# Properties

 Label 950.2.h.e 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 11q^{2} - q^{3} - 11q^{4} - 5q^{5} + q^{6} + 28q^{7} + 11q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 11q^{2} - q^{3} - 11q^{4} - 5q^{5} + q^{6} + 28q^{7} + 11q^{8} - 8q^{9} + 5q^{10} - 6q^{12} - 10q^{13} + 12q^{14} - 11q^{16} - 20q^{17} - 42q^{18} - 11q^{19} + 5q^{20} - 3q^{21} - 10q^{22} - 6q^{23} - 14q^{24} - 15q^{25} - 40q^{26} + 5q^{27} - 2q^{28} + 6q^{29} - 5q^{31} - 44q^{32} - 36q^{33} - 10q^{34} - 8q^{36} - 10q^{37} + 11q^{38} + 39q^{39} - 22q^{41} + 3q^{42} + 68q^{43} + 10q^{44} + 20q^{45} + 6q^{46} + 19q^{47} - 6q^{48} + 40q^{49} - 30q^{50} + 86q^{51} - 10q^{52} + 30q^{54} + 2q^{56} + 14q^{57} - 6q^{58} - 4q^{59} + 15q^{60} + 26q^{61} - 15q^{62} - 41q^{63} - 11q^{64} + 30q^{65} - 4q^{66} - 59q^{67} + 20q^{68} - 59q^{69} - 25q^{70} + 30q^{71} + 13q^{72} - 38q^{73} - 50q^{74} - 15q^{75} + 44q^{76} + 29q^{77} + 16q^{78} + 3q^{79} + 5q^{80} - 54q^{81} - 8q^{82} + 9q^{83} + 7q^{84} + 12q^{86} - 43q^{87} + 33q^{89} - 6q^{91} - 6q^{92} + 84q^{93} - 19q^{94} + q^{96} + 30q^{97} - 15q^{98} - 54q^{99} + O(q^{100})$$

## 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 −0.864167 2.65963i 0.309017 + 0.951057i −0.315348 + 2.21372i 0.864167 2.65963i 4.09905 −0.309017 + 0.951057i −3.89981 + 2.83338i −1.55631 + 1.60558i
191.2 0.809017 + 0.587785i −0.575164 1.77017i 0.309017 + 0.951057i −2.23087 + 0.152345i 0.575164 1.77017i 2.55490 −0.309017 + 0.951057i −0.375647 + 0.272924i −1.89436 1.18802i
191.3 0.809017 + 0.587785i −0.457139 1.40693i 0.309017 + 0.951057i 1.65863 1.49965i 0.457139 1.40693i 0.448991 −0.309017 + 0.951057i 0.656575 0.477030i 2.22333 0.238322i
191.4 0.809017 + 0.587785i −0.432606 1.33142i 0.309017 + 0.951057i −1.38217 1.75773i 0.432606 1.33142i −3.95270 −0.309017 + 0.951057i 0.841508 0.611391i −0.0850285 2.23445i
191.5 0.809017 + 0.587785i 0.140614 + 0.432766i 0.309017 + 0.951057i 0.187653 + 2.22818i −0.140614 + 0.432766i 3.42213 −0.309017 + 0.951057i 2.25954 1.64165i −1.15788 + 1.91294i
191.6 0.809017 + 0.587785i 0.188387 + 0.579797i 0.309017 + 0.951057i 1.27289 1.83841i −0.188387 + 0.579797i 0.873784 −0.309017 + 0.951057i 2.12638 1.54490i 2.11038 0.739116i
191.7 0.809017 + 0.587785i 0.401228 + 1.23485i 0.309017 + 0.951057i 1.49048 + 1.66687i −0.401228 + 1.23485i −2.35656 −0.309017 + 0.951057i 1.06317 0.772440i 0.226063 + 2.22461i
191.8 0.809017 + 0.587785i 0.625820 + 1.92608i 0.309017 + 0.951057i −0.243465 2.22277i −0.625820 + 1.92608i 4.25141 −0.309017 + 0.951057i −0.891064 + 0.647396i 1.10955 1.94137i
191.9 0.809017 + 0.587785i 0.688874 + 2.12014i 0.309017 + 0.951057i −0.561048 2.16454i −0.688874 + 2.12014i 0.565964 −0.309017 + 0.951057i −1.59338 + 1.15766i 0.818386 2.08092i
191.10 0.809017 + 0.587785i 0.739754 + 2.27673i 0.309017 + 0.951057i −2.19207 0.441377i −0.739754 + 2.27673i −0.427253 −0.309017 + 0.951057i −2.20921 + 1.60508i −1.51399 1.64555i
191.11 0.809017 + 0.587785i 0.971450 + 2.98981i 0.309017 + 0.951057i 1.62433 + 1.53673i −0.971450 + 2.98981i −0.243654 −0.309017 + 0.951057i −5.56823 + 4.04555i 0.410843 + 2.19800i
381.1 −0.309017 0.951057i −2.63401 + 1.91372i −0.809017 + 0.587785i −1.39221 + 1.74978i 2.63401 + 1.91372i −3.22496 0.809017 + 0.587785i 2.34863 7.22835i 2.09436 + 0.783360i
381.2 −0.309017 0.951057i −2.47282 + 1.79661i −0.809017 + 0.587785i 2.01410 0.971294i 2.47282 + 1.79661i 3.79510 0.809017 + 0.587785i 1.95997 6.03218i −1.54615 1.61537i
381.3 −0.309017 0.951057i −1.75602 + 1.27582i −0.809017 + 0.587785i 1.11250 + 1.93968i 1.75602 + 1.27582i −1.18687 0.809017 + 0.587785i 0.528832 1.62758i 1.50096 1.65744i
381.4 −0.309017 0.951057i −1.05744 + 0.768272i −0.809017 + 0.587785i −2.23141 + 0.144237i 1.05744 + 0.768272i −0.741348 0.809017 + 0.587785i −0.399123 + 1.22837i 0.826721 + 2.07763i
381.5 −0.309017 0.951057i −0.444175 + 0.322712i −0.809017 + 0.587785i −0.351941 + 2.20820i 0.444175 + 0.322712i 4.87281 0.809017 + 0.587785i −0.833903 + 2.56649i 2.20888 0.347655i
381.6 −0.309017 0.951057i −0.417797 + 0.303547i −0.809017 + 0.587785i −1.20950 1.88072i 0.417797 + 0.303547i 0.938984 0.809017 + 0.587785i −0.844638 + 2.59953i −1.41492 + 1.73148i
381.7 −0.309017 0.951057i 0.573480 0.416658i −0.809017 + 0.587785i −0.120367 2.23283i −0.573480 0.416658i −2.96937 0.809017 + 0.587785i −0.771775 + 2.37528i −2.08635 + 0.804457i
381.8 −0.309017 0.951057i 0.685786 0.498253i −0.809017 + 0.587785i 2.20717 0.358335i −0.685786 0.498253i 3.58997 0.809017 + 0.587785i −0.705004 + 2.16978i −1.02285 1.98841i
381.9 −0.309017 0.951057i 1.40669 1.02202i −0.809017 + 0.587785i −2.00793 + 0.983990i −1.40669 1.02202i 0.690753 0.809017 + 0.587785i 0.00719592 0.0221468i 1.55631 + 1.60558i
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.e 44
25.d even 5 1 inner 950.2.h.e 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.h.e 44 1.a even 1 1 trivial
950.2.h.e 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$.