Properties

Label 1148.2.n.a
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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{3} + ( -\beta_{1} + \beta_{6} ) q^{5} + \beta_{2} q^{7} + ( 2 + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{3} + ( -\beta_{1} + \beta_{6} ) q^{5} + \beta_{2} q^{7} + ( 2 + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{9} + ( -\beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{11} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{6} ) q^{15} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{17} + ( 3 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{21} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{23} + ( -2 \beta_{2} - 2 \beta_{7} ) q^{25} + ( 5 - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{27} + ( 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{33} + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{35} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{37} + ( 3 - 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{39} + ( -3 + 4 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{41} + ( -5 \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{43} + ( 1 + \beta_{2} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{45} + ( 3 + 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{47} + \beta_{4} q^{49} + ( 4 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} + 4 \beta_{7} ) q^{51} + ( -2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{6} ) q^{53} + ( 3 + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{55} + ( 3 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{7} ) q^{57} + ( -2 + 3 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{59} + ( 2 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{61} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{7} ) q^{63} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{65} + ( -4 + \beta_{1} - 4 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{67} + ( 2 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{69} + ( \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{71} + ( -8 - 2 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 6 - 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - 2 \beta_{7} ) q^{75} + ( 1 + \beta_{2} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{77} + ( -11 - 3 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{79} + ( -2 - 4 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} ) q^{81} + ( -3 + 5 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} ) q^{83} + ( -6 - 4 \beta_{3} - 5 \beta_{4} - 5 \beta_{6} - 4 \beta_{7} ) q^{85} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -2 - 7 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{89} + ( 2 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( 9 \beta_{2} + 9 \beta_{6} ) q^{93} + ( -6 - 2 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} ) q^{95} + ( 1 + \beta_{2} + 3 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{97} + ( -5 \beta_{1} - 4 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} - 5 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} - 2q^{5} - 2q^{7} + 12q^{9} + O(q^{10}) \) \( 8q - 4q^{3} - 2q^{5} - 2q^{7} + 12q^{9} - 2q^{15} - 14q^{17} + 12q^{19} - 4q^{21} - 8q^{23} + 4q^{25} + 32q^{27} + 2q^{29} + 18q^{31} + 14q^{33} - 2q^{35} + 14q^{37} - 6q^{39} - 22q^{41} + 20q^{43} + 10q^{45} + 10q^{47} - 2q^{49} + 24q^{51} - 8q^{53} + 12q^{55} + 34q^{57} - 16q^{59} + 14q^{61} - 8q^{63} + 2q^{65} - 26q^{67} + 28q^{69} + 14q^{71} - 40q^{73} + 32q^{75} - 72q^{79} - 16q^{81} - 8q^{83} - 28q^{85} - 14q^{87} + 2q^{89} + 20q^{91} - 36q^{93} - 8q^{95} + 18q^{97} - 14q^{99} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{6}\)

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
−1.14412 0.831254i
1.14412 + 0.831254i
−1.14412 + 0.831254i
1.14412 0.831254i
−0.437016 + 1.34500i
0.437016 1.34500i
−0.437016 1.34500i
0.437016 + 1.34500i
0 −2.49207 0 0.335106 + 0.243469i 0 0.309017 + 0.951057i 0 3.21039 0
57.2 0 −0.744002 0 −1.95314 1.41904i 0 0.309017 + 0.951057i 0 −2.44646 0
141.1 0 −2.49207 0 0.335106 0.243469i 0 0.309017 0.951057i 0 3.21039 0
141.2 0 −0.744002 0 −1.95314 + 1.41904i 0 0.309017 0.951057i 0 −2.44646 0
365.1 0 −1.67021 0 0.746033 2.29605i 0 −0.809017 0.587785i 0 −0.210393 0
365.2 0 2.90628 0 −0.127999 + 0.393941i 0 −0.809017 0.587785i 0 5.44646 0
953.1 0 −1.67021 0 0.746033 + 2.29605i 0 −0.809017 + 0.587785i 0 −0.210393 0
953.2 0 2.90628 0 −0.127999 0.393941i 0 −0.809017 + 0.587785i 0 5.44646 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.a 8
41.d even 5 1 inner 1148.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.n.a 8 1.a even 1 1 trivial
1148.2.n.a 8 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 2 T_{3}^{3} - 7 T_{3}^{2} - 18 T_{3} - 9 \) acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -9 - 18 T - 7 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$5$ \( 1 - 2 T + 5 T^{2} - 12 T^{3} + 29 T^{4} + 12 T^{5} + 5 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$11$ \( 1681 - 2870 T + 2027 T^{2} - 160 T^{3} + 164 T^{4} + 50 T^{5} + 18 T^{6} + T^{8} \)
$13$ \( 81 + 270 T + 387 T^{2} + 180 T^{3} + 214 T^{4} + 110 T^{5} + 28 T^{6} + T^{8} \)
$17$ \( 81 + 432 T + 3996 T^{2} + 4686 T^{3} + 2614 T^{4} + 788 T^{5} + 139 T^{6} + 14 T^{7} + T^{8} \)
$19$ \( 201601 - 8082 T + 15943 T^{2} + 3096 T^{3} + 780 T^{4} - 354 T^{5} + 98 T^{6} - 12 T^{7} + T^{8} \)
$23$ \( 256 + 256 T + 1344 T^{2} - 768 T^{3} + 64 T^{4} + 144 T^{5} + 56 T^{6} + 8 T^{7} + T^{8} \)
$29$ \( 1681 - 738 T + 2715 T^{2} + 972 T^{3} + 429 T^{4} + 28 T^{5} - 5 T^{6} - 2 T^{7} + T^{8} \)
$31$ \( 531441 - 472392 T + 196830 T^{2} - 42282 T^{3} + 10044 T^{4} - 1728 T^{5} + 225 T^{6} - 18 T^{7} + T^{8} \)
$37$ \( 1681 + 1066 T + 2975 T^{2} + 624 T^{3} + 1309 T^{4} + 464 T^{5} + 55 T^{6} - 14 T^{7} + T^{8} \)
$41$ \( 2825761 + 1516262 T + 425293 T^{2} + 78064 T^{3} + 12465 T^{4} + 1904 T^{5} + 253 T^{6} + 22 T^{7} + T^{8} \)
$43$ \( 229441 - 340090 T + 229533 T^{2} - 80040 T^{3} + 18004 T^{4} - 2730 T^{5} + 302 T^{6} - 20 T^{7} + T^{8} \)
$47$ \( 271441 - 468900 T + 342378 T^{2} - 85130 T^{3} + 14564 T^{4} - 1640 T^{5} + 167 T^{6} - 10 T^{7} + T^{8} \)
$53$ \( 18800896 - 2289408 T + 630080 T^{2} + 4352 T^{3} + 1344 T^{4} - 112 T^{5} + 120 T^{6} + 8 T^{7} + T^{8} \)
$59$ \( 1681 + 4018 T + 30051 T^{2} - 8416 T^{3} + 884 T^{4} + 122 T^{5} + 114 T^{6} + 16 T^{7} + T^{8} \)
$61$ \( 2307361 - 2317994 T + 1026305 T^{2} - 201936 T^{3} + 26749 T^{4} - 2656 T^{5} + 265 T^{6} - 14 T^{7} + T^{8} \)
$67$ \( 6241 + 9638 T + 14351 T^{2} + 9904 T^{3} + 5119 T^{4} + 1612 T^{5} + 299 T^{6} + 26 T^{7} + T^{8} \)
$71$ \( 1185921 - 71874 T + 59895 T^{2} + 9504 T^{3} + 1519 T^{4} - 916 T^{5} + 195 T^{6} - 14 T^{7} + T^{8} \)
$73$ \( ( -2416 - 880 T + 36 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$79$ \( ( -18401 - 1152 T + 320 T^{2} + 36 T^{3} + T^{4} )^{2} \)
$83$ \( ( 1359 - 528 T - 160 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$89$ \( 641601 - 543078 T + 203265 T^{2} - 32688 T^{3} + 5389 T^{4} - 832 T^{5} + 105 T^{6} - 2 T^{7} + T^{8} \)
$97$ \( 58081 - 113752 T + 86580 T^{2} + 2318 T^{3} + 14334 T^{4} - 508 T^{5} + 155 T^{6} - 18 T^{7} + T^{8} \)
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