Properties

Label 22.9.b.a
Level $22$
Weight $9$
Character orbit 22.b
Analytic conductor $8.962$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 22.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.96232942134\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 7944x^{6} + 15215349x^{4} + 1757611988x^{2} + 38177252100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_1 + 23) q^{3} - 128 q^{4} + (\beta_{2} + \beta_1 - 176) q^{5} + ( - \beta_{5} - 23 \beta_{4}) q^{6} + ( - \beta_{7} + 12 \beta_{4}) q^{7} + 128 \beta_{4} q^{8} + (\beta_{3} + 3 \beta_{2} - 16 \beta_1 + 5569) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_1 + 23) q^{3} - 128 q^{4} + (\beta_{2} + \beta_1 - 176) q^{5} + ( - \beta_{5} - 23 \beta_{4}) q^{6} + ( - \beta_{7} + 12 \beta_{4}) q^{7} + 128 \beta_{4} q^{8} + (\beta_{3} + 3 \beta_{2} - 16 \beta_1 + 5569) q^{9} + (2 \beta_{7} - 2 \beta_{6} + 177 \beta_{4}) q^{10} + ( - 2 \beta_{7} + 3 \beta_{6} - 6 \beta_{5} + 348 \beta_{4} - \beta_{3} + \cdots + 714) q^{11}+ \cdots + ( - 561 \beta_{7} + 41481 \beta_{6} - 61908 \beta_{5} + \cdots - 41774183) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 182 q^{3} - 1024 q^{4} - 1410 q^{5} + 44582 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 182 q^{3} - 1024 q^{4} - 1410 q^{5} + 44582 q^{9} + 5808 q^{11} - 23296 q^{12} + 12288 q^{14} + 87958 q^{15} + 131072 q^{16} + 180480 q^{20} + 359040 q^{22} - 802026 q^{23} + 999558 q^{25} - 1321728 q^{26} - 1561354 q^{27} + 196726 q^{31} - 4286722 q^{33} - 701952 q^{34} - 5706496 q^{36} + 8627998 q^{37} + 9288960 q^{38} + 4366848 q^{42} - 743424 q^{44} + 1146988 q^{45} - 14335392 q^{47} + 2981888 q^{48} - 6714712 q^{49} + 55946352 q^{53} - 10078442 q^{55} - 1572864 q^{56} - 23226624 q^{58} + 21793110 q^{59} - 11258624 q^{60} - 16777216 q^{64} - 44242176 q^{66} - 113809034 q^{67} - 171636914 q^{69} + 137817600 q^{70} + 16741974 q^{71} + 346496844 q^{75} - 137074080 q^{77} - 57993216 q^{78} - 23101440 q^{80} + 85282724 q^{81} + 47480832 q^{82} + 49839360 q^{86} - 45957120 q^{88} + 42055422 q^{89} - 146801952 q^{91} + 102659328 q^{92} + 253251118 q^{93} + 100034782 q^{97} - 333541978 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7944x^{6} + 15215349x^{4} + 1757611988x^{2} + 38177252100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 104101\nu^{6} + 824180014\nu^{4} + 1543517240941\nu^{2} + 79735595295834 ) / 341414886816 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2473\nu^{6} + 19351282\nu^{4} + 36437327137\nu^{2} + 3299729276088 ) / 2370936714 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1298539\nu^{6} - 9261809842\nu^{4} - 14218174807843\nu^{2} + 681237033157242 ) / 341414886816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4961\nu^{7} - 39431894\nu^{5} - 76538930009\nu^{3} - 12562713512898\nu ) / 4928269410090 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -72713347\nu^{7} - 573243568578\nu^{5} - 1072455149052603\nu^{3} - 64716434342432246\nu ) / 154419108182820 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 180562407 \nu^{7} - 1433706219178 \nu^{5} + \cdots - 26\!\cdots\!86 \nu ) / 308838216365640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4432187\nu^{7} + 35007933278\nu^{5} + 65818394352143\nu^{3} + 4964043628901166\nu ) / 5939196468570 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 6\beta_{7} + 2\beta_{6} + 7\beta_{5} + 12\beta_{4} ) / 352 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 15\beta_{3} + 466\beta_{2} - 1407\beta _1 - 349884 ) / 176 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -24498\beta_{7} - 5350\beta_{6} - 29021\beta_{5} - 1478688\beta_{4} ) / 352 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3789\beta_{3} - 575756\beta_{2} + 1922301\beta _1 + 359920446 ) / 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 104706102\beta_{7} + 5378210\beta_{6} + 138311983\beta_{5} + 9799783896\beta_{4} ) / 352 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -102414831\beta_{3} + 11323880830\beta_{2} - 39437377953\beta _1 - 6345168935316 ) / 176 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -469479034254\beta_{7} + 34727845574\beta_{6} - 669342100259\beta_{5} - 55459049761608\beta_{4} ) / 352 \) Copy content Toggle raw display

Character values

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

\(n\) \(13\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
9.64354i
53.7858i
70.2090i
5.36544i
9.64354i
53.7858i
70.2090i
5.36544i
11.3137i −143.261 −128.000 −309.991 1620.81i 1256.92i 1448.15i 13962.6 3507.15i
21.2 11.3137i −1.48475 −128.000 −214.559 16.7981i 2223.81i 1448.15i −6558.80 2427.46i
21.3 11.3137i 107.356 −128.000 −1065.90 1214.60i 3358.46i 1448.15i 4964.40 12059.2i
21.4 11.3137i 128.389 −128.000 885.446 1452.56i 2934.64i 1448.15i 9922.76 10017.7i
21.5 11.3137i −143.261 −128.000 −309.991 1620.81i 1256.92i 1448.15i 13962.6 3507.15i
21.6 11.3137i −1.48475 −128.000 −214.559 16.7981i 2223.81i 1448.15i −6558.80 2427.46i
21.7 11.3137i 107.356 −128.000 −1065.90 1214.60i 3358.46i 1448.15i 4964.40 12059.2i
21.8 11.3137i 128.389 −128.000 885.446 1452.56i 2934.64i 1448.15i 9922.76 10017.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.9.b.a 8
3.b odd 2 1 198.9.d.a 8
4.b odd 2 1 176.9.h.e 8
11.b odd 2 1 inner 22.9.b.a 8
33.d even 2 1 198.9.d.a 8
44.c even 2 1 176.9.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.9.b.a 8 1.a even 1 1 trivial
22.9.b.a 8 11.b odd 2 1 inner
176.9.h.e 8 4.b odd 2 1
176.9.h.e 8 44.c even 2 1
198.9.d.a 8 3.b odd 2 1
198.9.d.a 8 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 91 T^{3} - 20127 T^{2} + \cdots + 2931822)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 705 T^{3} + \cdots - 62773004450)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 26416560 T^{6} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{8} - 5808 T^{7} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{8} + 6134398128 T^{6} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + 18249994416 T^{6} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + 72786821808 T^{6} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{4} + 401013 T^{3} + \cdots + 63\!\cdots\!86)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 1885158744960 T^{6} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{4} - 98363 T^{3} + \cdots - 54\!\cdots\!26)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4313999 T^{3} + \cdots - 50\!\cdots\!02)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 26808983898672 T^{6} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{8} + 74059600768320 T^{6} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{4} + 7167696 T^{3} + \cdots - 14\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 27973176 T^{3} + \cdots - 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 10896555 T^{3} + \cdots + 77\!\cdots\!50)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 695753774140224 T^{6} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + 56904517 T^{3} + \cdots - 12\!\cdots\!14)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8370987 T^{3} + \cdots + 55\!\cdots\!34)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 854608068817968 T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 92\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} - 21027711 T^{3} + \cdots - 11\!\cdots\!58)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 50017391 T^{3} + \cdots - 28\!\cdots\!94)^{2} \) Copy content Toggle raw display
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