0.624919i	$0.214868\pi$
$487$	608.671i	1.24984i	−0.780690	+	0.624919i	$0.785132\pi$
$488$	0	0
$489$	128.498	0.262778
$490$	0	0
$491$	570.555i	1.16203i	0.813894	+	0.581013i	$0.197344\pi$
$491$	570.555i	1.16203i	−0.813894	+	0.581013i	$0.802656\pi$
$492$	0	0
$493$	717.994	1.45638
$494$	0	0
$495$	28.7236i	0.0580275i
$496$	0	0
$497$	434.675	0.874597
$498$	0	0
$499$	− 33.3916i	− 0.0669170i	−0.999440	−	0.0334585i	$-0.989348\pi$
$499$	− 33.3916i	− 0.0669170i	0.999440	−	0.0334585i	$-0.0106522\pi$
$500$	0	0
$501$	−463.830	−0.925808
$502$	0	0
$503$	− 330.237i	− 0.656535i	−0.944585	−	0.328268i	$-0.893535\pi$
$503$	− 330.237i	− 0.656535i	0.944585	−	0.328268i	$-0.106465\pi$
$504$	0	0
$505$	−432.492	−0.856420
$506$	0	0
$507$	− 139.138i	− 0.274434i
$508$	0	0
$509$	408.079	0.801727	0.400863	−	0.916138i	$-0.368710\pi$
$509$	408.079	0.801727	0.400863	+	0.916138i	$0.368710\pi$
$510$	0	0
$511$	438.885i	0.858874i
$512$	0	0
$513$	152.498	0.297268
$514$	0	0
$515$	− 87.8975i	− 0.170675i
$516$	0	0
$517$	73.3375	0.141852
$518$	0	0
$519$	− 324.036i	− 0.624346i
$520$	0	0
$521$	236.164	0.453290	0.226645	−	0.973977i	$-0.427224\pi$
$521$	236.164	0.453290	0.226645	+	0.973977i	$0.427224\pi$
$522$	0	0
$523$	349.772i	0.668781i	0.942435	+	0.334390i	$0.108530\pi$
$523$	349.772i	0.668781i	−0.942435	+	0.334390i	$0.891470\pi$
$524$	0	0
$525$	−37.0820	−0.0706325
$526$	0	0
$527$	− 41.5692i	− 0.0788790i
$528$	0	0
$529$	455.663	0.861366
$530$	0	0
$531$	− 137.553i	− 0.259046i
$532$	0	0
$533$	−395.489	−0.742006
$534$	0	0
$535$	− 344.480i	− 0.643887i
$536$	0	0
$537$	−206.420	−0.384394
$538$	0	0
$539$	131.306i	0.243611i
$540$	0	0
$541$	598.158	1.10565	0.552826	−	0.833297i	$-0.313549\pi$
$541$	598.158	1.10565	0.552826	+	0.833297i	$0.313549\pi$
$542$	0	0
$543$	− 431.559i	− 0.794769i
$544$	0	0
$545$	225.469	0.413705
$546$	0	0
$547$	− 364.696i	− 0.666720i	−0.942800	−	0.333360i	$-0.891818\pi$
$547$	− 364.696i	− 0.666720i	0.942800	−	0.333360i	$-0.108182\pi$
$548$	0	0
$549$	−338.498	−0.616573
$550$	0	0
$551$	− 829.068i	− 1.50466i
$552$	0	0
$553$	66.3344	0.119954
$554$	0	0
$555$	102.958i	0.185510i
$556$	0	0
$557$	−500.420	−0.898419	−0.449210	−	0.893426i	$-0.648294\pi$
$557$	−500.420	−0.898419	−0.449210	+	0.893426i	$0.648294\pi$
$558$	0	0
$559$	− 648.752i	− 1.16056i
$560$	0	0
$561$	188.498	0.336004
$562$	0	0
$563$	− 328.897i	− 0.584186i	−0.956390	−	0.292093i	$-0.905648\pi$
$563$	− 328.897i	− 0.584186i	0.956390	−	0.292093i	$-0.0943516\pi$
$564$	0	0
$565$	263.666	0.466665
$566$	0	0
$567$	38.5368i	0.0679661i
$568$	0	0
$569$	359.155	0.631203	0.315602	−	0.948892i	$-0.397794\pi$
$569$	359.155	0.631203	0.315602	+	0.948892i	$0.397794\pi$
$570$	0	0
$571$	− 251.823i	− 0.441021i	−0.975385	−	0.220511i	$-0.929228\pi$
$571$	− 251.823i	− 0.441021i	0.975385	−	0.220511i	$-0.0707723\pi$
$572$	0	0
$573$	−548.498	−0.957240
$574$	0	0
$575$	− 42.8187i	− 0.0744672i
$576$	0	0
$577$	986.316	1.70939	0.854693	−	0.519134i	$-0.173745\pi$
$577$	986.316	1.70939	0.854693	+	0.519134i	$0.173745\pi$
$578$	0	0
$579$	185.619i	0.320586i
$580$	0	0
$581$	554.991	0.955234
$582$	0	0
$583$	− 33.1878i	− 0.0569259i
$584$	0	0
$585$	−63.1672	−0.107978
$586$	0	0
$587$	491.607i	0.837491i	0.908104	+	0.418746i	$0.137530\pi$
$587$	491.607i	0.837491i	−0.908104	+	0.418746i	$0.862470\pi$
$588$	0	0
$589$	−48.0000	−0.0814941
$590$	0	0
$591$	50.0874i	0.0847502i
$592$	0	0
$593$	−328.249	−0.553540	−0.276770	−	0.960936i	$-0.589264\pi$
$593$	−328.249	−0.553540	−0.276770	+	0.960936i	$0.589264\pi$
$594$	0	0
$595$	243.350i	0.408992i
$596$	0	0
$597$	549.994	0.921263
$598$	0	0
$599$	51.3824i	0.0857803i	0.999080	+	0.0428901i	$0.0136565\pi$
$599$	51.3824i	0.0857803i	−0.999080	+	0.0428901i	$0.986343\pi$
$600$	0	0
$601$	138.997	0.231276	0.115638	−	0.993291i	$-0.463109\pi$
$601$	138.997	0.231276	0.115638	+	0.993291i	$0.463109\pi$
$602$	0	0
$603$	− 222.566i	− 0.369098i
$604$	0	0
$605$	−229.567	−0.379450
$606$	0	0
$607$	− 267.554i	− 0.440780i	−0.975412	−	0.220390i	$-0.929267\pi$
$607$	− 267.554i	− 0.440780i	0.975412	−	0.220390i	$-0.0707330\pi$
$608$	0	0
$609$	209.508	0.344019
$610$	0	0
$611$	161.279i	0.263959i
$612$	0	0
$613$	−170.584	−0.278277	−0.139138	−	0.990273i	$-0.544433\pi$
$613$	−170.584	−0.278277	−0.139138	+	0.990273i	$0.544433\pi$
$614$	0	0
$615$	− 162.665i	− 0.264496i
$616$	0	0
$617$	247.240	0.400713	0.200356	−	0.979723i	$-0.435790\pi$
$617$	247.240	0.400713	0.200356	+	0.979723i	$0.435790\pi$
$618$	0	0
$619$	1133.52i	1.83122i	0.402072	+	0.915608i	$0.368290\pi$
$619$	1133.52i	1.83122i	−0.402072	+	0.915608i	$0.631710\pi$
$620$	0	0
$621$	−44.4984	−0.0716561
$622$	0	0
$623$	− 167.709i	− 0.269195i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 217.659i	− 0.347144i
$628$	0	0
$629$	675.659	1.07418
$630$	0	0
$631$	− 600.732i	− 0.952032i	−0.879437	−	0.476016i	$-0.842080\pi$
$631$	− 600.732i	− 0.952032i	0.879437	−	0.476016i	$-0.157920\pi$
$632$	0	0
$633$	655.988	1.03632
$634$	0	0
$635$	− 157.180i	− 0.247527i
$636$	0	0
$637$	−288.760	−0.453313
$638$	0	0
$639$	304.546i	0.476598i
$640$	0	0
$641$	63.0093	0.0982985	0.0491492	−	0.998791i	$-0.484349\pi$
$641$	63.0093	0.0982985	0.0491492	+	0.998791i	$0.484349\pi$
$642$	0	0
$643$	− 109.988i	− 0.171054i	−0.996336	−	0.0855272i	$-0.972743\pi$
$643$	− 109.988i	− 0.171054i	0.996336	−	0.0855272i	$-0.0272574\pi$
$644$	0	0
$645$	266.833	0.413694
$646$	0	0
$647$	− 366.991i	− 0.567219i	−0.958940	−	0.283610i	$-0.908468\pi$
$647$	− 366.991i	− 0.567219i	0.958940	−	0.283610i	$-0.0915320\pi$
$648$	0	0
$649$	−196.328	−0.302509
$650$	0	0
$651$	− 12.1297i	− 0.0186325i
$652$	0	0
$653$	1183.91	1.81303	0.906515	−	0.422174i	$-0.138733\pi$
$653$	1183.91	1.81303	0.906515	+	0.422174i	$0.138733\pi$
$654$	0	0
$655$	− 274.868i	− 0.419646i
$656$	0	0
$657$	−307.495	−0.468029
$658$	0	0
$659$	− 28.7236i	− 0.0435867i	−0.999762	−	0.0217933i	$-0.993062\pi$
$659$	− 28.7236i	− 0.0435867i	0.999762	−	0.0217933i	$-0.00693759\pi$
$660$	0	0
$661$	638.158	0.965443	0.482722	−	0.875774i	$-0.339648\pi$
$661$	638.158	0.965443	0.482722	+	0.875774i	$0.339648\pi$
$662$	0	0
$663$	414.534i	0.625240i
$664$	0	0
$665$	280.997	0.422552
$666$	0	0
$667$	241.919i	0.362697i
$668$	0	0
$669$	−565.082	−0.844667
$670$	0	0
$671$	483.135i	0.720022i
$672$	0	0
$673$	−36.8328	−0.0547293	−0.0273646	−	0.999626i	$-0.508712\pi$
$673$	−36.8328	−0.0547293	−0.0273646	+	0.999626i	$0.508712\pi$
$674$	0	0
$675$	− 25.9808i	− 0.0384900i
$676$	0	0
$677$	−815.410	−1.20445	−0.602223	−	0.798328i	$-0.705718\pi$
$677$	−815.410	−1.20445	−0.602223	+	0.798328i	$0.705718\pi$
$678$	0	0
$679$	− 138.451i	− 0.203905i
$680$	0	0
$681$	489.325	0.718539
$682$	0	0
$683$	1010.25i	1.47913i	0.673084	+	0.739566i	$0.264969\pi$
$683$	1010.25i	1.47913i	−0.673084	+	0.739566i	$0.735031\pi$
$684$	0	0
$685$	317.331	0.463257
$686$	0	0
$687$	− 261.534i	− 0.380690i
$688$	0	0
$689$	72.9845	0.105928
$690$	0	0
$691$	295.891i	0.428207i	0.976811	+	0.214104i	$0.0686830\pi$
$691$	295.891i	0.428207i	−0.976811	+	0.214104i	$0.931317\pi$
$692$	0	0
$693$	55.0031	0.0793696
$694$	0	0
$695$	− 392.886i	− 0.565304i
$696$	0	0
$697$	−1067.49	−1.53155
$698$	0	0
$699$	− 200.486i	− 0.286819i
$700$	0	0
$701$	52.0914	0.0743101	0.0371550	−	0.999310i	$-0.488170\pi$
$701$	52.0914	0.0743101	0.0371550	+	0.999310i	$0.488170\pi$
$702$	0	0
$703$	− 780.184i	− 1.10979i
$704$	0	0
$705$	−66.3344	−0.0940913
$706$	0	0
$707$	828.183i	1.17140i
$708$	0	0
$709$	−87.3375	−0.123184	−0.0615920	−	0.998101i	$-0.519618\pi$
$709$	−87.3375	−0.123184	−0.0615920	+	0.998101i	$0.519618\pi$
$710$	0	0
$711$	46.4758i	0.0653668i
$712$	0	0
$713$	14.0062	0.0196441
$714$	0	0
$715$	90.1578i	0.126095i
$716$	0	0
$717$	−353.823	−0.493478
$718$	0	0
$719$	− 347.365i	− 0.483122i	−0.970386	−	0.241561i	$-0.922341\pi$
$719$	− 347.365i	− 0.483122i	0.970386	−	0.241561i	$-0.0776594\pi$
$720$	0	0
$721$	−168.316	−0.233448
$722$	0	0
$723$	149.525i	0.206812i
$724$	0	0
$725$	−141.246	−0.194822
$726$	0	0
$727$	− 508.882i	− 0.699976i	−0.936754	−	0.349988i	$-0.886186\pi$
$727$	− 508.882i	− 0.699976i	0.936754	−	0.349988i	$-0.113814\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1751.09i	− 2.39547i
$732$	0	0
$733$	−105.927	−0.144512	−0.0722560	−	0.997386i	$-0.523020\pi$
$733$	−105.927	−0.144512	−0.0722560	+	0.997386i	$0.523020\pi$
$734$	0	0
$735$	− 118.767i	− 0.161588i
$736$	0	0
$737$	−317.666	−0.431025
$738$	0	0
$739$	119.801i	0.162112i	0.996710	+	0.0810562i	$0.0258293\pi$
$739$	119.801i	0.162112i	−0.996710	+	0.0810562i	$0.974171\pi$
$740$	0	0
$741$	478.663	0.645968
$742$	0	0
$743$	990.529i	1.33315i	0.745439	+	0.666574i	$0.232240\pi$
$743$	990.529i	1.33315i	−0.745439	+	0.666574i	$0.767760\pi$
$744$	0	0
$745$	−102.669	−0.137810
$746$	0	0
$747$	388.843i	0.520539i
$748$	0	0
$749$	−659.647	−0.880704
$750$	0	0
$751$	− 541.445i	− 0.720966i	−0.932766	−	0.360483i	$-0.882612\pi$
$751$	− 541.445i	− 0.720966i	0.932766	−	0.360483i	$-0.117388\pi$
$752$	0	0
$753$	−267.915	−0.355797
$754$	0	0
$755$	363.629i	0.481628i
$756$	0	0
$757$	−559.568	−0.739192	−0.369596	−	0.929193i	$-0.620504\pi$
$757$	−559.568	−0.739192	−0.369596	+	0.929193i	$0.620504\pi$
$758$	0	0
$759$	63.5121i	0.0836787i
$760$	0	0
$761$	672.492	0.883695	0.441848	−	0.897090i	$-0.354323\pi$
$761$	672.492	0.883695	0.441848	+	0.897090i	$0.354323\pi$
$762$	0	0
$763$	− 431.753i	− 0.565862i
$764$	0	0
$765$	−170.498	−0.222874
$766$	0	0
$767$	− 431.753i	− 0.562911i
$768$	0	0
$769$	−771.981	−1.00388	−0.501938	−	0.864903i	$-0.667380\pi$
$769$	−771.981	−1.00388	−0.501938	+	0.864903i	$0.667380\pi$
$770$	0	0
$771$	820.550i	1.06427i
$772$	0	0
$773$	−40.0914	−0.0518646	−0.0259323	−	0.999664i	$-0.508255\pi$
$773$	−40.0914	−0.0518646	−0.0259323	+	0.999664i	$0.508255\pi$
$774$	0	0
$775$	8.17763i	0.0105518i
$776$	0	0
$777$	197.155	0.253738
$778$	0	0
$779$	1232.63i	1.58232i
$780$	0	0
$781$	434.675	0.556562
$782$	0	0
$783$	146.787i	0.187468i
$784$	0	0
$785$	452.610	0.576573
$786$	0	0
$787$	63.4210i	0.0805857i	0.999188	+	0.0402929i	$0.0128291\pi$
$787$	63.4210i	0.0805857i	−0.999188	+	0.0402929i	$0.987171\pi$
$788$	0	0
$789$	552.827	0.700667
$790$	0	0
$791$	− 504.895i	− 0.638300i
$792$	0	0
$793$	−1062.48	−1.33982
$794$	0	0
$795$	30.0186i	0.0377593i
$796$	0	0
$797$	−1240.41	−1.55635	−0.778173	−	0.628051i	$-0.783853\pi$
$797$	−1240.41	−1.55635	−0.778173	+	0.628051i	$0.783853\pi$
$798$	0	0
$799$	435.319i	0.544829i
$800$	0	0
$801$	117.502	0.146694
$802$	0	0
$803$	438.885i	0.546556i
$804$	0	0
$805$	−81.9938	−0.101856
$806$	0	0
$807$	789.952i	0.978875i
$808$	0	0
$809$	−1458.98	−1.80344	−0.901721	−	0.432319i	$-0.857696\pi$
$809$	−1458.98	−1.80344	−0.901721	+	0.432319i	$0.857696\pi$
$810$	0	0
$811$	− 516.344i	− 0.636676i	−0.947977	−	0.318338i	$-0.896875\pi$
$811$	− 516.344i	− 0.636676i	0.947977	−	0.318338i	$-0.103125\pi$
$812$	0	0
$813$	−250.663	−0.308318
$814$	0	0
$815$	165.891i	0.203547i
$816$	0	0
$817$	−2021.98	−2.47489
$818$	0	0
$819$	120.959i	0.147692i
$820$	0	0
$821$	241.416	0.294052	0.147026	−	0.989133i	$-0.453030\pi$
$821$	241.416	0.294052	0.147026	+	0.989133i	$0.453030\pi$
$822$	0	0
$823$	390.422i	0.474389i	0.971462	+	0.237194i	$0.0762278\pi$
$823$	390.422i	0.474389i	−0.971462	+	0.237194i	$0.923772\pi$
$824$	0	0
$825$	−37.0820	−0.0449479
$826$	0	0
$827$	− 231.130i	− 0.279480i	−0.990188	−	0.139740i	$-0.955373\pi$
$827$	− 231.130i	− 0.279480i	0.990188	−	0.139740i	$-0.0446266\pi$
$828$	0	0
$829$	277.817	0.335123	0.167562	−	0.985862i	$-0.446411\pi$
$829$	277.817	0.335123	0.167562	+	0.985862i	$0.446411\pi$
$830$	0	0
$831$	63.0804i	0.0759091i
$832$	0	0
$833$	−779.410	−0.935667
$834$	0	0
$835$	− 598.802i	− 0.717128i
$836$	0	0
$837$	8.49845	0.0101535
$838$	0	0
$839$	975.901i	1.16317i	0.813485	+	0.581586i	$0.197567\pi$
$839$	975.901i	1.16317i	−0.813485	+	0.581586i	$0.802433\pi$
$840$	0	0
$841$	−42.9814	−0.0511075
$842$	0	0
$843$	− 150.978i	− 0.179096i
$844$	0	0
$845$	179.626	0.212575
$846$	0	0
$847$	439.600i	0.519009i
$848$	0	0
$849$	413.823	0.487425
$850$	0	0
$851$	227.655i	0.267514i
$852$	0	0
$853$	−286.571	−0.335957	−0.167978	−	0.985791i	$-0.553724\pi$
$853$	−286.571	−0.335957	−0.167978	+	0.985791i	$0.553724\pi$
$854$	0	0
$855$	196.875i	0.230263i
$856$	0	0
$857$	826.231	0.964096	0.482048	−	0.876145i	$-0.339893\pi$
$857$	826.231	0.964096	0.482048	+	0.876145i	$0.339893\pi$
$858$	0	0
$859$	− 704.746i	− 0.820426i	−0.911990	−	0.410213i	$-0.865454\pi$
$859$	− 704.746i	− 0.820426i	0.911990	−	0.410213i	$-0.134546\pi$
$860$	0	0
$861$	−311.489	−0.361776
$862$	0	0
$863$	191.968i	0.222443i	0.993796	+	0.111221i	$0.0354763\pi$
$863$	191.968i	0.222443i	−0.993796	+	0.111221i	$0.964524\pi$
$864$	0	0
$865$	418.328	0.483616
$866$	0	0
$867$	618.331i	0.713185i
$868$	0	0
$869$	66.3344	0.0763341
$870$	0	0
$871$	− 698.590i	− 0.802055i
$872$	0	0
$873$	97.0031	0.111115
$874$	0	0
$875$	− 47.8727i	− 0.0547117i
$876$	0	0
$877$	928.577	1.05881	0.529406	−	0.848369i	$-0.322415\pi$
$877$	928.577	1.05881	0.529406	+	0.848369i	$0.322415\pi$
$878$	0	0
$879$	378.008i	0.430043i
$880$	0	0
$881$	−183.167	−0.207908	−0.103954	−	0.994582i	$-0.533150\pi$
$881$	−183.167	−0.207908	−0.103954	+	0.994582i	$0.533150\pi$
$882$	0	0
$883$	− 547.805i	− 0.620391i	−0.950673	−	0.310196i	$-0.899605\pi$
$883$	− 547.805i	− 0.620391i	0.950673	−	0.310196i	$-0.100395\pi$
$884$	0	0
$885$	177.580	0.200656
$886$	0	0
$887$	1224.07i	1.38001i	0.723806	+	0.690004i	$0.242391\pi$
$887$	1224.07i	1.38001i	−0.723806	+	0.690004i	$0.757609\pi$
$888$	0	0
$889$	−300.984	−0.338565
$890$	0	0
$891$	38.5368i	0.0432512i
$892$	0	0
$893$	502.663	0.562892
$894$	0	0
$895$	− 266.486i	− 0.297750i
$896$	0	0
$897$	−139.672	−0.155710
$898$	0	0
$899$	− 46.2024i	− 0.0513931i
$900$	0	0
$901$	196.997	0.218643
$902$	0	0
$903$	− 510.960i	− 0.565848i
$904$	0	0
$905$	557.141	0.615625
$906$	0	0
$907$	32.1422i	0.0354379i	0.999843	+	0.0177189i	$0.00564041\pi$
$907$	32.1422i	0.0354379i	−0.999843	+	0.0177189i	$0.994360\pi$
$908$	0	0
$909$	−580.249	−0.638338
$910$	0	0
$911$	425.688i	0.467275i	0.972324	+	0.233638i	$0.0750629\pi$
$911$	425.688i	0.467275i	−0.972324	+	0.233638i	$0.924937\pi$
$912$	0	0
$913$	554.991	0.607876
$914$	0	0
$915$	− 437.000i	− 0.477595i
$916$	0	0
$917$	−526.347	−0.573988
$918$	0	0
$919$	− 1141.31i	− 1.24191i	−0.783847	−	0.620954i	$-0.786745\pi$
$919$	− 1141.31i	− 1.24191i	0.783847	−	0.620954i	$-0.213255\pi$
$920$	0	0
$921$	801.325	0.870060
$922$	0	0
$923$	955.910i	1.03566i
$924$	0	0
$925$	−132.918	−0.143695
$926$	0	0
$927$	− 117.927i	− 0.127214i
$928$	0	0
$929$	−499.495	−0.537670	−0.268835	−	0.963186i	$-0.586639\pi$
$929$	−499.495	−0.537670	−0.268835	+	0.963186i	$0.586639\pi$
$930$	0	0
$931$	899.985i	0.966687i
$932$	0	0
$933$	−993.167	−1.06449
$934$	0	0
$935$	243.350i	0.260268i
$936$	0	0
$937$	−1309.66	−1.39772	−0.698858	−	0.715261i	$-0.746308\pi$
$937$	−1309.66	−1.39772	−0.698858	+	0.715261i	$0.746308\pi$
$938$	0	0
$939$	454.661i	0.484197i
$940$	0	0
$941$	619.240	0.658066	0.329033	−	0.944318i	$-0.393277\pi$
$941$	619.240	0.658066	0.329033	+	0.944318i	$0.393277\pi$
$942$	0	0
$943$	− 359.677i	− 0.381417i
$944$	0	0
$945$	−49.7508	−0.0526463
$946$	0	0
$947$	− 610.250i	− 0.644404i	−0.946671	−	0.322202i	$-0.895577\pi$
$947$	− 610.250i	− 0.644404i	0.946671	−	0.322202i	$-0.104423\pi$
$948$	0	0
$949$	−965.167	−1.01704
$950$	0	0
$951$	37.9576i	0.0399134i
$952$	0	0
$953$	1697.74	1.78147	0.890737	−	0.454519i	$-0.150189\pi$
$953$	1697.74	1.78147	0.890737	+	0.454519i	$0.150189\pi$
$954$	0	0
$955$	− 708.108i	− 0.741475i
$956$	0	0
$957$	209.508	0.218921
$958$	0	0
$959$	− 607.660i	− 0.633639i
$960$	0	0
$961$	958.325	0.997216
$962$	0	0
$963$	− 462.168i	− 0.479925i
$964$	0	0
$965$	−239.633	−0.248324
$966$	0	0
$967$	1573.51i	1.62721i	0.581420	+	0.813603i	$0.302497\pi$
$967$	1573.51i	1.62721i	−0.581420	+	0.813603i	$0.697503\pi$
$968$	0	0
$969$	1291.99	1.33332
$970$	0	0
$971$	− 1508.09i	− 1.55313i	−0.630038	−	0.776564i	$-0.716961\pi$
$971$	− 1508.09i	− 1.55313i	0.630038	−	0.776564i	$-0.283039\pi$
$972$	0	0
$973$	−752.341	−0.773217
$974$	0	0
$975$	− 81.5485i	− 0.0836395i
$976$	0	0
$977$	859.593	0.879829	0.439914	−	0.898040i	$-0.355009\pi$
$977$	859.593	0.879829	0.439914	+	0.898040i	$0.355009\pi$
$978$	0	0
$979$	− 167.709i	− 0.171306i
$980$	0	0
$981$	302.498	0.308357
$982$	0	0
$983$	38.8881i	0.0395606i	0.999804	+	0.0197803i	$0.00629668\pi$
$983$	38.8881i	0.0395606i	−0.999804	+	0.0197803i	$0.993703\pi$
$984$	0	0
$985$	−64.6625	−0.0656472
$986$	0	0
$987$	127.024i	0.128697i
$988$	0	0
$989$	590.006	0.596568
$990$	0	0
$991$	602.571i	0.608044i	0.952665	+	0.304022i	$0.0983296\pi$
$991$	602.571i	0.608044i	−0.952665	+	0.304022i	$0.901670\pi$
$992$	0	0
$993$	100.170	0.100876
$994$	0	0
$995$	710.039i	0.713607i
$996$	0	0
$997$	1720.89	1.72607	0.863036	−	0.505143i	$-0.168560\pi$
$997$	1720.89	1.72607	0.863036	+	0.505143i	$0.168560\pi$
$998$	0	0
$999$	138.132i	0.138271i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.e.b.31.3	yes	4
3.2	odd	2		720.3.e.e.271.4		4
4.3	odd	2	inner	240.3.e.b.31.1	✓	4
5.2	odd	4		1200.3.j.e.799.6		8
5.3	odd	4		1200.3.j.e.799.4		8
5.4	even	2		1200.3.e.j.751.1		4
8.3	odd	2		960.3.e.a.511.4		4
8.5	even	2		960.3.e.a.511.2		4
12.11	even	2		720.3.e.e.271.3		4
15.2	even	4		3600.3.j.p.1999.3		8
15.8	even	4		3600.3.j.p.1999.5		8
15.14	odd	2		3600.3.e.z.3151.2		4
20.3	even	4		1200.3.j.e.799.5		8
20.7	even	4		1200.3.j.e.799.3		8
20.19	odd	2		1200.3.e.j.751.4		4
24.5	odd	2		2880.3.e.d.2431.2		4
24.11	even	2		2880.3.e.d.2431.1		4
60.23	odd	4		3600.3.j.p.1999.4		8
60.47	odd	4		3600.3.j.p.1999.6		8
60.59	even	2		3600.3.e.z.3151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.e.b.31.1	✓	4	4.3	odd	2	inner
240.3.e.b.31.3	yes	4	1.1	even	1	trivial
720.3.e.e.271.3		4	12.11	even	2
720.3.e.e.271.4		4	3.2	odd	2
960.3.e.a.511.2		4	8.5	even	2
960.3.e.a.511.4		4	8.3	odd	2
1200.3.e.j.751.1		4	5.4	even	2
1200.3.e.j.751.4		4	20.19	odd	2
1200.3.j.e.799.3		8	20.7	even	4
1200.3.j.e.799.4		8	5.3	odd	4
1200.3.j.e.799.5		8	20.3	even	4
1200.3.j.e.799.6		8	5.2	odd	4
2880.3.e.d.2431.1		4	24.11	even	2
2880.3.e.d.2431.2		4	24.5	odd	2
3600.3.e.z.3151.2		4	15.14	odd	2
3600.3.e.z.3151.3		4	60.59	even	2
3600.3.j.p.1999.3		8	15.2	even	4
3600.3.j.p.1999.4		8	60.23	odd	4
3600.3.j.p.1999.5		8	15.8	even	4
3600.3.j.p.1999.6		8	60.47	odd	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -2.23607 q^{5} +4.28187i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -2.23607 q^{5} +4.28187i q^{7} -3.00000 q^{9} +4.28187i q^{11} -9.41641 q^{13} -3.87298i q^{15} -25.4164 q^{17} +29.3483i q^{19} -7.41641 q^{21} -8.56373i q^{23} +5.00000 q^{25} -5.19615i q^{27} -28.2492 q^{29} +1.63553i q^{31} -7.41641 q^{33} -9.57454i q^{35} -26.5836 q^{37} -16.3097i q^{39} +42.0000 q^{41} +68.8959i q^{43} +6.70820 q^{45} -17.1275i q^{47} +30.6656 q^{49} -44.0225i q^{51} -7.75078 q^{53} -9.57454i q^{55} -50.8328 q^{57} +45.8511i q^{59} +112.833 q^{61} -12.8456i q^{63} +21.0557 q^{65} +74.1886i q^{67} +14.8328 q^{69} -101.515i q^{71} +102.498 q^{73} +8.66025i q^{75} -18.3344 q^{77} -15.4919i q^{79} +9.00000 q^{81} -129.614i q^{83} +56.8328 q^{85} -48.9291i q^{87} -39.1672 q^{89} -40.3198i q^{91} -2.83282 q^{93} -65.6249i q^{95} -32.3344 q^{97} -12.8456i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 16 q^{13} - 48 q^{17} + 24 q^{21} + 20 q^{25} + 48 q^{29} + 24 q^{33} - 160 q^{37} + 168 q^{41} - 92 q^{49} - 192 q^{53} - 96 q^{57} + 344 q^{61} + 120 q^{65} - 48 q^{69} + 88 q^{73} - 288 q^{77} + 36 q^{81} + 120 q^{85} - 264 q^{89} + 96 q^{93} - 344 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 16 * q^13 - 48 * q^17 + 24 * q^21 + 20 * q^25 + 48 * q^29 + 24 * q^33 - 160 * q^37 + 168 * q^41 - 92 * q^49 - 192 * q^53 - 96 * q^57 + 344 * q^61 + 120 * q^65 - 48 * q^69 + 88 * q^73 - 288 * q^77 + 36 * q^81 + 120 * q^85 - 264 * q^89 + 96 * q^93 - 344 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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