# Properties

 Label 128.5.c.a Level 128 Weight 5 Character orbit 128.c Analytic conductor 13.231 Analytic rank 0 Dimension 8 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 128.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$13.2313552747$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.205520896.4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{39}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{3}$$ $$+ ( -6 - \beta_{2} ) q^{5}$$ $$+ ( -\beta_{1} - \beta_{4} ) q^{7}$$ $$+ ( -27 - \beta_{2} - \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{3}$$ $$+ ( -6 - \beta_{2} ) q^{5}$$ $$+ ( -\beta_{1} - \beta_{4} ) q^{7}$$ $$+ ( -27 - \beta_{2} - \beta_{3} ) q^{9}$$ $$+ ( -2 \beta_{1} + \beta_{4} + \beta_{6} ) q^{11}$$ $$+ ( -30 - \beta_{3} - \beta_{7} ) q^{13}$$ $$+ ( -11 \beta_{1} + \beta_{5} + 2 \beta_{6} ) q^{15}$$ $$+ ( 30 - 3 \beta_{2} + \beta_{3} + 2 \beta_{7} ) q^{17}$$ $$+ ( -10 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{19}$$ $$+ ( 152 + \beta_{2} + 7 \beta_{3} - \beta_{7} ) q^{21}$$ $$+ ( -7 \beta_{1} - 2 \beta_{4} - \beta_{5} + 6 \beta_{6} ) q^{23}$$ $$+ ( 83 + 24 \beta_{2} - 4 \beta_{3} + 2 \beta_{7} ) q^{25}$$ $$+ ( -31 \beta_{1} - 21 \beta_{4} - 2 \beta_{5} + 5 \beta_{6} ) q^{27}$$ $$+ ( -54 + 3 \beta_{2} + 4 \beta_{3} - 4 \beta_{7} ) q^{29}$$ $$+ ( 64 \beta_{1} - 3 \beta_{4} + 7 \beta_{5} + 6 \beta_{6} ) q^{31}$$ $$+ ( 124 + 21 \beta_{2} - 3 \beta_{3} + 4 \beta_{7} ) q^{33}$$ $$+ ( -10 \beta_{1} + 12 \beta_{4} + 10 \beta_{5} + 10 \beta_{6} ) q^{35}$$ $$+ ( -350 - 12 \beta_{2} - 5 \beta_{3} - 5 \beta_{7} ) q^{37}$$ $$+ ( -103 \beta_{1} - 15 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} ) q^{39}$$ $$+ ( -366 - 76 \beta_{2} + 6 \beta_{7} ) q^{41}$$ $$+ ( \beta_{1} + 10 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} ) q^{43}$$ $$+ ( 610 - 18 \beta_{2} + 9 \beta_{3} + \beta_{7} ) q^{45}$$ $$+ ( -242 \beta_{1} + 9 \beta_{4} + 17 \beta_{5} + 2 \beta_{6} ) q^{47}$$ $$+ ( -719 - 36 \beta_{2} - 20 \beta_{3} - 8 \beta_{7} ) q^{49}$$ $$+ ( 81 \beta_{1} + 9 \beta_{4} + 16 \beta_{5} - 7 \beta_{6} ) q^{51}$$ $$+ ( -222 + 2 \beta_{2} - 27 \beta_{3} - 3 \beta_{7} ) q^{53}$$ $$+ ( 175 \beta_{1} - 41 \beta_{4} - 24 \beta_{5} - 24 \beta_{6} ) q^{55}$$ $$+ ( 1076 + 95 \beta_{2} + 23 \beta_{3} - 4 \beta_{7} ) q^{57}$$ $$+ ( -29 \beta_{1} + 2 \beta_{4} - 26 \beta_{5} - 12 \beta_{6} ) q^{59}$$ $$+ ( -1582 - 36 \beta_{2} - 5 \beta_{3} + 19 \beta_{7} ) q^{61}$$ $$+ ( 643 \beta_{1} + 72 \beta_{4} + 15 \beta_{5} - 18 \beta_{6} ) q^{63}$$ $$+ ( 84 + 36 \beta_{2} + 8 \beta_{3} - 14 \beta_{7} ) q^{65}$$ $$+ ( 174 \beta_{1} + 15 \beta_{4} + 12 \beta_{5} - 45 \beta_{6} ) q^{67}$$ $$+ ( 552 + 107 \beta_{2} + 29 \beta_{3} + 21 \beta_{7} ) q^{69}$$ $$+ ( -221 \beta_{1} + 30 \beta_{4} - 31 \beta_{5} - 22 \beta_{6} ) q^{71}$$ $$+ ( 70 - 21 \beta_{2} + 27 \beta_{3} - 24 \beta_{7} ) q^{73}$$ $$+ ( -131 \beta_{1} - 96 \beta_{4} - 26 \beta_{5} - 46 \beta_{6} ) q^{75}$$ $$+ ( 3912 + 119 \beta_{2} - 15 \beta_{3} + 9 \beta_{7} ) q^{77}$$ $$+ ( -514 \beta_{1} + 58 \beta_{4} + 4 \beta_{5} - 48 \beta_{6} ) q^{79}$$ $$+ ( 1837 + 17 \beta_{2} + 89 \beta_{3} + 4 \beta_{7} ) q^{81}$$ $$+ ( -111 \beta_{1} + 110 \beta_{4} + 14 \beta_{5} - 8 \beta_{6} ) q^{83}$$ $$+ ( 1804 - 51 \beta_{2} - 15 \beta_{3} + 33 \beta_{7} ) q^{85}$$ $$+ ( 309 \beta_{1} + 108 \beta_{4} - 11 \beta_{5} + 2 \beta_{6} ) q^{87}$$ $$+ ( -2874 - 181 \beta_{2} - 53 \beta_{3} - 16 \beta_{7} ) q^{89}$$ $$+ ( 378 \beta_{1} + 128 \beta_{4} - 14 \beta_{5} - 42 \beta_{6} ) q^{91}$$ $$+ ( -7040 + 148 \beta_{2} - 68 \beta_{3} - 20 \beta_{7} ) q^{93}$$ $$+ ( 661 \beta_{1} - 127 \beta_{4} + 4 \beta_{5} - 24 \beta_{6} ) q^{95}$$ $$+ ( -466 - 183 \beta_{2} + 13 \beta_{3} - 14 \beta_{7} ) q^{97}$$ $$+ ( -201 \beta_{1} - 6 \beta_{4} - 10 \beta_{5} + 28 \beta_{6} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 48q^{5}$$ $$\mathstrut -\mathstrut 216q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 48q^{5}$$ $$\mathstrut -\mathstrut 216q^{9}$$ $$\mathstrut -\mathstrut 240q^{13}$$ $$\mathstrut +\mathstrut 240q^{17}$$ $$\mathstrut +\mathstrut 1216q^{21}$$ $$\mathstrut +\mathstrut 664q^{25}$$ $$\mathstrut -\mathstrut 432q^{29}$$ $$\mathstrut +\mathstrut 992q^{33}$$ $$\mathstrut -\mathstrut 2800q^{37}$$ $$\mathstrut -\mathstrut 2928q^{41}$$ $$\mathstrut +\mathstrut 4880q^{45}$$ $$\mathstrut -\mathstrut 5752q^{49}$$ $$\mathstrut -\mathstrut 1776q^{53}$$ $$\mathstrut +\mathstrut 8608q^{57}$$ $$\mathstrut -\mathstrut 12656q^{61}$$ $$\mathstrut +\mathstrut 672q^{65}$$ $$\mathstrut +\mathstrut 4416q^{69}$$ $$\mathstrut +\mathstrut 560q^{73}$$ $$\mathstrut +\mathstrut 31296q^{77}$$ $$\mathstrut +\mathstrut 14696q^{81}$$ $$\mathstrut +\mathstrut 14432q^{85}$$ $$\mathstrut -\mathstrut 22992q^{89}$$ $$\mathstrut -\mathstrut 56320q^{93}$$ $$\mathstrut -\mathstrut 3728q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut -\mathstrut$$ $$4$$ $$x^{7}\mathstrut +\mathstrut$$ $$12$$ $$x^{6}\mathstrut -\mathstrut$$ $$12$$ $$x^{5}\mathstrut -\mathstrut$$ $$8$$ $$x^{4}\mathstrut +\mathstrut$$ $$12$$ $$x^{3}\mathstrut +\mathstrut$$ $$12$$ $$x^{2}\mathstrut +\mathstrut$$ $$4$$ $$x\mathstrut +\mathstrut$$ $$1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$152 \nu^{7} - 638 \nu^{6} + 1996 \nu^{5} - 2362 \nu^{4} - 268 \nu^{3} + 1570 \nu^{2} + 1528 \nu + 344$$$$)/51$$ $$\beta_{2}$$ $$=$$ $$($$$$144 \nu^{7} - 676 \nu^{6} + 2120 \nu^{5} - 2864 \nu^{4} - 168 \nu^{3} + 2740 \nu^{2} + 832 \nu - 1120$$$$)/51$$ $$\beta_{3}$$ $$=$$ $$($$$$-176 \nu^{7} + 524 \nu^{6} - 1624 \nu^{5} + 720 \nu^{4} + 1656 \nu^{3} + 580 \nu^{2} + 192 \nu - 2016$$$$)/51$$ $$\beta_{4}$$ $$=$$ $$($$$$-824 \nu^{7} + 3226 \nu^{6} - 9396 \nu^{5} + 8406 \nu^{4} + 9620 \nu^{3} - 10662 \nu^{2} - 12120 \nu - 2824$$$$)/51$$ $$\beta_{5}$$ $$=$$ $$($$$$1288 \nu^{7} - 3742 \nu^{6} + 8732 \nu^{5} + 6430 \nu^{4} - 42620 \nu^{3} + 24674 \nu^{2} + 32360 \nu + 7768$$$$)/51$$ $$\beta_{6}$$ $$=$$ $$($$$$1488 \nu^{7} - 7008 \nu^{6} + 22768 \nu^{5} - 33720 \nu^{4} + 11184 \nu^{3} + 11200 \nu^{2} + 8144 \nu + 1664$$$$)/51$$ $$\beta_{7}$$ $$=$$ $$($$$$-4672 \nu^{7} + 20784 \nu^{6} - 64672 \nu^{5} + 81920 \nu^{4} + 10528 \nu^{3} - 73968 \nu^{2} - 22400 \nu - 896$$$$)/51$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$26$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$94$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$512$$$$)/1024$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{7}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$52$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$256$$$$)/256$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$3$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$7$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$19$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$13$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$85$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1408$$$$)/256$$ $$\nu^{4}$$ $$=$$ $$($$$$60$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$43$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$111$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$350$$ $$\beta_{1}$$$$)/256$$ $$\nu^{5}$$ $$=$$ $$($$$$27$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$67$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$49$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$130$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$101$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$749$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$363$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$12928$$$$)/256$$ $$\nu^{6}$$ $$=$$ $$($$$$61$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$390$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$254$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$592$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$233$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1689$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2766$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$28928$$$$)/256$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$394$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$5960$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3949$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$9397$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1218$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$10978$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$40910$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$200192$$$$)/1024$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.55050 − 0.642239i 1.12964 − 2.72719i −0.550501 − 0.228025i −0.129640 − 0.312979i −0.129640 + 0.312979i −0.550501 + 0.228025i 1.12964 + 2.72719i 1.55050 + 0.642239i
0 16.7135i 0 −10.1891 0 80.4325i 0 −198.341 0
127.2 0 7.95199i 0 −46.0525 0 57.5735i 0 17.7658 0
127.3 0 7.05664i 0 9.50277 0 6.49140i 0 31.2038 0
127.4 0 6.29514i 0 22.7388 0 51.5147i 0 41.3713 0
127.5 0 6.29514i 0 22.7388 0 51.5147i 0 41.3713 0
127.6 0 7.05664i 0 9.50277 0 6.49140i 0 31.2038 0
127.7 0 7.95199i 0 −46.0525 0 57.5735i 0 17.7658 0
127.8 0 16.7135i 0 −10.1891 0 80.4325i 0 −198.341 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{5}^{4}$$ $$\mathstrut +\mathstrut 24 T_{5}^{3}$$ $$\mathstrut -\mathstrut 1128 T_{5}^{2}$$ $$\mathstrut -\mathstrut 2976 T_{5}$$ $$\mathstrut +\mathstrut 101392$$ acting on $$S_{5}^{\mathrm{new}}(128, [\chi])$$.