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Properties

Label 75.2.g.b
Level $75$
Weight $2$
Character orbit 75.g
Analytic conductor $0.599$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,2,Mod(16,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.g (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{3}) q^{2} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{5} + (\beta_{5} - \beta_1) q^{6} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 5 \beta_1 + 1) q^{8} - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{3}) q^{2} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{3} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{5} + (\beta_{5} - \beta_1) q^{6} + (\beta_{7} - 2 \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 5 \beta_1 + 1) q^{8} - \beta_{6} q^{9} + (\beta_{7} + \beta_{4} - 4 \beta_1 + 1) q^{10} + (3 \beta_{6} + \beta_{4} + 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{12} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{13} + ( - \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - \beta_1) q^{14} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{15} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 2) q^{16} - \beta_{5} q^{17} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{18} + (2 \beta_{5} + \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{20} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{21} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 2) q^{22} + (\beta_{7} + 2 \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{24} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{25} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{26} - \beta_{3} q^{27} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_1 - 2) q^{28} + ( - 4 \beta_{6} + \beta_{4} - 5 \beta_{3} - 5 \beta_1 + 4) q^{29} + (\beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 3) q^{30} + (2 \beta_{3} - \beta_{2} - 5 \beta_1 - 2) q^{31} + (2 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + \cdots - 1) q^{32}+ \cdots + ( - 3 \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 2 q^{3} + q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} - 2 q^{3} + q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 2 q^{9} + 16 q^{11} - 9 q^{12} - 8 q^{13} - 8 q^{14} + 5 q^{15} - 17 q^{16} - q^{17} + 4 q^{18} - 5 q^{19} - 10 q^{20} - 11 q^{21} + 13 q^{22} + 7 q^{23} + 30 q^{24} - 15 q^{25} + 6 q^{26} - 2 q^{27} - 17 q^{28} + 5 q^{29} + 30 q^{30} - 19 q^{31} + 24 q^{32} - 9 q^{33} + 12 q^{34} - 10 q^{35} - 9 q^{36} - q^{37} - 10 q^{38} - 8 q^{39} + 25 q^{40} - 14 q^{41} - 8 q^{42} + 32 q^{43} - 3 q^{44} - 5 q^{45} + 16 q^{46} - q^{47} - 17 q^{48} + 16 q^{49} + 10 q^{50} + 4 q^{51} - 6 q^{52} - 3 q^{53} - q^{54} + 15 q^{55} - 15 q^{56} + 10 q^{57} + 5 q^{58} + 30 q^{59} - 15 q^{60} - 14 q^{61} - 17 q^{62} + 9 q^{63} - 44 q^{64} + 25 q^{65} - 7 q^{66} + 4 q^{67} - 22 q^{68} - 8 q^{69} - 15 q^{70} + 21 q^{71} + 2 q^{73} - 38 q^{74} - 15 q^{75} + 80 q^{76} - 37 q^{77} - 14 q^{78} - 30 q^{79} - 50 q^{80} - 2 q^{81} - 12 q^{82} + 2 q^{83} + 8 q^{84} - 30 q^{85} - 34 q^{86} + 15 q^{87} + 70 q^{88} - 5 q^{90} + 21 q^{91} + 9 q^{92} + 46 q^{93} - 33 q^{94} + 65 q^{95} + 34 q^{96} - 6 q^{97} + 73 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1} + \beta_{3} + \beta_{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
1.33631 + 0.462894i
−0.0272949 − 1.41395i
1.40799 + 0.132563i
−1.21700 − 0.720348i
1.40799 − 0.132563i
−1.21700 + 0.720348i
1.33631 − 0.462894i
−0.0272949 + 1.41395i
−2.18949 − 1.59076i 0.309017 + 0.951057i 1.64533 + 5.06380i 0.336312 + 2.21063i 0.836312 − 2.57390i 0.470294 2.78023 − 8.55667i −0.809017 + 0.587785i 2.78023 − 5.37515i
16.2 1.38048 + 1.00297i 0.309017 + 0.951057i 0.281722 + 0.867051i −1.02729 − 1.98612i −0.527295 + 1.62285i −3.94243 0.573870 − 1.76619i −0.809017 + 0.587785i 0.573870 − 3.77214i
31.1 −0.346820 − 1.06740i −0.809017 + 0.587785i 0.598970 − 0.435177i 0.407987 − 2.19853i 0.907987 + 0.659691i 1.11373 −2.48822 − 1.80780i 0.309017 − 0.951057i −2.48822 + 0.327009i
31.2 0.655837 + 2.01846i −0.809017 + 0.587785i −2.02602 + 1.47199i −2.21700 − 0.291365i −1.71700 − 1.24748i 4.35840 −0.865884 − 0.629102i 0.309017 − 0.951057i −0.865884 − 4.66602i
46.1 −0.346820 + 1.06740i −0.809017 − 0.587785i 0.598970 + 0.435177i 0.407987 + 2.19853i 0.907987 − 0.659691i 1.11373 −2.48822 + 1.80780i 0.309017 + 0.951057i −2.48822 − 0.327009i
46.2 0.655837 − 2.01846i −0.809017 − 0.587785i −2.02602 − 1.47199i −2.21700 + 0.291365i −1.71700 + 1.24748i 4.35840 −0.865884 + 0.629102i 0.309017 + 0.951057i −0.865884 + 4.66602i
61.1 −2.18949 + 1.59076i 0.309017 − 0.951057i 1.64533 − 5.06380i 0.336312 − 2.21063i 0.836312 + 2.57390i 0.470294 2.78023 + 8.55667i −0.809017 − 0.587785i 2.78023 + 5.37515i
61.2 1.38048 − 1.00297i 0.309017 − 0.951057i 0.281722 − 0.867051i −1.02729 + 1.98612i −0.527295 − 1.62285i −3.94243 0.573870 + 1.76619i −0.809017 − 0.587785i 0.573870 + 3.77214i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.2
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 75.2.g.b ✓ 8
3.b odd 2 1 225.2.h.c 8
5.b even 2 1 375.2.g.b 8
5.c odd 4 2 375.2.i.b 16
25.d even 5 1 inner 75.2.g.b ✓ 8
25.d even 5 1 1875.2.a.h 4
25.e even 10 1 375.2.g.b 8
25.e even 10 1 1875.2.a.e 4
25.f odd 20 2 375.2.i.b 16
25.f odd 20 2 1875.2.b.c 8
75.h odd 10 1 5625.2.a.n 4
75.j odd 10 1 225.2.h.c 8
75.j odd 10 1 5625.2.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b ✓ 8 1.a even 1 1 trivial
75.2.g.b ✓ 8 25.d even 5 1 inner
225.2.h.c 8 3.b odd 2 1
225.2.h.c 8 75.j odd 10 1
375.2.g.b 8 5.b even 2 1
375.2.g.b 8 25.e even 10 1
375.2.i.b 16 5.c odd 4 2
375.2.i.b 16 25.f odd 20 2
1875.2.a.e 4 25.e even 10 1
1875.2.a.h 4 25.d even 5 1
1875.2.b.c 8 25.f odd 20 2
5625.2.a.i 4 75.j odd 10 1
5625.2.a.n 4 75.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} + 2T_{2}^{6} + 3T_{2}^{5} + 25T_{2}^{4} - 43T_{2}^{3} + 82T_{2}^{2} - 11T_{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + 2 T^{6} + 3 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{7} + 20 T^{6} + 65 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 16 T^{2} + 27 T - 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 16 T^{7} + 127 T^{6} + \cdots + 11881 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + 43 T^{6} + 131 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + 7 T^{6} + 13 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + 5 T^{6} + \cdots + 75625 \) Copy content Toggle raw display
$23$ \( T^{8} - 7 T^{7} + 58 T^{6} - 169 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{8} - 5 T^{7} + 70 T^{6} + \cdots + 164025 \) Copy content Toggle raw display
$31$ \( T^{8} + 19 T^{7} + 172 T^{6} + \cdots + 505521 \) Copy content Toggle raw display
$37$ \( T^{8} + T^{7} + 77 T^{6} + 153 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} + 7 T^{3} + 69 T^{2} + 143 T + 121)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + 61 T^{2} + 24 T - 99)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} + 32 T^{6} + 63 T^{5} + \cdots + 96721 \) Copy content Toggle raw display
$53$ \( T^{8} + 3 T^{7} + 43 T^{6} + \cdots + 68121 \) Copy content Toggle raw display
$59$ \( T^{8} - 30 T^{7} + 495 T^{6} + \cdots + 13286025 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} + 157 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{7} + 37 T^{6} + \cdots + 29241 \) Copy content Toggle raw display
$71$ \( T^{8} - 21 T^{7} + 447 T^{6} + \cdots + 829921 \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} - 17 T^{6} + \cdots + 23707161 \) Copy content Toggle raw display
$79$ \( T^{8} + 30 T^{7} + 605 T^{6} + \cdots + 96924025 \) Copy content Toggle raw display
$83$ \( T^{8} - 2 T^{7} + 43 T^{6} + \cdots + 958441 \) Copy content Toggle raw display
$89$ \( T^{8} + 255 T^{6} + \cdots + 10923025 \) Copy content Toggle raw display
$97$ \( T^{8} + 6 T^{7} + 7 T^{6} + \cdots + 4414201 \) Copy content Toggle raw display
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