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Properties

Label 98.4.g.b
Level $98$
Weight $4$
Character orbit 98.g
Analytic conductor $5.782$
Analytic rank $0$
Dimension $84$
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] = [98,4,Mod(9,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 98.g (of order \(21\), degree \(12\), 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: \(5.78218718056\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 14 q^{2} + 2 q^{3} + 28 q^{4} - 9 q^{5} + 34 q^{6} + 28 q^{7} - 112 q^{8} - 175 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 14 q^{2} + 2 q^{3} + 28 q^{4} - 9 q^{5} + 34 q^{6} + 28 q^{7} - 112 q^{8} - 175 q^{9} - 18 q^{10} - 133 q^{11} - 20 q^{12} + 70 q^{13} - 70 q^{14} - 161 q^{15} + 112 q^{16} - 173 q^{17} - 742 q^{18} + 383 q^{19} + 184 q^{20} - 392 q^{21} - 154 q^{22} + 357 q^{23} - 40 q^{24} + 154 q^{25} - 406 q^{26} + 746 q^{27} - 252 q^{28} - 161 q^{29} + 14 q^{30} + 891 q^{31} + 224 q^{32} - 625 q^{33} - 190 q^{34} + 784 q^{35} - 364 q^{36} + 3668 q^{37} - 1068 q^{38} + 1239 q^{39} + 264 q^{40} + 308 q^{41} + 1162 q^{42} + 490 q^{43} - 728 q^{44} - 1830 q^{45} - 1638 q^{46} - 730 q^{47} - 1184 q^{48} - 2562 q^{49} + 4284 q^{50} - 2513 q^{51} - 1232 q^{52} - 1218 q^{53} - 1390 q^{54} - 935 q^{55} + 504 q^{56} - 2562 q^{57} + 896 q^{58} + 1360 q^{59} + 1204 q^{60} + 4478 q^{61} + 2400 q^{62} + 2723 q^{63} - 896 q^{64} + 434 q^{65} + 2264 q^{66} - 1505 q^{67} + 1268 q^{68} + 3427 q^{69} - 1470 q^{70} - 4697 q^{71} + 560 q^{72} + 681 q^{73} - 1876 q^{74} - 1544 q^{75} - 40 q^{76} - 4137 q^{77} + 1120 q^{78} + 1687 q^{79} + 976 q^{80} + 2310 q^{81} - 308 q^{82} + 7441 q^{83} + 364 q^{84} + 1267 q^{85} + 2254 q^{86} - 12015 q^{87} - 1456 q^{88} - 3593 q^{89} - 3558 q^{90} - 1981 q^{91} + 1260 q^{92} - 6370 q^{93} + 3132 q^{94} + 2247 q^{95} - 160 q^{96} - 18662 q^{97} - 4550 q^{98} + 14966 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.46610 − 1.36035i −5.79301 − 0.873156i 0.298920 + 3.98882i 0.000237039 0 0.000603965i 7.30536 + 9.16063i 0.150726 + 18.5196i 4.98792 − 6.25465i 6.99610 + 2.15801i 0.000474078 − 0.00120793i
9.2 −1.46610 − 1.36035i −5.07649 − 0.765157i 0.298920 + 3.98882i 4.25782 + 10.8487i 6.40178 + 8.02757i 10.4612 − 15.2828i 4.98792 − 6.25465i −0.615219 − 0.189770i 8.51563 − 21.6975i
9.3 −1.46610 − 1.36035i −2.91630 − 0.439561i 0.298920 + 3.98882i −5.72213 − 14.5797i 3.67764 + 4.61161i 18.2029 + 3.41371i 4.98792 − 6.25465i −17.4889 − 5.39461i −11.4443 + 29.1595i
9.4 −1.46610 − 1.36035i 1.04126 + 0.156944i 0.298920 + 3.98882i 4.89065 + 12.4612i −1.31309 − 1.64657i −18.0383 − 4.19769i 4.98792 − 6.25465i −24.7409 − 7.63155i 9.78130 − 24.9223i
9.5 −1.46610 − 1.36035i 3.58788 + 0.540786i 0.298920 + 3.98882i −5.75757 − 14.6700i −4.52455 − 5.67361i −16.0668 + 9.21182i 4.98792 − 6.25465i −13.2200 − 4.07784i −11.5151 + 29.3401i
9.6 −1.46610 − 1.36035i 6.60186 + 0.995070i 0.298920 + 3.98882i −1.48581 − 3.78578i −8.32537 − 10.4397i 2.76568 − 18.3126i 4.98792 − 6.25465i 16.7939 + 5.18025i −2.97162 + 7.57157i
9.7 −1.46610 − 1.36035i 9.21030 + 1.38823i 0.298920 + 3.98882i 2.03964 + 5.19691i −11.6148 − 14.5645i 8.54554 + 16.4309i 4.98792 − 6.25465i 57.1019 + 17.6136i 4.07928 − 10.3938i
11.1 −1.46610 + 1.36035i −5.79301 + 0.873156i 0.298920 − 3.98882i 0.000237039 0 0.000603965i 7.30536 − 9.16063i 0.150726 − 18.5196i 4.98792 + 6.25465i 6.99610 − 2.15801i 0.000474078 0.00120793i
11.2 −1.46610 + 1.36035i −5.07649 + 0.765157i 0.298920 − 3.98882i 4.25782 − 10.8487i 6.40178 − 8.02757i 10.4612 + 15.2828i 4.98792 + 6.25465i −0.615219 + 0.189770i 8.51563 + 21.6975i
11.3 −1.46610 + 1.36035i −2.91630 + 0.439561i 0.298920 − 3.98882i −5.72213 + 14.5797i 3.67764 − 4.61161i 18.2029 − 3.41371i 4.98792 + 6.25465i −17.4889 + 5.39461i −11.4443 − 29.1595i
11.4 −1.46610 + 1.36035i 1.04126 − 0.156944i 0.298920 − 3.98882i 4.89065 − 12.4612i −1.31309 + 1.64657i −18.0383 + 4.19769i 4.98792 + 6.25465i −24.7409 + 7.63155i 9.78130 + 24.9223i
11.5 −1.46610 + 1.36035i 3.58788 − 0.540786i 0.298920 − 3.98882i −5.75757 + 14.6700i −4.52455 + 5.67361i −16.0668 − 9.21182i 4.98792 + 6.25465i −13.2200 + 4.07784i −11.5151 − 29.3401i
11.6 −1.46610 + 1.36035i 6.60186 − 0.995070i 0.298920 − 3.98882i −1.48581 + 3.78578i −8.32537 + 10.4397i 2.76568 + 18.3126i 4.98792 + 6.25465i 16.7939 − 5.18025i −2.97162 − 7.57157i
11.7 −1.46610 + 1.36035i 9.21030 − 1.38823i 0.298920 − 3.98882i 2.03964 − 5.19691i −11.6148 + 14.5645i 8.54554 − 16.4309i 4.98792 + 6.25465i 57.1019 − 17.6136i 4.07928 + 10.3938i
23.1 0.149460 − 1.99441i −8.31950 + 2.56623i −3.95532 − 0.596169i 11.0231 + 10.2280i 3.87467 + 16.9760i −3.11346 − 18.2567i −1.78017 + 7.79942i 40.3202 − 27.4898i 22.0463 − 20.4560i
23.2 0.149460 − 1.99441i −7.35559 + 2.26890i −3.95532 − 0.596169i −8.97094 − 8.32382i 3.42574 + 15.0092i 15.6884 + 9.84254i −1.78017 + 7.79942i 26.6484 − 18.1686i −17.9419 + 16.6476i
23.3 0.149460 − 1.99441i −1.84353 + 0.568655i −3.95532 − 0.596169i 2.38053 + 2.20881i 0.858594 + 3.76175i −18.2272 + 3.28180i −1.78017 + 7.79942i −19.2332 + 13.1130i 4.76106 − 4.41762i
23.4 0.149460 − 1.99441i −0.574046 + 0.177070i −3.95532 − 0.596169i −5.69805 − 5.28701i 0.267352 + 1.17135i 13.1474 − 13.0440i −1.78017 + 7.79942i −22.0103 + 15.0063i −11.3961 + 10.5740i
23.5 0.149460 − 1.99441i 1.45066 − 0.447469i −3.95532 − 0.596169i 12.3189 + 11.4302i −0.675619 − 2.96008i 13.3348 + 12.8524i −1.78017 + 7.79942i −20.4043 + 13.9114i 24.6378 − 22.8605i
23.6 0.149460 − 1.99441i 4.57005 − 1.40967i −3.95532 − 0.596169i −15.8386 − 14.6961i −2.12842 − 9.32524i −12.2083 + 13.9269i −1.78017 + 7.79942i −3.41026 + 2.32507i −31.6773 + 29.3922i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.g.b ✓ 84
49.g even 21 1 inner 98.4.g.b ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.g.b ✓ 84 1.a even 1 1 trivial
98.4.g.b ✓ 84 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} - 2 T_{3}^{83} - 5 T_{3}^{82} - 416 T_{3}^{81} - 3678 T_{3}^{80} + 20444 T_{3}^{79} + \cdots + 15\!\cdots\!49 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display