Properties

 Label 20.11.d.c Level 20 Weight 11 Character orbit 20.d Analytic conductor 12.707 Analytic rank 0 Dimension 2 CM disc. -4 Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 20.d (of order $$2$$ and degree $$1$$)

Newform invariants

 Self dual: No Analytic conductor: $$12.7071450535$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ 8 \beta q^{2}$$ $$-1024 q^{4}$$ $$+ ( -237 + 779 \beta ) q^{5}$$ $$-8192 \beta q^{8}$$ $$-59049 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ 8 \beta q^{2}$$ $$-1024 q^{4}$$ $$+ ( -237 + 779 \beta ) q^{5}$$ $$-8192 \beta q^{8}$$ $$-59049 q^{9}$$ $$+ ( -99712 - 1896 \beta ) q^{10}$$ $$-72834 \beta q^{13}$$ $$+ 1048576 q^{16}$$ $$-452884 \beta q^{17}$$ $$-472392 \beta q^{18}$$ $$+ ( 242688 - 797696 \beta ) q^{20}$$ $$+ ( -9653287 - 369246 \beta ) q^{25}$$ $$+ 9322752 q^{26}$$ $$-32319798 q^{29}$$ $$+ 8388608 \beta q^{32}$$ $$+ 57969152 q^{34}$$ $$+ 60466176 q^{36}$$ $$+ 34559166 \beta q^{37}$$ $$+ ( 102105088 + 1941504 \beta ) q^{40}$$ $$-207190098 q^{41}$$ $$+ ( 13994613 - 45999171 \beta ) q^{45}$$ $$-282475249 q^{49}$$ $$+ ( 47263488 - 77226296 \beta ) q^{50}$$ $$+ 74582016 \beta q^{52}$$ $$-73384934 \beta q^{53}$$ $$-258558384 \beta q^{58}$$ $$+ 1330090102 q^{61}$$ $$-1073741824 q^{64}$$ $$+ ( 907802976 + 17261658 \beta ) q^{65}$$ $$+ 463753216 \beta q^{68}$$ $$+ 483729408 \beta q^{72}$$ $$+ 447227016 \beta q^{73}$$ $$-4423573248 q^{74}$$ $$+ ( -248512512 + 816840704 \beta ) q^{80}$$ $$+ 3486784401 q^{81}$$ $$-1657520784 \beta q^{82}$$ $$+ ( 5644746176 + 107333508 \beta ) q^{85}$$ $$-8562016398 q^{89}$$ $$+ ( 5887893888 + 111956904 \beta ) q^{90}$$ $$-3704292684 \beta q^{97}$$ $$-2259801992 \beta q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 2048q^{4}$$ $$\mathstrut -\mathstrut 474q^{5}$$ $$\mathstrut -\mathstrut 118098q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 2048q^{4}$$ $$\mathstrut -\mathstrut 474q^{5}$$ $$\mathstrut -\mathstrut 118098q^{9}$$ $$\mathstrut -\mathstrut 199424q^{10}$$ $$\mathstrut +\mathstrut 2097152q^{16}$$ $$\mathstrut +\mathstrut 485376q^{20}$$ $$\mathstrut -\mathstrut 19306574q^{25}$$ $$\mathstrut +\mathstrut 18645504q^{26}$$ $$\mathstrut -\mathstrut 64639596q^{29}$$ $$\mathstrut +\mathstrut 115938304q^{34}$$ $$\mathstrut +\mathstrut 120932352q^{36}$$ $$\mathstrut +\mathstrut 204210176q^{40}$$ $$\mathstrut -\mathstrut 414380196q^{41}$$ $$\mathstrut +\mathstrut 27989226q^{45}$$ $$\mathstrut -\mathstrut 564950498q^{49}$$ $$\mathstrut +\mathstrut 94526976q^{50}$$ $$\mathstrut +\mathstrut 2660180204q^{61}$$ $$\mathstrut -\mathstrut 2147483648q^{64}$$ $$\mathstrut +\mathstrut 1815605952q^{65}$$ $$\mathstrut -\mathstrut 8847146496q^{74}$$ $$\mathstrut -\mathstrut 497025024q^{80}$$ $$\mathstrut +\mathstrut 6973568802q^{81}$$ $$\mathstrut +\mathstrut 11289492352q^{85}$$ $$\mathstrut -\mathstrut 17124032796q^{89}$$ $$\mathstrut +\mathstrut 11775787776q^{90}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
32.0000i 0 −1024.00 −237.000 3116.00i 0 0 32768.0i −59049.0 −99712.0 + 7584.00i
19.2 32.0000i 0 −1024.00 −237.000 + 3116.00i 0 0 32768.0i −59049.0 −99712.0 7584.00i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by $$\Q(\sqrt{-1})$$ yes
5.b Even 1 yes
20.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{11}^{\mathrm{new}}(20, [\chi])$$.