Properties

Label 650.2.y.a
Level $650$
Weight $2$
Character orbit 650.y
Analytic conductor $5.190$
Analytic rank $0$
Dimension $136$
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] = [650,2,Mod(61,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([24, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.y (of order \(15\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(17\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 17 q^{2} + 17 q^{4} - 2 q^{5} + 16 q^{7} + 34 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 17 q^{2} + 17 q^{4} - 2 q^{5} + 16 q^{7} + 34 q^{8} + 9 q^{9} - q^{10} - 8 q^{13} - 8 q^{14} - 16 q^{15} + 17 q^{16} + 4 q^{17} - 112 q^{18} - 21 q^{19} - 4 q^{20} + 12 q^{21} - 10 q^{23} - 26 q^{25} + 16 q^{26} + 6 q^{27} - 4 q^{28} + 4 q^{29} - 4 q^{30} + 68 q^{32} - 38 q^{33} - 12 q^{34} - 21 q^{35} + 9 q^{36} - 4 q^{37} + 28 q^{38} - 40 q^{39} + 2 q^{40} + 23 q^{41} + 6 q^{42} + 6 q^{43} - 29 q^{45} - 10 q^{46} - 18 q^{47} - 44 q^{49} + 7 q^{50} + 136 q^{51} + 4 q^{52} + 62 q^{53} - 12 q^{54} + 12 q^{55} + 4 q^{56} - 104 q^{57} - 4 q^{58} - 10 q^{59} - 8 q^{60} + 8 q^{61} + 18 q^{63} - 34 q^{64} + 6 q^{65} + 4 q^{66} - 10 q^{67} + 4 q^{68} + 59 q^{69} - 72 q^{70} + 19 q^{71} - 19 q^{72} + 44 q^{73} - 56 q^{74} - 40 q^{75} + 14 q^{76} - 60 q^{77} + 38 q^{78} - 8 q^{79} + q^{80} + 91 q^{81} + 32 q^{82} - 66 q^{83} + 9 q^{84} + 27 q^{85} - 8 q^{86} - 40 q^{87} + 53 q^{89} + 82 q^{90} + 22 q^{91} - 20 q^{92} + 70 q^{93} - 9 q^{94} + 39 q^{95} - 50 q^{97} - q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 0.104528 + 0.994522i −1.84531 2.04942i −0.978148 + 0.207912i −1.97854 + 1.04182i 1.84531 2.04942i 1.12184 + 1.94308i −0.309017 0.951057i −0.481384 + 4.58006i −1.24292 1.85880i
61.2 0.104528 + 0.994522i −1.80469 2.00431i −0.978148 + 0.207912i 0.181318 2.22870i 1.80469 2.00431i 0.411646 + 0.712993i −0.309017 0.951057i −0.446769 + 4.25072i 2.23545 0.0526380i
61.3 0.104528 + 0.994522i −1.69496 1.88244i −0.978148 + 0.207912i 1.93624 + 1.11847i 1.69496 1.88244i 1.74958 + 3.03037i −0.309017 0.951057i −0.357118 + 3.39775i −0.909955 + 2.04254i
61.4 0.104528 + 0.994522i −1.44396 1.60367i −0.978148 + 0.207912i −2.15698 + 0.589430i 1.44396 1.60367i −1.31241 2.27315i −0.309017 0.951057i −0.173181 + 1.64770i −0.811667 2.08355i
61.5 0.104528 + 0.994522i −1.16207 1.29060i −0.978148 + 0.207912i 2.15683 0.589990i 1.16207 1.29060i −2.20900 3.82609i −0.309017 0.951057i −0.00167837 + 0.0159687i 0.812208 + 2.08334i
61.6 0.104528 + 0.994522i −0.818230 0.908737i −0.978148 + 0.207912i 0.204628 2.22669i 0.818230 0.908737i 0.874706 + 1.51504i −0.309017 0.951057i 0.157284 1.49645i 2.23588 0.0292447i
61.7 0.104528 + 0.994522i −0.734834 0.816116i −0.978148 + 0.207912i 0.210056 + 2.22618i 0.734834 0.816116i 0.371959 + 0.644252i −0.309017 0.951057i 0.187521 1.78415i −2.19203 + 0.441605i
61.8 0.104528 + 0.994522i −0.0628260 0.0697754i −0.978148 + 0.207912i −0.479060 + 2.18415i 0.0628260 0.0697754i −1.01992 1.76655i −0.309017 0.951057i 0.312664 2.97480i −2.22226 0.248130i
61.9 0.104528 + 0.994522i 0.159164 + 0.176770i −0.978148 + 0.207912i −2.06983 0.846046i −0.159164 + 0.176770i 1.07880 + 1.86853i −0.309017 0.951057i 0.307671 2.92729i 0.625055 2.14693i
61.10 0.104528 + 0.994522i 0.242087 + 0.268865i −0.978148 + 0.207912i 2.22771 0.193159i −0.242087 + 0.268865i −1.08600 1.88101i −0.309017 0.951057i 0.299903 2.85339i 0.424960 + 2.19532i
61.11 0.104528 + 0.994522i 0.464735 + 0.516140i −0.978148 + 0.207912i 1.57630 1.58596i −0.464735 + 0.516140i 0.0784084 + 0.135807i −0.309017 0.951057i 0.263163 2.50383i 1.74204 + 1.40189i
61.12 0.104528 + 0.994522i 0.915893 + 1.01720i −0.978148 + 0.207912i −0.917899 + 2.03899i −0.915893 + 1.01720i 2.47815 + 4.29228i −0.309017 0.951057i 0.117745 1.12027i −2.12376 0.699739i
61.13 0.104528 + 0.994522i 1.16060 + 1.28897i −0.978148 + 0.207912i 2.00803 + 0.983773i −1.16060 + 1.28897i 1.86603 + 3.23205i −0.309017 0.951057i −0.000881936 0.00839106i −0.768487 + 2.09986i
61.14 0.104528 + 0.994522i 1.22272 + 1.35797i −0.978148 + 0.207912i −0.743561 2.10882i −1.22272 + 1.35797i 0.434351 + 0.752319i −0.309017 0.951057i −0.0354483 + 0.337268i 2.01954 0.959920i
61.15 0.104528 + 0.994522i 1.43374 + 1.59233i −0.978148 + 0.207912i −0.730288 2.11345i −1.43374 + 1.59233i −2.26305 3.91972i −0.309017 0.951057i −0.166320 + 1.58243i 2.02554 0.947203i
61.16 0.104528 + 0.994522i 1.91593 + 2.12785i −0.978148 + 0.207912i 1.01743 + 1.99119i −1.91593 + 2.12785i −0.958044 1.65938i −0.309017 0.951057i −0.543394 + 5.17005i −1.87393 + 1.22000i
61.17 0.104528 + 0.994522i 2.05200 + 2.27897i −0.978148 + 0.207912i −2.13337 + 0.669877i −2.05200 + 2.27897i 0.382954 + 0.663295i −0.309017 0.951057i −0.669443 + 6.36932i −0.889205 2.05166i
81.1 −0.669131 + 0.743145i −2.81704 1.25423i −0.104528 0.994522i −2.21687 + 0.292359i 2.81704 1.25423i 0.737278 1.27700i 0.809017 + 0.587785i 4.35524 + 4.83699i 1.26611 1.84308i
81.2 −0.669131 + 0.743145i −2.65692 1.18294i −0.104528 0.994522i 1.90774 1.16642i 2.65692 1.18294i −0.454802 + 0.787741i 0.809017 + 0.587785i 3.65248 + 4.05649i −0.409703 + 2.19821i
81.3 −0.669131 + 0.743145i −2.46839 1.09900i −0.104528 0.994522i 1.53068 + 1.63004i 2.46839 1.09900i 0.489810 0.848375i 0.809017 + 0.587785i 2.87775 + 3.19607i −2.23558 + 0.0468064i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
25.d even 5 1 inner
325.y even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.y.a 136
13.c even 3 1 inner 650.2.y.a 136
25.d even 5 1 inner 650.2.y.a 136
325.y even 15 1 inner 650.2.y.a 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.y.a 136 1.a even 1 1 trivial
650.2.y.a 136 13.c even 3 1 inner
650.2.y.a 136 25.d even 5 1 inner
650.2.y.a 136 325.y even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{136} - 30 T_{3}^{134} - 2 T_{3}^{133} + 308 T_{3}^{132} + 148 T_{3}^{131} + 1527 T_{3}^{130} - 3375 T_{3}^{129} - 74534 T_{3}^{128} + 34148 T_{3}^{127} + 834026 T_{3}^{126} + 19162 T_{3}^{125} - 2388757 T_{3}^{124} + \cdots + 15\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display