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Properties

Label 750.2.g.d
Level $750$
Weight $2$
Character orbit 750.g
Analytic conductor $5.989$
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] = [750,2,Mod(151,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.g (of order \(5\), degree \(4\), not 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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1064390625.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 3x^{6} - 5x^{5} + 36x^{4} - 35x^{3} + 23x^{2} - 171x + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 150)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} - 1) q^{4} + \beta_{4} q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} - \beta_{4} q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{3} + (\beta_{4} + \beta_{3} + \beta_{2} - 1) q^{4} + \beta_{4} q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} - \beta_{4} q^{8} - \beta_{2} q^{9} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{11}+ \cdots + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{7} - 2 q^{8} - 2 q^{9} - 5 q^{11} + 2 q^{12} - 6 q^{13} + 2 q^{14} - 2 q^{16} + 2 q^{17} + 8 q^{18} + 8 q^{19} + 3 q^{21} + 20 q^{23} - 8 q^{24} + 14 q^{26} + 2 q^{27} - 3 q^{28} - 18 q^{29} + 9 q^{31} + 8 q^{32} - 3 q^{34} - 2 q^{36} - 21 q^{37} - 12 q^{38} + 6 q^{39} + 2 q^{41} + 3 q^{42} + 32 q^{43} - 20 q^{46} + 10 q^{47} + 2 q^{48} + 22 q^{49} - 2 q^{51} - 6 q^{52} - 7 q^{53} + 2 q^{54} - 3 q^{56} - 8 q^{57} - 18 q^{58} - 25 q^{59} + 10 q^{61} - 6 q^{62} + 2 q^{63} - 2 q^{64} + 5 q^{66} + 2 q^{67} + 2 q^{68} + 20 q^{69} - 2 q^{72} + 24 q^{73} - 26 q^{74} + 8 q^{76} - 35 q^{77} + q^{78} - 6 q^{79} - 2 q^{81} + 42 q^{82} - 11 q^{83} - 2 q^{84} + 2 q^{86} - 7 q^{87} - 5 q^{88} + 9 q^{89} - 4 q^{91} - 20 q^{92} + 6 q^{93} + 10 q^{94} + 2 q^{96} - q^{97} + 7 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 3x^{6} - 5x^{5} + 36x^{4} - 35x^{3} + 23x^{2} - 171x + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 9895 \nu^{7} - 2250 \nu^{6} - 82415 \nu^{5} + 417410 \nu^{4} + 424661 \nu^{3} - 65671 \nu^{2} + \cdots + 3933095 ) / 2119279 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27571 \nu^{7} - 156 \nu^{6} + 50800 \nu^{5} - 197116 \nu^{4} + 261151 \nu^{3} - 1281772 \nu^{2} + \cdots - 4276691 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45697 \nu^{7} - 18958 \nu^{6} + 87368 \nu^{5} + 389890 \nu^{4} - 351515 \nu^{3} + \cdots + 5880747 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 60697 \nu^{7} - 29865 \nu^{6} + 107003 \nu^{5} + 736723 \nu^{4} - 989612 \nu^{3} + \cdots + 13281190 ) / 4238558 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7961 \nu^{7} - 3952 \nu^{6} + 22802 \nu^{5} + 62920 \nu^{4} - 113815 \nu^{3} + 24054 \nu^{2} + \cdots + 930161 ) / 223082 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4351\nu^{7} - 4822\nu^{6} + 5807\nu^{5} + 53289\nu^{4} - 178\nu^{3} - 2826\nu^{2} - 62405\nu + 584505 ) / 111541 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 279061 \nu^{7} + 17407 \nu^{6} - 723449 \nu^{5} - 2023635 \nu^{4} + 3885332 \nu^{3} + \cdots - 35481474 ) / 4238558 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + 4\beta_{5} - 3\beta_{4} - \beta_{3} - \beta_{2} + 2\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} + 2\beta_{5} + \beta_{4} + 22\beta_{3} + 2\beta_{2} + \beta _1 - 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} + 3\beta_{5} - 31\beta_{4} + 3\beta_{3} + 8\beta_{2} + 9\beta _1 + 31 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} - 4\beta_{6} + 5\beta_{5} + 5\beta_{4} + 12\beta_{3} + 9\beta_{2} + 7\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} - 16\beta_{6} - 16\beta_{5} + 2\beta_{4} + 209\beta_{3} + 209\beta_{2} + 2\beta _1 + 28 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -72\beta_{7} - 123\beta_{6} + 102\beta_{5} - 254\beta_{4} - 58\beta_{3} + 72\beta_{2} + 246\beta _1 + 109 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 242\beta_{7} - 242\beta_{6} + 113\beta_{5} + 634\beta_{4} + 1078\beta_{3} + 763\beta_{2} + 129\beta _1 - 129 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1 + \beta_{2} + \beta_{3} + \beta_{4}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
−0.815575 + 1.64827i
1.31557 − 1.28500i
2.36886 + 0.0809628i
−1.86886 + 1.45788i
2.36886 − 0.0809628i
−1.86886 − 1.45788i
−0.815575 − 1.64827i
1.31557 + 1.28500i
0.309017 + 0.951057i 0.809017 − 0.587785i −0.809017 + 0.587785i 0 0.809017 + 0.587785i −2.63925 −0.809017 − 0.587785i 0.309017 − 0.951057i 0
151.2 0.309017 + 0.951057i 0.809017 − 0.587785i −0.809017 + 0.587785i 0 0.809017 + 0.587785i 4.25729 −0.809017 − 0.587785i 0.309017 − 0.951057i 0
301.1 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 − 0.951057i 0 −0.309017 − 0.951057i −2.92807 0.309017 + 0.951057i −0.809017 − 0.587785i 0
301.2 −0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 − 0.951057i 0 −0.309017 − 0.951057i 2.31003 0.309017 + 0.951057i −0.809017 − 0.587785i 0
451.1 −0.809017 − 0.587785i −0.309017 − 0.951057i 0.309017 + 0.951057i 0 −0.309017 + 0.951057i −2.92807 0.309017 − 0.951057i −0.809017 + 0.587785i 0
451.2 −0.809017 − 0.587785i −0.309017 − 0.951057i 0.309017 + 0.951057i 0 −0.309017 + 0.951057i 2.31003 0.309017 − 0.951057i −0.809017 + 0.587785i 0
601.1 0.309017 − 0.951057i 0.809017 + 0.587785i −0.809017 − 0.587785i 0 0.809017 − 0.587785i −2.63925 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
601.2 0.309017 − 0.951057i 0.809017 + 0.587785i −0.809017 − 0.587785i 0 0.809017 − 0.587785i 4.25729 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.2
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 750.2.g.d 8
5.b even 2 1 150.2.g.c ✓ 8
5.c odd 4 2 750.2.h.e 16
15.d odd 2 1 450.2.h.d 8
25.d even 5 1 inner 750.2.g.d 8
25.d even 5 1 3750.2.a.q 4
25.e even 10 1 150.2.g.c ✓ 8
25.e even 10 1 3750.2.a.l 4
25.f odd 20 2 750.2.h.e 16
25.f odd 20 2 3750.2.c.h 8
75.h odd 10 1 450.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.c ✓ 8 5.b even 2 1
150.2.g.c ✓ 8 25.e even 10 1
450.2.h.d 8 15.d odd 2 1
450.2.h.d 8 75.h odd 10 1
750.2.g.d 8 1.a even 1 1 trivial
750.2.g.d 8 25.d even 5 1 inner
750.2.h.e 16 5.c odd 4 2
750.2.h.e 16 25.f odd 20 2
3750.2.a.l 4 25.e even 10 1
3750.2.a.q 4 25.d even 5 1
3750.2.c.h 8 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - T_{7}^{3} - 19T_{7}^{2} + 4T_{7} + 76 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} - 19 T^{2} + \cdots + 76)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 5776 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + 16 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 18 T^{7} + \cdots + 1860496 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$37$ \( T^{8} + 21 T^{7} + \cdots + 11182336 \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{4} - 16 T^{3} + \cdots - 1424)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 774400 \) Copy content Toggle raw display
$53$ \( T^{8} + 7 T^{7} + \cdots + 85359121 \) Copy content Toggle raw display
$59$ \( T^{8} + 25 T^{7} + \cdots + 2310400 \) Copy content Toggle raw display
$61$ \( T^{8} - 10 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$67$ \( T^{8} - 2 T^{7} + \cdots + 6885376 \) Copy content Toggle raw display
$71$ \( T^{8} - 40 T^{6} + \cdots + 102400 \) Copy content Toggle raw display
$73$ \( T^{8} - 24 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( T^{8} + 6 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$83$ \( T^{8} + 11 T^{7} + \cdots + 1008016 \) Copy content Toggle raw display
$89$ \( T^{8} - 9 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + \cdots + 130321 \) Copy content Toggle raw display
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