LMFDB - Embedded newform 4851.2.a.be.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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{5} +1.58579 q^{8} +O(q^{10})$	$q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{5} +1.58579 q^{8} -0.828427 q^{10} -1.00000 q^{11} +4.82843 q^{13} +3.00000 q^{16} +1.58579 q^{17} -1.24264 q^{19} -3.65685 q^{20} +0.414214 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +5.24264 q^{29} -5.65685 q^{31} -4.41421 q^{32} -0.656854 q^{34} +7.48528 q^{37} +0.514719 q^{38} +3.17157 q^{40} +6.82843 q^{41} -11.2426 q^{43} +1.82843 q^{44} -2.89949 q^{46} -4.17157 q^{47} +0.414214 q^{50} -8.82843 q^{52} +12.8284 q^{53} -2.00000 q^{55} -2.17157 q^{58} -2.65685 q^{59} -4.00000 q^{61} +2.34315 q^{62} -4.17157 q^{64} +9.65685 q^{65} -8.82843 q^{67} -2.89949 q^{68} +9.82843 q^{71} +11.6569 q^{73} -3.10051 q^{74} +2.27208 q^{76} -9.31371 q^{79} +6.00000 q^{80} -2.82843 q^{82} -2.82843 q^{83} +3.17157 q^{85} +4.65685 q^{86} -1.58579 q^{88} -14.1421 q^{89} -12.7990 q^{92} +1.72792 q^{94} -2.48528 q^{95} +5.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^10 - 2 * q^11 + 4 * q^13 + 6 * q^16 + 6 * q^17 + 6 * q^19 + 4 * q^20 - 2 * q^22 + 14 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^29 - 6 * q^32 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 12 * q^40 + 8 * q^41 - 14 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 2 * q^50 - 12 * q^52 + 20 * q^53 - 4 * q^55 - 10 * q^58 + 6 * q^59 - 8 * q^61 + 16 * q^62 - 14 * q^64 + 8 * q^65 - 12 * q^67 + 14 * q^68 + 14 * q^71 + 12 * q^73 - 26 * q^74 + 30 * q^76 + 4 * q^79 + 12 * q^80 + 12 * q^85 - 2 * q^86 - 6 * q^88 + 14 * q^92 - 22 * q^94 + 12 * q^95 - 6 * 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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.58579	0.560660
$9$	0	0
$10$	−0.828427	−0.261972
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	1.58579	0.384610	0.192305	−	0.981335i	$-0.438404\pi$
$17$	1.58579	0.384610	0.192305	+	0.981335i	$0.438404\pi$
$18$	0	0
$19$	−1.24264	−0.285081	−0.142541	−	0.989789i	$-0.545527\pi$
$19$	−1.24264	−0.285081	−0.142541	+	0.989789i	$0.545527\pi$
$20$	−3.65685	−0.817697
$21$	0	0
$22$	0.414214	0.0883106
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	5.24264	0.973534	0.486767	−	0.873532i	$-0.338176\pi$
$29$	5.24264	0.973534	0.486767	+	0.873532i	$0.338176\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	−0.656854	−0.112650
$35$	0	0
$36$	0	0
$37$	7.48528	1.23057	0.615286	−	0.788304i	$-0.289040\pi$
$37$	7.48528	1.23057	0.615286	+	0.788304i	$0.289040\pi$
$38$	0.514719	0.0834984
$39$	0	0
$40$	3.17157	0.501470
$41$	6.82843	1.06642	0.533211	−	0.845983i	$-0.320985\pi$
$41$	6.82843	1.06642	0.533211	+	0.845983i	$0.320985\pi$
$42$	0	0
$43$	−11.2426	−1.71449	−0.857243	−	0.514912i	$-0.827825\pi$
$43$	−11.2426	−1.71449	−0.857243	+	0.514912i	$0.827825\pi$
$44$	1.82843	0.275646
$45$	0	0
$46$	−2.89949	−0.427507
$47$	−4.17157	−0.608486	−0.304243	−	0.952594i	$-0.598404\pi$
$47$	−4.17157	−0.608486	−0.304243	+	0.952594i	$0.598404\pi$
$48$	0	0
$49$	0	0
$50$	0.414214	0.0585786
$51$	0	0
$52$	−8.82843	−1.22428
$53$	12.8284	1.76212	0.881060	−	0.473005i	$-0.156831\pi$
$53$	12.8284	1.76212	0.881060	+	0.473005i	$0.156831\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	−2.17157	−0.285141
$59$	−2.65685	−0.345893	−0.172946	−	0.984931i	$-0.555329\pi$
$59$	−2.65685	−0.345893	−0.172946	+	0.984931i	$0.555329\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	2.34315	0.297580
$63$	0	0
$64$	−4.17157	−0.521447
$65$	9.65685	1.19779
$66$	0	0
$67$	−8.82843	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$68$	−2.89949	−0.351615
$69$	0	0
$70$	0	0
$71$	9.82843	1.16642	0.583210	−	0.812322i	$-0.301797\pi$
$71$	9.82843	1.16642	0.583210	+	0.812322i	$0.301797\pi$
$72$	0	0
$73$	11.6569	1.36433	0.682166	−	0.731198i	$-0.261038\pi$
$73$	11.6569	1.36433	0.682166	+	0.731198i	$0.261038\pi$
$74$	−3.10051	−0.360426
$75$	0	0
$76$	2.27208	0.260625
$77$	0	0
$78$	0	0
$79$	−9.31371	−1.04787	−0.523937	−	0.851757i	$-0.675537\pi$
$79$	−9.31371	−1.04787	−0.523937	+	0.851757i	$0.675537\pi$
$80$	6.00000	0.670820
$81$	0	0
$82$	−2.82843	−0.312348
$83$	−2.82843	−0.310460	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	−2.82843	−0.310460	−0.155230	+	0.987878i	$0.549612\pi$
$84$	0	0
$85$	3.17157	0.344005
$86$	4.65685	0.502162
$87$	0	0
$88$	−1.58579	−0.169045
$89$	−14.1421	−1.49906	−0.749532	−	0.661968i	$-0.769721\pi$
$89$	−14.1421	−1.49906	−0.749532	+	0.661968i	$0.769721\pi$
$90$	0	0
$91$	0	0
$92$	−12.7990	−1.33439
$93$	0	0
$94$	1.72792	0.178222
$95$	−2.48528	−0.254984
$96$	0	0
$97$	5.48528	0.556946	0.278473	−	0.960444i	$-0.410172\pi$
$97$	5.48528	0.556946	0.278473	+	0.960444i	$0.410172\pi$
$98$	0	0
$99$	0	0
$100$	1.82843	0.182843
$101$	14.8995	1.48256	0.741278	−	0.671199i	$-0.234220\pi$
$101$	14.8995	1.48256	0.741278	+	0.671199i	$0.234220\pi$
$102$	0	0
$103$	−4.48528	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$103$	−4.48528	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$104$	7.65685	0.750816
$105$	0	0
$106$	−5.31371	−0.516113
$107$	1.65685	0.160174	0.0800871	−	0.996788i	$-0.474480\pi$
$107$	1.65685	0.160174	0.0800871	+	0.996788i	$0.474480\pi$
$108$	0	0
$109$	−1.17157	−0.112216	−0.0561082	−	0.998425i	$-0.517869\pi$
$109$	−1.17157	−0.112216	−0.0561082	+	0.998425i	$0.517869\pi$
$110$	0.828427	0.0789874
$111$	0	0
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	14.0000	1.30551
$116$	−9.58579	−0.890018
$117$	0	0
$118$	1.10051	0.101310
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.65685	0.150005
$123$	0	0
$124$	10.3431	0.928842
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−1.92893	−0.171165	−0.0855825	−	0.996331i	$-0.527275\pi$
$127$	−1.92893	−0.171165	−0.0855825	+	0.996331i	$0.527275\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	−4.00000	−0.350823
$131$	10.8284	0.946084	0.473042	−	0.881040i	$-0.343156\pi$
$131$	10.8284	0.946084	0.473042	+	0.881040i	$0.343156\pi$
$132$	0	0
$133$	0	0
$134$	3.65685	0.315904
$135$	0	0
$136$	2.51472	0.215635
$137$	16.4853	1.40843	0.704216	−	0.709985i	$-0.251299\pi$
$137$	16.4853	1.40843	0.704216	+	0.709985i	$0.251299\pi$
$138$	0	0
$139$	5.58579	0.473780	0.236890	−	0.971536i	$-0.423872\pi$
$139$	5.58579	0.473780	0.236890	+	0.971536i	$0.423872\pi$
$140$	0	0
$141$	0	0
$142$	−4.07107	−0.341636
$143$	−4.82843	−0.403773
$144$	0	0
$145$	10.4853	0.870755
$146$	−4.82843	−0.399603
$147$	0	0
$148$	−13.6863	−1.12501
$149$	−20.2132	−1.65593	−0.827965	−	0.560780i	$-0.810501\pi$
$149$	−20.2132	−1.65593	−0.827965	+	0.560780i	$0.810501\pi$
$150$	0	0
$151$	0.899495	0.0731999	0.0365999	−	0.999330i	$-0.488347\pi$
$151$	0.899495	0.0731999	0.0365999	+	0.999330i	$0.488347\pi$
$152$	−1.97056	−0.159834
$153$	0	0
$154$	0	0
$155$	−11.3137	−0.908739
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	3.85786	0.306915
$159$	0	0
$160$	−8.82843	−0.697948
$161$	0	0
$162$	0	0
$163$	12.9706	1.01593	0.507966	−	0.861377i	$-0.330397\pi$
$163$	12.9706	1.01593	0.507966	+	0.861377i	$0.330397\pi$
$164$	−12.4853	−0.974937
$165$	0	0
$166$	1.17157	0.0909317
$167$	−10.8284	−0.837929	−0.418964	−	0.908003i	$-0.637607\pi$
$167$	−10.8284	−0.837929	−0.418964	+	0.908003i	$0.637607\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	−1.31371	−0.100757
$171$	0	0
$172$	20.5563	1.56741
$173$	9.17157	0.697302	0.348651	−	0.937253i	$-0.386640\pi$
$173$	9.17157	0.697302	0.348651	+	0.937253i	$0.386640\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	5.85786	0.439065
$179$	−12.1716	−0.909746	−0.454873	−	0.890556i	$-0.650315\pi$
$179$	−12.1716	−0.909746	−0.454873	+	0.890556i	$0.650315\pi$
$180$	0	0
$181$	−0.343146	−0.0255058	−0.0127529	−	0.999919i	$-0.504059\pi$
$181$	−0.343146	−0.0255058	−0.0127529	+	0.999919i	$0.504059\pi$
$182$	0	0
$183$	0	0
$184$	11.1005	0.818340
$185$	14.9706	1.10066
$186$	0	0
$187$	−1.58579	−0.115964
$188$	7.62742	0.556287
$189$	0	0
$190$	1.02944	0.0746832
$191$	10.3431	0.748404	0.374202	−	0.927347i	$-0.377917\pi$
$191$	10.3431	0.748404	0.374202	+	0.927347i	$0.377917\pi$
$192$	0	0
$193$	16.1421	1.16194	0.580968	−	0.813926i	$-0.302674\pi$
$193$	16.1421	1.16194	0.580968	+	0.813926i	$0.302674\pi$
$194$	−2.27208	−0.163126
$195$	0	0
$196$	0	0
$197$	13.5858	0.967947	0.483974	−	0.875083i	$-0.339193\pi$
$197$	13.5858	0.967947	0.483974	+	0.875083i	$0.339193\pi$
$198$	0	0
$199$	19.7990	1.40351	0.701757	−	0.712417i	$-0.252399\pi$
$199$	19.7990	1.40351	0.701757	+	0.712417i	$0.252399\pi$
$200$	−1.58579	−0.112132
$201$	0	0
$202$	−6.17157	−0.434230
$203$	0	0
$204$	0	0
$205$	13.6569	0.953836
$206$	1.85786	0.129444
$207$	0	0
$208$	14.4853	1.00437
$209$	1.24264	0.0859553
$210$	0	0
$211$	−18.9706	−1.30599	−0.652994	−	0.757363i	$-0.726487\pi$
$211$	−18.9706	−1.30599	−0.652994	+	0.757363i	$0.726487\pi$
$212$	−23.4558	−1.61095
$213$	0	0
$214$	−0.686292	−0.0469139
$215$	−22.4853	−1.53348
$216$	0	0
$217$	0	0
$218$	0.485281	0.0328674
$219$	0	0
$220$	3.65685	0.246545
$221$	7.65685	0.515056
$222$	0	0
$223$	10.9706	0.734643	0.367322	−	0.930094i	$-0.380275\pi$
$223$	10.9706	0.734643	0.367322	+	0.930094i	$0.380275\pi$
$224$	0	0
$225$	0	0
$226$	1.51472	0.100758
$227$	18.9706	1.25912	0.629560	−	0.776952i	$-0.283235\pi$
$227$	18.9706	1.25912	0.629560	+	0.776952i	$0.283235\pi$
$228$	0	0
$229$	−4.34315	−0.287003	−0.143502	−	0.989650i	$-0.545836\pi$
$229$	−4.34315	−0.287003	−0.143502	+	0.989650i	$0.545836\pi$
$230$	−5.79899	−0.382374
$231$	0	0
$232$	8.31371	0.545822
$233$	−7.58579	−0.496961	−0.248481	−	0.968637i	$-0.579931\pi$
$233$	−7.58579	−0.496961	−0.248481	+	0.968637i	$0.579931\pi$
$234$	0	0
$235$	−8.34315	−0.544247
$236$	4.85786	0.316220
$237$	0	0
$238$	0	0
$239$	−10.4853	−0.678236	−0.339118	−	0.940744i	$-0.610129\pi$
$239$	−10.4853	−0.678236	−0.339118	+	0.940744i	$0.610129\pi$
$240$	0	0
$241$	−15.3137	−0.986443	−0.493221	−	0.869904i	$-0.664181\pi$
$241$	−15.3137	−0.986443	−0.493221	+	0.869904i	$0.664181\pi$
$242$	−0.414214	−0.0266267
$243$	0	0
$244$	7.31371	0.468212
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	−8.97056	−0.569631
$249$	0	0
$250$	4.97056	0.314366
$251$	19.4853	1.22990	0.614950	−	0.788566i	$-0.289176\pi$
$251$	19.4853	1.22990	0.614950	+	0.788566i	$0.289176\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	0.798990	0.0501331
$255$	0	0
$256$	3.97056	0.248160
$257$	29.4558	1.83741	0.918703	−	0.394950i	$-0.129238\pi$
$257$	29.4558	1.83741	0.918703	+	0.394950i	$0.129238\pi$
$258$	0	0
$259$	0	0
$260$	−17.6569	−1.09503
$261$	0	0
$262$	−4.48528	−0.277102
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	25.6569	1.57609
$266$	0	0
$267$	0	0
$268$	16.1421	0.986038
$269$	−11.7990	−0.719397	−0.359699	−	0.933069i	$-0.617120\pi$
$269$	−11.7990	−0.719397	−0.359699	+	0.933069i	$0.617120\pi$
$270$	0	0
$271$	10.9706	0.666414	0.333207	−	0.942854i	$-0.391869\pi$
$271$	10.9706	0.666414	0.333207	+	0.942854i	$0.391869\pi$
$272$	4.75736	0.288457
$273$	0	0
$274$	−6.82843	−0.412520
$275$	1.00000	0.0603023
$276$	0	0
$277$	−14.3431	−0.861796	−0.430898	−	0.902401i	$-0.641803\pi$
$277$	−14.3431	−0.861796	−0.430898	+	0.902401i	$0.641803\pi$
$278$	−2.31371	−0.138767
$279$	0	0
$280$	0	0
$281$	−11.5858	−0.691150	−0.345575	−	0.938391i	$-0.612316\pi$
$281$	−11.5858	−0.691150	−0.345575	+	0.938391i	$0.612316\pi$
$282$	0	0
$283$	31.6569	1.88180	0.940902	−	0.338678i	$-0.109980\pi$
$283$	31.6569	1.88180	0.940902	+	0.338678i	$0.109980\pi$
$284$	−17.9706	−1.06636
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	−14.4853	−0.852075
$290$	−4.34315	−0.255038
$291$	0	0
$292$	−21.3137	−1.24729
$293$	19.7279	1.15252	0.576259	−	0.817267i	$-0.304512\pi$
$293$	19.7279	1.15252	0.576259	+	0.817267i	$0.304512\pi$
$294$	0	0
$295$	−5.31371	−0.309376
$296$	11.8701	0.689933
$297$	0	0
$298$	8.37258	0.485011
$299$	33.7990	1.95465
$300$	0	0
$301$	0	0
$302$	−0.372583	−0.0214397
$303$	0	0
$304$	−3.72792	−0.213811
$305$	−8.00000	−0.458079
$306$	0	0
$307$	5.31371	0.303269	0.151635	−	0.988437i	$-0.451546\pi$
$307$	5.31371	0.303269	0.151635	+	0.988437i	$0.451546\pi$
$308$	0	0
$309$	0	0
$310$	4.68629	0.266163
$311$	15.6274	0.886150	0.443075	−	0.896485i	$-0.353888\pi$
$311$	15.6274	0.886150	0.443075	+	0.896485i	$0.353888\pi$
$312$	0	0
$313$	−34.7990	−1.96696	−0.983478	−	0.181030i	$-0.942057\pi$
$313$	−34.7990	−1.96696	−0.983478	+	0.181030i	$0.942057\pi$
$314$	−2.89949	−0.163628
$315$	0	0
$316$	17.0294	0.957981
$317$	2.68629	0.150877	0.0754386	−	0.997150i	$-0.475964\pi$
$317$	2.68629	0.150877	0.0754386	+	0.997150i	$0.475964\pi$
$318$	0	0
$319$	−5.24264	−0.293532
$320$	−8.34315	−0.466396
$321$	0	0
$322$	0	0
$323$	−1.97056	−0.109645
$324$	0	0
$325$	−4.82843	−0.267833
$326$	−5.37258	−0.297560
$327$	0	0
$328$	10.8284	0.597900
$329$	0	0
$330$	0	0
$331$	5.51472	0.303116	0.151558	−	0.988448i	$-0.451571\pi$
$331$	5.51472	0.303116	0.151558	+	0.988448i	$0.451571\pi$
$332$	5.17157	0.283827
$333$	0	0
$334$	4.48528	0.245424
$335$	−17.6569	−0.964697
$336$	0	0
$337$	32.1421	1.75089	0.875447	−	0.483314i	$-0.160567\pi$
$337$	32.1421	1.75089	0.875447	+	0.483314i	$0.160567\pi$
$338$	−4.27208	−0.232370
$339$	0	0
$340$	−5.79899	−0.314494
$341$	5.65685	0.306336
$342$	0	0
$343$	0	0
$344$	−17.8284	−0.961244
$345$	0	0
$346$	−3.79899	−0.204235
$347$	18.9706	1.01839	0.509197	−	0.860650i	$-0.329943\pi$
$347$	18.9706	1.01839	0.509197	+	0.860650i	$0.329943\pi$
$348$	0	0
$349$	−22.9706	−1.22959	−0.614793	−	0.788688i	$-0.710760\pi$
$349$	−22.9706	−1.22959	−0.614793	+	0.788688i	$0.710760\pi$
$350$	0	0
$351$	0	0
$352$	4.41421	0.235278
$353$	9.31371	0.495719	0.247859	−	0.968796i	$-0.420273\pi$
$353$	9.31371	0.495719	0.247859	+	0.968796i	$0.420273\pi$
$354$	0	0
$355$	19.6569	1.04328
$356$	25.8579	1.37046
$357$	0	0
$358$	5.04163	0.266458
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−17.4558	−0.918729
$362$	0.142136	0.00747048
$363$	0	0
$364$	0	0
$365$	23.3137	1.22030
$366$	0	0
$367$	24.1421	1.26021	0.630105	−	0.776510i	$-0.283012\pi$
$367$	24.1421	1.26021	0.630105	+	0.776510i	$0.283012\pi$
$368$	21.0000	1.09470
$369$	0	0
$370$	−6.20101	−0.322375
$371$	0	0
$372$	0	0
$373$	9.65685	0.500013	0.250006	−	0.968244i	$-0.419567\pi$
$373$	9.65685	0.500013	0.250006	+	0.968244i	$0.419567\pi$
$374$	0.656854	0.0339651
$375$	0	0
$376$	−6.61522	−0.341154
$377$	25.3137	1.30372
$378$	0	0
$379$	13.1716	0.676578	0.338289	−	0.941042i	$-0.390152\pi$
$379$	13.1716	0.676578	0.338289	+	0.941042i	$0.390152\pi$
$380$	4.54416	0.233110
$381$	0	0
$382$	−4.28427	−0.219202
$383$	20.3137	1.03798	0.518991	−	0.854780i	$-0.326308\pi$
$383$	20.3137	1.03798	0.518991	+	0.854780i	$0.326308\pi$
$384$	0	0
$385$	0	0
$386$	−6.68629	−0.340323
$387$	0	0
$388$	−10.0294	−0.509168
$389$	−27.7990	−1.40946	−0.704732	−	0.709473i	$-0.748933\pi$
$389$	−27.7990	−1.40946	−0.704732	+	0.709473i	$0.748933\pi$
$390$	0	0
$391$	11.1005	0.561377
$392$	0	0
$393$	0	0
$394$	−5.62742	−0.283505
$395$	−18.6274	−0.937247
$396$	0	0
$397$	0.514719	0.0258330	0.0129165	−	0.999917i	$-0.495888\pi$
$397$	0.514719	0.0258330	0.0129165	+	0.999917i	$0.495888\pi$
$398$	−8.20101	−0.411079
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−24.4853	−1.22274	−0.611368	−	0.791346i	$-0.709381\pi$
$401$	−24.4853	−1.22274	−0.611368	+	0.791346i	$0.709381\pi$
$402$	0	0
$403$	−27.3137	−1.36059
$404$	−27.2426	−1.35537
$405$	0	0
$406$	0	0
$407$	−7.48528	−0.371032
$408$	0	0
$409$	31.7990	1.57236	0.786179	−	0.617998i	$-0.212056\pi$
$409$	31.7990	1.57236	0.786179	+	0.617998i	$0.212056\pi$
$410$	−5.65685	−0.279372
$411$	0	0
$412$	8.20101	0.404035
$413$	0	0
$414$	0	0
$415$	−5.65685	−0.277684
$416$	−21.3137	−1.04499
$417$	0	0
$418$	−0.514719	−0.0251757
$419$	24.7990	1.21151	0.605755	−	0.795651i	$-0.292871\pi$
$419$	24.7990	1.21151	0.605755	+	0.795651i	$0.292871\pi$
$420$	0	0
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	7.85786	0.382515
$423$	0	0
$424$	20.3431	0.987950
$425$	−1.58579	−0.0769219
$426$	0	0
$427$	0	0
$428$	−3.02944	−0.146433
$429$	0	0
$430$	9.31371	0.449147
$431$	26.9706	1.29913	0.649563	−	0.760308i	$-0.274952\pi$
$431$	26.9706	1.29913	0.649563	+	0.760308i	$0.274952\pi$
$432$	0	0
$433$	29.1421	1.40048	0.700241	−	0.713907i	$-0.253076\pi$
$433$	29.1421	1.40048	0.700241	+	0.713907i	$0.253076\pi$
$434$	0	0
$435$	0	0
$436$	2.14214	0.102590
$437$	−8.69848	−0.416105
$438$	0	0
$439$	11.1005	0.529798	0.264899	−	0.964276i	$-0.414661\pi$
$439$	11.1005	0.529798	0.264899	+	0.964276i	$0.414661\pi$
$440$	−3.17157	−0.151199
$441$	0	0
$442$	−3.17157	−0.150856
$443$	11.6274	0.552435	0.276218	−	0.961095i	$-0.410919\pi$
$443$	11.6274	0.552435	0.276218	+	0.961095i	$0.410919\pi$
$444$	0	0
$445$	−28.2843	−1.34080
$446$	−4.54416	−0.215172
$447$	0	0
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	−6.82843	−0.321538
$452$	6.68629	0.314497
$453$	0	0
$454$	−7.85786	−0.368788
$455$	0	0
$456$	0	0
$457$	13.1716	0.616140	0.308070	−	0.951364i	$-0.400317\pi$
$457$	13.1716	0.616140	0.308070	+	0.951364i	$0.400317\pi$
$458$	1.79899	0.0840613
$459$	0	0
$460$	−25.5980	−1.19351
$461$	−15.2426	−0.709921	−0.354960	−	0.934881i	$-0.615506\pi$
$461$	−15.2426	−0.709921	−0.354960	+	0.934881i	$0.615506\pi$
$462$	0	0
$463$	−26.6274	−1.23748	−0.618741	−	0.785595i	$-0.712357\pi$
$463$	−26.6274	−1.23748	−0.618741	+	0.785595i	$0.712357\pi$
$464$	15.7279	0.730150
$465$	0	0
$466$	3.14214	0.145557
$467$	−14.3137	−0.662359	−0.331180	−	0.943568i	$-0.607447\pi$
$467$	−14.3137	−0.662359	−0.331180	+	0.943568i	$0.607447\pi$
$468$	0	0
$469$	0	0
$470$	3.45584	0.159406
$471$	0	0
$472$	−4.21320	−0.193928
$473$	11.2426	0.516937
$474$	0	0
$475$	1.24264	0.0570163
$476$	0	0
$477$	0	0
$478$	4.34315	0.198651
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	0	0
$481$	36.1421	1.64794
$482$	6.34315	0.288922
$483$	0	0
$484$	−1.82843	−0.0831103
$485$	10.9706	0.498148
$486$	0	0
$487$	12.6274	0.572203	0.286101	−	0.958199i	$-0.407641\pi$
$487$	12.6274	0.572203	0.286101	+	0.958199i	$0.407641\pi$
$488$	−6.34315	−0.287141
$489$	0	0
$490$	0	0
$491$	18.4853	0.834229	0.417115	−	0.908854i	$-0.363041\pi$
$491$	18.4853	0.834229	0.417115	+	0.908854i	$0.363041\pi$
$492$	0	0
$493$	8.31371	0.374431
$494$	2.48528	0.111818
$495$	0	0
$496$	−16.9706	−0.762001
$497$	0	0
$498$	0	0
$499$	7.79899	0.349131	0.174565	−	0.984646i	$-0.444148\pi$
$499$	7.79899	0.349131	0.174565	+	0.984646i	$0.444148\pi$
$500$	21.9411	0.981237
$501$	0	0
$502$	−8.07107	−0.360229
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	29.7990	1.32604
$506$	2.89949	0.128898
$507$	0	0
$508$	3.52691	0.156481
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	0	0
$512$	−22.7574	−1.00574
$513$	0	0
$514$	−12.2010	−0.538163
$515$	−8.97056	−0.395290
$516$	0	0
$517$	4.17157	0.183466
$518$	0	0
$519$	0	0
$520$	15.3137	0.671551
$521$	−36.8284	−1.61348	−0.806741	−	0.590905i	$-0.798771\pi$
$521$	−36.8284	−1.61348	−0.806741	+	0.590905i	$0.798771\pi$
$522$	0	0
$523$	34.2843	1.49915	0.749573	−	0.661921i	$-0.230259\pi$
$523$	34.2843	1.49915	0.749573	+	0.661921i	$0.230259\pi$
$524$	−19.7990	−0.864923
$525$	0	0
$526$	7.45584	0.325090
$527$	−8.97056	−0.390764
$528$	0	0
$529$	26.0000	1.13043
$530$	−10.6274	−0.461625
$531$	0	0
$532$	0	0
$533$	32.9706	1.42811
$534$	0	0
$535$	3.31371	0.143264
$536$	−14.0000	−0.604708
$537$	0	0
$538$	4.88730	0.210707
$539$	0	0
$540$	0	0
$541$	8.34315	0.358700	0.179350	−	0.983785i	$-0.442601\pi$
$541$	8.34315	0.358700	0.179350	+	0.983785i	$0.442601\pi$
$542$	−4.54416	−0.195188
$543$	0	0
$544$	−7.00000	−0.300123
$545$	−2.34315	−0.100369
$546$	0	0
$547$	18.2132	0.778740	0.389370	−	0.921081i	$-0.372693\pi$
$547$	18.2132	0.778740	0.389370	+	0.921081i	$0.372693\pi$
$548$	−30.1421	−1.28761
$549$	0	0
$550$	−0.414214	−0.0176621
$551$	−6.51472	−0.277536
$552$	0	0
$553$	0	0
$554$	5.94113	0.252414
$555$	0	0
$556$	−10.2132	−0.433136
$557$	−22.5563	−0.955743	−0.477872	−	0.878430i	$-0.658592\pi$
$557$	−22.5563	−0.955743	−0.477872	+	0.878430i	$0.658592\pi$
$558$	0	0
$559$	−54.2843	−2.29598
$560$	0	0
$561$	0	0
$562$	4.79899	0.202433
$563$	−19.1716	−0.807985	−0.403993	−	0.914762i	$-0.632378\pi$
$563$	−19.1716	−0.807985	−0.403993	+	0.914762i	$0.632378\pi$
$564$	0	0
$565$	−7.31371	−0.307690
$566$	−13.1127	−0.551168
$567$	0	0
$568$	15.5858	0.653965
$569$	7.24264	0.303627	0.151814	−	0.988409i	$-0.451489\pi$
$569$	7.24264	0.303627	0.151814	+	0.988409i	$0.451489\pi$
$570$	0	0
$571$	−18.0711	−0.756251	−0.378125	−	0.925754i	$-0.623431\pi$
$571$	−18.0711	−0.756251	−0.378125	+	0.925754i	$0.623431\pi$
$572$	8.82843	0.369135
$573$	0	0
$574$	0	0
$575$	−7.00000	−0.291920
$576$	0	0
$577$	−3.65685	−0.152237	−0.0761184	−	0.997099i	$-0.524253\pi$
$577$	−3.65685	−0.152237	−0.0761184	+	0.997099i	$0.524253\pi$
$578$	6.00000	0.249567
$579$	0	0
$580$	−19.1716	−0.796056
$581$	0	0
$582$	0	0
$583$	−12.8284	−0.531299
$584$	18.4853	0.764926
$585$	0	0
$586$	−8.17157	−0.337565
$587$	−31.3137	−1.29246	−0.646228	−	0.763145i	$-0.723654\pi$
$587$	−31.3137	−1.29246	−0.646228	+	0.763145i	$0.723654\pi$
$588$	0	0
$589$	7.02944	0.289643
$590$	2.20101	0.0906142
$591$	0	0
$592$	22.4558	0.922930
$593$	47.7279	1.95995	0.979975	−	0.199118i	$-0.0638079\pi$
$593$	47.7279	1.95995	0.979975	+	0.199118i	$0.0638079\pi$
$594$	0	0
$595$	0	0
$596$	36.9584	1.51387
$597$	0	0
$598$	−14.0000	−0.572503
$599$	46.6274	1.90514	0.952572	−	0.304312i	$-0.0984267\pi$
$599$	46.6274	1.90514	0.952572	+	0.304312i	$0.0984267\pi$
$600$	0	0
$601$	2.48528	0.101377	0.0506884	−	0.998715i	$-0.483858\pi$
$601$	2.48528	0.101377	0.0506884	+	0.998715i	$0.483858\pi$
$602$	0	0
$603$	0	0
$604$	−1.64466	−0.0669203
$605$	2.00000	0.0813116
$606$	0	0
$607$	−41.3137	−1.67687	−0.838436	−	0.545000i	$-0.816530\pi$
$607$	−41.3137	−1.67687	−0.838436	+	0.545000i	$0.816530\pi$
$608$	5.48528	0.222458
$609$	0	0
$610$	3.31371	0.134168
$611$	−20.1421	−0.814864
$612$	0	0
$613$	30.8284	1.24515	0.622574	−	0.782561i	$-0.286087\pi$
$613$	30.8284	1.24515	0.622574	+	0.782561i	$0.286087\pi$
$614$	−2.20101	−0.0888255
$615$	0	0
$616$	0	0
$617$	−18.8284	−0.758004	−0.379002	−	0.925396i	$-0.623733\pi$
$617$	−18.8284	−0.758004	−0.379002	+	0.925396i	$0.623733\pi$
$618$	0	0
$619$	−10.6274	−0.427152	−0.213576	−	0.976926i	$-0.568511\pi$
$619$	−10.6274	−0.427152	−0.213576	+	0.976926i	$0.568511\pi$
$620$	20.6863	0.830781
$621$	0	0
$622$	−6.47309	−0.259547
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	14.4142	0.576108
$627$	0	0
$628$	−12.7990	−0.510735
$629$	11.8701	0.473290
$630$	0	0
$631$	−30.2843	−1.20560	−0.602799	−	0.797893i	$-0.705948\pi$
$631$	−30.2843	−1.20560	−0.602799	+	0.797893i	$0.705948\pi$
$632$	−14.7696	−0.587501
$633$	0	0
$634$	−1.11270	−0.0441909
$635$	−3.85786	−0.153095
$636$	0	0
$637$	0	0
$638$	2.17157	0.0859734
$639$	0	0
$640$	21.1127	0.834553
$641$	−4.48528	−0.177158	−0.0885790	−	0.996069i	$-0.528233\pi$
$641$	−4.48528	−0.177158	−0.0885790	+	0.996069i	$0.528233\pi$
$642$	0	0
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	0	0
$646$	0.816234	0.0321143
$647$	−3.31371	−0.130275	−0.0651377	−	0.997876i	$-0.520749\pi$
$647$	−3.31371	−0.130275	−0.0651377	+	0.997876i	$0.520749\pi$
$648$	0	0
$649$	2.65685	0.104291
$650$	2.00000	0.0784465
$651$	0	0
$652$	−23.7157	−0.928780
$653$	31.1127	1.21753	0.608767	−	0.793349i	$-0.291664\pi$
$653$	31.1127	1.21753	0.608767	+	0.793349i	$0.291664\pi$
$654$	0	0
$655$	21.6569	0.846203
$656$	20.4853	0.799816
$657$	0	0
$658$	0	0
$659$	−17.5147	−0.682277	−0.341138	−	0.940013i	$-0.610812\pi$
$659$	−17.5147	−0.682277	−0.341138	+	0.940013i	$0.610812\pi$
$660$	0	0
$661$	−21.9706	−0.854556	−0.427278	−	0.904120i	$-0.640527\pi$
$661$	−21.9706	−0.854556	−0.427278	+	0.904120i	$0.640527\pi$
$662$	−2.28427	−0.0887807
$663$	0	0
$664$	−4.48528	−0.174063
$665$	0	0
$666$	0	0
$667$	36.6985	1.42097
$668$	19.7990	0.766046
$669$	0	0
$670$	7.31371	0.282553
$671$	4.00000	0.154418
$672$	0	0
$673$	32.8284	1.26544	0.632721	−	0.774379i	$-0.281938\pi$
$673$	32.8284	1.26544	0.632721	+	0.774379i	$0.281938\pi$
$674$	−13.3137	−0.512825
$675$	0	0
$676$	−18.8579	−0.725302
$677$	−50.0122	−1.92212	−0.961062	−	0.276332i	$-0.910881\pi$
$677$	−50.0122	−1.92212	−0.961062	+	0.276332i	$0.910881\pi$
$678$	0	0
$679$	0	0
$680$	5.02944	0.192870
$681$	0	0
$682$	−2.34315	−0.0897237
$683$	−42.7990	−1.63766	−0.818829	−	0.574038i	$-0.805376\pi$
$683$	−42.7990	−1.63766	−0.818829	+	0.574038i	$0.805376\pi$
$684$	0	0
$685$	32.9706	1.25974
$686$	0	0
$687$	0	0
$688$	−33.7279	−1.28586
$689$	61.9411	2.35977
$690$	0	0
$691$	39.4558	1.50097	0.750486	−	0.660887i	$-0.229820\pi$
$691$	39.4558	1.50097	0.750486	+	0.660887i	$0.229820\pi$
$692$	−16.7696	−0.637483
$693$	0	0
$694$	−7.85786	−0.298280
$695$	11.1716	0.423762
$696$	0	0
$697$	10.8284	0.410156
$698$	9.51472	0.360137
$699$	0	0
$700$	0	0
$701$	−16.8995	−0.638285	−0.319143	−	0.947707i	$-0.603395\pi$
$701$	−16.8995	−0.638285	−0.319143	+	0.947707i	$0.603395\pi$
$702$	0	0
$703$	−9.30152	−0.350813
$704$	4.17157	0.157222
$705$	0	0
$706$	−3.85786	−0.145193
$707$	0	0
$708$	0	0
$709$	−47.7696	−1.79402	−0.897012	−	0.442007i	$-0.854267\pi$
$709$	−47.7696	−1.79402	−0.897012	+	0.442007i	$0.854267\pi$
$710$	−8.14214	−0.305569
$711$	0	0
$712$	−22.4264	−0.840465
$713$	−39.5980	−1.48296
$714$	0	0
$715$	−9.65685	−0.361146
$716$	22.2548	0.831702
$717$	0	0
$718$	−3.31371	−0.123667
$719$	−50.7990	−1.89448	−0.947241	−	0.320521i	$-0.896142\pi$
$719$	−50.7990	−1.89448	−0.947241	+	0.320521i	$0.896142\pi$
$720$	0	0
$721$	0	0
$722$	7.23045	0.269089
$723$	0	0
$724$	0.627417	0.0233178
$725$	−5.24264	−0.194707
$726$	0	0
$727$	−50.0833	−1.85749	−0.928743	−	0.370725i	$-0.879109\pi$
$727$	−50.0833	−1.85749	−0.928743	+	0.370725i	$0.879109\pi$
$728$	0	0
$729$	0	0
$730$	−9.65685	−0.357416
$731$	−17.8284	−0.659408
$732$	0	0
$733$	−1.65685	−0.0611973	−0.0305987	−	0.999532i	$-0.509741\pi$
$733$	−1.65685	−0.0611973	−0.0305987	+	0.999532i	$0.509741\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−30.8995	−1.13897
$737$	8.82843	0.325199
$738$	0	0
$739$	−28.3431	−1.04262	−0.521310	−	0.853368i	$-0.674556\pi$
$739$	−28.3431	−1.04262	−0.521310	+	0.853368i	$0.674556\pi$
$740$	−27.3726	−1.00624
$741$	0	0
$742$	0	0
$743$	5.79899	0.212744	0.106372	−	0.994326i	$-0.466077\pi$
$743$	5.79899	0.212744	0.106372	+	0.994326i	$0.466077\pi$
$744$	0	0
$745$	−40.4264	−1.48111
$746$	−4.00000	−0.146450
$747$	0	0
$748$	2.89949	0.106016
$749$	0	0
$750$	0	0
$751$	−6.68629	−0.243986	−0.121993	−	0.992531i	$-0.538929\pi$
$751$	−6.68629	−0.243986	−0.121993	+	0.992531i	$0.538929\pi$
$752$	−12.5147	−0.456365
$753$	0	0
$754$	−10.4853	−0.381851
$755$	1.79899	0.0654719
$756$	0	0
$757$	−30.3137	−1.10177	−0.550885	−	0.834581i	$-0.685710\pi$
$757$	−30.3137	−1.10177	−0.550885	+	0.834581i	$0.685710\pi$
$758$	−5.45584	−0.198165
$759$	0	0
$760$	−3.94113	−0.142960
$761$	39.7990	1.44271	0.721356	−	0.692564i	$-0.243519\pi$
$761$	39.7990	1.44271	0.721356	+	0.692564i	$0.243519\pi$
$762$	0	0
$763$	0	0
$764$	−18.9117	−0.684201
$765$	0	0
$766$	−8.41421	−0.304018
$767$	−12.8284	−0.463208
$768$	0	0
$769$	−5.79899	−0.209117	−0.104558	−	0.994519i	$-0.533343\pi$
$769$	−5.79899	−0.209117	−0.104558	+	0.994519i	$0.533343\pi$
$770$	0	0
$771$	0	0
$772$	−29.5147	−1.06226
$773$	40.3431	1.45104	0.725521	−	0.688200i	$-0.241599\pi$
$773$	40.3431	1.45104	0.725521	+	0.688200i	$0.241599\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	8.69848	0.312257
$777$	0	0
$778$	11.5147	0.412823
$779$	−8.48528	−0.304017
$780$	0	0
$781$	−9.82843	−0.351689
$782$	−4.59798	−0.164423
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−35.5269	−1.26640	−0.633199	−	0.773989i	$-0.718258\pi$
$787$	−35.5269	−1.26640	−0.633199	+	0.773989i	$0.718258\pi$
$788$	−24.8406	−0.884910
$789$	0	0
$790$	7.71573	0.274513
$791$	0	0
$792$	0	0
$793$	−19.3137	−0.685850
$794$	−0.213203	−0.00756631
$795$	0	0
$796$	−36.2010	−1.28311
$797$	−16.8284	−0.596093	−0.298047	−	0.954551i	$-0.596335\pi$
$797$	−16.8284	−0.596093	−0.298047	+	0.954551i	$0.596335\pi$
$798$	0	0
$799$	−6.61522	−0.234030
$800$	4.41421	0.156066
$801$	0	0
$802$	10.1421	0.358131
$803$	−11.6569	−0.411361
$804$	0	0
$805$	0	0
$806$	11.3137	0.398508
$807$	0	0
$808$	23.6274	0.831210
$809$	−5.17157	−0.181823	−0.0909114	−	0.995859i	$-0.528978\pi$
$809$	−5.17157	−0.181823	−0.0909114	+	0.995859i	$0.528978\pi$
$810$	0	0
$811$	−18.9706	−0.666147	−0.333073	−	0.942901i	$-0.608086\pi$
$811$	−18.9706	−0.666147	−0.333073	+	0.942901i	$0.608086\pi$
$812$	0	0
$813$	0	0
$814$	3.10051	0.108673
$815$	25.9411	0.908678
$816$	0	0
$817$	13.9706	0.488768
$818$	−13.1716	−0.460533
$819$	0	0
$820$	−24.9706	−0.872010
$821$	−33.1716	−1.15770	−0.578848	−	0.815436i	$-0.696498\pi$
$821$	−33.1716	−1.15770	−0.578848	+	0.815436i	$0.696498\pi$
$822$	0	0
$823$	−38.8284	−1.35347	−0.676737	−	0.736225i	$-0.736607\pi$
$823$	−38.8284	−1.35347	−0.676737	+	0.736225i	$0.736607\pi$
$824$	−7.11270	−0.247783
$825$	0	0
$826$	0	0
$827$	14.6863	0.510692	0.255346	−	0.966850i	$-0.417811\pi$
$827$	14.6863	0.510692	0.255346	+	0.966850i	$0.417811\pi$
$828$	0	0
$829$	−11.8284	−0.410818	−0.205409	−	0.978676i	$-0.565853\pi$
$829$	−11.8284	−0.410818	−0.205409	+	0.978676i	$0.565853\pi$
$830$	2.34315	0.0813318
$831$	0	0
$832$	−20.1421	−0.698303
$833$	0	0
$834$	0	0
$835$	−21.6569	−0.749466
$836$	−2.27208	−0.0785815
$837$	0	0
$838$	−10.2721	−0.354843
$839$	24.9706	0.862080	0.431040	−	0.902333i	$-0.358147\pi$
$839$	24.9706	0.862080	0.431040	+	0.902333i	$0.358147\pi$
$840$	0	0
$841$	−1.51472	−0.0522317
$842$	2.89949	0.0999232
$843$	0	0
$844$	34.6863	1.19395
$845$	20.6274	0.709605
$846$	0	0
$847$	0	0
$848$	38.4853	1.32159
$849$	0	0
$850$	0.656854	0.0225299
$851$	52.3970	1.79614
$852$	0	0
$853$	4.48528	0.153573	0.0767866	−	0.997048i	$-0.475534\pi$
$853$	4.48528	0.153573	0.0767866	+	0.997048i	$0.475534\pi$
$854$	0	0
$855$	0	0
$856$	2.62742	0.0898033
$857$	−26.3553	−0.900281	−0.450141	−	0.892958i	$-0.648626\pi$
$857$	−26.3553	−0.900281	−0.450141	+	0.892958i	$0.648626\pi$
$858$	0	0
$859$	−7.51472	−0.256399	−0.128199	−	0.991748i	$-0.540920\pi$
$859$	−7.51472	−0.256399	−0.128199	+	0.991748i	$0.540920\pi$
$860$	41.1127	1.40193
$861$	0	0
$862$	−11.1716	−0.380505
$863$	42.6274	1.45105	0.725527	−	0.688194i	$-0.241596\pi$
$863$	42.6274	1.45105	0.725527	+	0.688194i	$0.241596\pi$
$864$	0	0
$865$	18.3431	0.623686
$866$	−12.0711	−0.410192
$867$	0	0
$868$	0	0
$869$	9.31371	0.315946
$870$	0	0
$871$	−42.6274	−1.44437
$872$	−1.85786	−0.0629152
$873$	0	0
$874$	3.60303	0.121874
$875$	0	0
$876$	0	0
$877$	−14.4853	−0.489133	−0.244567	−	0.969632i	$-0.578646\pi$
$877$	−14.4853	−0.489133	−0.244567	+	0.969632i	$0.578646\pi$
$878$	−4.59798	−0.155174
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−39.3137	−1.32451	−0.662256	−	0.749277i	$-0.730401\pi$
$881$	−39.3137	−1.32451	−0.662256	+	0.749277i	$0.730401\pi$
$882$	0	0
$883$	28.3431	0.953823	0.476911	−	0.878951i	$-0.341756\pi$
$883$	28.3431	0.953823	0.476911	+	0.878951i	$0.341756\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	−4.81623	−0.161805
$887$	−9.31371	−0.312724	−0.156362	−	0.987700i	$-0.549977\pi$
$887$	−9.31371	−0.312724	−0.156362	+	0.987700i	$0.549977\pi$
$888$	0	0
$889$	0	0
$890$	11.7157	0.392712
$891$	0	0
$892$	−20.0589	−0.671621
$893$	5.18377	0.173468
$894$	0	0
$895$	−24.3431	−0.813702
$896$	0	0
$897$	0	0
$898$	−1.65685	−0.0552899
$899$	−29.6569	−0.989111
$900$	0	0
$901$	20.3431	0.677728
$902$	2.82843	0.0941763
$903$	0	0
$904$	−5.79899	−0.192872
$905$	−0.686292	−0.0228131
$906$	0	0
$907$	−34.4853	−1.14506	−0.572532	−	0.819882i	$-0.694039\pi$
$907$	−34.4853	−1.14506	−0.572532	+	0.819882i	$0.694039\pi$
$908$	−34.6863	−1.15111
$909$	0	0
$910$	0	0
$911$	21.4853	0.711839	0.355920	−	0.934517i	$-0.384168\pi$
$911$	21.4853	0.711839	0.355920	+	0.934517i	$0.384168\pi$
$912$	0	0
$913$	2.82843	0.0936073
$914$	−5.45584	−0.180463
$915$	0	0
$916$	7.94113	0.262382
$917$	0	0
$918$	0	0
$919$	−14.3553	−0.473539	−0.236769	−	0.971566i	$-0.576089\pi$
$919$	−14.3553	−0.473539	−0.236769	+	0.971566i	$0.576089\pi$
$920$	22.2010	0.731946
$921$	0	0
$922$	6.31371	0.207931
$923$	47.4558	1.56203
$924$	0	0
$925$	−7.48528	−0.246115
$926$	11.0294	0.362450
$927$	0	0
$928$	−23.1421	−0.759678
$929$	17.1716	0.563381	0.281691	−	0.959505i	$-0.409105\pi$
$929$	17.1716	0.563381	0.281691	+	0.959505i	$0.409105\pi$
$930$	0	0
$931$	0	0
$932$	13.8701	0.454329
$933$	0	0
$934$	5.92893	0.194001
$935$	−3.17157	−0.103722
$936$	0	0
$937$	34.1421	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$937$	34.1421	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$938$	0	0
$939$	0	0
$940$	15.2548	0.497558
$941$	−2.07107	−0.0675149	−0.0337574	−	0.999430i	$-0.510747\pi$
$941$	−2.07107	−0.0675149	−0.0337574	+	0.999430i	$0.510747\pi$
$942$	0	0
$943$	47.7990	1.55655
$944$	−7.97056	−0.259420
$945$	0	0
$946$	−4.65685	−0.151407
$947$	−20.5147	−0.666639	−0.333319	−	0.942814i	$-0.608169\pi$
$947$	−20.5147	−0.666639	−0.333319	+	0.942814i	$0.608169\pi$
$948$	0	0
$949$	56.2843	1.82706
$950$	−0.514719	−0.0166997
$951$	0	0
$952$	0	0
$953$	−14.1421	−0.458109	−0.229054	−	0.973414i	$-0.573563\pi$
$953$	−14.1421	−0.458109	−0.229054	+	0.973414i	$0.573563\pi$
$954$	0	0
$955$	20.6863	0.669393
$956$	19.1716	0.620053
$957$	0	0
$958$	5.79899	0.187357
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	−14.9706	−0.482670
$963$	0	0
$964$	28.0000	0.901819
$965$	32.2843	1.03927
$966$	0	0
$967$	27.0416	0.869600	0.434800	−	0.900527i	$-0.356819\pi$
$967$	27.0416	0.869600	0.434800	+	0.900527i	$0.356819\pi$
$968$	1.58579	0.0509691
$969$	0	0
$970$	−4.54416	−0.145904
$971$	28.9706	0.929710	0.464855	−	0.885387i	$-0.346107\pi$
$971$	28.9706	0.929710	0.464855	+	0.885387i	$0.346107\pi$
$972$	0	0
$973$	0	0
$974$	−5.23045	−0.167594
$975$	0	0
$976$	−12.0000	−0.384111
$977$	7.17157	0.229439	0.114719	−	0.993398i	$-0.463403\pi$
$977$	7.17157	0.229439	0.114719	+	0.993398i	$0.463403\pi$
$978$	0	0
$979$	14.1421	0.451985
$980$	0	0
$981$	0	0
$982$	−7.65685	−0.244340
$983$	35.9706	1.14728	0.573641	−	0.819107i	$-0.305530\pi$
$983$	35.9706	1.14728	0.573641	+	0.819107i	$0.305530\pi$
$984$	0	0
$985$	27.1716	0.865758
$986$	−3.44365	−0.109668
$987$	0	0
$988$	10.9706	0.349020
$989$	−78.6985	−2.50247
$990$	0	0
$991$	−47.3137	−1.50297	−0.751485	−	0.659750i	$-0.770662\pi$
$991$	−47.3137	−1.50297	−0.751485	+	0.659750i	$0.770662\pi$
$992$	24.9706	0.792816
$993$	0	0
$994$	0	0
$995$	39.5980	1.25534
$996$	0	0
$997$	38.1421	1.20797	0.603987	−	0.796994i	$-0.293578\pi$
$997$	38.1421	1.20797	0.603987	+	0.796994i	$0.293578\pi$
$998$	−3.23045	−0.102258
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.be.1.1		2
3.2	odd	2		1617.2.a.n.1.2		2
7.2	even	3		693.2.i.f.298.2		4
7.4	even	3		693.2.i.f.100.2		4
7.6	odd	2		4851.2.a.bd.1.1		2
21.2	odd	6		231.2.i.d.67.1	✓	4
21.11	odd	6		231.2.i.d.100.1	yes	4
21.20	even	2		1617.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.i.d.67.1	✓	4	21.2	odd	6
231.2.i.d.100.1	yes	4	21.11	odd	6
693.2.i.f.100.2		4	7.4	even	3
693.2.i.f.298.2		4	7.2	even	3
1617.2.a.m.1.2		2	21.20	even	2
1617.2.a.n.1.2		2	3.2	odd	2
4851.2.a.bd.1.1		2	7.6	odd	2
4851.2.a.be.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)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{5} +1.58579 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{5} +1.58579 q^{8} -0.828427 q^{10} -1.00000 q^{11} +4.82843 q^{13} +3.00000 q^{16} +1.58579 q^{17} -1.24264 q^{19} -3.65685 q^{20} +0.414214 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +5.24264 q^{29} -5.65685 q^{31} -4.41421 q^{32} -0.656854 q^{34} +7.48528 q^{37} +0.514719 q^{38} +3.17157 q^{40} +6.82843 q^{41} -11.2426 q^{43} +1.82843 q^{44} -2.89949 q^{46} -4.17157 q^{47} +0.414214 q^{50} -8.82843 q^{52} +12.8284 q^{53} -2.00000 q^{55} -2.17157 q^{58} -2.65685 q^{59} -4.00000 q^{61} +2.34315 q^{62} -4.17157 q^{64} +9.65685 q^{65} -8.82843 q^{67} -2.89949 q^{68} +9.82843 q^{71} +11.6569 q^{73} -3.10051 q^{74} +2.27208 q^{76} -9.31371 q^{79} +6.00000 q^{80} -2.82843 q^{82} -2.82843 q^{83} +3.17157 q^{85} +4.65685 q^{86} -1.58579 q^{88} -14.1421 q^{89} -12.7990 q^{92} +1.72792 q^{94} -2.48528 q^{95} +5.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 6 * q^8 + 4 * q^10 - 2 * q^11 + 4 * q^13 + 6 * q^16 + 6 * q^17 + 6 * q^19 + 4 * q^20 - 2 * q^22 + 14 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^29 - 6 * q^32 + 10 * q^34 - 2 * q^37 + 18 * q^38 + 12 * q^40 + 8 * q^41 - 14 * q^43 - 2 * q^44 + 14 * q^46 - 14 * q^47 - 2 * q^50 - 12 * q^52 + 20 * q^53 - 4 * q^55 - 10 * q^58 + 6 * q^59 - 8 * q^61 + 16 * q^62 - 14 * q^64 + 8 * q^65 - 12 * q^67 + 14 * q^68 + 14 * q^71 + 12 * q^73 - 26 * q^74 + 30 * q^76 + 4 * q^79 + 12 * q^80 + 12 * q^85 - 2 * q^86 - 6 * q^88 + 14 * q^92 - 22 * q^94 + 12 * q^95 - 6 * q^97

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