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Properties

Label 75.18.b.c.49.4
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $4$
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 = [4,0,0,-353032] 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: \(4\)
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: \( x^{4} + 7285x^{2} + 13264164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(60.8511i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.c.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+659.106i q^{2} -6561.00i q^{3} -303349. q^{4} +4.32440e6 q^{6} +1.59855e6i q^{7} -1.13549e8i q^{8} -4.30467e7 q^{9} -4.47381e8 q^{11} +1.99027e9i q^{12} +2.48486e9i q^{13} -1.05361e9 q^{14} +3.50803e10 q^{16} -2.48745e10i q^{17} -2.83724e10i q^{18} -8.23042e10 q^{19} +1.04881e10 q^{21} -2.94872e11i q^{22} -6.43083e11i q^{23} -7.44995e11 q^{24} -1.63779e12 q^{26} +2.82430e11i q^{27} -4.84918e11i q^{28} +9.82213e11 q^{29} +3.28632e12 q^{31} +8.23853e12i q^{32} +2.93527e12i q^{33} +1.63949e13 q^{34} +1.30582e13 q^{36} +2.63492e13i q^{37} -5.42472e13i q^{38} +1.63032e13 q^{39} -3.33007e13 q^{41} +6.91276e12i q^{42} -9.83107e13i q^{43} +1.35713e14 q^{44} +4.23860e14 q^{46} -1.62068e14i q^{47} -2.30162e14i q^{48} +2.30075e14 q^{49} -1.63201e14 q^{51} -7.53780e14i q^{52} +1.40921e14i q^{53} -1.86151e14 q^{54} +1.81514e14 q^{56} +5.39998e14i q^{57} +6.47383e14i q^{58} +9.80930e13 q^{59} +1.37376e15 q^{61} +2.16603e15i q^{62} -6.88123e13i q^{63} -8.32030e14 q^{64} -1.93465e15 q^{66} -1.85816e15i q^{67} +7.54565e15i q^{68} -4.21927e15 q^{69} -6.17500e15 q^{71} +4.88791e15i q^{72} +1.30214e16i q^{73} -1.73670e16 q^{74} +2.49669e16 q^{76} -7.15160e14i q^{77} +1.07455e16i q^{78} -1.27538e16 q^{79} +1.85302e15 q^{81} -2.19487e16i q^{82} +1.42886e16i q^{83} -3.18155e15 q^{84} +6.47972e16 q^{86} -6.44430e15i q^{87} +5.07996e16i q^{88} +3.77818e16 q^{89} -3.97217e15 q^{91} +1.95079e17i q^{92} -2.15615e16i q^{93} +1.06820e17 q^{94} +5.40530e16 q^{96} +1.09404e17i q^{97} +1.51644e17i q^{98} +1.92583e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 353032 q^{4} + 7794468 q^{6} - 172186884 q^{9} - 1975107024 q^{11} + 871130592 q^{14} + 101222647840 q^{16} - 160107084368 q^{19} + 321115915296 q^{21} - 1710324524304 q^{24} - 3271074310152 q^{26}+ \cdots + 85\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 659.106i 1.82054i 0.414014 + 0.910271i \(0.364127\pi\)
−0.414014 + 0.910271i \(0.635873\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) −303349. −2.31437
\(5\) 0 0
\(6\) 4.32440e6 1.05109
\(7\) 1.59855e6i 0.104808i 0.998626 + 0.0524038i \(0.0166883\pi\)
−0.998626 + 0.0524038i \(0.983312\pi\)
\(8\) − 1.13549e8i − 2.39287i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −4.47381e8 −0.629274 −0.314637 − 0.949212i \(-0.601883\pi\)
−0.314637 + 0.949212i \(0.601883\pi\)
\(12\) 1.99027e9i 1.33620i
\(13\) 2.48486e9i 0.844857i 0.906396 + 0.422428i \(0.138822\pi\)
−0.906396 + 0.422428i \(0.861178\pi\)
\(14\) −1.05361e9 −0.190806
\(15\) 0 0
\(16\) 3.50803e10 2.04194
\(17\) − 2.48745e10i − 0.864844i −0.901671 − 0.432422i \(-0.857659\pi\)
0.901671 − 0.432422i \(-0.142341\pi\)
\(18\) − 2.83724e10i − 0.606847i
\(19\) −8.23042e10 −1.11177 −0.555887 − 0.831258i \(-0.687621\pi\)
−0.555887 + 0.831258i \(0.687621\pi\)
\(20\) 0 0
\(21\) 1.04881e10 0.0605106
\(22\) − 2.94872e11i − 1.14562i
\(23\) − 6.43083e11i − 1.71230i −0.516726 − 0.856151i \(-0.672850\pi\)
0.516726 − 0.856151i \(-0.327150\pi\)
\(24\) −7.44995e11 −1.38152
\(25\) 0 0
\(26\) −1.63779e12 −1.53810
\(27\) 2.82430e11i 0.192450i
\(28\) − 4.84918e11i − 0.242563i
\(29\) 9.82213e11 0.364605 0.182302 − 0.983243i \(-0.441645\pi\)
0.182302 + 0.983243i \(0.441645\pi\)
\(30\) 0 0
\(31\) 3.28632e12 0.692046 0.346023 − 0.938226i \(-0.387532\pi\)
0.346023 + 0.938226i \(0.387532\pi\)
\(32\) 8.23853e12i 1.32457i
\(33\) 2.93527e12i 0.363311i
\(34\) 1.63949e13 1.57448
\(35\) 0 0
\(36\) 1.30582e13 0.771457
\(37\) 2.63492e13i 1.23326i 0.787254 + 0.616628i \(0.211502\pi\)
−0.787254 + 0.616628i \(0.788498\pi\)
\(38\) − 5.42472e13i − 2.02403i
\(39\) 1.63032e13 0.487778
\(40\) 0 0
\(41\) −3.33007e13 −0.651314 −0.325657 − 0.945488i \(-0.605585\pi\)
−0.325657 + 0.945488i \(0.605585\pi\)
\(42\) 6.91276e12i 0.110162i
\(43\) − 9.83107e13i − 1.28268i −0.767256 − 0.641341i \(-0.778378\pi\)
0.767256 − 0.641341i \(-0.221622\pi\)
\(44\) 1.35713e14 1.45637
\(45\) 0 0
\(46\) 4.23860e14 3.11732
\(47\) − 1.62068e14i − 0.992811i −0.868091 − 0.496406i \(-0.834653\pi\)
0.868091 − 0.496406i \(-0.165347\pi\)
\(48\) − 2.30162e14i − 1.17891i
\(49\) 2.30075e14 0.989015
\(50\) 0 0
\(51\) −1.63201e14 −0.499318
\(52\) − 7.53780e14i − 1.95531i
\(53\) 1.40921e14i 0.310907i 0.987843 + 0.155453i \(0.0496838\pi\)
−0.987843 + 0.155453i \(0.950316\pi\)
\(54\) −1.86151e14 −0.350363
\(55\) 0 0
\(56\) 1.81514e14 0.250790
\(57\) 5.39998e14i 0.641882i
\(58\) 6.47383e14i 0.663778i
\(59\) 9.80930e13 0.0869753 0.0434876 − 0.999054i \(-0.486153\pi\)
0.0434876 + 0.999054i \(0.486153\pi\)
\(60\) 0 0
\(61\) 1.37376e15 0.917503 0.458751 − 0.888565i \(-0.348297\pi\)
0.458751 + 0.888565i \(0.348297\pi\)
\(62\) 2.16603e15i 1.25990i
\(63\) − 6.88123e13i − 0.0349358i
\(64\) −8.32030e14 −0.369495
\(65\) 0 0
\(66\) −1.93465e15 −0.661423
\(67\) − 1.85816e15i − 0.559046i −0.960139 − 0.279523i \(-0.909824\pi\)
0.960139 − 0.279523i \(-0.0901763\pi\)
\(68\) 7.54565e15i 2.00157i
\(69\) −4.21927e15 −0.988598
\(70\) 0 0
\(71\) −6.17500e15 −1.13486 −0.567428 − 0.823423i \(-0.692061\pi\)
−0.567428 + 0.823423i \(0.692061\pi\)
\(72\) 4.88791e15i 0.797622i
\(73\) 1.30214e16i 1.88979i 0.327371 + 0.944896i \(0.393837\pi\)
−0.327371 + 0.944896i \(0.606163\pi\)
\(74\) −1.73670e16 −2.24519
\(75\) 0 0
\(76\) 2.49669e16 2.57305
\(77\) − 7.15160e14i − 0.0659526i
\(78\) 1.07455e16i 0.888021i
\(79\) −1.27538e16 −0.945824 −0.472912 − 0.881110i \(-0.656797\pi\)
−0.472912 + 0.881110i \(0.656797\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 2.19487e16i − 1.18574i
\(83\) 1.42886e16i 0.696348i 0.937430 + 0.348174i \(0.113198\pi\)
−0.937430 + 0.348174i \(0.886802\pi\)
\(84\) −3.18155e15 −0.140044
\(85\) 0 0
\(86\) 6.47972e16 2.33517
\(87\) − 6.44430e15i − 0.210505i
\(88\) 5.07996e16i 1.50577i
\(89\) 3.77818e16 1.01734 0.508672 − 0.860961i \(-0.330137\pi\)
0.508672 + 0.860961i \(0.330137\pi\)
\(90\) 0 0
\(91\) −3.97217e15 −0.0885473
\(92\) 1.95079e17i 3.96290i
\(93\) − 2.15615e16i − 0.399553i
\(94\) 1.06820e17 1.80745
\(95\) 0 0
\(96\) 5.40530e16 0.764741
\(97\) 1.09404e17i 1.41733i 0.705543 + 0.708667i \(0.250703\pi\)
−0.705543 + 0.708667i \(0.749297\pi\)
\(98\) 1.51644e17i 1.80054i
\(99\) 1.92583e16 0.209758
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.c.49.4 4
5.2 odd 4 75.18.a.b.1.1 2
5.3 odd 4 3.18.a.b.1.2 ✓ 2
5.4 even 2 inner 75.18.b.c.49.1 4
15.8 even 4 9.18.a.c.1.1 2
20.3 even 4 48.18.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.2 ✓ 2 5.3 odd 4
9.18.a.c.1.1 2 15.8 even 4
48.18.a.h.1.1 2 20.3 even 4
75.18.a.b.1.1 2 5.2 odd 4
75.18.b.c.49.1 4 5.4 even 2 inner
75.18.b.c.49.4 4 1.1 even 1 trivial