# Properties

 Label 304.7.e.d Level $304$ Weight $7$ Character orbit 304.e Analytic conductor $69.936$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$69.9364414204$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 483 x^{6} + 75582 x^{4} + 4242376 x^{2} + 71047680$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{11}\cdot 29$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 13 - \beta_{5} ) q^{5} + ( 18 - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -131 + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 13 - \beta_{5} ) q^{5} + ( 18 - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -131 + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{9} + ( 253 + 7 \beta_{3} - 6 \beta_{4} ) q^{11} + ( 6 \beta_{1} + 3 \beta_{2} + \beta_{7} ) q^{13} + ( 41 \beta_{1} - 12 \beta_{2} + \beta_{6} - \beta_{7} ) q^{15} + ( 745 - 42 \beta_{3} - 14 \beta_{4} - 12 \beta_{5} ) q^{17} + ( -2575 + 26 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} + 44 \beta_{4} - 12 \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( -10 \beta_{1} + 28 \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{21} + ( 6275 + 93 \beta_{3} + 103 \beta_{4} - 13 \beta_{5} ) q^{23} + ( 9832 + 170 \beta_{3} + 70 \beta_{4} + 41 \beta_{5} ) q^{25} + ( 504 \beta_{1} + 38 \beta_{2} + \beta_{6} + \beta_{7} ) q^{27} + ( 402 \beta_{1} + 40 \beta_{2} + 5 \beta_{6} + 12 \beta_{7} ) q^{29} + ( -201 \beta_{1} - 98 \beta_{2} + 3 \beta_{6} + 11 \beta_{7} ) q^{31} + ( 308 \beta_{1} - 91 \beta_{2} + 7 \beta_{6} + 13 \beta_{7} ) q^{33} + ( -26345 - 135 \beta_{3} + 230 \beta_{4} + 10 \beta_{5} ) q^{35} + ( 1360 \beta_{1} - 147 \beta_{2} - 5 \beta_{6} - 13 \beta_{7} ) q^{37} + ( -5575 - 233 \beta_{3} + 147 \beta_{4} - 219 \beta_{5} ) q^{39} + ( 356 \beta_{1} - 109 \beta_{2} - 27 \beta_{6} + \beta_{7} ) q^{41} + ( -32611 - 409 \beta_{3} + 6 \beta_{4} - 22 \beta_{5} ) q^{43} + ( -23897 - 370 \beta_{3} + 50 \beta_{4} - 41 \beta_{5} ) q^{45} + ( 12543 + 1133 \beta_{3} + 340 \beta_{4} + 24 \beta_{5} ) q^{47} + ( -37838 - 46 \beta_{3} - 686 \beta_{4} - 208 \beta_{5} ) q^{49} + ( 1851 \beta_{1} - 298 \beta_{2} - 30 \beta_{6} - 40 \beta_{7} ) q^{51} + ( -1458 \beta_{1} - 132 \beta_{2} + 15 \beta_{6} + 54 \beta_{7} ) q^{53} + ( 6305 + 15 \beta_{3} - 1400 \beta_{4} - 510 \beta_{5} ) q^{55} + ( -23638 - 3240 \beta_{1} + 469 \beta_{2} + 544 \beta_{3} + 252 \beta_{4} - 561 \beta_{5} + 15 \beta_{6} - 53 \beta_{7} ) q^{57} + ( -48 \beta_{1} - 752 \beta_{2} + 61 \beta_{6} + 119 \beta_{7} ) q^{59} + ( -7105 + 1434 \beta_{3} + 2082 \beta_{4} - 573 \beta_{5} ) q^{61} + ( 17075 + 1033 \beta_{3} - 164 \beta_{4} - 202 \beta_{5} ) q^{63} + ( -5884 \beta_{1} - 1832 \beta_{2} - 24 \beta_{6} - 66 \beta_{7} ) q^{65} + ( -3277 \beta_{1} - 1298 \beta_{2} - 100 \beta_{6} - 102 \beta_{7} ) q^{67} + ( 3350 \beta_{1} + 1193 \beta_{2} + 106 \beta_{6} - 23 \beta_{7} ) q^{69} + ( -3620 \beta_{1} + 354 \beta_{2} + 28 \beta_{6} - 154 \beta_{7} ) q^{71} + ( 58875 - 3088 \beta_{3} - 1368 \beta_{4} - 2242 \beta_{5} ) q^{73} + ( 5274 \beta_{1} + 1302 \beta_{2} + 129 \beta_{6} + 141 \beta_{7} ) q^{75} + ( -218759 - 542 \beta_{3} + 2210 \beta_{4} + 1307 \beta_{5} ) q^{77} + ( -5311 \beta_{1} + 3264 \beta_{2} - \beta_{6} - 79 \beta_{7} ) q^{79} + ( -534571 + 1084 \beta_{3} + 856 \beta_{4} + 770 \beta_{5} ) q^{81} + ( -59824 + 480 \beta_{3} + 3718 \beta_{4} + 1112 \beta_{5} ) q^{83} + ( 264995 + 4700 \beta_{3} + 2240 \beta_{4} + 2045 \beta_{5} ) q^{85} + ( -350793 - 6111 \beta_{3} + 881 \beta_{4} - 923 \beta_{5} ) q^{87} + ( 13024 \beta_{1} + 303 \beta_{2} + 55 \beta_{6} - 295 \beta_{7} ) q^{89} + ( -1000 \beta_{1} + 1016 \beta_{2} - 13 \beta_{6} + 89 \beta_{7} ) q^{91} + ( 188516 - 4148 \beta_{3} + 5760 \beta_{4} + 150 \beta_{5} ) q^{93} + ( 299263 - 16560 \beta_{1} - 2070 \beta_{2} - 515 \beta_{3} + 6760 \beta_{4} + 2554 \beta_{5} - 50 \beta_{6} - 140 \beta_{7} ) q^{95} + ( -5784 \beta_{1} - 1898 \beta_{2} + 396 \beta_{6} + 152 \beta_{7} ) q^{97} + ( -65357 - 1675 \beta_{3} + 1532 \beta_{4} + 1342 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 108q^{5} + 140q^{7} - 1052q^{9} + O(q^{10})$$ $$8q + 108q^{5} + 140q^{7} - 1052q^{9} + 2024q^{11} + 6008q^{17} - 20552q^{19} + 50252q^{23} + 78492q^{25} - 210800q^{35} - 43724q^{39} - 260800q^{43} - 191012q^{45} + 100248q^{47} - 301872q^{49} + 52480q^{55} - 186860q^{57} - 54548q^{61} + 137408q^{63} + 479968q^{73} - 1755300q^{77} - 4279648q^{81} - 483040q^{83} + 2111780q^{85} - 2802652q^{87} + 1507528q^{93} + 2383888q^{95} - 528224q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 483 x^{6} + 75582 x^{4} + 4242376 x^{2} + 71047680$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-11 \nu^{7} - 5377 \nu^{5} - 753578 \nu^{3} - 25301016 \nu$$$$)/1200192$$ $$\beta_{2}$$ $$=$$ $$($$$$-11 \nu^{7} - 5377 \nu^{5} - 753578 \nu^{3} - 6097944 \nu$$$$)/1200192$$ $$\beta_{3}$$ $$=$$ $$($$$$-11 \nu^{6} - 5377 \nu^{4} - 903602 \nu^{2} - 42403752$$$$)/150024$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{6} - 5377 \nu^{4} - 703570 \nu^{2} - 18249888$$$$)/50008$$ $$\beta_{5}$$ $$=$$ $$($$$$107 \nu^{6} + 38665 \nu^{4} + 3811514 \nu^{2} + 91312392$$$$)/150024$$ $$\beta_{6}$$ $$=$$ $$($$$$319 \nu^{7} + 155933 \nu^{5} + 26654530 \nu^{3} + 1540258488 \nu$$$$)/1200192$$ $$\beta_{7}$$ $$=$$ $$($$$$269 \nu^{7} + 104215 \nu^{5} + 11090870 \nu^{3} + 255721320 \nu$$$$)/300048$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} - 3 \beta_{3} - 483$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} - 42 \beta_{2} + 71 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-44 \beta_{5} - 365 \beta_{4} + 667 \beta_{3} + 82103$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-44 \beta_{7} - 269 \beta_{6} + 7971 \beta_{2} - 20076 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$21508 \beta_{5} + 96273 \beta_{4} - 134159 \beta_{3} - 15876643$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$21508 \beta_{7} + 62985 \beta_{6} - 1594095 \beta_{2} + 5088100 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 5.43437i 14.9269i − 8.08057i − 12.8592i 12.8592i 8.08057i − 14.9269i − 5.43437i
0 33.7437i 0 −51.7597 0 313.488 0 −409.636 0
113.2 0 33.0827i 0 159.622 0 −445.794 0 −365.466 0
113.3 0 27.2115i 0 162.986 0 68.0674 0 −11.4646 0
113.4 0 21.6433i 0 −216.848 0 134.238 0 260.567 0
113.5 0 21.6433i 0 −216.848 0 134.238 0 260.567 0
113.6 0 27.2115i 0 162.986 0 68.0674 0 −11.4646 0
113.7 0 33.0827i 0 159.622 0 −445.794 0 −365.466 0
113.8 0 33.7437i 0 −51.7597 0 313.488 0 −409.636 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.8 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 304.7.e.d 8
4.b odd 2 1 19.7.b.b 8
12.b even 2 1 171.7.c.d 8
19.b odd 2 1 inner 304.7.e.d 8
76.d even 2 1 19.7.b.b 8
228.b odd 2 1 171.7.c.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.7.b.b 8 4.b odd 2 1
19.7.b.b 8 76.d even 2 1
171.7.c.d 8 12.b even 2 1
171.7.c.d 8 228.b odd 2 1
304.7.e.d 8 1.a even 1 1 trivial
304.7.e.d 8 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{8} + 3442 T_{3}^{6} + 4292649 T_{3}^{4} + 2281096296 T_{3}^{2} + 432254085120$$ $$T_{5}^{4} - 54 T_{5}^{3} - 49415 T_{5}^{2} + 3367200 T_{5} + 292006000$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$432254085120 + 2281096296 T^{2} + 4292649 T^{4} + 3442 T^{6} + T^{8}$$
$5$ $$( 292006000 + 3367200 T - 49415 T^{2} - 54 T^{3} + T^{4} )^{2}$$
$7$ $$( -1276939885 + 29481334 T - 157380 T^{2} - 70 T^{3} + T^{4} )^{2}$$
$11$ $$( 401057409740 + 456204412 T - 1577997 T^{2} - 1012 T^{3} + T^{4} )^{2}$$
$13$ $$25\!\cdots\!80$$$$+ 9427729558012913056 T^{2} + 44831312024289 T^{4} + 14425362 T^{6} + T^{8}$$
$17$ $$( -21923949558015 + 54981558852 T - 27529338 T^{2} - 3004 T^{3} + T^{4} )^{2}$$
$19$ $$48\!\cdots\!21$$$$+$$$$21\!\cdots\!32$$$$T +$$$$31\!\cdots\!56$$$$T^{2} - 4770870492621376968 T^{3} - 5869847316568962 T^{4} - 101408888328 T^{5} + 144430096 T^{6} + 20552 T^{7} + T^{8}$$
$23$ $$( -7335568675200580 + 2996194244464 T - 56925063 T^{2} - 25126 T^{3} + T^{4} )^{2}$$
$29$ $$79\!\cdots\!00$$$$+$$$$17\!\cdots\!40$$$$T^{2} + 1722370266650575953 T^{4} + 2760432378 T^{6} + T^{8}$$
$31$ $$18\!\cdots\!00$$$$+$$$$42\!\cdots\!40$$$$T^{2} + 2319750675886043088 T^{4} + 2963275728 T^{6} + T^{8}$$
$37$ $$21\!\cdots\!20$$$$+$$$$49\!\cdots\!84$$$$T^{2} + 37605062047833382608 T^{4} + 10947702288 T^{6} + T^{8}$$
$41$ $$15\!\cdots\!00$$$$+$$$$72\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!12$$$$T^{4} + 22588953936 T^{6} + T^{8}$$
$43$ $$( -1733938754887040560 - 57659975836784 T + 3772254615 T^{2} + 130400 T^{3} + T^{4} )^{2}$$
$47$ $$( -36330072548540490980 + 1650087163302204 T - 17694876341 T^{2} - 50124 T^{3} + T^{4} )^{2}$$
$53$ $$44\!\cdots\!20$$$$+$$$$30\!\cdots\!96$$$$T^{2} +$$$$65\!\cdots\!09$$$$T^{4} + 47671898922 T^{6} + T^{8}$$
$59$ $$41\!\cdots\!00$$$$+$$$$82\!\cdots\!40$$$$T^{2} +$$$$26\!\cdots\!77$$$$T^{4} + 296237152074 T^{6} + T^{8}$$
$61$ $$( -$$$$10\!\cdots\!80$$$$- 10167316552637744 T - 121704664935 T^{2} + 27274 T^{3} + T^{4} )^{2}$$
$67$ $$86\!\cdots\!80$$$$+$$$$53\!\cdots\!04$$$$T^{2} +$$$$92\!\cdots\!49$$$$T^{4} + 555571931250 T^{6} + T^{8}$$
$71$ $$87\!\cdots\!00$$$$+$$$$41\!\cdots\!60$$$$T^{2} +$$$$68\!\cdots\!72$$$$T^{4} + 448388036040 T^{6} + T^{8}$$
$73$ $$( -$$$$18\!\cdots\!15$$$$+ 160405740236954736 T - 346509736338 T^{2} - 239984 T^{3} + T^{4} )^{2}$$
$79$ $$59\!\cdots\!00$$$$+$$$$10\!\cdots\!40$$$$T^{2} +$$$$65\!\cdots\!08$$$$T^{4} + 1476186591600 T^{6} + T^{8}$$
$83$ $$($$$$12\!\cdots\!60$$$$+ 3762431272496032 T - 321702333684 T^{2} + 241520 T^{3} + T^{4} )^{2}$$
$89$ $$10\!\cdots\!00$$$$+$$$$40\!\cdots\!60$$$$T^{2} +$$$$49\!\cdots\!48$$$$T^{4} + 2032849953192 T^{6} + T^{8}$$
$97$ $$14\!\cdots\!00$$$$+$$$$80\!\cdots\!00$$$$T^{2} +$$$$53\!\cdots\!60$$$$T^{4} + 4640339680008 T^{6} + T^{8}$$