# 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

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$9$$ over $$\Q(\zeta_{15})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 9q^{2} + q^{3} + 9q^{4} + 2q^{6} + 8q^{7} + 18q^{8} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 9q^{2} + q^{3} + 9q^{4} + 2q^{6} + 8q^{7} + 18q^{8} + 20q^{9} - 10q^{10} - 12q^{11} - 4q^{12} - 8q^{13} + 11q^{14} + 16q^{15} + 9q^{16} - 30q^{17} + 50q^{18} + 2q^{19} + 10q^{20} - 20q^{21} + 26q^{22} - 10q^{23} - 6q^{24} - 8q^{25} - 14q^{26} + 46q^{27} - 2q^{28} - 10q^{29} - 12q^{30} + 9q^{31} + 36q^{32} - 13q^{33} + 20q^{34} + 6q^{35} - 40q^{36} + 11q^{37} - 12q^{38} - 27q^{39} + 5q^{40} - 34q^{41} + 2q^{42} - 32q^{43} + 13q^{44} - 7q^{45} - 15q^{46} + 8q^{47} + 8q^{48} + 64q^{49} - 46q^{50} - 86q^{51} + 4q^{52} - 33q^{53} + 13q^{54} - 38q^{55} + 2q^{56} - 108q^{57} - 5q^{58} - q^{59} + 2q^{60} - 19q^{61} - 22q^{62} - 20q^{63} - 18q^{64} + 3q^{65} + 8q^{66} + 40q^{68} + 64q^{69} + 34q^{70} - 10q^{71} - 5q^{72} - 14q^{73} + 4q^{74} - 16q^{75} + 56q^{76} - 70q^{77} + 46q^{78} + 2q^{79} - 60q^{81} + 28q^{82} - 56q^{83} - 28q^{84} + 52q^{85} + 19q^{86} - 8q^{87} + 12q^{88} + 8q^{89} - 164q^{90} + 29q^{91} - 30q^{92} - 44q^{93} - 8q^{94} + 27q^{95} - q^{96} + 14q^{97} - 20q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 331.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{3}^{72} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database