LMFDB - Embedded newform 160.6.c.c.129.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(129,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.c (of order $2$, degree $1$, 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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 165x^{10} + 9528x^{8} + 254984x^{6} + 3245664x^{4} + 15975501x^{2} + 588289 $ x^12 + 165x^10 + 9528x^8 + 254984x^6 + 3245664x^4 + 15975501*x^2 + 588289
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{44}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.5
Root			$-0.192622i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.6.c.c.129.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.81633i q^{3} +(-16.9634 - 53.2658i) q^{5} -222.821i q^{7} +146.640 q^{9} +O(q^{10})$	$q-9.81633i q^{3} +(-16.9634 - 53.2658i) q^{5} -222.821i q^{7} +146.640 q^{9} +407.674 q^{11} -465.665i q^{13} +(-522.874 + 166.519i) q^{15} +284.781i q^{17} -323.952 q^{19} -2187.28 q^{21} +12.0533i q^{23} +(-2549.48 + 1807.14i) q^{25} -3824.83i q^{27} -4381.26 q^{29} +10140.5 q^{31} -4001.86i q^{33} +(-11868.7 + 3779.81i) q^{35} -6906.96i q^{37} -4571.12 q^{39} -10733.8 q^{41} +6002.36i q^{43} +(-2487.51 - 7810.87i) q^{45} +25033.9i q^{47} -32842.1 q^{49} +2795.51 q^{51} -10570.8i q^{53} +(-6915.56 - 21715.1i) q^{55} +3180.02i q^{57} +30921.1 q^{59} -34366.3 q^{61} -32674.4i q^{63} +(-24804.0 + 7899.27i) q^{65} -45890.9i q^{67} +118.319 q^{69} -58026.5 q^{71} +68664.7i q^{73} +(17739.5 + 25026.6i) q^{75} -90838.3i q^{77} +65066.0 q^{79} -1912.37 q^{81} +51017.2i q^{83} +(15169.1 - 4830.87i) q^{85} +43007.9i q^{87} +59409.1 q^{89} -103760. q^{91} -99542.4i q^{93} +(5495.34 + 17255.5i) q^{95} +73186.2i q^{97} +59781.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 60 q^{5} - 1676 q^{9}+O(q^{10}) $ 12 * q - 60 * q^5 - 1676 * q^9	$ 12 q - 60 q^{5} - 1676 q^{9} + 688 q^{21} + 1500 q^{25} - 21304 q^{29} - 109672 q^{41} - 75140 q^{45} - 32348 q^{49} + 64552 q^{61} - 160 q^{65} - 166352 q^{69} - 173764 q^{81} + 151360 q^{85} - 3720 q^{89}+O(q^{100}) $ 12 * q - 60 * q^5 - 1676 * q^9 + 688 * q^21 + 1500 * q^25 - 21304 * q^29 - 109672 * q^41 - 75140 * q^45 - 32348 * q^49 + 64552 * q^61 - 160 * q^65 - 166352 * q^69 - 173764 * q^81 + 151360 * q^85 - 3720 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	9.81633i		−	0.629718i	−0.949138	−	0.314859i	$-0.898043\pi$
$3$		−	9.81633i		−	0.629718i	0.949138	−	0.314859i	$-0.101957\pi$
$4$		0			0
$5$	−16.9634	−	53.2658i	−0.303451	−	0.952847i
$5$	−16.9634	−	53.2658i	−0.303451	−	0.952847i
$6$		0			0
$7$		−	222.821i		−	1.71874i	−0.511353	−	0.859371i	$-0.670855\pi$
$7$		−	222.821i		−	1.71874i	0.511353	−	0.859371i	$-0.329145\pi$
$8$		0			0
$9$	146.640			0.603455
$10$		0			0
$11$	407.674			1.01585			0.507927	−	0.861400i	$-0.330412\pi$
$11$	407.674			1.01585			0.507927	+	0.861400i	$0.330412\pi$
$12$		0			0
$13$		−	465.665i		−	0.764214i	−0.924118	−	0.382107i	$-0.875199\pi$
$13$		−	465.665i		−	0.764214i	0.924118	−	0.382107i	$-0.124801\pi$
$14$		0			0
$15$	−522.874	+	166.519i	−0.600025	+	0.191089i
$16$		0			0
$17$			284.781i			0.238995i	0.992834	+	0.119498i	$0.0381284\pi$
$17$			284.781i			0.238995i	−0.992834	+	0.119498i	$0.961872\pi$
$18$		0			0
$19$	−323.952			−0.205872			−0.102936	−	0.994688i	$-0.532824\pi$
$19$	−323.952			−0.205872			−0.102936	+	0.994688i	$0.532824\pi$
$20$		0			0
$21$	−2187.28			−1.08232
$22$		0			0
$23$			12.0533i			0.00475102i	0.999997	+	0.00237551i	$0.000756149\pi$
$23$			12.0533i			0.00475102i	−0.999997	+	0.00237551i	$0.999244\pi$
$24$		0			0
$25$	−2549.48	+	1807.14i	−0.815835	+	0.578285i
$26$		0			0
$27$		−	3824.83i		−	1.00972i
$28$		0			0
$29$	−4381.26			−0.967395			−0.483697	−	0.875235i	$-0.660706\pi$
$29$	−4381.26			−0.967395			−0.483697	+	0.875235i	$0.660706\pi$
$30$		0			0
$31$	10140.5			1.89520			0.947599	−	0.319461i	$-0.103502\pi$
$31$	10140.5			1.89520			0.947599	+	0.319461i	$0.103502\pi$
$32$		0			0
$33$		−	4001.86i		−	0.639702i
$34$		0			0
$35$	−11868.7	+	3779.81i	−1.63770	+	0.521554i
$36$		0			0
$37$		−	6906.96i		−	0.829435i	−0.909950	−	0.414718i	$-0.863880\pi$
$37$		−	6906.96i		−	0.829435i	0.909950	−	0.414718i	$-0.136120\pi$
$38$		0			0
$39$	−4571.12			−0.481239
$40$		0			0
$41$	−10733.8			−0.997231			−0.498615	−	0.866823i	$-0.666158\pi$
$41$	−10733.8			−0.997231			−0.498615	+	0.866823i	$0.666158\pi$
$42$		0			0
$43$			6002.36i			0.495052i	0.968881	+	0.247526i	$0.0796175\pi$
$43$			6002.36i			0.495052i	−0.968881	+	0.247526i	$0.920382\pi$
$44$		0			0
$45$	−2487.51	−	7810.87i	−0.183119	−	0.575001i
$46$		0			0
$47$			25033.9i			1.65304i	0.562905	+	0.826522i	$0.309684\pi$
$47$			25033.9i			1.65304i	−0.562905	+	0.826522i	$0.690316\pi$
$48$		0			0
$49$	−32842.1			−1.95407
$50$		0			0
$51$	2795.51			0.150499
$52$		0			0
$53$		−	10570.8i		−	0.516913i	−0.966023	−	0.258457i	$-0.916786\pi$
$53$		−	10570.8i		−	0.516913i	0.966023	−	0.258457i	$-0.0832139\pi$
$54$		0			0
$55$	−6915.56	−	21715.1i	−0.308262	−	0.967954i
$56$		0			0
$57$			3180.02i			0.129641i
$58$		0			0
$59$	30921.1			1.15644			0.578222	−	0.815880i	$-0.303747\pi$
$59$	30921.1			1.15644			0.578222	+	0.815880i	$0.303747\pi$
$60$		0			0
$61$	−34366.3			−1.18252			−0.591260	−	0.806481i	$-0.701369\pi$
$61$	−34366.3			−1.18252			−0.591260	+	0.806481i	$0.701369\pi$
$62$		0			0
$63$		−	32674.4i		−	1.03718i
$64$		0			0
$65$	−24804.0	+	7899.27i	−0.728179	+	0.231902i
$66$		0			0
$67$		−	45890.9i		−	1.24894i	−0.781051	−	0.624468i	$-0.785316\pi$
$67$		−	45890.9i		−	1.24894i	0.781051	−	0.624468i	$-0.214684\pi$
$68$		0			0
$69$	118.319			0.00299180
$70$		0			0
$71$	−58026.5			−1.36609			−0.683047	−	0.730375i	$-0.739345\pi$
$71$	−58026.5			−1.36609			−0.683047	+	0.730375i	$0.739345\pi$
$72$		0			0
$73$			68664.7i			1.50809i	0.656825	+	0.754043i	$0.271899\pi$
$73$			68664.7i			1.50809i	−0.656825	+	0.754043i	$0.728101\pi$
$74$		0			0
$75$	17739.5	+	25026.6i	0.364156	+	0.513746i
$76$		0			0
$77$		−	90838.3i		−	1.74599i
$78$		0			0
$79$	65066.0			1.17297			0.586484	−	0.809961i	$-0.300512\pi$
$79$	65066.0			1.17297			0.586484	+	0.809961i	$0.300512\pi$
$80$		0			0
$81$	−1912.37			−0.0323861
$82$		0			0
$83$			51017.2i			0.812871i	0.913679	+	0.406436i	$0.133228\pi$
$83$			51017.2i			0.812871i	−0.913679	+	0.406436i	$0.866772\pi$
$84$		0			0
$85$	15169.1	−	4830.87i	0.227726	−	0.0725233i
$86$		0			0
$87$			43007.9i			0.609186i
$88$		0			0
$89$	59409.1			0.795019			0.397510	−	0.917598i	$-0.369874\pi$
$89$	59409.1			0.795019			0.397510	+	0.917598i	$0.369874\pi$
$90$		0			0
$91$	−103760.			−1.31349
$92$		0			0
$93$		−	99542.4i		−	1.19344i
$94$		0			0
$95$	5495.34	+	17255.5i	0.0624720	+	0.196164i
$96$		0			0
$97$			73186.2i			0.789768i	0.918731	+	0.394884i	$0.129215\pi$
$97$			73186.2i			0.789768i	−0.918731	+	0.394884i	$0.870785\pi$
$98$		0			0
$99$	59781.2			0.613023
$100$		0			0
$101$	162003.			1.58023			0.790116	−	0.612958i	$-0.210021\pi$
$101$	162003.			1.58023			0.790116	+	0.612958i	$0.210021\pi$
$102$		0			0
$103$		−	23887.3i		−	0.221857i	−0.993828	−	0.110928i	$-0.964618\pi$
$103$		−	23887.3i		−	0.221857i	0.993828	−	0.110928i	$-0.0353825\pi$
$104$		0			0
$105$	37103.8	+	116507.i	0.328432	+	1.03129i
$106$		0			0
$107$		−	88952.0i		−	0.751098i	−0.926803	−	0.375549i	$-0.877454\pi$
$107$		−	88952.0i		−	0.751098i	0.926803	−	0.375549i	$-0.122546\pi$
$108$		0			0
$109$	73145.1			0.589683			0.294842	−	0.955546i	$-0.404733\pi$
$109$	73145.1			0.589683			0.294842	+	0.955546i	$0.404733\pi$
$110$		0			0
$111$	−67801.0			−0.522310
$112$		0			0
$113$			162814.i			1.19949i	0.800192	+	0.599744i	$0.204731\pi$
$113$			162814.i			1.19949i	−0.800192	+	0.599744i	$0.795269\pi$
$114$		0			0
$115$	642.029	−	204.466i	0.00452700	−	0.00144170i
$116$		0			0
$117$		−	68284.9i		−	0.461169i
$118$		0			0
$119$	63455.2			0.410771
$120$		0			0
$121$	5147.29			0.0319606
$122$		0			0
$123$			105367.i			0.627974i
$124$		0			0
$125$	139507.	+	105145.i	0.798583	+	0.601884i
$126$		0			0
$127$			48080.1i			0.264518i	0.991215	+	0.132259i	$0.0422231\pi$
$127$			48080.1i			0.264518i	−0.991215	+	0.132259i	$0.957777\pi$
$128$		0			0
$129$	58921.1			0.311743
$130$		0			0
$131$	245815.			1.25150			0.625750	−	0.780024i	$-0.284793\pi$
$131$	245815.			1.25150			0.625750	+	0.780024i	$0.284793\pi$
$132$		0			0
$133$			72183.2i			0.353840i
$134$		0			0
$135$	−203733.	+	64882.3i	−0.962113	+	0.306402i
$136$		0			0
$137$		−	188160.i		−	0.856496i	−0.903661	−	0.428248i	$-0.859131\pi$
$137$		−	188160.i		−	0.856496i	0.903661	−	0.428248i	$-0.140869\pi$
$138$		0			0
$139$	−277146.			−1.21666			−0.608332	−	0.793682i	$-0.708161\pi$
$139$	−277146.			−1.21666			−0.608332	+	0.793682i	$0.708161\pi$
$140$		0			0
$141$	245741.			1.04095
$142$		0			0
$143$		−	189839.i		−	0.776330i
$144$		0			0
$145$	74321.2	+	233371.i	0.293557	+	0.921779i
$146$		0			0
$147$			322389.i			1.23051i
$148$		0			0
$149$	139067.			0.513167			0.256584	−	0.966522i	$-0.417403\pi$
$149$	139067.			0.513167			0.256584	+	0.966522i	$0.417403\pi$
$150$		0			0
$151$	514519.			1.83636			0.918182	−	0.396158i	$-0.129657\pi$
$151$	514519.			1.83636			0.918182	+	0.396158i	$0.129657\pi$
$152$		0			0
$153$			41760.2i			0.144223i
$154$		0			0
$155$	−172018.	−	540141.i	−0.575100	−	1.80583i
$156$		0			0
$157$		−	578669.i		−	1.87362i	−0.349840	−	0.936809i	$-0.613764\pi$
$157$		−	578669.i		−	1.87362i	0.349840	−	0.936809i	$-0.386236\pi$
$158$		0			0
$159$	−103766.			−0.325509
$160$		0			0
$161$	2685.73			0.00816578
$162$		0			0
$163$		−	269341.i		−	0.794025i	−0.917813	−	0.397012i	$-0.870047\pi$
$163$		−	269341.i		−	0.794025i	0.917813	−	0.397012i	$-0.129953\pi$
$164$		0			0
$165$	−213162.	+	67885.4i	−0.609538	+	0.194118i
$166$		0			0
$167$		−	449114.i		−	1.24613i	−0.782168	−	0.623067i	$-0.785886\pi$
$167$		−	449114.i		−	1.24613i	0.782168	−	0.623067i	$-0.214114\pi$
$168$		0			0
$169$	154449.			0.415977
$170$		0			0
$171$	−47504.2			−0.124234
$172$		0			0
$173$			367584.i			0.933772i	0.884317	+	0.466886i	$0.154624\pi$
$173$			367584.i			0.933772i	−0.884317	+	0.466886i	$0.845376\pi$
$174$		0			0
$175$	402669.	+	568078.i	0.993923	+	1.40221i
$176$		0			0
$177$		−	303531.i		−	0.728233i
$178$		0			0
$179$	55367.6			0.129159			0.0645793	−	0.997913i	$-0.479429\pi$
$179$	55367.6			0.129159			0.0645793	+	0.997913i	$0.479429\pi$
$180$		0			0
$181$	−652653.			−1.48076			−0.740382	−	0.672186i	$-0.765355\pi$
$181$	−652653.			−1.48076			−0.740382	+	0.672186i	$0.765355\pi$
$182$		0			0
$183$			337351.i			0.744653i
$184$		0			0
$185$	−367904.	+	117166.i	−0.790325	+	0.251693i
$186$		0			0
$187$			116098.i			0.242784i
$188$		0			0
$189$	−852252.			−1.73546
$190$		0			0
$191$	−307365.			−0.609636			−0.304818	−	0.952411i	$-0.598596\pi$
$191$	−307365.			−0.609636			−0.304818	+	0.952411i	$0.598596\pi$
$192$		0			0
$193$			91632.5i			0.177075i	0.996073	+	0.0885373i	$0.0282193\pi$
$193$			91632.5i			0.177075i	−0.996073	+	0.0885373i	$0.971781\pi$
$194$		0			0
$195$	77541.9	+	243484.i	0.146033	+	0.458547i
$196$		0			0
$197$		−	256405.i		−	0.470718i	−0.971908	−	0.235359i	$-0.924373\pi$
$197$		−	256405.i		−	0.470718i	0.971908	−	0.235359i	$-0.0756266\pi$
$198$		0			0
$199$	−994011.			−1.77934			−0.889669	−	0.456605i	$-0.849065\pi$
$199$	−994011.			−1.77934			−0.889669	+	0.456605i	$0.849065\pi$
$200$		0			0
$201$	−450481.			−0.786477
$202$		0			0
$203$			976235.i			1.66270i
$204$		0			0
$205$	182083.	+	571746.i	0.302611	+	0.950208i
$206$		0			0
$207$			1767.49i			0.00286703i
$208$		0			0
$209$	−132067.			−0.209136
$210$		0			0
$211$	−81503.0			−0.126028			−0.0630140	−	0.998013i	$-0.520071\pi$
$211$	−81503.0			−0.126028			−0.0630140	+	0.998013i	$0.520071\pi$
$212$		0			0
$213$			569607.i			0.860253i
$214$		0			0
$215$	319720.	−	101821.i	0.471709	−	0.150224i
$216$		0			0
$217$		−	2.25951e6i		−	3.25736i
$218$		0			0
$219$	674035.			0.949669
$220$		0			0
$221$	132612.			0.182643
$222$		0			0
$223$		−	944128.i		−	1.27136i	−0.771953	−	0.635680i	$-0.780720\pi$
$223$		−	944128.i		−	1.27136i	0.771953	−	0.635680i	$-0.219280\pi$
$224$		0			0
$225$	−373855.	+	264999.i	−0.492320	+	0.348969i
$226$		0			0
$227$		−	225503.i		−	0.290462i	−0.989398	−	0.145231i	$-0.953608\pi$
$227$		−	225503.i		−	0.290462i	0.989398	−	0.145231i	$-0.0463925\pi$
$228$		0			0
$229$	−1.08157e6			−1.36290			−0.681450	−	0.731865i	$-0.738650\pi$
$229$	−1.08157e6			−1.36290			−0.681450	+	0.731865i	$0.738650\pi$
$230$		0			0
$231$	−891699.			−1.09948
$232$		0			0
$233$		−	768283.i		−	0.927110i	−0.886068	−	0.463555i	$-0.846574\pi$
$233$		−	768283.i		−	0.927110i	0.886068	−	0.463555i	$-0.153426\pi$
$234$		0			0
$235$	1.33345e6	−	424661.i	1.57510	−	0.501618i
$236$		0			0
$237$		−	638709.i		−	0.738639i
$238$		0			0
$239$	683119.			0.773573			0.386787	−	0.922169i	$-0.373585\pi$
$239$	683119.			0.773573			0.386787	+	0.922169i	$0.373585\pi$
$240$		0			0
$241$	−304420.			−0.337622			−0.168811	−	0.985648i	$-0.553993\pi$
$241$	−304420.			−0.337622			−0.168811	+	0.985648i	$0.553993\pi$
$242$		0			0
$243$		−	910662.i		−	0.989330i
$244$		0			0
$245$	557115.	+	1.74936e6i	0.592966	+	1.86193i
$246$		0			0
$247$			150853.i			0.157330i
$248$		0			0
$249$	500802.			0.511880
$250$		0			0
$251$	−630802.			−0.631988			−0.315994	−	0.948761i	$-0.602338\pi$
$251$	−630802.			−0.631988			−0.315994	+	0.948761i	$0.602338\pi$
$252$		0			0
$253$			4913.83i			0.00482635i
$254$		0			0
$255$	−47421.4	−	148905.i	−0.0456692	−	0.143403i
$256$		0			0
$257$		−	847796.i		−	0.800679i	−0.916367	−	0.400340i	$-0.868892\pi$
$257$		−	847796.i		−	0.800679i	0.916367	−	0.400340i	$-0.131108\pi$
$258$		0			0
$259$	−1.53901e6			−1.42559
$260$		0			0
$261$	−642466.			−0.583780
$262$		0			0
$263$			709631.i			0.632620i	0.948656	+	0.316310i	$0.102444\pi$
$263$			709631.i			0.632620i	−0.948656	+	0.316310i	$0.897556\pi$
$264$		0			0
$265$	−563061.	+	179317.i	−0.492539	+	0.156858i
$266$		0			0
$267$		−	583179.i		−	0.500638i
$268$		0			0
$269$	424943.			0.358055			0.179028	−	0.983844i	$-0.442705\pi$
$269$	424943.			0.358055			0.179028	+	0.983844i	$0.442705\pi$
$270$		0			0
$271$	833724.			0.689603			0.344801	−	0.938676i	$-0.387946\pi$
$271$	833724.			0.689603			0.344801	+	0.938676i	$0.387946\pi$
$272$		0			0
$273$			1.01854e6i			0.827126i
$274$		0			0
$275$	−1.03936e6	+	736725.i	−0.828769	+	0.587454i
$276$		0			0
$277$			424122.i			0.332117i	0.986116	+	0.166058i	$0.0531040\pi$
$277$			424122.i			0.332117i	−0.986116	+	0.166058i	$0.946896\pi$
$278$		0			0
$279$	1.48700e6			1.14367
$280$		0			0
$281$	1.72721e6			1.30491			0.652453	−	0.757830i	$-0.273740\pi$
$281$	1.72721e6			1.30491			0.652453	+	0.757830i	$0.273740\pi$
$282$		0			0
$283$			823843.i			0.611475i	0.952116	+	0.305737i	$0.0989029\pi$
$283$			823843.i			0.611475i	−0.952116	+	0.305737i	$0.901097\pi$
$284$		0			0
$285$	169386.	−	53944.0i	0.123528	−	0.0393397i
$286$		0			0
$287$			2.39172e6i			1.71398i
$288$		0			0
$289$	1.33876e6			0.942881
$290$		0			0
$291$	718419.			0.497331
$292$		0			0
$293$			2.60970e6i			1.77591i	0.459930	+	0.887955i	$0.347875\pi$
$293$			2.60970e6i			1.77591i	−0.459930	+	0.887955i	$0.652125\pi$
$294$		0			0
$295$	−524528.	−	1.64703e6i	−0.350924	−	1.10191i
$296$		0			0
$297$		−	1.55929e6i		−	1.02573i
$298$		0			0
$299$	5612.80			0.00363080
$300$		0			0
$301$	1.33745e6			0.850866
$302$		0			0
$303$		−	1.59028e6i		−	0.995100i
$304$		0			0
$305$	582971.	+	1.83055e6i	0.358837	+	1.12676i
$306$		0			0
$307$		−	747194.i		−	0.452467i	−0.974073	−	0.226234i	$-0.927359\pi$
$307$		−	747194.i		−	0.452467i	0.974073	−	0.226234i	$-0.0726413\pi$
$308$		0			0
$309$	−234485.			−0.139707
$310$		0			0
$311$	1.52486e6			0.893984			0.446992	−	0.894538i	$-0.352495\pi$
$311$	1.52486e6			0.893984			0.446992	+	0.894538i	$0.352495\pi$
$312$		0			0
$313$		−	285672.i		−	0.164819i	−0.996599	−	0.0824094i	$-0.973738\pi$
$313$		−	285672.i		−	0.164819i	0.996599	−	0.0824094i	$-0.0262615\pi$
$314$		0			0
$315$	−1.74043e6	+	554270.i	−0.988278	+	0.314735i
$316$		0			0
$317$		−	3.06761e6i		−	1.71456i	−0.514854	−	0.857278i	$-0.672154\pi$
$317$		−	3.06761e6i		−	1.71456i	0.514854	−	0.857278i	$-0.327846\pi$
$318$		0			0
$319$	−1.78613e6			−0.982733
$320$		0			0
$321$	−873182.			−0.472980
$322$		0			0
$323$		−	92255.4i		−	0.0492023i
$324$		0			0
$325$	841522.	+	1.18720e6i	0.441933	+	0.623472i
$326$		0			0
$327$		−	718016.i		−	0.371334i
$328$		0			0
$329$	5.57808e6			2.84116
$330$		0			0
$331$	2.05250e6			1.02971			0.514854	−	0.857278i	$-0.327846\pi$
$331$	2.05250e6			1.02971			0.514854	+	0.857278i	$0.327846\pi$
$332$		0			0
$333$		−	1.01283e6i		−	0.500527i
$334$		0			0
$335$	−2.44442e6	+	778468.i	−1.19004	+	0.378991i
$336$		0			0
$337$		−	271975.i		−	0.130453i	−0.997870	−	0.0652265i	$-0.979223\pi$
$337$		−	271975.i		−	0.130453i	0.997870	−	0.0652265i	$-0.0207770\pi$
$338$		0			0
$339$	1.59824e6			0.755339
$340$		0			0
$341$	4.13402e6			1.92525
$342$		0			0
$343$			3.57296e6i			1.63981i
$344$		0			0
$345$	−2007.10	−	6302.37i	−0.000907866	−	0.00285073i
$346$		0			0
$347$		−	1.43593e6i		−	0.640192i	−0.947385	−	0.320096i	$-0.896285\pi$
$347$		−	1.43593e6i		−	0.640192i	0.947385	−	0.320096i	$-0.103715\pi$
$348$		0			0
$349$	−2.95015e6			−1.29653			−0.648263	−	0.761417i	$-0.724504\pi$
$349$	−2.95015e6			−1.29653			−0.648263	+	0.761417i	$0.724504\pi$
$350$		0			0
$351$	−1.78109e6			−0.771645
$352$		0			0
$353$			3.45504e6i			1.47576i	0.674931	+	0.737881i	$0.264173\pi$
$353$			3.45504e6i			1.47576i	−0.674931	+	0.737881i	$0.735827\pi$
$354$		0			0
$355$	984328.	+	3.09082e6i	0.414543	+	1.30168i
$356$		0			0
$357$		−	622897.i		−	0.258670i
$358$		0			0
$359$	2.38425e6			0.976372			0.488186	−	0.872740i	$-0.337659\pi$
$359$	2.38425e6			0.976372			0.488186	+	0.872740i	$0.337659\pi$
$360$		0			0
$361$	−2.37115e6			−0.957617
$362$		0			0
$363$		−	50527.4i		−	0.0201262i
$364$		0			0
$365$	3.65748e6	−	1.16479e6i	1.43698	−	0.457631i
$366$		0			0
$367$		−	631924.i		−	0.244906i	−0.992474	−	0.122453i	$-0.960924\pi$
$367$		−	631924.i		−	0.244906i	0.992474	−	0.122453i	$-0.0390761\pi$
$368$		0			0
$369$	−1.57401e6			−0.601784
$370$		0			0
$371$	−2.35539e6			−0.888440
$372$		0			0
$373$			2.09617e6i			0.780107i	0.920792	+	0.390053i	$0.127543\pi$
$373$			2.09617e6i			0.780107i	−0.920792	+	0.390053i	$0.872457\pi$
$374$		0			0
$375$	1.03214e6	−	1.36944e6i	0.379017	−	0.502882i
$376$		0			0
$377$			2.04020e6i			0.739297i
$378$		0			0
$379$	15578.4			0.00557088			0.00278544	−	0.999996i	$-0.499113\pi$
$379$	15578.4			0.00557088			0.00278544	+	0.999996i	$0.499113\pi$
$380$		0			0
$381$	471970.			0.166572
$382$		0			0
$383$		−	2.67131e6i		−	0.930524i	−0.885173	−	0.465262i	$-0.845960\pi$
$383$		−	2.67131e6i		−	0.930524i	0.885173	−	0.465262i	$-0.154040\pi$
$384$		0			0
$385$	−4.83857e6	+	1.54093e6i	−1.66366	+	0.529823i
$386$		0			0
$387$			880183.i			0.298742i
$388$		0			0
$389$	−2.15353e6			−0.721568			−0.360784	−	0.932649i	$-0.617491\pi$
$389$	−2.15353e6			−0.721568			−0.360784	+	0.932649i	$0.617491\pi$
$390$		0			0
$391$	−3432.56			−0.00113547
$392$		0			0
$393$		−	2.41300e6i		−	0.788092i
$394$		0			0
$395$	−1.10374e6	−	3.46579e6i	−0.355939	−	1.11766i
$396$		0			0
$397$		−	1.79394e6i		−	0.571256i	−0.958341	−	0.285628i	$-0.907798\pi$
$397$		−	1.79394e6i		−	0.571256i	0.958341	−	0.285628i	$-0.0922021\pi$
$398$		0			0
$399$	708574.			0.222819
$400$		0			0
$401$	−1.26450e6			−0.392696			−0.196348	−	0.980534i	$-0.562908\pi$
$401$	−1.26450e6			−0.392696			−0.196348	+	0.980534i	$0.562908\pi$
$402$		0			0
$403$		−	4.72207e6i		−	1.44834i
$404$		0			0
$405$	32440.3	+	101864.i	0.00982760	+	0.0308590i
$406$		0			0
$407$		−	2.81579e6i		−	0.842586i
$408$		0			0
$409$	−476802.			−0.140939			−0.0704693	−	0.997514i	$-0.522450\pi$
$409$	−476802.			−0.140939			−0.0704693	+	0.997514i	$0.522450\pi$
$410$		0			0
$411$	−1.84704e6			−0.539351
$412$		0			0
$413$		−	6.88986e6i		−	1.98763i
$414$		0			0
$415$	2.71747e6	−	865428.i	0.774542	−	0.246667i
$416$		0			0
$417$			2.72055e6i			0.766156i
$418$		0			0
$419$	4.75187e6			1.32230			0.661149	−	0.750255i	$-0.270069\pi$
$419$	4.75187e6			1.32230			0.661149	+	0.750255i	$0.270069\pi$
$420$		0			0
$421$	−2.82485e6			−0.776765			−0.388383	−	0.921498i	$-0.626966\pi$
$421$	−2.82485e6			−0.776765			−0.388383	+	0.921498i	$0.626966\pi$
$422$		0			0
$423$			3.67097e6i			0.997538i
$424$		0			0
$425$	−514640.	−	726045.i	−0.138207	−	0.194980i
$426$		0			0
$427$			7.65753e6i			2.03245i
$428$		0			0
$429$	−1.86353e6			−0.488869
$430$		0			0
$431$	−2.09727e6			−0.543827			−0.271914	−	0.962322i	$-0.587656\pi$
$431$	−2.09727e6			−0.543827			−0.271914	+	0.962322i	$0.587656\pi$
$432$		0			0
$433$			2.22584e6i			0.570524i	0.958450	+	0.285262i	$0.0920806\pi$
$433$			2.22584e6i			0.570524i	−0.958450	+	0.285262i	$0.907919\pi$
$434$		0			0
$435$	2.29085e6	−	729561.i	0.580461	−	0.184858i
$436$		0			0
$437$		−	3904.69i		−	0.000978100i
$438$		0			0
$439$	−4.60019e6			−1.13924			−0.569618	−	0.821909i	$-0.692909\pi$
$439$	−4.60019e6			−1.13924			−0.569618	+	0.821909i	$0.692909\pi$
$440$		0			0
$441$	−4.81596e6			−1.17920
$442$		0			0
$443$			95739.0i			0.0231782i	0.999933	+	0.0115891i	$0.00368901\pi$
$443$			95739.0i			0.0231782i	−0.999933	+	0.0115891i	$0.996311\pi$
$444$		0			0
$445$	−1.00778e6	−	3.16447e6i	−0.241250	−	0.757532i
$446$		0			0
$447$		−	1.36513e6i		−	0.323150i
$448$		0			0
$449$	6.03855e6			1.41357			0.706784	−	0.707430i	$-0.250145\pi$
$449$	6.03855e6			1.41357			0.706784	+	0.707430i	$0.250145\pi$
$450$		0			0
$451$	−4.37591e6			−1.01304
$452$		0			0
$453$		−	5.05069e6i		−	1.15639i
$454$		0			0
$455$	1.76012e6	+	5.52684e6i	0.398579	+	1.25155i
$456$		0			0
$457$		−	2.79696e6i		−	0.626464i	−0.949677	−	0.313232i	$-0.898588\pi$
$457$		−	2.79696e6i		−	0.626464i	0.949677	−	0.313232i	$-0.101412\pi$
$458$		0			0
$459$	1.08924e6			0.241319
$460$		0			0
$461$	1.65757e6			0.363261			0.181631	−	0.983367i	$-0.441862\pi$
$461$	1.65757e6			0.363261			0.181631	+	0.983367i	$0.441862\pi$
$462$		0			0
$463$			6.11162e6i			1.32496i	0.749078	+	0.662481i	$0.230497\pi$
$463$			6.11162e6i			1.32496i	−0.749078	+	0.662481i	$0.769503\pi$
$464$		0			0
$465$	−5.30220e6	+	1.68858e6i	−1.13717	+	0.362151i
$466$		0			0
$467$			3.00858e6i			0.638366i	0.947693	+	0.319183i	$0.103409\pi$
$467$			3.00858e6i			0.638366i	−0.947693	+	0.319183i	$0.896591\pi$
$468$		0			0
$469$	−1.02255e7			−2.14660
$470$		0			0
$471$	−5.68041e6			−1.17985
$472$		0			0
$473$			2.44701e6i			0.502901i
$474$		0			0
$475$	825910.	−	585427.i	0.167957	−	0.119052i
$476$		0			0
$477$		−	1.55010e6i		−	0.311934i
$478$		0			0
$479$	−233940.			−0.0465872			−0.0232936	−	0.999729i	$-0.507415\pi$
$479$	−233940.			−0.0465872			−0.0232936	+	0.999729i	$0.507415\pi$
$480$		0			0
$481$	−3.21633e6			−0.633866
$482$		0			0
$483$		−	26364.0i		−	0.00514214i
$484$		0			0
$485$	3.89832e6	−	1.24149e6i	0.752528	−	0.239656i
$486$		0			0
$487$		−	4.85784e6i		−	0.928156i	−0.885794	−	0.464078i	$-0.846386\pi$
$487$		−	4.85784e6i		−	0.928156i	0.885794	−	0.464078i	$-0.153614\pi$
$488$		0			0
$489$	−2.64394e6			−0.500011
$490$		0			0
$491$	7.32174e6			1.37060			0.685300	−	0.728261i	$-0.259671\pi$
$491$	7.32174e6			1.37060			0.685300	+	0.728261i	$0.259671\pi$
$492$		0			0
$493$		−	1.24770e6i		−	0.231203i
$494$		0			0
$495$	−1.01410e6	−	3.18429e6i	−0.186023	−	0.584117i
$496$		0			0
$497$			1.29295e7i			2.34796i
$498$		0			0
$499$	−8.86841e6			−1.59439			−0.797195	−	0.603722i	$-0.793684\pi$
$499$	−8.86841e6			−1.59439			−0.797195	+	0.603722i	$0.793684\pi$
$500$		0			0
$501$	−4.40865e6			−0.784713
$502$		0			0
$503$		−	7.27944e6i		−	1.28286i	−0.767183	−	0.641428i	$-0.778342\pi$
$503$		−	7.27944e6i		−	1.28286i	0.767183	−	0.641428i	$-0.221658\pi$
$504$		0			0
$505$	−2.74813e6	−	8.62923e6i	−0.479523	−	1.50572i
$506$		0			0
$507$		−	1.51613e6i		−	0.261948i
$508$		0			0
$509$	598060.			0.102318			0.0511588	−	0.998691i	$-0.483709\pi$
$509$	598060.			0.102318			0.0511588	+	0.998691i	$0.483709\pi$
$510$		0			0
$511$	1.52999e7			2.59201
$512$		0			0
$513$			1.23906e6i			0.207874i
$514$		0			0
$515$	−1.27237e6	+	405210.i	−0.211396	+	0.0673228i
$516$		0			0
$517$			1.02057e7i			1.67925i
$518$		0			0
$519$	3.60832e6			0.588013
$520$		0			0
$521$	4.12658e6			0.666034			0.333017	−	0.942921i	$-0.391933\pi$
$521$	4.12658e6			0.666034			0.333017	+	0.942921i	$0.391933\pi$
$522$		0			0
$523$		−	5.94827e6i		−	0.950903i	−0.879742	−	0.475451i	$-0.842285\pi$
$523$		−	5.94827e6i		−	0.950903i	0.879742	−	0.475451i	$-0.157715\pi$
$524$		0			0
$525$	5.57644e6	−	3.95273e6i	0.882996	−	0.625891i
$526$		0			0
$527$			2.88782e6i			0.452943i
$528$		0			0
$529$	6.43620e6			0.999977
$530$		0			0
$531$	4.53426e6			0.697862
$532$		0			0
$533$			4.99837e6i			0.762097i
$534$		0			0
$535$	−4.73810e6	+	1.50893e6i	−0.715681	+	0.227921i
$536$		0			0
$537$		−	543506.i		−	0.0813334i
$538$		0			0
$539$	−1.33889e7			−1.98505
$540$		0			0
$541$	8.27787e6			1.21598			0.607989	−	0.793946i	$-0.291977\pi$
$541$	8.27787e6			1.21598			0.607989	+	0.793946i	$0.291977\pi$
$542$		0			0
$543$			6.40666e6i			0.932464i
$544$		0			0
$545$	−1.24079e6	−	3.89613e6i	−0.178940	−	0.561878i
$546$		0			0
$547$		−	1.18115e7i		−	1.68786i	−0.536453	−	0.843930i	$-0.680236\pi$
$547$		−	1.18115e7i		−	1.68786i	0.536453	−	0.843930i	$-0.319764\pi$
$548$		0			0
$549$	−5.03946e6			−0.713598
$550$		0			0
$551$	1.41932e6			0.199159
$552$		0			0
$553$		−	1.44981e7i		−	2.01603i
$554$		0			0
$555$	1.15014e6	+	3.61147e6i	0.158496	+	0.497682i
$556$		0			0
$557$			7.81776e6i			1.06769i	0.845583	+	0.533844i	$0.179253\pi$
$557$			7.81776e6i			1.06769i	−0.845583	+	0.533844i	$0.820747\pi$
$558$		0			0
$559$	2.79508e6			0.378325
$560$		0			0
$561$	1.13966e6			0.152886
$562$		0			0
$563$			1.16975e7i			1.55533i	0.628677	+	0.777666i	$0.283597\pi$
$563$			1.16975e7i			1.55533i	−0.628677	+	0.777666i	$0.716403\pi$
$564$		0			0
$565$	8.67242e6	−	2.76189e6i	1.14293	−	0.363986i
$566$		0			0
$567$			426115.i			0.0556634i
$568$		0			0
$569$	1.48869e7			1.92763			0.963813	−	0.266578i	$-0.0858931\pi$
$569$	1.48869e7			1.92763			0.963813	+	0.266578i	$0.0858931\pi$
$570$		0			0
$571$	−995656.			−0.127797			−0.0638983	−	0.997956i	$-0.520353\pi$
$571$	−995656.			−0.127797			−0.0638983	+	0.997956i	$0.520353\pi$
$572$		0			0
$573$			3.01719e6i			0.383899i
$574$		0			0
$575$	−21782.0	−	30729.7i	−0.00274745	−	0.00387605i
$576$		0			0
$577$			1.00279e6i			0.125392i	0.998033	+	0.0626961i	$0.0199699\pi$
$577$			1.00279e6i			0.125392i	−0.998033	+	0.0626961i	$0.980030\pi$
$578$		0			0
$579$	899495.			0.111507
$580$		0			0
$581$	1.13677e7			1.39712
$582$		0			0
$583$		−	4.30944e6i		−	0.525109i
$584$		0			0
$585$	−3.63725e6	+	1.15835e6i	−0.439423	+	0.139942i
$586$		0			0
$587$			3.54906e6i			0.425127i	0.977147	+	0.212563i	$0.0681812\pi$
$587$			3.54906e6i			0.425127i	−0.977147	+	0.212563i	$0.931819\pi$
$588$		0			0
$589$	−3.28503e6			−0.390168
$590$		0			0
$591$	−2.51696e6			−0.296420
$592$		0			0
$593$		−	4.20405e6i		−	0.490944i	−0.969404	−	0.245472i	$-0.921057\pi$
$593$		−	4.20405e6i		−	0.490944i	0.969404	−	0.245472i	$-0.0789429\pi$
$594$		0			0
$595$	−1.07642e6	−	3.37999e6i	−0.124649	−	0.391402i
$596$		0			0
$597$			9.75754e6i			1.12048i
$598$		0			0
$599$	−1.21200e7			−1.38017			−0.690087	−	0.723726i	$-0.742428\pi$
$599$	−1.21200e7			−1.38017			−0.690087	+	0.723726i	$0.742428\pi$
$600$		0			0
$601$	1.15778e6			0.130749			0.0653745	−	0.997861i	$-0.479176\pi$
$601$	1.15778e6			0.130749			0.0653745	+	0.997861i	$0.479176\pi$
$602$		0			0
$603$		−	6.72943e6i		−	0.753677i
$604$		0			0
$605$	−87315.7	−	274174.i	−0.00969848	−	0.0304536i
$606$		0			0
$607$			6.62196e6i			0.729482i	0.931109	+	0.364741i	$0.118842\pi$
$607$			6.62196e6i			0.729482i	−0.931109	+	0.364741i	$0.881158\pi$
$608$		0			0
$609$	9.58305e6			1.04703
$610$		0			0
$611$	1.16574e7			1.26328
$612$		0			0
$613$			8.15388e6i			0.876421i	0.898872	+	0.438211i	$0.144388\pi$
$613$			8.15388e6i			0.876421i	−0.898872	+	0.438211i	$0.855612\pi$
$614$		0			0
$615$	5.61245e6	−	1.78739e6i	0.598363	−	0.190559i
$616$		0			0
$617$			8.07144e6i			0.853568i	0.904353	+	0.426784i	$0.140354\pi$
$617$			8.07144e6i			0.853568i	−0.904353	+	0.426784i	$0.859646\pi$
$618$		0			0
$619$	1.78058e7			1.86782			0.933910	−	0.357509i	$-0.116374\pi$
$619$	1.78058e7			1.86782			0.933910	+	0.357509i	$0.116374\pi$
$620$		0			0
$621$	46101.9			0.00479722
$622$		0			0
$623$		−	1.32376e7i		−	1.36643i
$624$		0			0
$625$	3.23411e6	−	9.21455e6i	0.331173	−	0.943570i
$626$		0			0
$627$			1.29641e6i			0.131696i
$628$		0			0
$629$	1.96697e6			0.198231
$630$		0			0
$631$	−5.93229e6			−0.593129			−0.296564	−	0.955013i	$-0.595841\pi$
$631$	−5.93229e6			−0.593129			−0.296564	+	0.955013i	$0.595841\pi$
$632$		0			0
$633$			800060.i			0.0793621i
$634$		0			0
$635$	2.56102e6	−	815604.i	0.252046	−	0.0802684i
$636$		0			0
$637$			1.52934e7i			1.49333i
$638$		0			0
$639$	−8.50898e6			−0.824376
$640$		0			0
$641$	584130.			0.0561519			0.0280760	−	0.999606i	$-0.491062\pi$
$641$	584130.			0.0561519			0.0280760	+	0.999606i	$0.491062\pi$
$642$		0			0
$643$			1.65155e7i			1.57530i	0.616123	+	0.787650i	$0.288702\pi$
$643$			1.65155e7i			1.57530i	−0.616123	+	0.787650i	$0.711298\pi$
$644$		0			0
$645$	−999504.	−	3.13848e6i	−0.0945988	−	0.297043i
$646$		0			0
$647$			1.35366e7i			1.27130i	0.771976	+	0.635652i	$0.219269\pi$
$647$			1.35366e7i			1.27130i	−0.771976	+	0.635652i	$0.780731\pi$
$648$		0			0
$649$	1.26057e7			1.17478
$650$		0			0
$651$	−2.21801e7			−2.05122
$652$		0			0
$653$		−	8.32959e6i		−	0.764435i	−0.924072	−	0.382218i	$-0.875160\pi$
$653$		−	8.32959e6i		−	0.764435i	0.924072	−	0.382218i	$-0.124840\pi$
$654$		0			0
$655$	−4.16987e6	−	1.30935e7i	−0.379769	−	1.19249i
$656$		0			0
$657$			1.00690e7i			0.910063i
$658$		0			0
$659$	6.89776e6			0.618720			0.309360	−	0.950945i	$-0.399885\pi$
$659$	6.89776e6			0.618720			0.309360	+	0.950945i	$0.399885\pi$
$660$		0			0
$661$	−1.00381e7			−0.893606			−0.446803	−	0.894632i	$-0.647438\pi$
$661$	−1.00381e7			−0.893606			−0.446803	+	0.894632i	$0.647438\pi$
$662$		0			0
$663$		−	1.30177e6i		−	0.115014i
$664$		0			0
$665$	3.84489e6	−	1.22448e6i	0.337156	−	0.107373i
$666$		0			0
$667$		−	52808.7i		−	0.00459611i
$668$		0			0
$669$	−9.26787e6			−0.800598
$670$		0			0
$671$	−1.40103e7			−1.20127
$672$		0			0
$673$		−	1.76094e7i		−	1.49867i	−0.662190	−	0.749336i	$-0.730373\pi$
$673$		−	1.76094e7i		−	1.49867i	0.662190	−	0.749336i	$-0.269627\pi$
$674$		0			0
$675$	6.91201e6	+	9.75134e6i	0.583909	+	0.823768i
$676$		0			0
$677$			1.55786e7i			1.30634i	0.757209	+	0.653172i	$0.226562\pi$
$677$			1.55786e7i			1.30634i	−0.757209	+	0.653172i	$0.773438\pi$
$678$		0			0
$679$	1.63074e7			1.35741
$680$		0			0
$681$	−2.21362e6			−0.182909
$682$		0			0
$683$			2.77212e6i			0.227384i	0.993516	+	0.113692i	$0.0362677\pi$
$683$			2.77212e6i			0.227384i	−0.993516	+	0.113692i	$0.963732\pi$
$684$		0			0
$685$	−1.00225e7	+	3.19184e6i	−0.816109	+	0.259905i
$686$		0			0
$687$			1.06170e7i			0.858242i
$688$		0			0
$689$	−4.92244e6			−0.395032
$690$		0			0
$691$	−5.49060e6			−0.437446			−0.218723	−	0.975787i	$-0.570189\pi$
$691$	−5.49060e6			−0.437446			−0.218723	+	0.975787i	$0.570189\pi$
$692$		0			0
$693$		−	1.33205e7i		−	1.05363i
$694$		0			0
$695$	4.70134e6	+	1.47624e7i	0.369198	+	1.15930i
$696$		0			0
$697$		−	3.05680e6i		−	0.238333i
$698$		0			0
$699$	−7.54172e6			−0.583818
$700$		0			0
$701$	1.01658e7			0.781351			0.390676	−	0.920528i	$-0.372241\pi$
$701$	1.01658e7			0.781351			0.390676	+	0.920528i	$0.372241\pi$
$702$		0			0
$703$			2.23752e6i			0.170757i
$704$		0			0
$705$	−4.16862e6	−	1.30896e7i	−0.315878	−	0.991867i
$706$		0			0
$707$		−	3.60977e7i		−	2.71601i
$708$		0			0
$709$	−1.01024e7			−0.754761			−0.377380	−	0.926058i	$-0.623175\pi$
$709$	−1.01024e7			−0.754761			−0.377380	+	0.926058i	$0.623175\pi$
$710$		0			0
$711$	9.54125e6			0.707834
$712$		0			0
$713$			122227.i			0.00900413i
$714$		0			0
$715$	−1.01119e7	+	3.22033e6i	−0.739724	+	0.235578i
$716$		0			0
$717$		−	6.70572e6i		−	0.487133i
$718$		0			0
$719$	1.48567e7			1.07176			0.535882	−	0.844293i	$-0.319979\pi$
$719$	1.48567e7			1.07176			0.535882	+	0.844293i	$0.319979\pi$
$720$		0			0
$721$	−5.32258e6			−0.381315
$722$		0			0
$723$			2.98829e6i			0.212607i
$724$		0			0
$725$	1.11699e7	−	7.91755e6i	0.789234	−	0.559430i
$726$		0			0
$727$		−	1.24685e7i		−	0.874939i	−0.899233	−	0.437469i	$-0.855875\pi$
$727$		−	1.24685e7i		−	0.874939i	0.899233	−	0.437469i	$-0.144125\pi$
$728$		0			0
$729$	−9.40406e6			−0.655385
$730$		0			0
$731$	−1.70936e6			−0.118315
$732$		0			0
$733$			2.01823e7i			1.38743i	0.720251	+	0.693713i	$0.244027\pi$
$733$			2.01823e7i			1.38743i	−0.720251	+	0.693713i	$0.755973\pi$
$734$		0			0
$735$	1.71723e7	−	5.46883e6i	1.17249	−	0.373401i
$736$		0			0
$737$		−	1.87086e7i		−	1.26874i
$738$		0			0
$739$	1.18948e7			0.801209			0.400605	−	0.916251i	$-0.368800\pi$
$739$	1.18948e7			0.801209			0.400605	+	0.916251i	$0.368800\pi$
$740$		0			0
$741$	1.48082e6			0.0990735
$742$		0			0
$743$		−	1.70571e7i		−	1.13353i	−0.823879	−	0.566766i	$-0.808195\pi$
$743$		−	1.70571e7i		−	1.13353i	0.823879	−	0.566766i	$-0.191805\pi$
$744$		0			0
$745$	−2.35906e6	−	7.40752e6i	−0.155721	−	0.488970i
$746$		0			0
$747$			7.48115e6i			0.490532i
$748$		0			0
$749$	−1.98204e7			−1.29094
$750$		0			0
$751$	−1.69025e7			−1.09358			−0.546792	−	0.837268i	$-0.684151\pi$
$751$	−1.69025e7			−1.09358			−0.546792	+	0.837268i	$0.684151\pi$
$752$		0			0
$753$			6.19216e6i			0.397974i
$754$		0			0
$755$	−8.72801e6	−	2.74062e7i	−0.557247	−	1.74977i
$756$		0			0
$757$			2.47458e6i			0.156950i	0.996916	+	0.0784750i	$0.0250051\pi$
$757$			2.47458e6i			0.156950i	−0.996916	+	0.0784750i	$0.974995\pi$
$758$		0			0
$759$	48235.7			0.00303924
$760$		0			0
$761$	−2.38180e7			−1.49088			−0.745442	−	0.666570i	$-0.767762\pi$
$761$	−2.38180e7			−1.49088			−0.745442	+	0.666570i	$0.767762\pi$
$762$		0			0
$763$		−	1.62982e7i		−	1.01351i
$764$		0			0
$765$	2.22439e6	−	708397.i	0.137422	−	0.0437646i
$766$		0			0
$767$		−	1.43989e7i		−	0.883770i
$768$		0			0
$769$	−6.78796e6			−0.413927			−0.206963	−	0.978349i	$-0.566358\pi$
$769$	−6.78796e6			−0.413927			−0.206963	+	0.978349i	$0.566358\pi$
$770$		0			0
$771$	−8.32224e6			−0.504202
$772$		0			0
$773$		−	1.43751e7i		−	0.865291i	−0.901564	−	0.432646i	$-0.857580\pi$
$773$		−	1.43751e7i		−	0.865291i	0.901564	−	0.432646i	$-0.142420\pi$
$774$		0			0
$775$	−2.58530e7	+	1.83253e7i	−1.54617	+	1.09597i
$776$		0			0
$777$			1.51075e7i			0.897716i
$778$		0			0
$779$	3.47725e6			0.205301
$780$		0			0
$781$	−2.36559e7			−1.38775
$782$		0			0
$783$			1.67576e7i			0.976802i
$784$		0			0
$785$	−3.08233e7	+	9.81622e6i	−1.78527	+	0.568552i
$786$		0			0
$787$			2.78490e7i			1.60277i	0.598146	+	0.801387i	$0.295904\pi$
$787$			2.78490e7i			1.60277i	−0.598146	+	0.801387i	$0.704096\pi$
$788$		0			0
$789$	6.96597e6			0.398372
$790$		0			0
$791$	3.62784e7			2.06161
$792$		0			0
$793$			1.60032e7i			0.903697i
$794$		0			0
$795$	1.76023e6	+	5.52719e6i	0.0987762	+	0.310161i
$796$		0			0
$797$		−	3.40150e6i		−	0.189681i	−0.995492	−	0.0948406i	$-0.969766\pi$
$797$		−	3.40150e6i		−	0.189681i	0.995492	−	0.0948406i	$-0.0302341\pi$
$798$		0			0
$799$	−7.12919e6			−0.395069
$800$		0			0
$801$	8.71173e6			0.479759
$802$		0			0
$803$			2.79928e7i			1.53200i
$804$		0			0
$805$	−45559.2	−	143057.i	−0.00247792	−	0.00778074i
$806$		0			0
$807$		−	4.17138e6i		−	0.225474i
$808$		0			0
$809$	−1.06065e7			−0.569773			−0.284887	−	0.958561i	$-0.591956\pi$
$809$	−1.06065e7			−0.569773			−0.284887	+	0.958561i	$0.591956\pi$
$810$		0			0
$811$	3.49545e7			1.86617			0.933084	−	0.359658i	$-0.117107\pi$
$811$	3.49545e7			1.86617			0.933084	+	0.359658i	$0.117107\pi$
$812$		0			0
$813$		−	8.18411e6i		−	0.434255i
$814$		0			0
$815$	−1.43467e7	+	4.56895e6i	−0.756584	+	0.240948i
$816$		0			0
$817$		−	1.94447e6i		−	0.101917i
$818$		0			0
$819$	−1.52153e7			−0.792630
$820$		0			0
$821$	−1.06256e7			−0.550170			−0.275085	−	0.961420i	$-0.588706\pi$
$821$	−1.06256e7			−0.550170			−0.275085	+	0.961420i	$0.588706\pi$
$822$		0			0
$823$			2.25239e6i			0.115916i	0.998319	+	0.0579582i	$0.0184590\pi$
$823$			2.25239e6i			0.115916i	−0.998319	+	0.0579582i	$0.981541\pi$
$824$		0			0
$825$	7.23193e6	+	1.02027e7i	0.369930	+	0.521891i
$826$		0			0
$827$			2.38273e7i			1.21147i	0.795668	+	0.605733i	$0.207120\pi$
$827$			2.38273e7i			1.21147i	−0.795668	+	0.605733i	$0.792880\pi$
$828$		0			0
$829$	885468.			0.0447493			0.0223747	−	0.999750i	$-0.492877\pi$
$829$	885468.			0.0447493			0.0223747	+	0.999750i	$0.492877\pi$
$830$		0			0
$831$	4.16332e6			0.209140
$832$		0			0
$833$		−	9.35281e6i		−	0.467014i
$834$		0			0
$835$	−2.39224e7	+	7.61851e6i	−1.18738	+	0.378141i
$836$		0			0
$837$		−	3.87857e7i		−	1.91363i
$838$		0			0
$839$	−2.09943e7			−1.02966			−0.514832	−	0.857291i	$-0.672146\pi$
$839$	−2.09943e7			−1.02966			−0.514832	+	0.857291i	$0.672146\pi$
$840$		0			0
$841$	−1.31573e6			−0.0641470
$842$		0			0
$843$		−	1.69548e7i		−	0.821722i
$844$		0			0
$845$	−2.61999e6	−	8.22687e6i	−0.126229	−	0.396363i
$846$		0			0
$847$		−	1.14692e6i		−	0.0549320i
$848$		0			0
$849$	8.08712e6			0.385056
$850$		0			0
$851$	83251.8			0.00394067
$852$		0			0
$853$		−	342279.i		−	0.0161067i	−0.999968	−	0.00805336i	$-0.997437\pi$
$853$		−	342279.i		−	0.0161067i	0.999968	−	0.00805336i	$-0.00256349\pi$
$854$		0			0
$855$	805834.	+	2.53035e6i	0.0376991	+	0.118376i
$856$		0			0
$857$			2.13031e7i			0.990811i	0.868662	+	0.495405i	$0.164980\pi$
$857$			2.13031e7i			0.990811i	−0.868662	+	0.495405i	$0.835020\pi$
$858$		0			0
$859$	1.20032e7			0.555028			0.277514	−	0.960722i	$-0.410490\pi$
$859$	1.20032e7			0.555028			0.277514	+	0.960722i	$0.410490\pi$
$860$		0			0
$861$	2.34779e7			1.07932
$862$		0			0
$863$			1.11350e7i			0.508937i	0.967081	+	0.254469i	$0.0819005\pi$
$863$			1.11350e7i			0.508937i	−0.967081	+	0.254469i	$0.918099\pi$
$864$		0			0
$865$	1.95796e7	−	6.23548e6i	0.889742	−	0.283354i
$866$		0			0
$867$		−	1.31417e7i		−	0.593749i
$868$		0			0
$869$	2.65257e7			1.19157
$870$		0			0
$871$	−2.13698e7			−0.954454
$872$		0			0
$873$			1.07320e7i			0.476590i
$874$		0			0
$875$	2.34285e7	−	3.10850e7i	1.03448	−	1.37256i
$876$		0			0
$877$			3.41443e6i			0.149906i	0.997187	+	0.0749530i	$0.0238807\pi$
$877$			3.41443e6i			0.149906i	−0.997187	+	0.0749530i	$0.976119\pi$
$878$		0			0
$879$	2.56177e7			1.11832
$880$		0			0
$881$	1.89727e7			0.823548			0.411774	−	0.911286i	$-0.364909\pi$
$881$	1.89727e7			0.823548			0.411774	+	0.911286i	$0.364909\pi$
$882$		0			0
$883$		−	3.46909e7i		−	1.49732i	−0.662957	−	0.748658i	$-0.730699\pi$
$883$		−	3.46909e7i		−	1.49732i	0.662957	−	0.748658i	$-0.269301\pi$
$884$		0			0
$885$	−1.61678e7	+	5.14894e6i	−0.693895	+	0.220983i
$886$		0			0
$887$		−	4.95749e6i		−	0.211569i	−0.994389	−	0.105785i	$-0.966265\pi$
$887$		−	4.95749e6i		−	0.211569i	0.994389	−	0.105785i	$-0.0337354\pi$
$888$		0			0
$889$	1.07132e7			0.454639
$890$		0			0
$891$	−779623.			−0.0328996
$892$		0			0
$893$		−	8.10979e6i		−	0.340315i
$894$		0			0
$895$	−939225.	−	2.94920e6i	−0.0391933	−	0.123068i
$896$		0			0
$897$		−	55097.1i		−	0.00228638i
$898$		0			0
$899$	−4.44281e7			−1.83341
$900$		0			0
$901$	3.01036e6			0.123540
$902$		0			0
$903$		−	1.31288e7i		−	0.535806i
$904$		0			0
$905$	1.10712e7	+	3.47641e7i	0.449340	+	1.41094i
$906$		0			0
$907$			2.33839e7i			0.943841i	0.881641	+	0.471921i	$0.156439\pi$
$907$			2.33839e7i			0.943841i	−0.881641	+	0.471921i	$0.843561\pi$
$908$		0			0
$909$	2.37561e7			0.953599
$910$		0			0
$911$	2.03543e7			0.812569			0.406285	−	0.913747i	$-0.366824\pi$
$911$	2.03543e7			0.812569			0.406285	+	0.913747i	$0.366824\pi$
$912$		0			0
$913$			2.07984e7i			0.825759i
$914$		0			0
$915$	1.79693e7	−	5.72263e6i	0.709541	−	0.225966i
$916$		0			0
$917$		−	5.47728e7i		−	2.15101i
$918$		0			0
$919$	2.90338e7			1.13401			0.567003	−	0.823716i	$-0.308103\pi$
$919$	2.90338e7			1.13401			0.567003	+	0.823716i	$0.308103\pi$
$920$		0			0
$921$	−7.33470e6			−0.284927
$922$		0			0
$923$			2.70209e7i			1.04399i
$924$		0			0
$925$	1.24818e7	+	1.76092e7i	0.479650	+	0.676682i
$926$		0			0
$927$		−	3.50282e6i		−	0.133881i
$928$		0			0
$929$	−3.90013e7			−1.48266			−0.741328	−	0.671143i	$-0.765804\pi$
$929$	−3.90013e7			−1.48266			−0.741328	+	0.671143i	$0.765804\pi$
$930$		0			0
$931$	1.06393e7			0.402288
$932$		0			0
$933$		−	1.49686e7i		−	0.562958i
$934$		0			0
$935$	6.18405e6	−	1.96942e6i	0.231336	−	0.0736732i
$936$		0			0
$937$		−	2.32825e7i		−	0.866324i	−0.901316	−	0.433162i	$-0.857398\pi$
$937$		−	2.32825e7i		−	0.866324i	0.901316	−	0.433162i	$-0.142602\pi$
$938$		0			0
$939$	−2.80425e6			−0.103789
$940$		0			0
$941$	−3.90746e6			−0.143853			−0.0719267	−	0.997410i	$-0.522915\pi$
$941$	−3.90746e6			−0.143853			−0.0719267	+	0.997410i	$0.522915\pi$
$942$		0			0
$943$		−	129378.i		−	0.00473786i
$944$		0			0
$945$	1.44571e7	+	4.53959e7i	0.526626	+	1.65362i
$946$		0			0
$947$			1.52048e7i			0.550943i	0.961309	+	0.275471i	$0.0888339\pi$
$947$			1.52048e7i			0.550943i	−0.961309	+	0.275471i	$0.911166\pi$
$948$		0			0
$949$	3.19747e7			1.15250
$950$		0			0
$951$	−3.01127e7			−1.07969
$952$		0			0
$953$			3.60102e7i			1.28438i	0.766545	+	0.642190i	$0.221974\pi$
$953$			3.60102e7i			1.28438i	−0.766545	+	0.642190i	$0.778026\pi$
$954$		0			0
$955$	5.21396e6	+	1.63720e7i	0.184995	+	0.580890i
$956$		0			0
$957$			1.75332e7i			0.618844i
$958$		0			0
$959$	−4.19259e7			−1.47210
$960$		0			0
$961$	7.42004e7			2.59178
$962$		0			0
$963$		−	1.30439e7i		−	0.453254i
$964$		0			0
$965$	4.88088e6	−	1.55440e6i	0.168725	−	0.0537335i
$966$		0			0
$967$		−	2.39347e7i		−	0.823119i	−0.911383	−	0.411559i	$-0.864984\pi$
$967$		−	2.39347e7i		−	0.823119i	0.911383	−	0.411559i	$-0.135016\pi$
$968$		0			0
$969$	−905609.			−0.0309836
$970$		0			0
$971$	−2.87801e7			−0.979589			−0.489794	−	0.871838i	$-0.662928\pi$
$971$	−2.87801e7			−0.979589			−0.489794	+	0.871838i	$0.662928\pi$
$972$		0			0
$973$			6.17538e7i			2.09113i
$974$		0			0
$975$	1.16540e7	−	8.26065e6i	0.392612	−	0.278293i
$976$		0			0
$977$		−	1.96671e7i		−	0.659181i	−0.944124	−	0.329590i	$-0.893089\pi$
$977$		−	1.96671e7i		−	0.659181i	0.944124	−	0.329590i	$-0.106911\pi$
$978$		0			0
$979$	2.42195e7			0.807624
$980$		0			0
$981$	1.07260e7			0.355848
$982$		0			0
$983$			4.03974e7i			1.33343i	0.745314	+	0.666713i	$0.232300\pi$
$983$			4.03974e7i			1.33343i	−0.745314	+	0.666713i	$0.767700\pi$
$984$		0			0
$985$	−1.36576e7	+	4.34951e6i	−0.448522	+	0.142840i
$986$		0			0
$987$		−	5.47563e7i		−	1.78913i
$988$		0			0
$989$	−72348.3			−0.00235200
$990$		0			0
$991$	−1.08921e7			−0.352313			−0.176156	−	0.984362i	$-0.556366\pi$
$991$	−1.08921e7			−0.352313			−0.176156	+	0.984362i	$0.556366\pi$
$992$		0			0
$993$		−	2.01481e7i		−	0.648425i
$994$		0			0
$995$	1.68618e7	+	5.29468e7i	0.539943	+	1.69544i
$996$		0			0
$997$		−	3.25183e7i		−	1.03607i	−0.855359	−	0.518035i	$-0.826664\pi$
$997$		−	3.25183e7i		−	1.03607i	0.855359	−	0.518035i	$-0.173336\pi$
$998$		0			0
$999$	−2.64180e7			−0.837501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.c.c.129.5	✓	12
4.3	odd	2	inner	160.6.c.c.129.7	yes	12
5.2	odd	4		800.6.a.ba.1.3		6
5.3	odd	4		800.6.a.z.1.4		6
5.4	even	2	inner	160.6.c.c.129.8	yes	12
8.3	odd	2		320.6.c.k.129.6		12
8.5	even	2		320.6.c.k.129.8		12
20.3	even	4		800.6.a.z.1.3		6
20.7	even	4		800.6.a.ba.1.4		6
20.19	odd	2	inner	160.6.c.c.129.6	yes	12
40.19	odd	2		320.6.c.k.129.7		12
40.29	even	2		320.6.c.k.129.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.c.129.5	✓	12	1.1	even	1	trivial
160.6.c.c.129.6	yes	12	20.19	odd	2	inner
160.6.c.c.129.7	yes	12	4.3	odd	2	inner
160.6.c.c.129.8	yes	12	5.4	even	2	inner
320.6.c.k.129.5		12	40.29	even	2
320.6.c.k.129.6		12	8.3	odd	2
320.6.c.k.129.7		12	40.19	odd	2
320.6.c.k.129.8		12	8.5	even	2
800.6.a.z.1.3		6	20.3	even	4
800.6.a.z.1.4		6	5.3	odd	4
800.6.a.ba.1.3		6	5.2	odd	4
800.6.a.ba.1.4		6	20.7	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.81633i q^{3} +(-16.9634 - 53.2658i) q^{5} -222.821i q^{7} +146.640 q^{9} +O(q^{10})\)	\(q-9.81633i q^{3} +(-16.9634 - 53.2658i) q^{5} -222.821i q^{7} +146.640 q^{9} +407.674 q^{11} -465.665i q^{13} +(-522.874 + 166.519i) q^{15} +284.781i q^{17} -323.952 q^{19} -2187.28 q^{21} +12.0533i q^{23} +(-2549.48 + 1807.14i) q^{25} -3824.83i q^{27} -4381.26 q^{29} +10140.5 q^{31} -4001.86i q^{33} +(-11868.7 + 3779.81i) q^{35} -6906.96i q^{37} -4571.12 q^{39} -10733.8 q^{41} +6002.36i q^{43} +(-2487.51 - 7810.87i) q^{45} +25033.9i q^{47} -32842.1 q^{49} +2795.51 q^{51} -10570.8i q^{53} +(-6915.56 - 21715.1i) q^{55} +3180.02i q^{57} +30921.1 q^{59} -34366.3 q^{61} -32674.4i q^{63} +(-24804.0 + 7899.27i) q^{65} -45890.9i q^{67} +118.319 q^{69} -58026.5 q^{71} +68664.7i q^{73} +(17739.5 + 25026.6i) q^{75} -90838.3i q^{77} +65066.0 q^{79} -1912.37 q^{81} +51017.2i q^{83} +(15169.1 - 4830.87i) q^{85} +43007.9i q^{87} +59409.1 q^{89} -103760. q^{91} -99542.4i q^{93} +(5495.34 + 17255.5i) q^{95} +73186.2i q^{97} +59781.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 60 q^{5} - 1676 q^{9}+O(q^{10}) \) 12 * q - 60 * q^5 - 1676 * q^9	\( 12 q - 60 q^{5} - 1676 q^{9} + 688 q^{21} + 1500 q^{25} - 21304 q^{29} - 109672 q^{41} - 75140 q^{45} - 32348 q^{49} + 64552 q^{61} - 160 q^{65} - 166352 q^{69} - 173764 q^{81} + 151360 q^{85} - 3720 q^{89}+O(q^{100}) \) 12 * q - 60 * q^5 - 1676 * q^9 + 688 * q^21 + 1500 * q^25 - 21304 * q^29 - 109672 * q^41 - 75140 * q^45 - 32348 * q^49 + 64552 * q^61 - 160 * q^65 - 166352 * q^69 - 173764 * q^81 + 151360 * q^85 - 3720 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more