Properties

Label 841.2.b.a
Level $841$
Weight $2$
Character orbit 841.b
Analytic conductor $6.715$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 841 = 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 841.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.71541880999\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_1 q^{3} + ( - \beta_{3} - 1) q^{4} + q^{5} + (\beta_{3} + 3) q^{6} - \beta_{3} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_1 q^{3} + ( - \beta_{3} - 1) q^{4} + q^{5} + (\beta_{3} + 3) q^{6} - \beta_{3} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - \beta_{3} q^{9} + \beta_1 q^{10} - \beta_{2} q^{11} + (\beta_{2} + 4 \beta_1) q^{12} + (\beta_{3} + 1) q^{13} + ( - \beta_{2} - 3 \beta_1) q^{14} - \beta_1 q^{15} + 3 q^{16} + 2 \beta_{2} q^{17} + ( - \beta_{2} - 3 \beta_1) q^{18} + ( - 3 \beta_{2} - 3 \beta_1) q^{19} + ( - \beta_{3} - 1) q^{20} + (\beta_{2} + 3 \beta_1) q^{21} - q^{22} + (2 \beta_{3} - 2) q^{23} + ( - 2 \beta_{3} - 5) q^{24} - 4 q^{25} + (\beta_{2} + 4 \beta_1) q^{26} + \beta_{2} q^{27} + (\beta_{3} + 8) q^{28} + (\beta_{3} + 3) q^{30} + (\beta_{2} - 4 \beta_1) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + q^{33} + 2 q^{34} - \beta_{3} q^{35} + (\beta_{3} + 8) q^{36} + ( - 2 \beta_{2} - 2 \beta_1) q^{37} + (3 \beta_{3} + 6) q^{38} + ( - \beta_{2} - 4 \beta_1) q^{39} + ( - \beta_{2} - 2 \beta_1) q^{40} + (5 \beta_{2} - \beta_1) q^{41} + ( - 3 \beta_{3} - 8) q^{42} + ( - 3 \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{2} - \beta_1) q^{44} - \beta_{3} q^{45} + (2 \beta_{2} + 4 \beta_1) q^{46} + (2 \beta_{2} - \beta_1) q^{47} - 3 \beta_1 q^{48} + q^{49} - 4 \beta_1 q^{50} - 2 q^{51} + ( - 2 \beta_{3} - 9) q^{52} + (3 \beta_{3} + 1) q^{53} + q^{54} - \beta_{2} q^{55} + ( - \beta_{2} + 5 \beta_1) q^{56} + ( - 3 \beta_{3} - 6) q^{57} + ( - 2 \beta_{3} + 2) q^{59} + (\beta_{2} + 4 \beta_1) q^{60} + 2 \beta_1 q^{61} + (4 \beta_{3} + 13) q^{62} + 8 q^{63} + (\beta_{3} + 7) q^{64} + (\beta_{3} + 1) q^{65} + \beta_1 q^{66} - 2 \beta_{3} q^{67} + (4 \beta_{2} + 2 \beta_1) q^{68} + ( - 2 \beta_{2} - 4 \beta_1) q^{69} + ( - \beta_{2} - 3 \beta_1) q^{70} + (\beta_{3} + 6) q^{71} + ( - \beta_{2} + 5 \beta_1) q^{72} + (2 \beta_{2} + 2 \beta_1) q^{73} + (2 \beta_{3} + 4) q^{74} + 4 \beta_1 q^{75} + ( - 3 \beta_{2} + 9 \beta_1) q^{76} + ( - 3 \beta_{2} - \beta_1) q^{77} + (4 \beta_{3} + 11) q^{78} + \beta_1 q^{79} + 3 q^{80} + ( - 3 \beta_{3} - 1) q^{81} + (\beta_{3} + 8) q^{82} + (2 \beta_{3} + 2) q^{83} + ( - \beta_{2} - 11 \beta_1) q^{84} + 2 \beta_{2} q^{85} + (2 \beta_{3} + 3) q^{86} + (\beta_{3} - 1) q^{88} + ( - \beta_{2} + 5 \beta_1) q^{89} + ( - \beta_{2} - 3 \beta_1) q^{90} + ( - \beta_{3} - 8) q^{91} - 14 q^{92} + ( - 4 \beta_{3} - 13) q^{93} + (\beta_{3} + 5) q^{94} + ( - 3 \beta_{2} - 3 \beta_1) q^{95} + ( - \beta_{3} - 1) q^{96} + ( - 5 \beta_{2} + \beta_1) q^{97} + \beta_1 q^{98} + ( - 3 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5} + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} + 12 q^{6} + 4 q^{13} + 12 q^{16} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 20 q^{24} - 16 q^{25} + 32 q^{28} + 12 q^{30} + 4 q^{33} + 8 q^{34} + 32 q^{36} + 24 q^{38} - 32 q^{42} + 4 q^{49} - 8 q^{51} - 36 q^{52} + 4 q^{53} + 4 q^{54} - 24 q^{57} + 8 q^{59} + 52 q^{62} + 32 q^{63} + 28 q^{64} + 4 q^{65} + 24 q^{71} + 16 q^{74} + 44 q^{78} + 12 q^{80} - 4 q^{81} + 32 q^{82} + 8 q^{83} + 12 q^{86} - 4 q^{88} - 32 q^{91} - 56 q^{92} - 52 q^{93} + 20 q^{94} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/841\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
840.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 2.41421i −3.82843 1.00000 5.82843 −2.82843 4.41421i −2.82843 2.41421i
840.2 0.414214i 0.414214i 1.82843 1.00000 0.171573 2.82843 1.58579i 2.82843 0.414214i
840.3 0.414214i 0.414214i 1.82843 1.00000 0.171573 2.82843 1.58579i 2.82843 0.414214i
840.4 2.41421i 2.41421i −3.82843 1.00000 5.82843 −2.82843 4.41421i −2.82843 2.41421i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.b.a 4
29.b even 2 1 inner 841.2.b.a 4
29.c odd 4 1 29.2.a.a 2
29.c odd 4 1 841.2.a.d 2
29.d even 7 6 841.2.e.k 24
29.e even 14 6 841.2.e.k 24
29.f odd 28 6 841.2.d.f 12
29.f odd 28 6 841.2.d.j 12
87.f even 4 1 261.2.a.d 2
87.f even 4 1 7569.2.a.c 2
116.e even 4 1 464.2.a.h 2
145.e even 4 1 725.2.b.b 4
145.f odd 4 1 725.2.a.b 2
145.j even 4 1 725.2.b.b 4
203.g even 4 1 1421.2.a.j 2
232.k even 4 1 1856.2.a.w 2
232.l odd 4 1 1856.2.a.r 2
319.f even 4 1 3509.2.a.j 2
348.k odd 4 1 4176.2.a.bq 2
377.i odd 4 1 4901.2.a.g 2
435.l even 4 1 6525.2.a.o 2
493.h odd 4 1 8381.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 29.c odd 4 1
261.2.a.d 2 87.f even 4 1
464.2.a.h 2 116.e even 4 1
725.2.a.b 2 145.f odd 4 1
725.2.b.b 4 145.e even 4 1
725.2.b.b 4 145.j even 4 1
841.2.a.d 2 29.c odd 4 1
841.2.b.a 4 1.a even 1 1 trivial
841.2.b.a 4 29.b even 2 1 inner
841.2.d.f 12 29.f odd 28 6
841.2.d.j 12 29.f odd 28 6
841.2.e.k 24 29.d even 7 6
841.2.e.k 24 29.e even 14 6
1421.2.a.j 2 203.g even 4 1
1856.2.a.r 2 232.l odd 4 1
1856.2.a.w 2 232.k even 4 1
3509.2.a.j 2 319.f even 4 1
4176.2.a.bq 2 348.k odd 4 1
4901.2.a.g 2 377.i odd 4 1
6525.2.a.o 2 435.l even 4 1
7569.2.a.c 2 87.f even 4 1
8381.2.a.e 2 493.h odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(841, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 118T^{2} + 1681 \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
$43$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$47$ \( T^{4} + 38T^{2} + 289 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T - 71)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$67$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$83$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
$97$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
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