# Properties

 Label 22.14.c.b Level $22$ Weight $14$ Character orbit 22.c Analytic conductor $23.591$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Learn more

## 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: $$28$$ Relative dimension: $$7$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9}+O(q^{10})$$ 28 * q - 448 * q^2 - 1248 * q^3 - 28672 * q^4 + 71554 * q^5 + 99008 * q^6 + 361734 * q^7 - 1835008 * q^8 - 8615367 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 448 q^{2} - 1248 q^{3} - 28672 q^{4} + 71554 q^{5} + 99008 q^{6} + 361734 q^{7} - 1835008 q^{8} - 8615367 q^{9} + 2898176 q^{10} + 8699239 q^{11} - 2449408 q^{12} + 32373940 q^{13} + 23150976 q^{14} - 192099788 q^{15} - 117440512 q^{16} + 288881082 q^{17} - 275612928 q^{18} - 98588517 q^{19} + 293085184 q^{20} + 1362925660 q^{21} - 44352704 q^{22} - 233099500 q^{23} + 405536768 q^{24} + 1117600307 q^{25} - 2827224960 q^{26} + 3575804241 q^{27} - 1485029376 q^{28} - 16868227562 q^{29} - 12294386432 q^{30} - 10874790260 q^{31} + 30064771072 q^{32} + 53369498501 q^{33} + 6456111488 q^{34} + 18823205444 q^{35} - 17639227392 q^{36} + 47951509642 q^{37} + 14611374592 q^{38} - 161088016778 q^{39} - 24692916224 q^{40} - 155498856970 q^{41} - 47678239360 q^{42} + 156723459918 q^{43} + 31490678784 q^{44} + 75277819092 q^{45} + 46604369920 q^{46} + 394942115908 q^{47} - 20937965568 q^{48} - 10631171627 q^{49} - 421539129152 q^{50} - 468879528009 q^{51} - 180942397440 q^{52} + 608978721416 q^{53} + 286529794304 q^{54} + 82053078738 q^{55} + 430964736 q^{56} - 655027400217 q^{57} - 1079566563968 q^{58} + 381384884765 q^{59} + 528673062912 q^{60} + 1151615204532 q^{61} - 101484696320 q^{62} + 5117893076470 q^{63} - 481036337152 q^{64} - 5194662408432 q^{65} - 2892007512576 q^{66} + 833522448542 q^{67} + 1183256911872 q^{68} - 4423665001426 q^{69} + 835597274496 q^{70} + 3476544683262 q^{71} - 2258466766848 q^{72} - 3737607886538 q^{73} + 3068896617088 q^{74} + 9622942863695 q^{75} - 1062618816512 q^{76} - 4431389843152 q^{77} - 2335610564352 q^{78} + 7224103591056 q^{79} - 1580346638336 q^{80} - 9164585352662 q^{81} + 1269347381440 q^{82} + 9607066728079 q^{83} + 260135567360 q^{84} + 25585834186966 q^{85} - 3541381155008 q^{86} - 34193079605396 q^{87} - 4585813704704 q^{88} + 19963863911122 q^{89} + 15393877925888 q^{90} - 11356103019586 q^{91} - 2505291898880 q^{92} + 6475155937894 q^{93} - 39623598881408 q^{94} - 51625880722484 q^{95} - 1340029796352 q^{96} + 4169205798475 q^{97} + 38124564609792 q^{98} + 71382923151637 q^{99}+O(q^{100})$$ 28 * q - 448 * q^2 - 1248 * q^3 - 28672 * q^4 + 71554 * q^5 + 99008 * q^6 + 361734 * q^7 - 1835008 * q^8 - 8615367 * q^9 + 2898176 * q^10 + 8699239 * q^11 - 2449408 * q^12 + 32373940 * q^13 + 23150976 * q^14 - 192099788 * q^15 - 117440512 * q^16 + 288881082 * q^17 - 275612928 * q^18 - 98588517 * q^19 + 293085184 * q^20 + 1362925660 * q^21 - 44352704 * q^22 - 233099500 * q^23 + 405536768 * q^24 + 1117600307 * q^25 - 2827224960 * q^26 + 3575804241 * q^27 - 1485029376 * q^28 - 16868227562 * q^29 - 12294386432 * q^30 - 10874790260 * q^31 + 30064771072 * q^32 + 53369498501 * q^33 + 6456111488 * q^34 + 18823205444 * q^35 - 17639227392 * q^36 + 47951509642 * q^37 + 14611374592 * q^38 - 161088016778 * q^39 - 24692916224 * q^40 - 155498856970 * q^41 - 47678239360 * q^42 + 156723459918 * q^43 + 31490678784 * q^44 + 75277819092 * q^45 + 46604369920 * q^46 + 394942115908 * q^47 - 20937965568 * q^48 - 10631171627 * q^49 - 421539129152 * q^50 - 468879528009 * q^51 - 180942397440 * q^52 + 608978721416 * q^53 + 286529794304 * q^54 + 82053078738 * q^55 + 430964736 * q^56 - 655027400217 * q^57 - 1079566563968 * q^58 + 381384884765 * q^59 + 528673062912 * q^60 + 1151615204532 * q^61 - 101484696320 * q^62 + 5117893076470 * q^63 - 481036337152 * q^64 - 5194662408432 * q^65 - 2892007512576 * q^66 + 833522448542 * q^67 + 1183256911872 * q^68 - 4423665001426 * q^69 + 835597274496 * q^70 + 3476544683262 * q^71 - 2258466766848 * q^72 - 3737607886538 * q^73 + 3068896617088 * q^74 + 9622942863695 * q^75 - 1062618816512 * q^76 - 4431389843152 * q^77 - 2335610564352 * q^78 + 7224103591056 * q^79 - 1580346638336 * q^80 - 9164585352662 * q^81 + 1269347381440 * q^82 + 9607066728079 * q^83 + 260135567360 * q^84 + 25585834186966 * q^85 - 3541381155008 * q^86 - 34193079605396 * q^87 - 4585813704704 * q^88 + 19963863911122 * q^89 + 15393877925888 * q^90 - 11356103019586 * q^91 - 2505291898880 * q^92 + 6475155937894 * q^93 - 39623598881408 * q^94 - 51625880722484 * q^95 - 1340029796352 * q^96 + 4169205798475 * q^97 + 38124564609792 * q^98 + 71382923151637 * 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 −675.486 2078.93i 1265.73 3895.53i −36563.1 26564.6i 113181. + 82230.5i −161776. + 497897.i 81007.0 + 249314.i −2.57584e6 + 1.87146e6i 2.89245e6
3.2 −51.7771 + 37.6183i −630.449 1940.32i 1265.73 3895.53i 52985.6 + 38496.3i 105634. + 76747.8i 86177.6 265227.i 81007.0 + 249314.i −2.07755e6 + 1.50943e6i −4.19160e6
3.3 −51.7771 + 37.6183i −365.266 1124.17i 1265.73 3895.53i −21575.3 15675.4i 61201.8 + 44465.7i 145958. 449212.i 81007.0 + 249314.i 159488. 115875.i 1.70679e6
3.4 −51.7771 + 37.6183i 41.6348 + 128.139i 1265.73 3895.53i 2573.60 + 1869.83i −6976.08 5068.42i −27076.8 + 83333.8i 81007.0 + 249314.i 1.27515e6 926449.i −203593.
3.5 −51.7771 + 37.6183i 226.298 + 696.474i 1265.73 3895.53i 39506.7 + 28703.3i −37917.2 27548.5i −100433. + 309100.i 81007.0 + 249314.i 855969. 621898.i −3.12531e6
3.6 −51.7771 + 37.6183i 412.870 + 1270.68i 1265.73 3895.53i −41240.7 29963.1i −69178.0 50260.8i 25390.0 78142.3i 81007.0 + 249314.i −154338. + 112133.i 3.26248e6
3.7 −51.7771 + 37.6183i 751.070 + 2311.56i 1265.73 3895.53i 25138.8 + 18264.4i −125845. 91431.7i 81934.7 252169.i 81007.0 + 249314.i −3.48935e6 + 2.53516e6i −1.98869e6
5.1 19.7771 + 60.8676i −2003.44 1455.58i −3313.73 + 2407.57i 12586.7 38738.0i 48975.7 150732.i 345957. 251352.i −212079. 154084.i 1.40237e6 + 4.31604e6i 2.60682e6
5.2 19.7771 + 60.8676i −1229.94 893.604i −3313.73 + 2407.57i −7708.37 + 23723.9i 30066.9 92536.4i −267598. + 194422.i −212079. 154084.i 221552. + 681866.i −1.59647e6
5.3 19.7771 + 60.8676i −501.424 364.306i −3313.73 + 2407.57i 17458.1 53730.4i 12257.7 37725.4i −385628. + 280175.i −212079. 154084.i −373966. 1.15095e6i 3.61571e6
5.4 19.7771 + 60.8676i −306.982 223.036i −3313.73 + 2407.57i −5178.33 + 15937.3i 7504.43 23096.3i 291527. 211807.i −212079. 154084.i −448180. 1.37936e6i −1.07248e6
5.5 19.7771 + 60.8676i 697.531 + 506.786i −3313.73 + 2407.57i 7802.80 24014.5i −17051.7 + 52479.8i 9628.39 6995.43i −212079. 154084.i −262955. 809293.i 1.61602e6
5.6 19.7771 + 60.8676i 1193.16 + 866.884i −3313.73 + 2407.57i −16233.3 + 49960.9i −29167.9 + 89769.5i −203084. + 147550.i −212079. 154084.i 179478. + 552378.i −3.36205e6
5.7 19.7771 + 60.8676i 1766.42 + 1283.38i −3313.73 + 2407.57i 6223.79 19154.8i −43181.5 + 132899.i 339891. 246945.i −212079. 154084.i 980497. + 3.01766e6i 1.28900e6
9.1 19.7771 60.8676i −2003.44 + 1455.58i −3313.73 2407.57i 12586.7 + 38738.0i 48975.7 + 150732.i 345957. + 251352.i −212079. + 154084.i 1.40237e6 4.31604e6i 2.60682e6
9.2 19.7771 60.8676i −1229.94 + 893.604i −3313.73 2407.57i −7708.37 23723.9i 30066.9 + 92536.4i −267598. 194422.i −212079. + 154084.i 221552. 681866.i −1.59647e6
9.3 19.7771 60.8676i −501.424 + 364.306i −3313.73 2407.57i 17458.1 + 53730.4i 12257.7 + 37725.4i −385628. 280175.i −212079. + 154084.i −373966. + 1.15095e6i 3.61571e6
9.4 19.7771 60.8676i −306.982 + 223.036i −3313.73 2407.57i −5178.33 15937.3i 7504.43 + 23096.3i 291527. + 211807.i −212079. + 154084.i −448180. + 1.37936e6i −1.07248e6
9.5 19.7771 60.8676i 697.531 506.786i −3313.73 2407.57i 7802.80 + 24014.5i −17051.7 52479.8i 9628.39 + 6995.43i −212079. + 154084.i −262955. + 809293.i 1.61602e6
9.6 19.7771 60.8676i 1193.16 866.884i −3313.73 2407.57i −16233.3 49960.9i −29167.9 89769.5i −203084. 147550.i −212079. + 154084.i 179478. 552378.i −3.36205e6
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 15.7 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.b 28
11.c even 5 1 inner 22.14.c.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.14.c.b 28 1.a even 1 1 trivial
22.14.c.b 28 11.c even 5 1 inner