LMFDB - Embedded newform 2400.3.p.a.1999.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,3,Mod(1999,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2400.1999");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2400.p (of order $2$, degree $1$, not 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:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.3
Root			$1.20036 - 0.747754i$ of defining polynomial
Character	$\chi$	$=$	2400.1999
Dual form			2400.3.p.a.1999.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +2.13878 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +2.13878 q^{7} -3.00000 q^{9} +8.00000 q^{11} +11.6865 q^{13} +11.8564i q^{17} +14.9282 q^{19} -3.70447i q^{21} +4.27756 q^{23} +5.19615i q^{27} -0.573084i q^{29} +57.4399i q^{31} -13.8564i q^{33} -27.6506 q^{37} -20.2416i q^{39} -31.5692 q^{41} +28.7846i q^{43} +59.5787 q^{47} -44.4256 q^{49} +20.5359 q^{51} +31.3550 q^{53} -25.8564i q^{57} -52.7846 q^{59} -59.5787i q^{61} -6.41634 q^{63} +84.7846i q^{67} -7.40895i q^{69} +42.4685i q^{71} +5.42563i q^{73} +17.1102 q^{77} +44.6072i q^{79} +9.00000 q^{81} +67.7128i q^{83} -0.992611 q^{87} +133.138 q^{89} +24.9948 q^{91} +99.4888 q^{93} +97.1384i q^{97} -24.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 64 q^{11} + 64 q^{19} + 80 q^{41} + 88 q^{49} + 192 q^{51} - 256 q^{59} + 72 q^{81} + 400 q^{89} - 576 q^{91} - 192 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 64 * q^11 + 64 * q^19 + 80 * q^41 + 88 * q^49 + 192 * q^51 - 256 * q^59 + 72 * q^81 + 400 * q^89 - 576 * q^91 - 192 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.13878	0.305540	0.152770	−	0.988262i	$-0.451181\pi$
$7$	2.13878	0.305540	0.152770	+	0.988262i	$0.451181\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	8.00000	0.727273	0.363636	−	0.931541i	$-0.381535\pi$
$11$	8.00000	0.727273	0.363636	+	0.931541i	$0.381535\pi$
$12$	0	0
$13$	11.6865	0.898962	0.449481	−	0.893290i	$-0.351609\pi$
$13$	11.6865	0.898962	0.449481	+	0.893290i	$0.351609\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11.8564i	0.697436i	0.937228	+	0.348718i	$0.113383\pi$
$17$	11.8564i	0.697436i	−0.937228	+	0.348718i	$0.886617\pi$
$18$	0	0
$19$	14.9282	0.785695	0.392847	−	0.919604i	$-0.371490\pi$
$19$	14.9282	0.785695	0.392847	+	0.919604i	$0.371490\pi$
$20$	0	0
$21$	− 3.70447i	− 0.176403i
$22$	0	0
$23$	4.27756	0.185981	0.0929904	−	0.995667i	$-0.470357\pi$
$23$	4.27756	0.185981	0.0929904	+	0.995667i	$0.470357\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	− 0.573084i	− 0.0197615i	−0.999951	−	0.00988076i	$-0.996855\pi$
$29$	− 0.573084i	− 0.0197615i	0.999951	−	0.00988076i	$-0.00314519\pi$
$30$	0	0
$31$	57.4399i	1.85290i	0.376417	+	0.926450i	$0.377156\pi$
$31$	57.4399i	1.85290i	−0.376417	+	0.926450i	$0.622844\pi$
$32$	0	0
$33$	− 13.8564i	− 0.419891i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−27.6506	−0.747313	−0.373656	−	0.927567i	$-0.621896\pi$
$37$	−27.6506	−0.747313	−0.373656	+	0.927567i	$0.621896\pi$
$38$	0	0
$39$	− 20.2416i	− 0.519016i
$40$	0	0
$41$	−31.5692	−0.769981	−0.384990	−	0.922921i	$-0.625795\pi$
$41$	−31.5692	−0.769981	−0.384990	+	0.922921i	$0.625795\pi$
$42$	0	0
$43$	28.7846i	0.669410i	0.942323	+	0.334705i	$0.108637\pi$
$43$	28.7846i	0.669410i	−0.942323	+	0.334705i	$0.891363\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	59.5787	1.26763	0.633816	−	0.773484i	$-0.281488\pi$
$47$	59.5787	1.26763	0.633816	+	0.773484i	$0.281488\pi$
$48$	0	0
$49$	−44.4256	−0.906645
$50$	0	0
$51$	20.5359	0.402665
$52$	0	0
$53$	31.3550	0.591604	0.295802	−	0.955249i	$-0.404413\pi$
$53$	31.3550	0.591604	0.295802	+	0.955249i	$0.404413\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 25.8564i	− 0.453621i
$58$	0	0
$59$	−52.7846	−0.894654	−0.447327	−	0.894370i	$-0.647624\pi$
$59$	−52.7846	−0.894654	−0.447327	+	0.894370i	$0.647624\pi$
$60$	0	0
$61$	− 59.5787i	− 0.976700i	−0.872648	−	0.488350i	$-0.837599\pi$
$61$	− 59.5787i	− 0.976700i	0.872648	−	0.488350i	$-0.162401\pi$
$62$	0	0
$63$	−6.41634	−0.101847
$64$	0	0
$65$	0	0
$66$	0	0
$67$	84.7846i	1.26544i	0.774380	+	0.632721i	$0.218062\pi$
$67$	84.7846i	1.26544i	−0.774380	+	0.632721i	$0.781938\pi$
$68$	0	0
$69$	− 7.40895i	− 0.107376i
$70$	0	0
$71$	42.4685i	0.598147i	0.954230	+	0.299074i	$0.0966776\pi$
$71$	42.4685i	0.598147i	−0.954230	+	0.299074i	$0.903322\pi$
$72$	0	0
$73$	5.42563i	0.0743236i	0.999309	+	0.0371618i	$0.0118317\pi$
$73$	5.42563i	0.0743236i	−0.999309	+	0.0371618i	$0.988168\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.1102	0.222211
$78$	0	0
$79$	44.6072i	0.564649i	0.959319	+	0.282324i	$0.0911054\pi$
$79$	44.6072i	0.564649i	−0.959319	+	0.282324i	$0.908895\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	67.7128i	0.815817i	0.913023	+	0.407909i	$0.133742\pi$
$83$	67.7128i	0.815817i	−0.913023	+	0.407909i	$0.866258\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.992611	−0.0114093
$88$	0	0
$89$	133.138	1.49594	0.747969	−	0.663734i	$-0.231029\pi$
$89$	133.138	1.49594	0.747969	+	0.663734i	$0.231029\pi$
$90$	0	0
$91$	24.9948	0.274669
$92$	0	0
$93$	99.4888	1.06977
$94$	0	0
$95$	0	0
$96$	0	0
$97$	97.1384i	1.00143i	0.865613	+	0.500714i	$0.166929\pi$
$97$	97.1384i	1.00143i	−0.865613	+	0.500714i	$0.833071\pi$
$98$	0	0
$99$	−24.0000	−0.242424
$100$	0	0
$101$	− 62.1370i	− 0.615218i	−0.951513	−	0.307609i	$-0.900471\pi$
$101$	− 62.1370i	− 0.615218i	0.951513	−	0.307609i	$-0.0995288\pi$
$102$	0	0
$103$	27.8041	0.269943	0.134971	−	0.990849i	$-0.456906\pi$
$103$	27.8041	0.269943	0.134971	+	0.990849i	$0.456906\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 37.7795i	− 0.353079i	−0.984294	−	0.176540i	$-0.943510\pi$
$107$	− 37.7795i	− 0.353079i	0.984294	−	0.176540i	$-0.0564903\pi$
$108$	0	0
$109$	− 141.691i	− 1.29992i	−0.759968	−	0.649960i	$-0.774786\pi$
$109$	− 141.691i	− 1.29992i	0.759968	−	0.649960i	$-0.225214\pi$
$110$	0	0
$111$	47.8922i	0.431461i
$112$	0	0
$113$	58.2872i	0.515816i	0.966170	+	0.257908i	$0.0830331\pi$
$113$	58.2872i	0.515816i	−0.966170	+	0.257908i	$0.916967\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−35.0595	−0.299654
$118$	0	0
$119$	25.3582i	0.213094i
$120$	0	0
$121$	−57.0000	−0.471074
$122$	0	0
$123$	54.6795i	0.444549i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	185.152	1.45789	0.728946	−	0.684571i	$-0.240010\pi$
$127$	185.152	1.45789	0.728946	+	0.684571i	$0.240010\pi$
$128$	0	0
$129$	49.8564	0.386484
$130$	0	0
$131$	−125.359	−0.956939	−0.478469	−	0.878104i	$-0.658808\pi$
$131$	−125.359	−0.956939	−0.478469	+	0.878104i	$0.658808\pi$
$132$	0	0
$133$	31.9281	0.240061
$134$	0	0
$135$	0	0
$136$	0	0
$137$	99.5692i	0.726783i	0.931637	+	0.363391i	$0.118381\pi$
$137$	99.5692i	0.726783i	−0.931637	+	0.363391i	$0.881619\pi$
$138$	0	0
$139$	177.492	1.27692	0.638461	−	0.769654i	$-0.279571\pi$
$139$	177.492	1.27692	0.638461	+	0.769654i	$0.279571\pi$
$140$	0	0
$141$	− 103.193i	− 0.731867i
$142$	0	0
$143$	93.4920	0.653790
$144$	0	0
$145$	0	0
$146$	0	0
$147$	76.9474i	0.523452i
$148$	0	0
$149$	− 87.8023i	− 0.589277i	−0.955609	−	0.294639i	$-0.904801\pi$
$149$	− 87.8023i	− 0.589277i	0.955609	−	0.294639i	$-0.0951993\pi$
$150$	0	0
$151$	219.066i	1.45077i	0.688345	+	0.725383i	$0.258337\pi$
$151$	219.066i	1.45077i	−0.688345	+	0.725383i	$0.741663\pi$
$152$	0	0
$153$	− 35.5692i	− 0.232479i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	253.440	1.61427	0.807133	−	0.590370i	$-0.201018\pi$
$157$	253.440	1.61427	0.807133	+	0.590370i	$0.201018\pi$
$158$	0	0
$159$	− 54.3085i	− 0.341563i
$160$	0	0
$161$	9.14875	0.0568245
$162$	0	0
$163$	− 102.354i	− 0.627938i	−0.949433	−	0.313969i	$-0.898341\pi$
$163$	− 102.354i	− 0.627938i	0.949433	−	0.313969i	$-0.101659\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	281.090	1.68318	0.841588	−	0.540120i	$-0.181621\pi$
$167$	281.090	1.68318	0.841588	+	0.540120i	$0.181621\pi$
$168$	0	0
$169$	−32.4256	−0.191868
$170$	0	0
$171$	−44.7846	−0.261898
$172$	0	0
$173$	−242.858	−1.40381	−0.701903	−	0.712273i	$-0.747666\pi$
$173$	−242.858	−1.40381	−0.701903	+	0.712273i	$0.747666\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	91.4256i	0.516529i
$178$	0	0
$179$	318.354	1.77851	0.889257	−	0.457409i	$-0.151222\pi$
$179$	318.354	1.77851	0.889257	+	0.457409i	$0.151222\pi$
$180$	0	0
$181$	79.5132i	0.439299i	0.975579	+	0.219650i	$0.0704914\pi$
$181$	79.5132i	0.439299i	−0.975579	+	0.219650i	$0.929509\pi$
$182$	0	0
$183$	−103.193	−0.563898
$184$	0	0
$185$	0	0
$186$	0	0
$187$	94.8513i	0.507226i
$188$	0	0
$189$	11.1134i	0.0588012i
$190$	0	0
$191$	− 352.887i	− 1.84758i	−0.382902	−	0.923789i	$-0.625075\pi$
$191$	− 352.887i	− 1.84758i	0.382902	−	0.923789i	$-0.374925\pi$
$192$	0	0
$193$	284.277i	1.47294i	0.676472	+	0.736469i	$0.263508\pi$
$193$	284.277i	1.47294i	−0.676472	+	0.736469i	$0.736492\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−75.8087	−0.384816	−0.192408	−	0.981315i	$-0.561630\pi$
$197$	−75.8087	−0.384816	−0.192408	+	0.981315i	$0.561630\pi$
$198$	0	0
$199$	− 104.186i	− 0.523547i	−0.965129	−	0.261774i	$-0.915693\pi$
$199$	− 104.186i	− 0.523547i	0.965129	−	0.261774i	$-0.0843074\pi$
$200$	0	0
$201$	146.851	0.730603
$202$	0	0
$203$	− 1.22570i	− 0.00603793i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−12.8327	−0.0619936
$208$	0	0
$209$	119.426	0.571414
$210$	0	0
$211$	−136.918	−0.648900	−0.324450	−	0.945903i	$-0.605179\pi$
$211$	−136.918	−0.648900	−0.324450	+	0.945903i	$0.605179\pi$
$212$	0	0
$213$	73.5575	0.345341
$214$	0	0
$215$	0	0
$216$	0	0
$217$	122.851i	0.566135i
$218$	0	0
$219$	9.39746	0.0429108
$220$	0	0
$221$	138.560i	0.626968i
$222$	0	0
$223$	53.1624	0.238396	0.119198	−	0.992870i	$-0.461968\pi$
$223$	53.1624	0.238396	0.119198	+	0.992870i	$0.461968\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	119.846i	0.527956i	0.964529	+	0.263978i	$0.0850347\pi$
$227$	119.846i	0.527956i	−0.964529	+	0.263978i	$0.914965\pi$
$228$	0	0
$229$	− 214.103i	− 0.934946i	−0.884007	−	0.467473i	$-0.845164\pi$
$229$	− 214.103i	− 0.934946i	0.884007	−	0.467473i	$-0.154836\pi$
$230$	0	0
$231$	− 29.6358i	− 0.128293i
$232$	0	0
$233$	127.436i	0.546935i	0.961881	+	0.273468i	$0.0881707\pi$
$233$	127.436i	0.546935i	−0.961881	+	0.273468i	$0.911829\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	77.2620	0.326000
$238$	0	0
$239$	− 319.281i	− 1.33590i	−0.744204	−	0.667952i	$-0.767171\pi$
$239$	− 319.281i	− 1.33590i	0.744204	−	0.667952i	$-0.232829\pi$
$240$	0	0
$241$	−247.415	−1.02662	−0.513310	−	0.858203i	$-0.671581\pi$
$241$	−247.415	−1.02662	−0.513310	+	0.858203i	$0.671581\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	174.459	0.706310
$248$	0	0
$249$	117.282	0.471012
$250$	0	0
$251$	214.851	0.855981	0.427991	−	0.903783i	$-0.359222\pi$
$251$	214.851	0.855981	0.427991	+	0.903783i	$0.359222\pi$
$252$	0	0
$253$	34.2205	0.135259
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 84.2769i	− 0.327926i	−0.986467	−	0.163963i	$-0.947572\pi$
$257$	− 84.2769i	− 0.327926i	0.986467	−	0.163963i	$-0.0524277\pi$
$258$	0	0
$259$	−59.1384	−0.228334
$260$	0	0
$261$	1.71925i	0.00658717i
$262$	0	0
$263$	277.120	1.05369	0.526844	−	0.849962i	$-0.323375\pi$
$263$	277.120	1.05369	0.526844	+	0.849962i	$0.323375\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 230.603i	− 0.863680i
$268$	0	0
$269$	− 123.701i	− 0.459855i	−0.973208	−	0.229927i	$-0.926151\pi$
$269$	− 123.701i	− 0.459855i	0.973208	−	0.229927i	$-0.0738489\pi$
$270$	0	0
$271$	197.985i	0.730572i	0.930895	+	0.365286i	$0.119029\pi$
$271$	197.985i	0.730572i	−0.930895	+	0.365286i	$0.880971\pi$
$272$	0	0
$273$	− 43.2923i	− 0.158580i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	247.709	0.894256	0.447128	−	0.894470i	$-0.352447\pi$
$277$	247.709	0.894256	0.447128	+	0.894470i	$0.352447\pi$
$278$	0	0
$279$	− 172.320i	− 0.617633i
$280$	0	0
$281$	443.128	1.57697	0.788484	−	0.615055i	$-0.210866\pi$
$281$	443.128	1.57697	0.788484	+	0.615055i	$0.210866\pi$
$282$	0	0
$283$	− 294.620i	− 1.04106i	−0.853843	−	0.520531i	$-0.825734\pi$
$283$	− 294.620i	− 1.04106i	0.853843	−	0.520531i	$-0.174266\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−67.5196	−0.235260
$288$	0	0
$289$	148.426	0.513583
$290$	0	0
$291$	168.249	0.578174
$292$	0	0
$293$	−66.7217	−0.227719	−0.113859	−	0.993497i	$-0.536321\pi$
$293$	−66.7217	−0.227719	−0.113859	+	0.993497i	$0.536321\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	41.5692i	0.139964i
$298$	0	0
$299$	49.9897	0.167190
$300$	0	0
$301$	61.5639i	0.204531i
$302$	0	0
$303$	−107.624	−0.355196
$304$	0	0
$305$	0	0
$306$	0	0
$307$	524.210i	1.70753i	0.520662	+	0.853763i	$0.325685\pi$
$307$	524.210i	1.70753i	−0.520662	+	0.853763i	$0.674315\pi$
$308$	0	0
$309$	− 48.1582i	− 0.155852i
$310$	0	0
$311$	− 362.057i	− 1.16417i	−0.813128	−	0.582085i	$-0.802237\pi$
$311$	− 362.057i	− 1.16417i	0.813128	−	0.582085i	$-0.197763\pi$
$312$	0	0
$313$	− 252.277i	− 0.805996i	−0.915201	−	0.402998i	$-0.867968\pi$
$313$	− 252.277i	− 0.805996i	0.915201	−	0.402998i	$-0.132032\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	80.3934	0.253607	0.126803	−	0.991928i	$-0.459528\pi$
$317$	80.3934	0.253607	0.126803	+	0.991928i	$0.459528\pi$
$318$	0	0
$319$	− 4.58467i	− 0.0143720i
$320$	0	0
$321$	−65.4359	−0.203850
$322$	0	0
$323$	176.995i	0.547972i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−245.417	−0.750509
$328$	0	0
$329$	127.426	0.387312
$330$	0	0
$331$	−172.056	−0.519808	−0.259904	−	0.965635i	$-0.583691\pi$
$331$	−172.056	−0.519808	−0.259904	+	0.965635i	$0.583691\pi$
$332$	0	0
$333$	82.9517	0.249104
$334$	0	0
$335$	0	0
$336$	0	0
$337$	564.277i	1.67441i	0.546888	+	0.837206i	$0.315813\pi$
$337$	564.277i	1.67441i	−0.546888	+	0.837206i	$0.684187\pi$
$338$	0	0
$339$	100.956	0.297806
$340$	0	0
$341$	459.519i	1.34756i
$342$	0	0
$343$	−199.817	−0.582556
$344$	0	0
$345$	0	0
$346$	0	0
$347$	286.123i	0.824562i	0.911057	+	0.412281i	$0.135268\pi$
$347$	286.123i	0.824562i	−0.911057	+	0.412281i	$0.864732\pi$
$348$	0	0
$349$	421.021i	1.20636i	0.797603	+	0.603182i	$0.206101\pi$
$349$	421.021i	1.20636i	−0.797603	+	0.603182i	$0.793899\pi$
$350$	0	0
$351$	60.7249i	0.173005i
$352$	0	0
$353$	− 429.138i	− 1.21569i	−0.794056	−	0.607845i	$-0.792034\pi$
$353$	− 429.138i	− 1.21569i	0.794056	−	0.607845i	$-0.207966\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	43.9217	0.123030
$358$	0	0
$359$	− 263.673i	− 0.734465i	−0.930129	−	0.367233i	$-0.880305\pi$
$359$	− 263.673i	− 0.734465i	0.930129	−	0.367233i	$-0.119695\pi$
$360$	0	0
$361$	−138.149	−0.382684
$362$	0	0
$363$	98.7269i	0.271975i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	129.544	0.352981	0.176491	−	0.984302i	$-0.443525\pi$
$367$	129.544	0.352981	0.176491	+	0.984302i	$0.443525\pi$
$368$	0	0
$369$	94.7077	0.256660
$370$	0	0
$371$	67.0615	0.180759
$372$	0	0
$373$	302.478	0.810933	0.405467	−	0.914110i	$-0.367109\pi$
$373$	302.478	0.810933	0.405467	+	0.914110i	$0.367109\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 6.69735i	− 0.0177649i
$378$	0	0
$379$	−116.210	−0.306623	−0.153312	−	0.988178i	$-0.548994\pi$
$379$	−116.210	−0.306623	−0.153312	+	0.988178i	$0.548994\pi$
$380$	0	0
$381$	− 320.693i	− 0.841715i
$382$	0	0
$383$	566.151	1.47820	0.739101	−	0.673595i	$-0.235251\pi$
$383$	566.151	1.47820	0.739101	+	0.673595i	$0.235251\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 86.3538i	− 0.223137i
$388$	0	0
$389$	− 350.104i	− 0.900011i	−0.893026	−	0.450006i	$-0.851422\pi$
$389$	− 350.104i	− 0.900011i	0.893026	−	0.450006i	$-0.148578\pi$
$390$	0	0
$391$	50.7165i	0.129710i
$392$	0	0
$393$	217.128i	0.552489i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	544.149	1.37065	0.685326	−	0.728236i	$-0.259660\pi$
$397$	544.149	1.37065	0.685326	+	0.728236i	$0.259660\pi$
$398$	0	0
$399$	− 55.3011i	− 0.138599i
$400$	0	0
$401$	−296.431	−0.739229	−0.369614	−	0.929185i	$-0.620510\pi$
$401$	−296.431	−0.739229	−0.369614	+	0.929185i	$0.620510\pi$
$402$	0	0
$403$	671.272i	1.66569i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−221.205	−0.543500
$408$	0	0
$409$	247.415	0.604927	0.302464	−	0.953161i	$-0.402191\pi$
$409$	247.415	0.604927	0.302464	+	0.953161i	$0.402191\pi$
$410$	0	0
$411$	172.459	0.419608
$412$	0	0
$413$	−112.895	−0.273353
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 307.426i	− 0.737232i
$418$	0	0
$419$	−92.1333	−0.219889	−0.109944	−	0.993938i	$-0.535067\pi$
$419$	−92.1333	−0.219889	−0.109944	+	0.993938i	$0.535067\pi$
$420$	0	0
$421$	445.540i	1.05829i	0.848531	+	0.529145i	$0.177487\pi$
$421$	445.540i	1.05829i	−0.848531	+	0.529145i	$0.822513\pi$
$422$	0	0
$423$	−178.736	−0.422544
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 127.426i	− 0.298421i
$428$	0	0
$429$	− 161.933i	− 0.377466i
$430$	0	0
$431$	− 186.677i	− 0.433125i	−0.976269	−	0.216563i	$-0.930515\pi$
$431$	− 186.677i	− 0.433125i	0.976269	−	0.216563i	$-0.0694845\pi$
$432$	0	0
$433$	− 291.128i	− 0.672351i	−0.941799	−	0.336176i	$-0.890866\pi$
$433$	− 291.128i	− 0.672351i	0.941799	−	0.336176i	$-0.109134\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	63.8562	0.146124
$438$	0	0
$439$	− 87.6899i	− 0.199749i	−0.995000	−	0.0998746i	$-0.968156\pi$
$439$	− 87.6899i	− 0.199749i	0.995000	−	0.0998746i	$-0.0318442\pi$
$440$	0	0
$441$	133.277	0.302215
$442$	0	0
$443$	20.7077i	0.0467441i	0.999727	+	0.0233721i	$0.00744024\pi$
$443$	20.7077i	0.0467441i	−0.999727	+	0.0233721i	$0.992560\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−152.078	−0.340219
$448$	0	0
$449$	−584.410	−1.30158	−0.650791	−	0.759257i	$-0.725563\pi$
$449$	−584.410	−1.30158	−0.650791	+	0.759257i	$0.725563\pi$
$450$	0	0
$451$	−252.554	−0.559986
$452$	0	0
$453$	379.433	0.837600
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 269.692i	− 0.590136i	−0.955476	−	0.295068i	$-0.904658\pi$
$457$	− 269.692i	− 0.590136i	0.955476	−	0.295068i	$-0.0953423\pi$
$458$	0	0
$459$	−61.6077	−0.134222
$460$	0	0
$461$	44.4948i	0.0965181i	0.998835	+	0.0482590i	$0.0153673\pi$
$461$	44.4948i	0.0965181i	−0.998835	+	0.0482590i	$0.984633\pi$
$462$	0	0
$463$	−611.065	−1.31980	−0.659898	−	0.751355i	$-0.729400\pi$
$463$	−611.065	−1.31980	−0.659898	+	0.751355i	$0.729400\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 146.410i	− 0.313512i	−0.987637	−	0.156756i	$-0.949896\pi$
$467$	− 146.410i	− 0.313512i	0.987637	−	0.156756i	$-0.0501036\pi$
$468$	0	0
$469$	181.336i	0.386643i
$470$	0	0
$471$	− 438.971i	− 0.931997i
$472$	0	0
$473$	230.277i	0.486843i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−94.0651	−0.197201
$478$	0	0
$479$	191.876i	0.400576i	0.979737	+	0.200288i	$0.0641878\pi$
$479$	191.876i	0.400576i	−0.979737	+	0.200288i	$0.935812\pi$
$480$	0	0
$481$	−323.138	−0.671805
$482$	0	0
$483$	− 15.8461i	− 0.0328077i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−610.758	−1.25412	−0.627062	−	0.778969i	$-0.715743\pi$
$487$	−610.758	−1.25412	−0.627062	+	0.778969i	$0.715743\pi$
$488$	0	0
$489$	−177.282	−0.362540
$490$	0	0
$491$	142.354	0.289926	0.144963	−	0.989437i	$-0.453694\pi$
$491$	142.354	0.289926	0.144963	+	0.989437i	$0.453694\pi$
$492$	0	0
$493$	6.79472	0.0137824
$494$	0	0
$495$	0	0
$496$	0	0
$497$	90.8306i	0.182758i
$498$	0	0
$499$	91.3693	0.183105	0.0915524	−	0.995800i	$-0.470817\pi$
$499$	91.3693	0.183105	0.0915524	+	0.995800i	$0.470817\pi$
$500$	0	0
$501$	− 486.863i	− 0.971782i
$502$	0	0
$503$	−230.067	−0.457389	−0.228695	−	0.973498i	$-0.573446\pi$
$503$	−230.067	−0.457389	−0.228695	+	0.973498i	$0.573446\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	56.1628i	0.110775i
$508$	0	0
$509$	527.387i	1.03612i	0.855343	+	0.518062i	$0.173346\pi$
$509$	527.387i	1.03612i	−0.855343	+	0.518062i	$0.826654\pi$
$510$	0	0
$511$	11.6042i	0.0227088i
$512$	0	0
$513$	77.5692i	0.151207i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	476.630	0.921914
$518$	0	0
$519$	420.643i	0.810487i
$520$	0	0
$521$	191.856	0.368246	0.184123	−	0.982903i	$-0.441055\pi$
$521$	191.856	0.368246	0.184123	+	0.982903i	$0.441055\pi$
$522$	0	0
$523$	105.492i	0.201706i	0.994901	+	0.100853i	$0.0321572\pi$
$523$	105.492i	0.201706i	−0.994901	+	0.100853i	$0.967843\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−681.031	−1.29228
$528$	0	0
$529$	−510.703	−0.965411
$530$	0	0
$531$	158.354	0.298218
$532$	0	0
$533$	−368.934	−0.692184
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 551.405i	− 1.02682i
$538$	0	0
$539$	−355.405	−0.659378
$540$	0	0
$541$	459.744i	0.849804i	0.905239	+	0.424902i	$0.139691\pi$
$541$	459.744i	0.849804i	−0.905239	+	0.424902i	$0.860309\pi$
$542$	0	0
$543$	137.721	0.253630
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 67.3693i	− 0.123161i	−0.998102	−	0.0615807i	$-0.980386\pi$
$547$	− 67.3693i	− 0.123161i	0.998102	−	0.0615807i	$-0.0196142\pi$
$548$	0	0
$549$	178.736i	0.325567i
$550$	0	0
$551$	− 8.55511i	− 0.0155265i
$552$	0	0
$553$	95.4050i	0.172523i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−476.671	−0.855782	−0.427891	−	0.903830i	$-0.640743\pi$
$557$	−476.671	−0.855782	−0.427891	+	0.903830i	$0.640743\pi$
$558$	0	0
$559$	336.391i	0.601774i
$560$	0	0
$561$	164.287	0.292847
$562$	0	0
$563$	− 910.123i	− 1.61656i	−0.588799	−	0.808280i	$-0.700399\pi$
$563$	− 910.123i	− 1.61656i	0.588799	−	0.808280i	$-0.299601\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	19.2490	0.0339489
$568$	0	0
$569$	124.123	0.218142	0.109071	−	0.994034i	$-0.465212\pi$
$569$	124.123	0.218142	0.109071	+	0.994034i	$0.465212\pi$
$570$	0	0
$571$	−945.031	−1.65504	−0.827522	−	0.561433i	$-0.810250\pi$
$571$	−945.031	−1.65504	−0.827522	+	0.561433i	$0.810250\pi$
$572$	0	0
$573$	−611.219	−1.06670
$574$	0	0
$575$	0	0
$576$	0	0
$577$	215.682i	0.373799i	0.982379	+	0.186899i	$0.0598438\pi$
$577$	215.682i	0.373799i	−0.982379	+	0.186899i	$0.940156\pi$
$578$	0	0
$579$	492.382	0.850401
$580$	0	0
$581$	144.823i	0.249265i
$582$	0	0
$583$	250.840	0.430258
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 900.785i	− 1.53456i	−0.641314	−	0.767278i	$-0.721611\pi$
$587$	− 900.785i	− 1.53456i	0.641314	−	0.767278i	$-0.278389\pi$
$588$	0	0
$589$	857.475i	1.45581i
$590$	0	0
$591$	131.305i	0.222174i
$592$	0	0
$593$	508.585i	0.857647i	0.903388	+	0.428824i	$0.141072\pi$
$593$	508.585i	0.857647i	−0.903388	+	0.428824i	$0.858928\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−180.455	−0.302270
$598$	0	0
$599$	846.934i	1.41391i	0.707257	+	0.706957i	$0.249933\pi$
$599$	846.934i	1.41391i	−0.707257	+	0.706957i	$0.750067\pi$
$600$	0	0
$601$	−406.000	−0.675541	−0.337770	−	0.941229i	$-0.609673\pi$
$601$	−406.000	−0.675541	−0.337770	+	0.941229i	$0.609673\pi$
$602$	0	0
$603$	− 254.354i	− 0.421814i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	771.156	1.27044	0.635219	−	0.772332i	$-0.280910\pi$
$607$	771.156	1.27044	0.635219	+	0.772332i	$0.280910\pi$
$608$	0	0
$609$	−2.12297	−0.00348600
$610$	0	0
$611$	696.267	1.13955
$612$	0	0
$613$	336.699	0.549264	0.274632	−	0.961549i	$-0.411444\pi$
$613$	336.699	0.549264	0.274632	+	0.961549i	$0.411444\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 908.831i	− 1.47298i	−0.676447	−	0.736492i	$-0.736481\pi$
$617$	− 908.831i	− 1.47298i	0.676447	−	0.736492i	$-0.263519\pi$
$618$	0	0
$619$	−1047.77	−1.69268	−0.846340	−	0.532643i	$-0.821199\pi$
$619$	−1047.77	−1.69268	−0.846340	+	0.532643i	$0.821199\pi$
$620$	0	0
$621$	22.2268i	0.0357920i
$622$	0	0
$623$	284.754	0.457068
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 206.851i	− 0.329906i
$628$	0	0
$629$	− 327.836i	− 0.521202i
$630$	0	0
$631$	610.758i	0.967921i	0.875090	+	0.483961i	$0.160802\pi$
$631$	610.758i	0.967921i	−0.875090	+	0.483961i	$0.839198\pi$
$632$	0	0
$633$	237.149i	0.374643i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−519.180	−0.815040
$638$	0	0
$639$	− 127.405i	− 0.199382i
$640$	0	0
$641$	9.60015	0.0149768	0.00748842	−	0.999972i	$-0.497616\pi$
$641$	9.60015	0.0149768	0.00748842	+	0.999972i	$0.497616\pi$
$642$	0	0
$643$	86.1999i	0.134059i	0.997751	+	0.0670295i	$0.0213522\pi$
$643$	86.1999i	0.134059i	−0.997751	+	0.0670295i	$0.978648\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	352.580	0.544946	0.272473	−	0.962163i	$-0.412158\pi$
$647$	352.580	0.544946	0.272473	+	0.962163i	$0.412158\pi$
$648$	0	0
$649$	−422.277	−0.650658
$650$	0	0
$651$	212.785	0.326858
$652$	0	0
$653$	319.322	0.489008	0.244504	−	0.969648i	$-0.421375\pi$
$653$	319.322	0.489008	0.244504	+	0.969648i	$0.421375\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 16.2769i	− 0.0247745i
$658$	0	0
$659$	275.328	0.417797	0.208898	−	0.977937i	$-0.433012\pi$
$659$	275.328	0.417797	0.208898	+	0.977937i	$0.433012\pi$
$660$	0	0
$661$	133.668i	0.202221i	0.994875	+	0.101111i	$0.0322396\pi$
$661$	133.668i	0.202221i	−0.994875	+	0.101111i	$0.967760\pi$
$662$	0	0
$663$	239.993	0.361980
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.45140i	− 0.00367526i
$668$	0	0
$669$	− 92.0799i	− 0.137638i
$670$	0	0
$671$	− 476.630i	− 0.710327i
$672$	0	0
$673$	− 187.703i	− 0.278904i	−0.990229	−	0.139452i	$-0.955466\pi$
$673$	− 187.703i	− 0.278904i	0.990229	−	0.139452i	$-0.0445341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1169.84	−1.72797	−0.863986	−	0.503515i	$-0.832040\pi$
$677$	−1169.84	−1.72797	−0.863986	+	0.503515i	$0.832040\pi$
$678$	0	0
$679$	207.758i	0.305976i
$680$	0	0
$681$	207.580	0.304816
$682$	0	0
$683$	89.4566i	0.130976i	0.997853	+	0.0654880i	$0.0208604\pi$
$683$	89.4566i	0.130976i	−0.997853	+	0.0654880i	$0.979140\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−370.837	−0.539791
$688$	0	0
$689$	366.431	0.531830
$690$	0	0
$691$	−139.103	−0.201306	−0.100653	−	0.994922i	$-0.532093\pi$
$691$	−139.103	−0.201306	−0.100653	+	0.994922i	$0.532093\pi$
$692$	0	0
$693$	−51.3307	−0.0740703
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 374.297i	− 0.537012i
$698$	0	0
$699$	220.726	0.315773
$700$	0	0
$701$	− 1218.34i	− 1.73801i	−0.494804	−	0.869004i	$-0.664760\pi$
$701$	− 1218.34i	− 1.73801i	0.494804	−	0.869004i	$-0.335240\pi$
$702$	0	0
$703$	−412.773	−0.587160
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 132.897i	− 0.187974i
$708$	0	0
$709$	1080.13i	1.52346i	0.647895	+	0.761730i	$0.275649\pi$
$709$	1080.13i	1.52346i	−0.647895	+	0.761730i	$0.724351\pi$
$710$	0	0
$711$	− 133.822i	− 0.188216i
$712$	0	0
$713$	245.703i	0.344604i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−553.011	−0.771285
$718$	0	0
$719$	− 20.7736i	− 0.0288923i	−0.999896	−	0.0144461i	$-0.995401\pi$
$719$	− 20.7736i	− 0.0288923i	0.999896	−	0.0144461i	$-0.00459851\pi$
$720$	0	0
$721$	59.4669	0.0824783
$722$	0	0
$723$	428.536i	0.592719i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1031.17	1.41838	0.709192	−	0.705015i	$-0.249060\pi$
$727$	1031.17	1.41838	0.709192	+	0.705015i	$0.249060\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	−341.282	−0.466870
$732$	0	0
$733$	−881.072	−1.20201	−0.601004	−	0.799246i	$-0.705233\pi$
$733$	−881.072	−1.20201	−0.601004	+	0.799246i	$0.705233\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	678.277i	0.920321i
$738$	0	0
$739$	−671.195	−0.908247	−0.454124	−	0.890939i	$-0.650048\pi$
$739$	−671.195	−0.908247	−0.454124	+	0.890939i	$0.650048\pi$
$740$	0	0
$741$	− 302.171i	− 0.407788i
$742$	0	0
$743$	254.197	0.342122	0.171061	−	0.985260i	$-0.445281\pi$
$743$	254.197	0.342122	0.171061	+	0.985260i	$0.445281\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 203.138i	− 0.271939i
$748$	0	0
$749$	− 80.8019i	− 0.107880i
$750$	0	0
$751$	728.994i	0.970698i	0.874320	+	0.485349i	$0.161307\pi$
$751$	728.994i	0.970698i	−0.874320	+	0.485349i	$0.838693\pi$
$752$	0	0
$753$	− 372.133i	− 0.494201i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−372.679	−0.492311	−0.246155	−	0.969230i	$-0.579167\pi$
$757$	−372.679	−0.492311	−0.246155	+	0.969230i	$0.579167\pi$
$758$	0	0
$759$	− 59.2716i	− 0.0780917i
$760$	0	0
$761$	1257.80	1.65283	0.826416	−	0.563060i	$-0.190376\pi$
$761$	1257.80	1.65283	0.826416	+	0.563060i	$0.190376\pi$
$762$	0	0
$763$	− 303.046i	− 0.397177i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−616.868	−0.804260
$768$	0	0
$769$	−247.703	−0.322110	−0.161055	−	0.986945i	$-0.551490\pi$
$769$	−247.703	−0.322110	−0.161055	+	0.986945i	$0.551490\pi$
$770$	0	0
$771$	−145.972	−0.189328
$772$	0	0
$773$	−587.805	−0.760420	−0.380210	−	0.924900i	$-0.624148\pi$
$773$	−587.805	−0.760420	−0.380210	+	0.924900i	$0.624148\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	102.431i	0.131829i
$778$	0	0
$779$	−471.272	−0.604970
$780$	0	0
$781$	339.748i	0.435016i
$782$	0	0
$783$	2.97783	0.00380311
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.0821i	0.0394944i	0.999805	+	0.0197472i	$0.00628614\pi$
$787$	31.0821i	0.0394944i	−0.999805	+	0.0197472i	$0.993714\pi$
$788$	0	0
$789$	− 479.986i	− 0.608347i
$790$	0	0
$791$	124.663i	0.157602i
$792$	0	0
$793$	− 696.267i	− 0.878016i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−490.260	−0.615132	−0.307566	−	0.951527i	$-0.599514\pi$
$797$	−490.260	−0.615132	−0.307566	+	0.951527i	$0.599514\pi$
$798$	0	0
$799$	706.389i	0.884092i
$800$	0	0
$801$	−399.415	−0.498646
$802$	0	0
$803$	43.4050i	0.0540536i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−214.256	−0.265497
$808$	0	0
$809$	−676.102	−0.835726	−0.417863	−	0.908510i	$-0.637221\pi$
$809$	−676.102	−0.835726	−0.417863	+	0.908510i	$0.637221\pi$
$810$	0	0
$811$	74.1793	0.0914665	0.0457332	−	0.998954i	$-0.485438\pi$
$811$	74.1793	0.0914665	0.0457332	+	0.998954i	$0.485438\pi$
$812$	0	0
$813$	342.920	0.421796
$814$	0	0
$815$	0	0
$816$	0	0
$817$	429.703i	0.525952i
$818$	0	0
$819$	−74.9845	−0.0915562
$820$	0	0
$821$	− 1130.58i	− 1.37708i	−0.725198	−	0.688540i	$-0.758252\pi$
$821$	− 1130.58i	− 1.37708i	0.725198	−	0.688540i	$-0.241748\pi$
$822$	0	0
$823$	82.1839	0.0998589	0.0499295	−	0.998753i	$-0.484100\pi$
$823$	82.1839	0.0998589	0.0499295	+	0.998753i	$0.484100\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1504.57i	− 1.81931i	−0.415365	−	0.909655i	$-0.636346\pi$
$827$	− 1504.57i	− 1.81931i	0.415365	−	0.909655i	$-0.363654\pi$
$828$	0	0
$829$	− 1409.65i	− 1.70042i	−0.526445	−	0.850209i	$-0.676475\pi$
$829$	− 1409.65i	− 1.70042i	0.526445	−	0.850209i	$-0.323525\pi$
$830$	0	0
$831$	− 429.044i	− 0.516299i
$832$	0	0
$833$	− 526.728i	− 0.632327i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−298.467	−0.356591
$838$	0	0
$839$	− 288.110i	− 0.343397i	−0.985150	−	0.171698i	$-0.945075\pi$
$839$	− 288.110i	− 0.343397i	0.985150	−	0.171698i	$-0.0549254\pi$
$840$	0	0
$841$	840.672	0.999609
$842$	0	0
$843$	− 767.520i	− 0.910463i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−121.910	−0.143932
$848$	0	0
$849$	−510.297	−0.601057
$850$	0	0
$851$	−118.277	−0.138986
$852$	0	0
$853$	−645.132	−0.756310	−0.378155	−	0.925742i	$-0.623441\pi$
$853$	−645.132	−0.756310	−0.378155	+	0.925742i	$0.623441\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 995.549i	− 1.16167i	−0.814022	−	0.580833i	$-0.802727\pi$
$857$	− 995.549i	− 1.16167i	0.814022	−	0.580833i	$-0.197273\pi$
$858$	0	0
$859$	−774.354	−0.901460	−0.450730	−	0.892660i	$-0.648836\pi$
$859$	−774.354	−0.901460	−0.450730	+	0.892660i	$0.648836\pi$
$860$	0	0
$861$	116.947i	0.135827i
$862$	0	0
$863$	−1007.33	−1.16724	−0.583622	−	0.812025i	$-0.698365\pi$
$863$	−1007.33	−1.16724	−0.583622	+	0.812025i	$0.698365\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 257.081i	− 0.296518i
$868$	0	0
$869$	356.858i	0.410654i
$870$	0	0
$871$	990.836i	1.13758i
$872$	0	0
$873$	− 291.415i	− 0.333809i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−681.645	−0.777246	−0.388623	−	0.921397i	$-0.627049\pi$
$877$	−681.645	−0.777246	−0.388623	+	0.921397i	$0.627049\pi$
$878$	0	0
$879$	115.565i	0.131474i
$880$	0	0
$881$	−679.108	−0.770837	−0.385419	−	0.922742i	$-0.625943\pi$
$881$	−679.108	−0.770837	−0.385419	+	0.922742i	$0.625943\pi$
$882$	0	0
$883$	− 1059.44i	− 1.19982i	−0.800068	−	0.599910i	$-0.795203\pi$
$883$	− 1059.44i	− 1.19982i	0.800068	−	0.599910i	$-0.204797\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−856.411	−0.965514	−0.482757	−	0.875754i	$-0.660365\pi$
$887$	−856.411	−0.965514	−0.482757	+	0.875754i	$0.660365\pi$
$888$	0	0
$889$	396.000	0.445444
$890$	0	0
$891$	72.0000	0.0808081
$892$	0	0
$893$	889.403	0.995972
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 86.5847i	− 0.0965270i
$898$	0	0
$899$	32.9179	0.0366161
$900$	0	0
$901$	371.758i	0.412606i
$902$	0	0
$903$	106.632	0.118086
$904$	0	0
$905$	0	0
$906$	0	0
$907$	761.492i	0.839573i	0.907623	+	0.419786i	$0.137895\pi$
$907$	761.492i	0.839573i	−0.907623	+	0.419786i	$0.862105\pi$
$908$	0	0
$909$	186.411i	0.205073i
$910$	0	0
$911$	897.344i	0.985009i	0.870310	+	0.492505i	$0.163919\pi$
$911$	897.344i	0.985009i	−0.870310	+	0.492505i	$0.836081\pi$
$912$	0	0
$913$	541.703i	0.593321i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−268.115	−0.292383
$918$	0	0
$919$	62.6388i	0.0681598i	0.999419	+	0.0340799i	$0.0108501\pi$
$919$	62.6388i	0.0681598i	−0.999419	+	0.0340799i	$0.989150\pi$
$920$	0	0
$921$	907.959	0.985840
$922$	0	0
$923$	496.308i	0.537712i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−83.4124	−0.0899810
$928$	0	0
$929$	−253.313	−0.272673	−0.136336	−	0.990663i	$-0.543533\pi$
$929$	−253.313	−0.272673	−0.136336	+	0.990663i	$0.543533\pi$
$930$	0	0
$931$	−663.195	−0.712347
$932$	0	0
$933$	−627.101	−0.672134
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 30.5538i	− 0.0326081i	−0.999867	−	0.0163040i	$-0.994810\pi$
$937$	− 30.5538i	− 0.0326081i	0.999867	−	0.0163040i	$-0.00518996\pi$
$938$	0	0
$939$	−436.956	−0.465342
$940$	0	0
$941$	− 388.745i	− 0.413119i	−0.978434	−	0.206559i	$-0.933773\pi$
$941$	− 388.745i	− 0.413119i	0.978434	−	0.206559i	$-0.0662267\pi$
$942$	0	0
$943$	−135.039	−0.143202
$944$	0	0
$945$	0	0
$946$	0	0
$947$	263.615i	0.278369i	0.990266	+	0.139184i	$0.0444481\pi$
$947$	263.615i	0.278369i	−0.990266	+	0.139184i	$0.955552\pi$
$948$	0	0
$949$	63.4066i	0.0668141i
$950$	0	0
$951$	− 139.245i	− 0.146420i
$952$	0	0
$953$	− 1292.41i	− 1.35615i	−0.734993	−	0.678075i	$-0.762815\pi$
$953$	− 1292.41i	− 1.35615i	0.734993	−	0.678075i	$-0.237185\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.94089	−0.00829769
$958$	0	0
$959$	212.957i	0.222061i
$960$	0	0
$961$	−2338.34	−2.43324
$962$	0	0
$963$	113.338i	0.117693i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1110.00	−1.14788	−0.573942	−	0.818896i	$-0.694587\pi$
$967$	−1110.00	−1.14788	−0.573942	+	0.818896i	$0.694587\pi$
$968$	0	0
$969$	306.564	0.316372
$970$	0	0
$971$	1341.66	1.38173	0.690866	−	0.722983i	$-0.257230\pi$
$971$	1341.66	1.38173	0.690866	+	0.722983i	$0.257230\pi$
$972$	0	0
$973$	379.617	0.390151
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 920.431i	− 0.942099i	−0.882107	−	0.471050i	$-0.843875\pi$
$977$	− 920.431i	− 0.942099i	0.882107	−	0.471050i	$-0.156125\pi$
$978$	0	0
$979$	1065.11	1.08795
$980$	0	0
$981$	425.074i	0.433307i
$982$	0	0
$983$	−1697.84	−1.72720	−0.863601	−	0.504176i	$-0.831796\pi$
$983$	−1697.84	−1.72720	−0.863601	+	0.504176i	$0.831796\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 220.708i	− 0.223615i
$988$	0	0
$989$	123.128i	0.124497i
$990$	0	0
$991$	− 284.765i	− 0.287351i	−0.989625	−	0.143675i	$-0.954108\pi$
$991$	− 284.765i	− 0.287351i	0.989625	−	0.143675i	$-0.0458921\pi$
$992$	0	0
$993$	298.010i	0.300111i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1093.80	−1.09710	−0.548548	−	0.836119i	$-0.684819\pi$
$997$	−1093.80	−1.09710	−0.548548	+	0.836119i	$0.684819\pi$
$998$	0	0
$999$	− 143.677i	− 0.143820i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.p.a.1999.3		8
4.3	odd	2		600.3.p.a.499.4		8
5.2	odd	4		2400.3.g.a.751.2		4
5.3	odd	4		96.3.b.a.79.3		4
5.4	even	2	inner	2400.3.p.a.1999.6		8
8.3	odd	2	inner	2400.3.p.a.1999.2		8
8.5	even	2		600.3.p.a.499.6		8
15.8	even	4		288.3.b.b.271.4		4
20.3	even	4		24.3.b.a.19.4	yes	4
20.7	even	4		600.3.g.a.451.1		4
20.19	odd	2		600.3.p.a.499.5		8
40.3	even	4		96.3.b.a.79.4		4
40.13	odd	4		24.3.b.a.19.3	✓	4
40.19	odd	2	inner	2400.3.p.a.1999.7		8
40.27	even	4		2400.3.g.a.751.1		4
40.29	even	2		600.3.p.a.499.3		8
40.37	odd	4		600.3.g.a.451.2		4
60.23	odd	4		72.3.b.b.19.1		4
80.3	even	4		768.3.g.h.511.1		8
80.13	odd	4		768.3.g.h.511.5		8
80.43	even	4		768.3.g.h.511.8		8
80.53	odd	4		768.3.g.h.511.4		8
120.53	even	4		72.3.b.b.19.2		4
120.83	odd	4		288.3.b.b.271.1		4
240.53	even	4		2304.3.g.z.1279.2		8
240.83	odd	4		2304.3.g.z.1279.7		8
240.173	even	4		2304.3.g.z.1279.8		8
240.203	odd	4		2304.3.g.z.1279.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.b.a.19.3	✓	4	40.13	odd	4
24.3.b.a.19.4	yes	4	20.3	even	4
72.3.b.b.19.1		4	60.23	odd	4
72.3.b.b.19.2		4	120.53	even	4
96.3.b.a.79.3		4	5.3	odd	4
96.3.b.a.79.4		4	40.3	even	4
288.3.b.b.271.1		4	120.83	odd	4
288.3.b.b.271.4		4	15.8	even	4
600.3.g.a.451.1		4	20.7	even	4
600.3.g.a.451.2		4	40.37	odd	4
600.3.p.a.499.3		8	40.29	even	2
600.3.p.a.499.4		8	4.3	odd	2
600.3.p.a.499.5		8	20.19	odd	2
600.3.p.a.499.6		8	8.5	even	2
768.3.g.h.511.1		8	80.3	even	4
768.3.g.h.511.4		8	80.53	odd	4
768.3.g.h.511.5		8	80.13	odd	4
768.3.g.h.511.8		8	80.43	even	4
2304.3.g.z.1279.1		8	240.203	odd	4
2304.3.g.z.1279.2		8	240.53	even	4
2304.3.g.z.1279.7		8	240.83	odd	4
2304.3.g.z.1279.8		8	240.173	even	4
2400.3.g.a.751.1		4	40.27	even	4
2400.3.g.a.751.2		4	5.2	odd	4
2400.3.p.a.1999.2		8	8.3	odd	2	inner
2400.3.p.a.1999.3		8	1.1	even	1	trivial
2400.3.p.a.1999.6		8	5.4	even	2	inner
2400.3.p.a.1999.7		8	40.19	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2400.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +2.13878 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +2.13878 q^{7} -3.00000 q^{9} +8.00000 q^{11} +11.6865 q^{13} +11.8564i q^{17} +14.9282 q^{19} -3.70447i q^{21} +4.27756 q^{23} +5.19615i q^{27} -0.573084i q^{29} +57.4399i q^{31} -13.8564i q^{33} -27.6506 q^{37} -20.2416i q^{39} -31.5692 q^{41} +28.7846i q^{43} +59.5787 q^{47} -44.4256 q^{49} +20.5359 q^{51} +31.3550 q^{53} -25.8564i q^{57} -52.7846 q^{59} -59.5787i q^{61} -6.41634 q^{63} +84.7846i q^{67} -7.40895i q^{69} +42.4685i q^{71} +5.42563i q^{73} +17.1102 q^{77} +44.6072i q^{79} +9.00000 q^{81} +67.7128i q^{83} -0.992611 q^{87} +133.138 q^{89} +24.9948 q^{91} +99.4888 q^{93} +97.1384i q^{97} -24.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 64 q^{11} + 64 q^{19} + 80 q^{41} + 88 q^{49} + 192 q^{51} - 256 q^{59} + 72 q^{81} + 400 q^{89} - 576 q^{91} - 192 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 64 * q^11 + 64 * q^19 + 80 * q^41 + 88 * q^49 + 192 * q^51 - 256 * q^59 + 72 * q^81 + 400 * q^89 - 576 * q^91 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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