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Properties

Label 31.9.e.a
Level $31$
Weight $9$
Character orbit 31.e
Analytic conductor $12.629$
Analytic rank $0$
Dimension $40$
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] = [31,9,Mod(6,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.6");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 31.e (of order \(6\), degree \(2\), 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: \(12.6287369119\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{2} - 84 q^{3} + 4128 q^{4} + 146 q^{5} + 7482 q^{6} + 272 q^{7} + 17656 q^{8} + 45088 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{2} - 84 q^{3} + 4128 q^{4} + 146 q^{5} + 7482 q^{6} + 272 q^{7} + 17656 q^{8} + 45088 q^{9} + 12448 q^{10} - 12540 q^{11} - 61926 q^{12} - 33492 q^{13} + 107434 q^{14} + 451400 q^{16} + 67092 q^{17} - 336658 q^{18} + 3470 q^{19} - 235110 q^{20} + 43092 q^{21} - 487644 q^{22} + 1895706 q^{24} - 145968 q^{25} + 46596 q^{26} - 937326 q^{28} + 2008366 q^{31} + 4420212 q^{32} - 1658556 q^{33} - 3396300 q^{34} + 7810396 q^{35} + 5279544 q^{36} - 10470252 q^{37} - 5161808 q^{38} - 3234012 q^{39} + 6134366 q^{40} + 407222 q^{41} - 23718240 q^{42} - 12791892 q^{43} + 15133770 q^{44} + 10756748 q^{45} - 15743548 q^{47} - 17441484 q^{48} - 24492036 q^{49} + 21862824 q^{50} - 876018 q^{51} - 29690352 q^{52} + 16532034 q^{53} + 19037814 q^{55} + 42785018 q^{56} - 67655958 q^{57} + 5896664 q^{59} + 124600532 q^{62} + 28717316 q^{63} + 85694608 q^{64} + 12630390 q^{65} + 136383864 q^{66} + 15193700 q^{67} - 18870444 q^{68} + 39312270 q^{69} - 85917208 q^{70} + 95239622 q^{71} - 121999958 q^{72} + 86815806 q^{73} - 176382186 q^{74} + 85869744 q^{75} - 69283200 q^{76} - 347046300 q^{78} + 88945134 q^{79} - 270806366 q^{80} - 18370100 q^{81} - 17702138 q^{82} - 206513688 q^{83} + 186807684 q^{84} - 284190882 q^{86} + 181365936 q^{87} - 343782624 q^{88} - 299472386 q^{90} + 595622406 q^{93} + 314981452 q^{94} + 412793992 q^{95} + 485999670 q^{96} + 368521172 q^{97} - 118338462 q^{98} + 502689990 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −28.8089 1.91658 − 1.10654i 573.955 204.293 − 353.847i −55.2147 + 31.8782i 191.390 + 331.498i −9159.95 −3278.05 + 5677.75i −5885.48 + 10193.9i
6.2 −28.4801 −132.315 + 76.3924i 555.114 −393.261 + 681.148i 3768.35 − 2175.66i −1732.42 − 3000.63i −8518.78 8391.09 − 14533.8i 11200.1 − 19399.1i
6.3 −24.5577 96.1791 − 55.5290i 347.081 −336.439 + 582.730i −2361.94 + 1363.67i −450.359 − 780.045i −2236.74 2886.45 − 4999.47i 8262.18 − 14310.5i
6.4 −19.8052 −39.1650 + 22.6119i 136.245 −167.508 + 290.133i 775.671 − 447.834i 1248.42 + 2162.32i 2371.76 −2257.90 + 3910.80i 3317.54 − 5746.14i
6.5 −16.7344 89.2194 − 51.5108i 24.0412 552.897 − 957.646i −1493.04 + 862.004i −704.202 − 1219.71i 3881.70 2026.23 − 3509.53i −9252.42 + 16025.7i
6.6 −16.6586 −114.556 + 66.1389i 21.5080 413.548 − 716.285i 1908.34 − 1101.78i 1578.33 + 2733.75i 3906.30 5468.22 − 9471.23i −6889.11 + 11932.3i
6.7 −13.2036 −56.8917 + 32.8465i −81.6661 204.200 − 353.685i 751.173 − 433.690i −2184.65 − 3783.92i 4458.39 −1122.72 + 1944.61i −2696.16 + 4669.89i
6.8 −9.42582 −13.2074 + 7.62530i −167.154 −445.942 + 772.394i 124.491 − 71.8747i −597.559 − 1035.00i 3988.57 −3164.21 + 5480.57i 4203.37 − 7280.44i
6.9 −8.57074 101.323 − 58.4988i −182.542 −55.2679 + 95.7269i −868.413 + 501.378i 2168.18 + 3755.41i 3758.63 3563.72 − 6172.55i 473.687 − 820.451i
6.10 −1.32207 36.0373 − 20.8062i −254.252 72.3675 − 125.344i −47.6439 + 27.5072i 204.328 + 353.906i 674.589 −2414.71 + 4182.40i −95.6750 + 165.714i
6.11 3.57800 −113.299 + 65.4130i −243.198 −424.686 + 735.578i −405.382 + 234.047i 973.500 + 1686.15i −1786.13 5277.22 − 9140.41i −1519.52 + 2631.89i
6.12 5.30875 106.844 − 61.6866i −227.817 −199.683 + 345.861i 567.210 − 327.479i −2086.89 − 3614.60i −2568.47 4329.98 − 7499.74i −1060.07 + 1836.09i
6.13 6.13894 −89.7260 + 51.8034i −218.313 242.008 − 419.170i −550.823 + 318.018i −452.725 − 784.143i −2911.78 2086.67 − 3614.23i 1485.67 − 2573.26i
6.14 8.14444 0.794873 − 0.458920i −189.668 376.155 − 651.520i 6.47379 − 3.73765i 867.251 + 1502.12i −3629.72 −3280.08 + 5681.26i 3063.57 − 5306.26i
6.15 17.6238 18.4634 − 10.6599i 54.6000 −416.023 + 720.573i 325.397 − 187.868i 513.408 + 889.248i −3549.44 −3053.23 + 5288.36i −7331.93 + 12699.3i
6.16 20.1646 127.319 − 73.5076i 150.611 92.4092 − 160.057i 2567.34 − 1482.25i 1413.19 + 2447.72i −2125.13 7526.24 − 13035.8i 1863.39 − 3227.49i
6.17 22.5873 −84.4161 + 48.7376i 254.186 −83.3100 + 144.297i −1906.73 + 1100.85i −1468.75 − 2543.95i −40.9753 1470.21 − 2546.49i −1881.75 + 3259.28i
6.18 22.9050 47.8498 − 27.6261i 268.637 467.868 − 810.371i 1096.00 − 632.774i −1098.17 − 1902.09i 289.450 −1754.10 + 3038.19i 10716.5 − 18561.5i
6.19 27.6520 −80.4090 + 46.4242i 508.635 206.262 − 357.255i −2223.47 + 1283.72i 2267.29 + 3927.07i 6985.88 1029.91 − 1783.85i 5703.55 − 9878.84i
6.20 31.4642 56.0387 − 32.3540i 733.997 −236.886 + 410.298i 1763.21 − 1017.99i −513.570 − 889.530i 15039.8 −1186.94 + 2055.84i −7453.43 + 12909.7i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.9.e.a ✓ 40
31.e odd 6 1 inner 31.9.e.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.9.e.a ✓ 40 1.a even 1 1 trivial
31.9.e.a ✓ 40 31.e odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(31, [\chi])\).