# Properties

 Label 1450.4.a.g Level $1450$ Weight $4$ Character orbit 1450.a Self dual yes Analytic conductor $85.553$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$85.5527695083$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (2 \beta + 2) q^{6} + (8 \beta + 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9}+O(q^{10})$$ q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + (2*b + 2) * q^6 + (8*b + 8) * q^7 + 8 * q^8 + (2*b - 20) * q^9 $$q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (2 \beta + 2) q^{6} + (8 \beta + 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9} + ( - 3 \beta - 45) q^{11} + (4 \beta + 4) q^{12} + ( - 8 \beta + 25) q^{13} + (16 \beta + 16) q^{14} + 16 q^{16} + ( - 16 \beta + 22) q^{17} + (4 \beta - 40) q^{18} + ( - 4 \beta + 54) q^{19} + (16 \beta + 56) q^{21} + ( - 6 \beta - 90) q^{22} + (78 \beta + 14) q^{23} + (8 \beta + 8) q^{24} + ( - 16 \beta + 50) q^{26} + ( - 45 \beta - 35) q^{27} + (32 \beta + 32) q^{28} + 29 q^{29} + (109 \beta + 33) q^{31} + 32 q^{32} + ( - 48 \beta - 63) q^{33} + ( - 32 \beta + 44) q^{34} + (8 \beta - 80) q^{36} + ( - 4 \beta - 20) q^{37} + ( - 8 \beta + 108) q^{38} + (17 \beta - 23) q^{39} + ( - 80 \beta + 152) q^{41} + (32 \beta + 112) q^{42} + ( - 53 \beta + 65) q^{43} + ( - 12 \beta - 180) q^{44} + (156 \beta + 28) q^{46} + (99 \beta + 257) q^{47} + (16 \beta + 16) q^{48} + (128 \beta + 105) q^{49} + (6 \beta - 74) q^{51} + ( - 32 \beta + 100) q^{52} + ( - 52 \beta + 479) q^{53} + ( - 90 \beta - 70) q^{54} + (64 \beta + 64) q^{56} + (50 \beta + 30) q^{57} + 58 q^{58} + (250 \beta - 90) q^{59} + ( - 12 \beta + 514) q^{61} + (218 \beta + 66) q^{62} + ( - 144 \beta - 64) q^{63} + 64 q^{64} + ( - 96 \beta - 126) q^{66} + (20 \beta + 456) q^{67} + ( - 64 \beta + 88) q^{68} + (92 \beta + 482) q^{69} + (34 \beta + 398) q^{71} + (16 \beta - 160) q^{72} + ( - 176 \beta + 428) q^{73} + ( - 8 \beta - 40) q^{74} + ( - 16 \beta + 216) q^{76} + ( - 384 \beta - 504) q^{77} + (34 \beta - 46) q^{78} + ( - 361 \beta - 159) q^{79} + ( - 134 \beta + 235) q^{81} + ( - 160 \beta + 304) q^{82} + (38 \beta + 914) q^{83} + (64 \beta + 224) q^{84} + ( - 106 \beta + 130) q^{86} + (29 \beta + 29) q^{87} + ( - 24 \beta - 360) q^{88} + (72 \beta - 472) q^{89} + (136 \beta - 184) q^{91} + (312 \beta + 56) q^{92} + (142 \beta + 687) q^{93} + (198 \beta + 514) q^{94} + (32 \beta + 32) q^{96} + ( - 100 \beta - 184) q^{97} + (256 \beta + 210) q^{98} + ( - 30 \beta + 864) q^{99}+O(q^{100})$$ q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + (2*b + 2) * q^6 + (8*b + 8) * q^7 + 8 * q^8 + (2*b - 20) * q^9 + (-3*b - 45) * q^11 + (4*b + 4) * q^12 + (-8*b + 25) * q^13 + (16*b + 16) * q^14 + 16 * q^16 + (-16*b + 22) * q^17 + (4*b - 40) * q^18 + (-4*b + 54) * q^19 + (16*b + 56) * q^21 + (-6*b - 90) * q^22 + (78*b + 14) * q^23 + (8*b + 8) * q^24 + (-16*b + 50) * q^26 + (-45*b - 35) * q^27 + (32*b + 32) * q^28 + 29 * q^29 + (109*b + 33) * q^31 + 32 * q^32 + (-48*b - 63) * q^33 + (-32*b + 44) * q^34 + (8*b - 80) * q^36 + (-4*b - 20) * q^37 + (-8*b + 108) * q^38 + (17*b - 23) * q^39 + (-80*b + 152) * q^41 + (32*b + 112) * q^42 + (-53*b + 65) * q^43 + (-12*b - 180) * q^44 + (156*b + 28) * q^46 + (99*b + 257) * q^47 + (16*b + 16) * q^48 + (128*b + 105) * q^49 + (6*b - 74) * q^51 + (-32*b + 100) * q^52 + (-52*b + 479) * q^53 + (-90*b - 70) * q^54 + (64*b + 64) * q^56 + (50*b + 30) * q^57 + 58 * q^58 + (250*b - 90) * q^59 + (-12*b + 514) * q^61 + (218*b + 66) * q^62 + (-144*b - 64) * q^63 + 64 * q^64 + (-96*b - 126) * q^66 + (20*b + 456) * q^67 + (-64*b + 88) * q^68 + (92*b + 482) * q^69 + (34*b + 398) * q^71 + (16*b - 160) * q^72 + (-176*b + 428) * q^73 + (-8*b - 40) * q^74 + (-16*b + 216) * q^76 + (-384*b - 504) * q^77 + (34*b - 46) * q^78 + (-361*b - 159) * q^79 + (-134*b + 235) * q^81 + (-160*b + 304) * q^82 + (38*b + 914) * q^83 + (64*b + 224) * q^84 + (-106*b + 130) * q^86 + (29*b + 29) * q^87 + (-24*b - 360) * q^88 + (72*b - 472) * q^89 + (136*b - 184) * q^91 + (312*b + 56) * q^92 + (142*b + 687) * q^93 + (198*b + 514) * q^94 + (32*b + 32) * q^96 + (-100*b - 184) * q^97 + (256*b + 210) * q^98 + (-30*b + 864) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{7} + 16 q^{8} - 40 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^6 + 16 * q^7 + 16 * q^8 - 40 * q^9 $$2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} + 4 q^{6} + 16 q^{7} + 16 q^{8} - 40 q^{9} - 90 q^{11} + 8 q^{12} + 50 q^{13} + 32 q^{14} + 32 q^{16} + 44 q^{17} - 80 q^{18} + 108 q^{19} + 112 q^{21} - 180 q^{22} + 28 q^{23} + 16 q^{24} + 100 q^{26} - 70 q^{27} + 64 q^{28} + 58 q^{29} + 66 q^{31} + 64 q^{32} - 126 q^{33} + 88 q^{34} - 160 q^{36} - 40 q^{37} + 216 q^{38} - 46 q^{39} + 304 q^{41} + 224 q^{42} + 130 q^{43} - 360 q^{44} + 56 q^{46} + 514 q^{47} + 32 q^{48} + 210 q^{49} - 148 q^{51} + 200 q^{52} + 958 q^{53} - 140 q^{54} + 128 q^{56} + 60 q^{57} + 116 q^{58} - 180 q^{59} + 1028 q^{61} + 132 q^{62} - 128 q^{63} + 128 q^{64} - 252 q^{66} + 912 q^{67} + 176 q^{68} + 964 q^{69} + 796 q^{71} - 320 q^{72} + 856 q^{73} - 80 q^{74} + 432 q^{76} - 1008 q^{77} - 92 q^{78} - 318 q^{79} + 470 q^{81} + 608 q^{82} + 1828 q^{83} + 448 q^{84} + 260 q^{86} + 58 q^{87} - 720 q^{88} - 944 q^{89} - 368 q^{91} + 112 q^{92} + 1374 q^{93} + 1028 q^{94} + 64 q^{96} - 368 q^{97} + 420 q^{98} + 1728 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 + 4 * q^6 + 16 * q^7 + 16 * q^8 - 40 * q^9 - 90 * q^11 + 8 * q^12 + 50 * q^13 + 32 * q^14 + 32 * q^16 + 44 * q^17 - 80 * q^18 + 108 * q^19 + 112 * q^21 - 180 * q^22 + 28 * q^23 + 16 * q^24 + 100 * q^26 - 70 * q^27 + 64 * q^28 + 58 * q^29 + 66 * q^31 + 64 * q^32 - 126 * q^33 + 88 * q^34 - 160 * q^36 - 40 * q^37 + 216 * q^38 - 46 * q^39 + 304 * q^41 + 224 * q^42 + 130 * q^43 - 360 * q^44 + 56 * q^46 + 514 * q^47 + 32 * q^48 + 210 * q^49 - 148 * q^51 + 200 * q^52 + 958 * q^53 - 140 * q^54 + 128 * q^56 + 60 * q^57 + 116 * q^58 - 180 * q^59 + 1028 * q^61 + 132 * q^62 - 128 * q^63 + 128 * q^64 - 252 * q^66 + 912 * q^67 + 176 * q^68 + 964 * q^69 + 796 * q^71 - 320 * q^72 + 856 * q^73 - 80 * q^74 + 432 * q^76 - 1008 * q^77 - 92 * q^78 - 318 * q^79 + 470 * q^81 + 608 * q^82 + 1828 * q^83 + 448 * q^84 + 260 * q^86 + 58 * q^87 - 720 * q^88 - 944 * q^89 - 368 * q^91 + 112 * q^92 + 1374 * q^93 + 1028 * q^94 + 64 * q^96 - 368 * q^97 + 420 * q^98 + 1728 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.44949 2.44949
2.00000 −1.44949 4.00000 0 −2.89898 −11.5959 8.00000 −24.8990 0
1.2 2.00000 3.44949 4.00000 0 6.89898 27.5959 8.00000 −15.1010 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$29$$ $$-1$$

## 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 1450.4.a.g 2
5.b even 2 1 58.4.a.c 2
15.d odd 2 1 522.4.a.j 2
20.d odd 2 1 464.4.a.e 2
40.e odd 2 1 1856.4.a.i 2
40.f even 2 1 1856.4.a.l 2
145.d even 2 1 1682.4.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 5.b even 2 1
464.4.a.e 2 20.d odd 2 1
522.4.a.j 2 15.d odd 2 1
1450.4.a.g 2 1.a even 1 1 trivial
1682.4.a.c 2 145.d even 2 1
1856.4.a.i 2 40.e odd 2 1
1856.4.a.l 2 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1450))$$:

 $$T_{3}^{2} - 2T_{3} - 5$$ T3^2 - 2*T3 - 5 $$T_{7}^{2} - 16T_{7} - 320$$ T7^2 - 16*T7 - 320

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} - 2T - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 16T - 320$$
$11$ $$T^{2} + 90T + 1971$$
$13$ $$T^{2} - 50T + 241$$
$17$ $$T^{2} - 44T - 1052$$
$19$ $$T^{2} - 108T + 2820$$
$23$ $$T^{2} - 28T - 36308$$
$29$ $$(T - 29)^{2}$$
$31$ $$T^{2} - 66T - 70197$$
$37$ $$T^{2} + 40T + 304$$
$41$ $$T^{2} - 304T - 15296$$
$43$ $$T^{2} - 130T - 12629$$
$47$ $$T^{2} - 514T + 7243$$
$53$ $$T^{2} - 958T + 213217$$
$59$ $$T^{2} + 180T - 366900$$
$61$ $$T^{2} - 1028 T + 263332$$
$67$ $$T^{2} - 912T + 205536$$
$71$ $$T^{2} - 796T + 151468$$
$73$ $$T^{2} - 856T - 2672$$
$79$ $$T^{2} + 318T - 756645$$
$83$ $$T^{2} - 1828 T + 826732$$
$89$ $$T^{2} + 944T + 191680$$
$97$ $$T^{2} + 368T - 26144$$