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Properties

Label 1840.3.k.b
Level $1840$
Weight $3$
Character orbit 1840.k
Analytic conductor $50.136$
Analytic rank $0$
Dimension $10$
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] = [1840,3,Mod(321,1840)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1840.k (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: \(50.1363686423\)
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: \( x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 115)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{3} q^{5} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{7} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{3} q^{5} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{7} - 2) q^{9} + ( - \beta_{8} - \beta_{2}) q^{11} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{4}) q^{13}+ \cdots + ( - 4 \beta_{9} + 9 \beta_{8} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18413999298 \nu^{9} + 6940583162233 \nu^{8} + 2184040960024 \nu^{7} - 55776313196156 \nu^{6} + \cdots + 64\!\cdots\!66 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3178911074 \nu^{9} + 78987068911 \nu^{8} + 430137087328 \nu^{7} - 1072934217052 \nu^{6} + \cdots + 34\!\cdots\!42 ) / 251783039496280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27126561061 \nu^{9} + 36976343259 \nu^{8} + 207292153512 \nu^{7} - 662996871978 \nu^{6} + \cdots + 16\!\cdots\!88 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61680118787 \nu^{9} - 90970570937 \nu^{8} + 725904561704 \nu^{7} - 1238323970566 \nu^{6} + \cdots - 48\!\cdots\!74 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 468602611879 \nu^{9} - 217704046159 \nu^{8} + 6293663580928 \nu^{7} + \cdots + 20\!\cdots\!62 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 498310598539 \nu^{9} - 877397075979 \nu^{8} + 7538944599648 \nu^{7} + \cdots - 72\!\cdots\!98 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41025221057 \nu^{9} + 13161849303 \nu^{8} + 712152635904 \nu^{7} + \cdots + 15\!\cdots\!06 ) / 251783039496280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 236841500106 \nu^{9} - 458127503999 \nu^{8} - 2135744859512 \nu^{7} + 6675104951458 \nu^{6} + \cdots - 19\!\cdots\!38 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1050188941786 \nu^{9} - 2989519571129 \nu^{8} - 19681413465312 \nu^{7} + \cdots - 13\!\cdots\!38 ) / 57\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} - 2\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{9} - 3\beta_{8} - 2\beta_{7} + 7\beta_{6} - 5\beta_{5} - 5\beta_{4} - 35\beta_{3} - 3\beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{9} - 20\beta_{8} - 19\beta_{6} + 11\beta_{5} - 13\beta_{4} - 4\beta_{3} + 20\beta_{2} - 8\beta _1 - 259 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{9} - 35 \beta_{8} + 64 \beta_{7} + 198 \beta_{6} - 270 \beta_{5} - 272 \beta_{4} + \cdots - 102 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 94 \beta_{9} + 116 \beta_{8} + \beta_{7} - 172 \beta_{6} + 395 \beta_{5} - 332 \beta_{4} + \cdots - 1868 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 658 \beta_{9} + 798 \beta_{8} + 1534 \beta_{7} + 1163 \beta_{6} - 2001 \beta_{5} - 1831 \beta_{4} + \cdots - 2261 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4984 \beta_{9} + 6968 \beta_{8} - 240 \beta_{7} + 1923 \beta_{6} + 3469 \beta_{5} - 3745 \beta_{4} + \cdots + 6511 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5253 \beta_{9} + 22665 \beta_{8} - 9346 \beta_{7} - 29275 \beta_{6} + 37373 \beta_{5} + 37817 \beta_{4} + \cdots + 1529 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
2.24810 − 2.23607i
2.24810 + 2.23607i
−2.67869 − 2.23607i
−2.67869 + 2.23607i
3.44206 − 2.23607i
3.44206 + 2.23607i
1.32878 − 2.23607i
1.32878 + 2.23607i
−3.34025 − 2.23607i
−3.34025 + 2.23607i
0 −3.98928 0 − 2.23607i 0 − 6.68423i 0 6.91435 0
321.2 0 −3.98928 0 2.23607i 0 6.68423i 0 6.91435 0
321.3 0 −1.23330 0 − 2.23607i 0 − 0.521669i 0 −7.47898 0
321.4 0 −1.23330 0 2.23607i 0 0.521669i 0 −7.47898 0
321.5 0 0.0436725 0 − 2.23607i 0 2.33372i 0 −8.99809 0
321.6 0 0.0436725 0 2.23607i 0 − 2.33372i 0 −8.99809 0
321.7 0 2.60299 0 − 2.23607i 0 8.05652i 0 −2.22446 0
321.8 0 2.60299 0 2.23607i 0 − 8.05652i 0 −2.22446 0
321.9 0 3.57592 0 − 2.23607i 0 10.2321i 0 3.78719 0
321.10 0 3.57592 0 2.23607i 0 − 10.2321i 0 3.78719 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.3.k.b 10
4.b odd 2 1 115.3.d.b ✓ 10
12.b even 2 1 1035.3.g.b 10
20.d odd 2 1 575.3.d.g 10
20.e even 4 2 575.3.c.d 20
23.b odd 2 1 inner 1840.3.k.b 10
92.b even 2 1 115.3.d.b ✓ 10
276.h odd 2 1 1035.3.g.b 10
460.g even 2 1 575.3.d.g 10
460.k odd 4 2 575.3.c.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.d.b ✓ 10 4.b odd 2 1
115.3.d.b ✓ 10 92.b even 2 1
575.3.c.d 20 20.e even 4 2
575.3.c.d 20 460.k odd 4 2
575.3.d.g 10 20.d odd 2 1
575.3.d.g 10 460.g even 2 1
1035.3.g.b 10 12.b even 2 1
1035.3.g.b 10 276.h odd 2 1
1840.3.k.b 10 1.a even 1 1 trivial
1840.3.k.b 10 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - T_{3}^{4} - 18T_{3}^{3} + 19T_{3}^{2} + 45T_{3} - 2 \) acting on \(S_{3}^{\mathrm{new}}(1840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{5} - T^{4} - 18 T^{3} + \cdots - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 220 T^{8} + \cdots + 450000 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 5056200000 \) Copy content Toggle raw display
$13$ \( (T^{5} + T^{4} - 148 T^{3} + \cdots + 2272)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 32080050000 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 1154881800000 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( (T^{5} + 23 T^{4} + \cdots + 482044)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 8 T^{4} + \cdots - 4585671)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 632002759200000 \) Copy content Toggle raw display
$41$ \( (T^{5} + 42 T^{4} + \cdots + 137382991)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{5} + 56 T^{4} + \cdots + 75512)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 550998028800000 \) Copy content Toggle raw display
$59$ \( (T^{5} - 131 T^{4} + \cdots - 547216)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 118 T^{4} + \cdots + 2383066649)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 84 T^{4} + \cdots + 181129672)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
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