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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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [19,5,Mod(2,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.2");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 2.6
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 \(S_{5}^{\mathrm{new}}(19, [\chi])\).