LMFDB - Embedded newform 240.3.e.a.31.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.23607 q^{5} +11.2101i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -2.23607 q^{5} +11.2101i q^{7} -3.00000 q^{9} +4.28187i q^{11} -21.4164 q^{13} +3.87298i q^{15} +25.4164 q^{17} +22.4201i q^{19} +19.4164 q^{21} +37.9121i q^{23} +5.00000 q^{25} +5.19615i q^{27} +1.41641 q^{29} -19.1491i q^{31} +7.41641 q^{33} -25.0665i q^{35} -20.2492 q^{37} +37.0943i q^{39} -30.0000 q^{41} -24.0557i q^{43} +6.70820 q^{45} +70.9176i q^{47} -76.6656 q^{49} -44.0225i q^{51} +64.2492 q^{53} -9.57454i q^{55} +38.8328 q^{57} -88.6697i q^{59} -66.4984 q^{61} -33.6302i q^{63} +47.8885 q^{65} +36.6626i q^{67} +65.6656 q^{69} -133.271i q^{71} -28.8328 q^{73} -8.66025i q^{75} -48.0000 q^{77} +60.7183i q^{79} +9.00000 q^{81} -4.90658i q^{83} -56.8328 q^{85} -2.45329i q^{87} +32.8328 q^{89} -240.079i q^{91} -33.1672 q^{93} -50.1329i q^{95} -14.0000 q^{97} -12.8456i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 32 q^{13} + 48 q^{17} + 24 q^{21} + 20 q^{25} - 48 q^{29} - 24 q^{33} + 80 q^{37} - 120 q^{41} - 92 q^{49} + 96 q^{53} + 48 q^{57} + 56 q^{61} + 120 q^{65} + 48 q^{69} - 8 q^{73} - 192 q^{77} + 36 q^{81} - 120 q^{85} + 24 q^{89} - 240 q^{93} - 56 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 - 32 * q^13 + 48 * q^17 + 24 * q^21 + 20 * q^25 - 48 * q^29 - 24 * q^33 + 80 * q^37 - 120 * q^41 - 92 * q^49 + 96 * q^53 + 48 * q^57 + 56 * q^61 + 120 * q^65 + 48 * q^69 - 8 * q^73 - 192 * q^77 + 36 * q^81 - 120 * q^85 + 24 * q^89 - 240 * q^93 - 56 * 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
$7$	11.2101i	1.60144i	0.599040	+	0.800719i	$0.295549\pi$
$7$	11.2101i	1.60144i	−0.599040	+	0.800719i	$0.704451\pi$
$8$	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
$13$	−21.4164	−1.64742	−0.823708	−	0.567014i	$-0.808098\pi$
$13$	−21.4164	−1.64742	−0.823708	+	0.567014i	$0.808098\pi$
$14$	0	0
$15$	3.87298i	0.258199i
$16$	0	0
$17$	25.4164	1.49508	0.747541	−	0.664215i	$-0.231234\pi$
$17$	25.4164	1.49508	0.747541	+	0.664215i	$0.231234\pi$
$18$	0	0
$19$	22.4201i	1.18001i	0.807401	+	0.590004i	$0.200874\pi$
$19$	22.4201i	1.18001i	−0.807401	+	0.590004i	$0.799126\pi$
$20$	0	0
$21$	19.4164	0.924591
$22$	0	0
$23$	37.9121i	1.64835i	0.566335	+	0.824175i	$0.308361\pi$
$23$	37.9121i	1.64835i	−0.566335	+	0.824175i	$0.691639\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	1.41641	0.0488417	0.0244208	−	0.999702i	$-0.492226\pi$
$29$	1.41641	0.0488417	0.0244208	+	0.999702i	$0.492226\pi$
$30$	0	0
$31$	− 19.1491i	− 0.617712i	−0.951109	−	0.308856i	$-0.900054\pi$
$31$	− 19.1491i	− 0.617712i	0.951109	−	0.308856i	$-0.0999462\pi$
$32$	0	0
$33$	7.41641	0.224740
$34$	0	0
$35$	− 25.0665i	− 0.716185i
$36$	0	0
$37$	−20.2492	−0.547276	−0.273638	−	0.961833i	$-0.588227\pi$
$37$	−20.2492	−0.547276	−0.273638	+	0.961833i	$0.588227\pi$
$38$	0	0
$39$	37.0943i	0.951136i
$40$	0	0
$41$	−30.0000	−0.731707	−0.365854	−	0.930672i	$-0.619223\pi$
$41$	−30.0000	−0.731707	−0.365854	+	0.930672i	$0.619223\pi$
$42$	0	0
$43$	− 24.0557i	− 0.559434i	−0.960082	−	0.279717i	$-0.909759\pi$
$43$	− 24.0557i	− 0.559434i	0.960082	−	0.279717i	$-0.0902406\pi$
$44$	0	0
$45$	6.70820	0.149071
$46$	0	0
$47$	70.9176i	1.50888i	0.656367	+	0.754442i	$0.272092\pi$
$47$	70.9176i	1.50888i	−0.656367	+	0.754442i	$0.727908\pi$
$48$	0	0
$49$	−76.6656	−1.56460
$50$	0	0
$51$	− 44.0225i	− 0.863186i
$52$	0	0
$53$	64.2492	1.21225	0.606125	−	0.795370i	$-0.292723\pi$
$53$	64.2492	1.21225	0.606125	+	0.795370i	$0.292723\pi$
$54$	0	0
$55$	− 9.57454i	− 0.174083i
$56$	0	0
$57$	38.8328	0.681277
$58$	0	0
$59$	− 88.6697i	− 1.50288i	−0.659803	−	0.751438i	$-0.729360\pi$
$59$	− 88.6697i	− 1.50288i	0.659803	−	0.751438i	$-0.270640\pi$
$60$	0	0
$61$	−66.4984	−1.09014	−0.545069	−	0.838391i	$-0.683497\pi$
$61$	−66.4984	−1.09014	−0.545069	+	0.838391i	$0.683497\pi$
$62$	0	0
$63$	− 33.6302i	− 0.533813i
$64$	0	0
$65$	47.8885	0.736747
$66$	0	0
$67$	36.6626i	0.547204i	0.961843	+	0.273602i	$0.0882150\pi$
$67$	36.6626i	0.547204i	−0.961843	+	0.273602i	$0.911785\pi$
$68$	0	0
$69$	65.6656	0.951676
$70$	0	0
$71$	− 133.271i	− 1.87706i	−0.345195	−	0.938531i	$-0.612187\pi$
$71$	− 133.271i	− 1.87706i	0.345195	−	0.938531i	$-0.387813\pi$
$72$	0	0
$73$	−28.8328	−0.394970	−0.197485	−	0.980306i	$-0.563277\pi$
$73$	−28.8328	−0.394970	−0.197485	+	0.980306i	$0.563277\pi$
$74$	0	0
$75$	− 8.66025i	− 0.115470i
$76$	0	0
$77$	−48.0000	−0.623377
$78$	0	0
$79$	60.7183i	0.768586i	0.923211	+	0.384293i	$0.125555\pi$
$79$	60.7183i	0.768586i	−0.923211	+	0.384293i	$0.874445\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 4.90658i	− 0.0591154i	−0.999563	−	0.0295577i	$-0.990590\pi$
$83$	− 4.90658i	− 0.0591154i	0.999563	−	0.0295577i	$-0.00940988\pi$
$84$	0	0
$85$	−56.8328	−0.668621
$86$	0	0
$87$	− 2.45329i	− 0.0281987i
$88$	0	0
$89$	32.8328	0.368908	0.184454	−	0.982841i	$-0.440948\pi$
$89$	32.8328	0.368908	0.184454	+	0.982841i	$0.440948\pi$
$90$	0	0
$91$	− 240.079i	− 2.63824i
$92$	0	0
$93$	−33.1672	−0.356636
$94$	0	0
$95$	− 50.1329i	− 0.527715i
$96$	0	0
$97$	−14.0000	−0.144330	−0.0721649	−	0.997393i	$-0.522991\pi$
$97$	−14.0000	−0.144330	−0.0721649	+	0.997393i	$0.522991\pi$
$98$	0	0
$99$	− 12.8456i	− 0.129753i
$100$	0	0
$101$	108.748	1.07671	0.538355	−	0.842718i	$-0.319046\pi$
$101$	108.748	1.07671	0.538355	+	0.842718i	$0.319046\pi$
$102$	0	0
$103$	14.4811i	0.140593i	0.997526	+	0.0702967i	$0.0223946\pi$
$103$	14.4811i	0.140593i	−0.997526	+	0.0702967i	$0.977605\pi$
$104$	0	0
$105$	−43.4164	−0.413490
$106$	0	0
$107$	19.5352i	0.182572i	0.995825	+	0.0912859i	$0.0290977\pi$
$107$	19.5352i	0.182572i	−0.995825	+	0.0912859i	$0.970902\pi$
$108$	0	0
$109$	43.1672	0.396029	0.198015	−	0.980199i	$-0.436551\pi$
$109$	43.1672	0.396029	0.198015	+	0.980199i	$0.436551\pi$
$110$	0	0
$111$	35.0727i	0.315970i
$112$	0	0
$113$	76.9180	0.680690	0.340345	−	0.940301i	$-0.389456\pi$
$113$	76.9180	0.680690	0.340345	+	0.940301i	$0.389456\pi$
$114$	0	0
$115$	− 84.7740i	− 0.737165i
$116$	0	0
$117$	64.2492	0.549139
$118$	0	0
$119$	284.920i	2.39428i
$120$	0	0
$121$	102.666	0.848476
$122$	0	0
$123$	51.9615i	0.422451i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	184.029i	1.44905i	0.689250	+	0.724524i	$0.257940\pi$
$127$	184.029i	1.44905i	−0.689250	+	0.724524i	$0.742060\pi$
$128$	0	0
$129$	−41.6656	−0.322989
$130$	0	0
$131$	− 53.1654i	− 0.405843i	−0.979195	−	0.202921i	$-0.934956\pi$
$131$	− 53.1654i	− 0.405843i	0.979195	−	0.202921i	$-0.0650436\pi$
$132$	0	0
$133$	−251.331	−1.88971
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	−133.416	−0.973842	−0.486921	−	0.873446i	$-0.661880\pi$
$137$	−133.416	−0.973842	−0.486921	+	0.873446i	$0.661880\pi$
$138$	0	0
$139$	− 86.4095i	− 0.621651i	−0.950467	−	0.310826i	$-0.899395\pi$
$139$	− 86.4095i	− 0.621651i	0.950467	−	0.310826i	$-0.100605\pi$
$140$	0	0
$141$	122.833	0.871155
$142$	0	0
$143$	− 91.7022i	− 0.641274i
$144$	0	0
$145$	−3.16718	−0.0218427
$146$	0	0
$147$	132.789i	0.903325i
$148$	0	0
$149$	−43.0820	−0.289141	−0.144571	−	0.989494i	$-0.546180\pi$
$149$	−43.0820	−0.289141	−0.144571	+	0.989494i	$0.546180\pi$
$150$	0	0
$151$	57.9245i	0.383606i	0.981433	+	0.191803i	$0.0614334\pi$
$151$	57.9245i	0.383606i	−0.981433	+	0.191803i	$0.938567\pi$
$152$	0	0
$153$	−76.2492	−0.498361
$154$	0	0
$155$	42.8187i	0.276249i
$156$	0	0
$157$	−5.41641	−0.0344994	−0.0172497	−	0.999851i	$-0.505491\pi$
$157$	−5.41641	−0.0344994	−0.0172497	+	0.999851i	$0.505491\pi$
$158$	0	0
$159$	− 111.283i	− 0.699893i
$160$	0	0
$161$	−424.997	−2.63973
$162$	0	0
$163$	264.135i	1.62046i	0.586112	+	0.810230i	$0.300658\pi$
$163$	264.135i	1.62046i	−0.586112	+	0.810230i	$0.699342\pi$
$164$	0	0
$165$	−16.5836	−0.100507
$166$	0	0
$167$	231.130i	1.38401i	0.721893	+	0.692005i	$0.243272\pi$
$167$	231.130i	1.38401i	−0.721893	+	0.692005i	$0.756728\pi$
$168$	0	0
$169$	289.663	1.71398
$170$	0	0
$171$	− 67.2604i	− 0.393336i
$172$	0	0
$173$	333.915	1.93014	0.965072	−	0.261985i	$-0.0843772\pi$
$173$	333.915	1.93014	0.965072	+	0.261985i	$0.0843772\pi$
$174$	0	0
$175$	56.0503i	0.320288i
$176$	0	0
$177$	−153.580	−0.867686
$178$	0	0
$179$	148.616i	0.830256i	0.909763	+	0.415128i	$0.136263\pi$
$179$	148.616i	0.830256i	−0.909763	+	0.415128i	$0.863737\pi$
$180$	0	0
$181$	168.833	0.932778	0.466389	−	0.884580i	$-0.345555\pi$
$181$	168.833	0.932778	0.466389	+	0.884580i	$0.345555\pi$
$182$	0	0
$183$	115.179i	0.629392i
$184$	0	0
$185$	45.2786	0.244749
$186$	0	0
$187$	108.830i	0.581977i
$188$	0	0
$189$	−58.2492	−0.308197
$190$	0	0
$191$	− 284.920i	− 1.49173i	−0.666099	−	0.745863i	$-0.732037\pi$
$191$	− 284.920i	− 1.49173i	0.666099	−	0.745863i	$-0.267963\pi$
$192$	0	0
$193$	338.827	1.75558	0.877789	−	0.479047i	$-0.159018\pi$
$193$	338.827	1.75558	0.877789	+	0.479047i	$0.159018\pi$
$194$	0	0
$195$	− 82.9454i	− 0.425361i
$196$	0	0
$197$	45.9149	0.233070	0.116535	−	0.993187i	$-0.462821\pi$
$197$	45.9149	0.233070	0.116535	+	0.993187i	$0.462821\pi$
$198$	0	0
$199$	124.708i	0.626672i	0.949642	+	0.313336i	$0.101447\pi$
$199$	124.708i	0.626672i	−0.949642	+	0.313336i	$0.898553\pi$
$200$	0	0
$201$	63.5016	0.315928
$202$	0	0
$203$	15.8780i	0.0782169i
$204$	0	0
$205$	67.0820	0.327229
$206$	0	0
$207$	− 113.736i	− 0.549450i
$208$	0	0
$209$	−96.0000	−0.459330
$210$	0	0
$211$	269.042i	1.27508i	0.770418	+	0.637539i	$0.220048\pi$
$211$	269.042i	1.27508i	−0.770418	+	0.637539i	$0.779952\pi$
$212$	0	0
$213$	−230.833	−1.08372
$214$	0	0
$215$	53.7901i	0.250187i
$216$	0	0
$217$	214.663	0.989228
$218$	0	0
$219$	49.9399i	0.228036i
$220$	0	0
$221$	−544.328	−2.46302
$222$	0	0
$223$	− 254.561i	− 1.14153i	−0.821115	−	0.570763i	$-0.806647\pi$
$223$	− 254.561i	− 1.14153i	0.821115	−	0.570763i	$-0.193353\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	− 138.178i	− 0.608714i	−0.952558	−	0.304357i	$-0.901559\pi$
$227$	− 138.178i	− 0.608714i	0.952558	−	0.304357i	$-0.0984415\pi$
$228$	0	0
$229$	−366.997	−1.60261	−0.801303	−	0.598258i	$-0.795860\pi$
$229$	−366.997	−1.60261	−0.801303	+	0.598258i	$0.795860\pi$
$230$	0	0
$231$	83.1384i	0.359907i
$232$	0	0
$233$	206.085	0.884486	0.442243	−	0.896895i	$-0.354183\pi$
$233$	206.085	0.884486	0.442243	+	0.896895i	$0.354183\pi$
$234$	0	0
$235$	− 158.576i	− 0.674794i
$236$	0	0
$237$	105.167	0.443743
$238$	0	0
$239$	− 11.0626i	− 0.0462870i	−0.999732	−	0.0231435i	$-0.992633\pi$
$239$	− 11.0626i	− 0.0462870i	0.999732	−	0.0231435i	$-0.00736746\pi$
$240$	0	0
$241$	326.328	1.35406	0.677029	−	0.735956i	$-0.263267\pi$
$241$	326.328	1.35406	0.677029	+	0.735956i	$0.263267\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	171.430	0.699713
$246$	0	0
$247$	− 480.159i	− 1.94396i
$248$	0	0
$249$	−8.49845	−0.0341303
$250$	0	0
$251$	362.527i	1.44433i	0.691721	+	0.722165i	$0.256853\pi$
$251$	362.527i	1.44433i	−0.691721	+	0.722165i	$0.743147\pi$
$252$	0	0
$253$	−162.334	−0.641638
$254$	0	0
$255$	98.4373i	0.386029i
$256$	0	0
$257$	−263.410	−1.02494	−0.512471	−	0.858704i	$-0.671270\pi$
$257$	−263.410	−1.02494	−0.512471	+	0.858704i	$0.671270\pi$
$258$	0	0
$259$	− 226.995i	− 0.876429i
$260$	0	0
$261$	−4.24922	−0.0162806
$262$	0	0
$263$	− 262.886i	− 0.999565i	−0.866151	−	0.499783i	$-0.833413\pi$
$263$	− 262.886i	− 0.999565i	0.866151	−	0.499783i	$-0.166587\pi$
$264$	0	0
$265$	−143.666	−0.542134
$266$	0	0
$267$	− 56.8681i	− 0.212989i
$268$	0	0
$269$	79.0820	0.293985	0.146993	−	0.989138i	$-0.453041\pi$
$269$	79.0820	0.293985	0.146993	+	0.989138i	$0.453041\pi$
$270$	0	0
$271$	− 461.487i	− 1.70290i	−0.524433	−	0.851452i	$-0.675723\pi$
$271$	− 461.487i	− 1.70290i	0.524433	−	0.851452i	$-0.324277\pi$
$272$	0	0
$273$	−415.830	−1.52319
$274$	0	0
$275$	21.4093i	0.0778521i
$276$	0	0
$277$	377.416	1.36251	0.681257	−	0.732044i	$-0.261434\pi$
$277$	377.416	1.36251	0.681257	+	0.732044i	$0.261434\pi$
$278$	0	0
$279$	57.4472i	0.205904i
$280$	0	0
$281$	133.161	0.473882	0.236941	−	0.971524i	$-0.423855\pi$
$281$	133.161	0.473882	0.236941	+	0.971524i	$0.423855\pi$
$282$	0	0
$283$	− 62.8311i	− 0.222018i	−0.993819	−	0.111009i	$-0.964592\pi$
$283$	− 62.8311i	− 0.222018i	0.993819	−	0.111009i	$-0.0354082\pi$
$284$	0	0
$285$	−86.8328	−0.304677
$286$	0	0
$287$	− 336.302i	− 1.17178i
$288$	0	0
$289$	356.994	1.23527
$290$	0	0
$291$	24.2487i	0.0833289i
$292$	0	0
$293$	−95.4102	−0.325632	−0.162816	−	0.986656i	$-0.552058\pi$
$293$	−95.4102	−0.325632	−0.162816	+	0.986656i	$0.552058\pi$
$294$	0	0
$295$	198.272i	0.672107i
$296$	0	0
$297$	−22.2492	−0.0749132
$298$	0	0
$299$	− 811.940i	− 2.71552i
$300$	0	0
$301$	269.666	0.895899
$302$	0	0
$303$	− 188.356i	− 0.621639i
$304$	0	0
$305$	148.695	0.487525
$306$	0	0
$307$	261.341i	0.851274i	0.904894	+	0.425637i	$0.139950\pi$
$307$	261.341i	0.851274i	−0.904894	+	0.425637i	$0.860050\pi$
$308$	0	0
$309$	25.0820	0.0811716
$310$	0	0
$311$	147.900i	0.475563i	0.971319	+	0.237781i	$0.0764202\pi$
$311$	147.900i	0.475563i	−0.971319	+	0.237781i	$0.923580\pi$
$312$	0	0
$313$	−426.498	−1.36261	−0.681307	−	0.731997i	$-0.738588\pi$
$313$	−426.498	−1.36261	−0.681307	+	0.731997i	$0.738588\pi$
$314$	0	0
$315$	75.1994i	0.238728i
$316$	0	0
$317$	−164.420	−0.518674	−0.259337	−	0.965787i	$-0.583504\pi$
$317$	−164.420	−0.518674	−0.259337	+	0.965787i	$0.583504\pi$
$318$	0	0
$319$	6.06487i	0.0190121i
$320$	0	0
$321$	33.8359	0.105408
$322$	0	0
$323$	569.839i	1.76421i
$324$	0	0
$325$	−107.082	−0.329483
$326$	0	0
$327$	− 74.7678i	− 0.228648i
$328$	0	0
$329$	−794.991	−2.41639
$330$	0	0
$331$	428.299i	1.29396i	0.762509	+	0.646978i	$0.223967\pi$
$331$	428.299i	1.29396i	−0.762509	+	0.646978i	$0.776033\pi$
$332$	0	0
$333$	60.7477	0.182425
$334$	0	0
$335$	− 81.9802i	− 0.244717i
$336$	0	0
$337$	359.495	1.06675	0.533376	−	0.845878i	$-0.320923\pi$
$337$	359.495	1.06675	0.533376	+	0.845878i	$0.320923\pi$
$338$	0	0
$339$	− 133.226i	− 0.392997i
$340$	0	0
$341$	81.9938	0.240451
$342$	0	0
$343$	− 310.134i	− 0.904180i
$344$	0	0
$345$	−146.833	−0.425602
$346$	0	0
$347$	− 301.956i	− 0.870190i	−0.900385	−	0.435095i	$-0.856715\pi$
$347$	− 301.956i	− 0.870190i	0.900385	−	0.435095i	$-0.143285\pi$
$348$	0	0
$349$	40.8328	0.116999	0.0584997	−	0.998287i	$-0.481368\pi$
$349$	40.8328	0.116999	0.0584997	+	0.998287i	$0.481368\pi$
$350$	0	0
$351$	− 111.283i	− 0.317045i
$352$	0	0
$353$	557.745	1.58001	0.790006	−	0.613099i	$-0.210077\pi$
$353$	557.745	1.58001	0.790006	+	0.613099i	$0.210077\pi$
$354$	0	0
$355$	298.004i	0.839448i
$356$	0	0
$357$	493.495	1.38234
$358$	0	0
$359$	− 17.1275i	− 0.0477088i	−0.999715	−	0.0238544i	$-0.992406\pi$
$359$	− 17.1275i	− 0.0477088i	0.999715	−	0.0238544i	$-0.00759381\pi$
$360$	0	0
$361$	−141.663	−0.392417
$362$	0	0
$363$	− 177.822i	− 0.489868i
$364$	0	0
$365$	64.4721	0.176636
$366$	0	0
$367$	− 520.331i	− 1.41780i	−0.705311	−	0.708898i	$-0.749193\pi$
$367$	− 520.331i	− 1.41780i	0.705311	−	0.708898i	$-0.250807\pi$
$368$	0	0
$369$	90.0000	0.243902
$370$	0	0
$371$	720.238i	1.94134i
$372$	0	0
$373$	−238.413	−0.639178	−0.319589	−	0.947556i	$-0.603545\pi$
$373$	−238.413	−0.639178	−0.319589	+	0.947556i	$0.603545\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−30.3344	−0.0804625
$378$	0	0
$379$	80.8219i	0.213250i	0.994299	+	0.106625i	$0.0340045\pi$
$379$	80.8219i	0.213250i	−0.994299	+	0.106625i	$0.965996\pi$
$380$	0	0
$381$	318.748	0.836608
$382$	0	0
$383$	− 53.7901i	− 0.140444i	−0.997531	−	0.0702221i	$-0.977629\pi$
$383$	− 53.7901i	− 0.140444i	0.997531	−	0.0702221i	$-0.0223708\pi$
$384$	0	0
$385$	107.331	0.278783
$386$	0	0
$387$	72.1670i	0.186478i
$388$	0	0
$389$	667.240	1.71527	0.857635	−	0.514259i	$-0.171933\pi$
$389$	667.240	1.71527	0.857635	+	0.514259i	$0.171933\pi$
$390$	0	0
$391$	963.589i	2.46442i
$392$	0	0
$393$	−92.0851	−0.234313
$394$	0	0
$395$	− 135.770i	− 0.343722i
$396$	0	0
$397$	−630.243	−1.58751	−0.793757	−	0.608235i	$-0.791878\pi$
$397$	−630.243	−1.58751	−0.793757	+	0.608235i	$0.791878\pi$
$398$	0	0
$399$	435.319i	1.09102i
$400$	0	0
$401$	−666.158	−1.66124	−0.830621	−	0.556839i	$-0.812014\pi$
$401$	−666.158	−1.66124	−0.830621	+	0.556839i	$0.812014\pi$
$402$	0	0
$403$	410.105i	1.01763i
$404$	0	0
$405$	−20.1246	−0.0496904
$406$	0	0
$407$	− 86.7044i	− 0.213033i
$408$	0	0
$409$	155.337	0.379798	0.189899	−	0.981804i	$-0.439184\pi$
$409$	155.337	0.379798	0.189899	+	0.981804i	$0.439184\pi$
$410$	0	0
$411$	231.084i	0.562248i
$412$	0	0
$413$	993.994	2.40676
$414$	0	0
$415$	10.9714i	0.0264372i
$416$	0	0
$417$	−149.666	−0.358910
$418$	0	0
$419$	− 40.8534i	− 0.0975020i	−0.998811	−	0.0487510i	$-0.984476\pi$
$419$	− 40.8534i	− 0.0975020i	0.998811	−	0.0487510i	$-0.0155241\pi$
$420$	0	0
$421$	−496.991	−1.18050	−0.590250	−	0.807220i	$-0.700971\pi$
$421$	−496.991	−1.18050	−0.590250	+	0.807220i	$0.700971\pi$
$422$	0	0
$423$	− 212.753i	− 0.502961i
$424$	0	0
$425$	127.082	0.299017
$426$	0	0
$427$	− 745.452i	− 1.74579i
$428$	0	0
$429$	−158.833	−0.370240
$430$	0	0
$431$	344.866i	0.800153i	0.916482	+	0.400076i	$0.131016\pi$
$431$	344.866i	0.800153i	−0.916482	+	0.400076i	$0.868984\pi$
$432$	0	0
$433$	−186.498	−0.430712	−0.215356	−	0.976536i	$-0.569091\pi$
$433$	−186.498	−0.430712	−0.215356	+	0.976536i	$0.569091\pi$
$434$	0	0
$435$	5.48572i	0.0126109i
$436$	0	0
$437$	−849.994	−1.94507
$438$	0	0
$439$	265.771i	0.605400i	0.953086	+	0.302700i	$0.0978880\pi$
$439$	265.771i	0.605400i	−0.953086	+	0.302700i	$0.902112\pi$
$440$	0	0
$441$	229.997	0.521535
$442$	0	0
$443$	524.613i	1.18423i	0.805854	+	0.592114i	$0.201706\pi$
$443$	524.613i	1.18423i	−0.805854	+	0.592114i	$0.798294\pi$
$444$	0	0
$445$	−73.4164	−0.164981
$446$	0	0
$447$	74.6203i	0.166936i
$448$	0	0
$449$	−33.5016	−0.0746137	−0.0373069	−	0.999304i	$-0.511878\pi$
$449$	−33.5016	−0.0746137	−0.0373069	+	0.999304i	$0.511878\pi$
$450$	0	0
$451$	− 128.456i	− 0.284825i
$452$	0	0
$453$	100.328	0.221475
$454$	0	0
$455$	536.834i	1.17985i
$456$	0	0
$457$	−461.659	−1.01020	−0.505098	−	0.863062i	$-0.668544\pi$
$457$	−461.659	−1.01020	−0.505098	+	0.863062i	$0.668544\pi$
$458$	0	0
$459$	132.068i	0.287729i
$460$	0	0
$461$	−559.751	−1.21421	−0.607105	−	0.794622i	$-0.707669\pi$
$461$	−559.751	−1.21421	−0.607105	+	0.794622i	$0.707669\pi$
$462$	0	0
$463$	− 157.861i	− 0.340952i	−0.985362	−	0.170476i	$-0.945470\pi$
$463$	− 157.861i	− 0.340952i	0.985362	−	0.170476i	$-0.0545305\pi$
$464$	0	0
$465$	74.1641	0.159493
$466$	0	0
$467$	457.353i	0.979342i	0.871907	+	0.489671i	$0.162883\pi$
$467$	457.353i	0.979342i	−0.871907	+	0.489671i	$0.837117\pi$
$468$	0	0
$469$	−410.991	−0.876313
$470$	0	0
$471$	9.38149i	0.0199182i
$472$	0	0
$473$	103.003	0.217766
$474$	0	0
$475$	112.101i	0.236001i
$476$	0	0
$477$	−192.748	−0.404083
$478$	0	0
$479$	37.8209i	0.0789581i	0.999220	+	0.0394790i	$0.0125698\pi$
$479$	37.8209i	0.0789581i	−0.999220	+	0.0394790i	$0.987430\pi$
$480$	0	0
$481$	433.666	0.901592
$482$	0	0
$483$	736.116i	1.52405i
$484$	0	0
$485$	31.3050	0.0645463
$486$	0	0
$487$	767.430i	1.57583i	0.615783	+	0.787916i	$0.288840\pi$
$487$	767.430i	1.57583i	−0.615783	+	0.787916i	$0.711160\pi$
$488$	0	0
$489$	457.495	0.935573
$490$	0	0
$491$	108.114i	0.220191i	0.993921	+	0.110095i	$0.0351157\pi$
$491$	108.114i	0.220191i	−0.993921	+	0.110095i	$0.964884\pi$
$492$	0	0
$493$	36.0000	0.0730223
$494$	0	0
$495$	28.7236i	0.0580275i
$496$	0	0
$497$	1493.98	3.00600
$498$	0	0
$499$	− 675.398i	− 1.35350i	−0.736212	−	0.676751i	$-0.763387\pi$
$499$	− 675.398i	− 1.35350i	0.736212	−	0.676751i	$-0.236613\pi$
$500$	0	0
$501$	400.328	0.799058
$502$	0	0
$503$	− 721.396i	− 1.43419i	−0.696977	−	0.717094i	$-0.745472\pi$
$503$	− 721.396i	− 1.43419i	0.696977	−	0.717094i	$-0.254528\pi$
$504$	0	0
$505$	−243.167	−0.481519
$506$	0	0
$507$	− 501.710i	− 0.989566i
$508$	0	0
$509$	458.754	0.901285	0.450642	−	0.892705i	$-0.351195\pi$
$509$	458.754	0.901285	0.450642	+	0.892705i	$0.351195\pi$
$510$	0	0
$511$	− 323.218i	− 0.632520i
$512$	0	0
$513$	−116.498	−0.227092
$514$	0	0
$515$	− 32.3808i	− 0.0628753i
$516$	0	0
$517$	−303.659	−0.587349
$518$	0	0
$519$	− 578.357i	− 1.11437i
$520$	0	0
$521$	151.495	0.290778	0.145389	−	0.989375i	$-0.453557\pi$
$521$	151.495	0.290778	0.145389	+	0.989375i	$0.453557\pi$
$522$	0	0
$523$	− 5.86106i	− 0.0112066i	−0.999984	−	0.00560331i	$-0.998216\pi$
$523$	− 5.86106i	− 0.0112066i	0.999984	−	0.00560331i	$-0.00178360\pi$
$524$	0	0
$525$	97.0820	0.184918
$526$	0	0
$527$	− 486.701i	− 0.923531i
$528$	0	0
$529$	−908.325	−1.71706
$530$	0	0
$531$	266.009i	0.500959i
$532$	0	0
$533$	642.492	1.20543
$534$	0	0
$535$	− 43.6820i	− 0.0816486i
$536$	0	0
$537$	257.410	0.479349
$538$	0	0
$539$	− 328.272i	− 0.609039i
$540$	0	0
$541$	239.495	0.442690	0.221345	−	0.975196i	$-0.428955\pi$
$541$	239.495	0.442690	0.221345	+	0.975196i	$0.428955\pi$
$542$	0	0
$543$	− 292.427i	− 0.538540i
$544$	0	0
$545$	−96.5248	−0.177110
$546$	0	0
$547$	− 117.007i	− 0.213907i	−0.994264	−	0.106954i	$-0.965890\pi$
$547$	− 117.007i	− 0.213907i	0.994264	−	0.106954i	$-0.0341096\pi$
$548$	0	0
$549$	199.495	0.363379
$550$	0	0
$551$	31.7561i	0.0576335i
$552$	0	0
$553$	−680.656	−1.23084
$554$	0	0
$555$	− 78.4249i	− 0.141306i
$556$	0	0
$557$	−530.085	−0.951679	−0.475839	−	0.879532i	$-0.657856\pi$
$557$	−530.085	−0.951679	−0.475839	+	0.879532i	$0.657856\pi$
$558$	0	0
$559$	515.186i	0.921621i
$560$	0	0
$561$	188.498	0.336004
$562$	0	0
$563$	− 957.615i	− 1.70091i	−0.526044	−	0.850457i	$-0.676325\pi$
$563$	− 957.615i	− 1.70091i	0.526044	−	0.850457i	$-0.323675\pi$
$564$	0	0
$565$	−171.994	−0.304414
$566$	0	0
$567$	100.891i	0.177938i
$568$	0	0
$569$	592.152	1.04069	0.520344	−	0.853957i	$-0.325804\pi$
$569$	592.152	1.04069	0.520344	+	0.853957i	$0.325804\pi$
$570$	0	0
$571$	633.829i	1.11003i	0.831839	+	0.555016i	$0.187288\pi$
$571$	633.829i	1.11003i	−0.831839	+	0.555016i	$0.812712\pi$
$572$	0	0
$573$	−493.495	−0.861248
$574$	0	0
$575$	189.560i	0.329670i
$576$	0	0
$577$	−286.000	−0.495667	−0.247834	−	0.968803i	$-0.579719\pi$
$577$	−286.000	−0.495667	−0.247834	+	0.968803i	$0.579719\pi$
$578$	0	0
$579$	− 586.865i	− 1.01358i
$580$	0	0
$581$	55.0031	0.0946697
$582$	0	0
$583$	275.107i	0.471881i
$584$	0	0
$585$	−143.666	−0.245582
$586$	0	0
$587$	− 303.388i	− 0.516844i	−0.966032	−	0.258422i	$-0.916797\pi$
$587$	− 303.388i	− 0.516844i	0.966032	−	0.258422i	$-0.0832026\pi$
$588$	0	0
$589$	429.325	0.728905
$590$	0	0
$591$	− 79.5269i	− 0.134563i
$592$	0	0
$593$	−1031.41	−1.73931	−0.869654	−	0.493661i	$-0.835658\pi$
$593$	−1031.41	−1.73931	−0.869654	+	0.493661i	$0.835658\pi$
$594$	0	0
$595$	− 637.100i	− 1.07076i
$596$	0	0
$597$	216.000	0.361809
$598$	0	0
$599$	1056.54i	1.76384i	0.471399	+	0.881920i	$0.343749\pi$
$599$	1056.54i	1.76384i	−0.471399	+	0.881920i	$0.656251\pi$
$600$	0	0
$601$	249.003	0.414315	0.207157	−	0.978308i	$-0.433579\pi$
$601$	249.003	0.414315	0.207157	+	0.978308i	$0.433579\pi$
$602$	0	0
$603$	− 109.988i	− 0.182401i
$604$	0	0
$605$	−229.567	−0.379450
$606$	0	0
$607$	− 92.0319i	− 0.151618i	−0.997122	−	0.0758088i	$-0.975846\pi$
$607$	− 92.0319i	− 0.151618i	0.997122	−	0.0758088i	$-0.0241539\pi$
$608$	0	0
$609$	27.5016	0.0451585
$610$	0	0
$611$	− 1518.80i	− 2.48576i
$612$	0	0
$613$	−109.246	−0.178216	−0.0891078	−	0.996022i	$-0.528402\pi$
$613$	−109.246	−0.178216	−0.0891078	+	0.996022i	$0.528402\pi$
$614$	0	0
$615$	− 116.190i	− 0.188926i
$616$	0	0
$617$	−252.906	−0.409896	−0.204948	−	0.978773i	$-0.565702\pi$
$617$	−252.906	−0.409896	−0.204948	+	0.978773i	$0.565702\pi$
$618$	0	0
$619$	− 1130.82i	− 1.82685i	−0.407007	−	0.913425i	$-0.633428\pi$
$619$	− 1130.82i	− 1.82685i	0.407007	−	0.913425i	$-0.366572\pi$
$620$	0	0
$621$	−196.997	−0.317225
$622$	0	0
$623$	368.058i	0.590783i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	166.277i	0.265194i
$628$	0	0
$629$	−514.663	−0.818223
$630$	0	0
$631$	− 627.287i	− 0.994115i	−0.867718	−	0.497058i	$-0.834414\pi$
$631$	− 627.287i	− 0.994115i	0.867718	−	0.497058i	$-0.165586\pi$
$632$	0	0
$633$	465.994	0.736167
$634$	0	0
$635$	− 411.501i	− 0.648034i
$636$	0	0
$637$	1641.90	2.57755
$638$	0	0
$639$	399.814i	0.625687i
$640$	0	0
$641$	351.325	0.548089	0.274045	−	0.961717i	$-0.411638\pi$
$641$	351.325	0.548089	0.274045	+	0.961717i	$0.411638\pi$
$642$	0	0
$643$	− 107.671i	− 0.167452i	−0.996489	−	0.0837258i	$-0.973318\pi$
$643$	− 107.671i	− 0.167452i	0.996489	−	0.0837258i	$-0.0266820\pi$
$644$	0	0
$645$	93.1672	0.144445
$646$	0	0
$647$	− 767.963i	− 1.18696i	−0.804849	−	0.593480i	$-0.797754\pi$
$647$	− 767.963i	− 1.18696i	0.804849	−	0.593480i	$-0.202246\pi$
$648$	0	0
$649$	379.672	0.585011
$650$	0	0
$651$	− 371.806i	− 0.571131i
$652$	0	0
$653$	−476.735	−0.730069	−0.365035	−	0.930994i	$-0.618943\pi$
$653$	−476.735	−0.730069	−0.365035	+	0.930994i	$0.618943\pi$
$654$	0	0
$655$	118.881i	0.181498i
$656$	0	0
$657$	86.4984	0.131657
$658$	0	0
$659$	976.434i	1.48169i	0.671676	+	0.740845i	$0.265575\pi$
$659$	976.434i	1.48169i	−0.671676	+	0.740845i	$0.734425\pi$
$660$	0	0
$661$	−1060.49	−1.60438	−0.802188	−	0.597072i	$-0.796331\pi$
$661$	−1060.49	−1.60438	−0.802188	+	0.597072i	$0.796331\pi$
$662$	0	0
$663$	942.804i	1.42203i
$664$	0	0
$665$	561.994	0.845103
$666$	0	0
$667$	53.6990i	0.0805082i
$668$	0	0
$669$	−440.912	−0.659061
$670$	0	0
$671$	− 284.737i	− 0.424348i
$672$	0	0
$673$	272.492	0.404892	0.202446	−	0.979293i	$-0.435111\pi$
$673$	272.492	0.404892	0.202446	+	0.979293i	$0.435111\pi$
$674$	0	0
$675$	25.9808i	0.0384900i
$676$	0	0
$677$	201.246	0.297262	0.148631	−	0.988893i	$-0.452513\pi$
$677$	201.246	0.297262	0.148631	+	0.988893i	$0.452513\pi$
$678$	0	0
$679$	− 156.941i	− 0.231135i
$680$	0	0
$681$	−239.331	−0.351441
$682$	0	0
$683$	− 93.0427i	− 0.136227i	−0.997678	−	0.0681133i	$-0.978302\pi$
$683$	− 93.0427i	− 0.136227i	0.997678	−	0.0681133i	$-0.0216979\pi$
$684$	0	0
$685$	298.328	0.435516
$686$	0	0
$687$	635.657i	0.925265i
$688$	0	0
$689$	−1375.99	−1.99708
$690$	0	0
$691$	− 370.375i	− 0.535998i	−0.963419	−	0.267999i	$-0.913638\pi$
$691$	− 370.375i	− 0.535998i	0.963419	−	0.267999i	$-0.0863624\pi$
$692$	0	0
$693$	144.000	0.207792
$694$	0	0
$695$	193.217i	0.278011i
$696$	0	0
$697$	−762.492	−1.09396
$698$	0	0
$699$	− 356.950i	− 0.510658i
$700$	0	0
$701$	840.079	1.19840	0.599200	−	0.800599i	$-0.295485\pi$
$701$	840.079	1.19840	0.599200	+	0.800599i	$0.295485\pi$
$702$	0	0
$703$	− 453.990i	− 0.645790i
$704$	0	0
$705$	−274.663	−0.389592
$706$	0	0
$707$	1219.07i	1.72428i
$708$	0	0
$709$	−484.006	−0.682660	−0.341330	−	0.939943i	$-0.610877\pi$
$709$	−484.006	−0.682660	−0.341330	+	0.939943i	$0.610877\pi$
$710$	0	0
$711$	− 182.155i	− 0.256195i
$712$	0	0
$713$	725.981	1.01821
$714$	0	0
$715$	205.052i	0.286786i
$716$	0	0
$717$	−19.1610	−0.0267238
$718$	0	0
$719$	46.3847i	0.0645127i	0.999480	+	0.0322564i	$0.0102693\pi$
$719$	46.3847i	0.0645127i	−0.999480	+	0.0322564i	$0.989731\pi$
$720$	0	0
$721$	−162.334	−0.225152
$722$	0	0
$723$	− 565.217i	− 0.781766i
$724$	0	0
$725$	7.08204	0.00976833
$726$	0	0
$727$	287.271i	0.395146i	0.980288	+	0.197573i	$0.0633059\pi$
$727$	287.271i	0.395146i	−0.980288	+	0.197573i	$0.936694\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 611.409i	− 0.836400i
$732$	0	0
$733$	57.0758	0.0778661	0.0389330	−	0.999242i	$-0.487604\pi$
$733$	57.0758	0.0778661	0.0389330	+	0.999242i	$0.487604\pi$
$734$	0	0
$735$	− 296.925i	− 0.403979i
$736$	0	0
$737$	−156.984	−0.213005
$738$	0	0
$739$	265.293i	0.358990i	0.983759	+	0.179495i	$0.0574463\pi$
$739$	265.293i	0.358990i	−0.983759	+	0.179495i	$0.942554\pi$
$740$	0	0
$741$	−831.659	−1.12235
$742$	0	0
$743$	237.377i	0.319484i	0.987159	+	0.159742i	$0.0510663\pi$
$743$	237.377i	0.319484i	−0.987159	+	0.159742i	$0.948934\pi$
$744$	0	0
$745$	96.3344	0.129308
$746$	0	0
$747$	14.7197i	0.0197051i
$748$	0	0
$749$	−218.991	−0.292377
$750$	0	0
$751$	− 658.088i	− 0.876283i	−0.898906	−	0.438141i	$-0.855637\pi$
$751$	− 658.088i	− 0.876283i	0.898906	−	0.438141i	$-0.144363\pi$
$752$	0	0
$753$	627.915	0.833884
$754$	0	0
$755$	− 129.523i	− 0.171554i
$756$	0	0
$757$	422.741	0.558443	0.279222	−	0.960227i	$-0.409924\pi$
$757$	422.741	0.558443	0.279222	+	0.960227i	$0.409924\pi$
$758$	0	0
$759$	281.171i	0.370450i
$760$	0	0
$761$	−522.158	−0.686147	−0.343074	−	0.939309i	$-0.611468\pi$
$761$	−522.158	−0.686147	−0.343074	+	0.939309i	$0.611468\pi$
$762$	0	0
$763$	483.907i	0.634216i
$764$	0	0
$765$	170.498	0.222874
$766$	0	0
$767$	1898.99i	2.47586i
$768$	0	0
$769$	1351.98	1.75810	0.879052	−	0.476727i	$-0.158177\pi$
$769$	1351.98	1.75810	0.879052	+	0.476727i	$0.158177\pi$
$770$	0	0
$771$	456.240i	0.591751i
$772$	0	0
$773$	408.906	0.528985	0.264493	−	0.964388i	$-0.414796\pi$
$773$	408.906	0.528985	0.264493	+	0.964388i	$0.414796\pi$
$774$	0	0
$775$	− 95.7454i	− 0.123542i
$776$	0	0
$777$	−393.167	−0.506007
$778$	0	0
$779$	− 672.604i	− 0.863420i
$780$	0	0
$781$	570.650	0.730666
$782$	0	0
$783$	7.35987i	0.00939958i
$784$	0	0
$785$	12.1115	0.0154286
$786$	0	0
$787$	1238.90i	1.57420i	0.616823	+	0.787102i	$0.288420\pi$
$787$	1238.90i	1.57420i	−0.616823	+	0.787102i	$0.711580\pi$
$788$	0	0
$789$	−455.331	−0.577099
$790$	0	0
$791$	862.256i	1.09008i
$792$	0	0
$793$	1424.16	1.79591
$794$	0	0
$795$	248.836i	0.313001i
$796$	0	0
$797$	−694.073	−0.870857	−0.435428	−	0.900223i	$-0.643403\pi$
$797$	−694.073	−0.870857	−0.435428	+	0.900223i	$0.643403\pi$
$798$	0	0
$799$	1802.47i	2.25591i
$800$	0	0
$801$	−98.4984	−0.122969
$802$	0	0
$803$	− 123.458i	− 0.153746i
$804$	0	0
$805$	950.322	1.18052
$806$	0	0
$807$	− 136.974i	− 0.169732i
$808$	0	0
$809$	735.325	0.908931	0.454465	−	0.890764i	$-0.349830\pi$
$809$	735.325	0.908931	0.454465	+	0.890764i	$0.349830\pi$
$810$	0	0
$811$	− 121.437i	− 0.149737i	−0.997193	−	0.0748684i	$-0.976146\pi$
$811$	− 121.437i	− 0.149737i	0.997193	−	0.0748684i	$-0.0238537\pi$
$812$	0	0
$813$	−799.319	−0.983172
$814$	0	0
$815$	− 590.624i	− 0.724692i
$816$	0	0
$817$	539.331	0.660136
$818$	0	0
$819$	720.238i	0.879412i
$820$	0	0
$821$	237.088	0.288780	0.144390	−	0.989521i	$-0.453878\pi$
$821$	237.088	0.288780	0.144390	+	0.989521i	$0.453878\pi$
$822$	0	0
$823$	737.990i	0.896708i	0.893856	+	0.448354i	$0.147990\pi$
$823$	737.990i	0.896708i	−0.893856	+	0.448354i	$0.852010\pi$
$824$	0	0
$825$	37.0820	0.0449479
$826$	0	0
$827$	593.305i	0.717418i	0.933449	+	0.358709i	$0.116783\pi$
$827$	593.305i	0.717418i	−0.933449	+	0.358709i	$0.883217\pi$
$828$	0	0
$829$	650.486	0.784663	0.392332	−	0.919824i	$-0.371669\pi$
$829$	650.486	0.784663	0.392332	+	0.919824i	$0.371669\pi$
$830$	0	0
$831$	− 653.704i	− 0.786648i
$832$	0	0
$833$	−1948.56	−2.33921
$834$	0	0
$835$	− 516.821i	− 0.618948i
$836$	0	0
$837$	99.5016	0.118879
$838$	0	0
$839$	− 112.396i	− 0.133964i	−0.997754	−	0.0669819i	$-0.978663\pi$
$839$	− 112.396i	− 0.133964i	0.997754	−	0.0669819i	$-0.0213370\pi$
$840$	0	0
$841$	−838.994	−0.997614
$842$	0	0
$843$	− 230.642i	− 0.273596i
$844$	0	0
$845$	−647.705	−0.766515
$846$	0	0
$847$	1150.89i	1.35878i
$848$	0	0
$849$	−108.827	−0.128182
$850$	0	0
$851$	− 767.690i	− 0.902103i
$852$	0	0
$853$	653.404	0.766007	0.383004	−	0.923747i	$-0.374890\pi$
$853$	653.404	0.766007	0.383004	+	0.923747i	$0.374890\pi$
$854$	0	0
$855$	150.399i	0.175905i
$856$	0	0
$857$	1131.07	1.31980	0.659901	−	0.751353i	$-0.270598\pi$
$857$	1131.07	1.31980	0.659901	+	0.751353i	$0.270598\pi$
$858$	0	0
$859$	1329.81i	1.54809i	0.633132	+	0.774044i	$0.281769\pi$
$859$	1329.81i	1.54809i	−0.633132	+	0.774044i	$0.718231\pi$
$860$	0	0
$861$	−582.492	−0.676530
$862$	0	0
$863$	487.859i	0.565306i	0.959222	+	0.282653i	$0.0912145\pi$
$863$	487.859i	0.565306i	−0.959222	+	0.282653i	$0.908785\pi$
$864$	0	0
$865$	−746.656	−0.863186
$866$	0	0
$867$	− 618.331i	− 0.713185i
$868$	0	0
$869$	−259.988	−0.299180
$870$	0	0
$871$	− 785.182i	− 0.901472i
$872$	0	0
$873$	42.0000	0.0481100
$874$	0	0
$875$	− 125.332i	− 0.143237i
$876$	0	0
$877$	654.231	0.745987	0.372993	−	0.927834i	$-0.378331\pi$
$877$	654.231	0.745987	0.372993	+	0.927834i	$0.378331\pi$
$878$	0	0
$879$	165.255i	0.188004i
$880$	0	0
$881$	371.508	0.421689	0.210844	−	0.977520i	$-0.432379\pi$
$881$	371.508	0.421689	0.210844	+	0.977520i	$0.432379\pi$
$882$	0	0
$883$	949.823i	1.07568i	0.843048	+	0.537839i	$0.180759\pi$
$883$	949.823i	1.07568i	−0.843048	+	0.537839i	$0.819241\pi$
$884$	0	0
$885$	343.416	0.388041
$886$	0	0
$887$	− 485.725i	− 0.547604i	−0.961786	−	0.273802i	$-0.911719\pi$
$887$	− 485.725i	− 0.547604i	0.961786	−	0.273802i	$-0.0882813\pi$
$888$	0	0
$889$	−2062.98	−2.32056
$890$	0	0
$891$	38.5368i	0.0432512i
$892$	0	0
$893$	−1589.98	−1.78049
$894$	0	0
$895$	− 332.315i	− 0.371302i
$896$	0	0
$897$	−1406.32	−1.56781
$898$	0	0
$899$	− 27.1229i	− 0.0301701i
$900$	0	0
$901$	1632.98	1.81241
$902$	0	0
$903$	− 467.075i	− 0.517248i
$904$	0	0
$905$	−377.522	−0.417151
$906$	0	0
$907$	− 296.368i	− 0.326757i	−0.986563	−	0.163378i	$-0.947761\pi$
$907$	− 296.368i	− 0.326757i	0.986563	−	0.163378i	$-0.0522391\pi$
$908$	0	0
$909$	−326.243	−0.358903
$910$	0	0
$911$	− 332.371i	− 0.364842i	−0.983220	−	0.182421i	$-0.941607\pi$
$911$	− 332.371i	− 0.364842i	0.983220	−	0.182421i	$-0.0583934\pi$
$912$	0	0
$913$	21.0093	0.0230113
$914$	0	0
$915$	− 257.547i	− 0.281473i
$916$	0	0
$917$	595.988	0.649932
$918$	0	0
$919$	− 663.268i	− 0.721728i	−0.932618	−	0.360864i	$-0.882482\pi$
$919$	− 663.268i	− 0.721728i	0.932618	−	0.360864i	$-0.117518\pi$
$920$	0	0
$921$	452.656	0.491484
$922$	0	0
$923$	2854.19i	3.09230i
$924$	0	0
$925$	−101.246	−0.109455
$926$	0	0
$927$	− 43.4434i	− 0.0468645i
$928$	0	0
$929$	729.149	0.784875	0.392437	−	0.919779i	$-0.371632\pi$
$929$	729.149	0.784875	0.392437	+	0.919779i	$0.371632\pi$
$930$	0	0
$931$	− 1718.85i	− 1.84624i
$932$	0	0
$933$	256.170	0.274566
$934$	0	0
$935$	− 243.350i	− 0.260268i
$936$	0	0
$937$	−1671.00	−1.78335	−0.891674	−	0.452678i	$-0.850469\pi$
$937$	−1671.00	−1.78335	−0.891674	+	0.452678i	$0.850469\pi$
$938$	0	0
$939$	738.717i	0.786706i
$940$	0	0
$941$	−584.067	−0.620687	−0.310344	−	0.950624i	$-0.600444\pi$
$941$	−584.067	−0.620687	−0.310344	+	0.950624i	$0.600444\pi$
$942$	0	0
$943$	− 1137.36i	− 1.20611i
$944$	0	0
$945$	130.249	0.137830
$946$	0	0
$947$	− 495.356i	− 0.523079i	−0.965193	−	0.261539i	$-0.915770\pi$
$947$	− 495.356i	− 0.523079i	0.965193	−	0.261539i	$-0.0842302\pi$
$948$	0	0
$949$	617.495	0.650680
$950$	0	0
$951$	284.783i	0.299456i
$952$	0	0
$953$	−894.413	−0.938524	−0.469262	−	0.883059i	$-0.655480\pi$
$953$	−894.413	−0.938524	−0.469262	+	0.883059i	$0.655480\pi$
$954$	0	0
$955$	637.100i	0.667120i
$956$	0	0
$957$	10.5047	0.0109767
$958$	0	0
$959$	− 1495.61i	− 1.55955i
$960$	0	0
$961$	594.313	0.618431
$962$	0	0
$963$	− 58.6055i	− 0.0608573i
$964$	0	0
$965$	−757.639	−0.785118
$966$	0	0
$967$	− 174.693i	− 0.180655i	−0.995912	−	0.0903274i	$-0.971209\pi$
$967$	− 174.693i	− 0.180655i	0.995912	−	0.0903274i	$-0.0287913\pi$
$968$	0	0
$969$	986.991	1.01857
$970$	0	0
$971$	445.665i	0.458976i	0.973312	+	0.229488i	$0.0737051\pi$
$971$	445.665i	0.458976i	−0.973312	+	0.229488i	$0.926295\pi$
$972$	0	0
$973$	968.656	0.995536
$974$	0	0
$975$	185.472i	0.190227i
$976$	0	0
$977$	859.751	0.879991	0.439995	−	0.898000i	$-0.354980\pi$
$977$	859.751	0.879991	0.439995	+	0.898000i	$0.354980\pi$
$978$	0	0
$979$	140.586i	0.143601i
$980$	0	0
$981$	−129.502	−0.132010
$982$	0	0
$983$	− 1291.60i	− 1.31394i	−0.753918	−	0.656969i	$-0.771838\pi$
$983$	− 1291.60i	− 1.31394i	0.753918	−	0.656969i	$-0.228162\pi$
$984$	0	0
$985$	−102.669	−0.104232
$986$	0	0
$987$	1376.96i	1.39510i
$988$	0	0
$989$	912.000	0.922144
$990$	0	0
$991$	− 165.322i	− 0.166824i	−0.996515	−	0.0834119i	$-0.973418\pi$
$991$	− 165.322i	− 0.166824i	0.996515	−	0.0834119i	$-0.0265817\pi$
$992$	0	0
$993$	741.836	0.747065
$994$	0	0
$995$	− 278.855i	− 0.280256i
$996$	0	0
$997$	506.924	0.508450	0.254225	−	0.967145i	$-0.418180\pi$
$997$	506.924	0.508450	0.254225	+	0.967145i	$0.418180\pi$
$998$	0	0
$999$	− 105.218i	− 0.105323i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.3.e.a.31.1	✓	4
3.2	odd	2		720.3.e.a.271.4		4
4.3	odd	2	inner	240.3.e.a.31.3	yes	4
5.2	odd	4		1200.3.j.f.799.2		8
5.3	odd	4		1200.3.j.f.799.8		8
5.4	even	2		1200.3.e.n.751.3		4
8.3	odd	2		960.3.e.b.511.2		4
8.5	even	2		960.3.e.b.511.4		4
12.11	even	2		720.3.e.a.271.3		4
15.2	even	4		3600.3.j.l.1999.1		8
15.8	even	4		3600.3.j.l.1999.7		8
15.14	odd	2		3600.3.e.bh.3151.1		4
20.3	even	4		1200.3.j.f.799.1		8
20.7	even	4		1200.3.j.f.799.7		8
20.19	odd	2		1200.3.e.n.751.2		4
24.5	odd	2		2880.3.e.f.2431.2		4
24.11	even	2		2880.3.e.f.2431.1		4
60.23	odd	4		3600.3.j.l.1999.2		8
60.47	odd	4		3600.3.j.l.1999.8		8
60.59	even	2		3600.3.e.bh.3151.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.e.a.31.1	✓	4	1.1	even	1	trivial
240.3.e.a.31.3	yes	4	4.3	odd	2	inner
720.3.e.a.271.3		4	12.11	even	2
720.3.e.a.271.4		4	3.2	odd	2
960.3.e.b.511.2		4	8.3	odd	2
960.3.e.b.511.4		4	8.5	even	2
1200.3.e.n.751.2		4	20.19	odd	2
1200.3.e.n.751.3		4	5.4	even	2
1200.3.j.f.799.1		8	20.3	even	4
1200.3.j.f.799.2		8	5.2	odd	4
1200.3.j.f.799.7		8	20.7	even	4
1200.3.j.f.799.8		8	5.3	odd	4
2880.3.e.f.2431.1		4	24.11	even	2
2880.3.e.f.2431.2		4	24.5	odd	2
3600.3.e.bh.3151.1		4	15.14	odd	2
3600.3.e.bh.3151.4		4	60.59	even	2
3600.3.j.l.1999.1		8	15.2	even	4
3600.3.j.l.1999.2		8	60.23	odd	4
3600.3.j.l.1999.7		8	15.8	even	4
3600.3.j.l.1999.8		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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.23607 q^{5} +11.2101i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -2.23607 q^{5} +11.2101i q^{7} -3.00000 q^{9} +4.28187i q^{11} -21.4164 q^{13} +3.87298i q^{15} +25.4164 q^{17} +22.4201i q^{19} +19.4164 q^{21} +37.9121i q^{23} +5.00000 q^{25} +5.19615i q^{27} +1.41641 q^{29} -19.1491i q^{31} +7.41641 q^{33} -25.0665i q^{35} -20.2492 q^{37} +37.0943i q^{39} -30.0000 q^{41} -24.0557i q^{43} +6.70820 q^{45} +70.9176i q^{47} -76.6656 q^{49} -44.0225i q^{51} +64.2492 q^{53} -9.57454i q^{55} +38.8328 q^{57} -88.6697i q^{59} -66.4984 q^{61} -33.6302i q^{63} +47.8885 q^{65} +36.6626i q^{67} +65.6656 q^{69} -133.271i q^{71} -28.8328 q^{73} -8.66025i q^{75} -48.0000 q^{77} +60.7183i q^{79} +9.00000 q^{81} -4.90658i q^{83} -56.8328 q^{85} -2.45329i q^{87} +32.8328 q^{89} -240.079i q^{91} -33.1672 q^{93} -50.1329i q^{95} -14.0000 q^{97} -12.8456i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 32 q^{13} + 48 q^{17} + 24 q^{21} + 20 q^{25} - 48 q^{29} - 24 q^{33} + 80 q^{37} - 120 q^{41} - 92 q^{49} + 96 q^{53} + 48 q^{57} + 56 q^{61} + 120 q^{65} + 48 q^{69} - 8 q^{73} - 192 q^{77} + 36 q^{81} - 120 q^{85} + 24 q^{89} - 240 q^{93} - 56 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 - 32 * q^13 + 48 * q^17 + 24 * q^21 + 20 * q^25 - 48 * q^29 - 24 * q^33 + 80 * q^37 - 120 * q^41 - 92 * q^49 + 96 * q^53 + 48 * q^57 + 56 * q^61 + 120 * q^65 + 48 * q^69 - 8 * q^73 - 192 * q^77 + 36 * q^81 - 120 * q^85 + 24 * q^89 - 240 * q^93 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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