# Properties

 Label 105.4.a.e Level 105 Weight 4 Character orbit 105.a Self dual yes Analytic conductor 6.195 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.19520055060$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \beta ) q^{2} -3 q^{3} + ( 1 - 4 \beta ) q^{4} + 5 q^{5} + ( 3 - 6 \beta ) q^{6} -7 q^{7} + ( -9 - 10 \beta ) q^{8} + 9 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \beta ) q^{2} -3 q^{3} + ( 1 - 4 \beta ) q^{4} + 5 q^{5} + ( 3 - 6 \beta ) q^{6} -7 q^{7} + ( -9 - 10 \beta ) q^{8} + 9 q^{9} + ( -5 + 10 \beta ) q^{10} + ( -8 - 40 \beta ) q^{11} + ( -3 + 12 \beta ) q^{12} + ( -38 + 4 \beta ) q^{13} + ( 7 - 14 \beta ) q^{14} -15 q^{15} + ( -39 + 24 \beta ) q^{16} + ( -62 + 4 \beta ) q^{17} + ( -9 + 18 \beta ) q^{18} + ( -48 + 32 \beta ) q^{19} + ( 5 - 20 \beta ) q^{20} + 21 q^{21} + ( -152 + 24 \beta ) q^{22} + ( -8 + 68 \beta ) q^{23} + ( 27 + 30 \beta ) q^{24} + 25 q^{25} + ( 54 - 80 \beta ) q^{26} -27 q^{27} + ( -7 + 28 \beta ) q^{28} + ( 94 + 108 \beta ) q^{29} + ( 15 - 30 \beta ) q^{30} + ( -60 - 36 \beta ) q^{31} + ( 207 - 22 \beta ) q^{32} + ( 24 + 120 \beta ) q^{33} + ( 78 - 128 \beta ) q^{34} -35 q^{35} + ( 9 - 36 \beta ) q^{36} + ( -66 + 132 \beta ) q^{37} + ( 176 - 128 \beta ) q^{38} + ( 114 - 12 \beta ) q^{39} + ( -45 - 50 \beta ) q^{40} + ( 50 - 160 \beta ) q^{41} + ( -21 + 42 \beta ) q^{42} + ( -268 - 124 \beta ) q^{43} + ( 312 - 8 \beta ) q^{44} + 45 q^{45} + ( 280 - 84 \beta ) q^{46} + ( -464 + 84 \beta ) q^{47} + ( 117 - 72 \beta ) q^{48} + 49 q^{49} + ( -25 + 50 \beta ) q^{50} + ( 186 - 12 \beta ) q^{51} + ( -70 + 156 \beta ) q^{52} + ( 442 - 128 \beta ) q^{53} + ( 27 - 54 \beta ) q^{54} + ( -40 - 200 \beta ) q^{55} + ( 63 + 70 \beta ) q^{56} + ( 144 - 96 \beta ) q^{57} + ( 338 + 80 \beta ) q^{58} + ( 52 + 408 \beta ) q^{59} + ( -15 + 60 \beta ) q^{60} + ( -234 + 84 \beta ) q^{61} + ( -84 - 84 \beta ) q^{62} -63 q^{63} + ( 17 + 244 \beta ) q^{64} + ( -190 + 20 \beta ) q^{65} + ( 456 - 72 \beta ) q^{66} + ( -844 - 76 \beta ) q^{67} + ( -94 + 252 \beta ) q^{68} + ( 24 - 204 \beta ) q^{69} + ( 35 - 70 \beta ) q^{70} + ( -68 + 300 \beta ) q^{71} + ( -81 - 90 \beta ) q^{72} + ( 254 - 644 \beta ) q^{73} + ( 594 - 264 \beta ) q^{74} -75 q^{75} + ( -304 + 224 \beta ) q^{76} + ( 56 + 280 \beta ) q^{77} + ( -162 + 240 \beta ) q^{78} + ( -216 + 464 \beta ) q^{79} + ( -195 + 120 \beta ) q^{80} + 81 q^{81} + ( -690 + 260 \beta ) q^{82} + ( -292 + 168 \beta ) q^{83} + ( 21 - 84 \beta ) q^{84} + ( -310 + 20 \beta ) q^{85} + ( -228 - 412 \beta ) q^{86} + ( -282 - 324 \beta ) q^{87} + ( 872 + 440 \beta ) q^{88} + ( -702 - 224 \beta ) q^{89} + ( -45 + 90 \beta ) q^{90} + ( 266 - 28 \beta ) q^{91} + ( -552 + 100 \beta ) q^{92} + ( 180 + 108 \beta ) q^{93} + ( 800 - 1012 \beta ) q^{94} + ( -240 + 160 \beta ) q^{95} + ( -621 + 66 \beta ) q^{96} + ( -594 - 92 \beta ) q^{97} + ( -49 + 98 \beta ) q^{98} + ( -72 - 360 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} - 18q^{8} + 18q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 6q^{3} + 2q^{4} + 10q^{5} + 6q^{6} - 14q^{7} - 18q^{8} + 18q^{9} - 10q^{10} - 16q^{11} - 6q^{12} - 76q^{13} + 14q^{14} - 30q^{15} - 78q^{16} - 124q^{17} - 18q^{18} - 96q^{19} + 10q^{20} + 42q^{21} - 304q^{22} - 16q^{23} + 54q^{24} + 50q^{25} + 108q^{26} - 54q^{27} - 14q^{28} + 188q^{29} + 30q^{30} - 120q^{31} + 414q^{32} + 48q^{33} + 156q^{34} - 70q^{35} + 18q^{36} - 132q^{37} + 352q^{38} + 228q^{39} - 90q^{40} + 100q^{41} - 42q^{42} - 536q^{43} + 624q^{44} + 90q^{45} + 560q^{46} - 928q^{47} + 234q^{48} + 98q^{49} - 50q^{50} + 372q^{51} - 140q^{52} + 884q^{53} + 54q^{54} - 80q^{55} + 126q^{56} + 288q^{57} + 676q^{58} + 104q^{59} - 30q^{60} - 468q^{61} - 168q^{62} - 126q^{63} + 34q^{64} - 380q^{65} + 912q^{66} - 1688q^{67} - 188q^{68} + 48q^{69} + 70q^{70} - 136q^{71} - 162q^{72} + 508q^{73} + 1188q^{74} - 150q^{75} - 608q^{76} + 112q^{77} - 324q^{78} - 432q^{79} - 390q^{80} + 162q^{81} - 1380q^{82} - 584q^{83} + 42q^{84} - 620q^{85} - 456q^{86} - 564q^{87} + 1744q^{88} - 1404q^{89} - 90q^{90} + 532q^{91} - 1104q^{92} + 360q^{93} + 1600q^{94} - 480q^{95} - 1242q^{96} - 1188q^{97} - 98q^{98} - 144q^{99} + O(q^{100})$$

## 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
−3.82843 −3.00000 6.65685 5.00000 11.4853 −7.00000 5.14214 9.00000 −19.1421
1.2 1.82843 −3.00000 −4.65685 5.00000 −5.48528 −7.00000 −23.1421 9.00000 9.14214
 $$n$$: 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 105.4.a.e 2
3.b odd 2 1 315.4.a.k 2
4.b odd 2 1 1680.4.a.bo 2
5.b even 2 1 525.4.a.l 2
5.c odd 4 2 525.4.d.l 4
7.b odd 2 1 735.4.a.o 2
15.d odd 2 1 1575.4.a.q 2
21.c even 2 1 2205.4.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 1.a even 1 1 trivial
315.4.a.k 2 3.b odd 2 1
525.4.a.l 2 5.b even 2 1
525.4.d.l 4 5.c odd 4 2
735.4.a.o 2 7.b odd 2 1
1575.4.a.q 2 15.d odd 2 1
1680.4.a.bo 2 4.b odd 2 1
2205.4.a.bb 2 21.c even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2 T_{2} - 7$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(105))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 9 T^{2} + 16 T^{3} + 64 T^{4}$$
$3$ $$( 1 + 3 T )^{2}$$
$5$ $$( 1 - 5 T )^{2}$$
$7$ $$( 1 + 7 T )^{2}$$
$11$ $$1 + 16 T - 474 T^{2} + 21296 T^{3} + 1771561 T^{4}$$
$13$ $$1 + 76 T + 5806 T^{2} + 166972 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 124 T + 13638 T^{2} + 609212 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 96 T + 13974 T^{2} + 658464 T^{3} + 47045881 T^{4}$$
$23$ $$1 + 16 T + 15150 T^{2} + 194672 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 188 T + 34286 T^{2} - 4585132 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 132 T + 70814 T^{2} + 6686196 T^{3} + 2565726409 T^{4}$$
$41$ $$1 - 100 T + 89142 T^{2} - 6892100 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 536 T + 200086 T^{2} + 42615752 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 928 T + 408830 T^{2} + 96347744 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 884 T + 460350 T^{2} - 131607268 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 104 T + 80534 T^{2} - 21359416 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 1688 T + 1302310 T^{2} + 507687944 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 136 T + 540446 T^{2} + 48675896 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 508 T + 13078 T^{2} - 197620636 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 432 T + 602142 T^{2} + 212992848 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 584 T + 1172390 T^{2} + 333923608 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 1404 T + 1802390 T^{2} + 989776476 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 1188 T + 2161254 T^{2} + 1084255524 T^{3} + 832972004929 T^{4}$$