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Properties

Label 75.18.b.d.49.5
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,-596296] 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: \( x^{6} - 175350x^{3} + 346592689x^{2} - 3264490950x + 15373811250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.5
Root \(4.72071 + 4.72071i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.d.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+442.567i q^{2} +6561.00i q^{3} -64793.8 q^{4} -2.90368e6 q^{6} +2.47993e7i q^{7} +2.93326e7i q^{8} -4.30467e7 q^{9} -7.66684e6 q^{11} -4.25112e8i q^{12} +3.40086e9i q^{13} -1.09754e10 q^{14} -2.14743e10 q^{16} -5.35306e10i q^{17} -1.90511e10i q^{18} +1.29760e11 q^{19} -1.62708e11 q^{21} -3.39309e9i q^{22} -2.11422e11i q^{23} -1.92451e11 q^{24} -1.50511e12 q^{26} -2.82430e11i q^{27} -1.60684e12i q^{28} -4.13975e12 q^{29} -5.31377e12 q^{31} -5.65914e12i q^{32} -5.03022e10i q^{33} +2.36909e13 q^{34} +2.78916e12 q^{36} +1.85634e13i q^{37} +5.74274e13i q^{38} -2.23130e13 q^{39} -5.57404e13 q^{41} -7.20094e13i q^{42} +7.31648e13i q^{43} +4.96764e11 q^{44} +9.35684e13 q^{46} -1.22507e14i q^{47} -1.40893e14i q^{48} -3.82376e14 q^{49} +3.51215e14 q^{51} -2.20355e14i q^{52} -2.74251e14i q^{53} +1.24994e14 q^{54} -7.27428e14 q^{56} +8.51354e14i q^{57} -1.83212e15i q^{58} -6.31977e14 q^{59} -1.14361e15 q^{61} -2.35170e15i q^{62} -1.06753e15i q^{63} -3.10129e14 q^{64} +2.22621e13 q^{66} +6.08292e14i q^{67} +3.46845e15i q^{68} +1.38714e15 q^{69} +1.25518e15 q^{71} -1.26267e15i q^{72} -7.71376e15i q^{73} -8.21553e15 q^{74} -8.40763e15 q^{76} -1.90133e14i q^{77} -9.87502e15i q^{78} +2.63076e16 q^{79} +1.85302e15 q^{81} -2.46689e16i q^{82} -1.45466e16i q^{83} +1.05425e16 q^{84} -3.23803e16 q^{86} -2.71609e16i q^{87} -2.24888e14i q^{88} -2.61450e16 q^{89} -8.43391e16 q^{91} +1.36988e16i q^{92} -3.48637e16i q^{93} +5.42175e16 q^{94} +3.71296e16 q^{96} -9.64972e15i q^{97} -1.69227e17i q^{98} +3.30032e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 596296 q^{4} + 5799924 q^{6} - 258280326 q^{9} + 2099699440 q^{11} - 55172056656 q^{14} + 44785594912 q^{16} + 223792235992 q^{19} - 65119814568 q^{21} - 1428092349264 q^{24} - 2185023682088 q^{26}+ \cdots - 90\!\cdots\!40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 442.567i 1.22243i 0.791464 + 0.611215i \(0.209319\pi\)
−0.791464 + 0.611215i \(0.790681\pi\)
\(3\) 6561.00i 0.577350i
\(4\) −64793.8 −0.494337
\(5\) 0 0
\(6\) −2.90368e6 −0.705771
\(7\) 2.47993e7i 1.62595i 0.582300 + 0.812974i \(0.302153\pi\)
−0.582300 + 0.812974i \(0.697847\pi\)
\(8\) 2.93326e7i 0.618138i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −7.66684e6 −0.0107840 −0.00539199 − 0.999985i \(-0.501716\pi\)
−0.00539199 + 0.999985i \(0.501716\pi\)
\(12\) − 4.25112e8i − 0.285406i
\(13\) 3.40086e9i 1.15630i 0.815931 + 0.578150i \(0.196225\pi\)
−0.815931 + 0.578150i \(0.803775\pi\)
\(14\) −1.09754e10 −1.98761
\(15\) 0 0
\(16\) −2.14743e10 −1.24997
\(17\) − 5.35306e10i − 1.86117i −0.366073 − 0.930586i \(-0.619298\pi\)
0.366073 − 0.930586i \(-0.380702\pi\)
\(18\) − 1.90511e10i − 0.407477i
\(19\) 1.29760e11 1.75281 0.876404 − 0.481576i \(-0.159935\pi\)
0.876404 + 0.481576i \(0.159935\pi\)
\(20\) 0 0
\(21\) −1.62708e11 −0.938742
\(22\) − 3.39309e9i − 0.0131827i
\(23\) − 2.11422e11i − 0.562941i −0.959570 − 0.281471i \(-0.909178\pi\)
0.959570 − 0.281471i \(-0.0908222\pi\)
\(24\) −1.92451e11 −0.356882
\(25\) 0 0
\(26\) −1.50511e12 −1.41350
\(27\) − 2.82430e11i − 0.192450i
\(28\) − 1.60684e12i − 0.803767i
\(29\) −4.13975e12 −1.53671 −0.768354 − 0.640026i \(-0.778924\pi\)
−0.768354 + 0.640026i \(0.778924\pi\)
\(30\) 0 0
\(31\) −5.31377e12 −1.11900 −0.559498 − 0.828832i \(-0.689006\pi\)
−0.559498 + 0.828832i \(0.689006\pi\)
\(32\) − 5.65914e12i − 0.909862i
\(33\) − 5.03022e10i − 0.00622613i
\(34\) 2.36909e13 2.27515
\(35\) 0 0
\(36\) 2.78916e12 0.164779
\(37\) 1.85634e13i 0.868844i 0.900709 + 0.434422i \(0.143047\pi\)
−0.900709 + 0.434422i \(0.856953\pi\)
\(38\) 5.74274e13i 2.14269i
\(39\) −2.23130e13 −0.667590
\(40\) 0 0
\(41\) −5.57404e13 −1.09020 −0.545101 − 0.838370i \(-0.683509\pi\)
−0.545101 + 0.838370i \(0.683509\pi\)
\(42\) − 7.20094e13i − 1.14755i
\(43\) 7.31648e13i 0.954597i 0.878741 + 0.477299i \(0.158384\pi\)
−0.878741 + 0.477299i \(0.841616\pi\)
\(44\) 4.96764e11 0.00533092
\(45\) 0 0
\(46\) 9.35684e13 0.688157
\(47\) − 1.22507e14i − 0.750461i −0.926931 − 0.375231i \(-0.877564\pi\)
0.926931 − 0.375231i \(-0.122436\pi\)
\(48\) − 1.40893e14i − 0.721669i
\(49\) −3.82376e14 −1.64371
\(50\) 0 0
\(51\) 3.51215e14 1.07455
\(52\) − 2.20355e14i − 0.571602i
\(53\) − 2.74251e14i − 0.605068i −0.953139 − 0.302534i \(-0.902168\pi\)
0.953139 − 0.302534i \(-0.0978325\pi\)
\(54\) 1.24994e14 0.235257
\(55\) 0 0
\(56\) −7.27428e14 −1.00506
\(57\) 8.51354e14i 1.01198i
\(58\) − 1.83212e15i − 1.87852i
\(59\) −6.31977e14 −0.560350 −0.280175 − 0.959949i \(-0.590392\pi\)
−0.280175 + 0.959949i \(0.590392\pi\)
\(60\) 0 0
\(61\) −1.14361e15 −0.763791 −0.381895 − 0.924206i \(-0.624729\pi\)
−0.381895 + 0.924206i \(0.624729\pi\)
\(62\) − 2.35170e15i − 1.36790i
\(63\) − 1.06753e15i − 0.541983i
\(64\) −3.10129e14 −0.137725
\(65\) 0 0
\(66\) 2.22621e13 0.00761101
\(67\) 6.08292e14i 0.183011i 0.995805 + 0.0915053i \(0.0291679\pi\)
−0.995805 + 0.0915053i \(0.970832\pi\)
\(68\) 3.46845e15i 0.920047i
\(69\) 1.38714e15 0.325014
\(70\) 0 0
\(71\) 1.25518e15 0.230681 0.115340 − 0.993326i \(-0.463204\pi\)
0.115340 + 0.993326i \(0.463204\pi\)
\(72\) − 1.26267e15i − 0.206046i
\(73\) − 7.71376e15i − 1.11949i −0.828664 − 0.559747i \(-0.810898\pi\)
0.828664 − 0.559747i \(-0.189102\pi\)
\(74\) −8.21553e15 −1.06210
\(75\) 0 0
\(76\) −8.40763e15 −0.866479
\(77\) − 1.90133e14i − 0.0175342i
\(78\) − 9.87502e15i − 0.816082i
\(79\) 2.63076e16 1.95097 0.975486 − 0.220061i \(-0.0706257\pi\)
0.975486 + 0.220061i \(0.0706257\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 2.46689e16i − 1.33270i
\(83\) − 1.45466e16i − 0.708922i −0.935071 − 0.354461i \(-0.884664\pi\)
0.935071 − 0.354461i \(-0.115336\pi\)
\(84\) 1.05425e16 0.464055
\(85\) 0 0
\(86\) −3.23803e16 −1.16693
\(87\) − 2.71609e16i − 0.887218i
\(88\) − 2.24888e14i − 0.00666598i
\(89\) −2.61450e16 −0.704001 −0.352000 − 0.936000i \(-0.614498\pi\)
−0.352000 + 0.936000i \(0.614498\pi\)
\(90\) 0 0
\(91\) −8.43391e16 −1.88008
\(92\) 1.36988e16i 0.278283i
\(93\) − 3.48637e16i − 0.646053i
\(94\) 5.42175e16 0.917387
\(95\) 0 0
\(96\) 3.71296e16 0.525309
\(97\) − 9.64972e15i − 0.125013i −0.998045 − 0.0625064i \(-0.980091\pi\)
0.998045 − 0.0625064i \(-0.0199094\pi\)
\(98\) − 1.69227e17i − 2.00932i
\(99\) 3.30032e14 0.00359466
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.d.49.5 6
5.2 odd 4 75.18.a.e.1.1 3
5.3 odd 4 15.18.a.b.1.3 ✓ 3
5.4 even 2 inner 75.18.b.d.49.2 6
15.8 even 4 45.18.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.3 ✓ 3 5.3 odd 4
45.18.a.e.1.1 3 15.8 even 4
75.18.a.e.1.1 3 5.2 odd 4
75.18.b.d.49.2 6 5.4 even 2 inner
75.18.b.d.49.5 6 1.1 even 1 trivial