LMFDB - Embedded newform 2400.3.p.a.1999.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,3,Mod(1999,2400)] 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("2400.1999"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-24,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$65.3952634465$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.8
Root			$1.40994 - 0.109843i$ of defining polynomial
Character	$\chi$	$=$	2400.1999
Dual form			2400.3.p.a.1999.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +10.7436 q^{7} -3.00000 q^{9} +8.00000 q^{11} -15.7298 q^{13} -15.8564i q^{17} +1.07180 q^{19} +18.6085i q^{21} +21.4873 q^{23} -5.19615i q^{27} -40.0958i q^{29} -9.20092i q^{31} +13.8564i q^{33} +9.97227 q^{37} -27.2448i q^{39} +51.5692 q^{41} -12.7846i q^{43} +1.54272 q^{47} +66.4256 q^{49} +27.4641 q^{51} -28.5808 q^{53} +1.85641i q^{57} -11.2154 q^{59} -1.54272i q^{61} -32.2309 q^{63} +43.2154i q^{67} +37.2170i q^{69} -84.4063i q^{71} -105.426i q^{73} +85.9491 q^{77} -73.6627i q^{79} +9.00000 q^{81} +12.2872i q^{83} +69.4479 q^{87} -33.1384 q^{89} -168.995 q^{91} +15.9365 q^{93} -69.1384i q^{97} -24.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9} + 64 q^{11} + 64 q^{19} + 80 q^{41} + 88 q^{49} + 192 q^{51} - 256 q^{59} + 72 q^{81} + 400 q^{89} - 576 q^{91} - 192 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 64 * q^11 + 64 * q^19 + 80 * q^41 + 88 * q^49 + 192 * q^51 - 256 * q^59 + 72 * q^81 + 400 * q^89 - 576 * q^91 - 192 * q^99

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.7436	1.53480	0.767402	−	0.641166i	$-0.221549\pi$
$7$	10.7436	1.53480	0.767402	+	0.641166i	$0.221549\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	8.00000	0.727273	0.363636	−	0.931541i	$-0.381535\pi$
$11$	8.00000	0.727273	0.363636	+	0.931541i	$0.381535\pi$
$12$	0	0
$13$	−15.7298	−1.20998	−0.604991	−	0.796232i	$-0.706823\pi$
$13$	−15.7298	−1.20998	−0.604991	+	0.796232i	$0.706823\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 15.8564i	− 0.932730i	−0.884592	−	0.466365i	$-0.845563\pi$
$17$	− 15.8564i	− 0.932730i	0.884592	−	0.466365i	$-0.154437\pi$
$18$	0	0
$19$	1.07180	0.0564104	0.0282052	−	0.999602i	$-0.491021\pi$
$19$	1.07180	0.0564104	0.0282052	+	0.999602i	$0.491021\pi$
$20$	0	0
$21$	18.6085i	0.886120i
$22$	0	0
$23$	21.4873	0.934229	0.467114	−	0.884197i	$-0.345294\pi$
$23$	21.4873	0.934229	0.467114	+	0.884197i	$0.345294\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	− 40.0958i	− 1.38261i	−0.722562	−	0.691307i	$-0.757035\pi$
$29$	− 40.0958i	− 1.38261i	0.722562	−	0.691307i	$-0.242965\pi$
$30$	0	0
$31$	− 9.20092i	− 0.296804i	−0.988927	−	0.148402i	$-0.952587\pi$
$31$	− 9.20092i	− 0.296804i	0.988927	−	0.148402i	$-0.0474129\pi$
$32$	0	0
$33$	13.8564i	0.419891i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.97227	0.269521	0.134760	−	0.990878i	$-0.456974\pi$
$37$	9.97227	0.269521	0.134760	+	0.990878i	$0.456974\pi$
$38$	0	0
$39$	− 27.2448i	− 0.698584i
$40$	0	0
$41$	51.5692	1.25779	0.628893	−	0.777492i	$-0.283508\pi$
$41$	51.5692	1.25779	0.628893	+	0.777492i	$0.283508\pi$
$42$	0	0
$43$	− 12.7846i	− 0.297317i	−0.988889	−	0.148658i	$-0.952505\pi$
$43$	− 12.7846i	− 0.297317i	0.988889	−	0.148658i	$-0.0474954\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.54272	0.0328237	0.0164119	−	0.999865i	$-0.494776\pi$
$47$	1.54272	0.0328237	0.0164119	+	0.999865i	$0.494776\pi$
$48$	0	0
$49$	66.4256	1.35563
$50$	0	0
$51$	27.4641	0.538512
$52$	0	0
$53$	−28.5808	−0.539260	−0.269630	−	0.962964i	$-0.586901\pi$
$53$	−28.5808	−0.539260	−0.269630	+	0.962964i	$0.586901\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.85641i	0.0325685i
$58$	0	0
$59$	−11.2154	−0.190091	−0.0950457	−	0.995473i	$-0.530300\pi$
$59$	−11.2154	−0.190091	−0.0950457	+	0.995473i	$0.530300\pi$
$60$	0	0
$61$	− 1.54272i	− 0.0252904i	−0.999920	−	0.0126452i	$-0.995975\pi$
$61$	− 1.54272i	− 0.0252904i	0.999920	−	0.0126452i	$-0.00402520\pi$
$62$	0	0
$63$	−32.2309	−0.511602
$64$	0	0
$65$	0	0
$66$	0	0
$67$	43.2154i	0.645006i	0.946568	+	0.322503i	$0.104524\pi$
$67$	43.2154i	0.645006i	−0.946568	+	0.322503i	$0.895476\pi$
$68$	0	0
$69$	37.2170i	0.539377i
$70$	0	0
$71$	− 84.4063i	− 1.18882i	−0.804162	−	0.594411i	$-0.797385\pi$
$71$	− 84.4063i	− 1.18882i	0.804162	−	0.594411i	$-0.202615\pi$
$72$	0	0
$73$	− 105.426i	− 1.44419i	−0.691796	−	0.722093i	$-0.743180\pi$
$73$	− 105.426i	− 1.44419i	0.691796	−	0.722093i	$-0.256820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	85.9491	1.11622
$78$	0	0
$79$	− 73.6627i	− 0.932439i	−0.884669	−	0.466220i	$-0.845616\pi$
$79$	− 73.6627i	− 0.932439i	0.884669	−	0.466220i	$-0.154384\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	12.2872i	0.148038i	0.997257	+	0.0740192i	$0.0235826\pi$
$83$	12.2872i	0.148038i	−0.997257	+	0.0740192i	$0.976417\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	69.4479	0.798252
$88$	0	0
$89$	−33.1384	−0.372342	−0.186171	−	0.982517i	$-0.559608\pi$
$89$	−33.1384	−0.372342	−0.186171	+	0.982517i	$0.559608\pi$
$90$	0	0
$91$	−168.995	−1.85709
$92$	0	0
$93$	15.9365	0.171360
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 69.1384i	− 0.712767i	−0.934340	−	0.356384i	$-0.884010\pi$
$97$	− 69.1384i	− 0.712767i	0.934340	−	0.356384i	$-0.115990\pi$
$98$	0	0
$99$	−24.0000	−0.242424
$100$	0	0
$101$	97.2574i	0.962944i	0.876461	+	0.481472i	$0.159898\pi$
$101$	97.2574i	0.962944i	−0.876461	+	0.481472i	$0.840102\pi$
$102$	0	0
$103$	139.667	1.35599	0.677996	−	0.735065i	$-0.262849\pi$
$103$	139.667	1.35599	0.677996	+	0.735065i	$0.262849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	197.779i	1.84841i	0.381901	+	0.924203i	$0.375269\pi$
$107$	197.779i	1.84841i	−0.381901	+	0.924203i	$0.624731\pi$
$108$	0	0
$109$	− 190.713i	− 1.74966i	−0.484427	−	0.874832i	$-0.660972\pi$
$109$	− 190.713i	− 1.74966i	0.484427	−	0.874832i	$-0.339028\pi$
$110$	0	0
$111$	17.2725i	0.155608i
$112$	0	0
$113$	113.713i	1.00631i	0.864197	+	0.503154i	$0.167827\pi$
$113$	113.713i	1.00631i	−0.864197	+	0.503154i	$0.832173\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	47.1893	0.403327
$118$	0	0
$119$	− 170.355i	− 1.43156i
$120$	0	0
$121$	−57.0000	−0.471074
$122$	0	0
$123$	89.3205i	0.726183i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	36.8590	0.290229	0.145114	−	0.989415i	$-0.453645\pi$
$127$	36.8590	0.290229	0.145114	+	0.989415i	$0.453645\pi$
$128$	0	0
$129$	22.1436	0.171656
$130$	0	0
$131$	−194.641	−1.48581	−0.742905	−	0.669397i	$-0.766552\pi$
$131$	−194.641	−1.48581	−0.742905	+	0.669397i	$0.766552\pi$
$132$	0	0
$133$	11.5150	0.0865789
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.4308i	0.119933i	0.998200	+	0.0599664i	$0.0190993\pi$
$137$	16.4308i	0.119933i	−0.998200	+	0.0599664i	$0.980901\pi$
$138$	0	0
$139$	−113.492	−0.816491	−0.408246	−	0.912872i	$-0.633859\pi$
$139$	−113.492	−0.816491	−0.408246	+	0.912872i	$0.633859\pi$
$140$	0	0
$141$	2.67206i	0.0189508i
$142$	0	0
$143$	−125.838	−0.879987
$144$	0	0
$145$	0	0
$146$	0	0
$147$	115.053i	0.782670i
$148$	0	0
$149$	− 31.6662i	− 0.212525i	−0.994338	−	0.106262i	$-0.966112\pi$
$149$	− 31.6662i	− 0.212525i	0.994338	−	0.106262i	$-0.0338884\pi$
$150$	0	0
$151$	− 90.5218i	− 0.599482i	−0.954021	−	0.299741i	$-0.903100\pi$
$151$	− 90.5218i	− 0.599482i	0.954021	−	0.299741i	$-0.0969003\pi$
$152$	0	0
$153$	47.5692i	0.310910i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	231.016	1.47144	0.735719	−	0.677287i	$-0.236844\pi$
$157$	231.016	1.47144	0.735719	+	0.677287i	$0.236844\pi$
$158$	0	0
$159$	− 49.5034i	− 0.311342i
$160$	0	0
$161$	230.851	1.43386
$162$	0	0
$163$	22.3538i	0.137140i	0.997646	+	0.0685700i	$0.0218437\pi$
$163$	22.3538i	0.137140i	−0.997646	+	0.0685700i	$0.978156\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	221.044	1.32361	0.661807	−	0.749674i	$-0.269790\pi$
$167$	221.044	1.32361	0.661807	+	0.749674i	$0.269790\pi$
$168$	0	0
$169$	78.4256	0.464057
$170$	0	0
$171$	−3.21539	−0.0188035
$172$	0	0
$173$	231.525	1.33830	0.669148	−	0.743130i	$-0.266659\pi$
$173$	231.525	1.33830	0.669148	+	0.743130i	$0.266659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 19.4256i	− 0.109749i
$178$	0	0
$179$	193.646	1.08182	0.540911	−	0.841080i	$-0.318080\pi$
$179$	193.646	1.08182	0.540911	+	0.841080i	$0.318080\pi$
$180$	0	0
$181$	− 270.492i	− 1.49443i	−0.664583	−	0.747214i	$-0.731391\pi$
$181$	− 270.492i	− 1.49443i	0.664583	−	0.747214i	$-0.268609\pi$
$182$	0	0
$183$	2.67206	0.0146014
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 126.851i	− 0.678349i
$188$	0	0
$189$	− 55.8255i	− 0.295373i
$190$	0	0
$191$	311.510i	1.63094i	0.578798	+	0.815471i	$0.303522\pi$
$191$	311.510i	1.63094i	−0.578798	+	0.815471i	$0.696478\pi$
$192$	0	0
$193$	− 48.2769i	− 0.250139i	−0.992148	−	0.125070i	$-0.960085\pi$
$193$	− 48.2769i	− 0.250139i	0.992148	−	0.125070i	$-0.0399154\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	251.883	1.27859	0.639297	−	0.768960i	$-0.279225\pi$
$197$	251.883	1.27859	0.639297	+	0.768960i	$0.279225\pi$
$198$	0	0
$199$	72.1200i	0.362412i	0.983445	+	0.181206i	$0.0580001\pi$
$199$	72.1200i	0.362412i	−0.983445	+	0.181206i	$0.942000\pi$
$200$	0	0
$201$	−74.8513	−0.372394
$202$	0	0
$203$	− 430.774i	− 2.12204i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−64.4618	−0.311410
$208$	0	0
$209$	8.57437	0.0410257
$210$	0	0
$211$	264.918	1.25554	0.627768	−	0.778401i	$-0.283969\pi$
$211$	264.918	1.25554	0.627768	+	0.778401i	$0.283969\pi$
$212$	0	0
$213$	146.196	0.686367
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 98.8513i	− 0.455536i
$218$	0	0
$219$	182.603	0.833802
$220$	0	0
$221$	249.418i	1.12859i
$222$	0	0
$223$	−30.6882	−0.137615	−0.0688076	−	0.997630i	$-0.521919\pi$
$223$	−30.6882	−0.137615	−0.0688076	+	0.997630i	$0.521919\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 295.846i	− 1.30329i	−0.758525	−	0.651643i	$-0.774080\pi$
$227$	− 295.846i	− 1.30329i	0.758525	−	0.651643i	$-0.225920\pi$
$228$	0	0
$229$	− 256.718i	− 1.12104i	−0.828141	−	0.560519i	$-0.810602\pi$
$229$	− 256.718i	− 1.12104i	0.828141	−	0.560519i	$-0.189398\pi$
$230$	0	0
$231$	148.868i	0.644451i
$232$	0	0
$233$	404.564i	1.73633i	0.496279	+	0.868163i	$0.334699\pi$
$233$	404.564i	1.73633i	−0.496279	+	0.868163i	$0.665301\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	127.588	0.538344
$238$	0	0
$239$	− 115.150i	− 0.481799i	−0.970550	−	0.240899i	$-0.922558\pi$
$239$	− 115.150i	− 0.481799i	0.970550	−	0.240899i	$-0.0774424\pi$
$240$	0	0
$241$	251.415	1.04322	0.521609	−	0.853185i	$-0.325332\pi$
$241$	251.415	1.04322	0.521609	+	0.853185i	$0.325332\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.8591	−0.0682555
$248$	0	0
$249$	−21.2820	−0.0854700
$250$	0	0
$251$	−6.85125	−0.0272958	−0.0136479	−	0.999907i	$-0.504344\pi$
$251$	−6.85125	−0.0272958	−0.0136479	+	0.999907i	$0.504344\pi$
$252$	0	0
$253$	171.898	0.679439
$254$	0	0
$255$	0	0
$256$	0	0
$257$	248.277i	0.966058i	0.875604	+	0.483029i	$0.160463\pi$
$257$	248.277i	0.966058i	−0.875604	+	0.483029i	$0.839537\pi$
$258$	0	0
$259$	107.138	0.413662
$260$	0	0
$261$	120.287i	0.460871i
$262$	0	0
$263$	498.835	1.89671	0.948356	−	0.317208i	$-0.102745\pi$
$263$	498.835	1.89671	0.948356	+	0.317208i	$0.102745\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 57.3975i	− 0.214972i
$268$	0	0
$269$	234.611i	0.872158i	0.899908	+	0.436079i	$0.143633\pi$
$269$	234.611i	0.872158i	−0.899908	+	0.436079i	$0.856367\pi$
$270$	0	0
$271$	101.321i	0.373878i	0.982372	+	0.186939i	$0.0598566\pi$
$271$	101.321i	0.373878i	−0.982372	+	0.186939i	$0.940143\pi$
$272$	0	0
$273$	− 292.708i	− 1.07219i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−169.942	−0.613509	−0.306755	−	0.951789i	$-0.599243\pi$
$277$	−169.942	−0.613509	−0.306755	+	0.951789i	$0.599243\pi$
$278$	0	0
$279$	27.6027i	0.0989346i
$280$	0	0
$281$	−111.128	−0.395474	−0.197737	−	0.980255i	$-0.563359\pi$
$281$	−111.128	−0.395474	−0.197737	+	0.980255i	$0.563359\pi$
$282$	0	0
$283$	550.620i	1.94566i	0.231531	+	0.972828i	$0.425627\pi$
$283$	550.620i	1.94566i	−0.231531	+	0.972828i	$0.574373\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	554.041	1.93046
$288$	0	0
$289$	37.5744	0.130015
$290$	0	0
$291$	119.751	0.411516
$292$	0	0
$293$	−223.509	−0.762829	−0.381414	−	0.924404i	$-0.624563\pi$
$293$	−223.509	−0.762829	−0.381414	+	0.924404i	$0.624563\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 41.5692i	− 0.139964i
$298$	0	0
$299$	−337.990	−1.13040
$300$	0	0
$301$	− 137.353i	− 0.456323i
$302$	0	0
$303$	−168.455	−0.555956
$304$	0	0
$305$	0	0
$306$	0	0
$307$	371.790i	1.21104i	0.795829	+	0.605521i	$0.207035\pi$
$307$	371.790i	1.21104i	−0.795829	+	0.605521i	$0.792965\pi$
$308$	0	0
$309$	241.911i	0.782883i
$310$	0	0
$311$	− 330.023i	− 1.06117i	−0.847633	−	0.530583i	$-0.821973\pi$
$311$	− 330.023i	− 1.06117i	0.847633	−	0.530583i	$-0.178027\pi$
$312$	0	0
$313$	80.2769i	0.256476i	0.991743	+	0.128238i	$0.0409321\pi$
$313$	80.2769i	0.256476i	−0.991743	+	0.128238i	$0.959068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	68.8833	0.217297	0.108649	−	0.994080i	$-0.465348\pi$
$317$	68.8833	0.217297	0.108649	+	0.994080i	$0.465348\pi$
$318$	0	0
$319$	− 320.766i	− 1.00554i
$320$	0	0
$321$	−342.564	−1.06718
$322$	0	0
$323$	− 16.9948i	− 0.0526156i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	330.325	1.01017
$328$	0	0
$329$	16.5744	0.0503780
$330$	0	0
$331$	396.056	1.19654	0.598272	−	0.801293i	$-0.295854\pi$
$331$	396.056	1.19654	0.598272	+	0.801293i	$0.295854\pi$
$332$	0	0
$333$	−29.9168	−0.0898403
$334$	0	0
$335$	0	0
$336$	0	0
$337$	231.723i	0.687606i	0.939042	+	0.343803i	$0.111715\pi$
$337$	231.723i	0.687606i	−0.939042	+	0.343803i	$0.888285\pi$
$338$	0	0
$339$	−196.956	−0.580992
$340$	0	0
$341$	− 73.6073i	− 0.215857i
$342$	0	0
$343$	187.215	0.545815
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 462.123i	− 1.33177i	−0.746056	−	0.665883i	$-0.768055\pi$
$347$	− 462.123i	− 1.33177i	0.746056	−	0.665883i	$-0.231945\pi$
$348$	0	0
$349$	− 266.993i	− 0.765022i	−0.923951	−	0.382511i	$-0.875059\pi$
$349$	− 266.993i	− 0.765022i	0.923951	−	0.382511i	$-0.124941\pi$
$350$	0	0
$351$	81.7343i	0.232861i
$352$	0	0
$353$	− 262.862i	− 0.744650i	−0.928102	−	0.372325i	$-0.878561\pi$
$353$	− 262.862i	− 0.744650i	0.928102	−	0.372325i	$-0.121439\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	295.064	0.826510
$358$	0	0
$359$	164.185i	0.457339i	0.973504	+	0.228669i	$0.0734374\pi$
$359$	164.185i	0.457339i	−0.973504	+	0.228669i	$0.926563\pi$
$360$	0	0
$361$	−359.851	−0.996818
$362$	0	0
$363$	− 98.7269i	− 0.271975i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−242.475	−0.660696	−0.330348	−	0.943859i	$-0.607166\pi$
$367$	−242.475	−0.660696	−0.330348	+	0.943859i	$0.607166\pi$
$368$	0	0
$369$	−154.708	−0.419262
$370$	0	0
$371$	−307.061	−0.827659
$372$	0	0
$373$	328.480	0.880643	0.440321	−	0.897840i	$-0.354864\pi$
$373$	328.480	0.880643	0.440321	+	0.897840i	$0.354864\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	630.697i	1.67294i
$378$	0	0
$379$	36.2102	0.0955415	0.0477708	−	0.998858i	$-0.484788\pi$
$379$	36.2102	0.0955415	0.0477708	+	0.998858i	$0.484788\pi$
$380$	0	0
$381$	63.8417i	0.167564i
$382$	0	0
$383$	164.295	0.428969	0.214485	−	0.976727i	$-0.431193\pi$
$383$	164.295	0.428969	0.214485	+	0.976727i	$0.431193\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	38.3538i	0.0991055i
$388$	0	0
$389$	− 604.936i	− 1.55510i	−0.628819	−	0.777552i	$-0.716461\pi$
$389$	− 604.936i	− 1.55510i	0.628819	−	0.777552i	$-0.283539\pi$
$390$	0	0
$391$	− 340.711i	− 0.871383i
$392$	0	0
$393$	− 337.128i	− 0.857832i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−541.699	−1.36448	−0.682241	−	0.731128i	$-0.738994\pi$
$397$	−541.699	−1.36448	−0.682241	+	0.731128i	$0.738994\pi$
$398$	0	0
$399$	19.9445i	0.0499863i
$400$	0	0
$401$	−379.569	−0.946557	−0.473278	−	0.880913i	$-0.656930\pi$
$401$	−379.569	−0.946557	−0.473278	+	0.880913i	$0.656930\pi$
$402$	0	0
$403$	144.728i	0.359127i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	79.7782	0.196015
$408$	0	0
$409$	−251.415	−0.614707	−0.307354	−	0.951595i	$-0.599443\pi$
$409$	−251.415	−0.614707	−0.307354	+	0.951595i	$0.599443\pi$
$410$	0	0
$411$	−28.4589	−0.0692432
$412$	0	0
$413$	−120.494	−0.291753
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 196.574i	− 0.471401i
$418$	0	0
$419$	268.133	0.639936	0.319968	−	0.947428i	$-0.396328\pi$
$419$	268.133	0.639936	0.319968	+	0.947428i	$0.396328\pi$
$420$	0	0
$421$	− 218.261i	− 0.518434i	−0.965819	−	0.259217i	$-0.916536\pi$
$421$	− 218.261i	− 0.518434i	0.965819	−	0.259217i	$-0.0834645\pi$
$422$	0	0
$423$	−4.62815	−0.0109412
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16.5744i	− 0.0388159i
$428$	0	0
$429$	− 217.958i	− 0.508061i
$430$	0	0
$431$	550.955i	1.27832i	0.769074	+	0.639159i	$0.220718\pi$
$431$	550.955i	1.27832i	−0.769074	+	0.639159i	$0.779282\pi$
$432$	0	0
$433$	263.128i	0.607686i	0.952722	+	0.303843i	$0.0982699\pi$
$433$	263.128i	0.607686i	−0.952722	+	0.303843i	$0.901730\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.0300	0.0527002
$438$	0	0
$439$	− 440.489i	− 1.00339i	−0.865044	−	0.501696i	$-0.832710\pi$
$439$	− 440.489i	− 1.00339i	0.865044	−	0.501696i	$-0.167290\pi$
$440$	0	0
$441$	−199.277	−0.451875
$442$	0	0
$443$	− 228.708i	− 0.516270i	−0.966109	−	0.258135i	$-0.916892\pi$
$443$	− 228.708i	− 0.516270i	0.966109	−	0.258135i	$-0.0831080\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	54.8475	0.122701
$448$	0	0
$449$	108.410	0.241448	0.120724	−	0.992686i	$-0.461478\pi$
$449$	108.410	0.241448	0.120724	+	0.992686i	$0.461478\pi$
$450$	0	0
$451$	412.554	0.914753
$452$	0	0
$453$	156.788	0.346111
$454$	0	0
$455$	0	0
$456$	0	0
$457$	561.692i	1.22909i	0.788883	+	0.614543i	$0.210660\pi$
$457$	561.692i	1.22909i	−0.788883	+	0.614543i	$0.789340\pi$
$458$	0	0
$459$	−82.3923	−0.179504
$460$	0	0
$461$	335.160i	0.727028i	0.931589	+	0.363514i	$0.118423\pi$
$461$	335.160i	0.727028i	−0.931589	+	0.363514i	$0.881577\pi$
$462$	0	0
$463$	−389.912	−0.842142	−0.421071	−	0.907028i	$-0.638346\pi$
$463$	−389.912	−0.842142	−0.421071	+	0.907028i	$0.638346\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	546.410i	1.17004i	0.811018	+	0.585022i	$0.198914\pi$
$467$	546.410i	1.17004i	−0.811018	+	0.585022i	$0.801086\pi$
$468$	0	0
$469$	464.290i	0.989958i
$470$	0	0
$471$	400.131i	0.849535i
$472$	0	0
$473$	− 102.277i	− 0.216230i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	85.7424	0.179753
$478$	0	0
$479$	368.369i	0.769037i	0.923117	+	0.384519i	$0.125633\pi$
$479$	368.369i	0.769037i	−0.923117	+	0.384519i	$0.874367\pi$
$480$	0	0
$481$	−156.862	−0.326116
$482$	0	0
$483$	399.846i	0.827839i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−90.6326	−0.186104	−0.0930519	−	0.995661i	$-0.529662\pi$
$487$	−90.6326	−0.186104	−0.0930519	+	0.995661i	$0.529662\pi$
$488$	0	0
$489$	−38.7180	−0.0791778
$490$	0	0
$491$	17.6462	0.0359392	0.0179696	−	0.999839i	$-0.494280\pi$
$491$	17.6462	0.0359392	0.0179696	+	0.999839i	$0.494280\pi$
$492$	0	0
$493$	−635.775	−1.28960
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 906.831i	− 1.82461i
$498$	0	0
$499$	548.631	1.09946	0.549730	−	0.835342i	$-0.314731\pi$
$499$	548.631	1.09946	0.549730	+	0.835342i	$0.314731\pi$
$500$	0	0
$501$	382.859i	0.764189i
$502$	0	0
$503$	−262.475	−0.521820	−0.260910	−	0.965363i	$-0.584023\pi$
$503$	−262.475	−0.521820	−0.260910	+	0.965363i	$0.584023\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	135.837i	0.267923i
$508$	0	0
$509$	230.093i	0.452049i	0.974122	+	0.226025i	$0.0725730\pi$
$509$	230.093i	0.452049i	−0.974122	+	0.226025i	$0.927427\pi$
$510$	0	0
$511$	− 1132.65i	− 2.21654i
$512$	0	0
$513$	− 5.56922i	− 0.0108562i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.3417	0.0238718
$518$	0	0
$519$	401.013i	0.772665i
$520$	0	0
$521$	164.144	0.315055	0.157527	−	0.987515i	$-0.449648\pi$
$521$	164.144	0.315055	0.157527	+	0.987515i	$0.449648\pi$
$522$	0	0
$523$	− 185.492i	− 0.354670i	−0.984151	−	0.177335i	$-0.943252\pi$
$523$	− 185.492i	− 0.354670i	0.984151	−	0.177335i	$-0.0567476\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−145.893	−0.276838
$528$	0	0
$529$	−67.2975	−0.127216
$530$	0	0
$531$	33.6462	0.0633638
$532$	0	0
$533$	−811.172	−1.52190
$534$	0	0
$535$	0	0
$536$	0	0
$537$	335.405i	0.624590i
$538$	0	0
$539$	531.405	0.985909
$540$	0	0
$541$	− 891.253i	− 1.64742i	−0.567013	−	0.823709i	$-0.691901\pi$
$541$	− 891.253i	− 1.64742i	0.567013	−	0.823709i	$-0.308099\pi$
$542$	0	0
$543$	468.505	0.862809
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 524.631i	− 0.959105i	−0.877513	−	0.479553i	$-0.840799\pi$
$547$	− 524.631i	− 0.959105i	0.877513	−	0.479553i	$-0.159201\pi$
$548$	0	0
$549$	4.62815i	0.00843014i
$550$	0	0
$551$	− 42.9745i	− 0.0779937i
$552$	0	0
$553$	− 791.405i	− 1.43111i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−570.804	−1.02478	−0.512391	−	0.858752i	$-0.671240\pi$
$557$	−570.804	−1.02478	−0.512391	+	0.858752i	$0.671240\pi$
$558$	0	0
$559$	201.099i	0.359748i
$560$	0	0
$561$	219.713	0.391645
$562$	0	0
$563$	− 161.877i	− 0.287526i	−0.989612	−	0.143763i	$-0.954080\pi$
$563$	− 161.877i	− 0.287526i	0.989612	−	0.143763i	$-0.0459203\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	96.6927	0.170534
$568$	0	0
$569$	−624.123	−1.09688	−0.548438	−	0.836191i	$-0.684778\pi$
$569$	−624.123	−1.09688	−0.548438	+	0.836191i	$0.684778\pi$
$570$	0	0
$571$	593.031	1.03858	0.519291	−	0.854597i	$-0.326196\pi$
$571$	593.031	1.03858	0.519291	+	0.854597i	$0.326196\pi$
$572$	0	0
$573$	−539.551	−0.941625
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1003.68i	− 1.73948i	−0.493507	−	0.869742i	$-0.664285\pi$
$577$	− 1003.68i	− 1.73948i	0.493507	−	0.869742i	$-0.335715\pi$
$578$	0	0
$579$	83.6180	0.144418
$580$	0	0
$581$	132.009i	0.227210i
$582$	0	0
$583$	−228.646	−0.392189
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 859.215i	− 1.46374i	−0.681444	−	0.731870i	$-0.738648\pi$
$587$	− 859.215i	− 1.46374i	0.681444	−	0.731870i	$-0.261352\pi$
$588$	0	0
$589$	− 9.86151i	− 0.0167428i
$590$	0	0
$591$	436.274i	0.738196i
$592$	0	0
$593$	1007.42i	1.69885i	0.527713	+	0.849423i	$0.323050\pi$
$593$	1007.42i	1.69885i	−0.527713	+	0.849423i	$0.676950\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−124.915	−0.209239
$598$	0	0
$599$	86.0598i	0.143672i	0.997416	+	0.0718362i	$0.0228859\pi$
$599$	86.0598i	0.143672i	−0.997416	+	0.0718362i	$0.977114\pi$
$600$	0	0
$601$	−406.000	−0.675541	−0.337770	−	0.941229i	$-0.609673\pi$
$601$	−406.000	−0.675541	−0.337770	+	0.941229i	$0.609673\pi$
$602$	0	0
$603$	− 129.646i	− 0.215002i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1187.80	−1.95684	−0.978422	−	0.206617i	$-0.933755\pi$
$607$	−1187.80	−1.95684	−0.978422	+	0.206617i	$0.933755\pi$
$608$	0	0
$609$	746.123	1.22516
$610$	0	0
$611$	−24.2666	−0.0397162
$612$	0	0
$613$	500.378	0.816277	0.408139	−	0.912920i	$-0.366178\pi$
$613$	500.378	0.816277	0.408139	+	0.912920i	$0.366178\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	88.8306i	0.143972i	0.997406	+	0.0719859i	$0.0229337\pi$
$617$	88.8306i	0.143972i	−0.997406	+	0.0719859i	$0.977066\pi$
$618$	0	0
$619$	−424.231	−0.685349	−0.342674	−	0.939454i	$-0.611333\pi$
$619$	−424.231	−0.685349	−0.342674	+	0.939454i	$0.611333\pi$
$620$	0	0
$621$	− 111.651i	− 0.179792i
$622$	0	0
$623$	−356.027	−0.571472
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.8513i	0.0236862i
$628$	0	0
$629$	− 158.124i	− 0.251390i
$630$	0	0
$631$	90.6326i	0.143633i	0.997418	+	0.0718166i	$0.0228796\pi$
$631$	90.6326i	0.143633i	−0.997418	+	0.0718166i	$0.977120\pi$
$632$	0	0
$633$	458.851i	0.724883i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1044.86	−1.64028
$638$	0	0
$639$	253.219i	0.396274i
$640$	0	0
$641$	1090.40	1.70109	0.850546	−	0.525901i	$-0.176272\pi$
$641$	1090.40	1.70109	0.850546	+	0.525901i	$0.176272\pi$
$642$	0	0
$643$	− 454.200i	− 0.706376i	−0.935552	−	0.353188i	$-0.885098\pi$
$643$	− 454.200i	− 0.706376i	0.935552	−	0.353188i	$-0.114902\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−610.789	−0.944032	−0.472016	−	0.881590i	$-0.656474\pi$
$647$	−610.789	−0.944032	−0.472016	+	0.881590i	$0.656474\pi$
$648$	0	0
$649$	−89.7231	−0.138248
$650$	0	0
$651$	171.215	0.263004
$652$	0	0
$653$	673.612	1.03157	0.515783	−	0.856720i	$-0.327501\pi$
$653$	673.612	1.03157	0.515783	+	0.856720i	$0.327501\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	316.277i	0.481396i
$658$	0	0
$659$	−819.328	−1.24329	−0.621645	−	0.783299i	$-0.713535\pi$
$659$	−819.328	−1.24329	−0.621645	+	0.783299i	$0.713535\pi$
$660$	0	0
$661$	− 370.628i	− 0.560707i	−0.959897	−	0.280354i	$-0.909548\pi$
$661$	− 370.628i	− 0.560707i	0.959897	−	0.280354i	$-0.0904518\pi$
$662$	0	0
$663$	−432.004	−0.651590
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 861.549i	− 1.29168i
$668$	0	0
$669$	− 53.1535i	− 0.0794521i
$670$	0	0
$671$	− 12.3417i	− 0.0183930i
$672$	0	0
$673$	255.703i	0.379944i	0.981789	+	0.189972i	$0.0608398\pi$
$673$	255.703i	0.379944i	−0.981789	+	0.189972i	$0.939160\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	934.323	1.38009	0.690047	−	0.723765i	$-0.257590\pi$
$677$	934.323	1.38009	0.690047	+	0.723765i	$0.257590\pi$
$678$	0	0
$679$	− 742.798i	− 1.09396i
$680$	0	0
$681$	512.420	0.752453
$682$	0	0
$683$	1142.54i	1.67283i	0.548096	+	0.836415i	$0.315353\pi$
$683$	1142.54i	1.67283i	−0.548096	+	0.836415i	$0.684647\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	444.648	0.647232
$688$	0	0
$689$	449.569	0.652495
$690$	0	0
$691$	−1316.90	−1.90578	−0.952892	−	0.303309i	$-0.901909\pi$
$691$	−1316.90	−1.90578	−0.952892	+	0.303309i	$0.901909\pi$
$692$	0	0
$693$	−257.847	−0.372074
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 817.703i	− 1.17317i
$698$	0	0
$699$	−700.726	−1.00247
$700$	0	0
$701$	318.493i	0.454341i	0.973855	+	0.227170i	$0.0729474\pi$
$701$	318.493i	0.454341i	−0.973855	+	0.227170i	$0.927053\pi$
$702$	0	0
$703$	10.6883	0.0152038
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1044.90i	1.47793i
$708$	0	0
$709$	289.831i	0.408788i	0.978889	+	0.204394i	$0.0655224\pi$
$709$	289.831i	0.408788i	−0.978889	+	0.204394i	$0.934478\pi$
$710$	0	0
$711$	220.988i	0.310813i
$712$	0	0
$713$	− 197.703i	− 0.277283i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	199.445	0.278167
$718$	0	0
$719$	491.122i	0.683062i	0.939870	+	0.341531i	$0.110945\pi$
$719$	491.122i	0.683062i	−0.939870	+	0.341531i	$0.889055\pi$
$720$	0	0
$721$	1500.53	2.08118
$722$	0	0
$723$	435.464i	0.602302i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−774.918	−1.06591	−0.532956	−	0.846143i	$-0.678919\pi$
$727$	−774.918	−1.06591	−0.532956	+	0.846143i	$0.678919\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	−202.718	−0.277316
$732$	0	0
$733$	858.966	1.17185	0.585925	−	0.810365i	$-0.300731\pi$
$733$	858.966	1.17185	0.585925	+	0.810365i	$0.300731\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	345.723i	0.469095i
$738$	0	0
$739$	63.1948	0.0855139	0.0427569	−	0.999086i	$-0.486386\pi$
$739$	63.1948	0.0855139	0.0427569	+	0.999086i	$0.486386\pi$
$740$	0	0
$741$	− 29.2008i	− 0.0394073i
$742$	0	0
$743$	−1105.00	−1.48721	−0.743604	−	0.668620i	$-0.766885\pi$
$743$	−1105.00	−1.48721	−0.743604	+	0.668620i	$0.766885\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 36.8616i	− 0.0493461i
$748$	0	0
$749$	2124.87i	2.83694i
$750$	0	0
$751$	− 804.119i	− 1.07073i	−0.844621	−	0.535365i	$-0.820174\pi$
$751$	− 804.119i	− 1.07073i	0.844621	−	0.535365i	$-0.179826\pi$
$752$	0	0
$753$	− 11.8667i	− 0.0157593i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1351.03	−1.78471	−0.892355	−	0.451334i	$-0.850948\pi$
$757$	−1351.03	−1.78471	−0.892355	+	0.451334i	$0.850948\pi$
$758$	0	0
$759$	297.736i	0.392274i
$760$	0	0
$761$	−709.805	−0.932726	−0.466363	−	0.884593i	$-0.654436\pi$
$761$	−709.805	−0.932726	−0.466363	+	0.884593i	$0.654436\pi$
$762$	0	0
$763$	− 2048.95i	− 2.68539i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	176.415	0.230007
$768$	0	0
$769$	195.703	0.254490	0.127245	−	0.991871i	$-0.459387\pi$
$769$	195.703	0.254490	0.127245	+	0.991871i	$0.459387\pi$
$770$	0	0
$771$	−430.028	−0.557754
$772$	0	0
$773$	−12.5484	−0.0162334	−0.00811670	−	0.999967i	$-0.502584\pi$
$773$	−12.5484	−0.0162334	−0.00811670	+	0.999967i	$0.502584\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	185.569i	0.238828i
$778$	0	0
$779$	55.2717	0.0709521
$780$	0	0
$781$	− 675.251i	− 0.864598i
$782$	0	0
$783$	−208.344	−0.266084
$784$	0	0
$785$	0	0
$786$	0	0
$787$	432.918i	0.550086i	0.961432	+	0.275043i	$0.0886921\pi$
$787$	432.918i	0.550086i	−0.961432	+	0.275043i	$0.911308\pi$
$788$	0	0
$789$	864.008i	1.09507i
$790$	0	0
$791$	1221.69i	1.54449i
$792$	0	0
$793$	24.2666i	0.0306010i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	700.746	0.879230	0.439615	−	0.898186i	$-0.355115\pi$
$797$	700.746	0.879230	0.439615	+	0.898186i	$0.355115\pi$
$798$	0	0
$799$	− 24.4619i	− 0.0306157i
$800$	0	0
$801$	99.4153	0.124114
$802$	0	0
$803$	− 843.405i	− 1.05032i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−406.357	−0.503541
$808$	0	0
$809$	848.102	1.04833	0.524167	−	0.851615i	$-0.324377\pi$
$809$	848.102	1.04833	0.524167	+	0.851615i	$0.324377\pi$
$810$	0	0
$811$	−1242.18	−1.53166	−0.765832	−	0.643041i	$-0.777673\pi$
$811$	−1242.18	−1.53166	−0.765832	+	0.643041i	$0.777673\pi$
$812$	0	0
$813$	−175.493	−0.215858
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 13.7025i	− 0.0167717i
$818$	0	0
$819$	506.985	0.619029
$820$	0	0
$821$	− 208.303i	− 0.253719i	−0.991921	−	0.126859i	$-0.959510\pi$
$821$	− 208.303i	− 0.253719i	0.991921	−	0.126859i	$-0.0404897\pi$
$822$	0	0
$823$	−778.114	−0.945461	−0.472730	−	0.881207i	$-0.656732\pi$
$823$	−778.114	−0.945461	−0.472730	+	0.881207i	$0.656732\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1280.57i	1.54845i	0.632910	+	0.774225i	$0.281860\pi$
$827$	1280.57i	1.54845i	−0.632910	+	0.774225i	$0.718140\pi$
$828$	0	0
$829$	− 9.78043i	− 0.0117979i	−0.999983	−	0.00589893i	$-0.998122\pi$
$829$	− 9.78043i	− 0.0117979i	0.999983	−	0.00589893i	$-0.00187770\pi$
$830$	0	0
$831$	− 294.348i	− 0.354210i
$832$	0	0
$833$	− 1053.27i	− 1.26443i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−47.8094	−0.0571199
$838$	0	0
$839$	1232.38i	1.46886i	0.678682	+	0.734432i	$0.262551\pi$
$839$	1232.38i	1.46886i	−0.678682	+	0.734432i	$0.737449\pi$
$840$	0	0
$841$	−766.672	−0.911619
$842$	0	0
$843$	− 192.480i	− 0.228327i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−612.387	−0.723007
$848$	0	0
$849$	−953.703	−1.12332
$850$	0	0
$851$	214.277	0.251794
$852$	0	0
$853$	−412.170	−0.483201	−0.241600	−	0.970376i	$-0.577672\pi$
$853$	−412.170	−0.483201	−0.241600	+	0.970376i	$0.577672\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 136.451i	− 0.159220i	−0.996826	−	0.0796099i	$-0.974633\pi$
$857$	− 136.451i	− 0.159220i	0.996826	−	0.0796099i	$-0.0253675\pi$
$858$	0	0
$859$	−649.646	−0.756282	−0.378141	−	0.925748i	$-0.623437\pi$
$859$	−649.646	−0.756282	−0.378141	+	0.925748i	$0.623437\pi$
$860$	0	0
$861$	959.627i	1.11455i
$862$	0	0
$863$	1192.38	1.38167	0.690833	−	0.723015i	$-0.257244\pi$
$863$	1192.38	1.38167	0.690833	+	0.723015i	$0.257244\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	65.0807i	0.0750643i
$868$	0	0
$869$	− 589.302i	− 0.678138i
$870$	0	0
$871$	− 679.768i	− 0.780446i
$872$	0	0
$873$	207.415i	0.237589i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−744.451	−0.848861	−0.424431	−	0.905460i	$-0.639526\pi$
$877$	−744.451	−0.848861	−0.424431	+	0.905460i	$0.639526\pi$
$878$	0	0
$879$	− 387.129i	− 0.440419i
$880$	0	0
$881$	651.108	0.739055	0.369528	−	0.929220i	$-0.379520\pi$
$881$	651.108	0.739055	0.369528	+	0.929220i	$0.379520\pi$
$882$	0	0
$883$	1171.44i	1.32666i	0.748327	+	0.663330i	$0.230857\pi$
$883$	1171.44i	1.32666i	−0.748327	+	0.663330i	$0.769143\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1026.87	−1.15769	−0.578845	−	0.815437i	$-0.696496\pi$
$887$	−1026.87	−1.15769	−0.578845	+	0.815437i	$0.696496\pi$
$888$	0	0
$889$	396.000	0.445444
$890$	0	0
$891$	72.0000	0.0808081
$892$	0	0
$893$	1.65348	0.00185160
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 585.415i	− 0.652637i
$898$	0	0
$899$	−368.918	−0.410365
$900$	0	0
$901$	453.189i	0.502984i
$902$	0	0
$903$	237.903	0.263458
$904$	0	0
$905$	0	0
$906$	0	0
$907$	470.508i	0.518752i	0.965776	+	0.259376i	$0.0835168\pi$
$907$	470.508i	0.518752i	−0.965776	+	0.259376i	$0.916483\pi$
$908$	0	0
$909$	− 291.772i	− 0.320981i
$910$	0	0
$911$	− 553.930i	− 0.608046i	−0.952665	−	0.304023i	$-0.901670\pi$
$911$	− 553.930i	− 0.608046i	0.952665	−	0.304023i	$-0.0983300\pi$
$912$	0	0
$913$	98.2975i	0.107664i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2091.15	−2.28043
$918$	0	0
$919$	910.123i	0.990341i	0.868796	+	0.495170i	$0.164894\pi$
$919$	910.123i	0.990341i	−0.868796	+	0.495170i	$0.835106\pi$
$920$	0	0
$921$	−643.959	−0.699195
$922$	0	0
$923$	1327.69i	1.43845i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−419.002	−0.451997
$928$	0	0
$929$	−1278.69	−1.37641	−0.688206	−	0.725515i	$-0.741602\pi$
$929$	−1278.69	−1.37641	−0.688206	+	0.725515i	$0.741602\pi$
$930$	0	0
$931$	71.1948	0.0764713
$932$	0	0
$933$	571.616	0.612664
$934$	0	0
$935$	0	0
$936$	0	0
$937$	634.554i	0.677219i	0.940927	+	0.338609i	$0.109956\pi$
$937$	634.554i	0.677219i	−0.940927	+	0.338609i	$0.890044\pi$
$938$	0	0
$939$	−139.044	−0.148076
$940$	0	0
$941$	1136.25i	1.20749i	0.797177	+	0.603745i	$0.206326\pi$
$941$	1136.25i	1.20749i	−0.797177	+	0.603745i	$0.793674\pi$
$942$	0	0
$943$	1108.08	1.17506
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 775.615i	− 0.819023i	−0.912305	−	0.409512i	$-0.865699\pi$
$947$	− 775.615i	− 0.819023i	0.912305	−	0.409512i	$-0.134301\pi$
$948$	0	0
$949$	1658.32i	1.74744i
$950$	0	0
$951$	119.309i	0.125457i
$952$	0	0
$953$	− 599.590i	− 0.629160i	−0.949231	−	0.314580i	$-0.898136\pi$
$953$	− 599.590i	− 0.629160i	0.949231	−	0.314580i	$-0.101864\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	555.583	0.580547
$958$	0	0
$959$	176.526i	0.184073i
$960$	0	0
$961$	876.343	0.911908
$962$	0	0
$963$	− 593.338i	− 0.616135i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1407.53	−1.45556	−0.727780	−	0.685811i	$-0.759448\pi$
$967$	−1407.53	−1.45556	−0.727780	+	0.685811i	$0.759448\pi$
$968$	0	0
$969$	29.4359	0.0303776
$970$	0	0
$971$	−653.661	−0.673184	−0.336592	−	0.941651i	$-0.609274\pi$
$971$	−653.661	−0.673184	−0.336592	+	0.941651i	$0.609274\pi$
$972$	0	0
$973$	−1219.32	−1.25315
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1003.57i	− 1.02719i	−0.858031	−	0.513597i	$-0.828313\pi$
$977$	− 1003.57i	− 1.02719i	0.858031	−	0.513597i	$-0.171687\pi$
$978$	0	0
$979$	−265.108	−0.270794
$980$	0	0
$981$	572.140i	0.583221i
$982$	0	0
$983$	105.672	0.107500	0.0537498	−	0.998554i	$-0.482883\pi$
$983$	105.672	0.107500	0.0537498	+	0.998554i	$0.482883\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	28.7077i	0.0290858i
$988$	0	0
$989$	− 274.706i	− 0.277762i
$990$	0	0
$991$	− 1728.18i	− 1.74388i	−0.489615	−	0.871938i	$-0.662863\pi$
$991$	− 1728.18i	− 1.74388i	0.489615	−	0.871938i	$-0.337137\pi$
$992$	0	0
$993$	685.990i	0.690825i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−135.205	−0.135612	−0.0678060	−	0.997699i	$-0.521600\pi$
$997$	−135.205	−0.135612	−0.0678060	+	0.997699i	$0.521600\pi$
$998$	0	0
$999$	− 51.8175i	− 0.0518693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.3.p.a.1999.8		8
4.3	odd	2		600.3.p.a.499.7		8
5.2	odd	4		2400.3.g.a.751.4		4
5.3	odd	4		96.3.b.a.79.1		4
5.4	even	2	inner	2400.3.p.a.1999.1		8
8.3	odd	2	inner	2400.3.p.a.1999.5		8
8.5	even	2		600.3.p.a.499.1		8
15.8	even	4		288.3.b.b.271.3		4
20.3	even	4		24.3.b.a.19.1	✓	4
20.7	even	4		600.3.g.a.451.4		4
20.19	odd	2		600.3.p.a.499.2		8
40.3	even	4		96.3.b.a.79.2		4
40.13	odd	4		24.3.b.a.19.2	yes	4
40.19	odd	2	inner	2400.3.p.a.1999.4		8
40.27	even	4		2400.3.g.a.751.3		4
40.29	even	2		600.3.p.a.499.8		8
40.37	odd	4		600.3.g.a.451.3		4
60.23	odd	4		72.3.b.b.19.4		4
80.3	even	4		768.3.g.h.511.6		8
80.13	odd	4		768.3.g.h.511.2		8
80.43	even	4		768.3.g.h.511.3		8
80.53	odd	4		768.3.g.h.511.7		8
120.53	even	4		72.3.b.b.19.3		4
120.83	odd	4		288.3.b.b.271.2		4
240.53	even	4		2304.3.g.z.1279.4		8
240.83	odd	4		2304.3.g.z.1279.5		8
240.173	even	4		2304.3.g.z.1279.6		8
240.203	odd	4		2304.3.g.z.1279.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.b.a.19.1	✓	4	20.3	even	4
24.3.b.a.19.2	yes	4	40.13	odd	4
72.3.b.b.19.3		4	120.53	even	4
72.3.b.b.19.4		4	60.23	odd	4
96.3.b.a.79.1		4	5.3	odd	4
96.3.b.a.79.2		4	40.3	even	4
288.3.b.b.271.2		4	120.83	odd	4
288.3.b.b.271.3		4	15.8	even	4
600.3.g.a.451.3		4	40.37	odd	4
600.3.g.a.451.4		4	20.7	even	4
600.3.p.a.499.1		8	8.5	even	2
600.3.p.a.499.2		8	20.19	odd	2
600.3.p.a.499.7		8	4.3	odd	2
600.3.p.a.499.8		8	40.29	even	2
768.3.g.h.511.2		8	80.13	odd	4
768.3.g.h.511.3		8	80.43	even	4
768.3.g.h.511.6		8	80.3	even	4
768.3.g.h.511.7		8	80.53	odd	4
2304.3.g.z.1279.3		8	240.203	odd	4
2304.3.g.z.1279.4		8	240.53	even	4
2304.3.g.z.1279.5		8	240.83	odd	4
2304.3.g.z.1279.6		8	240.173	even	4
2400.3.g.a.751.3		4	40.27	even	4
2400.3.g.a.751.4		4	5.2	odd	4
2400.3.p.a.1999.1		8	5.4	even	2	inner
2400.3.p.a.1999.4		8	40.19	odd	2	inner
2400.3.p.a.1999.5		8	8.3	odd	2	inner
2400.3.p.a.1999.8		8	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +10.7436 q^{7} -3.00000 q^{9} +8.00000 q^{11} -15.7298 q^{13} -15.8564i q^{17} +1.07180 q^{19} +18.6085i q^{21} +21.4873 q^{23} -5.19615i q^{27} -40.0958i q^{29} -9.20092i q^{31} +13.8564i q^{33} +9.97227 q^{37} -27.2448i q^{39} +51.5692 q^{41} -12.7846i q^{43} +1.54272 q^{47} +66.4256 q^{49} +27.4641 q^{51} -28.5808 q^{53} +1.85641i q^{57} -11.2154 q^{59} -1.54272i q^{61} -32.2309 q^{63} +43.2154i q^{67} +37.2170i q^{69} -84.4063i q^{71} -105.426i q^{73} +85.9491 q^{77} -73.6627i q^{79} +9.00000 q^{81} +12.2872i q^{83} +69.4479 q^{87} -33.1384 q^{89} -168.995 q^{91} +15.9365 q^{93} -69.1384i q^{97} -24.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9} + 64 q^{11} + 64 q^{19} + 80 q^{41} + 88 q^{49} + 192 q^{51} - 256 q^{59} + 72 q^{81} + 400 q^{89} - 576 q^{91} - 192 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 64 * q^11 + 64 * q^19 + 80 * q^41 + 88 * q^49 + 192 * q^51 - 256 * q^59 + 72 * q^81 + 400 * q^89 - 576 * q^91 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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