LMFDB - Embedded newform 1200.4.f.j.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.j.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} -12.0000 q^{11} -38.0000i q^{13} -126.000i q^{17} +20.0000 q^{19} +48.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -30.0000 q^{29} +88.0000 q^{31} +36.0000i q^{33} +254.000i q^{37} -114.000 q^{39} +42.0000 q^{41} -52.0000i q^{43} +96.0000i q^{47} +87.0000 q^{49} -378.000 q^{51} -198.000i q^{53} -60.0000i q^{57} -660.000 q^{59} -538.000 q^{61} -144.000i q^{63} -884.000i q^{67} +504.000 q^{69} -792.000 q^{71} -218.000i q^{73} -192.000i q^{77} -520.000 q^{79} +81.0000 q^{81} -492.000i q^{83} +90.0000i q^{87} -810.000 q^{89} +608.000 q^{91} -264.000i q^{93} +1154.00i q^{97} +108.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 24 q^{11} + 40 q^{19} + 96 q^{21} - 60 q^{29} + 176 q^{31} - 228 q^{39} + 84 q^{41} + 174 q^{49} - 756 q^{51} - 1320 q^{59} - 1076 q^{61} + 1008 q^{69} - 1584 q^{71} - 1040 q^{79} + 162 q^{81} - 1620 q^{89} + 1216 q^{91} + 216 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 24 * q^11 + 40 * q^19 + 96 * q^21 - 60 * q^29 + 176 * q^31 - 228 * q^39 + 84 * q^41 + 174 * q^49 - 756 * q^51 - 1320 * q^59 - 1076 * q^61 + 1008 * q^69 - 1584 * q^71 - 1040 * q^79 + 162 * q^81 - 1620 * q^89 + 1216 * q^91 + 216 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$7$	16.0000i	0.863919i	0.901893	+	0.431959i	$0.142178\pi$
$7$	16.0000i	0.863919i	−0.901893	+	0.431959i	$0.857822\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	− 38.0000i	− 0.810716i	−0.914158	−	0.405358i	$-0.867147\pi$
$13$	− 38.0000i	− 0.810716i	0.914158	−	0.405358i	$-0.132853\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 126.000i	− 1.79762i	−0.438342	−	0.898808i	$-0.644434\pi$
$17$	− 126.000i	− 1.79762i	0.438342	−	0.898808i	$-0.355566\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	48.0000	0.498784
$22$	0	0
$23$	168.000i	1.52306i	0.648129	+	0.761531i	$0.275552\pi$
$23$	168.000i	1.52306i	−0.648129	+	0.761531i	$0.724448\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−30.0000	−0.192099	−0.0960493	−	0.995377i	$-0.530621\pi$
$29$	−30.0000	−0.192099	−0.0960493	+	0.995377i	$0.530621\pi$
$30$	0	0
$31$	88.0000	0.509847	0.254924	−	0.966961i	$-0.417950\pi$
$31$	88.0000	0.509847	0.254924	+	0.966961i	$0.417950\pi$
$32$	0	0
$33$	36.0000i	0.189903i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	254.000i	1.12858i	0.825578	+	0.564288i	$0.190849\pi$
$37$	254.000i	1.12858i	−0.825578	+	0.564288i	$0.809151\pi$
$38$	0	0
$39$	−114.000	−0.468067
$40$	0	0
$41$	42.0000	0.159983	0.0799914	−	0.996796i	$-0.474511\pi$
$41$	42.0000	0.159983	0.0799914	+	0.996796i	$0.474511\pi$
$42$	0	0
$43$	− 52.0000i	− 0.184417i	−0.995740	−	0.0922084i	$-0.970607\pi$
$43$	− 52.0000i	− 0.184417i	0.995740	−	0.0922084i	$-0.0293926\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	96.0000i	0.297937i	0.988842	+	0.148969i	$0.0475953\pi$
$47$	96.0000i	0.297937i	−0.988842	+	0.148969i	$0.952405\pi$
$48$	0	0
$49$	87.0000	0.253644
$50$	0	0
$51$	−378.000	−1.03785
$52$	0	0
$53$	− 198.000i	− 0.513158i	−0.966523	−	0.256579i	$-0.917405\pi$
$53$	− 198.000i	− 0.513158i	0.966523	−	0.256579i	$-0.0825954\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 60.0000i	− 0.139424i
$58$	0	0
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	0	0
$61$	−538.000	−1.12924	−0.564622	−	0.825350i	$-0.690978\pi$
$61$	−538.000	−1.12924	−0.564622	+	0.825350i	$0.690978\pi$
$62$	0	0
$63$	− 144.000i	− 0.287973i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 884.000i	− 1.61191i	−0.591979	−	0.805954i	$-0.701653\pi$
$67$	− 884.000i	− 1.61191i	0.591979	−	0.805954i	$-0.298347\pi$
$68$	0	0
$69$	504.000	0.879340
$70$	0	0
$71$	−792.000	−1.32385	−0.661923	−	0.749572i	$-0.730260\pi$
$71$	−792.000	−1.32385	−0.661923	+	0.749572i	$0.730260\pi$
$72$	0	0
$73$	− 218.000i	− 0.349520i	−0.984611	−	0.174760i	$-0.944085\pi$
$73$	− 218.000i	− 0.349520i	0.984611	−	0.174760i	$-0.0559150\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 192.000i	− 0.284161i
$78$	0	0
$79$	−520.000	−0.740564	−0.370282	−	0.928919i	$-0.620739\pi$
$79$	−520.000	−0.740564	−0.370282	+	0.928919i	$0.620739\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 492.000i	− 0.650651i	−0.945602	−	0.325325i	$-0.894526\pi$
$83$	− 492.000i	− 0.650651i	0.945602	−	0.325325i	$-0.105474\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	90.0000i	0.110908i
$88$	0	0
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	608.000	0.700393
$92$	0	0
$93$	− 264.000i	− 0.294360i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1154.00i	1.20795i	0.797004	+	0.603974i	$0.206417\pi$
$97$	1154.00i	1.20795i	−0.797004	+	0.603974i	$0.793583\pi$
$98$	0	0
$99$	108.000	0.109640
$100$	0	0
$101$	−618.000	−0.608845	−0.304422	−	0.952537i	$-0.598463\pi$
$101$	−618.000	−0.608845	−0.304422	+	0.952537i	$0.598463\pi$
$102$	0	0
$103$	128.000i	0.122449i	0.998124	+	0.0612243i	$0.0195005\pi$
$103$	128.000i	0.122449i	−0.998124	+	0.0612243i	$0.980499\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1476.00i	1.33355i	0.745257	+	0.666777i	$0.232327\pi$
$107$	1476.00i	1.33355i	−0.745257	+	0.666777i	$0.767673\pi$
$108$	0	0
$109$	−1190.00	−1.04570	−0.522850	−	0.852425i	$-0.675131\pi$
$109$	−1190.00	−1.04570	−0.522850	+	0.852425i	$0.675131\pi$
$110$	0	0
$111$	762.000	0.651584
$112$	0	0
$113$	462.000i	0.384613i	0.981335	+	0.192307i	$0.0615968\pi$
$113$	462.000i	0.384613i	−0.981335	+	0.192307i	$0.938403\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	342.000i	0.270239i
$118$	0	0
$119$	2016.00	1.55300
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	− 126.000i	− 0.0923662i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2536.00i	1.77192i	0.463763	+	0.885959i	$0.346499\pi$
$127$	2536.00i	1.77192i	−0.463763	+	0.885959i	$0.653501\pi$
$128$	0	0
$129$	−156.000	−0.106473
$130$	0	0
$131$	−2292.00	−1.52865	−0.764324	−	0.644832i	$-0.776927\pi$
$131$	−2292.00	−1.52865	−0.764324	+	0.644832i	$0.776927\pi$
$132$	0	0
$133$	320.000i	0.208628i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 726.000i	− 0.452747i	−0.974041	−	0.226374i	$-0.927313\pi$
$137$	− 726.000i	− 0.452747i	0.974041	−	0.226374i	$-0.0726870\pi$
$138$	0	0
$139$	380.000	0.231879	0.115939	−	0.993256i	$-0.463012\pi$
$139$	380.000	0.231879	0.115939	+	0.993256i	$0.463012\pi$
$140$	0	0
$141$	288.000	0.172014
$142$	0	0
$143$	456.000i	0.266662i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 261.000i	− 0.146442i
$148$	0	0
$149$	−1590.00	−0.874214	−0.437107	−	0.899410i	$-0.643997\pi$
$149$	−1590.00	−0.874214	−0.437107	+	0.899410i	$0.643997\pi$
$150$	0	0
$151$	−2432.00	−1.31068	−0.655342	−	0.755332i	$-0.727476\pi$
$151$	−2432.00	−1.31068	−0.655342	+	0.755332i	$0.727476\pi$
$152$	0	0
$153$	1134.00i	0.599206i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	614.000i	0.312118i	0.987748	+	0.156059i	$0.0498790\pi$
$157$	614.000i	0.312118i	−0.987748	+	0.156059i	$0.950121\pi$
$158$	0	0
$159$	−594.000	−0.296272
$160$	0	0
$161$	−2688.00	−1.31580
$162$	0	0
$163$	− 1852.00i	− 0.889938i	−0.895546	−	0.444969i	$-0.853215\pi$
$163$	− 1852.00i	− 0.889938i	0.895546	−	0.444969i	$-0.146785\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2136.00i	0.989752i	0.868964	+	0.494876i	$0.164787\pi$
$167$	2136.00i	0.989752i	−0.868964	+	0.494876i	$0.835213\pi$
$168$	0	0
$169$	753.000	0.342740
$170$	0	0
$171$	−180.000	−0.0804967
$172$	0	0
$173$	− 1758.00i	− 0.772591i	−0.922375	−	0.386296i	$-0.873754\pi$
$173$	− 1758.00i	− 0.772591i	0.922375	−	0.386296i	$-0.126246\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1980.00i	0.840824i
$178$	0	0
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	0	0
$181$	1982.00	0.813928	0.406964	−	0.913444i	$-0.366588\pi$
$181$	1982.00	0.813928	0.406964	+	0.913444i	$0.366588\pi$
$182$	0	0
$183$	1614.00i	0.651969i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1512.00i	0.591275i
$188$	0	0
$189$	−432.000	−0.166261
$190$	0	0
$191$	2688.00	1.01831	0.509154	−	0.860675i	$-0.329958\pi$
$191$	2688.00	1.01831	0.509154	+	0.860675i	$0.329958\pi$
$192$	0	0
$193$	2302.00i	0.858557i	0.903172	+	0.429279i	$0.141232\pi$
$193$	2302.00i	0.858557i	−0.903172	+	0.429279i	$0.858768\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4374.00i	1.58190i	0.611880	+	0.790951i	$0.290414\pi$
$197$	4374.00i	1.58190i	−0.611880	+	0.790951i	$0.709586\pi$
$198$	0	0
$199$	−1600.00	−0.569955	−0.284977	−	0.958534i	$-0.591986\pi$
$199$	−1600.00	−0.569955	−0.284977	+	0.958534i	$0.591986\pi$
$200$	0	0
$201$	−2652.00	−0.930635
$202$	0	0
$203$	− 480.000i	− 0.165958i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1512.00i	− 0.507687i
$208$	0	0
$209$	−240.000	−0.0794313
$210$	0	0
$211$	−3332.00	−1.08713	−0.543565	−	0.839367i	$-0.682926\pi$
$211$	−3332.00	−1.08713	−0.543565	+	0.839367i	$0.682926\pi$
$212$	0	0
$213$	2376.00i	0.764323i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1408.00i	0.440467i
$218$	0	0
$219$	−654.000	−0.201796
$220$	0	0
$221$	−4788.00	−1.45736
$222$	0	0
$223$	2648.00i	0.795171i	0.917565	+	0.397586i	$0.130152\pi$
$223$	2648.00i	0.795171i	−0.917565	+	0.397586i	$0.869848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2244.00i	− 0.656121i	−0.944657	−	0.328061i	$-0.893605\pi$
$227$	− 2244.00i	− 0.656121i	0.944657	−	0.328061i	$-0.106395\pi$
$228$	0	0
$229$	5650.00	1.63040	0.815202	−	0.579177i	$-0.196626\pi$
$229$	5650.00	1.63040	0.815202	+	0.579177i	$0.196626\pi$
$230$	0	0
$231$	−576.000	−0.164061
$232$	0	0
$233$	− 4698.00i	− 1.32093i	−0.750858	−	0.660464i	$-0.770360\pi$
$233$	− 4698.00i	− 1.32093i	0.750858	−	0.660464i	$-0.229640\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1560.00i	0.427565i
$238$	0	0
$239$	−1200.00	−0.324776	−0.162388	−	0.986727i	$-0.551920\pi$
$239$	−1200.00	−0.324776	−0.162388	+	0.986727i	$0.551920\pi$
$240$	0	0
$241$	−718.000	−0.191911	−0.0959553	−	0.995386i	$-0.530591\pi$
$241$	−718.000	−0.191911	−0.0959553	+	0.995386i	$0.530591\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 760.000i	− 0.195780i
$248$	0	0
$249$	−1476.00	−0.375653
$250$	0	0
$251$	−6012.00	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$251$	−6012.00	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$252$	0	0
$253$	− 2016.00i	− 0.500968i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2046.00i	− 0.496599i	−0.968683	−	0.248300i	$-0.920128\pi$
$257$	− 2046.00i	− 0.496599i	0.968683	−	0.248300i	$-0.0798717\pi$
$258$	0	0
$259$	−4064.00	−0.974999
$260$	0	0
$261$	270.000	0.0640329
$262$	0	0
$263$	− 6072.00i	− 1.42363i	−0.702365	−	0.711817i	$-0.747873\pi$
$263$	− 6072.00i	− 1.42363i	0.702365	−	0.711817i	$-0.252127\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2430.00i	0.556980i
$268$	0	0
$269$	6930.00	1.57074	0.785371	−	0.619025i	$-0.212472\pi$
$269$	6930.00	1.57074	0.785371	+	0.619025i	$0.212472\pi$
$270$	0	0
$271$	−1352.00	−0.303056	−0.151528	−	0.988453i	$-0.548419\pi$
$271$	−1352.00	−0.303056	−0.151528	+	0.988453i	$0.548419\pi$
$272$	0	0
$273$	− 1824.00i	− 0.404372i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1186.00i	− 0.257256i	−0.991693	−	0.128628i	$-0.958943\pi$
$277$	− 1186.00i	− 0.257256i	0.991693	−	0.128628i	$-0.0410573\pi$
$278$	0	0
$279$	−792.000	−0.169949
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	0	0
$283$	2828.00i	0.594018i	0.954875	+	0.297009i	$0.0959892\pi$
$283$	2828.00i	0.594018i	−0.954875	+	0.297009i	$0.904011\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	672.000i	0.138212i
$288$	0	0
$289$	−10963.0	−2.23143
$290$	0	0
$291$	3462.00	0.697409
$292$	0	0
$293$	− 4758.00i	− 0.948687i	−0.880340	−	0.474344i	$-0.842685\pi$
$293$	− 4758.00i	− 0.948687i	0.880340	−	0.474344i	$-0.157315\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 324.000i	− 0.0633010i
$298$	0	0
$299$	6384.00	1.23477
$300$	0	0
$301$	832.000	0.159321
$302$	0	0
$303$	1854.00i	0.351517i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8476.00i	1.57574i	0.615844	+	0.787868i	$0.288815\pi$
$307$	8476.00i	1.57574i	−0.615844	+	0.787868i	$0.711185\pi$
$308$	0	0
$309$	384.000	0.0706958
$310$	0	0
$311$	−4632.00	−0.844555	−0.422278	−	0.906467i	$-0.638769\pi$
$311$	−4632.00	−0.844555	−0.422278	+	0.906467i	$0.638769\pi$
$312$	0	0
$313$	4822.00i	0.870785i	0.900241	+	0.435392i	$0.143390\pi$
$313$	4822.00i	0.870785i	−0.900241	+	0.435392i	$0.856610\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3426.00i	− 0.607014i	−0.952829	−	0.303507i	$-0.901842\pi$
$317$	− 3426.00i	− 0.607014i	0.952829	−	0.303507i	$-0.0981575\pi$
$318$	0	0
$319$	360.000	0.0631854
$320$	0	0
$321$	4428.00	0.769928
$322$	0	0
$323$	− 2520.00i	− 0.434107i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3570.00i	0.603735i
$328$	0	0
$329$	−1536.00	−0.257393
$330$	0	0
$331$	2788.00	0.462968	0.231484	−	0.972839i	$-0.425642\pi$
$331$	2788.00	0.462968	0.231484	+	0.972839i	$0.425642\pi$
$332$	0	0
$333$	− 2286.00i	− 0.376192i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	434.000i	0.0701528i	0.999385	+	0.0350764i	$0.0111675\pi$
$337$	434.000i	0.0701528i	−0.999385	+	0.0350764i	$0.988833\pi$
$338$	0	0
$339$	1386.00	0.222057
$340$	0	0
$341$	−1056.00	−0.167700
$342$	0	0
$343$	6880.00i	1.08305i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6684.00i	− 1.03405i	−0.855970	−	0.517026i	$-0.827039\pi$
$347$	− 6684.00i	− 1.03405i	0.855970	−	0.517026i	$-0.172961\pi$
$348$	0	0
$349$	−2630.00	−0.403383	−0.201692	−	0.979449i	$-0.564644\pi$
$349$	−2630.00	−0.403383	−0.201692	+	0.979449i	$0.564644\pi$
$350$	0	0
$351$	1026.00	0.156022
$352$	0	0
$353$	7422.00i	1.11907i	0.828805	+	0.559537i	$0.189021\pi$
$353$	7422.00i	1.11907i	−0.828805	+	0.559537i	$0.810979\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6048.00i	− 0.896622i
$358$	0	0
$359$	−10440.0	−1.53482	−0.767412	−	0.641154i	$-0.778456\pi$
$359$	−10440.0	−1.53482	−0.767412	+	0.641154i	$0.778456\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	3561.00i	0.514887i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10424.0i	− 1.48264i	−0.671153	−	0.741319i	$-0.734200\pi$
$367$	− 10424.0i	− 1.48264i	0.671153	−	0.741319i	$-0.265800\pi$
$368$	0	0
$369$	−378.000	−0.0533276
$370$	0	0
$371$	3168.00	0.443327
$372$	0	0
$373$	− 3278.00i	− 0.455036i	−0.973774	−	0.227518i	$-0.926939\pi$
$373$	− 3278.00i	− 0.455036i	0.973774	−	0.227518i	$-0.0730610\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1140.00i	0.155737i
$378$	0	0
$379$	6140.00	0.832165	0.416083	−	0.909327i	$-0.363403\pi$
$379$	6140.00	0.832165	0.416083	+	0.909327i	$0.363403\pi$
$380$	0	0
$381$	7608.00	1.02302
$382$	0	0
$383$	− 3072.00i	− 0.409848i	−0.978778	−	0.204924i	$-0.934305\pi$
$383$	− 3072.00i	− 0.409848i	0.978778	−	0.204924i	$-0.0656948\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	468.000i	0.0614723i
$388$	0	0
$389$	−6150.00	−0.801587	−0.400794	−	0.916168i	$-0.631266\pi$
$389$	−6150.00	−0.801587	−0.400794	+	0.916168i	$0.631266\pi$
$390$	0	0
$391$	21168.0	2.73788
$392$	0	0
$393$	6876.00i	0.882566i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 106.000i	− 0.0134005i	−0.999978	−	0.00670024i	$-0.997867\pi$
$397$	− 106.000i	− 0.0134005i	0.999978	−	0.00670024i	$-0.00213277\pi$
$398$	0	0
$399$	960.000	0.120451
$400$	0	0
$401$	−1758.00	−0.218929	−0.109464	−	0.993991i	$-0.534914\pi$
$401$	−1758.00	−0.218929	−0.109464	+	0.993991i	$0.534914\pi$
$402$	0	0
$403$	− 3344.00i	− 0.413341i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3048.00i	− 0.371213i
$408$	0	0
$409$	3670.00	0.443691	0.221846	−	0.975082i	$-0.428792\pi$
$409$	3670.00	0.443691	0.221846	+	0.975082i	$0.428792\pi$
$410$	0	0
$411$	−2178.00	−0.261394
$412$	0	0
$413$	− 10560.0i	− 1.25817i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1140.00i	− 0.133875i
$418$	0	0
$419$	−9660.00	−1.12631	−0.563153	−	0.826353i	$-0.690412\pi$
$419$	−9660.00	−1.12631	−0.563153	+	0.826353i	$0.690412\pi$
$420$	0	0
$421$	8462.00	0.979602	0.489801	−	0.871834i	$-0.337069\pi$
$421$	8462.00	0.979602	0.489801	+	0.871834i	$0.337069\pi$
$422$	0	0
$423$	− 864.000i	− 0.0993123i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8608.00i	− 0.975575i
$428$	0	0
$429$	1368.00	0.153957
$430$	0	0
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	0	0
$433$	7342.00i	0.814859i	0.913237	+	0.407430i	$0.133575\pi$
$433$	7342.00i	0.814859i	−0.913237	+	0.407430i	$0.866425\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3360.00i	0.367805i
$438$	0	0
$439$	10640.0	1.15676	0.578382	−	0.815766i	$-0.303684\pi$
$439$	10640.0	1.15676	0.578382	+	0.815766i	$0.303684\pi$
$440$	0	0
$441$	−783.000	−0.0845481
$442$	0	0
$443$	− 17412.0i	− 1.86742i	−0.358024	−	0.933712i	$-0.616549\pi$
$443$	− 17412.0i	− 1.86742i	0.358024	−	0.933712i	$-0.383451\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4770.00i	0.504728i
$448$	0	0
$449$	1710.00	0.179732	0.0898662	−	0.995954i	$-0.471356\pi$
$449$	1710.00	0.179732	0.0898662	+	0.995954i	$0.471356\pi$
$450$	0	0
$451$	−504.000	−0.0526218
$452$	0	0
$453$	7296.00i	0.756724i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 646.000i	− 0.0661239i	−0.999453	−	0.0330619i	$-0.989474\pi$
$457$	− 646.000i	− 0.0661239i	0.999453	−	0.0330619i	$-0.0105259\pi$
$458$	0	0
$459$	3402.00	0.345952
$460$	0	0
$461$	−6018.00	−0.607996	−0.303998	−	0.952673i	$-0.598322\pi$
$461$	−6018.00	−0.607996	−0.303998	+	0.952673i	$0.598322\pi$
$462$	0	0
$463$	− 6712.00i	− 0.673722i	−0.941554	−	0.336861i	$-0.890635\pi$
$463$	− 6712.00i	− 0.673722i	0.941554	−	0.336861i	$-0.109365\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5364.00i	− 0.531512i	−0.964040	−	0.265756i	$-0.914378\pi$
$467$	− 5364.00i	− 0.531512i	0.964040	−	0.265756i	$-0.0856216\pi$
$468$	0	0
$469$	14144.0	1.39256
$470$	0	0
$471$	1842.00	0.180201
$472$	0	0
$473$	624.000i	0.0606587i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1782.00i	0.171053i
$478$	0	0
$479$	9840.00	0.938624	0.469312	−	0.883032i	$-0.344502\pi$
$479$	9840.00	0.938624	0.469312	+	0.883032i	$0.344502\pi$
$480$	0	0
$481$	9652.00	0.914955
$482$	0	0
$483$	8064.00i	0.759678i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1424.00i	− 0.132500i	−0.997803	−	0.0662501i	$-0.978896\pi$
$487$	− 1424.00i	− 0.132500i	0.997803	−	0.0662501i	$-0.0211035\pi$
$488$	0	0
$489$	−5556.00	−0.513806
$490$	0	0
$491$	4548.00	0.418021	0.209011	−	0.977913i	$-0.432976\pi$
$491$	4548.00	0.418021	0.209011	+	0.977913i	$0.432976\pi$
$492$	0	0
$493$	3780.00i	0.345320i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12672.0i	− 1.14370i
$498$	0	0
$499$	6500.00	0.583126	0.291563	−	0.956552i	$-0.405825\pi$
$499$	6500.00	0.583126	0.291563	+	0.956552i	$0.405825\pi$
$500$	0	0
$501$	6408.00	0.571434
$502$	0	0
$503$	12168.0i	1.07862i	0.842108	+	0.539308i	$0.181314\pi$
$503$	12168.0i	1.07862i	−0.842108	+	0.539308i	$0.818686\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2259.00i	− 0.197881i
$508$	0	0
$509$	21090.0	1.83654	0.918269	−	0.395957i	$-0.129587\pi$
$509$	21090.0	1.83654	0.918269	+	0.395957i	$0.129587\pi$
$510$	0	0
$511$	3488.00	0.301957
$512$	0	0
$513$	540.000i	0.0464748i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1152.00i	− 0.0979979i
$518$	0	0
$519$	−5274.00	−0.446056
$520$	0	0
$521$	−5238.00	−0.440462	−0.220231	−	0.975448i	$-0.570681\pi$
$521$	−5238.00	−0.440462	−0.220231	+	0.975448i	$0.570681\pi$
$522$	0	0
$523$	8588.00i	0.718025i	0.933333	+	0.359012i	$0.116886\pi$
$523$	8588.00i	0.718025i	−0.933333	+	0.359012i	$0.883114\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 11088.0i	− 0.916510i
$528$	0	0
$529$	−16057.0	−1.31972
$530$	0	0
$531$	5940.00	0.485450
$532$	0	0
$533$	− 1596.00i	− 0.129701i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1620.00i	0.130183i
$538$	0	0
$539$	−1044.00	−0.0834291
$540$	0	0
$541$	3062.00	0.243338	0.121669	−	0.992571i	$-0.461175\pi$
$541$	3062.00	0.243338	0.121669	+	0.992571i	$0.461175\pi$
$542$	0	0
$543$	− 5946.00i	− 0.469921i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8476.00i	0.662537i	0.943537	+	0.331268i	$0.107477\pi$
$547$	8476.00i	0.662537i	−0.943537	+	0.331268i	$0.892523\pi$
$548$	0	0
$549$	4842.00	0.376414
$550$	0	0
$551$	−600.000	−0.0463899
$552$	0	0
$553$	− 8320.00i	− 0.639787i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12546.0i	− 0.954383i	−0.878799	−	0.477191i	$-0.841655\pi$
$557$	− 12546.0i	− 0.954383i	0.878799	−	0.477191i	$-0.158345\pi$
$558$	0	0
$559$	−1976.00	−0.149510
$560$	0	0
$561$	4536.00	0.341373
$562$	0	0
$563$	− 12.0000i	0 0.000898294i	−1.00000		0.000449147i	$-0.999857\pi$
$563$	− 12.0000i	0 0.000898294i	1.00000		0.000449147i	$-0.000142968\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1296.00i	0.0959910i
$568$	0	0
$569$	−19290.0	−1.42123	−0.710614	−	0.703582i	$-0.751583\pi$
$569$	−19290.0	−1.42123	−0.710614	+	0.703582i	$0.751583\pi$
$570$	0	0
$571$	12148.0	0.890329	0.445165	−	0.895449i	$-0.353145\pi$
$571$	12148.0	0.890329	0.445165	+	0.895449i	$0.353145\pi$
$572$	0	0
$573$	− 8064.00i	− 0.587920i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10366.0i	− 0.747907i	−0.927447	−	0.373953i	$-0.878002\pi$
$577$	− 10366.0i	− 0.747907i	0.927447	−	0.373953i	$-0.121998\pi$
$578$	0	0
$579$	6906.00	0.495688
$580$	0	0
$581$	7872.00	0.562109
$582$	0	0
$583$	2376.00i	0.168789i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7644.00i	− 0.537482i	−0.963213	−	0.268741i	$-0.913393\pi$
$587$	− 7644.00i	− 0.537482i	0.963213	−	0.268741i	$-0.0866075\pi$
$588$	0	0
$589$	1760.00	0.123123
$590$	0	0
$591$	13122.0	0.913311
$592$	0	0
$593$	− 8658.00i	− 0.599564i	−0.954008	−	0.299782i	$-0.903086\pi$
$593$	− 8658.00i	− 0.599564i	0.954008	−	0.299782i	$-0.0969139\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4800.00i	0.329064i
$598$	0	0
$599$	25800.0	1.75987	0.879933	−	0.475098i	$-0.157587\pi$
$599$	25800.0	1.75987	0.879933	+	0.475098i	$0.157587\pi$
$600$	0	0
$601$	16202.0	1.09966	0.549828	−	0.835278i	$-0.314693\pi$
$601$	16202.0	1.09966	0.549828	+	0.835278i	$0.314693\pi$
$602$	0	0
$603$	7956.00i	0.537302i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24136.0i	1.61392i	0.590605	+	0.806960i	$0.298889\pi$
$607$	24136.0i	1.61392i	−0.590605	+	0.806960i	$0.701111\pi$
$608$	0	0
$609$	−1440.00	−0.0958157
$610$	0	0
$611$	3648.00	0.241542
$612$	0	0
$613$	4642.00i	0.305854i	0.988237	+	0.152927i	$0.0488700\pi$
$613$	4642.00i	0.305854i	−0.988237	+	0.152927i	$0.951130\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6726.00i	− 0.438863i	−0.975628	−	0.219432i	$-0.929580\pi$
$617$	− 6726.00i	− 0.438863i	0.975628	−	0.219432i	$-0.0704203\pi$
$618$	0	0
$619$	−21220.0	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$619$	−21220.0	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$620$	0	0
$621$	−4536.00	−0.293113
$622$	0	0
$623$	− 12960.0i	− 0.833437i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	720.000i	0.0458597i
$628$	0	0
$629$	32004.0	2.02875
$630$	0	0
$631$	−29792.0	−1.87956	−0.939779	−	0.341783i	$-0.888969\pi$
$631$	−29792.0	−1.87956	−0.939779	+	0.341783i	$0.888969\pi$
$632$	0	0
$633$	9996.00i	0.627655i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3306.00i	− 0.205633i
$638$	0	0
$639$	7128.00	0.441282
$640$	0	0
$641$	−10158.0	−0.625923	−0.312962	−	0.949766i	$-0.601321\pi$
$641$	−10158.0	−0.625923	−0.312962	+	0.949766i	$0.601321\pi$
$642$	0	0
$643$	29828.0i	1.82940i	0.404138	+	0.914698i	$0.367571\pi$
$643$	29828.0i	1.82940i	−0.404138	+	0.914698i	$0.632429\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1944.00i	− 0.118124i	−0.998254	−	0.0590622i	$-0.981189\pi$
$647$	− 1944.00i	− 0.118124i	0.998254	−	0.0590622i	$-0.0188110\pi$
$648$	0	0
$649$	7920.00	0.479025
$650$	0	0
$651$	4224.00	0.254304
$652$	0	0
$653$	− 26718.0i	− 1.60116i	−0.599227	−	0.800579i	$-0.704525\pi$
$653$	− 26718.0i	− 1.60116i	0.599227	−	0.800579i	$-0.295475\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1962.00i	0.116507i
$658$	0	0
$659$	4260.00	0.251815	0.125907	−	0.992042i	$-0.459816\pi$
$659$	4260.00	0.251815	0.125907	+	0.992042i	$0.459816\pi$
$660$	0	0
$661$	22862.0	1.34528	0.672639	−	0.739971i	$-0.265161\pi$
$661$	22862.0	1.34528	0.672639	+	0.739971i	$0.265161\pi$
$662$	0	0
$663$	14364.0i	0.841405i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5040.00i	− 0.292578i
$668$	0	0
$669$	7944.00	0.459092
$670$	0	0
$671$	6456.00	0.371432
$672$	0	0
$673$	32542.0i	1.86390i	0.362592	+	0.931948i	$0.381892\pi$
$673$	32542.0i	1.86390i	−0.362592	+	0.931948i	$0.618108\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14214.0i	0.806925i	0.914996	+	0.403463i	$0.132193\pi$
$677$	14214.0i	0.806925i	−0.914996	+	0.403463i	$0.867807\pi$
$678$	0	0
$679$	−18464.0	−1.04357
$680$	0	0
$681$	−6732.00	−0.378812
$682$	0	0
$683$	− 7092.00i	− 0.397317i	−0.980069	−	0.198659i	$-0.936341\pi$
$683$	− 7092.00i	− 0.397317i	0.980069	−	0.198659i	$-0.0636585\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 16950.0i	− 0.941314i
$688$	0	0
$689$	−7524.00	−0.416026
$690$	0	0
$691$	13228.0	0.728244	0.364122	−	0.931351i	$-0.381369\pi$
$691$	13228.0	0.728244	0.364122	+	0.931351i	$0.381369\pi$
$692$	0	0
$693$	1728.00i	0.0947205i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5292.00i	− 0.287588i
$698$	0	0
$699$	−14094.0	−0.762638
$700$	0	0
$701$	28062.0	1.51196	0.755982	−	0.654592i	$-0.227160\pi$
$701$	28062.0	1.51196	0.755982	+	0.654592i	$0.227160\pi$
$702$	0	0
$703$	5080.00i	0.272540i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9888.00i	− 0.525992i
$708$	0	0
$709$	27250.0	1.44343	0.721717	−	0.692188i	$-0.243353\pi$
$709$	27250.0	1.44343	0.721717	+	0.692188i	$0.243353\pi$
$710$	0	0
$711$	4680.00	0.246855
$712$	0	0
$713$	14784.0i	0.776529i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3600.00i	0.187510i
$718$	0	0
$719$	−14400.0	−0.746912	−0.373456	−	0.927648i	$-0.621827\pi$
$719$	−14400.0	−0.746912	−0.373456	+	0.927648i	$0.621827\pi$
$720$	0	0
$721$	−2048.00	−0.105786
$722$	0	0
$723$	2154.00i	0.110800i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17984.0i	− 0.917455i	−0.888577	−	0.458727i	$-0.848305\pi$
$727$	− 17984.0i	− 0.917455i	0.888577	−	0.458727i	$-0.151695\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−6552.00	−0.331511
$732$	0	0
$733$	− 16598.0i	− 0.836373i	−0.908361	−	0.418186i	$-0.862666\pi$
$733$	− 16598.0i	− 0.836373i	0.908361	−	0.418186i	$-0.137334\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10608.0i	0.530191i
$738$	0	0
$739$	1460.00	0.0726752	0.0363376	−	0.999340i	$-0.488431\pi$
$739$	1460.00	0.0726752	0.0363376	+	0.999340i	$0.488431\pi$
$740$	0	0
$741$	−2280.00	−0.113034
$742$	0	0
$743$	− 30072.0i	− 1.48484i	−0.669936	−	0.742419i	$-0.733678\pi$
$743$	− 30072.0i	− 1.48484i	0.669936	−	0.742419i	$-0.266322\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4428.00i	0.216884i
$748$	0	0
$749$	−23616.0	−1.15208
$750$	0	0
$751$	18088.0	0.878882	0.439441	−	0.898271i	$-0.355177\pi$
$751$	18088.0	0.878882	0.439441	+	0.898271i	$0.355177\pi$
$752$	0	0
$753$	18036.0i	0.872866i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	24734.0i	1.18755i	0.804633	+	0.593773i	$0.202362\pi$
$757$	24734.0i	1.18755i	−0.804633	+	0.593773i	$0.797638\pi$
$758$	0	0
$759$	−6048.00	−0.289234
$760$	0	0
$761$	−22278.0	−1.06120	−0.530602	−	0.847621i	$-0.678034\pi$
$761$	−22278.0	−1.06120	−0.530602	+	0.847621i	$0.678034\pi$
$762$	0	0
$763$	− 19040.0i	− 0.903400i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25080.0i	1.18069i
$768$	0	0
$769$	−16130.0	−0.756388	−0.378194	−	0.925726i	$-0.623455\pi$
$769$	−16130.0	−0.756388	−0.378194	+	0.925726i	$0.623455\pi$
$770$	0	0
$771$	−6138.00	−0.286712
$772$	0	0
$773$	− 29718.0i	− 1.38277i	−0.722486	−	0.691386i	$-0.757001\pi$
$773$	− 29718.0i	− 1.38277i	0.722486	−	0.691386i	$-0.242999\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12192.0i	0.562916i
$778$	0	0
$779$	840.000	0.0386343
$780$	0	0
$781$	9504.00	0.435442
$782$	0	0
$783$	− 810.000i	− 0.0369694i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9524.00i	− 0.431377i	−0.976462	−	0.215689i	$-0.930800\pi$
$787$	− 9524.00i	− 0.431377i	0.976462	−	0.215689i	$-0.0691996\pi$
$788$	0	0
$789$	−18216.0	−0.821935
$790$	0	0
$791$	−7392.00	−0.332275
$792$	0	0
$793$	20444.0i	0.915495i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 33906.0i	− 1.50692i	−0.657496	−	0.753458i	$-0.728384\pi$
$797$	− 33906.0i	− 1.50692i	0.657496	−	0.753458i	$-0.271616\pi$
$798$	0	0
$799$	12096.0	0.535577
$800$	0	0
$801$	7290.00	0.321572
$802$	0	0
$803$	2616.00i	0.114965i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 20790.0i	− 0.906868i
$808$	0	0
$809$	630.000	0.0273790	0.0136895	−	0.999906i	$-0.495642\pi$
$809$	630.000	0.0273790	0.0136895	+	0.999906i	$0.495642\pi$
$810$	0	0
$811$	20788.0	0.900081	0.450040	−	0.893008i	$-0.351410\pi$
$811$	20788.0	0.900081	0.450040	+	0.893008i	$0.351410\pi$
$812$	0	0
$813$	4056.00i	0.174969i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1040.00i	− 0.0445349i
$818$	0	0
$819$	−5472.00	−0.233464
$820$	0	0
$821$	−43098.0	−1.83207	−0.916036	−	0.401097i	$-0.868629\pi$
$821$	−43098.0	−1.83207	−0.916036	+	0.401097i	$0.868629\pi$
$822$	0	0
$823$	− 14272.0i	− 0.604484i	−0.953231	−	0.302242i	$-0.902265\pi$
$823$	− 14272.0i	− 0.604484i	0.953231	−	0.302242i	$-0.0977351\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13644.0i	− 0.573698i	−0.957976	−	0.286849i	$-0.907392\pi$
$827$	− 13644.0i	− 0.573698i	0.957976	−	0.286849i	$-0.0926078\pi$
$828$	0	0
$829$	2410.00	0.100968	0.0504842	−	0.998725i	$-0.483924\pi$
$829$	2410.00	0.100968	0.0504842	+	0.998725i	$0.483924\pi$
$830$	0	0
$831$	−3558.00	−0.148527
$832$	0	0
$833$	− 10962.0i	− 0.455955i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2376.00i	0.0981202i
$838$	0	0
$839$	23160.0	0.953006	0.476503	−	0.879173i	$-0.341904\pi$
$839$	23160.0	0.953006	0.476503	+	0.879173i	$0.341904\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	0	0
$843$	− 7326.00i	− 0.299313i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 18992.0i	− 0.770452i
$848$	0	0
$849$	8484.00	0.342957
$850$	0	0
$851$	−42672.0	−1.71889
$852$	0	0
$853$	− 32078.0i	− 1.28761i	−0.765190	−	0.643804i	$-0.777355\pi$
$853$	− 32078.0i	− 1.28761i	0.765190	−	0.643804i	$-0.222645\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14406.0i	− 0.574212i	−0.957899	−	0.287106i	$-0.907307\pi$
$857$	− 14406.0i	− 0.574212i	0.957899	−	0.287106i	$-0.0926932\pi$
$858$	0	0
$859$	30620.0	1.21623	0.608115	−	0.793849i	$-0.291926\pi$
$859$	30620.0	1.21623	0.608115	+	0.793849i	$0.291926\pi$
$860$	0	0
$861$	2016.00	0.0797969
$862$	0	0
$863$	17568.0i	0.692957i	0.938058	+	0.346478i	$0.112623\pi$
$863$	17568.0i	0.692957i	−0.938058	+	0.346478i	$0.887377\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	32889.0i	1.28831i
$868$	0	0
$869$	6240.00	0.243587
$870$	0	0
$871$	−33592.0	−1.30680
$872$	0	0
$873$	− 10386.0i	− 0.402649i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 21706.0i	− 0.835758i	−0.908503	−	0.417879i	$-0.862774\pi$
$877$	− 21706.0i	− 0.835758i	0.908503	−	0.417879i	$-0.137226\pi$
$878$	0	0
$879$	−14274.0	−0.547725
$880$	0	0
$881$	−14958.0	−0.572018	−0.286009	−	0.958227i	$-0.592329\pi$
$881$	−14958.0	−0.572018	−0.286009	+	0.958227i	$0.592329\pi$
$882$	0	0
$883$	− 32812.0i	− 1.25052i	−0.780415	−	0.625261i	$-0.784992\pi$
$883$	− 32812.0i	− 1.25052i	0.780415	−	0.625261i	$-0.215008\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38856.0i	1.47086i	0.677598	+	0.735432i	$0.263021\pi$
$887$	38856.0i	1.47086i	−0.677598	+	0.735432i	$0.736979\pi$
$888$	0	0
$889$	−40576.0	−1.53079
$890$	0	0
$891$	−972.000	−0.0365468
$892$	0	0
$893$	1920.00i	0.0719489i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 19152.0i	− 0.712895i
$898$	0	0
$899$	−2640.00	−0.0979410
$900$	0	0
$901$	−24948.0	−0.922462
$902$	0	0
$903$	− 2496.00i	− 0.0919841i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28276.0i	1.03516i	0.855635	+	0.517579i	$0.173167\pi$
$907$	28276.0i	1.03516i	−0.855635	+	0.517579i	$0.826833\pi$
$908$	0	0
$909$	5562.00	0.202948
$910$	0	0
$911$	−8112.00	−0.295019	−0.147510	−	0.989061i	$-0.547126\pi$
$911$	−8112.00	−0.295019	−0.147510	+	0.989061i	$0.547126\pi$
$912$	0	0
$913$	5904.00i	0.214013i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 36672.0i	− 1.32063i
$918$	0	0
$919$	−26080.0	−0.936126	−0.468063	−	0.883695i	$-0.655048\pi$
$919$	−26080.0	−0.936126	−0.468063	+	0.883695i	$0.655048\pi$
$920$	0	0
$921$	25428.0	0.909751
$922$	0	0
$923$	30096.0i	1.07326i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1152.00i	− 0.0408162i
$928$	0	0
$929$	−49170.0	−1.73651	−0.868254	−	0.496120i	$-0.834757\pi$
$929$	−49170.0	−1.73651	−0.868254	+	0.496120i	$0.834757\pi$
$930$	0	0
$931$	1740.00	0.0612526
$932$	0	0
$933$	13896.0i	0.487604i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48314.0i	1.68447i	0.539110	+	0.842236i	$0.318761\pi$
$937$	48314.0i	1.68447i	−0.539110	+	0.842236i	$0.681239\pi$
$938$	0	0
$939$	14466.0	0.502748
$940$	0	0
$941$	34782.0	1.20495	0.602477	−	0.798137i	$-0.294181\pi$
$941$	34782.0	1.20495	0.602477	+	0.798137i	$0.294181\pi$
$942$	0	0
$943$	7056.00i	0.243664i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25116.0i	0.861838i	0.902391	+	0.430919i	$0.141810\pi$
$947$	25116.0i	0.861838i	−0.902391	+	0.430919i	$0.858190\pi$
$948$	0	0
$949$	−8284.00	−0.283361
$950$	0	0
$951$	−10278.0	−0.350460
$952$	0	0
$953$	15462.0i	0.525565i	0.964855	+	0.262782i	$0.0846401\pi$
$953$	15462.0i	0.525565i	−0.964855	+	0.262782i	$0.915360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1080.00i	− 0.0364801i
$958$	0	0
$959$	11616.0	0.391137
$960$	0	0
$961$	−22047.0	−0.740056
$962$	0	0
$963$	− 13284.0i	− 0.444518i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	736.000i	0.0244759i	0.999925	+	0.0122379i	$0.00389555\pi$
$967$	736.000i	0.0244759i	−0.999925	+	0.0122379i	$0.996104\pi$
$968$	0	0
$969$	−7560.00	−0.250632
$970$	0	0
$971$	29268.0	0.967307	0.483653	−	0.875260i	$-0.339310\pi$
$971$	29268.0	0.967307	0.483653	+	0.875260i	$0.339310\pi$
$972$	0	0
$973$	6080.00i	0.200325i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16674.0i	0.546007i	0.962013	+	0.273003i	$0.0880170\pi$
$977$	16674.0i	0.546007i	−0.962013	+	0.273003i	$0.911983\pi$
$978$	0	0
$979$	9720.00	0.317316
$980$	0	0
$981$	10710.0	0.348567
$982$	0	0
$983$	− 31272.0i	− 1.01467i	−0.861749	−	0.507336i	$-0.830630\pi$
$983$	− 31272.0i	− 1.01467i	0.861749	−	0.507336i	$-0.169370\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4608.00i	0.148606i
$988$	0	0
$989$	8736.00	0.280878
$990$	0	0
$991$	15928.0	0.510565	0.255282	−	0.966867i	$-0.417832\pi$
$991$	15928.0	0.510565	0.255282	+	0.966867i	$0.417832\pi$
$992$	0	0
$993$	− 8364.00i	− 0.267295i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42014.0i	1.33460i	0.744789	+	0.667300i	$0.232550\pi$
$997$	42014.0i	1.33460i	−0.744789	+	0.667300i	$0.767450\pi$
$998$	0	0
$999$	−6858.00	−0.217195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.j.49.1		2
4.3	odd	2		150.4.c.d.49.1		2
5.2	odd	4		1200.4.a.b.1.1		1
5.3	odd	4		48.4.a.c.1.1		1
5.4	even	2	inner	1200.4.f.j.49.2		2
12.11	even	2		450.4.c.e.199.2		2
15.8	even	4		144.4.a.c.1.1		1
20.3	even	4		6.4.a.a.1.1	✓	1
20.7	even	4		150.4.a.i.1.1		1
20.19	odd	2		150.4.c.d.49.2		2
35.13	even	4		2352.4.a.e.1.1		1
40.3	even	4		192.4.a.i.1.1		1
40.13	odd	4		192.4.a.c.1.1		1
60.23	odd	4		18.4.a.a.1.1		1
60.47	odd	4		450.4.a.h.1.1		1
60.59	even	2		450.4.c.e.199.1		2
80.3	even	4		768.4.d.n.385.1		2
80.13	odd	4		768.4.d.c.385.2		2
80.43	even	4		768.4.d.n.385.2		2
80.53	odd	4		768.4.d.c.385.1		2
120.53	even	4		576.4.a.r.1.1		1
120.83	odd	4		576.4.a.q.1.1		1
140.3	odd	12		294.4.e.g.79.1		2
140.23	even	12		294.4.e.h.67.1		2
140.83	odd	4		294.4.a.e.1.1		1
140.103	odd	12		294.4.e.g.67.1		2
140.123	even	12		294.4.e.h.79.1		2
180.23	odd	12		162.4.c.c.55.1		2
180.43	even	12		162.4.c.f.109.1		2
180.83	odd	12		162.4.c.c.109.1		2
180.103	even	12		162.4.c.f.55.1		2
220.43	odd	4		726.4.a.f.1.1		1
260.83	odd	4		1014.4.b.d.337.2		2
260.103	even	4		1014.4.a.g.1.1		1
260.203	odd	4		1014.4.b.d.337.1		2
340.203	even	4		1734.4.a.d.1.1		1
380.303	odd	4		2166.4.a.i.1.1		1
420.23	odd	12		882.4.g.i.361.1		2
420.83	even	4		882.4.a.n.1.1		1
420.143	even	12		882.4.g.f.667.1		2
420.263	odd	12		882.4.g.i.667.1		2
420.383	even	12		882.4.g.f.361.1		2
660.263	even	4		2178.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.4.a.a.1.1	✓	1	20.3	even	4
18.4.a.a.1.1		1	60.23	odd	4
48.4.a.c.1.1		1	5.3	odd	4
144.4.a.c.1.1		1	15.8	even	4
150.4.a.i.1.1		1	20.7	even	4
150.4.c.d.49.1		2	4.3	odd	2
150.4.c.d.49.2		2	20.19	odd	2
162.4.c.c.55.1		2	180.23	odd	12
162.4.c.c.109.1		2	180.83	odd	12
162.4.c.f.55.1		2	180.103	even	12
162.4.c.f.109.1		2	180.43	even	12
192.4.a.c.1.1		1	40.13	odd	4
192.4.a.i.1.1		1	40.3	even	4
294.4.a.e.1.1		1	140.83	odd	4
294.4.e.g.67.1		2	140.103	odd	12
294.4.e.g.79.1		2	140.3	odd	12
294.4.e.h.67.1		2	140.23	even	12
294.4.e.h.79.1		2	140.123	even	12
450.4.a.h.1.1		1	60.47	odd	4
450.4.c.e.199.1		2	60.59	even	2
450.4.c.e.199.2		2	12.11	even	2
576.4.a.q.1.1		1	120.83	odd	4
576.4.a.r.1.1		1	120.53	even	4
726.4.a.f.1.1		1	220.43	odd	4
768.4.d.c.385.1		2	80.53	odd	4
768.4.d.c.385.2		2	80.13	odd	4
768.4.d.n.385.1		2	80.3	even	4
768.4.d.n.385.2		2	80.43	even	4
882.4.a.n.1.1		1	420.83	even	4
882.4.g.f.361.1		2	420.383	even	12
882.4.g.f.667.1		2	420.143	even	12
882.4.g.i.361.1		2	420.23	odd	12
882.4.g.i.667.1		2	420.263	odd	12
1014.4.a.g.1.1		1	260.103	even	4
1014.4.b.d.337.1		2	260.203	odd	4
1014.4.b.d.337.2		2	260.83	odd	4
1200.4.a.b.1.1		1	5.2	odd	4
1200.4.f.j.49.1		2	1.1	even	1	trivial
1200.4.f.j.49.2		2	5.4	even	2	inner
1734.4.a.d.1.1		1	340.203	even	4
2166.4.a.i.1.1		1	380.303	odd	4
2178.4.a.e.1.1		1	660.263	even	4
2352.4.a.e.1.1		1	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} -12.0000 q^{11} -38.0000i q^{13} -126.000i q^{17} +20.0000 q^{19} +48.0000 q^{21} +168.000i q^{23} +27.0000i q^{27} -30.0000 q^{29} +88.0000 q^{31} +36.0000i q^{33} +254.000i q^{37} -114.000 q^{39} +42.0000 q^{41} -52.0000i q^{43} +96.0000i q^{47} +87.0000 q^{49} -378.000 q^{51} -198.000i q^{53} -60.0000i q^{57} -660.000 q^{59} -538.000 q^{61} -144.000i q^{63} -884.000i q^{67} +504.000 q^{69} -792.000 q^{71} -218.000i q^{73} -192.000i q^{77} -520.000 q^{79} +81.0000 q^{81} -492.000i q^{83} +90.0000i q^{87} -810.000 q^{89} +608.000 q^{91} -264.000i q^{93} +1154.00i q^{97} +108.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 24 q^{11} + 40 q^{19} + 96 q^{21} - 60 q^{29} + 176 q^{31} - 228 q^{39} + 84 q^{41} + 174 q^{49} - 756 q^{51} - 1320 q^{59} - 1076 q^{61} + 1008 q^{69} - 1584 q^{71} - 1040 q^{79} + 162 q^{81} - 1620 q^{89} + 1216 q^{91} + 216 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 24 * q^11 + 40 * q^19 + 96 * q^21 - 60 * q^29 + 176 * q^31 - 228 * q^39 + 84 * q^41 + 174 * q^49 - 756 * q^51 - 1320 * q^59 - 1076 * q^61 + 1008 * q^69 - 1584 * q^71 - 1040 * q^79 + 162 * q^81 - 1620 * q^89 + 1216 * q^91 + 216 * q^99

Introduction

L-functions

Modular forms

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Representations

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