Properties

Label 22.3.d.a
Level $22$
Weight $3$
Character orbit 22.d
Analytic conductor $0.599$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 22.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.599456581593\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{6}+ \cdots + (\beta_{6} + 4 \beta_{5} - 4 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{6}+ \cdots + (33 \beta_{7} - 11 \beta_{6} - 33 \beta_{5} + 22 \beta_{4} + 22 \beta_{3} + \cdots - 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} - 20 q^{6} - 30 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} - 20 q^{6} - 30 q^{7} - 4 q^{9} - 4 q^{11} + 24 q^{12} + 30 q^{13} + 16 q^{14} + 42 q^{15} - 8 q^{16} + 30 q^{17} + 40 q^{18} - 30 q^{19} - 4 q^{20} + 24 q^{22} - 104 q^{23} - 40 q^{24} - 12 q^{25} - 96 q^{26} - 26 q^{27} - 40 q^{28} - 10 q^{29} - 60 q^{30} + 46 q^{31} - 14 q^{33} + 112 q^{34} + 70 q^{35} - 12 q^{36} + 6 q^{37} + 108 q^{38} + 130 q^{39} + 80 q^{40} + 250 q^{41} + 56 q^{42} - 12 q^{44} - 136 q^{45} - 160 q^{46} - 54 q^{47} - 8 q^{48} - 144 q^{49} - 80 q^{50} - 30 q^{51} - 40 q^{52} - 274 q^{53} - 26 q^{55} + 48 q^{56} - 130 q^{57} + 64 q^{58} + 50 q^{59} + 116 q^{60} + 50 q^{61} + 20 q^{62} - 20 q^{63} + 16 q^{64} - 136 q^{66} + 112 q^{67} + 60 q^{68} + 76 q^{69} + 4 q^{70} + 54 q^{71} - 80 q^{72} - 70 q^{73} - 40 q^{74} + 318 q^{75} + 266 q^{77} + 104 q^{78} + 370 q^{79} + 48 q^{80} + 180 q^{81} - 96 q^{82} - 150 q^{83} - 120 q^{84} - 330 q^{85} - 72 q^{86} + 72 q^{88} + 24 q^{89} + 160 q^{90} - 294 q^{91} - 112 q^{92} - 134 q^{93} - 20 q^{94} - 330 q^{95} - 18 q^{97} - 308 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.831254 + 1.14412i
0.831254 1.14412i
−1.34500 0.437016i
1.34500 + 0.437016i
−1.34500 + 0.437016i
1.34500 0.437016i
−0.831254 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i −1.32276 + 4.07104i −0.618034 1.90211i 6.27955 4.56236i −3.55822 4.89747i −2.67724 + 0.869888i 2.68999 + 0.874032i −7.54250 5.47994i 10.9771i
7.2 0.831254 1.14412i −0.295274 + 0.908759i −0.618034 1.90211i −2.42545 + 1.76219i 0.794285 + 1.09324i −3.70473 + 1.20374i −2.68999 0.874032i 6.54250 + 4.75340i 4.23984i
13.1 −1.34500 0.437016i 2.48527 1.80565i 1.61803 + 1.17557i −0.399565 1.22973i −4.13178 + 1.34250i −6.48527 + 8.92621i −1.66251 2.28825i 0.135021 0.415553i 1.82860i
13.2 1.34500 + 0.437016i −1.86723 + 1.35662i 1.61803 + 1.17557i −2.45454 7.55429i −3.10429 + 1.00865i −2.13277 + 2.93550i 1.66251 + 2.28825i −1.13502 + 3.49324i 11.2332i
17.1 −1.34500 + 0.437016i 2.48527 + 1.80565i 1.61803 1.17557i −0.399565 + 1.22973i −4.13178 1.34250i −6.48527 8.92621i −1.66251 + 2.28825i 0.135021 + 0.415553i 1.82860i
17.2 1.34500 0.437016i −1.86723 1.35662i 1.61803 1.17557i −2.45454 + 7.55429i −3.10429 1.00865i −2.13277 2.93550i 1.66251 2.28825i −1.13502 3.49324i 11.2332i
19.1 −0.831254 1.14412i −1.32276 4.07104i −0.618034 + 1.90211i 6.27955 + 4.56236i −3.55822 + 4.89747i −2.67724 0.869888i 2.68999 0.874032i −7.54250 + 5.47994i 10.9771i
19.2 0.831254 + 1.14412i −0.295274 0.908759i −0.618034 + 1.90211i −2.42545 1.76219i 0.794285 1.09324i −3.70473 1.20374i −2.68999 + 0.874032i 6.54250 4.75340i 4.23984i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.3.d.a 8
3.b odd 2 1 198.3.j.a 8
4.b odd 2 1 176.3.n.b 8
11.b odd 2 1 242.3.d.c 8
11.c even 5 1 242.3.b.d 8
11.c even 5 1 242.3.d.c 8
11.c even 5 1 242.3.d.d 8
11.c even 5 1 242.3.d.e 8
11.d odd 10 1 inner 22.3.d.a 8
11.d odd 10 1 242.3.b.d 8
11.d odd 10 1 242.3.d.d 8
11.d odd 10 1 242.3.d.e 8
33.f even 10 1 198.3.j.a 8
33.f even 10 1 2178.3.d.l 8
33.h odd 10 1 2178.3.d.l 8
44.g even 10 1 176.3.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.d.a 8 1.a even 1 1 trivial
22.3.d.a 8 11.d odd 10 1 inner
176.3.n.b 8 4.b odd 2 1
176.3.n.b 8 44.g even 10 1
198.3.j.a 8 3.b odd 2 1
198.3.j.a 8 33.f even 10 1
242.3.b.d 8 11.c even 5 1
242.3.b.d 8 11.d odd 10 1
242.3.d.c 8 11.b odd 2 1
242.3.d.c 8 11.c even 5 1
242.3.d.d 8 11.c even 5 1
242.3.d.d 8 11.d odd 10 1
242.3.d.e 8 11.c even 5 1
242.3.d.e 8 11.d odd 10 1
2178.3.d.l 8 33.f even 10 1
2178.3.d.l 8 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + 13 T^{6} - 16 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + 33 T^{6} + \cdots + 57121 \) Copy content Toggle raw display
$7$ \( T^{8} + 30 T^{7} + 473 T^{6} + \cdots + 192721 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} - 484 T^{5} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{8} - 30 T^{7} + \cdots + 2268521641 \) Copy content Toggle raw display
$17$ \( T^{8} - 30 T^{7} + 97 T^{6} + \cdots + 22934521 \) Copy content Toggle raw display
$19$ \( T^{8} + 30 T^{7} + \cdots + 3189877441 \) Copy content Toggle raw display
$23$ \( (T^{4} + 52 T^{3} + 124 T^{2} + \cdots - 90224)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 10 T^{7} + \cdots + 206760274681 \) Copy content Toggle raw display
$31$ \( T^{8} - 46 T^{7} + \cdots + 996728041 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + 317 T^{6} + \cdots + 20079361 \) Copy content Toggle raw display
$41$ \( T^{8} - 250 T^{7} + \cdots + 405257161 \) Copy content Toggle raw display
$43$ \( T^{8} + 3632 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$47$ \( T^{8} + 54 T^{7} + \cdots + 7428543721 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 189900554154481 \) Copy content Toggle raw display
$59$ \( T^{8} - 50 T^{7} + \cdots + 47196831300025 \) Copy content Toggle raw display
$61$ \( T^{8} - 50 T^{7} + \cdots + 4097365398025 \) Copy content Toggle raw display
$67$ \( (T^{4} - 56 T^{3} - 344 T^{2} + \cdots + 112576)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 54 T^{7} + \cdots + 14204072031241 \) Copy content Toggle raw display
$73$ \( T^{8} + 70 T^{7} + \cdots + 36635874025 \) Copy content Toggle raw display
$79$ \( T^{8} - 370 T^{7} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{8} + 150 T^{7} + \cdots + 4768279282321 \) Copy content Toggle raw display
$89$ \( (T^{4} - 12 T^{3} - 19836 T^{2} + \cdots - 22808304)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 18 T^{7} + \cdots + 185279454481 \) Copy content Toggle raw display
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