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Properties

Label 350.2.q.b
Level $350$
Weight $2$
Character orbit 350.q
Analytic conductor $2.795$
Analytic rank $0$
Dimension $72$
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] = [350,2,Mod(11,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, 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("350.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.q (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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 9 q^{2} + q^{3} + 9 q^{4} + 2 q^{6} + 8 q^{7} + 18 q^{8} + 20 q^{9} - 10 q^{10} - 12 q^{11} - 4 q^{12} - 8 q^{13} + 11 q^{14} + 16 q^{15} + 9 q^{16} - 30 q^{17} + 50 q^{18} + 2 q^{19} + 10 q^{20} - 20 q^{21} + 26 q^{22} - 10 q^{23} - 6 q^{24} - 8 q^{25} - 14 q^{26} + 46 q^{27} - 2 q^{28} - 10 q^{29} - 12 q^{30} + 9 q^{31} + 36 q^{32} - 13 q^{33} + 20 q^{34} + 6 q^{35} - 40 q^{36} + 11 q^{37} - 12 q^{38} - 27 q^{39} + 5 q^{40} - 34 q^{41} + 2 q^{42} - 32 q^{43} + 13 q^{44} - 7 q^{45} - 15 q^{46} + 8 q^{47} + 8 q^{48} + 64 q^{49} - 46 q^{50} - 86 q^{51} + 4 q^{52} - 33 q^{53} + 13 q^{54} - 38 q^{55} + 2 q^{56} - 108 q^{57} - 5 q^{58} - q^{59} + 2 q^{60} - 19 q^{61} - 22 q^{62} - 20 q^{63} - 18 q^{64} + 3 q^{65} + 8 q^{66} + 40 q^{68} + 64 q^{69} + 34 q^{70} - 10 q^{71} - 5 q^{72} - 14 q^{73} + 4 q^{74} - 16 q^{75} + 56 q^{76} - 70 q^{77} + 46 q^{78} + 2 q^{79} - 60 q^{81} + 28 q^{82} - 56 q^{83} - 28 q^{84} + 52 q^{85} + 19 q^{86} - 8 q^{87} + 12 q^{88} + 8 q^{89} - 164 q^{90} + 29 q^{91} - 30 q^{92} - 44 q^{93} - 8 q^{94} + 27 q^{95} - q^{96} + 14 q^{97} - 20 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.913545 − 0.406737i −2.28812 − 2.54121i 0.669131 + 0.743145i −1.49184 − 1.66566i 1.05669 + 3.25217i −2.47512 − 0.934763i −0.309017 − 0.951057i −0.908690 + 8.64561i 0.685379 + 2.12844i
11.2 −0.913545 − 0.406737i −2.03944 − 2.26502i 0.669131 + 0.743145i 2.23454 − 0.0827180i 0.941850 + 2.89872i 2.10598 + 1.60151i −0.309017 − 0.951057i −0.657446 + 6.25519i −2.07500 − 0.833302i
11.3 −0.913545 − 0.406737i −1.27054 − 1.41107i 0.669131 + 0.743145i −0.881031 + 2.05518i 0.586758 + 1.80585i 1.87755 − 1.86408i −0.309017 − 0.951057i −0.0632808 + 0.602076i 1.64078 − 1.51916i
11.4 −0.913545 − 0.406737i −0.848696 − 0.942572i 0.669131 + 0.743145i −2.15057 + 0.612414i 0.391944 + 1.20628i 0.891313 + 2.49110i −0.309017 − 0.951057i 0.145428 − 1.38365i 2.21373 + 0.315248i
11.5 −0.913545 − 0.406737i −0.318061 − 0.353243i 0.669131 + 0.743145i 1.85994 − 1.24123i 0.146887 + 0.452070i −2.60446 − 0.465583i −0.309017 − 0.951057i 0.289968 − 2.75886i −2.20399 + 0.377412i
11.6 −0.913545 − 0.406737i 0.798609 + 0.886946i 0.669131 + 0.743145i 0.469061 + 2.18632i −0.368813 − 1.13509i −1.16626 + 2.37483i −0.309017 − 0.951057i 0.164690 − 1.56692i 0.460747 − 2.18808i
11.7 −0.913545 − 0.406737i 0.880627 + 0.978035i 0.669131 + 0.743145i −1.52041 − 1.63962i −0.406690 − 1.25166i 2.30318 − 1.30207i −0.309017 − 0.951057i 0.132536 − 1.26100i 0.722064 + 2.11628i
11.8 −0.913545 − 0.406737i 1.22735 + 1.36311i 0.669131 + 0.743145i 2.14068 − 0.646136i −0.566812 − 1.74447i 2.32768 + 1.25774i −0.309017 − 0.951057i −0.0380947 + 0.362447i −2.21842 − 0.280418i
11.9 −0.913545 − 0.406737i 2.10646 + 2.33946i 0.669131 + 0.743145i −0.760329 − 2.10283i −0.972801 − 2.99397i −2.25986 + 1.37587i −0.309017 − 0.951057i −0.722315 + 6.87236i −0.160704 + 2.23029i
81.1 −0.669131 + 0.743145i −0.244482 − 2.32609i −0.104528 − 0.994522i −1.85594 − 1.24719i 1.89221 + 1.37477i −2.38175 + 1.15208i 0.809017 + 0.587785i −2.41648 + 0.513640i 2.16871 − 0.544700i
81.2 −0.669131 + 0.743145i −0.244226 − 2.32365i −0.104528 − 0.994522i 2.23442 + 0.0857073i 1.89023 + 1.37333i 1.81635 + 1.92376i 0.809017 + 0.587785i −2.40527 + 0.511257i −1.55881 + 1.60315i
81.3 −0.669131 + 0.743145i −0.190370 − 1.81125i −0.104528 − 0.994522i 0.565569 − 2.16336i 1.47340 + 1.07049i 0.983361 − 2.45622i 0.809017 + 0.587785i −0.309943 + 0.0658805i 1.22925 + 1.86787i
81.4 −0.669131 + 0.743145i −0.156346 − 1.48754i −0.104528 − 0.994522i −0.0739496 + 2.23484i 1.21007 + 0.879169i −1.32245 − 2.29153i 0.809017 + 0.587785i 0.746121 − 0.158593i −1.61133 − 1.55036i
81.5 −0.669131 + 0.743145i 0.0508913 + 0.484199i −0.104528 − 0.994522i 0.866210 + 2.06148i −0.393883 − 0.286173i −1.17653 + 2.36976i 0.809017 + 0.587785i 2.70258 − 0.574452i −2.11158 − 0.735677i
81.6 −0.669131 + 0.743145i 0.0698073 + 0.664172i −0.104528 − 0.994522i −1.87316 + 1.22117i −0.540286 − 0.392541i 2.57149 − 0.622454i 0.809017 + 0.587785i 2.49819 − 0.531007i 0.345886 − 2.20915i
81.7 −0.669131 + 0.743145i 0.143555 + 1.36584i −0.104528 − 0.994522i 2.23569 + 0.0410536i −1.11107 − 0.807241i 2.33546 − 1.24323i 0.809017 + 0.587785i 1.08954 − 0.231589i −1.52648 + 1.63397i
81.8 −0.669131 + 0.743145i 0.300861 + 2.86250i −0.104528 − 0.994522i −1.53794 − 1.62319i −2.32857 − 1.69180i 0.721052 + 2.54560i 0.809017 + 0.587785i −5.16894 + 1.09869i 2.23535 − 0.0567861i
81.9 −0.669131 + 0.743145i 0.310236 + 2.95170i −0.104528 − 0.994522i 2.20441 + 0.374925i −2.40113 − 1.74452i −2.54698 − 0.716179i 0.809017 + 0.587785i −5.68183 + 1.20771i −1.75366 + 1.38732i
121.1 −0.669131 − 0.743145i −0.244482 + 2.32609i −0.104528 + 0.994522i −1.85594 + 1.24719i 1.89221 − 1.37477i −2.38175 − 1.15208i 0.809017 − 0.587785i −2.41648 − 0.513640i 2.16871 + 0.544700i
121.2 −0.669131 − 0.743145i −0.244226 + 2.32365i −0.104528 + 0.994522i 2.23442 − 0.0857073i 1.89023 − 1.37333i 1.81635 − 1.92376i 0.809017 − 0.587785i −2.40527 − 0.511257i −1.55881 − 1.60315i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.q.b ✓ 72
7.c even 3 1 inner 350.2.q.b ✓ 72
25.d even 5 1 inner 350.2.q.b ✓ 72
175.q even 15 1 inner 350.2.q.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.q.b ✓ 72 1.a even 1 1 trivial
350.2.q.b ✓ 72 7.c even 3 1 inner
350.2.q.b ✓ 72 25.d even 5 1 inner
350.2.q.b ✓ 72 175.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - T_{3}^{71} - 23 T_{3}^{70} - 8 T_{3}^{69} + 259 T_{3}^{68} + 210 T_{3}^{67} + \cdots + 14\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display