LMFDB - Embedded newform 4851.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, 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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1617)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86081 q^{2} +1.46260 q^{4} -1.32340 q^{5} -1.00000 q^{8} +O(q^{10})$	$q+1.86081 q^{2} +1.46260 q^{4} -1.32340 q^{5} -1.00000 q^{8} -2.46260 q^{10} -1.00000 q^{11} +0.398207 q^{13} -4.78600 q^{16} +6.64681 q^{17} +1.32340 q^{19} -1.93561 q^{20} -1.86081 q^{22} -3.24860 q^{25} +0.740987 q^{26} -1.47301 q^{29} -4.79641 q^{31} -6.90582 q^{32} +12.3684 q^{34} -1.60179 q^{37} +2.46260 q^{38} +1.32340 q^{40} -4.79641 q^{41} -4.79641 q^{43} -1.46260 q^{44} -3.04502 q^{47} -6.04502 q^{50} +0.582418 q^{52} -8.64681 q^{53} +1.32340 q^{55} -2.74099 q^{58} +13.6918 q^{59} -3.20359 q^{61} -8.92520 q^{62} -3.27839 q^{64} -0.526989 q^{65} -10.6170 q^{67} +9.72161 q^{68} -15.9404 q^{71} +1.45219 q^{73} -2.98062 q^{74} +1.93561 q^{76} +3.44322 q^{79} +6.33382 q^{80} -8.92520 q^{82} +0.796415 q^{83} -8.79641 q^{85} -8.92520 q^{86} +1.00000 q^{88} +10.6468 q^{89} -5.66618 q^{94} -1.75140 q^{95} -12.2396 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} + 4 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 + 4 * q^5 - 3 * q^8	$ 3 q + 2 q^{4} + 4 q^{5} - 3 q^{8} - 5 q^{10} - 3 q^{11} - 2 q^{13} - 4 q^{16} + 4 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{25} + 11 q^{26} - 6 q^{29} - 8 q^{31} + 4 q^{32} + 10 q^{34} - 8 q^{37} + 5 q^{38} - 4 q^{40} - 8 q^{41} - 8 q^{43} - 2 q^{44} + 10 q^{47} + q^{50} - 15 q^{52} - 10 q^{53} - 4 q^{55} - 17 q^{58} + 6 q^{59} - 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} - 2 q^{73} + 11 q^{74} + 5 q^{76} - 12 q^{79} + 15 q^{80} - 22 q^{82} - 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} + 16 q^{89} - 21 q^{94} - 18 q^{95} - 8 q^{97}+O(q^{100}) $ 3 * q + 2 * q^4 + 4 * q^5 - 3 * q^8 - 5 * q^10 - 3 * q^11 - 2 * q^13 - 4 * q^16 + 4 * q^17 - 4 * q^19 - 5 * q^20 + 3 * q^25 + 11 * q^26 - 6 * q^29 - 8 * q^31 + 4 * q^32 + 10 * q^34 - 8 * q^37 + 5 * q^38 - 4 * q^40 - 8 * q^41 - 8 * q^43 - 2 * q^44 + 10 * q^47 + q^50 - 15 * q^52 - 10 * q^53 - 4 * q^55 - 17 * q^58 + 6 * q^59 - 16 * q^61 - 22 * q^62 - 21 * q^64 + 8 * q^67 + 18 * q^68 - 2 * q^73 + 11 * q^74 + 5 * q^76 - 12 * q^79 + 15 * q^80 - 22 * q^82 - 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 + 16 * q^89 - 21 * q^94 - 18 * q^95 - 8 * 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 $

$2$	1.86081	1.31579	0.657894	−	0.753110i	$-0.271447\pi$
$2$	1.86081	1.31579	0.657894	+	0.753110i	$0.271447\pi$
$3$	0	0
$3$	0	0
$4$	1.46260	0.731299
$5$	−1.32340	−0.591844	−0.295922	−	0.955212i	$-0.595627\pi$
$5$	−1.32340	−0.591844	−0.295922	+	0.955212i	$0.595627\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.46260	−0.778742
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.398207	0.110443	0.0552214	−	0.998474i	$-0.482414\pi$
$13$	0.398207	0.110443	0.0552214	+	0.998474i	$0.482414\pi$
$14$	0	0
$15$	0	0
$16$	−4.78600	−1.19650
$17$	6.64681	1.61209	0.806044	−	0.591856i	$-0.201604\pi$
$17$	6.64681	1.61209	0.806044	+	0.591856i	$0.201604\pi$
$18$	0	0
$19$	1.32340	0.303610	0.151805	−	0.988410i	$-0.451491\pi$
$19$	1.32340	0.303610	0.151805	+	0.988410i	$0.451491\pi$
$20$	−1.93561	−0.432815
$21$	0	0
$22$	−1.86081	−0.396725
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−3.24860	−0.649720
$26$	0.740987	0.145319
$27$	0	0
$28$	0	0
$29$	−1.47301	−0.273531	−0.136766	−	0.990603i	$-0.543671\pi$
$29$	−1.47301	−0.273531	−0.136766	+	0.990603i	$0.543671\pi$
$30$	0	0
$31$	−4.79641	−0.861462	−0.430731	−	0.902480i	$-0.641744\pi$
$31$	−4.79641	−0.861462	−0.430731	+	0.902480i	$0.641744\pi$
$32$	−6.90582	−1.22079
$33$	0	0
$34$	12.3684	2.12117
$35$	0	0
$36$	0	0
$37$	−1.60179	−0.263333	−0.131667	−	0.991294i	$-0.542033\pi$
$37$	−1.60179	−0.263333	−0.131667	+	0.991294i	$0.542033\pi$
$38$	2.46260	0.399486
$39$	0	0
$40$	1.32340	0.209249
$41$	−4.79641	−0.749074	−0.374537	−	0.927212i	$-0.622198\pi$
$41$	−4.79641	−0.749074	−0.374537	+	0.927212i	$0.622198\pi$
$42$	0	0
$43$	−4.79641	−0.731446	−0.365723	−	0.930724i	$-0.619178\pi$
$43$	−4.79641	−0.731446	−0.365723	+	0.930724i	$0.619178\pi$
$44$	−1.46260	−0.220495
$45$	0	0
$46$	0	0
$47$	−3.04502	−0.444161	−0.222081	−	0.975028i	$-0.571285\pi$
$47$	−3.04502	−0.444161	−0.222081	+	0.975028i	$0.571285\pi$
$48$	0	0
$49$	0	0
$50$	−6.04502	−0.854894
$51$	0	0
$52$	0.582418	0.0807668
$53$	−8.64681	−1.18773	−0.593865	−	0.804565i	$-0.702399\pi$
$53$	−8.64681	−1.18773	−0.593865	+	0.804565i	$0.702399\pi$
$54$	0	0
$55$	1.32340	0.178448
$56$	0	0
$57$	0	0
$58$	−2.74099	−0.359909
$59$	13.6918	1.78252	0.891262	−	0.453489i	$-0.149821\pi$
$59$	13.6918	1.78252	0.891262	+	0.453489i	$0.149821\pi$
$60$	0	0
$61$	−3.20359	−0.410177	−0.205089	−	0.978743i	$-0.565748\pi$
$61$	−3.20359	−0.410177	−0.205089	+	0.978743i	$0.565748\pi$
$62$	−8.92520	−1.13350
$63$	0	0
$64$	−3.27839	−0.409799
$65$	−0.526989	−0.0653650
$66$	0	0
$67$	−10.6170	−1.29708	−0.648538	−	0.761182i	$-0.724619\pi$
$67$	−10.6170	−1.29708	−0.648538	+	0.761182i	$0.724619\pi$
$68$	9.72161	1.17892
$69$	0	0
$70$	0	0
$71$	−15.9404	−1.89178	−0.945890	−	0.324487i	$-0.894808\pi$
$71$	−15.9404	−1.89178	−0.945890	+	0.324487i	$0.894808\pi$
$72$	0	0
$73$	1.45219	0.169966	0.0849828	−	0.996382i	$-0.472916\pi$
$73$	1.45219	0.169966	0.0849828	+	0.996382i	$0.472916\pi$
$74$	−2.98062	−0.346491
$75$	0	0
$76$	1.93561	0.222030
$77$	0	0
$78$	0	0
$79$	3.44322	0.387393	0.193696	−	0.981062i	$-0.437952\pi$
$79$	3.44322	0.387393	0.193696	+	0.981062i	$0.437952\pi$
$80$	6.33382	0.708142
$81$	0	0
$82$	−8.92520	−0.985623
$83$	0.796415	0.0874179	0.0437089	−	0.999044i	$-0.486083\pi$
$83$	0.796415	0.0874179	0.0437089	+	0.999044i	$0.486083\pi$
$84$	0	0
$85$	−8.79641	−0.954105
$86$	−8.92520	−0.962429
$87$	0	0
$88$	1.00000	0.106600
$89$	10.6468	1.12856	0.564280	−	0.825584i	$-0.309154\pi$
$89$	10.6468	1.12856	0.564280	+	0.825584i	$0.309154\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−5.66618	−0.584422
$95$	−1.75140	−0.179690
$96$	0	0
$97$	−12.2396	−1.24275	−0.621373	−	0.783515i	$-0.713425\pi$
$97$	−12.2396	−1.24275	−0.621373	+	0.783515i	$0.713425\pi$
$98$	0	0
$99$	0	0
$100$	−4.75140	−0.475140
$101$	−16.4972	−1.64153	−0.820766	−	0.571264i	$-0.806453\pi$
$101$	−16.4972	−1.64153	−0.820766	+	0.571264i	$0.806453\pi$
$102$	0	0
$103$	−13.8504	−1.36472	−0.682360	−	0.731016i	$-0.739046\pi$
$103$	−13.8504	−1.36472	−0.682360	+	0.731016i	$0.739046\pi$
$104$	−0.398207	−0.0390475
$105$	0	0
$106$	−16.0900	−1.56280
$107$	5.69182	0.550249	0.275125	−	0.961409i	$-0.411281\pi$
$107$	5.69182	0.550249	0.275125	+	0.961409i	$0.411281\pi$
$108$	0	0
$109$	−4.14961	−0.397460	−0.198730	−	0.980054i	$-0.563682\pi$
$109$	−4.14961	−0.397460	−0.198730	+	0.980054i	$0.563682\pi$
$110$	2.46260	0.234800
$111$	0	0
$112$	0	0
$113$	−0.946021	−0.0889942	−0.0444971	−	0.999010i	$-0.514169\pi$
$113$	−0.946021	−0.0889942	−0.0444971	+	0.999010i	$0.514169\pi$
$114$	0	0
$115$	0	0
$116$	−2.15442	−0.200033
$117$	0	0
$118$	25.4778	2.34542
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.96125	−0.539706
$123$	0	0
$124$	−7.01523	−0.629986
$125$	10.9162	0.976378
$126$	0	0
$127$	−4.55678	−0.404349	−0.202174	−	0.979350i	$-0.564801\pi$
$127$	−4.55678	−0.404349	−0.202174	+	0.979350i	$0.564801\pi$
$128$	7.71120	0.681580
$129$	0	0
$130$	−0.980625	−0.0860065
$131$	−1.05398	−0.0920866	−0.0460433	−	0.998939i	$-0.514661\pi$
$131$	−1.05398	−0.0920866	−0.0460433	+	0.998939i	$0.514661\pi$
$132$	0	0
$133$	0	0
$134$	−19.7562	−1.70668
$135$	0	0
$136$	−6.64681	−0.569959
$137$	−1.44322	−0.123303	−0.0616514	−	0.998098i	$-0.519637\pi$
$137$	−1.44322	−0.123303	−0.0616514	+	0.998098i	$0.519637\pi$
$138$	0	0
$139$	−18.5872	−1.57655	−0.788274	−	0.615324i	$-0.789025\pi$
$139$	−18.5872	−1.57655	−0.788274	+	0.615324i	$0.789025\pi$
$140$	0	0
$141$	0	0
$142$	−29.6620	−2.48918
$143$	−0.398207	−0.0332998
$144$	0	0
$145$	1.94939	0.161888
$146$	2.70224	0.223639
$147$	0	0
$148$	−2.34278	−0.192575
$149$	−3.88018	−0.317877	−0.158938	−	0.987289i	$-0.550807\pi$
$149$	−3.88018	−0.317877	−0.158938	+	0.987289i	$0.550807\pi$
$150$	0	0
$151$	10.6468	0.866425	0.433212	−	0.901292i	$-0.357380\pi$
$151$	10.6468	0.866425	0.433212	+	0.901292i	$0.357380\pi$
$152$	−1.32340	−0.107342
$153$	0	0
$154$	0	0
$155$	6.34760	0.509851
$156$	0	0
$157$	−16.7368	−1.33575	−0.667873	−	0.744276i	$-0.732795\pi$
$157$	−16.7368	−1.33575	−0.667873	+	0.744276i	$0.732795\pi$
$158$	6.40717	0.509727
$159$	0	0
$160$	9.13919	0.722517
$161$	0	0
$162$	0	0
$163$	2.67660	0.209647	0.104824	−	0.994491i	$-0.466572\pi$
$163$	2.67660	0.209647	0.104824	+	0.994491i	$0.466572\pi$
$164$	−7.01523	−0.547797
$165$	0	0
$166$	1.48197	0.115023
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−12.8414	−0.987802
$170$	−16.3684	−1.25540
$171$	0	0
$172$	−7.01523	−0.534906
$173$	12.7368	0.968364	0.484182	−	0.874967i	$-0.339117\pi$
$173$	12.7368	0.968364	0.484182	+	0.874967i	$0.339117\pi$
$174$	0	0
$175$	0	0
$176$	4.78600	0.360759
$177$	0	0
$178$	19.8116	1.48495
$179$	−5.85039	−0.437279	−0.218639	−	0.975806i	$-0.570162\pi$
$179$	−5.85039	−0.437279	−0.218639	+	0.975806i	$0.570162\pi$
$180$	0	0
$181$	−16.7964	−1.24847	−0.624234	−	0.781238i	$-0.714589\pi$
$181$	−16.7964	−1.24847	−0.624234	+	0.781238i	$0.714589\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.11982	0.155852
$186$	0	0
$187$	−6.64681	−0.486063
$188$	−4.45364	−0.324815
$189$	0	0
$190$	−3.25901	−0.236434
$191$	16.7368	1.21104	0.605518	−	0.795832i	$-0.292966\pi$
$191$	16.7368	1.21104	0.605518	+	0.795832i	$0.292966\pi$
$192$	0	0
$193$	9.94043	0.715527	0.357764	−	0.933812i	$-0.383539\pi$
$193$	9.94043	0.715527	0.357764	+	0.933812i	$0.383539\pi$
$194$	−22.7756	−1.63519
$195$	0	0
$196$	0	0
$197$	−15.8504	−1.12929	−0.564647	−	0.825333i	$-0.690988\pi$
$197$	−15.8504	−1.12929	−0.564647	+	0.825333i	$0.690988\pi$
$198$	0	0
$199$	6.38924	0.452922	0.226461	−	0.974020i	$-0.427284\pi$
$199$	6.38924	0.452922	0.226461	+	0.974020i	$0.427284\pi$
$200$	3.24860	0.229711
$201$	0	0
$202$	−30.6981	−2.15991
$203$	0	0
$204$	0	0
$205$	6.34760	0.443335
$206$	−25.7729	−1.79568
$207$	0	0
$208$	−1.90582	−0.132145
$209$	−1.32340	−0.0915418
$210$	0	0
$211$	−26.0900	−1.79611	−0.898056	−	0.439881i	$-0.855021\pi$
$211$	−26.0900	−1.79611	−0.898056	+	0.439881i	$0.855021\pi$
$212$	−12.6468	−0.868586
$213$	0	0
$214$	10.5914	0.724012
$215$	6.34760	0.432902
$216$	0	0
$217$	0	0
$218$	−7.72161	−0.522974
$219$	0	0
$220$	1.93561	0.130499
$221$	2.64681	0.178044
$222$	0	0
$223$	−18.0900	−1.21140	−0.605699	−	0.795694i	$-0.707106\pi$
$223$	−18.0900	−1.21140	−0.605699	+	0.795694i	$0.707106\pi$
$224$	0	0
$225$	0	0
$226$	−1.76036	−0.117098
$227$	28.1801	1.87038	0.935188	−	0.354151i	$-0.115230\pi$
$227$	28.1801	1.87038	0.935188	+	0.354151i	$0.115230\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	0	0
$231$	0	0
$232$	1.47301	0.0967079
$233$	21.4432	1.40479	0.702396	−	0.711786i	$-0.252114\pi$
$233$	21.4432	1.40479	0.702396	+	0.711786i	$0.252114\pi$
$234$	0	0
$235$	4.02979	0.262874
$236$	20.0256	1.30356
$237$	0	0
$238$	0	0
$239$	−11.5422	−0.746604	−0.373302	−	0.927710i	$-0.621774\pi$
$239$	−11.5422	−0.746604	−0.373302	+	0.927710i	$0.621774\pi$
$240$	0	0
$241$	12.9550	0.834504	0.417252	−	0.908791i	$-0.362993\pi$
$241$	12.9550	0.834504	0.417252	+	0.908791i	$0.362993\pi$
$242$	1.86081	0.119617
$243$	0	0
$244$	−4.68556	−0.299962
$245$	0	0
$246$	0	0
$247$	0.526989	0.0335315
$248$	4.79641	0.304573
$249$	0	0
$250$	20.3130	1.28471
$251$	28.8954	1.82386	0.911931	−	0.410343i	$-0.134591\pi$
$251$	28.8954	1.82386	0.911931	+	0.410343i	$0.134591\pi$
$252$	0	0
$253$	0	0
$254$	−8.47928	−0.532037
$255$	0	0
$256$	20.9058	1.30661
$257$	12.2278	0.762748	0.381374	−	0.924421i	$-0.375451\pi$
$257$	12.2278	0.762748	0.381374	+	0.924421i	$0.375451\pi$
$258$	0	0
$259$	0	0
$260$	−0.770774	−0.0478014
$261$	0	0
$262$	−1.96125	−0.121166
$263$	0.954984	0.0588868	0.0294434	−	0.999566i	$-0.490627\pi$
$263$	0.954984	0.0588868	0.0294434	+	0.999566i	$0.490627\pi$
$264$	0	0
$265$	11.4432	0.702952
$266$	0	0
$267$	0	0
$268$	−15.5284	−0.948550
$269$	−20.2396	−1.23403	−0.617016	−	0.786950i	$-0.711659\pi$
$269$	−20.2396	−1.23403	−0.617016	+	0.786950i	$0.711659\pi$
$270$	0	0
$271$	16.4674	1.00032	0.500162	−	0.865932i	$-0.333274\pi$
$271$	16.4674	1.00032	0.500162	+	0.865932i	$0.333274\pi$
$272$	−31.8116	−1.92886
$273$	0	0
$274$	−2.68556	−0.162241
$275$	3.24860	0.195898
$276$	0	0
$277$	18.1801	1.09233	0.546167	−	0.837676i	$-0.316086\pi$
$277$	18.1801	1.09233	0.546167	+	0.837676i	$0.316086\pi$
$278$	−34.5872	−2.07440
$279$	0	0
$280$	0	0
$281$	−24.3178	−1.45068	−0.725339	−	0.688391i	$-0.758317\pi$
$281$	−24.3178	−1.45068	−0.725339	+	0.688391i	$0.758317\pi$
$282$	0	0
$283$	21.5035	1.27825	0.639124	−	0.769103i	$-0.279297\pi$
$283$	21.5035	1.27825	0.639124	+	0.769103i	$0.279297\pi$
$284$	−23.3144	−1.38346
$285$	0	0
$286$	−0.740987	−0.0438155
$287$	0	0
$288$	0	0
$289$	27.1801	1.59883
$290$	3.62743	0.213010
$291$	0	0
$292$	2.12397	0.124296
$293$	5.59283	0.326737	0.163368	−	0.986565i	$-0.447764\pi$
$293$	5.59283	0.326737	0.163368	+	0.986565i	$0.447764\pi$
$294$	0	0
$295$	−18.1198	−1.05498
$296$	1.60179	0.0931023
$297$	0	0
$298$	−7.22026	−0.418259
$299$	0	0
$300$	0	0
$301$	0	0
$302$	19.8116	1.14003
$303$	0	0
$304$	−6.33382	−0.363269
$305$	4.23964	0.242761
$306$	0	0
$307$	10.5872	0.604245	0.302123	−	0.953269i	$-0.402305\pi$
$307$	10.5872	0.604245	0.302123	+	0.953269i	$0.402305\pi$
$308$	0	0
$309$	0	0
$310$	11.8116	0.670856
$311$	5.59283	0.317140	0.158570	−	0.987348i	$-0.449312\pi$
$311$	5.59283	0.317140	0.158570	+	0.987348i	$0.449312\pi$
$312$	0	0
$313$	−1.90997	−0.107958	−0.0539789	−	0.998542i	$-0.517190\pi$
$313$	−1.90997	−0.107958	−0.0539789	+	0.998542i	$0.517190\pi$
$314$	−31.1440	−1.75756
$315$	0	0
$316$	5.03605	0.283300
$317$	−9.44322	−0.530384	−0.265192	−	0.964196i	$-0.585435\pi$
$317$	−9.44322	−0.530384	−0.265192	+	0.964196i	$0.585435\pi$
$318$	0	0
$319$	1.47301	0.0824728
$320$	4.33863	0.242537
$321$	0	0
$322$	0	0
$323$	8.79641	0.489446
$324$	0	0
$325$	−1.29362	−0.0717570
$326$	4.98062	0.275851
$327$	0	0
$328$	4.79641	0.264838
$329$	0	0
$330$	0	0
$331$	−20.4793	−1.12564	−0.562821	−	0.826579i	$-0.690284\pi$
$331$	−20.4793	−1.12564	−0.562821	+	0.826579i	$0.690284\pi$
$332$	1.16484	0.0639286
$333$	0	0
$334$	29.7729	1.62910
$335$	14.0506	0.767667
$336$	0	0
$337$	17.3836	0.946948	0.473474	−	0.880808i	$-0.343000\pi$
$337$	17.3836	0.946948	0.473474	+	0.880808i	$0.343000\pi$
$338$	−23.8954	−1.29974
$339$	0	0
$340$	−12.8656	−0.697736
$341$	4.79641	0.259740
$342$	0	0
$343$	0	0
$344$	4.79641	0.258605
$345$	0	0
$346$	23.7008	1.27416
$347$	30.5872	1.64201	0.821004	−	0.570922i	$-0.193414\pi$
$347$	30.5872	1.64201	0.821004	+	0.570922i	$0.193414\pi$
$348$	0	0
$349$	32.8775	1.75989	0.879946	−	0.475074i	$-0.157579\pi$
$349$	32.8775	1.75989	0.879946	+	0.475074i	$0.157579\pi$
$350$	0	0
$351$	0	0
$352$	6.90582	0.368082
$353$	−33.0063	−1.75675	−0.878373	−	0.477976i	$-0.841371\pi$
$353$	−33.0063	−1.75675	−0.878373	+	0.477976i	$0.841371\pi$
$354$	0	0
$355$	21.0956	1.11964
$356$	15.5720	0.825315
$357$	0	0
$358$	−10.8864	−0.575367
$359$	19.7008	1.03977	0.519884	−	0.854237i	$-0.325975\pi$
$359$	19.7008	1.03977	0.519884	+	0.854237i	$0.325975\pi$
$360$	0	0
$361$	−17.2486	−0.907821
$362$	−31.2549	−1.64272
$363$	0	0
$364$	0	0
$365$	−1.92183	−0.100593
$366$	0	0
$367$	−31.1261	−1.62477	−0.812384	−	0.583123i	$-0.801831\pi$
$367$	−31.1261	−1.62477	−0.812384	+	0.583123i	$0.801831\pi$
$368$	0	0
$369$	0	0
$370$	3.94457	0.205069
$371$	0	0
$372$	0	0
$373$	−8.94602	−0.463207	−0.231604	−	0.972810i	$-0.574397\pi$
$373$	−8.94602	−0.463207	−0.231604	+	0.972810i	$0.574397\pi$
$374$	−12.3684	−0.639556
$375$	0	0
$376$	3.04502	0.157035
$377$	−0.586564	−0.0302096
$378$	0	0
$379$	4.20985	0.216246	0.108123	−	0.994138i	$-0.465516\pi$
$379$	4.20985	0.216246	0.108123	+	0.994138i	$0.465516\pi$
$380$	−2.56159	−0.131407
$381$	0	0
$382$	31.1440	1.59347
$383$	−14.5872	−0.745373	−0.372686	−	0.927957i	$-0.621563\pi$
$383$	−14.5872	−0.745373	−0.372686	+	0.927957i	$0.621563\pi$
$384$	0	0
$385$	0	0
$386$	18.4972	0.941483
$387$	0	0
$388$	−17.9017	−0.908820
$389$	30.4793	1.54536	0.772680	−	0.634795i	$-0.218916\pi$
$389$	30.4793	1.54536	0.772680	+	0.634795i	$0.218916\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−29.4945	−1.48591
$395$	−4.55678	−0.229276
$396$	0	0
$397$	−10.9044	−0.547275	−0.273637	−	0.961833i	$-0.588227\pi$
$397$	−10.9044	−0.547275	−0.273637	+	0.961833i	$0.588227\pi$
$398$	11.8891	0.595949
$399$	0	0
$400$	15.5478	0.777391
$401$	−12.4072	−0.619585	−0.309792	−	0.950804i	$-0.600260\pi$
$401$	−12.4072	−0.619585	−0.309792	+	0.950804i	$0.600260\pi$
$402$	0	0
$403$	−1.90997	−0.0951423
$404$	−24.1288	−1.20045
$405$	0	0
$406$	0	0
$407$	1.60179	0.0793979
$408$	0	0
$409$	31.3836	1.55182	0.775911	−	0.630843i	$-0.217291\pi$
$409$	31.3836	1.55182	0.775911	+	0.630843i	$0.217291\pi$
$410$	11.8116	0.583336
$411$	0	0
$412$	−20.2576	−0.998019
$413$	0	0
$414$	0	0
$415$	−1.05398	−0.0517378
$416$	−2.74995	−0.134827
$417$	0	0
$418$	−2.46260	−0.120450
$419$	−30.1711	−1.47395	−0.736977	−	0.675917i	$-0.763748\pi$
$419$	−30.1711	−1.47395	−0.736977	+	0.675917i	$0.763748\pi$
$420$	0	0
$421$	−21.2847	−1.03735	−0.518675	−	0.854971i	$-0.673575\pi$
$421$	−21.2847	−1.03735	−0.518675	+	0.854971i	$0.673575\pi$
$422$	−48.5485	−2.36330
$423$	0	0
$424$	8.64681	0.419926
$425$	−21.5928	−1.04741
$426$	0	0
$427$	0	0
$428$	8.32485	0.402397
$429$	0	0
$430$	11.8116	0.569608
$431$	14.7279	0.709417	0.354708	−	0.934977i	$-0.384580\pi$
$431$	14.7279	0.709417	0.354708	+	0.934977i	$0.384580\pi$
$432$	0	0
$433$	−14.1496	−0.679987	−0.339993	−	0.940428i	$-0.610425\pi$
$433$	−14.1496	−0.679987	−0.339993	+	0.940428i	$0.610425\pi$
$434$	0	0
$435$	0	0
$436$	−6.06921	−0.290662
$437$	0	0
$438$	0	0
$439$	−26.0007	−1.24094	−0.620472	−	0.784228i	$-0.713059\pi$
$439$	−26.0007	−1.24094	−0.620472	+	0.784228i	$0.713059\pi$
$440$	−1.32340	−0.0630908
$441$	0	0
$442$	4.92520	0.234268
$443$	32.9765	1.56676	0.783380	−	0.621543i	$-0.213494\pi$
$443$	32.9765	1.56676	0.783380	+	0.621543i	$0.213494\pi$
$444$	0	0
$445$	−14.0900	−0.667932
$446$	−33.6620	−1.59394
$447$	0	0
$448$	0	0
$449$	−4.96395	−0.234263	−0.117132	−	0.993116i	$-0.537370\pi$
$449$	−4.96395	−0.234263	−0.117132	+	0.993116i	$0.537370\pi$
$450$	0	0
$451$	4.79641	0.225854
$452$	−1.38365	−0.0650814
$453$	0	0
$454$	52.4376	2.46102
$455$	0	0
$456$	0	0
$457$	−22.1980	−1.03838	−0.519189	−	0.854659i	$-0.673766\pi$
$457$	−22.1980	−1.03838	−0.519189	+	0.854659i	$0.673766\pi$
$458$	14.8864	0.695598
$459$	0	0
$460$	0	0
$461$	−12.5389	−0.583993	−0.291996	−	0.956419i	$-0.594319\pi$
$461$	−12.5389	−0.583993	−0.291996	+	0.956419i	$0.594319\pi$
$462$	0	0
$463$	17.5214	0.814288	0.407144	−	0.913364i	$-0.366525\pi$
$463$	17.5214	0.814288	0.407144	+	0.913364i	$0.366525\pi$
$464$	7.04983	0.327280
$465$	0	0
$466$	39.9017	1.84841
$467$	−11.7819	−0.545199	−0.272600	−	0.962128i	$-0.587883\pi$
$467$	−11.7819	−0.545199	−0.272600	+	0.962128i	$0.587883\pi$
$468$	0	0
$469$	0	0
$470$	7.49865	0.345887
$471$	0	0
$472$	−13.6918	−0.630217
$473$	4.79641	0.220539
$474$	0	0
$475$	−4.29921	−0.197261
$476$	0	0
$477$	0	0
$478$	−21.4778	−0.982373
$479$	−5.03605	−0.230103	−0.115052	−	0.993360i	$-0.536703\pi$
$479$	−5.03605	−0.230103	−0.115052	+	0.993360i	$0.536703\pi$
$480$	0	0
$481$	−0.637846	−0.0290833
$482$	24.1067	1.09803
$483$	0	0
$484$	1.46260	0.0664817
$485$	16.1980	0.735513
$486$	0	0
$487$	20.1801	0.914446	0.457223	−	0.889352i	$-0.348844\pi$
$487$	20.1801	0.914446	0.457223	+	0.889352i	$0.348844\pi$
$488$	3.20359	0.145019
$489$	0	0
$490$	0	0
$491$	25.3926	1.14595	0.572976	−	0.819572i	$-0.305789\pi$
$491$	25.3926	1.14595	0.572976	+	0.819572i	$0.305789\pi$
$492$	0	0
$493$	−9.79082	−0.440956
$494$	0.980625	0.0441204
$495$	0	0
$496$	22.9557	1.03074
$497$	0	0
$498$	0	0
$499$	14.9162	0.667742	0.333871	−	0.942619i	$-0.391645\pi$
$499$	14.9162	0.667742	0.333871	+	0.942619i	$0.391645\pi$
$500$	15.9661	0.714024
$501$	0	0
$502$	53.7687	2.39982
$503$	−28.9765	−1.29200	−0.645999	−	0.763339i	$-0.723559\pi$
$503$	−28.9765	−1.29200	−0.645999	+	0.763339i	$0.723559\pi$
$504$	0	0
$505$	21.8325	0.971532
$506$	0	0
$507$	0	0
$508$	−6.66473	−0.295700
$509$	1.05398	0.0467168	0.0233584	−	0.999727i	$-0.492564\pi$
$509$	1.05398	0.0467168	0.0233584	+	0.999727i	$0.492564\pi$
$510$	0	0
$511$	0	0
$512$	23.4793	1.03765
$513$	0	0
$514$	22.7535	1.00361
$515$	18.3297	0.807702
$516$	0	0
$517$	3.04502	0.133920
$518$	0	0
$519$	0	0
$520$	0.526989	0.0231100
$521$	14.2999	0.626489	0.313245	−	0.949672i	$-0.398584\pi$
$521$	14.2999	0.626489	0.313245	+	0.949672i	$0.398584\pi$
$522$	0	0
$523$	34.8567	1.52418	0.762088	−	0.647474i	$-0.224175\pi$
$523$	34.8567	1.52418	0.762088	+	0.647474i	$0.224175\pi$
$524$	−1.54155	−0.0673428
$525$	0	0
$526$	1.77704	0.0774826
$527$	−31.8809	−1.38875
$528$	0	0
$529$	−23.0000	−1.00000
$530$	21.2936	0.924936
$531$	0	0
$532$	0	0
$533$	−1.90997	−0.0827299
$534$	0	0
$535$	−7.53258	−0.325662
$536$	10.6170	0.458585
$537$	0	0
$538$	−37.6620	−1.62373
$539$	0	0
$540$	0	0
$541$	39.2161	1.68603	0.843016	−	0.537888i	$-0.180778\pi$
$541$	39.2161	1.68603	0.843016	+	0.537888i	$0.180778\pi$
$542$	30.6427	1.31622
$543$	0	0
$544$	−45.9017	−1.96802
$545$	5.49161	0.235235
$546$	0	0
$547$	−38.5277	−1.64732	−0.823662	−	0.567081i	$-0.808073\pi$
$547$	−38.5277	−1.64732	−0.823662	+	0.567081i	$0.808073\pi$
$548$	−2.11086	−0.0901713
$549$	0	0
$550$	6.04502	0.257760
$551$	−1.94939	−0.0830467
$552$	0	0
$553$	0	0
$554$	33.8296	1.43728
$555$	0	0
$556$	−27.1857	−1.15293
$557$	−43.7610	−1.85421	−0.927107	−	0.374796i	$-0.877713\pi$
$557$	−43.7610	−1.85421	−0.927107	+	0.374796i	$0.877713\pi$
$558$	0	0
$559$	−1.90997	−0.0807830
$560$	0	0
$561$	0	0
$562$	−45.2507	−1.90879
$563$	−19.9821	−0.842144	−0.421072	−	0.907027i	$-0.638346\pi$
$563$	−19.9821	−0.842144	−0.421072	+	0.907027i	$0.638346\pi$
$564$	0	0
$565$	1.25197	0.0526707
$566$	40.0138	1.68190
$567$	0	0
$568$	15.9404	0.668845
$569$	−31.7312	−1.33024	−0.665121	−	0.746735i	$-0.731620\pi$
$569$	−31.7312	−1.33024	−0.665121	+	0.746735i	$0.731620\pi$
$570$	0	0
$571$	−10.9460	−0.458077	−0.229038	−	0.973417i	$-0.573558\pi$
$571$	−10.9460	−0.458077	−0.229038	+	0.973417i	$0.573558\pi$
$572$	−0.582418	−0.0243521
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.24523	0.135101	0.0675504	−	0.997716i	$-0.478482\pi$
$577$	3.24523	0.135101	0.0675504	+	0.997716i	$0.478482\pi$
$578$	50.5768	2.10372
$579$	0	0
$580$	2.85117	0.118388
$581$	0	0
$582$	0	0
$583$	8.64681	0.358114
$584$	−1.45219	−0.0600919
$585$	0	0
$586$	10.4072	0.429916
$587$	26.7875	1.10564	0.552818	−	0.833302i	$-0.313552\pi$
$587$	26.7875	1.10564	0.552818	+	0.833302i	$0.313552\pi$
$588$	0	0
$589$	−6.34760	−0.261548
$590$	−33.7175	−1.38813
$591$	0	0
$592$	7.66618	0.315078
$593$	32.2396	1.32392	0.661962	−	0.749538i	$-0.269724\pi$
$593$	32.2396	1.32392	0.661962	+	0.749538i	$0.269724\pi$
$594$	0	0
$595$	0	0
$596$	−5.67515	−0.232463
$597$	0	0
$598$	0	0
$599$	7.14401	0.291896	0.145948	−	0.989292i	$-0.453377\pi$
$599$	7.14401	0.291896	0.145948	+	0.989292i	$0.453377\pi$
$600$	0	0
$601$	21.7514	0.887258	0.443629	−	0.896211i	$-0.353691\pi$
$601$	21.7514	0.887258	0.443629	+	0.896211i	$0.353691\pi$
$602$	0	0
$603$	0	0
$604$	15.5720	0.633616
$605$	−1.32340	−0.0538040
$606$	0	0
$607$	−39.4134	−1.59974	−0.799871	−	0.600172i	$-0.795099\pi$
$607$	−39.4134	−1.59974	−0.799871	+	0.600172i	$0.795099\pi$
$608$	−9.13919	−0.370643
$609$	0	0
$610$	7.88914	0.319422
$611$	−1.21255	−0.0490544
$612$	0	0
$613$	−22.1980	−0.896568	−0.448284	−	0.893891i	$-0.647965\pi$
$613$	−22.1980	−0.896568	−0.448284	+	0.893891i	$0.647965\pi$
$614$	19.7008	0.795059
$615$	0	0
$616$	0	0
$617$	37.4432	1.50741	0.753704	−	0.657214i	$-0.228265\pi$
$617$	37.4432	1.50741	0.753704	+	0.657214i	$0.228265\pi$
$618$	0	0
$619$	−15.7008	−0.631068	−0.315534	−	0.948914i	$-0.602184\pi$
$619$	−15.7008	−0.631068	−0.315534	+	0.948914i	$0.602184\pi$
$620$	9.28398	0.372854
$621$	0	0
$622$	10.4072	0.417290
$623$	0	0
$624$	0	0
$625$	1.79641	0.0718566
$626$	−3.55408	−0.142050
$627$	0	0
$628$	−24.4793	−0.976829
$629$	−10.6468	−0.424516
$630$	0	0
$631$	35.0665	1.39598	0.697988	−	0.716110i	$-0.254079\pi$
$631$	35.0665	1.39598	0.697988	+	0.716110i	$0.254079\pi$
$632$	−3.44322	−0.136964
$633$	0	0
$634$	−17.5720	−0.697873
$635$	6.03046	0.239311
$636$	0	0
$637$	0	0
$638$	2.74099	0.108517
$639$	0	0
$640$	−10.2050	−0.403389
$641$	30.1801	1.19204	0.596020	−	0.802969i	$-0.296748\pi$
$641$	30.1801	1.19204	0.596020	+	0.802969i	$0.296748\pi$
$642$	0	0
$643$	−36.7964	−1.45111	−0.725554	−	0.688165i	$-0.758417\pi$
$643$	−36.7964	−1.45111	−0.725554	+	0.688165i	$0.758417\pi$
$644$	0	0
$645$	0	0
$646$	16.3684	0.644007
$647$	33.4522	1.31514	0.657571	−	0.753393i	$-0.271584\pi$
$647$	33.4522	1.31514	0.657571	+	0.753393i	$0.271584\pi$
$648$	0	0
$649$	−13.6918	−0.537451
$650$	−2.40717	−0.0944170
$651$	0	0
$652$	3.91478	0.153315
$653$	−5.50280	−0.215341	−0.107671	−	0.994187i	$-0.534339\pi$
$653$	−5.50280	−0.215341	−0.107671	+	0.994187i	$0.534339\pi$
$654$	0	0
$655$	1.39484	0.0545009
$656$	22.9557	0.896268
$657$	0	0
$658$	0	0
$659$	−33.1946	−1.29308	−0.646539	−	0.762881i	$-0.723784\pi$
$659$	−33.1946	−1.29308	−0.646539	+	0.762881i	$0.723784\pi$
$660$	0	0
$661$	43.3241	1.68511	0.842556	−	0.538609i	$-0.181050\pi$
$661$	43.3241	1.68511	0.842556	+	0.538609i	$0.181050\pi$
$662$	−38.1080	−1.48111
$663$	0	0
$664$	−0.796415	−0.0309069
$665$	0	0
$666$	0	0
$667$	0	0
$668$	23.4016	0.905434
$669$	0	0
$670$	26.1455	1.01009
$671$	3.20359	0.123673
$672$	0	0
$673$	−28.0721	−1.08210	−0.541050	−	0.840990i	$-0.681973\pi$
$673$	−28.0721	−1.08210	−0.541050	+	0.840990i	$0.681973\pi$
$674$	32.3476	1.24598
$675$	0	0
$676$	−18.7819	−0.722379
$677$	−22.8448	−0.877997	−0.438998	−	0.898488i	$-0.644667\pi$
$677$	−22.8448	−0.877997	−0.438998	+	0.898488i	$0.644667\pi$
$678$	0	0
$679$	0	0
$680$	8.79641	0.337327
$681$	0	0
$682$	8.92520	0.341763
$683$	−11.4016	−0.436269	−0.218135	−	0.975919i	$-0.569997\pi$
$683$	−11.4016	−0.436269	−0.218135	+	0.975919i	$0.569997\pi$
$684$	0	0
$685$	1.90997	0.0729761
$686$	0	0
$687$	0	0
$688$	22.9557	0.875176
$689$	−3.44322	−0.131176
$690$	0	0
$691$	−11.9404	−0.454235	−0.227118	−	0.973867i	$-0.572930\pi$
$691$	−11.9404	−0.454235	−0.227118	+	0.973867i	$0.572930\pi$
$692$	18.6289	0.708164
$693$	0	0
$694$	56.9169	2.16054
$695$	24.5984	0.933071
$696$	0	0
$697$	−31.8809	−1.20757
$698$	61.1786	2.31564
$699$	0	0
$700$	0	0
$701$	−17.1440	−0.647520	−0.323760	−	0.946139i	$-0.604947\pi$
$701$	−17.1440	−0.647520	−0.323760	+	0.946139i	$0.604947\pi$
$702$	0	0
$703$	−2.11982	−0.0799505
$704$	3.27839	0.123559
$705$	0	0
$706$	−61.4183	−2.31151
$707$	0	0
$708$	0	0
$709$	−50.7583	−1.90627	−0.953135	−	0.302546i	$-0.902163\pi$
$709$	−50.7583	−1.90627	−0.953135	+	0.302546i	$0.902163\pi$
$710$	39.2549	1.47321
$711$	0	0
$712$	−10.6468	−0.399006
$713$	0	0
$714$	0	0
$715$	0.526989	0.0197083
$716$	−8.55678	−0.319782
$717$	0	0
$718$	36.6593	1.36811
$719$	−4.35656	−0.162472	−0.0812361	−	0.996695i	$-0.525887\pi$
$719$	−4.35656	−0.162472	−0.0812361	+	0.996695i	$0.525887\pi$
$720$	0	0
$721$	0	0
$722$	−32.0963	−1.19450
$723$	0	0
$724$	−24.5664	−0.913003
$725$	4.78522	0.177719
$726$	0	0
$727$	−33.1745	−1.23037	−0.615186	−	0.788382i	$-0.710919\pi$
$727$	−33.1745	−1.23037	−0.615186	+	0.788382i	$0.710919\pi$
$728$	0	0
$729$	0	0
$730$	−3.57615	−0.132359
$731$	−31.8809	−1.17916
$732$	0	0
$733$	−31.3836	−1.15918	−0.579591	−	0.814908i	$-0.696788\pi$
$733$	−31.3836	−1.15918	−0.579591	+	0.814908i	$0.696788\pi$
$734$	−57.9196	−2.13785
$735$	0	0
$736$	0	0
$737$	10.6170	0.391083
$738$	0	0
$739$	29.4737	1.08421	0.542103	−	0.840312i	$-0.317628\pi$
$739$	29.4737	1.08421	0.542103	+	0.840312i	$0.317628\pi$
$740$	3.10044	0.113975
$741$	0	0
$742$	0	0
$743$	15.8414	0.581166	0.290583	−	0.956850i	$-0.406151\pi$
$743$	15.8414	0.581166	0.290583	+	0.956850i	$0.406151\pi$
$744$	0	0
$745$	5.13505	0.188134
$746$	−16.6468	−0.609483
$747$	0	0
$748$	−9.72161	−0.355457
$749$	0	0
$750$	0	0
$751$	44.1919	1.61259	0.806293	−	0.591516i	$-0.201470\pi$
$751$	44.1919	1.61259	0.806293	+	0.591516i	$0.201470\pi$
$752$	14.5735	0.531439
$753$	0	0
$754$	−1.09148	−0.0397494
$755$	−14.0900	−0.512789
$756$	0	0
$757$	44.1711	1.60543	0.802713	−	0.596366i	$-0.203389\pi$
$757$	44.1711	1.60543	0.802713	+	0.596366i	$0.203389\pi$
$758$	7.83372	0.284533
$759$	0	0
$760$	1.75140	0.0635299
$761$	41.0361	1.48756	0.743778	−	0.668427i	$-0.233032\pi$
$761$	41.0361	1.48756	0.743778	+	0.668427i	$0.233032\pi$
$762$	0	0
$763$	0	0
$764$	24.4793	0.885629
$765$	0	0
$766$	−27.1440	−0.980753
$767$	5.45219	0.196867
$768$	0	0
$769$	52.8358	1.90531	0.952654	−	0.304055i	$-0.0983407\pi$
$769$	52.8358	1.90531	0.952654	+	0.304055i	$0.0983407\pi$
$770$	0	0
$771$	0	0
$772$	14.5389	0.523265
$773$	−32.0907	−1.15422	−0.577111	−	0.816666i	$-0.695820\pi$
$773$	−32.0907	−1.15422	−0.577111	+	0.816666i	$0.695820\pi$
$774$	0	0
$775$	15.5816	0.559709
$776$	12.2396	0.439377
$777$	0	0
$778$	56.7160	2.03337
$779$	−6.34760	−0.227426
$780$	0	0
$781$	15.9404	0.570393
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.1496	0.790553
$786$	0	0
$787$	2.37738	0.0847446	0.0423723	−	0.999102i	$-0.486508\pi$
$787$	2.37738	0.0847446	0.0423723	+	0.999102i	$0.486508\pi$
$788$	−23.1828	−0.825852
$789$	0	0
$790$	−8.47928	−0.301679
$791$	0	0
$792$	0	0
$793$	−1.27569	−0.0453011
$794$	−20.2909	−0.720098
$795$	0	0
$796$	9.34490	0.331221
$797$	20.5091	0.726468	0.363234	−	0.931698i	$-0.381673\pi$
$797$	20.5091	0.726468	0.363234	+	0.931698i	$0.381673\pi$
$798$	0	0
$799$	−20.2396	−0.716027
$800$	22.4343	0.793171
$801$	0	0
$802$	−23.0873	−0.815242
$803$	−1.45219	−0.0512465
$804$	0	0
$805$	0	0
$806$	−3.55408	−0.125187
$807$	0	0
$808$	16.4972	0.580370
$809$	8.85666	0.311384	0.155692	−	0.987806i	$-0.450239\pi$
$809$	8.85666	0.311384	0.155692	+	0.987806i	$0.450239\pi$
$810$	0	0
$811$	8.26943	0.290379	0.145189	−	0.989404i	$-0.453621\pi$
$811$	8.26943	0.290379	0.145189	+	0.989404i	$0.453621\pi$
$812$	0	0
$813$	0	0
$814$	2.98062	0.104471
$815$	−3.54222	−0.124078
$816$	0	0
$817$	−6.34760	−0.222074
$818$	58.3989	2.04187
$819$	0	0
$820$	9.28398	0.324211
$821$	32.3595	1.12935	0.564676	−	0.825312i	$-0.309001\pi$
$821$	32.3595	1.12935	0.564676	+	0.825312i	$0.309001\pi$
$822$	0	0
$823$	−45.2459	−1.57717	−0.788587	−	0.614924i	$-0.789187\pi$
$823$	−45.2459	−1.57717	−0.788587	+	0.614924i	$0.789187\pi$
$824$	13.8504	0.482501
$825$	0	0
$826$	0	0
$827$	−20.8954	−0.726605	−0.363302	−	0.931671i	$-0.618351\pi$
$827$	−20.8954	−0.726605	−0.363302	+	0.931671i	$0.618351\pi$
$828$	0	0
$829$	44.6952	1.55233	0.776164	−	0.630531i	$-0.217163\pi$
$829$	44.6952	1.55233	0.776164	+	0.630531i	$0.217163\pi$
$830$	−1.96125	−0.0680760
$831$	0	0
$832$	−1.30548	−0.0452593
$833$	0	0
$834$	0	0
$835$	−21.1745	−0.732773
$836$	−1.93561	−0.0669444
$837$	0	0
$838$	−56.1426	−1.93941
$839$	−0.457782	−0.0158044	−0.00790219	−	0.999969i	$-0.502515\pi$
$839$	−0.457782	−0.0158044	−0.00790219	+	0.999969i	$0.502515\pi$
$840$	0	0
$841$	−26.8302	−0.925181
$842$	−39.6066	−1.36493
$843$	0	0
$844$	−38.1592	−1.31350
$845$	16.9944	0.584625
$846$	0	0
$847$	0	0
$848$	41.3836	1.42112
$849$	0	0
$850$	−40.1801	−1.37816
$851$	0	0
$852$	0	0
$853$	26.0900	0.893306	0.446653	−	0.894707i	$-0.352616\pi$
$853$	26.0900	0.893306	0.446653	+	0.894707i	$0.352616\pi$
$854$	0	0
$855$	0	0
$856$	−5.69182	−0.194543
$857$	6.38924	0.218252	0.109126	−	0.994028i	$-0.465195\pi$
$857$	6.38924	0.218252	0.109126	+	0.994028i	$0.465195\pi$
$858$	0	0
$859$	3.26316	0.111338	0.0556688	−	0.998449i	$-0.482271\pi$
$859$	3.26316	0.111338	0.0556688	+	0.998449i	$0.482271\pi$
$860$	9.28398	0.316581
$861$	0	0
$862$	27.4057	0.933443
$863$	12.6164	0.429466	0.214733	−	0.976673i	$-0.431112\pi$
$863$	12.6164	0.429466	0.214733	+	0.976673i	$0.431112\pi$
$864$	0	0
$865$	−16.8560	−0.573121
$866$	−26.3297	−0.894719
$867$	0	0
$868$	0	0
$869$	−3.44322	−0.116803
$870$	0	0
$871$	−4.22778	−0.143253
$872$	4.14961	0.140523
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.7964	−0.769780	−0.384890	−	0.922962i	$-0.625761\pi$
$877$	−22.7964	−0.769780	−0.384890	+	0.922962i	$0.625761\pi$
$878$	−48.3822	−1.63282
$879$	0	0
$880$	−6.33382	−0.213513
$881$	−15.9881	−0.538654	−0.269327	−	0.963049i	$-0.586801\pi$
$881$	−15.9881	−0.538654	−0.269327	+	0.963049i	$0.586801\pi$
$882$	0	0
$883$	30.9162	1.04041	0.520207	−	0.854040i	$-0.325855\pi$
$883$	30.9162	1.04041	0.520207	+	0.854040i	$0.325855\pi$
$884$	3.87122	0.130203
$885$	0	0
$886$	61.3628	2.06152
$887$	27.7008	0.930101	0.465051	−	0.885284i	$-0.346036\pi$
$887$	27.7008	0.930101	0.465051	+	0.885284i	$0.346036\pi$
$888$	0	0
$889$	0	0
$890$	−26.2188	−0.878857
$891$	0	0
$892$	−26.4585	−0.885895
$893$	−4.02979	−0.134852
$894$	0	0
$895$	7.74244	0.258801
$896$	0	0
$897$	0	0
$898$	−9.23694	−0.308241
$899$	7.06517	0.235637
$900$	0	0
$901$	−57.4737	−1.91473
$902$	8.92520	0.297177
$903$	0	0
$904$	0.946021	0.0314642
$905$	22.2284	0.738899
$906$	0	0
$907$	17.2936	0.574225	0.287113	−	0.957897i	$-0.407305\pi$
$907$	17.2936	0.574225	0.287113	+	0.957897i	$0.407305\pi$
$908$	41.2161	1.36780
$909$	0	0
$910$	0	0
$911$	−8.85599	−0.293412	−0.146706	−	0.989180i	$-0.546867\pi$
$911$	−8.85599	−0.293412	−0.146706	+	0.989180i	$0.546867\pi$
$912$	0	0
$913$	−0.796415	−0.0263575
$914$	−41.3061	−1.36629
$915$	0	0
$916$	11.7008	0.386605
$917$	0	0
$918$	0	0
$919$	54.0305	1.78230	0.891150	−	0.453708i	$-0.149899\pi$
$919$	54.0305	1.78230	0.891150	+	0.453708i	$0.149899\pi$
$920$	0	0
$921$	0	0
$922$	−23.3324	−0.768411
$923$	−6.34760	−0.208934
$924$	0	0
$925$	5.20359	0.171093
$926$	32.6039	1.07143
$927$	0	0
$928$	10.1723	0.333924
$929$	20.7666	0.681331	0.340665	−	0.940185i	$-0.389348\pi$
$929$	20.7666	0.681331	0.340665	+	0.940185i	$0.389348\pi$
$930$	0	0
$931$	0	0
$932$	31.3628	1.02732
$933$	0	0
$934$	−21.9237	−0.717367
$935$	8.79641	0.287674
$936$	0	0
$937$	3.20359	0.104657	0.0523283	−	0.998630i	$-0.483336\pi$
$937$	3.20359	0.104657	0.0523283	+	0.998630i	$0.483336\pi$
$938$	0	0
$939$	0	0
$940$	5.89396	0.192240
$941$	−43.0249	−1.40257	−0.701285	−	0.712881i	$-0.747390\pi$
$941$	−43.0249	−1.40257	−0.701285	+	0.712881i	$0.747390\pi$
$942$	0	0
$943$	0	0
$944$	−65.5291	−2.13279
$945$	0	0
$946$	8.92520	0.290183
$947$	−17.5749	−0.571108	−0.285554	−	0.958363i	$-0.592178\pi$
$947$	−17.5749	−0.571108	−0.285554	+	0.958363i	$0.592178\pi$
$948$	0	0
$949$	0.578271	0.0187715
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	14.7846	0.478919	0.239459	−	0.970906i	$-0.423030\pi$
$953$	14.7846	0.478919	0.239459	+	0.970906i	$0.423030\pi$
$954$	0	0
$955$	−22.1496	−0.716744
$956$	−16.8816	−0.545991
$957$	0	0
$958$	−9.37112	−0.302767
$959$	0	0
$960$	0	0
$961$	−7.99440	−0.257884
$962$	−1.18691	−0.0382674
$963$	0	0
$964$	18.9479	0.610272
$965$	−13.1552	−0.423481
$966$	0	0
$967$	−23.1440	−0.744261	−0.372131	−	0.928180i	$-0.621373\pi$
$967$	−23.1440	−0.744261	−0.372131	+	0.928180i	$0.621373\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	30.1413	0.967779
$971$	−36.5783	−1.17385	−0.586926	−	0.809640i	$-0.699662\pi$
$971$	−36.5783	−1.17385	−0.586926	+	0.809640i	$0.699662\pi$
$972$	0	0
$973$	0	0
$974$	37.5512	1.20322
$975$	0	0
$976$	15.3324	0.490777
$977$	−54.1621	−1.73280	−0.866400	−	0.499350i	$-0.833572\pi$
$977$	−54.1621	−1.73280	−0.866400	+	0.499350i	$0.833572\pi$
$978$	0	0
$979$	−10.6468	−0.340273
$980$	0	0
$981$	0	0
$982$	47.2507	1.50783
$983$	−1.29362	−0.0412600	−0.0206300	−	0.999787i	$-0.506567\pi$
$983$	−1.29362	−0.0412600	−0.0206300	+	0.999787i	$0.506567\pi$
$984$	0	0
$985$	20.9765	0.668366
$986$	−18.2188	−0.580205
$987$	0	0
$988$	0.770774	0.0245216
$989$	0	0
$990$	0	0
$991$	−15.9523	−0.506741	−0.253371	−	0.967369i	$-0.581539\pi$
$991$	−15.9523	−0.506741	−0.253371	+	0.967369i	$0.581539\pi$
$992$	33.1232	1.05166
$993$	0	0
$994$	0	0
$995$	−8.45555	−0.268059
$996$	0	0
$997$	−23.3836	−0.740568	−0.370284	−	0.928919i	$-0.620740\pi$
$997$	−23.3836	−0.740568	−0.370284	+	0.928919i	$0.620740\pi$
$998$	27.7562	0.878608
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bm.1.3		3
3.2	odd	2		1617.2.a.q.1.1	✓	3
7.6	odd	2		4851.2.a.bl.1.3		3
21.20	even	2		1617.2.a.r.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.a.q.1.1	✓	3	3.2	odd	2
1617.2.a.r.1.1	yes	3	21.20	even	2
4851.2.a.bl.1.3		3	7.6	odd	2
4851.2.a.bm.1.3		3	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:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+1.86081 q^{2} +1.46260 q^{4} -1.32340 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q+1.86081 q^{2} +1.46260 q^{4} -1.32340 q^{5} -1.00000 q^{8} -2.46260 q^{10} -1.00000 q^{11} +0.398207 q^{13} -4.78600 q^{16} +6.64681 q^{17} +1.32340 q^{19} -1.93561 q^{20} -1.86081 q^{22} -3.24860 q^{25} +0.740987 q^{26} -1.47301 q^{29} -4.79641 q^{31} -6.90582 q^{32} +12.3684 q^{34} -1.60179 q^{37} +2.46260 q^{38} +1.32340 q^{40} -4.79641 q^{41} -4.79641 q^{43} -1.46260 q^{44} -3.04502 q^{47} -6.04502 q^{50} +0.582418 q^{52} -8.64681 q^{53} +1.32340 q^{55} -2.74099 q^{58} +13.6918 q^{59} -3.20359 q^{61} -8.92520 q^{62} -3.27839 q^{64} -0.526989 q^{65} -10.6170 q^{67} +9.72161 q^{68} -15.9404 q^{71} +1.45219 q^{73} -2.98062 q^{74} +1.93561 q^{76} +3.44322 q^{79} +6.33382 q^{80} -8.92520 q^{82} +0.796415 q^{83} -8.79641 q^{85} -8.92520 q^{86} +1.00000 q^{88} +10.6468 q^{89} -5.66618 q^{94} -1.75140 q^{95} -12.2396 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} + 4 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 + 4 * q^5 - 3 * q^8	\( 3 q + 2 q^{4} + 4 q^{5} - 3 q^{8} - 5 q^{10} - 3 q^{11} - 2 q^{13} - 4 q^{16} + 4 q^{17} - 4 q^{19} - 5 q^{20} + 3 q^{25} + 11 q^{26} - 6 q^{29} - 8 q^{31} + 4 q^{32} + 10 q^{34} - 8 q^{37} + 5 q^{38} - 4 q^{40} - 8 q^{41} - 8 q^{43} - 2 q^{44} + 10 q^{47} + q^{50} - 15 q^{52} - 10 q^{53} - 4 q^{55} - 17 q^{58} + 6 q^{59} - 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} - 2 q^{73} + 11 q^{74} + 5 q^{76} - 12 q^{79} + 15 q^{80} - 22 q^{82} - 4 q^{83} - 20 q^{85} - 22 q^{86} + 3 q^{88} + 16 q^{89} - 21 q^{94} - 18 q^{95} - 8 q^{97}+O(q^{100}) \) 3 * q + 2 * q^4 + 4 * q^5 - 3 * q^8 - 5 * q^10 - 3 * q^11 - 2 * q^13 - 4 * q^16 + 4 * q^17 - 4 * q^19 - 5 * q^20 + 3 * q^25 + 11 * q^26 - 6 * q^29 - 8 * q^31 + 4 * q^32 + 10 * q^34 - 8 * q^37 + 5 * q^38 - 4 * q^40 - 8 * q^41 - 8 * q^43 - 2 * q^44 + 10 * q^47 + q^50 - 15 * q^52 - 10 * q^53 - 4 * q^55 - 17 * q^58 + 6 * q^59 - 16 * q^61 - 22 * q^62 - 21 * q^64 + 8 * q^67 + 18 * q^68 - 2 * q^73 + 11 * q^74 + 5 * q^76 - 12 * q^79 + 15 * q^80 - 22 * q^82 - 4 * q^83 - 20 * q^85 - 22 * q^86 + 3 * q^88 + 16 * q^89 - 21 * q^94 - 18 * q^95 - 8 * q^97

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