Properties

Label 1840.2.i.a
Level $1840$
Weight $2$
Character orbit 1840.i
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1471,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.i (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: \(14.6924739719\)
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: \( x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{3}) q^{7} + ( - \beta_{6} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{3}) q^{7} + ( - \beta_{6} - 2) q^{9} + (\beta_{12} - 2 \beta_{10}) q^{11} + (\beta_{6} + 1) q^{13} - \beta_{3} q^{15} + \beta_{13} q^{17} + (\beta_{15} + \beta_{12} - \beta_{10}) q^{19} + (\beta_{13} + 4 \beta_{8} - \beta_{4}) q^{21} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{23}+ \cdots + ( - 3 \beta_{15} - \beta_{12} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 7141 \nu^{14} + 639364 \nu^{12} - 14917580 \nu^{10} + 151546566 \nu^{8} - 729577504 \nu^{6} + \cdots - 188557796 ) / 547177568 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5547 \nu^{14} + 173356 \nu^{12} - 2080516 \nu^{10} + 11582258 \nu^{8} - 29821744 \nu^{6} + \cdots + 5858020 ) / 18868192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 165589 \nu^{15} + 5706972 \nu^{13} - 79149040 \nu^{11} + 555881878 \nu^{9} + \cdots - 262952740 \nu ) / 1094355136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7343 \nu^{15} + 214244 \nu^{13} - 2331136 \nu^{11} + 10946626 \nu^{9} - 19800248 \nu^{7} + \cdots - 11846476 \nu ) / 37736384 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 971 \nu^{14} - 32540 \nu^{12} + 432756 \nu^{10} - 2840762 \nu^{8} + 9652416 \nu^{6} + \cdots + 1863596 ) / 2088464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9687 \nu^{14} + 298300 \nu^{12} - 3520980 \nu^{10} + 19188922 \nu^{8} - 48266192 \nu^{6} + \cdots - 80965164 ) / 18868192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20175 \nu^{14} + 647232 \nu^{12} - 8072756 \nu^{10} + 47481542 \nu^{8} - 129690636 \nu^{6} + \cdots - 86948092 ) / 18868192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9589 \nu^{15} + 302708 \nu^{13} - 3710656 \nu^{11} + 21553958 \nu^{9} - 60743728 \nu^{7} + \cdots - 37094388 \nu ) / 18868192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 760831 \nu^{14} - 24021912 \nu^{12} + 294082508 \nu^{10} - 1701925438 \nu^{8} + 4773358420 \nu^{6} + \cdots + 692456380 ) / 547177568 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 345 \nu^{15} + 11264 \nu^{13} - 145224 \nu^{11} + 918310 \nu^{9} - 3009724 \nu^{7} + \cdots - 189996 \nu ) / 425488 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 169285 \nu^{14} + 5459985 \nu^{12} - 69284106 \nu^{10} + 428229415 \nu^{8} - 1354547795 \nu^{6} + \cdots - 239476132 ) / 68397196 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1996395 \nu^{15} - 64300516 \nu^{13} + 811056936 \nu^{11} - 4930874706 \nu^{9} + \cdots + 361303484 \nu ) / 1094355136 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 80267 \nu^{15} - 2562948 \nu^{13} + 31894496 \nu^{11} - 189379626 \nu^{9} + 550748136 \nu^{7} + \cdots + 336396572 \nu ) / 37736384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 81585 \nu^{15} - 2642028 \nu^{13} + 33591704 \nu^{11} - 206610582 \nu^{9} + 635359128 \nu^{7} + \cdots + 390867284 \nu ) / 37736384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2530093 \nu^{15} + 81970636 \nu^{13} - 1045043272 \nu^{11} + 6493543742 \nu^{9} + \cdots - 1062631524 \nu ) / 547177568 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{8} - 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + \beta_{5} - 2\beta_{2} + 2\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 3\beta_{13} + 2\beta_{12} + 13\beta_{10} - 14\beta_{8} + 3\beta_{4} - 14\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} - 4\beta_{9} + 14\beta_{6} + 25\beta_{5} - 22\beta_{2} + 30\beta _1 + 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{15} - 40\beta_{13} + 12\beta_{12} + 127\beta_{10} - 191\beta_{8} + 60\beta_{4} - 106\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 47\beta_{11} - 46\beta_{9} + 8\beta_{7} + 91\beta_{6} + 402\beta_{5} - 183\beta_{2} + 394\beta _1 + 484 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 122 \beta_{15} - 28 \beta_{14} - 462 \beta_{13} + 56 \beta_{12} + 953 \beta_{10} + \cdots - 706 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 712\beta_{11} - 448\beta_{9} + 80\beta_{7} + 470\beta_{6} + 5259\beta_{5} - 1118\beta_{2} + 4734\beta _1 + 2731 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 607 \beta_{15} - 528 \beta_{14} - 5061 \beta_{13} + 126 \beta_{12} + 4259 \beta_{10} + \cdots - 2826 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8714 \beta_{11} - 4332 \beta_{9} + 280 \beta_{7} + 118 \beta_{6} + 60643 \beta_{5} - 1502 \beta_{2} + \cdots + 2249 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2634 \beta_{15} - 6688 \beta_{14} - 52580 \beta_{13} - 2468 \beta_{12} - 24711 \beta_{10} + \cdots + 21314 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 92865 \beta_{11} - 41314 \beta_{9} - 4744 \beta_{7} - 48755 \beta_{6} + 630646 \beta_{5} + 98679 \beta_{2} + \cdots - 261836 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 125478 \beta_{15} - 69836 \beta_{14} - 507910 \beta_{13} - 62296 \beta_{12} - 998513 \beta_{10} + \cdots + 756434 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 876628 \beta_{11} - 372008 \beta_{9} - 129248 \beta_{7} - 1009946 \beta_{6} + 5895185 \beta_{5} + \cdots - 5610087 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2251575 \beta_{15} - 626000 \beta_{14} - 4417867 \beta_{13} - 1002078 \beta_{12} + \cdots + 13032306 \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
−2.25010 + 0.500000i
2.25010 0.500000i
−3.23546 + 0.500000i
3.23546 0.500000i
−1.19389 + 0.500000i
1.19389 0.500000i
0.208533 + 0.500000i
−0.208533 0.500000i
−0.208533 + 0.500000i
0.208533 0.500000i
1.19389 + 0.500000i
−1.19389 0.500000i
3.23546 + 0.500000i
−3.23546 0.500000i
2.25010 + 0.500000i
−2.25010 0.500000i
0 3.11612i 0 1.00000i 0 1.38407 0 −6.71022 0
1471.2 0 3.11612i 0 1.00000i 0 −1.38407 0 −6.71022 0
1471.3 0 2.36943i 0 1.00000i 0 4.10148 0 −2.61421 0
1471.4 0 2.36943i 0 1.00000i 0 −4.10148 0 −2.61421 0
1471.5 0 2.05992i 0 1.00000i 0 0.327869 0 −1.24327 0
1471.6 0 2.05992i 0 1.00000i 0 −0.327869 0 −1.24327 0
1471.7 0 0.657492i 0 1.00000i 0 −1.07456 0 2.56770 0
1471.8 0 0.657492i 0 1.00000i 0 1.07456 0 2.56770 0
1471.9 0 0.657492i 0 1.00000i 0 1.07456 0 2.56770 0
1471.10 0 0.657492i 0 1.00000i 0 −1.07456 0 2.56770 0
1471.11 0 2.05992i 0 1.00000i 0 −0.327869 0 −1.24327 0
1471.12 0 2.05992i 0 1.00000i 0 0.327869 0 −1.24327 0
1471.13 0 2.36943i 0 1.00000i 0 −4.10148 0 −2.61421 0
1471.14 0 2.36943i 0 1.00000i 0 4.10148 0 −2.61421 0
1471.15 0 3.11612i 0 1.00000i 0 −1.38407 0 −6.71022 0
1471.16 0 3.11612i 0 1.00000i 0 1.38407 0 −6.71022 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.i.a 16
4.b odd 2 1 inner 1840.2.i.a 16
23.b odd 2 1 inner 1840.2.i.a 16
92.b even 2 1 inner 1840.2.i.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.a 16 1.a even 1 1 trivial
1840.2.i.a 16 4.b odd 2 1 inner
1840.2.i.a 16 23.b odd 2 1 inner
1840.2.i.a 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 20T_{3}^{6} + 128T_{3}^{4} + 283T_{3}^{2} + 100 \) acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 20 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 20 T^{6} + 56 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 65 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots - 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 44 T^{6} + \cdots + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 133 T^{6} + \cdots + 462400)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{4} - 18 T^{3} + \cdots - 1172)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 255 T^{6} + \cdots + 6487209)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 137 T^{6} + \cdots + 153664)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + T^{3} - 58 T^{2} + \cdots - 89)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 80 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 43 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 129 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 415 T^{6} + \cdots + 462400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 247 T^{6} + \cdots + 4129024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 383 T^{6} + \cdots + 20070400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 191 T^{6} + \cdots + 1466521)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 5 T^{3} + \cdots - 1640)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 640 T^{6} + \cdots + 333865984)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 383 T^{6} + \cdots + 20070400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 240 T^{6} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 431 T^{6} + \cdots + 1993744)^{2} \) Copy content Toggle raw display
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