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Properties

Label 381.3.r.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 252 q + 5 q^{2} - 63 q^{3} - 65 q^{4} - 3 q^{6} - 52 q^{7} + 42 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 252 q + 5 q^{2} - 63 q^{3} - 65 q^{4} - 3 q^{6} - 52 q^{7} + 42 q^{8} - 63 q^{9} + 126 q^{10} - 38 q^{11} - 171 q^{12} + 11 q^{13} - 53 q^{14} - 141 q^{16} - 5 q^{17} - 3 q^{18} + 74 q^{19} - 108 q^{21} + 187 q^{22} - 8 q^{23} - 42 q^{24} + 174 q^{25} + 332 q^{26} + 309 q^{28} + 267 q^{29} + 24 q^{30} - 24 q^{31} - 67 q^{32} - 273 q^{33} - 623 q^{34} + 123 q^{35} - 129 q^{36} + 38 q^{37} + 55 q^{38} + 33 q^{39} - 210 q^{40} + 296 q^{41} - 207 q^{42} - 25 q^{43} + 318 q^{44} + 56 q^{46} + 73 q^{47} - 117 q^{48} + 199 q^{49} + 118 q^{50} - 298 q^{52} + 249 q^{53} - 63 q^{54} + 609 q^{55} - 278 q^{56} - 36 q^{57} - 68 q^{58} + 153 q^{59} + 468 q^{60} - 89 q^{61} + 97 q^{62} + 446 q^{64} + 125 q^{65} - 693 q^{66} - 116 q^{67} - 23 q^{68} + 384 q^{69} + 32 q^{70} + 79 q^{71} + 63 q^{72} - 583 q^{73} - 355 q^{74} - 1482 q^{75} + 35 q^{76} - 441 q^{77} - 180 q^{78} - 588 q^{79} + 637 q^{80} + 189 q^{81} + 650 q^{82} + 445 q^{83} + 57 q^{84} - 389 q^{85} + 1023 q^{86} - 432 q^{87} + 81 q^{88} + 1148 q^{89} - 54 q^{90} - 465 q^{91} + 149 q^{92} + 96 q^{93} - 198 q^{94} + 532 q^{95} + 15 q^{96} + 238 q^{97} + 154 q^{98} + 30 q^{99}+O(q^{100}) \) 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 −3.27262 + 1.57601i −1.17809 − 1.26968i 5.73228 − 7.18805i −6.05356 + 4.82755i 5.85649 + 2.29850i −10.1509 − 3.98394i −4.19805 + 18.3929i −0.224190 + 2.99161i 12.2027 − 25.3392i
10.2 −3.23025 + 1.55561i −1.17809 − 1.26968i 5.52066 − 6.92269i 3.95714 − 3.15572i 5.78067 + 2.26875i 5.15584 + 2.02352i −3.87292 + 16.9684i −0.224190 + 2.99161i −7.87352 + 16.3495i
10.3 −3.18302 + 1.53286i −1.17809 − 1.26968i 5.28799 − 6.63092i −4.55239 + 3.63041i 5.69614 + 2.23557i 11.1136 + 4.36176i −3.52291 + 15.4349i −0.224190 + 2.99161i 8.92543 − 18.5338i
10.4 −2.68471 + 1.29289i −1.17809 − 1.26968i 3.04215 − 3.81474i 1.95031 − 1.55532i 4.80440 + 1.88559i −2.73524 − 1.07350i −0.582992 + 2.55425i −0.224190 + 2.99161i −3.22516 + 6.69712i
10.5 −2.25916 + 1.08795i −1.17809 − 1.26968i 1.42619 − 1.78839i 2.46300 − 1.96418i 4.04286 + 1.58670i −8.98881 − 3.52785i 0.955552 − 4.18655i −0.224190 + 2.99161i −3.42737 + 7.11701i
10.6 −2.18808 + 1.05373i −1.17809 − 1.26968i 1.18341 − 1.48395i −2.78403 + 2.22019i 3.91567 + 1.53679i 2.06019 + 0.808564i 1.13592 − 4.97679i −0.224190 + 2.99161i 3.75222 − 7.79156i
10.7 −1.53655 + 0.739966i −1.17809 − 1.26968i −0.680509 + 0.853331i 5.83991 − 4.65717i 2.74973 + 1.07919i 5.44413 + 2.13666i 1.93219 − 8.46549i −0.224190 + 2.99161i −5.52719 + 11.4773i
10.8 −1.21475 + 0.584995i −1.17809 − 1.26968i −1.36055 + 1.70607i −2.93059 + 2.33706i 2.17385 + 0.853175i 12.5606 + 4.92967i 1.85477 − 8.12627i −0.224190 + 2.99161i 2.19277 − 4.55334i
10.9 −1.03660 + 0.499201i −1.17809 − 1.26968i −1.66862 + 2.09238i −3.00091 + 2.39314i 1.85504 + 0.728050i −6.57484 − 2.58043i 1.70925 − 7.48872i −0.224190 + 2.99161i 1.91609 − 3.97879i
10.10 −0.504735 + 0.243067i −1.17809 − 1.26968i −2.29828 + 2.88196i 2.78604 − 2.22180i 0.903243 + 0.354497i −12.2886 − 4.82290i 0.958150 − 4.19793i −0.224190 + 2.99161i −0.866166 + 1.79861i
10.11 0.107318 − 0.0516817i −1.17809 − 1.26968i −2.48511 + 3.11623i −6.45211 + 5.14539i −0.192050 − 0.0753741i 1.20160 + 0.471592i −0.211667 + 0.927374i −0.224190 + 2.99161i −0.426506 + 0.885650i
10.12 0.254841 − 0.122725i −1.17809 − 1.26968i −2.44408 + 3.06478i 0.572938 − 0.456903i −0.456048 − 0.178985i 3.43616 + 1.34859i −0.498488 + 2.18402i −0.224190 + 2.99161i 0.0899346 − 0.186751i
10.13 0.970790 − 0.467508i −1.17809 − 1.26968i −1.77009 + 2.21962i −0.322490 + 0.257177i −1.73727 − 0.681827i −2.78793 − 1.09418i −1.63976 + 7.18424i −0.224190 + 2.99161i −0.192838 + 0.400431i
10.14 1.24638 − 0.600223i −1.17809 − 1.26968i −1.30077 + 1.63112i 4.70591 − 3.75284i −2.23044 − 0.875384i 9.80724 + 3.84906i −1.87354 + 8.20850i −0.224190 + 2.99161i 3.61280 − 7.50205i
10.15 1.87723 − 0.904025i −1.17809 − 1.26968i 0.212764 − 0.266798i −0.138618 + 0.110544i −3.35938 − 1.31846i −4.87282 − 1.91244i −1.69633 + 7.43212i −0.224190 + 2.99161i −0.160282 + 0.332830i
10.16 2.19500 − 1.05705i −1.17809 − 1.26968i 1.20669 − 1.51314i 7.43325 − 5.92782i −3.92804 − 1.54164i −7.47410 − 2.93337i −1.11927 + 4.90385i −0.224190 + 2.99161i 10.0499 − 20.8689i
10.17 2.41119 − 1.16117i −1.17809 − 1.26968i 1.97156 − 2.47225i −5.43841 + 4.33699i −4.31492 − 1.69348i 6.13017 + 2.40591i −0.498962 + 2.18609i −0.224190 + 2.99161i −8.07706 + 16.7722i
10.18 2.53865 − 1.22255i −1.17809 − 1.26968i 2.45614 − 3.07991i −4.91205 + 3.91723i −4.54301 − 1.78300i −0.473666 − 0.185900i −0.0380281 + 0.166612i −0.224190 + 2.99161i −7.68095 + 15.9497i
10.19 2.85225 − 1.37357i −1.17809 − 1.26968i 3.75469 − 4.70823i 2.48579 − 1.98235i −5.10422 − 2.00326i 11.1778 + 4.38698i 1.42443 − 6.24084i −0.224190 + 2.99161i 4.36720 − 9.06858i
10.20 3.37738 − 1.62646i −1.17809 − 1.26968i 6.26739 − 7.85906i 4.44325 − 3.54337i −6.04397 − 2.37208i −1.05899 − 0.415622i 5.04834 − 22.1182i −0.224190 + 2.99161i 9.24340 − 19.1941i
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.a ✓ 252
127.j odd 42 1 inner 381.3.r.a ✓ 252
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.3.r.a ✓ 252 1.a even 1 1 trivial
381.3.r.a ✓ 252 127.j odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{252} - 5 T_{2}^{251} + 129 T_{2}^{250} - 612 T_{2}^{249} + 8941 T_{2}^{248} + \cdots + 26\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display