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Properties

Label 2300.3.f.c
Level $2300$
Weight $3$
Character orbit 2300.f
Analytic conductor $62.670$
Analytic rank $0$
Dimension $16$
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] = [2300,3,Mod(1701,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2300.f (of order \(2\), degree \(1\), 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: \(62.6704608029\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 487 x^{14} + 91703 x^{12} + 8599549 x^{10} + 437649516 x^{8} + 12136718132 x^{6} + \cdots + 1845424439296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{7} + (\beta_{3} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{7} + (\beta_{3} + 4) q^{9} - \beta_{8} q^{11} + \beta_{4} q^{13} + \beta_{9} q^{17} - \beta_{10} q^{19} + (\beta_{11} - \beta_{9} - \beta_1) q^{21} - \beta_{13} q^{23} + ( - \beta_{6} + \beta_{4} + 3 \beta_{2} - 4) q^{27} + ( - \beta_{7} - \beta_{6} + \cdots + 2 \beta_{2}) q^{29}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{13} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{9} - 6 q^{13} + 8 q^{23} - 66 q^{27} - 6 q^{29} + 28 q^{31} + 74 q^{39} - 90 q^{41} - 40 q^{47} - 190 q^{49} - 174 q^{59} + 50 q^{69} + 116 q^{71} - 110 q^{73} - 198 q^{77} + 56 q^{81} + 362 q^{87} + 50 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 487 x^{14} + 91703 x^{12} + 8599549 x^{10} + 437649516 x^{8} + 12136718132 x^{6} + \cdots + 1845424439296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!43 \nu^{14} + \cdots + 12\!\cdots\!16 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44\!\cdots\!63 \nu^{14} + \cdots - 30\!\cdots\!52 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!07 \nu^{14} + \cdots - 13\!\cdots\!04 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74\!\cdots\!11 \nu^{14} + \cdots + 82\!\cdots\!32 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!67 \nu^{14} + \cdots - 63\!\cdots\!04 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20\!\cdots\!71 \nu^{14} + \cdots - 16\!\cdots\!52 ) / 27\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!31 \nu^{15} + \cdots + 88\!\cdots\!12 \nu ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21\!\cdots\!61 \nu^{15} + \cdots - 20\!\cdots\!96 \nu ) / 57\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 95\!\cdots\!43 \nu^{15} + \cdots + 14\!\cdots\!96 \nu ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!09 \nu^{15} + \cdots - 48\!\cdots\!72 \nu ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!91 \nu^{15} + \cdots - 22\!\cdots\!12 \nu ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24\!\cdots\!25 \nu^{15} + \cdots + 11\!\cdots\!44 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24\!\cdots\!25 \nu^{15} + \cdots + 11\!\cdots\!24 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!89 \nu^{15} + \cdots - 61\!\cdots\!32 \nu ) / 20\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 3\beta_{2} - 61 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 4 \beta_{15} + \beta_{14} - \beta_{13} + 4 \beta_{12} + 4 \beta_{11} - 9 \beta_{10} - 12 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 47 \beta_{14} - 47 \beta_{13} - 161 \beta_{7} + 144 \beta_{6} - 165 \beta_{5} - 235 \beta_{4} + \cdots + 6708 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 803 \beta_{15} - 176 \beta_{14} + 176 \beta_{13} - 930 \beta_{12} - 868 \beta_{11} + 1568 \beta_{10} + \cdots + 176 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9216 \beta_{14} + 9216 \beta_{13} + 24471 \beta_{7} - 20987 \beta_{6} + 25655 \beta_{5} + 40301 \beta_{4} + \cdots - 911337 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 129522 \beta_{15} + 30903 \beta_{14} - 30903 \beta_{13} + 155288 \beta_{12} + 145528 \beta_{11} + \cdots - 30903 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1473995 \beta_{14} - 1473995 \beta_{13} - 3700101 \beta_{7} + 3136698 \beta_{6} - 3911771 \beta_{5} + \cdots + 133408434 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20023257 \beta_{15} - 5037528 \beta_{14} + 5037528 \beta_{13} - 24080654 \beta_{12} - 23021916 \beta_{11} + \cdots + 5037528 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 224792680 \beta_{14} + 224792680 \beta_{13} + 559153291 \beta_{7} - 474591829 \beta_{6} + \cdots - 20005703991 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3067773436 \beta_{15} + 789011287 \beta_{14} - 789011287 \beta_{13} + 3659131140 \beta_{12} + \cdots - 789011287 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 33830062119 \beta_{14} - 33830062119 \beta_{13} - 84497741981 \beta_{7} + 72170307584 \beta_{6} + \cdots + 3022700140252 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 469108962235 \beta_{15} - 121542433320 \beta_{14} + 121542433320 \beta_{13} - 552701812810 \beta_{12} + \cdots + 121542433320 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5070747434244 \beta_{14} + 5070747434244 \beta_{13} + 12769526229439 \beta_{7} - 10995961378411 \beta_{6} + \cdots - 457785934466401 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 71702872764434 \beta_{15} + 18604333793515 \beta_{14} - 18604333793515 \beta_{13} + \cdots - 18604333793515 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1701.1
− 12.3431i
12.3431i
− 6.27459i
6.27459i
− 1.98337i
1.98337i
− 5.85973i
5.85973i
− 7.17264i
7.17264i
− 12.2819i
12.2819i
− 2.45527i
2.45527i
− 6.97776i
6.97776i
0 −5.43352 0 0 0 − 12.3431i 0 20.5231 0
1701.2 0 −5.43352 0 0 0 12.3431i 0 20.5231 0
1701.3 0 −3.89154 0 0 0 − 6.27459i 0 6.14407 0
1701.4 0 −3.89154 0 0 0 6.27459i 0 6.14407 0
1701.5 0 −3.10848 0 0 0 − 1.98337i 0 0.662652 0
1701.6 0 −3.10848 0 0 0 1.98337i 0 0.662652 0
1701.7 0 −0.285226 0 0 0 − 5.85973i 0 −8.91865 0
1701.8 0 −0.285226 0 0 0 5.85973i 0 −8.91865 0
1701.9 0 1.54203 0 0 0 − 7.17264i 0 −6.62215 0
1701.10 0 1.54203 0 0 0 7.17264i 0 −6.62215 0
1701.11 0 1.82726 0 0 0 − 12.2819i 0 −5.66110 0
1701.12 0 1.82726 0 0 0 12.2819i 0 −5.66110 0
1701.13 0 4.38686 0 0 0 − 2.45527i 0 10.2446 0
1701.14 0 4.38686 0 0 0 2.45527i 0 10.2446 0
1701.15 0 4.96261 0 0 0 − 6.97776i 0 15.6275 0
1701.16 0 4.96261 0 0 0 6.97776i 0 15.6275 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1701.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.3.f.c ✓ 16
5.b even 2 1 2300.3.f.d yes 16
5.c odd 4 2 2300.3.d.c 32
23.b odd 2 1 inner 2300.3.f.c ✓ 16
115.c odd 2 1 2300.3.f.d yes 16
115.e even 4 2 2300.3.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.3.d.c 32 5.c odd 4 2
2300.3.d.c 32 115.e even 4 2
2300.3.f.c ✓ 16 1.a even 1 1 trivial
2300.3.f.c ✓ 16 23.b odd 2 1 inner
2300.3.f.d yes 16 5.b even 2 1
2300.3.f.d yes 16 115.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 52T_{3}^{6} + 11T_{3}^{5} + 805T_{3}^{4} - 321T_{3}^{3} - 3634T_{3}^{2} + 3040T_{3} + 1150 \) acting on \(S_{3}^{\mathrm{new}}(2300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 52 T^{6} + \cdots + 1150)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 1845424439296 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 2050624000000 \) Copy content Toggle raw display
$13$ \( (T^{8} + 3 T^{7} + \cdots + 112174750)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 141514816000000 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 3 T^{7} + \cdots - 74437957820)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots - 153039584)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + 45 T^{7} + \cdots - 40064976535)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{8} + 20 T^{7} + \cdots - 824220885700)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{8} + 87 T^{7} + \cdots + 953953376800)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 7291959658160)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 55 T^{7} + \cdots - 45897589375)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
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