# Properties

 Label 384.7.h.e Level $384$ Weight $7$ Character orbit 384.h Analytic conductor $88.341$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 384.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$88.3407681100$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 172x^{6} + 13179x^{4} - 522628x^{2} + 8755681$$ x^8 - 172*x^6 + 13179*x^4 - 522628*x^2 + 8755681 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{26}\cdot 3^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - 9) q^{3} - \beta_{6} q^{5} + (\beta_{6} + \beta_{3}) q^{7} + (3 \beta_{7} - 21 \beta_{5} - 15 \beta_{4} - 207) q^{9}+O(q^{10})$$ q + (b5 - 9) * q^3 - b6 * q^5 + (b6 + b3) * q^7 + (3*b7 - 21*b5 - 15*b4 - 207) * q^9 $$q + (\beta_{5} - 9) q^{3} - \beta_{6} q^{5} + (\beta_{6} + \beta_{3}) q^{7} + (3 \beta_{7} - 21 \beta_{5} - 15 \beta_{4} - 207) q^{9} + ( - 13 \beta_{5} - 13 \beta_{4} - 206) q^{11} - \beta_1 q^{13} + (9 \beta_{6} - 5 \beta_{3} + \beta_{2} + \beta_1) q^{15} + (22 \beta_{7} + 30 \beta_{5} - 30 \beta_{4}) q^{17} + (4 \beta_{7} + 51 \beta_{5} - 51 \beta_{4}) q^{19} + (27 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} - \beta_1) q^{21} + (6 \beta_{2} - 2 \beta_1) q^{23} + (20 \beta_{5} + 20 \beta_{4} - 25) q^{25} + ( - 108 \beta_{7} - 306 \beta_{5} - 189 \beta_{4} - 2673) q^{27} + (99 \beta_{6} - 88 \beta_{3}) q^{29} + (125 \beta_{6} - 23 \beta_{3}) q^{31} + ( - 39 \beta_{7} - 167 \beta_{5} + 195 \beta_{4} - 2826) q^{33} + ( - 1580 \beta_{5} - 1580 \beta_{4} - 17040) q^{35} + ( - 24 \beta_{2} - 19 \beta_1) q^{37} + ( - 252 \beta_{6} + 66 \beta_{3} - 3 \beta_{2} + 21 \beta_1) q^{39} + ( - 280 \beta_{7} - 468 \beta_{5} + 468 \beta_{4}) q^{41} + ( - 572 \beta_{7} + 561 \beta_{5} - 561 \beta_{4}) q^{43} + (207 \beta_{6} + 180 \beta_{3} + 12 \beta_{2} - 18 \beta_1) q^{45} + ( - 48 \beta_{2} - 40 \beta_1) q^{47} + (3284 \beta_{5} + 3284 \beta_{4} + 13151) q^{49} + ( - 240 \beta_{7} - 2214 \beta_{5} - 4146 \beta_{4} + \cdots - 34416) q^{51}+ \cdots + (84 \beta_{7} + 3741 \beta_{5} + 9057 \beta_{4} + 211122) q^{99}+O(q^{100})$$ q + (b5 - 9) * q^3 - b6 * q^5 + (b6 + b3) * q^7 + (3*b7 - 21*b5 - 15*b4 - 207) * q^9 + (-13*b5 - 13*b4 - 206) * q^11 - b1 * q^13 + (9*b6 - 5*b3 + b2 + b1) * q^15 + (22*b7 + 30*b5 - 30*b4) * q^17 + (4*b7 + 51*b5 - 51*b4) * q^19 + (27*b6 - 4*b3 - 4*b2 - b1) * q^21 + (6*b2 - 2*b1) * q^23 + (20*b5 + 20*b4 - 25) * q^25 + (-108*b7 - 306*b5 - 189*b4 - 2673) * q^27 + (99*b6 - 88*b3) * q^29 + (125*b6 - 23*b3) * q^31 + (-39*b7 - 167*b5 + 195*b4 - 2826) * q^33 + (-1580*b5 - 1580*b4 - 17040) * q^35 + (-24*b2 - 19*b1) * q^37 + (-252*b6 + 66*b3 - 3*b2 + 21*b1) * q^39 + (-280*b7 - 468*b5 + 468*b4) * q^41 + (-572*b7 + 561*b5 - 561*b4) * q^43 + (207*b6 + 180*b3 + 12*b2 - 18*b1) * q^45 + (-48*b2 - 40*b1) * q^47 + (3284*b5 + 3284*b4 + 13151) * q^49 + (-240*b7 - 2214*b5 - 4146*b4 - 34416) * q^51 + (-773*b6 - 272*b3) * q^53 + (206*b6 + 130*b3) * q^55 + (93*b7 - 1359*b5 - 1437*b4 - 48888) * q^57 + (-7527*b5 - 7527*b4 - 104578) * q^59 + (72*b2 + 95*b1) * q^61 + (-1503*b6 - 387*b3 - 30*b2 - 54*b1) * q^63 + (1500*b7 - 4560*b5 + 4560*b4) * q^65 + (-1856*b7 + 789*b5 - 789*b4) * q^67 + (792*b6 + 1068*b3 + 12*b2 + 114*b1) * q^69 + (102*b2 + 150*b1) * q^71 + (-3880*b5 - 3880*b4 - 245618) * q^73 + (60*b7 - 85*b5 - 300*b4 + 7425) * q^75 + (-1142*b6 - 336*b3) * q^77 + (5769*b6 - 43*b3) * q^79 + (702*b7 + 7074*b5 + 22734*b4 + 20817) * q^81 + (2435*b5 + 2435*b4 + 73378) * q^83 + (192*b2 - 28*b1) * q^85 + (-4059*b6 + 1287*b3 + 165*b2 - 99*b1) * q^87 + (-4086*b7 - 12810*b5 + 12810*b4) * q^89 + (3036*b7 - 4368*b5 + 4368*b4) * q^91 + (-1953*b6 + 832*b3 - 56*b2 - 125*b1) * q^93 + (126*b2 + 86*b1) * q^95 + (50092*b5 + 50092*b4 - 184654) * q^97 + (84*b7 + 3741*b5 + 9057*b4 + 211122) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 72 q^{3} - 1656 q^{9}+O(q^{10})$$ 8 * q - 72 * q^3 - 1656 * q^9 $$8 q - 72 q^{3} - 1656 q^{9} - 1648 q^{11} - 200 q^{25} - 21384 q^{27} - 22608 q^{33} - 136320 q^{35} + 105208 q^{49} - 275328 q^{51} - 391104 q^{57} - 836624 q^{59} - 1964944 q^{73} + 59400 q^{75} + 166536 q^{81} + 587024 q^{83} - 1477232 q^{97} + 1688976 q^{99}+O(q^{100})$$ 8 * q - 72 * q^3 - 1656 * q^9 - 1648 * q^11 - 200 * q^25 - 21384 * q^27 - 22608 * q^33 - 136320 * q^35 + 105208 * q^49 - 275328 * q^51 - 391104 * q^57 - 836624 * q^59 - 1964944 * q^73 + 59400 * q^75 + 166536 * q^81 + 587024 * q^83 - 1477232 * q^97 + 1688976 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 172x^{6} + 13179x^{4} - 522628x^{2} + 8755681$$ :

 $$\beta_{1}$$ $$=$$ $$( 4304\nu^{6} - 469248\nu^{4} + 18250656\nu^{2} - 251351264 ) / 97081$$ (4304*v^6 - 469248*v^4 + 18250656*v^2 - 251351264) / 97081 $$\beta_{2}$$ $$=$$ $$( 10144\nu^{6} - 1244544\nu^{4} + 74403840\nu^{2} - 1830375232 ) / 97081$$ (10144*v^6 - 1244544*v^4 + 74403840*v^2 - 1830375232) / 97081 $$\beta_{3}$$ $$=$$ $$( -31800\nu^{7} + 5753664\nu^{5} - 475691952\nu^{3} + 27076321320\nu ) / 287262679$$ (-31800*v^7 + 5753664*v^5 - 475691952*v^3 + 27076321320*v) / 287262679 $$\beta_{4}$$ $$=$$ $$( 37990 \nu^{7} - 47344 \nu^{6} - 4705618 \nu^{5} + 9433292 \nu^{4} + 279940646 \nu^{3} - 593741104 \nu^{2} - 6688318664 \nu + 16218154722 ) / 287262679$$ (37990*v^7 - 47344*v^6 - 4705618*v^5 + 9433292*v^4 + 279940646*v^3 - 593741104*v^2 - 6688318664*v + 16218154722) / 287262679 $$\beta_{5}$$ $$=$$ $$( - 37990 \nu^{7} - 47344 \nu^{6} + 4705618 \nu^{5} + 9433292 \nu^{4} - 279940646 \nu^{3} - 593741104 \nu^{2} + 6688318664 \nu + 16218154722 ) / 287262679$$ (-37990*v^7 - 47344*v^6 + 4705618*v^5 + 9433292*v^4 - 279940646*v^3 - 593741104*v^2 + 6688318664*v + 16218154722) / 287262679 $$\beta_{6}$$ $$=$$ $$( -293636\nu^{7} + 47925144\nu^{5} - 2781189072\nu^{3} + 71694324452\nu ) / 861788037$$ (-293636*v^7 + 47925144*v^5 - 2781189072*v^3 + 71694324452*v) / 861788037 $$\beta_{7}$$ $$=$$ $$( -240320\nu^{7} + 26137616\nu^{5} - 1288141264\nu^{3} + 26931123856\nu ) / 287262679$$ (-240320*v^7 + 26137616*v^5 - 1288141264*v^3 + 26931123856*v) / 287262679
 $$\nu$$ $$=$$ $$( \beta_{7} - 4\beta_{5} + 4\beta_{4} + 2\beta_{3} ) / 96$$ (b7 - 4*b5 + 4*b4 + 2*b3) / 96 $$\nu^{2}$$ $$=$$ $$( 48\beta_{5} + 48\beta_{4} + \beta_{2} - 2\beta _1 + 8256 ) / 192$$ (48*b5 + 48*b4 + b2 - 2*b1 + 8256) / 192 $$\nu^{3}$$ $$=$$ $$( 127\beta_{7} + 90\beta_{6} - 544\beta_{5} + 544\beta_{4} + 63\beta_{3} ) / 96$$ (127*b7 + 90*b6 - 544*b5 + 544*b4 + 63*b3) / 96 $$\nu^{4}$$ $$=$$ $$( 2718\beta_{5} + 2718\beta_{4} + 23\beta_{2} - 34\beta _1 + 38712 ) / 48$$ (2718*b5 + 2718*b4 + 23*b2 - 34*b1 + 38712) / 48 $$\nu^{5}$$ $$=$$ $$( 5056\beta_{7} + 11970\beta_{6} - 31372\beta_{5} + 31372\beta_{4} - 95\beta_{3} ) / 96$$ (5056*b7 + 11970*b6 - 31372*b5 + 31372*b4 - 95*b3) / 96 $$\nu^{6}$$ $$=$$ $$( 163632\beta_{5} + 163632\beta_{4} + 965\beta_{2} - 336\beta _1 - 1152256 ) / 32$$ (163632*b5 + 163632*b4 + 965*b2 - 336*b1 - 1152256) / 32 $$\nu^{7}$$ $$=$$ $$( -133523\beta_{7} + 819468\beta_{6} - 944428\beta_{5} + 944428\beta_{4} - 123892\beta_{3} ) / 96$$ (-133523*b7 + 819468*b6 - 944428*b5 + 944428*b4 - 123892*b3) / 96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −6.86097 + 3.28347i 6.86097 + 3.28347i −6.86097 − 3.28347i 6.86097 − 3.28347i 7.10120 + 0.847848i −7.10120 + 0.847848i 7.10120 − 0.847848i −7.10120 − 0.847848i
0 −22.4164 15.0501i 0 −122.733 0 −206.594 0 275.991 + 674.737i 0
65.2 0 −22.4164 15.0501i 0 122.733 0 206.594 0 275.991 + 674.737i 0
65.3 0 −22.4164 + 15.0501i 0 −122.733 0 −206.594 0 275.991 674.737i 0
65.4 0 −22.4164 + 15.0501i 0 122.733 0 206.594 0 275.991 674.737i 0
65.5 0 4.41641 26.6364i 0 −127.030 0 467.888 0 −689.991 235.274i 0
65.6 0 4.41641 26.6364i 0 127.030 0 −467.888 0 −689.991 235.274i 0
65.7 0 4.41641 + 26.6364i 0 −127.030 0 467.888 0 −689.991 + 235.274i 0
65.8 0 4.41641 + 26.6364i 0 127.030 0 −467.888 0 −689.991 + 235.274i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
12.b even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.7.h.e 8
3.b odd 2 1 384.7.h.f yes 8
4.b odd 2 1 384.7.h.f yes 8
8.b even 2 1 384.7.h.f yes 8
8.d odd 2 1 inner 384.7.h.e 8
12.b even 2 1 inner 384.7.h.e 8
24.f even 2 1 384.7.h.f yes 8
24.h odd 2 1 inner 384.7.h.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.7.h.e 8 1.a even 1 1 trivial
384.7.h.e 8 8.d odd 2 1 inner
384.7.h.e 8 12.b even 2 1 inner
384.7.h.e 8 24.h odd 2 1 inner
384.7.h.f yes 8 3.b odd 2 1
384.7.h.f yes 8 4.b odd 2 1
384.7.h.f yes 8 8.b even 2 1
384.7.h.f yes 8 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{4} - 31200T_{5}^{2} + 243072000$$ T5^4 - 31200*T5^2 + 243072000 $$T_{11}^{2} + 412T_{11} - 79244$$ T11^2 + 412*T11 - 79244

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 36 T^{3} + 1062 T^{2} + \cdots + 531441)^{2}$$
$5$ $$(T^{4} - 31200 T^{2} + \cdots + 243072000)^{2}$$
$7$ $$(T^{4} - 261600 T^{2} + \cdots + 9343687680)^{2}$$
$11$ $$(T^{2} + 412 T - 79244)^{4}$$
$13$ $$(T^{4} + 12372480 T^{2} + \cdots + 1562505707520)^{2}$$
$17$ $$(T^{4} + 31653504 T^{2} + \cdots + 231275569987584)^{2}$$
$19$ $$(T^{4} + 11092896 T^{2} + \cdots + 6011206401024)^{2}$$
$23$ $$(T^{4} + 590284800 T^{2} + \cdots + 87\!\cdots\!20)^{2}$$
$29$ $$(T^{4} - 1995222240 T^{2} + \cdots + 51\!\cdots\!80)^{2}$$
$31$ $$(T^{4} - 589774560 T^{2} + \cdots + 33\!\cdots\!80)^{2}$$
$37$ $$(T^{4} + 7027776000 T^{2} + \cdots + 45\!\cdots\!00)^{2}$$
$41$ $$(T^{4} + 5457148416 T^{2} + \cdots + 62\!\cdots\!44)^{2}$$
$43$ $$(T^{4} + 18530946720 T^{2} + \cdots + 34\!\cdots\!00)^{2}$$
$47$ $$(T^{4} + 29138780160 T^{2} + \cdots + 93\!\cdots\!20)^{2}$$
$53$ $$(T^{4} - 36473745120 T^{2} + \cdots + 10\!\cdots\!80)^{2}$$
$59$ $$(T^{2} + 209156 T - 29855566796)^{4}$$
$61$ $$(T^{4} + 108994152960 T^{2} + \cdots + 28\!\cdots\!20)^{2}$$
$67$ $$(T^{4} + 189436899744 T^{2} + \cdots + 60\!\cdots\!04)^{2}$$
$71$ $$(T^{4} + 258245314560 T^{2} + \cdots + 16\!\cdots\!20)^{2}$$
$73$ $$(T^{2} + 491236 T + 49489033924)^{4}$$
$79$ $$(T^{4} - 1037364956640 T^{2} + \cdots + 26\!\cdots\!80)^{2}$$
$83$ $$(T^{2} - 146756 T + 1115288884)^{4}$$
$89$ $$(T^{4} + 1658159600256 T^{2} + \cdots + 25\!\cdots\!64)^{2}$$
$97$ $$(T^{2} + 369308 T - 1772532994364)^{4}$$