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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,18,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,338] 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: yes
Analytic conductor: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 580318x^{4} + 45393344x^{3} + 72695152416x^{2} - 6623241804288x - 149217035286528 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 5^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(464.046\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q+520.046 q^{2} -6561.00 q^{3} +139375. q^{4} -3.41202e6 q^{6} -2.58253e7 q^{7} +4.31818e6 q^{8} +4.30467e7 q^{9} -8.94002e8 q^{11} -9.14442e8 q^{12} -4.54853e9 q^{13} -1.34304e10 q^{14} -1.60226e10 q^{16} -2.94030e10 q^{17} +2.23863e10 q^{18} +1.11377e11 q^{19} +1.69440e11 q^{21} -4.64922e11 q^{22} +4.88923e11 q^{23} -2.83316e10 q^{24} -2.36544e12 q^{26} -2.82430e11 q^{27} -3.59942e12 q^{28} +1.62091e12 q^{29} +4.41122e12 q^{31} -8.89846e12 q^{32} +5.86555e12 q^{33} -1.52909e13 q^{34} +5.99966e12 q^{36} -1.33885e13 q^{37} +5.79211e13 q^{38} +2.98429e13 q^{39} +6.15755e13 q^{41} +8.81165e13 q^{42} -1.02385e14 q^{43} -1.24602e14 q^{44} +2.54262e14 q^{46} +7.22660e13 q^{47} +1.05124e14 q^{48} +4.34317e14 q^{49} +1.92913e14 q^{51} -6.33953e14 q^{52} -4.48592e14 q^{53} -1.46876e14 q^{54} -1.11518e14 q^{56} -7.30744e14 q^{57} +8.42945e14 q^{58} -8.64700e14 q^{59} -4.71750e14 q^{61} +2.29404e15 q^{62} -1.11170e15 q^{63} -2.52749e15 q^{64} +3.05035e15 q^{66} +4.21073e15 q^{67} -4.09806e15 q^{68} -3.20782e15 q^{69} -2.09672e15 q^{71} +1.85883e14 q^{72} +3.12151e15 q^{73} -6.96263e15 q^{74} +1.55232e16 q^{76} +2.30879e16 q^{77} +1.55197e16 q^{78} -1.72600e16 q^{79} +1.85302e15 q^{81} +3.20220e16 q^{82} -6.18766e15 q^{83} +2.36158e16 q^{84} -5.32449e16 q^{86} -1.06348e16 q^{87} -3.86046e15 q^{88} -2.42408e16 q^{89} +1.17467e17 q^{91} +6.81439e16 q^{92} -2.89420e16 q^{93} +3.75816e16 q^{94} +5.83828e16 q^{96} -3.04115e16 q^{97} +2.25865e17 q^{98} -3.84838e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 338 q^{2} - 39366 q^{3} + 393248 q^{4} - 2217618 q^{6} - 20999794 q^{7} - 25242756 q^{8} + 258280326 q^{9} + 85907324 q^{11} - 2580100128 q^{12} - 344649098 q^{13} + 4962466602 q^{14} + 13788291320 q^{16}+ \cdots + 36\!\cdots\!04 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 520.046 1.43644 0.718218 − 0.695818i \(-0.244958\pi\)
0.718218 + 0.695818i \(0.244958\pi\)
\(3\) −6561.00 −0.577350
\(4\) 139375. 1.06335
\(5\) 0 0
\(6\) −3.41202e6 −0.829327
\(7\) −2.58253e7 −1.69322 −0.846608 − 0.532216i \(-0.821359\pi\)
−0.846608 + 0.532216i \(0.821359\pi\)
\(8\) 4.31818e6 0.0909988
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −8.94002e8 −1.25748 −0.628739 − 0.777616i \(-0.716429\pi\)
−0.628739 + 0.777616i \(0.716429\pi\)
\(12\) −9.14442e8 −0.613926
\(13\) −4.54853e9 −1.54651 −0.773255 − 0.634095i \(-0.781373\pi\)
−0.773255 + 0.634095i \(0.781373\pi\)
\(14\) −1.34304e10 −2.43220
\(15\) 0 0
\(16\) −1.60226e10 −0.932636
\(17\) −2.94030e10 −1.02230 −0.511148 − 0.859493i \(-0.670780\pi\)
−0.511148 + 0.859493i \(0.670780\pi\)
\(18\) 2.23863e10 0.478812
\(19\) 1.11377e11 1.50449 0.752245 − 0.658883i \(-0.228971\pi\)
0.752245 + 0.658883i \(0.228971\pi\)
\(20\) 0 0
\(21\) 1.69440e11 0.977579
\(22\) −4.64922e11 −1.80629
\(23\) 4.88923e11 1.30183 0.650915 − 0.759151i \(-0.274386\pi\)
0.650915 + 0.759151i \(0.274386\pi\)
\(24\) −2.83316e10 −0.0525382
\(25\) 0 0
\(26\) −2.36544e12 −2.22146
\(27\) −2.82430e11 −0.192450
\(28\) −3.59942e12 −1.80048
\(29\) 1.62091e12 0.601693 0.300846 − 0.953673i \(-0.402731\pi\)
0.300846 + 0.953673i \(0.402731\pi\)
\(30\) 0 0
\(31\) 4.41122e12 0.928933 0.464467 − 0.885591i \(-0.346246\pi\)
0.464467 + 0.885591i \(0.346246\pi\)
\(32\) −8.89846e12 −1.43067
\(33\) 5.86555e12 0.726006
\(34\) −1.52909e13 −1.46846
\(35\) 0 0
\(36\) 5.99966e12 0.354450
\(37\) −1.33885e13 −0.626639 −0.313319 − 0.949648i \(-0.601441\pi\)
−0.313319 + 0.949648i \(0.601441\pi\)
\(38\) 5.79211e13 2.16111
\(39\) 2.98429e13 0.892878
\(40\) 0 0
\(41\) 6.15755e13 1.20433 0.602164 − 0.798372i \(-0.294305\pi\)
0.602164 + 0.798372i \(0.294305\pi\)
\(42\) 8.81165e13 1.40423
\(43\) −1.02385e14 −1.33584 −0.667920 − 0.744233i \(-0.732815\pi\)
−0.667920 + 0.744233i \(0.732815\pi\)
\(44\) −1.24602e14 −1.33714
\(45\) 0 0
\(46\) 2.54262e14 1.86999
\(47\) 7.22660e13 0.442692 0.221346 − 0.975195i \(-0.428955\pi\)
0.221346 + 0.975195i \(0.428955\pi\)
\(48\) 1.05124e14 0.538458
\(49\) 4.34317e14 1.86698
\(50\) 0 0
\(51\) 1.92913e14 0.590223
\(52\) −6.33953e14 −1.64448
\(53\) −4.48592e14 −0.989708 −0.494854 − 0.868976i \(-0.664778\pi\)
−0.494854 + 0.868976i \(0.664778\pi\)
\(54\) −1.46876e14 −0.276442
\(55\) 0 0
\(56\) −1.11518e14 −0.154081
\(57\) −7.30744e14 −0.868618
\(58\) 8.42945e14 0.864293
\(59\) −8.64700e14 −0.766696 −0.383348 − 0.923604i \(-0.625229\pi\)
−0.383348 + 0.923604i \(0.625229\pi\)
\(60\) 0 0
\(61\) −4.71750e14 −0.315071 −0.157535 − 0.987513i \(-0.550355\pi\)
−0.157535 + 0.987513i \(0.550355\pi\)
\(62\) 2.29404e15 1.33435
\(63\) −1.11170e15 −0.564406
\(64\) −2.52749e15 −1.12243
\(65\) 0 0
\(66\) 3.05035e15 1.04286
\(67\) 4.21073e15 1.26684 0.633420 − 0.773808i \(-0.281651\pi\)
0.633420 + 0.773808i \(0.281651\pi\)
\(68\) −4.09806e15 −1.08706
\(69\) −3.20782e15 −0.751611
\(70\) 0 0
\(71\) −2.09672e15 −0.385340 −0.192670 − 0.981264i \(-0.561715\pi\)
−0.192670 + 0.981264i \(0.561715\pi\)
\(72\) 1.85883e14 0.0303329
\(73\) 3.12151e15 0.453023 0.226512 − 0.974008i \(-0.427268\pi\)
0.226512 + 0.974008i \(0.427268\pi\)
\(74\) −6.96263e15 −0.900127
\(75\) 0 0
\(76\) 1.55232e16 1.59980
\(77\) 2.30879e16 2.12918
\(78\) 1.55197e16 1.28256
\(79\) −1.72600e16 −1.28000 −0.640000 − 0.768375i \(-0.721066\pi\)
−0.640000 + 0.768375i \(0.721066\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 3.20220e16 1.72994
\(83\) −6.18766e15 −0.301552 −0.150776 − 0.988568i \(-0.548177\pi\)
−0.150776 + 0.988568i \(0.548177\pi\)
\(84\) 2.36158e16 1.03951
\(85\) 0 0
\(86\) −5.32449e16 −1.91885
\(87\) −1.06348e16 −0.347387
\(88\) −3.86046e15 −0.114429
\(89\) −2.42408e16 −0.652727 −0.326364 − 0.945244i \(-0.605823\pi\)
−0.326364 + 0.945244i \(0.605823\pi\)
\(90\) 0 0
\(91\) 1.17467e17 2.61858
\(92\) 6.81439e16 1.38430
\(93\) −2.89420e16 −0.536320
\(94\) 3.75816e16 0.635900
\(95\) 0 0
\(96\) 5.83828e16 0.825999
\(97\) −3.04115e16 −0.393983 −0.196992 − 0.980405i \(-0.563117\pi\)
−0.196992 + 0.980405i \(0.563117\pi\)
\(98\) 2.25865e17 2.68180
\(99\) −3.84838e16 −0.419160
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 75.18.a.j.1.5 yes 6
5.2 odd 4 75.18.b.h.49.10 12
5.3 odd 4 75.18.b.h.49.3 12
5.4 even 2 75.18.a.i.1.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.2 ✓ 6 5.4 even 2
75.18.a.j.1.5 yes 6 1.1 even 1 trivial
75.18.b.h.49.3 12 5.3 odd 4
75.18.b.h.49.10 12 5.2 odd 4