LMFDB - Embedded newform 69.8.c.a.68.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.5
Root			$1.33454 - 0.467979i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.8.c.a.68.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.9248i q^{2} +(37.2603 + 28.2607i) q^{3} -125.600 q^{4} +(-450.047 + 593.365i) q^{6} +38.2170i q^{8} +(589.667 + 2106.01i) q^{9} +O(q^{10})$	\(q+15.9248i q^{2} +(37.2603 + 28.2607i) q^{3} -125.600 q^{4} +(-450.047 + 593.365i) q^{6} +38.2170i q^{8} +(589.667 + 2106.01i) q^{9} +(-4679.91 - 3549.55i) q^{12} -13940.2 q^{13} -16685.4 q^{16} +(-33537.8 + 9390.34i) q^{18} +58350.9i q^{23} +(-1080.04 + 1423.98i) q^{24} -78125.0 q^{25} -221995. i q^{26} +(-37546.0 + 95134.9i) q^{27} +2688.67i q^{29} +325469. q^{31} -260821. i q^{32} +(-74062.2 - 264515. i) q^{36} +(-519417. - 393960. i) q^{39} -525376. i q^{41} -929228. q^{46} +487652. i q^{47} +(-621705. - 471541. i) q^{48} +823543. q^{49} -1.24413e6i q^{50} +1.75089e6 q^{52} +(-1.51501e6 - 597914. i) q^{54} -42816.6 q^{58} +3.15500e6i q^{59} +5.18304e6i q^{62} +2.01779e6 q^{64} +(-1.64904e6 + 2.17417e6i) q^{69} +3.47759e6i q^{71} +(-80485.2 + 22535.3i) q^{72} -600932. q^{73} +(-2.91096e6 - 2.20787e6i) q^{75} +(6.27374e6 - 8.27163e6i) q^{78} +(-4.08756e6 + 2.48368e6i) q^{81} +8.36652e6 q^{82} +(-75983.6 + 100181. i) q^{87} -7.32888e6i q^{92} +(1.21271e7 + 9.19798e6i) q^{93} -7.76577e6 q^{94} +(7.37097e6 - 9.71827e6i) q^{96} +1.31148e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 768 q^{4} - 3147 q^{6}+O(q^{10}) $ 6 * q - 768 * q^4 - 3147 * q^6	$ 6 q - 768 q^{4} - 3147 q^{6} - 21435 q^{12} + 98304 q^{16} - 72177 q^{18} + 402816 q^{24} - 468750 q^{25} - 225276 q^{27} + 1621587 q^{36} - 2196456 q^{39} - 866433 q^{48} + 4941258 q^{49} + 5680578 q^{52} - 12795270 q^{58} - 13290570 q^{64} + 9238656 q^{72} + 8715747 q^{78} + 59301462 q^{82} - 29588784 q^{87} + 30939042 q^{93} - 87793602 q^{94} - 59843631 q^{96}+O(q^{100}) $ 6 * q - 768 * q^4 - 3147 * q^6 - 21435 * q^12 + 98304 * q^16 - 72177 * q^18 + 402816 * q^24 - 468750 * q^25 - 225276 * q^27 + 1621587 * q^36 - 2196456 * q^39 - 866433 * q^48 + 4941258 * q^49 + 5680578 * q^52 - 12795270 * q^58 - 13290570 * q^64 + 9238656 * q^72 + 8715747 * q^78 + 59301462 * q^82 - 29588784 * q^87 + 30939042 * q^93 - 87793602 * q^94 - 59843631 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$			15.9248i			1.40757i	0.710413	+	0.703785i	$0.248508\pi$
$2$			15.9248i			1.40757i	−0.710413	+	0.703785i	$0.751492\pi$
$3$	37.2603	+	28.2607i	0.796751	+	0.604308i
$3$	37.2603	+	28.2607i	0.796751	+	0.604308i
$4$	−125.600			−0.981251
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−450.047	+	593.365i	−0.850605	+	1.12148i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			38.2170i			0.0263901i
$9$	589.667	+	2106.01i	0.269624	+	0.962966i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−4679.91	−	3549.55i	−0.781813	−	0.592978i
$13$	−13940.2			−1.75982			−0.879908	−	0.475143i	$-0.842396\pi$
$13$	−13940.2			−1.75982			−0.879908	+	0.475143i	$0.842396\pi$
$14$		0			0
$15$		0			0
$16$	−16685.4			−1.01840
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−33537.8	+	9390.34i	−1.35544	+	0.379514i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			58350.9i			1.00000i
$23$			58350.9i			1.00000i
$24$	−1080.04	+	1423.98i	−0.0159478	+	0.0210264i
$25$	−78125.0			−1.00000
$26$		−	221995.i		−	2.47706i
$27$	−37546.0	+	95134.9i	−0.367105	+	0.930179i
$28$		0			0
$29$			2688.67i			0.0204712i	0.999948	+	0.0102356i	$0.00325815\pi$
$29$			2688.67i			0.0204712i	−0.999948	+	0.0102356i	$0.996742\pi$
$30$		0			0
$31$	325469.			1.96220			0.981101	−	0.193498i	$-0.0619832\pi$
$31$	325469.			1.96220			0.981101	+	0.193498i	$0.0619832\pi$
$32$		−	260821.i		−	1.40707i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−74062.2	−	264515.i	−0.264568	−	0.944911i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−519417.	−	393960.i	−1.40214	−	1.06347i
$40$		0			0
$41$		−	525376.i		−	1.19049i	−0.803543	−	0.595246i	$-0.797054\pi$
$41$		−	525376.i		−	1.19049i	0.803543	−	0.595246i	$-0.202946\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−929228.			−1.40757
$47$			487652.i			0.685121i	0.939496	+	0.342561i	$0.111294\pi$
$47$			487652.i			0.685121i	−0.939496	+	0.342561i	$0.888706\pi$
$48$	−621705.	−	471541.i	−0.811409	−	0.615426i
$49$	823543.			1.00000
$50$		−	1.24413e6i		−	1.40757i
$51$		0			0
$52$	1.75089e6			1.72682
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−1.51501e6	−	597914.i	−1.30929	−	0.516726i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−42816.6			−0.0288147
$59$			3.15500e6i			1.99994i	0.00783115	+	0.999969i	$0.497507\pi$
$59$			3.15500e6i			1.99994i	−0.00783115	+	0.999969i	$0.502493\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$			5.18304e6i			2.76193i
$63$		0			0
$64$	2.01779e6			0.962158
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−1.64904e6	+	2.17417e6i	−0.604308	+	0.796751i
$70$		0			0
$71$			3.47759e6i			1.15312i	0.817055	+	0.576559i	$0.195605\pi$
$71$			3.47759e6i			1.15312i	−0.817055	+	0.576559i	$0.804395\pi$
$72$	−80485.2	+	22535.3i	−0.0254128	+	0.00711540i
$73$	−600932.			−0.180799			−0.0903993	−	0.995906i	$-0.528814\pi$
$73$	−600932.			−0.180799			−0.0903993	+	0.995906i	$0.528814\pi$
$74$		0			0
$75$	−2.91096e6	−	2.20787e6i	−0.796751	−	0.604308i
$76$		0			0
$77$		0			0
$78$	6.27374e6	−	8.27163e6i	1.49691	−	1.97360i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−4.08756e6	+	2.48368e6i	−0.854606	+	0.519277i
$82$	8.36652e6			1.67570
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−75983.6	+	100181.i	−0.0123709	+	0.0163105i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		−	7.32888e6i		−	0.981251i
$93$	1.21271e7	+	9.19798e6i	1.56339	+	1.18577i
$94$	−7.76577e6			−0.964356
$95$		0			0
$96$	7.37097e6	−	9.71827e6i	0.850306	−	1.12109i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			1.31148e7i			1.40757i
$99$		0			0
$100$	9.81251e6			0.981251
$101$			1.08442e7i			1.04730i	0.851933	+	0.523650i	$0.175430\pi$
$101$			1.08442e7i			1.04730i	−0.851933	+	0.523650i	$0.824570\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		−	532753.i		−	0.0464418i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	4.71578e6	−	1.19490e7i	0.360222	−	0.912740i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	337697.i		−	0.0200874i
$117$	−8.22008e6	−	2.93582e7i	−0.474488	−	1.69464i
$118$	−5.02428e7			−2.81505
$119$		0			0
$120$		0			0
$121$	−1.94872e7			−1.00000
$122$		0			0
$123$	1.48475e7	−	1.95757e7i	0.719424	−	0.948526i
$124$	−4.08790e7			−1.92541
$125$		0			0
$126$		0			0
$127$	−2.08122e7			−0.901580			−0.450790	−	0.892630i	$-0.648858\pi$
$127$	−2.08122e7			−0.901580			−0.450790	+	0.892630i	$0.648858\pi$
$128$		−	1.25207e6i		−	0.0527710i
$129$		0			0
$130$		0			0
$131$			3.90964e7i			1.51945i	0.650243	+	0.759727i	$0.274667\pi$
$131$			3.90964e7i			1.51945i	−0.650243	+	0.759727i	$0.725333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−3.46233e7	−	2.62606e7i	−1.12148	−	0.850605i
$139$	6.16654e7			1.94755			0.973777	−	0.227505i	$-0.0730566\pi$
$139$	6.16654e7			1.94755			0.973777	+	0.227505i	$0.0730566\pi$
$140$		0			0
$141$	−1.37814e7	+	1.81701e7i	−0.414024	+	0.545871i
$142$	−5.53800e7			−1.62309
$143$		0			0
$144$	−9.83884e6	−	3.51396e7i	−0.274584	−	0.980682i
$145$		0			0
$146$		−	9.56973e6i		−	0.254487i
$147$	3.06855e7	+	2.32739e7i	0.796751	+	0.604308i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	3.51599e7	−	4.63566e7i	0.850605	−	1.12148i
$151$	7.35151e7			1.73763			0.868815	−	0.495137i	$-0.164882\pi$
$151$	7.35151e7			1.73763			0.868815	+	0.495137i	$0.164882\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	6.52388e7	+	4.94814e7i	1.37585	+	1.04353i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−3.95522e7	−	6.50936e7i	−0.730918	−	1.20292i
$163$	1.01017e8			1.82699			0.913497	−	0.406845i	$-0.133371\pi$
$163$	1.01017e8			1.82699			0.913497	+	0.406845i	$0.133371\pi$
$164$			6.59873e7i			1.16817i
$165$		0			0
$166$		0			0
$167$		−	5.56023e7i		−	0.923815i	−0.886928	−	0.461907i	$-0.847165\pi$
$167$		−	5.56023e7i		−	0.923815i	0.886928	−	0.461907i	$-0.152835\pi$
$168$		0			0
$169$	1.31581e8			2.09696
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	7.26618e7i		−	1.06695i	−0.845815	−	0.533476i	$-0.820886\pi$
$173$		−	7.26618e7i		−	1.06695i	0.845815	−	0.533476i	$-0.179114\pi$
$174$	−1.59536e6	−	1.21003e6i	−0.0229581	−	0.0174129i
$175$		0			0
$176$		0			0
$177$	−8.91623e7	+	1.17556e8i	−1.20858	+	1.59345i
$178$		0			0
$179$			1.42600e8i			1.85837i	0.369611	+	0.929186i	$0.379491\pi$
$179$			1.42600e8i			1.85837i	−0.369611	+	0.929186i	$0.620509\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−2.23000e6			−0.0263901
$185$		0			0
$186$	−1.46476e8	+	1.93122e8i	−1.66906	+	2.20057i
$187$		0			0
$188$		−	6.12492e7i		−	0.672276i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	7.51836e7	+	5.70242e7i	0.766600	+	0.581440i
$193$	1.32545e7			0.132712			0.0663562	−	0.997796i	$-0.478863\pi$
$193$	1.32545e7			0.132712			0.0663562	+	0.997796i	$0.478863\pi$
$194$		0			0
$195$		0			0
$196$	−1.03437e8			−0.981251
$197$		−	4.99985e7i		−	0.465935i	−0.972484	−	0.232968i	$-0.925156\pi$
$197$		−	4.99985e7i		−	0.465935i	0.972484	−	0.232968i	$-0.0748436\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	2.98570e6i		−	0.0263901i
$201$		0			0
$202$	−1.72691e8			−1.47415
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.22887e8	+	3.44076e7i	−0.962966	+	0.269624i
$208$	2.32598e8			1.79219
$209$		0			0
$210$		0			0
$211$	2.24635e8			1.64623			0.823113	−	0.567878i	$-0.192235\pi$
$211$	2.24635e8			1.64623			0.823113	+	0.567878i	$0.192235\pi$
$212$		0			0
$213$	−9.82791e7	+	1.29576e8i	−0.696839	+	0.918748i
$214$		0			0
$215$		0			0
$216$	−3.63577e6	−	1.43490e6i	−0.0245476	−	0.00968796i
$217$		0			0
$218$		0			0
$219$	−2.23909e7	−	1.69827e7i	−0.144051	−	0.109258i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.13612e8			−1.89376			−0.946881	−	0.321583i	$-0.895785\pi$
$223$	−3.13612e8			−1.89376			−0.946881	+	0.321583i	$0.895785\pi$
$224$		0			0
$225$	−4.60677e7	−	1.64532e8i	−0.269624	−	0.962966i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−102753.			−0.000540239
$233$			2.26864e8i			1.17495i	0.809241	+	0.587476i	$0.199878\pi$
$233$			2.26864e8i			1.17495i	−0.809241	+	0.587476i	$0.800122\pi$
$234$	4.67524e8	−	1.30903e8i	2.38533	−	0.667875i
$235$		0			0
$236$		−	3.96268e8i		−	1.96244i
$237$		0			0
$238$		0			0
$239$		−	3.97272e8i		−	1.88233i	−0.337948	−	0.941165i	$-0.609733\pi$
$239$		−	3.97272e8i		−	1.88233i	0.337948	−	0.941165i	$-0.390267\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	3.10330e8i		−	1.40757i
$243$	−2.22494e8	−	2.29742e7i	−0.994711	−	0.102711i
$244$		0			0
$245$		0			0
$246$	3.11740e8	+	2.36444e8i	1.33512	+	1.01264i
$247$		0			0
$248$			1.24384e7i			0.0517828i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		−	3.31430e8i		−	1.26904i
$255$		0			0
$256$	2.78216e8			1.03644
$257$		−	5.41979e8i		−	1.99167i	−0.0911942	−	0.995833i	$-0.529068\pi$
$257$		−	5.41979e8i		−	1.99167i	0.0911942	−	0.995833i	$-0.470932\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−5.66235e6	+	1.58542e6i	−0.0197131	+	0.00551953i
$262$	−6.22604e8			−2.13874
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6.16125e8i			1.92990i	0.262427	+	0.964952i	$0.415477\pi$
$269$			6.16125e8i			1.92990i	−0.262427	+	0.964952i	$0.584523\pi$
$270$		0			0
$271$	−2.61606e8			−0.798464			−0.399232	−	0.916850i	$-0.630723\pi$
$271$	−2.61606e8			−0.798464			−0.399232	+	0.916850i	$0.630723\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	2.07119e8	−	2.73077e8i	0.592978	−	0.781813i
$277$	−7.06389e8			−1.99694			−0.998469	−	0.0553055i	$-0.982387\pi$
$277$	−7.06389e8			−1.99694			−0.998469	+	0.0553055i	$0.982387\pi$
$278$			9.82010e8i			2.74132i
$279$	1.91918e8	+	6.85440e8i	0.529056	+	1.88953i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−2.89355e8	−	2.19466e8i	−0.768351	−	0.582768i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	4.36786e8i		−	1.13150i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	5.49290e8	−	1.53797e8i	1.35496	−	0.379380i
$289$	−4.10339e8			−1.00000
$290$		0			0
$291$		0			0
$292$	7.54771e7			0.177409
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−3.70633e8	+	4.88661e8i	−0.850605	+	1.12148i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	8.13423e8i		−	1.75982i
$300$	3.65618e8	+	2.77308e8i	0.781813	+	0.592978i
$301$		0			0
$302$			1.17072e9i			2.44583i
$303$	−3.06463e8	+	4.04057e8i	−0.632892	+	0.834437i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	9.08266e8			1.79155			0.895775	−	0.444509i	$-0.146622\pi$
$307$	9.08266e8			1.79155			0.895775	+	0.444509i	$0.146622\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	2.69412e8i		−	0.507873i	−0.967221	−	0.253937i	$-0.918275\pi$
$311$		−	2.69412e8i		−	0.507873i	0.967221	−	0.253937i	$-0.0817255\pi$
$312$	1.50560e7	−	1.98506e7i	0.0280652	−	0.0370025i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8.45302e8i		−	1.49041i	−0.666838	−	0.745203i	$-0.732353\pi$
$317$		−	8.45302e8i		−	1.49041i	0.666838	−	0.745203i	$-0.267647\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	5.13398e8	−	3.11951e8i	0.838583	−	0.509541i
$325$	1.08908e9			1.75982
$326$			1.60868e9i			2.57162i
$327$		0			0
$328$	2.00783e7			0.0314173
$329$		0			0
$330$		0			0
$331$	−6.53397e8			−0.990329			−0.495164	−	0.868799i	$-0.664892\pi$
$331$	−6.53397e8			−0.990329			−0.495164	+	0.868799i	$0.664892\pi$
$332$		0			0
$333$		0			0
$334$	8.85457e8			1.30033
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			2.09540e9i			2.95161i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.15713e9			1.50181
$347$		−	1.50749e9i		−	1.93688i	−0.249251	−	0.968439i	$-0.580184\pi$
$347$		−	1.50749e9i		−	1.93688i	0.249251	−	0.968439i	$-0.419816\pi$
$348$	9.54355e6	−	1.25827e7i	0.0121390	−	0.0160047i
$349$	1.46751e9			1.84796			0.923978	−	0.382445i	$-0.124918\pi$
$349$	1.46751e9			1.84796			0.923978	+	0.382445i	$0.124918\pi$
$350$		0			0
$351$	5.23399e8	−	1.32620e9i	0.646038	−	1.63695i
$352$		0			0
$353$			1.49711e9i			1.81152i	0.423792	+	0.905760i	$0.360699\pi$
$353$			1.49711e9i			1.81152i	−0.423792	+	0.905760i	$0.639301\pi$
$354$	−1.87206e9	−	1.41990e9i	−2.24290	−	1.70116i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−2.27088e9			−2.61579
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	8.93872e8			1.00000
$362$		0			0
$363$	−7.26099e8	−	5.50721e8i	−0.796751	−	0.604308i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	9.73609e8i		−	1.01840i
$369$	1.10645e9	−	3.09797e8i	1.14640	−	0.320985i
$370$		0			0
$371$		0			0
$372$	−1.52316e9	−	1.15527e9i	−1.53407	−	1.16354i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−1.86366e7			−0.0180804
$377$		−	3.74806e7i		−	0.0360256i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−7.75469e8	−	5.88166e8i	−0.718335	−	0.544832i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	3.53845e7	−	4.66527e7i	0.0318899	−	0.0420453i
$385$		0			0
$386$			2.11075e8i			0.186802i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$			3.14733e7i			0.0263901i
$393$	−1.10489e9	+	1.45675e9i	−0.918218	+	1.21063i
$394$	7.96218e8			0.655836
$395$		0			0
$396$		0			0
$397$	−1.88167e9			−1.50930			−0.754651	−	0.656126i	$-0.772194\pi$
$397$	−1.88167e9			−1.50930			−0.754651	+	0.656126i	$0.772194\pi$
$398$		0			0
$399$		0			0
$400$	1.30355e9			1.01840
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−4.53710e9			−3.45312
$404$		−	1.36203e9i		−	1.02766i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.71588e9			1.96282			0.981408	−	0.191935i	$-0.0614762\pi$
$409$	2.71588e9			1.96282			0.981408	+	0.191935i	$0.0614762\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−5.47935e8	−	1.95696e9i	−0.379514	−	1.35544i
$415$		0			0
$416$			3.63589e9i			2.47619i
$417$	2.29767e9	+	1.74271e9i	1.55172	+	1.17692i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			3.57728e9i			2.31718i
$423$	−1.02700e9	+	2.87552e8i	−0.659748	+	0.184725i
$424$		0			0
$425$		0			0
$426$	−2.06348e9	−	1.56508e9i	−1.29320	−	0.980849i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	6.26471e8	−	1.58737e9i	0.373859	−	0.947292i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	2.70447e8	−	3.56572e8i	0.153788	−	0.202762i
$439$	−2.88478e9			−1.62737			−0.813685	−	0.581306i	$-0.802542\pi$
$439$	−2.88478e9			−1.62737			−0.813685	+	0.581306i	$0.802542\pi$
$440$		0			0
$441$	4.85616e8	+	1.73439e9i	0.269624	+	0.962966i
$442$		0			0
$443$		−	3.59756e9i		−	1.96605i	−0.183462	−	0.983027i	$-0.558730\pi$
$443$		−	3.59756e9i		−	1.96605i	0.183462	−	0.983027i	$-0.441270\pi$
$444$		0			0
$445$		0			0
$446$		−	4.99422e9i		−	2.66560i
$447$		0			0
$448$		0			0
$449$		−	3.82648e9i		−	1.99498i	−0.0708395	−	0.997488i	$-0.522568\pi$
$449$		−	3.82648e9i		−	1.99498i	0.0708395	−	0.997488i	$-0.477432\pi$
$450$	2.62014e9	−	7.33621e8i	1.35544	−	0.379514i
$451$		0			0
$452$		0			0
$453$	2.73920e9	+	2.07759e9i	1.38446	+	1.05006i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.27874e9i			1.55867i	0.626609	+	0.779334i	$0.284442\pi$
$461$			3.27874e9i			1.55867i	−0.626609	+	0.779334i	$0.715558\pi$
$462$		0			0
$463$	1.71939e9			0.805085			0.402543	−	0.915401i	$-0.368126\pi$
$463$	1.71939e9			0.805085			0.402543	+	0.915401i	$0.368126\pi$
$464$		−	4.48615e7i		−	0.0208478i
$465$		0			0
$466$	−3.61277e9			−1.65383
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	1.03244e9	+	3.68739e9i	0.465592	+	1.66287i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.20574e8			−0.0527787
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	6.32649e9			2.64951
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.44759e9			0.981251
$485$		0			0
$486$	3.65860e8	−	3.54318e9i	0.144573	−	1.40012i
$487$	−2.19042e9			−0.859362			−0.429681	−	0.902981i	$-0.641374\pi$
$487$	−2.19042e9			−0.859362			−0.429681	+	0.902981i	$0.641374\pi$
$488$		0			0
$489$	3.76392e9	+	2.85481e9i	1.45566	+	1.10407i
$490$		0			0
$491$			3.53523e9i			1.34782i	0.738812	+	0.673912i	$0.235387\pi$
$491$			3.53523e9i			1.34782i	−0.738812	+	0.673912i	$0.764613\pi$
$492$	−1.86485e9	+	2.45871e9i	−0.705936	+	0.930743i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−5.43059e9			−1.99830
$497$		0			0
$498$		0			0
$499$	−1.41765e9			−0.510761			−0.255380	−	0.966841i	$-0.582201\pi$
$499$	−1.41765e9			−0.510761			−0.255380	+	0.966841i	$0.582201\pi$
$500$		0			0
$501$	1.57136e9	−	2.07176e9i	0.558269	−	0.736050i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.90275e9	+	3.71857e9i	1.67075	+	1.26721i
$508$	2.61401e9			0.884677
$509$			1.77233e9i			0.595707i	0.954612	+	0.297853i	$0.0962706\pi$
$509$			1.77233e9i			0.595707i	−0.954612	+	0.297853i	$0.903729\pi$
$510$		0			0
$511$		0			0
$512$			4.27028e9i			1.40609i
$513$		0			0
$514$	8.63092e9			2.80341
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.05347e9	−	2.70740e9i	0.644767	−	0.850094i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−2.52475e7	−	9.01719e7i	−0.00776912	−	0.0277476i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		−	4.91052e9i		−	1.49097i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.40483e9			−1.00000
$530$		0			0
$531$	−6.64444e9	+	1.86040e9i	−1.92587	+	0.539231i
$532$		0			0
$533$			7.32385e9i			2.09505i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−4.02996e9	+	5.31331e9i	−1.12303	+	1.48066i
$538$	−9.81168e9			−2.71647
$539$		0			0
$540$		0			0
$541$	−6.37199e8			−0.173015			−0.0865076	−	0.996251i	$-0.527571\pi$
$541$	−6.37199e8			−0.173015			−0.0865076	+	0.996251i	$0.527571\pi$
$542$		−	4.16603e9i		−	1.12389i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.56469e9			0.408765			0.204382	−	0.978891i	$-0.434481\pi$
$547$	1.56469e9			0.408765			0.204382	+	0.978891i	$0.434481\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−8.30904e7	−	6.30212e7i	−0.0210264	−	0.0159478i
$553$		0			0
$554$		−	1.12491e10i		−	2.81083i
$555$		0			0
$556$	−7.74518e9			−1.91104
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−1.09155e10	+	3.05626e9i	−2.65965	+	0.744683i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	1.73094e9	−	2.28216e9i	0.406262	−	0.535637i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1.32903e8			−0.0304310
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.55866e9i		−	1.00000i
$576$	1.18982e9	+	4.24948e9i	0.259420	+	0.926525i
$577$	8.05760e9			1.74619			0.873093	−	0.487554i	$-0.162111\pi$
$577$	8.05760e9			1.74619			0.873093	+	0.487554i	$0.162111\pi$
$578$		−	6.53457e9i		−	1.40757i
$579$	4.93866e8	+	3.74580e8i	0.105739	+	0.0801991i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	2.29658e7i		−	0.00477130i
$585$		0			0
$586$		0			0
$587$		−	6.31368e9i		−	1.28840i	−0.764859	−	0.644198i	$-0.777191\pi$
$587$		−	6.31368e9i		−	1.28840i	0.764859	−	0.644198i	$-0.222809\pi$
$588$	−3.85410e9	−	2.92320e9i	−0.781813	−	0.592978i
$589$		0			0
$590$		0			0
$591$	1.41299e9	−	1.86296e9i	0.281568	−	0.371234i
$592$		0			0
$593$		−	6.80116e8i		−	0.133934i	−0.997755	−	0.0669671i	$-0.978668\pi$
$593$		−	6.80116e8i		−	0.133934i	0.997755	−	0.0669671i	$-0.0213323\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	1.29536e10			2.47706
$599$		−	1.03305e10i		−	1.96393i	−0.189052	−	0.981967i	$-0.560541\pi$
$599$		−	1.03305e10i		−	1.96393i	0.189052	−	0.981967i	$-0.439459\pi$
$600$	8.43780e7	−	1.11248e8i	0.0159478	−	0.0210264i
$601$	9.07724e9			1.70566			0.852831	−	0.522187i	$-0.174884\pi$
$601$	9.07724e9			1.70566			0.852831	+	0.522187i	$0.174884\pi$
$602$		0			0
$603$		0			0
$604$	−9.23351e9			−1.70505
$605$		0			0
$606$	−6.43454e9	−	4.88038e9i	−1.17453	−	0.890839i
$607$	3.74889e9			0.680365			0.340183	−	0.940359i	$-0.389511\pi$
$607$	3.74889e9			0.680365			0.340183	+	0.940359i	$0.389511\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	6.79797e9i		−	1.20569i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			1.44640e10i			2.52173i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−5.55121e9	−	2.19084e9i	−0.930179	−	0.367105i
$622$	4.29034e9			0.714867
$623$		0			0
$624$	8.66669e9	+	6.57339e9i	1.42793	+	1.08304i
$625$	6.10352e9			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	8.36999e9	+	6.34835e9i	1.31163	+	0.994828i
$634$	1.34613e10			2.09785
$635$		0			0
$636$		0			0
$637$	−1.14804e10			−1.75982
$638$		0			0
$639$	−7.32383e9	+	2.05062e9i	−1.11041	+	0.310908i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.06840e10i		−	1.55085i	−0.631441	−	0.775424i	$-0.717536\pi$
$647$		−	1.06840e10i		−	1.55085i	0.631441	−	0.775424i	$-0.282464\pi$
$648$	−9.49190e7	−	1.56214e8i	−0.0137038	−	0.0225532i
$649$		0			0
$650$			1.73434e10i			2.47706i
$651$		0			0
$652$	−1.26877e10			−1.79274
$653$			1.57744e9i			0.221696i	0.993837	+	0.110848i	$0.0353566\pi$
$653$			1.57744e9i			0.221696i	−0.993837	+	0.110848i	$0.964643\pi$
$654$		0			0
$655$		0			0
$656$			8.76612e9i			1.21239i
$657$	−3.54349e8	−	1.26557e9i	−0.0487476	−	0.174103i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	1.04052e10i		−	1.39396i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.56886e8			−0.0204712
$668$			6.98365e9i			0.906494i
$669$	−1.16853e10	−	8.86289e9i	−1.50886	−	1.14442i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.94077e9			−0.245426			−0.122713	−	0.992442i	$-0.539160\pi$
$673$	−1.94077e9			−0.245426			−0.122713	+	0.992442i	$0.539160\pi$
$674$		0			0
$675$	2.93328e9	−	7.43241e9i	0.367105	−	0.930179i
$676$	−1.65266e10			−2.05764
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.10695e10i			1.32940i	0.747109	+	0.664701i	$0.231441\pi$
$683$			1.10695e10i			1.32940i	−0.747109	+	0.664701i	$0.768559\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.06109e10			−1.22343			−0.611716	−	0.791078i	$-0.709520\pi$
$691$	−1.06109e10			−1.22343			−0.611716	+	0.791078i	$0.709520\pi$
$692$			9.12633e9i			1.04695i
$693$		0			0
$694$	2.40066e10			2.72629
$695$		0			0
$696$	−3.82860e6	−	2.90386e6i	−0.000430436	−	0.000326471i
$697$		0			0
$698$			2.33698e10i			2.60113i
$699$	−6.41134e9	+	8.45304e9i	−0.710033	+	0.936144i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	2.11195e10	+	8.33504e9i	2.30411	+	0.909343i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−2.38413e10			−2.54984
$707$		0			0
$708$	1.11988e10	−	1.47651e10i	1.18592	−	1.56358i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1.89914e10i			1.96220i
$714$		0			0
$715$		0			0
$716$		−	1.79105e10i		−	1.82353i
$717$	1.12272e10	−	1.48025e10i	1.13751	−	1.49975i
$718$		0			0
$719$			1.98207e10i			1.98870i	0.106173	+	0.994348i	$0.466140\pi$
$719$			1.98207e10i			1.98870i	−0.106173	+	0.994348i	$0.533860\pi$
$720$		0			0
$721$		0			0
$722$			1.42348e10i			1.40757i
$723$		0			0
$724$		0			0
$725$		−	2.10052e8i		−	0.0204712i
$726$	8.77014e9	−	1.15630e10i	0.850605	−	1.12148i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−7.64095e9	−	7.14387e9i	−0.730468	−	0.682947i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	1.52191e10			1.40707
$737$		0			0
$738$	4.93346e9	+	1.76200e10i	0.451809	+	1.61364i
$739$	1.98085e10			1.80549			0.902746	−	0.430174i	$-0.141548\pi$
$739$	1.98085e10			1.80549			0.902746	+	0.430174i	$0.141548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−3.51519e8	+	4.63461e8i	−0.0312927	+	0.0412580i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	8.13668e9i		−	0.697726i
$753$		0			0
$754$	5.96872e8			0.0507086
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	4.37612e9i		−	0.359951i	−0.983671	−	0.179975i	$-0.942398\pi$
$761$		−	4.37612e9i		−	0.359951i	0.983671	−	0.179975i	$-0.0576018\pi$
$762$	9.36645e9	−	1.23492e10i	0.766889	−	1.01111i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	4.39813e10i		−	3.51953i
$768$	1.03664e10	+	7.86258e9i	0.825782	+	0.626327i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	1.53167e10	−	2.01943e10i	1.20358	−	1.58686i
$772$	−1.66476e9			−0.130224
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−2.54273e10			−1.96220
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−2.55786e8	−	1.00949e8i	−0.0190419	−	0.00751510i
$784$	−1.37412e10			−1.01840
$785$		0			0
$786$	−2.31984e10	−	1.75952e10i	−1.70404	−	1.29246i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$			6.27982e9i			0.457200i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		−	2.99652e10i		−	2.12445i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$			2.03766e10i			1.40707i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	7.22526e10i		−	4.86050i
$807$	−1.74121e10	+	2.29570e10i	−1.16626	+	1.53765i
$808$	−4.14431e8			−0.0276384
$809$			2.68654e10i			1.78391i	0.452125	+	0.891955i	$0.350666\pi$
$809$			2.68654e10i			1.78391i	−0.452125	+	0.891955i	$0.649334\pi$
$810$		0			0
$811$	−2.33600e10			−1.53780			−0.768898	−	0.639371i	$-0.779195\pi$
$811$	−2.33600e10			−1.53780			−0.768898	+	0.639371i	$0.779195\pi$
$812$		0			0
$813$	−9.74754e9	−	7.39317e9i	−0.636177	−	0.482519i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			4.32500e10i			2.76280i
$819$		0			0
$820$		0			0
$821$			2.16676e10i			1.36650i	0.730185	+	0.683249i	$0.239434\pi$
$821$			2.16676e10i			1.36650i	−0.730185	+	0.683249i	$0.760566\pi$
$822$		0			0
$823$	2.98569e10			1.86700			0.933501	−	0.358574i	$-0.116737\pi$
$823$	2.98569e10			1.86700			0.933501	+	0.358574i	$0.116737\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	1.54347e10	−	4.32160e9i	0.944911	−	0.264568i
$829$	−7.89933e9			−0.481559			−0.240780	−	0.970580i	$-0.577403\pi$
$829$	−7.89933e9			−0.481559			−0.240780	+	0.970580i	$0.577403\pi$
$830$		0			0
$831$	−2.63203e10	−	1.99630e10i	−1.59106	−	1.20677i
$832$	−2.81284e10			−1.69322
$833$		0			0
$834$	−2.77523e10	+	3.65900e10i	−1.65660	+	2.18415i
$835$		0			0
$836$		0			0
$837$	−1.22201e10	+	3.09635e10i	−0.720334	+	1.82520i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.72426e10			0.999581
$842$		0			0
$843$		0			0
$844$	−2.82142e10			−1.61536
$845$		0			0
$846$	−4.57922e9	−	1.63548e10i	−0.260013	−	0.928641i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	1.23439e10	−	1.62748e10i	0.683774	−	0.901523i
$853$	3.38461e10			1.86718			0.933592	−	0.358338i	$-0.116656\pi$
$853$	3.38461e10			1.86718			0.933592	+	0.358338i	$0.116656\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	3.52479e10i		−	1.91294i	−0.291834	−	0.956469i	$-0.594266\pi$
$857$		−	3.52479e10i		−	1.91294i	0.291834	−	0.956469i	$-0.405734\pi$
$858$		0			0
$859$	−1.75816e10			−0.946416			−0.473208	−	0.880951i	$-0.656904\pi$
$859$	−1.75816e10			−0.946416			−0.473208	+	0.880951i	$0.656904\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			3.75427e10i			1.98832i	0.107897	+	0.994162i	$0.465588\pi$
$863$			3.75427e10i			1.98832i	−0.107897	+	0.994162i	$0.534412\pi$
$864$	2.48132e10	+	9.79277e9i	1.30883	+	0.516544i
$865$		0			0
$866$		0			0
$867$	−1.52894e10	−	1.15965e10i	−0.796751	−	0.604308i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	2.81230e9	+	2.13304e9i	0.141351	+	0.107210i
$877$	−3.97530e10			−1.99008			−0.995042	−	0.0994600i	$-0.968288\pi$
$877$	−3.97530e10			−1.99008			−0.995042	+	0.0994600i	$0.968288\pi$
$878$		−	4.59396e10i		−	2.29064i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−2.76198e10	+	7.73335e9i	−1.35544	+	0.379514i
$883$	3.77437e10			1.84494			0.922469	−	0.386072i	$-0.126168\pi$
$883$	3.77437e10			1.84494			0.922469	+	0.386072i	$0.126168\pi$
$884$		0			0
$885$		0			0
$886$	5.72906e10			2.76736
$887$			1.69785e10i			0.816896i	0.912781	+	0.408448i	$0.133930\pi$
$887$			1.69785e10i			0.816896i	−0.912781	+	0.408448i	$0.866070\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	3.93897e10			1.85826
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	2.29879e10	−	3.03084e10i	1.06347	−	1.40214i
$898$	6.09361e10			2.80807
$899$			8.75078e8i			0.0401687i
$900$	5.78611e9	+	2.06652e10i	0.264568	+	0.944911i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−3.30852e10	+	4.36213e10i	−1.47804	+	1.94872i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−2.28379e10	+	6.39444e9i	−1.00851	+	0.282377i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	3.38423e10	+	2.56682e10i	1.42742	+	1.08265i
$922$	−5.22134e10			−2.19393
$923$		−	4.84783e10i		−	2.02928i
$924$		0			0
$925$		0			0
$926$			2.73811e10i			1.13321i
$927$		0			0
$928$	7.01260e8			0.0288046
$929$			4.53220e10i			1.85462i	0.374300	+	0.927308i	$0.377883\pi$
$929$			4.53220e10i			1.85462i	−0.374300	+	0.927308i	$0.622117\pi$
$930$		0			0
$931$		0			0
$932$		−	2.84942e10i		−	1.15292i
$933$	7.61377e9	−	1.00384e10i	0.306912	−	0.404648i
$934$		0			0
$935$		0			0
$936$	1.12198e9	−	3.14147e8i	0.0447219	−	0.0125218i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	3.06562e10			1.19049
$944$		−	5.26424e10i		−	2.03673i
$945$		0			0
$946$		0			0
$947$			5.04672e10i			1.93101i	0.260385	+	0.965505i	$0.416151\pi$
$947$			5.04672e10i			1.93101i	−0.260385	+	0.965505i	$0.583849\pi$
$948$		0			0
$949$	8.37711e9			0.318172
$950$		0			0
$951$	2.38888e10	−	3.14962e10i	0.900664	−	1.18748i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			4.98975e10i			1.84704i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.84174e10			2.85023
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.82955e10			−0.650658			−0.325329	−	0.945601i	$-0.605475\pi$
$967$	−1.82955e10			−0.650658			−0.325329	+	0.945601i	$0.605475\pi$
$968$		−	7.44741e8i		−	0.0263901i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	2.79453e10	+	2.88556e9i	0.976062	+	0.100786i
$973$		0			0
$974$		−	3.48821e10i		−	1.20961i
$975$	4.05794e10	+	3.07781e10i	1.40214	+	1.06347i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−4.54623e10	+	5.99398e10i	−1.55405	+	2.04894i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−5.62980e10			−1.89715
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	7.48124e8	+	5.67427e8i	0.0250317	+	0.0189857i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.02745e10			−1.96732			−0.983662	−	0.180023i	$-0.942383\pi$
$991$	−6.02745e10			−1.96732			−0.983662	+	0.180023i	$0.942383\pi$
$992$		−	8.48890e10i		−	2.76096i
$993$	−2.43458e10	−	1.84655e10i	−0.789045	−	0.598464i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.06654e10			−0.340835			−0.170417	−	0.985372i	$-0.554512\pi$
$997$	−1.06654e10			−0.340835			−0.170417	+	0.985372i	$0.554512\pi$
$998$		−	2.25758e10i		−	0.718931i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.c.a.68.5	yes	6
3.2	odd	2	inner	69.8.c.a.68.2	✓	6
23.22	odd	2	CM	69.8.c.a.68.5	yes	6
69.68	even	2	inner	69.8.c.a.68.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.c.a.68.2	✓	6	3.2	odd	2	inner
69.8.c.a.68.2	✓	6	69.68	even	2	inner
69.8.c.a.68.5	yes	6	1.1	even	1	trivial
69.8.c.a.68.5	yes	6	23.22	odd	2	CM

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+15.9248i q^{2} +(37.2603 + 28.2607i) q^{3} -125.600 q^{4} +(-450.047 + 593.365i) q^{6} +38.2170i q^{8} +(589.667 + 2106.01i) q^{9} +O(q^{10})\)	\(q+15.9248i q^{2} +(37.2603 + 28.2607i) q^{3} -125.600 q^{4} +(-450.047 + 593.365i) q^{6} +38.2170i q^{8} +(589.667 + 2106.01i) q^{9} +(-4679.91 - 3549.55i) q^{12} -13940.2 q^{13} -16685.4 q^{16} +(-33537.8 + 9390.34i) q^{18} +58350.9i q^{23} +(-1080.04 + 1423.98i) q^{24} -78125.0 q^{25} -221995. i q^{26} +(-37546.0 + 95134.9i) q^{27} +2688.67i q^{29} +325469. q^{31} -260821. i q^{32} +(-74062.2 - 264515. i) q^{36} +(-519417. - 393960. i) q^{39} -525376. i q^{41} -929228. q^{46} +487652. i q^{47} +(-621705. - 471541. i) q^{48} +823543. q^{49} -1.24413e6i q^{50} +1.75089e6 q^{52} +(-1.51501e6 - 597914. i) q^{54} -42816.6 q^{58} +3.15500e6i q^{59} +5.18304e6i q^{62} +2.01779e6 q^{64} +(-1.64904e6 + 2.17417e6i) q^{69} +3.47759e6i q^{71} +(-80485.2 + 22535.3i) q^{72} -600932. q^{73} +(-2.91096e6 - 2.20787e6i) q^{75} +(6.27374e6 - 8.27163e6i) q^{78} +(-4.08756e6 + 2.48368e6i) q^{81} +8.36652e6 q^{82} +(-75983.6 + 100181. i) q^{87} -7.32888e6i q^{92} +(1.21271e7 + 9.19798e6i) q^{93} -7.76577e6 q^{94} +(7.37097e6 - 9.71827e6i) q^{96} +1.31148e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 768 q^{4} - 3147 q^{6}+O(q^{10}) \) 6 * q - 768 * q^4 - 3147 * q^6	\( 6 q - 768 q^{4} - 3147 q^{6} - 21435 q^{12} + 98304 q^{16} - 72177 q^{18} + 402816 q^{24} - 468750 q^{25} - 225276 q^{27} + 1621587 q^{36} - 2196456 q^{39} - 866433 q^{48} + 4941258 q^{49} + 5680578 q^{52} - 12795270 q^{58} - 13290570 q^{64} + 9238656 q^{72} + 8715747 q^{78} + 59301462 q^{82} - 29588784 q^{87} + 30939042 q^{93} - 87793602 q^{94} - 59843631 q^{96}+O(q^{100}) \) 6 * q - 768 * q^4 - 3147 * q^6 - 21435 * q^12 + 98304 * q^16 - 72177 * q^18 + 402816 * q^24 - 468750 * q^25 - 225276 * q^27 + 1621587 * q^36 - 2196456 * q^39 - 866433 * q^48 + 4941258 * q^49 + 5680578 * q^52 - 12795270 * q^58 - 13290570 * q^64 + 9238656 * q^72 + 8715747 * q^78 + 59301462 * q^82 - 29588784 * q^87 + 30939042 * q^93 - 87793602 * q^94 - 59843631 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more