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Properties

Label 931.2.cb.a
Level $931$
Weight $2$
Character orbit 931.cb
Analytic conductor $7.434$
Analytic rank $0$
Dimension $3276$
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] = [931,2,Mod(9,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(126))
 
chi = DirichletCharacter(H, H._module([6, 56]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.cb (of order \(63\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(3276\)
Relative dimension: \(91\) over \(\Q(\zeta_{63})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{63}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3276 q - 39 q^{2} - 39 q^{3} - 39 q^{4} - 39 q^{5} - 30 q^{6} - 21 q^{7} - 15 q^{8} - 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3276 q - 39 q^{2} - 39 q^{3} - 39 q^{4} - 39 q^{5} - 30 q^{6} - 21 q^{7} - 15 q^{8} - 39 q^{9} - 30 q^{10} - 63 q^{11} - 48 q^{12} - 30 q^{13} - 30 q^{14} - 21 q^{15} - 33 q^{16} - 51 q^{17} - 36 q^{18} + 9 q^{19} - 36 q^{21} - 18 q^{22} - 39 q^{23} - 69 q^{24} - 39 q^{25} + 54 q^{26} - 51 q^{27} - 3 q^{28} - 42 q^{29} - 36 q^{30} - 468 q^{31} - 282 q^{32} - 57 q^{33} - 18 q^{34} - 9 q^{35} - 51 q^{36} - 90 q^{37} - 87 q^{38} - 60 q^{39} - 12 q^{41} - 45 q^{42} - 54 q^{43} + 15 q^{44} + 78 q^{45} - 39 q^{46} - 75 q^{47} - 63 q^{48} - 27 q^{49} - 69 q^{50} - 39 q^{51} - 585 q^{52} - 75 q^{53} + 144 q^{54} - 15 q^{55} - 96 q^{56} - 36 q^{57} - 78 q^{58} - 12 q^{59} - 84 q^{60} - 159 q^{61} - 57 q^{62} - 129 q^{63} + 195 q^{64} + 21 q^{65} - 441 q^{66} - 54 q^{67} - 72 q^{68} - 63 q^{69} - 30 q^{70} + 15 q^{71} + 369 q^{72} - 108 q^{73} - 21 q^{74} - 141 q^{75} - 84 q^{76} - 102 q^{77} - 231 q^{78} - 12 q^{79} - 90 q^{80} - 33 q^{81} - 9 q^{82} - 3 q^{83} + 408 q^{84} + 162 q^{85} - 420 q^{86} - 12 q^{87} - 111 q^{88} - 30 q^{89} - 15 q^{90} - 453 q^{91} - 24 q^{92} + 102 q^{93} - 366 q^{94} - 3 q^{95} + 561 q^{96} - 87 q^{97} - 21 q^{98} - 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.51237 + 2.34161i 2.26071 − 1.38149i −2.37329 − 5.25975i 0.0314077 + 1.25941i −0.184122 + 7.38304i 2.20500 − 1.46219i 10.3928 + 1.56646i 1.83367 − 3.57671i −2.99655 − 1.83115i
9.2 −1.45455 + 2.25208i −1.51394 + 0.925147i −2.13357 − 4.72848i 0.0532967 + 2.13713i 0.118586 − 4.75517i 0.950567 − 2.46909i 8.45024 + 1.27367i 0.0674732 − 0.131611i −4.89051 − 2.98853i
9.3 −1.44001 + 2.22958i −1.74795 + 1.06815i −2.07480 − 4.59824i 0.0452745 + 1.81545i 0.135549 − 5.43534i 0.691645 + 2.55375i 7.99082 + 1.20442i 0.545751 − 1.06453i −4.11289 − 2.51333i
9.4 −1.43702 + 2.22494i −2.37432 + 1.45092i −2.06276 − 4.57155i −0.0797583 − 3.19821i 0.183739 − 7.36772i −1.65232 − 2.06636i 7.89750 + 1.19036i 2.16360 − 4.22027i 7.23043 + 4.41843i
9.5 −1.40933 + 2.18207i −0.0355117 + 0.0217007i −1.95264 − 4.32750i −0.0753958 − 3.02328i 0.00269516 − 0.108072i 2.62882 + 0.298830i 7.05761 + 1.06376i −1.36784 + 2.66807i 6.70326 + 4.09627i
9.6 −1.39992 + 2.16751i 1.22558 − 0.748939i −1.91573 − 4.24568i 0.0713603 + 2.86146i −0.0923949 + 3.70492i −0.627724 + 2.57021i 6.78149 + 1.02215i −0.427486 + 0.833842i −6.30213 − 3.85115i
9.7 −1.39502 + 2.15992i 0.790067 − 0.482800i −1.89658 − 4.20326i −0.0205701 − 0.824835i −0.0593534 + 2.38000i −1.92695 − 1.81297i 6.63940 + 1.00073i −0.977521 + 1.90673i 1.81027 + 1.10623i
9.8 −1.35756 + 2.10192i 1.36476 − 0.833989i −1.75250 − 3.88395i −0.0774530 − 3.10577i −0.0997738 + 4.00080i −1.42129 + 2.23157i 5.59434 + 0.843212i −0.201594 + 0.393224i 6.63321 + 4.05347i
9.9 −1.27387 + 1.97234i 2.56694 − 1.56863i −1.44481 − 3.20203i −0.0968019 − 3.88163i −0.176093 + 7.06112i −0.486276 − 2.60068i 3.51254 + 0.529430i 2.75997 − 5.38352i 7.77923 + 4.75379i
9.10 −1.27040 + 1.96697i −1.24225 + 0.759123i −1.43247 − 3.17468i −0.0549201 − 2.20223i 0.0849868 − 3.40786i −1.01138 + 2.44481i 3.43351 + 0.517518i −0.401714 + 0.783573i 4.40148 + 2.68969i
9.11 −1.20727 + 1.86922i 2.91339 − 1.78034i −1.21390 − 2.69029i 0.00986981 + 0.395767i −0.189410 + 7.59511i −1.99213 + 1.74110i 2.09357 + 0.315555i 3.94961 − 7.70401i −0.751691 − 0.459349i
9.12 −1.18691 + 1.83770i −1.24264 + 0.759362i −1.14581 − 2.53937i 0.0761776 + 3.05463i 0.0794264 − 3.18490i −2.59512 + 0.515141i 1.70010 + 0.256249i −0.401106 + 0.782386i −5.70391 − 3.48558i
9.13 −1.16244 + 1.79980i 0.302248 − 0.184700i −1.06546 − 2.36130i 0.0659114 + 2.64296i −0.0189205 + 0.758689i −1.43135 − 2.22514i 1.25114 + 0.188580i −1.31139 + 2.55796i −4.83343 − 2.95365i
9.14 −1.12155 + 1.73649i 0.145486 − 0.0889047i −0.934972 − 2.07211i −0.00792121 − 0.317631i −0.00878698 + 0.352347i 1.81386 + 1.92612i 0.558619 + 0.0841983i −1.35537 + 2.64375i 0.560448 + 0.342483i
9.15 −1.11733 + 1.72996i −2.29280 + 1.40110i −0.921779 − 2.04287i −0.0308093 − 1.23541i 0.137958 − 5.53195i 2.62868 − 0.300094i 0.491193 + 0.0740355i 1.92520 − 3.75525i 2.17165 + 1.32707i
9.16 −1.10491 + 1.71073i 2.17008 − 1.32611i −0.883200 − 1.95737i 0.0210166 + 0.842737i −0.129122 + 5.17764i 2.48754 + 0.901204i 0.296840 + 0.0447414i 1.58205 − 3.08591i −1.46492 − 0.895191i
9.17 −1.09175 + 1.69036i −1.72555 + 1.05446i −0.842819 − 1.86788i −0.0376047 − 1.50790i 0.101449 − 4.06800i −2.52874 + 0.778121i 0.0979432 + 0.0147626i 0.496995 − 0.969425i 2.58995 + 1.58268i
9.18 −1.08408 + 1.67849i 0.127338 − 0.0778147i −0.819511 − 1.81622i −0.0181264 − 0.726846i −0.00743397 + 0.298093i 1.81786 − 1.92234i −0.0147164 − 0.00221813i −1.35847 + 2.64980i 1.23965 + 0.757536i
9.19 −1.03951 + 1.60948i −2.12044 + 1.29577i −0.687263 − 1.52313i 0.0477581 + 1.91504i 0.118701 − 4.75977i 1.78684 − 1.95121i −0.623309 − 0.0939486i 1.44860 − 2.82561i −3.13186 − 1.91384i
9.20 −0.870346 + 1.34756i 1.67967 − 1.02642i −0.235840 − 0.522675i −0.0769447 − 3.08539i −0.0787255 + 3.15679i 2.15435 − 1.53583i −2.26294 − 0.341084i 0.399102 − 0.778477i 4.22471 + 2.58167i
See next 80 embeddings (of 3276 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.91
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
931.cb even 63 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.cb.a yes 3276
19.e even 9 1 931.2.ca.a ✓ 3276
49.g even 21 1 931.2.ca.a ✓ 3276
931.cb even 63 1 inner 931.2.cb.a yes 3276
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.ca.a ✓ 3276 19.e even 9 1
931.2.ca.a ✓ 3276 49.g even 21 1
931.2.cb.a yes 3276 1.a even 1 1 trivial
931.2.cb.a yes 3276 931.cb even 63 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(931, [\chi])\).