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Properties

Label 425.2.k.c
Level $425$
Weight $2$
Character orbit 425.k
Analytic conductor $3.394$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(86,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.86");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.k (of order \(5\), degree \(4\), 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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 2 q^{2} + 2 q^{3} - 20 q^{4} - 17 q^{6} - 44 q^{7} + 15 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 2 q^{2} + 2 q^{3} - 20 q^{4} - 17 q^{6} - 44 q^{7} + 15 q^{8} - 22 q^{9} - 2 q^{10} + 4 q^{11} + 14 q^{12} + 6 q^{13} - 10 q^{14} + 14 q^{15} - 32 q^{16} + 20 q^{17} - 62 q^{18} - 3 q^{19} + 16 q^{21} + 17 q^{22} - 5 q^{23} + 30 q^{24} - 14 q^{25} + 18 q^{26} + 17 q^{27} + 32 q^{28} + 20 q^{29} - 114 q^{30} - 21 q^{31} - 88 q^{32} - 2 q^{34} - 2 q^{35} + 47 q^{36} + 6 q^{37} + 22 q^{38} - 46 q^{39} + 59 q^{40} - 26 q^{41} - 3 q^{42} - 102 q^{43} - 44 q^{44} + 74 q^{45} + 2 q^{46} + 2 q^{47} + 72 q^{48} + 108 q^{49} + 7 q^{50} + 18 q^{51} + 65 q^{52} - 18 q^{53} - 29 q^{54} - 49 q^{55} - 117 q^{56} + 2 q^{57} + 42 q^{58} + 21 q^{59} + 112 q^{60} + 4 q^{61} + 15 q^{62} + 41 q^{63} - 11 q^{64} + 8 q^{65} - 156 q^{66} + 21 q^{67} - 80 q^{68} - 67 q^{69} + 32 q^{70} - 4 q^{71} + 83 q^{72} - 2 q^{73} + 78 q^{74} - 68 q^{75} + 16 q^{76} + 2 q^{77} - 2 q^{78} - 53 q^{79} - 54 q^{80} - 21 q^{81} - 72 q^{82} - 42 q^{83} - 52 q^{84} + 42 q^{86} - 8 q^{87} + 68 q^{88} - 73 q^{89} + 97 q^{90} - 83 q^{91} + 117 q^{92} - 120 q^{93} + 80 q^{94} + 16 q^{95} + 88 q^{96} + 57 q^{97} + 17 q^{98} + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −2.14699 + 1.55988i −0.173809 + 0.534928i 1.55831 − 4.79598i 0.731696 − 2.11297i −0.461258 − 1.41961i −2.73519 2.49532 + 7.67982i 2.17111 + 1.57741i 1.72503 + 5.67787i
86.2 −1.99102 + 1.44656i −0.829805 + 2.55388i 1.25360 − 3.85817i 2.22585 + 0.213537i −2.04218 − 6.28519i 1.62088 1.56414 + 4.81394i −3.40666 − 2.47508i −4.74061 + 2.79467i
86.3 −1.72275 + 1.25165i −0.0844546 + 0.259925i 0.783211 − 2.41047i −2.13220 − 0.673604i −0.179841 − 0.553494i 0.609651 0.351734 + 1.08253i 2.36662 + 1.71945i 4.51637 − 1.50832i
86.4 −1.57449 + 1.14393i 0.469321 − 1.44442i 0.552396 − 1.70010i 1.75698 + 1.38313i 0.913380 + 2.81109i −0.139793 −0.127742 − 0.393149i 0.560961 + 0.407562i −4.34854 − 0.167852i
86.5 −1.52182 + 1.10567i 0.721199 − 2.21962i 0.475405 − 1.46315i −0.865877 + 2.06162i 1.35663 + 4.17527i −1.25766 −0.268296 − 0.825730i −1.97954 − 1.43822i −0.961753 − 4.09478i
86.6 −0.993656 + 0.721933i −0.866619 + 2.66718i −0.151870 + 0.467407i −0.180116 − 2.22880i −1.06440 − 3.27590i 1.50469 −0.945616 − 2.91031i −3.93577 − 2.85950i 1.78802 + 2.08463i
86.7 −0.762747 + 0.554168i −0.324602 + 0.999022i −0.343353 + 1.05673i −1.44286 + 1.70826i −0.306037 − 0.941886i −2.98759 −0.906403 − 2.78962i 1.53437 + 1.11479i 0.153870 − 2.10256i
86.8 −0.422163 + 0.306719i 0.519515 − 1.59890i −0.533889 + 1.64314i 0.360883 − 2.20675i 0.271094 + 0.834342i 3.01247 −0.601098 − 1.84999i 0.140459 + 0.102050i 0.524502 + 1.04230i
86.9 −0.135768 + 0.0986414i −0.267848 + 0.824353i −0.609331 + 1.87533i 1.96067 + 1.07508i −0.0449500 − 0.138342i 2.33732 −0.205975 − 0.633926i 1.81924 + 1.32175i −0.372243 + 0.0474416i
86.10 −0.0822620 + 0.0597668i −0.947924 + 2.91741i −0.614839 + 1.89228i 1.94015 − 1.11167i −0.0963862 − 0.296646i −3.50410 −0.125360 − 0.385819i −5.18567 − 3.76761i −0.0931599 + 0.207405i
86.11 0.364820 − 0.265058i −0.632459 + 1.94651i −0.555196 + 1.70872i −2.23395 + 0.0971993i 0.285203 + 0.877764i 4.23898 0.529060 + 1.62828i −0.961836 − 0.698814i −0.789229 + 0.627587i
86.12 0.419644 − 0.304889i 1.00213 − 3.08423i −0.534891 + 1.64622i −2.22841 + 0.184895i −0.519812 − 1.59982i −3.33055 0.598031 + 1.84055i −6.08119 − 4.41824i −0.878766 + 0.757007i
86.13 0.545657 − 0.396443i 0.335346 − 1.03209i −0.477459 + 1.46947i 1.35235 + 1.78077i −0.226180 − 0.696112i −5.02070 0.738877 + 2.27403i 1.47430 + 1.07114i 1.44389 + 0.435563i
86.14 1.02591 − 0.745371i 0.735912 − 2.26490i −0.121110 + 0.372740i 1.39980 − 1.74372i −0.933210 − 2.87212i −0.795166 0.937309 + 2.88474i −2.16117 − 1.57018i 0.136361 − 2.83228i
86.15 1.15223 − 0.837145i −0.390637 + 1.20226i 0.00879129 − 0.0270568i −0.580418 − 2.15942i 0.556359 + 1.71230i 0.714392 0.867706 + 2.67052i 1.13423 + 0.824064i −2.47653 − 2.00226i
86.16 1.58361 − 1.15056i 0.0586727 − 0.180576i 0.565999 − 1.74196i 0.00528892 + 2.23606i −0.114849 − 0.353468i 2.20909 0.101857 + 0.313482i 2.39789 + 1.74217i 2.58110 + 3.53497i
86.17 1.64574 − 1.19570i −0.929019 + 2.85923i 0.660733 − 2.03353i −2.22620 − 0.209856i 1.88986 + 5.81638i −4.33298 −0.0868601 − 0.267328i −4.88505 − 3.54920i −3.91468 + 2.31650i
86.18 2.03533 − 1.47876i 0.0964356 − 0.296798i 1.33783 − 4.11742i 1.48564 − 1.67119i −0.242614 − 0.746689i −1.80284 −1.81087 − 5.57328i 2.34826 + 1.70611i 0.552492 − 5.59833i
86.19 2.08118 − 1.51206i 0.601438 − 1.85103i 1.42693 − 4.39163i −1.88021 + 1.21030i −1.54718 − 4.76174i −1.15333 −2.08086 − 6.40424i −0.637551 − 0.463208i −2.08299 + 5.36184i
86.20 2.11757 − 1.53851i −0.828858 + 2.55096i 1.49908 − 4.61370i 0.550931 + 2.16714i 2.16951 + 6.67705i 2.04850 −2.30611 − 7.09748i −3.39335 − 2.46541i 4.50079 + 3.74146i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.k.c ✓ 80
25.d even 5 1 inner 425.2.k.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.k.c ✓ 80 1.a even 1 1 trivial
425.2.k.c ✓ 80 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 2 T_{2}^{79} + 32 T_{2}^{78} - 69 T_{2}^{77} + 604 T_{2}^{76} - 1244 T_{2}^{75} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display