Show commands: Magma / Pari/GP / SageMath

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 = [4,33] 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: \(4\)
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: \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-265.574\) 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)
 
\(f(q)\) \(=\) \(q-257.574 q^{2} +6561.00 q^{3} -64727.8 q^{4} -1.68994e6 q^{6} -8.43659e6 q^{7} +5.04329e7 q^{8} +4.30467e7 q^{9} -1.27592e9 q^{11} -4.24679e8 q^{12} -2.16831e9 q^{13} +2.17304e9 q^{14} -4.50617e9 q^{16} -3.94044e10 q^{17} -1.10877e10 q^{18} +1.12109e11 q^{19} -5.53525e10 q^{21} +3.28642e11 q^{22} -5.33219e11 q^{23} +3.30890e11 q^{24} +5.58501e11 q^{26} +2.82430e11 q^{27} +5.46082e11 q^{28} -1.70330e12 q^{29} -1.16317e11 q^{31} -5.44967e12 q^{32} -8.37128e12 q^{33} +1.01495e13 q^{34} -2.78632e12 q^{36} -3.69532e13 q^{37} -2.88763e13 q^{38} -1.42263e13 q^{39} +6.01774e13 q^{41} +1.42573e13 q^{42} +3.10214e13 q^{43} +8.25872e13 q^{44} +1.37343e14 q^{46} +2.34600e14 q^{47} -2.95650e13 q^{48} -1.61454e14 q^{49} -2.58532e14 q^{51} +1.40350e14 q^{52} -5.42186e14 q^{53} -7.27464e13 q^{54} -4.25482e14 q^{56} +7.35546e14 q^{57} +4.38725e14 q^{58} -1.85401e14 q^{59} -1.71339e15 q^{61} +2.99603e13 q^{62} -3.63168e14 q^{63} +1.99432e15 q^{64} +2.15622e15 q^{66} +6.75194e12 q^{67} +2.55056e15 q^{68} -3.49845e15 q^{69} +9.21473e13 q^{71} +2.17097e15 q^{72} -1.14713e16 q^{73} +9.51816e15 q^{74} -7.25656e15 q^{76} +1.07644e16 q^{77} +3.66432e15 q^{78} +8.98946e15 q^{79} +1.85302e15 q^{81} -1.55001e16 q^{82} -4.25580e15 q^{83} +3.58285e15 q^{84} -7.99030e15 q^{86} -1.11753e16 q^{87} -6.43481e16 q^{88} -3.54476e16 q^{89} +1.82932e16 q^{91} +3.45141e16 q^{92} -7.63158e14 q^{93} -6.04269e16 q^{94} -3.57553e16 q^{96} -3.08428e16 q^{97} +4.15864e16 q^{98} -5.49240e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} + 26244 q^{3} + 439357 q^{4} + 216513 q^{6} - 17583104 q^{7} - 63651621 q^{8} + 172186884 q^{9} - 575495184 q^{11} + 2882621277 q^{12} + 5049645832 q^{13} - 6699316032 q^{14} + 7512683905 q^{16}+ \cdots - 24\!\cdots\!64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −257.574 −0.711453 −0.355727 − 0.934590i \(-0.615767\pi\)
−0.355727 + 0.934590i \(0.615767\pi\)
\(3\) 6561.00 0.577350
\(4\) −64727.8 −0.493834
\(5\) 0 0
\(6\) −1.68994e6 −0.410758
\(7\) −8.43659e6 −0.553138 −0.276569 − 0.960994i \(-0.589197\pi\)
−0.276569 + 0.960994i \(0.589197\pi\)
\(8\) 5.04329e7 1.06279
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −1.27592e9 −1.79467 −0.897334 − 0.441353i \(-0.854499\pi\)
−0.897334 + 0.441353i \(0.854499\pi\)
\(12\) −4.24679e8 −0.285115
\(13\) −2.16831e9 −0.737231 −0.368616 − 0.929582i \(-0.620168\pi\)
−0.368616 + 0.929582i \(0.620168\pi\)
\(14\) 2.17304e9 0.393532
\(15\) 0 0
\(16\) −4.50617e9 −0.262294
\(17\) −3.94044e10 −1.37003 −0.685013 − 0.728531i \(-0.740203\pi\)
−0.685013 + 0.728531i \(0.740203\pi\)
\(18\) −1.10877e10 −0.237151
\(19\) 1.12109e11 1.51438 0.757189 − 0.653196i \(-0.226572\pi\)
0.757189 + 0.653196i \(0.226572\pi\)
\(20\) 0 0
\(21\) −5.53525e10 −0.319354
\(22\) 3.28642e11 1.27682
\(23\) −5.33219e11 −1.41977 −0.709886 − 0.704317i \(-0.751254\pi\)
−0.709886 + 0.704317i \(0.751254\pi\)
\(24\) 3.30890e11 0.613604
\(25\) 0 0
\(26\) 5.58501e11 0.524506
\(27\) 2.82430e11 0.192450
\(28\) 5.46082e11 0.273159
\(29\) −1.70330e12 −0.632278 −0.316139 − 0.948713i \(-0.602387\pi\)
−0.316139 + 0.948713i \(0.602387\pi\)
\(30\) 0 0
\(31\) −1.16317e11 −0.0244946 −0.0122473 − 0.999925i \(-0.503899\pi\)
−0.0122473 + 0.999925i \(0.503899\pi\)
\(32\) −5.44967e12 −0.876184
\(33\) −8.37128e12 −1.03615
\(34\) 1.01495e13 0.974709
\(35\) 0 0
\(36\) −2.78632e12 −0.164611
\(37\) −3.69532e13 −1.72957 −0.864783 − 0.502146i \(-0.832544\pi\)
−0.864783 + 0.502146i \(0.832544\pi\)
\(38\) −2.88763e13 −1.07741
\(39\) −1.42263e13 −0.425641
\(40\) 0 0
\(41\) 6.01774e13 1.17698 0.588492 − 0.808503i \(-0.299722\pi\)
0.588492 + 0.808503i \(0.299722\pi\)
\(42\) 1.42573e13 0.227206
\(43\) 3.10214e13 0.404744 0.202372 − 0.979309i \(-0.435135\pi\)
0.202372 + 0.979309i \(0.435135\pi\)
\(44\) 8.25872e13 0.886268
\(45\) 0 0
\(46\) 1.37343e14 1.01010
\(47\) 2.34600e14 1.43713 0.718567 − 0.695458i \(-0.244799\pi\)
0.718567 + 0.695458i \(0.244799\pi\)
\(48\) −2.95650e13 −0.151435
\(49\) −1.61454e14 −0.694038
\(50\) 0 0
\(51\) −2.58532e14 −0.790985
\(52\) 1.40350e14 0.364070
\(53\) −5.42186e14 −1.19620 −0.598100 − 0.801422i \(-0.704077\pi\)
−0.598100 + 0.801422i \(0.704077\pi\)
\(54\) −7.27464e13 −0.136919
\(55\) 0 0
\(56\) −4.25482e14 −0.587872
\(57\) 7.35546e14 0.874327
\(58\) 4.38725e14 0.449836
\(59\) −1.85401e14 −0.164388 −0.0821941 − 0.996616i \(-0.526193\pi\)
−0.0821941 + 0.996616i \(0.526193\pi\)
\(60\) 0 0
\(61\) −1.71339e15 −1.14433 −0.572167 − 0.820137i \(-0.693897\pi\)
−0.572167 + 0.820137i \(0.693897\pi\)
\(62\) 2.99603e13 0.0174268
\(63\) −3.63168e14 −0.184379
\(64\) 1.99432e15 0.885657
\(65\) 0 0
\(66\) 2.15622e15 0.737173
\(67\) 6.75194e12 0.00203139 0.00101569 − 0.999999i \(-0.499677\pi\)
0.00101569 + 0.999999i \(0.499677\pi\)
\(68\) 2.55056e15 0.676566
\(69\) −3.49845e15 −0.819706
\(70\) 0 0
\(71\) 9.21473e13 0.0169351 0.00846753 − 0.999964i \(-0.497305\pi\)
0.00846753 + 0.999964i \(0.497305\pi\)
\(72\) 2.17097e15 0.354264
\(73\) −1.14713e16 −1.66483 −0.832415 − 0.554153i \(-0.813042\pi\)
−0.832415 + 0.554153i \(0.813042\pi\)
\(74\) 9.51816e15 1.23050
\(75\) 0 0
\(76\) −7.25656e15 −0.747852
\(77\) 1.07644e16 0.992699
\(78\) 3.66432e15 0.302824
\(79\) 8.98946e15 0.666658 0.333329 − 0.942810i \(-0.391828\pi\)
0.333329 + 0.942810i \(0.391828\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −1.55001e16 −0.837370
\(83\) −4.25580e15 −0.207404 −0.103702 − 0.994608i \(-0.533069\pi\)
−0.103702 + 0.994608i \(0.533069\pi\)
\(84\) 3.58285e15 0.157708
\(85\) 0 0
\(86\) −7.99030e15 −0.287956
\(87\) −1.11753e16 −0.365046
\(88\) −6.43481e16 −1.90736
\(89\) −3.54476e16 −0.954490 −0.477245 − 0.878770i \(-0.658365\pi\)
−0.477245 + 0.878770i \(0.658365\pi\)
\(90\) 0 0
\(91\) 1.82932e16 0.407791
\(92\) 3.45141e16 0.701132
\(93\) −7.63158e14 −0.0141420
\(94\) −6.04269e16 −1.02245
\(95\) 0 0
\(96\) −3.57553e16 −0.505865
\(97\) −3.08428e16 −0.399570 −0.199785 − 0.979840i \(-0.564024\pi\)
−0.199785 + 0.979840i \(0.564024\pi\)
\(98\) 4.15864e16 0.493776
\(99\) −5.49240e16 −0.598222
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.f.1.2 4
5.2 odd 4 75.18.b.f.49.4 8
5.3 odd 4 75.18.b.f.49.5 8
5.4 even 2 15.18.a.d.1.3 ✓ 4
15.14 odd 2 45.18.a.f.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.3 ✓ 4 5.4 even 2
45.18.a.f.1.2 4 15.14 odd 2
75.18.a.f.1.2 4 1.1 even 1 trivial
75.18.b.f.49.4 8 5.2 odd 4
75.18.b.f.49.5 8 5.3 odd 4