LMFDB - Embedded newform 4014.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.a (trivial)

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:	yes
Analytic conductor:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.232773917.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} - x^{3} + 33x^{2} + 5x - 22 $ x^6 - 12x^4 - x^3 + 33x^2 + 5*x - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.29309$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.29309 q^{5} +0.862607 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.29309 q^{5} +0.862607 q^{7} -1.00000 q^{8} +2.29309 q^{10} -2.58766 q^{11} -2.74336 q^{13} -0.862607 q^{14} +1.00000 q^{16} +0.677825 q^{17} +4.05298 q^{19} -2.29309 q^{20} +2.58766 q^{22} +6.02686 q^{23} +0.258243 q^{25} +2.74336 q^{26} +0.862607 q^{28} +0.415427 q^{29} -3.11543 q^{31} -1.00000 q^{32} -0.677825 q^{34} -1.97803 q^{35} +3.92798 q^{37} -4.05298 q^{38} +2.29309 q^{40} +6.70469 q^{41} +2.84048 q^{43} -2.58766 q^{44} -6.02686 q^{46} -0.984949 q^{47} -6.25591 q^{49} -0.258243 q^{50} -2.74336 q^{52} -0.697334 q^{53} +5.93373 q^{55} -0.862607 q^{56} -0.415427 q^{58} +2.32644 q^{59} +11.9828 q^{61} +3.11543 q^{62} +1.00000 q^{64} +6.29075 q^{65} -7.04341 q^{67} +0.677825 q^{68} +1.97803 q^{70} -6.61468 q^{71} -6.52276 q^{73} -3.92798 q^{74} +4.05298 q^{76} -2.23214 q^{77} -13.9286 q^{79} -2.29309 q^{80} -6.70469 q^{82} -5.98379 q^{83} -1.55431 q^{85} -2.84048 q^{86} +2.58766 q^{88} +5.18792 q^{89} -2.36644 q^{91} +6.02686 q^{92} +0.984949 q^{94} -9.29384 q^{95} -1.81522 q^{97} +6.25591 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} + 5 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 + 5 * q^7 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} + 5 q^{7} - 6 q^{8} + 6 q^{10} - q^{11} + 6 q^{13} - 5 q^{14} + 6 q^{16} - 10 q^{17} - 4 q^{19} - 6 q^{20} + q^{22} - 2 q^{23} - 6 q^{26} + 5 q^{28} - 6 q^{29} - 5 q^{31} - 6 q^{32} + 10 q^{34} + 2 q^{35} + q^{37} + 4 q^{38} + 6 q^{40} - 12 q^{41} - 11 q^{43} - q^{44} + 2 q^{46} - 5 q^{47} - q^{49} + 6 q^{52} - 4 q^{53} - 15 q^{55} - 5 q^{56} + 6 q^{58} - q^{59} - 8 q^{61} + 5 q^{62} + 6 q^{64} - 5 q^{65} - 6 q^{67} - 10 q^{68} - 2 q^{70} - q^{71} - 10 q^{73} - q^{74} - 4 q^{76} + 8 q^{77} - 10 q^{79} - 6 q^{80} + 12 q^{82} + 6 q^{83} - 2 q^{85} + 11 q^{86} + q^{88} + 3 q^{89} - 16 q^{91} - 2 q^{92} + 5 q^{94} + 10 q^{95} + 3 q^{97} + q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 + 5 * q^7 - 6 * q^8 + 6 * q^10 - q^11 + 6 * q^13 - 5 * q^14 + 6 * q^16 - 10 * q^17 - 4 * q^19 - 6 * q^20 + q^22 - 2 * q^23 - 6 * q^26 + 5 * q^28 - 6 * q^29 - 5 * q^31 - 6 * q^32 + 10 * q^34 + 2 * q^35 + q^37 + 4 * q^38 + 6 * q^40 - 12 * q^41 - 11 * q^43 - q^44 + 2 * q^46 - 5 * q^47 - q^49 + 6 * q^52 - 4 * q^53 - 15 * q^55 - 5 * q^56 + 6 * q^58 - q^59 - 8 * q^61 + 5 * q^62 + 6 * q^64 - 5 * q^65 - 6 * q^67 - 10 * q^68 - 2 * q^70 - q^71 - 10 * q^73 - q^74 - 4 * q^76 + 8 * q^77 - 10 * q^79 - 6 * q^80 + 12 * q^82 + 6 * q^83 - 2 * q^85 + 11 * q^86 + q^88 + 3 * q^89 - 16 * q^91 - 2 * q^92 + 5 * q^94 + 10 * q^95 + 3 * q^97 + q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.29309	−1.02550	−0.512750	−	0.858538i	$-0.671373\pi$
$5$	−2.29309	−1.02550	−0.512750	+	0.858538i	$0.671373\pi$
$6$	0	0
$7$	0.862607	0.326035	0.163017	−	0.986623i	$-0.447877\pi$
$7$	0.862607	0.326035	0.163017	+	0.986623i	$0.447877\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.29309	0.725137
$11$	−2.58766	−0.780210	−0.390105	−	0.920770i	$-0.627561\pi$
$11$	−2.58766	−0.780210	−0.390105	+	0.920770i	$0.627561\pi$
$12$	0	0
$13$	−2.74336	−0.760870	−0.380435	−	0.924808i	$-0.624226\pi$
$13$	−2.74336	−0.760870	−0.380435	+	0.924808i	$0.624226\pi$
$14$	−0.862607	−0.230541
$15$	0	0
$16$	1.00000	0.250000
$17$	0.677825	0.164397	0.0821984	−	0.996616i	$-0.473806\pi$
$17$	0.677825	0.164397	0.0821984	+	0.996616i	$0.473806\pi$
$18$	0	0
$19$	4.05298	0.929819	0.464909	−	0.885358i	$-0.346087\pi$
$19$	4.05298	0.929819	0.464909	+	0.885358i	$0.346087\pi$
$20$	−2.29309	−0.512750
$21$	0	0
$22$	2.58766	0.551692
$23$	6.02686	1.25669	0.628344	−	0.777936i	$-0.283733\pi$
$23$	6.02686	1.25669	0.628344	+	0.777936i	$0.283733\pi$
$24$	0	0
$25$	0.258243	0.0516486
$26$	2.74336	0.538016
$27$	0	0
$28$	0.862607	0.163017
$29$	0.415427	0.0771429	0.0385715	−	0.999256i	$-0.487719\pi$
$29$	0.415427	0.0771429	0.0385715	+	0.999256i	$0.487719\pi$
$30$	0	0
$31$	−3.11543	−0.559547	−0.279773	−	0.960066i	$-0.590259\pi$
$31$	−3.11543	−0.559547	−0.279773	+	0.960066i	$0.590259\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.677825	−0.116246
$35$	−1.97803	−0.334349
$36$	0	0
$37$	3.92798	0.645756	0.322878	−	0.946441i	$-0.395350\pi$
$37$	3.92798	0.645756	0.322878	+	0.946441i	$0.395350\pi$
$38$	−4.05298	−0.657481
$39$	0	0
$40$	2.29309	0.362569
$41$	6.70469	1.04710	0.523548	−	0.851996i	$-0.324608\pi$
$41$	6.70469	1.04710	0.523548	+	0.851996i	$0.324608\pi$
$42$	0	0
$43$	2.84048	0.433169	0.216585	−	0.976264i	$-0.430508\pi$
$43$	2.84048	0.433169	0.216585	+	0.976264i	$0.430508\pi$
$44$	−2.58766	−0.390105
$45$	0	0
$46$	−6.02686	−0.888612
$47$	−0.984949	−0.143670	−0.0718348	−	0.997417i	$-0.522885\pi$
$47$	−0.984949	−0.143670	−0.0718348	+	0.997417i	$0.522885\pi$
$48$	0	0
$49$	−6.25591	−0.893701
$50$	−0.258243	−0.0365211
$51$	0	0
$52$	−2.74336	−0.380435
$53$	−0.697334	−0.0957862	−0.0478931	−	0.998852i	$-0.515251\pi$
$53$	−0.697334	−0.0957862	−0.0478931	+	0.998852i	$0.515251\pi$
$54$	0	0
$55$	5.93373	0.800105
$56$	−0.862607	−0.115271
$57$	0	0
$58$	−0.415427	−0.0545483
$59$	2.32644	0.302876	0.151438	−	0.988467i	$-0.451610\pi$
$59$	2.32644	0.302876	0.151438	+	0.988467i	$0.451610\pi$
$60$	0	0
$61$	11.9828	1.53424	0.767118	−	0.641506i	$-0.221690\pi$
$61$	11.9828	1.53424	0.767118	+	0.641506i	$0.221690\pi$
$62$	3.11543	0.395659
$63$	0	0
$64$	1.00000	0.125000
$65$	6.29075	0.780272
$66$	0	0
$67$	−7.04341	−0.860489	−0.430244	−	0.902712i	$-0.641573\pi$
$67$	−7.04341	−0.860489	−0.430244	+	0.902712i	$0.641573\pi$
$68$	0.677825	0.0821984
$69$	0	0
$70$	1.97803	0.236420
$71$	−6.61468	−0.785019	−0.392509	−	0.919748i	$-0.628393\pi$
$71$	−6.61468	−0.785019	−0.392509	+	0.919748i	$0.628393\pi$
$72$	0	0
$73$	−6.52276	−0.763432	−0.381716	−	0.924280i	$-0.624667\pi$
$73$	−6.52276	−0.763432	−0.381716	+	0.924280i	$0.624667\pi$
$74$	−3.92798	−0.456618
$75$	0	0
$76$	4.05298	0.464909
$77$	−2.23214	−0.254376
$78$	0	0
$79$	−13.9286	−1.56709	−0.783544	−	0.621337i	$-0.786590\pi$
$79$	−13.9286	−1.56709	−0.783544	+	0.621337i	$0.786590\pi$
$80$	−2.29309	−0.256375
$81$	0	0
$82$	−6.70469	−0.740409
$83$	−5.98379	−0.656806	−0.328403	−	0.944538i	$-0.606510\pi$
$83$	−5.98379	−0.656806	−0.328403	+	0.944538i	$0.606510\pi$
$84$	0	0
$85$	−1.55431	−0.168589
$86$	−2.84048	−0.306297
$87$	0	0
$88$	2.58766	0.275846
$89$	5.18792	0.549918	0.274959	−	0.961456i	$-0.411336\pi$
$89$	5.18792	0.549918	0.274959	+	0.961456i	$0.411336\pi$
$90$	0	0
$91$	−2.36644	−0.248070
$92$	6.02686	0.628344
$93$	0	0
$94$	0.984949	0.101590
$95$	−9.29384	−0.953528
$96$	0	0
$97$	−1.81522	−0.184307	−0.0921537	−	0.995745i	$-0.529375\pi$
$97$	−1.81522	−0.184307	−0.0921537	+	0.995745i	$0.529375\pi$
$98$	6.25591	0.631942
$99$	0	0
$100$	0.258243	0.0258243
$101$	−18.5428	−1.84508	−0.922541	−	0.385900i	$-0.873891\pi$
$101$	−18.5428	−1.84508	−0.922541	+	0.385900i	$0.873891\pi$
$102$	0	0
$103$	−4.72506	−0.465574	−0.232787	−	0.972528i	$-0.574784\pi$
$103$	−4.72506	−0.465574	−0.232787	+	0.972528i	$0.574784\pi$
$104$	2.74336	0.269008
$105$	0	0
$106$	0.697334	0.0677311
$107$	−0.914259	−0.0883848	−0.0441924	−	0.999023i	$-0.514071\pi$
$107$	−0.914259	−0.0883848	−0.0441924	+	0.999023i	$0.514071\pi$
$108$	0	0
$109$	11.4008	1.09200	0.546000	−	0.837785i	$-0.316150\pi$
$109$	11.4008	1.09200	0.546000	+	0.837785i	$0.316150\pi$
$110$	−5.93373	−0.565759
$111$	0	0
$112$	0.862607	0.0815087
$113$	−1.83027	−0.172177	−0.0860886	−	0.996287i	$-0.527437\pi$
$113$	−1.83027	−0.172177	−0.0860886	+	0.996287i	$0.527437\pi$
$114$	0	0
$115$	−13.8201	−1.28873
$116$	0.415427	0.0385715
$117$	0	0
$118$	−2.32644	−0.214166
$119$	0.584697	0.0535991
$120$	0	0
$121$	−4.30400	−0.391273
$122$	−11.9828	−1.08487
$123$	0	0
$124$	−3.11543	−0.279773
$125$	10.8733	0.972534
$126$	0	0
$127$	11.7664	1.04410	0.522049	−	0.852915i	$-0.325168\pi$
$127$	11.7664	1.04410	0.522049	+	0.852915i	$0.325168\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.29075	−0.551735
$131$	10.0356	0.876814	0.438407	−	0.898776i	$-0.355543\pi$
$131$	10.0356	0.876814	0.438407	+	0.898776i	$0.355543\pi$
$132$	0	0
$133$	3.49613	0.303153
$134$	7.04341	0.608458
$135$	0	0
$136$	−0.677825	−0.0581230
$137$	−11.1811	−0.955263	−0.477631	−	0.878560i	$-0.658505\pi$
$137$	−11.1811	−0.955263	−0.477631	+	0.878560i	$0.658505\pi$
$138$	0	0
$139$	−8.05032	−0.682819	−0.341410	−	0.939915i	$-0.610904\pi$
$139$	−8.05032	−0.682819	−0.341410	+	0.939915i	$0.610904\pi$
$140$	−1.97803	−0.167174
$141$	0	0
$142$	6.61468	0.555092
$143$	7.09888	0.593638
$144$	0	0
$145$	−0.952611	−0.0791100
$146$	6.52276	0.539828
$147$	0	0
$148$	3.92798	0.322878
$149$	9.92769	0.813308	0.406654	−	0.913582i	$-0.366696\pi$
$149$	9.92769	0.813308	0.406654	+	0.913582i	$0.366696\pi$
$150$	0	0
$151$	−23.7693	−1.93432	−0.967160	−	0.254168i	$-0.918198\pi$
$151$	−23.7693	−1.93432	−0.967160	+	0.254168i	$0.918198\pi$
$152$	−4.05298	−0.328740
$153$	0	0
$154$	2.23214	0.179871
$155$	7.14394	0.573815
$156$	0	0
$157$	9.94651	0.793818	0.396909	−	0.917858i	$-0.370083\pi$
$157$	9.94651	0.793818	0.396909	+	0.917858i	$0.370083\pi$
$158$	13.9286	1.10810
$159$	0	0
$160$	2.29309	0.181284
$161$	5.19882	0.409724
$162$	0	0
$163$	6.29931	0.493400	0.246700	−	0.969092i	$-0.420654\pi$
$163$	6.29931	0.493400	0.246700	+	0.969092i	$0.420654\pi$
$164$	6.70469	0.523548
$165$	0	0
$166$	5.98379	0.464432
$167$	−7.87383	−0.609295	−0.304648	−	0.952465i	$-0.598539\pi$
$167$	−7.87383	−0.609295	−0.304648	+	0.952465i	$0.598539\pi$
$168$	0	0
$169$	−5.47400	−0.421077
$170$	1.55431	0.119210
$171$	0	0
$172$	2.84048	0.216585
$173$	−20.1495	−1.53194	−0.765968	−	0.642878i	$-0.777740\pi$
$173$	−20.1495	−1.53194	−0.765968	+	0.642878i	$0.777740\pi$
$174$	0	0
$175$	0.222762	0.0168392
$176$	−2.58766	−0.195052
$177$	0	0
$178$	−5.18792	−0.388851
$179$	−1.60552	−0.120002	−0.0600012	−	0.998198i	$-0.519110\pi$
$179$	−1.60552	−0.120002	−0.0600012	+	0.998198i	$0.519110\pi$
$180$	0	0
$181$	−11.9410	−0.887565	−0.443782	−	0.896135i	$-0.646364\pi$
$181$	−11.9410	−0.887565	−0.443782	+	0.896135i	$0.646364\pi$
$182$	2.36644	0.175412
$183$	0	0
$184$	−6.02686	−0.444306
$185$	−9.00720	−0.662222
$186$	0	0
$187$	−1.75398	−0.128264
$188$	−0.984949	−0.0718348
$189$	0	0
$190$	9.29384	0.674246
$191$	13.2911	0.961709	0.480854	−	0.876800i	$-0.340327\pi$
$191$	13.2911	0.961709	0.480854	+	0.876800i	$0.340327\pi$
$192$	0	0
$193$	6.31592	0.454630	0.227315	−	0.973821i	$-0.427005\pi$
$193$	6.31592	0.454630	0.227315	+	0.973821i	$0.427005\pi$
$194$	1.81522	0.130325
$195$	0	0
$196$	−6.25591	−0.446851
$197$	−11.2953	−0.804759	−0.402380	−	0.915473i	$-0.631817\pi$
$197$	−11.2953	−0.804759	−0.402380	+	0.915473i	$0.631817\pi$
$198$	0	0
$199$	−10.4408	−0.740129	−0.370064	−	0.929006i	$-0.620664\pi$
$199$	−10.4408	−0.740129	−0.370064	+	0.929006i	$0.620664\pi$
$200$	−0.258243	−0.0182605
$201$	0	0
$202$	18.5428	1.30467
$203$	0.358351	0.0251513
$204$	0	0
$205$	−15.3744	−1.07380
$206$	4.72506	0.329210
$207$	0	0
$208$	−2.74336	−0.190218
$209$	−10.4878	−0.725454
$210$	0	0
$211$	−24.2735	−1.67105	−0.835527	−	0.549449i	$-0.814838\pi$
$211$	−24.2735	−1.67105	−0.835527	+	0.549449i	$0.814838\pi$
$212$	−0.697334	−0.0478931
$213$	0	0
$214$	0.914259	0.0624975
$215$	−6.51347	−0.444215
$216$	0	0
$217$	−2.68739	−0.182432
$218$	−11.4008	−0.772161
$219$	0	0
$220$	5.93373	0.400052
$221$	−1.85952	−0.125085
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−0.862607	−0.0576354
$225$	0	0
$226$	1.83027	0.121748
$227$	4.11424	0.273071	0.136536	−	0.990635i	$-0.456403\pi$
$227$	4.11424	0.273071	0.136536	+	0.990635i	$0.456403\pi$
$228$	0	0
$229$	9.36331	0.618745	0.309372	−	0.950941i	$-0.399881\pi$
$229$	9.36331	0.618745	0.309372	+	0.950941i	$0.399881\pi$
$230$	13.8201	0.911271
$231$	0	0
$232$	−0.415427	−0.0272741
$233$	−6.20340	−0.406398	−0.203199	−	0.979137i	$-0.565134\pi$
$233$	−6.20340	−0.406398	−0.203199	+	0.979137i	$0.565134\pi$
$234$	0	0
$235$	2.25857	0.147333
$236$	2.32644	0.151438
$237$	0	0
$238$	−0.584697	−0.0379003
$239$	−5.35388	−0.346313	−0.173157	−	0.984894i	$-0.555397\pi$
$239$	−5.35388	−0.346313	−0.173157	+	0.984894i	$0.555397\pi$
$240$	0	0
$241$	−4.29821	−0.276872	−0.138436	−	0.990371i	$-0.544208\pi$
$241$	−4.29821	−0.276872	−0.138436	+	0.990371i	$0.544208\pi$
$242$	4.30400	0.276672
$243$	0	0
$244$	11.9828	0.767118
$245$	14.3453	0.916490
$246$	0	0
$247$	−11.1188	−0.707471
$248$	3.11543	0.197830
$249$	0	0
$250$	−10.8733	−0.687685
$251$	8.43192	0.532218	0.266109	−	0.963943i	$-0.414262\pi$
$251$	8.43192	0.532218	0.266109	+	0.963943i	$0.414262\pi$
$252$	0	0
$253$	−15.5955	−0.980480
$254$	−11.7664	−0.738289
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.8675	−0.802653	−0.401326	−	0.915935i	$-0.631451\pi$
$257$	−12.8675	−0.802653	−0.401326	+	0.915935i	$0.631451\pi$
$258$	0	0
$259$	3.38830	0.210539
$260$	6.29075	0.390136
$261$	0	0
$262$	−10.0356	−0.620001
$263$	9.12974	0.562964	0.281482	−	0.959567i	$-0.409174\pi$
$263$	9.12974	0.562964	0.281482	+	0.959567i	$0.409174\pi$
$264$	0	0
$265$	1.59905	0.0982287
$266$	−3.49613	−0.214362
$267$	0	0
$268$	−7.04341	−0.430244
$269$	−5.81453	−0.354518	−0.177259	−	0.984164i	$-0.556723\pi$
$269$	−5.81453	−0.354518	−0.177259	+	0.984164i	$0.556723\pi$
$270$	0	0
$271$	−17.3702	−1.05516	−0.527581	−	0.849504i	$-0.676901\pi$
$271$	−17.3702	−1.05516	−0.527581	+	0.849504i	$0.676901\pi$
$272$	0.677825	0.0410992
$273$	0	0
$274$	11.1811	0.675473
$275$	−0.668245	−0.0402967
$276$	0	0
$277$	1.99724	0.120002	0.0600012	−	0.998198i	$-0.480890\pi$
$277$	1.99724	0.120002	0.0600012	+	0.998198i	$0.480890\pi$
$278$	8.05032	0.482826
$279$	0	0
$280$	1.97803	0.118210
$281$	18.6697	1.11374	0.556872	−	0.830599i	$-0.312001\pi$
$281$	18.6697	1.11374	0.556872	+	0.830599i	$0.312001\pi$
$282$	0	0
$283$	−18.7106	−1.11223	−0.556114	−	0.831106i	$-0.687708\pi$
$283$	−18.7106	−1.11223	−0.556114	+	0.831106i	$0.687708\pi$
$284$	−6.61468	−0.392509
$285$	0	0
$286$	−7.09888	−0.419766
$287$	5.78351	0.341390
$288$	0	0
$289$	−16.5406	−0.972974
$290$	0.952611	0.0559392
$291$	0	0
$292$	−6.52276	−0.381716
$293$	9.71453	0.567529	0.283765	−	0.958894i	$-0.408417\pi$
$293$	9.71453	0.567529	0.283765	+	0.958894i	$0.408417\pi$
$294$	0	0
$295$	−5.33472	−0.310599
$296$	−3.92798	−0.228309
$297$	0	0
$298$	−9.92769	−0.575095
$299$	−16.5338	−0.956176
$300$	0	0
$301$	2.45022	0.141228
$302$	23.7693	1.36777
$303$	0	0
$304$	4.05298	0.232455
$305$	−27.4775	−1.57336
$306$	0	0
$307$	−6.55685	−0.374219	−0.187109	−	0.982339i	$-0.559912\pi$
$307$	−6.55685	−0.374219	−0.187109	+	0.982339i	$0.559912\pi$
$308$	−2.23214	−0.127188
$309$	0	0
$310$	−7.14394	−0.405748
$311$	−23.0685	−1.30809	−0.654047	−	0.756454i	$-0.726930\pi$
$311$	−23.0685	−1.30809	−0.654047	+	0.756454i	$0.726930\pi$
$312$	0	0
$313$	−16.9827	−0.959917	−0.479959	−	0.877291i	$-0.659348\pi$
$313$	−16.9827	−0.959917	−0.479959	+	0.877291i	$0.659348\pi$
$314$	−9.94651	−0.561314
$315$	0	0
$316$	−13.9286	−0.783544
$317$	−0.536123	−0.0301117	−0.0150558	−	0.999887i	$-0.504793\pi$
$317$	−0.536123	−0.0301117	−0.0150558	+	0.999887i	$0.504793\pi$
$318$	0	0
$319$	−1.07499	−0.0601877
$320$	−2.29309	−0.128187
$321$	0	0
$322$	−5.19882	−0.289719
$323$	2.74722	0.152859
$324$	0	0
$325$	−0.708452	−0.0392979
$326$	−6.29931	−0.348887
$327$	0	0
$328$	−6.70469	−0.370204
$329$	−0.849624	−0.0468413
$330$	0	0
$331$	25.2564	1.38821	0.694107	−	0.719872i	$-0.255799\pi$
$331$	25.2564	1.38821	0.694107	+	0.719872i	$0.255799\pi$
$332$	−5.98379	−0.328403
$333$	0	0
$334$	7.87383	0.430837
$335$	16.1511	0.882431
$336$	0	0
$337$	−9.84506	−0.536295	−0.268147	−	0.963378i	$-0.586411\pi$
$337$	−9.84506	−0.536295	−0.268147	+	0.963378i	$0.586411\pi$
$338$	5.47400	0.297746
$339$	0	0
$340$	−1.55431	−0.0842944
$341$	8.06167	0.436564
$342$	0	0
$343$	−11.4346	−0.617413
$344$	−2.84048	−0.153148
$345$	0	0
$346$	20.1495	1.08324
$347$	−6.40121	−0.343635	−0.171818	−	0.985129i	$-0.554964\pi$
$347$	−6.40121	−0.343635	−0.171818	+	0.985129i	$0.554964\pi$
$348$	0	0
$349$	−11.7812	−0.630632	−0.315316	−	0.948987i	$-0.602111\pi$
$349$	−11.7812	−0.630632	−0.315316	+	0.948987i	$0.602111\pi$
$350$	−0.222762	−0.0119071
$351$	0	0
$352$	2.58766	0.137923
$353$	−30.8673	−1.64290	−0.821451	−	0.570279i	$-0.806835\pi$
$353$	−30.8673	−1.64290	−0.821451	+	0.570279i	$0.806835\pi$
$354$	0	0
$355$	15.1680	0.805036
$356$	5.18792	0.274959
$357$	0	0
$358$	1.60552	0.0848545
$359$	5.33510	0.281576	0.140788	−	0.990040i	$-0.455036\pi$
$359$	5.33510	0.281576	0.140788	+	0.990040i	$0.455036\pi$
$360$	0	0
$361$	−2.57331	−0.135438
$362$	11.9410	0.627603
$363$	0	0
$364$	−2.36644	−0.124035
$365$	14.9573	0.782899
$366$	0	0
$367$	11.4126	0.595734	0.297867	−	0.954607i	$-0.403725\pi$
$367$	11.4126	0.595734	0.297867	+	0.954607i	$0.403725\pi$
$368$	6.02686	0.314172
$369$	0	0
$370$	9.00720	0.468262
$371$	−0.601526	−0.0312297
$372$	0	0
$373$	0.887429	0.0459493	0.0229747	−	0.999736i	$-0.492686\pi$
$373$	0.887429	0.0459493	0.0229747	+	0.999736i	$0.492686\pi$
$374$	1.75398	0.0906963
$375$	0	0
$376$	0.984949	0.0507949
$377$	−1.13967	−0.0586958
$378$	0	0
$379$	13.1494	0.675440	0.337720	−	0.941247i	$-0.390344\pi$
$379$	13.1494	0.675440	0.337720	+	0.941247i	$0.390344\pi$
$380$	−9.29384	−0.476764
$381$	0	0
$382$	−13.2911	−0.680031
$383$	−5.86267	−0.299568	−0.149784	−	0.988719i	$-0.547858\pi$
$383$	−5.86267	−0.299568	−0.149784	+	0.988719i	$0.547858\pi$
$384$	0	0
$385$	5.11848	0.260862
$386$	−6.31592	−0.321472
$387$	0	0
$388$	−1.81522	−0.0921537
$389$	19.8457	1.00622	0.503108	−	0.864224i	$-0.332190\pi$
$389$	19.8457	1.00622	0.503108	+	0.864224i	$0.332190\pi$
$390$	0	0
$391$	4.08516	0.206595
$392$	6.25591	0.315971
$393$	0	0
$394$	11.2953	0.569051
$395$	31.9394	1.60705
$396$	0	0
$397$	33.1439	1.66344	0.831722	−	0.555192i	$-0.187355\pi$
$397$	33.1439	1.66344	0.831722	+	0.555192i	$0.187355\pi$
$398$	10.4408	0.523350
$399$	0	0
$400$	0.258243	0.0129121
$401$	−12.2423	−0.611352	−0.305676	−	0.952136i	$-0.598882\pi$
$401$	−12.2423	−0.611352	−0.305676	+	0.952136i	$0.598882\pi$
$402$	0	0
$403$	8.54672	0.425743
$404$	−18.5428	−0.922541
$405$	0	0
$406$	−0.358351	−0.0177846
$407$	−10.1643	−0.503825
$408$	0	0
$409$	12.6371	0.624864	0.312432	−	0.949940i	$-0.398856\pi$
$409$	12.6371	0.624864	0.312432	+	0.949940i	$0.398856\pi$
$410$	15.3744	0.759289
$411$	0	0
$412$	−4.72506	−0.232787
$413$	2.00680	0.0987483
$414$	0	0
$415$	13.7213	0.673554
$416$	2.74336	0.134504
$417$	0	0
$418$	10.4878	0.512973
$419$	−18.7140	−0.914238	−0.457119	−	0.889406i	$-0.651119\pi$
$419$	−18.7140	−0.914238	−0.457119	+	0.889406i	$0.651119\pi$
$420$	0	0
$421$	0.909269	0.0443151	0.0221575	−	0.999754i	$-0.492946\pi$
$421$	0.909269	0.0443151	0.0221575	+	0.999754i	$0.492946\pi$
$422$	24.2735	1.18161
$423$	0	0
$424$	0.697334	0.0338655
$425$	0.175044	0.00849086
$426$	0	0
$427$	10.3364	0.500215
$428$	−0.914259	−0.0441924
$429$	0	0
$430$	6.51347	0.314107
$431$	2.94728	0.141966	0.0709828	−	0.997478i	$-0.477386\pi$
$431$	2.94728	0.141966	0.0709828	+	0.997478i	$0.477386\pi$
$432$	0	0
$433$	−1.66531	−0.0800296	−0.0400148	−	0.999199i	$-0.512741\pi$
$433$	−1.66531	−0.0800296	−0.0400148	+	0.999199i	$0.512741\pi$
$434$	2.68739	0.128999
$435$	0	0
$436$	11.4008	0.546000
$437$	24.4268	1.16849
$438$	0	0
$439$	−27.0565	−1.29134	−0.645668	−	0.763618i	$-0.723421\pi$
$439$	−27.0565	−1.29134	−0.645668	+	0.763618i	$0.723421\pi$
$440$	−5.93373	−0.282880
$441$	0	0
$442$	1.85952	0.0884482
$443$	−12.2932	−0.584068	−0.292034	−	0.956408i	$-0.594332\pi$
$443$	−12.2932	−0.584068	−0.292034	+	0.956408i	$0.594332\pi$
$444$	0	0
$445$	−11.8963	−0.563941
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	0.862607	0.0407544
$449$	−26.2543	−1.23901	−0.619507	−	0.784991i	$-0.712668\pi$
$449$	−26.2543	−1.23901	−0.619507	+	0.784991i	$0.712668\pi$
$450$	0	0
$451$	−17.3495	−0.816955
$452$	−1.83027	−0.0860886
$453$	0	0
$454$	−4.11424	−0.193091
$455$	5.42645	0.254396
$456$	0	0
$457$	−10.6761	−0.499405	−0.249703	−	0.968323i	$-0.580333\pi$
$457$	−10.6761	−0.499405	−0.249703	+	0.968323i	$0.580333\pi$
$458$	−9.36331	−0.437519
$459$	0	0
$460$	−13.8201	−0.644366
$461$	−27.4971	−1.28067	−0.640334	−	0.768097i	$-0.721204\pi$
$461$	−27.4971	−1.28067	−0.640334	+	0.768097i	$0.721204\pi$
$462$	0	0
$463$	−6.65356	−0.309217	−0.154609	−	0.987976i	$-0.549412\pi$
$463$	−6.65356	−0.309217	−0.154609	+	0.987976i	$0.549412\pi$
$464$	0.415427	0.0192857
$465$	0	0
$466$	6.20340	0.287367
$467$	6.89367	0.319001	0.159501	−	0.987198i	$-0.449012\pi$
$467$	6.89367	0.319001	0.159501	+	0.987198i	$0.449012\pi$
$468$	0	0
$469$	−6.07569	−0.280549
$470$	−2.25857	−0.104180
$471$	0	0
$472$	−2.32644	−0.107083
$473$	−7.35021	−0.337963
$474$	0	0
$475$	1.04665	0.0480238
$476$	0.584697	0.0267995
$477$	0	0
$478$	5.35388	0.244881
$479$	−17.7582	−0.811395	−0.405697	−	0.914007i	$-0.632971\pi$
$479$	−17.7582	−0.811395	−0.405697	+	0.914007i	$0.632971\pi$
$480$	0	0
$481$	−10.7758	−0.491336
$482$	4.29821	0.195778
$483$	0	0
$484$	−4.30400	−0.195636
$485$	4.16245	0.189007
$486$	0	0
$487$	10.5533	0.478214	0.239107	−	0.970993i	$-0.423145\pi$
$487$	10.5533	0.478214	0.239107	+	0.970993i	$0.423145\pi$
$488$	−11.9828	−0.542434
$489$	0	0
$490$	−14.3453	−0.648056
$491$	−22.4411	−1.01275	−0.506377	−	0.862312i	$-0.669016\pi$
$491$	−22.4411	−1.01275	−0.506377	+	0.862312i	$0.669016\pi$
$492$	0	0
$493$	0.281587	0.0126821
$494$	11.1188	0.500258
$495$	0	0
$496$	−3.11543	−0.139887
$497$	−5.70587	−0.255943
$498$	0	0
$499$	1.36860	0.0612671	0.0306336	−	0.999531i	$-0.490248\pi$
$499$	1.36860	0.0612671	0.0306336	+	0.999531i	$0.490248\pi$
$500$	10.8733	0.486267
$501$	0	0
$502$	−8.43192	−0.376335
$503$	28.8437	1.28608	0.643039	−	0.765834i	$-0.277673\pi$
$503$	28.8437	1.28608	0.643039	+	0.765834i	$0.277673\pi$
$504$	0	0
$505$	42.5203	1.89213
$506$	15.5955	0.693304
$507$	0	0
$508$	11.7664	0.522049
$509$	−34.2373	−1.51754	−0.758771	−	0.651357i	$-0.774200\pi$
$509$	−34.2373	−1.51754	−0.758771	+	0.651357i	$0.774200\pi$
$510$	0	0
$511$	−5.62658	−0.248905
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.8675	0.567561
$515$	10.8350	0.477445
$516$	0	0
$517$	2.54872	0.112092
$518$	−3.38830	−0.148874
$519$	0	0
$520$	−6.29075	−0.275868
$521$	12.6208	0.552929	0.276464	−	0.961024i	$-0.410837\pi$
$521$	12.6208	0.552929	0.276464	+	0.961024i	$0.410837\pi$
$522$	0	0
$523$	−14.2376	−0.622569	−0.311284	−	0.950317i	$-0.600759\pi$
$523$	−14.2376	−0.622569	−0.311284	+	0.950317i	$0.600759\pi$
$524$	10.0356	0.438407
$525$	0	0
$526$	−9.12974	−0.398076
$527$	−2.11171	−0.0919877
$528$	0	0
$529$	13.3231	0.579264
$530$	−1.59905	−0.0694582
$531$	0	0
$532$	3.49613	0.151577
$533$	−18.3933	−0.796704
$534$	0	0
$535$	2.09647	0.0906385
$536$	7.04341	0.304229
$537$	0	0
$538$	5.81453	0.250682
$539$	16.1882	0.697275
$540$	0	0
$541$	−11.1016	−0.477295	−0.238648	−	0.971106i	$-0.576704\pi$
$541$	−11.1016	−0.477295	−0.238648	+	0.971106i	$0.576704\pi$
$542$	17.3702	0.746113
$543$	0	0
$544$	−0.677825	−0.0290615
$545$	−26.1430	−1.11984
$546$	0	0
$547$	−24.9146	−1.06527	−0.532635	−	0.846345i	$-0.678798\pi$
$547$	−24.9146	−1.06527	−0.532635	+	0.846345i	$0.678798\pi$
$548$	−11.1811	−0.477631
$549$	0	0
$550$	0.668245	0.0284941
$551$	1.68372	0.0717289
$552$	0	0
$553$	−12.0149	−0.510925
$554$	−1.99724	−0.0848545
$555$	0	0
$556$	−8.05032	−0.341410
$557$	−25.9495	−1.09952	−0.549758	−	0.835324i	$-0.685280\pi$
$557$	−25.9495	−1.09952	−0.549758	+	0.835324i	$0.685280\pi$
$558$	0	0
$559$	−7.79245	−0.329586
$560$	−1.97803	−0.0835871
$561$	0	0
$562$	−18.6697	−0.787536
$563$	15.7475	0.663678	0.331839	−	0.943336i	$-0.392331\pi$
$563$	15.7475	0.663678	0.331839	+	0.943336i	$0.392331\pi$
$564$	0	0
$565$	4.19696	0.176568
$566$	18.7106	0.786464
$567$	0	0
$568$	6.61468	0.277546
$569$	−0.756325	−0.0317068	−0.0158534	−	0.999874i	$-0.505047\pi$
$569$	−0.756325	−0.0317068	−0.0158534	+	0.999874i	$0.505047\pi$
$570$	0	0
$571$	24.5127	1.02582	0.512912	−	0.858441i	$-0.328567\pi$
$571$	24.5127	1.02582	0.512912	+	0.858441i	$0.328567\pi$
$572$	7.09888	0.296819
$573$	0	0
$574$	−5.78351	−0.241399
$575$	1.55639	0.0649061
$576$	0	0
$577$	25.0143	1.04136	0.520680	−	0.853752i	$-0.325678\pi$
$577$	25.0143	1.04136	0.520680	+	0.853752i	$0.325678\pi$
$578$	16.5406	0.687996
$579$	0	0
$580$	−0.952611	−0.0395550
$581$	−5.16166	−0.214142
$582$	0	0
$583$	1.80447	0.0747334
$584$	6.52276	0.269914
$585$	0	0
$586$	−9.71453	−0.401304
$587$	18.8574	0.778329	0.389165	−	0.921168i	$-0.372764\pi$
$587$	18.8574	0.778329	0.389165	+	0.921168i	$0.372764\pi$
$588$	0	0
$589$	−12.6268	−0.520277
$590$	5.33472	0.219627
$591$	0	0
$592$	3.92798	0.161439
$593$	−6.06588	−0.249096	−0.124548	−	0.992214i	$-0.539748\pi$
$593$	−6.06588	−0.249096	−0.124548	+	0.992214i	$0.539748\pi$
$594$	0	0
$595$	−1.34076	−0.0549658
$596$	9.92769	0.406654
$597$	0	0
$598$	16.5338	0.676119
$599$	−13.6208	−0.556531	−0.278266	−	0.960504i	$-0.589760\pi$
$599$	−13.6208	−0.556531	−0.278266	+	0.960504i	$0.589760\pi$
$600$	0	0
$601$	1.95077	0.0795737	0.0397868	−	0.999208i	$-0.487332\pi$
$601$	1.95077	0.0795737	0.0397868	+	0.999208i	$0.487332\pi$
$602$	−2.45022	−0.0998635
$603$	0	0
$604$	−23.7693	−0.967160
$605$	9.86944	0.401250
$606$	0	0
$607$	23.1287	0.938767	0.469383	−	0.882995i	$-0.344476\pi$
$607$	23.1287	0.938767	0.469383	+	0.882995i	$0.344476\pi$
$608$	−4.05298	−0.164370
$609$	0	0
$610$	27.4775	1.11253
$611$	2.70207	0.109314
$612$	0	0
$613$	19.8577	0.802045	0.401023	−	0.916068i	$-0.368655\pi$
$613$	19.8577	0.802045	0.401023	+	0.916068i	$0.368655\pi$
$614$	6.55685	0.264613
$615$	0	0
$616$	2.23214	0.0899354
$617$	−20.9990	−0.845389	−0.422694	−	0.906272i	$-0.638916\pi$
$617$	−20.9990	−0.845389	−0.422694	+	0.906272i	$0.638916\pi$
$618$	0	0
$619$	26.4184	1.06184	0.530922	−	0.847421i	$-0.321846\pi$
$619$	26.4184	1.06184	0.530922	+	0.847421i	$0.321846\pi$
$620$	7.14394	0.286907
$621$	0	0
$622$	23.0685	0.924963
$623$	4.47514	0.179293
$624$	0	0
$625$	−26.2245	−1.04898
$626$	16.9827	0.678764
$627$	0	0
$628$	9.94651	0.396909
$629$	2.66248	0.106160
$630$	0	0
$631$	−12.3947	−0.493425	−0.246712	−	0.969089i	$-0.579350\pi$
$631$	−12.3947	−0.493425	−0.246712	+	0.969089i	$0.579350\pi$
$632$	13.9286	0.554049
$633$	0	0
$634$	0.536123	0.0212922
$635$	−26.9813	−1.07072
$636$	0	0
$637$	17.1622	0.679991
$638$	1.07499	0.0425591
$639$	0	0
$640$	2.29309	0.0906422
$641$	−38.4742	−1.51964	−0.759819	−	0.650135i	$-0.774712\pi$
$641$	−38.4742	−1.51964	−0.759819	+	0.650135i	$0.774712\pi$
$642$	0	0
$643$	20.3953	0.804312	0.402156	−	0.915571i	$-0.368261\pi$
$643$	20.3953	0.804312	0.402156	+	0.915571i	$0.368261\pi$
$644$	5.19882	0.204862
$645$	0	0
$646$	−2.74722	−0.108088
$647$	14.3041	0.562351	0.281175	−	0.959656i	$-0.409276\pi$
$647$	14.3041	0.562351	0.281175	+	0.959656i	$0.409276\pi$
$648$	0	0
$649$	−6.02004	−0.236307
$650$	0.708452	0.0277878
$651$	0	0
$652$	6.29931	0.246700
$653$	−20.2231	−0.791390	−0.395695	−	0.918382i	$-0.629496\pi$
$653$	−20.2231	−0.791390	−0.395695	+	0.918382i	$0.629496\pi$
$654$	0	0
$655$	−23.0125	−0.899172
$656$	6.70469	0.261774
$657$	0	0
$658$	0.849624	0.0331218
$659$	−14.7596	−0.574954	−0.287477	−	0.957788i	$-0.592816\pi$
$659$	−14.7596	−0.574954	−0.287477	+	0.957788i	$0.592816\pi$
$660$	0	0
$661$	16.6874	0.649063	0.324532	−	0.945875i	$-0.394793\pi$
$661$	16.6874	0.649063	0.324532	+	0.945875i	$0.394793\pi$
$662$	−25.2564	−0.981616
$663$	0	0
$664$	5.98379	0.232216
$665$	−8.01694	−0.310883
$666$	0	0
$667$	2.50372	0.0969446
$668$	−7.87383	−0.304648
$669$	0	0
$670$	−16.1511	−0.623973
$671$	−31.0074	−1.19703
$672$	0	0
$673$	−28.4686	−1.09738	−0.548692	−	0.836024i	$-0.684874\pi$
$673$	−28.4686	−1.09738	−0.548692	+	0.836024i	$0.684874\pi$
$674$	9.84506	0.379218
$675$	0	0
$676$	−5.47400	−0.210538
$677$	−10.1269	−0.389207	−0.194603	−	0.980882i	$-0.562342\pi$
$677$	−10.1269	−0.389207	−0.194603	+	0.980882i	$0.562342\pi$
$678$	0	0
$679$	−1.56582	−0.0600907
$680$	1.55431	0.0596051
$681$	0	0
$682$	−8.06167	−0.308697
$683$	6.34863	0.242924	0.121462	−	0.992596i	$-0.461242\pi$
$683$	6.34863	0.242924	0.121462	+	0.992596i	$0.461242\pi$
$684$	0	0
$685$	25.6391	0.979621
$686$	11.4346	0.436577
$687$	0	0
$688$	2.84048	0.108292
$689$	1.91304	0.0728809
$690$	0	0
$691$	−21.4667	−0.816633	−0.408316	−	0.912841i	$-0.633884\pi$
$691$	−21.4667	−0.816633	−0.408316	+	0.912841i	$0.633884\pi$
$692$	−20.1495	−0.765968
$693$	0	0
$694$	6.40121	0.242987
$695$	18.4601	0.700231
$696$	0	0
$697$	4.54461	0.172139
$698$	11.7812	0.445924
$699$	0	0
$700$	0.222762	0.00841962
$701$	−34.8304	−1.31553	−0.657763	−	0.753225i	$-0.728497\pi$
$701$	−34.8304	−1.31553	−0.657763	+	0.753225i	$0.728497\pi$
$702$	0	0
$703$	15.9200	0.600436
$704$	−2.58766	−0.0975262
$705$	0	0
$706$	30.8673	1.16171
$707$	−15.9952	−0.601561
$708$	0	0
$709$	36.4155	1.36761	0.683806	−	0.729664i	$-0.260323\pi$
$709$	36.4155	1.36761	0.683806	+	0.729664i	$0.260323\pi$
$710$	−15.1680	−0.569246
$711$	0	0
$712$	−5.18792	−0.194425
$713$	−18.7762	−0.703176
$714$	0	0
$715$	−16.2783	−0.608776
$716$	−1.60552	−0.0600012
$717$	0	0
$718$	−5.33510	−0.199104
$719$	−10.3944	−0.387645	−0.193822	−	0.981037i	$-0.562089\pi$
$719$	−10.3944	−0.387645	−0.193822	+	0.981037i	$0.562089\pi$
$720$	0	0
$721$	−4.07587	−0.151793
$722$	2.57331	0.0957688
$723$	0	0
$724$	−11.9410	−0.443782
$725$	0.107281	0.00398432
$726$	0	0
$727$	30.1277	1.11737	0.558687	−	0.829378i	$-0.311305\pi$
$727$	30.1277	1.11737	0.558687	+	0.829378i	$0.311305\pi$
$728$	2.36644	0.0877061
$729$	0	0
$730$	−14.9573	−0.553593
$731$	1.92535	0.0712117
$732$	0	0
$733$	32.0016	1.18201	0.591004	−	0.806669i	$-0.298732\pi$
$733$	32.0016	1.18201	0.591004	+	0.806669i	$0.298732\pi$
$734$	−11.4126	−0.421247
$735$	0	0
$736$	−6.02686	−0.222153
$737$	18.2260	0.671362
$738$	0	0
$739$	−5.56435	−0.204688	−0.102344	−	0.994749i	$-0.532634\pi$
$739$	−5.56435	−0.204688	−0.102344	+	0.994749i	$0.532634\pi$
$740$	−9.00720	−0.331111
$741$	0	0
$742$	0.601526	0.0220827
$743$	21.9762	0.806227	0.403114	−	0.915150i	$-0.367928\pi$
$743$	21.9762	0.806227	0.403114	+	0.915150i	$0.367928\pi$
$744$	0	0
$745$	−22.7650	−0.834046
$746$	−0.887429	−0.0324911
$747$	0	0
$748$	−1.75398	−0.0641320
$749$	−0.788647	−0.0288165
$750$	0	0
$751$	−39.5059	−1.44159	−0.720795	−	0.693149i	$-0.756223\pi$
$751$	−39.5059	−1.44159	−0.720795	+	0.693149i	$0.756223\pi$
$752$	−0.984949	−0.0359174
$753$	0	0
$754$	1.13967	0.0415042
$755$	54.5051	1.98364
$756$	0	0
$757$	33.3947	1.21375	0.606875	−	0.794797i	$-0.292423\pi$
$757$	33.3947	1.21375	0.606875	+	0.794797i	$0.292423\pi$
$758$	−13.1494	−0.477608
$759$	0	0
$760$	9.29384	0.337123
$761$	25.4699	0.923281	0.461641	−	0.887067i	$-0.347261\pi$
$761$	25.4699	0.923281	0.461641	+	0.887067i	$0.347261\pi$
$762$	0	0
$763$	9.83442	0.356030
$764$	13.2911	0.480854
$765$	0	0
$766$	5.86267	0.211827
$767$	−6.38225	−0.230450
$768$	0	0
$769$	30.7758	1.10980	0.554901	−	0.831916i	$-0.312756\pi$
$769$	30.7758	1.10980	0.554901	+	0.831916i	$0.312756\pi$
$770$	−5.11848	−0.184457
$771$	0	0
$772$	6.31592	0.227315
$773$	18.0416	0.648910	0.324455	−	0.945901i	$-0.394819\pi$
$773$	18.0416	0.648910	0.324455	+	0.945901i	$0.394819\pi$
$774$	0	0
$775$	−0.804536	−0.0288998
$776$	1.81522	0.0651625
$777$	0	0
$778$	−19.8457	−0.711502
$779$	27.1740	0.973610
$780$	0	0
$781$	17.1166	0.612479
$782$	−4.08516	−0.146085
$783$	0	0
$784$	−6.25591	−0.223425
$785$	−22.8082	−0.814060
$786$	0	0
$787$	−29.3139	−1.04493	−0.522463	−	0.852662i	$-0.674987\pi$
$787$	−29.3139	−1.04493	−0.522463	+	0.852662i	$0.674987\pi$
$788$	−11.2953	−0.402380
$789$	0	0
$790$	−31.9394	−1.13635
$791$	−1.57880	−0.0561358
$792$	0	0
$793$	−32.8730	−1.16735
$794$	−33.1439	−1.17623
$795$	0	0
$796$	−10.4408	−0.370064
$797$	−7.94610	−0.281465	−0.140733	−	0.990048i	$-0.544946\pi$
$797$	−7.94610	−0.281465	−0.140733	+	0.990048i	$0.544946\pi$
$798$	0	0
$799$	−0.667623	−0.0236188
$800$	−0.258243	−0.00913026
$801$	0	0
$802$	12.2423	0.432291
$803$	16.8787	0.595637
$804$	0	0
$805$	−11.9213	−0.420172
$806$	−8.54672	−0.301045
$807$	0	0
$808$	18.5428	0.652335
$809$	−8.27028	−0.290767	−0.145384	−	0.989375i	$-0.546442\pi$
$809$	−8.27028	−0.290767	−0.145384	+	0.989375i	$0.546442\pi$
$810$	0	0
$811$	47.3436	1.66246	0.831229	−	0.555930i	$-0.187638\pi$
$811$	47.3436	1.66246	0.831229	+	0.555930i	$0.187638\pi$
$812$	0.358351	0.0125756
$813$	0	0
$814$	10.1643	0.356258
$815$	−14.4449	−0.505982
$816$	0	0
$817$	11.5124	0.402769
$818$	−12.6371	−0.441845
$819$	0	0
$820$	−15.3744	−0.536898
$821$	−7.56334	−0.263962	−0.131981	−	0.991252i	$-0.542134\pi$
$821$	−7.56334	−0.263962	−0.131981	+	0.991252i	$0.542134\pi$
$822$	0	0
$823$	45.3990	1.58251	0.791255	−	0.611487i	$-0.209428\pi$
$823$	45.3990	1.58251	0.791255	+	0.611487i	$0.209428\pi$
$824$	4.72506	0.164605
$825$	0	0
$826$	−2.00680	−0.0698256
$827$	−18.3356	−0.637591	−0.318796	−	0.947824i	$-0.603278\pi$
$827$	−18.3356	−0.637591	−0.318796	+	0.947824i	$0.603278\pi$
$828$	0	0
$829$	48.5382	1.68580	0.842901	−	0.538069i	$-0.180846\pi$
$829$	48.5382	1.68580	0.842901	+	0.538069i	$0.180846\pi$
$830$	−13.7213	−0.476275
$831$	0	0
$832$	−2.74336	−0.0951088
$833$	−4.24041	−0.146922
$834$	0	0
$835$	18.0554	0.624832
$836$	−10.4878	−0.362727
$837$	0	0
$838$	18.7140	0.646464
$839$	20.6719	0.713673	0.356837	−	0.934167i	$-0.383855\pi$
$839$	20.6719	0.713673	0.356837	+	0.934167i	$0.383855\pi$
$840$	0	0
$841$	−28.8274	−0.994049
$842$	−0.909269	−0.0313355
$843$	0	0
$844$	−24.2735	−0.835527
$845$	12.5523	0.431814
$846$	0	0
$847$	−3.71266	−0.127569
$848$	−0.697334	−0.0239466
$849$	0	0
$850$	−0.175044	−0.00600394
$851$	23.6734	0.811513
$852$	0	0
$853$	57.7616	1.97772	0.988859	−	0.148853i	$-0.0475583\pi$
$853$	57.7616	1.97772	0.988859	+	0.148853i	$0.0475583\pi$
$854$	−10.3364	−0.353705
$855$	0	0
$856$	0.914259	0.0312487
$857$	2.45333	0.0838042	0.0419021	−	0.999122i	$-0.486658\pi$
$857$	2.45333	0.0838042	0.0419021	+	0.999122i	$0.486658\pi$
$858$	0	0
$859$	−11.5279	−0.393326	−0.196663	−	0.980471i	$-0.563010\pi$
$859$	−11.5279	−0.393326	−0.196663	+	0.980471i	$0.563010\pi$
$860$	−6.51347	−0.222107
$861$	0	0
$862$	−2.94728	−0.100385
$863$	21.4569	0.730403	0.365201	−	0.930929i	$-0.381000\pi$
$863$	21.4569	0.730403	0.365201	+	0.930929i	$0.381000\pi$
$864$	0	0
$865$	46.2045	1.57100
$866$	1.66531	0.0565895
$867$	0	0
$868$	−2.68739	−0.0912159
$869$	36.0425	1.22266
$870$	0	0
$871$	19.3226	0.654720
$872$	−11.4008	−0.386080
$873$	0	0
$874$	−24.4268	−0.826248
$875$	9.37935	0.317080
$876$	0	0
$877$	24.6974	0.833972	0.416986	−	0.908913i	$-0.363086\pi$
$877$	24.6974	0.833972	0.416986	+	0.908913i	$0.363086\pi$
$878$	27.0565	0.913113
$879$	0	0
$880$	5.93373	0.200026
$881$	15.4266	0.519736	0.259868	−	0.965644i	$-0.416321\pi$
$881$	15.4266	0.519736	0.259868	+	0.965644i	$0.416321\pi$
$882$	0	0
$883$	7.10548	0.239118	0.119559	−	0.992827i	$-0.461852\pi$
$883$	7.10548	0.239118	0.119559	+	0.992827i	$0.461852\pi$
$884$	−1.85952	−0.0625423
$885$	0	0
$886$	12.2932	0.412998
$887$	−24.3281	−0.816859	−0.408430	−	0.912790i	$-0.633923\pi$
$887$	−24.3281	−0.816859	−0.408430	+	0.912790i	$0.633923\pi$
$888$	0	0
$889$	10.1498	0.340413
$890$	11.8963	0.398766
$891$	0	0
$892$	1.00000	0.0334825
$893$	−3.99198	−0.133587
$894$	0	0
$895$	3.68160	0.123062
$896$	−0.862607	−0.0288177
$897$	0	0
$898$	26.2543	0.876116
$899$	−1.29423	−0.0431651
$900$	0	0
$901$	−0.472671	−0.0157469
$902$	17.3495	0.577674
$903$	0	0
$904$	1.83027	0.0608739
$905$	27.3817	0.910197
$906$	0	0
$907$	2.68607	0.0891895	0.0445947	−	0.999005i	$-0.485800\pi$
$907$	2.68607	0.0891895	0.0445947	+	0.999005i	$0.485800\pi$
$908$	4.11424	0.136536
$909$	0	0
$910$	−5.42645	−0.179885
$911$	−26.5728	−0.880395	−0.440198	−	0.897901i	$-0.645092\pi$
$911$	−26.5728	−0.880395	−0.440198	+	0.897901i	$0.645092\pi$
$912$	0	0
$913$	15.4840	0.512446
$914$	10.6761	0.353133
$915$	0	0
$916$	9.36331	0.309372
$917$	8.65678	0.285872
$918$	0	0
$919$	14.1083	0.465391	0.232696	−	0.972550i	$-0.425245\pi$
$919$	14.1083	0.465391	0.232696	+	0.972550i	$0.425245\pi$
$920$	13.8201	0.455636
$921$	0	0
$922$	27.4971	0.905569
$923$	18.1464	0.597297
$924$	0	0
$925$	1.01437	0.0333524
$926$	6.65356	0.218650
$927$	0	0
$928$	−0.415427	−0.0136371
$929$	−46.7984	−1.53540	−0.767702	−	0.640807i	$-0.778600\pi$
$929$	−46.7984	−1.53540	−0.767702	+	0.640807i	$0.778600\pi$
$930$	0	0
$931$	−25.3551	−0.830980
$932$	−6.20340	−0.203199
$933$	0	0
$934$	−6.89367	−0.225568
$935$	4.02204	0.131535
$936$	0	0
$937$	30.2057	0.986778	0.493389	−	0.869809i	$-0.335758\pi$
$937$	30.2057	0.986778	0.493389	+	0.869809i	$0.335758\pi$
$938$	6.07569	0.198378
$939$	0	0
$940$	2.25857	0.0736665
$941$	−25.6503	−0.836175	−0.418087	−	0.908407i	$-0.637299\pi$
$941$	−25.6503	−0.836175	−0.418087	+	0.908407i	$0.637299\pi$
$942$	0	0
$943$	40.4082	1.31587
$944$	2.32644	0.0757191
$945$	0	0
$946$	7.35021	0.238976
$947$	−1.75208	−0.0569348	−0.0284674	−	0.999595i	$-0.509063\pi$
$947$	−1.75208	−0.0569348	−0.0284674	+	0.999595i	$0.509063\pi$
$948$	0	0
$949$	17.8943	0.580872
$950$	−1.04665	−0.0339579
$951$	0	0
$952$	−0.584697	−0.0189501
$953$	31.3143	1.01437	0.507184	−	0.861838i	$-0.330686\pi$
$953$	31.3143	1.01437	0.507184	+	0.861838i	$0.330686\pi$
$954$	0	0
$955$	−30.4776	−0.986232
$956$	−5.35388	−0.173157
$957$	0	0
$958$	17.7582	0.573743
$959$	−9.64487	−0.311449
$960$	0	0
$961$	−21.2941	−0.686907
$962$	10.7758	0.347427
$963$	0	0
$964$	−4.29821	−0.138436
$965$	−14.4829	−0.466222
$966$	0	0
$967$	3.77700	0.121460	0.0607300	−	0.998154i	$-0.480657\pi$
$967$	3.77700	0.121460	0.0607300	+	0.998154i	$0.480657\pi$
$968$	4.30400	0.138336
$969$	0	0
$970$	−4.16245	−0.133648
$971$	25.6811	0.824146	0.412073	−	0.911151i	$-0.364805\pi$
$971$	25.6811	0.824146	0.412073	+	0.911151i	$0.364805\pi$
$972$	0	0
$973$	−6.94427	−0.222623
$974$	−10.5533	−0.338148
$975$	0	0
$976$	11.9828	0.383559
$977$	−16.4655	−0.526779	−0.263390	−	0.964690i	$-0.584840\pi$
$977$	−16.4655	−0.526779	−0.263390	+	0.964690i	$0.584840\pi$
$978$	0	0
$979$	−13.4246	−0.429052
$980$	14.3453	0.458245
$981$	0	0
$982$	22.4411	0.716125
$983$	−24.2214	−0.772543	−0.386272	−	0.922385i	$-0.626237\pi$
$983$	−24.2214	−0.772543	−0.386272	+	0.922385i	$0.626237\pi$
$984$	0	0
$985$	25.9012	0.825280
$986$	−0.281587	−0.00896757
$987$	0	0
$988$	−11.1188	−0.353736
$989$	17.1192	0.544359
$990$	0	0
$991$	29.4146	0.934387	0.467193	−	0.884155i	$-0.345265\pi$
$991$	29.4146	0.934387	0.467193	+	0.884155i	$0.345265\pi$
$992$	3.11543	0.0989149
$993$	0	0
$994$	5.70587	0.180979
$995$	23.9416	0.759001
$996$	0	0
$997$	−20.7179	−0.656144	−0.328072	−	0.944653i	$-0.606399\pi$
$997$	−20.7179	−0.656144	−0.328072	+	0.944653i	$0.606399\pi$
$998$	−1.36860	−0.0433224
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.s.1.3		6
3.2	odd	2		1338.2.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.i.1.4	✓	6	3.2	odd	2
4014.2.a.s.1.3		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.29309 q^{5} +0.862607 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.29309 q^{5} +0.862607 q^{7} -1.00000 q^{8} +2.29309 q^{10} -2.58766 q^{11} -2.74336 q^{13} -0.862607 q^{14} +1.00000 q^{16} +0.677825 q^{17} +4.05298 q^{19} -2.29309 q^{20} +2.58766 q^{22} +6.02686 q^{23} +0.258243 q^{25} +2.74336 q^{26} +0.862607 q^{28} +0.415427 q^{29} -3.11543 q^{31} -1.00000 q^{32} -0.677825 q^{34} -1.97803 q^{35} +3.92798 q^{37} -4.05298 q^{38} +2.29309 q^{40} +6.70469 q^{41} +2.84048 q^{43} -2.58766 q^{44} -6.02686 q^{46} -0.984949 q^{47} -6.25591 q^{49} -0.258243 q^{50} -2.74336 q^{52} -0.697334 q^{53} +5.93373 q^{55} -0.862607 q^{56} -0.415427 q^{58} +2.32644 q^{59} +11.9828 q^{61} +3.11543 q^{62} +1.00000 q^{64} +6.29075 q^{65} -7.04341 q^{67} +0.677825 q^{68} +1.97803 q^{70} -6.61468 q^{71} -6.52276 q^{73} -3.92798 q^{74} +4.05298 q^{76} -2.23214 q^{77} -13.9286 q^{79} -2.29309 q^{80} -6.70469 q^{82} -5.98379 q^{83} -1.55431 q^{85} -2.84048 q^{86} +2.58766 q^{88} +5.18792 q^{89} -2.36644 q^{91} +6.02686 q^{92} +0.984949 q^{94} -9.29384 q^{95} -1.81522 q^{97} +6.25591 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} + 5 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 + 5 * q^7 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} + 5 q^{7} - 6 q^{8} + 6 q^{10} - q^{11} + 6 q^{13} - 5 q^{14} + 6 q^{16} - 10 q^{17} - 4 q^{19} - 6 q^{20} + q^{22} - 2 q^{23} - 6 q^{26} + 5 q^{28} - 6 q^{29} - 5 q^{31} - 6 q^{32} + 10 q^{34} + 2 q^{35} + q^{37} + 4 q^{38} + 6 q^{40} - 12 q^{41} - 11 q^{43} - q^{44} + 2 q^{46} - 5 q^{47} - q^{49} + 6 q^{52} - 4 q^{53} - 15 q^{55} - 5 q^{56} + 6 q^{58} - q^{59} - 8 q^{61} + 5 q^{62} + 6 q^{64} - 5 q^{65} - 6 q^{67} - 10 q^{68} - 2 q^{70} - q^{71} - 10 q^{73} - q^{74} - 4 q^{76} + 8 q^{77} - 10 q^{79} - 6 q^{80} + 12 q^{82} + 6 q^{83} - 2 q^{85} + 11 q^{86} + q^{88} + 3 q^{89} - 16 q^{91} - 2 q^{92} + 5 q^{94} + 10 q^{95} + 3 q^{97} + q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 + 5 * q^7 - 6 * q^8 + 6 * q^10 - q^11 + 6 * q^13 - 5 * q^14 + 6 * q^16 - 10 * q^17 - 4 * q^19 - 6 * q^20 + q^22 - 2 * q^23 - 6 * q^26 + 5 * q^28 - 6 * q^29 - 5 * q^31 - 6 * q^32 + 10 * q^34 + 2 * q^35 + q^37 + 4 * q^38 + 6 * q^40 - 12 * q^41 - 11 * q^43 - q^44 + 2 * q^46 - 5 * q^47 - q^49 + 6 * q^52 - 4 * q^53 - 15 * q^55 - 5 * q^56 + 6 * q^58 - q^59 - 8 * q^61 + 5 * q^62 + 6 * q^64 - 5 * q^65 - 6 * q^67 - 10 * q^68 - 2 * q^70 - q^71 - 10 * q^73 - q^74 - 4 * q^76 + 8 * q^77 - 10 * q^79 - 6 * q^80 + 12 * q^82 + 6 * q^83 - 2 * q^85 + 11 * q^86 + q^88 + 3 * q^89 - 16 * q^91 - 2 * q^92 + 5 * q^94 + 10 * q^95 + 3 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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