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Properties

Label 381.3.v.b
Level $381$
Weight $3$
Character orbit 381.v
Analytic conductor $10.381$
Analytic rank $0$
Dimension $792$
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(7,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(126))
 
chi = DirichletCharacter(H, H._module([0, 115]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 381.v (of order \(126\), degree \(36\), 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: \(792\)
Relative dimension: \(22\) over \(\Q(\zeta_{126})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{126}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 792 q - 324 q^{4} + 18 q^{5} + 6 q^{7} - 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 792 q - 324 q^{4} + 18 q^{5} + 6 q^{7} - 75 q^{8} + 27 q^{11} + 27 q^{12} + 75 q^{13} + 114 q^{14} + 36 q^{15} - 756 q^{16} - 12 q^{17} + 9 q^{18} + 144 q^{20} + 18 q^{21} + 255 q^{22} - 441 q^{23} - 348 q^{25} - 504 q^{26} + 594 q^{27} - 186 q^{28} - 15 q^{29} + 18 q^{30} - 24 q^{31} + 90 q^{32} + 315 q^{33} + 420 q^{34} - 309 q^{35} - 54 q^{36} - 447 q^{37} + 681 q^{38} - 9 q^{39} - 72 q^{40} - 153 q^{41} + 318 q^{43} - 909 q^{44} - 54 q^{45} + 267 q^{46} + 135 q^{47} - 99 q^{48} + 18 q^{49} + 72 q^{50} + 852 q^{52} + 213 q^{53} - 1341 q^{55} + 1710 q^{56} - 207 q^{57} + 168 q^{58} + 84 q^{59} + 216 q^{60} + 480 q^{61} - 168 q^{62} - 993 q^{64} + 459 q^{65} - 405 q^{66} + 414 q^{67} - 1137 q^{68} - 135 q^{69} + 72 q^{70} + 246 q^{71} + 36 q^{72} - 90 q^{73} + 3075 q^{74} - 162 q^{75} + 204 q^{76} - 252 q^{78} - 150 q^{79} + 720 q^{80} - 348 q^{82} - 306 q^{83} + 315 q^{84} + 783 q^{85} + 465 q^{86} - 459 q^{88} - 294 q^{89} + 198 q^{90} - 849 q^{91} + 1632 q^{92} + 765 q^{93} + 468 q^{94} - 441 q^{95} - 99 q^{96} + 138 q^{97} - 384 q^{98} + 243 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.46239 + 3.08774i 1.66731 − 0.469109i −2.58068 − 11.3067i 0.304558 + 0.987352i −2.65709 + 6.30336i −10.9812 + 1.37612i 27.0338 + 13.0188i 2.55987 − 1.56431i −3.79862 − 1.49085i
7.2 −2.21424 + 2.77657i 1.66731 − 0.469109i −1.91639 − 8.39624i −1.94825 − 6.31606i −2.38932 + 5.66813i 8.01839 − 1.00483i 14.7574 + 7.10679i 2.55987 − 1.56431i 21.8508 + 8.57582i
7.3 −2.04588 + 2.56545i 1.66731 − 0.469109i −1.50583 − 6.59746i 1.86383 + 6.04237i −2.20764 + 5.23715i 5.47548 − 0.686167i 8.18067 + 3.93960i 2.55987 − 1.56431i −19.3145 − 7.58040i
7.4 −1.96937 + 2.46951i 1.66731 − 0.469109i −1.32998 − 5.82703i 1.94282 + 6.29848i −2.12508 + 5.04129i 4.26303 − 0.534227i 5.62584 + 2.70926i 2.55987 − 1.56431i −19.3803 − 7.60620i
7.5 −1.55029 + 1.94401i 1.66731 − 0.469109i −0.485667 − 2.12785i −2.35070 − 7.62078i −1.67287 + 3.96853i −6.54648 + 0.820380i −4.07148 − 1.96072i 2.55987 − 1.56431i 18.4591 + 7.24467i
7.6 −1.47440 + 1.84884i 1.66731 − 0.469109i −0.354264 − 1.55213i 0.165729 + 0.537281i −1.59098 + 3.77425i −10.0801 + 1.26321i −5.13031 − 2.47062i 2.55987 − 1.56431i −1.23769 − 0.485759i
7.7 −1.38899 + 1.74174i 1.66731 − 0.469109i −0.214272 − 0.938787i −0.700479 − 2.27090i −1.49881 + 3.55561i 6.89608 − 0.864191i −6.09584 − 2.93560i 2.55987 − 1.56431i 4.92826 + 1.93420i
7.8 −0.983880 + 1.23375i 1.66731 − 0.469109i 0.335973 + 1.47199i −1.04623 − 3.39178i −1.06168 + 2.51859i 7.35038 − 0.921122i −7.83362 − 3.77247i 2.55987 − 1.56431i 5.21396 + 2.04633i
7.9 −0.746286 + 0.935813i 1.66731 − 0.469109i 0.571280 + 2.50294i 1.27293 + 4.12673i −0.805295 + 1.91038i −2.95530 + 0.370347i −7.08229 − 3.41065i 2.55987 − 1.56431i −4.81181 − 1.88850i
7.10 −0.394086 + 0.494168i 1.66731 − 0.469109i 0.801185 + 3.51022i 1.89367 + 6.13913i −0.425246 + 1.00880i −1.69729 + 0.212698i −4.32826 − 2.08438i 2.55987 − 1.56431i −3.78003 − 1.48355i
7.11 −0.166971 + 0.209375i 1.66731 − 0.469109i 0.874125 + 3.82979i −1.60613 − 5.20696i −0.180173 + 0.427421i −5.44516 + 0.682367i −1.91294 − 0.921221i 2.55987 − 1.56431i 1.35838 + 0.533127i
7.12 0.284269 − 0.356462i 1.66731 − 0.469109i 0.843828 + 3.69705i −1.90335 − 6.17051i 0.306746 − 0.727687i −10.6825 + 1.33870i 3.20085 + 1.54145i 2.55987 − 1.56431i −2.74061 − 1.07561i
7.13 0.351369 − 0.440603i 1.66731 − 0.469109i 0.819413 + 3.59008i −0.121050 − 0.392434i 0.379152 − 0.899454i 7.65614 − 0.959438i 3.90069 + 1.87847i 2.55987 − 1.56431i −0.215441 − 0.0845543i
7.14 0.351685 − 0.440999i 1.66731 − 0.469109i 0.819286 + 3.58953i 2.39225 + 7.75547i 0.379492 − 0.900262i 13.2427 − 1.65953i 3.90391 + 1.88002i 2.55987 − 1.56431i 4.26147 + 1.67250i
7.15 0.917948 − 1.15107i 1.66731 − 0.469109i 0.407749 + 1.78647i 2.07723 + 6.73420i 0.990530 − 2.34981i −11.9526 + 1.49786i 7.73653 + 3.72572i 2.55987 − 1.56431i 9.65832 + 3.79061i
7.16 0.929327 − 1.16534i 1.66731 − 0.469109i 0.395717 + 1.73375i −0.279189 − 0.905107i 1.00281 − 2.37894i 1.73918 − 0.217947i 7.75982 + 3.73693i 2.55987 − 1.56431i −1.31421 − 0.515791i
7.17 1.35556 − 1.69981i 1.66731 − 0.469109i −0.161751 − 0.708678i −2.88980 − 9.36849i 1.46274 − 3.47003i 7.16028 − 0.897299i 6.41146 + 3.08760i 2.55987 − 1.56431i −19.8420 − 7.78740i
7.18 1.50666 − 1.88930i 1.66731 − 0.469109i −0.409322 − 1.79336i 0.297673 + 0.965032i 1.62579 − 3.85684i 1.67720 − 0.210181i 4.70387 + 2.26527i 2.55987 − 1.56431i 2.27172 + 0.891586i
7.19 1.91521 − 2.40159i 1.66731 − 0.469109i −1.20955 − 5.29939i 2.29268 + 7.43267i 2.06664 − 4.90265i 1.97537 − 0.247546i −3.97330 − 1.91344i 2.55987 − 1.56431i 22.2412 + 8.72903i
7.20 2.10708 − 2.64219i 1.66731 − 0.469109i −1.65132 − 7.23490i −1.67536 − 5.43138i 2.27368 − 5.39381i −8.04303 + 1.00792i −10.4162 − 5.01617i 2.55987 − 1.56431i −17.8808 − 7.01772i
See next 80 embeddings (of 792 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.l odd 126 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.3.v.b ✓ 792
127.l odd 126 1 inner 381.3.v.b ✓ 792
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.3.v.b ✓ 792 1.a even 1 1 trivial
381.3.v.b ✓ 792 127.l odd 126 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{792} + 426 T_{2}^{790} + 25 T_{2}^{789} + 92955 T_{2}^{788} + 10536 T_{2}^{787} + \cdots + 10\!\cdots\!36 \) acting on \(S_{3}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display