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Properties

Label 75.18.b.e.49.2
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [75,18,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 = [6,0,0,14174] 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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 364793x^{4} + 33276143056x^{2} + 15375182054400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(-436.958i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.e.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)
 
\(f(q)\) \(=\) \(q-352.958i q^{2} +6561.00i q^{3} +6492.31 q^{4} +2.31576e6 q^{6} -1.07009e7i q^{7} -4.85545e7i q^{8} -4.30467e7 q^{9} +1.00429e9 q^{11} +4.25961e7i q^{12} +3.43757e9i q^{13} -3.77697e9 q^{14} -1.62868e10 q^{16} -4.81590e10i q^{17} +1.51937e10i q^{18} -8.08158e9 q^{19} +7.02085e10 q^{21} -3.54473e11i q^{22} -3.08039e11i q^{23} +3.18566e11 q^{24} +1.21332e12 q^{26} -2.82430e11i q^{27} -6.94735e10i q^{28} -7.97487e11 q^{29} -3.17371e12 q^{31} -6.15585e11i q^{32} +6.58916e12i q^{33} -1.69981e13 q^{34} -2.79473e11 q^{36} -1.61437e13i q^{37} +2.85246e12i q^{38} -2.25539e13 q^{39} +7.63188e13 q^{41} -2.47807e13i q^{42} +1.29020e14i q^{43} +6.52018e12 q^{44} -1.08725e14 q^{46} +2.44935e14i q^{47} -1.06857e14i q^{48} +1.18121e14 q^{49} +3.15971e14 q^{51} +2.23178e13i q^{52} -8.60208e14i q^{53} -9.96859e13 q^{54} -5.19576e14 q^{56} -5.30232e13i q^{57} +2.81480e14i q^{58} -1.21582e14 q^{59} -4.27987e14 q^{61} +1.12019e15i q^{62} +4.60638e14i q^{63} -2.35201e15 q^{64} +2.32570e15 q^{66} +1.16734e15i q^{67} -3.12664e14i q^{68} +2.02104e15 q^{69} +3.01633e15 q^{71} +2.09011e15i q^{72} -7.32396e15i q^{73} -5.69806e15 q^{74} -5.24682e13 q^{76} -1.07468e16i q^{77} +7.96059e15i q^{78} +7.07645e15 q^{79} +1.85302e15 q^{81} -2.69374e16i q^{82} -1.18761e16i q^{83} +4.55816e14 q^{84} +4.55388e16 q^{86} -5.23231e15i q^{87} -4.87629e16i q^{88} -5.69376e16 q^{89} +3.67851e16 q^{91} -1.99989e15i q^{92} -2.08227e16i q^{93} +8.64519e16 q^{94} +4.03885e15 q^{96} +8.11777e16i q^{97} -4.16920e16i q^{98} -4.32315e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 14174 q^{4} - 3319866 q^{6} - 258280326 q^{9} + 1887127360 q^{11} - 3694967976 q^{14} - 43887742718 q^{16} + 156244985992 q^{19} - 56850855048 q^{21} + 212425667586 q^{24} + 5331480861772 q^{26} - 11550537176156 q^{29}+ \cdots - 81\!\cdots\!60 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 352.958i − 0.974919i −0.873146 − 0.487460i \(-0.837924\pi\)
0.873146 − 0.487460i \(-0.162076\pi\)
\(3\) 6561.00i 0.577350i
\(4\) 6492.31 0.0495324
\(5\) 0 0
\(6\) 2.31576e6 0.562870
\(7\) − 1.07009e7i − 0.701595i −0.936451 − 0.350798i \(-0.885911\pi\)
0.936451 − 0.350798i \(-0.114089\pi\)
\(8\) − 4.85545e7i − 1.02321i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 1.00429e9 1.41261 0.706305 − 0.707908i \(-0.250361\pi\)
0.706305 + 0.707908i \(0.250361\pi\)
\(12\) 4.25961e7i 0.0285976i
\(13\) 3.43757e9i 1.16878i 0.811473 + 0.584391i \(0.198666\pi\)
−0.811473 + 0.584391i \(0.801334\pi\)
\(14\) −3.77697e9 −0.683999
\(15\) 0 0
\(16\) −1.62868e10 −0.948014
\(17\) − 4.81590e10i − 1.67441i −0.546889 − 0.837205i \(-0.684188\pi\)
0.546889 − 0.837205i \(-0.315812\pi\)
\(18\) 1.51937e10i 0.324973i
\(19\) −8.08158e9 −0.109167 −0.0545834 − 0.998509i \(-0.517383\pi\)
−0.0545834 + 0.998509i \(0.517383\pi\)
\(20\) 0 0
\(21\) 7.02085e10 0.405066
\(22\) − 3.54473e11i − 1.37718i
\(23\) − 3.08039e11i − 0.820199i −0.912041 − 0.410099i \(-0.865494\pi\)
0.912041 − 0.410099i \(-0.134506\pi\)
\(24\) 3.18566e11 0.590750
\(25\) 0 0
\(26\) 1.21332e12 1.13947
\(27\) − 2.82430e11i − 0.192450i
\(28\) − 6.94735e10i − 0.0347517i
\(29\) −7.97487e11 −0.296033 −0.148017 − 0.988985i \(-0.547289\pi\)
−0.148017 + 0.988985i \(0.547289\pi\)
\(30\) 0 0
\(31\) −3.17371e12 −0.668334 −0.334167 − 0.942514i \(-0.608455\pi\)
−0.334167 + 0.942514i \(0.608455\pi\)
\(32\) − 6.15585e11i − 0.0989722i
\(33\) 6.58916e12i 0.815570i
\(34\) −1.69981e13 −1.63241
\(35\) 0 0
\(36\) −2.79473e11 −0.0165108
\(37\) − 1.61437e13i − 0.755595i −0.925888 − 0.377797i \(-0.876682\pi\)
0.925888 − 0.377797i \(-0.123318\pi\)
\(38\) 2.85246e12i 0.106429i
\(39\) −2.25539e13 −0.674796
\(40\) 0 0
\(41\) 7.63188e13 1.49269 0.746344 − 0.665560i \(-0.231807\pi\)
0.746344 + 0.665560i \(0.231807\pi\)
\(42\) − 2.47807e13i − 0.394907i
\(43\) 1.29020e14i 1.68336i 0.539978 + 0.841679i \(0.318432\pi\)
−0.539978 + 0.841679i \(0.681568\pi\)
\(44\) 6.52018e12 0.0699700
\(45\) 0 0
\(46\) −1.08725e14 −0.799628
\(47\) 2.44935e14i 1.50044i 0.661187 + 0.750221i \(0.270053\pi\)
−0.661187 + 0.750221i \(0.729947\pi\)
\(48\) − 1.06857e14i − 0.547336i
\(49\) 1.18121e14 0.507764
\(50\) 0 0
\(51\) 3.15971e14 0.966721
\(52\) 2.23178e13i 0.0578926i
\(53\) − 8.60208e14i − 1.89784i −0.315523 − 0.948918i \(-0.602180\pi\)
0.315523 − 0.948918i \(-0.397820\pi\)
\(54\) −9.96859e13 −0.187623
\(55\) 0 0
\(56\) −5.19576e14 −0.717879
\(57\) − 5.30232e13i − 0.0630275i
\(58\) 2.81480e14i 0.288609i
\(59\) −1.21582e14 −0.107802 −0.0539012 − 0.998546i \(-0.517166\pi\)
−0.0539012 + 0.998546i \(0.517166\pi\)
\(60\) 0 0
\(61\) −4.27987e14 −0.285842 −0.142921 − 0.989734i \(-0.545650\pi\)
−0.142921 + 0.989734i \(0.545650\pi\)
\(62\) 1.12019e15i 0.651572i
\(63\) 4.60638e14i 0.233865i
\(64\) −2.35201e15 −1.04450
\(65\) 0 0
\(66\) 2.32570e15 0.795115
\(67\) 1.16734e15i 0.351206i 0.984461 + 0.175603i \(0.0561874\pi\)
−0.984461 + 0.175603i \(0.943813\pi\)
\(68\) − 3.12664e14i − 0.0829376i
\(69\) 2.02104e15 0.473542
\(70\) 0 0
\(71\) 3.01633e15 0.554348 0.277174 − 0.960820i \(-0.410602\pi\)
0.277174 + 0.960820i \(0.410602\pi\)
\(72\) 2.09011e15i 0.341070i
\(73\) − 7.32396e15i − 1.06292i −0.847082 − 0.531461i \(-0.821643\pi\)
0.847082 − 0.531461i \(-0.178357\pi\)
\(74\) −5.69806e15 −0.736644
\(75\) 0 0
\(76\) −5.24682e13 −0.00540730
\(77\) − 1.07468e16i − 0.991080i
\(78\) 7.96059e15i 0.657872i
\(79\) 7.07645e15 0.524789 0.262395 − 0.964961i \(-0.415488\pi\)
0.262395 + 0.964961i \(0.415488\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 2.69374e16i − 1.45525i
\(83\) − 1.18761e16i − 0.578773i −0.957212 − 0.289387i \(-0.906549\pi\)
0.957212 − 0.289387i \(-0.0934513\pi\)
\(84\) 4.55816e14 0.0200639
\(85\) 0 0
\(86\) 4.55388e16 1.64114
\(87\) − 5.23231e15i − 0.170915i
\(88\) − 4.87629e16i − 1.44540i
\(89\) −5.69376e16 −1.53315 −0.766574 − 0.642156i \(-0.778040\pi\)
−0.766574 + 0.642156i \(0.778040\pi\)
\(90\) 0 0
\(91\) 3.67851e16 0.820011
\(92\) − 1.99989e15i − 0.0406264i
\(93\) − 2.08227e16i − 0.385863i
\(94\) 8.64519e16 1.46281
\(95\) 0 0
\(96\) 4.03885e15 0.0571416
\(97\) 8.11777e16i 1.05166i 0.850588 + 0.525832i \(0.176246\pi\)
−0.850588 + 0.525832i \(0.823754\pi\)
\(98\) − 4.16920e16i − 0.495029i
\(99\) −4.32315e16 −0.470870
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.e.49.2 6
5.2 odd 4 15.18.a.c.1.3 ✓ 3
5.3 odd 4 75.18.a.d.1.1 3
5.4 even 2 inner 75.18.b.e.49.5 6
15.2 even 4 45.18.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.3 ✓ 3 5.2 odd 4
45.18.a.d.1.1 3 15.2 even 4
75.18.a.d.1.1 3 5.3 odd 4
75.18.b.e.49.2 6 1.1 even 1 trivial
75.18.b.e.49.5 6 5.4 even 2 inner