LMFDB - Embedded newform 4014.2.a.q.1.1

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:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$-1.38266$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.38266 q^{5} +1.61326 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.38266 q^{5} +1.61326 q^{7} +1.00000 q^{8} -2.38266 q^{10} -1.29442 q^{11} -6.29035 q^{13} +1.61326 q^{14} +1.00000 q^{16} +6.73683 q^{17} -4.10296 q^{19} -2.38266 q^{20} -1.29442 q^{22} +8.82507 q^{23} +0.677084 q^{25} -6.29035 q^{26} +1.61326 q^{28} -3.74090 q^{29} +2.10296 q^{31} +1.00000 q^{32} +6.73683 q^{34} -3.84386 q^{35} -9.37859 q^{37} -4.10296 q^{38} -2.38266 q^{40} -8.42180 q^{41} -3.32292 q^{43} -1.29442 q^{44} +8.82507 q^{46} +1.48970 q^{47} -4.39738 q^{49} +0.677084 q^{50} -6.29035 q^{52} -7.88889 q^{53} +3.08417 q^{55} +1.61326 q^{56} -3.74090 q^{58} +11.0704 q^{59} +4.37859 q^{61} +2.10296 q^{62} +1.00000 q^{64} +14.9878 q^{65} -3.61734 q^{67} +6.73683 q^{68} -3.84386 q^{70} -2.01472 q^{71} +7.03507 q^{73} -9.37859 q^{74} -4.10296 q^{76} -2.08824 q^{77} -16.2716 q^{79} -2.38266 q^{80} -8.42180 q^{82} -4.90361 q^{83} -16.0516 q^{85} -3.32292 q^{86} -1.29442 q^{88} -18.3704 q^{89} -10.1480 q^{91} +8.82507 q^{92} +1.48970 q^{94} +9.77597 q^{95} -15.7613 q^{97} -4.39738 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8} - 2 q^{10} - 4 q^{11} - 5 q^{13} - 5 q^{14} + 4 q^{16} + 2 q^{17} - 7 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23} - 6 q^{25} - 5 q^{26} - 5 q^{28} - 9 q^{29} - q^{31} + 4 q^{32} + 2 q^{34} - 11 q^{37} - 7 q^{38} - 2 q^{40} - 14 q^{41} - 22 q^{43} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 7 q^{49} - 6 q^{50} - 5 q^{52} - 3 q^{53} - 13 q^{55} - 5 q^{56} - 9 q^{58} + 6 q^{59} - 9 q^{61} - q^{62} + 4 q^{64} + 3 q^{65} - 22 q^{67} + 2 q^{68} - 5 q^{71} - 3 q^{73} - 11 q^{74} - 7 q^{76} - 2 q^{77} - 29 q^{79} - 2 q^{80} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 22 q^{86} - 4 q^{88} - 9 q^{89} - 18 q^{91} + 4 q^{92} + 8 q^{94} + 2 q^{95} - 29 q^{97} - 7 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8 - 2 * q^10 - 4 * q^11 - 5 * q^13 - 5 * q^14 + 4 * q^16 + 2 * q^17 - 7 * q^19 - 2 * q^20 - 4 * q^22 + 4 * q^23 - 6 * q^25 - 5 * q^26 - 5 * q^28 - 9 * q^29 - q^31 + 4 * q^32 + 2 * q^34 - 11 * q^37 - 7 * q^38 - 2 * q^40 - 14 * q^41 - 22 * q^43 - 4 * q^44 + 4 * q^46 + 8 * q^47 - 7 * q^49 - 6 * q^50 - 5 * q^52 - 3 * q^53 - 13 * q^55 - 5 * q^56 - 9 * q^58 + 6 * q^59 - 9 * q^61 - q^62 + 4 * q^64 + 3 * q^65 - 22 * q^67 + 2 * q^68 - 5 * q^71 - 3 * q^73 - 11 * q^74 - 7 * q^76 - 2 * q^77 - 29 * q^79 - 2 * q^80 - 14 * q^82 + 12 * q^83 - 10 * q^85 - 22 * q^86 - 4 * q^88 - 9 * q^89 - 18 * q^91 + 4 * q^92 + 8 * q^94 + 2 * q^95 - 29 * q^97 - 7 * 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
$5$	−2.38266	−1.06556	−0.532780	−	0.846254i	$-0.678853\pi$
$5$	−2.38266	−1.06556	−0.532780	+	0.846254i	$0.678853\pi$
$6$	0	0
$7$	1.61326	0.609756	0.304878	−	0.952391i	$-0.401384\pi$
$7$	1.61326	0.609756	0.304878	+	0.952391i	$0.401384\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.38266	−0.753464
$11$	−1.29442	−0.390283	−0.195141	−	0.980775i	$-0.562517\pi$
$11$	−1.29442	−0.390283	−0.195141	+	0.980775i	$0.562517\pi$
$12$	0	0
$13$	−6.29035	−1.74463	−0.872314	−	0.488946i	$-0.837382\pi$
$13$	−6.29035	−1.74463	−0.872314	+	0.488946i	$0.837382\pi$
$14$	1.61326	0.431163
$15$	0	0
$16$	1.00000	0.250000
$17$	6.73683	1.63392	0.816961	−	0.576693i	$-0.195657\pi$
$17$	6.73683	1.63392	0.816961	+	0.576693i	$0.195657\pi$
$18$	0	0
$19$	−4.10296	−0.941283	−0.470642	−	0.882324i	$-0.655978\pi$
$19$	−4.10296	−0.941283	−0.470642	+	0.882324i	$0.655978\pi$
$20$	−2.38266	−0.532780
$21$	0	0
$22$	−1.29442	−0.275971
$23$	8.82507	1.84016	0.920078	−	0.391736i	$-0.128125\pi$
$23$	8.82507	1.84016	0.920078	+	0.391736i	$0.128125\pi$
$24$	0	0
$25$	0.677084	0.135417
$26$	−6.29035	−1.23364
$27$	0	0
$28$	1.61326	0.304878
$29$	−3.74090	−0.694669	−0.347334	−	0.937741i	$-0.612913\pi$
$29$	−3.74090	−0.694669	−0.347334	+	0.937741i	$0.612913\pi$
$30$	0	0
$31$	2.10296	0.377703	0.188851	−	0.982006i	$-0.439524\pi$
$31$	2.10296	0.377703	0.188851	+	0.982006i	$0.439524\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.73683	1.15536
$35$	−3.84386	−0.649732
$36$	0	0
$37$	−9.37859	−1.54183	−0.770915	−	0.636938i	$-0.780201\pi$
$37$	−9.37859	−1.54183	−0.770915	+	0.636938i	$0.780201\pi$
$38$	−4.10296	−0.665588
$39$	0	0
$40$	−2.38266	−0.376732
$41$	−8.42180	−1.31526	−0.657632	−	0.753339i	$-0.728442\pi$
$41$	−8.42180	−1.31526	−0.657632	+	0.753339i	$0.728442\pi$
$42$	0	0
$43$	−3.32292	−0.506740	−0.253370	−	0.967369i	$-0.581539\pi$
$43$	−3.32292	−0.506740	−0.253370	+	0.967369i	$0.581539\pi$
$44$	−1.29442	−0.195141
$45$	0	0
$46$	8.82507	1.30119
$47$	1.48970	0.217294	0.108647	−	0.994080i	$-0.465348\pi$
$47$	1.48970	0.217294	0.108647	+	0.994080i	$0.465348\pi$
$48$	0	0
$49$	−4.39738	−0.628197
$50$	0.677084	0.0957542
$51$	0	0
$52$	−6.29035	−0.872314
$53$	−7.88889	−1.08362	−0.541812	−	0.840500i	$-0.682261\pi$
$53$	−7.88889	−1.08362	−0.541812	+	0.840500i	$0.682261\pi$
$54$	0	0
$55$	3.08417	0.415869
$56$	1.61326	0.215581
$57$	0	0
$58$	−3.74090	−0.491205
$59$	11.0704	1.44124	0.720621	−	0.693329i	$-0.243857\pi$
$59$	11.0704	1.44124	0.720621	+	0.693329i	$0.243857\pi$
$60$	0	0
$61$	4.37859	0.560621	0.280311	−	0.959909i	$-0.409563\pi$
$61$	4.37859	0.560621	0.280311	+	0.959909i	$0.409563\pi$
$62$	2.10296	0.267076
$63$	0	0
$64$	1.00000	0.125000
$65$	14.9878	1.85901
$66$	0	0
$67$	−3.61734	−0.441928	−0.220964	−	0.975282i	$-0.570920\pi$
$67$	−3.61734	−0.441928	−0.220964	+	0.975282i	$0.570920\pi$
$68$	6.73683	0.816961
$69$	0	0
$70$	−3.84386	−0.459430
$71$	−2.01472	−0.239103	−0.119551	−	0.992828i	$-0.538146\pi$
$71$	−2.01472	−0.239103	−0.119551	+	0.992828i	$0.538146\pi$
$72$	0	0
$73$	7.03507	0.823392	0.411696	−	0.911321i	$-0.364937\pi$
$73$	7.03507	0.823392	0.411696	+	0.911321i	$0.364937\pi$
$74$	−9.37859	−1.09024
$75$	0	0
$76$	−4.10296	−0.470642
$77$	−2.08824	−0.237977
$78$	0	0
$79$	−16.2716	−1.83069	−0.915347	−	0.402667i	$-0.868083\pi$
$79$	−16.2716	−1.83069	−0.915347	+	0.402667i	$0.868083\pi$
$80$	−2.38266	−0.266390
$81$	0	0
$82$	−8.42180	−0.930032
$83$	−4.90361	−0.538241	−0.269121	−	0.963106i	$-0.586733\pi$
$83$	−4.90361	−0.538241	−0.269121	+	0.963106i	$0.586733\pi$
$84$	0	0
$85$	−16.0516	−1.74104
$86$	−3.32292	−0.358319
$87$	0	0
$88$	−1.29442	−0.137986
$89$	−18.3704	−1.94726	−0.973632	−	0.228126i	$-0.926740\pi$
$89$	−18.3704	−1.94726	−0.973632	+	0.228126i	$0.926740\pi$
$90$	0	0
$91$	−10.1480	−1.06380
$92$	8.82507	0.920078
$93$	0	0
$94$	1.48970	0.153650
$95$	9.77597	1.00299
$96$	0	0
$97$	−15.7613	−1.60031	−0.800156	−	0.599791i	$-0.795250\pi$
$97$	−15.7613	−1.60031	−0.800156	+	0.599791i	$0.795250\pi$
$98$	−4.39738	−0.444202
$99$	0	0
$100$	0.677084	0.0677084
$101$	−2.17267	−0.216189	−0.108094	−	0.994141i	$-0.534475\pi$
$101$	−2.17267	−0.216189	−0.108094	+	0.994141i	$0.534475\pi$
$102$	0	0
$103$	4.33131	0.426776	0.213388	−	0.976968i	$-0.431550\pi$
$103$	4.33131	0.426776	0.213388	+	0.976968i	$0.431550\pi$
$104$	−6.29035	−0.616819
$105$	0	0
$106$	−7.88889	−0.766237
$107$	−13.7531	−1.32956	−0.664782	−	0.747038i	$-0.731475\pi$
$107$	−13.7531	−1.32956	−0.664782	+	0.747038i	$0.731475\pi$
$108$	0	0
$109$	7.03125	0.673472	0.336736	−	0.941599i	$-0.390677\pi$
$109$	7.03125	0.673472	0.336736	+	0.941599i	$0.390677\pi$
$110$	3.08417	0.294064
$111$	0	0
$112$	1.61326	0.152439
$113$	−8.71215	−0.819570	−0.409785	−	0.912182i	$-0.634396\pi$
$113$	−8.71215	−0.819570	−0.409785	+	0.912182i	$0.634396\pi$
$114$	0	0
$115$	−21.0272	−1.96079
$116$	−3.74090	−0.347334
$117$	0	0
$118$	11.0704	1.01911
$119$	10.8683	0.996294
$120$	0	0
$121$	−9.32447	−0.847679
$122$	4.37859	0.396419
$123$	0	0
$124$	2.10296	0.188851
$125$	10.3001	0.921265
$126$	0	0
$127$	−9.49940	−0.842935	−0.421468	−	0.906843i	$-0.638485\pi$
$127$	−9.49940	−0.842935	−0.421468	+	0.906843i	$0.638485\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	14.9878	1.31452
$131$	14.3460	1.25342	0.626709	−	0.779253i	$-0.284402\pi$
$131$	14.3460	1.25342	0.626709	+	0.779253i	$0.284402\pi$
$132$	0	0
$133$	−6.61916	−0.573954
$134$	−3.61734	−0.312490
$135$	0	0
$136$	6.73683	0.577679
$137$	5.10296	0.435975	0.217988	−	0.975952i	$-0.430051\pi$
$137$	5.10296	0.435975	0.217988	+	0.975952i	$0.430051\pi$
$138$	0	0
$139$	−5.63795	−0.478204	−0.239102	−	0.970994i	$-0.576853\pi$
$139$	−5.63795	−0.478204	−0.239102	+	0.970994i	$0.576853\pi$
$140$	−3.84386	−0.324866
$141$	0	0
$142$	−2.01472	−0.169071
$143$	8.14236	0.680898
$144$	0	0
$145$	8.91332	0.740211
$146$	7.03507	0.582226
$147$	0	0
$148$	−9.37859	−0.770915
$149$	5.67119	0.464602	0.232301	−	0.972644i	$-0.425375\pi$
$149$	5.67119	0.464602	0.232301	+	0.972644i	$0.425375\pi$
$150$	0	0
$151$	12.4343	1.01189	0.505943	−	0.862567i	$-0.331145\pi$
$151$	12.4343	1.01189	0.505943	+	0.862567i	$0.331145\pi$
$152$	−4.10296	−0.332794
$153$	0	0
$154$	−2.08824	−0.168275
$155$	−5.01064	−0.402465
$156$	0	0
$157$	8.23623	0.657323	0.328661	−	0.944448i	$-0.393402\pi$
$157$	8.23623	0.657323	0.328661	+	0.944448i	$0.393402\pi$
$158$	−16.2716	−1.29450
$159$	0	0
$160$	−2.38266	−0.188366
$161$	14.2372	1.12205
$162$	0	0
$163$	−7.27000	−0.569430	−0.284715	−	0.958612i	$-0.591899\pi$
$163$	−7.27000	−0.569430	−0.284715	+	0.958612i	$0.591899\pi$
$164$	−8.42180	−0.657632
$165$	0	0
$166$	−4.90361	−0.380594
$167$	9.90361	0.766364	0.383182	−	0.923673i	$-0.374828\pi$
$167$	9.90361	0.766364	0.383182	+	0.923673i	$0.374828\pi$
$168$	0	0
$169$	26.5685	2.04373
$170$	−16.0516	−1.23110
$171$	0	0
$172$	−3.32292	−0.253370
$173$	−23.2252	−1.76578	−0.882890	−	0.469580i	$-0.844405\pi$
$173$	−23.2252	−1.76578	−0.882890	+	0.469580i	$0.844405\pi$
$174$	0	0
$175$	1.09232	0.0825713
$176$	−1.29442	−0.0975707
$177$	0	0
$178$	−18.3704	−1.37692
$179$	21.2499	1.58829	0.794146	−	0.607727i	$-0.207919\pi$
$179$	21.2499	1.58829	0.794146	+	0.607727i	$0.207919\pi$
$180$	0	0
$181$	−20.7569	−1.54285	−0.771425	−	0.636320i	$-0.780456\pi$
$181$	−20.7569	−1.54285	−0.771425	+	0.636320i	$0.780456\pi$
$182$	−10.1480	−0.752219
$183$	0	0
$184$	8.82507	0.650593
$185$	22.3460	1.64291
$186$	0	0
$187$	−8.72030	−0.637691
$188$	1.48970	0.108647
$189$	0	0
$190$	9.77597	0.709223
$191$	−19.2018	−1.38940	−0.694698	−	0.719301i	$-0.744462\pi$
$191$	−19.2018	−1.38940	−0.694698	+	0.719301i	$0.744462\pi$
$192$	0	0
$193$	10.2433	0.737331	0.368665	−	0.929562i	$-0.379815\pi$
$193$	10.2433	0.737331	0.368665	+	0.929562i	$0.379815\pi$
$194$	−15.7613	−1.13159
$195$	0	0
$196$	−4.39738	−0.314099
$197$	−9.76507	−0.695732	−0.347866	−	0.937544i	$-0.613094\pi$
$197$	−9.76507	−0.695732	−0.347866	+	0.937544i	$0.613094\pi$
$198$	0	0
$199$	−3.06564	−0.217317	−0.108659	−	0.994079i	$-0.534656\pi$
$199$	−3.06564	−0.217317	−0.108659	+	0.994079i	$0.534656\pi$
$200$	0.677084	0.0478771
$201$	0	0
$202$	−2.17267	−0.152869
$203$	−6.03507	−0.423579
$204$	0	0
$205$	20.0663	1.40149
$206$	4.33131	0.301776
$207$	0	0
$208$	−6.29035	−0.436157
$209$	5.31096	0.367367
$210$	0	0
$211$	0.240565	0.0165612	0.00828059	−	0.999966i	$-0.497364\pi$
$211$	0.240565	0.0165612	0.00828059	+	0.999966i	$0.497364\pi$
$212$	−7.88889	−0.541812
$213$	0	0
$214$	−13.7531	−0.940143
$215$	7.91739	0.539961
$216$	0	0
$217$	3.39263	0.230307
$218$	7.03125	0.476217
$219$	0	0
$220$	3.08417	0.207935
$221$	−42.3770	−2.85059
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	1.61326	0.107791
$225$	0	0
$226$	−8.71215	−0.579524
$227$	14.8523	0.985779	0.492889	−	0.870092i	$-0.335941\pi$
$227$	14.8523	0.985779	0.492889	+	0.870092i	$0.335941\pi$
$228$	0	0
$229$	5.82639	0.385019	0.192509	−	0.981295i	$-0.438337\pi$
$229$	5.82639	0.385019	0.192509	+	0.981295i	$0.438337\pi$
$230$	−21.0272	−1.38649
$231$	0	0
$232$	−3.74090	−0.245602
$233$	−17.0770	−1.11875	−0.559375	−	0.828915i	$-0.688959\pi$
$233$	−17.0770	−1.11875	−0.559375	+	0.828915i	$0.688959\pi$
$234$	0	0
$235$	−3.54944	−0.231540
$236$	11.0704	0.720621
$237$	0	0
$238$	10.8683	0.704486
$239$	22.0091	1.42365	0.711824	−	0.702358i	$-0.247869\pi$
$239$	22.0091	1.42365	0.711824	+	0.702358i	$0.247869\pi$
$240$	0	0
$241$	13.7948	0.888599	0.444299	−	0.895878i	$-0.353453\pi$
$241$	13.7948	0.888599	0.444299	+	0.895878i	$0.353453\pi$
$242$	−9.32447	−0.599400
$243$	0	0
$244$	4.37859	0.280311
$245$	10.4775	0.669381
$246$	0	0
$247$	25.8090	1.64219
$248$	2.10296	0.133538
$249$	0	0
$250$	10.3001	0.651433
$251$	12.2127	0.770862	0.385431	−	0.922737i	$-0.374053\pi$
$251$	12.2127	0.770862	0.385431	+	0.922737i	$0.374053\pi$
$252$	0	0
$253$	−11.4234	−0.718181
$254$	−9.49940	−0.596045
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.2616	0.951992	0.475996	−	0.879447i	$-0.342088\pi$
$257$	15.2616	0.951992	0.475996	+	0.879447i	$0.342088\pi$
$258$	0	0
$259$	−15.1301	−0.940141
$260$	14.9878	0.929503
$261$	0	0
$262$	14.3460	0.886300
$263$	−29.3341	−1.80882	−0.904408	−	0.426669i	$-0.859687\pi$
$263$	−29.3341	−1.80882	−0.904408	+	0.426669i	$0.859687\pi$
$264$	0	0
$265$	18.7966	1.15466
$266$	−6.61916	−0.405846
$267$	0	0
$268$	−3.61734	−0.220964
$269$	−5.04597	−0.307658	−0.153829	−	0.988097i	$-0.549161\pi$
$269$	−5.04597	−0.307658	−0.153829	+	0.988097i	$0.549161\pi$
$270$	0	0
$271$	−17.5932	−1.06871	−0.534354	−	0.845261i	$-0.679445\pi$
$271$	−17.5932	−1.06871	−0.534354	+	0.845261i	$0.679445\pi$
$272$	6.73683	0.408480
$273$	0	0
$274$	5.10296	0.308281
$275$	−0.876432	−0.0528508
$276$	0	0
$277$	−21.1091	−1.26832	−0.634161	−	0.773201i	$-0.718654\pi$
$277$	−21.1091	−1.26832	−0.634161	+	0.773201i	$0.718654\pi$
$278$	−5.63795	−0.338141
$279$	0	0
$280$	−3.84386	−0.229715
$281$	−24.8464	−1.48221	−0.741105	−	0.671389i	$-0.765698\pi$
$281$	−24.8464	−1.48221	−0.741105	+	0.671389i	$0.765698\pi$
$282$	0	0
$283$	15.2459	0.906277	0.453138	−	0.891440i	$-0.350304\pi$
$283$	15.2459	0.906277	0.453138	+	0.891440i	$0.350304\pi$
$284$	−2.01472	−0.119551
$285$	0	0
$286$	8.14236	0.481468
$287$	−13.5866	−0.801991
$288$	0	0
$289$	28.3849	1.66970
$290$	8.91332	0.523408
$291$	0	0
$292$	7.03507	0.411696
$293$	−16.2334	−0.948363	−0.474181	−	0.880427i	$-0.657256\pi$
$293$	−16.2334	−0.948363	−0.474181	+	0.880427i	$0.657256\pi$
$294$	0	0
$295$	−26.3770	−1.53573
$296$	−9.37859	−0.545119
$297$	0	0
$298$	5.67119	0.328523
$299$	−55.5128	−3.21039
$300$	0	0
$301$	−5.36074	−0.308988
$302$	12.4343	0.715512
$303$	0	0
$304$	−4.10296	−0.235321
$305$	−10.4327	−0.597375
$306$	0	0
$307$	8.09889	0.462228	0.231114	−	0.972927i	$-0.425763\pi$
$307$	8.09889	0.462228	0.231114	+	0.972927i	$0.425763\pi$
$308$	−2.08824	−0.118989
$309$	0	0
$310$	−5.01064	−0.284585
$311$	23.2534	1.31858	0.659291	−	0.751888i	$-0.270857\pi$
$311$	23.2534	1.31858	0.659291	+	0.751888i	$0.270857\pi$
$312$	0	0
$313$	7.91176	0.447199	0.223599	−	0.974681i	$-0.428219\pi$
$313$	7.91176	0.447199	0.223599	+	0.974681i	$0.428219\pi$
$314$	8.23623	0.464797
$315$	0	0
$316$	−16.2716	−0.915347
$317$	2.05975	0.115687	0.0578435	−	0.998326i	$-0.481578\pi$
$317$	2.05975	0.115687	0.0578435	+	0.998326i	$0.481578\pi$
$318$	0	0
$319$	4.84231	0.271117
$320$	−2.38266	−0.133195
$321$	0	0
$322$	14.2372	0.793407
$323$	−27.6409	−1.53798
$324$	0	0
$325$	−4.25910	−0.236252
$326$	−7.27000	−0.402648
$327$	0	0
$328$	−8.42180	−0.465016
$329$	2.40327	0.132497
$330$	0	0
$331$	6.59629	0.362565	0.181283	−	0.983431i	$-0.441975\pi$
$331$	6.59629	0.362565	0.181283	+	0.983431i	$0.441975\pi$
$332$	−4.90361	−0.269121
$333$	0	0
$334$	9.90361	0.541902
$335$	8.61890	0.470901
$336$	0	0
$337$	13.5444	0.737812	0.368906	−	0.929467i	$-0.379732\pi$
$337$	13.5444	0.737812	0.368906	+	0.929467i	$0.379732\pi$
$338$	26.5685	1.44513
$339$	0	0
$340$	−16.0516	−0.870520
$341$	−2.72211	−0.147411
$342$	0	0
$343$	−18.3870	−0.992804
$344$	−3.32292	−0.179160
$345$	0	0
$346$	−23.2252	−1.24860
$347$	11.3204	0.607711	0.303855	−	0.952718i	$-0.401726\pi$
$347$	11.3204	0.607711	0.303855	+	0.952718i	$0.401726\pi$
$348$	0	0
$349$	−23.1517	−1.23928	−0.619641	−	0.784886i	$-0.712722\pi$
$349$	−23.1517	−1.23928	−0.619641	+	0.784886i	$0.712722\pi$
$350$	1.09232	0.0583867
$351$	0	0
$352$	−1.29442	−0.0689929
$353$	31.6382	1.68393	0.841965	−	0.539531i	$-0.181399\pi$
$353$	31.6382	1.68393	0.841965	+	0.539531i	$0.181399\pi$
$354$	0	0
$355$	4.80039	0.254778
$356$	−18.3704	−0.973632
$357$	0	0
$358$	21.2499	1.12309
$359$	30.8685	1.62918	0.814589	−	0.580038i	$-0.196962\pi$
$359$	30.8685	1.62918	0.814589	+	0.580038i	$0.196962\pi$
$360$	0	0
$361$	−2.16572	−0.113986
$362$	−20.7569	−1.09096
$363$	0	0
$364$	−10.1480	−0.531899
$365$	−16.7622	−0.877373
$366$	0	0
$367$	18.5835	0.970048	0.485024	−	0.874501i	$-0.338811\pi$
$367$	18.5835	0.970048	0.485024	+	0.874501i	$0.338811\pi$
$368$	8.82507	0.460039
$369$	0	0
$370$	22.3460	1.16171
$371$	−12.7269	−0.660746
$372$	0	0
$373$	4.22351	0.218685	0.109343	−	0.994004i	$-0.465125\pi$
$373$	4.22351	0.218685	0.109343	+	0.994004i	$0.465125\pi$
$374$	−8.72030	−0.450916
$375$	0	0
$376$	1.48970	0.0768252
$377$	23.5316	1.21194
$378$	0	0
$379$	9.04753	0.464740	0.232370	−	0.972627i	$-0.425352\pi$
$379$	9.04753	0.464740	0.232370	+	0.972627i	$0.425352\pi$
$380$	9.77597	0.501497
$381$	0	0
$382$	−19.2018	−0.982452
$383$	−14.8525	−0.758928	−0.379464	−	0.925207i	$-0.623892\pi$
$383$	−14.8525	−0.758928	−0.379464	+	0.925207i	$0.623892\pi$
$384$	0	0
$385$	4.97558	0.253579
$386$	10.2433	0.521371
$387$	0	0
$388$	−15.7613	−0.800156
$389$	4.34446	0.220273	0.110137	−	0.993916i	$-0.464871\pi$
$389$	4.34446	0.220273	0.110137	+	0.993916i	$0.464871\pi$
$390$	0	0
$391$	59.4530	3.00667
$392$	−4.39738	−0.222101
$393$	0	0
$394$	−9.76507	−0.491957
$395$	38.7696	1.95071
$396$	0	0
$397$	−36.6902	−1.84143	−0.920715	−	0.390236i	$-0.872393\pi$
$397$	−36.6902	−1.84143	−0.920715	+	0.390236i	$0.872393\pi$
$398$	−3.06564	−0.153667
$399$	0	0
$400$	0.677084	0.0338542
$401$	−7.96441	−0.397724	−0.198862	−	0.980028i	$-0.563725\pi$
$401$	−7.96441	−0.397724	−0.198862	+	0.980028i	$0.563725\pi$
$402$	0	0
$403$	−13.2283	−0.658951
$404$	−2.17267	−0.108094
$405$	0	0
$406$	−6.03507	−0.299515
$407$	12.1398	0.601750
$408$	0	0
$409$	−5.89271	−0.291376	−0.145688	−	0.989331i	$-0.546540\pi$
$409$	−5.89271	−0.291376	−0.145688	+	0.989331i	$0.546540\pi$
$410$	20.0663	0.991005
$411$	0	0
$412$	4.33131	0.213388
$413$	17.8595	0.878807
$414$	0	0
$415$	11.6837	0.573528
$416$	−6.29035	−0.308410
$417$	0	0
$418$	5.31096	0.259767
$419$	12.3962	0.605593	0.302797	−	0.953055i	$-0.402080\pi$
$419$	12.3962	0.605593	0.302797	+	0.953055i	$0.402080\pi$
$420$	0	0
$421$	−27.5556	−1.34298	−0.671488	−	0.741015i	$-0.734345\pi$
$421$	−27.5556	−1.34298	−0.671488	+	0.741015i	$0.734345\pi$
$422$	0.240565	0.0117105
$423$	0	0
$424$	−7.88889	−0.383119
$425$	4.56140	0.221261
$426$	0	0
$427$	7.06382	0.341842
$428$	−13.7531	−0.664782
$429$	0	0
$430$	7.91739	0.381810
$431$	−19.1468	−0.922268	−0.461134	−	0.887330i	$-0.652557\pi$
$431$	−19.1468	−0.922268	−0.461134	+	0.887330i	$0.652557\pi$
$432$	0	0
$433$	−30.9452	−1.48713	−0.743567	−	0.668662i	$-0.766868\pi$
$433$	−30.9452	−1.48713	−0.743567	+	0.668662i	$0.766868\pi$
$434$	3.39263	0.162851
$435$	0	0
$436$	7.03125	0.336736
$437$	−36.2089	−1.73211
$438$	0	0
$439$	22.6235	1.07976	0.539880	−	0.841742i	$-0.318470\pi$
$439$	22.6235	1.07976	0.539880	+	0.841742i	$0.318470\pi$
$440$	3.08417	0.147032
$441$	0	0
$442$	−42.3770	−2.01567
$443$	0.982585	0.0466840	0.0233420	−	0.999728i	$-0.492569\pi$
$443$	0.982585	0.0466840	0.0233420	+	0.999728i	$0.492569\pi$
$444$	0	0
$445$	43.7706	2.07492
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	1.61326	0.0762195
$449$	−5.66487	−0.267342	−0.133671	−	0.991026i	$-0.542676\pi$
$449$	−5.66487	−0.267342	−0.133671	+	0.991026i	$0.542676\pi$
$450$	0	0
$451$	10.9014	0.513325
$452$	−8.71215	−0.409785
$453$	0	0
$454$	14.8523	0.697051
$455$	24.1792	1.13354
$456$	0	0
$457$	−7.75756	−0.362883	−0.181442	−	0.983402i	$-0.558076\pi$
$457$	−7.75756	−0.362883	−0.181442	+	0.983402i	$0.558076\pi$
$458$	5.82639	0.272249
$459$	0	0
$460$	−21.0272	−0.980397
$461$	28.0453	1.30620	0.653099	−	0.757272i	$-0.273468\pi$
$461$	28.0453	1.30620	0.653099	+	0.757272i	$0.273468\pi$
$462$	0	0
$463$	−2.07628	−0.0964931	−0.0482465	−	0.998835i	$-0.515363\pi$
$463$	−2.07628	−0.0964931	−0.0482465	+	0.998835i	$0.515363\pi$
$464$	−3.74090	−0.173667
$465$	0	0
$466$	−17.0770	−0.791075
$467$	−23.7477	−1.09891	−0.549457	−	0.835522i	$-0.685165\pi$
$467$	−23.7477	−1.09891	−0.549457	+	0.835522i	$0.685165\pi$
$468$	0	0
$469$	−5.83572	−0.269468
$470$	−3.54944	−0.163724
$471$	0	0
$472$	11.0704	0.509556
$473$	4.30125	0.197772
$474$	0	0
$475$	−2.77805	−0.127466
$476$	10.8683	0.498147
$477$	0	0
$478$	22.0091	1.00667
$479$	16.7167	0.763807	0.381903	−	0.924202i	$-0.375269\pi$
$479$	16.7167	0.763807	0.381903	+	0.924202i	$0.375269\pi$
$480$	0	0
$481$	58.9946	2.68992
$482$	13.7948	0.628334
$483$	0	0
$484$	−9.32447	−0.423840
$485$	37.5538	1.70523
$486$	0	0
$487$	6.73683	0.305275	0.152637	−	0.988282i	$-0.451223\pi$
$487$	6.73683	0.305275	0.152637	+	0.988282i	$0.451223\pi$
$488$	4.37859	0.198209
$489$	0	0
$490$	10.4775	0.473324
$491$	−17.0566	−0.769754	−0.384877	−	0.922968i	$-0.625756\pi$
$491$	−17.0566	−0.769754	−0.384877	+	0.922968i	$0.625756\pi$
$492$	0	0
$493$	−25.2018	−1.13503
$494$	25.8090	1.16120
$495$	0	0
$496$	2.10296	0.0944257
$497$	−3.25027	−0.145795
$498$	0	0
$499$	35.2150	1.57644	0.788220	−	0.615394i	$-0.211003\pi$
$499$	35.2150	1.57644	0.788220	+	0.615394i	$0.211003\pi$
$500$	10.3001	0.460632
$501$	0	0
$502$	12.2127	0.545082
$503$	5.47723	0.244218	0.122109	−	0.992517i	$-0.461034\pi$
$503$	5.47723	0.244218	0.122109	+	0.992517i	$0.461034\pi$
$504$	0	0
$505$	5.17674	0.230362
$506$	−11.4234	−0.507830
$507$	0	0
$508$	−9.49940	−0.421468
$509$	37.9674	1.68288	0.841438	−	0.540354i	$-0.181710\pi$
$509$	37.9674	1.68288	0.841438	+	0.540354i	$0.181710\pi$
$510$	0	0
$511$	11.3494	0.502069
$512$	1.00000	0.0441942
$513$	0	0
$514$	15.2616	0.673160
$515$	−10.3200	−0.454755
$516$	0	0
$517$	−1.92829	−0.0848062
$518$	−15.1301	−0.664780
$519$	0	0
$520$	14.9878	0.657258
$521$	19.1242	0.837848	0.418924	−	0.908021i	$-0.362407\pi$
$521$	19.1242	0.837848	0.418924	+	0.908021i	$0.362407\pi$
$522$	0	0
$523$	−43.0530	−1.88258	−0.941289	−	0.337602i	$-0.890384\pi$
$523$	−43.0530	−1.88258	−0.941289	+	0.337602i	$0.890384\pi$
$524$	14.3460	0.626709
$525$	0	0
$526$	−29.3341	−1.27903
$527$	14.1673	0.617137
$528$	0	0
$529$	54.8819	2.38617
$530$	18.7966	0.816471
$531$	0	0
$532$	−6.61916	−0.286977
$533$	52.9761	2.29465
$534$	0	0
$535$	32.7690	1.41673
$536$	−3.61734	−0.156245
$537$	0	0
$538$	−5.04597	−0.217547
$539$	5.69206	0.245174
$540$	0	0
$541$	−19.2809	−0.828950	−0.414475	−	0.910061i	$-0.636035\pi$
$541$	−19.2809	−0.828950	−0.414475	+	0.910061i	$0.636035\pi$
$542$	−17.5932	−0.755691
$543$	0	0
$544$	6.73683	0.288839
$545$	−16.7531	−0.717624
$546$	0	0
$547$	20.1958	0.863509	0.431755	−	0.901991i	$-0.357895\pi$
$547$	20.1958	0.863509	0.431755	+	0.901991i	$0.357895\pi$
$548$	5.10296	0.217988
$549$	0	0
$550$	−0.876432	−0.0373712
$551$	15.3488	0.653880
$552$	0	0
$553$	−26.2503	−1.11628
$554$	−21.1091	−0.896839
$555$	0	0
$556$	−5.63795	−0.239102
$557$	−15.2462	−0.646002	−0.323001	−	0.946399i	$-0.604692\pi$
$557$	−15.2462	−0.646002	−0.323001	+	0.946399i	$0.604692\pi$
$558$	0	0
$559$	20.9023	0.884073
$560$	−3.84386	−0.162433
$561$	0	0
$562$	−24.8464	−1.04808
$563$	12.8131	0.540008	0.270004	−	0.962859i	$-0.412975\pi$
$563$	12.8131	0.540008	0.270004	+	0.962859i	$0.412975\pi$
$564$	0	0
$565$	20.7581	0.873301
$566$	15.2459	0.640835
$567$	0	0
$568$	−2.01472	−0.0845356
$569$	−14.6814	−0.615476	−0.307738	−	0.951471i	$-0.599572\pi$
$569$	−14.6814	−0.615476	−0.307738	+	0.951471i	$0.599572\pi$
$570$	0	0
$571$	10.8880	0.455647	0.227823	−	0.973702i	$-0.426839\pi$
$571$	10.8880	0.455647	0.227823	+	0.973702i	$0.426839\pi$
$572$	8.14236	0.340449
$573$	0	0
$574$	−13.5866	−0.567093
$575$	5.97532	0.249188
$576$	0	0
$577$	26.2349	1.09217	0.546087	−	0.837729i	$-0.316117\pi$
$577$	26.2349	1.09217	0.546087	+	0.837729i	$0.316117\pi$
$578$	28.3849	1.18066
$579$	0	0
$580$	8.91332	0.370105
$581$	−7.91082	−0.328196
$582$	0	0
$583$	10.2116	0.422919
$584$	7.03507	0.291113
$585$	0	0
$586$	−16.2334	−0.670594
$587$	10.5850	0.436891	0.218445	−	0.975849i	$-0.429901\pi$
$587$	10.5850	0.436891	0.218445	+	0.975849i	$0.429901\pi$
$588$	0	0
$589$	−8.62836	−0.355525
$590$	−26.3770	−1.08592
$591$	0	0
$592$	−9.37859	−0.385458
$593$	24.4252	1.00302	0.501512	−	0.865151i	$-0.332777\pi$
$593$	24.4252	1.00302	0.501512	+	0.865151i	$0.332777\pi$
$594$	0	0
$595$	−25.8955	−1.06161
$596$	5.67119	0.232301
$597$	0	0
$598$	−55.5128	−2.27009
$599$	−39.2278	−1.60281	−0.801403	−	0.598125i	$-0.795913\pi$
$599$	−39.2278	−1.60281	−0.801403	+	0.598125i	$0.795913\pi$
$600$	0	0
$601$	34.6552	1.41361	0.706807	−	0.707407i	$-0.250135\pi$
$601$	34.6552	1.41361	0.706807	+	0.707407i	$0.250135\pi$
$602$	−5.36074	−0.218487
$603$	0	0
$604$	12.4343	0.505943
$605$	22.2171	0.903253
$606$	0	0
$607$	1.28853	0.0522998	0.0261499	−	0.999658i	$-0.491675\pi$
$607$	1.28853	0.0522998	0.0261499	+	0.999658i	$0.491675\pi$
$608$	−4.10296	−0.166397
$609$	0	0
$610$	−10.4327	−0.422408
$611$	−9.37070	−0.379098
$612$	0	0
$613$	31.4868	1.27174	0.635870	−	0.771796i	$-0.280641\pi$
$613$	31.4868	1.27174	0.635870	+	0.771796i	$0.280641\pi$
$614$	8.09889	0.326844
$615$	0	0
$616$	−2.08824	−0.0841377
$617$	20.5693	0.828088	0.414044	−	0.910257i	$-0.364116\pi$
$617$	20.5693	0.828088	0.414044	+	0.910257i	$0.364116\pi$
$618$	0	0
$619$	−0.765762	−0.0307786	−0.0153893	−	0.999882i	$-0.504899\pi$
$619$	−0.765762	−0.0307786	−0.0153893	+	0.999882i	$0.504899\pi$
$620$	−5.01064	−0.201232
$621$	0	0
$622$	23.2534	0.932378
$623$	−29.6364	−1.18736
$624$	0	0
$625$	−27.9270	−1.11708
$626$	7.91176	0.316217
$627$	0	0
$628$	8.23623	0.328661
$629$	−63.1820	−2.51923
$630$	0	0
$631$	−25.8176	−1.02778	−0.513892	−	0.857855i	$-0.671797\pi$
$631$	−25.8176	−1.02778	−0.513892	+	0.857855i	$0.671797\pi$
$632$	−16.2716	−0.647248
$633$	0	0
$634$	2.05975	0.0818030
$635$	22.6339	0.898198
$636$	0	0
$637$	27.6611	1.09597
$638$	4.84231	0.191709
$639$	0	0
$640$	−2.38266	−0.0941830
$641$	−23.2933	−0.920029	−0.460015	−	0.887911i	$-0.652156\pi$
$641$	−23.2933	−0.920029	−0.460015	+	0.887911i	$0.652156\pi$
$642$	0	0
$643$	−34.0373	−1.34230	−0.671150	−	0.741321i	$-0.734199\pi$
$643$	−34.0373	−1.34230	−0.671150	+	0.741321i	$0.734199\pi$
$644$	14.2372	0.561023
$645$	0	0
$646$	−27.6409	−1.08752
$647$	37.4626	1.47281	0.736404	−	0.676542i	$-0.236522\pi$
$647$	37.4626	1.47281	0.736404	+	0.676542i	$0.236522\pi$
$648$	0	0
$649$	−14.3297	−0.562492
$650$	−4.25910	−0.167055
$651$	0	0
$652$	−7.27000	−0.284715
$653$	−36.2540	−1.41873	−0.709363	−	0.704843i	$-0.751017\pi$
$653$	−36.2540	−1.41873	−0.709363	+	0.704843i	$0.751017\pi$
$654$	0	0
$655$	−34.1817	−1.33559
$656$	−8.42180	−0.328816
$657$	0	0
$658$	2.40327	0.0936893
$659$	20.3589	0.793071	0.396535	−	0.918019i	$-0.370212\pi$
$659$	20.3589	0.793071	0.396535	+	0.918019i	$0.370212\pi$
$660$	0	0
$661$	38.1758	1.48487	0.742434	−	0.669919i	$-0.233671\pi$
$661$	38.1758	1.48487	0.742434	+	0.669919i	$0.233671\pi$
$662$	6.59629	0.256372
$663$	0	0
$664$	−4.90361	−0.190297
$665$	15.7712	0.611582
$666$	0	0
$667$	−33.0138	−1.27830
$668$	9.90361	0.383182
$669$	0	0
$670$	8.61890	0.332977
$671$	−5.66774	−0.218801
$672$	0	0
$673$	20.4071	0.786635	0.393318	−	0.919403i	$-0.371327\pi$
$673$	20.4071	0.786635	0.393318	+	0.919403i	$0.371327\pi$
$674$	13.5444	0.521712
$675$	0	0
$676$	26.5685	1.02186
$677$	34.5293	1.32707	0.663535	−	0.748145i	$-0.269055\pi$
$677$	34.5293	1.32707	0.663535	+	0.748145i	$0.269055\pi$
$678$	0	0
$679$	−25.4271	−0.975801
$680$	−16.0516	−0.615551
$681$	0	0
$682$	−2.72211	−0.104235
$683$	7.23805	0.276956	0.138478	−	0.990365i	$-0.455779\pi$
$683$	7.23805	0.276956	0.138478	+	0.990365i	$0.455779\pi$
$684$	0	0
$685$	−12.1586	−0.464558
$686$	−18.3870	−0.702018
$687$	0	0
$688$	−3.32292	−0.126685
$689$	49.6239	1.89052
$690$	0	0
$691$	−26.0593	−0.991343	−0.495671	−	0.868510i	$-0.665078\pi$
$691$	−26.0593	−0.991343	−0.495671	+	0.868510i	$0.665078\pi$
$692$	−23.2252	−0.882890
$693$	0	0
$694$	11.3204	0.429717
$695$	13.4333	0.509555
$696$	0	0
$697$	−56.7363	−2.14904
$698$	−23.1517	−0.876304
$699$	0	0
$700$	1.09232	0.0412856
$701$	3.75011	0.141640	0.0708198	−	0.997489i	$-0.477438\pi$
$701$	3.75011	0.141640	0.0708198	+	0.997489i	$0.477438\pi$
$702$	0	0
$703$	38.4800	1.45130
$704$	−1.29442	−0.0487853
$705$	0	0
$706$	31.6382	1.19072
$707$	−3.50509	−0.131823
$708$	0	0
$709$	−24.7368	−0.929011	−0.464506	−	0.885570i	$-0.653768\pi$
$709$	−24.7368	−0.929011	−0.464506	+	0.885570i	$0.653768\pi$
$710$	4.80039	0.180156
$711$	0	0
$712$	−18.3704	−0.688461
$713$	18.5588	0.695031
$714$	0	0
$715$	−19.4005	−0.725537
$716$	21.2499	0.794146
$717$	0	0
$718$	30.8685	1.15200
$719$	−23.8382	−0.889015	−0.444508	−	0.895775i	$-0.646621\pi$
$719$	−23.8382	−0.889015	−0.444508	+	0.895775i	$0.646621\pi$
$720$	0	0
$721$	6.98754	0.260229
$722$	−2.16572	−0.0805999
$723$	0	0
$724$	−20.7569	−0.771425
$725$	−2.53291	−0.0940698
$726$	0	0
$727$	−21.3209	−0.790749	−0.395375	−	0.918520i	$-0.629385\pi$
$727$	−21.3209	−0.790749	−0.395375	+	0.918520i	$0.629385\pi$
$728$	−10.1480	−0.376110
$729$	0	0
$730$	−16.7622	−0.620397
$731$	−22.3859	−0.827973
$732$	0	0
$733$	41.0982	1.51800	0.758998	−	0.651093i	$-0.225689\pi$
$733$	41.0982	1.51800	0.758998	+	0.651093i	$0.225689\pi$
$734$	18.5835	0.685928
$735$	0	0
$736$	8.82507	0.325297
$737$	4.68236	0.172477
$738$	0	0
$739$	−15.0093	−0.552127	−0.276064	−	0.961139i	$-0.589030\pi$
$739$	−15.0093	−0.552127	−0.276064	+	0.961139i	$0.589030\pi$
$740$	22.3460	0.821456
$741$	0	0
$742$	−12.7269	−0.467218
$743$	−37.6777	−1.38226	−0.691131	−	0.722729i	$-0.742887\pi$
$743$	−37.6777	−1.38226	−0.691131	+	0.722729i	$0.742887\pi$
$744$	0	0
$745$	−13.5125	−0.495061
$746$	4.22351	0.154634
$747$	0	0
$748$	−8.72030	−0.318846
$749$	−22.1874	−0.810710
$750$	0	0
$751$	0.0365249	0.00133281	0.000666406	−	1.00000i	$-0.499788\pi$
$751$	0.0365249	0.00133281	0.000666406		1.00000i	$0.499788\pi$
$752$	1.48970	0.0543236
$753$	0	0
$754$	23.5316	0.856970
$755$	−29.6267	−1.07822
$756$	0	0
$757$	9.40958	0.341997	0.170999	−	0.985271i	$-0.445301\pi$
$757$	9.40958	0.341997	0.170999	+	0.985271i	$0.445301\pi$
$758$	9.04753	0.328621
$759$	0	0
$760$	9.77597	0.354612
$761$	−4.60658	−0.166988	−0.0834941	−	0.996508i	$-0.526608\pi$
$761$	−4.60658	−0.166988	−0.0834941	+	0.996508i	$0.526608\pi$
$762$	0	0
$763$	11.3433	0.410654
$764$	−19.2018	−0.694698
$765$	0	0
$766$	−14.8525	−0.536643
$767$	−69.6366	−2.51443
$768$	0	0
$769$	53.1560	1.91685	0.958427	−	0.285337i	$-0.0921055\pi$
$769$	53.1560	1.91685	0.958427	+	0.285337i	$0.0921055\pi$
$770$	4.97558	0.179307
$771$	0	0
$772$	10.2433	0.368665
$773$	38.6451	1.38997	0.694984	−	0.719025i	$-0.255411\pi$
$773$	38.6451	1.38997	0.694984	+	0.719025i	$0.255411\pi$
$774$	0	0
$775$	1.42388	0.0511473
$776$	−15.7613	−0.565796
$777$	0	0
$778$	4.34446	0.155757
$779$	34.5543	1.23804
$780$	0	0
$781$	2.60789	0.0933177
$782$	59.4530	2.12604
$783$	0	0
$784$	−4.39738	−0.157049
$785$	−19.6242	−0.700417
$786$	0	0
$787$	49.8025	1.77527	0.887634	−	0.460550i	$-0.152348\pi$
$787$	49.8025	1.77527	0.887634	+	0.460550i	$0.152348\pi$
$788$	−9.76507	−0.347866
$789$	0	0
$790$	38.7696	1.37936
$791$	−14.0550	−0.499738
$792$	0	0
$793$	−27.5429	−0.978075
$794$	−36.6902	−1.30209
$795$	0	0
$796$	−3.06564	−0.108659
$797$	36.2893	1.28543	0.642716	−	0.766104i	$-0.277807\pi$
$797$	36.2893	1.28543	0.642716	+	0.766104i	$0.277807\pi$
$798$	0	0
$799$	10.0358	0.355042
$800$	0.677084	0.0239385
$801$	0	0
$802$	−7.96441	−0.281233
$803$	−9.10634	−0.321356
$804$	0	0
$805$	−33.9224	−1.19561
$806$	−13.2283	−0.465949
$807$	0	0
$808$	−2.17267	−0.0764343
$809$	21.1471	0.743493	0.371746	−	0.928334i	$-0.378759\pi$
$809$	21.1471	0.743493	0.371746	+	0.928334i	$0.378759\pi$
$810$	0	0
$811$	−33.5906	−1.17953	−0.589764	−	0.807576i	$-0.700779\pi$
$811$	−33.5906	−1.17953	−0.589764	+	0.807576i	$0.700779\pi$
$812$	−6.03507	−0.211789
$813$	0	0
$814$	12.1398	0.425501
$815$	17.3220	0.606762
$816$	0	0
$817$	13.6338	0.476986
$818$	−5.89271	−0.206034
$819$	0	0
$820$	20.0663	0.700746
$821$	39.7957	1.38888	0.694439	−	0.719551i	$-0.255652\pi$
$821$	39.7957	1.38888	0.694439	+	0.719551i	$0.255652\pi$
$822$	0	0
$823$	−33.5506	−1.16950	−0.584751	−	0.811213i	$-0.698808\pi$
$823$	−33.5506	−1.16950	−0.584751	+	0.811213i	$0.698808\pi$
$824$	4.33131	0.150888
$825$	0	0
$826$	17.8595	0.621410
$827$	37.2737	1.29613	0.648066	−	0.761584i	$-0.275578\pi$
$827$	37.2737	1.29613	0.648066	+	0.761584i	$0.275578\pi$
$828$	0	0
$829$	1.16245	0.0403734	0.0201867	−	0.999796i	$-0.493574\pi$
$829$	1.16245	0.0403734	0.0201867	+	0.999796i	$0.493574\pi$
$830$	11.6837	0.405546
$831$	0	0
$832$	−6.29035	−0.218079
$833$	−29.6244	−1.02643
$834$	0	0
$835$	−23.5970	−0.816607
$836$	5.31096	0.183683
$837$	0	0
$838$	12.3962	0.428219
$839$	19.5397	0.674587	0.337293	−	0.941400i	$-0.390489\pi$
$839$	19.5397	0.674587	0.337293	+	0.941400i	$0.390489\pi$
$840$	0	0
$841$	−15.0056	−0.517436
$842$	−27.5556	−0.949628
$843$	0	0
$844$	0.240565	0.00828059
$845$	−63.3037	−2.17771
$846$	0	0
$847$	−15.0428	−0.516878
$848$	−7.88889	−0.270906
$849$	0	0
$850$	4.56140	0.156455
$851$	−82.7668	−2.83721
$852$	0	0
$853$	19.5682	0.670004	0.335002	−	0.942217i	$-0.391263\pi$
$853$	19.5682	0.670004	0.335002	+	0.942217i	$0.391263\pi$
$854$	7.06382	0.241719
$855$	0	0
$856$	−13.7531	−0.470072
$857$	−25.5547	−0.872932	−0.436466	−	0.899721i	$-0.643770\pi$
$857$	−25.5547	−0.872932	−0.436466	+	0.899721i	$0.643770\pi$
$858$	0	0
$859$	−17.9939	−0.613945	−0.306973	−	0.951718i	$-0.599316\pi$
$859$	−17.9939	−0.613945	−0.306973	+	0.951718i	$0.599316\pi$
$860$	7.91739	0.269981
$861$	0	0
$862$	−19.1468	−0.652142
$863$	25.7633	0.876993	0.438497	−	0.898733i	$-0.355511\pi$
$863$	25.7633	0.876993	0.438497	+	0.898733i	$0.355511\pi$
$864$	0	0
$865$	55.3379	1.88154
$866$	−30.9452	−1.05156
$867$	0	0
$868$	3.39263	0.115153
$869$	21.0622	0.714488
$870$	0	0
$871$	22.7543	0.771000
$872$	7.03125	0.238108
$873$	0	0
$874$	−36.2089	−1.22478
$875$	16.6167	0.561747
$876$	0	0
$877$	14.5703	0.492004	0.246002	−	0.969269i	$-0.420883\pi$
$877$	14.5703	0.492004	0.246002	+	0.969269i	$0.420883\pi$
$878$	22.6235	0.763505
$879$	0	0
$880$	3.08417	0.103967
$881$	−22.2400	−0.749286	−0.374643	−	0.927169i	$-0.622235\pi$
$881$	−22.2400	−0.749286	−0.374643	+	0.927169i	$0.622235\pi$
$882$	0	0
$883$	−36.4675	−1.22723	−0.613615	−	0.789606i	$-0.710285\pi$
$883$	−36.4675	−1.22723	−0.613615	+	0.789606i	$0.710285\pi$
$884$	−42.3770	−1.42529
$885$	0	0
$886$	0.982585	0.0330106
$887$	−15.1332	−0.508122	−0.254061	−	0.967188i	$-0.581766\pi$
$887$	−15.1332	−0.508122	−0.254061	+	0.967188i	$0.581766\pi$
$888$	0	0
$889$	−15.3250	−0.513985
$890$	43.7706	1.46719
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−6.11216	−0.204536
$894$	0	0
$895$	−50.6313	−1.69242
$896$	1.61326	0.0538954
$897$	0	0
$898$	−5.66487	−0.189039
$899$	−7.86697	−0.262378
$900$	0	0
$901$	−53.1462	−1.77056
$902$	10.9014	0.362975
$903$	0	0
$904$	−8.71215	−0.289762
$905$	49.4568	1.64400
$906$	0	0
$907$	50.7958	1.68665	0.843323	−	0.537407i	$-0.180596\pi$
$907$	50.7958	1.68665	0.843323	+	0.537407i	$0.180596\pi$
$908$	14.8523	0.492889
$909$	0	0
$910$	24.1792	0.801534
$911$	−50.6064	−1.67667	−0.838333	−	0.545159i	$-0.816469\pi$
$911$	−50.6064	−1.67667	−0.838333	+	0.545159i	$0.816469\pi$
$912$	0	0
$913$	6.34734	0.210066
$914$	−7.75756	−0.256597
$915$	0	0
$916$	5.82639	0.192509
$917$	23.1439	0.764279
$918$	0	0
$919$	−1.34015	−0.0442074	−0.0221037	−	0.999756i	$-0.507036\pi$
$919$	−1.34015	−0.0442074	−0.0221037	+	0.999756i	$0.507036\pi$
$920$	−21.0272	−0.693246
$921$	0	0
$922$	28.0453	0.923622
$923$	12.6733	0.417146
$924$	0	0
$925$	−6.35010	−0.208790
$926$	−2.07628	−0.0682309
$927$	0	0
$928$	−3.74090	−0.122801
$929$	55.9142	1.83448	0.917242	−	0.398330i	$-0.130410\pi$
$929$	55.9142	1.83448	0.917242	+	0.398330i	$0.130410\pi$
$930$	0	0
$931$	18.0423	0.591312
$932$	−17.0770	−0.559375
$933$	0	0
$934$	−23.7477	−0.777049
$935$	20.7775	0.679498
$936$	0	0
$937$	27.2834	0.891310	0.445655	−	0.895205i	$-0.352971\pi$
$937$	27.2834	0.891310	0.445655	+	0.895205i	$0.352971\pi$
$938$	−5.83572	−0.190543
$939$	0	0
$940$	−3.54944	−0.115770
$941$	−22.9411	−0.747859	−0.373929	−	0.927457i	$-0.621990\pi$
$941$	−22.9411	−0.747859	−0.373929	+	0.927457i	$0.621990\pi$
$942$	0	0
$943$	−74.3230	−2.42029
$944$	11.0704	0.360311
$945$	0	0
$946$	4.30125	0.139846
$947$	−42.6806	−1.38693	−0.693467	−	0.720488i	$-0.743918\pi$
$947$	−42.6806	−1.38693	−0.693467	+	0.720488i	$0.743918\pi$
$948$	0	0
$949$	−44.2530	−1.43651
$950$	−2.77805	−0.0901318
$951$	0	0
$952$	10.8683	0.352243
$953$	26.7188	0.865507	0.432753	−	0.901512i	$-0.357542\pi$
$953$	26.7188	0.865507	0.432753	+	0.901512i	$0.357542\pi$
$954$	0	0
$955$	45.7515	1.48048
$956$	22.0091	0.711824
$957$	0	0
$958$	16.7167	0.540093
$959$	8.23242	0.265839
$960$	0	0
$961$	−26.5776	−0.857341
$962$	58.9946	1.90206
$963$	0	0
$964$	13.7948	0.444299
$965$	−24.4064	−0.785669
$966$	0	0
$967$	−55.9573	−1.79946	−0.899732	−	0.436442i	$-0.856238\pi$
$967$	−55.9573	−1.79946	−0.899732	+	0.436442i	$0.856238\pi$
$968$	−9.32447	−0.299700
$969$	0	0
$970$	37.5538	1.20578
$971$	5.50174	0.176559	0.0882796	−	0.996096i	$-0.471863\pi$
$971$	5.50174	0.176559	0.0882796	+	0.996096i	$0.471863\pi$
$972$	0	0
$973$	−9.09549	−0.291588
$974$	6.73683	0.215862
$975$	0	0
$976$	4.37859	0.140155
$977$	15.0995	0.483076	0.241538	−	0.970391i	$-0.422348\pi$
$977$	15.0995	0.483076	0.241538	+	0.970391i	$0.422348\pi$
$978$	0	0
$979$	23.7791	0.759983
$980$	10.4775	0.334691
$981$	0	0
$982$	−17.0566	−0.544298
$983$	−48.5862	−1.54966	−0.774830	−	0.632170i	$-0.782164\pi$
$983$	−48.5862	−1.54966	−0.774830	+	0.632170i	$0.782164\pi$
$984$	0	0
$985$	23.2669	0.741344
$986$	−25.2018	−0.802590
$987$	0	0
$988$	25.8090	0.821095
$989$	−29.3250	−0.932480
$990$	0	0
$991$	33.9605	1.07879	0.539395	−	0.842053i	$-0.318653\pi$
$991$	33.9605	1.07879	0.539395	+	0.842053i	$0.318653\pi$
$992$	2.10296	0.0667690
$993$	0	0
$994$	−3.25027	−0.103092
$995$	7.30439	0.231565
$996$	0	0
$997$	8.69148	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$997$	8.69148	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$998$	35.2150	1.11471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.q.1.1		4
3.2	odd	2		1338.2.a.g.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.g.1.4	✓	4	3.2	odd	2
4014.2.a.q.1.1		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.38266 q^{5} +1.61326 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.38266 q^{5} +1.61326 q^{7} +1.00000 q^{8} -2.38266 q^{10} -1.29442 q^{11} -6.29035 q^{13} +1.61326 q^{14} +1.00000 q^{16} +6.73683 q^{17} -4.10296 q^{19} -2.38266 q^{20} -1.29442 q^{22} +8.82507 q^{23} +0.677084 q^{25} -6.29035 q^{26} +1.61326 q^{28} -3.74090 q^{29} +2.10296 q^{31} +1.00000 q^{32} +6.73683 q^{34} -3.84386 q^{35} -9.37859 q^{37} -4.10296 q^{38} -2.38266 q^{40} -8.42180 q^{41} -3.32292 q^{43} -1.29442 q^{44} +8.82507 q^{46} +1.48970 q^{47} -4.39738 q^{49} +0.677084 q^{50} -6.29035 q^{52} -7.88889 q^{53} +3.08417 q^{55} +1.61326 q^{56} -3.74090 q^{58} +11.0704 q^{59} +4.37859 q^{61} +2.10296 q^{62} +1.00000 q^{64} +14.9878 q^{65} -3.61734 q^{67} +6.73683 q^{68} -3.84386 q^{70} -2.01472 q^{71} +7.03507 q^{73} -9.37859 q^{74} -4.10296 q^{76} -2.08824 q^{77} -16.2716 q^{79} -2.38266 q^{80} -8.42180 q^{82} -4.90361 q^{83} -16.0516 q^{85} -3.32292 q^{86} -1.29442 q^{88} -18.3704 q^{89} -10.1480 q^{91} +8.82507 q^{92} +1.48970 q^{94} +9.77597 q^{95} -15.7613 q^{97} -4.39738 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 2 q^{5} - 5 q^{7} + 4 q^{8} - 2 q^{10} - 4 q^{11} - 5 q^{13} - 5 q^{14} + 4 q^{16} + 2 q^{17} - 7 q^{19} - 2 q^{20} - 4 q^{22} + 4 q^{23} - 6 q^{25} - 5 q^{26} - 5 q^{28} - 9 q^{29} - q^{31} + 4 q^{32} + 2 q^{34} - 11 q^{37} - 7 q^{38} - 2 q^{40} - 14 q^{41} - 22 q^{43} - 4 q^{44} + 4 q^{46} + 8 q^{47} - 7 q^{49} - 6 q^{50} - 5 q^{52} - 3 q^{53} - 13 q^{55} - 5 q^{56} - 9 q^{58} + 6 q^{59} - 9 q^{61} - q^{62} + 4 q^{64} + 3 q^{65} - 22 q^{67} + 2 q^{68} - 5 q^{71} - 3 q^{73} - 11 q^{74} - 7 q^{76} - 2 q^{77} - 29 q^{79} - 2 q^{80} - 14 q^{82} + 12 q^{83} - 10 q^{85} - 22 q^{86} - 4 q^{88} - 9 q^{89} - 18 q^{91} + 4 q^{92} + 8 q^{94} + 2 q^{95} - 29 q^{97} - 7 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 2 * q^5 - 5 * q^7 + 4 * q^8 - 2 * q^10 - 4 * q^11 - 5 * q^13 - 5 * q^14 + 4 * q^16 + 2 * q^17 - 7 * q^19 - 2 * q^20 - 4 * q^22 + 4 * q^23 - 6 * q^25 - 5 * q^26 - 5 * q^28 - 9 * q^29 - q^31 + 4 * q^32 + 2 * q^34 - 11 * q^37 - 7 * q^38 - 2 * q^40 - 14 * q^41 - 22 * q^43 - 4 * q^44 + 4 * q^46 + 8 * q^47 - 7 * q^49 - 6 * q^50 - 5 * q^52 - 3 * q^53 - 13 * q^55 - 5 * q^56 - 9 * q^58 + 6 * q^59 - 9 * q^61 - q^62 + 4 * q^64 + 3 * q^65 - 22 * q^67 + 2 * q^68 - 5 * q^71 - 3 * q^73 - 11 * q^74 - 7 * q^76 - 2 * q^77 - 29 * q^79 - 2 * q^80 - 14 * q^82 + 12 * q^83 - 10 * q^85 - 22 * q^86 - 4 * q^88 - 9 * q^89 - 18 * q^91 + 4 * q^92 + 8 * q^94 + 2 * q^95 - 29 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more