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Properties

Label 608.4.a.j
Level $608$
Weight $4$
Character orbit 608.a
Self dual yes
Analytic conductor $35.873$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,4,Mod(1,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.a (trivial)

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: yes
Analytic conductor: \(35.8731612835\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{3} - 2) q^{5} + ( - \beta_{5} + \beta_{2} - 6) q^{7} + (\beta_{6} + \beta_{5} - 2 \beta_{2} + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{3} - 2) q^{5} + ( - \beta_{5} + \beta_{2} - 6) q^{7} + (\beta_{6} + \beta_{5} - 2 \beta_{2} + 12) q^{9} + ( - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{11}+ \cdots + (26 \beta_{6} + 10 \beta_{5} + \cdots + 266) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 3 q^{3} - 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 3 q^{3} - 17 q^{5} - 42 q^{7} + 86 q^{9} + 33 q^{11} - 35 q^{13} - 120 q^{15} + 66 q^{17} + 133 q^{19} + 33 q^{21} - 389 q^{23} + 44 q^{25} + 39 q^{27} + 233 q^{29} - 158 q^{31} - 206 q^{33} - 123 q^{35} - 436 q^{37} - 807 q^{39} - 94 q^{41} - 645 q^{43} + 103 q^{45} - 1451 q^{47} + 93 q^{49} - 1741 q^{51} + 3 q^{53} - 1971 q^{55} - 57 q^{57} - 297 q^{59} + 93 q^{61} - 2999 q^{63} - 788 q^{65} - 1641 q^{67} + 945 q^{69} - 2392 q^{71} + 324 q^{73} - 1909 q^{75} - 711 q^{77} - 2492 q^{79} + 143 q^{81} - 310 q^{83} + 2353 q^{85} - 4795 q^{87} - 440 q^{89} + 107 q^{91} + 900 q^{93} - 323 q^{95} - 532 q^{97} + 1591 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31441 \nu^{6} - 53905 \nu^{5} - 3640731 \nu^{4} + 5610384 \nu^{3} + 129105226 \nu^{2} + \cdots - 1374812288 ) / 46945020 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 39878 \nu^{6} - 170155 \nu^{5} + 4088763 \nu^{4} + 14312913 \nu^{3} - 123786608 \nu^{2} + \cdots + 1133044114 ) / 23472510 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 41606 \nu^{6} - 149275 \nu^{5} + 4108191 \nu^{4} + 9529701 \nu^{3} - 122864216 \nu^{2} + \cdots + 1155407968 ) / 23472510 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 134983 \nu^{6} + 651005 \nu^{5} - 12792033 \nu^{4} - 43729188 \nu^{3} + 362351638 \nu^{2} + \cdots - 3246819584 ) / 46945020 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 320173 \nu^{6} - 369335 \nu^{5} + 34597563 \nu^{4} + 19633308 \nu^{3} - 1044937738 \nu^{2} + \cdots + 8164814204 ) / 46945020 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 169831 \nu^{6} - 229925 \nu^{5} + 21008001 \nu^{4} + 17685276 \nu^{3} - 766255246 \nu^{2} + \cdots + 7496458508 ) / 23472510 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - \beta_{2} + 5\beta _1 + 70 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{6} - 2\beta_{5} + 54\beta_{4} + 76\beta_{3} + 40\beta_{2} + 18\beta _1 - 104 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 39\beta_{6} + 126\beta_{5} - 36\beta_{4} - 182\beta_{3} - 226\beta_{2} + 804\beta _1 + 6554 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -178\beta_{6} - 62\beta_{5} + 1652\beta_{4} + 1821\beta_{3} + 1683\beta_{2} - 558\beta _1 - 5331 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4441\beta_{6} + 6522\beta_{5} - 5178\beta_{4} - 14256\beta_{3} - 19324\beta_{2} + 55846\beta _1 + 359232 ) / 4 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.42108
7.37394
6.59304
−4.00212
4.17107
−6.10834
3.39349
0 −8.54092 0 13.1578 0 −19.9667 0 45.9472 0
1.2 0 −6.88542 0 4.69330 0 −3.62949 0 20.4091 0
1.3 0 −5.59727 0 −20.9584 0 −13.9085 0 4.32941 0
1.4 0 0.923414 0 −0.670967 0 28.6780 0 −26.1473 0
1.5 0 2.43763 0 −9.42708 0 9.58919 0 −21.0580 0
1.6 0 5.20712 0 7.79991 0 −26.8592 0 0.114065 0
1.7 0 9.45545 0 −11.5945 0 −15.9032 0 62.4055 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.4.a.j ✓ 7
4.b odd 2 1 608.4.a.k yes 7
8.b even 2 1 1216.4.a.bg 7
8.d odd 2 1 1216.4.a.bf 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.j ✓ 7 1.a even 1 1 trivial
608.4.a.k yes 7 4.b odd 2 1
1216.4.a.bf 7 8.d odd 2 1
1216.4.a.bg 7 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(608))\):

\( T_{3}^{7} + 3T_{3}^{6} - 133T_{3}^{5} - 367T_{3}^{4} + 4632T_{3}^{3} + 6688T_{3}^{2} - 49248T_{3} + 36480 \) Copy content Toggle raw display
\( T_{5}^{7} + 17T_{5}^{6} - 315T_{5}^{5} - 4077T_{5}^{4} + 28950T_{5}^{3} + 216728T_{5}^{2} - 972192T_{5} - 740352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 3 T^{6} + \cdots + 36480 \) Copy content Toggle raw display
$5$ \( T^{7} + 17 T^{6} + \cdots - 740352 \) Copy content Toggle raw display
$7$ \( T^{7} + 42 T^{6} + \cdots + 118397740 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots + 47353183200 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 361563654528 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 821802019554 \) Copy content Toggle raw display
$19$ \( (T - 19)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 74757540326400 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 758502349773120 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 542529415987200 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 82\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 883998869667840 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 71\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 43\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 52\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots - 28\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 67\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 22\!\cdots\!06 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 35\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 86\!\cdots\!12 \) Copy content Toggle raw display
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