LMFDB - Embedded newform 240.6.f.f.49.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,6,Mod(49,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("240.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	240.f (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:	$38.4921167551$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 142x^{6} + 5761x^{4} + 52020x^{2} + 129600 $ x^8 + 142x^6 + 5761x^4 + 52020*x^2 + 129600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{2}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.8
Root			$7.82911i$ of defining polynomial
Character	$\chi$	$=$	240.49
Dual form			240.6.f.f.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +(55.6058 - 5.74389i) q^{5} -98.1720i q^{7} -81.0000 q^{9} -430.039 q^{11} +756.588i q^{13} +(51.6950 + 500.452i) q^{15} +542.278i q^{17} +731.455 q^{19} +883.548 q^{21} +1993.90i q^{23} +(3059.02 - 638.788i) q^{25} -729.000i q^{27} +7208.45 q^{29} -7156.48 q^{31} -3870.35i q^{33} +(-563.889 - 5458.93i) q^{35} +8199.54i q^{37} -6809.29 q^{39} +3623.74 q^{41} +14899.5i q^{43} +(-4504.07 + 465.255i) q^{45} +3274.60i q^{47} +7169.26 q^{49} -4880.51 q^{51} +30198.3i q^{53} +(-23912.7 + 2470.10i) q^{55} +6583.09i q^{57} +13968.2 q^{59} -19116.3 q^{61} +7951.93i q^{63} +(4345.76 + 42070.7i) q^{65} +44151.1i q^{67} -17945.1 q^{69} -28917.7 q^{71} -34137.8i q^{73} +(5749.09 + 27531.1i) q^{75} +42217.8i q^{77} +76490.6 q^{79} +6561.00 q^{81} +34315.0i q^{83} +(3114.79 + 30153.8i) q^{85} +64876.1i q^{87} -148918. q^{89} +74275.7 q^{91} -64408.3i q^{93} +(40673.1 - 4201.40i) q^{95} +40117.3i q^{97} +34833.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 66 q^{5} - 648 q^{9} - 684 q^{11} + 738 q^{15} + 3040 q^{19} + 108 q^{21} - 564 q^{25} - 8004 q^{29} - 6024 q^{31} + 4476 q^{35} - 3996 q^{39} - 37800 q^{41} + 5346 q^{45} - 34368 q^{49} - 11124 q^{51}+ \cdots + 55404 q^{99}+O(q^{100}) $ 8 * q - 66 * q^5 - 648 * q^9 - 684 * q^11 + 738 * q^15 + 3040 * q^19 + 108 * q^21 - 564 * q^25 - 8004 * q^29 - 6024 * q^31 + 4476 * q^35 - 3996 * q^39 - 37800 * q^41 + 5346 * q^45 - 34368 * q^49 - 11124 * q^51 - 39968 * q^55 + 86028 * q^59 - 38960 * q^61 + 84204 * q^65 - 13536 * q^69 - 162072 * q^71 - 31968 * q^75 + 169976 * q^79 + 52488 * q^81 + 240588 * q^85 - 452976 * q^89 - 610456 * q^91 - 42048 * q^95 + 55404 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			9.00000i			0.577350i
$3$			9.00000i			0.577350i
$4$		0			0
$5$	55.6058	−	5.74389i	0.994707	−	0.102750i
$5$	55.6058	−	5.74389i	0.994707	−	0.102750i
$6$		0			0
$7$		−	98.1720i		−	0.757255i	−0.925549	−	0.378628i	$-0.876396\pi$
$7$		−	98.1720i		−	0.757255i	0.925549	−	0.378628i	$-0.123604\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−430.039			−1.07158			−0.535792	−	0.844350i	$-0.679987\pi$
$11$	−430.039			−1.07158			−0.535792	+	0.844350i	$0.679987\pi$
$12$		0			0
$13$			756.588i			1.24165i	0.783947	+	0.620827i	$0.213203\pi$
$13$			756.588i			1.24165i	−0.783947	+	0.620827i	$0.786797\pi$
$14$		0			0
$15$	51.6950	+	500.452i	0.0593227	+	0.574294i
$16$		0			0
$17$			542.278i			0.455093i	0.973767	+	0.227546i	$0.0730704\pi$
$17$			542.278i			0.455093i	−0.973767	+	0.227546i	$0.926930\pi$
$18$		0			0
$19$	731.455			0.464840			0.232420	−	0.972616i	$-0.425336\pi$
$19$	731.455			0.464840			0.232420	+	0.972616i	$0.425336\pi$
$20$		0			0
$21$	883.548			0.437202
$22$		0			0
$23$			1993.90i			0.785931i	0.919553	+	0.392966i	$0.128551\pi$
$23$			1993.90i			0.785931i	−0.919553	+	0.392966i	$0.871449\pi$
$24$		0			0
$25$	3059.02	−	638.788i	0.978885	−	0.204412i
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$	7208.45			1.59165			0.795824	−	0.605528i	$-0.207038\pi$
$29$	7208.45			1.59165			0.795824	+	0.605528i	$0.207038\pi$
$30$		0			0
$31$	−7156.48			−1.33750			−0.668752	−	0.743485i	$-0.733171\pi$
$31$	−7156.48			−1.33750			−0.668752	+	0.743485i	$0.733171\pi$
$32$		0			0
$33$		−	3870.35i		−	0.618680i
$34$		0			0
$35$	−563.889	−	5458.93i	−0.0778079	−	0.753247i
$36$		0			0
$37$			8199.54i			0.984657i	0.870409	+	0.492329i	$0.163854\pi$
$37$			8199.54i			0.984657i	−0.870409	+	0.492329i	$0.836146\pi$
$38$		0			0
$39$	−6809.29			−0.716870
$40$		0			0
$41$	3623.74			0.336664			0.168332	−	0.985730i	$-0.446162\pi$
$41$	3623.74			0.336664			0.168332	+	0.985730i	$0.446162\pi$
$42$		0			0
$43$			14899.5i			1.22886i	0.788973	+	0.614428i	$0.210613\pi$
$43$			14899.5i			1.22886i	−0.788973	+	0.614428i	$0.789387\pi$
$44$		0			0
$45$	−4504.07	+	465.255i	−0.331569	+	0.0342500i
$46$		0			0
$47$			3274.60i			0.216229i	0.994138	+	0.108115i	$0.0344813\pi$
$47$			3274.60i			0.216229i	−0.994138	+	0.108115i	$0.965519\pi$
$48$		0			0
$49$	7169.26			0.426564
$50$		0			0
$51$	−4880.51			−0.262748
$52$		0			0
$53$			30198.3i			1.47670i	0.674416	+	0.738352i	$0.264395\pi$
$53$			30198.3i			1.47670i	−0.674416	+	0.738352i	$0.735605\pi$
$54$		0			0
$55$	−23912.7	+	2470.10i	−1.06591	+	0.110105i
$56$		0			0
$57$			6583.09i			0.268376i
$58$		0			0
$59$	13968.2			0.522409			0.261205	−	0.965283i	$-0.415880\pi$
$59$	13968.2			0.522409			0.261205	+	0.965283i	$0.415880\pi$
$60$		0			0
$61$	−19116.3			−0.657778			−0.328889	−	0.944369i	$-0.606674\pi$
$61$	−19116.3			−0.657778			−0.328889	+	0.944369i	$0.606674\pi$
$62$		0			0
$63$			7951.93i			0.252418i
$64$		0			0
$65$	4345.76	+	42070.7i	0.127580	+	1.23508i
$66$		0			0
$67$			44151.1i			1.20159i	0.799405	+	0.600793i	$0.205148\pi$
$67$			44151.1i			1.20159i	−0.799405	+	0.600793i	$0.794852\pi$
$68$		0			0
$69$	−17945.1			−0.453758
$70$		0			0
$71$	−28917.7			−0.680798			−0.340399	−	0.940281i	$-0.610562\pi$
$71$	−28917.7			−0.680798			−0.340399	+	0.940281i	$0.610562\pi$
$72$		0			0
$73$		−	34137.8i		−	0.749770i	−0.927071	−	0.374885i	$-0.877682\pi$
$73$		−	34137.8i		−	0.749770i	0.927071	−	0.374885i	$-0.122318\pi$
$74$		0			0
$75$	5749.09	+	27531.1i	0.118017	+	0.565159i
$76$		0			0
$77$			42217.8i			0.811463i
$78$		0			0
$79$	76490.6			1.37892			0.689462	−	0.724322i	$-0.257847\pi$
$79$	76490.6			1.37892			0.689462	+	0.724322i	$0.257847\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$			34315.0i			0.546750i	0.961908	+	0.273375i	$0.0881400\pi$
$83$			34315.0i			0.546750i	−0.961908	+	0.273375i	$0.911860\pi$
$84$		0			0
$85$	3114.79	+	30153.8i	0.0467607	+	0.452684i
$86$		0			0
$87$			64876.1i			0.918939i
$88$		0			0
$89$	−148918.			−1.99284			−0.996422	−	0.0845227i	$-0.973063\pi$
$89$	−148918.			−1.99284			−0.996422	+	0.0845227i	$0.973063\pi$
$90$		0			0
$91$	74275.7			0.940250
$92$		0			0
$93$		−	64408.3i		−	0.772208i
$94$		0			0
$95$	40673.1	−	4201.40i	0.462380	−	0.0477623i
$96$		0			0
$97$			40117.3i			0.432915i	0.976292	+	0.216458i	$0.0694503\pi$
$97$			40117.3i			0.432915i	−0.976292	+	0.216458i	$0.930550\pi$
$98$		0			0
$99$	34833.2			0.357195
$100$		0			0
$101$	−76979.5			−0.750882			−0.375441	−	0.926846i	$-0.622509\pi$
$101$	−76979.5			−0.750882			−0.375441	+	0.926846i	$0.622509\pi$
$102$		0			0
$103$			7245.94i			0.0672980i	0.999434	+	0.0336490i	$0.0107128\pi$
$103$			7245.94i			0.0672980i	−0.999434	+	0.0336490i	$0.989287\pi$
$104$		0			0
$105$	49130.4	−	5075.00i	0.434888	−	0.0449224i
$106$		0			0
$107$		−	213563.i		−	1.80329i	−0.432474	−	0.901646i	$-0.642359\pi$
$107$		−	213563.i		−	1.80329i	0.432474	−	0.901646i	$-0.357641\pi$
$108$		0			0
$109$	84468.7			0.680973			0.340486	−	0.940249i	$-0.389408\pi$
$109$	84468.7			0.680973			0.340486	+	0.940249i	$0.389408\pi$
$110$		0			0
$111$	−73795.9			−0.568492
$112$		0			0
$113$			203010.i			1.49562i	0.663911	+	0.747811i	$0.268895\pi$
$113$			203010.i			1.49562i	−0.663911	+	0.747811i	$0.731105\pi$
$114$		0			0
$115$	11452.8	+	110873.i	0.0807543	+	0.781771i
$116$		0			0
$117$		−	61283.6i		−	0.413885i
$118$		0			0
$119$	53236.5			0.344622
$120$		0			0
$121$	23882.8			0.148293
$122$		0			0
$123$			32613.7i			0.194373i
$124$		0			0
$125$	166430.	−	53091.0i	0.952701	−	0.303911i
$126$		0			0
$127$			224433.i			1.23475i	0.786670	+	0.617374i	$0.211803\pi$
$127$			224433.i			1.23475i	−0.786670	+	0.617374i	$0.788197\pi$
$128$		0			0
$129$	−134096.			−0.709481
$130$		0			0
$131$	−190194.			−0.968319			−0.484159	−	0.874980i	$-0.660874\pi$
$131$	−190194.			−0.968319			−0.484159	+	0.874980i	$0.660874\pi$
$132$		0			0
$133$		−	71808.4i		−	0.352003i
$134$		0			0
$135$	−4187.30	−	40536.6i	−0.0197742	−	0.191431i
$136$		0			0
$137$			136236.i			0.620141i	0.950714	+	0.310070i	$0.100353\pi$
$137$			136236.i			0.620141i	−0.950714	+	0.310070i	$0.899647\pi$
$138$		0			0
$139$	319350.			1.40194			0.700971	−	0.713189i	$-0.252750\pi$
$139$	319350.			1.40194			0.700971	+	0.713189i	$0.252750\pi$
$140$		0			0
$141$	−29471.4			−0.124840
$142$		0			0
$143$		−	325362.i		−	1.33054i
$144$		0			0
$145$	400832.	−	41404.6i	1.58322	−	0.163542i
$146$		0			0
$147$			64523.4i			0.246277i
$148$		0			0
$149$	−217019.			−0.800813			−0.400407	−	0.916338i	$-0.631131\pi$
$149$	−217019.			−0.800813			−0.400407	+	0.916338i	$0.631131\pi$
$150$		0			0
$151$	−398484.			−1.42223			−0.711113	−	0.703078i	$-0.751808\pi$
$151$	−398484.			−1.42223			−0.711113	+	0.703078i	$0.751808\pi$
$152$		0			0
$153$		−	43924.6i		−	0.151698i
$154$		0			0
$155$	−397942.	+	41106.1i	−1.33043	+	0.137428i
$156$		0			0
$157$		−	408704.i		−	1.32330i	−0.749811	−	0.661652i	$-0.769855\pi$
$157$		−	408704.i		−	1.32330i	0.749811	−	0.661652i	$-0.230145\pi$
$158$		0			0
$159$	−271785.			−0.852575
$160$		0			0
$161$	195745.			0.595151
$162$		0			0
$163$		−	282690.i		−	0.833376i	−0.909050	−	0.416688i	$-0.863191\pi$
$163$		−	282690.i		−	0.833376i	0.909050	−	0.416688i	$-0.136809\pi$
$164$		0			0
$165$	−22230.9	−	215214.i	−0.0635693	−	0.615405i
$166$		0			0
$167$		−	236370.i		−	0.655844i	−0.944705	−	0.327922i	$-0.893652\pi$
$167$		−	236370.i		−	0.655844i	0.944705	−	0.327922i	$-0.106348\pi$
$168$		0			0
$169$	−201132.			−0.541707
$170$		0			0
$171$	−59247.8			−0.154947
$172$		0			0
$173$		−	491653.i		−	1.24894i	−0.781047	−	0.624472i	$-0.785314\pi$
$173$		−	491653.i		−	1.24894i	0.781047	−	0.624472i	$-0.214686\pi$
$174$		0			0
$175$	−62711.1	−	300310.i	−0.154792	−	0.741266i
$176$		0			0
$177$			125714.i			0.301613i
$178$		0			0
$179$	123617.			0.288367			0.144183	−	0.989551i	$-0.453945\pi$
$179$	123617.			0.288367			0.144183	+	0.989551i	$0.453945\pi$
$180$		0			0
$181$	824818.			1.87138			0.935689	−	0.352825i	$-0.114779\pi$
$181$	824818.			1.87138			0.935689	+	0.352825i	$0.114779\pi$
$182$		0			0
$183$		−	172047.i		−	0.379768i
$184$		0			0
$185$	47097.3	+	455942.i	0.101173	+	0.979446i
$186$		0			0
$187$		−	233201.i		−	0.487670i
$188$		0			0
$189$	−71567.4			−0.145734
$190$		0			0
$191$	12418.8			0.0246318			0.0123159	−	0.999924i	$-0.496080\pi$
$191$	12418.8			0.0246318			0.0123159	+	0.999924i	$0.496080\pi$
$192$		0			0
$193$		−	114127.i		−	0.220544i	−0.993901	−	0.110272i	$-0.964828\pi$
$193$		−	114127.i		−	0.220544i	0.993901	−	0.110272i	$-0.0351722\pi$
$194$		0			0
$195$	−378636.	+	39111.8i	−0.713076	+	0.0736583i
$196$		0			0
$197$		−	703659.i		−	1.29180i	−0.763421	−	0.645902i	$-0.776482\pi$
$197$		−	703659.i		−	1.29180i	0.763421	−	0.645902i	$-0.223518\pi$
$198$		0			0
$199$	1.06174e6			1.90057			0.950285	−	0.311383i	$-0.100792\pi$
$199$	1.06174e6			1.90057			0.950285	+	0.311383i	$0.100792\pi$
$200$		0			0
$201$	−397360.			−0.693736
$202$		0			0
$203$		−	707668.i		−	1.20528i
$204$		0			0
$205$	201501.	−	20814.4i	0.334883	−	0.0345922i
$206$		0			0
$207$		−	161506.i		−	0.261977i
$208$		0			0
$209$	−314554.			−0.498115
$210$		0			0
$211$	704961.			1.09008			0.545040	−	0.838410i	$-0.316514\pi$
$211$	704961.			1.09008			0.545040	+	0.838410i	$0.316514\pi$
$212$		0			0
$213$		−	260259.i		−	0.393059i
$214$		0			0
$215$	85581.3	+	828500.i	0.126265	+	1.22235i
$216$		0			0
$217$			702566.i			1.01283i
$218$		0			0
$219$	307240.			0.432880
$220$		0			0
$221$	−410281.			−0.565068
$222$		0			0
$223$		−	564437.i		−	0.760070i	−0.924972	−	0.380035i	$-0.875912\pi$
$223$		−	564437.i		−	0.760070i	0.924972	−	0.380035i	$-0.124088\pi$
$224$		0			0
$225$	−247780.	+	51741.8i	−0.326295	+	0.0681374i
$226$		0			0
$227$			899921.i			1.15915i	0.814919	+	0.579575i	$0.196781\pi$
$227$			899921.i			1.15915i	−0.814919	+	0.579575i	$0.803219\pi$
$228$		0			0
$229$	−496492.			−0.625639			−0.312820	−	0.949813i	$-0.601274\pi$
$229$	−496492.			−0.625639			−0.312820	+	0.949813i	$0.601274\pi$
$230$		0			0
$231$	−379960.			−0.468498
$232$		0			0
$233$		−	1.29202e6i		−	1.55912i	−0.626325	−	0.779562i	$-0.715442\pi$
$233$		−	1.29202e6i		−	1.55912i	0.626325	−	0.779562i	$-0.284558\pi$
$234$		0			0
$235$	18809.0	+	182087.i	0.0222175	+	0.215085i
$236$		0			0
$237$			688415.i			0.796122i
$238$		0			0
$239$	963704.			1.09131			0.545656	−	0.838009i	$-0.316281\pi$
$239$	963704.			1.09131			0.545656	+	0.838009i	$0.316281\pi$
$240$		0			0
$241$	356398.			0.395269			0.197635	−	0.980276i	$-0.436674\pi$
$241$	356398.			0.395269			0.197635	+	0.980276i	$0.436674\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	398653.	−	41179.5i	0.424306	−	0.0438294i
$246$		0			0
$247$			553410.i			0.577171i
$248$		0			0
$249$	−308835.			−0.315666
$250$		0			0
$251$	−738019.			−0.739406			−0.369703	−	0.929150i	$-0.620541\pi$
$251$	−738019.			−0.739406			−0.369703	+	0.929150i	$0.620541\pi$
$252$		0			0
$253$		−	857457.i		−	0.842192i
$254$		0			0
$255$	−271385.	+	28033.1i	−0.261357	+	0.0269973i
$256$		0			0
$257$			891006.i			0.841488i	0.907179	+	0.420744i	$0.138231\pi$
$257$			891006.i			0.841488i	−0.907179	+	0.420744i	$0.861769\pi$
$258$		0			0
$259$	804965.			0.745637
$260$		0			0
$261$	−583885.			−0.530549
$262$		0			0
$263$			1.23488e6i			1.10087i	0.834877	+	0.550436i	$0.185539\pi$
$263$			1.23488e6i			1.10087i	−0.834877	+	0.550436i	$0.814461\pi$
$264$		0			0
$265$	173456.	+	1.67920e6i	0.151731	+	1.46889i
$266$		0			0
$267$		−	1.34026e6i		−	1.15057i
$268$		0			0
$269$	−1.52614e6			−1.28592			−0.642960	−	0.765900i	$-0.722294\pi$
$269$	−1.52614e6			−1.28592			−0.642960	+	0.765900i	$0.722294\pi$
$270$		0			0
$271$	1.26507e6			1.04639			0.523193	−	0.852214i	$-0.324741\pi$
$271$	1.26507e6			1.04639			0.523193	+	0.852214i	$0.324741\pi$
$272$		0			0
$273$			668481.i			0.542854i
$274$		0			0
$275$	−1.31550e6	+	274704.i	−1.04896	+	0.219045i
$276$		0			0
$277$		−	593191.i		−	0.464510i	−0.972655	−	0.232255i	$-0.925390\pi$
$277$		−	593191.i		−	0.464510i	0.972655	−	0.232255i	$-0.0746104\pi$
$278$		0			0
$279$	579675.			0.445835
$280$		0			0
$281$	−643782.			−0.486377			−0.243189	−	0.969979i	$-0.578193\pi$
$281$	−643782.			−0.486377			−0.243189	+	0.969979i	$0.578193\pi$
$282$		0			0
$283$		−	464286.i		−	0.344603i	−0.985044	−	0.172302i	$-0.944880\pi$
$283$		−	464286.i		−	0.344603i	0.985044	−	0.172302i	$-0.0551203\pi$
$284$		0			0
$285$	37812.6	+	366058.i	0.0275756	+	0.266955i
$286$		0			0
$287$		−	355750.i		−	0.254941i
$288$		0			0
$289$	1.12579e6			0.792890
$290$		0			0
$291$	−361056.			−0.249944
$292$		0			0
$293$		−	1.87511e6i		−	1.27602i	−0.770028	−	0.638010i	$-0.779758\pi$
$293$		−	1.87511e6i		−	1.27602i	0.770028	−	0.638010i	$-0.220242\pi$
$294$		0			0
$295$	776714.	−	80232.0i	0.519644	−	0.0536775i
$296$		0			0
$297$			313499.i			0.206227i
$298$		0			0
$299$	−1.50856e6			−0.975855
$300$		0			0
$301$	1.46272e6			0.930558
$302$		0			0
$303$		−	692816.i		−	0.433522i
$304$		0			0
$305$	−1.06298e6	+	109802.i	−0.654296	+	0.0675866i
$306$		0			0
$307$			2.58035e6i			1.56254i	0.624190	+	0.781272i	$0.285429\pi$
$307$			2.58035e6i			1.56254i	−0.624190	+	0.781272i	$0.714571\pi$
$308$		0			0
$309$	−65213.5			−0.0388545
$310$		0			0
$311$	−796125.			−0.466746			−0.233373	−	0.972387i	$-0.574976\pi$
$311$	−796125.			−0.466746			−0.233373	+	0.972387i	$0.574976\pi$
$312$		0			0
$313$			2.81857e6i			1.62618i	0.582138	+	0.813090i	$0.302216\pi$
$313$			2.81857e6i			1.62618i	−0.582138	+	0.813090i	$0.697784\pi$
$314$		0			0
$315$	45675.0	+	442174.i	0.0259360	+	0.251082i
$316$		0			0
$317$			148948.i			0.0832506i	0.999133	+	0.0416253i	$0.0132536\pi$
$317$			148948.i			0.0832506i	−0.999133	+	0.0416253i	$0.986746\pi$
$318$		0			0
$319$	−3.09992e6			−1.70559
$320$		0			0
$321$	1.92207e6			1.04113
$322$		0			0
$323$			396652.i			0.211545i
$324$		0			0
$325$	483299.	+	2.31441e6i	0.253809	+	1.21544i
$326$		0			0
$327$			760218.i			0.393160i
$328$		0			0
$329$	321474.			0.163741
$330$		0			0
$331$	538832.			0.270323			0.135162	−	0.990824i	$-0.456845\pi$
$331$	538832.			0.270323			0.135162	+	0.990824i	$0.456845\pi$
$332$		0			0
$333$		−	664163.i		−	0.328219i
$334$		0			0
$335$	253599.	+	2.45506e6i	0.123463	+	1.19523i
$336$		0			0
$337$			1.61866e6i			0.776391i	0.921577	+	0.388195i	$0.126901\pi$
$337$			1.61866e6i			0.776391i	−0.921577	+	0.388195i	$0.873099\pi$
$338$		0			0
$339$	−1.82709e6			−0.863498
$340$		0			0
$341$	3.07757e6			1.43325
$342$		0			0
$343$		−	2.35380e6i		−	1.08027i
$344$		0			0
$345$	−997854.	+	103075.i	−0.451356	+	0.0466235i
$346$		0			0
$347$			2.47998e6i			1.10567i	0.833291	+	0.552834i	$0.186454\pi$
$347$			2.47998e6i			1.10567i	−0.833291	+	0.552834i	$0.813546\pi$
$348$		0			0
$349$	829390.			0.364498			0.182249	−	0.983252i	$-0.441662\pi$
$349$	829390.			0.364498			0.182249	+	0.983252i	$0.441662\pi$
$350$		0			0
$351$	551552.			0.238957
$352$		0			0
$353$			1.22488e6i			0.523186i	0.965178	+	0.261593i	$0.0842478\pi$
$353$			1.22488e6i			0.523186i	−0.965178	+	0.261593i	$0.915752\pi$
$354$		0			0
$355$	−1.60799e6	+	166100.i	−0.677195	+	0.0699519i
$356$		0			0
$357$			479129.i			0.198967i
$358$		0			0
$359$	−4.18958e6			−1.71567			−0.857836	−	0.513924i	$-0.828191\pi$
$359$	−4.18958e6			−1.71567			−0.857836	+	0.513924i	$0.828191\pi$
$360$		0			0
$361$	−1.94107e6			−0.783924
$362$		0			0
$363$			214945.i			0.0856172i
$364$		0			0
$365$	−196084.	−	1.89826e6i	−0.0770388	−	0.745802i
$366$		0			0
$367$			2.18983e6i			0.848684i	0.905502	+	0.424342i	$0.139495\pi$
$367$			2.18983e6i			0.848684i	−0.905502	+	0.424342i	$0.860505\pi$
$368$		0			0
$369$	−293523.			−0.112221
$370$		0			0
$371$	2.96463e6			1.11824
$372$		0			0
$373$		−	5.07922e6i		−	1.89027i	−0.326673	−	0.945137i	$-0.605928\pi$
$373$		−	5.07922e6i		−	1.89027i	0.326673	−	0.945137i	$-0.394072\pi$
$374$		0			0
$375$	477819.	+	1.49787e6i	0.175463	+	0.550042i
$376$		0			0
$377$			5.45383e6i			1.97628i
$378$		0			0
$379$	3.29383e6			1.17789			0.588943	−	0.808175i	$-0.299544\pi$
$379$	3.29383e6			1.17789			0.588943	+	0.808175i	$0.299544\pi$
$380$		0			0
$381$	−2.01990e6			−0.712882
$382$		0			0
$383$		−	2.20927e6i		−	0.769576i	−0.923005	−	0.384788i	$-0.874275\pi$
$383$		−	2.20927e6i		−	0.769576i	0.923005	−	0.384788i	$-0.125725\pi$
$384$		0			0
$385$	242495.	+	2.34756e6i	0.0833778	+	0.807168i
$386$		0			0
$387$		−	1.20686e6i		−	0.409619i
$388$		0			0
$389$	2.83486e6			0.949856			0.474928	−	0.880025i	$-0.342474\pi$
$389$	2.83486e6			0.949856			0.474928	+	0.880025i	$0.342474\pi$
$390$		0			0
$391$	−1.08125e6			−0.357672
$392$		0			0
$393$		−	1.71174e6i		−	0.559059i
$394$		0			0
$395$	4.25332e6	−	439354.i	1.37163	−	0.141684i
$396$		0			0
$397$		−	5.64498e6i		−	1.79757i	−0.438388	−	0.898786i	$-0.644450\pi$
$397$		−	5.64498e6i		−	1.79757i	0.438388	−	0.898786i	$-0.355550\pi$
$398$		0			0
$399$	646275.			0.203229
$400$		0			0
$401$	773543.			0.240228			0.120114	−	0.992760i	$-0.461674\pi$
$401$	773543.			0.240228			0.120114	+	0.992760i	$0.461674\pi$
$402$		0			0
$403$		−	5.41450e6i		−	1.66072i
$404$		0			0
$405$	364830.	−	37685.7i	0.110523	−	0.0114167i
$406$		0			0
$407$		−	3.52612e6i		−	1.05514i
$408$		0			0
$409$	1.34855e6			0.398621			0.199310	−	0.979936i	$-0.436130\pi$
$409$	1.34855e6			0.398621			0.199310	+	0.979936i	$0.436130\pi$
$410$		0			0
$411$	−1.22612e6			−0.358038
$412$		0			0
$413$		−	1.37129e6i		−	0.395597i
$414$		0			0
$415$	197102.	+	1.90811e6i	0.0561785	+	0.543856i
$416$		0			0
$417$			2.87415e6i			0.809412i
$418$		0			0
$419$	2.51681e6			0.700349			0.350175	−	0.936684i	$-0.386122\pi$
$419$	2.51681e6			0.700349			0.350175	+	0.936684i	$0.386122\pi$
$420$		0			0
$421$	−3.09143e6			−0.850070			−0.425035	−	0.905177i	$-0.639738\pi$
$421$	−3.09143e6			−0.850070			−0.425035	+	0.905177i	$0.639738\pi$
$422$		0			0
$423$		−	265243.i		−	0.0720764i
$424$		0			0
$425$	346401.	+	1.65884e6i	0.0930265	+	0.445484i
$426$		0			0
$427$			1.87669e6i			0.498106i
$428$		0			0
$429$	2.92826e6			0.768187
$430$		0			0
$431$	4.22770e6			1.09625			0.548127	−	0.836395i	$-0.315341\pi$
$431$	4.22770e6			1.09625			0.548127	+	0.836395i	$0.315341\pi$
$432$		0			0
$433$		−	1.49557e6i		−	0.383344i	−0.981459	−	0.191672i	$-0.938609\pi$
$433$		−	1.49557e6i		−	0.383344i	0.981459	−	0.191672i	$-0.0613909\pi$
$434$		0			0
$435$	372641.	+	3.60749e6i	0.0944208	+	0.914075i
$436$		0			0
$437$			1.45845e6i			0.365332i
$438$		0			0
$439$	−4.20364e6			−1.04103			−0.520516	−	0.853852i	$-0.674260\pi$
$439$	−4.20364e6			−1.04103			−0.520516	+	0.853852i	$0.674260\pi$
$440$		0			0
$441$	−580710.			−0.142188
$442$		0			0
$443$		−	6.99329e6i		−	1.69306i	−0.532342	−	0.846529i	$-0.678688\pi$
$443$		−	6.99329e6i		−	1.69306i	0.532342	−	0.846529i	$-0.321312\pi$
$444$		0			0
$445$	−8.28073e6	+	855371.i	−1.98230	+	0.204764i
$446$		0			0
$447$		−	1.95317e6i		−	0.462350i
$448$		0			0
$449$	−2.13051e6			−0.498732			−0.249366	−	0.968409i	$-0.580222\pi$
$449$	−2.13051e6			−0.498732			−0.249366	+	0.968409i	$0.580222\pi$
$450$		0			0
$451$	−1.55835e6			−0.360764
$452$		0			0
$453$		−	3.58636e6i		−	0.821122i
$454$		0			0
$455$	4.13016e6	−	426632.i	0.935273	−	0.0966106i
$456$		0			0
$457$		−	7.46601e6i		−	1.67224i	−0.548548	−	0.836119i	$-0.684819\pi$
$457$		−	7.46601e6i		−	1.67224i	0.548548	−	0.836119i	$-0.315181\pi$
$458$		0			0
$459$	395321.			0.0875827
$460$		0			0
$461$	−8.35600e6			−1.83124			−0.915622	−	0.402040i	$-0.868301\pi$
$461$	−8.35600e6			−1.83124			−0.915622	+	0.402040i	$0.868301\pi$
$462$		0			0
$463$			10090.1i			0.00218748i	0.999999	+	0.00109374i	$0.000348148\pi$
$463$			10090.1i			0.00218748i	−0.999999	+	0.00109374i	$0.999652\pi$
$464$		0			0
$465$	−369955.	−	3.58148e6i	−0.0793443	−	0.768121i
$466$		0			0
$467$		−	1.30989e6i		−	0.277934i	−0.990297	−	0.138967i	$-0.955622\pi$
$467$		−	1.30989e6i		−	0.277934i	0.990297	−	0.138967i	$-0.0443782\pi$
$468$		0			0
$469$	4.33440e6			0.909907
$470$		0			0
$471$	3.67834e6			0.764010
$472$		0			0
$473$		−	6.40738e6i		−	1.31682i
$474$		0			0
$475$	2.23753e6	−	467244.i	0.455025	−	0.0950189i
$476$		0			0
$477$		−	2.44606e6i		−	0.492234i
$478$		0			0
$479$	−5.05350e6			−1.00636			−0.503180	−	0.864181i	$-0.667837\pi$
$479$	−5.05350e6			−1.00636			−0.503180	+	0.864181i	$0.667837\pi$
$480$		0			0
$481$	−6.20367e6			−1.22260
$482$		0			0
$483$			1.76171e6i			0.343610i
$484$		0			0
$485$	230430.	+	2.23076e6i	0.0444820	+	0.430624i
$486$		0			0
$487$			5.19810e6i			0.993167i	0.867989	+	0.496583i	$0.165412\pi$
$487$			5.19810e6i			0.993167i	−0.867989	+	0.496583i	$0.834588\pi$
$488$		0			0
$489$	2.54421e6			0.481150
$490$		0			0
$491$	−3.45964e6			−0.647630			−0.323815	−	0.946120i	$-0.604966\pi$
$491$	−3.45964e6			−0.647630			−0.323815	+	0.946120i	$0.604966\pi$
$492$		0			0
$493$			3.90899e6i			0.724348i
$494$		0			0
$495$	1.93693e6	−	200078.i	0.355304	−	0.0367017i
$496$		0			0
$497$			2.83891e6i			0.515538i
$498$		0			0
$499$	223651.			0.0402086			0.0201043	−	0.999798i	$-0.493600\pi$
$499$	223651.			0.0402086			0.0201043	+	0.999798i	$0.493600\pi$
$500$		0			0
$501$	2.12733e6			0.378651
$502$		0			0
$503$		−	7.20345e6i		−	1.26946i	−0.772732	−	0.634732i	$-0.781111\pi$
$503$		−	7.20345e6i		−	1.26946i	0.772732	−	0.634732i	$-0.218889\pi$
$504$		0			0
$505$	−4.28051e6	+	442162.i	−0.746908	+	0.0771530i
$506$		0			0
$507$		−	1.81019e6i		−	0.312755i
$508$		0			0
$509$	3.81092e6			0.651982			0.325991	−	0.945373i	$-0.394302\pi$
$509$	3.81092e6			0.651982			0.325991	+	0.945373i	$0.394302\pi$
$510$		0			0
$511$	−3.35137e6			−0.567768
$512$		0			0
$513$		−	533231.i		−	0.0894585i
$514$		0			0
$515$	41619.9	+	402917.i	0.00691486	+	0.0669418i
$516$		0			0
$517$		−	1.40821e6i		−	0.231708i
$518$		0			0
$519$	4.42487e6			0.721079
$520$		0			0
$521$	4.26468e6			0.688322			0.344161	−	0.938911i	$-0.388163\pi$
$521$	4.26468e6			0.688322			0.344161	+	0.938911i	$0.388163\pi$
$522$		0			0
$523$			2.86274e6i			0.457644i	0.973468	+	0.228822i	$0.0734874\pi$
$523$			2.86274e6i			0.457644i	−0.973468	+	0.228822i	$0.926513\pi$
$524$		0			0
$525$	2.70279e6	−	564400.i	0.427970	−	0.0893693i
$526$		0			0
$527$		−	3.88080e6i		−	0.608689i
$528$		0			0
$529$	2.46069e6			0.382312
$530$		0			0
$531$	−1.13143e6			−0.174136
$532$		0			0
$533$			2.74168e6i			0.418021i
$534$		0			0
$535$	−1.22668e6	−	1.18753e7i	−0.185288	−	1.79375i
$536$		0			0
$537$			1.11255e6i			0.166489i
$538$		0			0
$539$	−3.08307e6			−0.457100
$540$		0			0
$541$	4.59224e6			0.674576			0.337288	−	0.941401i	$-0.390490\pi$
$541$	4.59224e6			0.674576			0.337288	+	0.941401i	$0.390490\pi$
$542$		0			0
$543$			7.42336e6i			1.08044i
$544$		0			0
$545$	4.69695e6	−	485179.i	0.677368	−	0.0699699i
$546$		0			0
$547$			32883.2i			0.00469900i	0.999997	+	0.00234950i	$0.000747870\pi$
$547$			32883.2i			0.00469900i	−0.999997	+	0.00234950i	$0.999252\pi$
$548$		0			0
$549$	1.54842e6			0.219259
$550$		0			0
$551$	5.27266e6			0.739862
$552$		0			0
$553$		−	7.50923e6i		−	1.04420i
$554$		0			0
$555$	−4.10348e6	+	423876.i	−0.565483	+	0.0584125i
$556$		0			0
$557$			2.37300e6i			0.324085i	0.986784	+	0.162042i	$0.0518082\pi$
$557$			2.37300e6i			0.324085i	−0.986784	+	0.162042i	$0.948192\pi$
$558$		0			0
$559$	−1.12728e7			−1.52582
$560$		0			0
$561$	2.09881e6			0.281557
$562$		0			0
$563$			5.20699e6i			0.692334i	0.938173	+	0.346167i	$0.112517\pi$
$563$			5.20699e6i			0.692334i	−0.938173	+	0.346167i	$0.887483\pi$
$564$		0			0
$565$	1.16607e6	+	1.12886e7i	0.153675	+	1.48771i
$566$		0			0
$567$		−	644106.i		−	0.0841395i
$568$		0			0
$569$	4.63746e6			0.600482			0.300241	−	0.953863i	$-0.402933\pi$
$569$	4.63746e6			0.600482			0.300241	+	0.953863i	$0.402933\pi$
$570$		0			0
$571$	−6.26440e6			−0.804061			−0.402031	−	0.915626i	$-0.631695\pi$
$571$	−6.26440e6			−0.804061			−0.402031	+	0.915626i	$0.631695\pi$
$572$		0			0
$573$			111769.i			0.0142212i
$574$		0			0
$575$	1.27368e6	+	6.09938e6i	0.160654	+	0.769336i
$576$		0			0
$577$		−	9.42064e6i		−	1.17799i	−0.808137	−	0.588994i	$-0.799524\pi$
$577$		−	9.42064e6i		−	1.17799i	0.808137	−	0.588994i	$-0.200476\pi$
$578$		0			0
$579$	1.02714e6			0.127331
$580$		0			0
$581$	3.36877e6			0.414030
$582$		0			0
$583$		−	1.29865e7i		−	1.58241i
$584$		0			0
$585$	−352007.	−	3.40773e6i	−0.0425266	−	0.411694i
$586$		0			0
$587$		−	1.31865e7i		−	1.57955i	−0.613394	−	0.789777i	$-0.710196\pi$
$587$		−	1.31865e7i		−	1.57955i	0.613394	−	0.789777i	$-0.289804\pi$
$588$		0			0
$589$	−5.23464e6			−0.621726
$590$		0			0
$591$	6.33293e6			0.745823
$592$		0			0
$593$		−	9.77475e6i		−	1.14148i	−0.821130	−	0.570741i	$-0.806656\pi$
$593$		−	9.77475e6i		−	1.14148i	0.821130	−	0.570741i	$-0.193344\pi$
$594$		0			0
$595$	2.96026e6	−	305785.i	0.342798	−	0.0354098i
$596$		0			0
$597$			9.55562e6i			1.09729i
$598$		0			0
$599$	1.76512e6			0.201005			0.100503	−	0.994937i	$-0.467955\pi$
$599$	1.76512e6			0.201005			0.100503	+	0.994937i	$0.467955\pi$
$600$		0			0
$601$	−5.21462e6			−0.588893			−0.294446	−	0.955668i	$-0.595135\pi$
$601$	−5.21462e6			−0.588893			−0.294446	+	0.955668i	$0.595135\pi$
$602$		0			0
$603$		−	3.57624e6i		−	0.400529i
$604$		0			0
$605$	1.32802e6	−	137180.i	0.147508	−	0.0152371i
$606$		0			0
$607$			1.78378e7i			1.96503i	0.186188	+	0.982514i	$0.440387\pi$
$607$			1.78378e7i			1.96503i	−0.186188	+	0.982514i	$0.559613\pi$
$608$		0			0
$609$	6.36901e6			0.695871
$610$		0			0
$611$	−2.47753e6			−0.268482
$612$		0			0
$613$			933298.i			0.100316i	0.998741	+	0.0501579i	$0.0159725\pi$
$613$			933298.i			0.100316i	−0.998741	+	0.0501579i	$0.984028\pi$
$614$		0			0
$615$	187329.	+	1.81351e6i	0.0199718	+	0.193345i
$616$		0			0
$617$			1.16331e7i			1.23022i	0.788442	+	0.615109i	$0.210888\pi$
$617$			1.16331e7i			1.23022i	−0.788442	+	0.615109i	$0.789112\pi$
$618$		0			0
$619$	2.70476e6			0.283728			0.141864	−	0.989886i	$-0.454690\pi$
$619$	2.70476e6			0.283728			0.141864	+	0.989886i	$0.454690\pi$
$620$		0			0
$621$	1.45356e6			0.151253
$622$		0			0
$623$			1.46196e7i			1.50909i
$624$		0			0
$625$	8.94953e6	−	3.90812e6i	0.916431	−	0.400192i
$626$		0			0
$627$		−	2.83099e6i		−	0.287587i
$628$		0			0
$629$	−4.44643e6			−0.448111
$630$		0			0
$631$	−1.10675e7			−1.10657			−0.553284	−	0.832993i	$-0.686625\pi$
$631$	−1.10675e7			−1.10657			−0.553284	+	0.832993i	$0.686625\pi$
$632$		0			0
$633$			6.34464e6i			0.629358i
$634$		0			0
$635$	1.28912e6	+	1.24798e7i	0.126870	+	1.22821i
$636$		0			0
$637$			5.42418e6i			0.529646i
$638$		0			0
$639$	2.34234e6			0.226933
$640$		0			0
$641$	6.17000e6			0.593117			0.296559	−	0.955015i	$-0.404161\pi$
$641$	6.17000e6			0.593117			0.296559	+	0.955015i	$0.404161\pi$
$642$		0			0
$643$			2.89248e6i			0.275895i	0.990440	+	0.137947i	$0.0440504\pi$
$643$			2.89248e6i			0.275895i	−0.990440	+	0.137947i	$0.955950\pi$
$644$		0			0
$645$	−7.45650e6	+	770231.i	−0.705725	+	0.0728990i
$646$		0			0
$647$			7.80156e6i			0.732691i	0.930479	+	0.366345i	$0.119391\pi$
$647$			7.80156e6i			0.732691i	−0.930479	+	0.366345i	$0.880609\pi$
$648$		0			0
$649$	−6.00688e6			−0.559806
$650$		0			0
$651$	−6.32309e6			−0.584759
$652$		0			0
$653$			1.17587e7i			1.07914i	0.841942	+	0.539568i	$0.181413\pi$
$653$			1.17587e7i			1.07914i	−0.841942	+	0.539568i	$0.818587\pi$
$654$		0			0
$655$	−1.05759e7	+	1.09245e6i	−0.963194	+	0.0994946i
$656$		0			0
$657$			2.76516e6i			0.249923i
$658$		0			0
$659$	−9.18873e6			−0.824218			−0.412109	−	0.911135i	$-0.635208\pi$
$659$	−9.18873e6			−0.824218			−0.412109	+	0.911135i	$0.635208\pi$
$660$		0			0
$661$	−1.01023e7			−0.899323			−0.449662	−	0.893199i	$-0.648455\pi$
$661$	−1.01023e7			−0.899323			−0.449662	+	0.893199i	$0.648455\pi$
$662$		0			0
$663$		−	3.69253e6i		−	0.326242i
$664$		0			0
$665$	−412460.	−	3.99296e6i	−0.0361682	−	0.350140i
$666$		0			0
$667$			1.43730e7i			1.25093i
$668$		0			0
$669$	5.07993e6			0.438826
$670$		0			0
$671$	8.22076e6			0.704865
$672$		0			0
$673$			1.87926e7i			1.59937i	0.600419	+	0.799686i	$0.295000\pi$
$673$			1.87926e7i			1.59937i	−0.600419	+	0.799686i	$0.705000\pi$
$674$		0			0
$675$	−465676.	−	2.23002e6i	−0.0393391	−	0.188386i
$676$		0			0
$677$		−	5.30871e6i		−	0.445161i	−0.974914	−	0.222581i	$-0.928552\pi$
$677$		−	5.30871e6i		−	0.445161i	0.974914	−	0.222581i	$-0.0714481\pi$
$678$		0			0
$679$	3.93840e6			0.327827
$680$		0			0
$681$	−8.09929e6			−0.669236
$682$		0			0
$683$			9.15255e6i			0.750742i	0.926875	+	0.375371i	$0.122485\pi$
$683$			9.15255e6i			0.750742i	−0.926875	+	0.375371i	$0.877515\pi$
$684$		0			0
$685$	782524.	+	7.57551e6i	0.0637194	+	0.616858i
$686$		0			0
$687$		−	4.46843e6i		−	0.361213i
$688$		0			0
$689$	−2.28477e7			−1.83356
$690$		0			0
$691$	1.67488e7			1.33441			0.667203	−	0.744876i	$-0.267492\pi$
$691$	1.67488e7			1.33441			0.667203	+	0.744876i	$0.267492\pi$
$692$		0			0
$693$		−	3.41964e6i		−	0.270488i
$694$		0			0
$695$	1.77577e7	−	1.83431e6i	1.39452	−	0.144049i
$696$		0			0
$697$			1.96508e6i			0.153214i
$698$		0			0
$699$	1.16282e7			0.900161
$700$		0			0
$701$	6.07936e6			0.467265			0.233632	−	0.972325i	$-0.424939\pi$
$701$	6.07936e6			0.467265			0.233632	+	0.972325i	$0.424939\pi$
$702$		0			0
$703$			5.99759e6i			0.457708i
$704$		0			0
$705$	−1.63878e6	+	169281.i	−0.124179	+	0.0128273i
$706$		0			0
$707$			7.55723e6i			0.568609i
$708$		0			0
$709$	−2.31099e7			−1.72657			−0.863283	−	0.504721i	$-0.831595\pi$
$709$	−2.31099e7			−1.72657			−0.863283	+	0.504721i	$0.831595\pi$
$710$		0			0
$711$	−6.19574e6			−0.459641
$712$		0			0
$713$		−	1.42693e7i		−	1.05119i
$714$		0			0
$715$	−1.86885e6	−	1.80920e7i	−0.136713	−	1.32350i
$716$		0			0
$717$			8.67333e6i			0.630069i
$718$		0			0
$719$	1.04614e7			0.754687			0.377344	−	0.926073i	$-0.376838\pi$
$719$	1.04614e7			0.754687			0.377344	+	0.926073i	$0.376838\pi$
$720$		0			0
$721$	711349.			0.0509618
$722$		0			0
$723$			3.20758e6i			0.228209i
$724$		0			0
$725$	2.20508e7	−	4.60467e6i	1.55804	−	0.325352i
$726$		0			0
$727$		−	2.41813e7i		−	1.69685i	−0.529312	−	0.848427i	$-0.677550\pi$
$727$		−	2.41813e7i		−	1.69685i	0.529312	−	0.848427i	$-0.322450\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$	−8.07969e6			−0.559244
$732$		0			0
$733$			2.85033e7i			1.95945i	0.200339	+	0.979727i	$0.435796\pi$
$733$			2.85033e7i			1.95945i	−0.200339	+	0.979727i	$0.564204\pi$
$734$		0			0
$735$	370615.	+	3.58788e6i	0.0253049	+	0.244973i
$736$		0			0
$737$		−	1.89867e7i		−	1.28760i
$738$		0			0
$739$	2.38372e7			1.60562			0.802811	−	0.596233i	$-0.203337\pi$
$739$	2.38372e7			1.60562			0.802811	+	0.596233i	$0.203337\pi$
$740$		0			0
$741$	−4.98069e6			−0.333230
$742$		0			0
$743$			8.90275e6i			0.591633i	0.955245	+	0.295816i	$0.0955916\pi$
$743$			8.90275e6i			0.591633i	−0.955245	+	0.295816i	$0.904408\pi$
$744$		0			0
$745$	−1.20675e7	+	1.24653e6i	−0.796575	+	0.0822835i
$746$		0			0
$747$		−	2.77952e6i		−	0.182250i
$748$		0			0
$749$	−2.09659e7			−1.36555
$750$		0			0
$751$	−4.81345e6			−0.311427			−0.155714	−	0.987802i	$-0.549768\pi$
$751$	−4.81345e6			−0.311427			−0.155714	+	0.987802i	$0.549768\pi$
$752$		0			0
$753$		−	6.64217e6i		−	0.426896i
$754$		0			0
$755$	−2.21580e7	+	2.28885e6i	−1.41470	+	0.146134i
$756$		0			0
$757$		−	2.20098e7i		−	1.39597i	−0.716111	−	0.697987i	$-0.754079\pi$
$757$		−	2.20098e7i		−	1.39597i	0.716111	−	0.697987i	$-0.245921\pi$
$758$		0			0
$759$	7.71711e6			0.486240
$760$		0			0
$761$	3.57721e6			0.223915			0.111957	−	0.993713i	$-0.464288\pi$
$761$	3.57721e6			0.223915			0.111957	+	0.993713i	$0.464288\pi$
$762$		0			0
$763$		−	8.29246e6i		−	0.515670i
$764$		0			0
$765$	−252298.	−	2.44246e6i	−0.0155869	−	0.150895i
$766$		0			0
$767$			1.05682e7i			0.648652i
$768$		0			0
$769$	7.00739e6			0.427308			0.213654	−	0.976909i	$-0.431464\pi$
$769$	7.00739e6			0.427308			0.213654	+	0.976909i	$0.431464\pi$
$770$		0			0
$771$	−8.01906e6			−0.485833
$772$		0			0
$773$			3.24711e7i			1.95456i	0.211955	+	0.977279i	$0.432017\pi$
$773$			3.24711e7i			1.95456i	−0.211955	+	0.977279i	$0.567983\pi$
$774$		0			0
$775$	−2.18918e7	+	4.57147e6i	−1.30926	+	0.273402i
$776$		0			0
$777$			7.24468e6i			0.430494i
$778$		0			0
$779$	2.65060e6			0.156495
$780$		0			0
$781$	1.24358e7			0.729532
$782$		0			0
$783$		−	5.25496e6i		−	0.306313i
$784$		0			0
$785$	−2.34755e6	−	2.27263e7i	−0.135969	−	1.31630i
$786$		0			0
$787$			2.07810e7i			1.19600i	0.801498	+	0.597998i	$0.204037\pi$
$787$			2.07810e7i			1.19600i	−0.801498	+	0.597998i	$0.795963\pi$
$788$		0			0
$789$	−1.11140e7			−0.635589
$790$		0			0
$791$	1.99299e7			1.13257
$792$		0			0
$793$		−	1.44632e7i		−	0.816733i
$794$		0			0
$795$	−1.51128e7	+	1.56110e6i	−0.848062	+	0.0876020i
$796$		0			0
$797$		−	1.78055e7i		−	0.992907i	−0.868063	−	0.496454i	$-0.834635\pi$
$797$		−	1.78055e7i		−	0.992907i	0.868063	−	0.496454i	$-0.165365\pi$
$798$		0			0
$799$	−1.77575e6			−0.0984043
$800$		0			0
$801$	1.20624e7			0.664281
$802$		0			0
$803$			1.46806e7i			0.803442i
$804$		0			0
$805$	1.08846e7	−	1.12434e6i	0.592001	−	0.0611517i
$806$		0			0
$807$		−	1.37353e7i		−	0.742426i
$808$		0			0
$809$	439992.			0.0236360			0.0118180	−	0.999930i	$-0.496238\pi$
$809$	439992.			0.0236360			0.0118180	+	0.999930i	$0.496238\pi$
$810$		0			0
$811$	2.69214e7			1.43730			0.718648	−	0.695374i	$-0.244761\pi$
$811$	2.69214e7			1.43730			0.718648	+	0.695374i	$0.244761\pi$
$812$		0			0
$813$			1.13857e7i			0.604132i
$814$		0			0
$815$	−1.62374e6	−	1.57192e7i	−0.0856293	−	0.828965i
$816$		0			0
$817$			1.08983e7i			0.571222i
$818$		0			0
$819$	−6.01633e6			−0.313417
$820$		0			0
$821$	1.68848e7			0.874253			0.437127	−	0.899400i	$-0.355996\pi$
$821$	1.68848e7			0.874253			0.437127	+	0.899400i	$0.355996\pi$
$822$		0			0
$823$			6.16037e6i			0.317035i	0.987356	+	0.158517i	$0.0506714\pi$
$823$			6.16037e6i			0.317035i	−0.987356	+	0.158517i	$0.949329\pi$
$824$		0			0
$825$	−2.47234e6	−	1.18395e7i	−0.126466	−	0.605616i
$826$		0			0
$827$		−	3.05061e7i		−	1.55104i	−0.631324	−	0.775519i	$-0.717488\pi$
$827$		−	3.05061e7i		−	1.55104i	0.631324	−	0.775519i	$-0.282512\pi$
$828$		0			0
$829$	3.55918e7			1.79872			0.899360	−	0.437209i	$-0.144033\pi$
$829$	3.55918e7			1.79872			0.899360	+	0.437209i	$0.144033\pi$
$830$		0			0
$831$	5.33872e6			0.268185
$832$		0			0
$833$			3.88774e6i			0.194126i
$834$		0			0
$835$	−1.35768e6	−	1.31435e7i	−0.0673879	−	0.652372i
$836$		0			0
$837$			5.21707e6i			0.257403i
$838$		0			0
$839$	−1.13197e7			−0.555175			−0.277588	−	0.960700i	$-0.589535\pi$
$839$	−1.13197e7			−0.555175			−0.277588	+	0.960700i	$0.589535\pi$
$840$		0			0
$841$	3.14507e7			1.53334
$842$		0			0
$843$		−	5.79404e6i		−	0.280810i
$844$		0			0
$845$	−1.11841e7	+	1.15528e6i	−0.538840	+	0.0556603i
$846$		0			0
$847$		−	2.34462e6i		−	0.112296i
$848$		0			0
$849$	4.17857e6			0.198957
$850$		0			0
$851$	−1.63491e7			−0.773873
$852$		0			0
$853$			1.56521e7i			0.736546i	0.929718	+	0.368273i	$0.120051\pi$
$853$			1.56521e7i			0.736546i	−0.929718	+	0.368273i	$0.879949\pi$
$854$		0			0
$855$	−3.29452e6	+	340313.i	−0.154127	+	0.0159208i
$856$		0			0
$857$		−	974421.i		−	0.0453205i	−0.999743	−	0.0226602i	$-0.992786\pi$
$857$		−	974421.i		−	0.0453205i	0.999743	−	0.0226602i	$-0.00721360\pi$
$858$		0			0
$859$	4.18796e7			1.93651			0.968254	−	0.249967i	$-0.0804198\pi$
$859$	4.18796e7			1.93651			0.968254	+	0.249967i	$0.0804198\pi$
$860$		0			0
$861$	3.20175e6			0.147190
$862$		0			0
$863$			1.57024e7i			0.717695i	0.933396	+	0.358848i	$0.116830\pi$
$863$			1.57024e7i			0.717695i	−0.933396	+	0.358848i	$0.883170\pi$
$864$		0			0
$865$	−2.82400e6	−	2.73388e7i	−0.128329	−	1.24233i
$866$		0			0
$867$			1.01321e7i			0.457776i
$868$		0			0
$869$	−3.28940e7			−1.47763
$870$		0			0
$871$	−3.34042e7			−1.49195
$872$		0			0
$873$		−	3.24951e6i		−	0.144305i
$874$		0			0
$875$	−5.21205e6	−	1.63388e7i	−0.230138	−	0.721438i
$876$		0			0
$877$		−	2.33389e7i		−	1.02467i	−0.858787	−	0.512333i	$-0.828781\pi$
$877$		−	2.33389e7i		−	1.02467i	0.858787	−	0.512333i	$-0.171219\pi$
$878$		0			0
$879$	1.68760e7			0.736710
$880$		0			0
$881$	−2.87114e6			−0.124628			−0.0623138	−	0.998057i	$-0.519848\pi$
$881$	−2.87114e6			−0.124628			−0.0623138	+	0.998057i	$0.519848\pi$
$882$		0			0
$883$			1.69806e7i			0.732913i	0.930435	+	0.366456i	$0.119429\pi$
$883$			1.69806e7i			0.732913i	−0.930435	+	0.366456i	$0.880571\pi$
$884$		0			0
$885$	722088.	+	6.99043e6i	0.0309907	+	0.300017i
$886$		0			0
$887$		−	3.41812e7i		−	1.45874i	−0.684119	−	0.729371i	$-0.739813\pi$
$887$		−	3.41812e7i		−	1.45874i	0.684119	−	0.729371i	$-0.260187\pi$
$888$		0			0
$889$	2.20331e7			0.935020
$890$		0			0
$891$	−2.82149e6			−0.119065
$892$		0			0
$893$			2.39523e6i			0.100512i
$894$		0			0
$895$	6.87381e6	−	710042.i	0.286840	−	0.0296296i
$896$		0			0
$897$		−	1.35771e7i		−	0.563410i
$898$		0			0
$899$	−5.15872e7			−2.12884
$900$		0			0
$901$	−1.63759e7			−0.672037
$902$		0			0
$903$			1.31644e7i			0.537258i
$904$		0			0
$905$	4.58647e7	−	4.73767e6i	1.86147	−	0.192284i
$906$		0			0
$907$		−	4.27598e7i		−	1.72591i	−0.505283	−	0.862954i	$-0.668612\pi$
$907$		−	4.27598e7i		−	1.72591i	0.505283	−	0.862954i	$-0.331388\pi$
$908$		0			0
$909$	6.23534e6			0.250294
$910$		0			0
$911$	−3.98612e7			−1.59131			−0.795654	−	0.605751i	$-0.792873\pi$
$911$	−3.98612e7			−1.59131			−0.795654	+	0.605751i	$0.792873\pi$
$912$		0			0
$913$		−	1.47568e7i		−	0.585889i
$914$		0			0
$915$	−988218.	−	9.56680e6i	−0.0390212	−	0.377758i
$916$		0			0
$917$			1.86717e7i			0.733265i
$918$		0			0
$919$	4.59245e6			0.179373			0.0896863	−	0.995970i	$-0.471414\pi$
$919$	4.59245e6			0.179373			0.0896863	+	0.995970i	$0.471414\pi$
$920$		0			0
$921$	−2.32231e7			−0.902136
$922$		0			0
$923$		−	2.18788e7i		−	0.845316i
$924$		0			0
$925$	5.23777e6	+	2.50825e7i	0.201276	+	0.963866i
$926$		0			0
$927$		−	586921.i		−	0.0224327i
$928$		0			0
$929$	−3.54063e7			−1.34599			−0.672994	−	0.739648i	$-0.734992\pi$
$929$	−3.54063e7			−1.34599			−0.672994	+	0.739648i	$0.734992\pi$
$930$		0			0
$931$	5.24399e6			0.198284
$932$		0			0
$933$		−	7.16512e6i		−	0.269476i
$934$		0			0
$935$	−1.33948e6	−	1.29673e7i	−0.0501081	−	0.485089i
$936$		0			0
$937$		−	1.01939e7i		−	0.379306i	−0.981851	−	0.189653i	$-0.939264\pi$
$937$		−	1.01939e7i		−	0.379306i	0.981851	−	0.189653i	$-0.0607363\pi$
$938$		0			0
$939$	−2.53672e7			−0.938875
$940$		0			0
$941$	6.06141e6			0.223152			0.111576	−	0.993756i	$-0.464410\pi$
$941$	6.06141e6			0.223152			0.111576	+	0.993756i	$0.464410\pi$
$942$		0			0
$943$			7.22538e6i			0.264595i
$944$		0			0
$945$	−3.97956e6	+	411075.i	−0.144963	+	0.0149741i
$946$		0			0
$947$			6.61820e6i			0.239809i	0.992785	+	0.119904i	$0.0382588\pi$
$947$			6.61820e6i			0.239809i	−0.992785	+	0.119904i	$0.961741\pi$
$948$		0			0
$949$	2.58282e7			0.930956
$950$		0			0
$951$	−1.34053e6			−0.0480648
$952$		0			0
$953$		−	2.15434e7i		−	0.768391i	−0.923252	−	0.384196i	$-0.874479\pi$
$953$		−	2.15434e7i		−	0.768391i	0.923252	−	0.384196i	$-0.125521\pi$
$954$		0			0
$955$	690558.	−	71332.4i	0.0245015	−	0.00253092i
$956$		0			0
$957$		−	2.78993e7i		−	0.984720i
$958$		0			0
$959$	1.33745e7			0.469605
$960$		0			0
$961$	2.25860e7			0.788918
$962$		0			0
$963$			1.72986e7i			0.601098i
$964$		0			0
$965$	−655533.	−	6.34612e6i	−0.0226608	−	0.219376i
$966$		0			0
$967$		−	2.96402e6i		−	0.101933i	−0.998700	−	0.0509666i	$-0.983770\pi$
$967$		−	2.96402e6i		−	0.101933i	0.998700	−	0.0509666i	$-0.0162302\pi$
$968$		0			0
$969$	−3.56987e6			−0.122136
$970$		0			0
$971$	4.76132e7			1.62061			0.810306	−	0.586007i	$-0.199301\pi$
$971$	4.76132e7			1.62061			0.810306	+	0.586007i	$0.199301\pi$
$972$		0			0
$973$		−	3.13512e7i		−	1.06163i
$974$		0			0
$975$	−2.08297e7	+	4.34969e6i	−0.701733	+	0.146537i
$976$		0			0
$977$			3.65168e7i			1.22393i	0.790886	+	0.611964i	$0.209620\pi$
$977$			3.65168e7i			1.22393i	−0.790886	+	0.611964i	$0.790380\pi$
$978$		0			0
$979$	6.40407e7			2.13550
$980$		0			0
$981$	−6.84197e6			−0.226991
$982$		0			0
$983$		−	3.76835e7i		−	1.24385i	−0.783078	−	0.621924i	$-0.786351\pi$
$983$		−	3.76835e7i		−	1.24385i	0.783078	−	0.621924i	$-0.213649\pi$
$984$		0			0
$985$	−4.04174e6	−	3.91275e7i	−0.132733	−	1.28497i
$986$		0			0
$987$			2.89327e6i			0.0945357i
$988$		0			0
$989$	−2.97082e7			−0.965796
$990$		0			0
$991$	−2.51273e7			−0.812758			−0.406379	−	0.913705i	$-0.633209\pi$
$991$	−2.51273e7			−0.812758			−0.406379	+	0.913705i	$0.633209\pi$
$992$		0			0
$993$			4.84949e6i			0.156071i
$994$		0			0
$995$	5.90387e7	−	6.09850e6i	1.89051	−	0.195283i
$996$		0			0
$997$		−	1.37723e7i		−	0.438803i	−0.975635	−	0.219401i	$-0.929590\pi$
$997$		−	1.37723e7i		−	0.438803i	0.975635	−	0.219401i	$-0.0704104\pi$
$998$		0			0
$999$	5.97746e6			0.189497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.6.f.f.49.8		8
3.2	odd	2		720.6.f.o.289.2		8
4.3	odd	2		120.6.f.b.49.4	✓	8
5.4	even	2	inner	240.6.f.f.49.4		8
12.11	even	2		360.6.f.c.289.2		8
15.14	odd	2		720.6.f.o.289.1		8
20.3	even	4		600.6.a.w.1.3		4
20.7	even	4		600.6.a.v.1.2		4
20.19	odd	2		120.6.f.b.49.8	yes	8
60.59	even	2		360.6.f.c.289.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.f.b.49.4	✓	8	4.3	odd	2
120.6.f.b.49.8	yes	8	20.19	odd	2
240.6.f.f.49.4		8	5.4	even	2	inner
240.6.f.f.49.8		8	1.1	even	1	trivial
360.6.f.c.289.1		8	60.59	even	2
360.6.f.c.289.2		8	12.11	even	2
600.6.a.v.1.2		4	20.7	even	4
600.6.a.w.1.3		4	20.3	even	4
720.6.f.o.289.1		8	15.14	odd	2
720.6.f.o.289.2		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	240.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +(55.6058 - 5.74389i) q^{5} -98.1720i q^{7} -81.0000 q^{9} -430.039 q^{11} +756.588i q^{13} +(51.6950 + 500.452i) q^{15} +542.278i q^{17} +731.455 q^{19} +883.548 q^{21} +1993.90i q^{23} +(3059.02 - 638.788i) q^{25} -729.000i q^{27} +7208.45 q^{29} -7156.48 q^{31} -3870.35i q^{33} +(-563.889 - 5458.93i) q^{35} +8199.54i q^{37} -6809.29 q^{39} +3623.74 q^{41} +14899.5i q^{43} +(-4504.07 + 465.255i) q^{45} +3274.60i q^{47} +7169.26 q^{49} -4880.51 q^{51} +30198.3i q^{53} +(-23912.7 + 2470.10i) q^{55} +6583.09i q^{57} +13968.2 q^{59} -19116.3 q^{61} +7951.93i q^{63} +(4345.76 + 42070.7i) q^{65} +44151.1i q^{67} -17945.1 q^{69} -28917.7 q^{71} -34137.8i q^{73} +(5749.09 + 27531.1i) q^{75} +42217.8i q^{77} +76490.6 q^{79} +6561.00 q^{81} +34315.0i q^{83} +(3114.79 + 30153.8i) q^{85} +64876.1i q^{87} -148918. q^{89} +74275.7 q^{91} -64408.3i q^{93} +(40673.1 - 4201.40i) q^{95} +40117.3i q^{97} +34833.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 66 q^{5} - 648 q^{9} - 684 q^{11} + 738 q^{15} + 3040 q^{19} + 108 q^{21} - 564 q^{25} - 8004 q^{29} - 6024 q^{31} + 4476 q^{35} - 3996 q^{39} - 37800 q^{41} + 5346 q^{45} - 34368 q^{49} - 11124 q^{51}+ \cdots + 55404 q^{99}+O(q^{100}) \) 8 * q - 66 * q^5 - 648 * q^9 - 684 * q^11 + 738 * q^15 + 3040 * q^19 + 108 * q^21 - 564 * q^25 - 8004 * q^29 - 6024 * q^31 + 4476 * q^35 - 3996 * q^39 - 37800 * q^41 + 5346 * q^45 - 34368 * q^49 - 11124 * q^51 - 39968 * q^55 + 86028 * q^59 - 38960 * q^61 + 84204 * q^65 - 13536 * q^69 - 162072 * q^71 - 31968 * q^75 + 169976 * q^79 + 52488 * q^81 + 240588 * q^85 - 452976 * q^89 - 610456 * q^91 - 42048 * q^95 + 55404 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more