Properties

Label 320.10.d.b
Level $320$
Weight $10$
Character orbit 320.d
Analytic conductor $164.811$
Analytic rank $0$
Dimension $12$
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] = [320,10,Mod(161,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.161");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 320.d (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: \(164.811467572\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 38483 x^{10} - 394584 x^{9} + 506716006 x^{8} + 9440196504 x^{7} - 2647129433407 x^{6} - 62475307207056 x^{5} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{2}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 27 \beta_1) q^{3} - 625 \beta_1 q^{5} + (\beta_{5} - 7 \beta_{2} + 1425) q^{7} + (\beta_{6} - \beta_{5} - \beta_{4} + 23 \beta_{2} - 6702) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 27 \beta_1) q^{3} - 625 \beta_1 q^{5} + (\beta_{5} - 7 \beta_{2} + 1425) q^{7} + (\beta_{6} - \beta_{5} - \beta_{4} + 23 \beta_{2} - 6702) q^{9} + (\beta_{11} + \beta_{9} + 2 \beta_{8} + 104 \beta_{3} - 3592 \beta_1) q^{11} + (\beta_{11} - 9 \beta_{10} + \beta_{9} - 8 \beta_{8} + 19 \beta_{3} - 3400 \beta_1) q^{13} + (625 \beta_{2} - 16875) q^{15} + (3 \beta_{7} + 19 \beta_{6} + 3 \beta_{4} - 559 \beta_{2} + 19094) q^{17} + ( - 3 \beta_{11} - 52 \beta_{10} - \beta_{9} - 54 \beta_{8} + 994 \beta_{3} + \cdots - 41292 \beta_1) q^{19}+ \cdots + (6853 \beta_{11} + 31236 \beta_{10} - 29133 \beta_{9} + \cdots + 455361094 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 17100 q^{7} - 80428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 17100 q^{7} - 80428 q^{9} - 202500 q^{15} + 229152 q^{17} + 3526700 q^{23} - 4687500 q^{25} + 11020200 q^{31} - 33371624 q^{33} - 5674600 q^{39} - 92870424 q^{41} - 99341900 q^{47} - 32900324 q^{49} - 26940000 q^{55} - 311503112 q^{57} - 599850500 q^{63} - 25500000 q^{65} + 12096600 q^{71} + 22901936 q^{73} - 677569200 q^{79} + 1245124588 q^{81} - 324941600 q^{87} + 2297486712 q^{89} - 309695000 q^{95} - 29517040 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 38483 x^{10} - 394584 x^{9} + 506716006 x^{8} + 9440196504 x^{7} - 2647129433407 x^{6} - 62475307207056 x^{5} + \cdots + 90\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 32\!\cdots\!09 \nu^{11} + \cdots + 16\!\cdots\!75 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32\!\cdots\!09 \nu^{11} + \cdots - 16\!\cdots\!75 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 82\!\cdots\!31 \nu^{11} + \cdots - 28\!\cdots\!75 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97\!\cdots\!57 \nu^{11} + \cdots - 24\!\cdots\!50 ) / 99\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 73\!\cdots\!97 \nu^{11} + \cdots - 12\!\cdots\!25 ) / 59\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!65 \nu^{11} + \cdots - 67\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52\!\cdots\!03 \nu^{11} + \cdots - 25\!\cdots\!75 ) / 59\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!58 \nu^{11} + \cdots + 30\!\cdots\!75 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!83 \nu^{11} + \cdots - 26\!\cdots\!75 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 72\!\cdots\!97 \nu^{11} + \cdots - 32\!\cdots\!75 ) / 59\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!93 \nu^{11} + \cdots + 12\!\cdots\!25 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 31\beta_{2} + 25655 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} - 11 \beta_{7} + 31 \beta_{6} - 384 \beta_{5} + 80 \beta_{4} + 93 \beta_{3} + 48447 \beta_{2} + 76967 \beta _1 + 789145 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{11} - 31 \beta_{10} - 80 \beta_{9} + 384 \beta_{8} - 1542 \beta_{7} - 15394 \beta_{6} + 14878 \beta_{5} + 17086 \beta_{4} + 48449 \beta_{3} + 749650 \beta_{2} + 789145 \beta _1 + 311667670 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7710 \beta_{11} + 76975 \beta_{10} - 85435 \beta_{9} - 74395 \beta_{8} - 279621 \beta_{7} - 145923 \beta_{6} - 6869010 \beta_{5} + 1782744 \beta_{4} + 3748405 \beta_{3} + \cdots + 19148097519 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 419459 \beta_{11} + 218807 \beta_{10} - 2674316 \beta_{9} + 10304475 \beta_{8} - 30750873 \beta_{7} - 224887960 \beta_{6} + 180138856 \beta_{5} + 268389208 \beta_{4} + \cdots + 4444014485687 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 107641548 \beta_{11} + 787242568 \beta_{10} - 939511741 \beta_{9} - 630616189 \beta_{8} - 2618701494 \beta_{7} - 4574192925 \beta_{6} + \cdots + 195066430357899 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10476763477 \beta_{11} + 18297792727 \beta_{10} - 67480896604 \beta_{9} + 203907333294 \beta_{8} - 508890904044 \beta_{7} - 3335043767700 \beta_{6} + \cdots + 66\!\cdots\!58 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4581309867354 \beta_{11} + 30024841143417 \beta_{10} - 37511442120021 \beta_{9} - 17831340540825 \beta_{8} - 91294936319759 \beta_{7} + \cdots + 74\!\cdots\!41 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 228315919461743 \beta_{11} + 563638589205989 \beta_{10} + \cdots + 10\!\cdots\!64 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 89\!\cdots\!72 \beta_{11} + \cdots + 13\!\cdots\!41 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−114.459 + 0.500000i
127.519 0.500000i
−60.0333 + 0.500000i
64.4188 0.500000i
−33.3883 + 0.500000i
15.9428 0.500000i
15.9428 + 0.500000i
−33.3883 0.500000i
64.4188 + 0.500000i
−60.0333 0.500000i
127.519 + 0.500000i
−114.459 0.500000i
0 255.917i 0 625.000i 0 10540.4 0 −45810.7 0
161.2 0 228.037i 0 625.000i 0 −2403.46 0 −32317.9 0
161.3 0 147.067i 0 625.000i 0 −7622.41 0 −1945.56 0
161.4 0 101.838i 0 625.000i 0 6926.31 0 9312.09 0
161.5 0 93.7766i 0 625.000i 0 −471.106 0 10889.0 0
161.6 0 4.88562i 0 625.000i 0 1580.23 0 19659.1 0
161.7 0 4.88562i 0 625.000i 0 1580.23 0 19659.1 0
161.8 0 93.7766i 0 625.000i 0 −471.106 0 10889.0 0
161.9 0 101.838i 0 625.000i 0 6926.31 0 9312.09 0
161.10 0 147.067i 0 625.000i 0 −7622.41 0 −1945.56 0
161.11 0 228.037i 0 625.000i 0 −2403.46 0 −32317.9 0
161.12 0 255.917i 0 625.000i 0 10540.4 0 −45810.7 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.10.d.b yes 12
4.b odd 2 1 320.10.d.a 12
8.b even 2 1 inner 320.10.d.b yes 12
8.d odd 2 1 320.10.d.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.10.d.a 12 4.b odd 2 1
320.10.d.a 12 8.d odd 2 1
320.10.d.b yes 12 1.a even 1 1 trivial
320.10.d.b yes 12 8.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 \(S_{10}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{12} + 158312 T_{3}^{10} + 8708242348 T_{3}^{8} + 200530699308864 T_{3}^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{6} - 8550 T_{7}^{5} - 76284490 T_{7}^{4} + 510434694000 T_{7}^{3} + 942955371844500 T_{7}^{2} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 158312 T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} + 390625)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 8550 T^{5} + \cdots - 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 15047724680 T^{10} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{12} + 60201895880 T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} - 114576 T^{5} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 1909525122520 T^{10} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} - 1763350 T^{5} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 118417890953600 T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} - 5510100 T^{5} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + 46435212 T^{5} + \cdots - 27\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + 49670950 T^{5} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} - 6048300 T^{5} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 11450968 T^{5} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 338784600 T^{5} + \cdots - 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} - 1148743356 T^{5} + \cdots + 14\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 14758520 T^{5} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
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