Properties

Label 2320.2.d.h
Level $2320$
Weight $2$
Character orbit 2320.d
Analytic conductor $18.525$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2320,2,Mod(929,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2320.929");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.d (of order \(2\), degree \(1\), not 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: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 290)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + \beta_{5} q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + \beta_{5} q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + (\beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{13} + (\beta_{8} - \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{15} + ( - \beta_{9} - \beta_{8} - 3 \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{17} - 2 \beta_{4} q^{19} + ( - 4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{21} + ( - \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{2} - \beta_1) q^{23} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{25} + (\beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{27} - q^{29} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 1) q^{31} + (2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{5} + \beta_{3}) q^{33} + (\beta_{9} - 3 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{9} - 2 \beta_{8} - 4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{37} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{39} + ( - 2 \beta_{6} + 2 \beta_{2} - 2 \beta_1) q^{41} + ( - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 3) q^{45} + (2 \beta_{9} - 2 \beta_{8} - \beta_{2} - \beta_1) q^{47} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{49} + ( - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{51} + (2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2} - \beta_1) q^{53} + (2 \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{55} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{57} + (\beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - 3) q^{59} + (5 \beta_{6} - \beta_{5} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{61} + ( - 3 \beta_{9} + 3 \beta_{8} - \beta_{5} + \beta_{3}) q^{63} + ( - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + \cdots + 2) q^{65}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 20 q^{9} - 18 q^{15} - 8 q^{19} - 12 q^{21} - 4 q^{25} - 10 q^{29} - 10 q^{31} - 18 q^{35} + 10 q^{39} - 8 q^{41} + 26 q^{45} - 32 q^{49} + 64 q^{51} - 16 q^{55} - 18 q^{59} + 10 q^{61} + 20 q^{65} - 14 q^{69} + 12 q^{71} - 20 q^{75} - 34 q^{79} - 6 q^{81} + 18 q^{85} - 20 q^{89} + 44 q^{91} + 8 q^{95} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 27 \nu^{9} + 20 \nu^{8} - 688 \nu^{7} + 352 \nu^{6} - 4808 \nu^{5} + 544 \nu^{4} - 11267 \nu^{3} - 76 \nu^{2} - 8136 \nu + 2176 ) / 1216 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27 \nu^{9} - 20 \nu^{8} - 688 \nu^{7} - 352 \nu^{6} - 4808 \nu^{5} - 544 \nu^{4} - 11267 \nu^{3} + 76 \nu^{2} - 8136 \nu - 2176 ) / 1216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{9} + 200 \nu^{8} - 224 \nu^{7} + 4128 \nu^{6} - 968 \nu^{5} + 16384 \nu^{4} - 3059 \nu^{3} + 17480 \nu^{2} - 5240 \nu + 1696 ) / 1216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{8} - 632\nu^{6} - 3008\nu^{4} - 4237\nu^{2} - 784 ) / 152 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11 \nu^{9} + 200 \nu^{8} + 224 \nu^{7} + 4128 \nu^{6} + 968 \nu^{5} + 16384 \nu^{4} + 3059 \nu^{3} + 17480 \nu^{2} + 5240 \nu + 1696 ) / 1216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\nu^{8} + 360\nu^{6} + 1572\nu^{4} + 2033\nu^{2} + 360 ) / 76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -45\nu^{9} - 944\nu^{7} - 3960\nu^{5} - 4389\nu^{3} + 1032\nu ) / 608 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55\nu^{9} + 1120\nu^{7} + 4232\nu^{5} + 4351\nu^{3} + 1272\nu ) / 608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45\nu^{9} + 944\nu^{7} + 3960\nu^{5} + 4389\nu^{3} - 424\nu ) / 304 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + 2\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{4} - \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{9} + 8\beta_{8} - 16\beta_{7} - 4\beta_{5} + 4\beta_{3} - 2\beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 23\beta_{6} - 4\beta_{5} + 18\beta_{4} - 4\beta_{3} + 12\beta_{2} - 12\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 193\beta_{9} - 146\beta_{8} + 206\beta_{7} + 82\beta_{5} - 82\beta_{3} + 36\beta_{2} + 36\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -384\beta_{6} + 73\beta_{5} - 294\beta_{4} + 73\beta_{3} - 176\beta_{2} + 176\beta _1 - 531 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3009\beta_{9} + 2352\beta_{8} - 3122\beta_{7} - 1356\beta_{5} + 1356\beta_{3} - 588\beta_{2} - 588\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6129\beta_{6} - 1176\beta_{5} + 4681\beta_{4} - 1176\beta_{3} + 2737\beta_{2} - 2737\beta _1 + 8188 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 47429 \beta_{9} - 37272 \beta_{8} + 48944 \beta_{7} + 21620 \beta_{5} - 21620 \beta_{3} + 9362 \beta_{2} + 9362 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1
3.97530i
1.81278i
1.35864i
0.485804i
1.68193i
1.68193i
0.485804i
1.35864i
1.81278i
3.97530i
0 2.97530i 0 −0.639550 2.14266i 0 3.57331i 0 −5.85241 0
929.2 0 2.81278i 0 −1.63193 + 1.52866i 0 0.647892i 0 −4.91176 0
929.3 0 2.35864i 0 1.32739 1.79945i 0 4.21797i 0 −2.56319 0
929.4 0 1.48580i 0 −1.29150 1.82539i 0 4.37538i 0 0.792385 0
929.5 0 0.681929i 0 2.23558 + 0.0464742i 0 0.936197i 0 2.53497 0
929.6 0 0.681929i 0 2.23558 0.0464742i 0 0.936197i 0 2.53497 0
929.7 0 1.48580i 0 −1.29150 + 1.82539i 0 4.37538i 0 0.792385 0
929.8 0 2.35864i 0 1.32739 + 1.79945i 0 4.21797i 0 −2.56319 0
929.9 0 2.81278i 0 −1.63193 1.52866i 0 0.647892i 0 −4.91176 0
929.10 0 2.97530i 0 −0.639550 + 2.14266i 0 3.57331i 0 −5.85241 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 929.10
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 2320.2.d.h 10
4.b odd 2 1 290.2.b.b 10
5.b even 2 1 inner 2320.2.d.h 10
12.b even 2 1 2610.2.e.i 10
20.d odd 2 1 290.2.b.b 10
20.e even 4 1 1450.2.a.t 5
20.e even 4 1 1450.2.a.u 5
60.h even 2 1 2610.2.e.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.b 10 4.b odd 2 1
290.2.b.b 10 20.d odd 2 1
1450.2.a.t 5 20.e even 4 1
1450.2.a.u 5 20.e even 4 1
2320.2.d.h 10 1.a even 1 1 trivial
2320.2.d.h 10 5.b even 2 1 inner
2610.2.e.i 10 12.b even 2 1
2610.2.e.i 10 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2320, [\chi])\):

\( T_{3}^{10} + 25T_{3}^{8} + 224T_{3}^{6} + 849T_{3}^{4} + 1209T_{3}^{2} + 400 \) Copy content Toggle raw display
\( T_{7}^{10} + 51T_{7}^{8} + 877T_{7}^{6} + 5420T_{7}^{4} + 5936T_{7}^{2} + 1600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 25 T^{8} + 224 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{8} - 18 T^{7} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 51 T^{8} + 877 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$11$ \( (T^{5} - 27 T^{3} - 4 T^{2} + 172 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 53 T^{8} + 356 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{10} + 119 T^{8} + 5325 T^{6} + \cdots + 2930944 \) Copy content Toggle raw display
$19$ \( (T^{5} + 4 T^{4} - 56 T^{3} - 144 T^{2} + \cdots + 640)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 167 T^{8} + 9813 T^{6} + \cdots + 2611456 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} + 5 T^{4} - 22 T^{3} - 175 T^{2} + \cdots - 184)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 212 T^{8} + 12608 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( (T^{5} + 4 T^{4} - 76 T^{3} - 504 T^{2} + \cdots + 608)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 205 T^{8} + 11856 T^{6} + \cdots + 440896 \) Copy content Toggle raw display
$47$ \( T^{10} + 194 T^{8} + 11057 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{10} + 109 T^{8} + 1084 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( (T^{5} + 9 T^{4} - 79 T^{3} - 448 T^{2} + \cdots + 4000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 5 T^{4} - 169 T^{3} + 998 T^{2} + \cdots - 9808)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 564 T^{8} + 112832 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$71$ \( (T^{5} - 6 T^{4} - 208 T^{3} + 1192 T^{2} + \cdots - 62464)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 543 T^{8} + \cdots + 3008303104 \) Copy content Toggle raw display
$79$ \( (T^{5} + 17 T^{4} + 48 T^{3} - 471 T^{2} + \cdots - 3020)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 684 T^{8} + \cdots + 3399356416 \) Copy content Toggle raw display
$89$ \( (T^{5} + 10 T^{4} - 96 T^{3} - 1088 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 623 T^{8} + \cdots + 3351946816 \) Copy content Toggle raw display
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