Properties

Label 19.5.f.a
Level $19$
Weight $5$
Character orbit 19.f
Analytic conductor $1.964$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.96402929859\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q - 6q^{2} - 18q^{3} - 48q^{4} - 6q^{5} - 48q^{7} - 9q^{8} + 246q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q - 6q^{2} - 18q^{3} - 48q^{4} - 6q^{5} - 48q^{7} - 9q^{8} + 246q^{9} - 153q^{10} - 48q^{11} + 855q^{12} + 555q^{13} - 399q^{14} - 1647q^{15} - 2112q^{16} + 39q^{17} - 12q^{19} + 2886q^{20} + 912q^{21} + 1386q^{22} + 1074q^{23} + 168q^{24} + 2448q^{25} - 3675q^{26} + 1215q^{27} + 2316q^{28} - 1743q^{29} + 2580q^{30} - 2817q^{31} - 6039q^{32} - 7368q^{33} - 8460q^{34} - 1326q^{35} - 4581q^{36} + 2892q^{38} + 7164q^{39} + 4482q^{40} + 6150q^{41} + 22299q^{42} + 2091q^{43} + 34053q^{44} + 3165q^{45} + 7326q^{46} + 4539q^{47} - 44943q^{48} + 102q^{49} - 51876q^{50} - 25836q^{51} - 3279q^{52} - 11607q^{53} + 6069q^{54} - 7044q^{55} + 2532q^{57} + 13644q^{58} - 681q^{59} + 63654q^{60} + 44346q^{61} + 14796q^{62} + 53388q^{63} + 939q^{64} + 3636q^{65} + 11148q^{66} - 52089q^{67} + 5262q^{68} - 54954q^{69} - 69999q^{70} - 24504q^{71} - 101118q^{72} - 6666q^{73} - 9675q^{74} - 6462q^{76} + 48918q^{77} + 56871q^{78} + 41043q^{79} + 65649q^{80} + 2919q^{81} + 114711q^{82} - 17598q^{83} + 11115q^{84} + 52344q^{85} - 55716q^{86} + 2121q^{87} - 69057q^{88} - 11706q^{89} - 41394q^{90} - 84249q^{91} - 88668q^{92} - 60321q^{93} + 93405q^{95} + 167166q^{96} + 4401q^{97} + 69381q^{98} + 58386q^{99} + O(q^{100}) \)

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 −7.01187 1.23638i −2.17032 + 2.58649i 32.6027 + 11.8664i 26.1793 9.52849i 18.4159 15.4528i 22.6666 + 39.2597i −115.276 66.5547i 12.0859 + 68.5424i −195.347 + 34.4449i
2.2 −3.73687 0.658911i 0.577174 0.687849i −1.50506 0.547798i −39.5293 + 14.3875i −2.61005 + 2.19009i −34.7004 60.1028i 57.8416 + 33.3949i 13.9255 + 78.9754i 157.196 27.7179i
2.3 −1.06853 0.188411i 6.31184 7.52216i −13.9288 5.06968i 33.8670 12.3266i −8.16166 + 6.84845i −8.52110 14.7590i 28.9626 + 16.7216i −2.67805 15.1880i −38.5105 + 6.79043i
2.4 −0.301706 0.0531989i −9.51563 + 11.3403i −14.9469 5.44022i −0.554323 + 0.201757i 3.47421 2.91521i 33.9928 + 58.8772i 8.46520 + 4.88738i −23.9894 136.050i 0.177976 0.0313819i
2.5 4.43376 + 0.781791i 6.29743 7.50499i 4.01191 + 1.46022i −26.3200 + 9.57971i 33.7886 28.3520i 37.8762 + 65.6035i −45.7374 26.4065i −2.60170 14.7550i −124.186 + 21.8973i
2.6 5.74553 + 1.01309i −4.05101 + 4.82781i 16.9497 + 6.16919i 13.0485 4.74927i −28.1662 + 23.6343i −38.0385 65.8845i 10.2945 + 5.94354i 7.16848 + 40.6545i 79.7821 14.0677i
3.1 −5.01717 + 5.97923i 4.42557 + 12.1592i −7.80084 44.2408i −1.37942 + 7.82309i −94.9062 34.5430i −2.12885 3.68728i 195.510 + 112.878i −66.2097 + 55.5565i −39.8553 47.4977i
3.2 −3.41023 + 4.06416i −4.67230 12.8370i −2.10931 11.9625i 3.16499 17.9496i 68.1054 + 24.7883i −22.5528 39.0626i −17.7027 10.2207i −80.9096 + 67.8913i 62.1565 + 74.0752i
3.3 −1.35909 + 1.61970i 0.767327 + 2.10821i 2.00207 + 11.3543i −3.09216 + 17.5365i −4.45754 1.62241i 14.2495 + 24.6809i −50.4090 29.1037i 58.1938 48.8304i −24.2013 28.8420i
3.4 1.68606 2.00936i 4.63257 + 12.7279i 1.58361 + 8.98112i 4.60964 26.1426i 33.3857 + 12.1514i −34.5022 59.7596i 57.0623 + 32.9449i −78.4888 + 65.8599i −44.7579 53.3404i
3.5 2.50472 2.98501i −3.27675 9.00281i 0.141711 + 0.803685i 2.69721 15.2966i −35.0808 12.7684i 18.9504 + 32.8230i 56.7476 + 32.7632i −8.26383 + 6.93418i −38.9048 46.3649i
3.6 4.76936 5.68391i 1.42559 + 3.91678i −6.78159 38.4603i −7.62340 + 43.2344i 29.0618 + 10.5776i −9.09343 15.7503i −148.137 85.5267i 48.7408 40.8984i 209.382 + 249.531i
10.1 −7.01187 + 1.23638i −2.17032 2.58649i 32.6027 11.8664i 26.1793 + 9.52849i 18.4159 + 15.4528i 22.6666 39.2597i −115.276 + 66.5547i 12.0859 68.5424i −195.347 34.4449i
10.2 −3.73687 + 0.658911i 0.577174 + 0.687849i −1.50506 + 0.547798i −39.5293 14.3875i −2.61005 2.19009i −34.7004 + 60.1028i 57.8416 33.3949i 13.9255 78.9754i 157.196 + 27.7179i
10.3 −1.06853 + 0.188411i 6.31184 + 7.52216i −13.9288 + 5.06968i 33.8670 + 12.3266i −8.16166 6.84845i −8.52110 + 14.7590i 28.9626 16.7216i −2.67805 + 15.1880i −38.5105 6.79043i
10.4 −0.301706 + 0.0531989i −9.51563 11.3403i −14.9469 + 5.44022i −0.554323 0.201757i 3.47421 + 2.91521i 33.9928 58.8772i 8.46520 4.88738i −23.9894 + 136.050i 0.177976 + 0.0313819i
10.5 4.43376 0.781791i 6.29743 + 7.50499i 4.01191 1.46022i −26.3200 9.57971i 33.7886 + 28.3520i 37.8762 65.6035i −45.7374 + 26.4065i −2.60170 + 14.7550i −124.186 21.8973i
10.6 5.74553 1.01309i −4.05101 4.82781i 16.9497 6.16919i 13.0485 + 4.74927i −28.1662 23.6343i −38.0385 + 65.8845i 10.2945 5.94354i 7.16848 40.6545i 79.7821 + 14.0677i
13.1 −5.01717 5.97923i 4.42557 12.1592i −7.80084 + 44.2408i −1.37942 7.82309i −94.9062 + 34.5430i −2.12885 + 3.68728i 195.510 112.878i −66.2097 55.5565i −39.8553 + 47.4977i
13.2 −3.41023 4.06416i −4.67230 + 12.8370i −2.10931 + 11.9625i 3.16499 + 17.9496i 68.1054 24.7883i −22.5528 + 39.0626i −17.7027 + 10.2207i −80.9096 67.8913i 62.1565 74.0752i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.5.f.a 36
19.f odd 18 1 inner 19.5.f.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.5.f.a 36 1.a even 1 1 trivial
19.5.f.a 36 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(19, [\chi])\).