↑

Properties

Label 75.18.b.d.49.4
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

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 49.4
Root \(-98.7543 + 98.7543i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.d.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)
 
\(f(q)\) \(=\) \(q+213.954i q^{2} -6561.00i q^{3} +85295.6 q^{4} +1.40375e6 q^{6} -7.24373e6i q^{7} +4.62928e7i q^{8} -4.30467e7 q^{9} +4.10790e7 q^{11} -5.59624e8i q^{12} +1.35746e9i q^{13} +1.54983e9 q^{14} +1.27533e9 q^{16} -1.84008e10i q^{17} -9.21003e9i q^{18} -1.42200e9 q^{19} -4.75261e10 q^{21} +8.78904e9i q^{22} -1.96071e10i q^{23} +3.03727e11 q^{24} -2.90434e11 q^{26} +2.82430e11i q^{27} -6.17858e11i q^{28} +3.22761e12 q^{29} -1.93640e12 q^{31} +6.34055e12i q^{32} -2.69520e11i q^{33} +3.93694e12 q^{34} -3.67169e12 q^{36} +2.47063e13i q^{37} -3.04244e11i q^{38} +8.90629e12 q^{39} +1.91037e13 q^{41} -1.01684e13i q^{42} -9.80229e12i q^{43} +3.50386e12 q^{44} +4.19503e12 q^{46} -6.08880e13i q^{47} -8.36742e12i q^{48} +1.80159e14 q^{49} -1.20728e14 q^{51} +1.15785e14i q^{52} +3.72760e14i q^{53} -6.04270e13 q^{54} +3.35332e14 q^{56} +9.32977e12i q^{57} +6.90561e14i q^{58} -1.78795e15 q^{59} +1.98485e15 q^{61} -4.14302e14i q^{62} +3.11819e14i q^{63} -1.18943e15 q^{64} +5.76649e13 q^{66} +4.22235e15i q^{67} -1.56951e15i q^{68} -1.28642e14 q^{69} -1.65823e15 q^{71} -1.99275e15i q^{72} -1.26706e16i q^{73} -5.28602e15 q^{74} -1.21291e14 q^{76} -2.97565e14i q^{77} +1.90554e15i q^{78} +1.63691e16 q^{79} +1.85302e15 q^{81} +4.08732e15i q^{82} -2.45355e16i q^{83} -4.05377e15 q^{84} +2.09724e15 q^{86} -2.11763e16i q^{87} +1.90166e15i q^{88} +4.52816e15 q^{89} +9.83307e15 q^{91} -1.67240e15i q^{92} +1.27047e16i q^{93} +1.30272e16 q^{94} +4.16003e16 q^{96} -3.59936e16i q^{97} +3.85458e16i q^{98} -1.76832e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 596296 q^{4} + 5799924 q^{6} - 258280326 q^{9} + 2099699440 q^{11} - 55172056656 q^{14} + 44785594912 q^{16} + 223792235992 q^{19} - 65119814568 q^{21} - 1428092349264 q^{24} - 2185023682088 q^{26}+ \cdots - 90\!\cdots\!40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 213.954i 0.590971i 0.955347 + 0.295485i \(0.0954814\pi\)
−0.955347 + 0.295485i \(0.904519\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) 85295.6 0.650754
\(5\) 0 0
\(6\) 1.40375e6 0.341197
\(7\) − 7.24373e6i − 0.474929i −0.971396 − 0.237465i \(-0.923684\pi\)
0.971396 − 0.237465i \(-0.0763164\pi\)
\(8\) 4.62928e7i 0.975547i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 4.10790e7 0.0577807 0.0288903 − 0.999583i \(-0.490803\pi\)
0.0288903 + 0.999583i \(0.490803\pi\)
\(12\) − 5.59624e8i − 0.375713i
\(13\) 1.35746e9i 0.461539i 0.973008 + 0.230769i \(0.0741243\pi\)
−0.973008 + 0.230769i \(0.925876\pi\)
\(14\) 1.54983e9 0.280669
\(15\) 0 0
\(16\) 1.27533e9 0.0742338
\(17\) − 1.84008e10i − 0.639766i −0.947457 − 0.319883i \(-0.896356\pi\)
0.947457 − 0.319883i \(-0.103644\pi\)
\(18\) − 9.21003e9i − 0.196990i
\(19\) −1.42200e9 −0.0192086 −0.00960429 − 0.999954i \(-0.503057\pi\)
−0.00960429 + 0.999954i \(0.503057\pi\)
\(20\) 0 0
\(21\) −4.75261e10 −0.274201
\(22\) 8.78904e9i 0.0341467i
\(23\) − 1.96071e10i − 0.0522068i −0.999659 − 0.0261034i \(-0.991690\pi\)
0.999659 − 0.0261034i \(-0.00830992\pi\)
\(24\) 3.03727e11 0.563232
\(25\) 0 0
\(26\) −2.90434e11 −0.272756
\(27\) 2.82430e11i 0.192450i
\(28\) − 6.17858e11i − 0.309062i
\(29\) 3.22761e12 1.19811 0.599057 − 0.800707i \(-0.295542\pi\)
0.599057 + 0.800707i \(0.295542\pi\)
\(30\) 0 0
\(31\) −1.93640e12 −0.407776 −0.203888 − 0.978994i \(-0.565358\pi\)
−0.203888 + 0.978994i \(0.565358\pi\)
\(32\) 6.34055e12i 1.01942i
\(33\) − 2.69520e11i − 0.0333597i
\(34\) 3.93694e12 0.378083
\(35\) 0 0
\(36\) −3.67169e12 −0.216918
\(37\) 2.47063e13i 1.15636i 0.815909 + 0.578180i \(0.196237\pi\)
−0.815909 + 0.578180i \(0.803763\pi\)
\(38\) − 3.04244e11i − 0.0113517i
\(39\) 8.90629e12 0.266470
\(40\) 0 0
\(41\) 1.91037e13 0.373641 0.186821 − 0.982394i \(-0.440182\pi\)
0.186821 + 0.982394i \(0.440182\pi\)
\(42\) − 1.01684e13i − 0.162044i
\(43\) − 9.80229e12i − 0.127893i −0.997953 − 0.0639463i \(-0.979631\pi\)
0.997953 − 0.0639463i \(-0.0203686\pi\)
\(44\) 3.50386e12 0.0376010
\(45\) 0 0
\(46\) 4.19503e12 0.0308527
\(47\) − 6.08880e13i − 0.372992i −0.982456 − 0.186496i \(-0.940287\pi\)
0.982456 − 0.186496i \(-0.0597132\pi\)
\(48\) − 8.36742e12i − 0.0428589i
\(49\) 1.80159e14 0.774442
\(50\) 0 0
\(51\) −1.20728e14 −0.369369
\(52\) 1.15785e14i 0.300348i
\(53\) 3.72760e14i 0.822402i 0.911545 + 0.411201i \(0.134891\pi\)
−0.911545 + 0.411201i \(0.865109\pi\)
\(54\) −6.04270e13 −0.113732
\(55\) 0 0
\(56\) 3.35332e14 0.463316
\(57\) 9.32977e12i 0.0110901i
\(58\) 6.90561e14i 0.708050i
\(59\) −1.78795e15 −1.58530 −0.792652 − 0.609675i \(-0.791300\pi\)
−0.792652 + 0.609675i \(0.791300\pi\)
\(60\) 0 0
\(61\) 1.98485e15 1.32563 0.662817 − 0.748781i \(-0.269361\pi\)
0.662817 + 0.748781i \(0.269361\pi\)
\(62\) − 4.14302e14i − 0.240983i
\(63\) 3.11819e14i 0.158310i
\(64\) −1.18943e15 −0.528212
\(65\) 0 0
\(66\) 5.76649e13 0.0197146
\(67\) 4.22235e15i 1.27033i 0.772375 + 0.635167i \(0.219069\pi\)
−0.772375 + 0.635167i \(0.780931\pi\)
\(68\) − 1.56951e15i − 0.416330i
\(69\) −1.28642e14 −0.0301416
\(70\) 0 0
\(71\) −1.65823e15 −0.304754 −0.152377 − 0.988322i \(-0.548693\pi\)
−0.152377 + 0.988322i \(0.548693\pi\)
\(72\) − 1.99275e15i − 0.325182i
\(73\) − 1.26706e16i − 1.83888i −0.393233 − 0.919439i \(-0.628643\pi\)
0.393233 − 0.919439i \(-0.371357\pi\)
\(74\) −5.28602e15 −0.683375
\(75\) 0 0
\(76\) −1.21291e14 −0.0125001
\(77\) − 2.97565e14i − 0.0274417i
\(78\) 1.90554e15i 0.157476i
\(79\) 1.63691e16 1.21393 0.606965 − 0.794728i \(-0.292387\pi\)
0.606965 + 0.794728i \(0.292387\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 4.08732e15i 0.220811i
\(83\) − 2.45355e16i − 1.19573i −0.801598 − 0.597863i \(-0.796017\pi\)
0.801598 − 0.597863i \(-0.203983\pi\)
\(84\) −4.05377e15 −0.178437
\(85\) 0 0
\(86\) 2.09724e15 0.0755808
\(87\) − 2.11763e16i − 0.691731i
\(88\) 1.90166e15i 0.0563678i
\(89\) 4.52816e15 0.121929 0.0609645 − 0.998140i \(-0.480582\pi\)
0.0609645 + 0.998140i \(0.480582\pi\)
\(90\) 0 0
\(91\) 9.83307e15 0.219198
\(92\) − 1.67240e15i − 0.0339738i
\(93\) 1.27047e16i 0.235429i
\(94\) 1.30272e16 0.220427
\(95\) 0 0
\(96\) 4.16003e16 0.588561
\(97\) − 3.59936e16i − 0.466301i −0.972441 − 0.233150i \(-0.925097\pi\)
0.972441 − 0.233150i \(-0.0749034\pi\)
\(98\) 3.85458e16i 0.457673i
\(99\) −1.76832e15 −0.0192602
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.49.4 6
5.2 odd 4 15.18.a.b.1.2 ✓ 3
5.3 odd 4 75.18.a.e.1.2 3
5.4 even 2 inner 75.18.b.d.49.3 6
15.2 even 4 45.18.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.2 ✓ 3 5.2 odd 4
45.18.a.e.1.2 3 15.2 even 4
75.18.a.e.1.2 3 5.3 odd 4
75.18.b.d.49.3 6 5.4 even 2 inner
75.18.b.d.49.4 6 1.1 even 1 trivial