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Properties

Label 75.18.b.f.49.7
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $8$
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 = [8,0,0,-878714] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 963373x^{6} + 304121840676x^{4} + 35407691713753600x^{2} + 1297706673500416000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{8}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.7
Root \(569.922i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.f.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)
 
\(f(q)\) \(=\) \(q+577.922i q^{2} -6561.00i q^{3} -202922. q^{4} +3.79175e6 q^{6} -2.79117e7i q^{7} -4.15238e7i q^{8} -4.30467e7 q^{9} +6.58150e8 q^{11} +1.33137e9i q^{12} +4.25443e8i q^{13} +1.61308e10 q^{14} -2.59987e9 q^{16} +4.12181e10i q^{17} -2.48777e10i q^{18} -1.11193e11 q^{19} -1.83129e11 q^{21} +3.80360e11i q^{22} +2.41101e11i q^{23} -2.72438e11 q^{24} -2.45873e11 q^{26} +2.82430e11i q^{27} +5.66390e12i q^{28} +9.10346e11 q^{29} +6.61433e12 q^{31} -6.94513e12i q^{32} -4.31812e12i q^{33} -2.38209e13 q^{34} +8.73514e12 q^{36} +2.53610e13i q^{37} -6.42609e13i q^{38} +2.79133e12 q^{39} +9.96460e12 q^{41} -1.05834e14i q^{42} -4.20716e12i q^{43} -1.33553e14 q^{44} -1.39338e14 q^{46} +1.54313e14i q^{47} +1.70577e13i q^{48} -5.46431e14 q^{49} +2.70432e14 q^{51} -8.63318e13i q^{52} -7.54932e14i q^{53} -1.63222e14 q^{54} -1.15900e15 q^{56} +7.29537e14i q^{57} +5.26110e14i q^{58} +6.28211e14 q^{59} +3.95755e14 q^{61} +3.82257e15i q^{62} +1.20151e15i q^{63} +3.67298e15 q^{64} +2.49554e15 q^{66} -6.34779e15i q^{67} -8.36407e15i q^{68} +1.58186e15 q^{69} +1.00514e16 q^{71} +1.78746e15i q^{72} +8.47690e14i q^{73} -1.46567e16 q^{74} +2.25635e16 q^{76} -1.83701e16i q^{77} +1.61317e15i q^{78} -4.80312e15 q^{79} +1.85302e15 q^{81} +5.75877e15i q^{82} -6.98391e15i q^{83} +3.71608e16 q^{84} +2.43141e15 q^{86} -5.97278e15i q^{87} -2.73289e16i q^{88} +1.08446e16 q^{89} +1.18748e16 q^{91} -4.89247e16i q^{92} -4.33966e16i q^{93} -8.91808e16 q^{94} -4.55670e16 q^{96} -7.39778e16i q^{97} -3.15795e17i q^{98} -2.83312e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 878714 q^{4} + 433026 q^{6} - 344373768 q^{9} - 1150990368 q^{11} + 13398632064 q^{14} + 15025367810 q^{16} - 397827528736 q^{19} - 230725490688 q^{21} + 835236570762 q^{24} - 1055057698044 q^{26}+ \cdots + 49\!\cdots\!28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 577.922i 1.59630i 0.602459 + 0.798150i \(0.294188\pi\)
−0.602459 + 0.798150i \(0.705812\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) −202922. −1.54817
\(5\) 0 0
\(6\) 3.79175e6 0.921624
\(7\) − 2.79117e7i − 1.83001i −0.403446 − 0.915003i \(-0.632188\pi\)
0.403446 − 0.915003i \(-0.367812\pi\)
\(8\) − 4.15238e7i − 0.875049i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 6.58150e8 0.925736 0.462868 − 0.886427i \(-0.346820\pi\)
0.462868 + 0.886427i \(0.346820\pi\)
\(12\) 1.33137e9i 0.893838i
\(13\) 4.25443e8i 0.144651i 0.997381 + 0.0723257i \(0.0230421\pi\)
−0.997381 + 0.0723257i \(0.976958\pi\)
\(14\) 1.61308e10 2.92124
\(15\) 0 0
\(16\) −2.59987e9 −0.151332
\(17\) 4.12181e10i 1.43309i 0.697543 + 0.716543i \(0.254277\pi\)
−0.697543 + 0.716543i \(0.745723\pi\)
\(18\) − 2.48777e10i − 0.532100i
\(19\) −1.11193e11 −1.50201 −0.751003 − 0.660298i \(-0.770430\pi\)
−0.751003 + 0.660298i \(0.770430\pi\)
\(20\) 0 0
\(21\) −1.83129e11 −1.05655
\(22\) 3.80360e11i 1.47775i
\(23\) 2.41101e11i 0.641966i 0.947085 + 0.320983i \(0.104013\pi\)
−0.947085 + 0.320983i \(0.895987\pi\)
\(24\) −2.72438e11 −0.505210
\(25\) 0 0
\(26\) −2.45873e11 −0.230907
\(27\) 2.82430e11i 0.192450i
\(28\) 5.66390e12i 2.83317i
\(29\) 9.10346e11 0.337928 0.168964 − 0.985622i \(-0.445958\pi\)
0.168964 + 0.985622i \(0.445958\pi\)
\(30\) 0 0
\(31\) 6.61433e12 1.39287 0.696437 − 0.717618i \(-0.254768\pi\)
0.696437 + 0.717618i \(0.254768\pi\)
\(32\) − 6.94513e12i − 1.11662i
\(33\) − 4.31812e12i − 0.534474i
\(34\) −2.38209e13 −2.28763
\(35\) 0 0
\(36\) 8.73514e12 0.516058
\(37\) 2.53610e13i 1.18700i 0.804833 + 0.593502i \(0.202255\pi\)
−0.804833 + 0.593502i \(0.797745\pi\)
\(38\) − 6.42609e13i − 2.39765i
\(39\) 2.79133e12 0.0835146
\(40\) 0 0
\(41\) 9.96460e12 0.194893 0.0974467 − 0.995241i \(-0.468932\pi\)
0.0974467 + 0.995241i \(0.468932\pi\)
\(42\) − 1.05834e14i − 1.68658i
\(43\) − 4.20716e12i − 0.0548918i −0.999623 − 0.0274459i \(-0.991263\pi\)
0.999623 − 0.0274459i \(-0.00873740\pi\)
\(44\) −1.33553e14 −1.43320
\(45\) 0 0
\(46\) −1.39338e14 −1.02477
\(47\) 1.54313e14i 0.945301i 0.881250 + 0.472651i \(0.156703\pi\)
−0.881250 + 0.472651i \(0.843297\pi\)
\(48\) 1.70577e13i 0.0873717i
\(49\) −5.46431e14 −2.34892
\(50\) 0 0
\(51\) 2.70432e14 0.827393
\(52\) − 8.63318e13i − 0.223946i
\(53\) − 7.54932e14i − 1.66557i −0.553597 − 0.832785i \(-0.686745\pi\)
0.553597 − 0.832785i \(-0.313255\pi\)
\(54\) −1.63222e14 −0.307208
\(55\) 0 0
\(56\) −1.15900e15 −1.60135
\(57\) 7.29537e14i 0.867184i
\(58\) 5.26110e14i 0.539434i
\(59\) 6.28211e14 0.557011 0.278505 − 0.960435i \(-0.410161\pi\)
0.278505 + 0.960435i \(0.410161\pi\)
\(60\) 0 0
\(61\) 3.95755e14 0.264315 0.132158 − 0.991229i \(-0.457809\pi\)
0.132158 + 0.991229i \(0.457809\pi\)
\(62\) 3.82257e15i 2.22344i
\(63\) 1.20151e15i 0.610002i
\(64\) 3.67298e15 1.63113
\(65\) 0 0
\(66\) 2.49554e15 0.853181
\(67\) − 6.34779e15i − 1.90980i −0.296935 − 0.954898i \(-0.595964\pi\)
0.296935 − 0.954898i \(-0.404036\pi\)
\(68\) − 8.36407e15i − 2.21867i
\(69\) 1.58186e15 0.370639
\(70\) 0 0
\(71\) 1.00514e16 1.84727 0.923636 − 0.383270i \(-0.125202\pi\)
0.923636 + 0.383270i \(0.125202\pi\)
\(72\) 1.78746e15i 0.291683i
\(73\) 8.47690e14i 0.123025i 0.998106 + 0.0615124i \(0.0195924\pi\)
−0.998106 + 0.0615124i \(0.980408\pi\)
\(74\) −1.46567e16 −1.89481
\(75\) 0 0
\(76\) 2.25635e16 2.32537
\(77\) − 1.83701e16i − 1.69410i
\(78\) 1.61317e15i 0.133314i
\(79\) −4.80312e15 −0.356200 −0.178100 − 0.984012i \(-0.556995\pi\)
−0.178100 + 0.984012i \(0.556995\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 5.75877e15i 0.311108i
\(83\) − 6.98391e15i − 0.340357i −0.985413 − 0.170179i \(-0.945566\pi\)
0.985413 − 0.170179i \(-0.0544345\pi\)
\(84\) 3.71608e16 1.63573
\(85\) 0 0
\(86\) 2.43141e15 0.0876238
\(87\) − 5.97278e15i − 0.195103i
\(88\) − 2.73289e16i − 0.810065i
\(89\) 1.08446e16 0.292011 0.146005 − 0.989284i \(-0.453358\pi\)
0.146005 + 0.989284i \(0.453358\pi\)
\(90\) 0 0
\(91\) 1.18748e16 0.264713
\(92\) − 4.89247e16i − 0.993875i
\(93\) − 4.33966e16i − 0.804176i
\(94\) −8.91808e16 −1.50898
\(95\) 0 0
\(96\) −4.55670e16 −0.644681
\(97\) − 7.39778e16i − 0.958389i −0.877709 − 0.479195i \(-0.840929\pi\)
0.877709 − 0.479195i \(-0.159071\pi\)
\(98\) − 3.15795e17i − 3.74959i
\(99\) −2.83312e16 −0.308579
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.f.49.7 8
5.2 odd 4 15.18.a.d.1.1 ✓ 4
5.3 odd 4 75.18.a.f.1.4 4
5.4 even 2 inner 75.18.b.f.49.2 8
15.2 even 4 45.18.a.f.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.1 ✓ 4 5.2 odd 4
45.18.a.f.1.4 4 15.2 even 4
75.18.a.f.1.4 4 5.3 odd 4
75.18.b.f.49.2 8 5.4 even 2 inner
75.18.b.f.49.7 8 1.1 even 1 trivial