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Properties

Label 50.6.d.a
Level $50$
Weight $6$
Character orbit 50.d
Analytic conductor $8.019$
Analytic rank $0$
Dimension $24$
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] = [50,6,Mod(11,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.11");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), 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: \(8.01919099065\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{2} - 11 q^{3} - 96 q^{4} - 120 q^{5} - 44 q^{6} + 548 q^{7} - 384 q^{8} - 893 q^{9} + 320 q^{10} - 327 q^{11} - 96 q^{12} - 1161 q^{13} - 948 q^{14} + 760 q^{15} - 1536 q^{16} + 83 q^{17} + 7368 q^{18} - 6250 q^{19} + 2320 q^{20} + 6193 q^{21} + 852 q^{22} + 2069 q^{23} + 2176 q^{24} - 14380 q^{25} + 15976 q^{26} - 18935 q^{27} - 592 q^{28} - 11775 q^{29} - 9840 q^{30} + 2103 q^{31} + 24576 q^{32} - 11747 q^{33} + 972 q^{34} + 33235 q^{35} - 14288 q^{36} + 4148 q^{37} + 14260 q^{38} - 12576 q^{39} + 1600 q^{40} - 26427 q^{41} + 24772 q^{42} + 13024 q^{43} + 3408 q^{44} - 65535 q^{45} - 16964 q^{46} + 29803 q^{47} - 1536 q^{48} + 38368 q^{49} - 14020 q^{50} + 194058 q^{51} - 18576 q^{52} - 29011 q^{53} + 11820 q^{54} - 4185 q^{55} - 2368 q^{56} - 78040 q^{57} - 47100 q^{58} + 31430 q^{59} + 28160 q^{60} - 18362 q^{61} - 15828 q^{62} - 95886 q^{63} - 24576 q^{64} - 193280 q^{65} + 28992 q^{66} - 71657 q^{67} - 10432 q^{68} + 245919 q^{69} - 107460 q^{70} - 153827 q^{71} - 1792 q^{72} + 108264 q^{73} - 27648 q^{74} + 126020 q^{75} + 85920 q^{76} + 96141 q^{77} - 55724 q^{78} + 66670 q^{79} - 33280 q^{80} + 17709 q^{81} - 42408 q^{82} - 30621 q^{83} - 140352 q^{84} - 64895 q^{85} - 94444 q^{86} - 265360 q^{87} - 20928 q^{88} + 351835 q^{89} - 101720 q^{90} + 19323 q^{91} - 67856 q^{92} + 707998 q^{93} + 119212 q^{94} + 158570 q^{95} - 11264 q^{96} + 11058 q^{97} + 18532 q^{98} - 455966 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.23607 + 2.35114i −6.86375 + 21.1244i 4.94427 − 15.2169i −38.8214 − 40.2232i −27.4550 − 84.4978i −78.8464 19.7771 + 60.8676i −202.540 − 147.154i 220.199 + 38.8904i
11.2 −3.23607 + 2.35114i −4.88978 + 15.0492i 4.94427 − 15.2169i 51.0006 − 22.8897i −19.5591 − 60.1968i 188.761 19.7771 + 60.8676i −5.97726 − 4.34273i −111.224 + 193.982i
11.3 −3.23607 + 2.35114i −3.10515 + 9.55668i 4.94427 − 15.2169i −0.387302 + 55.9004i −12.4206 − 38.2267i −100.787 19.7771 + 60.8676i 114.903 + 83.4819i −130.176 − 181.808i
11.4 −3.23607 + 2.35114i 2.33322 − 7.18091i 4.94427 − 15.2169i −54.1589 + 13.8497i 9.33287 + 28.7236i 54.5999 19.7771 + 60.8676i 150.470 + 109.323i 142.699 − 172.154i
11.5 −3.23607 + 2.35114i 5.67913 − 17.4786i 4.94427 − 15.2169i 6.69834 − 55.4989i 22.7165 + 69.9143i −86.1311 19.7771 + 60.8676i −76.6568 − 55.6945i 108.810 + 195.347i
11.6 −3.23607 + 2.35114i 9.12748 − 28.0915i 4.94427 − 15.2169i 30.2654 + 47.0001i 36.5099 + 112.366i 204.125 19.7771 + 60.8676i −509.231 − 369.978i −208.445 − 80.9373i
21.1 1.23607 − 3.80423i −19.7905 − 14.3786i −12.9443 − 9.40456i −30.4129 + 46.9047i −79.1620 + 57.5145i 49.2262 −51.7771 + 37.6183i 109.827 + 338.014i 140.844 + 173.675i
21.2 1.23607 − 3.80423i −16.6068 − 12.0655i −12.9443 − 9.40456i 2.13641 − 55.8609i −66.4272 + 48.2622i 12.6460 −51.7771 + 37.6183i 55.1171 + 169.633i −209.867 − 77.1752i
21.3 1.23607 − 3.80423i −0.732003 − 0.531831i −12.9443 − 9.40456i 54.7131 + 11.4663i −2.92801 + 2.12732i 133.265 −51.7771 + 37.6183i −74.8381 − 230.328i 111.249 − 193.968i
21.4 1.23607 − 3.80423i 1.10044 + 0.799520i −12.9443 − 9.40456i −0.782112 + 55.8962i 4.40178 − 3.19808i −204.325 −51.7771 + 37.6183i −74.5194 − 229.347i 211.675 + 72.0669i
21.5 1.23607 − 3.80423i 8.60489 + 6.25182i −12.9443 − 9.40456i −38.0198 − 40.9817i 34.4196 − 25.0073i −106.678 −51.7771 + 37.6183i −40.1323 − 123.514i −202.899 + 93.9797i
21.6 1.23607 − 3.80423i 19.6428 + 14.2713i −12.9443 − 9.40456i −42.2315 + 36.6265i 78.5712 − 57.0853i 208.145 −51.7771 + 37.6183i 107.078 + 329.551i 87.1346 + 205.931i
31.1 1.23607 + 3.80423i −19.7905 + 14.3786i −12.9443 + 9.40456i −30.4129 − 46.9047i −79.1620 − 57.5145i 49.2262 −51.7771 − 37.6183i 109.827 − 338.014i 140.844 − 173.675i
31.2 1.23607 + 3.80423i −16.6068 + 12.0655i −12.9443 + 9.40456i 2.13641 + 55.8609i −66.4272 − 48.2622i 12.6460 −51.7771 − 37.6183i 55.1171 − 169.633i −209.867 + 77.1752i
31.3 1.23607 + 3.80423i −0.732003 + 0.531831i −12.9443 + 9.40456i 54.7131 − 11.4663i −2.92801 − 2.12732i 133.265 −51.7771 − 37.6183i −74.8381 + 230.328i 111.249 + 193.968i
31.4 1.23607 + 3.80423i 1.10044 − 0.799520i −12.9443 + 9.40456i −0.782112 − 55.8962i 4.40178 + 3.19808i −204.325 −51.7771 − 37.6183i −74.5194 + 229.347i 211.675 − 72.0669i
31.5 1.23607 + 3.80423i 8.60489 − 6.25182i −12.9443 + 9.40456i −38.0198 + 40.9817i 34.4196 + 25.0073i −106.678 −51.7771 − 37.6183i −40.1323 + 123.514i −202.899 − 93.9797i
31.6 1.23607 + 3.80423i 19.6428 − 14.2713i −12.9443 + 9.40456i −42.2315 − 36.6265i 78.5712 + 57.0853i 208.145 −51.7771 − 37.6183i 107.078 − 329.551i 87.1346 − 205.931i
41.1 −3.23607 − 2.35114i −6.86375 − 21.1244i 4.94427 + 15.2169i −38.8214 + 40.2232i −27.4550 + 84.4978i −78.8464 19.7771 − 60.8676i −202.540 + 147.154i 220.199 − 38.8904i
41.2 −3.23607 − 2.35114i −4.88978 − 15.0492i 4.94427 + 15.2169i 51.0006 + 22.8897i −19.5591 + 60.1968i 188.761 19.7771 − 60.8676i −5.97726 + 4.34273i −111.224 − 193.982i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.6.d.a ✓ 24
25.d even 5 1 inner 50.6.d.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.6.d.a ✓ 24 1.a even 1 1 trivial
50.6.d.a ✓ 24 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 11 T_{3}^{23} + 1236 T_{3}^{22} + 20436 T_{3}^{21} + 1020571 T_{3}^{20} + \cdots + 53\!\cdots\!96 \) acting on \(S_{6}^{\mathrm{new}}(50, [\chi])\). Copy content Toggle raw display