LMFDB - Embedded newform 4028.2.a.f.1.2

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			1.2
Root			$-2.63586$ 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)

$f(q)$	$=$	$q-2.63586 q^{3} -2.63773 q^{5} -2.07266 q^{7} +3.94775 q^{9} +O(q^{10})$	$q-2.63586 q^{3} -2.63773 q^{5} -2.07266 q^{7} +3.94775 q^{9} -0.242814 q^{11} +2.69158 q^{13} +6.95269 q^{15} -1.25625 q^{17} -1.00000 q^{19} +5.46324 q^{21} -1.78981 q^{23} +1.95763 q^{25} -2.49813 q^{27} -7.92876 q^{29} -3.30730 q^{31} +0.640023 q^{33} +5.46712 q^{35} -10.9102 q^{37} -7.09461 q^{39} -7.94545 q^{41} +7.02769 q^{43} -10.4131 q^{45} -1.81990 q^{47} -2.70408 q^{49} +3.31129 q^{51} -1.00000 q^{53} +0.640478 q^{55} +2.63586 q^{57} +3.44995 q^{59} -11.3553 q^{61} -8.18234 q^{63} -7.09966 q^{65} +1.35536 q^{67} +4.71767 q^{69} -5.54018 q^{71} -8.98019 q^{73} -5.16003 q^{75} +0.503271 q^{77} +6.67097 q^{79} -5.25854 q^{81} -9.43314 q^{83} +3.31365 q^{85} +20.8991 q^{87} -9.13237 q^{89} -5.57872 q^{91} +8.71756 q^{93} +2.63773 q^{95} -1.58967 q^{97} -0.958569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.63586	−1.52181	−0.760907	−	0.648861i	$-0.775245\pi$
$3$	−2.63586	−1.52181	−0.760907	+	0.648861i	$0.775245\pi$
$4$	0	0
$5$	−2.63773	−1.17963	−0.589815	−	0.807539i	$-0.700799\pi$
$5$	−2.63773	−1.17963	−0.589815	+	0.807539i	$0.700799\pi$
$6$	0	0
$7$	−2.07266	−0.783392	−0.391696	−	0.920095i	$-0.628112\pi$
$7$	−2.07266	−0.783392	−0.391696	+	0.920095i	$0.628112\pi$
$8$	0	0
$9$	3.94775	1.31592
$10$	0	0
$11$	−0.242814	−0.0732112	−0.0366056	−	0.999330i	$-0.511655\pi$
$11$	−0.242814	−0.0732112	−0.0366056	+	0.999330i	$0.511655\pi$
$12$	0	0
$13$	2.69158	0.746509	0.373254	−	0.927729i	$-0.378242\pi$
$13$	2.69158	0.746509	0.373254	+	0.927729i	$0.378242\pi$
$14$	0	0
$15$	6.95269	1.79518
$16$	0	0
$17$	−1.25625	−0.304685	−0.152342	−	0.988328i	$-0.548682\pi$
$17$	−1.25625	−0.304685	−0.152342	+	0.988328i	$0.548682\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.46324	1.19218
$22$	0	0
$23$	−1.78981	−0.373200	−0.186600	−	0.982436i	$-0.559747\pi$
$23$	−1.78981	−0.373200	−0.186600	+	0.982436i	$0.559747\pi$
$24$	0	0
$25$	1.95763	0.391526
$26$	0	0
$27$	−2.49813	−0.480764
$28$	0	0
$29$	−7.92876	−1.47233	−0.736167	−	0.676800i	$-0.763366\pi$
$29$	−7.92876	−1.47233	−0.736167	+	0.676800i	$0.763366\pi$
$30$	0	0
$31$	−3.30730	−0.594008	−0.297004	−	0.954876i	$-0.595987\pi$
$31$	−3.30730	−0.594008	−0.297004	+	0.954876i	$0.595987\pi$
$32$	0	0
$33$	0.640023	0.111414
$34$	0	0
$35$	5.46712	0.924112
$36$	0	0
$37$	−10.9102	−1.79363	−0.896814	−	0.442407i	$-0.854125\pi$
$37$	−10.9102	−1.79363	−0.896814	+	0.442407i	$0.854125\pi$
$38$	0	0
$39$	−7.09461	−1.13605
$40$	0	0
$41$	−7.94545	−1.24087	−0.620435	−	0.784258i	$-0.713044\pi$
$41$	−7.94545	−1.24087	−0.620435	+	0.784258i	$0.713044\pi$
$42$	0	0
$43$	7.02769	1.07171	0.535856	−	0.844309i	$-0.319989\pi$
$43$	7.02769	1.07171	0.535856	+	0.844309i	$0.319989\pi$
$44$	0	0
$45$	−10.4131	−1.55229
$46$	0	0
$47$	−1.81990	−0.265460	−0.132730	−	0.991152i	$-0.542374\pi$
$47$	−1.81990	−0.265460	−0.132730	+	0.991152i	$0.542374\pi$
$48$	0	0
$49$	−2.70408	−0.386297
$50$	0	0
$51$	3.31129	0.463674
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	0.640478	0.0863621
$56$	0	0
$57$	2.63586	0.349128
$58$	0	0
$59$	3.44995	0.449145	0.224572	−	0.974457i	$-0.427901\pi$
$59$	3.44995	0.449145	0.224572	+	0.974457i	$0.427901\pi$
$60$	0	0
$61$	−11.3553	−1.45389	−0.726945	−	0.686695i	$-0.759061\pi$
$61$	−11.3553	−1.45389	−0.726945	+	0.686695i	$0.759061\pi$
$62$	0	0
$63$	−8.18234	−1.03088
$64$	0	0
$65$	−7.09966	−0.880604
$66$	0	0
$67$	1.35536	0.165584	0.0827918	−	0.996567i	$-0.473616\pi$
$67$	1.35536	0.165584	0.0827918	+	0.996567i	$0.473616\pi$
$68$	0	0
$69$	4.71767	0.567941
$70$	0	0
$71$	−5.54018	−0.657499	−0.328749	−	0.944417i	$-0.606627\pi$
$71$	−5.54018	−0.657499	−0.328749	+	0.944417i	$0.606627\pi$
$72$	0	0
$73$	−8.98019	−1.05105	−0.525526	−	0.850778i	$-0.676131\pi$
$73$	−8.98019	−1.05105	−0.525526	+	0.850778i	$0.676131\pi$
$74$	0	0
$75$	−5.16003	−0.595829
$76$	0	0
$77$	0.503271	0.0573531
$78$	0	0
$79$	6.67097	0.750542	0.375271	−	0.926915i	$-0.377550\pi$
$79$	6.67097	0.750542	0.375271	+	0.926915i	$0.377550\pi$
$80$	0	0
$81$	−5.25854	−0.584282
$82$	0	0
$83$	−9.43314	−1.03542	−0.517711	−	0.855556i	$-0.673216\pi$
$83$	−9.43314	−1.03542	−0.517711	+	0.855556i	$0.673216\pi$
$84$	0	0
$85$	3.31365	0.359415
$86$	0	0
$87$	20.8991	2.24062
$88$	0	0
$89$	−9.13237	−0.968029	−0.484015	−	0.875060i	$-0.660822\pi$
$89$	−9.13237	−0.968029	−0.484015	+	0.875060i	$0.660822\pi$
$90$	0	0
$91$	−5.57872	−0.584809
$92$	0	0
$93$	8.71756	0.903969
$94$	0	0
$95$	2.63773	0.270626
$96$	0	0
$97$	−1.58967	−0.161407	−0.0807033	−	0.996738i	$-0.525717\pi$
$97$	−1.58967	−0.161407	−0.0807033	+	0.996738i	$0.525717\pi$
$98$	0	0
$99$	−0.958569	−0.0963398
$100$	0	0
$101$	14.3338	1.42627	0.713134	−	0.701028i	$-0.247275\pi$
$101$	14.3338	1.42627	0.713134	+	0.701028i	$0.247275\pi$
$102$	0	0
$103$	5.47831	0.539794	0.269897	−	0.962889i	$-0.413010\pi$
$103$	5.47831	0.539794	0.269897	+	0.962889i	$0.413010\pi$
$104$	0	0
$105$	−14.4106	−1.40633
$106$	0	0
$107$	2.01903	0.195187	0.0975933	−	0.995226i	$-0.468886\pi$
$107$	2.01903	0.195187	0.0975933	+	0.995226i	$0.468886\pi$
$108$	0	0
$109$	17.6950	1.69487	0.847437	−	0.530896i	$-0.178145\pi$
$109$	17.6950	1.69487	0.847437	+	0.530896i	$0.178145\pi$
$110$	0	0
$111$	28.7578	2.72957
$112$	0	0
$113$	−8.29809	−0.780619	−0.390309	−	0.920684i	$-0.627632\pi$
$113$	−8.29809	−0.780619	−0.390309	+	0.920684i	$0.627632\pi$
$114$	0	0
$115$	4.72103	0.440238
$116$	0	0
$117$	10.6257	0.982343
$118$	0	0
$119$	2.60378	0.238688
$120$	0	0
$121$	−10.9410	−0.994640
$122$	0	0
$123$	20.9431	1.88837
$124$	0	0
$125$	8.02496	0.717774
$126$	0	0
$127$	8.47952	0.752436	0.376218	−	0.926531i	$-0.377224\pi$
$127$	8.47952	0.752436	0.376218	+	0.926531i	$0.377224\pi$
$128$	0	0
$129$	−18.5240	−1.63095
$130$	0	0
$131$	−12.6442	−1.10473	−0.552364	−	0.833603i	$-0.686274\pi$
$131$	−12.6442	−1.10473	−0.552364	+	0.833603i	$0.686274\pi$
$132$	0	0
$133$	2.07266	0.179722
$134$	0	0
$135$	6.58938	0.567124
$136$	0	0
$137$	−0.502502	−0.0429317	−0.0214658	−	0.999770i	$-0.506833\pi$
$137$	−0.502502	−0.0429317	−0.0214658	+	0.999770i	$0.506833\pi$
$138$	0	0
$139$	9.35770	0.793710	0.396855	−	0.917881i	$-0.370102\pi$
$139$	9.35770	0.793710	0.396855	+	0.917881i	$0.370102\pi$
$140$	0	0
$141$	4.79701	0.403981
$142$	0	0
$143$	−0.653553	−0.0546528
$144$	0	0
$145$	20.9139	1.73681
$146$	0	0
$147$	7.12757	0.587872
$148$	0	0
$149$	17.0368	1.39571	0.697853	−	0.716241i	$-0.254139\pi$
$149$	17.0368	1.39571	0.697853	+	0.716241i	$0.254139\pi$
$150$	0	0
$151$	10.1826	0.828647	0.414323	−	0.910130i	$-0.364018\pi$
$151$	10.1826	0.828647	0.414323	+	0.910130i	$0.364018\pi$
$152$	0	0
$153$	−4.95935	−0.400940
$154$	0	0
$155$	8.72376	0.700709
$156$	0	0
$157$	−3.38988	−0.270542	−0.135271	−	0.990809i	$-0.543190\pi$
$157$	−3.38988	−0.270542	−0.135271	+	0.990809i	$0.543190\pi$
$158$	0	0
$159$	2.63586	0.209037
$160$	0	0
$161$	3.70966	0.292362
$162$	0	0
$163$	−7.35412	−0.576019	−0.288010	−	0.957627i	$-0.592994\pi$
$163$	−7.35412	−0.576019	−0.288010	+	0.957627i	$0.592994\pi$
$164$	0	0
$165$	−1.68821	−0.131427
$166$	0	0
$167$	−15.5234	−1.20123	−0.600617	−	0.799537i	$-0.705078\pi$
$167$	−15.5234	−1.20123	−0.600617	+	0.799537i	$0.705078\pi$
$168$	0	0
$169$	−5.75542	−0.442725
$170$	0	0
$171$	−3.94775	−0.301892
$172$	0	0
$173$	2.44929	0.186216	0.0931082	−	0.995656i	$-0.470320\pi$
$173$	2.44929	0.186216	0.0931082	+	0.995656i	$0.470320\pi$
$174$	0	0
$175$	−4.05750	−0.306718
$176$	0	0
$177$	−9.09357	−0.683515
$178$	0	0
$179$	−8.43608	−0.630542	−0.315271	−	0.949002i	$-0.602095\pi$
$179$	−8.43608	−0.630542	−0.315271	+	0.949002i	$0.602095\pi$
$180$	0	0
$181$	18.0591	1.34232	0.671161	−	0.741312i	$-0.265796\pi$
$181$	18.0591	1.34232	0.671161	+	0.741312i	$0.265796\pi$
$182$	0	0
$183$	29.9308	2.21255
$184$	0	0
$185$	28.7782	2.11582
$186$	0	0
$187$	0.305035	0.0223064
$188$	0	0
$189$	5.17777	0.376627
$190$	0	0
$191$	−5.41107	−0.391531	−0.195766	−	0.980651i	$-0.562719\pi$
$191$	−5.41107	−0.391531	−0.195766	+	0.980651i	$0.562719\pi$
$192$	0	0
$193$	15.8725	1.14253	0.571263	−	0.820767i	$-0.306454\pi$
$193$	15.8725	1.14253	0.571263	+	0.820767i	$0.306454\pi$
$194$	0	0
$195$	18.7137	1.34011
$196$	0	0
$197$	−16.1515	−1.15075	−0.575374	−	0.817891i	$-0.695143\pi$
$197$	−16.1515	−1.15075	−0.575374	+	0.817891i	$0.695143\pi$
$198$	0	0
$199$	−10.6999	−0.758497	−0.379248	−	0.925295i	$-0.623817\pi$
$199$	−10.6999	−0.758497	−0.379248	+	0.925295i	$0.623817\pi$
$200$	0	0
$201$	−3.57254	−0.251987
$202$	0	0
$203$	16.4336	1.15341
$204$	0	0
$205$	20.9580	1.46377
$206$	0	0
$207$	−7.06570	−0.491100
$208$	0	0
$209$	0.242814	0.0167958
$210$	0	0
$211$	4.70025	0.323578	0.161789	−	0.986825i	$-0.448274\pi$
$211$	4.70025	0.323578	0.161789	+	0.986825i	$0.448274\pi$
$212$	0	0
$213$	14.6031	1.00059
$214$	0	0
$215$	−18.5372	−1.26422
$216$	0	0
$217$	6.85490	0.465341
$218$	0	0
$219$	23.6705	1.59950
$220$	0	0
$221$	−3.38129	−0.227450
$222$	0	0
$223$	−6.77024	−0.453369	−0.226684	−	0.973968i	$-0.572789\pi$
$223$	−6.77024	−0.453369	−0.226684	+	0.973968i	$0.572789\pi$
$224$	0	0
$225$	7.72822	0.515215
$226$	0	0
$227$	13.3031	0.882955	0.441478	−	0.897272i	$-0.354454\pi$
$227$	13.3031	0.882955	0.441478	+	0.897272i	$0.354454\pi$
$228$	0	0
$229$	4.05862	0.268201	0.134101	−	0.990968i	$-0.457185\pi$
$229$	4.05862	0.268201	0.134101	+	0.990968i	$0.457185\pi$
$230$	0	0
$231$	−1.32655	−0.0872807
$232$	0	0
$233$	15.2200	0.997097	0.498548	−	0.866862i	$-0.333867\pi$
$233$	15.2200	0.997097	0.498548	+	0.866862i	$0.333867\pi$
$234$	0	0
$235$	4.80042	0.313145
$236$	0	0
$237$	−17.5837	−1.14219
$238$	0	0
$239$	8.41031	0.544018	0.272009	−	0.962295i	$-0.412312\pi$
$239$	8.41031	0.544018	0.272009	+	0.962295i	$0.412312\pi$
$240$	0	0
$241$	−25.7918	−1.66139	−0.830697	−	0.556725i	$-0.812058\pi$
$241$	−25.7918	−1.66139	−0.830697	+	0.556725i	$0.812058\pi$
$242$	0	0
$243$	21.3551	1.36993
$244$	0	0
$245$	7.13263	0.455687
$246$	0	0
$247$	−2.69158	−0.171261
$248$	0	0
$249$	24.8644	1.57572
$250$	0	0
$251$	−28.5930	−1.80478	−0.902388	−	0.430924i	$-0.858188\pi$
$251$	−28.5930	−1.80478	−0.902388	+	0.430924i	$0.858188\pi$
$252$	0	0
$253$	0.434590	0.0273225
$254$	0	0
$255$	−8.73430	−0.546963
$256$	0	0
$257$	−14.2387	−0.888187	−0.444093	−	0.895980i	$-0.646474\pi$
$257$	−14.2387	−0.888187	−0.444093	+	0.895980i	$0.646474\pi$
$258$	0	0
$259$	22.6132	1.40511
$260$	0	0
$261$	−31.3007	−1.93747
$262$	0	0
$263$	8.22574	0.507221	0.253610	−	0.967306i	$-0.418382\pi$
$263$	8.22574	0.507221	0.253610	+	0.967306i	$0.418382\pi$
$264$	0	0
$265$	2.63773	0.162035
$266$	0	0
$267$	24.0716	1.47316
$268$	0	0
$269$	−10.9419	−0.667138	−0.333569	−	0.942726i	$-0.608253\pi$
$269$	−10.9419	−0.667138	−0.333569	+	0.942726i	$0.608253\pi$
$270$	0	0
$271$	26.1119	1.58618	0.793092	−	0.609101i	$-0.208470\pi$
$271$	26.1119	1.58618	0.793092	+	0.609101i	$0.208470\pi$
$272$	0	0
$273$	14.7047	0.889970
$274$	0	0
$275$	−0.475340	−0.0286641
$276$	0	0
$277$	10.2742	0.617315	0.308657	−	0.951173i	$-0.400120\pi$
$277$	10.2742	0.617315	0.308657	+	0.951173i	$0.400120\pi$
$278$	0	0
$279$	−13.0564	−0.781664
$280$	0	0
$281$	−0.0376216	−0.00224432	−0.00112216	−	0.999999i	$-0.500357\pi$
$281$	−0.0376216	−0.00224432	−0.00112216	+	0.999999i	$0.500357\pi$
$282$	0	0
$283$	14.7233	0.875212	0.437606	−	0.899167i	$-0.355827\pi$
$283$	14.7233	0.875212	0.437606	+	0.899167i	$0.355827\pi$
$284$	0	0
$285$	−6.95269	−0.411842
$286$	0	0
$287$	16.4682	0.972088
$288$	0	0
$289$	−15.4218	−0.907167
$290$	0	0
$291$	4.19014	0.245631
$292$	0	0
$293$	−8.28325	−0.483912	−0.241956	−	0.970287i	$-0.577789\pi$
$293$	−8.28325	−0.483912	−0.241956	+	0.970287i	$0.577789\pi$
$294$	0	0
$295$	−9.10003	−0.529825
$296$	0	0
$297$	0.606580	0.0351973
$298$	0	0
$299$	−4.81740	−0.278597
$300$	0	0
$301$	−14.5660	−0.839571
$302$	0	0
$303$	−37.7819	−2.17051
$304$	0	0
$305$	29.9521	1.71505
$306$	0	0
$307$	27.5521	1.57248	0.786242	−	0.617919i	$-0.212024\pi$
$307$	27.5521	1.57248	0.786242	+	0.617919i	$0.212024\pi$
$308$	0	0
$309$	−14.4400	−0.821465
$310$	0	0
$311$	9.58731	0.543647	0.271823	−	0.962347i	$-0.412373\pi$
$311$	9.58731	0.543647	0.271823	+	0.962347i	$0.412373\pi$
$312$	0	0
$313$	−15.2168	−0.860103	−0.430052	−	0.902804i	$-0.641505\pi$
$313$	−15.2168	−0.860103	−0.430052	+	0.902804i	$0.641505\pi$
$314$	0	0
$315$	21.5828	1.21605
$316$	0	0
$317$	18.7580	1.05355	0.526776	−	0.850004i	$-0.323401\pi$
$317$	18.7580	1.05355	0.526776	+	0.850004i	$0.323401\pi$
$318$	0	0
$319$	1.92522	0.107791
$320$	0	0
$321$	−5.32187	−0.297038
$322$	0	0
$323$	1.25625	0.0698995
$324$	0	0
$325$	5.26911	0.292277
$326$	0	0
$327$	−46.6415	−2.57928
$328$	0	0
$329$	3.77204	0.207959
$330$	0	0
$331$	−19.1177	−1.05081	−0.525403	−	0.850853i	$-0.676086\pi$
$331$	−19.1177	−1.05081	−0.525403	+	0.850853i	$0.676086\pi$
$332$	0	0
$333$	−43.0708	−2.36026
$334$	0	0
$335$	−3.57508	−0.195327
$336$	0	0
$337$	7.07410	0.385351	0.192675	−	0.981263i	$-0.438284\pi$
$337$	7.07410	0.385351	0.192675	+	0.981263i	$0.438284\pi$
$338$	0	0
$339$	21.8726	1.18796
$340$	0	0
$341$	0.803058	0.0434880
$342$	0	0
$343$	20.1133	1.08601
$344$	0	0
$345$	−12.4440	−0.669960
$346$	0	0
$347$	2.06499	0.110854	0.0554272	−	0.998463i	$-0.482348\pi$
$347$	2.06499	0.110854	0.0554272	+	0.998463i	$0.482348\pi$
$348$	0	0
$349$	−16.9587	−0.907780	−0.453890	−	0.891058i	$-0.649964\pi$
$349$	−16.9587	−0.907780	−0.453890	+	0.891058i	$0.649964\pi$
$350$	0	0
$351$	−6.72389	−0.358895
$352$	0	0
$353$	31.5585	1.67969	0.839844	−	0.542827i	$-0.182646\pi$
$353$	31.5585	1.67969	0.839844	+	0.542827i	$0.182646\pi$
$354$	0	0
$355$	14.6135	0.775605
$356$	0	0
$357$	−6.86318	−0.363238
$358$	0	0
$359$	−1.64169	−0.0866453	−0.0433227	−	0.999061i	$-0.513794\pi$
$359$	−1.64169	−0.0866453	−0.0433227	+	0.999061i	$0.513794\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.8390	1.51366
$364$	0	0
$365$	23.6873	1.23985
$366$	0	0
$367$	14.0121	0.731427	0.365713	−	0.930728i	$-0.380825\pi$
$367$	14.0121	0.731427	0.365713	+	0.930728i	$0.380825\pi$
$368$	0	0
$369$	−31.3666	−1.63288
$370$	0	0
$371$	2.07266	0.107607
$372$	0	0
$373$	14.2902	0.739917	0.369958	−	0.929048i	$-0.379372\pi$
$373$	14.2902	0.739917	0.369958	+	0.929048i	$0.379372\pi$
$374$	0	0
$375$	−21.1527	−1.09232
$376$	0	0
$377$	−21.3409	−1.09911
$378$	0	0
$379$	35.1611	1.80610	0.903051	−	0.429533i	$-0.141322\pi$
$379$	35.1611	1.80610	0.903051	+	0.429533i	$0.141322\pi$
$380$	0	0
$381$	−22.3508	−1.14507
$382$	0	0
$383$	8.59929	0.439403	0.219702	−	0.975567i	$-0.429492\pi$
$383$	8.59929	0.439403	0.219702	+	0.975567i	$0.429492\pi$
$384$	0	0
$385$	−1.32749	−0.0676554
$386$	0	0
$387$	27.7435	1.41028
$388$	0	0
$389$	19.3780	0.982505	0.491252	−	0.871017i	$-0.336539\pi$
$389$	19.3780	0.982505	0.491252	+	0.871017i	$0.336539\pi$
$390$	0	0
$391$	2.24844	0.113709
$392$	0	0
$393$	33.3283	1.68119
$394$	0	0
$395$	−17.5962	−0.885362
$396$	0	0
$397$	2.52327	0.126639	0.0633197	−	0.997993i	$-0.479831\pi$
$397$	2.52327	0.126639	0.0633197	+	0.997993i	$0.479831\pi$
$398$	0	0
$399$	−5.46324	−0.273504
$400$	0	0
$401$	−20.6714	−1.03228	−0.516141	−	0.856503i	$-0.672632\pi$
$401$	−20.6714	−1.03228	−0.516141	+	0.856503i	$0.672632\pi$
$402$	0	0
$403$	−8.90184	−0.443432
$404$	0	0
$405$	13.8706	0.689236
$406$	0	0
$407$	2.64915	0.131314
$408$	0	0
$409$	−13.7174	−0.678281	−0.339140	−	0.940736i	$-0.610136\pi$
$409$	−13.7174	−0.678281	−0.339140	+	0.940736i	$0.610136\pi$
$410$	0	0
$411$	1.32452	0.0653340
$412$	0	0
$413$	−7.15057	−0.351857
$414$	0	0
$415$	24.8821	1.22141
$416$	0	0
$417$	−24.6656	−1.20788
$418$	0	0
$419$	−32.4734	−1.58643	−0.793214	−	0.608943i	$-0.791594\pi$
$419$	−32.4734	−1.58643	−0.793214	+	0.608943i	$0.791594\pi$
$420$	0	0
$421$	−17.4990	−0.852851	−0.426425	−	0.904523i	$-0.640227\pi$
$421$	−17.4990	−0.852851	−0.426425	+	0.904523i	$0.640227\pi$
$422$	0	0
$423$	−7.18452	−0.349323
$424$	0	0
$425$	−2.45927	−0.119292
$426$	0	0
$427$	23.5356	1.13897
$428$	0	0
$429$	1.72267	0.0831714
$430$	0	0
$431$	−12.2712	−0.591082	−0.295541	−	0.955330i	$-0.595500\pi$
$431$	−12.2712	−0.591082	−0.295541	+	0.955330i	$0.595500\pi$
$432$	0	0
$433$	−28.8543	−1.38665	−0.693325	−	0.720625i	$-0.743855\pi$
$433$	−28.8543	−1.38665	−0.693325	+	0.720625i	$0.743855\pi$
$434$	0	0
$435$	−55.1262	−2.64310
$436$	0	0
$437$	1.78981	0.0856180
$438$	0	0
$439$	−34.2011	−1.63233	−0.816166	−	0.577818i	$-0.803904\pi$
$439$	−34.2011	−1.63233	−0.816166	+	0.577818i	$0.803904\pi$
$440$	0	0
$441$	−10.6750	−0.508334
$442$	0	0
$443$	−7.25383	−0.344640	−0.172320	−	0.985041i	$-0.555126\pi$
$443$	−7.25383	−0.344640	−0.172320	+	0.985041i	$0.555126\pi$
$444$	0	0
$445$	24.0887	1.14192
$446$	0	0
$447$	−44.9065	−2.12400
$448$	0	0
$449$	14.2512	0.672555	0.336278	−	0.941763i	$-0.390832\pi$
$449$	14.2512	0.672555	0.336278	+	0.941763i	$0.390832\pi$
$450$	0	0
$451$	1.92927	0.0908456
$452$	0	0
$453$	−26.8398	−1.26105
$454$	0	0
$455$	14.7152	0.689858
$456$	0	0
$457$	31.4773	1.47245	0.736223	−	0.676739i	$-0.236607\pi$
$457$	31.4773	1.47245	0.736223	+	0.676739i	$0.236607\pi$
$458$	0	0
$459$	3.13827	0.146482
$460$	0	0
$461$	8.91003	0.414981	0.207491	−	0.978237i	$-0.433470\pi$
$461$	8.91003	0.414981	0.207491	+	0.978237i	$0.433470\pi$
$462$	0	0
$463$	18.1183	0.842030	0.421015	−	0.907054i	$-0.361674\pi$
$463$	18.1183	0.842030	0.421015	+	0.907054i	$0.361674\pi$
$464$	0	0
$465$	−22.9946	−1.06635
$466$	0	0
$467$	40.4175	1.87030	0.935149	−	0.354255i	$-0.115266\pi$
$467$	40.4175	1.87030	0.935149	+	0.354255i	$0.115266\pi$
$468$	0	0
$469$	−2.80920	−0.129717
$470$	0	0
$471$	8.93523	0.411714
$472$	0	0
$473$	−1.70642	−0.0784614
$474$	0	0
$475$	−1.95763	−0.0898222
$476$	0	0
$477$	−3.94775	−0.180755
$478$	0	0
$479$	−12.6930	−0.579959	−0.289980	−	0.957033i	$-0.593649\pi$
$479$	−12.6930	−0.579959	−0.289980	+	0.957033i	$0.593649\pi$
$480$	0	0
$481$	−29.3657	−1.33896
$482$	0	0
$483$	−9.77814	−0.444921
$484$	0	0
$485$	4.19312	0.190400
$486$	0	0
$487$	26.2589	1.18990	0.594952	−	0.803761i	$-0.297171\pi$
$487$	26.2589	1.18990	0.594952	+	0.803761i	$0.297171\pi$
$488$	0	0
$489$	19.3844	0.876594
$490$	0	0
$491$	−15.8449	−0.715071	−0.357535	−	0.933900i	$-0.616383\pi$
$491$	−15.8449	−0.715071	−0.357535	+	0.933900i	$0.616383\pi$
$492$	0	0
$493$	9.96049	0.448598
$494$	0	0
$495$	2.52845	0.113645
$496$	0	0
$497$	11.4829	0.515079
$498$	0	0
$499$	19.2622	0.862296	0.431148	−	0.902281i	$-0.358109\pi$
$499$	19.2622	0.862296	0.431148	+	0.902281i	$0.358109\pi$
$500$	0	0
$501$	40.9174	1.82805
$502$	0	0
$503$	−2.39550	−0.106810	−0.0534051	−	0.998573i	$-0.517007\pi$
$503$	−2.39550	−0.106810	−0.0534051	+	0.998573i	$0.517007\pi$
$504$	0	0
$505$	−37.8087	−1.68247
$506$	0	0
$507$	15.1705	0.673744
$508$	0	0
$509$	−39.8086	−1.76449	−0.882243	−	0.470794i	$-0.843967\pi$
$509$	−39.8086	−1.76449	−0.882243	+	0.470794i	$0.843967\pi$
$510$	0	0
$511$	18.6129	0.823385
$512$	0	0
$513$	2.49813	0.110295
$514$	0	0
$515$	−14.4503	−0.636757
$516$	0	0
$517$	0.441898	0.0194347
$518$	0	0
$519$	−6.45599	−0.283387
$520$	0	0
$521$	16.3142	0.714740	0.357370	−	0.933963i	$-0.383674\pi$
$521$	16.3142	0.714740	0.357370	+	0.933963i	$0.383674\pi$
$522$	0	0
$523$	−32.7364	−1.43146	−0.715731	−	0.698376i	$-0.753906\pi$
$523$	−32.7364	−1.43146	−0.715731	+	0.698376i	$0.753906\pi$
$524$	0	0
$525$	10.6950	0.466768
$526$	0	0
$527$	4.15479	0.180985
$528$	0	0
$529$	−19.7966	−0.860721
$530$	0	0
$531$	13.6195	0.591037
$532$	0	0
$533$	−21.3858	−0.926321
$534$	0	0
$535$	−5.32565	−0.230248
$536$	0	0
$537$	22.2363	0.959568
$538$	0	0
$539$	0.656588	0.0282813
$540$	0	0
$541$	3.29786	0.141786	0.0708931	−	0.997484i	$-0.477415\pi$
$541$	3.29786	0.141786	0.0708931	+	0.997484i	$0.477415\pi$
$542$	0	0
$543$	−47.6012	−2.04276
$544$	0	0
$545$	−46.6747	−1.99932
$546$	0	0
$547$	11.7057	0.500501	0.250250	−	0.968181i	$-0.419487\pi$
$547$	11.7057	0.500501	0.250250	+	0.968181i	$0.419487\pi$
$548$	0	0
$549$	−44.8277	−1.91320
$550$	0	0
$551$	7.92876	0.337777
$552$	0	0
$553$	−13.8266	−0.587969
$554$	0	0
$555$	−75.8553	−3.21988
$556$	0	0
$557$	32.5814	1.38052	0.690259	−	0.723562i	$-0.257497\pi$
$557$	32.5814	1.38052	0.690259	+	0.723562i	$0.257497\pi$
$558$	0	0
$559$	18.9156	0.800043
$560$	0	0
$561$	−0.804028	−0.0339461
$562$	0	0
$563$	26.0778	1.09905	0.549523	−	0.835478i	$-0.314809\pi$
$563$	26.0778	1.09905	0.549523	+	0.835478i	$0.314809\pi$
$564$	0	0
$565$	21.8881	0.920841
$566$	0	0
$567$	10.8992	0.457722
$568$	0	0
$569$	−9.58472	−0.401812	−0.200906	−	0.979610i	$-0.564389\pi$
$569$	−9.58472	−0.401812	−0.200906	+	0.979610i	$0.564389\pi$
$570$	0	0
$571$	22.2935	0.932956	0.466478	−	0.884533i	$-0.345523\pi$
$571$	22.2935	0.932956	0.466478	+	0.884533i	$0.345523\pi$
$572$	0	0
$573$	14.2628	0.595837
$574$	0	0
$575$	−3.50378	−0.146118
$576$	0	0
$577$	26.3503	1.09698	0.548490	−	0.836157i	$-0.315203\pi$
$577$	26.3503	1.09698	0.548490	+	0.836157i	$0.315203\pi$
$578$	0	0
$579$	−41.8376	−1.73871
$580$	0	0
$581$	19.5517	0.811141
$582$	0	0
$583$	0.242814	0.0100563
$584$	0	0
$585$	−28.0276	−1.15880
$586$	0	0
$587$	18.2999	0.755317	0.377659	−	0.925945i	$-0.376729\pi$
$587$	18.2999	0.755317	0.377659	+	0.925945i	$0.376729\pi$
$588$	0	0
$589$	3.30730	0.136275
$590$	0	0
$591$	42.5731	1.75122
$592$	0	0
$593$	−12.5775	−0.516497	−0.258248	−	0.966079i	$-0.583145\pi$
$593$	−12.5775	−0.516497	−0.258248	+	0.966079i	$0.583145\pi$
$594$	0	0
$595$	−6.86806	−0.281563
$596$	0	0
$597$	28.2035	1.15429
$598$	0	0
$599$	21.2907	0.869914	0.434957	−	0.900451i	$-0.356764\pi$
$599$	21.2907	0.869914	0.434957	+	0.900451i	$0.356764\pi$
$600$	0	0
$601$	13.8966	0.566855	0.283427	−	0.958994i	$-0.408528\pi$
$601$	13.8966	0.566855	0.283427	+	0.958994i	$0.408528\pi$
$602$	0	0
$603$	5.35062	0.217894
$604$	0	0
$605$	28.8595	1.17331
$606$	0	0
$607$	17.2176	0.698842	0.349421	−	0.936966i	$-0.386378\pi$
$607$	17.2176	0.698842	0.349421	+	0.936966i	$0.386378\pi$
$608$	0	0
$609$	−43.3167	−1.75528
$610$	0	0
$611$	−4.89841	−0.198168
$612$	0	0
$613$	41.3419	1.66978	0.834892	−	0.550414i	$-0.185530\pi$
$613$	41.3419	1.66978	0.834892	+	0.550414i	$0.185530\pi$
$614$	0	0
$615$	−55.2422	−2.22758
$616$	0	0
$617$	−12.2857	−0.494603	−0.247302	−	0.968939i	$-0.579544\pi$
$617$	−12.2857	−0.494603	−0.247302	+	0.968939i	$0.579544\pi$
$618$	0	0
$619$	−31.3037	−1.25820	−0.629102	−	0.777323i	$-0.716577\pi$
$619$	−31.3037	−1.25820	−0.629102	+	0.777323i	$0.716577\pi$
$620$	0	0
$621$	4.47116	0.179421
$622$	0	0
$623$	18.9283	0.758346
$624$	0	0
$625$	−30.9558	−1.23823
$626$	0	0
$627$	−0.640023	−0.0255601
$628$	0	0
$629$	13.7059	0.546492
$630$	0	0
$631$	−33.7044	−1.34175	−0.670876	−	0.741569i	$-0.734082\pi$
$631$	−33.7044	−1.34175	−0.670876	+	0.741569i	$0.734082\pi$
$632$	0	0
$633$	−12.3892	−0.492426
$634$	0	0
$635$	−22.3667	−0.887595
$636$	0	0
$637$	−7.27823	−0.288374
$638$	0	0
$639$	−21.8712	−0.865213
$640$	0	0
$641$	−17.6936	−0.698857	−0.349428	−	0.936963i	$-0.613624\pi$
$641$	−17.6936	−0.698857	−0.349428	+	0.936963i	$0.613624\pi$
$642$	0	0
$643$	−30.9959	−1.22236	−0.611179	−	0.791492i	$-0.709304\pi$
$643$	−30.9959	−1.22236	−0.611179	+	0.791492i	$0.709304\pi$
$644$	0	0
$645$	48.8613	1.92391
$646$	0	0
$647$	−16.4685	−0.647443	−0.323721	−	0.946152i	$-0.604934\pi$
$647$	−16.4685	−0.647443	−0.323721	+	0.946152i	$0.604934\pi$
$648$	0	0
$649$	−0.837696	−0.0328824
$650$	0	0
$651$	−18.0686	−0.708162
$652$	0	0
$653$	8.52177	0.333483	0.166741	−	0.986001i	$-0.446676\pi$
$653$	8.52177	0.333483	0.166741	+	0.986001i	$0.446676\pi$
$654$	0	0
$655$	33.3520	1.30317
$656$	0	0
$657$	−35.4515	−1.38309
$658$	0	0
$659$	−3.42762	−0.133521	−0.0667606	−	0.997769i	$-0.521266\pi$
$659$	−3.42762	−0.133521	−0.0667606	+	0.997769i	$0.521266\pi$
$660$	0	0
$661$	−4.64060	−0.180499	−0.0902493	−	0.995919i	$-0.528766\pi$
$661$	−4.64060	−0.180499	−0.0902493	+	0.995919i	$0.528766\pi$
$662$	0	0
$663$	8.91259	0.346136
$664$	0	0
$665$	−5.46712	−0.212006
$666$	0	0
$667$	14.1909	0.549476
$668$	0	0
$669$	17.8454	0.689943
$670$	0	0
$671$	2.75721	0.106441
$672$	0	0
$673$	−11.4890	−0.442869	−0.221434	−	0.975175i	$-0.571074\pi$
$673$	−11.4890	−0.442869	−0.221434	+	0.975175i	$0.571074\pi$
$674$	0	0
$675$	−4.89040	−0.188232
$676$	0	0
$677$	−35.4714	−1.36328	−0.681638	−	0.731690i	$-0.738732\pi$
$677$	−35.4714	−1.36328	−0.681638	+	0.731690i	$0.738732\pi$
$678$	0	0
$679$	3.29485	0.126445
$680$	0	0
$681$	−35.0650	−1.34369
$682$	0	0
$683$	−14.9939	−0.573728	−0.286864	−	0.957971i	$-0.592613\pi$
$683$	−14.9939	−0.573728	−0.286864	+	0.957971i	$0.592613\pi$
$684$	0	0
$685$	1.32547	0.0506435
$686$	0	0
$687$	−10.6979	−0.408152
$688$	0	0
$689$	−2.69158	−0.102541
$690$	0	0
$691$	43.8909	1.66969	0.834844	−	0.550487i	$-0.185558\pi$
$691$	43.8909	1.66969	0.834844	+	0.550487i	$0.185558\pi$
$692$	0	0
$693$	1.98679	0.0754718
$694$	0	0
$695$	−24.6831	−0.936284
$696$	0	0
$697$	9.98145	0.378074
$698$	0	0
$699$	−40.1178	−1.51740
$700$	0	0
$701$	−42.9682	−1.62289	−0.811443	−	0.584432i	$-0.801317\pi$
$701$	−42.9682	−1.62289	−0.811443	+	0.584432i	$0.801317\pi$
$702$	0	0
$703$	10.9102	0.411487
$704$	0	0
$705$	−12.6532	−0.476548
$706$	0	0
$707$	−29.7091	−1.11733
$708$	0	0
$709$	1.11964	0.0420490	0.0210245	−	0.999779i	$-0.493307\pi$
$709$	1.11964	0.0420490	0.0210245	+	0.999779i	$0.493307\pi$
$710$	0	0
$711$	26.3353	0.987650
$712$	0	0
$713$	5.91942	0.221684
$714$	0	0
$715$	1.72390	0.0644701
$716$	0	0
$717$	−22.1684	−0.827894
$718$	0	0
$719$	−30.4802	−1.13672	−0.568360	−	0.822780i	$-0.692422\pi$
$719$	−30.4802	−1.13672	−0.568360	+	0.822780i	$0.692422\pi$
$720$	0	0
$721$	−11.3547	−0.422870
$722$	0	0
$723$	67.9834	2.52833
$724$	0	0
$725$	−15.5216	−0.576457
$726$	0	0
$727$	33.2815	1.23434	0.617171	−	0.786829i	$-0.288279\pi$
$727$	33.2815	1.23434	0.617171	+	0.786829i	$0.288279\pi$
$728$	0	0
$729$	−40.5135	−1.50050
$730$	0	0
$731$	−8.82852	−0.326535
$732$	0	0
$733$	−8.77678	−0.324178	−0.162089	−	0.986776i	$-0.551823\pi$
$733$	−8.77678	−0.324178	−0.162089	+	0.986776i	$0.551823\pi$
$734$	0	0
$735$	−18.8006	−0.693471
$736$	0	0
$737$	−0.329101	−0.0121226
$738$	0	0
$739$	18.7675	0.690375	0.345188	−	0.938534i	$-0.387815\pi$
$739$	18.7675	0.690375	0.345188	+	0.938534i	$0.387815\pi$
$740$	0	0
$741$	7.09461	0.260627
$742$	0	0
$743$	21.0744	0.773145	0.386572	−	0.922259i	$-0.373659\pi$
$743$	21.0744	0.773145	0.386572	+	0.922259i	$0.373659\pi$
$744$	0	0
$745$	−44.9384	−1.64642
$746$	0	0
$747$	−37.2397	−1.36253
$748$	0	0
$749$	−4.18476	−0.152908
$750$	0	0
$751$	−23.9860	−0.875260	−0.437630	−	0.899155i	$-0.644182\pi$
$751$	−23.9860	−0.875260	−0.437630	+	0.899155i	$0.644182\pi$
$752$	0	0
$753$	75.3672	2.74653
$754$	0	0
$755$	−26.8589	−0.977496
$756$	0	0
$757$	24.9693	0.907525	0.453763	−	0.891123i	$-0.350081\pi$
$757$	24.9693	0.907525	0.453763	+	0.891123i	$0.350081\pi$
$758$	0	0
$759$	−1.14552	−0.0415797
$760$	0	0
$761$	42.0608	1.52470	0.762351	−	0.647164i	$-0.224045\pi$
$761$	42.0608	1.52470	0.762351	+	0.647164i	$0.224045\pi$
$762$	0	0
$763$	−36.6757	−1.32775
$764$	0	0
$765$	13.0814	0.472960
$766$	0	0
$767$	9.28579	0.335291
$768$	0	0
$769$	33.5960	1.21150	0.605752	−	0.795654i	$-0.292872\pi$
$769$	33.5960	1.21150	0.605752	+	0.795654i	$0.292872\pi$
$770$	0	0
$771$	37.5312	1.35165
$772$	0	0
$773$	20.8594	0.750261	0.375130	−	0.926972i	$-0.377598\pi$
$773$	20.8594	0.750261	0.375130	+	0.926972i	$0.377598\pi$
$774$	0	0
$775$	−6.47446	−0.232569
$776$	0	0
$777$	−59.6051	−2.13832
$778$	0	0
$779$	7.94545	0.284675
$780$	0	0
$781$	1.34523	0.0481363
$782$	0	0
$783$	19.8070	0.707846
$784$	0	0
$785$	8.94158	0.319139
$786$	0	0
$787$	−10.7282	−0.382421	−0.191210	−	0.981549i	$-0.561241\pi$
$787$	−10.7282	−0.382421	−0.191210	+	0.981549i	$0.561241\pi$
$788$	0	0
$789$	−21.6819	−0.771895
$790$	0	0
$791$	17.1991	0.611531
$792$	0	0
$793$	−30.5635	−1.08534
$794$	0	0
$795$	−6.95269	−0.246586
$796$	0	0
$797$	−26.4043	−0.935290	−0.467645	−	0.883916i	$-0.654897\pi$
$797$	−26.4043	−0.935290	−0.467645	+	0.883916i	$0.654897\pi$
$798$	0	0
$799$	2.28625	0.0808818
$800$	0	0
$801$	−36.0523	−1.27384
$802$	0	0
$803$	2.18052	0.0769487
$804$	0	0
$805$	−9.78509	−0.344879
$806$	0	0
$807$	28.8412	1.01526
$808$	0	0
$809$	−17.9924	−0.632577	−0.316289	−	0.948663i	$-0.602437\pi$
$809$	−17.9924	−0.632577	−0.316289	+	0.948663i	$0.602437\pi$
$810$	0	0
$811$	24.3274	0.854250	0.427125	−	0.904193i	$-0.359526\pi$
$811$	24.3274	0.854250	0.427125	+	0.904193i	$0.359526\pi$
$812$	0	0
$813$	−68.8272	−2.41388
$814$	0	0
$815$	19.3982	0.679489
$816$	0	0
$817$	−7.02769	−0.245868
$818$	0	0
$819$	−22.0234	−0.769559
$820$	0	0
$821$	31.7933	1.10959	0.554797	−	0.831986i	$-0.312796\pi$
$821$	31.7933	1.10959	0.554797	+	0.831986i	$0.312796\pi$
$822$	0	0
$823$	32.4067	1.12963	0.564813	−	0.825219i	$-0.308948\pi$
$823$	32.4067	1.12963	0.564813	+	0.825219i	$0.308948\pi$
$824$	0	0
$825$	1.25293	0.0436214
$826$	0	0
$827$	20.0643	0.697703	0.348851	−	0.937178i	$-0.386572\pi$
$827$	20.0643	0.697703	0.348851	+	0.937178i	$0.386572\pi$
$828$	0	0
$829$	−49.6995	−1.72614	−0.863068	−	0.505087i	$-0.831460\pi$
$829$	−49.6995	−1.72614	−0.863068	+	0.505087i	$0.831460\pi$
$830$	0	0
$831$	−27.0812	−0.939438
$832$	0	0
$833$	3.39699	0.117699
$834$	0	0
$835$	40.9465	1.41701
$836$	0	0
$837$	8.26204	0.285578
$838$	0	0
$839$	−20.8192	−0.718758	−0.359379	−	0.933192i	$-0.617011\pi$
$839$	−20.8192	−0.718758	−0.359379	+	0.933192i	$0.617011\pi$
$840$	0	0
$841$	33.8653	1.16777
$842$	0	0
$843$	0.0991652	0.00341543
$844$	0	0
$845$	15.1813	0.522251
$846$	0	0
$847$	22.6771	0.779193
$848$	0	0
$849$	−38.8086	−1.33191
$850$	0	0
$851$	19.5272	0.669383
$852$	0	0
$853$	23.5478	0.806261	0.403130	−	0.915143i	$-0.367922\pi$
$853$	23.5478	0.806261	0.403130	+	0.915143i	$0.367922\pi$
$854$	0	0
$855$	10.4131	0.356120
$856$	0	0
$857$	35.1902	1.20208	0.601038	−	0.799220i	$-0.294754\pi$
$857$	35.1902	1.20208	0.601038	+	0.799220i	$0.294754\pi$
$858$	0	0
$859$	−7.61222	−0.259725	−0.129863	−	0.991532i	$-0.541454\pi$
$859$	−7.61222	−0.259725	−0.129863	+	0.991532i	$0.541454\pi$
$860$	0	0
$861$	−43.4079	−1.47934
$862$	0	0
$863$	−25.1750	−0.856966	−0.428483	−	0.903550i	$-0.640952\pi$
$863$	−25.1750	−0.856966	−0.428483	+	0.903550i	$0.640952\pi$
$864$	0	0
$865$	−6.46058	−0.219666
$866$	0	0
$867$	40.6498	1.38054
$868$	0	0
$869$	−1.61980	−0.0549481
$870$	0	0
$871$	3.64806	0.123610
$872$	0	0
$873$	−6.27561	−0.212397
$874$	0	0
$875$	−16.6330	−0.562299
$876$	0	0
$877$	30.2000	1.01978	0.509891	−	0.860239i	$-0.329686\pi$
$877$	30.2000	1.01978	0.509891	+	0.860239i	$0.329686\pi$
$878$	0	0
$879$	21.8335	0.736424
$880$	0	0
$881$	−26.2513	−0.884428	−0.442214	−	0.896909i	$-0.645807\pi$
$881$	−26.2513	−0.884428	−0.442214	+	0.896909i	$0.645807\pi$
$882$	0	0
$883$	−36.3335	−1.22272	−0.611360	−	0.791353i	$-0.709377\pi$
$883$	−36.3335	−1.22272	−0.611360	+	0.791353i	$0.709377\pi$
$884$	0	0
$885$	23.9864	0.806294
$886$	0	0
$887$	−5.13976	−0.172576	−0.0862881	−	0.996270i	$-0.527501\pi$
$887$	−5.13976	−0.172576	−0.0862881	+	0.996270i	$0.527501\pi$
$888$	0	0
$889$	−17.5752	−0.589452
$890$	0	0
$891$	1.27685	0.0427760
$892$	0	0
$893$	1.81990	0.0609008
$894$	0	0
$895$	22.2521	0.743806
$896$	0	0
$897$	12.6980	0.423973
$898$	0	0
$899$	26.2228	0.874578
$900$	0	0
$901$	1.25625	0.0418517
$902$	0	0
$903$	38.3939	1.27767
$904$	0	0
$905$	−47.6350	−1.58344
$906$	0	0
$907$	−35.5627	−1.18084	−0.590421	−	0.807096i	$-0.701038\pi$
$907$	−35.5627	−1.18084	−0.590421	+	0.807096i	$0.701038\pi$
$908$	0	0
$909$	56.5862	1.87685
$910$	0	0
$911$	−52.6337	−1.74383	−0.871916	−	0.489655i	$-0.837123\pi$
$911$	−52.6337	−1.74383	−0.871916	+	0.489655i	$0.837123\pi$
$912$	0	0
$913$	2.29050	0.0758045
$914$	0	0
$915$	−78.9495	−2.60999
$916$	0	0
$917$	26.2071	0.865435
$918$	0	0
$919$	20.3019	0.669699	0.334850	−	0.942272i	$-0.391315\pi$
$919$	20.3019	0.669699	0.334850	+	0.942272i	$0.391315\pi$
$920$	0	0
$921$	−72.6235	−2.39303
$922$	0	0
$923$	−14.9118	−0.490829
$924$	0	0
$925$	−21.3582	−0.702252
$926$	0	0
$927$	21.6270	0.710323
$928$	0	0
$929$	−32.1380	−1.05441	−0.527207	−	0.849737i	$-0.676761\pi$
$929$	−32.1380	−1.05441	−0.527207	+	0.849737i	$0.676761\pi$
$930$	0	0
$931$	2.70408	0.0886226
$932$	0	0
$933$	−25.2708	−0.827329
$934$	0	0
$935$	−0.804600	−0.0263132
$936$	0	0
$937$	42.8770	1.40073	0.700365	−	0.713785i	$-0.253021\pi$
$937$	42.8770	1.40073	0.700365	+	0.713785i	$0.253021\pi$
$938$	0	0
$939$	40.1093	1.30892
$940$	0	0
$941$	−9.77619	−0.318695	−0.159347	−	0.987223i	$-0.550939\pi$
$941$	−9.77619	−0.318695	−0.159347	+	0.987223i	$0.550939\pi$
$942$	0	0
$943$	14.2208	0.463093
$944$	0	0
$945$	−13.6576	−0.444280
$946$	0	0
$947$	10.3083	0.334976	0.167488	−	0.985874i	$-0.446434\pi$
$947$	10.3083	0.334976	0.167488	+	0.985874i	$0.446434\pi$
$948$	0	0
$949$	−24.1709	−0.784619
$950$	0	0
$951$	−49.4433	−1.60331
$952$	0	0
$953$	−32.1822	−1.04248	−0.521242	−	0.853409i	$-0.674531\pi$
$953$	−32.1822	−1.04248	−0.521242	+	0.853409i	$0.674531\pi$
$954$	0	0
$955$	14.2729	0.461862
$956$	0	0
$957$	−5.07459	−0.164038
$958$	0	0
$959$	1.04152	0.0336323
$960$	0	0
$961$	−20.0618	−0.647154
$962$	0	0
$963$	7.97061	0.256849
$964$	0	0
$965$	−41.8673	−1.34776
$966$	0	0
$967$	31.0361	0.998052	0.499026	−	0.866587i	$-0.333691\pi$
$967$	31.0361	0.998052	0.499026	+	0.866587i	$0.333691\pi$
$968$	0	0
$969$	−3.31129	−0.106374
$970$	0	0
$971$	19.1321	0.613979	0.306990	−	0.951713i	$-0.400678\pi$
$971$	19.1321	0.613979	0.306990	+	0.951713i	$0.400678\pi$
$972$	0	0
$973$	−19.3953	−0.621786
$974$	0	0
$975$	−13.8886	−0.444792
$976$	0	0
$977$	−21.4334	−0.685716	−0.342858	−	0.939387i	$-0.611395\pi$
$977$	−21.4334	−0.685716	−0.342858	+	0.939387i	$0.611395\pi$
$978$	0	0
$979$	2.21747	0.0708706
$980$	0	0
$981$	69.8554	2.23031
$982$	0	0
$983$	−25.9065	−0.826288	−0.413144	−	0.910666i	$-0.635569\pi$
$983$	−25.9065	−0.826288	−0.413144	+	0.910666i	$0.635569\pi$
$984$	0	0
$985$	42.6033	1.35746
$986$	0	0
$987$	−9.94257	−0.316476
$988$	0	0
$989$	−12.5782	−0.399964
$990$	0	0
$991$	1.91309	0.0607712	0.0303856	−	0.999538i	$-0.490326\pi$
$991$	1.91309	0.0607712	0.0303856	+	0.999538i	$0.490326\pi$
$992$	0	0
$993$	50.3917	1.59913
$994$	0	0
$995$	28.2235	0.894745
$996$	0	0
$997$	7.64819	0.242221	0.121110	−	0.992639i	$-0.461355\pi$
$997$	7.64819	0.242221	0.121110	+	0.992639i	$0.461355\pi$
$998$	0	0
$999$	27.2551	0.862313

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-2.63586 q^{3} -2.63773 q^{5} -2.07266 q^{7} +3.94775 q^{9} +O(q^{10})\)	\(q-2.63586 q^{3} -2.63773 q^{5} -2.07266 q^{7} +3.94775 q^{9} -0.242814 q^{11} +2.69158 q^{13} +6.95269 q^{15} -1.25625 q^{17} -1.00000 q^{19} +5.46324 q^{21} -1.78981 q^{23} +1.95763 q^{25} -2.49813 q^{27} -7.92876 q^{29} -3.30730 q^{31} +0.640023 q^{33} +5.46712 q^{35} -10.9102 q^{37} -7.09461 q^{39} -7.94545 q^{41} +7.02769 q^{43} -10.4131 q^{45} -1.81990 q^{47} -2.70408 q^{49} +3.31129 q^{51} -1.00000 q^{53} +0.640478 q^{55} +2.63586 q^{57} +3.44995 q^{59} -11.3553 q^{61} -8.18234 q^{63} -7.09966 q^{65} +1.35536 q^{67} +4.71767 q^{69} -5.54018 q^{71} -8.98019 q^{73} -5.16003 q^{75} +0.503271 q^{77} +6.67097 q^{79} -5.25854 q^{81} -9.43314 q^{83} +3.31365 q^{85} +20.8991 q^{87} -9.13237 q^{89} -5.57872 q^{91} +8.71756 q^{93} +2.63773 q^{95} -1.58967 q^{97} -0.958569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 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