LMFDB - Embedded newform 1710.2.c.c.1709.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,2,Mod(1709,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1710.1709");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1710.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:	$13.6544187456$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1709.6
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1710.1709
Dual form			1710.2.c.c.1709.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-1.73205 + 1.41421i) q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +(-1.73205 + 1.41421i) q^{5} -1.00000i q^{8} +(-1.41421 - 1.73205i) q^{10} -2.82843i q^{11} +1.00000 q^{16} +(4.00000 - 1.73205i) q^{19} +(1.73205 - 1.41421i) q^{20} +2.82843 q^{22} -3.46410 q^{23} +(1.00000 - 4.89898i) q^{25} -2.44949 q^{29} -3.46410i q^{31} +1.00000i q^{32} +8.48528 q^{37} +(1.73205 + 4.00000i) q^{38} +(1.41421 + 1.73205i) q^{40} +9.79796 q^{41} -2.44949i q^{43} +2.82843i q^{44} -3.46410i q^{46} -10.3923 q^{47} +7.00000 q^{49} +(4.89898 + 1.00000i) q^{50} +6.00000i q^{53} +(4.00000 + 4.89898i) q^{55} -2.44949i q^{58} +12.2474 q^{59} -2.00000 q^{61} +3.46410 q^{62} -1.00000 q^{64} +8.48528 q^{67} +4.89898 q^{71} -4.89898i q^{73} +8.48528i q^{74} +(-4.00000 + 1.73205i) q^{76} +10.3923i q^{79} +(-1.73205 + 1.41421i) q^{80} +9.79796i q^{82} -10.3923 q^{83} +2.44949 q^{86} -2.82843 q^{88} +9.79796 q^{89} +3.46410 q^{92} -10.3923i q^{94} +(-4.47871 + 8.65685i) q^{95} +4.24264 q^{97} +7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} + 32 q^{19} + 8 q^{25} + 56 q^{49} + 32 q^{55} - 16 q^{61} - 8 q^{64} - 32 q^{76}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 + 32 * q^19 + 8 * q^25 + 56 * q^49 + 32 * q^55 - 16 * q^61 - 8 * q^64 - 32 * q^76

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$5$	−1.73205	+	1.41421i	−0.774597	+	0.632456i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.41421	−	1.73205i	−0.447214	−	0.547723i
$11$		−	2.82843i		−	0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$		−	2.82843i		−	0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$	1.73205	−	1.41421i	0.387298	−	0.316228i
$21$		0			0
$22$	2.82843			0.603023
$23$	−3.46410			−0.722315			−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410			−0.722315			−0.361158	+	0.932505i	$0.617618\pi$
$24$		0			0
$25$	1.00000	−	4.89898i	0.200000	−	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.44949			−0.454859			−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949			−0.454859			−0.227429	+	0.973795i	$0.573032\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.48528			1.39497			0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528			1.39497			0.697486	+	0.716599i	$0.254302\pi$
$38$	1.73205	+	4.00000i	0.280976	+	0.648886i
$39$		0			0
$40$	1.41421	+	1.73205i	0.223607	+	0.273861i
$41$	9.79796			1.53018			0.765092	−	0.643921i	$-0.222693\pi$
$41$	9.79796			1.53018			0.765092	+	0.643921i	$0.222693\pi$
$42$		0			0
$43$		−	2.44949i		−	0.373544i	−0.982403	−	0.186772i	$-0.940197\pi$
$43$		−	2.44949i		−	0.373544i	0.982403	−	0.186772i	$-0.0598025\pi$
$44$			2.82843i			0.426401i
$45$		0			0
$46$		−	3.46410i		−	0.510754i
$47$	−10.3923			−1.51587			−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923			−1.51587			−0.757937	+	0.652328i	$0.773792\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	4.89898	+	1.00000i	0.692820	+	0.141421i
$51$		0			0
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	4.00000	+	4.89898i	0.539360	+	0.660578i
$56$		0			0
$57$		0			0
$58$		−	2.44949i		−	0.321634i
$59$	12.2474			1.59448			0.797241	−	0.603661i	$-0.206292\pi$
$59$	12.2474			1.59448			0.797241	+	0.603661i	$0.206292\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.46410			0.439941
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	8.48528			1.03664			0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528			1.03664			0.518321	+	0.855186i	$0.326557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.89898			0.581402			0.290701	−	0.956814i	$-0.406112\pi$
$71$	4.89898			0.581402			0.290701	+	0.956814i	$0.406112\pi$
$72$		0			0
$73$		−	4.89898i		−	0.573382i	−0.958023	−	0.286691i	$-0.907445\pi$
$73$		−	4.89898i		−	0.573382i	0.958023	−	0.286691i	$-0.0925553\pi$
$74$			8.48528i			0.986394i
$75$		0			0
$76$	−4.00000	+	1.73205i	−0.458831	+	0.198680i
$77$		0			0
$78$		0			0
$79$			10.3923i			1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$			10.3923i			1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$	−1.73205	+	1.41421i	−0.193649	+	0.158114i
$81$		0			0
$82$			9.79796i			1.08200i
$83$	−10.3923			−1.14070			−0.570352	−	0.821401i	$-0.693193\pi$
$83$	−10.3923			−1.14070			−0.570352	+	0.821401i	$0.693193\pi$
$84$		0			0
$85$		0			0
$86$	2.44949			0.264135
$87$		0			0
$88$	−2.82843			−0.301511
$89$	9.79796			1.03858			0.519291	−	0.854598i	$-0.326196\pi$
$89$	9.79796			1.03858			0.519291	+	0.854598i	$0.326196\pi$
$90$		0			0
$91$		0			0
$92$	3.46410			0.361158
$93$		0			0
$94$		−	10.3923i		−	1.07188i
$95$	−4.47871	+	8.65685i	−0.459506	+	0.888175i
$96$		0			0
$97$	4.24264			0.430775			0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264			0.430775			0.215387	+	0.976529i	$0.430899\pi$
$98$			7.00000i			0.707107i
$99$		0			0
$100$	−1.00000	+	4.89898i	−0.100000	+	0.489898i
$101$		−	2.82843i		−	0.281439i	−0.990050	−	0.140720i	$-0.955058\pi$
$101$		−	2.82843i		−	0.281439i	0.990050	−	0.140720i	$-0.0449416\pi$
$102$		0			0
$103$	−4.24264			−0.418040			−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264			−0.418040			−0.209020	+	0.977911i	$0.567027\pi$
$104$		0			0
$105$		0			0
$106$	−6.00000			−0.582772
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		−	13.8564i		−	1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$		−	13.8564i		−	1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$	−4.89898	+	4.00000i	−0.467099	+	0.381385i
$111$		0			0
$112$		0			0
$113$			12.0000i			1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$			12.0000i			1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$		0			0
$115$	6.00000	−	4.89898i	0.559503	−	0.456832i
$116$	2.44949			0.227429
$117$		0			0
$118$			12.2474i			1.12747i
$119$		0			0
$120$		0			0
$121$	3.00000			0.272727
$122$		−	2.00000i		−	0.181071i
$123$		0			0
$124$			3.46410i			0.311086i
$125$	5.19615	+	9.89949i	0.464758	+	0.885438i
$126$		0			0
$127$	12.7279			1.12942			0.564710	−	0.825289i	$-0.308988\pi$
$127$	12.7279			1.12942			0.564710	+	0.825289i	$0.308988\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		0			0
$131$		−	5.65685i		−	0.494242i	−0.968985	−	0.247121i	$-0.920516\pi$
$131$		−	5.65685i		−	0.494242i	0.968985	−	0.247121i	$-0.0794845\pi$
$132$		0			0
$133$		0			0
$134$			8.48528i			0.733017i
$135$		0			0
$136$		0			0
$137$	20.7846			1.77575			0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846			1.77575			0.887875	+	0.460086i	$0.152181\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$			4.89898i			0.411113i
$143$		0			0
$144$		0			0
$145$	4.24264	−	3.46410i	0.352332	−	0.287678i
$146$	4.89898			0.405442
$147$		0			0
$148$	−8.48528			−0.697486
$149$		−	2.82843i		−	0.231714i	−0.993266	−	0.115857i	$-0.963039\pi$
$149$		−	2.82843i		−	0.231714i	0.993266	−	0.115857i	$-0.0369614\pi$
$150$		0			0
$151$		−	17.3205i		−	1.40952i	−0.709444	−	0.704761i	$-0.751054\pi$
$151$		−	17.3205i		−	1.40952i	0.709444	−	0.704761i	$-0.248946\pi$
$152$	−1.73205	−	4.00000i	−0.140488	−	0.324443i
$153$		0			0
$154$		0			0
$155$	4.89898	+	6.00000i	0.393496	+	0.481932i
$156$		0			0
$157$			7.34847i			0.586472i	0.956040	+	0.293236i	$0.0947321\pi$
$157$			7.34847i			0.586472i	−0.956040	+	0.293236i	$0.905268\pi$
$158$	−10.3923			−0.826767
$159$		0			0
$160$	−1.41421	−	1.73205i	−0.111803	−	0.136931i
$161$		0			0
$162$		0			0
$163$		−	17.1464i		−	1.34301i	−0.740999	−	0.671506i	$-0.765648\pi$
$163$		−	17.1464i		−	1.34301i	0.740999	−	0.671506i	$-0.234352\pi$
$164$	−9.79796			−0.765092
$165$		0			0
$166$		−	10.3923i		−	0.806599i
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$			2.44949i			0.186772i
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$		0			0
$176$		−	2.82843i		−	0.213201i
$177$		0			0
$178$			9.79796i			0.734388i
$179$	−17.1464			−1.28158			−0.640792	−	0.767714i	$-0.721394\pi$
$179$	−17.1464			−1.28158			−0.640792	+	0.767714i	$0.721394\pi$
$180$		0			0
$181$		−	3.46410i		−	0.257485i	−0.991678	−	0.128742i	$-0.958906\pi$
$181$		−	3.46410i		−	0.257485i	0.991678	−	0.128742i	$-0.0410940\pi$
$182$		0			0
$183$		0			0
$184$			3.46410i			0.255377i
$185$	−14.6969	+	12.0000i	−1.08054	+	0.882258i
$186$		0			0
$187$		0			0
$188$	10.3923			0.757937
$189$		0			0
$190$	−8.65685	−	4.47871i	−0.628034	−	0.324920i
$191$		−	9.89949i		−	0.716302i	−0.933664	−	0.358151i	$-0.883407\pi$
$191$		−	9.89949i		−	0.716302i	0.933664	−	0.358151i	$-0.116593\pi$
$192$		0			0
$193$	−4.24264			−0.305392			−0.152696	−	0.988273i	$-0.548796\pi$
$193$	−4.24264			−0.305392			−0.152696	+	0.988273i	$0.548796\pi$
$194$			4.24264i			0.304604i
$195$		0			0
$196$	−7.00000			−0.500000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$	−4.89898	−	1.00000i	−0.346410	−	0.0707107i
$201$		0			0
$202$	2.82843			0.199007
$203$		0			0
$204$		0			0
$205$	−16.9706	+	13.8564i	−1.18528	+	0.967773i
$206$		−	4.24264i		−	0.295599i
$207$		0			0
$208$		0			0
$209$	−4.89898	−	11.3137i	−0.338869	−	0.782586i
$210$		0			0
$211$			13.8564i			0.953914i	0.878927	+	0.476957i	$0.158260\pi$
$211$			13.8564i			0.953914i	−0.878927	+	0.476957i	$0.841740\pi$
$212$		−	6.00000i		−	0.412082i
$213$		0			0
$214$		0			0
$215$	3.46410	+	4.24264i	0.236250	+	0.289346i
$216$		0			0
$217$		0			0
$218$	13.8564			0.938474
$219$		0			0
$220$	−4.00000	−	4.89898i	−0.269680	−	0.330289i
$221$		0			0
$222$		0			0
$223$	−4.24264			−0.284108			−0.142054	−	0.989859i	$-0.545371\pi$
$223$	−4.24264			−0.284108			−0.142054	+	0.989859i	$0.545371\pi$
$224$		0			0
$225$		0			0
$226$	−12.0000			−0.798228
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$	4.89898	+	6.00000i	0.323029	+	0.395628i
$231$		0			0
$232$			2.44949i			0.160817i
$233$	6.92820			0.453882			0.226941	−	0.973909i	$-0.427128\pi$
$233$	6.92820			0.453882			0.226941	+	0.973909i	$0.427128\pi$
$234$		0			0
$235$	18.0000	−	14.6969i	1.17419	−	0.958723i
$236$	−12.2474			−0.797241
$237$		0			0
$238$		0			0
$239$		−	9.89949i		−	0.640345i	−0.947359	−	0.320173i	$-0.896259\pi$
$239$		−	9.89949i		−	0.640345i	0.947359	−	0.320173i	$-0.103741\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$			3.00000i			0.192847i
$243$		0			0
$244$	2.00000			0.128037
$245$	−12.1244	+	9.89949i	−0.774597	+	0.632456i
$246$		0			0
$247$		0			0
$248$	−3.46410			−0.219971
$249$		0			0
$250$	−9.89949	+	5.19615i	−0.626099	+	0.328634i
$251$			11.3137i			0.714115i	0.934082	+	0.357057i	$0.116220\pi$
$251$			11.3137i			0.714115i	−0.934082	+	0.357057i	$0.883780\pi$
$252$		0			0
$253$			9.79796i			0.615992i
$254$			12.7279i			0.798621i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	24.0000i		−	1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$		−	24.0000i		−	1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	5.65685			0.349482
$263$	−3.46410			−0.213606			−0.106803	−	0.994280i	$-0.534061\pi$
$263$	−3.46410			−0.213606			−0.106803	+	0.994280i	$0.534061\pi$
$264$		0			0
$265$	−8.48528	−	10.3923i	−0.521247	−	0.638394i
$266$		0			0
$267$		0			0
$268$	−8.48528			−0.518321
$269$	12.2474			0.746740			0.373370	−	0.927682i	$-0.378202\pi$
$269$	12.2474			0.746740			0.373370	+	0.927682i	$0.378202\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$		0			0
$273$		0			0
$274$			20.7846i			1.25564i
$275$	−13.8564	−	2.82843i	−0.835573	−	0.170561i
$276$		0			0
$277$			7.34847i			0.441527i	0.975327	+	0.220763i	$0.0708548\pi$
$277$			7.34847i			0.441527i	−0.975327	+	0.220763i	$0.929145\pi$
$278$			4.00000i			0.239904i
$279$		0			0
$280$		0			0
$281$	−14.6969			−0.876746			−0.438373	−	0.898793i	$-0.644445\pi$
$281$	−14.6969			−0.876746			−0.438373	+	0.898793i	$0.644445\pi$
$282$		0			0
$283$			7.34847i			0.436821i	0.975857	+	0.218411i	$0.0700872\pi$
$283$			7.34847i			0.436821i	−0.975857	+	0.218411i	$0.929913\pi$
$284$	−4.89898			−0.290701
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$	3.46410	+	4.24264i	0.203419	+	0.249136i
$291$		0			0
$292$			4.89898i			0.286691i
$293$		−	30.0000i		−	1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$		−	30.0000i		−	1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$		0			0
$295$	−21.2132	+	17.3205i	−1.23508	+	1.00844i
$296$		−	8.48528i		−	0.493197i
$297$		0			0
$298$	2.82843			0.163846
$299$		0			0
$300$		0			0
$301$		0			0
$302$	17.3205			0.996683
$303$		0			0
$304$	4.00000	−	1.73205i	0.229416	−	0.0993399i
$305$	3.46410	−	2.82843i	0.198354	−	0.161955i
$306$		0			0
$307$	25.4558			1.45284			0.726421	−	0.687250i	$-0.241182\pi$
$307$	25.4558			1.45284			0.726421	+	0.687250i	$0.241182\pi$
$308$		0			0
$309$		0			0
$310$	−6.00000	+	4.89898i	−0.340777	+	0.278243i
$311$		−	24.0416i		−	1.36328i	−0.731690	−	0.681638i	$-0.761268\pi$
$311$		−	24.0416i		−	1.36328i	0.731690	−	0.681638i	$-0.238732\pi$
$312$		0			0
$313$		−	9.79796i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$		−	9.79796i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$	−7.34847			−0.414698
$315$		0			0
$316$		−	10.3923i		−	0.584613i
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$			6.92820i			0.387905i
$320$	1.73205	−	1.41421i	0.0968246	−	0.0790569i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	17.1464			0.949653
$327$		0			0
$328$		−	9.79796i		−	0.541002i
$329$		0			0
$330$		0			0
$331$			34.6410i			1.90404i	0.306032	+	0.952021i	$0.400999\pi$
$331$			34.6410i			1.90404i	−0.306032	+	0.952021i	$0.599001\pi$
$332$	10.3923			0.570352
$333$		0			0
$334$	18.0000			0.984916
$335$	−14.6969	+	12.0000i	−0.802980	+	0.655630i
$336$		0			0
$337$	21.2132			1.15556			0.577778	−	0.816194i	$-0.303920\pi$
$337$	21.2132			1.15556			0.577778	+	0.816194i	$0.303920\pi$
$338$		−	13.0000i		−	0.707107i
$339$		0			0
$340$		0			0
$341$	−9.79796			−0.530589
$342$		0			0
$343$		0			0
$344$	−2.44949			−0.132068
$345$		0			0
$346$	−6.00000			−0.322562
$347$	6.92820			0.371925			0.185963	−	0.982557i	$-0.440460\pi$
$347$	6.92820			0.371925			0.185963	+	0.982557i	$0.440460\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$	2.82843			0.150756
$353$	−6.92820			−0.368751			−0.184376	−	0.982856i	$-0.559026\pi$
$353$	−6.92820			−0.368751			−0.184376	+	0.982856i	$0.559026\pi$
$354$		0			0
$355$	−8.48528	+	6.92820i	−0.450352	+	0.367711i
$356$	−9.79796			−0.519291
$357$		0			0
$358$		−	17.1464i		−	0.906217i
$359$		−	7.07107i		−	0.373197i	−0.982436	−	0.186598i	$-0.940254\pi$
$359$		−	7.07107i		−	0.373197i	0.982436	−	0.186598i	$-0.0597463\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$	3.46410			0.182069
$363$		0			0
$364$		0			0
$365$	6.92820	+	8.48528i	0.362639	+	0.444140i
$366$		0			0
$367$		−	24.4949i		−	1.27862i	−0.768948	−	0.639312i	$-0.779219\pi$
$367$		−	24.4949i		−	1.27862i	0.768948	−	0.639312i	$-0.220781\pi$
$368$	−3.46410			−0.180579
$369$		0			0
$370$	−12.0000	−	14.6969i	−0.623850	−	0.764057i
$371$		0			0
$372$		0			0
$373$	−33.9411			−1.75740			−0.878702	−	0.477370i	$-0.841590\pi$
$373$	−33.9411			−1.75740			−0.878702	+	0.477370i	$0.841590\pi$
$374$		0			0
$375$		0			0
$376$			10.3923i			0.535942i
$377$		0			0
$378$		0			0
$379$		−	10.3923i		−	0.533817i	−0.963722	−	0.266908i	$-0.913998\pi$
$379$		−	10.3923i		−	0.533817i	0.963722	−	0.266908i	$-0.0860021\pi$
$380$	4.47871	−	8.65685i	0.229753	−	0.444087i
$381$		0			0
$382$	9.89949			0.506502
$383$			6.00000i			0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$			6.00000i			0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$		0			0
$385$		0			0
$386$		−	4.24264i		−	0.215945i
$387$		0			0
$388$	−4.24264			−0.215387
$389$		−	5.65685i		−	0.286814i	−0.989664	−	0.143407i	$-0.954194\pi$
$389$		−	5.65685i		−	0.286814i	0.989664	−	0.143407i	$-0.0458058\pi$
$390$		0			0
$391$		0			0
$392$		−	7.00000i		−	0.353553i
$393$		0			0
$394$		0			0
$395$	−14.6969	−	18.0000i	−0.739483	−	0.905678i
$396$		0			0
$397$			31.8434i			1.59817i	0.601216	+	0.799086i	$0.294683\pi$
$397$			31.8434i			1.59817i	−0.601216	+	0.799086i	$0.705317\pi$
$398$			14.0000i			0.701757i
$399$		0			0
$400$	1.00000	−	4.89898i	0.0500000	−	0.244949i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$			2.82843i			0.140720i
$405$		0			0
$406$		0			0
$407$		−	24.0000i		−	1.18964i
$408$		0			0
$409$			20.7846i			1.02773i	0.857870	+	0.513866i	$0.171787\pi$
$409$			20.7846i			1.02773i	−0.857870	+	0.513866i	$0.828213\pi$
$410$	−13.8564	−	16.9706i	−0.684319	−	0.838116i
$411$		0			0
$412$	4.24264			0.209020
$413$		0			0
$414$		0			0
$415$	18.0000	−	14.6969i	0.883585	−	0.721444i
$416$		0			0
$417$		0			0
$418$	11.3137	−	4.89898i	0.553372	−	0.239617i
$419$			2.82843i			0.138178i	0.997611	+	0.0690889i	$0.0220092\pi$
$419$			2.82843i			0.138178i	−0.997611	+	0.0690889i	$0.977991\pi$
$420$		0			0
$421$			3.46410i			0.168830i	0.996431	+	0.0844150i	$0.0269021\pi$
$421$			3.46410i			0.168830i	−0.996431	+	0.0844150i	$0.973098\pi$
$422$	−13.8564			−0.674519
$423$		0			0
$424$	6.00000			0.291386
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−4.24264	+	3.46410i	−0.204598	+	0.167054i
$431$	19.5959			0.943902			0.471951	−	0.881625i	$-0.343550\pi$
$431$	19.5959			0.943902			0.471951	+	0.881625i	$0.343550\pi$
$432$		0			0
$433$	4.24264			0.203888			0.101944	−	0.994790i	$-0.467494\pi$
$433$	4.24264			0.203888			0.101944	+	0.994790i	$0.467494\pi$
$434$		0			0
$435$		0			0
$436$			13.8564i			0.663602i
$437$	−13.8564	+	6.00000i	−0.662842	+	0.287019i
$438$		0			0
$439$			3.46410i			0.165333i	0.996577	+	0.0826663i	$0.0263436\pi$
$439$			3.46410i			0.165333i	−0.996577	+	0.0826663i	$0.973656\pi$
$440$	4.89898	−	4.00000i	0.233550	−	0.190693i
$441$		0			0
$442$		0			0
$443$	17.3205			0.822922			0.411461	−	0.911427i	$-0.365019\pi$
$443$	17.3205			0.822922			0.411461	+	0.911427i	$0.365019\pi$
$444$		0			0
$445$	−16.9706	+	13.8564i	−0.804482	+	0.656857i
$446$		−	4.24264i		−	0.200895i
$447$		0			0
$448$		0			0
$449$	−4.89898			−0.231197			−0.115599	−	0.993296i	$-0.536879\pi$
$449$	−4.89898			−0.231197			−0.115599	+	0.993296i	$0.536879\pi$
$450$		0			0
$451$		−	27.7128i		−	1.30495i
$452$		−	12.0000i		−	0.564433i
$453$		0			0
$454$	12.0000			0.563188
$455$		0			0
$456$		0			0
$457$			4.89898i			0.229165i	0.993414	+	0.114582i	$0.0365530\pi$
$457$			4.89898i			0.229165i	−0.993414	+	0.114582i	$0.963447\pi$
$458$		−	14.0000i		−	0.654177i
$459$		0			0
$460$	−6.00000	+	4.89898i	−0.279751	+	0.228416i
$461$			11.3137i			0.526932i	0.964669	+	0.263466i	$0.0848657\pi$
$461$			11.3137i			0.526932i	−0.964669	+	0.263466i	$0.915134\pi$
$462$		0			0
$463$		−	29.3939i		−	1.36605i	−0.730395	−	0.683025i	$-0.760664\pi$
$463$		−	29.3939i		−	1.36605i	0.730395	−	0.683025i	$-0.239336\pi$
$464$	−2.44949			−0.113715
$465$		0			0
$466$			6.92820i			0.320943i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$	14.6969	+	18.0000i	0.677919	+	0.830278i
$471$		0			0
$472$		−	12.2474i		−	0.563735i
$473$	−6.92820			−0.318559
$474$		0			0
$475$	−4.48528	−	21.3280i	−0.205799	−	0.978594i
$476$		0			0
$477$		0			0
$478$	9.89949			0.452792
$479$		−	15.5563i		−	0.710788i	−0.934717	−	0.355394i	$-0.884347\pi$
$479$		−	15.5563i		−	0.710788i	0.934717	−	0.355394i	$-0.115653\pi$
$480$		0			0
$481$		0			0
$482$	−13.8564			−0.631142
$483$		0			0
$484$	−3.00000			−0.136364
$485$	−7.34847	+	6.00000i	−0.333677	+	0.272446i
$486$		0			0
$487$	29.6985			1.34577			0.672883	−	0.739749i	$-0.265056\pi$
$487$	29.6985			1.34577			0.672883	+	0.739749i	$0.265056\pi$
$488$			2.00000i			0.0905357i
$489$		0			0
$490$	−9.89949	−	12.1244i	−0.447214	−	0.547723i
$491$		−	5.65685i		−	0.255290i	−0.991820	−	0.127645i	$-0.959258\pi$
$491$		−	5.65685i		−	0.255290i	0.991820	−	0.127645i	$-0.0407419\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	3.46410i		−	0.155543i
$497$		0			0
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	−5.19615	−	9.89949i	−0.232379	−	0.442719i
$501$		0			0
$502$	−11.3137			−0.504956
$503$	−24.2487			−1.08120			−0.540598	−	0.841281i	$-0.681802\pi$
$503$	−24.2487			−1.08120			−0.540598	+	0.841281i	$0.681802\pi$
$504$		0			0
$505$	4.00000	+	4.89898i	0.177998	+	0.218002i
$506$	−9.79796			−0.435572
$507$		0			0
$508$	−12.7279			−0.564710
$509$	−36.7423			−1.62858			−0.814288	−	0.580461i	$-0.802872\pi$
$509$	−36.7423			−1.62858			−0.814288	+	0.580461i	$0.802872\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	24.0000			1.05859
$515$	7.34847	−	6.00000i	0.323812	−	0.264392i
$516$		0			0
$517$			29.3939i			1.29274i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−44.0908			−1.93165			−0.965827	−	0.259188i	$-0.916545\pi$
$521$	−44.0908			−1.93165			−0.965827	+	0.259188i	$0.916545\pi$
$522$		0			0
$523$	−8.48528			−0.371035			−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528			−0.371035			−0.185518	+	0.982641i	$0.559396\pi$
$524$			5.65685i			0.247121i
$525$		0			0
$526$		−	3.46410i		−	0.151042i
$527$		0			0
$528$		0			0
$529$	−11.0000			−0.478261
$530$	10.3923	−	8.48528i	0.451413	−	0.368577i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		−	8.48528i		−	0.366508i
$537$		0			0
$538$			12.2474i			0.528025i
$539$		−	19.7990i		−	0.852803i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		−	14.0000i		−	0.601351i
$543$		0			0
$544$		0			0
$545$	19.5959	+	24.0000i	0.839397	+	1.02805i
$546$		0			0
$547$	−8.48528			−0.362804			−0.181402	−	0.983409i	$-0.558064\pi$
$547$	−8.48528			−0.362804			−0.181402	+	0.983409i	$0.558064\pi$
$548$	−20.7846			−0.887875
$549$		0			0
$550$	2.82843	−	13.8564i	0.120605	−	0.590839i
$551$	−9.79796	+	4.24264i	−0.417407	+	0.180743i
$552$		0			0
$553$		0			0
$554$	−7.34847			−0.312207
$555$		0			0
$556$	−4.00000			−0.169638
$557$	27.7128			1.17423			0.587115	−	0.809504i	$-0.300264\pi$
$557$	27.7128			1.17423			0.587115	+	0.809504i	$0.300264\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		−	14.6969i		−	0.619953i
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	−16.9706	−	20.7846i	−0.713957	−	0.874415i
$566$	−7.34847			−0.308879
$567$		0			0
$568$		−	4.89898i		−	0.205557i
$569$	24.4949			1.02688			0.513440	−	0.858126i	$-0.328371\pi$
$569$	24.4949			1.02688			0.513440	+	0.858126i	$0.328371\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.46410	+	16.9706i	−0.144463	+	0.707721i
$576$		0			0
$577$		−	29.3939i		−	1.22368i	−0.790980	−	0.611842i	$-0.790429\pi$
$577$		−	29.3939i		−	1.22368i	0.790980	−	0.611842i	$-0.209571\pi$
$578$		−	17.0000i		−	0.707107i
$579$		0			0
$580$	−4.24264	+	3.46410i	−0.176166	+	0.143839i
$581$		0			0
$582$		0			0
$583$	16.9706			0.702849
$584$	−4.89898			−0.202721
$585$		0			0
$586$	30.0000			1.23929
$587$	6.92820			0.285958			0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820			0.285958			0.142979	+	0.989726i	$0.454332\pi$
$588$		0			0
$589$	−6.00000	−	13.8564i	−0.247226	−	0.570943i
$590$	−17.3205	−	21.2132i	−0.713074	−	0.873334i
$591$		0			0
$592$	8.48528			0.348743
$593$	34.6410			1.42254			0.711268	−	0.702921i	$-0.248121\pi$
$593$	34.6410			1.42254			0.711268	+	0.702921i	$0.248121\pi$
$594$		0			0
$595$		0			0
$596$			2.82843i			0.115857i
$597$		0			0
$598$		0			0
$599$	−9.79796			−0.400334			−0.200167	−	0.979762i	$-0.564148\pi$
$599$	−9.79796			−0.400334			−0.200167	+	0.979762i	$0.564148\pi$
$600$		0			0
$601$			13.8564i			0.565215i	0.959236	+	0.282607i	$0.0911993\pi$
$601$			13.8564i			0.565215i	−0.959236	+	0.282607i	$0.908801\pi$
$602$		0			0
$603$		0			0
$604$			17.3205i			0.704761i
$605$	−5.19615	+	4.24264i	−0.211254	+	0.172488i
$606$		0			0
$607$	−4.24264			−0.172203			−0.0861017	−	0.996286i	$-0.527441\pi$
$607$	−4.24264			−0.172203			−0.0861017	+	0.996286i	$0.527441\pi$
$608$	1.73205	+	4.00000i	0.0702439	+	0.162221i
$609$		0			0
$610$	2.82843	+	3.46410i	0.114520	+	0.140257i
$611$		0			0
$612$		0			0
$613$			46.5403i			1.87975i	0.341525	+	0.939873i	$0.389057\pi$
$613$			46.5403i			1.87975i	−0.341525	+	0.939873i	$0.610943\pi$
$614$			25.4558i			1.02731i
$615$		0			0
$616$		0			0
$617$	13.8564			0.557838			0.278919	−	0.960315i	$-0.410024\pi$
$617$	13.8564			0.557838			0.278919	+	0.960315i	$0.410024\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$	−4.89898	−	6.00000i	−0.196748	−	0.240966i
$621$		0			0
$622$	24.0416			0.963982
$623$		0			0
$624$		0			0
$625$	−23.0000	−	9.79796i	−0.920000	−	0.391918i
$626$	9.79796			0.391605
$627$		0			0
$628$		−	7.34847i		−	0.293236i
$629$		0			0
$630$		0			0
$631$	−2.00000			−0.0796187			−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000			−0.0796187			−0.0398094	+	0.999207i	$0.512675\pi$
$632$	10.3923			0.413384
$633$		0			0
$634$	6.00000			0.238290
$635$	−22.0454	+	18.0000i	−0.874845	+	0.714308i
$636$		0			0
$637$		0			0
$638$	−6.92820			−0.274290
$639$		0			0
$640$	1.41421	+	1.73205i	0.0559017	+	0.0684653i
$641$	34.2929			1.35449			0.677243	−	0.735759i	$-0.263174\pi$
$641$	34.2929			1.35449			0.677243	+	0.735759i	$0.263174\pi$
$642$		0			0
$643$			2.44949i			0.0965984i	0.998833	+	0.0482992i	$0.0153801\pi$
$643$			2.44949i			0.0965984i	−0.998833	+	0.0482992i	$0.984620\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.3205			0.680939			0.340470	−	0.940255i	$-0.389414\pi$
$647$	17.3205			0.680939			0.340470	+	0.940255i	$0.389414\pi$
$648$		0			0
$649$		−	34.6410i		−	1.35978i
$650$		0			0
$651$		0			0
$652$			17.1464i			0.671506i
$653$	27.7128			1.08449			0.542243	−	0.840222i	$-0.317575\pi$
$653$	27.7128			1.08449			0.542243	+	0.840222i	$0.317575\pi$
$654$		0			0
$655$	8.00000	+	9.79796i	0.312586	+	0.382838i
$656$	9.79796			0.382546
$657$		0			0
$658$		0			0
$659$	22.0454			0.858767			0.429384	−	0.903122i	$-0.358731\pi$
$659$	22.0454			0.858767			0.429384	+	0.903122i	$0.358731\pi$
$660$		0			0
$661$			3.46410i			0.134738i	0.997728	+	0.0673690i	$0.0214605\pi$
$661$			3.46410i			0.134738i	−0.997728	+	0.0673690i	$0.978540\pi$
$662$	−34.6410			−1.34636
$663$		0			0
$664$			10.3923i			0.403300i
$665$		0			0
$666$		0			0
$667$	8.48528			0.328551
$668$			18.0000i			0.696441i
$669$		0			0
$670$	−12.0000	−	14.6969i	−0.463600	−	0.567792i
$671$			5.65685i			0.218380i
$672$		0			0
$673$	−21.2132			−0.817709			−0.408854	−	0.912600i	$-0.634072\pi$
$673$	−21.2132			−0.817709			−0.408854	+	0.912600i	$0.634072\pi$
$674$			21.2132i			0.817102i
$675$		0			0
$676$	13.0000			0.500000
$677$		−	18.0000i		−	0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$		−	18.0000i		−	0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		−	9.79796i		−	0.375183i
$683$			48.0000i			1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$			48.0000i			1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$		0			0
$685$	−36.0000	+	29.3939i	−1.37549	+	1.12308i
$686$		0			0
$687$		0			0
$688$		−	2.44949i		−	0.0933859i
$689$		0			0
$690$		0			0
$691$	16.0000			0.608669			0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000			0.608669			0.304334	+	0.952565i	$0.401566\pi$
$692$		−	6.00000i		−	0.228086i
$693$		0			0
$694$			6.92820i			0.262991i
$695$	−6.92820	+	5.65685i	−0.262802	+	0.214577i
$696$		0			0
$697$		0			0
$698$		−	10.0000i		−	0.378506i
$699$		0			0
$700$		0			0
$701$			48.0833i			1.81608i	0.418884	+	0.908040i	$0.362421\pi$
$701$			48.0833i			1.81608i	−0.418884	+	0.908040i	$0.637579\pi$
$702$		0			0
$703$	33.9411	−	14.6969i	1.28011	−	0.554306i
$704$			2.82843i			0.106600i
$705$		0			0
$706$		−	6.92820i		−	0.260746i
$707$		0			0
$708$		0			0
$709$	−34.0000			−1.27690			−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000			−1.27690			−0.638448	+	0.769665i	$0.720423\pi$
$710$	−6.92820	−	8.48528i	−0.260011	−	0.318447i
$711$		0			0
$712$		−	9.79796i		−	0.367194i
$713$			12.0000i			0.449404i
$714$		0			0
$715$		0			0
$716$	17.1464			0.640792
$717$		0			0
$718$	7.07107			0.263890
$719$		−	32.5269i		−	1.21305i	−0.795065	−	0.606525i	$-0.792563\pi$
$719$		−	32.5269i		−	1.21305i	0.795065	−	0.606525i	$-0.207437\pi$
$720$		0			0
$721$		0			0
$722$	13.8564	+	13.0000i	0.515682	+	0.483810i
$723$		0			0
$724$			3.46410i			0.128742i
$725$	−2.44949	+	12.0000i	−0.0909718	+	0.445669i
$726$		0			0
$727$			34.2929i			1.27185i	0.771750	+	0.635926i	$0.219382\pi$
$727$			34.2929i			1.27185i	−0.771750	+	0.635926i	$0.780618\pi$
$728$		0			0
$729$		0			0
$730$	−8.48528	+	6.92820i	−0.314054	+	0.256424i
$731$		0			0
$732$		0			0
$733$			26.9444i			0.995214i	0.867403	+	0.497607i	$0.165788\pi$
$733$			26.9444i			0.995214i	−0.867403	+	0.497607i	$0.834212\pi$
$734$	24.4949			0.904123
$735$		0			0
$736$		−	3.46410i		−	0.127688i
$737$		−	24.0000i		−	0.884051i
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$	14.6969	−	12.0000i	0.540270	−	0.441129i
$741$		0			0
$742$		0			0
$743$			6.00000i			0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$			6.00000i			0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$		0			0
$745$	4.00000	+	4.89898i	0.146549	+	0.179485i
$746$		−	33.9411i		−	1.24267i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			31.1769i			1.13766i	0.822455	+	0.568831i	$0.192604\pi$
$751$			31.1769i			1.13766i	−0.822455	+	0.568831i	$0.807396\pi$
$752$	−10.3923			−0.378968
$753$		0			0
$754$		0			0
$755$	24.4949	+	30.0000i	0.891461	+	1.09181i
$756$		0			0
$757$			31.8434i			1.15737i	0.815552	+	0.578683i	$0.196433\pi$
$757$			31.8434i			1.15737i	−0.815552	+	0.578683i	$0.803567\pi$
$758$	10.3923			0.377466
$759$		0			0
$760$	8.65685	+	4.47871i	0.314017	+	0.162460i
$761$			52.3259i			1.89681i	0.317058	+	0.948406i	$0.397305\pi$
$761$			52.3259i			1.89681i	−0.317058	+	0.948406i	$0.602695\pi$
$762$		0			0
$763$		0			0
$764$			9.89949i			0.358151i
$765$		0			0
$766$	−6.00000			−0.216789
$767$		0			0
$768$		0			0
$769$	16.0000			0.576975			0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000			0.576975			0.288487	+	0.957484i	$0.406848\pi$
$770$		0			0
$771$		0			0
$772$	4.24264			0.152696
$773$			30.0000i			1.07903i	0.841978	+	0.539513i	$0.181391\pi$
$773$			30.0000i			1.07903i	−0.841978	+	0.539513i	$0.818609\pi$
$774$		0			0
$775$	−16.9706	−	3.46410i	−0.609601	−	0.124434i
$776$		−	4.24264i		−	0.152302i
$777$		0			0
$778$	5.65685			0.202808
$779$	39.1918	−	16.9706i	1.40419	−	0.608034i
$780$		0			0
$781$		−	13.8564i		−	0.495821i
$782$		0			0
$783$		0			0
$784$	7.00000			0.250000
$785$	−10.3923	−	12.7279i	−0.370917	−	0.454279i
$786$		0			0
$787$	−33.9411			−1.20987			−0.604935	−	0.796275i	$-0.706801\pi$
$787$	−33.9411			−1.20987			−0.604935	+	0.796275i	$0.706801\pi$
$788$		0			0
$789$		0			0
$790$	18.0000	−	14.6969i	0.640411	−	0.522894i
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−31.8434			−1.13008
$795$		0			0
$796$	−14.0000			−0.496217
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$		0			0
$800$	4.89898	+	1.00000i	0.173205	+	0.0353553i
$801$		0			0
$802$		0			0
$803$	−13.8564			−0.488982
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−2.82843			−0.0995037
$809$			32.5269i			1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$809$			32.5269i			1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$810$		0			0
$811$		−	34.6410i		−	1.21641i	−0.793780	−	0.608205i	$-0.791890\pi$
$811$		−	34.6410i		−	1.21641i	0.793780	−	0.608205i	$-0.208110\pi$
$812$		0			0
$813$		0			0
$814$	24.0000			0.841200
$815$	24.2487	+	29.6985i	0.849395	+	1.04029i
$816$		0			0
$817$	−4.24264	−	9.79796i	−0.148431	−	0.342787i
$818$	−20.7846			−0.726717
$819$		0			0
$820$	16.9706	−	13.8564i	0.592638	−	0.483887i
$821$			53.7401i			1.87554i	0.347253	+	0.937771i	$0.387115\pi$
$821$			53.7401i			1.87554i	−0.347253	+	0.937771i	$0.612885\pi$
$822$		0			0
$823$		−	34.2929i		−	1.19537i	−0.801730	−	0.597687i	$-0.796087\pi$
$823$		−	34.2929i		−	1.19537i	0.801730	−	0.597687i	$-0.203913\pi$
$824$			4.24264i			0.147799i
$825$		0			0
$826$		0			0
$827$		−	12.0000i		−	0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$		−	12.0000i		−	0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$		0			0
$829$		−	27.7128i		−	0.962506i	−0.876582	−	0.481253i	$-0.840182\pi$
$829$		−	27.7128i		−	0.962506i	0.876582	−	0.481253i	$-0.159818\pi$
$830$	14.6969	+	18.0000i	0.510138	+	0.624789i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	25.4558	+	31.1769i	0.880936	+	1.07892i
$836$	4.89898	+	11.3137i	0.169435	+	0.391293i
$837$		0			0
$838$	−2.82843			−0.0977064
$839$	53.8888			1.86045			0.930224	−	0.366993i	$-0.119613\pi$
$839$	53.8888			1.86045			0.930224	+	0.366993i	$0.119613\pi$
$840$		0			0
$841$	−23.0000			−0.793103
$842$	−3.46410			−0.119381
$843$		0			0
$844$		−	13.8564i		−	0.476957i
$845$	22.5167	−	18.3848i	0.774597	−	0.632456i
$846$		0			0
$847$		0			0
$848$			6.00000i			0.206041i
$849$		0			0
$850$		0			0
$851$	−29.3939			−1.00761
$852$		0			0
$853$			31.8434i			1.09030i	0.838340	+	0.545148i	$0.183527\pi$
$853$			31.8434i			1.09030i	−0.838340	+	0.545148i	$0.816473\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	−40.0000			−1.36478			−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000			−1.36478			−0.682391	+	0.730987i	$0.739060\pi$
$860$	−3.46410	−	4.24264i	−0.118125	−	0.144673i
$861$		0			0
$862$			19.5959i			0.667440i
$863$		−	48.0000i		−	1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$		−	48.0000i		−	1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$		0			0
$865$	−8.48528	−	10.3923i	−0.288508	−	0.353349i
$866$			4.24264i			0.144171i
$867$		0			0
$868$		0			0
$869$	29.3939			0.997119
$870$		0			0
$871$		0			0
$872$	−13.8564			−0.469237
$873$		0			0
$874$	−6.00000	−	13.8564i	−0.202953	−	0.468700i
$875$		0			0
$876$		0			0
$877$	25.4558			0.859583			0.429791	−	0.902928i	$-0.358587\pi$
$877$	25.4558			0.859583			0.429791	+	0.902928i	$0.358587\pi$
$878$	−3.46410			−0.116908
$879$		0			0
$880$	4.00000	+	4.89898i	0.134840	+	0.165145i
$881$			9.89949i			0.333522i	0.985997	+	0.166761i	$0.0533309\pi$
$881$			9.89949i			0.333522i	−0.985997	+	0.166761i	$0.946669\pi$
$882$		0			0
$883$			22.0454i			0.741887i	0.928655	+	0.370944i	$0.120966\pi$
$883$			22.0454i			0.741887i	−0.928655	+	0.370944i	$0.879034\pi$
$884$		0			0
$885$		0			0
$886$			17.3205i			0.581894i
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$		0			0
$890$	−13.8564	−	16.9706i	−0.464468	−	0.568855i
$891$		0			0
$892$	4.24264			0.142054
$893$	−41.5692	+	18.0000i	−1.39106	+	0.602347i
$894$		0			0
$895$	29.6985	−	24.2487i	0.992711	−	0.810545i
$896$		0			0
$897$		0			0
$898$		−	4.89898i		−	0.163481i
$899$			8.48528i			0.283000i
$900$		0			0
$901$		0			0
$902$	27.7128			0.922736
$903$		0			0
$904$	12.0000			0.399114
$905$	4.89898	+	6.00000i	0.162848	+	0.199447i
$906$		0			0
$907$	42.4264			1.40875			0.704373	−	0.709830i	$-0.251228\pi$
$907$	42.4264			1.40875			0.704373	+	0.709830i	$0.251228\pi$
$908$			12.0000i			0.398234i
$909$		0			0
$910$		0			0
$911$	34.2929			1.13617			0.568087	−	0.822969i	$-0.307684\pi$
$911$	34.2929			1.13617			0.568087	+	0.822969i	$0.307684\pi$
$912$		0			0
$913$			29.3939i			0.972795i
$914$	−4.89898			−0.162044
$915$		0			0
$916$	14.0000			0.462573
$917$		0			0
$918$		0			0
$919$	38.0000			1.25350			0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000			1.25350			0.626752	+	0.779219i	$0.284384\pi$
$920$	−4.89898	−	6.00000i	−0.161515	−	0.197814i
$921$		0			0
$922$	−11.3137			−0.372597
$923$		0			0
$924$		0			0
$925$	8.48528	−	41.5692i	0.278994	−	1.36679i
$926$	29.3939			0.965943
$927$		0			0
$928$		−	2.44949i		−	0.0804084i
$929$		−	35.3553i		−	1.15997i	−0.814627	−	0.579986i	$-0.803058\pi$
$929$		−	35.3553i		−	1.15997i	0.814627	−	0.579986i	$-0.196942\pi$
$930$		0			0
$931$	28.0000	−	12.1244i	0.917663	−	0.397360i
$932$	−6.92820			−0.226941
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			53.8888i			1.76047i	0.474538	+	0.880235i	$0.342615\pi$
$937$			53.8888i			1.76047i	−0.474538	+	0.880235i	$0.657385\pi$
$938$		0			0
$939$		0			0
$940$	−18.0000	+	14.6969i	−0.587095	+	0.479361i
$941$	−51.4393			−1.67687			−0.838436	−	0.544999i	$-0.816530\pi$
$941$	−51.4393			−1.67687			−0.838436	+	0.544999i	$0.816530\pi$
$942$		0			0
$943$	−33.9411			−1.10528
$944$	12.2474			0.398621
$945$		0			0
$946$		−	6.92820i		−	0.225255i
$947$	−13.8564			−0.450273			−0.225136	−	0.974327i	$-0.572283\pi$
$947$	−13.8564			−0.450273			−0.225136	+	0.974327i	$0.572283\pi$
$948$		0			0
$949$		0			0
$950$	21.3280	−	4.48528i	0.691971	−	0.145522i
$951$		0			0
$952$		0			0
$953$			42.0000i			1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$			42.0000i			1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$		0			0
$955$	14.0000	+	17.1464i	0.453029	+	0.554845i
$956$			9.89949i			0.320173i
$957$		0			0
$958$	15.5563			0.502603
$959$		0			0
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		0			0
$964$		−	13.8564i		−	0.446285i
$965$	7.34847	−	6.00000i	0.236556	−	0.193147i
$966$		0			0
$967$		−	4.89898i		−	0.157541i	−0.996893	−	0.0787703i	$-0.974901\pi$
$967$		−	4.89898i		−	0.157541i	0.996893	−	0.0787703i	$-0.0250994\pi$
$968$		−	3.00000i		−	0.0964237i
$969$		0			0
$970$	−6.00000	−	7.34847i	−0.192648	−	0.235945i
$971$	−2.44949			−0.0786079			−0.0393039	−	0.999227i	$-0.512514\pi$
$971$	−2.44949			−0.0786079			−0.0393039	+	0.999227i	$0.512514\pi$
$972$		0			0
$973$		0			0
$974$			29.6985i			0.951601i
$975$		0			0
$976$	−2.00000			−0.0640184
$977$			24.0000i			0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$			24.0000i			0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$		0			0
$979$		−	27.7128i		−	0.885705i
$980$	12.1244	−	9.89949i	0.387298	−	0.316228i
$981$		0			0
$982$	5.65685			0.180517
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			8.48528i			0.269816i
$990$		0			0
$991$		−	38.1051i		−	1.21045i	−0.796055	−	0.605224i	$-0.793083\pi$
$991$		−	38.1051i		−	1.21045i	0.796055	−	0.605224i	$-0.206917\pi$
$992$	3.46410			0.109985
$993$		0			0
$994$		0			0
$995$	−24.2487	+	19.7990i	−0.768736	+	0.627670i
$996$		0			0
$997$		−	26.9444i		−	0.853337i	−0.904408	−	0.426669i	$-0.859687\pi$
$997$		−	26.9444i		−	0.853337i	0.904408	−	0.426669i	$-0.140313\pi$
$998$			4.00000i			0.126618i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.2.c.c.1709.6	yes	8
3.2	odd	2	inner	1710.2.c.c.1709.3	yes	8
5.4	even	2	inner	1710.2.c.c.1709.4	yes	8
15.14	odd	2	inner	1710.2.c.c.1709.5	yes	8
19.18	odd	2	inner	1710.2.c.c.1709.2	yes	8
57.56	even	2	inner	1710.2.c.c.1709.7	yes	8
95.94	odd	2	inner	1710.2.c.c.1709.8	yes	8
285.284	even	2	inner	1710.2.c.c.1709.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.c.c.1709.1	✓	8	285.284	even	2	inner
1710.2.c.c.1709.2	yes	8	19.18	odd	2	inner
1710.2.c.c.1709.3	yes	8	3.2	odd	2	inner
1710.2.c.c.1709.4	yes	8	5.4	even	2	inner
1710.2.c.c.1709.5	yes	8	15.14	odd	2	inner
1710.2.c.c.1709.6	yes	8	1.1	even	1	trivial
1710.2.c.c.1709.7	yes	8	57.56	even	2	inner
1710.2.c.c.1709.8	yes	8	95.94	odd	2	inner

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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.73205 + 1.41421i) q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.73205 + 1.41421i) q^{5} -1.00000i q^{8} +(-1.41421 - 1.73205i) q^{10} -2.82843i q^{11} +1.00000 q^{16} +(4.00000 - 1.73205i) q^{19} +(1.73205 - 1.41421i) q^{20} +2.82843 q^{22} -3.46410 q^{23} +(1.00000 - 4.89898i) q^{25} -2.44949 q^{29} -3.46410i q^{31} +1.00000i q^{32} +8.48528 q^{37} +(1.73205 + 4.00000i) q^{38} +(1.41421 + 1.73205i) q^{40} +9.79796 q^{41} -2.44949i q^{43} +2.82843i q^{44} -3.46410i q^{46} -10.3923 q^{47} +7.00000 q^{49} +(4.89898 + 1.00000i) q^{50} +6.00000i q^{53} +(4.00000 + 4.89898i) q^{55} -2.44949i q^{58} +12.2474 q^{59} -2.00000 q^{61} +3.46410 q^{62} -1.00000 q^{64} +8.48528 q^{67} +4.89898 q^{71} -4.89898i q^{73} +8.48528i q^{74} +(-4.00000 + 1.73205i) q^{76} +10.3923i q^{79} +(-1.73205 + 1.41421i) q^{80} +9.79796i q^{82} -10.3923 q^{83} +2.44949 q^{86} -2.82843 q^{88} +9.79796 q^{89} +3.46410 q^{92} -10.3923i q^{94} +(-4.47871 + 8.65685i) q^{95} +4.24264 q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} + 32 q^{19} + 8 q^{25} + 56 q^{49} + 32 q^{55} - 16 q^{61} - 8 q^{64} - 32 q^{76}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 + 32 * q^19 + 8 * q^25 + 56 * q^49 + 32 * q^55 - 16 * q^61 - 8 * q^64 - 32 * q^76

Introduction

L-functions

Modular forms

Varieties

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Database

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