LMFDB - Embedded newform 4014.2.a.r.1.2

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:	$5$
Coefficient field:	5.5.356173.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 9x^{2} + 14x - 4 $ x^5 - 2x^4 - 7x^3 + 9x^2 + 14x - 4
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			1.2
Root			$2.75496$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.54501 q^{5} -1.28984 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.54501 q^{5} -1.28984 q^{7} +1.00000 q^{8} -1.54501 q^{10} +2.49412 q^{11} -2.65927 q^{13} -1.28984 q^{14} +1.00000 q^{16} -1.92011 q^{17} +4.69562 q^{19} -1.54501 q^{20} +2.49412 q^{22} -3.63334 q^{23} -2.61296 q^{25} -2.65927 q^{26} -1.28984 q^{28} -9.08085 q^{29} +5.96966 q^{31} +1.00000 q^{32} -1.92011 q^{34} +1.99281 q^{35} +5.70604 q^{37} +4.69562 q^{38} -1.54501 q^{40} -6.53614 q^{41} +6.44584 q^{43} +2.49412 q^{44} -3.63334 q^{46} -12.1071 q^{47} -5.33631 q^{49} -2.61296 q^{50} -2.65927 q^{52} -2.97407 q^{53} -3.85342 q^{55} -1.28984 q^{56} -9.08085 q^{58} +12.6372 q^{59} -10.5220 q^{61} +5.96966 q^{62} +1.00000 q^{64} +4.10859 q^{65} +4.51299 q^{67} -1.92011 q^{68} +1.99281 q^{70} -6.01319 q^{71} -9.46377 q^{73} +5.70604 q^{74} +4.69562 q^{76} -3.21701 q^{77} -9.23357 q^{79} -1.54501 q^{80} -6.53614 q^{82} -12.9523 q^{83} +2.96659 q^{85} +6.44584 q^{86} +2.49412 q^{88} +0.168223 q^{89} +3.43004 q^{91} -3.63334 q^{92} -12.1071 q^{94} -7.25476 q^{95} +1.05018 q^{97} -5.33631 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8} - 5 q^{10} - 9 q^{11} - q^{14} + 5 q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{20} - 9 q^{22} - 16 q^{23} + 8 q^{25} - q^{28} - 8 q^{29} - q^{31} + 5 q^{32} - 6 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} - 5 q^{40} - 4 q^{41} + 3 q^{43} - 9 q^{44} - 16 q^{46} - 18 q^{47} + 2 q^{49} + 8 q^{50} - 26 q^{53} + q^{55} - q^{56} - 8 q^{58} - 21 q^{59} - 20 q^{61} - q^{62} + 5 q^{64} + 3 q^{65} - 5 q^{67} - 6 q^{68} - 22 q^{70} - 17 q^{71} + 5 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{77} - 21 q^{79} - 5 q^{80} - 4 q^{82} - 11 q^{83} - 12 q^{85} + 3 q^{86} - 9 q^{88} + 5 q^{89} - 10 q^{91} - 16 q^{92} - 18 q^{94} - 10 q^{95} - 11 q^{97} + 2 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8 - 5 * q^10 - 9 * q^11 - q^14 + 5 * q^16 - 6 * q^17 - 4 * q^19 - 5 * q^20 - 9 * q^22 - 16 * q^23 + 8 * q^25 - q^28 - 8 * q^29 - q^31 + 5 * q^32 - 6 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 - 5 * q^40 - 4 * q^41 + 3 * q^43 - 9 * q^44 - 16 * q^46 - 18 * q^47 + 2 * q^49 + 8 * q^50 - 26 * q^53 + q^55 - q^56 - 8 * q^58 - 21 * q^59 - 20 * q^61 - q^62 + 5 * q^64 + 3 * q^65 - 5 * q^67 - 6 * q^68 - 22 * q^70 - 17 * q^71 + 5 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^77 - 21 * q^79 - 5 * q^80 - 4 * q^82 - 11 * q^83 - 12 * q^85 + 3 * q^86 - 9 * q^88 + 5 * q^89 - 10 * q^91 - 16 * q^92 - 18 * q^94 - 10 * q^95 - 11 * q^97 + 2 * 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
$5$	−1.54501	−0.690947	−0.345474	−	0.938428i	$-0.612282\pi$
$5$	−1.54501	−0.690947	−0.345474	+	0.938428i	$0.612282\pi$
$6$	0	0
$7$	−1.28984	−0.487514	−0.243757	−	0.969836i	$-0.578380\pi$
$7$	−1.28984	−0.487514	−0.243757	+	0.969836i	$0.578380\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.54501	−0.488574
$11$	2.49412	0.752005	0.376002	−	0.926619i	$-0.377298\pi$
$11$	2.49412	0.752005	0.376002	+	0.926619i	$0.377298\pi$
$12$	0	0
$13$	−2.65927	−0.737549	−0.368775	−	0.929519i	$-0.620223\pi$
$13$	−2.65927	−0.737549	−0.368775	+	0.929519i	$0.620223\pi$
$14$	−1.28984	−0.344724
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.92011	−0.465696	−0.232848	−	0.972513i	$-0.574805\pi$
$17$	−1.92011	−0.465696	−0.232848	+	0.972513i	$0.574805\pi$
$18$	0	0
$19$	4.69562	1.07725	0.538624	−	0.842546i	$-0.318944\pi$
$19$	4.69562	1.07725	0.538624	+	0.842546i	$0.318944\pi$
$20$	−1.54501	−0.345474
$21$	0	0
$22$	2.49412	0.531747
$23$	−3.63334	−0.757604	−0.378802	−	0.925478i	$-0.623664\pi$
$23$	−3.63334	−0.757604	−0.378802	+	0.925478i	$0.623664\pi$
$24$	0	0
$25$	−2.61296	−0.522592
$26$	−2.65927	−0.521526
$27$	0	0
$28$	−1.28984	−0.243757
$29$	−9.08085	−1.68627	−0.843136	−	0.537700i	$-0.819293\pi$
$29$	−9.08085	−1.68627	−0.843136	+	0.537700i	$0.819293\pi$
$30$	0	0
$31$	5.96966	1.07218	0.536091	−	0.844160i	$-0.319900\pi$
$31$	5.96966	1.07218	0.536091	+	0.844160i	$0.319900\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.92011	−0.329297
$35$	1.99281	0.336846
$36$	0	0
$37$	5.70604	0.938067	0.469033	−	0.883180i	$-0.344602\pi$
$37$	5.70604	0.938067	0.469033	+	0.883180i	$0.344602\pi$
$38$	4.69562	0.761730
$39$	0	0
$40$	−1.54501	−0.244287
$41$	−6.53614	−1.02077	−0.510387	−	0.859945i	$-0.670498\pi$
$41$	−6.53614	−1.02077	−0.510387	+	0.859945i	$0.670498\pi$
$42$	0	0
$43$	6.44584	0.982981	0.491491	−	0.870883i	$-0.336452\pi$
$43$	6.44584	0.982981	0.491491	+	0.870883i	$0.336452\pi$
$44$	2.49412	0.376002
$45$	0	0
$46$	−3.63334	−0.535707
$47$	−12.1071	−1.76600	−0.882999	−	0.469374i	$-0.844479\pi$
$47$	−12.1071	−1.76600	−0.882999	+	0.469374i	$0.844479\pi$
$48$	0	0
$49$	−5.33631	−0.762330
$50$	−2.61296	−0.369528
$51$	0	0
$52$	−2.65927	−0.368775
$53$	−2.97407	−0.408520	−0.204260	−	0.978917i	$-0.565479\pi$
$53$	−2.97407	−0.408520	−0.204260	+	0.978917i	$0.565479\pi$
$54$	0	0
$55$	−3.85342	−0.519595
$56$	−1.28984	−0.172362
$57$	0	0
$58$	−9.08085	−1.19237
$59$	12.6372	1.64523	0.822615	−	0.568599i	$-0.192514\pi$
$59$	12.6372	1.64523	0.822615	+	0.568599i	$0.192514\pi$
$60$	0	0
$61$	−10.5220	−1.34721	−0.673603	−	0.739094i	$-0.735254\pi$
$61$	−10.5220	−1.34721	−0.673603	+	0.739094i	$0.735254\pi$
$62$	5.96966	0.758147
$63$	0	0
$64$	1.00000	0.125000
$65$	4.10859	0.509608
$66$	0	0
$67$	4.51299	0.551349	0.275675	−	0.961251i	$-0.411099\pi$
$67$	4.51299	0.551349	0.275675	+	0.961251i	$0.411099\pi$
$68$	−1.92011	−0.232848
$69$	0	0
$70$	1.99281	0.238186
$71$	−6.01319	−0.713635	−0.356817	−	0.934174i	$-0.616138\pi$
$71$	−6.01319	−0.713635	−0.356817	+	0.934174i	$0.616138\pi$
$72$	0	0
$73$	−9.46377	−1.10765	−0.553825	−	0.832633i	$-0.686832\pi$
$73$	−9.46377	−1.10765	−0.553825	+	0.832633i	$0.686832\pi$
$74$	5.70604	0.663313
$75$	0	0
$76$	4.69562	0.538624
$77$	−3.21701	−0.366613
$78$	0	0
$79$	−9.23357	−1.03886	−0.519429	−	0.854514i	$-0.673855\pi$
$79$	−9.23357	−1.03886	−0.519429	+	0.854514i	$0.673855\pi$
$80$	−1.54501	−0.172737
$81$	0	0
$82$	−6.53614	−0.721796
$83$	−12.9523	−1.42170	−0.710852	−	0.703342i	$-0.751690\pi$
$83$	−12.9523	−1.42170	−0.710852	+	0.703342i	$0.751690\pi$
$84$	0	0
$85$	2.96659	0.321772
$86$	6.44584	0.695073
$87$	0	0
$88$	2.49412	0.265874
$89$	0.168223	0.0178316	0.00891580	−	0.999960i	$-0.497162\pi$
$89$	0.168223	0.0178316	0.00891580	+	0.999960i	$0.497162\pi$
$90$	0	0
$91$	3.43004	0.359565
$92$	−3.63334	−0.378802
$93$	0	0
$94$	−12.1071	−1.24875
$95$	−7.25476	−0.744322
$96$	0	0
$97$	1.05018	0.106630	0.0533151	−	0.998578i	$-0.483021\pi$
$97$	1.05018	0.106630	0.0533151	+	0.998578i	$0.483021\pi$
$98$	−5.33631	−0.539049
$99$	0	0
$100$	−2.61296	−0.261296
$101$	4.99849	0.497368	0.248684	−	0.968585i	$-0.420002\pi$
$101$	4.99849	0.497368	0.248684	+	0.968585i	$0.420002\pi$
$102$	0	0
$103$	15.2450	1.50213	0.751065	−	0.660228i	$-0.229540\pi$
$103$	15.2450	1.50213	0.751065	+	0.660228i	$0.229540\pi$
$104$	−2.65927	−0.260763
$105$	0	0
$106$	−2.97407	−0.288867
$107$	11.7155	1.13258	0.566288	−	0.824207i	$-0.308379\pi$
$107$	11.7155	1.13258	0.566288	+	0.824207i	$0.308379\pi$
$108$	0	0
$109$	−16.5487	−1.58508	−0.792541	−	0.609818i	$-0.791242\pi$
$109$	−16.5487	−1.58508	−0.792541	+	0.609818i	$0.791242\pi$
$110$	−3.85342	−0.367409
$111$	0	0
$112$	−1.28984	−0.121878
$113$	−2.38720	−0.224569	−0.112284	−	0.993676i	$-0.535817\pi$
$113$	−2.38720	−0.224569	−0.112284	+	0.993676i	$0.535817\pi$
$114$	0	0
$115$	5.61353	0.523465
$116$	−9.08085	−0.843136
$117$	0	0
$118$	12.6372	1.16335
$119$	2.47664	0.227033
$120$	0	0
$121$	−4.77938	−0.434489
$122$	−10.5220	−0.952618
$123$	0	0
$124$	5.96966	0.536091
$125$	11.7621	1.05203
$126$	0	0
$127$	−14.2746	−1.26667	−0.633333	−	0.773880i	$-0.718314\pi$
$127$	−14.2746	−1.26667	−0.633333	+	0.773880i	$0.718314\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.10859	0.360347
$131$	−19.9949	−1.74696	−0.873480	−	0.486859i	$-0.838142\pi$
$131$	−19.9949	−1.74696	−0.873480	+	0.486859i	$0.838142\pi$
$132$	0	0
$133$	−6.05660	−0.525174
$134$	4.51299	0.389863
$135$	0	0
$136$	−1.92011	−0.164648
$137$	−10.4458	−0.892448	−0.446224	−	0.894921i	$-0.647232\pi$
$137$	−10.4458	−0.892448	−0.446224	+	0.894921i	$0.647232\pi$
$138$	0	0
$139$	12.4043	1.05212	0.526058	−	0.850449i	$-0.323669\pi$
$139$	12.4043	1.05212	0.526058	+	0.850449i	$0.323669\pi$
$140$	1.99281	0.168423
$141$	0	0
$142$	−6.01319	−0.504616
$143$	−6.63253	−0.554640
$144$	0	0
$145$	14.0300	1.16513
$146$	−9.46377	−0.783227
$147$	0	0
$148$	5.70604	0.469033
$149$	−2.70297	−0.221436	−0.110718	−	0.993852i	$-0.535315\pi$
$149$	−2.70297	−0.221436	−0.110718	+	0.993852i	$0.535315\pi$
$150$	0	0
$151$	−22.0553	−1.79483	−0.897415	−	0.441187i	$-0.854558\pi$
$151$	−22.0553	−1.79483	−0.897415	+	0.441187i	$0.854558\pi$
$152$	4.69562	0.380865
$153$	0	0
$154$	−3.21701	−0.259234
$155$	−9.22315	−0.740821
$156$	0	0
$157$	7.38327	0.589249	0.294625	−	0.955613i	$-0.404805\pi$
$157$	7.38327	0.589249	0.294625	+	0.955613i	$0.404805\pi$
$158$	−9.23357	−0.734583
$159$	0	0
$160$	−1.54501	−0.122143
$161$	4.68643	0.369343
$162$	0	0
$163$	−4.26962	−0.334423	−0.167211	−	0.985921i	$-0.553476\pi$
$163$	−4.26962	−0.334423	−0.167211	+	0.985921i	$0.553476\pi$
$164$	−6.53614	−0.510387
$165$	0	0
$166$	−12.9523	−1.00530
$167$	8.38192	0.648613	0.324306	−	0.945952i	$-0.394869\pi$
$167$	8.38192	0.648613	0.324306	+	0.945952i	$0.394869\pi$
$168$	0	0
$169$	−5.92827	−0.456021
$170$	2.96659	0.227527
$171$	0	0
$172$	6.44584	0.491491
$173$	6.01306	0.457164	0.228582	−	0.973525i	$-0.426591\pi$
$173$	6.01306	0.457164	0.228582	+	0.973525i	$0.426591\pi$
$174$	0	0
$175$	3.37030	0.254771
$176$	2.49412	0.188001
$177$	0	0
$178$	0.168223	0.0126089
$179$	−23.1043	−1.72690	−0.863449	−	0.504435i	$-0.831701\pi$
$179$	−23.1043	−1.72690	−0.863449	+	0.504435i	$0.831701\pi$
$180$	0	0
$181$	−20.2695	−1.50662	−0.753310	−	0.657666i	$-0.771544\pi$
$181$	−20.2695	−1.50662	−0.753310	+	0.657666i	$0.771544\pi$
$182$	3.43004	0.254251
$183$	0	0
$184$	−3.63334	−0.267854
$185$	−8.81586	−0.648155
$186$	0	0
$187$	−4.78899	−0.350206
$188$	−12.1071	−0.882999
$189$	0	0
$190$	−7.25476	−0.526315
$191$	−5.79518	−0.419325	−0.209662	−	0.977774i	$-0.567236\pi$
$191$	−5.79518	−0.419325	−0.209662	+	0.977774i	$0.567236\pi$
$192$	0	0
$193$	12.3097	0.886070	0.443035	−	0.896504i	$-0.353902\pi$
$193$	12.3097	0.886070	0.443035	+	0.896504i	$0.353902\pi$
$194$	1.05018	0.0753989
$195$	0	0
$196$	−5.33631	−0.381165
$197$	−18.7257	−1.33415	−0.667077	−	0.744989i	$-0.732455\pi$
$197$	−18.7257	−1.33415	−0.667077	+	0.744989i	$0.732455\pi$
$198$	0	0
$199$	3.65510	0.259103	0.129552	−	0.991573i	$-0.458646\pi$
$199$	3.65510	0.259103	0.129552	+	0.991573i	$0.458646\pi$
$200$	−2.61296	−0.184764
$201$	0	0
$202$	4.99849	0.351692
$203$	11.7128	0.822081
$204$	0	0
$205$	10.0984	0.705301
$206$	15.2450	1.06217
$207$	0	0
$208$	−2.65927	−0.184387
$209$	11.7114	0.810096
$210$	0	0
$211$	26.2300	1.80575	0.902873	−	0.429907i	$-0.141454\pi$
$211$	26.2300	1.80575	0.902873	+	0.429907i	$0.141454\pi$
$212$	−2.97407	−0.204260
$213$	0	0
$214$	11.7155	0.800852
$215$	−9.95885	−0.679188
$216$	0	0
$217$	−7.69990	−0.522703
$218$	−16.5487	−1.12082
$219$	0	0
$220$	−3.85342	−0.259798
$221$	5.10611	0.343474
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−1.28984	−0.0861811
$225$	0	0
$226$	−2.38720	−0.158794
$227$	27.2280	1.80718	0.903592	−	0.428395i	$-0.140921\pi$
$227$	27.2280	1.80718	0.903592	+	0.428395i	$0.140921\pi$
$228$	0	0
$229$	−2.43949	−0.161206	−0.0806029	−	0.996746i	$-0.525685\pi$
$229$	−2.43949	−0.161206	−0.0806029	+	0.996746i	$0.525685\pi$
$230$	5.61353	0.370145
$231$	0	0
$232$	−9.08085	−0.596187
$233$	11.3011	0.740358	0.370179	−	0.928961i	$-0.379296\pi$
$233$	11.3011	0.740358	0.370179	+	0.928961i	$0.379296\pi$
$234$	0	0
$235$	18.7055	1.22021
$236$	12.6372	0.822615
$237$	0	0
$238$	2.47664	0.160537
$239$	8.55819	0.553584	0.276792	−	0.960930i	$-0.410729\pi$
$239$	8.55819	0.553584	0.276792	+	0.960930i	$0.410729\pi$
$240$	0	0
$241$	17.4393	1.12336	0.561681	−	0.827354i	$-0.310155\pi$
$241$	17.4393	1.12336	0.561681	+	0.827354i	$0.310155\pi$
$242$	−4.77938	−0.307230
$243$	0	0
$244$	−10.5220	−0.673603
$245$	8.24463	0.526730
$246$	0	0
$247$	−12.4869	−0.794524
$248$	5.96966	0.379074
$249$	0	0
$250$	11.7621	0.743898
$251$	−0.239068	−0.0150898	−0.00754492	−	0.999972i	$-0.502402\pi$
$251$	−0.239068	−0.0150898	−0.00754492	+	0.999972i	$0.502402\pi$
$252$	0	0
$253$	−9.06198	−0.569722
$254$	−14.2746	−0.895668
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.145523	−0.00907746	−0.00453873	−	0.999990i	$-0.501445\pi$
$257$	−0.145523	−0.00907746	−0.00453873	+	0.999990i	$0.501445\pi$
$258$	0	0
$259$	−7.35988	−0.457320
$260$	4.10859	0.254804
$261$	0	0
$262$	−19.9949	−1.23529
$263$	8.17409	0.504036	0.252018	−	0.967723i	$-0.418906\pi$
$263$	8.17409	0.504036	0.252018	+	0.967723i	$0.418906\pi$
$264$	0	0
$265$	4.59496	0.282266
$266$	−6.05660	−0.371354
$267$	0	0
$268$	4.51299	0.275675
$269$	21.6988	1.32300	0.661501	−	0.749944i	$-0.269920\pi$
$269$	21.6988	1.32300	0.661501	+	0.749944i	$0.269920\pi$
$270$	0	0
$271$	−31.1848	−1.89434	−0.947170	−	0.320732i	$-0.896071\pi$
$271$	−31.1848	−1.89434	−0.947170	+	0.320732i	$0.896071\pi$
$272$	−1.92011	−0.116424
$273$	0	0
$274$	−10.4458	−0.631056
$275$	−6.51703	−0.392991
$276$	0	0
$277$	19.0999	1.14760	0.573802	−	0.818994i	$-0.305468\pi$
$277$	19.0999	1.14760	0.573802	+	0.818994i	$0.305468\pi$
$278$	12.4043	0.743959
$279$	0	0
$280$	1.99281	0.119093
$281$	−7.56711	−0.451416	−0.225708	−	0.974195i	$-0.572469\pi$
$281$	−7.56711	−0.451416	−0.225708	+	0.974195i	$0.572469\pi$
$282$	0	0
$283$	−5.87036	−0.348956	−0.174478	−	0.984661i	$-0.555824\pi$
$283$	−5.87036	−0.348956	−0.174478	+	0.984661i	$0.555824\pi$
$284$	−6.01319	−0.356817
$285$	0	0
$286$	−6.63253	−0.392190
$287$	8.43058	0.497641
$288$	0	0
$289$	−13.3132	−0.783127
$290$	14.0300	0.823868
$291$	0	0
$292$	−9.46377	−0.553825
$293$	16.2606	0.949954	0.474977	−	0.879998i	$-0.342456\pi$
$293$	16.2606	0.949954	0.474977	+	0.879998i	$0.342456\pi$
$294$	0	0
$295$	−19.5246	−1.13677
$296$	5.70604	0.331657
$297$	0	0
$298$	−2.70297	−0.156579
$299$	9.66205	0.558771
$300$	0	0
$301$	−8.31410	−0.479217
$302$	−22.0553	−1.26914
$303$	0	0
$304$	4.69562	0.269312
$305$	16.2566	0.930848
$306$	0	0
$307$	−7.70583	−0.439795	−0.219897	−	0.975523i	$-0.570572\pi$
$307$	−7.70583	−0.439795	−0.219897	+	0.975523i	$0.570572\pi$
$308$	−3.21701	−0.183306
$309$	0	0
$310$	−9.22315	−0.523840
$311$	−8.35322	−0.473668	−0.236834	−	0.971550i	$-0.576110\pi$
$311$	−8.35322	−0.473668	−0.236834	+	0.971550i	$0.576110\pi$
$312$	0	0
$313$	−12.8779	−0.727900	−0.363950	−	0.931418i	$-0.618572\pi$
$313$	−12.8779	−0.727900	−0.363950	+	0.931418i	$0.618572\pi$
$314$	7.38327	0.416662
$315$	0	0
$316$	−9.23357	−0.519429
$317$	−17.0655	−0.958493	−0.479246	−	0.877680i	$-0.659090\pi$
$317$	−17.0655	−0.958493	−0.479246	+	0.877680i	$0.659090\pi$
$318$	0	0
$319$	−22.6487	−1.26808
$320$	−1.54501	−0.0863684
$321$	0	0
$322$	4.68643	0.261165
$323$	−9.01613	−0.501671
$324$	0	0
$325$	6.94857	0.385437
$326$	−4.26962	−0.236472
$327$	0	0
$328$	−6.53614	−0.360898
$329$	15.6162	0.860949
$330$	0	0
$331$	−19.9208	−1.09495	−0.547473	−	0.836823i	$-0.684410\pi$
$331$	−19.9208	−1.09495	−0.547473	+	0.836823i	$0.684410\pi$
$332$	−12.9523	−0.710852
$333$	0	0
$334$	8.38192	0.458638
$335$	−6.97259	−0.380953
$336$	0	0
$337$	4.53383	0.246973	0.123487	−	0.992346i	$-0.460592\pi$
$337$	4.53383	0.246973	0.123487	+	0.992346i	$0.460592\pi$
$338$	−5.92827	−0.322456
$339$	0	0
$340$	2.96659	0.160886
$341$	14.8890	0.806286
$342$	0	0
$343$	15.9119	0.859160
$344$	6.44584	0.347536
$345$	0	0
$346$	6.01306	0.323264
$347$	34.7965	1.86797	0.933986	−	0.357309i	$-0.116306\pi$
$347$	34.7965	1.86797	0.933986	+	0.357309i	$0.116306\pi$
$348$	0	0
$349$	−21.9889	−1.17704	−0.588518	−	0.808484i	$-0.700288\pi$
$349$	−21.9889	−1.17704	−0.588518	+	0.808484i	$0.700288\pi$
$350$	3.37030	0.180150
$351$	0	0
$352$	2.49412	0.132937
$353$	5.13896	0.273519	0.136760	−	0.990604i	$-0.456331\pi$
$353$	5.13896	0.273519	0.136760	+	0.990604i	$0.456331\pi$
$354$	0	0
$355$	9.29041	0.493084
$356$	0.168223	0.00891580
$357$	0	0
$358$	−23.1043	−1.22110
$359$	−11.4094	−0.602163	−0.301081	−	0.953598i	$-0.597348\pi$
$359$	−11.4094	−0.602163	−0.301081	+	0.953598i	$0.597348\pi$
$360$	0	0
$361$	3.04884	0.160465
$362$	−20.2695	−1.06534
$363$	0	0
$364$	3.43004	0.179783
$365$	14.6216	0.765328
$366$	0	0
$367$	23.4709	1.22517	0.612585	−	0.790405i	$-0.290130\pi$
$367$	23.4709	1.22517	0.612585	+	0.790405i	$0.290130\pi$
$368$	−3.63334	−0.189401
$369$	0	0
$370$	−8.81586	−0.458315
$371$	3.83608	0.199159
$372$	0	0
$373$	14.7702	0.764773	0.382387	−	0.924002i	$-0.375102\pi$
$373$	14.7702	0.764773	0.382387	+	0.924002i	$0.375102\pi$
$374$	−4.78899	−0.247633
$375$	0	0
$376$	−12.1071	−0.624375
$377$	24.1485	1.24371
$378$	0	0
$379$	9.14574	0.469785	0.234893	−	0.972021i	$-0.424526\pi$
$379$	9.14574	0.469785	0.234893	+	0.972021i	$0.424526\pi$
$380$	−7.25476	−0.372161
$381$	0	0
$382$	−5.79518	−0.296507
$383$	15.2947	0.781520	0.390760	−	0.920493i	$-0.372212\pi$
$383$	15.2947	0.781520	0.390760	+	0.920493i	$0.372212\pi$
$384$	0	0
$385$	4.97030	0.253310
$386$	12.3097	0.626546
$387$	0	0
$388$	1.05018	0.0533151
$389$	37.7337	1.91317	0.956586	−	0.291449i	$-0.0941375\pi$
$389$	37.7337	1.91317	0.956586	+	0.291449i	$0.0941375\pi$
$390$	0	0
$391$	6.97644	0.352814
$392$	−5.33631	−0.269524
$393$	0	0
$394$	−18.7257	−0.943390
$395$	14.2659	0.717796
$396$	0	0
$397$	−26.6903	−1.33955	−0.669774	−	0.742565i	$-0.733609\pi$
$397$	−26.6903	−1.33955	−0.669774	+	0.742565i	$0.733609\pi$
$398$	3.65510	0.183214
$399$	0	0
$400$	−2.61296	−0.130648
$401$	31.3412	1.56510	0.782552	−	0.622585i	$-0.213918\pi$
$401$	31.3412	1.56510	0.782552	+	0.622585i	$0.213918\pi$
$402$	0	0
$403$	−15.8749	−0.790787
$404$	4.99849	0.248684
$405$	0	0
$406$	11.7128	0.581299
$407$	14.2315	0.705430
$408$	0	0
$409$	38.0880	1.88333	0.941666	−	0.336550i	$-0.109260\pi$
$409$	38.0880	1.88333	0.941666	+	0.336550i	$0.109260\pi$
$410$	10.0984	0.498723
$411$	0	0
$412$	15.2450	0.751065
$413$	−16.3000	−0.802072
$414$	0	0
$415$	20.0114	0.982323
$416$	−2.65927	−0.130382
$417$	0	0
$418$	11.7114	0.572824
$419$	−22.7446	−1.11115	−0.555574	−	0.831467i	$-0.687501\pi$
$419$	−22.7446	−1.11115	−0.555574	+	0.831467i	$0.687501\pi$
$420$	0	0
$421$	−21.6762	−1.05643	−0.528217	−	0.849109i	$-0.677139\pi$
$421$	−21.6762	−1.05643	−0.528217	+	0.849109i	$0.677139\pi$
$422$	26.2300	1.27686
$423$	0	0
$424$	−2.97407	−0.144434
$425$	5.01718	0.243369
$426$	0	0
$427$	13.5717	0.656781
$428$	11.7155	0.566288
$429$	0	0
$430$	−9.95885	−0.480259
$431$	24.7372	1.19155	0.595775	−	0.803151i	$-0.296845\pi$
$431$	24.7372	1.19155	0.595775	+	0.803151i	$0.296845\pi$
$432$	0	0
$433$	29.6583	1.42529	0.712645	−	0.701525i	$-0.247497\pi$
$433$	29.6583	1.42529	0.712645	+	0.701525i	$0.247497\pi$
$434$	−7.69990	−0.369607
$435$	0	0
$436$	−16.5487	−0.792541
$437$	−17.0608	−0.816128
$438$	0	0
$439$	27.9533	1.33414	0.667069	−	0.744996i	$-0.267548\pi$
$439$	27.9533	1.33414	0.667069	+	0.744996i	$0.267548\pi$
$440$	−3.85342	−0.183705
$441$	0	0
$442$	5.10611	0.242873
$443$	−21.5451	−1.02364	−0.511819	−	0.859093i	$-0.671028\pi$
$443$	−21.5451	−1.02364	−0.511819	+	0.859093i	$0.671028\pi$
$444$	0	0
$445$	−0.259905	−0.0123207
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−1.28984	−0.0609392
$449$	−38.3133	−1.80812	−0.904059	−	0.427407i	$-0.859427\pi$
$449$	−38.3133	−1.80812	−0.904059	+	0.427407i	$0.859427\pi$
$450$	0	0
$451$	−16.3019	−0.767627
$452$	−2.38720	−0.112284
$453$	0	0
$454$	27.2280	1.27787
$455$	−5.29942	−0.248441
$456$	0	0
$457$	−8.77811	−0.410623	−0.205311	−	0.978697i	$-0.565821\pi$
$457$	−8.77811	−0.410623	−0.205311	+	0.978697i	$0.565821\pi$
$458$	−2.43949	−0.113990
$459$	0	0
$460$	5.61353	0.261732
$461$	14.4028	0.670807	0.335404	−	0.942075i	$-0.391127\pi$
$461$	14.4028	0.670807	0.335404	+	0.942075i	$0.391127\pi$
$462$	0	0
$463$	13.6803	0.635779	0.317889	−	0.948128i	$-0.397026\pi$
$463$	13.6803	0.635779	0.317889	+	0.948128i	$0.397026\pi$
$464$	−9.08085	−0.421568
$465$	0	0
$466$	11.3011	0.523512
$467$	−8.37193	−0.387407	−0.193703	−	0.981060i	$-0.562050\pi$
$467$	−8.37193	−0.387407	−0.193703	+	0.981060i	$0.562050\pi$
$468$	0	0
$469$	−5.82103	−0.268790
$470$	18.7055	0.862820
$471$	0	0
$472$	12.6372	0.581676
$473$	16.0767	0.739206
$474$	0	0
$475$	−12.2695	−0.562961
$476$	2.47664	0.113517
$477$	0	0
$478$	8.55819	0.391443
$479$	21.7260	0.992688	0.496344	−	0.868126i	$-0.334675\pi$
$479$	21.7260	0.992688	0.496344	+	0.868126i	$0.334675\pi$
$480$	0	0
$481$	−15.1739	−0.691870
$482$	17.4393	0.794337
$483$	0	0
$484$	−4.77938	−0.217245
$485$	−1.62254	−0.0736758
$486$	0	0
$487$	2.38053	0.107872	0.0539360	−	0.998544i	$-0.482823\pi$
$487$	2.38053	0.107872	0.0539360	+	0.998544i	$0.482823\pi$
$488$	−10.5220	−0.476309
$489$	0	0
$490$	8.24463	0.372454
$491$	−20.8266	−0.939890	−0.469945	−	0.882696i	$-0.655726\pi$
$491$	−20.8266	−0.939890	−0.469945	+	0.882696i	$0.655726\pi$
$492$	0	0
$493$	17.4363	0.785291
$494$	−12.4869	−0.561813
$495$	0	0
$496$	5.96966	0.268045
$497$	7.75606	0.347907
$498$	0	0
$499$	18.7813	0.840765	0.420382	−	0.907347i	$-0.361896\pi$
$499$	18.7813	0.840765	0.420382	+	0.907347i	$0.361896\pi$
$500$	11.7621	0.526015
$501$	0	0
$502$	−0.239068	−0.0106701
$503$	11.5451	0.514771	0.257385	−	0.966309i	$-0.417139\pi$
$503$	11.5451	0.514771	0.257385	+	0.966309i	$0.417139\pi$
$504$	0	0
$505$	−7.72269	−0.343655
$506$	−9.06198	−0.402854
$507$	0	0
$508$	−14.2746	−0.633333
$509$	−21.8205	−0.967177	−0.483589	−	0.875295i	$-0.660667\pi$
$509$	−21.8205	−0.967177	−0.483589	+	0.875295i	$0.660667\pi$
$510$	0	0
$511$	12.2068	0.539995
$512$	1.00000	0.0441942
$513$	0	0
$514$	−0.145523	−0.00641873
$515$	−23.5535	−1.03789
$516$	0	0
$517$	−30.1965	−1.32804
$518$	−7.35988	−0.323374
$519$	0	0
$520$	4.10859	0.180174
$521$	5.31636	0.232914	0.116457	−	0.993196i	$-0.462846\pi$
$521$	5.31636	0.232914	0.116457	+	0.993196i	$0.462846\pi$
$522$	0	0
$523$	31.5525	1.37969	0.689847	−	0.723955i	$-0.257678\pi$
$523$	31.5525	1.37969	0.689847	+	0.723955i	$0.257678\pi$
$524$	−19.9949	−0.873480
$525$	0	0
$526$	8.17409	0.356407
$527$	−11.4624	−0.499311
$528$	0	0
$529$	−9.79882	−0.426036
$530$	4.59496	0.199592
$531$	0	0
$532$	−6.05660	−0.262587
$533$	17.3814	0.752871
$534$	0	0
$535$	−18.1004	−0.782550
$536$	4.51299	0.194931
$537$	0	0
$538$	21.6988	0.935504
$539$	−13.3094	−0.573276
$540$	0	0
$541$	−10.2150	−0.439179	−0.219590	−	0.975592i	$-0.570472\pi$
$541$	−10.2150	−0.439179	−0.219590	+	0.975592i	$0.570472\pi$
$542$	−31.1848	−1.33950
$543$	0	0
$544$	−1.92011	−0.0823242
$545$	25.5679	1.09521
$546$	0	0
$547$	6.88780	0.294501	0.147251	−	0.989099i	$-0.452958\pi$
$547$	6.88780	0.294501	0.147251	+	0.989099i	$0.452958\pi$
$548$	−10.4458	−0.446224
$549$	0	0
$550$	−6.51703	−0.277887
$551$	−42.6402	−1.81653
$552$	0	0
$553$	11.9098	0.506457
$554$	19.0999	0.811478
$555$	0	0
$556$	12.4043	0.526058
$557$	25.1326	1.06490	0.532451	−	0.846461i	$-0.321271\pi$
$557$	25.1326	1.06490	0.532451	+	0.846461i	$0.321271\pi$
$558$	0	0
$559$	−17.1412	−0.724997
$560$	1.99281	0.0842116
$561$	0	0
$562$	−7.56711	−0.319199
$563$	−36.3836	−1.53339	−0.766693	−	0.642013i	$-0.778099\pi$
$563$	−36.3836	−1.53339	−0.766693	+	0.642013i	$0.778099\pi$
$564$	0	0
$565$	3.68824	0.155165
$566$	−5.87036	−0.246749
$567$	0	0
$568$	−6.01319	−0.252308
$569$	40.7614	1.70881	0.854404	−	0.519610i	$-0.173923\pi$
$569$	40.7614	1.70881	0.854404	+	0.519610i	$0.173923\pi$
$570$	0	0
$571$	−12.3995	−0.518903	−0.259451	−	0.965756i	$-0.583542\pi$
$571$	−12.3995	−0.518903	−0.259451	+	0.965756i	$0.583542\pi$
$572$	−6.63253	−0.277320
$573$	0	0
$574$	8.43058	0.351886
$575$	9.49378	0.395918
$576$	0	0
$577$	−9.50735	−0.395796	−0.197898	−	0.980223i	$-0.563412\pi$
$577$	−9.50735	−0.395796	−0.197898	+	0.980223i	$0.563412\pi$
$578$	−13.3132	−0.553754
$579$	0	0
$580$	14.0300	0.582563
$581$	16.7064	0.693100
$582$	0	0
$583$	−7.41768	−0.307209
$584$	−9.46377	−0.391614
$585$	0	0
$586$	16.2606	0.671719
$587$	28.0628	1.15827	0.579137	−	0.815230i	$-0.303390\pi$
$587$	28.0628	1.15827	0.579137	+	0.815230i	$0.303390\pi$
$588$	0	0
$589$	28.0312	1.15501
$590$	−19.5246	−0.803815
$591$	0	0
$592$	5.70604	0.234517
$593$	38.8259	1.59439	0.797194	−	0.603723i	$-0.206317\pi$
$593$	38.8259	1.59439	0.797194	+	0.603723i	$0.206317\pi$
$594$	0	0
$595$	−3.82642	−0.156868
$596$	−2.70297	−0.110718
$597$	0	0
$598$	9.66205	0.395110
$599$	−40.9391	−1.67273	−0.836363	−	0.548176i	$-0.815322\pi$
$599$	−40.9391	−1.67273	−0.836363	+	0.548176i	$0.815322\pi$
$600$	0	0
$601$	−20.9510	−0.854608	−0.427304	−	0.904108i	$-0.640537\pi$
$601$	−20.9510	−0.854608	−0.427304	+	0.904108i	$0.640537\pi$
$602$	−8.31410	−0.338857
$603$	0	0
$604$	−22.0553	−0.897415
$605$	7.38417	0.300209
$606$	0	0
$607$	18.1587	0.737039	0.368519	−	0.929620i	$-0.379865\pi$
$607$	18.1587	0.737039	0.368519	+	0.929620i	$0.379865\pi$
$608$	4.69562	0.190432
$609$	0	0
$610$	16.2566	0.658209
$611$	32.1960	1.30251
$612$	0	0
$613$	33.1238	1.33786	0.668928	−	0.743327i	$-0.266753\pi$
$613$	33.1238	1.33786	0.668928	+	0.743327i	$0.266753\pi$
$614$	−7.70583	−0.310982
$615$	0	0
$616$	−3.21701	−0.129617
$617$	−20.7781	−0.836493	−0.418247	−	0.908333i	$-0.637355\pi$
$617$	−20.7781	−0.836493	−0.418247	+	0.908333i	$0.637355\pi$
$618$	0	0
$619$	−44.0401	−1.77012	−0.885060	−	0.465476i	$-0.845883\pi$
$619$	−44.0401	−1.77012	−0.885060	+	0.465476i	$0.845883\pi$
$620$	−9.22315	−0.370411
$621$	0	0
$622$	−8.35322	−0.334934
$623$	−0.216981	−0.00869315
$624$	0	0
$625$	−5.10765	−0.204306
$626$	−12.8779	−0.514703
$627$	0	0
$628$	7.38327	0.294625
$629$	−10.9562	−0.436854
$630$	0	0
$631$	−42.0837	−1.67533	−0.837664	−	0.546186i	$-0.816079\pi$
$631$	−42.0837	−1.67533	−0.837664	+	0.546186i	$0.816079\pi$
$632$	−9.23357	−0.367292
$633$	0	0
$634$	−17.0655	−0.677757
$635$	22.0543	0.875199
$636$	0	0
$637$	14.1907	0.562256
$638$	−22.6487	−0.896671
$639$	0	0
$640$	−1.54501	−0.0610717
$641$	−25.3047	−0.999475	−0.499738	−	0.866177i	$-0.666570\pi$
$641$	−25.3047	−0.999475	−0.499738	+	0.866177i	$0.666570\pi$
$642$	0	0
$643$	33.5383	1.32262	0.661310	−	0.750113i	$-0.270001\pi$
$643$	33.5383	1.32262	0.661310	+	0.750113i	$0.270001\pi$
$644$	4.68643	0.184671
$645$	0	0
$646$	−9.01613	−0.354735
$647$	−26.8197	−1.05439	−0.527195	−	0.849744i	$-0.676756\pi$
$647$	−26.8197	−1.05439	−0.527195	+	0.849744i	$0.676756\pi$
$648$	0	0
$649$	31.5188	1.23722
$650$	6.94857	0.272545
$651$	0	0
$652$	−4.26962	−0.167211
$653$	−33.5965	−1.31473	−0.657366	−	0.753571i	$-0.728329\pi$
$653$	−33.5965	−1.31473	−0.657366	+	0.753571i	$0.728329\pi$
$654$	0	0
$655$	30.8922	1.20706
$656$	−6.53614	−0.255193
$657$	0	0
$658$	15.6162	0.608783
$659$	−7.35401	−0.286471	−0.143236	−	0.989689i	$-0.545751\pi$
$659$	−7.35401	−0.286471	−0.143236	+	0.989689i	$0.545751\pi$
$660$	0	0
$661$	−25.7191	−1.00036	−0.500179	−	0.865922i	$-0.666732\pi$
$661$	−25.7191	−1.00036	−0.500179	+	0.865922i	$0.666732\pi$
$662$	−19.9208	−0.774243
$663$	0	0
$664$	−12.9523	−0.502648
$665$	9.35747	0.362867
$666$	0	0
$667$	32.9939	1.27753
$668$	8.38192	0.324306
$669$	0	0
$670$	−6.97259	−0.269375
$671$	−26.2431	−1.01310
$672$	0	0
$673$	18.1684	0.700340	0.350170	−	0.936686i	$-0.386124\pi$
$673$	18.1684	0.700340	0.350170	+	0.936686i	$0.386124\pi$
$674$	4.53383	0.174636
$675$	0	0
$676$	−5.92827	−0.228011
$677$	13.2991	0.511125	0.255563	−	0.966793i	$-0.417739\pi$
$677$	13.2991	0.511125	0.255563	+	0.966793i	$0.417739\pi$
$678$	0	0
$679$	−1.35457	−0.0519836
$680$	2.96659	0.113763
$681$	0	0
$682$	14.8890	0.570130
$683$	25.8252	0.988175	0.494087	−	0.869412i	$-0.335502\pi$
$683$	25.8252	0.988175	0.494087	+	0.869412i	$0.335502\pi$
$684$	0	0
$685$	16.1389	0.616635
$686$	15.9119	0.607518
$687$	0	0
$688$	6.44584	0.245745
$689$	7.90886	0.301304
$690$	0	0
$691$	−30.4652	−1.15895	−0.579476	−	0.814989i	$-0.696743\pi$
$691$	−30.4652	−1.15895	−0.579476	+	0.814989i	$0.696743\pi$
$692$	6.01306	0.228582
$693$	0	0
$694$	34.7965	1.32086
$695$	−19.1647	−0.726957
$696$	0	0
$697$	12.5501	0.475371
$698$	−21.9889	−0.832290
$699$	0	0
$700$	3.37030	0.127385
$701$	5.22755	0.197442	0.0987209	−	0.995115i	$-0.468525\pi$
$701$	5.22755	0.197442	0.0987209	+	0.995115i	$0.468525\pi$
$702$	0	0
$703$	26.7934	1.01053
$704$	2.49412	0.0940006
$705$	0	0
$706$	5.13896	0.193407
$707$	−6.44725	−0.242474
$708$	0	0
$709$	30.7149	1.15352	0.576760	−	0.816913i	$-0.304317\pi$
$709$	30.7149	1.15352	0.576760	+	0.816913i	$0.304317\pi$
$710$	9.29041	0.348663
$711$	0	0
$712$	0.168223	0.00630443
$713$	−21.6898	−0.812290
$714$	0	0
$715$	10.2473	0.383227
$716$	−23.1043	−0.863449
$717$	0	0
$718$	−11.4094	−0.425793
$719$	19.0574	0.710722	0.355361	−	0.934729i	$-0.384358\pi$
$719$	19.0574	0.710722	0.355361	+	0.934729i	$0.384358\pi$
$720$	0	0
$721$	−19.6636	−0.732309
$722$	3.04884	0.113466
$723$	0	0
$724$	−20.2695	−0.753310
$725$	23.7279	0.881232
$726$	0	0
$727$	39.1391	1.45159	0.725794	−	0.687913i	$-0.241473\pi$
$727$	39.1391	1.45159	0.725794	+	0.687913i	$0.241473\pi$
$728$	3.43004	0.127126
$729$	0	0
$730$	14.6216	0.541169
$731$	−12.3768	−0.457771
$732$	0	0
$733$	−18.2484	−0.674020	−0.337010	−	0.941501i	$-0.609416\pi$
$733$	−18.2484	−0.674020	−0.337010	+	0.941501i	$0.609416\pi$
$734$	23.4709	0.866326
$735$	0	0
$736$	−3.63334	−0.133927
$737$	11.2559	0.414617
$738$	0	0
$739$	−20.2613	−0.745323	−0.372661	−	0.927967i	$-0.621555\pi$
$739$	−20.2613	−0.745323	−0.372661	+	0.927967i	$0.621555\pi$
$740$	−8.81586	−0.324077
$741$	0	0
$742$	3.83608	0.140827
$743$	−6.60499	−0.242314	−0.121157	−	0.992633i	$-0.538660\pi$
$743$	−6.60499	−0.242314	−0.121157	+	0.992633i	$0.538660\pi$
$744$	0	0
$745$	4.17610	0.153001
$746$	14.7702	0.540776
$747$	0	0
$748$	−4.78899	−0.175103
$749$	−15.1111	−0.552146
$750$	0	0
$751$	29.0359	1.05953	0.529767	−	0.848143i	$-0.322279\pi$
$751$	29.0359	1.05953	0.529767	+	0.848143i	$0.322279\pi$
$752$	−12.1071	−0.441500
$753$	0	0
$754$	24.1485	0.879435
$755$	34.0755	1.24013
$756$	0	0
$757$	−27.4584	−0.997993	−0.498997	−	0.866604i	$-0.666298\pi$
$757$	−27.4584	−0.997993	−0.498997	+	0.866604i	$0.666298\pi$
$758$	9.14574	0.332188
$759$	0	0
$760$	−7.25476	−0.263158
$761$	−6.91985	−0.250844	−0.125422	−	0.992103i	$-0.540029\pi$
$761$	−6.91985	−0.250844	−0.125422	+	0.992103i	$0.540029\pi$
$762$	0	0
$763$	21.3452	0.772750
$764$	−5.79518	−0.209662
$765$	0	0
$766$	15.2947	0.552618
$767$	−33.6059	−1.21344
$768$	0	0
$769$	−11.4749	−0.413796	−0.206898	−	0.978362i	$-0.566337\pi$
$769$	−11.4749	−0.413796	−0.206898	+	0.978362i	$0.566337\pi$
$770$	4.97030	0.179117
$771$	0	0
$772$	12.3097	0.443035
$773$	−21.8130	−0.784559	−0.392279	−	0.919846i	$-0.628313\pi$
$773$	−21.8130	−0.784559	−0.392279	+	0.919846i	$0.628313\pi$
$774$	0	0
$775$	−15.5985	−0.560313
$776$	1.05018	0.0376994
$777$	0	0
$778$	37.7337	1.35282
$779$	−30.6912	−1.09963
$780$	0	0
$781$	−14.9976	−0.536657
$782$	6.97644	0.249477
$783$	0	0
$784$	−5.33631	−0.190583
$785$	−11.4072	−0.407140
$786$	0	0
$787$	14.3504	0.511538	0.255769	−	0.966738i	$-0.417671\pi$
$787$	14.3504	0.511538	0.255769	+	0.966738i	$0.417671\pi$
$788$	−18.7257	−0.667077
$789$	0	0
$790$	14.2659	0.507558
$791$	3.07911	0.109480
$792$	0	0
$793$	27.9809	0.993630
$794$	−26.6903	−0.947204
$795$	0	0
$796$	3.65510	0.129552
$797$	2.16335	0.0766298	0.0383149	−	0.999266i	$-0.487801\pi$
$797$	2.16335	0.0766298	0.0383149	+	0.999266i	$0.487801\pi$
$798$	0	0
$799$	23.2470	0.822419
$800$	−2.61296	−0.0923821
$801$	0	0
$802$	31.3412	1.10670
$803$	−23.6038	−0.832958
$804$	0	0
$805$	−7.24056	−0.255196
$806$	−15.8749	−0.559171
$807$	0	0
$808$	4.99849	0.175846
$809$	−7.90979	−0.278093	−0.139047	−	0.990286i	$-0.544404\pi$
$809$	−7.90979	−0.278093	−0.139047	+	0.990286i	$0.544404\pi$
$810$	0	0
$811$	−23.1783	−0.813900	−0.406950	−	0.913450i	$-0.633408\pi$
$811$	−23.1783	−0.813900	−0.406950	+	0.913450i	$0.633408\pi$
$812$	11.7128	0.411040
$813$	0	0
$814$	14.2315	0.498815
$815$	6.59659	0.231068
$816$	0	0
$817$	30.2672	1.05892
$818$	38.0880	1.33172
$819$	0	0
$820$	10.0984	0.352650
$821$	13.0185	0.454347	0.227174	−	0.973854i	$-0.427051\pi$
$821$	13.0185	0.454347	0.227174	+	0.973854i	$0.427051\pi$
$822$	0	0
$823$	−23.0296	−0.802762	−0.401381	−	0.915911i	$-0.631470\pi$
$823$	−23.0296	−0.802762	−0.401381	+	0.915911i	$0.631470\pi$
$824$	15.2450	0.531083
$825$	0	0
$826$	−16.3000	−0.567150
$827$	48.7420	1.69492	0.847462	−	0.530856i	$-0.178129\pi$
$827$	48.7420	1.69492	0.847462	+	0.530856i	$0.178129\pi$
$828$	0	0
$829$	−3.02728	−0.105142	−0.0525709	−	0.998617i	$-0.516742\pi$
$829$	−3.02728	−0.105142	−0.0525709	+	0.998617i	$0.516742\pi$
$830$	20.0114	0.694607
$831$	0	0
$832$	−2.65927	−0.0921937
$833$	10.2463	0.355014
$834$	0	0
$835$	−12.9501	−0.448157
$836$	11.7114	0.405048
$837$	0	0
$838$	−22.7446	−0.785700
$839$	21.2196	0.732583	0.366292	−	0.930500i	$-0.380627\pi$
$839$	21.2196	0.732583	0.366292	+	0.930500i	$0.380627\pi$
$840$	0	0
$841$	53.4619	1.84351
$842$	−21.6762	−0.747012
$843$	0	0
$844$	26.2300	0.902873
$845$	9.15921	0.315087
$846$	0	0
$847$	6.16464	0.211819
$848$	−2.97407	−0.102130
$849$	0	0
$850$	5.01718	0.172088
$851$	−20.7320	−0.710684
$852$	0	0
$853$	−23.0700	−0.789901	−0.394950	−	0.918703i	$-0.629238\pi$
$853$	−23.0700	−0.789901	−0.394950	+	0.918703i	$0.629238\pi$
$854$	13.5717	0.464414
$855$	0	0
$856$	11.7155	0.400426
$857$	−2.90304	−0.0991661	−0.0495831	−	0.998770i	$-0.515789\pi$
$857$	−2.90304	−0.0991661	−0.0495831	+	0.998770i	$0.515789\pi$
$858$	0	0
$859$	30.4007	1.03726	0.518629	−	0.854999i	$-0.326443\pi$
$859$	30.4007	1.03726	0.518629	+	0.854999i	$0.326443\pi$
$860$	−9.95885	−0.339594
$861$	0	0
$862$	24.7372	0.842553
$863$	−38.0933	−1.29671	−0.648356	−	0.761337i	$-0.724543\pi$
$863$	−38.0933	−1.29671	−0.648356	+	0.761337i	$0.724543\pi$
$864$	0	0
$865$	−9.29021	−0.315877
$866$	29.6583	1.00783
$867$	0	0
$868$	−7.69990	−0.261352
$869$	−23.0296	−0.781226
$870$	0	0
$871$	−12.0013	−0.406647
$872$	−16.5487	−0.560411
$873$	0	0
$874$	−17.0608	−0.577090
$875$	−15.1712	−0.512879
$876$	0	0
$877$	−45.3907	−1.53274	−0.766368	−	0.642402i	$-0.777938\pi$
$877$	−45.3907	−1.53274	−0.766368	+	0.642402i	$0.777938\pi$
$878$	27.9533	0.943379
$879$	0	0
$880$	−3.85342	−0.129899
$881$	−17.8676	−0.601976	−0.300988	−	0.953628i	$-0.597316\pi$
$881$	−17.8676	−0.601976	−0.300988	+	0.953628i	$0.597316\pi$
$882$	0	0
$883$	25.3660	0.853634	0.426817	−	0.904338i	$-0.359635\pi$
$883$	25.3660	0.853634	0.426817	+	0.904338i	$0.359635\pi$
$884$	5.10611	0.171737
$885$	0	0
$886$	−21.5451	−0.723822
$887$	31.6442	1.06251	0.531255	−	0.847212i	$-0.321721\pi$
$887$	31.6442	1.06251	0.531255	+	0.847212i	$0.321721\pi$
$888$	0	0
$889$	18.4119	0.617517
$890$	−0.259905	−0.00871205
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−56.8502	−1.90242
$894$	0	0
$895$	35.6963	1.19320
$896$	−1.28984	−0.0430905
$897$	0	0
$898$	−38.3133	−1.27853
$899$	−54.2096	−1.80799
$900$	0	0
$901$	5.71056	0.190246
$902$	−16.3019	−0.542794
$903$	0	0
$904$	−2.38720	−0.0793971
$905$	31.3165	1.04099
$906$	0	0
$907$	27.9750	0.928894	0.464447	−	0.885601i	$-0.346253\pi$
$907$	27.9750	0.928894	0.464447	+	0.885601i	$0.346253\pi$
$908$	27.2280	0.903592
$909$	0	0
$910$	−5.29942	−0.175674
$911$	−8.88228	−0.294283	−0.147142	−	0.989115i	$-0.547007\pi$
$911$	−8.88228	−0.294283	−0.147142	+	0.989115i	$0.547007\pi$
$912$	0	0
$913$	−32.3046	−1.06913
$914$	−8.77811	−0.290354
$915$	0	0
$916$	−2.43949	−0.0806029
$917$	25.7902	0.851667
$918$	0	0
$919$	−38.5695	−1.27229	−0.636146	−	0.771569i	$-0.719472\pi$
$919$	−38.5695	−1.27229	−0.636146	+	0.771569i	$0.719472\pi$
$920$	5.61353	0.185073
$921$	0	0
$922$	14.4028	0.474332
$923$	15.9907	0.526341
$924$	0	0
$925$	−14.9096	−0.490226
$926$	13.6803	0.449563
$927$	0	0
$928$	−9.08085	−0.298094
$929$	16.4905	0.541037	0.270518	−	0.962715i	$-0.412805\pi$
$929$	16.4905	0.541037	0.270518	+	0.962715i	$0.412805\pi$
$930$	0	0
$931$	−25.0573	−0.821220
$932$	11.3011	0.370179
$933$	0	0
$934$	−8.37193	−0.273938
$935$	7.39902	0.241974
$936$	0	0
$937$	−19.5436	−0.638462	−0.319231	−	0.947677i	$-0.603425\pi$
$937$	−19.5436	−0.638462	−0.319231	+	0.947677i	$0.603425\pi$
$938$	−5.82103	−0.190063
$939$	0	0
$940$	18.7055	0.610106
$941$	58.8382	1.91807	0.959035	−	0.283287i	$-0.0914250\pi$
$941$	58.8382	1.91807	0.959035	+	0.283287i	$0.0914250\pi$
$942$	0	0
$943$	23.7480	0.773343
$944$	12.6372	0.411307
$945$	0	0
$946$	16.0767	0.522698
$947$	19.1164	0.621199	0.310600	−	0.950541i	$-0.399470\pi$
$947$	19.1164	0.621199	0.310600	+	0.950541i	$0.399470\pi$
$948$	0	0
$949$	25.1667	0.816947
$950$	−12.2695	−0.398074
$951$	0	0
$952$	2.47664	0.0802684
$953$	8.48256	0.274777	0.137389	−	0.990517i	$-0.456129\pi$
$953$	8.48256	0.274777	0.137389	+	0.990517i	$0.456129\pi$
$954$	0	0
$955$	8.95359	0.289731
$956$	8.55819	0.276792
$957$	0	0
$958$	21.7260	0.701936
$959$	13.4735	0.435081
$960$	0	0
$961$	4.63679	0.149574
$962$	−15.1739	−0.489226
$963$	0	0
$964$	17.4393	0.561681
$965$	−19.0185	−0.612228
$966$	0	0
$967$	−6.04992	−0.194552	−0.0972762	−	0.995257i	$-0.531013\pi$
$967$	−6.04992	−0.194552	−0.0972762	+	0.995257i	$0.531013\pi$
$968$	−4.77938	−0.153615
$969$	0	0
$970$	−1.62254	−0.0520967
$971$	−8.71377	−0.279638	−0.139819	−	0.990177i	$-0.544652\pi$
$971$	−8.71377	−0.279638	−0.139819	+	0.990177i	$0.544652\pi$
$972$	0	0
$973$	−15.9995	−0.512921
$974$	2.38053	0.0762770
$975$	0	0
$976$	−10.5220	−0.336801
$977$	59.1766	1.89323	0.946614	−	0.322370i	$-0.104479\pi$
$977$	59.1766	1.89323	0.946614	+	0.322370i	$0.104479\pi$
$978$	0	0
$979$	0.419568	0.0134095
$980$	8.24463	0.263365
$981$	0	0
$982$	−20.8266	−0.664603
$983$	−28.0539	−0.894780	−0.447390	−	0.894339i	$-0.647646\pi$
$983$	−28.0539	−0.894780	−0.447390	+	0.894339i	$0.647646\pi$
$984$	0	0
$985$	28.9314	0.921830
$986$	17.4363	0.555284
$987$	0	0
$988$	−12.4869	−0.397262
$989$	−23.4199	−0.744711
$990$	0	0
$991$	5.76653	0.183180	0.0915899	−	0.995797i	$-0.470805\pi$
$991$	5.76653	0.183180	0.0915899	+	0.995797i	$0.470805\pi$
$992$	5.96966	0.189537
$993$	0	0
$994$	7.75606	0.246007
$995$	−5.64715	−0.179027
$996$	0	0
$997$	−27.9499	−0.885183	−0.442592	−	0.896723i	$-0.645941\pi$
$997$	−27.9499	−0.885183	−0.442592	+	0.896723i	$0.645941\pi$
$998$	18.7813	0.594511
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.r.1.2		5
3.2	odd	2		1338.2.a.h.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.h.1.4	✓	5	3.2	odd	2
4014.2.a.r.1.2		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.54501 q^{5} -1.28984 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.54501 q^{5} -1.28984 q^{7} +1.00000 q^{8} -1.54501 q^{10} +2.49412 q^{11} -2.65927 q^{13} -1.28984 q^{14} +1.00000 q^{16} -1.92011 q^{17} +4.69562 q^{19} -1.54501 q^{20} +2.49412 q^{22} -3.63334 q^{23} -2.61296 q^{25} -2.65927 q^{26} -1.28984 q^{28} -9.08085 q^{29} +5.96966 q^{31} +1.00000 q^{32} -1.92011 q^{34} +1.99281 q^{35} +5.70604 q^{37} +4.69562 q^{38} -1.54501 q^{40} -6.53614 q^{41} +6.44584 q^{43} +2.49412 q^{44} -3.63334 q^{46} -12.1071 q^{47} -5.33631 q^{49} -2.61296 q^{50} -2.65927 q^{52} -2.97407 q^{53} -3.85342 q^{55} -1.28984 q^{56} -9.08085 q^{58} +12.6372 q^{59} -10.5220 q^{61} +5.96966 q^{62} +1.00000 q^{64} +4.10859 q^{65} +4.51299 q^{67} -1.92011 q^{68} +1.99281 q^{70} -6.01319 q^{71} -9.46377 q^{73} +5.70604 q^{74} +4.69562 q^{76} -3.21701 q^{77} -9.23357 q^{79} -1.54501 q^{80} -6.53614 q^{82} -12.9523 q^{83} +2.96659 q^{85} +6.44584 q^{86} +2.49412 q^{88} +0.168223 q^{89} +3.43004 q^{91} -3.63334 q^{92} -12.1071 q^{94} -7.25476 q^{95} +1.05018 q^{97} -5.33631 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} - 5 q^{5} - q^{7} + 5 q^{8} - 5 q^{10} - 9 q^{11} - q^{14} + 5 q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{20} - 9 q^{22} - 16 q^{23} + 8 q^{25} - q^{28} - 8 q^{29} - q^{31} + 5 q^{32} - 6 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} - 5 q^{40} - 4 q^{41} + 3 q^{43} - 9 q^{44} - 16 q^{46} - 18 q^{47} + 2 q^{49} + 8 q^{50} - 26 q^{53} + q^{55} - q^{56} - 8 q^{58} - 21 q^{59} - 20 q^{61} - q^{62} + 5 q^{64} + 3 q^{65} - 5 q^{67} - 6 q^{68} - 22 q^{70} - 17 q^{71} + 5 q^{73} - 2 q^{74} - 4 q^{76} - 2 q^{77} - 21 q^{79} - 5 q^{80} - 4 q^{82} - 11 q^{83} - 12 q^{85} + 3 q^{86} - 9 q^{88} + 5 q^{89} - 10 q^{91} - 16 q^{92} - 18 q^{94} - 10 q^{95} - 11 q^{97} + 2 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 - 5 * q^5 - q^7 + 5 * q^8 - 5 * q^10 - 9 * q^11 - q^14 + 5 * q^16 - 6 * q^17 - 4 * q^19 - 5 * q^20 - 9 * q^22 - 16 * q^23 + 8 * q^25 - q^28 - 8 * q^29 - q^31 + 5 * q^32 - 6 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 - 5 * q^40 - 4 * q^41 + 3 * q^43 - 9 * q^44 - 16 * q^46 - 18 * q^47 + 2 * q^49 + 8 * q^50 - 26 * q^53 + q^55 - q^56 - 8 * q^58 - 21 * q^59 - 20 * q^61 - q^62 + 5 * q^64 + 3 * q^65 - 5 * q^67 - 6 * q^68 - 22 * q^70 - 17 * q^71 + 5 * q^73 - 2 * q^74 - 4 * q^76 - 2 * q^77 - 21 * q^79 - 5 * q^80 - 4 * q^82 - 11 * q^83 - 12 * q^85 + 3 * q^86 - 9 * q^88 + 5 * q^89 - 10 * q^91 - 16 * q^92 - 18 * q^94 - 10 * q^95 - 11 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more