Properties

Label 2300.1.bh.a
Level $2300$
Weight $1$
Character orbit 2300.bh
Analytic conductor $1.148$
Analytic rank $0$
Dimension $40$
Projective image $D_{22}$
CM discriminant -20
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,1,Mod(7,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 11, 38]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.7");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2300.bh (of order \(44\), degree \(20\), 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: \(1.14784952906\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(2\) over \(\Q(\zeta_{44})\)
Coefficient field: \(\Q(\zeta_{88})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{88} q^{2} + ( - \zeta_{88}^{31} - \zeta_{88}^{7}) q^{3} + \zeta_{88}^{2} q^{4} + (\zeta_{88}^{32} + \zeta_{88}^{8}) q^{6} + ( - \zeta_{88}^{41} + \zeta_{88}^{33}) q^{7} - \zeta_{88}^{3} q^{8} + (\zeta_{88}^{38} - \zeta_{88}^{18} + \zeta_{88}^{14}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{88} q^{2} + ( - \zeta_{88}^{31} - \zeta_{88}^{7}) q^{3} + \zeta_{88}^{2} q^{4} + (\zeta_{88}^{32} + \zeta_{88}^{8}) q^{6} + ( - \zeta_{88}^{41} + \zeta_{88}^{33}) q^{7} - \zeta_{88}^{3} q^{8} + (\zeta_{88}^{38} - \zeta_{88}^{18} + \zeta_{88}^{14}) q^{9} + ( - \zeta_{88}^{33} - \zeta_{88}^{9}) q^{12} + (\zeta_{88}^{42} - \zeta_{88}^{34}) q^{14} + \zeta_{88}^{4} q^{16} + ( - \zeta_{88}^{39} + \zeta_{88}^{19} - \zeta_{88}^{15}) q^{18} + ( - \zeta_{88}^{40} - \zeta_{88}^{28} + \zeta_{88}^{20} - \zeta_{88}^{4}) q^{21} - \zeta_{88}^{23} q^{23} + (\zeta_{88}^{34} + \zeta_{88}^{10}) q^{24} + ( - \zeta_{88}^{25} - \zeta_{88}^{21} - \zeta_{88}^{5} - \zeta_{88}) q^{27} + ( - \zeta_{88}^{43} + \zeta_{88}^{35}) q^{28} + (\zeta_{88}^{26} - \zeta_{88}^{14}) q^{29} - \zeta_{88}^{5} q^{32} + (\zeta_{88}^{40} - \zeta_{88}^{20} + \zeta_{88}^{16}) q^{36} + ( - \zeta_{88}^{32} - \zeta_{88}^{24}) q^{41} + (\zeta_{88}^{41} + \zeta_{88}^{29} - \zeta_{88}^{21} + \zeta_{88}^{5}) q^{42} + ( - \zeta_{88}^{43} - \zeta_{88}^{39}) q^{43} + \zeta_{88}^{24} q^{46} + ( - \zeta_{88}^{17} + \zeta_{88}^{5}) q^{47} + ( - \zeta_{88}^{35} - \zeta_{88}^{11}) q^{48} + ( - \zeta_{88}^{38} + \zeta_{88}^{30} - \zeta_{88}^{22}) q^{49} + ( - \zeta_{88}^{26} + \zeta_{88}^{22} + \zeta_{88}^{6} - \zeta_{88}^{2}) q^{54} + ( - \zeta_{88}^{36} - 1) q^{56} + ( - \zeta_{88}^{27} + \zeta_{88}^{15}) q^{58} + ( - \zeta_{88}^{16} - \zeta_{88}^{12}) q^{61} + (\zeta_{88}^{35} - \zeta_{88}^{27} - \zeta_{88}^{15} + \zeta_{88}^{11} + \zeta_{88}^{7} - \zeta_{88}^{3}) q^{63} + \zeta_{88}^{6} q^{64} + (\zeta_{88}^{37} + \zeta_{88}^{9}) q^{67} + (\zeta_{88}^{30} - \zeta_{88}^{10}) q^{69} + ( - \zeta_{88}^{41} + \zeta_{88}^{21} - \zeta_{88}^{17}) q^{72} + (\zeta_{88}^{36} + \zeta_{88}^{32} + \zeta_{88}^{28} - \zeta_{88}^{12} + \zeta_{88}^{8}) q^{81} + (\zeta_{88}^{33} + \zeta_{88}^{25}) q^{82} + ( - \zeta_{88}^{35} + \zeta_{88}^{19}) q^{83} + ( - \zeta_{88}^{42} - \zeta_{88}^{30} + \zeta_{88}^{22} - \zeta_{88}^{6}) q^{84} + (\zeta_{88}^{40} - 1) q^{86} + ( - \zeta_{88}^{33} + \zeta_{88}^{21} + \zeta_{88}^{13} - \zeta_{88}) q^{87} + ( - \zeta_{88}^{42} + \zeta_{88}^{18}) q^{89} - \zeta_{88}^{25} q^{92} + (\zeta_{88}^{18} - \zeta_{88}^{6}) q^{94} + (\zeta_{88}^{36} + \zeta_{88}^{12}) q^{96} + (\zeta_{88}^{39} - \zeta_{88}^{31} + \zeta_{88}^{23}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 8 q^{6} + 4 q^{16} - 12 q^{36} + 8 q^{41} - 4 q^{46} - 44 q^{56} + 20 q^{81} - 44 q^{86} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(\zeta_{88}^{22}\) \(-1\) \(\zeta_{88}^{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.0713392 + 0.997452i
−0.0713392 0.997452i
0.599278 0.800541i
−0.599278 + 0.800541i
0.599278 + 0.800541i
−0.599278 0.800541i
0.479249 + 0.877679i
−0.479249 0.877679i
0.212565 + 0.977147i
−0.212565 0.977147i
0.877679 + 0.479249i
−0.877679 0.479249i
0.349464 + 0.936950i
−0.349464 0.936950i
0.997452 0.0713392i
−0.997452 + 0.0713392i
0.800541 0.599278i
−0.800541 + 0.599278i
0.997452 + 0.0713392i
−0.997452 0.0713392i
−0.0713392 0.997452i 1.27979 + 0.278401i −0.989821 + 0.142315i 0 0.186393 1.29639i 0.494541 + 0.270040i 0.212565 + 0.977147i 0.650724 + 0.297176i 0
7.2 0.0713392 + 0.997452i −1.27979 0.278401i −0.989821 + 0.142315i 0 0.186393 1.29639i −0.494541 0.270040i −0.212565 0.977147i 0.650724 + 0.297176i 0
43.1 −0.599278 + 0.800541i −0.0994679 0.266684i −0.281733 0.959493i 0 0.273100 + 0.0801894i −0.229843 + 1.05657i 0.936950 + 0.349464i 0.694523 0.601808i 0
43.2 0.599278 0.800541i 0.0994679 + 0.266684i −0.281733 0.959493i 0 0.273100 + 0.0801894i 0.229843 1.05657i −0.936950 0.349464i 0.694523 0.601808i 0
107.1 −0.599278 0.800541i −0.0994679 + 0.266684i −0.281733 + 0.959493i 0 0.273100 0.0801894i −0.229843 1.05657i 0.936950 0.349464i 0.694523 + 0.601808i 0
107.2 0.599278 + 0.800541i 0.0994679 0.266684i −0.281733 + 0.959493i 0 0.273100 0.0801894i 0.229843 + 1.05657i −0.936950 + 0.349464i 0.694523 + 0.601808i 0
143.1 −0.479249 0.877679i −0.136899 1.91410i −0.540641 + 0.841254i 0 −1.61435 + 1.03748i −1.70456 0.635768i 0.997452 + 0.0713392i −2.65520 + 0.381761i 0
143.2 0.479249 + 0.877679i 0.136899 + 1.91410i −0.540641 + 0.841254i 0 −1.61435 + 1.03748i 1.70456 + 0.635768i −0.997452 0.0713392i −2.65520 + 0.381761i 0
343.1 −0.212565 0.977147i 1.34692 + 1.00829i −0.909632 + 0.415415i 0 0.698939 1.53046i 0.107829 + 1.50765i 0.599278 + 0.800541i 0.515804 + 1.75667i 0
343.2 0.212565 + 0.977147i −1.34692 1.00829i −0.909632 + 0.415415i 0 0.698939 1.53046i −0.107829 1.50765i −0.599278 0.800541i 0.515804 + 1.75667i 0
543.1 −0.877679 0.479249i 1.91410 + 0.136899i 0.540641 + 0.841254i 0 −1.61435 1.03748i −0.635768 1.70456i −0.0713392 0.997452i 2.65520 + 0.381761i 0
543.2 0.877679 + 0.479249i −1.91410 0.136899i 0.540641 + 0.841254i 0 −1.61435 1.03748i 0.635768 + 1.70456i 0.0713392 + 0.997452i 2.65520 + 0.381761i 0
707.1 −0.349464 0.936950i −0.398174 0.729202i −0.755750 + 0.654861i 0 −0.544078 + 0.627899i −1.58479 + 1.18636i 0.877679 + 0.479249i 0.167448 0.260554i 0
707.2 0.349464 + 0.936950i 0.398174 + 0.729202i −0.755750 + 0.654861i 0 −0.544078 + 0.627899i 1.58479 1.18636i −0.877679 0.479249i 0.167448 0.260554i 0
743.1 −0.997452 + 0.0713392i −0.278401 + 1.27979i 0.989821 0.142315i 0 0.186393 1.29639i 0.270040 0.494541i −0.977147 + 0.212565i −0.650724 0.297176i 0
743.2 0.997452 0.0713392i 0.278401 1.27979i 0.989821 0.142315i 0 0.186393 1.29639i −0.270040 + 0.494541i 0.977147 0.212565i −0.650724 0.297176i 0
843.1 −0.800541 + 0.599278i −0.266684 0.0994679i 0.281733 0.959493i 0 0.273100 0.0801894i −1.05657 + 0.229843i 0.349464 + 0.936950i −0.694523 0.601808i 0
843.2 0.800541 0.599278i 0.266684 + 0.0994679i 0.281733 0.959493i 0 0.273100 0.0801894i 1.05657 0.229843i −0.349464 0.936950i −0.694523 0.601808i 0
907.1 −0.997452 0.0713392i −0.278401 1.27979i 0.989821 + 0.142315i 0 0.186393 + 1.29639i 0.270040 + 0.494541i −0.977147 0.212565i −0.650724 + 0.297176i 0
907.2 0.997452 + 0.0713392i 0.278401 + 1.27979i 0.989821 + 0.142315i 0 0.186393 + 1.29639i −0.270040 0.494541i 0.977147 + 0.212565i −0.650724 + 0.297176i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.e even 4 2 inner
23.d odd 22 1 inner
92.h even 22 1 inner
115.i odd 22 1 inner
115.l even 44 2 inner
460.o even 22 1 inner
460.v odd 44 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.1.bh.a 40
4.b odd 2 1 inner 2300.1.bh.a 40
5.b even 2 1 inner 2300.1.bh.a 40
5.c odd 4 2 inner 2300.1.bh.a 40
20.d odd 2 1 CM 2300.1.bh.a 40
20.e even 4 2 inner 2300.1.bh.a 40
23.d odd 22 1 inner 2300.1.bh.a 40
92.h even 22 1 inner 2300.1.bh.a 40
115.i odd 22 1 inner 2300.1.bh.a 40
115.l even 44 2 inner 2300.1.bh.a 40
460.o even 22 1 inner 2300.1.bh.a 40
460.v odd 44 2 inner 2300.1.bh.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.1.bh.a 40 1.a even 1 1 trivial
2300.1.bh.a 40 4.b odd 2 1 inner
2300.1.bh.a 40 5.b even 2 1 inner
2300.1.bh.a 40 5.c odd 4 2 inner
2300.1.bh.a 40 20.d odd 2 1 CM
2300.1.bh.a 40 20.e even 4 2 inner
2300.1.bh.a 40 23.d odd 22 1 inner
2300.1.bh.a 40 92.h even 22 1 inner
2300.1.bh.a 40 115.i odd 22 1 inner
2300.1.bh.a 40 115.l even 44 2 inner
2300.1.bh.a 40 460.o even 22 1 inner
2300.1.bh.a 40 460.v odd 44 2 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{40} - T^{36} + T^{32} - T^{28} + T^{24} - T^{20} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{40} - 16 T^{36} - 52 T^{32} + 1074 T^{28} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{40} \) Copy content Toggle raw display
$7$ \( T^{40} + 11 T^{36} + 55 T^{32} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( T^{40} \) Copy content Toggle raw display
$13$ \( T^{40} \) Copy content Toggle raw display
$17$ \( T^{40} \) Copy content Toggle raw display
$19$ \( T^{40} \) Copy content Toggle raw display
$23$ \( T^{40} - T^{36} + T^{32} - T^{28} + T^{24} - T^{20} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{20} - 4 T^{18} + 16 T^{16} - 9 T^{14} + 36 T^{12} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{40} \) Copy content Toggle raw display
$37$ \( T^{40} \) Copy content Toggle raw display
$41$ \( (T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} - 6 T^{6} + \cdots + 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{40} + 1815 T^{28} + 31460 T^{24} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( (T^{20} + 25 T^{16} + 184 T^{12} + 403 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{40} \) Copy content Toggle raw display
$59$ \( T^{40} \) Copy content Toggle raw display
$61$ \( (T^{10} - 11 T^{7} + 33 T^{4} + 11 T^{3} + \cdots + 11)^{4} \) Copy content Toggle raw display
$67$ \( T^{40} + 1815 T^{28} + 31460 T^{24} + \cdots + 14641 \) Copy content Toggle raw display
$71$ \( T^{40} \) Copy content Toggle raw display
$73$ \( T^{40} \) Copy content Toggle raw display
$79$ \( T^{40} \) Copy content Toggle raw display
$83$ \( T^{40} + 44 T^{32} - 2904 T^{28} + \cdots + 14641 \) Copy content Toggle raw display
$89$ \( (T^{20} - 22 T^{14} + 154 T^{12} - 22 T^{10} + \cdots + 121)^{2} \) Copy content Toggle raw display
$97$ \( T^{40} \) Copy content Toggle raw display
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