↑

Properties

Label 35.5.l.a.32.10
Level $35$
Weight $5$
Character 35.32
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
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] = [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 32.10
Character \(\chi\) \(=\) 35.32
Dual form 35.5.l.a.23.10

$q$-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+(0.823393 - 3.07295i) q^{2} +(2.62596 + 9.80023i) q^{3} +(5.09139 + 2.93952i) q^{4} +(-13.4213 + 21.0919i) q^{5} +32.2778 q^{6} +(-16.0958 + 46.2809i) q^{7} +(49.2180 - 49.2180i) q^{8} +(-19.0008 + 10.9701i) q^{9} +(53.7632 + 58.6099i) q^{10} +(63.4350 - 109.873i) q^{11} +(-15.4381 + 57.6159i) 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} +(-150.434 + 40.3088i) q^{17} +(18.0654 + 67.4211i) 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} +(407.336 + 109.145i) q^{23} +(611.593 + 353.103i) q^{24} +(-264.737 - 566.162i) q^{25} +(-217.047 - 375.936i) q^{26} +(423.711 + 423.711i) q^{27} +(-217.994 + 188.320i) q^{28} -1292.44i q^{29} +(-433.210 + 680.800i) q^{30} +(520.308 - 901.199i) q^{31} +(684.320 - 183.363i) q^{32} +(1243.35 + 333.156i) q^{33} +495.466i q^{34} +(-760.126 - 960.642i) q^{35} -128.987 q^{36} +(-560.325 + 2091.16i) q^{37} +(237.160 + 885.095i) q^{38} +(1198.94 + 692.206i) q^{39} +(377.531 + 1698.67i) q^{40} -194.875 q^{41} +(-519.537 + 1493.85i) q^{42} +(-816.338 + 816.338i) q^{43} +(645.944 - 372.936i) q^{44} +(23.6350 - 547.996i) q^{45} +(670.795 - 1161.85i) q^{46} +(844.721 - 3152.54i) 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} +(774.858 - 207.623i) q^{52} +(508.552 + 1897.94i) 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} +(-3971.61 - 1064.19i) q^{58} +(1284.10 + 741.375i) q^{59} +(-1008.03 - 1098.90i) q^{60} +(-2035.24 - 3525.14i) q^{61} +(-2340.92 - 2340.92i) q^{62} +(-201.873 - 1055.95i) q^{63} -4291.82i q^{64} +(740.093 + 3329.99i) q^{65} +(2047.54 - 3546.44i) q^{66} +(-5889.98 + 1578.22i) q^{67} +(-884.408 - 236.977i) q^{68} +4278.59i q^{69} +(-3577.88 + 1544.84i) q^{70} +2072.09 q^{71} +(-395.254 + 1475.11i) q^{72} +(968.940 + 3616.13i) q^{73} +(5964.65 + 3443.69i) q^{74} +(4853.33 - 4081.20i) q^{75} -1693.33 q^{76} +(4063.97 + 4704.32i) q^{77} +(3114.31 - 3114.31i) q^{78} +(-2548.98 + 1471.65i) q^{79} +(3181.35 + 137.212i) q^{80} +(-3928.39 + 6804.17i) q^{81} +(-160.459 + 598.841i) 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} +(12666.3 - 3393.91i) q^{87} +(-2285.57 - 8529.85i) 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} +(10198.3 + 2732.62i) q^{93} +(-8992.05 - 5191.56i) q^{94} +(310.277 - 7194.01i) q^{95} +(3594.00 + 6224.99i) q^{96} +(4562.76 + 4562.76i) q^{97} +(-6128.58 + 4559.15i) 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.823393 − 3.07295i 0.205848 − 0.768236i −0.783341 − 0.621592i \(-0.786486\pi\)
0.989189 − 0.146644i \(-0.0468472\pi\)
\(3\) 2.62596 + 9.80023i 0.291774 + 1.08891i 0.943746 + 0.330672i \(0.107275\pi\)
−0.651972 + 0.758243i \(0.726058\pi\)
\(4\) 5.09139 + 2.93952i 0.318212 + 0.183720i
\(5\) −13.4213 + 21.0919i −0.536853 + 0.843676i
\(6\) 32.2778 0.896605
\(7\) −16.0958 + 46.2809i −0.328486 + 0.944509i
\(8\) 49.2180 − 49.2180i 0.769032 − 0.769032i
\(9\) −19.0008 + 10.9701i −0.234578 + 0.135433i
\(10\) 53.7632 + 58.6099i 0.537632 + 0.586099i
\(11\) 63.4350 − 109.873i 0.524256 − 0.908038i −0.475345 − 0.879799i \(-0.657677\pi\)
0.999601 − 0.0282384i \(-0.00898977\pi\)
\(12\) −15.4381 + 57.6159i −0.107209 + 0.400110i
\(13\) 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
\(17\) −150.434 + 40.3088i −0.520534 + 0.139477i −0.509514 − 0.860463i \(-0.670175\pi\)
−0.0110204 + 0.999939i \(0.503508\pi\)
\(18\) 18.0654 + 67.4211i 0.0557575 + 0.208090i
\(19\) −249.440 + 144.014i −0.690969 + 0.398931i −0.803975 − 0.594663i \(-0.797285\pi\)
0.113006 + 0.993594i \(0.463952\pi\)
\(20\) −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
\(23\) 407.336 + 109.145i 0.770010 + 0.206324i 0.622376 − 0.782718i \(-0.286167\pi\)
0.147634 + 0.989042i \(0.452834\pi\)
\(24\) 611.593 + 353.103i 1.06179 + 0.613026i
\(25\) −264.737 − 566.162i −0.423579 − 0.905859i
\(26\) −217.047 − 375.936i −0.321075 − 0.556119i
\(27\) 423.711 + 423.711i 0.581222 + 0.581222i
\(28\) −217.994 + 188.320i −0.278053 + 0.240205i
\(29\) − 1292.44i − 1.53679i −0.639973 − 0.768397i \(-0.721054\pi\)
0.639973 − 0.768397i \(-0.278946\pi\)
\(30\) −433.210 + 680.800i −0.481345 + 0.756444i
\(31\) 520.308 − 901.199i 0.541423 − 0.937772i −0.457400 − 0.889261i \(-0.651219\pi\)
0.998823 − 0.0485111i \(-0.0154476\pi\)
\(32\) 684.320 − 183.363i 0.668282 − 0.179066i
\(33\) 1243.35 + 333.156i 1.14174 + 0.305928i
\(34\) 495.466i 0.428604i
\(35\) −760.126 − 960.642i −0.620511 − 0.784198i
\(36\) −128.987 −0.0995272
\(37\) −560.325 + 2091.16i −0.409295 + 1.52751i 0.386699 + 0.922206i \(0.373615\pi\)
−0.795994 + 0.605304i \(0.793052\pi\)
\(38\) 237.160 + 885.095i 0.164238 + 0.612946i
\(39\) 1198.94 + 692.206i 0.788255 + 0.455099i
\(40\) 377.531 + 1698.67i 0.235957 + 1.06167i
\(41\) −194.875 −0.115928 −0.0579641 − 0.998319i \(-0.518461\pi\)
−0.0579641 + 0.998319i \(0.518461\pi\)
\(42\) −519.537 + 1493.85i −0.294522 + 0.846851i
\(43\) −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
\(45\) 23.6350 − 547.996i 0.0116716 − 0.270615i
\(46\) 670.795 − 1161.85i 0.317011 − 0.549079i
\(47\) 844.721 − 3152.54i 0.382400 − 1.42714i −0.459825 − 0.888010i \(-0.652088\pi\)
0.842225 − 0.539126i \(-0.181245\pi\)
\(48\) 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
\(52\) 774.858 − 207.623i 0.286560 − 0.0767835i
\(53\) 508.552 + 1897.94i 0.181044 + 0.675664i 0.995443 + 0.0953594i \(0.0304000\pi\)
−0.814399 + 0.580305i \(0.802933\pi\)
\(54\) 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
\(58\) −3971.61 − 1064.19i −1.18062 − 0.316347i
\(59\) 1284.10 + 741.375i 0.368888 + 0.212978i 0.672973 − 0.739667i \(-0.265017\pi\)
−0.304084 + 0.952645i \(0.598351\pi\)
\(60\) −1008.03 − 1098.90i −0.280008 − 0.305250i
\(61\) −2035.24 − 3525.14i −0.546961 − 0.947365i −0.998481 − 0.0551033i \(-0.982451\pi\)
0.451519 − 0.892261i \(-0.350882\pi\)
\(62\) −2340.92 − 2340.92i −0.608980 − 0.608980i
\(63\) −201.873 − 1055.95i −0.0508625 − 0.266049i
\(64\) − 4291.82i − 1.04781i
\(65\) 740.093 + 3329.99i 0.175170 + 0.788164i
\(66\) 2047.54 − 3546.44i 0.470050 − 0.814151i
\(67\) −5889.98 + 1578.22i −1.31209 + 0.351574i −0.846010 − 0.533168i \(-0.821002\pi\)
−0.466082 + 0.884741i \(0.654335\pi\)
\(68\) −884.408 − 236.977i −0.191265 − 0.0512492i
\(69\) 4278.59i 0.898675i
\(70\) −3577.88 + 1544.84i −0.730180 + 0.315273i
\(71\) 2072.09 0.411047 0.205524 − 0.978652i \(-0.434110\pi\)
0.205524 + 0.978652i \(0.434110\pi\)
\(72\) −395.254 + 1475.11i −0.0762450 + 0.284550i
\(73\) 968.940 + 3616.13i 0.181824 + 0.678576i 0.995288 + 0.0969612i \(0.0309123\pi\)
−0.813464 + 0.581615i \(0.802421\pi\)
\(74\) 5964.65 + 3443.69i 1.08924 + 0.628871i
\(75\) 4853.33 − 4081.20i 0.862814 − 0.725547i
\(76\) −1693.33 −0.293166
\(77\) 4063.97 + 4704.32i 0.685439 + 0.793442i
\(78\) 3114.31 − 3114.31i 0.511885 − 0.511885i
\(79\) −2548.98 + 1471.65i −0.408425 + 0.235804i −0.690113 − 0.723702i \(-0.742439\pi\)
0.281688 + 0.959506i \(0.409106\pi\)
\(80\) 3181.35 + 137.212i 0.497087 + 0.0214393i
\(81\) −3928.39 + 6804.17i −0.598749 + 1.03706i
\(82\) −160.459 + 598.841i −0.0238636 + 0.0890603i
\(83\) 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
\(87\) 12666.3 − 3393.91i 1.67344 − 0.448396i
\(88\) −2285.57 − 8529.85i −0.295140 − 1.10148i
\(89\) −11308.9 + 6529.17i −1.42770 + 0.824286i −0.996939 − 0.0781781i \(-0.975090\pi\)
−0.430765 + 0.902464i \(0.641756\pi\)
\(90\) −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
\(93\) 10198.3 + 2732.62i 1.17913 + 0.315946i
\(94\) −8992.05 − 5191.56i −1.01766 − 0.587547i
\(95\) 310.277 − 7194.01i 0.0343798 − 0.797121i
\(96\) 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 \(0.638590\pi\)
\(98\) −6128.58 + 4559.15i −0.638128 + 0.474714i
\(99\) 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.32.10 yes 56
5.3 odd 4 inner 35.5.l.a.18.5 yes 56
7.2 even 3 inner 35.5.l.a.2.5 ✓ 56
35.23 odd 12 inner 35.5.l.a.23.10 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.5 ✓ 56 7.2 even 3 inner
35.5.l.a.18.5 yes 56 5.3 odd 4 inner
35.5.l.a.23.10 yes 56 35.23 odd 12 inner
35.5.l.a.32.10 yes 56 1.1 even 1 trivial