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Properties

Label 1004.2.a.b
Level $1004$
Weight $2$
Character orbit 1004.a
Self dual yes
Analytic conductor $8.017$
Analytic rank $0$
Dimension $14$
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] = [1004,2,Mod(1,1004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1004 = 2^{2} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1004.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: \(8.01698036294\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} + ( - \beta_{9} + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} + ( - \beta_{9} + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{8} + 1) q^{11} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{13}+ \cdots + (\beta_{13} - \beta_{12} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{3} - 2 q^{5} + 8 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{3} - 2 q^{5} + 8 q^{7} + 21 q^{9} + 9 q^{11} - q^{13} + 14 q^{15} + 27 q^{19} - 3 q^{21} + 13 q^{23} + 26 q^{25} + 15 q^{27} + 25 q^{31} + 16 q^{33} + 21 q^{35} - q^{37} + 33 q^{39} + 10 q^{41} + 35 q^{43} - 4 q^{45} + 6 q^{47} + 36 q^{49} + 48 q^{51} - q^{53} + 41 q^{55} + 14 q^{57} + 30 q^{59} + 3 q^{61} + 31 q^{63} + 7 q^{65} + 22 q^{67} - 17 q^{69} + 6 q^{71} + 5 q^{73} + 4 q^{75} - 14 q^{77} + 56 q^{79} + 26 q^{81} - 28 q^{83} - 23 q^{85} + 11 q^{87} - 24 q^{89} + 38 q^{91} - 55 q^{93} - 4 q^{95} + 6 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 458652 \nu^{13} + 10207449 \nu^{12} + 8865462 \nu^{11} - 298701457 \nu^{10} + \cdots + 1861877352 ) / 762886946 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 458652 \nu^{13} - 10207449 \nu^{12} - 8865462 \nu^{11} + 298701457 \nu^{10} + 45787181 \nu^{9} + \cdots - 4913425136 ) / 762886946 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8519764 \nu^{13} + 100995904 \nu^{12} - 475048121 \nu^{11} - 2501409580 \nu^{10} + \cdots + 9359448304 ) / 4577321676 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4195687 \nu^{13} + 59312328 \nu^{12} - 28844005 \nu^{11} - 1424333823 \nu^{10} + \cdots + 2862099808 ) / 1525773892 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5336880 \nu^{13} - 865584 \nu^{12} + 175902437 \nu^{11} + 7885974 \nu^{10} + \cdots + 1054324184 ) / 1525773892 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 34205344 \nu^{13} - 272353484 \nu^{12} - 487766789 \nu^{11} + 6852436190 \nu^{10} + \cdots - 18482722832 ) / 4577321676 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5967155 \nu^{13} - 22669783 \nu^{12} - 134567812 \nu^{11} + 542735784 \nu^{10} + \cdots - 2832505686 ) / 762886946 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13587249 \nu^{13} + 82033945 \nu^{12} + 220632596 \nu^{11} - 2025748304 \nu^{10} + \cdots + 6261708580 ) / 1525773892 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42935729 \nu^{13} - 93792808 \nu^{12} - 1159294432 \nu^{11} + 2336791369 \nu^{10} + \cdots - 22751872012 ) / 4577321676 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20741705 \nu^{13} - 78046665 \nu^{12} - 457715319 \nu^{11} + 1856372484 \nu^{10} + \cdots - 46956196 ) / 1525773892 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 118858355 \nu^{13} - 427006354 \nu^{12} - 2693628022 \nu^{11} + 10377729199 \nu^{10} + \cdots - 7920902056 ) / 4577321676 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40998083 \nu^{13} - 152031178 \nu^{12} - 851304577 \nu^{11} + 3546579085 \nu^{10} + \cdots - 3159888728 ) / 1525773892 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 4 \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 15 \beta_{11} + 13 \beta_{10} + 17 \beta_{9} + 13 \beta_{7} - 19 \beta_{6} - 18 \beta_{5} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + 13 \beta_{12} - \beta_{11} - 65 \beta_{10} + 31 \beta_{9} + 22 \beta_{8} + 16 \beta_{7} + \cdots + 227 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{13} + 6 \beta_{12} - 186 \beta_{11} + 146 \beta_{10} + 220 \beta_{9} + 8 \beta_{8} + 135 \beta_{7} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7 \beta_{13} + 158 \beta_{12} - 7 \beta_{11} - 849 \beta_{10} + 365 \beta_{9} + 326 \beta_{8} + \cdots + 2187 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 119 \beta_{13} + 143 \beta_{12} - 2190 \beta_{11} + 1644 \beta_{10} + 2590 \beta_{9} + 171 \beta_{8} + \cdots - 913 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 50 \beta_{13} + 1940 \beta_{12} + 64 \beta_{11} - 10374 \beta_{10} + 3908 \beta_{9} + 4210 \beta_{8} + \cdots + 22786 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 920 \beta_{13} + 2294 \beta_{12} - 25342 \beta_{11} + 18962 \beta_{10} + 29328 \beta_{9} + 2457 \beta_{8} + \cdots - 16315 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2101 \beta_{13} + 23661 \beta_{12} + 2860 \beta_{11} - 123400 \beta_{10} + 40080 \beta_{9} + \cdots + 247897 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6170 \beta_{13} + 31190 \beta_{12} - 291368 \beta_{11} + 223100 \beta_{10} + 326994 \beta_{9} + \cdots - 238643 \) 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
−3.43087
−2.20631
−1.93078
−1.64615
−1.48367
−0.788858
−0.194346
0.224635
1.46572
1.90340
2.37559
2.53364
2.87232
3.30568
0 −3.43087 0 −0.723735 0 3.63900 0 8.77086 0
1.2 0 −2.20631 0 −0.868402 0 −2.87616 0 1.86782 0
1.3 0 −1.93078 0 3.14832 0 3.83969 0 0.727926 0
1.4 0 −1.64615 0 −4.00715 0 2.94760 0 −0.290201 0
1.5 0 −1.48367 0 −3.74411 0 −4.87233 0 −0.798715 0
1.6 0 −0.788858 0 0.735420 0 1.11520 0 −2.37770 0
1.7 0 −0.194346 0 2.87489 0 0.213356 0 −2.96223 0
1.8 0 0.224635 0 −2.42875 0 −2.24830 0 −2.94954 0
1.9 0 1.46572 0 0.817924 0 4.79612 0 −0.851657 0
1.10 0 1.90340 0 3.78264 0 −0.441155 0 0.622940 0
1.11 0 2.37559 0 −4.08952 0 3.67423 0 2.64345 0
1.12 0 2.53364 0 2.70770 0 1.66309 0 3.41934 0
1.13 0 2.87232 0 −0.801100 0 −4.32848 0 5.25020 0
1.14 0 3.30568 0 0.595872 0 0.878141 0 7.92752 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(1\)

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 1004.2.a.b ✓ 14
3.b odd 2 1 9036.2.a.m 14
4.b odd 2 1 4016.2.a.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1004.2.a.b ✓ 14 1.a even 1 1 trivial
4016.2.a.j 14 4.b odd 2 1
9036.2.a.m 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 3 T_{3}^{13} - 27 T_{3}^{12} + 79 T_{3}^{11} + 274 T_{3}^{10} - 747 T_{3}^{9} - 1422 T_{3}^{8} + \cdots - 196 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1004))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} + \cdots - 196 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots - 2493 \) Copy content Toggle raw display
$7$ \( T^{14} - 8 T^{13} + \cdots - 15173 \) Copy content Toggle raw display
$11$ \( T^{14} - 9 T^{13} + \cdots - 82944 \) Copy content Toggle raw display
$13$ \( T^{14} + T^{13} + \cdots + 152194 \) Copy content Toggle raw display
$17$ \( T^{14} - 134 T^{12} + \cdots + 2192193 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 292774912 \) Copy content Toggle raw display
$23$ \( T^{14} - 13 T^{13} + \cdots - 46617849 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 309728256 \) Copy content Toggle raw display
$31$ \( T^{14} - 25 T^{13} + \cdots - 2981 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 3935804416 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 2984712723 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 254630912 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 197148672 \) Copy content Toggle raw display
$53$ \( T^{14} + T^{13} + \cdots - 83321856 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 112057344 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 7827757056 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 65870265429 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 2571959282688 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15414402577 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 3667218167 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 190266048 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 8007145007814 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 12261356544 \) Copy content Toggle raw display
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