LMFDB - Embedded newform 3330.2.d.m.1999.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1999,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3330.1999");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.d (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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 370)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1999.2
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3330.1999
Dual form			3330.2.d.m.1999.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +4.44949i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +4.44949i q^{7} +1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} -4.89898 q^{11} -4.00000i q^{13} +4.44949 q^{14} +1.00000 q^{16} +4.89898i q^{17} -3.55051 q^{19} +(-2.00000 + 1.00000i) q^{20} +4.89898i q^{22} -8.89898i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{26} -4.44949i q^{28} -1.55051 q^{31} -1.00000i q^{32} +4.89898 q^{34} +(4.44949 + 8.89898i) q^{35} +1.00000i q^{37} +3.55051i q^{38} +(1.00000 + 2.00000i) q^{40} -2.00000 q^{41} -4.00000i q^{43} +4.89898 q^{44} -8.89898 q^{46} -4.44949i q^{47} -12.7980 q^{49} +(-4.00000 - 3.00000i) q^{50} +4.00000i q^{52} -11.7980i q^{53} +(-9.79796 + 4.89898i) q^{55} -4.44949 q^{56} -3.55051 q^{59} +12.0000 q^{61} +1.55051i q^{62} -1.00000 q^{64} +(-4.00000 - 8.00000i) q^{65} -5.55051i q^{67} -4.89898i q^{68} +(8.89898 - 4.44949i) q^{70} -4.89898 q^{71} -4.00000i q^{73} +1.00000 q^{74} +3.55051 q^{76} -21.7980i q^{77} -6.44949 q^{79} +(2.00000 - 1.00000i) q^{80} +2.00000i q^{82} -9.55051i q^{83} +(4.89898 + 9.79796i) q^{85} -4.00000 q^{86} -4.89898i q^{88} +15.7980 q^{89} +17.7980 q^{91} +8.89898i q^{92} -4.44949 q^{94} +(-7.10102 + 3.55051i) q^{95} -2.00000i q^{97} +12.7980i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{5}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^5	$ 4 q - 4 q^{4} + 8 q^{5} - 4 q^{10} + 8 q^{14} + 4 q^{16} - 24 q^{19} - 8 q^{20} + 12 q^{25} - 16 q^{26} - 16 q^{31} + 8 q^{35} + 4 q^{40} - 8 q^{41} - 16 q^{46} - 12 q^{49} - 16 q^{50} - 8 q^{56} - 24 q^{59} + 48 q^{61} - 4 q^{64} - 16 q^{65} + 16 q^{70} + 4 q^{74} + 24 q^{76} - 16 q^{79} + 8 q^{80} - 16 q^{86} + 24 q^{89} + 32 q^{91} - 8 q^{94} - 48 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^5 - 4 * q^10 + 8 * q^14 + 4 * q^16 - 24 * q^19 - 8 * q^20 + 12 * q^25 - 16 * q^26 - 16 * q^31 + 8 * q^35 + 4 * q^40 - 8 * q^41 - 16 * q^46 - 12 * q^49 - 16 * q^50 - 8 * q^56 - 24 * q^59 + 48 * q^61 - 4 * q^64 - 16 * q^65 + 16 * q^70 + 4 * q^74 + 24 * q^76 - 16 * q^79 + 8 * q^80 - 16 * q^86 + 24 * q^89 + 32 * q^91 - 8 * q^94 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$			4.44949i			1.68175i	0.541230	+	0.840875i	$0.317959\pi$
$7$			4.44949i			1.68175i	−0.541230	+	0.840875i	$0.682041\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$	−4.89898			−1.47710			−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898			−1.47710			−0.738549	+	0.674200i	$0.764489\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	4.44949			1.18918
$15$		0			0
$16$	1.00000			0.250000
$17$			4.89898i			1.18818i	0.804400	+	0.594089i	$0.202487\pi$
$17$			4.89898i			1.18818i	−0.804400	+	0.594089i	$0.797513\pi$
$18$		0			0
$19$	−3.55051			−0.814543			−0.407271	−	0.913307i	$-0.633520\pi$
$19$	−3.55051			−0.814543			−0.407271	+	0.913307i	$0.633520\pi$
$20$	−2.00000	+	1.00000i	−0.447214	+	0.223607i
$21$		0			0
$22$			4.89898i			1.04447i
$23$		−	8.89898i		−	1.85557i	−0.373121	−	0.927783i	$-0.621712\pi$
$23$		−	8.89898i		−	1.85557i	0.373121	−	0.927783i	$-0.378288\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	−4.00000			−0.784465
$27$		0			0
$28$		−	4.44949i		−	0.840875i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−1.55051			−0.278480			−0.139240	−	0.990259i	$-0.544466\pi$
$31$	−1.55051			−0.278480			−0.139240	+	0.990259i	$0.544466\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	4.89898			0.840168
$35$	4.44949	+	8.89898i	0.752101	+	1.50420i
$36$		0			0
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$			3.55051i			0.575969i
$39$		0			0
$40$	1.00000	+	2.00000i	0.158114	+	0.316228i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	4.89898			0.738549
$45$		0			0
$46$	−8.89898			−1.31208
$47$		−	4.44949i		−	0.649025i	−0.945881	−	0.324512i	$-0.894800\pi$
$47$		−	4.44949i		−	0.649025i	0.945881	−	0.324512i	$-0.105200\pi$
$48$		0			0
$49$	−12.7980			−1.82828
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$		−	11.7980i		−	1.62057i	−0.586033	−	0.810287i	$-0.699311\pi$
$53$		−	11.7980i		−	1.62057i	0.586033	−	0.810287i	$-0.300689\pi$
$54$		0			0
$55$	−9.79796	+	4.89898i	−1.32116	+	0.660578i
$56$	−4.44949			−0.594588
$57$		0			0
$58$		0			0
$59$	−3.55051			−0.462237			−0.231119	−	0.972926i	$-0.574239\pi$
$59$	−3.55051			−0.462237			−0.231119	+	0.972926i	$0.574239\pi$
$60$		0			0
$61$	12.0000			1.53644			0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000			1.53644			0.768221	+	0.640184i	$0.221142\pi$
$62$			1.55051i			0.196915i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000	−	8.00000i	−0.496139	−	0.992278i
$66$		0			0
$67$		−	5.55051i		−	0.678103i	−0.940768	−	0.339051i	$-0.889894\pi$
$67$		−	5.55051i		−	0.678103i	0.940768	−	0.339051i	$-0.110106\pi$
$68$		−	4.89898i		−	0.594089i
$69$		0			0
$70$	8.89898	−	4.44949i	1.06363	−	0.531816i
$71$	−4.89898			−0.581402			−0.290701	−	0.956814i	$-0.593888\pi$
$71$	−4.89898			−0.581402			−0.290701	+	0.956814i	$0.593888\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	1.00000			0.116248
$75$		0			0
$76$	3.55051			0.407271
$77$		−	21.7980i		−	2.48411i
$78$		0			0
$79$	−6.44949			−0.725624			−0.362812	−	0.931862i	$-0.618183\pi$
$79$	−6.44949			−0.725624			−0.362812	+	0.931862i	$0.618183\pi$
$80$	2.00000	−	1.00000i	0.223607	−	0.111803i
$81$		0			0
$82$			2.00000i			0.220863i
$83$		−	9.55051i		−	1.04830i	−0.851625	−	0.524152i	$-0.824382\pi$
$83$		−	9.55051i		−	1.04830i	0.851625	−	0.524152i	$-0.175618\pi$
$84$		0			0
$85$	4.89898	+	9.79796i	0.531369	+	1.06274i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		−	4.89898i		−	0.522233i
$89$	15.7980			1.67458			0.837290	−	0.546759i	$-0.184139\pi$
$89$	15.7980			1.67458			0.837290	+	0.546759i	$0.184139\pi$
$90$		0			0
$91$	17.7980			1.86573
$92$			8.89898i			0.927783i
$93$		0			0
$94$	−4.44949			−0.458930
$95$	−7.10102	+	3.55051i	−0.728549	+	0.364275i
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$			12.7980i			1.29279i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−10.6969			−1.06439			−0.532193	−	0.846623i	$-0.678632\pi$
$101$	−10.6969			−1.06439			−0.532193	+	0.846623i	$0.678632\pi$
$102$		0			0
$103$		−	9.79796i		−	0.965422i	−0.875780	−	0.482711i	$-0.839652\pi$
$103$		−	9.79796i		−	0.965422i	0.875780	−	0.482711i	$-0.160348\pi$
$104$	4.00000			0.392232
$105$		0			0
$106$	−11.7980			−1.14592
$107$			5.55051i			0.536588i	0.963337	+	0.268294i	$0.0864599\pi$
$107$			5.55051i			0.536588i	−0.963337	+	0.268294i	$0.913540\pi$
$108$		0			0
$109$	5.79796			0.555344			0.277672	−	0.960676i	$-0.410437\pi$
$109$	5.79796			0.555344			0.277672	+	0.960676i	$0.410437\pi$
$110$	4.89898	+	9.79796i	0.467099	+	0.934199i
$111$		0			0
$112$			4.44949i			0.420437i
$113$		−	3.10102i		−	0.291719i	−0.989305	−	0.145860i	$-0.953405\pi$
$113$		−	3.10102i		−	0.291719i	0.989305	−	0.145860i	$-0.0465948\pi$
$114$		0			0
$115$	−8.89898	−	17.7980i	−0.829834	−	1.65967i
$116$		0			0
$117$		0			0
$118$			3.55051i			0.326851i
$119$	−21.7980			−1.99822
$120$		0			0
$121$	13.0000			1.18182
$122$		−	12.0000i		−	1.08643i
$123$		0			0
$124$	1.55051			0.139240
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		−	2.65153i		−	0.235285i	−0.993056	−	0.117643i	$-0.962466\pi$
$127$		−	2.65153i		−	0.235285i	0.993056	−	0.117643i	$-0.0375337\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−8.00000	+	4.00000i	−0.701646	+	0.350823i
$131$	10.2474			0.895324			0.447662	−	0.894203i	$-0.352257\pi$
$131$	10.2474			0.895324			0.447662	+	0.894203i	$0.352257\pi$
$132$		0			0
$133$		−	15.7980i		−	1.36986i
$134$	−5.55051			−0.479491
$135$		0			0
$136$	−4.89898			−0.420084
$137$		−	19.5959i		−	1.67419i	−0.547056	−	0.837096i	$-0.684251\pi$
$137$		−	19.5959i		−	1.67419i	0.547056	−	0.837096i	$-0.315749\pi$
$138$		0			0
$139$	5.79796			0.491776			0.245888	−	0.969298i	$-0.420920\pi$
$139$	5.79796			0.491776			0.245888	+	0.969298i	$0.420920\pi$
$140$	−4.44949	−	8.89898i	−0.376051	−	0.752101i
$141$		0			0
$142$			4.89898i			0.411113i
$143$			19.5959i			1.63869i
$144$		0			0
$145$		0			0
$146$	−4.00000			−0.331042
$147$		0			0
$148$		−	1.00000i		−	0.0821995i
$149$	−18.6969			−1.53171			−0.765856	−	0.643012i	$-0.777685\pi$
$149$	−18.6969			−1.53171			−0.765856	+	0.643012i	$0.777685\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		−	3.55051i		−	0.287984i
$153$		0			0
$154$	−21.7980			−1.75653
$155$	−3.10102	+	1.55051i	−0.249080	+	0.124540i
$156$		0			0
$157$		−	7.79796i		−	0.622345i	−0.950353	−	0.311172i	$-0.899278\pi$
$157$		−	7.79796i		−	0.622345i	0.950353	−	0.311172i	$-0.100722\pi$
$158$			6.44949i			0.513094i
$159$		0			0
$160$	−1.00000	−	2.00000i	−0.0790569	−	0.158114i
$161$	39.5959			3.12060
$162$		0			0
$163$			21.7980i			1.70735i	0.520808	+	0.853674i	$0.325631\pi$
$163$			21.7980i			1.70735i	−0.520808	+	0.853674i	$0.674369\pi$
$164$	2.00000			0.156174
$165$		0			0
$166$	−9.55051			−0.741263
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$	9.79796	−	4.89898i	0.751469	−	0.375735i
$171$		0			0
$172$			4.00000i			0.304997i
$173$			19.7980i			1.50521i	0.658472	+	0.752605i	$0.271203\pi$
$173$			19.7980i			1.50521i	−0.658472	+	0.752605i	$0.728797\pi$
$174$		0			0
$175$	17.7980	+	13.3485i	1.34540	+	1.00905i
$176$	−4.89898			−0.369274
$177$		0			0
$178$		−	15.7980i		−	1.18411i
$179$	−9.34847			−0.698737			−0.349369	−	0.936985i	$-0.613604\pi$
$179$	−9.34847			−0.698737			−0.349369	+	0.936985i	$0.613604\pi$
$180$		0			0
$181$	10.6969			0.795097			0.397549	−	0.917581i	$-0.369861\pi$
$181$	10.6969			0.795097			0.397549	+	0.917581i	$0.369861\pi$
$182$		−	17.7980i		−	1.31927i
$183$		0			0
$184$	8.89898			0.656041
$185$	1.00000	+	2.00000i	0.0735215	+	0.147043i
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$			4.44949i			0.324512i
$189$		0			0
$190$	3.55051	+	7.10102i	0.257581	+	0.515162i
$191$	−11.3485			−0.821146			−0.410573	−	0.911828i	$-0.634671\pi$
$191$	−11.3485			−0.821146			−0.410573	+	0.911828i	$0.634671\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−2.00000			−0.143592
$195$		0			0
$196$	12.7980			0.914140
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−6.44949			−0.457192			−0.228596	−	0.973521i	$-0.573414\pi$
$199$	−6.44949			−0.457192			−0.228596	+	0.973521i	$0.573414\pi$
$200$	4.00000	+	3.00000i	0.282843	+	0.212132i
$201$		0			0
$202$			10.6969i			0.752634i
$203$		0			0
$204$		0			0
$205$	−4.00000	+	2.00000i	−0.279372	+	0.139686i
$206$	−9.79796			−0.682656
$207$		0			0
$208$		−	4.00000i		−	0.277350i
$209$	17.3939			1.20316
$210$		0			0
$211$	−6.69694			−0.461036			−0.230518	−	0.973068i	$-0.574042\pi$
$211$	−6.69694			−0.461036			−0.230518	+	0.973068i	$0.574042\pi$
$212$			11.7980i			0.810287i
$213$		0			0
$214$	5.55051			0.379425
$215$	−4.00000	−	8.00000i	−0.272798	−	0.545595i
$216$		0			0
$217$		−	6.89898i		−	0.468333i
$218$		−	5.79796i		−	0.392687i
$219$		0			0
$220$	9.79796	−	4.89898i	0.660578	−	0.330289i
$221$	19.5959			1.31816
$222$		0			0
$223$		−	0.449490i		−	0.0301001i	−0.999887	−	0.0150500i	$-0.995209\pi$
$223$		−	0.449490i		−	0.0301001i	0.999887	−	0.0150500i	$-0.00479075\pi$
$224$	4.44949			0.297294
$225$		0			0
$226$	−3.10102			−0.206277
$227$			24.8990i			1.65260i	0.563228	+	0.826302i	$0.309559\pi$
$227$			24.8990i			1.65260i	−0.563228	+	0.826302i	$0.690441\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	−17.7980	+	8.89898i	−1.17356	+	0.586781i
$231$		0			0
$232$		0			0
$233$			9.79796i			0.641886i	0.947099	+	0.320943i	$0.104000\pi$
$233$			9.79796i			0.641886i	−0.947099	+	0.320943i	$0.896000\pi$
$234$		0			0
$235$	−4.44949	−	8.89898i	−0.290253	−	0.580505i
$236$	3.55051			0.231119
$237$		0			0
$238$			21.7980i			1.41295i
$239$	−18.0454			−1.16726			−0.583630	−	0.812020i	$-0.698368\pi$
$239$	−18.0454			−1.16726			−0.583630	+	0.812020i	$0.698368\pi$
$240$		0			0
$241$	7.79796			0.502311			0.251155	−	0.967947i	$-0.419189\pi$
$241$	7.79796			0.502311			0.251155	+	0.967947i	$0.419189\pi$
$242$		−	13.0000i		−	0.835672i
$243$		0			0
$244$	−12.0000			−0.768221
$245$	−25.5959	+	12.7980i	−1.63526	+	0.817632i
$246$		0			0
$247$			14.2020i			0.903654i
$248$		−	1.55051i		−	0.0984575i
$249$		0			0
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	−21.3485			−1.34750			−0.673752	−	0.738958i	$-0.735318\pi$
$251$	−21.3485			−1.34750			−0.673752	+	0.738958i	$0.735318\pi$
$252$		0			0
$253$			43.5959i			2.74085i
$254$	−2.65153			−0.166372
$255$		0			0
$256$	1.00000			0.0625000
$257$			4.89898i			0.305590i	0.988258	+	0.152795i	$0.0488274\pi$
$257$			4.89898i			0.305590i	−0.988258	+	0.152795i	$0.951173\pi$
$258$		0			0
$259$	−4.44949			−0.276478
$260$	4.00000	+	8.00000i	0.248069	+	0.496139i
$261$		0			0
$262$		−	10.2474i		−	0.633089i
$263$			1.75255i			0.108067i	0.998539	+	0.0540335i	$0.0172078\pi$
$263$			1.75255i			0.108067i	−0.998539	+	0.0540335i	$0.982792\pi$
$264$		0			0
$265$	−11.7980	−	23.5959i	−0.724743	−	1.44949i
$266$	−15.7980			−0.968635
$267$		0			0
$268$			5.55051i			0.339051i
$269$	−18.6969			−1.13997			−0.569986	−	0.821654i	$-0.693051\pi$
$269$	−18.6969			−1.13997			−0.569986	+	0.821654i	$0.693051\pi$
$270$		0			0
$271$	−32.4949			−1.97392			−0.986962	−	0.160952i	$-0.948544\pi$
$271$	−32.4949			−1.97392			−0.986962	+	0.160952i	$0.948544\pi$
$272$			4.89898i			0.297044i
$273$		0			0
$274$	−19.5959			−1.18383
$275$	−14.6969	+	19.5959i	−0.886259	+	1.18168i
$276$		0			0
$277$			19.5959i			1.17740i	0.808350	+	0.588702i	$0.200361\pi$
$277$			19.5959i			1.17740i	−0.808350	+	0.588702i	$0.799639\pi$
$278$		−	5.79796i		−	0.347738i
$279$		0			0
$280$	−8.89898	+	4.44949i	−0.531816	+	0.265908i
$281$	−27.7980			−1.65829			−0.829144	−	0.559036i	$-0.811171\pi$
$281$	−27.7980			−1.65829			−0.829144	+	0.559036i	$0.811171\pi$
$282$		0			0
$283$		−	15.5959i		−	0.927081i	−0.886076	−	0.463541i	$-0.846579\pi$
$283$		−	15.5959i		−	0.927081i	0.886076	−	0.463541i	$-0.153421\pi$
$284$	4.89898			0.290701
$285$		0			0
$286$	19.5959			1.15873
$287$		−	8.89898i		−	0.525290i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		0			0
$292$			4.00000i			0.234082i
$293$			25.5959i			1.49533i	0.664076	+	0.747665i	$0.268825\pi$
$293$			25.5959i			1.49533i	−0.664076	+	0.747665i	$0.731175\pi$
$294$		0			0
$295$	−7.10102	+	3.55051i	−0.413437	+	0.206719i
$296$	−1.00000			−0.0581238
$297$		0			0
$298$			18.6969i			1.08308i
$299$	−35.5959			−2.05857
$300$		0			0
$301$	17.7980			1.02586
$302$			8.00000i			0.460348i
$303$		0			0
$304$	−3.55051			−0.203636
$305$	24.0000	−	12.0000i	1.37424	−	0.687118i
$306$		0			0
$307$			13.1464i			0.750306i	0.926963	+	0.375153i	$0.122410\pi$
$307$			13.1464i			0.750306i	−0.926963	+	0.375153i	$0.877590\pi$
$308$			21.7980i			1.24205i
$309$		0			0
$310$	1.55051	+	3.10102i	0.0880631	+	0.176126i
$311$	7.34847			0.416693			0.208347	−	0.978055i	$-0.433192\pi$
$311$	7.34847			0.416693			0.208347	+	0.978055i	$0.433192\pi$
$312$		0			0
$313$			17.5959i			0.994580i	0.867584	+	0.497290i	$0.165672\pi$
$313$			17.5959i			0.994580i	−0.867584	+	0.497290i	$0.834328\pi$
$314$	−7.79796			−0.440064
$315$		0			0
$316$	6.44949			0.362812
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$	−2.00000	+	1.00000i	−0.111803	+	0.0559017i
$321$		0			0
$322$		−	39.5959i		−	2.20659i
$323$		−	17.3939i		−	0.967821i
$324$		0			0
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	21.7980			1.20728
$327$		0			0
$328$		−	2.00000i		−	0.110432i
$329$	19.7980			1.09150
$330$		0			0
$331$	14.2474			0.783111			0.391555	−	0.920155i	$-0.371937\pi$
$331$	14.2474			0.783111			0.391555	+	0.920155i	$0.371937\pi$
$332$			9.55051i			0.524152i
$333$		0			0
$334$	−8.00000			−0.437741
$335$	−5.55051	−	11.1010i	−0.303257	−	0.606514i
$336$		0			0
$337$		−	3.59592i		−	0.195882i	−0.995192	−	0.0979411i	$-0.968774\pi$
$337$		−	3.59592i		−	0.195882i	0.995192	−	0.0979411i	$-0.0312257\pi$
$338$			3.00000i			0.163178i
$339$		0			0
$340$	−4.89898	−	9.79796i	−0.265684	−	0.531369i
$341$	7.59592			0.411342
$342$		0			0
$343$		−	25.7980i		−	1.39296i
$344$	4.00000			0.215666
$345$		0			0
$346$	19.7980			1.06434
$347$		−	2.20204i		−	0.118212i	−0.998252	−	0.0591059i	$-0.981175\pi$
$347$		−	2.20204i		−	0.118212i	0.998252	−	0.0591059i	$-0.0188250\pi$
$348$		0			0
$349$	−7.10102			−0.380109			−0.190054	−	0.981774i	$-0.560866\pi$
$349$	−7.10102			−0.380109			−0.190054	+	0.981774i	$0.560866\pi$
$350$	13.3485	−	17.7980i	0.713506	−	0.951341i
$351$		0			0
$352$			4.89898i			0.261116i
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$	−9.79796	+	4.89898i	−0.520022	+	0.260011i
$356$	−15.7980			−0.837290
$357$		0			0
$358$			9.34847i			0.494082i
$359$	−12.8990			−0.680782			−0.340391	−	0.940284i	$-0.610559\pi$
$359$	−12.8990			−0.680782			−0.340391	+	0.940284i	$0.610559\pi$
$360$		0			0
$361$	−6.39388			−0.336520
$362$		−	10.6969i		−	0.562219i
$363$		0			0
$364$	−17.7980			−0.932867
$365$	−4.00000	−	8.00000i	−0.209370	−	0.418739i
$366$		0			0
$367$		−	2.65153i		−	0.138409i	−0.997603	−	0.0692044i	$-0.977954\pi$
$367$		−	2.65153i		−	0.138409i	0.997603	−	0.0692044i	$-0.0220461\pi$
$368$		−	8.89898i		−	0.463891i
$369$		0			0
$370$	2.00000	−	1.00000i	0.103975	−	0.0519875i
$371$	52.4949			2.72540
$372$		0			0
$373$			11.7980i			0.610875i	0.952212	+	0.305438i	$0.0988027\pi$
$373$			11.7980i			0.610875i	−0.952212	+	0.305438i	$0.901197\pi$
$374$	−24.0000			−1.24101
$375$		0			0
$376$	4.44949			0.229465
$377$		0			0
$378$		0			0
$379$	31.5959			1.62297			0.811487	−	0.584371i	$-0.198659\pi$
$379$	31.5959			1.62297			0.811487	+	0.584371i	$0.198659\pi$
$380$	7.10102	−	3.55051i	0.364275	−	0.182137i
$381$		0			0
$382$			11.3485i			0.580638i
$383$		−	34.6969i		−	1.77293i	−0.462795	−	0.886465i	$-0.653153\pi$
$383$		−	34.6969i		−	1.77293i	0.462795	−	0.886465i	$-0.346847\pi$
$384$		0			0
$385$	−21.7980	−	43.5959i	−1.11093	−	2.22185i
$386$	−14.0000			−0.712581
$387$		0			0
$388$			2.00000i			0.101535i
$389$	−25.7980			−1.30801			−0.654004	−	0.756491i	$-0.726912\pi$
$389$	−25.7980			−1.30801			−0.654004	+	0.756491i	$0.726912\pi$
$390$		0			0
$391$	43.5959			2.20474
$392$		−	12.7980i		−	0.646395i
$393$		0			0
$394$	2.00000			0.100759
$395$	−12.8990	+	6.44949i	−0.649018	+	0.324509i
$396$		0			0
$397$		−	16.2020i		−	0.813157i	−0.913616	−	0.406579i	$-0.866722\pi$
$397$		−	16.2020i		−	0.813157i	0.913616	−	0.406579i	$-0.133278\pi$
$398$			6.44949i			0.323284i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	23.7980			1.18841			0.594207	−	0.804312i	$-0.297466\pi$
$401$	23.7980			1.18841			0.594207	+	0.804312i	$0.297466\pi$
$402$		0			0
$403$			6.20204i			0.308946i
$404$	10.6969			0.532193
$405$		0			0
$406$		0			0
$407$		−	4.89898i		−	0.242833i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	2.00000	+	4.00000i	0.0987730	+	0.197546i
$411$		0			0
$412$			9.79796i			0.482711i
$413$		−	15.7980i		−	0.777367i
$414$		0			0
$415$	−9.55051	−	19.1010i	−0.468816	−	0.937632i
$416$	−4.00000			−0.196116
$417$		0			0
$418$		−	17.3939i		−	0.850762i
$419$	5.79796			0.283249			0.141624	−	0.989920i	$-0.454767\pi$
$419$	5.79796			0.283249			0.141624	+	0.989920i	$0.454767\pi$
$420$		0			0
$421$	−19.5959			−0.955047			−0.477523	−	0.878619i	$-0.658465\pi$
$421$	−19.5959			−0.955047			−0.477523	+	0.878619i	$0.658465\pi$
$422$			6.69694i			0.326002i
$423$		0			0
$424$	11.7980			0.572960
$425$	19.5959	+	14.6969i	0.950542	+	0.712906i
$426$		0			0
$427$			53.3939i			2.58391i
$428$		−	5.55051i		−	0.268294i
$429$		0			0
$430$	−8.00000	+	4.00000i	−0.385794	+	0.192897i
$431$	15.7526			0.758774			0.379387	−	0.925238i	$-0.376135\pi$
$431$	15.7526			0.758774			0.379387	+	0.925238i	$0.376135\pi$
$432$		0			0
$433$		−	21.3939i		−	1.02812i	−0.857753	−	0.514062i	$-0.828140\pi$
$433$		−	21.3939i		−	1.02812i	0.857753	−	0.514062i	$-0.171860\pi$
$434$	−6.89898			−0.331162
$435$		0			0
$436$	−5.79796			−0.277672
$437$			31.5959i			1.51144i
$438$		0			0
$439$	−20.6515			−0.985644			−0.492822	−	0.870130i	$-0.664035\pi$
$439$	−20.6515			−0.985644			−0.492822	+	0.870130i	$0.664035\pi$
$440$	−4.89898	−	9.79796i	−0.233550	−	0.467099i
$441$		0			0
$442$		−	19.5959i		−	0.932083i
$443$			24.6515i			1.17123i	0.810589	+	0.585615i	$0.199147\pi$
$443$			24.6515i			1.17123i	−0.810589	+	0.585615i	$0.800853\pi$
$444$		0			0
$445$	31.5959	−	15.7980i	1.49779	−	0.748895i
$446$	−0.449490			−0.0212840
$447$		0			0
$448$		−	4.44949i		−	0.210219i
$449$	−15.7980			−0.745552			−0.372776	−	0.927921i	$-0.621594\pi$
$449$	−15.7980			−0.745552			−0.372776	+	0.927921i	$0.621594\pi$
$450$		0			0
$451$	9.79796			0.461368
$452$			3.10102i			0.145860i
$453$		0			0
$454$	24.8990			1.16857
$455$	35.5959	−	17.7980i	1.66876	−	0.834381i
$456$		0			0
$457$		−	2.00000i		−	0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$		−	2.00000i		−	0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$		−	10.0000i		−	0.467269i
$459$		0			0
$460$	8.89898	+	17.7980i	0.414917	+	0.829834i
$461$	33.7980			1.57413			0.787064	−	0.616871i	$-0.211600\pi$
$461$	33.7980			1.57413			0.787064	+	0.616871i	$0.211600\pi$
$462$		0			0
$463$			20.4949i			0.952479i	0.879316	+	0.476239i	$0.158000\pi$
$463$			20.4949i			0.952479i	−0.879316	+	0.476239i	$0.842000\pi$
$464$		0			0
$465$		0			0
$466$	9.79796			0.453882
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$	24.6969			1.14040
$470$	−8.89898	+	4.44949i	−0.410479	+	0.205240i
$471$		0			0
$472$		−	3.55051i		−	0.163425i
$473$			19.5959i			0.901021i
$474$		0			0
$475$	−10.6515	+	14.2020i	−0.488726	+	0.651634i
$476$	21.7980			0.999108
$477$		0			0
$478$			18.0454i			0.825378i
$479$	5.14643			0.235146			0.117573	−	0.993064i	$-0.462489\pi$
$479$	5.14643			0.235146			0.117573	+	0.993064i	$0.462489\pi$
$480$		0			0
$481$	4.00000			0.182384
$482$		−	7.79796i		−	0.355187i
$483$		0			0
$484$	−13.0000			−0.590909
$485$	−2.00000	−	4.00000i	−0.0908153	−	0.181631i
$486$		0			0
$487$		−	29.3939i		−	1.33196i	−0.745968	−	0.665982i	$-0.768013\pi$
$487$		−	29.3939i		−	1.33196i	0.745968	−	0.665982i	$-0.231987\pi$
$488$			12.0000i			0.543214i
$489$		0			0
$490$	12.7980	+	25.5959i	0.578153	+	1.15631i
$491$	9.30306			0.419841			0.209921	−	0.977718i	$-0.432679\pi$
$491$	9.30306			0.419841			0.209921	+	0.977718i	$0.432679\pi$
$492$		0			0
$493$		0			0
$494$	14.2020			0.638980
$495$		0			0
$496$	−1.55051			−0.0696200
$497$		−	21.7980i		−	0.977772i
$498$		0			0
$499$	10.6515			0.476828			0.238414	−	0.971164i	$-0.423372\pi$
$499$	10.6515			0.476828			0.238414	+	0.971164i	$0.423372\pi$
$500$	−2.00000	+	11.0000i	−0.0894427	+	0.491935i
$501$		0			0
$502$			21.3485i			0.952829i
$503$		−	1.79796i		−	0.0801670i	−0.999196	−	0.0400835i	$-0.987238\pi$
$503$		−	1.79796i		−	0.0801670i	0.999196	−	0.0400835i	$-0.0127624\pi$
$504$		0			0
$505$	−21.3939	+	10.6969i	−0.952015	+	0.476008i
$506$	43.5959			1.93807
$507$		0			0
$508$			2.65153i			0.117643i
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	17.7980			0.787335
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	4.89898			0.216085
$515$	−9.79796	−	19.5959i	−0.431750	−	0.863499i
$516$		0			0
$517$			21.7980i			0.958673i
$518$			4.44949i			0.195499i
$519$		0			0
$520$	8.00000	−	4.00000i	0.350823	−	0.175412i
$521$	−17.7980			−0.779743			−0.389871	−	0.920869i	$-0.627481\pi$
$521$	−17.7980			−0.779743			−0.389871	+	0.920869i	$0.627481\pi$
$522$		0			0
$523$			8.89898i			0.389125i	0.980890	+	0.194563i	$0.0623287\pi$
$523$			8.89898i			0.389125i	−0.980890	+	0.194563i	$0.937671\pi$
$524$	−10.2474			−0.447662
$525$		0			0
$526$	1.75255			0.0764149
$527$		−	7.59592i		−	0.330883i
$528$		0			0
$529$	−56.1918			−2.44312
$530$	−23.5959	+	11.7980i	−1.02494	+	0.512471i
$531$		0			0
$532$			15.7980i			0.684928i
$533$			8.00000i			0.346518i
$534$		0			0
$535$	5.55051	+	11.1010i	0.239969	+	0.479939i
$536$	5.55051			0.239746
$537$		0			0
$538$			18.6969i			0.806082i
$539$	62.6969			2.70055
$540$		0			0
$541$	0.404082			0.0173728			0.00868642	−	0.999962i	$-0.497235\pi$
$541$	0.404082			0.0173728			0.00868642	+	0.999962i	$0.497235\pi$
$542$			32.4949i			1.39578i
$543$		0			0
$544$	4.89898			0.210042
$545$	11.5959	−	5.79796i	0.496715	−	0.248357i
$546$		0			0
$547$		−	10.6969i		−	0.457368i	−0.973501	−	0.228684i	$-0.926558\pi$
$547$		−	10.6969i		−	0.457368i	0.973501	−	0.228684i	$-0.0734423\pi$
$548$			19.5959i			0.837096i
$549$		0			0
$550$	19.5959	+	14.6969i	0.835573	+	0.626680i
$551$		0			0
$552$		0			0
$553$		−	28.6969i		−	1.22032i
$554$	19.5959			0.832551
$555$		0			0
$556$	−5.79796			−0.245888
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$	4.44949	+	8.89898i	0.188025	+	0.376051i
$561$		0			0
$562$			27.7980i			1.17259i
$563$		−	34.6969i		−	1.46230i	−0.682216	−	0.731151i	$-0.738984\pi$
$563$		−	34.6969i		−	1.46230i	0.682216	−	0.731151i	$-0.261016\pi$
$564$		0			0
$565$	−3.10102	−	6.20204i	−0.130461	−	0.260922i
$566$	−15.5959			−0.655545
$567$		0			0
$568$		−	4.89898i		−	0.205557i
$569$	21.5959			0.905348			0.452674	−	0.891676i	$-0.350470\pi$
$569$	21.5959			0.905348			0.452674	+	0.891676i	$0.350470\pi$
$570$		0			0
$571$	−25.3939			−1.06270			−0.531350	−	0.847152i	$-0.678315\pi$
$571$	−25.3939			−1.06270			−0.531350	+	0.847152i	$0.678315\pi$
$572$		−	19.5959i		−	0.819346i
$573$		0			0
$574$	−8.89898			−0.371436
$575$	−35.5959	−	26.6969i	−1.48445	−	1.11334i
$576$		0			0
$577$		−	30.6969i		−	1.27793i	−0.769236	−	0.638965i	$-0.779363\pi$
$577$		−	30.6969i		−	1.27793i	0.769236	−	0.638965i	$-0.220637\pi$
$578$			7.00000i			0.291162i
$579$		0			0
$580$		0			0
$581$	42.4949			1.76299
$582$		0			0
$583$			57.7980i			2.39375i
$584$	4.00000			0.165521
$585$		0			0
$586$	25.5959			1.05736
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	5.50510			0.226834
$590$	3.55051	+	7.10102i	0.146172	+	0.292344i
$591$		0			0
$592$			1.00000i			0.0410997i
$593$			24.0000i			0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$			24.0000i			0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$		0			0
$595$	−43.5959	+	21.7980i	−1.78726	+	0.893629i
$596$	18.6969			0.765856
$597$		0			0
$598$			35.5959i			1.45563i
$599$	11.5959			0.473796			0.236898	−	0.971534i	$-0.423869\pi$
$599$	11.5959			0.473796			0.236898	+	0.971534i	$0.423869\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		−	17.7980i		−	0.725391i
$603$		0			0
$604$	8.00000			0.325515
$605$	26.0000	−	13.0000i	1.05705	−	0.528525i
$606$		0			0
$607$			26.6969i			1.08360i	0.840509	+	0.541798i	$0.182256\pi$
$607$			26.6969i			1.08360i	−0.840509	+	0.541798i	$0.817744\pi$
$608$			3.55051i			0.143992i
$609$		0			0
$610$	−12.0000	−	24.0000i	−0.485866	−	0.971732i
$611$	−17.7980			−0.720028
$612$		0			0
$613$		−	39.7980i		−	1.60742i	−0.595018	−	0.803712i	$-0.702855\pi$
$613$		−	39.7980i		−	1.60742i	0.595018	−	0.803712i	$-0.297145\pi$
$614$	13.1464			0.530547
$615$		0			0
$616$	21.7980			0.878265
$617$		−	19.5959i		−	0.788902i	−0.918917	−	0.394451i	$-0.870935\pi$
$617$		−	19.5959i		−	0.788902i	0.918917	−	0.394451i	$-0.129065\pi$
$618$		0			0
$619$	12.8990			0.518454			0.259227	−	0.965816i	$-0.416532\pi$
$619$	12.8990			0.518454			0.259227	+	0.965816i	$0.416532\pi$
$620$	3.10102	−	1.55051i	0.124540	−	0.0622700i
$621$		0			0
$622$		−	7.34847i		−	0.294647i
$623$			70.2929i			2.81622i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	17.5959			0.703274
$627$		0			0
$628$			7.79796i			0.311172i
$629$	−4.89898			−0.195335
$630$		0			0
$631$	−20.2474			−0.806038			−0.403019	−	0.915192i	$-0.632039\pi$
$631$	−20.2474			−0.806038			−0.403019	+	0.915192i	$0.632039\pi$
$632$		−	6.44949i		−	0.256547i
$633$		0			0
$634$	−18.0000			−0.714871
$635$	−2.65153	−	5.30306i	−0.105223	−	0.210446i
$636$		0			0
$637$			51.1918i			2.02829i
$638$		0			0
$639$		0			0
$640$	1.00000	+	2.00000i	0.0395285	+	0.0790569i
$641$	−17.7980			−0.702977			−0.351489	−	0.936192i	$-0.614324\pi$
$641$	−17.7980			−0.702977			−0.351489	+	0.936192i	$0.614324\pi$
$642$		0			0
$643$			23.1010i			0.911015i	0.890232	+	0.455508i	$0.150542\pi$
$643$			23.1010i			0.911015i	−0.890232	+	0.455508i	$0.849458\pi$
$644$	−39.5959			−1.56030
$645$		0			0
$646$	−17.3939			−0.684353
$647$			32.0000i			1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$			32.0000i			1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$		0			0
$649$	17.3939			0.682769
$650$	−12.0000	+	16.0000i	−0.470679	+	0.627572i
$651$		0			0
$652$		−	21.7980i		−	0.853674i
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$	20.4949	−	10.2474i	0.800802	−	0.400401i
$656$	−2.00000			−0.0780869
$657$		0			0
$658$		−	19.7980i		−	0.771805i
$659$	31.5959			1.23080			0.615401	−	0.788214i	$-0.288994\pi$
$659$	31.5959			1.23080			0.615401	+	0.788214i	$0.288994\pi$
$660$		0			0
$661$	−31.1918			−1.21322			−0.606611	−	0.794999i	$-0.707471\pi$
$661$	−31.1918			−1.21322			−0.606611	+	0.794999i	$0.707471\pi$
$662$		−	14.2474i		−	0.553743i
$663$		0			0
$664$	9.55051			0.370632
$665$	−15.7980	−	31.5959i	−0.612619	−	1.22524i
$666$		0			0
$667$		0			0
$668$			8.00000i			0.309529i
$669$		0			0
$670$	−11.1010	+	5.55051i	−0.428870	+	0.214435i
$671$	−58.7878			−2.26948
$672$		0			0
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$	−3.59592			−0.138510
$675$		0			0
$676$	3.00000			0.115385
$677$		−	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
$679$	8.89898			0.341511
$680$	−9.79796	+	4.89898i	−0.375735	+	0.187867i
$681$		0			0
$682$		−	7.59592i		−	0.290863i
$683$			15.5959i			0.596761i	0.954447	+	0.298381i	$0.0964465\pi$
$683$			15.5959i			0.596761i	−0.954447	+	0.298381i	$0.903554\pi$
$684$		0			0
$685$	−19.5959	−	39.1918i	−0.748722	−	1.49744i
$686$	−25.7980			−0.984971
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	−47.1918			−1.79787
$690$		0			0
$691$	37.7980			1.43790			0.718951	−	0.695061i	$-0.244623\pi$
$691$	37.7980			1.43790			0.718951	+	0.695061i	$0.244623\pi$
$692$		−	19.7980i		−	0.752605i
$693$		0			0
$694$	−2.20204			−0.0835883
$695$	11.5959	−	5.79796i	0.439858	−	0.219929i
$696$		0			0
$697$		−	9.79796i		−	0.371124i
$698$			7.10102i			0.268778i
$699$		0			0
$700$	−17.7980	−	13.3485i	−0.672700	−	0.504525i
$701$	13.7980			0.521142			0.260571	−	0.965455i	$-0.416089\pi$
$701$	13.7980			0.521142			0.260571	+	0.965455i	$0.416089\pi$
$702$		0			0
$703$		−	3.55051i		−	0.133910i
$704$	4.89898			0.184637
$705$		0			0
$706$	−6.00000			−0.225813
$707$		−	47.5959i		−	1.79003i
$708$		0			0
$709$	−5.79796			−0.217747			−0.108873	−	0.994056i	$-0.534724\pi$
$709$	−5.79796			−0.217747			−0.108873	+	0.994056i	$0.534724\pi$
$710$	4.89898	+	9.79796i	0.183855	+	0.367711i
$711$		0			0
$712$			15.7980i			0.592054i
$713$			13.7980i			0.516738i
$714$		0			0
$715$	19.5959	+	39.1918i	0.732846	+	1.46569i
$716$	9.34847			0.349369
$717$		0			0
$718$			12.8990i			0.481386i
$719$	51.5959			1.92420			0.962102	−	0.272692i	$-0.0879138\pi$
$719$	51.5959			1.92420			0.962102	+	0.272692i	$0.0879138\pi$
$720$		0			0
$721$	43.5959			1.62360
$722$			6.39388i			0.237955i
$723$		0			0
$724$	−10.6969			−0.397549
$725$		0			0
$726$		0			0
$727$			0.898979i			0.0333413i	0.999861	+	0.0166707i	$0.00530668\pi$
$727$			0.898979i			0.0333413i	−0.999861	+	0.0166707i	$0.994693\pi$
$728$			17.7980i			0.659636i
$729$		0			0
$730$	−8.00000	+	4.00000i	−0.296093	+	0.148047i
$731$	19.5959			0.724781
$732$		0			0
$733$			17.5959i			0.649920i	0.945728	+	0.324960i	$0.105351\pi$
$733$			17.5959i			0.649920i	−0.945728	+	0.324960i	$0.894649\pi$
$734$	−2.65153			−0.0978698
$735$		0			0
$736$	−8.89898			−0.328021
$737$			27.1918i			1.00162i
$738$		0			0
$739$	45.7980			1.68471			0.842353	−	0.538927i	$-0.181170\pi$
$739$	45.7980			1.68471			0.842353	+	0.538927i	$0.181170\pi$
$740$	−1.00000	−	2.00000i	−0.0367607	−	0.0735215i
$741$		0			0
$742$		−	52.4949i		−	1.92715i
$743$		−	13.7526i		−	0.504532i	−0.967658	−	0.252266i	$-0.918824\pi$
$743$		−	13.7526i		−	0.504532i	0.967658	−	0.252266i	$-0.0811758\pi$
$744$		0			0
$745$	−37.3939	+	18.6969i	−1.37001	+	0.685003i
$746$	11.7980			0.431954
$747$		0			0
$748$			24.0000i			0.877527i
$749$	−24.6969			−0.902406
$750$		0			0
$751$	30.6969			1.12015			0.560074	−	0.828443i	$-0.310773\pi$
$751$	30.6969			1.12015			0.560074	+	0.828443i	$0.310773\pi$
$752$		−	4.44949i		−	0.162256i
$753$		0			0
$754$		0			0
$755$	−16.0000	+	8.00000i	−0.582300	+	0.291150i
$756$		0			0
$757$		−	33.5959i		−	1.22106i	−0.791991	−	0.610532i	$-0.790956\pi$
$757$		−	33.5959i		−	1.22106i	0.791991	−	0.610532i	$-0.209044\pi$
$758$		−	31.5959i		−	1.14762i
$759$		0			0
$760$	−3.55051	−	7.10102i	−0.128791	−	0.257581i
$761$	−25.1918			−0.913203			−0.456602	−	0.889671i	$-0.650934\pi$
$761$	−25.1918			−0.913203			−0.456602	+	0.889671i	$0.650934\pi$
$762$		0			0
$763$			25.7980i			0.933949i
$764$	11.3485			0.410573
$765$		0			0
$766$	−34.6969			−1.25365
$767$			14.2020i			0.512806i
$768$		0			0
$769$	35.7980			1.29091			0.645454	−	0.763799i	$-0.276668\pi$
$769$	35.7980			1.29091			0.645454	+	0.763799i	$0.276668\pi$
$770$	−43.5959	+	21.7980i	−1.57109	+	0.785544i
$771$		0			0
$772$			14.0000i			0.503871i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$	−4.65153	+	6.20204i	−0.167088	+	0.222784i
$776$	2.00000			0.0717958
$777$		0			0
$778$			25.7980i			0.924902i
$779$	7.10102			0.254420
$780$		0			0
$781$	24.0000			0.858788
$782$		−	43.5959i		−	1.55899i
$783$		0			0
$784$	−12.7980			−0.457070
$785$	−7.79796	−	15.5959i	−0.278321	−	0.556642i
$786$		0			0
$787$		−	28.7423i		−	1.02455i	−0.858820	−	0.512277i	$-0.828802\pi$
$787$		−	28.7423i		−	1.02455i	0.858820	−	0.512277i	$-0.171198\pi$
$788$		−	2.00000i		−	0.0712470i
$789$		0			0
$790$	6.44949	+	12.8990i	0.229463	+	0.458925i
$791$	13.7980			0.490599
$792$		0			0

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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +4.44949i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(2.00000 - 1.00000i) q^{5} +4.44949i q^{7} +1.00000i q^{8} +(-1.00000 - 2.00000i) q^{10} -4.89898 q^{11} -4.00000i q^{13} +4.44949 q^{14} +1.00000 q^{16} +4.89898i q^{17} -3.55051 q^{19} +(-2.00000 + 1.00000i) q^{20} +4.89898i q^{22} -8.89898i q^{23} +(3.00000 - 4.00000i) q^{25} -4.00000 q^{26} -4.44949i q^{28} -1.55051 q^{31} -1.00000i q^{32} +4.89898 q^{34} +(4.44949 + 8.89898i) q^{35} +1.00000i q^{37} +3.55051i q^{38} +(1.00000 + 2.00000i) q^{40} -2.00000 q^{41} -4.00000i q^{43} +4.89898 q^{44} -8.89898 q^{46} -4.44949i q^{47} -12.7980 q^{49} +(-4.00000 - 3.00000i) q^{50} +4.00000i q^{52} -11.7980i q^{53} +(-9.79796 + 4.89898i) q^{55} -4.44949 q^{56} -3.55051 q^{59} +12.0000 q^{61} +1.55051i q^{62} -1.00000 q^{64} +(-4.00000 - 8.00000i) q^{65} -5.55051i q^{67} -4.89898i q^{68} +(8.89898 - 4.44949i) q^{70} -4.89898 q^{71} -4.00000i q^{73} +1.00000 q^{74} +3.55051 q^{76} -21.7980i q^{77} -6.44949 q^{79} +(2.00000 - 1.00000i) q^{80} +2.00000i q^{82} -9.55051i q^{83} +(4.89898 + 9.79796i) q^{85} -4.00000 q^{86} -4.89898i q^{88} +15.7980 q^{89} +17.7980 q^{91} +8.89898i q^{92} -4.44949 q^{94} +(-7.10102 + 3.55051i) q^{95} -2.00000i q^{97} +12.7980i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{5}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^5	\( 4 q - 4 q^{4} + 8 q^{5} - 4 q^{10} + 8 q^{14} + 4 q^{16} - 24 q^{19} - 8 q^{20} + 12 q^{25} - 16 q^{26} - 16 q^{31} + 8 q^{35} + 4 q^{40} - 8 q^{41} - 16 q^{46} - 12 q^{49} - 16 q^{50} - 8 q^{56} - 24 q^{59} + 48 q^{61} - 4 q^{64} - 16 q^{65} + 16 q^{70} + 4 q^{74} + 24 q^{76} - 16 q^{79} + 8 q^{80} - 16 q^{86} + 24 q^{89} + 32 q^{91} - 8 q^{94} - 48 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^5 - 4 * q^10 + 8 * q^14 + 4 * q^16 - 24 * q^19 - 8 * q^20 + 12 * q^25 - 16 * q^26 - 16 * q^31 + 8 * q^35 + 4 * q^40 - 8 * q^41 - 16 * q^46 - 12 * q^49 - 16 * q^50 - 8 * q^56 - 24 * q^59 + 48 * q^61 - 4 * q^64 - 16 * q^65 + 16 * q^70 + 4 * q^74 + 24 * q^76 - 16 * q^79 + 8 * q^80 - 16 * q^86 + 24 * q^89 + 32 * q^91 - 8 * q^94 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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