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Properties

Label 8030.2.a.bc
Level $8030$
Weight $2$
Character orbit 8030.a
Self dual yes
Analytic conductor $64.120$
Analytic rank $1$
Dimension $11$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, 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("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.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: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 10 x^{9} + 71 x^{8} + 28 x^{7} - 360 x^{6} - 60 x^{5} + 788 x^{4} + 309 x^{3} + \cdots - 95 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5 x^{10} - 10 x^{9} + 71 x^{8} + 28 x^{7} - 360 x^{6} - 60 x^{5} + 788 x^{4} + 309 x^{3} + \cdots - 95 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{10} + 17 \nu^{9} + 33 \nu^{8} - 200 \nu^{7} - 25 \nu^{6} + 730 \nu^{5} - 121 \nu^{4} + \cdots + 573 ) / 79 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9 \nu^{10} + 58 \nu^{9} + 15 \nu^{8} - 687 \nu^{7} + 635 \nu^{6} + 2630 \nu^{5} - 2820 \nu^{4} + \cdots + 993 ) / 79 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 35 \nu^{10} - 129 \nu^{9} - 427 \nu^{8} + 1671 \nu^{7} + 1779 \nu^{6} - 6980 \nu^{5} - 3385 \nu^{4} + \cdots - 649 ) / 158 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19 \nu^{10} - 140 \nu^{9} + 21 \nu^{8} + 1740 \nu^{7} - 2113 \nu^{6} - 7299 \nu^{5} + 9640 \nu^{4} + \cdots - 1438 ) / 158 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39 \nu^{10} - 225 \nu^{9} - 223 \nu^{8} + 2977 \nu^{7} - 1119 \nu^{6} - 13714 \nu^{5} + 7717 \nu^{4} + \cdots - 6199 ) / 158 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41 \nu^{10} - 273 \nu^{9} - 121 \nu^{8} + 3551 \nu^{7} - 2489 \nu^{6} - 15896 \nu^{5} + 12557 \nu^{4} + \cdots - 6209 ) / 158 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 44 \nu^{10} - 266 \nu^{9} - 205 \nu^{8} + 3464 \nu^{7} - 1779 \nu^{6} - 15535 \nu^{5} + 10337 \nu^{4} + \cdots - 6066 ) / 79 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 44 \nu^{10} - 266 \nu^{9} - 205 \nu^{8} + 3464 \nu^{7} - 1779 \nu^{6} - 15535 \nu^{5} + 10337 \nu^{4} + \cdots - 6303 ) / 79 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{3} + 7\beta_{2} + 9\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{10} - 10\beta_{9} - \beta_{8} - 3\beta_{7} + \beta_{4} + 3\beta_{2} + 43\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{10} - 13 \beta_{9} - 15 \beta_{8} - 18 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 123 \beta_{10} - 85 \beta_{9} - 23 \beta_{8} - 51 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 16 \beta_{4} + \cdots + 304 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 337 \beta_{10} - 138 \beta_{9} - 167 \beta_{8} - 216 \beta_{7} - 26 \beta_{6} - 17 \beta_{5} + \cdots + 1349 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1195 \beta_{10} - 708 \beta_{9} - 331 \beta_{8} - 609 \beta_{7} - 32 \beta_{6} - 40 \beta_{5} + \cdots + 2808 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3418 \beta_{10} - 1378 \beta_{9} - 1693 \beta_{8} - 2225 \beta_{7} - 238 \beta_{6} - 204 \beta_{5} + \cdots + 10851 \) 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
3.03286
2.97446
2.54167
2.29138
1.62999
−0.328152
−0.722992
−0.858907
−1.10547
−2.03519
−2.41966
−1.00000 −3.03286 1.00000 1.00000 3.03286 −1.46211 −1.00000 6.19822 −1.00000
1.2 −1.00000 −2.97446 1.00000 1.00000 2.97446 −1.89665 −1.00000 5.84742 −1.00000
1.3 −1.00000 −2.54167 1.00000 1.00000 2.54167 1.74685 −1.00000 3.46009 −1.00000
1.4 −1.00000 −2.29138 1.00000 1.00000 2.29138 3.99723 −1.00000 2.25044 −1.00000
1.5 −1.00000 −1.62999 1.00000 1.00000 1.62999 −2.80536 −1.00000 −0.343117 −1.00000
1.6 −1.00000 0.328152 1.00000 1.00000 −0.328152 −2.28236 −1.00000 −2.89232 −1.00000
1.7 −1.00000 0.722992 1.00000 1.00000 −0.722992 3.72469 −1.00000 −2.47728 −1.00000
1.8 −1.00000 0.858907 1.00000 1.00000 −0.858907 0.0500818 −1.00000 −2.26228 −1.00000
1.9 −1.00000 1.10547 1.00000 1.00000 −1.10547 −0.823192 −1.00000 −1.77794 −1.00000
1.10 −1.00000 2.03519 1.00000 1.00000 −2.03519 −2.33086 −1.00000 1.14199 −1.00000
1.11 −1.00000 2.41966 1.00000 1.00000 −2.41966 1.08168 −1.00000 2.85477 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(73\) \(1\)

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 8030.2.a.bc ✓ 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.bc ✓ 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{11} + 5 T_{3}^{10} - 10 T_{3}^{9} - 71 T_{3}^{8} + 28 T_{3}^{7} + 360 T_{3}^{6} - 60 T_{3}^{5} + \cdots + 95 \) Copy content Toggle raw display
\( T_{7}^{11} + T_{7}^{10} - 29 T_{7}^{9} - 51 T_{7}^{8} + 250 T_{7}^{7} + 626 T_{7}^{6} - 433 T_{7}^{5} + \cdots - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{11} \) Copy content Toggle raw display
$3$ \( T^{11} + 5 T^{10} + \cdots + 95 \) Copy content Toggle raw display
$5$ \( (T - 1)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + T^{10} + \cdots - 48 \) Copy content Toggle raw display
$11$ \( (T - 1)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} + 8 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$17$ \( T^{11} + 15 T^{10} + \cdots - 2028 \) Copy content Toggle raw display
$19$ \( T^{11} - 5 T^{10} + \cdots - 119092 \) Copy content Toggle raw display
$23$ \( T^{11} + 24 T^{10} + \cdots + 323152 \) Copy content Toggle raw display
$29$ \( T^{11} - 10 T^{10} + \cdots + 11044004 \) Copy content Toggle raw display
$31$ \( T^{11} - 7 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{11} + 24 T^{10} + \cdots + 20140 \) Copy content Toggle raw display
$41$ \( T^{11} - 10 T^{10} + \cdots - 89056 \) Copy content Toggle raw display
$43$ \( T^{11} + 14 T^{10} + \cdots + 706080 \) Copy content Toggle raw display
$47$ \( T^{11} + 8 T^{10} + \cdots - 5111552 \) Copy content Toggle raw display
$53$ \( T^{11} + 30 T^{10} + \cdots + 3414000 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 152225280 \) Copy content Toggle raw display
$61$ \( T^{11} + 26 T^{10} + \cdots - 7382700 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 10687459588 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 907247069 \) Copy content Toggle raw display
$73$ \( (T + 1)^{11} \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 1146154880 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 2470810982 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots + 278516605 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 3673547840 \) Copy content Toggle raw display
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