Properties

Label 950.2.h.b
Level $950$
Weight $2$
Character orbit 950.h
Analytic conductor $7.586$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 10q^{2} + 5q^{3} - 10q^{4} + 5q^{6} - 12q^{7} - 10q^{8} - 3q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 10q^{2} + 5q^{3} - 10q^{4} + 5q^{6} - 12q^{7} - 10q^{8} - 3q^{9} - 6q^{11} + 16q^{13} - 2q^{14} + 16q^{15} - 10q^{16} - 2q^{17} + 22q^{18} - 10q^{19} - 5q^{20} + 3q^{21} + 14q^{22} + 4q^{23} - 10q^{24} - 2q^{25} - 34q^{26} + 29q^{27} + 8q^{28} + 16q^{30} + 19q^{31} + 40q^{32} - 16q^{33} + 3q^{34} - 24q^{35} - 3q^{36} + q^{37} - 10q^{38} - 7q^{39} + 5q^{40} + 20q^{41} + 3q^{42} - 48q^{43} + 14q^{44} + 53q^{45} + 4q^{46} + 29q^{47} + 28q^{49} + 3q^{50} - 122q^{51} + 16q^{52} - 5q^{53} - 6q^{54} + 56q^{55} + 8q^{56} - 10q^{57} + 20q^{59} - 19q^{60} + 42q^{61} - 21q^{62} + 9q^{63} - 10q^{64} + 35q^{65} + 24q^{66} - 3q^{67} - 2q^{68} - 9q^{69} - 19q^{70} + 18q^{71} - 8q^{72} + 8q^{73} - 64q^{74} + 7q^{75} + 40q^{76} + 35q^{77} - 2q^{78} + q^{79} + 59q^{81} - 30q^{82} + 11q^{83} - 7q^{84} - 125q^{85} + 32q^{86} - 31q^{87} - 6q^{88} - 34q^{89} - 7q^{90} + 10q^{91} + 4q^{92} + 24q^{93} + 29q^{94} + 5q^{95} + 5q^{96} + 90q^{97} - 12q^{98} + 10q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 −0.809017 0.587785i −1.01321 3.11835i 0.309017 + 0.951057i −2.01188 0.975874i −1.01321 + 3.11835i −3.17274 0.309017 0.951057i −6.27043 + 4.55573i 1.05404 + 1.97205i
191.2 −0.809017 0.587785i −0.853804 2.62774i 0.309017 + 0.951057i −0.950820 + 2.02384i −0.853804 + 2.62774i 1.85248 0.309017 0.951057i −3.74897 + 2.72379i 1.95881 1.07845i
191.3 −0.809017 0.587785i −0.410709 1.26403i 0.309017 + 0.951057i 0.971992 2.01376i −0.410709 + 1.26403i −2.95140 0.309017 0.951057i 0.997953 0.725055i −1.97002 + 1.05784i
191.4 −0.809017 0.587785i −0.199467 0.613896i 0.309017 + 0.951057i 1.52772 + 1.63281i −0.199467 + 0.613896i 4.95147 0.309017 0.951057i 2.08997 1.51845i −0.276209 2.21894i
191.5 −0.809017 0.587785i −0.0472563 0.145440i 0.309017 + 0.951057i 1.99007 + 1.01962i −0.0472563 + 0.145440i −0.777730 0.309017 0.951057i 2.40813 1.74961i −1.01068 1.99462i
191.6 −0.809017 0.587785i −0.0202803 0.0624164i 0.309017 + 0.951057i −0.536379 2.17078i −0.0202803 + 0.0624164i 2.69704 0.309017 0.951057i 2.42357 1.76082i −0.842014 + 2.07148i
191.7 −0.809017 0.587785i 0.272680 + 0.839224i 0.309017 + 0.951057i 0.404965 + 2.19909i 0.272680 0.839224i −4.83673 0.309017 0.951057i 1.79711 1.30568i 0.964970 2.01713i
191.8 −0.809017 0.587785i 0.416614 + 1.28220i 0.309017 + 0.951057i −1.66523 + 1.49231i 0.416614 1.28220i 0.369301 0.309017 0.951057i 0.956568 0.694988i 2.22436 0.228508i
191.9 −0.809017 0.587785i 0.558576 + 1.71912i 0.309017 + 0.951057i −1.95557 1.08432i 0.558576 1.71912i 1.04781 0.309017 0.951057i −0.216318 + 0.157164i 0.944740 + 2.02669i
191.10 −0.809017 0.587785i 0.869807 + 2.67699i 0.309017 + 0.951057i 2.22514 0.220827i 0.869807 2.67699i 0.0565513 0.309017 0.951057i −3.98267 + 2.89358i −1.92997 1.12925i
381.1 0.309017 + 0.951057i −2.17406 + 1.57955i −0.809017 + 0.587785i 1.08520 + 1.95508i −2.17406 1.57955i −3.41021 −0.809017 0.587785i 1.30452 4.01490i −1.52404 + 1.63624i
381.2 0.309017 + 0.951057i −1.69612 + 1.23231i −0.809017 + 0.587785i −2.05239 0.887532i −1.69612 1.23231i −0.189957 −0.809017 0.587785i 0.431206 1.32712i 0.209871 2.22620i
381.3 0.309017 + 0.951057i −1.03616 + 0.752812i −0.809017 + 0.587785i −0.232130 2.22399i −1.03616 0.752812i −0.203240 −0.809017 0.587785i −0.420156 + 1.29311i 2.04340 0.908018i
381.4 0.309017 + 0.951057i −0.346529 + 0.251768i −0.809017 + 0.587785i −1.41712 + 1.72968i −0.346529 0.251768i 1.40627 −0.809017 0.587785i −0.870356 + 2.67868i −2.08293 0.813259i
381.5 0.309017 + 0.951057i −0.0867222 + 0.0630074i −0.809017 + 0.587785i 2.07960 0.821736i −0.0867222 0.0630074i −1.31483 −0.809017 0.587785i −0.923500 + 2.84224i 1.42415 + 1.72389i
381.6 0.309017 + 0.951057i 0.786708 0.571577i −0.809017 + 0.587785i −2.18231 0.487343i 0.786708 + 0.571577i 4.32575 −0.809017 0.587785i −0.634842 + 1.95384i −0.210881 2.22610i
381.7 0.309017 + 0.951057i 1.41542 1.02837i −0.809017 + 0.587785i −1.29625 + 1.82202i 1.41542 + 1.02837i −2.41426 −0.809017 0.587785i 0.0188385 0.0579789i −2.13340 0.669769i
381.8 0.309017 + 0.951057i 1.59293 1.15733i −0.809017 + 0.587785i 2.11352 + 0.730086i 1.59293 + 1.15733i −1.09272 −0.809017 0.587785i 0.270952 0.833904i −0.0412386 + 2.23569i
381.9 0.309017 + 0.951057i 2.18620 1.58837i −0.809017 + 0.587785i 1.58887 + 1.57337i 2.18620 + 1.58837i 2.75740 −0.809017 0.587785i 1.32952 4.09183i −1.00537 + 1.99731i
381.10 0.309017 + 0.951057i 2.28538 1.66043i −0.809017 + 0.587785i 0.312990 2.21405i 2.28538 + 1.66043i −5.10025 −0.809017 0.587785i 1.53890 4.73626i 2.20241 0.386510i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 761.10
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 950.2.h.b 40
25.d even 5 1 inner 950.2.h.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.h.b 40 1.a even 1 1 trivial
950.2.h.b 40 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{40} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).