LMFDB - Embedded newform 600.4.f.b.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.f (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:	$35.4011460034$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.4.f.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} -28.0000 q^{11} -74.0000i q^{13} -82.0000i q^{17} -92.0000 q^{19} -72.0000 q^{21} +8.00000i q^{23} -27.0000i q^{27} +138.000 q^{29} +80.0000 q^{31} -84.0000i q^{33} -30.0000i q^{37} +222.000 q^{39} +282.000 q^{41} +4.00000i q^{43} -240.000i q^{47} -233.000 q^{49} +246.000 q^{51} -130.000i q^{53} -276.000i q^{57} -596.000 q^{59} -218.000 q^{61} -216.000i q^{63} +436.000i q^{67} -24.0000 q^{69} +856.000 q^{71} -998.000i q^{73} -672.000i q^{77} +32.0000 q^{79} +81.0000 q^{81} -1508.00i q^{83} +414.000i q^{87} +246.000 q^{89} +1776.00 q^{91} +240.000i q^{93} -866.000i q^{97} +252.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 56 q^{11} - 184 q^{19} - 144 q^{21} + 276 q^{29} + 160 q^{31} + 444 q^{39} + 564 q^{41} - 466 q^{49} + 492 q^{51} - 1192 q^{59} - 436 q^{61} - 48 q^{69} + 1712 q^{71} + 64 q^{79} + 162 q^{81} + 492 q^{89} + 3552 q^{91} + 504 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 56 * q^11 - 184 * q^19 - 144 * q^21 + 276 * q^29 + 160 * q^31 + 444 * q^39 + 564 * q^41 - 466 * q^49 + 492 * q^51 - 1192 * q^59 - 436 * q^61 - 48 * q^69 + 1712 * q^71 + 64 * q^79 + 162 * q^81 + 492 * q^89 + 3552 * q^91 + 504 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	24.0000i	1.29588i	0.761692	+	0.647939i	$0.224369\pi$
$7$	24.0000i	1.29588i	−0.761692	+	0.647939i	$0.775631\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	0	0
$13$	− 74.0000i	− 1.57876i	−0.613904	−	0.789381i	$-0.710402\pi$
$13$	− 74.0000i	− 1.57876i	0.613904	−	0.789381i	$-0.289598\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 82.0000i	− 1.16988i	−0.811077	−	0.584939i	$-0.801118\pi$
$17$	− 82.0000i	− 1.16988i	0.811077	−	0.584939i	$-0.198882\pi$
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	−72.0000	−0.748176
$22$	0	0
$23$	8.00000i	0.0725268i	0.999342	+	0.0362634i	$0.0115455\pi$
$23$	8.00000i	0.0725268i	−0.999342	+	0.0362634i	$0.988454\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	138.000	0.883654	0.441827	−	0.897100i	$-0.354331\pi$
$29$	138.000	0.883654	0.441827	+	0.897100i	$0.354331\pi$
$30$	0	0
$31$	80.0000	0.463498	0.231749	−	0.972776i	$-0.425555\pi$
$31$	80.0000	0.463498	0.231749	+	0.972776i	$0.425555\pi$
$32$	0	0
$33$	− 84.0000i	− 0.443107i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 30.0000i	− 0.133296i	−0.997777	−	0.0666482i	$-0.978769\pi$
$37$	− 30.0000i	− 0.133296i	0.997777	−	0.0666482i	$-0.0212305\pi$
$38$	0	0
$39$	222.000	0.911499
$40$	0	0
$41$	282.000	1.07417	0.537085	−	0.843528i	$-0.319525\pi$
$41$	282.000	1.07417	0.537085	+	0.843528i	$0.319525\pi$
$42$	0	0
$43$	4.00000i	0.0141859i	0.999975	+	0.00709296i	$0.00225778\pi$
$43$	4.00000i	0.0141859i	−0.999975	+	0.00709296i	$0.997742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 240.000i	− 0.744843i	−0.928064	−	0.372421i	$-0.878528\pi$
$47$	− 240.000i	− 0.744843i	0.928064	−	0.372421i	$-0.121472\pi$
$48$	0	0
$49$	−233.000	−0.679300
$50$	0	0
$51$	246.000	0.675429
$52$	0	0
$53$	− 130.000i	− 0.336922i	−0.985708	−	0.168461i	$-0.946120\pi$
$53$	− 130.000i	− 0.336922i	0.985708	−	0.168461i	$-0.0538797\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 276.000i	− 0.641353i
$58$	0	0
$59$	−596.000	−1.31513	−0.657564	−	0.753398i	$-0.728413\pi$
$59$	−596.000	−1.31513	−0.657564	+	0.753398i	$0.728413\pi$
$60$	0	0
$61$	−218.000	−0.457574	−0.228787	−	0.973476i	$-0.573476\pi$
$61$	−218.000	−0.457574	−0.228787	+	0.973476i	$0.573476\pi$
$62$	0	0
$63$	− 216.000i	− 0.431959i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	436.000i	0.795013i	0.917599	+	0.397507i	$0.130124\pi$
$67$	436.000i	0.795013i	−0.917599	+	0.397507i	$0.869876\pi$
$68$	0	0
$69$	−24.0000	−0.0418733
$70$	0	0
$71$	856.000	1.43082	0.715412	−	0.698703i	$-0.246239\pi$
$71$	856.000	1.43082	0.715412	+	0.698703i	$0.246239\pi$
$72$	0	0
$73$	− 998.000i	− 1.60010i	−0.599935	−	0.800048i	$-0.704807\pi$
$73$	− 998.000i	− 1.60010i	0.599935	−	0.800048i	$-0.295193\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 672.000i	− 0.994565i
$78$	0	0
$79$	32.0000	0.0455732	0.0227866	−	0.999740i	$-0.492746\pi$
$79$	32.0000	0.0455732	0.0227866	+	0.999740i	$0.492746\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1508.00i	− 1.99427i	−0.0756351	−	0.997136i	$-0.524098\pi$
$83$	− 1508.00i	− 1.99427i	0.0756351	−	0.997136i	$-0.475902\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	414.000i	0.510178i
$88$	0	0
$89$	246.000	0.292988	0.146494	−	0.989212i	$-0.453201\pi$
$89$	246.000	0.292988	0.146494	+	0.989212i	$0.453201\pi$
$90$	0	0
$91$	1776.00	2.04588
$92$	0	0
$93$	240.000i	0.267600i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 866.000i	− 0.906484i	−0.891387	−	0.453242i	$-0.850267\pi$
$97$	− 866.000i	− 0.906484i	0.891387	−	0.453242i	$-0.149733\pi$
$98$	0	0
$99$	252.000	0.255828
$100$	0	0
$101$	270.000	0.266000	0.133000	−	0.991116i	$-0.457539\pi$
$101$	270.000	0.266000	0.133000	+	0.991116i	$0.457539\pi$
$102$	0	0
$103$	− 1496.00i	− 1.43112i	−0.698552	−	0.715560i	$-0.746172\pi$
$103$	− 1496.00i	− 1.43112i	0.698552	−	0.715560i	$-0.253828\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1692.00i	1.52871i	0.644797	+	0.764354i	$0.276942\pi$
$107$	1692.00i	1.52871i	−0.644797	+	0.764354i	$0.723058\pi$
$108$	0	0
$109$	−406.000	−0.356768	−0.178384	−	0.983961i	$-0.557087\pi$
$109$	−406.000	−0.356768	−0.178384	+	0.983961i	$0.557087\pi$
$110$	0	0
$111$	90.0000	0.0769588
$112$	0	0
$113$	786.000i	0.654342i	0.944965	+	0.327171i	$0.106095\pi$
$113$	786.000i	0.654342i	−0.944965	+	0.327171i	$0.893905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	666.000i	0.526254i
$118$	0	0
$119$	1968.00	1.51602
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	846.000i	0.620173i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1744.00i	− 1.21854i	−0.792962	−	0.609272i	$-0.791462\pi$
$127$	− 1744.00i	− 1.21854i	0.792962	−	0.609272i	$-0.208538\pi$
$128$	0	0
$129$	−12.0000	−0.00819024
$130$	0	0
$131$	652.000	0.434851	0.217426	−	0.976077i	$-0.430234\pi$
$131$	652.000	0.434851	0.217426	+	0.976077i	$0.430234\pi$
$132$	0	0
$133$	− 2208.00i	− 1.43953i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1530.00i	− 0.954137i	−0.878866	−	0.477068i	$-0.841699\pi$
$137$	− 1530.00i	− 0.954137i	0.878866	−	0.477068i	$-0.158301\pi$
$138$	0	0
$139$	−516.000	−0.314867	−0.157434	−	0.987530i	$-0.550322\pi$
$139$	−516.000	−0.314867	−0.157434	+	0.987530i	$0.550322\pi$
$140$	0	0
$141$	720.000	0.430035
$142$	0	0
$143$	2072.00i	1.21167i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 699.000i	− 0.392194i
$148$	0	0
$149$	−1342.00	−0.737859	−0.368929	−	0.929457i	$-0.620276\pi$
$149$	−1342.00	−0.737859	−0.368929	+	0.929457i	$0.620276\pi$
$150$	0	0
$151$	−424.000	−0.228507	−0.114254	−	0.993452i	$-0.536448\pi$
$151$	−424.000	−0.228507	−0.114254	+	0.993452i	$0.536448\pi$
$152$	0	0
$153$	738.000i	0.389959i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 262.000i	− 0.133184i	−0.997780	−	0.0665920i	$-0.978787\pi$
$157$	− 262.000i	− 0.133184i	0.997780	−	0.0665920i	$-0.0212126\pi$
$158$	0	0
$159$	390.000	0.194522
$160$	0	0
$161$	−192.000	−0.0939858
$162$	0	0
$163$	− 2292.00i	− 1.10137i	−0.834713	−	0.550685i	$-0.814367\pi$
$163$	− 2292.00i	− 1.10137i	0.834713	−	0.550685i	$-0.185633\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1896.00i	0.878544i	0.898354	+	0.439272i	$0.144764\pi$
$167$	1896.00i	0.878544i	−0.898354	+	0.439272i	$0.855236\pi$
$168$	0	0
$169$	−3279.00	−1.49249
$170$	0	0
$171$	828.000	0.370285
$172$	0	0
$173$	− 2874.00i	− 1.26304i	−0.775359	−	0.631521i	$-0.782431\pi$
$173$	− 2874.00i	− 1.26304i	0.775359	−	0.631521i	$-0.217569\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1788.00i	− 0.759290i
$178$	0	0
$179$	1188.00	0.496063	0.248032	−	0.968752i	$-0.420216\pi$
$179$	1188.00	0.496063	0.248032	+	0.968752i	$0.420216\pi$
$180$	0	0
$181$	−3474.00	−1.42663	−0.713316	−	0.700843i	$-0.752808\pi$
$181$	−3474.00	−1.42663	−0.713316	+	0.700843i	$0.752808\pi$
$182$	0	0
$183$	− 654.000i	− 0.264181i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2296.00i	0.897862i
$188$	0	0
$189$	648.000	0.249392
$190$	0	0
$191$	192.000	0.0727363	0.0363681	−	0.999338i	$-0.488421\pi$
$191$	192.000	0.0727363	0.0363681	+	0.999338i	$0.488421\pi$
$192$	0	0
$193$	4802.00i	1.79096i	0.445100	+	0.895481i	$0.353168\pi$
$193$	4802.00i	1.79096i	−0.445100	+	0.895481i	$0.646832\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1518.00i	− 0.549000i	−0.961587	−	0.274500i	$-0.911488\pi$
$197$	− 1518.00i	− 0.549000i	0.961587	−	0.274500i	$-0.0885123\pi$
$198$	0	0
$199$	−5128.00	−1.82670	−0.913352	−	0.407170i	$-0.866516\pi$
$199$	−5128.00	−1.82670	−0.913352	+	0.407170i	$0.866516\pi$
$200$	0	0
$201$	−1308.00	−0.459001
$202$	0	0
$203$	3312.00i	1.14511i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 72.0000i	− 0.0241756i
$208$	0	0
$209$	2576.00	0.852563
$210$	0	0
$211$	1084.00	0.353676	0.176838	−	0.984240i	$-0.443413\pi$
$211$	1084.00	0.353676	0.176838	+	0.984240i	$0.443413\pi$
$212$	0	0
$213$	2568.00i	0.826087i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1920.00i	0.600636i
$218$	0	0
$219$	2994.00	0.923816
$220$	0	0
$221$	−6068.00	−1.84696
$222$	0	0
$223$	688.000i	0.206600i	0.994650	+	0.103300i	$0.0329402\pi$
$223$	688.000i	0.206600i	−0.994650	+	0.103300i	$0.967060\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4812.00i	− 1.40698i	−0.710707	−	0.703488i	$-0.751625\pi$
$227$	− 4812.00i	− 1.40698i	0.710707	−	0.703488i	$-0.248375\pi$
$228$	0	0
$229$	−2494.00	−0.719686	−0.359843	−	0.933013i	$-0.617170\pi$
$229$	−2494.00	−0.719686	−0.359843	+	0.933013i	$0.617170\pi$
$230$	0	0
$231$	2016.00	0.574212
$232$	0	0
$233$	698.000i	0.196255i	0.995174	+	0.0981277i	$0.0312854\pi$
$233$	698.000i	0.196255i	−0.995174	+	0.0981277i	$0.968715\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	96.0000i	0.0263117i
$238$	0	0
$239$	6320.00	1.71049	0.855244	−	0.518225i	$-0.173407\pi$
$239$	6320.00	1.71049	0.855244	+	0.518225i	$0.173407\pi$
$240$	0	0
$241$	−6510.00	−1.74002	−0.870012	−	0.493030i	$-0.835889\pi$
$241$	−6510.00	−1.74002	−0.870012	+	0.493030i	$0.835889\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6808.00i	1.75378i
$248$	0	0
$249$	4524.00	1.15139
$250$	0	0
$251$	628.000	0.157924	0.0789622	−	0.996878i	$-0.474839\pi$
$251$	628.000	0.157924	0.0789622	+	0.996878i	$0.474839\pi$
$252$	0	0
$253$	− 224.000i	− 0.0556631i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4862.00i	1.18009i	0.807370	+	0.590045i	$0.200890\pi$
$257$	4862.00i	1.18009i	−0.807370	+	0.590045i	$0.799110\pi$
$258$	0	0
$259$	720.000	0.172736
$260$	0	0
$261$	−1242.00	−0.294551
$262$	0	0
$263$	5816.00i	1.36361i	0.731533	+	0.681806i	$0.238805\pi$
$263$	5816.00i	1.36361i	−0.731533	+	0.681806i	$0.761195\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	738.000i	0.169157i
$268$	0	0
$269$	−3526.00	−0.799197	−0.399599	−	0.916690i	$-0.630850\pi$
$269$	−3526.00	−0.799197	−0.399599	+	0.916690i	$0.630850\pi$
$270$	0	0
$271$	−256.000	−0.0573834	−0.0286917	−	0.999588i	$-0.509134\pi$
$271$	−256.000	−0.0573834	−0.0286917	+	0.999588i	$0.509134\pi$
$272$	0	0
$273$	5328.00i	1.18119i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 142.000i	− 0.0308013i	−0.999881	−	0.0154006i	$-0.995098\pi$
$277$	− 142.000i	− 0.0308013i	0.999881	−	0.0154006i	$-0.00490237\pi$
$278$	0	0
$279$	−720.000	−0.154499
$280$	0	0
$281$	8842.00	1.87712	0.938558	−	0.345122i	$-0.112162\pi$
$281$	8842.00	1.87712	0.938558	+	0.345122i	$0.112162\pi$
$282$	0	0
$283$	− 7180.00i	− 1.50815i	−0.656788	−	0.754075i	$-0.728085\pi$
$283$	− 7180.00i	− 1.50815i	0.656788	−	0.754075i	$-0.271915\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6768.00i	1.39199i
$288$	0	0
$289$	−1811.00	−0.368614
$290$	0	0
$291$	2598.00	0.523359
$292$	0	0
$293$	7374.00i	1.47029i	0.677912	+	0.735143i	$0.262885\pi$
$293$	7374.00i	1.47029i	−0.677912	+	0.735143i	$0.737115\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	756.000i	0.147702i
$298$	0	0
$299$	592.000	0.114502
$300$	0	0
$301$	−96.0000	−0.0183832
$302$	0	0
$303$	810.000i	0.153575i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1500.00i	− 0.278858i	−0.990232	−	0.139429i	$-0.955473\pi$
$307$	− 1500.00i	− 0.278858i	0.990232	−	0.139429i	$-0.0445268\pi$
$308$	0	0
$309$	4488.00	0.826257
$310$	0	0
$311$	−7608.00	−1.38717	−0.693585	−	0.720374i	$-0.743970\pi$
$311$	−7608.00	−1.38717	−0.693585	+	0.720374i	$0.743970\pi$
$312$	0	0
$313$	− 4758.00i	− 0.859227i	−0.903013	−	0.429614i	$-0.858650\pi$
$313$	− 4758.00i	− 0.859227i	0.903013	−	0.429614i	$-0.141350\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4374.00i	− 0.774979i	−0.921874	−	0.387489i	$-0.873342\pi$
$317$	− 4374.00i	− 0.774979i	0.921874	−	0.387489i	$-0.126658\pi$
$318$	0	0
$319$	−3864.00	−0.678190
$320$	0	0
$321$	−5076.00	−0.882600
$322$	0	0
$323$	7544.00i	1.29956i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1218.00i	− 0.205980i
$328$	0	0
$329$	5760.00	0.965225
$330$	0	0
$331$	−7804.00	−1.29591	−0.647956	−	0.761678i	$-0.724376\pi$
$331$	−7804.00	−1.29591	−0.647956	+	0.761678i	$0.724376\pi$
$332$	0	0
$333$	270.000i	0.0444322i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5106.00i	− 0.825346i	−0.910879	−	0.412673i	$-0.864595\pi$
$337$	− 5106.00i	− 0.825346i	0.910879	−	0.412673i	$-0.135405\pi$
$338$	0	0
$339$	−2358.00	−0.377785
$340$	0	0
$341$	−2240.00	−0.355727
$342$	0	0
$343$	2640.00i	0.415588i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4716.00i	0.729591i	0.931088	+	0.364796i	$0.118861\pi$
$347$	4716.00i	0.729591i	−0.931088	+	0.364796i	$0.881139\pi$
$348$	0	0
$349$	−7302.00	−1.11996	−0.559982	−	0.828505i	$-0.689192\pi$
$349$	−7302.00	−1.11996	−0.559982	+	0.828505i	$0.689192\pi$
$350$	0	0
$351$	−1998.00	−0.303833
$352$	0	0
$353$	− 4382.00i	− 0.660709i	−0.943857	−	0.330355i	$-0.892832\pi$
$353$	− 4382.00i	− 0.660709i	0.943857	−	0.330355i	$-0.107168\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5904.00i	0.875274i
$358$	0	0
$359$	−7224.00	−1.06203	−0.531014	−	0.847363i	$-0.678189\pi$
$359$	−7224.00	−1.06203	−0.531014	+	0.847363i	$0.678189\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	− 1641.00i	− 0.237273i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1408.00i	− 0.200264i	−0.994974	−	0.100132i	$-0.968073\pi$
$367$	− 1408.00i	− 0.200264i	0.994974	−	0.100132i	$-0.0319266\pi$
$368$	0	0
$369$	−2538.00	−0.358057
$370$	0	0
$371$	3120.00	0.436610
$372$	0	0
$373$	− 1714.00i	− 0.237929i	−0.992899	−	0.118965i	$-0.962043\pi$
$373$	− 1714.00i	− 0.237929i	0.992899	−	0.118965i	$-0.0379575\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 10212.0i	− 1.39508i
$378$	0	0
$379$	−884.000	−0.119810	−0.0599051	−	0.998204i	$-0.519080\pi$
$379$	−884.000	−0.119810	−0.0599051	+	0.998204i	$0.519080\pi$
$380$	0	0
$381$	5232.00	0.703526
$382$	0	0
$383$	10368.0i	1.38324i	0.722263	+	0.691619i	$0.243102\pi$
$383$	10368.0i	1.38324i	−0.722263	+	0.691619i	$0.756898\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 36.0000i	− 0.00472864i
$388$	0	0
$389$	−398.000	−0.0518751	−0.0259375	−	0.999664i	$-0.508257\pi$
$389$	−398.000	−0.0518751	−0.0259375	+	0.999664i	$0.508257\pi$
$390$	0	0
$391$	656.000	0.0848474
$392$	0	0
$393$	1956.00i	0.251061i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5098.00i	0.644487i	0.946657	+	0.322243i	$0.104437\pi$
$397$	5098.00i	0.644487i	−0.946657	+	0.322243i	$0.895563\pi$
$398$	0	0
$399$	6624.00	0.831115
$400$	0	0
$401$	10002.0	1.24558	0.622788	−	0.782391i	$-0.286000\pi$
$401$	10002.0	1.24558	0.622788	+	0.782391i	$0.286000\pi$
$402$	0	0
$403$	− 5920.00i	− 0.731752i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	840.000i	0.102303i
$408$	0	0
$409$	9270.00	1.12071	0.560357	−	0.828251i	$-0.310664\pi$
$409$	9270.00	1.12071	0.560357	+	0.828251i	$0.310664\pi$
$410$	0	0
$411$	4590.00	0.550871
$412$	0	0
$413$	− 14304.0i	− 1.70425i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1548.00i	− 0.181789i
$418$	0	0
$419$	6516.00	0.759731	0.379866	−	0.925042i	$-0.375970\pi$
$419$	6516.00	0.759731	0.379866	+	0.925042i	$0.375970\pi$
$420$	0	0
$421$	−2626.00	−0.303999	−0.151999	−	0.988381i	$-0.548571\pi$
$421$	−2626.00	−0.303999	−0.151999	+	0.988381i	$0.548571\pi$
$422$	0	0
$423$	2160.00i	0.248281i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5232.00i	− 0.592961i
$428$	0	0
$429$	−6216.00	−0.699560
$430$	0	0
$431$	−4304.00	−0.481012	−0.240506	−	0.970648i	$-0.577313\pi$
$431$	−4304.00	−0.481012	−0.240506	+	0.970648i	$0.577313\pi$
$432$	0	0
$433$	11794.0i	1.30897i	0.756076	+	0.654484i	$0.227114\pi$
$433$	11794.0i	1.30897i	−0.756076	+	0.654484i	$0.772886\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 736.000i	− 0.0805667i
$438$	0	0
$439$	5544.00	0.602735	0.301368	−	0.953508i	$-0.402557\pi$
$439$	5544.00	0.602735	0.301368	+	0.953508i	$0.402557\pi$
$440$	0	0
$441$	2097.00	0.226433
$442$	0	0
$443$	− 3788.00i	− 0.406260i	−0.979152	−	0.203130i	$-0.934889\pi$
$443$	− 3788.00i	− 0.406260i	0.979152	−	0.203130i	$-0.0651115\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4026.00i	− 0.426003i
$448$	0	0
$449$	13342.0	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$449$	13342.0	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$450$	0	0
$451$	−7896.00	−0.824408
$452$	0	0
$453$	− 1272.00i	− 0.131929i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4390.00i	0.449356i	0.974433	+	0.224678i	$0.0721330\pi$
$457$	4390.00i	0.449356i	−0.974433	+	0.224678i	$0.927867\pi$
$458$	0	0
$459$	−2214.00	−0.225143
$460$	0	0
$461$	5798.00	0.585770	0.292885	−	0.956148i	$-0.405385\pi$
$461$	5798.00	0.585770	0.292885	+	0.956148i	$0.405385\pi$
$462$	0	0
$463$	− 14656.0i	− 1.47111i	−0.677467	−	0.735553i	$-0.736922\pi$
$463$	− 14656.0i	− 1.47111i	0.677467	−	0.735553i	$-0.263078\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8412.00i	− 0.833535i	−0.909013	−	0.416768i	$-0.863163\pi$
$467$	− 8412.00i	− 0.833535i	0.909013	−	0.416768i	$-0.136837\pi$
$468$	0	0
$469$	−10464.0	−1.03024
$470$	0	0
$471$	786.000	0.0768938
$472$	0	0
$473$	− 112.000i	− 0.0108875i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1170.00i	0.112307i
$478$	0	0
$479$	−14848.0	−1.41633	−0.708165	−	0.706047i	$-0.750477\pi$
$479$	−14848.0	−1.41633	−0.708165	+	0.706047i	$0.750477\pi$
$480$	0	0
$481$	−2220.00	−0.210443
$482$	0	0
$483$	− 576.000i	− 0.0542627i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 18568.0i	− 1.72771i	−0.503738	−	0.863857i	$-0.668042\pi$
$487$	− 18568.0i	− 1.72771i	0.503738	−	0.863857i	$-0.331958\pi$
$488$	0	0
$489$	6876.00	0.635876
$490$	0	0
$491$	−14364.0	−1.32024	−0.660120	−	0.751160i	$-0.729495\pi$
$491$	−14364.0	−1.32024	−0.660120	+	0.751160i	$0.729495\pi$
$492$	0	0
$493$	− 11316.0i	− 1.03377i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20544.0i	1.85417i
$498$	0	0
$499$	−21660.0	−1.94316	−0.971578	−	0.236720i	$-0.923928\pi$
$499$	−21660.0	−1.94316	−0.971578	+	0.236720i	$0.923928\pi$
$500$	0	0
$501$	−5688.00	−0.507228
$502$	0	0
$503$	− 17112.0i	− 1.51687i	−0.651748	−	0.758436i	$-0.725964\pi$
$503$	− 17112.0i	− 1.51687i	0.651748	−	0.758436i	$-0.274036\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9837.00i	− 0.861689i
$508$	0	0
$509$	−11478.0	−0.999516	−0.499758	−	0.866165i	$-0.666578\pi$
$509$	−11478.0	−0.999516	−0.499758	+	0.866165i	$0.666578\pi$
$510$	0	0
$511$	23952.0	2.07353
$512$	0	0
$513$	2484.00i	0.213784i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6720.00i	0.571654i
$518$	0	0
$519$	8622.00	0.729217
$520$	0	0
$521$	13114.0	1.10275	0.551377	−	0.834256i	$-0.314103\pi$
$521$	13114.0	1.10275	0.551377	+	0.834256i	$0.314103\pi$
$522$	0	0
$523$	− 4508.00i	− 0.376905i	−0.982082	−	0.188452i	$-0.939653\pi$
$523$	− 4508.00i	− 0.376905i	0.982082	−	0.188452i	$-0.0603471\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6560.00i	− 0.542235i
$528$	0	0
$529$	12103.0	0.994740
$530$	0	0
$531$	5364.00	0.438376
$532$	0	0
$533$	− 20868.0i	− 1.69586i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3564.00i	0.286402i
$538$	0	0
$539$	6524.00	0.521352
$540$	0	0
$541$	22950.0	1.82384	0.911920	−	0.410368i	$-0.134600\pi$
$541$	22950.0	1.82384	0.911920	+	0.410368i	$0.134600\pi$
$542$	0	0
$543$	− 10422.0i	− 0.823666i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6580.00i	0.514334i	0.966367	+	0.257167i	$0.0827890\pi$
$547$	6580.00i	0.514334i	−0.966367	+	0.257167i	$0.917211\pi$
$548$	0	0
$549$	1962.00	0.152525
$550$	0	0
$551$	−12696.0	−0.981611
$552$	0	0
$553$	768.000i	0.0590573i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7046.00i	− 0.535994i	−0.963420	−	0.267997i	$-0.913638\pi$
$557$	− 7046.00i	− 0.535994i	0.963420	−	0.267997i	$-0.0863617\pi$
$558$	0	0
$559$	296.000	0.0223962
$560$	0	0
$561$	−6888.00	−0.518381
$562$	0	0
$563$	8252.00i	0.617727i	0.951106	+	0.308864i	$0.0999486\pi$
$563$	8252.00i	0.617727i	−0.951106	+	0.308864i	$0.900051\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1944.00i	0.143986i
$568$	0	0
$569$	6838.00	0.503803	0.251901	−	0.967753i	$-0.418944\pi$
$569$	6838.00	0.503803	0.251901	+	0.967753i	$0.418944\pi$
$570$	0	0
$571$	23316.0	1.70883	0.854417	−	0.519588i	$-0.173915\pi$
$571$	23316.0	1.70883	0.854417	+	0.519588i	$0.173915\pi$
$572$	0	0
$573$	576.000i	0.0419943i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10558.0i	0.761760i	0.924625	+	0.380880i	$0.124379\pi$
$577$	10558.0i	0.761760i	−0.924625	+	0.380880i	$0.875621\pi$
$578$	0	0
$579$	−14406.0	−1.03401
$580$	0	0
$581$	36192.0	2.58433
$582$	0	0
$583$	3640.00i	0.258582i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1028.00i	− 0.0722830i	−0.999347	−	0.0361415i	$-0.988493\pi$
$587$	− 1028.00i	− 0.0722830i	0.999347	−	0.0361415i	$-0.0115067\pi$
$588$	0	0
$589$	−7360.00	−0.514879
$590$	0	0
$591$	4554.00	0.316965
$592$	0	0
$593$	1202.00i	0.0832382i	0.999134	+	0.0416191i	$0.0132516\pi$
$593$	1202.00i	0.0832382i	−0.999134	+	0.0416191i	$0.986748\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 15384.0i	− 1.05465i
$598$	0	0
$599$	3576.00	0.243926	0.121963	−	0.992535i	$-0.461081\pi$
$599$	3576.00	0.243926	0.121963	+	0.992535i	$0.461081\pi$
$600$	0	0
$601$	8650.00	0.587090	0.293545	−	0.955945i	$-0.405165\pi$
$601$	8650.00	0.587090	0.293545	+	0.955945i	$0.405165\pi$
$602$	0	0
$603$	− 3924.00i	− 0.265004i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 12656.0i	− 0.846279i	−0.906065	−	0.423139i	$-0.860928\pi$
$607$	− 12656.0i	− 0.846279i	0.906065	−	0.423139i	$-0.139072\pi$
$608$	0	0
$609$	−9936.00	−0.661128
$610$	0	0
$611$	−17760.0	−1.17593
$612$	0	0
$613$	− 3298.00i	− 0.217300i	−0.994080	−	0.108650i	$-0.965347\pi$
$613$	− 3298.00i	− 0.217300i	0.994080	−	0.108650i	$-0.0346528\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5370.00i	− 0.350386i	−0.984534	−	0.175193i	$-0.943945\pi$
$617$	− 5370.00i	− 0.350386i	0.984534	−	0.175193i	$-0.0560549\pi$
$618$	0	0
$619$	16220.0	1.05321	0.526605	−	0.850110i	$-0.323465\pi$
$619$	16220.0	1.05321	0.526605	+	0.850110i	$0.323465\pi$
$620$	0	0
$621$	216.000	0.0139578
$622$	0	0
$623$	5904.00i	0.379677i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7728.00i	0.492227i
$628$	0	0
$629$	−2460.00	−0.155941
$630$	0	0
$631$	−20360.0	−1.28450	−0.642249	−	0.766496i	$-0.721999\pi$
$631$	−20360.0	−1.28450	−0.642249	+	0.766496i	$0.721999\pi$
$632$	0	0
$633$	3252.00i	0.204195i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	17242.0i	1.07245i
$638$	0	0
$639$	−7704.00	−0.476941
$640$	0	0
$641$	14498.0	0.893349	0.446674	−	0.894697i	$-0.352608\pi$
$641$	14498.0	0.893349	0.446674	+	0.894697i	$0.352608\pi$
$642$	0	0
$643$	21612.0i	1.32550i	0.748842	+	0.662748i	$0.230610\pi$
$643$	21612.0i	1.32550i	−0.748842	+	0.662748i	$0.769390\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12184.0i	− 0.740344i	−0.928963	−	0.370172i	$-0.879299\pi$
$647$	− 12184.0i	− 0.740344i	0.928963	−	0.370172i	$-0.120701\pi$
$648$	0	0
$649$	16688.0	1.00934
$650$	0	0
$651$	−5760.00	−0.346778
$652$	0	0
$653$	− 28122.0i	− 1.68530i	−0.538464	−	0.842648i	$-0.680995\pi$
$653$	− 28122.0i	− 1.68530i	0.538464	−	0.842648i	$-0.319005\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8982.00i	0.533366i
$658$	0	0
$659$	5700.00	0.336935	0.168468	−	0.985707i	$-0.446118\pi$
$659$	5700.00	0.336935	0.168468	+	0.985707i	$0.446118\pi$
$660$	0	0
$661$	−29458.0	−1.73341	−0.866705	−	0.498822i	$-0.833766\pi$
$661$	−29458.0	−1.73341	−0.866705	+	0.498822i	$0.833766\pi$
$662$	0	0
$663$	− 18204.0i	− 1.06634i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1104.00i	0.0640885i
$668$	0	0
$669$	−2064.00	−0.119281
$670$	0	0
$671$	6104.00	0.351181
$672$	0	0
$673$	19810.0i	1.13465i	0.823494	+	0.567325i	$0.192022\pi$
$673$	19810.0i	1.13465i	−0.823494	+	0.567325i	$0.807978\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10450.0i	0.593244i	0.954995	+	0.296622i	$0.0958601\pi$
$677$	10450.0i	0.593244i	−0.954995	+	0.296622i	$0.904140\pi$
$678$	0	0
$679$	20784.0	1.17469
$680$	0	0
$681$	14436.0	0.812318
$682$	0	0
$683$	23300.0i	1.30534i	0.757641	+	0.652672i	$0.226352\pi$
$683$	23300.0i	1.30534i	−0.757641	+	0.652672i	$0.773648\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7482.00i	− 0.415511i
$688$	0	0
$689$	−9620.00	−0.531920
$690$	0	0
$691$	−14212.0	−0.782417	−0.391208	−	0.920302i	$-0.627943\pi$
$691$	−14212.0	−0.782417	−0.391208	+	0.920302i	$0.627943\pi$
$692$	0	0
$693$	6048.00i	0.331522i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 23124.0i	− 1.25665i
$698$	0	0
$699$	−2094.00	−0.113308
$700$	0	0
$701$	−15978.0	−0.860885	−0.430443	−	0.902618i	$-0.641643\pi$
$701$	−15978.0	−0.860885	−0.430443	+	0.902618i	$0.641643\pi$
$702$	0	0
$703$	2760.00i	0.148073i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6480.00i	0.344704i
$708$	0	0
$709$	8866.00	0.469633	0.234816	−	0.972040i	$-0.424551\pi$
$709$	8866.00	0.469633	0.234816	+	0.972040i	$0.424551\pi$
$710$	0	0
$711$	−288.000	−0.0151911
$712$	0	0
$713$	640.000i	0.0336160i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	18960.0i	0.987551i
$718$	0	0
$719$	−7760.00	−0.402502	−0.201251	−	0.979540i	$-0.564501\pi$
$719$	−7760.00	−0.402502	−0.201251	+	0.979540i	$0.564501\pi$
$720$	0	0
$721$	35904.0	1.85456
$722$	0	0
$723$	− 19530.0i	− 1.00460i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13080.0i	− 0.667277i	−0.942701	−	0.333638i	$-0.891724\pi$
$727$	− 13080.0i	− 0.667277i	0.942701	−	0.333638i	$-0.108276\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	328.000	0.0165958
$732$	0	0
$733$	16934.0i	0.853304i	0.904416	+	0.426652i	$0.140307\pi$
$733$	16934.0i	0.853304i	−0.904416	+	0.426652i	$0.859693\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12208.0i	− 0.610159i
$738$	0	0
$739$	7060.00	0.351429	0.175715	−	0.984441i	$-0.443776\pi$
$739$	7060.00	0.351429	0.175715	+	0.984441i	$0.443776\pi$
$740$	0	0
$741$	−20424.0	−1.01254
$742$	0	0
$743$	− 12520.0i	− 0.618189i	−0.951031	−	0.309094i	$-0.899974\pi$
$743$	− 12520.0i	− 0.618189i	0.951031	−	0.309094i	$-0.100026\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13572.0i	0.664757i
$748$	0	0
$749$	−40608.0	−1.98102
$750$	0	0
$751$	−9792.00	−0.475786	−0.237893	−	0.971291i	$-0.576457\pi$
$751$	−9792.00	−0.475786	−0.237893	+	0.971291i	$0.576457\pi$
$752$	0	0
$753$	1884.00i	0.0911777i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 13166.0i	− 0.632135i	−0.948737	−	0.316068i	$-0.897637\pi$
$757$	− 13166.0i	− 0.632135i	0.948737	−	0.316068i	$-0.102363\pi$
$758$	0	0
$759$	672.000	0.0321371
$760$	0	0
$761$	−23222.0	−1.10617	−0.553086	−	0.833124i	$-0.686550\pi$
$761$	−23222.0	−1.10617	−0.553086	+	0.833124i	$0.686550\pi$
$762$	0	0
$763$	− 9744.00i	− 0.462328i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	44104.0i	2.07628i
$768$	0	0
$769$	39934.0	1.87264	0.936318	−	0.351154i	$-0.114211\pi$
$769$	39934.0	1.87264	0.936318	+	0.351154i	$0.114211\pi$
$770$	0	0
$771$	−14586.0	−0.681325
$772$	0	0
$773$	− 17106.0i	− 0.795938i	−0.917399	−	0.397969i	$-0.869715\pi$
$773$	− 17106.0i	− 0.795938i	0.917399	−	0.397969i	$-0.130285\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2160.00i	0.0997292i
$778$	0	0
$779$	−25944.0	−1.19325
$780$	0	0
$781$	−23968.0	−1.09813
$782$	0	0
$783$	− 3726.00i	− 0.170059i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9956.00i	0.450944i	0.974250	+	0.225472i	$0.0723924\pi$
$787$	9956.00i	0.450944i	−0.974250	+	0.225472i	$0.927608\pi$
$788$	0	0
$789$	−17448.0	−0.787282
$790$	0	0
$791$	−18864.0	−0.847948
$792$	0	0
$793$	16132.0i	0.722401i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9130.00i	0.405773i	0.979202	+	0.202887i	$0.0650323\pi$
$797$	9130.00i	0.405773i	−0.979202	+	0.202887i	$0.934968\pi$
$798$	0	0
$799$	−19680.0	−0.871375
$800$	0	0
$801$	−2214.00	−0.0976627
$802$	0	0
$803$	27944.0i	1.22805i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10578.0i	− 0.461417i
$808$	0	0
$809$	−11482.0	−0.498993	−0.249497	−	0.968376i	$-0.580265\pi$
$809$	−11482.0	−0.498993	−0.249497	+	0.968376i	$0.580265\pi$
$810$	0	0
$811$	4612.00	0.199691	0.0998454	−	0.995003i	$-0.468165\pi$
$811$	4612.00	0.199691	0.0998454	+	0.995003i	$0.468165\pi$
$812$	0	0
$813$	− 768.000i	− 0.0331303i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 368.000i	− 0.0157585i
$818$	0	0
$819$	−15984.0	−0.681961
$820$	0	0
$821$	−35010.0	−1.48826	−0.744128	−	0.668038i	$-0.767135\pi$
$821$	−35010.0	−1.48826	−0.744128	+	0.668038i	$0.767135\pi$
$822$	0	0
$823$	13688.0i	0.579749i	0.957065	+	0.289875i	$0.0936136\pi$
$823$	13688.0i	0.579749i	−0.957065	+	0.289875i	$0.906386\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 11668.0i	− 0.490612i	−0.969446	−	0.245306i	$-0.921112\pi$
$827$	− 11668.0i	− 0.490612i	0.969446	−	0.245306i	$-0.0788884\pi$
$828$	0	0
$829$	29306.0	1.22779	0.613896	−	0.789387i	$-0.289601\pi$
$829$	29306.0	1.22779	0.613896	+	0.789387i	$0.289601\pi$
$830$	0	0
$831$	426.000	0.0177831
$832$	0	0
$833$	19106.0i	0.794698i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2160.00i	− 0.0892001i
$838$	0	0
$839$	2664.00	0.109620	0.0548102	−	0.998497i	$-0.482545\pi$
$839$	2664.00	0.109620	0.0548102	+	0.998497i	$0.482545\pi$
$840$	0	0
$841$	−5345.00	−0.219156
$842$	0	0
$843$	26526.0i	1.08375i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 13128.0i	− 0.532566i
$848$	0	0
$849$	21540.0	0.870731
$850$	0	0
$851$	240.000	0.00966756
$852$	0	0
$853$	26030.0i	1.04484i	0.852688	+	0.522421i	$0.174971\pi$
$853$	26030.0i	1.04484i	−0.852688	+	0.522421i	$0.825029\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 44202.0i	− 1.76186i	−0.473249	−	0.880929i	$-0.656919\pi$
$857$	− 44202.0i	− 1.76186i	0.473249	−	0.880929i	$-0.343081\pi$
$858$	0	0
$859$	32748.0	1.30075	0.650377	−	0.759612i	$-0.274611\pi$
$859$	32748.0	1.30075	0.650377	+	0.759612i	$0.274611\pi$
$860$	0	0
$861$	−20304.0	−0.803668
$862$	0	0
$863$	45344.0i	1.78856i	0.447507	+	0.894280i	$0.352312\pi$
$863$	45344.0i	1.78856i	−0.447507	+	0.894280i	$0.647688\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5433.00i	− 0.212819i
$868$	0	0
$869$	−896.000	−0.0349767
$870$	0	0
$871$	32264.0	1.25514
$872$	0	0
$873$	7794.00i	0.302161i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8778.00i	0.337984i	0.985617	+	0.168992i	$0.0540512\pi$
$877$	8778.00i	0.337984i	−0.985617	+	0.168992i	$0.945949\pi$
$878$	0	0
$879$	−22122.0	−0.848870
$880$	0	0
$881$	−4142.00	−0.158397	−0.0791984	−	0.996859i	$-0.525236\pi$
$881$	−4142.00	−0.158397	−0.0791984	+	0.996859i	$0.525236\pi$
$882$	0	0
$883$	22076.0i	0.841355i	0.907210	+	0.420678i	$0.138208\pi$
$883$	22076.0i	0.841355i	−0.907210	+	0.420678i	$0.861792\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40376.0i	1.52840i	0.644978	+	0.764201i	$0.276867\pi$
$887$	40376.0i	1.52840i	−0.644978	+	0.764201i	$0.723133\pi$
$888$	0	0
$889$	41856.0	1.57908
$890$	0	0
$891$	−2268.00	−0.0852759
$892$	0	0
$893$	22080.0i	0.827412i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1776.00i	0.0661080i
$898$	0	0
$899$	11040.0	0.409571
$900$	0	0
$901$	−10660.0	−0.394158
$902$	0	0
$903$	− 288.000i	− 0.0106136i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26396.0i	0.966334i	0.875528	+	0.483167i	$0.160514\pi$
$907$	26396.0i	0.966334i	−0.875528	+	0.483167i	$0.839486\pi$
$908$	0	0
$909$	−2430.00	−0.0886667
$910$	0	0
$911$	24368.0	0.886222	0.443111	−	0.896467i	$-0.353875\pi$
$911$	24368.0	0.886222	0.443111	+	0.896467i	$0.353875\pi$
$912$	0	0
$913$	42224.0i	1.53057i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15648.0i	0.563514i
$918$	0	0
$919$	5096.00	0.182918	0.0914589	−	0.995809i	$-0.470847\pi$
$919$	5096.00	0.182918	0.0914589	+	0.995809i	$0.470847\pi$
$920$	0	0
$921$	4500.00	0.160999
$922$	0	0
$923$	− 63344.0i	− 2.25893i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13464.0i	0.477040i
$928$	0	0
$929$	18494.0	0.653142	0.326571	−	0.945173i	$-0.394107\pi$
$929$	18494.0	0.653142	0.326571	+	0.945173i	$0.394107\pi$
$930$	0	0
$931$	21436.0	0.754604
$932$	0	0
$933$	− 22824.0i	− 0.800883i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33222.0i	1.15829i	0.815225	+	0.579144i	$0.196613\pi$
$937$	33222.0i	1.15829i	−0.815225	+	0.579144i	$0.803387\pi$
$938$	0	0
$939$	14274.0	0.496075
$940$	0	0
$941$	27846.0	0.964669	0.482335	−	0.875987i	$-0.339789\pi$
$941$	27846.0	0.964669	0.482335	+	0.875987i	$0.339789\pi$
$942$	0	0
$943$	2256.00i	0.0779061i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41052.0i	− 1.40867i	−0.709868	−	0.704335i	$-0.751245\pi$
$947$	− 41052.0i	− 1.40867i	0.709868	−	0.704335i	$-0.248755\pi$
$948$	0	0
$949$	−73852.0	−2.52617
$950$	0	0
$951$	13122.0	0.447434
$952$	0	0
$953$	5706.00i	0.193951i	0.995287	+	0.0969756i	$0.0309169\pi$
$953$	5706.00i	0.193951i	−0.995287	+	0.0969756i	$0.969083\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 11592.0i	− 0.391553i
$958$	0	0
$959$	36720.0	1.23644
$960$	0	0
$961$	−23391.0	−0.785170
$962$	0	0
$963$	− 15228.0i	− 0.509570i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	39352.0i	1.30866i	0.756209	+	0.654330i	$0.227049\pi$
$967$	39352.0i	1.30866i	−0.756209	+	0.654330i	$0.772951\pi$
$968$	0	0
$969$	−22632.0	−0.750304
$970$	0	0
$971$	−33180.0	−1.09660	−0.548299	−	0.836282i	$-0.684724\pi$
$971$	−33180.0	−1.09660	−0.548299	+	0.836282i	$0.684724\pi$
$972$	0	0
$973$	− 12384.0i	− 0.408030i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4014.00i	0.131442i	0.997838	+	0.0657212i	$0.0209348\pi$
$977$	4014.00i	0.131442i	−0.997838	+	0.0657212i	$0.979065\pi$
$978$	0	0
$979$	−6888.00	−0.224864
$980$	0	0
$981$	3654.00	0.118923
$982$	0	0
$983$	20328.0i	0.659575i	0.944055	+	0.329788i	$0.106977\pi$
$983$	20328.0i	0.659575i	−0.944055	+	0.329788i	$0.893023\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17280.0i	0.557273i
$988$	0	0
$989$	−32.0000	−0.00102886
$990$	0	0
$991$	11728.0	0.375936	0.187968	−	0.982175i	$-0.439810\pi$
$991$	11728.0	0.375936	0.187968	+	0.982175i	$0.439810\pi$
$992$	0	0
$993$	− 23412.0i	− 0.748195i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 50974.0i	− 1.61922i	−0.586968	−	0.809610i	$-0.699679\pi$
$997$	− 50974.0i	− 1.61922i	0.586968	−	0.809610i	$-0.300321\pi$
$998$	0	0
$999$	−810.000	−0.0256529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.f.b.49.2		2
3.2	odd	2		1800.4.f.q.649.2		2
4.3	odd	2		1200.4.f.p.49.1		2
5.2	odd	4		24.4.a.a.1.1	✓	1
5.3	odd	4		600.4.a.h.1.1		1
5.4	even	2	inner	600.4.f.b.49.1		2
15.2	even	4		72.4.a.b.1.1		1
15.8	even	4		1800.4.a.bg.1.1		1
15.14	odd	2		1800.4.f.q.649.1		2
20.3	even	4		1200.4.a.u.1.1		1
20.7	even	4		48.4.a.b.1.1		1
20.19	odd	2		1200.4.f.p.49.2		2
35.27	even	4		1176.4.a.a.1.1		1
40.27	even	4		192.4.a.g.1.1		1
40.37	odd	4		192.4.a.a.1.1		1
45.2	even	12		648.4.i.k.433.1		2
45.7	odd	12		648.4.i.b.433.1		2
45.22	odd	12		648.4.i.b.217.1		2
45.32	even	12		648.4.i.k.217.1		2
60.47	odd	4		144.4.a.b.1.1		1
80.27	even	4		768.4.d.b.385.2		2
80.37	odd	4		768.4.d.o.385.1		2
80.67	even	4		768.4.d.b.385.1		2
80.77	odd	4		768.4.d.o.385.2		2
120.77	even	4		576.4.a.u.1.1		1
120.107	odd	4		576.4.a.v.1.1		1
140.27	odd	4		2352.4.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.a.a.1.1	✓	1	5.2	odd	4
48.4.a.b.1.1		1	20.7	even	4
72.4.a.b.1.1		1	15.2	even	4
144.4.a.b.1.1		1	60.47	odd	4
192.4.a.a.1.1		1	40.37	odd	4
192.4.a.g.1.1		1	40.27	even	4
576.4.a.u.1.1		1	120.77	even	4
576.4.a.v.1.1		1	120.107	odd	4
600.4.a.h.1.1		1	5.3	odd	4
600.4.f.b.49.1		2	5.4	even	2	inner
600.4.f.b.49.2		2	1.1	even	1	trivial
648.4.i.b.217.1		2	45.22	odd	12
648.4.i.b.433.1		2	45.7	odd	12
648.4.i.k.217.1		2	45.32	even	12
648.4.i.k.433.1		2	45.2	even	12
768.4.d.b.385.1		2	80.67	even	4
768.4.d.b.385.2		2	80.27	even	4
768.4.d.o.385.1		2	80.37	odd	4
768.4.d.o.385.2		2	80.77	odd	4
1176.4.a.a.1.1		1	35.27	even	4
1200.4.a.u.1.1		1	20.3	even	4
1200.4.f.p.49.1		2	4.3	odd	2
1200.4.f.p.49.2		2	20.19	odd	2
1800.4.a.bg.1.1		1	15.8	even	4
1800.4.f.q.649.1		2	15.14	odd	2
1800.4.f.q.649.2		2	3.2	odd	2
2352.4.a.w.1.1		1	140.27	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} -28.0000 q^{11} -74.0000i q^{13} -82.0000i q^{17} -92.0000 q^{19} -72.0000 q^{21} +8.00000i q^{23} -27.0000i q^{27} +138.000 q^{29} +80.0000 q^{31} -84.0000i q^{33} -30.0000i q^{37} +222.000 q^{39} +282.000 q^{41} +4.00000i q^{43} -240.000i q^{47} -233.000 q^{49} +246.000 q^{51} -130.000i q^{53} -276.000i q^{57} -596.000 q^{59} -218.000 q^{61} -216.000i q^{63} +436.000i q^{67} -24.0000 q^{69} +856.000 q^{71} -998.000i q^{73} -672.000i q^{77} +32.0000 q^{79} +81.0000 q^{81} -1508.00i q^{83} +414.000i q^{87} +246.000 q^{89} +1776.00 q^{91} +240.000i q^{93} -866.000i q^{97} +252.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 56 q^{11} - 184 q^{19} - 144 q^{21} + 276 q^{29} + 160 q^{31} + 444 q^{39} + 564 q^{41} - 466 q^{49} + 492 q^{51} - 1192 q^{59} - 436 q^{61} - 48 q^{69} + 1712 q^{71} + 64 q^{79} + 162 q^{81} + 492 q^{89} + 3552 q^{91} + 504 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 56 * q^11 - 184 * q^19 - 144 * q^21 + 276 * q^29 + 160 * q^31 + 444 * q^39 + 564 * q^41 - 466 * q^49 + 492 * q^51 - 1192 * q^59 - 436 * q^61 - 48 * q^69 + 1712 * q^71 + 64 * q^79 + 162 * q^81 + 492 * q^89 + 3552 * q^91 + 504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more