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Properties

Label 2310.2.e.o
Level $2310$
Weight $2$
Character orbit 2310.e
Analytic conductor $18.445$
Analytic rank $0$
Dimension $8$
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] = [2310,2,Mod(1849,2310)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2310.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.e (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: \(18.4454428669\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.309760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{4} q^{3} - q^{4} + \beta_{7} q^{5} - q^{6} + \beta_{4} q^{7} + \beta_{4} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{4} q^{3} - q^{4} + \beta_{7} q^{5} - q^{6} + \beta_{4} q^{7} + \beta_{4} q^{8} - q^{9} - \beta_{3} q^{10} - q^{11} + \beta_{4} q^{12} + \beta_{6} q^{13} + q^{14} - \beta_{3} q^{15} + q^{16} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + 1) q^{17} + \beta_{4} q^{18} + (\beta_{7} + \beta_{5} + \beta_1 - 1) q^{19} - \beta_{7} q^{20} + q^{21} + \beta_{4} q^{22} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{23}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 4 q^{5} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 4 q^{5} - 8 q^{6} - 8 q^{9} - 8 q^{11} + 8 q^{14} + 8 q^{16} - 8 q^{19} - 4 q^{20} + 8 q^{21} + 8 q^{24} - 16 q^{25} + 16 q^{29} - 4 q^{30} + 32 q^{31} + 16 q^{34} + 8 q^{36} - 8 q^{41} + 8 q^{44} - 4 q^{45} + 8 q^{46} - 8 q^{49} + 16 q^{51} + 8 q^{54} - 4 q^{55} - 8 q^{56} + 8 q^{61} - 8 q^{64} + 8 q^{65} + 8 q^{66} + 8 q^{69} + 4 q^{70} - 40 q^{71} + 8 q^{74} + 8 q^{76} - 24 q^{79} + 4 q^{80} + 8 q^{81} - 8 q^{84} + 56 q^{85} - 40 q^{86} - 32 q^{89} - 8 q^{94} + 16 q^{95} - 8 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26\nu^{7} - 94\nu^{6} + 202\nu^{5} - 110\nu^{4} + 286\nu^{3} - 138\nu^{2} - 60\nu + 638 ) / 245 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -10\nu^{7} + 55\nu^{6} - 138\nu^{5} + 129\nu^{4} - 12\nu^{3} - 11\nu^{2} - 188\nu + 75 ) / 49 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -72\nu^{7} + 249\nu^{6} - 386\nu^{5} - 257\nu^{4} + 384\nu^{3} + 9\nu^{2} - 942\nu - 685 ) / 245 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 74\nu^{7} - 309\nu^{6} + 639\nu^{5} - 249\nu^{4} - 19\nu^{3} - 291\nu^{2} + 1401\nu + 229 ) / 245 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -34\nu^{7} + 138\nu^{6} - 283\nu^{5} + 76\nu^{4} + 67\nu^{3} - 8\nu^{2} - 649\nu - 186 ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 174\nu^{7} - 712\nu^{6} + 1480\nu^{5} - 608\nu^{4} + 248\nu^{3} - 720\nu^{2} + 3722\nu + 606 ) / 245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 186\nu^{7} - 778\nu^{6} + 1675\nu^{5} - 802\nu^{4} + 37\nu^{3} - 60\nu^{2} + 3193\nu + 369 ) / 245 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{4} - \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 4\beta_{4} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} - 5\beta_{4} - \beta_{3} - 2\beta_{2} + 2\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} - 7\beta_{5} + 2\beta_{3} - 2\beta_{2} + 6\beta _1 - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{7} - 21\beta_{6} - 34\beta_{5} + 48\beta_{4} + 34\beta_{3} + 16\beta_{2} + 21\beta _1 - 82 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{7} - 35\beta_{6} - 17\beta_{5} + 78\beta_{4} + 47\beta_{3} + 47\beta_{2} - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 232\beta_{7} - 121\beta_{6} + 104\beta_{5} + 254\beta_{4} + 104\beta_{3} + 232\beta_{2} - 121\beta _1 + 358 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2310\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(661\) \(1387\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.16407 − 1.16407i
−0.164066 + 0.164066i
−0.748606 + 0.748606i
1.74861 − 1.74861i
−0.164066 − 0.164066i
1.16407 + 1.16407i
1.74861 + 1.74861i
−0.748606 − 0.748606i
− 1.00000i − 1.00000i −1.00000 −0.618034 − 2.14896i −1.00000 1.00000i 1.00000i −1.00000 −2.14896 + 0.618034i
1849.2 − 1.00000i − 1.00000i −1.00000 −0.618034 + 2.14896i −1.00000 1.00000i 1.00000i −1.00000 2.14896 + 0.618034i
1849.3 − 1.00000i − 1.00000i −1.00000 1.61803 − 1.54336i −1.00000 1.00000i 1.00000i −1.00000 −1.54336 − 1.61803i
1849.4 − 1.00000i − 1.00000i −1.00000 1.61803 + 1.54336i −1.00000 1.00000i 1.00000i −1.00000 1.54336 − 1.61803i
1849.5 1.00000i 1.00000i −1.00000 −0.618034 − 2.14896i −1.00000 − 1.00000i − 1.00000i −1.00000 2.14896 − 0.618034i
1849.6 1.00000i 1.00000i −1.00000 −0.618034 + 2.14896i −1.00000 − 1.00000i − 1.00000i −1.00000 −2.14896 − 0.618034i
1849.7 1.00000i 1.00000i −1.00000 1.61803 − 1.54336i −1.00000 − 1.00000i − 1.00000i −1.00000 1.54336 + 1.61803i
1849.8 1.00000i 1.00000i −1.00000 1.61803 + 1.54336i −1.00000 − 1.00000i − 1.00000i −1.00000 −1.54336 + 1.61803i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2310.2.e.o ✓ 8
5.b even 2 1 inner 2310.2.e.o ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.e.o ✓ 8 1.a even 1 1 trivial
2310.2.e.o ✓ 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2310, [\chi])\):

\( T_{13}^{4} + 32T_{13}^{2} + 176 \) Copy content Toggle raw display
\( T_{17}^{8} + 72T_{17}^{6} + 1456T_{17}^{4} + 7040T_{17}^{2} + 6400 \) Copy content Toggle raw display
\( T_{19}^{4} + 4T_{19}^{3} - 36T_{19}^{2} - 160T_{19} - 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{3} + 6 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 176)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 72 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} - 36 T^{2} + \cdots - 80)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 80 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 16 T^{3} + \cdots - 3280)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 80 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 224 T^{6} + \cdots + 3936256 \) Copy content Toggle raw display
$47$ \( T^{8} + 88 T^{6} + \cdots + 6400 \) Copy content Toggle raw display
$53$ \( T^{8} + 272 T^{6} + \cdots + 5382400 \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} - 56 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 560 T^{6} + \cdots + 139806976 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 536 T^{6} + \cdots + 49336576 \) Copy content Toggle raw display
$79$ \( (T^{4} + 12 T^{3} + \cdots - 4720)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 472 T^{6} + \cdots + 4000000 \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + \cdots - 80)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 288 T^{6} + \cdots + 774400 \) Copy content Toggle raw display
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