↑

Properties

Label 825.2.m.c
Level $825$
Weight $2$
Character orbit 825.m
Analytic conductor $6.588$
Analytic rank $0$
Dimension $116$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(16,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.m (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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(29\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 2 q^{2} - 29 q^{3} - 30 q^{4} - q^{5} - 2 q^{6} - 13 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q - 2 q^{2} - 29 q^{3} - 30 q^{4} - q^{5} - 2 q^{6} - 13 q^{7} - 6 q^{8} - 29 q^{9} + 6 q^{10} - 6 q^{11} - 30 q^{12} - 8 q^{13} - 4 q^{14} - q^{15} - 20 q^{16} - 8 q^{17} - 2 q^{18} - 23 q^{20} - 8 q^{21} - 19 q^{22} + q^{23} - 6 q^{24} + 35 q^{25} - 32 q^{26} - 29 q^{27} + 62 q^{28} + 24 q^{29} - 14 q^{30} - 10 q^{31} + 36 q^{32} + 9 q^{33} - 16 q^{34} - 21 q^{35} - 30 q^{36} - 41 q^{37} - 40 q^{38} - 8 q^{39} - q^{40} - 22 q^{41} + 6 q^{42} + 58 q^{43} - 15 q^{44} - q^{45} - 7 q^{47} + 80 q^{48} - 66 q^{49} + 31 q^{50} - 8 q^{51} - 28 q^{52} + 17 q^{53} - 2 q^{54} - 85 q^{55} - 13 q^{56} - 15 q^{57} + q^{58} + 12 q^{60} - 26 q^{61} - 8 q^{62} + 42 q^{63} - 30 q^{64} - 14 q^{65} + 11 q^{66} - 19 q^{67} + 10 q^{68} + q^{69} + 7 q^{70} - 44 q^{71} + 24 q^{72} + 114 q^{73} + 58 q^{74} - 15 q^{75} + 56 q^{76} + 111 q^{77} - 7 q^{78} - 83 q^{79} + 12 q^{80} - 29 q^{81} - 14 q^{82} - 8 q^{83} - 18 q^{84} - 58 q^{85} - 82 q^{86} + 24 q^{87} - 8 q^{88} + 3 q^{89} - 4 q^{90} - 102 q^{91} - 7 q^{92} + 30 q^{93} - 79 q^{94} + 36 q^{95} - 9 q^{96} - q^{97} - 96 q^{98} - 11 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −0.804956 + 2.47740i 0.309017 − 0.951057i −3.87152 − 2.81282i −2.10387 − 0.757453i 2.10740 + 1.53112i −0.773803 + 0.562201i 5.87008 − 4.26487i −0.809017 − 0.587785i 3.57004 − 4.60241i
16.2 −0.792144 + 2.43797i 0.309017 − 0.951057i −3.69817 − 2.68688i −0.246863 + 2.22240i 2.07386 + 1.50675i −1.46145 + 1.06181i 5.33228 − 3.87413i −0.809017 − 0.587785i −5.22259 − 2.36230i
16.3 −0.791764 + 2.43680i 0.309017 − 0.951057i −3.69306 − 2.68317i 1.86074 − 1.24001i 2.07286 + 1.50602i −2.43303 + 1.76770i 5.31664 − 3.86276i −0.809017 − 0.587785i 1.54839 + 5.51606i
16.4 −0.646651 + 1.99019i 0.309017 − 0.951057i −1.92465 − 1.39834i 1.32708 − 1.79969i 1.69295 + 1.23000i 1.12591 − 0.818024i 0.641630 − 0.466172i −0.809017 − 0.587785i 2.72356 + 3.80490i
16.5 −0.609491 + 1.87582i 0.309017 − 0.951057i −1.52919 − 1.11102i 2.01010 + 0.979547i 1.59567 + 1.15932i 3.12033 − 2.26705i −0.175224 + 0.127308i −0.809017 − 0.587785i −3.06259 + 3.17356i
16.6 −0.580354 + 1.78615i 0.309017 − 0.951057i −1.23548 − 0.897627i −2.06744 − 0.851887i 1.51939 + 1.10390i 1.57692 − 1.14570i −0.718467 + 0.521997i −0.809017 − 0.587785i 2.72144 − 3.19835i
16.7 −0.537784 + 1.65513i 0.309017 − 0.951057i −0.832208 − 0.604634i 1.64516 + 1.51442i 1.40794 + 1.02293i −2.49392 + 1.81194i −1.36758 + 0.993604i −0.809017 − 0.587785i −3.39129 + 1.90852i
16.8 −0.470563 + 1.44824i 0.309017 − 0.951057i −0.257948 − 0.187410i −0.902458 − 2.04587i 1.23195 + 0.895064i 0.190008 − 0.138049i −2.07110 + 1.50474i −0.809017 − 0.587785i 3.38758 − 0.344270i
16.9 −0.365237 + 1.12408i 0.309017 − 0.951057i 0.487870 + 0.354458i −0.965326 + 2.01696i 0.956202 + 0.694721i −0.249732 + 0.181441i −2.48903 + 1.80839i −0.809017 − 0.587785i −1.91466 − 1.82178i
16.10 −0.353970 + 1.08941i 0.309017 − 0.951057i 0.556520 + 0.404335i −0.00714460 + 2.23606i 0.926705 + 0.673291i 1.88201 − 1.36736i −2.49089 + 1.80974i −0.809017 − 0.587785i −2.43345 − 0.799280i
16.11 −0.233387 + 0.718290i 0.309017 − 0.951057i 1.15656 + 0.840292i −2.15772 + 0.586707i 0.611014 + 0.443928i −2.11047 + 1.53334i −2.09553 + 1.52249i −0.809017 − 0.587785i 0.0821578 − 1.68680i
16.12 −0.232438 + 0.715372i 0.309017 − 0.951057i 1.16031 + 0.843011i −0.526666 − 2.17316i 0.608531 + 0.442124i −3.40758 + 2.47575i −2.08983 + 1.51835i −0.809017 − 0.587785i 1.67703 + 0.128364i
16.13 −0.0969709 + 0.298446i 0.309017 − 0.951057i 1.53837 + 1.11769i −1.79296 − 1.33614i 0.253873 + 0.184450i 3.15086 − 2.28924i −0.990493 + 0.719635i −0.809017 − 0.587785i 0.572632 − 0.405535i
16.14 −0.0943615 + 0.290415i 0.309017 − 0.951057i 1.54260 + 1.12076i 2.12707 − 0.689623i 0.247042 + 0.179486i −0.125177 + 0.0909464i −0.965131 + 0.701209i −0.809017 − 0.587785i −0.000436476 0.682806i
16.15 −0.0820556 + 0.252541i 0.309017 − 0.951057i 1.56099 + 1.13413i 2.15157 + 0.608873i 0.214824 + 0.156079i −3.12161 + 2.26799i −0.844150 + 0.613311i −0.809017 − 0.587785i −0.330314 + 0.493400i
16.16 0.145110 − 0.446603i 0.309017 − 0.951057i 1.43964 + 1.04596i −2.05557 + 0.880141i −0.379903 − 0.276016i 0.279206 − 0.202855i 1.43584 − 1.04320i −0.809017 − 0.587785i 0.0947900 + 1.04574i
16.17 0.207541 − 0.638745i 0.309017 − 0.951057i 1.25311 + 0.910439i −1.99371 + 1.01248i −0.543349 − 0.394766i −2.55400 + 1.85559i 1.92831 − 1.40100i −0.809017 − 0.587785i 0.232938 + 1.48360i
16.18 0.293245 − 0.902515i 0.309017 − 0.951057i 0.889493 + 0.646255i 1.63470 + 1.52570i −0.767725 − 0.557785i 2.30889 − 1.67750i 2.37954 − 1.72884i −0.809017 − 0.587785i 1.85633 − 1.02793i
16.19 0.343229 − 1.05635i 0.309017 − 0.951057i 0.619967 + 0.450432i 1.19249 − 1.89155i −0.898584 − 0.652859i 0.429431 − 0.312000i 2.48577 − 1.80602i −0.809017 − 0.587785i −1.58885 − 1.90892i
16.20 0.412444 − 1.26937i 0.309017 − 0.951057i 0.176842 + 0.128483i 2.16539 − 0.557739i −1.07979 − 0.784514i −1.15722 + 0.840771i 2.39561 − 1.74051i −0.809017 − 0.587785i 0.185125 − 2.97872i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.g even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.m.c ✓ 116
11.c even 5 1 825.2.o.c yes 116
25.d even 5 1 825.2.o.c yes 116
275.g even 5 1 inner 825.2.m.c ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.m.c ✓ 116 1.a even 1 1 trivial
825.2.m.c ✓ 116 275.g even 5 1 inner
825.2.o.c yes 116 11.c even 5 1
825.2.o.c yes 116 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} + 2 T_{2}^{115} + 46 T_{2}^{114} + 94 T_{2}^{113} + 1174 T_{2}^{112} + 2348 T_{2}^{111} + \cdots + 36019202560000 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display