Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3] 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: \(38.6276877123\)
Analytic rank: \(1\)
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: \( x^{4} - x^{3} - 469x^{2} + 4449x - 5580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-25.1000\) 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)
 
\(f(q)\) \(=\) \(q-33.3847 q^{2} +81.0000 q^{3} +602.537 q^{4} -2704.16 q^{6} +176.118 q^{7} -3022.54 q^{8} +6561.00 q^{9} +7384.01 q^{11} +48805.5 q^{12} -104513. q^{13} -5879.66 q^{14} -207592. q^{16} -511883. q^{17} -219037. q^{18} +833220. q^{19} +14265.6 q^{21} -246513. q^{22} +868467. q^{23} -244826. q^{24} +3.48914e6 q^{26} +531441. q^{27} +106118. q^{28} +5.35363e6 q^{29} -4.02269e6 q^{31} +8.47794e6 q^{32} +598105. q^{33} +1.70891e7 q^{34} +3.95324e6 q^{36} -1.22004e7 q^{37} -2.78168e7 q^{38} -8.46556e6 q^{39} +3.24801e7 q^{41} -476252. q^{42} -1.40745e7 q^{43} +4.44914e6 q^{44} -2.89935e7 q^{46} -4.74310e7 q^{47} -1.68150e7 q^{48} -4.03226e7 q^{49} -4.14625e7 q^{51} -6.29730e7 q^{52} +4.00971e7 q^{53} -1.77420e7 q^{54} -532326. q^{56} +6.74908e7 q^{57} -1.78729e8 q^{58} -1.14018e8 q^{59} -4.22275e7 q^{61} +1.34296e8 q^{62} +1.15551e6 q^{63} -1.76746e8 q^{64} -1.99675e7 q^{66} -1.12743e8 q^{67} -3.08428e8 q^{68} +7.03458e7 q^{69} -2.94605e8 q^{71} -1.98309e7 q^{72} -2.45468e8 q^{73} +4.07307e8 q^{74} +5.02046e8 q^{76} +1.30046e6 q^{77} +2.82620e8 q^{78} -4.95230e7 q^{79} +4.30467e7 q^{81} -1.08434e9 q^{82} +1.96219e8 q^{83} +8.59554e6 q^{84} +4.69872e8 q^{86} +4.33644e8 q^{87} -2.23185e7 q^{88} +1.02076e8 q^{89} -1.84067e7 q^{91} +5.23284e8 q^{92} -3.25838e8 q^{93} +1.58347e9 q^{94} +6.86713e8 q^{96} -1.08777e8 q^{97} +1.34616e9 q^{98} +4.84465e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 324 q^{3} + 597 q^{4} + 243 q^{6} - 9834 q^{7} - 7671 q^{8} + 26244 q^{9} - 35994 q^{11} + 48357 q^{12} - 79998 q^{13} - 208182 q^{14} - 752815 q^{16} - 667878 q^{17} + 19683 q^{18} - 425792 q^{19}+ \cdots - 236156634 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −33.3847 −1.47541 −0.737704 − 0.675124i \(-0.764090\pi\)
−0.737704 + 0.675124i \(0.764090\pi\)
\(3\) 81.0000 0.577350
\(4\) 602.537 1.17683
\(5\) 0 0
\(6\) −2704.16 −0.851827
\(7\) 176.118 0.0277245 0.0138622 − 0.999904i \(-0.495587\pi\)
0.0138622 + 0.999904i \(0.495587\pi\)
\(8\) −3022.54 −0.260896
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 7384.01 0.152064 0.0760318 − 0.997105i \(-0.475775\pi\)
0.0760318 + 0.997105i \(0.475775\pi\)
\(12\) 48805.5 0.679443
\(13\) −104513. −1.01491 −0.507453 − 0.861679i \(-0.669413\pi\)
−0.507453 + 0.861679i \(0.669413\pi\)
\(14\) −5879.66 −0.0409049
\(15\) 0 0
\(16\) −207592. −0.791901
\(17\) −511883. −1.48645 −0.743226 − 0.669041i \(-0.766705\pi\)
−0.743226 + 0.669041i \(0.766705\pi\)
\(18\) −219037. −0.491803
\(19\) 833220. 1.46679 0.733396 − 0.679802i \(-0.237934\pi\)
0.733396 + 0.679802i \(0.237934\pi\)
\(20\) 0 0
\(21\) 14265.6 0.0160067
\(22\) −246513. −0.224356
\(23\) 868467. 0.647110 0.323555 − 0.946209i \(-0.395122\pi\)
0.323555 + 0.946209i \(0.395122\pi\)
\(24\) −244826. −0.150628
\(25\) 0 0
\(26\) 3.48914e6 1.49740
\(27\) 531441. 0.192450
\(28\) 106118. 0.0326270
\(29\) 5.35363e6 1.40559 0.702793 − 0.711395i \(-0.251936\pi\)
0.702793 + 0.711395i \(0.251936\pi\)
\(30\) 0 0
\(31\) −4.02269e6 −0.782328 −0.391164 − 0.920321i \(-0.627927\pi\)
−0.391164 + 0.920321i \(0.627927\pi\)
\(32\) 8.47794e6 1.42927
\(33\) 598105. 0.0877939
\(34\) 1.70891e7 2.19312
\(35\) 0 0
\(36\) 3.95324e6 0.392277
\(37\) −1.22004e7 −1.07020 −0.535102 − 0.844787i \(-0.679727\pi\)
−0.535102 + 0.844787i \(0.679727\pi\)
\(38\) −2.78168e7 −2.16412
\(39\) −8.46556e6 −0.585956
\(40\) 0 0
\(41\) 3.24801e7 1.79511 0.897554 − 0.440905i \(-0.145342\pi\)
0.897554 + 0.440905i \(0.145342\pi\)
\(42\) −476252. −0.0236165
\(43\) −1.40745e7 −0.627804 −0.313902 − 0.949455i \(-0.601636\pi\)
−0.313902 + 0.949455i \(0.601636\pi\)
\(44\) 4.44914e6 0.178953
\(45\) 0 0
\(46\) −2.89935e7 −0.954752
\(47\) −4.74310e7 −1.41782 −0.708911 − 0.705298i \(-0.750813\pi\)
−0.708911 + 0.705298i \(0.750813\pi\)
\(48\) −1.68150e7 −0.457204
\(49\) −4.03226e7 −0.999231
\(50\) 0 0
\(51\) −4.14625e7 −0.858203
\(52\) −6.29730e7 −1.19437
\(53\) 4.00971e7 0.698026 0.349013 − 0.937118i \(-0.386517\pi\)
0.349013 + 0.937118i \(0.386517\pi\)
\(54\) −1.77420e7 −0.283942
\(55\) 0 0
\(56\) −532326. −0.00723321
\(57\) 6.74908e7 0.846853
\(58\) −1.78729e8 −2.07381
\(59\) −1.14018e8 −1.22501 −0.612503 − 0.790468i \(-0.709837\pi\)
−0.612503 + 0.790468i \(0.709837\pi\)
\(60\) 0 0
\(61\) −4.22275e7 −0.390491 −0.195245 − 0.980754i \(-0.562550\pi\)
−0.195245 + 0.980754i \(0.562550\pi\)
\(62\) 1.34296e8 1.15425
\(63\) 1.15551e6 0.00924150
\(64\) −1.76746e8 −1.31686
\(65\) 0 0
\(66\) −1.99675e7 −0.129532
\(67\) −1.12743e8 −0.683521 −0.341760 − 0.939787i \(-0.611023\pi\)
−0.341760 + 0.939787i \(0.611023\pi\)
\(68\) −3.08428e8 −1.74930
\(69\) 7.03458e7 0.373609
\(70\) 0 0
\(71\) −2.94605e8 −1.37587 −0.687936 − 0.725771i \(-0.741483\pi\)
−0.687936 + 0.725771i \(0.741483\pi\)
\(72\) −1.98309e7 −0.0869654
\(73\) −2.45468e8 −1.01168 −0.505838 − 0.862629i \(-0.668817\pi\)
−0.505838 + 0.862629i \(0.668817\pi\)
\(74\) 4.07307e8 1.57899
\(75\) 0 0
\(76\) 5.02046e8 1.72616
\(77\) 1.30046e6 0.00421589
\(78\) 2.82620e8 0.864525
\(79\) −4.95230e7 −0.143049 −0.0715245 − 0.997439i \(-0.522786\pi\)
−0.0715245 + 0.997439i \(0.522786\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) −1.08434e9 −2.64852
\(83\) 1.96219e8 0.453826 0.226913 − 0.973915i \(-0.427137\pi\)
0.226913 + 0.973915i \(0.427137\pi\)
\(84\) 8.59554e6 0.0188372
\(85\) 0 0
\(86\) 4.69872e8 0.926267
\(87\) 4.33644e8 0.811515
\(88\) −2.23185e7 −0.0396728
\(89\) 1.02076e8 0.172452 0.0862259 − 0.996276i \(-0.472519\pi\)
0.0862259 + 0.996276i \(0.472519\pi\)
\(90\) 0 0
\(91\) −1.84067e7 −0.0281377
\(92\) 5.23284e8 0.761538
\(93\) −3.25838e8 −0.451677
\(94\) 1.58347e9 2.09187
\(95\) 0 0
\(96\) 6.86713e8 0.825192
\(97\) −1.08777e8 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(98\) 1.34616e9 1.47427
\(99\) 4.84465e7 0.0506879
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 75.10.a.l.1.1 4
3.2 odd 2 225.10.a.q.1.4 4
5.2 odd 4 15.10.b.a.4.1 ✓ 8
5.3 odd 4 15.10.b.a.4.8 yes 8
5.4 even 2 75.10.a.i.1.4 4
15.2 even 4 45.10.b.c.19.8 8
15.8 even 4 45.10.b.c.19.1 8
15.14 odd 2 225.10.a.u.1.1 4
20.3 even 4 240.10.f.c.49.1 8
20.7 even 4 240.10.f.c.49.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.b.a.4.1 ✓ 8 5.2 odd 4
15.10.b.a.4.8 yes 8 5.3 odd 4
45.10.b.c.19.1 8 15.8 even 4
45.10.b.c.19.8 8 15.2 even 4
75.10.a.i.1.4 4 5.4 even 2
75.10.a.l.1.1 4 1.1 even 1 trivial
225.10.a.q.1.4 4 3.2 odd 2
225.10.a.u.1.1 4 15.14 odd 2
240.10.f.c.49.1 8 20.3 even 4
240.10.f.c.49.5 8 20.7 even 4