LMFDB - Embedded newform 4028.2.a.f.1.15

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			1.15
Root			$1.95331$ 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)

$f(q)$	$=$	$q+1.95331 q^{3} -3.65783 q^{5} -1.44461 q^{7} +0.815417 q^{9} +O(q^{10})$	$q+1.95331 q^{3} -3.65783 q^{5} -1.44461 q^{7} +0.815417 q^{9} -5.75850 q^{11} +3.96143 q^{13} -7.14488 q^{15} +1.99113 q^{17} -1.00000 q^{19} -2.82177 q^{21} -4.14177 q^{23} +8.37975 q^{25} -4.26717 q^{27} -7.62907 q^{29} +5.72892 q^{31} -11.2481 q^{33} +5.28415 q^{35} +10.4472 q^{37} +7.73789 q^{39} +0.760606 q^{41} +11.7183 q^{43} -2.98266 q^{45} +10.8762 q^{47} -4.91310 q^{49} +3.88928 q^{51} -1.00000 q^{53} +21.0636 q^{55} -1.95331 q^{57} -3.29327 q^{59} -0.205821 q^{61} -1.17796 q^{63} -14.4902 q^{65} +13.6393 q^{67} -8.09016 q^{69} -2.74232 q^{71} -6.40893 q^{73} +16.3683 q^{75} +8.31879 q^{77} +15.3022 q^{79} -10.7813 q^{81} +4.83898 q^{83} -7.28321 q^{85} -14.9019 q^{87} +12.5175 q^{89} -5.72272 q^{91} +11.1904 q^{93} +3.65783 q^{95} -0.842775 q^{97} -4.69558 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.95331	1.12774	0.563872	−	0.825862i	$-0.309311\pi$
$3$	1.95331	1.12774	0.563872	+	0.825862i	$0.309311\pi$
$4$	0	0
$5$	−3.65783	−1.63583	−0.817917	−	0.575337i	$-0.804871\pi$
$5$	−3.65783	−1.63583	−0.817917	+	0.575337i	$0.804871\pi$
$6$	0	0
$7$	−1.44461	−0.546012	−0.273006	−	0.962012i	$-0.588018\pi$
$7$	−1.44461	−0.546012	−0.273006	+	0.962012i	$0.588018\pi$
$8$	0	0
$9$	0.815417	0.271806
$10$	0	0
$11$	−5.75850	−1.73625	−0.868126	−	0.496343i	$-0.834676\pi$
$11$	−5.75850	−1.73625	−0.868126	+	0.496343i	$0.834676\pi$
$12$	0	0
$13$	3.96143	1.09870	0.549351	−	0.835592i	$-0.314875\pi$
$13$	3.96143	1.09870	0.549351	+	0.835592i	$0.314875\pi$
$14$	0	0
$15$	−7.14488	−1.84480
$16$	0	0
$17$	1.99113	0.482919	0.241460	−	0.970411i	$-0.422374\pi$
$17$	1.99113	0.482919	0.241460	+	0.970411i	$0.422374\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.82177	−0.615761
$22$	0	0
$23$	−4.14177	−0.863619	−0.431809	−	0.901965i	$-0.642125\pi$
$23$	−4.14177	−0.863619	−0.431809	+	0.901965i	$0.642125\pi$
$24$	0	0
$25$	8.37975	1.67595
$26$	0	0
$27$	−4.26717	−0.821217
$28$	0	0
$29$	−7.62907	−1.41668	−0.708342	−	0.705870i	$-0.750556\pi$
$29$	−7.62907	−1.41668	−0.708342	+	0.705870i	$0.750556\pi$
$30$	0	0
$31$	5.72892	1.02894	0.514472	−	0.857507i	$-0.327988\pi$
$31$	5.72892	1.02894	0.514472	+	0.857507i	$0.327988\pi$
$32$	0	0
$33$	−11.2481	−1.95805
$34$	0	0
$35$	5.28415	0.893184
$36$	0	0
$37$	10.4472	1.71751	0.858754	−	0.512388i	$-0.171239\pi$
$37$	10.4472	1.71751	0.858754	+	0.512388i	$0.171239\pi$
$38$	0	0
$39$	7.73789	1.23905
$40$	0	0
$41$	0.760606	0.118787	0.0593934	−	0.998235i	$-0.481083\pi$
$41$	0.760606	0.118787	0.0593934	+	0.998235i	$0.481083\pi$
$42$	0	0
$43$	11.7183	1.78702	0.893510	−	0.449044i	$-0.148235\pi$
$43$	11.7183	1.78702	0.893510	+	0.449044i	$0.148235\pi$
$44$	0	0
$45$	−2.98266	−0.444629
$46$	0	0
$47$	10.8762	1.58646	0.793229	−	0.608924i	$-0.208398\pi$
$47$	10.8762	1.58646	0.793229	+	0.608924i	$0.208398\pi$
$48$	0	0
$49$	−4.91310	−0.701871
$50$	0	0
$51$	3.88928	0.544609
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	21.0636	2.84022
$56$	0	0
$57$	−1.95331	−0.258722
$58$	0	0
$59$	−3.29327	−0.428748	−0.214374	−	0.976752i	$-0.568771\pi$
$59$	−3.29327	−0.428748	−0.214374	+	0.976752i	$0.568771\pi$
$60$	0	0
$61$	−0.205821	−0.0263527	−0.0131764	−	0.999913i	$-0.504194\pi$
$61$	−0.205821	−0.0263527	−0.0131764	+	0.999913i	$0.504194\pi$
$62$	0	0
$63$	−1.17796	−0.148409
$64$	0	0
$65$	−14.4902	−1.79729
$66$	0	0
$67$	13.6393	1.66631	0.833155	−	0.553040i	$-0.186532\pi$
$67$	13.6393	1.66631	0.833155	+	0.553040i	$0.186532\pi$
$68$	0	0
$69$	−8.09016	−0.973940
$70$	0	0
$71$	−2.74232	−0.325453	−0.162727	−	0.986671i	$-0.552029\pi$
$71$	−2.74232	−0.325453	−0.162727	+	0.986671i	$0.552029\pi$
$72$	0	0
$73$	−6.40893	−0.750109	−0.375054	−	0.927003i	$-0.622376\pi$
$73$	−6.40893	−0.750109	−0.375054	+	0.927003i	$0.622376\pi$
$74$	0	0
$75$	16.3683	1.89004
$76$	0	0
$77$	8.31879	0.948015
$78$	0	0
$79$	15.3022	1.72163	0.860814	−	0.508920i	$-0.169955\pi$
$79$	15.3022	1.72163	0.860814	+	0.508920i	$0.169955\pi$
$80$	0	0
$81$	−10.7813	−1.19793
$82$	0	0
$83$	4.83898	0.531147	0.265574	−	0.964091i	$-0.414439\pi$
$83$	4.83898	0.531147	0.265574	+	0.964091i	$0.414439\pi$
$84$	0	0
$85$	−7.28321	−0.789975
$86$	0	0
$87$	−14.9019	−1.59766
$88$	0	0
$89$	12.5175	1.32685	0.663425	−	0.748243i	$-0.269102\pi$
$89$	12.5175	1.32685	0.663425	+	0.748243i	$0.269102\pi$
$90$	0	0
$91$	−5.72272	−0.599904
$92$	0	0
$93$	11.1904	1.16039
$94$	0	0
$95$	3.65783	0.375286
$96$	0	0
$97$	−0.842775	−0.0855709	−0.0427854	−	0.999084i	$-0.513623\pi$
$97$	−0.842775	−0.0855709	−0.0427854	+	0.999084i	$0.513623\pi$
$98$	0	0
$99$	−4.69558	−0.471923
$100$	0	0
$101$	8.12793	0.808759	0.404380	−	0.914591i	$-0.367487\pi$
$101$	8.12793	0.808759	0.404380	+	0.914591i	$0.367487\pi$
$102$	0	0
$103$	−12.3473	−1.21662	−0.608309	−	0.793700i	$-0.708152\pi$
$103$	−12.3473	−1.21662	−0.608309	+	0.793700i	$0.708152\pi$
$104$	0	0
$105$	10.3216	1.00728
$106$	0	0
$107$	−18.6163	−1.79970	−0.899851	−	0.436198i	$-0.856325\pi$
$107$	−18.6163	−1.79970	−0.899851	+	0.436198i	$0.856325\pi$
$108$	0	0
$109$	2.93534	0.281155	0.140577	−	0.990070i	$-0.455104\pi$
$109$	2.93534	0.281155	0.140577	+	0.990070i	$0.455104\pi$
$110$	0	0
$111$	20.4066	1.93691
$112$	0	0
$113$	−0.516753	−0.0486120	−0.0243060	−	0.999705i	$-0.507738\pi$
$113$	−0.516753	−0.0486120	−0.0243060	+	0.999705i	$0.507738\pi$
$114$	0	0
$115$	15.1499	1.41274
$116$	0	0
$117$	3.23021	0.298633
$118$	0	0
$119$	−2.87640	−0.263680
$120$	0	0
$121$	22.1603	2.01457
$122$	0	0
$123$	1.48570	0.133961
$124$	0	0
$125$	−12.3626	−1.10574
$126$	0	0
$127$	−10.2841	−0.912564	−0.456282	−	0.889835i	$-0.650819\pi$
$127$	−10.2841	−0.912564	−0.456282	+	0.889835i	$0.650819\pi$
$128$	0	0
$129$	22.8894	2.01530
$130$	0	0
$131$	11.8436	1.03478	0.517388	−	0.855751i	$-0.326904\pi$
$131$	11.8436	1.03478	0.517388	+	0.855751i	$0.326904\pi$
$132$	0	0
$133$	1.44461	0.125264
$134$	0	0
$135$	15.6086	1.34337
$136$	0	0
$137$	−8.17927	−0.698802	−0.349401	−	0.936973i	$-0.613615\pi$
$137$	−8.17927	−0.698802	−0.349401	+	0.936973i	$0.613615\pi$
$138$	0	0
$139$	4.21843	0.357802	0.178901	−	0.983867i	$-0.442746\pi$
$139$	4.21843	0.357802	0.178901	+	0.983867i	$0.442746\pi$
$140$	0	0
$141$	21.2446	1.78912
$142$	0	0
$143$	−22.8119	−1.90762
$144$	0	0
$145$	27.9059	2.31746
$146$	0	0
$147$	−9.59680	−0.791531
$148$	0	0
$149$	−0.0264631	−0.00216794	−0.00108397	−	0.999999i	$-0.500345\pi$
$149$	−0.0264631	−0.00216794	−0.00108397	+	0.999999i	$0.500345\pi$
$150$	0	0
$151$	−16.2907	−1.32572	−0.662858	−	0.748745i	$-0.730657\pi$
$151$	−16.2907	−1.32572	−0.662858	+	0.748745i	$0.730657\pi$
$152$	0	0
$153$	1.62360	0.131260
$154$	0	0
$155$	−20.9555	−1.68318
$156$	0	0
$157$	19.3202	1.54192	0.770961	−	0.636882i	$-0.219776\pi$
$157$	19.3202	1.54192	0.770961	+	0.636882i	$0.219776\pi$
$158$	0	0
$159$	−1.95331	−0.154907
$160$	0	0
$161$	5.98325	0.471546
$162$	0	0
$163$	1.74702	0.136837	0.0684187	−	0.997657i	$-0.478205\pi$
$163$	1.74702	0.136837	0.0684187	+	0.997657i	$0.478205\pi$
$164$	0	0
$165$	41.1438	3.20304
$166$	0	0
$167$	13.2638	1.02638	0.513191	−	0.858274i	$-0.328463\pi$
$167$	13.2638	1.02638	0.513191	+	0.858274i	$0.328463\pi$
$168$	0	0
$169$	2.69289	0.207146
$170$	0	0
$171$	−0.815417	−0.0623565
$172$	0	0
$173$	1.64487	0.125057	0.0625285	−	0.998043i	$-0.480084\pi$
$173$	1.64487	0.125057	0.0625285	+	0.998043i	$0.480084\pi$
$174$	0	0
$175$	−12.1055	−0.915089
$176$	0	0
$177$	−6.43278	−0.483517
$178$	0	0
$179$	23.1025	1.72676	0.863382	−	0.504551i	$-0.168342\pi$
$179$	23.1025	1.72676	0.863382	+	0.504551i	$0.168342\pi$
$180$	0	0
$181$	6.03857	0.448843	0.224422	−	0.974492i	$-0.427951\pi$
$181$	6.03857	0.448843	0.224422	+	0.974492i	$0.427951\pi$
$182$	0	0
$183$	−0.402033	−0.0297191
$184$	0	0
$185$	−38.2141	−2.80956
$186$	0	0
$187$	−11.4659	−0.838470
$188$	0	0
$189$	6.16440	0.448394
$190$	0	0
$191$	19.5953	1.41787	0.708935	−	0.705274i	$-0.249176\pi$
$191$	19.5953	1.41787	0.708935	+	0.705274i	$0.249176\pi$
$192$	0	0
$193$	−23.6969	−1.70574	−0.852868	−	0.522126i	$-0.825139\pi$
$193$	−23.6969	−1.70574	−0.852868	+	0.522126i	$0.825139\pi$
$194$	0	0
$195$	−28.3039	−2.02689
$196$	0	0
$197$	−7.85059	−0.559331	−0.279666	−	0.960097i	$-0.590224\pi$
$197$	−7.85059	−0.559331	−0.279666	+	0.960097i	$0.590224\pi$
$198$	0	0
$199$	−5.23479	−0.371085	−0.185542	−	0.982636i	$-0.559404\pi$
$199$	−5.23479	−0.371085	−0.185542	+	0.982636i	$0.559404\pi$
$200$	0	0
$201$	26.6418	1.87917
$202$	0	0
$203$	11.0210	0.773526
$204$	0	0
$205$	−2.78217	−0.194315
$206$	0	0
$207$	−3.37727	−0.234736
$208$	0	0
$209$	5.75850	0.398324
$210$	0	0
$211$	2.36283	0.162664	0.0813319	−	0.996687i	$-0.474083\pi$
$211$	2.36283	0.162664	0.0813319	+	0.996687i	$0.474083\pi$
$212$	0	0
$213$	−5.35659	−0.367028
$214$	0	0
$215$	−42.8635	−2.92327
$216$	0	0
$217$	−8.27607	−0.561816
$218$	0	0
$219$	−12.5186	−0.845930
$220$	0	0
$221$	7.88770	0.530584
$222$	0	0
$223$	16.5102	1.10561	0.552804	−	0.833311i	$-0.313558\pi$
$223$	16.5102	1.10561	0.552804	+	0.833311i	$0.313558\pi$
$224$	0	0
$225$	6.83299	0.455533
$226$	0	0
$227$	4.66642	0.309721	0.154861	−	0.987936i	$-0.450507\pi$
$227$	4.66642	0.309721	0.154861	+	0.987936i	$0.450507\pi$
$228$	0	0
$229$	5.71391	0.377586	0.188793	−	0.982017i	$-0.439543\pi$
$229$	5.71391	0.377586	0.188793	+	0.982017i	$0.439543\pi$
$230$	0	0
$231$	16.2492	1.06912
$232$	0	0
$233$	12.5702	0.823501	0.411750	−	0.911297i	$-0.364918\pi$
$233$	12.5702	0.823501	0.411750	+	0.911297i	$0.364918\pi$
$234$	0	0
$235$	−39.7834	−2.59518
$236$	0	0
$237$	29.8899	1.94155
$238$	0	0
$239$	18.2878	1.18294	0.591471	−	0.806327i	$-0.298548\pi$
$239$	18.2878	1.18294	0.591471	+	0.806327i	$0.298548\pi$
$240$	0	0
$241$	−20.2721	−1.30584	−0.652920	−	0.757427i	$-0.726456\pi$
$241$	−20.2721	−1.30584	−0.652920	+	0.757427i	$0.726456\pi$
$242$	0	0
$243$	−8.25780	−0.529738
$244$	0	0
$245$	17.9713	1.14814
$246$	0	0
$247$	−3.96143	−0.252059
$248$	0	0
$249$	9.45203	0.598998
$250$	0	0
$251$	13.8952	0.877057	0.438529	−	0.898717i	$-0.355500\pi$
$251$	13.8952	0.877057	0.438529	+	0.898717i	$0.355500\pi$
$252$	0	0
$253$	23.8504	1.49946
$254$	0	0
$255$	−14.2264	−0.890889
$256$	0	0
$257$	2.57327	0.160516	0.0802579	−	0.996774i	$-0.474426\pi$
$257$	2.57327	0.160516	0.0802579	+	0.996774i	$0.474426\pi$
$258$	0	0
$259$	−15.0921	−0.937780
$260$	0	0
$261$	−6.22087	−0.385062
$262$	0	0
$263$	7.58598	0.467772	0.233886	−	0.972264i	$-0.424856\pi$
$263$	7.58598	0.467772	0.233886	+	0.972264i	$0.424856\pi$
$264$	0	0
$265$	3.65783	0.224699
$266$	0	0
$267$	24.4505	1.49635
$268$	0	0
$269$	17.9099	1.09199	0.545993	−	0.837790i	$-0.316153\pi$
$269$	17.9099	1.09199	0.545993	+	0.837790i	$0.316153\pi$
$270$	0	0
$271$	29.5018	1.79211	0.896054	−	0.443945i	$-0.146421\pi$
$271$	29.5018	1.79211	0.896054	+	0.443945i	$0.146421\pi$
$272$	0	0
$273$	−11.1782	−0.676538
$274$	0	0
$275$	−48.2548	−2.90987
$276$	0	0
$277$	−15.4330	−0.927278	−0.463639	−	0.886024i	$-0.653456\pi$
$277$	−15.4330	−0.927278	−0.463639	+	0.886024i	$0.653456\pi$
$278$	0	0
$279$	4.67146	0.279673
$280$	0	0
$281$	−26.6725	−1.59115	−0.795574	−	0.605857i	$-0.792830\pi$
$281$	−26.6725	−1.59115	−0.795574	+	0.605857i	$0.792830\pi$
$282$	0	0
$283$	−18.4307	−1.09559	−0.547796	−	0.836612i	$-0.684533\pi$
$283$	−18.4307	−1.09559	−0.547796	+	0.836612i	$0.684533\pi$
$284$	0	0
$285$	7.14488	0.423226
$286$	0	0
$287$	−1.09878	−0.0648590
$288$	0	0
$289$	−13.0354	−0.766789
$290$	0	0
$291$	−1.64620	−0.0965020
$292$	0	0
$293$	20.9248	1.22244	0.611219	−	0.791462i	$-0.290680\pi$
$293$	20.9248	1.22244	0.611219	+	0.791462i	$0.290680\pi$
$294$	0	0
$295$	12.0463	0.701360
$296$	0	0
$297$	24.5725	1.42584
$298$	0	0
$299$	−16.4073	−0.948859
$300$	0	0
$301$	−16.9283	−0.975734
$302$	0	0
$303$	15.8764	0.912073
$304$	0	0
$305$	0.752860	0.0431087
$306$	0	0
$307$	16.0425	0.915596	0.457798	−	0.889056i	$-0.348638\pi$
$307$	16.0425	0.915596	0.457798	+	0.889056i	$0.348638\pi$
$308$	0	0
$309$	−24.1181	−1.37203
$310$	0	0
$311$	−20.2086	−1.14592	−0.572962	−	0.819582i	$-0.694206\pi$
$311$	−20.2086	−1.14592	−0.572962	+	0.819582i	$0.694206\pi$
$312$	0	0
$313$	−21.3003	−1.20396	−0.601981	−	0.798510i	$-0.705622\pi$
$313$	−21.3003	−1.20396	−0.601981	+	0.798510i	$0.705622\pi$
$314$	0	0
$315$	4.30878	0.242772
$316$	0	0
$317$	13.0770	0.734479	0.367240	−	0.930126i	$-0.380303\pi$
$317$	13.0770	0.734479	0.367240	+	0.930126i	$0.380303\pi$
$318$	0	0
$319$	43.9320	2.45972
$320$	0	0
$321$	−36.3633	−2.02960
$322$	0	0
$323$	−1.99113	−0.110789
$324$	0	0
$325$	33.1958	1.84137
$326$	0	0
$327$	5.73363	0.317071
$328$	0	0
$329$	−15.7119	−0.866225
$330$	0	0
$331$	−15.1075	−0.830385	−0.415192	−	0.909734i	$-0.636286\pi$
$331$	−15.1075	−0.830385	−0.415192	+	0.909734i	$0.636286\pi$
$332$	0	0
$333$	8.51882	0.466828
$334$	0	0
$335$	−49.8904	−2.72581
$336$	0	0
$337$	−6.60548	−0.359823	−0.179912	−	0.983683i	$-0.557581\pi$
$337$	−6.60548	−0.359823	−0.179912	+	0.983683i	$0.557581\pi$
$338$	0	0
$339$	−1.00938	−0.0548219
$340$	0	0
$341$	−32.9900	−1.78651
$342$	0	0
$343$	17.2098	0.929242
$344$	0	0
$345$	29.5925	1.59320
$346$	0	0
$347$	−5.14295	−0.276088	−0.138044	−	0.990426i	$-0.544082\pi$
$347$	−5.14295	−0.276088	−0.138044	+	0.990426i	$0.544082\pi$
$348$	0	0
$349$	20.4025	1.09212	0.546061	−	0.837745i	$-0.316127\pi$
$349$	20.4025	1.09212	0.546061	+	0.837745i	$0.316127\pi$
$350$	0	0
$351$	−16.9041	−0.902272
$352$	0	0
$353$	5.57750	0.296860	0.148430	−	0.988923i	$-0.452578\pi$
$353$	5.57750	0.296860	0.148430	+	0.988923i	$0.452578\pi$
$354$	0	0
$355$	10.0309	0.532387
$356$	0	0
$357$	−5.61851	−0.297363
$358$	0	0
$359$	−16.2952	−0.860027	−0.430014	−	0.902822i	$-0.641491\pi$
$359$	−16.2952	−0.860027	−0.430014	+	0.902822i	$0.641491\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	43.2859	2.27192
$364$	0	0
$365$	23.4428	1.22705
$366$	0	0
$367$	−36.9492	−1.92873	−0.964366	−	0.264571i	$-0.914770\pi$
$367$	−36.9492	−1.92873	−0.964366	+	0.264571i	$0.914770\pi$
$368$	0	0
$369$	0.620211	0.0322869
$370$	0	0
$371$	1.44461	0.0750005
$372$	0	0
$373$	−18.8850	−0.977829	−0.488915	−	0.872332i	$-0.662607\pi$
$373$	−18.8850	−0.977829	−0.488915	+	0.872332i	$0.662607\pi$
$374$	0	0
$375$	−24.1479	−1.24699
$376$	0	0
$377$	−30.2220	−1.55651
$378$	0	0
$379$	16.1756	0.830886	0.415443	−	0.909619i	$-0.363627\pi$
$379$	16.1756	0.830886	0.415443	+	0.909619i	$0.363627\pi$
$380$	0	0
$381$	−20.0880	−1.02914
$382$	0	0
$383$	30.4616	1.55651	0.778257	−	0.627946i	$-0.216104\pi$
$383$	30.4616	1.55651	0.778257	+	0.627946i	$0.216104\pi$
$384$	0	0
$385$	−30.4288	−1.55079
$386$	0	0
$387$	9.55527	0.485722
$388$	0	0
$389$	−11.5024	−0.583192	−0.291596	−	0.956541i	$-0.594186\pi$
$389$	−11.5024	−0.583192	−0.291596	+	0.956541i	$0.594186\pi$
$390$	0	0
$391$	−8.24678	−0.417058
$392$	0	0
$393$	23.1341	1.16696
$394$	0	0
$395$	−55.9728	−2.81630
$396$	0	0
$397$	−13.9334	−0.699296	−0.349648	−	0.936881i	$-0.613699\pi$
$397$	−13.9334	−0.699296	−0.349648	+	0.936881i	$0.613699\pi$
$398$	0	0
$399$	2.82177	0.141265
$400$	0	0
$401$	−0.442692	−0.0221070	−0.0110535	−	0.999939i	$-0.503519\pi$
$401$	−0.442692	−0.0221070	−0.0110535	+	0.999939i	$0.503519\pi$
$402$	0	0
$403$	22.6947	1.13050
$404$	0	0
$405$	39.4364	1.95961
$406$	0	0
$407$	−60.1602	−2.98203
$408$	0	0
$409$	−12.6402	−0.625017	−0.312508	−	0.949915i	$-0.601169\pi$
$409$	−12.6402	−0.625017	−0.312508	+	0.949915i	$0.601169\pi$
$410$	0	0
$411$	−15.9766	−0.788070
$412$	0	0
$413$	4.75750	0.234101
$414$	0	0
$415$	−17.7002	−0.868868
$416$	0	0
$417$	8.23989	0.403509
$418$	0	0
$419$	−15.5759	−0.760933	−0.380467	−	0.924795i	$-0.624237\pi$
$419$	−15.5759	−0.760933	−0.380467	+	0.924795i	$0.624237\pi$
$420$	0	0
$421$	27.0087	1.31632	0.658161	−	0.752877i	$-0.271335\pi$
$421$	27.0087	1.31632	0.658161	+	0.752877i	$0.271335\pi$
$422$	0	0
$423$	8.86864	0.431208
$424$	0	0
$425$	16.6851	0.809349
$426$	0	0
$427$	0.297332	0.0143889
$428$	0	0
$429$	−44.5586	−2.15131
$430$	0	0
$431$	28.5384	1.37465	0.687323	−	0.726352i	$-0.258786\pi$
$431$	28.5384	1.37465	0.687323	+	0.726352i	$0.258786\pi$
$432$	0	0
$433$	28.9392	1.39073	0.695365	−	0.718657i	$-0.255243\pi$
$433$	28.9392	1.39073	0.695365	+	0.718657i	$0.255243\pi$
$434$	0	0
$435$	54.5088	2.61350
$436$	0	0
$437$	4.14177	0.198128
$438$	0	0
$439$	6.24168	0.297899	0.148949	−	0.988845i	$-0.452411\pi$
$439$	6.24168	0.297899	0.148949	+	0.988845i	$0.452411\pi$
$440$	0	0
$441$	−4.00622	−0.190772
$442$	0	0
$443$	−30.0131	−1.42597	−0.712983	−	0.701182i	$-0.752656\pi$
$443$	−30.0131	−1.42597	−0.712983	+	0.701182i	$0.752656\pi$
$444$	0	0
$445$	−45.7868	−2.17050
$446$	0	0
$447$	−0.0516906	−0.00244488
$448$	0	0
$449$	12.5736	0.593384	0.296692	−	0.954973i	$-0.404117\pi$
$449$	12.5736	0.593384	0.296692	+	0.954973i	$0.404117\pi$
$450$	0	0
$451$	−4.37995	−0.206244
$452$	0	0
$453$	−31.8207	−1.49507
$454$	0	0
$455$	20.9328	0.981343
$456$	0	0
$457$	−34.5749	−1.61735	−0.808673	−	0.588258i	$-0.799814\pi$
$457$	−34.5749	−1.61735	−0.808673	+	0.588258i	$0.799814\pi$
$458$	0	0
$459$	−8.49647	−0.396581
$460$	0	0
$461$	19.1171	0.890371	0.445185	−	0.895438i	$-0.353138\pi$
$461$	19.1171	0.890371	0.445185	+	0.895438i	$0.353138\pi$
$462$	0	0
$463$	25.8614	1.20188	0.600941	−	0.799293i	$-0.294793\pi$
$463$	25.8614	1.20188	0.600941	+	0.799293i	$0.294793\pi$
$464$	0	0
$465$	−40.9325	−1.89820
$466$	0	0
$467$	−41.2699	−1.90974	−0.954871	−	0.297020i	$-0.904007\pi$
$467$	−41.2699	−1.90974	−0.954871	+	0.297020i	$0.904007\pi$
$468$	0	0
$469$	−19.7035	−0.909825
$470$	0	0
$471$	37.7384	1.73889
$472$	0	0
$473$	−67.4796	−3.10272
$474$	0	0
$475$	−8.37975	−0.384489
$476$	0	0
$477$	−0.815417	−0.0373354
$478$	0	0
$479$	7.17542	0.327853	0.163927	−	0.986473i	$-0.447584\pi$
$479$	7.17542	0.327853	0.163927	+	0.986473i	$0.447584\pi$
$480$	0	0
$481$	41.3858	1.88703
$482$	0	0
$483$	11.6871	0.531783
$484$	0	0
$485$	3.08273	0.139980
$486$	0	0
$487$	−3.04039	−0.137773	−0.0688866	−	0.997624i	$-0.521945\pi$
$487$	−3.04039	−0.137773	−0.0688866	+	0.997624i	$0.521945\pi$
$488$	0	0
$489$	3.41247	0.154317
$490$	0	0
$491$	−16.1195	−0.727465	−0.363733	−	0.931503i	$-0.618498\pi$
$491$	−16.1195	−0.727465	−0.363733	+	0.931503i	$0.618498\pi$
$492$	0	0
$493$	−15.1905	−0.684143
$494$	0	0
$495$	17.1756	0.771988
$496$	0	0
$497$	3.96158	0.177701
$498$	0	0
$499$	17.4369	0.780584	0.390292	−	0.920691i	$-0.372374\pi$
$499$	17.4369	0.780584	0.390292	+	0.920691i	$0.372374\pi$
$500$	0	0
$501$	25.9083	1.15750
$502$	0	0
$503$	37.7619	1.68372	0.841861	−	0.539695i	$-0.181460\pi$
$503$	37.7619	1.68372	0.841861	+	0.539695i	$0.181460\pi$
$504$	0	0
$505$	−29.7306	−1.32300
$506$	0	0
$507$	5.26005	0.233607
$508$	0	0
$509$	4.99587	0.221438	0.110719	−	0.993852i	$-0.464685\pi$
$509$	4.99587	0.221438	0.110719	+	0.993852i	$0.464685\pi$
$510$	0	0
$511$	9.25842	0.409568
$512$	0	0
$513$	4.26717	0.188400
$514$	0	0
$515$	45.1645	1.99018
$516$	0	0
$517$	−62.6306	−2.75449
$518$	0	0
$519$	3.21294	0.141032
$520$	0	0
$521$	−8.83819	−0.387208	−0.193604	−	0.981080i	$-0.562018\pi$
$521$	−8.83819	−0.387208	−0.193604	+	0.981080i	$0.562018\pi$
$522$	0	0
$523$	16.0869	0.703432	0.351716	−	0.936107i	$-0.385598\pi$
$523$	16.0869	0.703432	0.351716	+	0.936107i	$0.385598\pi$
$524$	0	0
$525$	−23.6458	−1.03199
$526$	0	0
$527$	11.4070	0.496897
$528$	0	0
$529$	−5.84575	−0.254163
$530$	0	0
$531$	−2.68539	−0.116536
$532$	0	0
$533$	3.01308	0.130511
$534$	0	0
$535$	68.0952	2.94401
$536$	0	0
$537$	45.1264	1.94735
$538$	0	0
$539$	28.2921	1.21863
$540$	0	0
$541$	−24.1131	−1.03670	−0.518351	−	0.855168i	$-0.673454\pi$
$541$	−24.1131	−1.03670	−0.518351	+	0.855168i	$0.673454\pi$
$542$	0	0
$543$	11.7952	0.506180
$544$	0	0
$545$	−10.7370	−0.459923
$546$	0	0
$547$	5.43822	0.232522	0.116261	−	0.993219i	$-0.462909\pi$
$547$	5.43822	0.232522	0.116261	+	0.993219i	$0.462909\pi$
$548$	0	0
$549$	−0.167830	−0.00716282
$550$	0	0
$551$	7.62907	0.325010
$552$	0	0
$553$	−22.1057	−0.940029
$554$	0	0
$555$	−74.6440	−3.16846
$556$	0	0
$557$	27.3949	1.16076	0.580379	−	0.814346i	$-0.302904\pi$
$557$	27.3949	1.16076	0.580379	+	0.814346i	$0.302904\pi$
$558$	0	0
$559$	46.4211	1.96340
$560$	0	0
$561$	−22.3964	−0.945579
$562$	0	0
$563$	27.7033	1.16756	0.583778	−	0.811914i	$-0.301574\pi$
$563$	27.7033	1.16756	0.583778	+	0.811914i	$0.301574\pi$
$564$	0	0
$565$	1.89020	0.0795212
$566$	0	0
$567$	15.5749	0.654082
$568$	0	0
$569$	22.1932	0.930389	0.465194	−	0.885209i	$-0.345984\pi$
$569$	22.1932	0.930389	0.465194	+	0.885209i	$0.345984\pi$
$570$	0	0
$571$	28.9292	1.21065	0.605324	−	0.795979i	$-0.293043\pi$
$571$	28.9292	1.21065	0.605324	+	0.795979i	$0.293043\pi$
$572$	0	0
$573$	38.2758	1.59899
$574$	0	0
$575$	−34.7070	−1.44738
$576$	0	0
$577$	−10.5822	−0.440541	−0.220270	−	0.975439i	$-0.570694\pi$
$577$	−10.5822	−0.440541	−0.220270	+	0.975439i	$0.570694\pi$
$578$	0	0
$579$	−46.2873	−1.92363
$580$	0	0
$581$	−6.99045	−0.290013
$582$	0	0
$583$	5.75850	0.238493
$584$	0	0
$585$	−11.8156	−0.488514
$586$	0	0
$587$	−32.0585	−1.32320	−0.661598	−	0.749859i	$-0.730122\pi$
$587$	−32.0585	−1.32320	−0.661598	+	0.749859i	$0.730122\pi$
$588$	0	0
$589$	−5.72892	−0.236056
$590$	0	0
$591$	−15.3346	−0.630782
$592$	0	0
$593$	26.4213	1.08499	0.542497	−	0.840058i	$-0.317479\pi$
$593$	26.4213	1.08499	0.542497	+	0.840058i	$0.317479\pi$
$594$	0	0
$595$	10.5214	0.431336
$596$	0	0
$597$	−10.2252	−0.418488
$598$	0	0
$599$	−38.9003	−1.58942	−0.794712	−	0.606987i	$-0.792378\pi$
$599$	−38.9003	−1.58942	−0.794712	+	0.606987i	$0.792378\pi$
$600$	0	0
$601$	−27.8460	−1.13586	−0.567930	−	0.823077i	$-0.692256\pi$
$601$	−27.8460	−1.13586	−0.567930	+	0.823077i	$0.692256\pi$
$602$	0	0
$603$	11.1217	0.452912
$604$	0	0
$605$	−81.0588	−3.29551
$606$	0	0
$607$	−26.5827	−1.07896	−0.539480	−	0.841999i	$-0.681379\pi$
$607$	−26.5827	−1.07896	−0.539480	+	0.841999i	$0.681379\pi$
$608$	0	0
$609$	21.5275	0.872339
$610$	0	0
$611$	43.0853	1.74304
$612$	0	0
$613$	30.3586	1.22617	0.613086	−	0.790016i	$-0.289928\pi$
$613$	30.3586	1.22617	0.613086	+	0.790016i	$0.289928\pi$
$614$	0	0
$615$	−5.43444	−0.219138
$616$	0	0
$617$	21.5237	0.866510	0.433255	−	0.901271i	$-0.357365\pi$
$617$	21.5237	0.866510	0.433255	+	0.901271i	$0.357365\pi$
$618$	0	0
$619$	−15.4809	−0.622229	−0.311115	−	0.950372i	$-0.600702\pi$
$619$	−15.4809	−0.622229	−0.311115	+	0.950372i	$0.600702\pi$
$620$	0	0
$621$	17.6736	0.709218
$622$	0	0
$623$	−18.0829	−0.724476
$624$	0	0
$625$	3.32151	0.132860
$626$	0	0
$627$	11.2481	0.449207
$628$	0	0
$629$	20.8017	0.829417
$630$	0	0
$631$	2.47755	0.0986297	0.0493149	−	0.998783i	$-0.484296\pi$
$631$	2.47755	0.0986297	0.0493149	+	0.998783i	$0.484296\pi$
$632$	0	0
$633$	4.61533	0.183443
$634$	0	0
$635$	37.6174	1.49280
$636$	0	0
$637$	−19.4629	−0.771147
$638$	0	0
$639$	−2.23613	−0.0884600
$640$	0	0
$641$	−0.951088	−0.0375657	−0.0187829	−	0.999824i	$-0.505979\pi$
$641$	−0.951088	−0.0375657	−0.0187829	+	0.999824i	$0.505979\pi$
$642$	0	0
$643$	−27.8805	−1.09950	−0.549749	−	0.835330i	$-0.685277\pi$
$643$	−27.8805	−1.09950	−0.549749	+	0.835330i	$0.685277\pi$
$644$	0	0
$645$	−83.7257	−3.29669
$646$	0	0
$647$	−40.2223	−1.58130	−0.790651	−	0.612267i	$-0.790258\pi$
$647$	−40.2223	−1.58130	−0.790651	+	0.612267i	$0.790258\pi$
$648$	0	0
$649$	18.9643	0.744414
$650$	0	0
$651$	−16.1657	−0.633584
$652$	0	0
$653$	5.78818	0.226509	0.113255	−	0.993566i	$-0.463872\pi$
$653$	5.78818	0.226509	0.113255	+	0.993566i	$0.463872\pi$
$654$	0	0
$655$	−43.3218	−1.69272
$656$	0	0
$657$	−5.22595	−0.203884
$658$	0	0
$659$	33.2227	1.29417	0.647086	−	0.762417i	$-0.275987\pi$
$659$	33.2227	1.29417	0.647086	+	0.762417i	$0.275987\pi$
$660$	0	0
$661$	2.10733	0.0819657	0.0409829	−	0.999160i	$-0.486951\pi$
$661$	2.10733	0.0819657	0.0409829	+	0.999160i	$0.486951\pi$
$662$	0	0
$663$	15.4071	0.598363
$664$	0	0
$665$	−5.28415	−0.204911
$666$	0	0
$667$	31.5979	1.22347
$668$	0	0
$669$	32.2496	1.24684
$670$	0	0
$671$	1.18522	0.0457550
$672$	0	0
$673$	3.29248	0.126916	0.0634579	−	0.997985i	$-0.479787\pi$
$673$	3.29248	0.126916	0.0634579	+	0.997985i	$0.479787\pi$
$674$	0	0
$675$	−35.7578	−1.37632
$676$	0	0
$677$	39.9350	1.53483	0.767413	−	0.641153i	$-0.221544\pi$
$677$	39.9350	1.53483	0.767413	+	0.641153i	$0.221544\pi$
$678$	0	0
$679$	1.21748	0.0467227
$680$	0	0
$681$	9.11496	0.349286
$682$	0	0
$683$	28.8427	1.10364	0.551818	−	0.833965i	$-0.313934\pi$
$683$	28.8427	1.10364	0.551818	+	0.833965i	$0.313934\pi$
$684$	0	0
$685$	29.9184	1.14312
$686$	0	0
$687$	11.1610	0.425820
$688$	0	0
$689$	−3.96143	−0.150918
$690$	0	0
$691$	−7.69233	−0.292630	−0.146315	−	0.989238i	$-0.546741\pi$
$691$	−7.69233	−0.292630	−0.146315	+	0.989238i	$0.546741\pi$
$692$	0	0
$693$	6.78328	0.257676
$694$	0	0
$695$	−15.4303	−0.585305
$696$	0	0
$697$	1.51446	0.0573644
$698$	0	0
$699$	24.5535	0.928698
$700$	0	0
$701$	−10.2542	−0.387297	−0.193649	−	0.981071i	$-0.562032\pi$
$701$	−10.2542	−0.387297	−0.193649	+	0.981071i	$0.562032\pi$
$702$	0	0
$703$	−10.4472	−0.394023
$704$	0	0
$705$	−77.7092	−2.92670
$706$	0	0
$707$	−11.7417	−0.441592
$708$	0	0
$709$	−33.6241	−1.26278	−0.631390	−	0.775465i	$-0.717515\pi$
$709$	−33.6241	−1.26278	−0.631390	+	0.775465i	$0.717515\pi$
$710$	0	0
$711$	12.4776	0.467948
$712$	0	0
$713$	−23.7279	−0.888616
$714$	0	0
$715$	83.4420	3.12055
$716$	0	0
$717$	35.7218	1.33405
$718$	0	0
$719$	34.7079	1.29439	0.647193	−	0.762326i	$-0.275943\pi$
$719$	34.7079	1.29439	0.647193	+	0.762326i	$0.275943\pi$
$720$	0	0
$721$	17.8371	0.664288
$722$	0	0
$723$	−39.5977	−1.47265
$724$	0	0
$725$	−63.9298	−2.37429
$726$	0	0
$727$	−28.9369	−1.07321	−0.536605	−	0.843834i	$-0.680293\pi$
$727$	−28.9369	−1.07321	−0.536605	+	0.843834i	$0.680293\pi$
$728$	0	0
$729$	16.2140	0.600518
$730$	0	0
$731$	23.3326	0.862986
$732$	0	0
$733$	47.3788	1.74998	0.874988	−	0.484145i	$-0.160869\pi$
$733$	47.3788	1.74998	0.874988	+	0.484145i	$0.160869\pi$
$734$	0	0
$735$	35.1035	1.29481
$736$	0	0
$737$	−78.5421	−2.89313
$738$	0	0
$739$	−33.1730	−1.22029	−0.610143	−	0.792291i	$-0.708888\pi$
$739$	−33.1730	−1.22029	−0.610143	+	0.792291i	$0.708888\pi$
$740$	0	0
$741$	−7.73789	−0.284258
$742$	0	0
$743$	30.6315	1.12376	0.561880	−	0.827219i	$-0.310078\pi$
$743$	30.6315	1.12376	0.561880	+	0.827219i	$0.310078\pi$
$744$	0	0
$745$	0.0967975	0.00354639
$746$	0	0
$747$	3.94579	0.144369
$748$	0	0
$749$	26.8933	0.982659
$750$	0	0
$751$	29.5689	1.07898	0.539492	−	0.841991i	$-0.318616\pi$
$751$	29.5689	1.07898	0.539492	+	0.841991i	$0.318616\pi$
$752$	0	0
$753$	27.1416	0.989096
$754$	0	0
$755$	59.5886	2.16865
$756$	0	0
$757$	32.9103	1.19615	0.598073	−	0.801442i	$-0.295933\pi$
$757$	32.9103	1.19615	0.598073	+	0.801442i	$0.295933\pi$
$758$	0	0
$759$	46.5872	1.69101
$760$	0	0
$761$	7.92148	0.287153	0.143577	−	0.989639i	$-0.454140\pi$
$761$	7.92148	0.287153	0.143577	+	0.989639i	$0.454140\pi$
$762$	0	0
$763$	−4.24043	−0.153514
$764$	0	0
$765$	−5.93885	−0.214720
$766$	0	0
$767$	−13.0461	−0.471066
$768$	0	0
$769$	6.35287	0.229090	0.114545	−	0.993418i	$-0.463459\pi$
$769$	6.35287	0.229090	0.114545	+	0.993418i	$0.463459\pi$
$770$	0	0
$771$	5.02638	0.181021
$772$	0	0
$773$	7.30817	0.262856	0.131428	−	0.991326i	$-0.458044\pi$
$773$	7.30817	0.262856	0.131428	+	0.991326i	$0.458044\pi$
$774$	0	0
$775$	48.0070	1.72446
$776$	0	0
$777$	−29.4796	−1.05757
$778$	0	0
$779$	−0.760606	−0.0272515
$780$	0	0
$781$	15.7916	0.565069
$782$	0	0
$783$	32.5545	1.16340
$784$	0	0
$785$	−70.6702	−2.52233
$786$	0	0
$787$	34.7722	1.23949	0.619747	−	0.784801i	$-0.287235\pi$
$787$	34.7722	1.23949	0.619747	+	0.784801i	$0.287235\pi$
$788$	0	0
$789$	14.8178	0.527527
$790$	0	0
$791$	0.746507	0.0265427
$792$	0	0
$793$	−0.815346	−0.0289538
$794$	0	0
$795$	7.14488	0.253403
$796$	0	0
$797$	15.5525	0.550897	0.275448	−	0.961316i	$-0.411174\pi$
$797$	15.5525	0.550897	0.275448	+	0.961316i	$0.411174\pi$
$798$	0	0
$799$	21.6559	0.766131
$800$	0	0
$801$	10.2070	0.360645
$802$	0	0
$803$	36.9058	1.30238
$804$	0	0
$805$	−21.8857	−0.771371
$806$	0	0
$807$	34.9836	1.23148
$808$	0	0
$809$	19.1408	0.672956	0.336478	−	0.941691i	$-0.390764\pi$
$809$	19.1408	0.672956	0.336478	+	0.941691i	$0.390764\pi$
$810$	0	0
$811$	43.2446	1.51852	0.759261	−	0.650786i	$-0.225560\pi$
$811$	43.2446	1.51852	0.759261	+	0.650786i	$0.225560\pi$
$812$	0	0
$813$	57.6262	2.02104
$814$	0	0
$815$	−6.39032	−0.223843
$816$	0	0
$817$	−11.7183	−0.409970
$818$	0	0
$819$	−4.66640	−0.163057
$820$	0	0
$821$	−26.7313	−0.932929	−0.466464	−	0.884540i	$-0.654472\pi$
$821$	−26.7313	−0.932929	−0.466464	+	0.884540i	$0.654472\pi$
$822$	0	0
$823$	3.22328	0.112356	0.0561782	−	0.998421i	$-0.482109\pi$
$823$	3.22328	0.112356	0.0561782	+	0.998421i	$0.482109\pi$
$824$	0	0
$825$	−94.2566	−3.28159
$826$	0	0
$827$	−3.96629	−0.137921	−0.0689607	−	0.997619i	$-0.521968\pi$
$827$	−3.96629	−0.137921	−0.0689607	+	0.997619i	$0.521968\pi$
$828$	0	0
$829$	−17.4461	−0.605928	−0.302964	−	0.953002i	$-0.597976\pi$
$829$	−17.4461	−0.605928	−0.302964	+	0.953002i	$0.597976\pi$
$830$	0	0
$831$	−30.1454	−1.04573
$832$	0	0
$833$	−9.78260	−0.338947
$834$	0	0
$835$	−48.5167	−1.67899
$836$	0	0
$837$	−24.4463	−0.844987
$838$	0	0
$839$	34.5372	1.19236	0.596179	−	0.802852i	$-0.296685\pi$
$839$	34.5372	1.19236	0.596179	+	0.802852i	$0.296685\pi$
$840$	0	0
$841$	29.2028	1.00699
$842$	0	0
$843$	−52.0996	−1.79441
$844$	0	0
$845$	−9.85015	−0.338856
$846$	0	0
$847$	−32.0130	−1.09998
$848$	0	0
$849$	−36.0009	−1.23555
$850$	0	0
$851$	−43.2699	−1.48327
$852$	0	0
$853$	−4.74018	−0.162301	−0.0811503	−	0.996702i	$-0.525859\pi$
$853$	−4.74018	−0.162301	−0.0811503	+	0.996702i	$0.525859\pi$
$854$	0	0
$855$	2.98266	0.102005
$856$	0	0
$857$	−45.6554	−1.55956	−0.779779	−	0.626054i	$-0.784669\pi$
$857$	−45.6554	−1.55956	−0.779779	+	0.626054i	$0.784669\pi$
$858$	0	0
$859$	54.3182	1.85331	0.926657	−	0.375909i	$-0.122669\pi$
$859$	54.3182	1.85331	0.926657	+	0.375909i	$0.122669\pi$
$860$	0	0
$861$	−2.14626	−0.0731443
$862$	0	0
$863$	−27.1765	−0.925098	−0.462549	−	0.886594i	$-0.653065\pi$
$863$	−27.1765	−0.925098	−0.462549	+	0.886594i	$0.653065\pi$
$864$	0	0
$865$	−6.01666	−0.204573
$866$	0	0
$867$	−25.4622	−0.864742
$868$	0	0
$869$	−88.1175	−2.98918
$870$	0	0
$871$	54.0312	1.83078
$872$	0	0
$873$	−0.687213	−0.0232586
$874$	0	0
$875$	17.8591	0.603749
$876$	0	0
$877$	−4.48040	−0.151292	−0.0756462	−	0.997135i	$-0.524102\pi$
$877$	−4.48040	−0.151292	−0.0756462	+	0.997135i	$0.524102\pi$
$878$	0	0
$879$	40.8725	1.37860
$880$	0	0
$881$	25.0110	0.842643	0.421321	−	0.906911i	$-0.361566\pi$
$881$	25.0110	0.842643	0.421321	+	0.906911i	$0.361566\pi$
$882$	0	0
$883$	20.5437	0.691350	0.345675	−	0.938354i	$-0.387650\pi$
$883$	20.5437	0.691350	0.345675	+	0.938354i	$0.387650\pi$
$884$	0	0
$885$	23.5301	0.790954
$886$	0	0
$887$	−31.9978	−1.07438	−0.537190	−	0.843461i	$-0.680514\pi$
$887$	−31.9978	−1.07438	−0.537190	+	0.843461i	$0.680514\pi$
$888$	0	0
$889$	14.8565	0.498271
$890$	0	0
$891$	62.0844	2.07990
$892$	0	0
$893$	−10.8762	−0.363958
$894$	0	0
$895$	−84.5052	−2.82470
$896$	0	0
$897$	−32.0485	−1.07007
$898$	0	0
$899$	−43.7064	−1.45769
$900$	0	0
$901$	−1.99113	−0.0663340
$902$	0	0
$903$	−33.0663	−1.10038
$904$	0	0
$905$	−22.0881	−0.734233
$906$	0	0
$907$	41.0256	1.36223	0.681116	−	0.732176i	$-0.261495\pi$
$907$	41.0256	1.36223	0.681116	+	0.732176i	$0.261495\pi$
$908$	0	0
$909$	6.62765	0.219825
$910$	0	0
$911$	−40.9142	−1.35555	−0.677773	−	0.735271i	$-0.737055\pi$
$911$	−40.9142	−1.35555	−0.677773	+	0.735271i	$0.737055\pi$
$912$	0	0
$913$	−27.8653	−0.922206
$914$	0	0
$915$	1.47057	0.0486155
$916$	0	0
$917$	−17.1093	−0.565000
$918$	0	0
$919$	43.7491	1.44315	0.721575	−	0.692337i	$-0.243419\pi$
$919$	43.7491	1.44315	0.721575	+	0.692337i	$0.243419\pi$
$920$	0	0
$921$	31.3360	1.03256
$922$	0	0
$923$	−10.8635	−0.357576
$924$	0	0
$925$	87.5449	2.87846
$926$	0	0
$927$	−10.0682	−0.330684
$928$	0	0
$929$	32.0077	1.05014	0.525069	−	0.851059i	$-0.324039\pi$
$929$	32.0077	1.05014	0.525069	+	0.851059i	$0.324039\pi$
$930$	0	0
$931$	4.91310	0.161020
$932$	0	0
$933$	−39.4736	−1.29231
$934$	0	0
$935$	41.9404	1.37160
$936$	0	0
$937$	8.64753	0.282503	0.141251	−	0.989974i	$-0.454887\pi$
$937$	8.64753	0.282503	0.141251	+	0.989974i	$0.454887\pi$
$938$	0	0
$939$	−41.6060	−1.35776
$940$	0	0
$941$	−5.54020	−0.180605	−0.0903026	−	0.995914i	$-0.528783\pi$
$941$	−5.54020	−0.180605	−0.0903026	+	0.995914i	$0.528783\pi$
$942$	0	0
$943$	−3.15026	−0.102586
$944$	0	0
$945$	−22.5483	−0.733498
$946$	0	0
$947$	23.4116	0.760776	0.380388	−	0.924827i	$-0.375790\pi$
$947$	23.4116	0.760776	0.380388	+	0.924827i	$0.375790\pi$
$948$	0	0
$949$	−25.3885	−0.824146
$950$	0	0
$951$	25.5435	0.828304
$952$	0	0
$953$	−29.1978	−0.945809	−0.472905	−	0.881114i	$-0.656795\pi$
$953$	−29.1978	−0.945809	−0.472905	+	0.881114i	$0.656795\pi$
$954$	0	0
$955$	−71.6765	−2.31940
$956$	0	0
$957$	85.8128	2.77393
$958$	0	0
$959$	11.8159	0.381554
$960$	0	0
$961$	1.82055	0.0587274
$962$	0	0
$963$	−15.1800	−0.489169
$964$	0	0
$965$	86.6792	2.79030
$966$	0	0
$967$	−49.7682	−1.60044	−0.800219	−	0.599708i	$-0.795283\pi$
$967$	−49.7682	−1.60044	−0.800219	+	0.599708i	$0.795283\pi$
$968$	0	0
$969$	−3.88928	−0.124942
$970$	0	0
$971$	17.0405	0.546854	0.273427	−	0.961893i	$-0.411843\pi$
$971$	17.0405	0.546854	0.273427	+	0.961893i	$0.411843\pi$
$972$	0	0
$973$	−6.09399	−0.195364
$974$	0	0
$975$	64.8416	2.07659
$976$	0	0
$977$	−32.5435	−1.04116	−0.520580	−	0.853813i	$-0.674284\pi$
$977$	−32.5435	−1.04116	−0.520580	+	0.853813i	$0.674284\pi$
$978$	0	0
$979$	−72.0819	−2.30375
$980$	0	0
$981$	2.39353	0.0764195
$982$	0	0
$983$	39.9426	1.27397	0.636986	−	0.770876i	$-0.280181\pi$
$983$	39.9426	1.27397	0.636986	+	0.770876i	$0.280181\pi$
$984$	0	0
$985$	28.7162	0.914973
$986$	0	0
$987$	−30.6902	−0.976879
$988$	0	0
$989$	−48.5344	−1.54330
$990$	0	0
$991$	−25.6557	−0.814980	−0.407490	−	0.913210i	$-0.633596\pi$
$991$	−25.6557	−0.814980	−0.407490	+	0.913210i	$0.633596\pi$
$992$	0	0
$993$	−29.5097	−0.936461
$994$	0	0
$995$	19.1480	0.607033
$996$	0	0
$997$	−39.5713	−1.25324	−0.626618	−	0.779326i	$-0.715561\pi$
$997$	−39.5713	−1.25324	−0.626618	+	0.779326i	$0.715561\pi$
$998$	0	0
$999$	−44.5799	−1.41045

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.15	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.15	✓	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)

\(f(q)\)	\(=\)	\(q+1.95331 q^{3} -3.65783 q^{5} -1.44461 q^{7} +0.815417 q^{9} +O(q^{10})\)	\(q+1.95331 q^{3} -3.65783 q^{5} -1.44461 q^{7} +0.815417 q^{9} -5.75850 q^{11} +3.96143 q^{13} -7.14488 q^{15} +1.99113 q^{17} -1.00000 q^{19} -2.82177 q^{21} -4.14177 q^{23} +8.37975 q^{25} -4.26717 q^{27} -7.62907 q^{29} +5.72892 q^{31} -11.2481 q^{33} +5.28415 q^{35} +10.4472 q^{37} +7.73789 q^{39} +0.760606 q^{41} +11.7183 q^{43} -2.98266 q^{45} +10.8762 q^{47} -4.91310 q^{49} +3.88928 q^{51} -1.00000 q^{53} +21.0636 q^{55} -1.95331 q^{57} -3.29327 q^{59} -0.205821 q^{61} -1.17796 q^{63} -14.4902 q^{65} +13.6393 q^{67} -8.09016 q^{69} -2.74232 q^{71} -6.40893 q^{73} +16.3683 q^{75} +8.31879 q^{77} +15.3022 q^{79} -10.7813 q^{81} +4.83898 q^{83} -7.28321 q^{85} -14.9019 q^{87} +12.5175 q^{89} -5.72272 q^{91} +11.1904 q^{93} +3.65783 q^{95} -0.842775 q^{97} -4.69558 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 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

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