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Properties

Label 91.8.bc.a
Level $91$
Weight $8$
Character orbit 91.bc
Analytic conductor $28.427$
Analytic rank $0$
Dimension $256$
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] = [91,8,Mod(6,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.6");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.bc (of order \(12\), degree \(4\), 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: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 8 q^{2} - 12 q^{4} - 2576 q^{7} - 520 q^{8} + 90392 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 8 q^{2} - 12 q^{4} - 2576 q^{7} - 520 q^{8} + 90392 q^{9} + 140 q^{11} + 18952 q^{14} - 5144 q^{15} + 524284 q^{16} - 49780 q^{18} + 105576 q^{21} - 16068 q^{22} - 12 q^{23} - 82212 q^{28} - 515084 q^{29} - 807756 q^{30} + 664108 q^{32} - 828764 q^{35} - 907920 q^{36} + 336116 q^{37} + 1230908 q^{39} - 2861572 q^{42} + 1445760 q^{43} + 6188328 q^{44} + 7251012 q^{46} - 150672 q^{49} + 6746164 q^{50} - 13243992 q^{53} + 12393168 q^{56} - 6634220 q^{57} - 12046944 q^{58} + 10702124 q^{60} - 13227040 q^{63} + 2971492 q^{65} + 17955336 q^{67} + 14991480 q^{70} + 29025828 q^{71} - 24700280 q^{72} + 2492856 q^{74} - 49919332 q^{78} + 15493808 q^{79} - 47979668 q^{81} - 57449120 q^{84} + 44075300 q^{85} + 52959360 q^{86} + 52919340 q^{88} + 21040608 q^{91} - 132973464 q^{92} + 113964348 q^{93} - 30386388 q^{95} - 71663372 q^{98} + 7655092 q^{99}+O(q^{100}) \) 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 −5.58754 − 20.8530i −28.3234 − 16.3525i −292.775 + 169.034i 294.812 + 294.812i −182.741 + 681.998i 579.003 − 698.784i 3206.77 + 3206.77i −558.689 − 967.677i 4500.44 − 7794.98i
6.2 −5.58754 − 20.8530i 28.3234 + 16.3525i −292.775 + 169.034i −294.812 − 294.812i 182.741 − 681.998i −152.039 − 894.666i 3206.77 + 3206.77i −558.689 − 967.677i −4500.44 + 7794.98i
6.3 −5.20663 − 19.4314i −33.3188 − 19.2366i −239.619 + 138.344i −132.154 − 132.154i −200.316 + 747.590i −555.772 + 717.399i 2115.06 + 2115.06i −353.404 − 612.114i −1879.86 + 3256.01i
6.4 −5.20663 − 19.4314i 33.3188 + 19.2366i −239.619 + 138.344i 132.154 + 132.154i 200.316 − 747.590i 122.613 + 899.171i 2115.06 + 2115.06i −353.404 − 612.114i 1879.86 − 3256.01i
6.5 −5.18506 − 19.3509i −43.5873 − 25.1651i −236.722 + 136.671i −48.6975 − 48.6975i −260.966 + 973.937i 754.522 + 504.221i 2058.90 + 2058.90i 173.069 + 299.765i −689.841 + 1194.84i
6.6 −5.18506 − 19.3509i 43.5873 + 25.1651i −236.722 + 136.671i 48.6975 + 48.6975i 260.966 − 973.937i −905.546 + 59.4071i 2058.90 + 2058.90i 173.069 + 299.765i 689.841 − 1194.84i
6.7 −4.93901 − 18.4326i −71.3292 − 41.1819i −204.517 + 118.078i −205.442 − 205.442i −406.796 + 1518.18i −481.943 − 768.944i 1459.41 + 1459.41i 2298.40 + 3980.95i −2772.16 + 4801.52i
6.8 −4.93901 − 18.4326i 71.3292 + 41.1819i −204.517 + 118.078i 205.442 + 205.442i 406.796 − 1518.18i 801.846 − 424.953i 1459.41 + 1459.41i 2298.40 + 3980.95i 2772.16 − 4801.52i
6.9 −4.39405 − 16.3988i −1.89548 − 1.09436i −138.762 + 80.1143i 215.230 + 215.230i −9.61732 + 35.8923i −870.796 − 255.457i 386.896 + 386.896i −1091.10 − 1889.85i 2583.79 − 4475.25i
6.10 −4.39405 − 16.3988i 1.89548 + 1.09436i −138.762 + 80.1143i −215.230 − 215.230i 9.61732 − 35.8923i 881.859 + 214.166i 386.896 + 386.896i −1091.10 − 1889.85i −2583.79 + 4475.25i
6.11 −4.35987 − 16.2713i −78.9325 − 45.5717i −134.894 + 77.8812i 338.507 + 338.507i −397.373 + 1483.02i −338.835 + 841.863i 330.690 + 330.690i 3060.05 + 5300.17i 4032.08 − 6983.78i
6.12 −4.35987 − 16.2713i 78.9325 + 45.5717i −134.894 + 77.8812i −338.507 − 338.507i 397.373 − 1483.02i −127.492 + 898.492i 330.690 + 330.690i 3060.05 + 5300.17i −4032.08 + 6983.78i
6.13 −3.88048 − 14.4821i −25.0579 − 14.4672i −83.8228 + 48.3951i 190.771 + 190.771i −112.279 + 419.031i −429.219 − 799.571i −330.875 − 330.875i −674.902 − 1168.96i 2022.49 − 3503.05i
6.14 −3.88048 − 14.4821i 25.0579 + 14.4672i −83.8228 + 48.3951i −190.771 − 190.771i 112.279 − 419.031i 771.500 − 477.840i −330.875 − 330.875i −674.902 − 1168.96i −2022.49 + 3503.05i
6.15 −3.41580 − 12.7480i −50.2427 − 29.0077i −39.9914 + 23.0890i −75.5410 − 75.5410i −198.169 + 739.577i 439.074 − 794.202i −763.575 − 763.575i 589.389 + 1020.85i −704.960 + 1221.03i
6.16 −3.41580 − 12.7480i 50.2427 + 29.0077i −39.9914 + 23.0890i 75.5410 + 75.5410i 198.169 − 739.577i 16.8520 − 907.336i −763.575 − 763.575i 589.389 + 1020.85i 704.960 − 1221.03i
6.17 −3.09983 − 11.5687i −10.9907 − 6.34548i −13.3753 + 7.72225i −345.991 − 345.991i −39.3398 + 146.818i −891.928 + 167.352i −953.221 − 953.221i −1012.97 − 1754.52i −2930.16 + 5075.19i
6.18 −3.09983 − 11.5687i 10.9907 + 6.34548i −13.3753 + 7.72225i 345.991 + 345.991i 39.3398 − 146.818i 688.756 + 590.896i −953.221 − 953.221i −1012.97 − 1754.52i 2930.16 − 5075.19i
6.19 −2.96607 − 11.0695i −36.3025 − 20.9592i −2.88572 + 1.66607i −3.37228 − 3.37228i −124.333 + 464.018i −601.519 + 679.498i −1010.24 − 1010.24i −214.921 − 372.253i −27.3271 + 47.3320i
6.20 −2.96607 − 11.0695i 36.3025 + 20.9592i −2.88572 + 1.66607i 3.37228 + 3.37228i 124.333 − 464.018i 181.182 + 889.222i −1010.24 − 1010.24i −214.921 − 372.253i 27.3271 − 47.3320i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.f odd 12 1 inner
91.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.bc.a ✓ 256
7.b odd 2 1 inner 91.8.bc.a ✓ 256
13.f odd 12 1 inner 91.8.bc.a ✓ 256
91.bc even 12 1 inner 91.8.bc.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.bc.a ✓ 256 1.a even 1 1 trivial
91.8.bc.a ✓ 256 7.b odd 2 1 inner
91.8.bc.a ✓ 256 13.f odd 12 1 inner
91.8.bc.a ✓ 256 91.bc even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(91, [\chi])\).