LMFDB - Embedded newform 4851.2.a.bp.1.2

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:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
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			1.2
Root			$-0.167449$ 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)

$f(q)$	$=$	$q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +O(q^{10})$	$q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +0.637062 q^{10} -1.00000 q^{11} -3.80451 q^{13} +3.83255 q^{16} +0.334898 q^{17} -8.13941 q^{19} -7.50235 q^{20} -0.167449 q^{22} +1.66510 q^{23} +9.47431 q^{25} -0.637062 q^{26} -0.195488 q^{29} +9.94392 q^{31} +1.97196 q^{32} +0.0560785 q^{34} -4.47431 q^{37} -1.36294 q^{38} -2.53039 q^{40} -6.27882 q^{41} +2.33490 q^{43} +1.97196 q^{44} +0.278820 q^{46} -12.1394 q^{47} +1.58647 q^{50} +7.50235 q^{52} -7.94392 q^{53} -3.80451 q^{55} -0.0327344 q^{58} +3.74843 q^{59} -6.00000 q^{61} +1.66510 q^{62} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} -0.660406 q^{68} -4.66980 q^{71} -4.19549 q^{73} -0.749219 q^{74} +16.0506 q^{76} +3.33020 q^{79} +14.5810 q^{80} -1.05138 q^{82} -13.9439 q^{83} +1.27412 q^{85} +0.390977 q^{86} +0.665102 q^{88} -9.88784 q^{89} -3.28352 q^{92} -2.03273 q^{94} -30.9665 q^{95} -0.0560785 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 - 3 * q^8	$ 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 24 * 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 $

$2$	0.167449	0.118404	0.0592022	−	0.998246i	$-0.481144\pi$
$2$	0.167449	0.118404	0.0592022	+	0.998246i	$0.481144\pi$
$3$	0	0
$3$	0	0
$4$	−1.97196	−0.985980
$5$	3.80451	1.70143	0.850715	−	0.525628i	$-0.176170\pi$
$5$	3.80451	1.70143	0.850715	+	0.525628i	$0.176170\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−0.665102	−0.235149
$9$	0	0
$10$	0.637062	0.201457
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.80451	−1.05518	−0.527591	−	0.849499i	$-0.676905\pi$
$13$	−3.80451	−1.05518	−0.527591	+	0.849499i	$0.676905\pi$
$14$	0	0
$15$	0	0
$16$	3.83255	0.958138
$17$	0.334898	0.0812248	0.0406124	−	0.999175i	$-0.487069\pi$
$17$	0.334898	0.0812248	0.0406124	+	0.999175i	$0.487069\pi$
$18$	0	0
$19$	−8.13941	−1.86731	−0.933654	−	0.358175i	$-0.883399\pi$
$19$	−8.13941	−1.86731	−0.933654	+	0.358175i	$0.883399\pi$
$20$	−7.50235	−1.67758
$21$	0	0
$22$	−0.167449	−0.0357003
$23$	1.66510	0.347198	0.173599	−	0.984816i	$-0.444460\pi$
$23$	1.66510	0.347198	0.173599	+	0.984816i	$0.444460\pi$
$24$	0	0
$25$	9.47431	1.89486
$26$	−0.637062	−0.124938
$27$	0	0
$28$	0	0
$29$	−0.195488	−0.0363013	−0.0181506	−	0.999835i	$-0.505778\pi$
$29$	−0.195488	−0.0363013	−0.0181506	+	0.999835i	$0.505778\pi$
$30$	0	0
$31$	9.94392	1.78598	0.892991	−	0.450075i	$-0.148603\pi$
$31$	9.94392	1.78598	0.892991	+	0.450075i	$0.148603\pi$
$32$	1.97196	0.348597
$33$	0	0
$34$	0.0560785	0.00961738
$35$	0	0
$36$	0	0
$37$	−4.47431	−0.735572	−0.367786	−	0.929911i	$-0.619884\pi$
$37$	−4.47431	−0.735572	−0.367786	+	0.929911i	$0.619884\pi$
$38$	−1.36294	−0.221098
$39$	0	0
$40$	−2.53039	−0.400089
$41$	−6.27882	−0.980587	−0.490293	−	0.871557i	$-0.663110\pi$
$41$	−6.27882	−0.980587	−0.490293	+	0.871557i	$0.663110\pi$
$42$	0	0
$43$	2.33490	0.356069	0.178034	−	0.984024i	$-0.443026\pi$
$43$	2.33490	0.356069	0.178034	+	0.984024i	$0.443026\pi$
$44$	1.97196	0.297284
$45$	0	0
$46$	0.278820	0.0411098
$47$	−12.1394	−1.77071	−0.885357	−	0.464911i	$-0.846086\pi$
$47$	−12.1394	−1.77071	−0.885357	+	0.464911i	$0.846086\pi$
$48$	0	0
$49$	0	0
$50$	1.58647	0.224360
$51$	0	0
$52$	7.50235	1.04039
$53$	−7.94392	−1.09118	−0.545591	−	0.838052i	$-0.683695\pi$
$53$	−7.94392	−1.09118	−0.545591	+	0.838052i	$0.683695\pi$
$54$	0	0
$55$	−3.80451	−0.513000
$56$	0	0
$57$	0	0
$58$	−0.0327344	−0.00429823
$59$	3.74843	0.488004	0.244002	−	0.969775i	$-0.421540\pi$
$59$	3.74843	0.488004	0.244002	+	0.969775i	$0.421540\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	1.66510	0.211468
$63$	0	0
$64$	−7.33490	−0.916862
$65$	−14.4743	−1.79532
$66$	0	0
$67$	−0.139410	−0.0170316	−0.00851582	−	0.999964i	$-0.502711\pi$
$67$	−0.139410	−0.0170316	−0.00851582	+	0.999964i	$0.502711\pi$
$68$	−0.660406	−0.0800860
$69$	0	0
$70$	0	0
$71$	−4.66980	−0.554203	−0.277101	−	0.960841i	$-0.589374\pi$
$71$	−4.66980	−0.554203	−0.277101	+	0.960841i	$0.589374\pi$
$72$	0	0
$73$	−4.19549	−0.491045	−0.245522	−	0.969391i	$-0.578959\pi$
$73$	−4.19549	−0.491045	−0.245522	+	0.969391i	$0.578959\pi$
$74$	−0.749219	−0.0870950
$75$	0	0
$76$	16.0506	1.84113
$77$	0	0
$78$	0	0
$79$	3.33020	0.374677	0.187339	−	0.982295i	$-0.440014\pi$
$79$	3.33020	0.374677	0.187339	+	0.982295i	$0.440014\pi$
$80$	14.5810	1.63020
$81$	0	0
$82$	−1.05138	−0.116106
$83$	−13.9439	−1.53054	−0.765272	−	0.643707i	$-0.777396\pi$
$83$	−13.9439	−1.53054	−0.765272	+	0.643707i	$0.777396\pi$
$84$	0	0
$85$	1.27412	0.138198
$86$	0.390977	0.0421601
$87$	0	0
$88$	0.665102	0.0709001
$89$	−9.88784	−1.04811	−0.524055	−	0.851685i	$-0.675581\pi$
$89$	−9.88784	−1.04811	−0.524055	+	0.851685i	$0.675581\pi$
$90$	0	0
$91$	0	0
$92$	−3.28352	−0.342330
$93$	0	0
$94$	−2.03273	−0.209661
$95$	−30.9665	−3.17709
$96$	0	0
$97$	−0.0560785	−0.00569391	−0.00284695	−	0.999996i	$-0.500906\pi$
$97$	−0.0560785	−0.00569391	−0.00284695	+	0.999996i	$0.500906\pi$
$98$	0	0
$99$	0	0
$100$	−18.6830	−1.86830
$101$	−18.8831	−1.87894	−0.939472	−	0.342626i	$-0.888683\pi$
$101$	−18.8831	−1.87894	−0.939472	+	0.342626i	$0.888683\pi$
$102$	0	0
$103$	8.27882	0.815736	0.407868	−	0.913041i	$-0.366272\pi$
$103$	8.27882	0.815736	0.407868	+	0.913041i	$0.366272\pi$
$104$	2.53039	0.248125
$105$	0	0
$106$	−1.33020	−0.129201
$107$	8.13941	0.786866	0.393433	−	0.919353i	$-0.371287\pi$
$107$	8.13941	0.786866	0.393433	+	0.919353i	$0.371287\pi$
$108$	0	0
$109$	11.5529	1.10657	0.553286	−	0.832992i	$-0.313374\pi$
$109$	11.5529	1.10657	0.553286	+	0.832992i	$0.313374\pi$
$110$	−0.637062	−0.0607415
$111$	0	0
$112$	0	0
$113$	1.33020	0.125135	0.0625675	−	0.998041i	$-0.480071\pi$
$113$	1.33020	0.125135	0.0625675	+	0.998041i	$0.480071\pi$
$114$	0	0
$115$	6.33490	0.590732
$116$	0.385496	0.0357924
$117$	0	0
$118$	0.627672	0.0577819
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.00470	−0.0909608
$123$	0	0
$124$	−19.6090	−1.76094
$125$	17.0226	1.52254
$126$	0	0
$127$	14.6137	1.29676	0.648379	−	0.761318i	$-0.275447\pi$
$127$	14.6137	1.29676	0.648379	+	0.761318i	$0.275447\pi$
$128$	−5.17214	−0.457157
$129$	0	0
$130$	−2.42371	−0.212574
$131$	20.5576	1.79613	0.898065	−	0.439863i	$-0.144973\pi$
$131$	20.5576	1.79613	0.898065	+	0.439863i	$0.144973\pi$
$132$	0	0
$133$	0	0
$134$	−0.0233441	−0.00201662
$135$	0	0
$136$	−0.222741	−0.0190999
$137$	16.2227	1.38600	0.693001	−	0.720936i	$-0.256288\pi$
$137$	16.2227	1.38600	0.693001	+	0.720936i	$0.256288\pi$
$138$	0	0
$139$	−22.5482	−1.91252	−0.956259	−	0.292522i	$-0.905506\pi$
$139$	−22.5482	−1.91252	−0.956259	+	0.292522i	$0.905506\pi$
$140$	0	0
$141$	0	0
$142$	−0.781954	−0.0656201
$143$	3.80451	0.318149
$144$	0	0
$145$	−0.743738	−0.0617641
$146$	−0.702531	−0.0581419
$147$	0	0
$148$	8.82316	0.725259
$149$	−8.19549	−0.671401	−0.335700	−	0.941969i	$-0.608973\pi$
$149$	−8.19549	−0.671401	−0.335700	+	0.941969i	$0.608973\pi$
$150$	0	0
$151$	−13.2741	−1.08023	−0.540116	−	0.841590i	$-0.681620\pi$
$151$	−13.2741	−1.08023	−0.540116	+	0.841590i	$0.681620\pi$
$152$	5.41353	0.439096
$153$	0	0
$154$	0	0
$155$	37.8318	3.03872
$156$	0	0
$157$	16.9392	1.35190	0.675949	−	0.736949i	$-0.263734\pi$
$157$	16.9392	1.35190	0.675949	+	0.736949i	$0.263734\pi$
$158$	0.557640	0.0443634
$159$	0	0
$160$	7.50235	0.593113
$161$	0	0
$162$	0	0
$163$	−6.79982	−0.532603	−0.266301	−	0.963890i	$-0.585802\pi$
$163$	−6.79982	−0.532603	−0.266301	+	0.963890i	$0.585802\pi$
$164$	12.3816	0.966839
$165$	0	0
$166$	−2.33490	−0.181223
$167$	−18.2227	−1.41012	−0.705059	−	0.709149i	$-0.749080\pi$
$167$	−18.2227	−1.41012	−0.705059	+	0.709149i	$0.749080\pi$
$168$	0	0
$169$	1.47431	0.113408
$170$	0.213351	0.0163633
$171$	0	0
$172$	−4.60433	−0.351077
$173$	−1.72118	−0.130859	−0.0654294	−	0.997857i	$-0.520842\pi$
$173$	−1.72118	−0.130859	−0.0654294	+	0.997857i	$0.520842\pi$
$174$	0	0
$175$	0	0
$176$	−3.83255	−0.288889
$177$	0	0
$178$	−1.65571	−0.124101
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−0.725875	−0.0539539	−0.0269769	−	0.999636i	$-0.508588\pi$
$181$	−0.725875	−0.0539539	−0.0269769	+	0.999636i	$0.508588\pi$
$182$	0	0
$183$	0	0
$184$	−1.10746	−0.0816432
$185$	−17.0226	−1.25152
$186$	0	0
$187$	−0.334898	−0.0244902
$188$	23.9384	1.74589
$189$	0	0
$190$	−5.18531	−0.376182
$191$	−5.27412	−0.381622	−0.190811	−	0.981627i	$-0.561112\pi$
$191$	−5.27412	−0.381622	−0.190811	+	0.981627i	$0.561112\pi$
$192$	0	0
$193$	−19.8318	−1.42752	−0.713761	−	0.700390i	$-0.753010\pi$
$193$	−19.8318	−1.42752	−0.713761	+	0.700390i	$0.753010\pi$
$194$	−0.00939029	−0.000674184 0
$195$	0	0
$196$	0	0
$197$	−2.66980	−0.190215	−0.0951076	−	0.995467i	$-0.530320\pi$
$197$	−2.66980	−0.190215	−0.0951076	+	0.995467i	$0.530320\pi$
$198$	0	0
$199$	−13.5529	−0.960743	−0.480371	−	0.877065i	$-0.659498\pi$
$199$	−13.5529	−0.960743	−0.480371	+	0.877065i	$0.659498\pi$
$200$	−6.30138	−0.445575
$201$	0	0
$202$	−3.16197	−0.222475
$203$	0	0
$204$	0	0
$205$	−23.8878	−1.66840
$206$	1.38628	0.0965868
$207$	0	0
$208$	−14.5810	−1.01101
$209$	8.13941	0.563015
$210$	0	0
$211$	−4.27882	−0.294566	−0.147283	−	0.989094i	$-0.547053\pi$
$211$	−4.27882	−0.294566	−0.147283	+	0.989094i	$0.547053\pi$
$212$	15.6651	1.07588
$213$	0	0
$214$	1.36294	0.0931685
$215$	8.88315	0.605826
$216$	0	0
$217$	0	0
$218$	1.93453	0.131023
$219$	0	0
$220$	7.50235	0.505808
$221$	−1.27412	−0.0857069
$222$	0	0
$223$	10.2694	0.687692	0.343846	−	0.939026i	$-0.388270\pi$
$223$	10.2694	0.687692	0.343846	+	0.939026i	$0.388270\pi$
$224$	0	0
$225$	0	0
$226$	0.222741	0.0148165
$227$	−0.390977	−0.0259500	−0.0129750	−	0.999916i	$-0.504130\pi$
$227$	−0.390977	−0.0259500	−0.0129750	+	0.999916i	$0.504130\pi$
$228$	0	0
$229$	−7.94392	−0.524949	−0.262475	−	0.964939i	$-0.584539\pi$
$229$	−7.94392	−0.524949	−0.262475	+	0.964939i	$0.584539\pi$
$230$	1.06077	0.0699453
$231$	0	0
$232$	0.130020	0.00853621
$233$	−26.5576	−1.73985	−0.869924	−	0.493185i	$-0.835833\pi$
$233$	−26.5576	−1.73985	−0.869924	+	0.493185i	$0.835833\pi$
$234$	0	0
$235$	−46.1845	−3.01275
$236$	−7.39176	−0.481163
$237$	0	0
$238$	0	0
$239$	−10.7998	−0.698582	−0.349291	−	0.937014i	$-0.613578\pi$
$239$	−10.7998	−0.698582	−0.349291	+	0.937014i	$0.613578\pi$
$240$	0	0
$241$	12.0833	0.778356	0.389178	−	0.921163i	$-0.372759\pi$
$241$	12.0833	0.778356	0.389178	+	0.921163i	$0.372759\pi$
$242$	0.167449	0.0107640
$243$	0	0
$244$	11.8318	0.757451
$245$	0	0
$246$	0	0
$247$	30.9665	1.97035
$248$	−6.61372	−0.419972
$249$	0	0
$250$	2.85041	0.180276
$251$	4.80921	0.303554	0.151777	−	0.988415i	$-0.451500\pi$
$251$	4.80921	0.303554	0.151777	+	0.988415i	$0.451500\pi$
$252$	0	0
$253$	−1.66510	−0.104684
$254$	2.44706	0.153542
$255$	0	0
$256$	13.8037	0.862733
$257$	−16.7531	−1.04503	−0.522516	−	0.852630i	$-0.675006\pi$
$257$	−16.7531	−1.04503	−0.522516	+	0.852630i	$0.675006\pi$
$258$	0	0
$259$	0	0
$260$	28.5428	1.77015
$261$	0	0
$262$	3.44236	0.212670
$263$	12.1394	0.748548	0.374274	−	0.927318i	$-0.377892\pi$
$263$	12.1394	0.748548	0.374274	+	0.927318i	$0.377892\pi$
$264$	0	0
$265$	−30.2227	−1.85657
$266$	0	0
$267$	0	0
$268$	0.274911	0.0167929
$269$	−25.2180	−1.53757	−0.768786	−	0.639506i	$-0.779139\pi$
$269$	−25.2180	−1.53757	−0.768786	+	0.639506i	$0.779139\pi$
$270$	0	0
$271$	−4.13941	−0.251451	−0.125726	−	0.992065i	$-0.540126\pi$
$271$	−4.13941	−0.251451	−0.125726	+	0.992065i	$0.540126\pi$
$272$	1.28352	0.0778245
$273$	0	0
$274$	2.71648	0.164109
$275$	−9.47431	−0.571322
$276$	0	0
$277$	−18.2788	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$277$	−18.2788	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$278$	−3.77569	−0.226451
$279$	0	0
$280$	0	0
$281$	−6.74374	−0.402298	−0.201149	−	0.979561i	$-0.564467\pi$
$281$	−6.74374	−0.402298	−0.201149	+	0.979561i	$0.564467\pi$
$282$	0	0
$283$	−23.3575	−1.38846	−0.694228	−	0.719755i	$-0.744254\pi$
$283$	−23.3575	−1.38846	−0.694228	+	0.719755i	$0.744254\pi$
$284$	9.20866	0.546433
$285$	0	0
$286$	0.637062	0.0376703
$287$	0	0
$288$	0	0
$289$	−16.8878	−0.993403
$290$	−0.124538	−0.00731314
$291$	0	0
$292$	8.27334	0.484161
$293$	−14.1667	−0.827625	−0.413813	−	0.910362i	$-0.635803\pi$
$293$	−14.1667	−0.827625	−0.413813	+	0.910362i	$0.635803\pi$
$294$	0	0
$295$	14.2610	0.830305
$296$	2.97587	0.172969
$297$	0	0
$298$	−1.37233	−0.0794968
$299$	−6.33490	−0.366357
$300$	0	0
$301$	0	0
$302$	−2.22274	−0.127904
$303$	0	0
$304$	−31.1947	−1.78914
$305$	−22.8271	−1.30707
$306$	0	0
$307$	12.5576	0.716702	0.358351	−	0.933587i	$-0.383339\pi$
$307$	12.5576	0.716702	0.358351	+	0.933587i	$0.383339\pi$
$308$	0	0
$309$	0	0
$310$	6.33490	0.359798
$311$	9.33959	0.529600	0.264800	−	0.964303i	$-0.414694\pi$
$311$	9.33959	0.529600	0.264800	+	0.964303i	$0.414694\pi$
$312$	0	0
$313$	2.99530	0.169305	0.0846523	−	0.996411i	$-0.473022\pi$
$313$	2.99530	0.169305	0.0846523	+	0.996411i	$0.473022\pi$
$314$	2.83646	0.160071
$315$	0	0
$316$	−6.56703	−0.369424
$317$	−9.93453	−0.557979	−0.278989	−	0.960294i	$-0.589999\pi$
$317$	−9.93453	−0.557979	−0.278989	+	0.960294i	$0.589999\pi$
$318$	0	0
$319$	0.195488	0.0109453
$320$	−27.9057	−1.55998
$321$	0	0
$322$	0	0
$323$	−2.72588	−0.151672
$324$	0	0
$325$	−36.0451	−1.99942
$326$	−1.13862	−0.0630625
$327$	0	0
$328$	4.17605	0.230584
$329$	0	0
$330$	0	0
$331$	22.5482	1.23936	0.619682	−	0.784853i	$-0.287262\pi$
$331$	22.5482	1.23936	0.619682	+	0.784853i	$0.287262\pi$
$332$	27.4969	1.50909
$333$	0	0
$334$	−3.05138	−0.166964
$335$	−0.530387	−0.0289781
$336$	0	0
$337$	13.2835	0.723599	0.361800	−	0.932256i	$-0.382162\pi$
$337$	13.2835	0.723599	0.361800	+	0.932256i	$0.382162\pi$
$338$	0.246872	0.0134281
$339$	0	0
$340$	−2.51252	−0.136261
$341$	−9.94392	−0.538494
$342$	0	0
$343$	0	0
$344$	−1.55294	−0.0837292
$345$	0	0
$346$	−0.288210	−0.0154943
$347$	27.7757	1.49108	0.745538	−	0.666463i	$-0.232192\pi$
$347$	27.7757	1.49108	0.745538	+	0.666463i	$0.232192\pi$
$348$	0	0
$349$	2.85589	0.152873	0.0764363	−	0.997074i	$-0.475646\pi$
$349$	2.85589	0.152873	0.0764363	+	0.997074i	$0.475646\pi$
$350$	0	0
$351$	0	0
$352$	−1.97196	−0.105106
$353$	24.6410	1.31151	0.655753	−	0.754975i	$-0.272351\pi$
$353$	24.6410	1.31151	0.655753	+	0.754975i	$0.272351\pi$
$354$	0	0
$355$	−17.7663	−0.942937
$356$	19.4984	1.03342
$357$	0	0
$358$	2.00939	0.106200
$359$	11.8878	0.627416	0.313708	−	0.949519i	$-0.398429\pi$
$359$	11.8878	0.627416	0.313708	+	0.949519i	$0.398429\pi$
$360$	0	0
$361$	47.2500	2.48684
$362$	−0.121547	−0.00638838
$363$	0	0
$364$	0	0
$365$	−15.9618	−0.835478
$366$	0	0
$367$	−3.60902	−0.188389	−0.0941947	−	0.995554i	$-0.530028\pi$
$367$	−3.60902	−0.188389	−0.0941947	+	0.995554i	$0.530028\pi$
$368$	6.38159	0.332663
$369$	0	0
$370$	−2.85041	−0.148186
$371$	0	0
$372$	0	0
$373$	12.3349	0.638677	0.319338	−	0.947641i	$-0.396539\pi$
$373$	12.3349	0.638677	0.319338	+	0.947641i	$0.396539\pi$
$374$	−0.0560785	−0.00289975
$375$	0	0
$376$	8.07394	0.416382
$377$	0.743738	0.0383045
$378$	0	0
$379$	23.3575	1.19979	0.599896	−	0.800078i	$-0.295209\pi$
$379$	23.3575	1.19979	0.599896	+	0.800078i	$0.295209\pi$
$380$	61.0647	3.13255
$381$	0	0
$382$	−0.883148	−0.0451858
$383$	−15.2180	−0.777606	−0.388803	−	0.921321i	$-0.627111\pi$
$383$	−15.2180	−0.777606	−0.388803	+	0.921321i	$0.627111\pi$
$384$	0	0
$385$	0	0
$386$	−3.32081	−0.169025
$387$	0	0
$388$	0.110585	0.00561408
$389$	−3.73057	−0.189147	−0.0945737	−	0.995518i	$-0.530149\pi$
$389$	−3.73057	−0.189147	−0.0945737	+	0.995518i	$0.530149\pi$
$390$	0	0
$391$	0.557640	0.0282011
$392$	0	0
$393$	0	0
$394$	−0.447055	−0.0225223
$395$	12.6698	0.637487
$396$	0	0
$397$	−20.8925	−1.04857	−0.524283	−	0.851544i	$-0.675667\pi$
$397$	−20.8925	−1.04857	−0.524283	+	0.851544i	$0.675667\pi$
$398$	−2.26943	−0.113756
$399$	0	0
$400$	36.3108	1.81554
$401$	−23.2741	−1.16225	−0.581127	−	0.813813i	$-0.697388\pi$
$401$	−23.2741	−1.16225	−0.581127	+	0.813813i	$0.697388\pi$
$402$	0	0
$403$	−37.8318	−1.88453
$404$	37.2368	1.85260
$405$	0	0
$406$	0	0
$407$	4.47431	0.221783
$408$	0	0
$409$	23.8972	1.18164	0.590821	−	0.806803i	$-0.298804\pi$
$409$	23.8972	1.18164	0.590821	+	0.806803i	$0.298804\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	−16.3255	−0.804300
$413$	0	0
$414$	0	0
$415$	−53.0498	−2.60411
$416$	−7.50235	−0.367833
$417$	0	0
$418$	1.36294	0.0666635
$419$	−30.2967	−1.48009	−0.740045	−	0.672557i	$-0.765196\pi$
$419$	−30.2967	−1.48009	−0.740045	+	0.672557i	$0.765196\pi$
$420$	0	0
$421$	−15.5257	−0.756676	−0.378338	−	0.925668i	$-0.623504\pi$
$421$	−15.5257	−0.756676	−0.378338	+	0.925668i	$0.623504\pi$
$422$	−0.716485	−0.0348779
$423$	0	0
$424$	5.28352	0.256590
$425$	3.17293	0.153910
$426$	0	0
$427$	0	0
$428$	−16.0506	−0.775835
$429$	0	0
$430$	1.48748	0.0717325
$431$	−3.07864	−0.148293	−0.0741463	−	0.997247i	$-0.523623\pi$
$431$	−3.07864	−0.148293	−0.0741463	+	0.997247i	$0.523623\pi$
$432$	0	0
$433$	−16.9392	−0.814047	−0.407024	−	0.913418i	$-0.633433\pi$
$433$	−16.9392	−0.814047	−0.407024	+	0.913418i	$0.633433\pi$
$434$	0	0
$435$	0	0
$436$	−22.7820	−1.09106
$437$	−13.5529	−0.648325
$438$	0	0
$439$	11.0786	0.528754	0.264377	−	0.964419i	$-0.414834\pi$
$439$	11.0786	0.528754	0.264377	+	0.964419i	$0.414834\pi$
$440$	2.53039	0.120631
$441$	0	0
$442$	−0.213351	−0.0101481
$443$	14.5482	0.691208	0.345604	−	0.938380i	$-0.387674\pi$
$443$	14.5482	0.691208	0.345604	+	0.938380i	$0.387674\pi$
$444$	0	0
$445$	−37.6184	−1.78328
$446$	1.71961	0.0814258
$447$	0	0
$448$	0	0
$449$	−27.4875	−1.29721	−0.648607	−	0.761123i	$-0.724648\pi$
$449$	−27.4875	−1.29721	−0.648607	+	0.761123i	$0.724648\pi$
$450$	0	0
$451$	6.27882	0.295658
$452$	−2.62311	−0.123381
$453$	0	0
$454$	−0.0654688	−0.00307260
$455$	0	0
$456$	0	0
$457$	−7.94392	−0.371601	−0.185800	−	0.982587i	$-0.559488\pi$
$457$	−7.94392	−0.371601	−0.185800	+	0.982587i	$0.559488\pi$
$458$	−1.33020	−0.0621563
$459$	0	0
$460$	−12.4922	−0.582450
$461$	8.26943	0.385146	0.192573	−	0.981283i	$-0.438317\pi$
$461$	8.26943	0.385146	0.192573	+	0.981283i	$0.438317\pi$
$462$	0	0
$463$	9.73904	0.452612	0.226306	−	0.974056i	$-0.427335\pi$
$463$	9.73904	0.452612	0.226306	+	0.974056i	$0.427335\pi$
$464$	−0.749219	−0.0347816
$465$	0	0
$466$	−4.44706	−0.206006
$467$	−40.6970	−1.88323	−0.941617	−	0.336685i	$-0.890694\pi$
$467$	−40.6970	−1.88323	−0.941617	+	0.336685i	$0.890694\pi$
$468$	0	0
$469$	0	0
$470$	−7.73356	−0.356723
$471$	0	0
$472$	−2.49309	−0.114754
$473$	−2.33490	−0.107359
$474$	0	0
$475$	−77.1153	−3.53829
$476$	0	0
$477$	0	0
$478$	−1.80842	−0.0827152
$479$	37.8318	1.72858	0.864289	−	0.502996i	$-0.167769\pi$
$479$	37.8318	1.72858	0.864289	+	0.502996i	$0.167769\pi$
$480$	0	0
$481$	17.0226	0.776162
$482$	2.02334	0.0921608
$483$	0	0
$484$	−1.97196	−0.0896346
$485$	−0.213351	−0.00968778
$486$	0	0
$487$	40.5576	1.83784	0.918921	−	0.394442i	$-0.129062\pi$
$487$	40.5576	1.83784	0.918921	+	0.394442i	$0.129062\pi$
$488$	3.99061	0.180646
$489$	0	0
$490$	0	0
$491$	−31.0786	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$491$	−31.0786	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$492$	0	0
$493$	−0.0654688	−0.00294856
$494$	5.18531	0.233298
$495$	0	0
$496$	38.1106	1.71122
$497$	0	0
$498$	0	0
$499$	15.3575	0.687494	0.343747	−	0.939062i	$-0.388304\pi$
$499$	15.3575	0.687494	0.343747	+	0.939062i	$0.388304\pi$
$500$	−33.5678	−1.50120
$501$	0	0
$502$	0.805298	0.0359422
$503$	14.8925	0.664025	0.332013	−	0.943275i	$-0.392272\pi$
$503$	14.8925	0.664025	0.332013	+	0.943275i	$0.392272\pi$
$504$	0	0
$505$	−71.8412	−3.19689
$506$	−0.278820	−0.0123951
$507$	0	0
$508$	−28.8177	−1.27858
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	0	0
$512$	12.6557	0.559309
$513$	0	0
$514$	−2.80530	−0.123736
$515$	31.4969	1.38792
$516$	0	0
$517$	12.1394	0.533891
$518$	0	0
$519$	0	0
$520$	9.62689	0.422167
$521$	27.6924	1.21322	0.606612	−	0.794998i	$-0.292528\pi$
$521$	27.6924	1.21322	0.606612	+	0.794998i	$0.292528\pi$
$522$	0	0
$523$	−18.4088	−0.804962	−0.402481	−	0.915428i	$-0.631852\pi$
$523$	−18.4088	−0.804962	−0.402481	+	0.915428i	$0.631852\pi$
$524$	−40.5389	−1.77095
$525$	0	0
$526$	2.03273	0.0886314
$527$	3.33020	0.145066
$528$	0	0
$529$	−20.2274	−0.879454
$530$	−5.06077	−0.219826
$531$	0	0
$532$	0	0
$533$	23.8878	1.03470
$534$	0	0
$535$	30.9665	1.33880
$536$	0.0927218	0.00400497
$537$	0	0
$538$	−4.22274	−0.182055
$539$	0	0
$540$	0	0
$541$	−4.26943	−0.183557	−0.0917786	−	0.995779i	$-0.529255\pi$
$541$	−4.26943	−0.183557	−0.0917786	+	0.995779i	$0.529255\pi$
$542$	−0.693141	−0.0297729
$543$	0	0
$544$	0.660406	0.0283147
$545$	43.9533	1.88275
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−31.9906	−1.36657
$549$	0	0
$550$	−1.58647	−0.0676471
$551$	1.59116	0.0677857
$552$	0	0
$553$	0	0
$554$	−3.06077	−0.130040
$555$	0	0
$556$	44.4643	1.88570
$557$	12.4743	0.528553	0.264277	−	0.964447i	$-0.414867\pi$
$557$	12.4743	0.528553	0.264277	+	0.964447i	$0.414867\pi$
$558$	0	0
$559$	−8.88315	−0.375717
$560$	0	0
$561$	0	0
$562$	−1.12923	−0.0476338
$563$	−32.7710	−1.38113	−0.690566	−	0.723269i	$-0.742639\pi$
$563$	−32.7710	−1.38113	−0.690566	+	0.723269i	$0.742639\pi$
$564$	0	0
$565$	5.06077	0.212908
$566$	−3.91119	−0.164399
$567$	0	0
$568$	3.10589	0.130320
$569$	29.2180	1.22488	0.612442	−	0.790515i	$-0.290187\pi$
$569$	29.2180	1.22488	0.612442	+	0.790515i	$0.290187\pi$
$570$	0	0
$571$	32.4455	1.35780	0.678901	−	0.734230i	$-0.262457\pi$
$571$	32.4455	1.35780	0.678901	+	0.734230i	$0.262457\pi$
$572$	−7.50235	−0.313689
$573$	0	0
$574$	0	0
$575$	15.7757	0.657892
$576$	0	0
$577$	14.8831	0.619594	0.309797	−	0.950803i	$-0.399739\pi$
$577$	14.8831	0.619594	0.309797	+	0.950803i	$0.399739\pi$
$578$	−2.82786	−0.117623
$579$	0	0
$580$	1.46662	0.0608982
$581$	0	0
$582$	0	0
$583$	7.94392	0.329004
$584$	2.79043	0.115469
$585$	0	0
$586$	−2.37220	−0.0979945
$587$	−4.53039	−0.186989	−0.0934945	−	0.995620i	$-0.529804\pi$
$587$	−4.53039	−0.186989	−0.0934945	+	0.995620i	$0.529804\pi$
$588$	0	0
$589$	−80.9377	−3.33498
$590$	2.38799	0.0983118
$591$	0	0
$592$	−17.1480	−0.704779
$593$	8.28821	0.340356	0.170178	−	0.985413i	$-0.445566\pi$
$593$	8.28821	0.340356	0.170178	+	0.985413i	$0.445566\pi$
$594$	0	0
$595$	0	0
$596$	16.1612	0.661988
$597$	0	0
$598$	−1.06077	−0.0433783
$599$	−1.99061	−0.0813341	−0.0406671	−	0.999173i	$-0.512948\pi$
$599$	−1.99061	−0.0813341	−0.0406671	+	0.999173i	$0.512948\pi$
$600$	0	0
$601$	8.47431	0.345674	0.172837	−	0.984950i	$-0.444707\pi$
$601$	8.47431	0.345674	0.172837	+	0.984950i	$0.444707\pi$
$602$	0	0
$603$	0	0
$604$	26.1761	1.06509
$605$	3.80451	0.154675
$606$	0	0
$607$	22.9665	0.932181	0.466090	−	0.884737i	$-0.345662\pi$
$607$	22.9665	0.932181	0.466090	+	0.884737i	$0.345662\pi$
$608$	−16.0506	−0.650938
$609$	0	0
$610$	−3.82237	−0.154763
$611$	46.1845	1.86843
$612$	0	0
$613$	3.55294	0.143502	0.0717510	−	0.997423i	$-0.477141\pi$
$613$	3.55294	0.143502	0.0717510	+	0.997423i	$0.477141\pi$
$614$	2.10277	0.0848608
$615$	0	0
$616$	0	0
$617$	−22.6698	−0.912652	−0.456326	−	0.889813i	$-0.650835\pi$
$617$	−22.6698	−0.912652	−0.456326	+	0.889813i	$0.650835\pi$
$618$	0	0
$619$	−8.71648	−0.350345	−0.175173	−	0.984538i	$-0.556048\pi$
$619$	−8.71648	−0.350345	−0.175173	+	0.984538i	$0.556048\pi$
$620$	−74.6028	−2.99612
$621$	0	0
$622$	1.56391	0.0627070
$623$	0	0
$624$	0	0
$625$	17.3910	0.695639
$626$	0.501561	0.0200464
$627$	0	0
$628$	−33.4035	−1.33294
$629$	−1.49844	−0.0597467
$630$	0	0
$631$	−11.8878	−0.473248	−0.236624	−	0.971601i	$-0.576041\pi$
$631$	−11.8878	−0.473248	−0.236624	+	0.971601i	$0.576041\pi$
$632$	−2.21492	−0.0881049
$633$	0	0
$634$	−1.66353	−0.0660672
$635$	55.5981	2.20634
$636$	0	0
$637$	0	0
$638$	0.0327344	0.00129597
$639$	0	0
$640$	−19.6775	−0.777821
$641$	4.89254	0.193244	0.0966218	−	0.995321i	$-0.469196\pi$
$641$	4.89254	0.193244	0.0966218	+	0.995321i	$0.469196\pi$
$642$	0	0
$643$	−22.7804	−0.898371	−0.449185	−	0.893439i	$-0.648286\pi$
$643$	−22.7804	−0.898371	−0.449185	+	0.893439i	$0.648286\pi$
$644$	0	0
$645$	0	0
$646$	−0.456446	−0.0179586
$647$	31.2453	1.22838	0.614190	−	0.789158i	$-0.289483\pi$
$647$	31.2453	1.22838	0.614190	+	0.789158i	$0.289483\pi$
$648$	0	0
$649$	−3.74843	−0.147139
$650$	−6.03573	−0.236741
$651$	0	0
$652$	13.4090	0.525136
$653$	−28.2882	−1.10700	−0.553502	−	0.832848i	$-0.686709\pi$
$653$	−28.2882	−1.10700	−0.553502	+	0.832848i	$0.686709\pi$
$654$	0	0
$655$	78.2118	3.05599
$656$	−24.0639	−0.939537
$657$	0	0
$658$	0	0
$659$	−30.2967	−1.18019	−0.590096	−	0.807333i	$-0.700910\pi$
$659$	−30.2967	−1.18019	−0.590096	+	0.807333i	$0.700910\pi$
$660$	0	0
$661$	35.7196	1.38933	0.694666	−	0.719333i	$-0.255552\pi$
$661$	35.7196	1.38933	0.694666	+	0.719333i	$0.255552\pi$
$662$	3.77569	0.146746
$663$	0	0
$664$	9.27412	0.359906
$665$	0	0
$666$	0	0
$667$	−0.325508	−0.0126037
$668$	35.9345	1.39035
$669$	0	0
$670$	−0.0888128	−0.00343114
$671$	6.00000	0.231627
$672$	0	0
$673$	34.9377	1.34675	0.673374	−	0.739302i	$-0.264844\pi$
$673$	34.9377	1.34675	0.673374	+	0.739302i	$0.264844\pi$
$674$	2.22431	0.0856774
$675$	0	0
$676$	−2.90728	−0.111818
$677$	20.0094	0.769023	0.384512	−	0.923120i	$-0.374370\pi$
$677$	20.0094	0.769023	0.384512	+	0.923120i	$0.374370\pi$
$678$	0	0
$679$	0	0
$680$	−0.847422	−0.0324972
$681$	0	0
$682$	−1.66510	−0.0637600
$683$	−25.0498	−0.958504	−0.479252	−	0.877677i	$-0.659092\pi$
$683$	−25.0498	−0.958504	−0.479252	+	0.877677i	$0.659092\pi$
$684$	0	0
$685$	61.7196	2.35818
$686$	0	0
$687$	0	0
$688$	8.94862	0.341163
$689$	30.2227	1.15139
$690$	0	0
$691$	38.8271	1.47705	0.738526	−	0.674225i	$-0.235522\pi$
$691$	38.8271	1.47705	0.738526	+	0.674225i	$0.235522\pi$
$692$	3.39410	0.129024
$693$	0	0
$694$	4.65102	0.176550
$695$	−85.7851	−3.25401
$696$	0	0
$697$	−2.10277	−0.0796480
$698$	0.478217	0.0181008
$699$	0	0
$700$	0	0
$701$	41.7757	1.57785	0.788923	−	0.614492i	$-0.210639\pi$
$701$	41.7757	1.57785	0.788923	+	0.614492i	$0.210639\pi$
$702$	0	0
$703$	36.4182	1.37354
$704$	7.33490	0.276444
$705$	0	0
$706$	4.12611	0.155288
$707$	0	0
$708$	0	0
$709$	−13.4041	−0.503403	−0.251702	−	0.967805i	$-0.580990\pi$
$709$	−13.4041	−0.503403	−0.251702	+	0.967805i	$0.580990\pi$
$710$	−2.97495	−0.111648
$711$	0	0
$712$	6.57642	0.246462
$713$	16.5576	0.620088
$714$	0	0
$715$	14.4743	0.541308
$716$	−23.6635	−0.884348
$717$	0	0
$718$	1.99061	0.0742889
$719$	5.47900	0.204332	0.102166	−	0.994767i	$-0.467423\pi$
$719$	5.47900	0.204332	0.102166	+	0.994767i	$0.467423\pi$
$720$	0	0
$721$	0	0
$722$	7.91197	0.294453
$723$	0	0
$724$	1.43140	0.0531975
$725$	−1.85212	−0.0687859
$726$	0	0
$727$	32.8831	1.21957	0.609784	−	0.792567i	$-0.291256\pi$
$727$	32.8831	1.21957	0.609784	+	0.792567i	$0.291256\pi$
$728$	0	0
$729$	0	0
$730$	−2.67279	−0.0989243
$731$	0.781954	0.0289216
$732$	0	0
$733$	35.8972	1.32589	0.662947	−	0.748666i	$-0.269305\pi$
$733$	35.8972	1.32589	0.662947	+	0.748666i	$0.269305\pi$
$734$	−0.604328	−0.0223062
$735$	0	0
$736$	3.28352	0.121032
$737$	0.139410	0.00513523
$738$	0	0
$739$	29.6651	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$739$	29.6651	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$740$	33.5678	1.23398
$741$	0	0
$742$	0	0
$743$	18.7998	0.689698	0.344849	−	0.938658i	$-0.387930\pi$
$743$	18.7998	0.689698	0.344849	+	0.938658i	$0.387930\pi$
$744$	0	0
$745$	−31.1798	−1.14234
$746$	2.06547	0.0756222
$747$	0	0
$748$	0.660406	0.0241469
$749$	0	0
$750$	0	0
$751$	−9.59116	−0.349986	−0.174993	−	0.984570i	$-0.555990\pi$
$751$	−9.59116	−0.349986	−0.174993	+	0.984570i	$0.555990\pi$
$752$	−46.5249	−1.69659
$753$	0	0
$754$	0.124538	0.00453542
$755$	−50.5016	−1.83794
$756$	0	0
$757$	2.74374	0.0997229	0.0498614	−	0.998756i	$-0.484122\pi$
$757$	2.74374	0.0997229	0.0498614	+	0.998756i	$0.484122\pi$
$758$	3.91119	0.142061
$759$	0	0
$760$	20.5959	0.747090
$761$	6.39098	0.231673	0.115836	−	0.993268i	$-0.463045\pi$
$761$	6.39098	0.231673	0.115836	+	0.993268i	$0.463045\pi$
$762$	0	0
$763$	0	0
$764$	10.4004	0.376272
$765$	0	0
$766$	−2.54825	−0.0920720
$767$	−14.2610	−0.514933
$768$	0	0
$769$	−17.7951	−0.641708	−0.320854	−	0.947129i	$-0.603970\pi$
$769$	−17.7951	−0.641708	−0.320854	+	0.947129i	$0.603970\pi$
$770$	0	0
$771$	0	0
$772$	39.1075	1.40751
$773$	−31.5257	−1.13390	−0.566950	−	0.823752i	$-0.691877\pi$
$773$	−31.5257	−1.13390	−0.566950	+	0.823752i	$0.691877\pi$
$774$	0	0
$775$	94.2118	3.38419
$776$	0.0372979	0.00133892
$777$	0	0
$778$	−0.624681	−0.0223959
$779$	51.1059	1.83106
$780$	0	0
$781$	4.66980	0.167098
$782$	0.0933763	0.00333913
$783$	0	0
$784$	0	0
$785$	64.4455	2.30016
$786$	0	0
$787$	−31.8606	−1.13571	−0.567854	−	0.823130i	$-0.692226\pi$
$787$	−31.8606	−1.13571	−0.567854	+	0.823130i	$0.692226\pi$
$788$	5.26473	0.187548
$789$	0	0
$790$	2.12155	0.0754813
$791$	0	0
$792$	0	0
$793$	22.8271	0.810613
$794$	−3.49844	−0.124155
$795$	0	0
$796$	26.7259	0.947274
$797$	43.8590	1.55357	0.776783	−	0.629768i	$-0.216850\pi$
$797$	43.8590	1.55357	0.776783	+	0.629768i	$0.216850\pi$
$798$	0	0
$799$	−4.06547	−0.143826
$800$	18.6830	0.660543
$801$	0	0
$802$	−3.89723	−0.137616
$803$	4.19549	0.148056
$804$	0	0
$805$	0	0
$806$	−6.33490	−0.223137
$807$	0	0
$808$	12.5592	0.441832
$809$	−6.74374	−0.237097	−0.118549	−	0.992948i	$-0.537824\pi$
$809$	−6.74374	−0.237097	−0.118549	+	0.992948i	$0.537824\pi$
$810$	0	0
$811$	−3.97275	−0.139502	−0.0697510	−	0.997564i	$-0.522220\pi$
$811$	−3.97275	−0.139502	−0.0697510	+	0.997564i	$0.522220\pi$
$812$	0	0
$813$	0	0
$814$	0.749219	0.0262601
$815$	−25.8700	−0.906186
$816$	0	0
$817$	−19.0047	−0.664890
$818$	4.00157	0.139912
$819$	0	0
$820$	47.1059	1.64501
$821$	−49.5896	−1.73069	−0.865344	−	0.501178i	$-0.832900\pi$
$821$	−49.5896	−1.73069	−0.865344	+	0.501178i	$0.832900\pi$
$822$	0	0
$823$	23.1908	0.808380	0.404190	−	0.914675i	$-0.367553\pi$
$823$	23.1908	0.808380	0.404190	+	0.914675i	$0.367553\pi$
$824$	−5.50626	−0.191820
$825$	0	0
$826$	0	0
$827$	−27.7484	−0.964908	−0.482454	−	0.875921i	$-0.660254\pi$
$827$	−27.7484	−0.964908	−0.482454	+	0.875921i	$0.660254\pi$
$828$	0	0
$829$	−0.269430	−0.00935768	−0.00467884	−	0.999989i	$-0.501489\pi$
$829$	−0.269430	−0.00935768	−0.00467884	+	0.999989i	$0.501489\pi$
$830$	−8.88315	−0.308339
$831$	0	0
$832$	27.9057	0.967456
$833$	0	0
$834$	0	0
$835$	−69.3286	−2.39922
$836$	−16.0506	−0.555122
$837$	0	0
$838$	−5.07316	−0.175249
$839$	−27.3575	−0.944484	−0.472242	−	0.881469i	$-0.656555\pi$
$839$	−27.3575	−0.944484	−0.472242	+	0.881469i	$0.656555\pi$
$840$	0	0
$841$	−28.9618	−0.998682
$842$	−2.59976	−0.0895938
$843$	0	0
$844$	8.43767	0.290436
$845$	5.60902	0.192956
$846$	0	0
$847$	0	0
$848$	−30.4455	−1.04550
$849$	0	0
$850$	0.531305	0.0182236
$851$	−7.45018	−0.255389
$852$	0	0
$853$	28.5482	0.977473	0.488737	−	0.872431i	$-0.337458\pi$
$853$	28.5482	0.977473	0.488737	+	0.872431i	$0.337458\pi$
$854$	0	0
$855$	0	0
$856$	−5.41353	−0.185031
$857$	−39.8972	−1.36286	−0.681432	−	0.731882i	$-0.738642\pi$
$857$	−39.8972	−1.36286	−0.681432	+	0.731882i	$0.738642\pi$
$858$	0	0
$859$	−20.5576	−0.701418	−0.350709	−	0.936485i	$-0.614059\pi$
$859$	−20.5576	−0.701418	−0.350709	+	0.936485i	$0.614059\pi$
$860$	−17.5172	−0.597332
$861$	0	0
$862$	−0.515515	−0.0175585
$863$	9.66510	0.329004	0.164502	−	0.986377i	$-0.447398\pi$
$863$	9.66510	0.329004	0.164502	+	0.986377i	$0.447398\pi$
$864$	0	0
$865$	−6.54825	−0.222647
$866$	−2.83646	−0.0963868
$867$	0	0
$868$	0	0
$869$	−3.33020	−0.112969
$870$	0	0
$871$	0.530387	0.0179715
$872$	−7.68388	−0.260209
$873$	0	0
$874$	−2.26943	−0.0767646
$875$	0	0
$876$	0	0
$877$	−25.7212	−0.868543	−0.434271	−	0.900782i	$-0.642994\pi$
$877$	−25.7212	−0.868543	−0.434271	+	0.900782i	$0.642994\pi$
$878$	1.85511	0.0626069
$879$	0	0
$880$	−14.5810	−0.491525
$881$	−14.6316	−0.492950	−0.246475	−	0.969149i	$-0.579272\pi$
$881$	−14.6316	−0.492950	−0.246475	+	0.969149i	$0.579272\pi$
$882$	0	0
$883$	18.6877	0.628890	0.314445	−	0.949276i	$-0.398182\pi$
$883$	18.6877	0.628890	0.314445	+	0.949276i	$0.398182\pi$
$884$	2.51252	0.0845053
$885$	0	0
$886$	2.43609	0.0818421
$887$	38.6137	1.29652	0.648261	−	0.761418i	$-0.275497\pi$
$887$	38.6137	1.29652	0.648261	+	0.761418i	$0.275497\pi$
$888$	0	0
$889$	0	0
$890$	−6.29917	−0.211149
$891$	0	0
$892$	−20.2509	−0.678051
$893$	98.8076	3.30647
$894$	0	0
$895$	45.6541	1.52605
$896$	0	0
$897$	0	0
$898$	−4.60276	−0.153596
$899$	−1.94392	−0.0648334
$900$	0	0
$901$	−2.66041	−0.0886310
$902$	1.05138	0.0350072
$903$	0	0
$904$	−0.884720	−0.0294254
$905$	−2.76160	−0.0917987
$906$	0	0
$907$	49.0965	1.63022	0.815111	−	0.579304i	$-0.196676\pi$
$907$	49.0965	1.63022	0.815111	+	0.579304i	$0.196676\pi$
$908$	0.770991	0.0255862
$909$	0	0
$910$	0	0
$911$	−29.3753	−0.973248	−0.486624	−	0.873612i	$-0.661772\pi$
$911$	−29.3753	−0.973248	−0.486624	+	0.873612i	$0.661772\pi$
$912$	0	0
$913$	13.9439	0.461476
$914$	−1.33020	−0.0439992
$915$	0	0
$916$	15.6651	0.517590
$917$	0	0
$918$	0	0
$919$	−27.0047	−0.890803	−0.445401	−	0.895331i	$-0.646939\pi$
$919$	−27.0047	−0.890803	−0.445401	+	0.895331i	$0.646939\pi$
$920$	−4.21335	−0.138910
$921$	0	0
$922$	1.38471	0.0456030
$923$	17.7663	0.584785
$924$	0	0
$925$	−42.3910	−1.39381
$926$	1.63079	0.0535912
$927$	0	0
$928$	−0.385496	−0.0126545
$929$	57.8496	1.89798	0.948992	−	0.315299i	$-0.102105\pi$
$929$	57.8496	1.89798	0.948992	+	0.315299i	$0.102105\pi$
$930$	0	0
$931$	0	0
$932$	52.3706	1.71546
$933$	0	0
$934$	−6.81469	−0.222983
$935$	−1.27412	−0.0416683
$936$	0	0
$937$	−25.2180	−0.823838	−0.411919	−	0.911221i	$-0.635141\pi$
$937$	−25.2180	−0.823838	−0.411919	+	0.911221i	$0.635141\pi$
$938$	0	0
$939$	0	0
$940$	91.0741	2.97051
$941$	−34.2788	−1.11746	−0.558729	−	0.829350i	$-0.688711\pi$
$941$	−34.2788	−1.11746	−0.558729	+	0.829350i	$0.688711\pi$
$942$	0	0
$943$	−10.4549	−0.340458
$944$	14.3661	0.467575
$945$	0	0
$946$	−0.390977	−0.0127118
$947$	18.1573	0.590032	0.295016	−	0.955492i	$-0.404675\pi$
$947$	18.1573	0.590032	0.295016	+	0.955492i	$0.404675\pi$
$948$	0	0
$949$	15.9618	0.518141
$950$	−12.9129	−0.418950
$951$	0	0
$952$	0	0
$953$	19.8045	0.641531	0.320766	−	0.947159i	$-0.396060\pi$
$953$	19.8045	0.641531	0.320766	+	0.947159i	$0.396060\pi$
$954$	0	0
$955$	−20.0655	−0.649303
$956$	21.2968	0.688788
$957$	0	0
$958$	6.33490	0.204671
$959$	0	0
$960$	0	0
$961$	67.8816	2.18973
$962$	2.85041	0.0919010
$963$	0	0
$964$	−23.8279	−0.767444
$965$	−75.4502	−2.42883
$966$	0	0
$967$	−0.278820	−0.00896624	−0.00448312	−	0.999990i	$-0.501427\pi$
$967$	−0.278820	−0.00896624	−0.00448312	+	0.999990i	$0.501427\pi$
$968$	−0.665102	−0.0213772
$969$	0	0
$970$	−0.0357255	−0.00114708
$971$	25.3481	0.813458	0.406729	−	0.913549i	$-0.366669\pi$
$971$	25.3481	0.813458	0.406729	+	0.913549i	$0.366669\pi$
$972$	0	0
$973$	0	0
$974$	6.79134	0.217609
$975$	0	0
$976$	−22.9953	−0.736062
$977$	−6.71648	−0.214879	−0.107440	−	0.994212i	$-0.534265\pi$
$977$	−6.71648	−0.214879	−0.107440	+	0.994212i	$0.534265\pi$
$978$	0	0
$979$	9.88784	0.316017
$980$	0	0
$981$	0	0
$982$	−5.20409	−0.166069
$983$	9.99061	0.318651	0.159325	−	0.987226i	$-0.449068\pi$
$983$	9.99061	0.318651	0.159325	+	0.987226i	$0.449068\pi$
$984$	0	0
$985$	−10.1573	−0.323638
$986$	−0.0109627	−0.000349123 0
$987$	0	0
$988$	−61.0647	−1.94273
$989$	3.88784	0.123626
$990$	0	0
$991$	−26.7064	−0.848358	−0.424179	−	0.905578i	$-0.639437\pi$
$991$	−26.7064	−0.848358	−0.424179	+	0.905578i	$0.639437\pi$
$992$	19.6090	0.622587
$993$	0	0
$994$	0	0
$995$	−51.5623	−1.63464
$996$	0	0
$997$	−54.3333	−1.72075	−0.860377	−	0.509658i	$-0.829772\pi$
$997$	−54.3333	−1.72075	−0.860377	+	0.509658i	$0.829772\pi$
$998$	2.57159	0.0814024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bp.1.2		3
3.2	odd	2		1617.2.a.s.1.2		3
7.6	odd	2		693.2.a.m.1.2		3
21.20	even	2		231.2.a.d.1.2	✓	3
77.76	even	2		7623.2.a.cb.1.2		3
84.83	odd	2		3696.2.a.bp.1.3		3
105.104	even	2		5775.2.a.bw.1.2		3
231.230	odd	2		2541.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.2	✓	3	21.20	even	2
693.2.a.m.1.2		3	7.6	odd	2
1617.2.a.s.1.2		3	3.2	odd	2
2541.2.a.bi.1.2		3	231.230	odd	2
3696.2.a.bp.1.3		3	84.83	odd	2
4851.2.a.bp.1.2		3	1.1	even	1	trivial
5775.2.a.bw.1.2		3	105.104	even	2
7623.2.a.cb.1.2		3	77.76	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +O(q^{10})\)	\(q+0.167449 q^{2} -1.97196 q^{4} +3.80451 q^{5} -0.665102 q^{8} +0.637062 q^{10} -1.00000 q^{11} -3.80451 q^{13} +3.83255 q^{16} +0.334898 q^{17} -8.13941 q^{19} -7.50235 q^{20} -0.167449 q^{22} +1.66510 q^{23} +9.47431 q^{25} -0.637062 q^{26} -0.195488 q^{29} +9.94392 q^{31} +1.97196 q^{32} +0.0560785 q^{34} -4.47431 q^{37} -1.36294 q^{38} -2.53039 q^{40} -6.27882 q^{41} +2.33490 q^{43} +1.97196 q^{44} +0.278820 q^{46} -12.1394 q^{47} +1.58647 q^{50} +7.50235 q^{52} -7.94392 q^{53} -3.80451 q^{55} -0.0327344 q^{58} +3.74843 q^{59} -6.00000 q^{61} +1.66510 q^{62} -7.33490 q^{64} -14.4743 q^{65} -0.139410 q^{67} -0.660406 q^{68} -4.66980 q^{71} -4.19549 q^{73} -0.749219 q^{74} +16.0506 q^{76} +3.33020 q^{79} +14.5810 q^{80} -1.05138 q^{82} -13.9439 q^{83} +1.27412 q^{85} +0.390977 q^{86} +0.665102 q^{88} -9.88784 q^{89} -3.28352 q^{92} -2.03273 q^{94} -30.9665 q^{95} -0.0560785 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 - 3 * q^8	\( 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 24 * q^97

Introduction

L-functions

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