Properties

Label 1840.2.e.e
Level $1840$
Weight $2$
Character orbit 1840.e
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.11574317056.3
Defining polynomial: \(x^{8} + 45 x^{4} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( \beta_{2} - \beta_{5} ) q^{3} + ( \beta_{3} + \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} - \beta_{5} ) q^{3} + ( \beta_{3} + \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{9} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{13} + ( 2 - \beta_{2} - \beta_{5} + 2 \beta_{7} ) q^{15} + ( -\beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{17} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + \beta_{7} ) q^{19} + ( 2 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{21} + \beta_{2} q^{23} + ( -2 + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{27} + ( -2 \beta_{4} + 2 \beta_{7} ) q^{29} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{33} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{35} + ( 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{37} + ( -\beta_{1} - \beta_{3} + \beta_{7} ) q^{39} + ( 5 - 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} + 4 \beta_{6} ) q^{43} + ( -3 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{45} + ( -2 \beta_{5} + 2 \beta_{6} ) q^{47} + ( 2 + \beta_{4} + 2 \beta_{7} ) q^{49} + ( -1 - \beta_{1} - \beta_{3} - 3 \beta_{4} + 3 \beta_{7} ) q^{51} + ( \beta_{1} - \beta_{3} + 4 \beta_{6} ) q^{53} + ( 2 - \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( 3 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{57} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} ) q^{59} + ( 3 + 3 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{7} ) q^{61} + ( \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{63} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{65} + ( 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{67} + ( -1 + \beta_{4} ) q^{69} + ( -5 + \beta_{4} - 2 \beta_{7} ) q^{71} + ( 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{6} ) q^{73} + ( 4 + 4 \beta_{1} - 7 \beta_{2} + 3 \beta_{5} ) q^{75} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{77} + ( -2 - 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 6 \beta_{7} ) q^{79} + ( 5 - \beta_{1} - \beta_{3} - 4 \beta_{4} + 5 \beta_{7} ) q^{81} + ( -3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -4 - 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{85} + ( -2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} ) q^{87} + ( -8 - 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{7} ) q^{89} + ( 3 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{91} + ( -\beta_{1} + 6 \beta_{2} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} ) q^{93} + ( -2 + \beta_{1} + \beta_{2} + 5 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{95} + ( -3 \beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{97} + ( -7 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 14q^{9} + O(q^{10}) \) \( 8q - 14q^{9} - 10q^{11} + 16q^{15} - 2q^{19} + 12q^{21} - 12q^{25} - 4q^{29} - 10q^{31} + 16q^{35} + 46q^{41} - 26q^{45} + 18q^{49} - 14q^{51} + 18q^{55} + 32q^{59} + 18q^{61} - 16q^{65} - 6q^{69} - 38q^{71} + 32q^{75} - 12q^{79} + 32q^{81} - 24q^{85} - 60q^{89} + 26q^{91} - 18q^{95} - 54q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 45 x^{4} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 47 \nu^{2} \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 47 \nu^{3} \)\()/14\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{7} - 2 \nu^{5} - 4 \nu^{4} + 221 \nu^{3} - 66 \nu - 76 \)\()/56\)
\(\beta_{5}\)\(=\)\((\)\( 5 \nu^{7} + 6 \nu^{6} + 2 \nu^{5} + 221 \nu^{3} + 254 \nu^{2} + 66 \nu \)\()/56\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} + 2 \nu^{5} + 315 \nu^{3} + 94 \nu \)\()/28\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{7} + 2 \nu^{5} - 315 \nu^{3} + 94 \nu \)\()/28\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{5} + \beta_{3} - 3 \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} - 7 \beta_{3}\)
\(\nu^{4}\)\(=\)\(-7 \beta_{7} - 14 \beta_{4} + 7 \beta_{3} + 7 \beta_{1} - 19\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} + 7 \beta_{6} - 47 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-47 \beta_{6} + 94 \beta_{5} - 47 \beta_{3} + 127 \beta_{2} + 47 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-47 \beta_{7} + 47 \beta_{6} + 315 \beta_{3}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
369.1
0.386289 + 0.386289i
1.83051 1.83051i
−0.386289 0.386289i
−1.83051 1.83051i
−1.83051 + 1.83051i
−0.386289 + 0.386289i
1.83051 + 1.83051i
0.386289 0.386289i
0 3.25886i 0 0.386289 + 2.20245i 0 1.44270i 0 −7.62018 0
369.2 0 2.40815i 0 1.83051 + 1.28422i 0 0.706585i 0 −2.79917 0
369.3 0 1.44270i 0 −0.386289 2.20245i 0 3.25886i 0 0.918614 0
369.4 0 0.706585i 0 −1.83051 + 1.28422i 0 2.40815i 0 2.50074 0
369.5 0 0.706585i 0 −1.83051 1.28422i 0 2.40815i 0 2.50074 0
369.6 0 1.44270i 0 −0.386289 + 2.20245i 0 3.25886i 0 0.918614 0
369.7 0 2.40815i 0 1.83051 1.28422i 0 0.706585i 0 −2.79917 0
369.8 0 3.25886i 0 0.386289 2.20245i 0 1.44270i 0 −7.62018 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 369.8
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 1840.2.e.e 8
4.b odd 2 1 230.2.b.b 8
5.b even 2 1 inner 1840.2.e.e 8
5.c odd 4 1 9200.2.a.cj 4
5.c odd 4 1 9200.2.a.cr 4
12.b even 2 1 2070.2.d.f 8
20.d odd 2 1 230.2.b.b 8
20.e even 4 1 1150.2.a.r 4
20.e even 4 1 1150.2.a.s 4
60.h even 2 1 2070.2.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.b 8 4.b odd 2 1
230.2.b.b 8 20.d odd 2 1
1150.2.a.r 4 20.e even 4 1
1150.2.a.s 4 20.e even 4 1
1840.2.e.e 8 1.a even 1 1 trivial
1840.2.e.e 8 5.b even 2 1 inner
2070.2.d.f 8 12.b even 2 1
2070.2.d.f 8 60.h even 2 1
9200.2.a.cj 4 5.c odd 4 1
9200.2.a.cr 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\):

\( T_{3}^{8} + 19 T_{3}^{6} + 105 T_{3}^{4} + 176 T_{3}^{2} + 64 \)
\( T_{7}^{8} + 19 T_{7}^{6} + 105 T_{7}^{4} + 176 T_{7}^{2} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 64 + 176 T^{2} + 105 T^{4} + 19 T^{6} + T^{8} \)
$5$ \( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} \)
$7$ \( 64 + 176 T^{2} + 105 T^{4} + 19 T^{6} + T^{8} \)
$11$ \( ( -10 - 18 T - 3 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$13$ \( 16 + 72 T^{2} + 73 T^{4} + 23 T^{6} + T^{8} \)
$17$ \( 400 + 9976 T^{2} + 1441 T^{4} + 67 T^{6} + T^{8} \)
$19$ \( ( 370 - 42 T - 53 T^{2} + T^{3} + T^{4} )^{2} \)
$23$ \( ( 1 + T^{2} )^{4} \)
$29$ \( ( -400 - 360 T - 72 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( ( 100 - 32 T - 25 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$37$ \( 817216 + 128944 T^{2} + 6820 T^{4} + 144 T^{6} + T^{8} \)
$41$ \( ( -4016 + 552 T + 115 T^{2} - 23 T^{3} + T^{4} )^{2} \)
$43$ \( 341056 + 108528 T^{2} + 7364 T^{4} + 168 T^{6} + T^{8} \)
$47$ \( 256 + 1600 T^{2} + 2368 T^{4} + 100 T^{6} + T^{8} \)
$53$ \( ( 4232 + 158 T^{2} + T^{4} )^{2} \)
$59$ \( ( 1600 + 672 T - 20 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$61$ \( ( -226 + 258 T - 61 T^{2} - 9 T^{3} + T^{4} )^{2} \)
$67$ \( 27962944 + 2019120 T^{2} + 46084 T^{4} + 376 T^{6} + T^{8} \)
$71$ \( ( 32 + 144 T + 93 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$73$ \( 54405376 + 3354368 T^{2} + 61408 T^{4} + 432 T^{6} + T^{8} \)
$79$ \( ( 5920 - 304 T - 188 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$83$ \( 15178816 + 3996208 T^{2} + 74308 T^{4} + 472 T^{6} + T^{8} \)
$89$ \( ( -5840 - 744 T + 200 T^{2} + 30 T^{3} + T^{4} )^{2} \)
$97$ \( 16 + 2424 T^{2} + 1273 T^{4} + 91 T^{6} + T^{8} \)
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