LMFDB - Embedded newform 4028.2.a.d.1.16

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			1.16
Root			$-1.94972$ 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)

$f(q)$	$=$	$q+1.94972 q^{3} +1.42435 q^{5} -4.01869 q^{7} +0.801405 q^{9} +O(q^{10})$	$q+1.94972 q^{3} +1.42435 q^{5} -4.01869 q^{7} +0.801405 q^{9} -1.76924 q^{11} -0.162788 q^{13} +2.77708 q^{15} +3.69306 q^{17} -1.00000 q^{19} -7.83532 q^{21} +1.52669 q^{23} -2.97122 q^{25} -4.28664 q^{27} +2.66959 q^{29} -10.7471 q^{31} -3.44952 q^{33} -5.72403 q^{35} +1.29232 q^{37} -0.317391 q^{39} -7.80202 q^{41} +4.54569 q^{43} +1.14148 q^{45} -8.83786 q^{47} +9.14987 q^{49} +7.20043 q^{51} +1.00000 q^{53} -2.52002 q^{55} -1.94972 q^{57} -2.95882 q^{59} -7.23691 q^{61} -3.22060 q^{63} -0.231867 q^{65} +14.9269 q^{67} +2.97662 q^{69} -11.1361 q^{71} +0.331709 q^{73} -5.79305 q^{75} +7.11002 q^{77} -4.60353 q^{79} -10.7620 q^{81} -8.46798 q^{83} +5.26021 q^{85} +5.20494 q^{87} +3.52631 q^{89} +0.654194 q^{91} -20.9539 q^{93} -1.42435 q^{95} +14.8111 q^{97} -1.41788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.94972	1.12567	0.562835	−	0.826569i	$-0.309710\pi$
$3$	1.94972	1.12567	0.562835	+	0.826569i	$0.309710\pi$
$4$	0	0
$5$	1.42435	0.636989	0.318495	−	0.947925i	$-0.396823\pi$
$5$	1.42435	0.636989	0.318495	+	0.947925i	$0.396823\pi$
$6$	0	0
$7$	−4.01869	−1.51892	−0.759461	−	0.650553i	$-0.774537\pi$
$7$	−4.01869	−1.51892	−0.759461	+	0.650553i	$0.774537\pi$
$8$	0	0
$9$	0.801405	0.267135
$10$	0	0
$11$	−1.76924	−0.533445	−0.266723	−	0.963773i	$-0.585941\pi$
$11$	−1.76924	−0.533445	−0.266723	+	0.963773i	$0.585941\pi$
$12$	0	0
$13$	−0.162788	−0.0451492	−0.0225746	−	0.999745i	$-0.507186\pi$
$13$	−0.162788	−0.0451492	−0.0225746	+	0.999745i	$0.507186\pi$
$14$	0	0
$15$	2.77708	0.717040
$16$	0	0
$17$	3.69306	0.895698	0.447849	−	0.894109i	$-0.352190\pi$
$17$	3.69306	0.895698	0.447849	+	0.894109i	$0.352190\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−7.83532	−1.70981
$22$	0	0
$23$	1.52669	0.318337	0.159169	−	0.987251i	$-0.449119\pi$
$23$	1.52669	0.318337	0.159169	+	0.987251i	$0.449119\pi$
$24$	0	0
$25$	−2.97122	−0.594245
$26$	0	0
$27$	−4.28664	−0.824965
$28$	0	0
$29$	2.66959	0.495730	0.247865	−	0.968795i	$-0.420271\pi$
$29$	2.66959	0.495730	0.247865	+	0.968795i	$0.420271\pi$
$30$	0	0
$31$	−10.7471	−1.93024	−0.965119	−	0.261810i	$-0.915681\pi$
$31$	−10.7471	−1.93024	−0.965119	+	0.261810i	$0.915681\pi$
$32$	0	0
$33$	−3.44952	−0.600484
$34$	0	0
$35$	−5.72403	−0.967537
$36$	0	0
$37$	1.29232	0.212455	0.106228	−	0.994342i	$-0.466123\pi$
$37$	1.29232	0.212455	0.106228	+	0.994342i	$0.466123\pi$
$38$	0	0
$39$	−0.317391	−0.0508232
$40$	0	0
$41$	−7.80202	−1.21847	−0.609235	−	0.792989i	$-0.708524\pi$
$41$	−7.80202	−1.21847	−0.609235	+	0.792989i	$0.708524\pi$
$42$	0	0
$43$	4.54569	0.693211	0.346606	−	0.938011i	$-0.387334\pi$
$43$	4.54569	0.693211	0.346606	+	0.938011i	$0.387334\pi$
$44$	0	0
$45$	1.14148	0.170162
$46$	0	0
$47$	−8.83786	−1.28913	−0.644567	−	0.764548i	$-0.722962\pi$
$47$	−8.83786	−1.28913	−0.644567	+	0.764548i	$0.722962\pi$
$48$	0	0
$49$	9.14987	1.30712
$50$	0	0
$51$	7.20043	1.00826
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−2.52002	−0.339799
$56$	0	0
$57$	−1.94972	−0.258247
$58$	0	0
$59$	−2.95882	−0.385205	−0.192603	−	0.981277i	$-0.561693\pi$
$59$	−2.95882	−0.385205	−0.192603	+	0.981277i	$0.561693\pi$
$60$	0	0
$61$	−7.23691	−0.926592	−0.463296	−	0.886204i	$-0.653333\pi$
$61$	−7.23691	−0.926592	−0.463296	+	0.886204i	$0.653333\pi$
$62$	0	0
$63$	−3.22060	−0.405757
$64$	0	0
$65$	−0.231867	−0.0287596
$66$	0	0
$67$	14.9269	1.82361	0.911806	−	0.410621i	$-0.134688\pi$
$67$	14.9269	1.82361	0.911806	+	0.410621i	$0.134688\pi$
$68$	0	0
$69$	2.97662	0.358343
$70$	0	0
$71$	−11.1361	−1.32161	−0.660807	−	0.750556i	$-0.729786\pi$
$71$	−11.1361	−1.32161	−0.660807	+	0.750556i	$0.729786\pi$
$72$	0	0
$73$	0.331709	0.0388236	0.0194118	−	0.999812i	$-0.493821\pi$
$73$	0.331709	0.0388236	0.0194118	+	0.999812i	$0.493821\pi$
$74$	0	0
$75$	−5.79305	−0.668924
$76$	0	0
$77$	7.11002	0.810262
$78$	0	0
$79$	−4.60353	−0.517937	−0.258969	−	0.965886i	$-0.583383\pi$
$79$	−4.60353	−0.517937	−0.258969	+	0.965886i	$0.583383\pi$
$80$	0	0
$81$	−10.7620	−1.19577
$82$	0	0
$83$	−8.46798	−0.929482	−0.464741	−	0.885447i	$-0.653853\pi$
$83$	−8.46798	−0.929482	−0.464741	+	0.885447i	$0.653853\pi$
$84$	0	0
$85$	5.26021	0.570550
$86$	0	0
$87$	5.20494	0.558029
$88$	0	0
$89$	3.52631	0.373788	0.186894	−	0.982380i	$-0.440158\pi$
$89$	3.52631	0.373788	0.186894	+	0.982380i	$0.440158\pi$
$90$	0	0
$91$	0.654194	0.0685782
$92$	0	0
$93$	−20.9539	−2.17281
$94$	0	0
$95$	−1.42435	−0.146135
$96$	0	0
$97$	14.8111	1.50384	0.751918	−	0.659257i	$-0.229129\pi$
$97$	14.8111	1.50384	0.751918	+	0.659257i	$0.229129\pi$
$98$	0	0
$99$	−1.41788	−0.142502
$100$	0	0
$101$	−4.70443	−0.468109	−0.234054	−	0.972224i	$-0.575199\pi$
$101$	−4.70443	−0.468109	−0.234054	+	0.972224i	$0.575199\pi$
$102$	0	0
$103$	−3.54468	−0.349267	−0.174634	−	0.984633i	$-0.555874\pi$
$103$	−3.54468	−0.349267	−0.174634	+	0.984633i	$0.555874\pi$
$104$	0	0
$105$	−11.1602	−1.08913
$106$	0	0
$107$	−4.86249	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$107$	−4.86249	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$108$	0	0
$109$	−3.01434	−0.288722	−0.144361	−	0.989525i	$-0.546113\pi$
$109$	−3.01434	−0.288722	−0.144361	+	0.989525i	$0.546113\pi$
$110$	0	0
$111$	2.51965	0.239155
$112$	0	0
$113$	−2.73218	−0.257022	−0.128511	−	0.991708i	$-0.541020\pi$
$113$	−2.73218	−0.257022	−0.128511	+	0.991708i	$0.541020\pi$
$114$	0	0
$115$	2.17455	0.202777
$116$	0	0
$117$	−0.130459	−0.0120609
$118$	0	0
$119$	−14.8413	−1.36050
$120$	0	0
$121$	−7.86980	−0.715436
$122$	0	0
$123$	−15.2117	−1.37160
$124$	0	0
$125$	−11.3538	−1.01552
$126$	0	0
$127$	−11.3238	−1.00482	−0.502411	−	0.864629i	$-0.667554\pi$
$127$	−11.3238	−1.00482	−0.502411	+	0.864629i	$0.667554\pi$
$128$	0	0
$129$	8.86282	0.780328
$130$	0	0
$131$	−19.2810	−1.68459	−0.842295	−	0.539017i	$-0.818796\pi$
$131$	−19.2810	−1.68459	−0.842295	+	0.539017i	$0.818796\pi$
$132$	0	0
$133$	4.01869	0.348465
$134$	0	0
$135$	−6.10569	−0.525494
$136$	0	0
$137$	−3.20964	−0.274218	−0.137109	−	0.990556i	$-0.543781\pi$
$137$	−3.20964	−0.274218	−0.137109	+	0.990556i	$0.543781\pi$
$138$	0	0
$139$	2.54096	0.215521	0.107761	−	0.994177i	$-0.465632\pi$
$139$	2.54096	0.215521	0.107761	+	0.994177i	$0.465632\pi$
$140$	0	0
$141$	−17.2313	−1.45114
$142$	0	0
$143$	0.288010	0.0240846
$144$	0	0
$145$	3.80243	0.315774
$146$	0	0
$147$	17.8397	1.47139
$148$	0	0
$149$	12.5830	1.03084	0.515422	−	0.856937i	$-0.327635\pi$
$149$	12.5830	1.03084	0.515422	+	0.856937i	$0.327635\pi$
$150$	0	0
$151$	−0.0949678	−0.00772837	−0.00386419	−	0.999993i	$-0.501230\pi$
$151$	−0.0949678	−0.00772837	−0.00386419	+	0.999993i	$0.501230\pi$
$152$	0	0
$153$	2.95963	0.239272
$154$	0	0
$155$	−15.3077	−1.22954
$156$	0	0
$157$	3.06340	0.244486	0.122243	−	0.992500i	$-0.460991\pi$
$157$	3.06340	0.244486	0.122243	+	0.992500i	$0.460991\pi$
$158$	0	0
$159$	1.94972	0.154623
$160$	0	0
$161$	−6.13530	−0.483530
$162$	0	0
$163$	9.09905	0.712693	0.356346	−	0.934354i	$-0.384022\pi$
$163$	9.09905	0.712693	0.356346	+	0.934354i	$0.384022\pi$
$164$	0	0
$165$	−4.91332	−0.382502
$166$	0	0
$167$	4.22356	0.326829	0.163415	−	0.986557i	$-0.447749\pi$
$167$	4.22356	0.326829	0.163415	+	0.986557i	$0.447749\pi$
$168$	0	0
$169$	−12.9735	−0.997962
$170$	0	0
$171$	−0.801405	−0.0612850
$172$	0	0
$173$	23.7089	1.80256	0.901279	−	0.433240i	$-0.142630\pi$
$173$	23.7089	1.80256	0.901279	+	0.433240i	$0.142630\pi$
$174$	0	0
$175$	11.9404	0.902612
$176$	0	0
$177$	−5.76887	−0.433615
$178$	0	0
$179$	−9.81120	−0.733323	−0.366662	−	0.930354i	$-0.619499\pi$
$179$	−9.81120	−0.733323	−0.366662	+	0.930354i	$0.619499\pi$
$180$	0	0
$181$	6.32977	0.470488	0.235244	−	0.971936i	$-0.424411\pi$
$181$	6.32977	0.470488	0.235244	+	0.971936i	$0.424411\pi$
$182$	0	0
$183$	−14.1099	−1.04304
$184$	0	0
$185$	1.84071	0.135332
$186$	0	0
$187$	−6.53390	−0.477806
$188$	0	0
$189$	17.2267	1.25306
$190$	0	0
$191$	22.6326	1.63764	0.818820	−	0.574050i	$-0.194628\pi$
$191$	22.6326	1.63764	0.818820	+	0.574050i	$0.194628\pi$
$192$	0	0
$193$	−11.4711	−0.825706	−0.412853	−	0.910798i	$-0.635468\pi$
$193$	−11.4711	−0.825706	−0.412853	+	0.910798i	$0.635468\pi$
$194$	0	0
$195$	−0.452076	−0.0323738
$196$	0	0
$197$	−13.9329	−0.992679	−0.496340	−	0.868128i	$-0.665323\pi$
$197$	−13.9329	−0.992679	−0.496340	+	0.868128i	$0.665323\pi$
$198$	0	0
$199$	−10.0427	−0.711907	−0.355954	−	0.934504i	$-0.615844\pi$
$199$	−10.0427	−0.711907	−0.355954	+	0.934504i	$0.615844\pi$
$200$	0	0
$201$	29.1033	2.05279
$202$	0	0
$203$	−10.7282	−0.752975
$204$	0	0
$205$	−11.1128	−0.776153
$206$	0	0
$207$	1.22350	0.0850390
$208$	0	0
$209$	1.76924	0.122381
$210$	0	0
$211$	−0.736794	−0.0507230	−0.0253615	−	0.999678i	$-0.508074\pi$
$211$	−0.736794	−0.0507230	−0.0253615	+	0.999678i	$0.508074\pi$
$212$	0	0
$213$	−21.7123	−1.48770
$214$	0	0
$215$	6.47466	0.441568
$216$	0	0
$217$	43.1893	2.93188
$218$	0	0
$219$	0.646740	0.0437026
$220$	0	0
$221$	−0.601185	−0.0404401
$222$	0	0
$223$	−1.32103	−0.0884629	−0.0442315	−	0.999021i	$-0.514084\pi$
$223$	−1.32103	−0.0884629	−0.0442315	+	0.999021i	$0.514084\pi$
$224$	0	0
$225$	−2.38115	−0.158744
$226$	0	0
$227$	10.2954	0.683332	0.341666	−	0.939821i	$-0.389009\pi$
$227$	10.2954	0.683332	0.341666	+	0.939821i	$0.389009\pi$
$228$	0	0
$229$	14.9132	0.985491	0.492745	−	0.870174i	$-0.335993\pi$
$229$	14.9132	0.985491	0.492745	+	0.870174i	$0.335993\pi$
$230$	0	0
$231$	13.8625	0.912088
$232$	0	0
$233$	2.93685	0.192399	0.0961997	−	0.995362i	$-0.469331\pi$
$233$	2.93685	0.192399	0.0961997	+	0.995362i	$0.469331\pi$
$234$	0	0
$235$	−12.5882	−0.821165
$236$	0	0
$237$	−8.97559	−0.583027
$238$	0	0
$239$	−2.93200	−0.189655	−0.0948276	−	0.995494i	$-0.530230\pi$
$239$	−2.93200	−0.189655	−0.0948276	+	0.995494i	$0.530230\pi$
$240$	0	0
$241$	22.5700	1.45386	0.726930	−	0.686711i	$-0.240946\pi$
$241$	22.5700	1.45386	0.726930	+	0.686711i	$0.240946\pi$
$242$	0	0
$243$	−8.12288	−0.521083
$244$	0	0
$245$	13.0326	0.832624
$246$	0	0
$247$	0.162788	0.0103579
$248$	0	0
$249$	−16.5102	−1.04629
$250$	0	0
$251$	15.4083	0.972562	0.486281	−	0.873802i	$-0.338353\pi$
$251$	15.4083	0.972562	0.486281	+	0.873802i	$0.338353\pi$
$252$	0	0
$253$	−2.70108	−0.169815
$254$	0	0
$255$	10.2559	0.642252
$256$	0	0
$257$	0.167907	0.0104737	0.00523686	−	0.999986i	$-0.498333\pi$
$257$	0.167907	0.0104737	0.00523686	+	0.999986i	$0.498333\pi$
$258$	0	0
$259$	−5.19342	−0.322703
$260$	0	0
$261$	2.13942	0.132427
$262$	0	0
$263$	−8.62095	−0.531590	−0.265795	−	0.964030i	$-0.585635\pi$
$263$	−8.62095	−0.531590	−0.265795	+	0.964030i	$0.585635\pi$
$264$	0	0
$265$	1.42435	0.0874972
$266$	0	0
$267$	6.87531	0.420762
$268$	0	0
$269$	−20.5238	−1.25136	−0.625678	−	0.780081i	$-0.715178\pi$
$269$	−20.5238	−1.25136	−0.625678	+	0.780081i	$0.715178\pi$
$270$	0	0
$271$	−12.1308	−0.736892	−0.368446	−	0.929649i	$-0.620110\pi$
$271$	−12.1308	−0.736892	−0.368446	+	0.929649i	$0.620110\pi$
$272$	0	0
$273$	1.27549	0.0771964
$274$	0	0
$275$	5.25680	0.316997
$276$	0	0
$277$	19.2451	1.15633	0.578163	−	0.815921i	$-0.303770\pi$
$277$	19.2451	1.15633	0.578163	+	0.815921i	$0.303770\pi$
$278$	0	0
$279$	−8.61279	−0.515634
$280$	0	0
$281$	−18.6933	−1.11515	−0.557574	−	0.830127i	$-0.688268\pi$
$281$	−18.6933	−1.11515	−0.557574	+	0.830127i	$0.688268\pi$
$282$	0	0
$283$	7.24216	0.430502	0.215251	−	0.976559i	$-0.430943\pi$
$283$	7.24216	0.430502	0.215251	+	0.976559i	$0.430943\pi$
$284$	0	0
$285$	−2.77708	−0.164500
$286$	0	0
$287$	31.3539	1.85076
$288$	0	0
$289$	−3.36132	−0.197725
$290$	0	0
$291$	28.8774	1.69282
$292$	0	0
$293$	28.5009	1.66504	0.832521	−	0.553994i	$-0.186897\pi$
$293$	28.5009	1.66504	0.832521	+	0.553994i	$0.186897\pi$
$294$	0	0
$295$	−4.21440	−0.245372
$296$	0	0
$297$	7.58409	0.440073
$298$	0	0
$299$	−0.248527	−0.0143727
$300$	0	0
$301$	−18.2677	−1.05293
$302$	0	0
$303$	−9.17232	−0.526936
$304$	0	0
$305$	−10.3079	−0.590229
$306$	0	0
$307$	−4.60309	−0.262712	−0.131356	−	0.991335i	$-0.541933\pi$
$307$	−4.60309	−0.262712	−0.131356	+	0.991335i	$0.541933\pi$
$308$	0	0
$309$	−6.91112	−0.393160
$310$	0	0
$311$	1.69986	0.0963901	0.0481951	−	0.998838i	$-0.484653\pi$
$311$	1.69986	0.0963901	0.0481951	+	0.998838i	$0.484653\pi$
$312$	0	0
$313$	−1.68364	−0.0951649	−0.0475825	−	0.998867i	$-0.515152\pi$
$313$	−1.68364	−0.0951649	−0.0475825	+	0.998867i	$0.515152\pi$
$314$	0	0
$315$	−4.58726	−0.258463
$316$	0	0
$317$	−19.4273	−1.09115	−0.545573	−	0.838063i	$-0.683688\pi$
$317$	−19.4273	−1.09115	−0.545573	+	0.838063i	$0.683688\pi$
$318$	0	0
$319$	−4.72313	−0.264445
$320$	0	0
$321$	−9.48048	−0.529149
$322$	0	0
$323$	−3.69306	−0.205487
$324$	0	0
$325$	0.483679	0.0268297
$326$	0	0
$327$	−5.87712	−0.325006
$328$	0	0
$329$	35.5166	1.95809
$330$	0	0
$331$	12.1416	0.667365	0.333682	−	0.942686i	$-0.391709\pi$
$331$	12.1416	0.667365	0.333682	+	0.942686i	$0.391709\pi$
$332$	0	0
$333$	1.03567	0.0567543
$334$	0	0
$335$	21.2612	1.16162
$336$	0	0
$337$	11.7602	0.640619	0.320309	−	0.947313i	$-0.396213\pi$
$337$	11.7602	0.640619	0.320309	+	0.947313i	$0.396213\pi$
$338$	0	0
$339$	−5.32699	−0.289323
$340$	0	0
$341$	19.0142	1.02968
$342$	0	0
$343$	−8.63967	−0.466498
$344$	0	0
$345$	4.23975	0.228261
$346$	0	0
$347$	20.8677	1.12024	0.560118	−	0.828413i	$-0.310756\pi$
$347$	20.8677	1.12024	0.560118	+	0.828413i	$0.310756\pi$
$348$	0	0
$349$	33.9337	1.81643	0.908216	−	0.418502i	$-0.137445\pi$
$349$	33.9337	1.81643	0.908216	+	0.418502i	$0.137445\pi$
$350$	0	0
$351$	0.697813	0.0372465
$352$	0	0
$353$	8.87569	0.472405	0.236203	−	0.971704i	$-0.424097\pi$
$353$	8.87569	0.472405	0.236203	+	0.971704i	$0.424097\pi$
$354$	0	0
$355$	−15.8617	−0.841854
$356$	0	0
$357$	−28.9363	−1.53147
$358$	0	0
$359$	−3.06841	−0.161945	−0.0809723	−	0.996716i	$-0.525803\pi$
$359$	−3.06841	−0.161945	−0.0809723	+	0.996716i	$0.525803\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−15.3439	−0.805346
$364$	0	0
$365$	0.472470	0.0247302
$366$	0	0
$367$	11.7441	0.613038	0.306519	−	0.951865i	$-0.400836\pi$
$367$	11.7441	0.613038	0.306519	+	0.951865i	$0.400836\pi$
$368$	0	0
$369$	−6.25258	−0.325496
$370$	0	0
$371$	−4.01869	−0.208640
$372$	0	0
$373$	−19.9768	−1.03436	−0.517179	−	0.855877i	$-0.673018\pi$
$373$	−19.9768	−1.03436	−0.517179	+	0.855877i	$0.673018\pi$
$374$	0	0
$375$	−22.1368	−1.14314
$376$	0	0
$377$	−0.434576	−0.0223818
$378$	0	0
$379$	5.66779	0.291135	0.145567	−	0.989348i	$-0.453499\pi$
$379$	5.66779	0.291135	0.145567	+	0.989348i	$0.453499\pi$
$380$	0	0
$381$	−22.0782	−1.13110
$382$	0	0
$383$	6.72159	0.343457	0.171728	−	0.985144i	$-0.445065\pi$
$383$	6.72159	0.343457	0.171728	+	0.985144i	$0.445065\pi$
$384$	0	0
$385$	10.1272	0.516128
$386$	0	0
$387$	3.64294	0.185181
$388$	0	0
$389$	−9.08084	−0.460417	−0.230208	−	0.973141i	$-0.573941\pi$
$389$	−9.08084	−0.460417	−0.230208	+	0.973141i	$0.573941\pi$
$390$	0	0
$391$	5.63816	0.285134
$392$	0	0
$393$	−37.5926	−1.89629
$394$	0	0
$395$	−6.55704	−0.329921
$396$	0	0
$397$	−10.9518	−0.549657	−0.274828	−	0.961493i	$-0.588621\pi$
$397$	−10.9518	−0.549657	−0.274828	+	0.961493i	$0.588621\pi$
$398$	0	0
$399$	7.83532	0.392257
$400$	0	0
$401$	−1.26592	−0.0632169	−0.0316084	−	0.999500i	$-0.510063\pi$
$401$	−1.26592	−0.0632169	−0.0316084	+	0.999500i	$0.510063\pi$
$402$	0	0
$403$	1.74950	0.0871488
$404$	0	0
$405$	−15.3288	−0.761695
$406$	0	0
$407$	−2.28641	−0.113333
$408$	0	0
$409$	17.2219	0.851570	0.425785	−	0.904824i	$-0.359998\pi$
$409$	17.2219	0.851570	0.425785	+	0.904824i	$0.359998\pi$
$410$	0	0
$411$	−6.25789	−0.308679
$412$	0	0
$413$	11.8906	0.585097
$414$	0	0
$415$	−12.0614	−0.592070
$416$	0	0
$417$	4.95415	0.242606
$418$	0	0
$419$	19.0193	0.929152	0.464576	−	0.885533i	$-0.346207\pi$
$419$	19.0193	0.929152	0.464576	+	0.885533i	$0.346207\pi$
$420$	0	0
$421$	−27.6185	−1.34604	−0.673021	−	0.739623i	$-0.735004\pi$
$421$	−27.6185	−1.34604	−0.673021	+	0.739623i	$0.735004\pi$
$422$	0	0
$423$	−7.08270	−0.344373
$424$	0	0
$425$	−10.9729	−0.532264
$426$	0	0
$427$	29.0829	1.40742
$428$	0	0
$429$	0.561539	0.0271114
$430$	0	0
$431$	−15.3626	−0.739990	−0.369995	−	0.929034i	$-0.620641\pi$
$431$	−15.3626	−0.739990	−0.369995	+	0.929034i	$0.620641\pi$
$432$	0	0
$433$	−6.98336	−0.335599	−0.167799	−	0.985821i	$-0.553666\pi$
$433$	−6.98336	−0.335599	−0.167799	+	0.985821i	$0.553666\pi$
$434$	0	0
$435$	7.41367	0.355458
$436$	0	0
$437$	−1.52669	−0.0730316
$438$	0	0
$439$	−31.0799	−1.48336	−0.741682	−	0.670752i	$-0.765971\pi$
$439$	−31.0799	−1.48336	−0.741682	+	0.670752i	$0.765971\pi$
$440$	0	0
$441$	7.33275	0.349179
$442$	0	0
$443$	13.0078	0.618018	0.309009	−	0.951059i	$-0.400003\pi$
$443$	13.0078	0.618018	0.309009	+	0.951059i	$0.400003\pi$
$444$	0	0
$445$	5.02270	0.238099
$446$	0	0
$447$	24.5334	1.16039
$448$	0	0
$449$	10.7300	0.506381	0.253191	−	0.967416i	$-0.418520\pi$
$449$	10.7300	0.506381	0.253191	+	0.967416i	$0.418520\pi$
$450$	0	0
$451$	13.8036	0.649987
$452$	0	0
$453$	−0.185161	−0.00869960
$454$	0	0
$455$	0.931802	0.0436836
$456$	0	0
$457$	−30.6499	−1.43374	−0.716871	−	0.697206i	$-0.754426\pi$
$457$	−30.6499	−1.43374	−0.716871	+	0.697206i	$0.754426\pi$
$458$	0	0
$459$	−15.8308	−0.738919
$460$	0	0
$461$	−1.50835	−0.0702507	−0.0351254	−	0.999383i	$-0.511183\pi$
$461$	−1.50835	−0.0702507	−0.0351254	+	0.999383i	$0.511183\pi$
$462$	0	0
$463$	−13.4523	−0.625181	−0.312591	−	0.949888i	$-0.601197\pi$
$463$	−13.4523	−0.625181	−0.312591	+	0.949888i	$0.601197\pi$
$464$	0	0
$465$	−29.8457	−1.38406
$466$	0	0
$467$	33.4845	1.54948	0.774738	−	0.632282i	$-0.217882\pi$
$467$	33.4845	1.54948	0.774738	+	0.632282i	$0.217882\pi$
$468$	0	0
$469$	−59.9866	−2.76993
$470$	0	0
$471$	5.97278	0.275211
$472$	0	0
$473$	−8.04241	−0.369790
$474$	0	0
$475$	2.97122	0.136329
$476$	0	0
$477$	0.801405	0.0366938
$478$	0	0
$479$	−3.80274	−0.173752	−0.0868758	−	0.996219i	$-0.527688\pi$
$479$	−3.80274	−0.173752	−0.0868758	+	0.996219i	$0.527688\pi$
$480$	0	0
$481$	−0.210373	−0.00959220
$482$	0	0
$483$	−11.9621	−0.544295
$484$	0	0
$485$	21.0962	0.957927
$486$	0	0
$487$	−41.6493	−1.88731	−0.943655	−	0.330932i	$-0.892637\pi$
$487$	−41.6493	−1.88731	−0.943655	+	0.330932i	$0.892637\pi$
$488$	0	0
$489$	17.7406	0.802258
$490$	0	0
$491$	23.7177	1.07036	0.535182	−	0.844737i	$-0.320243\pi$
$491$	23.7177	1.07036	0.535182	+	0.844737i	$0.320243\pi$
$492$	0	0
$493$	9.85894	0.444024
$494$	0	0
$495$	−2.01955	−0.0907721
$496$	0	0
$497$	44.7526	2.00743
$498$	0	0
$499$	16.2505	0.727472	0.363736	−	0.931502i	$-0.381501\pi$
$499$	16.2505	0.727472	0.363736	+	0.931502i	$0.381501\pi$
$500$	0	0
$501$	8.23476	0.367902
$502$	0	0
$503$	24.0712	1.07328	0.536641	−	0.843811i	$-0.319693\pi$
$503$	24.0712	1.07328	0.536641	+	0.843811i	$0.319693\pi$
$504$	0	0
$505$	−6.70076	−0.298180
$506$	0	0
$507$	−25.2947	−1.12338
$508$	0	0
$509$	−32.1638	−1.42564	−0.712819	−	0.701348i	$-0.752582\pi$
$509$	−32.1638	−1.42564	−0.712819	+	0.701348i	$0.752582\pi$
$510$	0	0
$511$	−1.33304	−0.0589701
$512$	0	0
$513$	4.28664	0.189260
$514$	0	0
$515$	−5.04886	−0.222479
$516$	0	0
$517$	15.6363	0.687682
$518$	0	0
$519$	46.2258	2.02909
$520$	0	0
$521$	−24.6898	−1.08168	−0.540840	−	0.841125i	$-0.681894\pi$
$521$	−24.6898	−1.08168	−0.540840	+	0.841125i	$0.681894\pi$
$522$	0	0
$523$	32.2724	1.41117	0.705587	−	0.708623i	$-0.250683\pi$
$523$	32.2724	1.41117	0.705587	+	0.708623i	$0.250683\pi$
$524$	0	0
$525$	23.2805	1.01604
$526$	0	0
$527$	−39.6897	−1.72891
$528$	0	0
$529$	−20.6692	−0.898661
$530$	0	0
$531$	−2.37121	−0.102902
$532$	0	0
$533$	1.27007	0.0550130
$534$	0	0
$535$	−6.92589	−0.299432
$536$	0	0
$537$	−19.1291	−0.825481
$538$	0	0
$539$	−16.1883	−0.697279
$540$	0	0
$541$	15.4223	0.663055	0.331528	−	0.943445i	$-0.392436\pi$
$541$	15.4223	0.663055	0.331528	+	0.943445i	$0.392436\pi$
$542$	0	0
$543$	12.3413	0.529614
$544$	0	0
$545$	−4.29348	−0.183913
$546$	0	0
$547$	27.9367	1.19449	0.597245	−	0.802059i	$-0.296262\pi$
$547$	27.9367	1.19449	0.597245	+	0.802059i	$0.296262\pi$
$548$	0	0
$549$	−5.79970	−0.247525
$550$	0	0
$551$	−2.66959	−0.113728
$552$	0	0
$553$	18.5002	0.786707
$554$	0	0
$555$	3.58887	0.152339
$556$	0	0
$557$	−29.4559	−1.24809	−0.624044	−	0.781389i	$-0.714511\pi$
$557$	−29.4559	−1.24809	−0.624044	+	0.781389i	$0.714511\pi$
$558$	0	0
$559$	−0.739983	−0.0312980
$560$	0	0
$561$	−12.7393	−0.537852
$562$	0	0
$563$	−28.3290	−1.19392	−0.596962	−	0.802269i	$-0.703626\pi$
$563$	−28.3290	−1.19392	−0.596962	+	0.802269i	$0.703626\pi$
$564$	0	0
$565$	−3.89159	−0.163720
$566$	0	0
$567$	43.2490	1.81629
$568$	0	0
$569$	33.5542	1.40666	0.703332	−	0.710862i	$-0.251695\pi$
$569$	33.5542	1.40666	0.703332	+	0.710862i	$0.251695\pi$
$570$	0	0
$571$	43.1494	1.80574	0.902872	−	0.429909i	$-0.141454\pi$
$571$	43.1494	1.80574	0.902872	+	0.429909i	$0.141454\pi$
$572$	0	0
$573$	44.1273	1.84344
$574$	0	0
$575$	−4.53614	−0.189170
$576$	0	0
$577$	3.79949	0.158175	0.0790873	−	0.996868i	$-0.474799\pi$
$577$	3.79949	0.158175	0.0790873	+	0.996868i	$0.474799\pi$
$578$	0	0
$579$	−22.3654	−0.929473
$580$	0	0
$581$	34.0302	1.41181
$582$	0	0
$583$	−1.76924	−0.0732743
$584$	0	0
$585$	−0.185819	−0.00768269
$586$	0	0
$587$	−6.94946	−0.286835	−0.143417	−	0.989662i	$-0.545809\pi$
$587$	−6.94946	−0.286835	−0.143417	+	0.989662i	$0.545809\pi$
$588$	0	0
$589$	10.7471	0.442827
$590$	0	0
$591$	−27.1653	−1.11743
$592$	0	0
$593$	−17.2668	−0.709063	−0.354532	−	0.935044i	$-0.615360\pi$
$593$	−17.2668	−0.709063	−0.354532	+	0.935044i	$0.615360\pi$
$594$	0	0
$595$	−21.1392	−0.866621
$596$	0	0
$597$	−19.5804	−0.801373
$598$	0	0
$599$	−9.97441	−0.407543	−0.203772	−	0.979018i	$-0.565320\pi$
$599$	−9.97441	−0.407543	−0.203772	+	0.979018i	$0.565320\pi$
$600$	0	0
$601$	−1.42390	−0.0580821	−0.0290410	−	0.999578i	$-0.509245\pi$
$601$	−1.42390	−0.0580821	−0.0290410	+	0.999578i	$0.509245\pi$
$602$	0	0
$603$	11.9625	0.487151
$604$	0	0
$605$	−11.2094	−0.455725
$606$	0	0
$607$	−30.0325	−1.21898	−0.609490	−	0.792794i	$-0.708626\pi$
$607$	−30.0325	−1.21898	−0.609490	+	0.792794i	$0.708626\pi$
$608$	0	0
$609$	−20.9171	−0.847602
$610$	0	0
$611$	1.43870	0.0582034
$612$	0	0
$613$	43.3035	1.74901	0.874505	−	0.485016i	$-0.161186\pi$
$613$	43.3035	1.74901	0.874505	+	0.485016i	$0.161186\pi$
$614$	0	0
$615$	−21.6669	−0.873693
$616$	0	0
$617$	−34.5602	−1.39134	−0.695670	−	0.718362i	$-0.744892\pi$
$617$	−34.5602	−1.39134	−0.695670	+	0.718362i	$0.744892\pi$
$618$	0	0
$619$	13.9032	0.558816	0.279408	−	0.960172i	$-0.409862\pi$
$619$	13.9032	0.558816	0.279408	+	0.960172i	$0.409862\pi$
$620$	0	0
$621$	−6.54438	−0.262617
$622$	0	0
$623$	−14.1711	−0.567755
$624$	0	0
$625$	−1.31571	−0.0526285
$626$	0	0
$627$	3.44952	0.137760
$628$	0	0
$629$	4.77260	0.190296
$630$	0	0
$631$	−29.8687	−1.18906	−0.594528	−	0.804075i	$-0.702661\pi$
$631$	−29.8687	−1.18906	−0.594528	+	0.804075i	$0.702661\pi$
$632$	0	0
$633$	−1.43654	−0.0570974
$634$	0	0
$635$	−16.1290	−0.640061
$636$	0	0
$637$	−1.48949	−0.0590157
$638$	0	0
$639$	−8.92454	−0.353049
$640$	0	0
$641$	23.0211	0.909280	0.454640	−	0.890675i	$-0.349768\pi$
$641$	23.0211	0.909280	0.454640	+	0.890675i	$0.349768\pi$
$642$	0	0
$643$	−13.5428	−0.534076	−0.267038	−	0.963686i	$-0.586045\pi$
$643$	−13.5428	−0.534076	−0.267038	+	0.963686i	$0.586045\pi$
$644$	0	0
$645$	12.6238	0.497060
$646$	0	0
$647$	−30.7319	−1.20819	−0.604097	−	0.796911i	$-0.706466\pi$
$647$	−30.7319	−1.20819	−0.604097	+	0.796911i	$0.706466\pi$
$648$	0	0
$649$	5.23485	0.205486
$650$	0	0
$651$	84.2071	3.30034
$652$	0	0
$653$	17.3399	0.678564	0.339282	−	0.940685i	$-0.389816\pi$
$653$	17.3399	0.678564	0.339282	+	0.940685i	$0.389816\pi$
$654$	0	0
$655$	−27.4629	−1.07307
$656$	0	0
$657$	0.265833	0.0103711
$658$	0	0
$659$	13.4149	0.522571	0.261286	−	0.965262i	$-0.415854\pi$
$659$	13.4149	0.522571	0.261286	+	0.965262i	$0.415854\pi$
$660$	0	0
$661$	5.94449	0.231214	0.115607	−	0.993295i	$-0.463119\pi$
$661$	5.94449	0.231214	0.115607	+	0.993295i	$0.463119\pi$
$662$	0	0
$663$	−1.17214	−0.0455222
$664$	0	0
$665$	5.72403	0.221968
$666$	0	0
$667$	4.07564	0.157809
$668$	0	0
$669$	−2.57564	−0.0995801
$670$	0	0
$671$	12.8038	0.494286
$672$	0	0
$673$	33.1464	1.27770	0.638849	−	0.769332i	$-0.279411\pi$
$673$	33.1464	1.27770	0.638849	+	0.769332i	$0.279411\pi$
$674$	0	0
$675$	12.7366	0.490231
$676$	0	0
$677$	34.0632	1.30915	0.654577	−	0.755996i	$-0.272847\pi$
$677$	34.0632	1.30915	0.654577	+	0.755996i	$0.272847\pi$
$678$	0	0
$679$	−59.5211	−2.28421
$680$	0	0
$681$	20.0732	0.769207
$682$	0	0
$683$	−15.3969	−0.589146	−0.294573	−	0.955629i	$-0.595177\pi$
$683$	−15.3969	−0.589146	−0.294573	+	0.955629i	$0.595177\pi$
$684$	0	0
$685$	−4.57165	−0.174674
$686$	0	0
$687$	29.0765	1.10934
$688$	0	0
$689$	−0.162788	−0.00620172
$690$	0	0
$691$	−23.1007	−0.878793	−0.439396	−	0.898293i	$-0.644808\pi$
$691$	−23.1007	−0.878793	−0.439396	+	0.898293i	$0.644808\pi$
$692$	0	0
$693$	5.69800	0.216449
$694$	0	0
$695$	3.61921	0.137285
$696$	0	0
$697$	−28.8133	−1.09138
$698$	0	0
$699$	5.72603	0.216579
$700$	0	0
$701$	31.9502	1.20674	0.603371	−	0.797461i	$-0.293824\pi$
$701$	31.9502	1.20674	0.603371	+	0.797461i	$0.293824\pi$
$702$	0	0
$703$	−1.29232	−0.0487406
$704$	0	0
$705$	−24.5435	−0.924361
$706$	0	0
$707$	18.9057	0.711020
$708$	0	0
$709$	27.7244	1.04121	0.520605	−	0.853797i	$-0.325706\pi$
$709$	27.7244	1.04121	0.520605	+	0.853797i	$0.325706\pi$
$710$	0	0
$711$	−3.68929	−0.138359
$712$	0	0
$713$	−16.4075	−0.614467
$714$	0	0
$715$	0.410228	0.0153417
$716$	0	0
$717$	−5.71657	−0.213489
$718$	0	0
$719$	−41.8691	−1.56146	−0.780728	−	0.624871i	$-0.785151\pi$
$719$	−41.8691	−1.56146	−0.780728	+	0.624871i	$0.785151\pi$
$720$	0	0
$721$	14.2450	0.530510
$722$	0	0
$723$	44.0051	1.63657
$724$	0	0
$725$	−7.93194	−0.294585
$726$	0	0
$727$	−35.9844	−1.33459	−0.667293	−	0.744795i	$-0.732547\pi$
$727$	−35.9844	−1.33459	−0.667293	+	0.744795i	$0.732547\pi$
$728$	0	0
$729$	16.4486	0.609206
$730$	0	0
$731$	16.7875	0.620908
$732$	0	0
$733$	−17.0063	−0.628144	−0.314072	−	0.949399i	$-0.601693\pi$
$733$	−17.0063	−0.628144	−0.314072	+	0.949399i	$0.601693\pi$
$734$	0	0
$735$	25.4100	0.937261
$736$	0	0
$737$	−26.4092	−0.972797
$738$	0	0
$739$	11.8096	0.434422	0.217211	−	0.976125i	$-0.430304\pi$
$739$	11.8096	0.434422	0.217211	+	0.976125i	$0.430304\pi$
$740$	0	0
$741$	0.317391	0.0116596
$742$	0	0
$743$	−40.6894	−1.49275	−0.746375	−	0.665525i	$-0.768208\pi$
$743$	−40.6894	−1.49275	−0.746375	+	0.665525i	$0.768208\pi$
$744$	0	0
$745$	17.9227	0.656636
$746$	0	0
$747$	−6.78628	−0.248297
$748$	0	0
$749$	19.5408	0.714006
$750$	0	0
$751$	−2.94419	−0.107435	−0.0537175	−	0.998556i	$-0.517107\pi$
$751$	−2.94419	−0.107435	−0.0537175	+	0.998556i	$0.517107\pi$
$752$	0	0
$753$	30.0418	1.09479
$754$	0	0
$755$	−0.135268	−0.00492289
$756$	0	0
$757$	−10.8143	−0.393052	−0.196526	−	0.980499i	$-0.562966\pi$
$757$	−10.8143	−0.393052	−0.196526	+	0.980499i	$0.562966\pi$
$758$	0	0
$759$	−5.26635	−0.191156
$760$	0	0
$761$	−23.6050	−0.855679	−0.427839	−	0.903855i	$-0.640725\pi$
$761$	−23.6050	−0.855679	−0.427839	+	0.903855i	$0.640725\pi$
$762$	0	0
$763$	12.1137	0.438546
$764$	0	0
$765$	4.21556	0.152414
$766$	0	0
$767$	0.481660	0.0173917
$768$	0	0
$769$	26.5819	0.958568	0.479284	−	0.877660i	$-0.340896\pi$
$769$	26.5819	0.958568	0.479284	+	0.877660i	$0.340896\pi$
$770$	0	0
$771$	0.327371	0.0117900
$772$	0	0
$773$	−41.0455	−1.47630	−0.738151	−	0.674635i	$-0.764301\pi$
$773$	−41.0455	−1.47630	−0.738151	+	0.674635i	$0.764301\pi$
$774$	0	0
$775$	31.9321	1.14703
$776$	0	0
$777$	−10.1257	−0.363258
$778$	0	0
$779$	7.80202	0.279536
$780$	0	0
$781$	19.7024	0.705008
$782$	0	0
$783$	−11.4436	−0.408960
$784$	0	0
$785$	4.36336	0.155735
$786$	0	0
$787$	−14.3395	−0.511147	−0.255573	−	0.966790i	$-0.582264\pi$
$787$	−14.3395	−0.511147	−0.255573	+	0.966790i	$0.582264\pi$
$788$	0	0
$789$	−16.8084	−0.598396
$790$	0	0
$791$	10.9798	0.390397
$792$	0	0
$793$	1.17808	0.0418349
$794$	0	0
$795$	2.77708	0.0984931
$796$	0	0
$797$	−6.29350	−0.222927	−0.111464	−	0.993769i	$-0.535554\pi$
$797$	−6.29350	−0.222927	−0.111464	+	0.993769i	$0.535554\pi$
$798$	0	0
$799$	−32.6387	−1.15467
$800$	0	0
$801$	2.82600	0.0998518
$802$	0	0
$803$	−0.586872	−0.0207103
$804$	0	0
$805$	−8.73883	−0.308003
$806$	0	0
$807$	−40.0156	−1.40862
$808$	0	0
$809$	6.44386	0.226554	0.113277	−	0.993563i	$-0.463865\pi$
$809$	6.44386	0.226554	0.113277	+	0.993563i	$0.463865\pi$
$810$	0	0
$811$	−3.73893	−0.131291	−0.0656457	−	0.997843i	$-0.520911\pi$
$811$	−3.73893	−0.131291	−0.0656457	+	0.997843i	$0.520911\pi$
$812$	0	0
$813$	−23.6516	−0.829498
$814$	0	0
$815$	12.9602	0.453978
$816$	0	0
$817$	−4.54569	−0.159034
$818$	0	0
$819$	0.524274	0.0183196
$820$	0	0
$821$	−2.42599	−0.0846677	−0.0423339	−	0.999104i	$-0.513479\pi$
$821$	−2.42599	−0.0846677	−0.0423339	+	0.999104i	$0.513479\pi$
$822$	0	0
$823$	−7.88782	−0.274952	−0.137476	−	0.990505i	$-0.543899\pi$
$823$	−7.88782	−0.274952	−0.137476	+	0.990505i	$0.543899\pi$
$824$	0	0
$825$	10.2493	0.356834
$826$	0	0
$827$	−19.4720	−0.677109	−0.338554	−	0.940947i	$-0.609938\pi$
$827$	−19.4720	−0.677109	−0.338554	+	0.940947i	$0.609938\pi$
$828$	0	0
$829$	1.52092	0.0528239	0.0264119	−	0.999651i	$-0.491592\pi$
$829$	1.52092	0.0528239	0.0264119	+	0.999651i	$0.491592\pi$
$830$	0	0
$831$	37.5225	1.30164
$832$	0	0
$833$	33.7910	1.17079
$834$	0	0
$835$	6.01584	0.208187
$836$	0	0
$837$	46.0690	1.59238
$838$	0	0
$839$	−32.8160	−1.13293	−0.566467	−	0.824084i	$-0.691690\pi$
$839$	−32.8160	−1.13293	−0.566467	+	0.824084i	$0.691690\pi$
$840$	0	0
$841$	−21.8733	−0.754252
$842$	0	0
$843$	−36.4467	−1.25529
$844$	0	0
$845$	−18.4788	−0.635691
$846$	0	0
$847$	31.6263	1.08669
$848$	0	0
$849$	14.1202	0.484603
$850$	0	0
$851$	1.97297	0.0676325
$852$	0	0
$853$	−37.4185	−1.28119	−0.640593	−	0.767881i	$-0.721311\pi$
$853$	−37.4185	−1.28119	−0.640593	+	0.767881i	$0.721311\pi$
$854$	0	0
$855$	−1.14148	−0.0390379
$856$	0	0
$857$	23.2022	0.792571	0.396285	−	0.918127i	$-0.370299\pi$
$857$	23.2022	0.792571	0.396285	+	0.918127i	$0.370299\pi$
$858$	0	0
$859$	−20.1674	−0.688103	−0.344051	−	0.938951i	$-0.611799\pi$
$859$	−20.1674	−0.688103	−0.344051	+	0.938951i	$0.611799\pi$
$860$	0	0
$861$	61.1313	2.08335
$862$	0	0
$863$	−33.0397	−1.12468	−0.562342	−	0.826905i	$-0.690099\pi$
$863$	−33.0397	−1.12468	−0.562342	+	0.826905i	$0.690099\pi$
$864$	0	0
$865$	33.7699	1.14821
$866$	0	0
$867$	−6.55364	−0.222573
$868$	0	0
$869$	8.14473	0.276291
$870$	0	0
$871$	−2.42992	−0.0823347
$872$	0	0
$873$	11.8697	0.401727
$874$	0	0
$875$	45.6275	1.54249
$876$	0	0
$877$	36.0491	1.21729	0.608645	−	0.793443i	$-0.291713\pi$
$877$	36.0491	1.21729	0.608645	+	0.793443i	$0.291713\pi$
$878$	0	0
$879$	55.5688	1.87429
$880$	0	0
$881$	30.4686	1.02651	0.513257	−	0.858235i	$-0.328439\pi$
$881$	30.4686	1.02651	0.513257	+	0.858235i	$0.328439\pi$
$882$	0	0
$883$	−28.5425	−0.960530	−0.480265	−	0.877123i	$-0.659460\pi$
$883$	−28.5425	−0.960530	−0.480265	+	0.877123i	$0.659460\pi$
$884$	0	0
$885$	−8.21689	−0.276208
$886$	0	0
$887$	43.9704	1.47638	0.738191	−	0.674591i	$-0.235680\pi$
$887$	43.9704	1.47638	0.738191	+	0.674591i	$0.235680\pi$
$888$	0	0
$889$	45.5068	1.52625
$890$	0	0
$891$	19.0405	0.637880
$892$	0	0
$893$	8.83786	0.295748
$894$	0	0
$895$	−13.9746	−0.467119
$896$	0	0
$897$	−0.484558	−0.0161789
$898$	0	0
$899$	−28.6904	−0.956877
$900$	0	0
$901$	3.69306	0.123034
$902$	0	0
$903$	−35.6169	−1.18526
$904$	0	0
$905$	9.01581	0.299696
$906$	0	0
$907$	−3.54621	−0.117750	−0.0588750	−	0.998265i	$-0.518751\pi$
$907$	−3.54621	−0.117750	−0.0588750	+	0.998265i	$0.518751\pi$
$908$	0	0
$909$	−3.77016	−0.125048
$910$	0	0
$911$	45.2186	1.49816	0.749080	−	0.662480i	$-0.230496\pi$
$911$	45.2186	1.49816	0.749080	+	0.662480i	$0.230496\pi$
$912$	0	0
$913$	14.9819	0.495827
$914$	0	0
$915$	−20.0975	−0.664404
$916$	0	0
$917$	77.4844	2.55876
$918$	0	0
$919$	35.3502	1.16610	0.583048	−	0.812438i	$-0.301860\pi$
$919$	35.3502	1.16610	0.583048	+	0.812438i	$0.301860\pi$
$920$	0	0
$921$	−8.97473	−0.295727
$922$	0	0
$923$	1.81282	0.0596698
$924$	0	0
$925$	−3.83976	−0.126251
$926$	0	0
$927$	−2.84072	−0.0933015
$928$	0	0
$929$	−3.67818	−0.120677	−0.0603385	−	0.998178i	$-0.519218\pi$
$929$	−3.67818	−0.120677	−0.0603385	+	0.998178i	$0.519218\pi$
$930$	0	0
$931$	−9.14987	−0.299875
$932$	0	0
$933$	3.31425	0.108504
$934$	0	0
$935$	−9.30656	−0.304357
$936$	0	0
$937$	3.59956	0.117592	0.0587962	−	0.998270i	$-0.481274\pi$
$937$	3.59956	0.117592	0.0587962	+	0.998270i	$0.481274\pi$
$938$	0	0
$939$	−3.28262	−0.107124
$940$	0	0
$941$	12.4926	0.407246	0.203623	−	0.979049i	$-0.434728\pi$
$941$	12.4926	0.407246	0.203623	+	0.979049i	$0.434728\pi$
$942$	0	0
$943$	−11.9113	−0.387885
$944$	0	0
$945$	24.5369	0.798184
$946$	0	0
$947$	49.8307	1.61928	0.809640	−	0.586926i	$-0.199662\pi$
$947$	49.8307	1.61928	0.809640	+	0.586926i	$0.199662\pi$
$948$	0	0
$949$	−0.0539982	−0.00175286
$950$	0	0
$951$	−37.8778	−1.22827
$952$	0	0
$953$	9.62384	0.311747	0.155873	−	0.987777i	$-0.450181\pi$
$953$	9.62384	0.311747	0.155873	+	0.987777i	$0.450181\pi$
$954$	0	0
$955$	32.2368	1.04316
$956$	0	0
$957$	−9.20878	−0.297678
$958$	0	0
$959$	12.8985	0.416515
$960$	0	0
$961$	84.5005	2.72582
$962$	0	0
$963$	−3.89682	−0.125573
$964$	0	0
$965$	−16.3388	−0.525966
$966$	0	0
$967$	12.6937	0.408203	0.204102	−	0.978950i	$-0.434573\pi$
$967$	12.6937	0.408203	0.204102	+	0.978950i	$0.434573\pi$
$968$	0	0
$969$	−7.20043	−0.231311
$970$	0	0
$971$	4.07752	0.130854	0.0654269	−	0.997857i	$-0.479159\pi$
$971$	4.07752	0.130854	0.0654269	+	0.997857i	$0.479159\pi$
$972$	0	0
$973$	−10.2113	−0.327360
$974$	0	0
$975$	0.943038	0.0302014
$976$	0	0
$977$	43.4477	1.39001	0.695007	−	0.719003i	$-0.255401\pi$
$977$	43.4477	1.39001	0.695007	+	0.719003i	$0.255401\pi$
$978$	0	0
$979$	−6.23888	−0.199395
$980$	0	0
$981$	−2.41571	−0.0771277
$982$	0	0
$983$	5.09640	0.162550	0.0812749	−	0.996692i	$-0.474101\pi$
$983$	5.09640	0.162550	0.0812749	+	0.996692i	$0.474101\pi$
$984$	0	0
$985$	−19.8454	−0.632326
$986$	0	0
$987$	69.2474	2.20417
$988$	0	0
$989$	6.93987	0.220675
$990$	0	0
$991$	−0.598040	−0.0189974	−0.00949868	−	0.999955i	$-0.503024\pi$
$991$	−0.598040	−0.0189974	−0.00949868	+	0.999955i	$0.503024\pi$
$992$	0	0
$993$	23.6728	0.751233
$994$	0	0
$995$	−14.3043	−0.453477
$996$	0	0
$997$	41.6843	1.32016	0.660078	−	0.751197i	$-0.270523\pi$
$997$	41.6843	1.32016	0.660078	+	0.751197i	$0.270523\pi$
$998$	0	0
$999$	−5.53970	−0.175268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.16	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+1.94972 q^{3} +1.42435 q^{5} -4.01869 q^{7} +0.801405 q^{9} +O(q^{10})\)	\(q+1.94972 q^{3} +1.42435 q^{5} -4.01869 q^{7} +0.801405 q^{9} -1.76924 q^{11} -0.162788 q^{13} +2.77708 q^{15} +3.69306 q^{17} -1.00000 q^{19} -7.83532 q^{21} +1.52669 q^{23} -2.97122 q^{25} -4.28664 q^{27} +2.66959 q^{29} -10.7471 q^{31} -3.44952 q^{33} -5.72403 q^{35} +1.29232 q^{37} -0.317391 q^{39} -7.80202 q^{41} +4.54569 q^{43} +1.14148 q^{45} -8.83786 q^{47} +9.14987 q^{49} +7.20043 q^{51} +1.00000 q^{53} -2.52002 q^{55} -1.94972 q^{57} -2.95882 q^{59} -7.23691 q^{61} -3.22060 q^{63} -0.231867 q^{65} +14.9269 q^{67} +2.97662 q^{69} -11.1361 q^{71} +0.331709 q^{73} -5.79305 q^{75} +7.11002 q^{77} -4.60353 q^{79} -10.7620 q^{81} -8.46798 q^{83} +5.26021 q^{85} +5.20494 q^{87} +3.52631 q^{89} +0.654194 q^{91} -20.9539 q^{93} -1.42435 q^{95} +14.8111 q^{97} -1.41788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 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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