# Properties

 Label 76.3.h.a Level $76$ Weight $3$ Character orbit 76.h Analytic conductor $2.071$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} + 56 x^{6} - 154 x^{5} + 917 x^{4} - 1582 x^{3} + 4294 x^{2} - 3528 x + 4971$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} - \beta_{4} ) q^{3} + ( \beta_{5} - \beta_{6} ) q^{5} + ( -2 + \beta_{2} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} - \beta_{4} ) q^{3} + ( \beta_{5} - \beta_{6} ) q^{5} + ( -2 + \beta_{2} ) q^{7} + ( -1 + \beta_{1} - \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{7} ) q^{9} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{11} + ( 1 - 3 \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{13} + ( 7 - \beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{15} + ( 6 + 6 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( -8 + 4 \beta_{1} - \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{19} + ( 3 + 4 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{21} + ( -1 + \beta_{1} + \beta_{2} + 7 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{23} + ( 4 - 4 \beta_{1} - \beta_{2} + 19 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{25} + ( -11 + 2 \beta_{1} - \beta_{2} - 20 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{27} + ( -18 - 6 \beta_{1} - 12 \beta_{3} - 6 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{29} + ( 4 - 6 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{7} ) q^{31} + ( -6 + 13 \beta_{3} + 7 \beta_{4} + \beta_{5} + \beta_{6} ) q^{33} + ( -21 + 18 \beta_{1} - 12 \beta_{3} + 9 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{35} + ( 14 + 3 \beta_{1} + 31 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} ) q^{37} + ( 28 - \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} ) q^{39} + ( 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{41} + ( -7 - 10 \beta_{1} - 12 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{43} + ( 22 - 10 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - 20 \beta_{4} ) q^{45} + ( -7 + 7 \beta_{1} + 3 \beta_{2} - 29 \beta_{3} - 7 \beta_{4} - \beta_{5} - 3 \beta_{7} ) q^{47} + ( 41 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{49} + ( -2 - 10 \beta_{1} + \beta_{2} - 6 \beta_{3} - 10 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{51} + ( 3 + 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{53} + ( 12 + 12 \beta_{1} + 18 \beta_{3} + 6 \beta_{4} + 5 \beta_{7} ) q^{55} + ( -59 + 8 \beta_{1} + 5 \beta_{2} - 14 \beta_{3} + 6 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{57} + ( 33 - 2 \beta_{2} - 27 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{59} + ( -4 + 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 4 \beta_{7} ) q^{61} + ( 6 - 6 \beta_{1} + 4 \beta_{2} + 96 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 4 \beta_{7} ) q^{63} + ( -36 + 9 \beta_{1} + 4 \beta_{2} - 63 \beta_{3} + 10 \beta_{5} - 5 \beta_{6} - 8 \beta_{7} ) q^{65} + ( -70 - 12 \beta_{1} - \beta_{2} - 41 \beta_{3} - 12 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{67} + ( -16 - 8 \beta_{1} - 4 \beta_{2} - 40 \beta_{3} - 2 \beta_{5} + \beta_{6} + 8 \beta_{7} ) q^{69} + ( -21 + 18 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{71} + ( 2 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - 12 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} ) q^{73} + ( 57 - 15 \beta_{1} + \beta_{2} + 99 \beta_{3} - 10 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{75} + ( 72 + 3 \beta_{1} + 3 \beta_{3} + 6 \beta_{4} - 10 \beta_{6} ) q^{77} + ( 5 - 8 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} ) q^{79} + ( -24 + 18 \beta_{1} - 15 \beta_{3} + 9 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} - \beta_{7} ) q^{81} + ( -7 + 10 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} + 20 \beta_{4} + 3 \beta_{6} ) q^{83} + ( 5 - 5 \beta_{1} + 2 \beta_{2} - 23 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} ) q^{85} + ( 56 + 7 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} + 14 \beta_{4} + 3 \beta_{6} ) q^{87} + ( -30 - 6 \beta_{1} - 5 \beta_{2} - 18 \beta_{3} - 6 \beta_{4} + \beta_{5} - 2 \beta_{6} - 5 \beta_{7} ) q^{89} + ( 34 - 6 \beta_{1} - 2 \beta_{2} + 14 \beta_{3} - 6 \beta_{4} - 10 \beta_{5} + 20 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 81 - 18 \beta_{1} + 72 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{93} + ( -65 - 4 \beta_{1} - 7 \beta_{2} - 46 \beta_{3} + \beta_{4} - 13 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{95} + ( 15 + 8 \beta_{2} - 9 \beta_{3} + 6 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{97} + ( 5 - 5 \beta_{1} + \beta_{2} + 103 \beta_{3} + 5 \beta_{4} + 12 \beta_{5} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 6q^{3} - q^{5} - 12q^{7} + 16q^{9} + O(q^{10})$$ $$8q + 6q^{3} - q^{5} - 12q^{7} + 16q^{9} - 10q^{11} + 9q^{13} + 33q^{15} + 23q^{17} - 33q^{19} - 31q^{23} - 73q^{25} - 105q^{29} - 111q^{33} - 68q^{35} + 234q^{39} + 18q^{41} - 41q^{43} + 200q^{45} + 107q^{47} + 312q^{49} - 9q^{51} + 39q^{53} + 70q^{55} - 381q^{57} + 348q^{59} - 45q^{61} - 358q^{63} - 432q^{67} - 243q^{71} + 16q^{73} + 544q^{77} + 75q^{79} - 68q^{81} - 82q^{83} + 109q^{85} + 414q^{87} - 213q^{89} + 222q^{91} + 288q^{93} - 385q^{95} + 144q^{97} - 388q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 56 x^{6} - 154 x^{5} + 917 x^{4} - 1582 x^{3} + 4294 x^{2} - 3528 x + 4971$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu + 13$$ $$\beta_{3}$$ $$=$$ $$($$$$8 \nu^{7} - 28 \nu^{6} + 434 \nu^{5} - 1015 \nu^{4} + 6314 \nu^{3} - 8470 \nu^{2} + 16740 \nu - 7506$$$$)/1029$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{7} + 14 \nu^{6} - 217 \nu^{5} + 1022 \nu^{4} - 4186 \nu^{3} + 17612 \nu^{2} - 21747 \nu + 39768$$$$)/1029$$ $$\beta_{5}$$ $$=$$ $$($$$$-47 \nu^{7} - 7 \nu^{6} - 1778 \nu^{5} - 2569 \nu^{4} - 14714 \nu^{3} - 57169 \nu^{2} + 21531 \nu - 187170$$$$)/2058$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{5} - 46 \nu^{4} + 87 \nu^{3} - 562 \nu^{2} + 519 \nu - 1251$$$$)/6$$ $$\beta_{7}$$ $$=$$ $$($$$$34 \nu^{7} - 119 \nu^{6} + 1673 \nu^{5} - 3885 \nu^{4} + 22204 \nu^{3} - 29309 \nu^{2} + 56739 \nu - 21439$$$$)/343$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} - 13$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - 2 \beta_{6} + 4 \beta_{5} - \beta_{3} + \beta_{2} - 17 \beta_{1} - 11$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} + 2 \beta_{4} - \beta_{3} - 24 \beta_{2} - 34 \beta_{1} + 246$$ $$\nu^{5}$$ $$=$$ $$-24 \beta_{7} + 44 \beta_{6} - 88 \beta_{5} + 5 \beta_{4} + 50 \beta_{3} - 48 \beta_{2} + 329 \beta_{1} + 466$$ $$\nu^{6}$$ $$=$$ $$-77 \beta_{7} + 136 \beta_{6} - 284 \beta_{5} - 77 \beta_{4} + 109 \beta_{3} + 485 \beta_{2} + 1029 \beta_{1} - 4820$$ $$\nu^{7}$$ $$=$$ $$497 \beta_{7} - 840 \beta_{6} + 1638 \beta_{5} - 287 \beta_{4} - 1540 \beta_{3} + 1526 \beta_{2} - 6177 \beta_{1} - 15083$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$

## 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
 0.5 − 4.68383i 0.5 − 1.68338i 0.5 + 1.77696i 0.5 + 4.59025i 0.5 + 4.68383i 0.5 + 1.68338i 0.5 − 1.77696i 0.5 − 4.59025i
0 −3.30632 1.90890i 0 −1.55796 + 2.69846i 0 −11.1883 0 2.78782 + 4.82865i 0
65.2 0 −0.707846 0.408675i 0 2.50973 4.34699i 0 7.91625 0 −4.16597 7.21567i 0
65.3 0 2.28889 + 1.32149i 0 −4.81674 + 8.34284i 0 7.59243 0 −1.00732 1.74474i 0
65.4 0 4.72527 + 2.72814i 0 3.36497 5.82829i 0 −10.3204 0 10.3855 + 17.9882i 0
69.1 0 −3.30632 + 1.90890i 0 −1.55796 2.69846i 0 −11.1883 0 2.78782 4.82865i 0
69.2 0 −0.707846 + 0.408675i 0 2.50973 + 4.34699i 0 7.91625 0 −4.16597 + 7.21567i 0
69.3 0 2.28889 1.32149i 0 −4.81674 8.34284i 0 7.59243 0 −1.00732 + 1.74474i 0
69.4 0 4.72527 2.72814i 0 3.36497 + 5.82829i 0 −10.3204 0 10.3855 17.9882i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.h.a 8
3.b odd 2 1 684.3.y.h 8
4.b odd 2 1 304.3.r.c 8
19.c even 3 1 1444.3.c.b 8
19.d odd 6 1 inner 76.3.h.a 8
19.d odd 6 1 1444.3.c.b 8
57.f even 6 1 684.3.y.h 8
76.f even 6 1 304.3.r.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.h.a 8 1.a even 1 1 trivial
76.3.h.a 8 19.d odd 6 1 inner
304.3.r.c 8 4.b odd 2 1
304.3.r.c 8 76.f even 6 1
684.3.y.h 8 3.b odd 2 1
684.3.y.h 8 57.f even 6 1
1444.3.c.b 8 19.c even 3 1
1444.3.c.b 8 19.d odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$2025 + 3240 T + 828 T^{2} - 1440 T^{3} + 211 T^{4} + 120 T^{5} - 8 T^{6} - 6 T^{7} + T^{8}$$
$5$ $$1028196 + 79092 T + 93288 T^{2} - 8736 T^{3} + 6304 T^{4} - 242 T^{5} + 87 T^{6} + T^{7} + T^{8}$$
$7$ $$( 6940 - 498 T - 158 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$11$ $$( -4746 - 2001 T - 209 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$13$ $$46785600 - 14035680 T - 1099872 T^{2} + 751032 T^{3} + 133272 T^{4} + 3294 T^{5} - 339 T^{6} - 9 T^{7} + T^{8}$$
$17$ $$56250000 - 13500000 T + 3315000 T^{2} - 327000 T^{3} + 49000 T^{4} - 3830 T^{5} + 519 T^{6} - 23 T^{7} + T^{8}$$
$19$ $$16983563041 + 1552514073 T + 35056349 T^{2} - 2386932 T^{3} - 228798 T^{4} - 6612 T^{5} + 269 T^{6} + 33 T^{7} + T^{8}$$
$23$ $$21126622500 - 1656990000 T + 212228100 T^{2} - 2559300 T^{3} + 528406 T^{4} + 5254 T^{5} + 1527 T^{6} + 31 T^{7} + T^{8}$$
$29$ $$21068100 - 124177860 T + 246010932 T^{2} - 12011976 T^{3} - 754344 T^{4} + 46620 T^{5} + 4119 T^{6} + 105 T^{7} + T^{8}$$
$31$ $$2425365504 + 163312524 T^{2} + 1572732 T^{4} + 2688 T^{6} + T^{8}$$
$37$ $$409702406400 + 3845107692 T^{2} + 7272720 T^{4} + 4800 T^{6} + T^{8}$$
$41$ $$74733890625 + 8901636750 T - 69756552 T^{2} - 50405976 T^{3} + 1927557 T^{4} + 27864 T^{5} - 1440 T^{6} - 18 T^{7} + T^{8}$$
$43$ $$386933761600 + 58553869280 T + 7304489344 T^{2} + 286525544 T^{3} + 10741456 T^{4} + 85682 T^{5} + 4183 T^{6} + 41 T^{7} + T^{8}$$
$47$ $$105819282922500 - 4227771907800 T + 156978390144 T^{2} - 2678131980 T^{3} + 55608166 T^{4} - 697856 T^{5} + 12609 T^{6} - 107 T^{7} + T^{8}$$
$53$ $$143608586342400 + 885258408960 T - 87770967552 T^{2} - 552267072 T^{3} + 42946560 T^{4} + 291564 T^{5} - 6969 T^{6} - 39 T^{7} + T^{8}$$
$59$ $$8199855604521 + 1011751326558 T + 5394103956 T^{2} - 4468816656 T^{3} + 203820795 T^{4} - 4401504 T^{5} + 53016 T^{6} - 348 T^{7} + T^{8}$$
$61$ $$4850565316 - 726686364 T + 286187072 T^{2} + 20296824 T^{3} + 6882000 T^{4} - 93702 T^{5} + 4571 T^{6} + 45 T^{7} + T^{8}$$
$67$ $$669597133730625 + 1725708786750 T - 441317432700 T^{2} - 1141199280 T^{3} + 309093759 T^{4} + 7392384 T^{5} + 79320 T^{6} + 432 T^{7} + T^{8}$$
$71$ $$918515225664 + 63656396640 T - 2859476256 T^{2} - 300085560 T^{3} + 15990696 T^{4} + 1097874 T^{5} + 24201 T^{6} + 243 T^{7} + T^{8}$$
$73$ $$1139611003515625 + 3788336787500 T + 414247499650 T^{2} - 254933560 T^{3} + 109599799 T^{4} - 34072 T^{5} + 12154 T^{6} - 16 T^{7} + T^{8}$$
$79$ $$771333601536 - 1261526918400 T + 699847174656 T^{2} - 19787846400 T^{3} + 154746432 T^{4} + 1033200 T^{5} - 11901 T^{6} - 75 T^{7} + T^{8}$$
$83$ $$( -897750 + 218025 T - 9455 T^{2} + 41 T^{3} + T^{4} )^{2}$$
$89$ $$498714472231056 - 37989000962928 T + 756634674096 T^{2} + 15840717696 T^{3} - 56397240 T^{4} - 1983456 T^{5} + 5811 T^{6} + 213 T^{7} + T^{8}$$
$97$ $$258895984160025 - 2575340253720 T - 225766839978 T^{2} + 2330735472 T^{3} + 203644287 T^{4} + 2096928 T^{5} - 7650 T^{6} - 144 T^{7} + T^{8}$$