# Properties

 Label 13.4.a.b Level 13 Weight 4 Character orbit 13.a Self dual yes Analytic conductor 0.767 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 13.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.767024830075$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( 4 - 3 \beta ) q^{3} + ( -4 + \beta ) q^{4} + ( -2 + \beta ) q^{5} + ( -12 + \beta ) q^{6} + ( -10 + 11 \beta ) q^{7} + ( 4 - 11 \beta ) q^{8} + ( 25 - 15 \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( 4 - 3 \beta ) q^{3} + ( -4 + \beta ) q^{4} + ( -2 + \beta ) q^{5} + ( -12 + \beta ) q^{6} + ( -10 + 11 \beta ) q^{7} + ( 4 - 11 \beta ) q^{8} + ( 25 - 15 \beta ) q^{9} + ( 4 - \beta ) q^{10} + ( 34 + 12 \beta ) q^{11} + ( -28 + 13 \beta ) q^{12} -13 q^{13} + ( 44 + \beta ) q^{14} + ( -20 + 7 \beta ) q^{15} + ( -12 - 15 \beta ) q^{16} + ( 18 - 17 \beta ) q^{17} + ( -60 + 10 \beta ) q^{18} + ( -26 - 32 \beta ) q^{19} + ( 12 - 5 \beta ) q^{20} + ( -172 + 41 \beta ) q^{21} + ( 48 + 46 \beta ) q^{22} + ( 104 - 12 \beta ) q^{23} + ( 148 - 23 \beta ) q^{24} + ( -117 - 3 \beta ) q^{25} -13 \beta q^{26} + ( 172 - 9 \beta ) q^{27} + ( 84 - 43 \beta ) q^{28} + ( -70 + 96 \beta ) q^{29} + ( 28 - 13 \beta ) q^{30} + ( -26 - 34 \beta ) q^{31} + ( -92 + 61 \beta ) q^{32} + ( -8 - 90 \beta ) q^{33} + ( -68 + \beta ) q^{34} + ( 64 - 21 \beta ) q^{35} + ( -160 + 70 \beta ) q^{36} + ( 102 + 5 \beta ) q^{37} + ( -128 - 58 \beta ) q^{38} + ( -52 + 39 \beta ) q^{39} + ( -52 + 15 \beta ) q^{40} + ( -126 + 22 \beta ) q^{41} + ( 164 - 131 \beta ) q^{42} + ( 72 + 143 \beta ) q^{43} + ( -88 - 2 \beta ) q^{44} + ( -110 + 40 \beta ) q^{45} + ( -48 + 92 \beta ) q^{46} + ( 278 - 121 \beta ) q^{47} + ( 132 + 21 \beta ) q^{48} + ( 241 - 99 \beta ) q^{49} + ( -12 - 120 \beta ) q^{50} + ( 276 - 71 \beta ) q^{51} + ( 52 - 13 \beta ) q^{52} + ( -74 + 30 \beta ) q^{53} + ( -36 + 163 \beta ) q^{54} + ( -20 + 22 \beta ) q^{55} + ( -524 + 33 \beta ) q^{56} + ( 280 + 46 \beta ) q^{57} + ( 384 + 26 \beta ) q^{58} + ( -246 + 124 \beta ) q^{59} + ( 108 - 41 \beta ) q^{60} + ( -434 - 190 \beta ) q^{61} + ( -136 - 60 \beta ) q^{62} + ( -910 + 260 \beta ) q^{63} + ( 340 + 89 \beta ) q^{64} + ( 26 - 13 \beta ) q^{65} + ( -360 - 98 \beta ) q^{66} + ( 150 - 232 \beta ) q^{67} + ( -140 + 69 \beta ) q^{68} + ( 560 - 324 \beta ) q^{69} + ( -84 + 43 \beta ) q^{70} + ( 50 - 231 \beta ) q^{71} + ( 760 - 170 \beta ) q^{72} + ( 98 + 260 \beta ) q^{73} + ( 20 + 107 \beta ) q^{74} + ( -432 + 348 \beta ) q^{75} + ( -24 + 70 \beta ) q^{76} + ( 188 + 386 \beta ) q^{77} + ( 156 - 13 \beta ) q^{78} + ( -524 + 40 \beta ) q^{79} + ( -36 + 3 \beta ) q^{80} + ( 121 - 120 \beta ) q^{81} + ( 88 - 104 \beta ) q^{82} + ( 1070 - 182 \beta ) q^{83} + ( 852 - 295 \beta ) q^{84} + ( -104 + 35 \beta ) q^{85} + ( 572 + 215 \beta ) q^{86} + ( -1432 + 306 \beta ) q^{87} + ( -392 - 458 \beta ) q^{88} + ( -166 - 388 \beta ) q^{89} + ( 160 - 70 \beta ) q^{90} + ( 130 - 143 \beta ) q^{91} + ( -464 + 140 \beta ) q^{92} + ( 304 + 44 \beta ) q^{93} + ( -484 + 157 \beta ) q^{94} + ( -76 + 6 \beta ) q^{95} + ( -1100 + 337 \beta ) q^{96} + ( -718 + 508 \beta ) q^{97} + ( -396 + 142 \beta ) q^{98} + ( 130 - 390 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 5q^{3} - 7q^{4} - 3q^{5} - 23q^{6} - 9q^{7} - 3q^{8} + 35q^{9} + O(q^{10})$$ $$2q + q^{2} + 5q^{3} - 7q^{4} - 3q^{5} - 23q^{6} - 9q^{7} - 3q^{8} + 35q^{9} + 7q^{10} + 80q^{11} - 43q^{12} - 26q^{13} + 89q^{14} - 33q^{15} - 39q^{16} + 19q^{17} - 110q^{18} - 84q^{19} + 19q^{20} - 303q^{21} + 142q^{22} + 196q^{23} + 273q^{24} - 237q^{25} - 13q^{26} + 335q^{27} + 125q^{28} - 44q^{29} + 43q^{30} - 86q^{31} - 123q^{32} - 106q^{33} - 135q^{34} + 107q^{35} - 250q^{36} + 209q^{37} - 314q^{38} - 65q^{39} - 89q^{40} - 230q^{41} + 197q^{42} + 287q^{43} - 178q^{44} - 180q^{45} - 4q^{46} + 435q^{47} + 285q^{48} + 383q^{49} - 144q^{50} + 481q^{51} + 91q^{52} - 118q^{53} + 91q^{54} - 18q^{55} - 1015q^{56} + 606q^{57} + 794q^{58} - 368q^{59} + 175q^{60} - 1058q^{61} - 332q^{62} - 1560q^{63} + 769q^{64} + 39q^{65} - 818q^{66} + 68q^{67} - 211q^{68} + 796q^{69} - 125q^{70} - 131q^{71} + 1350q^{72} + 456q^{73} + 147q^{74} - 516q^{75} + 22q^{76} + 762q^{77} + 299q^{78} - 1008q^{79} - 69q^{80} + 122q^{81} + 72q^{82} + 1958q^{83} + 1409q^{84} - 173q^{85} + 1359q^{86} - 2558q^{87} - 1242q^{88} - 720q^{89} + 250q^{90} + 117q^{91} - 788q^{92} + 652q^{93} - 811q^{94} - 146q^{95} - 1863q^{96} - 928q^{97} - 650q^{98} - 130q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
−1.56155 8.68466 −5.56155 −3.56155 −13.5616 −27.1771 21.1771 48.4233 5.56155
1.2 2.56155 −3.68466 −1.43845 0.561553 −9.43845 18.1771 −24.1771 −13.4233 1.43845
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 13.4.a.b 2
3.b odd 2 1 117.4.a.d 2
4.b odd 2 1 208.4.a.h 2
5.b even 2 1 325.4.a.f 2
5.c odd 4 2 325.4.b.e 4
7.b odd 2 1 637.4.a.b 2
8.b even 2 1 832.4.a.s 2
8.d odd 2 1 832.4.a.z 2
11.b odd 2 1 1573.4.a.b 2
12.b even 2 1 1872.4.a.bb 2
13.b even 2 1 169.4.a.g 2
13.c even 3 2 169.4.c.g 4
13.d odd 4 2 169.4.b.f 4
13.e even 6 2 169.4.c.j 4
13.f odd 12 4 169.4.e.f 8
39.d odd 2 1 1521.4.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 1.a even 1 1 trivial
117.4.a.d 2 3.b odd 2 1
169.4.a.g 2 13.b even 2 1
169.4.b.f 4 13.d odd 4 2
169.4.c.g 4 13.c even 3 2
169.4.c.j 4 13.e even 6 2
169.4.e.f 8 13.f odd 12 4
208.4.a.h 2 4.b odd 2 1
325.4.a.f 2 5.b even 2 1
325.4.b.e 4 5.c odd 4 2
637.4.a.b 2 7.b odd 2 1
832.4.a.s 2 8.b even 2 1
832.4.a.z 2 8.d odd 2 1
1521.4.a.r 2 39.d odd 2 1
1573.4.a.b 2 11.b odd 2 1
1872.4.a.bb 2 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(13))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 12 T^{2} - 8 T^{3} + 64 T^{4}$$
$3$ $$1 - 5 T + 22 T^{2} - 135 T^{3} + 729 T^{4}$$
$5$ $$1 + 3 T + 248 T^{2} + 375 T^{3} + 15625 T^{4}$$
$7$ $$1 + 9 T + 192 T^{2} + 3087 T^{3} + 117649 T^{4}$$
$11$ $$1 - 80 T + 3650 T^{2} - 106480 T^{3} + 1771561 T^{4}$$
$13$ $$( 1 + 13 T )^{2}$$
$17$ $$1 - 19 T + 8688 T^{2} - 93347 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 84 T + 11130 T^{2} + 576156 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 196 T + 33326 T^{2} - 2384732 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 44 T + 10094 T^{2} + 1073116 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 86 T + 56518 T^{2} + 2562026 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 209 T + 112120 T^{2} - 10586477 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 230 T + 149010 T^{2} + 15851830 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 287 T + 92698 T^{2} - 22818509 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 435 T + 192728 T^{2} - 45163005 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 118 T + 297410 T^{2} + 17567486 T^{3} + 22164361129 T^{4}$$
$59$ $$1 + 368 T + 379266 T^{2} + 75579472 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 1058 T + 580378 T^{2} + 240145898 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 68 T + 373930 T^{2} - 20451884 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 131 T + 493328 T^{2} + 46886341 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 456 T + 542718 T^{2} - 177391752 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 1008 T + 1233294 T^{2} + 496983312 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 1958 T + 1961238 T^{2} - 1119558946 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 720 T + 899726 T^{2} + 507577680 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 928 T + 943870 T^{2} + 846960544 T^{3} + 832972004929 T^{4}$$