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Properties

Label 75.18.b.a.49.1
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $2$
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 = [2,0,0,178912] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.a.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)
 
\(f(q)\) \(=\) \(q-204.000i q^{2} -6561.00i q^{3} +89456.0 q^{4} -1.33844e6 q^{6} +2.08466e7i q^{7} -4.49877e7i q^{8} -4.30467e7 q^{9} +8.17372e8 q^{11} -5.86921e8i q^{12} +2.99590e8i q^{13} +4.25270e9 q^{14} +2.54768e9 q^{16} +4.47756e10i q^{17} +8.78153e9i q^{18} -7.87487e10 q^{19} +1.36774e11 q^{21} -1.66744e11i q^{22} -7.04672e11i q^{23} -2.95164e11 q^{24} +6.11163e10 q^{26} +2.82430e11i q^{27} +1.86485e12i q^{28} +1.63794e11 q^{29} +1.04986e12 q^{31} -6.41636e12i q^{32} -5.36278e12i q^{33} +9.13422e12 q^{34} -3.85079e12 q^{36} +1.98057e13i q^{37} +1.60647e13i q^{38} +1.96561e12 q^{39} +1.46600e13 q^{41} -2.79020e13i q^{42} +1.16039e14i q^{43} +7.31189e13 q^{44} -1.43753e14 q^{46} +1.76607e14i q^{47} -1.67154e13i q^{48} -2.01949e14 q^{49} +2.93773e14 q^{51} +2.68001e13i q^{52} +1.52863e14i q^{53} +5.76156e13 q^{54} +9.37839e14 q^{56} +5.16670e14i q^{57} -3.34139e13i q^{58} +2.62797e14 q^{59} -1.35855e15 q^{61} -2.14172e14i q^{62} -8.97376e14i q^{63} -9.75007e14 q^{64} -1.09401e15 q^{66} -4.44864e14i q^{67} +4.00545e15i q^{68} -4.62335e15 q^{69} -4.00327e15 q^{71} +1.93657e15i q^{72} +9.24833e14i q^{73} +4.04037e15 q^{74} -7.04454e15 q^{76} +1.70394e16i q^{77} -4.00984e14i q^{78} -1.47473e16 q^{79} +1.85302e15 q^{81} -2.99065e15i q^{82} +2.64230e16i q^{83} +1.22353e16 q^{84} +2.36719e16 q^{86} -1.07465e15i q^{87} -3.67717e16i q^{88} +3.88837e16 q^{89} -6.24542e15 q^{91} -6.30371e16i q^{92} -6.88814e15i q^{93} +3.60277e16 q^{94} -4.20977e16 q^{96} +2.53744e16i q^{97} +4.11975e16i q^{98} -3.51852e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 178912 q^{4} - 2676888 q^{6} - 86093442 q^{9} + 1634744712 q^{11} + 8505396480 q^{14} + 5095367168 q^{16} - 157497303928 q^{19} + 273548560320 q^{21} - 590328756864 q^{24} + 122232621264 q^{26} + 327587570484 q^{29}+ \cdots - 70\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 204.000i − 0.563476i −0.959491 − 0.281738i \(-0.909089\pi\)
0.959491 − 0.281738i \(-0.0909108\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) 89456.0 0.682495
\(5\) 0 0
\(6\) −1.33844e6 −0.325323
\(7\) 2.08466e7i 1.36679i 0.730050 + 0.683394i \(0.239497\pi\)
−0.730050 + 0.683394i \(0.760503\pi\)
\(8\) − 4.49877e7i − 0.948045i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 8.17372e8 1.14969 0.574847 − 0.818261i \(-0.305062\pi\)
0.574847 + 0.818261i \(0.305062\pi\)
\(12\) − 5.86921e8i − 0.394039i
\(13\) 2.99590e8i 0.101861i 0.998702 + 0.0509306i \(0.0162187\pi\)
−0.998702 + 0.0509306i \(0.983781\pi\)
\(14\) 4.25270e9 0.770152
\(15\) 0 0
\(16\) 2.54768e9 0.148295
\(17\) 4.47756e10i 1.55677i 0.627785 + 0.778387i \(0.283962\pi\)
−0.627785 + 0.778387i \(0.716038\pi\)
\(18\) 8.78153e9i 0.187825i
\(19\) −7.87487e10 −1.06374 −0.531872 − 0.846824i \(-0.678511\pi\)
−0.531872 + 0.846824i \(0.678511\pi\)
\(20\) 0 0
\(21\) 1.36774e11 0.789115
\(22\) − 1.66744e11i − 0.647824i
\(23\) − 7.04672e11i − 1.87629i −0.346240 − 0.938146i \(-0.612542\pi\)
0.346240 − 0.938146i \(-0.387458\pi\)
\(24\) −2.95164e11 −0.547354
\(25\) 0 0
\(26\) 6.11163e10 0.0573963
\(27\) 2.82430e11i 0.192450i
\(28\) 1.86485e12i 0.932826i
\(29\) 1.63794e11 0.0608015 0.0304008 − 0.999538i \(-0.490322\pi\)
0.0304008 + 0.999538i \(0.490322\pi\)
\(30\) 0 0
\(31\) 1.04986e12 0.221084 0.110542 − 0.993871i \(-0.464741\pi\)
0.110542 + 0.993871i \(0.464741\pi\)
\(32\) − 6.41636e12i − 1.03161i
\(33\) − 5.36278e12i − 0.663776i
\(34\) 9.13422e12 0.877204
\(35\) 0 0
\(36\) −3.85079e12 −0.227498
\(37\) 1.98057e13i 0.926993i 0.886099 + 0.463496i \(0.153405\pi\)
−0.886099 + 0.463496i \(0.846595\pi\)
\(38\) 1.60647e13i 0.599394i
\(39\) 1.96561e12 0.0588095
\(40\) 0 0
\(41\) 1.46600e13 0.286729 0.143365 − 0.989670i \(-0.454208\pi\)
0.143365 + 0.989670i \(0.454208\pi\)
\(42\) − 2.79020e13i − 0.444647i
\(43\) 1.16039e14i 1.51399i 0.653424 + 0.756993i \(0.273332\pi\)
−0.653424 + 0.756993i \(0.726668\pi\)
\(44\) 7.31189e13 0.784660
\(45\) 0 0
\(46\) −1.43753e14 −1.05724
\(47\) 1.76607e14i 1.08187i 0.841064 + 0.540935i \(0.181930\pi\)
−0.841064 + 0.540935i \(0.818070\pi\)
\(48\) − 1.67154e13i − 0.0856180i
\(49\) −2.01949e14 −0.868109
\(50\) 0 0
\(51\) 2.93773e14 0.898804
\(52\) 2.68001e13i 0.0695197i
\(53\) 1.52863e14i 0.337255i 0.985680 + 0.168628i \(0.0539336\pi\)
−0.985680 + 0.168628i \(0.946066\pi\)
\(54\) 5.76156e13 0.108441
\(55\) 0 0
\(56\) 9.37839e14 1.29578
\(57\) 5.16670e14i 0.614153i
\(58\) − 3.34139e13i − 0.0342602i
\(59\) 2.62797e14 0.233012 0.116506 − 0.993190i \(-0.462831\pi\)
0.116506 + 0.993190i \(0.462831\pi\)
\(60\) 0 0
\(61\) −1.35855e15 −0.907346 −0.453673 − 0.891168i \(-0.649887\pi\)
−0.453673 + 0.891168i \(0.649887\pi\)
\(62\) − 2.14172e14i − 0.124575i
\(63\) − 8.97376e14i − 0.455596i
\(64\) −9.75007e14 −0.432990
\(65\) 0 0
\(66\) −1.09401e15 −0.374022
\(67\) − 4.44864e14i − 0.133842i −0.997758 − 0.0669208i \(-0.978683\pi\)
0.997758 − 0.0669208i \(-0.0213175\pi\)
\(68\) 4.00545e15i 1.06249i
\(69\) −4.62335e15 −1.08328
\(70\) 0 0
\(71\) −4.00327e15 −0.735731 −0.367865 − 0.929879i \(-0.619911\pi\)
−0.367865 + 0.929879i \(0.619911\pi\)
\(72\) 1.93657e15i 0.316015i
\(73\) 9.24833e14i 0.134220i 0.997746 + 0.0671102i \(0.0213779\pi\)
−0.997746 + 0.0671102i \(0.978622\pi\)
\(74\) 4.04037e15 0.522338
\(75\) 0 0
\(76\) −7.04454e15 −0.726001
\(77\) 1.70394e16i 1.57139i
\(78\) − 4.00984e14i − 0.0331378i
\(79\) −1.47473e16 −1.09366 −0.546830 − 0.837244i \(-0.684166\pi\)
−0.546830 + 0.837244i \(0.684166\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 2.99065e15i − 0.161565i
\(83\) 2.64230e16i 1.28771i 0.765148 + 0.643855i \(0.222666\pi\)
−0.765148 + 0.643855i \(0.777334\pi\)
\(84\) 1.22353e16 0.538567
\(85\) 0 0
\(86\) 2.36719e16 0.853094
\(87\) − 1.07465e15i − 0.0351038i
\(88\) − 3.67717e16i − 1.08996i
\(89\) 3.88837e16 1.04702 0.523508 − 0.852021i \(-0.324623\pi\)
0.523508 + 0.852021i \(0.324623\pi\)
\(90\) 0 0
\(91\) −6.24542e15 −0.139223
\(92\) − 6.30371e16i − 1.28056i
\(93\) − 6.88814e15i − 0.127643i
\(94\) 3.60277e16 0.609608
\(95\) 0 0
\(96\) −4.20977e16 −0.595598
\(97\) 2.53744e16i 0.328727i 0.986400 + 0.164364i \(0.0525571\pi\)
−0.986400 + 0.164364i \(0.947443\pi\)
\(98\) 4.11975e16i 0.489158i
\(99\) −3.51852e16 −0.383231
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.a.49.1 2
5.2 odd 4 3.18.a.a.1.1 ✓ 1
5.3 odd 4 75.18.a.a.1.1 1
5.4 even 2 inner 75.18.b.a.49.2 2
15.2 even 4 9.18.a.a.1.1 1
20.7 even 4 48.18.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.a.1.1 ✓ 1 5.2 odd 4
9.18.a.a.1.1 1 15.2 even 4
48.18.a.e.1.1 1 20.7 even 4
75.18.a.a.1.1 1 5.3 odd 4
75.18.b.a.49.1 2 1.1 even 1 trivial
75.18.b.a.49.2 2 5.4 even 2 inner