Properties

Label 128.4.e.b
Level 128
Weight 4
Character orbit 128.e
Analytic conductor 7.552
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 128 = 2^{7} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 128.e (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{20} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{5} q^{3} \) \( -\beta_{3} q^{5} \) \( + ( -3 \beta_{1} + \beta_{4} ) q^{7} \) \( + ( 5 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{5} q^{3} \) \( -\beta_{3} q^{5} \) \( + ( -3 \beta_{1} + \beta_{4} ) q^{7} \) \( + ( 5 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{7} + \beta_{9} ) q^{11} \) \( + ( \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{13} \) \( + ( -13 - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{15} \) \( + ( -4 + 5 \beta_{2} - 2 \beta_{3} + 5 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{17} \) \( + ( 1 + \beta_{1} + 2 \beta_{4} + \beta_{5} + 6 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{19} \) \( + ( -8 + 8 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{21} \) \( + ( 23 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{9} ) q^{23} \) \( + ( -\beta_{1} + 10 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} ) q^{25} \) \( + ( -19 + 19 \beta_{1} - 6 \beta_{3} + 2 \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{27} \) \( + ( 16 + 16 \beta_{1} - 2 \beta_{4} + 20 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{29} \) \( + ( 38 + 4 \beta_{2} + 6 \beta_{3} + 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{31} \) \( + ( 4 - 15 \beta_{2} - 4 \beta_{3} - 15 \beta_{5} + 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{33} \) \( + ( -46 - 46 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 2 \beta_{8} ) q^{35} \) \( + ( 8 - 8 \beta_{1} - 30 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{37} \) \( + ( -71 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} ) q^{39} \) \( + ( -8 \beta_{1} - 24 \beta_{2} - 2 \beta_{3} + 24 \beta_{5} - 2 \beta_{6} - 4 \beta_{9} ) q^{41} \) \( + ( 84 - 84 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} + 4 \beta_{7} - 4 \beta_{9} ) q^{43} \) \( + ( -8 - 8 \beta_{1} - \beta_{4} - 46 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - 5 \beta_{9} ) q^{45} \) \( + ( -98 - 6 \beta_{3} + 6 \beta_{6} + 10 \beta_{7} + 4 \beta_{8} ) q^{47} \) \( + ( 7 + 26 \beta_{2} + 12 \beta_{3} + 26 \beta_{5} - 12 \beta_{6} - 6 \beta_{7} + 10 \beta_{8} ) q^{49} \) \( + ( 153 + 153 \beta_{1} - 12 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} + 12 \beta_{8} + 5 \beta_{9} ) q^{51} \) \( + ( 32 - 32 \beta_{1} + 42 \beta_{2} + 7 \beta_{3} + 13 \beta_{4} + 7 \beta_{7} + 13 \beta_{8} - 7 \beta_{9} ) q^{53} \) \( + ( 157 \beta_{1} + 28 \beta_{2} - 12 \beta_{3} - 3 \beta_{4} - 28 \beta_{5} - 12 \beta_{6} - 12 \beta_{9} ) q^{55} \) \( + ( -4 \beta_{1} + 25 \beta_{2} + 16 \beta_{3} - 7 \beta_{4} - 25 \beta_{5} + 16 \beta_{6} - 9 \beta_{9} ) q^{57} \) \( + ( -174 + 174 \beta_{1} - \beta_{2} - 14 \beta_{4} + 4 \beta_{7} - 14 \beta_{8} - 4 \beta_{9} ) q^{59} \) \( + ( -96 - 96 \beta_{1} + 9 \beta_{4} + 30 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} - 9 \beta_{8} - 11 \beta_{9} ) q^{61} \) \( + ( 271 - 32 \beta_{2} - 6 \beta_{3} - 32 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} - \beta_{8} ) q^{63} \) \( + ( -22 - 40 \beta_{2} + 26 \beta_{3} - 40 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} ) q^{65} \) \( + ( -189 - 189 \beta_{1} - 12 \beta_{4} - 11 \beta_{5} - 2 \beta_{6} - \beta_{7} + 12 \beta_{8} - \beta_{9} ) q^{67} \) \( + ( -48 + 48 \beta_{1} - 50 \beta_{2} + 6 \beta_{3} - \beta_{4} + 5 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{69} \) \( + ( -337 \beta_{1} + 28 \beta_{2} + 6 \beta_{3} - 7 \beta_{4} - 28 \beta_{5} + 6 \beta_{6} + 2 \beta_{9} ) q^{71} \) \( + ( -50 \beta_{1} - 31 \beta_{2} + 12 \beta_{3} + 17 \beta_{4} + 31 \beta_{5} + 12 \beta_{6} - \beta_{9} ) q^{73} \) \( + ( 288 - 288 \beta_{1} + 9 \beta_{2} - 28 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{75} \) \( + ( 32 + 32 \beta_{1} - 15 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 15 \beta_{8} + 5 \beta_{9} ) q^{77} \) \( + ( -428 + 4 \beta_{2} + 32 \beta_{3} + 4 \beta_{5} - 32 \beta_{6} - 8 \beta_{7} + 4 \beta_{8} ) q^{79} \) \( + ( 35 + 9 \beta_{2} - 24 \beta_{3} + 9 \beta_{5} + 24 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} ) q^{81} \) \( + ( 260 + 260 \beta_{1} + 22 \beta_{4} - 11 \beta_{5} - 28 \beta_{6} - 2 \beta_{7} - 22 \beta_{8} - 2 \beta_{9} ) q^{83} \) \( + ( 8 - 8 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} - 7 \beta_{4} - 17 \beta_{7} - 7 \beta_{8} + 17 \beta_{9} ) q^{85} \) \( + ( 575 \beta_{1} - 76 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} + 76 \beta_{5} - 10 \beta_{6} + 18 \beta_{9} ) q^{87} \) \( + ( 18 \beta_{1} + 15 \beta_{2} - 52 \beta_{3} - \beta_{4} - 15 \beta_{5} - 52 \beta_{6} + \beta_{9} ) q^{89} \) \( + ( -320 + 320 \beta_{1} + 2 \beta_{2} + 36 \beta_{3} + 34 \beta_{4} + 2 \beta_{7} + 34 \beta_{8} - 2 \beta_{9} ) q^{91} \) \( + ( 216 + 216 \beta_{1} - 10 \beta_{4} + 20 \beta_{5} - 8 \beta_{6} + 14 \beta_{7} + 10 \beta_{8} + 14 \beta_{9} ) q^{93} \) \( + ( 641 + 104 \beta_{2} - 24 \beta_{3} + 104 \beta_{5} + 24 \beta_{6} - 8 \beta_{7} + 3 \beta_{8} ) q^{95} \) \( + ( -36 + 25 \beta_{2} - 58 \beta_{3} + 25 \beta_{5} + 58 \beta_{6} - 3 \beta_{7} - 15 \beta_{8} ) q^{97} \) \( + ( -518 - 518 \beta_{1} + 22 \beta_{4} + 21 \beta_{5} + 64 \beta_{6} + 4 \beta_{7} - 22 \beta_{8} + 4 \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(10q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(10q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 124q^{15} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 26q^{19} \) \(\mathstrut -\mathstrut 52q^{21} \) \(\mathstrut -\mathstrut 184q^{27} \) \(\mathstrut +\mathstrut 202q^{29} \) \(\mathstrut +\mathstrut 368q^{31} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 476q^{35} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 838q^{43} \) \(\mathstrut -\mathstrut 194q^{45} \) \(\mathstrut -\mathstrut 944q^{47} \) \(\mathstrut +\mathstrut 94q^{49} \) \(\mathstrut +\mathstrut 1500q^{51} \) \(\mathstrut +\mathstrut 378q^{53} \) \(\mathstrut -\mathstrut 1706q^{59} \) \(\mathstrut -\mathstrut 910q^{61} \) \(\mathstrut +\mathstrut 2628q^{63} \) \(\mathstrut -\mathstrut 492q^{65} \) \(\mathstrut -\mathstrut 1942q^{67} \) \(\mathstrut -\mathstrut 580q^{69} \) \(\mathstrut +\mathstrut 2954q^{75} \) \(\mathstrut +\mathstrut 268q^{77} \) \(\mathstrut -\mathstrut 4416q^{79} \) \(\mathstrut +\mathstrut 482q^{81} \) \(\mathstrut +\mathstrut 2562q^{83} \) \(\mathstrut +\mathstrut 12q^{85} \) \(\mathstrut -\mathstrut 3332q^{91} \) \(\mathstrut +\mathstrut 2192q^{93} \) \(\mathstrut +\mathstrut 6900q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 4958q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(2\) \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut +\mathstrut \) \(6\) \(x^{7}\mathstrut +\mathstrut \) \(14\) \(x^{6}\mathstrut -\mathstrut \) \(80\) \(x^{5}\mathstrut +\mathstrut \) \(56\) \(x^{4}\mathstrut +\mathstrut \) \(96\) \(x^{3}\mathstrut -\mathstrut \) \(64\) \(x^{2}\mathstrut -\mathstrut \) \(512\) \(x\mathstrut +\mathstrut \) \(1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{9} + 14 \nu^{8} - 7 \nu^{7} - 82 \nu^{6} + 170 \nu^{5} + 120 \nu^{4} - 536 \nu^{3} - 384 \nu^{2} + 2752 \nu - 3072 \)\()/1280\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{9} + 2 \nu^{8} - 101 \nu^{7} + 114 \nu^{6} - 210 \nu^{5} + 120 \nu^{4} - 8 \nu^{3} + 3008 \nu^{2} - 3264 \nu + 7424 \)\()/1280\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{9} - 122 \nu^{8} + 341 \nu^{7} - 634 \nu^{6} + 130 \nu^{5} + 120 \nu^{4} + 3848 \nu^{3} - 9728 \nu^{2} + 16064 \nu - 2304 \)\()/1280\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 174 \nu^{6} - 870 \nu^{5} + 1240 \nu^{4} + 2152 \nu^{3} - 5632 \nu^{2} - 15424 \nu + 31744 \)\()/1280\)
\(\beta_{5}\)\(=\)\((\)\( -17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864 \)\()/1280\)
\(\beta_{6}\)\(=\)\((\)\( -37 \nu^{9} + 198 \nu^{8} - 419 \nu^{7} + 166 \nu^{6} + 50 \nu^{5} + 1560 \nu^{4} - 7992 \nu^{3} + 11392 \nu^{2} - 6976 \nu + 256 \)\()/1280\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{9} + 2 \nu^{8} + \nu^{7} - 6 \nu^{6} - 14 \nu^{5} + 80 \nu^{4} - 56 \nu^{3} - 96 \nu^{2} + 320 \nu + 416 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{9} + 10 \nu^{8} + 3 \nu^{7} - 6 \nu^{6} - 90 \nu^{5} + 184 \nu^{4} - 56 \nu^{3} - 128 \nu^{2} - 896 \nu + 1728 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( 39 \nu^{9} - 66 \nu^{8} - 47 \nu^{7} + 414 \nu^{6} - 294 \nu^{5} - 1288 \nu^{4} + 1704 \nu^{3} + 2944 \nu^{2} - 11072 \nu + 11264 \)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut -\mathstrut \) \(9\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(23\) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(43\) \(\beta_{1}\mathstrut -\mathstrut \) \(117\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(7\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(43\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(67\) \(\beta_{1}\mathstrut +\mathstrut \) \(301\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(15\) \(\beta_{8}\mathstrut +\mathstrut \) \(28\) \(\beta_{7}\mathstrut -\mathstrut \) \(27\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(49\) \(\beta_{3}\mathstrut -\mathstrut \) \(43\) \(\beta_{2}\mathstrut -\mathstrut \) \(99\) \(\beta_{1}\mathstrut +\mathstrut \) \(117\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(24\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(33\) \(\beta_{6}\mathstrut +\mathstrut \) \(133\) \(\beta_{5}\mathstrut -\mathstrut \) \(45\) \(\beta_{4}\mathstrut +\mathstrut \) \(61\) \(\beta_{3}\mathstrut -\mathstrut \) \(233\) \(\beta_{2}\mathstrut -\mathstrut \) \(155\) \(\beta_{1}\mathstrut +\mathstrut \) \(859\)\()/16\)
\(\nu^{8}\)\(=\)\((\)\(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(119\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(197\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{5}\mathstrut -\mathstrut \) \(122\) \(\beta_{4}\mathstrut -\mathstrut \) \(191\) \(\beta_{3}\mathstrut +\mathstrut \) \(91\) \(\beta_{2}\mathstrut +\mathstrut \) \(1043\) \(\beta_{1}\mathstrut +\mathstrut \) \(987\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(208\) \(\beta_{9}\mathstrut +\mathstrut \) \(146\) \(\beta_{8}\mathstrut +\mathstrut \) \(121\) \(\beta_{7}\mathstrut +\mathstrut \) \(31\) \(\beta_{6}\mathstrut +\mathstrut \) \(155\) \(\beta_{5}\mathstrut -\mathstrut \) \(75\) \(\beta_{4}\mathstrut +\mathstrut \) \(211\) \(\beta_{3}\mathstrut -\mathstrut \) \(71\) \(\beta_{2}\mathstrut +\mathstrut \) \(3019\) \(\beta_{1}\mathstrut -\mathstrut \) \(4187\)\()/16\)

Character Values

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

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.28199 + 1.53509i
0.932438 1.76934i
−1.62580 + 1.16481i
−1.56339 1.24732i
1.97476 + 0.316760i
1.28199 1.53509i
0.932438 + 1.76934i
−1.62580 1.16481i
−1.56339 + 1.24732i
1.97476 0.316760i
0 −5.49618 5.49618i 0 4.66372 4.66372i 0 24.8965i 0 33.4160i 0
33.2 0 −1.98356 1.98356i 0 0.596848 0.596848i 0 29.0828i 0 19.1310i 0
33.3 0 −0.756776 0.756776i 0 −8.22587 + 8.22587i 0 2.67171i 0 25.8546i 0
33.4 0 3.27139 + 3.27139i 0 12.6449 12.6449i 0 13.8754i 0 5.59607i 0
33.5 0 5.96513 + 5.96513i 0 −8.67959 + 8.67959i 0 1.63924i 0 44.1656i 0
97.1 0 −5.49618 + 5.49618i 0 4.66372 + 4.66372i 0 24.8965i 0 33.4160i 0
97.2 0 −1.98356 + 1.98356i 0 0.596848 + 0.596848i 0 29.0828i 0 19.1310i 0
97.3 0 −0.756776 + 0.756776i 0 −8.22587 8.22587i 0 2.67171i 0 25.8546i 0
97.4 0 3.27139 3.27139i 0 12.6449 + 12.6449i 0 13.8754i 0 5.59607i 0
97.5 0 5.96513 5.96513i 0 −8.67959 8.67959i 0 1.63924i 0 44.1656i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
16.e Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{10} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(128, [\chi])\).