LMFDB - Embedded newform 4028.2.a.d.1.6

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			1.6
Root			$1.96979$ 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)

$f(q)$	$=$	$q-1.96979 q^{3} -1.29651 q^{5} -5.21755 q^{7} +0.880060 q^{9} +O(q^{10})$	$q-1.96979 q^{3} -1.29651 q^{5} -5.21755 q^{7} +0.880060 q^{9} -5.81982 q^{11} +4.89321 q^{13} +2.55386 q^{15} +5.22872 q^{17} -1.00000 q^{19} +10.2775 q^{21} +4.72779 q^{23} -3.31905 q^{25} +4.17583 q^{27} +6.72475 q^{29} +3.44639 q^{31} +11.4638 q^{33} +6.76463 q^{35} +1.61079 q^{37} -9.63857 q^{39} -4.72814 q^{41} -1.03265 q^{43} -1.14101 q^{45} +1.95545 q^{47} +20.2228 q^{49} -10.2995 q^{51} +1.00000 q^{53} +7.54548 q^{55} +1.96979 q^{57} -11.9904 q^{59} +10.5960 q^{61} -4.59175 q^{63} -6.34411 q^{65} -4.14317 q^{67} -9.31274 q^{69} -0.689352 q^{71} -9.72589 q^{73} +6.53782 q^{75} +30.3652 q^{77} +2.39759 q^{79} -10.8657 q^{81} +0.0104429 q^{83} -6.77911 q^{85} -13.2463 q^{87} -8.42399 q^{89} -25.5305 q^{91} -6.78865 q^{93} +1.29651 q^{95} -12.3552 q^{97} -5.12179 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.96979	−1.13726	−0.568628	−	0.822594i	$-0.692526\pi$
$3$	−1.96979	−1.13726	−0.568628	+	0.822594i	$0.692526\pi$
$4$	0	0
$5$	−1.29651	−0.579819	−0.289909	−	0.957054i	$-0.593625\pi$
$5$	−1.29651	−0.579819	−0.289909	+	0.957054i	$0.593625\pi$
$6$	0	0
$7$	−5.21755	−1.97205	−0.986024	−	0.166603i	$-0.946720\pi$
$7$	−5.21755	−1.97205	−0.986024	+	0.166603i	$0.946720\pi$
$8$	0	0
$9$	0.880060	0.293353
$10$	0	0
$11$	−5.81982	−1.75474	−0.877372	−	0.479812i	$-0.840705\pi$
$11$	−5.81982	−1.75474	−0.877372	+	0.479812i	$0.840705\pi$
$12$	0	0
$13$	4.89321	1.35713	0.678566	−	0.734540i	$-0.262602\pi$
$13$	4.89321	1.35713	0.678566	+	0.734540i	$0.262602\pi$
$14$	0	0
$15$	2.55386	0.659403
$16$	0	0
$17$	5.22872	1.26815	0.634075	−	0.773271i	$-0.281381\pi$
$17$	5.22872	1.26815	0.634075	+	0.773271i	$0.281381\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	10.2775	2.24273
$22$	0	0
$23$	4.72779	0.985812	0.492906	−	0.870083i	$-0.335935\pi$
$23$	4.72779	0.985812	0.492906	+	0.870083i	$0.335935\pi$
$24$	0	0
$25$	−3.31905	−0.663810
$26$	0	0
$27$	4.17583	0.803639
$28$	0	0
$29$	6.72475	1.24876	0.624378	−	0.781123i	$-0.285353\pi$
$29$	6.72475	1.24876	0.624378	+	0.781123i	$0.285353\pi$
$30$	0	0
$31$	3.44639	0.618989	0.309495	−	0.950901i	$-0.399840\pi$
$31$	3.44639	0.618989	0.309495	+	0.950901i	$0.399840\pi$
$32$	0	0
$33$	11.4638	1.99559
$34$	0	0
$35$	6.76463	1.14343
$36$	0	0
$37$	1.61079	0.264812	0.132406	−	0.991196i	$-0.457730\pi$
$37$	1.61079	0.264812	0.132406	+	0.991196i	$0.457730\pi$
$38$	0	0
$39$	−9.63857	−1.54341
$40$	0	0
$41$	−4.72814	−0.738411	−0.369206	−	0.929348i	$-0.620370\pi$
$41$	−4.72814	−0.738411	−0.369206	+	0.929348i	$0.620370\pi$
$42$	0	0
$43$	−1.03265	−0.157477	−0.0787386	−	0.996895i	$-0.525089\pi$
$43$	−1.03265	−0.157477	−0.0787386	+	0.996895i	$0.525089\pi$
$44$	0	0
$45$	−1.14101	−0.170092
$46$	0	0
$47$	1.95545	0.285232	0.142616	−	0.989778i	$-0.454449\pi$
$47$	1.95545	0.285232	0.142616	+	0.989778i	$0.454449\pi$
$48$	0	0
$49$	20.2228	2.88897
$50$	0	0
$51$	−10.2995	−1.44221
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	7.54548	1.01743
$56$	0	0
$57$	1.96979	0.260905
$58$	0	0
$59$	−11.9904	−1.56102	−0.780510	−	0.625143i	$-0.785041\pi$
$59$	−11.9904	−1.56102	−0.780510	+	0.625143i	$0.785041\pi$
$60$	0	0
$61$	10.5960	1.35668	0.678340	−	0.734749i	$-0.262700\pi$
$61$	10.5960	1.35668	0.678340	+	0.734749i	$0.262700\pi$
$62$	0	0
$63$	−4.59175	−0.578507
$64$	0	0
$65$	−6.34411	−0.786890
$66$	0	0
$67$	−4.14317	−0.506169	−0.253085	−	0.967444i	$-0.581445\pi$
$67$	−4.14317	−0.506169	−0.253085	+	0.967444i	$0.581445\pi$
$68$	0	0
$69$	−9.31274	−1.12112
$70$	0	0
$71$	−0.689352	−0.0818110	−0.0409055	−	0.999163i	$-0.513024\pi$
$71$	−0.689352	−0.0818110	−0.0409055	+	0.999163i	$0.513024\pi$
$72$	0	0
$73$	−9.72589	−1.13833	−0.569165	−	0.822224i	$-0.692733\pi$
$73$	−9.72589	−1.13833	−0.569165	+	0.822224i	$0.692733\pi$
$74$	0	0
$75$	6.53782	0.754923
$76$	0	0
$77$	30.3652	3.46044
$78$	0	0
$79$	2.39759	0.269749	0.134875	−	0.990863i	$-0.456937\pi$
$79$	2.39759	0.269749	0.134875	+	0.990863i	$0.456937\pi$
$80$	0	0
$81$	−10.8657	−1.20730
$82$	0	0
$83$	0.0104429	0.00114626	0.000573130	−	1.00000i	$-0.499818\pi$
$83$	0.0104429	0.00114626	0.000573130		1.00000i	$0.499818\pi$
$84$	0	0
$85$	−6.77911	−0.735297
$86$	0	0
$87$	−13.2463	−1.42016
$88$	0	0
$89$	−8.42399	−0.892941	−0.446470	−	0.894798i	$-0.647319\pi$
$89$	−8.42399	−0.892941	−0.446470	+	0.894798i	$0.647319\pi$
$90$	0	0
$91$	−25.5305	−2.67633
$92$	0	0
$93$	−6.78865	−0.703950
$94$	0	0
$95$	1.29651	0.133020
$96$	0	0
$97$	−12.3552	−1.25448	−0.627240	−	0.778826i	$-0.715816\pi$
$97$	−12.3552	−1.25448	−0.627240	+	0.778826i	$0.715816\pi$
$98$	0	0
$99$	−5.12179	−0.514760
$100$	0	0
$101$	−13.3851	−1.33187	−0.665933	−	0.746012i	$-0.731966\pi$
$101$	−13.3851	−1.33187	−0.665933	+	0.746012i	$0.731966\pi$
$102$	0	0
$103$	−6.37542	−0.628188	−0.314094	−	0.949392i	$-0.601701\pi$
$103$	−6.37542	−0.628188	−0.314094	+	0.949392i	$0.601701\pi$
$104$	0	0
$105$	−13.3249	−1.30037
$106$	0	0
$107$	0.657226	0.0635365	0.0317682	−	0.999495i	$-0.489886\pi$
$107$	0.657226	0.0635365	0.0317682	+	0.999495i	$0.489886\pi$
$108$	0	0
$109$	−3.90451	−0.373985	−0.186992	−	0.982361i	$-0.559874\pi$
$109$	−3.90451	−0.373985	−0.186992	+	0.982361i	$0.559874\pi$
$110$	0	0
$111$	−3.17291	−0.301159
$112$	0	0
$113$	12.3708	1.16375	0.581873	−	0.813280i	$-0.302320\pi$
$113$	12.3708	1.16375	0.581873	+	0.813280i	$0.302320\pi$
$114$	0	0
$115$	−6.12965	−0.571592
$116$	0	0
$117$	4.30631	0.398119
$118$	0	0
$119$	−27.2811	−2.50085
$120$	0	0
$121$	22.8704	2.07912
$122$	0	0
$123$	9.31342	0.839763
$124$	0	0
$125$	10.7858	0.964708
$126$	0	0
$127$	13.4231	1.19111	0.595554	−	0.803316i	$-0.296933\pi$
$127$	13.4231	1.19111	0.595554	+	0.803316i	$0.296933\pi$
$128$	0	0
$129$	2.03410	0.179092
$130$	0	0
$131$	13.9258	1.21670	0.608352	−	0.793667i	$-0.291831\pi$
$131$	13.9258	1.21670	0.608352	+	0.793667i	$0.291831\pi$
$132$	0	0
$133$	5.21755	0.452419
$134$	0	0
$135$	−5.41402	−0.465965
$136$	0	0
$137$	6.06657	0.518302	0.259151	−	0.965837i	$-0.416557\pi$
$137$	6.06657	0.518302	0.259151	+	0.965837i	$0.416557\pi$
$138$	0	0
$139$	1.40671	0.119315	0.0596576	−	0.998219i	$-0.480999\pi$
$139$	1.40671	0.119315	0.0596576	+	0.998219i	$0.480999\pi$
$140$	0	0
$141$	−3.85182	−0.324382
$142$	0	0
$143$	−28.4776	−2.38142
$144$	0	0
$145$	−8.71874	−0.724052
$146$	0	0
$147$	−39.8346	−3.28550
$148$	0	0
$149$	−4.00989	−0.328503	−0.164251	−	0.986419i	$-0.552521\pi$
$149$	−4.00989	−0.328503	−0.164251	+	0.986419i	$0.552521\pi$
$150$	0	0
$151$	−2.16067	−0.175833	−0.0879164	−	0.996128i	$-0.528021\pi$
$151$	−2.16067	−0.175833	−0.0879164	+	0.996128i	$0.528021\pi$
$152$	0	0
$153$	4.60158	0.372016
$154$	0	0
$155$	−4.46829	−0.358902
$156$	0	0
$157$	−19.5293	−1.55861	−0.779303	−	0.626647i	$-0.784427\pi$
$157$	−19.5293	−1.55861	−0.779303	+	0.626647i	$0.784427\pi$
$158$	0	0
$159$	−1.96979	−0.156214
$160$	0	0
$161$	−24.6675	−1.94407
$162$	0	0
$163$	−0.608060	−0.0476269	−0.0238135	−	0.999716i	$-0.507581\pi$
$163$	−0.608060	−0.0476269	−0.0238135	+	0.999716i	$0.507581\pi$
$164$	0	0
$165$	−14.8630	−1.15708
$166$	0	0
$167$	19.2419	1.48898	0.744492	−	0.667632i	$-0.232692\pi$
$167$	19.2419	1.48898	0.744492	+	0.667632i	$0.232692\pi$
$168$	0	0
$169$	10.9435	0.841805
$170$	0	0
$171$	−0.880060	−0.0672998
$172$	0	0
$173$	−16.1152	−1.22522	−0.612610	−	0.790386i	$-0.709880\pi$
$173$	−16.1152	−1.22522	−0.612610	+	0.790386i	$0.709880\pi$
$174$	0	0
$175$	17.3173	1.30907
$176$	0	0
$177$	23.6186	1.77528
$178$	0	0
$179$	−2.72804	−0.203903	−0.101952	−	0.994789i	$-0.532509\pi$
$179$	−2.72804	−0.203903	−0.101952	+	0.994789i	$0.532509\pi$
$180$	0	0
$181$	13.5398	1.00640	0.503202	−	0.864169i	$-0.332155\pi$
$181$	13.5398	1.00640	0.503202	+	0.864169i	$0.332155\pi$
$182$	0	0
$183$	−20.8719	−1.54289
$184$	0	0
$185$	−2.08841	−0.153543
$186$	0	0
$187$	−30.4302	−2.22528
$188$	0	0
$189$	−21.7876	−1.58481
$190$	0	0
$191$	16.7158	1.20951	0.604757	−	0.796410i	$-0.293270\pi$
$191$	16.7158	1.20951	0.604757	+	0.796410i	$0.293270\pi$
$192$	0	0
$193$	21.9979	1.58344	0.791721	−	0.610883i	$-0.209185\pi$
$193$	21.9979	1.58344	0.791721	+	0.610883i	$0.209185\pi$
$194$	0	0
$195$	12.4965	0.894896
$196$	0	0
$197$	17.3672	1.23736	0.618681	−	0.785642i	$-0.287667\pi$
$197$	17.3672	1.23736	0.618681	+	0.785642i	$0.287667\pi$
$198$	0	0
$199$	9.55530	0.677357	0.338679	−	0.940902i	$-0.390020\pi$
$199$	9.55530	0.677357	0.338679	+	0.940902i	$0.390020\pi$
$200$	0	0
$201$	8.16117	0.575645
$202$	0	0
$203$	−35.0867	−2.46261
$204$	0	0
$205$	6.13010	0.428145
$206$	0	0
$207$	4.16074	0.289191
$208$	0	0
$209$	5.81982	0.402566
$210$	0	0
$211$	−18.2772	−1.25825	−0.629127	−	0.777302i	$-0.716588\pi$
$211$	−18.2772	−1.25825	−0.629127	+	0.777302i	$0.716588\pi$
$212$	0	0
$213$	1.35788	0.0930401
$214$	0	0
$215$	1.33884	0.0913083
$216$	0	0
$217$	−17.9817	−1.22068
$218$	0	0
$219$	19.1579	1.29457
$220$	0	0
$221$	25.5852	1.72105
$222$	0	0
$223$	17.0794	1.14372	0.571861	−	0.820351i	$-0.306222\pi$
$223$	17.0794	1.14372	0.571861	+	0.820351i	$0.306222\pi$
$224$	0	0
$225$	−2.92096	−0.194731
$226$	0	0
$227$	18.7796	1.24645	0.623225	−	0.782043i	$-0.285822\pi$
$227$	18.7796	1.24645	0.623225	+	0.782043i	$0.285822\pi$
$228$	0	0
$229$	4.97105	0.328496	0.164248	−	0.986419i	$-0.447480\pi$
$229$	4.97105	0.328496	0.164248	+	0.986419i	$0.447480\pi$
$230$	0	0
$231$	−59.8130	−3.93541
$232$	0	0
$233$	−24.8924	−1.63076	−0.815379	−	0.578927i	$-0.803471\pi$
$233$	−24.8924	−1.63076	−0.815379	+	0.578927i	$0.803471\pi$
$234$	0	0
$235$	−2.53527	−0.165383
$236$	0	0
$237$	−4.72273	−0.306774
$238$	0	0
$239$	6.31306	0.408358	0.204179	−	0.978934i	$-0.434548\pi$
$239$	6.31306	0.408358	0.204179	+	0.978934i	$0.434548\pi$
$240$	0	0
$241$	−20.9988	−1.35265	−0.676325	−	0.736603i	$-0.736429\pi$
$241$	−20.9988	−1.35265	−0.676325	+	0.736603i	$0.736429\pi$
$242$	0	0
$243$	8.87557	0.569368
$244$	0	0
$245$	−26.2192	−1.67508
$246$	0	0
$247$	−4.89321	−0.311347
$248$	0	0
$249$	−0.0205704	−0.00130359
$250$	0	0
$251$	10.2553	0.647308	0.323654	−	0.946176i	$-0.395089\pi$
$251$	10.2553	0.647308	0.323654	+	0.946176i	$0.395089\pi$
$252$	0	0
$253$	−27.5149	−1.72985
$254$	0	0
$255$	13.3534	0.836222
$256$	0	0
$257$	−22.7615	−1.41982	−0.709912	−	0.704290i	$-0.751265\pi$
$257$	−22.7615	−1.41982	−0.709912	+	0.704290i	$0.751265\pi$
$258$	0	0
$259$	−8.40436	−0.522222
$260$	0	0
$261$	5.91818	0.366326
$262$	0	0
$263$	−32.4166	−1.99889	−0.999446	−	0.0332770i	$-0.989406\pi$
$263$	−32.4166	−1.99889	−0.999446	+	0.0332770i	$0.989406\pi$
$264$	0	0
$265$	−1.29651	−0.0796442
$266$	0	0
$267$	16.5935	1.01550
$268$	0	0
$269$	−5.64412	−0.344128	−0.172064	−	0.985086i	$-0.555044\pi$
$269$	−5.64412	−0.344128	−0.172064	+	0.985086i	$0.555044\pi$
$270$	0	0
$271$	5.44087	0.330509	0.165255	−	0.986251i	$-0.447155\pi$
$271$	5.44087	0.330509	0.165255	+	0.986251i	$0.447155\pi$
$272$	0	0
$273$	50.2897	3.04367
$274$	0	0
$275$	19.3163	1.16482
$276$	0	0
$277$	12.7312	0.764941	0.382471	−	0.923968i	$-0.375073\pi$
$277$	12.7312	0.764941	0.382471	+	0.923968i	$0.375073\pi$
$278$	0	0
$279$	3.03303	0.181583
$280$	0	0
$281$	17.6263	1.05150	0.525749	−	0.850640i	$-0.323785\pi$
$281$	17.6263	1.05150	0.525749	+	0.850640i	$0.323785\pi$
$282$	0	0
$283$	−1.86621	−0.110935	−0.0554674	−	0.998461i	$-0.517665\pi$
$283$	−1.86621	−0.110935	−0.0554674	+	0.998461i	$0.517665\pi$
$284$	0	0
$285$	−2.55386	−0.151277
$286$	0	0
$287$	24.6693	1.45618
$288$	0	0
$289$	10.3395	0.608205
$290$	0	0
$291$	24.3371	1.42667
$292$	0	0
$293$	−14.3860	−0.840441	−0.420220	−	0.907422i	$-0.638047\pi$
$293$	−14.3860	−0.840441	−0.420220	+	0.907422i	$0.638047\pi$
$294$	0	0
$295$	15.5458	0.905109
$296$	0	0
$297$	−24.3026	−1.41018
$298$	0	0
$299$	23.1340	1.33788
$300$	0	0
$301$	5.38789	0.310553
$302$	0	0
$303$	26.3658	1.51467
$304$	0	0
$305$	−13.7379	−0.786628
$306$	0	0
$307$	−29.0997	−1.66081	−0.830405	−	0.557161i	$-0.811891\pi$
$307$	−29.0997	−1.66081	−0.830405	+	0.557161i	$0.811891\pi$
$308$	0	0
$309$	12.5582	0.714412
$310$	0	0
$311$	28.0564	1.59093	0.795467	−	0.605996i	$-0.207225\pi$
$311$	28.0564	1.59093	0.795467	+	0.605996i	$0.207225\pi$
$312$	0	0
$313$	−9.68353	−0.547345	−0.273673	−	0.961823i	$-0.588239\pi$
$313$	−9.68353	−0.547345	−0.273673	+	0.961823i	$0.588239\pi$
$314$	0	0
$315$	5.95327	0.335429
$316$	0	0
$317$	−14.1958	−0.797313	−0.398656	−	0.917100i	$-0.630523\pi$
$317$	−14.1958	−0.797313	−0.398656	+	0.917100i	$0.630523\pi$
$318$	0	0
$319$	−39.1369	−2.19124
$320$	0	0
$321$	−1.29460	−0.0722573
$322$	0	0
$323$	−5.22872	−0.290934
$324$	0	0
$325$	−16.2408	−0.900878
$326$	0	0
$327$	7.69106	0.425317
$328$	0	0
$329$	−10.2027	−0.562491
$330$	0	0
$331$	14.3822	0.790516	0.395258	−	0.918570i	$-0.370655\pi$
$331$	14.3822	0.790516	0.395258	+	0.918570i	$0.370655\pi$
$332$	0	0
$333$	1.41759	0.0776834
$334$	0	0
$335$	5.37168	0.293486
$336$	0	0
$337$	−9.30426	−0.506835	−0.253418	−	0.967357i	$-0.581555\pi$
$337$	−9.30426	−0.506835	−0.253418	+	0.967357i	$0.581555\pi$
$338$	0	0
$339$	−24.3678	−1.32348
$340$	0	0
$341$	−20.0574	−1.08617
$342$	0	0
$343$	−68.9907	−3.72515
$344$	0	0
$345$	12.0741	0.650047
$346$	0	0
$347$	30.1963	1.62102	0.810510	−	0.585725i	$-0.199190\pi$
$347$	30.1963	1.62102	0.810510	+	0.585725i	$0.199190\pi$
$348$	0	0
$349$	−1.27305	−0.0681450	−0.0340725	−	0.999419i	$-0.510848\pi$
$349$	−1.27305	−0.0681450	−0.0340725	+	0.999419i	$0.510848\pi$
$350$	0	0
$351$	20.4332	1.09064
$352$	0	0
$353$	3.54421	0.188639	0.0943197	−	0.995542i	$-0.469932\pi$
$353$	3.54421	0.188639	0.0943197	+	0.995542i	$0.469932\pi$
$354$	0	0
$355$	0.893754	0.0474356
$356$	0	0
$357$	53.7379	2.84411
$358$	0	0
$359$	−8.41021	−0.443874	−0.221937	−	0.975061i	$-0.571238\pi$
$359$	−8.41021	−0.443874	−0.221937	+	0.975061i	$0.571238\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−45.0497	−2.36450
$364$	0	0
$365$	12.6098	0.660025
$366$	0	0
$367$	−6.39992	−0.334073	−0.167037	−	0.985951i	$-0.553420\pi$
$367$	−6.39992	−0.334073	−0.167037	+	0.985951i	$0.553420\pi$
$368$	0	0
$369$	−4.16104	−0.216615
$370$	0	0
$371$	−5.21755	−0.270882
$372$	0	0
$373$	−31.9953	−1.65665	−0.828327	−	0.560246i	$-0.810707\pi$
$373$	−31.9953	−1.65665	−0.828327	+	0.560246i	$0.810707\pi$
$374$	0	0
$375$	−21.2457	−1.09712
$376$	0	0
$377$	32.9056	1.69472
$378$	0	0
$379$	−26.8210	−1.37770	−0.688851	−	0.724903i	$-0.741885\pi$
$379$	−26.8210	−1.37770	−0.688851	+	0.724903i	$0.741885\pi$
$380$	0	0
$381$	−26.4406	−1.35459
$382$	0	0
$383$	0.876156	0.0447695	0.0223847	−	0.999749i	$-0.492874\pi$
$383$	0.876156	0.0447695	0.0223847	+	0.999749i	$0.492874\pi$
$384$	0	0
$385$	−39.3689	−2.00643
$386$	0	0
$387$	−0.908791	−0.0461965
$388$	0	0
$389$	3.38270	0.171510	0.0857549	−	0.996316i	$-0.472670\pi$
$389$	3.38270	0.171510	0.0857549	+	0.996316i	$0.472670\pi$
$390$	0	0
$391$	24.7203	1.25016
$392$	0	0
$393$	−27.4309	−1.38370
$394$	0	0
$395$	−3.10850	−0.156406
$396$	0	0
$397$	−21.4280	−1.07544	−0.537721	−	0.843123i	$-0.680715\pi$
$397$	−21.4280	−1.07544	−0.537721	+	0.843123i	$0.680715\pi$
$398$	0	0
$399$	−10.2775	−0.514516
$400$	0	0
$401$	18.4808	0.922886	0.461443	−	0.887170i	$-0.347332\pi$
$401$	18.4808	0.922886	0.461443	+	0.887170i	$0.347332\pi$
$402$	0	0
$403$	16.8639	0.840050
$404$	0	0
$405$	14.0875	0.700013
$406$	0	0
$407$	−9.37450	−0.464677
$408$	0	0
$409$	−39.1287	−1.93479	−0.967395	−	0.253271i	$-0.918494\pi$
$409$	−39.1287	−1.93479	−0.967395	+	0.253271i	$0.918494\pi$
$410$	0	0
$411$	−11.9499	−0.589443
$412$	0	0
$413$	62.5606	3.07841
$414$	0	0
$415$	−0.0135394	−0.000664624 0
$416$	0	0
$417$	−2.77091	−0.135692
$418$	0	0
$419$	−18.2905	−0.893547	−0.446774	−	0.894647i	$-0.647427\pi$
$419$	−18.2905	−0.893547	−0.446774	+	0.894647i	$0.647427\pi$
$420$	0	0
$421$	−21.9161	−1.06812	−0.534062	−	0.845445i	$-0.679335\pi$
$421$	−21.9161	−1.06812	−0.534062	+	0.845445i	$0.679335\pi$
$422$	0	0
$423$	1.72091	0.0836737
$424$	0	0
$425$	−17.3544	−0.841811
$426$	0	0
$427$	−55.2852	−2.67544
$428$	0	0
$429$	56.0948	2.70828
$430$	0	0
$431$	−28.6450	−1.37978	−0.689891	−	0.723914i	$-0.742341\pi$
$431$	−28.6450	−1.37978	−0.689891	+	0.723914i	$0.742341\pi$
$432$	0	0
$433$	29.2106	1.40377	0.701886	−	0.712289i	$-0.252341\pi$
$433$	29.2106	1.40377	0.701886	+	0.712289i	$0.252341\pi$
$434$	0	0
$435$	17.1741	0.823433
$436$	0	0
$437$	−4.72779	−0.226161
$438$	0	0
$439$	−12.5734	−0.600096	−0.300048	−	0.953924i	$-0.597003\pi$
$439$	−12.5734	−0.600096	−0.300048	+	0.953924i	$0.597003\pi$
$440$	0	0
$441$	17.7973	0.847490
$442$	0	0
$443$	3.00267	0.142661	0.0713306	−	0.997453i	$-0.477275\pi$
$443$	3.00267	0.142661	0.0713306	+	0.997453i	$0.477275\pi$
$444$	0	0
$445$	10.9218	0.517744
$446$	0	0
$447$	7.89862	0.373592
$448$	0	0
$449$	−25.8614	−1.22047	−0.610237	−	0.792219i	$-0.708926\pi$
$449$	−25.8614	−1.22047	−0.610237	+	0.792219i	$0.708926\pi$
$450$	0	0
$451$	27.5169	1.29572
$452$	0	0
$453$	4.25606	0.199967
$454$	0	0
$455$	33.1007	1.55178
$456$	0	0
$457$	24.2264	1.13327	0.566633	−	0.823971i	$-0.308246\pi$
$457$	24.2264	1.13327	0.566633	+	0.823971i	$0.308246\pi$
$458$	0	0
$459$	21.8342	1.01913
$460$	0	0
$461$	−19.0271	−0.886180	−0.443090	−	0.896477i	$-0.646118\pi$
$461$	−19.0271	−0.886180	−0.443090	+	0.896477i	$0.646118\pi$
$462$	0	0
$463$	5.91912	0.275085	0.137542	−	0.990496i	$-0.456080\pi$
$463$	5.91912	0.275085	0.137542	+	0.990496i	$0.456080\pi$
$464$	0	0
$465$	8.80158	0.408163
$466$	0	0
$467$	−10.5350	−0.487503	−0.243752	−	0.969838i	$-0.578378\pi$
$467$	−10.5350	−0.487503	−0.243752	+	0.969838i	$0.578378\pi$
$468$	0	0
$469$	21.6172	0.998190
$470$	0	0
$471$	38.4685	1.77254
$472$	0	0
$473$	6.00983	0.276332
$474$	0	0
$475$	3.31905	0.152289
$476$	0	0
$477$	0.880060	0.0402952
$478$	0	0
$479$	42.0433	1.92101	0.960503	−	0.278269i	$-0.0897608\pi$
$479$	42.0433	1.92101	0.960503	+	0.278269i	$0.0897608\pi$
$480$	0	0
$481$	7.88192	0.359384
$482$	0	0
$483$	48.5897	2.21091
$484$	0	0
$485$	16.0187	0.727372
$486$	0	0
$487$	−42.2443	−1.91427	−0.957136	−	0.289640i	$-0.906465\pi$
$487$	−42.2443	−1.91427	−0.957136	+	0.289640i	$0.906465\pi$
$488$	0	0
$489$	1.19775	0.0541641
$490$	0	0
$491$	−32.0263	−1.44533	−0.722663	−	0.691201i	$-0.757082\pi$
$491$	−32.0263	−1.44533	−0.722663	+	0.691201i	$0.757082\pi$
$492$	0	0
$493$	35.1618	1.58361
$494$	0	0
$495$	6.64048	0.298467
$496$	0	0
$497$	3.59673	0.161335
$498$	0	0
$499$	32.3743	1.44927	0.724637	−	0.689131i	$-0.242007\pi$
$499$	32.3743	1.44927	0.724637	+	0.689131i	$0.242007\pi$
$500$	0	0
$501$	−37.9025	−1.69336
$502$	0	0
$503$	−25.3227	−1.12908	−0.564542	−	0.825404i	$-0.690947\pi$
$503$	−25.3227	−1.12908	−0.564542	+	0.825404i	$0.690947\pi$
$504$	0	0
$505$	17.3540	0.772241
$506$	0	0
$507$	−21.5563	−0.957349
$508$	0	0
$509$	7.74919	0.343477	0.171738	−	0.985143i	$-0.445062\pi$
$509$	7.74919	0.343477	0.171738	+	0.985143i	$0.445062\pi$
$510$	0	0
$511$	50.7453	2.24484
$512$	0	0
$513$	−4.17583	−0.184367
$514$	0	0
$515$	8.26582	0.364235
$516$	0	0
$517$	−11.3804	−0.500509
$518$	0	0
$519$	31.7436	1.39339
$520$	0	0
$521$	23.2359	1.01798	0.508991	−	0.860772i	$-0.330019\pi$
$521$	23.2359	1.01798	0.508991	+	0.860772i	$0.330019\pi$
$522$	0	0
$523$	−14.7581	−0.645324	−0.322662	−	0.946514i	$-0.604578\pi$
$523$	−14.7581	−0.645324	−0.322662	+	0.946514i	$0.604578\pi$
$524$	0	0
$525$	−34.1114	−1.48874
$526$	0	0
$527$	18.0202	0.784972
$528$	0	0
$529$	−0.648008	−0.0281743
$530$	0	0
$531$	−10.5523	−0.457930
$532$	0	0
$533$	−23.1358	−1.00212
$534$	0	0
$535$	−0.852103	−0.0368396
$536$	0	0
$537$	5.37366	0.231891
$538$	0	0
$539$	−117.693	−5.06941
$540$	0	0
$541$	2.11018	0.0907238	0.0453619	−	0.998971i	$-0.485556\pi$
$541$	2.11018	0.0907238	0.0453619	+	0.998971i	$0.485556\pi$
$542$	0	0
$543$	−26.6705	−1.14454
$544$	0	0
$545$	5.06226	0.216843
$546$	0	0
$547$	−9.81048	−0.419466	−0.209733	−	0.977759i	$-0.567259\pi$
$547$	−9.81048	−0.419466	−0.209733	+	0.977759i	$0.567259\pi$
$548$	0	0
$549$	9.32511	0.397986
$550$	0	0
$551$	−6.72475	−0.286484
$552$	0	0
$553$	−12.5095	−0.531959
$554$	0	0
$555$	4.11372	0.174618
$556$	0	0
$557$	23.9508	1.01483	0.507414	−	0.861702i	$-0.330601\pi$
$557$	23.9508	1.01483	0.507414	+	0.861702i	$0.330601\pi$
$558$	0	0
$559$	−5.05296	−0.213717
$560$	0	0
$561$	59.9410	2.53071
$562$	0	0
$563$	26.0745	1.09891	0.549455	−	0.835524i	$-0.314836\pi$
$563$	26.0745	1.09891	0.549455	+	0.835524i	$0.314836\pi$
$564$	0	0
$565$	−16.0389	−0.674762
$566$	0	0
$567$	56.6922	2.38085
$568$	0	0
$569$	−22.3815	−0.938282	−0.469141	−	0.883123i	$-0.655436\pi$
$569$	−22.3815	−0.938282	−0.469141	+	0.883123i	$0.655436\pi$
$570$	0	0
$571$	−7.62134	−0.318943	−0.159472	−	0.987203i	$-0.550979\pi$
$571$	−7.62134	−0.318943	−0.159472	+	0.987203i	$0.550979\pi$
$572$	0	0
$573$	−32.9266	−1.37553
$574$	0	0
$575$	−15.6918	−0.654392
$576$	0	0
$577$	24.6818	1.02751	0.513757	−	0.857936i	$-0.328253\pi$
$577$	24.6818	1.02751	0.513757	+	0.857936i	$0.328253\pi$
$578$	0	0
$579$	−43.3311	−1.80078
$580$	0	0
$581$	−0.0544865	−0.00226048
$582$	0	0
$583$	−5.81982	−0.241033
$584$	0	0
$585$	−5.58320	−0.230837
$586$	0	0
$587$	−40.6158	−1.67639	−0.838197	−	0.545368i	$-0.816390\pi$
$587$	−40.6158	−1.67639	−0.838197	+	0.545368i	$0.816390\pi$
$588$	0	0
$589$	−3.44639	−0.142006
$590$	0	0
$591$	−34.2097	−1.40720
$592$	0	0
$593$	10.9958	0.451543	0.225772	−	0.974180i	$-0.427510\pi$
$593$	10.9958	0.451543	0.225772	+	0.974180i	$0.427510\pi$
$594$	0	0
$595$	35.3703	1.45004
$596$	0	0
$597$	−18.8219	−0.770329
$598$	0	0
$599$	20.0389	0.818769	0.409384	−	0.912362i	$-0.365743\pi$
$599$	20.0389	0.818769	0.409384	+	0.912362i	$0.365743\pi$
$600$	0	0
$601$	34.3763	1.40224	0.701119	−	0.713044i	$-0.252684\pi$
$601$	34.3763	1.40224	0.701119	+	0.713044i	$0.252684\pi$
$602$	0	0
$603$	−3.64624	−0.148486
$604$	0	0
$605$	−29.6517	−1.20551
$606$	0	0
$607$	16.6434	0.675536	0.337768	−	0.941229i	$-0.390328\pi$
$607$	16.6434	0.675536	0.337768	+	0.941229i	$0.390328\pi$
$608$	0	0
$609$	69.1134	2.80061
$610$	0	0
$611$	9.56842	0.387097
$612$	0	0
$613$	−40.6343	−1.64120	−0.820602	−	0.571500i	$-0.806362\pi$
$613$	−40.6343	−1.64120	−0.820602	+	0.571500i	$0.806362\pi$
$614$	0	0
$615$	−12.0750	−0.486910
$616$	0	0
$617$	−7.51663	−0.302608	−0.151304	−	0.988487i	$-0.548347\pi$
$617$	−7.51663	−0.302608	−0.151304	+	0.988487i	$0.548347\pi$
$618$	0	0
$619$	30.9390	1.24354	0.621771	−	0.783199i	$-0.286413\pi$
$619$	30.9390	1.24354	0.621771	+	0.783199i	$0.286413\pi$
$620$	0	0
$621$	19.7424	0.792237
$622$	0	0
$623$	43.9526	1.76092
$624$	0	0
$625$	2.61136	0.104454
$626$	0	0
$627$	−11.4638	−0.457821
$628$	0	0
$629$	8.42235	0.335821
$630$	0	0
$631$	−45.0085	−1.79176	−0.895881	−	0.444295i	$-0.853454\pi$
$631$	−45.0085	−1.79176	−0.895881	+	0.444295i	$0.853454\pi$
$632$	0	0
$633$	36.0022	1.43096
$634$	0	0
$635$	−17.4032	−0.690626
$636$	0	0
$637$	98.9544	3.92072
$638$	0	0
$639$	−0.606671	−0.0239995
$640$	0	0
$641$	−2.61776	−0.103395	−0.0516976	−	0.998663i	$-0.516463\pi$
$641$	−2.61776	−0.103395	−0.0516976	+	0.998663i	$0.516463\pi$
$642$	0	0
$643$	−40.6569	−1.60335	−0.801675	−	0.597760i	$-0.796058\pi$
$643$	−40.6569	−1.60335	−0.801675	+	0.597760i	$0.796058\pi$
$644$	0	0
$645$	−2.63723	−0.103841
$646$	0	0
$647$	17.2443	0.677944	0.338972	−	0.940796i	$-0.389921\pi$
$647$	17.2443	0.677944	0.338972	+	0.940796i	$0.389921\pi$
$648$	0	0
$649$	69.7822	2.73919
$650$	0	0
$651$	35.4201	1.38822
$652$	0	0
$653$	−24.2210	−0.947839	−0.473920	−	0.880568i	$-0.657161\pi$
$653$	−24.2210	−0.947839	−0.473920	+	0.880568i	$0.657161\pi$
$654$	0	0
$655$	−18.0550	−0.705468
$656$	0	0
$657$	−8.55936	−0.333932
$658$	0	0
$659$	−0.0140813	−0.000548531 0	−0.000274265	−	1.00000i	$-0.500087\pi$
$659$	−0.0140813	−0.000548531 0	−0.000274265		1.00000i	$0.500087\pi$
$660$	0	0
$661$	−21.6053	−0.840347	−0.420173	−	0.907444i	$-0.638031\pi$
$661$	−21.6053	−0.840347	−0.420173	+	0.907444i	$0.638031\pi$
$662$	0	0
$663$	−50.3974	−1.95727
$664$	0	0
$665$	−6.76463	−0.262321
$666$	0	0
$667$	31.7932	1.23104
$668$	0	0
$669$	−33.6428	−1.30071
$670$	0	0
$671$	−61.6669	−2.38062
$672$	0	0
$673$	41.7802	1.61051	0.805254	−	0.592930i	$-0.202029\pi$
$673$	41.7802	1.61051	0.805254	+	0.592930i	$0.202029\pi$
$674$	0	0
$675$	−13.8598	−0.533464
$676$	0	0
$677$	−0.267033	−0.0102629	−0.00513146	−	0.999987i	$-0.501633\pi$
$677$	−0.267033	−0.0102629	−0.00513146	+	0.999987i	$0.501633\pi$
$678$	0	0
$679$	64.4639	2.47390
$680$	0	0
$681$	−36.9919	−1.41753
$682$	0	0
$683$	−22.0413	−0.843388	−0.421694	−	0.906738i	$-0.638564\pi$
$683$	−22.0413	−0.843388	−0.421694	+	0.906738i	$0.638564\pi$
$684$	0	0
$685$	−7.86540	−0.300521
$686$	0	0
$687$	−9.79190	−0.373584
$688$	0	0
$689$	4.89321	0.186416
$690$	0	0
$691$	17.9762	0.683847	0.341924	−	0.939728i	$-0.388922\pi$
$691$	17.9762	0.683847	0.341924	+	0.939728i	$0.388922\pi$
$692$	0	0
$693$	26.7232	1.01513
$694$	0	0
$695$	−1.82381	−0.0691812
$696$	0	0
$697$	−24.7221	−0.936416
$698$	0	0
$699$	49.0328	1.85459
$700$	0	0
$701$	−36.8121	−1.39037	−0.695187	−	0.718829i	$-0.744679\pi$
$701$	−36.8121	−1.39037	−0.695187	+	0.718829i	$0.744679\pi$
$702$	0	0
$703$	−1.61079	−0.0607520
$704$	0	0
$705$	4.99394	0.188083
$706$	0	0
$707$	69.8373	2.62650
$708$	0	0
$709$	2.29726	0.0862754	0.0431377	−	0.999069i	$-0.486265\pi$
$709$	2.29726	0.0862754	0.0431377	+	0.999069i	$0.486265\pi$
$710$	0	0
$711$	2.11002	0.0791319
$712$	0	0
$713$	16.2938	0.610207
$714$	0	0
$715$	36.9216	1.38079
$716$	0	0
$717$	−12.4354	−0.464408
$718$	0	0
$719$	−6.90092	−0.257361	−0.128680	−	0.991686i	$-0.541074\pi$
$719$	−6.90092	−0.257361	−0.128680	+	0.991686i	$0.541074\pi$
$720$	0	0
$721$	33.2640	1.23882
$722$	0	0
$723$	41.3631	1.53831
$724$	0	0
$725$	−22.3198	−0.828937
$726$	0	0
$727$	−39.1083	−1.45045	−0.725223	−	0.688514i	$-0.758263\pi$
$727$	−39.1083	−1.45045	−0.725223	+	0.688514i	$0.758263\pi$
$728$	0	0
$729$	15.1140	0.559779
$730$	0	0
$731$	−5.39942	−0.199705
$732$	0	0
$733$	−45.5880	−1.68383	−0.841915	−	0.539610i	$-0.818572\pi$
$733$	−45.5880	−1.68383	−0.841915	+	0.539610i	$0.818572\pi$
$734$	0	0
$735$	51.6462	1.90500
$736$	0	0
$737$	24.1126	0.888197
$738$	0	0
$739$	−12.7034	−0.467302	−0.233651	−	0.972321i	$-0.575067\pi$
$739$	−12.7034	−0.467302	−0.233651	+	0.972321i	$0.575067\pi$
$740$	0	0
$741$	9.63857	0.354082
$742$	0	0
$743$	28.2085	1.03487	0.517434	−	0.855723i	$-0.326887\pi$
$743$	28.2085	1.03487	0.517434	+	0.855723i	$0.326887\pi$
$744$	0	0
$745$	5.19887	0.190472
$746$	0	0
$747$	0.00919040	0.000336259 0
$748$	0	0
$749$	−3.42911	−0.125297
$750$	0	0
$751$	−5.41814	−0.197711	−0.0988553	−	0.995102i	$-0.531518\pi$
$751$	−5.41814	−0.197711	−0.0988553	+	0.995102i	$0.531518\pi$
$752$	0	0
$753$	−20.2007	−0.736155
$754$	0	0
$755$	2.80134	0.101951
$756$	0	0
$757$	−27.2837	−0.991642	−0.495821	−	0.868425i	$-0.665133\pi$
$757$	−27.2837	−0.991642	−0.495821	+	0.868425i	$0.665133\pi$
$758$	0	0
$759$	54.1985	1.96728
$760$	0	0
$761$	26.0819	0.945470	0.472735	−	0.881205i	$-0.343267\pi$
$761$	26.0819	0.945470	0.472735	+	0.881205i	$0.343267\pi$
$762$	0	0
$763$	20.3720	0.737516
$764$	0	0
$765$	−5.96602	−0.215702
$766$	0	0
$767$	−58.6716	−2.11851
$768$	0	0
$769$	18.4272	0.664501	0.332250	−	0.943191i	$-0.392192\pi$
$769$	18.4272	0.664501	0.332250	+	0.943191i	$0.392192\pi$
$770$	0	0
$771$	44.8353	1.61470
$772$	0	0
$773$	1.48208	0.0533068	0.0266534	−	0.999645i	$-0.491515\pi$
$773$	1.48208	0.0533068	0.0266534	+	0.999645i	$0.491515\pi$
$774$	0	0
$775$	−11.4387	−0.410892
$776$	0	0
$777$	16.5548	0.593900
$778$	0	0
$779$	4.72814	0.169403
$780$	0	0
$781$	4.01191	0.143557
$782$	0	0
$783$	28.0814	1.00355
$784$	0	0
$785$	25.3200	0.903709
$786$	0	0
$787$	−21.8288	−0.778114	−0.389057	−	0.921214i	$-0.627199\pi$
$787$	−21.8288	−0.778114	−0.389057	+	0.921214i	$0.627199\pi$
$788$	0	0
$789$	63.8538	2.27325
$790$	0	0
$791$	−64.5452	−2.29496
$792$	0	0
$793$	51.8484	1.84119
$794$	0	0
$795$	2.55386	0.0905759
$796$	0	0
$797$	12.8962	0.456807	0.228404	−	0.973567i	$-0.426649\pi$
$797$	12.8962	0.456807	0.228404	+	0.973567i	$0.426649\pi$
$798$	0	0
$799$	10.2245	0.361717
$800$	0	0
$801$	−7.41361	−0.261947
$802$	0	0
$803$	56.6030	1.99748
$804$	0	0
$805$	31.9817	1.12721
$806$	0	0
$807$	11.1177	0.391362
$808$	0	0
$809$	−52.3210	−1.83951	−0.919754	−	0.392495i	$-0.871612\pi$
$809$	−52.3210	−1.83951	−0.919754	+	0.392495i	$0.871612\pi$
$810$	0	0
$811$	49.3916	1.73437	0.867186	−	0.497984i	$-0.165926\pi$
$811$	49.3916	1.73437	0.867186	+	0.497984i	$0.165926\pi$
$812$	0	0
$813$	−10.7173	−0.375874
$814$	0	0
$815$	0.788359	0.0276150
$816$	0	0
$817$	1.03265	0.0361278
$818$	0	0
$819$	−22.4684	−0.785109
$820$	0	0
$821$	34.4543	1.20246	0.601231	−	0.799075i	$-0.294677\pi$
$821$	34.4543	1.20246	0.601231	+	0.799075i	$0.294677\pi$
$822$	0	0
$823$	−51.3415	−1.78965	−0.894827	−	0.446413i	$-0.852701\pi$
$823$	−51.3415	−1.78965	−0.894827	+	0.446413i	$0.852701\pi$
$824$	0	0
$825$	−38.0490	−1.32470
$826$	0	0
$827$	−38.2623	−1.33051	−0.665255	−	0.746617i	$-0.731677\pi$
$827$	−38.2623	−1.33051	−0.665255	+	0.746617i	$0.731677\pi$
$828$	0	0
$829$	26.0998	0.906484	0.453242	−	0.891387i	$-0.350267\pi$
$829$	26.0998	0.906484	0.453242	+	0.891387i	$0.350267\pi$
$830$	0	0
$831$	−25.0777	−0.869935
$832$	0	0
$833$	105.739	3.66365
$834$	0	0
$835$	−24.9474	−0.863341
$836$	0	0
$837$	14.3915	0.497444
$838$	0	0
$839$	51.3229	1.77186	0.885932	−	0.463815i	$-0.153520\pi$
$839$	51.3229	1.77186	0.885932	+	0.463815i	$0.153520\pi$
$840$	0	0
$841$	16.2223	0.559390
$842$	0	0
$843$	−34.7201	−1.19582
$844$	0	0
$845$	−14.1884	−0.488094
$846$	0	0
$847$	−119.327	−4.10013
$848$	0	0
$849$	3.67604	0.126161
$850$	0	0
$851$	7.61546	0.261055
$852$	0	0
$853$	48.1042	1.64706	0.823528	−	0.567276i	$-0.192003\pi$
$853$	48.1042	1.64706	0.823528	+	0.567276i	$0.192003\pi$
$854$	0	0
$855$	1.14101	0.0390217
$856$	0	0
$857$	14.9577	0.510944	0.255472	−	0.966816i	$-0.417769\pi$
$857$	14.9577	0.510944	0.255472	+	0.966816i	$0.417769\pi$
$858$	0	0
$859$	38.1349	1.30114	0.650572	−	0.759444i	$-0.274529\pi$
$859$	38.1349	1.30114	0.650572	+	0.759444i	$0.274529\pi$
$860$	0	0
$861$	−48.5932	−1.65605
$862$	0	0
$863$	17.8644	0.608110	0.304055	−	0.952655i	$-0.401659\pi$
$863$	17.8644	0.608110	0.304055	+	0.952655i	$0.401659\pi$
$864$	0	0
$865$	20.8936	0.710405
$866$	0	0
$867$	−20.3666	−0.691685
$868$	0	0
$869$	−13.9535	−0.473341
$870$	0	0
$871$	−20.2734	−0.686938
$872$	0	0
$873$	−10.8733	−0.368006
$874$	0	0
$875$	−56.2753	−1.90245
$876$	0	0
$877$	4.31446	0.145689	0.0728445	−	0.997343i	$-0.476792\pi$
$877$	4.31446	0.145689	0.0728445	+	0.997343i	$0.476792\pi$
$878$	0	0
$879$	28.3374	0.955797
$880$	0	0
$881$	31.6397	1.06597	0.532985	−	0.846125i	$-0.321070\pi$
$881$	31.6397	1.06597	0.532985	+	0.846125i	$0.321070\pi$
$882$	0	0
$883$	−36.3038	−1.22172	−0.610861	−	0.791738i	$-0.709177\pi$
$883$	−36.3038	−1.22172	−0.610861	+	0.791738i	$0.709177\pi$
$884$	0	0
$885$	−30.6218	−1.02934
$886$	0	0
$887$	42.6516	1.43210	0.716050	−	0.698049i	$-0.245948\pi$
$887$	42.6516	1.43210	0.716050	+	0.698049i	$0.245948\pi$
$888$	0	0
$889$	−70.0357	−2.34892
$890$	0	0
$891$	63.2363	2.11850
$892$	0	0
$893$	−1.95545	−0.0654367
$894$	0	0
$895$	3.53695	0.118227
$896$	0	0
$897$	−45.5691	−1.52151
$898$	0	0
$899$	23.1761	0.772966
$900$	0	0
$901$	5.22872	0.174194
$902$	0	0
$903$	−10.6130	−0.353178
$904$	0	0
$905$	−17.5545	−0.583532
$906$	0	0
$907$	4.74377	0.157514	0.0787571	−	0.996894i	$-0.474905\pi$
$907$	4.74377	0.157514	0.0787571	+	0.996894i	$0.474905\pi$
$908$	0	0
$909$	−11.7797	−0.390707
$910$	0	0
$911$	−27.2390	−0.902467	−0.451234	−	0.892406i	$-0.649016\pi$
$911$	−27.2390	−0.902467	−0.451234	+	0.892406i	$0.649016\pi$
$912$	0	0
$913$	−0.0607760	−0.00201139
$914$	0	0
$915$	27.0607	0.894598
$916$	0	0
$917$	−72.6586	−2.39940
$918$	0	0
$919$	−26.2709	−0.866597	−0.433299	−	0.901250i	$-0.642650\pi$
$919$	−26.2709	−0.866597	−0.433299	+	0.901250i	$0.642650\pi$
$920$	0	0
$921$	57.3203	1.88877
$922$	0	0
$923$	−3.37314	−0.111028
$924$	0	0
$925$	−5.34629	−0.175785
$926$	0	0
$927$	−5.61075	−0.184281
$928$	0	0
$929$	49.4545	1.62255	0.811275	−	0.584665i	$-0.198774\pi$
$929$	49.4545	1.62255	0.811275	+	0.584665i	$0.198774\pi$
$930$	0	0
$931$	−20.2228	−0.662776
$932$	0	0
$933$	−55.2652	−1.80930
$934$	0	0
$935$	39.4532	1.29026
$936$	0	0
$937$	−18.2135	−0.595008	−0.297504	−	0.954721i	$-0.596154\pi$
$937$	−18.2135	−0.595008	−0.297504	+	0.954721i	$0.596154\pi$
$938$	0	0
$939$	19.0745	0.622472
$940$	0	0
$941$	33.0321	1.07682	0.538408	−	0.842684i	$-0.319026\pi$
$941$	33.0321	1.07682	0.538408	+	0.842684i	$0.319026\pi$
$942$	0	0
$943$	−22.3536	−0.727935
$944$	0	0
$945$	28.2479	0.918905
$946$	0	0
$947$	−44.3381	−1.44080	−0.720398	−	0.693561i	$-0.756041\pi$
$947$	−44.3381	−1.44080	−0.720398	+	0.693561i	$0.756041\pi$
$948$	0	0
$949$	−47.5908	−1.54486
$950$	0	0
$951$	27.9626	0.906750
$952$	0	0
$953$	17.3032	0.560506	0.280253	−	0.959926i	$-0.409582\pi$
$953$	17.3032	0.560506	0.280253	+	0.959926i	$0.409582\pi$
$954$	0	0
$955$	−21.6723	−0.701299
$956$	0	0
$957$	77.0913	2.49201
$958$	0	0
$959$	−31.6526	−1.02212
$960$	0	0
$961$	−19.1224	−0.616852
$962$	0	0
$963$	0.578398	0.0186386
$964$	0	0
$965$	−28.5206	−0.918109
$966$	0	0
$967$	−37.7864	−1.21513	−0.607565	−	0.794270i	$-0.707854\pi$
$967$	−37.7864	−1.21513	−0.607565	+	0.794270i	$0.707854\pi$
$968$	0	0
$969$	10.2995	0.330866
$970$	0	0
$971$	4.19841	0.134733	0.0673666	−	0.997728i	$-0.478540\pi$
$971$	4.19841	0.134733	0.0673666	+	0.997728i	$0.478540\pi$
$972$	0	0
$973$	−7.33956	−0.235295
$974$	0	0
$975$	31.9909	1.02453
$976$	0	0
$977$	−5.66411	−0.181211	−0.0906054	−	0.995887i	$-0.528880\pi$
$977$	−5.66411	−0.181211	−0.0906054	+	0.995887i	$0.528880\pi$
$978$	0	0
$979$	49.0261	1.56688
$980$	0	0
$981$	−3.43620	−0.109710
$982$	0	0
$983$	6.09691	0.194461	0.0972306	−	0.995262i	$-0.469002\pi$
$983$	6.09691	0.194461	0.0972306	+	0.995262i	$0.469002\pi$
$984$	0	0
$985$	−22.5168	−0.717446
$986$	0	0
$987$	20.0971	0.639696
$988$	0	0
$989$	−4.88214	−0.155243
$990$	0	0
$991$	36.9040	1.17229	0.586147	−	0.810205i	$-0.300644\pi$
$991$	36.9040	1.17229	0.586147	+	0.810205i	$0.300644\pi$
$992$	0	0
$993$	−28.3298	−0.899020
$994$	0	0
$995$	−12.3886	−0.392744
$996$	0	0
$997$	11.7984	0.373660	0.186830	−	0.982392i	$-0.440179\pi$
$997$	11.7984	0.373660	0.186830	+	0.982392i	$0.440179\pi$
$998$	0	0
$999$	6.72638	0.212813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.6	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-1.96979 q^{3} -1.29651 q^{5} -5.21755 q^{7} +0.880060 q^{9} +O(q^{10})\)	\(q-1.96979 q^{3} -1.29651 q^{5} -5.21755 q^{7} +0.880060 q^{9} -5.81982 q^{11} +4.89321 q^{13} +2.55386 q^{15} +5.22872 q^{17} -1.00000 q^{19} +10.2775 q^{21} +4.72779 q^{23} -3.31905 q^{25} +4.17583 q^{27} +6.72475 q^{29} +3.44639 q^{31} +11.4638 q^{33} +6.76463 q^{35} +1.61079 q^{37} -9.63857 q^{39} -4.72814 q^{41} -1.03265 q^{43} -1.14101 q^{45} +1.95545 q^{47} +20.2228 q^{49} -10.2995 q^{51} +1.00000 q^{53} +7.54548 q^{55} +1.96979 q^{57} -11.9904 q^{59} +10.5960 q^{61} -4.59175 q^{63} -6.34411 q^{65} -4.14317 q^{67} -9.31274 q^{69} -0.689352 q^{71} -9.72589 q^{73} +6.53782 q^{75} +30.3652 q^{77} +2.39759 q^{79} -10.8657 q^{81} +0.0104429 q^{83} -6.77911 q^{85} -13.2463 q^{87} -8.42399 q^{89} -25.5305 q^{91} -6.78865 q^{93} +1.29651 q^{95} -12.3552 q^{97} -5.12179 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 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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