Properties

Label 22.8.c.a
Level $22$
Weight $8$
Character orbit 22.c
Analytic conductor $6.872$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.87247056065\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} + 718 x^{10} - 5886 x^{9} + 371698 x^{8} + 6317266 x^{7} + 171079947 x^{6} + 3212289784 x^{5} + 91377152203 x^{4} + \cdots + 475277889155161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (8 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 8) q^{2} + ( - \beta_{9} + 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{3} - 64 \beta_{3} q^{4} + (\beta_{11} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \cdots + 3) q^{5}+ \cdots + ( - 15 \beta_{11} - 12 \beta_{9} + 17 \beta_{8} + 15 \beta_{7} + 17 \beta_{5} + \cdots - 276) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (8 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 8) q^{2} + ( - \beta_{9} + 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{3} - 64 \beta_{3} q^{4} + (\beta_{11} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \cdots + 3) q^{5}+ \cdots + ( - 4818 \beta_{11} + 29832 \beta_{10} + 24992 \beta_{9} + \cdots + 5872196) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{2} + 44 q^{3} - 192 q^{4} + 220 q^{5} - 248 q^{6} + 534 q^{7} - 1536 q^{8} - 3745 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{2} + 44 q^{3} - 192 q^{4} + 220 q^{5} - 248 q^{6} + 534 q^{7} - 1536 q^{8} - 3745 q^{9} - 6080 q^{10} - 12881 q^{11} - 1664 q^{12} - 1834 q^{13} + 4272 q^{14} - 50174 q^{15} - 12288 q^{16} + 49128 q^{17} + 28720 q^{18} - 24453 q^{19} + 14080 q^{20} + 135280 q^{21} - 10648 q^{22} + 1212 q^{23} - 15872 q^{24} + 233893 q^{25} - 2352 q^{26} + 24029 q^{27} + 25216 q^{28} + 410548 q^{29} - 401392 q^{30} - 285894 q^{31} + 393216 q^{32} - 561803 q^{33} - 416016 q^{34} + 40188 q^{35} + 229760 q^{36} - 229440 q^{37} + 559696 q^{38} + 102742 q^{39} + 81920 q^{40} - 838292 q^{41} - 547200 q^{42} - 774698 q^{43} + 30976 q^{44} - 3155228 q^{45} + 438176 q^{46} + 1107540 q^{47} + 180224 q^{48} + 4281299 q^{49} - 625416 q^{50} + 2561801 q^{51} - 18816 q^{52} - 3013626 q^{53} - 2622208 q^{54} - 4505512 q^{55} - 950272 q^{56} + 10084223 q^{57} + 3284384 q^{58} + 2583885 q^{59} + 3746304 q^{60} + 418618 q^{61} + 1193008 q^{62} + 4045264 q^{63} - 786432 q^{64} - 11397748 q^{65} + 1076416 q^{66} - 18354554 q^{67} + 3144192 q^{68} + 11135006 q^{69} + 207744 q^{70} + 22870340 q^{71} - 1917440 q^{72} - 7374584 q^{73} - 1835520 q^{74} - 19671977 q^{75} - 5825152 q^{76} - 2121482 q^{77} - 20124544 q^{78} + 4715278 q^{79} + 655360 q^{80} - 745714 q^{81} + 12958184 q^{82} - 13257515 q^{83} + 48640 q^{84} + 8622056 q^{85} - 6992984 q^{86} + 6788612 q^{87} + 613888 q^{88} - 34864794 q^{89} + 11604816 q^{90} + 31631268 q^{91} - 3544192 q^{92} + 41624776 q^{93} + 6007840 q^{94} + 2925362 q^{95} + 1441792 q^{96} - 28563397 q^{97} - 5044288 q^{98} + 28188523 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 718 x^{10} - 5886 x^{9} + 371698 x^{8} + 6317266 x^{7} + 171079947 x^{6} + 3212289784 x^{5} + 91377152203 x^{4} + \cdots + 475277889155161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 69\!\cdots\!30 \nu^{11} + \cdots + 22\!\cdots\!29 ) / 50\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!83 \nu^{11} + \cdots - 41\!\cdots\!92 ) / 50\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!04 \nu^{11} + \cdots - 10\!\cdots\!02 ) / 50\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!20 \nu^{11} + \cdots + 11\!\cdots\!48 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!62 \nu^{11} + \cdots - 48\!\cdots\!83 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46\!\cdots\!97 \nu^{11} + \cdots + 22\!\cdots\!81 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35\!\cdots\!43 \nu^{11} + \cdots - 29\!\cdots\!13 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!69 \nu^{11} + \cdots - 71\!\cdots\!88 ) / 11\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!12 \nu^{11} + \cdots - 15\!\cdots\!01 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!12 \nu^{11} + \cdots - 10\!\cdots\!89 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71\!\cdots\!61 \nu^{11} + \cdots - 83\!\cdots\!31 ) / 40\!\cdots\!75 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - \beta_{11} - \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 8 \beta_{4} - 8 \beta_{2} - 10 \beta _1 + 8 ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 41 \beta_{10} - 82 \beta_{9} - 41 \beta_{7} - 2 \beta_{6} - 24 \beta_{5} - 24 \beta_{4} - 778 \beta_{3} - 8942 \beta_{2} - 778 \beta_1 ) / 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 62 \beta_{11} + 153 \beta_{10} - 316 \beta_{9} + 134 \beta_{8} + 153 \beta_{6} - 316 \beta_{5} + 134 \beta_{4} + 8594 \beta_{3} - 24 \beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 38283 \beta_{11} - 32979 \beta_{10} - 1362 \beta_{9} - 70736 \beta_{8} + 38283 \beta_{7} - 70736 \beta_{5} - 69374 \beta_{4} + 4581740 \beta_{3} + 5344384 \beta_{2} + 5344384 \beta _1 - 4581740 ) / 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 574908 \beta_{11} - 1181551 \beta_{8} + 237038 \beta_{7} - 574908 \beta_{6} + 471140 \beta_{5} - 4462385 \beta_{3} - 4462385 \beta_{2} - 44220397 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2255449 \beta_{11} + 2255449 \beta_{10} - 838488 \beta_{9} + 838488 \beta_{8} + 494509 \beta_{7} - 494509 \beta_{6} + 5812094 \beta_{4} - 60764054 \beta_{2} - 333823380 \beta _1 + 60764054 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 841719090 \beta_{10} + 1668887490 \beta_{9} + 841719090 \beta_{7} - 233649187 \beta_{6} - 672489776 \beta_{5} - 672489776 \beta_{4} + 13175260376 \beta_{3} + \cdots + 13175260376 \beta_1 ) / 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 19409340873 \beta_{11} - 23058631461 \beta_{10} + 41345264592 \beta_{9} - 12409667796 \beta_{8} - 23058631461 \beta_{6} + 41345264592 \beta_{5} + \cdots - 583022731398 ) / 22 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 57368220294 \beta_{11} + 46995525029 \beta_{10} - 47734913938 \beta_{9} + 109143640188 \beta_{8} - 57368220294 \beta_{7} + 109143640188 \beta_{5} + \cdots + 4791567788420 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8728251913465 \beta_{11} + 15485537475455 \beta_{8} - 976960321745 \beta_{7} + 8728251913465 \beta_{6} - 6232337674740 \beta_{5} + \cdots + 749761760338304 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 428027268620764 \beta_{11} - 428027268620764 \beta_{10} + 428594973439378 \beta_{9} - 428594973439378 \beta_{8} - 49500845231363 \beta_{7} + \cdots - 13\!\cdots\!38 ) / 22 \) Copy content Toggle raw display

Character values

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

\(n\) \(13\)
\(\chi(n)\) \(-\beta_{3}\)

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
−7.43190 5.39959i
−13.3823 9.72279i
21.3142 + 15.4857i
−8.68746 + 26.7373i
2.97963 9.17036i
6.20783 19.1057i
−8.68746 26.7373i
2.97963 + 9.17036i
6.20783 + 19.1057i
−7.43190 + 5.39959i
−13.3823 + 9.72279i
21.3142 15.4857i
−6.47214 + 4.70228i −19.8857 61.2018i 19.7771 60.8676i −37.7557 27.4311i 416.491 + 302.598i −204.857 + 630.484i 158.217 + 486.941i −1580.90 + 1148.59i 373.349
3.2 −6.47214 + 4.70228i 3.49129 + 10.7451i 19.7771 60.8676i −150.057 109.023i −73.1224 53.1266i 47.2510 145.424i 158.217 + 486.941i 1666.05 1210.46i 1483.84
3.3 −6.47214 + 4.70228i 19.5681 + 60.2246i 19.7771 60.8676i 352.380 + 256.019i −409.841 297.767i 23.8954 73.5425i 158.217 + 486.941i −1474.77 + 1071.48i −3484.52
5.1 2.47214 + 7.60845i −26.1922 19.0297i −51.7771 + 37.6183i 40.3475 124.177i 80.0362 246.326i 972.268 706.394i −414.217 300.946i −351.920 1083.10i 1044.54
5.2 2.47214 + 7.60845i −5.52734 4.01585i −51.7771 + 37.6183i −14.6685 + 45.1450i 16.8900 51.9822i −1183.62 + 859.952i −414.217 300.946i −661.396 2035.57i −379.746
5.3 2.47214 + 7.60845i 50.5458 + 36.7237i −51.7771 + 37.6183i −80.2463 + 246.973i −154.454 + 475.361i 612.065 444.691i −414.217 300.946i 530.429 + 1632.49i −2077.46
9.1 2.47214 7.60845i −26.1922 + 19.0297i −51.7771 37.6183i 40.3475 + 124.177i 80.0362 + 246.326i 972.268 + 706.394i −414.217 + 300.946i −351.920 + 1083.10i 1044.54
9.2 2.47214 7.60845i −5.52734 + 4.01585i −51.7771 37.6183i −14.6685 45.1450i 16.8900 + 51.9822i −1183.62 859.952i −414.217 + 300.946i −661.396 + 2035.57i −379.746
9.3 2.47214 7.60845i 50.5458 36.7237i −51.7771 37.6183i −80.2463 246.973i −154.454 475.361i 612.065 + 444.691i −414.217 + 300.946i 530.429 1632.49i −2077.46
15.1 −6.47214 4.70228i −19.8857 + 61.2018i 19.7771 + 60.8676i −37.7557 + 27.4311i 416.491 302.598i −204.857 630.484i 158.217 486.941i −1580.90 1148.59i 373.349
15.2 −6.47214 4.70228i 3.49129 10.7451i 19.7771 + 60.8676i −150.057 + 109.023i −73.1224 + 53.1266i 47.2510 + 145.424i 158.217 486.941i 1666.05 + 1210.46i 1483.84
15.3 −6.47214 4.70228i 19.5681 60.2246i 19.7771 + 60.8676i 352.380 256.019i −409.841 + 297.767i 23.8954 + 73.5425i 158.217 486.941i −1474.77 1071.48i −3484.52
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.3
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 22.8.c.a 12
11.c even 5 1 inner 22.8.c.a 12
11.c even 5 1 242.8.a.q 6
11.d odd 10 1 242.8.a.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.8.c.a 12 1.a even 1 1 trivial
22.8.c.a 12 11.c even 5 1 inner
242.8.a.o 6 11.d odd 10 1
242.8.a.q 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 44 T_{3}^{11} + 6121 T_{3}^{10} - 203741 T_{3}^{9} + 18257077 T_{3}^{8} - 95237091 T_{3}^{7} + 21982078431 T_{3}^{6} + 1714736709702 T_{3}^{5} + 76543472730339 T_{3}^{4} + \cdots + 40\!\cdots\!61 \) acting on \(S_{8}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{3} + 64 T^{2} + 512 T + 4096)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - 44 T^{11} + \cdots + 40\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{12} - 220 T^{11} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} - 534 T^{11} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + 12881 T^{11} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{12} + 1834 T^{11} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} - 49128 T^{11} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{12} + 24453 T^{11} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( (T^{6} - 606 T^{5} + \cdots - 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 410548 T^{11} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + 285894 T^{11} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + 229440 T^{11} + \cdots + 75\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{12} + 838292 T^{11} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{6} + 387349 T^{5} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 1107540 T^{11} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + 3013626 T^{11} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} - 2583885 T^{11} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} - 418618 T^{11} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + 9177277 T^{5} + \cdots - 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 22870340 T^{11} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + 7374584 T^{11} + \cdots + 38\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{12} - 4715278 T^{11} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + 13257515 T^{11} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{6} + 17432397 T^{5} + \cdots - 34\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 28563397 T^{11} + \cdots + 46\!\cdots\!01 \) Copy content Toggle raw display
show more
show less