Properties

Label 930.4.a.n
Level $930$
Weight $4$
Character orbit 930.a
Self dual yes
Analytic conductor $54.872$
Analytic rank $0$
Dimension $4$
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] = [930,4,Mod(1,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, 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("930.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 930.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: \(54.8717763053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 128x^{2} + 114x + 3030 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 6 q^{6} + ( - \beta_{2} - 1) q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 6 q^{6} + ( - \beta_{2} - 1) q^{7} + 8 q^{8} + 9 q^{9} - 10 q^{10} + ( - \beta_{3} - \beta_{2} - \beta_1 + 11) q^{11} - 12 q^{12} + (\beta_{3} + 3 \beta_1 - 6) q^{13} + ( - 2 \beta_{2} - 2) q^{14} + 15 q^{15} + 16 q^{16} + ( - 3 \beta_{2} + 5 \beta_1 - 9) q^{17} + 18 q^{18} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 17) q^{19} - 20 q^{20} + (3 \beta_{2} + 3) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 22) q^{22} + (\beta_{3} + 2 \beta_{2} - 14 \beta_1 - 8) q^{23} - 24 q^{24} + 25 q^{25} + (2 \beta_{3} + 6 \beta_1 - 12) q^{26} - 27 q^{27} + ( - 4 \beta_{2} - 4) q^{28} + (\beta_{3} + 3 \beta_{2} + 10 \beta_1 + 49) q^{29} + 30 q^{30} + 31 q^{31} + 32 q^{32} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 33) q^{33} + ( - 6 \beta_{2} + 10 \beta_1 - 18) q^{34} + (5 \beta_{2} + 5) q^{35} + 36 q^{36} + (3 \beta_{3} - 2 \beta_{2} - 25 \beta_1 - 34) q^{37} + ( - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 34) q^{38} + ( - 3 \beta_{3} - 9 \beta_1 + 18) q^{39} - 40 q^{40} + (4 \beta_{3} + 10 \beta_{2} - 2 \beta_1 + 34) q^{41} + (6 \beta_{2} + 6) q^{42} + ( - 11 \beta_{3} - \beta_{2} + 11 \beta_1 + 171) q^{43} + ( - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 44) q^{44} - 45 q^{45} + (2 \beta_{3} + 4 \beta_{2} - 28 \beta_1 - 16) q^{46} + (2 \beta_{3} - 7 \beta_{2} + 31 \beta_1 + 29) q^{47} - 48 q^{48} + ( - 9 \beta_{3} - 2 \beta_{2} + 8 \beta_1 + 219) q^{49} + 50 q^{50} + (9 \beta_{2} - 15 \beta_1 + 27) q^{51} + (4 \beta_{3} + 12 \beta_1 - 24) q^{52} + (13 \beta_{3} - 2 \beta_{2} + 8 \beta_1 + 142) q^{53} - 54 q^{54} + (5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 55) q^{55} + ( - 8 \beta_{2} - 8) q^{56} + (3 \beta_{3} - 3 \beta_{2} + 9 \beta_1 + 51) q^{57} + (2 \beta_{3} + 6 \beta_{2} + 20 \beta_1 + 98) q^{58} + ( - 3 \beta_{3} - 7 \beta_{2} - 20 \beta_1 + 57) q^{59} + 60 q^{60} + ( - 4 \beta_{3} + 2 \beta_{2} + 30 \beta_1 + 316) q^{61} + 62 q^{62} + ( - 9 \beta_{2} - 9) q^{63} + 64 q^{64} + ( - 5 \beta_{3} - 15 \beta_1 + 30) q^{65} + (6 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 66) q^{66} + (3 \beta_{3} + 2 \beta_{2} - 35 \beta_1 + 326) q^{67} + ( - 12 \beta_{2} + 20 \beta_1 - 36) q^{68} + ( - 3 \beta_{3} - 6 \beta_{2} + 42 \beta_1 + 24) q^{69} + (10 \beta_{2} + 10) q^{70} + (2 \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 614) q^{71} + 72 q^{72} + (6 \beta_{3} + 29 \beta_{2} + 52 \beta_1 - 97) q^{73} + (6 \beta_{3} - 4 \beta_{2} - 50 \beta_1 - 68) q^{74} - 75 q^{75} + ( - 4 \beta_{3} + 4 \beta_{2} - 12 \beta_1 - 68) q^{76} + (\beta_{3} - 27 \beta_{2} + 87 \beta_1 + 439) q^{77} + ( - 6 \beta_{3} - 18 \beta_1 + 36) q^{78} + ( - 5 \beta_{3} - 4 \beta_{2} - 58 \beta_1 + 528) q^{79} - 80 q^{80} + 81 q^{81} + (8 \beta_{3} + 20 \beta_{2} - 4 \beta_1 + 68) q^{82} + ( - 2 \beta_{3} + 3 \beta_{2} - 37 \beta_1 + 259) q^{83} + (12 \beta_{2} + 12) q^{84} + (15 \beta_{2} - 25 \beta_1 + 45) q^{85} + ( - 22 \beta_{3} - 2 \beta_{2} + 22 \beta_1 + 342) q^{86} + ( - 3 \beta_{3} - 9 \beta_{2} - 30 \beta_1 - 147) q^{87} + ( - 8 \beta_{3} - 8 \beta_{2} - 8 \beta_1 + 88) q^{88} + ( - 8 \beta_{3} + 10 \beta_{2} - 95 \beta_1 + 292) q^{89} - 90 q^{90} + ( - 20 \beta_{3} + 15 \beta_{2} - 79 \beta_1 + 125) q^{91} + (4 \beta_{3} + 8 \beta_{2} - 56 \beta_1 - 32) q^{92} - 93 q^{93} + (4 \beta_{3} - 14 \beta_{2} + 62 \beta_1 + 58) q^{94} + (5 \beta_{3} - 5 \beta_{2} + 15 \beta_1 + 85) q^{95} - 96 q^{96} + ( - 14 \beta_{3} + 16 \beta_{2} + 64 \beta_1 + 732) q^{97} + ( - 18 \beta_{3} - 4 \beta_{2} + 16 \beta_1 + 438) q^{98} + ( - 9 \beta_{3} - 9 \beta_{2} - 9 \beta_1 + 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} - 24 q^{6} - 3 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 20 q^{5} - 24 q^{6} - 3 q^{7} + 32 q^{8} + 36 q^{9} - 40 q^{10} + 43 q^{11} - 48 q^{12} - 20 q^{13} - 6 q^{14} + 60 q^{15} + 64 q^{16} - 28 q^{17} + 72 q^{18} - 73 q^{19} - 80 q^{20} + 9 q^{21} + 86 q^{22} - 47 q^{23} - 96 q^{24} + 100 q^{25} - 40 q^{26} - 108 q^{27} - 12 q^{28} + 204 q^{29} + 120 q^{30} + 124 q^{31} + 128 q^{32} - 129 q^{33} - 56 q^{34} + 15 q^{35} + 144 q^{36} - 156 q^{37} - 146 q^{38} + 60 q^{39} - 160 q^{40} + 128 q^{41} + 18 q^{42} + 685 q^{43} + 172 q^{44} - 180 q^{45} - 94 q^{46} + 156 q^{47} - 192 q^{48} + 877 q^{49} + 200 q^{50} + 84 q^{51} - 80 q^{52} + 591 q^{53} - 216 q^{54} - 215 q^{55} - 24 q^{56} + 219 q^{57} + 408 q^{58} + 212 q^{59} + 240 q^{60} + 1288 q^{61} + 248 q^{62} - 27 q^{63} + 256 q^{64} + 100 q^{65} - 258 q^{66} + 1270 q^{67} - 112 q^{68} + 141 q^{69} + 30 q^{70} + 2465 q^{71} + 288 q^{72} - 359 q^{73} - 312 q^{74} - 300 q^{75} - 292 q^{76} + 1871 q^{77} + 120 q^{78} + 2053 q^{79} - 320 q^{80} + 324 q^{81} + 256 q^{82} + 994 q^{83} + 36 q^{84} + 140 q^{85} + 1370 q^{86} - 612 q^{87} + 344 q^{88} + 1055 q^{89} - 360 q^{90} + 386 q^{91} - 188 q^{92} - 372 q^{93} + 312 q^{94} + 365 q^{95} - 384 q^{96} + 2962 q^{97} + 1754 q^{98} + 387 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 128x^{2} + 114x + 3030 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - 81\nu - 57 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 9\beta_{2} + 81\beta _1 - 7 \) Copy content Toggle raw display

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
−5.01222
9.66700
−9.94367
6.28889
2.00000 −3.00000 4.00000 −5.00000 −6.00000 −28.5770 8.00000 9.00000 −10.0000
1.2 2.00000 −3.00000 4.00000 −5.00000 −6.00000 −18.4237 8.00000 9.00000 −10.0000
1.3 2.00000 −3.00000 4.00000 −5.00000 −6.00000 14.0981 8.00000 9.00000 −10.0000
1.4 2.00000 −3.00000 4.00000 −5.00000 −6.00000 29.9026 8.00000 9.00000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(31\) \(-1\)

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 930.4.a.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.4.a.n 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 3T_{7}^{3} - 1120T_{7}^{2} - 3352T_{7} + 221954 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(930))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} - 1120 T^{2} + \cdots + 221954 \) Copy content Toggle raw display
$11$ \( T^{4} - 43 T^{3} - 2290 T^{2} + \cdots - 87048 \) Copy content Toggle raw display
$13$ \( T^{4} + 20 T^{3} - 3038 T^{2} + \cdots - 34740 \) Copy content Toggle raw display
$17$ \( T^{4} + 28 T^{3} - 13232 T^{2} + \cdots - 2335440 \) Copy content Toggle raw display
$19$ \( T^{4} + 73 T^{3} - 2684 T^{2} + \cdots - 5882560 \) Copy content Toggle raw display
$23$ \( T^{4} + 47 T^{3} + \cdots + 153604380 \) Copy content Toggle raw display
$29$ \( T^{4} - 204 T^{3} - 7824 T^{2} + \cdots + 3115368 \) Copy content Toggle raw display
$31$ \( (T - 31)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 156 T^{3} + \cdots - 1266846400 \) Copy content Toggle raw display
$41$ \( T^{4} - 128 T^{3} + \cdots - 823625856 \) Copy content Toggle raw display
$43$ \( T^{4} - 685 T^{3} + \cdots + 5476990600 \) Copy content Toggle raw display
$47$ \( T^{4} - 156 T^{3} + \cdots + 4286240520 \) Copy content Toggle raw display
$53$ \( T^{4} - 591 T^{3} + \cdots + 9055700772 \) Copy content Toggle raw display
$59$ \( T^{4} - 212 T^{3} + \cdots - 1590405900 \) Copy content Toggle raw display
$61$ \( T^{4} - 1288 T^{3} + \cdots - 16176506480 \) Copy content Toggle raw display
$67$ \( T^{4} - 1270 T^{3} + \cdots - 1068288812 \) Copy content Toggle raw display
$71$ \( T^{4} - 2465 T^{3} + \cdots + 136879857138 \) Copy content Toggle raw display
$73$ \( T^{4} + 359 T^{3} + \cdots + 155582080054 \) Copy content Toggle raw display
$79$ \( T^{4} - 2053 T^{3} + \cdots - 91345005380 \) Copy content Toggle raw display
$83$ \( T^{4} - 994 T^{3} + \cdots + 570567120 \) Copy content Toggle raw display
$89$ \( T^{4} - 1055 T^{3} + \cdots + 314857727442 \) Copy content Toggle raw display
$97$ \( T^{4} - 2962 T^{3} + \cdots - 1059198695360 \) Copy content Toggle raw display
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