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Properties

Label 381.3.r.b
Level $381$
Weight $3$
Character orbit 381.r
Analytic conductor $10.381$
Analytic rank $0$
Dimension $252$
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] = [381,3,Mod(10,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 381.r (of order \(42\), degree \(12\), 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: \(10.3814980721\)
Analytic rank: \(0\)
Dimension: \(252\)
Relative dimension: \(21\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 252 q + 5 q^{2} + 63 q^{3} - 65 q^{4} + 3 q^{6} + 68 q^{7} + 42 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 252 q + 5 q^{2} + 63 q^{3} - 65 q^{4} + 3 q^{6} + 68 q^{7} + 42 q^{8} - 63 q^{9} - 126 q^{10} - 20 q^{11} + 87 q^{12} - q^{13} - 77 q^{14} - 237 q^{16} + 13 q^{17} - 3 q^{18} + 170 q^{19} - 12 q^{21} - 89 q^{22} + 124 q^{23} + 42 q^{24} + 66 q^{25} + 332 q^{26} + 189 q^{28} - 255 q^{29} + 36 q^{30} - 12 q^{31} - 67 q^{32} - 231 q^{33} - 623 q^{34} + 321 q^{35} - 129 q^{36} + 110 q^{37} + 115 q^{38} + 3 q^{39} + 210 q^{40} - 76 q^{41} + 63 q^{42} + 83 q^{43} - 672 q^{44} - 832 q^{46} + 49 q^{47} + 933 q^{48} - 449 q^{49} - 110 q^{50} - 274 q^{52} + 45 q^{53} + 63 q^{54} + 741 q^{55} - 482 q^{56} - 108 q^{57} + 100 q^{58} - 231 q^{59} + 348 q^{60} - 113 q^{61} + 73 q^{62} - 610 q^{64} - 1123 q^{65} + 273 q^{66} - 128 q^{67} + 451 q^{68} - 306 q^{69} - 160 q^{70} - 281 q^{71} + 63 q^{72} + 53 q^{73} - 1417 q^{74} + 2022 q^{75} + 707 q^{76} + 399 q^{77} + 180 q^{78} - 768 q^{79} + 637 q^{80} + 189 q^{81} + 110 q^{82} + 73 q^{83} + 63 q^{84} + 439 q^{85} - 387 q^{86} - 36 q^{87} + 777 q^{88} - 364 q^{89} - 234 q^{90} - 189 q^{91} - 859 q^{92} + 204 q^{93} + 66 q^{94} + 532 q^{95} - 15 q^{96} - 158 q^{97} - 590 q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.56571 + 1.71715i 1.17809 + 1.26968i 7.27168 − 9.11840i −2.99131 + 2.38549i −6.38098 − 2.50435i 1.54430 + 0.606092i −6.74836 + 29.5665i −0.224190 + 2.99161i 6.56987 − 13.6425i
10.2 −2.97086 + 1.43069i 1.17809 + 1.26968i 4.28518 − 5.37345i 3.64355 − 2.90563i −5.31648 − 2.08656i −8.50074 − 3.33629i −2.10797 + 9.23562i −0.224190 + 2.99161i −6.66741 + 13.8450i
10.3 −2.96977 + 1.43017i 1.17809 + 1.26968i 4.28022 − 5.36722i 7.20416 − 5.74513i −5.31453 − 2.08580i 11.5780 + 4.54402i −2.10135 + 9.20662i −0.224190 + 2.99161i −13.1782 + 27.3649i
10.4 −2.86232 + 1.37842i 1.17809 + 1.26968i 3.79889 − 4.76366i 0.975176 − 0.777677i −5.12225 − 2.01033i −0.837732 − 0.328786i −1.47958 + 6.48247i −0.224190 + 2.99161i −1.71930 + 3.57017i
10.5 −2.37393 + 1.14322i 1.17809 + 1.26968i 1.83462 − 2.30054i −6.55192 + 5.22498i −4.24824 − 1.66731i 6.48980 + 2.54706i 0.620026 − 2.71651i −0.224190 + 2.99161i 9.58047 − 19.8941i
10.6 −1.82523 + 0.878985i 1.17809 + 1.26968i 0.0648975 − 0.0813789i −1.41850 + 1.13122i −3.26633 − 1.28194i 2.57789 + 1.01175i 1.75626 − 7.69467i −0.224190 + 2.99161i 1.59477 − 3.31157i
10.7 −1.32172 + 0.636509i 1.17809 + 1.26968i −1.15215 + 1.44475i 3.65295 − 2.91313i −2.36528 − 0.928304i 2.52744 + 0.991946i 1.90898 − 8.36380i −0.224190 + 2.99161i −2.97396 + 6.17550i
10.8 −1.19917 + 0.577492i 1.17809 + 1.26968i −1.38944 + 1.74230i 3.17272 − 2.53016i −2.14597 − 0.842231i −3.61170 − 1.41749i 1.84470 − 8.08216i −0.224190 + 2.99161i −2.34350 + 4.86633i
10.9 −0.701250 + 0.337704i 1.17809 + 1.26968i −2.11625 + 2.65370i −5.35077 + 4.26710i −1.25491 − 0.492518i −6.19298 − 2.43057i 1.28063 − 5.61083i −0.224190 + 2.99161i 2.31121 − 4.79927i
10.10 −0.180011 + 0.0866887i 1.17809 + 1.26968i −2.46907 + 3.09612i 4.09439 − 3.26516i −0.322137 − 0.126429i −10.8677 − 4.26525i 0.353897 − 1.55053i −0.224190 + 2.99161i −0.453981 + 0.942702i
10.11 0.116892 − 0.0562921i 1.17809 + 1.26968i −2.48346 + 3.11417i −1.96552 + 1.56745i 0.209183 + 0.0820981i 10.4169 + 4.08834i −0.230473 + 1.00977i −0.224190 + 2.99161i −0.141518 + 0.293866i
10.12 0.314806 − 0.151603i 1.17809 + 1.26968i −2.41784 + 3.03188i 6.99676 − 5.57973i 0.563358 + 0.221102i 7.41549 + 2.91037i −0.612513 + 2.68359i −0.224190 + 2.99161i 1.35672 − 2.81726i
10.13 0.440897 − 0.212325i 1.17809 + 1.26968i −2.34465 + 2.94010i −6.12145 + 4.88170i 0.789004 + 0.309661i 1.81055 + 0.710588i −0.845065 + 3.70247i −0.224190 + 2.99161i −1.66243 + 3.45207i
10.14 1.03453 − 0.498201i 1.17809 + 1.26968i −1.67192 + 2.09652i 1.45323 − 1.15891i 1.85133 + 0.726592i −3.32807 − 1.30617i −1.70718 + 7.47965i −0.224190 + 2.99161i 0.926031 − 1.92292i
10.15 1.49427 − 0.719601i 1.17809 + 1.26968i −0.778954 + 0.976777i −1.40037 + 1.11676i 2.67405 + 1.04949i −7.84216 − 3.07782i −1.93729 + 8.48781i −0.224190 + 2.99161i −1.28891 + 2.67644i
10.16 1.95668 − 0.942288i 1.17809 + 1.26968i 0.446734 − 0.560187i 3.56999 − 2.84697i 3.50156 + 1.37426i 2.41072 + 0.946140i −1.58678 + 6.95215i −0.224190 + 2.99161i 4.30266 − 8.93457i
10.17 2.31199 − 1.11340i 1.17809 + 1.26968i 1.61169 − 2.02099i −2.47099 + 1.97055i 4.13740 + 1.62381i 8.46499 + 3.32226i −0.808018 + 3.54016i −0.224190 + 2.99161i −3.51890 + 7.30708i
10.18 2.68745 − 1.29421i 1.17809 + 1.26968i 3.05347 − 3.82893i −7.42954 + 5.92486i 4.80931 + 1.88751i −6.75443 − 2.65092i 0.595636 − 2.60965i −0.224190 + 2.99161i −12.2985 + 25.5382i
10.19 3.08322 − 1.48480i 1.17809 + 1.26968i 4.80765 − 6.02860i 7.08776 − 5.65230i 5.51755 + 2.16548i 4.49074 + 1.76249i 2.82580 − 12.3806i −0.224190 + 2.99161i 13.4606 − 27.9512i
10.20 3.12365 − 1.50427i 1.17809 + 1.26968i 5.00041 − 6.27032i 2.12135 − 1.69172i 5.58990 + 2.19388i −4.41073 − 1.73108i 3.10138 − 13.5880i −0.224190 + 2.99161i 4.08155 − 8.47543i
See next 80 embeddings (of 252 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.j odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.3.r.b ✓ 252
127.j odd 42 1 inner 381.3.r.b ✓ 252
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.3.r.b ✓ 252 1.a even 1 1 trivial
381.3.r.b ✓ 252 127.j odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{252} - 5 T_{2}^{251} + 129 T_{2}^{250} - 612 T_{2}^{249} + 8965 T_{2}^{248} + \cdots + 14\!\cdots\!49 \) acting on \(S_{3}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display