# Properties

 Label 469.1.bl.a Level $469$ Weight $1$ Character orbit 469.bl Analytic conductor $0.234$ Analytic rank $0$ Dimension $20$ Projective image $D_{33}$ CM discriminant -7 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$469 = 7 \cdot 67$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 469.bl (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.234061490925$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{33})$$ Defining polynomial: $$x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{33}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{33} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{66}^{10} + \zeta_{66}^{16} ) q^{2} + ( \zeta_{66}^{20} + \zeta_{66}^{26} + \zeta_{66}^{32} ) q^{4} + \zeta_{66}^{2} q^{7} + ( -\zeta_{66}^{3} - \zeta_{66}^{9} - \zeta_{66}^{15} + \zeta_{66}^{30} ) q^{8} + \zeta_{66}^{24} q^{9} +O(q^{10})$$ $$q + ( \zeta_{66}^{10} + \zeta_{66}^{16} ) q^{2} + ( \zeta_{66}^{20} + \zeta_{66}^{26} + \zeta_{66}^{32} ) q^{4} + \zeta_{66}^{2} q^{7} + ( -\zeta_{66}^{3} - \zeta_{66}^{9} - \zeta_{66}^{15} + \zeta_{66}^{30} ) q^{8} + \zeta_{66}^{24} q^{9} + ( -\zeta_{66}^{21} + \zeta_{66}^{28} ) q^{11} + ( \zeta_{66}^{12} + \zeta_{66}^{18} ) q^{14} + ( -\zeta_{66}^{7} - \zeta_{66}^{13} - \zeta_{66}^{19} - \zeta_{66}^{25} - \zeta_{66}^{31} ) q^{16} + ( -\zeta_{66} - \zeta_{66}^{7} ) q^{18} + ( \zeta_{66}^{4} - \zeta_{66}^{5} - \zeta_{66}^{11} - \zeta_{66}^{31} ) q^{22} + ( \zeta_{66}^{6} - \zeta_{66}^{29} ) q^{23} -\zeta_{66}^{27} q^{25} + ( -\zeta_{66} + \zeta_{66}^{22} + \zeta_{66}^{28} ) q^{28} + ( -\zeta_{66}^{9} - \zeta_{66}^{13} ) q^{29} + ( \zeta_{66}^{2} + \zeta_{66}^{8} + \zeta_{66}^{14} - \zeta_{66}^{17} - \zeta_{66}^{23} - \zeta_{66}^{29} ) q^{32} + ( -\zeta_{66}^{11} - \zeta_{66}^{17} - \zeta_{66}^{23} ) q^{36} + ( -\zeta_{66}^{3} + \zeta_{66}^{8} ) q^{37} + ( \zeta_{66}^{14} + \zeta_{66}^{22} ) q^{43} + ( \zeta_{66}^{8} + \zeta_{66}^{14} - \zeta_{66}^{15} + \zeta_{66}^{20} - \zeta_{66}^{21} - \zeta_{66}^{27} ) q^{44} + ( \zeta_{66}^{6} + \zeta_{66}^{12} + \zeta_{66}^{16} + \zeta_{66}^{22} ) q^{46} + \zeta_{66}^{4} q^{49} + ( \zeta_{66}^{4} + \zeta_{66}^{10} ) q^{50} + ( -\zeta_{66}^{5} - \zeta_{66}^{25} ) q^{53} + ( -\zeta_{66}^{5} - \zeta_{66}^{11} - \zeta_{66}^{17} + \zeta_{66}^{32} ) q^{56} + ( -\zeta_{66}^{19} - \zeta_{66}^{23} - \zeta_{66}^{25} - \zeta_{66}^{29} ) q^{58} + \zeta_{66}^{26} q^{63} + ( 1 + \zeta_{66}^{6} + \zeta_{66}^{12} + \zeta_{66}^{18} + \zeta_{66}^{24} - \zeta_{66}^{27} + \zeta_{66}^{30} ) q^{64} -\zeta_{66}^{19} q^{67} + ( -\zeta_{66} + \zeta_{66}^{18} ) q^{71} + ( 1 + \zeta_{66}^{6} - \zeta_{66}^{21} - \zeta_{66}^{27} ) q^{72} + ( -\zeta_{66}^{13} + \zeta_{66}^{18} - \zeta_{66}^{19} + \zeta_{66}^{24} ) q^{74} + ( -\zeta_{66}^{23} + \zeta_{66}^{30} ) q^{77} + ( -\zeta_{66}^{17} + \zeta_{66}^{26} ) q^{79} -\zeta_{66}^{15} q^{81} + ( -\zeta_{66}^{5} + \zeta_{66}^{24} + \zeta_{66}^{30} + \zeta_{66}^{32} ) q^{86} + ( -\zeta_{66}^{3} + \zeta_{66}^{4} + \zeta_{66}^{10} + \zeta_{66}^{18} + \zeta_{66}^{24} - \zeta_{66}^{25} + \zeta_{66}^{30} - \zeta_{66}^{31} ) q^{88} + ( -\zeta_{66}^{5} + \zeta_{66}^{16} + \zeta_{66}^{22} + \zeta_{66}^{26} + \zeta_{66}^{28} + \zeta_{66}^{32} ) q^{92} + ( \zeta_{66}^{14} + \zeta_{66}^{20} ) q^{98} + ( \zeta_{66}^{12} - \zeta_{66}^{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 2q^{2} + 3q^{4} + q^{7} - 8q^{8} - 2q^{9} + O(q^{10})$$ $$20q + 2q^{2} + 3q^{4} + q^{7} - 8q^{8} - 2q^{9} - q^{11} - 4q^{14} + 5q^{16} + 2q^{18} - 7q^{22} - q^{23} - 2q^{25} - 8q^{28} - q^{29} + 6q^{32} - 8q^{36} - q^{37} - 9q^{43} - 3q^{44} - 13q^{46} + q^{49} + 2q^{50} + 2q^{53} - 7q^{56} + 4q^{58} + q^{63} + 8q^{64} + q^{67} - q^{71} + 14q^{72} - 2q^{74} - q^{77} + 2q^{79} - 2q^{81} - 2q^{86} - 4q^{88} - 5q^{92} + 2q^{98} - q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/469\mathbb{Z}\right)^\times$$.

 $$n$$ $$269$$ $$337$$ $$\chi(n)$$ $$-1$$ $$\zeta_{66}^{26}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 −0.995472 + 0.0950560i −0.327068 − 0.945001i 0.580057 − 0.814576i 0.928368 − 0.371662i 0.235759 + 0.971812i 0.723734 + 0.690079i 0.0475819 + 0.998867i 0.235759 − 0.971812i −0.786053 − 0.618159i 0.981929 + 0.189251i 0.981929 − 0.189251i −0.888835 + 0.458227i −0.327068 + 0.945001i 0.723734 − 0.690079i 0.580057 + 0.814576i 0.928368 + 0.371662i 0.0475819 − 0.998867i −0.995472 − 0.0950560i −0.786053 + 0.618159i −0.888835 − 0.458227i
0.627639 1.81344i 0 −2.10859 1.65822i 0 0 0.981929 0.189251i −2.71616 + 1.74557i −0.654861 0.755750i 0
55.1 1.56199 + 0.625325i 0 1.32503 + 1.26342i 0 0 −0.786053 + 0.618159i 0.580699 + 1.27155i −0.142315 0.989821i 0
83.1 −1.88431 0.363170i 0 2.49035 + 0.996987i 0 0 −0.327068 0.945001i −2.71616 1.74557i −0.654861 + 0.755750i 0
90.1 0.195876 + 0.807410i 0 0.275291 0.141923i 0 0 0.723734 0.690079i 0.712591 + 0.822373i −0.959493 0.281733i 0
132.1 −0.0623191 + 1.30824i 0 −0.712131 0.0680003i 0 0 −0.888835 + 0.458227i −0.0530529 + 0.368991i 0.841254 + 0.540641i 0
153.1 1.16413 + 0.600149i 0 0.414955 + 0.582723i 0 0 0.0475819 + 0.998867i −0.0530529 0.368991i 0.841254 0.540641i 0
160.1 −0.165101 0.231852i 0 0.300571 0.868442i 0 0 −0.995472 + 0.0950560i −0.524075 + 0.153882i 0.415415 0.909632i 0
167.1 −0.0623191 1.30824i 0 −0.712131 + 0.0680003i 0 0 −0.888835 0.458227i −0.0530529 0.368991i 0.841254 0.540641i 0
181.1 0.601300 0.573338i 0 −0.0147371 + 0.309371i 0 0 0.235759 + 0.971812i 0.712591 + 0.822373i −0.959493 0.281733i 0
188.1 −1.32254 + 1.04006i 0 0.431635 1.77922i 0 0 0.928368 + 0.371662i 0.580699 + 1.27155i −0.142315 0.989821i 0
237.1 −1.32254 1.04006i 0 0.431635 + 1.77922i 0 0 0.928368 0.371662i 0.580699 1.27155i −0.142315 + 0.989821i 0
272.1 0.283341 + 0.0270558i 0 −0.902379 0.173919i 0 0 0.580057 0.814576i −0.524075 0.153882i 0.415415 + 0.909632i 0
307.1 1.56199 0.625325i 0 1.32503 1.26342i 0 0 −0.786053 0.618159i 0.580699 1.27155i −0.142315 + 0.989821i 0
328.1 1.16413 0.600149i 0 0.414955 0.582723i 0 0 0.0475819 0.998867i −0.0530529 + 0.368991i 0.841254 + 0.540641i 0
356.1 −1.88431 + 0.363170i 0 2.49035 0.996987i 0 0 −0.327068 + 0.945001i −2.71616 + 1.74557i −0.654861 0.755750i 0
370.1 0.195876 0.807410i 0 0.275291 + 0.141923i 0 0 0.723734 + 0.690079i 0.712591 0.822373i −0.959493 + 0.281733i 0
384.1 −0.165101 + 0.231852i 0 0.300571 + 0.868442i 0 0 −0.995472 0.0950560i −0.524075 0.153882i 0.415415 + 0.909632i 0
391.1 0.627639 + 1.81344i 0 −2.10859 + 1.65822i 0 0 0.981929 + 0.189251i −2.71616 1.74557i −0.654861 + 0.755750i 0
412.1 0.601300 + 0.573338i 0 −0.0147371 0.309371i 0 0 0.235759 0.971812i 0.712591 0.822373i −0.959493 + 0.281733i 0
419.1 0.283341 0.0270558i 0 −0.902379 + 0.173919i 0 0 0.580057 + 0.814576i −0.524075 + 0.153882i 0.415415 0.909632i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
67.g even 33 1 inner
469.bl odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 469.1.bl.a 20
7.b odd 2 1 CM 469.1.bl.a 20
7.c even 3 1 3283.1.bw.a 20
7.c even 3 1 3283.1.cd.a 20
7.d odd 6 1 3283.1.bw.a 20
7.d odd 6 1 3283.1.cd.a 20
67.g even 33 1 inner 469.1.bl.a 20
469.y even 33 1 3283.1.cd.a 20
469.bb even 33 1 3283.1.bw.a 20
469.bc odd 66 1 3283.1.cd.a 20
469.bj odd 66 1 3283.1.bw.a 20
469.bl odd 66 1 inner 469.1.bl.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
469.1.bl.a 20 1.a even 1 1 trivial
469.1.bl.a 20 7.b odd 2 1 CM
469.1.bl.a 20 67.g even 33 1 inner
469.1.bl.a 20 469.bl odd 66 1 inner
3283.1.bw.a 20 7.c even 3 1
3283.1.bw.a 20 7.d odd 6 1
3283.1.bw.a 20 469.bb even 33 1
3283.1.bw.a 20 469.bj odd 66 1
3283.1.cd.a 20 7.c even 3 1
3283.1.cd.a 20 7.d odd 6 1
3283.1.cd.a 20 469.y even 33 1
3283.1.cd.a 20 469.bc odd 66 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(469, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T + 11 T^{2} - 48 T^{3} + 266 T^{4} - 671 T^{5} + 1116 T^{6} - 1207 T^{7} + 869 T^{8} - 284 T^{9} - 45 T^{10} + 94 T^{11} - 33 T^{12} - 29 T^{13} + 42 T^{14} - 16 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$3$ $$T^{20}$$
$5$ $$T^{20}$$
$7$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$11$ $$1 + 12 T + 154 T^{2} + 626 T^{3} + 934 T^{4} + 253 T^{5} - 560 T^{6} - 87 T^{7} + 605 T^{8} - 155 T^{9} + 87 T^{10} + 241 T^{11} + 11 T^{12} + 89 T^{13} + 78 T^{14} + 10 T^{16} + 10 T^{17} + T^{19} + T^{20}$$
$13$ $$T^{20}$$
$17$ $$T^{20}$$
$19$ $$T^{20}$$
$23$ $$1 - 10 T + 88 T^{2} + 197 T^{3} + 384 T^{4} + 792 T^{5} + 188 T^{6} - 1451 T^{7} - 792 T^{8} + 417 T^{9} + 483 T^{10} + 65 T^{11} + 88 T^{12} + 34 T^{13} - 43 T^{14} - 44 T^{15} - T^{16} - T^{17} + T^{19} + T^{20}$$
$29$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$31$ $$T^{20}$$
$37$ $$1 + 12 T + 132 T^{2} + 230 T^{3} + 703 T^{4} + 550 T^{5} + 2025 T^{6} + 1431 T^{7} + 2673 T^{8} + 1220 T^{9} + 1935 T^{10} + 714 T^{11} + 968 T^{12} + 254 T^{13} + 320 T^{14} + 66 T^{15} + 76 T^{16} + 10 T^{17} + 11 T^{18} + T^{19} + T^{20}$$
$41$ $$T^{20}$$
$43$ $$1 - 13 T + 36 T^{2} + 381 T^{3} + 742 T^{4} + 874 T^{5} + 1965 T^{6} + 3578 T^{7} + 5069 T^{8} + 6194 T^{9} + 6633 T^{10} + 6194 T^{11} + 5047 T^{12} + 3567 T^{13} + 2174 T^{14} + 1127 T^{15} + 489 T^{16} + 172 T^{17} + 47 T^{18} + 9 T^{19} + T^{20}$$
$47$ $$T^{20}$$
$53$ $$1 + 9 T + 58 T^{2} - 367 T^{3} + 1171 T^{4} - 2129 T^{5} + 2119 T^{6} - 1383 T^{7} + 779 T^{8} + 67 T^{9} - 66 T^{10} + 67 T^{11} + 185 T^{12} - 118 T^{13} + 51 T^{14} + 27 T^{15} - 17 T^{16} + 7 T^{17} + 3 T^{18} - 2 T^{19} + T^{20}$$
$59$ $$T^{20}$$
$61$ $$T^{20}$$
$67$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20}$$
$71$ $$1 - 10 T + 143 T^{2} - 595 T^{3} + 835 T^{4} - 22 T^{5} - 395 T^{6} - 274 T^{7} - 121 T^{8} + 32 T^{9} + 395 T^{10} + 362 T^{11} + 77 T^{12} - 10 T^{13} + 12 T^{14} + 22 T^{15} + 10 T^{16} - T^{17} + T^{19} + T^{20}$$
$73$ $$T^{20}$$
$79$ $$1 + 5 T + 11 T^{2} + 62 T^{3} + 178 T^{4} + 77 T^{5} + 49 T^{6} - 19 T^{7} - 154 T^{8} - 130 T^{9} + 164 T^{10} - 49 T^{11} + 11 T^{12} - 40 T^{13} + 42 T^{14} - 11 T^{15} - 5 T^{16} + 8 T^{17} - 2 T^{19} + T^{20}$$
$83$ $$T^{20}$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$