Properties

Label 69.5.h.a
Level $69$
Weight $5$
Character orbit 69.h
Analytic conductor $7.133$
Analytic rank $0$
Dimension $300$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 69.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.13252745279\)
Analytic rank: \(0\)
Dimension: \(300\)
Relative dimension: \(30\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 300q - 19q^{3} + 234q^{4} - 10q^{6} - 18q^{7} - 111q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 300q - 19q^{3} + 234q^{4} - 10q^{6} - 18q^{7} - 111q^{9} + 18q^{10} + 426q^{12} + 82q^{13} - 1400q^{15} - 2046q^{16} + 3624q^{18} + 426q^{19} + 636q^{21} - 2264q^{22} - 6518q^{24} + 4436q^{25} - 5794q^{27} + 2606q^{28} + 1623q^{30} - 3038q^{31} + 5918q^{33} - 11500q^{34} + 442q^{36} + 6818q^{37} + 2527q^{39} + 36544q^{40} + 6305q^{42} + 8786q^{43} + 8024q^{45} - 8030q^{46} - 14146q^{48} - 48856q^{49} - 5201q^{51} - 41374q^{52} + 39645q^{54} + 18588q^{55} + 5238q^{57} + 38840q^{58} - 70087q^{60} - 29934q^{61} - 31600q^{63} - 15644q^{64} - 29948q^{66} - 3168q^{67} + 7524q^{69} + 42356q^{70} + 70033q^{72} + 6544q^{73} - 11440q^{75} + 57622q^{76} + 21112q^{78} - 19986q^{79} + 98277q^{81} - 29792q^{82} + 42585q^{84} - 44480q^{85} + 77891q^{87} + 32678q^{88} + 39986q^{90} + 4792q^{91} - 78912q^{93} + 42448q^{94} - 107223q^{96} - 83108q^{97} - 153089q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −4.11610 + 6.40478i 8.97488 0.671914i −17.4323 38.1713i 6.99421 23.8201i −32.6381 + 60.2478i −31.6181 + 36.4893i 195.658 + 28.1313i 80.0971 12.0607i 123.774 + 142.842i
2.2 −4.08077 + 6.34980i −1.82217 + 8.81361i −17.0207 37.2701i −1.39862 + 4.76328i −48.5289 47.5367i 54.0931 62.4268i 186.576 + 26.8256i −74.3594 32.1197i −24.5384 28.3188i
2.3 −3.63251 + 5.65229i −6.90793 5.76893i −12.1066 26.5099i 2.68502 9.14434i 57.7008 18.0899i 3.93980 4.54677i 87.4108 + 12.5678i 14.4389 + 79.7027i 41.9331 + 48.3934i
2.4 −3.39074 + 5.27609i 3.68875 8.20933i −9.69338 21.2255i −7.37394 + 25.1133i 30.8056 + 47.2979i −0.916617 + 1.05783i 45.5297 + 6.54619i −53.7862 60.5644i −107.497 124.058i
2.5 −3.15045 + 4.90219i −7.93709 + 4.24295i −7.45954 16.3341i −12.0836 + 41.1529i 4.20561 52.2764i −46.0495 + 53.1439i 11.2870 + 1.62282i 44.9947 67.3534i −163.671 188.886i
2.6 −2.72930 + 4.24687i 2.47487 + 8.65304i −3.94020 8.62784i 4.64119 15.8064i −43.5030 13.1063i −37.0863 + 42.7998i −32.5548 4.68067i −68.7501 + 42.8302i 54.4607 + 62.8510i
2.7 −2.52768 + 3.93314i 7.84620 + 4.40876i −2.43382 5.32931i −7.84851 + 26.7296i −37.1729 + 19.7163i 16.6015 19.1591i −46.9311 6.74767i 42.1256 + 69.1841i −85.2928 98.4331i
2.8 −2.30231 + 3.58247i −7.95344 + 4.21222i −0.886802 1.94183i 8.56609 29.1734i 3.22114 38.1908i 24.5435 28.3247i −58.4441 8.40299i 45.5143 67.0033i 84.7911 + 97.8541i
2.9 −2.18249 + 3.39602i 8.74164 2.14095i −0.123067 0.269480i 7.29897 24.8580i −11.8079 + 34.3594i 47.9514 55.3388i −62.7486 9.02189i 71.8327 37.4308i 68.4884 + 79.0399i
2.10 −1.85051 + 2.87944i 0.149878 8.99875i 1.77982 + 3.89726i 13.0909 44.5836i 25.6340 + 17.0838i −46.3292 + 53.4668i −68.7229 9.88086i −80.9551 2.69742i 104.151 + 120.197i
2.11 −1.18989 + 1.85151i −8.40626 3.21477i 4.63439 + 10.1479i −4.40450 + 15.0003i 15.9547 11.7391i 8.45761 9.76060i −59.1593 8.50582i 60.3306 + 54.0483i −22.5324 26.0038i
2.12 −1.16876 + 1.81863i −1.49992 8.87413i 4.70524 + 10.3030i −4.27625 + 14.5636i 17.8918 + 7.64395i 33.8030 39.0108i −58.4735 8.40722i −76.5005 + 26.6209i −21.4878 24.7982i
2.13 −0.842718 + 1.31130i 7.53602 4.92020i 5.63732 + 12.3440i −5.53880 + 18.8634i 0.101095 + 14.0283i −41.7057 + 48.1309i −45.6233 6.55964i 32.5832 74.1575i −20.0679 23.1596i
2.14 −0.386142 + 0.600848i −1.66494 + 8.84466i 6.43473 + 14.0901i −11.2060 + 38.1641i −4.67140 4.41567i 37.6947 43.5020i −22.2621 3.20080i −75.4560 29.4516i −18.6037 21.4699i
2.15 −0.152702 + 0.237609i −5.50899 + 7.11696i 6.61350 + 14.4815i 1.03882 3.53791i −0.849819 2.39576i −24.2369 + 27.9709i −8.92398 1.28308i −20.3021 78.4144i 0.682008 + 0.787079i
2.16 0.152702 0.237609i 7.29091 + 5.27661i 6.61350 + 14.4815i −1.03882 + 3.53791i 2.36711 0.926638i −24.2369 + 27.9709i 8.92398 + 1.28308i 25.3148 + 76.9426i 0.682008 + 0.787079i
2.17 0.386142 0.600848i 4.08932 + 8.01732i 6.43473 + 14.0901i 11.2060 38.1641i 6.39625 + 0.638760i 37.6947 43.5020i 22.2621 + 3.20080i −47.5549 + 65.5708i −18.6037 21.4699i
2.18 0.842718 1.31130i −8.61694 2.59776i 5.63732 + 12.3440i 5.53880 18.8634i −10.6681 + 9.11018i −41.7057 + 48.1309i 45.6233 + 6.55964i 67.5033 + 44.7694i −20.0679 23.1596i
2.19 1.16876 1.81863i −1.06097 8.93724i 4.70524 + 10.3030i 4.27625 14.5636i −17.4935 8.51599i 33.8030 39.0108i 58.4735 + 8.40722i −78.7487 + 18.9643i −21.4878 24.7982i
2.20 1.18989 1.85151i 7.16005 5.45286i 4.63439 + 10.1479i 4.40450 15.0003i −1.57634 19.7452i 8.45761 9.76060i 59.1593 + 8.50582i 21.5326 78.0855i −22.5324 26.0038i
See next 80 embeddings (of 300 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 62.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.5.h.a 300
3.b odd 2 1 inner 69.5.h.a 300
23.c even 11 1 inner 69.5.h.a 300
69.h odd 22 1 inner 69.5.h.a 300
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.5.h.a 300 1.a even 1 1 trivial
69.5.h.a 300 3.b odd 2 1 inner
69.5.h.a 300 23.c even 11 1 inner
69.5.h.a 300 69.h odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(69, [\chi])\).