# Properties

 Label 1045.2.b.e Level $1045$ Weight $2$ Character orbit 1045.b Analytic conductor $8.344$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$30$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 42 q^{4} + 12 q^{6} - 40 q^{9}+O(q^{10})$$ 30 * q - 42 * q^4 + 12 * q^6 - 40 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 42 q^{4} + 12 q^{6} - 40 q^{9} + 10 q^{10} + 30 q^{11} + 4 q^{14} + 4 q^{15} + 66 q^{16} - 30 q^{19} + 10 q^{20} + 14 q^{21} - 22 q^{24} - 6 q^{25} - 30 q^{29} + 14 q^{30} + 26 q^{31} - 12 q^{34} + 6 q^{35} + 78 q^{36} - 64 q^{39} - 20 q^{40} + 22 q^{41} - 42 q^{44} + 6 q^{45} + 28 q^{46} - 60 q^{49} + 64 q^{51} - 62 q^{54} - 32 q^{56} + 14 q^{59} - 28 q^{60} + 78 q^{61} - 90 q^{64} + 40 q^{65} + 12 q^{66} + 28 q^{69} + 12 q^{70} + 20 q^{71} - 42 q^{74} + 50 q^{75} + 42 q^{76} - 102 q^{79} - 40 q^{80} + 42 q^{81} - 98 q^{84} - 2 q^{85} - 52 q^{86} + 8 q^{89} + 22 q^{90} + 56 q^{91} - 40 q^{94} - 74 q^{96} - 40 q^{99}+O(q^{100})$$ 30 * q - 42 * q^4 + 12 * q^6 - 40 * q^9 + 10 * q^10 + 30 * q^11 + 4 * q^14 + 4 * q^15 + 66 * q^16 - 30 * q^19 + 10 * q^20 + 14 * q^21 - 22 * q^24 - 6 * q^25 - 30 * q^29 + 14 * q^30 + 26 * q^31 - 12 * q^34 + 6 * q^35 + 78 * q^36 - 64 * q^39 - 20 * q^40 + 22 * q^41 - 42 * q^44 + 6 * q^45 + 28 * q^46 - 60 * q^49 + 64 * q^51 - 62 * q^54 - 32 * q^56 + 14 * q^59 - 28 * q^60 + 78 * q^61 - 90 * q^64 + 40 * q^65 + 12 * q^66 + 28 * q^69 + 12 * q^70 + 20 * q^71 - 42 * q^74 + 50 * q^75 + 42 * q^76 - 102 * q^79 - 40 * q^80 + 42 * q^81 - 98 * q^84 - 2 * q^85 - 52 * q^86 + 8 * q^89 + 22 * q^90 + 56 * q^91 - 40 * q^94 - 74 * q^96 - 40 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1 2.79219i 1.83665i −5.79631 −2.23016 + 0.162474i 5.12828 0.971452i 10.6000i −0.373298 0.453657 + 6.22702i
419.2 2.77451i 2.74433i −5.69792 0.316248 + 2.21359i −7.61418 3.20844i 10.2599i −4.53135 6.14164 0.877434i
419.3 2.44393i 0.661177i −3.97280 1.55248 + 1.60929i 1.61587 2.05193i 4.82139i 2.56284 3.93299 3.79415i
419.4 2.39831i 0.881728i −3.75187 −0.259771 2.22093i −2.11465 2.42713i 4.20151i 2.22256 −5.32646 + 0.623010i
419.5 2.39110i 3.31399i −3.71736 1.04940 1.97453i 7.92409 0.907652i 4.10637i −7.98255 −4.72129 2.50922i
419.6 2.14697i 2.93729i −2.60947 −0.584477 + 2.15833i 6.30628 3.64055i 1.30851i −5.62770 4.63386 + 1.25485i
419.7 1.98136i 2.35379i −1.92578 0.201454 2.22697i −4.66371 1.46477i 0.147060i −2.54034 −4.41243 0.399153i
419.8 1.66433i 1.25863i −0.769991 −1.73458 + 1.41111i −2.09478 4.13429i 2.04714i 1.41584 2.34855 + 2.88691i
419.9 1.65738i 1.46266i −0.746921 2.23602 + 0.0141055i 2.42418 5.18504i 2.07683i 0.860633 0.0233783 3.70595i
419.10 1.45151i 0.0791235i −0.106883 −2.19310 + 0.436262i −0.114849 2.96150i 2.74788i 2.99374 0.633239 + 3.18330i
419.11 0.961626i 0.510054i 1.07528 1.58322 + 1.57905i 0.490481 0.688271i 2.95726i 2.73984 1.51846 1.52247i
419.12 0.661223i 3.01131i 1.56278 −1.75827 1.38148i −1.99115 0.592449i 2.35579i −6.06799 −0.913465 + 1.16261i
419.13 0.598278i 1.31473i 1.64206 1.14673 + 1.91964i −0.786572 0.976473i 2.17897i 1.27149 1.14848 0.686061i
419.14 0.379302i 2.60759i 1.85613 2.20907 0.346449i 0.989065 4.15654i 1.46264i −3.79954 −0.131409 0.837903i
419.15 0.202376i 2.47875i 1.95904 −1.53426 1.62666i 0.501638 5.06314i 0.801215i −3.14418 −0.329197 + 0.310498i
419.16 0.202376i 2.47875i 1.95904 −1.53426 + 1.62666i 0.501638 5.06314i 0.801215i −3.14418 −0.329197 0.310498i
419.17 0.379302i 2.60759i 1.85613 2.20907 + 0.346449i 0.989065 4.15654i 1.46264i −3.79954 −0.131409 + 0.837903i
419.18 0.598278i 1.31473i 1.64206 1.14673 1.91964i −0.786572 0.976473i 2.17897i 1.27149 1.14848 + 0.686061i
419.19 0.661223i 3.01131i 1.56278 −1.75827 + 1.38148i −1.99115 0.592449i 2.35579i −6.06799 −0.913465 1.16261i
419.20 0.961626i 0.510054i 1.07528 1.58322 1.57905i 0.490481 0.688271i 2.95726i 2.73984 1.51846 + 1.52247i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.e 30
5.b even 2 1 inner 1045.2.b.e 30
5.c odd 4 2 5225.2.a.bc 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.e 30 1.a even 1 1 trivial
1045.2.b.e 30 5.b even 2 1 inner
5225.2.a.bc 30 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{30} + 51 T_{2}^{28} + 1161 T_{2}^{26} + 15571 T_{2}^{24} + 136754 T_{2}^{22} + 826847 T_{2}^{20} + 3521942 T_{2}^{18} + 10632298 T_{2}^{16} + 22578273 T_{2}^{14} + 33035145 T_{2}^{12} + 32143513 T_{2}^{10} + \cdots + 2916$$ acting on $$S_{2}^{\mathrm{new}}(1045, [\chi])$$.