LMFDB - Embedded newform 560.6.g.f.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(449,560)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("560.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$89.8149390953$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2727 x^{14} + 2931087 x^{12} + 1572724517 x^{10} + 437533757892 x^{8} + 58220937475176 x^{6} + \cdots + 17\!\cdots\!00 $ x^16 + 2727x^14 + 2931087x^12 + 1572724517x^10 + 437533757892x^8 + 58220937475176x^6 + 2846440955743600x^4 + 47679368531490000*x^2 + 172239110289000000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2}\cdot 5^{8}\cdot 7^{10} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-27.3503i$ of defining polynomial
Character	$\chi$	$=$	560.449
Dual form			560.6.g.f.449.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.3503i q^{3} +(-55.3209 + 8.03755i) q^{5} -49.0000i q^{7} -399.638 q^{9} +141.450 q^{11} -297.641i q^{13} +(203.754 + 1402.40i) q^{15} +1226.74i q^{17} +1777.30 q^{19} -1242.16 q^{21} +3158.14i q^{23} +(2995.80 - 889.288i) q^{25} +3970.81i q^{27} +7419.42 q^{29} -970.605 q^{31} -3585.80i q^{33} +(393.840 + 2710.72i) q^{35} +2337.19i q^{37} -7545.29 q^{39} -561.020 q^{41} +1974.49i q^{43} +(22108.3 - 3212.11i) q^{45} +10976.5i q^{47} -2401.00 q^{49} +31098.1 q^{51} -33500.0i q^{53} +(-7825.14 + 1136.91i) q^{55} -45055.2i q^{57} +32227.0 q^{59} -34441.8 q^{61} +19582.2i q^{63} +(2392.30 + 16465.8i) q^{65} +12670.3i q^{67} +80059.9 q^{69} +83271.0 q^{71} +89632.0i q^{73} +(-22543.7 - 75944.3i) q^{75} -6931.05i q^{77} +4871.34 q^{79} +3549.22 q^{81} +67688.7i q^{83} +(-9859.95 - 67864.1i) q^{85} -188085. i q^{87} +86637.4 q^{89} -14584.4 q^{91} +24605.1i q^{93} +(-98322.0 + 14285.2i) q^{95} +6607.75i q^{97} -56528.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{5} - 1606 q^{9} - 294 q^{11} - 754 q^{15} + 3512 q^{19} + 1274 q^{21} + 7500 q^{25} - 20870 q^{29} - 4028 q^{31} - 3038 q^{35} + 25358 q^{39} + 24952 q^{41} - 10650 q^{45} - 38416 q^{49} + 12302 q^{51}+ \cdots - 293964 q^{99}+O(q^{100}) $ 16 * q - 2 * q^5 - 1606 * q^9 - 294 * q^11 - 754 * q^15 + 3512 * q^19 + 1274 * q^21 + 7500 * q^25 - 20870 * q^29 - 4028 * q^31 - 3038 * q^35 + 25358 * q^39 + 24952 * q^41 - 10650 * q^45 - 38416 * q^49 + 12302 * q^51 + 7612 * q^55 - 36156 * q^59 - 14556 * q^61 - 72446 * q^65 + 99340 * q^69 + 47992 * q^71 + 155016 * q^75 - 430 * q^79 + 156208 * q^81 - 195046 * q^85 + 304180 * q^89 - 16954 * q^91 - 149080 * q^95 - 293964 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		−	25.3503i		−	1.62622i	−0.582108	−	0.813111i	$-0.697772\pi$
$3$		−	25.3503i		−	1.62622i	0.582108	−	0.813111i	$-0.302228\pi$
$4$		0			0
$5$	−55.3209	+	8.03755i	−0.989610	+	0.143780i
$5$	−55.3209	+	8.03755i	−0.989610	+	0.143780i
$6$		0			0
$7$		−	49.0000i		−	0.377964i
$7$		−	49.0000i		−	0.377964i
$8$		0			0
$9$	−399.638			−1.64460
$10$		0			0
$11$	141.450			0.352469			0.176235	−	0.984348i	$-0.443608\pi$
$11$	141.450			0.352469			0.176235	+	0.984348i	$0.443608\pi$
$12$		0			0
$13$		−	297.641i		−	0.488466i	−0.969717	−	0.244233i	$-0.921464\pi$
$13$		−	297.641i		−	0.488466i	0.969717	−	0.244233i	$-0.0785362\pi$
$14$		0			0
$15$	203.754	+	1402.40i	0.233818	+	1.60933i
$16$		0			0
$17$			1226.74i			1.02951i	0.857339	+	0.514753i	$0.172116\pi$
$17$			1226.74i			1.02951i	−0.857339	+	0.514753i	$0.827884\pi$
$18$		0			0
$19$	1777.30			1.12948			0.564739	−	0.825270i	$-0.308977\pi$
$19$	1777.30			1.12948			0.564739	+	0.825270i	$0.308977\pi$
$20$		0			0
$21$	−1242.16			−0.614654
$22$		0			0
$23$			3158.14i			1.24484i	0.782685	+	0.622418i	$0.213850\pi$
$23$			3158.14i			1.24484i	−0.782685	+	0.622418i	$0.786150\pi$
$24$		0			0
$25$	2995.80	−	889.288i	0.958655	−	0.284572i
$26$		0			0
$27$			3970.81i			1.04826i
$28$		0			0
$29$	7419.42			1.63823			0.819115	−	0.573629i	$-0.194465\pi$
$29$	7419.42			1.63823			0.819115	+	0.573629i	$0.194465\pi$
$30$		0			0
$31$	−970.605			−0.181400			−0.0907002	−	0.995878i	$-0.528910\pi$
$31$	−970.605			−0.181400			−0.0907002	+	0.995878i	$0.528910\pi$
$32$		0			0
$33$		−	3585.80i		−	0.573193i
$34$		0			0
$35$	393.840	+	2710.72i	0.0543438	+	0.374037i
$36$		0			0
$37$			2337.19i			0.280666i	0.990104	+	0.140333i	$0.0448173\pi$
$37$			2337.19i			0.280666i	−0.990104	+	0.140333i	$0.955183\pi$
$38$		0			0
$39$	−7545.29			−0.794354
$40$		0			0
$41$	−561.020			−0.0521217			−0.0260609	−	0.999660i	$-0.508296\pi$
$41$	−561.020			−0.0521217			−0.0260609	+	0.999660i	$0.508296\pi$
$42$		0			0
$43$			1974.49i			0.162848i	0.996680	+	0.0814242i	$0.0259469\pi$
$43$			1974.49i			0.162848i	−0.996680	+	0.0814242i	$0.974053\pi$
$44$		0			0
$45$	22108.3	−	3212.11i	1.62751	−	0.236461i
$46$		0			0
$47$			10976.5i			0.724802i	0.932022	+	0.362401i	$0.118043\pi$
$47$			10976.5i			0.724802i	−0.932022	+	0.362401i	$0.881957\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	31098.1			1.67421
$52$		0			0
$53$		−	33500.0i		−	1.63816i	−0.573681	−	0.819079i	$-0.694485\pi$
$53$		−	33500.0i		−	1.63816i	0.573681	−	0.819079i	$-0.305515\pi$
$54$		0			0
$55$	−7825.14	+	1136.91i	−0.348807	+	0.0506781i
$56$		0			0
$57$		−	45055.2i		−	1.83678i
$58$		0			0
$59$	32227.0			1.20529			0.602643	−	0.798011i	$-0.294114\pi$
$59$	32227.0			1.20529			0.602643	+	0.798011i	$0.294114\pi$
$60$		0			0
$61$	−34441.8			−1.18512			−0.592558	−	0.805528i	$-0.701882\pi$
$61$	−34441.8			−1.18512			−0.592558	+	0.805528i	$0.701882\pi$
$62$		0			0
$63$			19582.2i			0.621600i
$64$		0			0
$65$	2392.30	+	16465.8i	0.0702316	+	0.483391i
$66$		0			0
$67$			12670.3i			0.344827i	0.985025	+	0.172413i	$0.0551565\pi$
$67$			12670.3i			0.344827i	−0.985025	+	0.172413i	$0.944844\pi$
$68$		0			0
$69$	80059.9			2.02438
$70$		0			0
$71$	83271.0			1.96042			0.980208	−	0.197972i	$-0.0634355\pi$
$71$	83271.0			1.96042			0.980208	+	0.197972i	$0.0634355\pi$
$72$		0			0
$73$			89632.0i			1.96859i	0.176521	+	0.984297i	$0.443516\pi$
$73$			89632.0i			1.96859i	−0.176521	+	0.984297i	$0.556484\pi$
$74$		0			0
$75$	−22543.7	−	75944.3i	−0.462778	−	1.55899i
$76$		0			0
$77$		−	6931.05i		−	0.133221i
$78$		0			0
$79$	4871.34			0.0878174			0.0439087	−	0.999036i	$-0.486019\pi$
$79$	4871.34			0.0878174			0.0439087	+	0.999036i	$0.486019\pi$
$80$		0			0
$81$	3549.22			0.0601064
$82$		0			0
$83$			67688.7i			1.07850i	0.842145	+	0.539251i	$0.181293\pi$
$83$			67688.7i			1.07850i	−0.842145	+	0.539251i	$0.818707\pi$
$84$		0			0
$85$	−9859.95	−	67864.1i	−0.148022	−	1.01881i
$86$		0			0
$87$		−	188085.i		−	2.66413i
$88$		0			0
$89$	86637.4			1.15939			0.579696	−	0.814833i	$-0.303171\pi$
$89$	86637.4			1.15939			0.579696	+	0.814833i	$0.303171\pi$
$90$		0			0
$91$	−14584.4			−0.184623
$92$		0			0
$93$			24605.1i			0.294997i
$94$		0			0
$95$	−98322.0	+	14285.2i	−1.11774	+	0.162396i
$96$		0			0
$97$			6607.75i			0.0713057i	0.999364	+	0.0356528i	$0.0113511\pi$
$97$			6607.75i			0.0713057i	−0.999364	+	0.0356528i	$0.988649\pi$
$98$		0			0
$99$	−56528.7			−0.579671
$100$		0			0
$101$	−38774.9			−0.378223			−0.189111	−	0.981956i	$-0.560561\pi$
$101$	−38774.9			−0.378223			−0.189111	+	0.981956i	$0.560561\pi$
$102$		0			0
$103$		−	1658.22i		−	0.0154010i	−0.999970	−	0.00770049i	$-0.997549\pi$
$103$		−	1658.22i		−	0.0154010i	0.999970	−	0.00770049i	$-0.00245117\pi$
$104$		0			0
$105$	68717.6	−	9983.96i	0.608268	−	0.0883750i
$106$		0			0
$107$		−	192852.i		−	1.62842i	−0.580573	−	0.814208i	$-0.697171\pi$
$107$		−	192852.i		−	1.62842i	0.580573	−	0.814208i	$-0.302829\pi$
$108$		0			0
$109$	−125324.			−1.01034			−0.505171	−	0.863019i	$-0.668571\pi$
$109$	−125324.			−1.01034			−0.505171	+	0.863019i	$0.668571\pi$
$110$		0			0
$111$	59248.4			0.456425
$112$		0			0
$113$		−	118360.i		−	0.871984i	−0.899951	−	0.435992i	$-0.856398\pi$
$113$		−	118360.i		−	0.871984i	0.899951	−	0.435992i	$-0.143602\pi$
$114$		0			0
$115$	−25383.7	−	174711.i	−0.178983	−	1.23190i
$116$		0			0
$117$			118948.i			0.803330i
$118$		0			0
$119$	60110.1			0.389117
$120$		0			0
$121$	−141043.			−0.875765
$122$		0			0
$123$			14222.0i			0.0847615i
$124$		0			0
$125$	−158582.	+	73275.1i	−0.907778	+	0.419451i
$126$		0			0
$127$		−	78506.1i		−	0.431911i	−0.976403	−	0.215955i	$-0.930713\pi$
$127$		−	78506.1i		−	0.431911i	0.976403	−	0.215955i	$-0.0692866\pi$
$128$		0			0
$129$	50053.9			0.264828
$130$		0			0
$131$	21998.0			0.111996			0.0559982	−	0.998431i	$-0.482166\pi$
$131$	21998.0			0.111996			0.0559982	+	0.998431i	$0.482166\pi$
$132$		0			0
$133$		−	87087.9i		−	0.426902i
$134$		0			0
$135$	−31915.6	−	219668.i	−0.150719	−	1.03737i
$136$		0			0
$137$		−	330457.i		−	1.50423i	−0.659034	−	0.752113i	$-0.729035\pi$
$137$		−	330457.i		−	1.50423i	0.659034	−	0.752113i	$-0.270965\pi$
$138$		0			0
$139$	403251.			1.77027			0.885133	−	0.465337i	$-0.154067\pi$
$139$	403251.			1.77027			0.885133	+	0.465337i	$0.154067\pi$
$140$		0			0
$141$	278258.			1.17869
$142$		0			0
$143$		−	42101.3i		−	0.172169i
$144$		0			0
$145$	−410449.	+	59634.0i	−1.62121	+	0.235545i
$146$		0			0
$147$			60866.1i			0.232317i
$148$		0			0
$149$	−441499.			−1.62916			−0.814581	−	0.580050i	$-0.803033\pi$
$149$	−441499.			−1.62916			−0.814581	+	0.580050i	$0.803033\pi$
$150$		0			0
$151$	−424395.			−1.51471			−0.757353	−	0.653006i	$-0.773508\pi$
$151$	−424395.			−1.51471			−0.757353	+	0.653006i	$0.773508\pi$
$152$		0			0
$153$		−	490250.i		−	1.69312i
$154$		0			0
$155$	53694.7	−	7801.28i	0.179516	−	0.0260817i
$156$		0			0
$157$			154250.i			0.499431i	0.968319	+	0.249715i	$0.0803371\pi$
$157$			154250.i			0.499431i	−0.968319	+	0.249715i	$0.919663\pi$
$158$		0			0
$159$	−849236.			−2.66401
$160$		0			0
$161$	154749.			0.470504
$162$		0			0
$163$			156114.i			0.460227i	0.973164	+	0.230113i	$0.0739097\pi$
$163$			156114.i			0.460227i	−0.973164	+	0.230113i	$0.926090\pi$
$164$		0			0
$165$	28821.0	+	198370.i	0.0824138	+	0.567238i
$166$		0			0
$167$			84036.3i			0.233172i	0.993181	+	0.116586i	$0.0371950\pi$
$167$			84036.3i			0.233172i	−0.993181	+	0.116586i	$0.962805\pi$
$168$		0			0
$169$	282703.			0.761401
$170$		0			0
$171$	−710277.			−1.85754
$172$		0			0
$173$		−	155939.i		−	0.396133i	−0.980189	−	0.198066i	$-0.936534\pi$
$173$		−	155939.i		−	0.396133i	0.980189	−	0.198066i	$-0.0634661\pi$
$174$		0			0
$175$	−43575.1	−	146794.i	−0.107558	−	0.362337i
$176$		0			0
$177$		−	816965.i		−	1.96006i
$178$		0			0
$179$	485375.			1.13226			0.566129	−	0.824317i	$-0.308440\pi$
$179$	485375.			1.13226			0.566129	+	0.824317i	$0.308440\pi$
$180$		0			0
$181$	9996.05			0.0226794			0.0113397	−	0.999936i	$-0.496390\pi$
$181$	9996.05			0.0226794			0.0113397	+	0.999936i	$0.496390\pi$
$182$		0			0
$183$			873109.i			1.92726i
$184$		0			0
$185$	−18785.3	−	129295.i	−0.0403542	−	0.277750i
$186$		0			0
$187$			173522.i			0.362869i
$188$		0			0
$189$	194570.			0.396205
$190$		0			0
$191$	−398572.			−0.790540			−0.395270	−	0.918565i	$-0.629349\pi$
$191$	−398572.			−0.790540			−0.395270	+	0.918565i	$0.629349\pi$
$192$		0			0
$193$		−	669914.i		−	1.29457i	−0.762248	−	0.647286i	$-0.775904\pi$
$193$		−	669914.i		−	1.29457i	0.762248	−	0.647286i	$-0.224096\pi$
$194$		0			0
$195$	417412.	−	60645.6i	0.786100	−	0.114212i
$196$		0			0
$197$			937695.i			1.72146i	0.509065	+	0.860728i	$0.329991\pi$
$197$			937695.i			1.72146i	−0.509065	+	0.860728i	$0.670009\pi$
$198$		0			0
$199$	104138.			0.186413			0.0932066	−	0.995647i	$-0.470288\pi$
$199$	104138.			0.186413			0.0932066	+	0.995647i	$0.470288\pi$
$200$		0			0
$201$	321197.			0.560765
$202$		0			0
$203$		−	363552.i		−	0.619193i
$204$		0			0
$205$	31036.1	−	4509.22i	0.0515801	−	0.00749406i
$206$		0			0
$207$		−	1.26211e6i		−	2.04726i
$208$		0			0
$209$	251400.			0.398106
$210$		0			0
$211$	1.11877e6			1.72995			0.864975	−	0.501815i	$-0.167334\pi$
$211$	1.11877e6			1.72995			0.864975	+	0.501815i	$0.167334\pi$
$212$		0			0
$213$		−	2.11095e6i		−	3.18807i
$214$		0			0
$215$	−15870.1	−	109230.i	−0.0234144	−	0.161156i
$216$		0			0
$217$			47559.6i			0.0685629i
$218$		0			0
$219$	2.27220e6			3.20137
$220$		0			0
$221$	365127.			0.502879
$222$		0			0
$223$		−	972345.i		−	1.30936i	−0.755907	−	0.654679i	$-0.772804\pi$
$223$		−	972345.i		−	1.30936i	0.755907	−	0.654679i	$-0.227196\pi$
$224$		0			0
$225$	−1.19723e6	+	355393.i	−1.57660	+	0.468007i
$226$		0			0
$227$			378096.i			0.487010i	0.969900	+	0.243505i	$0.0782972\pi$
$227$			378096.i			0.487010i	−0.969900	+	0.243505i	$0.921703\pi$
$228$		0			0
$229$	1.18847e6			1.49761			0.748807	−	0.662788i	$-0.230627\pi$
$229$	1.18847e6			1.49761			0.748807	+	0.662788i	$0.230627\pi$
$230$		0			0
$231$	−175704.			−0.216647
$232$		0			0
$233$		−	1.18457e6i		−	1.42946i	−0.699402	−	0.714729i	$-0.746550\pi$
$233$		−	1.18457e6i		−	1.42946i	0.699402	−	0.714729i	$-0.253450\pi$
$234$		0			0
$235$	−88224.2	−	607230.i	−0.104212	−	0.717271i
$236$		0			0
$237$		−	123490.i		−	0.142811i
$238$		0			0
$239$	803487.			0.909880			0.454940	−	0.890522i	$-0.349661\pi$
$239$	803487.			0.909880			0.454940	+	0.890522i	$0.349661\pi$
$240$		0			0
$241$	344716.			0.382313			0.191156	−	0.981560i	$-0.438776\pi$
$241$	344716.			0.382313			0.191156	+	0.981560i	$0.438776\pi$
$242$		0			0
$243$			874932.i			0.950515i
$244$		0			0
$245$	132825.	−	19298.2i	0.141373	−	0.0205400i
$246$		0			0
$247$		−	528998.i		−	0.551711i
$248$		0			0
$249$	1.71593e6			1.75388
$250$		0			0
$251$	76632.3			0.0767764			0.0383882	−	0.999263i	$-0.487778\pi$
$251$	76632.3			0.0767764			0.0383882	+	0.999263i	$0.487778\pi$
$252$		0			0
$253$			446719.i			0.438766i
$254$		0			0
$255$	−1.72037e6	+	249953.i	−1.65681	+	0.240717i
$256$		0			0
$257$		−	734767.i		−	0.693931i	−0.937878	−	0.346966i	$-0.887212\pi$
$257$		−	734767.i		−	0.693931i	0.937878	−	0.346966i	$-0.112788\pi$
$258$		0			0
$259$	114522.			0.106082
$260$		0			0
$261$	−2.96508e6			−2.69423
$262$		0			0
$263$		−	448950.i		−	0.400229i	−0.979772	−	0.200115i	$-0.935869\pi$
$263$		−	448950.i		−	0.400229i	0.979772	−	0.200115i	$-0.0641315\pi$
$264$		0			0
$265$	269258.	+	1.85325e6i	0.235534	+	1.62114i
$266$		0			0
$267$		−	2.19628e6i		−	1.88543i
$268$		0			0
$269$	1.66975e6			1.40692			0.703462	−	0.710733i	$-0.251637\pi$
$269$	1.66975e6			1.40692			0.703462	+	0.710733i	$0.251637\pi$
$270$		0			0
$271$	−466892.			−0.386183			−0.193092	−	0.981181i	$-0.561851\pi$
$271$	−466892.			−0.386183			−0.193092	+	0.981181i	$0.561851\pi$
$272$		0			0
$273$			369719.i			0.300238i
$274$		0			0
$275$	423755.	−	125790.i	0.337896	−	0.100303i
$276$		0			0
$277$			1.86028e6i			1.45673i	0.685190	+	0.728364i	$0.259719\pi$
$277$			1.86028e6i			1.45673i	−0.685190	+	0.728364i	$0.740281\pi$
$278$		0			0
$279$	387890.			0.298331
$280$		0			0
$281$	384958.			0.290836			0.145418	−	0.989370i	$-0.453547\pi$
$281$	384958.			0.290836			0.145418	+	0.989370i	$0.453547\pi$
$282$		0			0
$283$		−	255926.i		−	0.189954i	−0.995479	−	0.0949769i	$-0.969722\pi$
$283$		−	255926.i		−	0.189954i	0.995479	−	0.0949769i	$-0.0302777\pi$
$284$		0			0
$285$	362133.	+	2.49249e6i	0.264093	+	1.81770i
$286$		0			0
$287$			27490.0i			0.0197002i
$288$		0			0
$289$	−85024.4			−0.0598824
$290$		0			0
$291$	167508.			0.115959
$292$		0			0
$293$			345511.i			0.235122i	0.993066	+	0.117561i	$0.0375075\pi$
$293$			345511.i			0.235122i	−0.993066	+	0.117561i	$0.962492\pi$
$294$		0			0
$295$	−1.78283e6	+	259026.i	−1.19276	+	0.173296i
$296$		0			0
$297$			561671.i			0.369480i
$298$		0			0
$299$	939992.			0.608060
$300$		0			0
$301$	96750.0			0.0615509
$302$		0			0
$303$			982956.i			0.615074i
$304$		0			0
$305$	1.90535e6	−	276828.i	1.17280	−	0.170396i
$306$		0			0
$307$			1.38493e6i			0.838651i	0.907836	+	0.419326i	$0.137733\pi$
$307$			1.38493e6i			0.838651i	−0.907836	+	0.419326i	$0.862267\pi$
$308$		0			0
$309$	−42036.3			−0.0250454
$310$		0			0
$311$	−900004.			−0.527647			−0.263824	−	0.964571i	$-0.584984\pi$
$311$	−900004.			−0.527647			−0.263824	+	0.964571i	$0.584984\pi$
$312$		0			0
$313$			2.80649e6i			1.61921i	0.586978	+	0.809603i	$0.300318\pi$
$313$			2.80649e6i			1.61921i	−0.586978	+	0.809603i	$0.699682\pi$
$314$		0			0
$315$	−157393.	−	1.08331e6i	−0.0893737	−	0.615141i
$316$		0			0
$317$		−	2.45481e6i		−	1.37205i	−0.727579	−	0.686024i	$-0.759355\pi$
$317$		−	2.45481e6i		−	1.37205i	0.727579	−	0.686024i	$-0.240645\pi$
$318$		0			0
$319$	1.04948e6			0.577426
$320$		0			0
$321$	−4.88886e6			−2.64817
$322$		0			0
$323$			2.18028e6i			1.16280i
$324$		0			0
$325$	−264689.	−	891671.i	−0.139004	−	0.468270i
$326$		0			0
$327$			3.17700e6i			1.64304i
$328$		0			0
$329$	537849.			0.273949
$330$		0			0
$331$	3.87397e6			1.94351			0.971755	−	0.235993i	$-0.0758341\pi$
$331$	3.87397e6			1.94351			0.971755	+	0.235993i	$0.0758341\pi$
$332$		0			0
$333$		−	934029.i		−	0.461583i
$334$		0			0
$335$	−101838.	−	700934.i	−0.0495792	−	0.341244i
$336$		0			0
$337$			3.04334e6i			1.45974i	0.683584	+	0.729871i	$0.260420\pi$
$337$			3.04334e6i			1.45974i	−0.683584	+	0.729871i	$0.739580\pi$
$338$		0			0
$339$	−3.00046e6			−1.41804
$340$		0			0
$341$	−137292.			−0.0639380
$342$		0			0
$343$			117649.i			0.0539949i
$344$		0			0
$345$	−4.42898e6	+	643485.i	−2.00335	+	0.291065i
$346$		0			0
$347$		−	1.78331e6i		−	0.795068i	−0.917587	−	0.397534i	$-0.869866\pi$
$347$		−	1.78331e6i		−	0.795068i	0.917587	−	0.397534i	$-0.130134\pi$
$348$		0			0
$349$	−1.76269e6			−0.774661			−0.387330	−	0.921941i	$-0.626603\pi$
$349$	−1.76269e6			−0.774661			−0.387330	+	0.921941i	$0.626603\pi$
$350$		0			0
$351$	1.18187e6			0.512040
$352$		0			0
$353$		−	675787.i		−	0.288651i	−0.989530	−	0.144325i	$-0.953899\pi$
$353$		−	675787.i		−	0.288651i	0.989530	−	0.144325i	$-0.0461012\pi$
$354$		0			0
$355$	−4.60663e6	+	669295.i	−1.94005	+	0.281869i
$356$		0			0
$357$		−	1.52381e6i		−	0.632790i
$358$		0			0
$359$	−1.77828e6			−0.728223			−0.364111	−	0.931355i	$-0.618627\pi$
$359$	−1.77828e6			−0.728223			−0.364111	+	0.931355i	$0.618627\pi$
$360$		0			0
$361$	682710.			0.275720
$362$		0			0
$363$			3.57548e6i			1.42419i
$364$		0			0
$365$	−720422.	−	4.95852e6i	−0.283044	−	1.94814i
$366$		0			0
$367$		−	4.35734e6i		−	1.68871i	−0.535781	−	0.844357i	$-0.679983\pi$
$367$		−	4.35734e6i		−	1.68871i	0.535781	−	0.844357i	$-0.320017\pi$
$368$		0			0
$369$	224205.			0.0857193
$370$		0			0
$371$	−1.64150e6			−0.619166
$372$		0			0
$373$			2.87221e6i			1.06892i	0.845195	+	0.534458i	$0.179484\pi$
$373$			2.87221e6i			1.06892i	−0.845195	+	0.534458i	$0.820516\pi$
$374$		0			0
$375$	1.85754e6	+	4.02011e6i	0.682120	+	1.47625i
$376$		0			0
$377$		−	2.20832e6i		−	0.800220i
$378$		0			0
$379$	2.11884e6			0.757705			0.378853	−	0.925457i	$-0.376319\pi$
$379$	2.11884e6			0.757705			0.378853	+	0.925457i	$0.376319\pi$
$380$		0			0
$381$	−1.99015e6			−0.702383
$382$		0			0
$383$			3.78103e6i			1.31708i	0.752544	+	0.658542i	$0.228826\pi$
$383$			3.78103e6i			1.31708i	−0.752544	+	0.658542i	$0.771174\pi$
$384$		0			0
$385$	55708.7	+	383432.i	0.0191545	+	0.131837i
$386$		0			0
$387$		−	789080.i		−	0.267820i
$388$		0			0
$389$	3.66654e6			1.22852			0.614261	−	0.789103i	$-0.289454\pi$
$389$	3.66654e6			1.22852			0.614261	+	0.789103i	$0.289454\pi$
$390$		0			0
$391$	−3.87421e6			−1.28157
$392$		0			0
$393$		−	557655.i		−	0.182131i
$394$		0			0
$395$	−269487.	+	39153.6i	−0.0869049	+	0.0126264i
$396$		0			0
$397$			2.32579e6i			0.740618i	0.928909	+	0.370309i	$0.120748\pi$
$397$			2.32579e6i			0.740618i	−0.928909	+	0.370309i	$0.879252\pi$
$398$		0			0
$399$	−2.20770e6			−0.694238
$400$		0			0
$401$	1.74464e6			0.541807			0.270904	−	0.962607i	$-0.412678\pi$
$401$	1.74464e6			0.541807			0.270904	+	0.962607i	$0.412678\pi$
$402$		0			0
$403$			288892.i			0.0886079i
$404$		0			0
$405$	−196346.	+	28527.1i	−0.0594819	+	0.00864211i
$406$		0			0
$407$			330595.i			0.0989261i
$408$		0			0
$409$	669116.			0.197785			0.0988924	−	0.995098i	$-0.468470\pi$
$409$	669116.			0.197785			0.0988924	+	0.995098i	$0.468470\pi$
$410$		0			0
$411$	−8.37718e6			−2.44621
$412$		0			0
$413$		−	1.57912e6i		−	0.455555i
$414$		0			0
$415$	−544051.	−	3.74460e6i	−0.155067	−	1.06730i
$416$		0			0
$417$		−	1.02225e7i		−	2.87885i
$418$		0			0
$419$	4.67106e6			1.29981			0.649906	−	0.760014i	$-0.274808\pi$
$419$	4.67106e6			1.29981			0.649906	+	0.760014i	$0.274808\pi$
$420$		0			0
$421$	4.55778e6			1.25328			0.626640	−	0.779309i	$-0.284430\pi$
$421$	4.55778e6			1.25328			0.626640	+	0.779309i	$0.284430\pi$
$422$		0			0
$423$		−	4.38662e6i		−	1.19201i
$424$		0			0
$425$	1.09092e6	+	3.67505e6i	0.292969	+	0.986941i
$426$		0			0
$427$			1.68765e6i			0.447932i
$428$		0			0
$429$	−1.06728e6			−0.279985
$430$		0			0
$431$	−426321.			−0.110546			−0.0552731	−	0.998471i	$-0.517603\pi$
$431$	−426321.			−0.110546			−0.0552731	+	0.998471i	$0.517603\pi$
$432$		0			0
$433$			3.32358e6i			0.851896i	0.904748	+	0.425948i	$0.140059\pi$
$433$			3.32358e6i			0.851896i	−0.904748	+	0.425948i	$0.859941\pi$
$434$		0			0
$435$	1.51174e6	+	1.04050e7i	0.383048	+	2.63645i
$436$		0			0
$437$			5.61298e6i			1.40601i
$438$		0			0
$439$	4.11543e6			1.01919			0.509594	−	0.860415i	$-0.329796\pi$
$439$	4.11543e6			1.01919			0.509594	+	0.860415i	$0.329796\pi$
$440$		0			0
$441$	959530.			0.234943
$442$		0			0
$443$		−	2.88152e6i		−	0.697610i	−0.937195	−	0.348805i	$-0.886588\pi$
$443$		−	2.88152e6i		−	0.697610i	0.937195	−	0.348805i	$-0.113412\pi$
$444$		0			0
$445$	−4.79286e6	+	696353.i	−1.14735	+	0.166698i
$446$		0			0
$447$			1.11921e7i			2.64938i
$448$		0			0
$449$	−4.35526e6			−1.01952			−0.509762	−	0.860315i	$-0.670267\pi$
$449$	−4.35526e6			−1.01952			−0.509762	+	0.860315i	$0.670267\pi$
$450$		0			0
$451$	−79356.3			−0.0183713
$452$		0			0
$453$			1.07585e7i			2.46325i
$454$		0			0
$455$	806822.	−	117223.i	0.182704	−	0.0265451i
$456$		0			0
$457$		−	7.79301e6i		−	1.74548i	−0.488186	−	0.872739i	$-0.662341\pi$
$457$		−	7.79301e6i		−	1.74548i	0.488186	−	0.872739i	$-0.337659\pi$
$458$		0			0
$459$	−4.87113e6			−1.07919
$460$		0			0
$461$	−2.48971e6			−0.545627			−0.272814	−	0.962067i	$-0.587954\pi$
$461$	−2.48971e6			−0.545627			−0.272814	+	0.962067i	$0.587954\pi$
$462$		0			0
$463$			6.12486e6i			1.32783i	0.747807	+	0.663917i	$0.231107\pi$
$463$			6.12486e6i			1.32783i	−0.747807	+	0.663917i	$0.768893\pi$
$464$		0			0
$465$	−197765.	−	1.36118e6i	−0.0424147	−	0.291932i
$466$		0			0
$467$			6.48662e6i			1.37634i	0.725549	+	0.688171i	$0.241586\pi$
$467$			6.48662e6i			1.37634i	−0.725549	+	0.688171i	$0.758414\pi$
$468$		0			0
$469$	620846.			0.130332
$470$		0			0
$471$	3.91028e6			0.812186
$472$		0			0
$473$			279291.i			0.0573991i
$474$		0			0
$475$	5.32444e6	−	1.58054e6i	1.08278	−	0.321418i
$476$		0			0
$477$			1.33879e7i			2.69411i
$478$		0			0
$479$	4.61538e6			0.919112			0.459556	−	0.888149i	$-0.348009\pi$
$479$	4.61538e6			0.919112			0.459556	+	0.888149i	$0.348009\pi$
$480$		0			0
$481$	695643.			0.137096
$482$		0			0
$483$		−	3.92293e6i		−	0.765144i
$484$		0			0
$485$	−53110.1	−	365546.i	−0.0102523	−	0.0705648i
$486$		0			0
$487$		−	8.66187e6i		−	1.65497i	−0.561490	−	0.827484i	$-0.689772\pi$
$487$		−	8.66187e6i		−	1.65497i	0.561490	−	0.827484i	$-0.310228\pi$
$488$		0			0
$489$	3.95753e6			0.748431
$490$		0			0
$491$	−1.87139e6			−0.350316			−0.175158	−	0.984540i	$-0.556044\pi$
$491$	−1.87139e6			−0.350316			−0.175158	+	0.984540i	$0.556044\pi$
$492$		0			0
$493$			9.10167e6i			1.68657i
$494$		0			0
$495$	3.12722e6	−	454352.i	0.573648	−	0.0833451i
$496$		0			0
$497$		−	4.08028e6i		−	0.740967i
$498$		0			0
$499$	6.42247e6			1.15465			0.577325	−	0.816514i	$-0.304096\pi$
$499$	6.42247e6			1.15465			0.577325	+	0.816514i	$0.304096\pi$
$500$		0			0
$501$	2.13034e6			0.379189
$502$		0			0
$503$		−	6.83277e6i		−	1.20414i	−0.798443	−	0.602070i	$-0.794343\pi$
$503$		−	6.83277e6i		−	1.20414i	0.798443	−	0.602070i	$-0.205657\pi$
$504$		0			0
$505$	2.14506e6	−	311655.i	0.374293	−	0.0543809i
$506$		0			0
$507$		−	7.16660e6i		−	1.23821i
$508$		0			0
$509$	−9.16537e6			−1.56803			−0.784017	−	0.620740i	$-0.786832\pi$
$509$	−9.16537e6			−1.56803			−0.784017	+	0.620740i	$0.786832\pi$
$510$		0			0
$511$	4.39197e6			0.744058
$512$		0			0
$513$			7.05733e6i			1.18399i
$514$		0			0
$515$	13328.0	+	91734.0i	0.00221435	+	0.0152410i
$516$		0			0
$517$			1.55263e6i			0.255470i
$518$		0			0
$519$	−3.95311e6			−0.644200
$520$		0			0
$521$	6.71054e6			1.08309			0.541543	−	0.840673i	$-0.317840\pi$
$521$	6.71054e6			1.08309			0.541543	+	0.840673i	$0.317840\pi$
$522$		0			0
$523$		−	3.31400e6i		−	0.529783i	−0.964278	−	0.264892i	$-0.914664\pi$
$523$		−	3.31400e6i		−	0.529783i	0.964278	−	0.264892i	$-0.0853362\pi$
$524$		0			0
$525$	−3.72127e6	+	1.10464e6i	−0.589241	+	0.174914i
$526$		0			0
$527$		−	1.19068e6i		−	0.186753i
$528$		0			0
$529$	−3.53752e6			−0.549617
$530$		0			0
$531$	−1.28791e7			−1.98221
$532$		0			0
$533$			166982.i			0.0254597i
$534$		0			0
$535$	1.55006e6	+	1.06688e7i	0.234134	+	1.61150i
$536$		0			0
$537$		−	1.23044e7i		−	1.84130i
$538$		0			0
$539$	−339621.			−0.0503528
$540$		0			0
$541$	4.39605e6			0.645757			0.322879	−	0.946440i	$-0.395349\pi$
$541$	4.39605e6			0.645757			0.322879	+	0.946440i	$0.395349\pi$
$542$		0			0
$543$		−	253403.i		−	0.0368818i
$544$		0			0
$545$	6.93304e6	−	1.00730e6i	0.999845	−	0.145267i
$546$		0			0
$547$		−	1.03221e7i		−	1.47503i	−0.675331	−	0.737515i	$-0.735999\pi$
$547$		−	1.03221e7i		−	1.47503i	0.675331	−	0.737515i	$-0.264001\pi$
$548$		0			0
$549$	1.37642e7			1.94904
$550$		0			0
$551$	1.31866e7			1.85035
$552$		0			0
$553$		−	238695.i		−	0.0331918i
$554$		0			0
$555$	−3.27768e6	+	476212.i	−0.451683	+	0.0656248i
$556$		0			0
$557$			4.02433e6i			0.549611i	0.961500	+	0.274805i	$0.0886134\pi$
$557$			4.02433e6i			0.549611i	−0.961500	+	0.274805i	$0.911387\pi$
$558$		0			0
$559$	587689.			0.0795459
$560$		0			0
$561$	4.39883e6			0.590106
$562$		0			0
$563$			7.76463e6i			1.03240i	0.856467	+	0.516202i	$0.172655\pi$
$563$			7.76463e6i			1.03240i	−0.856467	+	0.516202i	$0.827345\pi$
$564$		0			0
$565$	951324.	+	6.54777e6i	0.125374	+	0.862924i
$566$		0			0
$567$		−	173912.i		−	0.0227181i
$568$		0			0
$569$	−962147.			−0.124584			−0.0622918	−	0.998058i	$-0.519841\pi$
$569$	−962147.			−0.124584			−0.0622918	+	0.998058i	$0.519841\pi$
$570$		0			0
$571$	−7.97240e6			−1.02329			−0.511645	−	0.859197i	$-0.670964\pi$
$571$	−7.97240e6			−1.02329			−0.511645	+	0.859197i	$0.670964\pi$
$572$		0			0
$573$			1.01039e7i			1.28559i
$574$		0			0
$575$	2.80850e6	+	9.46115e6i	0.354246	+	1.19337i
$576$		0			0
$577$		−	3.13108e6i		−	0.391521i	−0.980652	−	0.195761i	$-0.937282\pi$
$577$		−	3.13108e6i		−	0.391521i	0.980652	−	0.195761i	$-0.0627175\pi$
$578$		0			0
$579$	−1.69825e7			−2.10526
$580$		0			0
$581$	3.31675e6			0.407635
$582$		0			0
$583$		−	4.73858e6i		−	0.577400i
$584$		0			0
$585$	−956054.	−	6.58033e6i	−0.115503	−	0.794984i
$586$		0			0
$587$		−	5.27415e6i		−	0.631767i	−0.948798	−	0.315884i	$-0.897699\pi$
$587$		−	5.27415e6i		−	0.631767i	0.948798	−	0.315884i	$-0.102301\pi$
$588$		0			0
$589$	−1.72506e6			−0.204888
$590$		0			0
$591$	2.37708e7			2.79947
$592$		0			0
$593$			4.94938e6i			0.577981i	0.957332	+	0.288991i	$0.0933196\pi$
$593$			4.94938e6i			0.577981i	−0.957332	+	0.288991i	$0.906680\pi$
$594$		0			0
$595$	−3.32534e6	+	483138.i	−0.385074	+	0.0559472i
$596$		0			0
$597$		−	2.63993e6i		−	0.303149i
$598$		0			0
$599$	2.70972e6			0.308573			0.154286	−	0.988026i	$-0.450692\pi$
$599$	2.70972e6			0.308573			0.154286	+	0.988026i	$0.450692\pi$
$600$		0			0
$601$	−992330.			−0.112065			−0.0560325	−	0.998429i	$-0.517845\pi$
$601$	−992330.			−0.112065			−0.0560325	+	0.998429i	$0.517845\pi$
$602$		0			0
$603$		−	5.06354e6i		−	0.567102i
$604$		0			0
$605$	7.80261e6	−	1.13364e6i	0.866666	−	0.125918i
$606$		0			0
$607$			1.35542e7i			1.49315i	0.665302	+	0.746574i	$0.268303\pi$
$607$			1.35542e7i			1.49315i	−0.665302	+	0.746574i	$0.731697\pi$
$608$		0			0
$609$	−9.21614e6			−1.00695
$610$		0			0
$611$	3.26706e6			0.354041
$612$		0			0
$613$			8.37239e6i			0.899908i	0.893052	+	0.449954i	$0.148560\pi$
$613$			8.37239e6i			0.899908i	−0.893052	+	0.449954i	$0.851440\pi$
$614$		0			0
$615$	−114310.	−	786774.i	−0.0121870	−	0.0838808i
$616$		0			0
$617$			6.49138e6i			0.686474i	0.939249	+	0.343237i	$0.111523\pi$
$617$			6.49138e6i			0.686474i	−0.939249	+	0.343237i	$0.888477\pi$
$618$		0			0
$619$	8.98202e6			0.942210			0.471105	−	0.882077i	$-0.343855\pi$
$619$	8.98202e6			0.942210			0.471105	+	0.882077i	$0.343855\pi$
$620$		0			0
$621$	−1.25404e7			−1.30491
$622$		0			0
$623$		−	4.24523e6i		−	0.438209i
$624$		0			0
$625$	8.18396e6	−	5.32825e6i	0.838037	−	0.545613i
$626$		0			0
$627$		−	6.37306e6i		−	0.647409i
$628$		0			0
$629$	−2.86711e6			−0.288947
$630$		0			0
$631$	1.68639e7			1.68611			0.843053	−	0.537831i	$-0.180756\pi$
$631$	1.68639e7			1.68611			0.843053	+	0.537831i	$0.180756\pi$
$632$		0			0
$633$		−	2.83611e7i		−	2.81328i
$634$		0			0
$635$	630996.	+	4.34302e6i	0.0621001	+	0.427423i
$636$		0			0
$637$			714636.i			0.0697808i
$638$		0			0
$639$	−3.32782e7			−3.22410
$640$		0			0
$641$	−1.35136e7			−1.29905			−0.649524	−	0.760341i	$-0.725032\pi$
$641$	−1.35136e7			−1.29905			−0.649524	+	0.760341i	$0.725032\pi$
$642$		0			0
$643$		−	9.15022e6i		−	0.872779i	−0.899758	−	0.436390i	$-0.856257\pi$
$643$		−	9.15022e6i		−	0.872779i	0.899758	−	0.436390i	$-0.143743\pi$
$644$		0			0
$645$	−2.76902e6	+	402311.i	−0.262076	+	0.0380769i
$646$		0			0
$647$			3.76700e6i			0.353781i	0.984231	+	0.176891i	$0.0566039\pi$
$647$			3.76700e6i			0.353781i	−0.984231	+	0.176891i	$0.943396\pi$
$648$		0			0
$649$	4.55851e6			0.424826
$650$		0			0
$651$	1.20565e6			0.111498
$652$		0			0
$653$		−	1.36182e7i		−	1.24979i	−0.780708	−	0.624897i	$-0.785141\pi$
$653$		−	1.36182e7i		−	1.24979i	0.780708	−	0.624897i	$-0.214859\pi$
$654$		0			0
$655$	−1.21695e6	+	176810.i	−0.110833	+	0.0161029i
$656$		0			0
$657$		−	3.58203e7i		−	3.23755i
$658$		0			0
$659$	1.63668e6			0.146808			0.0734039	−	0.997302i	$-0.476614\pi$
$659$	1.63668e6			0.146808			0.0734039	+	0.997302i	$0.476614\pi$
$660$		0			0
$661$	−1.43398e7			−1.27656			−0.638278	−	0.769806i	$-0.720353\pi$
$661$	−1.43398e7			−1.27656			−0.638278	+	0.769806i	$0.720353\pi$
$662$		0			0
$663$		−	9.25607e6i		−	0.817792i
$664$		0			0
$665$	699973.	+	4.81778e6i	0.0613801	+	0.422467i
$666$		0			0
$667$			2.34316e7i			2.03933i
$668$		0			0
$669$	−2.46492e7			−2.12931
$670$		0			0
$671$	−4.87179e6			−0.417717
$672$		0			0
$673$		−	9.54325e6i		−	0.812192i	−0.913830	−	0.406096i	$-0.866890\pi$
$673$		−	9.54325e6i		−	0.812192i	0.913830	−	0.406096i	$-0.133110\pi$
$674$		0			0
$675$	3.53119e6	+	1.18957e7i	0.298306	+	1.00492i
$676$		0			0
$677$		−	1.76523e7i		−	1.48023i	−0.672479	−	0.740116i	$-0.734771\pi$
$677$		−	1.76523e7i		−	1.48023i	0.672479	−	0.740116i	$-0.265229\pi$
$678$		0			0
$679$	323780.			0.0269510
$680$		0			0
$681$	9.58485e6			0.791986
$682$		0			0
$683$		−	3.15865e6i		−	0.259089i	−0.991574	−	0.129545i	$-0.958648\pi$
$683$		−	3.15865e6i		−	0.259089i	0.991574	−	0.129545i	$-0.0413515\pi$
$684$		0			0
$685$	2.65606e6	+	1.82812e7i	0.216278	+	1.48860i
$686$		0			0
$687$		−	3.01281e7i		−	2.43545i
$688$		0			0
$689$	−9.97099e6			−0.800184
$690$		0			0
$691$	−624449.			−0.0497510			−0.0248755	−	0.999691i	$-0.507919\pi$
$691$	−624449.			−0.0497510			−0.0248755	+	0.999691i	$0.507919\pi$
$692$		0			0
$693$			2.76991e6i			0.219095i
$694$		0			0
$695$	−2.23082e7	+	3.24115e6i	−1.75187	+	0.254529i
$696$		0			0
$697$		−	688223.i		−	0.0536596i
$698$		0			0
$699$	−3.00292e7			−2.32462
$700$		0			0
$701$	1.57395e7			1.20975			0.604875	−	0.796320i	$-0.293223\pi$
$701$	1.57395e7			1.20975			0.604875	+	0.796320i	$0.293223\pi$
$702$		0			0
$703$			4.15390e6i			0.317006i
$704$		0			0
$705$	−1.53935e7	+	2.23651e6i	−1.16644	+	0.169472i
$706$		0			0
$707$			1.89997e6i			0.142955i
$708$		0			0
$709$	5.74101e6			0.428917			0.214458	−	0.976733i	$-0.431201\pi$
$709$	5.74101e6			0.428917			0.214458	+	0.976733i	$0.431201\pi$
$710$		0			0
$711$	−1.94677e6			−0.144424
$712$		0			0
$713$		−	3.06531e6i		−	0.225814i
$714$		0			0
$715$	338391.	+	2.32908e6i	0.0247545	+	0.170380i
$716$		0			0
$717$		−	2.03686e7i		−	1.47967i
$718$		0			0
$719$	−3.79854e6			−0.274028			−0.137014	−	0.990569i	$-0.543750\pi$
$719$	−3.79854e6			−0.274028			−0.137014	+	0.990569i	$0.543750\pi$
$720$		0			0
$721$	−81252.6			−0.00582102
$722$		0			0
$723$		−	8.73865e6i		−	0.621725i
$724$		0			0
$725$	2.22271e7	−	6.59801e6i	1.57050	−	0.466195i
$726$		0			0
$727$		−	1.68558e7i		−	1.18280i	−0.806377	−	0.591402i	$-0.798575\pi$
$727$		−	1.68558e7i		−	1.18280i	0.806377	−	0.591402i	$-0.201425\pi$
$728$		0			0
$729$	2.30423e7			1.60585
$730$		0			0
$731$	−2.42218e6			−0.167653
$732$		0			0
$733$		−	2.16490e7i		−	1.48826i	−0.668035	−	0.744130i	$-0.732864\pi$
$733$		−	2.16490e7i		−	1.48826i	0.668035	−	0.744130i	$-0.267136\pi$
$734$		0			0
$735$	−489214.	−	3.36716e6i	−0.0334026	−	0.229904i
$736$		0			0
$737$			1.79222e6i			0.121541i
$738$		0			0
$739$	1.64235e7			1.10625			0.553125	−	0.833098i	$-0.313435\pi$
$739$	1.64235e7			1.10625			0.553125	+	0.833098i	$0.313435\pi$
$740$		0			0
$741$	−1.34103e7			−0.897205
$742$		0			0
$743$		−	5.84269e6i		−	0.388276i	−0.980974	−	0.194138i	$-0.937809\pi$
$743$		−	5.84269e6i		−	0.388276i	0.980974	−	0.194138i	$-0.0621910\pi$
$744$		0			0
$745$	2.44241e7	−	3.54857e6i	1.61223	−	0.234241i
$746$		0			0
$747$		−	2.70509e7i		−	1.77370i
$748$		0			0
$749$	−9.44977e6			−0.615484
$750$		0			0
$751$	−1.02447e7			−0.662823			−0.331412	−	0.943486i	$-0.607525\pi$
$751$	−1.02447e7			−0.662823			−0.331412	+	0.943486i	$0.607525\pi$
$752$		0			0
$753$		−	1.94265e6i		−	0.124855i
$754$		0			0
$755$	2.34779e7	−	3.41110e6i	1.49897	−	0.217784i
$756$		0			0
$757$		−	1.30935e7i		−	0.830452i	−0.909718	−	0.415226i	$-0.863703\pi$
$757$		−	1.30935e7i		−	0.830452i	0.909718	−	0.415226i	$-0.136297\pi$
$758$		0			0
$759$	1.13245e7			0.713532
$760$		0			0
$761$	−6.99157e6			−0.437636			−0.218818	−	0.975766i	$-0.570220\pi$
$761$	−6.99157e6			−0.437636			−0.218818	+	0.975766i	$0.570220\pi$
$762$		0			0
$763$			6.14088e6i			0.381874i
$764$		0			0
$765$	3.94041e6	+	2.71210e7i	0.243437	+	1.67553i
$766$		0			0
$767$		−	9.59208e6i		−	0.588741i
$768$		0			0
$769$	−1.46387e7			−0.892660			−0.446330	−	0.894868i	$-0.647269\pi$
$769$	−1.46387e7			−0.892660			−0.446330	+	0.894868i	$0.647269\pi$
$770$		0			0
$771$	−1.86265e7			−1.12849
$772$		0			0
$773$			2.11451e7i			1.27280i	0.771357	+	0.636402i	$0.219578\pi$
$773$			2.11451e7i			1.27280i	−0.771357	+	0.636402i	$0.780422\pi$
$774$		0			0
$775$	−2.90773e6	+	863147.i	−0.173900	+	0.0516215i
$776$		0			0
$777$		−	2.90317e6i		−	0.172512i
$778$		0			0
$779$	−997103.			−0.0588703
$780$		0			0
$781$	1.17787e7			0.690986
$782$		0			0
$783$			2.94611e7i			1.71729i
$784$		0			0
$785$	−1.23979e6	−	8.53323e6i	−0.0718082	−	0.494242i
$786$		0			0
$787$		−	1.80434e7i		−	1.03844i	−0.854641	−	0.519219i	$-0.826223\pi$
$787$		−	1.80434e7i		−	1.03844i	0.854641	−	0.519219i	$-0.173777\pi$
$788$		0			0
$789$	−1.13810e7			−0.650862
$790$		0			0
$791$	−5.79964e6			−0.329579
$792$		0			0
$793$			1.02513e7i			0.578889i
$794$		0			0
$795$	4.69805e7	−	6.82578e6i	2.63633	−	0.383031i
$796$		0			0
$797$			2.14077e7i			1.19378i	0.802323	+	0.596890i	$0.203597\pi$
$797$			2.14077e7i			1.19378i	−0.802323	+	0.596890i	$0.796403\pi$
$798$		0			0
$799$	−1.34653e7			−0.746188
$800$		0			0
$801$	−3.46236e7			−1.90674
$802$		0			0
$803$			1.26785e7i			0.693869i
$804$		0			0
$805$	−8.56085e6	+	1.24380e6i	−0.465615	+	0.0676491i
$806$		0			0
$807$		−	4.23286e7i		−	2.28797i
$808$		0			0
$809$	3.85342e6			0.207002			0.103501	−	0.994629i	$-0.466995\pi$
$809$	3.85342e6			0.207002			0.103501	+	0.994629i	$0.466995\pi$
$810$		0			0
$811$	−1.82776e7			−0.975814			−0.487907	−	0.872896i	$-0.662239\pi$
$811$	−1.82776e7			−0.975814			−0.487907	+	0.872896i	$0.662239\pi$
$812$		0			0
$813$			1.18359e7i			0.628020i
$814$		0			0
$815$	−1.25477e6	−	8.63634e6i	−0.0661714	−	0.455445i
$816$		0			0
$817$			3.50927e6i			0.183934i
$818$		0			0
$819$	5.82848e6			0.303630
$820$		0			0
$821$	2.63108e7			1.36231			0.681156	−	0.732138i	$-0.261477\pi$
$821$	2.63108e7			1.36231			0.681156	+	0.732138i	$0.261477\pi$
$822$		0			0
$823$			3.23546e7i			1.66509i	0.553960	+	0.832543i	$0.313116\pi$
$823$			3.23546e7i			1.66509i	−0.553960	+	0.832543i	$0.686884\pi$
$824$		0			0
$825$	−3.18881e6	−	1.07423e7i	−0.163115	−	0.549494i
$826$		0			0
$827$		−	5.30031e6i		−	0.269487i	−0.990881	−	0.134743i	$-0.956979\pi$
$827$		−	5.30031e6i		−	0.269487i	0.990881	−	0.134743i	$-0.0430210\pi$
$828$		0			0
$829$	−2.12766e7			−1.07526			−0.537632	−	0.843180i	$-0.680681\pi$
$829$	−2.12766e7			−1.07526			−0.537632	+	0.843180i	$0.680681\pi$
$830$		0			0
$831$	4.71586e7			2.36896
$832$		0			0
$833$		−	2.94539e6i		−	0.147072i
$834$		0			0
$835$	−675446.	−	4.64896e6i	−0.0335254	−	0.230749i
$836$		0			0
$837$		−	3.85408e6i		−	0.190155i
$838$		0			0
$839$	−2.68060e6			−0.131470			−0.0657350	−	0.997837i	$-0.520939\pi$
$839$	−2.68060e6			−0.131470			−0.0657350	+	0.997837i	$0.520939\pi$
$840$		0			0
$841$	3.45367e7			1.68380
$842$		0			0
$843$		−	9.75881e6i		−	0.472964i
$844$		0			0
$845$	−1.56394e7	+	2.27224e6i	−0.753490	+	0.109474i
$846$		0			0
$847$			6.91110e6i			0.331008i
$848$		0			0
$849$	−6.48780e6			−0.308907
$850$		0			0
$851$	−7.38118e6			−0.349383
$852$		0			0
$853$			5.65172e6i			0.265955i	0.991119	+	0.132977i	$0.0424538\pi$
$853$			5.65172e6i			0.265955i	−0.991119	+	0.132977i	$0.957546\pi$
$854$		0			0
$855$	3.92931e7	−	5.70889e6i	1.83824	−	0.267077i
$856$		0			0
$857$			2.76367e7i			1.28539i	0.766123	+	0.642694i	$0.222183\pi$
$857$			2.76367e7i			1.28539i	−0.766123	+	0.642694i	$0.777817\pi$
$858$		0			0
$859$	−2.44636e6			−0.113119			−0.0565597	−	0.998399i	$-0.518013\pi$
$859$	−2.44636e6			−0.113119			−0.0565597	+	0.998399i	$0.518013\pi$
$860$		0			0
$861$	696879.			0.0320368
$862$		0			0
$863$			3.06078e7i			1.39896i	0.714652	+	0.699481i	$0.246585\pi$
$863$			3.06078e7i			1.39896i	−0.714652	+	0.699481i	$0.753415\pi$
$864$		0			0
$865$	1.25337e6	+	8.62670e6i	0.0569560	+	0.392017i
$866$		0			0
$867$			2.15539e6i			0.0973821i
$868$		0			0
$869$	689051.			0.0309529
$870$		0			0
$871$	3.77121e6			0.168436
$872$		0			0
$873$		−	2.64070e6i		−	0.117269i
$874$		0			0
$875$	3.59048e6	+	7.77053e6i	0.158538	+	0.343108i
$876$		0			0
$877$		−	2.14914e7i		−	0.943552i	−0.881718	−	0.471776i	$-0.843613\pi$
$877$		−	2.14914e7i		−	0.943552i	0.881718	−	0.471776i	$-0.156387\pi$
$878$		0			0
$879$	8.75881e6			0.382360
$880$		0			0
$881$	2.79915e7			1.21503			0.607514	−	0.794309i	$-0.292167\pi$
$881$	2.79915e7			1.21503			0.607514	+	0.794309i	$0.292167\pi$
$882$		0			0
$883$			2.40091e7i			1.03627i	0.855298	+	0.518136i	$0.173374\pi$
$883$			2.40091e7i			1.03627i	−0.855298	+	0.518136i	$0.826626\pi$
$884$		0			0
$885$	6.56639e6	+	4.51952e7i	0.281818	+	1.93970i
$886$		0			0
$887$		−	4.01622e7i		−	1.71399i	−0.515326	−	0.856995i	$-0.672329\pi$
$887$		−	4.01622e7i		−	1.71399i	0.515326	−	0.856995i	$-0.327671\pi$
$888$		0			0
$889$	−3.84680e6			−0.163247
$890$		0			0
$891$	502038.			0.0211857
$892$		0			0
$893$			1.95086e7i			0.818648i
$894$		0			0
$895$	−2.68514e7	+	3.90123e6i	−1.12049	+	0.162796i
$896$		0			0
$897$		−	2.38291e7i		−	0.988841i
$898$		0			0
$899$	−7.20132e6			−0.297176
$900$		0			0
$901$	4.10957e7			1.68649
$902$		0			0
$903$		−	2.45264e6i		−	0.100095i
$904$		0			0
$905$	−552990.	+	80343.8i	−0.0224438	+	0.00326085i
$906$		0			0
$907$			2.19331e7i			0.885283i	0.896699	+	0.442641i	$0.145959\pi$
$907$			2.19331e7i			0.885283i	−0.896699	+	0.442641i	$0.854041\pi$
$908$		0			0
$909$	1.54959e7			0.622025
$910$		0			0
$911$	2.21245e7			0.883236			0.441618	−	0.897203i	$-0.354405\pi$
$911$	2.21245e7			0.883236			0.441618	+	0.897203i	$0.354405\pi$
$912$		0			0
$913$			9.57457e6i			0.380139i
$914$		0			0
$915$	−7.01766e6	−	4.83012e7i	−0.277102	−	1.90724i
$916$		0			0
$917$		−	1.07790e6i		−	0.0423307i
$918$		0			0
$919$	−1.82158e7			−0.711476			−0.355738	−	0.934586i	$-0.615770\pi$
$919$	−1.82158e7			−0.711476			−0.355738	+	0.934586i	$0.615770\pi$
$920$		0			0
$921$	3.51084e7			1.36383
$922$		0			0
$923$		−	2.47849e7i		−	0.957596i
$924$		0			0
$925$	2.07844e6	+	7.00174e6i	0.0798697	+	0.269062i
$926$		0			0
$927$			662686.i			0.0253284i
$928$		0			0
$929$	−1.48086e7			−0.562958			−0.281479	−	0.959567i	$-0.590825\pi$
$929$	−1.48086e7			−0.562958			−0.281479	+	0.959567i	$0.590825\pi$
$930$		0			0
$931$	−4.26731e6			−0.161354
$932$		0			0
$933$			2.28154e7i			0.858071i
$934$		0			0
$935$	−1.39469e6	−	9.59938e6i	−0.0521734	−	0.359099i
$936$		0			0
$937$		−	3.96760e7i		−	1.47631i	−0.674629	−	0.738157i	$-0.735696\pi$
$937$		−	3.96760e7i		−	1.47631i	0.674629	−	0.738157i	$-0.264304\pi$
$938$		0			0
$939$	7.11453e7			2.63319
$940$		0			0
$941$	−2.74007e7			−1.00876			−0.504380	−	0.863482i	$-0.668279\pi$
$941$	−2.74007e7			−1.00876			−0.504380	+	0.863482i	$0.668279\pi$
$942$		0			0
$943$		−	1.77178e6i		−	0.0648830i
$944$		0			0
$945$	−1.07638e7	+	1.56386e6i	−0.392089	+	0.0569664i
$946$		0			0
$947$		−	3.70369e7i		−	1.34202i	−0.741448	−	0.671011i	$-0.765860\pi$
$947$		−	3.70369e7i		−	1.34202i	0.741448	−	0.671011i	$-0.234140\pi$
$948$		0			0
$949$	2.66782e7			0.961591
$950$		0			0
$951$	−6.22301e7			−2.23125
$952$		0			0
$953$		−	3.01349e7i		−	1.07482i	−0.843320	−	0.537412i	$-0.819402\pi$
$953$		−	3.01349e7i		−	1.07482i	0.843320	−	0.537412i	$-0.180598\pi$
$954$		0			0
$955$	2.20494e7	−	3.20354e6i	0.782326	−	0.113664i
$956$		0			0
$957$		−	2.66046e7i		−	0.939023i
$958$		0			0
$959$	−1.61924e7			−0.568544
$960$		0			0
$961$	−2.76871e7			−0.967094
$962$		0			0
$963$			7.70710e7i			2.67809i
$964$		0			0
$965$	5.38447e6	+	3.70602e7i	0.186134	+	1.28112i
$966$		0			0
$967$			5.27724e7i			1.81485i	0.420213	+	0.907425i	$0.361955\pi$
$967$			5.27724e7i			1.81485i	−0.420213	+	0.907425i	$0.638045\pi$
$968$		0			0
$969$	5.52708e7			1.89098
$970$		0			0
$971$	−2.09055e7			−0.711560			−0.355780	−	0.934570i	$-0.615785\pi$
$971$	−2.09055e7			−0.711560			−0.355780	+	0.934570i	$0.615785\pi$
$972$		0			0
$973$		−	1.97593e7i		−	0.669098i
$974$		0			0
$975$	−2.26041e7	+	6.70993e6i	−0.761511	+	0.226051i
$976$		0			0
$977$			609167.i			0.0204174i	0.999948	+	0.0102087i	$0.00324959\pi$
$977$			609167.i			0.0204174i	−0.999948	+	0.0102087i	$0.996750\pi$
$978$		0			0
$979$	1.22549e7			0.408650
$980$		0			0
$981$	5.00842e7			1.66161
$982$		0			0
$983$			4.01952e6i			0.132676i	0.997797	+	0.0663378i	$0.0211315\pi$
$983$			4.01952e6i			0.132676i	−0.997797	+	0.0663378i	$0.978869\pi$
$984$		0			0
$985$	−7.53677e6	−	5.18741e7i	−0.247511	−	1.70357i
$986$		0			0
$987$		−	1.36346e7i		−	0.445503i
$988$		0			0
$989$	−6.23572e6			−0.202720
$990$		0			0
$991$	−2.08792e6			−0.0675351			−0.0337676	−	0.999430i	$-0.510751\pi$
$991$	−2.08792e6			−0.0675351			−0.0337676	+	0.999430i	$0.510751\pi$
$992$		0			0
$993$		−	9.82064e7i		−	3.16058i
$994$		0			0
$995$	−5.76101e6	+	837014.i	−0.184476	+	0.0268025i
$996$		0			0
$997$			6.09104e6i			0.194068i	0.995281	+	0.0970340i	$0.0309356\pi$
$997$			6.09104e6i			0.194068i	−0.995281	+	0.0970340i	$0.969064\pi$
$998$		0			0
$999$	−9.28053e6			−0.294211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.g.f.449.2		16
4.3	odd	2		140.6.e.a.29.15	yes	16
5.4	even	2	inner	560.6.g.f.449.15		16
20.3	even	4		700.6.a.p.1.1		8
20.7	even	4		700.6.a.o.1.8		8
20.19	odd	2		140.6.e.a.29.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.e.a.29.2	✓	16	20.19	odd	2
140.6.e.a.29.15	yes	16	4.3	odd	2
560.6.g.f.449.2		16	1.1	even	1	trivial
560.6.g.f.449.15		16	5.4	even	2	inner
700.6.a.o.1.8		8	20.7	even	4
700.6.a.p.1.1		8	20.3	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-25.3503i q^{3} +(-55.3209 + 8.03755i) q^{5} -49.0000i q^{7} -399.638 q^{9} +141.450 q^{11} -297.641i q^{13} +(203.754 + 1402.40i) q^{15} +1226.74i q^{17} +1777.30 q^{19} -1242.16 q^{21} +3158.14i q^{23} +(2995.80 - 889.288i) q^{25} +3970.81i q^{27} +7419.42 q^{29} -970.605 q^{31} -3585.80i q^{33} +(393.840 + 2710.72i) q^{35} +2337.19i q^{37} -7545.29 q^{39} -561.020 q^{41} +1974.49i q^{43} +(22108.3 - 3212.11i) q^{45} +10976.5i q^{47} -2401.00 q^{49} +31098.1 q^{51} -33500.0i q^{53} +(-7825.14 + 1136.91i) q^{55} -45055.2i q^{57} +32227.0 q^{59} -34441.8 q^{61} +19582.2i q^{63} +(2392.30 + 16465.8i) q^{65} +12670.3i q^{67} +80059.9 q^{69} +83271.0 q^{71} +89632.0i q^{73} +(-22543.7 - 75944.3i) q^{75} -6931.05i q^{77} +4871.34 q^{79} +3549.22 q^{81} +67688.7i q^{83} +(-9859.95 - 67864.1i) q^{85} -188085. i q^{87} +86637.4 q^{89} -14584.4 q^{91} +24605.1i q^{93} +(-98322.0 + 14285.2i) q^{95} +6607.75i q^{97} -56528.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{5} - 1606 q^{9} - 294 q^{11} - 754 q^{15} + 3512 q^{19} + 1274 q^{21} + 7500 q^{25} - 20870 q^{29} - 4028 q^{31} - 3038 q^{35} + 25358 q^{39} + 24952 q^{41} - 10650 q^{45} - 38416 q^{49} + 12302 q^{51}+ \cdots - 293964 q^{99}+O(q^{100}) \) 16 * q - 2 * q^5 - 1606 * q^9 - 294 * q^11 - 754 * q^15 + 3512 * q^19 + 1274 * q^21 + 7500 * q^25 - 20870 * q^29 - 4028 * q^31 - 3038 * q^35 + 25358 * q^39 + 24952 * q^41 - 10650 * q^45 - 38416 * q^49 + 12302 * q^51 + 7612 * q^55 - 36156 * q^59 - 14556 * q^61 - 72446 * q^65 + 99340 * q^69 + 47992 * q^71 + 155016 * q^75 - 430 * q^79 + 156208 * q^81 - 195046 * q^85 + 304180 * q^89 - 16954 * q^91 - 149080 * q^95 - 293964 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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