# Properties

 Label 380.3.e.a Level $380$ Weight $3$ Character orbit 380.e Analytic conductor $10.354$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 380.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3542500457$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600$$ x^12 + 62*x^10 + 1445*x^8 + 15924*x^6 + 83244*x^4 + 170640*x^2 + 55600 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{9}\cdot 5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7} + (\beta_{7} - \beta_{6} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + b6 * q^5 + (-b11 + b6 - 1) * q^7 + (b7 - b6 - 1) * q^9 $$q + \beta_1 q^{3} + \beta_{6} q^{5} + ( - \beta_{11} + \beta_{6} - 1) q^{7} + (\beta_{7} - \beta_{6} - 1) q^{9} + ( - \beta_{11} - \beta_{8} + \beta_{7} - 2) q^{11} + ( - \beta_{4} - \beta_1) q^{13} + \beta_{5} q^{15} + (\beta_{11} - \beta_{8} + 2 \beta_{7} - \beta_{6}) q^{17} + (\beta_{10} + \beta_{8} - \beta_{2} - \beta_1 + 2) q^{19} + (\beta_{9} + \beta_{5} + \beta_{3} - 2 \beta_1) q^{21} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 3 \beta_{6} + 1) q^{23} + 5 q^{25} + (\beta_{9} - 2 \beta_{5} + 3 \beta_1) q^{27} + (\beta_{9} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{29} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{31} + (2 \beta_{9} + \beta_{4} - 6 \beta_1) q^{33} + (\beta_{10} - \beta_{8} - \beta_{6} + 4) q^{35} + (\beta_{9} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{37} + ( - 4 \beta_{8} + \beta_{7} + \beta_{6} + 12) q^{39} + (2 \beta_{9} + \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_1) q^{41} + (2 \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 15) q^{43} + (\beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - 2 \beta_{6} - 3) q^{45} + (5 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 7) q^{47} + (\beta_{11} + 2 \beta_{10} - 5 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} + 1) q^{49} + (\beta_{9} - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} - 7 \beta_1) q^{51} + (2 \beta_{9} - 2 \beta_{5} + 4 \beta_{3} + \beta_1) q^{53} + ( - \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{55} + (\beta_{11} + 4 \beta_{10} + \beta_{9} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 13) q^{57}+ \cdots + (\beta_{11} - 4 \beta_{10} + \beta_{8} - 9 \beta_{7} + 6 \beta_{6} + 28) q^{99}+O(q^{100})$$ q + b1 * q^3 + b6 * q^5 + (-b11 + b6 - 1) * q^7 + (b7 - b6 - 1) * q^9 + (-b11 - b8 + b7 - 2) * q^11 + (-b4 - b1) * q^13 + b5 * q^15 + (b11 - b8 + 2*b7 - b6) * q^17 + (b10 + b8 - b2 - b1 + 2) * q^19 + (b9 + b5 + b3 - 2*b1) * q^21 + (-b11 - 2*b10 - 2*b8 + 2*b7 + 3*b6 + 1) * q^23 + 5 * q^25 + (b9 - 2*b5 + 3*b1) * q^27 + (b9 - b4 - 2*b3 - 2*b2 + b1) * q^29 + (b5 - b4 + b3 + b1) * q^31 + (2*b9 + b4 - 6*b1) * q^33 + (b10 - b8 - b6 + 4) * q^35 + (b9 + 2*b5 - b4 - 2*b3 - 2*b1) * q^37 + (-4*b8 + b7 + b6 + 12) * q^39 + (2*b9 + b5 + b4 + b3 - 3*b1) * q^41 + (2*b11 - 2*b10 + b8 + 2*b7 + 2*b6 - 15) * q^43 + (b11 + b10 + b8 + b7 - 2*b6 - 3) * q^45 + (5*b8 + 2*b7 - 2*b6 - 7) * q^47 + (b11 + 2*b10 - 5*b8 + 2*b7 - 5*b6 + 1) * q^49 + (b9 - 2*b5 + b4 - 2*b3 - 7*b1) * q^51 + (2*b9 - 2*b5 + 4*b3 + b1) * q^53 + (-b11 + 2*b10 + b8 - b7 - 2*b6) * q^55 + (b11 + 4*b10 + b9 - 3*b6 - 2*b5 + 2*b3 + 2*b2 - 2*b1 + 13) * q^57 + (-b9 + 5*b5 - b3 - 2*b2) * q^59 + (b11 - 4*b10 - b8 + b7 + 6*b6 + 12) * q^61 + (-2*b10 - b8 - 2*b7 - 2*b6 + 3) * q^63 + (-b5 - b4 - 2*b3) * q^65 + (-2*b9 - b4 + 2*b3 + 4*b2 - b1) * q^67 + (b9 + 5*b5 - 3*b3 - 4*b2 - 2*b1) * q^69 + (-4*b9 - b5 - 5*b4 - b3 + 11*b1) * q^71 + (-5*b11 - 2*b10 + 3*b8 - 5*b6 - 14) * q^73 + 5*b1 * q^75 + (-4*b11 - 2*b8 + 14*b6 + 32) * q^77 + (-4*b9 + 5*b5 + b4 + 3*b3 - 5*b1) * q^79 + (3*b11 - 4*b10 + b8 + 5*b7 + 10*b6 - 39) * q^81 + (-8*b11 - 2*b10 - 3*b8 - 6*b7 + 14*b6 - 19) * q^83 + (b10 + 4*b8 - b6 - 1) * q^85 + (b11 + 8*b10 + 9*b8 - 8*b7 - 5*b6 - 10) * q^87 + (-10*b5 - 4*b2 + 8*b1) * q^89 + (-3*b9 - 9*b5 + 4*b4 - 3*b3 + 12*b1) * q^91 + (4*b11 - 8*b8 + 8*b7 - 14*b6 - 10) * q^93 + (-b11 - 2*b9 - 2*b8 + 4*b7 + b6 - b5 + b4 + b2 + 2*b1 + 2) * q^95 + (-3*b9 + 8*b5 + b4 + 4*b2 + 16*b1) * q^97 + (b11 - 4*b10 + b8 - 9*b7 + 6*b6 + 28) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{7} - 16 q^{9}+O(q^{10})$$ 12 * q - 12 * q^7 - 16 * q^9 $$12 q - 12 q^{7} - 16 q^{9} - 32 q^{11} - 12 q^{17} + 24 q^{19} + 4 q^{23} + 60 q^{25} + 40 q^{35} + 124 q^{39} - 176 q^{43} - 40 q^{45} - 72 q^{47} - 24 q^{49} + 140 q^{57} + 152 q^{61} + 48 q^{63} - 148 q^{73} + 376 q^{77} - 468 q^{81} - 208 q^{83} - 84 q^{87} - 184 q^{93} + 392 q^{99}+O(q^{100})$$ 12 * q - 12 * q^7 - 16 * q^9 - 32 * q^11 - 12 * q^17 + 24 * q^19 + 4 * q^23 + 60 * q^25 + 40 * q^35 + 124 * q^39 - 176 * q^43 - 40 * q^45 - 72 * q^47 - 24 * q^49 + 140 * q^57 + 152 * q^61 + 48 * q^63 - 148 * q^73 + 376 * q^77 - 468 * q^81 - 208 * q^83 - 84 * q^87 - 184 * q^93 + 392 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 59\nu^{11} + 5868\nu^{9} + 184075\nu^{7} + 2290126\nu^{5} + 10898236\nu^{3} + 13285720\nu ) / 407880$$ (59*v^11 + 5868*v^9 + 184075*v^7 + 2290126*v^5 + 10898236*v^3 + 13285720*v) / 407880 $$\beta_{3}$$ $$=$$ $$( -239\nu^{11} - 15993\nu^{9} - 373210\nu^{7} - 3571816\nu^{5} - 13082536\nu^{3} - 14157280\nu ) / 407880$$ (-239*v^11 - 15993*v^9 - 373210*v^7 - 3571816*v^5 - 13082536*v^3 - 14157280*v) / 407880 $$\beta_{4}$$ $$=$$ $$( -401\nu^{11} - 20007\nu^{9} - 344590\nu^{7} - 2594164\nu^{5} - 8665084\nu^{3} - 10740520\nu ) / 203940$$ (-401*v^11 - 20007*v^9 - 344590*v^7 - 2594164*v^5 - 8665084*v^3 - 10740520*v) / 203940 $$\beta_{5}$$ $$=$$ $$( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1502992\nu^{3} - 417160\nu ) / 37080$$ (-113*v^11 - 5661*v^9 - 95740*v^7 - 650992*v^5 - 1502992*v^3 - 417160*v) / 37080 $$\beta_{6}$$ $$=$$ $$( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1502992\nu^{2} - 417160 ) / 37080$$ (-113*v^10 - 5661*v^8 - 95740*v^6 - 650992*v^4 - 1502992*v^2 - 417160) / 37080 $$\beta_{7}$$ $$=$$ $$( -113\nu^{10} - 5661\nu^{8} - 95740\nu^{6} - 650992\nu^{4} - 1465912\nu^{2} - 46360 ) / 37080$$ (-113*v^10 - 5661*v^8 - 95740*v^6 - 650992*v^4 - 1465912*v^2 - 46360) / 37080 $$\beta_{8}$$ $$=$$ $$( 301\nu^{10} + 14382\nu^{8} + 228185\nu^{6} + 1462904\nu^{4} + 3463424\nu^{2} + 1849460 ) / 67980$$ (301*v^10 + 14382*v^8 + 228185*v^6 + 1462904*v^4 + 3463424*v^2 + 1849460) / 67980 $$\beta_{9}$$ $$=$$ $$( -113\nu^{11} - 5661\nu^{9} - 95740\nu^{7} - 650992\nu^{5} - 1484452\nu^{3} - 139060\nu ) / 18540$$ (-113*v^11 - 5661*v^9 - 95740*v^7 - 650992*v^5 - 1484452*v^3 - 139060*v) / 18540 $$\beta_{10}$$ $$=$$ $$( -361\nu^{10} - 17757\nu^{8} - 291230\nu^{6} - 1890134\nu^{4} - 4123544\nu^{2} - 1392200 ) / 67980$$ (-361*v^10 - 17757*v^8 - 291230*v^6 - 1890134*v^4 - 4123544*v^2 - 1392200) / 67980 $$\beta_{11}$$ $$=$$ $$( 545\nu^{10} + 28107\nu^{8} + 495898\nu^{6} + 3578728\nu^{4} + 9148096\nu^{2} + 3524896 ) / 81576$$ (545*v^10 + 28107*v^8 + 495898*v^6 + 3578728*v^4 + 9148096*v^2 + 3524896) / 81576
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{6} - 10$$ b7 - b6 - 10 $$\nu^{3}$$ $$=$$ $$\beta_{9} - 2\beta_{5} - 15\beta_1$$ b9 - 2*b5 - 15*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{11} - 4\beta_{10} + \beta_{8} - 22\beta_{7} + 37\beta_{6} + 150$$ 3*b11 - 4*b10 + b8 - 22*b7 + 37*b6 + 150 $$\nu^{5}$$ $$=$$ $$-29\beta_{9} + 62\beta_{5} - 5\beta_{4} - 6\beta_{3} - 8\beta_{2} + 269\beta_1$$ -29*b9 + 62*b5 - 5*b4 - 6*b3 - 8*b2 + 269*b1 $$\nu^{6}$$ $$=$$ $$-93\beta_{11} + 158\beta_{10} - 15\beta_{8} + 470\beta_{7} - 971\beta_{6} - 2674$$ -93*b11 + 158*b10 - 15*b8 + 470*b7 - 971*b6 - 2674 $$\nu^{7}$$ $$=$$ $$721\beta_{9} - 1584\beta_{5} + 173\beta_{4} + 236\beta_{3} + 316\beta_{2} - 5403\beta_1$$ 721*b9 - 1584*b5 + 173*b4 + 236*b3 + 316*b2 - 5403*b1 $$\nu^{8}$$ $$=$$ $$2413\beta_{11} - 4526\beta_{10} + 91\beta_{8} - 10310\beta_{7} + 23619\beta_{6} + 53398$$ 2413*b11 - 4526*b10 + 91*b8 - 10310*b7 + 23619*b6 + 53398 $$\nu^{9}$$ $$=$$ $$-17249\beta_{9} + 38364\beta_{5} - 4617\beta_{4} - 6848\beta_{3} - 9052\beta_{2} + 116231\beta_1$$ -17249*b9 + 38364*b5 - 4617*b4 - 6848*b3 - 9052*b2 + 116231*b1 $$\nu^{10}$$ $$=$$ $$-59373\beta_{11} + 115918\beta_{10} + 2389\beta_{8} + 231734\beta_{7} - 560747\beta_{6} - 1144366$$ -59373*b11 + 115918*b10 + 2389*b8 + 231734*b7 - 560747*b6 - 1144366 $$\nu^{11}$$ $$=$$ $$407025\beta_{9} - 910788\beta_{5} + 113529\beta_{4} + 177680\beta_{3} + 231836\beta_{2} - 2599023\beta_1$$ 407025*b9 - 910788*b5 + 113529*b4 + 177680*b3 + 231836*b2 - 2599023*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 − 4.83157i − 3.53755i − 3.42504i − 3.14288i − 2.03369i − 0.630185i 0.630185i 2.03369i 3.14288i 3.42504i 3.53755i 4.83157i
0 4.83157i 0 2.23607 0 −3.72856 0 −14.3441 0
341.2 0 3.53755i 0 −2.23607 0 0.468611 0 −3.51429 0
341.3 0 3.42504i 0 2.23607 0 9.18599 0 −2.73093 0
341.4 0 3.14288i 0 −2.23607 0 −12.2192 0 −0.877687 0
341.5 0 2.03369i 0 −2.23607 0 4.27843 0 4.86411 0
341.6 0 0.630185i 0 2.23607 0 −3.98530 0 8.60287 0
341.7 0 0.630185i 0 2.23607 0 −3.98530 0 8.60287 0
341.8 0 2.03369i 0 −2.23607 0 4.27843 0 4.86411 0
341.9 0 3.14288i 0 −2.23607 0 −12.2192 0 −0.877687 0
341.10 0 3.42504i 0 2.23607 0 9.18599 0 −2.73093 0
341.11 0 3.53755i 0 −2.23607 0 0.468611 0 −3.51429 0
341.12 0 4.83157i 0 2.23607 0 −3.72856 0 −14.3441 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.e.a 12
3.b odd 2 1 3420.3.o.a 12
4.b odd 2 1 1520.3.h.b 12
5.b even 2 1 1900.3.e.f 12
5.c odd 4 2 1900.3.g.c 24
19.b odd 2 1 inner 380.3.e.a 12
57.d even 2 1 3420.3.o.a 12
76.d even 2 1 1520.3.h.b 12
95.d odd 2 1 1900.3.e.f 12
95.g even 4 2 1900.3.g.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.e.a 12 1.a even 1 1 trivial
380.3.e.a 12 19.b odd 2 1 inner
1520.3.h.b 12 4.b odd 2 1
1520.3.h.b 12 76.d even 2 1
1900.3.e.f 12 5.b even 2 1
1900.3.e.f 12 95.d odd 2 1
1900.3.g.c 24 5.c odd 4 2
1900.3.g.c 24 95.g even 4 2
3420.3.o.a 12 3.b odd 2 1
3420.3.o.a 12 57.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 62 T^{10} + 1445 T^{8} + \cdots + 55600$$
$5$ $$(T^{2} - 5)^{6}$$
$7$ $$(T^{6} + 6 T^{5} - 123 T^{4} - 448 T^{3} + \cdots - 3344)^{2}$$
$11$ $$(T^{6} + 16 T^{5} - 164 T^{4} + \cdots - 43264)^{2}$$
$13$ $$T^{12} + 1174 T^{10} + \cdots + 18127879600$$
$17$ $$(T^{6} + 6 T^{5} - 703 T^{4} + \cdots + 818576)^{2}$$
$19$ $$T^{12} - 24 T^{11} + \cdots + 22\!\cdots\!61$$
$23$ $$(T^{6} - 2 T^{5} - 1379 T^{4} + \cdots - 752400)^{2}$$
$29$ $$T^{12} + 6442 T^{10} + \cdots + 24\!\cdots\!00$$
$31$ $$T^{12} + \cdots + 119374584217600$$
$37$ $$T^{12} + 9204 T^{10} + \cdots + 25\!\cdots\!00$$
$41$ $$T^{12} + 7120 T^{10} + \cdots + 12\!\cdots\!00$$
$43$ $$(T^{6} + 88 T^{5} + 460 T^{4} + \cdots - 2451136)^{2}$$
$47$ $$(T^{6} + 36 T^{5} - 7500 T^{4} + \cdots - 9167899456)^{2}$$
$53$ $$T^{12} + 28390 T^{10} + \cdots + 31\!\cdots\!00$$
$59$ $$T^{12} + \cdots + 492787956121600$$
$61$ $$(T^{6} - 76 T^{5} - 2952 T^{4} + \cdots - 1301500864)^{2}$$
$67$ $$T^{12} + 19710 T^{10} + \cdots + 39\!\cdots\!00$$
$71$ $$T^{12} + 52160 T^{10} + \cdots + 33\!\cdots\!00$$
$73$ $$(T^{6} + 74 T^{5} - 7103 T^{4} + \cdots + 5637612816)^{2}$$
$79$ $$T^{12} + 49000 T^{10} + \cdots + 36\!\cdots\!00$$
$83$ $$(T^{6} + 104 T^{5} - 12064 T^{4} + \cdots - 2227337984)^{2}$$
$89$ $$T^{12} + 42776 T^{10} + \cdots + 10\!\cdots\!00$$
$97$ $$T^{12} + 55476 T^{10} + \cdots + 23\!\cdots\!00$$