# Properties

 Label 1950.2.bc.g Level $1950$ Weight $2$ Character orbit 1950.bc Analytic conductor $15.571$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.bc (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.17284886784.1 Defining polynomial: $$x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{5} ) q^{2} + ( 1 - \beta_{6} ) q^{3} + \beta_{6} q^{4} -\beta_{5} q^{6} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} -\beta_{2} q^{8} -\beta_{6} q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{5} ) q^{2} + ( 1 - \beta_{6} ) q^{3} + \beta_{6} q^{4} -\beta_{5} q^{6} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} -\beta_{2} q^{8} -\beta_{6} q^{9} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + q^{12} + ( 1 + \beta_{3} + 2 \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{14} + ( -1 + \beta_{6} ) q^{16} -4 \beta_{6} q^{17} + \beta_{2} q^{18} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{19} + ( \beta_{3} - \beta_{7} ) q^{21} + ( -1 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{22} + ( -\beta_{1} - \beta_{4} ) q^{23} + ( -\beta_{2} - \beta_{5} ) q^{24} + ( -2 - \beta_{2} - \beta_{5} + \beta_{7} ) q^{26} - q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{28} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{31} + \beta_{5} q^{32} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{33} + 4 \beta_{2} q^{34} + ( 1 - \beta_{6} ) q^{36} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{38} + ( 2 - \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{39} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{41} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{42} + ( 1 - \beta_{1} + 8 \beta_{2} - \beta_{3} + 3 \beta_{5} - 4 \beta_{6} ) q^{43} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{44} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{46} + ( -2 - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{6} - 2 \beta_{7} ) q^{47} + \beta_{6} q^{48} + ( 4 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{49} -4 q^{51} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{52} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - \beta_{7} ) q^{53} + ( \beta_{2} + \beta_{5} ) q^{54} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{56} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{57} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{58} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( -4 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} ) q^{61} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{62} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{63} - q^{64} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{66} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( 4 - 4 \beta_{6} ) q^{68} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{69} + ( -2 - 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{5} ) q^{71} -\beta_{5} q^{72} + ( -4 - 2 \beta_{2} - 2 \beta_{3} + 8 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -1 + 5 \beta_{2} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{74} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{76} + ( -2 - 2 \beta_{1} - 8 \beta_{2} + \beta_{3} + 2 \beta_{4} - 14 \beta_{5} - \beta_{7} ) q^{77} + ( -1 - \beta_{1} - 2 \beta_{5} + \beta_{6} ) q^{78} + ( -1 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{79} + ( -1 + \beta_{6} ) q^{81} + ( -2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{82} + ( -4 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{84} + ( 4 + 3 \beta_{2} + \beta_{3} - 8 \beta_{6} - \beta_{7} ) q^{86} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{87} + ( \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{88} + ( 5 + \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{89} + ( 1 - 2 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} ) q^{91} + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{92} + ( -4 - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{93} + ( -1 - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{94} -\beta_{2} q^{96} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -8 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{98} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{3} + 4q^{4} - 4q^{9} + O(q^{10})$$ $$8q + 4q^{3} + 4q^{4} - 4q^{9} + 6q^{11} + 8q^{12} + 12q^{13} + 4q^{14} - 4q^{16} - 16q^{17} - 6q^{19} - 2q^{22} - 4q^{23} - 12q^{26} - 8q^{27} - 8q^{29} + 6q^{33} + 4q^{36} - 30q^{37} + 6q^{39} + 2q^{42} - 14q^{43} - 6q^{46} + 4q^{48} + 14q^{49} - 32q^{51} + 6q^{52} - 16q^{53} + 2q^{56} + 6q^{58} + 24q^{59} - 16q^{61} - 4q^{62} - 8q^{64} - 4q^{66} - 24q^{67} + 16q^{68} + 4q^{69} - 12q^{71} + 10q^{74} - 6q^{76} - 16q^{77} - 6q^{78} - 20q^{79} - 4q^{81} - 4q^{82} + 8q^{87} + 2q^{88} + 42q^{89} - 10q^{91} - 8q^{92} - 30q^{93} - 8q^{94} + 24q^{97} - 48q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 2 x^{6} + 30 x^{5} + 185 x^{4} + 36 x^{3} + 8 x^{2} + 208 x + 2704$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + 258164 \nu - 320632$$$$)/20561424$$ $$\beta_{3}$$ $$=$$ $$($$$$-1777 \nu^{7} - 125666 \nu^{6} + 800522 \nu^{5} - 2370014 \nu^{4} - 784473 \nu^{3} - 5277648 \nu^{2} + 34618460 \nu - 62031216$$$$)/6853808$$ $$\beta_{4}$$ $$=$$ $$($$$$-299 \nu^{7} + 7801 \nu^{6} - 39200 \nu^{5} + 84154 \nu^{4} + 80915 \nu^{3} + 360425 \nu^{2} - 1640788 \nu + 2274116$$$$)/395412$$ $$\beta_{5}$$ $$=$$ $$($$$$43733 \nu^{7} - 71918 \nu^{6} - 318186 \nu^{5} + 3350390 \nu^{4} + 3714597 \nu^{3} - 2633192 \nu^{2} - 18392236 \nu + 94417440$$$$)/20561424$$ $$\beta_{6}$$ $$=$$ $$($$$$-151 \nu^{7} + 822 \nu^{6} - 2018 \nu^{5} - 2554 \nu^{4} - 7135 \nu^{3} + 37828 \nu^{2} - 46500 \nu - 3328$$$$)/51792$$ $$\beta_{7}$$ $$=$$ $$($$$$-123974 \nu^{7} + 362933 \nu^{6} - 209286 \nu^{5} - 5793110 \nu^{4} - 10547568 \nu^{3} + 7562729 \nu^{2} + 3493156 \nu - 117816036$$$$)/10280712$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} + 9 \beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$6 \beta_{7} - 10 \beta_{6} + 26 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 18 \beta_{2} + 3 \beta_{1} - 18$$ $$\nu^{4}$$ $$=$$ $$23 \beta_{7} + 139 \beta_{5} + 5 \beta_{4} + 28 \beta_{3} + 67 \beta_{2} - 5 \beta_{1} - 149$$ $$\nu^{5}$$ $$=$$ $$250 \beta_{6} + 322 \beta_{5} - 72 \beta_{4} + 72 \beta_{3} - 76 \beta_{2} - 149 \beta_{1} - 398$$ $$\nu^{6}$$ $$=$$ $$-471 \beta_{7} + 1748 \beta_{6} - 619 \beta_{5} - 619 \beta_{4} - 148 \beta_{3} - 2095 \beta_{2} - 619 \beta_{1} - 255$$ $$\nu^{7}$$ $$=$$ $$-2986 \beta_{7} + 5302 \beta_{6} - 11002 \beta_{5} - 2714 \beta_{4} - 2714 \beta_{3} - 10118 \beta_{2} - 1493 \beta_{1} + 7530$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
751.1
 1.33404 − 1.33404i −1.70006 + 1.70006i 3.17270 + 3.17270i −1.80668 − 1.80668i 1.33404 + 1.33404i −1.70006 − 1.70006i 3.17270 − 3.17270i −1.80668 + 1.80668i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i −4.02239 2.32233i 1.00000i −0.500000 + 0.866025i 0
751.2 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0 −0.866025 0.500000i 3.15637 + 1.82233i 1.00000i −0.500000 + 0.866025i 0
751.3 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i −1.14539 0.661290i 1.00000i −0.500000 + 0.866025i 0
751.4 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 0 0.866025 + 0.500000i 2.01141 + 1.16129i 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −4.02239 + 2.32233i 1.00000i −0.500000 0.866025i 0
901.2 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 3.15637 1.82233i 1.00000i −0.500000 0.866025i 0
901.3 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i −1.14539 + 0.661290i 1.00000i −0.500000 0.866025i 0
901.4 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 0 0.866025 0.500000i 2.01141 1.16129i 1.00000i −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.bc.g 8
5.b even 2 1 390.2.bb.c 8
5.c odd 4 1 1950.2.y.j 8
5.c odd 4 1 1950.2.y.k 8
13.e even 6 1 inner 1950.2.bc.g 8
15.d odd 2 1 1170.2.bs.f 8
65.l even 6 1 390.2.bb.c 8
65.l even 6 1 5070.2.b.ba 8
65.n even 6 1 5070.2.b.ba 8
65.r odd 12 1 1950.2.y.j 8
65.r odd 12 1 1950.2.y.k 8
65.s odd 12 1 5070.2.a.bz 4
65.s odd 12 1 5070.2.a.ca 4
195.y odd 6 1 1170.2.bs.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c 8 5.b even 2 1
390.2.bb.c 8 65.l even 6 1
1170.2.bs.f 8 15.d odd 2 1
1170.2.bs.f 8 195.y odd 6 1
1950.2.y.j 8 5.c odd 4 1
1950.2.y.j 8 65.r odd 12 1
1950.2.y.k 8 5.c odd 4 1
1950.2.y.k 8 65.r odd 12 1
1950.2.bc.g 8 1.a even 1 1 trivial
1950.2.bc.g 8 13.e even 6 1 inner
5070.2.a.bz 4 65.s odd 12 1
5070.2.a.ca 4 65.s odd 12 1
5070.2.b.ba 8 65.l even 6 1
5070.2.b.ba 8 65.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 21 T_{7}^{6} + 389 T_{7}^{4} - 504 T_{7}^{3} - 900 T_{7}^{2} + 1248 T_{7} + 2704$$ acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{2}$$
$3$ $$( 1 - T + T^{2} )^{4}$$
$5$ $$T^{8}$$
$7$ $$2704 + 1248 T - 900 T^{2} - 504 T^{3} + 389 T^{4} - 21 T^{6} + T^{8}$$
$11$ $$32761 + 22806 T - 138 T^{2} - 3780 T^{3} + 467 T^{4} + 180 T^{5} - 18 T^{6} - 6 T^{7} + T^{8}$$
$13$ $$28561 - 26364 T + 7605 T^{2} + 312 T^{3} - 556 T^{4} + 24 T^{5} + 45 T^{6} - 12 T^{7} + T^{8}$$
$17$ $$( 16 + 4 T + T^{2} )^{4}$$
$19$ $$219024 - 84240 T - 15876 T^{2} + 10260 T^{3} + 2421 T^{4} - 342 T^{5} - 45 T^{6} + 6 T^{7} + T^{8}$$
$23$ $$2704 - 832 T + 2492 T^{2} + 272 T^{3} + 1861 T^{4} - 140 T^{5} + 59 T^{6} + 4 T^{7} + T^{8}$$
$29$ $$141376 - 55648 T + 36568 T^{2} - 244 T^{3} + 2329 T^{4} - 16 T^{5} + 103 T^{6} + 8 T^{7} + T^{8}$$
$31$ $$913936 + 158088 T^{2} + 8777 T^{4} + 174 T^{6} + T^{8}$$
$37$ $$644809 + 216810 T - 28698 T^{2} - 17820 T^{3} + 2459 T^{4} + 1980 T^{5} + 366 T^{6} + 30 T^{7} + T^{8}$$
$41$ $$692224 - 159744 T - 57600 T^{2} + 16128 T^{3} + 6224 T^{4} - 84 T^{6} + T^{8}$$
$43$ $$327184 + 478192 T + 674300 T^{2} + 51964 T^{3} + 14125 T^{4} + 1070 T^{5} + 239 T^{6} + 14 T^{7} + T^{8}$$
$47$ $$1216609 + 421284 T^{2} + 16934 T^{4} + 228 T^{6} + T^{8}$$
$53$ $$( 52 - 4 T - 75 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$59$ $$80656 - 23856 T - 10996 T^{2} + 3948 T^{3} + 1821 T^{4} - 1128 T^{5} + 239 T^{6} - 24 T^{7} + T^{8}$$
$61$ $$( 16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$67$ $$123904 - 118272 T + 27776 T^{2} + 9408 T^{3} - 2256 T^{4} - 672 T^{5} + 164 T^{6} + 24 T^{7} + T^{8}$$
$71$ $$25240576 + 19051008 T + 5817984 T^{2} + 773568 T^{3} + 31472 T^{4} - 2448 T^{5} - 156 T^{6} + 12 T^{7} + T^{8}$$
$73$ $$20647936 + 1859072 T^{2} + 47760 T^{4} + 392 T^{6} + T^{8}$$
$79$ $$( 3508 - 860 T - 147 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$83$ $$25240576 + 2743296 T^{2} + 61904 T^{4} + 456 T^{6} + T^{8}$$
$89$ $$1008016 + 349392 T - 131316 T^{2} - 59508 T^{3} + 35117 T^{4} - 7182 T^{5} + 759 T^{6} - 42 T^{7} + T^{8}$$
$97$ $$29246464 - 23622144 T + 7289984 T^{2} - 751296 T^{3} + 48 T^{4} + 4128 T^{5} + 20 T^{6} - 24 T^{7} + T^{8}$$