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Properties

Label 1300.2.bb.g
Level $1300$
Weight $2$
Character orbit 1300.bb
Analytic conductor $10.381$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(549,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bb (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 19 x^{18} + 238 x^{16} - 1699 x^{14} + 8746 x^{12} - 27082 x^{10} + 60616 x^{8} - 82792 x^{6} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{15} + \beta_{9}) q^{7} + (\beta_{10} + \beta_{4} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{15} + \beta_{9}) q^{7} + (\beta_{10} + \beta_{4} + \beta_{2} + 1) q^{9} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{11}+ \cdots + ( - \beta_{11} - 3 \beta_{8} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 8 q^{9} - 14 q^{11} + 8 q^{19} + 16 q^{21} + 6 q^{29} + 32 q^{31} - 28 q^{39} + 20 q^{41} + 22 q^{49} + 16 q^{51} - 12 q^{59} - 18 q^{61} - 48 q^{69} + 10 q^{71} + 22 q^{81} - 44 q^{89} - 14 q^{91} - 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 19 x^{18} + 238 x^{16} - 1699 x^{14} + 8746 x^{12} - 27082 x^{10} + 60616 x^{8} - 82792 x^{6} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4588309165 \nu^{18} + 81403211798 \nu^{16} - 991263888570 \nu^{14} + \cdots - 550174301003214 ) / 153928509753411 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22395361106 \nu^{18} - 820925317053 \nu^{16} + 12067793582290 \nu^{14} + \cdots - 32\!\cdots\!96 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29967872890 \nu^{18} - 555624657415 \nu^{16} + 6888144112426 \nu^{14} + \cdots - 650414083119138 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34853684523 \nu^{18} - 979288358159 \nu^{16} + 13996213364080 \nu^{14} + \cdots + 124603098806367 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41864801864 \nu^{18} + 619582430608 \nu^{16} - 6239472153402 \nu^{14} + \cdots - 18\!\cdots\!04 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 44943573871 \nu^{18} + 980595088301 \nu^{16} - 12745555504185 \nu^{14} + \cdots + 428601940086132 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 64108 \nu^{18} - 986777 \nu^{16} + 10966845 \nu^{14} - 56668807 \nu^{12} + 206132693 \nu^{10} + \cdots + 633310731 ) / 602218719 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 78223815665 \nu^{19} + 2333170830878 \nu^{17} - 34252631522330 \nu^{15} + \cdots + 10\!\cdots\!10 \nu ) / 41\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 106106564065 \nu^{18} + 1978288994266 \nu^{16} - 24578784783994 \nu^{14} + \cdots + 24\!\cdots\!62 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 309243842772 \nu^{18} + 5448597495772 \nu^{16} - 65582805694403 \nu^{14} + \cdots - 338453802428976 ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29967872890 \nu^{19} - 555624657415 \nu^{17} + 6888144112426 \nu^{15} + \cdots - 650414083119138 \nu ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 65325703412 \nu^{19} - 1196244790957 \nu^{17} + 14566922323755 \nu^{15} + \cdots + 611937886751262 \nu ) / 461785529260233 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 731579482450 \nu^{19} - 13754455172164 \nu^{17} + 174497237687371 \nu^{15} + \cdots + 77\!\cdots\!38 \nu ) / 41\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 755321431064 \nu^{19} - 16142690922863 \nu^{17} + 212695607714555 \nu^{15} + \cdots - 37\!\cdots\!87 \nu ) / 41\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1029569601065 \nu^{19} - 18095298818045 \nu^{17} + 217831988351630 \nu^{15} + \cdots - 37\!\cdots\!29 \nu ) / 41\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1470638839991 \nu^{19} - 25417609031165 \nu^{17} + 306997287052430 \nu^{15} + \cdots + 11\!\cdots\!83 \nu ) / 41\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18577864069 \nu^{19} - 338635173832 \nu^{17} + 4123631141880 \nu^{15} + \cdots + 265636265042807 \nu ) / 51309503251137 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1671432885658 \nu^{19} - 31663707320890 \nu^{17} + 392581936487461 \nu^{15} + \cdots - 42\!\cdots\!79 \nu ) / 41\!\cdots\!97 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 4\beta_{4} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{16} - \beta_{15} + 6\beta_{12} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + 9\beta_{10} - \beta_{8} + \beta_{6} - \beta_{5} + 25\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - 9\beta_{17} + \beta_{16} - 11\beta_{15} + 10\beta_{14} + 41\beta_{12} - 6\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} - 17\beta_{8} + 17\beta_{7} - 2\beta_{6} - 10\beta_{5} + 12\beta_{3} - 71\beta_{2} - 175 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -69\beta_{19} + 102\beta_{18} - 81\beta_{17} + 81\beta_{16} + 69\beta_{14} - 90\beta_{13} - 295\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81 \beta_{11} - 547 \beta_{10} + 201 \beta_{7} - 114 \beta_{6} + 33 \beta_{5} - 1273 \beta_{4} + \cdots - 1273 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 628 \beta_{19} + 895 \beta_{18} - 114 \beta_{17} + 514 \beta_{16} + 895 \beta_{15} + \cdots - 2172 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1009\beta_{11} - 4209\beta_{10} + 2047\beta_{8} - 628\beta_{6} + 1009\beta_{5} - 9430\beta_{4} - 628\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1009 \beta_{19} + 3828 \beta_{17} - 1009 \beta_{16} + 7646 \beta_{15} - 4837 \beta_{14} + \cdots + 9825 \beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3818 \beta_{11} + 19271 \beta_{8} - 19271 \beta_{7} + 3818 \beta_{6} + 4837 \beta_{5} - 8655 \beta_{3} + \cdots + 70690 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 28704 \beta_{19} - 64266 \beta_{18} + 37359 \beta_{17} - 37359 \beta_{16} - 28704 \beta_{14} + \cdots + 122242 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 37359 \beta_{11} + 252571 \beta_{10} - 173094 \beta_{7} + 72921 \beta_{6} - 35562 \beta_{5} + \cdots + 534982 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 289930 \beta_{19} - 534148 \beta_{18} + 72921 \beta_{17} - 217009 \beta_{16} - 534148 \beta_{15} + \cdots + 929844 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 607069 \beta_{11} + 1970931 \beta_{10} - 1508101 \beta_{8} + 289930 \beta_{6} + \cdots + 289930 \beta_{3} \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 607069 \beta_{19} - 1653792 \beta_{17} + 607069 \beta_{16} - 4403240 \beta_{15} + \cdots - 6929403 \beta_{9} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2749448 \beta_{11} - 12867953 \beta_{8} + 12867953 \beta_{7} - 2749448 \beta_{6} - 2260861 \beta_{5} + \cdots - 31376290 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 12695535 \beta_{19} + 36072693 \beta_{18} - 17705844 \beta_{17} + 17705844 \beta_{16} + \cdots - 54995086 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
−2.44987 + 1.41443i
−2.20779 + 1.27467i
−1.29317 + 0.746615i
−1.19413 + 0.689434i
−0.524909 + 0.303056i
0.524909 − 0.303056i
1.19413 − 0.689434i
1.29317 − 0.746615i
2.20779 − 1.27467i
2.44987 − 1.41443i
−2.44987 − 1.41443i
−2.20779 − 1.27467i
−1.29317 − 0.746615i
−1.19413 − 0.689434i
−0.524909 − 0.303056i
0.524909 + 0.303056i
1.19413 + 0.689434i
1.29317 + 0.746615i
2.20779 + 1.27467i
2.44987 + 1.41443i
0 −2.44987 + 1.41443i 0 0 0 −4.33228 − 2.50124i 0 2.50124 − 4.33228i 0
549.2 0 −2.20779 + 1.27467i 0 0 0 3.03034 + 1.74957i 0 1.74957 − 3.03034i 0
549.3 0 −1.29317 + 0.746615i 0 0 0 0.667069 + 0.385132i 0 −0.385132 + 0.667069i 0
549.4 0 −1.19413 + 0.689434i 0 0 0 −0.951523 − 0.549362i 0 −0.549362 + 0.951523i 0
549.5 0 −0.524909 + 0.303056i 0 0 0 2.27992 + 1.31631i 0 −1.31631 + 2.27992i 0
549.6 0 0.524909 − 0.303056i 0 0 0 −2.27992 − 1.31631i 0 −1.31631 + 2.27992i 0
549.7 0 1.19413 − 0.689434i 0 0 0 0.951523 + 0.549362i 0 −0.549362 + 0.951523i 0
549.8 0 1.29317 − 0.746615i 0 0 0 −0.667069 − 0.385132i 0 −0.385132 + 0.667069i 0
549.9 0 2.20779 − 1.27467i 0 0 0 −3.03034 − 1.74957i 0 1.74957 − 3.03034i 0
549.10 0 2.44987 − 1.41443i 0 0 0 4.33228 + 2.50124i 0 2.50124 − 4.33228i 0
1049.1 0 −2.44987 − 1.41443i 0 0 0 −4.33228 + 2.50124i 0 2.50124 + 4.33228i 0
1049.2 0 −2.20779 − 1.27467i 0 0 0 3.03034 − 1.74957i 0 1.74957 + 3.03034i 0
1049.3 0 −1.29317 − 0.746615i 0 0 0 0.667069 − 0.385132i 0 −0.385132 − 0.667069i 0
1049.4 0 −1.19413 − 0.689434i 0 0 0 −0.951523 + 0.549362i 0 −0.549362 − 0.951523i 0
1049.5 0 −0.524909 − 0.303056i 0 0 0 2.27992 − 1.31631i 0 −1.31631 − 2.27992i 0
1049.6 0 0.524909 + 0.303056i 0 0 0 −2.27992 + 1.31631i 0 −1.31631 − 2.27992i 0
1049.7 0 1.19413 + 0.689434i 0 0 0 0.951523 − 0.549362i 0 −0.549362 − 0.951523i 0
1049.8 0 1.29317 + 0.746615i 0 0 0 −0.667069 + 0.385132i 0 −0.385132 − 0.667069i 0
1049.9 0 2.20779 + 1.27467i 0 0 0 −3.03034 + 1.74957i 0 1.74957 + 3.03034i 0
1049.10 0 2.44987 + 1.41443i 0 0 0 4.33228 − 2.50124i 0 2.50124 + 4.33228i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 549.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.bb.g 20
5.b even 2 1 inner 1300.2.bb.g 20
5.c odd 4 1 1300.2.i.g ✓ 10
5.c odd 4 1 1300.2.i.h yes 10
13.c even 3 1 inner 1300.2.bb.g 20
65.n even 6 1 inner 1300.2.bb.g 20
65.q odd 12 1 1300.2.i.g ✓ 10
65.q odd 12 1 1300.2.i.h yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1300.2.i.g ✓ 10 5.c odd 4 1
1300.2.i.g ✓ 10 65.q odd 12 1
1300.2.i.h yes 10 5.c odd 4 1
1300.2.i.h yes 10 65.q odd 12 1
1300.2.bb.g 20 1.a even 1 1 trivial
1300.2.bb.g 20 5.b even 2 1 inner
1300.2.bb.g 20 13.c even 3 1 inner
1300.2.bb.g 20 65.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1300, [\chi])\):

\( T_{3}^{20} - 19 T_{3}^{18} + 238 T_{3}^{16} - 1699 T_{3}^{14} + 8746 T_{3}^{12} - 27082 T_{3}^{10} + \cdots + 6561 \) Copy content Toggle raw display
\( T_{7}^{20} - 46 T_{7}^{18} + 1471 T_{7}^{16} - 23326 T_{7}^{14} + 265885 T_{7}^{12} - 1658485 T_{7}^{10} + \cdots + 2313441 \) Copy content Toggle raw display
\( T_{19}^{10} - 4 T_{19}^{9} + 64 T_{19}^{8} - 110 T_{19}^{7} + 2348 T_{19}^{6} - 4211 T_{19}^{5} + \cdots + 2082249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 19 T^{18} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 46 T^{18} + \cdots + 2313441 \) Copy content Toggle raw display
$11$ \( (T^{10} + 7 T^{9} + \cdots + 164025)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 187388721 \) Copy content Toggle raw display
$19$ \( (T^{10} - 4 T^{9} + \cdots + 2082249)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 2593262550321 \) Copy content Toggle raw display
$29$ \( (T^{10} - 3 T^{9} + \cdots + 5349969)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots + 1131)^{4} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 424125260001 \) Copy content Toggle raw display
$41$ \( (T^{10} - 10 T^{9} + \cdots + 13024881)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 34695975728241 \) Copy content Toggle raw display
$47$ \( (T^{10} + 196 T^{8} + \cdots + 9979281)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 442 T^{8} + \cdots + 3995124849)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + 6 T^{9} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 9 T^{9} + \cdots + 725494225)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 829710108386721 \) Copy content Toggle raw display
$71$ \( (T^{10} - 5 T^{9} + \cdots + 34963569)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 446 T^{8} + \cdots + 163302841)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 181 T^{3} + \cdots + 67)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + 171 T^{8} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 22 T^{9} + \cdots + 187388721)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} - 281 T^{18} + \cdots + 923521 \) Copy content Toggle raw display
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