LMFDB - Embedded newform 4851.2.a.bh.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - 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			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.381966 q^{2} -1.85410 q^{4} -2.23607 q^{5} -1.47214 q^{8} +O(q^{10})$	$q+0.381966 q^{2} -1.85410 q^{4} -2.23607 q^{5} -1.47214 q^{8} -0.854102 q^{10} +1.00000 q^{11} -5.47214 q^{13} +3.14590 q^{16} -6.00000 q^{17} +0.236068 q^{19} +4.14590 q^{20} +0.381966 q^{22} -6.47214 q^{23} -2.09017 q^{26} +5.76393 q^{29} -0.472136 q^{31} +4.14590 q^{32} -2.29180 q^{34} -9.47214 q^{37} +0.0901699 q^{38} +3.29180 q^{40} -6.00000 q^{41} +8.47214 q^{43} -1.85410 q^{44} -2.47214 q^{46} -2.52786 q^{47} +10.1459 q^{52} -4.94427 q^{53} -2.23607 q^{55} +2.20163 q^{58} -5.94427 q^{59} +10.9443 q^{61} -0.180340 q^{62} -4.70820 q^{64} +12.2361 q^{65} -12.7082 q^{67} +11.1246 q^{68} +4.47214 q^{71} -1.00000 q^{73} -3.61803 q^{74} -0.437694 q^{76} +6.47214 q^{79} -7.03444 q^{80} -2.29180 q^{82} -3.52786 q^{83} +13.4164 q^{85} +3.23607 q^{86} -1.47214 q^{88} -13.4164 q^{89} +12.0000 q^{92} -0.965558 q^{94} -0.527864 q^{95} +10.9443 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + 5 q^{10} + 2 q^{11} - 2 q^{13} + 13 q^{16} - 12 q^{17} - 4 q^{19} + 15 q^{20} + 3 q^{22} - 4 q^{23} + 7 q^{26} + 16 q^{29} + 8 q^{31} + 15 q^{32} - 18 q^{34} - 10 q^{37} - 11 q^{38} + 20 q^{40} - 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} + 27 q^{52} + 8 q^{53} + 29 q^{58} + 6 q^{59} + 4 q^{61} + 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} - 18 q^{68} - 2 q^{73} - 5 q^{74} - 21 q^{76} + 4 q^{79} + 15 q^{80} - 18 q^{82} - 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} - 31 q^{94} - 10 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 + 5 * q^10 + 2 * q^11 - 2 * q^13 + 13 * q^16 - 12 * q^17 - 4 * q^19 + 15 * q^20 + 3 * q^22 - 4 * q^23 + 7 * q^26 + 16 * q^29 + 8 * q^31 + 15 * q^32 - 18 * q^34 - 10 * q^37 - 11 * q^38 + 20 * q^40 - 12 * q^41 + 8 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 + 27 * q^52 + 8 * q^53 + 29 * q^58 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 + 20 * q^65 - 12 * q^67 - 18 * q^68 - 2 * q^73 - 5 * q^74 - 21 * q^76 + 4 * q^79 + 15 * q^80 - 18 * q^82 - 16 * q^83 + 2 * q^86 + 6 * q^88 + 24 * q^92 - 31 * q^94 - 10 * q^95 + 4 * 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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.47214	−0.520479
$9$	0	0
$10$	−0.854102	−0.270091
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	0	0
$16$	3.14590	0.786475
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	0.236068	0.0541577	0.0270789	−	0.999633i	$-0.491379\pi$
$19$	0.236068	0.0541577	0.0270789	+	0.999633i	$0.491379\pi$
$20$	4.14590	0.927051
$21$	0	0
$22$	0.381966	0.0814354
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	0	0
$26$	−2.09017	−0.409916
$27$	0	0
$28$	0	0
$29$	5.76393	1.07034	0.535168	−	0.844746i	$-0.320248\pi$
$29$	5.76393	1.07034	0.535168	+	0.844746i	$0.320248\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	4.14590	0.732898
$33$	0	0
$34$	−2.29180	−0.393040
$35$	0	0
$36$	0	0
$37$	−9.47214	−1.55721	−0.778605	−	0.627515i	$-0.784072\pi$
$37$	−9.47214	−1.55721	−0.778605	+	0.627515i	$0.784072\pi$
$38$	0.0901699	0.0146275
$39$	0	0
$40$	3.29180	0.520479
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.47214	1.29199	0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214	1.29199	0.645994	+	0.763342i	$0.276443\pi$
$44$	−1.85410	−0.279516
$45$	0	0
$46$	−2.47214	−0.364497
$47$	−2.52786	−0.368727	−0.184363	−	0.982858i	$-0.559022\pi$
$47$	−2.52786	−0.368727	−0.184363	+	0.982858i	$0.559022\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	10.1459	1.40698
$53$	−4.94427	−0.679148	−0.339574	−	0.940579i	$-0.610283\pi$
$53$	−4.94427	−0.679148	−0.339574	+	0.940579i	$0.610283\pi$
$54$	0	0
$55$	−2.23607	−0.301511
$56$	0	0
$57$	0	0
$58$	2.20163	0.289088
$59$	−5.94427	−0.773878	−0.386939	−	0.922105i	$-0.626468\pi$
$59$	−5.94427	−0.773878	−0.386939	+	0.922105i	$0.626468\pi$
$60$	0	0
$61$	10.9443	1.40127	0.700635	−	0.713520i	$-0.252900\pi$
$61$	10.9443	1.40127	0.700635	+	0.713520i	$0.252900\pi$
$62$	−0.180340	−0.0229032
$63$	0	0
$64$	−4.70820	−0.588525
$65$	12.2361	1.51770
$66$	0	0
$67$	−12.7082	−1.55255	−0.776277	−	0.630392i	$-0.782894\pi$
$67$	−12.7082	−1.55255	−0.776277	+	0.630392i	$0.782894\pi$
$68$	11.1246	1.34906
$69$	0	0
$70$	0	0
$71$	4.47214	0.530745	0.265372	−	0.964146i	$-0.414505\pi$
$71$	4.47214	0.530745	0.265372	+	0.964146i	$0.414505\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−3.61803	−0.420588
$75$	0	0
$76$	−0.437694	−0.0502070
$77$	0	0
$78$	0	0
$79$	6.47214	0.728172	0.364086	−	0.931365i	$-0.381381\pi$
$79$	6.47214	0.728172	0.364086	+	0.931365i	$0.381381\pi$
$80$	−7.03444	−0.786475
$81$	0	0
$82$	−2.29180	−0.253087
$83$	−3.52786	−0.387233	−0.193617	−	0.981077i	$-0.562022\pi$
$83$	−3.52786	−0.387233	−0.193617	+	0.981077i	$0.562022\pi$
$84$	0	0
$85$	13.4164	1.45521
$86$	3.23607	0.348954
$87$	0	0
$88$	−1.47214	−0.156930
$89$	−13.4164	−1.42214	−0.711068	−	0.703123i	$-0.751788\pi$
$89$	−13.4164	−1.42214	−0.711068	+	0.703123i	$0.751788\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	−0.965558	−0.0995897
$95$	−0.527864	−0.0541577
$96$	0	0
$97$	10.9443	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$97$	10.9443	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.4721	−1.24102	−0.620512	−	0.784197i	$-0.713075\pi$
$101$	−12.4721	−1.24102	−0.620512	+	0.784197i	$0.713075\pi$
$102$	0	0
$103$	19.4164	1.91316	0.956578	−	0.291477i	$-0.0941468\pi$
$103$	19.4164	1.91316	0.956578	+	0.291477i	$0.0941468\pi$
$104$	8.05573	0.789929
$105$	0	0
$106$	−1.88854	−0.183432
$107$	−2.05573	−0.198735	−0.0993674	−	0.995051i	$-0.531682\pi$
$107$	−2.05573	−0.198735	−0.0993674	+	0.995051i	$0.531682\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−0.854102	−0.0814354
$111$	0	0
$112$	0	0
$113$	2.47214	0.232559	0.116279	−	0.993217i	$-0.462903\pi$
$113$	2.47214	0.232559	0.116279	+	0.993217i	$0.462903\pi$
$114$	0	0
$115$	14.4721	1.34953
$116$	−10.6869	−0.992255
$117$	0	0
$118$	−2.27051	−0.209017
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.18034	0.378470
$123$	0	0
$124$	0.875388	0.0786122
$125$	11.1803	1.00000
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−10.0902	−0.891853
$129$	0	0
$130$	4.67376	0.409916
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	−4.85410	−0.419331
$135$	0	0
$136$	8.83282	0.757408
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	1.70820	0.143349
$143$	−5.47214	−0.457603
$144$	0	0
$145$	−12.8885	−1.07034
$146$	−0.381966	−0.0316117
$147$	0	0
$148$	17.5623	1.44361
$149$	15.6525	1.28230	0.641150	−	0.767415i	$-0.278457\pi$
$149$	15.6525	1.28230	0.641150	+	0.767415i	$0.278457\pi$
$150$	0	0
$151$	10.9443	0.890632	0.445316	−	0.895373i	$-0.353091\pi$
$151$	10.9443	0.890632	0.445316	+	0.895373i	$0.353091\pi$
$152$	−0.347524	−0.0281879
$153$	0	0
$154$	0	0
$155$	1.05573	0.0847981
$156$	0	0
$157$	−23.4164	−1.86883	−0.934416	−	0.356183i	$-0.884078\pi$
$157$	−23.4164	−1.86883	−0.934416	+	0.356183i	$0.884078\pi$
$158$	2.47214	0.196673
$159$	0	0
$160$	−9.27051	−0.732898
$161$	0	0
$162$	0	0
$163$	22.1246	1.73293	0.866467	−	0.499235i	$-0.166386\pi$
$163$	22.1246	1.73293	0.866467	+	0.499235i	$0.166386\pi$
$164$	11.1246	0.868686
$165$	0	0
$166$	−1.34752	−0.104588
$167$	−9.52786	−0.737288	−0.368644	−	0.929571i	$-0.620178\pi$
$167$	−9.52786	−0.737288	−0.368644	+	0.929571i	$0.620178\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	5.12461	0.393040
$171$	0	0
$172$	−15.7082	−1.19774
$173$	−14.4721	−1.10030	−0.550148	−	0.835067i	$-0.685429\pi$
$173$	−14.4721	−1.10030	−0.550148	+	0.835067i	$0.685429\pi$
$174$	0	0
$175$	0	0
$176$	3.14590	0.237131
$177$	0	0
$178$	−5.12461	−0.384106
$179$	14.4721	1.08170	0.540849	−	0.841120i	$-0.318103\pi$
$179$	14.4721	1.08170	0.540849	+	0.841120i	$0.318103\pi$
$180$	0	0
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	0	0
$183$	0	0
$184$	9.52786	0.702403
$185$	21.1803	1.55721
$186$	0	0
$187$	−6.00000	−0.438763
$188$	4.68692	0.341829
$189$	0	0
$190$	−0.201626	−0.0146275
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−9.88854	−0.711793	−0.355896	−	0.934525i	$-0.615824\pi$
$193$	−9.88854	−0.711793	−0.355896	+	0.934525i	$0.615824\pi$
$194$	4.18034	0.300131
$195$	0	0
$196$	0	0
$197$	−0.472136	−0.0336383	−0.0168191	−	0.999859i	$-0.505354\pi$
$197$	−0.472136	−0.0336383	−0.0168191	+	0.999859i	$0.505354\pi$
$198$	0	0
$199$	15.5279	1.10074	0.550371	−	0.834921i	$-0.314486\pi$
$199$	15.5279	1.10074	0.550371	+	0.834921i	$0.314486\pi$
$200$	0	0
$201$	0	0
$202$	−4.76393	−0.335189
$203$	0	0
$204$	0	0
$205$	13.4164	0.937043
$206$	7.41641	0.516726
$207$	0	0
$208$	−17.2148	−1.19363
$209$	0.236068	0.0163292
$210$	0	0
$211$	1.05573	0.0726793	0.0363397	−	0.999339i	$-0.488430\pi$
$211$	1.05573	0.0726793	0.0363397	+	0.999339i	$0.488430\pi$
$212$	9.16718	0.629605
$213$	0	0
$214$	−0.785218	−0.0536764
$215$	−18.9443	−1.29199
$216$	0	0
$217$	0	0
$218$	2.29180	0.155220
$219$	0	0
$220$	4.14590	0.279516
$221$	32.8328	2.20857
$222$	0	0
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	0	0
$225$	0	0
$226$	0.944272	0.0628120
$227$	26.8328	1.78096	0.890478	−	0.455026i	$-0.150370\pi$
$227$	26.8328	1.78096	0.890478	+	0.455026i	$0.150370\pi$
$228$	0	0
$229$	−11.4164	−0.754417	−0.377209	−	0.926128i	$-0.623116\pi$
$229$	−11.4164	−0.754417	−0.377209	+	0.926128i	$0.623116\pi$
$230$	5.52786	0.364497
$231$	0	0
$232$	−8.48529	−0.557087
$233$	13.4164	0.878938	0.439469	−	0.898258i	$-0.355167\pi$
$233$	13.4164	0.878938	0.439469	+	0.898258i	$0.355167\pi$
$234$	0	0
$235$	5.65248	0.368727
$236$	11.0213	0.717425
$237$	0	0
$238$	0	0
$239$	27.3607	1.76982	0.884908	−	0.465767i	$-0.154221\pi$
$239$	27.3607	1.76982	0.884908	+	0.465767i	$0.154221\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	0.381966	0.0245537
$243$	0	0
$244$	−20.2918	−1.29905
$245$	0	0
$246$	0	0
$247$	−1.29180	−0.0821950
$248$	0.695048	0.0441356
$249$	0	0
$250$	4.27051	0.270091
$251$	13.9443	0.880155	0.440077	−	0.897960i	$-0.354951\pi$
$251$	13.9443	0.880155	0.440077	+	0.897960i	$0.354951\pi$
$252$	0	0
$253$	−6.47214	−0.406900
$254$	6.11146	0.383467
$255$	0	0
$256$	5.56231	0.347644
$257$	12.2361	0.763265	0.381632	−	0.924314i	$-0.375362\pi$
$257$	12.2361	0.763265	0.381632	+	0.924314i	$0.375362\pi$
$258$	0	0
$259$	0	0
$260$	−22.6869	−1.40698
$261$	0	0
$262$	−1.70820	−0.105533
$263$	−22.4164	−1.38225	−0.691127	−	0.722733i	$-0.742886\pi$
$263$	−22.4164	−1.38225	−0.691127	+	0.722733i	$0.742886\pi$
$264$	0	0
$265$	11.0557	0.679148
$266$	0	0
$267$	0	0
$268$	23.5623	1.43930
$269$	7.52786	0.458982	0.229491	−	0.973311i	$-0.426294\pi$
$269$	7.52786	0.458982	0.229491	+	0.973311i	$0.426294\pi$
$270$	0	0
$271$	18.7082	1.13644	0.568221	−	0.822876i	$-0.307632\pi$
$271$	18.7082	1.13644	0.568221	+	0.822876i	$0.307632\pi$
$272$	−18.8754	−1.14449
$273$	0	0
$274$	−2.29180	−0.138452
$275$	0	0
$276$	0	0
$277$	−1.05573	−0.0634326	−0.0317163	−	0.999497i	$-0.510097\pi$
$277$	−1.05573	−0.0634326	−0.0317163	+	0.999497i	$0.510097\pi$
$278$	3.05573	0.183270
$279$	0	0
$280$	0	0
$281$	17.1803	1.02489	0.512447	−	0.858719i	$-0.328739\pi$
$281$	17.1803	1.02489	0.512447	+	0.858719i	$0.328739\pi$
$282$	0	0
$283$	−8.70820	−0.517649	−0.258824	−	0.965924i	$-0.583335\pi$
$283$	−8.70820	−0.517649	−0.258824	+	0.965924i	$0.583335\pi$
$284$	−8.29180	−0.492028
$285$	0	0
$286$	−2.09017	−0.123594
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	−4.92299	−0.289088
$291$	0	0
$292$	1.85410	0.108503
$293$	−25.8885	−1.51242	−0.756212	−	0.654326i	$-0.772952\pi$
$293$	−25.8885	−1.51242	−0.756212	+	0.654326i	$0.772952\pi$
$294$	0	0
$295$	13.2918	0.773878
$296$	13.9443	0.810494
$297$	0	0
$298$	5.97871	0.346338
$299$	35.4164	2.04818
$300$	0	0
$301$	0	0
$302$	4.18034	0.240552
$303$	0	0
$304$	0.742646	0.0425937
$305$	−24.4721	−1.40127
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	0.403252	0.0229032
$311$	−4.94427	−0.280364	−0.140182	−	0.990126i	$-0.544769\pi$
$311$	−4.94427	−0.280364	−0.140182	+	0.990126i	$0.544769\pi$
$312$	0	0
$313$	−18.3607	−1.03781	−0.518903	−	0.854833i	$-0.673660\pi$
$313$	−18.3607	−1.03781	−0.518903	+	0.854833i	$0.673660\pi$
$314$	−8.94427	−0.504754
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−9.05573	−0.508620	−0.254310	−	0.967123i	$-0.581848\pi$
$317$	−9.05573	−0.508620	−0.254310	+	0.967123i	$0.581848\pi$
$318$	0	0
$319$	5.76393	0.322718
$320$	10.5279	0.588525
$321$	0	0
$322$	0	0
$323$	−1.41641	−0.0788110
$324$	0	0
$325$	0	0
$326$	8.45085	0.468049
$327$	0	0
$328$	8.83282	0.487711
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	6.54102	0.358985
$333$	0	0
$334$	−3.63932	−0.199135
$335$	28.4164	1.55255
$336$	0	0
$337$	−15.4164	−0.839785	−0.419893	−	0.907574i	$-0.637932\pi$
$337$	−15.4164	−0.839785	−0.419893	+	0.907574i	$0.637932\pi$
$338$	6.47214	0.352038
$339$	0	0
$340$	−24.8754	−1.34906
$341$	−0.472136	−0.0255676
$342$	0	0
$343$	0	0
$344$	−12.4721	−0.672453
$345$	0	0
$346$	−5.52786	−0.297180
$347$	16.9443	0.909616	0.454808	−	0.890589i	$-0.349708\pi$
$347$	16.9443	0.909616	0.454808	+	0.890589i	$0.349708\pi$
$348$	0	0
$349$	24.4164	1.30698	0.653490	−	0.756935i	$-0.273304\pi$
$349$	24.4164	1.30698	0.653490	+	0.756935i	$0.273304\pi$
$350$	0	0
$351$	0	0
$352$	4.14590	0.220977
$353$	−0.236068	−0.0125646	−0.00628232	−	0.999980i	$-0.502000\pi$
$353$	−0.236068	−0.0125646	−0.00628232	+	0.999980i	$0.502000\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	24.8754	1.31839
$357$	0	0
$358$	5.52786	0.292157
$359$	3.05573	0.161275	0.0806376	−	0.996743i	$-0.474304\pi$
$359$	3.05573	0.161275	0.0806376	+	0.996743i	$0.474304\pi$
$360$	0	0
$361$	−18.9443	−0.997067
$362$	−3.23607	−0.170084
$363$	0	0
$364$	0	0
$365$	2.23607	0.117041
$366$	0	0
$367$	1.41641	0.0739359	0.0369679	−	0.999316i	$-0.488230\pi$
$367$	1.41641	0.0739359	0.0369679	+	0.999316i	$0.488230\pi$
$368$	−20.3607	−1.06137
$369$	0	0
$370$	8.09017	0.420588
$371$	0	0
$372$	0	0
$373$	0.944272	0.0488925	0.0244463	−	0.999701i	$-0.492218\pi$
$373$	0.944272	0.0488925	0.0244463	+	0.999701i	$0.492218\pi$
$374$	−2.29180	−0.118506
$375$	0	0
$376$	3.72136	0.191914
$377$	−31.5410	−1.62445
$378$	0	0
$379$	5.65248	0.290348	0.145174	−	0.989406i	$-0.453626\pi$
$379$	5.65248	0.290348	0.145174	+	0.989406i	$0.453626\pi$
$380$	0.978714	0.0502070
$381$	0	0
$382$	−4.58359	−0.234517
$383$	8.94427	0.457031	0.228515	−	0.973540i	$-0.426613\pi$
$383$	8.94427	0.457031	0.228515	+	0.973540i	$0.426613\pi$
$384$	0	0
$385$	0	0
$386$	−3.77709	−0.192249
$387$	0	0
$388$	−20.2918	−1.03016
$389$	32.8328	1.66469	0.832345	−	0.554258i	$-0.186998\pi$
$389$	32.8328	1.66469	0.832345	+	0.554258i	$0.186998\pi$
$390$	0	0
$391$	38.8328	1.96386
$392$	0	0
$393$	0	0
$394$	−0.180340	−0.00908539
$395$	−14.4721	−0.728172
$396$	0	0
$397$	−17.4164	−0.874104	−0.437052	−	0.899436i	$-0.643978\pi$
$397$	−17.4164	−0.874104	−0.437052	+	0.899436i	$0.643978\pi$
$398$	5.93112	0.297300
$399$	0	0
$400$	0	0
$401$	−0.472136	−0.0235773	−0.0117887	−	0.999931i	$-0.503753\pi$
$401$	−0.472136	−0.0235773	−0.0117887	+	0.999931i	$0.503753\pi$
$402$	0	0
$403$	2.58359	0.128698
$404$	23.1246	1.15049
$405$	0	0
$406$	0	0
$407$	−9.47214	−0.469516
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	5.12461	0.253087
$411$	0	0
$412$	−36.0000	−1.77359
$413$	0	0
$414$	0	0
$415$	7.88854	0.387233
$416$	−22.6869	−1.11232
$417$	0	0
$418$	0.0901699	0.00441036
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	20.5279	1.00047	0.500233	−	0.865891i	$-0.333248\pi$
$421$	20.5279	1.00047	0.500233	+	0.865891i	$0.333248\pi$
$422$	0.403252	0.0196300
$423$	0	0
$424$	7.27864	0.353482
$425$	0	0
$426$	0	0
$427$	0	0
$428$	3.81153	0.184237
$429$	0	0
$430$	−7.23607	−0.348954
$431$	−27.3607	−1.31792	−0.658959	−	0.752179i	$-0.729003\pi$
$431$	−27.3607	−1.31792	−0.658959	+	0.752179i	$0.729003\pi$
$432$	0	0
$433$	−39.4164	−1.89423	−0.947116	−	0.320892i	$-0.896017\pi$
$433$	−39.4164	−1.89423	−0.947116	+	0.320892i	$0.896017\pi$
$434$	0	0
$435$	0	0
$436$	−11.1246	−0.532772
$437$	−1.52786	−0.0730876
$438$	0	0
$439$	−32.1246	−1.53322	−0.766612	−	0.642111i	$-0.778059\pi$
$439$	−32.1246	−1.53322	−0.766612	+	0.642111i	$0.778059\pi$
$440$	3.29180	0.156930
$441$	0	0
$442$	12.5410	0.596515
$443$	−26.9443	−1.28016	−0.640080	−	0.768308i	$-0.721099\pi$
$443$	−26.9443	−1.28016	−0.640080	+	0.768308i	$0.721099\pi$
$444$	0	0
$445$	30.0000	1.42214
$446$	6.87539	0.325559
$447$	0	0
$448$	0	0
$449$	−1.41641	−0.0668444	−0.0334222	−	0.999441i	$-0.510641\pi$
$449$	−1.41641	−0.0668444	−0.0334222	+	0.999441i	$0.510641\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	−4.58359	−0.215594
$453$	0	0
$454$	10.2492	0.481020
$455$	0	0
$456$	0	0
$457$	15.4164	0.721149	0.360575	−	0.932730i	$-0.382581\pi$
$457$	15.4164	0.721149	0.360575	+	0.932730i	$0.382581\pi$
$458$	−4.36068	−0.203761
$459$	0	0
$460$	−26.8328	−1.25109
$461$	−7.52786	−0.350608	−0.175304	−	0.984514i	$-0.556091\pi$
$461$	−7.52786	−0.350608	−0.175304	+	0.984514i	$0.556091\pi$
$462$	0	0
$463$	10.7082	0.497652	0.248826	−	0.968548i	$-0.419955\pi$
$463$	10.7082	0.497652	0.248826	+	0.968548i	$0.419955\pi$
$464$	18.1327	0.841791
$465$	0	0
$466$	5.12461	0.237393
$467$	29.8328	1.38050	0.690249	−	0.723572i	$-0.257501\pi$
$467$	29.8328	1.38050	0.690249	+	0.723572i	$0.257501\pi$
$468$	0	0
$469$	0	0
$470$	2.15905	0.0995897
$471$	0	0
$472$	8.75078	0.402787
$473$	8.47214	0.389549
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	10.4508	0.478011
$479$	13.8885	0.634584	0.317292	−	0.948328i	$-0.397227\pi$
$479$	13.8885	0.634584	0.317292	+	0.948328i	$0.397227\pi$
$480$	0	0
$481$	51.8328	2.36337
$482$	−7.25735	−0.330563
$483$	0	0
$484$	−1.85410	−0.0842774
$485$	−24.4721	−1.11122
$486$	0	0
$487$	13.8885	0.629350	0.314675	−	0.949199i	$-0.398104\pi$
$487$	13.8885	0.629350	0.314675	+	0.949199i	$0.398104\pi$
$488$	−16.1115	−0.729331
$489$	0	0
$490$	0	0
$491$	−24.8885	−1.12320	−0.561602	−	0.827407i	$-0.689815\pi$
$491$	−24.8885	−1.12320	−0.561602	+	0.827407i	$0.689815\pi$
$492$	0	0
$493$	−34.5836	−1.55757
$494$	−0.493422	−0.0222001
$495$	0	0
$496$	−1.48529	−0.0666916
$497$	0	0
$498$	0	0
$499$	−18.1246	−0.811369	−0.405685	−	0.914013i	$-0.632967\pi$
$499$	−18.1246	−0.811369	−0.405685	+	0.914013i	$0.632967\pi$
$500$	−20.7295	−0.927051
$501$	0	0
$502$	5.32624	0.237722
$503$	−37.4164	−1.66832	−0.834158	−	0.551526i	$-0.814046\pi$
$503$	−37.4164	−1.66832	−0.834158	+	0.551526i	$0.814046\pi$
$504$	0	0
$505$	27.8885	1.24102
$506$	−2.47214	−0.109900
$507$	0	0
$508$	−29.6656	−1.31620
$509$	−44.4721	−1.97119	−0.985596	−	0.169115i	$-0.945909\pi$
$509$	−44.4721	−1.97119	−0.985596	+	0.169115i	$0.945909\pi$
$510$	0	0
$511$	0	0
$512$	22.3050	0.985749
$513$	0	0
$514$	4.67376	0.206151
$515$	−43.4164	−1.91316
$516$	0	0
$517$	−2.52786	−0.111175
$518$	0	0
$519$	0	0
$520$	−18.0132	−0.789929
$521$	28.2361	1.23704	0.618522	−	0.785767i	$-0.287732\pi$
$521$	28.2361	1.23704	0.618522	+	0.785767i	$0.287732\pi$
$522$	0	0
$523$	16.7082	0.730599	0.365299	−	0.930890i	$-0.380967\pi$
$523$	16.7082	0.730599	0.365299	+	0.930890i	$0.380967\pi$
$524$	8.29180	0.362229
$525$	0	0
$526$	−8.56231	−0.373334
$527$	2.83282	0.123399
$528$	0	0
$529$	18.8885	0.821241
$530$	4.22291	0.183432
$531$	0	0
$532$	0	0
$533$	32.8328	1.42215
$534$	0	0
$535$	4.59675	0.198735
$536$	18.7082	0.808071
$537$	0	0
$538$	2.87539	0.123967
$539$	0	0
$540$	0	0
$541$	−5.41641	−0.232870	−0.116435	−	0.993198i	$-0.537147\pi$
$541$	−5.41641	−0.232870	−0.116435	+	0.993198i	$0.537147\pi$
$542$	7.14590	0.306943
$543$	0	0
$544$	−24.8754	−1.06652
$545$	−13.4164	−0.574696
$546$	0	0
$547$	13.4164	0.573644	0.286822	−	0.957984i	$-0.407401\pi$
$547$	13.4164	0.573644	0.286822	+	0.957984i	$0.407401\pi$
$548$	11.1246	0.475220
$549$	0	0
$550$	0	0
$551$	1.36068	0.0579669
$552$	0	0
$553$	0	0
$554$	−0.403252	−0.0171325
$555$	0	0
$556$	−14.8328	−0.629052
$557$	−32.1246	−1.36116	−0.680582	−	0.732672i	$-0.738273\pi$
$557$	−32.1246	−1.36116	−0.680582	+	0.732672i	$0.738273\pi$
$558$	0	0
$559$	−46.3607	−1.96085
$560$	0	0
$561$	0	0
$562$	6.56231	0.276814
$563$	−4.47214	−0.188478	−0.0942390	−	0.995550i	$-0.530042\pi$
$563$	−4.47214	−0.188478	−0.0942390	+	0.995550i	$0.530042\pi$
$564$	0	0
$565$	−5.52786	−0.232559
$566$	−3.32624	−0.139812
$567$	0	0
$568$	−6.58359	−0.276241
$569$	3.52786	0.147896	0.0739479	−	0.997262i	$-0.476440\pi$
$569$	3.52786	0.147896	0.0739479	+	0.997262i	$0.476440\pi$
$570$	0	0
$571$	−37.4164	−1.56583	−0.782914	−	0.622130i	$-0.786268\pi$
$571$	−37.4164	−1.56583	−0.782914	+	0.622130i	$0.786268\pi$
$572$	10.1459	0.424221
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.0557	0.710039	0.355020	−	0.934859i	$-0.384474\pi$
$577$	17.0557	0.710039	0.355020	+	0.934859i	$0.384474\pi$
$578$	7.25735	0.301866
$579$	0	0
$580$	23.8967	0.992255
$581$	0	0
$582$	0	0
$583$	−4.94427	−0.204771
$584$	1.47214	0.0609174
$585$	0	0
$586$	−9.88854	−0.408492
$587$	−31.9443	−1.31848	−0.659241	−	0.751932i	$-0.729122\pi$
$587$	−31.9443	−1.31848	−0.659241	+	0.751932i	$0.729122\pi$
$588$	0	0
$589$	−0.111456	−0.00459247
$590$	5.07701	0.209017
$591$	0	0
$592$	−29.7984	−1.22471
$593$	10.5836	0.434616	0.217308	−	0.976103i	$-0.430272\pi$
$593$	10.5836	0.434616	0.217308	+	0.976103i	$0.430272\pi$
$594$	0	0
$595$	0	0
$596$	−29.0213	−1.18876
$597$	0	0
$598$	13.5279	0.553195
$599$	37.3050	1.52424	0.762120	−	0.647436i	$-0.224159\pi$
$599$	37.3050	1.52424	0.762120	+	0.647436i	$0.224159\pi$
$600$	0	0
$601$	−47.8328	−1.95114	−0.975571	−	0.219686i	$-0.929497\pi$
$601$	−47.8328	−1.95114	−0.975571	+	0.219686i	$0.929497\pi$
$602$	0	0
$603$	0	0
$604$	−20.2918	−0.825661
$605$	−2.23607	−0.0909091
$606$	0	0
$607$	−9.29180	−0.377142	−0.188571	−	0.982060i	$-0.560386\pi$
$607$	−9.29180	−0.377142	−0.188571	+	0.982060i	$0.560386\pi$
$608$	0.978714	0.0396921
$609$	0	0
$610$	−9.34752	−0.378470
$611$	13.8328	0.559616
$612$	0	0
$613$	−7.41641	−0.299546	−0.149773	−	0.988720i	$-0.547854\pi$
$613$	−7.41641	−0.299546	−0.149773	+	0.988720i	$0.547854\pi$
$614$	6.11146	0.246638
$615$	0	0
$616$	0	0
$617$	−40.2492	−1.62037	−0.810186	−	0.586172i	$-0.800634\pi$
$617$	−40.2492	−1.62037	−0.810186	+	0.586172i	$0.800634\pi$
$618$	0	0
$619$	0.583592	0.0234565	0.0117283	−	0.999931i	$-0.496267\pi$
$619$	0.583592	0.0234565	0.0117283	+	0.999931i	$0.496267\pi$
$620$	−1.95743	−0.0786122
$621$	0	0
$622$	−1.88854	−0.0757237
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−7.01316	−0.280302
$627$	0	0
$628$	43.4164	1.73250
$629$	56.8328	2.26607
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	−9.52786	−0.378998
$633$	0	0
$634$	−3.45898	−0.137374
$635$	−35.7771	−1.41977
$636$	0	0
$637$	0	0
$638$	2.20163	0.0871632
$639$	0	0
$640$	22.5623	0.891853
$641$	−40.4721	−1.59855	−0.799277	−	0.600963i	$-0.794784\pi$
$641$	−40.4721	−1.59855	−0.799277	+	0.600963i	$0.794784\pi$
$642$	0	0
$643$	−21.4164	−0.844581	−0.422290	−	0.906461i	$-0.638774\pi$
$643$	−21.4164	−0.844581	−0.422290	+	0.906461i	$0.638774\pi$
$644$	0	0
$645$	0	0
$646$	−0.541020	−0.0212861
$647$	−22.4164	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$647$	−22.4164	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$648$	0	0
$649$	−5.94427	−0.233333
$650$	0	0
$651$	0	0
$652$	−41.0213	−1.60652
$653$	29.8885	1.16963	0.584815	−	0.811167i	$-0.301167\pi$
$653$	29.8885	1.16963	0.584815	+	0.811167i	$0.301167\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	−18.8754	−0.736960
$657$	0	0
$658$	0	0
$659$	2.88854	0.112522	0.0562608	−	0.998416i	$-0.482082\pi$
$659$	2.88854	0.112522	0.0562608	+	0.998416i	$0.482082\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	−6.11146	−0.237528
$663$	0	0
$664$	5.19350	0.201547
$665$	0	0
$666$	0	0
$667$	−37.3050	−1.44445
$668$	17.6656	0.683504
$669$	0	0
$670$	10.8541	0.419331
$671$	10.9443	0.422499
$672$	0	0
$673$	−45.4164	−1.75067	−0.875337	−	0.483513i	$-0.839360\pi$
$673$	−45.4164	−1.75067	−0.875337	+	0.483513i	$0.839360\pi$
$674$	−5.88854	−0.226818
$675$	0	0
$676$	−31.4164	−1.20832
$677$	−16.5836	−0.637359	−0.318680	−	0.947862i	$-0.603239\pi$
$677$	−16.5836	−0.637359	−0.318680	+	0.947862i	$0.603239\pi$
$678$	0	0
$679$	0	0
$680$	−19.7508	−0.757408
$681$	0	0
$682$	−0.180340	−0.00690557
$683$	−18.4721	−0.706817	−0.353408	−	0.935469i	$-0.614977\pi$
$683$	−18.4721	−0.706817	−0.353408	+	0.935469i	$0.614977\pi$
$684$	0	0
$685$	13.4164	0.512615
$686$	0	0
$687$	0	0
$688$	26.6525	1.01612
$689$	27.0557	1.03074
$690$	0	0
$691$	33.4164	1.27122	0.635610	−	0.772010i	$-0.280749\pi$
$691$	33.4164	1.27122	0.635610	+	0.772010i	$0.280749\pi$
$692$	26.8328	1.02003
$693$	0	0
$694$	6.47214	0.245679
$695$	−17.8885	−0.678551
$696$	0	0
$697$	36.0000	1.36360
$698$	9.32624	0.353003
$699$	0	0
$700$	0	0
$701$	−13.4164	−0.506731	−0.253365	−	0.967371i	$-0.581537\pi$
$701$	−13.4164	−0.506731	−0.253365	+	0.967371i	$0.581537\pi$
$702$	0	0
$703$	−2.23607	−0.0843349
$704$	−4.70820	−0.177447
$705$	0	0
$706$	−0.0901699	−0.00339359
$707$	0	0
$708$	0	0
$709$	−4.41641	−0.165862	−0.0829308	−	0.996555i	$-0.526428\pi$
$709$	−4.41641	−0.165862	−0.0829308	+	0.996555i	$0.526428\pi$
$710$	−3.81966	−0.143349
$711$	0	0
$712$	19.7508	0.740192
$713$	3.05573	0.114438
$714$	0	0
$715$	12.2361	0.457603
$716$	−26.8328	−1.00279
$717$	0	0
$718$	1.16718	0.0435589
$719$	49.3607	1.84084	0.920421	−	0.390928i	$-0.127846\pi$
$719$	49.3607	1.84084	0.920421	+	0.390928i	$0.127846\pi$
$720$	0	0
$721$	0	0
$722$	−7.23607	−0.269299
$723$	0	0
$724$	15.7082	0.583791
$725$	0	0
$726$	0	0
$727$	31.4164	1.16517	0.582585	−	0.812770i	$-0.302041\pi$
$727$	31.4164	1.16517	0.582585	+	0.812770i	$0.302041\pi$
$728$	0	0
$729$	0	0
$730$	0.854102	0.0316117
$731$	−50.8328	−1.88012
$732$	0	0
$733$	46.9443	1.73393	0.866963	−	0.498372i	$-0.166069\pi$
$733$	46.9443	1.73393	0.866963	+	0.498372i	$0.166069\pi$
$734$	0.541020	0.0199694
$735$	0	0
$736$	−26.8328	−0.989071
$737$	−12.7082	−0.468113
$738$	0	0
$739$	16.5836	0.610037	0.305019	−	0.952346i	$-0.401337\pi$
$739$	16.5836	0.610037	0.305019	+	0.952346i	$0.401337\pi$
$740$	−39.2705	−1.44361
$741$	0	0
$742$	0	0
$743$	50.3050	1.84551	0.922755	−	0.385387i	$-0.125932\pi$
$743$	50.3050	1.84551	0.922755	+	0.385387i	$0.125932\pi$
$744$	0	0
$745$	−35.0000	−1.28230
$746$	0.360680	0.0132054
$747$	0	0
$748$	11.1246	0.406756
$749$	0	0
$750$	0	0
$751$	−28.1246	−1.02628	−0.513141	−	0.858304i	$-0.671518\pi$
$751$	−28.1246	−1.02628	−0.513141	+	0.858304i	$0.671518\pi$
$752$	−7.95240	−0.289994
$753$	0	0
$754$	−12.0476	−0.438748
$755$	−24.4721	−0.890632
$756$	0	0
$757$	−38.4164	−1.39627	−0.698134	−	0.715967i	$-0.745986\pi$
$757$	−38.4164	−1.39627	−0.698134	+	0.715967i	$0.745986\pi$
$758$	2.15905	0.0784204
$759$	0	0
$760$	0.777088	0.0281879
$761$	−22.4721	−0.814614	−0.407307	−	0.913291i	$-0.633532\pi$
$761$	−22.4721	−0.814614	−0.407307	+	0.913291i	$0.633532\pi$
$762$	0	0
$763$	0	0
$764$	22.2492	0.804949
$765$	0	0
$766$	3.41641	0.123440
$767$	32.5279	1.17451
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	18.3344	0.659868
$773$	51.6525	1.85781	0.928905	−	0.370318i	$-0.120751\pi$
$773$	51.6525	1.85781	0.928905	+	0.370318i	$0.120751\pi$
$774$	0	0
$775$	0	0
$776$	−16.1115	−0.578368
$777$	0	0
$778$	12.5410	0.449617
$779$	−1.41641	−0.0507481
$780$	0	0
$781$	4.47214	0.160026
$782$	14.8328	0.530420
$783$	0	0
$784$	0	0
$785$	52.3607	1.86883
$786$	0	0
$787$	−2.12461	−0.0757342	−0.0378671	−	0.999283i	$-0.512056\pi$
$787$	−2.12461	−0.0757342	−0.0378671	+	0.999283i	$0.512056\pi$
$788$	0.875388	0.0311844
$789$	0	0
$790$	−5.52786	−0.196673
$791$	0	0
$792$	0	0
$793$	−59.8885	−2.12670
$794$	−6.65248	−0.236088
$795$	0	0
$796$	−28.7902	−1.02044
$797$	50.2361	1.77945	0.889726	−	0.456494i	$-0.150895\pi$
$797$	50.2361	1.77945	0.889726	+	0.456494i	$0.150895\pi$
$798$	0	0
$799$	15.1672	0.536576
$800$	0	0
$801$	0	0
$802$	−0.180340	−0.00636802
$803$	−1.00000	−0.0352892
$804$	0	0
$805$	0	0
$806$	0.986844	0.0347601
$807$	0	0
$808$	18.3607	0.645926
$809$	37.1803	1.30719	0.653596	−	0.756844i	$-0.273260\pi$
$809$	37.1803	1.30719	0.653596	+	0.756844i	$0.273260\pi$
$810$	0	0
$811$	39.5410	1.38847	0.694236	−	0.719747i	$-0.255742\pi$
$811$	39.5410	1.38847	0.694236	+	0.719747i	$0.255742\pi$
$812$	0	0
$813$	0	0
$814$	−3.61803	−0.126812
$815$	−49.4721	−1.73293
$816$	0	0
$817$	2.00000	0.0699711
$818$	5.34752	0.186972
$819$	0	0
$820$	−24.8754	−0.868686
$821$	−33.5410	−1.17059	−0.585295	−	0.810821i	$-0.699021\pi$
$821$	−33.5410	−1.17059	−0.585295	+	0.810821i	$0.699021\pi$
$822$	0	0
$823$	21.5410	0.750873	0.375436	−	0.926848i	$-0.377493\pi$
$823$	21.5410	0.750873	0.375436	+	0.926848i	$0.377493\pi$
$824$	−28.5836	−0.995757
$825$	0	0
$826$	0	0
$827$	−53.8328	−1.87195	−0.935975	−	0.352066i	$-0.885479\pi$
$827$	−53.8328	−1.87195	−0.935975	+	0.352066i	$0.885479\pi$
$828$	0	0
$829$	18.8328	0.654091	0.327045	−	0.945009i	$-0.393947\pi$
$829$	18.8328	0.654091	0.327045	+	0.945009i	$0.393947\pi$
$830$	3.01316	0.104588
$831$	0	0
$832$	25.7639	0.893204
$833$	0	0
$834$	0	0
$835$	21.3050	0.737288
$836$	−0.437694	−0.0151380
$837$	0	0
$838$	−1.14590	−0.0395844
$839$	29.4721	1.01749	0.508746	−	0.860917i	$-0.330109\pi$
$839$	29.4721	1.01749	0.508746	+	0.860917i	$0.330109\pi$
$840$	0	0
$841$	4.22291	0.145618
$842$	7.84095	0.270217
$843$	0	0
$844$	−1.95743	−0.0673774
$845$	−37.8885	−1.30341
$846$	0	0
$847$	0	0
$848$	−15.5542	−0.534133
$849$	0	0
$850$	0	0
$851$	61.3050	2.10151
$852$	0	0
$853$	34.7214	1.18884	0.594418	−	0.804156i	$-0.297382\pi$
$853$	34.7214	1.18884	0.594418	+	0.804156i	$0.297382\pi$
$854$	0	0
$855$	0	0
$856$	3.02631	0.103437
$857$	−38.7214	−1.32270	−0.661348	−	0.750079i	$-0.730015\pi$
$857$	−38.7214	−1.32270	−0.661348	+	0.750079i	$0.730015\pi$
$858$	0	0
$859$	−11.4164	−0.389523	−0.194761	−	0.980851i	$-0.562393\pi$
$859$	−11.4164	−0.389523	−0.194761	+	0.980851i	$0.562393\pi$
$860$	35.1246	1.19774
$861$	0	0
$862$	−10.4508	−0.355957
$863$	27.3050	0.929471	0.464736	−	0.885449i	$-0.346149\pi$
$863$	27.3050	0.929471	0.464736	+	0.885449i	$0.346149\pi$
$864$	0	0
$865$	32.3607	1.10030
$866$	−15.0557	−0.511614
$867$	0	0
$868$	0	0
$869$	6.47214	0.219552
$870$	0	0
$871$	69.5410	2.35631
$872$	−8.83282	−0.299117
$873$	0	0
$874$	−0.583592	−0.0197403
$875$	0	0
$876$	0	0
$877$	13.8885	0.468983	0.234491	−	0.972118i	$-0.424658\pi$
$877$	13.8885	0.468983	0.234491	+	0.972118i	$0.424658\pi$
$878$	−12.2705	−0.414110
$879$	0	0
$880$	−7.03444	−0.237131
$881$	−26.5967	−0.896067	−0.448034	−	0.894017i	$-0.647876\pi$
$881$	−26.5967	−0.896067	−0.448034	+	0.894017i	$0.647876\pi$
$882$	0	0
$883$	−0.236068	−0.00794432	−0.00397216	−	0.999992i	$-0.501264\pi$
$883$	−0.236068	−0.00794432	−0.00397216	+	0.999992i	$0.501264\pi$
$884$	−60.8754	−2.04746
$885$	0	0
$886$	−10.2918	−0.345760
$887$	−16.3607	−0.549338	−0.274669	−	0.961539i	$-0.588568\pi$
$887$	−16.3607	−0.549338	−0.274669	+	0.961539i	$0.588568\pi$
$888$	0	0
$889$	0	0
$890$	11.4590	0.384106
$891$	0	0
$892$	−33.3738	−1.11744
$893$	−0.596748	−0.0199694
$894$	0	0
$895$	−32.3607	−1.08170
$896$	0	0
$897$	0	0
$898$	−0.541020	−0.0180541
$899$	−2.72136	−0.0907624
$900$	0	0
$901$	29.6656	0.988305
$902$	−2.29180	−0.0763085
$903$	0	0
$904$	−3.63932	−0.121042
$905$	18.9443	0.629729
$906$	0	0
$907$	−49.8885	−1.65652	−0.828261	−	0.560343i	$-0.810669\pi$
$907$	−49.8885	−1.65652	−0.828261	+	0.560343i	$0.810669\pi$
$908$	−49.7508	−1.65104
$909$	0	0
$910$	0	0
$911$	4.58359	0.151861	0.0759306	−	0.997113i	$-0.475807\pi$
$911$	4.58359	0.151861	0.0759306	+	0.997113i	$0.475807\pi$
$912$	0	0
$913$	−3.52786	−0.116755
$914$	5.88854	0.194776
$915$	0	0
$916$	21.1672	0.699383
$917$	0	0
$918$	0	0
$919$	−33.8885	−1.11788	−0.558940	−	0.829208i	$-0.688792\pi$
$919$	−33.8885	−1.11788	−0.558940	+	0.829208i	$0.688792\pi$
$920$	−21.3050	−0.702403
$921$	0	0
$922$	−2.87539	−0.0946959
$923$	−24.4721	−0.805510
$924$	0	0
$925$	0	0
$926$	4.09017	0.134411
$927$	0	0
$928$	23.8967	0.784447
$929$	32.4853	1.06581	0.532904	−	0.846176i	$-0.321101\pi$
$929$	32.4853	1.06581	0.532904	+	0.846176i	$0.321101\pi$
$930$	0	0
$931$	0	0
$932$	−24.8754	−0.814820
$933$	0	0
$934$	11.3951	0.372860
$935$	13.4164	0.438763
$936$	0	0
$937$	−46.7214	−1.52632	−0.763160	−	0.646209i	$-0.776353\pi$
$937$	−46.7214	−1.52632	−0.763160	+	0.646209i	$0.776353\pi$
$938$	0	0
$939$	0	0
$940$	−10.4803	−0.341829
$941$	47.1935	1.53846	0.769232	−	0.638970i	$-0.220639\pi$
$941$	47.1935	1.53846	0.769232	+	0.638970i	$0.220639\pi$
$942$	0	0
$943$	38.8328	1.26457
$944$	−18.7001	−0.608636
$945$	0	0
$946$	3.23607	0.105214
$947$	−35.8885	−1.16622	−0.583110	−	0.812393i	$-0.698165\pi$
$947$	−35.8885	−1.16622	−0.583110	+	0.812393i	$0.698165\pi$
$948$	0	0
$949$	5.47214	0.177633
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56.4853	−1.82974	−0.914869	−	0.403751i	$-0.867706\pi$
$953$	−56.4853	−1.82974	−0.914869	+	0.403751i	$0.867706\pi$
$954$	0	0
$955$	26.8328	0.868290
$956$	−50.7295	−1.64071
$957$	0	0
$958$	5.30495	0.171395
$959$	0	0
$960$	0	0
$961$	−30.7771	−0.992809
$962$	19.7984	0.638325
$963$	0	0
$964$	35.2279	1.13462
$965$	22.1115	0.711793
$966$	0	0
$967$	−34.2492	−1.10138	−0.550690	−	0.834710i	$-0.685635\pi$
$967$	−34.2492	−1.10138	−0.550690	+	0.834710i	$0.685635\pi$
$968$	−1.47214	−0.0473162
$969$	0	0
$970$	−9.34752	−0.300131
$971$	10.8885	0.349430	0.174715	−	0.984619i	$-0.444100\pi$
$971$	10.8885	0.349430	0.174715	+	0.984619i	$0.444100\pi$
$972$	0	0
$973$	0	0
$974$	5.30495	0.169982
$975$	0	0
$976$	34.4296	1.10206
$977$	22.4721	0.718947	0.359474	−	0.933155i	$-0.382956\pi$
$977$	22.4721	0.718947	0.359474	+	0.933155i	$0.382956\pi$
$978$	0	0
$979$	−13.4164	−0.428790
$980$	0	0
$981$	0	0
$982$	−9.50658	−0.303367
$983$	14.8328	0.473093	0.236547	−	0.971620i	$-0.423984\pi$
$983$	14.8328	0.473093	0.236547	+	0.971620i	$0.423984\pi$
$984$	0	0
$985$	1.05573	0.0336383
$986$	−13.2098	−0.420684
$987$	0	0
$988$	2.39512	0.0761990
$989$	−54.8328	−1.74358
$990$	0	0
$991$	26.7082	0.848414	0.424207	−	0.905565i	$-0.360553\pi$
$991$	26.7082	0.848414	0.424207	+	0.905565i	$0.360553\pi$
$992$	−1.95743	−0.0621484
$993$	0	0
$994$	0	0
$995$	−34.7214	−1.10074
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−6.92299	−0.219143
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bh.1.1		2
3.2	odd	2		1617.2.a.k.1.2	✓	2
7.6	odd	2		4851.2.a.bg.1.1		2
21.20	even	2		1617.2.a.l.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.2.a.k.1.2	✓	2	3.2	odd	2
1617.2.a.l.1.2	yes	2	21.20	even	2
4851.2.a.bg.1.1		2	7.6	odd	2
4851.2.a.bh.1.1		2	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)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} -1.85410 q^{4} -2.23607 q^{5} -1.47214 q^{8} +O(q^{10})\)	\(q+0.381966 q^{2} -1.85410 q^{4} -2.23607 q^{5} -1.47214 q^{8} -0.854102 q^{10} +1.00000 q^{11} -5.47214 q^{13} +3.14590 q^{16} -6.00000 q^{17} +0.236068 q^{19} +4.14590 q^{20} +0.381966 q^{22} -6.47214 q^{23} -2.09017 q^{26} +5.76393 q^{29} -0.472136 q^{31} +4.14590 q^{32} -2.29180 q^{34} -9.47214 q^{37} +0.0901699 q^{38} +3.29180 q^{40} -6.00000 q^{41} +8.47214 q^{43} -1.85410 q^{44} -2.47214 q^{46} -2.52786 q^{47} +10.1459 q^{52} -4.94427 q^{53} -2.23607 q^{55} +2.20163 q^{58} -5.94427 q^{59} +10.9443 q^{61} -0.180340 q^{62} -4.70820 q^{64} +12.2361 q^{65} -12.7082 q^{67} +11.1246 q^{68} +4.47214 q^{71} -1.00000 q^{73} -3.61803 q^{74} -0.437694 q^{76} +6.47214 q^{79} -7.03444 q^{80} -2.29180 q^{82} -3.52786 q^{83} +13.4164 q^{85} +3.23607 q^{86} -1.47214 q^{88} -13.4164 q^{89} +12.0000 q^{92} -0.965558 q^{94} -0.527864 q^{95} +10.9443 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} + 6 q^{8} + 5 q^{10} + 2 q^{11} - 2 q^{13} + 13 q^{16} - 12 q^{17} - 4 q^{19} + 15 q^{20} + 3 q^{22} - 4 q^{23} + 7 q^{26} + 16 q^{29} + 8 q^{31} + 15 q^{32} - 18 q^{34} - 10 q^{37} - 11 q^{38} + 20 q^{40} - 12 q^{41} + 8 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} + 27 q^{52} + 8 q^{53} + 29 q^{58} + 6 q^{59} + 4 q^{61} + 22 q^{62} + 4 q^{64} + 20 q^{65} - 12 q^{67} - 18 q^{68} - 2 q^{73} - 5 q^{74} - 21 q^{76} + 4 q^{79} + 15 q^{80} - 18 q^{82} - 16 q^{83} + 2 q^{86} + 6 q^{88} + 24 q^{92} - 31 q^{94} - 10 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 + 6 * q^8 + 5 * q^10 + 2 * q^11 - 2 * q^13 + 13 * q^16 - 12 * q^17 - 4 * q^19 + 15 * q^20 + 3 * q^22 - 4 * q^23 + 7 * q^26 + 16 * q^29 + 8 * q^31 + 15 * q^32 - 18 * q^34 - 10 * q^37 - 11 * q^38 + 20 * q^40 - 12 * q^41 + 8 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 + 27 * q^52 + 8 * q^53 + 29 * q^58 + 6 * q^59 + 4 * q^61 + 22 * q^62 + 4 * q^64 + 20 * q^65 - 12 * q^67 - 18 * q^68 - 2 * q^73 - 5 * q^74 - 21 * q^76 + 4 * q^79 + 15 * q^80 - 18 * q^82 - 16 * q^83 + 2 * q^86 + 6 * q^88 + 24 * q^92 - 31 * q^94 - 10 * q^95 + 4 * q^97

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