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Properties

Label 31.4.g.a
Level $31$
Weight $4$
Character orbit 31.g
Analytic conductor $1.829$
Analytic rank $0$
Dimension $56$
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] = [31,4,Mod(7,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([28]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 31.g (of order \(15\), degree \(8\), 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: \(1.82905921018\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{2} + q^{3} - 42 q^{4} - 53 q^{6} - 71 q^{7} - 7 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{2} + q^{3} - 42 q^{4} - 53 q^{6} - 71 q^{7} - 7 q^{8} + 4 q^{9} + 48 q^{10} - 25 q^{11} + 319 q^{12} - 45 q^{13} - 50 q^{14} + 217 q^{15} - 322 q^{16} + 61 q^{17} + 285 q^{18} - 143 q^{19} + 95 q^{20} - 257 q^{21} - 1147 q^{22} - 244 q^{23} - 226 q^{24} - 96 q^{25} - 1019 q^{26} + 124 q^{27} + 1920 q^{28} - 142 q^{29} + 2430 q^{30} + 112 q^{31} + 618 q^{32} + 134 q^{33} + 2488 q^{34} + 813 q^{35} - 857 q^{36} - 866 q^{37} - 1186 q^{38} - 1090 q^{39} - 663 q^{40} - 333 q^{41} - 3347 q^{42} + 703 q^{43} + 1327 q^{44} - 1201 q^{45} - 818 q^{46} + 1886 q^{47} - 3046 q^{48} + 1346 q^{49} - 4015 q^{50} - 3179 q^{51} - 4447 q^{52} - 1613 q^{53} + 1276 q^{54} + 2486 q^{55} + 4558 q^{56} + 104 q^{57} + 415 q^{58} + 863 q^{59} + 9839 q^{60} + 6180 q^{61} + 427 q^{62} + 3128 q^{63} + 2807 q^{64} + 5398 q^{65} + 7976 q^{66} + 1026 q^{67} - 4992 q^{68} - 4882 q^{69} - 7786 q^{70} + 583 q^{71} - 299 q^{72} - 3709 q^{73} - 5029 q^{74} - 5668 q^{75} - 9928 q^{76} - 9614 q^{77} - 4597 q^{78} + 2323 q^{79} - 2582 q^{80} - 1715 q^{81} - 4314 q^{82} - 2399 q^{83} - 1740 q^{84} + 733 q^{85} + 11232 q^{86} + 4919 q^{87} + 3611 q^{88} - 2244 q^{89} + 10029 q^{90} + 9776 q^{91} + 15424 q^{92} - 2975 q^{93} + 988 q^{94} + 5791 q^{95} + 20204 q^{96} + 10170 q^{97} + 2332 q^{98} - 6383 q^{99}+O(q^{100}) \) 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 −1.65450 − 5.09202i −3.03615 + 0.645354i −16.7192 + 12.1472i 4.36814 − 7.56584i 8.30946 + 14.3924i −10.2190 − 4.54979i 54.8634 + 39.8606i −15.8640 + 7.06311i −45.7525 − 9.72500i
7.2 −1.00392 − 3.08973i 6.59870 − 1.40260i −2.06647 + 1.50138i −2.34783 + 4.06657i −10.9582 − 18.9801i −2.01209 − 0.895840i −14.3129 − 10.3989i 16.9098 − 7.52874i 14.9216 + 3.17169i
7.3 −0.625521 − 1.92515i −8.62815 + 1.83397i 3.15719 − 2.29384i −9.74442 + 16.8778i 8.92776 + 15.4633i −17.1531 − 7.63704i −19.4919 − 14.1617i 46.4158 − 20.6656i 38.5878 + 8.20208i
7.4 −0.272594 − 0.838959i −2.78212 + 0.591358i 5.84259 − 4.24489i 7.92204 − 13.7214i 1.25451 + 2.17288i 8.72475 + 3.88451i −10.8632 − 7.89260i −17.2752 + 7.69143i −13.6712 − 2.90590i
7.5 0.529206 + 1.62873i 3.53631 − 0.751666i 4.09944 − 2.97842i −3.32346 + 5.75640i 3.09570 + 5.36190i −2.35128 − 1.04686i 18.1043 + 13.1535i −12.7252 + 5.66564i −11.1344 − 2.36669i
7.6 1.26435 + 3.89127i −5.86148 + 1.24590i −7.07124 + 5.13756i −2.32632 + 4.02930i −12.2591 − 21.2333i 19.5539 + 8.70595i −2.45125 − 1.78094i 8.13897 − 3.62370i −18.6204 − 3.95788i
7.7 1.57199 + 4.83809i 5.55581 − 1.18092i −14.4638 + 10.5086i 7.62837 − 13.2127i 14.4471 + 25.0231i −27.7661 − 12.3623i −40.6542 − 29.5370i 4.80669 − 2.14008i 75.9161 + 16.1365i
9.1 −1.65450 + 5.09202i −3.03615 − 0.645354i −16.7192 − 12.1472i 4.36814 + 7.56584i 8.30946 − 14.3924i −10.2190 + 4.54979i 54.8634 − 39.8606i −15.8640 − 7.06311i −45.7525 + 9.72500i
9.2 −1.00392 + 3.08973i 6.59870 + 1.40260i −2.06647 − 1.50138i −2.34783 − 4.06657i −10.9582 + 18.9801i −2.01209 + 0.895840i −14.3129 + 10.3989i 16.9098 + 7.52874i 14.9216 − 3.17169i
9.3 −0.625521 + 1.92515i −8.62815 − 1.83397i 3.15719 + 2.29384i −9.74442 − 16.8778i 8.92776 − 15.4633i −17.1531 + 7.63704i −19.4919 + 14.1617i 46.4158 + 20.6656i 38.5878 − 8.20208i
9.4 −0.272594 + 0.838959i −2.78212 − 0.591358i 5.84259 + 4.24489i 7.92204 + 13.7214i 1.25451 − 2.17288i 8.72475 − 3.88451i −10.8632 + 7.89260i −17.2752 − 7.69143i −13.6712 + 2.90590i
9.5 0.529206 − 1.62873i 3.53631 + 0.751666i 4.09944 + 2.97842i −3.32346 − 5.75640i 3.09570 − 5.36190i −2.35128 + 1.04686i 18.1043 − 13.1535i −12.7252 − 5.66564i −11.1344 + 2.36669i
9.6 1.26435 − 3.89127i −5.86148 − 1.24590i −7.07124 − 5.13756i −2.32632 − 4.02930i −12.2591 + 21.2333i 19.5539 − 8.70595i −2.45125 + 1.78094i 8.13897 + 3.62370i −18.6204 + 3.95788i
9.7 1.57199 − 4.83809i 5.55581 + 1.18092i −14.4638 − 10.5086i 7.62837 + 13.2127i 14.4471 − 25.0231i −27.7661 + 12.3623i −40.6542 + 29.5370i 4.80669 + 2.14008i 75.9161 − 16.1365i
10.1 −4.13779 − 3.00628i −1.02796 − 9.78042i 5.61145 + 17.2703i −1.20661 − 2.08991i −25.1492 + 43.5597i 9.84945 + 2.09357i 16.0563 − 49.4162i −68.1899 + 14.4942i −1.29016 + 12.2750i
10.2 −3.25803 − 2.36710i 0.539766 + 5.13553i 2.53947 + 7.81570i 4.24115 + 7.34588i 10.3977 − 18.0094i 22.3519 + 4.75104i 0.271177 − 0.834598i 0.327669 − 0.0696482i 3.57063 − 33.9723i
10.3 −2.29158 − 1.66493i 0.0959507 + 0.912910i 0.00720215 + 0.0221659i −6.04591 − 10.4718i 1.30005 − 2.25175i −31.4535 − 6.68564i −6.98203 + 21.4885i 25.5858 − 5.43843i −3.58017 + 34.0630i
10.4 0.125300 + 0.0910361i −0.485362 − 4.61791i −2.46472 − 7.58564i −0.723817 − 1.25369i 0.359581 − 0.622812i 13.3162 + 2.83044i 0.764620 − 2.35326i 5.32043 − 1.13089i 0.0234363 − 0.222981i
10.5 1.24677 + 0.905833i 0.665097 + 6.32798i −1.73823 − 5.34972i 6.49781 + 11.2545i −4.90287 + 8.49201i −5.97991 − 1.27107i 6.48857 − 19.9698i −13.1909 + 2.80382i −2.09344 + 19.9178i
10.6 3.38160 + 2.45688i 0.294099 + 2.79817i 2.92686 + 9.00793i −6.85368 − 11.8709i −5.88023 + 10.1849i 1.00919 + 0.214509i −1.90068 + 5.84969i 18.6667 − 3.96774i 5.98899 − 56.9814i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.4.g.a ✓ 56
31.g even 15 1 inner 31.4.g.a ✓ 56
31.g even 15 1 961.4.a.m 28
31.h odd 30 1 961.4.a.l 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.4.g.a ✓ 56 1.a even 1 1 trivial
31.4.g.a ✓ 56 31.g even 15 1 inner
961.4.a.l 28 31.h odd 30 1
961.4.a.m 28 31.g even 15 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(31, [\chi])\).