LMFDB - Embedded newform 4851.2.a.w.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:	$2$
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 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +O(q^{10})$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +0.618034 q^{10} -1.00000 q^{11} -3.47214 q^{13} +1.85410 q^{16} +5.23607 q^{17} +6.70820 q^{19} -1.61803 q^{20} -0.618034 q^{22} -5.70820 q^{23} -4.00000 q^{25} -2.14590 q^{26} -5.00000 q^{29} +5.23607 q^{31} +5.61803 q^{32} +3.23607 q^{34} -7.00000 q^{37} +4.14590 q^{38} -2.23607 q^{40} -2.47214 q^{41} +5.70820 q^{43} +1.61803 q^{44} -3.52786 q^{46} +0.236068 q^{47} -2.47214 q^{50} +5.61803 q^{52} +12.1803 q^{53} -1.00000 q^{55} -3.09017 q^{58} -11.1803 q^{59} -2.00000 q^{61} +3.23607 q^{62} -0.236068 q^{64} -3.47214 q^{65} -9.76393 q^{67} -8.47214 q^{68} +2.47214 q^{71} -4.52786 q^{73} -4.32624 q^{74} -10.8541 q^{76} -14.4721 q^{79} +1.85410 q^{80} -1.52786 q^{82} +6.76393 q^{83} +5.23607 q^{85} +3.52786 q^{86} +2.23607 q^{88} -4.47214 q^{89} +9.23607 q^{92} +0.145898 q^{94} +6.70820 q^{95} -9.70820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5	$ 2 q - q^{2} - q^{4} + 2 q^{5} - q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{16} + 6 q^{17} - q^{20} + q^{22} + 2 q^{23} - 8 q^{25} - 11 q^{26} - 10 q^{29} + 6 q^{31} + 9 q^{32} + 2 q^{34} - 14 q^{37} + 15 q^{38} + 4 q^{41} - 2 q^{43} + q^{44} - 16 q^{46} - 4 q^{47} + 4 q^{50} + 9 q^{52} + 2 q^{53} - 2 q^{55} + 5 q^{58} - 4 q^{61} + 2 q^{62} + 4 q^{64} + 2 q^{65} - 24 q^{67} - 8 q^{68} - 4 q^{71} - 18 q^{73} + 7 q^{74} - 15 q^{76} - 20 q^{79} - 3 q^{80} - 12 q^{82} + 18 q^{83} + 6 q^{85} + 16 q^{86} + 14 q^{92} + 7 q^{94} - 6 q^{97}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - q^10 - 2 * q^11 + 2 * q^13 - 3 * q^16 + 6 * q^17 - q^20 + q^22 + 2 * q^23 - 8 * q^25 - 11 * q^26 - 10 * q^29 + 6 * q^31 + 9 * q^32 + 2 * q^34 - 14 * q^37 + 15 * q^38 + 4 * q^41 - 2 * q^43 + q^44 - 16 * q^46 - 4 * q^47 + 4 * q^50 + 9 * q^52 + 2 * q^53 - 2 * q^55 + 5 * q^58 - 4 * q^61 + 2 * q^62 + 4 * q^64 + 2 * q^65 - 24 * q^67 - 8 * q^68 - 4 * q^71 - 18 * q^73 + 7 * q^74 - 15 * q^76 - 20 * q^79 - 3 * q^80 - 12 * q^82 + 18 * q^83 + 6 * q^85 + 16 * q^86 + 14 * q^92 + 7 * q^94 - 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.47214	−0.962997	−0.481499	−	0.876447i	$-0.659907\pi$
$13$	−3.47214	−0.962997	−0.481499	+	0.876447i	$0.659907\pi$
$14$	0	0
$15$	0	0
$16$	1.85410	0.463525
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	−0.618034	−0.131765
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−2.14590	−0.420845
$27$	0	0
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	5.23607	0.940426	0.470213	−	0.882553i	$-0.344177\pi$
$31$	5.23607	0.940426	0.470213	+	0.882553i	$0.344177\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	3.23607	0.554981
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	4.14590	0.672553
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	0	0
$43$	5.70820	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$44$	1.61803	0.243928
$45$	0	0
$46$	−3.52786	−0.520155
$47$	0.236068	0.0344341	0.0172170	−	0.999852i	$-0.494519\pi$
$47$	0.236068	0.0344341	0.0172170	+	0.999852i	$0.494519\pi$
$48$	0	0
$49$	0	0
$50$	−2.47214	−0.349613
$51$	0	0
$52$	5.61803	0.779081
$53$	12.1803	1.67310	0.836549	−	0.547892i	$-0.184569\pi$
$53$	12.1803	1.67310	0.836549	+	0.547892i	$0.184569\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	−3.09017	−0.405759
$59$	−11.1803	−1.45556	−0.727778	−	0.685813i	$-0.759447\pi$
$59$	−11.1803	−1.45556	−0.727778	+	0.685813i	$0.759447\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.23607	0.410981
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−3.47214	−0.430665
$66$	0	0
$67$	−9.76393	−1.19285	−0.596427	−	0.802667i	$-0.703414\pi$
$67$	−9.76393	−1.19285	−0.596427	+	0.802667i	$0.703414\pi$
$68$	−8.47214	−1.02740
$69$	0	0
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	−4.52786	−0.529946	−0.264973	−	0.964256i	$-0.585363\pi$
$73$	−4.52786	−0.529946	−0.264973	+	0.964256i	$0.585363\pi$
$74$	−4.32624	−0.502915
$75$	0	0
$76$	−10.8541	−1.24505
$77$	0	0
$78$	0	0
$79$	−14.4721	−1.62824	−0.814121	−	0.580695i	$-0.802781\pi$
$79$	−14.4721	−1.62824	−0.814121	+	0.580695i	$0.802781\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	−1.52786	−0.168724
$83$	6.76393	0.742438	0.371219	−	0.928545i	$-0.378940\pi$
$83$	6.76393	0.742438	0.371219	+	0.928545i	$0.378940\pi$
$84$	0	0
$85$	5.23607	0.567931
$86$	3.52786	0.380419
$87$	0	0
$88$	2.23607	0.238366
$89$	−4.47214	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$90$	0	0
$91$	0	0
$92$	9.23607	0.962927
$93$	0	0
$94$	0.145898	0.0150482
$95$	6.70820	0.688247
$96$	0	0
$97$	−9.70820	−0.985719	−0.492859	−	0.870109i	$-0.664048\pi$
$97$	−9.70820	−0.985719	−0.492859	+	0.870109i	$0.664048\pi$
$98$	0	0
$99$	0	0
$100$	6.47214	0.647214
$101$	18.1803	1.80901	0.904506	−	0.426461i	$-0.140240\pi$
$101$	18.1803	1.80901	0.904506	+	0.426461i	$0.140240\pi$
$102$	0	0
$103$	−17.4164	−1.71609	−0.858045	−	0.513575i	$-0.828321\pi$
$103$	−17.4164	−1.71609	−0.858045	+	0.513575i	$0.828321\pi$
$104$	7.76393	0.761316
$105$	0	0
$106$	7.52786	0.731171
$107$	4.23607	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$107$	4.23607	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$108$	0	0
$109$	2.76393	0.264737	0.132368	−	0.991201i	$-0.457742\pi$
$109$	2.76393	0.264737	0.132368	+	0.991201i	$0.457742\pi$
$110$	−0.618034	−0.0589272
$111$	0	0
$112$	0	0
$113$	0.472136	0.0444148	0.0222074	−	0.999753i	$-0.492931\pi$
$113$	0.472136	0.0444148	0.0222074	+	0.999753i	$0.492931\pi$
$114$	0	0
$115$	−5.70820	−0.532293
$116$	8.09017	0.751153
$117$	0	0
$118$	−6.90983	−0.636101
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.23607	−0.111908
$123$	0	0
$124$	−8.47214	−0.760820
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−12.6525	−1.12273	−0.561363	−	0.827570i	$-0.689723\pi$
$127$	−12.6525	−1.12273	−0.561363	+	0.827570i	$0.689723\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	−2.14590	−0.188208
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	0	0
$134$	−6.03444	−0.521296
$135$	0	0
$136$	−11.7082	−1.00397
$137$	−19.7082	−1.68379	−0.841893	−	0.539645i	$-0.818559\pi$
$137$	−19.7082	−1.68379	−0.841893	+	0.539645i	$0.818559\pi$
$138$	0	0
$139$	−14.4721	−1.22751	−0.613755	−	0.789496i	$-0.710342\pi$
$139$	−14.4721	−1.22751	−0.613755	+	0.789496i	$0.710342\pi$
$140$	0	0
$141$	0	0
$142$	1.52786	0.128216
$143$	3.47214	0.290355
$144$	0	0
$145$	−5.00000	−0.415227
$146$	−2.79837	−0.231595
$147$	0	0
$148$	11.3262	0.931011
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	−14.1803	−1.15398	−0.576990	−	0.816751i	$-0.695773\pi$
$151$	−14.1803	−1.15398	−0.576990	+	0.816751i	$0.695773\pi$
$152$	−15.0000	−1.21666
$153$	0	0
$154$	0	0
$155$	5.23607	0.420571
$156$	0	0
$157$	15.4164	1.23036	0.615182	−	0.788385i	$-0.289083\pi$
$157$	15.4164	1.23036	0.615182	+	0.788385i	$0.289083\pi$
$158$	−8.94427	−0.711568
$159$	0	0
$160$	5.61803	0.444145
$161$	0	0
$162$	0	0
$163$	−22.7082	−1.77864	−0.889322	−	0.457282i	$-0.848823\pi$
$163$	−22.7082	−1.77864	−0.889322	+	0.457282i	$0.848823\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	4.18034	0.324457
$167$	−22.6525	−1.75290	−0.876451	−	0.481492i	$-0.840095\pi$
$167$	−22.6525	−1.75290	−0.876451	+	0.481492i	$0.840095\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	3.23607	0.248195
$171$	0	0
$172$	−9.23607	−0.704244
$173$	−1.52786	−0.116161	−0.0580807	−	0.998312i	$-0.518498\pi$
$173$	−1.52786	−0.116161	−0.0580807	+	0.998312i	$0.518498\pi$
$174$	0	0
$175$	0	0
$176$	−1.85410	−0.139758
$177$	0	0
$178$	−2.76393	−0.207165
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	0.763932	0.0567826	0.0283913	−	0.999597i	$-0.490962\pi$
$181$	0.763932	0.0567826	0.0283913	+	0.999597i	$0.490962\pi$
$182$	0	0
$183$	0	0
$184$	12.7639	0.940970
$185$	−7.00000	−0.514650
$186$	0	0
$187$	−5.23607	−0.382899
$188$	−0.381966	−0.0278577
$189$	0	0
$190$	4.14590	0.300775
$191$	10.7639	0.778851	0.389425	−	0.921058i	$-0.372674\pi$
$191$	10.7639	0.778851	0.389425	+	0.921058i	$0.372674\pi$
$192$	0	0
$193$	14.6525	1.05471	0.527354	−	0.849646i	$-0.323184\pi$
$193$	14.6525	1.05471	0.527354	+	0.849646i	$0.323184\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	0	0
$197$	16.4721	1.17359	0.586796	−	0.809735i	$-0.300389\pi$
$197$	16.4721	1.17359	0.586796	+	0.809735i	$0.300389\pi$
$198$	0	0
$199$	3.81966	0.270769	0.135384	−	0.990793i	$-0.456773\pi$
$199$	3.81966	0.270769	0.135384	+	0.990793i	$0.456773\pi$
$200$	8.94427	0.632456
$201$	0	0
$202$	11.2361	0.790567
$203$	0	0
$204$	0	0
$205$	−2.47214	−0.172661
$206$	−10.7639	−0.749959
$207$	0	0
$208$	−6.43769	−0.446374
$209$	−6.70820	−0.464016
$210$	0	0
$211$	5.41641	0.372881	0.186440	−	0.982466i	$-0.440305\pi$
$211$	5.41641	0.372881	0.186440	+	0.982466i	$0.440305\pi$
$212$	−19.7082	−1.35357
$213$	0	0
$214$	2.61803	0.178965
$215$	5.70820	0.389296
$216$	0	0
$217$	0	0
$218$	1.70820	0.115694
$219$	0	0
$220$	1.61803	0.109088
$221$	−18.1803	−1.22294
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	0	0
$226$	0.291796	0.0194100
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	0	0
$229$	−7.23607	−0.478173	−0.239086	−	0.970998i	$-0.576848\pi$
$229$	−7.23607	−0.478173	−0.239086	+	0.970998i	$0.576848\pi$
$230$	−3.52786	−0.232620
$231$	0	0
$232$	11.1803	0.734025
$233$	14.9443	0.979032	0.489516	−	0.871994i	$-0.337174\pi$
$233$	14.9443	0.979032	0.489516	+	0.871994i	$0.337174\pi$
$234$	0	0
$235$	0.236068	0.0153994
$236$	18.0902	1.17757
$237$	0	0
$238$	0	0
$239$	−10.1246	−0.654907	−0.327453	−	0.944867i	$-0.606190\pi$
$239$	−10.1246	−0.654907	−0.327453	+	0.944867i	$0.606190\pi$
$240$	0	0
$241$	−25.9443	−1.67122	−0.835609	−	0.549325i	$-0.814885\pi$
$241$	−25.9443	−1.67122	−0.835609	+	0.549325i	$0.814885\pi$
$242$	0.618034	0.0397287
$243$	0	0
$244$	3.23607	0.207168
$245$	0	0
$246$	0	0
$247$	−23.2918	−1.48202
$248$	−11.7082	−0.743472
$249$	0	0
$250$	−5.56231	−0.351791
$251$	12.1246	0.765299	0.382649	−	0.923894i	$-0.375012\pi$
$251$	12.1246	0.765299	0.382649	+	0.923894i	$0.375012\pi$
$252$	0	0
$253$	5.70820	0.358872
$254$	−7.81966	−0.490649
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	0	0
$260$	5.61803	0.348416
$261$	0	0
$262$	0.583592	0.0360544
$263$	26.1246	1.61091	0.805456	−	0.592655i	$-0.201920\pi$
$263$	26.1246	1.61091	0.805456	+	0.592655i	$0.201920\pi$
$264$	0	0
$265$	12.1803	0.748232
$266$	0	0
$267$	0	0
$268$	15.7984	0.965039
$269$	1.05573	0.0643689	0.0321844	−	0.999482i	$-0.489754\pi$
$269$	1.05573	0.0643689	0.0321844	+	0.999482i	$0.489754\pi$
$270$	0	0
$271$	−5.29180	−0.321454	−0.160727	−	0.986999i	$-0.551384\pi$
$271$	−5.29180	−0.321454	−0.160727	+	0.986999i	$0.551384\pi$
$272$	9.70820	0.588646
$273$	0	0
$274$	−12.1803	−0.735841
$275$	4.00000	0.241209
$276$	0	0
$277$	−6.47214	−0.388873	−0.194436	−	0.980915i	$-0.562288\pi$
$277$	−6.47214	−0.388873	−0.194436	+	0.980915i	$0.562288\pi$
$278$	−8.94427	−0.536442
$279$	0	0
$280$	0	0
$281$	−11.4721	−0.684370	−0.342185	−	0.939633i	$-0.611167\pi$
$281$	−11.4721	−0.684370	−0.342185	+	0.939633i	$0.611167\pi$
$282$	0	0
$283$	13.7639	0.818181	0.409090	−	0.912494i	$-0.365846\pi$
$283$	13.7639	0.818181	0.409090	+	0.912494i	$0.365846\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	2.14590	0.126890
$287$	0	0
$288$	0	0
$289$	10.4164	0.612730
$290$	−3.09017	−0.181461
$291$	0	0
$292$	7.32624	0.428736
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	−11.1803	−0.650945
$296$	15.6525	0.909782
$297$	0	0
$298$	−3.09017	−0.179009
$299$	19.8197	1.14620
$300$	0	0
$301$	0	0
$302$	−8.76393	−0.504308
$303$	0	0
$304$	12.4377	0.713351
$305$	−2.00000	−0.114520
$306$	0	0
$307$	20.9443	1.19535	0.597676	−	0.801737i	$-0.296091\pi$
$307$	20.9443	1.19535	0.597676	+	0.801737i	$0.296091\pi$
$308$	0	0
$309$	0	0
$310$	3.23607	0.183796
$311$	9.88854	0.560728	0.280364	−	0.959894i	$-0.409545\pi$
$311$	9.88854	0.560728	0.280364	+	0.959894i	$0.409545\pi$
$312$	0	0
$313$	−24.6525	−1.39344	−0.696720	−	0.717343i	$-0.745358\pi$
$313$	−24.6525	−1.39344	−0.696720	+	0.717343i	$0.745358\pi$
$314$	9.52786	0.537688
$315$	0	0
$316$	23.4164	1.31728
$317$	−24.1803	−1.35810	−0.679052	−	0.734091i	$-0.737609\pi$
$317$	−24.1803	−1.35810	−0.679052	+	0.734091i	$0.737609\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	0	0
$323$	35.1246	1.95439
$324$	0	0
$325$	13.8885	0.770398
$326$	−14.0344	−0.777296
$327$	0	0
$328$	5.52786	0.305225
$329$	0	0
$330$	0	0
$331$	−11.4164	−0.627503	−0.313751	−	0.949505i	$-0.601586\pi$
$331$	−11.4164	−0.627503	−0.313751	+	0.949505i	$0.601586\pi$
$332$	−10.9443	−0.600645
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−9.76393	−0.533461
$336$	0	0
$337$	−8.18034	−0.445612	−0.222806	−	0.974863i	$-0.571522\pi$
$337$	−8.18034	−0.445612	−0.222806	+	0.974863i	$0.571522\pi$
$338$	−0.583592	−0.0317432
$339$	0	0
$340$	−8.47214	−0.459466
$341$	−5.23607	−0.283549
$342$	0	0
$343$	0	0
$344$	−12.7639	−0.688185
$345$	0	0
$346$	−0.944272	−0.0507644
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	−1.58359	−0.0847677	−0.0423839	−	0.999101i	$-0.513495\pi$
$349$	−1.58359	−0.0847677	−0.0423839	+	0.999101i	$0.513495\pi$
$350$	0	0
$351$	0	0
$352$	−5.61803	−0.299442
$353$	24.5279	1.30549	0.652743	−	0.757579i	$-0.273618\pi$
$353$	24.5279	1.30549	0.652743	+	0.757579i	$0.273618\pi$
$354$	0	0
$355$	2.47214	0.131207
$356$	7.23607	0.383511
$357$	0	0
$358$	−5.52786	−0.292157
$359$	−23.4164	−1.23587	−0.617935	−	0.786229i	$-0.712031\pi$
$359$	−23.4164	−1.23587	−0.617935	+	0.786229i	$0.712031\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0.472136	0.0248149
$363$	0	0
$364$	0	0
$365$	−4.52786	−0.236999
$366$	0	0
$367$	19.8885	1.03817	0.519087	−	0.854722i	$-0.326272\pi$
$367$	19.8885	1.03817	0.519087	+	0.854722i	$0.326272\pi$
$368$	−10.5836	−0.551708
$369$	0	0
$370$	−4.32624	−0.224910
$371$	0	0
$372$	0	0
$373$	4.65248	0.240896	0.120448	−	0.992720i	$-0.461567\pi$
$373$	4.65248	0.240896	0.120448	+	0.992720i	$0.461567\pi$
$374$	−3.23607	−0.167333
$375$	0	0
$376$	−0.527864	−0.0272225
$377$	17.3607	0.894120
$378$	0	0
$379$	−31.1803	−1.60163	−0.800813	−	0.598914i	$-0.795599\pi$
$379$	−31.1803	−1.60163	−0.800813	+	0.598914i	$0.795599\pi$
$380$	−10.8541	−0.556804
$381$	0	0
$382$	6.65248	0.340370
$383$	32.9443	1.68337	0.841687	−	0.539966i	$-0.181563\pi$
$383$	32.9443	1.68337	0.841687	+	0.539966i	$0.181563\pi$
$384$	0	0
$385$	0	0
$386$	9.05573	0.460924
$387$	0	0
$388$	15.7082	0.797463
$389$	−11.0557	−0.560548	−0.280274	−	0.959920i	$-0.590425\pi$
$389$	−11.0557	−0.560548	−0.280274	+	0.959920i	$0.590425\pi$
$390$	0	0
$391$	−29.8885	−1.51153
$392$	0	0
$393$	0	0
$394$	10.1803	0.512878
$395$	−14.4721	−0.728172
$396$	0	0
$397$	−23.1246	−1.16059	−0.580295	−	0.814406i	$-0.697063\pi$
$397$	−23.1246	−1.16059	−0.580295	+	0.814406i	$0.697063\pi$
$398$	2.36068	0.118330
$399$	0	0
$400$	−7.41641	−0.370820
$401$	29.7082	1.48356	0.741778	−	0.670645i	$-0.233983\pi$
$401$	29.7082	1.48356	0.741778	+	0.670645i	$0.233983\pi$
$402$	0	0
$403$	−18.1803	−0.905627
$404$	−29.4164	−1.46352
$405$	0	0
$406$	0	0
$407$	7.00000	0.346977
$408$	0	0
$409$	−21.0557	−1.04114	−0.520569	−	0.853819i	$-0.674280\pi$
$409$	−21.0557	−1.04114	−0.520569	+	0.853819i	$0.674280\pi$
$410$	−1.52786	−0.0754558
$411$	0	0
$412$	28.1803	1.38835
$413$	0	0
$414$	0	0
$415$	6.76393	0.332028
$416$	−19.5066	−0.956389
$417$	0	0
$418$	−4.14590	−0.202783
$419$	−1.18034	−0.0576634	−0.0288317	−	0.999584i	$-0.509179\pi$
$419$	−1.18034	−0.0576634	−0.0288317	+	0.999584i	$0.509179\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	3.34752	0.162955
$423$	0	0
$424$	−27.2361	−1.32270
$425$	−20.9443	−1.01595
$426$	0	0
$427$	0	0
$428$	−6.85410	−0.331306
$429$	0	0
$430$	3.52786	0.170129
$431$	−8.70820	−0.419459	−0.209730	−	0.977759i	$-0.567258\pi$
$431$	−8.70820	−0.419459	−0.209730	+	0.977759i	$0.567258\pi$
$432$	0	0
$433$	10.4721	0.503259	0.251629	−	0.967824i	$-0.419034\pi$
$433$	10.4721	0.503259	0.251629	+	0.967824i	$0.419034\pi$
$434$	0	0
$435$	0	0
$436$	−4.47214	−0.214176
$437$	−38.2918	−1.83175
$438$	0	0
$439$	−11.1803	−0.533609	−0.266804	−	0.963751i	$-0.585968\pi$
$439$	−11.1803	−0.533609	−0.266804	+	0.963751i	$0.585968\pi$
$440$	2.23607	0.106600
$441$	0	0
$442$	−11.2361	−0.534445
$443$	−18.4721	−0.877638	−0.438819	−	0.898576i	$-0.644603\pi$
$443$	−18.4721	−0.877638	−0.438819	+	0.898576i	$0.644603\pi$
$444$	0	0
$445$	−4.47214	−0.212000
$446$	3.70820	0.175589
$447$	0	0
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	2.47214	0.116408
$452$	−0.763932	−0.0359323
$453$	0	0
$454$	−1.23607	−0.0580115
$455$	0	0
$456$	0	0
$457$	15.2361	0.712713	0.356357	−	0.934350i	$-0.384019\pi$
$457$	15.2361	0.712713	0.356357	+	0.934350i	$0.384019\pi$
$458$	−4.47214	−0.208969
$459$	0	0
$460$	9.23607	0.430634
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	−4.81966	−0.223989	−0.111994	−	0.993709i	$-0.535724\pi$
$463$	−4.81966	−0.223989	−0.111994	+	0.993709i	$0.535724\pi$
$464$	−9.27051	−0.430373
$465$	0	0
$466$	9.23607	0.427853
$467$	29.1803	1.35031	0.675153	−	0.737678i	$-0.264078\pi$
$467$	29.1803	1.35031	0.675153	+	0.737678i	$0.264078\pi$
$468$	0	0
$469$	0	0
$470$	0.145898	0.00672977
$471$	0	0
$472$	25.0000	1.15072
$473$	−5.70820	−0.262463
$474$	0	0
$475$	−26.8328	−1.23117
$476$	0	0
$477$	0	0
$478$	−6.25735	−0.286205
$479$	11.7082	0.534961	0.267481	−	0.963563i	$-0.413809\pi$
$479$	11.7082	0.534961	0.267481	+	0.963563i	$0.413809\pi$
$480$	0	0
$481$	24.3050	1.10821
$482$	−16.0344	−0.730349
$483$	0	0
$484$	−1.61803	−0.0735470
$485$	−9.70820	−0.440827
$486$	0	0
$487$	16.9443	0.767818	0.383909	−	0.923371i	$-0.374578\pi$
$487$	16.9443	0.767818	0.383909	+	0.923371i	$0.374578\pi$
$488$	4.47214	0.202444
$489$	0	0
$490$	0	0
$491$	18.1246	0.817952	0.408976	−	0.912545i	$-0.365886\pi$
$491$	18.1246	0.817952	0.408976	+	0.912545i	$0.365886\pi$
$492$	0	0
$493$	−26.1803	−1.17910
$494$	−14.3951	−0.647667
$495$	0	0
$496$	9.70820	0.435911
$497$	0	0
$498$	0	0
$499$	−2.23607	−0.100100	−0.0500501	−	0.998747i	$-0.515938\pi$
$499$	−2.23607	−0.100100	−0.0500501	+	0.998747i	$0.515938\pi$
$500$	14.5623	0.651246
$501$	0	0
$502$	7.49342	0.334448
$503$	2.29180	0.102186	0.0510931	−	0.998694i	$-0.483729\pi$
$503$	2.29180	0.102186	0.0510931	+	0.998694i	$0.483729\pi$
$504$	0	0
$505$	18.1803	0.809015
$506$	3.52786	0.156833
$507$	0	0
$508$	20.4721	0.908304
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	18.7082	0.826794
$513$	0	0
$514$	−4.32624	−0.190822
$515$	−17.4164	−0.767459
$516$	0	0
$517$	−0.236068	−0.0103823
$518$	0	0
$519$	0	0
$520$	7.76393	0.340471
$521$	38.3050	1.67817	0.839085	−	0.544000i	$-0.183091\pi$
$521$	38.3050	1.67817	0.839085	+	0.544000i	$0.183091\pi$
$522$	0	0
$523$	21.6525	0.946797	0.473398	−	0.880848i	$-0.343027\pi$
$523$	21.6525	0.946797	0.473398	+	0.880848i	$0.343027\pi$
$524$	−1.52786	−0.0667451
$525$	0	0
$526$	16.1459	0.703995
$527$	27.4164	1.19428
$528$	0	0
$529$	9.58359	0.416678
$530$	7.52786	0.326990
$531$	0	0
$532$	0	0
$533$	8.58359	0.371797
$534$	0	0
$535$	4.23607	0.183141
$536$	21.8328	0.943034
$537$	0	0
$538$	0.652476	0.0281302
$539$	0	0
$540$	0	0
$541$	−36.9443	−1.58836	−0.794179	−	0.607684i	$-0.792099\pi$
$541$	−36.9443	−1.58836	−0.794179	+	0.607684i	$0.792099\pi$
$542$	−3.27051	−0.140480
$543$	0	0
$544$	29.4164	1.26122
$545$	2.76393	0.118394
$546$	0	0
$547$	14.8328	0.634205	0.317103	−	0.948391i	$-0.397290\pi$
$547$	14.8328	0.634205	0.317103	+	0.948391i	$0.397290\pi$
$548$	31.8885	1.36221
$549$	0	0
$550$	2.47214	0.105412
$551$	−33.5410	−1.42890
$552$	0	0
$553$	0	0
$554$	−4.00000	−0.169944
$555$	0	0
$556$	23.4164	0.993077
$557$	12.5279	0.530823	0.265411	−	0.964135i	$-0.414492\pi$
$557$	12.5279	0.530823	0.265411	+	0.964135i	$0.414492\pi$
$558$	0	0
$559$	−19.8197	−0.838282
$560$	0	0
$561$	0	0
$562$	−7.09017	−0.299081
$563$	34.6525	1.46043	0.730214	−	0.683219i	$-0.239420\pi$
$563$	34.6525	1.46043	0.730214	+	0.683219i	$0.239420\pi$
$564$	0	0
$565$	0.472136	0.0198629
$566$	8.50658	0.357558
$567$	0	0
$568$	−5.52786	−0.231944
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	17.5279	0.733518	0.366759	−	0.930316i	$-0.380467\pi$
$571$	17.5279	0.733518	0.366759	+	0.930316i	$0.380467\pi$
$572$	−5.61803	−0.234902
$573$	0	0
$574$	0	0
$575$	22.8328	0.952194
$576$	0	0
$577$	−7.34752	−0.305881	−0.152941	−	0.988235i	$-0.548874\pi$
$577$	−7.34752	−0.305881	−0.152941	+	0.988235i	$0.548874\pi$
$578$	6.43769	0.267773
$579$	0	0
$580$	8.09017	0.335926
$581$	0	0
$582$	0	0
$583$	−12.1803	−0.504458
$584$	10.1246	0.418959
$585$	0	0
$586$	−9.88854	−0.408492
$587$	46.0132	1.89917	0.949583	−	0.313515i	$-0.101507\pi$
$587$	46.0132	1.89917	0.949583	+	0.313515i	$0.101507\pi$
$588$	0	0
$589$	35.1246	1.44728
$590$	−6.90983	−0.284473
$591$	0	0
$592$	−12.9787	−0.533422
$593$	−22.8328	−0.937631	−0.468816	−	0.883296i	$-0.655319\pi$
$593$	−22.8328	−0.937631	−0.468816	+	0.883296i	$0.655319\pi$
$594$	0	0
$595$	0	0
$596$	8.09017	0.331386
$597$	0	0
$598$	12.2492	0.500908
$599$	25.5279	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$599$	25.5279	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$600$	0	0
$601$	−27.0000	−1.10135	−0.550676	−	0.834719i	$-0.685630\pi$
$601$	−27.0000	−1.10135	−0.550676	+	0.834719i	$0.685630\pi$
$602$	0	0
$603$	0	0
$604$	22.9443	0.933589
$605$	1.00000	0.0406558
$606$	0	0
$607$	−6.81966	−0.276801	−0.138401	−	0.990376i	$-0.544196\pi$
$607$	−6.81966	−0.276801	−0.138401	+	0.990376i	$0.544196\pi$
$608$	37.6869	1.52841
$609$	0	0
$610$	−1.23607	−0.0500469
$611$	−0.819660	−0.0331599
$612$	0	0
$613$	13.5967	0.549167	0.274584	−	0.961563i	$-0.411460\pi$
$613$	13.5967	0.549167	0.274584	+	0.961563i	$0.411460\pi$
$614$	12.9443	0.522388
$615$	0	0
$616$	0	0
$617$	−32.4721	−1.30728	−0.653639	−	0.756806i	$-0.726759\pi$
$617$	−32.4721	−1.30728	−0.653639	+	0.756806i	$0.726759\pi$
$618$	0	0
$619$	44.0689	1.77128	0.885639	−	0.464374i	$-0.153721\pi$
$619$	44.0689	1.77128	0.885639	+	0.464374i	$0.153721\pi$
$620$	−8.47214	−0.340249
$621$	0	0
$622$	6.11146	0.245047
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	−15.2361	−0.608956
$627$	0	0
$628$	−24.9443	−0.995385
$629$	−36.6525	−1.46143
$630$	0	0
$631$	44.3607	1.76597	0.882985	−	0.469400i	$-0.155530\pi$
$631$	44.3607	1.76597	0.882985	+	0.469400i	$0.155530\pi$
$632$	32.3607	1.28724
$633$	0	0
$634$	−14.9443	−0.593513
$635$	−12.6525	−0.502098
$636$	0	0
$637$	0	0
$638$	3.09017	0.122341
$639$	0	0
$640$	−11.3820	−0.449912
$641$	46.5410	1.83826	0.919130	−	0.393955i	$-0.128893\pi$
$641$	46.5410	1.83826	0.919130	+	0.393955i	$0.128893\pi$
$642$	0	0
$643$	47.9574	1.89126	0.945628	−	0.325250i	$-0.105448\pi$
$643$	47.9574	1.89126	0.945628	+	0.325250i	$0.105448\pi$
$644$	0	0
$645$	0	0
$646$	21.7082	0.854098
$647$	12.3475	0.485431	0.242716	−	0.970097i	$-0.421962\pi$
$647$	12.3475	0.485431	0.242716	+	0.970097i	$0.421962\pi$
$648$	0	0
$649$	11.1803	0.438867
$650$	8.58359	0.336676
$651$	0	0
$652$	36.7426	1.43895
$653$	44.9443	1.75881	0.879403	−	0.476079i	$-0.157942\pi$
$653$	44.9443	1.75881	0.879403	+	0.476079i	$0.157942\pi$
$654$	0	0
$655$	0.944272	0.0368958
$656$	−4.58359	−0.178959
$657$	0	0
$658$	0	0
$659$	−23.5410	−0.917028	−0.458514	−	0.888687i	$-0.651618\pi$
$659$	−23.5410	−0.917028	−0.458514	+	0.888687i	$0.651618\pi$
$660$	0	0
$661$	−40.5410	−1.57686	−0.788431	−	0.615123i	$-0.789106\pi$
$661$	−40.5410	−1.57686	−0.788431	+	0.615123i	$0.789106\pi$
$662$	−7.05573	−0.274229
$663$	0	0
$664$	−15.1246	−0.586949
$665$	0	0
$666$	0	0
$667$	28.5410	1.10511
$668$	36.6525	1.41813
$669$	0	0
$670$	−6.03444	−0.233131
$671$	2.00000	0.0772091
$672$	0	0
$673$	3.59675	0.138644	0.0693222	−	0.997594i	$-0.477916\pi$
$673$	3.59675	0.138644	0.0693222	+	0.997594i	$0.477916\pi$
$674$	−5.05573	−0.194739
$675$	0	0
$676$	1.52786	0.0587640
$677$	19.3050	0.741950	0.370975	−	0.928643i	$-0.379024\pi$
$677$	19.3050	0.741950	0.370975	+	0.928643i	$0.379024\pi$
$678$	0	0
$679$	0	0
$680$	−11.7082	−0.448989
$681$	0	0
$682$	−3.23607	−0.123915
$683$	−37.0132	−1.41627	−0.708135	−	0.706078i	$-0.750463\pi$
$683$	−37.0132	−1.41627	−0.708135	+	0.706078i	$0.750463\pi$
$684$	0	0
$685$	−19.7082	−0.753012
$686$	0	0
$687$	0	0
$688$	10.5836	0.403496
$689$	−42.2918	−1.61119
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	2.47214	0.0939765
$693$	0	0
$694$	−17.3050	−0.656887
$695$	−14.4721	−0.548959
$696$	0	0
$697$	−12.9443	−0.490299
$698$	−0.978714	−0.0370449
$699$	0	0
$700$	0	0
$701$	−4.11146	−0.155288	−0.0776438	−	0.996981i	$-0.524740\pi$
$701$	−4.11146	−0.155288	−0.0776438	+	0.996981i	$0.524740\pi$
$702$	0	0
$703$	−46.9574	−1.77103
$704$	0.236068	0.00889715
$705$	0	0
$706$	15.1591	0.570519
$707$	0	0
$708$	0	0
$709$	−49.7214	−1.86732	−0.933662	−	0.358154i	$-0.883406\pi$
$709$	−49.7214	−1.86732	−0.933662	+	0.358154i	$0.883406\pi$
$710$	1.52786	0.0573397
$711$	0	0
$712$	10.0000	0.374766
$713$	−29.8885	−1.11933
$714$	0	0
$715$	3.47214	0.129851
$716$	14.4721	0.540849
$717$	0	0
$718$	−14.4721	−0.540095
$719$	7.76393	0.289546	0.144773	−	0.989465i	$-0.453755\pi$
$719$	7.76393	0.289546	0.144773	+	0.989465i	$0.453755\pi$
$720$	0	0
$721$	0	0
$722$	16.0689	0.598022
$723$	0	0
$724$	−1.23607	−0.0459381
$725$	20.0000	0.742781
$726$	0	0
$727$	−24.1803	−0.896799	−0.448400	−	0.893833i	$-0.648006\pi$
$727$	−24.1803	−0.896799	−0.448400	+	0.893833i	$0.648006\pi$
$728$	0	0
$729$	0	0
$730$	−2.79837	−0.103572
$731$	29.8885	1.10547
$732$	0	0
$733$	8.11146	0.299603	0.149802	−	0.988716i	$-0.452136\pi$
$733$	8.11146	0.299603	0.149802	+	0.988716i	$0.452136\pi$
$734$	12.2918	0.453698
$735$	0	0
$736$	−32.0689	−1.18207
$737$	9.76393	0.359659
$738$	0	0
$739$	34.0689	1.25324	0.626622	−	0.779323i	$-0.284437\pi$
$739$	34.0689	1.25324	0.626622	+	0.779323i	$0.284437\pi$
$740$	11.3262	0.416361
$741$	0	0
$742$	0	0
$743$	−2.81966	−0.103443	−0.0517216	−	0.998662i	$-0.516471\pi$
$743$	−2.81966	−0.103443	−0.0517216	+	0.998662i	$0.516471\pi$
$744$	0	0
$745$	−5.00000	−0.183186
$746$	2.87539	0.105275
$747$	0	0
$748$	8.47214	0.309772
$749$	0	0
$750$	0	0
$751$	39.7639	1.45101	0.725503	−	0.688219i	$-0.241607\pi$
$751$	39.7639	1.45101	0.725503	+	0.688219i	$0.241607\pi$
$752$	0.437694	0.0159611
$753$	0	0
$754$	10.7295	0.390745
$755$	−14.1803	−0.516075
$756$	0	0
$757$	−51.7214	−1.87984	−0.939922	−	0.341388i	$-0.889103\pi$
$757$	−51.7214	−1.87984	−0.939922	+	0.341388i	$0.889103\pi$
$758$	−19.2705	−0.699936
$759$	0	0
$760$	−15.0000	−0.544107
$761$	27.7771	1.00692	0.503459	−	0.864019i	$-0.332060\pi$
$761$	27.7771	1.00692	0.503459	+	0.864019i	$0.332060\pi$
$762$	0	0
$763$	0	0
$764$	−17.4164	−0.630104
$765$	0	0
$766$	20.3607	0.735661
$767$	38.8197	1.40170
$768$	0	0
$769$	−13.9443	−0.502843	−0.251422	−	0.967878i	$-0.580898\pi$
$769$	−13.9443	−0.502843	−0.251422	+	0.967878i	$0.580898\pi$
$770$	0	0
$771$	0	0
$772$	−23.7082	−0.853277
$773$	−5.47214	−0.196819	−0.0984095	−	0.995146i	$-0.531376\pi$
$773$	−5.47214	−0.196819	−0.0984095	+	0.995146i	$0.531376\pi$
$774$	0	0
$775$	−20.9443	−0.752340
$776$	21.7082	0.779279
$777$	0	0
$778$	−6.83282	−0.244968
$779$	−16.5836	−0.594169
$780$	0	0
$781$	−2.47214	−0.0884600
$782$	−18.4721	−0.660562
$783$	0	0
$784$	0	0
$785$	15.4164	0.550235
$786$	0	0
$787$	−3.65248	−0.130197	−0.0650984	−	0.997879i	$-0.520736\pi$
$787$	−3.65248	−0.130197	−0.0650984	+	0.997879i	$0.520736\pi$
$788$	−26.6525	−0.949455
$789$	0	0
$790$	−8.94427	−0.318223
$791$	0	0
$792$	0	0
$793$	6.94427	0.246598
$794$	−14.2918	−0.507197
$795$	0	0
$796$	−6.18034	−0.219056
$797$	−2.52786	−0.0895415	−0.0447708	−	0.998997i	$-0.514256\pi$
$797$	−2.52786	−0.0895415	−0.0447708	+	0.998997i	$0.514256\pi$
$798$	0	0
$799$	1.23607	0.0437289
$800$	−22.4721	−0.794510
$801$	0	0
$802$	18.3607	0.648338
$803$	4.52786	0.159785
$804$	0	0
$805$	0	0
$806$	−11.2361	−0.395774
$807$	0	0
$808$	−40.6525	−1.43015
$809$	56.3050	1.97958	0.989788	−	0.142545i	$-0.0455285\pi$
$809$	56.3050	1.97958	0.989788	+	0.142545i	$0.0455285\pi$
$810$	0	0
$811$	−15.2918	−0.536968	−0.268484	−	0.963284i	$-0.586523\pi$
$811$	−15.2918	−0.536968	−0.268484	+	0.963284i	$0.586523\pi$
$812$	0	0
$813$	0	0
$814$	4.32624	0.151635
$815$	−22.7082	−0.795434
$816$	0	0
$817$	38.2918	1.33966
$818$	−13.0132	−0.454994
$819$	0	0
$820$	4.00000	0.139686
$821$	7.47214	0.260779	0.130390	−	0.991463i	$-0.458377\pi$
$821$	7.47214	0.260779	0.130390	+	0.991463i	$0.458377\pi$
$822$	0	0
$823$	−17.1803	−0.598869	−0.299435	−	0.954117i	$-0.596798\pi$
$823$	−17.1803	−0.598869	−0.299435	+	0.954117i	$0.596798\pi$
$824$	38.9443	1.35669
$825$	0	0
$826$	0	0
$827$	−12.3475	−0.429365	−0.214683	−	0.976684i	$-0.568872\pi$
$827$	−12.3475	−0.429365	−0.214683	+	0.976684i	$0.568872\pi$
$828$	0	0
$829$	−35.7771	−1.24259	−0.621295	−	0.783577i	$-0.713393\pi$
$829$	−35.7771	−1.24259	−0.621295	+	0.783577i	$0.713393\pi$
$830$	4.18034	0.145102
$831$	0	0
$832$	0.819660	0.0284166
$833$	0	0
$834$	0	0
$835$	−22.6525	−0.783921
$836$	10.8541	0.375397
$837$	0	0
$838$	−0.729490	−0.0251998
$839$	−23.5410	−0.812726	−0.406363	−	0.913712i	$-0.633203\pi$
$839$	−23.5410	−0.812726	−0.406363	+	0.913712i	$0.633203\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−8.03444	−0.276885
$843$	0	0
$844$	−8.76393	−0.301667
$845$	−0.944272	−0.0324839
$846$	0	0
$847$	0	0
$848$	22.5836	0.775524
$849$	0	0
$850$	−12.9443	−0.443985
$851$	39.9574	1.36972
$852$	0	0
$853$	29.4164	1.00720	0.503599	−	0.863937i	$-0.332009\pi$
$853$	29.4164	1.00720	0.503599	+	0.863937i	$0.332009\pi$
$854$	0	0
$855$	0	0
$856$	−9.47214	−0.323751
$857$	0.111456	0.00380727	0.00190364	−	0.999998i	$-0.499394\pi$
$857$	0.111456	0.00380727	0.00190364	+	0.999998i	$0.499394\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−9.23607	−0.314947
$861$	0	0
$862$	−5.38197	−0.183310
$863$	43.2361	1.47177	0.735886	−	0.677105i	$-0.236766\pi$
$863$	43.2361	1.47177	0.735886	+	0.677105i	$0.236766\pi$
$864$	0	0
$865$	−1.52786	−0.0519489
$866$	6.47214	0.219932
$867$	0	0
$868$	0	0
$869$	14.4721	0.490934
$870$	0	0
$871$	33.9017	1.14872
$872$	−6.18034	−0.209293
$873$	0	0
$874$	−23.6656	−0.800502
$875$	0	0
$876$	0	0
$877$	4.58359	0.154777	0.0773885	−	0.997001i	$-0.475342\pi$
$877$	4.58359	0.154777	0.0773885	+	0.997001i	$0.475342\pi$
$878$	−6.90983	−0.233195
$879$	0	0
$880$	−1.85410	−0.0625018
$881$	54.8885	1.84924	0.924621	−	0.380888i	$-0.124382\pi$
$881$	54.8885	1.84924	0.924621	+	0.380888i	$0.124382\pi$
$882$	0	0
$883$	−14.8197	−0.498721	−0.249361	−	0.968411i	$-0.580220\pi$
$883$	−14.8197	−0.498721	−0.249361	+	0.968411i	$0.580220\pi$
$884$	29.4164	0.989381
$885$	0	0
$886$	−11.4164	−0.383542
$887$	10.7639	0.361417	0.180709	−	0.983537i	$-0.442161\pi$
$887$	10.7639	0.361417	0.180709	+	0.983537i	$0.442161\pi$
$888$	0	0
$889$	0	0
$890$	−2.76393	−0.0926472
$891$	0	0
$892$	−9.70820	−0.325055
$893$	1.58359	0.0529929
$894$	0	0
$895$	−8.94427	−0.298974
$896$	0	0
$897$	0	0
$898$	−12.3607	−0.412481
$899$	−26.1803	−0.873163
$900$	0	0
$901$	63.7771	2.12472
$902$	1.52786	0.0508723
$903$	0	0
$904$	−1.05573	−0.0351130
$905$	0.763932	0.0253940
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	3.23607	0.107393
$909$	0	0
$910$	0	0
$911$	14.5836	0.483176	0.241588	−	0.970379i	$-0.422332\pi$
$911$	14.5836	0.483176	0.241588	+	0.970379i	$0.422332\pi$
$912$	0	0
$913$	−6.76393	−0.223853
$914$	9.41641	0.311467
$915$	0	0
$916$	11.7082	0.386850
$917$	0	0
$918$	0	0
$919$	39.5967	1.30618	0.653088	−	0.757282i	$-0.273473\pi$
$919$	39.5967	1.30618	0.653088	+	0.757282i	$0.273473\pi$
$920$	12.7639	0.420814
$921$	0	0
$922$	−17.3050	−0.569908
$923$	−8.58359	−0.282532
$924$	0	0
$925$	28.0000	0.920634
$926$	−2.97871	−0.0978866
$927$	0	0
$928$	−28.0902	−0.922105
$929$	12.8885	0.422859	0.211430	−	0.977393i	$-0.432188\pi$
$929$	12.8885	0.422859	0.211430	+	0.977393i	$0.432188\pi$
$930$	0	0
$931$	0	0
$932$	−24.1803	−0.792053
$933$	0	0
$934$	18.0344	0.590105
$935$	−5.23607	−0.171238
$936$	0	0
$937$	39.8885	1.30310	0.651551	−	0.758605i	$-0.274119\pi$
$937$	39.8885	1.30310	0.651551	+	0.758605i	$0.274119\pi$
$938$	0	0
$939$	0	0
$940$	−0.381966	−0.0124584
$941$	−60.3607	−1.96770	−0.983851	−	0.178990i	$-0.942717\pi$
$941$	−60.3607	−1.96770	−0.983851	+	0.178990i	$0.942717\pi$
$942$	0	0
$943$	14.1115	0.459532
$944$	−20.7295	−0.674687
$945$	0	0
$946$	−3.52786	−0.114701
$947$	5.41641	0.176010	0.0880048	−	0.996120i	$-0.471951\pi$
$947$	5.41641	0.176010	0.0880048	+	0.996120i	$0.471951\pi$
$948$	0	0
$949$	15.7214	0.510337
$950$	−16.5836	−0.538043
$951$	0	0
$952$	0	0
$953$	−44.7771	−1.45047	−0.725236	−	0.688500i	$-0.758269\pi$
$953$	−44.7771	−1.45047	−0.725236	+	0.688500i	$0.758269\pi$
$954$	0	0
$955$	10.7639	0.348313
$956$	16.3820	0.529831
$957$	0	0
$958$	7.23607	0.233787
$959$	0	0
$960$	0	0
$961$	−3.58359	−0.115600
$962$	15.0213	0.484306
$963$	0	0
$964$	41.9787	1.35204
$965$	14.6525	0.471680
$966$	0	0
$967$	−61.1935	−1.96785	−0.983925	−	0.178582i	$-0.942849\pi$
$967$	−61.1935	−1.96785	−0.983925	+	0.178582i	$0.942849\pi$
$968$	−2.23607	−0.0718699
$969$	0	0
$970$	−6.00000	−0.192648
$971$	32.1246	1.03093	0.515464	−	0.856911i	$-0.327620\pi$
$971$	32.1246	1.03093	0.515464	+	0.856911i	$0.327620\pi$
$972$	0	0
$973$	0	0
$974$	10.4721	0.335549
$975$	0	0
$976$	−3.70820	−0.118697
$977$	−4.18034	−0.133741	−0.0668705	−	0.997762i	$-0.521301\pi$
$977$	−4.18034	−0.133741	−0.0668705	+	0.997762i	$0.521301\pi$
$978$	0	0
$979$	4.47214	0.142930
$980$	0	0
$981$	0	0
$982$	11.2016	0.357458
$983$	27.4164	0.874448	0.437224	−	0.899353i	$-0.355962\pi$
$983$	27.4164	0.874448	0.437224	+	0.899353i	$0.355962\pi$
$984$	0	0
$985$	16.4721	0.524846
$986$	−16.1803	−0.515287
$987$	0	0
$988$	37.6869	1.19898
$989$	−32.5836	−1.03610
$990$	0	0
$991$	−29.1803	−0.926944	−0.463472	−	0.886112i	$-0.653397\pi$
$991$	−29.1803	−0.926944	−0.463472	+	0.886112i	$0.653397\pi$
$992$	29.4164	0.933972
$993$	0	0
$994$	0	0
$995$	3.81966	0.121091
$996$	0	0
$997$	−26.9443	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$997$	−26.9443	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$998$	−1.38197	−0.0437454
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.w.1.2		2
3.2	odd	2		1617.2.a.p.1.1		2
7.6	odd	2		693.2.a.f.1.2		2
21.20	even	2		231.2.a.c.1.1	✓	2
77.76	even	2		7623.2.a.bm.1.1		2
84.83	odd	2		3696.2.a.be.1.2		2
105.104	even	2		5775.2.a.be.1.2		2
231.230	odd	2		2541.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.c.1.1	✓	2	21.20	even	2
693.2.a.f.1.2		2	7.6	odd	2
1617.2.a.p.1.1		2	3.2	odd	2
2541.2.a.t.1.2		2	231.230	odd	2
3696.2.a.be.1.2		2	84.83	odd	2
4851.2.a.w.1.2		2	1.1	even	1	trivial
5775.2.a.be.1.2		2	105.104	even	2
7623.2.a.bm.1.1		2	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)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +0.618034 q^{10} -1.00000 q^{11} -3.47214 q^{13} +1.85410 q^{16} +5.23607 q^{17} +6.70820 q^{19} -1.61803 q^{20} -0.618034 q^{22} -5.70820 q^{23} -4.00000 q^{25} -2.14590 q^{26} -5.00000 q^{29} +5.23607 q^{31} +5.61803 q^{32} +3.23607 q^{34} -7.00000 q^{37} +4.14590 q^{38} -2.23607 q^{40} -2.47214 q^{41} +5.70820 q^{43} +1.61803 q^{44} -3.52786 q^{46} +0.236068 q^{47} -2.47214 q^{50} +5.61803 q^{52} +12.1803 q^{53} -1.00000 q^{55} -3.09017 q^{58} -11.1803 q^{59} -2.00000 q^{61} +3.23607 q^{62} -0.236068 q^{64} -3.47214 q^{65} -9.76393 q^{67} -8.47214 q^{68} +2.47214 q^{71} -4.52786 q^{73} -4.32624 q^{74} -10.8541 q^{76} -14.4721 q^{79} +1.85410 q^{80} -1.52786 q^{82} +6.76393 q^{83} +5.23607 q^{85} +3.52786 q^{86} +2.23607 q^{88} -4.47214 q^{89} +9.23607 q^{92} +0.145898 q^{94} +6.70820 q^{95} -9.70820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5	\( 2 q - q^{2} - q^{4} + 2 q^{5} - q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{16} + 6 q^{17} - q^{20} + q^{22} + 2 q^{23} - 8 q^{25} - 11 q^{26} - 10 q^{29} + 6 q^{31} + 9 q^{32} + 2 q^{34} - 14 q^{37} + 15 q^{38} + 4 q^{41} - 2 q^{43} + q^{44} - 16 q^{46} - 4 q^{47} + 4 q^{50} + 9 q^{52} + 2 q^{53} - 2 q^{55} + 5 q^{58} - 4 q^{61} + 2 q^{62} + 4 q^{64} + 2 q^{65} - 24 q^{67} - 8 q^{68} - 4 q^{71} - 18 q^{73} + 7 q^{74} - 15 q^{76} - 20 q^{79} - 3 q^{80} - 12 q^{82} + 18 q^{83} + 6 q^{85} + 16 q^{86} + 14 q^{92} + 7 q^{94} - 6 q^{97}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - q^10 - 2 * q^11 + 2 * q^13 - 3 * q^16 + 6 * q^17 - q^20 + q^22 + 2 * q^23 - 8 * q^25 - 11 * q^26 - 10 * q^29 + 6 * q^31 + 9 * q^32 + 2 * q^34 - 14 * q^37 + 15 * q^38 + 4 * q^41 - 2 * q^43 + q^44 - 16 * q^46 - 4 * q^47 + 4 * q^50 + 9 * q^52 + 2 * q^53 - 2 * q^55 + 5 * q^58 - 4 * q^61 + 2 * q^62 + 4 * q^64 + 2 * q^65 - 24 * q^67 - 8 * q^68 - 4 * q^71 - 18 * q^73 + 7 * q^74 - 15 * q^76 - 20 * q^79 - 3 * q^80 - 12 * q^82 + 18 * q^83 + 6 * q^85 + 16 * q^86 + 14 * q^92 + 7 * q^94 - 6 * q^97

Introduction

L-functions

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