Properties

Label 75.10.a.j.1.1
Level $75$
Weight $10$
Character 75.1
Self dual yes
Analytic conductor $38.628$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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] = [75,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2] 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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5^{2}\cdot 23 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(33.1381\) 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)
 
\(f(q)\) \(=\) \(q-44.2736 q^{2} +81.0000 q^{3} +1448.15 q^{4} -3586.16 q^{6} +3879.20 q^{7} -41447.0 q^{8} +6561.00 q^{9} +84809.6 q^{11} +117301. q^{12} +58662.8 q^{13} -171746. q^{14} +1.09355e6 q^{16} +176199. q^{17} -290479. q^{18} +800041. q^{19} +314215. q^{21} -3.75483e6 q^{22} -1.65723e6 q^{23} -3.35720e6 q^{24} -2.59721e6 q^{26} +531441. q^{27} +5.61768e6 q^{28} -4.70073e6 q^{29} +5.05198e6 q^{31} -2.71947e7 q^{32} +6.86958e6 q^{33} -7.80099e6 q^{34} +9.50134e6 q^{36} +4.68753e6 q^{37} -3.54207e7 q^{38} +4.75168e6 q^{39} -4.71307e6 q^{41} -1.39114e7 q^{42} -1.65548e7 q^{43} +1.22817e8 q^{44} +7.33718e7 q^{46} +1.79860e7 q^{47} +8.85778e7 q^{48} -2.53054e7 q^{49} +1.42722e7 q^{51} +8.49527e7 q^{52} +1.50727e7 q^{53} -2.35288e7 q^{54} -1.60781e8 q^{56} +6.48033e7 q^{57} +2.08118e8 q^{58} +1.15379e7 q^{59} -3.48684e7 q^{61} -2.23669e8 q^{62} +2.54514e7 q^{63} +6.44110e8 q^{64} -3.04141e8 q^{66} +1.51756e7 q^{67} +2.55164e8 q^{68} -1.34236e8 q^{69} -3.98346e7 q^{71} -2.71934e8 q^{72} -7.02902e7 q^{73} -2.07534e8 q^{74} +1.15858e9 q^{76} +3.28993e8 q^{77} -2.10374e8 q^{78} +1.16781e8 q^{79} +4.30467e7 q^{81} +2.08665e8 q^{82} +7.69619e8 q^{83} +4.55032e8 q^{84} +7.32941e8 q^{86} -3.80759e8 q^{87} -3.51510e9 q^{88} -9.57076e8 q^{89} +2.27564e8 q^{91} -2.39993e9 q^{92} +4.09210e8 q^{93} -7.96305e8 q^{94} -2.20277e9 q^{96} +4.93418e8 q^{97} +1.12036e9 q^{98} +5.56436e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 324 q^{3} + 1792 q^{4} - 162 q^{6} + 13036 q^{7} - 24636 q^{8} + 26244 q^{9} + 104696 q^{11} + 145152 q^{12} + 140812 q^{13} - 181062 q^{14} + 1319800 q^{16} + 489352 q^{17} - 13122 q^{18}+ \cdots + 686910456 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −44.2736 −1.95664 −0.978318 − 0.207107i \(-0.933595\pi\)
−0.978318 + 0.207107i \(0.933595\pi\)
\(3\) 81.0000 0.577350
\(4\) 1448.15 2.82843
\(5\) 0 0
\(6\) −3586.16 −1.12966
\(7\) 3879.20 0.610662 0.305331 − 0.952246i \(-0.401233\pi\)
0.305331 + 0.952246i \(0.401233\pi\)
\(8\) −41447.0 −3.57757
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 84809.6 1.74654 0.873269 − 0.487239i \(-0.161996\pi\)
0.873269 + 0.487239i \(0.161996\pi\)
\(12\) 117301. 1.63299
\(13\) 58662.8 0.569662 0.284831 − 0.958578i \(-0.408062\pi\)
0.284831 + 0.958578i \(0.408062\pi\)
\(14\) −171746. −1.19484
\(15\) 0 0
\(16\) 1.09355e6 4.17157
\(17\) 176199. 0.511663 0.255832 − 0.966721i \(-0.417651\pi\)
0.255832 + 0.966721i \(0.417651\pi\)
\(18\) −290479. −0.652212
\(19\) 800041. 1.40838 0.704192 − 0.710009i \(-0.251309\pi\)
0.704192 + 0.710009i \(0.251309\pi\)
\(20\) 0 0
\(21\) 314215. 0.352566
\(22\) −3.75483e6 −3.41734
\(23\) −1.65723e6 −1.23483 −0.617417 − 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(24\) −3.35720e6 −2.06551
\(25\) 0 0
\(26\) −2.59721e6 −1.11462
\(27\) 531441. 0.192450
\(28\) 5.61768e6 1.72721
\(29\) −4.70073e6 −1.23417 −0.617084 − 0.786897i \(-0.711686\pi\)
−0.617084 + 0.786897i \(0.711686\pi\)
\(30\) 0 0
\(31\) 5.05198e6 0.982503 0.491251 − 0.871018i \(-0.336540\pi\)
0.491251 + 0.871018i \(0.336540\pi\)
\(32\) −2.71947e7 −4.58469
\(33\) 6.86958e6 1.00836
\(34\) −7.80099e6 −1.00114
\(35\) 0 0
\(36\) 9.50134e6 0.942809
\(37\) 4.68753e6 0.411184 0.205592 − 0.978638i \(-0.434088\pi\)
0.205592 + 0.978638i \(0.434088\pi\)
\(38\) −3.54207e7 −2.75570
\(39\) 4.75168e6 0.328895
\(40\) 0 0
\(41\) −4.71307e6 −0.260481 −0.130241 − 0.991482i \(-0.541575\pi\)
−0.130241 + 0.991482i \(0.541575\pi\)
\(42\) −1.39114e7 −0.689843
\(43\) −1.65548e7 −0.738441 −0.369221 − 0.929342i \(-0.620375\pi\)
−0.369221 + 0.929342i \(0.620375\pi\)
\(44\) 1.22817e8 4.93995
\(45\) 0 0
\(46\) 7.33718e7 2.41612
\(47\) 1.79860e7 0.537643 0.268821 − 0.963190i \(-0.413366\pi\)
0.268821 + 0.963190i \(0.413366\pi\)
\(48\) 8.85778e7 2.40846
\(49\) −2.53054e7 −0.627092
\(50\) 0 0
\(51\) 1.42722e7 0.295409
\(52\) 8.49527e7 1.61125
\(53\) 1.50727e7 0.262391 0.131195 − 0.991357i \(-0.458118\pi\)
0.131195 + 0.991357i \(0.458118\pi\)
\(54\) −2.35288e7 −0.376555
\(55\) 0 0
\(56\) −1.60781e8 −2.18468
\(57\) 6.48033e7 0.813131
\(58\) 2.08118e8 2.41482
\(59\) 1.15379e7 0.123963 0.0619814 − 0.998077i \(-0.480258\pi\)
0.0619814 + 0.998077i \(0.480258\pi\)
\(60\) 0 0
\(61\) −3.48684e7 −0.322439 −0.161219 − 0.986919i \(-0.551543\pi\)
−0.161219 + 0.986919i \(0.551543\pi\)
\(62\) −2.23669e8 −1.92240
\(63\) 2.54514e7 0.203554
\(64\) 6.44110e8 4.79899
\(65\) 0 0
\(66\) −3.04141e8 −1.97300
\(67\) 1.51756e7 0.0920044 0.0460022 − 0.998941i \(-0.485352\pi\)
0.0460022 + 0.998941i \(0.485352\pi\)
\(68\) 2.55164e8 1.44720
\(69\) −1.34236e8 −0.712932
\(70\) 0 0
\(71\) −3.98346e7 −0.186036 −0.0930182 − 0.995664i \(-0.529651\pi\)
−0.0930182 + 0.995664i \(0.529651\pi\)
\(72\) −2.71934e8 −1.19252
\(73\) −7.02902e7 −0.289696 −0.144848 − 0.989454i \(-0.546269\pi\)
−0.144848 + 0.989454i \(0.546269\pi\)
\(74\) −2.07534e8 −0.804538
\(75\) 0 0
\(76\) 1.15858e9 3.98351
\(77\) 3.28993e8 1.06654
\(78\) −2.10374e8 −0.643527
\(79\) 1.16781e8 0.337327 0.168663 − 0.985674i \(-0.446055\pi\)
0.168663 + 0.985674i \(0.446055\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 2.08665e8 0.509667
\(83\) 7.69619e8 1.78002 0.890009 − 0.455943i \(-0.150698\pi\)
0.890009 + 0.455943i \(0.150698\pi\)
\(84\) 4.55032e8 0.997207
\(85\) 0 0
\(86\) 7.32941e8 1.44486
\(87\) −3.80759e8 −0.712547
\(88\) −3.51510e9 −6.24836
\(89\) −9.57076e8 −1.61693 −0.808466 − 0.588543i \(-0.799702\pi\)
−0.808466 + 0.588543i \(0.799702\pi\)
\(90\) 0 0
\(91\) 2.27564e8 0.347871
\(92\) −2.39993e9 −3.49264
\(93\) 4.09210e8 0.567248
\(94\) −7.96305e8 −1.05197
\(95\) 0 0
\(96\) −2.20277e9 −2.64697
\(97\) 4.93418e8 0.565903 0.282952 − 0.959134i \(-0.408686\pi\)
0.282952 + 0.959134i \(0.408686\pi\)
\(98\) 1.12036e9 1.22699
\(99\) 5.56436e8 0.582179
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 75.10.a.j.1.1 ✓ 4
3.2 odd 2 225.10.a.t.1.4 4
5.2 odd 4 75.10.b.h.49.1 8
5.3 odd 4 75.10.b.h.49.8 8
5.4 even 2 75.10.a.k.1.4 yes 4
15.2 even 4 225.10.b.n.199.8 8
15.8 even 4 225.10.b.n.199.1 8
15.14 odd 2 225.10.a.r.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.1 ✓ 4 1.1 even 1 trivial
75.10.a.k.1.4 yes 4 5.4 even 2
75.10.b.h.49.1 8 5.2 odd 4
75.10.b.h.49.8 8 5.3 odd 4
225.10.a.r.1.1 4 15.14 odd 2
225.10.a.t.1.4 4 3.2 odd 2
225.10.b.n.199.1 8 15.8 even 4
225.10.b.n.199.8 8 15.2 even 4