Properties

Label 41.11.e.a
Level $41$
Weight $11$
Character orbit 41.e
Analytic conductor $26.050$
Analytic rank $0$
Dimension $136$
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] = [41,11,Mod(3,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 41.e (of order \(8\), 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: \(26.0496473596\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(34\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 68 q^{2} - 92 q^{3} - 4 q^{5} - 34432 q^{6} - 4 q^{7} + 102396 q^{8} - 242988 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 68 q^{2} - 92 q^{3} - 4 q^{5} - 34432 q^{6} - 4 q^{7} + 102396 q^{8} - 242988 q^{9} - 8 q^{10} - 4 q^{11} + 258164 q^{12} + 1717336 q^{13} + 1175656 q^{14} + 2282768 q^{15} - 32505864 q^{16} + 5033020 q^{17} - 8 q^{18} + 6048236 q^{19} - 1636120 q^{20} + 3986728 q^{21} - 24419324 q^{22} - 36267708 q^{24} - 9974084 q^{26} + 12095464 q^{27} + 4194300 q^{28} + 20806536 q^{29} - 9731960 q^{30} + 361448336 q^{32} + 24464216 q^{33} - 110069848 q^{34} - 80795444 q^{35} - 596884800 q^{36} + 81957384 q^{37} + 186467668 q^{38} + 166599276 q^{39} - 238032092 q^{41} - 976047640 q^{42} - 502015480 q^{43} - 310259716 q^{44} + 1694145940 q^{46} + 603097452 q^{47} + 1587981888 q^{48} - 1056170764 q^{49} - 687037864 q^{50} - 2603416824 q^{51} + 3527070328 q^{52} + 51810832 q^{53} - 589938412 q^{54} + 1637216340 q^{55} + 1542812032 q^{56} - 4986495040 q^{57} - 4931743400 q^{58} - 3176564552 q^{59} + 2876106548 q^{60} - 1788511500 q^{61} + 318795208 q^{62} - 12784962700 q^{63} + 7321931752 q^{65} - 3945409468 q^{67} + 8344683180 q^{68} + 12448497964 q^{69} + 13605682700 q^{70} + 4700517652 q^{71} + 937095136 q^{73} - 10288042020 q^{74} + 20795196852 q^{75} - 18422762696 q^{76} + 18458397928 q^{77} + 11820197552 q^{78} + 550856776 q^{79} + 9755435892 q^{80} + 2010567116 q^{82} - 6499037104 q^{83} - 2819557676 q^{84} + 917552948 q^{85} - 8974302400 q^{87} - 10275500540 q^{88} + 9679887592 q^{89} + 45805512072 q^{90} - 6621203168 q^{91} - 66325590688 q^{92} - 9704387424 q^{93} - 79962089204 q^{94} - 61876018160 q^{95} + 11006701356 q^{96} - 34836533432 q^{97} + 94516412136 q^{98} - 35821160260 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −43.4948 + 43.4948i −173.386 + 418.590i 2759.59i 2003.37 + 2003.37i −10665.1 25747.8i 9569.84 23103.6i 75489.1 + 75489.1i −103401. 103401.i −174272.
3.2 −43.3059 + 43.3059i −34.7089 + 83.7946i 2726.80i −932.314 932.314i −2125.70 5131.90i −11739.6 + 28341.9i 73741.1 + 73741.1i 35937.1 + 35937.1i 80749.3
3.3 −40.7233 + 40.7233i 165.169 398.754i 2292.78i 3636.89 + 3636.89i 9512.34 + 22964.8i −288.015 + 695.329i 51669.1 + 51669.1i −89969.6 89969.6i −296213.
3.4 −40.5638 + 40.5638i 75.2761 181.733i 2266.84i −2614.09 2614.09i 4318.28 + 10425.3i 5340.28 12892.6i 50414.5 + 50414.5i 14393.7 + 14393.7i 212075.
3.5 −35.7906 + 35.7906i −5.95702 + 14.3815i 1537.93i 2505.68 + 2505.68i −301.517 727.927i 470.064 1134.83i 18393.8 + 18393.8i 41582.6 + 41582.6i −179359.
3.6 −30.9351 + 30.9351i −97.3324 + 234.981i 889.964i −3921.72 3921.72i −4258.18 10280.2i 9122.15 22022.8i −4146.43 4146.43i −3988.62 3988.62i 242638.
3.7 −30.5648 + 30.5648i −40.2742 + 97.2305i 844.416i 1121.32 + 1121.32i −1740.86 4202.81i 2970.09 7170.43i −5488.94 5488.94i 33922.2 + 33922.2i −68546.1
3.8 −29.6057 + 29.6057i −137.161 + 331.137i 728.991i −1675.81 1675.81i −5742.77 13864.3i −7692.51 + 18571.4i −8733.93 8733.93i −49084.3 49084.3i 99227.1
3.9 −29.2016 + 29.2016i 132.555 320.015i 681.461i −876.164 876.164i 5474.13 + 13215.7i −1512.10 + 3650.54i −10002.7 10002.7i −43084.9 43084.9i 51170.7
3.10 −21.1119 + 21.1119i 89.4836 216.032i 132.575i −2323.82 2323.82i 2671.69 + 6450.02i −6803.92 + 16426.1i −24417.5 24417.5i 3091.24 + 3091.24i 98120.7
3.11 −17.0387 + 17.0387i 51.4488 124.208i 443.364i 3686.39 + 3686.39i 1239.73 + 2992.98i −11841.4 + 28587.6i −25002.0 25002.0i 28973.2 + 28973.2i −125623.
3.12 −16.4049 + 16.4049i −144.811 + 349.605i 485.760i 2748.40 + 2748.40i −3359.62 8110.85i −2544.07 + 6141.94i −24767.4 24767.4i −59499.6 59499.6i −90174.2
3.13 −14.7946 + 14.7946i 94.8180 228.911i 586.238i 1671.69 + 1671.69i 1983.85 + 4789.45i 11564.6 27919.4i −23822.9 23822.9i −1655.79 1655.79i −49463.9
3.14 −10.0851 + 10.0851i −107.923 + 260.549i 820.583i 877.673 + 877.673i −1539.24 3716.05i 3602.01 8696.03i −18602.7 18602.7i −14484.3 14484.3i −17702.8
3.15 −9.84548 + 9.84548i −22.0412 + 53.2121i 830.133i −1874.03 1874.03i −306.893 740.905i 5342.50 12897.9i −18254.8 18254.8i 39408.2 + 39408.2i 36901.4
3.16 −6.24980 + 6.24980i −14.1891 + 34.2556i 945.880i −2714.12 2714.12i −125.411 302.770i −7786.00 + 18797.1i −12311.4 12311.4i 40781.8 + 40781.8i 33925.5
3.17 −3.04539 + 3.04539i 181.284 437.659i 1005.45i 815.394 + 815.394i 780.761 + 1884.92i −2428.27 + 5862.36i −6180.46 6180.46i −116928. 116928.i −4966.38
3.18 4.21015 4.21015i 35.1231 84.7947i 988.549i 2200.24 + 2200.24i −209.125 504.871i −2906.06 + 7015.84i 8473.13 + 8473.13i 35797.4 + 35797.4i 18526.7
3.19 6.82166 6.82166i 129.778 313.312i 930.930i −4122.43 4122.43i −1252.01 3022.61i 4925.95 11892.3i 13335.9 + 13335.9i −39568.2 39568.2i −56243.6
3.20 7.44843 7.44843i −179.027 + 432.210i 913.042i −2627.98 2627.98i 1885.81 + 4552.76i 6311.03 15236.2i 14427.9 + 14427.9i −113001. 113001.i −39148.7
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.e odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.11.e.a 136
41.e odd 8 1 inner 41.11.e.a 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.11.e.a 136 1.a even 1 1 trivial
41.11.e.a 136 41.e odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(41, [\chi])\).