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Properties

Label 946.2.bc.a
Level $946$
Weight $2$
Character orbit 946.bc
Analytic conductor $7.554$
Analytic rank $0$
Dimension $1008$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [946,2,Mod(9,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(210))
 
chi = DirichletCharacter(H, H._module([126, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.bc (of order \(105\), degree \(48\), 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: \(7.55384803121\)
Analytic rank: \(0\)
Dimension: \(1008\)
Relative dimension: \(21\) over \(\Q(\zeta_{105})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{105}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 42 q^{2} + 42 q^{4} - 4 q^{5} - 3 q^{7} - 42 q^{8} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 42 q^{2} + 42 q^{4} - 4 q^{5} - 3 q^{7} - 42 q^{8} - 19 q^{9} + 4 q^{10} - 8 q^{11} + 18 q^{13} + 24 q^{14} + 45 q^{15} + 42 q^{16} + 25 q^{17} + 19 q^{18} - 11 q^{19} - 4 q^{20} - 16 q^{21} + 5 q^{22} + 16 q^{23} - 29 q^{25} + 37 q^{26} - 42 q^{27} + 2 q^{28} - 17 q^{29} - 17 q^{30} - 6 q^{31} + 168 q^{32} + 165 q^{33} - 30 q^{34} + 8 q^{35} + 114 q^{36} + 30 q^{37} - 122 q^{38} - 54 q^{39} + 15 q^{40} - 32 q^{41} - 4 q^{42} - 232 q^{43} - 12 q^{44} - 74 q^{45} - 8 q^{46} - 12 q^{47} + 81 q^{49} - 106 q^{50} - 42 q^{51} + 12 q^{52} - 49 q^{53} + 12 q^{54} + 4 q^{55} - 2 q^{56} - 32 q^{57} + 3 q^{58} + 72 q^{59} - 8 q^{60} + q^{61} + 43 q^{62} + 59 q^{63} + 42 q^{64} - 90 q^{65} - 156 q^{66} - 78 q^{67} - 24 q^{68} - 66 q^{69} - 8 q^{70} - 45 q^{71} + 19 q^{72} + 150 q^{73} - 2 q^{74} + 159 q^{75} + 16 q^{76} + 33 q^{77} - 124 q^{78} + 16 q^{79} + 6 q^{80} - 5 q^{81} + 42 q^{82} + 109 q^{83} + 4 q^{84} - 322 q^{85} + 75 q^{86} + 204 q^{87} + 5 q^{88} + 54 q^{89} - 74 q^{90} + 44 q^{91} - 16 q^{92} - 184 q^{93} - 27 q^{94} - 62 q^{95} + 56 q^{97} - 70 q^{98} - 180 q^{99}+O(q^{100}) \) 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 −0.936235 + 0.351375i −2.80638 + 1.45647i 0.753071 − 0.657939i 0.00593905 − 0.0203288i 2.11567 − 2.34969i −1.32816 + 0.591336i −0.473869 + 0.880596i 4.02763 − 5.72164i 0.00158268 + 0.0211193i
9.2 −0.936235 + 0.351375i −2.25456 + 1.17008i 0.753071 − 0.657939i −1.17237 + 4.01289i 1.69966 − 1.88766i −1.19300 + 0.531157i −0.473869 + 0.880596i 1.98710 − 2.82286i −0.312420 − 4.16895i
9.3 −0.936235 + 0.351375i −2.14414 + 1.11277i 0.753071 − 0.657939i 1.18140 − 4.04382i 1.61642 − 1.79522i −2.82789 + 1.25906i −0.473869 + 0.880596i 1.63223 − 2.31874i 0.314827 + 4.20108i
9.4 −0.936235 + 0.351375i −2.06671 + 1.07259i 0.753071 − 0.657939i 0.800440 − 2.73983i 1.55804 − 1.73038i 2.98052 − 1.32701i −0.473869 + 0.880596i 1.39398 − 1.98028i 0.213307 + 2.84638i
9.5 −0.936235 + 0.351375i −1.97352 + 1.02422i 0.753071 − 0.657939i −0.515364 + 1.76404i 1.48779 − 1.65236i −1.91908 + 0.854430i −0.473869 + 0.880596i 1.11889 − 1.58949i −0.137337 − 1.83264i
9.6 −0.936235 + 0.351375i −1.52663 + 0.792295i 0.753071 − 0.657939i 0.0480560 − 0.164491i 1.15089 − 1.27819i 3.46998 − 1.54493i −0.473869 + 0.880596i −0.0239898 + 0.0340798i 0.0128063 + 0.170888i
9.7 −0.936235 + 0.351375i −1.31379 + 0.681836i 0.753071 − 0.657939i −0.828802 + 2.83691i 0.990437 − 1.09999i 2.50621 − 1.11584i −0.473869 + 0.880596i −0.465704 + 0.661578i −0.220865 − 2.94723i
9.8 −0.936235 + 0.351375i −0.983255 + 0.510293i 0.753071 − 0.657939i 0.472522 − 1.61740i 0.741254 − 0.823246i −0.573333 + 0.255265i −0.473869 + 0.880596i −1.02046 + 1.44966i 0.125921 + 1.68030i
9.9 −0.936235 + 0.351375i −0.820151 + 0.425645i 0.753071 − 0.657939i −0.0350788 + 0.120071i 0.618293 − 0.686684i −3.75393 + 1.67136i −0.473869 + 0.880596i −1.23538 + 1.75497i −0.00934803 − 0.124741i
9.10 −0.936235 + 0.351375i −0.350405 + 0.181854i 0.753071 − 0.657939i −0.397862 + 1.36184i 0.264162 − 0.293382i 2.19242 − 0.976128i −0.473869 + 0.880596i −1.63714 + 2.32571i −0.106025 − 1.41480i
9.11 −0.936235 + 0.351375i −0.0206216 + 0.0107023i 0.753071 − 0.657939i 0.664682 − 2.27514i 0.0155462 − 0.0172658i −0.870753 + 0.387684i −0.473869 + 0.880596i −1.72654 + 2.45272i 0.177129 + 2.36362i
9.12 −0.936235 + 0.351375i 0.200325 − 0.103965i 0.753071 − 0.657939i 0.531861 − 1.82051i −0.151020 + 0.167725i −2.78058 + 1.23800i −0.473869 + 0.880596i −1.69753 + 2.41150i 0.141734 + 1.89131i
9.13 −0.936235 + 0.351375i 0.671558 − 0.348527i 0.753071 − 0.657939i −0.888743 + 3.04208i −0.506272 + 0.562272i 1.37747 − 0.613289i −0.473869 + 0.880596i −1.39733 + 1.98505i −0.236838 − 3.16038i
9.14 −0.936235 + 0.351375i 0.965550 − 0.501105i 0.753071 − 0.657939i −0.130939 + 0.448191i −0.727906 + 0.808422i −0.915530 + 0.407620i −0.473869 + 0.880596i −1.04567 + 1.48547i −0.0348935 − 0.465621i
9.15 −0.936235 + 0.351375i 1.11442 − 0.578365i 0.753071 − 0.657939i −0.288762 + 0.988403i −0.840135 + 0.933065i 3.48740 − 1.55269i −0.473869 + 0.880596i −0.819427 + 1.16407i −0.0769511 − 1.02684i
9.16 −0.936235 + 0.351375i 1.79745 − 0.932847i 0.753071 − 0.657939i 0.815273 − 2.79060i −1.35506 + 1.50494i 4.01606 − 1.78806i −0.473869 + 0.880596i 0.633771 − 0.900333i 0.217259 + 2.89912i
9.17 −0.936235 + 0.351375i 1.82407 − 0.946664i 0.753071 − 0.657939i 0.589978 − 2.01944i −1.37513 + 1.52723i 1.57026 − 0.699127i −0.473869 + 0.880596i 0.704221 − 1.00041i 0.157221 + 2.09797i
9.18 −0.936235 + 0.351375i 2.12595 − 1.10334i 0.753071 − 0.657939i −0.771737 + 2.64158i −1.60271 + 1.77999i −0.121070 + 0.0539039i −0.473869 + 0.880596i 1.57548 − 2.23812i −0.205657 − 2.74431i
9.19 −0.936235 + 0.351375i 2.20229 − 1.14295i 0.753071 − 0.657939i 0.208790 − 0.714668i −1.66026 + 1.84390i −1.98616 + 0.884295i −0.473869 + 0.880596i 1.81689 − 2.58107i 0.0556398 + 0.742461i
9.20 −0.936235 + 0.351375i 2.44617 − 1.26952i 0.753071 − 0.657939i −0.486332 + 1.66467i −1.84411 + 2.04809i −4.27204 + 1.90204i −0.473869 + 0.880596i 2.64520 − 3.75775i −0.129601 − 1.72940i
See next 80 embeddings (of 1008 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
43.g even 21 1 inner
473.bc even 105 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.bc.a ✓ 1008
11.c even 5 1 inner 946.2.bc.a ✓ 1008
43.g even 21 1 inner 946.2.bc.a ✓ 1008
473.bc even 105 1 inner 946.2.bc.a ✓ 1008
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.bc.a ✓ 1008 1.a even 1 1 trivial
946.2.bc.a ✓ 1008 11.c even 5 1 inner
946.2.bc.a ✓ 1008 43.g even 21 1 inner
946.2.bc.a ✓ 1008 473.bc even 105 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{1008} + 41 T_{3}^{1006} + 14 T_{3}^{1005} + 979 T_{3}^{1004} + 590 T_{3}^{1003} + \cdots + 24\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display