Properties

Label 22.4.c.b
Level $22$
Weight $4$
Character orbit 22.c
Analytic conductor $1.298$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.29804202013\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} + 71x^{6} - 141x^{5} + 2911x^{4} + 2710x^{3} + 75340x^{2} + 169400x + 5856400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{3} + 4 \beta_{2} q^{4} + ( - \beta_{6} + \beta_{5} + 4 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - 5) q^{5} + ( - 2 \beta_{7} - 2 \beta_{4} + 2) q^{6} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{7} + 8 \beta_{4} q^{8} + (2 \beta_{7} - 2 \beta_{5} + 28 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{3} + 4 \beta_{2} q^{4} + ( - \beta_{6} + \beta_{5} + 4 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - 5) q^{5} + ( - 2 \beta_{7} - 2 \beta_{4} + 2) q^{6} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{7} + 8 \beta_{4} q^{8} + (2 \beta_{7} - 2 \beta_{5} + 28 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 7) q^{9} + (2 \beta_{7} - 8 \beta_{4} + 8 \beta_{2} - 2 \beta_1 - 8) q^{10} + (2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 15 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + \cdots - 13) q^{11}+ \cdots + (57 \beta_{7} + 23 \beta_{6} + 104 \beta_{5} + 583 \beta_{4} - 770 \beta_{3} + \cdots - 736) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 5 q^{5} + 14 q^{6} - q^{7} + 16 q^{8} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 3 q^{3} - 8 q^{4} + 5 q^{5} + 14 q^{6} - q^{7} + 16 q^{8} - 21 q^{9} - 100 q^{10} - 155 q^{11} + 32 q^{12} + 7 q^{13} + 2 q^{14} + 211 q^{15} - 32 q^{16} + 161 q^{17} + 162 q^{18} - 272 q^{19} + 20 q^{20} - 50 q^{21} + 628 q^{23} + 56 q^{24} - 17 q^{25} + 96 q^{26} - 528 q^{27} + 16 q^{28} + 33 q^{29} - 422 q^{30} + 323 q^{31} - 256 q^{32} - 1144 q^{33} + 208 q^{34} - 697 q^{35} - 324 q^{36} + 49 q^{37} - 576 q^{38} + 391 q^{39} + 240 q^{40} + 361 q^{41} + 1430 q^{42} + 1442 q^{43} + 620 q^{44} + 2652 q^{45} - 416 q^{46} - 1069 q^{47} + 48 q^{48} - 709 q^{49} - 76 q^{50} - 1332 q^{51} - 192 q^{52} - 281 q^{53} - 1144 q^{54} - 7 q^{55} + 48 q^{56} - 438 q^{57} - 66 q^{58} - 128 q^{59} - 1116 q^{60} - 617 q^{61} + 1044 q^{62} + 694 q^{63} - 128 q^{64} - 138 q^{65} + 1248 q^{66} + 578 q^{67} + 644 q^{68} - 310 q^{69} + 34 q^{70} + 115 q^{71} + 168 q^{72} - 1487 q^{73} - 98 q^{74} - 1852 q^{75} - 128 q^{76} + 553 q^{77} - 4152 q^{78} + 71 q^{79} - 480 q^{80} + 1630 q^{81} + 658 q^{82} + 1942 q^{83} + 2960 q^{84} - 329 q^{85} + 2426 q^{86} + 2122 q^{87} + 560 q^{88} - 2202 q^{89} + 1286 q^{90} + 4523 q^{91} - 2088 q^{92} + 6019 q^{93} - 1332 q^{94} - 793 q^{95} - 96 q^{96} - 5128 q^{97} - 3292 q^{98} - 2213 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 71x^{6} - 141x^{5} + 2911x^{4} + 2710x^{3} + 75340x^{2} + 169400x + 5856400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22554025143 \nu^{7} + 151013926876 \nu^{6} - 1356924294556 \nu^{5} + 28230146638036 \nu^{4} - 111071208987556 \nu^{3} + \cdots + 48\!\cdots\!00 ) / 34\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14112299722 \nu^{7} + 955320989289 \nu^{6} + 1256420838616 \nu^{5} - 5250112809736 \nu^{4} + 12754848436776 \nu^{3} + \cdots - 66\!\cdots\!20 ) / 17\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 123788106187 \nu^{7} + 5732748190805 \nu^{6} - 11662655347575 \nu^{5} + 374269556401525 \nu^{4} + \cdots + 42\!\cdots\!20 ) / 68\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15778904729 \nu^{7} - 268932734519 \nu^{6} + 2855478562109 \nu^{5} - 16065997834439 \nu^{4} + 722720174659769 \nu^{3} + \cdots - 12\!\cdots\!00 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15728307097 \nu^{7} - 23011705883 \nu^{6} - 3489045673392 \nu^{5} - 619537249643 \nu^{4} - 38031215360567 \nu^{3} + \cdots - 22\!\cdots\!05 ) / 77\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 254952731119 \nu^{7} - 130622718559 \nu^{6} + 16218883337689 \nu^{5} - 18738040845679 \nu^{4} + 629078544513089 \nu^{3} + \cdots + 32\!\cdots\!00 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{5} - 8\beta_{4} + 8\beta_{3} + 54\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{6} + 47\beta_{5} + 46\beta_{4} + 8\beta_{3} - 8\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 109\beta_{7} + 16\beta_{6} - 16\beta_{5} + 2226\beta_{4} - 3034\beta_{3} - 3034\beta_{2} - 2210 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -824\beta_{7} - 3143\beta_{6} - 1608\beta_{4} + 1608\beta_{2} + 824\beta _1 - 7549 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2432\beta_{7} + 2432\beta_{5} + 176314\beta_{3} + 63048\beta_{2} - 6725\beta _1 + 63048 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 185471 \beta_{7} + 185471 \beta_{6} - 119991 \beta_{5} + 185128 \beta_{4} - 185128 \beta_{3} - 513934 \beta_{2} - 119991 \beta _1 + 185471 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−4.79501 + 3.48378i
5.60402 4.07156i
−2.53202 7.79275i
2.22300 + 6.84169i
−2.53202 + 7.79275i
2.22300 6.84169i
−4.79501 3.48378i
5.60402 + 4.07156i
1.61803 1.17557i −1.33153 4.09803i 1.23607 3.80423i 6.52241 + 4.73881i −6.97198 5.06544i −8.05890 + 24.8027i −2.47214 7.60845i 6.82261 4.95692i 16.1243
3.2 1.61803 1.17557i 2.64055 + 8.12677i 1.23607 3.80423i −10.3036 7.48598i 13.8261 + 10.0452i 7.24988 22.3128i −2.47214 7.60845i −37.2284 + 27.0480i −25.4718
5.1 −0.618034 1.90211i −6.12891 4.45291i −3.23607 + 2.35114i 1.67119 5.14341i −4.68207 + 14.4099i 17.9196 13.0193i 6.47214 + 4.70228i 9.39163 + 28.9045i −10.8162
5.2 −0.618034 1.90211i 6.31989 + 4.59167i −3.23607 + 2.35114i 4.60996 14.1880i 4.82797 14.8590i −17.6106 + 12.7948i 6.47214 + 4.70228i 10.5141 + 32.3592i −29.8363
9.1 −0.618034 + 1.90211i −6.12891 + 4.45291i −3.23607 2.35114i 1.67119 + 5.14341i −4.68207 14.4099i 17.9196 + 13.0193i 6.47214 4.70228i 9.39163 28.9045i −10.8162
9.2 −0.618034 + 1.90211i 6.31989 4.59167i −3.23607 2.35114i 4.60996 + 14.1880i 4.82797 + 14.8590i −17.6106 12.7948i 6.47214 4.70228i 10.5141 32.3592i −29.8363
15.1 1.61803 + 1.17557i −1.33153 + 4.09803i 1.23607 + 3.80423i 6.52241 4.73881i −6.97198 + 5.06544i −8.05890 24.8027i −2.47214 + 7.60845i 6.82261 + 4.95692i 16.1243
15.2 1.61803 + 1.17557i 2.64055 8.12677i 1.23607 + 3.80423i −10.3036 + 7.48598i 13.8261 10.0452i 7.24988 + 22.3128i −2.47214 + 7.60845i −37.2284 27.0480i −25.4718
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.4.c.b 8
3.b odd 2 1 198.4.f.d 8
4.b odd 2 1 176.4.m.b 8
11.b odd 2 1 242.4.c.q 8
11.c even 5 1 inner 22.4.c.b 8
11.c even 5 1 242.4.a.n 4
11.c even 5 2 242.4.c.r 8
11.d odd 10 1 242.4.a.o 4
11.d odd 10 2 242.4.c.n 8
11.d odd 10 1 242.4.c.q 8
33.f even 10 1 2178.4.a.bt 4
33.h odd 10 1 198.4.f.d 8
33.h odd 10 1 2178.4.a.by 4
44.g even 10 1 1936.4.a.bm 4
44.h odd 10 1 176.4.m.b 8
44.h odd 10 1 1936.4.a.bn 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 1.a even 1 1 trivial
22.4.c.b 8 11.c even 5 1 inner
176.4.m.b 8 4.b odd 2 1
176.4.m.b 8 44.h odd 10 1
198.4.f.d 8 3.b odd 2 1
198.4.f.d 8 33.h odd 10 1
242.4.a.n 4 11.c even 5 1
242.4.a.o 4 11.d odd 10 1
242.4.c.n 8 11.d odd 10 2
242.4.c.q 8 11.b odd 2 1
242.4.c.q 8 11.d odd 10 1
242.4.c.r 8 11.c even 5 2
1936.4.a.bm 4 44.g even 10 1
1936.4.a.bn 4 44.h odd 10 1
2178.4.a.bt 4 33.f even 10 1
2178.4.a.by 4 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 3T_{3}^{7} + 42T_{3}^{6} + 185T_{3}^{5} + 1931T_{3}^{4} - 11455T_{3}^{3} + 224168T_{3}^{2} + 368251T_{3} + 4748041 \) acting on \(S_{4}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + 42 T^{6} + \cdots + 4748041 \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} + 146 T^{6} + \cdots + 68624656 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + 698 T^{6} + \cdots + 87027360016 \) Copy content Toggle raw display
$11$ \( T^{8} + 155 T^{7} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{8} - 7 T^{7} + \cdots + 16022496400 \) Copy content Toggle raw display
$17$ \( T^{8} - 161 T^{7} + \cdots + 4078198497025 \) Copy content Toggle raw display
$19$ \( T^{8} + 272 T^{7} + \cdots + 3649418019025 \) Copy content Toggle raw display
$23$ \( (T^{4} - 314 T^{3} + 13148 T^{2} + \cdots - 257882816)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 33 T^{7} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} - 323 T^{7} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} - 49 T^{7} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{8} - 361 T^{7} + \cdots + 60\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{4} - 721 T^{3} + 102089 T^{2} + \cdots - 2713875120)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 1069 T^{7} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + 281 T^{7} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + 128 T^{7} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + 617 T^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} - 289 T^{3} + \cdots + 35027256944)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 115 T^{7} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{8} + 1487 T^{7} + \cdots + 32\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{8} - 71 T^{7} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} - 1942 T^{7} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1101 T^{3} + \cdots - 64943655580)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 5128 T^{7} + \cdots + 46\!\cdots\!81 \) Copy content Toggle raw display
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