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Properties

Label 91.8.k.a
Level $91$
Weight $8$
Character orbit 91.k
Analytic conductor $28.427$
Analytic rank $0$
Dimension $126$
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] = [91,8,Mod(4,91)]
 
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, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.k (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: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(126\)
Relative dimension: \(63\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 126 q + 82 q^{3} - 7682 q^{4} - 6 q^{6} + 2009 q^{7} - 42855 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 126 q + 82 q^{3} - 7682 q^{4} - 6 q^{6} + 2009 q^{7} - 42855 q^{9} - 3743 q^{10} + 4539 q^{11} - 18328 q^{12} + 1816 q^{13} - 6668 q^{14} - 11817 q^{15} + 474438 q^{16} + 57938 q^{17} - 75423 q^{18} - 90798 q^{19} + 12342 q^{20} + 54000 q^{21} + 51388 q^{22} - 34134 q^{23} + 7326 q^{24} + 810665 q^{25} - 254220 q^{26} - 528170 q^{27} - 749453 q^{28} + 191096 q^{29} + 216080 q^{30} + 259995 q^{31} + 377481 q^{33} + 52743 q^{35} + 1965131 q^{36} - 1061954 q^{38} + 1397885 q^{39} + 345362 q^{40} - 636567 q^{41} + 1978917 q^{42} - 659976 q^{43} - 2519268 q^{44} + 294840 q^{47} + 4726652 q^{48} + 1441579 q^{49} - 2908992 q^{50} - 502094 q^{51} - 815239 q^{52} - 434723 q^{53} + 2152196 q^{55} + 2105913 q^{56} + 11163681 q^{58} + 3768141 q^{60} + 2427458 q^{61} - 1713436 q^{62} + 4932312 q^{63} - 40270472 q^{64} + 2921652 q^{65} + 8270835 q^{66} - 3848622 q^{67} - 1062758 q^{68} - 311689 q^{69} - 5298534 q^{70} + 1505877 q^{71} + 21241725 q^{72} + 13207461 q^{73} - 7461054 q^{74} - 11249902 q^{75} + 25297146 q^{76} + 11107478 q^{77} - 9362803 q^{78} + 5012226 q^{79} + 9241518 q^{80} - 22994883 q^{81} + 17023394 q^{82} - 42775845 q^{84} + 19854117 q^{85} - 5226210 q^{86} - 34663466 q^{87} - 12174155 q^{88} + 90517324 q^{90} + 23781225 q^{91} - 8502078 q^{92} + 3855208 q^{94} - 29621818 q^{95} - 41499936 q^{96} - 60232128 q^{97} + 11120340 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 22.4878i 4.91018 + 8.50469i −377.703 268.738 − 155.156i 191.252 − 110.419i 462.735 + 780.653i 5615.28i 1045.28 − 1810.48i −3489.13 − 6043.35i
4.2 − 21.5972i 40.9143 + 70.8657i −338.439 −42.3342 + 24.4417i 1530.50 − 883.635i 473.722 − 774.035i 4544.89i −2254.46 + 3904.84i 527.872 + 914.301i
4.3 − 21.3491i −18.9734 − 32.8629i −327.785 −411.268 + 237.445i −701.593 + 405.065i 227.741 − 878.452i 4265.24i 373.522 − 646.959i 5069.25 + 8780.20i
4.4 − 20.0980i −33.6434 − 58.2720i −275.930 13.7956 − 7.96491i −1171.15 + 676.165i −771.267 + 478.215i 2973.10i −1170.25 + 2026.94i −160.079 − 277.265i
4.5 − 19.2917i 18.6461 + 32.2960i −244.168 −25.2662 + 14.5875i 623.044 − 359.715i −846.087 − 328.146i 2241.08i 398.145 − 689.608i 281.416 + 487.428i
4.6 − 19.2412i −25.3490 − 43.9057i −242.222 428.668 − 247.491i −844.797 + 487.744i −155.598 − 894.054i 2197.77i −191.642 + 331.934i −4762.02 − 8248.07i
4.7 − 18.9619i 14.9174 + 25.8377i −231.552 −406.721 + 234.820i 489.930 − 282.861i −430.042 + 799.129i 1963.54i 648.444 − 1123.14i 4452.63 + 7712.18i
4.8 − 18.0981i 6.75116 + 11.6934i −199.542 99.0313 − 57.1757i 211.628 − 122.183i 719.918 + 552.504i 1294.77i 1002.34 − 1736.11i −1034.77 − 1792.28i
4.9 − 17.5147i −7.98479 − 13.8301i −178.764 59.9286 − 34.5998i −242.229 + 139.851i 429.489 − 799.426i 889.110i 965.986 − 1673.14i −606.004 − 1049.63i
4.10 − 17.5064i −33.9814 − 58.8575i −178.475 −154.359 + 89.1194i −1030.39 + 594.893i 760.471 + 495.204i 883.636i −1215.97 + 2106.13i 1560.16 + 2702.28i
4.11 − 17.0376i 31.2189 + 54.0726i −162.279 428.751 − 247.539i 921.266 − 531.893i −904.500 + 73.6394i 584.023i −855.734 + 1482.17i −4217.47 − 7304.87i
4.12 − 15.5459i 43.1042 + 74.6586i −113.676 −122.410 + 70.6734i 1160.64 − 670.094i 82.2302 + 903.759i − 222.687i −2622.44 + 4542.19i 1098.68 + 1902.97i
4.13 − 14.2843i 25.9053 + 44.8693i −76.0422 −189.087 + 109.169i 640.928 − 370.040i 819.936 − 388.907i − 742.183i −248.671 + 430.710i 1559.41 + 2700.98i
4.14 − 12.9731i −14.7980 − 25.6309i −40.3019 169.575 − 97.9040i −332.513 + 191.977i −701.054 + 576.252i − 1137.72i 655.537 − 1135.42i −1270.12 − 2199.91i
4.15 − 12.7020i 28.2624 + 48.9519i −33.3420 371.359 − 214.404i 621.789 − 358.990i 769.685 − 480.758i − 1202.35i −504.025 + 872.997i −2723.38 − 4717.02i
4.16 − 12.4760i −0.875577 − 1.51654i −27.6513 −129.006 + 74.4818i −18.9204 + 10.9237i −665.294 − 617.192i − 1251.95i 1091.97 − 1891.34i 929.237 + 1609.48i
4.17 − 12.1973i −38.6895 − 67.0121i −20.7751 388.438 − 224.265i −817.369 + 471.908i 893.657 + 157.863i − 1307.86i −1900.25 + 3291.33i −2735.44 − 4737.91i
4.18 − 11.5117i −44.2677 − 76.6740i −4.51871 −146.660 + 84.6740i −882.646 + 509.596i −546.773 − 724.281i − 1421.48i −2825.77 + 4894.37i 974.740 + 1688.30i
4.19 − 11.0879i −8.56212 − 14.8300i 5.05843 −357.050 + 206.143i −164.434 + 94.9360i 851.091 + 314.941i − 1475.34i 946.880 − 1640.04i 2285.69 + 3958.93i
4.20 − 9.66535i −18.8752 − 32.6928i 34.5810 −377.122 + 217.731i −315.987 + 182.435i −761.759 + 493.220i − 1571.40i 380.956 − 659.834i 2104.45 + 3645.01i
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.63
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.8.k.a ✓ 126
7.c even 3 1 91.8.u.a yes 126
13.e even 6 1 91.8.u.a yes 126
91.k even 6 1 inner 91.8.k.a ✓ 126
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.k.a ✓ 126 1.a even 1 1 trivial
91.8.k.a ✓ 126 91.k even 6 1 inner
91.8.u.a yes 126 7.c even 3 1
91.8.u.a yes 126 13.e even 6 1

Hecke kernels

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