# Properties

 Label 260.2.j.a Level $260$ Weight $2$ Character orbit 260.j Analytic conductor $2.076$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$28$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 12 q^{6} + 12 q^{8} - 56 q^{9}+O(q^{10})$$ 56 * q - 12 * q^6 + 12 * q^8 - 56 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 12 q^{6} + 12 q^{8} - 56 q^{9} + 16 q^{14} - 12 q^{18} - 8 q^{20} - 16 q^{21} - 40 q^{24} - 16 q^{26} - 44 q^{28} + 40 q^{32} - 4 q^{34} + 16 q^{37} + 8 q^{41} + 8 q^{42} + 28 q^{44} - 12 q^{46} + 104 q^{48} + 56 q^{52} - 16 q^{53} + 20 q^{54} - 48 q^{57} - 4 q^{58} + 16 q^{61} - 8 q^{65} + 64 q^{66} + 24 q^{68} - 8 q^{70} - 32 q^{72} + 48 q^{73} - 136 q^{74} - 88 q^{76} + 52 q^{78} - 32 q^{80} + 56 q^{81} - 20 q^{84} - 64 q^{86} - 8 q^{89} - 88 q^{92} - 48 q^{93} - 16 q^{94} - 4 q^{96} - 32 q^{97} + 16 q^{98}+O(q^{100})$$ 56 * q - 12 * q^6 + 12 * q^8 - 56 * q^9 + 16 * q^14 - 12 * q^18 - 8 * q^20 - 16 * q^21 - 40 * q^24 - 16 * q^26 - 44 * q^28 + 40 * q^32 - 4 * q^34 + 16 * q^37 + 8 * q^41 + 8 * q^42 + 28 * q^44 - 12 * q^46 + 104 * q^48 + 56 * q^52 - 16 * q^53 + 20 * q^54 - 48 * q^57 - 4 * q^58 + 16 * q^61 - 8 * q^65 + 64 * q^66 + 24 * q^68 - 8 * q^70 - 32 * q^72 + 48 * q^73 - 136 * q^74 - 88 * q^76 + 52 * q^78 - 32 * q^80 + 56 * q^81 - 20 * q^84 - 64 * q^86 - 8 * q^89 - 88 * q^92 - 48 * q^93 - 16 * q^94 - 4 * q^96 - 32 * q^97 + 16 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.41372 0.0374414i 1.74175i 1.99720 + 0.105863i −0.707107 + 0.707107i 0.0652137 2.46234i −2.26642 + 2.26642i −2.81951 0.224439i −0.0336953 1.02612 0.973174i
31.2 −1.36090 0.384637i 1.17966i 1.70411 + 1.04691i −0.707107 + 0.707107i −0.453740 + 1.60540i 0.171101 0.171101i −1.91645 2.08020i 1.60841 1.23428 0.690324i
31.3 −1.25736 + 0.647340i 0.759638i 1.16190 1.62788i 0.707107 0.707107i 0.491744 + 0.955138i −2.17387 + 2.17387i −0.407140 + 2.79897i 2.42295 −0.431349 + 1.34683i
31.4 −1.25602 0.649941i 2.18560i 1.15515 + 1.63267i 0.707107 0.707107i −1.42051 + 2.74515i 1.49904 1.49904i −0.389750 2.80145i −1.77684 −1.34772 + 0.428560i
31.5 −1.22023 + 0.714871i 1.31885i 0.977920 1.74461i −0.707107 + 0.707107i −0.942808 1.60930i 3.16584 3.16584i 0.0538847 + 2.82791i 1.26063 0.357343 1.36832i
31.6 −1.20609 + 0.738473i 3.43622i 0.909314 1.78133i 0.707107 0.707107i −2.53756 4.14440i −1.29013 + 1.29013i 0.218751 + 2.81996i −8.80762 −0.330656 + 1.37502i
31.7 −1.08931 0.901894i 0.947478i 0.373176 + 1.96488i 0.707107 0.707107i 0.854524 1.03209i −2.28468 + 2.28468i 1.36561 2.47692i 2.10229 −1.40799 + 0.132521i
31.8 −0.853677 1.12749i 2.83163i −0.542471 + 1.92503i −0.707107 + 0.707107i −3.19263 + 2.41729i −2.55134 + 2.55134i 2.63354 1.03172i −5.01810 1.40090 + 0.193616i
31.9 −0.738473 + 1.20609i 3.43622i −0.909314 1.78133i 0.707107 0.707107i 4.14440 + 2.53756i 1.29013 1.29013i 2.81996 + 0.218751i −8.80762 0.330656 + 1.37502i
31.10 −0.714871 + 1.22023i 1.31885i −0.977920 1.74461i −0.707107 + 0.707107i 1.60930 + 0.942808i −3.16584 + 3.16584i 2.82791 + 0.0538847i 1.26063 −0.357343 1.36832i
31.11 −0.647340 + 1.25736i 0.759638i −1.16190 1.62788i 0.707107 0.707107i −0.955138 0.491744i 2.17387 2.17387i 2.79897 0.407140i 2.42295 0.431349 + 1.34683i
31.12 −0.551623 1.30220i 0.224099i −1.39142 + 1.43664i −0.707107 + 0.707107i 0.291821 0.123618i −0.228458 + 0.228458i 2.63833 + 1.01942i 2.94978 1.31085 + 0.530735i
31.13 −0.329884 1.37520i 0.0805516i −1.78235 + 0.907313i 0.707107 0.707107i 0.110775 0.0265727i 2.41026 2.41026i 1.83571 + 2.15179i 2.99351 −1.20568 0.739151i
31.14 0.0374414 + 1.41372i 1.74175i −1.99720 + 0.105863i −0.707107 + 0.707107i 2.46234 0.0652137i 2.26642 2.26642i −0.224439 2.81951i −0.0336953 −1.02612 0.973174i
31.15 0.0957733 1.41097i 2.67523i −1.98165 0.270266i −0.707107 + 0.707107i 3.77466 + 0.256215i 0.0140243 0.0140243i −0.571126 + 2.77017i −4.15684 0.929982 + 1.06543i
31.16 0.270865 1.38803i 2.58563i −1.85326 0.751939i 0.707107 0.707107i −3.58893 0.700357i −1.49185 + 1.49185i −1.54570 + 2.36871i −3.68547 −0.789956 1.17302i
31.17 0.384637 + 1.36090i 1.17966i −1.70411 + 1.04691i −0.707107 + 0.707107i −1.60540 + 0.453740i −0.171101 + 0.171101i −2.08020 1.91645i 1.60841 −1.23428 0.690324i
31.18 0.400070 1.35645i 2.57103i −1.67989 1.08535i −0.707107 + 0.707107i −3.48746 1.02859i 3.59259 3.59259i −2.14428 + 1.84446i −3.61018 0.676260 + 1.24204i
31.19 0.649941 + 1.25602i 2.18560i −1.15515 + 1.63267i 0.707107 0.707107i −2.74515 + 1.42051i −1.49904 + 1.49904i −2.80145 0.389750i −1.77684 1.34772 + 0.428560i
31.20 0.723264 1.21527i 1.80245i −0.953778 1.75793i 0.707107 0.707107i 2.19047 + 1.30365i 1.20463 1.20463i −2.82620 0.112344i −0.248815 −0.347903 1.37075i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 151.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.d odd 4 1 inner
52.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.j.a 56
4.b odd 2 1 inner 260.2.j.a 56
13.d odd 4 1 inner 260.2.j.a 56
52.f even 4 1 inner 260.2.j.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.j.a 56 1.a even 1 1 trivial
260.2.j.a 56 4.b odd 2 1 inner
260.2.j.a 56 13.d odd 4 1 inner
260.2.j.a 56 52.f even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(260, [\chi])$$.