# Properties

 Label 503.2.a.f Level $503$ Weight $2$ Character orbit 503.a Self dual yes Analytic conductor $4.016$ Analytic rank $0$ Dimension $26$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 503.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.01647522167$$ Analytic rank: $$0$$ Dimension: $$26$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$26q + 4q^{2} + 4q^{3} + 36q^{4} + 9q^{5} - 4q^{6} + 11q^{7} + 18q^{8} + 42q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$26q + 4q^{2} + 4q^{3} + 36q^{4} + 9q^{5} - 4q^{6} + 11q^{7} + 18q^{8} + 42q^{9} + 4q^{10} - 17q^{11} + 19q^{12} + 14q^{13} + q^{14} + 18q^{15} + 48q^{16} + 17q^{17} - 10q^{18} - 22q^{19} - 19q^{20} - 16q^{21} + 38q^{22} + 27q^{23} - 9q^{24} + 93q^{25} + q^{26} + 31q^{27} - 9q^{28} + 13q^{29} - 28q^{30} + 26q^{31} + 5q^{32} + 6q^{33} - 32q^{34} - 22q^{35} + 52q^{36} + 55q^{37} - 24q^{38} - 15q^{39} - 7q^{40} + 24q^{41} - 50q^{42} + 20q^{43} - 27q^{44} - 8q^{45} + 6q^{46} - 25q^{47} + 29q^{48} + 65q^{49} - 16q^{50} + 7q^{51} + 32q^{52} + 30q^{53} - 82q^{54} + 25q^{55} + 3q^{56} + 9q^{57} + 58q^{58} - 26q^{59} - 68q^{60} + 15q^{61} - 12q^{62} - 19q^{63} + 44q^{64} + 20q^{65} - 55q^{66} - 20q^{67} - 4q^{68} - 27q^{69} + 2q^{70} - 35q^{71} - 26q^{72} + 38q^{73} - 59q^{74} + 2q^{75} - 42q^{76} - 6q^{77} - 47q^{78} + 21q^{79} - 100q^{80} + 70q^{81} - 59q^{82} - 48q^{83} - 116q^{84} + 6q^{85} - 7q^{86} - 9q^{87} + 106q^{88} - 5q^{89} - 118q^{90} - 24q^{91} + 26q^{92} - 8q^{93} - 22q^{94} + 43q^{95} - 100q^{96} + 142q^{97} - 38q^{98} - 50q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.73065 0.249551 5.45647 −3.41933 −0.681437 −3.14640 −9.43843 −2.93772 9.33700
1.2 −2.69742 3.34519 5.27606 2.36922 −9.02336 −0.269938 −8.83689 8.19027 −6.39078
1.3 −2.26823 −1.83158 3.14489 −3.37084 4.15444 3.93290 −2.59688 0.354669 7.64586
1.4 −2.16310 −2.85594 2.67899 1.91587 6.17767 1.96852 −1.46872 5.15637 −4.14421
1.5 −1.89206 1.34769 1.57988 4.23270 −2.54991 −3.43664 0.794895 −1.18373 −8.00851
1.6 −1.79231 2.94588 1.21237 −4.09318 −5.27994 3.85397 1.41167 5.67824 7.33625
1.7 −1.44158 −1.65492 0.0781455 1.16845 2.38569 −5.22170 2.77050 −0.261242 −1.68442
1.8 −1.43135 2.32813 0.0487521 1.62138 −3.33236 3.43484 2.79291 2.42020 −2.32076
1.9 −1.28997 −1.08298 −0.335977 3.81229 1.39701 4.19464 3.01334 −1.82716 −4.91774
1.10 −0.462244 3.26409 −1.78633 1.15565 −1.50881 −1.19216 1.75021 7.65430 −0.534193
1.11 −0.425199 −1.16576 −1.81921 −3.04539 0.495677 0.946556 1.62392 −1.64102 1.29490
1.12 −0.349081 1.74581 −1.87814 1.36534 −0.609428 −0.0430977 1.35379 0.0478420 −0.476616
1.13 −0.328801 −2.39019 −1.89189 −3.16539 0.785896 −3.22876 1.27966 2.71299 1.04078
1.14 0.342177 −2.49378 −1.88292 4.14215 −0.853311 −0.810419 −1.32864 3.21891 1.41735
1.15 0.477293 −0.705033 −1.77219 −0.869095 −0.336507 5.04945 −1.80044 −2.50293 −0.414813
1.16 0.738482 2.27911 −1.45464 3.79116 1.68308 4.03115 −2.55119 2.19436 2.79970
1.17 1.26316 −3.11338 −0.404427 −4.34050 −3.93270 1.70318 −3.03718 6.69315 −5.48274
1.18 1.61798 2.76613 0.617860 3.34245 4.47554 −5.14301 −2.23628 4.65146 5.40802
1.19 1.81497 0.577924 1.29413 2.10638 1.04892 1.73516 −1.28114 −2.66600 3.82302
1.20 1.88946 −1.08803 1.57004 3.67682 −2.05579 1.62501 −0.812389 −1.81619 6.94719
See all 26 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.26 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$503$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 503.2.a.f 26
3.b odd 2 1 4527.2.a.o 26
4.b odd 2 1 8048.2.a.u 26

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.f 26 1.a even 1 1 trivial
4527.2.a.o 26 3.b odd 2 1
8048.2.a.u 26 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(503))$$:

 $$T_{2}^{26} - \cdots$$ $$T_{3}^{26} - \cdots$$