LMFDB - Embedded newform 5148.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.90996.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 2 $ x^4 - x^3 - 11x^2 - 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.63415$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22388 q^{5} -0.874887 q^{7} +O(q^{10})$	$q-3.22388 q^{5} -0.874887 q^{7} -1.00000 q^{11} +1.00000 q^{13} +1.16953 q^{17} -0.874887 q^{19} -8.84117 q^{23} +5.39341 q^{25} -1.65101 q^{29} -1.22388 q^{31} +2.82053 q^{35} -0.820532 q^{37} -11.6617 q^{41} +4.91930 q^{43} +12.4140 q^{47} -6.23457 q^{49} -4.44776 q^{53} +3.22388 q^{55} +6.44776 q^{59} +11.2683 q^{61} -3.22388 q^{65} -10.0107 q^{67} +7.71605 q^{71} +5.57287 q^{73} +0.874887 q^{77} +2.47154 q^{79} -16.9843 q^{83} -3.77041 q^{85} -14.0444 q^{89} -0.874887 q^{91} +2.82053 q^{95} +10.2165 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 + 2 * q^7	$ 4 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 4 q^{25} - 12 q^{29} + 6 q^{31} + 10 q^{35} - 2 q^{37} - 6 q^{41} + 2 q^{43} - 6 q^{47} + 32 q^{49} + 4 q^{53} + 2 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} + 6 q^{67} - 14 q^{71} + 6 q^{73} - 2 q^{77} + 14 q^{79} + 34 q^{85} - 44 q^{89} + 2 q^{91} + 10 q^{95} + 18 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 + 4 * q^13 - 2 * q^17 + 2 * q^19 + 4 * q^23 + 4 * q^25 - 12 * q^29 + 6 * q^31 + 10 * q^35 - 2 * q^37 - 6 * q^41 + 2 * q^43 - 6 * q^47 + 32 * q^49 + 4 * q^53 + 2 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 + 6 * q^67 - 14 * q^71 + 6 * q^73 - 2 * q^77 + 14 * q^79 + 34 * q^85 - 44 * q^89 + 2 * q^91 + 10 * q^95 + 18 * 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
$5$	−3.22388	−1.44176	−0.720882	−	0.693058i	$-0.756263\pi$
$5$	−3.22388	−1.44176	−0.720882	+	0.693058i	$0.756263\pi$
$6$	0	0
$7$	−0.874887	−0.330676	−0.165338	−	0.986237i	$-0.552872\pi$
$7$	−0.874887	−0.330676	−0.165338	+	0.986237i	$0.552872\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.16953	0.283652	0.141826	−	0.989892i	$-0.454703\pi$
$17$	1.16953	0.283652	0.141826	+	0.989892i	$0.454703\pi$
$18$	0	0
$19$	−0.874887	−0.200713	−0.100356	−	0.994952i	$-0.531998\pi$
$19$	−0.874887	−0.200713	−0.100356	+	0.994952i	$0.531998\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.84117	−1.84351	−0.921755	−	0.387772i	$-0.873245\pi$
$23$	−8.84117	−1.84351	−0.921755	+	0.387772i	$0.873245\pi$
$24$	0	0
$25$	5.39341	1.07868
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.65101	−0.306584	−0.153292	−	0.988181i	$-0.548988\pi$
$29$	−1.65101	−0.306584	−0.153292	+	0.988181i	$0.548988\pi$
$30$	0	0
$31$	−1.22388	−0.219815	−0.109908	−	0.993942i	$-0.535056\pi$
$31$	−1.22388	−0.219815	−0.109908	+	0.993942i	$0.535056\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82053	0.476757
$36$	0	0
$37$	−0.820532	−0.134895	−0.0674473	−	0.997723i	$-0.521485\pi$
$37$	−0.820532	−0.134895	−0.0674473	+	0.997723i	$0.521485\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.6617	−1.82125	−0.910626	−	0.413232i	$-0.864400\pi$
$41$	−11.6617	−1.82125	−0.910626	+	0.413232i	$0.864400\pi$
$42$	0	0
$43$	4.91930	0.750186	0.375093	−	0.926987i	$-0.377611\pi$
$43$	4.91930	0.750186	0.375093	+	0.926987i	$0.377611\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.4140	1.81077	0.905387	−	0.424587i	$-0.139581\pi$
$47$	12.4140	1.81077	0.905387	+	0.424587i	$0.139581\pi$
$48$	0	0
$49$	−6.23457	−0.890653
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.44776	−0.610947	−0.305473	−	0.952201i	$-0.598815\pi$
$53$	−4.44776	−0.610947	−0.305473	+	0.952201i	$0.598815\pi$
$54$	0	0
$55$	3.22388	0.434708
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.44776	0.839427	0.419713	−	0.907657i	$-0.362131\pi$
$59$	6.44776	0.839427	0.419713	+	0.907657i	$0.362131\pi$
$60$	0	0
$61$	11.2683	1.44276	0.721379	−	0.692541i	$-0.243509\pi$
$61$	11.2683	1.44276	0.721379	+	0.692541i	$0.243509\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.22388	−0.399873
$66$	0	0
$67$	−10.0107	−1.22300	−0.611500	−	0.791244i	$-0.709434\pi$
$67$	−10.0107	−1.22300	−0.611500	+	0.791244i	$0.709434\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.71605	0.915727	0.457864	−	0.889022i	$-0.348615\pi$
$71$	7.71605	0.915727	0.457864	+	0.889022i	$0.348615\pi$
$72$	0	0
$73$	5.57287	0.652256	0.326128	−	0.945326i	$-0.394256\pi$
$73$	5.57287	0.652256	0.326128	+	0.945326i	$0.394256\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.874887	0.0997026
$78$	0	0
$79$	2.47154	0.278070	0.139035	−	0.990287i	$-0.455600\pi$
$79$	2.47154	0.278070	0.139035	+	0.990287i	$0.455600\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.9843	−1.86427	−0.932137	−	0.362105i	$-0.882058\pi$
$83$	−16.9843	−1.86427	−0.932137	+	0.362105i	$0.882058\pi$
$84$	0	0
$85$	−3.77041	−0.408958
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0444	−1.48870	−0.744352	−	0.667787i	$-0.767242\pi$
$89$	−14.0444	−1.48870	−0.744352	+	0.667787i	$0.767242\pi$
$90$	0	0
$91$	−0.874887	−0.0917131
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.82053	0.289380
$96$	0	0
$97$	10.2165	1.03733	0.518664	−	0.854978i	$-0.326429\pi$
$97$	10.2165	1.03733	0.518664	+	0.854978i	$0.326429\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9193	−1.08651	−0.543255	−	0.839567i	$-0.682808\pi$
$101$	−10.9193	−1.08651	−0.543255	+	0.839567i	$0.682808\pi$
$102$	0	0
$103$	−7.14575	−0.704091	−0.352046	−	0.935983i	$-0.614514\pi$
$103$	−7.14575	−0.704091	−0.352046	+	0.935983i	$0.614514\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.44776	0.236634	0.118317	−	0.992976i	$-0.462250\pi$
$107$	2.44776	0.236634	0.118317	+	0.992976i	$0.462250\pi$
$108$	0	0
$109$	11.3226	1.08451	0.542257	−	0.840213i	$-0.317570\pi$
$109$	11.3226	1.08451	0.542257	+	0.840213i	$0.317570\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.19754	0.583015	0.291508	−	0.956569i	$-0.405843\pi$
$113$	6.19754	0.583015	0.291508	+	0.956569i	$0.405843\pi$
$114$	0	0
$115$	28.5029	2.65791
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.02320	−0.0937968
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.26829	−0.113440
$126$	0	0
$127$	16.5465	1.46827	0.734134	−	0.679005i	$-0.237589\pi$
$127$	16.5465	1.46827	0.734134	+	0.679005i	$0.237589\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.7868	1.46667	0.733335	−	0.679867i	$-0.237963\pi$
$131$	16.7868	1.46667	0.733335	+	0.679867i	$0.237963\pi$
$132$	0	0
$133$	0.765428	0.0663710
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.45514	0.124321	0.0621603	−	0.998066i	$-0.480201\pi$
$137$	1.45514	0.124321	0.0621603	+	0.998066i	$0.480201\pi$
$138$	0	0
$139$	6.79675	0.576493	0.288247	−	0.957556i	$-0.406928\pi$
$139$	6.79675	0.576493	0.288247	+	0.957556i	$0.406928\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	5.32265	0.442022
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.66170	0.299978	0.149989	−	0.988688i	$-0.452076\pi$
$149$	3.66170	0.299978	0.149989	+	0.988688i	$0.452076\pi$
$150$	0	0
$151$	−2.51595	−0.204745	−0.102373	−	0.994746i	$-0.532643\pi$
$151$	−2.51595	−0.204745	−0.102373	+	0.994746i	$0.532643\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.94564	0.316922
$156$	0	0
$157$	−1.35637	−0.108250	−0.0541250	−	0.998534i	$-0.517237\pi$
$157$	−1.35637	−0.108250	−0.0541250	+	0.998534i	$0.517237\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.73502	0.609605
$162$	0	0
$163$	−13.6379	−1.06820	−0.534102	−	0.845420i	$-0.679350\pi$
$163$	−13.6379	−1.06820	−0.534102	+	0.845420i	$0.679350\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.25023	−0.483657	−0.241828	−	0.970319i	$-0.577747\pi$
$167$	−6.25023	−0.483657	−0.241828	+	0.970319i	$0.577747\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.7630	1.12241	0.561206	−	0.827676i	$-0.310337\pi$
$173$	14.7630	1.12241	0.561206	+	0.827676i	$0.310337\pi$
$174$	0	0
$175$	−4.71862	−0.356694
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.8255	−1.63132	−0.815658	−	0.578535i	$-0.803625\pi$
$179$	−21.8255	−1.63132	−0.815658	+	0.578535i	$0.803625\pi$
$180$	0	0
$181$	5.35637	0.398136	0.199068	−	0.979986i	$-0.436209\pi$
$181$	5.35637	0.398136	0.199068	+	0.979986i	$0.436209\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.64530	0.194486
$186$	0	0
$187$	−1.16953	−0.0855241
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.4478	1.62426	0.812131	−	0.583474i	$-0.198307\pi$
$191$	22.4478	1.62426	0.812131	+	0.583474i	$0.198307\pi$
$192$	0	0
$193$	9.68158	0.696896	0.348448	−	0.937328i	$-0.386709\pi$
$193$	9.68158	0.696896	0.348448	+	0.937328i	$0.386709\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.5729	0.824533	0.412267	−	0.911063i	$-0.364737\pi$
$197$	11.5729	0.824533	0.412267	+	0.911063i	$0.364737\pi$
$198$	0	0
$199$	20.2864	1.43806	0.719031	−	0.694978i	$-0.244586\pi$
$199$	20.2864	1.43806	0.719031	+	0.694978i	$0.244586\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.44444	0.101380
$204$	0	0
$205$	37.5959	2.62581
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.874887	0.0605172
$210$	0	0
$211$	24.1539	1.66282	0.831411	−	0.555659i	$-0.187534\pi$
$211$	24.1539	1.66282	0.831411	+	0.555659i	$0.187534\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−15.8592	−1.08159
$216$	0	0
$217$	1.07076	0.0726877
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.16953	0.0786708
$222$	0	0
$223$	0.723429	0.0484444	0.0242222	−	0.999707i	$-0.492289\pi$
$223$	0.723429	0.0484444	0.0242222	+	0.999707i	$0.492289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.3440	−1.15116	−0.575582	−	0.817744i	$-0.695224\pi$
$227$	−17.3440	−1.15116	−0.575582	+	0.817744i	$0.695224\pi$
$228$	0	0
$229$	12.0551	0.796624	0.398312	−	0.917250i	$-0.369596\pi$
$229$	12.0551	0.796624	0.398312	+	0.917250i	$0.369596\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.3333	0.742472	0.371236	−	0.928539i	$-0.378934\pi$
$233$	11.3333	0.742472	0.371236	+	0.928539i	$0.378934\pi$
$234$	0	0
$235$	−40.0214	−2.61071
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.4485	1.06397	0.531983	−	0.846755i	$-0.321447\pi$
$239$	16.4485	1.06397	0.531983	+	0.846755i	$0.321447\pi$
$240$	0	0
$241$	−6.94308	−0.447243	−0.223621	−	0.974676i	$-0.571788\pi$
$241$	−6.94308	−0.447243	−0.223621	+	0.974676i	$0.571788\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.0995	1.28411
$246$	0	0
$247$	−0.874887	−0.0556677
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.94821	0.438567	0.219284	−	0.975661i	$-0.429628\pi$
$251$	6.94821	0.438567	0.219284	+	0.975661i	$0.429628\pi$
$252$	0	0
$253$	8.84117	0.555839
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.5366	1.28104	0.640519	−	0.767943i	$-0.278719\pi$
$257$	20.5366	1.28104	0.640519	+	0.767943i	$0.278719\pi$
$258$	0	0
$259$	0.717873	0.0446064
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.80670	0.543044	0.271522	−	0.962432i	$-0.412473\pi$
$263$	8.80670	0.543044	0.271522	+	0.962432i	$0.412473\pi$
$264$	0	0
$265$	14.3391	0.880841
$266$	0	0
$267$	0	0
$268$	0	0
$269$	23.0931	1.40801	0.704004	−	0.710196i	$-0.251394\pi$
$269$	23.0931	1.40801	0.704004	+	0.710196i	$0.251394\pi$
$270$	0	0
$271$	−11.2552	−0.683705	−0.341853	−	0.939754i	$-0.611054\pi$
$271$	−11.2552	−0.683705	−0.341853	+	0.939754i	$0.611054\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.39341	−0.325235
$276$	0	0
$277$	−9.14575	−0.549515	−0.274757	−	0.961514i	$-0.588598\pi$
$277$	−9.14575	−0.549515	−0.274757	+	0.961514i	$0.588598\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.57287	0.451760	0.225880	−	0.974155i	$-0.427474\pi$
$281$	7.57287	0.451760	0.225880	+	0.974155i	$0.427474\pi$
$282$	0	0
$283$	8.34899	0.496296	0.248148	−	0.968722i	$-0.420178\pi$
$283$	8.34899	0.496296	0.248148	+	0.968722i	$0.420178\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.2027	0.602245
$288$	0	0
$289$	−15.6322	−0.919542
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.5729	1.37714	0.688571	−	0.725169i	$-0.258238\pi$
$293$	23.5729	1.37714	0.688571	+	0.725169i	$0.258238\pi$
$294$	0	0
$295$	−20.7868	−1.21025
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.84117	−0.511298
$300$	0	0
$301$	−4.30383	−0.248069
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−36.3276	−2.08011
$306$	0	0
$307$	−6.62466	−0.378089	−0.189045	−	0.981968i	$-0.560539\pi$
$307$	−6.62466	−0.378089	−0.189045	+	0.981968i	$0.560539\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.69542	−0.0961384	−0.0480692	−	0.998844i	$-0.515307\pi$
$311$	−1.69542	−0.0961384	−0.0480692	+	0.998844i	$0.515307\pi$
$312$	0	0
$313$	−12.5909	−0.711682	−0.355841	−	0.934546i	$-0.615806\pi$
$313$	−12.5909	−0.711682	−0.355841	+	0.934546i	$0.615806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.5967	0.875995	0.437998	−	0.898976i	$-0.355688\pi$
$317$	15.5967	0.875995	0.437998	+	0.898976i	$0.355688\pi$
$318$	0	0
$319$	1.65101	0.0924386
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.02320	−0.0569325
$324$	0	0
$325$	5.39341	0.299172
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.8609	−0.598780
$330$	0	0
$331$	5.44039	0.299031	0.149515	−	0.988759i	$-0.452229\pi$
$331$	5.44039	0.299031	0.149515	+	0.988759i	$0.452229\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	32.2733	1.76328
$336$	0	0
$337$	−12.4478	−0.678073	−0.339036	−	0.940773i	$-0.610101\pi$
$337$	−12.4478	−0.678073	−0.339036	+	0.940773i	$0.610101\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.22388	0.0662768
$342$	0	0
$343$	11.5788	0.625194
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.9663	1.60867	0.804337	−	0.594173i	$-0.202521\pi$
$347$	29.9663	1.60867	0.804337	+	0.594173i	$0.202521\pi$
$348$	0	0
$349$	4.40649	0.235874	0.117937	−	0.993021i	$-0.462372\pi$
$349$	4.40649	0.235874	0.117937	+	0.993021i	$0.462372\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.76047	0.519497	0.259749	−	0.965676i	$-0.416360\pi$
$353$	9.76047	0.519497	0.259749	+	0.965676i	$0.416360\pi$
$354$	0	0
$355$	−24.8756	−1.32026
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.9162	1.20947	0.604734	−	0.796427i	$-0.293279\pi$
$359$	22.9162	1.20947	0.604734	+	0.796427i	$0.293279\pi$
$360$	0	0
$361$	−18.2346	−0.959714
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.9663	−0.940398
$366$	0	0
$367$	9.48480	0.495102	0.247551	−	0.968875i	$-0.420374\pi$
$367$	9.48480	0.495102	0.247551	+	0.968875i	$0.420374\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.89129	0.202026
$372$	0	0
$373$	−22.0888	−1.14372	−0.571858	−	0.820353i	$-0.693777\pi$
$373$	−22.0888	−1.14372	−0.571858	+	0.820353i	$0.693777\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.65101	−0.0850312
$378$	0	0
$379$	17.1375	0.880293	0.440146	−	0.897926i	$-0.354927\pi$
$379$	17.1375	0.880293	0.440146	+	0.897926i	$0.354927\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.33905	−0.119520	−0.0597599	−	0.998213i	$-0.519034\pi$
$383$	−2.33905	−0.119520	−0.0597599	+	0.998213i	$0.519034\pi$
$384$	0	0
$385$	−2.82053	−0.143748
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.12586	−0.0570834	−0.0285417	−	0.999593i	$-0.509086\pi$
$389$	−1.12586	−0.0570834	−0.0285417	+	0.999593i	$0.509086\pi$
$390$	0	0
$391$	−10.3400	−0.522915
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.96794	−0.400911
$396$	0	0
$397$	−17.2156	−0.864026	−0.432013	−	0.901867i	$-0.642197\pi$
$397$	−17.2156	−0.864026	−0.432013	+	0.901867i	$0.642197\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.9580	−1.29628	−0.648140	−	0.761521i	$-0.724453\pi$
$401$	−25.9580	−1.29628	−0.648140	+	0.761521i	$0.724453\pi$
$402$	0	0
$403$	−1.22388	−0.0609658
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.820532	0.0406723
$408$	0	0
$409$	−34.6873	−1.71518	−0.857589	−	0.514336i	$-0.828038\pi$
$409$	−34.6873	−1.71518	−0.857589	+	0.514336i	$0.828038\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.64106	−0.277579
$414$	0	0
$415$	54.7555	2.68784
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.78424	0.0871660	0.0435830	−	0.999050i	$-0.486123\pi$
$419$	1.78424	0.0871660	0.0435830	+	0.999050i	$0.486123\pi$
$420$	0	0
$421$	−18.2864	−0.891223	−0.445611	−	0.895227i	$-0.647014\pi$
$421$	−18.2864	−0.891223	−0.445611	+	0.895227i	$0.647014\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.30772	0.305970
$426$	0	0
$427$	−9.85849	−0.477086
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0413	−1.15803	−0.579014	−	0.815318i	$-0.696562\pi$
$431$	−24.0413	−1.15803	−0.579014	+	0.815318i	$0.696562\pi$
$432$	0	0
$433$	−16.2915	−0.782919	−0.391460	−	0.920195i	$-0.628030\pi$
$433$	−16.2915	−0.782919	−0.391460	+	0.920195i	$0.628030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.73502	0.370016
$438$	0	0
$439$	21.4198	1.02231	0.511154	−	0.859489i	$-0.329218\pi$
$439$	21.4198	1.02231	0.511154	+	0.859489i	$0.329218\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.50045	0.403869	0.201934	−	0.979399i	$-0.435277\pi$
$443$	8.50045	0.403869	0.201934	+	0.979399i	$0.435277\pi$
$444$	0	0
$445$	45.2775	2.14636
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.4058	−1.52932	−0.764661	−	0.644432i	$-0.777094\pi$
$449$	−32.4058	−1.52932	−0.764661	+	0.644432i	$0.777094\pi$
$450$	0	0
$451$	11.6617	0.549128
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.82053	0.132229
$456$	0	0
$457$	−35.6823	−1.66915	−0.834575	−	0.550895i	$-0.814286\pi$
$457$	−35.6823	−1.66915	−0.834575	+	0.550895i	$0.814286\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.9851	1.44312	0.721560	−	0.692352i	$-0.243425\pi$
$461$	30.9851	1.44312	0.721560	+	0.692352i	$0.243425\pi$
$462$	0	0
$463$	5.50783	0.255970	0.127985	−	0.991776i	$-0.459149\pi$
$463$	5.50783	0.255970	0.127985	+	0.991776i	$0.459149\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.8056	1.56434	0.782169	−	0.623066i	$-0.214113\pi$
$467$	33.8056	1.56434	0.782169	+	0.623066i	$0.214113\pi$
$468$	0	0
$469$	8.75823	0.404417
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.91930	−0.226190
$474$	0	0
$475$	−4.71862	−0.216505
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.7704	−1.26886	−0.634431	−	0.772979i	$-0.718766\pi$
$479$	−27.7704	−1.26886	−0.634431	+	0.772979i	$0.718766\pi$
$480$	0	0
$481$	−0.820532	−0.0374130
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−32.9368	−1.49558
$486$	0	0
$487$	−1.39912	−0.0634000	−0.0317000	−	0.999497i	$-0.510092\pi$
$487$	−1.39912	−0.0634000	−0.0317000	+	0.999497i	$0.510092\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.7892	1.47976	0.739879	−	0.672741i	$-0.234883\pi$
$491$	32.7892	1.47976	0.739879	+	0.672741i	$0.234883\pi$
$492$	0	0
$493$	−1.93089	−0.0869631
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.75068	−0.302809
$498$	0	0
$499$	−40.1555	−1.79761	−0.898804	−	0.438350i	$-0.855563\pi$
$499$	−40.1555	−1.79761	−0.898804	+	0.438350i	$0.855563\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.72989	−0.433834	−0.216917	−	0.976190i	$-0.569600\pi$
$503$	−9.72989	−0.433834	−0.216917	+	0.976190i	$0.569600\pi$
$504$	0	0
$505$	35.2025	1.56649
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.44039	0.329789	0.164895	−	0.986311i	$-0.447272\pi$
$509$	7.44039	0.329789	0.164895	+	0.986311i	$0.447272\pi$
$510$	0	0
$511$	−4.87564	−0.215685
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.0370	1.01513
$516$	0	0
$517$	−12.4140	−0.545969
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.3951	−0.981146	−0.490573	−	0.871400i	$-0.663212\pi$
$521$	−22.3951	−0.981146	−0.490573	+	0.871400i	$0.663212\pi$
$522$	0	0
$523$	3.32579	0.145427	0.0727133	−	0.997353i	$-0.476834\pi$
$523$	3.32579	0.145427	0.0727133	+	0.997353i	$0.476834\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.43136	−0.0623510
$528$	0	0
$529$	55.1662	2.39853
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.6617	−0.505124
$534$	0	0
$535$	−7.89129	−0.341170
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.23457	0.268542
$540$	0	0
$541$	−36.9696	−1.58945	−0.794724	−	0.606972i	$-0.792384\pi$
$541$	−36.9696	−1.58945	−0.794724	+	0.606972i	$0.792384\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−36.5029	−1.56361
$546$	0	0
$547$	31.6724	1.35421	0.677107	−	0.735885i	$-0.263233\pi$
$547$	31.6724	1.35421	0.677107	+	0.735885i	$0.263233\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.44444	0.0615354
$552$	0	0
$553$	−2.16232	−0.0919511
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.88002	0.206773	0.103387	−	0.994641i	$-0.467032\pi$
$557$	4.88002	0.206773	0.103387	+	0.994641i	$0.467032\pi$
$558$	0	0
$559$	4.91930	0.208064
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.60916	−0.362833	−0.181416	−	0.983406i	$-0.558068\pi$
$563$	−8.60916	−0.362833	−0.181416	+	0.983406i	$0.558068\pi$
$564$	0	0
$565$	−19.9801	−0.840570
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−33.5646	−1.40710	−0.703551	−	0.710645i	$-0.748403\pi$
$569$	−33.5646	−1.40710	−0.703551	+	0.710645i	$0.748403\pi$
$570$	0	0
$571$	45.8362	1.91819	0.959093	−	0.283092i	$-0.0913602\pi$
$571$	45.8362	1.91819	0.959093	+	0.283092i	$0.0913602\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−47.6840	−1.98856
$576$	0	0
$577$	33.7185	1.40372	0.701859	−	0.712316i	$-0.252354\pi$
$577$	33.7185	1.40372	0.701859	+	0.712316i	$0.252354\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14.8594	0.616471
$582$	0	0
$583$	4.44776	0.184207
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.8067	0.693687	0.346843	−	0.937923i	$-0.387254\pi$
$587$	16.8067	0.693687	0.346843	+	0.937923i	$0.387254\pi$
$588$	0	0
$589$	1.07076	0.0441198
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.6090	0.723115	0.361558	−	0.932350i	$-0.382245\pi$
$593$	17.6090	0.723115	0.361558	+	0.932350i	$0.382245\pi$
$594$	0	0
$595$	3.29868	0.135233
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.5218	−0.838499	−0.419250	−	0.907871i	$-0.637707\pi$
$599$	−20.5218	−0.838499	−0.419250	+	0.907871i	$0.637707\pi$
$600$	0	0
$601$	11.3580	0.463304	0.231652	−	0.972799i	$-0.425587\pi$
$601$	11.3580	0.463304	0.231652	+	0.972799i	$0.425587\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.22388	−0.131069
$606$	0	0
$607$	−34.8181	−1.41322	−0.706612	−	0.707601i	$-0.749777\pi$
$607$	−34.8181	−1.41322	−0.706612	+	0.707601i	$0.749777\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.4140	0.502218
$612$	0	0
$613$	−41.3967	−1.67200	−0.835999	−	0.548731i	$-0.815111\pi$
$613$	−41.3967	−1.67200	−0.835999	+	0.548731i	$0.815111\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.7647	−1.88268	−0.941338	−	0.337465i	$-0.890430\pi$
$617$	−46.7647	−1.88268	−0.941338	+	0.337465i	$0.890430\pi$
$618$	0	0
$619$	17.0288	0.684444	0.342222	−	0.939619i	$-0.388821\pi$
$619$	17.0288	0.684444	0.342222	+	0.939619i	$0.388821\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.2873	0.492279
$624$	0	0
$625$	−22.8782	−0.915128
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.959633	−0.0382631
$630$	0	0
$631$	22.5810	0.898935	0.449468	−	0.893297i	$-0.351614\pi$
$631$	22.5810	0.898935	0.449468	+	0.893297i	$0.351614\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−53.3440	−2.11689
$636$	0	0
$637$	−6.23457	−0.247023
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−31.6149	−1.24871	−0.624357	−	0.781139i	$-0.714639\pi$
$641$	−31.6149	−1.24871	−0.624357	+	0.781139i	$0.714639\pi$
$642$	0	0
$643$	−8.06338	−0.317989	−0.158994	−	0.987279i	$-0.550825\pi$
$643$	−8.06338	−0.317989	−0.158994	+	0.987279i	$0.550825\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.15627	0.320656	0.160328	−	0.987064i	$-0.448745\pi$
$647$	8.15627	0.320656	0.160328	+	0.987064i	$0.448745\pi$
$648$	0	0
$649$	−6.44776	−0.253097
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.9317	0.975651	0.487826	−	0.872941i	$-0.337790\pi$
$653$	24.9317	0.975651	0.487826	+	0.872941i	$0.337790\pi$
$654$	0	0
$655$	−54.1187	−2.11459
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.9558	1.43959	0.719796	−	0.694186i	$-0.244235\pi$
$659$	36.9558	1.43959	0.719796	+	0.694186i	$0.244235\pi$
$660$	0	0
$661$	0.906942	0.0352760	0.0176380	−	0.999844i	$-0.494385\pi$
$661$	0.906942	0.0352760	0.0176380	+	0.999844i	$0.494385\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.46765	−0.0956913
$666$	0	0
$667$	14.5968	0.565191
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.2683	−0.435008
$672$	0	0
$673$	25.7573	0.992872	0.496436	−	0.868073i	$-0.334642\pi$
$673$	25.7573	0.992872	0.496436	+	0.868073i	$0.334642\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.6871	0.602906	0.301453	−	0.953481i	$-0.402528\pi$
$677$	15.6871	0.602906	0.301453	+	0.953481i	$0.402528\pi$
$678$	0	0
$679$	−8.93829	−0.343020
$680$	0	0
$681$	0	0
$682$	0	0
$683$	32.6116	1.24785	0.623924	−	0.781485i	$-0.285538\pi$
$683$	32.6116	1.24785	0.623924	+	0.781485i	$0.285538\pi$
$684$	0	0
$685$	−4.69119	−0.179241
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.44776	−0.169446
$690$	0	0
$691$	−27.2100	−1.03512	−0.517559	−	0.855647i	$-0.673159\pi$
$691$	−27.2100	−1.03512	−0.517559	+	0.855647i	$0.673159\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.9119	−0.831167
$696$	0	0
$697$	−13.6386	−0.516601
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−32.3153	−1.22053	−0.610266	−	0.792197i	$-0.708937\pi$
$701$	−32.3153	−1.22053	−0.610266	+	0.792197i	$0.708937\pi$
$702$	0	0
$703$	0.717873	0.0270751
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.55316	0.359283
$708$	0	0
$709$	49.4923	1.85872	0.929362	−	0.369170i	$-0.120358\pi$
$709$	49.4923	1.85872	0.929362	+	0.369170i	$0.120358\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.8205	0.405232
$714$	0	0
$715$	3.22388	0.120566
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.0825	−1.49483	−0.747413	−	0.664359i	$-0.768704\pi$
$719$	−40.0825	−1.49483	−0.747413	+	0.664359i	$0.768704\pi$
$720$	0	0
$721$	6.25172	0.232826
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.90455	−0.330707
$726$	0	0
$727$	−2.98948	−0.110874	−0.0554369	−	0.998462i	$-0.517655\pi$
$727$	−2.98948	−0.110874	−0.0554369	+	0.998462i	$0.517655\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.75324	0.212791
$732$	0	0
$733$	−9.28651	−0.343005	−0.171503	−	0.985184i	$-0.554862\pi$
$733$	−9.28651	−0.343005	−0.171503	+	0.985184i	$0.554862\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0107	0.368749
$738$	0	0
$739$	−31.6304	−1.16354	−0.581771	−	0.813352i	$-0.697640\pi$
$739$	−31.6304	−1.16354	−0.581771	+	0.813352i	$0.697640\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.3752	0.747493	0.373747	−	0.927531i	$-0.378073\pi$
$743$	20.3752	0.747493	0.373747	+	0.927531i	$0.378073\pi$
$744$	0	0
$745$	−11.8049	−0.432497
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.14151	−0.0782492
$750$	0	0
$751$	−14.6254	−0.533689	−0.266844	−	0.963740i	$-0.585981\pi$
$751$	−14.6254	−0.533689	−0.266844	+	0.963740i	$0.585981\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.11112	0.295194
$756$	0	0
$757$	34.5762	1.25669	0.628347	−	0.777934i	$-0.283732\pi$
$757$	34.5762	1.25669	0.628347	+	0.777934i	$0.283732\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.4314	−0.486886	−0.243443	−	0.969915i	$-0.578277\pi$
$761$	−13.4314	−0.486886	−0.243443	+	0.969915i	$0.578277\pi$
$762$	0	0
$763$	−9.90604	−0.358623
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.44776	0.232815
$768$	0	0
$769$	−6.21742	−0.224206	−0.112103	−	0.993697i	$-0.535759\pi$
$769$	−6.21742	−0.224206	−0.112103	+	0.993697i	$0.535759\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−48.0805	−1.72934	−0.864669	−	0.502343i	$-0.832472\pi$
$773$	−48.0805	−1.72934	−0.864669	+	0.502343i	$0.832472\pi$
$774$	0	0
$775$	−6.60088	−0.237111
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.2027	0.365549
$780$	0	0
$781$	−7.71605	−0.276102
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.37277	0.156071
$786$	0	0
$787$	−8.76618	−0.312480	−0.156240	−	0.987719i	$-0.549937\pi$
$787$	−8.76618	−0.312480	−0.156240	+	0.987719i	$0.549937\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.42214	−0.192789
$792$	0	0
$793$	11.2683	0.400149
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.7185	0.910995	0.455497	−	0.890237i	$-0.349461\pi$
$797$	25.7185	0.910995	0.455497	+	0.890237i	$0.349461\pi$
$798$	0	0
$799$	14.5185	0.513629
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.57287	−0.196662
$804$	0	0
$805$	−24.9368	−0.878906
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41.6272	1.46354	0.731768	−	0.681554i	$-0.238695\pi$
$809$	41.6272	1.46354	0.731768	+	0.681554i	$0.238695\pi$
$810$	0	0
$811$	−32.1983	−1.13063	−0.565317	−	0.824874i	$-0.691246\pi$
$811$	−32.1983	−1.13063	−0.565317	+	0.824874i	$0.691246\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	43.9670	1.54010
$816$	0	0
$817$	−4.30383	−0.150572
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.8948	0.938634	0.469317	−	0.883030i	$-0.344500\pi$
$821$	26.8948	0.938634	0.469317	+	0.883030i	$0.344500\pi$
$822$	0	0
$823$	0.442626	0.0154290	0.00771448	−	0.999970i	$-0.497544\pi$
$823$	0.442626	0.0154290	0.00771448	+	0.999970i	$0.497544\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.5891	−0.750727	−0.375364	−	0.926878i	$-0.622482\pi$
$827$	−21.5891	−0.750727	−0.375364	+	0.926878i	$0.622482\pi$
$828$	0	0
$829$	−20.4691	−0.710923	−0.355461	−	0.934691i	$-0.615676\pi$
$829$	−20.4691	−0.710923	−0.355461	+	0.934691i	$0.615676\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.29149	−0.252635
$834$	0	0
$835$	20.1500	0.697319
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.0015	0.690528	0.345264	−	0.938506i	$-0.387789\pi$
$839$	20.0015	0.690528	0.345264	+	0.938506i	$0.387789\pi$
$840$	0	0
$841$	−26.2742	−0.906006
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.22388	−0.110905
$846$	0	0
$847$	−0.874887	−0.0300615
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.25446	0.248680
$852$	0	0
$853$	26.7194	0.914854	0.457427	−	0.889247i	$-0.348771\pi$
$853$	26.7194	0.914854	0.457427	+	0.889247i	$0.348771\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.45257	0.152097	0.0760484	−	0.997104i	$-0.475770\pi$
$857$	4.45257	0.152097	0.0760484	+	0.997104i	$0.475770\pi$
$858$	0	0
$859$	1.59351	0.0543698	0.0271849	−	0.999630i	$-0.491346\pi$
$859$	1.59351	0.0543698	0.0271849	+	0.999630i	$0.491346\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.9154	0.439646	0.219823	−	0.975540i	$-0.429452\pi$
$863$	12.9154	0.439646	0.219823	+	0.975540i	$0.429452\pi$
$864$	0	0
$865$	−47.5943	−1.61825
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.47154	−0.0838412
$870$	0	0
$871$	−10.0107	−0.339199
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.10961	0.0375118
$876$	0	0
$877$	−4.37882	−0.147862	−0.0739312	−	0.997263i	$-0.523555\pi$
$877$	−4.37882	−0.147862	−0.0739312	+	0.997263i	$0.523555\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.42638	−0.149128	−0.0745642	−	0.997216i	$-0.523757\pi$
$881$	−4.42638	−0.149128	−0.0745642	+	0.997216i	$0.523757\pi$
$882$	0	0
$883$	33.8998	1.14082	0.570409	−	0.821361i	$-0.306785\pi$
$883$	33.8998	1.14082	0.570409	+	0.821361i	$0.306785\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.6995	−1.60159	−0.800796	−	0.598937i	$-0.795590\pi$
$887$	−47.6995	−1.60159	−0.800796	+	0.598937i	$0.795590\pi$
$888$	0	0
$889$	−14.4763	−0.485521
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.8609	−0.363446
$894$	0	0
$895$	70.3628	2.35197
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.02063	0.0673919
$900$	0	0
$901$	−5.20177	−0.173296
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.2683	−0.574017
$906$	0	0
$907$	26.1629	0.868725	0.434362	−	0.900738i	$-0.356974\pi$
$907$	26.1629	0.868725	0.434362	+	0.900738i	$0.356974\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−10.6929	−0.354270	−0.177135	−	0.984187i	$-0.556683\pi$
$911$	−10.6929	−0.354270	−0.177135	+	0.984187i	$0.556683\pi$
$912$	0	0
$913$	16.9843	0.562100
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.6866	−0.484993
$918$	0	0
$919$	4.72418	0.155836	0.0779181	−	0.996960i	$-0.475173\pi$
$919$	4.72418	0.155836	0.0779181	+	0.996960i	$0.475173\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7.71605	0.253977
$924$	0	0
$925$	−4.42546	−0.145508
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.75715	−0.287313	−0.143656	−	0.989628i	$-0.545886\pi$
$929$	−8.75715	−0.287313	−0.143656	+	0.989628i	$0.545886\pi$
$930$	0	0
$931$	5.45455	0.178766
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.77041	0.123306
$936$	0	0
$937$	−4.64288	−0.151676	−0.0758382	−	0.997120i	$-0.524163\pi$
$937$	−4.64288	−0.151676	−0.0758382	+	0.997120i	$0.524163\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.4536	−0.471175	−0.235588	−	0.971853i	$-0.575701\pi$
$941$	−14.4536	−0.471175	−0.235588	+	0.971853i	$0.575701\pi$
$942$	0	0
$943$	103.103	3.35750
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.1638	0.980192	0.490096	−	0.871668i	$-0.336962\pi$
$947$	30.1638	0.980192	0.490096	+	0.871668i	$0.336962\pi$
$948$	0	0
$949$	5.57287	0.180903
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0.0987677	0.00319940	0.00159970	−	0.999999i	$-0.499491\pi$
$953$	0.0987677	0.00319940	0.00159970	+	0.999999i	$0.499491\pi$
$954$	0	0
$955$	−72.3689	−2.34180
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.27308	−0.0411099
$960$	0	0
$961$	−29.5021	−0.951681
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−31.2123	−1.00476
$966$	0	0
$967$	34.4684	1.10843	0.554214	−	0.832374i	$-0.313019\pi$
$967$	34.4684	1.10843	0.554214	+	0.832374i	$0.313019\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.9818	0.705429	0.352714	−	0.935731i	$-0.385259\pi$
$971$	21.9818	0.705429	0.352714	+	0.935731i	$0.385259\pi$
$972$	0	0
$973$	−5.94639	−0.190633
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.8725	1.49958	0.749792	−	0.661674i	$-0.230154\pi$
$977$	46.8725	1.49958	0.749792	+	0.661674i	$0.230154\pi$
$978$	0	0
$979$	14.0444	0.448861
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.1060	0.481806	0.240903	−	0.970549i	$-0.422556\pi$
$983$	15.1060	0.481806	0.240903	+	0.970549i	$0.422556\pi$
$984$	0	0
$985$	−37.3096	−1.18878
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43.4923	−1.38298
$990$	0	0
$991$	37.6348	1.19551	0.597754	−	0.801680i	$-0.296060\pi$
$991$	37.6348	1.19551	0.597754	+	0.801680i	$0.296060\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−65.4008	−2.07334
$996$	0	0
$997$	−6.52183	−0.206549	−0.103274	−	0.994653i	$-0.532932\pi$
$997$	−6.52183	−0.206549	−0.103274	+	0.994653i	$0.532932\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.q.1.1		4
3.2	odd	2		1716.2.a.i.1.4	✓	4
12.11	even	2		6864.2.a.cb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.i.1.4	✓	4	3.2	odd	2
5148.2.a.q.1.1		4	1.1	even	1	trivial
6864.2.a.cb.1.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-3.22388 q^{5} -0.874887 q^{7} +O(q^{10})\)	\(q-3.22388 q^{5} -0.874887 q^{7} -1.00000 q^{11} +1.00000 q^{13} +1.16953 q^{17} -0.874887 q^{19} -8.84117 q^{23} +5.39341 q^{25} -1.65101 q^{29} -1.22388 q^{31} +2.82053 q^{35} -0.820532 q^{37} -11.6617 q^{41} +4.91930 q^{43} +12.4140 q^{47} -6.23457 q^{49} -4.44776 q^{53} +3.22388 q^{55} +6.44776 q^{59} +11.2683 q^{61} -3.22388 q^{65} -10.0107 q^{67} +7.71605 q^{71} +5.57287 q^{73} +0.874887 q^{77} +2.47154 q^{79} -16.9843 q^{83} -3.77041 q^{85} -14.0444 q^{89} -0.874887 q^{91} +2.82053 q^{95} +10.2165 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 + 2 * q^7	\( 4 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 4 q^{25} - 12 q^{29} + 6 q^{31} + 10 q^{35} - 2 q^{37} - 6 q^{41} + 2 q^{43} - 6 q^{47} + 32 q^{49} + 4 q^{53} + 2 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} + 6 q^{67} - 14 q^{71} + 6 q^{73} - 2 q^{77} + 14 q^{79} + 34 q^{85} - 44 q^{89} + 2 q^{91} + 10 q^{95} + 18 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 + 4 * q^13 - 2 * q^17 + 2 * q^19 + 4 * q^23 + 4 * q^25 - 12 * q^29 + 6 * q^31 + 10 * q^35 - 2 * q^37 - 6 * q^41 + 2 * q^43 - 6 * q^47 + 32 * q^49 + 4 * q^53 + 2 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 + 6 * q^67 - 14 * q^71 + 6 * q^73 - 2 * q^77 + 14 * q^79 + 34 * q^85 - 44 * q^89 + 2 * q^91 + 10 * q^95 + 18 * q^97

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