LMFDB - Embedded newform 4028.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.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:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + \cdots + 139 $ x^19 - 4x^18 - 30x^17 + 124x^16 + 364x^15 - 1554x^14 - 2310x^13 + 10113x^12 + 8368x^11 - 36567x^10 - 18074x^9 + 72868x^8 + 23819x^7 - 73816x^6 - 17877x^5 + 30973x^4 + 5074x^3 - 4299x^2 - 454x + 139
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.828283$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.828283 q^{3} +0.378111 q^{5} -3.91567 q^{7} -2.31395 q^{9} +O(q^{10})$	$q-0.828283 q^{3} +0.378111 q^{5} -3.91567 q^{7} -2.31395 q^{9} +4.10431 q^{11} -0.0890774 q^{13} -0.313183 q^{15} +2.16847 q^{17} -1.00000 q^{19} +3.24328 q^{21} -3.36192 q^{23} -4.85703 q^{25} +4.40145 q^{27} +0.979419 q^{29} +0.866138 q^{31} -3.39953 q^{33} -1.48056 q^{35} +2.32034 q^{37} +0.0737813 q^{39} -8.03809 q^{41} -4.18676 q^{43} -0.874929 q^{45} +4.66697 q^{47} +8.33247 q^{49} -1.79611 q^{51} -1.00000 q^{53} +1.55189 q^{55} +0.828283 q^{57} -0.0391736 q^{59} -12.6790 q^{61} +9.06066 q^{63} -0.0336811 q^{65} -2.22129 q^{67} +2.78462 q^{69} +14.5178 q^{71} -5.23443 q^{73} +4.02300 q^{75} -16.0711 q^{77} -4.14237 q^{79} +3.29620 q^{81} +6.30308 q^{83} +0.819923 q^{85} -0.811236 q^{87} +11.8949 q^{89} +0.348798 q^{91} -0.717407 q^{93} -0.378111 q^{95} +2.09440 q^{97} -9.49717 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.828283	−0.478209	−0.239105	−	0.970994i	$-0.576854\pi$
$3$	−0.828283	−0.478209	−0.239105	+	0.970994i	$0.576854\pi$
$4$	0	0
$5$	0.378111	0.169096	0.0845482	−	0.996419i	$-0.473055\pi$
$5$	0.378111	0.169096	0.0845482	+	0.996419i	$0.473055\pi$
$6$	0	0
$7$	−3.91567	−1.47998	−0.739992	−	0.672616i	$-0.765171\pi$
$7$	−3.91567	−1.47998	−0.739992	+	0.672616i	$0.765171\pi$
$8$	0	0
$9$	−2.31395	−0.771316
$10$	0	0
$11$	4.10431	1.23750	0.618748	−	0.785589i	$-0.287640\pi$
$11$	4.10431	1.23750	0.618748	+	0.785589i	$0.287640\pi$
$12$	0	0
$13$	−0.0890774	−0.0247056	−0.0123528	−	0.999924i	$-0.503932\pi$
$13$	−0.0890774	−0.0247056	−0.0123528	+	0.999924i	$0.503932\pi$
$14$	0	0
$15$	−0.313183	−0.0808634
$16$	0	0
$17$	2.16847	0.525932	0.262966	−	0.964805i	$-0.415299\pi$
$17$	2.16847	0.525932	0.262966	+	0.964805i	$0.415299\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.24328	0.707742
$22$	0	0
$23$	−3.36192	−0.701009	−0.350505	−	0.936561i	$-0.613990\pi$
$23$	−3.36192	−0.701009	−0.350505	+	0.936561i	$0.613990\pi$
$24$	0	0
$25$	−4.85703	−0.971406
$26$	0	0
$27$	4.40145	0.847060
$28$	0	0
$29$	0.979419	0.181873	0.0909367	−	0.995857i	$-0.471014\pi$
$29$	0.979419	0.181873	0.0909367	+	0.995857i	$0.471014\pi$
$30$	0	0
$31$	0.866138	0.155563	0.0777815	−	0.996970i	$-0.475216\pi$
$31$	0.866138	0.155563	0.0777815	+	0.996970i	$0.475216\pi$
$32$	0	0
$33$	−3.39953	−0.591782
$34$	0	0
$35$	−1.48056	−0.250260
$36$	0	0
$37$	2.32034	0.381461	0.190731	−	0.981642i	$-0.438914\pi$
$37$	2.32034	0.381461	0.190731	+	0.981642i	$0.438914\pi$
$38$	0	0
$39$	0.0737813	0.0118145
$40$	0	0
$41$	−8.03809	−1.25534	−0.627670	−	0.778480i	$-0.715991\pi$
$41$	−8.03809	−1.25534	−0.627670	+	0.778480i	$0.715991\pi$
$42$	0	0
$43$	−4.18676	−0.638475	−0.319237	−	0.947675i	$-0.603427\pi$
$43$	−4.18676	−0.638475	−0.319237	+	0.947675i	$0.603427\pi$
$44$	0	0
$45$	−0.874929	−0.130427
$46$	0	0
$47$	4.66697	0.680747	0.340373	−	0.940290i	$-0.389447\pi$
$47$	4.66697	0.680747	0.340373	+	0.940290i	$0.389447\pi$
$48$	0	0
$49$	8.33247	1.19035
$50$	0	0
$51$	−1.79611	−0.251505
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	1.55189	0.209256
$56$	0	0
$57$	0.828283	0.109709
$58$	0	0
$59$	−0.0391736	−0.00509997	−0.00254998	−	0.999997i	$-0.500812\pi$
$59$	−0.0391736	−0.00509997	−0.00254998	+	0.999997i	$0.500812\pi$
$60$	0	0
$61$	−12.6790	−1.62338	−0.811691	−	0.584086i	$-0.801453\pi$
$61$	−12.6790	−1.62338	−0.811691	+	0.584086i	$0.801453\pi$
$62$	0	0
$63$	9.06066	1.14154
$64$	0	0
$65$	−0.0336811	−0.00417763
$66$	0	0
$67$	−2.22129	−0.271374	−0.135687	−	0.990752i	$-0.543324\pi$
$67$	−2.22129	−0.271374	−0.135687	+	0.990752i	$0.543324\pi$
$68$	0	0
$69$	2.78462	0.335229
$70$	0	0
$71$	14.5178	1.72294	0.861471	−	0.507807i	$-0.169544\pi$
$71$	14.5178	1.72294	0.861471	+	0.507807i	$0.169544\pi$
$72$	0	0
$73$	−5.23443	−0.612644	−0.306322	−	0.951928i	$-0.599098\pi$
$73$	−5.23443	−0.612644	−0.306322	+	0.951928i	$0.599098\pi$
$74$	0	0
$75$	4.02300	0.464536
$76$	0	0
$77$	−16.0711	−1.83148
$78$	0	0
$79$	−4.14237	−0.466053	−0.233026	−	0.972470i	$-0.574863\pi$
$79$	−4.14237	−0.466053	−0.233026	+	0.972470i	$0.574863\pi$
$80$	0	0
$81$	3.29620	0.366244
$82$	0	0
$83$	6.30308	0.691852	0.345926	−	0.938262i	$-0.387565\pi$
$83$	6.30308	0.691852	0.345926	+	0.938262i	$0.387565\pi$
$84$	0	0
$85$	0.819923	0.0889331
$86$	0	0
$87$	−0.811236	−0.0869736
$88$	0	0
$89$	11.8949	1.26086	0.630430	−	0.776246i	$-0.282879\pi$
$89$	11.8949	1.26086	0.630430	+	0.776246i	$0.282879\pi$
$90$	0	0
$91$	0.348798	0.0365639
$92$	0	0
$93$	−0.717407	−0.0743917
$94$	0	0
$95$	−0.378111	−0.0387934
$96$	0	0
$97$	2.09440	0.212655	0.106327	−	0.994331i	$-0.466091\pi$
$97$	2.09440	0.212655	0.106327	+	0.994331i	$0.466091\pi$
$98$	0	0
$99$	−9.49717	−0.954501
$100$	0	0
$101$	16.2276	1.61470	0.807351	−	0.590071i	$-0.200900\pi$
$101$	16.2276	1.61470	0.807351	+	0.590071i	$0.200900\pi$
$102$	0	0
$103$	16.6250	1.63811	0.819053	−	0.573718i	$-0.194500\pi$
$103$	16.6250	1.63811	0.819053	+	0.573718i	$0.194500\pi$
$104$	0	0
$105$	1.22632	0.119677
$106$	0	0
$107$	13.4730	1.30248	0.651242	−	0.758870i	$-0.274248\pi$
$107$	13.4730	1.30248	0.651242	+	0.758870i	$0.274248\pi$
$108$	0	0
$109$	−7.44655	−0.713250	−0.356625	−	0.934248i	$-0.616073\pi$
$109$	−7.44655	−0.713250	−0.356625	+	0.934248i	$0.616073\pi$
$110$	0	0
$111$	−1.92190	−0.182418
$112$	0	0
$113$	10.5070	0.988412	0.494206	−	0.869345i	$-0.335459\pi$
$113$	10.5070	0.988412	0.494206	+	0.869345i	$0.335459\pi$
$114$	0	0
$115$	−1.27118	−0.118538
$116$	0	0
$117$	0.206120	0.0190558
$118$	0	0
$119$	−8.49102	−0.778370
$120$	0	0
$121$	5.84539	0.531399
$122$	0	0
$123$	6.65781	0.600315
$124$	0	0
$125$	−3.72705	−0.333358
$126$	0	0
$127$	18.5308	1.64434	0.822171	−	0.569240i	$-0.192762\pi$
$127$	18.5308	1.64434	0.822171	+	0.569240i	$0.192762\pi$
$128$	0	0
$129$	3.46782	0.305324
$130$	0	0
$131$	−17.3973	−1.52001	−0.760003	−	0.649919i	$-0.774803\pi$
$131$	−17.3973	−1.52001	−0.760003	+	0.649919i	$0.774803\pi$
$132$	0	0
$133$	3.91567	0.339532
$134$	0	0
$135$	1.66424	0.143235
$136$	0	0
$137$	20.6976	1.76832	0.884158	−	0.467189i	$-0.154733\pi$
$137$	20.6976	1.76832	0.884158	+	0.467189i	$0.154733\pi$
$138$	0	0
$139$	22.5713	1.91447	0.957236	−	0.289307i	$-0.0934250\pi$
$139$	22.5713	1.91447	0.957236	+	0.289307i	$0.0934250\pi$
$140$	0	0
$141$	−3.86557	−0.325539
$142$	0	0
$143$	−0.365602	−0.0305731
$144$	0	0
$145$	0.370329	0.0307541
$146$	0	0
$147$	−6.90164	−0.569238
$148$	0	0
$149$	−1.72496	−0.141315	−0.0706573	−	0.997501i	$-0.522510\pi$
$149$	−1.72496	−0.141315	−0.0706573	+	0.997501i	$0.522510\pi$
$150$	0	0
$151$	−16.7100	−1.35984	−0.679921	−	0.733285i	$-0.737986\pi$
$151$	−16.7100	−1.35984	−0.679921	+	0.733285i	$0.737986\pi$
$152$	0	0
$153$	−5.01773	−0.405659
$154$	0	0
$155$	0.327496	0.0263051
$156$	0	0
$157$	13.7580	1.09801	0.549004	−	0.835820i	$-0.315007\pi$
$157$	13.7580	1.09801	0.549004	+	0.835820i	$0.315007\pi$
$158$	0	0
$159$	0.828283	0.0656871
$160$	0	0
$161$	13.1642	1.03748
$162$	0	0
$163$	−7.20956	−0.564696	−0.282348	−	0.959312i	$-0.591113\pi$
$163$	−7.20956	−0.564696	−0.282348	+	0.959312i	$0.591113\pi$
$164$	0	0
$165$	−1.28540	−0.100068
$166$	0	0
$167$	14.4919	1.12142	0.560710	−	0.828012i	$-0.310528\pi$
$167$	14.4919	1.12142	0.560710	+	0.828012i	$0.310528\pi$
$168$	0	0
$169$	−12.9921	−0.999390
$170$	0	0
$171$	2.31395	0.176952
$172$	0	0
$173$	3.15543	0.239903	0.119951	−	0.992780i	$-0.461726\pi$
$173$	3.15543	0.239903	0.119951	+	0.992780i	$0.461726\pi$
$174$	0	0
$175$	19.0185	1.43767
$176$	0	0
$177$	0.0324468	0.00243885
$178$	0	0
$179$	20.9889	1.56878	0.784391	−	0.620267i	$-0.212976\pi$
$179$	20.9889	1.56878	0.784391	+	0.620267i	$0.212976\pi$
$180$	0	0
$181$	−26.4201	−1.96379	−0.981894	−	0.189430i	$-0.939336\pi$
$181$	−26.4201	−1.96379	−0.981894	+	0.189430i	$0.939336\pi$
$182$	0	0
$183$	10.5018	0.776317
$184$	0	0
$185$	0.877345	0.0645037
$186$	0	0
$187$	8.90009	0.650839
$188$	0	0
$189$	−17.2346	−1.25363
$190$	0	0
$191$	7.64649	0.553280	0.276640	−	0.960974i	$-0.410779\pi$
$191$	7.64649	0.553280	0.276640	+	0.960974i	$0.410779\pi$
$192$	0	0
$193$	−13.2158	−0.951295	−0.475647	−	0.879636i	$-0.657786\pi$
$193$	−13.2158	−0.951295	−0.475647	+	0.879636i	$0.657786\pi$
$194$	0	0
$195$	0.0278975	0.00199778
$196$	0	0
$197$	22.9886	1.63787	0.818934	−	0.573888i	$-0.194566\pi$
$197$	22.9886	1.63787	0.818934	+	0.573888i	$0.194566\pi$
$198$	0	0
$199$	17.3189	1.22771	0.613853	−	0.789421i	$-0.289619\pi$
$199$	17.3189	1.22771	0.613853	+	0.789421i	$0.289619\pi$
$200$	0	0
$201$	1.83986	0.129774
$202$	0	0
$203$	−3.83508	−0.269170
$204$	0	0
$205$	−3.03929	−0.212273
$206$	0	0
$207$	7.77931	0.540700
$208$	0	0
$209$	−4.10431	−0.283901
$210$	0	0
$211$	−12.4812	−0.859239	−0.429619	−	0.903010i	$-0.641352\pi$
$211$	−12.4812	−0.859239	−0.429619	+	0.903010i	$0.641352\pi$
$212$	0	0
$213$	−12.0248	−0.823927
$214$	0	0
$215$	−1.58306	−0.107964
$216$	0	0
$217$	−3.39151	−0.230231
$218$	0	0
$219$	4.33559	0.292972
$220$	0	0
$221$	−0.193162	−0.0129935
$222$	0	0
$223$	9.93320	0.665176	0.332588	−	0.943072i	$-0.392078\pi$
$223$	9.93320	0.665176	0.332588	+	0.943072i	$0.392078\pi$
$224$	0	0
$225$	11.2389	0.749261
$226$	0	0
$227$	−14.4363	−0.958172	−0.479086	−	0.877768i	$-0.659032\pi$
$227$	−14.4363	−0.958172	−0.479086	+	0.877768i	$0.659032\pi$
$228$	0	0
$229$	5.48752	0.362626	0.181313	−	0.983425i	$-0.441965\pi$
$229$	5.48752	0.362626	0.181313	+	0.983425i	$0.441965\pi$
$230$	0	0
$231$	13.3114	0.875829
$232$	0	0
$233$	5.43196	0.355860	0.177930	−	0.984043i	$-0.443060\pi$
$233$	5.43196	0.355860	0.177930	+	0.984043i	$0.443060\pi$
$234$	0	0
$235$	1.76463	0.115112
$236$	0	0
$237$	3.43105	0.222871
$238$	0	0
$239$	9.48749	0.613695	0.306847	−	0.951759i	$-0.400726\pi$
$239$	9.48749	0.613695	0.306847	+	0.951759i	$0.400726\pi$
$240$	0	0
$241$	9.91700	0.638810	0.319405	−	0.947618i	$-0.396517\pi$
$241$	9.91700	0.638810	0.319405	+	0.947618i	$0.396517\pi$
$242$	0	0
$243$	−15.9345	−1.02220
$244$	0	0
$245$	3.15060	0.201284
$246$	0	0
$247$	0.0890774	0.00566786
$248$	0	0
$249$	−5.22073	−0.330850
$250$	0	0
$251$	−0.679568	−0.0428940	−0.0214470	−	0.999770i	$-0.506827\pi$
$251$	−0.679568	−0.0428940	−0.0214470	+	0.999770i	$0.506827\pi$
$252$	0	0
$253$	−13.7984	−0.867497
$254$	0	0
$255$	−0.679128	−0.0425286
$256$	0	0
$257$	−8.90519	−0.555491	−0.277745	−	0.960655i	$-0.589587\pi$
$257$	−8.90519	−0.555491	−0.277745	+	0.960655i	$0.589587\pi$
$258$	0	0
$259$	−9.08568	−0.564557
$260$	0	0
$261$	−2.26632	−0.140282
$262$	0	0
$263$	22.9695	1.41636	0.708180	−	0.706032i	$-0.249516\pi$
$263$	22.9695	1.41636	0.708180	+	0.706032i	$0.249516\pi$
$264$	0	0
$265$	−0.378111	−0.0232272
$266$	0	0
$267$	−9.85236	−0.602955
$268$	0	0
$269$	−4.42674	−0.269903	−0.134952	−	0.990852i	$-0.543088\pi$
$269$	−4.42674	−0.269903	−0.134952	+	0.990852i	$0.543088\pi$
$270$	0	0
$271$	−11.6785	−0.709421	−0.354710	−	0.934976i	$-0.615421\pi$
$271$	−11.6785	−0.709421	−0.354710	+	0.934976i	$0.615421\pi$
$272$	0	0
$273$	−0.288903	−0.0174852
$274$	0	0
$275$	−19.9348	−1.20211
$276$	0	0
$277$	−22.4871	−1.35112	−0.675560	−	0.737305i	$-0.736098\pi$
$277$	−22.4871	−1.35112	−0.675560	+	0.737305i	$0.736098\pi$
$278$	0	0
$279$	−2.00420	−0.119988
$280$	0	0
$281$	−9.52034	−0.567936	−0.283968	−	0.958834i	$-0.591651\pi$
$281$	−9.52034	−0.567936	−0.283968	+	0.958834i	$0.591651\pi$
$282$	0	0
$283$	−1.35147	−0.0803366	−0.0401683	−	0.999193i	$-0.512789\pi$
$283$	−1.35147	−0.0803366	−0.0401683	+	0.999193i	$0.512789\pi$
$284$	0	0
$285$	0.313183	0.0185513
$286$	0	0
$287$	31.4745	1.85788
$288$	0	0
$289$	−12.2977	−0.723396
$290$	0	0
$291$	−1.73476	−0.101693
$292$	0	0
$293$	22.6722	1.32453	0.662263	−	0.749272i	$-0.269596\pi$
$293$	22.6722	1.32453	0.662263	+	0.749272i	$0.269596\pi$
$294$	0	0
$295$	−0.0148120	−0.000862386 0
$296$	0	0
$297$	18.0649	1.04823
$298$	0	0
$299$	0.299471	0.0173189
$300$	0	0
$301$	16.3940	0.944932
$302$	0	0
$303$	−13.4410	−0.772166
$304$	0	0
$305$	−4.79408	−0.274508
$306$	0	0
$307$	13.3942	0.764450	0.382225	−	0.924069i	$-0.375158\pi$
$307$	13.3942	0.764450	0.382225	+	0.924069i	$0.375158\pi$
$308$	0	0
$309$	−13.7702	−0.783357
$310$	0	0
$311$	2.77309	0.157247	0.0786237	−	0.996904i	$-0.474947\pi$
$311$	2.77309	0.157247	0.0786237	+	0.996904i	$0.474947\pi$
$312$	0	0
$313$	−9.73670	−0.550351	−0.275175	−	0.961394i	$-0.588736\pi$
$313$	−9.73670	−0.550351	−0.275175	+	0.961394i	$0.588736\pi$
$314$	0	0
$315$	3.42593	0.193029
$316$	0	0
$317$	9.43509	0.529927	0.264964	−	0.964258i	$-0.414640\pi$
$317$	9.43509	0.529927	0.264964	+	0.964258i	$0.414640\pi$
$318$	0	0
$319$	4.01984	0.225068
$320$	0	0
$321$	−11.1595	−0.622860
$322$	0	0
$323$	−2.16847	−0.120657
$324$	0	0
$325$	0.432652	0.0239992
$326$	0	0
$327$	6.16785	0.341083
$328$	0	0
$329$	−18.2743	−1.00749
$330$	0	0
$331$	4.95225	0.272200	0.136100	−	0.990695i	$-0.456543\pi$
$331$	4.95225	0.272200	0.136100	+	0.990695i	$0.456543\pi$
$332$	0	0
$333$	−5.36914	−0.294227
$334$	0	0
$335$	−0.839895	−0.0458884
$336$	0	0
$337$	−17.9644	−0.978582	−0.489291	−	0.872121i	$-0.662744\pi$
$337$	−17.9644	−0.978582	−0.489291	+	0.872121i	$0.662744\pi$
$338$	0	0
$339$	−8.70273	−0.472668
$340$	0	0
$341$	3.55490	0.192509
$342$	0	0
$343$	−5.21752	−0.281720
$344$	0	0
$345$	1.05290	0.0566860
$346$	0	0
$347$	−10.8617	−0.583086	−0.291543	−	0.956558i	$-0.594169\pi$
$347$	−10.8617	−0.583086	−0.291543	+	0.956558i	$0.594169\pi$
$348$	0	0
$349$	7.31215	0.391410	0.195705	−	0.980663i	$-0.437300\pi$
$349$	7.31215	0.391410	0.195705	+	0.980663i	$0.437300\pi$
$350$	0	0
$351$	−0.392070	−0.0209271
$352$	0	0
$353$	−21.0465	−1.12019	−0.560095	−	0.828429i	$-0.689235\pi$
$353$	−21.0465	−1.12019	−0.560095	+	0.828429i	$0.689235\pi$
$354$	0	0
$355$	5.48932	0.291343
$356$	0	0
$357$	7.03296	0.372224
$358$	0	0
$359$	12.7695	0.673948	0.336974	−	0.941514i	$-0.390596\pi$
$359$	12.7695	0.673948	0.336974	+	0.941514i	$0.390596\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.84163	−0.254120
$364$	0	0
$365$	−1.97919	−0.103596
$366$	0	0
$367$	10.7010	0.558586	0.279293	−	0.960206i	$-0.409900\pi$
$367$	10.7010	0.558586	0.279293	+	0.960206i	$0.409900\pi$
$368$	0	0
$369$	18.5997	0.968263
$370$	0	0
$371$	3.91567	0.203291
$372$	0	0
$373$	−3.09464	−0.160235	−0.0801173	−	0.996785i	$-0.525529\pi$
$373$	−3.09464	−0.160235	−0.0801173	+	0.996785i	$0.525529\pi$
$374$	0	0
$375$	3.08705	0.159415
$376$	0	0
$377$	−0.0872441	−0.00449330
$378$	0	0
$379$	37.0291	1.90206	0.951030	−	0.309100i	$-0.100028\pi$
$379$	37.0291	1.90206	0.951030	+	0.309100i	$0.100028\pi$
$380$	0	0
$381$	−15.3487	−0.786340
$382$	0	0
$383$	−20.7727	−1.06143	−0.530717	−	0.847549i	$-0.678077\pi$
$383$	−20.7727	−1.06143	−0.530717	+	0.847549i	$0.678077\pi$
$384$	0	0
$385$	−6.07667	−0.309696
$386$	0	0
$387$	9.68794	0.492466
$388$	0	0
$389$	0.567083	0.0287523	0.0143761	−	0.999897i	$-0.495424\pi$
$389$	0.567083	0.0287523	0.0143761	+	0.999897i	$0.495424\pi$
$390$	0	0
$391$	−7.29023	−0.368683
$392$	0	0
$393$	14.4099	0.726881
$394$	0	0
$395$	−1.56627	−0.0788078
$396$	0	0
$397$	6.40546	0.321481	0.160740	−	0.986997i	$-0.448612\pi$
$397$	6.40546	0.321481	0.160740	+	0.986997i	$0.448612\pi$
$398$	0	0
$399$	−3.24328	−0.162367
$400$	0	0
$401$	14.1075	0.704496	0.352248	−	0.935907i	$-0.385417\pi$
$401$	14.1075	0.704496	0.352248	+	0.935907i	$0.385417\pi$
$402$	0	0
$403$	−0.0771533	−0.00384328
$404$	0	0
$405$	1.24633	0.0619305
$406$	0	0
$407$	9.52340	0.472057
$408$	0	0
$409$	32.1865	1.59152	0.795761	−	0.605611i	$-0.207071\pi$
$409$	32.1865	1.59152	0.795761	+	0.605611i	$0.207071\pi$
$410$	0	0
$411$	−17.1435	−0.845625
$412$	0	0
$413$	0.153391	0.00754787
$414$	0	0
$415$	2.38326	0.116990
$416$	0	0
$417$	−18.6954	−0.915518
$418$	0	0
$419$	3.55307	0.173579	0.0867895	−	0.996227i	$-0.472339\pi$
$419$	3.55307	0.173579	0.0867895	+	0.996227i	$0.472339\pi$
$420$	0	0
$421$	23.2282	1.13208	0.566038	−	0.824379i	$-0.308476\pi$
$421$	23.2282	1.13208	0.566038	+	0.824379i	$0.308476\pi$
$422$	0	0
$423$	−10.7991	−0.525071
$424$	0	0
$425$	−10.5323	−0.510893
$426$	0	0
$427$	49.6469	2.40258
$428$	0	0
$429$	0.302822	0.0146204
$430$	0	0
$431$	−18.4613	−0.889250	−0.444625	−	0.895717i	$-0.646663\pi$
$431$	−18.4613	−0.889250	−0.444625	+	0.895717i	$0.646663\pi$
$432$	0	0
$433$	20.5262	0.986426	0.493213	−	0.869909i	$-0.335822\pi$
$433$	20.5262	0.986426	0.493213	+	0.869909i	$0.335822\pi$
$434$	0	0
$435$	−0.306737	−0.0147069
$436$	0	0
$437$	3.36192	0.160823
$438$	0	0
$439$	31.0875	1.48373	0.741863	−	0.670552i	$-0.233943\pi$
$439$	31.0875	1.48373	0.741863	+	0.670552i	$0.233943\pi$
$440$	0	0
$441$	−19.2809	−0.918138
$442$	0	0
$443$	25.5002	1.21155	0.605775	−	0.795636i	$-0.292863\pi$
$443$	25.5002	1.21155	0.605775	+	0.795636i	$0.292863\pi$
$444$	0	0
$445$	4.49760	0.213207
$446$	0	0
$447$	1.42876	0.0675779
$448$	0	0
$449$	−8.82141	−0.416308	−0.208154	−	0.978096i	$-0.566746\pi$
$449$	−8.82141	−0.416308	−0.208154	+	0.978096i	$0.566746\pi$
$450$	0	0
$451$	−32.9908	−1.55348
$452$	0	0
$453$	13.8406	0.650289
$454$	0	0
$455$	0.131884	0.00618283
$456$	0	0
$457$	−29.3945	−1.37502	−0.687508	−	0.726177i	$-0.741295\pi$
$457$	−29.3945	−1.37502	−0.687508	+	0.726177i	$0.741295\pi$
$458$	0	0
$459$	9.54442	0.445495
$460$	0	0
$461$	−13.8725	−0.646106	−0.323053	−	0.946381i	$-0.604709\pi$
$461$	−13.8725	−0.646106	−0.323053	+	0.946381i	$0.604709\pi$
$462$	0	0
$463$	7.38550	0.343233	0.171617	−	0.985164i	$-0.445101\pi$
$463$	7.38550	0.343233	0.171617	+	0.985164i	$0.445101\pi$
$464$	0	0
$465$	−0.271259	−0.0125794
$466$	0	0
$467$	3.35080	0.155056	0.0775282	−	0.996990i	$-0.475297\pi$
$467$	3.35080	0.155056	0.0775282	+	0.996990i	$0.475297\pi$
$468$	0	0
$469$	8.69785	0.401629
$470$	0	0
$471$	−11.3955	−0.525077
$472$	0	0
$473$	−17.1838	−0.790110
$474$	0	0
$475$	4.85703	0.222856
$476$	0	0
$477$	2.31395	0.105948
$478$	0	0
$479$	−20.8708	−0.953612	−0.476806	−	0.879009i	$-0.658205\pi$
$479$	−20.8708	−0.953612	−0.476806	+	0.879009i	$0.658205\pi$
$480$	0	0
$481$	−0.206690	−0.00942425
$482$	0	0
$483$	−10.9037	−0.496134
$484$	0	0
$485$	0.791917	0.0359591
$486$	0	0
$487$	16.2128	0.734670	0.367335	−	0.930089i	$-0.380270\pi$
$487$	16.2128	0.734670	0.367335	+	0.930089i	$0.380270\pi$
$488$	0	0
$489$	5.97155	0.270043
$490$	0	0
$491$	−28.3249	−1.27828	−0.639142	−	0.769089i	$-0.720710\pi$
$491$	−28.3249	−1.27828	−0.639142	+	0.769089i	$0.720710\pi$
$492$	0	0
$493$	2.12384	0.0956530
$494$	0	0
$495$	−3.59098	−0.161403
$496$	0	0
$497$	−56.8468	−2.54993
$498$	0	0
$499$	25.4137	1.13768	0.568838	−	0.822450i	$-0.307393\pi$
$499$	25.4137	1.13768	0.568838	+	0.822450i	$0.307393\pi$
$500$	0	0
$501$	−12.0034	−0.536274
$502$	0	0
$503$	42.4706	1.89367	0.946836	−	0.321716i	$-0.104260\pi$
$503$	42.4706	1.89367	0.946836	+	0.321716i	$0.104260\pi$
$504$	0	0
$505$	6.13582	0.273040
$506$	0	0
$507$	10.7611	0.477917
$508$	0	0
$509$	−0.233397	−0.0103451	−0.00517257	−	0.999987i	$-0.501646\pi$
$509$	−0.233397	−0.0103451	−0.00517257	+	0.999987i	$0.501646\pi$
$510$	0	0
$511$	20.4963	0.906703
$512$	0	0
$513$	−4.40145	−0.194329
$514$	0	0
$515$	6.28608	0.276998
$516$	0	0
$517$	19.1547	0.842422
$518$	0	0
$519$	−2.61359	−0.114724
$520$	0	0
$521$	−34.8104	−1.52507	−0.762536	−	0.646946i	$-0.776046\pi$
$521$	−34.8104	−1.52507	−0.762536	+	0.646946i	$0.776046\pi$
$522$	0	0
$523$	−40.3998	−1.76656	−0.883279	−	0.468848i	$-0.844669\pi$
$523$	−40.3998	−1.76656	−0.883279	+	0.468848i	$0.844669\pi$
$524$	0	0
$525$	−15.7527	−0.687505
$526$	0	0
$527$	1.87820	0.0818155
$528$	0	0
$529$	−11.6975	−0.508586
$530$	0	0
$531$	0.0906456	0.00393369
$532$	0	0
$533$	0.716013	0.0310139
$534$	0	0
$535$	5.09429	0.220245
$536$	0	0
$537$	−17.3847	−0.750206
$538$	0	0
$539$	34.1991	1.47306
$540$	0	0
$541$	−12.5704	−0.540444	−0.270222	−	0.962798i	$-0.587097\pi$
$541$	−12.5704	−0.540444	−0.270222	+	0.962798i	$0.587097\pi$
$542$	0	0
$543$	21.8833	0.939102
$544$	0	0
$545$	−2.81562	−0.120608
$546$	0	0
$547$	34.0452	1.45567	0.727833	−	0.685755i	$-0.240528\pi$
$547$	34.0452	1.45567	0.727833	+	0.685755i	$0.240528\pi$
$548$	0	0
$549$	29.3386	1.25214
$550$	0	0
$551$	−0.979419	−0.0417246
$552$	0	0
$553$	16.2202	0.689751
$554$	0	0
$555$	−0.726690	−0.0308463
$556$	0	0
$557$	−6.77271	−0.286969	−0.143484	−	0.989653i	$-0.545831\pi$
$557$	−6.77271	−0.286969	−0.143484	+	0.989653i	$0.545831\pi$
$558$	0	0
$559$	0.372946	0.0157739
$560$	0	0
$561$	−7.37179	−0.311237
$562$	0	0
$563$	26.1954	1.10401	0.552003	−	0.833842i	$-0.313864\pi$
$563$	26.1954	1.10401	0.552003	+	0.833842i	$0.313864\pi$
$564$	0	0
$565$	3.97280	0.167137
$566$	0	0
$567$	−12.9068	−0.542036
$568$	0	0
$569$	6.13417	0.257158	0.128579	−	0.991699i	$-0.458958\pi$
$569$	6.13417	0.257158	0.128579	+	0.991699i	$0.458958\pi$
$570$	0	0
$571$	−39.5467	−1.65498	−0.827490	−	0.561481i	$-0.810232\pi$
$571$	−39.5467	−1.65498	−0.827490	+	0.561481i	$0.810232\pi$
$572$	0	0
$573$	−6.33345	−0.264584
$574$	0	0
$575$	16.3290	0.680965
$576$	0	0
$577$	−2.68114	−0.111617	−0.0558086	−	0.998441i	$-0.517774\pi$
$577$	−2.68114	−0.111617	−0.0558086	+	0.998441i	$0.517774\pi$
$578$	0	0
$579$	10.9464	0.454918
$580$	0	0
$581$	−24.6808	−1.02393
$582$	0	0
$583$	−4.10431	−0.169983
$584$	0	0
$585$	0.0779364	0.00322227
$586$	0	0
$587$	−32.2307	−1.33030	−0.665151	−	0.746709i	$-0.731633\pi$
$587$	−32.2307	−1.33030	−0.665151	+	0.746709i	$0.731633\pi$
$588$	0	0
$589$	−0.866138	−0.0356886
$590$	0	0
$591$	−19.0410	−0.783243
$592$	0	0
$593$	29.5006	1.21144	0.605722	−	0.795677i	$-0.292885\pi$
$593$	29.5006	1.21144	0.605722	+	0.795677i	$0.292885\pi$
$594$	0	0
$595$	−3.21055	−0.131620
$596$	0	0
$597$	−14.3450	−0.587100
$598$	0	0
$599$	33.2119	1.35700	0.678501	−	0.734600i	$-0.262630\pi$
$599$	33.2119	1.35700	0.678501	+	0.734600i	$0.262630\pi$
$600$	0	0
$601$	−4.02015	−0.163985	−0.0819927	−	0.996633i	$-0.526128\pi$
$601$	−4.02015	−0.163985	−0.0819927	+	0.996633i	$0.526128\pi$
$602$	0	0
$603$	5.13996	0.209315
$604$	0	0
$605$	2.21020	0.0898575
$606$	0	0
$607$	−31.1206	−1.26314	−0.631572	−	0.775317i	$-0.717590\pi$
$607$	−31.1206	−1.26314	−0.631572	+	0.775317i	$0.717590\pi$
$608$	0	0
$609$	3.17653	0.128720
$610$	0	0
$611$	−0.415721	−0.0168183
$612$	0	0
$613$	19.1723	0.774362	0.387181	−	0.922004i	$-0.373449\pi$
$613$	19.1723	0.774362	0.387181	+	0.922004i	$0.373449\pi$
$614$	0	0
$615$	2.51739	0.101511
$616$	0	0
$617$	40.9128	1.64709	0.823544	−	0.567252i	$-0.191993\pi$
$617$	40.9128	1.64709	0.823544	+	0.567252i	$0.191993\pi$
$618$	0	0
$619$	19.2162	0.772365	0.386182	−	0.922422i	$-0.373794\pi$
$619$	19.2162	0.772365	0.386182	+	0.922422i	$0.373794\pi$
$620$	0	0
$621$	−14.7973	−0.593797
$622$	0	0
$623$	−46.5766	−1.86605
$624$	0	0
$625$	22.8759	0.915037
$626$	0	0
$627$	3.39953	0.135764
$628$	0	0
$629$	5.03159	0.200623
$630$	0	0
$631$	−25.2949	−1.00697	−0.503486	−	0.864003i	$-0.667950\pi$
$631$	−25.2949	−1.00697	−0.503486	+	0.864003i	$0.667950\pi$
$632$	0	0
$633$	10.3379	0.410896
$634$	0	0
$635$	7.00670	0.278052
$636$	0	0
$637$	−0.742235	−0.0294084
$638$	0	0
$639$	−33.5933	−1.32893
$640$	0	0
$641$	−28.1814	−1.11310	−0.556548	−	0.830815i	$-0.687875\pi$
$641$	−28.1814	−1.11310	−0.556548	+	0.830815i	$0.687875\pi$
$642$	0	0
$643$	30.5905	1.20637	0.603185	−	0.797601i	$-0.293898\pi$
$643$	30.5905	1.20637	0.603185	+	0.797601i	$0.293898\pi$
$644$	0	0
$645$	1.31122	0.0516292
$646$	0	0
$647$	0.431673	0.0169708	0.00848541	−	0.999964i	$-0.497299\pi$
$647$	0.431673	0.0169708	0.00848541	+	0.999964i	$0.497299\pi$
$648$	0	0
$649$	−0.160781	−0.00631119
$650$	0	0
$651$	2.80913	0.110098
$652$	0	0
$653$	44.2956	1.73342	0.866711	−	0.498811i	$-0.166230\pi$
$653$	44.2956	1.73342	0.866711	+	0.498811i	$0.166230\pi$
$654$	0	0
$655$	−6.57810	−0.257027
$656$	0	0
$657$	12.1122	0.472542
$658$	0	0
$659$	−9.80516	−0.381955	−0.190977	−	0.981594i	$-0.561166\pi$
$659$	−9.80516	−0.381955	−0.190977	+	0.981594i	$0.561166\pi$
$660$	0	0
$661$	4.22817	0.164457	0.0822284	−	0.996614i	$-0.473796\pi$
$661$	4.22817	0.164457	0.0822284	+	0.996614i	$0.473796\pi$
$662$	0	0
$663$	0.159993	0.00621360
$664$	0	0
$665$	1.48056	0.0574135
$666$	0	0
$667$	−3.29273	−0.127495
$668$	0	0
$669$	−8.22750	−0.318094
$670$	0	0
$671$	−52.0387	−2.00893
$672$	0	0
$673$	3.85472	0.148589	0.0742943	−	0.997236i	$-0.476330\pi$
$673$	3.85472	0.148589	0.0742943	+	0.997236i	$0.476330\pi$
$674$	0	0
$675$	−21.3780	−0.822839
$676$	0	0
$677$	25.2356	0.969883	0.484941	−	0.874547i	$-0.338841\pi$
$677$	25.2356	0.969883	0.484941	+	0.874547i	$0.338841\pi$
$678$	0	0
$679$	−8.20100	−0.314725
$680$	0	0
$681$	11.9574	0.458207
$682$	0	0
$683$	−7.52795	−0.288049	−0.144025	−	0.989574i	$-0.546004\pi$
$683$	−7.52795	−0.288049	−0.144025	+	0.989574i	$0.546004\pi$
$684$	0	0
$685$	7.82599	0.299016
$686$	0	0
$687$	−4.54522	−0.173411
$688$	0	0
$689$	0.0890774	0.00339358
$690$	0	0
$691$	−16.9361	−0.644278	−0.322139	−	0.946692i	$-0.604402\pi$
$691$	−16.9361	−0.644278	−0.322139	+	0.946692i	$0.604402\pi$
$692$	0	0
$693$	37.1878	1.41265
$694$	0	0
$695$	8.53445	0.323730
$696$	0	0
$697$	−17.4304	−0.660223
$698$	0	0
$699$	−4.49920	−0.170175
$700$	0	0
$701$	−36.4557	−1.37691	−0.688456	−	0.725278i	$-0.741711\pi$
$701$	−36.4557	−1.37691	−0.688456	+	0.725278i	$0.741711\pi$
$702$	0	0
$703$	−2.32034	−0.0875133
$704$	0	0
$705$	−1.46161	−0.0550475
$706$	0	0
$707$	−63.5418	−2.38973
$708$	0	0
$709$	−26.4031	−0.991589	−0.495795	−	0.868440i	$-0.665123\pi$
$709$	−26.4031	−0.991589	−0.495795	+	0.868440i	$0.665123\pi$
$710$	0	0
$711$	9.58523	0.359474
$712$	0	0
$713$	−2.91189	−0.109051
$714$	0	0
$715$	−0.138238	−0.00516981
$716$	0	0
$717$	−7.85832	−0.293474
$718$	0	0
$719$	4.11539	0.153478	0.0767390	−	0.997051i	$-0.475549\pi$
$719$	4.11539	0.153478	0.0767390	+	0.997051i	$0.475549\pi$
$720$	0	0
$721$	−65.0978	−2.42437
$722$	0	0
$723$	−8.21408	−0.305485
$724$	0	0
$725$	−4.75707	−0.176673
$726$	0	0
$727$	−18.0533	−0.669558	−0.334779	−	0.942297i	$-0.608662\pi$
$727$	−18.0533	−0.669558	−0.334779	+	0.942297i	$0.608662\pi$
$728$	0	0
$729$	3.30971	0.122582
$730$	0	0
$731$	−9.07886	−0.335794
$732$	0	0
$733$	−14.0327	−0.518311	−0.259156	−	0.965836i	$-0.583444\pi$
$733$	−14.0327	−0.518311	−0.259156	+	0.965836i	$0.583444\pi$
$734$	0	0
$735$	−2.60959	−0.0962560
$736$	0	0
$737$	−9.11688	−0.335825
$738$	0	0
$739$	−37.5790	−1.38237	−0.691183	−	0.722679i	$-0.742910\pi$
$739$	−37.5790	−1.38237	−0.691183	+	0.722679i	$0.742910\pi$
$740$	0	0
$741$	−0.0737813	−0.00271042
$742$	0	0
$743$	−31.8613	−1.16888	−0.584440	−	0.811437i	$-0.698686\pi$
$743$	−31.8613	−1.16888	−0.584440	+	0.811437i	$0.698686\pi$
$744$	0	0
$745$	−0.652227	−0.0238958
$746$	0	0
$747$	−14.5850	−0.533637
$748$	0	0
$749$	−52.7558	−1.92766
$750$	0	0
$751$	−12.3356	−0.450131	−0.225065	−	0.974344i	$-0.572260\pi$
$751$	−12.3356	−0.450131	−0.225065	+	0.974344i	$0.572260\pi$
$752$	0	0
$753$	0.562875	0.0205123
$754$	0	0
$755$	−6.31824	−0.229944
$756$	0	0
$757$	46.5111	1.69048	0.845238	−	0.534391i	$-0.179459\pi$
$757$	46.5111	1.69048	0.845238	+	0.534391i	$0.179459\pi$
$758$	0	0
$759$	11.4290	0.414845
$760$	0	0
$761$	−23.9076	−0.866650	−0.433325	−	0.901238i	$-0.642660\pi$
$761$	−23.9076	−0.866650	−0.433325	+	0.901238i	$0.642660\pi$
$762$	0	0
$763$	29.1582	1.05560
$764$	0	0
$765$	−1.89726	−0.0685955
$766$	0	0
$767$	0.00348948	0.000125998 0
$768$	0	0
$769$	−41.1394	−1.48352	−0.741762	−	0.670664i	$-0.766009\pi$
$769$	−41.1394	−1.48352	−0.741762	+	0.670664i	$0.766009\pi$
$770$	0	0
$771$	7.37602	0.265641
$772$	0	0
$773$	−40.0330	−1.43989	−0.719944	−	0.694032i	$-0.755833\pi$
$773$	−40.0330	−1.43989	−0.719944	+	0.694032i	$0.755833\pi$
$774$	0	0
$775$	−4.20686	−0.151115
$776$	0	0
$777$	7.52551	0.269976
$778$	0	0
$779$	8.03809	0.287995
$780$	0	0
$781$	59.5854	2.13213
$782$	0	0
$783$	4.31086	0.154058
$784$	0	0
$785$	5.20204	0.185669
$786$	0	0
$787$	−0.210961	−0.00751993	−0.00375997	−	0.999993i	$-0.501197\pi$
$787$	−0.210961	−0.00751993	−0.00375997	+	0.999993i	$0.501197\pi$
$788$	0	0
$789$	−19.0252	−0.677317
$790$	0	0
$791$	−41.1418	−1.46283
$792$	0	0
$793$	1.12941	0.0401067
$794$	0	0
$795$	0.313183	0.0111074
$796$	0	0
$797$	−15.5773	−0.551776	−0.275888	−	0.961190i	$-0.588972\pi$
$797$	−15.5773	−0.551776	−0.275888	+	0.961190i	$0.588972\pi$
$798$	0	0
$799$	10.1202	0.358026
$800$	0	0
$801$	−27.5242	−0.972521
$802$	0	0
$803$	−21.4837	−0.758145
$804$	0	0
$805$	4.97752	0.175434
$806$	0	0
$807$	3.66659	0.129070
$808$	0	0
$809$	−39.9480	−1.40450	−0.702249	−	0.711931i	$-0.747821\pi$
$809$	−39.9480	−1.40450	−0.702249	+	0.711931i	$0.747821\pi$
$810$	0	0
$811$	52.5129	1.84398	0.921988	−	0.387219i	$-0.126564\pi$
$811$	52.5129	1.84398	0.921988	+	0.387219i	$0.126564\pi$
$812$	0	0
$813$	9.67313	0.339251
$814$	0	0
$815$	−2.72601	−0.0954880
$816$	0	0
$817$	4.18676	0.146476
$818$	0	0
$819$	−0.807100	−0.0282024
$820$	0	0
$821$	17.6124	0.614677	0.307339	−	0.951600i	$-0.400562\pi$
$821$	17.6124	0.614677	0.307339	+	0.951600i	$0.400562\pi$
$822$	0	0
$823$	12.9225	0.450451	0.225225	−	0.974307i	$-0.427688\pi$
$823$	12.9225	0.450451	0.225225	+	0.974307i	$0.427688\pi$
$824$	0	0
$825$	16.5116	0.574861
$826$	0	0
$827$	−48.4882	−1.68610	−0.843050	−	0.537835i	$-0.819242\pi$
$827$	−48.4882	−1.68610	−0.843050	+	0.537835i	$0.819242\pi$
$828$	0	0
$829$	32.4763	1.12795	0.563974	−	0.825793i	$-0.309272\pi$
$829$	32.4763	1.12795	0.563974	+	0.825793i	$0.309272\pi$
$830$	0	0
$831$	18.6257	0.646118
$832$	0	0
$833$	18.0687	0.626044
$834$	0	0
$835$	5.47956	0.189628
$836$	0	0
$837$	3.81226	0.131771
$838$	0	0
$839$	−42.8691	−1.48001	−0.740003	−	0.672604i	$-0.765176\pi$
$839$	−42.8691	−1.48001	−0.740003	+	0.672604i	$0.765176\pi$
$840$	0	0
$841$	−28.0407	−0.966922
$842$	0	0
$843$	7.88553	0.271592
$844$	0	0
$845$	−4.91244	−0.168993
$846$	0	0
$847$	−22.8886	−0.786462
$848$	0	0
$849$	1.11940	0.0384177
$850$	0	0
$851$	−7.80080	−0.267408
$852$	0	0
$853$	−4.35815	−0.149220	−0.0746101	−	0.997213i	$-0.523771\pi$
$853$	−4.35815	−0.149220	−0.0746101	+	0.997213i	$0.523771\pi$
$854$	0	0
$855$	0.874929	0.0299219
$856$	0	0
$857$	20.1432	0.688077	0.344038	−	0.938956i	$-0.388205\pi$
$857$	20.1432	0.688077	0.344038	+	0.938956i	$0.388205\pi$
$858$	0	0
$859$	−16.9311	−0.577680	−0.288840	−	0.957377i	$-0.593270\pi$
$859$	−16.9311	−0.577680	−0.288840	+	0.957377i	$0.593270\pi$
$860$	0	0
$861$	−26.0698	−0.888456
$862$	0	0
$863$	−38.4896	−1.31020	−0.655101	−	0.755542i	$-0.727374\pi$
$863$	−38.4896	−1.31020	−0.655101	+	0.755542i	$0.727374\pi$
$864$	0	0
$865$	1.19310	0.0405666
$866$	0	0
$867$	10.1860	0.345935
$868$	0	0
$869$	−17.0016	−0.576739
$870$	0	0
$871$	0.197867	0.00670447
$872$	0	0
$873$	−4.84634	−0.164024
$874$	0	0
$875$	14.5939	0.493364
$876$	0	0
$877$	−44.7280	−1.51036	−0.755178	−	0.655519i	$-0.772450\pi$
$877$	−44.7280	−1.51036	−0.755178	+	0.655519i	$0.772450\pi$
$878$	0	0
$879$	−18.7790	−0.633400
$880$	0	0
$881$	18.6811	0.629383	0.314692	−	0.949194i	$-0.398099\pi$
$881$	18.6811	0.629383	0.314692	+	0.949194i	$0.398099\pi$
$882$	0	0
$883$	11.7652	0.395930	0.197965	−	0.980209i	$-0.436567\pi$
$883$	11.7652	0.395930	0.197965	+	0.980209i	$0.436567\pi$
$884$	0	0
$885$	0.0122685	0.000412401 0
$886$	0	0
$887$	11.0450	0.370853	0.185427	−	0.982658i	$-0.440633\pi$
$887$	11.0450	0.370853	0.185427	+	0.982658i	$0.440633\pi$
$888$	0	0
$889$	−72.5605	−2.43360
$890$	0	0
$891$	13.5286	0.453226
$892$	0	0
$893$	−4.66697	−0.156174
$894$	0	0
$895$	7.93612	0.265275
$896$	0	0
$897$	−0.248047	−0.00828205
$898$	0	0
$899$	0.848312	0.0282928
$900$	0	0
$901$	−2.16847	−0.0722423
$902$	0	0
$903$	−13.5788	−0.451875
$904$	0	0
$905$	−9.98972	−0.332069
$906$	0	0
$907$	−50.0673	−1.66246	−0.831229	−	0.555930i	$-0.812362\pi$
$907$	−50.0673	−1.66246	−0.831229	+	0.555930i	$0.812362\pi$
$908$	0	0
$909$	−37.5497	−1.24545
$910$	0	0
$911$	−4.70657	−0.155935	−0.0779677	−	0.996956i	$-0.524843\pi$
$911$	−4.70657	−0.155935	−0.0779677	+	0.996956i	$0.524843\pi$
$912$	0	0
$913$	25.8698	0.856165
$914$	0	0
$915$	3.97085	0.131272
$916$	0	0
$917$	68.1220	2.24959
$918$	0	0
$919$	51.8282	1.70965	0.854827	−	0.518914i	$-0.173663\pi$
$919$	51.8282	1.70965	0.854827	+	0.518914i	$0.173663\pi$
$920$	0	0
$921$	−11.0942	−0.365567
$922$	0	0
$923$	−1.29320	−0.0425664
$924$	0	0
$925$	−11.2700	−0.370554
$926$	0	0
$927$	−38.4693	−1.26350
$928$	0	0
$929$	18.9487	0.621688	0.310844	−	0.950461i	$-0.399388\pi$
$929$	18.9487	0.621688	0.310844	+	0.950461i	$0.399388\pi$
$930$	0	0
$931$	−8.33247	−0.273086
$932$	0	0
$933$	−2.29690	−0.0751972
$934$	0	0
$935$	3.36522	0.110054
$936$	0	0
$937$	32.0597	1.04735	0.523673	−	0.851919i	$-0.324561\pi$
$937$	32.0597	1.04735	0.523673	+	0.851919i	$0.324561\pi$
$938$	0	0
$939$	8.06474	0.263183
$940$	0	0
$941$	−51.2150	−1.66956	−0.834781	−	0.550582i	$-0.814406\pi$
$941$	−51.2150	−1.66956	−0.834781	+	0.550582i	$0.814406\pi$
$942$	0	0
$943$	27.0234	0.880004
$944$	0	0
$945$	−6.51660	−0.211985
$946$	0	0
$947$	20.4255	0.663739	0.331870	−	0.943325i	$-0.392321\pi$
$947$	20.4255	0.663739	0.331870	+	0.943325i	$0.392321\pi$
$948$	0	0
$949$	0.466269	0.0151357
$950$	0	0
$951$	−7.81492	−0.253416
$952$	0	0
$953$	32.5365	1.05396	0.526980	−	0.849878i	$-0.323324\pi$
$953$	32.5365	1.05396	0.526980	+	0.849878i	$0.323324\pi$
$954$	0	0
$955$	2.89122	0.0935576
$956$	0	0
$957$	−3.32956	−0.107630
$958$	0	0
$959$	−81.0450	−2.61708
$960$	0	0
$961$	−30.2498	−0.975800
$962$	0	0
$963$	−31.1758	−1.00463
$964$	0	0
$965$	−4.99704	−0.160860
$966$	0	0
$967$	−34.9576	−1.12416	−0.562080	−	0.827083i	$-0.689999\pi$
$967$	−34.9576	−1.12416	−0.562080	+	0.827083i	$0.689999\pi$
$968$	0	0
$969$	1.79611	0.0576993
$970$	0	0
$971$	−27.8485	−0.893700	−0.446850	−	0.894609i	$-0.647454\pi$
$971$	−27.8485	−0.893700	−0.446850	+	0.894609i	$0.647454\pi$
$972$	0	0
$973$	−88.3818	−2.83339
$974$	0	0
$975$	−0.358358	−0.0114766
$976$	0	0
$977$	−6.52941	−0.208894	−0.104447	−	0.994530i	$-0.533307\pi$
$977$	−6.52941	−0.208894	−0.104447	+	0.994530i	$0.533307\pi$
$978$	0	0
$979$	48.8205	1.56031
$980$	0	0
$981$	17.2309	0.550141
$982$	0	0
$983$	9.75863	0.311252	0.155626	−	0.987816i	$-0.450261\pi$
$983$	9.75863	0.311252	0.155626	+	0.987816i	$0.450261\pi$
$984$	0	0
$985$	8.69222	0.276957
$986$	0	0
$987$	15.1363	0.481793
$988$	0	0
$989$	14.0756	0.447576
$990$	0	0
$991$	30.7301	0.976174	0.488087	−	0.872795i	$-0.337695\pi$
$991$	30.7301	0.976174	0.488087	+	0.872795i	$0.337695\pi$
$992$	0	0
$993$	−4.10186	−0.130169
$994$	0	0
$995$	6.54847	0.207600
$996$	0	0
$997$	45.0757	1.42756	0.713781	−	0.700369i	$-0.246981\pi$
$997$	45.0757	1.42756	0.713781	+	0.700369i	$0.246981\pi$
$998$	0	0
$999$	10.2129	0.323121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.7	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.7	✓	19	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q-0.828283 q^{3} +0.378111 q^{5} -3.91567 q^{7} -2.31395 q^{9} +O(q^{10})\)	\(q-0.828283 q^{3} +0.378111 q^{5} -3.91567 q^{7} -2.31395 q^{9} +4.10431 q^{11} -0.0890774 q^{13} -0.313183 q^{15} +2.16847 q^{17} -1.00000 q^{19} +3.24328 q^{21} -3.36192 q^{23} -4.85703 q^{25} +4.40145 q^{27} +0.979419 q^{29} +0.866138 q^{31} -3.39953 q^{33} -1.48056 q^{35} +2.32034 q^{37} +0.0737813 q^{39} -8.03809 q^{41} -4.18676 q^{43} -0.874929 q^{45} +4.66697 q^{47} +8.33247 q^{49} -1.79611 q^{51} -1.00000 q^{53} +1.55189 q^{55} +0.828283 q^{57} -0.0391736 q^{59} -12.6790 q^{61} +9.06066 q^{63} -0.0336811 q^{65} -2.22129 q^{67} +2.78462 q^{69} +14.5178 q^{71} -5.23443 q^{73} +4.02300 q^{75} -16.0711 q^{77} -4.14237 q^{79} +3.29620 q^{81} +6.30308 q^{83} +0.819923 q^{85} -0.811236 q^{87} +11.8949 q^{89} +0.348798 q^{91} -0.717407 q^{93} -0.378111 q^{95} +2.09440 q^{97} -9.49717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more