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Properties

Label 400.2.bl.a
Level $400$
Weight $2$
Character orbit 400.bl
Analytic conductor $3.194$
Analytic rank $0$
Dimension $464$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,2,Mod(29,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 15, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.bl (of order \(20\), degree \(8\), 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(464\)
Relative dimension: \(58\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 464 q - 10 q^{2} - 10 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} - 40 q^{8} - 2 q^{10} - 6 q^{11} - 10 q^{12} - 10 q^{13} - 18 q^{14} - 16 q^{15} - 6 q^{16} - 20 q^{17} - 6 q^{19} + 6 q^{20} - 24 q^{21} - 10 q^{22} - 16 q^{24} - 36 q^{26} - 10 q^{27} - 50 q^{28} - 6 q^{29} + 14 q^{30} - 36 q^{31} - 20 q^{33} - 46 q^{34} - 36 q^{35} + 38 q^{36} - 10 q^{37} + 60 q^{38} - 84 q^{40} - 70 q^{42} + 36 q^{44} - 24 q^{45} + 2 q^{46} - 20 q^{47} - 140 q^{48} + 336 q^{49} - 46 q^{50} - 28 q^{51} + 80 q^{52} - 10 q^{53} - 30 q^{54} - 48 q^{56} - 10 q^{58} - 6 q^{59} + 112 q^{60} - 6 q^{61} - 10 q^{62} - 160 q^{63} - 24 q^{64} - 48 q^{65} - 104 q^{66} - 70 q^{67} + 6 q^{69} - 100 q^{70} - 10 q^{72} + 28 q^{74} - 46 q^{75} + 44 q^{76} - 80 q^{77} - 10 q^{78} - 52 q^{79} - 2 q^{80} + 72 q^{81} - 10 q^{83} + 28 q^{84} - 18 q^{85} + 34 q^{86} + 100 q^{88} + 74 q^{90} - 48 q^{91} + 80 q^{92} - 6 q^{94} - 40 q^{95} + 54 q^{96} - 20 q^{97} - 100 q^{98} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41418 − 0.0103712i 1.06269 − 0.168314i 1.99978 + 0.0293334i −2.11616 + 0.722412i −1.50458 + 0.227004i −0.232144 −2.82774 − 0.0622227i −1.75218 + 0.569319i 3.00011 − 0.999671i
29.2 −1.41356 − 0.0431277i −3.09356 + 0.489972i 1.99628 + 0.121927i −2.23084 + 0.152807i 4.39405 − 0.559185i 2.16271 −2.81659 − 0.258445i 6.47689 − 2.10447i 3.16001 − 0.119790i
29.3 −1.41316 − 0.0546657i 1.83259 − 0.290254i 1.99402 + 0.154502i 0.995469 + 2.00226i −2.60560 + 0.309994i −0.288575 −2.80942 − 0.327341i 0.420967 − 0.136781i −1.29730 − 2.88392i
29.4 −1.39795 + 0.213865i 3.11943 − 0.494069i 1.90852 − 0.597945i 1.43675 − 1.71340i −4.25514 + 1.35782i −3.50228 −2.54014 + 1.24406i 6.63355 − 2.15537i −1.64207 + 2.70252i
29.5 −1.38613 − 0.280454i −0.894740 + 0.141713i 1.84269 + 0.777490i 0.571921 − 2.16169i 1.27997 + 0.0545018i 4.11279 −2.33615 − 1.59449i −2.07269 + 0.673458i −1.39901 + 2.83598i
29.6 −1.37753 + 0.320006i −0.593985 + 0.0940780i 1.79519 − 0.881639i −0.779471 − 2.09581i 0.788128 − 0.319675i −1.40656 −2.19081 + 1.78896i −2.50920 + 0.815289i 1.74442 + 2.63761i
29.7 −1.35280 − 0.412237i −1.98405 + 0.314242i 1.66012 + 1.11535i −0.149735 + 2.23105i 2.81356 + 0.392793i −4.59495 −1.78602 − 2.19320i 0.984533 − 0.319894i 1.12228 − 2.95643i
29.8 −1.29143 + 0.576366i −0.752396 + 0.119168i 1.33560 − 1.48868i 0.231849 + 2.22402i 0.902985 − 0.587553i 2.56554 −0.866820 + 2.69233i −2.30127 + 0.747728i −1.58127 − 2.73854i
29.9 −1.27798 + 0.605618i −2.41193 + 0.382013i 1.26645 − 1.54793i 2.20655 − 0.362152i 2.85104 − 1.94891i −1.42770 −0.681042 + 2.74521i 2.81831 − 0.915726i −2.60059 + 1.79915i
29.10 −1.26619 − 0.629895i 2.62418 − 0.415629i 1.20646 + 1.59513i −0.886089 − 2.05301i −3.58451 − 1.12669i 3.34726 −0.522846 − 2.77968i 3.86039 − 1.25432i −0.171225 + 3.15764i
29.11 −1.21289 − 0.727254i −0.266037 + 0.0421361i 0.942204 + 1.76416i 1.58641 − 1.57585i 0.353317 + 0.142370i −4.44042 0.140201 − 2.82495i −2.78417 + 0.904631i −3.07018 + 0.757610i
29.12 −1.09755 + 0.891847i 1.78947 − 0.283424i 0.409216 − 1.95769i 2.13879 − 0.652354i −1.71125 + 1.90700i 4.96308 1.29682 + 2.51361i 0.268690 − 0.0873027i −1.76562 + 2.62347i
29.13 −1.09639 + 0.893266i 2.76199 − 0.437457i 0.404150 − 1.95874i −1.79261 + 1.33662i −2.63746 + 2.94682i 0.291269 1.30657 + 2.50856i 4.58407 − 1.48945i 0.771442 − 3.06674i
29.14 −1.08196 − 0.910696i 1.78850 − 0.283271i 0.341265 + 1.97067i 2.01441 + 0.970637i −2.19306 − 1.32229i 1.01538 1.42545 − 2.44297i 0.265328 − 0.0862102i −1.29556 − 2.88471i
29.15 −1.01647 − 0.983253i −0.224268 + 0.0355205i 0.0664253 + 1.99890i −1.20871 + 1.88123i 0.262887 + 0.184406i 2.48678 1.89790 − 2.09713i −2.80414 + 0.911119i 3.07834 − 0.723748i
29.16 −1.01050 + 0.989384i 0.847034 − 0.134157i 0.0422379 − 1.99955i −1.16539 − 1.90836i −0.723199 + 0.973608i −1.50111 1.93565 + 2.06235i −2.15370 + 0.699780i 3.06574 + 0.775388i
29.17 −0.919307 − 1.07465i −1.25267 + 0.198404i −0.309751 + 1.97587i −2.17568 − 0.516161i 1.36480 + 1.16379i −0.199333 2.40813 − 1.48355i −1.32335 + 0.429981i 1.44542 + 2.81261i
29.18 −0.919044 − 1.07488i −3.33287 + 0.527874i −0.310718 + 1.97572i 0.778461 − 2.09619i 3.63045 + 3.09728i −0.463968 2.40921 − 1.48179i 7.97618 − 2.59162i −2.96858 + 1.08974i
29.19 −0.842684 + 1.13573i −2.41612 + 0.382676i −0.579767 − 1.91412i −2.22343 + 0.237412i 1.60141 − 3.06654i −1.53357 2.66249 + 0.954542i 2.83804 − 0.922135i 1.60401 − 2.72528i
29.20 −0.667125 + 1.24697i 0.284389 − 0.0450428i −1.10989 − 1.66378i 2.23360 − 0.104931i −0.133556 + 0.384675i −1.82171 2.81512 − 0.274055i −2.77432 + 0.901432i −1.35925 + 2.85525i
See next 80 embeddings (of 464 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
25.e even 10 1 inner
400.bl even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.bl.a ✓ 464
16.e even 4 1 inner 400.2.bl.a ✓ 464
25.e even 10 1 inner 400.2.bl.a ✓ 464
400.bl even 20 1 inner 400.2.bl.a ✓ 464
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.bl.a ✓ 464 1.a even 1 1 trivial
400.2.bl.a ✓ 464 16.e even 4 1 inner
400.2.bl.a ✓ 464 25.e even 10 1 inner
400.2.bl.a ✓ 464 400.bl even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(400, [\chi])\).