Properties

Label 98.5.h.a
Level $98$
Weight $5$
Character orbit 98.h
Analytic conductor $10.130$
Analytic rank $0$
Dimension $216$
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] = [98,5,Mod(3,98)]
 
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([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 98.h (of order \(42\), 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: \(10.1302563822\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 18 q^{3} + 144 q^{4} - 54 q^{5} + 224 q^{6} + 28 q^{7} - 560 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q + 18 q^{3} + 144 q^{4} - 54 q^{5} + 224 q^{6} + 28 q^{7} - 560 q^{9} + 96 q^{10} - 312 q^{11} - 144 q^{12} - 12 q^{15} + 1152 q^{16} + 552 q^{17} - 256 q^{18} - 30 q^{19} - 1008 q^{20} + 378 q^{21} + 1408 q^{22} + 5088 q^{23} + 768 q^{24} - 1376 q^{25} + 384 q^{26} - 6090 q^{27} - 1456 q^{28} + 1584 q^{29} - 1856 q^{30} + 546 q^{31} - 1062 q^{33} + 6090 q^{35} + 6272 q^{36} - 17364 q^{37} - 2400 q^{38} - 8930 q^{39} + 2816 q^{40} + 12936 q^{41} + 31136 q^{42} - 4016 q^{43} + 7248 q^{44} + 31810 q^{45} + 12480 q^{46} + 3120 q^{47} - 1652 q^{49} - 3840 q^{50} - 27642 q^{51} + 3856 q^{52} - 41292 q^{53} - 34848 q^{54} - 58870 q^{55} - 10752 q^{56} - 10090 q^{57} - 33920 q^{58} - 25806 q^{59} + 6096 q^{60} + 44408 q^{61} + 28224 q^{62} + 31710 q^{63} - 18432 q^{64} + 2352 q^{65} + 3456 q^{66} + 702 q^{67} + 7344 q^{68} - 7504 q^{69} - 23968 q^{70} + 31674 q^{71} - 2048 q^{72} + 34536 q^{73} + 26784 q^{74} + 7068 q^{75} - 32004 q^{77} + 40576 q^{78} - 5858 q^{79} + 3456 q^{80} - 63718 q^{81} - 9984 q^{82} + 81060 q^{83} - 23520 q^{84} - 39940 q^{85} + 37536 q^{86} + 132956 q^{87} + 8704 q^{88} - 23730 q^{89} + 51072 q^{90} + 71288 q^{91} - 6816 q^{92} + 30652 q^{93} - 63328 q^{94} + 26406 q^{95} - 6144 q^{96} - 64512 q^{98} - 60040 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.03334 + 2.63291i −13.7798 1.03265i −5.86441 5.44138i 4.87396 7.14878i 16.9581 35.2139i −23.3515 43.0779i 20.3866 9.81767i 108.722 + 16.3872i 13.7856 + 20.2198i
3.2 −1.03334 + 2.63291i −9.18673 0.688451i −5.86441 5.44138i −15.1525 + 22.2246i 11.3057 23.4764i −28.8684 + 39.5931i 20.3866 9.81767i 3.82683 + 0.576802i −42.8576 62.8606i
3.3 −1.03334 + 2.63291i −9.09005 0.681205i −5.86441 5.44138i 10.9258 16.0251i 11.1867 23.2294i 40.8166 + 27.1109i 20.3866 9.81767i 2.06973 + 0.311961i 30.9027 + 45.3259i
3.4 −1.03334 + 2.63291i −0.341292 0.0255763i −5.86441 5.44138i −7.38538 + 10.8324i 0.420011 0.872161i 15.9874 46.3185i 20.3866 9.81767i −79.9795 12.0550i −20.8890 30.6385i
3.5 −1.03334 + 2.63291i 2.64055 + 0.197882i −5.86441 5.44138i 23.2214 34.0595i −3.24960 + 6.74786i −48.1538 + 9.06681i 20.3866 9.81767i −73.1619 11.0274i 65.6800 + 96.3349i
3.6 −1.03334 + 2.63291i 6.56959 + 0.492323i −5.86441 5.44138i −19.9016 + 29.1903i −8.08486 + 16.7884i 46.9335 + 14.0801i 20.3866 9.81767i −37.1782 5.60371i −56.2903 82.5627i
3.7 −1.03334 + 2.63291i 8.86092 + 0.664034i −5.86441 5.44138i 13.8478 20.3110i −10.9047 + 22.6438i 2.81979 + 48.9188i 20.3866 9.81767i −2.02032 0.304515i 39.1675 + 57.4481i
3.8 −1.03334 + 2.63291i 9.41284 + 0.705395i −5.86441 5.44138i −10.9686 + 16.0880i −11.5839 + 24.0542i −48.4695 7.19094i 20.3866 9.81767i 8.00874 + 1.20712i −31.0238 45.5036i
3.9 −1.03334 + 2.63291i 16.4510 + 1.23283i −5.86441 5.44138i 8.38583 12.2998i −20.2455 + 42.0401i 6.27978 48.5959i 20.3866 9.81767i 189.021 + 28.4904i 23.7187 + 34.7890i
3.10 1.03334 2.63291i −14.1719 1.06204i −5.86441 5.44138i −9.39032 + 13.7731i −17.4407 + 36.2160i −46.9328 14.0822i −20.3866 + 9.81767i 119.621 + 18.0299i 26.5598 + 38.9561i
3.11 1.03334 2.63291i −13.2432 0.992441i −5.86441 5.44138i 25.8416 37.9027i −16.2977 + 33.8426i 23.3218 43.0940i −20.3866 + 9.81767i 94.3023 + 14.2138i −73.0912 107.205i
3.12 1.03334 2.63291i −11.5689 0.866972i −5.86441 5.44138i −9.61510 + 14.1028i −14.2373 + 29.5641i 43.1303 + 23.2546i −20.3866 + 9.81767i 52.9935 + 7.98749i 27.1956 + 39.8886i
3.13 1.03334 2.63291i −3.49449 0.261876i −5.86441 5.44138i 9.11024 13.3623i −4.30049 + 8.93006i −34.1673 + 35.1226i −20.3866 + 9.81767i −67.9524 10.2422i −25.7676 37.7942i
3.14 1.03334 2.63291i −1.48521 0.111301i −5.86441 5.44138i −10.4504 + 15.3280i −1.82778 + 3.79542i 33.1945 36.0434i −20.3866 + 9.81767i −77.9018 11.7418i 29.5583 + 43.3540i
3.15 1.03334 2.63291i 7.10780 + 0.532656i −5.86441 5.44138i −17.6874 + 25.9427i 8.74721 18.1638i −35.5979 + 33.6717i −20.3866 + 9.81767i −29.8583 4.50041i 50.0276 + 73.3770i
3.16 1.03334 2.63291i 8.32236 + 0.623675i −5.86441 5.44138i 9.25164 13.5697i 10.2419 21.2676i −16.2717 46.2194i −20.3866 + 9.81767i −11.2225 1.69152i −26.1676 38.3808i
3.17 1.03334 2.63291i 12.2032 + 0.914503i −5.86441 5.44138i −15.7636 + 23.1209i 15.0179 31.1849i 45.1496 + 19.0398i −20.3866 + 9.81767i 67.9864 + 10.2473i 44.5861 + 65.3959i
3.18 1.03334 2.63291i 15.1566 + 1.13583i −5.86441 5.44138i 24.6502 36.1553i 18.6525 38.7324i 32.5086 + 36.6632i −20.3866 + 9.81767i 148.338 + 22.3584i −69.7214 102.263i
5.1 −2.79684 + 0.421555i −9.54857 + 14.0052i 7.64458 2.35804i −30.6644 2.29798i 20.8018 43.1954i −42.9807 + 23.5299i −20.3866 + 9.81767i −75.3773 192.058i 86.7321 6.49967i
5.2 −2.79684 + 0.421555i −7.75332 + 11.3720i 7.64458 2.35804i 20.2326 + 1.51622i 16.8908 35.0742i 21.8955 43.8359i −20.3866 + 9.81767i −39.6166 100.941i −57.2263 + 4.28852i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.h odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.5.h.a 216
49.h odd 42 1 inner 98.5.h.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.5.h.a 216 1.a even 1 1 trivial
98.5.h.a 216 49.h odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(98, [\chi])\).