Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,3,Mod(7,75)] 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("75.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.43
Dual form 75.3.f.c.7.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.22474 - 2.22474i) q^{2} +(-1.22474 - 1.22474i) q^{3} -5.89898i q^{4} -5.44949 q^{6} +(1.44949 - 1.44949i) q^{7} +(-4.22474 - 4.22474i) q^{8} +3.00000i q^{9} -3.34847 q^{11} +(-7.22474 + 7.22474i) q^{12} +(10.4495 + 10.4495i) q^{13} -6.44949i q^{14} +4.79796 q^{16} +(2.65153 - 2.65153i) q^{17} +(6.67423 + 6.67423i) q^{18} +20.6969i q^{19} -3.55051 q^{21} +(-7.44949 + 7.44949i) q^{22} +(-16.4495 - 16.4495i) q^{23} +10.3485i q^{24} +46.4949 q^{26} +(3.67423 - 3.67423i) q^{27} +(-8.55051 - 8.55051i) q^{28} -0.853572i q^{29} -18.6969 q^{31} +(27.5732 - 27.5732i) q^{32} +(4.10102 + 4.10102i) q^{33} -11.7980i q^{34} +17.6969 q^{36} +(-38.0454 + 38.0454i) q^{37} +(46.0454 + 46.0454i) q^{38} -25.5959i q^{39} -28.6969 q^{41} +(-7.89898 + 7.89898i) q^{42} +(-22.4949 - 22.4949i) q^{43} +19.7526i q^{44} -73.1918 q^{46} +(-19.7526 + 19.7526i) q^{47} +(-5.87628 - 5.87628i) q^{48} +44.7980i q^{49} -6.49490 q^{51} +(61.6413 - 61.6413i) q^{52} +(-28.6969 - 28.6969i) q^{53} -16.3485i q^{54} -12.2474 q^{56} +(25.3485 - 25.3485i) q^{57} +(-1.89898 - 1.89898i) q^{58} -111.934i q^{59} +94.0908 q^{61} +(-41.5959 + 41.5959i) q^{62} +(4.34847 + 4.34847i) q^{63} -103.495i q^{64} +18.2474 q^{66} +(54.8990 - 54.8990i) q^{67} +(-15.6413 - 15.6413i) q^{68} +40.2929i q^{69} -68.0000 q^{71} +(12.6742 - 12.6742i) q^{72} +(39.7878 + 39.7878i) q^{73} +169.283i q^{74} +122.091 q^{76} +(-4.85357 + 4.85357i) q^{77} +(-56.9444 - 56.9444i) q^{78} -24.4949i q^{79} -9.00000 q^{81} +(-63.8434 + 63.8434i) q^{82} +(21.1464 + 21.1464i) q^{83} +20.9444i q^{84} -100.091 q^{86} +(-1.04541 + 1.04541i) q^{87} +(14.1464 + 14.1464i) q^{88} +94.1816i q^{89} +30.2929 q^{91} +(-97.0352 + 97.0352i) q^{92} +(22.8990 + 22.8990i) q^{93} +87.8888i q^{94} -67.5403 q^{96} +(-14.5959 + 14.5959i) q^{97} +(99.6640 + 99.6640i) q^{98} -10.0454i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 12 q^{6} - 4 q^{7} - 12 q^{8} + 16 q^{11} - 24 q^{12} + 32 q^{13} - 20 q^{16} + 40 q^{17} + 12 q^{18} - 24 q^{21} - 20 q^{22} - 56 q^{23} + 88 q^{26} - 44 q^{28} - 16 q^{31} + 76 q^{32}+ \cdots + 188 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.22474 − 2.22474i 1.11237 − 1.11237i 0.119543 − 0.992829i \(-0.461857\pi\)
0.992829 − 0.119543i \(-0.0381431\pi\)
\(3\) −1.22474 − 1.22474i −0.408248 − 0.408248i
\(4\) − 5.89898i − 1.47474i
\(5\) 0 0
\(6\) −5.44949 −0.908248
\(7\) 1.44949 − 1.44949i 0.207070 − 0.207070i −0.595951 − 0.803021i \(-0.703225\pi\)
0.803021 + 0.595951i \(0.203225\pi\)
\(8\) −4.22474 − 4.22474i −0.528093 − 0.528093i
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −3.34847 −0.304406 −0.152203 − 0.988349i \(-0.548637\pi\)
−0.152203 + 0.988349i \(0.548637\pi\)
\(12\) −7.22474 + 7.22474i −0.602062 + 0.602062i
\(13\) 10.4495 + 10.4495i 0.803807 + 0.803807i 0.983688 − 0.179881i \(-0.0575714\pi\)
−0.179881 + 0.983688i \(0.557571\pi\)
\(14\) − 6.44949i − 0.460678i
\(15\) 0 0
\(16\) 4.79796 0.299872
\(17\) 2.65153 − 2.65153i 0.155972 − 0.155972i −0.624807 − 0.780779i \(-0.714822\pi\)
0.780779 + 0.624807i \(0.214822\pi\)
\(18\) 6.67423 + 6.67423i 0.370791 + 0.370791i
\(19\) 20.6969i 1.08931i 0.838659 + 0.544656i \(0.183340\pi\)
−0.838659 + 0.544656i \(0.816660\pi\)
\(20\) 0 0
\(21\) −3.55051 −0.169072
\(22\) −7.44949 + 7.44949i −0.338613 + 0.338613i
\(23\) −16.4495 − 16.4495i −0.715195 − 0.715195i 0.252422 − 0.967617i \(-0.418773\pi\)
−0.967617 + 0.252422i \(0.918773\pi\)
\(24\) 10.3485i 0.431186i
\(25\) 0 0
\(26\) 46.4949 1.78827
\(27\) 3.67423 − 3.67423i 0.136083 − 0.136083i
\(28\) −8.55051 − 8.55051i −0.305375 − 0.305375i
\(29\) − 0.853572i − 0.0294335i −0.999892 − 0.0147168i \(-0.995315\pi\)
0.999892 − 0.0147168i \(-0.00468466\pi\)
\(30\) 0 0
\(31\) −18.6969 −0.603127 −0.301564 − 0.953446i \(-0.597509\pi\)
−0.301564 + 0.953446i \(0.597509\pi\)
\(32\) 27.5732 − 27.5732i 0.861663 − 0.861663i
\(33\) 4.10102 + 4.10102i 0.124273 + 0.124273i
\(34\) − 11.7980i − 0.346999i
\(35\) 0 0
\(36\) 17.6969 0.491582
\(37\) −38.0454 + 38.0454i −1.02825 + 1.02825i −0.0286652 + 0.999589i \(0.509126\pi\)
−0.999589 + 0.0286652i \(0.990874\pi\)
\(38\) 46.0454 + 46.0454i 1.21172 + 1.21172i
\(39\) − 25.5959i − 0.656306i
\(40\) 0 0
\(41\) −28.6969 −0.699925 −0.349963 − 0.936764i \(-0.613806\pi\)
−0.349963 + 0.936764i \(0.613806\pi\)
\(42\) −7.89898 + 7.89898i −0.188071 + 0.188071i
\(43\) −22.4949 − 22.4949i −0.523137 − 0.523137i 0.395380 − 0.918517i \(-0.370613\pi\)
−0.918517 + 0.395380i \(0.870613\pi\)
\(44\) 19.7526i 0.448922i
\(45\) 0 0
\(46\) −73.1918 −1.59113
\(47\) −19.7526 + 19.7526i −0.420267 + 0.420267i −0.885296 − 0.465029i \(-0.846044\pi\)
0.465029 + 0.885296i \(0.346044\pi\)
\(48\) −5.87628 − 5.87628i −0.122422 − 0.122422i
\(49\) 44.7980i 0.914244i
\(50\) 0 0
\(51\) −6.49490 −0.127351
\(52\) 61.6413 − 61.6413i 1.18541 − 1.18541i
\(53\) −28.6969 − 28.6969i −0.541452 − 0.541452i 0.382503 − 0.923954i \(-0.375062\pi\)
−0.923954 + 0.382503i \(0.875062\pi\)
\(54\) − 16.3485i − 0.302749i
\(55\) 0 0
\(56\) −12.2474 −0.218704
\(57\) 25.3485 − 25.3485i 0.444710 − 0.444710i
\(58\) −1.89898 − 1.89898i −0.0327410 − 0.0327410i
\(59\) − 111.934i − 1.89719i −0.316493 − 0.948595i \(-0.602505\pi\)
0.316493 − 0.948595i \(-0.397495\pi\)
\(60\) 0 0
\(61\) 94.0908 1.54247 0.771236 − 0.636549i \(-0.219639\pi\)
0.771236 + 0.636549i \(0.219639\pi\)
\(62\) −41.5959 + 41.5959i −0.670902 + 0.670902i
\(63\) 4.34847 + 4.34847i 0.0690233 + 0.0690233i
\(64\) − 103.495i − 1.61711i
\(65\) 0 0
\(66\) 18.2474 0.276476
\(67\) 54.8990 − 54.8990i 0.819388 − 0.819388i −0.166631 − 0.986019i \(-0.553289\pi\)
0.986019 + 0.166631i \(0.0532890\pi\)
\(68\) −15.6413 − 15.6413i −0.230019 − 0.230019i
\(69\) 40.2929i 0.583954i
\(70\) 0 0
\(71\) −68.0000 −0.957746 −0.478873 − 0.877884i \(-0.658955\pi\)
−0.478873 + 0.877884i \(0.658955\pi\)
\(72\) 12.6742 − 12.6742i 0.176031 − 0.176031i
\(73\) 39.7878 + 39.7878i 0.545038 + 0.545038i 0.925001 − 0.379964i \(-0.124064\pi\)
−0.379964 + 0.925001i \(0.624064\pi\)
\(74\) 169.283i 2.28760i
\(75\) 0 0
\(76\) 122.091 1.60646
\(77\) −4.85357 + 4.85357i −0.0630334 + 0.0630334i
\(78\) −56.9444 − 56.9444i −0.730056 − 0.730056i
\(79\) − 24.4949i − 0.310062i −0.987910 − 0.155031i \(-0.950452\pi\)
0.987910 − 0.155031i \(-0.0495477\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) −63.8434 + 63.8434i −0.778578 + 0.778578i
\(83\) 21.1464 + 21.1464i 0.254776 + 0.254776i 0.822926 − 0.568149i \(-0.192340\pi\)
−0.568149 + 0.822926i \(0.692340\pi\)
\(84\) 20.9444i 0.249338i
\(85\) 0 0
\(86\) −100.091 −1.16385
\(87\) −1.04541 + 1.04541i −0.0120162 + 0.0120162i
\(88\) 14.1464 + 14.1464i 0.160755 + 0.160755i
\(89\) 94.1816i 1.05822i 0.848553 + 0.529110i \(0.177474\pi\)
−0.848553 + 0.529110i \(0.822526\pi\)
\(90\) 0 0
\(91\) 30.2929 0.332889
\(92\) −97.0352 + 97.0352i −1.05473 + 1.05473i
\(93\) 22.8990 + 22.8990i 0.246226 + 0.246226i
\(94\) 87.8888i 0.934987i
\(95\) 0 0
\(96\) −67.5403 −0.703545
\(97\) −14.5959 + 14.5959i −0.150473 + 0.150473i −0.778329 − 0.627856i \(-0.783933\pi\)
0.627856 + 0.778329i \(0.283933\pi\)
\(98\) 99.6640 + 99.6640i 1.01698 + 1.01698i
\(99\) − 10.0454i − 0.101469i
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 75.3.f.c.43.2 4
3.2 odd 2 225.3.g.a.118.1 4
4.3 odd 2 1200.3.bg.k.193.2 4
5.2 odd 4 inner 75.3.f.c.7.2 4
5.3 odd 4 15.3.f.a.7.1 ✓ 4
5.4 even 2 15.3.f.a.13.1 yes 4
15.2 even 4 225.3.g.a.82.1 4
15.8 even 4 45.3.g.b.37.2 4
15.14 odd 2 45.3.g.b.28.2 4
20.3 even 4 240.3.bg.a.97.1 4
20.7 even 4 1200.3.bg.k.1057.2 4
20.19 odd 2 240.3.bg.a.193.1 4
40.3 even 4 960.3.bg.h.577.2 4
40.13 odd 4 960.3.bg.i.577.1 4
40.19 odd 2 960.3.bg.h.193.2 4
40.29 even 2 960.3.bg.i.193.1 4
45.4 even 6 405.3.l.h.298.1 8
45.13 odd 12 405.3.l.h.217.2 8
45.14 odd 6 405.3.l.f.298.2 8
45.23 even 12 405.3.l.f.217.1 8
45.29 odd 6 405.3.l.f.28.1 8
45.34 even 6 405.3.l.h.28.2 8
45.38 even 12 405.3.l.f.352.2 8
45.43 odd 12 405.3.l.h.352.1 8
60.23 odd 4 720.3.bh.k.577.1 4
60.59 even 2 720.3.bh.k.433.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.3.f.a.7.1 ✓ 4 5.3 odd 4
15.3.f.a.13.1 yes 4 5.4 even 2
45.3.g.b.28.2 4 15.14 odd 2
45.3.g.b.37.2 4 15.8 even 4
75.3.f.c.7.2 4 5.2 odd 4 inner
75.3.f.c.43.2 4 1.1 even 1 trivial
225.3.g.a.82.1 4 15.2 even 4
225.3.g.a.118.1 4 3.2 odd 2
240.3.bg.a.97.1 4 20.3 even 4
240.3.bg.a.193.1 4 20.19 odd 2
405.3.l.f.28.1 8 45.29 odd 6
405.3.l.f.217.1 8 45.23 even 12
405.3.l.f.298.2 8 45.14 odd 6
405.3.l.f.352.2 8 45.38 even 12
405.3.l.h.28.2 8 45.34 even 6
405.3.l.h.217.2 8 45.13 odd 12
405.3.l.h.298.1 8 45.4 even 6
405.3.l.h.352.1 8 45.43 odd 12
720.3.bh.k.433.1 4 60.59 even 2
720.3.bh.k.577.1 4 60.23 odd 4
960.3.bg.h.193.2 4 40.19 odd 2
960.3.bg.h.577.2 4 40.3 even 4
960.3.bg.i.193.1 4 40.29 even 2
960.3.bg.i.577.1 4 40.13 odd 4
1200.3.bg.k.193.2 4 4.3 odd 2
1200.3.bg.k.1057.2 4 20.7 even 4