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Properties

Label 750.4.c.e
Level $750$
Weight $4$
Character orbit 750.c
Analytic conductor $44.251$
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] = [750,4,Mod(499,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.499");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 750.c (of order \(2\), degree \(1\), 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: \(44.2514325043\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 99x^{6} + 3541x^{4} + 54684x^{2} + 309136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 3 \beta_1 q^{3} - 4 q^{4} - 6 q^{6} + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 3 \beta_1 q^{3} - 4 q^{4} - 6 q^{6} + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + (9 \beta_{6} + 27 \beta_{4} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} - 48 q^{6} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} - 48 q^{6} - 72 q^{9} - 30 q^{11} + 12 q^{14} + 128 q^{16} + 128 q^{19} + 18 q^{21} + 192 q^{24} - 516 q^{26} + 252 q^{29} + 174 q^{31} + 52 q^{34} + 288 q^{36} - 774 q^{39} - 78 q^{41} + 120 q^{44} - 520 q^{46} - 238 q^{49} + 78 q^{51} + 432 q^{54} - 48 q^{56} + 1530 q^{59} + 1310 q^{61} - 512 q^{64} + 180 q^{66} - 780 q^{69} - 3780 q^{71} + 3924 q^{74} - 512 q^{76} + 1974 q^{79} + 648 q^{81} - 72 q^{84} - 3408 q^{86} + 2814 q^{89} - 3884 q^{91} + 520 q^{94} - 768 q^{96} + 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 99x^{6} + 3541x^{4} + 54684x^{2} + 309136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -35\nu^{7} - 2909\nu^{5} - 76675\nu^{3} - 643480\nu ) / 36696 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} - 211\nu^{4} - 4413\nu^{2} - 27800 ) / 66 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -335\nu^{7} - 25659\nu^{5} - 609385\nu^{3} - 4562310\nu ) / 18348 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{6} + 1165\nu^{4} + 28335\nu^{2} + 219344 ) / 132 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 233\nu^{5} - 5667\nu^{3} - 44212\nu ) / 132 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 77\nu^{4} + 1841\nu^{2} + 13894 ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1703\nu^{7} - 132457\nu^{5} - 3227527\nu^{3} - 25041872\nu ) / 36696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{6} + 4\beta_{4} - \beta_{2} - 121 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -38\beta_{7} + 51\beta_{5} + 26\beta_{3} + 136\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 171\beta_{6} - 222\beta_{4} + 72\beta_{2} + 3245 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 287\beta_{7} - 289\beta_{5} - 288\beta_{3} - 1565\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7614\beta_{6} + 9730\beta_{4} - 3703\beta_{2} - 96574 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -54407\beta_{7} + 45144\beta_{5} + 62726\beta_{3} + 365575\beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
− 4.12081i
4.58789i
− 6.20593i
4.73884i
− 4.73884i
6.20593i
− 4.58789i
4.12081i
− 2.00000i − 3.00000i −4.00000 0 −6.00000 − 30.2138i 8.00000i −9.00000 0
499.2 − 2.00000i − 3.00000i −4.00000 0 −6.00000 1.17164i 8.00000i −9.00000 0
499.3 − 2.00000i − 3.00000i −4.00000 0 −6.00000 10.3907i 8.00000i −9.00000 0
499.4 − 2.00000i − 3.00000i −4.00000 0 −6.00000 21.6515i 8.00000i −9.00000 0
499.5 2.00000i 3.00000i −4.00000 0 −6.00000 − 21.6515i − 8.00000i −9.00000 0
499.6 2.00000i 3.00000i −4.00000 0 −6.00000 − 10.3907i − 8.00000i −9.00000 0
499.7 2.00000i 3.00000i −4.00000 0 −6.00000 − 1.17164i − 8.00000i −9.00000 0
499.8 2.00000i 3.00000i −4.00000 0 −6.00000 30.2138i − 8.00000i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.4.c.e 8
5.b even 2 1 inner 750.4.c.e 8
5.c odd 4 1 750.4.a.l ✓ 4
5.c odd 4 1 750.4.a.m yes 4
15.e even 4 1 2250.4.a.m 4
15.e even 4 1 2250.4.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.4.a.l ✓ 4 5.c odd 4 1
750.4.a.m yes 4 5.c odd 4 1
750.4.c.e 8 1.a even 1 1 trivial
750.4.c.e 8 5.b even 2 1 inner
2250.4.a.m 4 15.e even 4 1
2250.4.a.z 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 1491T_{7}^{6} + 579161T_{7}^{4} + 46995576T_{7}^{2} + 63425296 \) acting on \(S_{4}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 1491 T^{6} + \cdots + 63425296 \) Copy content Toggle raw display
$11$ \( (T^{4} + 15 T^{3} + \cdots + 4564900)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 211169839875856 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 940674973456 \) Copy content Toggle raw display
$19$ \( (T^{4} - 64 T^{3} + \cdots + 23326411)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{4} - 126 T^{3} + \cdots - 7108544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 87 T^{3} + \cdots + 189319696)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + 39 T^{3} + \cdots - 349203404)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 87\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( (T^{4} - 765 T^{3} + \cdots + 4735044400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 655 T^{3} + \cdots - 5893747100)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1890 T^{3} + \cdots + 4604628400)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{4} - 987 T^{3} + \cdots - 126778756004)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1407 T^{3} + \cdots - 45454135964)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
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