# Properties

 Label 22.14.c.a Level $22$ Weight $14$ Character orbit 22.c Analytic conductor $23.591$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 22.c (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.5908043694$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9}+O(q^{10})$$ 24 * q + 384 * q^2 + 2708 * q^3 - 24576 * q^4 + 32920 * q^5 + 47488 * q^6 - 509572 * q^7 + 1572864 * q^8 - 3517894 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 384 q^{2} + 2708 q^{3} - 24576 q^{4} + 32920 q^{5} + 47488 q^{6} - 509572 q^{7} + 1572864 q^{8} - 3517894 q^{9} + 9574400 q^{10} - 17933212 q^{11} - 16105472 q^{12} - 8592392 q^{13} + 32612608 q^{14} + 39493882 q^{15} - 100663296 q^{16} + 94365328 q^{17} + 160304896 q^{18} - 152599556 q^{19} + 134840320 q^{20} - 1160906408 q^{21} + 103665408 q^{22} - 3452143608 q^{23} + 194510848 q^{24} - 4256252342 q^{25} - 292945152 q^{26} + 4686820568 q^{27} + 2444976128 q^{28} - 5215924524 q^{29} - 2527608448 q^{30} + 3405899922 q^{31} - 25769803776 q^{32} - 23679929966 q^{33} - 8202684672 q^{34} - 99104472424 q^{35} - 10259513344 q^{36} + 40846533764 q^{37} + 28549809024 q^{38} + 64855004884 q^{39} - 10978590720 q^{40} - 51540853640 q^{41} + 24490870272 q^{42} + 124560388560 q^{43} - 74477658112 q^{44} - 204284993840 q^{45} - 109655514368 q^{46} + 19947862428 q^{47} + 45432700928 q^{48} - 175112415622 q^{49} - 80504514432 q^{50} + 474711827630 q^{51} + 18748489728 q^{52} + 558965931900 q^{53} + 965595769088 q^{54} + 441854256778 q^{55} + 45794459648 q^{56} - 418297095718 q^{57} + 333819169536 q^{58} + 328926408084 q^{59} + 90834665472 q^{60} - 564751916716 q^{61} + 263779911552 q^{62} - 1395614157158 q^{63} - 412316860416 q^{64} + 4138087857488 q^{65} - 538298241536 q^{66} + 1362842206448 q^{67} + 386520383488 q^{68} + 711220837976 q^{69} - 5066472500864 q^{70} - 4941518810510 q^{71} + 922194804736 q^{72} + 6897669499740 q^{73} - 2614178160896 q^{74} - 5379948655580 q^{75} + 4904471117824 q^{76} - 1193035831856 q^{77} + 131225586944 q^{78} - 6326717823822 q^{79} + 702629806080 q^{80} - 21467454228532 q^{81} - 3106749998720 q^{82} + 18781365306578 q^{83} + 3944952020992 q^{84} - 11645336492276 q^{85} - 3434691211520 q^{86} + 10865107308368 q^{87} - 728560566272 q^{88} - 13785696202276 q^{89} - 6776795347200 q^{90} + 8137957243966 q^{91} + 52037189632 q^{92} - 27108762213068 q^{93} - 2207923445632 q^{94} + 15370245327434 q^{95} - 2907692859392 q^{96} + 20175078383702 q^{97} + 20871981500928 q^{98} + 37377269966674 q^{99}+O(q^{100})$$ 24 * q + 384 * q^2 + 2708 * q^3 - 24576 * q^4 + 32920 * q^5 + 47488 * q^6 - 509572 * q^7 + 1572864 * q^8 - 3517894 * q^9 + 9574400 * q^10 - 17933212 * q^11 - 16105472 * q^12 - 8592392 * q^13 + 32612608 * q^14 + 39493882 * q^15 - 100663296 * q^16 + 94365328 * q^17 + 160304896 * q^18 - 152599556 * q^19 + 134840320 * q^20 - 1160906408 * q^21 + 103665408 * q^22 - 3452143608 * q^23 + 194510848 * q^24 - 4256252342 * q^25 - 292945152 * q^26 + 4686820568 * q^27 + 2444976128 * q^28 - 5215924524 * q^29 - 2527608448 * q^30 + 3405899922 * q^31 - 25769803776 * q^32 - 23679929966 * q^33 - 8202684672 * q^34 - 99104472424 * q^35 - 10259513344 * q^36 + 40846533764 * q^37 + 28549809024 * q^38 + 64855004884 * q^39 - 10978590720 * q^40 - 51540853640 * q^41 + 24490870272 * q^42 + 124560388560 * q^43 - 74477658112 * q^44 - 204284993840 * q^45 - 109655514368 * q^46 + 19947862428 * q^47 + 45432700928 * q^48 - 175112415622 * q^49 - 80504514432 * q^50 + 474711827630 * q^51 + 18748489728 * q^52 + 558965931900 * q^53 + 965595769088 * q^54 + 441854256778 * q^55 + 45794459648 * q^56 - 418297095718 * q^57 + 333819169536 * q^58 + 328926408084 * q^59 + 90834665472 * q^60 - 564751916716 * q^61 + 263779911552 * q^62 - 1395614157158 * q^63 - 412316860416 * q^64 + 4138087857488 * q^65 - 538298241536 * q^66 + 1362842206448 * q^67 + 386520383488 * q^68 + 711220837976 * q^69 - 5066472500864 * q^70 - 4941518810510 * q^71 + 922194804736 * q^72 + 6897669499740 * q^73 - 2614178160896 * q^74 - 5379948655580 * q^75 + 4904471117824 * q^76 - 1193035831856 * q^77 + 131225586944 * q^78 - 6326717823822 * q^79 + 702629806080 * q^80 - 21467454228532 * q^81 - 3106749998720 * q^82 + 18781365306578 * q^83 + 3944952020992 * q^84 - 11645336492276 * q^85 - 3434691211520 * q^86 + 10865107308368 * q^87 - 728560566272 * q^88 - 13785696202276 * q^89 - 6776795347200 * q^90 + 8137957243966 * q^91 + 52037189632 * q^92 - 27108762213068 * q^93 - 2207923445632 * q^94 + 15370245327434 * q^95 - 2907692859392 * q^96 + 20175078383702 * q^97 + 20871981500928 * q^98 + 37377269966674 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 51.7771 37.6183i −663.076 2040.74i 1265.73 3895.53i 18873.2 + 13712.2i −111101. 80719.7i −130921. + 402934.i −81007.0 249314.i −2.43511e6 + 1.76921e6i 1.49303e6
3.2 51.7771 37.6183i −417.662 1285.43i 1265.73 3895.53i −34004.8 24705.9i −69981.0 50844.2i 96600.4 297306.i −81007.0 249314.i −188057. + 136631.i −2.69006e6
3.3 51.7771 37.6183i −252.702 777.736i 1265.73 3895.53i 25133.6 + 18260.6i −42341.2 30762.7i 29224.9 89945.0i −81007.0 249314.i 748819. 544049.i 1.98828e6
3.4 51.7771 37.6183i 346.581 + 1066.67i 1265.73 3895.53i −24923.9 18108.3i 58071.1 + 42191.1i −7979.10 + 24557.2i −81007.0 249314.i 272175. 197747.i −1.97169e6
3.5 51.7771 37.6183i 460.202 + 1416.35i 1265.73 3895.53i 46818.1 + 34015.4i 77108.7 + 56022.7i −157481. + 484676.i −81007.0 249314.i −504441. + 366498.i 3.70370e6
3.6 51.7771 37.6183i 461.283 + 1419.68i 1265.73 3895.53i −3259.92 2368.47i 77289.8 + 56154.3i 80603.6 248072.i −81007.0 249314.i −512880. + 372629.i −257887.
5.1 −19.7771 60.8676i −1602.93 1164.60i −3313.73 + 2407.57i 100.214 308.427i −39185.0 + 120599.i 51623.9 37507.0i 212079. + 154084.i 720428. + 2.21725e6i −20755.1
5.2 −19.7771 60.8676i −351.775 255.580i −3313.73 + 2407.57i −3087.88 + 9503.52i −8599.44 + 26466.3i 327454. 237910.i 212079. + 154084.i −434248. 1.33648e6i 639526.
5.3 −19.7771 60.8676i −269.653 195.914i −3313.73 + 2407.57i −21195.1 + 65231.8i −6591.88 + 20287.7i −348859. + 253461.i 212079. + 154084.i −458343. 1.41063e6i 4.38968e6
5.4 −19.7771 60.8676i 719.209 + 522.536i −3313.73 + 2407.57i 552.241 1699.62i 17581.7 54110.8i −85138.9 + 61857.0i 212079. + 154084.i −248455. 764667.i −114374.
5.5 −19.7771 60.8676i 950.343 + 690.465i −3313.73 + 2407.57i 20485.5 63047.8i 23231.9 71500.5i 4712.51 3423.84i 212079. + 154084.i −66262.0 203934.i −4.24271e6
5.6 −19.7771 60.8676i 1974.18 + 1434.33i −3313.73 + 2407.57i −9031.32 + 27795.5i 48260.5 148531.i −114626. + 83280.8i 212079. + 154084.i 1.34743e6 + 4.14695e6i 1.87046e6
9.1 −19.7771 + 60.8676i −1602.93 + 1164.60i −3313.73 2407.57i 100.214 + 308.427i −39185.0 120599.i 51623.9 + 37507.0i 212079. 154084.i 720428. 2.21725e6i −20755.1
9.2 −19.7771 + 60.8676i −351.775 + 255.580i −3313.73 2407.57i −3087.88 9503.52i −8599.44 26466.3i 327454. + 237910.i 212079. 154084.i −434248. + 1.33648e6i 639526.
9.3 −19.7771 + 60.8676i −269.653 + 195.914i −3313.73 2407.57i −21195.1 65231.8i −6591.88 20287.7i −348859. 253461.i 212079. 154084.i −458343. + 1.41063e6i 4.38968e6
9.4 −19.7771 + 60.8676i 719.209 522.536i −3313.73 2407.57i 552.241 + 1699.62i 17581.7 + 54110.8i −85138.9 61857.0i 212079. 154084.i −248455. + 764667.i −114374.
9.5 −19.7771 + 60.8676i 950.343 690.465i −3313.73 2407.57i 20485.5 + 63047.8i 23231.9 + 71500.5i 4712.51 + 3423.84i 212079. 154084.i −66262.0 + 203934.i −4.24271e6
9.6 −19.7771 + 60.8676i 1974.18 1434.33i −3313.73 2407.57i −9031.32 27795.5i 48260.5 + 148531.i −114626. 83280.8i 212079. 154084.i 1.34743e6 4.14695e6i 1.87046e6
15.1 51.7771 + 37.6183i −663.076 + 2040.74i 1265.73 + 3895.53i 18873.2 13712.2i −111101. + 80719.7i −130921. 402934.i −81007.0 + 249314.i −2.43511e6 1.76921e6i 1.49303e6
15.2 51.7771 + 37.6183i −417.662 + 1285.43i 1265.73 + 3895.53i −34004.8 + 24705.9i −69981.0 + 50844.2i 96600.4 + 297306.i −81007.0 + 249314.i −188057. 136631.i −2.69006e6
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 15.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 22.14.c.a 24
11.c even 5 1 inner 22.14.c.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.14.c.a 24 1.a even 1 1 trivial
22.14.c.a 24 11.c even 5 1 inner