↑

Properties

Label 41.10.f.a
Level $41$
Weight $10$
Character orbit 41.f
Analytic conductor $21.116$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [41,10,Mod(4,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 41.f (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: \(21.1164692827\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 3 q^{2} - 7171 q^{4} + 1023 q^{5} + 10155 q^{6} - 5 q^{7} + 15352 q^{8} - 690166 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 3 q^{2} - 7171 q^{4} + 1023 q^{5} + 10155 q^{6} - 5 q^{7} + 15352 q^{8} - 690166 q^{9} + 20552 q^{10} - 125880 q^{11} + 98410 q^{12} - 5 q^{13} + 595750 q^{15} - 2229131 q^{16} + 479475 q^{17} + 652204 q^{18} + 984820 q^{19} - 2240400 q^{20} + 266734 q^{21} - 1702085 q^{22} - 454413 q^{23} - 5822870 q^{24} - 984687 q^{25} + 4512235 q^{26} + 13622230 q^{28} - 9783835 q^{29} + 11998410 q^{30} + 6573049 q^{31} - 33907764 q^{32} - 52676383 q^{33} + 73409645 q^{34} - 19738185 q^{35} + 143971574 q^{36} + 36031667 q^{37} + 26127560 q^{39} - 213125458 q^{40} - 63886523 q^{41} - 80605752 q^{42} + 3387128 q^{43} - 37375547 q^{45} + 212658216 q^{46} + 151485145 q^{47} - 306800725 q^{48} - 34551163 q^{49} - 200992420 q^{50} + 58633928 q^{51} - 171448075 q^{52} + 308415095 q^{53} - 298017350 q^{54} + 296090405 q^{56} + 320931662 q^{57} - 534470810 q^{58} + 227064721 q^{59} + 316518900 q^{60} + 192202465 q^{61} - 294479892 q^{62} + 65357295 q^{63} + 369256402 q^{64} - 642135075 q^{65} - 344874770 q^{66} + 1413985360 q^{67} - 213501040 q^{69} + 751241615 q^{70} + 188804325 q^{71} - 1147896117 q^{72} + 859717758 q^{73} - 892419615 q^{74} - 393239545 q^{75} + 2958110630 q^{76} - 816677909 q^{77} + 610119700 q^{78} - 791902551 q^{80} + 4963674308 q^{81} - 2213326995 q^{82} - 2949468614 q^{83} + 2623597750 q^{84} + 1050979875 q^{86} + 813362402 q^{87} + 791170260 q^{88} - 4279742275 q^{89} - 5906513280 q^{90} - 496112006 q^{91} - 1214777649 q^{92} - 5450081350 q^{93} - 5913467170 q^{94} + 4615127835 q^{95} - 3924102745 q^{97} + 3053082146 q^{98} - 4774775920 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −33.1388 + 24.0768i − 179.086i 360.274 − 1108.81i 395.503 − 1217.23i 4311.80 + 5934.69i −5863.08 + 8069.83i 8276.65 + 25472.9i −12388.7 16200.5 + 49860.1i
4.2 −33.0862 + 24.0385i 65.9199i 358.628 − 1103.74i −409.145 + 1259.22i −1584.62 − 2181.04i −3518.05 + 4842.18i 8196.19 + 25225.3i 15337.6 −16732.7 − 51498.0i
4.3 −32.9990 + 23.9751i − 88.9402i 355.907 − 1095.37i 152.664 − 469.853i 2132.36 + 2934.94i 7238.90 − 9963.49i 8063.58 + 24817.1i 11772.6 6227.02 + 19164.8i
4.4 −31.9788 + 23.2339i 218.588i 324.609 − 999.045i 91.9118 − 282.875i −5078.65 − 6990.16i 1155.47 − 1590.37i 6577.16 + 20242.4i −28097.5 3633.08 + 11181.5i
4.5 −26.0020 + 18.8916i − 63.7430i 160.996 − 495.496i −610.224 + 1878.08i 1204.20 + 1657.44i 311.647 − 428.946i 89.3386 + 274.956i 15619.8 −19612.7 − 60361.8i
4.6 −25.1488 + 18.2717i 110.926i 140.391 − 432.080i 679.773 − 2092.13i −2026.80 − 2789.65i 1732.16 − 2384.11i −554.114 − 1705.39i 7378.44 21131.2 + 65035.0i
4.7 −22.2795 + 16.1870i − 264.101i 76.1406 − 234.337i −242.400 + 746.031i 4275.00 + 5884.04i 934.770 − 1286.60i −2260.29 − 6956.47i −50066.3 −6675.46 − 20545.0i
4.8 −19.9906 + 14.5240i 72.5029i 30.4601 − 93.7465i 403.444 − 1241.67i −1053.03 − 1449.38i −6048.85 + 8325.52i −3156.83 − 9715.73i 14426.3 9968.98 + 30681.4i
4.9 −18.8766 + 13.7146i − 112.606i 10.0170 − 30.8290i 427.260 − 1314.97i 1544.35 + 2125.62i 538.411 − 741.059i −3457.90 − 10642.3i 7002.79 9969.13 + 30681.8i
4.10 −17.9243 + 13.0228i 201.317i −6.52793 + 20.0909i −448.757 + 1381.13i −2621.70 − 3608.47i −699.414 + 962.660i −3650.03 − 11233.6i −20845.3 −9942.53 − 30600.0i
4.11 −10.9485 + 7.95454i 72.6057i −101.622 + 312.760i −228.290 + 702.604i −577.545 − 794.922i 6205.84 − 8541.61i −3516.42 − 10822.4i 14411.4 −3089.46 − 9508.39i
4.12 −8.69416 + 6.31668i − 114.112i −122.529 + 377.104i 191.786 − 590.258i 720.809 + 992.108i −1189.12 + 1636.69i −3017.05 − 9285.53i 6661.46 2061.05 + 6343.25i
4.13 −5.08226 + 3.69248i − 119.559i −146.022 + 449.409i −608.182 + 1871.79i 441.469 + 607.630i −4407.62 + 6066.57i −1911.23 − 5882.17i 5388.67 −3820.62 − 11758.6i
4.14 −2.95607 + 2.14771i 276.274i −154.091 + 474.243i 526.214 − 1619.52i −593.355 − 816.683i 1730.66 − 2382.05i −1141.14 − 3512.07i −56644.2 1922.73 + 5917.56i
4.15 −0.784459 + 0.569943i − 201.574i −157.926 + 486.047i 652.251 − 2007.42i 114.886 + 158.127i 3359.38 − 4623.79i −306.546 − 943.453i −20949.2 632.452 + 1946.49i
4.16 −0.311789 + 0.226528i 131.998i −158.171 + 486.800i −84.7993 + 260.985i −29.9011 − 41.1554i −6514.03 + 8965.79i −121.933 − 375.272i 2259.63 −32.6810 − 100.582i
4.17 3.75145 − 2.72559i 87.4036i −151.572 + 466.491i 408.748 − 1258.00i 238.226 + 327.890i 754.854 − 1038.97i 1436.50 + 4421.11i 12043.6 −1895.38 − 5833.39i
4.18 7.93798 − 5.76728i − 214.547i −128.467 + 395.380i −572.420 + 1761.73i −1237.35 − 1703.07i 5227.79 − 7195.43i 2812.90 + 8657.23i −26347.4 5616.51 + 17285.9i
4.19 8.46184 − 6.14789i − 2.54745i −124.410 + 382.896i −309.582 + 952.796i −15.6614 − 21.5561i 2985.67 − 4109.42i 2956.11 + 9097.98i 19676.5 3238.05 + 9965.68i
4.20 13.2721 − 9.64278i 200.593i −75.0500 + 230.980i −676.667 + 2082.57i 1934.27 + 2662.30i 554.095 − 762.647i 3826.80 + 11777.7i −20554.5 11100.9 + 34165.1i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.10.f.a ✓ 120
41.f even 10 1 inner 41.10.f.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.10.f.a ✓ 120 1.a even 1 1 trivial
41.10.f.a ✓ 120 41.f even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(41, [\chi])\).