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Properties

Label 1300.2.i.h
Level $1300$
Weight $2$
Character orbit 1300.i
Analytic conductor $10.381$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1300,2,Mod(601,1300)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.601");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.i (of order \(3\), degree \(2\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 10x^{8} - 5x^{7} + 74x^{6} - 44x^{5} + 166x^{4} - 64x^{3} + 259x^{2} - 126x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{9} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 2) q^{11}+ \cdots + ( - \beta_{9} + \beta_{8} - \beta_{6} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 4 q^{7} - 4 q^{9} - 7 q^{11} + 6 q^{13} + 8 q^{17} - 4 q^{19} + 8 q^{21} + q^{23} - 2 q^{27} - 3 q^{29} + 16 q^{31} + 10 q^{33} - q^{37} + 14 q^{39} + 10 q^{41} + 16 q^{43} - 8 q^{47} - 11 q^{49} + 8 q^{51} - 32 q^{53} + 6 q^{59} - 9 q^{61} + 46 q^{63} + 2 q^{67} + 24 q^{69} + 5 q^{71} + 16 q^{73} - 56 q^{77} + 11 q^{81} - 30 q^{83} - 19 q^{87} + 22 q^{89} - 7 q^{91} - 3 q^{93} - 21 q^{97} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 10x^{8} - 5x^{7} + 74x^{6} - 44x^{5} + 166x^{4} - 64x^{3} + 259x^{2} - 126x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1119 \nu^{9} + 9464 \nu^{8} - 33800 \nu^{7} + 86345 \nu^{6} - 224432 \nu^{5} + 585416 \nu^{4} + \cdots + 1449144 ) / 2300421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8345 \nu^{9} + 22610 \nu^{8} - 80750 \nu^{7} + 141626 \nu^{6} - 536180 \nu^{5} + 1398590 \nu^{4} + \cdots + 9111045 ) / 2300421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34865 \nu^{9} + 230458 \nu^{8} - 275345 \nu^{7} + 1774767 \nu^{6} - 754761 \nu^{5} + \cdots + 24799140 ) / 6901263 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53672 \nu^{9} - 50315 \nu^{8} + 508328 \nu^{7} - 166960 \nu^{6} + 3712693 \nu^{5} - 1688272 \nu^{4} + \cdots - 5119992 ) / 6901263 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 136198 \nu^{9} + 223373 \nu^{8} - 1345480 \nu^{7} + 1717500 \nu^{6} - 10007517 \nu^{5} + \cdots + 40853673 ) / 6901263 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 189653 \nu^{9} - 133430 \nu^{8} + 1791062 \nu^{7} - 242962 \nu^{6} + 13242232 \nu^{5} + \cdots + 6853167 ) / 6901263 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 253084 \nu^{9} + 31923 \nu^{8} - 2195344 \nu^{7} - 699587 \nu^{6} - 16969522 \nu^{5} + \cdots + 12477744 ) / 6901263 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 283056 \nu^{9} + 364903 \nu^{8} - 2836839 \nu^{7} + 2204512 \nu^{6} - 20155519 \nu^{5} + \cdots + 36028341 ) / 6901263 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 4\beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + 9\beta_{7} - \beta_{6} - 25\beta_{5} - \beta_{4} - \beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{9} + 10\beta_{8} + 11\beta_{7} - 9\beta_{6} - 6\beta_{5} + 9\beta_{4} - 11\beta_{3} + 41\beta_{2} - 41\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{9} + 2\beta_{8} + 10\beta_{6} + 12\beta_{4} - 71\beta_{3} + 17\beta_{2} + 175 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -81\beta_{9} - 12\beta_{8} - 102\beta_{7} + 81\beta_{6} + 90\beta_{5} + 12\beta_{4} + 295\beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 81 \beta_{9} - 114 \beta_{8} - 547 \beta_{7} + 33 \beta_{6} + 1273 \beta_{5} - 33 \beta_{4} + \cdots + 201 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 514\beta_{9} - 514\beta_{8} - 114\beta_{6} - 628\beta_{4} + 895\beta_{3} - 2172\beta_{2} - 999 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−1.27467 − 2.20779i
−0.689434 − 1.19413i
0.303056 + 0.524909i
0.746615 + 1.29317i
1.41443 + 2.44987i
−1.27467 + 2.20779i
−0.689434 + 1.19413i
0.303056 − 0.524909i
0.746615 − 1.29317i
1.41443 − 2.44987i
0 −1.27467 − 2.20779i 0 0 0 1.74957 − 3.03034i 0 −1.74957 + 3.03034i 0
601.2 0 −0.689434 − 1.19413i 0 0 0 −0.549362 + 0.951523i 0 0.549362 − 0.951523i 0
601.3 0 0.303056 + 0.524909i 0 0 0 −1.31631 + 2.27992i 0 1.31631 − 2.27992i 0
601.4 0 0.746615 + 1.29317i 0 0 0 −0.385132 + 0.667069i 0 0.385132 − 0.667069i 0
601.5 0 1.41443 + 2.44987i 0 0 0 2.50124 − 4.33228i 0 −2.50124 + 4.33228i 0
1101.1 0 −1.27467 + 2.20779i 0 0 0 1.74957 + 3.03034i 0 −1.74957 − 3.03034i 0
1101.2 0 −0.689434 + 1.19413i 0 0 0 −0.549362 − 0.951523i 0 0.549362 + 0.951523i 0
1101.3 0 0.303056 − 0.524909i 0 0 0 −1.31631 − 2.27992i 0 1.31631 + 2.27992i 0
1101.4 0 0.746615 − 1.29317i 0 0 0 −0.385132 − 0.667069i 0 0.385132 + 0.667069i 0
1101.5 0 1.41443 − 2.44987i 0 0 0 2.50124 + 4.33228i 0 −2.50124 − 4.33228i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{10} - T_{3}^{9} + 10T_{3}^{8} - 5T_{3}^{7} + 74T_{3}^{6} - 44T_{3}^{5} + 166T_{3}^{4} - 64T_{3}^{3} + 259T_{3}^{2} - 126T_{3} + 81 \) 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

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 4 T^{9} + \cdots + 1521 \) Copy content Toggle raw display
$11$ \( T^{10} + 7 T^{9} + \cdots + 164025 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} - 8 T^{9} + \cdots + 13689 \) Copy content Toggle raw display
$19$ \( T^{10} + 4 T^{9} + \cdots + 2082249 \) Copy content Toggle raw display
$23$ \( T^{10} - T^{9} + \cdots + 1610361 \) Copy content Toggle raw display
$29$ \( T^{10} + 3 T^{9} + \cdots + 5349969 \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots + 1131)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + T^{9} + \cdots + 651249 \) Copy content Toggle raw display
$41$ \( T^{10} - 10 T^{9} + \cdots + 13024881 \) Copy content Toggle raw display
$43$ \( T^{10} - 16 T^{9} + \cdots + 5890329 \) Copy content Toggle raw display
$47$ \( (T^{5} + 4 T^{4} + \cdots + 3159)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 16 T^{4} + \cdots + 63207)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 6 T^{9} + \cdots + 2025 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 725494225 \) Copy content Toggle raw display
$67$ \( T^{10} - 2 T^{9} + \cdots + 28804689 \) Copy content Toggle raw display
$71$ \( T^{10} - 5 T^{9} + \cdots + 34963569 \) Copy content Toggle raw display
$73$ \( (T^{5} - 8 T^{4} + \cdots - 12779)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 181 T^{3} + \cdots - 67)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} + 15 T^{4} + \cdots - 81)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 187388721 \) Copy content Toggle raw display
$97$ \( T^{10} + 21 T^{9} + \cdots + 961 \) Copy content Toggle raw display
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