# Properties

 Label 950.2.h.d Level $950$ Weight $2$ Character orbit 950.h Analytic conductor $7.586$ Analytic rank $0$ Dimension $44$ CM no Inner twists $2$

# Related objects

## 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: $$44$$ Relative dimension: $$11$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 11q^{2} - q^{3} - 11q^{4} - 11q^{5} + q^{6} - 10q^{7} + 11q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 11q^{2} - q^{3} - 11q^{4} - 11q^{5} + q^{6} - 10q^{7} + 11q^{8} - 4q^{10} + 7q^{11} + 4q^{12} + 10q^{13} - 5q^{14} - 20q^{15} - 11q^{16} + 11q^{17} - 50q^{18} + 11q^{19} + 4q^{20} + 9q^{21} + 8q^{22} - 7q^{23} + 6q^{24} - 3q^{25} + 10q^{26} + 5q^{27} - 20q^{29} - 10q^{30} - 11q^{31} - 44q^{32} - 2q^{33} - q^{34} + 5q^{35} + 43q^{37} - 11q^{38} - 33q^{39} - 4q^{40} - 32q^{41} - 9q^{42} - 74q^{43} - 8q^{44} + 10q^{45} - 8q^{46} - 19q^{47} + 4q^{48} + 54q^{49} + 18q^{50} + 34q^{51} + 10q^{52} + 23q^{53} - 40q^{54} + 2q^{55} + 6q^{57} + 20q^{58} + 24q^{59} - 35q^{60} - 12q^{61} + q^{62} - 44q^{63} - 11q^{64} + 43q^{65} - 8q^{66} + 35q^{67} - 24q^{68} + 29q^{69} - 5q^{70} - 2q^{71} + 25q^{72} + 22q^{73} + 72q^{74} + 7q^{75} - 44q^{76} - 3q^{77} - 22q^{78} - 19q^{79} - q^{80} - 26q^{81} - 58q^{82} - 24q^{83} - 21q^{84} - 18q^{85} - 6q^{86} + 83q^{87} - 7q^{88} - 8q^{89} + 60q^{90} - 30q^{91} + 8q^{92} - 72q^{93} + 19q^{94} - 4q^{95} + q^{96} - 16q^{97} + q^{98} + 84q^{99} + O(q^{100})$$

## 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 −1.00220 3.08444i 0.309017 + 0.951057i −2.22762 + 0.194172i 1.00220 3.08444i −1.89338 −0.309017 + 0.951057i −6.08232 + 4.41906i −1.91632 1.15227i
191.2 0.809017 + 0.587785i −0.806394 2.48183i 0.309017 + 0.951057i 1.11301 1.93938i 0.806394 2.48183i −1.68962 −0.309017 + 0.951057i −3.08213 + 2.23930i 2.04039 0.914782i
191.3 0.809017 + 0.587785i −0.734690 2.26114i 0.309017 + 0.951057i −0.796273 2.08949i 0.734690 2.26114i 3.12738 −0.309017 + 0.951057i −2.14595 + 1.55912i 0.583970 2.15847i
191.4 0.809017 + 0.587785i −0.320919 0.987687i 0.309017 + 0.951057i −0.846301 + 2.06973i 0.320919 0.987687i −2.76904 −0.309017 + 0.951057i 1.55451 1.12942i −1.90123 + 1.17700i
191.5 0.809017 + 0.587785i −0.183900 0.565985i 0.309017 + 0.951057i 2.14212 + 0.641332i 0.183900 0.565985i 2.23803 −0.309017 + 0.951057i 2.14053 1.55519i 1.35605 + 1.77796i
191.6 0.809017 + 0.587785i −0.00648263 0.0199515i 0.309017 + 0.951057i 2.20515 + 0.370553i 0.00648263 0.0199515i −4.68808 −0.309017 + 0.951057i 2.42669 1.76310i 1.56620 + 1.59594i
191.7 0.809017 + 0.587785i 0.0375929 + 0.115699i 0.309017 + 0.951057i −2.23367 + 0.103463i −0.0375929 + 0.115699i 1.64271 −0.309017 + 0.951057i 2.41508 1.75466i −1.86789 1.22922i
191.8 0.809017 + 0.587785i 0.154308 + 0.474912i 0.309017 + 0.951057i −1.19349 1.89092i −0.154308 + 0.474912i −2.61594 −0.309017 + 0.951057i 2.22532 1.61679i 0.145902 2.23130i
191.9 0.809017 + 0.587785i 0.578674 + 1.78097i 0.309017 + 0.951057i −1.13510 + 1.92654i −0.578674 + 1.78097i −2.72132 −0.309017 + 0.951057i −0.409956 + 0.297850i −2.05071 + 0.891406i
191.10 0.809017 + 0.587785i 0.611239 + 1.88120i 0.309017 + 0.951057i 2.16649 0.553476i −0.611239 + 1.88120i 2.35142 −0.309017 + 0.951057i −0.738252 + 0.536372i 2.07805 + 0.825657i
191.11 0.809017 + 0.587785i 0.863749 + 2.65835i 0.309017 + 0.951057i −1.38530 + 1.75527i −0.863749 + 2.65835i 3.39982 −0.309017 + 0.951057i −3.89370 + 2.82894i −2.15245 + 0.605783i
381.1 −0.309017 0.951057i −2.30507 + 1.67473i −0.809017 + 0.587785i −0.482704 2.18335i 2.30507 + 1.67473i −2.79249 0.809017 + 0.587785i 1.58157 4.86758i −1.92732 + 1.13377i
381.2 −0.309017 0.951057i −2.00675 + 1.45799i −0.809017 + 0.587785i −2.21649 + 0.295244i 2.00675 + 1.45799i 2.78117 0.809017 + 0.587785i 0.974256 2.99845i 0.965727 + 2.01677i
381.3 −0.309017 0.951057i −1.70816 + 1.24105i −0.809017 + 0.587785i 2.12953 0.681987i 1.70816 + 1.24105i −1.90564 0.809017 + 0.587785i 0.450556 1.38667i −1.30667 1.81456i
381.4 −0.309017 0.951057i −1.20914 + 0.878491i −0.809017 + 0.587785i 0.176849 2.22906i 1.20914 + 0.878491i 2.76699 0.809017 + 0.587785i −0.236780 + 0.728734i −2.17461 + 0.520625i
381.5 −0.309017 0.951057i −0.628367 + 0.456535i −0.809017 + 0.587785i −0.766450 + 2.10061i 0.628367 + 0.456535i −4.62972 0.809017 + 0.587785i −0.740631 + 2.27943i 2.23464 + 0.0798131i
381.6 −0.309017 0.951057i −0.591887 + 0.430031i −0.809017 + 0.587785i 1.69555 + 1.45777i 0.591887 + 0.430031i 1.38478 0.809017 + 0.587785i −0.761648 + 2.34411i 0.862467 2.06304i
381.7 −0.309017 0.951057i 0.616737 0.448085i −0.809017 + 0.587785i −1.35137 + 1.78152i −0.616737 0.448085i −0.0296482 0.809017 + 0.587785i −0.747467 + 2.30047i 2.11192 + 0.734710i
381.8 −0.309017 0.951057i 0.823288 0.598153i −0.809017 + 0.587785i −1.34910 1.78323i −0.823288 0.598153i 3.63885 0.809017 + 0.587785i −0.607036 + 1.86826i −1.27906 + 1.83412i
381.9 −0.309017 0.951057i 2.31854 1.68452i −0.809017 + 0.587785i 1.05411 1.97202i −2.31854 1.68452i −0.662081 0.809017 + 0.587785i 1.61098 4.95809i −2.20124 0.393129i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 761.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d 44
25.d even 5 1 inner 950.2.h.d 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.h.d 44 1.a even 1 1 trivial
950.2.h.d 44 25.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{44} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.