# Properties

 Label 1421.4.a.c Level $1421$ Weight $4$ Character orbit 1421.a Self dual yes Analytic conductor $83.842$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1421 = 7^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1421.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$83.8417141182$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 5 + 3 \beta ) q^{3} + ( -5 - 2 \beta ) q^{4} + ( 5 - 4 \beta ) q^{5} + ( 1 + 2 \beta ) q^{6} + ( 9 - 11 \beta ) q^{8} + ( 16 + 30 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 5 + 3 \beta ) q^{3} + ( -5 - 2 \beta ) q^{4} + ( 5 - 4 \beta ) q^{5} + ( 1 + 2 \beta ) q^{6} + ( 9 - 11 \beta ) q^{8} + ( 16 + 30 \beta ) q^{9} + ( -13 + 9 \beta ) q^{10} + ( -13 - 37 \beta ) q^{11} + ( -37 - 25 \beta ) q^{12} + ( 13 + 26 \beta ) q^{13} + ( 1 - 5 \beta ) q^{15} + ( 9 + 36 \beta ) q^{16} + ( -30 - 18 \beta ) q^{17} + ( 44 - 14 \beta ) q^{18} + ( 110 - 32 \beta ) q^{19} + ( -9 + 10 \beta ) q^{20} + ( -61 + 24 \beta ) q^{22} + ( 26 + 48 \beta ) q^{23} + ( -21 - 28 \beta ) q^{24} + ( -68 - 40 \beta ) q^{25} + ( 39 - 13 \beta ) q^{26} + ( 125 + 117 \beta ) q^{27} + 29 q^{29} + ( -11 + 6 \beta ) q^{30} + ( 147 + 63 \beta ) q^{31} + ( -9 + 61 \beta ) q^{32} + ( -287 - 224 \beta ) q^{33} + ( -6 - 12 \beta ) q^{34} + ( -200 - 182 \beta ) q^{36} + ( 156 - 56 \beta ) q^{37} + ( -174 + 142 \beta ) q^{38} + ( 221 + 169 \beta ) q^{39} + ( 133 - 91 \beta ) q^{40} + ( -20 - 138 \beta ) q^{41} + ( -161 + 171 \beta ) q^{43} + ( 213 + 211 \beta ) q^{44} + ( -160 + 86 \beta ) q^{45} + ( 70 - 22 \beta ) q^{46} + ( 65 + 207 \beta ) q^{47} + ( 261 + 207 \beta ) q^{48} + ( -12 - 28 \beta ) q^{50} + ( -258 - 180 \beta ) q^{51} + ( -169 - 156 \beta ) q^{52} + ( 501 - 122 \beta ) q^{53} + ( 109 + 8 \beta ) q^{54} + ( 231 - 133 \beta ) q^{55} + ( 358 + 170 \beta ) q^{57} + ( -29 + 29 \beta ) q^{58} + ( 450 - 248 \beta ) q^{59} + ( 15 + 23 \beta ) q^{60} + ( 474 + 178 \beta ) q^{61} + ( -21 + 84 \beta ) q^{62} + ( 59 - 358 \beta ) q^{64} + ( -143 + 78 \beta ) q^{65} + ( -161 - 63 \beta ) q^{66} + ( 160 + 484 \beta ) q^{67} + ( 222 + 150 \beta ) q^{68} + ( 418 + 318 \beta ) q^{69} + ( -330 - 34 \beta ) q^{71} + ( -516 + 94 \beta ) q^{72} + ( -324 + 640 \beta ) q^{73} + ( -268 + 212 \beta ) q^{74} + ( -580 - 404 \beta ) q^{75} + ( -422 - 60 \beta ) q^{76} + ( 117 + 52 \beta ) q^{78} + ( 129 - 341 \beta ) q^{79} + ( -243 + 144 \beta ) q^{80} + ( 895 + 150 \beta ) q^{81} + ( -256 + 118 \beta ) q^{82} + ( -606 - 64 \beta ) q^{83} + ( -6 + 30 \beta ) q^{85} + ( 503 - 332 \beta ) q^{86} + ( 145 + 87 \beta ) q^{87} + ( 697 - 190 \beta ) q^{88} + ( -380 - 522 \beta ) q^{89} + ( 332 - 246 \beta ) q^{90} + ( -322 - 292 \beta ) q^{92} + ( 1113 + 756 \beta ) q^{93} + ( 349 - 142 \beta ) q^{94} + ( 806 - 600 \beta ) q^{95} + ( 321 + 278 \beta ) q^{96} + ( -12 + 578 \beta ) q^{97} + ( -2428 - 982 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 10 q^{3} - 10 q^{4} + 10 q^{5} + 2 q^{6} + 18 q^{8} + 32 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + 10 q^{3} - 10 q^{4} + 10 q^{5} + 2 q^{6} + 18 q^{8} + 32 q^{9} - 26 q^{10} - 26 q^{11} - 74 q^{12} + 26 q^{13} + 2 q^{15} + 18 q^{16} - 60 q^{17} + 88 q^{18} + 220 q^{19} - 18 q^{20} - 122 q^{22} + 52 q^{23} - 42 q^{24} - 136 q^{25} + 78 q^{26} + 250 q^{27} + 58 q^{29} - 22 q^{30} + 294 q^{31} - 18 q^{32} - 574 q^{33} - 12 q^{34} - 400 q^{36} + 312 q^{37} - 348 q^{38} + 442 q^{39} + 266 q^{40} - 40 q^{41} - 322 q^{43} + 426 q^{44} - 320 q^{45} + 140 q^{46} + 130 q^{47} + 522 q^{48} - 24 q^{50} - 516 q^{51} - 338 q^{52} + 1002 q^{53} + 218 q^{54} + 462 q^{55} + 716 q^{57} - 58 q^{58} + 900 q^{59} + 30 q^{60} + 948 q^{61} - 42 q^{62} + 118 q^{64} - 286 q^{65} - 322 q^{66} + 320 q^{67} + 444 q^{68} + 836 q^{69} - 660 q^{71} - 1032 q^{72} - 648 q^{73} - 536 q^{74} - 1160 q^{75} - 844 q^{76} + 234 q^{78} + 258 q^{79} - 486 q^{80} + 1790 q^{81} - 512 q^{82} - 1212 q^{83} - 12 q^{85} + 1006 q^{86} + 290 q^{87} + 1394 q^{88} - 760 q^{89} + 664 q^{90} - 644 q^{92} + 2226 q^{93} + 698 q^{94} + 1612 q^{95} + 642 q^{96} - 24 q^{97} - 4856 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−2.41421 0.757359 −2.17157 10.6569 −1.82843 0 24.5563 −26.4264 −25.7279
1.2 0.414214 9.24264 −7.82843 −0.656854 3.82843 0 −6.55635 58.4264 −0.272078
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$29$$ $$-1$$

## 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 1421.4.a.c 2
7.b odd 2 1 29.4.a.a 2
21.c even 2 1 261.4.a.b 2
28.d even 2 1 464.4.a.f 2
35.c odd 2 1 725.4.a.b 2
56.e even 2 1 1856.4.a.h 2
56.h odd 2 1 1856.4.a.n 2
203.c odd 2 1 841.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.a 2 7.b odd 2 1
261.4.a.b 2 21.c even 2 1
464.4.a.f 2 28.d even 2 1
725.4.a.b 2 35.c odd 2 1
841.4.a.a 2 203.c odd 2 1
1421.4.a.c 2 1.a even 1 1 trivial
1856.4.a.h 2 56.e even 2 1
1856.4.a.n 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1421))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{3}^{2} - 10 T_{3} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$7 - 10 T + T^{2}$$
$5$ $$-7 - 10 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-2569 + 26 T + T^{2}$$
$13$ $$-1183 - 26 T + T^{2}$$
$17$ $$252 + 60 T + T^{2}$$
$19$ $$10052 - 220 T + T^{2}$$
$23$ $$-3932 - 52 T + T^{2}$$
$29$ $$( -29 + T )^{2}$$
$31$ $$13671 - 294 T + T^{2}$$
$37$ $$18064 - 312 T + T^{2}$$
$41$ $$-37688 + 40 T + T^{2}$$
$43$ $$-32561 + 322 T + T^{2}$$
$47$ $$-81473 - 130 T + T^{2}$$
$53$ $$221233 - 1002 T + T^{2}$$
$59$ $$79492 - 900 T + T^{2}$$
$61$ $$161308 - 948 T + T^{2}$$
$67$ $$-442912 - 320 T + T^{2}$$
$71$ $$106588 + 660 T + T^{2}$$
$73$ $$-714224 + 648 T + T^{2}$$
$79$ $$-215921 - 258 T + T^{2}$$
$83$ $$359044 + 1212 T + T^{2}$$
$89$ $$-400568 + 760 T + T^{2}$$
$97$ $$-668024 + 24 T + T^{2}$$