LMFDB - Embedded newform 1200.4.f.t.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,4,Mod(49,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,0,0,0,0,-18,0,112,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$70.8022920069$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
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.t.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} +56.0000 q^{11} +86.0000i q^{13} -106.000i q^{17} +4.00000 q^{19} -60.0000 q^{21} +136.000i q^{23} +27.0000i q^{27} +206.000 q^{29} +152.000 q^{31} -168.000i q^{33} +282.000i q^{37} +258.000 q^{39} -246.000 q^{41} +412.000i q^{43} -40.0000i q^{47} -57.0000 q^{49} -318.000 q^{51} +126.000i q^{53} -12.0000i q^{57} +56.0000 q^{59} -2.00000 q^{61} +180.000i q^{63} +388.000i q^{67} +408.000 q^{69} +672.000 q^{71} -1170.00i q^{73} -1120.00i q^{77} +408.000 q^{79} +81.0000 q^{81} +668.000i q^{83} -618.000i q^{87} -66.0000 q^{89} +1720.00 q^{91} -456.000i q^{93} -926.000i q^{97} -504.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 112 q^{11} + 8 q^{19} - 120 q^{21} + 412 q^{29} + 304 q^{31} + 516 q^{39} - 492 q^{41} - 114 q^{49} - 636 q^{51} + 112 q^{59} - 4 q^{61} + 816 q^{69} + 1344 q^{71} + 816 q^{79} + 162 q^{81}+ \cdots - 1008 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 112 * q^11 + 8 * q^19 - 120 * q^21 + 412 * q^29 + 304 * q^31 + 516 * q^39 - 492 * q^41 - 114 * q^49 - 636 * q^51 + 112 * q^59 - 4 * q^61 + 816 * q^69 + 1344 * q^71 + 816 * q^79 + 162 * q^81 - 132 * q^89 + 3440 * q^91 - 1008 * q^99

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
$7$	− 20.0000i	− 1.07990i	−0.841698	−	0.539949i	$-0.818443\pi$
$7$	− 20.0000i	− 1.07990i	0.841698	−	0.539949i	$-0.181557\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	56.0000	1.53497	0.767483	−	0.641069i	$-0.221509\pi$
$11$	56.0000	1.53497	0.767483	+	0.641069i	$0.221509\pi$
$12$	0	0
$13$	86.0000i	1.83478i	0.397992	+	0.917389i	$0.369707\pi$
$13$	86.0000i	1.83478i	−0.397992	+	0.917389i	$0.630293\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 106.000i	− 1.51228i	−0.654409	−	0.756140i	$-0.727083\pi$
$17$	− 106.000i	− 1.51228i	0.654409	−	0.756140i	$-0.272917\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	−60.0000	−0.623480
$22$	0	0
$23$	136.000i	1.23295i	0.787373	+	0.616477i	$0.211441\pi$
$23$	136.000i	1.23295i	−0.787373	+	0.616477i	$0.788559\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	206.000	1.31908	0.659539	−	0.751671i	$-0.270752\pi$
$29$	206.000	1.31908	0.659539	+	0.751671i	$0.270752\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	0	0
$33$	− 168.000i	− 0.886214i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	282.000i	1.25299i	0.779427	+	0.626493i	$0.215510\pi$
$37$	282.000i	1.25299i	−0.779427	+	0.626493i	$0.784490\pi$
$38$	0	0
$39$	258.000	1.05931
$40$	0	0
$41$	−246.000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−246.000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	412.000i	1.46115i	0.682833	+	0.730575i	$0.260748\pi$
$43$	412.000i	1.46115i	−0.682833	+	0.730575i	$0.739252\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 40.0000i	− 0.124140i	−0.998072	−	0.0620702i	$-0.980230\pi$
$47$	− 40.0000i	− 0.124140i	0.998072	−	0.0620702i	$-0.0197703\pi$
$48$	0	0
$49$	−57.0000	−0.166181
$50$	0	0
$51$	−318.000	−0.873116
$52$	0	0
$53$	126.000i	0.326555i	0.986580	+	0.163278i	$0.0522066\pi$
$53$	126.000i	0.326555i	−0.986580	+	0.163278i	$0.947793\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 12.0000i	− 0.0278849i
$58$	0	0
$59$	56.0000	0.123569	0.0617846	−	0.998090i	$-0.480321\pi$
$59$	56.0000	0.123569	0.0617846	+	0.998090i	$0.480321\pi$
$60$	0	0
$61$	−2.00000	−0.00419793	−0.00209897	−	0.999998i	$-0.500668\pi$
$61$	−2.00000	−0.00419793	−0.00209897	+	0.999998i	$0.500668\pi$
$62$	0	0
$63$	180.000i	0.359966i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	388.000i	0.707489i	0.935342	+	0.353744i	$0.115092\pi$
$67$	388.000i	0.707489i	−0.935342	+	0.353744i	$0.884908\pi$
$68$	0	0
$69$	408.000	0.711847
$70$	0	0
$71$	672.000	1.12326	0.561632	−	0.827387i	$-0.310174\pi$
$71$	672.000	1.12326	0.561632	+	0.827387i	$0.310174\pi$
$72$	0	0
$73$	− 1170.00i	− 1.87586i	−0.346818	−	0.937932i	$-0.612738\pi$
$73$	− 1170.00i	− 1.87586i	0.346818	−	0.937932i	$-0.387262\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1120.00i	− 1.65761i
$78$	0	0
$79$	408.000	0.581058	0.290529	−	0.956866i	$-0.406169\pi$
$79$	408.000	0.581058	0.290529	+	0.956866i	$0.406169\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	668.000i	0.883404i	0.897162	+	0.441702i	$0.145625\pi$
$83$	668.000i	0.883404i	−0.897162	+	0.441702i	$0.854375\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 618.000i	− 0.761570i
$88$	0	0
$89$	−66.0000	−0.0786066	−0.0393033	−	0.999227i	$-0.512514\pi$
$89$	−66.0000	−0.0786066	−0.0393033	+	0.999227i	$0.512514\pi$
$90$	0	0
$91$	1720.00	1.98137
$92$	0	0
$93$	− 456.000i	− 0.508441i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 926.000i	− 0.969289i	−0.874711	−	0.484645i	$-0.838949\pi$
$97$	− 926.000i	− 0.969289i	0.874711	−	0.484645i	$-0.161051\pi$
$98$	0	0
$99$	−504.000	−0.511656
$100$	0	0
$101$	−198.000	−0.195067	−0.0975333	−	0.995232i	$-0.531095\pi$
$101$	−198.000	−0.195067	−0.0975333	+	0.995232i	$0.531095\pi$
$102$	0	0
$103$	− 1532.00i	− 1.46556i	−0.680467	−	0.732779i	$-0.738223\pi$
$103$	− 1532.00i	− 1.46556i	0.680467	−	0.732779i	$-0.261777\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	444.000i	0.401150i	0.979678	+	0.200575i	$0.0642811\pi$
$107$	444.000i	0.401150i	−0.979678	+	0.200575i	$0.935719\pi$
$108$	0	0
$109$	−62.0000	−0.0544819	−0.0272409	−	0.999629i	$-0.508672\pi$
$109$	−62.0000	−0.0544819	−0.0272409	+	0.999629i	$0.508672\pi$
$110$	0	0
$111$	846.000	0.723412
$112$	0	0
$113$	− 414.000i	− 0.344653i	−0.985040	−	0.172327i	$-0.944872\pi$
$113$	− 414.000i	− 0.344653i	0.985040	−	0.172327i	$-0.0551285\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 774.000i	− 0.611593i
$118$	0	0
$119$	−2120.00	−1.63311
$120$	0	0
$121$	1805.00	1.35612
$122$	0	0
$123$	738.000i	0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	996.000i	0.695911i	0.937511	+	0.347956i	$0.113124\pi$
$127$	996.000i	0.695911i	−0.937511	+	0.347956i	$0.886876\pi$
$128$	0	0
$129$	1236.00	0.843595
$130$	0	0
$131$	264.000	0.176075	0.0880374	−	0.996117i	$-0.471941\pi$
$131$	264.000	0.176075	0.0880374	+	0.996117i	$0.471941\pi$
$132$	0	0
$133$	− 80.0000i	− 0.0521570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2278.00i	1.42060i	0.703897	+	0.710302i	$0.251442\pi$
$137$	2278.00i	1.42060i	−0.703897	+	0.710302i	$0.748558\pi$
$138$	0	0
$139$	1812.00	1.10570	0.552848	−	0.833282i	$-0.313541\pi$
$139$	1812.00	1.10570	0.552848	+	0.833282i	$0.313541\pi$
$140$	0	0
$141$	−120.000	−0.0716725
$142$	0	0
$143$	4816.00i	2.81632i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	171.000i	0.0959445i
$148$	0	0
$149$	1534.00	0.843424	0.421712	−	0.906730i	$-0.361429\pi$
$149$	1534.00	0.843424	0.421712	+	0.906730i	$0.361429\pi$
$150$	0	0
$151$	3016.00	1.62542	0.812711	−	0.582668i	$-0.197991\pi$
$151$	3016.00	1.62542	0.812711	+	0.582668i	$0.197991\pi$
$152$	0	0
$153$	954.000i	0.504094i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1814.00i	− 0.922121i	−0.887369	−	0.461060i	$-0.847469\pi$
$157$	− 1814.00i	− 0.922121i	0.887369	−	0.461060i	$-0.152531\pi$
$158$	0	0
$159$	378.000	0.188537
$160$	0	0
$161$	2720.00	1.33147
$162$	0	0
$163$	− 1844.00i	− 0.886093i	−0.896499	−	0.443047i	$-0.853898\pi$
$163$	− 1844.00i	− 0.886093i	0.896499	−	0.443047i	$-0.146102\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3768.00i	− 1.74597i	−0.487749	−	0.872984i	$-0.662182\pi$
$167$	− 3768.00i	− 1.74597i	0.487749	−	0.872984i	$-0.337818\pi$
$168$	0	0
$169$	−5199.00	−2.36641
$170$	0	0
$171$	−36.0000	−0.0160993
$172$	0	0
$173$	− 938.000i	− 0.412224i	−0.978528	−	0.206112i	$-0.933919\pi$
$173$	− 938.000i	− 0.412224i	0.978528	−	0.206112i	$-0.0660812\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 168.000i	− 0.0713427i
$178$	0	0
$179$	3968.00	1.65688	0.828442	−	0.560075i	$-0.189228\pi$
$179$	3968.00	1.65688	0.828442	+	0.560075i	$0.189228\pi$
$180$	0	0
$181$	−3514.00	−1.44306	−0.721529	−	0.692384i	$-0.756560\pi$
$181$	−3514.00	−1.44306	−0.721529	+	0.692384i	$0.756560\pi$
$182$	0	0
$183$	6.00000i	0.00242368i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5936.00i	− 2.32130i
$188$	0	0
$189$	540.000	0.207827
$190$	0	0
$191$	1480.00	0.560676	0.280338	−	0.959901i	$-0.409554\pi$
$191$	1480.00	0.560676	0.280338	+	0.959901i	$0.409554\pi$
$192$	0	0
$193$	2774.00i	1.03460i	0.855806	+	0.517298i	$0.173062\pi$
$193$	2774.00i	1.03460i	−0.855806	+	0.517298i	$0.826938\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3806.00i	− 1.37648i	−0.725484	−	0.688239i	$-0.758384\pi$
$197$	− 3806.00i	− 1.37648i	0.725484	−	0.688239i	$-0.241616\pi$
$198$	0	0
$199$	−856.000	−0.304926	−0.152463	−	0.988309i	$-0.548720\pi$
$199$	−856.000	−0.304926	−0.152463	+	0.988309i	$0.548720\pi$
$200$	0	0
$201$	1164.00	0.408469
$202$	0	0
$203$	− 4120.00i	− 1.42447i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1224.00i	− 0.410985i
$208$	0	0
$209$	224.000	0.0741359
$210$	0	0
$211$	−3020.00	−0.985334	−0.492667	−	0.870218i	$-0.663978\pi$
$211$	−3020.00	−0.985334	−0.492667	+	0.870218i	$0.663978\pi$
$212$	0	0
$213$	− 2016.00i	− 0.648517i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3040.00i	− 0.951008i
$218$	0	0
$219$	−3510.00	−1.08303
$220$	0	0
$221$	9116.00	2.77470
$222$	0	0
$223$	− 1684.00i	− 0.505690i	−0.967507	−	0.252845i	$-0.918634\pi$
$223$	− 1684.00i	− 0.505690i	0.967507	−	0.252845i	$-0.0813664\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2004.00i	− 0.585948i	−0.956120	−	0.292974i	$-0.905355\pi$
$227$	− 2004.00i	− 0.585948i	0.956120	−	0.292974i	$-0.0946449\pi$
$228$	0	0
$229$	5042.00	1.45496	0.727478	−	0.686131i	$-0.240693\pi$
$229$	5042.00	1.45496	0.727478	+	0.686131i	$0.240693\pi$
$230$	0	0
$231$	−3360.00	−0.957021
$232$	0	0
$233$	3090.00i	0.868810i	0.900718	+	0.434405i	$0.143041\pi$
$233$	3090.00i	0.868810i	−0.900718	+	0.434405i	$0.856959\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1224.00i	− 0.335474i
$238$	0	0
$239$	2136.00	0.578102	0.289051	−	0.957314i	$-0.406660\pi$
$239$	2136.00	0.578102	0.289051	+	0.957314i	$0.406660\pi$
$240$	0	0
$241$	98.0000	0.0261939	0.0130970	−	0.999914i	$-0.495831\pi$
$241$	98.0000	0.0261939	0.0130970	+	0.999914i	$0.495831\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	344.000i	0.0886162i
$248$	0	0
$249$	2004.00	0.510033
$250$	0	0
$251$	5040.00	1.26742	0.633709	−	0.773571i	$-0.281532\pi$
$251$	5040.00	1.26742	0.633709	+	0.773571i	$0.281532\pi$
$252$	0	0
$253$	7616.00i	1.89254i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1986.00i	− 0.482036i	−0.970521	−	0.241018i	$-0.922519\pi$
$257$	− 1986.00i	− 0.482036i	0.970521	−	0.241018i	$-0.0774813\pi$
$258$	0	0
$259$	5640.00	1.35310
$260$	0	0
$261$	−1854.00	−0.439692
$262$	0	0
$263$	1416.00i	0.331994i	0.986126	+	0.165997i	$0.0530841\pi$
$263$	1416.00i	0.331994i	−0.986126	+	0.165997i	$0.946916\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	198.000i	0.0453835i
$268$	0	0
$269$	6670.00	1.51181	0.755905	−	0.654681i	$-0.227197\pi$
$269$	6670.00	1.51181	0.755905	+	0.654681i	$0.227197\pi$
$270$	0	0
$271$	−48.0000	−0.0107594	−0.00537969	−	0.999986i	$-0.501712\pi$
$271$	−48.0000	−0.0107594	−0.00537969	+	0.999986i	$0.501712\pi$
$272$	0	0
$273$	− 5160.00i	− 1.14395i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6938.00i	1.50492i	0.658636	+	0.752462i	$0.271134\pi$
$277$	6938.00i	1.50492i	−0.658636	+	0.752462i	$0.728866\pi$
$278$	0	0
$279$	−1368.00	−0.293548
$280$	0	0
$281$	−1694.00	−0.359628	−0.179814	−	0.983701i	$-0.557550\pi$
$281$	−1694.00	−0.359628	−0.179814	+	0.983701i	$0.557550\pi$
$282$	0	0
$283$	− 6364.00i	− 1.33675i	−0.743824	−	0.668376i	$-0.766990\pi$
$283$	− 6364.00i	− 1.33675i	0.743824	−	0.668376i	$-0.233010\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4920.00i	1.01191i
$288$	0	0
$289$	−6323.00	−1.28699
$290$	0	0
$291$	−2778.00	−0.559619
$292$	0	0
$293$	3134.00i	0.624881i	0.949937	+	0.312441i	$0.101147\pi$
$293$	3134.00i	0.624881i	−0.949937	+	0.312441i	$0.898853\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1512.00i	0.295405i
$298$	0	0
$299$	−11696.0	−2.26220
$300$	0	0
$301$	8240.00	1.57789
$302$	0	0
$303$	594.000i	0.112622i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	236.000i	0.0438737i	0.999759	+	0.0219369i	$0.00698328\pi$
$307$	236.000i	0.0438737i	−0.999759	+	0.0219369i	$0.993017\pi$
$308$	0	0
$309$	−4596.00	−0.846140
$310$	0	0
$311$	−3776.00	−0.688480	−0.344240	−	0.938882i	$-0.611863\pi$
$311$	−3776.00	−0.688480	−0.344240	+	0.938882i	$0.611863\pi$
$312$	0	0
$313$	7918.00i	1.42988i	0.699187	+	0.714939i	$0.253546\pi$
$313$	7918.00i	1.42988i	−0.699187	+	0.714939i	$0.746454\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4362.00i	0.772853i	0.922320	+	0.386426i	$0.126291\pi$
$317$	4362.00i	0.772853i	−0.922320	+	0.386426i	$0.873709\pi$
$318$	0	0
$319$	11536.0	2.02474
$320$	0	0
$321$	1332.00	0.231604
$322$	0	0
$323$	− 424.000i	− 0.0730402i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	186.000i	0.0314551i
$328$	0	0
$329$	−800.000	−0.134059
$330$	0	0
$331$	−7980.00	−1.32514	−0.662569	−	0.749001i	$-0.730534\pi$
$331$	−7980.00	−1.32514	−0.662569	+	0.749001i	$0.730534\pi$
$332$	0	0
$333$	− 2538.00i	− 0.417662i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8294.00i	− 1.34066i	−0.742062	−	0.670331i	$-0.766152\pi$
$337$	− 8294.00i	− 1.34066i	0.742062	−	0.670331i	$-0.233848\pi$
$338$	0	0
$339$	−1242.00	−0.198986
$340$	0	0
$341$	8512.00	1.35176
$342$	0	0
$343$	− 5720.00i	− 0.900440i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	964.000i	0.149136i	0.997216	+	0.0745681i	$0.0237578\pi$
$347$	964.000i	0.149136i	−0.997216	+	0.0745681i	$0.976242\pi$
$348$	0	0
$349$	−8670.00	−1.32978	−0.664892	−	0.746940i	$-0.731522\pi$
$349$	−8670.00	−1.32978	−0.664892	+	0.746940i	$0.731522\pi$
$350$	0	0
$351$	−2322.00	−0.353103
$352$	0	0
$353$	2314.00i	0.348900i	0.984666	+	0.174450i	$0.0558148\pi$
$353$	2314.00i	0.348900i	−0.984666	+	0.174450i	$0.944185\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6360.00i	0.942876i
$358$	0	0
$359$	−1896.00	−0.278738	−0.139369	−	0.990240i	$-0.544507\pi$
$359$	−1896.00	−0.278738	−0.139369	+	0.990240i	$0.544507\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	− 5415.00i	− 0.782958i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1484.00i	− 0.211074i	−0.994415	−	0.105537i	$-0.966344\pi$
$367$	− 1484.00i	− 0.211074i	0.994415	−	0.105537i	$-0.0336562\pi$
$368$	0	0
$369$	2214.00	0.312348
$370$	0	0
$371$	2520.00	0.352647
$372$	0	0
$373$	− 12370.0i	− 1.71714i	−0.512694	−	0.858571i	$-0.671352\pi$
$373$	− 12370.0i	− 1.71714i	0.512694	−	0.858571i	$-0.328648\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17716.0i	2.42021i
$378$	0	0
$379$	5620.00	0.761689	0.380844	−	0.924639i	$-0.375633\pi$
$379$	5620.00	0.761689	0.380844	+	0.924639i	$0.375633\pi$
$380$	0	0
$381$	2988.00	0.401784
$382$	0	0
$383$	5880.00i	0.784475i	0.919864	+	0.392238i	$0.128299\pi$
$383$	5880.00i	0.784475i	−0.919864	+	0.392238i	$0.871701\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3708.00i	− 0.487050i
$388$	0	0
$389$	−2082.00	−0.271367	−0.135683	−	0.990752i	$-0.543323\pi$
$389$	−2082.00	−0.271367	−0.135683	+	0.990752i	$0.543323\pi$
$390$	0	0
$391$	14416.0	1.86457
$392$	0	0
$393$	− 792.000i	− 0.101657i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1742.00i	− 0.220223i	−0.993919	−	0.110111i	$-0.964879\pi$
$397$	− 1742.00i	− 0.220223i	0.993919	−	0.110111i	$-0.0351208\pi$
$398$	0	0
$399$	−240.000	−0.0301129
$400$	0	0
$401$	−3270.00	−0.407222	−0.203611	−	0.979052i	$-0.565268\pi$
$401$	−3270.00	−0.407222	−0.203611	+	0.979052i	$0.565268\pi$
$402$	0	0
$403$	13072.0i	1.61579i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15792.0i	1.92329i
$408$	0	0
$409$	6134.00	0.741581	0.370791	−	0.928716i	$-0.379087\pi$
$409$	6134.00	0.741581	0.370791	+	0.928716i	$0.379087\pi$
$410$	0	0
$411$	6834.00	0.820186
$412$	0	0
$413$	− 1120.00i	− 0.133442i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5436.00i	− 0.638374i
$418$	0	0
$419$	−10392.0	−1.21165	−0.605826	−	0.795597i	$-0.707157\pi$
$419$	−10392.0	−1.21165	−0.605826	+	0.795597i	$0.707157\pi$
$420$	0	0
$421$	−12690.0	−1.46906	−0.734528	−	0.678578i	$-0.762596\pi$
$421$	−12690.0	−1.46906	−0.734528	+	0.678578i	$0.762596\pi$
$422$	0	0
$423$	360.000i	0.0413801i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	40.0000i	0.00453334i
$428$	0	0
$429$	14448.0	1.62600
$430$	0	0
$431$	7408.00	0.827914	0.413957	−	0.910297i	$-0.364146\pi$
$431$	7408.00	0.827914	0.413957	+	0.910297i	$0.364146\pi$
$432$	0	0
$433$	5062.00i	0.561811i	0.959735	+	0.280906i	$0.0906348\pi$
$433$	5062.00i	0.561811i	−0.959735	+	0.280906i	$0.909365\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	544.000i	0.0595493i
$438$	0	0
$439$	−7160.00	−0.778424	−0.389212	−	0.921148i	$-0.627253\pi$
$439$	−7160.00	−0.778424	−0.389212	+	0.921148i	$0.627253\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	17100.0i	1.83396i	0.398930	+	0.916981i	$0.369382\pi$
$443$	17100.0i	1.83396i	−0.398930	+	0.916981i	$0.630618\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4602.00i	− 0.486951i
$448$	0	0
$449$	−8634.00	−0.907491	−0.453746	−	0.891131i	$-0.649913\pi$
$449$	−8634.00	−0.907491	−0.453746	+	0.891131i	$0.649913\pi$
$450$	0	0
$451$	−13776.0	−1.43833
$452$	0	0
$453$	− 9048.00i	− 0.938437i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2986.00i	0.305644i	0.988254	+	0.152822i	$0.0488361\pi$
$457$	2986.00i	0.305644i	−0.988254	+	0.152822i	$0.951164\pi$
$458$	0	0
$459$	2862.00	0.291039
$460$	0	0
$461$	−2406.00	−0.243077	−0.121539	−	0.992587i	$-0.538783\pi$
$461$	−2406.00	−0.243077	−0.121539	+	0.992587i	$0.538783\pi$
$462$	0	0
$463$	− 14316.0i	− 1.43698i	−0.695538	−	0.718489i	$-0.744834\pi$
$463$	− 14316.0i	− 1.43698i	0.695538	−	0.718489i	$-0.255166\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	292.000i	0.0289339i	0.999895	+	0.0144670i	$0.00460514\pi$
$467$	292.000i	0.0289339i	−0.999895	+	0.0144670i	$0.995395\pi$
$468$	0	0
$469$	7760.00	0.764016
$470$	0	0
$471$	−5442.00	−0.532387
$472$	0	0
$473$	23072.0i	2.24282i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1134.00i	− 0.108852i
$478$	0	0
$479$	14056.0	1.34078	0.670391	−	0.742008i	$-0.266126\pi$
$479$	14056.0	1.34078	0.670391	+	0.742008i	$0.266126\pi$
$480$	0	0
$481$	−24252.0	−2.29895
$482$	0	0
$483$	− 8160.00i	− 0.768722i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	11204.0i	1.04251i	0.853401	+	0.521254i	$0.174536\pi$
$487$	11204.0i	1.04251i	−0.853401	+	0.521254i	$0.825464\pi$
$488$	0	0
$489$	−5532.00	−0.511586
$490$	0	0
$491$	−4608.00	−0.423536	−0.211768	−	0.977320i	$-0.567922\pi$
$491$	−4608.00	−0.423536	−0.211768	+	0.977320i	$0.567922\pi$
$492$	0	0
$493$	− 21836.0i	− 1.99482i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13440.0i	− 1.21301i
$498$	0	0
$499$	2468.00	0.221409	0.110704	−	0.993853i	$-0.464689\pi$
$499$	2468.00	0.221409	0.110704	+	0.993853i	$0.464689\pi$
$500$	0	0
$501$	−11304.0	−1.00803
$502$	0	0
$503$	12192.0i	1.08074i	0.841426	+	0.540372i	$0.181717\pi$
$503$	12192.0i	1.08074i	−0.841426	+	0.540372i	$0.818283\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15597.0i	1.36625i
$508$	0	0
$509$	−1714.00	−0.149257	−0.0746284	−	0.997211i	$-0.523777\pi$
$509$	−1714.00	−0.149257	−0.0746284	+	0.997211i	$0.523777\pi$
$510$	0	0
$511$	−23400.0	−2.02574
$512$	0	0
$513$	108.000i	0.00929496i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2240.00i	− 0.190551i
$518$	0	0
$519$	−2814.00	−0.237998
$520$	0	0
$521$	−18014.0	−1.51479	−0.757397	−	0.652955i	$-0.773529\pi$
$521$	−18014.0	−1.51479	−0.757397	+	0.652955i	$0.773529\pi$
$522$	0	0
$523$	− 16748.0i	− 1.40027i	−0.714013	−	0.700133i	$-0.753124\pi$
$523$	− 16748.0i	− 1.40027i	0.714013	−	0.700133i	$-0.246876\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 16112.0i	− 1.33178i
$528$	0	0
$529$	−6329.00	−0.520178
$530$	0	0
$531$	−504.000	−0.0411897
$532$	0	0
$533$	− 21156.0i	− 1.71926i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11904.0i	− 0.956602i
$538$	0	0
$539$	−3192.00	−0.255082
$540$	0	0
$541$	−14018.0	−1.11401	−0.557006	−	0.830508i	$-0.688050\pi$
$541$	−14018.0	−1.11401	−0.557006	+	0.830508i	$0.688050\pi$
$542$	0	0
$543$	10542.0i	0.833150i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	412.000i	0.0322045i	0.999870	+	0.0161022i	$0.00512572\pi$
$547$	412.000i	0.0322045i	−0.999870	+	0.0161022i	$0.994874\pi$
$548$	0	0
$549$	18.0000	0.00139931
$550$	0	0
$551$	824.000	0.0637089
$552$	0	0
$553$	− 8160.00i	− 0.627484i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18218.0i	1.38586i	0.721007	+	0.692928i	$0.243679\pi$
$557$	18218.0i	1.38586i	−0.721007	+	0.692928i	$0.756321\pi$
$558$	0	0
$559$	−35432.0	−2.68088
$560$	0	0
$561$	−17808.0	−1.34020
$562$	0	0
$563$	23524.0i	1.76096i	0.474087	+	0.880478i	$0.342778\pi$
$563$	23524.0i	1.76096i	−0.474087	+	0.880478i	$0.657222\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1620.00i	− 0.119989i
$568$	0	0
$569$	−23330.0	−1.71888	−0.859442	−	0.511234i	$-0.829189\pi$
$569$	−23330.0	−1.71888	−0.859442	+	0.511234i	$0.829189\pi$
$570$	0	0
$571$	13124.0	0.961860	0.480930	−	0.876759i	$-0.340299\pi$
$571$	13124.0	0.961860	0.480930	+	0.876759i	$0.340299\pi$
$572$	0	0
$573$	− 4440.00i	− 0.323706i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11714.0i	0.845165i	0.906324	+	0.422582i	$0.138876\pi$
$577$	11714.0i	0.845165i	−0.906324	+	0.422582i	$0.861124\pi$
$578$	0	0
$579$	8322.00	0.597324
$580$	0	0
$581$	13360.0	0.953987
$582$	0	0
$583$	7056.00i	0.501252i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17628.0i	1.23950i	0.784800	+	0.619749i	$0.212766\pi$
$587$	17628.0i	1.23950i	−0.784800	+	0.619749i	$0.787234\pi$
$588$	0	0
$589$	608.000	0.0425335
$590$	0	0
$591$	−11418.0	−0.794710
$592$	0	0
$593$	2802.00i	0.194038i	0.995283	+	0.0970188i	$0.0309307\pi$
$593$	2802.00i	0.194038i	−0.995283	+	0.0970188i	$0.969069\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2568.00i	0.176049i
$598$	0	0
$599$	−2664.00	−0.181716	−0.0908582	−	0.995864i	$-0.528961\pi$
$599$	−2664.00	−0.181716	−0.0908582	+	0.995864i	$0.528961\pi$
$600$	0	0
$601$	23962.0	1.62634	0.813170	−	0.582026i	$-0.197740\pi$
$601$	23962.0	1.62634	0.813170	+	0.582026i	$0.197740\pi$
$602$	0	0
$603$	− 3492.00i	− 0.235830i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11940.0i	− 0.798401i	−0.916864	−	0.399201i	$-0.869288\pi$
$607$	− 11940.0i	− 0.798401i	0.916864	−	0.399201i	$-0.130712\pi$
$608$	0	0
$609$	−12360.0	−0.822418
$610$	0	0
$611$	3440.00	0.227770
$612$	0	0
$613$	− 16794.0i	− 1.10653i	−0.833005	−	0.553265i	$-0.813382\pi$
$613$	− 16794.0i	− 1.10653i	0.833005	−	0.553265i	$-0.186618\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20706.0i	− 1.35104i	−0.737341	−	0.675520i	$-0.763919\pi$
$617$	− 20706.0i	− 1.35104i	0.737341	−	0.675520i	$-0.236081\pi$
$618$	0	0
$619$	10724.0	0.696339	0.348170	−	0.937432i	$-0.386803\pi$
$619$	10724.0	0.696339	0.348170	+	0.937432i	$0.386803\pi$
$620$	0	0
$621$	−3672.00	−0.237282
$622$	0	0
$623$	1320.00i	0.0848871i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 672.000i	− 0.0428024i
$628$	0	0
$629$	29892.0	1.89487
$630$	0	0
$631$	5744.00	0.362385	0.181193	−	0.983448i	$-0.442004\pi$
$631$	5744.00	0.362385	0.181193	+	0.983448i	$0.442004\pi$
$632$	0	0
$633$	9060.00i	0.568883i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4902.00i	− 0.304905i
$638$	0	0
$639$	−6048.00	−0.374421
$640$	0	0
$641$	27906.0	1.71953	0.859767	−	0.510687i	$-0.170609\pi$
$641$	27906.0	1.71953	0.859767	+	0.510687i	$0.170609\pi$
$642$	0	0
$643$	20556.0i	1.26073i	0.776299	+	0.630365i	$0.217095\pi$
$643$	20556.0i	1.26073i	−0.776299	+	0.630365i	$0.782905\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10224.0i	0.621247i	0.950533	+	0.310624i	$0.100538\pi$
$647$	10224.0i	0.621247i	−0.950533	+	0.310624i	$0.899462\pi$
$648$	0	0
$649$	3136.00	0.189675
$650$	0	0
$651$	−9120.00	−0.549064
$652$	0	0
$653$	12982.0i	0.777986i	0.921241	+	0.388993i	$0.127177\pi$
$653$	12982.0i	0.777986i	−0.921241	+	0.388993i	$0.872823\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10530.0i	0.625288i
$658$	0	0
$659$	−1512.00	−0.0893766	−0.0446883	−	0.999001i	$-0.514229\pi$
$659$	−1512.00	−0.0893766	−0.0446883	+	0.999001i	$0.514229\pi$
$660$	0	0
$661$	16710.0	0.983273	0.491637	−	0.870800i	$-0.336399\pi$
$661$	16710.0	0.983273	0.491637	+	0.870800i	$0.336399\pi$
$662$	0	0
$663$	− 27348.0i	− 1.60197i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28016.0i	1.62636i
$668$	0	0
$669$	−5052.00	−0.291961
$670$	0	0
$671$	−112.000	−0.00644368
$672$	0	0
$673$	− 7962.00i	− 0.456036i	−0.973657	−	0.228018i	$-0.926775\pi$
$673$	− 7962.00i	− 0.456036i	0.973657	−	0.228018i	$-0.0732246\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12226.0i	0.694067i	0.937853	+	0.347033i	$0.112811\pi$
$677$	12226.0i	0.694067i	−0.937853	+	0.347033i	$0.887189\pi$
$678$	0	0
$679$	−18520.0	−1.04673
$680$	0	0
$681$	−6012.00	−0.338297
$682$	0	0
$683$	− 8748.00i	− 0.490092i	−0.969511	−	0.245046i	$-0.921197\pi$
$683$	− 8748.00i	− 0.490092i	0.969511	−	0.245046i	$-0.0788031\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15126.0i	− 0.840019i
$688$	0	0
$689$	−10836.0	−0.599156
$690$	0	0
$691$	7324.00	0.403210	0.201605	−	0.979467i	$-0.435384\pi$
$691$	7324.00	0.403210	0.201605	+	0.979467i	$0.435384\pi$
$692$	0	0
$693$	10080.0i	0.552536i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	26076.0i	1.41707i
$698$	0	0
$699$	9270.00	0.501607
$700$	0	0
$701$	−21934.0	−1.18179	−0.590896	−	0.806748i	$-0.701226\pi$
$701$	−21934.0	−1.18179	−0.590896	+	0.806748i	$0.701226\pi$
$702$	0	0
$703$	1128.00i	0.0605168i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3960.00i	0.210652i
$708$	0	0
$709$	10690.0	0.566250	0.283125	−	0.959083i	$-0.408629\pi$
$709$	10690.0	0.566250	0.283125	+	0.959083i	$0.408629\pi$
$710$	0	0
$711$	−3672.00	−0.193686
$712$	0	0
$713$	20672.0i	1.08580i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6408.00i	− 0.333767i
$718$	0	0
$719$	13792.0	0.715375	0.357688	−	0.933841i	$-0.383565\pi$
$719$	13792.0	0.715375	0.357688	+	0.933841i	$0.383565\pi$
$720$	0	0
$721$	−30640.0	−1.58265
$722$	0	0
$723$	− 294.000i	− 0.0151231i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24004.0i	1.22457i	0.790639	+	0.612283i	$0.209749\pi$
$727$	24004.0i	1.22457i	−0.790639	+	0.612283i	$0.790251\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	43672.0	2.20967
$732$	0	0
$733$	− 8562.00i	− 0.431439i	−0.976455	−	0.215719i	$-0.930790\pi$
$733$	− 8562.00i	− 0.431439i	0.976455	−	0.215719i	$-0.0692097\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21728.0i	1.08597i
$738$	0	0
$739$	−13836.0	−0.688722	−0.344361	−	0.938837i	$-0.611904\pi$
$739$	−13836.0	−0.688722	−0.344361	+	0.938837i	$0.611904\pi$
$740$	0	0
$741$	1032.00	0.0511626
$742$	0	0
$743$	− 22224.0i	− 1.09733i	−0.836041	−	0.548667i	$-0.815135\pi$
$743$	− 22224.0i	− 1.09733i	0.836041	−	0.548667i	$-0.184865\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6012.00i	− 0.294468i
$748$	0	0
$749$	8880.00	0.433202
$750$	0	0
$751$	−11544.0	−0.560914	−0.280457	−	0.959867i	$-0.590486\pi$
$751$	−11544.0	−0.560914	−0.280457	+	0.959867i	$0.590486\pi$
$752$	0	0
$753$	− 15120.0i	− 0.731744i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3814.00i	− 0.183120i	−0.995800	−	0.0915602i	$-0.970815\pi$
$757$	− 3814.00i	− 0.183120i	0.995800	−	0.0915602i	$-0.0291854\pi$
$758$	0	0
$759$	22848.0	1.09266
$760$	0	0
$761$	−25662.0	−1.22240	−0.611200	−	0.791476i	$-0.709313\pi$
$761$	−25662.0	−1.22240	−0.611200	+	0.791476i	$0.709313\pi$
$762$	0	0
$763$	1240.00i	0.0588349i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4816.00i	0.226722i
$768$	0	0
$769$	−30658.0	−1.43765	−0.718827	−	0.695189i	$-0.755321\pi$
$769$	−30658.0	−1.43765	−0.718827	+	0.695189i	$0.755321\pi$
$770$	0	0
$771$	−5958.00	−0.278304
$772$	0	0
$773$	30894.0i	1.43749i	0.695274	+	0.718745i	$0.255283\pi$
$773$	30894.0i	1.43749i	−0.695274	+	0.718745i	$0.744717\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16920.0i	− 0.781212i
$778$	0	0
$779$	−984.000	−0.0452573
$780$	0	0
$781$	37632.0	1.72417
$782$	0	0
$783$	5562.00i	0.253857i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 21596.0i	− 0.978163i	−0.872238	−	0.489081i	$-0.837332\pi$
$787$	− 21596.0i	− 0.978163i	0.872238	−	0.489081i	$-0.162668\pi$
$788$	0	0
$789$	4248.00	0.191677
$790$	0	0
$791$	−8280.00	−0.372191
$792$	0	0
$793$	− 172.000i	− 0.00770227i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8646.00i	− 0.384262i	−0.981369	−	0.192131i	$-0.938460\pi$
$797$	− 8646.00i	− 0.384262i	0.981369	−	0.192131i	$-0.0615399\pi$
$798$	0	0
$799$	−4240.00	−0.187735
$800$	0	0
$801$	594.000	0.0262022
$802$	0	0
$803$	− 65520.0i	− 2.87939i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 20010.0i	− 0.872844i
$808$	0	0
$809$	−24954.0	−1.08447	−0.542235	−	0.840227i	$-0.682422\pi$
$809$	−24954.0	−1.08447	−0.542235	+	0.840227i	$0.682422\pi$
$810$	0	0
$811$	−40004.0	−1.73210	−0.866048	−	0.499960i	$-0.833348\pi$
$811$	−40004.0	−1.73210	−0.866048	+	0.499960i	$0.833348\pi$
$812$	0	0
$813$	144.000i	0.00621193i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1648.00i	0.0705707i
$818$	0	0
$819$	−15480.0	−0.660458
$820$	0	0
$821$	16570.0	0.704381	0.352191	−	0.935928i	$-0.385437\pi$
$821$	16570.0	0.704381	0.352191	+	0.935928i	$0.385437\pi$
$822$	0	0
$823$	− 4388.00i	− 0.185852i	−0.995673	−	0.0929259i	$-0.970378\pi$
$823$	− 4388.00i	− 0.185852i	0.995673	−	0.0929259i	$-0.0296220\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 14364.0i	− 0.603972i	−0.953312	−	0.301986i	$-0.902350\pi$
$827$	− 14364.0i	− 0.603972i	0.953312	−	0.301986i	$-0.0976497\pi$
$828$	0	0
$829$	21170.0	0.886929	0.443465	−	0.896292i	$-0.353749\pi$
$829$	21170.0	0.886929	0.443465	+	0.896292i	$0.353749\pi$
$830$	0	0
$831$	20814.0	0.868868
$832$	0	0
$833$	6042.00i	0.251312i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4104.00i	0.169480i
$838$	0	0
$839$	10664.0	0.438811	0.219405	−	0.975634i	$-0.429588\pi$
$839$	10664.0	0.438811	0.219405	+	0.975634i	$0.429588\pi$
$840$	0	0
$841$	18047.0	0.739965
$842$	0	0
$843$	5082.00i	0.207632i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 36100.0i	− 1.46448i
$848$	0	0
$849$	−19092.0	−0.771774
$850$	0	0
$851$	−38352.0	−1.54488
$852$	0	0
$853$	3190.00i	0.128046i	0.997948	+	0.0640232i	$0.0203932\pi$
$853$	3190.00i	0.128046i	−0.997948	+	0.0640232i	$0.979607\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20814.0i	0.829630i	0.909906	+	0.414815i	$0.136154\pi$
$857$	20814.0i	0.829630i	−0.909906	+	0.414815i	$0.863846\pi$
$858$	0	0
$859$	−18988.0	−0.754205	−0.377103	−	0.926172i	$-0.623080\pi$
$859$	−18988.0	−0.754205	−0.377103	+	0.926172i	$0.623080\pi$
$860$	0	0
$861$	14760.0	0.584227
$862$	0	0
$863$	11664.0i	0.460078i	0.973181	+	0.230039i	$0.0738854\pi$
$863$	11664.0i	0.460078i	−0.973181	+	0.230039i	$0.926115\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18969.0i	0.743046i
$868$	0	0
$869$	22848.0	0.891905
$870$	0	0
$871$	−33368.0	−1.29808
$872$	0	0
$873$	8334.00i	0.323096i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 8246.00i	− 0.317500i	−0.987319	−	0.158750i	$-0.949254\pi$
$877$	− 8246.00i	− 0.317500i	0.987319	−	0.158750i	$-0.0507464\pi$
$878$	0	0
$879$	9402.00	0.360775
$880$	0	0
$881$	22890.0	0.875350	0.437675	−	0.899133i	$-0.355802\pi$
$881$	22890.0	0.875350	0.437675	+	0.899133i	$0.355802\pi$
$882$	0	0
$883$	33548.0i	1.27857i	0.768969	+	0.639287i	$0.220770\pi$
$883$	33548.0i	1.27857i	−0.768969	+	0.639287i	$0.779230\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32264.0i	1.22133i	0.791889	+	0.610665i	$0.209098\pi$
$887$	32264.0i	1.22133i	−0.791889	+	0.610665i	$0.790902\pi$
$888$	0	0
$889$	19920.0	0.751513
$890$	0	0
$891$	4536.00	0.170552
$892$	0	0
$893$	− 160.000i	− 0.00599574i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	35088.0i	1.30608i
$898$	0	0
$899$	31312.0	1.16164
$900$	0	0
$901$	13356.0	0.493843
$902$	0	0
$903$	− 24720.0i	− 0.910997i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 51228.0i	− 1.87541i	−0.347431	−	0.937706i	$-0.612946\pi$
$907$	− 51228.0i	− 1.87541i	0.347431	−	0.937706i	$-0.387054\pi$
$908$	0	0
$909$	1782.00	0.0650222
$910$	0	0
$911$	2144.00	0.0779735	0.0389868	−	0.999240i	$-0.487587\pi$
$911$	2144.00	0.0779735	0.0389868	+	0.999240i	$0.487587\pi$
$912$	0	0
$913$	37408.0i	1.35600i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5280.00i	− 0.190143i
$918$	0	0
$919$	33584.0	1.20548	0.602739	−	0.797939i	$-0.294076\pi$
$919$	33584.0	1.20548	0.602739	+	0.797939i	$0.294076\pi$
$920$	0	0
$921$	708.000	0.0253305
$922$	0	0
$923$	57792.0i	2.06094i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13788.0i	0.488519i
$928$	0	0
$929$	3590.00	0.126786	0.0633929	−	0.997989i	$-0.479808\pi$
$929$	3590.00	0.126786	0.0633929	+	0.997989i	$0.479808\pi$
$930$	0	0
$931$	−228.000	−0.00802621
$932$	0	0
$933$	11328.0i	0.397494i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 21686.0i	− 0.756084i	−0.925788	−	0.378042i	$-0.876597\pi$
$937$	− 21686.0i	− 0.756084i	0.925788	−	0.378042i	$-0.123403\pi$
$938$	0	0
$939$	23754.0	0.825540
$940$	0	0
$941$	−5174.00	−0.179243	−0.0896215	−	0.995976i	$-0.528566\pi$
$941$	−5174.00	−0.179243	−0.0896215	+	0.995976i	$0.528566\pi$
$942$	0	0
$943$	− 33456.0i	− 1.15533i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 35524.0i	− 1.21898i	−0.792793	−	0.609490i	$-0.791374\pi$
$947$	− 35524.0i	− 1.21898i	0.792793	−	0.609490i	$-0.208626\pi$
$948$	0	0
$949$	100620.	3.44179
$950$	0	0
$951$	13086.0	0.446207
$952$	0	0
$953$	16122.0i	0.547999i	0.961730	+	0.273999i	$0.0883466\pi$
$953$	16122.0i	0.547999i	−0.961730	+	0.273999i	$0.911653\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 34608.0i	− 1.16898i
$958$	0	0
$959$	45560.0	1.53411
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	− 3996.00i	− 0.133717i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 19188.0i	− 0.638102i	−0.947738	−	0.319051i	$-0.896636\pi$
$967$	− 19188.0i	− 0.638102i	0.947738	−	0.319051i	$-0.103364\pi$
$968$	0	0
$969$	−1272.00	−0.0421698
$970$	0	0
$971$	38464.0	1.27123	0.635617	−	0.772004i	$-0.280746\pi$
$971$	38464.0	1.27123	0.635617	+	0.772004i	$0.280746\pi$
$972$	0	0
$973$	− 36240.0i	− 1.19404i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43930.0i	− 1.43853i	−0.694735	−	0.719266i	$-0.744478\pi$
$977$	− 43930.0i	− 1.43853i	0.694735	−	0.719266i	$-0.255522\pi$
$978$	0	0
$979$	−3696.00	−0.120659
$980$	0	0
$981$	558.000	0.0181606
$982$	0	0
$983$	17328.0i	0.562235i	0.959673	+	0.281118i	$0.0907051\pi$
$983$	17328.0i	0.562235i	−0.959673	+	0.281118i	$0.909295\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2400.00i	0.0773990i
$988$	0	0
$989$	−56032.0	−1.80153
$990$	0	0
$991$	18160.0	0.582110	0.291055	−	0.956706i	$-0.405994\pi$
$991$	18160.0	0.582110	0.291055	+	0.956706i	$0.405994\pi$
$992$	0	0
$993$	23940.0i	0.765068i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9102.00i	− 0.289131i	−0.989495	−	0.144565i	$-0.953822\pi$
$997$	− 9102.00i	− 0.289131i	0.989495	−	0.144565i	$-0.0461784\pi$
$998$	0	0
$999$	−7614.00	−0.241137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.t.49.1		2
4.3	odd	2		600.4.f.a.49.2		2
5.2	odd	4		1200.4.a.p.1.1		1
5.3	odd	4		240.4.a.g.1.1		1
5.4	even	2	inner	1200.4.f.t.49.2		2
12.11	even	2		1800.4.f.v.649.2		2
15.8	even	4		720.4.a.q.1.1		1
20.3	even	4		120.4.a.b.1.1	✓	1
20.7	even	4		600.4.a.i.1.1		1
20.19	odd	2		600.4.f.a.49.1		2
40.3	even	4		960.4.a.bj.1.1		1
40.13	odd	4		960.4.a.k.1.1		1
60.23	odd	4		360.4.a.n.1.1		1
60.47	odd	4		1800.4.a.f.1.1		1
60.59	even	2		1800.4.f.v.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.b.1.1	✓	1	20.3	even	4
240.4.a.g.1.1		1	5.3	odd	4
360.4.a.n.1.1		1	60.23	odd	4
600.4.a.i.1.1		1	20.7	even	4
600.4.f.a.49.1		2	20.19	odd	2
600.4.f.a.49.2		2	4.3	odd	2
720.4.a.q.1.1		1	15.8	even	4
960.4.a.k.1.1		1	40.13	odd	4
960.4.a.bj.1.1		1	40.3	even	4
1200.4.a.p.1.1		1	5.2	odd	4
1200.4.f.t.49.1		2	1.1	even	1	trivial
1200.4.f.t.49.2		2	5.4	even	2	inner
1800.4.a.f.1.1		1	60.47	odd	4
1800.4.f.v.649.1		2	60.59	even	2
1800.4.f.v.649.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} +56.0000 q^{11} +86.0000i q^{13} -106.000i q^{17} +4.00000 q^{19} -60.0000 q^{21} +136.000i q^{23} +27.0000i q^{27} +206.000 q^{29} +152.000 q^{31} -168.000i q^{33} +282.000i q^{37} +258.000 q^{39} -246.000 q^{41} +412.000i q^{43} -40.0000i q^{47} -57.0000 q^{49} -318.000 q^{51} +126.000i q^{53} -12.0000i q^{57} +56.0000 q^{59} -2.00000 q^{61} +180.000i q^{63} +388.000i q^{67} +408.000 q^{69} +672.000 q^{71} -1170.00i q^{73} -1120.00i q^{77} +408.000 q^{79} +81.0000 q^{81} +668.000i q^{83} -618.000i q^{87} -66.0000 q^{89} +1720.00 q^{91} -456.000i q^{93} -926.000i q^{97} -504.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 112 q^{11} + 8 q^{19} - 120 q^{21} + 412 q^{29} + 304 q^{31} + 516 q^{39} - 492 q^{41} - 114 q^{49} - 636 q^{51} + 112 q^{59} - 4 q^{61} + 816 q^{69} + 1344 q^{71} + 816 q^{79} + 162 q^{81}+ \cdots - 1008 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 112 * q^11 + 8 * q^19 - 120 * q^21 + 412 * q^29 + 304 * q^31 + 516 * q^39 - 492 * q^41 - 114 * q^49 - 636 * q^51 + 112 * q^59 - 4 * q^61 + 816 * q^69 + 1344 * q^71 + 816 * q^79 + 162 * q^81 - 132 * q^89 + 3440 * q^91 - 1008 * q^99

Introduction

L-functions

Modular forms

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Database

Properties

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