Properties

Label 4030.2.a.r
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: \( x^{9} - 4x^{8} - 8x^{7} + 39x^{6} + 13x^{5} - 106x^{4} + 9x^{3} + 74x^{2} - 3x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 8x^{7} + 39x^{6} + 13x^{5} - 106x^{4} + 9x^{3} + 74x^{2} - 3x - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{8} + 45\nu^{7} - 39\nu^{6} - 438\nu^{5} + 665\nu^{4} + 1147\nu^{3} - 1692\nu^{2} - 524\nu + 445 ) / 146 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{8} - 44\nu^{7} - 103\nu^{6} + 365\nu^{5} + 96\nu^{4} - 617\nu^{3} + 647\nu^{2} - 229\nu - 500 ) / 146 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{8} - 35\nu^{7} - 140\nu^{6} + 365\nu^{5} + 521\nu^{4} - 1030\nu^{3} - 655\nu^{2} + 659\nu + 173 ) / 146 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{8} - 80\nu^{7} - 101\nu^{6} + 803\nu^{5} - 144\nu^{4} - 2177\nu^{3} + 1037\nu^{2} + 745\nu - 126 ) / 146 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\nu^{8} - 80\nu^{7} - 101\nu^{6} + 803\nu^{5} - 144\nu^{4} - 2323\nu^{3} + 1183\nu^{2} + 1767\nu - 418 ) / 146 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\nu^{8} - 149\nu^{7} - 523\nu^{6} + 1314\nu^{5} + 1951\nu^{4} - 2685\nu^{3} - 2778\nu^{2} + 580\nu + 749 ) / 292 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -95\nu^{8} + 417\nu^{7} + 573\nu^{6} - 3796\nu^{5} + 225\nu^{4} + 9091\nu^{3} - 3240\nu^{2} - 4554\nu + 279 ) / 292 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 3\beta _1 + 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{6} - 5\beta_{5} + 8\beta_{4} - 8\beta_{2} + 3\beta _1 + 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - \beta_{7} - 2\beta_{6} - 3\beta_{5} + 5\beta_{4} - \beta_{3} - 3\beta_{2} + 9\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{8} - 12\beta_{7} - 33\beta_{6} - 35\beta_{5} + 77\beta_{4} - 3\beta_{3} - 56\beta_{2} + 45\beta _1 + 116 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 48 \beta_{8} - 66 \beta_{7} - 93 \beta_{6} - 86 \beta_{5} + 191 \beta_{4} - 24 \beta_{3} - 80 \beta_{2} + 240 \beta _1 + 530 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 123 \beta_{8} - 237 \beta_{7} - 363 \beta_{6} - 298 \beta_{5} + 790 \beta_{4} - 18 \beta_{3} - 391 \beta_{2} + 531 \beta _1 + 1216 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 202 \beta_{8} - 322 \beta_{7} - 379 \beta_{6} - 282 \beta_{5} + 754 \beta_{4} - 37 \beta_{3} - 222 \beta_{2} + 743 \beta _1 + 1576 \) 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
−0.316108
−2.32727
1.72735
0.351548
3.27701
−1.78896
1.19574
2.70656
−0.825879
1.00000 −2.83087 1.00000 1.00000 −2.83087 3.35447 1.00000 5.01385 1.00000
1.2 1.00000 −2.15121 1.00000 1.00000 −2.15121 −1.07964 1.00000 1.62769 1.00000
1.3 1.00000 −1.50823 1.00000 1.00000 −1.50823 −3.09594 1.00000 −0.725248 1.00000
1.4 1.00000 −0.748457 1.00000 1.00000 −0.748457 4.59004 1.00000 −2.43981 1.00000
1.5 1.00000 0.574493 1.00000 1.00000 0.574493 2.65097 1.00000 −2.66996 1.00000
1.6 1.00000 1.40140 1.00000 1.00000 1.40140 −1.57433 1.00000 −1.03607 1.00000
1.7 1.00000 2.25079 1.00000 1.00000 2.25079 2.47698 1.00000 2.06607 1.00000
1.8 1.00000 2.79279 1.00000 1.00000 2.79279 3.17111 1.00000 4.79966 1.00000
1.9 1.00000 3.21929 1.00000 1.00000 3.21929 −1.49366 1.00000 7.36381 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.r 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4030.2.a.r 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 3T_{3}^{8} - 16T_{3}^{7} + 46T_{3}^{6} + 82T_{3}^{5} - 216T_{3}^{4} - 155T_{3}^{3} + 334T_{3}^{2} + 84T_{3} - 112 \) 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
$3$ \( T^{9} - 3 T^{8} - 16 T^{7} + 46 T^{6} + \cdots - 112 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 9 T^{8} + 5 T^{7} + 144 T^{6} + \cdots - 2520 \) Copy content Toggle raw display
$11$ \( T^{9} - 10 T^{8} + 5 T^{7} + 158 T^{6} + \cdots + 560 \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} - 50 T^{7} + T^{6} + \cdots - 954 \) Copy content Toggle raw display
$19$ \( T^{9} - 10 T^{8} - 29 T^{7} + \cdots - 6308 \) Copy content Toggle raw display
$23$ \( T^{9} - 8 T^{8} - 56 T^{7} + \cdots + 36008 \) Copy content Toggle raw display
$29$ \( T^{9} - 9 T^{8} - 50 T^{7} + \cdots + 19350 \) Copy content Toggle raw display
$31$ \( (T + 1)^{9} \) Copy content Toggle raw display
$37$ \( T^{9} + 3 T^{8} - 78 T^{7} + \cdots - 38102 \) Copy content Toggle raw display
$41$ \( T^{9} - 51 T^{7} + 17 T^{6} + 424 T^{5} + \cdots - 40 \) Copy content Toggle raw display
$43$ \( T^{9} - 7 T^{8} - 105 T^{7} + \cdots + 715568 \) Copy content Toggle raw display
$47$ \( T^{9} - 7 T^{8} - 153 T^{7} + \cdots + 166160 \) Copy content Toggle raw display
$53$ \( T^{9} - 8 T^{8} - 175 T^{7} + \cdots - 27640 \) Copy content Toggle raw display
$59$ \( T^{9} - 2 T^{8} - 316 T^{7} + \cdots + 8513900 \) Copy content Toggle raw display
$61$ \( T^{9} + 2 T^{8} - 221 T^{7} + \cdots + 42322 \) Copy content Toggle raw display
$67$ \( T^{9} - 18 T^{8} - 156 T^{7} + \cdots - 10707728 \) Copy content Toggle raw display
$71$ \( T^{9} - 14 T^{8} - 389 T^{7} + \cdots + 43268384 \) Copy content Toggle raw display
$73$ \( T^{9} - T^{8} - 269 T^{7} + \cdots + 5387144 \) Copy content Toggle raw display
$79$ \( T^{9} - 6 T^{8} - 227 T^{7} + \cdots - 152000 \) Copy content Toggle raw display
$83$ \( T^{9} - 7 T^{8} - 431 T^{7} + \cdots + 96523092 \) Copy content Toggle raw display
$89$ \( T^{9} + 19 T^{8} - 66 T^{7} + \cdots - 790346 \) Copy content Toggle raw display
$97$ \( T^{9} + 6 T^{8} - 434 T^{7} + \cdots - 10716566 \) Copy content Toggle raw display
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