Properties

Label 4030.2.a.q
Level $4030$
Weight $2$
Character orbit 4030.a
Self dual yes
Analytic conductor $32.180$
Analytic rank $0$
Dimension $9$
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] = [4030,2,Mod(1,4030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4030.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: \(32.1797120146\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 16x^{7} + 48x^{6} + 66x^{5} - 202x^{4} - 75x^{3} + 210x^{2} + 68x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{2} q^{7} + q^{8} + (\beta_{7} + \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + \beta_{2} q^{7} + q^{8} + (\beta_{7} + \beta_{6} + 1) q^{9} - q^{10} + (\beta_{7} + \beta_{3} + \beta_{2} - \beta_1) q^{11} + \beta_1 q^{12} + q^{13} + \beta_{2} q^{14} - \beta_1 q^{15} + q^{16} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{2}) q^{17} + (\beta_{7} + \beta_{6} + 1) q^{18} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{19} - q^{20} + (\beta_{6} - \beta_{4} + \beta_1 - 1) q^{21} + (\beta_{7} + \beta_{3} + \beta_{2} - \beta_1) q^{22} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} + 3) q^{23} + \beta_1 q^{24} + q^{25} + q^{26} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1 + 1) q^{27} + \beta_{2} q^{28} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{29} - \beta_1 q^{30} + q^{31} + q^{32} + ( - \beta_{8} - 3 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{33} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{2}) q^{34} - \beta_{2} q^{35} + (\beta_{7} + \beta_{6} + 1) q^{36} + ( - \beta_{6} - \beta_{3}) q^{37} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{38} + \beta_1 q^{39} - q^{40} + (\beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{41} + (\beta_{6} - \beta_{4} + \beta_1 - 1) q^{42} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{43} + (\beta_{7} + \beta_{3} + \beta_{2} - \beta_1) q^{44} + ( - \beta_{7} - \beta_{6} - 1) q^{45} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} + 3) q^{46} + (\beta_{8} + \beta_{7} + \beta_{4} - \beta_{2} - \beta_1 + 3) q^{47} + \beta_1 q^{48} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{49} + q^{50} + ( - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 4 \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{51} + q^{52} + ( - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + 3) q^{53} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1 + 1) q^{54} + ( - \beta_{7} - \beta_{3} - \beta_{2} + \beta_1) q^{55} + \beta_{2} q^{56} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{57} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{58} + ( - \beta_{7} - \beta_{4} + \beta_{2} - 1) q^{59} - \beta_1 q^{60} + (\beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2} + 2) q^{61} + q^{62} + ( - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{63} + q^{64} - q^{65} + ( - \beta_{8} - 3 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{66} + ( - \beta_{8} - \beta_{6} + \beta_{2}) q^{67} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{2}) q^{68} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{69} - \beta_{2} q^{70} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{71} + (\beta_{7} + \beta_{6} + 1) q^{72} + (\beta_{8} + 2 \beta_{7} + \beta_{5} + 2 \beta_{3} - 2 \beta_1 - 2) q^{73} + ( - \beta_{6} - \beta_{3}) q^{74} + \beta_1 q^{75} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{76} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{77} + \beta_1 q^{78} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{79} - q^{80} + (3 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} + 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{81} + (\beta_{6} + \beta_{5} - \beta_{2} + \beta_1) q^{82} + (\beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{83} + (\beta_{6} - \beta_{4} + \beta_1 - 1) q^{84} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2}) q^{85} + ( - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{86} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{87} + (\beta_{7} + \beta_{3} + \beta_{2} - \beta_1) q^{88} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{89} + ( - \beta_{7} - \beta_{6} - 1) q^{90} + \beta_{2} q^{91} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} + 3) q^{92} + \beta_1 q^{93} + (\beta_{8} + \beta_{7} + \beta_{4} - \beta_{2} - \beta_1 + 3) q^{94} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 1) q^{95} + \beta_1 q^{96} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 3) q^{97} + ( - \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{98} + (4 \beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_{3} - 6 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} + 3 q^{7} + 9 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 3 q^{6} + 3 q^{7} + 9 q^{8} + 14 q^{9} - 9 q^{10} + 6 q^{11} + 3 q^{12} + 9 q^{13} + 3 q^{14} - 3 q^{15} + 9 q^{16} + 3 q^{17} + 14 q^{18} + 6 q^{19} - 9 q^{20} - q^{21} + 6 q^{22} + 14 q^{23} + 3 q^{24} + 9 q^{25} + 9 q^{26} + 9 q^{27} + 3 q^{28} + 17 q^{29} - 3 q^{30} + 9 q^{31} + 9 q^{32} - 8 q^{33} + 3 q^{34} - 3 q^{35} + 14 q^{36} - 3 q^{37} + 6 q^{38} + 3 q^{39} - 9 q^{40} + 4 q^{41} - q^{42} + 5 q^{43} + 6 q^{44} - 14 q^{45} + 14 q^{46} + 25 q^{47} + 3 q^{48} + 8 q^{49} + 9 q^{50} + 15 q^{51} + 9 q^{52} + 18 q^{53} + 9 q^{54} - 6 q^{55} + 3 q^{56} + 13 q^{57} + 17 q^{58} - 6 q^{59} - 3 q^{60} + 26 q^{61} + 9 q^{62} + 8 q^{63} + 9 q^{64} - 9 q^{65} - 8 q^{66} - 2 q^{67} + 3 q^{68} + 8 q^{69} - 3 q^{70} + 18 q^{71} + 14 q^{72} - 5 q^{73} - 3 q^{74} + 3 q^{75} + 6 q^{76} + 19 q^{77} + 3 q^{78} + 18 q^{79} - 9 q^{80} + 41 q^{81} + 4 q^{82} + 13 q^{83} - q^{84} - 3 q^{85} + 5 q^{86} + 19 q^{87} + 6 q^{88} + 9 q^{89} - 14 q^{90} + 3 q^{91} + 14 q^{92} + 3 q^{93} + 25 q^{94} - 6 q^{95} + 3 q^{96} - 18 q^{97} + 8 q^{98} + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 16x^{7} + 48x^{6} + 66x^{5} - 202x^{4} - 75x^{3} + 210x^{2} + 68x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 125 \nu^{8} - 623 \nu^{7} + 3654 \nu^{6} + 11052 \nu^{5} - 31314 \nu^{4} - 52462 \nu^{3} + 78455 \nu^{2} + 59144 \nu - 10736 ) / 3732 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 377 \nu^{8} + 823 \nu^{7} + 6348 \nu^{6} - 12108 \nu^{5} - 27894 \nu^{4} + 40526 \nu^{3} + 23747 \nu^{2} - 9610 \nu + 10792 ) / 3732 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 202 \nu^{8} - 389 \nu^{7} - 3285 \nu^{6} + 5562 \nu^{5} + 13164 \nu^{4} - 17314 \nu^{3} - 7210 \nu^{2} + 791 \nu - 8282 ) / 1866 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 541 \nu^{8} - 1379 \nu^{7} - 8664 \nu^{6} + 21612 \nu^{5} + 34554 \nu^{4} - 87358 \nu^{3} - 28603 \nu^{2} + 79442 \nu + 16072 ) / 3732 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 99 \nu^{8} + 146 \nu^{7} + 1747 \nu^{6} - 1990 \nu^{5} - 8564 \nu^{4} + 5742 \nu^{3} + 11829 \nu^{2} - 731 \nu - 2472 ) / 622 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\nu^{8} - 146\nu^{7} - 1747\nu^{6} + 1990\nu^{5} + 8564\nu^{4} - 5742\nu^{3} - 11207\nu^{2} + 731\nu - 16 ) / 622 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 659 \nu^{8} - 1597 \nu^{7} - 10740 \nu^{6} + 23808 \nu^{5} + 45126 \nu^{4} - 83006 \nu^{3} - 45341 \nu^{2} + 36478 \nu + 16832 ) / 3732 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} + 13\beta_{7} + 14\beta_{6} + 4\beta_{3} - \beta_{2} - 2\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{8} - 13\beta_{7} + 18\beta_{6} + 4\beta_{5} + 15\beta_{4} + \beta_{3} - 16\beta_{2} + 74\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45\beta_{8} + 146\beta_{7} + 163\beta_{6} + \beta_{5} + 10\beta_{4} + 69\beta_{3} - 15\beta_{2} - 30\beta _1 + 246 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 149 \beta_{8} - 134 \beta_{7} + 245 \beta_{6} + 69 \beta_{5} + 196 \beta_{4} + 39 \beta_{3} - 201 \beta_{2} + 723 \beta _1 + 180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 551 \beta_{8} + 1577 \beta_{7} + 1836 \beta_{6} + 39 \beta_{5} + 222 \beta_{4} + 909 \beta_{3} - 211 \beta_{2} - 321 \beta _1 + 2434 \) 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
−3.17012
−2.15368
−0.959789
−0.522486
0.161390
1.68895
1.90335
2.63839
3.41400
1.00000 −3.17012 1.00000 −1.00000 −3.17012 −0.334308 1.00000 7.04967 −1.00000
1.2 1.00000 −2.15368 1.00000 −1.00000 −2.15368 1.26985 1.00000 1.63833 −1.00000
1.3 1.00000 −0.959789 1.00000 −1.00000 −0.959789 5.04132 1.00000 −2.07880 −1.00000
1.4 1.00000 −0.522486 1.00000 −1.00000 −0.522486 −4.13215 1.00000 −2.72701 −1.00000
1.5 1.00000 0.161390 1.00000 −1.00000 0.161390 0.164045 1.00000 −2.97395 −1.00000
1.6 1.00000 1.68895 1.00000 −1.00000 1.68895 2.51858 1.00000 −0.147452 −1.00000
1.7 1.00000 1.90335 1.00000 −1.00000 1.90335 −3.95405 1.00000 0.622734 −1.00000
1.8 1.00000 2.63839 1.00000 −1.00000 2.63839 2.17174 1.00000 3.96109 −1.00000
1.9 1.00000 3.41400 1.00000 −1.00000 3.41400 0.254978 1.00000 8.65540 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)
\(31\) \(-1\)

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 4030.2.a.q 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4030.2.a.q 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 3T_{3}^{8} - 16T_{3}^{7} + 48T_{3}^{6} + 66T_{3}^{5} - 202T_{3}^{4} - 75T_{3}^{3} + 210T_{3}^{2} + 68T_{3} - 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} - 16 T^{7} + 48 T^{6} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 3 T^{8} - 31 T^{7} + 94 T^{6} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{9} - 6 T^{8} - 31 T^{7} + \cdots + 1944 \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 3 T^{8} - 102 T^{7} + \cdots + 231444 \) Copy content Toggle raw display
$19$ \( T^{9} - 6 T^{8} - 67 T^{7} + \cdots - 4424 \) Copy content Toggle raw display
$23$ \( T^{9} - 14 T^{8} + 16 T^{7} + \cdots + 16794 \) Copy content Toggle raw display
$29$ \( T^{9} - 17 T^{8} + 38 T^{7} + \cdots - 52488 \) Copy content Toggle raw display
$31$ \( (T - 1)^{9} \) Copy content Toggle raw display
$37$ \( T^{9} + 3 T^{8} - 76 T^{7} + \cdots - 24074 \) Copy content Toggle raw display
$41$ \( T^{9} - 4 T^{8} - 97 T^{7} + \cdots - 26568 \) Copy content Toggle raw display
$43$ \( T^{9} - 5 T^{8} - 245 T^{7} + \cdots - 7574992 \) Copy content Toggle raw display
$47$ \( T^{9} - 25 T^{8} + 77 T^{7} + \cdots - 48492 \) Copy content Toggle raw display
$53$ \( T^{9} - 18 T^{8} - 3 T^{7} + \cdots + 41688 \) Copy content Toggle raw display
$59$ \( T^{9} + 6 T^{8} - 160 T^{7} + \cdots + 7776 \) Copy content Toggle raw display
$61$ \( T^{9} - 26 T^{8} + 195 T^{7} + \cdots + 6452 \) Copy content Toggle raw display
$67$ \( T^{9} + 2 T^{8} - 78 T^{7} + \cdots - 4016 \) Copy content Toggle raw display
$71$ \( T^{9} - 18 T^{8} + 15 T^{7} + \cdots - 2592 \) Copy content Toggle raw display
$73$ \( T^{9} + 5 T^{8} - 355 T^{7} + \cdots + 2032304 \) Copy content Toggle raw display
$79$ \( T^{9} - 18 T^{8} - 129 T^{7} + \cdots - 3994384 \) Copy content Toggle raw display
$83$ \( T^{9} - 13 T^{8} + \cdots - 300861216 \) Copy content Toggle raw display
$89$ \( T^{9} - 9 T^{8} + \cdots - 1760749326 \) Copy content Toggle raw display
$97$ \( T^{9} + 18 T^{8} - 52 T^{7} + \cdots - 507898 \) Copy content Toggle raw display
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