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Properties

Label 99.7.k.b
Level $99$
Weight $7$
Character orbit 99.k
Analytic conductor $22.775$
Analytic rank $0$
Dimension $48$
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] = [99,7,Mod(19,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 99.k (of order \(10\), degree \(4\), 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: \(22.7753542784\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 204 q^{4} + 224 q^{5} + 720 q^{7} + 3200 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 204 q^{4} + 224 q^{5} + 720 q^{7} + 3200 q^{8} - 6664 q^{11} + 3600 q^{13} + 14602 q^{14} - 13416 q^{16} + 7280 q^{17} + 11958 q^{20} - 19518 q^{22} - 64704 q^{23} - 9492 q^{25} - 89270 q^{26} - 149370 q^{28} + 128800 q^{29} + 79308 q^{31} + 355152 q^{34} - 380480 q^{35} + 236112 q^{37} + 318458 q^{38} - 1045170 q^{40} + 528360 q^{41} - 676754 q^{44} + 1215450 q^{46} - 141720 q^{47} + 215268 q^{49} + 679470 q^{50} - 1957890 q^{52} + 313488 q^{53} - 397656 q^{55} - 960220 q^{56} + 198558 q^{58} - 1236208 q^{59} - 1409280 q^{61} + 453280 q^{62} + 521784 q^{64} + 930528 q^{67} + 842240 q^{68} + 2765100 q^{70} + 3520024 q^{71} + 201780 q^{73} - 2051950 q^{74} - 2434584 q^{77} - 321960 q^{79} - 2395994 q^{80} + 2354178 q^{82} + 4816400 q^{83} - 1250040 q^{85} + 3323560 q^{86} - 1493928 q^{88} - 4157744 q^{89} - 1843512 q^{91} + 4448550 q^{92} + 14061810 q^{94} - 14355680 q^{95} + 2325888 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −8.96906 − 12.3449i 0 −52.1743 + 160.576i −137.717 − 100.057i 0 −91.3553 − 29.6831i 1521.46 − 494.352i 0 2597.52i
19.2 −8.69356 − 11.9657i 0 −47.8219 + 147.181i 118.124 + 85.8219i 0 185.111 + 60.1462i 1276.60 − 414.794i 0 − 2159.53i
19.3 −6.26570 − 8.62400i 0 −15.3373 + 47.2032i −161.919 − 117.641i 0 −310.129 − 100.767i −145.660 + 47.3278i 0 2133.50i
19.4 −6.02470 − 8.29229i 0 −12.6880 + 39.0496i 154.787 + 112.460i 0 −515.310 − 167.434i −223.631 + 72.6622i 0 − 1961.08i
19.5 −4.99839 − 6.87969i 0 −2.56917 + 7.90708i 1.33123 + 0.967196i 0 525.675 + 170.802i −450.364 + 146.332i 0 − 13.9929i
19.6 −2.37250 − 3.26547i 0 14.7426 − 45.3730i −26.5586 − 19.2960i 0 457.293 + 148.583i −428.823 + 139.333i 0 132.506i
19.7 −1.61569 − 2.22380i 0 17.4422 − 53.6817i 125.017 + 90.8304i 0 −357.347 − 116.109i −314.870 + 102.307i 0 − 424.767i
19.8 1.16939 + 1.60953i 0 18.5540 − 57.1033i 1.49745 + 1.08796i 0 −25.7895 − 8.37953i 234.702 − 76.2594i 0 3.68244i
19.9 5.06354 + 6.96937i 0 −3.15555 + 9.71180i 8.79332 + 6.38872i 0 −242.092 − 78.6605i 440.688 − 143.188i 0 93.6335i
19.10 6.32244 + 8.70209i 0 −15.9760 + 49.1691i 92.2594 + 67.0304i 0 −121.251 − 39.3968i 125.833 − 40.8857i 0 1226.64i
19.11 6.91015 + 9.51101i 0 −22.9320 + 70.5775i −174.946 − 127.106i 0 472.320 + 153.466i −114.152 + 37.0902i 0 − 2542.24i
19.12 8.29374 + 11.4154i 0 −41.7471 + 128.484i 55.3317 + 40.2008i 0 511.454 + 166.181i −954.081 + 310.000i 0 965.045i
28.1 −12.8846 + 4.18646i 0 96.7090 − 70.2632i −19.1112 + 58.8184i 0 −53.6679 − 73.8676i −442.262 + 608.722i 0 − 837.858i
28.2 −11.5908 + 3.76609i 0 68.3868 − 49.6859i −42.9151 + 132.079i 0 −91.9841 − 126.605i −147.072 + 202.428i 0 − 1692.53i
28.3 −7.56988 + 2.45960i 0 −0.523682 + 0.380478i 67.3727 − 207.352i 0 −377.234 − 519.218i 302.449 − 416.285i 0 1735.34i
28.4 −6.43087 + 2.08952i 0 −14.7870 + 10.7434i 39.7158 − 122.233i 0 156.717 + 215.703i 327.013 − 450.095i 0 869.050i
28.5 −4.51299 + 1.46636i 0 −33.5602 + 24.3829i −6.35291 + 19.5523i 0 355.989 + 489.977i 294.210 − 404.946i 0 − 97.5547i
28.6 0.674572 − 0.219182i 0 −51.3701 + 37.3225i 8.69665 − 26.7655i 0 −301.986 − 415.648i −53.1545 + 73.1609i 0 − 19.9614i
28.7 2.69988 − 0.877245i 0 −45.2573 + 32.8813i −71.7465 + 220.813i 0 166.082 + 228.592i −200.136 + 275.463i 0 659.108i
28.8 5.17091 − 1.68013i 0 −27.8616 + 20.2427i 25.6501 − 78.9429i 0 −29.3745 − 40.4305i −314.591 + 432.997i 0 − 451.302i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.7.k.b 48
3.b odd 2 1 33.7.g.a ✓ 48
11.d odd 10 1 inner 99.7.k.b 48
33.f even 10 1 33.7.g.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.g.a ✓ 48 3.b odd 2 1
33.7.g.a ✓ 48 33.f even 10 1
99.7.k.b 48 1.a even 1 1 trivial
99.7.k.b 48 11.d odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 486 T_{2}^{46} - 3200 T_{2}^{45} + 177420 T_{2}^{44} + 1555200 T_{2}^{43} + \cdots + 40\!\cdots\!16 \) acting on \(S_{7}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display