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Properties

Label 375.2.g.d
Level $375$
Weight $2$
Character orbit 375.g
Analytic conductor $2.994$
Analytic rank $0$
Dimension $16$
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] = [375,2,Mod(76,375)] 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("375.76"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(375, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-2] 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: no
Analytic conductor: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{3} q^{3} + ( - \beta_{11} + \beta_{8}) q^{4} + ( - \beta_{13} - \beta_{11} + \beta_{5} + \cdots + 1) q^{6} + ( - \beta_{15} - \beta_{12} - \beta_{9} + \cdots + 1) q^{7} + ( - \beta_{15} + \beta_{14} + \cdots - 2 \beta_{2}) q^{8}+ \cdots + ( - \beta_{15} - \beta_{12} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 4 q^{3} - 2 q^{4} + 2 q^{6} + 16 q^{7} - 6 q^{8} - 4 q^{9} - 6 q^{11} + 2 q^{12} - 8 q^{13} + 12 q^{14} - 10 q^{16} - 8 q^{17} + 8 q^{18} + 2 q^{19} + 4 q^{21} + 4 q^{22} - 2 q^{23} - 24 q^{24}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11 \nu^{15} + 22 \nu^{14} - 194 \nu^{13} + 388 \nu^{12} - 1257 \nu^{11} + 2515 \nu^{10} - 3731 \nu^{9} + \cdots + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{14} - 194\nu^{12} - 1257\nu^{10} - 3731\nu^{8} - 5277\nu^{6} - 3223\nu^{4} - 565\nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7 \nu^{15} + 32 \nu^{14} - 124 \nu^{13} + 564 \nu^{12} - 810 \nu^{11} + 3652 \nu^{10} - 2444 \nu^{9} + \cdots + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{14} + 282\nu^{12} + 1826\nu^{10} + 5419\nu^{8} + 7680\nu^{6} + 4732\nu^{4} + 865\nu^{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{15} - 22 \nu^{14} - 194 \nu^{13} - 388 \nu^{12} - 1257 \nu^{11} - 2515 \nu^{10} - 3731 \nu^{9} + \cdots - 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} + 34 \nu^{14} - 19 \nu^{13} + 600 \nu^{12} - 138 \nu^{11} + 3893 \nu^{10} - 490 \nu^{9} + \cdots + 21 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{15} + 32 \nu^{14} + 124 \nu^{13} + 564 \nu^{12} + 810 \nu^{11} + 3652 \nu^{10} + 2444 \nu^{9} + \cdots + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47 \nu^{15} + 11 \nu^{14} + 828 \nu^{13} + 194 \nu^{12} + 5356 \nu^{11} + 1257 \nu^{10} + 15855 \nu^{9} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 56 \nu^{14} + 19 \nu^{13} + 988 \nu^{12} + 138 \nu^{11} + 6407 \nu^{10} + 490 \nu^{9} + \cdots + 25 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29 \nu^{15} - 32 \nu^{14} - 512 \nu^{13} - 564 \nu^{12} - 3325 \nu^{11} - 3652 \nu^{10} - 9921 \nu^{9} + \cdots - 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 45 \nu^{15} + 23 \nu^{14} + 794 \nu^{13} + 406 \nu^{12} + 5150 \nu^{11} + 2635 \nu^{10} + 15324 \nu^{9} + \cdots + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 45 \nu^{15} - 64 \nu^{14} + 791 \nu^{13} - 1129 \nu^{12} + 5098 \nu^{11} - 7321 \nu^{10} + 14996 \nu^{9} + \cdots - 38 ) / 4 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -45\nu^{14} - 794\nu^{12} - 5150\nu^{10} - 15324\nu^{8} - 21803\nu^{6} - 13514\nu^{4} - 2485\nu^{2} - 17 ) / 2 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13 \nu^{15} - 91 \nu^{14} - 230 \nu^{13} - 1605 \nu^{12} - 1498 \nu^{11} - 10404 \nu^{10} - 4486 \nu^{9} + \cdots - 39 ) / 4 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 45 \nu^{15} - 64 \nu^{14} - 791 \nu^{13} - 1129 \nu^{12} - 5098 \nu^{11} - 7321 \nu^{10} - 14996 \nu^{9} + \cdots - 38 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{15} - 4 \beta_{14} - 2 \beta_{13} + 3 \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{9} + \beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 12 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} + 22 \beta_{11} + 6 \beta_{10} + \cdots - 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{13} - 7\beta_{9} - \beta_{7} - 7\beta_{6} + 2\beta_{4} - \beta_{3} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16 \beta_{15} - 54 \beta_{14} - 62 \beta_{13} + 38 \beta_{12} - 124 \beta_{11} - 22 \beta_{10} + \cdots + 73 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 43 \beta_{13} + 44 \beta_{9} + 9 \beta_{7} + 44 \beta_{6} + 3 \beta_{5} - 16 \beta_{4} + 9 \beta_{3} + \cdots - 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 103 \beta_{15} + 292 \beta_{14} + 366 \beta_{13} - 189 \beta_{12} + 732 \beta_{11} + 116 \beta_{10} + \cdots - 424 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{15} - 255 \beta_{13} - 4 \beta_{12} - 273 \beta_{9} - 66 \beta_{7} - 273 \beta_{6} + \cdots + 440 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 686 \beta_{15} - 1704 \beta_{14} - 2217 \beta_{13} + 1018 \beta_{12} - 4434 \beta_{11} - 752 \beta_{10} + \cdots + 2593 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 60 \beta_{15} + 1500 \beta_{13} + 60 \beta_{12} + 1696 \beta_{9} + 457 \beta_{7} + 1696 \beta_{6} + \cdots - 2621 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4573 \beta_{15} + 10262 \beta_{14} + 13611 \beta_{13} - 5689 \beta_{12} + 27222 \beta_{11} + \cdots - 16199 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 606 \beta_{15} - 8832 \beta_{13} - 606 \beta_{12} - 10573 \beta_{9} - 3096 \beta_{7} - 10573 \beta_{6} + \cdots + 15840 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 30351 \beta_{15} - 62714 \beta_{14} - 84222 \beta_{13} + 32363 \beta_{12} - 168444 \beta_{11} + \cdots + 102123 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5188 \beta_{15} + 52218 \beta_{13} + 5188 \beta_{12} + 66151 \beta_{9} + 20741 \beta_{7} + 66151 \beta_{6} + \cdots - 96580 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 200513 \beta_{15} + 386472 \beta_{14} + 523991 \beta_{13} - 185959 \beta_{12} + 1047982 \beta_{11} + \cdots - 646739 ) / 5 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-\beta_{3}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
2.53767i
− 1.08982i
0.0898194i
− 1.53767i
− 1.35083i
0.536547i
− 1.53655i
2.35083i
1.35083i
− 0.536547i
1.53655i
− 2.35083i
− 2.53767i
1.08982i
− 0.0898194i
1.53767i
−2.05302 − 1.49161i −0.309017 − 0.951057i 1.37197 + 4.22249i 0 −0.784184 + 2.41347i 1.04054 1.91324 − 5.88835i −0.809017 + 0.587785i 0
76.2 −0.881682 − 0.640580i −0.309017 − 0.951057i −0.251013 − 0.772537i 0 −0.336773 + 1.03648i −3.08724 −0.947104 + 2.91489i −0.809017 + 0.587785i 0
76.3 0.0726655 + 0.0527945i −0.309017 − 0.951057i −0.615541 − 1.89444i 0 0.0277557 − 0.0854234i 4.36070 0.110799 − 0.341004i −0.809017 + 0.587785i 0
76.4 1.24400 + 0.903822i −0.309017 − 0.951057i 0.112618 + 0.346603i 0 0.475167 − 1.46241i 1.68601 0.777165 − 2.39187i −0.809017 + 0.587785i 0
151.1 −0.417429 − 1.28472i 0.809017 − 0.587785i 0.141788 − 0.103015i 0 −1.09284 − 0.793998i 1.59580 −2.37722 − 1.72715i 0.309017 − 0.951057i 0
151.2 −0.165802 − 0.510286i 0.809017 − 0.587785i 1.38513 − 1.00636i 0 −0.434076 − 0.315374i −2.57318 −1.61134 − 1.17071i 0.309017 − 0.951057i 0
151.3 0.474819 + 1.46134i 0.809017 − 0.587785i −0.292036 + 0.212177i 0 1.24309 + 0.903160i 1.49550 2.03746 + 1.48030i 0.309017 − 0.951057i 0
151.4 0.726446 + 2.23577i 0.809017 − 0.587785i −2.85292 + 2.07277i 0 1.90186 + 1.38178i 3.48189 −2.90300 − 2.10915i 0.309017 − 0.951057i 0
226.1 −0.417429 + 1.28472i 0.809017 + 0.587785i 0.141788 + 0.103015i 0 −1.09284 + 0.793998i 1.59580 −2.37722 + 1.72715i 0.309017 + 0.951057i 0
226.2 −0.165802 + 0.510286i 0.809017 + 0.587785i 1.38513 + 1.00636i 0 −0.434076 + 0.315374i −2.57318 −1.61134 + 1.17071i 0.309017 + 0.951057i 0
226.3 0.474819 − 1.46134i 0.809017 + 0.587785i −0.292036 − 0.212177i 0 1.24309 − 0.903160i 1.49550 2.03746 − 1.48030i 0.309017 + 0.951057i 0
226.4 0.726446 − 2.23577i 0.809017 + 0.587785i −2.85292 − 2.07277i 0 1.90186 − 1.38178i 3.48189 −2.90300 + 2.10915i 0.309017 + 0.951057i 0
301.1 −2.05302 + 1.49161i −0.309017 + 0.951057i 1.37197 − 4.22249i 0 −0.784184 − 2.41347i 1.04054 1.91324 + 5.88835i −0.809017 − 0.587785i 0
301.2 −0.881682 + 0.640580i −0.309017 + 0.951057i −0.251013 + 0.772537i 0 −0.336773 − 1.03648i −3.08724 −0.947104 − 2.91489i −0.809017 − 0.587785i 0
301.3 0.0726655 − 0.0527945i −0.309017 + 0.951057i −0.615541 + 1.89444i 0 0.0277557 + 0.0854234i 4.36070 0.110799 + 0.341004i −0.809017 − 0.587785i 0
301.4 1.24400 − 0.903822i −0.309017 + 0.951057i 0.112618 − 0.346603i 0 0.475167 + 1.46241i 1.68601 0.777165 + 2.39187i −0.809017 − 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.g.d 16
5.b even 2 1 375.2.g.e 16
5.c odd 4 1 75.2.i.a ✓ 16
5.c odd 4 1 375.2.i.c 16
15.e even 4 1 225.2.m.b 16
25.d even 5 1 inner 375.2.g.d 16
25.d even 5 1 1875.2.a.p 8
25.e even 10 1 375.2.g.e 16
25.e even 10 1 1875.2.a.m 8
25.f odd 20 1 75.2.i.a ✓ 16
25.f odd 20 1 375.2.i.c 16
25.f odd 20 2 1875.2.b.h 16
75.h odd 10 1 5625.2.a.bd 8
75.j odd 10 1 5625.2.a.t 8
75.l even 20 1 225.2.m.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a ✓ 16 5.c odd 4 1
75.2.i.a ✓ 16 25.f odd 20 1
225.2.m.b 16 15.e even 4 1
225.2.m.b 16 75.l even 20 1
375.2.g.d 16 1.a even 1 1 trivial
375.2.g.d 16 25.d even 5 1 inner
375.2.g.e 16 5.b even 2 1
375.2.g.e 16 25.e even 10 1
375.2.i.c 16 5.c odd 4 1
375.2.i.c 16 25.f odd 20 1
1875.2.a.m 8 25.e even 10 1
1875.2.a.p 8 25.d even 5 1
1875.2.b.h 16 25.f odd 20 2
5625.2.a.t 8 75.j odd 10 1
5625.2.a.bd 8 75.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 2 T_{2}^{15} + 7 T_{2}^{14} + 16 T_{2}^{13} + 46 T_{2}^{12} + 22 T_{2}^{11} + 125 T_{2}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 8 T^{7} + \cdots + 505)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 6 T^{15} + \cdots + 27888961 \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{15} + \cdots + 78961 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + \cdots + 53860921 \) Copy content Toggle raw display
$19$ \( T^{16} - 2 T^{15} + \cdots + 6375625 \) Copy content Toggle raw display
$23$ \( T^{16} + 2 T^{15} + \cdots + 4389025 \) Copy content Toggle raw display
$29$ \( T^{16} + 16 T^{15} + \cdots + 156025 \) Copy content Toggle raw display
$31$ \( T^{16} - 6 T^{15} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 8653650625 \) Copy content Toggle raw display
$41$ \( T^{16} + 14 T^{15} + \cdots + 22137025 \) Copy content Toggle raw display
$43$ \( (T^{8} - 20 T^{7} + \cdots + 22961)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 36687479166361 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 40398990025 \) Copy content Toggle raw display
$59$ \( T^{16} + 12 T^{15} + \cdots + 12924025 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 275701483356121 \) Copy content Toggle raw display
$67$ \( T^{16} - 12 T^{15} + \cdots + 13980121 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 25529328841 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 757413387025 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 3940125750625 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2356228681 \) Copy content Toggle raw display
$89$ \( T^{16} + 18 T^{15} + \cdots + 25 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 216648961 \) Copy content Toggle raw display
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