Properties

Label 176.4.a.i
Level $176$
Weight $4$
Character orbit 176.a
Self dual yes
Analytic conductor $10.384$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3843361610\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( 1 + 2 \beta ) q^{5} + ( -10 + \beta ) q^{7} + ( 22 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( 1 + 2 \beta ) q^{5} + ( -10 + \beta ) q^{7} + ( 22 + 2 \beta ) q^{9} + 11 q^{11} + ( 40 - 5 \beta ) q^{13} + ( 97 + 3 \beta ) q^{15} + ( -62 + 3 \beta ) q^{17} + ( -36 - 15 \beta ) q^{19} + ( 38 - 9 \beta ) q^{21} + ( 49 + 9 \beta ) q^{23} + ( 68 + 4 \beta ) q^{25} + ( 91 - 3 \beta ) q^{27} + ( 72 - 14 \beta ) q^{29} + ( 17 - 7 \beta ) q^{31} + ( 11 + 11 \beta ) q^{33} + ( 86 - 19 \beta ) q^{35} + ( 27 - 2 \beta ) q^{37} + ( -200 + 35 \beta ) q^{39} + ( 268 - \beta ) q^{41} + ( 30 + 4 \beta ) q^{43} + ( 214 + 46 \beta ) q^{45} + ( 136 + 30 \beta ) q^{47} + ( -195 - 20 \beta ) q^{49} + ( 82 - 59 \beta ) q^{51} + ( -246 - 14 \beta ) q^{53} + ( 11 + 22 \beta ) q^{55} + ( -756 - 51 \beta ) q^{57} + ( -317 + 33 \beta ) q^{59} + ( 420 + 46 \beta ) q^{61} + ( -124 + 2 \beta ) q^{63} + ( -440 + 75 \beta ) q^{65} + ( -377 + 5 \beta ) q^{67} + ( 481 + 58 \beta ) q^{69} + ( 339 - 19 \beta ) q^{71} + ( -200 - 117 \beta ) q^{73} + ( 260 + 72 \beta ) q^{75} + ( -110 + 11 \beta ) q^{77} + ( -158 - 164 \beta ) q^{79} + ( -647 + 34 \beta ) q^{81} + ( -234 - 30 \beta ) q^{83} + ( 226 - 121 \beta ) q^{85} + ( -600 + 58 \beta ) q^{87} + ( -921 - 82 \beta ) q^{89} + ( -640 + 90 \beta ) q^{91} + ( -319 + 10 \beta ) q^{93} + ( -1476 - 87 \beta ) q^{95} + ( 1097 + 36 \beta ) q^{97} + ( 242 + 22 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 20 q^{7} + 44 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{5} - 20 q^{7} + 44 q^{9} + 22 q^{11} + 80 q^{13} + 194 q^{15} - 124 q^{17} - 72 q^{19} + 76 q^{21} + 98 q^{23} + 136 q^{25} + 182 q^{27} + 144 q^{29} + 34 q^{31} + 22 q^{33} + 172 q^{35} + 54 q^{37} - 400 q^{39} + 536 q^{41} + 60 q^{43} + 428 q^{45} + 272 q^{47} - 390 q^{49} + 164 q^{51} - 492 q^{53} + 22 q^{55} - 1512 q^{57} - 634 q^{59} + 840 q^{61} - 248 q^{63} - 880 q^{65} - 754 q^{67} + 962 q^{69} + 678 q^{71} - 400 q^{73} + 520 q^{75} - 220 q^{77} - 316 q^{79} - 1294 q^{81} - 468 q^{83} + 452 q^{85} - 1200 q^{87} - 1842 q^{89} - 1280 q^{91} - 638 q^{93} - 2952 q^{95} + 2194 q^{97} + 484 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −5.92820 0 −12.8564 0 −16.9282 0 8.14359 0
1.2 0 7.92820 0 14.8564 0 −3.07180 0 35.8564 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.4.a.i 2
3.b odd 2 1 1584.4.a.bc 2
4.b odd 2 1 11.4.a.a 2
8.b even 2 1 704.4.a.n 2
8.d odd 2 1 704.4.a.p 2
11.b odd 2 1 1936.4.a.w 2
12.b even 2 1 99.4.a.c 2
20.d odd 2 1 275.4.a.b 2
20.e even 4 2 275.4.b.c 4
28.d even 2 1 539.4.a.e 2
44.c even 2 1 121.4.a.c 2
44.g even 10 4 121.4.c.f 8
44.h odd 10 4 121.4.c.c 8
52.b odd 2 1 1859.4.a.a 2
60.h even 2 1 2475.4.a.q 2
132.d odd 2 1 1089.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 4.b odd 2 1
99.4.a.c 2 12.b even 2 1
121.4.a.c 2 44.c even 2 1
121.4.c.c 8 44.h odd 10 4
121.4.c.f 8 44.g even 10 4
176.4.a.i 2 1.a even 1 1 trivial
275.4.a.b 2 20.d odd 2 1
275.4.b.c 4 20.e even 4 2
539.4.a.e 2 28.d even 2 1
704.4.a.n 2 8.b even 2 1
704.4.a.p 2 8.d odd 2 1
1089.4.a.v 2 132.d odd 2 1
1584.4.a.bc 2 3.b odd 2 1
1859.4.a.a 2 52.b odd 2 1
1936.4.a.w 2 11.b odd 2 1
2475.4.a.q 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} - 47 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(176))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -47 - 2 T + T^{2} \)
$5$ \( -191 - 2 T + T^{2} \)
$7$ \( 52 + 20 T + T^{2} \)
$11$ \( ( -11 + T )^{2} \)
$13$ \( 400 - 80 T + T^{2} \)
$17$ \( 3412 + 124 T + T^{2} \)
$19$ \( -9504 + 72 T + T^{2} \)
$23$ \( -1487 - 98 T + T^{2} \)
$29$ \( -4224 - 144 T + T^{2} \)
$31$ \( -2063 - 34 T + T^{2} \)
$37$ \( 537 - 54 T + T^{2} \)
$41$ \( 71776 - 536 T + T^{2} \)
$43$ \( 132 - 60 T + T^{2} \)
$47$ \( -24704 - 272 T + T^{2} \)
$53$ \( 51108 + 492 T + T^{2} \)
$59$ \( 48217 + 634 T + T^{2} \)
$61$ \( 74832 - 840 T + T^{2} \)
$67$ \( 140929 + 754 T + T^{2} \)
$71$ \( 97593 - 678 T + T^{2} \)
$73$ \( -617072 + 400 T + T^{2} \)
$79$ \( -1266044 + 316 T + T^{2} \)
$83$ \( 11556 + 468 T + T^{2} \)
$89$ \( 525489 + 1842 T + T^{2} \)
$97$ \( 1141201 - 2194 T + T^{2} \)
show more
show less