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Properties

Label 475.2.e.d.201.1
Level $475$
Weight $2$
Character 475.201
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(26,475)] 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("475.26"); 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([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3518667.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) 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: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.1
Root \(-1.25351 + 2.17114i\) of defining polynomial
Character \(\chi\) \(=\) 475.201
Dual form 475.2.e.d.26.1

$q$-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+(-1.25351 + 2.17114i) q^{2} +(0.610938 - 1.05818i) q^{3} +(-2.14257 - 3.71104i) q^{4} +(1.53163 + 2.65287i) q^{6} +0.221876 q^{7} +5.72889 q^{8} +(0.753509 + 1.30512i) q^{9} -0.778124 q^{11} -5.23591 q^{12} +(-2.50000 - 4.33013i) q^{13} +(-0.278124 + 0.481725i) q^{14} +(-2.89608 + 5.01616i) q^{16} +(3.53865 - 6.12912i) q^{17} -3.77812 q^{18} +(1.33281 - 4.15013i) q^{19} +(0.135553 - 0.234784i) q^{21} +(0.975385 - 1.68942i) q^{22} +(4.03865 + 6.99515i) q^{23} +(3.50000 - 6.06218i) q^{24} +12.5351 q^{26} +5.50702 q^{27} +(-0.475385 - 0.823392i) q^{28} +(-0.110938 - 0.192150i) q^{29} +2.50702 q^{31} +(-1.53163 - 2.65287i) q^{32} +(-0.475385 + 0.823392i) q^{33} +(8.87147 + 15.3658i) q^{34} +(3.22889 - 5.59261i) q^{36} +1.90466 q^{37} +(7.33983 + 8.09596i) q^{38} -6.10938 q^{39} +(3.61796 - 6.26648i) q^{41} +(0.339833 + 0.588608i) q^{42} +(3.64959 - 6.32128i) q^{43} +(1.66719 + 2.88765i) q^{44} -20.2500 q^{46} +(-1.39608 - 2.41808i) q^{47} +(3.53865 + 6.12912i) q^{48} -6.95077 q^{49} +(-4.32379 - 7.48903i) q^{51} +(-10.7129 + 18.5552i) q^{52} +(2.19024 + 3.79361i) q^{53} +(-6.90310 + 11.9565i) q^{54} +1.27111 q^{56} +(-3.57730 - 3.94583i) q^{57} +0.556248 q^{58} +(1.39608 - 2.41808i) q^{59} +(6.29216 + 10.8983i) q^{61} +(-3.14257 + 5.44309i) q^{62} +(0.167186 + 0.289574i) q^{63} -3.90466 q^{64} +(-1.19180 - 2.06426i) q^{66} +(-5.28514 - 9.15414i) q^{67} -30.3273 q^{68} +9.86946 q^{69} +(4.92070 - 8.52289i) q^{71} +(4.31678 + 7.47687i) q^{72} +(-7.03865 + 12.1913i) q^{73} +(-2.38750 + 4.13528i) q^{74} +(-18.2570 + 3.94583i) q^{76} -0.172647 q^{77} +(7.65817 - 13.2643i) q^{78} +(-0.792161 + 1.37206i) q^{79} +(1.10392 - 1.91204i) q^{81} +(9.07028 + 15.7102i) q^{82} +9.52106 q^{83} -1.16172 q^{84} +(9.14959 + 15.8476i) q^{86} -0.271105 q^{87} -4.45779 q^{88} +(-1.57028 - 2.71981i) q^{89} +(-0.554690 - 0.960752i) q^{91} +(17.3062 - 29.9752i) q^{92} +(1.53163 - 2.65287i) q^{93} +7.00000 q^{94} -3.74293 q^{96} +(-3.18122 + 5.51004i) q^{97} +(8.71286 - 15.0911i) q^{98} +(-0.586324 - 1.01554i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} - 10 q^{11} + 8 q^{12} - 15 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} - 28 q^{18} + 12 q^{21} - 8 q^{22} + 4 q^{23} + 21 q^{24} - 10 q^{26}+ \cdots + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25351 + 2.17114i −0.886365 + 1.53523i −0.0422238 + 0.999108i \(0.513444\pi\)
−0.844141 + 0.536121i \(0.819889\pi\)
\(3\) 0.610938 − 1.05818i 0.352725 − 0.610938i −0.634001 − 0.773333i \(-0.718588\pi\)
0.986726 + 0.162394i \(0.0519217\pi\)
\(4\) −2.14257 − 3.71104i −1.07129 − 1.85552i
\(5\) 0 0
\(6\) 1.53163 + 2.65287i 0.625287 + 1.08303i
\(7\) 0.221876 0.0838613 0.0419307 − 0.999121i \(-0.486649\pi\)
0.0419307 + 0.999121i \(0.486649\pi\)
\(8\) 5.72889 2.02547
\(9\) 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 \(0.537426\pi\)
\(12\) −5.23591 −1.51148
\(13\) −2.50000 − 4.33013i −0.693375 − 1.20096i −0.970725 − 0.240192i \(-0.922790\pi\)
0.277350 − 0.960769i \(-0.410544\pi\)
\(14\) −0.278124 + 0.481725i −0.0743317 + 0.128746i
\(15\) 0 0
\(16\) −2.89608 + 5.01616i −0.724020 + 1.25404i
\(17\) 3.53865 − 6.12912i 0.858249 − 1.48653i −0.0153485 − 0.999882i \(-0.504886\pi\)
0.873598 − 0.486649i \(-0.161781\pi\)
\(18\) −3.77812 −0.890512
\(19\) 1.33281 − 4.15013i 0.305769 − 0.952106i
\(20\) 0 0
\(21\) 0.135553 − 0.234784i 0.0295800 − 0.0512341i
\(22\) 0.975385 − 1.68942i 0.207953 − 0.360185i
\(23\) 4.03865 + 6.99515i 0.842117 + 1.45859i 0.888101 + 0.459647i \(0.152024\pi\)
−0.0459843 + 0.998942i \(0.514642\pi\)
\(24\) 3.50000 − 6.06218i 0.714435 − 1.23744i
\(25\) 0 0
\(26\) 12.5351 2.45833
\(27\) 5.50702 1.05983
\(28\) −0.475385 − 0.823392i −0.0898394 − 0.155606i
\(29\) −0.110938 − 0.192150i −0.0206007 − 0.0356814i 0.855541 − 0.517735i \(-0.173225\pi\)
−0.876142 + 0.482053i \(0.839891\pi\)
\(30\) 0 0
\(31\) 2.50702 0.450274 0.225137 − 0.974327i \(-0.427717\pi\)
0.225137 + 0.974327i \(0.427717\pi\)
\(32\) −1.53163 − 2.65287i −0.270757 − 0.468965i
\(33\) −0.475385 + 0.823392i −0.0827540 + 0.143334i
\(34\) 8.87147 + 15.3658i 1.52144 + 2.63522i
\(35\) 0 0
\(36\) 3.22889 − 5.59261i 0.538149 − 0.932102i
\(37\) 1.90466 0.313124 0.156562 − 0.987668i \(-0.449959\pi\)
0.156562 + 0.987668i \(0.449959\pi\)
\(38\) 7.33983 + 8.09596i 1.19068 + 1.31334i
\(39\) −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 \(-0.0244687\pi\)
\(42\) 0.339833 + 0.588608i 0.0524374 + 0.0908242i
\(43\) 3.64959 − 6.32128i 0.556557 − 0.963985i −0.441223 − 0.897397i \(-0.645455\pi\)
0.997781 − 0.0665881i \(-0.0212113\pi\)
\(44\) 1.66719 + 2.88765i 0.251338 + 0.435330i
\(45\) 0 0
\(46\) −20.2500 −2.98569
\(47\) −1.39608 − 2.41808i −0.203639 − 0.352714i 0.746059 − 0.665880i \(-0.231944\pi\)
−0.949698 + 0.313166i \(0.898610\pi\)
\(48\) 3.53865 + 6.12912i 0.510760 + 0.884663i
\(49\) −6.95077 −0.992967
\(50\) 0 0
\(51\) −4.32379 − 7.48903i −0.605452 − 1.04867i
\(52\) −10.7129 + 18.5552i −1.48561 + 2.57314i
\(53\) 2.19024 + 3.79361i 0.300853 + 0.521093i 0.976329 − 0.216289i \(-0.0693953\pi\)
−0.675476 + 0.737382i \(0.736062\pi\)
\(54\) −6.90310 + 11.9565i −0.939393 + 1.62708i
\(55\) 0 0
\(56\) 1.27111 0.169859
\(57\) −3.57730 − 3.94583i −0.473825 − 0.522637i
\(58\) 0.556248 0.0730389
\(59\) 1.39608 − 2.41808i 0.181754 − 0.314808i −0.760724 − 0.649076i \(-0.775156\pi\)
0.942478 + 0.334268i \(0.108489\pi\)
\(60\) 0 0
\(61\) 6.29216 + 10.8983i 0.805629 + 1.39539i 0.915866 + 0.401484i \(0.131506\pi\)
−0.110237 + 0.993905i \(0.535161\pi\)
\(62\) −3.14257 + 5.44309i −0.399107 + 0.691274i
\(63\) 0.167186 + 0.289574i 0.0210634 + 0.0364829i
\(64\) −3.90466 −0.488082
\(65\) 0 0
\(66\) −1.19180 − 2.06426i −0.146700 − 0.254093i
\(67\) −5.28514 − 9.15414i −0.645683 − 1.11836i −0.984143 − 0.177375i \(-0.943239\pi\)
0.338460 − 0.940981i \(-0.390094\pi\)
\(68\) −30.3273 −3.67772
\(69\) 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 \(-0.0318383\pi\)
\(72\) 4.31678 + 7.47687i 0.508737 + 0.881158i
\(73\) −7.03865 + 12.1913i −0.823812 + 1.42688i 0.0790121 + 0.996874i \(0.474823\pi\)
−0.902824 + 0.430010i \(0.858510\pi\)
\(74\) −2.38750 + 4.13528i −0.277542 + 0.480716i
\(75\) 0 0
\(76\) −18.2570 + 3.94583i −2.09422 + 0.452617i
\(77\) −0.172647 −0.0196750
\(78\) 7.65817 − 13.2643i 0.867117 − 1.50189i
\(79\) −0.792161 + 1.37206i −0.0891251 + 0.154369i −0.907142 − 0.420826i \(-0.861740\pi\)
0.818016 + 0.575195i \(0.195074\pi\)
\(80\) 0 0
\(81\) 1.10392 − 1.91204i 0.122658 − 0.212449i
\(82\) 9.07028 + 15.7102i 1.00165 + 1.73490i
\(83\) 9.52106 1.04507 0.522536 − 0.852617i \(-0.324986\pi\)
0.522536 + 0.852617i \(0.324986\pi\)
\(84\) −1.16172 −0.126755
\(85\) 0 0
\(86\) 9.14959 + 15.8476i 0.986626 + 1.70889i
\(87\) −0.271105 −0.0290655
\(88\) −4.45779 −0.475202
\(89\) −1.57028 − 2.71981i −0.166450 − 0.288300i 0.770719 − 0.637175i \(-0.219897\pi\)
−0.937169 + 0.348875i \(0.886564\pi\)
\(90\) 0 0
\(91\) −0.554690 − 0.960752i −0.0581474 − 0.100714i
\(92\) 17.3062 − 29.9752i 1.80430 − 3.12513i
\(93\) 1.53163 − 2.65287i 0.158823 − 0.275089i
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) −3.74293 −0.382011
\(97\) −3.18122 + 5.51004i −0.323004 + 0.559460i −0.981106 − 0.193469i \(-0.938026\pi\)
0.658102 + 0.752929i \(0.271359\pi\)
\(98\) 8.71286 − 15.0911i 0.880131 − 1.52443i
\(99\) −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.e.d.201.1 6
5.2 odd 4 475.2.j.b.49.1 12
5.3 odd 4 475.2.j.b.49.6 12
5.4 even 2 95.2.e.b.11.3 ✓ 6
15.14 odd 2 855.2.k.g.676.1 6
19.7 even 3 inner 475.2.e.d.26.1 6
19.8 odd 6 9025.2.a.ba.1.1 3
19.11 even 3 9025.2.a.z.1.3 3
20.19 odd 2 1520.2.q.j.961.2 6
95.7 odd 12 475.2.j.b.349.6 12
95.49 even 6 1805.2.a.h.1.1 3
95.64 even 6 95.2.e.b.26.3 yes 6
95.83 odd 12 475.2.j.b.349.1 12
95.84 odd 6 1805.2.a.g.1.3 3
285.254 odd 6 855.2.k.g.406.1 6
380.159 odd 6 1520.2.q.j.881.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.b.11.3 ✓ 6 5.4 even 2
95.2.e.b.26.3 yes 6 95.64 even 6
475.2.e.d.26.1 6 19.7 even 3 inner
475.2.e.d.201.1 6 1.1 even 1 trivial
475.2.j.b.49.1 12 5.2 odd 4
475.2.j.b.49.6 12 5.3 odd 4
475.2.j.b.349.1 12 95.83 odd 12
475.2.j.b.349.6 12 95.7 odd 12
855.2.k.g.406.1 6 285.254 odd 6
855.2.k.g.676.1 6 15.14 odd 2
1520.2.q.j.881.2 6 380.159 odd 6
1520.2.q.j.961.2 6 20.19 odd 2
1805.2.a.g.1.3 3 95.84 odd 6
1805.2.a.h.1.1 3 95.49 even 6
9025.2.a.z.1.3 3 19.11 even 3
9025.2.a.ba.1.1 3 19.8 odd 6