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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,18,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, 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, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-878714] 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: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 963373x^{6} + 304121840676x^{4} + 35407691713753600x^{2} + 1297706673500416000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{8}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(-265.574i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.f.49.5

$q$-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)
 
\(f(q)\) \(=\) \(q-257.574i q^{2} -6561.00i q^{3} +64727.8 q^{4} -1.68994e6 q^{6} -8.43659e6i q^{7} -5.04329e7i q^{8} -4.30467e7 q^{9} -1.27592e9 q^{11} -4.24679e8i q^{12} +2.16831e9i q^{13} -2.17304e9 q^{14} -4.50617e9 q^{16} -3.94044e10i q^{17} +1.10877e10i q^{18} -1.12109e11 q^{19} -5.53525e10 q^{21} +3.28642e11i q^{22} +5.33219e11i q^{23} -3.30890e11 q^{24} +5.58501e11 q^{26} +2.82430e11i q^{27} -5.46082e11i q^{28} +1.70330e12 q^{29} -1.16317e11 q^{31} -5.44967e12i q^{32} +8.37128e12i q^{33} -1.01495e13 q^{34} -2.78632e12 q^{36} -3.69532e13i q^{37} +2.88763e13i q^{38} +1.42263e13 q^{39} +6.01774e13 q^{41} +1.42573e13i q^{42} -3.10214e13i q^{43} -8.25872e13 q^{44} +1.37343e14 q^{46} +2.34600e14i q^{47} +2.95650e13i q^{48} +1.61454e14 q^{49} -2.58532e14 q^{51} +1.40350e14i q^{52} +5.42186e14i q^{53} +7.27464e13 q^{54} -4.25482e14 q^{56} +7.35546e14i q^{57} -4.38725e14i q^{58} +1.85401e14 q^{59} -1.71339e15 q^{61} +2.99603e13i q^{62} +3.63168e14i q^{63} -1.99432e15 q^{64} +2.15622e15 q^{66} +6.75194e12i q^{67} -2.55056e15i q^{68} +3.49845e15 q^{69} +9.21473e13 q^{71} +2.17097e15i q^{72} +1.14713e16i q^{73} -9.51816e15 q^{74} -7.25656e15 q^{76} +1.07644e16i q^{77} -3.66432e15i q^{78} -8.98946e15 q^{79} +1.85302e15 q^{81} -1.55001e16i q^{82} +4.25580e15i q^{83} -3.58285e15 q^{84} -7.99030e15 q^{86} -1.11753e16i q^{87} +6.43481e16i q^{88} +3.54476e16 q^{89} +1.82932e16 q^{91} +3.45141e16i q^{92} +7.63158e14i q^{93} +6.04269e16 q^{94} -3.57553e16 q^{96} -3.08428e16i q^{97} -4.15864e16i q^{98} +5.49240e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 878714 q^{4} + 433026 q^{6} - 344373768 q^{9} - 1150990368 q^{11} + 13398632064 q^{14} + 15025367810 q^{16} - 397827528736 q^{19} - 230725490688 q^{21} + 835236570762 q^{24} - 1055057698044 q^{26}+ \cdots + 49\!\cdots\!28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 257.574i − 0.711453i −0.934590 − 0.355727i \(-0.884233\pi\)
0.934590 − 0.355727i \(-0.115767\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) 64727.8 0.493834
\(5\) 0 0
\(6\) −1.68994e6 −0.410758
\(7\) − 8.43659e6i − 0.553138i −0.960994 − 0.276569i \(-0.910803\pi\)
0.960994 − 0.276569i \(-0.0891975\pi\)
\(8\) − 5.04329e7i − 1.06279i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −1.27592e9 −1.79467 −0.897334 − 0.441353i \(-0.854499\pi\)
−0.897334 + 0.441353i \(0.854499\pi\)
\(12\) − 4.24679e8i − 0.285115i
\(13\) 2.16831e9i 0.737231i 0.929582 + 0.368616i \(0.120168\pi\)
−0.929582 + 0.368616i \(0.879832\pi\)
\(14\) −2.17304e9 −0.393532
\(15\) 0 0
\(16\) −4.50617e9 −0.262294
\(17\) − 3.94044e10i − 1.37003i −0.728531 − 0.685013i \(-0.759797\pi\)
0.728531 − 0.685013i \(-0.240203\pi\)
\(18\) 1.10877e10i 0.237151i
\(19\) −1.12109e11 −1.51438 −0.757189 − 0.653196i \(-0.773428\pi\)
−0.757189 + 0.653196i \(0.773428\pi\)
\(20\) 0 0
\(21\) −5.53525e10 −0.319354
\(22\) 3.28642e11i 1.27682i
\(23\) 5.33219e11i 1.41977i 0.704317 + 0.709886i \(0.251254\pi\)
−0.704317 + 0.709886i \(0.748746\pi\)
\(24\) −3.30890e11 −0.613604
\(25\) 0 0
\(26\) 5.58501e11 0.524506
\(27\) 2.82430e11i 0.192450i
\(28\) − 5.46082e11i − 0.273159i
\(29\) 1.70330e12 0.632278 0.316139 − 0.948713i \(-0.397613\pi\)
0.316139 + 0.948713i \(0.397613\pi\)
\(30\) 0 0
\(31\) −1.16317e11 −0.0244946 −0.0122473 − 0.999925i \(-0.503899\pi\)
−0.0122473 + 0.999925i \(0.503899\pi\)
\(32\) − 5.44967e12i − 0.876184i
\(33\) 8.37128e12i 1.03615i
\(34\) −1.01495e13 −0.974709
\(35\) 0 0
\(36\) −2.78632e12 −0.164611
\(37\) − 3.69532e13i − 1.72957i −0.502146 − 0.864783i \(-0.667456\pi\)
0.502146 − 0.864783i \(-0.332544\pi\)
\(38\) 2.88763e13i 1.07741i
\(39\) 1.42263e13 0.425641
\(40\) 0 0
\(41\) 6.01774e13 1.17698 0.588492 − 0.808503i \(-0.299722\pi\)
0.588492 + 0.808503i \(0.299722\pi\)
\(42\) 1.42573e13i 0.227206i
\(43\) − 3.10214e13i − 0.404744i −0.979309 − 0.202372i \(-0.935135\pi\)
0.979309 − 0.202372i \(-0.0648650\pi\)
\(44\) −8.25872e13 −0.886268
\(45\) 0 0
\(46\) 1.37343e14 1.01010
\(47\) 2.34600e14i 1.43713i 0.695458 + 0.718567i \(0.255201\pi\)
−0.695458 + 0.718567i \(0.744799\pi\)
\(48\) 2.95650e13i 0.151435i
\(49\) 1.61454e14 0.694038
\(50\) 0 0
\(51\) −2.58532e14 −0.790985
\(52\) 1.40350e14i 0.364070i
\(53\) 5.42186e14i 1.19620i 0.801422 + 0.598100i \(0.204077\pi\)
−0.801422 + 0.598100i \(0.795923\pi\)
\(54\) 7.27464e13 0.136919
\(55\) 0 0
\(56\) −4.25482e14 −0.587872
\(57\) 7.35546e14i 0.874327i
\(58\) − 4.38725e14i − 0.449836i
\(59\) 1.85401e14 0.164388 0.0821941 − 0.996616i \(-0.473807\pi\)
0.0821941 + 0.996616i \(0.473807\pi\)
\(60\) 0 0
\(61\) −1.71339e15 −1.14433 −0.572167 − 0.820137i \(-0.693897\pi\)
−0.572167 + 0.820137i \(0.693897\pi\)
\(62\) 2.99603e13i 0.0174268i
\(63\) 3.63168e14i 0.184379i
\(64\) −1.99432e15 −0.885657
\(65\) 0 0
\(66\) 2.15622e15 0.737173
\(67\) 6.75194e12i 0.00203139i 0.999999 + 0.00101569i \(0.000323306\pi\)
−0.999999 + 0.00101569i \(0.999677\pi\)
\(68\) − 2.55056e15i − 0.676566i
\(69\) 3.49845e15 0.819706
\(70\) 0 0
\(71\) 9.21473e13 0.0169351 0.00846753 − 0.999964i \(-0.497305\pi\)
0.00846753 + 0.999964i \(0.497305\pi\)
\(72\) 2.17097e15i 0.354264i
\(73\) 1.14713e16i 1.66483i 0.554153 + 0.832415i \(0.313042\pi\)
−0.554153 + 0.832415i \(0.686958\pi\)
\(74\) −9.51816e15 −1.23050
\(75\) 0 0
\(76\) −7.25656e15 −0.747852
\(77\) 1.07644e16i 0.992699i
\(78\) − 3.66432e15i − 0.302824i
\(79\) −8.98946e15 −0.666658 −0.333329 − 0.942810i \(-0.608172\pi\)
−0.333329 + 0.942810i \(0.608172\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 1.55001e16i − 0.837370i
\(83\) 4.25580e15i 0.207404i 0.994608 + 0.103702i \(0.0330689\pi\)
−0.994608 + 0.103702i \(0.966931\pi\)
\(84\) −3.58285e15 −0.157708
\(85\) 0 0
\(86\) −7.99030e15 −0.287956
\(87\) − 1.11753e16i − 0.365046i
\(88\) 6.43481e16i 1.90736i
\(89\) 3.54476e16 0.954490 0.477245 − 0.878770i \(-0.341635\pi\)
0.477245 + 0.878770i \(0.341635\pi\)
\(90\) 0 0
\(91\) 1.82932e16 0.407791
\(92\) 3.45141e16i 0.701132i
\(93\) 7.63158e14i 0.0141420i
\(94\) 6.04269e16 1.02245
\(95\) 0 0
\(96\) −3.57553e16 −0.505865
\(97\) − 3.08428e16i − 0.399570i −0.979840 − 0.199785i \(-0.935976\pi\)
0.979840 − 0.199785i \(-0.0640244\pi\)
\(98\) − 4.15864e16i − 0.493776i
\(99\) 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.b.f.49.4 8
5.2 odd 4 15.18.a.d.1.3 ✓ 4
5.3 odd 4 75.18.a.f.1.2 4
5.4 even 2 inner 75.18.b.f.49.5 8
15.2 even 4 45.18.a.f.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.3 ✓ 4 5.2 odd 4
45.18.a.f.1.2 4 15.2 even 4
75.18.a.f.1.2 4 5.3 odd 4
75.18.b.f.49.4 8 1.1 even 1 trivial
75.18.b.f.49.5 8 5.4 even 2 inner