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(1,75)] 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("75.1"); 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, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,33] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
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: \( x^{4} - x^{3} - 481686x^{2} + 26523040x + 36023696000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(569.922\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q+577.922 q^{2} +6561.00 q^{3} +202922. q^{4} +3.79175e6 q^{6} -2.79117e7 q^{7} +4.15238e7 q^{8} +4.30467e7 q^{9} +6.58150e8 q^{11} +1.33137e9 q^{12} -4.25443e8 q^{13} -1.61308e10 q^{14} -2.59987e9 q^{16} +4.12181e10 q^{17} +2.48777e10 q^{18} +1.11193e11 q^{19} -1.83129e11 q^{21} +3.80360e11 q^{22} -2.41101e11 q^{23} +2.72438e11 q^{24} -2.45873e11 q^{26} +2.82430e11 q^{27} -5.66390e12 q^{28} -9.10346e11 q^{29} +6.61433e12 q^{31} -6.94513e12 q^{32} +4.31812e12 q^{33} +2.38209e13 q^{34} +8.73514e12 q^{36} +2.53610e13 q^{37} +6.42609e13 q^{38} -2.79133e12 q^{39} +9.96460e12 q^{41} -1.05834e14 q^{42} +4.20716e12 q^{43} +1.33553e14 q^{44} -1.39338e14 q^{46} +1.54313e14 q^{47} -1.70577e13 q^{48} +5.46431e14 q^{49} +2.70432e14 q^{51} -8.63318e13 q^{52} +7.54932e14 q^{53} +1.63222e14 q^{54} -1.15900e15 q^{56} +7.29537e14 q^{57} -5.26110e14 q^{58} -6.28211e14 q^{59} +3.95755e14 q^{61} +3.82257e15 q^{62} -1.20151e15 q^{63} -3.67298e15 q^{64} +2.49554e15 q^{66} -6.34779e15 q^{67} +8.36407e15 q^{68} -1.58186e15 q^{69} +1.00514e16 q^{71} +1.78746e15 q^{72} -8.47690e14 q^{73} +1.46567e16 q^{74} +2.25635e16 q^{76} -1.83701e16 q^{77} -1.61317e15 q^{78} +4.80312e15 q^{79} +1.85302e15 q^{81} +5.75877e15 q^{82} +6.98391e15 q^{83} -3.71608e16 q^{84} +2.43141e15 q^{86} -5.97278e15 q^{87} +2.73289e16 q^{88} -1.08446e16 q^{89} +1.18748e16 q^{91} -4.89247e16 q^{92} +4.33966e16 q^{93} +8.91808e16 q^{94} -4.55670e16 q^{96} -7.39778e16 q^{97} +3.15795e17 q^{98} +2.83312e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} + 26244 q^{3} + 439357 q^{4} + 216513 q^{6} - 17583104 q^{7} - 63651621 q^{8} + 172186884 q^{9} - 575495184 q^{11} + 2882621277 q^{12} + 5049645832 q^{13} - 6699316032 q^{14} + 7512683905 q^{16}+ \cdots - 24\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 577.922 1.59630 0.798150 − 0.602459i \(-0.205812\pi\)
0.798150 + 0.602459i \(0.205812\pi\)
\(3\) 6561.00 0.577350
\(4\) 202922. 1.54817
\(5\) 0 0
\(6\) 3.79175e6 0.921624
\(7\) −2.79117e7 −1.83001 −0.915003 − 0.403446i \(-0.867812\pi\)
−0.915003 + 0.403446i \(0.867812\pi\)
\(8\) 4.15238e7 0.875049
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 6.58150e8 0.925736 0.462868 − 0.886427i \(-0.346820\pi\)
0.462868 + 0.886427i \(0.346820\pi\)
\(12\) 1.33137e9 0.893838
\(13\) −4.25443e8 −0.144651 −0.0723257 − 0.997381i \(-0.523042\pi\)
−0.0723257 + 0.997381i \(0.523042\pi\)
\(14\) −1.61308e10 −2.92124
\(15\) 0 0
\(16\) −2.59987e9 −0.151332
\(17\) 4.12181e10 1.43309 0.716543 − 0.697543i \(-0.245723\pi\)
0.716543 + 0.697543i \(0.245723\pi\)
\(18\) 2.48777e10 0.532100
\(19\) 1.11193e11 1.50201 0.751003 − 0.660298i \(-0.229570\pi\)
0.751003 + 0.660298i \(0.229570\pi\)
\(20\) 0 0
\(21\) −1.83129e11 −1.05655
\(22\) 3.80360e11 1.47775
\(23\) −2.41101e11 −0.641966 −0.320983 − 0.947085i \(-0.604013\pi\)
−0.320983 + 0.947085i \(0.604013\pi\)
\(24\) 2.72438e11 0.505210
\(25\) 0 0
\(26\) −2.45873e11 −0.230907
\(27\) 2.82430e11 0.192450
\(28\) −5.66390e12 −2.83317
\(29\) −9.10346e11 −0.337928 −0.168964 − 0.985622i \(-0.554042\pi\)
−0.168964 + 0.985622i \(0.554042\pi\)
\(30\) 0 0
\(31\) 6.61433e12 1.39287 0.696437 − 0.717618i \(-0.254768\pi\)
0.696437 + 0.717618i \(0.254768\pi\)
\(32\) −6.94513e12 −1.11662
\(33\) 4.31812e12 0.534474
\(34\) 2.38209e13 2.28763
\(35\) 0 0
\(36\) 8.73514e12 0.516058
\(37\) 2.53610e13 1.18700 0.593502 − 0.804833i \(-0.297745\pi\)
0.593502 + 0.804833i \(0.297745\pi\)
\(38\) 6.42609e13 2.39765
\(39\) −2.79133e12 −0.0835146
\(40\) 0 0
\(41\) 9.96460e12 0.194893 0.0974467 − 0.995241i \(-0.468932\pi\)
0.0974467 + 0.995241i \(0.468932\pi\)
\(42\) −1.05834e14 −1.68658
\(43\) 4.20716e12 0.0548918 0.0274459 − 0.999623i \(-0.491263\pi\)
0.0274459 + 0.999623i \(0.491263\pi\)
\(44\) 1.33553e14 1.43320
\(45\) 0 0
\(46\) −1.39338e14 −1.02477
\(47\) 1.54313e14 0.945301 0.472651 − 0.881250i \(-0.343297\pi\)
0.472651 + 0.881250i \(0.343297\pi\)
\(48\) −1.70577e13 −0.0873717
\(49\) 5.46431e14 2.34892
\(50\) 0 0
\(51\) 2.70432e14 0.827393
\(52\) −8.63318e13 −0.223946
\(53\) 7.54932e14 1.66557 0.832785 − 0.553597i \(-0.186745\pi\)
0.832785 + 0.553597i \(0.186745\pi\)
\(54\) 1.63222e14 0.307208
\(55\) 0 0
\(56\) −1.15900e15 −1.60135
\(57\) 7.29537e14 0.867184
\(58\) −5.26110e14 −0.539434
\(59\) −6.28211e14 −0.557011 −0.278505 − 0.960435i \(-0.589839\pi\)
−0.278505 + 0.960435i \(0.589839\pi\)
\(60\) 0 0
\(61\) 3.95755e14 0.264315 0.132158 − 0.991229i \(-0.457809\pi\)
0.132158 + 0.991229i \(0.457809\pi\)
\(62\) 3.82257e15 2.22344
\(63\) −1.20151e15 −0.610002
\(64\) −3.67298e15 −1.63113
\(65\) 0 0
\(66\) 2.49554e15 0.853181
\(67\) −6.34779e15 −1.90980 −0.954898 − 0.296935i \(-0.904036\pi\)
−0.954898 + 0.296935i \(0.904036\pi\)
\(68\) 8.36407e15 2.21867
\(69\) −1.58186e15 −0.370639
\(70\) 0 0
\(71\) 1.00514e16 1.84727 0.923636 − 0.383270i \(-0.125202\pi\)
0.923636 + 0.383270i \(0.125202\pi\)
\(72\) 1.78746e15 0.291683
\(73\) −8.47690e14 −0.123025 −0.0615124 − 0.998106i \(-0.519592\pi\)
−0.0615124 + 0.998106i \(0.519592\pi\)
\(74\) 1.46567e16 1.89481
\(75\) 0 0
\(76\) 2.25635e16 2.32537
\(77\) −1.83701e16 −1.69410
\(78\) −1.61317e15 −0.133314
\(79\) 4.80312e15 0.356200 0.178100 − 0.984012i \(-0.443005\pi\)
0.178100 + 0.984012i \(0.443005\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 5.75877e15 0.311108
\(83\) 6.98391e15 0.340357 0.170179 − 0.985413i \(-0.445566\pi\)
0.170179 + 0.985413i \(0.445566\pi\)
\(84\) −3.71608e16 −1.63573
\(85\) 0 0
\(86\) 2.43141e15 0.0876238
\(87\) −5.97278e15 −0.195103
\(88\) 2.73289e16 0.810065
\(89\) −1.08446e16 −0.292011 −0.146005 − 0.989284i \(-0.546642\pi\)
−0.146005 + 0.989284i \(0.546642\pi\)
\(90\) 0 0
\(91\) 1.18748e16 0.264713
\(92\) −4.89247e16 −0.993875
\(93\) 4.33966e16 0.804176
\(94\) 8.91808e16 1.50898
\(95\) 0 0
\(96\) −4.55670e16 −0.644681
\(97\) −7.39778e16 −0.958389 −0.479195 − 0.877709i \(-0.659071\pi\)
−0.479195 + 0.877709i \(0.659071\pi\)
\(98\) 3.15795e17 3.74959
\(99\) 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.a.f.1.4 4
5.2 odd 4 75.18.b.f.49.7 8
5.3 odd 4 75.18.b.f.49.2 8
5.4 even 2 15.18.a.d.1.1 ✓ 4
15.14 odd 2 45.18.a.f.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.d.1.1 ✓ 4 5.4 even 2
45.18.a.f.1.4 4 15.14 odd 2
75.18.a.f.1.4 4 1.1 even 1 trivial
75.18.b.f.49.2 8 5.3 odd 4
75.18.b.f.49.7 8 5.2 odd 4