# Properties

 Label 1859.2.a.t Level $1859$ Weight $2$ Character orbit 1859.a Self dual yes Analytic conductor $14.844$ Analytic rank $0$ Dimension $21$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1859.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8441897358$$ Analytic rank: $$0$$ Dimension: $$21$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$21 q + 6 q^{3} + 32 q^{4} + 7 q^{5} - 19 q^{6} + q^{7} - 3 q^{8} + 33 q^{9}+O(q^{10})$$ 21 * q + 6 * q^3 + 32 * q^4 + 7 * q^5 - 19 * q^6 + q^7 - 3 * q^8 + 33 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$21 q + 6 q^{3} + 32 q^{4} + 7 q^{5} - 19 q^{6} + q^{7} - 3 q^{8} + 33 q^{9} + 18 q^{10} + 21 q^{11} + 23 q^{12} + 20 q^{14} + 16 q^{15} + 50 q^{16} + 16 q^{17} + 3 q^{18} - 11 q^{19} + 24 q^{20} - 5 q^{21} - 9 q^{23} - 54 q^{24} + 36 q^{25} - 11 q^{28} + 28 q^{29} + 21 q^{30} + 15 q^{31} - 61 q^{32} + 6 q^{33} - 6 q^{34} - 3 q^{35} + 45 q^{36} - 12 q^{37} + q^{38} + 55 q^{40} - 4 q^{41} - 34 q^{42} + 17 q^{43} + 32 q^{44} + 9 q^{45} + 11 q^{46} + 36 q^{47} + 24 q^{48} + 72 q^{49} - 9 q^{50} + 2 q^{51} + 19 q^{53} + q^{54} + 7 q^{55} + 44 q^{56} - 4 q^{57} - 33 q^{58} + 54 q^{59} + 64 q^{60} + 98 q^{61} - 29 q^{62} - 81 q^{63} + 63 q^{64} - 19 q^{66} + 25 q^{67} + 4 q^{68} + 89 q^{69} + 65 q^{70} + 37 q^{71} + 55 q^{72} + 8 q^{73} - 11 q^{74} + 24 q^{75} + 13 q^{76} + q^{77} + 24 q^{79} + 26 q^{80} + 81 q^{81} + 26 q^{82} - 34 q^{83} - 103 q^{84} - 11 q^{85} + 30 q^{86} + 32 q^{87} - 3 q^{88} + 6 q^{89} + 47 q^{90} - 80 q^{92} + 41 q^{93} + 40 q^{94} + 20 q^{95} - 98 q^{96} - 5 q^{98} + 33 q^{99}+O(q^{100})$$ 21 * q + 6 * q^3 + 32 * q^4 + 7 * q^5 - 19 * q^6 + q^7 - 3 * q^8 + 33 * q^9 + 18 * q^10 + 21 * q^11 + 23 * q^12 + 20 * q^14 + 16 * q^15 + 50 * q^16 + 16 * q^17 + 3 * q^18 - 11 * q^19 + 24 * q^20 - 5 * q^21 - 9 * q^23 - 54 * q^24 + 36 * q^25 - 11 * q^28 + 28 * q^29 + 21 * q^30 + 15 * q^31 - 61 * q^32 + 6 * q^33 - 6 * q^34 - 3 * q^35 + 45 * q^36 - 12 * q^37 + q^38 + 55 * q^40 - 4 * q^41 - 34 * q^42 + 17 * q^43 + 32 * q^44 + 9 * q^45 + 11 * q^46 + 36 * q^47 + 24 * q^48 + 72 * q^49 - 9 * q^50 + 2 * q^51 + 19 * q^53 + q^54 + 7 * q^55 + 44 * q^56 - 4 * q^57 - 33 * q^58 + 54 * q^59 + 64 * q^60 + 98 * q^61 - 29 * q^62 - 81 * q^63 + 63 * q^64 - 19 * q^66 + 25 * q^67 + 4 * q^68 + 89 * q^69 + 65 * q^70 + 37 * q^71 + 55 * q^72 + 8 * q^73 - 11 * q^74 + 24 * q^75 + 13 * q^76 + q^77 + 24 * q^79 + 26 * q^80 + 81 * q^81 + 26 * q^82 - 34 * q^83 - 103 * q^84 - 11 * q^85 + 30 * q^86 + 32 * q^87 - 3 * q^88 + 6 * q^89 + 47 * q^90 - 80 * q^92 + 41 * q^93 + 40 * q^94 + 20 * q^95 - 98 * q^96 - 5 * q^98 + 33 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.76912 −0.0832308 5.66800 −1.06756 0.230476 −3.59787 −10.1571 −2.99307 2.95619
1.2 −2.70607 0.960218 5.32282 −3.57279 −2.59842 3.95601 −8.99178 −2.07798 9.66821
1.3 −2.68347 3.12668 5.20101 2.33476 −8.39034 0.266736 −8.58980 6.77612 −6.26526
1.4 −2.32475 2.00106 3.40448 3.04764 −4.65197 −5.08642 −3.26506 1.00425 −7.08500
1.5 −1.77144 2.92005 1.13800 −2.22672 −5.17269 0.178675 1.52699 5.52669 3.94449
1.6 −1.53678 −3.28188 0.361702 −1.21246 5.04353 −1.98662 2.51771 7.77071 1.86328
1.7 −1.50636 −0.544988 0.269129 3.40659 0.820949 −0.714897 2.60732 −2.70299 −5.13156
1.8 −0.930026 2.53271 −1.13505 −3.38691 −2.35548 −3.38196 2.91568 3.41460 3.14991
1.9 −0.603243 −0.154529 −1.63610 4.17643 0.0932187 2.73997 2.19345 −2.97612 −2.51940
1.10 −0.392741 1.69948 −1.84575 1.18432 −0.667456 5.07600 1.51039 −0.111771 −0.465133
1.11 −0.340522 −1.21327 −1.88404 −3.18769 0.413146 4.05402 1.32260 −1.52797 1.08548
1.12 0.149116 −3.21882 −1.97776 1.07607 −0.479977 −3.58386 −0.593149 7.36077 0.160459
1.13 0.776244 3.01008 −1.39745 0.0164567 2.33656 0.793722 −2.63725 6.06060 0.0127744
1.14 1.17546 −1.22926 −0.618298 −1.06938 −1.44494 −3.67782 −3.07770 −1.48892 −1.25701
1.15 1.43463 0.655312 0.0581570 1.45609 0.940129 3.21683 −2.78582 −2.57057 2.08894
1.16 2.13563 −2.20961 2.56090 −1.38469 −4.71890 5.12151 1.19787 1.88237 −2.95718
1.17 2.14442 −1.72478 2.59856 −3.54668 −3.69866 −2.57025 1.28356 −0.0251423 −7.60558
1.18 2.23194 1.36188 2.98154 4.28736 3.03963 0.384100 2.19073 −1.14528 9.56911
1.19 2.26490 3.17598 3.12978 2.32745 7.19329 −4.39568 2.55884 7.08687 5.27144
1.20 2.56428 1.12676 4.57554 1.24792 2.88933 2.50384 6.60441 −1.73041 3.20001
See all 21 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.21 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$
$$13$$ $$1$$

## 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 1859.2.a.t yes 21
13.b even 2 1 1859.2.a.s 21

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1859.2.a.s 21 13.b even 2 1
1859.2.a.t yes 21 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1859))$$:

 $$T_{2}^{21} - 37 T_{2}^{19} + T_{2}^{18} + 582 T_{2}^{17} - 20 T_{2}^{16} - 5075 T_{2}^{15} + 99 T_{2}^{14} + 26832 T_{2}^{13} + 510 T_{2}^{12} - 88400 T_{2}^{11} - 7129 T_{2}^{10} + 179407 T_{2}^{9} + 29118 T_{2}^{8} - 212824 T_{2}^{7} + \cdots + 448$$ T2^21 - 37*T2^19 + T2^18 + 582*T2^17 - 20*T2^16 - 5075*T2^15 + 99*T2^14 + 26832*T2^13 + 510*T2^12 - 88400*T2^11 - 7129*T2^10 + 179407*T2^9 + 29118*T2^8 - 212824*T2^7 - 55275*T2^6 + 130810*T2^5 + 48608*T2^4 - 30512*T2^3 - 15568*T2^2 - 224*T2 + 448 $$T_{7}^{21} - T_{7}^{20} - 109 T_{7}^{19} + 94 T_{7}^{18} + 4999 T_{7}^{17} - 3763 T_{7}^{16} - 125749 T_{7}^{15} + 85240 T_{7}^{14} + 1893974 T_{7}^{13} - 1215357 T_{7}^{12} - 17460834 T_{7}^{11} + 11281124 T_{7}^{10} + \cdots + 2981888$$ T7^21 - T7^20 - 109*T7^19 + 94*T7^18 + 4999*T7^17 - 3763*T7^16 - 125749*T7^15 + 85240*T7^14 + 1893974*T7^13 - 1215357*T7^12 - 17460834*T7^11 + 11281124*T7^10 + 96183712*T7^9 - 66488000*T7^8 - 290903456*T7^7 + 226843968*T7^6 + 384480768*T7^5 - 361342464*T7^4 - 64669696*T7^3 + 126300160*T7^2 - 35553280*T7 + 2981888