Properties

Label 1638.2.x.c
Level $1638$
Weight $2$
Character orbit 1638.x
Analytic conductor $13.079$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.836829184.2
Defining polynomial: \(x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} -\beta_{7} q^{4} + ( -\beta_{2} + \beta_{5} ) q^{5} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{3} q^{2} -\beta_{7} q^{4} + ( -\beta_{2} + \beta_{5} ) q^{5} + ( -1 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + \beta_{2} q^{8} + ( -\beta_{2} - \beta_{3} + \beta_{6} ) q^{10} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{13} + ( -\beta_{1} + \beta_{6} ) q^{14} - q^{16} + ( -2 + \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} + ( -\beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( -\beta_{3} + \beta_{4} ) q^{20} + ( -2 + \beta_{4} + \beta_{5} - \beta_{6} ) q^{22} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 5 \beta_{7} ) q^{23} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{25} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{26} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{28} + ( -4 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{29} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{31} + \beta_{3} q^{32} + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{34} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{35} + ( 3 \beta_{2} - \beta_{5} ) q^{37} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{38} + ( -\beta_{1} - \beta_{7} ) q^{40} + ( -1 - \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{41} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{7} ) q^{43} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{44} + ( 3 - \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{46} + 2 \beta_{3} q^{47} + ( 4 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{49} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{50} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{7} ) q^{52} + ( -6 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{53} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{7} ) q^{55} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{56} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{58} + ( 1 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 4 \beta_{6} - 5 \beta_{7} ) q^{59} + ( -5 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{61} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{62} + \beta_{7} q^{64} + ( 2 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{65} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{6} - 3 \beta_{7} ) q^{67} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{68} + ( -\beta_{1} - 4 \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{70} + ( 4 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{71} + ( -4 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} + 4 \beta_{6} - 4 \beta_{7} ) q^{73} + ( -2 + \beta_{2} + \beta_{3} - \beta_{6} ) q^{74} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{76} + ( -3 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{7} ) q^{77} + ( -\beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{79} + ( \beta_{2} - \beta_{5} ) q^{80} + ( 6 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{82} + ( -7 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - 6 \beta_{7} ) q^{83} + ( 2 - 3 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{85} + ( 2 - 3 \beta_{2} + \beta_{5} + 2 \beta_{7} ) q^{86} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{88} + ( -3 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{89} + ( -6 + \beta_{1} - 5 \beta_{2} - 6 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{91} + ( 4 - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{92} + 2 \beta_{7} q^{94} + ( -\beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{7} ) q^{95} + ( -4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{97} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{5} + O(q^{10}) \) \( 8q + 4q^{5} + 4q^{10} + 16q^{13} + 4q^{14} - 8q^{16} - 4q^{17} - 8q^{19} + 4q^{20} - 12q^{22} + 16q^{26} - 8q^{28} - 12q^{29} - 8q^{31} - 8q^{34} + 24q^{35} - 4q^{37} + 4q^{38} - 12q^{41} + 24q^{46} + 32q^{49} + 8q^{50} - 4q^{52} - 40q^{53} + 4q^{56} + 4q^{58} + 8q^{59} - 8q^{62} + 12q^{65} + 32q^{67} + 8q^{70} + 12q^{71} + 20q^{73} - 20q^{74} - 8q^{76} - 8q^{77} + 24q^{79} - 4q^{80} + 40q^{82} - 44q^{83} + 20q^{85} + 20q^{86} - 16q^{89} - 28q^{91} + 28q^{92} - 8q^{97} + 16q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + \nu^{4} + 11 \nu^{3} + 7 \nu^{2} + 26 \nu + 6 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 11 \nu^{3} + 7 \nu^{2} - 26 \nu + 6 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + \nu^{5} + 11 \nu^{4} + 7 \nu^{3} + 26 \nu^{2} + 6 \nu \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - \nu^{5} + 11 \nu^{4} - 7 \nu^{3} + 26 \nu^{2} - 6 \nu \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 11 \nu^{4} + 30 \nu^{2} + 12 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 12 \nu^{5} + 41 \nu^{3} + 38 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} - \beta_{4} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-7 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-11 \beta_{5} + 11 \beta_{4} + 7 \beta_{3} - 7 \beta_{2} + 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(51 \beta_{6} - 47 \beta_{5} - 47 \beta_{4} - 44 \beta_{3} - 44 \beta_{2} - 87\)
\(\nu^{7}\)\(=\)\(8 \beta_{7} + 91 \beta_{5} - 91 \beta_{4} - 43 \beta_{3} + 43 \beta_{2} - 181 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\beta_{7}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.222191i
2.63640i
1.65222i
2.06644i
0.222191i
2.63640i
1.65222i
2.06644i
−0.707107 0.707107i 0 1.00000i −0.864220 + 0.864220i 0 −2.02133 + 1.70711i 0.707107 0.707107i 0 1.22219
307.2 −0.707107 0.707107i 0 1.00000i 1.15711 1.15711i 0 2.02133 + 1.70711i 0.707107 0.707107i 0 −1.63640
307.3 0.707107 + 0.707107i 0 1.00000i −0.461191 + 0.461191i 0 −2.62949 + 0.292893i −0.707107 + 0.707107i 0 −0.652223
307.4 0.707107 + 0.707107i 0 1.00000i 2.16830 2.16830i 0 2.62949 + 0.292893i −0.707107 + 0.707107i 0 3.06644
811.1 −0.707107 + 0.707107i 0 1.00000i −0.864220 0.864220i 0 −2.02133 1.70711i 0.707107 + 0.707107i 0 1.22219
811.2 −0.707107 + 0.707107i 0 1.00000i 1.15711 + 1.15711i 0 2.02133 1.70711i 0.707107 + 0.707107i 0 −1.63640
811.3 0.707107 0.707107i 0 1.00000i −0.461191 0.461191i 0 −2.62949 0.292893i −0.707107 0.707107i 0 −0.652223
811.4 0.707107 0.707107i 0 1.00000i 2.16830 + 2.16830i 0 2.62949 0.292893i −0.707107 0.707107i 0 3.06644
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 811.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.x.c 8
3.b odd 2 1 546.2.o.b 8
7.b odd 2 1 1638.2.x.a 8
13.d odd 4 1 1638.2.x.a 8
21.c even 2 1 546.2.o.c yes 8
39.f even 4 1 546.2.o.c yes 8
91.i even 4 1 inner 1638.2.x.c 8
273.o odd 4 1 546.2.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.o.b 8 3.b odd 2 1
546.2.o.b 8 273.o odd 4 1
546.2.o.c yes 8 21.c even 2 1
546.2.o.c yes 8 39.f even 4 1
1638.2.x.a 8 7.b odd 2 1
1638.2.x.a 8 13.d odd 4 1
1638.2.x.c 8 1.a even 1 1 trivial
1638.2.x.c 8 91.i even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( 2401 - 784 T^{2} + 130 T^{4} - 16 T^{6} + T^{8} \)
$11$ \( 256 + 768 T + 1152 T^{2} + 816 T^{3} + 321 T^{4} + 48 T^{5} + T^{8} \)
$13$ \( 28561 - 35152 T + 19266 T^{2} - 6656 T^{3} + 1890 T^{4} - 512 T^{5} + 114 T^{6} - 16 T^{7} + T^{8} \)
$17$ \( ( -8 + 32 T - 23 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$19$ \( 4096 + 6144 T + 4608 T^{2} + 1376 T^{3} + 209 T^{4} + 24 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$23$ \( 454276 + 103396 T^{2} + 6669 T^{4} + 146 T^{6} + T^{8} \)
$29$ \( ( 98 - 224 T - 45 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$31$ \( 4096 - 4096 T + 2048 T^{2} + 256 T^{3} + 16 T^{4} - 32 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$37$ \( 64 - 192 T + 288 T^{2} + 328 T^{3} + 209 T^{4} - 36 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$41$ \( 4096 - 22528 T + 61952 T^{2} + 11200 T^{3} + 1028 T^{4} - 56 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} \)
$43$ \( 1024 + 6848 T^{2} + 1585 T^{4} + 86 T^{6} + T^{8} \)
$47$ \( ( 16 + T^{4} )^{2} \)
$53$ \( ( 128 - 288 T + 66 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$59$ \( 1993744 + 1107008 T + 307328 T^{2} - 136096 T^{3} + 32520 T^{4} + 720 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$61$ \( 134606404 + 6081228 T^{2} + 87229 T^{4} + 502 T^{6} + T^{8} \)
$67$ \( 10837264 - 3371008 T + 524288 T^{2} - 27520 T^{3} + 12360 T^{4} - 3456 T^{5} + 512 T^{6} - 32 T^{7} + T^{8} \)
$71$ \( 817216 - 1417472 T + 1229312 T^{2} - 208416 T^{3} + 17684 T^{4} - 56 T^{5} + 72 T^{6} - 12 T^{7} + T^{8} \)
$73$ \( 23078416 + 7571104 T + 1241888 T^{2} - 97768 T^{3} + 5521 T^{4} + 884 T^{5} + 200 T^{6} - 20 T^{7} + T^{8} \)
$79$ \( ( -1264 + 1008 T - 70 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$83$ \( 1459264 + 2126080 T + 1548800 T^{2} + 513568 T^{3} + 101268 T^{4} + 12408 T^{5} + 968 T^{6} + 44 T^{7} + T^{8} \)
$89$ \( 80656 + 99968 T + 61952 T^{2} + 14400 T^{3} + 1352 T^{4} - 96 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$97$ \( 295936 - 1027072 T + 1782272 T^{2} + 380800 T^{3} + 40528 T^{4} + 256 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
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