LMFDB - Embedded newform 4028.2.a.d.1.13

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:	$1$
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} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
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			1.13
Root			$-0.537191$ 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)

$f(q)$	$=$	$q+0.537191 q^{3} -2.35184 q^{5} -1.00979 q^{7} -2.71143 q^{9} +O(q^{10})$	$q+0.537191 q^{3} -2.35184 q^{5} -1.00979 q^{7} -2.71143 q^{9} +3.55138 q^{11} +0.172083 q^{13} -1.26339 q^{15} +7.68990 q^{17} -1.00000 q^{19} -0.542449 q^{21} -8.17344 q^{23} +0.531160 q^{25} -3.06813 q^{27} +3.95811 q^{29} +8.96845 q^{31} +1.90777 q^{33} +2.37486 q^{35} -9.28780 q^{37} +0.0924417 q^{39} -0.449703 q^{41} +9.15762 q^{43} +6.37684 q^{45} -9.50313 q^{47} -5.98033 q^{49} +4.13095 q^{51} +1.00000 q^{53} -8.35229 q^{55} -0.537191 q^{57} -7.97890 q^{59} +3.34325 q^{61} +2.73797 q^{63} -0.404713 q^{65} -4.16062 q^{67} -4.39070 q^{69} +4.34907 q^{71} +3.25468 q^{73} +0.285335 q^{75} -3.58614 q^{77} +5.21648 q^{79} +6.48611 q^{81} -1.01694 q^{83} -18.0854 q^{85} +2.12626 q^{87} -8.11647 q^{89} -0.173768 q^{91} +4.81777 q^{93} +2.35184 q^{95} +3.08376 q^{97} -9.62930 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + 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
$3$	0.537191	0.310147	0.155074	−	0.987903i	$-0.450438\pi$
$3$	0.537191	0.310147	0.155074	+	0.987903i	$0.450438\pi$
$4$	0	0
$5$	−2.35184	−1.05178	−0.525888	−	0.850554i	$-0.676267\pi$
$5$	−2.35184	−1.05178	−0.525888	+	0.850554i	$0.676267\pi$
$6$	0	0
$7$	−1.00979	−0.381664	−0.190832	−	0.981623i	$-0.561119\pi$
$7$	−1.00979	−0.381664	−0.190832	+	0.981623i	$0.561119\pi$
$8$	0	0
$9$	−2.71143	−0.903809
$10$	0	0
$11$	3.55138	1.07078	0.535391	−	0.844604i	$-0.320164\pi$
$11$	3.55138	1.07078	0.535391	+	0.844604i	$0.320164\pi$
$12$	0	0
$13$	0.172083	0.0477273	0.0238637	−	0.999715i	$-0.492403\pi$
$13$	0.172083	0.0477273	0.0238637	+	0.999715i	$0.492403\pi$
$14$	0	0
$15$	−1.26339	−0.326206
$16$	0	0
$17$	7.68990	1.86507	0.932537	−	0.361073i	$-0.117590\pi$
$17$	7.68990	1.86507	0.932537	+	0.361073i	$0.117590\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.542449	−0.118372
$22$	0	0
$23$	−8.17344	−1.70428	−0.852140	−	0.523313i	$-0.824696\pi$
$23$	−8.17344	−1.70428	−0.852140	+	0.523313i	$0.824696\pi$
$24$	0	0
$25$	0.531160	0.106232
$26$	0	0
$27$	−3.06813	−0.590461
$28$	0	0
$29$	3.95811	0.735003	0.367501	−	0.930023i	$-0.380213\pi$
$29$	3.95811	0.735003	0.367501	+	0.930023i	$0.380213\pi$
$30$	0	0
$31$	8.96845	1.61078	0.805391	−	0.592744i	$-0.201955\pi$
$31$	8.96845	1.61078	0.805391	+	0.592744i	$0.201955\pi$
$32$	0	0
$33$	1.90777	0.332100
$34$	0	0
$35$	2.37486	0.401425
$36$	0	0
$37$	−9.28780	−1.52690	−0.763452	−	0.645864i	$-0.776497\pi$
$37$	−9.28780	−1.52690	−0.763452	+	0.645864i	$0.776497\pi$
$38$	0	0
$39$	0.0924417	0.0148025
$40$	0	0
$41$	−0.449703	−0.0702318	−0.0351159	−	0.999383i	$-0.511180\pi$
$41$	−0.449703	−0.0702318	−0.0351159	+	0.999383i	$0.511180\pi$
$42$	0	0
$43$	9.15762	1.39652	0.698262	−	0.715842i	$-0.253957\pi$
$43$	9.15762	1.39652	0.698262	+	0.715842i	$0.253957\pi$
$44$	0	0
$45$	6.37684	0.950604
$46$	0	0
$47$	−9.50313	−1.38617	−0.693087	−	0.720854i	$-0.743750\pi$
$47$	−9.50313	−1.38617	−0.693087	+	0.720854i	$0.743750\pi$
$48$	0	0
$49$	−5.98033	−0.854333
$50$	0	0
$51$	4.13095	0.578448
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−8.35229	−1.12622
$56$	0	0
$57$	−0.537191	−0.0711527
$58$	0	0
$59$	−7.97890	−1.03876	−0.519382	−	0.854542i	$-0.673838\pi$
$59$	−7.97890	−1.03876	−0.519382	+	0.854542i	$0.673838\pi$
$60$	0	0
$61$	3.34325	0.428060	0.214030	−	0.976827i	$-0.431341\pi$
$61$	3.34325	0.428060	0.214030	+	0.976827i	$0.431341\pi$
$62$	0	0
$63$	2.73797	0.344951
$64$	0	0
$65$	−0.404713	−0.0501984
$66$	0	0
$67$	−4.16062	−0.508301	−0.254151	−	0.967165i	$-0.581796\pi$
$67$	−4.16062	−0.508301	−0.254151	+	0.967165i	$0.581796\pi$
$68$	0	0
$69$	−4.39070	−0.528578
$70$	0	0
$71$	4.34907	0.516140	0.258070	−	0.966126i	$-0.416914\pi$
$71$	4.34907	0.516140	0.258070	+	0.966126i	$0.416914\pi$
$72$	0	0
$73$	3.25468	0.380932	0.190466	−	0.981694i	$-0.439000\pi$
$73$	3.25468	0.380932	0.190466	+	0.981694i	$0.439000\pi$
$74$	0	0
$75$	0.285335	0.0329476
$76$	0	0
$77$	−3.58614	−0.408679
$78$	0	0
$79$	5.21648	0.586900	0.293450	−	0.955974i	$-0.405197\pi$
$79$	5.21648	0.586900	0.293450	+	0.955974i	$0.405197\pi$
$80$	0	0
$81$	6.48611	0.720678
$82$	0	0
$83$	−1.01694	−0.111624	−0.0558118	−	0.998441i	$-0.517775\pi$
$83$	−1.01694	−0.111624	−0.0558118	+	0.998441i	$0.517775\pi$
$84$	0	0
$85$	−18.0854	−1.96164
$86$	0	0
$87$	2.12626	0.227959
$88$	0	0
$89$	−8.11647	−0.860344	−0.430172	−	0.902747i	$-0.641547\pi$
$89$	−8.11647	−0.860344	−0.430172	+	0.902747i	$0.641547\pi$
$90$	0	0
$91$	−0.173768	−0.0182158
$92$	0	0
$93$	4.81777	0.499580
$94$	0	0
$95$	2.35184	0.241294
$96$	0	0
$97$	3.08376	0.313108	0.156554	−	0.987669i	$-0.449961\pi$
$97$	3.08376	0.313108	0.156554	+	0.987669i	$0.449961\pi$
$98$	0	0
$99$	−9.62930	−0.967781
$100$	0	0
$101$	−11.1511	−1.10958	−0.554789	−	0.831991i	$-0.687201\pi$
$101$	−11.1511	−1.10958	−0.554789	+	0.831991i	$0.687201\pi$
$102$	0	0
$103$	−14.6088	−1.43945	−0.719725	−	0.694260i	$-0.755732\pi$
$103$	−14.6088	−1.43945	−0.719725	+	0.694260i	$0.755732\pi$
$104$	0	0
$105$	1.27575	0.124501
$106$	0	0
$107$	15.2664	1.47586	0.737931	−	0.674876i	$-0.235803\pi$
$107$	15.2664	1.47586	0.737931	+	0.674876i	$0.235803\pi$
$108$	0	0
$109$	−7.20109	−0.689740	−0.344870	−	0.938651i	$-0.612077\pi$
$109$	−7.20109	−0.689740	−0.344870	+	0.938651i	$0.612077\pi$
$110$	0	0
$111$	−4.98932	−0.473566
$112$	0	0
$113$	−17.7284	−1.66775	−0.833873	−	0.551957i	$-0.813881\pi$
$113$	−17.7284	−1.66775	−0.833873	+	0.551957i	$0.813881\pi$
$114$	0	0
$115$	19.2226	1.79252
$116$	0	0
$117$	−0.466591	−0.0431364
$118$	0	0
$119$	−7.76517	−0.711832
$120$	0	0
$121$	1.61230	0.146573
$122$	0	0
$123$	−0.241576	−0.0217822
$124$	0	0
$125$	10.5100	0.940043
$126$	0	0
$127$	−9.27267	−0.822816	−0.411408	−	0.911451i	$-0.634963\pi$
$127$	−9.27267	−0.822816	−0.411408	+	0.911451i	$0.634963\pi$
$128$	0	0
$129$	4.91939	0.433128
$130$	0	0
$131$	−8.15882	−0.712839	−0.356420	−	0.934326i	$-0.616003\pi$
$131$	−8.15882	−0.712839	−0.356420	+	0.934326i	$0.616003\pi$
$132$	0	0
$133$	1.00979	0.0875597
$134$	0	0
$135$	7.21575	0.621033
$136$	0	0
$137$	−1.79834	−0.153643	−0.0768214	−	0.997045i	$-0.524477\pi$
$137$	−1.79834	−0.153643	−0.0768214	+	0.997045i	$0.524477\pi$
$138$	0	0
$139$	−2.34073	−0.198538	−0.0992689	−	0.995061i	$-0.531650\pi$
$139$	−2.34073	−0.198538	−0.0992689	+	0.995061i	$0.531650\pi$
$140$	0	0
$141$	−5.10500	−0.429918
$142$	0	0
$143$	0.611133	0.0511055
$144$	0	0
$145$	−9.30885	−0.773058
$146$	0	0
$147$	−3.21258	−0.264969
$148$	0	0
$149$	−9.11663	−0.746864	−0.373432	−	0.927658i	$-0.621819\pi$
$149$	−9.11663	−0.746864	−0.373432	+	0.927658i	$0.621819\pi$
$150$	0	0
$151$	−6.02151	−0.490024	−0.245012	−	0.969520i	$-0.578792\pi$
$151$	−6.02151	−0.490024	−0.245012	+	0.969520i	$0.578792\pi$
$152$	0	0
$153$	−20.8506	−1.68567
$154$	0	0
$155$	−21.0924	−1.69418
$156$	0	0
$157$	−21.0057	−1.67644	−0.838218	−	0.545336i	$-0.816402\pi$
$157$	−21.0057	−1.67644	−0.838218	+	0.545336i	$0.816402\pi$
$158$	0	0
$159$	0.537191	0.0426020
$160$	0	0
$161$	8.25345	0.650463
$162$	0	0
$163$	−12.5285	−0.981306	−0.490653	−	0.871355i	$-0.663242\pi$
$163$	−12.5285	−0.981306	−0.490653	+	0.871355i	$0.663242\pi$
$164$	0	0
$165$	−4.48677	−0.349295
$166$	0	0
$167$	−24.5039	−1.89617	−0.948083	−	0.318024i	$-0.896981\pi$
$167$	−24.5039	−1.89617	−0.948083	+	0.318024i	$0.896981\pi$
$168$	0	0
$169$	−12.9704	−0.997722
$170$	0	0
$171$	2.71143	0.207348
$172$	0	0
$173$	−20.5556	−1.56281	−0.781406	−	0.624023i	$-0.785497\pi$
$173$	−20.5556	−1.56281	−0.781406	+	0.624023i	$0.785497\pi$
$174$	0	0
$175$	−0.536359	−0.0405450
$176$	0	0
$177$	−4.28619	−0.322170
$178$	0	0
$179$	9.49690	0.709832	0.354916	−	0.934898i	$-0.384509\pi$
$179$	9.49690	0.709832	0.354916	+	0.934898i	$0.384509\pi$
$180$	0	0
$181$	12.0182	0.893305	0.446652	−	0.894708i	$-0.352616\pi$
$181$	12.0182	0.893305	0.446652	+	0.894708i	$0.352616\pi$
$182$	0	0
$183$	1.79597	0.132762
$184$	0	0
$185$	21.8434	1.60596
$186$	0	0
$187$	27.3098	1.99709
$188$	0	0
$189$	3.09816	0.225358
$190$	0	0
$191$	18.6439	1.34902	0.674512	−	0.738264i	$-0.264354\pi$
$191$	18.6439	1.34902	0.674512	+	0.738264i	$0.264354\pi$
$192$	0	0
$193$	4.92405	0.354441	0.177221	−	0.984171i	$-0.443289\pi$
$193$	4.92405	0.354441	0.177221	+	0.984171i	$0.443289\pi$
$194$	0	0
$195$	−0.217408	−0.0155689
$196$	0	0
$197$	−21.1462	−1.50661	−0.753303	−	0.657674i	$-0.771541\pi$
$197$	−21.1462	−1.50661	−0.753303	+	0.657674i	$0.771541\pi$
$198$	0	0
$199$	−23.8261	−1.68899	−0.844493	−	0.535566i	$-0.820098\pi$
$199$	−23.8261	−1.68899	−0.844493	+	0.535566i	$0.820098\pi$
$200$	0	0
$201$	−2.23505	−0.157648
$202$	0	0
$203$	−3.99685	−0.280524
$204$	0	0
$205$	1.05763	0.0738681
$206$	0	0
$207$	22.1617	1.54034
$208$	0	0
$209$	−3.55138	−0.245654
$210$	0	0
$211$	−5.46889	−0.376494	−0.188247	−	0.982122i	$-0.560281\pi$
$211$	−5.46889	−0.376494	−0.188247	+	0.982122i	$0.560281\pi$
$212$	0	0
$213$	2.33628	0.160079
$214$	0	0
$215$	−21.5373	−1.46883
$216$	0	0
$217$	−9.05624	−0.614778
$218$	0	0
$219$	1.74839	0.118145
$220$	0	0
$221$	1.32330	0.0890151
$222$	0	0
$223$	14.6634	0.981932	0.490966	−	0.871179i	$-0.336644\pi$
$223$	14.6634	0.981932	0.490966	+	0.871179i	$0.336644\pi$
$224$	0	0
$225$	−1.44020	−0.0960134
$226$	0	0
$227$	−18.2870	−1.21375	−0.606877	−	0.794796i	$-0.707578\pi$
$227$	−18.2870	−1.21375	−0.606877	+	0.794796i	$0.707578\pi$
$228$	0	0
$229$	−3.32252	−0.219558	−0.109779	−	0.993956i	$-0.535014\pi$
$229$	−3.32252	−0.219558	−0.109779	+	0.993956i	$0.535014\pi$
$230$	0	0
$231$	−1.92644	−0.126751
$232$	0	0
$233$	3.50426	0.229572	0.114786	−	0.993390i	$-0.463382\pi$
$233$	3.50426	0.229572	0.114786	+	0.993390i	$0.463382\pi$
$234$	0	0
$235$	22.3499	1.45794
$236$	0	0
$237$	2.80225	0.182026
$238$	0	0
$239$	−8.15493	−0.527499	−0.263749	−	0.964591i	$-0.584959\pi$
$239$	−8.15493	−0.527499	−0.263749	+	0.964591i	$0.584959\pi$
$240$	0	0
$241$	−5.36186	−0.345388	−0.172694	−	0.984976i	$-0.555247\pi$
$241$	−5.36186	−0.345388	−0.172694	+	0.984976i	$0.555247\pi$
$242$	0	0
$243$	12.6887	0.813978
$244$	0	0
$245$	14.0648	0.898566
$246$	0	0
$247$	−0.172083	−0.0109494
$248$	0	0
$249$	−0.546291	−0.0346198
$250$	0	0
$251$	25.3806	1.60201	0.801003	−	0.598660i	$-0.204300\pi$
$251$	25.3806	1.60201	0.801003	+	0.598660i	$0.204300\pi$
$252$	0	0
$253$	−29.0270	−1.82491
$254$	0	0
$255$	−9.71533	−0.608398
$256$	0	0
$257$	14.4397	0.900722	0.450361	−	0.892847i	$-0.351295\pi$
$257$	14.4397	0.900722	0.450361	+	0.892847i	$0.351295\pi$
$258$	0	0
$259$	9.37871	0.582765
$260$	0	0
$261$	−10.7321	−0.664302
$262$	0	0
$263$	24.7452	1.52585	0.762927	−	0.646484i	$-0.223761\pi$
$263$	24.7452	1.52585	0.762927	+	0.646484i	$0.223761\pi$
$264$	0	0
$265$	−2.35184	−0.144472
$266$	0	0
$267$	−4.36010	−0.266834
$268$	0	0
$269$	29.2054	1.78068	0.890342	−	0.455292i	$-0.150465\pi$
$269$	29.2054	1.78068	0.890342	+	0.455292i	$0.150465\pi$
$270$	0	0
$271$	7.60309	0.461855	0.230927	−	0.972971i	$-0.425824\pi$
$271$	7.60309	0.461855	0.230927	+	0.972971i	$0.425824\pi$
$272$	0	0
$273$	−0.0933465	−0.00564959
$274$	0	0
$275$	1.88635	0.113751
$276$	0	0
$277$	−5.89074	−0.353940	−0.176970	−	0.984216i	$-0.556630\pi$
$277$	−5.89074	−0.353940	−0.176970	+	0.984216i	$0.556630\pi$
$278$	0	0
$279$	−24.3173	−1.45584
$280$	0	0
$281$	3.42637	0.204400	0.102200	−	0.994764i	$-0.467412\pi$
$281$	3.42637	0.204400	0.102200	+	0.994764i	$0.467412\pi$
$282$	0	0
$283$	−3.35801	−0.199613	−0.0998064	−	0.995007i	$-0.531822\pi$
$283$	−3.35801	−0.199613	−0.0998064	+	0.995007i	$0.531822\pi$
$284$	0	0
$285$	1.26339	0.0748367
$286$	0	0
$287$	0.454104	0.0268049
$288$	0	0
$289$	42.1346	2.47850
$290$	0	0
$291$	1.65657	0.0971098
$292$	0	0
$293$	17.9446	1.04834	0.524168	−	0.851615i	$-0.324376\pi$
$293$	17.9446	1.04834	0.524168	+	0.851615i	$0.324376\pi$
$294$	0	0
$295$	18.7651	1.09255
$296$	0	0
$297$	−10.8961	−0.632255
$298$	0	0
$299$	−1.40651	−0.0813408
$300$	0	0
$301$	−9.24726	−0.533003
$302$	0	0
$303$	−5.99028	−0.344133
$304$	0	0
$305$	−7.86280	−0.450223
$306$	0	0
$307$	−24.2104	−1.38176	−0.690881	−	0.722968i	$-0.742777\pi$
$307$	−24.2104	−1.38176	−0.690881	+	0.722968i	$0.742777\pi$
$308$	0	0
$309$	−7.84773	−0.446442
$310$	0	0
$311$	−31.3781	−1.77929	−0.889643	−	0.456657i	$-0.849047\pi$
$311$	−31.3781	−1.77929	−0.889643	+	0.456657i	$0.849047\pi$
$312$	0	0
$313$	9.19990	0.520009	0.260004	−	0.965607i	$-0.416276\pi$
$313$	9.19990	0.520009	0.260004	+	0.965607i	$0.416276\pi$
$314$	0	0
$315$	−6.43926	−0.362811
$316$	0	0
$317$	−18.4653	−1.03712	−0.518559	−	0.855042i	$-0.673531\pi$
$317$	−18.4653	−1.03712	−0.518559	+	0.855042i	$0.673531\pi$
$318$	0	0
$319$	14.0568	0.787027
$320$	0	0
$321$	8.20099	0.457735
$322$	0	0
$323$	−7.68990	−0.427878
$324$	0	0
$325$	0.0914038	0.00507017
$326$	0	0
$327$	−3.86836	−0.213921
$328$	0	0
$329$	9.59615	0.529053
$330$	0	0
$331$	0.353909	0.0194526	0.00972629	−	0.999953i	$-0.496904\pi$
$331$	0.353909	0.0194526	0.00972629	+	0.999953i	$0.496904\pi$
$332$	0	0
$333$	25.1832	1.38003
$334$	0	0
$335$	9.78513	0.534619
$336$	0	0
$337$	−9.18426	−0.500298	−0.250149	−	0.968207i	$-0.580480\pi$
$337$	−9.18426	−0.500298	−0.250149	+	0.968207i	$0.580480\pi$
$338$	0	0
$339$	−9.52353	−0.517247
$340$	0	0
$341$	31.8504	1.72480
$342$	0	0
$343$	13.1074	0.707732
$344$	0	0
$345$	10.3262	0.555946
$346$	0	0
$347$	17.3237	0.929986	0.464993	−	0.885314i	$-0.346057\pi$
$347$	17.3237	0.929986	0.464993	+	0.885314i	$0.346057\pi$
$348$	0	0
$349$	−13.3943	−0.716980	−0.358490	−	0.933534i	$-0.616708\pi$
$349$	−13.3943	−0.716980	−0.358490	+	0.933534i	$0.616708\pi$
$350$	0	0
$351$	−0.527974	−0.0281811
$352$	0	0
$353$	25.5060	1.35755	0.678775	−	0.734346i	$-0.262511\pi$
$353$	25.5060	1.35755	0.678775	+	0.734346i	$0.262511\pi$
$354$	0	0
$355$	−10.2283	−0.542863
$356$	0	0
$357$	−4.17138	−0.220773
$358$	0	0
$359$	−0.721974	−0.0381043	−0.0190522	−	0.999818i	$-0.506065\pi$
$359$	−0.721974	−0.0381043	−0.0190522	+	0.999818i	$0.506065\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.866115	0.0454592
$364$	0	0
$365$	−7.65450	−0.400655
$366$	0	0
$367$	−23.0271	−1.20200	−0.601001	−	0.799248i	$-0.705231\pi$
$367$	−23.0271	−1.20200	−0.601001	+	0.799248i	$0.705231\pi$
$368$	0	0
$369$	1.21934	0.0634761
$370$	0	0
$371$	−1.00979	−0.0524256
$372$	0	0
$373$	2.10637	0.109063	0.0545317	−	0.998512i	$-0.482633\pi$
$373$	2.10637	0.109063	0.0545317	+	0.998512i	$0.482633\pi$
$374$	0	0
$375$	5.64588	0.291552
$376$	0	0
$377$	0.681125	0.0350797
$378$	0	0
$379$	−23.6695	−1.21582	−0.607911	−	0.794005i	$-0.707992\pi$
$379$	−23.6695	−1.21582	−0.607911	+	0.794005i	$0.707992\pi$
$380$	0	0
$381$	−4.98120	−0.255194
$382$	0	0
$383$	−26.5618	−1.35724	−0.678621	−	0.734488i	$-0.737422\pi$
$383$	−26.5618	−1.35724	−0.678621	+	0.734488i	$0.737422\pi$
$384$	0	0
$385$	8.43404	0.429838
$386$	0	0
$387$	−24.8302	−1.26219
$388$	0	0
$389$	6.63263	0.336288	0.168144	−	0.985762i	$-0.446223\pi$
$389$	6.63263	0.336288	0.168144	+	0.985762i	$0.446223\pi$
$390$	0	0
$391$	−62.8530	−3.17861
$392$	0	0
$393$	−4.38285	−0.221085
$394$	0	0
$395$	−12.2683	−0.617288
$396$	0	0
$397$	27.0956	1.35989	0.679945	−	0.733263i	$-0.262004\pi$
$397$	27.0956	1.35989	0.679945	+	0.733263i	$0.262004\pi$
$398$	0	0
$399$	0.542449	0.0271564
$400$	0	0
$401$	31.7468	1.58536	0.792679	−	0.609639i	$-0.208685\pi$
$401$	31.7468	1.58536	0.792679	+	0.609639i	$0.208685\pi$
$402$	0	0
$403$	1.54332	0.0768783
$404$	0	0
$405$	−15.2543	−0.757992
$406$	0	0
$407$	−32.9845	−1.63498
$408$	0	0
$409$	28.1464	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$409$	28.1464	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$410$	0	0
$411$	−0.966055	−0.0476520
$412$	0	0
$413$	8.05700	0.396459
$414$	0	0
$415$	2.39168	0.117403
$416$	0	0
$417$	−1.25742	−0.0615760
$418$	0	0
$419$	18.1647	0.887404	0.443702	−	0.896174i	$-0.353665\pi$
$419$	18.1647	0.887404	0.443702	+	0.896174i	$0.353665\pi$
$420$	0	0
$421$	−8.21227	−0.400242	−0.200121	−	0.979771i	$-0.564133\pi$
$421$	−8.21227	−0.400242	−0.200121	+	0.979771i	$0.564133\pi$
$422$	0	0
$423$	25.7670	1.25284
$424$	0	0
$425$	4.08457	0.198131
$426$	0	0
$427$	−3.37598	−0.163375
$428$	0	0
$429$	0.328295	0.0158503
$430$	0	0
$431$	−1.07761	−0.0519064	−0.0259532	−	0.999663i	$-0.508262\pi$
$431$	−1.07761	−0.0519064	−0.0259532	+	0.999663i	$0.508262\pi$
$432$	0	0
$433$	−15.0333	−0.722452	−0.361226	−	0.932478i	$-0.617642\pi$
$433$	−15.0333	−0.722452	−0.361226	+	0.932478i	$0.617642\pi$
$434$	0	0
$435$	−5.00063	−0.239762
$436$	0	0
$437$	8.17344	0.390989
$438$	0	0
$439$	−13.3808	−0.638629	−0.319315	−	0.947649i	$-0.603453\pi$
$439$	−13.3808	−0.638629	−0.319315	+	0.947649i	$0.603453\pi$
$440$	0	0
$441$	16.2152	0.772153
$442$	0	0
$443$	−9.81811	−0.466473	−0.233236	−	0.972420i	$-0.574932\pi$
$443$	−9.81811	−0.466473	−0.233236	+	0.972420i	$0.574932\pi$
$444$	0	0
$445$	19.0887	0.904889
$446$	0	0
$447$	−4.89737	−0.231638
$448$	0	0
$449$	−2.96158	−0.139766	−0.0698828	−	0.997555i	$-0.522263\pi$
$449$	−2.96158	−0.139766	−0.0698828	+	0.997555i	$0.522263\pi$
$450$	0	0
$451$	−1.59707	−0.0752029
$452$	0	0
$453$	−3.23470	−0.151980
$454$	0	0
$455$	0.408674	0.0191589
$456$	0	0
$457$	2.88861	0.135123	0.0675617	−	0.997715i	$-0.478478\pi$
$457$	2.88861	0.135123	0.0675617	+	0.997715i	$0.478478\pi$
$458$	0	0
$459$	−23.5936	−1.10125
$460$	0	0
$461$	15.8850	0.739840	0.369920	−	0.929064i	$-0.379385\pi$
$461$	15.8850	0.739840	0.369920	+	0.929064i	$0.379385\pi$
$462$	0	0
$463$	−2.39483	−0.111297	−0.0556486	−	0.998450i	$-0.517723\pi$
$463$	−2.39483	−0.111297	−0.0556486	+	0.998450i	$0.517723\pi$
$464$	0	0
$465$	−11.3306	−0.525446
$466$	0	0
$467$	29.8941	1.38334	0.691668	−	0.722216i	$-0.256876\pi$
$467$	29.8941	1.38334	0.691668	+	0.722216i	$0.256876\pi$
$468$	0	0
$469$	4.20135	0.194000
$470$	0	0
$471$	−11.2841	−0.519942
$472$	0	0
$473$	32.5222	1.49537
$474$	0	0
$475$	−0.531160	−0.0243713
$476$	0	0
$477$	−2.71143	−0.124148
$478$	0	0
$479$	−1.51674	−0.0693014	−0.0346507	−	0.999399i	$-0.511032\pi$
$479$	−1.51674	−0.0693014	−0.0346507	+	0.999399i	$0.511032\pi$
$480$	0	0
$481$	−1.59828	−0.0728751
$482$	0	0
$483$	4.43368	0.201739
$484$	0	0
$485$	−7.25252	−0.329320
$486$	0	0
$487$	2.66752	0.120877	0.0604385	−	0.998172i	$-0.480750\pi$
$487$	2.66752	0.120877	0.0604385	+	0.998172i	$0.480750\pi$
$488$	0	0
$489$	−6.73018	−0.304349
$490$	0	0
$491$	−27.2545	−1.22998	−0.614989	−	0.788536i	$-0.710840\pi$
$491$	−27.2545	−1.22998	−0.614989	+	0.788536i	$0.710840\pi$
$492$	0	0
$493$	30.4375	1.37084
$494$	0	0
$495$	22.6466	1.01789
$496$	0	0
$497$	−4.39164	−0.196992
$498$	0	0
$499$	−0.817949	−0.0366164	−0.0183082	−	0.999832i	$-0.505828\pi$
$499$	−0.817949	−0.0366164	−0.0183082	+	0.999832i	$0.505828\pi$
$500$	0	0
$501$	−13.1633	−0.588091
$502$	0	0
$503$	26.4923	1.18123	0.590616	−	0.806953i	$-0.298885\pi$
$503$	26.4923	1.18123	0.590616	+	0.806953i	$0.298885\pi$
$504$	0	0
$505$	26.2257	1.16703
$506$	0	0
$507$	−6.96758	−0.309441
$508$	0	0
$509$	28.9142	1.28160	0.640799	−	0.767709i	$-0.278603\pi$
$509$	28.9142	1.28160	0.640799	+	0.767709i	$0.278603\pi$
$510$	0	0
$511$	−3.28654	−0.145388
$512$	0	0
$513$	3.06813	0.135461
$514$	0	0
$515$	34.3576	1.51398
$516$	0	0
$517$	−33.7492	−1.48429
$518$	0	0
$519$	−11.0423	−0.484702
$520$	0	0
$521$	−3.94832	−0.172979	−0.0864894	−	0.996253i	$-0.527565\pi$
$521$	−3.94832	−0.172979	−0.0864894	+	0.996253i	$0.527565\pi$
$522$	0	0
$523$	−19.7903	−0.865368	−0.432684	−	0.901546i	$-0.642433\pi$
$523$	−19.7903	−0.865368	−0.432684	+	0.901546i	$0.642433\pi$
$524$	0	0
$525$	−0.288128	−0.0125749
$526$	0	0
$527$	68.9665	3.00423
$528$	0	0
$529$	43.8052	1.90457
$530$	0	0
$531$	21.6342	0.938844
$532$	0	0
$533$	−0.0773864	−0.00335197
$534$	0	0
$535$	−35.9042	−1.55228
$536$	0	0
$537$	5.10165	0.220153
$538$	0	0
$539$	−21.2384	−0.914803
$540$	0	0
$541$	−14.6271	−0.628866	−0.314433	−	0.949280i	$-0.601814\pi$
$541$	−14.6271	−0.628866	−0.314433	+	0.949280i	$0.601814\pi$
$542$	0	0
$543$	6.45606	0.277056
$544$	0	0
$545$	16.9358	0.725451
$546$	0	0
$547$	−15.3337	−0.655622	−0.327811	−	0.944743i	$-0.606311\pi$
$547$	−15.3337	−0.655622	−0.327811	+	0.944743i	$0.606311\pi$
$548$	0	0
$549$	−9.06498	−0.386884
$550$	0	0
$551$	−3.95811	−0.168621
$552$	0	0
$553$	−5.26754	−0.223999
$554$	0	0
$555$	11.7341	0.498085
$556$	0	0
$557$	12.6605	0.536442	0.268221	−	0.963357i	$-0.413564\pi$
$557$	12.6605	0.536442	0.268221	+	0.963357i	$0.413564\pi$
$558$	0	0
$559$	1.57587	0.0666524
$560$	0	0
$561$	14.6706	0.619392
$562$	0	0
$563$	23.3355	0.983476	0.491738	−	0.870743i	$-0.336362\pi$
$563$	23.3355	0.983476	0.491738	+	0.870743i	$0.336362\pi$
$564$	0	0
$565$	41.6943	1.75409
$566$	0	0
$567$	−6.54959	−0.275057
$568$	0	0
$569$	19.0758	0.799699	0.399850	−	0.916581i	$-0.369062\pi$
$569$	19.0758	0.799699	0.399850	+	0.916581i	$0.369062\pi$
$570$	0	0
$571$	−25.5288	−1.06835	−0.534174	−	0.845374i	$-0.679377\pi$
$571$	−25.5288	−1.06835	−0.534174	+	0.845374i	$0.679377\pi$
$572$	0	0
$573$	10.0153	0.418396
$574$	0	0
$575$	−4.34141	−0.181049
$576$	0	0
$577$	−16.0887	−0.669780	−0.334890	−	0.942257i	$-0.608699\pi$
$577$	−16.0887	−0.669780	−0.334890	+	0.942257i	$0.608699\pi$
$578$	0	0
$579$	2.64516	0.109929
$580$	0	0
$581$	1.02689	0.0426027
$582$	0	0
$583$	3.55138	0.147083
$584$	0	0
$585$	1.09735	0.0453698
$586$	0	0
$587$	4.13097	0.170503	0.0852517	−	0.996359i	$-0.472831\pi$
$587$	4.13097	0.170503	0.0852517	+	0.996359i	$0.472831\pi$
$588$	0	0
$589$	−8.96845	−0.369539
$590$	0	0
$591$	−11.3596	−0.467270
$592$	0	0
$593$	−30.1142	−1.23664	−0.618320	−	0.785926i	$-0.712187\pi$
$593$	−30.1142	−1.23664	−0.618320	+	0.785926i	$0.712187\pi$
$594$	0	0
$595$	18.2625	0.748688
$596$	0	0
$597$	−12.7992	−0.523835
$598$	0	0
$599$	−26.5899	−1.08643	−0.543217	−	0.839593i	$-0.682794\pi$
$599$	−26.5899	−1.08643	−0.543217	+	0.839593i	$0.682794\pi$
$600$	0	0
$601$	−43.1341	−1.75948	−0.879739	−	0.475457i	$-0.842283\pi$
$601$	−43.1341	−1.75948	−0.879739	+	0.475457i	$0.842283\pi$
$602$	0	0
$603$	11.2812	0.459407
$604$	0	0
$605$	−3.79188	−0.154162
$606$	0	0
$607$	−27.5219	−1.11708	−0.558540	−	0.829478i	$-0.688638\pi$
$607$	−27.5219	−1.11708	−0.558540	+	0.829478i	$0.688638\pi$
$608$	0	0
$609$	−2.14707	−0.0870039
$610$	0	0
$611$	−1.63533	−0.0661584
$612$	0	0
$613$	23.7133	0.957770	0.478885	−	0.877878i	$-0.341041\pi$
$613$	23.7133	0.957770	0.478885	+	0.877878i	$0.341041\pi$
$614$	0	0
$615$	0.568149	0.0229100
$616$	0	0
$617$	30.1084	1.21212	0.606058	−	0.795420i	$-0.292750\pi$
$617$	30.1084	1.21212	0.606058	+	0.795420i	$0.292750\pi$
$618$	0	0
$619$	−15.2290	−0.612104	−0.306052	−	0.952015i	$-0.599008\pi$
$619$	−15.2290	−0.612104	−0.306052	+	0.952015i	$0.599008\pi$
$620$	0	0
$621$	25.0772	1.00631
$622$	0	0
$623$	8.19592	0.328363
$624$	0	0
$625$	−27.3737	−1.09495
$626$	0	0
$627$	−1.90777	−0.0761890
$628$	0	0
$629$	−71.4223	−2.84779
$630$	0	0
$631$	18.5005	0.736494	0.368247	−	0.929728i	$-0.379958\pi$
$631$	18.5005	0.736494	0.368247	+	0.929728i	$0.379958\pi$
$632$	0	0
$633$	−2.93784	−0.116769
$634$	0	0
$635$	21.8079	0.865418
$636$	0	0
$637$	−1.02911	−0.0407750
$638$	0	0
$639$	−11.7922	−0.466492
$640$	0	0
$641$	2.17206	0.0857913	0.0428956	−	0.999080i	$-0.486342\pi$
$641$	2.17206	0.0857913	0.0428956	+	0.999080i	$0.486342\pi$
$642$	0	0
$643$	−3.98380	−0.157106	−0.0785530	−	0.996910i	$-0.525030\pi$
$643$	−3.98380	−0.157106	−0.0785530	+	0.996910i	$0.525030\pi$
$644$	0	0
$645$	−11.5696	−0.455554
$646$	0	0
$647$	44.9861	1.76859	0.884293	−	0.466932i	$-0.154641\pi$
$647$	44.9861	1.76859	0.884293	+	0.466932i	$0.154641\pi$
$648$	0	0
$649$	−28.3361	−1.11229
$650$	0	0
$651$	−4.86493	−0.190672
$652$	0	0
$653$	2.38843	0.0934665	0.0467333	−	0.998907i	$-0.485119\pi$
$653$	2.38843	0.0934665	0.0467333	+	0.998907i	$0.485119\pi$
$654$	0	0
$655$	19.1883	0.749747
$656$	0	0
$657$	−8.82483	−0.344290
$658$	0	0
$659$	−17.7243	−0.690440	−0.345220	−	0.938522i	$-0.612196\pi$
$659$	−17.7243	−0.690440	−0.345220	+	0.938522i	$0.612196\pi$
$660$	0	0
$661$	−3.26632	−0.127045	−0.0635226	−	0.997980i	$-0.520233\pi$
$661$	−3.26632	−0.127045	−0.0635226	+	0.997980i	$0.520233\pi$
$662$	0	0
$663$	0.710867	0.0276078
$664$	0	0
$665$	−2.37486	−0.0920932
$666$	0	0
$667$	−32.3514	−1.25265
$668$	0	0
$669$	7.87704	0.304544
$670$	0	0
$671$	11.8732	0.458358
$672$	0	0
$673$	15.0893	0.581651	0.290825	−	0.956776i	$-0.406070\pi$
$673$	15.0893	0.581651	0.290825	+	0.956776i	$0.406070\pi$
$674$	0	0
$675$	−1.62967	−0.0627259
$676$	0	0
$677$	−41.1222	−1.58045	−0.790227	−	0.612814i	$-0.790038\pi$
$677$	−41.1222	−1.58045	−0.790227	+	0.612814i	$0.790038\pi$
$678$	0	0
$679$	−3.11394	−0.119502
$680$	0	0
$681$	−9.82364	−0.376443
$682$	0	0
$683$	30.9665	1.18490	0.592450	−	0.805607i	$-0.298161\pi$
$683$	30.9665	1.18490	0.592450	+	0.805607i	$0.298161\pi$
$684$	0	0
$685$	4.22942	0.161598
$686$	0	0
$687$	−1.78483	−0.0680955
$688$	0	0
$689$	0.172083	0.00655585
$690$	0	0
$691$	−39.2013	−1.49129	−0.745644	−	0.666345i	$-0.767858\pi$
$691$	−39.2013	−1.49129	−0.745644	+	0.666345i	$0.767858\pi$
$692$	0	0
$693$	9.72356	0.369367
$694$	0	0
$695$	5.50502	0.208817
$696$	0	0
$697$	−3.45817	−0.130988
$698$	0	0
$699$	1.88246	0.0712011
$700$	0	0
$701$	0.0334055	0.00126171	0.000630854	−	1.00000i	$-0.499799\pi$
$701$	0.0334055	0.00126171	0.000630854		1.00000i	$0.499799\pi$
$702$	0	0
$703$	9.28780	0.350296
$704$	0	0
$705$	12.0061	0.452178
$706$	0	0
$707$	11.2603	0.423486
$708$	0	0
$709$	33.0691	1.24194	0.620968	−	0.783836i	$-0.286740\pi$
$709$	33.0691	1.24194	0.620968	+	0.783836i	$0.286740\pi$
$710$	0	0
$711$	−14.1441	−0.530446
$712$	0	0
$713$	−73.3032	−2.74522
$714$	0	0
$715$	−1.43729	−0.0537516
$716$	0	0
$717$	−4.38076	−0.163602
$718$	0	0
$719$	−31.2750	−1.16636	−0.583180	−	0.812343i	$-0.698192\pi$
$719$	−31.2750	−1.16636	−0.583180	+	0.812343i	$0.698192\pi$
$720$	0	0
$721$	14.7518	0.549386
$722$	0	0
$723$	−2.88034	−0.107121
$724$	0	0
$725$	2.10239	0.0780809
$726$	0	0
$727$	−30.0929	−1.11609	−0.558043	−	0.829812i	$-0.688447\pi$
$727$	−30.0929	−1.11609	−0.558043	+	0.829812i	$0.688447\pi$
$728$	0	0
$729$	−12.6421	−0.468225
$730$	0	0
$731$	70.4212	2.60462
$732$	0	0
$733$	−37.2170	−1.37464	−0.687320	−	0.726355i	$-0.741213\pi$
$733$	−37.2170	−1.37464	−0.687320	+	0.726355i	$0.741213\pi$
$734$	0	0
$735$	7.55548	0.278688
$736$	0	0
$737$	−14.7760	−0.544279
$738$	0	0
$739$	−7.09354	−0.260940	−0.130470	−	0.991452i	$-0.541649\pi$
$739$	−7.09354	−0.260940	−0.130470	+	0.991452i	$0.541649\pi$
$740$	0	0
$741$	−0.0924417	−0.00339593
$742$	0	0
$743$	−31.0530	−1.13922	−0.569612	−	0.821914i	$-0.692906\pi$
$743$	−31.0530	−1.13922	−0.569612	+	0.821914i	$0.692906\pi$
$744$	0	0
$745$	21.4409	0.785533
$746$	0	0
$747$	2.75736	0.100886
$748$	0	0
$749$	−15.4159	−0.563283
$750$	0	0
$751$	33.4721	1.22142	0.610708	−	0.791856i	$-0.290885\pi$
$751$	33.4721	1.22142	0.610708	+	0.791856i	$0.290885\pi$
$752$	0	0
$753$	13.6342	0.496858
$754$	0	0
$755$	14.1616	0.515395
$756$	0	0
$757$	27.1696	0.987496	0.493748	−	0.869605i	$-0.335626\pi$
$757$	27.1696	0.987496	0.493748	+	0.869605i	$0.335626\pi$
$758$	0	0
$759$	−15.5931	−0.565992
$760$	0	0
$761$	−15.4760	−0.561006	−0.280503	−	0.959853i	$-0.590501\pi$
$761$	−15.4760	−0.561006	−0.280503	+	0.959853i	$0.590501\pi$
$762$	0	0
$763$	7.27158	0.263249
$764$	0	0
$765$	49.0373	1.77295
$766$	0	0
$767$	−1.37304	−0.0495774
$768$	0	0
$769$	33.2946	1.20063	0.600317	−	0.799762i	$-0.295041\pi$
$769$	33.2946	1.20063	0.600317	+	0.799762i	$0.295041\pi$
$770$	0	0
$771$	7.75686	0.279357
$772$	0	0
$773$	40.6534	1.46220	0.731101	−	0.682270i	$-0.239007\pi$
$773$	40.6534	1.46220	0.731101	+	0.682270i	$0.239007\pi$
$774$	0	0
$775$	4.76369	0.171117
$776$	0	0
$777$	5.03816	0.180743
$778$	0	0
$779$	0.449703	0.0161123
$780$	0	0
$781$	15.4452	0.552673
$782$	0	0
$783$	−12.1440	−0.433991
$784$	0	0
$785$	49.4020	1.76323
$786$	0	0
$787$	9.10363	0.324509	0.162255	−	0.986749i	$-0.448123\pi$
$787$	9.10363	0.324509	0.162255	+	0.986749i	$0.448123\pi$
$788$	0	0
$789$	13.2929	0.473240
$790$	0	0
$791$	17.9019	0.636518
$792$	0	0
$793$	0.575318	0.0204301
$794$	0	0
$795$	−1.26339	−0.0448078
$796$	0	0
$797$	34.4289	1.21954	0.609768	−	0.792580i	$-0.291263\pi$
$797$	34.4289	1.21954	0.609768	+	0.792580i	$0.291263\pi$
$798$	0	0
$799$	−73.0781	−2.58532
$800$	0	0
$801$	22.0072	0.777587
$802$	0	0
$803$	11.5586	0.407895
$804$	0	0
$805$	−19.4108	−0.684141
$806$	0	0
$807$	15.6889	0.552275
$808$	0	0
$809$	−17.2597	−0.606819	−0.303409	−	0.952860i	$-0.598125\pi$
$809$	−17.2597	−0.606819	−0.303409	+	0.952860i	$0.598125\pi$
$810$	0	0
$811$	13.5516	0.475862	0.237931	−	0.971282i	$-0.423531\pi$
$811$	13.5516	0.475862	0.237931	+	0.971282i	$0.423531\pi$
$812$	0	0
$813$	4.08431	0.143243
$814$	0	0
$815$	29.4650	1.03211
$816$	0	0
$817$	−9.15762	−0.320385
$818$	0	0
$819$	0.471158	0.0164636
$820$	0	0
$821$	7.96460	0.277966	0.138983	−	0.990295i	$-0.455617\pi$
$821$	7.96460	0.277966	0.138983	+	0.990295i	$0.455617\pi$
$822$	0	0
$823$	9.74529	0.339700	0.169850	−	0.985470i	$-0.445672\pi$
$823$	9.74529	0.339700	0.169850	+	0.985470i	$0.445672\pi$
$824$	0	0
$825$	1.01333	0.0352797
$826$	0	0
$827$	40.5356	1.40956	0.704780	−	0.709426i	$-0.251046\pi$
$827$	40.5356	1.40956	0.704780	+	0.709426i	$0.251046\pi$
$828$	0	0
$829$	−12.8584	−0.446591	−0.223296	−	0.974751i	$-0.571681\pi$
$829$	−12.8584	−0.446591	−0.223296	+	0.974751i	$0.571681\pi$
$830$	0	0
$831$	−3.16445	−0.109774
$832$	0	0
$833$	−45.9881	−1.59339
$834$	0	0
$835$	57.6292	1.99434
$836$	0	0
$837$	−27.5164	−0.951105
$838$	0	0
$839$	−34.4598	−1.18968	−0.594842	−	0.803843i	$-0.702785\pi$
$839$	−34.4598	−1.18968	−0.594842	+	0.803843i	$0.702785\pi$
$840$	0	0
$841$	−13.3334	−0.459771
$842$	0	0
$843$	1.84062	0.0633942
$844$	0	0
$845$	30.5043	1.04938
$846$	0	0
$847$	−1.62808	−0.0559416
$848$	0	0
$849$	−1.80389	−0.0619094
$850$	0	0
$851$	75.9133	2.60228
$852$	0	0
$853$	12.9779	0.444355	0.222178	−	0.975006i	$-0.428684\pi$
$853$	12.9779	0.444355	0.222178	+	0.975006i	$0.428684\pi$
$854$	0	0
$855$	−6.37684	−0.218083
$856$	0	0
$857$	56.0715	1.91537	0.957684	−	0.287823i	$-0.0929315\pi$
$857$	56.0715	1.91537	0.957684	+	0.287823i	$0.0929315\pi$
$858$	0	0
$859$	−43.7921	−1.49417	−0.747084	−	0.664730i	$-0.768547\pi$
$859$	−43.7921	−1.49417	−0.747084	+	0.664730i	$0.768547\pi$
$860$	0	0
$861$	0.243941	0.00831348
$862$	0	0
$863$	23.0763	0.785527	0.392764	−	0.919639i	$-0.371519\pi$
$863$	23.0763	0.785527	0.392764	+	0.919639i	$0.371519\pi$
$864$	0	0
$865$	48.3435	1.64373
$866$	0	0
$867$	22.6343	0.768702
$868$	0	0
$869$	18.5257	0.628442
$870$	0	0
$871$	−0.715974	−0.0242599
$872$	0	0
$873$	−8.36139	−0.282990
$874$	0	0
$875$	−10.6129	−0.358781
$876$	0	0
$877$	−34.7098	−1.17207	−0.586033	−	0.810287i	$-0.699311\pi$
$877$	−34.7098	−1.17207	−0.586033	+	0.810287i	$0.699311\pi$
$878$	0	0
$879$	9.63969	0.325139
$880$	0	0
$881$	38.4105	1.29408	0.647041	−	0.762455i	$-0.276006\pi$
$881$	38.4105	1.29408	0.647041	+	0.762455i	$0.276006\pi$
$882$	0	0
$883$	−43.7436	−1.47209	−0.736045	−	0.676933i	$-0.763309\pi$
$883$	−43.7436	−1.47209	−0.736045	+	0.676933i	$0.763309\pi$
$884$	0	0
$885$	10.0805	0.338851
$886$	0	0
$887$	36.9634	1.24111	0.620555	−	0.784163i	$-0.286907\pi$
$887$	36.9634	1.24111	0.620555	+	0.784163i	$0.286907\pi$
$888$	0	0
$889$	9.36343	0.314039
$890$	0	0
$891$	23.0346	0.771689
$892$	0	0
$893$	9.50313	0.318010
$894$	0	0
$895$	−22.3352	−0.746584
$896$	0	0
$897$	−0.755567	−0.0252276
$898$	0	0
$899$	35.4981	1.18393
$900$	0	0
$901$	7.68990	0.256188
$902$	0	0
$903$	−4.96754	−0.165310
$904$	0	0
$905$	−28.2649	−0.939556
$906$	0	0
$907$	−55.3568	−1.83809	−0.919046	−	0.394149i	$-0.871039\pi$
$907$	−55.3568	−1.83809	−0.919046	+	0.394149i	$0.871039\pi$
$908$	0	0
$909$	30.2354	1.00285
$910$	0	0
$911$	−18.5500	−0.614590	−0.307295	−	0.951614i	$-0.599424\pi$
$911$	−18.5500	−0.614590	−0.307295	+	0.951614i	$0.599424\pi$
$912$	0	0
$913$	−3.61154	−0.119525
$914$	0	0
$915$	−4.22383	−0.139635
$916$	0	0
$917$	8.23868	0.272065
$918$	0	0
$919$	−19.7145	−0.650320	−0.325160	−	0.945659i	$-0.605418\pi$
$919$	−19.7145	−0.650320	−0.325160	+	0.945659i	$0.605418\pi$
$920$	0	0
$921$	−13.0056	−0.428550
$922$	0	0
$923$	0.748403	0.0246340
$924$	0	0
$925$	−4.93331	−0.162206
$926$	0	0
$927$	39.6107	1.30099
$928$	0	0
$929$	31.0600	1.01905	0.509523	−	0.860457i	$-0.329822\pi$
$929$	31.0600	1.01905	0.509523	+	0.860457i	$0.329822\pi$
$930$	0	0
$931$	5.98033	0.195997
$932$	0	0
$933$	−16.8560	−0.551841
$934$	0	0
$935$	−64.2282	−2.10049
$936$	0	0
$937$	52.5358	1.71627	0.858136	−	0.513423i	$-0.171623\pi$
$937$	52.5358	1.71627	0.858136	+	0.513423i	$0.171623\pi$
$938$	0	0
$939$	4.94210	0.161279
$940$	0	0
$941$	−23.6149	−0.769822	−0.384911	−	0.922954i	$-0.625768\pi$
$941$	−23.6149	−0.769822	−0.384911	+	0.922954i	$0.625768\pi$
$942$	0	0
$943$	3.67562	0.119695
$944$	0	0
$945$	−7.28638	−0.237026
$946$	0	0
$947$	−52.4192	−1.70339	−0.851697	−	0.524034i	$-0.824426\pi$
$947$	−52.4192	−1.70339	−0.851697	+	0.524034i	$0.824426\pi$
$948$	0	0
$949$	0.560077	0.0181809
$950$	0	0
$951$	−9.91942	−0.321659
$952$	0	0
$953$	−23.7899	−0.770630	−0.385315	−	0.922785i	$-0.625907\pi$
$953$	−23.7899	−0.770630	−0.385315	+	0.922785i	$0.625907\pi$
$954$	0	0
$955$	−43.8475	−1.41887
$956$	0	0
$957$	7.55117	0.244095
$958$	0	0
$959$	1.81595	0.0586400
$960$	0	0
$961$	49.4332	1.59462
$962$	0	0
$963$	−41.3938	−1.33390
$964$	0	0
$965$	−11.5806	−0.372793
$966$	0	0
$967$	42.0466	1.35213	0.676064	−	0.736843i	$-0.263684\pi$
$967$	42.0466	1.35213	0.676064	+	0.736843i	$0.263684\pi$
$968$	0	0
$969$	−4.13095	−0.132705
$970$	0	0
$971$	−4.01025	−0.128695	−0.0643475	−	0.997928i	$-0.520497\pi$
$971$	−4.01025	−0.128695	−0.0643475	+	0.997928i	$0.520497\pi$
$972$	0	0
$973$	2.36364	0.0757747
$974$	0	0
$975$	0.0491013	0.00157250
$976$	0	0
$977$	−33.0559	−1.05755	−0.528776	−	0.848761i	$-0.677349\pi$
$977$	−33.0559	−1.05755	−0.528776	+	0.848761i	$0.677349\pi$
$978$	0	0
$979$	−28.8247	−0.921241
$980$	0	0
$981$	19.5252	0.623393
$982$	0	0
$983$	−47.4817	−1.51443	−0.757215	−	0.653166i	$-0.773440\pi$
$983$	−47.4817	−1.51443	−0.757215	+	0.653166i	$0.773440\pi$
$984$	0	0
$985$	49.7326	1.58461
$986$	0	0
$987$	5.15497	0.164084
$988$	0	0
$989$	−74.8493	−2.38007
$990$	0	0
$991$	22.4450	0.712989	0.356495	−	0.934297i	$-0.383972\pi$
$991$	22.4450	0.712989	0.356495	+	0.934297i	$0.383972\pi$
$992$	0	0
$993$	0.190117	0.00603317
$994$	0	0
$995$	56.0352	1.77643
$996$	0	0
$997$	−46.5069	−1.47289	−0.736445	−	0.676498i	$-0.763497\pi$
$997$	−46.5069	−1.47289	−0.736445	+	0.676498i	$0.763497\pi$
$998$	0	0
$999$	28.4962	0.901578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.13	✓	19	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+0.537191 q^{3} -2.35184 q^{5} -1.00979 q^{7} -2.71143 q^{9} +O(q^{10})\)	\(q+0.537191 q^{3} -2.35184 q^{5} -1.00979 q^{7} -2.71143 q^{9} +3.55138 q^{11} +0.172083 q^{13} -1.26339 q^{15} +7.68990 q^{17} -1.00000 q^{19} -0.542449 q^{21} -8.17344 q^{23} +0.531160 q^{25} -3.06813 q^{27} +3.95811 q^{29} +8.96845 q^{31} +1.90777 q^{33} +2.37486 q^{35} -9.28780 q^{37} +0.0924417 q^{39} -0.449703 q^{41} +9.15762 q^{43} +6.37684 q^{45} -9.50313 q^{47} -5.98033 q^{49} +4.13095 q^{51} +1.00000 q^{53} -8.35229 q^{55} -0.537191 q^{57} -7.97890 q^{59} +3.34325 q^{61} +2.73797 q^{63} -0.404713 q^{65} -4.16062 q^{67} -4.39070 q^{69} +4.34907 q^{71} +3.25468 q^{73} +0.285335 q^{75} -3.58614 q^{77} +5.21648 q^{79} +6.48611 q^{81} -1.01694 q^{83} -18.0854 q^{85} +2.12626 q^{87} -8.11647 q^{89} -0.173768 q^{91} +4.81777 q^{93} +2.35184 q^{95} +3.08376 q^{97} -9.62930 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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