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Properties

Label 98.4.e.a
Level $98$
Weight $4$
Character orbit 98.e
Analytic conductor $5.782$
Analytic rank $0$
Dimension $42$
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(15,98)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(98, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([10]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("98.15");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 98.e (of order \(7\), degree \(6\), 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: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q - 14 q^{2} - 5 q^{3} - 28 q^{4} + 12 q^{5} - 10 q^{6} - 7 q^{7} - 56 q^{8} - 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q - 14 q^{2} - 5 q^{3} - 28 q^{4} + 12 q^{5} - 10 q^{6} - 7 q^{7} - 56 q^{8} - 98 q^{9} + 24 q^{10} + 28 q^{11} + 8 q^{12} + 14 q^{13} - 14 q^{14} + 161 q^{15} - 112 q^{16} + 338 q^{17} + 784 q^{18} - 842 q^{19} - 64 q^{20} + 371 q^{21} + 154 q^{22} - 168 q^{23} + 16 q^{24} - 217 q^{25} + 154 q^{26} + 355 q^{27} - 168 q^{28} + 161 q^{29} + 28 q^{30} - 1104 q^{31} - 224 q^{32} + 1006 q^{33} + 214 q^{34} - 385 q^{35} - 392 q^{36} + 490 q^{37} + 612 q^{38} + 693 q^{39} + 264 q^{40} - 14 q^{41} - 1666 q^{42} - 238 q^{43} - 280 q^{44} - 2208 q^{45} - 630 q^{46} - 737 q^{47} + 32 q^{48} - 1575 q^{49} + 2016 q^{50} - 1498 q^{51} + 476 q^{52} - 525 q^{53} + 346 q^{54} - 145 q^{55} - 168 q^{56} + 2226 q^{57} - 1148 q^{58} + 1871 q^{59} + 644 q^{60} - 275 q^{61} - 150 q^{62} + 2044 q^{63} - 448 q^{64} + 868 q^{65} + 1102 q^{66} - 3766 q^{67} - 3128 q^{68} + 677 q^{69} + 1512 q^{70} + 4697 q^{71} - 392 q^{72} + 156 q^{73} + 1078 q^{74} + 6275 q^{75} - 8 q^{76} - 3843 q^{77} - 280 q^{78} + 2114 q^{79} - 928 q^{80} + 3948 q^{81} - 28 q^{82} - 1897 q^{83} + 392 q^{84} - 1267 q^{85} - 2338 q^{86} - 66 q^{87} - 560 q^{88} + 4982 q^{89} + 2934 q^{90} - 2681 q^{91} - 1260 q^{92} - 2975 q^{93} - 2706 q^{94} + 1113 q^{95} + 64 q^{96} - 784 q^{97} + 686 q^{98} - 14966 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −0.445042 + 1.94986i −5.20139 + 6.52234i −3.60388 − 1.73553i −1.22909 + 1.54123i −10.4028 − 13.0447i −9.20628 + 16.0700i 4.98792 − 6.25465i −9.47836 − 41.5274i −2.45818 − 3.08246i
15.2 −0.445042 + 1.94986i −3.63082 + 4.55291i −3.60388 − 1.73553i 12.3250 − 15.4550i −7.26165 − 9.10582i −4.74678 − 17.9016i 4.98792 − 6.25465i −1.53803 − 6.73857i 24.6499 + 30.9100i
15.3 −0.445042 + 1.94986i −3.36657 + 4.22154i −3.60388 − 1.73553i −13.2280 + 16.5873i −6.73314 − 8.44309i 10.2119 − 15.4505i 4.98792 − 6.25465i −0.479583 − 2.10119i −26.4559 − 33.1747i
15.4 −0.445042 + 1.94986i −0.145956 + 0.183024i −3.60388 − 1.73553i 2.48380 − 3.11459i −0.291913 − 0.366047i 16.9271 + 7.51489i 4.98792 − 6.25465i 5.99587 + 26.2696i 4.96761 + 6.22918i
15.5 −0.445042 + 1.94986i 3.11606 − 3.90742i −3.60388 − 1.73553i 0.568161 − 0.712452i 6.23212 + 7.81484i 0.964798 − 18.4951i 4.98792 − 6.25465i 0.449990 + 1.97154i 1.13632 + 1.42490i
15.6 −0.445042 + 1.94986i 3.18644 − 3.99567i −3.60388 − 1.73553i −8.89709 + 11.1566i 6.37287 + 7.99133i −18.5074 + 0.691270i 4.98792 − 6.25465i 0.196101 + 0.859176i −17.7942 − 22.3132i
15.7 −0.445042 + 1.94986i 5.26476 − 6.60180i −3.60388 − 1.73553i 5.40383 − 6.77619i 10.5295 + 13.2036i −1.27168 + 18.4765i 4.98792 − 6.25465i −9.85802 − 43.1908i 10.8077 + 13.5524i
29.1 −1.80194 − 0.867767i −2.15175 − 9.42743i 2.49396 + 3.12733i 4.34097 + 19.0190i −4.30350 + 18.8549i 6.19281 + 17.4542i −1.78017 − 7.79942i −59.9202 + 28.8561i 8.68193 − 38.0380i
29.2 −1.80194 − 0.867767i −1.25435 − 5.49565i 2.49396 + 3.12733i −2.29871 − 10.0713i −2.50869 + 10.9913i −18.3537 − 2.47852i −1.78017 − 7.79942i −4.30260 + 2.07203i −4.59742 + 20.1426i
29.3 −1.80194 − 0.867767i −0.794208 − 3.47965i 2.49396 + 3.12733i 0.315483 + 1.38222i −1.58842 + 6.95931i 15.0023 − 10.8596i −1.78017 − 7.79942i 12.8489 − 6.18772i 0.630966 − 2.76444i
29.4 −1.80194 − 0.867767i −0.114684 − 0.502464i 2.49396 + 3.12733i −0.679639 − 2.97769i −0.229368 + 1.00493i 1.03833 + 18.4911i −1.78017 − 7.79942i 24.0868 − 11.5996i −1.35928 + 5.95538i
29.5 −1.80194 − 0.867767i 1.06874 + 4.68247i 2.49396 + 3.12733i 4.02646 + 17.6411i 2.13749 − 9.36494i −18.4086 − 2.03062i −1.78017 − 7.79942i 3.54285 − 1.70614i 8.05292 − 35.2822i
29.6 −1.80194 − 0.867767i 1.29034 + 5.65335i 2.49396 + 3.12733i −2.60649 − 11.4198i 2.58068 − 11.3067i −0.111382 − 18.5199i −1.78017 − 7.79942i −5.96920 + 2.87462i −5.21299 + 22.8396i
29.7 −1.80194 − 0.867767i 1.85687 + 8.13549i 2.49396 + 3.12733i 1.43996 + 6.30889i 3.71375 − 16.2710i 17.4802 + 6.11906i −1.78017 − 7.79942i −38.4121 + 18.4983i 2.87993 − 12.6178i
43.1 1.24698 − 1.56366i −7.90239 − 3.80559i −0.890084 − 3.89971i −9.01214 − 4.34002i −15.8048 + 7.61118i 18.2960 + 2.87351i −7.20775 − 3.47107i 31.1310 + 39.0371i −18.0243 + 8.68004i
43.2 1.24698 − 1.56366i −4.67537 − 2.25154i −0.890084 − 3.89971i 1.93905 + 0.933796i −9.35073 + 4.50308i −18.1813 + 3.52695i −7.20775 − 3.47107i −0.0445982 − 0.0559244i 3.87809 − 1.86759i
43.3 1.24698 − 1.56366i −4.11853 − 1.98338i −0.890084 − 3.89971i 19.6858 + 9.48017i −8.23706 + 3.96676i 4.51381 − 17.9618i −7.20775 − 3.47107i −3.80574 − 4.77225i 39.3715 − 18.9603i
43.4 1.24698 − 1.56366i 0.463878 + 0.223392i −0.890084 − 3.89971i −12.5797 − 6.05807i 0.927756 − 0.446784i −15.5361 + 10.0812i −7.20775 − 3.47107i −16.6689 − 20.9022i −25.1594 + 12.1161i
43.5 1.24698 − 1.56366i 1.32303 + 0.637137i −0.890084 − 3.89971i −8.65063 − 4.16592i 2.64606 − 1.27427i 9.07048 − 16.1470i −7.20775 − 3.47107i −15.4898 − 19.4235i −17.3013 + 8.33185i
43.6 1.24698 − 1.56366i 4.91348 + 2.36621i −0.890084 − 3.89971i 8.86074 + 4.26711i 9.82696 − 4.73242i 7.93934 + 16.7322i −7.20775 − 3.47107i 1.70914 + 2.14320i 17.7215 − 8.53422i
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 98.4.e.a ✓ 42
49.e even 7 1 inner 98.4.e.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.e.a ✓ 42 1.a even 1 1 trivial
98.4.e.a ✓ 42 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{42} + 5 T_{3}^{41} + 156 T_{3}^{40} + 584 T_{3}^{39} + 13901 T_{3}^{38} + 61980 T_{3}^{37} + \cdots + 60\!\cdots\!49 \) acting on \(S_{4}^{\mathrm{new}}(98, [\chi])\). Copy content Toggle raw display