↑

Properties

Label 475.2.j.b.49.6
Level $475$
Weight $2$
Character 475.49
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [475,2,Mod(49,475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("475.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.50712647503417344.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 13x^{10} + 119x^{8} - 552x^{6} + 1863x^{4} - 2450x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.6
Root \(2.17114 + 1.25351i\) of defining polynomial
Character \(\chi\) \(=\) 475.49
Dual form 475.2.j.b.349.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.17114 + 1.25351i) q^{2} +(1.05818 + 0.610938i) q^{3} +(2.14257 + 3.71104i) q^{4} +(1.53163 + 2.65287i) q^{6} -0.221876i q^{7} +5.72889i q^{8} +(-0.753509 - 1.30512i) q^{9} -0.778124 q^{11} +5.23591i q^{12} +(4.33013 - 2.50000i) q^{13} +(0.278124 - 0.481725i) q^{14} +(-2.89608 + 5.01616i) q^{16} +(-6.12912 - 3.53865i) q^{17} -3.77812i q^{18} +(-1.33281 + 4.15013i) q^{19} +(0.135553 - 0.234784i) q^{21} +(-1.68942 - 0.975385i) q^{22} +(-6.99515 + 4.03865i) q^{23} +(-3.50000 + 6.06218i) q^{24} +12.5351 q^{26} -5.50702i q^{27} +(0.823392 - 0.475385i) q^{28} +(0.110938 + 0.192150i) q^{29} +2.50702 q^{31} +(-2.65287 + 1.53163i) q^{32} +(-0.823392 - 0.475385i) q^{33} +(-8.87147 - 15.3658i) q^{34} +(3.22889 - 5.59261i) q^{36} -1.90466i q^{37} +(-8.09596 + 7.33983i) q^{38} +6.10938 q^{39} +(3.61796 - 6.26648i) q^{41} +(0.588608 - 0.339833i) q^{42} +(6.32128 + 3.64959i) q^{43} +(-1.66719 - 2.88765i) q^{44} -20.2500 q^{46} +(-2.41808 + 1.39608i) q^{47} +(-6.12912 + 3.53865i) q^{48} +6.95077 q^{49} +(-4.32379 - 7.48903i) q^{51} +(18.5552 + 10.7129i) q^{52} +(-3.79361 + 2.19024i) q^{53} +(6.90310 - 11.9565i) q^{54} +1.27111 q^{56} +(-3.94583 + 3.57730i) q^{57} +0.556248i q^{58} +(-1.39608 + 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(5.44309 + 3.14257i) q^{62} +(-0.289574 + 0.167186i) q^{63} +3.90466 q^{64} +(-1.19180 - 2.06426i) q^{66} +(-9.15414 + 5.28514i) q^{67} -30.3273i q^{68} -9.86946 q^{69} +(4.92070 - 8.52289i) q^{71} +(7.47687 - 4.31678i) q^{72} +(-12.1913 - 7.03865i) q^{73} +(2.38750 - 4.13528i) q^{74} +(-18.2570 + 3.94583i) q^{76} +0.172647i q^{77} +(13.2643 + 7.65817i) q^{78} +(0.792161 - 1.37206i) q^{79} +(1.10392 - 1.91204i) q^{81} +(15.7102 - 9.07028i) q^{82} +9.52106i q^{83} +1.16172 q^{84} +(9.14959 + 15.8476i) q^{86} +0.271105i q^{87} -4.45779i q^{88} +(1.57028 + 2.71981i) q^{89} +(-0.554690 - 0.960752i) q^{91} +(-29.9752 - 17.3062i) q^{92} +(2.65287 + 1.53163i) q^{93} -7.00000 q^{94} -3.74293 q^{96} +(5.51004 + 3.18122i) q^{97} +(15.0911 + 8.71286i) q^{98} +(0.586324 + 1.01554i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 14 q^{4} + 12 q^{6} + 8 q^{9} - 20 q^{11} + 14 q^{14} - 6 q^{16} + 24 q^{21} - 42 q^{24} - 20 q^{26} - 4 q^{29} - 4 q^{31} - 50 q^{34} - 6 q^{36} + 20 q^{39} + 4 q^{41} - 36 q^{44} - 96 q^{46} + 28 q^{49}+ \cdots - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.17114 + 1.25351i 1.53523 + 0.886365i 0.999108 + 0.0422238i \(0.0134442\pi\)
0.536121 + 0.844141i \(0.319889\pi\)
\(3\) 1.05818 + 0.610938i 0.610938 + 0.352725i 0.773333 − 0.634001i \(-0.218588\pi\)
−0.162394 + 0.986726i \(0.551922\pi\)
\(4\) 2.14257 + 3.71104i 1.07129 + 1.85552i
\(5\) 0 0
\(6\) 1.53163 + 2.65287i 0.625287 + 1.08303i
\(7\) − 0.221876i − 0.0838613i −0.999121 − 0.0419307i \(-0.986649\pi\)
0.999121 − 0.0419307i \(-0.0133509\pi\)
\(8\) 5.72889i 2.02547i
\(9\) −0.753509 − 1.30512i −0.251170 − 0.435039i
\(10\) 0 0
\(11\) −0.778124 −0.234613 −0.117307 − 0.993096i \(-0.537426\pi\)
−0.117307 + 0.993096i \(0.537426\pi\)
\(12\) 5.23591i 1.51148i
\(13\) 4.33013 − 2.50000i 1.20096 − 0.693375i 0.240192 − 0.970725i \(-0.422790\pi\)
0.960769 + 0.277350i \(0.0894562\pi\)
\(14\) 0.278124 − 0.481725i 0.0743317 − 0.128746i
\(15\) 0 0
\(16\) −2.89608 + 5.01616i −0.724020 + 1.25404i
\(17\) −6.12912 − 3.53865i −1.48653 − 0.858249i −0.486649 − 0.873598i \(-0.661781\pi\)
−0.999882 + 0.0153485i \(0.995114\pi\)
\(18\) − 3.77812i − 0.890512i
\(19\) −1.33281 + 4.15013i −0.305769 + 0.952106i
\(20\) 0 0
\(21\) 0.135553 − 0.234784i 0.0295800 − 0.0512341i
\(22\) −1.68942 − 0.975385i −0.360185 − 0.207953i
\(23\) −6.99515 + 4.03865i −1.45859 + 0.842117i −0.998942 − 0.0459843i \(-0.985358\pi\)
−0.459647 + 0.888101i \(0.652024\pi\)
\(24\) −3.50000 + 6.06218i −0.714435 + 1.23744i
\(25\) 0 0
\(26\) 12.5351 2.45833
\(27\) − 5.50702i − 1.05983i
\(28\) 0.823392 − 0.475385i 0.155606 − 0.0898394i
\(29\) 0.110938 + 0.192150i 0.0206007 + 0.0356814i 0.876142 − 0.482053i \(-0.160109\pi\)
−0.855541 + 0.517735i \(0.826775\pi\)
\(30\) 0 0
\(31\) 2.50702 0.450274 0.225137 − 0.974327i \(-0.427717\pi\)
0.225137 + 0.974327i \(0.427717\pi\)
\(32\) −2.65287 + 1.53163i −0.468965 + 0.270757i
\(33\) −0.823392 − 0.475385i −0.143334 − 0.0827540i
\(34\) −8.87147 − 15.3658i −1.52144 − 2.63522i
\(35\) 0 0
\(36\) 3.22889 − 5.59261i 0.538149 − 0.932102i
\(37\) − 1.90466i − 0.313124i −0.987668 − 0.156562i \(-0.949959\pi\)
0.987668 − 0.156562i \(-0.0500410\pi\)
\(38\) −8.09596 + 7.33983i −1.31334 + 1.19068i
\(39\) 6.10938 0.978284
\(40\) 0 0
\(41\) 3.61796 − 6.26648i 0.565030 − 0.978661i −0.432017 − 0.901865i \(-0.642198\pi\)
0.997047 − 0.0767950i \(-0.0244687\pi\)
\(42\) 0.588608 − 0.339833i 0.0908242 − 0.0524374i
\(43\) 6.32128 + 3.64959i 0.963985 + 0.556557i 0.897397 − 0.441223i \(-0.145455\pi\)
0.0665881 + 0.997781i \(0.478789\pi\)
\(44\) −1.66719 − 2.88765i −0.251338 − 0.435330i
\(45\) 0 0
\(46\) −20.2500 −2.98569
\(47\) −2.41808 + 1.39608i −0.352714 + 0.203639i −0.665880 − 0.746059i \(-0.731944\pi\)
0.313166 + 0.949698i \(0.398610\pi\)
\(48\) −6.12912 + 3.53865i −0.884663 + 0.510760i
\(49\) 6.95077 0.992967
\(50\) 0 0
\(51\) −4.32379 − 7.48903i −0.605452 − 1.04867i
\(52\) 18.5552 + 10.7129i 2.57314 + 1.48561i
\(53\) −3.79361 + 2.19024i −0.521093 + 0.300853i −0.737382 − 0.675476i \(-0.763938\pi\)
0.216289 + 0.976329i \(0.430605\pi\)
\(54\) 6.90310 − 11.9565i 0.939393 − 1.62708i
\(55\) 0 0
\(56\) 1.27111 0.169859
\(57\) −3.94583 + 3.57730i −0.522637 + 0.473825i
\(58\) 0.556248i 0.0730389i
\(59\) −1.39608 + 2.41808i −0.181754 + 0.314808i −0.942478 − 0.334268i \(-0.891511\pi\)
0.760724 + 0.649076i \(0.224844\pi\)
\(60\) 0 0
\(61\) 6.29216 + 10.8983i 0.805629 + 1.39539i 0.915866 + 0.401484i \(0.131506\pi\)
−0.110237 + 0.993905i \(0.535161\pi\)
\(62\) 5.44309 + 3.14257i 0.691274 + 0.399107i
\(63\) −0.289574 + 0.167186i −0.0364829 + 0.0210634i
\(64\) 3.90466 0.488082
\(65\) 0 0
\(66\) −1.19180 − 2.06426i −0.146700 − 0.254093i
\(67\) −9.15414 + 5.28514i −1.11836 + 0.645683i −0.940981 − 0.338460i \(-0.890094\pi\)
−0.177375 + 0.984143i \(0.556761\pi\)
\(68\) − 30.3273i − 3.67772i
\(69\) −9.86946 −1.18814
\(70\) 0 0
\(71\) 4.92070 − 8.52289i 0.583979 − 1.01148i −0.411023 − 0.911625i \(-0.634828\pi\)
0.995002 − 0.0998563i \(-0.0318383\pi\)
\(72\) 7.47687 − 4.31678i 0.881158 − 0.508737i
\(73\) −12.1913 − 7.03865i −1.42688 − 0.823812i −0.430010 − 0.902824i \(-0.641490\pi\)
−0.996874 + 0.0790121i \(0.974823\pi\)
\(74\) 2.38750 − 4.13528i 0.277542 − 0.480716i
\(75\) 0 0
\(76\) −18.2570 + 3.94583i −2.09422 + 0.452617i
\(77\) 0.172647i 0.0196750i
\(78\) 13.2643 + 7.65817i 1.50189 + 0.867117i
\(79\) 0.792161 − 1.37206i 0.0891251 − 0.154369i −0.818016 − 0.575195i \(-0.804926\pi\)
0.907142 + 0.420826i \(0.138260\pi\)
\(80\) 0 0
\(81\) 1.10392 − 1.91204i 0.122658 − 0.212449i
\(82\) 15.7102 − 9.07028i 1.73490 − 1.00165i
\(83\) 9.52106i 1.04507i 0.852617 + 0.522536i \(0.175014\pi\)
−0.852617 + 0.522536i \(0.824986\pi\)
\(84\) 1.16172 0.126755
\(85\) 0 0
\(86\) 9.14959 + 15.8476i 0.986626 + 1.70889i
\(87\) 0.271105i 0.0290655i
\(88\) − 4.45779i − 0.475202i
\(89\) 1.57028 + 2.71981i 0.166450 + 0.288300i 0.937169 − 0.348875i \(-0.113436\pi\)
−0.770719 + 0.637175i \(0.780103\pi\)
\(90\) 0 0
\(91\) −0.554690 − 0.960752i −0.0581474 − 0.100714i
\(92\) −29.9752 − 17.3062i −3.12513 − 1.80430i
\(93\) 2.65287 + 1.53163i 0.275089 + 0.158823i
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) −3.74293 −0.382011
\(97\) 5.51004 + 3.18122i 0.559460 + 0.323004i 0.752929 − 0.658102i \(-0.228641\pi\)
−0.193469 + 0.981106i \(0.561974\pi\)
\(98\) 15.0911 + 8.71286i 1.52443 + 0.880131i
\(99\) 0.586324 + 1.01554i 0.0589277 + 0.102066i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 475.2.j.b.49.6 12
5.2 odd 4 475.2.e.d.201.1 6
5.3 odd 4 95.2.e.b.11.3 ✓ 6
5.4 even 2 inner 475.2.j.b.49.1 12
15.8 even 4 855.2.k.g.676.1 6
19.7 even 3 inner 475.2.j.b.349.1 12
20.3 even 4 1520.2.q.j.961.2 6
95.7 odd 12 475.2.e.d.26.1 6
95.8 even 12 1805.2.a.g.1.3 3
95.27 even 12 9025.2.a.ba.1.1 3
95.64 even 6 inner 475.2.j.b.349.6 12
95.68 odd 12 1805.2.a.h.1.1 3
95.83 odd 12 95.2.e.b.26.3 yes 6
95.87 odd 12 9025.2.a.z.1.3 3
285.83 even 12 855.2.k.g.406.1 6
380.83 even 12 1520.2.q.j.881.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.b.11.3 ✓ 6 5.3 odd 4
95.2.e.b.26.3 yes 6 95.83 odd 12
475.2.e.d.26.1 6 95.7 odd 12
475.2.e.d.201.1 6 5.2 odd 4
475.2.j.b.49.1 12 5.4 even 2 inner
475.2.j.b.49.6 12 1.1 even 1 trivial
475.2.j.b.349.1 12 19.7 even 3 inner
475.2.j.b.349.6 12 95.64 even 6 inner
855.2.k.g.406.1 6 285.83 even 12
855.2.k.g.676.1 6 15.8 even 4
1520.2.q.j.881.2 6 380.83 even 12
1520.2.q.j.961.2 6 20.3 even 4
1805.2.a.g.1.3 3 95.8 even 12
1805.2.a.h.1.1 3 95.68 odd 12
9025.2.a.z.1.3 3 95.87 odd 12
9025.2.a.ba.1.1 3 95.27 even 12