LMFDB - Embedded newform 120.6.f.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,6,Mod(49,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, 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("120.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$120 = 2^{3} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	120.f (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:	$19.2460583776$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.25787221056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 61x^{4} + 852x^{2} + 576$ x^6 + 61x^4 + 852x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{4}\cdot 5^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$6.33429i$ of defining polynomial
Character	$\chi$	$=$	120.49
Dual form			120.6.f.a.49.5

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +(33.8706 + 44.4723i) q^{5} +85.8595i q^{7} -81.0000 q^{9} -488.087 q^{11} -38.5971i q^{13} +(400.250 - 304.835i) q^{15} -692.837i q^{17} -2489.26 q^{19} +772.735 q^{21} +4124.41i q^{23} +(-830.563 + 3012.61i) q^{25} +729.000i q^{27} +1886.71 q^{29} -2299.60 q^{31} +4392.78i q^{33} +(-3818.36 + 2908.11i) q^{35} +10628.6i q^{37} -347.373 q^{39} -15268.5 q^{41} +9468.20i q^{43} +(-2743.52 - 3602.25i) q^{45} -14323.3i q^{47} +9435.15 q^{49} -6235.53 q^{51} +13918.1i q^{53} +(-16531.8 - 21706.3i) q^{55} +22403.3i q^{57} -39692.5 q^{59} +38221.6 q^{61} -6954.62i q^{63} +(1716.50 - 1307.31i) q^{65} -1416.38i q^{67} +37119.6 q^{69} -11637.1 q^{71} +31082.1i q^{73} +(27113.4 + 7475.07i) q^{75} -41906.9i q^{77} +43476.3 q^{79} +6561.00 q^{81} -86699.9i q^{83} +(30812.0 - 23466.8i) q^{85} -16980.4i q^{87} +100834. q^{89} +3313.92 q^{91} +20696.4i q^{93} +(-84312.6 - 110703. i) q^{95} -39911.0i q^{97} +39535.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 50 q^{5} - 486 q^{9} - 664 q^{11} + 540 q^{15} + 288 q^{19} - 1872 q^{21} + 3750 q^{25} - 1892 q^{29} - 10248 q^{31} - 18880 q^{35} + 1584 q^{39} - 1324 q^{41} - 4050 q^{45} + 28410 q^{49} + 20088 q^{51}+ \cdots + 53784 q^{99}+O(q^{100})$ 6 * q + 50 * q^5 - 486 * q^9 - 664 * q^11 + 540 * q^15 + 288 * q^19 - 1872 * q^21 + 3750 * q^25 - 1892 * q^29 - 10248 * q^31 - 18880 * q^35 + 1584 * q^39 - 1324 * q^41 - 4050 * q^45 + 28410 * q^49 + 20088 * q^51 - 47160 * q^55 - 10296 * q^59 - 52116 * q^61 - 13480 * q^65 + 51624 * q^69 - 28288 * q^71 - 41400 * q^75 + 220200 * q^79 + 39366 * q^81 + 95880 * q^85 + 351420 * q^89 - 17088 * q^91 - 349120 * q^95 + 53784 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$ .

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	33.8706	+	44.4723i	0.605896	+	0.795544i
$5$	33.8706	+	44.4723i	0.605896	+	0.795544i
$6$		0			0
$7$			85.8595i			0.662282i	0.943581	+	0.331141i	$0.107434\pi$
$7$			85.8595i			0.662282i	−0.943581	+	0.331141i	$0.892566\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−488.087			−1.21623			−0.608114	−	0.793849i	$-0.708074\pi$
$11$	−488.087			−1.21623			−0.608114	+	0.793849i	$0.708074\pi$
$12$		0			0
$13$		−	38.5971i		−	0.0633426i	−0.999498	−	0.0316713i	$-0.989917\pi$
$13$		−	38.5971i		−	0.0633426i	0.999498	−	0.0316713i	$-0.0100830\pi$
$14$		0			0
$15$	400.250	−	304.835i	0.459308	−	0.349814i
$16$		0			0
$17$		−	692.837i		−	0.581445i	−0.956807	−	0.290723i	$-0.906104\pi$
$17$		−	692.837i		−	0.581445i	0.956807	−	0.290723i	$-0.0938957\pi$
$18$		0			0
$19$	−2489.26			−1.58192			−0.790962	−	0.611865i	$-0.790419\pi$
$19$	−2489.26			−1.58192			−0.790962	+	0.611865i	$0.790419\pi$
$20$		0			0
$21$	772.735			0.382369
$22$		0			0
$23$			4124.41i			1.62571i	0.582470	+	0.812853i	$0.302087\pi$
$23$			4124.41i			1.62571i	−0.582470	+	0.812853i	$0.697913\pi$
$24$		0			0
$25$	−830.563	+	3012.61i	−0.265780	+	0.964034i
$26$		0			0
$27$			729.000i			0.192450i
$28$		0			0
$29$	1886.71			0.416592			0.208296	−	0.978066i	$-0.433208\pi$
$29$	1886.71			0.416592			0.208296	+	0.978066i	$0.433208\pi$
$30$		0			0
$31$	−2299.60			−0.429781			−0.214891	−	0.976638i	$-0.568940\pi$
$31$	−2299.60			−0.429781			−0.214891	+	0.976638i	$0.568940\pi$
$32$		0			0
$33$			4392.78i			0.702190i
$34$		0			0
$35$	−3818.36	+	2908.11i	−0.526875	+	0.401274i
$36$		0			0
$37$			10628.6i			1.27636i	0.769888	+	0.638178i	$0.220312\pi$
$37$			10628.6i			1.27636i	−0.769888	+	0.638178i	$0.779688\pi$
$38$		0			0
$39$	−347.373			−0.0365709
$40$		0			0
$41$	−15268.5			−1.41853			−0.709264	−	0.704943i	$-0.750973\pi$
$41$	−15268.5			−1.41853			−0.709264	+	0.704943i	$0.750973\pi$
$42$		0			0
$43$			9468.20i			0.780901i	0.920624	+	0.390451i	$0.127681\pi$
$43$			9468.20i			0.780901i	−0.920624	+	0.390451i	$0.872319\pi$
$44$		0			0
$45$	−2743.52	−	3602.25i	−0.201965	−	0.265181i
$46$		0			0
$47$		−	14323.3i		−	0.945800i	−0.881116	−	0.472900i	$-0.843207\pi$
$47$		−	14323.3i		−	0.945800i	0.881116	−	0.472900i	$-0.156793\pi$
$48$		0			0
$49$	9435.15			0.561382
$50$		0			0
$51$	−6235.53			−0.335698
$52$		0			0
$53$			13918.1i			0.680596i	0.940318	+	0.340298i	$0.110528\pi$
$53$			13918.1i			0.680596i	−0.940318	+	0.340298i	$0.889472\pi$
$54$		0			0
$55$	−16531.8	−	21706.3i	−0.736908	−	0.967564i
$56$		0			0
$57$			22403.3i			0.913324i
$58$		0			0
$59$	−39692.5			−1.48449			−0.742247	−	0.670126i	$-0.766240\pi$
$59$	−39692.5			−1.48449			−0.742247	+	0.670126i	$0.766240\pi$
$60$		0			0
$61$	38221.6			1.31518			0.657588	−	0.753377i	$-0.271577\pi$
$61$	38221.6			1.31518			0.657588	+	0.753377i	$0.271577\pi$
$62$		0			0
$63$		−	6954.62i		−	0.220761i
$64$		0			0
$65$	1716.50	−	1307.31i	0.0503918	−	0.0383790i
$66$		0			0
$67$		−	1416.38i		−	0.0385472i	−0.999814	−	0.0192736i	$-0.993865\pi$
$67$		−	1416.38i		−	0.0385472i	0.999814	−	0.0192736i	$-0.00613536\pi$
$68$		0			0
$69$	37119.6			0.938601
$70$		0			0
$71$	−11637.1			−0.273966			−0.136983	−	0.990573i	$-0.543741\pi$
$71$	−11637.1			−0.273966			−0.136983	+	0.990573i	$0.543741\pi$
$72$		0			0
$73$			31082.1i			0.682657i	0.939944	+	0.341329i	$0.110877\pi$
$73$			31082.1i			0.682657i	−0.939944	+	0.341329i	$0.889123\pi$
$74$		0			0
$75$	27113.4	+	7475.07i	0.556585	+	0.153448i
$76$		0			0
$77$		−	41906.9i		−	0.805487i
$78$		0			0
$79$	43476.3			0.783763			0.391881	−	0.920016i	$-0.371824\pi$
$79$	43476.3			0.783763			0.391881	+	0.920016i	$0.371824\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$		−	86699.9i		−	1.38141i	−0.723135	−	0.690706i	$-0.757300\pi$
$83$		−	86699.9i		−	1.38141i	0.723135	−	0.690706i	$-0.242700\pi$
$84$		0			0
$85$	30812.0	−	23466.8i	0.462565	−	0.352295i
$86$		0			0
$87$		−	16980.4i		−	0.240519i
$88$		0			0
$89$	100834.			1.34937			0.674686	−	0.738105i	$-0.264279\pi$
$89$	100834.			1.34937			0.674686	+	0.738105i	$0.264279\pi$
$90$		0			0
$91$	3313.92			0.0419507
$92$		0			0
$93$			20696.4i			0.248134i
$94$		0			0
$95$	−84312.6	−	110703.i	−0.958481	−	1.25849i
$96$		0			0
$97$		−	39911.0i		−	0.430688i	−0.976538	−	0.215344i	$-0.930913\pi$
$97$		−	39911.0i		−	0.430688i	0.976538	−	0.215344i	$-0.0690873\pi$
$98$		0			0
$99$	39535.0			0.405410
$100$		0			0
$101$	−117147.			−1.14269			−0.571344	−	0.820711i	$-0.693578\pi$
$101$	−117147.			−1.14269			−0.571344	+	0.820711i	$0.693578\pi$
$102$		0			0
$103$		−	266.000i		−	0.00247052i	−0.999999	−	0.00123526i	$-0.999607\pi$
$103$		−	266.000i		−	0.00247052i	0.999999	−	0.00123526i	$-0.000393195\pi$
$104$		0			0
$105$	26173.0	+	34365.3i	0.231676	+	0.304191i
$106$		0			0
$107$		−	120151.i		−	1.01454i	−0.861787	−	0.507270i	$-0.830655\pi$
$107$		−	120151.i		−	1.01454i	0.861787	−	0.507270i	$-0.169345\pi$
$108$		0			0
$109$	135567.			1.09292			0.546458	−	0.837486i	$-0.315976\pi$
$109$	135567.			1.09292			0.546458	+	0.837486i	$0.315976\pi$
$110$		0			0
$111$	95657.5			0.736905
$112$		0			0
$113$		−	94060.5i		−	0.692965i	−0.938056	−	0.346482i	$-0.887376\pi$
$113$		−	94060.5i		−	0.692965i	0.938056	−	0.346482i	$-0.112624\pi$
$114$		0			0
$115$	−183422.	+	139696.i	−1.29332	+	0.985008i
$116$		0			0
$117$			3126.36i			0.0211142i
$118$		0			0
$119$	59486.6			0.385081
$120$		0			0
$121$	77177.7			0.479213
$122$		0			0
$123$			137417.i			0.818988i
$124$		0			0
$125$	−162109.	+	65101.7i	−0.927966	+	0.372664i
$126$		0			0
$127$			118472.i			0.651786i	0.945407	+	0.325893i	$0.105665\pi$
$127$			118472.i			0.651786i	−0.945407	+	0.325893i	$0.894335\pi$
$128$		0			0
$129$	85213.8			0.450854
$130$		0			0
$131$	−232998.			−1.18624			−0.593122	−	0.805113i	$-0.702105\pi$
$131$	−232998.			−1.18624			−0.593122	+	0.805113i	$0.702105\pi$
$132$		0			0
$133$		−	213726.i		−	1.04768i
$134$		0			0
$135$	−32420.3	+	24691.7i	−0.153103	+	0.116605i
$136$		0			0
$137$			139571.i			0.635321i	0.948204	+	0.317661i	$0.102897\pi$
$137$			139571.i			0.635321i	−0.948204	+	0.317661i	$0.897103\pi$
$138$		0			0
$139$	263812.			1.15813			0.579066	−	0.815280i	$-0.303417\pi$
$139$	263812.			1.15813			0.579066	+	0.815280i	$0.303417\pi$
$140$		0			0
$141$	−128910.			−0.546058
$142$		0			0
$143$			18838.7i			0.0770391i
$144$		0			0
$145$	63904.1	+	83906.3i	0.252411	+	0.331417i
$146$		0			0
$147$		−	84916.4i		−	0.324114i
$148$		0			0
$149$	−344172.			−1.27002			−0.635008	−	0.772505i	$-0.719003\pi$
$149$	−344172.			−1.27002			−0.635008	+	0.772505i	$0.719003\pi$
$150$		0			0
$151$	−12286.8			−0.0438527			−0.0219263	−	0.999760i	$-0.506980\pi$
$151$	−12286.8			−0.0438527			−0.0219263	+	0.999760i	$0.506980\pi$
$152$		0			0
$153$			56119.8i			0.193815i
$154$		0			0
$155$	−77888.8	−	102268.i	−0.260403	−	0.341910i
$156$		0			0
$157$			240704.i			0.779354i	0.920952	+	0.389677i	$0.127413\pi$
$157$			240704.i			0.779354i	−0.920952	+	0.389677i	$0.872587\pi$
$158$		0			0
$159$	125263.			0.392942
$160$		0			0
$161$	−354119.			−1.07668
$162$		0			0
$163$			565359.i			1.66669i	0.552752	+	0.833346i	$0.313578\pi$
$163$			565359.i			1.66669i	−0.552752	+	0.833346i	$0.686422\pi$
$164$		0			0
$165$	−195357.	+	148786.i	−0.558623	+	0.425454i
$166$		0			0
$167$			411038.i			1.14049i	0.821475	+	0.570244i	$0.193151\pi$
$167$			411038.i			1.14049i	−0.821475	+	0.570244i	$0.806849\pi$
$168$		0			0
$169$	369803.			0.995988
$170$		0			0
$171$	201630.			0.527308
$172$		0			0
$173$		−	188716.i		−	0.479395i	−0.970848	−	0.239698i	$-0.922952\pi$
$173$		−	188716.i		−	0.479395i	0.970848	−	0.239698i	$-0.0770483\pi$
$174$		0			0
$175$	−258661.	−	71311.7i	−0.638462	−	0.176022i
$176$		0			0
$177$			357233.i			0.857073i
$178$		0			0
$179$	601974.			1.40425			0.702126	−	0.712053i	$-0.252234\pi$
$179$	601974.			1.40425			0.702126	+	0.712053i	$0.252234\pi$
$180$		0			0
$181$	−836032.			−1.89682			−0.948410	−	0.317045i	$-0.897309\pi$
$181$	−836032.			−1.89682			−0.948410	+	0.317045i	$0.897309\pi$
$182$		0			0
$183$		−	343994.i		−	0.759318i
$184$		0			0
$185$	−472678.	+	359998.i	−1.01540	+	0.773339i
$186$		0			0
$187$			338165.i			0.707170i
$188$		0			0
$189$	−62591.6			−0.127456
$190$		0			0
$191$	502698.			0.997066			0.498533	−	0.866871i	$-0.333872\pi$
$191$	502698.			0.997066			0.498533	+	0.866871i	$0.333872\pi$
$192$		0			0
$193$		−	731154.i		−	1.41291i	−0.707756	−	0.706457i	$-0.750293\pi$
$193$		−	731154.i		−	1.41291i	0.707756	−	0.706457i	$-0.249707\pi$
$194$		0			0
$195$	−11765.8	−	15448.5i	−0.0221581	−	0.0290937i
$196$		0			0
$197$		−	871888.i		−	1.60065i	−0.599569	−	0.800323i	$-0.704661\pi$
$197$		−	871888.i		−	1.60065i	0.599569	−	0.800323i	$-0.295339\pi$
$198$		0			0
$199$	101311.			0.181352			0.0906760	−	0.995880i	$-0.471097\pi$
$199$	101311.			0.181352			0.0906760	+	0.995880i	$0.471097\pi$
$200$		0			0
$201$	−12747.4			−0.0222553
$202$		0			0
$203$			161992.i			0.275901i
$204$		0			0
$205$	−517155.	−	679027.i	−0.859481	−	1.12850i
$206$		0			0
$207$		−	334077.i		−	0.541902i
$208$		0			0
$209$	1.21497e6			1.92398
$210$		0			0
$211$	−725153.			−1.12130			−0.560652	−	0.828051i	$-0.689450\pi$
$211$	−725153.			−1.12130			−0.560652	+	0.828051i	$0.689450\pi$
$212$		0			0
$213$			104734.i			0.158175i
$214$		0			0
$215$	−421072.	+	320694.i	−0.621241	+	0.473145i
$216$		0			0
$217$		−	197442.i		−	0.284637i
$218$		0			0
$219$	279739.			0.394132
$220$		0			0
$221$	−26741.5			−0.0368302
$222$		0			0
$223$			680623.i			0.916526i	0.888817	+	0.458263i	$0.151528\pi$
$223$			680623.i			0.916526i	−0.888817	+	0.458263i	$0.848472\pi$
$224$		0			0
$225$	67275.6	−	244021.i	0.0885934	−	0.321345i
$226$		0			0
$227$		−	1.01003e6i		−	1.30098i	−0.759516	−	0.650488i	$-0.774564\pi$
$227$		−	1.01003e6i		−	1.30098i	0.759516	−	0.650488i	$-0.225436\pi$
$228$		0			0
$229$	1.22740e6			1.54667			0.773337	−	0.633996i	$-0.218586\pi$
$229$	1.22740e6			1.54667			0.773337	+	0.633996i	$0.218586\pi$
$230$		0			0
$231$	−377162.			−0.465048
$232$		0			0
$233$			599483.i			0.723415i	0.932292	+	0.361707i	$0.117806\pi$
$233$			599483.i			0.723415i	−0.932292	+	0.361707i	$0.882194\pi$
$234$		0			0
$235$	636991.	−	485140.i	0.752425	−	0.573056i
$236$		0			0
$237$		−	391287.i		−	0.452506i
$238$		0			0
$239$	1.37162e6			1.55324			0.776620	−	0.629970i	$-0.216933\pi$
$239$	1.37162e6			1.55324			0.776620	+	0.629970i	$0.216933\pi$
$240$		0			0
$241$	1.36857e6			1.51784			0.758918	−	0.651186i	$-0.225728\pi$
$241$	1.36857e6			1.51784			0.758918	+	0.651186i	$0.225728\pi$
$242$		0			0
$243$		−	59049.0i		−	0.0641500i
$244$		0			0
$245$	319574.	+	419602.i	0.340139	+	0.446604i
$246$		0			0
$247$			96078.0i			0.100203i
$248$		0			0
$249$	−780299.			−0.797559
$250$		0			0
$251$	730293.			0.731666			0.365833	−	0.930681i	$-0.380784\pi$
$251$	730293.			0.731666			0.365833	+	0.930681i	$0.380784\pi$
$252$		0			0
$253$		−	2.01307e6i		−	1.97723i
$254$		0			0
$255$	−211201.	−	277308.i	−0.203398	−	0.267062i
$256$		0			0
$257$			1.36832e6i			1.29228i	0.763221	+	0.646138i	$0.223617\pi$
$257$			1.36832e6i			1.29228i	−0.763221	+	0.646138i	$0.776383\pi$
$258$		0			0
$259$	−912567.			−0.845309
$260$		0			0
$261$	−152824.			−0.138864
$262$		0			0
$263$			1.26265e6i			1.12562i	0.826585	+	0.562811i	$0.190280\pi$
$263$			1.26265e6i			1.12562i	−0.826585	+	0.562811i	$0.809720\pi$
$264$		0			0
$265$	−618968.	+	471413.i	−0.541444	+	0.412370i
$266$		0			0
$267$		−	907505.i		−	0.779060i
$268$		0			0
$269$	−652896.			−0.550127			−0.275064	−	0.961426i	$-0.588699\pi$
$269$	−652896.			−0.550127			−0.275064	+	0.961426i	$0.588699\pi$
$270$		0			0
$271$	−649071.			−0.536870			−0.268435	−	0.963298i	$-0.586506\pi$
$271$	−649071.			−0.536870			−0.268435	+	0.963298i	$0.586506\pi$
$272$		0			0
$273$		−	29825.3i		−	0.0242202i
$274$		0			0
$275$	405387.	−	1.47041e6i	0.323250	−	1.17249i
$276$		0			0
$277$		−	668500.i		−	0.523482i	−0.965138	−	0.261741i	$-0.915703\pi$
$277$		−	668500.i		−	0.523482i	0.965138	−	0.261741i	$-0.0842967\pi$
$278$		0			0
$279$	186267.			0.143260
$280$		0			0
$281$	1.03323e6			0.780607			0.390304	−	0.920686i	$-0.372370\pi$
$281$	1.03323e6			0.780607			0.390304	+	0.920686i	$0.372370\pi$
$282$		0			0
$283$		−	384887.i		−	0.285672i	−0.989746	−	0.142836i	$-0.954378\pi$
$283$		−	384887.i		−	0.285672i	0.989746	−	0.142836i	$-0.0456221\pi$
$284$		0			0
$285$	−996326.	+	758814.i	−0.726590	+	0.553379i
$286$		0			0
$287$		−	1.31095e6i		−	0.939466i
$288$		0			0
$289$	939834.			0.661922
$290$		0			0
$291$	−359199.			−0.248658
$292$		0			0
$293$			74753.1i			0.0508698i	0.999676	+	0.0254349i	$0.00809705\pi$
$293$			74753.1i			0.0508698i	−0.999676	+	0.0254349i	$0.991903\pi$
$294$		0			0
$295$	−1.34441e6	−	1.76522e6i	−0.899449	−	1.18098i
$296$		0			0
$297$		−	355815.i		−	0.234063i
$298$		0			0
$299$	159190.			0.102976
$300$		0			0
$301$	−812934.			−0.517177
$302$		0			0
$303$			1.05432e6i			0.659731i
$304$		0			0
$305$	1.29459e6	+	1.69980e6i	0.796860	+	1.04628i
$306$		0			0
$307$			1.96989e6i			1.19288i	0.802659	+	0.596438i	$0.203418\pi$
$307$			1.96989e6i			1.19288i	−0.802659	+	0.596438i	$0.796582\pi$
$308$		0			0
$309$	−2394.00			−0.00142635
$310$		0			0
$311$	−1.92570e6			−1.12899			−0.564493	−	0.825438i	$-0.690928\pi$
$311$	−1.92570e6			−1.12899			−0.564493	+	0.825438i	$0.690928\pi$
$312$		0			0
$313$			1.05127e6i			0.606531i	0.952906	+	0.303265i	$0.0980768\pi$
$313$			1.05127e6i			0.606531i	−0.952906	+	0.303265i	$0.901923\pi$
$314$		0			0
$315$	309288.	−	235557.i	0.175625	−	0.133758i
$316$		0			0
$317$			2.68469e6i			1.50053i	0.661134	+	0.750267i	$0.270075\pi$
$317$			2.68469e6i			1.50053i	−0.661134	+	0.750267i	$0.729925\pi$
$318$		0			0
$319$	−920879.			−0.506671
$320$		0			0
$321$	−1.08136e6			−0.585745
$322$		0			0
$323$			1.72465e6i			0.919802i
$324$		0			0
$325$	116278.	+	32057.3i	0.0610644	+	0.0168352i
$326$		0			0
$327$		−	1.22010e6i		−	0.630996i
$328$		0			0
$329$	1.22979e6			0.626387
$330$		0			0
$331$	−1.50456e6			−0.754813			−0.377407	−	0.926048i	$-0.623184\pi$
$331$	−1.50456e6			−0.754813			−0.377407	+	0.926048i	$0.623184\pi$
$332$		0			0
$333$		−	860917.i		−	0.425452i
$334$		0			0
$335$	62989.7	−	47973.7i	0.0306660	−	0.0233556i
$336$		0			0
$337$			3.84030e6i			1.84200i	0.389559	+	0.921001i	$0.372627\pi$
$337$			3.84030e6i			1.84200i	−0.389559	+	0.921001i	$0.627373\pi$
$338$		0			0
$339$	−846544.			−0.400083
$340$		0			0
$341$	1.12240e6			0.522713
$342$		0			0
$343$			2.25314e6i			1.03408i
$344$		0			0
$345$	1.25727e6	+	1.65079e6i	0.568695	+	0.746699i
$346$		0			0
$347$			939488.i			0.418859i	0.977824	+	0.209429i	$0.0671606\pi$
$347$			939488.i			0.418859i	−0.977824	+	0.209429i	$0.932839\pi$
$348$		0			0
$349$	−3.56398e6			−1.56629			−0.783144	−	0.621841i	$-0.786385\pi$
$349$	−3.56398e6			−1.56629			−0.783144	+	0.621841i	$0.786385\pi$
$350$		0			0
$351$	28137.2			0.0121903
$352$		0			0
$353$		−	2.24033e6i		−	0.956919i	−0.878110	−	0.478459i	$-0.841195\pi$
$353$		−	2.24033e6i		−	0.956919i	0.878110	−	0.478459i	$-0.158805\pi$
$354$		0			0
$355$	−394154.	−	517526.i	−0.165995	−	0.217952i
$356$		0			0
$357$		−	535380.i		−	0.222327i
$358$		0			0
$359$	−290038.			−0.118773			−0.0593867	−	0.998235i	$-0.518915\pi$
$359$	−290038.			−0.118773			−0.0593867	+	0.998235i	$0.518915\pi$
$360$		0			0
$361$	3.72030e6			1.50248
$362$		0			0
$363$		−	694599.i		−	0.276674i
$364$		0			0
$365$	−1.38229e6	+	1.05277e6i	−0.543084	+	0.413619i
$366$		0			0
$367$		−	961035.i		−	0.372455i	−0.982507	−	0.186228i	$-0.940374\pi$
$367$		−	961035.i		−	0.372455i	0.982507	−	0.186228i	$-0.0596262\pi$
$368$		0			0
$369$	1.23675e6			0.472843
$370$		0			0
$371$	−1.19500e6			−0.450746
$372$		0			0
$373$		−	3.48605e6i		−	1.29736i	−0.761060	−	0.648681i	$-0.775321\pi$
$373$		−	3.48605e6i		−	1.29736i	0.761060	−	0.648681i	$-0.224679\pi$
$374$		0			0
$375$	585916.	+	1.45898e6i	0.215158	+	0.535762i
$376$		0			0
$377$		−	72821.5i		−	0.0263880i
$378$		0			0
$379$	−3.08939e6			−1.10478			−0.552389	−	0.833587i	$-0.686284\pi$
$379$	−3.08939e6			−1.10478			−0.552389	+	0.833587i	$0.686284\pi$
$380$		0			0
$381$	1.06624e6			0.376309
$382$		0			0
$383$			513758.i			0.178962i	0.995989	+	0.0894812i	$0.0285209\pi$
$383$			513758.i			0.178962i	−0.995989	+	0.0894812i	$0.971479\pi$
$384$		0			0
$385$	1.86369e6	−	1.41941e6i	0.640800	−	0.488041i
$386$		0			0
$387$		−	766924.i		−	0.260300i
$388$		0			0
$389$	−5.32273e6			−1.78345			−0.891725	−	0.452578i	$-0.850504\pi$
$389$	−5.32273e6			−1.78345			−0.891725	+	0.452578i	$0.850504\pi$
$390$		0			0
$391$	2.85754e6			0.945258
$392$		0			0
$393$			2.09698e6i			0.684878i
$394$		0			0
$395$	1.47257e6	+	1.93349e6i	0.474879	+	0.623518i
$396$		0			0
$397$			2.10400e6i			0.669990i	0.942220	+	0.334995i	$0.108735\pi$
$397$			2.10400e6i			0.669990i	−0.942220	+	0.334995i	$0.891265\pi$
$398$		0			0
$399$	−1.92354e6			−0.604878
$400$		0			0
$401$	1.41705e6			0.440072			0.220036	−	0.975492i	$-0.429382\pi$
$401$	1.41705e6			0.440072			0.220036	+	0.975492i	$0.429382\pi$
$402$		0			0
$403$			88757.7i			0.0272235i
$404$		0			0
$405$	222225.	+	291782.i	0.0673218	+	0.0883938i
$406$		0			0
$407$		−	5.18768e6i		−	1.55234i
$408$		0			0
$409$	−1.07044e6			−0.316414			−0.158207	−	0.987406i	$-0.550571\pi$
$409$	−1.07044e6			−0.316414			−0.158207	+	0.987406i	$0.550571\pi$
$410$		0			0
$411$	1.25614e6			0.366803
$412$		0			0
$413$		−	3.40798e6i		−	0.983154i
$414$		0			0
$415$	3.85574e6	−	2.93658e6i	1.09897	−	0.836992i
$416$		0			0
$417$		−	2.37431e6i		−	0.668648i
$418$		0			0
$419$	−2.75665e6			−0.767090			−0.383545	−	0.923522i	$-0.625297\pi$
$419$	−2.75665e6			−0.767090			−0.383545	+	0.923522i	$0.625297\pi$
$420$		0			0
$421$	−3.32521e6			−0.914352			−0.457176	−	0.889376i	$-0.651139\pi$
$421$	−3.32521e6			−0.914352			−0.457176	+	0.889376i	$0.651139\pi$
$422$		0			0
$423$			1.16019e6i			0.315267i
$424$		0			0
$425$	2.08724e6	+	575445.i	0.560533	+	0.154537i
$426$		0			0
$427$			3.28169e6i			0.871018i
$428$		0			0
$429$	169548.			0.0444785
$430$		0			0
$431$	1.26909e6			0.329078			0.164539	−	0.986371i	$-0.447386\pi$
$431$	1.26909e6			0.329078			0.164539	+	0.986371i	$0.447386\pi$
$432$		0			0
$433$		−	5.17143e6i		−	1.32553i	−0.748826	−	0.662767i	$-0.769382\pi$
$433$		−	5.17143e6i		−	1.32553i	0.748826	−	0.662767i	$-0.230618\pi$
$434$		0			0
$435$	755157.	−	575137.i	0.191344	−	0.145730i
$436$		0			0
$437$		−	1.02667e7i		−	2.57174i
$438$		0			0
$439$	6.78923e6			1.68136			0.840678	−	0.541536i	$-0.182157\pi$
$439$	6.78923e6			1.68136			0.840678	+	0.541536i	$0.182157\pi$
$440$		0			0
$441$	−764247.			−0.187127
$442$		0			0
$443$			3.16608e6i			0.766500i	0.923645	+	0.383250i	$0.125195\pi$
$443$			3.16608e6i			0.766500i	−0.923645	+	0.383250i	$0.874805\pi$
$444$		0			0
$445$	3.41531e6	+	4.48431e6i	0.817579	+	1.07348i
$446$		0			0
$447$			3.09755e6i			0.733245i
$448$		0			0
$449$	−4.91937e6			−1.15158			−0.575790	−	0.817598i	$-0.695305\pi$
$449$	−4.91937e6			−1.15158			−0.575790	+	0.817598i	$0.695305\pi$
$450$		0			0
$451$	7.45238e6			1.72526
$452$		0			0
$453$			110581.i			0.0253184i
$454$		0			0
$455$	112245.	+	147378.i	0.0254177	+	0.0333736i
$456$		0			0
$457$			7.30601e6i			1.63640i	0.574932	+	0.818201i	$0.305028\pi$
$457$			7.30601e6i			1.63640i	−0.574932	+	0.818201i	$0.694972\pi$
$458$		0			0
$459$	505078.			0.111899
$460$		0			0
$461$	−2.52155e6			−0.552606			−0.276303	−	0.961071i	$-0.589109\pi$
$461$	−2.52155e6			−0.552606			−0.276303	+	0.961071i	$0.589109\pi$
$462$		0			0
$463$		−	6.44859e6i		−	1.39802i	−0.715114	−	0.699008i	$-0.753625\pi$
$463$		−	6.44859e6i		−	1.39802i	0.715114	−	0.699008i	$-0.246375\pi$
$464$		0			0
$465$	−920415.	+	700999.i	−0.197402	+	0.150344i
$466$		0			0
$467$		−	5.39798e6i		−	1.14535i	−0.819782	−	0.572676i	$-0.805905\pi$
$467$		−	5.39798e6i		−	1.14535i	0.819782	−	0.572676i	$-0.194095\pi$
$468$		0			0
$469$	121610.			0.0255291
$470$		0			0
$471$	2.16634e6			0.449960
$472$		0			0
$473$		−	4.62130e6i		−	0.949755i
$474$		0			0
$475$	2.06749e6	−	7.49915e6i	0.420444	−	1.52503i
$476$		0			0
$477$		−	1.12736e6i		−	0.226865i
$478$		0			0
$479$	−5.25882e6			−1.04725			−0.523624	−	0.851950i	$-0.675420\pi$
$479$	−5.25882e6			−1.04725			−0.523624	+	0.851950i	$0.675420\pi$
$480$		0			0
$481$	410233.			0.0808477
$482$		0			0
$483$			3.18707e6i			0.621619i
$484$		0			0
$485$	1.77493e6	−	1.35181e6i	0.342631	−	0.260952i
$486$		0			0
$487$			5.61141e6i			1.07214i	0.844175	+	0.536068i	$0.180091\pi$
$487$			5.61141e6i			1.07214i	−0.844175	+	0.536068i	$0.819909\pi$
$488$		0			0
$489$	5.08823e6			0.962265
$490$		0			0
$491$	5.14176e6			0.962516			0.481258	−	0.876579i	$-0.340180\pi$
$491$	5.14176e6			0.962516			0.481258	+	0.876579i	$0.340180\pi$
$492$		0			0
$493$		−	1.30718e6i		−	0.242225i
$494$		0			0
$495$	1.33908e6	+	1.75821e6i	0.245636	+	0.322521i
$496$		0			0
$497$		−	999151.i		−	0.181443i
$498$		0			0
$499$	−4.53158e6			−0.814701			−0.407350	−	0.913272i	$-0.633547\pi$
$499$	−4.53158e6			−0.814701			−0.407350	+	0.913272i	$0.633547\pi$
$500$		0			0
$501$	3.69934e6			0.658461
$502$		0			0
$503$			3.93518e6i			0.693497i	0.937958	+	0.346749i	$0.112714\pi$
$503$			3.93518e6i			0.693497i	−0.937958	+	0.346749i	$0.887286\pi$
$504$		0			0
$505$	−3.96784e6	−	5.20979e6i	−0.692350	−	0.909058i
$506$		0			0
$507$		−	3.32823e6i		−	0.575034i
$508$		0			0
$509$	1.37582e6			0.235378			0.117689	−	0.993050i	$-0.462451\pi$
$509$	1.37582e6			0.235378			0.117689	+	0.993050i	$0.462451\pi$
$510$		0			0
$511$	−2.66869e6			−0.452112
$512$		0			0
$513$		−	1.81467e6i		−	0.304441i
$514$		0			0
$515$	11829.6	−	9009.57i	0.00196541	−	0.00149688i
$516$		0			0
$517$			6.99103e6i			1.15031i
$518$		0			0
$519$	−1.69844e6			−0.276779
$520$		0			0
$521$	−7.67174e6			−1.23822			−0.619112	−	0.785302i	$-0.712507\pi$
$521$	−7.67174e6			−1.23822			−0.619112	+	0.785302i	$0.712507\pi$
$522$		0			0
$523$			1.16971e7i			1.86993i	0.354743	+	0.934964i	$0.384568\pi$
$523$			1.16971e7i			1.86993i	−0.354743	+	0.934964i	$0.615432\pi$
$524$		0			0
$525$	−641806.	+	2.32795e6i	−0.101626	+	0.368616i
$526$		0			0
$527$			1.59325e6i			0.249894i
$528$		0			0
$529$	−1.05744e7			−1.64292
$530$		0			0
$531$	3.21509e6			0.494831
$532$		0			0
$533$			589321.i			0.0898533i
$534$		0			0
$535$	5.34341e6	−	4.06960e6i	0.807112	−	0.614706i
$536$		0			0
$537$		−	5.41777e6i		−	0.810745i
$538$		0			0
$539$	−4.60517e6			−0.682769
$540$		0			0
$541$	−3.26111e6			−0.479041			−0.239521	−	0.970891i	$-0.576990\pi$
$541$	−3.26111e6			−0.479041			−0.239521	+	0.970891i	$0.576990\pi$
$542$		0			0
$543$			7.52429e6i			1.09513i
$544$		0			0
$545$	4.59173e6	+	6.02896e6i	0.662194	+	0.869463i
$546$		0			0
$547$			707108.i			0.101046i	0.998723	+	0.0505228i	$0.0160888\pi$
$547$			707108.i			0.101046i	−0.998723	+	0.0505228i	$0.983911\pi$
$548$		0			0
$549$	−3.09595e6			−0.438392
$550$		0			0
$551$	−4.69651e6			−0.659016
$552$		0			0
$553$			3.73285e6i			0.519072i
$554$		0			0
$555$	3.23998e6	+	4.25410e6i	0.446488	+	0.586240i
$556$		0			0
$557$			2.55292e6i			0.348657i	0.984688	+	0.174329i	$0.0557755\pi$
$557$			2.55292e6i			0.348657i	−0.984688	+	0.174329i	$0.944224\pi$
$558$		0			0
$559$	365445.			0.0494643
$560$		0			0
$561$	3.04348e6			0.408285
$562$		0			0
$563$			1.60880e6i			0.213910i	0.994264	+	0.106955i	$0.0341101\pi$
$563$			1.60880e6i			0.213910i	−0.994264	+	0.106955i	$0.965890\pi$
$564$		0			0
$565$	4.18308e6	−	3.18589e6i	0.551284	−	0.419865i
$566$		0			0
$567$			563324.i			0.0735869i
$568$		0			0
$569$	−1.25953e7			−1.63091			−0.815453	−	0.578823i	$-0.803512\pi$
$569$	−1.25953e7			−1.63091			−0.815453	+	0.578823i	$0.803512\pi$
$570$		0			0
$571$	2.22289e6			0.285317			0.142659	−	0.989772i	$-0.454435\pi$
$571$	2.22289e6			0.285317			0.142659	+	0.989772i	$0.454435\pi$
$572$		0			0
$573$		−	4.52429e6i		−	0.575657i
$574$		0			0
$575$	−1.24252e7	−	3.42558e6i	−1.56723	−	0.432080i
$576$		0			0
$577$			8.71842e6i			1.09018i	0.838377	+	0.545090i	$0.183505\pi$
$577$			8.71842e6i			1.09018i	−0.838377	+	0.545090i	$0.816495\pi$
$578$		0			0
$579$	−6.58039e6			−0.815746
$580$		0			0
$581$	7.44401e6			0.914885
$582$		0			0
$583$		−	6.79322e6i		−	0.827760i
$584$		0			0
$585$	−139036.	+	105892.i	−0.0167973	+	0.0127930i
$586$		0			0
$587$		−	3.00682e6i		−	0.360174i	−0.983651	−	0.180087i	$-0.942362\pi$
$587$		−	3.00682e6i		−	0.360174i	0.983651	−	0.180087i	$-0.0576379\pi$
$588$		0			0
$589$	5.72429e6			0.679882
$590$		0			0
$591$	−7.84700e6			−0.924133
$592$		0			0
$593$		−	224622.i		−	0.0262311i	−0.999914	−	0.0131155i	$-0.995825\pi$
$593$		−	224622.i		−	0.0262311i	0.999914	−	0.0131155i	$-0.00417493\pi$
$594$		0			0
$595$	2.01485e6	+	2.64550e6i	0.233319	+	0.306349i
$596$		0			0
$597$		−	911795.i		−	0.104704i
$598$		0			0
$599$	1.14533e7			1.30426			0.652129	−	0.758108i	$-0.273876\pi$
$599$	1.14533e7			1.30426			0.652129	+	0.758108i	$0.273876\pi$
$600$		0			0
$601$	−1.17677e7			−1.32894			−0.664470	−	0.747315i	$-0.731343\pi$
$601$	−1.17677e7			−1.32894			−0.664470	+	0.747315i	$0.731343\pi$
$602$		0			0
$603$			114727.i			0.0128491i
$604$		0			0
$605$	2.61406e6	+	3.43227e6i	0.290353	+	0.381235i
$606$		0			0
$607$			1.24589e6i			0.137248i	0.997643	+	0.0686242i	$0.0218609\pi$
$607$			1.24589e6i			0.137248i	−0.997643	+	0.0686242i	$0.978139\pi$
$608$		0			0
$609$	1.45793e6			0.159292
$610$		0			0
$611$	−552838.			−0.0599094
$612$		0			0
$613$			1.45412e7i			1.56296i	0.623931	+	0.781480i	$0.285535\pi$
$613$			1.45412e7i			1.56296i	−0.623931	+	0.781480i	$0.714465\pi$
$614$		0			0
$615$	−6.11124e6	+	4.65440e6i	−0.651541	+	0.496221i
$616$		0			0
$617$		−	5.06609e6i		−	0.535747i	−0.963454	−	0.267874i	$-0.913679\pi$
$617$		−	5.06609e6i		−	0.535747i	0.963454	−	0.267874i	$-0.0863209\pi$
$618$		0			0
$619$	−1.03229e7			−1.08287			−0.541435	−	0.840742i	$-0.682119\pi$
$619$	−1.03229e7			−1.08287			−0.541435	+	0.840742i	$0.682119\pi$
$620$		0			0
$621$	−3.00669e6			−0.312867
$622$		0			0
$623$			8.65755e6i			0.893665i
$624$		0			0
$625$	−8.38595e6	−	5.00432e6i	−0.858722	−	0.512442i
$626$		0			0
$627$		−	1.09348e7i		−	1.11081i
$628$		0			0
$629$	7.36389e6			0.742132
$630$		0			0
$631$	1.27332e7			1.27310			0.636552	−	0.771234i	$-0.280360\pi$
$631$	1.27332e7			1.27310			0.636552	+	0.771234i	$0.280360\pi$
$632$		0			0
$633$			6.52638e6i			0.647386i
$634$		0			0
$635$	−5.26870e6	+	4.01270e6i	−0.518524	+	0.394914i
$636$		0			0
$637$		−	364169.i		−	0.0355594i
$638$		0			0
$639$	942602.			0.0913221
$640$		0			0
$641$	1.37535e7			1.32211			0.661057	−	0.750336i	$-0.270108\pi$
$641$	1.37535e7			1.32211			0.661057	+	0.750336i	$0.270108\pi$
$642$		0			0
$643$		−	1.62140e7i		−	1.54654i	−0.634075	−	0.773272i	$-0.718619\pi$
$643$		−	1.62140e7i		−	1.54654i	0.634075	−	0.773272i	$-0.281381\pi$
$644$		0			0
$645$	2.88624e6	+	3.78965e6i	0.273170	+	0.358674i
$646$		0			0
$647$			1.27009e7i			1.19282i	0.802680	+	0.596410i	$0.203407\pi$
$647$			1.27009e7i			1.19282i	−0.802680	+	0.596410i	$0.796593\pi$
$648$		0			0
$649$	1.93734e7			1.80548
$650$		0			0
$651$	−1.77698e6			−0.164335
$652$		0			0
$653$			1.73886e7i			1.59581i	0.602781	+	0.797907i	$0.294059\pi$
$653$			1.73886e7i			1.59581i	−0.602781	+	0.797907i	$0.705941\pi$
$654$		0			0
$655$	−7.89179e6	−	1.03620e7i	−0.718740	−	0.943709i
$656$		0			0
$657$		−	2.51765e6i		−	0.227552i
$658$		0			0
$659$	1.26691e7			1.13640			0.568200	−	0.822891i	$-0.307640\pi$
$659$	1.26691e7			1.13640			0.568200	+	0.822891i	$0.307640\pi$
$660$		0			0
$661$	6.69054e6			0.595604			0.297802	−	0.954628i	$-0.403746\pi$
$661$	6.69054e6			0.595604			0.297802	+	0.954628i	$0.403746\pi$
$662$		0			0
$663$			240673.i			0.0212639i
$664$		0			0
$665$	9.50489e6	−	7.23904e6i	0.833476	−	0.634785i
$666$		0			0
$667$			7.78157e6i			0.677255i
$668$		0			0
$669$	6.12561e6			0.529156
$670$		0			0
$671$	−1.86555e7			−1.59956
$672$		0			0
$673$		−	1.31246e7i		−	1.11699i	−0.829509	−	0.558494i	$-0.811379\pi$
$673$		−	1.31246e7i		−	1.11699i	0.829509	−	0.558494i	$-0.188621\pi$
$674$		0			0
$675$	−2.19619e6	−	605481.i	−0.185528	−	0.0511494i
$676$		0			0
$677$		−	6.74331e6i		−	0.565460i	−0.959200	−	0.282730i	$-0.908760\pi$
$677$		−	6.74331e6i		−	0.565460i	0.959200	−	0.282730i	$-0.0912400\pi$
$678$		0			0
$679$	3.42674e6			0.285237
$680$		0			0
$681$	−9.09027e6			−0.751119
$682$		0			0
$683$		−	1.57439e7i		−	1.29140i	−0.763593	−	0.645698i	$-0.776566\pi$
$683$		−	1.57439e7i		−	1.29140i	0.763593	−	0.645698i	$-0.223434\pi$
$684$		0			0
$685$	−6.20703e6	+	4.72735e6i	−0.505426	+	0.384939i
$686$		0			0
$687$		−	1.10466e7i		−	0.892972i
$688$		0			0
$689$	537196.			0.0431107
$690$		0			0
$691$	−1.89262e7			−1.50789			−0.753943	−	0.656940i	$-0.771850\pi$
$691$	−1.89262e7			−1.50789			−0.753943	+	0.656940i	$0.771850\pi$
$692$		0			0
$693$			3.39446e6i			0.268496i
$694$		0			0
$695$	8.93549e6	+	1.17323e7i	0.701708	+	0.921345i
$696$		0			0
$697$			1.05786e7i			0.824797i
$698$		0			0
$699$	5.39535e6			0.417664
$700$		0			0
$701$	1.21080e7			0.930628			0.465314	−	0.885146i	$-0.345941\pi$
$701$	1.21080e7			0.930628			0.465314	+	0.885146i	$0.345941\pi$
$702$		0			0
$703$		−	2.64573e7i		−	2.01910i
$704$		0			0
$705$	−4.36626e6	−	5.73292e6i	−0.330854	−	0.434413i
$706$		0			0
$707$		−	1.00582e7i		−	0.756782i
$708$		0			0
$709$	1.59350e6			0.119052			0.0595259	−	0.998227i	$-0.481041\pi$
$709$	1.59350e6			0.119052			0.0595259	+	0.998227i	$0.481041\pi$
$710$		0			0
$711$	−3.52158e6			−0.261254
$712$		0			0
$713$		−	9.48447e6i		−	0.698698i
$714$		0			0
$715$	−837800.	+	638079.i	−0.0612880	+	0.0466777i
$716$		0			0
$717$		−	1.23446e7i		−	0.896763i
$718$		0			0
$719$	1.65457e7			1.19361			0.596806	−	0.802386i	$-0.296436\pi$
$719$	1.65457e7			1.19361			0.596806	+	0.802386i	$0.296436\pi$
$720$		0			0
$721$	22838.6			0.00163618
$722$		0			0
$723$		−	1.23171e7i		−	0.876323i
$724$		0			0
$725$	−1.56703e6	+	5.68392e6i	−0.110722	+	0.401608i
$726$		0			0
$727$		−	1.71115e7i		−	1.20075i	−0.799720	−	0.600373i	$-0.795019\pi$
$727$		−	1.71115e7i		−	1.20075i	0.799720	−	0.600373i	$-0.204981\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$	6.55992e6			0.454051
$732$		0			0
$733$		−	1.21563e7i		−	0.835683i	−0.908520	−	0.417841i	$-0.862787\pi$
$733$		−	1.21563e7i		−	0.835683i	0.908520	−	0.417841i	$-0.137213\pi$
$734$		0			0
$735$	3.77642e6	−	2.87617e6i	0.257847	−	0.196379i
$736$		0			0
$737$			691317.i			0.0468823i
$738$		0			0
$739$	−981165.			−0.0660893			−0.0330446	−	0.999454i	$-0.510520\pi$
$739$	−981165.			−0.0660893			−0.0330446	+	0.999454i	$0.510520\pi$
$740$		0			0
$741$	864702.			0.0578523
$742$		0			0
$743$			2.75144e6i			0.182847i	0.995812	+	0.0914237i	$0.0291417\pi$
$743$			2.75144e6i			0.182847i	−0.995812	+	0.0914237i	$0.970858\pi$
$744$		0			0
$745$	−1.16573e7	−	1.53061e7i	−0.769498	−	1.01035i
$746$		0			0
$747$			7.02269e6i			0.460471i
$748$		0			0
$749$	1.03161e7			0.671912
$750$		0			0
$751$	2.03594e6			0.131724			0.0658622	−	0.997829i	$-0.479020\pi$
$751$	2.03594e6			0.131724			0.0658622	+	0.997829i	$0.479020\pi$
$752$		0			0
$753$		−	6.57263e6i		−	0.422427i
$754$		0			0
$755$	−416161.	−	546421.i	−0.0265702	−	0.0348867i
$756$		0			0
$757$			2.50989e7i			1.59189i	0.605366	+	0.795947i	$0.293027\pi$
$757$			2.50989e7i			1.59189i	−0.605366	+	0.795947i	$0.706973\pi$
$758$		0			0
$759$	−1.81176e7			−1.14155
$760$		0			0
$761$	715007.			0.0447557			0.0223779	−	0.999750i	$-0.492876\pi$
$761$	715007.			0.0447557			0.0223779	+	0.999750i	$0.492876\pi$
$762$		0			0
$763$			1.16397e7i			0.723819i
$764$		0			0
$765$	−2.49577e6	+	1.90081e6i	−0.154188	+	0.117432i
$766$		0			0
$767$			1.53201e6i			0.0940317i
$768$		0			0
$769$	2.97361e6			0.181329			0.0906646	−	0.995881i	$-0.471101\pi$
$769$	2.97361e6			0.181329			0.0906646	+	0.995881i	$0.471101\pi$
$770$		0			0
$771$	1.23149e7			0.746095
$772$		0			0
$773$			2.89819e7i			1.74453i	0.489035	+	0.872264i	$0.337349\pi$
$773$			2.89819e7i			1.74453i	−0.489035	+	0.872264i	$0.662651\pi$
$774$		0			0
$775$	1.90996e6	−	6.92778e6i	0.114227	−	0.414324i
$776$		0			0
$777$			8.21310e6i			0.488039i
$778$		0			0
$779$	3.80073e7			2.24400
$780$		0			0
$781$	5.67989e6			0.333206
$782$		0			0
$783$			1.37541e6i			0.0801731i
$784$		0			0
$785$	−1.07047e7	+	8.15280e6i	−0.620011	+	0.472208i
$786$		0			0
$787$			3.25612e7i			1.87397i	0.349364	+	0.936987i	$0.386398\pi$
$787$			3.25612e7i			1.87397i	−0.349364	+	0.936987i	$0.613602\pi$
$788$		0			0
$789$	1.13638e7			0.649879
$790$		0			0
$791$	8.07598e6			0.458938
$792$		0			0
$793$		−	1.47524e6i		−	0.0833067i
$794$		0			0
$795$	4.24272e6	+	5.57071e6i	0.238082	+	0.312603i
$796$		0			0
$797$		−	1.77148e7i		−	0.987848i	−0.869505	−	0.493924i	$-0.835562\pi$
$797$		−	1.77148e7i		−	0.987848i	0.869505	−	0.493924i	$-0.164438\pi$
$798$		0			0
$799$	−9.92373e6			−0.549931
$800$		0			0
$801$	−8.16755e6			−0.449791
$802$		0			0
$803$		−	1.51707e7i		−	0.830268i
$804$		0			0
$805$	−1.19942e7	−	1.57485e7i	−0.652353	−	0.856543i
$806$		0			0
$807$			5.87606e6i			0.317616i
$808$		0			0
$809$	2.42153e6			0.130083			0.0650413	−	0.997883i	$-0.479282\pi$
$809$	2.42153e6			0.130083			0.0650413	+	0.997883i	$0.479282\pi$
$810$		0			0
$811$	−6.47458e6			−0.345668			−0.172834	−	0.984951i	$-0.555292\pi$
$811$	−6.47458e6			−0.345668			−0.172834	+	0.984951i	$0.555292\pi$
$812$		0			0
$813$			5.84164e6i			0.309962i
$814$		0			0
$815$	−2.51428e7	+	1.91491e7i	−1.32593	+	1.00984i
$816$		0			0
$817$		−	2.35688e7i		−	1.23533i
$818$		0			0
$819$	−268428.			−0.0139836
$820$		0			0
$821$	−5.71963e6			−0.296149			−0.148074	−	0.988976i	$-0.547308\pi$
$821$	−5.71963e6			−0.296149			−0.148074	+	0.988976i	$0.547308\pi$
$822$		0			0
$823$			1.45747e7i			0.750066i	0.927011	+	0.375033i	$0.122369\pi$
$823$			1.45747e7i			0.750066i	−0.927011	+	0.375033i	$0.877631\pi$
$824$		0			0
$825$	−1.32337e7	−	3.64848e6i	−0.676935	−	0.186628i
$826$		0			0
$827$			5.71684e6i			0.290664i	0.989383	+	0.145332i	$0.0464251\pi$
$827$			5.71684e6i			0.290664i	−0.989383	+	0.145332i	$0.953575\pi$
$828$		0			0
$829$	3.16906e7			1.60156			0.800781	−	0.598958i	$-0.204418\pi$
$829$	3.16906e7			1.60156			0.800781	+	0.598958i	$0.204418\pi$
$830$		0			0
$831$	−6.01650e6			−0.302233
$832$		0			0
$833$		−	6.53702e6i		−	0.326413i
$834$		0			0
$835$	−1.82798e7	+	1.39221e7i	−0.907309	+	0.691017i
$836$		0			0
$837$		−	1.67641e6i		−	0.0827115i
$838$		0			0
$839$	1.30388e7			0.639487			0.319744	−	0.947504i	$-0.396403\pi$
$839$	1.30388e7			0.639487			0.319744	+	0.947504i	$0.396403\pi$
$840$		0			0
$841$	−1.69515e7			−0.826451
$842$		0			0
$843$		−	9.29910e6i		−	0.450684i
$844$		0			0
$845$	1.25255e7	+	1.64460e7i	0.603465	+	0.792352i
$846$		0			0
$847$			6.62644e6i			0.317374i
$848$		0			0
$849$	−3.46399e6			−0.164933
$850$		0			0
$851$	−4.38367e7			−2.07498
$852$		0			0
$853$		−	1.19404e6i		−	0.0561883i	−0.999605	−	0.0280941i	$-0.991056\pi$
$853$		−	1.19404e6i		−	0.0561883i	0.999605	−	0.0280941i	$-0.00894382\pi$
$854$		0			0
$855$	6.82932e6	+	8.96693e6i	0.319494	+	0.419497i
$856$		0			0
$857$			1.25697e7i			0.584617i	0.956324	+	0.292309i	$0.0944234\pi$
$857$			1.25697e7i			0.584617i	−0.956324	+	0.292309i	$0.905577\pi$
$858$		0			0
$859$	8.48043e6			0.392135			0.196067	−	0.980590i	$-0.437183\pi$
$859$	8.48043e6			0.392135			0.196067	+	0.980590i	$0.437183\pi$
$860$		0			0
$861$	−1.17985e7			−0.542401
$862$		0			0
$863$		−	1.53122e6i		−	0.0699859i	−0.999388	−	0.0349929i	$-0.988859\pi$
$863$		−	1.53122e6i		−	0.0699859i	0.999388	−	0.0349929i	$-0.0111409\pi$
$864$		0			0
$865$	8.39263e6	−	6.39193e6i	0.381380	−	0.290464i
$866$		0			0
$867$		−	8.45850e6i		−	0.382161i
$868$		0			0
$869$	−2.12202e7			−0.953235
$870$		0			0
$871$	−54668.1			−0.00244168
$872$		0			0
$873$			3.23279e6i			0.143563i
$874$		0			0
$875$	−5.58960e6	−	1.39186e7i	−0.246809	−	0.614576i
$876$		0			0
$877$		−	3.58663e7i		−	1.57466i	−0.616530	−	0.787331i	$-0.711462\pi$
$877$		−	3.58663e7i		−	1.57466i	0.616530	−	0.787331i	$-0.288538\pi$
$878$		0			0
$879$	672777.			0.0293697
$880$		0			0
$881$	−2.74419e7			−1.19117			−0.595586	−	0.803291i	$-0.703080\pi$
$881$	−2.74419e7			−1.19117			−0.595586	+	0.803291i	$0.703080\pi$
$882$		0			0
$883$			1.71743e7i			0.741273i	0.928778	+	0.370637i	$0.120861\pi$
$883$			1.71743e7i			0.741273i	−0.928778	+	0.370637i	$0.879139\pi$
$884$		0			0
$885$	−1.58869e7	+	1.20997e7i	−0.681839	+	0.519297i
$886$		0			0
$887$			9.60304e6i			0.409826i	0.978780	+	0.204913i	$0.0656912\pi$
$887$			9.60304e6i			0.409826i	−0.978780	+	0.204913i	$0.934309\pi$
$888$		0			0
$889$	−1.01719e7			−0.431666
$890$		0			0
$891$	−3.20234e6			−0.135137
$892$		0			0
$893$			3.56544e7i			1.49618i
$894$		0			0
$895$	2.03892e7	+	2.67711e7i	0.850831	+	1.11714i
$896$		0			0
$897$		−	1.43271e6i		−	0.0594534i
$898$		0			0
$899$	−4.33868e6			−0.179043
$900$		0			0
$901$	9.64295e6			0.395729
$902$		0			0
$903$			7.31641e6i			0.298592i
$904$		0			0
$905$	−2.83169e7	−	3.71802e7i	−1.14928	−	1.50900i
$906$		0			0
$907$			4.28904e7i			1.73118i	0.500754	+	0.865590i	$0.333056\pi$
$907$			4.28904e7i			1.73118i	−0.500754	+	0.865590i	$0.666944\pi$
$908$		0			0
$909$	9.48890e6			0.380896
$910$		0			0
$911$	2.43996e7			0.974061			0.487031	−	0.873385i	$-0.338080\pi$
$911$	2.43996e7			0.974061			0.487031	+	0.873385i	$0.338080\pi$
$912$		0			0
$913$			4.23171e7i			1.68011i
$914$		0			0
$915$	1.52982e7	−	1.16513e7i	0.604071	−	0.460068i
$916$		0			0
$917$		−	2.00051e7i		−	0.785628i
$918$		0			0
$919$	4.90096e7			1.91422			0.957112	−	0.289719i	$-0.0935617\pi$
$919$	4.90096e7			1.91422			0.957112	+	0.289719i	$0.0935617\pi$
$920$		0			0
$921$	1.77290e7			0.688707
$922$		0			0
$923$			449156.i			0.0173537i
$924$		0			0
$925$	−3.20198e7	−	8.82774e6i	−1.23045	−	0.339231i
$926$		0			0
$927$			21546.0i			0.000823506i
$928$		0			0
$929$	−4.34428e7			−1.65150			−0.825750	−	0.564036i	$-0.809248\pi$
$929$	−4.34428e7			−1.65150			−0.825750	+	0.564036i	$0.809248\pi$
$930$		0			0
$931$	−2.34865e7			−0.888064
$932$		0			0
$933$			1.73313e7i			0.651820i
$934$		0			0
$935$	−1.50389e7	+	1.14538e7i	−0.562585	+	0.428472i
$936$		0			0
$937$			2.45065e6i			0.0911868i	0.998960	+	0.0455934i	$0.0145179\pi$
$937$			2.45065e6i			0.0911868i	−0.998960	+	0.0455934i	$0.985482\pi$
$938$		0			0
$939$	9.46141e6			0.350181
$940$		0			0
$941$	9.14331e6			0.336612			0.168306	−	0.985735i	$-0.446170\pi$
$941$	9.14331e6			0.336612			0.168306	+	0.985735i	$0.446170\pi$
$942$		0			0
$943$		−	6.29737e7i		−	2.30611i
$944$		0			0
$945$	−2.12001e6	−	2.78359e6i	−0.0772252	−	0.101397i
$946$		0			0
$947$		−	1.87759e7i		−	0.680341i	−0.940364	−	0.340171i	$-0.889515\pi$
$947$		−	1.87759e7i		−	0.680341i	0.940364	−	0.340171i	$-0.110485\pi$
$948$		0			0
$949$	1.19968e6			0.0432413
$950$		0			0
$951$	2.41622e7			0.866334
$952$		0			0
$953$			5.76978e6i			0.205791i	0.994692	+	0.102896i	$0.0328108\pi$
$953$			5.76978e6i			0.205791i	−0.994692	+	0.102896i	$0.967189\pi$
$954$		0			0
$955$	1.70267e7	+	2.23561e7i	0.604118	+	0.793210i
$956$		0			0
$957$			8.28791e6i			0.292527i
$958$		0			0
$959$	−1.19835e7			−0.420762
$960$		0			0
$961$	−2.33410e7			−0.815288
$962$		0			0
$963$			9.73227e6i			0.338180i
$964$		0			0
$965$	3.25161e7	−	2.47646e7i	1.12403	−	0.856078i
$966$		0			0
$967$		−	2.51533e6i		−	0.0865024i	−0.999064	−	0.0432512i	$-0.986228\pi$
$967$		−	2.51533e6i		−	0.0865024i	0.999064	−	0.0432512i	$-0.0137716\pi$
$968$		0			0
$969$	1.55218e7			0.531048
$970$		0			0
$971$	3.98957e7			1.35793			0.678966	−	0.734170i	$-0.262428\pi$
$971$	3.98957e7			1.35793			0.678966	+	0.734170i	$0.262428\pi$
$972$		0			0
$973$			2.26508e7i			0.767011i
$974$		0			0
$975$	288516.	−	1.04650e6i	0.00971981	−	0.0352555i
$976$		0			0
$977$			5.40009e7i			1.80994i	0.425473	+	0.904971i	$0.360108\pi$
$977$			5.40009e7i			1.80994i	−0.425473	+	0.904971i	$0.639892\pi$
$978$		0			0
$979$	−4.92157e7			−1.64115
$980$		0			0
$981$	−1.09809e7			−0.364306
$982$		0			0
$983$		−	3.78231e7i		−	1.24846i	−0.781242	−	0.624228i	$-0.785414\pi$
$983$		−	3.78231e7i		−	1.24846i	0.781242	−	0.624228i	$-0.214586\pi$
$984$		0			0
$985$	3.87748e7	−	2.95314e7i	1.27338	−	0.969825i
$986$		0			0
$987$		−	1.10681e7i		−	0.361644i
$988$		0			0
$989$	−3.90507e7			−1.26952
$990$		0			0
$991$	5.61819e6			0.181724			0.0908620	−	0.995863i	$-0.471038\pi$
$991$	5.61819e6			0.181724			0.0908620	+	0.995863i	$0.471038\pi$
$992$		0			0
$993$			1.35410e7i			0.435792i
$994$		0			0
$995$	3.43145e6	+	4.50551e6i	0.109880	+	0.144273i
$996$		0			0
$997$		−	4.35983e7i		−	1.38910i	−0.719447	−	0.694548i	$-0.755605\pi$
$997$		−	4.35983e7i		−	1.38910i	0.719447	−	0.694548i	$-0.244395\pi$
$998$		0			0
$999$	−7.74826e6			−0.245635

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.6.f.a.49.2	✓	6
3.2	odd	2		360.6.f.a.289.3		6
4.3	odd	2		240.6.f.e.49.5		6
5.2	odd	4		600.6.a.r.1.1		3
5.3	odd	4		600.6.a.s.1.3		3
5.4	even	2	inner	120.6.f.a.49.5	yes	6
12.11	even	2		720.6.f.l.289.3		6
15.14	odd	2		360.6.f.a.289.4		6
20.19	odd	2		240.6.f.e.49.2		6
60.59	even	2		720.6.f.l.289.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.f.a.49.2	✓	6	1.1	even	1	trivial
120.6.f.a.49.5	yes	6	5.4	even	2	inner
240.6.f.e.49.2		6	20.19	odd	2
240.6.f.e.49.5		6	4.3	odd	2
360.6.f.a.289.3		6	3.2	odd	2
360.6.f.a.289.4		6	15.14	odd	2
600.6.a.r.1.1		3	5.2	odd	4
600.6.a.s.1.3		3	5.3	odd	4
720.6.f.l.289.3		6	12.11	even	2
720.6.f.l.289.4		6	60.59	even	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	120.6.f.a.49.2

Level	$120$
Weight	$6$
Character	120.49
Analytic conductor	$19.246$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$