Properties

Label 1148.2.n.b
Level $1148$
Weight $2$
Character orbit 1148.n
Analytic conductor $9.167$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.64000000.2
Defining polynomial: \(x^{8} + 2 x^{6} + 4 x^{4} + 8 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{5} ) q^{3} + ( \beta_{2} + \beta_{7} ) q^{5} + \beta_{4} q^{7} + 2 \beta_{5} q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{5} ) q^{3} + ( \beta_{2} + \beta_{7} ) q^{5} + \beta_{4} q^{7} + 2 \beta_{5} q^{9} + ( 3 \beta_{2} + 3 \beta_{4} ) q^{11} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{13} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{15} + ( \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{19} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{21} + ( -1 - \beta_{2} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{25} + ( 1 - \beta_{5} ) q^{27} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{29} + ( 1 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{33} + ( \beta_{1} + \beta_{6} ) q^{35} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{7} ) q^{37} + ( 6 + 2 \beta_{1} + 6 \beta_{2} + 4 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{39} + ( -2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{41} + ( -4 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 4 \beta_{2} + 2 \beta_{7} ) q^{45} + ( 3 + \beta_{1} + 3 \beta_{2} + \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{47} + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} ) q^{49} + ( 4 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{51} + ( -2 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{53} + ( -3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{55} + ( 4 \beta_{1} + 6 \beta_{2} + 3 \beta_{4} + 6 \beta_{6} + 4 \beta_{7} ) q^{57} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{59} + ( \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{61} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{63} + ( -3 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{65} + ( -4 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{67} + ( -5 + 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{69} + ( -4 + 2 \beta_{1} - 6 \beta_{2} - 6 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{71} + ( -4 - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{73} + 2 \beta_{4} q^{75} + ( -3 - 3 \beta_{2} - 3 \beta_{4} ) q^{77} + ( 1 + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{79} + ( -1 - 6 \beta_{5} ) q^{81} + ( 3 - 4 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{83} + ( -4 + 4 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 4 \beta_{7} ) q^{85} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{87} + ( -6 \beta_{1} - \beta_{2} + \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{89} + ( 1 - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 11 + 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 11 \beta_{6} ) q^{93} + ( -6 - 4 \beta_{1} - 6 \beta_{2} - 4 \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{95} + ( -4 - 2 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{7} ) q^{97} + ( -6 \beta_{1} - 6 \beta_{3} - 6 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} - 2q^{5} - 2q^{7} + O(q^{10}) \) \( 8q + 8q^{3} - 2q^{5} - 2q^{7} - 12q^{11} + 10q^{13} - 6q^{15} + 12q^{17} - 2q^{19} - 2q^{21} - 12q^{23} + 4q^{25} + 8q^{27} - 2q^{29} + 14q^{31} - 12q^{33} - 2q^{35} + 14q^{37} + 30q^{39} - 18q^{41} - 14q^{43} - 8q^{45} + 28q^{47} - 2q^{49} + 28q^{51} + 4q^{53} - 12q^{55} - 30q^{57} + 4q^{59} - 4q^{61} - 30q^{65} - 28q^{67} - 36q^{69} - 44q^{73} - 4q^{75} - 12q^{77} + 16q^{79} - 8q^{81} + 40q^{83} - 12q^{85} + 22q^{87} + 14q^{89} + 42q^{93} - 30q^{95} - 38q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 2 x^{6} + 4 x^{4} + 8 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(4 \beta_{4}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{6}\)
\(\nu^{7}\)\(=\)\(8 \beta_{7}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
−0.437016 + 1.34500i
0.437016 1.34500i
−0.437016 1.34500i
0.437016 + 1.34500i
1.14412 0.831254i
−1.14412 + 0.831254i
1.14412 + 0.831254i
−1.14412 0.831254i
0 −0.414214 0 0.335106 + 0.243469i 0 0.309017 + 0.951057i 0 −2.82843 0
57.2 0 2.41421 0 −1.95314 1.41904i 0 0.309017 + 0.951057i 0 2.82843 0
141.1 0 −0.414214 0 0.335106 0.243469i 0 0.309017 0.951057i 0 −2.82843 0
141.2 0 2.41421 0 −1.95314 + 1.41904i 0 0.309017 0.951057i 0 2.82843 0
365.1 0 −0.414214 0 −0.127999 + 0.393941i 0 −0.809017 0.587785i 0 −2.82843 0
365.2 0 2.41421 0 0.746033 2.29605i 0 −0.809017 0.587785i 0 2.82843 0
953.1 0 −0.414214 0 −0.127999 0.393941i 0 −0.809017 + 0.587785i 0 −2.82843 0
953.2 0 2.41421 0 0.746033 + 2.29605i 0 −0.809017 + 0.587785i 0 2.82843 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 953.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.n.b 8
41.d even 5 1 inner 1148.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.n.b 8 1.a even 1 1 trivial
1148.2.n.b 8 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -1 - 2 T + T^{2} )^{4} \)
$5$ \( 1 - 2 T + 5 T^{2} - 12 T^{3} + 29 T^{4} + 12 T^{5} + 5 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$11$ \( ( 81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$13$ \( 625 + 1250 T + 1625 T^{2} + 1000 T^{3} + 325 T^{4} - 200 T^{5} + 65 T^{6} - 10 T^{7} + T^{8} \)
$17$ \( 57121 + 18642 T + 1375 T^{2} - 1428 T^{3} + 1534 T^{4} - 462 T^{5} + 100 T^{6} - 12 T^{7} + T^{8} \)
$19$ \( 1615441 + 602454 T + 143889 T^{2} + 14728 T^{3} + 1589 T^{4} - 104 T^{5} + 41 T^{6} + 2 T^{7} + T^{8} \)
$23$ \( 1 + 38 T + 545 T^{2} - 452 T^{3} + 1454 T^{4} + 22 T^{5} + 60 T^{6} + 12 T^{7} + T^{8} \)
$29$ \( 6241 + 5688 T + 9340 T^{2} + 1778 T^{3} + 254 T^{4} - 28 T^{5} + 5 T^{6} + 2 T^{7} + T^{8} \)
$31$ \( 194481 + 148176 T + 238140 T^{2} + 21714 T^{3} + 214 T^{4} - 116 T^{5} + 115 T^{6} - 14 T^{7} + T^{8} \)
$37$ \( 1681 + 1066 T + 2975 T^{2} + 624 T^{3} + 1309 T^{4} + 464 T^{5} + 55 T^{6} - 14 T^{7} + T^{8} \)
$41$ \( 2825761 + 1240578 T + 189953 T^{2} + 11316 T^{3} + 385 T^{4} + 276 T^{5} + 113 T^{6} + 18 T^{7} + T^{8} \)
$43$ \( 8288641 + 650654 T + 176075 T^{2} - 6044 T^{3} + 2629 T^{4} + 676 T^{5} + 155 T^{6} + 14 T^{7} + T^{8} \)
$47$ \( 201601 + 101474 T + 251479 T^{2} - 54932 T^{3} + 11134 T^{4} - 2334 T^{5} + 356 T^{6} - 28 T^{7} + T^{8} \)
$53$ \( 1597696 - 970752 T + 295296 T^{2} - 55616 T^{3} + 14144 T^{4} - 2048 T^{5} + 164 T^{6} - 4 T^{7} + T^{8} \)
$59$ \( 160801 + 34486 T + 26535 T^{2} - 636 T^{3} + 334 T^{4} + 54 T^{5} + 20 T^{6} - 4 T^{7} + T^{8} \)
$61$ \( 125238481 + 25090222 T + 5642791 T^{2} + 411236 T^{3} + 21334 T^{4} - 402 T^{5} + 44 T^{6} + 4 T^{7} + T^{8} \)
$67$ \( 24760576 + 11225856 T + 2664384 T^{2} + 399872 T^{3} + 54224 T^{4} + 5504 T^{5} + 476 T^{6} + 28 T^{7} + T^{8} \)
$71$ \( 430336 - 367360 T + 129728 T^{2} - 5120 T^{3} + 2624 T^{4} + 400 T^{5} + 72 T^{6} + T^{8} \)
$73$ \( ( 41 + 238 T + 129 T^{2} + 22 T^{3} + T^{4} )^{2} \)
$79$ \( ( -919 + 472 T - 42 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$83$ \( ( -151 - 60 T + 106 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$89$ \( 25816561 + 18108684 T + 4358572 T^{2} - 660878 T^{3} + 99830 T^{4} - 6692 T^{5} + 407 T^{6} - 14 T^{7} + T^{8} \)
$97$ \( 8814961 + 219706 T + 654029 T^{2} + 117492 T^{3} + 21109 T^{4} + 3804 T^{5} + 621 T^{6} + 38 T^{7} + T^{8} \)
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