# Properties

 Label 1045.4.a.d Level $1045$ Weight $4$ Character orbit 1045.a Self dual yes Analytic conductor $61.657$ Analytic rank $1$ Dimension $22$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1045.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$61.6569959560$$ Analytic rank: $$1$$ Dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10})$$ 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100})$$ 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −5.35702 −10.2482 20.6977 5.00000 54.9000 −17.5818 −68.0218 78.0263 −26.7851
1.2 −4.84890 −0.335233 15.5118 5.00000 1.62551 −24.3429 −36.4242 −26.8876 −24.2445
1.3 −4.73877 1.86006 14.4559 5.00000 −8.81437 1.31072 −30.5930 −23.5402 −23.6938
1.4 −4.71998 6.85295 14.2783 5.00000 −32.3458 15.9495 −29.6333 19.9630 −23.5999
1.5 −3.16156 −8.60870 1.99549 5.00000 27.2170 32.5634 18.9837 47.1098 −15.8078
1.6 −3.11441 −0.179144 1.69958 5.00000 0.557928 16.3837 19.6221 −26.9679 −15.5721
1.7 −2.86531 −5.40234 0.209982 5.00000 15.4794 −24.6494 22.3208 2.18531 −14.3265
1.8 −2.52988 7.31543 −1.59971 5.00000 −18.5072 2.79284 24.2861 26.5155 −12.6494
1.9 −2.22182 6.66604 −3.06353 5.00000 −14.8107 −18.2528 24.5811 17.4361 −11.1091
1.10 −1.15290 −8.73139 −6.67082 5.00000 10.0664 10.3417 16.9140 49.2372 −5.76450
1.11 −0.0295893 −3.21150 −7.99912 5.00000 0.0950261 7.73734 0.473403 −16.6862 −0.147946
1.12 0.0768470 2.42732 −7.99409 5.00000 0.186533 −11.4153 −1.22910 −21.1081 0.384235
1.13 1.04826 7.96396 −6.90116 5.00000 8.34827 −11.0379 −15.6202 36.4247 5.24128
1.14 1.41332 3.57743 −6.00252 5.00000 5.05605 30.1984 −19.7901 −14.2020 7.06661
1.15 1.45119 −8.75662 −5.89406 5.00000 −12.7075 −31.7430 −20.1629 49.6783 7.25593
1.16 2.01284 −4.64592 −3.94849 5.00000 −9.35147 7.18147 −24.0504 −5.41547 10.0642
1.17 3.65397 3.36009 5.35150 5.00000 12.2777 −1.36266 −9.67753 −15.7098 18.2699
1.18 3.68170 2.35610 5.55493 5.00000 8.67444 −1.20638 −9.00201 −21.4488 18.4085
1.19 3.88965 −9.46724 7.12938 5.00000 −36.8243 13.8479 −3.38640 62.6286 19.4483
1.20 4.17470 5.44396 9.42811 5.00000 22.7269 −28.9642 5.96192 2.63670 20.8735
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$1$$

## 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 1045.4.a.d 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.d 22 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{22} + 4 T_{2}^{21} - 117 T_{2}^{20} - 442 T_{2}^{19} + 5863 T_{2}^{18} + 20719 T_{2}^{17} - 164424 T_{2}^{16} - 537203 T_{2}^{15} + 2831700 T_{2}^{14} + 8413712 T_{2}^{13} - 30998309 T_{2}^{12} + \cdots - 4958464$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1045))$$.