LMFDB - Embedded newform 504.6.s.d.289.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,6,Mod(289,504)] 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("504.289"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$80.8334451857$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 119 x^{8} - 521 x^{7} - 898 x^{6} + 27806 x^{5} + 657990 x^{4} + 3648839 x^{3} + \cdots + 92895579 $ x^10 - 3x^9 - 119x^8 - 521x^7 - 898x^6 + 27806x^5 + 657990x^4 + 3648839x^3 + 1863790x^2 - 25332057*x + 92895579
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.5
Root			$-5.12305 + 3.65205i$ of defining polynomial
Character	$\chi$	$=$	504.289
Dual form			504.6.s.d.361.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(40.3290 - 69.8518i) q^{5} +(-8.39664 - 129.370i) q^{7} +(-253.362 - 438.837i) q^{11} +391.283 q^{13} +(-449.706 - 778.913i) q^{17} +(-168.157 + 291.256i) q^{19} +(2436.06 - 4219.37i) q^{23} +(-1690.35 - 2927.78i) q^{25} -546.591 q^{29} +(-3049.51 - 5281.91i) q^{31} +(-9375.33 - 4630.82i) q^{35} +(-7404.84 + 12825.6i) q^{37} +9837.31 q^{41} +18661.0 q^{43} +(699.243 - 1211.12i) q^{47} +(-16666.0 + 2172.54i) q^{49} +(-9880.41 - 17113.4i) q^{53} -40871.4 q^{55} +(13585.4 + 23530.6i) q^{59} +(-7083.72 + 12269.4i) q^{61} +(15780.1 - 27331.9i) q^{65} +(25223.0 + 43687.6i) q^{67} +45279.2 q^{71} +(3783.41 + 6553.06i) q^{73} +(-54644.7 + 36462.2i) q^{77} +(-2414.96 + 4182.83i) q^{79} +14067.3 q^{83} -72544.7 q^{85} +(-44776.9 + 77555.9i) q^{89} +(-3285.47 - 50620.2i) q^{91} +(13563.2 + 23492.1i) q^{95} +7209.53 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 31 q^{5} - 92 q^{7} - 351 q^{11} - 108 q^{13} + 111 q^{17} - 1035 q^{19} + 3639 q^{23} - 1540 q^{25} + 1468 q^{29} - 7677 q^{31} - 19899 q^{35} - 13595 q^{37} - 10620 q^{41} + 1528 q^{43} + 6675 q^{47}+ \cdots + 270940 q^{97}+O(q^{100}) $ 10 * q + 31 * q^5 - 92 * q^7 - 351 * q^11 - 108 * q^13 + 111 * q^17 - 1035 * q^19 + 3639 * q^23 - 1540 * q^25 + 1468 * q^29 - 7677 * q^31 - 19899 * q^35 - 13595 * q^37 - 10620 * q^41 + 1528 * q^43 + 6675 * q^47 + 30122 * q^49 - 30753 * q^53 + 56534 * q^55 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 + 217440 * q^71 - 141659 * q^73 - 271471 * q^77 - 118919 * q^79 - 422008 * q^83 - 286758 * q^85 - 55861 * q^89 + 403912 * q^91 - 26279 * q^95 + 270940 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$5$	40.3290	−	69.8518i	0.721427	−	1.24955i	−0.239001	−	0.971019i	$-0.576820\pi$
$5$	40.3290	−	69.8518i	0.721427	−	1.24955i	0.960428	−	0.278528i	$-0.0898467\pi$
$6$		0			0
$7$	−8.39664	−	129.370i	−0.0647680	−	0.997900i
$7$	−8.39664	−	129.370i	−0.0647680	−	0.997900i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−253.362	−	438.837i	−0.631336	−	1.09351i	−0.987279	−	0.158998i	$-0.949174\pi$
$11$	−253.362	−	438.837i	−0.631336	−	1.09351i	0.355943	−	0.934508i	$-0.384160\pi$
$12$		0			0
$13$	391.283			0.642145			0.321072	−	0.947055i	$-0.395957\pi$
$13$	391.283			0.642145			0.321072	+	0.947055i	$0.395957\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−449.706	−	778.913i	−0.377404	−	0.653682i	0.613280	−	0.789866i	$-0.289850\pi$
$17$	−449.706	−	778.913i	−0.377404	−	0.653682i	−0.990684	+	0.136183i	$0.956516\pi$
$18$		0			0
$19$	−168.157	+	291.256i	−0.106864	+	0.185093i	−0.914498	−	0.404590i	$-0.867414\pi$
$19$	−168.157	+	291.256i	−0.106864	+	0.185093i	0.807634	+	0.589684i	$0.200748\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2436.06	−	4219.37i	0.960213	−	1.66314i	0.238252	−	0.971203i	$-0.423425\pi$
$23$	2436.06	−	4219.37i	0.960213	−	1.66314i	0.721960	−	0.691934i	$-0.243241\pi$
$24$		0			0
$25$	−1690.35	−	2927.78i	−0.540913	−	0.936889i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−546.591			−0.120689			−0.0603444	−	0.998178i	$-0.519220\pi$
$29$	−546.591			−0.120689			−0.0603444	+	0.998178i	$0.519220\pi$
$30$		0			0
$31$	−3049.51	−	5281.91i	−0.569936	−	0.987158i	−0.996572	−	0.0827346i	$-0.973635\pi$
$31$	−3049.51	−	5281.91i	−0.569936	−	0.987158i	0.426636	−	0.904424i	$-0.359699\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9375.33	−	4630.82i	−1.29365	−	0.638981i
$36$		0			0
$37$	−7404.84	+	12825.6i	−0.889224	+	1.54018i	−0.0484303	+	0.998827i	$0.515422\pi$
$37$	−7404.84	+	12825.6i	−0.889224	+	1.54018i	−0.840794	+	0.541355i	$0.817911\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	9837.31			0.913938			0.456969	−	0.889483i	$-0.348935\pi$
$41$	9837.31			0.913938			0.456969	+	0.889483i	$0.348935\pi$
$42$		0			0
$43$	18661.0			1.53909			0.769546	−	0.638591i	$-0.220482\pi$
$43$	18661.0			1.53909			0.769546	+	0.638591i	$0.220482\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	699.243	−	1211.12i	0.0461725	−	0.0799731i	−0.842016	−	0.539453i	$-0.818631\pi$
$47$	699.243	−	1211.12i	0.0461725	−	0.0799731i	0.888188	+	0.459480i	$0.151964\pi$
$48$		0			0
$49$	−16666.0	+	2172.54i	−0.991610	+	0.129264i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9880.41	−	17113.4i	−0.483154	−	0.836847i	0.516659	−	0.856191i	$-0.327175\pi$
$53$	−9880.41	−	17113.4i	−0.483154	−	0.836847i	−0.999813	+	0.0193444i	$0.993842\pi$
$54$		0			0
$55$	−40871.4			−1.82185
$56$		0			0
$57$		0			0
$58$		0			0
$59$	13585.4	+	23530.6i	0.508092	+	0.880041i	0.999956	+	0.00936932i	$0.00298239\pi$
$59$	13585.4	+	23530.6i	0.508092	+	0.880041i	−0.491864	+	0.870672i	$0.663684\pi$
$60$		0			0
$61$	−7083.72	+	12269.4i	−0.243746	+	0.422180i	−0.961778	−	0.273830i	$-0.911710\pi$
$61$	−7083.72	+	12269.4i	−0.243746	+	0.422180i	0.718033	+	0.696009i	$0.245043\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	15780.1	−	27331.9i	0.463260	−	0.802391i
$66$		0			0
$67$	25223.0	+	43687.6i	0.686453	+	1.18897i	0.972978	+	0.230898i	$0.0741664\pi$
$67$	25223.0	+	43687.6i	0.686453	+	1.18897i	−0.286525	+	0.958073i	$0.592500\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	45279.2			1.06599			0.532995	−	0.846118i	$-0.321066\pi$
$71$	45279.2			1.06599			0.532995	+	0.846118i	$0.321066\pi$
$72$		0			0
$73$	3783.41	+	6553.06i	0.0830953	+	0.143925i	0.904578	−	0.426308i	$-0.140186\pi$
$73$	3783.41	+	6553.06i	0.0830953	+	0.143925i	−0.821483	+	0.570233i	$0.806853\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−54644.7	+	36462.2i	−1.05032	+	0.700834i
$78$		0			0
$79$	−2414.96	+	4182.83i	−0.0435353	+	0.0754054i	−0.886972	−	0.461823i	$-0.847195\pi$
$79$	−2414.96	+	4182.83i	−0.0435353	+	0.0754054i	0.843437	+	0.537229i	$0.180529\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14067.3			0.224138			0.112069	−	0.993700i	$-0.464252\pi$
$83$	14067.3			0.224138			0.112069	+	0.993700i	$0.464252\pi$
$84$		0			0
$85$	−72544.7			−1.08908
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−44776.9	+	77555.9i	−0.599210	+	1.03786i	0.393727	+	0.919227i	$0.371185\pi$
$89$	−44776.9	+	77555.9i	−0.599210	+	1.03786i	−0.992938	+	0.118636i	$0.962148\pi$
$90$		0			0
$91$	−3285.47	−	50620.2i	−0.0415904	−	0.640797i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	13563.2	+	23492.1i	0.154189	+	0.267063i
$96$		0			0
$97$	7209.53			0.0777997			0.0388998	−	0.999243i	$-0.487615\pi$
$97$	7209.53			0.0777997			0.0388998	+	0.999243i	$0.487615\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−97609.2	−	169064.i	−0.952110	−	1.64910i	−0.740847	−	0.671674i	$-0.765576\pi$
$101$	−97609.2	−	169064.i	−0.952110	−	1.64910i	−0.211263	−	0.977429i	$-0.567758\pi$
$102$		0			0
$103$	11816.8	−	20467.2i	0.109750	−	0.190093i	−0.805919	−	0.592026i	$-0.798328\pi$
$103$	11816.8	−	20467.2i	0.109750	−	0.190093i	0.915669	+	0.401933i	$0.131662\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−53546.1	+	92744.5i	−0.452135	+	0.783121i	−0.998518	−	0.0544144i	$-0.982671\pi$
$107$	−53546.1	+	92744.5i	−0.452135	+	0.783121i	0.546383	+	0.837535i	$0.316004\pi$
$108$		0			0
$109$	−23110.7	−	40029.0i	−0.186315	−	0.322707i	0.757704	−	0.652598i	$-0.226321\pi$
$109$	−23110.7	−	40029.0i	−0.186315	−	0.322707i	−0.944019	+	0.329892i	$0.892988\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−55520.3			−0.409030			−0.204515	−	0.978863i	$-0.565562\pi$
$113$	−55520.3			−0.409030			−0.204515	+	0.978863i	$0.565562\pi$
$114$		0			0
$115$	−196487.	−	340326.i	−1.38545	−	2.39966i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−96991.7	+	64718.5i	−0.627866	+	0.418949i
$120$		0			0
$121$	−47859.5	+	82895.1i	−0.297170	+	0.514713i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−20624.8			−0.118063
$126$		0			0
$127$	−230185.			−1.26639			−0.633195	−	0.773992i	$-0.718257\pi$
$127$	−230185.			−1.26639			−0.633195	+	0.773992i	$0.718257\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	99500.5	−	172340.i	0.506579	−	0.877420i	−0.493392	−	0.869807i	$-0.664243\pi$
$131$	99500.5	−	172340.i	0.506579	−	0.877420i	0.999971	−	0.00761308i	$-0.00242334\pi$
$132$		0			0
$133$	39091.6	+	19308.8i	0.191626	+	0.0946512i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	63765.5	+	110445.i	0.290258	+	0.502741i	0.973871	−	0.227104i	$-0.0729256\pi$
$137$	63765.5	+	110445.i	0.290258	+	0.502741i	−0.683613	+	0.729845i	$0.739592\pi$
$138$		0			0
$139$	17259.8			0.0757703			0.0378851	−	0.999282i	$-0.487938\pi$
$139$	17259.8			0.0757703			0.0378851	+	0.999282i	$0.487938\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−99136.5	−	171709.i	−0.405409	−	0.702189i
$144$		0			0
$145$	−22043.4	+	38180.4i	−0.0870682	+	0.150806i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−417.263	+	722.720i	−0.00153973	+	0.00266689i	−0.866794	−	0.498666i	$-0.833823\pi$
$149$	−417.263	+	722.720i	−0.00153973	+	0.00266689i	0.865255	+	0.501333i	$0.167157\pi$
$150$		0			0
$151$	−20469.3	−	35453.9i	−0.0730569	−	0.126538i	0.827183	−	0.561933i	$-0.189942\pi$
$151$	−20469.3	−	35453.9i	−0.0730569	−	0.126538i	−0.900240	+	0.435395i	$0.856609\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−491935.			−1.64467
$156$		0			0
$157$	−16444.1	−	28482.0i	−0.0532428	−	0.0922193i	0.838176	−	0.545400i	$-0.183622\pi$
$157$	−16444.1	−	28482.0i	−0.0532428	−	0.0922193i	−0.891418	+	0.453181i	$0.850289\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−566313.	−	279723.i	−1.72184	−	0.850479i
$162$		0			0
$163$	35149.3	−	60880.4i	0.103621	−	0.179477i	−0.809553	−	0.587047i	$-0.800290\pi$
$163$	35149.3	−	60880.4i	0.103621	−	0.179477i	0.913174	+	0.407570i	$0.133624\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−227937.			−0.632447			−0.316224	−	0.948685i	$-0.602415\pi$
$167$	−227937.			−0.632447			−0.316224	+	0.948685i	$0.602415\pi$
$168$		0			0
$169$	−218190.			−0.587650
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16895.1	+	29263.2i	−0.0429186	+	0.0743372i	−0.886687	−	0.462371i	$-0.846999\pi$
$173$	−16895.1	+	29263.2i	−0.0429186	+	0.0743372i	0.843768	+	0.536708i	$0.180332\pi$
$174$		0			0
$175$	−364572.	+	243264.i	−0.899888	+	0.600458i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	307606.	+	532789.i	0.717567	+	1.24286i	0.961961	+	0.273186i	$0.0880775\pi$
$179$	307606.	+	532789.i	0.717567	+	1.24286i	−0.244394	+	0.969676i	$0.578589\pi$
$180$		0			0
$181$	−678787.			−1.54006			−0.770029	−	0.638008i	$-0.779758\pi$
$181$	−678787.			−1.54006			−0.770029	+	0.638008i	$0.779758\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	597259.	+	1.03448e6i	1.28302	+	2.22226i
$186$		0			0
$187$	−227877.	+	394695.i	−0.476537	+	0.825386i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	327588.	−	567398.i	0.649747	−	1.12539i	−0.333436	−	0.942773i	$-0.608208\pi$
$191$	327588.	−	567398.i	0.649747	−	1.12539i	0.983183	−	0.182622i	$-0.0584584\pi$
$192$		0			0
$193$	160230.	+	277526.i	0.309635	+	0.536303i	0.978282	−	0.207276i	$-0.0664598\pi$
$193$	160230.	+	277526.i	0.309635	+	0.536303i	−0.668648	+	0.743579i	$0.733126\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−202103.			−0.371029			−0.185514	−	0.982642i	$-0.559395\pi$
$197$	−202103.			−0.371029			−0.185514	+	0.982642i	$0.559395\pi$
$198$		0			0
$199$	−319170.	−	552819.i	−0.571333	−	0.989578i	−0.996429	−	0.0844301i	$-0.973093\pi$
$199$	−319170.	−	552819.i	−0.571333	−	0.989578i	0.425096	−	0.905148i	$-0.360240\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4589.53	+	70712.2i	0.00781678	+	0.120435i
$204$		0			0
$205$	396729.	−	687154.i	0.659339	−	1.14201i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	170418.			0.269868
$210$		0			0
$211$	−517370.			−0.800010			−0.400005	−	0.916513i	$-0.630992\pi$
$211$	−517370.			−0.800010			−0.400005	+	0.916513i	$0.630992\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	752581.	−	1.30351e6i	1.11034	−	1.92317i
$216$		0			0
$217$	−657713.	+	438865.i	−0.948172	+	0.632676i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−175962.	−	304776.i	−0.242348	−	0.419759i
$222$		0			0
$223$	755836.			1.01781			0.508904	−	0.860824i	$-0.330051\pi$
$223$	755836.			1.01781			0.508904	+	0.860824i	$0.330051\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−440624.	−	763184.i	−0.567550	−	0.983025i	−0.996807	−	0.0798426i	$-0.974558\pi$
$227$	−440624.	−	763184.i	−0.567550	−	0.983025i	0.429258	−	0.903182i	$-0.358775\pi$
$228$		0			0
$229$	139756.	−	242065.i	0.176109	−	0.305031i	−0.764435	−	0.644701i	$-0.776982\pi$
$229$	139756.	−	242065.i	0.176109	−	0.305031i	0.940545	+	0.339670i	$0.110315\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−533391.	+	923860.i	−0.643659	+	1.11485i	0.340950	+	0.940081i	$0.389251\pi$
$233$	−533391.	+	923860.i	−0.643659	+	1.11485i	−0.984609	+	0.174769i	$0.944082\pi$
$234$		0			0
$235$	−56399.5	−	97686.8i	−0.0666201	−	0.115389i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	270341.			0.306138			0.153069	−	0.988215i	$-0.451084\pi$
$239$	270341.			0.306138			0.153069	+	0.988215i	$0.451084\pi$
$240$		0			0
$241$	657034.	+	1.13802e6i	0.728694	+	1.26214i	0.957435	+	0.288648i	$0.0932058\pi$
$241$	657034.	+	1.13802e6i	0.728694	+	1.26214i	−0.228741	+	0.973487i	$0.573461\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−520367.	+	1.25177e6i	−0.553853	+	1.33232i
$246$		0			0
$247$	−65796.9	+	113964.i	−0.0686220	+	0.118857i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.31291e6			1.31538			0.657690	−	0.753289i	$-0.271534\pi$
$251$	1.31291e6			1.31538			0.657690	+	0.753289i	$0.271534\pi$
$252$		0			0
$253$	−2.46882e6			−2.42487
$254$		0			0
$255$		0			0
$256$		0			0
$257$	353720.	−	612661.i	0.334062	−	0.578612i	−0.649242	−	0.760582i	$-0.724914\pi$
$257$	353720.	−	612661.i	0.334062	−	0.578612i	0.983304	+	0.181970i	$0.0582472\pi$
$258$		0			0
$259$	1.72141e6	+	850270.i	1.59454	+	0.787603i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−369721.	−	640376.i	−0.329598	−	0.570881i	0.652834	−	0.757501i	$-0.273580\pi$
$263$	−369721.	−	640376.i	−0.329598	−	0.570881i	−0.982432	+	0.186620i	$0.940247\pi$
$264$		0			0
$265$	−1.59387e6			−1.39424
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−619595.	−	1.07317e6i	−0.522068	−	0.904249i	−0.999670	−	0.0256724i	$-0.991827\pi$
$269$	−619595.	−	1.07317e6i	−0.522068	−	0.904249i	0.477602	−	0.878576i	$-0.341506\pi$
$270$		0			0
$271$	257391.	−	445815.i	0.212898	−	0.368749i	−0.739723	−	0.672912i	$-0.765043\pi$
$271$	257391.	−	445815.i	0.212898	−	0.368749i	0.952620	+	0.304163i	$0.0983766\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−856544.	+	1.48358e6i	−0.682996	+	1.18298i
$276$		0			0
$277$	648520.	+	1.12327e6i	0.507837	+	0.879599i	0.999959	+	0.00907261i	$0.00288794\pi$
$277$	648520.	+	1.12327e6i	0.507837	+	0.879599i	−0.492122	+	0.870526i	$0.663779\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.61884e6			−1.97853			−0.989265	−	0.146133i	$-0.953317\pi$
$281$	−2.61884e6			−1.97853			−0.989265	+	0.146133i	$0.953317\pi$
$282$		0			0
$283$	−20922.9	−	36239.4i	−0.0155294	−	0.0268977i	0.858156	−	0.513389i	$-0.171610\pi$
$283$	−20922.9	−	36239.4i	−0.0155294	−	0.0268977i	−0.873686	+	0.486491i	$0.838277\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−82600.4	−	1.27265e6i	−0.0591940	−	0.912019i
$288$		0			0
$289$	305458.	−	529069.i	0.215133	−	0.372621i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.94837e6			1.32588			0.662938	−	0.748674i	$-0.269309\pi$
$293$	1.94837e6			1.32588			0.662938	+	0.748674i	$0.269309\pi$
$294$		0			0
$295$	2.19154e6			1.46620
$296$		0			0
$297$		0			0
$298$		0			0
$299$	953188.	−	1.65097e6i	0.616596	−	1.06798i
$300$		0			0
$301$	−156690.	−	2.41417e6i	−0.0996840	−	1.53586i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	571358.	+	989622.i	0.351689	+	0.609143i
$306$		0			0
$307$	−209113.			−0.126630			−0.0633149	−	0.997994i	$-0.520167\pi$
$307$	−209113.			−0.126630			−0.0633149	+	0.997994i	$0.520167\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−46862.0	−	81167.3i	−0.0274739	−	0.0475861i	0.851962	−	0.523604i	$-0.175413\pi$
$311$	−46862.0	−	81167.3i	−0.0274739	−	0.0475861i	−0.879435	+	0.476018i	$0.842080\pi$
$312$		0			0
$313$	250057.	−	433112.i	0.144271	−	0.249884i	−0.784830	−	0.619711i	$-0.787250\pi$
$313$	250057.	−	433112.i	0.144271	−	0.249884i	0.929101	+	0.369827i	$0.120583\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−737146.	+	1.27677e6i	−0.412007	+	0.713618i	−0.995109	−	0.0987813i	$-0.968506\pi$
$317$	−737146.	+	1.27677e6i	−0.412007	+	0.713618i	0.583102	+	0.812399i	$0.301839\pi$
$318$		0			0
$319$	138485.	+	239864.i	0.0761952	+	0.131974i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	302484.			0.161323
$324$		0			0
$325$	−661407.	−	1.14559e6i	−0.347345	−	0.601618i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−162554.	−	80291.4i	−0.0827957	−	0.0408958i
$330$		0			0
$331$	1.18818e6	−	2.05800e6i	0.596093	−	1.03246i	−0.397299	−	0.917689i	$-0.630052\pi$
$331$	1.18818e6	−	2.05800e6i	0.596093	−	1.03246i	0.993392	−	0.114774i	$-0.0366143\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.06888e6			1.98090
$336$		0			0
$337$	2.56857e6			1.23202			0.616008	−	0.787740i	$-0.288749\pi$
$337$	2.56857e6			1.23202			0.616008	+	0.787740i	$0.288749\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.54526e6	+	2.67647e6i	−0.719642	+	1.24646i
$342$		0			0
$343$	420999.	+	2.13783e6i	0.193217	+	0.981156i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−306273.	−	530480.i	−0.136548	−	0.236508i	0.789640	−	0.613571i	$-0.210267\pi$
$347$	−306273.	−	530480.i	−0.136548	−	0.236508i	−0.926188	+	0.377063i	$0.876934\pi$
$348$		0			0
$349$	3.83323e6			1.68462			0.842310	−	0.538994i	$-0.181195\pi$
$349$	3.83323e6			1.68462			0.842310	+	0.538994i	$0.181195\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.42925e6	+	2.47554e6i	0.610481	+	1.05738i	0.991159	+	0.132676i	$0.0423570\pi$
$353$	1.42925e6	+	2.47554e6i	0.610481	+	1.05738i	−0.380679	+	0.924707i	$0.624310\pi$
$354$		0			0
$355$	1.82606e6	−	3.16284e6i	0.769034	−	1.33201i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	193874.	−	335800.i	0.0793933	−	0.137513i	−0.823595	−	0.567178i	$-0.808035\pi$
$359$	193874.	−	335800.i	0.0793933	−	0.137513i	0.902988	+	0.429665i	$0.141368\pi$
$360$		0			0
$361$	1.18150e6	+	2.04641e6i	0.477160	+	0.826466i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	610325.			0.239789
$366$		0			0
$367$	−96154.9	−	166545.i	−0.0372654	−	0.0645456i	0.846791	−	0.531926i	$-0.178531\pi$
$367$	−96154.9	−	166545.i	−0.0372654	−	0.0645456i	−0.884057	+	0.467380i	$0.845198\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.13099e6	+	1.42192e6i	−0.803797	+	0.536340i
$372$		0			0
$373$	1.13262e6	−	1.96175e6i	0.421514	−	0.730083i	−0.574574	−	0.818453i	$-0.694832\pi$
$373$	1.13262e6	−	1.96175e6i	0.421514	−	0.730083i	0.996088	+	0.0883694i	$0.0281656\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−213872.			−0.0774997
$378$		0			0
$379$	4.46258e6			1.59583			0.797917	−	0.602767i	$-0.205935\pi$
$379$	4.46258e6			1.59583			0.797917	+	0.602767i	$0.205935\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−902218.	+	1.56269e6i	−0.314279	+	0.544346i	−0.979284	−	0.202492i	$-0.935096\pi$
$383$	−902218.	+	1.56269e6i	−0.314279	+	0.544346i	0.665005	+	0.746839i	$0.268429\pi$
$384$		0			0
$385$	343182.	+	5.28752e6i	0.117998	+	1.81803i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	55455.6	+	96051.9i	0.0185811	+	0.0321834i	0.875166	−	0.483822i	$-0.160752\pi$
$389$	55455.6	+	96051.9i	0.0185811	+	0.0321834i	−0.856585	+	0.516005i	$0.827418\pi$
$390$		0			0
$391$	−4.38203e6			−1.44955
$392$		0			0
$393$		0			0
$394$		0			0
$395$	194786.	+	337379.i	0.0628151	+	0.108799i
$396$		0			0
$397$	−2.74389e6	+	4.75256e6i	−0.873758	+	1.51339i	−0.0156779	+	0.999877i	$0.504991\pi$
$397$	−2.74389e6	+	4.75256e6i	−0.873758	+	1.51339i	−0.858080	+	0.513516i	$0.828343\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	467219.	−	809247.i	0.145097	−	0.251316i	−0.784312	−	0.620367i	$-0.786984\pi$
$401$	467219.	−	809247.i	0.145097	−	0.251316i	0.929409	+	0.369051i	$0.120317\pi$
$402$		0			0
$403$	−1.19322e6	−	2.06672e6i	−0.365982	−	0.633899i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.50443e6			2.24560
$408$		0			0
$409$	738463.	+	1.27905e6i	0.218283	+	0.378078i	0.954283	−	0.298904i	$-0.0966210\pi$
$409$	738463.	+	1.27905e6i	0.218283	+	0.378078i	−0.736000	+	0.676981i	$0.763288\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.93007e6	−	1.95512e6i	0.845285	−	0.564024i
$414$		0			0
$415$	567320.	−	982627.i	0.161699	−	0.280071i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.66235e6			−0.740849			−0.370424	−	0.928863i	$-0.620788\pi$
$419$	−2.66235e6			−0.740849			−0.370424	+	0.928863i	$0.620788\pi$
$420$		0			0
$421$	−1.02032e6			−0.280563			−0.140282	−	0.990112i	$-0.544801\pi$
$421$	−1.02032e6			−0.280563			−0.140282	+	0.990112i	$0.544801\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.52032e6	+	2.63328e6i	−0.408285	+	0.707170i
$426$		0			0
$427$	1.64676e6	+	813396.i	0.437080	+	0.215890i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−100134.	−	173436.i	−0.0259649	−	0.0449725i	0.852751	−	0.522318i	$-0.174932\pi$
$431$	−100134.	−	173436.i	−0.0259649	−	0.0449725i	−0.878716	+	0.477345i	$0.841599\pi$
$432$		0			0
$433$	5.84752e6			1.49883			0.749415	−	0.662101i	$-0.230335\pi$
$433$	5.84752e6			1.49883			0.749415	+	0.662101i	$0.230335\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	819278.	+	1.41903e6i	0.205224	+	0.355458i
$438$		0			0
$439$	−3.16278e6	+	5.47809e6i	−0.783262	+	1.35665i	0.146769	+	0.989171i	$0.453112\pi$
$439$	−3.16278e6	+	5.47809e6i	−0.783262	+	1.35665i	−0.930032	+	0.367479i	$0.880221\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.10193e6	−	1.90859e6i	0.266774	−	0.462066i	−0.701253	−	0.712913i	$-0.747376\pi$
$443$	1.10193e6	−	1.90859e6i	0.266774	−	0.462066i	0.968027	+	0.250846i	$0.0807089\pi$
$444$		0			0
$445$	3.61162e6	+	6.25550e6i	0.864573	+	1.49748i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.66634e6			0.624166			0.312083	−	0.950055i	$-0.398973\pi$
$449$	2.66634e6			0.624166			0.312083	+	0.950055i	$0.398973\pi$
$450$		0			0
$451$	−2.49241e6	−	4.31697e6i	−0.577002	−	0.999397i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.66841e6	−	1.81196e6i	−0.830710	−	0.410319i
$456$		0			0
$457$	3.04406e6	−	5.27247e6i	0.681809	−	1.18093i	−0.292619	−	0.956229i	$-0.594527\pi$
$457$	3.04406e6	−	5.27247e6i	0.681809	−	1.18093i	0.974428	−	0.224699i	$-0.0721400\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−381317.			−0.0835668			−0.0417834	−	0.999127i	$-0.513304\pi$
$461$	−381317.			−0.0835668			−0.0417834	+	0.999127i	$0.513304\pi$
$462$		0			0
$463$	−99697.8			−0.0216139			−0.0108069	−	0.999942i	$-0.503440\pi$
$463$	−99697.8			−0.0216139			−0.0108069	+	0.999942i	$0.503440\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2.81889e6	−	4.88246e6i	0.598116	−	1.03597i	−0.394983	−	0.918689i	$-0.629249\pi$
$467$	2.81889e6	−	4.88246e6i	0.598116	−	1.03597i	0.993099	−	0.117279i	$-0.0374172\pi$
$468$		0			0
$469$	5.44006e6	−	3.62993e6i	1.14201	−	0.762019i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.72801e6	−	8.18915e6i	−0.971685	−	1.68301i
$474$		0			0
$475$	1.13698e6			0.231216
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−870394.	−	1.50757e6i	−0.173331	−	0.300219i	0.766251	−	0.642541i	$-0.222120\pi$
$479$	−870394.	−	1.50757e6i	−0.173331	−	0.300219i	−0.939583	+	0.342322i	$0.888787\pi$
$480$		0			0
$481$	−2.89739e6	+	5.01843e6i	−0.571011	+	0.989020i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	290753.	−	503599.i	0.0561268	−	0.0972144i
$486$		0			0
$487$	−2.41235e6	−	4.17832e6i	−0.460913	−	0.798324i	0.538094	−	0.842885i	$-0.319145\pi$
$487$	−2.41235e6	−	4.17832e6i	−0.460913	−	0.798324i	−0.999007	+	0.0445607i	$0.985811\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.46085e6			0.835053			0.417527	−	0.908665i	$-0.362897\pi$
$491$	4.46085e6			0.835053			0.417527	+	0.908665i	$0.362897\pi$
$492$		0			0
$493$	245805.	+	425746.i	0.0455484	+	0.0788921i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−380193.	−	5.85775e6i	−0.0690420	−	1.06375i
$498$		0			0
$499$	370378.	−	641513.i	0.0665876	−	0.115333i	−0.830810	−	0.556557i	$-0.812122\pi$
$499$	370378.	−	641513.i	0.0665876	−	0.115333i	0.897397	+	0.441224i	$0.145455\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.74899e6			0.660686			0.330343	−	0.943861i	$-0.392836\pi$
$503$	3.74899e6			0.660686			0.330343	+	0.943861i	$0.392836\pi$
$504$		0			0
$505$	−1.57459e7			−2.74751
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.33228e6	+	2.30757e6i	−0.227929	+	0.394785i	−0.957194	−	0.289446i	$-0.906529\pi$
$509$	−1.33228e6	+	2.30757e6i	−0.227929	+	0.394785i	0.729265	+	0.684231i	$0.239862\pi$
$510$		0			0
$511$	815999.	−	544482.i	0.138241	−	0.0922426i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−953116.	−	1.65085e6i	−0.158353	−	0.274276i
$516$		0			0
$517$	−708647.			−0.116601
$518$		0			0
$519$		0			0
$520$		0			0
$521$	416312.	+	721073.i	0.0671930	+	0.116382i	0.897665	−	0.440679i	$-0.145262\pi$
$521$	416312.	+	721073.i	0.0671930	+	0.116382i	−0.830472	+	0.557061i	$0.811929\pi$
$522$		0			0
$523$	936507.	−	1.62208e6i	0.149712	−	0.259309i	−0.781409	−	0.624019i	$-0.785499\pi$
$523$	936507.	−	1.62208e6i	0.149712	−	0.259309i	0.931121	+	0.364710i	$0.118832\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.74277e6	+	4.75061e6i	−0.430192	+	0.745114i
$528$		0			0
$529$	−8.65056e6	−	1.49832e7i	−1.34402	−	2.32791i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.84918e6			0.586881
$534$		0			0
$535$	4.31892e6	+	7.48058e6i	0.652364	+	1.12993i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.17593e6	+	6.76321e6i	0.767390	+	1.00272i
$540$		0			0
$541$	−2.23793e6	+	3.87621e6i	−0.328741	+	0.569395i	−0.982262	−	0.187512i	$-0.939958\pi$
$541$	−2.23793e6	+	3.87621e6i	−0.328741	+	0.569395i	0.653522	+	0.756908i	$0.273291\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.72813e6			−0.537650
$546$		0			0
$547$	6.12226e6			0.874869			0.437435	−	0.899250i	$-0.355887\pi$
$547$	6.12226e6			0.874869			0.437435	+	0.899250i	$0.355887\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	91912.9	−	159198.i	0.0128973	−	0.0223387i
$552$		0			0
$553$	561409.	+	277300.i	0.0780668	+	0.0385601i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.76272e6	−	8.24927e6i	−0.650454	−	1.12662i	−0.983013	−	0.183537i	$-0.941245\pi$
$557$	−4.76272e6	−	8.24927e6i	−0.650454	−	1.12662i	0.332559	−	0.943083i	$-0.392088\pi$
$558$		0			0
$559$	7.30176e6			0.988321
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.04958e6	+	7.01408e6i	0.538442	+	0.932609i	0.998988	+	0.0449731i	$0.0143202\pi$
$563$	4.04958e6	+	7.01408e6i	0.538442	+	0.932609i	−0.460546	+	0.887636i	$0.652346\pi$
$564$		0			0
$565$	−2.23908e6	+	3.87819e6i	−0.295085	+	0.511103i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.69587e6	+	6.40144e6i	−0.478560	+	0.828890i	−0.999698	−	0.0245827i	$-0.992174\pi$
$569$	−3.69587e6	+	6.40144e6i	−0.478560	+	0.828890i	0.521138	+	0.853472i	$0.325508\pi$
$570$		0			0
$571$	−1.73172e6	−	2.99943e6i	−0.222273	−	0.384989i	0.733225	−	0.679986i	$-0.238014\pi$
$571$	−1.73172e6	−	2.99943e6i	−0.222273	−	0.384989i	−0.955498	+	0.294998i	$0.904681\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.64712e7			−2.07757
$576$		0			0
$577$	−2.58289e6	−	4.47369e6i	−0.322973	−	0.559405i	0.658127	−	0.752907i	$-0.271349\pi$
$577$	−2.58289e6	−	4.47369e6i	−0.322973	−	0.559405i	−0.981100	+	0.193502i	$0.938015\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−118118.	−	1.81988e6i	−0.0145170	−	0.223667i
$582$		0			0
$583$	−5.00665e6	+	8.67177e6i	−0.610065	+	1.05666i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.08305e6			−0.489090			−0.244545	−	0.969638i	$-0.578639\pi$
$587$	−4.08305e6			−0.489090			−0.244545	+	0.969638i	$0.578639\pi$
$588$		0			0
$589$	2.05118e6			0.243622
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.91092e6	+	6.77391e6i	−0.456712	+	0.791048i	−0.998785	−	0.0492833i	$-0.984306\pi$
$593$	−3.91092e6	+	6.77391e6i	−0.456712	+	0.791048i	0.542073	+	0.840331i	$0.317640\pi$
$594$		0			0
$595$	609132.	+	9.38508e6i	0.0705373	+	1.08679i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.15891e6	−	5.47139e6i	−0.359725	−	0.623061i	0.628190	−	0.778060i	$-0.283796\pi$
$599$	−3.15891e6	−	5.47139e6i	−0.359725	−	0.623061i	−0.987915	+	0.154999i	$0.950463\pi$
$600$		0			0
$601$	−1.68407e6			−0.190184			−0.0950919	−	0.995468i	$-0.530315\pi$
$601$	−1.68407e6			−0.190184			−0.0950919	+	0.995468i	$0.530315\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.86025e6	+	6.68615e6i	0.428773	+	0.742656i
$606$		0			0
$607$	−2.72757e6	+	4.72429e6i	−0.300472	+	0.520432i	−0.976243	−	0.216679i	$-0.930477\pi$
$607$	−2.72757e6	+	4.72429e6i	−0.300472	+	0.520432i	0.675771	+	0.737112i	$0.263811\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	273602.	−	473893.i	0.0296494	−	0.0513543i
$612$		0			0
$613$	4.06685e6	+	7.04399e6i	0.437127	+	0.757125i	0.997467	−	0.0711373i	$-0.0226629\pi$
$613$	4.06685e6	+	7.04399e6i	0.437127	+	0.757125i	−0.560340	+	0.828263i	$0.689330\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.83775e6			−0.300097			−0.150048	−	0.988679i	$-0.547943\pi$
$617$	−2.83775e6			−0.300097			−0.150048	+	0.988679i	$0.547943\pi$
$618$		0			0
$619$	−6.19545e6	−	1.07308e7i	−0.649900	−	1.12566i	−0.983146	−	0.182820i	$-0.941478\pi$
$619$	−6.19545e6	−	1.07308e7i	−0.649900	−	1.12566i	0.333247	−	0.942840i	$-0.391856\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.04094e7	+	5.14157e6i	1.07449	+	0.530732i
$624$		0			0
$625$	4.45058e6	−	7.70863e6i	0.455739	−	0.789363i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.33200e7			1.34239
$630$		0			0
$631$	3.25659e6			0.325604			0.162802	−	0.986659i	$-0.447947\pi$
$631$	3.25659e6			0.325604			0.162802	+	0.986659i	$0.447947\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.28312e6	+	1.60788e7i	−0.913608	+	1.58241i
$636$		0			0
$637$	−6.52113e6	+	850079.i	−0.636757	+	0.0830062i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.15075e6	+	1.99316e6i	0.110621	+	0.191601i	0.916021	−	0.401131i	$-0.131383\pi$
$641$	1.15075e6	+	1.99316e6i	0.110621	+	0.191601i	−0.805400	+	0.592732i	$0.798049\pi$
$642$		0			0
$643$	−8.68991e6			−0.828872			−0.414436	−	0.910078i	$-0.636021\pi$
$643$	−8.68991e6			−0.828872			−0.414436	+	0.910078i	$0.636021\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.14216e6	−	1.97828e6i	−0.107267	−	0.185792i	0.807395	−	0.590011i	$-0.200877\pi$
$647$	−1.14216e6	−	1.97828e6i	−0.107267	−	0.185792i	−0.914662	+	0.404219i	$0.867543\pi$
$648$		0			0
$649$	6.88406e6	−	1.19235e7i	0.641554	−	1.11120i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.08384e6	−	1.87726e6i	0.0994674	−	0.172283i	−0.811997	−	0.583662i	$-0.801619\pi$
$653$	1.08384e6	−	1.87726e6i	0.0994674	−	0.172283i	0.911464	+	0.411379i	$0.134953\pi$
$654$		0			0
$655$	−8.02550e6	−	1.39006e7i	−0.730919	−	1.26599i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.18596e7			−1.06380			−0.531898	−	0.846809i	$-0.678521\pi$
$659$	−1.18596e7			−1.06380			−0.531898	+	0.846809i	$0.678521\pi$
$660$		0			0
$661$	−713359.	−	1.23557e6i	−0.0635045	−	0.109993i	0.832525	−	0.553987i	$-0.186894\pi$
$661$	−713359.	−	1.23557e6i	−0.0635045	−	0.109993i	−0.896030	+	0.443994i	$0.853561\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.92528e6	−	1.95192e6i	0.256515	−	0.171162i
$666$		0			0
$667$	−1.33152e6	+	2.30627e6i	−0.115887	+	0.200722i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.17899e6			0.615541
$672$		0			0
$673$	−2.07459e7			−1.76561			−0.882806	−	0.469738i	$-0.844348\pi$
$673$	−2.07459e7			−1.76561			−0.882806	+	0.469738i	$0.844348\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.42895e6	+	1.11353e7i	−0.539098	+	0.933746i	0.459854	+	0.887994i	$0.347902\pi$
$677$	−6.42895e6	+	1.11353e7i	−0.539098	+	0.933746i	−0.998953	+	0.0457516i	$0.985432\pi$
$678$		0			0
$679$	−60535.9	−	932695.i	−0.00503893	−	0.0776363i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.64820e6	−	1.67112e7i	−0.791397	−	1.37074i	−0.925102	−	0.379719i	$-0.876021\pi$
$683$	−9.64820e6	−	1.67112e7i	−0.791397	−	1.37074i	0.133705	−	0.991021i	$-0.457313\pi$
$684$		0			0
$685$	1.02864e7			0.837599
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.86604e6	−	6.69618e6i	−0.310255	−	0.537377i
$690$		0			0
$691$	−9.14995e6	+	1.58482e7i	−0.728993	+	1.26265i	0.228316	+	0.973587i	$0.426678\pi$
$691$	−9.14995e6	+	1.58482e7i	−0.728993	+	1.26265i	−0.957309	+	0.289066i	$0.906655\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	696070.	−	1.20563e6i	0.0546627	−	0.0946786i
$696$		0			0
$697$	−4.42389e6	−	7.66241e6i	−0.344924	−	0.597425i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.61515e7			1.24142			0.620710	−	0.784040i	$-0.286844\pi$
$701$	1.61515e7			1.24142			0.620710	+	0.784040i	$0.286844\pi$
$702$		0			0
$703$	−2.49035e6	−	4.31341e6i	−0.190052	−	0.329179i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.10522e7	+	1.40472e7i	−1.58397	+	1.05692i
$708$		0			0
$709$	−749950.	+	1.29895e6i	−0.0560295	+	0.0970459i	−0.892680	−	0.450692i	$-0.851177\pi$
$709$	−749950.	+	1.29895e6i	−0.0560295	+	0.0970459i	0.836650	+	0.547738i	$0.184511\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.97151e7			−2.18904
$714$		0			0
$715$	−1.59923e7			−1.16989
$716$		0			0
$717$		0			0
$718$		0			0
$719$	8.78693e6	−	1.52194e7i	0.633892	−	1.09793i	−0.352857	−	0.935677i	$-0.614790\pi$
$719$	8.78693e6	−	1.52194e7i	0.633892	−	1.09793i	0.986749	−	0.162255i	$-0.0518768\pi$
$720$		0			0
$721$	−2.74706e6	−	1.35687e6i	−0.196802	−	0.0972078i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	923931.	+	1.60030e6i	0.0652822	+	0.113072i
$726$		0			0
$727$	−2.16304e7			−1.51785			−0.758923	−	0.651180i	$-0.774274\pi$
$727$	−2.16304e7			−1.51785			−0.758923	+	0.651180i	$0.774274\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.39198e6	−	1.45353e7i	−0.580859	−	1.00608i
$732$		0			0
$733$	6.27682e6	−	1.08718e7i	0.431499	−	0.747378i	−0.565504	−	0.824746i	$-0.691318\pi$
$733$	6.27682e6	−	1.08718e7i	0.431499	−	0.747378i	0.997003	+	0.0773678i	$0.0246516\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.27811e7	−	2.21376e7i	0.866764	−	1.50128i
$738$		0			0
$739$	378866.	+	656215.i	0.0255196	+	0.0442013i	0.878503	−	0.477737i	$-0.158543\pi$
$739$	378866.	+	656215.i	0.0255196	+	0.0442013i	−0.852984	+	0.521938i	$0.825209\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.40878e7			−1.60075			−0.800377	−	0.599497i	$-0.795367\pi$
$743$	−2.40878e7			−1.60075			−0.800377	+	0.599497i	$0.795367\pi$
$744$		0			0
$745$	33655.6	+	58293.2i	0.00222160	+	0.00384793i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.24479e7	+	6.14849e6i	0.810760	+	0.400464i
$750$		0			0
$751$	6.88436e6	−	1.19241e7i	0.445414	−	0.771480i	−0.552667	−	0.833402i	$-0.686390\pi$
$751$	6.88436e6	−	1.19241e7i	0.445414	−	0.771480i	0.998081	+	0.0619225i	$0.0197232\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.30203e6			−0.210821
$756$		0			0
$757$	2.43930e7			1.54713			0.773564	−	0.633718i	$-0.218472\pi$
$757$	2.43930e7			1.54713			0.773564	+	0.633718i	$0.218472\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.28001e7	−	2.21704e7i	0.801220	−	1.38775i	−0.117594	−	0.993062i	$-0.537518\pi$
$761$	1.28001e7	−	2.21704e7i	0.801220	−	1.38775i	0.918814	−	0.394692i	$-0.129149\pi$
$762$		0			0
$763$	−4.98448e6	+	3.32594e6i	−0.309962	+	0.206825i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.31574e6	+	9.20713e6i	0.326269	+	0.565114i
$768$		0			0
$769$	2.09807e7			1.27939			0.639697	−	0.768627i	$-0.279060\pi$
$769$	2.09807e7			1.27939			0.639697	+	0.768627i	$0.279060\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.64645e6	−	9.77993e6i	−0.339881	−	0.588691i	0.644529	−	0.764580i	$-0.277053\pi$
$773$	−5.64645e6	−	9.77993e6i	−0.339881	−	0.588691i	−0.984410	+	0.175889i	$0.943720\pi$
$774$		0			0
$775$	−1.03095e7	+	1.78566e7i	−0.616572	+	1.06793i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.65421e6	+	2.86518e6i	−0.0976668	+	0.169164i
$780$		0			0
$781$	−1.14721e7	−	1.98702e7i	−0.672998	−	1.16567i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.65270e6			−0.153643
$786$		0			0
$787$	1.18585e7	+	2.05395e7i	0.682482	+	1.18209i	0.974221	+	0.225596i	$0.0724329\pi$
$787$	1.18585e7	+	2.05395e7i	0.682482	+	1.18209i	−0.291739	+	0.956498i	$0.594234\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	466184.	+	7.18264e6i	0.0264921	+	0.408172i
$792$		0			0
$793$	−2.77174e6	+	4.80080e6i	−0.156520	+	0.271101i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.73492e7			1.52510			0.762550	−	0.646929i	$-0.223947\pi$
$797$	2.73492e7			1.52510			0.762550	+	0.646929i	$0.223947\pi$
$798$		0			0
$799$	−1.25781e6			−0.0697026
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.91715e6	−	3.32060e6i	0.104922	−	0.181730i
$804$		0			0
$805$	−4.23780e7	+	2.82771e7i	−2.30489	+	1.53796i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−5.46781e6	−	9.47052e6i	−0.293726	−	0.508748i	0.680962	−	0.732319i	$-0.261562\pi$
$809$	−5.46781e6	−	9.47052e6i	−0.293726	−	0.508748i	−0.974688	+	0.223571i	$0.928229\pi$
$810$		0			0
$811$	3.14053e7			1.67668			0.838342	−	0.545144i	$-0.183525\pi$
$811$	3.14053e7			1.67668			0.838342	+	0.545144i	$0.183525\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.83507e6	−	4.91049e6i	−0.149510	−	0.258959i
$816$		0			0
$817$	−3.13798e6	+	5.43514e6i	−0.164473	+	0.284876i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.56250e7	−	2.70634e7i	0.809028	−	1.40128i	−0.104510	−	0.994524i	$-0.533327\pi$
$821$	1.56250e7	−	2.70634e7i	0.809028	−	1.40128i	0.913538	−	0.406754i	$-0.133339\pi$
$822$		0			0
$823$	−1.83382e7	−	3.17627e7i	−0.943749	−	1.63462i	−0.758236	−	0.651980i	$-0.773939\pi$
$823$	−1.83382e7	−	3.17627e7i	−0.943749	−	1.63462i	−0.185513	−	0.982642i	$-0.559395\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.67620e7			0.852242			0.426121	−	0.904666i	$-0.359880\pi$
$827$	1.67620e7			0.852242			0.426121	+	0.904666i	$0.359880\pi$
$828$		0			0
$829$	595242.	+	1.03099e6i	0.0300820	+	0.0521036i	0.880674	−	0.473722i	$-0.157090\pi$
$829$	595242.	+	1.03099e6i	0.0300820	+	0.0521036i	−0.850592	+	0.525826i	$0.823756\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.18701e6	+	1.20044e7i	0.458735	+	0.599413i
$834$		0			0
$835$	−9.19248e6	+	1.59218e7i	−0.456264	+	0.790273i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−8.21562e6			−0.402935			−0.201468	−	0.979495i	$-0.564571\pi$
$839$	−8.21562e6			−0.402935			−0.201468	+	0.979495i	$0.564571\pi$
$840$		0			0
$841$	−2.02124e7			−0.985434
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.79939e6	+	1.52410e7i	−0.423946	+	0.734297i
$846$		0			0
$847$	1.11260e7	+	5.49553e6i	0.532880	+	0.263209i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.60772e7	+	6.24875e7i	1.70769	+	2.95780i
$852$		0			0
$853$	2.55133e7			1.20059			0.600295	−	0.799779i	$-0.295050\pi$
$853$	2.55133e7			1.20059			0.600295	+	0.799779i	$0.295050\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.43863e7	−	2.49178e7i	−0.669109	−	1.15893i	−0.978154	−	0.207884i	$-0.933342\pi$
$857$	−1.43863e7	−	2.49178e7i	−0.669109	−	1.15893i	0.309044	−	0.951048i	$-0.399991\pi$
$858$		0			0
$859$	−5.17972e6	+	8.97154e6i	−0.239510	+	0.414843i	−0.960574	−	0.278025i	$-0.910320\pi$
$859$	−5.17972e6	+	8.97154e6i	−0.239510	+	0.414843i	0.721064	+	0.692869i	$0.243653\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.87161e6	−	4.97377e6i	0.131250	−	0.227331i	−0.792909	−	0.609340i	$-0.791434\pi$
$863$	2.87161e6	−	4.97377e6i	0.131250	−	0.227331i	0.924159	+	0.382009i	$0.124768\pi$
$864$		0			0
$865$	1.36272e6	+	2.36031e6i	0.0619253	+	0.107258i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.44744e6			0.109942
$870$		0			0
$871$	9.86936e6	+	1.70942e7i	0.440802	+	0.763491i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	173179.	+	2.66822e6i	0.00764671	+	0.117815i
$876$		0			0
$877$	1.80935e7	−	3.13389e7i	0.794372	−	1.37589i	−0.128866	−	0.991662i	$-0.541134\pi$
$877$	1.80935e7	−	3.13389e7i	0.794372	−	1.37589i	0.923237	−	0.384230i	$-0.125533\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.20285e7			0.956192			0.478096	−	0.878308i	$-0.341327\pi$
$881$	2.20285e7			0.956192			0.478096	+	0.878308i	$0.341327\pi$
$882$		0			0
$883$	−3.82050e7			−1.64899			−0.824497	−	0.565867i	$-0.808542\pi$
$883$	−3.82050e7			−1.64899			−0.824497	+	0.565867i	$0.808542\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.87196e7	+	3.24233e7i	−0.798890	+	1.38372i	0.121449	+	0.992598i	$0.461246\pi$
$887$	−1.87196e7	+	3.24233e7i	−0.798890	+	1.38372i	−0.920339	+	0.391121i	$0.872087\pi$
$888$		0			0
$889$	1.93278e6	+	2.97789e7i	0.0820216	+	1.26373i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	235165.	+	407317.i	0.00986832	+	0.0170924i
$894$		0			0
$895$	4.96218e7			2.07069
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.66683e6	+	2.88704e6i	0.0687849	+	0.119139i
$900$		0			0
$901$	−8.88655e6	+	1.53920e7i	−0.364688	+	0.631658i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.73748e7	+	4.74145e7i	−1.11104	+	1.92438i
$906$		0			0
$907$	1.19207e7	+	2.06473e7i	0.481154	+	0.833383i	0.999766	−	0.0216263i	$-0.00688439\pi$
$907$	1.19207e7	+	2.06473e7i	0.481154	+	0.833383i	−0.518612	+	0.855010i	$0.673551\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.78434e7			−0.712332			−0.356166	−	0.934423i	$-0.615916\pi$
$911$	−1.78434e7			−0.712332			−0.356166	+	0.934423i	$0.615916\pi$
$912$		0			0
$913$	−3.56413e6	−	6.17325e6i	−0.141506	−	0.245096i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.31310e7	−	1.14253e7i	−0.908388	−	0.448686i
$918$		0			0
$919$	−8.21451e6	+	1.42280e7i	−0.320843	+	0.555717i	−0.980662	−	0.195708i	$-0.937300\pi$
$919$	−8.21451e6	+	1.42280e7i	−0.320843	+	0.555717i	0.659819	+	0.751425i	$0.270633\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.77170e7			0.684520
$924$		0			0
$925$	5.00672e7			1.92397
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.82722e6	+	1.18251e7i	−0.259540	+	0.449537i	−0.966119	−	0.258098i	$-0.916904\pi$
$929$	−6.82722e6	+	1.18251e7i	−0.259540	+	0.449537i	0.706579	+	0.707635i	$0.250238\pi$
$930$		0			0
$931$	2.16973e6	−	5.21940e6i	0.0820412	−	0.197354i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.83801e7	+	3.18353e7i	0.687573	+	1.19091i
$936$		0			0
$937$	3.81848e7			1.42083			0.710414	−	0.703784i	$-0.248508\pi$
$937$	3.81848e7			1.42083			0.710414	+	0.703784i	$0.248508\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.51554e6	+	6.08909e6i	0.129425	+	0.224170i	0.923454	−	0.383709i	$-0.125354\pi$
$941$	3.51554e6	+	6.08909e6i	0.129425	+	0.224170i	−0.794029	+	0.607880i	$0.792020\pi$
$942$		0			0
$943$	2.39642e7	−	4.15073e7i	0.877575	−	1.52000i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.10099e7	−	1.90698e7i	0.398942	−	0.690988i	−0.594653	−	0.803982i	$-0.702711\pi$
$947$	1.10099e7	−	1.90698e7i	0.398942	−	0.690988i	0.993596	+	0.112994i	$0.0360440\pi$
$948$		0			0
$949$	1.48039e6	+	2.56410e6i	0.0533592	+	0.0924209i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−578419.			−0.0206305			−0.0103153	−	0.999947i	$-0.503284\pi$
$953$	−578419.			−0.0206305			−0.0103153	+	0.999947i	$0.503284\pi$
$954$		0			0
$955$	−2.64226e7	−	4.57652e7i	−0.937489	−	1.62378i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.37528e7	−	9.17668e6i	0.482886	−	0.322210i
$960$		0			0
$961$	−4.28447e6	+	7.42093e6i	−0.149654	+	0.259209i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.58476e7			0.893516
$966$		0			0
$967$	2.00036e7			0.687925			0.343963	−	0.938983i	$-0.388231\pi$
$967$	2.00036e7			0.687925			0.343963	+	0.938983i	$0.388231\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.54560e7	−	4.40910e7i	0.866446	−	1.50073i	0.000840840	−	1.00000i	$-0.499732\pi$
$971$	2.54560e7	−	4.40910e7i	0.866446	−	1.50073i	0.865605	−	0.500728i	$-0.166934\pi$
$972$		0			0
$973$	−144924.	−	2.23289e6i	−0.00490749	−	0.0756112i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.69359e6	−	1.50577e7i	−0.291382	−	0.504688i	0.682755	−	0.730648i	$-0.260782\pi$
$977$	−8.69359e6	−	1.50577e7i	−0.291382	−	0.504688i	−0.974137	+	0.225959i	$0.927448\pi$
$978$		0			0
$979$	4.53792e7			1.51321
$980$		0			0
$981$		0			0
$982$		0			0
$983$	5.54285e6	+	9.60050e6i	0.182957	+	0.316891i	0.942886	−	0.333115i	$-0.108100\pi$
$983$	5.54285e6	+	9.60050e6i	0.182957	+	0.316891i	−0.759929	+	0.650006i	$0.774766\pi$
$984$		0			0
$985$	−8.15062e6	+	1.41173e7i	−0.267670	+	0.463618i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.54593e7	−	7.87379e7i	1.47786	−	2.55972i
$990$		0			0
$991$	−1.91462e7	−	3.31622e7i	−0.619297	−	1.07265i	−0.989614	−	0.143749i	$-0.954084\pi$
$991$	−1.91462e7	−	3.31622e7i	−0.619297	−	1.07265i	0.370317	−	0.928905i	$-0.379249\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.14872e7			−1.64870
$996$		0			0
$997$	−2.63586e7	−	4.56545e7i	−0.839818	−	1.45461i	−0.890047	−	0.455869i	$-0.849329\pi$
$997$	−2.63586e7	−	4.56545e7i	−0.839818	−	1.45461i	0.0502293	−	0.998738i	$-0.484005\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.6.s.d.289.5		10
3.2	odd	2		56.6.i.a.9.1	✓	10
7.4	even	3	inner	504.6.s.d.361.5		10
12.11	even	2		112.6.i.g.65.5		10
21.2	odd	6		392.6.a.l.1.5		5
21.5	even	6		392.6.a.i.1.1		5
21.11	odd	6		56.6.i.a.25.1	yes	10
21.17	even	6		392.6.i.p.361.5		10
21.20	even	2		392.6.i.p.177.5		10
84.11	even	6		112.6.i.g.81.5		10
84.23	even	6		784.6.a.bj.1.1		5
84.47	odd	6		784.6.a.bm.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.a.9.1	✓	10	3.2	odd	2
56.6.i.a.25.1	yes	10	21.11	odd	6
112.6.i.g.65.5		10	12.11	even	2
112.6.i.g.81.5		10	84.11	even	6
392.6.a.i.1.1		5	21.5	even	6
392.6.a.l.1.5		5	21.2	odd	6
392.6.i.p.177.5		10	21.20	even	2
392.6.i.p.361.5		10	21.17	even	6
504.6.s.d.289.5		10	1.1	even	1	trivial
504.6.s.d.361.5		10	7.4	even	3	inner
784.6.a.bj.1.1		5	84.23	even	6
784.6.a.bm.1.5		5	84.47	odd	6

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(40.3290 - 69.8518i) q^{5} +(-8.39664 - 129.370i) q^{7} +(-253.362 - 438.837i) q^{11} +391.283 q^{13} +(-449.706 - 778.913i) q^{17} +(-168.157 + 291.256i) q^{19} +(2436.06 - 4219.37i) q^{23} +(-1690.35 - 2927.78i) q^{25} -546.591 q^{29} +(-3049.51 - 5281.91i) q^{31} +(-9375.33 - 4630.82i) q^{35} +(-7404.84 + 12825.6i) q^{37} +9837.31 q^{41} +18661.0 q^{43} +(699.243 - 1211.12i) q^{47} +(-16666.0 + 2172.54i) q^{49} +(-9880.41 - 17113.4i) q^{53} -40871.4 q^{55} +(13585.4 + 23530.6i) q^{59} +(-7083.72 + 12269.4i) q^{61} +(15780.1 - 27331.9i) q^{65} +(25223.0 + 43687.6i) q^{67} +45279.2 q^{71} +(3783.41 + 6553.06i) q^{73} +(-54644.7 + 36462.2i) q^{77} +(-2414.96 + 4182.83i) q^{79} +14067.3 q^{83} -72544.7 q^{85} +(-44776.9 + 77555.9i) q^{89} +(-3285.47 - 50620.2i) q^{91} +(13563.2 + 23492.1i) q^{95} +7209.53 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 31 q^{5} - 92 q^{7} - 351 q^{11} - 108 q^{13} + 111 q^{17} - 1035 q^{19} + 3639 q^{23} - 1540 q^{25} + 1468 q^{29} - 7677 q^{31} - 19899 q^{35} - 13595 q^{37} - 10620 q^{41} + 1528 q^{43} + 6675 q^{47}+ \cdots + 270940 q^{97}+O(q^{100}) \) 10 * q + 31 * q^5 - 92 * q^7 - 351 * q^11 - 108 * q^13 + 111 * q^17 - 1035 * q^19 + 3639 * q^23 - 1540 * q^25 + 1468 * q^29 - 7677 * q^31 - 19899 * q^35 - 13595 * q^37 - 10620 * q^41 + 1528 * q^43 + 6675 * q^47 + 30122 * q^49 - 30753 * q^53 + 56534 * q^55 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 + 217440 * q^71 - 141659 * q^73 - 271471 * q^77 - 118919 * q^79 - 422008 * q^83 - 286758 * q^85 - 55861 * q^89 + 403912 * q^91 - 26279 * q^95 + 270940 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more