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 1.2
Root \(-26.5703\) 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)
 
\(f(q)\) \(=\) \(q-15.4348 q^{2} -81.0000 q^{3} -273.767 q^{4} +1250.22 q^{6} +8162.66 q^{7} +12128.2 q^{8} +6561.00 q^{9} -8297.50 q^{11} +22175.1 q^{12} -193920. q^{13} -125989. q^{14} -47027.4 q^{16} -465904. q^{17} -101268. q^{18} +489565. q^{19} -661176. q^{21} +128070. q^{22} -335362. q^{23} -982381. q^{24} +2.99313e6 q^{26} -531441. q^{27} -2.23466e6 q^{28} +4.89027e6 q^{29} -3.76904e6 q^{31} -5.48376e6 q^{32} +672098. q^{33} +7.19113e6 q^{34} -1.79618e6 q^{36} +1.61418e6 q^{37} -7.55634e6 q^{38} +1.57076e7 q^{39} -6.08529e6 q^{41} +1.02051e7 q^{42} +3.90755e6 q^{43} +2.27158e6 q^{44} +5.17625e6 q^{46} +5.28173e7 q^{47} +3.80922e6 q^{48} +2.62754e7 q^{49} +3.77382e7 q^{51} +5.30889e7 q^{52} -1.00862e8 q^{53} +8.20269e6 q^{54} +9.89981e7 q^{56} -3.96548e7 q^{57} -7.54805e7 q^{58} +1.08004e8 q^{59} +3.92523e7 q^{61} +5.81744e7 q^{62} +5.35552e7 q^{63} +1.08719e8 q^{64} -1.03737e7 q^{66} +2.20069e8 q^{67} +1.27549e8 q^{68} +2.71643e7 q^{69} +1.79330e8 q^{71} +7.95729e7 q^{72} -3.59772e8 q^{73} -2.49146e7 q^{74} -1.34026e8 q^{76} -6.77297e7 q^{77} -2.42443e8 q^{78} +1.72846e8 q^{79} +4.30467e7 q^{81} +9.39253e7 q^{82} +4.11145e8 q^{83} +1.81008e8 q^{84} -6.03123e7 q^{86} -3.96112e8 q^{87} -1.00633e8 q^{88} +9.60619e8 q^{89} -1.58291e9 q^{91} +9.18110e7 q^{92} +3.05292e8 q^{93} -8.15224e8 q^{94} +4.44184e8 q^{96} -1.13187e9 q^{97} -4.05556e8 q^{98} -5.44399e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9} + 104696 q^{11} - 145152 q^{12} - 140812 q^{13} - 181062 q^{14} + 1319800 q^{16} - 489352 q^{17} + 13122 q^{18}+ \cdots + 686910456 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.4348 −0.682129 −0.341064 − 0.940040i \(-0.610787\pi\)
−0.341064 + 0.940040i \(0.610787\pi\)
\(3\) −81.0000 −0.577350
\(4\) −273.767 −0.534700
\(5\) 0 0
\(6\) 1250.22 0.393827
\(7\) 8162.66 1.28496 0.642481 − 0.766301i \(-0.277905\pi\)
0.642481 + 0.766301i \(0.277905\pi\)
\(8\) 12128.2 1.04686
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −8297.50 −0.170876 −0.0854378 − 0.996344i \(-0.527229\pi\)
−0.0854378 + 0.996344i \(0.527229\pi\)
\(12\) 22175.1 0.308709
\(13\) −193920. −1.88312 −0.941561 − 0.336841i \(-0.890641\pi\)
−0.941561 + 0.336841i \(0.890641\pi\)
\(14\) −125989. −0.876510
\(15\) 0 0
\(16\) −47027.4 −0.179395
\(17\) −465904. −1.35293 −0.676466 − 0.736474i \(-0.736489\pi\)
−0.676466 + 0.736474i \(0.736489\pi\)
\(18\) −101268. −0.227376
\(19\) 489565. 0.861825 0.430912 − 0.902394i \(-0.358192\pi\)
0.430912 + 0.902394i \(0.358192\pi\)
\(20\) 0 0
\(21\) −661176. −0.741874
\(22\) 128070. 0.116559
\(23\) −335362. −0.249884 −0.124942 − 0.992164i \(-0.539875\pi\)
−0.124942 + 0.992164i \(0.539875\pi\)
\(24\) −982381. −0.604407
\(25\) 0 0
\(26\) 2.99313e6 1.28453
\(27\) −531441. −0.192450
\(28\) −2.23466e6 −0.687070
\(29\) 4.89027e6 1.28393 0.641966 − 0.766733i \(-0.278119\pi\)
0.641966 + 0.766733i \(0.278119\pi\)
\(30\) 0 0
\(31\) −3.76904e6 −0.732998 −0.366499 − 0.930418i \(-0.619444\pi\)
−0.366499 + 0.930418i \(0.619444\pi\)
\(32\) −5.48376e6 −0.924493
\(33\) 672098. 0.0986551
\(34\) 7.19113e6 0.922874
\(35\) 0 0
\(36\) −1.79618e6 −0.178233
\(37\) 1.61418e6 0.141594 0.0707970 − 0.997491i \(-0.477446\pi\)
0.0707970 + 0.997491i \(0.477446\pi\)
\(38\) −7.55634e6 −0.587876
\(39\) 1.57076e7 1.08722
\(40\) 0 0
\(41\) −6.08529e6 −0.336321 −0.168160 − 0.985760i \(-0.553783\pi\)
−0.168160 + 0.985760i \(0.553783\pi\)
\(42\) 1.02051e7 0.506053
\(43\) 3.90755e6 0.174300 0.0871498 − 0.996195i \(-0.472224\pi\)
0.0871498 + 0.996195i \(0.472224\pi\)
\(44\) 2.27158e6 0.0913673
\(45\) 0 0
\(46\) 5.17625e6 0.170453
\(47\) 5.28173e7 1.57883 0.789415 − 0.613860i \(-0.210384\pi\)
0.789415 + 0.613860i \(0.210384\pi\)
\(48\) 3.80922e6 0.103574
\(49\) 2.62754e7 0.651130
\(50\) 0 0
\(51\) 3.77382e7 0.781116
\(52\) 5.30889e7 1.00691
\(53\) −1.00862e8 −1.75585 −0.877926 − 0.478795i \(-0.841074\pi\)
−0.877926 + 0.478795i \(0.841074\pi\)
\(54\) 8.20269e6 0.131276
\(55\) 0 0
\(56\) 9.89981e7 1.34518
\(57\) −3.96548e7 −0.497575
\(58\) −7.54805e7 −0.875807
\(59\) 1.08004e8 1.16039 0.580196 − 0.814477i \(-0.302976\pi\)
0.580196 + 0.814477i \(0.302976\pi\)
\(60\) 0 0
\(61\) 3.92523e7 0.362979 0.181489 − 0.983393i \(-0.441908\pi\)
0.181489 + 0.983393i \(0.441908\pi\)
\(62\) 5.81744e7 0.499999
\(63\) 5.35552e7 0.428321
\(64\) 1.08719e8 0.810018
\(65\) 0 0
\(66\) −1.03737e7 −0.0672955
\(67\) 2.20069e8 1.33421 0.667103 − 0.744966i \(-0.267534\pi\)
0.667103 + 0.744966i \(0.267534\pi\)
\(68\) 1.27549e8 0.723413
\(69\) 2.71643e7 0.144271
\(70\) 0 0
\(71\) 1.79330e8 0.837513 0.418756 − 0.908099i \(-0.362466\pi\)
0.418756 + 0.908099i \(0.362466\pi\)
\(72\) 7.95729e7 0.348954
\(73\) −3.59772e8 −1.48277 −0.741387 − 0.671078i \(-0.765832\pi\)
−0.741387 + 0.671078i \(0.765832\pi\)
\(74\) −2.49146e7 −0.0965853
\(75\) 0 0
\(76\) −1.34026e8 −0.460818
\(77\) −6.77297e7 −0.219569
\(78\) −2.42443e8 −0.741625
\(79\) 1.72846e8 0.499272 0.249636 − 0.968340i \(-0.419689\pi\)
0.249636 + 0.968340i \(0.419689\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 9.39253e7 0.229414
\(83\) 4.11145e8 0.950920 0.475460 − 0.879737i \(-0.342282\pi\)
0.475460 + 0.879737i \(0.342282\pi\)
\(84\) 1.81008e8 0.396680
\(85\) 0 0
\(86\) −6.03123e7 −0.118895
\(87\) −3.96112e8 −0.741279
\(88\) −1.00633e8 −0.178883
\(89\) 9.60619e8 1.62292 0.811458 − 0.584410i \(-0.198674\pi\)
0.811458 + 0.584410i \(0.198674\pi\)
\(90\) 0 0
\(91\) −1.58291e9 −2.41974
\(92\) 9.18110e7 0.133613
\(93\) 3.05292e8 0.423197
\(94\) −8.15224e8 −1.07697
\(95\) 0 0
\(96\) 4.44184e8 0.533756
\(97\) −1.13187e9 −1.29814 −0.649071 − 0.760728i \(-0.724842\pi\)
−0.649071 + 0.760728i \(0.724842\pi\)
\(98\) −4.05556e8 −0.444154
\(99\) −5.44399e7 −0.0569586
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.k.1.2 yes 4
3.2 odd 2 225.10.a.r.1.3 4
5.2 odd 4 75.10.b.h.49.3 8
5.3 odd 4 75.10.b.h.49.6 8
5.4 even 2 75.10.a.j.1.3 ✓ 4
15.2 even 4 225.10.b.n.199.6 8
15.8 even 4 225.10.b.n.199.3 8
15.14 odd 2 225.10.a.t.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.3 ✓ 4 5.4 even 2
75.10.a.k.1.2 yes 4 1.1 even 1 trivial
75.10.b.h.49.3 8 5.2 odd 4
75.10.b.h.49.6 8 5.3 odd 4
225.10.a.r.1.3 4 3.2 odd 2
225.10.a.t.1.2 4 15.14 odd 2
225.10.b.n.199.3 8 15.8 even 4
225.10.b.n.199.6 8 15.2 even 4