LMFDB - Embedded newform 8028.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, 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("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1710888.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 3x^{2} + 12x + 4 $ x^5 - x^4 - 10x^3 + 3x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2676)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.466688$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.28552 q^{5} +3.47913 q^{7} +O(q^{10})$	$q+4.28552 q^{5} +3.47913 q^{7} +6.16942 q^{17} +3.78220 q^{19} +4.95826 q^{23} +13.3657 q^{25} -9.03773 q^{29} -6.10435 q^{31} +14.9099 q^{35} +5.63611 q^{37} +0.520869 q^{41} +5.79039 q^{43} +3.18272 q^{47} +5.10435 q^{49} -9.18767 q^{53} -0.704290 q^{59} -4.05999 q^{61} -5.81953 q^{67} +4.54615 q^{71} -14.9656 q^{73} -15.3239 q^{79} -3.00618 q^{83} +26.4392 q^{85} +7.41914 q^{89} +16.2087 q^{95} -1.60967 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q + 6 * q^5 + 3 * q^7	$ 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) $ 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.28552	1.91654	0.958272	−	0.285859i	$-0.0922791\pi$
$5$	4.28552	1.91654	0.958272	+	0.285859i	$0.0922791\pi$
$6$	0	0
$7$	3.47913	1.31499	0.657494	−	0.753460i	$-0.271617\pi$
$7$	3.47913	1.31499	0.657494	+	0.753460i	$0.271617\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.16942	1.49630	0.748152	−	0.663527i	$-0.230941\pi$
$17$	6.16942	1.49630	0.748152	+	0.663527i	$0.230941\pi$
$18$	0	0
$19$	3.78220	0.867697	0.433848	−	0.900986i	$-0.357155\pi$
$19$	3.78220	0.867697	0.433848	+	0.900986i	$0.357155\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.95826	1.03387	0.516935	−	0.856025i	$-0.327073\pi$
$23$	4.95826	1.03387	0.516935	+	0.856025i	$0.327073\pi$
$24$	0	0
$25$	13.3657	2.67314
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.03773	−1.67826	−0.839132	−	0.543928i	$-0.816937\pi$
$29$	−9.03773	−1.67826	−0.839132	+	0.543928i	$0.816937\pi$
$30$	0	0
$31$	−6.10435	−1.09637	−0.548187	−	0.836356i	$-0.684682\pi$
$31$	−6.10435	−1.09637	−0.548187	+	0.836356i	$0.684682\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.9099	2.52023
$36$	0	0
$37$	5.63611	0.926571	0.463285	−	0.886209i	$-0.346670\pi$
$37$	5.63611	0.926571	0.463285	+	0.886209i	$0.346670\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.520869	0.0813460	0.0406730	−	0.999173i	$-0.487050\pi$
$41$	0.520869	0.0813460	0.0406730	+	0.999173i	$0.487050\pi$
$42$	0	0
$43$	5.79039	0.883027	0.441513	−	0.897255i	$-0.354442\pi$
$43$	5.79039	0.883027	0.441513	+	0.897255i	$0.354442\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.18272	0.464248	0.232124	−	0.972686i	$-0.425433\pi$
$47$	3.18272	0.464248	0.232124	+	0.972686i	$0.425433\pi$
$48$	0	0
$49$	5.10435	0.729193
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.18767	−1.26202	−0.631012	−	0.775773i	$-0.717360\pi$
$53$	−9.18767	−1.26202	−0.631012	+	0.775773i	$0.717360\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.704290	−0.0916908	−0.0458454	−	0.998949i	$-0.514598\pi$
$59$	−0.704290	−0.0916908	−0.0458454	+	0.998949i	$0.514598\pi$
$60$	0	0
$61$	−4.05999	−0.519828	−0.259914	−	0.965632i	$-0.583694\pi$
$61$	−4.05999	−0.519828	−0.259914	+	0.965632i	$0.583694\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.81953	−0.710969	−0.355485	−	0.934682i	$-0.615684\pi$
$67$	−5.81953	−0.710969	−0.355485	+	0.934682i	$0.615684\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.54615	0.539529	0.269765	−	0.962926i	$-0.413054\pi$
$71$	4.54615	0.539529	0.269765	+	0.962926i	$0.413054\pi$
$72$	0	0
$73$	−14.9656	−1.75159	−0.875797	−	0.482680i	$-0.839663\pi$
$73$	−14.9656	−1.75159	−0.875797	+	0.482680i	$0.839663\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.3239	−1.72408	−0.862040	−	0.506841i	$-0.830813\pi$
$79$	−15.3239	−1.72408	−0.862040	+	0.506841i	$0.830813\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.00618	−0.329971	−0.164986	−	0.986296i	$-0.552758\pi$
$83$	−3.00618	−0.329971	−0.164986	+	0.986296i	$0.552758\pi$
$84$	0	0
$85$	26.4392	2.86773
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.41914	0.786428	0.393214	−	0.919447i	$-0.371363\pi$
$89$	7.41914	0.786428	0.393214	+	0.919447i	$0.371363\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.2087	1.66298
$96$	0	0
$97$	−1.60967	−0.163437	−0.0817186	−	0.996655i	$-0.526041\pi$
$97$	−1.60967	−0.163437	−0.0817186	+	0.996655i	$0.526041\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.22516	0.718930	0.359465	−	0.933159i	$-0.382959\pi$
$101$	7.22516	0.718930	0.359465	+	0.933159i	$0.382959\pi$
$102$	0	0
$103$	14.4257	1.42140	0.710702	−	0.703493i	$-0.248377\pi$
$103$	14.4257	1.42140	0.710702	+	0.703493i	$0.248377\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.08488	−0.201552	−0.100776	−	0.994909i	$-0.532133\pi$
$107$	−2.08488	−0.201552	−0.100776	+	0.994909i	$0.532133\pi$
$108$	0	0
$109$	−13.2114	−1.26542	−0.632712	−	0.774388i	$-0.718058\pi$
$109$	−13.2114	−1.26542	−0.632712	+	0.774388i	$0.718058\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.75486	−0.823588	−0.411794	−	0.911277i	$-0.635098\pi$
$113$	−8.75486	−0.823588	−0.411794	+	0.911277i	$0.635098\pi$
$114$	0	0
$115$	21.2487	1.98145
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.4642	1.96762
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	35.8513	3.20664
$126$	0	0
$127$	−6.36652	−0.564937	−0.282468	−	0.959277i	$-0.591153\pi$
$127$	−6.36652	−0.564937	−0.282468	+	0.959277i	$0.591153\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.2785	1.33489	0.667444	−	0.744660i	$-0.267388\pi$
$131$	15.2785	1.33489	0.667444	+	0.744660i	$0.267388\pi$
$132$	0	0
$133$	13.1588	1.14101
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.43702	−0.464516	−0.232258	−	0.972654i	$-0.574611\pi$
$137$	−5.43702	−0.464516	−0.232258	+	0.972654i	$0.574611\pi$
$138$	0	0
$139$	−0.834829	−0.0708093	−0.0354046	−	0.999373i	$-0.511272\pi$
$139$	−0.834829	−0.0708093	−0.0354046	+	0.999373i	$0.511272\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−38.7314	−3.21647
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.2656	1.49638	0.748188	−	0.663487i	$-0.230924\pi$
$149$	18.2656	1.49638	0.748188	+	0.663487i	$0.230924\pi$
$150$	0	0
$151$	−12.1935	−0.992291	−0.496146	−	0.868239i	$-0.665252\pi$
$151$	−12.1935	−0.992291	−0.496146	+	0.868239i	$0.665252\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−26.1603	−2.10125
$156$	0	0
$157$	−14.5608	−1.16208	−0.581040	−	0.813875i	$-0.697354\pi$
$157$	−14.5608	−1.16208	−0.581040	+	0.813875i	$0.697354\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	17.2504	1.35953
$162$	0	0
$163$	−16.6796	−1.30645	−0.653225	−	0.757163i	$-0.726585\pi$
$163$	−16.6796	−1.30645	−0.653225	+	0.757163i	$0.726585\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.9450	1.15648	0.578239	−	0.815867i	$-0.303740\pi$
$167$	14.9450	1.15648	0.578239	+	0.815867i	$0.303740\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.5924	−0.957381	−0.478690	−	0.877984i	$-0.658888\pi$
$173$	−12.5924	−0.957381	−0.478690	+	0.877984i	$0.658888\pi$
$174$	0	0
$175$	46.5010	3.51514
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.0720	−0.827558	−0.413779	−	0.910377i	$-0.635791\pi$
$179$	−11.0720	−0.827558	−0.413779	+	0.910377i	$0.635791\pi$
$180$	0	0
$181$	−6.43354	−0.478201	−0.239101	−	0.970995i	$-0.576853\pi$
$181$	−6.43354	−0.478201	−0.239101	+	0.970995i	$0.576853\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	24.1537	1.77581
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.61631	0.189309	0.0946546	−	0.995510i	$-0.469825\pi$
$191$	2.61631	0.189309	0.0946546	+	0.995510i	$0.469825\pi$
$192$	0	0
$193$	−20.4268	−1.47036	−0.735178	−	0.677874i	$-0.762901\pi$
$193$	−20.4268	−1.47036	−0.735178	+	0.677874i	$0.762901\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.1102	0.862813	0.431407	−	0.902158i	$-0.358017\pi$
$197$	12.1102	0.862813	0.431407	+	0.902158i	$0.358017\pi$
$198$	0	0
$199$	13.6305	0.966244	0.483122	−	0.875553i	$-0.339503\pi$
$199$	13.6305	0.966244	0.483122	+	0.875553i	$0.339503\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−31.4434	−2.20690
$204$	0	0
$205$	2.23219	0.155903
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	25.3848	1.74756	0.873780	−	0.486322i	$-0.161662\pi$
$211$	25.3848	1.74756	0.873780	+	0.486322i	$0.161662\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.8148	1.69236
$216$	0	0
$217$	−21.2378	−1.44172
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.70432	−0.511353	−0.255677	−	0.966762i	$-0.582298\pi$
$227$	−7.70432	−0.511353	−0.255677	+	0.966762i	$0.582298\pi$
$228$	0	0
$229$	−8.01236	−0.529472	−0.264736	−	0.964321i	$-0.585285\pi$
$229$	−8.01236	−0.529472	−0.264736	+	0.964321i	$0.585285\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.94852	−0.127652	−0.0638258	−	0.997961i	$-0.520330\pi$
$233$	−1.94852	−0.127652	−0.0638258	+	0.997961i	$0.520330\pi$
$234$	0	0
$235$	13.6396	0.889751
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.3228	−0.926463	−0.463231	−	0.886237i	$-0.653310\pi$
$239$	−14.3228	−0.926463	−0.463231	+	0.886237i	$0.653310\pi$
$240$	0	0
$241$	22.3878	1.44212	0.721062	−	0.692870i	$-0.243654\pi$
$241$	22.3878	1.44212	0.721062	+	0.692870i	$0.243654\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	21.8748	1.39753
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.4899	0.977713	0.488857	−	0.872364i	$-0.337414\pi$
$251$	15.4899	0.977713	0.488857	+	0.872364i	$0.337414\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.7025	−1.66565	−0.832827	−	0.553533i	$-0.813279\pi$
$257$	−26.7025	−1.66565	−0.832827	+	0.553533i	$0.813279\pi$
$258$	0	0
$259$	19.6088	1.21843
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.218922	0.0134993	0.00674965	−	0.999977i	$-0.497852\pi$
$263$	0.218922	0.0134993	0.00674965	+	0.999977i	$0.497852\pi$
$264$	0	0
$265$	−39.3740	−2.41872
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.9533	0.667835	0.333918	−	0.942602i	$-0.391629\pi$
$269$	10.9533	0.667835	0.333918	+	0.942602i	$0.391629\pi$
$270$	0	0
$271$	−10.3357	−0.627851	−0.313926	−	0.949448i	$-0.601644\pi$
$271$	−10.3357	−0.627851	−0.313926	+	0.949448i	$0.601644\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.2305	1.21553	0.607766	−	0.794116i	$-0.292066\pi$
$277$	20.2305	1.21553	0.607766	+	0.794116i	$0.292066\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.7505	−0.939598	−0.469799	−	0.882774i	$-0.655674\pi$
$281$	−15.7505	−0.939598	−0.469799	+	0.882774i	$0.655674\pi$
$282$	0	0
$283$	−17.2522	−1.02554	−0.512770	−	0.858526i	$-0.671381\pi$
$283$	−17.2522	−1.02554	−0.512770	+	0.858526i	$0.671381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.81217	0.106969
$288$	0	0
$289$	21.0618	1.23893
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.10387	0.531854	0.265927	−	0.963993i	$-0.414322\pi$
$293$	9.10387	0.531854	0.265927	+	0.963993i	$0.414322\pi$
$294$	0	0
$295$	−3.01825	−0.175729
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	20.1455	1.16117
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−17.3992	−0.996273
$306$	0	0
$307$	−7.00999	−0.400081	−0.200041	−	0.979788i	$-0.564107\pi$
$307$	−7.00999	−0.400081	−0.200041	+	0.979788i	$0.564107\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.5404	−1.27815	−0.639076	−	0.769144i	$-0.720683\pi$
$311$	−22.5404	−1.27815	−0.639076	+	0.769144i	$0.720683\pi$
$312$	0	0
$313$	−11.1905	−0.632522	−0.316261	−	0.948672i	$-0.602428\pi$
$313$	−11.1905	−0.632522	−0.316261	+	0.948672i	$0.602428\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.1167	−0.792874	−0.396437	−	0.918062i	$-0.629753\pi$
$317$	−14.1167	−0.792874	−0.396437	+	0.918062i	$0.629753\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.3340	1.29834
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	11.0731	0.610480
$330$	0	0
$331$	16.6497	0.915150	0.457575	−	0.889171i	$-0.348718\pi$
$331$	16.6497	0.915150	0.457575	+	0.889171i	$0.348718\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.9397	−1.36260
$336$	0	0
$337$	−1.28860	−0.0701947	−0.0350974	−	0.999384i	$-0.511174\pi$
$337$	−1.28860	−0.0701947	−0.0350974	+	0.999384i	$0.511174\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.59520	−0.356107
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.4049	1.68590	0.842951	−	0.537990i	$-0.180816\pi$
$347$	31.4049	1.68590	0.842951	+	0.537990i	$0.180816\pi$
$348$	0	0
$349$	−5.92682	−0.317256	−0.158628	−	0.987338i	$-0.550707\pi$
$349$	−5.92682	−0.317256	−0.158628	+	0.987338i	$0.550707\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.82544	−0.203608	−0.101804	−	0.994804i	$-0.532461\pi$
$353$	−3.82544	−0.203608	−0.101804	+	0.994804i	$0.532461\pi$
$354$	0	0
$355$	19.4826	1.03403
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.96602	−0.209319	−0.104659	−	0.994508i	$-0.533375\pi$
$359$	−3.96602	−0.209319	−0.104659	+	0.994508i	$0.533375\pi$
$360$	0	0
$361$	−4.69494	−0.247102
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−64.1355	−3.35701
$366$	0	0
$367$	−22.8054	−1.19043	−0.595217	−	0.803565i	$-0.702934\pi$
$367$	−22.8054	−1.19043	−0.595217	+	0.803565i	$0.702934\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−31.9651	−1.65955
$372$	0	0
$373$	−16.2905	−0.843489	−0.421744	−	0.906715i	$-0.638582\pi$
$373$	−16.2905	−0.843489	−0.421744	+	0.906715i	$0.638582\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.50174	0.128506	0.0642528	−	0.997934i	$-0.479534\pi$
$379$	2.50174	0.128506	0.0642528	+	0.997934i	$0.479534\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−35.4543	−1.81163	−0.905816	−	0.423672i	$-0.860741\pi$
$383$	−35.4543	−1.81163	−0.905816	+	0.423672i	$0.860741\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.42412	−0.427120	−0.213560	−	0.976930i	$-0.568506\pi$
$389$	−8.42412	−0.427120	−0.213560	+	0.976930i	$0.568506\pi$
$390$	0	0
$391$	30.5896	1.54698
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−65.6711	−3.30427
$396$	0	0
$397$	13.2122	0.663103	0.331552	−	0.943437i	$-0.392428\pi$
$397$	13.2122	0.663103	0.331552	+	0.943437i	$0.392428\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.9975	−0.748942	−0.374471	−	0.927239i	$-0.622176\pi$
$401$	−14.9975	−0.748942	−0.374471	+	0.927239i	$0.622176\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−31.9912	−1.58186	−0.790932	−	0.611904i	$-0.790404\pi$
$409$	−31.9912	−1.58186	−0.790932	+	0.611904i	$0.790404\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.45032	−0.120572
$414$	0	0
$415$	−12.8830	−0.632404
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.79162	0.136380	0.0681898	−	0.997672i	$-0.478278\pi$
$419$	2.79162	0.136380	0.0681898	+	0.997672i	$0.478278\pi$
$420$	0	0
$421$	14.1346	0.688881	0.344440	−	0.938808i	$-0.388069\pi$
$421$	14.1346	0.688881	0.344440	+	0.938808i	$0.388069\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	82.4586	3.99983
$426$	0	0
$427$	−14.1252	−0.683568
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.8917	1.77701	0.888506	−	0.458866i	$-0.151744\pi$
$431$	36.8917	1.77701	0.888506	+	0.458866i	$0.151744\pi$
$432$	0	0
$433$	4.87420	0.234239	0.117119	−	0.993118i	$-0.462634\pi$
$433$	4.87420	0.234239	0.117119	+	0.993118i	$0.462634\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.7532	0.897085
$438$	0	0
$439$	−19.2787	−0.920121	−0.460061	−	0.887887i	$-0.652172\pi$
$439$	−19.2787	−0.920121	−0.460061	+	0.887887i	$0.652172\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.4388	0.543475	0.271737	−	0.962371i	$-0.412402\pi$
$443$	11.4388	0.543475	0.271737	+	0.962371i	$0.412402\pi$
$444$	0	0
$445$	31.7949	1.50722
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0965	−0.476483	−0.238241	−	0.971206i	$-0.576571\pi$
$449$	−10.0965	−0.476483	−0.238241	+	0.971206i	$0.576571\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.5367	−0.680000	−0.340000	−	0.940425i	$-0.610427\pi$
$457$	−14.5367	−0.680000	−0.340000	+	0.940425i	$0.610427\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	37.9170	1.76597	0.882986	−	0.469399i	$-0.155530\pi$
$461$	37.9170	1.76597	0.882986	+	0.469399i	$0.155530\pi$
$462$	0	0
$463$	−27.9133	−1.29724	−0.648621	−	0.761112i	$-0.724654\pi$
$463$	−27.9133	−1.29724	−0.648621	+	0.761112i	$0.724654\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.0254	−1.48196	−0.740979	−	0.671528i	$-0.765638\pi$
$467$	−32.0254	−1.48196	−0.740979	+	0.671528i	$0.765638\pi$
$468$	0	0
$469$	−20.2469	−0.934916
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	50.5517	2.31947
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.2217	0.512730	0.256365	−	0.966580i	$-0.417475\pi$
$479$	11.2217	0.512730	0.256365	+	0.966580i	$0.417475\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.89827	−0.313234
$486$	0	0
$487$	33.6153	1.52325	0.761627	−	0.648015i	$-0.224401\pi$
$487$	33.6153	1.52325	0.761627	+	0.648015i	$0.224401\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.1501	0.819103	0.409552	−	0.912287i	$-0.365685\pi$
$491$	18.1501	0.819103	0.409552	+	0.912287i	$0.365685\pi$
$492$	0	0
$493$	−55.7576	−2.51120
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.8167	0.709475
$498$	0	0
$499$	17.0817	0.764682	0.382341	−	0.924021i	$-0.375118\pi$
$499$	17.0817	0.764682	0.382341	+	0.924021i	$0.375118\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.6994	1.36882	0.684409	−	0.729098i	$-0.260060\pi$
$503$	30.6994	1.36882	0.684409	+	0.729098i	$0.260060\pi$
$504$	0	0
$505$	30.9636	1.37786
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.3447	0.813116	0.406558	−	0.913625i	$-0.366729\pi$
$509$	18.3447	0.813116	0.406558	+	0.913625i	$0.366729\pi$
$510$	0	0
$511$	−52.0674	−2.30332
$512$	0	0
$513$	0	0
$514$	0	0
$515$	61.8215	2.72418
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.8201	1.65693	0.828465	−	0.560041i	$-0.189215\pi$
$521$	37.8201	1.65693	0.828465	+	0.560041i	$0.189215\pi$
$522$	0	0
$523$	38.5732	1.68669	0.843345	−	0.537373i	$-0.180583\pi$
$523$	38.5732	1.68669	0.843345	+	0.537373i	$0.180583\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.6603	−1.64051
$528$	0	0
$529$	1.58436	0.0688854
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.93478	−0.386284
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−28.4539	−1.22333	−0.611665	−	0.791117i	$-0.709500\pi$
$541$	−28.4539	−1.22333	−0.611665	+	0.791117i	$0.709500\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−56.6178	−2.42524
$546$	0	0
$547$	24.3069	1.03929	0.519643	−	0.854383i	$-0.326065\pi$
$547$	24.3069	1.03929	0.519643	+	0.854383i	$0.326065\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.1825	−1.45622
$552$	0	0
$553$	−53.3140	−2.26714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.3107	0.902965	0.451482	−	0.892280i	$-0.350895\pi$
$557$	21.3107	0.902965	0.451482	+	0.892280i	$0.350895\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.3996	0.859740	0.429870	−	0.902891i	$-0.358559\pi$
$563$	20.3996	0.859740	0.429870	+	0.902891i	$0.358559\pi$
$564$	0	0
$565$	−37.5191	−1.57844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	43.4245	1.82045	0.910225	−	0.414115i	$-0.135909\pi$
$569$	43.4245	1.82045	0.910225	+	0.414115i	$0.135909\pi$
$570$	0	0
$571$	27.8795	1.16672	0.583361	−	0.812213i	$-0.301737\pi$
$571$	27.8795	1.16672	0.583361	+	0.812213i	$0.301737\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	66.2706	2.76367
$576$	0	0
$577$	29.2425	1.21738	0.608691	−	0.793408i	$-0.291695\pi$
$577$	29.2425	1.21738	0.608691	+	0.793408i	$0.291695\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.4589	−0.433908
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.1755	−0.874005	−0.437003	−	0.899460i	$-0.643960\pi$
$587$	−21.1755	−0.874005	−0.437003	+	0.899460i	$0.643960\pi$
$588$	0	0
$589$	−23.0879	−0.951320
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−31.0396	−1.27464	−0.637321	−	0.770598i	$-0.719958\pi$
$593$	−31.0396	−1.27464	−0.637321	+	0.770598i	$0.719958\pi$
$594$	0	0
$595$	91.9854	3.77103
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.9783	1.06145	0.530723	−	0.847546i	$-0.321921\pi$
$599$	25.9783	1.06145	0.530723	+	0.847546i	$0.321921\pi$
$600$	0	0
$601$	−0.713707	−0.0291127	−0.0145564	−	0.999894i	$-0.504634\pi$
$601$	−0.713707	−0.0291127	−0.0145564	+	0.999894i	$0.504634\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−47.1407	−1.91654
$606$	0	0
$607$	20.4709	0.830890	0.415445	−	0.909618i	$-0.363626\pi$
$607$	20.4709	0.830890	0.415445	+	0.909618i	$0.363626\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	42.4140	1.71309	0.856544	−	0.516075i	$-0.172607\pi$
$613$	42.4140	1.71309	0.856544	+	0.516075i	$0.172607\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.7492	0.795074	0.397537	−	0.917586i	$-0.369865\pi$
$617$	19.7492	0.795074	0.397537	+	0.917586i	$0.369865\pi$
$618$	0	0
$619$	31.2405	1.25566	0.627830	−	0.778350i	$-0.283943\pi$
$619$	31.2405	1.25566	0.627830	+	0.778350i	$0.283943\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	25.8122	1.03414
$624$	0	0
$625$	86.8132	3.47253
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.7716	1.38643
$630$	0	0
$631$	38.6033	1.53677	0.768386	−	0.639987i	$-0.221060\pi$
$631$	38.6033	1.53677	0.768386	+	0.639987i	$0.221060\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27.2838	−1.08273
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.4738	0.729673	0.364837	−	0.931072i	$-0.381125\pi$
$641$	18.4738	0.729673	0.364837	+	0.931072i	$0.381125\pi$
$642$	0	0
$643$	19.4588	0.767381	0.383691	−	0.923462i	$-0.374653\pi$
$643$	19.4588	0.767381	0.383691	+	0.923462i	$0.374653\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6247	0.614272	0.307136	−	0.951666i	$-0.400629\pi$
$647$	15.6247	0.614272	0.307136	+	0.951666i	$0.400629\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.2327	−1.41789	−0.708947	−	0.705262i	$-0.750830\pi$
$653$	−36.2327	−1.41789	−0.708947	+	0.705262i	$0.750830\pi$
$654$	0	0
$655$	65.4763	2.55837
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−33.9765	−1.32354	−0.661768	−	0.749709i	$-0.730193\pi$
$659$	−33.9765	−1.32354	−0.661768	+	0.749709i	$0.730193\pi$
$660$	0	0
$661$	27.3540	1.06395	0.531973	−	0.846761i	$-0.321451\pi$
$661$	27.3540	1.06395	0.531973	+	0.846761i	$0.321451\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	56.3922	2.18680
$666$	0	0
$667$	−44.8114	−1.73511
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−18.6756	−0.719893	−0.359946	−	0.932973i	$-0.617205\pi$
$673$	−18.6756	−0.719893	−0.359946	+	0.932973i	$0.617205\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.4092	1.62992	0.814959	−	0.579518i	$-0.196759\pi$
$677$	42.4092	1.62992	0.814959	+	0.579518i	$0.196759\pi$
$678$	0	0
$679$	−5.60025	−0.214918
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23.5066	0.899455	0.449727	−	0.893166i	$-0.351521\pi$
$683$	23.5066	0.899455	0.449727	+	0.893166i	$0.351521\pi$
$684$	0	0
$685$	−23.3005	−0.890265
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−32.5308	−1.23753	−0.618765	−	0.785576i	$-0.712367\pi$
$691$	−32.5308	−1.23753	−0.618765	+	0.785576i	$0.712367\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.57768	−0.135709
$696$	0	0
$697$	3.21346	0.121718
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.0087	−1.13341	−0.566706	−	0.823920i	$-0.691783\pi$
$701$	−30.0087	−1.13341	−0.566706	+	0.823920i	$0.691783\pi$
$702$	0	0
$703$	21.3169	0.803983
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.1373	0.945385
$708$	0	0
$709$	−32.1523	−1.20751	−0.603753	−	0.797172i	$-0.706329\pi$
$709$	−32.1523	−1.20751	−0.603753	+	0.797172i	$0.706329\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−30.2670	−1.13351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.2379	0.792040	0.396020	−	0.918242i	$-0.370391\pi$
$719$	21.2379	0.792040	0.396020	+	0.918242i	$0.370391\pi$
$720$	0	0
$721$	50.1888	1.86913
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−120.795	−4.48623
$726$	0	0
$727$	40.8689	1.51574	0.757871	−	0.652404i	$-0.226239\pi$
$727$	40.8689	1.51574	0.757871	+	0.652404i	$0.226239\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	35.7234	1.32128
$732$	0	0
$733$	−23.1497	−0.855053	−0.427527	−	0.904003i	$-0.640615\pi$
$733$	−23.1497	−0.855053	−0.427527	+	0.904003i	$0.640615\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−10.5614	−0.388507	−0.194253	−	0.980951i	$-0.562228\pi$
$739$	−10.5614	−0.388507	−0.194253	+	0.980951i	$0.562228\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.9092	1.83099	0.915495	−	0.402329i	$-0.131799\pi$
$743$	49.9092	1.83099	0.915495	+	0.402329i	$0.131799\pi$
$744$	0	0
$745$	78.2775	2.86787
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.25355	−0.265039
$750$	0	0
$751$	−0.0238137	−0.000868975 0	−0.000434488	−	1.00000i	$-0.500138\pi$
$751$	−0.0238137	−0.000868975 0	−0.000434488		1.00000i	$0.500138\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−52.2554	−1.90177
$756$	0	0
$757$	−47.0366	−1.70957	−0.854787	−	0.518980i	$-0.826312\pi$
$757$	−47.0366	−1.70957	−0.854787	+	0.518980i	$0.826312\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26.5612	−0.962842	−0.481421	−	0.876489i	$-0.659879\pi$
$761$	−26.5612	−0.962842	−0.481421	+	0.876489i	$0.659879\pi$
$762$	0	0
$763$	−45.9642	−1.66402
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	7.31664	0.263845	0.131922	−	0.991260i	$-0.457885\pi$
$769$	7.31664	0.263845	0.131922	+	0.991260i	$0.457885\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.9384	0.501328	0.250664	−	0.968074i	$-0.419351\pi$
$773$	13.9384	0.501328	0.250664	+	0.968074i	$0.419351\pi$
$774$	0	0
$775$	−81.5889	−2.93076
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.97003	0.0705837
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−62.4007	−2.22718
$786$	0	0
$787$	−3.21173	−0.114486	−0.0572429	−	0.998360i	$-0.518231\pi$
$787$	−3.21173	−0.114486	−0.0572429	+	0.998360i	$0.518231\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−30.4593	−1.08301
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−41.0723	−1.45485	−0.727427	−	0.686185i	$-0.759284\pi$
$797$	−41.0723	−1.45485	−0.727427	+	0.686185i	$0.759284\pi$
$798$	0	0
$799$	19.6356	0.694656
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	73.9271	2.60559
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.5991	0.759383	0.379691	−	0.925113i	$-0.376030\pi$
$809$	21.5991	0.759383	0.379691	+	0.925113i	$0.376030\pi$
$810$	0	0
$811$	23.8643	0.837989	0.418994	−	0.907989i	$-0.362383\pi$
$811$	23.8643	0.837989	0.418994	+	0.907989i	$0.362383\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−71.4810	−2.50387
$816$	0	0
$817$	21.9004	0.766199
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.2160	0.356541	0.178270	−	0.983982i	$-0.442950\pi$
$821$	10.2160	0.356541	0.178270	+	0.983982i	$0.442950\pi$
$822$	0	0
$823$	−18.0718	−0.629945	−0.314972	−	0.949101i	$-0.601995\pi$
$823$	−18.0718	−0.629945	−0.314972	+	0.949101i	$0.601995\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.8100	−1.14092	−0.570458	−	0.821327i	$-0.693234\pi$
$827$	−32.8100	−1.14092	−0.570458	+	0.821327i	$0.693234\pi$
$828$	0	0
$829$	12.3069	0.427437	0.213718	−	0.976895i	$-0.431443\pi$
$829$	12.3069	0.427437	0.213718	+	0.976895i	$0.431443\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.4909	1.09110
$834$	0	0
$835$	64.0471	2.21644
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.8001	−1.09786	−0.548931	−	0.835867i	$-0.684965\pi$
$839$	−31.8001	−1.09786	−0.548931	+	0.835867i	$0.684965\pi$
$840$	0	0
$841$	52.6805	1.81657
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−55.7118	−1.91654
$846$	0	0
$847$	−38.2704	−1.31499
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27.9453	0.957953
$852$	0	0
$853$	7.57742	0.259446	0.129723	−	0.991550i	$-0.458591\pi$
$853$	7.57742	0.259446	0.129723	+	0.991550i	$0.458591\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.80251	0.129891	0.0649456	−	0.997889i	$-0.479313\pi$
$857$	3.80251	0.129891	0.0649456	+	0.997889i	$0.479313\pi$
$858$	0	0
$859$	41.0519	1.40067	0.700336	−	0.713813i	$-0.253033\pi$
$859$	41.0519	1.40067	0.700336	+	0.713813i	$0.253033\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.21443	−0.313663	−0.156831	−	0.987625i	$-0.550128\pi$
$863$	−9.21443	−0.313663	−0.156831	+	0.987625i	$0.550128\pi$
$864$	0	0
$865$	−53.9649	−1.83486
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	124.731	4.21669
$876$	0	0
$877$	21.8207	0.736832	0.368416	−	0.929661i	$-0.379900\pi$
$877$	21.8207	0.736832	0.368416	+	0.929661i	$0.379900\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−56.1390	−1.89137	−0.945686	−	0.325083i	$-0.894608\pi$
$881$	−56.1390	−1.89137	−0.945686	+	0.325083i	$0.894608\pi$
$882$	0	0
$883$	1.97707	0.0665336	0.0332668	−	0.999447i	$-0.489409\pi$
$883$	1.97707	0.0665336	0.0332668	+	0.999447i	$0.489409\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.1966	0.946750	0.473375	−	0.880861i	$-0.343035\pi$
$887$	28.1966	0.946750	0.473375	+	0.880861i	$0.343035\pi$
$888$	0	0
$889$	−22.1499	−0.742885
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.0377	0.402826
$894$	0	0
$895$	−47.4492	−1.58605
$896$	0	0
$897$	0	0
$898$	0	0
$899$	55.1695	1.84001
$900$	0	0
$901$	−56.6827	−1.88837
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−27.5711	−0.916493
$906$	0	0
$907$	−5.10557	−0.169528	−0.0847638	−	0.996401i	$-0.527014\pi$
$907$	−5.10557	−0.169528	−0.0847638	+	0.996401i	$0.527014\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.6542	−0.750566	−0.375283	−	0.926910i	$-0.622454\pi$
$911$	−22.6542	−0.750566	−0.375283	+	0.926910i	$0.622454\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	53.1559	1.75536
$918$	0	0
$919$	−14.7425	−0.486312	−0.243156	−	0.969987i	$-0.578183\pi$
$919$	−14.7425	−0.486312	−0.243156	+	0.969987i	$0.578183\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	75.3305	2.47685
$926$	0	0
$927$	0	0
$928$	0	0
$929$	58.4487	1.91764	0.958820	−	0.284016i	$-0.0916669\pi$
$929$	58.4487	1.91764	0.958820	+	0.284016i	$0.0916669\pi$
$930$	0	0
$931$	19.3057	0.632719
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.5061	1.25794	0.628970	−	0.777429i	$-0.283477\pi$
$937$	38.5061	1.25794	0.628970	+	0.777429i	$0.283477\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55.3428	1.80412	0.902061	−	0.431608i	$-0.142054\pi$
$941$	55.3428	1.80412	0.902061	+	0.431608i	$0.142054\pi$
$942$	0	0
$943$	2.58260	0.0841012
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.9346	−1.36269	−0.681345	−	0.731962i	$-0.738605\pi$
$947$	−41.9346	−1.36269	−0.681345	+	0.731962i	$0.738605\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−51.4757	−1.66746	−0.833730	−	0.552172i	$-0.813799\pi$
$953$	−51.4757	−1.66746	−0.833730	+	0.552172i	$0.813799\pi$
$954$	0	0
$955$	11.2122	0.362819
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.9161	−0.610833
$960$	0	0
$961$	6.26313	0.202037
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−87.5396	−2.81800
$966$	0	0
$967$	−53.1689	−1.70980	−0.854898	−	0.518795i	$-0.826381\pi$
$967$	−53.1689	−1.70980	−0.854898	+	0.518795i	$0.826381\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.0064	−0.609944	−0.304972	−	0.952361i	$-0.598647\pi$
$971$	−19.0064	−0.609944	−0.304972	+	0.952361i	$0.598647\pi$
$972$	0	0
$973$	−2.90448	−0.0931134
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.8349	0.954502	0.477251	−	0.878767i	$-0.341633\pi$
$977$	29.8349	0.954502	0.477251	+	0.878767i	$0.341633\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.30155	0.200988	0.100494	−	0.994938i	$-0.467958\pi$
$983$	6.30155	0.200988	0.100494	+	0.994938i	$0.467958\pi$
$984$	0	0
$985$	51.8983	1.65362
$986$	0	0
$987$	0	0
$988$	0	0
$989$	28.7103	0.912934
$990$	0	0
$991$	−18.8983	−0.600325	−0.300163	−	0.953888i	$-0.597041\pi$
$991$	−18.8983	−0.600325	−0.300163	+	0.953888i	$0.597041\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	58.4140	1.85185
$996$	0	0
$997$	−40.9803	−1.29786	−0.648929	−	0.760849i	$-0.724783\pi$
$997$	−40.9803	−1.29786	−0.648929	+	0.760849i	$0.724783\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.i.1.5		5
3.2	odd	2		2676.2.a.c.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2676.2.a.c.1.1	✓	5	3.2	odd	2
8028.2.a.i.1.5		5	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+4.28552 q^{5} +3.47913 q^{7} +O(q^{10})\)	\(q+4.28552 q^{5} +3.47913 q^{7} +6.16942 q^{17} +3.78220 q^{19} +4.95826 q^{23} +13.3657 q^{25} -9.03773 q^{29} -6.10435 q^{31} +14.9099 q^{35} +5.63611 q^{37} +0.520869 q^{41} +5.79039 q^{43} +3.18272 q^{47} +5.10435 q^{49} -9.18767 q^{53} -0.704290 q^{59} -4.05999 q^{61} -5.81953 q^{67} +4.54615 q^{71} -14.9656 q^{73} -15.3239 q^{79} -3.00618 q^{83} +26.4392 q^{85} +7.41914 q^{89} +16.2087 q^{95} -1.60967 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q + 6 * q^5 + 3 * q^7	\( 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) \) 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

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