LMFDB - Embedded newform 414.3.b.c.91.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(91,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("414.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.b (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:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1358954496.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 1 $ x^8 + 8x^6 + 20x^4 + 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.2
Root			$1.98289i$ of defining polynomial
Character	$\chi$	$=$	414.91
Dual form			414.3.b.c.91.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -4.92225i q^{5} +5.13851i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -4.92225i q^{5} +5.13851i q^{7} -2.82843 q^{8} +6.96111i q^{10} +7.71925i q^{11} +0.944060 q^{13} -7.26695i q^{14} +4.00000 q^{16} +27.6556i q^{17} -9.08137i q^{19} -9.84450i q^{20} -10.9167i q^{22} +(20.0652 - 11.2423i) q^{23} +0.771456 q^{25} -1.33510 q^{26} +10.2770i q^{28} +14.4550 q^{29} -0.830916 q^{31} -5.65685 q^{32} -39.1110i q^{34} +25.2930 q^{35} +23.8369i q^{37} +12.8430i q^{38} +13.9222i q^{40} +26.8431 q^{41} -26.7736i q^{43} +15.4385i q^{44} +(-28.3764 + 15.8990i) q^{46} +38.2280 q^{47} +22.5957 q^{49} -1.09100 q^{50} +1.88812 q^{52} +92.3727i q^{53} +37.9961 q^{55} -14.5339i q^{56} -20.4425 q^{58} +47.8457 q^{59} +121.824i q^{61} +1.17509 q^{62} +8.00000 q^{64} -4.64690i q^{65} +8.10477i q^{67} +55.3113i q^{68} -35.7698 q^{70} +23.8139 q^{71} +98.1035 q^{73} -33.7105i q^{74} -18.1627i q^{76} -39.6654 q^{77} -70.0030i q^{79} -19.6890i q^{80} -37.9618 q^{82} -50.7347i q^{83} +136.128 q^{85} +37.8636i q^{86} -21.8333i q^{88} -87.6727i q^{89} +4.85106i q^{91} +(40.1304 - 22.4845i) q^{92} -54.0626 q^{94} -44.7008 q^{95} +36.7344i q^{97} -31.9551 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 16 q^{13} + 32 q^{16} - 16 q^{23} + 72 q^{25} - 32 q^{26} + 144 q^{29} - 128 q^{31} + 112 q^{35} + 16 q^{41} - 80 q^{46} + 112 q^{47} + 40 q^{49} + 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} - 80 q^{59} + 96 q^{62} + 64 q^{64} - 144 q^{70} - 32 q^{71} + 64 q^{73} - 224 q^{77} + 48 q^{85} - 32 q^{92} - 16 q^{94} - 112 q^{95} - 224 q^{98}+O(q^{100}) $ 8 * q + 16 * q^4 + 16 * q^13 + 32 * q^16 - 16 * q^23 + 72 * q^25 - 32 * q^26 + 144 * q^29 - 128 * q^31 + 112 * q^35 + 16 * q^41 - 80 * q^46 + 112 * q^47 + 40 * q^49 + 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 - 80 * q^59 + 96 * q^62 + 64 * q^64 - 144 * q^70 - 32 * q^71 + 64 * q^73 - 224 * q^77 + 48 * q^85 - 32 * q^92 - 16 * q^94 - 112 * q^95 - 224 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$		−	4.92225i		−	0.984450i	−0.870468	−	0.492225i	$-0.836184\pi$
$5$		−	4.92225i		−	0.984450i	0.870468	−	0.492225i	$-0.163816\pi$
$6$		0			0
$7$			5.13851i			0.734073i	0.930206	+	0.367037i	$0.119628\pi$
$7$			5.13851i			0.734073i	−0.930206	+	0.367037i	$0.880372\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$			6.96111i			0.696111i
$11$			7.71925i			0.701750i	0.936422	+	0.350875i	$0.114116\pi$
$11$			7.71925i			0.701750i	−0.936422	+	0.350875i	$0.885884\pi$
$12$		0			0
$13$	0.944060			0.0726200			0.0363100	−	0.999341i	$-0.488440\pi$
$13$	0.944060			0.0726200			0.0363100	+	0.999341i	$0.488440\pi$
$14$		−	7.26695i		−	0.519068i
$15$		0			0
$16$	4.00000			0.250000
$17$			27.6556i			1.62680i	0.581703	+	0.813401i	$0.302387\pi$
$17$			27.6556i			1.62680i	−0.581703	+	0.813401i	$0.697613\pi$
$18$		0			0
$19$		−	9.08137i		−	0.477967i	−0.971024	−	0.238983i	$-0.923186\pi$
$19$		−	9.08137i		−	0.477967i	0.971024	−	0.238983i	$-0.0768141\pi$
$20$		−	9.84450i		−	0.492225i
$21$		0			0
$22$		−	10.9167i		−	0.496212i
$23$	20.0652	−	11.2423i	0.872399	−	0.488794i
$23$	20.0652	−	11.2423i	0.872399	−	0.488794i
$24$		0			0
$25$	0.771456			0.0308582
$26$	−1.33510			−0.0513501
$27$		0			0
$28$			10.2770i			0.367037i
$29$	14.4550			0.498449			0.249225	−	0.968446i	$-0.419824\pi$
$29$	14.4550			0.498449			0.249225	+	0.968446i	$0.419824\pi$
$30$		0			0
$31$	−0.830916			−0.0268037			−0.0134019	−	0.999910i	$-0.504266\pi$
$31$	−0.830916			−0.0268037			−0.0134019	+	0.999910i	$0.504266\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	39.1110i		−	1.15032i
$35$	25.2930			0.722658
$36$		0			0
$37$			23.8369i			0.644241i	0.946699	+	0.322120i	$0.104396\pi$
$37$			23.8369i			0.644241i	−0.946699	+	0.322120i	$0.895604\pi$
$38$			12.8430i			0.337973i
$39$		0			0
$40$			13.9222i			0.348056i
$41$	26.8431			0.654709			0.327354	−	0.944902i	$-0.393843\pi$
$41$	26.8431			0.654709			0.327354	+	0.944902i	$0.393843\pi$
$42$		0			0
$43$		−	26.7736i		−	0.622642i	−0.950305	−	0.311321i	$-0.899229\pi$
$43$		−	26.7736i		−	0.622642i	0.950305	−	0.311321i	$-0.100771\pi$
$44$			15.4385i			0.350875i
$45$		0			0
$46$	−28.3764	+	15.8990i	−0.616879	+	0.345630i
$47$	38.2280			0.813363			0.406681	−	0.913570i	$-0.366686\pi$
$47$	38.2280			0.813363			0.406681	+	0.913570i	$0.366686\pi$
$48$		0			0
$49$	22.5957			0.461136
$50$	−1.09100			−0.0218201
$51$		0			0
$52$	1.88812			0.0363100
$53$			92.3727i			1.74288i	0.490502	+	0.871440i	$0.336814\pi$
$53$			92.3727i			1.74288i	−0.490502	+	0.871440i	$0.663186\pi$
$54$		0			0
$55$	37.9961			0.690837
$56$		−	14.5339i		−	0.259534i
$57$		0			0
$58$	−20.4425			−0.352457
$59$	47.8457			0.810944			0.405472	−	0.914108i	$-0.367107\pi$
$59$	47.8457			0.810944			0.405472	+	0.914108i	$0.367107\pi$
$60$		0			0
$61$			121.824i			1.99711i	0.0537398	+	0.998555i	$0.482886\pi$
$61$			121.824i			1.99711i	−0.0537398	+	0.998555i	$0.517114\pi$
$62$	1.17509			0.0189531
$63$		0			0
$64$	8.00000			0.125000
$65$		−	4.64690i		−	0.0714907i
$66$		0			0
$67$			8.10477i			0.120967i	0.998169	+	0.0604834i	$0.0192642\pi$
$67$			8.10477i			0.120967i	−0.998169	+	0.0604834i	$0.980736\pi$
$68$			55.3113i			0.813401i
$69$		0			0
$70$	−35.7698			−0.510997
$71$	23.8139			0.335407			0.167704	−	0.985837i	$-0.446365\pi$
$71$	23.8139			0.335407			0.167704	+	0.985837i	$0.446365\pi$
$72$		0			0
$73$	98.1035			1.34388			0.671942	−	0.740604i	$-0.265460\pi$
$73$	98.1035			1.34388			0.671942	+	0.740604i	$0.265460\pi$
$74$		−	33.7105i		−	0.455547i
$75$		0			0
$76$		−	18.1627i		−	0.238983i
$77$	−39.6654			−0.515136
$78$		0			0
$79$		−	70.0030i		−	0.886114i	−0.896493	−	0.443057i	$-0.853894\pi$
$79$		−	70.0030i		−	0.886114i	0.896493	−	0.443057i	$-0.146106\pi$
$80$		−	19.6890i		−	0.246112i
$81$		0			0
$82$	−37.9618			−0.462949
$83$		−	50.7347i		−	0.611261i	−0.952150	−	0.305631i	$-0.901133\pi$
$83$		−	50.7347i		−	0.611261i	0.952150	−	0.305631i	$-0.0988672\pi$
$84$		0			0
$85$	136.128			1.60151
$86$			37.8636i			0.440274i
$87$		0			0
$88$		−	21.8333i		−	0.248106i
$89$		−	87.6727i		−	0.985086i	−0.870288	−	0.492543i	$-0.836067\pi$
$89$		−	87.6727i		−	0.985086i	0.870288	−	0.492543i	$-0.163933\pi$
$90$		0			0
$91$			4.85106i			0.0533084i
$92$	40.1304	−	22.4845i	0.436200	−	0.244397i
$93$		0			0
$94$	−54.0626			−0.575134
$95$	−44.7008			−0.470534
$96$		0			0
$97$			36.7344i			0.378705i	0.981909	+	0.189353i	$0.0606390\pi$
$97$			36.7344i			0.378705i	−0.981909	+	0.189353i	$0.939361\pi$
$98$	−31.9551			−0.326073
$99$		0			0
$100$	1.54291			0.0154291
$101$	−189.428			−1.87553			−0.937764	−	0.347273i	$-0.887108\pi$
$101$	−189.428			−1.87553			−0.937764	+	0.347273i	$0.887108\pi$
$102$		0			0
$103$			85.5225i			0.830316i	0.909749	+	0.415158i	$0.136274\pi$
$103$			85.5225i			0.830316i	−0.909749	+	0.415158i	$0.863726\pi$
$104$	−2.67020			−0.0256750
$105$		0			0
$106$		−	130.635i		−	1.23240i
$107$			138.422i			1.29366i	0.762634	+	0.646830i	$0.223906\pi$
$107$			138.422i			1.29366i	−0.762634	+	0.646830i	$0.776094\pi$
$108$		0			0
$109$			49.7589i			0.456503i	0.973602	+	0.228252i	$0.0733009\pi$
$109$			49.7589i			0.456503i	−0.973602	+	0.228252i	$0.926699\pi$
$110$	−53.7345			−0.488496
$111$		0			0
$112$			20.5541i			0.183518i
$113$		−	189.178i		−	1.67414i	−0.547096	−	0.837070i	$-0.684267\pi$
$113$		−	189.178i		−	1.67414i	0.547096	−	0.837070i	$-0.315733\pi$
$114$		0			0
$115$	−55.3373	−	98.7658i	−0.481193	−	0.858833i
$116$	28.9101			0.249225
$117$		0			0
$118$	−67.6640			−0.573424
$119$	−142.109			−1.19419
$120$		0			0
$121$	61.4133			0.507548
$122$		−	172.285i		−	1.41217i
$123$		0			0
$124$	−1.66183			−0.0134019
$125$		−	126.854i		−	1.01483i
$126$		0			0
$127$	−32.2016			−0.253556			−0.126778	−	0.991931i	$-0.540464\pi$
$127$	−32.2016			−0.253556			−0.126778	+	0.991931i	$0.540464\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$			6.57170i			0.0505516i
$131$	−88.6957			−0.677066			−0.338533	−	0.940954i	$-0.609931\pi$
$131$	−88.6957			−0.677066			−0.338533	+	0.940954i	$0.609931\pi$
$132$		0			0
$133$	46.6647			0.350863
$134$		−	11.4619i		−	0.0855364i
$135$		0			0
$136$		−	78.2220i		−	0.575162i
$137$			88.3772i			0.645089i	0.946554	+	0.322544i	$0.104538\pi$
$137$			88.3772i			0.645089i	−0.946554	+	0.322544i	$0.895462\pi$
$138$		0			0
$139$	−183.605			−1.32090			−0.660449	−	0.750871i	$-0.729634\pi$
$139$	−183.605			−1.32090			−0.660449	+	0.750871i	$0.729634\pi$
$140$	50.5861			0.361329
$141$		0			0
$142$	−33.6780			−0.237169
$143$			7.28743i			0.0509610i
$144$		0			0
$145$		−	71.1513i		−	0.490698i
$146$	−138.739			−0.950269
$147$		0			0
$148$			47.6738i			0.322120i
$149$			86.5480i			0.580859i	0.956896	+	0.290430i	$0.0937982\pi$
$149$			86.5480i			0.580859i	−0.956896	+	0.290430i	$0.906202\pi$
$150$		0			0
$151$	−263.765			−1.74679			−0.873393	−	0.487016i	$-0.838085\pi$
$151$	−263.765			−1.74679			−0.873393	+	0.487016i	$0.838085\pi$
$152$			25.6860i			0.168987i
$153$		0			0
$154$	56.0954			0.364256
$155$			4.08998i			0.0263869i
$156$		0			0
$157$		−	83.4996i		−	0.531845i	−0.963994	−	0.265922i	$-0.914324\pi$
$157$		−	83.4996i		−	0.531845i	0.963994	−	0.265922i	$-0.0856765\pi$
$158$			98.9992i			0.626577i
$159$		0			0
$160$			27.8445i			0.174028i
$161$	57.7685	+	103.105i	0.358811	+	0.640405i
$162$		0			0
$163$	−178.643			−1.09597			−0.547986	−	0.836487i	$-0.684605\pi$
$163$	−178.643			−1.09597			−0.547986	+	0.836487i	$0.684605\pi$
$164$	53.6861			0.327354
$165$		0			0
$166$			71.7497i			0.432227i
$167$	125.596			0.752072			0.376036	−	0.926605i	$-0.377287\pi$
$167$	125.596			0.752072			0.376036	+	0.926605i	$0.377287\pi$
$168$		0			0
$169$	−168.109			−0.994726
$170$	−192.514			−1.13244
$171$		0			0
$172$		−	53.5472i		−	0.311321i
$173$	−145.986			−0.843849			−0.421925	−	0.906631i	$-0.638645\pi$
$173$	−145.986			−0.843849			−0.421925	+	0.906631i	$0.638645\pi$
$174$		0			0
$175$			3.96414i			0.0226522i
$176$			30.8770i			0.175437i
$177$		0			0
$178$			123.988i			0.696561i
$179$	152.764			0.853428			0.426714	−	0.904387i	$-0.359671\pi$
$179$	152.764			0.853428			0.426714	+	0.904387i	$0.359671\pi$
$180$		0			0
$181$		−	133.516i		−	0.737658i	−0.929497	−	0.368829i	$-0.879759\pi$
$181$		−	133.516i		−	0.737658i	0.929497	−	0.368829i	$-0.120241\pi$
$182$		−	6.86044i		−	0.0376947i
$183$		0			0
$184$	−56.7529	+	31.7979i	−0.308440	+	0.172815i
$185$	117.331			0.634223
$186$		0			0
$187$	−213.481			−1.14161
$188$	76.4561			0.406681
$189$		0			0
$190$	63.2164			0.332718
$191$		−	271.502i		−	1.42148i	−0.703456	−	0.710739i	$-0.748361\pi$
$191$		−	271.502i		−	1.42148i	0.703456	−	0.710739i	$-0.251639\pi$
$192$		0			0
$193$	242.088			1.25434			0.627172	−	0.778881i	$-0.284212\pi$
$193$	242.088			1.25434			0.627172	+	0.778881i	$0.284212\pi$
$194$		−	51.9503i		−	0.267785i
$195$		0			0
$196$	45.1914			0.230568
$197$	251.316			1.27571			0.637857	−	0.770155i	$-0.279821\pi$
$197$	251.316			1.27571			0.637857	+	0.770155i	$0.279821\pi$
$198$		0			0
$199$		−	176.314i		−	0.885999i	−0.896522	−	0.443000i	$-0.853914\pi$
$199$		−	176.314i		−	0.885999i	0.896522	−	0.443000i	$-0.146086\pi$
$200$	−2.18201			−0.0109100
$201$		0			0
$202$	267.892			1.32620
$203$			74.2773i			0.365898i
$204$		0			0
$205$		−	132.128i		−	0.644528i
$206$		−	120.947i		−	0.587122i
$207$		0			0
$208$	3.77624			0.0181550
$209$	70.1013			0.335413
$210$		0			0
$211$	215.285			1.02031			0.510155	−	0.860082i	$-0.329588\pi$
$211$	215.285			1.02031			0.510155	+	0.860082i	$0.329588\pi$
$212$			184.745i			0.871440i
$213$		0			0
$214$		−	195.758i		−	0.914756i
$215$	−131.786			−0.612960
$216$		0			0
$217$		−	4.26967i		−	0.0196759i
$218$		−	70.3697i		−	0.322797i
$219$		0			0
$220$	75.9921			0.345419
$221$			26.1086i			0.118138i
$222$		0			0
$223$	237.892			1.06678			0.533391	−	0.845869i	$-0.320918\pi$
$223$	237.892			1.06678			0.533391	+	0.845869i	$0.320918\pi$
$224$		−	29.0678i		−	0.129767i
$225$		0			0
$226$			267.538i			1.18380i
$227$			1.80104i			0.00793410i	0.999992	+	0.00396705i	$0.00126275\pi$
$227$			1.80104i			0.00793410i	−0.999992	+	0.00396705i	$0.998737\pi$
$228$		0			0
$229$			230.994i			1.00871i	0.863497	+	0.504353i	$0.168269\pi$
$229$			230.994i			1.00871i	−0.863497	+	0.504353i	$0.831731\pi$
$230$	78.2587	+	139.676i	0.340255	+	0.607287i
$231$		0			0
$232$	−40.8850			−0.176228
$233$	50.0293			0.214718			0.107359	−	0.994220i	$-0.465761\pi$
$233$	50.0293			0.214718			0.107359	+	0.994220i	$0.465761\pi$
$234$		0			0
$235$		−	188.168i		−	0.800715i
$236$	95.6914			0.405472
$237$		0			0
$238$	200.972			0.844422
$239$	−405.258			−1.69564			−0.847820	−	0.530284i	$-0.822085\pi$
$239$	−405.258			−1.69564			−0.847820	+	0.530284i	$0.822085\pi$
$240$		0			0
$241$			348.532i			1.44619i	0.690748	+	0.723095i	$0.257281\pi$
$241$			348.532i			1.44619i	−0.690748	+	0.723095i	$0.742719\pi$
$242$	−86.8515			−0.358890
$243$		0			0
$244$			243.647i			0.998555i
$245$		−	111.222i		−	0.453966i
$246$		0			0
$247$		−	8.57335i		−	0.0347099i
$248$	2.35019			0.00947655
$249$		0			0
$250$			179.398i			0.717592i
$251$			5.76047i			0.0229501i	0.999934	+	0.0114750i	$0.00365270\pi$
$251$			5.76047i			0.0229501i	−0.999934	+	0.0114750i	$0.996347\pi$
$252$		0			0
$253$	86.7818	+	154.888i	0.343011	+	0.612206i
$254$	45.5399			0.179291
$255$		0			0
$256$	16.0000			0.0625000
$257$	−26.3305			−0.102453			−0.0512267	−	0.998687i	$-0.516313\pi$
$257$	−26.3305			−0.102453			−0.0512267	+	0.998687i	$0.516313\pi$
$258$		0			0
$259$	−122.486			−0.472920
$260$		−	9.29379i		−	0.0357454i
$261$		0			0
$262$	125.435			0.478758
$263$			61.0826i			0.232253i	0.993234	+	0.116127i	$0.0370478\pi$
$263$			61.0826i			0.232253i	−0.993234	+	0.116127i	$0.962952\pi$
$264$		0			0
$265$	454.681			1.71578
$266$	−65.9939			−0.248097
$267$		0			0
$268$			16.2095i			0.0604834i
$269$	8.00773			0.0297685			0.0148843	−	0.999889i	$-0.495262\pi$
$269$	8.00773			0.0297685			0.0148843	+	0.999889i	$0.495262\pi$
$270$		0			0
$271$	347.677			1.28294			0.641470	−	0.767148i	$-0.278325\pi$
$271$	347.677			1.28294			0.641470	+	0.767148i	$0.278325\pi$
$272$			110.623i			0.406701i
$273$		0			0
$274$		−	124.984i		−	0.456147i
$275$			5.95506i			0.0216547i
$276$		0			0
$277$	−320.160			−1.15581			−0.577907	−	0.816103i	$-0.696130\pi$
$277$	−320.160			−1.15581			−0.577907	+	0.816103i	$0.696130\pi$
$278$	259.657			0.934017
$279$		0			0
$280$	−71.5395			−0.255498
$281$		−	77.6750i		−	0.276424i	−0.990403	−	0.138212i	$-0.955865\pi$
$281$		−	77.6750i		−	0.276424i	0.990403	−	0.138212i	$-0.0441355\pi$
$282$		0			0
$283$			449.812i			1.58944i	0.606974	+	0.794721i	$0.292383\pi$
$283$			449.812i			1.58944i	−0.606974	+	0.794721i	$0.707617\pi$
$284$	47.6278			0.167704
$285$		0			0
$286$		−	10.3060i		−	0.0360349i
$287$			137.933i			0.480604i
$288$		0			0
$289$	−475.835			−1.64649
$290$			100.623i			0.346976i
$291$		0			0
$292$	196.207			0.671942
$293$			437.459i			1.49304i	0.665366	+	0.746518i	$0.268276\pi$
$293$			437.459i			1.49304i	−0.665366	+	0.746518i	$0.731724\pi$
$294$		0			0
$295$		−	235.508i		−	0.798334i
$296$		−	67.4209i		−	0.227773i
$297$		0			0
$298$		−	122.397i		−	0.410729i
$299$	18.9427	−	10.6134i	0.0633536	−	0.0354962i
$300$		0			0
$301$	137.577			0.457065
$302$	373.020			1.23516
$303$		0			0
$304$		−	36.3255i		−	0.119492i
$305$	599.647			1.96605
$306$		0			0
$307$	−465.359			−1.51583			−0.757914	−	0.652354i	$-0.773781\pi$
$307$	−465.359			−1.51583			−0.757914	+	0.652354i	$0.773781\pi$
$308$	−79.3309			−0.257568
$309$		0			0
$310$		−	5.78410i		−	0.0186584i
$311$	−596.299			−1.91736			−0.958679	−	0.284489i	$-0.908176\pi$
$311$	−596.299			−1.91736			−0.958679	+	0.284489i	$0.908176\pi$
$312$		0			0
$313$		−	411.536i		−	1.31481i	−0.753536	−	0.657406i	$-0.771654\pi$
$313$		−	411.536i		−	1.31481i	0.753536	−	0.657406i	$-0.228346\pi$
$314$			118.086i			0.376071i
$315$		0			0
$316$		−	140.006i		−	0.443057i
$317$	348.576			1.09961			0.549804	−	0.835293i	$-0.314702\pi$
$317$	348.576			1.09961			0.549804	+	0.835293i	$0.314702\pi$
$318$		0			0
$319$			111.582i			0.349786i
$320$		−	39.3780i		−	0.123056i
$321$		0			0
$322$	−81.6970	−	145.813i	−0.253718	−	0.452835i
$323$	251.151			0.777558
$324$		0			0
$325$	0.728300			0.00224092
$326$	252.640			0.774969
$327$		0			0
$328$	−75.9236			−0.231474
$329$			196.435i			0.597068i
$330$		0			0
$331$	−62.3646			−0.188413			−0.0942064	−	0.995553i	$-0.530031\pi$
$331$	−62.3646			−0.188413			−0.0942064	+	0.995553i	$0.530031\pi$
$332$		−	101.469i		−	0.305631i
$333$		0			0
$334$	−177.620			−0.531795
$335$	39.8937			0.119086
$336$		0			0
$337$			106.990i			0.317477i	0.987321	+	0.158738i	$0.0507427\pi$
$337$			106.990i			0.317477i	−0.987321	+	0.158738i	$0.949257\pi$
$338$	237.742			0.703378
$339$		0			0
$340$	272.256			0.800753
$341$		−	6.41404i		−	0.0188095i
$342$		0			0
$343$			367.895i			1.07258i
$344$			75.7272i			0.220137i
$345$		0			0
$346$	206.455			0.596692
$347$	319.649			0.921177			0.460589	−	0.887614i	$-0.347638\pi$
$347$	319.649			0.921177			0.460589	+	0.887614i	$0.347638\pi$
$348$		0			0
$349$	−156.934			−0.449667			−0.224834	−	0.974397i	$-0.572184\pi$
$349$	−156.934			−0.449667			−0.224834	+	0.974397i	$0.572184\pi$
$350$		−	5.60613i		−	0.0160175i
$351$		0			0
$352$		−	43.6666i		−	0.124053i
$353$	−157.767			−0.446933			−0.223466	−	0.974712i	$-0.571737\pi$
$353$	−157.767			−0.446933			−0.223466	+	0.974712i	$0.571737\pi$
$354$		0			0
$355$		−	117.218i		−	0.330192i
$356$		−	175.345i		−	0.492543i
$357$		0			0
$358$	−216.040			−0.603465
$359$		−	149.911i		−	0.417580i	−0.977961	−	0.208790i	$-0.933047\pi$
$359$		−	149.911i		−	0.417580i	0.977961	−	0.208790i	$-0.0669525\pi$
$360$		0			0
$361$	278.529			0.771548
$362$			188.820i			0.521603i
$363$		0			0
$364$			9.70212i			0.0266542i
$365$		−	482.890i		−	1.32299i
$366$		0			0
$367$		−	643.183i		−	1.75254i	−0.481818	−	0.876271i	$-0.660023\pi$
$367$		−	643.183i		−	1.75254i	0.481818	−	0.876271i	$-0.339977\pi$
$368$	80.2607	−	44.9691i	0.218100	−	0.122199i
$369$		0			0
$370$	−165.931			−0.448463
$371$	−474.658			−1.27940
$372$		0			0
$373$		−	240.542i		−	0.644884i	−0.946589	−	0.322442i	$-0.895496\pi$
$373$		−	240.542i		−	0.644884i	0.946589	−	0.322442i	$-0.104504\pi$
$374$	301.907			0.807239
$375$		0			0
$376$	−108.125			−0.287567
$377$	13.6464			0.0361974
$378$		0			0
$379$		−	690.895i		−	1.82294i	−0.411364	−	0.911471i	$-0.634948\pi$
$379$		−	690.895i		−	1.82294i	0.411364	−	0.911471i	$-0.365052\pi$
$380$	−89.4015			−0.235267
$381$		0			0
$382$			383.962i			1.00514i
$383$		−	249.591i		−	0.651674i	−0.945426	−	0.325837i	$-0.894354\pi$
$383$		−	249.591i		−	0.651674i	0.945426	−	0.325837i	$-0.105646\pi$
$384$		0			0
$385$			195.243i			0.507125i
$386$	−342.365			−0.886955
$387$		0			0
$388$			73.4689i			0.189353i
$389$			515.323i			1.32474i	0.749177	+	0.662369i	$0.230449\pi$
$389$			515.323i			1.32474i	−0.749177	+	0.662369i	$0.769551\pi$
$390$		0			0
$391$	310.912	+	554.915i	0.795172	+	1.41922i
$392$	−63.9103			−0.163036
$393$		0			0
$394$	−355.414			−0.902066
$395$	−344.572			−0.872335
$396$		0			0
$397$	−219.200			−0.552140			−0.276070	−	0.961138i	$-0.589032\pi$
$397$	−219.200			−0.552140			−0.276070	+	0.961138i	$0.589032\pi$
$398$			249.345i			0.626496i
$399$		0			0
$400$	3.08582			0.00771456
$401$			110.690i			0.276034i	0.990430	+	0.138017i	$0.0440729\pi$
$401$			110.690i			0.276034i	−0.990430	+	0.138017i	$0.955927\pi$
$402$		0			0
$403$	−0.784434			−0.00194649
$404$	−378.857			−0.937764
$405$		0			0
$406$		−	105.044i		−	0.258729i
$407$	−184.003			−0.452096
$408$		0			0
$409$	632.891			1.54741			0.773706	−	0.633545i	$-0.218401\pi$
$409$	632.891			1.54741			0.773706	+	0.633545i	$0.218401\pi$
$410$			186.858i			0.455750i
$411$		0			0
$412$			171.045i			0.415158i
$413$			245.856i			0.595292i
$414$		0			0
$415$	−249.729			−0.601756
$416$	−5.34041			−0.0128375
$417$		0			0
$418$	−99.1382			−0.237173
$419$		−	655.862i		−	1.56530i	−0.622461	−	0.782651i	$-0.713867\pi$
$419$		−	655.862i		−	1.56530i	0.622461	−	0.782651i	$-0.286133\pi$
$420$		0			0
$421$		−	257.748i		−	0.612228i	−0.951995	−	0.306114i	$-0.900971\pi$
$421$		−	257.748i		−	0.612228i	0.951995	−	0.306114i	$-0.0990288\pi$
$422$	−304.460			−0.721468
$423$		0			0
$424$		−	261.269i		−	0.616201i
$425$			21.3351i			0.0502003i
$426$		0			0
$427$	−625.993			−1.46603
$428$			276.843i			0.646830i
$429$		0			0
$430$	186.374			0.433428
$431$		−	703.840i		−	1.63304i	−0.577317	−	0.816520i	$-0.695900\pi$
$431$		−	703.840i		−	1.63304i	0.577317	−	0.816520i	$-0.304100\pi$
$432$		0			0
$433$			621.176i			1.43459i	0.696772	+	0.717293i	$0.254619\pi$
$433$			621.176i			1.43459i	−0.696772	+	0.717293i	$0.745381\pi$
$434$			6.03823i			0.0139130i
$435$		0			0
$436$			99.5177i			0.228252i
$437$	−102.095	−	182.219i	−0.233627	−	0.416978i
$438$		0			0
$439$	−292.224			−0.665659			−0.332830	−	0.942987i	$-0.608003\pi$
$439$	−292.224			−0.665659			−0.332830	+	0.942987i	$0.608003\pi$
$440$	−107.469			−0.244248
$441$		0			0
$442$		−	36.9231i		−	0.0835364i
$443$	541.206			1.22168			0.610842	−	0.791753i	$-0.290831\pi$
$443$	541.206			1.22168			0.610842	+	0.791753i	$0.290831\pi$
$444$		0			0
$445$	−431.547			−0.969768
$446$	−336.430			−0.754328
$447$		0			0
$448$			41.1081i			0.0917592i
$449$	−554.320			−1.23457			−0.617283	−	0.786741i	$-0.711767\pi$
$449$	−554.320			−1.23457			−0.617283	+	0.786741i	$0.711767\pi$
$450$		0			0
$451$			207.208i			0.459442i
$452$		−	378.356i		−	0.837070i
$453$		0			0
$454$		−	2.54706i		−	0.00561026i
$455$	23.8781			0.0524794
$456$		0			0
$457$		−	740.778i		−	1.62096i	−0.585767	−	0.810480i	$-0.699207\pi$
$457$		−	740.778i		−	1.62096i	0.585767	−	0.810480i	$-0.300793\pi$
$458$		−	326.674i		−	0.713263i
$459$		0			0
$460$	−110.675	−	197.532i	−0.240597	−	0.429417i
$461$	−74.1009			−0.160739			−0.0803697	−	0.996765i	$-0.525610\pi$
$461$	−74.1009			−0.160739			−0.0803697	+	0.996765i	$0.525610\pi$
$462$		0			0
$463$	704.351			1.52128			0.760638	−	0.649176i	$-0.224886\pi$
$463$	704.351			1.52128			0.760638	+	0.649176i	$0.224886\pi$
$464$	57.8201			0.124612
$465$		0			0
$466$	−70.7520			−0.151828
$467$			584.754i			1.25215i	0.779763	+	0.626075i	$0.215340\pi$
$467$			584.754i			1.25215i	−0.779763	+	0.626075i	$0.784660\pi$
$468$		0			0
$469$	−41.6465			−0.0887984
$470$			266.110i			0.566191i
$471$		0			0
$472$	−135.328			−0.286712
$473$	206.672			0.436939
$474$		0			0
$475$		−	7.00587i		−	0.0147492i
$476$	−284.218			−0.597096
$477$		0			0
$478$	573.122			1.19900
$479$		−	586.758i		−	1.22496i	−0.790485	−	0.612482i	$-0.790171\pi$
$479$		−	586.758i		−	1.22496i	0.790485	−	0.612482i	$-0.209829\pi$
$480$		0			0
$481$			22.5035i			0.0467847i
$482$		−	492.899i		−	1.02261i
$483$		0			0
$484$	122.827			0.253774
$485$	180.816			0.372817
$486$		0			0
$487$	−143.981			−0.295650			−0.147825	−	0.989014i	$-0.547227\pi$
$487$	−143.981			−0.295650			−0.147825	+	0.989014i	$0.547227\pi$
$488$		−	344.569i		−	0.706085i
$489$		0			0
$490$			157.291i			0.321002i
$491$	−672.591			−1.36984			−0.684920	−	0.728618i	$-0.740163\pi$
$491$	−672.591			−1.36984			−0.684920	+	0.728618i	$0.740163\pi$
$492$		0			0
$493$			399.763i			0.810879i
$494$			12.1245i			0.0245436i
$495$		0			0
$496$	−3.32366			−0.00670094
$497$			122.368i			0.246213i
$498$		0			0
$499$	110.196			0.220834			0.110417	−	0.993885i	$-0.464781\pi$
$499$	110.196			0.220834			0.110417	+	0.993885i	$0.464781\pi$
$500$		−	253.707i		−	0.507414i
$501$		0			0
$502$		−	8.14653i		−	0.0162281i
$503$			697.316i			1.38631i	0.720786	+	0.693157i	$0.243781\pi$
$503$			697.316i			1.38631i	−0.720786	+	0.693157i	$0.756219\pi$
$504$		0			0
$505$			932.414i			1.84636i
$506$	−122.728	−	219.045i	−0.242546	−	0.432895i
$507$		0			0
$508$	−64.4031			−0.126778
$509$	−702.717			−1.38058			−0.690292	−	0.723531i	$-0.742518\pi$
$509$	−702.717			−1.38058			−0.690292	+	0.723531i	$0.742518\pi$
$510$		0			0
$511$			504.106i			0.986509i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	37.2370			0.0724454
$515$	420.963			0.817405
$516$		0			0
$517$			295.092i			0.570777i
$518$	173.222			0.334405
$519$		0			0
$520$			13.1434i			0.0252758i
$521$		−	130.216i		−	0.249935i	−0.992161	−	0.124968i	$-0.960117\pi$
$521$		−	130.216i		−	0.249935i	0.992161	−	0.124968i	$-0.0398827\pi$
$522$		0			0
$523$		−	966.051i		−	1.84713i	−0.383436	−	0.923567i	$-0.625259\pi$
$523$		−	966.051i		−	1.84713i	0.383436	−	0.923567i	$-0.374741\pi$
$524$	−177.391			−0.338533
$525$		0			0
$526$		−	86.3838i		−	0.164228i
$527$		−	22.9795i		−	0.0436044i
$528$		0			0
$529$	276.223	−	451.156i	0.522160	−	0.852847i
$530$	−643.017			−1.21324
$531$		0			0
$532$	93.3294			0.175431
$533$	25.3414			0.0475449
$534$		0			0
$535$	681.346			1.27354
$536$		−	22.9237i		−	0.0427682i
$537$		0			0
$538$	−11.3246			−0.0210495
$539$			174.422i			0.323602i
$540$		0			0
$541$	250.233			0.462539			0.231269	−	0.972890i	$-0.425712\pi$
$541$	250.233			0.462539			0.231269	+	0.972890i	$0.425712\pi$
$542$	−491.689			−0.907175
$543$		0			0
$544$		−	156.444i		−	0.287581i
$545$	244.926			0.449405
$546$		0			0
$547$	226.518			0.414109			0.207055	−	0.978329i	$-0.433612\pi$
$547$	226.518			0.414109			0.207055	+	0.978329i	$0.433612\pi$
$548$			176.754i			0.322544i
$549$		0			0
$550$		−	8.42172i		−	0.0153122i
$551$		−	131.271i		−	0.238242i
$552$		0			0
$553$	359.711			0.650473
$554$	452.775			0.817284
$555$		0			0
$556$	−367.210			−0.660449
$557$			693.109i			1.24436i	0.782874	+	0.622181i	$0.213753\pi$
$557$			693.109i			1.24436i	−0.782874	+	0.622181i	$0.786247\pi$
$558$		0			0
$559$		−	25.2759i		−	0.0452162i
$560$	101.172			0.180665
$561$		0			0
$562$			109.849i			0.195461i
$563$			232.265i			0.412549i	0.978494	+	0.206275i	$0.0661340\pi$
$563$			232.265i			0.412549i	−0.978494	+	0.206275i	$0.933866\pi$
$564$		0			0
$565$	−931.180			−1.64811
$566$		−	636.131i		−	1.12391i
$567$		0			0
$568$	−67.3559			−0.118584
$569$		−	285.831i		−	0.502340i	−0.967943	−	0.251170i	$-0.919185\pi$
$569$		−	285.831i		−	0.502340i	0.967943	−	0.251170i	$-0.0808153\pi$
$570$		0			0
$571$		−	572.347i		−	1.00236i	−0.865344	−	0.501179i	$-0.832900\pi$
$571$		−	572.347i		−	1.00236i	0.865344	−	0.501179i	$-0.167100\pi$
$572$			14.5749i			0.0254805i
$573$		0			0
$574$		−	195.067i		−	0.339838i
$575$	15.4794	−	8.67291i	0.0269207	−	0.0150833i
$576$		0			0
$577$	157.233			0.272501			0.136251	−	0.990674i	$-0.456495\pi$
$577$	157.233			0.272501			0.136251	+	0.990674i	$0.456495\pi$
$578$	672.932			1.16424
$579$		0			0
$580$		−	142.303i		−	0.245349i
$581$	260.701			0.448710
$582$		0			0
$583$	−713.047			−1.22307
$584$	−277.479			−0.475135
$585$		0			0
$586$		−	618.661i		−	1.05574i
$587$	212.520			0.362044			0.181022	−	0.983479i	$-0.442059\pi$
$587$	212.520			0.362044			0.181022	+	0.983479i	$0.442059\pi$
$588$		0			0
$589$			7.54585i			0.0128113i
$590$			333.059i			0.564507i
$591$		0			0
$592$			95.3476i			0.161060i
$593$	937.090			1.58025			0.790127	−	0.612944i	$-0.210015\pi$
$593$	937.090			1.58025			0.790127	+	0.612944i	$0.210015\pi$
$594$		0			0
$595$			699.495i			1.17562i
$596$			173.096i			0.290430i
$597$		0			0
$598$	−26.7891	+	15.0096i	−0.0447978	+	0.0250996i
$599$	318.493			0.531708			0.265854	−	0.964013i	$-0.414346\pi$
$599$	318.493			0.531708			0.265854	+	0.964013i	$0.414346\pi$
$600$		0			0
$601$	799.628			1.33050			0.665248	−	0.746623i	$-0.268326\pi$
$601$	799.628			1.33050			0.665248	+	0.746623i	$0.268326\pi$
$602$	−194.563			−0.323194
$603$		0			0
$604$	−527.529			−0.873393
$605$		−	302.291i		−	0.499655i
$606$		0			0
$607$	−185.298			−0.305268			−0.152634	−	0.988283i	$-0.548776\pi$
$607$	−185.298			−0.305268			−0.152634	+	0.988283i	$0.548776\pi$
$608$			51.3720i			0.0844934i
$609$		0			0
$610$	−848.029			−1.39021
$611$	36.0896			0.0590664
$612$		0			0
$613$		−	399.963i		−	0.652469i	−0.945289	−	0.326234i	$-0.894220\pi$
$613$		−	399.963i		−	0.652469i	0.945289	−	0.326234i	$-0.105780\pi$
$614$	658.117			1.07185
$615$		0			0
$616$	112.191			0.182128
$617$		−	308.392i		−	0.499826i	−0.968268	−	0.249913i	$-0.919598\pi$
$617$		−	308.392i		−	0.499826i	0.968268	−	0.249913i	$-0.0804020\pi$
$618$		0			0
$619$			659.561i			1.06553i	0.846264	+	0.532763i	$0.178846\pi$
$619$			659.561i			1.06553i	−0.846264	+	0.532763i	$0.821154\pi$
$620$			8.17995i			0.0131935i
$621$		0			0
$622$	843.294			1.35578
$623$	450.507			0.723125
$624$		0			0
$625$	−605.118			−0.968190
$626$			582.000i			0.929713i
$627$		0			0
$628$		−	166.999i		−	0.265922i
$629$	−659.225			−1.04805
$630$		0			0
$631$			102.663i			0.162699i	0.996686	+	0.0813497i	$0.0259231\pi$
$631$			102.663i			0.162699i	−0.996686	+	0.0813497i	$0.974077\pi$
$632$			197.998i			0.313289i
$633$		0			0
$634$	−492.961			−0.777541
$635$			158.504i			0.249613i
$636$		0			0
$637$	21.3317			0.0334877
$638$		−	157.801i		−	0.247336i
$639$		0			0
$640$			55.6889i			0.0870139i
$641$			721.856i			1.12614i	0.826409	+	0.563070i	$0.190380\pi$
$641$			721.856i			1.12614i	−0.826409	+	0.563070i	$0.809620\pi$
$642$		0			0
$643$		−	177.237i		−	0.275641i	−0.990457	−	0.137821i	$-0.955990\pi$
$643$		−	177.237i		−	0.275641i	0.990457	−	0.137821i	$-0.0440097\pi$
$644$	115.537	+	206.210i	0.179405	+	0.320202i
$645$		0			0
$646$	−355.181			−0.549816
$647$	−568.014			−0.877920			−0.438960	−	0.898507i	$-0.644653\pi$
$647$	−568.014			−0.877920			−0.438960	+	0.898507i	$0.644653\pi$
$648$		0			0
$649$			369.333i			0.569079i
$650$	−1.02997			−0.00158457
$651$		0			0
$652$	−357.287			−0.547986
$653$	550.970			0.843752			0.421876	−	0.906653i	$-0.361372\pi$
$653$	550.970			0.843752			0.421876	+	0.906653i	$0.361372\pi$
$654$		0			0
$655$			436.582i			0.666538i
$656$	107.372			0.163677
$657$		0			0
$658$		−	277.801i		−	0.422191i
$659$			663.157i			1.00631i	0.864197	+	0.503154i	$0.167827\pi$
$659$			663.157i			1.00631i	−0.864197	+	0.503154i	$0.832173\pi$
$660$		0			0
$661$		−	893.783i		−	1.35217i	−0.736825	−	0.676084i	$-0.763676\pi$
$661$		−	893.783i		−	1.35217i	0.736825	−	0.676084i	$-0.236324\pi$
$662$	88.1969			0.133228
$663$		0			0
$664$			143.499i			0.216113i
$665$		−	229.695i		−	0.345407i
$666$		0			0
$667$	290.043	−	162.507i	0.434847	−	0.243639i
$668$	251.192			0.376036
$669$		0			0
$670$	−56.4182			−0.0842063
$671$	−940.387			−1.40147
$672$		0			0
$673$	1201.94			1.78594			0.892969	−	0.450118i	$-0.148618\pi$
$673$	1201.94			1.78594			0.892969	+	0.450118i	$0.148618\pi$
$674$		−	151.306i		−	0.224490i
$675$		0			0
$676$	−336.218			−0.497363
$677$		−	338.932i		−	0.500639i	−0.968163	−	0.250319i	$-0.919464\pi$
$677$		−	338.932i		−	0.500639i	0.968163	−	0.250319i	$-0.0805356\pi$
$678$		0			0
$679$	−188.760			−0.277998
$680$	−385.028			−0.566218
$681$		0			0
$682$			9.07083i			0.0133003i
$683$	951.288			1.39281			0.696404	−	0.717650i	$-0.254782\pi$
$683$	951.288			1.39281			0.696404	+	0.717650i	$0.254782\pi$
$684$		0			0
$685$	435.015			0.635058
$686$		−	520.283i		−	0.758429i
$687$		0			0
$688$		−	107.094i		−	0.155661i
$689$			87.2053i			0.126568i
$690$		0			0
$691$	−407.739			−0.590071			−0.295036	−	0.955486i	$-0.595332\pi$
$691$	−407.739			−0.590071			−0.295036	+	0.955486i	$0.595332\pi$
$692$	−291.972			−0.421925
$693$		0			0
$694$	−452.051			−0.651371
$695$			903.749i			1.30036i
$696$		0			0
$697$			742.362i			1.06508i
$698$	221.938			0.317963
$699$		0			0
$700$			7.92827i			0.0113261i
$701$		−	1016.66i		−	1.45029i	−0.688594	−	0.725147i	$-0.741772\pi$
$701$		−	1016.66i		−	1.45029i	0.688594	−	0.725147i	$-0.258228\pi$
$702$		0			0
$703$	216.472			0.307926
$704$			61.7540i			0.0877187i
$705$		0			0
$706$	223.117			0.316029
$707$		−	973.380i		−	1.37678i
$708$		0			0
$709$		−	121.061i		−	0.170749i	−0.996349	−	0.0853744i	$-0.972791\pi$
$709$		−	121.061i		−	0.170749i	0.996349	−	0.0853744i	$-0.0272086\pi$
$710$			165.771i			0.233481i
$711$		0			0
$712$			247.976i			0.348281i
$713$	−16.6725	+	9.34138i	−0.0233836	+	0.0131015i
$714$		0			0
$715$	35.8705			0.0501686
$716$	305.527			0.426714
$717$		0			0
$718$			212.006i			0.295274i
$719$	−160.481			−0.223200			−0.111600	−	0.993753i	$-0.535598\pi$
$719$	−160.481			−0.223200			−0.111600	+	0.993753i	$0.535598\pi$
$720$		0			0
$721$	−439.459			−0.609513
$722$	−393.899			−0.545567
$723$		0			0
$724$		−	267.032i		−	0.368829i
$725$	11.1514			0.0153813
$726$		0			0
$727$		−	328.928i		−	0.452446i	−0.974076	−	0.226223i	$-0.927362\pi$
$727$		−	328.928i		−	0.452446i	0.974076	−	0.226223i	$-0.0726378\pi$
$728$		−	13.7209i		−	0.0188474i
$729$		0			0
$730$			682.910i			0.935493i
$731$	740.441			1.01292
$732$		0			0
$733$			528.441i			0.720929i	0.932773	+	0.360465i	$0.117382\pi$
$733$			528.441i			0.720929i	−0.932773	+	0.360465i	$0.882618\pi$
$734$			909.598i			1.23923i
$735$		0			0
$736$	−113.506	+	63.5959i	−0.154220	+	0.0864074i
$737$	−62.5627			−0.0848883
$738$		0			0
$739$	816.397			1.10473			0.552366	−	0.833602i	$-0.313725\pi$
$739$	816.397			1.10473			0.552366	+	0.833602i	$0.313725\pi$
$740$	234.662			0.317111
$741$		0			0
$742$	671.268			0.904674
$743$		−	501.473i		−	0.674930i	−0.941338	−	0.337465i	$-0.890430\pi$
$743$		−	501.473i		−	0.674930i	0.941338	−	0.337465i	$-0.109570\pi$
$744$		0			0
$745$	426.011			0.571827
$746$			340.177i			0.456002i
$747$		0			0
$748$	−426.961			−0.570804
$749$	−711.281			−0.949641
$750$		0			0
$751$			706.401i			0.940614i	0.882503	+	0.470307i	$0.155857\pi$
$751$			706.401i			0.940614i	−0.882503	+	0.470307i	$0.844143\pi$
$752$	152.912			0.203341
$753$		0			0
$754$	−19.2989			−0.0255954
$755$			1298.32i			1.71962i
$756$		0			0
$757$		−	1154.74i		−	1.52542i	−0.646740	−	0.762711i	$-0.723868\pi$
$757$		−	1154.74i		−	1.52542i	0.646740	−	0.762711i	$-0.276132\pi$
$758$			977.073i			1.28901i
$759$		0			0
$760$	126.433			0.166359
$761$	−834.803			−1.09698			−0.548491	−	0.836157i	$-0.684797\pi$
$761$	−834.803			−1.09698			−0.548491	+	0.836157i	$0.684797\pi$
$762$		0			0
$763$	−255.687			−0.335107
$764$		−	543.004i		−	0.710739i
$765$		0			0
$766$			352.975i			0.460803i
$767$	45.1692			0.0588907
$768$		0			0
$769$			154.214i			0.200538i	0.994960	+	0.100269i	$0.0319703\pi$
$769$			154.214i			0.200538i	−0.994960	+	0.100269i	$0.968030\pi$
$770$		−	276.116i		−	0.358592i
$771$		0			0
$772$	484.177			0.627172
$773$		−	487.995i		−	0.631301i	−0.948876	−	0.315650i	$-0.897777\pi$
$773$		−	487.995i		−	0.631301i	0.948876	−	0.315650i	$-0.102223\pi$
$774$		0			0
$775$	−0.641015			−0.000827116
$776$		−	103.901i		−	0.133893i
$777$		0			0
$778$		−	728.777i		−	0.936732i
$779$		−	243.772i		−	0.312929i
$780$		0			0
$781$			183.825i			0.235372i
$782$	−439.696	−	784.769i	−0.562271	−	1.00354i
$783$		0			0
$784$	90.3827			0.115284
$785$	−411.006			−0.523574
$786$		0			0
$787$		−	193.960i		−	0.246455i	−0.992378	−	0.123228i	$-0.960675\pi$
$787$		−	193.960i		−	0.246455i	0.992378	−	0.123228i	$-0.0393246\pi$
$788$	502.631			0.637857
$789$		0			0
$790$	487.299			0.616834
$791$	972.092			1.22894
$792$		0			0
$793$			115.009i			0.145030i
$794$	309.995			0.390422
$795$		0			0
$796$		−	352.628i		−	0.443000i
$797$		−	916.068i		−	1.14939i	−0.818366	−	0.574697i	$-0.805120\pi$
$797$		−	916.068i		−	1.14939i	0.818366	−	0.574697i	$-0.194880\pi$
$798$		0			0
$799$			1057.22i			1.32318i
$800$	−4.36401			−0.00545502
$801$		0			0
$802$		−	156.539i		−	0.195186i
$803$			757.285i			0.943070i
$804$		0			0
$805$	507.509	−	284.351i	0.630447	−	0.353231i
$806$	1.10936			0.00137637
$807$		0			0
$808$	535.784			0.663099
$809$	81.4166			0.100639			0.0503193	−	0.998733i	$-0.483976\pi$
$809$	81.4166			0.100639			0.0503193	+	0.998733i	$0.483976\pi$
$810$		0			0
$811$	1503.38			1.85374			0.926870	−	0.375382i	$-0.122489\pi$
$811$	1503.38			1.85374			0.926870	+	0.375382i	$0.122489\pi$
$812$			148.555i			0.182949i
$813$		0			0
$814$	260.219			0.319680
$815$			879.328i			1.07893i
$816$		0			0
$817$	−243.141			−0.297602
$818$	−895.044			−1.09419
$819$		0			0
$820$		−	264.256i		−	0.322264i
$821$	136.557			0.166330			0.0831651	−	0.996536i	$-0.473497\pi$
$821$	136.557			0.166330			0.0831651	+	0.996536i	$0.473497\pi$
$822$		0			0
$823$	−134.188			−0.163048			−0.0815239	−	0.996671i	$-0.525979\pi$
$823$	−134.188			−0.163048			−0.0815239	+	0.996671i	$0.525979\pi$
$824$		−	241.894i		−	0.293561i
$825$		0			0
$826$		−	347.692i		−	0.420935i
$827$		−	1190.70i		−	1.43979i	−0.694085	−	0.719893i	$-0.744191\pi$
$827$		−	1190.70i		−	1.43979i	0.694085	−	0.719893i	$-0.255809\pi$
$828$		0			0
$829$	167.849			0.202471			0.101236	−	0.994862i	$-0.467720\pi$
$829$	167.849			0.202471			0.101236	+	0.994862i	$0.467720\pi$
$830$	353.170			0.425506
$831$		0			0
$832$	7.55248			0.00907750
$833$			624.898i			0.750178i
$834$		0			0
$835$		−	618.215i		−	0.740378i
$836$	140.203			0.167706
$837$		0			0
$838$			927.528i			1.10684i
$839$		−	620.476i		−	0.739542i	−0.929123	−	0.369771i	$-0.879436\pi$
$839$		−	620.476i		−	0.739542i	0.929123	−	0.369771i	$-0.120564\pi$
$840$		0			0
$841$	−632.052			−0.751548
$842$			364.511i			0.432910i
$843$		0			0
$844$	430.571			0.510155
$845$			827.473i			0.979258i
$846$		0			0
$847$			315.573i			0.372577i
$848$			369.491i			0.435720i
$849$		0			0
$850$		−	30.1724i		−	0.0354969i
$851$	267.981	+	478.292i	0.314901	+	0.562035i
$852$		0			0
$853$	639.619			0.749846			0.374923	−	0.927056i	$-0.377669\pi$
$853$	639.619			0.749846			0.374923	+	0.927056i	$0.377669\pi$
$854$	885.287			1.03664
$855$		0			0
$856$		−	391.516i		−	0.457378i
$857$	−351.336			−0.409960			−0.204980	−	0.978766i	$-0.565713\pi$
$857$	−351.336			−0.409960			−0.204980	+	0.978766i	$0.565713\pi$
$858$		0			0
$859$	−1548.57			−1.80276			−0.901382	−	0.433024i	$-0.857446\pi$
$859$	−1548.57			−1.80276			−0.901382	+	0.433024i	$0.857446\pi$
$860$	−263.573			−0.306480
$861$		0			0
$862$			995.381i			1.15473i
$863$	1502.94			1.74153			0.870763	−	0.491703i	$-0.163625\pi$
$863$	1502.94			1.74153			0.870763	+	0.491703i	$0.163625\pi$
$864$		0			0
$865$			718.579i			0.830727i
$866$		−	878.475i		−	1.01441i
$867$		0			0
$868$		−	8.53934i		−	0.00983795i
$869$	540.370			0.621830
$870$		0			0
$871$			7.65138i			0.00878460i
$872$		−	140.739i		−	0.161398i
$873$		0			0
$874$	144.384	+	257.697i	0.165200	+	0.294848i
$875$	651.839			0.744958
$876$		0			0
$877$	−474.173			−0.540676			−0.270338	−	0.962765i	$-0.587135\pi$
$877$	−474.173			−0.540676			−0.270338	+	0.962765i	$0.587135\pi$
$878$	413.268			0.470692
$879$		0			0
$880$	151.984			0.172709
$881$			727.978i			0.826309i	0.910661	+	0.413154i	$0.135573\pi$
$881$			727.978i			0.826309i	−0.910661	+	0.413154i	$0.864427\pi$
$882$		0			0
$883$	134.373			0.152178			0.0760890	−	0.997101i	$-0.475757\pi$
$883$	134.373			0.152178			0.0760890	+	0.997101i	$0.475757\pi$
$884$			52.2172i			0.0590692i
$885$		0			0
$886$	−765.381			−0.863861
$887$	−510.194			−0.575190			−0.287595	−	0.957752i	$-0.592856\pi$
$887$	−510.194			−0.575190			−0.287595	+	0.957752i	$0.592856\pi$
$888$		0			0
$889$		−	165.468i		−	0.186128i
$890$	610.299			0.685729
$891$		0			0
$892$	475.784			0.533391
$893$		−	347.163i		−	0.388760i
$894$		0			0
$895$		−	751.941i		−	0.840157i
$896$		−	58.1356i		−	0.0648835i
$897$		0			0
$898$	783.927			0.872970
$899$	−12.0109			−0.0133603
$900$		0			0
$901$	−2554.63			−2.83532
$902$		−	293.037i		−	0.324874i
$903$		0			0
$904$			535.076i			0.591898i
$905$	−657.199			−0.726187
$906$		0			0
$907$		−	594.907i		−	0.655907i	−0.944694	−	0.327953i	$-0.893641\pi$
$907$		−	594.907i		−	0.655907i	0.944694	−	0.327953i	$-0.106359\pi$
$908$			3.60208i			0.00396705i
$909$		0			0
$910$	−33.7688			−0.0371086
$911$			1025.46i			1.12564i	0.826579	+	0.562821i	$0.190284\pi$
$911$			1025.46i			1.12564i	−0.826579	+	0.562821i	$0.809716\pi$
$912$		0			0
$913$	391.633			0.428952
$914$			1047.62i			1.14619i
$915$		0			0
$916$			461.987i			0.504353i
$917$		−	455.764i		−	0.497016i
$918$		0			0
$919$		−	1333.78i		−	1.45133i	−0.688046	−	0.725667i	$-0.741531\pi$
$919$		−	1333.78i		−	1.45133i	0.688046	−	0.725667i	$-0.258469\pi$
$920$	156.517	+	279.352i	0.170128	+	0.303643i
$921$		0			0
$922$	104.794			0.113660
$923$	22.4817			0.0243573
$924$		0			0
$925$			18.3891i			0.0198801i
$926$	−996.103			−1.07570
$927$		0			0
$928$	−81.7700			−0.0881142
$929$	1146.22			1.23382			0.616911	−	0.787033i	$-0.288384\pi$
$929$	1146.22			1.23382			0.616911	+	0.787033i	$0.288384\pi$
$930$		0			0
$931$		−	205.200i		−	0.220408i
$932$	100.059			0.107359
$933$		0			0
$934$		−	826.968i		−	0.885404i
$935$			1050.81i			1.12386i
$936$		0			0
$937$			1095.51i			1.16916i	0.811334	+	0.584582i	$0.198742\pi$
$937$			1095.51i			1.16916i	−0.811334	+	0.584582i	$0.801258\pi$
$938$	58.8970			0.0627900
$939$		0			0
$940$		−	376.336i		−	0.400357i
$941$		−	422.732i		−	0.449237i	−0.974447	−	0.224618i	$-0.927886\pi$
$941$		−	422.732i		−	0.449237i	0.974447	−	0.224618i	$-0.0721135\pi$
$942$		0			0
$943$	538.611	−	301.777i	0.571167	−	0.320018i
$944$	191.383			0.202736
$945$		0			0
$946$	−292.278			−0.308962
$947$	−823.217			−0.869289			−0.434645	−	0.900602i	$-0.643126\pi$
$947$	−823.217			−0.869289			−0.434645	+	0.900602i	$0.643126\pi$
$948$		0			0
$949$	92.6156			0.0975928
$950$			9.90780i			0.0104293i
$951$		0			0
$952$	401.945			0.422211
$953$		−	397.062i		−	0.416644i	−0.978060	−	0.208322i	$-0.933200\pi$
$953$		−	397.062i		−	0.416644i	0.978060	−	0.208322i	$-0.0668002\pi$
$954$		0			0
$955$	−1336.40			−1.39937
$956$	−810.516			−0.847820
$957$		0			0
$958$			829.801i			0.866180i
$959$	−454.127			−0.473543
$960$		0			0
$961$	−960.310			−0.999282
$962$		−	31.8247i		−	0.0330818i
$963$		0			0
$964$			697.064i			0.723095i
$965$		−	1191.62i		−	1.23484i
$966$		0			0
$967$	−981.927			−1.01544			−0.507718	−	0.861523i	$-0.669511\pi$
$967$	−981.927			−1.01544			−0.507718	+	0.861523i	$0.669511\pi$
$968$	−173.703			−0.179445
$969$		0			0
$970$	−255.713			−0.263621
$971$		−	985.395i		−	1.01483i	−0.861703	−	0.507413i	$-0.830602\pi$
$971$		−	985.395i		−	1.01483i	0.861703	−	0.507413i	$-0.169398\pi$
$972$		0			0
$973$		−	943.456i		−	0.969637i
$974$	203.620			0.209056
$975$		0			0
$976$			487.295i			0.499277i
$977$			1772.59i			1.81432i	0.420783	+	0.907161i	$0.361755\pi$
$977$			1772.59i			1.81432i	−0.420783	+	0.907161i	$0.638245\pi$
$978$		0			0
$979$	676.767			0.691284
$980$		−	222.443i		−	0.226983i
$981$		0			0
$982$	951.188			0.968623
$983$			1328.52i			1.35149i	0.737135	+	0.675746i	$0.236178\pi$
$983$			1328.52i			1.35149i	−0.737135	+	0.675746i	$0.763822\pi$
$984$		0			0
$985$		−	1237.04i		−	1.25588i
$986$		−	565.350i		−	0.573378i
$987$		0			0
$988$		−	17.1467i		−	0.0173550i
$989$	−300.996	−	537.217i	−0.304344	−	0.543192i
$990$		0			0
$991$	−274.330			−0.276821			−0.138411	−	0.990375i	$-0.544199\pi$
$991$	−274.330			−0.276821			−0.138411	+	0.990375i	$0.544199\pi$
$992$	4.70037			0.00473828
$993$		0			0
$994$		−	173.055i		−	0.174099i
$995$	−867.861			−0.872222
$996$		0			0
$997$	−1081.51			−1.08476			−0.542380	−	0.840133i	$-0.682477\pi$
$997$	−1081.51			−1.08476			−0.542380	+	0.840133i	$0.682477\pi$
$998$	−155.841			−0.156153
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.b.c.91.2		8
3.2	odd	2		138.3.b.a.91.8	yes	8
12.11	even	2		1104.3.c.c.1057.4		8
23.22	odd	2	inner	414.3.b.c.91.3		8
69.68	even	2		138.3.b.a.91.7	✓	8
276.275	odd	2		1104.3.c.c.1057.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.b.a.91.7	✓	8	69.68	even	2
138.3.b.a.91.8	yes	8	3.2	odd	2
414.3.b.c.91.2		8	1.1	even	1	trivial
414.3.b.c.91.3		8	23.22	odd	2	inner
1104.3.c.c.1057.1		8	276.275	odd	2
1104.3.c.c.1057.4		8	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -4.92225i q^{5} +5.13851i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -4.92225i q^{5} +5.13851i q^{7} -2.82843 q^{8} +6.96111i q^{10} +7.71925i q^{11} +0.944060 q^{13} -7.26695i q^{14} +4.00000 q^{16} +27.6556i q^{17} -9.08137i q^{19} -9.84450i q^{20} -10.9167i q^{22} +(20.0652 - 11.2423i) q^{23} +0.771456 q^{25} -1.33510 q^{26} +10.2770i q^{28} +14.4550 q^{29} -0.830916 q^{31} -5.65685 q^{32} -39.1110i q^{34} +25.2930 q^{35} +23.8369i q^{37} +12.8430i q^{38} +13.9222i q^{40} +26.8431 q^{41} -26.7736i q^{43} +15.4385i q^{44} +(-28.3764 + 15.8990i) q^{46} +38.2280 q^{47} +22.5957 q^{49} -1.09100 q^{50} +1.88812 q^{52} +92.3727i q^{53} +37.9961 q^{55} -14.5339i q^{56} -20.4425 q^{58} +47.8457 q^{59} +121.824i q^{61} +1.17509 q^{62} +8.00000 q^{64} -4.64690i q^{65} +8.10477i q^{67} +55.3113i q^{68} -35.7698 q^{70} +23.8139 q^{71} +98.1035 q^{73} -33.7105i q^{74} -18.1627i q^{76} -39.6654 q^{77} -70.0030i q^{79} -19.6890i q^{80} -37.9618 q^{82} -50.7347i q^{83} +136.128 q^{85} +37.8636i q^{86} -21.8333i q^{88} -87.6727i q^{89} +4.85106i q^{91} +(40.1304 - 22.4845i) q^{92} -54.0626 q^{94} -44.7008 q^{95} +36.7344i q^{97} -31.9551 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 16 q^{13} + 32 q^{16} - 16 q^{23} + 72 q^{25} - 32 q^{26} + 144 q^{29} - 128 q^{31} + 112 q^{35} + 16 q^{41} - 80 q^{46} + 112 q^{47} + 40 q^{49} + 160 q^{50} + 32 q^{52} - 64 q^{55} + 128 q^{58} - 80 q^{59} + 96 q^{62} + 64 q^{64} - 144 q^{70} - 32 q^{71} + 64 q^{73} - 224 q^{77} + 48 q^{85} - 32 q^{92} - 16 q^{94} - 112 q^{95} - 224 q^{98}+O(q^{100}) \) 8 * q + 16 * q^4 + 16 * q^13 + 32 * q^16 - 16 * q^23 + 72 * q^25 - 32 * q^26 + 144 * q^29 - 128 * q^31 + 112 * q^35 + 16 * q^41 - 80 * q^46 + 112 * q^47 + 40 * q^49 + 160 * q^50 + 32 * q^52 - 64 * q^55 + 128 * q^58 - 80 * q^59 + 96 * q^62 + 64 * q^64 - 144 * q^70 - 32 * q^71 + 64 * q^73 - 224 * q^77 + 48 * q^85 - 32 * q^92 - 16 * q^94 - 112 * q^95 - 224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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