Properties

Label 322.2.e.c
Level $322$
Weight $2$
Character orbit 322.e
Analytic conductor $2.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1767277521.3
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + 1) q^{2} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4}) q^{3} - \beta_{6} q^{4} + (\beta_{7} + \beta_{6} + \beta_{3} - 1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} - q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + 1) q^{2} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4}) q^{3} - \beta_{6} q^{4} + (\beta_{7} + \beta_{6} + \beta_{3} - 1) q^{5} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{7} + \beta_{3} - \beta_1) q^{7} - q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{9} + (\beta_{7} + \beta_{6}) q^{10} + (2 \beta_{6} + \beta_{4} + \beta_{2} + \beta_1) q^{11} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{12} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{13} + (\beta_{7} + \beta_{5} - \beta_1) q^{14} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{15} + (\beta_{6} - 1) q^{16} + ( - \beta_{7} - 4 \beta_{6}) q^{17} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{18} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} + 1) q^{19} + ( - \beta_{3} + 1) q^{20} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 3) q^{21} + ( - \beta_{5} + \beta_{4} + \beta_1 + 2) q^{22} + (\beta_{6} - 1) q^{23} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4}) q^{24} + ( - 3 \beta_{7} + \beta_{5} + \beta_{4}) q^{25} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{26} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 1) q^{27} + (\beta_{5} - \beta_{3}) q^{28} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{29} + ( - 3 \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{30} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{31} + \beta_{6} q^{32} + ( - 3 \beta_{7} + \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{33} + (\beta_{3} - 4) q^{34} + ( - 2 \beta_{7} - 5 \beta_{6} + \beta_{5} + 1) q^{35} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{36} + (2 \beta_{7} + 6 \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 6) q^{37} + (\beta_{7} - \beta_{6} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{38} + (\beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{39} + ( - \beta_{7} - \beta_{6} - \beta_{3} + 1) q^{40} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1) q^{41} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{42} + (\beta_{5} - \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{43} + ( - 2 \beta_{6} - \beta_{5} - \beta_{2} + 2) q^{44} + ( - 3 \beta_{7} - 6 \beta_{6}) q^{45} + \beta_{6} q^{46} + (\beta_{7} + 5 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{47} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{48} + ( - \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{49} + (3 \beta_{3} - \beta_{2} + \beta_1) q^{50} + (6 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + \cdots - 6) q^{51}+ \cdots + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - q^{3} - 4 q^{4} - 3 q^{5} - 2 q^{6} - q^{7} - 8 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - q^{3} - 4 q^{4} - 3 q^{5} - 2 q^{6} - q^{7} - 8 q^{8} - 7 q^{9} + 3 q^{10} + 6 q^{11} - q^{12} - 2 q^{13} + q^{14} + 6 q^{15} - 4 q^{16} - 15 q^{17} + 7 q^{18} + q^{19} + 6 q^{20} - q^{21} + 12 q^{22} - 4 q^{23} + q^{24} + 5 q^{25} - q^{26} - 10 q^{27} + 2 q^{28} + 12 q^{29} + 3 q^{30} - 8 q^{31} + 4 q^{32} - 9 q^{33} - 30 q^{34} - 6 q^{35} + 14 q^{36} - 8 q^{37} - q^{38} + 25 q^{39} + 3 q^{40} + 18 q^{41} - 14 q^{42} + 28 q^{43} + 6 q^{44} - 21 q^{45} + 4 q^{46} - 9 q^{47} + 2 q^{48} + 5 q^{49} + 10 q^{50} - 6 q^{51} + q^{52} + 3 q^{53} - 5 q^{54} - 24 q^{55} + q^{56} + 46 q^{57} + 6 q^{58} - 12 q^{59} - 3 q^{60} - 11 q^{61} - 16 q^{62} - 19 q^{63} + 8 q^{64} + 9 q^{66} + q^{67} - 15 q^{68} + 2 q^{69} - 30 q^{70} - 6 q^{71} + 7 q^{72} + 4 q^{73} + 8 q^{74} - 22 q^{75} - 2 q^{76} - 9 q^{77} + 50 q^{78} - 5 q^{79} - 3 q^{80} + 8 q^{81} + 9 q^{82} + 24 q^{83} - 13 q^{84} + 48 q^{85} + 14 q^{86} + 9 q^{87} - 6 q^{88} - 27 q^{89} - 42 q^{90} - 26 q^{91} + 8 q^{92} + 25 q^{93} + 9 q^{94} + 3 q^{95} + q^{96} + 4 q^{97} - 26 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\nu^{7} - 55\nu^{6} + 338\nu^{5} - 909\nu^{4} + 2308\nu^{3} - 5301\nu^{2} + 11263\nu - 6620 ) / 8102 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -655\nu^{7} + 938\nu^{6} - 314\nu^{5} + 6885\nu^{4} - 22495\nu^{3} + 14321\nu^{2} - 38221\nu + 14204 ) / 8102 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -367\nu^{7} + 674\nu^{6} - 312\nu^{5} + 3332\nu^{4} - 13037\nu^{3} + 12372\nu^{2} - 18187\nu + 6734 ) / 4051 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 ) / 8102 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{6} - 4\beta_{4} + 2\beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 5\beta_{6} - 10\beta_{5} + 2\beta_{2} + 11\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{6} - 22\beta_{5} - 20\beta_{4} - 2\beta_{3} - 4\beta_{2} - 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{7} - 42\beta_{6} + \beta_{5} - 2\beta_{4} + 4\beta_{3} + 62\beta_{2} - 11\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} + 5\beta_{6} - 115\beta_{5} + 88\beta_{4} - 62\beta_{3} + 68\beta_{2} + 30\beta _1 - 118 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
0.373419 0.0835272i
−1.54162 1.88572i
0.0512865 + 1.21608i
2.11692 0.978886i
0.373419 + 0.0835272i
−1.54162 + 1.88572i
0.0512865 1.21608i
2.11692 + 0.978886i
0.500000 0.866025i −1.31242 2.27317i −0.500000 0.866025i −1.68584 + 2.91996i −2.62484 −1.55926 + 2.13746i −1.00000 −1.94488 + 3.36864i 1.68584 + 2.91996i
93.2 0.500000 0.866025i −0.937623 1.62401i −0.500000 0.866025i 0.603998 1.04616i −1.87525 2.64562 0.0264594i −1.00000 −0.258274 + 0.447344i −0.603998 1.04616i
93.3 0.500000 0.866025i 0.319548 + 0.553474i −0.500000 0.866025i 0.268262 0.464643i 0.639096 0.716975 2.54675i −1.00000 1.29578 2.24435i −0.268262 0.464643i
93.4 0.500000 0.866025i 1.43049 + 2.47769i −0.500000 0.866025i −0.686423 + 1.18892i 2.86099 −2.30334 + 1.30178i −1.00000 −2.59262 + 4.49055i 0.686423 + 1.18892i
277.1 0.500000 + 0.866025i −1.31242 + 2.27317i −0.500000 + 0.866025i −1.68584 2.91996i −2.62484 −1.55926 2.13746i −1.00000 −1.94488 3.36864i 1.68584 2.91996i
277.2 0.500000 + 0.866025i −0.937623 + 1.62401i −0.500000 + 0.866025i 0.603998 + 1.04616i −1.87525 2.64562 + 0.0264594i −1.00000 −0.258274 0.447344i −0.603998 + 1.04616i
277.3 0.500000 + 0.866025i 0.319548 0.553474i −0.500000 + 0.866025i 0.268262 + 0.464643i 0.639096 0.716975 + 2.54675i −1.00000 1.29578 + 2.24435i −0.268262 + 0.464643i
277.4 0.500000 + 0.866025i 1.43049 2.47769i −0.500000 + 0.866025i −0.686423 1.18892i 2.86099 −2.30334 1.30178i −1.00000 −2.59262 4.49055i 0.686423 1.18892i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.e.c 8
7.c even 3 1 inner 322.2.e.c 8
7.c even 3 1 2254.2.a.v 4
7.d odd 6 1 2254.2.a.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.c 8 1.a even 1 1 trivial
322.2.e.c 8 7.c even 3 1 inner
2254.2.a.q 4 7.d odd 6 1
2254.2.a.v 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + T_{3}^{7} + 10T_{3}^{6} + 9T_{3}^{5} + 81T_{3}^{4} + 63T_{3}^{3} + 162T_{3}^{2} - 81T_{3} + 81 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + 10 T^{6} + 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + 12 T^{6} + T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - 2 T^{6} - 17 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + 36 T^{6} + \cdots + 3969 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} - 24 T^{2} - 70 T - 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 15 T^{7} + 147 T^{6} + \cdots + 15129 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + 46 T^{6} - 5 T^{5} + \cdots + 3481 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 6 T^{3} - 15 T^{2} + 100 T - 81)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + 70 T^{6} + \cdots + 7569 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + 184 T^{6} + \cdots + 14432401 \) Copy content Toggle raw display
$41$ \( (T^{4} - 9 T^{3} - 18 T^{2} + 306 T - 567)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} - 18 T^{2} + 371 T + 767)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 9 T^{7} + 126 T^{6} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( T^{8} - 3 T^{7} + 174 T^{6} + \cdots + 11350161 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + 144 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{8} + 11 T^{7} + 319 T^{6} + \cdots + 58537801 \) Copy content Toggle raw display
$67$ \( T^{8} - T^{7} + 85 T^{6} + \cdots + 790321 \) Copy content Toggle raw display
$71$ \( (T^{4} + 3 T^{3} - 126 T^{2} + 420 T - 369)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 4 T^{7} + 106 T^{6} + \cdots + 2152089 \) Copy content Toggle raw display
$79$ \( T^{8} + 5 T^{7} + \cdots + 1315730529 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} - 21 T^{2} + 490 T - 1029)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 27 T^{7} + 558 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} - 219 T^{2} + 600 T + 2427)^{2} \) Copy content Toggle raw display
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