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(49,75)] 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("75.49"); 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, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-1082] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.f.49.4

$q$-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)
 
\(f(q)\) \(=\) \(q-38.7863i q^{2} -81.0000i q^{3} -992.374 q^{4} -3141.69 q^{6} -12272.1i q^{7} +18631.9i q^{8} -6561.00 q^{9} -65897.9 q^{11} +80382.3i q^{12} -112300. i q^{13} -475990. q^{14} +214567. q^{16} -96634.7i q^{17} +254477. i q^{18} +181332. q^{19} -994042. q^{21} +2.55593e6i q^{22} +244145. i q^{23} +1.50919e6 q^{24} -4.35569e6 q^{26} +531441. i q^{27} +1.21785e7i q^{28} +5.26461e6 q^{29} +1.83457e6 q^{31} +1.21730e6i q^{32} +5.33773e6i q^{33} -3.74810e6 q^{34} +6.51097e6 q^{36} -6.36194e6i q^{37} -7.03318e6i q^{38} -9.09628e6 q^{39} +1.57111e6 q^{41} +3.85552e7i q^{42} -1.99504e7i q^{43} +6.53953e7 q^{44} +9.46946e6 q^{46} -3.00961e7i q^{47} -1.73799e7i q^{48} -1.10251e8 q^{49} -7.82741e6 q^{51} +1.11443e8i q^{52} -2.57306e6i q^{53} +2.06126e7 q^{54} +2.28653e8 q^{56} -1.46879e7i q^{57} -2.04195e8i q^{58} +1.19004e8 q^{59} +1.92875e8 q^{61} -7.11559e7i q^{62} +8.05174e7i q^{63} +1.57073e8 q^{64} +2.07030e8 q^{66} +1.20193e8i q^{67} +9.58978e7i q^{68} +1.97757e7 q^{69} -699549. q^{71} -1.22244e8i q^{72} -8.91287e7i q^{73} -2.46756e8 q^{74} -1.79949e8 q^{76} +8.08707e8i q^{77} +3.52811e8i q^{78} -4.31205e8 q^{79} +4.30467e7 q^{81} -6.09375e7i q^{82} -1.69761e7i q^{83} +9.86461e8 q^{84} -7.73800e8 q^{86} -4.26433e8i q^{87} -1.22780e9i q^{88} +3.09863e6 q^{89} -1.37816e9 q^{91} -2.42283e8i q^{92} -1.48600e8i q^{93} -1.16732e9 q^{94} +9.86009e7 q^{96} -5.72609e8i q^{97} +4.27624e9i q^{98} +4.32356e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9} - 43024 q^{11} - 923328 q^{14} + 774530 q^{16} + 191792 q^{19} - 2286144 q^{21} + 4233546 q^{24} - 10838332 q^{26} + 5356424 q^{29} + 21564864 q^{31} - 11445244 q^{34}+ \cdots + 282280464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 38.7863i − 1.71413i −0.515211 − 0.857063i \(-0.672286\pi\)
0.515211 − 0.857063i \(-0.327714\pi\)
\(3\) − 81.0000i − 0.577350i
\(4\) −992.374 −1.93823
\(5\) 0 0
\(6\) −3141.69 −0.989652
\(7\) − 12272.1i − 1.93187i −0.258780 − 0.965936i \(-0.583320\pi\)
0.258780 − 0.965936i \(-0.416680\pi\)
\(8\) 18631.9i 1.60825i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) −65897.9 −1.35708 −0.678538 − 0.734565i \(-0.737386\pi\)
−0.678538 + 0.734565i \(0.737386\pi\)
\(12\) 80382.3i 1.11904i
\(13\) − 112300.i − 1.09052i −0.838267 − 0.545260i \(-0.816431\pi\)
0.838267 − 0.545260i \(-0.183569\pi\)
\(14\) −475990. −3.31147
\(15\) 0 0
\(16\) 214567. 0.818508
\(17\) − 96634.7i − 0.280616i −0.990108 − 0.140308i \(-0.955191\pi\)
0.990108 − 0.140308i \(-0.0448093\pi\)
\(18\) 254477.i 0.571376i
\(19\) 181332. 0.319214 0.159607 − 0.987181i \(-0.448977\pi\)
0.159607 + 0.987181i \(0.448977\pi\)
\(20\) 0 0
\(21\) −994042. −1.11537
\(22\) 2.55593e6i 2.32620i
\(23\) 244145.i 0.181916i 0.995855 + 0.0909582i \(0.0289930\pi\)
−0.995855 + 0.0909582i \(0.971007\pi\)
\(24\) 1.50919e6 0.928521
\(25\) 0 0
\(26\) −4.35569e6 −1.86929
\(27\) 531441.i 0.192450i
\(28\) 1.21785e7i 3.74442i
\(29\) 5.26461e6 1.38221 0.691107 − 0.722752i \(-0.257123\pi\)
0.691107 + 0.722752i \(0.257123\pi\)
\(30\) 0 0
\(31\) 1.83457e6 0.356784 0.178392 − 0.983959i \(-0.442910\pi\)
0.178392 + 0.983959i \(0.442910\pi\)
\(32\) 1.21730e6i 0.205221i
\(33\) 5.33773e6i 0.783508i
\(34\) −3.74810e6 −0.481012
\(35\) 0 0
\(36\) 6.51097e6 0.646077
\(37\) − 6.36194e6i − 0.558061i −0.960282 − 0.279030i \(-0.909987\pi\)
0.960282 − 0.279030i \(-0.0900130\pi\)
\(38\) − 7.03318e6i − 0.547174i
\(39\) −9.09628e6 −0.629612
\(40\) 0 0
\(41\) 1.57111e6 0.0868319 0.0434159 − 0.999057i \(-0.486176\pi\)
0.0434159 + 0.999057i \(0.486176\pi\)
\(42\) 3.85552e7i 1.91188i
\(43\) − 1.99504e7i − 0.889903i −0.895555 − 0.444952i \(-0.853221\pi\)
0.895555 − 0.444952i \(-0.146779\pi\)
\(44\) 6.53953e7 2.63033
\(45\) 0 0
\(46\) 9.46946e6 0.311828
\(47\) − 3.00961e7i − 0.899643i −0.893119 − 0.449821i \(-0.851488\pi\)
0.893119 − 0.449821i \(-0.148512\pi\)
\(48\) − 1.73799e7i − 0.472566i
\(49\) −1.10251e8 −2.73213
\(50\) 0 0
\(51\) −7.82741e6 −0.162014
\(52\) 1.11443e8i 2.11368i
\(53\) − 2.57306e6i − 0.0447929i −0.999749 − 0.0223964i \(-0.992870\pi\)
0.999749 − 0.0223964i \(-0.00712960\pi\)
\(54\) 2.06126e7 0.329884
\(55\) 0 0
\(56\) 2.28653e8 3.10693
\(57\) − 1.46879e7i − 0.184299i
\(58\) − 2.04195e8i − 2.36929i
\(59\) 1.19004e8 1.27858 0.639290 − 0.768965i \(-0.279228\pi\)
0.639290 + 0.768965i \(0.279228\pi\)
\(60\) 0 0
\(61\) 1.92875e8 1.78358 0.891790 − 0.452449i \(-0.149450\pi\)
0.891790 + 0.452449i \(0.149450\pi\)
\(62\) − 7.11559e7i − 0.611573i
\(63\) 8.05174e7i 0.643958i
\(64\) 1.57073e8 1.17028
\(65\) 0 0
\(66\) 2.07030e8 1.34303
\(67\) 1.20193e8i 0.728691i 0.931264 + 0.364345i \(0.118707\pi\)
−0.931264 + 0.364345i \(0.881293\pi\)
\(68\) 9.58978e7i 0.543899i
\(69\) 1.97757e7 0.105030
\(70\) 0 0
\(71\) −699549. −0.00326705 −0.00163352 − 0.999999i \(-0.500520\pi\)
−0.00163352 + 0.999999i \(0.500520\pi\)
\(72\) − 1.22244e8i − 0.536082i
\(73\) − 8.91287e7i − 0.367337i −0.982988 − 0.183669i \(-0.941203\pi\)
0.982988 − 0.183669i \(-0.0587973\pi\)
\(74\) −2.46756e8 −0.956587
\(75\) 0 0
\(76\) −1.79949e8 −0.618711
\(77\) 8.08707e8i 2.62170i
\(78\) 3.52811e8i 1.07924i
\(79\) −4.31205e8 −1.24555 −0.622776 − 0.782400i \(-0.713995\pi\)
−0.622776 + 0.782400i \(0.713995\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) − 6.09375e7i − 0.148841i
\(83\) − 1.69761e7i − 0.0392633i −0.999807 − 0.0196316i \(-0.993751\pi\)
0.999807 − 0.0196316i \(-0.00624935\pi\)
\(84\) 9.86461e8 2.16184
\(85\) 0 0
\(86\) −7.73800e8 −1.52541
\(87\) − 4.26433e8i − 0.798022i
\(88\) − 1.22780e9i − 2.18251i
\(89\) 3.09863e6 0.00523498 0.00261749 − 0.999997i \(-0.499167\pi\)
0.00261749 + 0.999997i \(0.499167\pi\)
\(90\) 0 0
\(91\) −1.37816e9 −2.10675
\(92\) − 2.42283e8i − 0.352596i
\(93\) − 1.48600e8i − 0.205989i
\(94\) −1.16732e9 −1.54210
\(95\) 0 0
\(96\) 9.86009e7 0.118484
\(97\) − 5.72609e8i − 0.656727i −0.944551 − 0.328364i \(-0.893503\pi\)
0.944551 − 0.328364i \(-0.106497\pi\)
\(98\) 4.27624e9i 4.68322i
\(99\) 4.32356e8 0.452359
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.b.f.49.1 4
3.2 odd 2 225.10.b.i.199.4 4
5.2 odd 4 15.10.a.d.1.2 ✓ 2
5.3 odd 4 75.10.a.f.1.1 2
5.4 even 2 inner 75.10.b.f.49.4 4
15.2 even 4 45.10.a.d.1.1 2
15.8 even 4 225.10.a.k.1.2 2
15.14 odd 2 225.10.b.i.199.1 4
20.7 even 4 240.10.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.d.1.2 ✓ 2 5.2 odd 4
45.10.a.d.1.1 2 15.2 even 4
75.10.a.f.1.1 2 5.3 odd 4
75.10.b.f.49.1 4 1.1 even 1 trivial
75.10.b.f.49.4 4 5.4 even 2 inner
225.10.a.k.1.2 2 15.8 even 4
225.10.b.i.199.1 4 15.14 odd 2
225.10.b.i.199.4 4 3.2 odd 2
240.10.a.r.1.1 2 20.7 even 4