Properties

Label 176.11.h.a
Level 176
Weight 11
Character orbit 176.h
Self dual yes
Analytic conductor 111.823
Analytic rank 0
Dimension 1
CM discriminant -11
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 176.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(111.822876471\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 475q^{3} - 3001q^{5} + 166576q^{9} + O(q^{10}) \) \( q - 475q^{3} - 3001q^{5} + 166576q^{9} + 161051q^{11} + 1425475q^{15} + 11910325q^{23} - 759624q^{25} - 51075325q^{27} - 3192323q^{31} - 76499225q^{33} - 137082625q^{37} - 499894576q^{45} - 151795250q^{47} + 282475249q^{49} + 375066650q^{53} - 483314051q^{55} + 813567973q^{59} - 2616638675q^{67} - 5657404375q^{69} - 783651827q^{71} + 360821400q^{75} + 14424633151q^{81} - 2870912977q^{89} + 1516353425q^{93} + 9454010975q^{97} + 26827231376q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0
0 −475.000 0 −3001.00 0 0 0 166576. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.11.h.a 1
4.b odd 2 1 11.11.b.a 1
11.b odd 2 1 CM 176.11.h.a 1
12.b even 2 1 99.11.c.a 1
44.c even 2 1 11.11.b.a 1
132.d odd 2 1 99.11.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.11.b.a 1 4.b odd 2 1
11.11.b.a 1 44.c even 2 1
99.11.c.a 1 12.b even 2 1
99.11.c.a 1 132.d odd 2 1
176.11.h.a 1 1.a even 1 1 trivial
176.11.h.a 1 11.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 475 \) acting on \(S_{11}^{\mathrm{new}}(176, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 475 T + 59049 T^{2} \)
$5$ \( 1 + 3001 T + 9765625 T^{2} \)
$7$ \( ( 1 - 16807 T )( 1 + 16807 T ) \)
$11$ \( 1 - 161051 T \)
$13$ \( ( 1 - 371293 T )( 1 + 371293 T ) \)
$17$ \( ( 1 - 1419857 T )( 1 + 1419857 T ) \)
$19$ \( ( 1 - 2476099 T )( 1 + 2476099 T ) \)
$23$ \( 1 - 11910325 T + 41426511213649 T^{2} \)
$29$ \( ( 1 - 20511149 T )( 1 + 20511149 T ) \)
$31$ \( 1 + 3192323 T + 819628286980801 T^{2} \)
$37$ \( 1 + 137082625 T + 4808584372417849 T^{2} \)
$41$ \( ( 1 - 115856201 T )( 1 + 115856201 T ) \)
$43$ \( ( 1 - 147008443 T )( 1 + 147008443 T ) \)
$47$ \( 1 + 151795250 T + 52599132235830049 T^{2} \)
$53$ \( 1 - 375066650 T + 174887470365513049 T^{2} \)
$59$ \( 1 - 813567973 T + 511116753300641401 T^{2} \)
$61$ \( ( 1 - 844596301 T )( 1 + 844596301 T ) \)
$67$ \( 1 + 2616638675 T + 1822837804551761449 T^{2} \)
$71$ \( 1 + 783651827 T + 3255243551009881201 T^{2} \)
$73$ \( ( 1 - 2073071593 T )( 1 + 2073071593 T ) \)
$79$ \( ( 1 - 3077056399 T )( 1 + 3077056399 T ) \)
$83$ \( ( 1 - 3939040643 T )( 1 + 3939040643 T ) \)
$89$ \( 1 + 2870912977 T + 31181719929966183601 T^{2} \)
$97$ \( 1 - 9454010975 T + 73742412689492826049 T^{2} \)
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