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Properties

Label 19.9.d.a
Level $19$
Weight $9$
Character orbit 19.d
Analytic conductor $7.740$
Analytic rank $0$
Dimension $26$
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] = [19,9,Mod(8,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.8");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 19.d (of order \(6\), degree \(2\), 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: \(7.74019359116\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 3 q^{2} - 171 q^{3} + 2029 q^{4} + 278 q^{5} - 275 q^{6} - 7504 q^{7} + 25574 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 3 q^{2} - 171 q^{3} + 2029 q^{4} + 278 q^{5} - 275 q^{6} - 7504 q^{7} + 25574 q^{9} + 2784 q^{10} - 26530 q^{11} - 88254 q^{13} + 122358 q^{14} + 246474 q^{15} - 375891 q^{16} + 14378 q^{17} + 481391 q^{19} - 627532 q^{20} + 43494 q^{21} + 680829 q^{22} - 60892 q^{23} + 162409 q^{24} - 1007371 q^{25} + 507252 q^{26} - 3333872 q^{28} - 109884 q^{29} + 6068704 q^{30} - 4485465 q^{32} - 1807311 q^{33} + 3137142 q^{34} + 542014 q^{35} - 10525926 q^{36} + 10867338 q^{38} + 3117892 q^{39} + 9687504 q^{40} + 16439685 q^{41} - 2916908 q^{42} - 7298466 q^{43} - 16490521 q^{44} - 32357068 q^{45} - 2654026 q^{47} + 71125161 q^{48} + 7642254 q^{49} - 68110752 q^{51} - 53430522 q^{52} + 9445632 q^{53} + 34820561 q^{54} - 21205876 q^{55} + 22269764 q^{57} - 5883596 q^{58} + 21006195 q^{59} + 209639046 q^{60} - 4744062 q^{61} - 58089324 q^{62} - 66326902 q^{63} - 101431266 q^{64} - 119841553 q^{66} + 102477003 q^{67} + 70594196 q^{68} - 122733774 q^{70} + 7935738 q^{71} + 46393554 q^{72} + 26449961 q^{73} - 104007498 q^{74} + 99727859 q^{76} - 38226872 q^{77} + 128944746 q^{78} + 148452324 q^{79} - 128268802 q^{80} + 87756319 q^{81} + 38787119 q^{82} - 143066578 q^{83} - 13463814 q^{85} + 259064682 q^{86} - 275422336 q^{87} + 62156112 q^{89} - 289880070 q^{90} + 299224752 q^{91} - 267189628 q^{92} + 264058076 q^{93} + 91155758 q^{95} - 508960698 q^{96} - 123441153 q^{97} - 579159765 q^{98} + 35656504 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −26.1432 − 15.0938i −106.716 − 61.6122i 327.646 + 567.500i −245.340 + 424.941i 1859.93 + 3221.49i −2946.80 − 12053.7i 4311.63 + 7467.97i 12828.0 − 7406.22i
8.2 −21.7992 − 12.5858i 123.984 + 71.5823i 188.802 + 327.015i −68.8102 + 119.183i −1801.84 − 3120.87i −3038.80 − 3060.98i 6967.56 + 12068.2i 3000.01 − 1732.06i
8.3 −20.8066 − 12.0127i 8.34450 + 4.81770i 160.610 + 278.184i 351.045 − 608.028i −115.747 − 200.480i 3502.01 − 1566.93i −3234.08 − 5601.59i −14608.1 + 8433.99i
8.4 −15.7546 − 9.09592i −1.65090 − 0.953147i 37.4715 + 64.9025i −325.497 + 563.777i 17.3395 + 30.0329i 511.341 3293.76i −3278.68 − 5678.85i 10256.1 − 5921.38i
8.5 −10.3491 − 5.97503i −77.3651 − 44.6668i −56.5981 − 98.0307i 550.431 − 953.374i 533.771 + 924.518i −3192.94 4411.91i 709.744 + 1229.31i −11392.9 + 6577.68i
8.6 −2.50545 − 1.44652i 39.0380 + 22.5386i −123.815 − 214.454i −239.375 + 414.610i −65.2053 − 112.939i −698.057 1457.03i −2264.52 − 3922.27i 1199.49 − 692.524i
8.7 −2.21055 − 1.27626i −112.712 − 65.0743i −124.742 − 216.060i −220.920 + 382.645i 166.103 + 287.700i 2295.81 1290.26i 5188.82 + 8987.29i 976.709 − 563.903i
8.8 0.481619 + 0.278063i 98.2973 + 56.7520i −127.845 − 221.435i 348.476 − 603.578i 31.5612 + 54.6657i 795.375 − 284.564i 3161.08 + 5475.15i 335.665 − 193.796i
8.9 12.2020 + 7.04482i −39.0166 − 22.5262i −28.7410 − 49.7809i 15.4573 − 26.7728i −317.387 − 549.730i −3198.31 − 4416.85i −2265.64 − 3924.20i 377.218 − 217.787i
8.10 15.6174 + 9.01669i −53.5408 − 30.9118i 34.6014 + 59.9313i 258.395 − 447.553i −557.445 − 965.522i 3990.32 − 3368.59i −1369.42 − 2371.90i 8070.90 − 4659.74i
8.11 17.8651 + 10.3144i 83.3526 + 48.1237i 84.7755 + 146.836i −381.822 + 661.334i 992.738 + 1719.47i 1105.97 − 1783.35i 1351.27 + 2340.47i −13642.6 + 7876.56i
8.12 25.4196 + 14.6760i 59.8378 + 34.5473i 302.772 + 524.417i 537.998 − 931.840i 1014.04 + 1756.36i −1769.29 10259.9i −893.462 − 1547.52i 27351.4 − 15791.4i
8.13 26.4829 + 15.2899i −107.354 − 61.9806i 339.563 + 588.140i −441.038 + 763.901i −1895.36 − 3282.85i −1108.63 12939.1i 4402.69 + 7625.69i −23359.9 + 13486.9i
12.1 −26.1432 + 15.0938i −106.716 + 61.6122i 327.646 − 567.500i −245.340 − 424.941i 1859.93 − 3221.49i −2946.80 12053.7i 4311.63 − 7467.97i 12828.0 + 7406.22i
12.2 −21.7992 + 12.5858i 123.984 − 71.5823i 188.802 − 327.015i −68.8102 − 119.183i −1801.84 + 3120.87i −3038.80 3060.98i 6967.56 − 12068.2i 3000.01 + 1732.06i
12.3 −20.8066 + 12.0127i 8.34450 − 4.81770i 160.610 − 278.184i 351.045 + 608.028i −115.747 + 200.480i 3502.01 1566.93i −3234.08 + 5601.59i −14608.1 − 8433.99i
12.4 −15.7546 + 9.09592i −1.65090 + 0.953147i 37.4715 − 64.9025i −325.497 − 563.777i 17.3395 − 30.0329i 511.341 − 3293.76i −3278.68 + 5678.85i 10256.1 + 5921.38i
12.5 −10.3491 + 5.97503i −77.3651 + 44.6668i −56.5981 + 98.0307i 550.431 + 953.374i 533.771 − 924.518i −3192.94 − 4411.91i 709.744 − 1229.31i −11392.9 − 6577.68i
12.6 −2.50545 + 1.44652i 39.0380 − 22.5386i −123.815 + 214.454i −239.375 − 414.610i −65.2053 + 112.939i −698.057 − 1457.03i −2264.52 + 3922.27i 1199.49 + 692.524i
12.7 −2.21055 + 1.27626i −112.712 + 65.0743i −124.742 + 216.060i −220.920 − 382.645i 166.103 − 287.700i 2295.81 − 1290.26i 5188.82 − 8987.29i 976.709 + 563.903i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.9.d.a ✓ 26
19.d odd 6 1 inner 19.9.d.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.9.d.a ✓ 26 1.a even 1 1 trivial
19.9.d.a ✓ 26 19.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(19, [\chi])\).