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Properties

Label 2300.4.a.i
Level $2300$
Weight $4$
Character orbit 2300.a
Self dual yes
Analytic conductor $135.704$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2300,4,Mod(1,2300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2300.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11,0,1,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(135.704393013\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - x^{10} - 208 x^{9} + 222 x^{8} + 14781 x^{7} - 13745 x^{6} - 409063 x^{5} + 214585 x^{4} + \cdots - 4413664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} + 1) q^{7} + (\beta_{2} + 11) q^{9} + ( - \beta_{8} + \beta_1 - 2) q^{11} + ( - \beta_{3} + 5) q^{13} + ( - \beta_{8} - \beta_{5} + \beta_{4} + \cdots - 2) q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{2} + \cdots + 24) q^{19}+ \cdots + (6 \beta_{10} + 15 \beta_{9} + \cdots + 108) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + q^{3} + 10 q^{7} + 120 q^{9} - 23 q^{11} + 60 q^{13} - 13 q^{17} + 257 q^{19} + 32 q^{21} + 253 q^{23} - 95 q^{27} + 222 q^{29} + 424 q^{31} + 405 q^{33} - 366 q^{37} - 112 q^{39} + 29 q^{41} - 352 q^{43}+ \cdots + 1176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 208 x^{9} + 222 x^{8} + 14781 x^{7} - 13745 x^{6} - 409063 x^{5} + 214585 x^{4} + \cdots - 4413664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2900617 \nu^{10} - 922198275 \nu^{9} - 3845744652 \nu^{8} + 168910733152 \nu^{7} + \cdots - 12\!\cdots\!68 ) / 3662390817486 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 283499895 \nu^{10} - 923570945 \nu^{9} - 64238311768 \nu^{8} + 152251115994 \nu^{7} + \cdots - 12\!\cdots\!76 ) / 10987172452458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 366188995 \nu^{10} + 1028907313 \nu^{9} - 81965928554 \nu^{8} - 276051342772 \nu^{7} + \cdots + 26\!\cdots\!94 ) / 10987172452458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 403650979 \nu^{10} - 949066049 \nu^{9} + 88536952920 \nu^{8} + 218431446832 \nu^{7} + \cdots - 17\!\cdots\!24 ) / 10987172452458 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 82995584 \nu^{10} - 831679899 \nu^{9} + 11894704333 \nu^{8} + 126251917707 \nu^{7} + \cdots - 91381360585112 ) / 1831195408743 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 473216698 \nu^{10} + 177988734 \nu^{9} - 96276295144 \nu^{8} - 28835430805 \nu^{7} + \cdots - 90299195952005 ) / 5493586226229 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1058021126 \nu^{10} + 1929505225 \nu^{9} - 201886164381 \nu^{8} - 259186073000 \nu^{7} + \cdots - 12\!\cdots\!50 ) / 5493586226229 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2707157905 \nu^{10} - 1518061819 \nu^{9} + 534905765924 \nu^{8} + 98048139442 \nu^{7} + \cdots + 69\!\cdots\!26 ) / 10987172452458 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} + 66\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 5\beta_{8} + \beta_{7} - 4\beta_{6} - 4\beta_{5} - 8\beta_{4} + 83\beta_{2} - 35\beta _1 + 2497 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{10} + 102 \beta_{9} - 189 \beta_{8} + 95 \beta_{7} - 7 \beta_{6} + 17 \beta_{5} + 35 \beta_{4} + \cdots - 2233 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 69 \beta_{10} - 178 \beta_{9} + 554 \beta_{8} + 102 \beta_{7} - 479 \beta_{6} - 392 \beta_{5} + \cdots + 181431 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 639 \beta_{10} + 9161 \beta_{9} - 15748 \beta_{8} + 8000 \beta_{7} - 1311 \beta_{6} + 1947 \beta_{5} + \cdots - 300372 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11508 \beta_{10} - 21663 \beta_{9} + 52596 \beta_{8} + 9073 \beta_{7} - 44393 \beta_{6} + \cdots + 13642651 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 88728 \beta_{10} + 797187 \beta_{9} - 1290012 \beta_{8} + 650435 \beta_{7} - 159127 \beta_{6} + \cdots - 33012750 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1394220 \beta_{10} - 2342799 \beta_{9} + 4831896 \beta_{8} + 740650 \beta_{7} - 3747617 \beta_{6} + \cdots + 1045166027 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−9.19180
−8.89016
−5.16894
−3.77328
−0.959068
−0.729604
2.69343
2.80447
7.75726
7.81313
8.64456
0 −9.19180 0 0 0 10.9535 0 57.4892 0
1.2 0 −8.89016 0 0 0 −19.6039 0 52.0350 0
1.3 0 −5.16894 0 0 0 19.4790 0 −0.282053 0
1.4 0 −3.77328 0 0 0 −11.4097 0 −12.7624 0
1.5 0 −0.959068 0 0 0 −17.4631 0 −26.0802 0
1.6 0 −0.729604 0 0 0 24.3488 0 −26.4677 0
1.7 0 2.69343 0 0 0 −6.75132 0 −19.7455 0
1.8 0 2.80447 0 0 0 14.1394 0 −19.1349 0
1.9 0 7.75726 0 0 0 2.60728 0 33.1750 0
1.10 0 7.81313 0 0 0 −35.8618 0 34.0450 0
1.11 0 8.64456 0 0 0 29.5619 0 47.7285 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(23\) \( -1 \)

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 2300.4.a.i yes 11
5.b even 2 1 2300.4.a.h ✓ 11
5.c odd 4 2 2300.4.c.g 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.4.a.h ✓ 11 5.b even 2 1
2300.4.a.i yes 11 1.a even 1 1 trivial
2300.4.c.g 22 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} - T_{3}^{10} - 208 T_{3}^{9} + 222 T_{3}^{8} + 14781 T_{3}^{7} - 13745 T_{3}^{6} + \cdots - 4413664 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2300))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} \) Copy content Toggle raw display
$3$ \( T^{11} - T^{10} + \cdots - 4413664 \) Copy content Toggle raw display
$5$ \( T^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots + 5354328377088 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots - 12\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots - 70\!\cdots\!50 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 37\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 73\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( (T - 23)^{11} \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 34\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots + 96\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 60\!\cdots\!63 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots - 47\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 81\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 29\!\cdots\!67 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 24\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
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