Properties

Label 8034.2.a.w
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta_{11} q^{5} - q^{6} -\beta_{9} q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + \beta_{11} q^{5} - q^{6} -\beta_{9} q^{7} - q^{8} + q^{9} -\beta_{11} q^{10} + ( -1 - \beta_{1} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{11} + q^{12} + q^{13} + \beta_{9} q^{14} + \beta_{11} q^{15} + q^{16} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{17} - q^{18} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{19} + \beta_{11} q^{20} -\beta_{9} q^{21} + ( 1 + \beta_{1} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{22} + ( -2 - \beta_{4} + \beta_{7} - \beta_{8} ) q^{23} - q^{24} + ( 1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{25} - q^{26} + q^{27} -\beta_{9} q^{28} + ( -2 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} ) q^{29} -\beta_{11} q^{30} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{31} - q^{32} + ( -1 - \beta_{1} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{33} + ( 3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{34} + ( -1 + \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{6} + 3 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{35} + q^{36} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{37} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{38} + q^{39} -\beta_{11} q^{40} + ( 3 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} + 3 \beta_{11} ) q^{41} + \beta_{9} q^{42} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{43} + ( -1 - \beta_{1} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{44} + \beta_{11} q^{45} + ( 2 + \beta_{4} - \beta_{7} + \beta_{8} ) q^{46} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{47} + q^{48} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{49} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} - \beta_{10} - \beta_{11} ) q^{50} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{51} + q^{52} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{53} - q^{54} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{55} + \beta_{9} q^{56} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{57} + ( 2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{10} ) q^{58} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{59} + \beta_{11} q^{60} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{61} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{62} -\beta_{9} q^{63} + q^{64} + \beta_{11} q^{65} + ( 1 + \beta_{1} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{66} + ( -\beta_{2} - \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{67} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{68} + ( -2 - \beta_{4} + \beta_{7} - \beta_{8} ) q^{69} + ( 1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{6} - 3 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{70} + ( -4 + \beta_{2} + 4 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{71} - q^{72} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{73} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{74} + ( 1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} + \beta_{10} + \beta_{11} ) q^{75} + ( -\beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{76} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{77} - q^{78} + ( -2 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + \beta_{11} q^{80} + q^{81} + ( -3 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{82} + ( -3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{83} -\beta_{9} q^{84} + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{85} + ( \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{86} + ( -2 + \beta_{1} - \beta_{2} + \beta_{6} - \beta_{10} ) q^{87} + ( 1 + \beta_{1} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{88} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{89} -\beta_{11} q^{90} -\beta_{9} q^{91} + ( -2 - \beta_{4} + \beta_{7} - \beta_{8} ) q^{92} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{93} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{94} + ( -5 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{95} - q^{96} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{10} ) q^{97} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{98} + ( -1 - \beta_{1} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 81 \nu^{11} - 874 \nu^{10} + 1400 \nu^{9} + 10622 \nu^{8} - 27271 \nu^{7} - 48976 \nu^{6} + 141564 \nu^{5} + 120931 \nu^{4} - 292064 \nu^{3} - 190830 \nu^{2} + 207480 \nu + 143896 \)\()/2936\)
\(\beta_{2}\)\(=\)\((\)\( 103 \nu^{11} - 835 \nu^{10} + 276 \nu^{9} + 11894 \nu^{8} - 17071 \nu^{7} - 64829 \nu^{6} + 108598 \nu^{5} + 175289 \nu^{4} - 254607 \nu^{3} - 253616 \nu^{2} + 199594 \nu + 165236 \)\()/1468\)
\(\beta_{3}\)\(=\)\((\)\( -75 \nu^{11} + 551 \nu^{10} - 5 \nu^{9} - 7606 \nu^{8} + 8967 \nu^{7} + 39381 \nu^{6} - 56903 \nu^{5} - 97565 \nu^{4} + 125519 \nu^{3} + 123561 \nu^{2} - 89922 \nu - 70548 \)\()/734\)
\(\beta_{4}\)\(=\)\((\)\( -268 \nu^{11} + 1827 \nu^{10} + 814 \nu^{9} - 26572 \nu^{8} + 21678 \nu^{7} + 146843 \nu^{6} - 158256 \nu^{5} - 389932 \nu^{4} + 369489 \nu^{3} + 514220 \nu^{2} - 276606 \nu - 281980 \)\()/1468\)
\(\beta_{5}\)\(=\)\((\)\( 176 \nu^{11} - 1156 \nu^{10} - 551 \nu^{9} + 16048 \nu^{8} - 13086 \nu^{7} - 82784 \nu^{6} + 93363 \nu^{5} + 199984 \nu^{4} - 211208 \nu^{3} - 240617 \nu^{2} + 152708 \nu + 131084 \)\()/734\)
\(\beta_{6}\)\(=\)\((\)\( 464 \nu^{11} - 3081 \nu^{10} - 1486 \nu^{9} + 43376 \nu^{8} - 34366 \nu^{7} - 228825 \nu^{6} + 247440 \nu^{5} + 569936 \nu^{4} - 561859 \nu^{3} - 702716 \nu^{2} + 402794 \nu + 376680 \)\()/1468\)
\(\beta_{7}\)\(=\)\((\)\( -927 \nu^{11} + 6414 \nu^{10} + 1920 \nu^{9} - 90898 \nu^{8} + 85377 \nu^{7} + 488408 \nu^{6} - 593500 \nu^{5} - 1267853 \nu^{4} + 1378000 \nu^{3} + 1679930 \nu^{2} - 1045464 \nu - 977728 \)\()/2936\)
\(\beta_{8}\)\(=\)\((\)\( -1151 \nu^{11} + 7952 \nu^{10} + 2688 \nu^{9} - 113458 \nu^{8} + 101765 \nu^{7} + 614922 \nu^{6} - 714928 \nu^{5} - 1610725 \nu^{4} + 1662758 \nu^{3} + 2140510 \nu^{2} - 1257836 \nu - 1227304 \)\()/2936\)
\(\beta_{9}\)\(=\)\((\)\( 1227 \nu^{11} - 8618 \nu^{10} - 1900 \nu^{9} + 121322 \nu^{8} - 121245 \nu^{7} - 645932 \nu^{6} + 821112 \nu^{5} + 1658113 \nu^{4} - 1877140 \nu^{3} - 2177110 \nu^{2} + 1387536 \nu + 1262856 \)\()/2936\)
\(\beta_{10}\)\(=\)\((\)\( -1415 \nu^{11} + 9686 \nu^{10} + 2964 \nu^{9} - 134594 \nu^{8} + 124697 \nu^{7} + 704600 \nu^{6} - 850752 \nu^{5} - 1763901 \nu^{4} + 1927456 \nu^{3} + 2233342 \nu^{2} - 1408360 \nu - 1251440 \)\()/2936\)
\(\beta_{11}\)\(=\)\((\)\( 1555 \nu^{11} - 10372 \nu^{10} - 4912 \nu^{9} + 147226 \nu^{8} - 118241 \nu^{7} - 788534 \nu^{6} + 860768 \nu^{5} + 2017465 \nu^{4} - 1999642 \nu^{3} - 2588182 \nu^{2} + 1485492 \nu + 1437152 \)\()/2936\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{11} - \beta_{9} + \beta_{8} - \beta_{4} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{11} - 3 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 4 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(8 \beta_{11} - 9 \beta_{9} + 10 \beta_{8} + 2 \beta_{6} + \beta_{5} - 11 \beta_{4} + 5 \beta_{3} + 11 \beta_{2} + 11 \beta_{1} + 34\)
\(\nu^{5}\)\(=\)\(11 \beta_{11} - 2 \beta_{10} - 24 \beta_{9} + 33 \beta_{8} - 8 \beta_{7} + 20 \beta_{6} + 3 \beta_{5} - 37 \beta_{4} + 36 \beta_{3} + 37 \beta_{2} + 34 \beta_{1} + 71\)
\(\nu^{6}\)\(=\)\(55 \beta_{11} - 5 \beta_{10} - 75 \beta_{9} + 90 \beta_{8} + 3 \beta_{7} + 29 \beta_{6} + 18 \beta_{5} - 111 \beta_{4} + 80 \beta_{3} + 109 \beta_{2} + 102 \beta_{1} + 271\)
\(\nu^{7}\)\(=\)\(95 \beta_{11} - 34 \beta_{10} - 207 \beta_{9} + 284 \beta_{8} - 23 \beta_{7} + 151 \beta_{6} + 56 \beta_{5} - 360 \beta_{4} + 352 \beta_{3} + 350 \beta_{2} + 305 \beta_{1} + 692\)
\(\nu^{8}\)\(=\)\(380 \beta_{11} - 98 \beta_{10} - 641 \beta_{9} + 820 \beta_{8} + 64 \beta_{7} + 313 \beta_{6} + 235 \beta_{5} - 1117 \beta_{4} + 964 \beta_{3} + 1062 \beta_{2} + 937 \beta_{1} + 2375\)
\(\nu^{9}\)\(=\)\(786 \beta_{11} - 434 \beta_{10} - 1845 \beta_{9} + 2568 \beta_{8} + 72 \beta_{7} + 1232 \beta_{6} + 749 \beta_{5} - 3584 \beta_{4} + 3542 \beta_{3} + 3356 \beta_{2} + 2835 \beta_{1} + 6734\)
\(\nu^{10}\)\(=\)\(2788 \beta_{11} - 1331 \beta_{10} - 5677 \beta_{9} + 7715 \beta_{8} + 920 \beta_{7} + 3041 \beta_{6} + 2744 \beta_{5} - 11286 \beta_{4} + 10557 \beta_{3} + 10348 \beta_{2} + 8791 \beta_{1} + 22049\)
\(\nu^{11}\)\(=\)\(6699 \beta_{11} - 4984 \beta_{10} - 16860 \beta_{9} + 24258 \beta_{8} + 2411 \beta_{7} + 10443 \beta_{6} + 8847 \beta_{5} - 36083 \beta_{4} + 35919 \beta_{3} + 32539 \beta_{2} + 27032 \beta_{1} + 66028\)

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
2.37328
−0.829811
1.78941
−2.08335
1.38143
3.10048
−1.02386
−1.61063
3.23048
−2.20360
2.65870
−0.782524
−1.00000 1.00000 1.00000 −3.83399 −1.00000 −3.29790 −1.00000 1.00000 3.83399
1.2 −1.00000 1.00000 1.00000 −3.54947 −1.00000 1.31288 −1.00000 1.00000 3.54947
1.3 −1.00000 1.00000 1.00000 −2.74714 −1.00000 3.39290 −1.00000 1.00000 2.74714
1.4 −1.00000 1.00000 1.00000 −1.71945 −1.00000 2.07930 −1.00000 1.00000 1.71945
1.5 −1.00000 1.00000 1.00000 −1.60911 −1.00000 2.84255 −1.00000 1.00000 1.60911
1.6 −1.00000 1.00000 1.00000 −1.18783 −1.00000 0.107097 −1.00000 1.00000 1.18783
1.7 −1.00000 1.00000 1.00000 −0.812175 −1.00000 −2.80411 −1.00000 1.00000 0.812175
1.8 −1.00000 1.00000 1.00000 1.40749 −1.00000 −0.998879 −1.00000 1.00000 −1.40749
1.9 −1.00000 1.00000 1.00000 1.42840 −1.00000 −2.28162 −1.00000 1.00000 −1.42840
1.10 −1.00000 1.00000 1.00000 2.09722 −1.00000 −0.0373018 −1.00000 1.00000 −2.09722
1.11 −1.00000 1.00000 1.00000 2.30797 −1.00000 4.49849 −1.00000 1.00000 −2.30797
1.12 −1.00000 1.00000 1.00000 4.21809 −1.00000 −4.81342 −1.00000 1.00000 −4.21809
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8034.2.a.w 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.w 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( ( -1 + T )^{12} \)
$5$ \( -4096 - 6656 T + 4208 T^{2} + 9852 T^{3} - 638 T^{4} - 5293 T^{5} - 585 T^{6} + 1255 T^{7} + 235 T^{8} - 127 T^{9} - 29 T^{10} + 4 T^{11} + T^{12} \)
$7$ \( 48 + 848 T - 11911 T^{2} - 3397 T^{3} + 14923 T^{4} + 1175 T^{5} - 5271 T^{6} - 144 T^{7} + 761 T^{8} + 6 T^{9} - 47 T^{10} + T^{12} \)
$11$ \( 8268 + 25690 T - 20235 T^{2} - 116985 T^{3} - 104707 T^{4} - 14592 T^{5} + 17046 T^{6} + 5901 T^{7} - 521 T^{8} - 427 T^{9} - 24 T^{10} + 9 T^{11} + T^{12} \)
$13$ \( ( -1 + T )^{12} \)
$17$ \( -1291856 - 2909408 T - 2010640 T^{2} + 8636 T^{3} + 631165 T^{4} + 262535 T^{5} - 1435 T^{6} - 28167 T^{7} - 7618 T^{8} - 501 T^{9} + 104 T^{10} + 20 T^{11} + T^{12} \)
$19$ \( -28672 - 13824 T + 169216 T^{2} - 75920 T^{3} - 139212 T^{4} + 104525 T^{5} + 2298 T^{6} - 16134 T^{7} + 2565 T^{8} + 480 T^{9} - 102 T^{10} - 4 T^{11} + T^{12} \)
$23$ \( 5057504 - 429392 T - 6055454 T^{2} - 1494541 T^{3} + 1708948 T^{4} + 847393 T^{5} + 14507 T^{6} - 60515 T^{7} - 11128 T^{8} + 359 T^{9} + 292 T^{10} + 30 T^{11} + T^{12} \)
$29$ \( -10293248 - 5933824 T + 25004288 T^{2} + 35299428 T^{3} + 18951160 T^{4} + 4591609 T^{5} + 229377 T^{6} - 127839 T^{7} - 26691 T^{8} - 1036 T^{9} + 233 T^{10} + 29 T^{11} + T^{12} \)
$31$ \( 155392 - 2762560 T + 6585808 T^{2} - 4785828 T^{3} + 199306 T^{4} + 808701 T^{5} - 129116 T^{6} - 45114 T^{7} + 8089 T^{8} + 980 T^{9} - 170 T^{10} - 6 T^{11} + T^{12} \)
$37$ \( 10528208 + 35086004 T + 10737714 T^{2} - 20477965 T^{3} + 1332153 T^{4} + 2034368 T^{5} - 241310 T^{6} - 77489 T^{7} + 10610 T^{8} + 1243 T^{9} - 178 T^{10} - 7 T^{11} + T^{12} \)
$41$ \( -123048656 + 341108764 T - 262426452 T^{2} + 13094359 T^{3} + 34683939 T^{4} - 3289191 T^{5} - 1557990 T^{6} + 129018 T^{7} + 30642 T^{8} - 1772 T^{9} - 282 T^{10} + 8 T^{11} + T^{12} \)
$43$ \( 5504384 - 87782416 T - 112480368 T^{2} + 61850335 T^{3} + 14256985 T^{4} - 5883523 T^{5} - 830215 T^{6} + 193404 T^{7} + 24196 T^{8} - 2214 T^{9} - 273 T^{10} + 8 T^{11} + T^{12} \)
$47$ \( -1167890192 + 63787036 T + 572990506 T^{2} + 102000311 T^{3} - 46451418 T^{4} - 12089465 T^{5} + 770025 T^{6} + 374180 T^{7} + 6407 T^{8} - 4304 T^{9} - 207 T^{10} + 16 T^{11} + T^{12} \)
$53$ \( 4951728 - 173386688 T - 42529760 T^{2} + 87274580 T^{3} + 8147769 T^{4} - 7782477 T^{5} - 722825 T^{6} + 229904 T^{7} + 22712 T^{8} - 2599 T^{9} - 272 T^{10} + 9 T^{11} + T^{12} \)
$59$ \( 48294912 - 60904448 T - 76036832 T^{2} + 120230752 T^{3} - 24168946 T^{4} - 7740613 T^{5} + 1558218 T^{6} + 266734 T^{7} - 28418 T^{8} - 5015 T^{9} + 43 T^{10} + 29 T^{11} + T^{12} \)
$61$ \( -108370312 - 3820620 T + 59553982 T^{2} + 2093051 T^{3} - 9716465 T^{4} - 813257 T^{5} + 608257 T^{6} + 74066 T^{7} - 14916 T^{8} - 2436 T^{9} + 79 T^{10} + 26 T^{11} + T^{12} \)
$67$ \( -418980608 + 855407296 T - 161195628 T^{2} - 135181918 T^{3} + 25010347 T^{4} + 8140300 T^{5} - 1202411 T^{6} - 220597 T^{7} + 26109 T^{8} + 2692 T^{9} - 263 T^{10} - 12 T^{11} + T^{12} \)
$71$ \( 12376064 + 358165248 T + 175401992 T^{2} - 71235288 T^{3} - 32134192 T^{4} + 3384449 T^{5} + 1852206 T^{6} + 3683 T^{7} - 42099 T^{8} - 2639 T^{9} + 283 T^{10} + 35 T^{11} + T^{12} \)
$73$ \( 27188417904 + 361621096 T - 16997367304 T^{2} - 2473531310 T^{3} + 738221637 T^{4} + 103218992 T^{5} - 12637982 T^{6} - 1472189 T^{7} + 111703 T^{8} + 8650 T^{9} - 522 T^{10} - 18 T^{11} + T^{12} \)
$79$ \( -519477248 + 2666457600 T - 2920891552 T^{2} + 950942928 T^{3} - 11483942 T^{4} - 38609307 T^{5} + 2816645 T^{6} + 678199 T^{7} - 44809 T^{8} - 7116 T^{9} + 157 T^{10} + 37 T^{11} + T^{12} \)
$83$ \( 776561664 - 2834486528 T + 1243902560 T^{2} + 413088216 T^{3} - 153511390 T^{4} - 34236299 T^{5} + 3679064 T^{6} + 933758 T^{7} - 12683 T^{8} - 9121 T^{9} - 243 T^{10} + 24 T^{11} + T^{12} \)
$89$ \( 180564352 + 66956352 T - 208842280 T^{2} - 26842984 T^{3} + 31872888 T^{4} + 4803923 T^{5} - 1357279 T^{6} - 205681 T^{7} + 24046 T^{8} + 3023 T^{9} - 227 T^{10} - 15 T^{11} + T^{12} \)
$97$ \( -83823616 - 55019904 T + 18793376 T^{2} + 16533076 T^{3} - 933968 T^{4} - 1774511 T^{5} - 48305 T^{6} + 83164 T^{7} + 5275 T^{8} - 1678 T^{9} - 138 T^{10} + 11 T^{11} + T^{12} \)
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