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Properties

Label 95.2.e.b.11.3
Level $95$
Weight $2$
Character 95.11
Analytic conductor $0.759$
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] = [95,2,Mod(11,95)] 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("95.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 95.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(0.758578819202\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 11.3
Root \(-1.25351 + 2.17114i\) of defining polynomial
Character \(\chi\) \(=\) 95.11
Dual form 95.2.e.b.26.3

$q$-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} +(0.500000 - 0.866025i) q^{5} +(1.53163 + 2.65287i) q^{6} -0.221876 q^{7} -5.72889 q^{8} +(0.753509 + 1.30512i) q^{9} +(-1.25351 - 2.17114i) q^{10} -0.778124 q^{11} +5.23591 q^{12} +(2.50000 + 4.33013i) q^{13} +(-0.278124 + 0.481725i) q^{14} +(0.610938 + 1.05818i) q^{15} +(-2.89608 + 5.01616i) q^{16} +(-3.53865 + 6.12912i) q^{17} +3.77812 q^{18} +(1.33281 - 4.15013i) q^{19} -4.28514 q^{20} +(0.135553 - 0.234784i) q^{21} +(-0.975385 + 1.68942i) q^{22} +(-4.03865 - 6.99515i) q^{23} +(3.50000 - 6.06218i) q^{24} +(-0.500000 - 0.866025i) q^{25} +12.5351 q^{26} -5.50702 q^{27} +(0.475385 + 0.823392i) q^{28} +(-0.110938 - 0.192150i) q^{29} +3.06327 q^{30} +2.50702 q^{31} +(1.53163 + 2.65287i) q^{32} +(0.475385 - 0.823392i) q^{33} +(8.87147 + 15.3658i) q^{34} +(-0.110938 + 0.192150i) q^{35} +(3.22889 - 5.59261i) q^{36} -1.90466 q^{37} +(-7.33983 - 8.09596i) q^{38} -6.10938 q^{39} +(-2.86445 + 4.96137i) q^{40} +(3.61796 - 6.26648i) q^{41} +(-0.339833 - 0.588608i) q^{42} +(-3.64959 + 6.32128i) q^{43} +(1.66719 + 2.88765i) q^{44} +1.50702 q^{45} -20.2500 q^{46} +(1.39608 + 2.41808i) q^{47} +(-3.53865 - 6.12912i) q^{48} -6.95077 q^{49} -2.50702 q^{50} +(-4.32379 - 7.48903i) q^{51} +(10.7129 - 18.5552i) q^{52} +(-2.19024 - 3.79361i) q^{53} +(-6.90310 + 11.9565i) q^{54} +(-0.389062 + 0.673875i) q^{55} +1.27111 q^{56} +(3.57730 + 3.94583i) q^{57} -0.556248 q^{58} +(1.39608 - 2.41808i) q^{59} +(2.61796 - 4.53443i) q^{60} +(6.29216 + 10.8983i) q^{61} +(3.14257 - 5.44309i) q^{62} +(-0.167186 - 0.289574i) q^{63} -3.90466 q^{64} +5.00000 q^{65} +(-1.19180 - 2.06426i) q^{66} +(5.28514 + 9.15414i) q^{67} +30.3273 q^{68} +9.86946 q^{69} +(0.278124 + 0.481725i) q^{70} +(4.92070 - 8.52289i) q^{71} +(-4.31678 - 7.47687i) q^{72} +(7.03865 - 12.1913i) q^{73} +(-2.38750 + 4.13528i) q^{74} +1.22188 q^{75} +(-18.2570 + 3.94583i) q^{76} +0.172647 q^{77} +(-7.65817 + 13.2643i) q^{78} +(-0.792161 + 1.37206i) q^{79} +(2.89608 + 5.01616i) q^{80} +(1.10392 - 1.91204i) q^{81} +(-9.07028 - 15.7102i) q^{82} -9.52106 q^{83} -1.16172 q^{84} +(3.53865 + 6.12912i) q^{85} +(9.14959 + 15.8476i) q^{86} +0.271105 q^{87} +4.45779 q^{88} +(-1.57028 - 2.71981i) q^{89} +(1.88906 - 3.27195i) q^{90} +(-0.554690 - 0.960752i) q^{91} +(-17.3062 + 29.9752i) q^{92} +(-1.53163 + 2.65287i) q^{93} +7.00000 q^{94} +(-2.92771 - 3.22932i) q^{95} -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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - q^{3} - 7 q^{4} + 3 q^{5} + 6 q^{6} + 4 q^{7} - 12 q^{8} - 4 q^{9} + q^{10} - 10 q^{11} - 8 q^{12} + 15 q^{13} - 7 q^{14} + q^{15} - 3 q^{16} - q^{17} + 28 q^{18} - 14 q^{20} + 12 q^{21}+ \cdots + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 \(-0.486556\pi\)
0.844141 − 0.536121i \(-0.180111\pi\)
\(3\) −0.610938 + 1.05818i −0.352725 + 0.610938i −0.986726 − 0.162394i \(-0.948078\pi\)
0.634001 + 0.773333i \(0.281412\pi\)
\(4\) −2.14257 − 3.71104i −1.07129 − 1.85552i
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 1.53163 + 2.65287i 0.625287 + 1.08303i
\(7\) −0.221876 −0.0838613 −0.0419307 − 0.999121i \(-0.513351\pi\)
−0.0419307 + 0.999121i \(0.513351\pi\)
\(8\) −5.72889 −2.02547
\(9\) 0.753509 + 1.30512i 0.251170 + 0.435039i
\(10\) −1.25351 − 2.17114i −0.396394 − 0.686575i
\(11\) −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 \(0.0772105\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −0.278124 + 0.481725i −0.0743317 + 0.128746i
\(15\) 0.610938 + 1.05818i 0.157744 + 0.273220i
\(16\) −2.89608 + 5.01616i −0.724020 + 1.25404i
\(17\) −3.53865 + 6.12912i −0.858249 + 1.48653i 0.0153485 + 0.999882i \(0.495114\pi\)
−0.873598 + 0.486649i \(0.838219\pi\)
\(18\) 3.77812 0.890512
\(19\) 1.33281 − 4.15013i 0.305769 − 0.952106i
\(20\) −4.28514 −0.958187
\(21\) 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 \(-0.847976\pi\)
0.0459843 − 0.998942i \(-0.485358\pi\)
\(24\) 3.50000 − 6.06218i 0.714435 − 1.23744i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 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 \(0.839891\pi\)
\(30\) 3.06327 0.559273
\(31\) 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
\(35\) −0.110938 + 0.192150i −0.0187520 + 0.0324793i
\(36\) 3.22889 − 5.59261i 0.538149 − 0.932102i
\(37\) −1.90466 −0.313124 −0.156562 − 0.987668i \(-0.550041\pi\)
−0.156562 + 0.987668i \(0.550041\pi\)
\(38\) −7.33983 − 8.09596i −1.19068 − 1.31334i
\(39\) −6.10938 −0.978284
\(40\) −2.86445 + 4.96137i −0.452909 + 0.784461i
\(41\) 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 \(0.354545\pi\)
−0.997781 + 0.0665881i \(0.978789\pi\)
\(44\) 1.66719 + 2.88765i 0.251338 + 0.435330i
\(45\) 1.50702 0.224653
\(46\) −20.2500 −2.98569
\(47\) 1.39608 + 2.41808i 0.203639 + 0.352714i 0.949698 − 0.313166i \(-0.101390\pi\)
−0.746059 + 0.665880i \(0.768056\pi\)
\(48\) −3.53865 − 6.12912i −0.510760 − 0.884663i
\(49\) −6.95077 −0.992967
\(50\) −2.50702 −0.354546
\(51\) −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 0.675476 − 0.737382i \(-0.263938\pi\)
−0.976329 + 0.216289i \(0.930605\pi\)
\(54\) −6.90310 + 11.9565i −0.939393 + 1.62708i
\(55\) −0.389062 + 0.673875i −0.0524611 + 0.0908653i
\(56\) 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 \(0.108489\pi\)
\(60\) 2.61796 − 4.53443i 0.337977 − 0.585393i
\(61\) 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
\(65\) 5.00000 0.620174
\(66\) −1.19180 − 2.06426i −0.146700 − 0.254093i
\(67\) 5.28514 + 9.15414i 0.645683 + 1.11836i 0.984143 + 0.177375i \(0.0567605\pi\)
−0.338460 + 0.940981i \(0.609906\pi\)
\(68\) 30.3273 3.67772
\(69\) 9.86946 1.18814
\(70\) 0.278124 + 0.481725i 0.0332422 + 0.0575771i
\(71\) 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 \(-0.525177\pi\)
0.902824 − 0.430010i \(-0.141490\pi\)
\(74\) −2.38750 + 4.13528i −0.277542 + 0.480716i
\(75\) 1.22188 0.141090
\(76\) −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 \(0.195074\pi\)
\(80\) 2.89608 + 5.01616i 0.323792 + 0.560824i
\(81\) 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 \(-0.675014\pi\)
−0.522536 + 0.852617i \(0.675014\pi\)
\(84\) −1.16172 −0.126755
\(85\) 3.53865 + 6.12912i 0.383821 + 0.664797i
\(86\) 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 \(0.886564\pi\)
\(90\) 1.88906 − 3.27195i 0.199125 − 0.344894i
\(91\) −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
\(95\) −2.92771 − 3.22932i −0.300377 − 0.331321i
\(96\) −3.74293 −0.382011
\(97\) 3.18122 − 5.51004i 0.323004 − 0.559460i −0.658102 − 0.752929i \(-0.728641\pi\)
0.981106 + 0.193469i \(0.0619738\pi\)
\(98\) −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 95.2.e.b.11.3 ✓ 6
3.2 odd 2 855.2.k.g.676.1 6
4.3 odd 2 1520.2.q.j.961.2 6
5.2 odd 4 475.2.j.b.49.6 12
5.3 odd 4 475.2.j.b.49.1 12
5.4 even 2 475.2.e.d.201.1 6
19.7 even 3 inner 95.2.e.b.26.3 yes 6
19.8 odd 6 1805.2.a.g.1.3 3
19.11 even 3 1805.2.a.h.1.1 3
57.26 odd 6 855.2.k.g.406.1 6
76.7 odd 6 1520.2.q.j.881.2 6
95.7 odd 12 475.2.j.b.349.1 12
95.49 even 6 9025.2.a.z.1.3 3
95.64 even 6 475.2.e.d.26.1 6
95.83 odd 12 475.2.j.b.349.6 12
95.84 odd 6 9025.2.a.ba.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.b.11.3 ✓ 6 1.1 even 1 trivial
95.2.e.b.26.3 yes 6 19.7 even 3 inner
475.2.e.d.26.1 6 95.64 even 6
475.2.e.d.201.1 6 5.4 even 2
475.2.j.b.49.1 12 5.3 odd 4
475.2.j.b.49.6 12 5.2 odd 4
475.2.j.b.349.1 12 95.7 odd 12
475.2.j.b.349.6 12 95.83 odd 12
855.2.k.g.406.1 6 57.26 odd 6
855.2.k.g.676.1 6 3.2 odd 2
1520.2.q.j.881.2 6 76.7 odd 6
1520.2.q.j.961.2 6 4.3 odd 2
1805.2.a.g.1.3 3 19.8 odd 6
1805.2.a.h.1.1 3 19.11 even 3
9025.2.a.z.1.3 3 95.49 even 6
9025.2.a.ba.1.1 3 95.84 odd 6