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Properties

Label 75.18.b.e.49.4
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 49.4
Root \(21.5502i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.e.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q+105.550i q^{2} +6561.00i q^{3} +119931. q^{4} -692515. q^{6} +2.39385e7i q^{7} +2.64934e7i q^{8} -4.30467e7 q^{9} -5.09412e8 q^{11} +7.86868e8i q^{12} +4.71682e9i q^{13} -2.52671e9 q^{14} +1.29232e10 q^{16} +4.44993e10i q^{17} -4.54359e9i q^{18} +4.25101e10 q^{19} -1.57060e11 q^{21} -5.37686e10i q^{22} +4.70013e11i q^{23} -1.73823e11 q^{24} -4.97862e11 q^{26} -2.82430e11i q^{27} +2.87097e12i q^{28} -3.15724e12 q^{29} +3.46790e12 q^{31} +4.83660e12i q^{32} -3.34226e12i q^{33} -4.69691e12 q^{34} -5.16264e12 q^{36} -3.18740e13i q^{37} +4.48695e12i q^{38} -3.09471e13 q^{39} +8.13537e13 q^{41} -1.65778e13i q^{42} -1.24496e14i q^{43} -6.10944e13 q^{44} -4.96099e13 q^{46} +7.05130e13i q^{47} +8.47893e13i q^{48} -3.40421e14 q^{49} -2.91960e14 q^{51} +5.65694e14i q^{52} +2.51778e14i q^{53} +2.98105e13 q^{54} -6.34213e14 q^{56} +2.78909e14i q^{57} -3.33247e14i q^{58} +6.39615e14 q^{59} -4.96148e13 q^{61} +3.66037e14i q^{62} -1.03047e15i q^{63} +1.18337e15 q^{64} +3.52776e14 q^{66} +2.75976e15i q^{67} +5.33686e15i q^{68} -3.08375e15 q^{69} +2.50612e15 q^{71} -1.14046e15i q^{72} +8.85474e14i q^{73} +3.36430e15 q^{74} +5.09829e15 q^{76} -1.21946e16i q^{77} -3.26647e15i q^{78} +6.85544e13 q^{79} +1.85302e15 q^{81} +8.58690e15i q^{82} +3.38359e16i q^{83} -1.88364e16 q^{84} +1.31405e16 q^{86} -2.07146e16i q^{87} -1.34961e16i q^{88} +3.37871e16 q^{89} -1.12914e17 q^{91} +5.63692e16i q^{92} +2.27529e16i q^{93} -7.44266e15 q^{94} -3.17329e16 q^{96} -6.48330e16i q^{97} -3.59315e16i q^{98} +2.19285e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 14174 q^{4} - 3319866 q^{6} - 258280326 q^{9} + 1887127360 q^{11} - 3694967976 q^{14} - 43887742718 q^{16} + 156244985992 q^{19} - 56850855048 q^{21} + 212425667586 q^{24} + 5331480861772 q^{26} - 11550537176156 q^{29}+ \cdots - 81\!\cdots\!60 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 105.550i 0.291544i 0.989318 + 0.145772i \(0.0465666\pi\)
−0.989318 + 0.145772i \(0.953433\pi\)
\(3\) 6561.00i 0.577350i
\(4\) 119931. 0.915002
\(5\) 0 0
\(6\) −692515. −0.168323
\(7\) 2.39385e7i 1.56951i 0.619807 + 0.784754i \(0.287211\pi\)
−0.619807 + 0.784754i \(0.712789\pi\)
\(8\) 2.64934e7i 0.558307i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −5.09412e8 −0.716526 −0.358263 − 0.933621i \(-0.616631\pi\)
−0.358263 + 0.933621i \(0.616631\pi\)
\(12\) 7.86868e8i 0.528277i
\(13\) 4.71682e9i 1.60373i 0.597506 + 0.801865i \(0.296159\pi\)
−0.597506 + 0.801865i \(0.703841\pi\)
\(14\) −2.52671e9 −0.457580
\(15\) 0 0
\(16\) 1.29232e10 0.752231
\(17\) 4.44993e10i 1.54717i 0.633693 + 0.773584i \(0.281538\pi\)
−0.633693 + 0.773584i \(0.718462\pi\)
\(18\) − 4.54359e9i − 0.0971813i
\(19\) 4.25101e10 0.574231 0.287116 − 0.957896i \(-0.407304\pi\)
0.287116 + 0.957896i \(0.407304\pi\)
\(20\) 0 0
\(21\) −1.57060e11 −0.906156
\(22\) − 5.37686e10i − 0.208899i
\(23\) 4.70013e11i 1.25148i 0.780033 + 0.625739i \(0.215202\pi\)
−0.780033 + 0.625739i \(0.784798\pi\)
\(24\) −1.73823e11 −0.322339
\(25\) 0 0
\(26\) −4.97862e11 −0.467558
\(27\) − 2.82430e11i − 0.192450i
\(28\) 2.87097e12i 1.43610i
\(29\) −3.15724e12 −1.17199 −0.585996 − 0.810314i \(-0.699296\pi\)
−0.585996 + 0.810314i \(0.699296\pi\)
\(30\) 0 0
\(31\) 3.46790e12 0.730285 0.365142 − 0.930952i \(-0.381020\pi\)
0.365142 + 0.930952i \(0.381020\pi\)
\(32\) 4.83660e12i 0.777616i
\(33\) − 3.34226e12i − 0.413686i
\(34\) −4.69691e12 −0.451068
\(35\) 0 0
\(36\) −5.16264e12 −0.305001
\(37\) − 3.18740e13i − 1.49184i −0.666037 − 0.745919i \(-0.732011\pi\)
0.666037 − 0.745919i \(-0.267989\pi\)
\(38\) 4.48695e12i 0.167414i
\(39\) −3.09471e13 −0.925914
\(40\) 0 0
\(41\) 8.13537e13 1.59116 0.795581 − 0.605847i \(-0.207166\pi\)
0.795581 + 0.605847i \(0.207166\pi\)
\(42\) − 1.65778e13i − 0.264184i
\(43\) − 1.24496e14i − 1.62432i −0.583433 − 0.812161i \(-0.698291\pi\)
0.583433 − 0.812161i \(-0.301709\pi\)
\(44\) −6.10944e13 −0.655623
\(45\) 0 0
\(46\) −4.96099e13 −0.364861
\(47\) 7.05130e13i 0.431954i 0.976398 + 0.215977i \(0.0692936\pi\)
−0.976398 + 0.215977i \(0.930706\pi\)
\(48\) 8.47893e13i 0.434301i
\(49\) −3.40421e14 −1.46336
\(50\) 0 0
\(51\) −2.91960e14 −0.893258
\(52\) 5.65694e14i 1.46742i
\(53\) 2.51778e14i 0.555485i 0.960656 + 0.277743i \(0.0895862\pi\)
−0.960656 + 0.277743i \(0.910414\pi\)
\(54\) 2.98105e13 0.0561076
\(55\) 0 0
\(56\) −6.34213e14 −0.876268
\(57\) 2.78909e14i 0.331533i
\(58\) − 3.33247e14i − 0.341687i
\(59\) 6.39615e14 0.567122 0.283561 − 0.958954i \(-0.408484\pi\)
0.283561 + 0.958954i \(0.408484\pi\)
\(60\) 0 0
\(61\) −4.96148e13 −0.0331366 −0.0165683 − 0.999863i \(-0.505274\pi\)
−0.0165683 + 0.999863i \(0.505274\pi\)
\(62\) 3.66037e14i 0.212910i
\(63\) − 1.03047e15i − 0.523169i
\(64\) 1.18337e15 0.525522
\(65\) 0 0
\(66\) 3.52776e14 0.120608
\(67\) 2.75976e15i 0.830300i 0.909753 + 0.415150i \(0.136271\pi\)
−0.909753 + 0.415150i \(0.863729\pi\)
\(68\) 5.33686e15i 1.41566i
\(69\) −3.08375e15 −0.722541
\(70\) 0 0
\(71\) 2.50612e15 0.460582 0.230291 − 0.973122i \(-0.426032\pi\)
0.230291 + 0.973122i \(0.426032\pi\)
\(72\) − 1.14046e15i − 0.186102i
\(73\) 8.85474e14i 0.128508i 0.997934 + 0.0642542i \(0.0204669\pi\)
−0.997934 + 0.0642542i \(0.979533\pi\)
\(74\) 3.36430e15 0.434936
\(75\) 0 0
\(76\) 5.09829e15 0.525423
\(77\) − 1.21946e16i − 1.12459i
\(78\) − 3.26647e15i − 0.269944i
\(79\) 6.85544e13 0.00508400 0.00254200 − 0.999997i \(-0.499191\pi\)
0.00254200 + 0.999997i \(0.499191\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 8.58690e15i 0.463894i
\(83\) 3.38359e16i 1.64897i 0.565881 + 0.824487i \(0.308536\pi\)
−0.565881 + 0.824487i \(0.691464\pi\)
\(84\) −1.88364e16 −0.829135
\(85\) 0 0
\(86\) 1.31405e16 0.473561
\(87\) − 2.07146e16i − 0.676650i
\(88\) − 1.34961e16i − 0.400041i
\(89\) 3.37871e16 0.909778 0.454889 − 0.890548i \(-0.349679\pi\)
0.454889 + 0.890548i \(0.349679\pi\)
\(90\) 0 0
\(91\) −1.12914e17 −2.51707
\(92\) 5.63692e16i 1.14510i
\(93\) 2.27529e16i 0.421630i
\(94\) −7.44266e15 −0.125934
\(95\) 0 0
\(96\) −3.17329e16 −0.448957
\(97\) − 6.48330e16i − 0.839917i −0.907543 − 0.419958i \(-0.862045\pi\)
0.907543 − 0.419958i \(-0.137955\pi\)
\(98\) − 3.59315e16i − 0.426632i
\(99\) 2.19285e16 0.238842
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.49.4 6
5.2 odd 4 15.18.a.c.1.2 ✓ 3
5.3 odd 4 75.18.a.d.1.2 3
5.4 even 2 inner 75.18.b.e.49.3 6
15.2 even 4 45.18.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.2 ✓ 3 5.2 odd 4
45.18.a.d.1.2 3 15.2 even 4
75.18.a.d.1.2 3 5.3 odd 4
75.18.b.e.49.3 6 5.4 even 2 inner
75.18.b.e.49.4 6 1.1 even 1 trivial