Properties

Label 8048.2.a.q
Level 8048
Weight 2
Character orbit 8048.a
Self dual Yes
Analytic conductor 64.264
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8048.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{9} q^{5} \) \( + ( -1 - \beta_{10} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{9} q^{5} \) \( + ( -1 - \beta_{10} ) q^{7} \) \( + ( 1 + \beta_{2} ) q^{9} \) \( + ( -2 + \beta_{8} ) q^{11} \) \( + \beta_{11} q^{13} \) \( + ( 1 + \beta_{3} + \beta_{4} + \beta_{10} ) q^{15} \) \( + ( 1 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{17} \) \( + ( -1 + \beta_{6} ) q^{19} \) \( + ( \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{21} \) \( + ( -1 - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} \) \( + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{25} \) \( + ( -1 - \beta_{4} - \beta_{5} - \beta_{6} ) q^{27} \) \( + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{29} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} ) q^{31} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - 3 \beta_{8} + \beta_{10} + \beta_{11} ) q^{33} \) \( + ( 2 \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{35} \) \( + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{37} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{39} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{41} \) \( + ( \beta_{1} + \beta_{2} - 2 \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{43} \) \( + ( -\beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{45} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{47} \) \( + ( 1 + \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{49} \) \( + ( \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{51} \) \( + ( 1 - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{10} ) q^{53} \) \( + ( -1 - \beta_{2} - \beta_{5} + 2 \beta_{8} + 2 \beta_{9} ) q^{55} \) \( + ( -\beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{57} \) \( + ( -1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{8} + \beta_{10} + \beta_{11} ) q^{59} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{61} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{63} \) \( + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{65} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{67} \) \( + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{69} \) \( + ( -2 - 3 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{71} \) \( + ( -2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{73} \) \( + ( 4 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{75} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{77} \) \( + ( -2 + \beta_{3} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{79} \) \( + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{81} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{83} \) \( + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{85} \) \( + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{87} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{10} ) q^{89} \) \( + ( 3 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 3 \beta_{11} ) q^{91} \) \( + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{93} \) \( + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{8} ) q^{95} \) \( + ( -7 + \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} \) \( + ( -\beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{10} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 15q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 5q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 15q^{9} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 2q^{15} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 20q^{29} \) \(\mathstrut -\mathstrut 7q^{31} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut q^{35} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 7q^{39} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 7q^{45} \) \(\mathstrut -\mathstrut 12q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut -\mathstrut 6q^{59} \) \(\mathstrut -\mathstrut 10q^{61} \) \(\mathstrut -\mathstrut 7q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 7q^{67} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut -\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 23q^{73} \) \(\mathstrut +\mathstrut 34q^{75} \) \(\mathstrut -\mathstrut 19q^{77} \) \(\mathstrut -\mathstrut 13q^{79} \) \(\mathstrut -\mathstrut 28q^{81} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut -\mathstrut 28q^{85} \) \(\mathstrut +\mathstrut 22q^{87} \) \(\mathstrut -\mathstrut 3q^{89} \) \(\mathstrut +\mathstrut 21q^{91} \) \(\mathstrut -\mathstrut 33q^{93} \) \(\mathstrut +\mathstrut 2q^{95} \) \(\mathstrut -\mathstrut 70q^{97} \) \(\mathstrut +\mathstrut 6q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(5\) \(x^{11}\mathstrut -\mathstrut \) \(13\) \(x^{10}\mathstrut +\mathstrut \) \(94\) \(x^{9}\mathstrut +\mathstrut \) \(15\) \(x^{8}\mathstrut -\mathstrut \) \(616\) \(x^{7}\mathstrut +\mathstrut \) \(368\) \(x^{6}\mathstrut +\mathstrut \) \(1643\) \(x^{5}\mathstrut -\mathstrut \) \(1463\) \(x^{4}\mathstrut -\mathstrut \) \(1556\) \(x^{3}\mathstrut +\mathstrut \) \(1284\) \(x^{2}\mathstrut +\mathstrut \) \(576\) \(x\mathstrut -\mathstrut \) \(208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + 543 \nu^{10} - 917 \nu^{9} - 9770 \nu^{8} + 14883 \nu^{7} + 57076 \nu^{6} - 82700 \nu^{5} - 116237 \nu^{4} + 172813 \nu^{3} + 42460 \nu^{2} - 79464 \nu + 1864 \)\()/7192\)
\(\beta_{4}\)\(=\)\((\)\( 115 \nu^{11} - 485 \nu^{10} - 2969 \nu^{9} + 10988 \nu^{8} + 28617 \nu^{7} - 87062 \nu^{6} - 124940 \nu^{5} + 283161 \nu^{4} + 244729 \nu^{3} - 325906 \nu^{2} - 184320 \nu + 56136 \)\()/7192\)
\(\beta_{5}\)\(=\)\((\)\( 577 \nu^{11} - 1339 \nu^{10} - 11285 \nu^{9} + 24612 \nu^{8} + 77557 \nu^{7} - 157540 \nu^{6} - 221932 \nu^{5} + 406595 \nu^{4} + 245943 \nu^{3} - 348940 \nu^{2} - 127588 \nu + 65052 \)\()/3596\)
\(\beta_{6}\)\(=\)\((\)\( -1269 \nu^{11} + 3163 \nu^{10} + 25539 \nu^{9} - 60212 \nu^{8} - 183731 \nu^{7} + 402142 \nu^{6} + 568804 \nu^{5} - 1096351 \nu^{4} - 729423 \nu^{3} + 1023786 \nu^{2} + 396344 \nu - 193432 \)\()/7192\)
\(\beta_{7}\)\(=\)\((\)\( -1533 \nu^{11} + 3651 \nu^{10} + 30291 \nu^{9} - 66456 \nu^{8} - 214779 \nu^{7} + 419298 \nu^{6} + 663784 \nu^{5} - 1062087 \nu^{4} - 869847 \nu^{3} + 882834 \nu^{2} + 471292 \nu - 131744 \)\()/7192\)
\(\beta_{8}\)\(=\)\((\)\( 969 \nu^{11} - 2445 \nu^{10} - 18341 \nu^{9} + 44290 \nu^{8} + 120335 \nu^{7} - 276728 \nu^{6} - 319484 \nu^{5} + 683499 \nu^{4} + 302929 \nu^{3} - 523112 \nu^{2} - 136116 \nu + 72944 \)\()/3596\)
\(\beta_{9}\)\(=\)\((\)\( 1163 \nu^{11} - 3185 \nu^{10} - 21833 \nu^{9} + 58386 \nu^{8} + 141625 \nu^{7} - 369564 \nu^{6} - 370074 \nu^{5} + 924969 \nu^{4} + 351689 \nu^{3} - 713098 \nu^{2} - 199010 \nu + 85748 \)\()/3596\)
\(\beta_{10}\)\(=\)\((\)\( 2572 \nu^{11} - 6743 \nu^{10} - 48993 \nu^{9} + 123571 \nu^{8} + 325094 \nu^{7} - 783065 \nu^{6} - 882020 \nu^{5} + 1969696 \nu^{4} + 887759 \nu^{3} - 1550579 \nu^{2} - 452248 \nu + 209308 \)\()/3596\)
\(\beta_{11}\)\(=\)\((\)\( -646 \nu^{11} + 1630 \nu^{10} + 12527 \nu^{9} - 30126 \nu^{8} - 85018 \nu^{7} + 192876 \nu^{6} + 237562 \nu^{5} - 490727 \nu^{4} - 249300 \nu^{3} + 391294 \nu^{2} + 125805 \nu - 53424 \)\()/899\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(24\)
\(\nu^{5}\)\(=\)\(-\)\(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(38\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)
\(\nu^{6}\)\(=\)\(-\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(18\) \(\beta_{10}\mathstrut +\mathstrut \) \(39\) \(\beta_{8}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(24\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(57\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(162\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut -\mathstrut \) \(42\) \(\beta_{9}\mathstrut +\mathstrut \) \(111\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(124\) \(\beta_{6}\mathstrut +\mathstrut \) \(71\) \(\beta_{5}\mathstrut +\mathstrut \) \(179\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(252\) \(\beta_{1}\mathstrut +\mathstrut \) \(116\)
\(\nu^{8}\)\(=\)\(-\)\(32\) \(\beta_{11}\mathstrut -\mathstrut \) \(220\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut +\mathstrut \) \(517\) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(116\) \(\beta_{6}\mathstrut +\mathstrut \) \(85\) \(\beta_{5}\mathstrut +\mathstrut \) \(353\) \(\beta_{4}\mathstrut -\mathstrut \) \(110\) \(\beta_{3}\mathstrut +\mathstrut \) \(407\) \(\beta_{2}\mathstrut -\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(1195\)
\(\nu^{9}\)\(=\)\(-\)\(21\) \(\beta_{11}\mathstrut -\mathstrut \) \(197\) \(\beta_{10}\mathstrut -\mathstrut \) \(440\) \(\beta_{9}\mathstrut +\mathstrut \) \(1468\) \(\beta_{8}\mathstrut +\mathstrut \) \(156\) \(\beta_{7}\mathstrut +\mathstrut \) \(1222\) \(\beta_{6}\mathstrut +\mathstrut \) \(563\) \(\beta_{5}\mathstrut +\mathstrut \) \(1943\) \(\beta_{4}\mathstrut +\mathstrut \) \(15\) \(\beta_{3}\mathstrut -\mathstrut \) \(20\) \(\beta_{2}\mathstrut +\mathstrut \) \(1751\) \(\beta_{1}\mathstrut +\mathstrut \) \(1222\)
\(\nu^{10}\)\(=\)\(-\)\(373\) \(\beta_{11}\mathstrut -\mathstrut \) \(2361\) \(\beta_{10}\mathstrut -\mathstrut \) \(167\) \(\beta_{9}\mathstrut +\mathstrut \) \(5950\) \(\beta_{8}\mathstrut +\mathstrut \) \(155\) \(\beta_{7}\mathstrut +\mathstrut \) \(1609\) \(\beta_{6}\mathstrut +\mathstrut \) \(741\) \(\beta_{5}\mathstrut +\mathstrut \) \(4347\) \(\beta_{4}\mathstrut -\mathstrut \) \(921\) \(\beta_{3}\mathstrut +\mathstrut \) \(2987\) \(\beta_{2}\mathstrut -\mathstrut \) \(60\) \(\beta_{1}\mathstrut +\mathstrut \) \(9512\)
\(\nu^{11}\)\(=\)\(-\)\(323\) \(\beta_{11}\mathstrut -\mathstrut \) \(2948\) \(\beta_{10}\mathstrut -\mathstrut \) \(4203\) \(\beta_{9}\mathstrut +\mathstrut \) \(17093\) \(\beta_{8}\mathstrut +\mathstrut \) \(1615\) \(\beta_{7}\mathstrut +\mathstrut \) \(11846\) \(\beta_{6}\mathstrut +\mathstrut \) \(4629\) \(\beta_{5}\mathstrut +\mathstrut \) \(20163\) \(\beta_{4}\mathstrut +\mathstrut \) \(142\) \(\beta_{3}\mathstrut -\mathstrut \) \(21\) \(\beta_{2}\mathstrut +\mathstrut \) \(12753\) \(\beta_{1}\mathstrut +\mathstrut \) \(12839\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.16835
2.49390
2.44300
2.31191
1.74625
1.29494
0.264006
−0.665271
−0.867509
−2.17256
−2.25121
−2.76581
0 −3.16835 0 0.310718 0 −0.631809 0 7.03843 0
1.2 0 −2.49390 0 1.94778 0 1.87622 0 3.21954 0
1.3 0 −2.44300 0 −1.37594 0 −4.82526 0 2.96827 0
1.4 0 −2.31191 0 0.313584 0 2.05708 0 2.34493 0
1.5 0 −1.74625 0 −2.58578 0 −0.635213 0 0.0494057 0
1.6 0 −1.29494 0 3.51641 0 −1.29880 0 −1.32313 0
1.7 0 −0.264006 0 1.70058 0 −2.77338 0 −2.93030 0
1.8 0 0.665271 0 2.52521 0 3.86663 0 −2.55741 0
1.9 0 0.867509 0 −3.26212 0 −1.33041 0 −2.24743 0
1.10 0 2.17256 0 2.95208 0 −2.27467 0 1.72004 0
1.11 0 2.25121 0 1.33593 0 −4.39363 0 2.06796 0
1.12 0 2.76581 0 −2.37847 0 2.36325 0 4.64970 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{13}^{12} - \cdots\)