Properties

Label 22.7.b.a
Level $22$
Weight $7$
Character orbit 22.b
Analytic conductor $5.061$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 22.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.06118983964\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_1 - 9) q^{3} - 32 q^{4} + (3 \beta_{3} - 2 \beta_1 + 61) q^{5} + ( - \beta_{5} - \beta_{4} - 9 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - 12 \beta_{2}) q^{7} - 32 \beta_{2} q^{8} + (7 \beta_{3} - 30 \beta_1 - 50) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_1 - 9) q^{3} - 32 q^{4} + (3 \beta_{3} - 2 \beta_1 + 61) q^{5} + ( - \beta_{5} - \beta_{4} - 9 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - 12 \beta_{2}) q^{7} - 32 \beta_{2} q^{8} + (7 \beta_{3} - 30 \beta_1 - 50) q^{9} + (11 \beta_{5} - \beta_{4} + 67 \beta_{2}) q^{10} + (11 \beta_{5} + 11 \beta_{3} + 33 \beta_1 - 209) q^{11} + ( - 32 \beta_1 + 288) q^{12} + ( - 24 \beta_{5} - 15 \beta_{4}) q^{13} + ( - 8 \beta_{3} - 88 \beta_1 + 400) q^{14} + ( - 2 \beta_{3} + 163 \beta_1 - 1139) q^{15} + 1024 q^{16} + ( - 26 \beta_{5} + 33 \beta_{4} - 168 \beta_{2}) q^{17} + (51 \beta_{5} + 23 \beta_{4} - 36 \beta_{2}) q^{18} + ( - 116 \beta_{5} + 9 \beta_{4} - 588 \beta_{2}) q^{19} + ( - 96 \beta_{3} + 64 \beta_1 - 1952) q^{20} + (126 \beta_{5} + 45 \beta_{4} + 1848 \beta_{2}) q^{21} + ( - 44 \beta_{4} - 88 \beta_{3} - 187 \beta_{2} + 88 \beta_1 + 176) q^{22} + ( - 68 \beta_{3} + 141 \beta_1 + 335) q^{23} + (32 \beta_{5} + 32 \beta_{4} + 288 \beta_{2}) q^{24} + (67 \beta_{3} + 62 \beta_1 + 9934) q^{25} + (72 \beta_{3} - 552 \beta_1 - 144) q^{26} + ( - 182 \beta_{3} - 9 \beta_1 - 9515) q^{27} + (64 \beta_{5} + 96 \beta_{4} + 384 \beta_{2}) q^{28} + (62 \beta_{5} - 30 \beta_{4} - 2616 \beta_{2}) q^{29} + ( - 169 \beta_{5} - 161 \beta_{4} - 1143 \beta_{2}) q^{30} + (416 \beta_{3} - 143 \beta_1 - 13169) q^{31} + 1024 \beta_{2} q^{32} + ( - 132 \beta_{5} - 99 \beta_{4} + 275 \beta_{3} - 1056 \beta_{2} + \cdots + 23837) q^{33}+ \cdots + ( - 2409 \beta_{5} + 2739 \beta_{4} - 11693 \beta_{3} + \cdots - 414458) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 52 q^{3} - 192 q^{4} + 368 q^{5} - 346 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 52 q^{3} - 192 q^{4} + 368 q^{5} - 346 q^{9} - 1166 q^{11} + 1664 q^{12} + 2208 q^{14} - 6512 q^{15} + 6144 q^{16} - 11776 q^{20} + 1056 q^{22} + 2156 q^{23} + 59862 q^{25} - 1824 q^{26} - 57472 q^{27} - 78468 q^{31} + 142208 q^{33} + 28704 q^{34} + 11072 q^{36} - 205920 q^{37} + 101472 q^{38} - 344160 q^{42} + 37312 q^{44} + 368716 q^{45} + 493460 q^{47} - 53248 q^{48} - 270762 q^{49} - 531700 q^{53} + 274956 q^{55} - 70656 q^{56} + 509184 q^{58} - 833380 q^{59} + 208384 q^{60} - 196608 q^{64} + 193248 q^{66} + 537420 q^{67} + 398860 q^{69} + 96096 q^{70} - 460372 q^{71} - 211428 q^{75} + 249744 q^{77} - 1866912 q^{78} + 376832 q^{80} + 485654 q^{81} + 428640 q^{82} + 1055808 q^{86} - 33792 q^{88} + 2377952 q^{89} - 5068656 q^{91} - 68992 q^{92} + 699868 q^{93} + 1351632 q^{97} - 2470930 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2707\nu^{5} - 22767\nu^{4} + 4050251\nu^{3} + 3625173\nu^{2} - 657163750\nu + 10557684654 ) / 963515190 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5414\nu^{5} + 45534\nu^{4} - 8100502\nu^{3} - 7250346\nu^{2} + 3241357880\nu - 21115369308 ) / 481757595 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 44423 \nu^{5} - 907518 \nu^{4} + 67178179 \nu^{3} + 1011971817 \nu^{2} - 24365375270 \nu - 111796059144 ) / 3372303165 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 95153 \nu^{5} - 18774978 \nu^{4} - 154836221 \nu^{3} + 16296294372 \nu^{2} + 108700698730 \nu - 1601025981984 ) / 6744606330 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46261 \nu^{5} - 187378 \nu^{4} + 90358871 \nu^{3} - 137288348 \nu^{2} - 45368513086 \nu + 315774403296 ) / 449640422 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} + 14\beta_{3} - 24\beta _1 + 1192 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 99\beta_{5} + 15\beta_{4} + 28\beta_{3} + 1834\beta_{2} + 3504\beta _1 - 23048 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -859\beta_{5} - 887\beta_{4} + 6181\beta_{3} - 5748\beta_{2} - 16062\beta _1 + 521678 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 174345\beta_{5} + 49605\beta_{4} - 128548\beta_{3} + 2694664\beta_{2} + 3324008\beta _1 - 33241672 ) / 4 \) 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
−32.5131 1.41421i
7.52788 1.41421i
25.9852 1.41421i
−32.5131 + 1.41421i
7.52788 + 1.41421i
25.9852 + 1.41421i
5.65685i −41.5131 −32.0000 155.573 234.833i 562.567i 181.019i 994.336 880.056i
21.2 5.65685i −1.47212 −32.0000 −147.340 8.32754i 44.7193i 181.019i −726.833 833.480i
21.3 5.65685i 16.9852 −32.0000 175.767 96.0828i 412.125i 181.019i −440.503 994.286i
21.4 5.65685i −41.5131 −32.0000 155.573 234.833i 562.567i 181.019i 994.336 880.056i
21.5 5.65685i −1.47212 −32.0000 −147.340 8.32754i 44.7193i 181.019i −726.833 833.480i
21.6 5.65685i 16.9852 −32.0000 175.767 96.0828i 412.125i 181.019i −440.503 994.286i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.6
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 22.7.b.a 6
3.b odd 2 1 198.7.d.a 6
4.b odd 2 1 176.7.h.e 6
11.b odd 2 1 inner 22.7.b.a 6
33.d even 2 1 198.7.d.a 6
44.c even 2 1 176.7.h.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.7.b.a 6 1.a even 1 1 trivial
22.7.b.a 6 11.b odd 2 1 inner
176.7.h.e 6 4.b odd 2 1
176.7.h.e 6 44.c even 2 1
198.7.d.a 6 3.b odd 2 1
198.7.d.a 6 33.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{3} \) Copy content Toggle raw display
$3$ \( (T^{3} + 26 T^{2} - 669 T - 1038)^{2} \) Copy content Toggle raw display
$5$ \( (T^{3} - 184 T^{2} - 21475 T + 4028950)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 107496740487168 \) Copy content Toggle raw display
$11$ \( T^{6} + 1166 T^{5} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{6} + 16642440 T^{4} + \cdots + 91\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{6} + 83290056 T^{4} + \cdots + 29\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{6} + 256000200 T^{4} + \cdots + 61\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{3} - 1078 T^{2} + \cdots + 6239437498)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 808758528 T^{4} + \cdots + 56\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( (T^{3} + 39234 T^{2} + \cdots - 1132011557222)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 102960 T^{2} + \cdots + 26364256495174)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 15110795400 T^{4} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{6} + 18236828832 T^{4} + \cdots + 19\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( (T^{3} - 246730 T^{2} + \cdots - 432230445439256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 265850 T^{2} + \cdots - 31\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 416690 T^{2} + \cdots - 90\!\cdots\!90)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 146302417824 T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{3} - 268710 T^{2} + \cdots + 18\!\cdots\!74)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 230186 T^{2} + \cdots + 97\!\cdots\!82)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 704846656392 T^{4} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + 882501800064 T^{4} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} + 1744310743296 T^{4} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( (T^{3} - 1188976 T^{2} + \cdots - 22\!\cdots\!14)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 675816 T^{2} + \cdots + 17\!\cdots\!14)^{2} \) Copy content Toggle raw display
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