LMFDB - Embedded newform 1008.5.f.h.433.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,5,Mod(433,1008)] 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("1008.433"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1008.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,76,0,0,0,360] 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:	$104.196922789$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.1308672.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 72x^{2} + 1278 $ x^4 + 72*x^2 + 1278
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.3
Root			$-6.34371i$ of defining polynomial
Character	$\chi$	$=$	1008.433
Dual form			1008.5.f.h.433.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+23.1980i q^{5} +(-6.45584 - 48.5729i) q^{7} +191.823 q^{11} -48.5729i q^{13} +181.977i q^{17} -599.915i q^{19} -469.529 q^{23} +86.8519 q^{25} +338.881 q^{29} -267.556i q^{31} +(1126.79 - 149.763i) q^{35} -668.530 q^{37} -1323.85i q^{41} -1940.23 q^{43} +2936.89i q^{47} +(-2317.64 + 627.158i) q^{49} +1460.94 q^{53} +4449.92i q^{55} -1730.83i q^{59} +246.343i q^{61} +1126.79 q^{65} +1076.59 q^{67} -2276.39 q^{71} -7106.94i q^{73} +(-1238.38 - 9317.41i) q^{77} -7012.38 q^{79} -1448.36i q^{83} -4221.52 q^{85} +2133.73i q^{89} +(-2359.32 + 313.579i) q^{91} +13916.8 q^{95} -5898.76i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 76 q^{7} + 360 q^{11} - 792 q^{23} - 2300 q^{25} - 1224 q^{29} + 4032 q^{35} - 3896 q^{37} - 3688 q^{43} - 1532 q^{49} - 5832 q^{53} + 4032 q^{65} + 1048 q^{67} - 21528 q^{71} - 3528 q^{77} - 12776 q^{79}+ \cdots + 36864 q^{95}+O(q^{100}) $ 4 * q + 76 * q^7 + 360 * q^11 - 792 * q^23 - 2300 * q^25 - 1224 * q^29 + 4032 * q^35 - 3896 * q^37 - 3688 * q^43 - 1532 * q^49 - 5832 * q^53 + 4032 * q^65 + 1048 * q^67 - 21528 * q^71 - 3528 * q^77 - 12776 * q^79 + 16512 * q^85 - 5568 * q^91 + 36864 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$			23.1980i			0.927921i	0.885856	+	0.463960i	$0.153572\pi$
$5$			23.1980i			0.927921i	−0.885856	+	0.463960i	$0.846428\pi$
$6$		0			0
$7$	−6.45584	−	48.5729i	−0.131752	−	0.991283i
$7$	−6.45584	−	48.5729i	−0.131752	−	0.991283i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	191.823			1.58532			0.792659	−	0.609666i	$-0.208696\pi$
$11$	191.823			1.58532			0.792659	+	0.609666i	$0.208696\pi$
$12$		0			0
$13$		−	48.5729i		−	0.287413i	−0.989620	−	0.143707i	$-0.954098\pi$
$13$		−	48.5729i		−	0.287413i	0.989620	−	0.143707i	$-0.0459022\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			181.977i			0.629680i	0.949145	+	0.314840i	$0.101951\pi$
$17$			181.977i			0.629680i	−0.949145	+	0.314840i	$0.898049\pi$
$18$		0			0
$19$		−	599.915i		−	1.66182i	−0.556410	−	0.830908i	$-0.687822\pi$
$19$		−	599.915i		−	1.66182i	0.556410	−	0.830908i	$-0.312178\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−469.529			−0.887578			−0.443789	−	0.896131i	$-0.646366\pi$
$23$	−469.529			−0.887578			−0.443789	+	0.896131i	$0.646366\pi$
$24$		0			0
$25$	86.8519			0.138963
$26$		0			0
$27$		0			0
$28$		0			0
$29$	338.881			0.402951			0.201475	−	0.979494i	$-0.435426\pi$
$29$	338.881			0.402951			0.201475	+	0.979494i	$0.435426\pi$
$30$		0			0
$31$		−	267.556i		−	0.278414i	−0.990263	−	0.139207i	$-0.955545\pi$
$31$		−	267.556i		−	0.278414i	0.990263	−	0.139207i	$-0.0444554\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1126.79	−	149.763i	0.919832	−	0.122255i
$36$		0			0
$37$	−668.530			−0.488334			−0.244167	−	0.969733i	$-0.578515\pi$
$37$	−668.530			−0.488334			−0.244167	+	0.969733i	$0.578515\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1323.85i		−	0.787534i	−0.919210	−	0.393767i	$-0.871172\pi$
$41$		−	1323.85i		−	0.787534i	0.919210	−	0.393767i	$-0.128828\pi$
$42$		0			0
$43$	−1940.23			−1.04934			−0.524671	−	0.851305i	$-0.675812\pi$
$43$	−1940.23			−1.04934			−0.524671	+	0.851305i	$0.675812\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2936.89i			1.32951i	0.747062	+	0.664755i	$0.231464\pi$
$47$			2936.89i			1.32951i	−0.747062	+	0.664755i	$0.768536\pi$
$48$		0			0
$49$	−2317.64	+	627.158i	−0.965283	+	0.261207i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1460.94			0.520091			0.260046	−	0.965596i	$-0.416262\pi$
$53$	1460.94			0.520091			0.260046	+	0.965596i	$0.416262\pi$
$54$		0			0
$55$			4449.92i			1.47105i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1730.83i		−	0.497223i	−0.968603	−	0.248612i	$-0.920026\pi$
$59$		−	1730.83i		−	0.497223i	0.968603	−	0.248612i	$-0.0799743\pi$
$60$		0			0
$61$			246.343i			0.0662034i	0.999452	+	0.0331017i	$0.0105385\pi$
$61$			246.343i			0.0662034i	−0.999452	+	0.0331017i	$0.989461\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1126.79			0.266697
$66$		0			0
$67$	1076.59			0.239828			0.119914	−	0.992784i	$-0.461738\pi$
$67$	1076.59			0.239828			0.119914	+	0.992784i	$0.461738\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2276.39			−0.451574			−0.225787	−	0.974177i	$-0.572495\pi$
$71$	−2276.39			−0.451574			−0.225787	+	0.974177i	$0.572495\pi$
$72$		0			0
$73$		−	7106.94i		−	1.33363i	−0.745221	−	0.666817i	$-0.767656\pi$
$73$		−	7106.94i		−	1.33363i	0.745221	−	0.666817i	$-0.232344\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1238.38	−	9317.41i	−0.208869	−	1.57150i
$78$		0			0
$79$	−7012.38			−1.12360			−0.561799	−	0.827274i	$-0.689891\pi$
$79$	−7012.38			−1.12360			−0.561799	+	0.827274i	$0.689891\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1448.36i		−	0.210243i	−0.994459	−	0.105121i	$-0.966477\pi$
$83$		−	1448.36i		−	0.210243i	0.994459	−	0.105121i	$-0.0335231\pi$
$84$		0			0
$85$	−4221.52			−0.584293
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2133.73i			0.269376i	0.990888	+	0.134688i	$0.0430032\pi$
$89$			2133.73i			0.269376i	−0.990888	+	0.134688i	$0.956997\pi$
$90$		0			0
$91$	−2359.32	+	313.579i	−0.284908	+	0.0378673i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13916.8			1.54203
$96$		0			0
$97$		−	5898.76i		−	0.626928i	−0.949600	−	0.313464i	$-0.898511\pi$
$97$		−	5898.76i		−	0.626928i	0.949600	−	0.313464i	$-0.101489\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	9172.07i		−	0.899135i	−0.893246	−	0.449567i	$-0.851578\pi$
$101$		−	9172.07i		−	0.899135i	0.893246	−	0.449567i	$-0.148422\pi$
$102$		0			0
$103$		−	3906.46i		−	0.368222i	−0.982905	−	0.184111i	$-0.941059\pi$
$103$		−	3906.46i		−	0.368222i	0.982905	−	0.184111i	$-0.0589406\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12141.3			−1.06047			−0.530233	−	0.847852i	$-0.677896\pi$
$107$	−12141.3			−1.06047			−0.530233	+	0.847852i	$0.677896\pi$
$108$		0			0
$109$	6808.34			0.573044			0.286522	−	0.958074i	$-0.407501\pi$
$109$	6808.34			0.573044			0.286522	+	0.958074i	$0.407501\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4764.20			0.373107			0.186553	−	0.982445i	$-0.440268\pi$
$113$	4764.20			0.373107			0.186553	+	0.982445i	$0.440268\pi$
$114$		0			0
$115$		−	10892.1i		−	0.823602i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8839.17	−	1174.82i	0.624191	−	0.0829615i
$120$		0			0
$121$	22155.2			1.51323
$122$		0			0
$123$		0			0
$124$		0			0
$125$			16513.6i			1.05687i
$126$		0			0
$127$	27968.9			1.73408			0.867038	−	0.498242i	$-0.166021\pi$
$127$	27968.9			1.73408			0.867038	+	0.498242i	$0.166021\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	24016.5i		−	1.39948i	−0.714397	−	0.699741i	$-0.753299\pi$
$131$		−	24016.5i		−	1.39948i	0.714397	−	0.699741i	$-0.246701\pi$
$132$		0			0
$133$	−29139.6	+	3872.96i	−1.64733	+	0.218947i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4162.00			0.221748			0.110874	−	0.993834i	$-0.464635\pi$
$137$	4162.00			0.221748			0.110874	+	0.993834i	$0.464635\pi$
$138$		0			0
$139$		−	26365.8i		−	1.36462i	−0.731064	−	0.682309i	$-0.760976\pi$
$139$		−	26365.8i		−	1.36462i	0.731064	−	0.682309i	$-0.239024\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	9317.41i		−	0.455641i
$144$		0			0
$145$			7861.38i			0.373906i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6576.57			0.296229			0.148114	−	0.988970i	$-0.452680\pi$
$149$	6576.57			0.296229			0.148114	+	0.988970i	$0.452680\pi$
$150$		0			0
$151$	22930.4			1.00568			0.502839	−	0.864380i	$-0.332289\pi$
$151$	22930.4			1.00568			0.502839	+	0.864380i	$0.332289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6206.77			0.258346
$156$		0			0
$157$			37292.9i			1.51296i	0.654017	+	0.756480i	$0.273082\pi$
$157$			37292.9i			1.51296i	−0.654017	+	0.756480i	$0.726918\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3031.21	+	22806.4i	0.116940	+	0.879841i
$162$		0			0
$163$	−40854.0			−1.53766			−0.768828	−	0.639455i	$-0.779160\pi$
$163$	−40854.0			−1.53766			−0.768828	+	0.639455i	$0.779160\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	34774.9i		−	1.24690i	−0.781862	−	0.623452i	$-0.785730\pi$
$167$		−	34774.9i		−	1.24690i	0.781862	−	0.623452i	$-0.214270\pi$
$168$		0			0
$169$	26201.7			0.917394
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	31600.1i		−	1.05583i	−0.849296	−	0.527917i	$-0.822973\pi$
$173$		−	31600.1i		−	1.05583i	0.849296	−	0.527917i	$-0.177027\pi$
$174$		0			0
$175$	−560.703	−	4218.65i	−0.0183087	−	0.137752i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22750.7			0.710048			0.355024	−	0.934857i	$-0.384473\pi$
$179$	22750.7			0.710048			0.355024	+	0.934857i	$0.384473\pi$
$180$		0			0
$181$		−	55434.4i		−	1.69208i	−0.533116	−	0.846042i	$-0.678979\pi$
$181$		−	55434.4i		−	1.69208i	0.533116	−	0.846042i	$-0.321021\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	15508.6i		−	0.453136i
$186$		0			0
$187$			34907.5i			0.998242i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−50817.6			−1.39299			−0.696494	−	0.717562i	$-0.745258\pi$
$191$	−50817.6			−1.39299			−0.696494	+	0.717562i	$0.745258\pi$
$192$		0			0
$193$	−1248.34			−0.0335134			−0.0167567	−	0.999860i	$-0.505334\pi$
$193$	−1248.34			−0.0335134			−0.0167567	+	0.999860i	$0.505334\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−64454.6			−1.66082			−0.830408	−	0.557155i	$-0.811893\pi$
$197$	−64454.6			−1.66082			−0.830408	+	0.557155i	$0.811893\pi$
$198$		0			0
$199$			2352.60i			0.0594076i	0.999559	+	0.0297038i	$0.00945641\pi$
$199$			2352.60i			0.0594076i	−0.999559	+	0.0297038i	$0.990544\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2187.77	−	16460.4i	−0.0530895	−	0.399438i
$204$		0			0
$205$	30710.6			0.730769
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	115078.i		−	2.63450i
$210$		0			0
$211$	65056.4			1.46125			0.730626	−	0.682778i	$-0.239228\pi$
$211$	65056.4			1.46125			0.730626	+	0.682778i	$0.239228\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	45009.6i		−	0.973706i
$216$		0			0
$217$	−12996.0	+	1727.30i	−0.275987	+	0.0366816i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8839.17			0.180978
$222$		0			0
$223$		−	30412.4i		−	0.611563i	−0.952102	−	0.305781i	$-0.901082\pi$
$223$		−	30412.4i		−	0.611563i	0.952102	−	0.305781i	$-0.0989177\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			52125.5i			1.01158i	0.862658	+	0.505788i	$0.168798\pi$
$227$			52125.5i			1.01158i	−0.862658	+	0.505788i	$0.831202\pi$
$228$		0			0
$229$		−	81280.2i		−	1.54994i	−0.632000	−	0.774968i	$-0.717766\pi$
$229$		−	81280.2i		−	1.54994i	0.632000	−	0.774968i	$-0.282234\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	41718.9			0.768459			0.384229	−	0.923238i	$-0.374467\pi$
$233$	41718.9			0.768459			0.384229	+	0.923238i	$0.374467\pi$
$234$		0			0
$235$	−68129.9			−1.23368
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3936.55			−0.0689160			−0.0344580	−	0.999406i	$-0.510970\pi$
$239$	−3936.55			−0.0689160			−0.0344580	+	0.999406i	$0.510970\pi$
$240$		0			0
$241$		−	70511.5i		−	1.21402i	−0.794694	−	0.607010i	$-0.792369\pi$
$241$		−	70511.5i		−	1.21402i	0.794694	−	0.607010i	$-0.207631\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14548.8	−	53764.8i	−0.242379	−	0.895706i
$246$		0			0
$247$	−29139.6			−0.477628
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	72042.3i		−	1.14351i	−0.820424	−	0.571755i	$-0.806263\pi$
$251$		−	72042.3i		−	1.14351i	0.820424	−	0.571755i	$-0.193737\pi$
$252$		0			0
$253$	−90066.6			−1.40709
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	65615.5i		−	0.993436i	−0.867912	−	0.496718i	$-0.834538\pi$
$257$		−	65615.5i		−	0.993436i	0.867912	−	0.496718i	$-0.165462\pi$
$258$		0			0
$259$	4315.92	+	32472.4i	0.0643390	+	0.484078i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−38706.1			−0.559588			−0.279794	−	0.960060i	$-0.590266\pi$
$263$	−38706.1			−0.559588			−0.279794	+	0.960060i	$0.590266\pi$
$264$		0			0
$265$			33890.8i			0.482604i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			87226.9i			1.20544i	0.797952	+	0.602721i	$0.205917\pi$
$269$			87226.9i			1.20544i	−0.797952	+	0.602721i	$0.794083\pi$
$270$		0			0
$271$		−	105362.i		−	1.43465i	−0.696739	−	0.717324i	$-0.745367\pi$
$271$		−	105362.i		−	1.43465i	0.696739	−	0.717324i	$-0.254633\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16660.2			0.220301
$276$		0			0
$277$	36178.9			0.471515			0.235758	−	0.971812i	$-0.424243\pi$
$277$	36178.9			0.471515			0.235758	+	0.971812i	$0.424243\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−99142.2			−1.25558			−0.627792	−	0.778381i	$-0.716041\pi$
$281$	−99142.2			−1.25558			−0.627792	+	0.778381i	$0.716041\pi$
$282$		0			0
$283$			4153.27i			0.0518581i	0.999664	+	0.0259291i	$0.00825441\pi$
$283$			4153.27i			0.0518581i	−0.999664	+	0.0259291i	$0.991746\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−64302.9	+	8546.54i	−0.780669	+	0.103759i
$288$		0			0
$289$	50405.2			0.603503
$290$		0			0
$291$		0			0
$292$		0			0
$293$			20239.4i			0.235756i	0.993028	+	0.117878i	$0.0376092\pi$
$293$			20239.4i			0.235756i	−0.993028	+	0.117878i	$0.962391\pi$
$294$		0			0
$295$	40151.9			0.461384
$296$		0			0
$297$		0			0
$298$		0			0
$299$			22806.4i			0.255102i
$300$		0			0
$301$	12525.8	+	94242.7i	0.138253	+	1.04019i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5714.66			−0.0614315
$306$		0			0
$307$		−	63269.8i		−	0.671305i	−0.941986	−	0.335652i	$-0.891043\pi$
$307$		−	63269.8i		−	0.671305i	0.941986	−	0.335652i	$-0.108957\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14375.4i		−	0.148627i	−0.997235	−	0.0743137i	$-0.976323\pi$
$311$		−	14375.4i		−	0.148627i	0.997235	−	0.0743137i	$-0.0236766\pi$
$312$		0			0
$313$			36763.0i			0.375252i	0.982241	+	0.187626i	$0.0600792\pi$
$313$			36763.0i			0.375252i	−0.982241	+	0.187626i	$0.939921\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	125556.			1.24945			0.624726	−	0.780844i	$-0.285211\pi$
$317$	125556.			1.24945			0.624726	+	0.780844i	$0.285211\pi$
$318$		0			0
$319$	65005.4			0.638804
$320$		0			0
$321$		0			0
$322$		0			0
$323$	109171.			1.04641
$324$		0			0
$325$		−	4218.65i		−	0.0399399i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	142653.	−	18960.1i	1.31792	−	0.175165i
$330$		0			0
$331$	−5376.54			−0.0490735			−0.0245367	−	0.999699i	$-0.507811\pi$
$331$	−5376.54			−0.0490735			−0.0245367	+	0.999699i	$0.507811\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			24974.7i			0.222541i
$336$		0			0
$337$	2202.27			0.0193914			0.00969572	−	0.999953i	$-0.496914\pi$
$337$	2202.27			0.0193914			0.00969572	+	0.999953i	$0.496914\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	51323.5i		−	0.441375i
$342$		0			0
$343$	45425.2	+	108526.i	0.386108	+	0.922454i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	222201.			1.84538			0.922691	−	0.385541i	$-0.125985\pi$
$347$	222201.			1.84538			0.922691	+	0.385541i	$0.125985\pi$
$348$		0			0
$349$		−	102679.i		−	0.843006i	−0.906827	−	0.421503i	$-0.861503\pi$
$349$		−	102679.i		−	0.843006i	0.906827	−	0.421503i	$-0.138497\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	62595.6i		−	0.502336i	−0.967943	−	0.251168i	$-0.919185\pi$
$353$		−	62595.6i		−	0.502336i	0.967943	−	0.251168i	$-0.0808147\pi$
$354$		0			0
$355$		−	52807.7i		−	0.419025i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	95505.9			0.741040			0.370520	−	0.928825i	$-0.379180\pi$
$359$	95505.9			0.741040			0.370520	+	0.928825i	$0.379180\pi$
$360$		0			0
$361$	−229577.			−1.76163
$362$		0			0
$363$		0			0
$364$		0			0
$365$	164867.			1.23751
$366$		0			0
$367$			82330.9i			0.611267i	0.952149	+	0.305633i	$0.0988682\pi$
$367$			82330.9i			0.611267i	−0.952149	+	0.305633i	$0.901132\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9431.58	−	70961.9i	−0.0685230	−	0.515558i
$372$		0			0
$373$	130223.			0.935991			0.467995	−	0.883731i	$-0.344976\pi$
$373$	130223.			0.935991			0.467995	+	0.883731i	$0.344976\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	16460.4i		−	0.115813i
$378$		0			0
$379$	−192349.			−1.33909			−0.669546	−	0.742770i	$-0.733511\pi$
$379$	−192349.			−1.33909			−0.669546	+	0.742770i	$0.733511\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	101933.i		−	0.694891i	−0.937700	−	0.347446i	$-0.887049\pi$
$383$		−	101933.i		−	0.694891i	0.937700	−	0.347446i	$-0.112951\pi$
$384$		0			0
$385$	216145.	−	28728.0i	1.45823	−	0.193813i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−191074.			−1.26271			−0.631353	−	0.775495i	$-0.717500\pi$
$389$	−191074.			−1.26271			−0.631353	+	0.775495i	$0.717500\pi$
$390$		0			0
$391$		−	85443.7i		−	0.558890i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	162673.i		−	1.04261i
$396$		0			0
$397$			201143.i			1.27622i	0.769947	+	0.638108i	$0.220283\pi$
$397$			201143.i			1.27622i	−0.769947	+	0.638108i	$0.779717\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	39978.4			0.248621			0.124310	−	0.992243i	$-0.460328\pi$
$401$	39978.4			0.248621			0.124310	+	0.992243i	$0.460328\pi$
$402$		0			0
$403$	−12996.0			−0.0800200
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−128240.			−0.774165
$408$		0			0
$409$		−	80655.7i		−	0.482157i	−0.970506	−	0.241078i	$-0.922499\pi$
$409$		−	80655.7i		−	0.482157i	0.970506	−	0.241078i	$-0.0775011\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−84071.6	+	11174.0i	−0.492889	+	0.0655101i
$414$		0			0
$415$	33599.1			0.195088
$416$		0			0
$417$		0			0
$418$		0			0
$419$			252034.i			1.43559i	0.696254	+	0.717795i	$0.254849\pi$
$419$			252034.i			1.43559i	−0.696254	+	0.717795i	$0.745151\pi$
$420$		0			0
$421$	−84439.3			−0.476410			−0.238205	−	0.971215i	$-0.576559\pi$
$421$	−84439.3			−0.476410			−0.238205	+	0.971215i	$0.576559\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			15805.1i			0.0875023i
$426$		0			0
$427$	11965.6	−	1590.35i	0.0656263	−	0.00872242i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−127512.			−0.686431			−0.343215	−	0.939257i	$-0.611516\pi$
$431$	−127512.			−0.686431			−0.343215	+	0.939257i	$0.611516\pi$
$432$		0			0
$433$		−	233539.i		−	1.24562i	−0.782375	−	0.622808i	$-0.785992\pi$
$433$		−	233539.i		−	1.24562i	0.782375	−	0.622808i	$-0.214008\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			281678.i			1.47499i
$438$		0			0
$439$			304238.i			1.57864i	0.613980	+	0.789322i	$0.289568\pi$
$439$			304238.i			1.57864i	−0.613980	+	0.789322i	$0.710432\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−87061.0			−0.443625			−0.221813	−	0.975089i	$-0.571197\pi$
$443$	−87061.0			−0.443625			−0.221813	+	0.975089i	$0.571197\pi$
$444$		0			0
$445$	−49498.2			−0.249960
$446$		0			0
$447$		0			0
$448$		0			0
$449$	91141.4			0.452088			0.226044	−	0.974117i	$-0.427421\pi$
$449$	91141.4			0.452088			0.226044	+	0.974117i	$0.427421\pi$
$450$		0			0
$451$		−	253944.i		−	1.24849i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7274.41	−	54731.6i	−0.0351378	−	0.264372i
$456$		0			0
$457$	−411928.			−1.97237			−0.986187	−	0.165634i	$-0.947033\pi$
$457$	−411928.			−1.97237			−0.986187	+	0.165634i	$0.947033\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			157397.i			0.740617i	0.928909	+	0.370309i	$0.120748\pi$
$461$			157397.i			0.740617i	−0.928909	+	0.370309i	$0.879252\pi$
$462$		0			0
$463$	−245557.			−1.14549			−0.572745	−	0.819734i	$-0.694121\pi$
$463$	−245557.			−1.14549			−0.572745	+	0.819734i	$0.694121\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	33069.2i		−	0.151632i	−0.997122	−	0.0758158i	$-0.975844\pi$
$467$		−	33069.2i		−	0.151632i	0.997122	−	0.0758158i	$-0.0241561\pi$
$468$		0			0
$469$	−6950.28	−	52292.9i	−0.0315978	−	0.237737i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−372182.			−1.66354
$474$		0			0
$475$		−	52103.8i		−	0.230931i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			384277.i			1.67484i	0.546559	+	0.837421i	$0.315937\pi$
$479$			384277.i			1.67484i	−0.546559	+	0.837421i	$0.684063\pi$
$480$		0			0
$481$			32472.4i			0.140354i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	136840.			0.581739
$486$		0			0
$487$	173526.			0.731657			0.365829	−	0.930682i	$-0.380786\pi$
$487$	173526.			0.731657			0.365829	+	0.930682i	$0.380786\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20006.3			−0.0829858			−0.0414929	−	0.999139i	$-0.513211\pi$
$491$	−20006.3			−0.0829858			−0.0414929	+	0.999139i	$0.513211\pi$
$492$		0			0
$493$			61668.8i			0.253730i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14696.0	+	110571.i	0.0594958	+	0.447638i
$498$		0			0
$499$	428368.			1.72035			0.860174	−	0.510000i	$-0.170355\pi$
$499$	428368.			1.72035			0.860174	+	0.510000i	$0.170355\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			36793.4i			0.145423i	0.997353	+	0.0727116i	$0.0231653\pi$
$503$			36793.4i			0.145423i	−0.997353	+	0.0727116i	$0.976835\pi$
$504$		0			0
$505$	212774.			0.834326
$506$		0			0
$507$		0			0
$508$		0			0
$509$			137334.i			0.530080i	0.964237	+	0.265040i	$0.0853852\pi$
$509$			137334.i			0.530080i	−0.964237	+	0.265040i	$0.914615\pi$
$510$		0			0
$511$	−345204.	+	45881.3i	−1.32201	+	0.175709i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	90622.2			0.341681
$516$		0			0
$517$			563363.i			2.10769i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	102775.i		−	0.378629i	−0.981916	−	0.189315i	$-0.939373\pi$
$521$		−	102775.i		−	0.378629i	0.981916	−	0.189315i	$-0.0606266\pi$
$522$		0			0
$523$			314194.i			1.14867i	0.818621	+	0.574334i	$0.194739\pi$
$523$			314194.i			1.14867i	−0.818621	+	0.574334i	$0.805261\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	48689.2			0.175312
$528$		0			0
$529$	−59383.5			−0.212204
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−64302.9			−0.226348
$534$		0			0
$535$		−	281654.i		−	0.984029i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−444578.	+	120303.i	−1.53028	+	0.414096i
$540$		0			0
$541$	298384.			1.01948			0.509742	−	0.860327i	$-0.329741\pi$
$541$	298384.			1.01948			0.509742	+	0.860327i	$0.329741\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			157940.i			0.531740i
$546$		0			0
$547$	−361462.			−1.20806			−0.604029	−	0.796963i	$-0.706439\pi$
$547$	−361462.			−1.20806			−0.604029	+	0.796963i	$0.706439\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	203300.i		−	0.669629i
$552$		0			0
$553$	45270.8	+	340611.i	0.148036	+	1.11380i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−112424.			−0.362367			−0.181183	−	0.983449i	$-0.557993\pi$
$557$	−112424.			−0.362367			−0.181183	+	0.983449i	$0.557993\pi$
$558$		0			0
$559$			94242.7i			0.301595i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			441530.i			1.39298i	0.717569	+	0.696488i	$0.245255\pi$
$563$			441530.i			1.39298i	−0.717569	+	0.696488i	$0.754745\pi$
$564$		0			0
$565$			110520.i			0.346214i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−397273.			−1.22706			−0.613529	−	0.789673i	$-0.710250\pi$
$569$	−397273.			−1.22706			−0.613529	+	0.789673i	$0.710250\pi$
$570$		0			0
$571$	67235.8			0.206219			0.103109	−	0.994670i	$-0.467121\pi$
$571$	67235.8			0.206219			0.103109	+	0.994670i	$0.467121\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−40779.5			−0.123341
$576$		0			0
$577$		−	64756.1i		−	0.194504i	−0.995260	−	0.0972521i	$-0.968995\pi$
$577$		−	64756.1i		−	0.194504i	0.995260	−	0.0972521i	$-0.0310053\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−70351.0	+	9350.39i	−0.208410	+	0.0276999i
$582$		0			0
$583$	280242.			0.824510
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	338967.i		−	0.983741i	−0.870668	−	0.491870i	$-0.836313\pi$
$587$		−	338967.i		−	0.983741i	0.870668	−	0.491870i	$-0.163687\pi$
$588$		0			0
$589$	−160511.			−0.462673
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	255668.i		−	0.727054i	−0.931584	−	0.363527i	$-0.881572\pi$
$593$		−	255668.i		−	0.727054i	0.931584	−	0.363527i	$-0.118428\pi$
$594$		0			0
$595$	27253.5	+	205051.i	0.0769817	+	0.579200i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	129521.			0.360982			0.180491	−	0.983577i	$-0.442231\pi$
$599$	129521.			0.360982			0.180491	+	0.983577i	$0.442231\pi$
$600$		0			0
$601$		−	377277.i		−	1.04451i	−0.852790	−	0.522254i	$-0.825091\pi$
$601$		−	377277.i		−	1.04451i	0.852790	−	0.522254i	$-0.174909\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			513957.i			1.40416i
$606$		0			0
$607$		−	421091.i		−	1.14287i	−0.820646	−	0.571437i	$-0.806386\pi$
$607$		−	421091.i		−	1.14287i	0.820646	−	0.571437i	$-0.193614\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	142653.			0.382119
$612$		0			0
$613$	412173.			1.09688			0.548440	−	0.836190i	$-0.315222\pi$
$613$	412173.			1.09688			0.548440	+	0.836190i	$0.315222\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	262676.			0.690002			0.345001	−	0.938602i	$-0.387879\pi$
$617$	262676.			0.690002			0.345001	+	0.938602i	$0.387879\pi$
$618$		0			0
$619$			373975.i			0.976026i	0.872836	+	0.488013i	$0.162278\pi$
$619$			373975.i			0.976026i	−0.872836	+	0.488013i	$0.837722\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	103641.	−	13775.0i	0.267028	−	0.0354908i
$624$		0			0
$625$	−328799.			−0.841726
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	121657.i		−	0.307494i
$630$		0			0
$631$	408746.			1.02659			0.513293	−	0.858214i	$-0.328426\pi$
$631$	408746.			1.02659			0.513293	+	0.858214i	$0.328426\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			648824.i			1.60909i
$636$		0			0
$637$	30462.8	+	112575.i	0.0750743	+	0.277435i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	528074.			1.28522			0.642612	−	0.766192i	$-0.277851\pi$
$641$	528074.			1.28522			0.642612	+	0.766192i	$0.277851\pi$
$642$		0			0
$643$			323445.i			0.782310i	0.920325	+	0.391155i	$0.127924\pi$
$643$			323445.i			0.782310i	−0.920325	+	0.391155i	$0.872076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			592372.i			1.41510i	0.706665	+	0.707548i	$0.250199\pi$
$647$			592372.i			1.41510i	−0.706665	+	0.707548i	$0.749801\pi$
$648$		0			0
$649$		−	332015.i		−	0.788257i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	329810.			0.773459			0.386730	−	0.922193i	$-0.373605\pi$
$653$	329810.			0.773459			0.386730	+	0.922193i	$0.373605\pi$
$654$		0			0
$655$	557135.			1.29861
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−526737.			−1.21290			−0.606448	−	0.795123i	$-0.707406\pi$
$659$	−526737.			−1.21290			−0.606448	+	0.795123i	$0.707406\pi$
$660$		0			0
$661$		−	144047.i		−	0.329686i	−0.986320	−	0.164843i	$-0.947288\pi$
$661$		−	144047.i		−	0.329686i	0.986320	−	0.164843i	$-0.0527117\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−89845.0	−	675981.i	−0.203166	−	1.52859i
$666$		0			0
$667$	−159115.			−0.357650
$668$		0			0
$669$		0			0
$670$		0			0
$671$			47254.3i			0.104953i
$672$		0			0
$673$	620117.			1.36913			0.684563	−	0.728953i	$-0.259993\pi$
$673$	620117.			1.36913			0.684563	+	0.728953i	$0.259993\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			626172.i			1.36621i	0.730322	+	0.683103i	$0.239370\pi$
$677$			626172.i			1.36621i	−0.730322	+	0.683103i	$0.760630\pi$
$678$		0			0
$679$	−286520.	+	38081.5i	−0.621463	+	0.0825989i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	185558.			0.397775			0.198888	−	0.980022i	$-0.436267\pi$
$683$	185558.			0.397775			0.198888	+	0.980022i	$0.436267\pi$
$684$		0			0
$685$			96550.1i			0.205765i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	70961.9i		−	0.149481i
$690$		0			0
$691$			548881.i			1.14953i	0.818317	+	0.574767i	$0.194907\pi$
$691$			548881.i			1.14953i	−0.818317	+	0.574767i	$0.805093\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	611634.			1.26626
$696$		0			0
$697$	240910.			0.495894
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−517501.			−1.05311			−0.526556	−	0.850140i	$-0.676517\pi$
$701$	−517501.			−1.05311			−0.526556	+	0.850140i	$0.676517\pi$
$702$		0			0
$703$			401061.i			0.811522i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−445514.	+	59213.5i	−0.891297	+	0.118463i
$708$		0			0
$709$	6035.96			0.0120075			0.00600377	−	0.999982i	$-0.498089\pi$
$709$	6035.96			0.0120075			0.00600377	+	0.999982i	$0.498089\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			125625.i			0.247115i
$714$		0			0
$715$	216145.			0.422799
$716$		0			0
$717$		0			0
$718$		0			0
$719$			748658.i			1.44819i	0.689700	+	0.724095i	$0.257742\pi$
$719$			748658.i			1.44819i	−0.689700	+	0.724095i	$0.742258\pi$
$720$		0			0
$721$	−189748.	+	25219.5i	−0.365012	+	0.0485139i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	29432.5			0.0559953
$726$		0			0
$727$			370335.i			0.700691i	0.936621	+	0.350345i	$0.113936\pi$
$727$			370335.i			0.700691i	−0.936621	+	0.350345i	$0.886064\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	353079.i		−	0.660750i
$732$		0			0
$733$			144545.i			0.269027i	0.990912	+	0.134513i	$0.0429471\pi$
$733$			144545.i			0.269027i	−0.990912	+	0.134513i	$0.957053\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	206515.			0.380203
$738$		0			0
$739$	−4650.26			−0.00851506			−0.00425753	−	0.999991i	$-0.501355\pi$
$739$	−4650.26			−0.00851506			−0.00425753	+	0.999991i	$0.501355\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	500204.			0.906087			0.453043	−	0.891489i	$-0.350338\pi$
$743$	500204.			0.906087			0.453043	+	0.891489i	$0.350338\pi$
$744$		0			0
$745$			152563.i			0.274877i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	78382.2	+	589737.i	0.139718	+	1.05122i
$750$		0			0
$751$	−944576.			−1.67478			−0.837389	−	0.546608i	$-0.815919\pi$
$751$	−944576.			−1.67478			−0.837389	+	0.546608i	$0.815919\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			531941.i			0.933189i
$756$		0			0
$757$	−654028.			−1.14131			−0.570656	−	0.821189i	$-0.693311\pi$
$757$	−654028.			−1.14131			−0.570656	+	0.821189i	$0.693311\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	643673.i		−	1.11147i	−0.831361	−	0.555733i	$-0.812438\pi$
$761$		−	643673.i		−	1.11147i	0.831361	−	0.555733i	$-0.187562\pi$
$762$		0			0
$763$	−43953.6	−	330700.i	−0.0754997	−	0.568049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−84071.6			−0.142909
$768$		0			0
$769$		−	103697.i		−	0.175354i	−0.996149	−	0.0876768i	$-0.972056\pi$
$769$		−	103697.i		−	0.175354i	0.996149	−	0.0876768i	$-0.0279443\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			796380.i			1.33279i	0.745599	+	0.666394i	$0.232163\pi$
$773$			796380.i			1.33279i	−0.745599	+	0.666394i	$0.767837\pi$
$774$		0			0
$775$		−	23237.8i		−	0.0386893i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−794195.			−1.30874
$780$		0			0
$781$	−436664.			−0.715889
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−865122.			−1.40391
$786$		0			0
$787$		−	1.16895e6i		−	1.88733i	−0.330905	−	0.943664i	$-0.607354\pi$
$787$		−	1.16895e6i		−	1.88733i	0.330905	−	0.943664i	$-0.392646\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−30756.9	−	231411.i	−0.0491575	−	0.369854i
$792$		0			0
$793$	11965.6			0.0190277
$794$		0			0
$795$		0			0
$796$		0			0
$797$			808048.i			1.27210i	0.771649	+	0.636049i	$0.219432\pi$
$797$			808048.i			1.27210i	−0.771649	+	0.636049i	$0.780568\pi$
$798$		0			0
$799$	−534447.			−0.837165
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.36328e6i		−	2.11423i
$804$		0			0
$805$	−529062.	+	70318.0i	−0.816423	+	0.108511i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	692708.			1.05841			0.529204	−	0.848495i	$-0.322491\pi$
$809$	692708.			1.05841			0.529204	+	0.848495i	$0.322491\pi$
$810$		0			0
$811$		−	382267.i		−	0.581200i	−0.956845	−	0.290600i	$-0.906145\pi$
$811$		−	382267.i		−	0.581200i	0.956845	−	0.290600i	$-0.0938548\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	947732.i		−	1.42682i
$816$		0			0
$817$			1.16398e6i			1.74381i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−116392.			−0.172678			−0.0863392	−	0.996266i	$-0.527517\pi$
$821$	−116392.			−0.172678			−0.0863392	+	0.996266i	$0.527517\pi$
$822$		0			0
$823$	−640526.			−0.945665			−0.472832	−	0.881152i	$-0.656768\pi$
$823$	−640526.			−0.945665			−0.472832	+	0.881152i	$0.656768\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−158299.			−0.231455			−0.115727	−	0.993281i	$-0.536920\pi$
$827$	−158299.			−0.231455			−0.115727	+	0.993281i	$0.536920\pi$
$828$		0			0
$829$		−	680501.i		−	0.990192i	−0.868838	−	0.495096i	$-0.835133\pi$
$829$		−	680501.i		−	0.990192i	0.868838	−	0.495096i	$-0.164867\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−114129.	−	421759.i	−0.164477	−	0.607819i
$834$		0			0
$835$	806709.			1.15703
$836$		0			0
$837$		0			0
$838$		0			0
$839$			622397.i			0.884186i	0.896969	+	0.442093i	$0.145764\pi$
$839$			622397.i			0.884186i	−0.896969	+	0.442093i	$0.854236\pi$
$840$		0			0
$841$	−592440.			−0.837631
$842$		0			0
$843$		0			0
$844$		0			0
$845$			607827.i			0.851269i
$846$		0			0
$847$	−143031.	−	1.07614e6i	−0.199371	−	1.50004i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	313894.			0.433435
$852$		0			0
$853$			826596.i			1.13604i	0.823013	+	0.568022i	$0.192291\pi$
$853$			826596.i			1.13604i	−0.823013	+	0.568022i	$0.807709\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	539076.i		−	0.733987i	−0.930223	−	0.366994i	$-0.880387\pi$
$857$		−	539076.i		−	0.733987i	0.930223	−	0.366994i	$-0.119613\pi$
$858$		0			0
$859$		−	499964.i		−	0.677567i	−0.940864	−	0.338783i	$-0.889985\pi$
$859$		−	499964.i		−	0.677567i	0.940864	−	0.338783i	$-0.110015\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.16944e6			1.57021			0.785103	−	0.619365i	$-0.212610\pi$
$863$	1.16944e6			1.57021			0.785103	+	0.619365i	$0.212610\pi$
$864$		0			0
$865$	733059.			0.979730
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.34514e6			−1.78126
$870$		0			0
$871$		−	52292.9i		−	0.0689297i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	802110.	−	106609.i	1.04765	−	0.139244i
$876$		0			0
$877$	187496.			0.243777			0.121888	−	0.992544i	$-0.461105\pi$
$877$	187496.			0.243777			0.121888	+	0.992544i	$0.461105\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			179673.i			0.231489i	0.993279	+	0.115745i	$0.0369254\pi$
$881$			179673.i			0.231489i	−0.993279	+	0.115745i	$0.963075\pi$
$882$		0			0
$883$	1.04658e6			1.34231			0.671155	−	0.741317i	$-0.265799\pi$
$883$	1.04658e6			1.34231			0.671155	+	0.741317i	$0.265799\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.30029e6i		−	1.65270i	−0.563158	−	0.826349i	$-0.690414\pi$
$887$		−	1.30029e6i		−	1.65270i	0.563158	−	0.826349i	$-0.309586\pi$
$888$		0			0
$889$	−180563.	−	1.35853e6i	−0.228468	−	1.71896i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.76188e6			2.20940
$894$		0			0
$895$			527770.i			0.658868i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	90669.8i		−	0.112187i
$900$		0			0
$901$			265858.i			0.327491i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.28597e6			1.57012
$906$		0			0
$907$	−1.18388e6			−1.43911			−0.719556	−	0.694435i	$-0.755654\pi$
$907$	−1.18388e6			−1.43911			−0.719556	+	0.694435i	$0.755654\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.30731e6			−1.57522			−0.787612	−	0.616172i	$-0.788683\pi$
$911$	−1.30731e6			−1.57522			−0.787612	+	0.616172i	$0.788683\pi$
$912$		0			0
$913$		−	277830.i		−	0.333301i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.16655e6	+	155047.i	−1.38728	+	0.184384i
$918$		0			0
$919$	−698775.			−0.827382			−0.413691	−	0.910417i	$-0.635761\pi$
$919$	−698775.			−0.827382			−0.413691	+	0.910417i	$0.635761\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			110571.i			0.129789i
$924$		0			0
$925$	−58063.1			−0.0678605
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1.39460e6i		−	1.61591i	−0.589244	−	0.807955i	$-0.700574\pi$
$929$		−	1.39460e6i		−	1.61591i	0.589244	−	0.807955i	$-0.299426\pi$
$930$		0			0
$931$	376241.	+	1.39039e6i	0.434077	+	1.60412i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−809786.			−0.926290
$936$		0			0
$937$			509380.i			0.580180i	0.956999	+	0.290090i	$0.0936852\pi$
$937$			509380.i			0.580180i	−0.956999	+	0.290090i	$0.906315\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.62505e6i		−	1.83522i	−0.397486	−	0.917608i	$-0.630117\pi$
$941$		−	1.62505e6i		−	1.83522i	0.397486	−	0.917608i	$-0.369883\pi$
$942$		0			0
$943$			621584.i			0.698998i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.22462e6			1.36553			0.682763	−	0.730640i	$-0.260778\pi$
$947$	1.22462e6			1.36553			0.682763	+	0.730640i	$0.260778\pi$
$948$		0			0
$949$	−345204.			−0.383304
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.14847e6			1.26454			0.632269	−	0.774749i	$-0.282124\pi$
$953$	1.14847e6			1.26454			0.632269	+	0.774749i	$0.282124\pi$
$954$		0			0
$955$		−	1.17887e6i		−	1.29258i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−26869.2	−	202160.i	−0.0292158	−	0.219815i
$960$		0			0
$961$	851935.			0.922485
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	28959.0i		−	0.0310978i
$966$		0			0
$967$	733668.			0.784597			0.392298	−	0.919838i	$-0.371680\pi$
$967$	733668.			0.784597			0.392298	+	0.919838i	$0.371680\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			786749.i			0.834445i	0.908804	+	0.417223i	$0.136996\pi$
$971$			786749.i			0.834445i	−0.908804	+	0.417223i	$0.863004\pi$
$972$		0			0
$973$	−1.28066e6	+	170213.i	−1.35272	+	0.179791i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−837039.			−0.876913			−0.438457	−	0.898752i	$-0.644475\pi$
$977$	−837039.			−0.876913			−0.438457	+	0.898752i	$0.644475\pi$
$978$		0			0
$979$			409299.i			0.427046i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			644129.i			0.666600i	0.942821	+	0.333300i	$0.108162\pi$
$983$			644129.i			0.666600i	−0.942821	+	0.333300i	$0.891838\pi$
$984$		0			0
$985$		−	1.49522e6i		−	1.54111i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	910996.			0.931374
$990$		0			0
$991$	−92517.5			−0.0942056			−0.0471028	−	0.998890i	$-0.514999\pi$
$991$	−92517.5			−0.0942056			−0.0471028	+	0.998890i	$0.514999\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−54575.7			−0.0551256
$996$		0			0
$997$			366383.i			0.368591i	0.982871	+	0.184295i	$0.0590003\pi$
$997$			366383.i			0.368591i	−0.982871	+	0.184295i	$0.941000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.5.f.h.433.3		4
3.2	odd	2		112.5.c.c.97.4		4
4.3	odd	2		126.5.c.a.55.4		4
7.6	odd	2	inner	1008.5.f.h.433.2		4
12.11	even	2		14.5.b.a.13.1	✓	4
21.20	even	2		112.5.c.c.97.1		4
24.5	odd	2		448.5.c.f.321.1		4
24.11	even	2		448.5.c.e.321.4		4
28.27	even	2		126.5.c.a.55.3		4
60.23	odd	4		350.5.d.a.349.8		8
60.47	odd	4		350.5.d.a.349.1		8
60.59	even	2		350.5.b.a.251.4		4
84.11	even	6		98.5.d.d.19.3		8
84.23	even	6		98.5.d.d.31.4		8
84.47	odd	6		98.5.d.d.31.3		8
84.59	odd	6		98.5.d.d.19.4		8
84.83	odd	2		14.5.b.a.13.2	yes	4
168.83	odd	2		448.5.c.e.321.1		4
168.125	even	2		448.5.c.f.321.4		4
420.83	even	4		350.5.d.a.349.5		8
420.167	even	4		350.5.d.a.349.4		8
420.419	odd	2		350.5.b.a.251.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.5.b.a.13.1	✓	4	12.11	even	2
14.5.b.a.13.2	yes	4	84.83	odd	2
98.5.d.d.19.3		8	84.11	even	6
98.5.d.d.19.4		8	84.59	odd	6
98.5.d.d.31.3		8	84.47	odd	6
98.5.d.d.31.4		8	84.23	even	6
112.5.c.c.97.1		4	21.20	even	2
112.5.c.c.97.4		4	3.2	odd	2
126.5.c.a.55.3		4	28.27	even	2
126.5.c.a.55.4		4	4.3	odd	2
350.5.b.a.251.3		4	420.419	odd	2
350.5.b.a.251.4		4	60.59	even	2
350.5.d.a.349.1		8	60.47	odd	4
350.5.d.a.349.4		8	420.167	even	4
350.5.d.a.349.5		8	420.83	even	4
350.5.d.a.349.8		8	60.23	odd	4
448.5.c.e.321.1		4	168.83	odd	2
448.5.c.e.321.4		4	24.11	even	2
448.5.c.f.321.1		4	24.5	odd	2
448.5.c.f.321.4		4	168.125	even	2
1008.5.f.h.433.2		4	7.6	odd	2	inner
1008.5.f.h.433.3		4	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+23.1980i q^{5} +(-6.45584 - 48.5729i) q^{7} +191.823 q^{11} -48.5729i q^{13} +181.977i q^{17} -599.915i q^{19} -469.529 q^{23} +86.8519 q^{25} +338.881 q^{29} -267.556i q^{31} +(1126.79 - 149.763i) q^{35} -668.530 q^{37} -1323.85i q^{41} -1940.23 q^{43} +2936.89i q^{47} +(-2317.64 + 627.158i) q^{49} +1460.94 q^{53} +4449.92i q^{55} -1730.83i q^{59} +246.343i q^{61} +1126.79 q^{65} +1076.59 q^{67} -2276.39 q^{71} -7106.94i q^{73} +(-1238.38 - 9317.41i) q^{77} -7012.38 q^{79} -1448.36i q^{83} -4221.52 q^{85} +2133.73i q^{89} +(-2359.32 + 313.579i) q^{91} +13916.8 q^{95} -5898.76i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 76 q^{7} + 360 q^{11} - 792 q^{23} - 2300 q^{25} - 1224 q^{29} + 4032 q^{35} - 3896 q^{37} - 3688 q^{43} - 1532 q^{49} - 5832 q^{53} + 4032 q^{65} + 1048 q^{67} - 21528 q^{71} - 3528 q^{77} - 12776 q^{79}+ \cdots + 36864 q^{95}+O(q^{100}) \) 4 * q + 76 * q^7 + 360 * q^11 - 792 * q^23 - 2300 * q^25 - 1224 * q^29 + 4032 * q^35 - 3896 * q^37 - 3688 * q^43 - 1532 * q^49 - 5832 * q^53 + 4032 * q^65 + 1048 * q^67 - 21528 * q^71 - 3528 * q^77 - 12776 * q^79 + 16512 * q^85 - 5568 * q^91 + 36864 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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