LMFDB - Embedded newform 4032.3.d.n.449.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4032,3,Mod(449,4032)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4032, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4032.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4032.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,-64] 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:	$109.864042590$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 $ x^12 + 44x^10 + 719x^8 + 5356x^6 + 17809x^4 + 20000*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.10
Root			$2.61146i$ of defining polynomial
Character	$\chi$	$=$	4032.449
Dual form			4032.3.d.n.449.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.81315i q^{5} +2.64575 q^{7} +1.61241i q^{11} -0.520800 q^{13} +26.9683i q^{17} +17.8871 q^{19} +12.1896i q^{23} -21.4191 q^{25} -47.9985i q^{29} +46.8270 q^{31} +18.0259i q^{35} +1.85849 q^{37} +41.6825i q^{41} -2.33064 q^{43} +91.9001i q^{47} +7.00000 q^{49} -30.2708i q^{53} -10.9856 q^{55} +72.1273i q^{59} +32.1019 q^{61} -3.54829i q^{65} +45.0801 q^{67} -111.454i q^{71} +99.9042 q^{73} +4.26603i q^{77} +3.78325 q^{79} +35.2017i q^{83} -183.739 q^{85} +48.7809i q^{89} -1.37791 q^{91} +121.868i q^{95} +12.1334 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{13} - 64 q^{19} - 124 q^{25} + 160 q^{31} - 56 q^{37} + 64 q^{43} + 84 q^{49} - 160 q^{55} - 104 q^{61} + 64 q^{67} - 64 q^{73} + 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) $ 12 * q - 16 * q^13 - 64 * q^19 - 124 * q^25 + 160 * q^31 - 56 * q^37 + 64 * q^43 + 84 * q^49 - 160 * q^55 - 104 * q^61 + 64 * q^67 - 64 * q^73 + 32 * q^79 + 184 * q^85 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.81315i	1.36263i	0.731990	+	0.681315i	$0.238592\pi$
$5$	6.81315i	1.36263i	−0.731990	+	0.681315i	$0.761408\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.61241i	0.146583i	0.997311	+	0.0732913i	$0.0233503\pi$
$11$	1.61241i	0.146583i	−0.997311	+	0.0732913i	$0.976650\pi$
$12$	0	0
$13$	−0.520800	−0.0400616	−0.0200308	−	0.999799i	$-0.506376\pi$
$13$	−0.520800	−0.0400616	−0.0200308	+	0.999799i	$0.506376\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.9683i	1.58637i	0.608980	+	0.793186i	$0.291579\pi$
$17$	26.9683i	1.58637i	−0.608980	+	0.793186i	$0.708421\pi$
$18$	0	0
$19$	17.8871	0.941427	0.470713	−	0.882286i	$-0.343997\pi$
$19$	17.8871	0.941427	0.470713	+	0.882286i	$0.343997\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	12.1896i	0.529982i	0.964251	+	0.264991i	$0.0853690\pi$
$23$	12.1896i	0.529982i	−0.964251	+	0.264991i	$0.914631\pi$
$24$	0	0
$25$	−21.4191	−0.856763
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 47.9985i	− 1.65512i	−0.561377	−	0.827560i	$-0.689728\pi$
$29$	− 47.9985i	− 1.65512i	0.561377	−	0.827560i	$-0.310272\pi$
$30$	0	0
$31$	46.8270	1.51055	0.755274	−	0.655409i	$-0.227504\pi$
$31$	46.8270	1.51055	0.755274	+	0.655409i	$0.227504\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	18.0259i	0.515026i
$36$	0	0
$37$	1.85849	0.0502295	0.0251148	−	0.999685i	$-0.492005\pi$
$37$	1.85849	0.0502295	0.0251148	+	0.999685i	$0.492005\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	41.6825i	1.01665i	0.861166	+	0.508324i	$0.169735\pi$
$41$	41.6825i	1.01665i	−0.861166	+	0.508324i	$0.830265\pi$
$42$	0	0
$43$	−2.33064	−0.0542009	−0.0271005	−	0.999633i	$-0.508627\pi$
$43$	−2.33064	−0.0542009	−0.0271005	+	0.999633i	$0.508627\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	91.9001i	1.95532i	0.210188	+	0.977661i	$0.432592\pi$
$47$	91.9001i	1.95532i	−0.210188	+	0.977661i	$0.567408\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 30.2708i	− 0.571147i	−0.958357	−	0.285574i	$-0.907816\pi$
$53$	− 30.2708i	− 0.571147i	0.958357	−	0.285574i	$-0.0921841\pi$
$54$	0	0
$55$	−10.9856	−0.199738
$56$	0	0
$57$	0	0
$58$	0	0
$59$	72.1273i	1.22250i	0.791439	+	0.611249i	$0.209332\pi$
$59$	72.1273i	1.22250i	−0.791439	+	0.611249i	$0.790668\pi$
$60$	0	0
$61$	32.1019	0.526261	0.263131	−	0.964760i	$-0.415245\pi$
$61$	32.1019	0.526261	0.263131	+	0.964760i	$0.415245\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.54829i	− 0.0545891i
$66$	0	0
$67$	45.0801	0.672837	0.336419	−	0.941713i	$-0.390784\pi$
$67$	45.0801	0.672837	0.336419	+	0.941713i	$0.390784\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 111.454i	− 1.56978i	−0.619635	−	0.784890i	$-0.712719\pi$
$71$	− 111.454i	− 1.56978i	0.619635	−	0.784890i	$-0.287281\pi$
$72$	0	0
$73$	99.9042	1.36855	0.684276	−	0.729223i	$-0.260119\pi$
$73$	99.9042	1.36855	0.684276	+	0.729223i	$0.260119\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.26603i	0.0554030i
$78$	0	0
$79$	3.78325	0.0478892	0.0239446	−	0.999713i	$-0.492377\pi$
$79$	3.78325	0.0478892	0.0239446	+	0.999713i	$0.492377\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	35.2017i	0.424117i	0.977257	+	0.212059i	$0.0680168\pi$
$83$	35.2017i	0.424117i	−0.977257	+	0.212059i	$0.931983\pi$
$84$	0	0
$85$	−183.739	−2.16164
$86$	0	0
$87$	0	0
$88$	0	0
$89$	48.7809i	0.548100i	0.961716	+	0.274050i	$0.0883633\pi$
$89$	48.7809i	0.548100i	−0.961716	+	0.274050i	$0.911637\pi$
$90$	0	0
$91$	−1.37791	−0.0151418
$92$	0	0
$93$	0	0
$94$	0	0
$95$	121.868i	1.28282i
$96$	0	0
$97$	12.1334	0.125087	0.0625434	−	0.998042i	$-0.480079\pi$
$97$	12.1334	0.125087	0.0625434	+	0.998042i	$0.480079\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.7469i	0.116306i	0.998308	+	0.0581529i	$0.0185211\pi$
$101$	11.7469i	0.116306i	−0.998308	+	0.0581529i	$0.981479\pi$
$102$	0	0
$103$	−153.740	−1.49262	−0.746310	−	0.665599i	$-0.768176\pi$
$103$	−153.740	−1.49262	−0.746310	+	0.665599i	$0.768176\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 54.3156i	− 0.507622i	−0.967254	−	0.253811i	$-0.918316\pi$
$107$	− 54.3156i	− 0.507622i	0.967254	−	0.253811i	$-0.0816841\pi$
$108$	0	0
$109$	−172.032	−1.57828	−0.789139	−	0.614215i	$-0.789473\pi$
$109$	−172.032	−1.57828	−0.789139	+	0.614215i	$0.789473\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 64.8360i	− 0.573770i	−0.957965	−	0.286885i	$-0.907380\pi$
$113$	− 64.8360i	− 0.573770i	0.957965	−	0.286885i	$-0.0926198\pi$
$114$	0	0
$115$	−83.0496	−0.722170
$116$	0	0
$117$	0	0
$118$	0	0
$119$	71.3515i	0.599592i
$120$	0	0
$121$	118.400	0.978514
$122$	0	0
$123$	0	0
$124$	0	0
$125$	24.3974i	0.195179i
$126$	0	0
$127$	185.124	1.45767	0.728834	−	0.684690i	$-0.240062\pi$
$127$	185.124	1.45767	0.728834	+	0.684690i	$0.240062\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	31.0619i	0.237114i	0.992947	+	0.118557i	$0.0378268\pi$
$131$	31.0619i	0.237114i	−0.992947	+	0.118557i	$0.962173\pi$
$132$	0	0
$133$	47.3248	0.355826
$134$	0	0
$135$	0	0
$136$	0	0
$137$	85.6561i	0.625227i	0.949880	+	0.312614i	$0.101204\pi$
$137$	85.6561i	0.625227i	−0.949880	+	0.312614i	$0.898796\pi$
$138$	0	0
$139$	2.59091	0.0186396	0.00931982	−	0.999957i	$-0.497033\pi$
$139$	2.59091	0.0186396	0.00931982	+	0.999957i	$0.497033\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 0.839742i	− 0.00587232i
$144$	0	0
$145$	327.021	2.25532
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 113.941i	− 0.764703i	−0.924017	−	0.382351i	$-0.875114\pi$
$149$	− 113.941i	− 0.764703i	0.924017	−	0.382351i	$-0.124886\pi$
$150$	0	0
$151$	11.2688	0.0746277	0.0373138	−	0.999304i	$-0.488120\pi$
$151$	11.2688	0.0746277	0.0373138	+	0.999304i	$0.488120\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	319.039i	2.05832i
$156$	0	0
$157$	−41.7543	−0.265951	−0.132976	−	0.991119i	$-0.542453\pi$
$157$	−41.7543	−0.265951	−0.132976	+	0.991119i	$0.542453\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	32.2506i	0.200314i
$162$	0	0
$163$	141.787	0.869860	0.434930	−	0.900464i	$-0.356773\pi$
$163$	141.787	0.869860	0.434930	+	0.900464i	$0.356773\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	166.816i	0.998897i	0.866344	+	0.499449i	$0.166464\pi$
$167$	166.816i	0.998897i	−0.866344	+	0.499449i	$0.833536\pi$
$168$	0	0
$169$	−168.729	−0.998395
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.0306i	0.0579805i	0.999580	+	0.0289903i	$0.00922918\pi$
$173$	10.0306i	0.0579805i	−0.999580	+	0.0289903i	$0.990771\pi$
$174$	0	0
$175$	−56.6695	−0.323826
$176$	0	0
$177$	0	0
$178$	0	0
$179$	99.4270i	0.555458i	0.960659	+	0.277729i	$0.0895817\pi$
$179$	99.4270i	0.555458i	−0.960659	+	0.277729i	$0.910418\pi$
$180$	0	0
$181$	−217.242	−1.20023	−0.600115	−	0.799914i	$-0.704878\pi$
$181$	−217.242	−1.20023	−0.600115	+	0.799914i	$0.704878\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.6622i	0.0684443i
$186$	0	0
$187$	−43.4839	−0.232534
$188$	0	0
$189$	0	0
$190$	0	0
$191$	237.181i	1.24179i	0.783895	+	0.620893i	$0.213230\pi$
$191$	237.181i	1.24179i	−0.783895	+	0.620893i	$0.786770\pi$
$192$	0	0
$193$	234.019	1.21253	0.606266	−	0.795262i	$-0.292667\pi$
$193$	234.019	1.21253	0.606266	+	0.795262i	$0.292667\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 277.878i	− 1.41055i	−0.708934	−	0.705274i	$-0.750824\pi$
$197$	− 277.878i	− 1.41055i	0.708934	−	0.705274i	$-0.249176\pi$
$198$	0	0
$199$	−169.646	−0.852494	−0.426247	−	0.904607i	$-0.640164\pi$
$199$	−169.646	−0.852494	−0.426247	+	0.904607i	$0.640164\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 126.992i	− 0.625576i
$204$	0	0
$205$	−283.990	−1.38531
$206$	0	0
$207$	0	0
$208$	0	0
$209$	28.8413i	0.137997i
$210$	0	0
$211$	−307.022	−1.45508	−0.727540	−	0.686066i	$-0.759336\pi$
$211$	−307.022	−1.45508	−0.727540	+	0.686066i	$0.759336\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 15.8790i	− 0.0738559i
$216$	0	0
$217$	123.893	0.570933
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 14.0451i	− 0.0635525i
$222$	0	0
$223$	−317.188	−1.42237	−0.711185	−	0.703005i	$-0.751841\pi$
$223$	−317.188	−1.42237	−0.711185	+	0.703005i	$0.751841\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	205.937i	0.907214i	0.891202	+	0.453607i	$0.149863\pi$
$227$	205.937i	0.907214i	−0.891202	+	0.453607i	$0.850137\pi$
$228$	0	0
$229$	55.1812	0.240966	0.120483	−	0.992715i	$-0.461556\pi$
$229$	55.1812	0.240966	0.120483	+	0.992715i	$0.461556\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.0717i	0.0646855i	0.999477	+	0.0323427i	$0.0102968\pi$
$233$	15.0717i	0.0646855i	−0.999477	+	0.0323427i	$0.989703\pi$
$234$	0	0
$235$	−626.130	−2.66438
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 106.732i	− 0.446576i	−0.974753	−	0.223288i	$-0.928321\pi$
$239$	− 106.732i	− 0.446576i	0.974753	−	0.223288i	$-0.0716790\pi$
$240$	0	0
$241$	21.9793	0.0912005	0.0456003	−	0.998960i	$-0.485480\pi$
$241$	21.9793	0.0912005	0.0456003	+	0.998960i	$0.485480\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	47.6921i	0.194662i
$246$	0	0
$247$	−9.31561	−0.0377150
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 389.193i	− 1.55057i	−0.631613	−	0.775284i	$-0.717607\pi$
$251$	− 389.193i	− 1.55057i	0.631613	−	0.775284i	$-0.282393\pi$
$252$	0	0
$253$	−19.6546	−0.0776861
$254$	0	0
$255$	0	0
$256$	0	0
$257$	409.681i	1.59409i	0.603920	+	0.797045i	$0.293605\pi$
$257$	409.681i	1.59409i	−0.603920	+	0.797045i	$0.706395\pi$
$258$	0	0
$259$	4.91711	0.0189850
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 288.396i	− 1.09656i	−0.836294	−	0.548281i	$-0.815282\pi$
$263$	− 288.396i	− 1.09656i	0.836294	−	0.548281i	$-0.184718\pi$
$264$	0	0
$265$	206.240	0.778263
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 264.656i	− 0.983852i	−0.870637	−	0.491926i	$-0.836293\pi$
$269$	− 264.656i	− 0.983852i	0.870637	−	0.491926i	$-0.163707\pi$
$270$	0	0
$271$	−437.788	−1.61545	−0.807727	−	0.589556i	$-0.799303\pi$
$271$	−437.788	−1.61545	−0.807727	+	0.589556i	$0.799303\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 34.5363i	− 0.125586i
$276$	0	0
$277$	438.941	1.58462	0.792312	−	0.610116i	$-0.208877\pi$
$277$	438.941	1.58462	0.792312	+	0.610116i	$0.208877\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	62.8497i	0.223664i	0.993727	+	0.111832i	$0.0356719\pi$
$281$	62.8497i	0.223664i	−0.993727	+	0.111832i	$0.964328\pi$
$282$	0	0
$283$	−427.199	−1.50954	−0.754769	−	0.655990i	$-0.772251\pi$
$283$	−427.199	−1.50954	−0.754769	+	0.655990i	$0.772251\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	110.282i	0.384256i
$288$	0	0
$289$	−438.290	−1.51657
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 375.101i	− 1.28021i	−0.768288	−	0.640104i	$-0.778891\pi$
$293$	− 375.101i	− 1.28021i	0.768288	−	0.640104i	$-0.221109\pi$
$294$	0	0
$295$	−491.415	−1.66581
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.34834i	− 0.0212319i
$300$	0	0
$301$	−6.16629	−0.0204860
$302$	0	0
$303$	0	0
$304$	0	0
$305$	218.716i	0.717100i
$306$	0	0
$307$	268.463	0.874473	0.437237	−	0.899347i	$-0.355957\pi$
$307$	268.463	0.874473	0.437237	+	0.899347i	$0.355957\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	156.940i	0.504629i	0.967645	+	0.252314i	$0.0811917\pi$
$311$	156.940i	0.504629i	−0.967645	+	0.252314i	$0.918808\pi$
$312$	0	0
$313$	−39.3100	−0.125591	−0.0627956	−	0.998026i	$-0.520002\pi$
$313$	−39.3100	−0.125591	−0.0627956	+	0.998026i	$0.520002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 491.176i	− 1.54945i	−0.632298	−	0.774725i	$-0.717888\pi$
$317$	− 491.176i	− 1.54945i	0.632298	−	0.774725i	$-0.282112\pi$
$318$	0	0
$319$	77.3931	0.242612
$320$	0	0
$321$	0	0
$322$	0	0
$323$	482.385i	1.49345i
$324$	0	0
$325$	11.1551	0.0343233
$326$	0	0
$327$	0	0
$328$	0	0
$329$	243.145i	0.739042i
$330$	0	0
$331$	−571.594	−1.72687	−0.863435	−	0.504460i	$-0.831692\pi$
$331$	−571.594	−1.72687	−0.863435	+	0.504460i	$0.831692\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	307.138i	0.916829i
$336$	0	0
$337$	176.364	0.523334	0.261667	−	0.965158i	$-0.415728\pi$
$337$	176.364	0.523334	0.261667	+	0.965158i	$0.415728\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	75.5042i	0.221420i
$342$	0	0
$343$	18.5203	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	518.445i	1.49408i	0.664780	+	0.747039i	$0.268525\pi$
$347$	518.445i	1.49408i	−0.664780	+	0.747039i	$0.731475\pi$
$348$	0	0
$349$	−2.55868	−0.00733145	−0.00366573	−	0.999993i	$-0.501167\pi$
$349$	−2.55868	−0.00733145	−0.00366573	+	0.999993i	$0.501167\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 44.1821i	− 0.125162i	−0.998040	−	0.0625809i	$-0.980067\pi$
$353$	− 44.1821i	− 0.125162i	0.998040	−	0.0625809i	$-0.0199332\pi$
$354$	0	0
$355$	759.356	2.13903
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.3091i	0.0677134i	0.999427	+	0.0338567i	$0.0107790\pi$
$359$	24.3091i	0.0677134i	−0.999427	+	0.0338567i	$0.989221\pi$
$360$	0	0
$361$	−41.0514	−0.113716
$362$	0	0
$363$	0	0
$364$	0	0
$365$	680.663i	1.86483i
$366$	0	0
$367$	271.257	0.739121	0.369560	−	0.929207i	$-0.379508\pi$
$367$	271.257	0.739121	0.369560	+	0.929207i	$0.379508\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 80.0890i	− 0.215873i
$372$	0	0
$373$	−479.051	−1.28432	−0.642159	−	0.766571i	$-0.721961\pi$
$373$	−479.051	−1.28432	−0.642159	+	0.766571i	$0.721961\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.9976i	0.0663067i
$378$	0	0
$379$	−286.046	−0.754740	−0.377370	−	0.926063i	$-0.623171\pi$
$379$	−286.046	−0.754740	−0.377370	+	0.926063i	$0.623171\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	453.990i	1.18535i	0.805440	+	0.592677i	$0.201929\pi$
$383$	453.990i	1.18535i	−0.805440	+	0.592677i	$0.798071\pi$
$384$	0	0
$385$	−29.0651	−0.0754938
$386$	0	0
$387$	0	0
$388$	0	0
$389$	556.266i	1.42999i	0.699129	+	0.714995i	$0.253571\pi$
$389$	556.266i	1.42999i	−0.699129	+	0.714995i	$0.746429\pi$
$390$	0	0
$391$	−328.733	−0.840749
$392$	0	0
$393$	0	0
$394$	0	0
$395$	25.7758i	0.0652553i
$396$	0	0
$397$	65.9114	0.166024	0.0830118	−	0.996549i	$-0.473546\pi$
$397$	65.9114	0.166024	0.0830118	+	0.996549i	$0.473546\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 706.777i	− 1.76253i	−0.472618	−	0.881267i	$-0.656691\pi$
$401$	− 706.777i	− 1.76253i	0.472618	−	0.881267i	$-0.343309\pi$
$402$	0	0
$403$	−24.3875	−0.0605149
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.99665i	0.00736277i
$408$	0	0
$409$	541.836	1.32478	0.662391	−	0.749159i	$-0.269542\pi$
$409$	541.836	1.32478	0.662391	+	0.749159i	$0.269542\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	190.831i	0.462061i
$414$	0	0
$415$	−239.835	−0.577915
$416$	0	0
$417$	0	0
$418$	0	0
$419$	529.656i	1.26410i	0.774930	+	0.632048i	$0.217785\pi$
$419$	529.656i	1.26410i	−0.774930	+	0.632048i	$0.782215\pi$
$420$	0	0
$421$	−251.002	−0.596205	−0.298103	−	0.954534i	$-0.596354\pi$
$421$	−251.002	−0.596205	−0.298103	+	0.954534i	$0.596354\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 577.636i	− 1.35914i
$426$	0	0
$427$	84.9338	0.198908
$428$	0	0
$429$	0	0
$430$	0	0
$431$	631.431i	1.46504i	0.680747	+	0.732519i	$0.261655\pi$
$431$	631.431i	1.46504i	−0.680747	+	0.732519i	$0.738345\pi$
$432$	0	0
$433$	−645.417	−1.49057	−0.745285	−	0.666746i	$-0.767687\pi$
$433$	−645.417	−1.49057	−0.745285	+	0.666746i	$0.767687\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	218.037i	0.498939i
$438$	0	0
$439$	−66.3355	−0.151106	−0.0755529	−	0.997142i	$-0.524072\pi$
$439$	−66.3355	−0.151106	−0.0755529	+	0.997142i	$0.524072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	52.3648i	0.118205i	0.998252	+	0.0591025i	$0.0188239\pi$
$443$	52.3648i	0.118205i	−0.998252	+	0.0591025i	$0.981176\pi$
$444$	0	0
$445$	−332.352	−0.746857
$446$	0	0
$447$	0	0
$448$	0	0
$449$	460.715i	1.02609i	0.858361	+	0.513045i	$0.171483\pi$
$449$	460.715i	1.02609i	−0.858361	+	0.513045i	$0.828517\pi$
$450$	0	0
$451$	−67.2092	−0.149023
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 9.38790i	− 0.0206327i
$456$	0	0
$457$	−609.343	−1.33335	−0.666677	−	0.745346i	$-0.732284\pi$
$457$	−609.343	−1.33335	−0.666677	+	0.745346i	$0.732284\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	244.657i	0.530709i	0.964151	+	0.265354i	$0.0854890\pi$
$461$	244.657i	0.530709i	−0.964151	+	0.265354i	$0.914511\pi$
$462$	0	0
$463$	241.775	0.522191	0.261096	−	0.965313i	$-0.415916\pi$
$463$	241.775	0.522191	0.261096	+	0.965313i	$0.415916\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	449.911i	0.963408i	0.876334	+	0.481704i	$0.159982\pi$
$467$	449.911i	0.963408i	−0.876334	+	0.481704i	$0.840018\pi$
$468$	0	0
$469$	119.271	0.254309
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3.75794i	− 0.00794491i
$474$	0	0
$475$	−383.125	−0.806579
$476$	0	0
$477$	0	0
$478$	0	0
$479$	159.741i	0.333489i	0.986000	+	0.166744i	$0.0533255\pi$
$479$	159.741i	0.333489i	−0.986000	+	0.166744i	$0.946675\pi$
$480$	0	0
$481$	−0.967903	−0.00201227
$482$	0	0
$483$	0	0
$484$	0	0
$485$	82.6669i	0.170447i
$486$	0	0
$487$	320.830	0.658788	0.329394	−	0.944193i	$-0.393156\pi$
$487$	320.830	0.658788	0.329394	+	0.944193i	$0.393156\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	786.947i	1.60274i	0.598167	+	0.801372i	$0.295896\pi$
$491$	786.947i	1.60274i	−0.598167	+	0.801372i	$0.704104\pi$
$492$	0	0
$493$	1294.44	2.62563
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 294.881i	− 0.593321i
$498$	0	0
$499$	−560.988	−1.12422	−0.562112	−	0.827061i	$-0.690011\pi$
$499$	−560.988	−1.12422	−0.562112	+	0.827061i	$0.690011\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 63.4623i	− 0.126168i	−0.998008	−	0.0630838i	$-0.979906\pi$
$503$	− 63.4623i	− 0.126168i	0.998008	−	0.0630838i	$-0.0200936\pi$
$504$	0	0
$505$	−80.0333	−0.158482
$506$	0	0
$507$	0	0
$508$	0	0
$509$	114.008i	0.223985i	0.993709	+	0.111992i	$0.0357233\pi$
$509$	114.008i	0.223985i	−0.993709	+	0.111992i	$0.964277\pi$
$510$	0	0
$511$	264.322	0.517264
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1047.45i	− 2.03389i
$516$	0	0
$517$	−148.180	−0.286616
$518$	0	0
$519$	0	0
$520$	0	0
$521$	179.557i	0.344640i	0.985041	+	0.172320i	$0.0551262\pi$
$521$	179.557i	0.344640i	−0.985041	+	0.172320i	$0.944874\pi$
$522$	0	0
$523$	378.737	0.724163	0.362082	−	0.932146i	$-0.382066\pi$
$523$	378.737	0.724163	0.362082	+	0.932146i	$0.382066\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1262.84i	2.39629i
$528$	0	0
$529$	380.414	0.719119
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 21.7083i	− 0.0407285i
$534$	0	0
$535$	370.060	0.691701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.2869i	0.0209404i
$540$	0	0
$541$	−466.858	−0.862954	−0.431477	−	0.902124i	$-0.642007\pi$
$541$	−466.858	−0.862954	−0.431477	+	0.902124i	$0.642007\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1172.08i	− 2.15061i
$546$	0	0
$547$	1068.48	1.95335	0.976675	−	0.214721i	$-0.0688843\pi$
$547$	1068.48	1.95335	0.976675	+	0.214721i	$0.0688843\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 858.554i	− 1.55817i
$552$	0	0
$553$	10.0095	0.0181004
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 374.378i	− 0.672132i	−0.941838	−	0.336066i	$-0.890903\pi$
$557$	− 374.378i	− 0.672132i	0.941838	−	0.336066i	$-0.109097\pi$
$558$	0	0
$559$	1.21380	0.00217137
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 74.2058i	− 0.131804i	−0.997826	−	0.0659021i	$-0.979007\pi$
$563$	− 74.2058i	− 0.131804i	0.997826	−	0.0659021i	$-0.0209925\pi$
$564$	0	0
$565$	441.738	0.781837
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 674.188i	− 1.18486i	−0.805620	−	0.592432i	$-0.798168\pi$
$569$	− 674.188i	− 1.18486i	0.805620	−	0.592432i	$-0.201832\pi$
$570$	0	0
$571$	193.792	0.339390	0.169695	−	0.985497i	$-0.445722\pi$
$571$	193.792	0.339390	0.169695	+	0.985497i	$0.445722\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 261.090i	− 0.454069i
$576$	0	0
$577$	458.303	0.794286	0.397143	−	0.917757i	$-0.370002\pi$
$577$	458.303	0.794286	0.397143	+	0.917757i	$0.370002\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	93.1351i	0.160301i
$582$	0	0
$583$	48.8089	0.0837202
$584$	0	0
$585$	0	0
$586$	0	0
$587$	980.515i	1.67038i	0.549959	+	0.835192i	$0.314643\pi$
$587$	980.515i	1.67038i	−0.549959	+	0.835192i	$0.685357\pi$
$588$	0	0
$589$	837.599	1.42207
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.9429i	0.0656710i	0.999461	+	0.0328355i	$0.0104537\pi$
$593$	38.9429i	0.0656710i	−0.999461	+	0.0328355i	$0.989546\pi$
$594$	0	0
$595$	−486.128	−0.817023
$596$	0	0
$597$	0	0
$598$	0	0
$599$	937.594i	1.56527i	0.622483	+	0.782633i	$0.286124\pi$
$599$	937.594i	1.56527i	−0.622483	+	0.782633i	$0.713876\pi$
$600$	0	0
$601$	−898.370	−1.49479	−0.747396	−	0.664379i	$-0.768696\pi$
$601$	−898.370	−1.49479	−0.747396	+	0.664379i	$0.768696\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	806.678i	1.33335i
$606$	0	0
$607$	−957.285	−1.57708	−0.788538	−	0.614986i	$-0.789162\pi$
$607$	−957.285	−1.57708	−0.788538	+	0.614986i	$0.789162\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 47.8616i	− 0.0783333i
$612$	0	0
$613$	602.677	0.983160	0.491580	−	0.870832i	$-0.336419\pi$
$613$	602.677	0.983160	0.491580	+	0.870832i	$0.336419\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 248.867i	− 0.403350i	−0.979453	−	0.201675i	$-0.935362\pi$
$617$	− 248.867i	− 0.403350i	0.979453	−	0.201675i	$-0.0646384\pi$
$618$	0	0
$619$	−63.7190	−0.102939	−0.0514693	−	0.998675i	$-0.516390\pi$
$619$	−63.7190	−0.102939	−0.0514693	+	0.998675i	$0.516390\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	129.062i	0.207162i
$624$	0	0
$625$	−701.700	−1.12272
$626$	0	0
$627$	0	0
$628$	0	0
$629$	50.1204i	0.0796827i
$630$	0	0
$631$	191.480	0.303455	0.151728	−	0.988422i	$-0.451516\pi$
$631$	191.480	0.303455	0.151728	+	0.988422i	$0.451516\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1261.28i	1.98626i
$636$	0	0
$637$	−3.64560	−0.00572308
$638$	0	0
$639$	0	0
$640$	0	0
$641$	276.684i	0.431644i	0.976433	+	0.215822i	$0.0692431\pi$
$641$	276.684i	0.431644i	−0.976433	+	0.215822i	$0.930757\pi$
$642$	0	0
$643$	−154.293	−0.239958	−0.119979	−	0.992776i	$-0.538283\pi$
$643$	−154.293	−0.239958	−0.119979	+	0.992776i	$0.538283\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 198.767i	− 0.307214i	−0.988132	−	0.153607i	$-0.950911\pi$
$647$	− 198.767i	− 0.307214i	0.988132	−	0.153607i	$-0.0490890\pi$
$648$	0	0
$649$	−116.299	−0.179197
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 703.030i	− 1.07662i	−0.842748	−	0.538308i	$-0.819064\pi$
$653$	− 703.030i	− 1.07662i	0.842748	−	0.538308i	$-0.180936\pi$
$654$	0	0
$655$	−211.629	−0.323098
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 964.242i	− 1.46319i	−0.681740	−	0.731595i	$-0.738776\pi$
$659$	− 964.242i	− 1.46319i	0.681740	−	0.731595i	$-0.261224\pi$
$660$	0	0
$661$	777.062	1.17559	0.587793	−	0.809011i	$-0.299997\pi$
$661$	777.062	1.17559	0.587793	+	0.809011i	$0.299997\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	322.431i	0.484859i
$666$	0	0
$667$	585.082	0.877184
$668$	0	0
$669$	0	0
$670$	0	0
$671$	51.7614i	0.0771407i
$672$	0	0
$673$	−249.025	−0.370022	−0.185011	−	0.982736i	$-0.559232\pi$
$673$	−249.025	−0.370022	−0.185011	+	0.982736i	$0.559232\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	500.574i	0.739401i	0.929151	+	0.369700i	$0.120540\pi$
$677$	500.574i	0.739401i	−0.929151	+	0.369700i	$0.879460\pi$
$678$	0	0
$679$	32.1020	0.0472784
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 549.601i	− 0.804687i	−0.915489	−	0.402344i	$-0.868196\pi$
$683$	− 549.601i	− 0.804687i	0.915489	−	0.402344i	$-0.131804\pi$
$684$	0	0
$685$	−583.588	−0.851954
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.7650i	0.0228811i
$690$	0	0
$691$	−206.279	−0.298522	−0.149261	−	0.988798i	$-0.547690\pi$
$691$	−206.279	−0.298522	−0.149261	+	0.988798i	$0.547690\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.6523i	0.0253990i
$696$	0	0
$697$	−1124.11	−1.61278
$698$	0	0
$699$	0	0
$700$	0	0
$701$	246.956i	0.352291i	0.984364	+	0.176146i	$0.0563630\pi$
$701$	246.956i	0.352291i	−0.984364	+	0.176146i	$0.943637\pi$
$702$	0	0
$703$	33.2430	0.0472874
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31.0793i	0.0439594i
$708$	0	0
$709$	1086.05	1.53180	0.765901	−	0.642958i	$-0.222293\pi$
$709$	1086.05	1.53180	0.765901	+	0.642958i	$0.222293\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	570.802i	0.800564i
$714$	0	0
$715$	5.72129	0.00800181
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 563.186i	− 0.783291i	−0.920116	−	0.391645i	$-0.871906\pi$
$719$	− 563.186i	− 0.783291i	0.920116	−	0.391645i	$-0.128094\pi$
$720$	0	0
$721$	−406.757	−0.564157
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1028.08i	1.41805i
$726$	0	0
$727$	619.737	0.852458	0.426229	−	0.904615i	$-0.359842\pi$
$727$	619.737	0.852458	0.426229	+	0.904615i	$0.359842\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 62.8534i	− 0.0859828i
$732$	0	0
$733$	807.509	1.10165	0.550825	−	0.834621i	$-0.314313\pi$
$733$	807.509	1.10165	0.550825	+	0.834621i	$0.314313\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	72.6875i	0.0986261i
$738$	0	0
$739$	−653.350	−0.884101	−0.442050	−	0.896990i	$-0.645749\pi$
$739$	−653.350	−0.884101	−0.442050	+	0.896990i	$0.645749\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	388.142i	0.522399i	0.965285	+	0.261199i	$0.0841180\pi$
$743$	388.142i	0.522399i	−0.965285	+	0.261199i	$0.915882\pi$
$744$	0	0
$745$	776.296	1.04201
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 143.705i	− 0.191863i
$750$	0	0
$751$	1366.13	1.81909	0.909543	−	0.415610i	$-0.136432\pi$
$751$	1366.13	1.81909	0.909543	+	0.415610i	$0.136432\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	76.7759i	0.101690i
$756$	0	0
$757$	−556.658	−0.735348	−0.367674	−	0.929955i	$-0.619846\pi$
$757$	−556.658	−0.735348	−0.367674	+	0.929955i	$0.619846\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	393.624i	0.517246i	0.965978	+	0.258623i	$0.0832687\pi$
$761$	393.624i	0.517246i	−0.965978	+	0.258623i	$0.916731\pi$
$762$	0	0
$763$	−455.154	−0.596533
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 37.5639i	− 0.0489752i
$768$	0	0
$769$	−1150.26	−1.49578	−0.747892	−	0.663821i	$-0.768934\pi$
$769$	−1150.26	−1.49578	−0.747892	+	0.663821i	$0.768934\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 961.972i	− 1.24447i	−0.782832	−	0.622233i	$-0.786226\pi$
$773$	− 961.972i	− 1.24447i	0.782832	−	0.622233i	$-0.213774\pi$
$774$	0	0
$775$	−1002.99	−1.29418
$776$	0	0
$777$	0	0
$778$	0	0
$779$	745.580i	0.957099i
$780$	0	0
$781$	179.710	0.230102
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 284.479i	− 0.362393i
$786$	0	0
$787$	512.863	0.651669	0.325834	−	0.945427i	$-0.394355\pi$
$787$	512.863	0.651669	0.325834	+	0.945427i	$0.394355\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 171.540i	− 0.216865i
$792$	0	0
$793$	−16.7187	−0.0210829
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 322.873i	− 0.405111i	−0.979271	−	0.202555i	$-0.935075\pi$
$797$	− 322.873i	− 0.405111i	0.979271	−	0.202555i	$-0.0649246\pi$
$798$	0	0
$799$	−2478.39	−3.10187
$800$	0	0
$801$	0	0
$802$	0	0
$803$	161.086i	0.200606i
$804$	0	0
$805$	−219.729	−0.272955
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 243.580i	− 0.301088i	−0.988603	−	0.150544i	$-0.951898\pi$
$809$	− 243.580i	− 0.301088i	0.988603	−	0.150544i	$-0.0481024\pi$
$810$	0	0
$811$	834.792	1.02934	0.514668	−	0.857389i	$-0.327915\pi$
$811$	834.792	1.02934	0.514668	+	0.857389i	$0.327915\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	966.018i	1.18530i
$816$	0	0
$817$	−41.6884	−0.0510262
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 465.749i	− 0.567295i	−0.958929	−	0.283647i	$-0.908456\pi$
$821$	− 465.749i	− 0.567295i	0.958929	−	0.283647i	$-0.0915445\pi$
$822$	0	0
$823$	1471.89	1.78844	0.894221	−	0.447626i	$-0.147730\pi$
$823$	1471.89	1.78844	0.894221	+	0.447626i	$0.147730\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 770.034i	− 0.931118i	−0.885017	−	0.465559i	$-0.845853\pi$
$827$	− 770.034i	− 0.931118i	0.885017	−	0.465559i	$-0.154147\pi$
$828$	0	0
$829$	1130.62	1.36383	0.681915	−	0.731431i	$-0.261147\pi$
$829$	1130.62	1.36383	0.681915	+	0.731431i	$0.261147\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	188.778i	0.226625i
$834$	0	0
$835$	−1136.54	−1.36113
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1258.96i	− 1.50054i	−0.661129	−	0.750272i	$-0.729922\pi$
$839$	− 1258.96i	− 1.50054i	0.661129	−	0.750272i	$-0.270078\pi$
$840$	0	0
$841$	−1462.85	−1.73942
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1149.58i	− 1.36044i
$846$	0	0
$847$	313.257	0.369843
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22.6543i	0.0266208i
$852$	0	0
$853$	−595.267	−0.697851	−0.348925	−	0.937150i	$-0.613453\pi$
$853$	−595.267	−0.697851	−0.348925	+	0.937150i	$0.613453\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1319.66i	− 1.53986i	−0.638130	−	0.769929i	$-0.720292\pi$
$857$	− 1319.66i	− 1.53986i	0.638130	−	0.769929i	$-0.279708\pi$
$858$	0	0
$859$	1519.85	1.76932	0.884662	−	0.466233i	$-0.154389\pi$
$859$	1519.85	1.76932	0.884662	+	0.466233i	$0.154389\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 921.628i	− 1.06793i	−0.845505	−	0.533967i	$-0.820701\pi$
$863$	− 921.628i	− 1.06793i	0.845505	−	0.533967i	$-0.179299\pi$
$864$	0	0
$865$	−68.3403	−0.0790061
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.10014i	0.00701972i
$870$	0	0
$871$	−23.4777	−0.0269549
$872$	0	0
$873$	0	0
$874$	0	0
$875$	64.5495i	0.0737709i
$876$	0	0
$877$	−300.963	−0.343173	−0.171587	−	0.985169i	$-0.554889\pi$
$877$	−300.963	−0.343173	−0.171587	+	0.985169i	$0.554889\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	950.730i	1.07915i	0.841938	+	0.539574i	$0.181415\pi$
$881$	950.730i	1.07915i	−0.841938	+	0.539574i	$0.818585\pi$
$882$	0	0
$883$	1486.55	1.68352	0.841761	−	0.539850i	$-0.181519\pi$
$883$	1486.55	1.68352	0.841761	+	0.539850i	$0.181519\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1500.35i	− 1.69149i	−0.533590	−	0.845743i	$-0.679157\pi$
$887$	− 1500.35i	− 1.69149i	0.533590	−	0.845743i	$-0.320843\pi$
$888$	0	0
$889$	489.792	0.550947
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1643.83i	1.84079i
$894$	0	0
$895$	−677.411	−0.756884
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2247.62i	− 2.50014i
$900$	0	0
$901$	816.353	0.906052
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1480.10i	− 1.63547i
$906$	0	0
$907$	−1353.09	−1.49183	−0.745914	−	0.666043i	$-0.767987\pi$
$907$	−1353.09	−1.49183	−0.745914	+	0.666043i	$0.767987\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	405.043i	0.444613i	0.974977	+	0.222307i	$0.0713586\pi$
$911$	405.043i	0.444613i	−0.974977	+	0.222307i	$0.928641\pi$
$912$	0	0
$913$	−56.7596	−0.0621682
$914$	0	0
$915$	0	0
$916$	0	0
$917$	82.1820i	0.0896205i
$918$	0	0
$919$	1523.01	1.65724	0.828621	−	0.559810i	$-0.189126\pi$
$919$	1523.01	1.65724	0.828621	+	0.559810i	$0.189126\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	58.0455i	0.0628879i
$924$	0	0
$925$	−39.8072	−0.0430348
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1044.12i	− 1.12391i	−0.827166	−	0.561957i	$-0.810049\pi$
$929$	− 1044.12i	− 1.12391i	0.827166	−	0.561957i	$-0.189951\pi$
$930$	0	0
$931$	125.210	0.134490
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 296.263i	− 0.316858i
$936$	0	0
$937$	1563.83	1.66897	0.834487	−	0.551027i	$-0.185764\pi$
$937$	1563.83	1.66897	0.834487	+	0.551027i	$0.185764\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	75.2579i	0.0799765i	0.999200	+	0.0399882i	$0.0127320\pi$
$941$	75.2579i	0.0799765i	−0.999200	+	0.0399882i	$0.987268\pi$
$942$	0	0
$943$	−508.093	−0.538805
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1441.19i	− 1.52185i	−0.648842	−	0.760923i	$-0.724747\pi$
$947$	− 1441.19i	− 1.52185i	0.648842	−	0.760923i	$-0.275253\pi$
$948$	0	0
$949$	−52.0302	−0.0548263
$950$	0	0
$951$	0	0
$952$	0	0
$953$	156.478i	0.164195i	0.996624	+	0.0820974i	$0.0261618\pi$
$953$	156.478i	0.164195i	−0.996624	+	0.0820974i	$0.973838\pi$
$954$	0	0
$955$	−1615.95	−1.69210
$956$	0	0
$957$	0	0
$958$	0	0
$959$	226.625i	0.236314i
$960$	0	0
$961$	1231.77	1.28175
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1594.41i	1.65223i
$966$	0	0
$967$	−704.516	−0.728558	−0.364279	−	0.931290i	$-0.618685\pi$
$967$	−704.516	−0.728558	−0.364279	+	0.931290i	$0.618685\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	984.744i	1.01415i	0.861901	+	0.507077i	$0.169274\pi$
$971$	984.744i	1.01415i	−0.861901	+	0.507077i	$0.830726\pi$
$972$	0	0
$973$	6.85490	0.00704512
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1235.23i	1.26431i	0.774841	+	0.632156i	$0.217830\pi$
$977$	1235.23i	1.26431i	−0.774841	+	0.632156i	$0.782170\pi$
$978$	0	0
$979$	−78.6546	−0.0803418
$980$	0	0
$981$	0	0
$982$	0	0
$983$	873.904i	0.889017i	0.895775	+	0.444509i	$0.146622\pi$
$983$	873.904i	0.889017i	−0.895775	+	0.444509i	$0.853378\pi$
$984$	0	0
$985$	1893.23	1.92206
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 28.4096i	− 0.0287255i
$990$	0	0
$991$	421.680	0.425509	0.212755	−	0.977106i	$-0.431756\pi$
$991$	421.680	0.425509	0.212755	+	0.977106i	$0.431756\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1155.83i	− 1.16163i
$996$	0	0
$997$	−1500.34	−1.50486	−0.752428	−	0.658674i	$-0.771117\pi$
$997$	−1500.34	−1.50486	−0.752428	+	0.658674i	$0.771117\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.3.d.n.449.10		12
3.2	odd	2	inner	4032.3.d.n.449.3		12
4.3	odd	2		4032.3.d.o.449.10		12
8.3	odd	2		2016.3.d.e.449.3	✓	12
8.5	even	2		2016.3.d.f.449.3	yes	12
12.11	even	2		4032.3.d.o.449.3		12
24.5	odd	2		2016.3.d.f.449.10	yes	12
24.11	even	2		2016.3.d.e.449.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.3.d.e.449.3	✓	12	8.3	odd	2
2016.3.d.e.449.10	yes	12	24.11	even	2
2016.3.d.f.449.3	yes	12	8.5	even	2
2016.3.d.f.449.10	yes	12	24.5	odd	2
4032.3.d.n.449.3		12	3.2	odd	2	inner
4032.3.d.n.449.10		12	1.1	even	1	trivial
4032.3.d.o.449.3		12	12.11	even	2
4032.3.d.o.449.10		12	4.3	odd	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4032.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.81315i q^{5} +2.64575 q^{7} +1.61241i q^{11} -0.520800 q^{13} +26.9683i q^{17} +17.8871 q^{19} +12.1896i q^{23} -21.4191 q^{25} -47.9985i q^{29} +46.8270 q^{31} +18.0259i q^{35} +1.85849 q^{37} +41.6825i q^{41} -2.33064 q^{43} +91.9001i q^{47} +7.00000 q^{49} -30.2708i q^{53} -10.9856 q^{55} +72.1273i q^{59} +32.1019 q^{61} -3.54829i q^{65} +45.0801 q^{67} -111.454i q^{71} +99.9042 q^{73} +4.26603i q^{77} +3.78325 q^{79} +35.2017i q^{83} -183.739 q^{85} +48.7809i q^{89} -1.37791 q^{91} +121.868i q^{95} +12.1334 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{13} - 64 q^{19} - 124 q^{25} + 160 q^{31} - 56 q^{37} + 64 q^{43} + 84 q^{49} - 160 q^{55} - 104 q^{61} + 64 q^{67} - 64 q^{73} + 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) \) 12 * q - 16 * q^13 - 64 * q^19 - 124 * q^25 + 160 * q^31 - 56 * q^37 + 64 * q^43 + 84 * q^49 - 160 * q^55 - 104 * q^61 + 64 * q^67 - 64 * q^73 + 32 * q^79 + 184 * q^85 - 96 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more