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Properties

Label 35.5.l.a.2.5
Level $35$
Weight $5$
Character 35.2
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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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] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 2.5
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.5

$q$-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+(-3.07295 + 0.823393i) q^{2} +(-9.80023 - 2.62596i) q^{3} +(-5.09139 + 2.93952i) q^{4} +(24.9768 + 1.07725i) q^{5} +32.2778 q^{6} +(46.2809 - 16.0958i) q^{7} +(49.2180 - 49.2180i) q^{8} +(19.0008 + 10.9701i) q^{9} +(-77.6393 + 17.2554i) q^{10} +(63.4350 + 109.873i) q^{11} +(57.6159 - 15.4381i) q^{12} +(96.4845 - 96.4845i) q^{13} +(-128.966 + 87.5690i) q^{14} +(-241.949 - 76.1454i) q^{15} +(-63.6862 + 110.308i) q^{16} +(40.3088 - 150.434i) q^{17} +(-67.4211 - 18.0654i) q^{18} +(249.440 + 144.014i) q^{19} +(-130.333 + 67.9350i) q^{20} +(-495.831 + 36.2108i) q^{21} +(-285.400 - 285.400i) q^{22} +(-109.145 - 407.336i) q^{23} +(-611.593 + 353.103i) q^{24} +(622.679 + 53.8123i) q^{25} +(-217.047 + 375.936i) q^{26} +(423.711 + 423.711i) q^{27} +(-188.320 + 217.994i) q^{28} -1292.44i q^{29} +(806.195 + 34.7711i) q^{30} +(520.308 + 901.199i) q^{31} +(-183.363 + 684.320i) q^{32} +(-333.156 - 1243.35i) q^{33} +495.466i q^{34} +(1173.29 - 352.166i) q^{35} -128.987 q^{36} +(2091.16 - 560.325i) q^{37} +(-885.095 - 237.160i) q^{38} +(-1198.94 + 692.206i) q^{39} +(1282.33 - 1176.29i) q^{40} -194.875 q^{41} +(1493.85 - 519.537i) q^{42} +(-816.338 + 816.338i) q^{43} +(-645.944 - 372.936i) q^{44} +(462.761 + 294.466i) q^{45} +(670.795 + 1161.85i) q^{46} +(-3152.54 + 844.721i) q^{47} +(913.804 - 913.804i) q^{48} +(1882.85 - 1489.86i) q^{49} +(-1957.77 + 347.347i) q^{50} +(-790.070 + 1368.44i) q^{51} +(-207.623 + 774.858i) q^{52} +(-1897.94 - 508.552i) q^{53} +(-1650.92 - 953.159i) q^{54} +(1466.04 + 2812.60i) q^{55} +(1485.65 - 3070.06i) q^{56} +(-2066.39 - 2066.39i) q^{57} +(1064.19 + 3971.61i) q^{58} +(-1284.10 + 741.375i) q^{59} +(1455.69 - 323.528i) q^{60} +(-2035.24 + 3525.14i) q^{61} +(-2340.92 - 2340.92i) q^{62} +(1055.95 + 201.873i) q^{63} -4291.82i q^{64} +(2513.81 - 2305.94i) q^{65} +(2047.54 + 3546.44i) q^{66} +(1578.22 - 5889.98i) q^{67} +(236.977 + 884.408i) q^{68} +4278.59i q^{69} +(-3315.48 + 2048.26i) q^{70} +2072.09 q^{71} +(1475.11 - 395.254i) q^{72} +(-3616.13 - 968.940i) q^{73} +(-5964.65 + 3443.69i) q^{74} +(-5961.09 - 2162.51i) q^{75} -1693.33 q^{76} +(4704.32 + 4063.97i) q^{77} +(3114.31 - 3114.31i) q^{78} +(2548.98 + 1471.65i) q^{79} +(-1709.51 + 2686.53i) q^{80} +(-3928.39 - 6804.17i) q^{81} +(598.841 - 160.459i) q^{82} +(3187.19 - 3187.19i) q^{83} +(2418.03 - 1641.87i) q^{84} +(1168.84 - 3713.94i) q^{85} +(1836.40 - 3180.73i) q^{86} +(-3393.91 + 12666.3i) q^{87} +(8529.85 + 2285.57i) q^{88} +(11308.9 + 6529.17i) q^{89} +(-1664.50 - 523.845i) q^{90} +(2912.40 - 6018.39i) q^{91} +(1753.07 + 1753.07i) q^{92} +(-2732.62 - 10198.3i) q^{93} +(8992.05 - 5191.56i) q^{94} +(6075.06 + 3865.72i) q^{95} +(3594.00 - 6224.99i) q^{96} +(4562.76 + 4562.76i) q^{97} +(-4559.15 + 6128.58i) q^{98} +2783.55i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.07295 + 0.823393i −0.768236 + 0.205848i −0.621592 − 0.783341i \(-0.713514\pi\)
−0.146644 + 0.989189i \(0.546847\pi\)
\(3\) −9.80023 − 2.62596i −1.08891 − 0.291774i −0.330672 − 0.943746i \(-0.607275\pi\)
−0.758243 + 0.651972i \(0.773942\pi\)
\(4\) −5.09139 + 2.93952i −0.318212 + 0.183720i
\(5\) 24.9768 + 1.07725i 0.999071 + 0.0430899i
\(6\) 32.2778 0.896605
\(7\) 46.2809 − 16.0958i 0.944509 − 0.328486i
\(8\) 49.2180 − 49.2180i 0.769032 − 0.769032i
\(9\) 19.0008 + 10.9701i 0.234578 + 0.135433i
\(10\) −77.6393 + 17.2554i −0.776393 + 0.172554i
\(11\) 63.4350 + 109.873i 0.524256 + 0.908038i 0.999601 + 0.0282384i \(0.00898977\pi\)
−0.475345 + 0.879799i \(0.657677\pi\)
\(12\) 57.6159 − 15.4381i 0.400110 − 0.107209i
\(13\) 96.4845 − 96.4845i 0.570914 − 0.570914i −0.361470 − 0.932384i \(-0.617725\pi\)
0.932384 + 0.361470i \(0.117725\pi\)
\(14\) −128.966 + 87.5690i −0.657988 + 0.446781i
\(15\) −241.949 − 76.1454i −1.07533 − 0.338424i
\(16\) −63.6862 + 110.308i −0.248774 + 0.430890i
\(17\) 40.3088 − 150.434i 0.139477 − 0.520534i −0.860463 − 0.509514i \(-0.829825\pi\)
0.999939 − 0.0110204i \(-0.00350799\pi\)
\(18\) −67.4211 − 18.0654i −0.208090 − 0.0557575i
\(19\) 249.440 + 144.014i 0.690969 + 0.398931i 0.803975 − 0.594663i \(-0.202715\pi\)
−0.113006 + 0.993594i \(0.536048\pi\)
\(20\) −130.333 + 67.9350i −0.325833 + 0.169837i
\(21\) −495.831 + 36.2108i −1.12433 + 0.0821105i
\(22\) −285.400 − 285.400i −0.589670 − 0.589670i
\(23\) −109.145 − 407.336i −0.206324 − 0.770010i −0.989042 − 0.147634i \(-0.952834\pi\)
0.782718 − 0.622376i \(-0.213833\pi\)
\(24\) −611.593 + 353.103i −1.06179 + 0.613026i
\(25\) 622.679 + 53.8123i 0.996287 + 0.0860997i
\(26\) −217.047 + 375.936i −0.321075 + 0.556119i
\(27\) 423.711 + 423.711i 0.581222 + 0.581222i
\(28\) −188.320 + 217.994i −0.240205 + 0.278053i
\(29\) − 1292.44i − 1.53679i −0.639973 − 0.768397i \(-0.721054\pi\)
0.639973 − 0.768397i \(-0.278946\pi\)
\(30\) 806.195 + 34.7711i 0.895772 + 0.0386346i
\(31\) 520.308 + 901.199i 0.541423 + 0.937772i 0.998823 + 0.0485111i \(0.0154476\pi\)
−0.457400 + 0.889261i \(0.651219\pi\)
\(32\) −183.363 + 684.320i −0.179066 + 0.668282i
\(33\) −333.156 − 1243.35i −0.305928 − 1.14174i
\(34\) 495.466i 0.428604i
\(35\) 1173.29 − 352.166i 0.957786 − 0.287482i
\(36\) −128.987 −0.0995272
\(37\) 2091.16 − 560.325i 1.52751 − 0.409295i 0.605304 − 0.795994i \(-0.293052\pi\)
0.922206 + 0.386699i \(0.126385\pi\)
\(38\) −885.095 − 237.160i −0.612946 − 0.164238i
\(39\) −1198.94 + 692.206i −0.788255 + 0.455099i
\(40\) 1282.33 − 1176.29i 0.801455 − 0.735180i
\(41\) −194.875 −0.115928 −0.0579641 − 0.998319i \(-0.518461\pi\)
−0.0579641 + 0.998319i \(0.518461\pi\)
\(42\) 1493.85 − 519.537i 0.846851 − 0.294522i
\(43\) −816.338 + 816.338i −0.441503 + 0.441503i −0.892517 − 0.451014i \(-0.851062\pi\)
0.451014 + 0.892517i \(0.351062\pi\)
\(44\) −645.944 − 372.936i −0.333649 − 0.192632i
\(45\) 462.761 + 294.466i 0.228524 + 0.145416i
\(46\) 670.795 + 1161.85i 0.317011 + 0.549079i
\(47\) −3152.54 + 844.721i −1.42714 + 0.382400i −0.888010 − 0.459825i \(-0.847912\pi\)
−0.539126 + 0.842225i \(0.681245\pi\)
\(48\) 913.804 − 913.804i 0.396616 − 0.396616i
\(49\) 1882.85 − 1489.86i 0.784194 − 0.620516i
\(50\) −1957.77 + 347.347i −0.783107 + 0.138939i
\(51\) −790.070 + 1368.44i −0.303756 + 0.526121i
\(52\) −207.623 + 774.858i −0.0767835 + 0.286560i
\(53\) −1897.94 − 508.552i −0.675664 − 0.181044i −0.0953594 − 0.995443i \(-0.530400\pi\)
−0.580305 + 0.814399i \(0.697067\pi\)
\(54\) −1650.92 − 953.159i −0.566159 − 0.326872i
\(55\) 1466.04 + 2812.60i 0.484642 + 0.929784i
\(56\) 1485.65 − 3070.06i 0.473741 − 0.978973i
\(57\) −2066.39 − 2066.39i −0.636008 − 0.636008i
\(58\) 1064.19 + 3971.61i 0.316347 + 1.18062i
\(59\) −1284.10 + 741.375i −0.368888 + 0.212978i −0.672973 − 0.739667i \(-0.734983\pi\)
0.304084 + 0.952645i \(0.401649\pi\)
\(60\) 1455.69 − 323.528i 0.404358 − 0.0898689i
\(61\) −2035.24 + 3525.14i −0.546961 + 0.947365i 0.451519 + 0.892261i \(0.350882\pi\)
−0.998481 + 0.0551033i \(0.982451\pi\)
\(62\) −2340.92 − 2340.92i −0.608980 − 0.608980i
\(63\) 1055.95 + 201.873i 0.266049 + 0.0508625i
\(64\) − 4291.82i − 1.04781i
\(65\) 2513.81 − 2305.94i 0.594985 − 0.545783i
\(66\) 2047.54 + 3546.44i 0.470050 + 0.814151i
\(67\) 1578.22 − 5889.98i 0.351574 − 1.31209i −0.533168 − 0.846010i \(-0.678998\pi\)
0.884741 − 0.466082i \(-0.154335\pi\)
\(68\) 236.977 + 884.408i 0.0512492 + 0.191265i
\(69\) 4278.59i 0.898675i
\(70\) −3315.48 + 2048.26i −0.676628 + 0.418013i
\(71\) 2072.09 0.411047 0.205524 − 0.978652i \(-0.434110\pi\)
0.205524 + 0.978652i \(0.434110\pi\)
\(72\) 1475.11 − 395.254i 0.284550 − 0.0762450i
\(73\) −3616.13 − 968.940i −0.678576 − 0.181824i −0.0969612 − 0.995288i \(-0.530912\pi\)
−0.581615 + 0.813464i \(0.697579\pi\)
\(74\) −5964.65 + 3443.69i −1.08924 + 0.628871i
\(75\) −5961.09 − 2162.51i −1.05975 − 0.384445i
\(76\) −1693.33 −0.293166
\(77\) 4704.32 + 4063.97i 0.793442 + 0.685439i
\(78\) 3114.31 − 3114.31i 0.511885 − 0.511885i
\(79\) 2548.98 + 1471.65i 0.408425 + 0.235804i 0.690113 − 0.723702i \(-0.257561\pi\)
−0.281688 + 0.959506i \(0.590894\pi\)
\(80\) −1709.51 + 2686.53i −0.267110 + 0.419770i
\(81\) −3928.39 − 6804.17i −0.598749 − 1.03706i
\(82\) 598.841 − 160.459i 0.0890603 − 0.0238636i
\(83\) 3187.19 − 3187.19i 0.462649 − 0.462649i −0.436874 − 0.899523i \(-0.643914\pi\)
0.899523 + 0.436874i \(0.143914\pi\)
\(84\) 2418.03 − 1641.87i 0.342691 − 0.232691i
\(85\) 1168.84 − 3713.94i 0.161777 − 0.514041i
\(86\) 1836.40 − 3180.73i 0.248296 − 0.430061i
\(87\) −3393.91 + 12666.3i −0.448396 + 1.67344i
\(88\) 8529.85 + 2285.57i 1.10148 + 0.295140i
\(89\) 11308.9 + 6529.17i 1.42770 + 0.824286i 0.996939 − 0.0781781i \(-0.0249103\pi\)
0.430765 + 0.902464i \(0.358244\pi\)
\(90\) −1664.50 − 523.845i −0.205494 − 0.0646722i
\(91\) 2912.40 − 6018.39i 0.351696 − 0.726771i
\(92\) 1753.07 + 1753.07i 0.207121 + 0.207121i
\(93\) −2732.62 − 10198.3i −0.315946 − 1.17913i
\(94\) 8992.05 − 5191.56i 1.01766 − 0.587547i
\(95\) 6075.06 + 3865.72i 0.673137 + 0.428334i
\(96\) 3594.00 − 6224.99i 0.389974 − 0.675455i
\(97\) 4562.76 + 4562.76i 0.484935 + 0.484935i 0.906704 − 0.421768i \(-0.138590\pi\)
−0.421768 + 0.906704i \(0.638590\pi\)
\(98\) −4559.15 + 6128.58i −0.474714 + 0.638128i
\(99\) 2783.55i 0.284007i
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 35.5.l.a.2.5 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.10 yes 56
7.4 even 3 inner 35.5.l.a.32.10 yes 56
35.18 odd 12 inner 35.5.l.a.18.5 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.5 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.5 yes 56 35.18 odd 12 inner
35.5.l.a.23.10 yes 56 5.3 odd 4 inner
35.5.l.a.32.10 yes 56 7.4 even 3 inner