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Properties

Label 75.18.b.h.49.4
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $12$
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 = [12,0,0,-786496] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1179680 x^{10} + 508533387652 x^{8} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{10}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(-354.696i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.h.49.9

$q$-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)
 
\(f(q)\) \(=\) \(q-354.696i q^{2} +6561.00i q^{3} +5262.54 q^{4} +2.32716e6 q^{6} -1.41469e7i q^{7} -4.83574e7i q^{8} -4.30467e7 q^{9} +5.09867e8 q^{11} +3.45275e7i q^{12} -4.37542e9i q^{13} -5.01784e9 q^{14} -1.64624e10 q^{16} +1.49113e10i q^{17} +1.52685e10i q^{18} -1.24854e11 q^{19} +9.28176e10 q^{21} -1.80848e11i q^{22} -8.35954e10i q^{23} +3.17273e11 q^{24} -1.55194e12 q^{26} -2.82430e11i q^{27} -7.44485e10i q^{28} -5.15565e12 q^{29} +3.28299e12 q^{31} -4.99142e11i q^{32} +3.34523e12i q^{33} +5.28897e12 q^{34} -2.26535e11 q^{36} +1.60563e13i q^{37} +4.42852e13i q^{38} +2.87071e13 q^{39} +4.59655e13 q^{41} -3.29221e13i q^{42} -7.93712e13i q^{43} +2.68319e12 q^{44} -2.96510e13 q^{46} -1.62631e13i q^{47} -1.08010e14i q^{48} +3.24966e13 q^{49} -9.78329e13 q^{51} -2.30258e13i q^{52} +2.27707e14i q^{53} -1.00177e14 q^{54} -6.84105e14 q^{56} -8.19166e14i q^{57} +1.82869e15i q^{58} +7.24114e14 q^{59} +2.08878e15 q^{61} -1.16446e15i q^{62} +6.08976e14i q^{63} -2.33480e15 q^{64} +1.18654e15 q^{66} -1.17513e15i q^{67} +7.84712e13i q^{68} +5.48469e14 q^{69} -5.82360e15 q^{71} +2.08163e15i q^{72} +4.08549e15i q^{73} +5.69510e15 q^{74} -6.57048e14 q^{76} -7.21302e15i q^{77} -1.01823e16i q^{78} -2.19279e16 q^{79} +1.85302e15 q^{81} -1.63038e16i q^{82} +2.80193e16i q^{83} +4.88456e14 q^{84} -2.81527e16 q^{86} -3.38262e16i q^{87} -2.46558e16i q^{88} -3.90478e16 q^{89} -6.18985e16 q^{91} -4.39924e14i q^{92} +2.15397e16i q^{93} -5.76847e15 q^{94} +3.27487e15 q^{96} -1.01770e17i q^{97} -1.15264e16i q^{98} -2.19481e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 786496 q^{4} - 4435236 q^{6} - 516560652 q^{9} + 171814648 q^{11} - 9924933204 q^{14} + 27576582640 q^{16} - 54387440804 q^{19} + 275559296868 q^{21} - 331235444232 q^{24} + 1222667429284 q^{26}+ \cdots - 73\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 354.696i − 0.979719i −0.871801 − 0.489860i \(-0.837048\pi\)
0.871801 − 0.489860i \(-0.162952\pi\)
\(3\) 6561.00i 0.577350i
\(4\) 5262.54 0.0401500
\(5\) 0 0
\(6\) 2.32716e6 0.565641
\(7\) − 1.41469e7i − 0.927528i −0.885959 − 0.463764i \(-0.846499\pi\)
0.885959 − 0.463764i \(-0.153501\pi\)
\(8\) − 4.83574e7i − 1.01906i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 5.09867e8 0.717164 0.358582 − 0.933498i \(-0.383260\pi\)
0.358582 + 0.933498i \(0.383260\pi\)
\(12\) 3.45275e7i 0.0231806i
\(13\) − 4.37542e9i − 1.48765i −0.668374 − 0.743826i \(-0.733009\pi\)
0.668374 − 0.743826i \(-0.266991\pi\)
\(14\) −5.01784e9 −0.908717
\(15\) 0 0
\(16\) −1.64624e10 −0.958238
\(17\) 1.49113e10i 0.518440i 0.965818 + 0.259220i \(0.0834656\pi\)
−0.965818 + 0.259220i \(0.916534\pi\)
\(18\) 1.52685e10i 0.326573i
\(19\) −1.24854e11 −1.68654 −0.843269 − 0.537492i \(-0.819372\pi\)
−0.843269 + 0.537492i \(0.819372\pi\)
\(20\) 0 0
\(21\) 9.28176e10 0.535508
\(22\) − 1.80848e11i − 0.702620i
\(23\) − 8.35954e10i − 0.222585i −0.993788 − 0.111292i \(-0.964501\pi\)
0.993788 − 0.111292i \(-0.0354990\pi\)
\(24\) 3.17273e11 0.588352
\(25\) 0 0
\(26\) −1.55194e12 −1.45748
\(27\) − 2.82430e11i − 0.192450i
\(28\) − 7.44485e10i − 0.0372402i
\(29\) −5.15565e12 −1.91382 −0.956909 − 0.290389i \(-0.906215\pi\)
−0.956909 + 0.290389i \(0.906215\pi\)
\(30\) 0 0
\(31\) 3.28299e12 0.691345 0.345673 − 0.938355i \(-0.387651\pi\)
0.345673 + 0.938355i \(0.387651\pi\)
\(32\) − 4.99142e11i − 0.0802508i
\(33\) 3.34523e12i 0.414055i
\(34\) 5.28897e12 0.507926
\(35\) 0 0
\(36\) −2.26535e11 −0.0133833
\(37\) 1.60563e13i 0.751501i 0.926721 + 0.375751i \(0.122615\pi\)
−0.926721 + 0.375751i \(0.877385\pi\)
\(38\) 4.42852e13i 1.65233i
\(39\) 2.87071e13 0.858896
\(40\) 0 0
\(41\) 4.59655e13 0.899020 0.449510 − 0.893275i \(-0.351599\pi\)
0.449510 + 0.893275i \(0.351599\pi\)
\(42\) − 3.29221e13i − 0.524648i
\(43\) − 7.93712e13i − 1.03557i −0.855510 − 0.517787i \(-0.826756\pi\)
0.855510 − 0.517787i \(-0.173244\pi\)
\(44\) 2.68319e12 0.0287941
\(45\) 0 0
\(46\) −2.96510e13 −0.218071
\(47\) − 1.62631e13i − 0.0996259i −0.998759 − 0.0498129i \(-0.984137\pi\)
0.998759 − 0.0498129i \(-0.0158625\pi\)
\(48\) − 1.08010e14i − 0.553239i
\(49\) 3.24966e13 0.139692
\(50\) 0 0
\(51\) −9.78329e13 −0.299322
\(52\) − 2.30258e13i − 0.0597292i
\(53\) 2.27707e14i 0.502380i 0.967938 + 0.251190i \(0.0808219\pi\)
−0.967938 + 0.251190i \(0.919178\pi\)
\(54\) −1.00177e14 −0.188547
\(55\) 0 0
\(56\) −6.84105e14 −0.945202
\(57\) − 8.19166e14i − 0.973723i
\(58\) 1.82869e15i 1.87500i
\(59\) 7.24114e14 0.642044 0.321022 − 0.947072i \(-0.395974\pi\)
0.321022 + 0.947072i \(0.395974\pi\)
\(60\) 0 0
\(61\) 2.08878e15 1.39505 0.697524 − 0.716562i \(-0.254285\pi\)
0.697524 + 0.716562i \(0.254285\pi\)
\(62\) − 1.16446e15i − 0.677324i
\(63\) 6.08976e14i 0.309176i
\(64\) −2.33480e15 −1.03686
\(65\) 0 0
\(66\) 1.18654e15 0.405658
\(67\) − 1.17513e15i − 0.353549i −0.984251 − 0.176774i \(-0.943434\pi\)
0.984251 − 0.176774i \(-0.0565663\pi\)
\(68\) 7.84712e13i 0.0208154i
\(69\) 5.48469e14 0.128509
\(70\) 0 0
\(71\) −5.82360e15 −1.07028 −0.535138 − 0.844765i \(-0.679740\pi\)
−0.535138 + 0.844765i \(0.679740\pi\)
\(72\) 2.08163e15i 0.339685i
\(73\) 4.08549e15i 0.592926i 0.955044 + 0.296463i \(0.0958071\pi\)
−0.955044 + 0.296463i \(0.904193\pi\)
\(74\) 5.69510e15 0.736260
\(75\) 0 0
\(76\) −6.57048e14 −0.0677145
\(77\) − 7.21302e15i − 0.665190i
\(78\) − 1.01823e16i − 0.841477i
\(79\) −2.19279e16 −1.62617 −0.813087 − 0.582142i \(-0.802215\pi\)
−0.813087 + 0.582142i \(0.802215\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 1.63038e16i − 0.880788i
\(83\) 2.80193e16i 1.36550i 0.730650 + 0.682752i \(0.239217\pi\)
−0.730650 + 0.682752i \(0.760783\pi\)
\(84\) 4.88456e14 0.0215007
\(85\) 0 0
\(86\) −2.81527e16 −1.01457
\(87\) − 3.38262e16i − 1.10494i
\(88\) − 2.46558e16i − 0.730830i
\(89\) −3.90478e16 −1.05143 −0.525717 − 0.850660i \(-0.676203\pi\)
−0.525717 + 0.850660i \(0.676203\pi\)
\(90\) 0 0
\(91\) −6.18985e16 −1.37984
\(92\) − 4.39924e14i − 0.00893678i
\(93\) 2.15397e16i 0.399148i
\(94\) −5.76847e15 −0.0976054
\(95\) 0 0
\(96\) 3.27487e15 0.0463328
\(97\) − 1.01770e17i − 1.31844i −0.751950 − 0.659220i \(-0.770886\pi\)
0.751950 − 0.659220i \(-0.229114\pi\)
\(98\) − 1.15264e16i − 0.136859i
\(99\) −2.19481e16 −0.239055
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.h.49.4 12
5.2 odd 4 75.18.a.i.1.5 ✓ 6
5.3 odd 4 75.18.a.j.1.2 yes 6
5.4 even 2 inner 75.18.b.h.49.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.5 ✓ 6 5.2 odd 4
75.18.a.j.1.2 yes 6 5.3 odd 4
75.18.b.h.49.4 12 1.1 even 1 trivial
75.18.b.h.49.9 12 5.4 even 2 inner