↑

Properties

Label 91.8.e.a
Level $91$
Weight $8$
Character orbit 91.e
Analytic conductor $28.427$
Analytic rank $0$
Dimension $54$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,8,Mod(53,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.53");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.e (of order \(3\), 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: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(27\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q + 17 q^{2} - 1509 q^{4} + 454 q^{5} + 1018 q^{6} - 1827 q^{7} - 6996 q^{8} - 16797 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q + 17 q^{2} - 1509 q^{4} + 454 q^{5} + 1018 q^{6} - 1827 q^{7} - 6996 q^{8} - 16797 q^{9} + 3029 q^{10} + 10777 q^{11} - 20225 q^{12} + 118638 q^{13} + 7960 q^{14} - 47928 q^{15} - 24781 q^{16} + 19613 q^{17} + 37688 q^{18} - 9769 q^{19} - 541594 q^{20} + 140084 q^{21} + 244852 q^{22} + 339878 q^{23} + 141702 q^{24} - 418663 q^{25} + 37349 q^{26} - 640656 q^{27} + 431823 q^{28} - 247470 q^{29} + 978620 q^{30} - 7970 q^{31} + 523571 q^{32} + 305124 q^{33} - 1753788 q^{34} + 5348 q^{35} + 1876484 q^{36} + 646244 q^{37} + 622642 q^{38} - 1877384 q^{40} - 119708 q^{41} + 1556136 q^{42} - 3404204 q^{43} + 1776744 q^{44} + 2031374 q^{45} - 355335 q^{46} + 374963 q^{47} + 4195950 q^{48} - 581499 q^{49} - 7875660 q^{50} + 2434456 q^{51} - 3315273 q^{52} + 3382159 q^{53} - 7078392 q^{54} + 13234336 q^{55} + 3947433 q^{56} - 13313344 q^{57} + 506135 q^{58} + 1218481 q^{59} + 4205254 q^{60} + 2001871 q^{61} + 3689432 q^{62} + 935669 q^{63} - 3653552 q^{64} + 997438 q^{65} + 1251563 q^{66} + 2318139 q^{67} - 4569190 q^{68} + 7267764 q^{69} - 15158614 q^{70} - 15188154 q^{71} + 10960138 q^{72} + 19349964 q^{73} + 5817783 q^{74} - 1332996 q^{75} + 23128660 q^{76} - 7709302 q^{77} + 2236546 q^{78} - 1081242 q^{79} + 26292797 q^{80} - 17601899 q^{81} - 17107466 q^{82} - 3621516 q^{83} - 56254406 q^{84} - 69959728 q^{85} + 36433820 q^{86} + 39961456 q^{87} + 1954557 q^{88} + 10184152 q^{89} + 46377518 q^{90} - 4013919 q^{91} - 112368162 q^{92} + 13114370 q^{93} + 30611806 q^{94} + 11984388 q^{95} + 26137821 q^{96} + 49839080 q^{97} - 65533609 q^{98} - 100761546 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −10.4385 − 18.0801i −13.2292 + 22.9136i −153.926 + 266.608i 111.568 + 193.241i 552.373 −257.953 + 870.059i 3754.80 743.478 + 1287.74i 2329.20 − 4034.30i
53.2 −9.49430 − 16.4446i 8.84975 − 15.3282i −116.283 + 201.409i −82.0184 − 142.060i −336.089 184.409 − 888.559i 1985.58 936.864 + 1622.70i −1557.41 + 2697.52i
53.3 −9.14617 − 15.8416i 19.7233 − 34.1618i −103.305 + 178.929i 21.3669 + 37.0086i −721.572 834.500 + 356.586i 1437.95 315.479 + 546.426i 390.851 − 676.974i
53.4 −8.60217 − 14.8994i −44.5246 + 77.1189i −83.9945 + 145.483i 211.631 + 366.556i 1532.03 −800.595 − 427.307i 687.984 −2871.38 − 4973.38i 3640.97 − 6306.34i
53.5 −7.74840 − 13.4206i 39.3636 − 68.1797i −56.0755 + 97.1256i −48.9657 − 84.8110i −1220.02 −901.549 + 103.698i −245.609 −2005.48 − 3473.59i −758.812 + 1314.30i
53.6 −7.11370 − 12.3213i −27.2334 + 47.1696i −37.2095 + 64.4488i −156.036 − 270.263i 774.920 526.955 + 738.825i −762.318 −389.813 − 675.176i −2219.99 + 3845.14i
53.7 −5.67433 − 9.82823i 16.6456 − 28.8309i −0.396011 + 0.685911i 246.906 + 427.654i −377.809 −742.030 − 522.431i −1443.64 539.351 + 934.183i 2802.05 − 4853.30i
53.8 −5.54093 − 9.59718i −28.9501 + 50.1430i 2.59610 − 4.49657i 26.3761 + 45.6847i 641.642 443.079 − 791.975i −1476.02 −582.716 − 1009.29i 292.296 − 506.272i
53.9 −4.15541 − 7.19738i 5.55308 − 9.61822i 29.4651 − 51.0351i −205.280 − 355.555i −92.3014 −902.192 + 97.9399i −1553.54 1031.83 + 1787.18i −1706.04 + 2954.95i
53.10 −3.73725 − 6.47311i −15.7770 + 27.3266i 36.0659 − 62.4680i 222.347 + 385.116i 235.851 262.001 + 868.849i −1495.89 595.670 + 1031.73i 1661.93 − 2878.55i
53.11 −3.07377 − 5.32392i 26.1369 − 45.2705i 45.1039 − 78.1223i 125.041 + 216.578i −321.355 827.516 − 372.505i −1341.44 −272.778 − 472.466i 768.695 − 1331.42i
53.12 −1.39957 − 2.42413i −28.3323 + 49.0729i 60.0824 − 104.066i −50.2742 − 87.0775i 158.612 −874.517 + 242.412i −694.650 −511.933 − 886.694i −140.725 + 243.743i
53.13 −1.14991 − 1.99171i 42.3845 − 73.4121i 61.3554 − 106.271i −210.002 − 363.734i −194.954 523.116 + 741.547i −576.591 −2499.39 − 4329.07i −482.968 + 836.525i
53.14 −0.235047 − 0.407113i −6.76003 + 11.7087i 63.8895 − 110.660i −159.773 − 276.735i 6.35570 782.544 − 459.529i −120.240 1002.10 + 1735.69i −75.1083 + 130.091i
53.15 2.28438 + 3.95666i 0.182525 − 0.316143i 53.5632 − 92.7742i 75.8741 + 131.418i 1.66783 −446.022 − 790.321i 1074.24 1093.43 + 1893.88i −346.650 + 600.416i
53.16 3.17916 + 5.50646i −34.5304 + 59.8084i 43.7859 − 75.8395i −93.1038 − 161.261i −439.110 725.299 + 545.421i 1370.67 −1291.20 − 2236.42i 591.983 − 1025.34i
53.17 4.30389 + 7.45456i 21.1765 − 36.6788i 26.9530 − 46.6839i 69.0611 + 119.617i 364.566 −657.627 + 625.356i 1565.81 196.610 + 340.539i −594.463 + 1029.64i
53.18 4.32377 + 7.48899i 45.0771 − 78.0759i 26.6100 − 46.0899i 76.9046 + 133.203i 779.613 351.585 − 836.619i 1567.11 −2970.39 − 5144.87i −665.036 + 1151.88i
53.19 4.62503 + 8.01079i −11.1858 + 19.3744i 21.2182 − 36.7510i 211.072 + 365.588i −206.939 64.9698 + 905.164i 1576.55 843.254 + 1460.56i −1952.43 + 3381.71i
53.20 4.84953 + 8.39963i 19.3096 − 33.4452i 16.9641 − 29.3827i −207.198 − 358.877i 374.570 552.809 − 719.684i 1570.55 347.779 + 602.371i 2009.62 − 3480.77i
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.e.a ✓ 54
7.c even 3 1 inner 91.8.e.a ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.e.a ✓ 54 1.a even 1 1 trivial
91.8.e.a ✓ 54 7.c even 3 1 inner