LMFDB - Embedded newform 320.6.c.l.129.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(129,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.c (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:	$51.3228223402$
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} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + \cdots + 1072562500 $ x^12 - 6x^11 + 18x^10 + 348x^9 + 21226x^8 - 87824x^7 + 205428x^6 + 2113880x^5 + 27827225x^4 - 59798850x^3 + 89646050x^2 + 438522500*x + 1072562500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{46}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$4.92310 + 4.92310i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.6.c.l.129.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.9825i q^{3} +(-12.3893 + 54.5115i) q^{5} -211.691i q^{7} -45.4059 q^{9} +O(q^{10})$	$q-16.9825i q^{3} +(-12.3893 + 54.5115i) q^{5} -211.691i q^{7} -45.4059 q^{9} +520.707 q^{11} -732.105i q^{13} +(925.743 + 210.402i) q^{15} +2269.82i q^{17} +2033.73 q^{19} -3595.05 q^{21} +974.449i q^{23} +(-2818.01 - 1350.72i) q^{25} -3355.65i q^{27} +5276.32 q^{29} -2001.75 q^{31} -8842.91i q^{33} +(11539.6 + 2622.71i) q^{35} +3652.25i q^{37} -12433.0 q^{39} +17113.3 q^{41} -4073.34i q^{43} +(562.548 - 2475.14i) q^{45} -13335.0i q^{47} -28006.2 q^{49} +38547.2 q^{51} -27482.0i q^{53} +(-6451.20 + 28384.5i) q^{55} -34537.9i q^{57} +3890.44 q^{59} -8737.66 q^{61} +9612.02i q^{63} +(39908.2 + 9070.28i) q^{65} -40950.5i q^{67} +16548.6 q^{69} -26897.5 q^{71} -9824.26i q^{73} +(-22938.6 + 47856.9i) q^{75} -110229. i q^{77} -60376.0 q^{79} -68020.9 q^{81} +2419.27i q^{83} +(-123731. - 28121.5i) q^{85} -89605.2i q^{87} -69364.1 q^{89} -154980. q^{91} +33994.7i q^{93} +(-25196.6 + 110862. i) q^{95} -76217.3i q^{97} -23643.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 60 q^{5} - 268 q^{9}+O(q^{10}) $ 12 * q + 60 * q^5 - 268 * q^9	$ 12 q + 60 q^{5} - 268 q^{9} - 12080 q^{21} - 16420 q^{25} + 7864 q^{29} + 65560 q^{41} + 60420 q^{45} + 17956 q^{49} + 118232 q^{61} + 11360 q^{65} - 204464 q^{69} - 346436 q^{81} - 96320 q^{85} + 354936 q^{89}+O(q^{100}) $ 12 * q + 60 * q^5 - 268 * q^9 - 12080 * q^21 - 16420 * q^25 + 7864 * q^29 + 65560 * q^41 + 60420 * q^45 + 17956 * q^49 + 118232 * q^61 + 11360 * q^65 - 204464 * q^69 - 346436 * q^81 - 96320 * q^85 + 354936 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	16.9825i		−	1.08943i	−0.838622	−	0.544714i	$-0.816638\pi$
$3$		−	16.9825i		−	1.08943i	0.838622	−	0.544714i	$-0.183362\pi$
$4$		0			0
$5$	−12.3893	+	54.5115i	−0.221627	+	0.975132i
$5$	−12.3893	+	54.5115i	−0.221627	+	0.975132i
$6$		0			0
$7$		−	211.691i		−	1.63289i	−0.577421	−	0.816447i	$-0.695941\pi$
$7$		−	211.691i		−	1.63289i	0.577421	−	0.816447i	$-0.304059\pi$
$8$		0			0
$9$	−45.4059			−0.186855
$10$		0			0
$11$	520.707			1.29751			0.648756	−	0.760997i	$-0.275290\pi$
$11$	520.707			1.29751			0.648756	+	0.760997i	$0.275290\pi$
$12$		0			0
$13$		−	732.105i		−	1.20148i	−0.799446	−	0.600738i	$-0.794874\pi$
$13$		−	732.105i		−	1.20148i	0.799446	−	0.600738i	$-0.205126\pi$
$14$		0			0
$15$	925.743	+	210.402i	1.06234	+	0.241447i
$16$		0			0
$17$			2269.82i			1.90488i	0.304721	+	0.952442i	$0.401437\pi$
$17$			2269.82i			1.90488i	−0.304721	+	0.952442i	$0.598563\pi$
$18$		0			0
$19$	2033.73			1.29244			0.646220	−	0.763152i	$-0.276349\pi$
$19$	2033.73			1.29244			0.646220	+	0.763152i	$0.276349\pi$
$20$		0			0
$21$	−3595.05			−1.77892
$22$		0			0
$23$			974.449i			0.384096i	0.981386	+	0.192048i	$0.0615129\pi$
$23$			974.449i			0.384096i	−0.981386	+	0.192048i	$0.938487\pi$
$24$		0			0
$25$	−2818.01	−	1350.72i	−0.901763	−	0.432231i
$26$		0			0
$27$		−	3355.65i		−	0.885863i
$28$		0			0
$29$	5276.32			1.16503			0.582514	−	0.812821i	$-0.302069\pi$
$29$	5276.32			1.16503			0.582514	+	0.812821i	$0.302069\pi$
$30$		0			0
$31$	−2001.75			−0.374115			−0.187058	−	0.982349i	$-0.559895\pi$
$31$	−2001.75			−0.374115			−0.187058	+	0.982349i	$0.559895\pi$
$32$		0			0
$33$		−	8842.91i		−	1.41355i
$34$		0			0
$35$	11539.6	+	2622.71i	1.59229	+	0.361893i
$36$		0			0
$37$			3652.25i			0.438587i	0.975659	+	0.219294i	$0.0703752\pi$
$37$			3652.25i			0.438587i	−0.975659	+	0.219294i	$0.929625\pi$
$38$		0			0
$39$	−12433.0			−1.30892
$40$		0			0
$41$	17113.3			1.58992			0.794960	−	0.606662i	$-0.207492\pi$
$41$	17113.3			1.58992			0.794960	+	0.606662i	$0.207492\pi$
$42$		0			0
$43$		−	4073.34i		−	0.335953i	−0.985791	−	0.167977i	$-0.946277\pi$
$43$		−	4073.34i		−	0.335953i	0.985791	−	0.167977i	$-0.0537233\pi$
$44$		0			0
$45$	562.548	−	2475.14i	0.0414122	−	0.182209i
$46$		0			0
$47$		−	13335.0i		−	0.880539i	−0.897866	−	0.440270i	$-0.854883\pi$
$47$		−	13335.0i		−	0.880539i	0.897866	−	0.440270i	$-0.145117\pi$
$48$		0			0
$49$	−28006.2			−1.66634
$50$		0			0
$51$	38547.2			2.07524
$52$		0			0
$53$		−	27482.0i		−	1.34388i	−0.740607	−	0.671938i	$-0.765462\pi$
$53$		−	27482.0i		−	1.34388i	0.740607	−	0.671938i	$-0.234538\pi$
$54$		0			0
$55$	−6451.20	+	28384.5i	−0.287564	+	1.26524i
$56$		0			0
$57$		−	34537.9i		−	1.40802i
$58$		0			0
$59$	3890.44			0.145502			0.0727509	−	0.997350i	$-0.476822\pi$
$59$	3890.44			0.145502			0.0727509	+	0.997350i	$0.476822\pi$
$60$		0			0
$61$	−8737.66			−0.300656			−0.150328	−	0.988636i	$-0.548033\pi$
$61$	−8737.66			−0.300656			−0.150328	+	0.988636i	$0.548033\pi$
$62$		0			0
$63$			9612.02i			0.305115i
$64$		0			0
$65$	39908.2	+	9070.28i	1.17160	+	0.266279i
$66$		0			0
$67$		−	40950.5i		−	1.11448i	−0.830351	−	0.557240i	$-0.811860\pi$
$67$		−	40950.5i		−	1.11448i	0.830351	−	0.557240i	$-0.188140\pi$
$68$		0			0
$69$	16548.6			0.418445
$70$		0			0
$71$	−26897.5			−0.633236			−0.316618	−	0.948553i	$-0.602547\pi$
$71$	−26897.5			−0.633236			−0.316618	+	0.948553i	$0.602547\pi$
$72$		0			0
$73$		−	9824.26i		−	0.215771i	−0.994163	−	0.107885i	$-0.965592\pi$
$73$		−	9824.26i		−	0.215771i	0.994163	−	0.107885i	$-0.0344080\pi$
$74$		0			0
$75$	−22938.6	+	47856.9i	−0.470885	+	0.982407i
$76$		0			0
$77$		−	110229.i		−	2.11870i
$78$		0			0
$79$	−60376.0			−1.08842			−0.544210	−	0.838949i	$-0.683171\pi$
$79$	−60376.0			−1.08842			−0.544210	+	0.838949i	$0.683171\pi$
$80$		0			0
$81$	−68020.9			−1.15194
$82$		0			0
$83$			2419.27i			0.0385468i	0.999814	+	0.0192734i	$0.00613529\pi$
$83$			2419.27i			0.0385468i	−0.999814	+	0.0192734i	$0.993865\pi$
$84$		0			0
$85$	−123731.	−	28121.5i	−1.85751	−	0.422173i
$86$		0			0
$87$		−	89605.2i		−	1.26921i
$88$		0			0
$89$	−69364.1			−0.928239			−0.464120	−	0.885773i	$-0.653629\pi$
$89$	−69364.1			−0.928239			−0.464120	+	0.885773i	$0.653629\pi$
$90$		0			0
$91$	−154980.			−1.96188
$92$		0			0
$93$			33994.7i			0.407572i
$94$		0			0
$95$	−25196.6	+	110862.i	−0.286439	+	1.26030i
$96$		0			0
$97$		−	76217.3i		−	0.822478i	−0.911527	−	0.411239i	$-0.865096\pi$
$97$		−	76217.3i		−	0.822478i	0.911527	−	0.411239i	$-0.134904\pi$
$98$		0			0
$99$	−23643.1			−0.242447
$100$		0			0
$101$	15273.9			0.148986			0.0744930	−	0.997222i	$-0.476266\pi$
$101$	15273.9			0.148986			0.0744930	+	0.997222i	$0.476266\pi$
$102$		0			0
$103$		−	37841.1i		−	0.351456i	−0.984439	−	0.175728i	$-0.943772\pi$
$103$		−	37841.1i		−	0.351456i	0.984439	−	0.175728i	$-0.0562279\pi$
$104$		0			0
$105$	44540.2	−	195972.i	0.394257	−	1.73468i
$106$		0			0
$107$			159135.i			1.34371i	0.740681	+	0.671857i	$0.234503\pi$
$107$			159135.i			1.34371i	−0.740681	+	0.671857i	$0.765497\pi$
$108$		0			0
$109$	93740.1			0.755717			0.377858	−	0.925863i	$-0.376661\pi$
$109$	93740.1			0.755717			0.377858	+	0.925863i	$0.376661\pi$
$110$		0			0
$111$	62024.3			0.477809
$112$		0			0
$113$		−	145638.i		−	1.07295i	−0.843916	−	0.536476i	$-0.819755\pi$
$113$		−	145638.i		−	1.07295i	0.843916	−	0.536476i	$-0.180245\pi$
$114$		0			0
$115$	−53118.7	−	12072.8i	−0.374544	−	0.0851259i
$116$		0			0
$117$			33241.9i			0.224502i
$118$		0			0
$119$	480500.			3.11047
$120$		0			0
$121$	110084.			0.683537
$122$		0			0
$123$		−	290628.i		−	1.73210i
$124$		0			0
$125$	108543.	−	136879.i	0.621337	−	0.783544i
$126$		0			0
$127$		−	8018.59i		−	0.0441153i	−0.999757	−	0.0220576i	$-0.992978\pi$
$127$		−	8018.59i		−	0.0441153i	0.999757	−	0.0220576i	$-0.00702173\pi$
$128$		0			0
$129$	−69175.5			−0.365997
$130$		0			0
$131$	110408.			0.562110			0.281055	−	0.959692i	$-0.409316\pi$
$131$	110408.			0.562110			0.281055	+	0.959692i	$0.409316\pi$
$132$		0			0
$133$		−	430524.i		−	2.11042i
$134$		0			0
$135$	182921.	+	41574.2i	0.863833	+	0.196331i
$136$		0			0
$137$			4536.88i			0.0206517i	0.999947	+	0.0103259i	$0.00328688\pi$
$137$			4536.88i			0.0206517i	−0.999947	+	0.0103259i	$0.996713\pi$
$138$		0			0
$139$	6371.93			0.0279727			0.0139863	−	0.999902i	$-0.495548\pi$
$139$	6371.93			0.0279727			0.0139863	+	0.999902i	$0.495548\pi$
$140$		0			0
$141$	−226462.			−0.959285
$142$		0			0
$143$		−	381212.i		−	1.55893i
$144$		0			0
$145$	−65370.0	+	287620.i	−0.258201	+	1.13606i
$146$		0			0
$147$			475615.i			1.81536i
$148$		0			0
$149$	−232367.			−0.857450			−0.428725	−	0.903435i	$-0.641037\pi$
$149$	−232367.			−0.857450			−0.428725	+	0.903435i	$0.641037\pi$
$150$		0			0
$151$	234269.			0.836128			0.418064	−	0.908418i	$-0.362709\pi$
$151$	234269.			0.836128			0.418064	+	0.908418i	$0.362709\pi$
$152$		0			0
$153$		−	103063.i		−	0.355938i
$154$		0			0
$155$	24800.3	−	109118.i	0.0829140	−	0.364812i
$156$		0			0
$157$		−	661.264i		−	0.00214105i	−0.999999	−	0.00107052i	$-0.999659\pi$
$157$		−	661.264i		−	0.00214105i	0.999999	−	0.00107052i	$-0.000340758\pi$
$158$		0			0
$159$	−466714.			−1.46406
$160$		0			0
$161$	206282.			0.627187
$162$		0			0
$163$		−	388467.i		−	1.14521i	−0.819831	−	0.572606i	$-0.805933\pi$
$163$		−	388467.i		−	1.14521i	0.819831	−	0.572606i	$-0.194067\pi$
$164$		0			0
$165$	482040.	+	109558.i	1.37839	+	0.313280i
$166$		0			0
$167$		−	35545.3i		−	0.0986258i	−0.998783	−	0.0493129i	$-0.984297\pi$
$167$		−	35545.3i		−	0.0986258i	0.998783	−	0.0493129i	$-0.0157031\pi$
$168$		0			0
$169$	−164685.			−0.443544
$170$		0			0
$171$	−92343.4			−0.241499
$172$		0			0
$173$			469613.i			1.19296i	0.802629	+	0.596479i	$0.203434\pi$
$173$			469613.i			1.19296i	−0.802629	+	0.596479i	$0.796566\pi$
$174$		0			0
$175$	−285936.	+	596548.i	−0.705787	+	1.47248i
$176$		0			0
$177$		−	66069.4i		−	0.158514i
$178$		0			0
$179$	−25442.9			−0.0593518			−0.0296759	−	0.999560i	$-0.509448\pi$
$179$	−25442.9			−0.0593518			−0.0296759	+	0.999560i	$0.509448\pi$
$180$		0			0
$181$	131312.			0.297926			0.148963	−	0.988843i	$-0.452406\pi$
$181$	131312.			0.297926			0.148963	+	0.988843i	$0.452406\pi$
$182$		0			0
$183$			148387.i			0.327544i
$184$		0			0
$185$	−199090.	−	45248.9i	−0.427680	−	0.0972027i
$186$		0			0
$187$			1.18191e6i			2.47161i
$188$		0			0
$189$	−710361.			−1.44652
$190$		0			0
$191$	−165447.			−0.328152			−0.164076	−	0.986448i	$-0.552464\pi$
$191$	−165447.			−0.328152			−0.164076	+	0.986448i	$0.552464\pi$
$192$		0			0
$193$			501316.i			0.968765i	0.874856	+	0.484383i	$0.160956\pi$
$193$			501316.i			0.968765i	−0.874856	+	0.484383i	$0.839044\pi$
$194$		0			0
$195$	154036.	−	677741.i	0.290092	−	1.27637i
$196$		0			0
$197$		−	142169.i		−	0.261000i	−0.991448	−	0.130500i	$-0.958342\pi$
$197$		−	142169.i		−	0.261000i	0.991448	−	0.130500i	$-0.0416583\pi$
$198$		0			0
$199$	−946508.			−1.69430			−0.847152	−	0.531350i	$-0.821685\pi$
$199$	−946508.			−1.69430			−0.847152	+	0.531350i	$0.821685\pi$
$200$		0			0
$201$	−695443.			−1.21415
$202$		0			0
$203$		−	1.11695e6i		−	1.90237i
$204$		0			0
$205$	−212023.	+	932874.i	−0.352369	+	1.55038i
$206$		0			0
$207$		−	44245.7i		−	0.0717704i
$208$		0			0
$209$	1.05898e6			1.67696
$210$		0			0
$211$	946154.			1.46304			0.731519	−	0.681821i	$-0.238812\pi$
$211$	946154.			1.46304			0.731519	+	0.681821i	$0.238812\pi$
$212$		0			0
$213$			456787.i			0.689866i
$214$		0			0
$215$	222044.	+	50465.9i	0.327599	+	0.0744563i
$216$		0			0
$217$			423753.i			0.610890i
$218$		0			0
$219$	−166841.			−0.235067
$220$		0			0
$221$	1.66174e6			2.28867
$222$		0			0
$223$		−	459405.i		−	0.618633i	−0.950959	−	0.309317i	$-0.899900\pi$
$223$		−	459405.i		−	0.618633i	0.950959	−	0.309317i	$-0.100100\pi$
$224$		0			0
$225$	127954.	+	61330.6i	0.168499	+	0.0807646i
$226$		0			0
$227$			493138.i			0.635190i	0.948226	+	0.317595i	$0.102875\pi$
$227$			493138.i			0.635190i	−0.948226	+	0.317595i	$0.897125\pi$
$228$		0			0
$229$	−210901.			−0.265760			−0.132880	−	0.991132i	$-0.542423\pi$
$229$	−210901.			−0.265760			−0.132880	+	0.991132i	$0.542423\pi$
$230$		0			0
$231$	−1.87197e6			−2.30817
$232$		0			0
$233$			7304.87i			0.00881501i	0.999990	+	0.00440750i	$0.00140296\pi$
$233$			7304.87i			0.00881501i	−0.999990	+	0.00440750i	$0.998597\pi$
$234$		0			0
$235$	726912.	+	165212.i	0.858642	+	0.195151i
$236$		0			0
$237$			1.02534e6i			1.18576i
$238$		0			0
$239$	868906.			0.983962			0.491981	−	0.870606i	$-0.336273\pi$
$239$	868906.			0.983962			0.491981	+	0.870606i	$0.336273\pi$
$240$		0			0
$241$	359511.			0.398722			0.199361	−	0.979926i	$-0.436113\pi$
$241$	359511.			0.398722			0.199361	+	0.979926i	$0.436113\pi$
$242$		0			0
$243$			339745.i			0.369094i
$244$		0			0
$245$	346977.	−	1.52666e6i	0.369306	−	1.62490i
$246$		0			0
$247$		−	1.48891e6i		−	1.55283i
$248$		0			0
$249$	41085.2			0.0419940
$250$		0			0
$251$	−999037.			−1.00092			−0.500458	−	0.865761i	$-0.666835\pi$
$251$	−999037.			−1.00092			−0.500458	+	0.865761i	$0.666835\pi$
$252$		0			0
$253$			507402.i			0.498369i
$254$		0			0
$255$	−477573.	+	2.10127e6i	−0.459928	+	2.02363i
$256$		0			0
$257$			1.58468e6i			1.49661i	0.663354	+	0.748306i	$0.269132\pi$
$257$			1.58468e6i			1.49661i	−0.663354	+	0.748306i	$0.730868\pi$
$258$		0			0
$259$	773149.			0.716166
$260$		0			0
$261$	−239576.			−0.217692
$262$		0			0
$263$			1.84241e6i			1.64247i	0.570589	+	0.821236i	$0.306715\pi$
$263$			1.84241e6i			1.64247i	−0.570589	+	0.821236i	$0.693285\pi$
$264$		0			0
$265$	1.49809e6	+	340484.i	1.31046	+	0.297839i
$266$		0			0
$267$			1.17798e6i			1.01125i
$268$		0			0
$269$	−1.28997e6			−1.08693			−0.543464	−	0.839433i	$-0.682887\pi$
$269$	−1.28997e6			−1.08693			−0.543464	+	0.839433i	$0.682887\pi$
$270$		0			0
$271$	1.88685e6			1.56068			0.780339	−	0.625357i	$-0.215047\pi$
$271$	1.88685e6			1.56068			0.780339	+	0.625357i	$0.215047\pi$
$272$		0			0
$273$			2.63195e6i			2.13733i
$274$		0			0
$275$	−1.46736e6	−	703329.i	−1.17005	−	0.560824i
$276$		0			0
$277$			593570.i			0.464807i	0.972619	+	0.232404i	$0.0746590\pi$
$277$			593570.i			0.464807i	−0.972619	+	0.232404i	$0.925341\pi$
$278$		0			0
$279$	90891.1			0.0699054
$280$		0			0
$281$	643035.			0.485813			0.242906	−	0.970050i	$-0.421899\pi$
$281$	643035.			0.485813			0.242906	+	0.970050i	$0.421899\pi$
$282$		0			0
$283$		−	1.62977e6i		−	1.20965i	−0.796358	−	0.604825i	$-0.793243\pi$
$283$		−	1.62977e6i		−	1.20965i	0.796358	−	0.604825i	$-0.206757\pi$
$284$		0			0
$285$	1.88271e6	+	427901.i	1.37301	+	0.312055i
$286$		0			0
$287$		−	3.62274e6i		−	2.59617i
$288$		0			0
$289$	−3.73221e6			−2.62858
$290$		0			0
$291$	−1.29436e6			−0.896032
$292$		0			0
$293$			266348.i			0.181251i	0.995885	+	0.0906254i	$0.0288866\pi$
$293$			266348.i			0.181251i	−0.995885	+	0.0906254i	$0.971113\pi$
$294$		0			0
$295$	−48199.8	+	212074.i	−0.0322471	+	0.141883i
$296$		0			0
$297$		−	1.74731e6i		−	1.14942i
$298$		0			0
$299$	713399.			0.461482
$300$		0			0
$301$	−862289.			−0.548576
$302$		0			0
$303$		−	259389.i		−	0.162310i
$304$		0			0
$305$	108254.	−	476303.i	0.0666335	−	0.293179i
$306$		0			0
$307$		−	1.64753e6i		−	0.997672i	−0.866696	−	0.498836i	$-0.833761\pi$
$307$		−	1.64753e6i		−	0.997672i	0.866696	−	0.498836i	$-0.166239\pi$
$308$		0			0
$309$	−642638.			−0.382886
$310$		0			0
$311$	3.04162e6			1.78322			0.891609	−	0.452806i	$-0.149577\pi$
$311$	3.04162e6			1.78322			0.891609	+	0.452806i	$0.149577\pi$
$312$		0			0
$313$		−	3.02897e6i		−	1.74757i	−0.486314	−	0.873784i	$-0.661659\pi$
$313$		−	3.02897e6i		−	1.74757i	0.486314	−	0.873784i	$-0.338341\pi$
$314$		0			0
$315$	−523966.	−	119086.i	−0.297527	−	0.0676217i
$316$		0			0
$317$			2.07750e6i			1.16116i	0.814202	+	0.580582i	$0.197175\pi$
$317$			2.07750e6i			1.16116i	−0.814202	+	0.580582i	$0.802825\pi$
$318$		0			0
$319$	2.74741e6			1.51164
$320$		0			0
$321$	2.70251e6			1.46388
$322$		0			0
$323$			4.61620e6i			2.46195i
$324$		0			0
$325$	−988870.	+	2.06308e6i	−0.519315	+	1.08345i
$326$		0			0
$327$		−	1.59194e6i		−	0.823300i
$328$		0			0
$329$	−2.82291e6			−1.43783
$330$		0			0
$331$	3.75626e6			1.88446			0.942228	−	0.334973i	$-0.108727\pi$
$331$	3.75626e6			1.88446			0.942228	+	0.334973i	$0.108727\pi$
$332$		0			0
$333$		−	165833.i		−	0.0819524i
$334$		0			0
$335$	2.23228e6	+	507349.i	1.08677	+	0.246999i
$336$		0			0
$337$			1.79503e6i			0.860985i	0.902594	+	0.430493i	$0.141660\pi$
$337$			1.79503e6i			0.860985i	−0.902594	+	0.430493i	$0.858340\pi$
$338$		0			0
$339$	−2.47331e6			−1.16890
$340$		0			0
$341$	−1.04232e6			−0.485419
$342$		0			0
$343$			2.37077e6i			1.08806i
$344$		0			0
$345$	−205026.	+	902089.i	−0.0927387	+	0.408039i
$346$		0			0
$347$			1.73738e6i			0.774591i	0.921956	+	0.387295i	$0.126591\pi$
$347$			1.73738e6i			0.774591i	−0.921956	+	0.387295i	$0.873409\pi$
$348$		0			0
$349$	3.63164e6			1.59602			0.798012	−	0.602642i	$-0.205885\pi$
$349$	3.63164e6			1.59602			0.798012	+	0.602642i	$0.205885\pi$
$350$		0			0
$351$	−2.45669e6			−1.06434
$352$		0			0
$353$		−	655592.i		−	0.280025i	−0.990150	−	0.140012i	$-0.955286\pi$
$353$		−	655592.i		−	0.280025i	0.990150	−	0.140012i	$-0.0447143\pi$
$354$		0			0
$355$	333241.	−	1.46622e6i	0.140342	−	0.617489i
$356$		0			0
$357$		−	8.16010e6i		−	3.38864i
$358$		0			0
$359$	1.17606e6			0.481609			0.240805	−	0.970574i	$-0.422589\pi$
$359$	1.17606e6			0.481609			0.240805	+	0.970574i	$0.422589\pi$
$360$		0			0
$361$	1.65997e6			0.670399
$362$		0			0
$363$		−	1.86951e6i		−	0.744665i
$364$		0			0
$365$	535535.	+	121716.i	0.210405	+	0.0478206i
$366$		0			0
$367$		−	788323.i		−	0.305520i	−0.988263	−	0.152760i	$-0.951184\pi$
$367$		−	788323.i		−	0.305520i	0.988263	−	0.152760i	$-0.0488161\pi$
$368$		0			0
$369$	−777046.			−0.297085
$370$		0			0
$371$	−5.81770e6			−2.19441
$372$		0			0
$373$		−	3.14542e6i		−	1.17060i	−0.810819	−	0.585298i	$-0.800978\pi$
$373$		−	3.14542e6i		−	1.17060i	0.810819	−	0.585298i	$-0.199022\pi$
$374$		0			0
$375$	−2.32456e6	−	1.84333e6i	−0.853615	−	0.676902i
$376$		0			0
$377$		−	3.86282e6i		−	1.39975i
$378$		0			0
$379$	−1.29034e6			−0.461430			−0.230715	−	0.973021i	$-0.574106\pi$
$379$	−1.29034e6			−0.461430			−0.230715	+	0.973021i	$0.574106\pi$
$380$		0			0
$381$	−136176.			−0.0480604
$382$		0			0
$383$			2.39221e6i			0.833303i	0.909066	+	0.416652i	$0.136796\pi$
$383$			2.39221e6i			0.833303i	−0.909066	+	0.416652i	$0.863204\pi$
$384$		0			0
$385$	6.00875e6	+	1.36566e6i	2.06601	+	0.469560i
$386$		0			0
$387$			184953.i			0.0627747i
$388$		0			0
$389$	−4.31206e6			−1.44481			−0.722406	−	0.691469i	$-0.756964\pi$
$389$	−4.31206e6			−1.44481			−0.722406	+	0.691469i	$0.756964\pi$
$390$		0			0
$391$	−2.21182e6			−0.731658
$392$		0			0
$393$		−	1.87500e6i		−	0.612378i
$394$		0			0
$395$	748017.	−	3.29119e6i	0.241223	−	1.06135i
$396$		0			0
$397$			2.22787e6i			0.709436i	0.934973	+	0.354718i	$0.115423\pi$
$397$			2.22787e6i			0.709436i	−0.934973	+	0.354718i	$0.884577\pi$
$398$		0			0
$399$	−7.31137e6			−2.29915
$400$		0			0
$401$	−4.52683e6			−1.40583			−0.702915	−	0.711274i	$-0.748119\pi$
$401$	−4.52683e6			−1.40583			−0.702915	+	0.711274i	$0.748119\pi$
$402$		0			0
$403$			1.46549e6i			0.449490i
$404$		0			0
$405$	842733.	−	3.70792e6i	0.255301	−	1.12329i
$406$		0			0
$407$			1.90175e6i			0.569072i
$408$		0			0
$409$	−4.73728e6			−1.40030			−0.700149	−	0.713997i	$-0.746883\pi$
$409$	−4.73728e6			−1.40030			−0.700149	+	0.713997i	$0.746883\pi$
$410$		0			0
$411$	77047.7			0.0224986
$412$		0			0
$413$		−	823571.i		−	0.237589i
$414$		0			0
$415$	−131878.	−	29973.1i	−0.0375882	−	0.00854301i
$416$		0			0
$417$		−	108211.i		−	0.0304743i
$418$		0			0
$419$	−1.14343e6			−0.318181			−0.159091	−	0.987264i	$-0.550856\pi$
$419$	−1.14343e6			−0.318181			−0.159091	+	0.987264i	$0.550856\pi$
$420$		0			0
$421$	6.03804e6			1.66032			0.830158	−	0.557528i	$-0.188250\pi$
$421$	6.03804e6			1.66032			0.830158	+	0.557528i	$0.188250\pi$
$422$		0			0
$423$			605488.i			0.164534i
$424$		0			0
$425$	3.06589e6	−	6.39636e6i	0.823349	−	1.71775i
$426$		0			0
$427$			1.84968e6i			0.490940i
$428$		0			0
$429$	−6.47394e6			−1.69834
$430$		0			0
$431$	−461842.			−0.119757			−0.0598783	−	0.998206i	$-0.519071\pi$
$431$	−461842.			−0.119757			−0.0598783	+	0.998206i	$0.519071\pi$
$432$		0			0
$433$			1.86278e6i			0.477464i	0.971085	+	0.238732i	$0.0767318\pi$
$433$			1.86278e6i			0.477464i	−0.971085	+	0.238732i	$0.923268\pi$
$434$		0			0
$435$	4.88451e6	+	1.11015e6i	1.23765	+	0.281292i
$436$		0			0
$437$			1.98177e6i			0.496420i
$438$		0			0
$439$	−5.85752e6			−1.45062			−0.725308	−	0.688424i	$-0.758303\pi$
$439$	−5.85752e6			−1.45062			−0.725308	+	0.688424i	$0.758303\pi$
$440$		0			0
$441$	1.27164e6			0.311365
$442$		0			0
$443$			4.28843e6i			1.03822i	0.854708	+	0.519109i	$0.173736\pi$
$443$			4.28843e6i			1.03822i	−0.854708	+	0.519109i	$0.826264\pi$
$444$		0			0
$445$	859374.	−	3.78114e6i	0.205723	−	0.905155i
$446$		0			0
$447$			3.94617e6i			0.934130i
$448$		0			0
$449$	−2.93198e6			−0.686348			−0.343174	−	0.939272i	$-0.611502\pi$
$449$	−2.93198e6			−0.686348			−0.343174	+	0.939272i	$0.611502\pi$
$450$		0			0
$451$	8.91103e6			2.06294
$452$		0			0
$453$		−	3.97848e6i		−	0.910902i
$454$		0			0
$455$	1.92010e6	−	8.44821e6i	0.434806	−	1.91309i
$456$		0			0
$457$		−	5.06303e6i		−	1.13402i	−0.823712	−	0.567009i	$-0.808101\pi$
$457$		−	5.06303e6i		−	1.13402i	0.823712	−	0.567009i	$-0.191899\pi$
$458$		0			0
$459$	7.61670e6			1.68747
$460$		0			0
$461$	2.43641e6			0.533947			0.266974	−	0.963704i	$-0.413976\pi$
$461$	2.43641e6			0.533947			0.266974	+	0.963704i	$0.413976\pi$
$462$		0			0
$463$		−	118770.i		−	0.0257485i	−0.999917	−	0.0128743i	$-0.995902\pi$
$463$		−	118770.i		−	0.0257485i	0.999917	−	0.0128743i	$-0.00409812\pi$
$464$		0			0
$465$	−1.85310e6	−	421172.i	−0.397436	−	0.0903289i
$466$		0			0
$467$		−	7.00735e6i		−	1.48683i	−0.668830	−	0.743415i	$-0.733205\pi$
$467$		−	7.00735e6i		−	1.48683i	0.668830	−	0.743415i	$-0.266795\pi$
$468$		0			0
$469$	−8.66887e6			−1.81983
$470$		0			0
$471$	−11229.9			−0.00233252
$472$		0			0
$473$		−	2.12101e6i		−	0.435903i
$474$		0			0
$475$	−5.73108e6	−	2.74701e6i	−1.16547	−	0.558632i
$476$		0			0
$477$			1.24785e6i			0.251110i
$478$		0			0
$479$	7.00020e6			1.39403			0.697014	−	0.717058i	$-0.254512\pi$
$479$	7.00020e6			1.39403			0.697014	+	0.717058i	$0.254512\pi$
$480$		0			0
$481$	2.67383e6			0.526952
$482$		0			0
$483$		−	3.50319e6i		−	0.683276i
$484$		0			0
$485$	4.15472e6	+	944281.i	0.802025	+	0.182283i
$486$		0			0
$487$			9.44869e6i			1.80530i	0.430375	+	0.902650i	$0.358381\pi$
$487$			9.44869e6i			1.80530i	−0.430375	+	0.902650i	$0.641619\pi$
$488$		0			0
$489$	−6.59715e6			−1.24763
$490$		0			0
$491$	1.54041e6			0.288358			0.144179	−	0.989552i	$-0.453946\pi$
$491$	1.54041e6			0.288358			0.144179	+	0.989552i	$0.453946\pi$
$492$		0			0
$493$			1.19763e7i			2.21924i
$494$		0			0
$495$	292922.	−	1.28882e6i	0.0537328	−	0.236418i
$496$		0			0
$497$			5.69396e6i			1.03401i
$498$		0			0
$499$	6.02185e6			1.08263			0.541313	−	0.840821i	$-0.317927\pi$
$499$	6.02185e6			1.08263			0.541313	+	0.840821i	$0.317927\pi$
$500$		0			0
$501$	−603648.			−0.107446
$502$		0			0
$503$		−	1.84251e6i		−	0.324705i	−0.986733	−	0.162352i	$-0.948092\pi$
$503$		−	1.84251e6i		−	0.324705i	0.986733	−	0.162352i	$-0.0519082\pi$
$504$		0			0
$505$	−189233.	+	832601.i	−0.0330193	+	0.145281i
$506$		0			0
$507$			2.79676e6i			0.483210i
$508$		0			0
$509$	−7.12602e6			−1.21914			−0.609568	−	0.792734i	$-0.708657\pi$
$509$	−7.12602e6			−1.21914			−0.609568	+	0.792734i	$0.708657\pi$
$510$		0			0
$511$	−2.07971e6			−0.352331
$512$		0			0
$513$		−	6.82449e6i		−	1.14492i
$514$		0			0
$515$	2.06278e6	+	468826.i	0.342716	+	0.0778921i
$516$		0			0
$517$		−	6.94363e6i		−	1.14251i
$518$		0			0
$519$	7.97522e6			1.29964
$520$		0			0
$521$	−2.26117e6			−0.364954			−0.182477	−	0.983210i	$-0.558412\pi$
$521$	−2.26117e6			−0.364954			−0.182477	+	0.983210i	$0.558412\pi$
$522$		0			0
$523$			748790.i			0.119703i	0.998207	+	0.0598516i	$0.0190628\pi$
$523$			748790.i			0.119703i	−0.998207	+	0.0598516i	$0.980937\pi$
$524$		0			0
$525$	1.01309e7	+	4.85591e6i	1.60417	+	0.768904i
$526$		0			0
$527$		−	4.54360e6i		−	0.712646i
$528$		0			0
$529$	5.48679e6			0.852470
$530$		0			0
$531$	−176649.			−0.0271878
$532$		0			0
$533$		−	1.25288e7i		−	1.91025i
$534$		0			0
$535$	−8.67470e6	−	1.97158e6i	−1.31030	−	0.297803i
$536$		0			0
$537$			432085.i			0.0646596i
$538$		0			0
$539$	−1.45830e7			−2.16210
$540$		0			0
$541$	−278009.			−0.0408381			−0.0204190	−	0.999792i	$-0.506500\pi$
$541$	−278009.			−0.0408381			−0.0204190	+	0.999792i	$0.506500\pi$
$542$		0			0
$543$		−	2.23001e6i		−	0.324570i
$544$		0			0
$545$	−1.16138e6	+	5.10991e6i	−0.167487	+	0.736923i
$546$		0			0
$547$		−	6.06047e6i		−	0.866040i	−0.901384	−	0.433020i	$-0.857448\pi$
$547$		−	6.06047e6i		−	0.866040i	0.901384	−	0.433020i	$-0.142552\pi$
$548$		0			0
$549$	396741.			0.0561792
$550$		0			0
$551$	1.07306e7			1.50573
$552$		0			0
$553$			1.27811e7i			1.77727i
$554$		0			0
$555$	−768439.	+	3.38104e6i	−0.105895	+	0.465927i
$556$		0			0
$557$		−	8.37326e6i		−	1.14355i	−0.820409	−	0.571777i	$-0.806254\pi$
$557$		−	8.37326e6i		−	1.14355i	0.820409	−	0.571777i	$-0.193746\pi$
$558$		0			0
$559$	−2.98211e6			−0.403640
$560$		0			0
$561$	2.00718e7			2.69264
$562$		0			0
$563$			9.87244e6i			1.31266i	0.754472	+	0.656332i	$0.227893\pi$
$563$			9.87244e6i			1.31266i	−0.754472	+	0.656332i	$0.772107\pi$
$564$		0			0
$565$	7.93897e6	+	1.80436e6i	1.04627	+	0.237795i
$566$		0			0
$567$			1.43994e7i			1.88100i
$568$		0			0
$569$	6.41803e6			0.831038			0.415519	−	0.909584i	$-0.363600\pi$
$569$	6.41803e6			0.831038			0.415519	+	0.909584i	$0.363600\pi$
$570$		0			0
$571$	−1.36965e7			−1.75801			−0.879003	−	0.476817i	$-0.841790\pi$
$571$	−1.36965e7			−1.75801			−0.879003	+	0.476817i	$0.841790\pi$
$572$		0			0
$573$			2.80970e6i			0.357498i
$574$		0			0
$575$	1.31621e6	−	2.74601e6i	0.166018	−	0.346363i
$576$		0			0
$577$			1.49348e6i			0.186749i	0.995631	+	0.0933747i	$0.0297655\pi$
$577$			1.49348e6i			0.186749i	−0.995631	+	0.0933747i	$0.970235\pi$
$578$		0			0
$579$	8.51361e6			1.05540
$580$		0			0
$581$	512137.			0.0629428
$582$		0			0
$583$		−	1.43101e7i		−	1.74369i
$584$		0			0
$585$	−1.81206e6	−	411844.i	−0.218919	−	0.0497557i
$586$		0			0
$587$		−	1.27999e7i		−	1.53324i	−0.642098	−	0.766622i	$-0.721936\pi$
$587$		−	1.27999e7i		−	1.53324i	0.642098	−	0.766622i	$-0.278064\pi$
$588$		0			0
$589$	−4.07102e6			−0.483521
$590$		0			0
$591$	−2.41439e6			−0.284341
$592$		0			0
$593$			6.38952e6i			0.746160i	0.927799	+	0.373080i	$0.121698\pi$
$593$			6.38952e6i			0.746160i	−0.927799	+	0.373080i	$0.878302\pi$
$594$		0			0
$595$	−5.95307e6	+	2.61928e7i	−0.689364	+	3.03312i
$596$		0			0
$597$			1.60741e7i			1.84582i
$598$		0			0
$599$	7.77509e6			0.885398			0.442699	−	0.896670i	$-0.354021\pi$
$599$	7.77509e6			0.885398			0.442699	+	0.896670i	$0.354021\pi$
$600$		0			0
$601$	−1.29871e7			−1.46665			−0.733323	−	0.679880i	$-0.762032\pi$
$601$	−1.29871e7			−1.46665			−0.733323	+	0.679880i	$0.762032\pi$
$602$		0			0
$603$			1.85939e6i			0.208247i
$604$		0			0
$605$	−1.36387e6	+	6.00086e6i	−0.151490	+	0.666538i
$606$		0			0
$607$		−	6.40031e6i		−	0.705066i	−0.935799	−	0.352533i	$-0.885321\pi$
$607$		−	6.40031e6i		−	0.705066i	0.935799	−	0.352533i	$-0.114679\pi$
$608$		0			0
$609$	−1.89686e7			−2.07249
$610$		0			0
$611$	−9.76263e6			−1.05795
$612$		0			0
$613$			1.47519e7i			1.58562i	0.609471	+	0.792808i	$0.291382\pi$
$613$			1.47519e7i			1.58562i	−0.609471	+	0.792808i	$0.708618\pi$
$614$		0			0
$615$	1.58426e7	+	3.60068e6i	1.68903	+	0.383881i
$616$		0			0
$617$		−	979056.i		−	0.103537i	−0.998659	−	0.0517684i	$-0.983514\pi$
$617$		−	979056.i		−	0.103537i	0.998659	−	0.0517684i	$-0.0164858\pi$
$618$		0			0
$619$	−3.60371e6			−0.378027			−0.189013	−	0.981974i	$-0.560529\pi$
$619$	−3.60371e6			−0.378027			−0.189013	+	0.981974i	$0.560529\pi$
$620$		0			0
$621$	3.26991e6			0.340256
$622$		0			0
$623$			1.46838e7i			1.51572i
$624$		0			0
$625$	6.11673e6	+	7.61269e6i	0.626353	+	0.779539i
$626$		0			0
$627$		−	1.79841e7i		−	1.82692i
$628$		0			0
$629$	−8.28993e6			−0.835457
$630$		0			0
$631$	−3.51546e6			−0.351486			−0.175743	−	0.984436i	$-0.556233\pi$
$631$	−3.51546e6			−0.351486			−0.175743	+	0.984436i	$0.556233\pi$
$632$		0			0
$633$		−	1.60681e7i		−	1.59388i
$634$		0			0
$635$	437106.	+	99344.9i	0.0430182	+	0.00977713i
$636$		0			0
$637$			2.05035e7i			2.00207i
$638$		0			0
$639$	1.22130e6			0.118324
$640$		0			0
$641$	−2.73523e6			−0.262935			−0.131468	−	0.991320i	$-0.541969\pi$
$641$	−2.73523e6			−0.262935			−0.131468	+	0.991320i	$0.541969\pi$
$642$		0			0
$643$		−	1.57847e7i		−	1.50560i	−0.658251	−	0.752799i	$-0.728703\pi$
$643$		−	1.57847e7i		−	1.50560i	0.658251	−	0.752799i	$-0.271297\pi$
$644$		0			0
$645$	857037.	−	3.77086e6i	0.0811148	−	0.356896i
$646$		0			0
$647$			4.62226e6i			0.434104i	0.976160	+	0.217052i	$0.0696441\pi$
$647$			4.62226e6i			0.434104i	−0.976160	+	0.217052i	$0.930356\pi$
$648$		0			0
$649$	2.02578e6			0.188790
$650$		0			0
$651$	7.19639e6			0.665521
$652$		0			0
$653$		−	4.60370e6i		−	0.422497i	−0.977432	−	0.211249i	$-0.932247\pi$
$653$		−	4.60370e6i		−	0.422497i	0.977432	−	0.211249i	$-0.0677529\pi$
$654$		0			0
$655$	−1.36788e6	+	6.01849e6i	−0.124579	+	0.548131i
$656$		0			0
$657$			446079.i			0.0403179i
$658$		0			0
$659$	−1.49631e7			−1.34217			−0.671087	−	0.741378i	$-0.734172\pi$
$659$	−1.49631e7			−1.34217			−0.671087	+	0.741378i	$0.734172\pi$
$660$		0			0
$661$	1.40490e7			1.25067			0.625333	−	0.780358i	$-0.284963\pi$
$661$	1.40490e7			1.25067			0.625333	+	0.780358i	$0.284963\pi$
$662$		0			0
$663$		−	2.82206e7i		−	2.49334i
$664$		0			0
$665$	2.34685e7	+	5.33389e6i	2.05793	+	0.467725i
$666$		0			0
$667$			5.14150e6i			0.447482i
$668$		0			0
$669$	−7.80185e6			−0.673957
$670$		0			0
$671$	−4.54975e6			−0.390105
$672$		0			0
$673$			1.81487e7i			1.54457i	0.635276	+	0.772285i	$0.280886\pi$
$673$			1.81487e7i			1.54457i	−0.635276	+	0.772285i	$0.719114\pi$
$674$		0			0
$675$	−4.53254e6	+	9.45624e6i	−0.382897	+	0.798839i
$676$		0			0
$677$			3.73842e6i			0.313485i	0.987640	+	0.156743i	$0.0500993\pi$
$677$			3.73842e6i			0.313485i	−0.987640	+	0.156743i	$0.949901\pi$
$678$		0			0
$679$	−1.61345e7			−1.34302
$680$		0			0
$681$	8.37472e6			0.691994
$682$		0			0
$683$			1.31301e7i			1.07700i	0.842624	+	0.538502i	$0.181010\pi$
$683$			1.31301e7i			1.07700i	−0.842624	+	0.538502i	$0.818990\pi$
$684$		0			0
$685$	−247312.	−	56208.9i	−0.0201381	−	0.00457698i
$686$		0			0
$687$			3.58163e6i			0.289527i
$688$		0			0
$689$	−2.01197e7			−1.61463
$690$		0			0
$691$	4.49754e6			0.358327			0.179164	−	0.983819i	$-0.442661\pi$
$691$	4.49754e6			0.358327			0.179164	+	0.983819i	$0.442661\pi$
$692$		0			0
$693$			5.00504e6i			0.395890i
$694$		0			0
$695$	−78943.9	+	347344.i	−0.00619950	+	0.0272770i
$696$		0			0
$697$			3.88441e7i			3.02861i
$698$		0			0
$699$	124055.			0.00960333
$700$		0			0
$701$	1.36524e7			1.04933			0.524666	−	0.851308i	$-0.324190\pi$
$701$	1.36524e7			1.04933			0.524666	+	0.851308i	$0.324190\pi$
$702$		0			0
$703$			7.42770e6i			0.566847i
$704$		0			0
$705$	2.80571e6	−	1.23448e7i	0.212603	−	0.935429i
$706$		0			0
$707$		−	3.23334e6i		−	0.243278i
$708$		0			0
$709$	−7.79465e6			−0.582346			−0.291173	−	0.956670i	$-0.594046\pi$
$709$	−7.79465e6			−0.582346			−0.291173	+	0.956670i	$0.594046\pi$
$710$		0			0
$711$	2.74142e6			0.203377
$712$		0			0
$713$		−	1.95060e6i		−	0.143696i
$714$		0			0
$715$	2.07804e7	+	4.72296e6i	1.52016	+	0.345501i
$716$		0			0
$717$		−	1.47562e7i		−	1.07196i
$718$		0			0
$719$	−1.98555e7			−1.43238			−0.716191	−	0.697904i	$-0.754116\pi$
$719$	−1.98555e7			−1.43238			−0.716191	+	0.697904i	$0.754116\pi$
$720$		0			0
$721$	−8.01064e6			−0.573890
$722$		0			0
$723$		−	6.10540e6i		−	0.434379i
$724$		0			0
$725$	−1.48687e7	−	7.12684e6i	−1.05058	−	0.503561i
$726$		0			0
$727$			2.12828e7i			1.49346i	0.665128	+	0.746730i	$0.268377\pi$
$727$			2.12828e7i			1.49346i	−0.665128	+	0.746730i	$0.731623\pi$
$728$		0			0
$729$	−1.07594e7			−0.749839
$730$		0			0
$731$	9.24572e6			0.639952
$732$		0			0
$733$		−	726358.i		−	0.0499334i	−0.999688	−	0.0249667i	$-0.992052\pi$
$733$		−	726358.i		−	0.0499334i	0.999688	−	0.0249667i	$-0.00794797\pi$
$734$		0			0
$735$	−2.59265e7	−	5.89255e6i	−1.77021	−	0.402332i
$736$		0			0
$737$		−	2.13232e7i		−	1.44605i
$738$		0			0
$739$	259167.			0.0174570			0.00872849	−	0.999962i	$-0.497222\pi$
$739$	259167.			0.0174570			0.00872849	+	0.999962i	$0.497222\pi$
$740$		0			0
$741$	−2.52854e7			−1.69170
$742$		0			0
$743$		−	5.47729e6i		−	0.363994i	−0.983299	−	0.181997i	$-0.941744\pi$
$743$		−	5.47729e6i		−	0.363994i	0.983299	−	0.181997i	$-0.0582560\pi$
$744$		0			0
$745$	2.87887e6	−	1.26667e7i	0.190034	−	0.836126i
$746$		0			0
$747$		−	109849.i		−	0.00720268i
$748$		0			0
$749$	3.36875e7			2.19414
$750$		0			0
$751$	1.96106e7			1.26879			0.634396	−	0.773008i	$-0.281249\pi$
$751$	1.96106e7			1.26879			0.634396	+	0.773008i	$0.281249\pi$
$752$		0			0
$753$			1.69662e7i			1.09043i
$754$		0			0
$755$	−2.90244e6	+	1.27704e7i	−0.185308	+	0.815335i
$756$		0			0
$757$		−	5.92552e6i		−	0.375826i	−0.982186	−	0.187913i	$-0.939828\pi$
$757$		−	5.92552e6i		−	0.375826i	0.982186	−	0.187913i	$-0.0601723\pi$
$758$		0			0
$759$	8.61696e6			0.542937
$760$		0			0
$761$	1.56740e7			0.981112			0.490556	−	0.871410i	$-0.336794\pi$
$761$	1.56740e7			0.981112			0.490556	+	0.871410i	$0.336794\pi$
$762$		0			0
$763$		−	1.98439e7i		−	1.23400i
$764$		0			0
$765$	5.61812e6	+	1.27688e6i	0.347086	+	0.0788854i
$766$		0			0
$767$		−	2.84821e6i		−	0.174817i
$768$		0			0
$769$	2.63779e7			1.60851			0.804255	−	0.594285i	$-0.202565\pi$
$769$	2.63779e7			1.60851			0.804255	+	0.594285i	$0.202565\pi$
$770$		0			0
$771$	2.69119e7			1.63045
$772$		0			0
$773$			1.84175e7i			1.10862i	0.832312	+	0.554308i	$0.187017\pi$
$773$			1.84175e7i			1.10862i	−0.832312	+	0.554308i	$0.812983\pi$
$774$		0			0
$775$	5.64095e6	+	2.70380e6i	0.337363	+	0.161704i
$776$		0			0
$777$		−	1.31300e7i		−	0.780212i
$778$		0			0
$779$	3.48040e7			2.05487
$780$		0			0
$781$	−1.40057e7			−0.821631
$782$		0			0
$783$		−	1.77055e7i		−	1.03206i
$784$		0			0
$785$	36046.5	+	8192.62i	0.00208780	+	0.000474513i
$786$		0			0
$787$		−	1.66821e7i		−	0.960091i	−0.877243	−	0.480046i	$-0.840620\pi$
$787$		−	1.66821e7i		−	0.960091i	0.877243	−	0.480046i	$-0.159380\pi$
$788$		0			0
$789$	3.12888e7			1.78936
$790$		0			0
$791$	−3.08304e7			−1.75201
$792$		0			0
$793$			6.39688e6i			0.361231i
$794$		0			0
$795$	5.78227e6	−	2.54413e7i	0.324474	−	1.42765i
$796$		0			0
$797$		−	2.61029e7i		−	1.45560i	−0.685789	−	0.727800i	$-0.740543\pi$
$797$		−	2.61029e7i		−	1.45560i	0.685789	−	0.727800i	$-0.259457\pi$
$798$		0			0
$799$	3.02680e7			1.67732
$800$		0			0
$801$	3.14954e6			0.173446
$802$		0			0
$803$		−	5.11556e6i		−	0.279965i
$804$		0			0
$805$	−2.55570e6	+	1.12448e7i	−0.139002	+	0.611590i
$806$		0			0
$807$			2.19070e7i			1.18413i
$808$		0			0
$809$	−1.68998e7			−0.907844			−0.453922	−	0.891041i	$-0.649975\pi$
$809$	−1.68998e7			−0.907844			−0.453922	+	0.891041i	$0.649975\pi$
$810$		0			0
$811$	−4.50687e6			−0.240615			−0.120308	−	0.992737i	$-0.538388\pi$
$811$	−4.50687e6			−0.240615			−0.120308	+	0.992737i	$0.538388\pi$
$812$		0			0
$813$		−	3.20434e7i		−	1.70025i
$814$		0			0
$815$	2.11759e7	+	4.81285e6i	1.11673	+	0.253810i
$816$		0			0
$817$		−	8.28408e6i		−	0.434199i
$818$		0			0
$819$	7.03701e6			0.366588
$820$		0			0
$821$	1.72987e7			0.895684			0.447842	−	0.894113i	$-0.352193\pi$
$821$	1.72987e7			0.895684			0.447842	+	0.894113i	$0.352193\pi$
$822$		0			0
$823$		−	4.38452e6i		−	0.225643i	−0.993615	−	0.112822i	$-0.964011\pi$
$823$		−	4.38452e6i		−	0.225643i	0.993615	−	0.112822i	$-0.0359889\pi$
$824$		0			0
$825$	−1.19443e7	+	2.49194e7i	−0.610978	+	1.27468i
$826$		0			0
$827$			2.67189e7i			1.35848i	0.733914	+	0.679242i	$0.237691\pi$
$827$			2.67189e7i			1.35848i	−0.733914	+	0.679242i	$0.762309\pi$
$828$		0			0
$829$	−1.60252e7			−0.809873			−0.404936	−	0.914345i	$-0.632706\pi$
$829$	−1.60252e7			−0.809873			−0.404936	+	0.914345i	$0.632706\pi$
$830$		0			0
$831$	1.00803e7			0.506374
$832$		0			0
$833$		−	6.35689e7i		−	3.17418i
$834$		0			0
$835$	1.93763e6	+	440381.i	0.0961731	+	0.0218581i
$836$		0			0
$837$			6.71716e6i			0.331415i
$838$		0			0
$839$	−2.65800e7			−1.30362			−0.651808	−	0.758384i	$-0.725989\pi$
$839$	−2.65800e7			−1.30362			−0.651808	+	0.758384i	$0.725989\pi$
$840$		0			0
$841$	7.32841e6			0.357289
$842$		0			0
$843$		−	1.09204e7i		−	0.529258i
$844$		0			0
$845$	2.04033e6	−	8.97722e6i	0.0983014	−	0.432514i
$846$		0			0
$847$		−	2.33039e7i		−	1.11614i
$848$		0			0
$849$	−2.76776e7			−1.31783
$850$		0			0
$851$	−3.55893e6			−0.168459
$852$		0			0
$853$			1.33996e7i			0.630548i	0.949001	+	0.315274i	$0.102096\pi$
$853$			1.33996e7i			0.630548i	−0.949001	+	0.315274i	$0.897904\pi$
$854$		0			0
$855$	1.14407e6	−	5.03378e6i	0.0535227	−	0.235493i
$856$		0			0
$857$		−	2.97622e7i		−	1.38425i	−0.721780	−	0.692123i	$-0.756676\pi$
$857$		−	2.97622e7i		−	1.38425i	0.721780	−	0.692123i	$-0.243324\pi$
$858$		0			0
$859$	6.89754e6			0.318942			0.159471	−	0.987203i	$-0.449021\pi$
$859$	6.89754e6			0.318942			0.159471	+	0.987203i	$0.449021\pi$
$860$		0			0
$861$	−6.15233e7			−2.82834
$862$		0			0
$863$			9.06032e6i			0.414111i	0.978329	+	0.207055i	$0.0663880\pi$
$863$			9.06032e6i			0.414111i	−0.978329	+	0.207055i	$0.933612\pi$
$864$		0			0
$865$	−2.55993e7	−	5.81819e6i	−1.16329	−	0.264392i
$866$		0			0
$867$			6.33823e7i			2.86365i
$868$		0			0
$869$	−3.14382e7			−1.41224
$870$		0			0
$871$	−2.99801e7			−1.33902
$872$		0			0
$873$			3.46071e6i			0.153684i
$874$		0			0
$875$	−2.89762e7	−	2.29776e7i	−1.27944	−	1.01458i
$876$		0			0
$877$			2.83641e7i			1.24529i	0.782504	+	0.622645i	$0.213942\pi$
$877$			2.83641e7i			1.24529i	−0.782504	+	0.622645i	$0.786058\pi$
$878$		0			0
$879$	4.52325e6			0.197460
$880$		0			0
$881$	1.27391e7			0.552966			0.276483	−	0.961019i	$-0.410831\pi$
$881$	1.27391e7			0.552966			0.276483	+	0.961019i	$0.410831\pi$
$882$		0			0
$883$			2.72031e6i			0.117413i	0.998275	+	0.0587066i	$0.0186976\pi$
$883$			2.72031e6i			0.117413i	−0.998275	+	0.0587066i	$0.981302\pi$
$884$		0			0
$885$	3.60154e6	+	818555.i	0.154572	+	0.0351309i
$886$		0			0
$887$			5.83636e6i			0.249077i	0.992215	+	0.124538i	$0.0397450\pi$
$887$			5.83636e6i			0.249077i	−0.992215	+	0.124538i	$0.960255\pi$
$888$		0			0
$889$	−1.69747e6			−0.0720355
$890$		0			0
$891$	−3.54189e7			−1.49466
$892$		0			0
$893$		−	2.71199e7i		−	1.13804i
$894$		0			0
$895$	315220.	−	1.38693e6i	0.0131540	−	0.0578759i
$896$		0			0
$897$		−	1.21153e7i		−	0.502752i
$898$		0			0
$899$	−1.05619e7			−0.435854
$900$		0			0
$901$	6.23792e7			2.55993
$902$		0			0
$903$			1.46438e7i			0.597634i
$904$		0			0
$905$	−1.62687e6	+	7.15803e6i	−0.0660285	+	0.290517i
$906$		0			0
$907$			1.25967e7i			0.508439i	0.967147	+	0.254220i	$0.0818186\pi$
$907$			1.25967e7i			0.508439i	−0.967147	+	0.254220i	$0.918181\pi$
$908$		0			0
$909$	−693523.			−0.0278388
$910$		0			0
$911$	1.28644e7			0.513561			0.256781	−	0.966470i	$-0.417338\pi$
$911$	1.28644e7			0.513561			0.256781	+	0.966470i	$0.417338\pi$
$912$		0			0
$913$			1.25973e6i			0.0500149i
$914$		0			0
$915$	−8.08882e6	−	1.83842e6i	−0.319398	−	0.0725925i
$916$		0			0
$917$		−	2.33723e7i		−	0.917865i
$918$		0			0
$919$	2.81974e7			1.10134			0.550669	−	0.834724i	$-0.314373\pi$
$919$	2.81974e7			1.10134			0.550669	+	0.834724i	$0.314373\pi$
$920$		0			0
$921$	−2.79792e7			−1.08689
$922$		0			0
$923$			1.96918e7i			0.760818i
$924$		0			0
$925$	4.93317e6	−	1.02921e7i	0.189571	−	0.395502i
$926$		0			0
$927$			1.71821e6i			0.0656715i
$928$		0			0
$929$	2.92018e7			1.11012			0.555061	−	0.831810i	$-0.312695\pi$
$929$	2.92018e7			1.11012			0.555061	+	0.831810i	$0.312695\pi$
$930$		0			0
$931$	−5.69571e7			−2.15364
$932$		0			0
$933$		−	5.16544e7i		−	1.94269i
$934$		0			0
$935$	−6.44276e7	−	1.46430e7i	−2.41014	−	0.547775i
$936$		0			0
$937$			2.99796e7i			1.11552i	0.830003	+	0.557759i	$0.188339\pi$
$937$			2.99796e7i			1.11552i	−0.830003	+	0.557759i	$0.811661\pi$
$938$		0			0
$939$	−5.14395e7			−1.90385
$940$		0			0
$941$	−1.20363e7			−0.443119			−0.221560	−	0.975147i	$-0.571115\pi$
$941$	−1.20363e7			−0.443119			−0.221560	+	0.975147i	$0.571115\pi$
$942$		0			0
$943$			1.66761e7i			0.610681i
$944$		0			0
$945$	8.80089e6	−	3.87228e7i	0.320588	−	1.41055i
$946$		0			0
$947$		−	4.86315e7i		−	1.76215i	−0.472977	−	0.881075i	$-0.656821\pi$
$947$		−	4.86315e7i		−	1.76215i	0.472977	−	0.881075i	$-0.343179\pi$
$948$		0			0
$949$	−7.19239e6			−0.259243
$950$		0			0
$951$	3.52813e7			1.26501
$952$		0			0
$953$			2.77941e7i			0.991333i	0.868513	+	0.495667i	$0.165076\pi$
$953$			2.77941e7i			0.991333i	−0.868513	+	0.495667i	$0.834924\pi$
$954$		0			0
$955$	2.04977e6	−	9.01875e6i	0.0727273	−	0.319991i
$956$		0			0
$957$		−	4.66580e7i		−	1.64682i
$958$		0			0
$959$	960419.			0.0337221
$960$		0			0
$961$	−2.46222e7			−0.860038
$962$		0			0
$963$		−	7.22567e6i		−	0.251080i
$964$		0			0
$965$	−2.73275e7	−	6.21097e6i	−0.944673	−	0.214704i
$966$		0			0
$967$		−	4.24365e7i		−	1.45940i	−0.683768	−	0.729699i	$-0.739660\pi$
$967$		−	4.24365e7i		−	1.45940i	0.683768	−	0.729699i	$-0.260340\pi$
$968$		0			0
$969$	7.83947e7			2.68212
$970$		0			0
$971$	2.23502e7			0.760736			0.380368	−	0.924835i	$-0.375797\pi$
$971$	2.23502e7			0.760736			0.380368	+	0.924835i	$0.375797\pi$
$972$		0			0
$973$		−	1.34888e6i		−	0.0456764i
$974$		0			0
$975$	3.50363e7	+	1.67935e7i	1.18034	+	0.565757i
$976$		0			0
$977$			3.02839e7i			1.01502i	0.861645	+	0.507511i	$0.169434\pi$
$977$			3.02839e7i			1.01502i	−0.861645	+	0.507511i	$0.830566\pi$
$978$		0			0
$979$	−3.61184e7			−1.20440
$980$		0			0
$981$	−4.25635e6			−0.141210
$982$		0			0
$983$			1.64282e7i			0.542257i	0.962543	+	0.271128i	$0.0873968\pi$
$983$			1.64282e7i			0.542257i	−0.962543	+	0.271128i	$0.912603\pi$
$984$		0			0
$985$	7.74987e6	+	1.76138e6i	0.254509	+	0.0578446i
$986$		0			0
$987$			4.79400e7i			1.56641i
$988$		0			0
$989$	3.96926e6			0.129038
$990$		0			0
$991$	2.23109e6			0.0721660			0.0360830	−	0.999349i	$-0.488512\pi$
$991$	2.23109e6			0.0721660			0.0360830	+	0.999349i	$0.488512\pi$
$992$		0			0
$993$		−	6.37908e7i		−	2.05298i
$994$		0			0
$995$	1.17266e7	−	5.15956e7i	0.375503	−	1.65217i
$996$		0			0
$997$			5.30762e7i			1.69107i	0.533919	+	0.845536i	$0.320719\pi$
$997$			5.30762e7i			1.69107i	−0.533919	+	0.845536i	$0.679281\pi$
$998$		0			0
$999$	1.22556e7			0.388528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.c.l.129.4		12
4.3	odd	2	inner	320.6.c.l.129.10		12
5.4	even	2	inner	320.6.c.l.129.9		12
8.3	odd	2		160.6.c.d.129.3	✓	12
8.5	even	2		160.6.c.d.129.9	yes	12
20.19	odd	2	inner	320.6.c.l.129.3		12
40.3	even	4		800.6.a.w.1.6		6
40.13	odd	4		800.6.a.bb.1.1		6
40.19	odd	2		160.6.c.d.129.10	yes	12
40.27	even	4		800.6.a.bb.1.2		6
40.29	even	2		160.6.c.d.129.4	yes	12
40.37	odd	4		800.6.a.w.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.d.129.3	✓	12	8.3	odd	2
160.6.c.d.129.4	yes	12	40.29	even	2
160.6.c.d.129.9	yes	12	8.5	even	2
160.6.c.d.129.10	yes	12	40.19	odd	2
320.6.c.l.129.3		12	20.19	odd	2	inner
320.6.c.l.129.4		12	1.1	even	1	trivial
320.6.c.l.129.9		12	5.4	even	2	inner
320.6.c.l.129.10		12	4.3	odd	2	inner
800.6.a.w.1.5		6	40.37	odd	4
800.6.a.w.1.6		6	40.3	even	4
800.6.a.bb.1.1		6	40.13	odd	4
800.6.a.bb.1.2		6	40.27	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-16.9825i q^{3} +(-12.3893 + 54.5115i) q^{5} -211.691i q^{7} -45.4059 q^{9} +O(q^{10})\)	\(q-16.9825i q^{3} +(-12.3893 + 54.5115i) q^{5} -211.691i q^{7} -45.4059 q^{9} +520.707 q^{11} -732.105i q^{13} +(925.743 + 210.402i) q^{15} +2269.82i q^{17} +2033.73 q^{19} -3595.05 q^{21} +974.449i q^{23} +(-2818.01 - 1350.72i) q^{25} -3355.65i q^{27} +5276.32 q^{29} -2001.75 q^{31} -8842.91i q^{33} +(11539.6 + 2622.71i) q^{35} +3652.25i q^{37} -12433.0 q^{39} +17113.3 q^{41} -4073.34i q^{43} +(562.548 - 2475.14i) q^{45} -13335.0i q^{47} -28006.2 q^{49} +38547.2 q^{51} -27482.0i q^{53} +(-6451.20 + 28384.5i) q^{55} -34537.9i q^{57} +3890.44 q^{59} -8737.66 q^{61} +9612.02i q^{63} +(39908.2 + 9070.28i) q^{65} -40950.5i q^{67} +16548.6 q^{69} -26897.5 q^{71} -9824.26i q^{73} +(-22938.6 + 47856.9i) q^{75} -110229. i q^{77} -60376.0 q^{79} -68020.9 q^{81} +2419.27i q^{83} +(-123731. - 28121.5i) q^{85} -89605.2i q^{87} -69364.1 q^{89} -154980. q^{91} +33994.7i q^{93} +(-25196.6 + 110862. i) q^{95} -76217.3i q^{97} -23643.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 60 q^{5} - 268 q^{9}+O(q^{10}) \) 12 * q + 60 * q^5 - 268 * q^9	\( 12 q + 60 q^{5} - 268 q^{9} - 12080 q^{21} - 16420 q^{25} + 7864 q^{29} + 65560 q^{41} + 60420 q^{45} + 17956 q^{49} + 118232 q^{61} + 11360 q^{65} - 204464 q^{69} - 346436 q^{81} - 96320 q^{85} + 354936 q^{89}+O(q^{100}) \) 12 * q + 60 * q^5 - 268 * q^9 - 12080 * q^21 - 16420 * q^25 + 7864 * q^29 + 65560 * q^41 + 60420 * q^45 + 17956 * q^49 + 118232 * q^61 + 11360 * q^65 - 204464 * q^69 - 346436 * q^81 - 96320 * q^85 + 354936 * q^89

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