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:

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

Atkin-Lehner signs

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

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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{12} \)
$3$ \( ( 1 - T )^{12} \)
$5$ \( 1 + 4 T + 31 T^{2} + 93 T^{3} + 435 T^{4} + 1040 T^{5} + 3690 T^{6} + 6832 T^{7} + 20687 T^{8} + 27902 T^{9} + 85823 T^{10} + 82249 T^{11} + 361034 T^{12} + 411245 T^{13} + 2145575 T^{14} + 3487750 T^{15} + 12929375 T^{16} + 21350000 T^{17} + 57656250 T^{18} + 81250000 T^{19} + 169921875 T^{20} + 181640625 T^{21} + 302734375 T^{22} + 195312500 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + 37 T^{2} + 6 T^{3} + 705 T^{4} + 234 T^{5} + 9170 T^{6} + 4703 T^{7} + 91608 T^{8} + 62424 T^{9} + 762310 T^{10} + 591697 T^{11} + 5608434 T^{12} + 4141879 T^{13} + 37353190 T^{14} + 21411432 T^{15} + 219950808 T^{16} + 79043321 T^{17} + 1078841330 T^{18} + 192709062 T^{19} + 4064184705 T^{20} + 242121642 T^{21} + 10451584213 T^{22} + 13841287201 T^{24} \)
$11$ \( 1 + 9 T + 108 T^{2} + 662 T^{3} + 4825 T^{4} + 23523 T^{5} + 133338 T^{6} + 556308 T^{7} + 2669196 T^{8} + 9818358 T^{9} + 41239643 T^{10} + 135341926 T^{11} + 506238982 T^{12} + 1488761186 T^{13} + 4989996803 T^{14} + 13068234498 T^{15} + 39079698636 T^{16} + 89593959708 T^{17} + 236216400618 T^{18} + 458396723433 T^{19} + 1034281600825 T^{20} + 1560961371442 T^{21} + 2801241856908 T^{22} + 2567805035499 T^{23} + 3138428376721 T^{24} \)
$13$ \( ( 1 - T )^{12} \)
$17$ \( 1 + 20 T + 308 T^{2} + 3239 T^{3} + 29136 T^{4} + 213080 T^{5} + 1395897 T^{6} + 7911158 T^{7} + 41497074 T^{8} + 195858496 T^{9} + 887388135 T^{10} + 3759932727 T^{11} + 15860100946 T^{12} + 63918856359 T^{13} + 256455171015 T^{14} + 962252790848 T^{15} + 3465877117554 T^{16} + 11232713064406 T^{17} + 33693560154393 T^{18} + 87434964442840 T^{19} + 203245668800976 T^{20} + 384106131973783 T^{21} + 620926121338292 T^{22} + 685437926152660 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 4 T + 126 T^{2} - 356 T^{3} + 7011 T^{4} - 13474 T^{5} + 244168 T^{6} - 330157 T^{7} + 6604515 T^{8} - 7926739 T^{9} + 156854122 T^{10} - 194231596 T^{11} + 3243988882 T^{12} - 3690400324 T^{13} + 56624338042 T^{14} - 54369502801 T^{15} + 860706999315 T^{16} - 817501417543 T^{17} + 11487098672008 T^{18} - 12044027811286 T^{19} + 119071760480451 T^{20} - 114876820409324 T^{21} + 772514348482926 T^{22} - 465961035592876 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + 30 T + 568 T^{2} + 7949 T^{3} + 90946 T^{4} + 886648 T^{5} + 7594755 T^{6} + 58167924 T^{7} + 403735953 T^{8} + 2561028471 T^{9} + 14941629527 T^{10} + 80475187848 T^{11} + 401092920340 T^{12} + 1850929320504 T^{13} + 7904122019783 T^{14} + 31160033406657 T^{15} + 112981872823473 T^{16} + 374388710461932 T^{17} + 1124296308162195 T^{18} + 3018881672931656 T^{19} + 7122070867365826 T^{20} + 14317362505969387 T^{21} + 23530258369352632 T^{22} + 28584292737417810 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 + 29 T + 581 T^{2} + 8215 T^{3} + 96385 T^{4} + 943160 T^{5} + 8220530 T^{6} + 63975721 T^{7} + 462363625 T^{8} + 3089568988 T^{9} + 19514878297 T^{10} + 115038001803 T^{11} + 641439793194 T^{12} + 3336102052287 T^{13} + 16412012647777 T^{14} + 75351498048332 T^{15} + 327021007053625 T^{16} + 1312215545813429 T^{17} + 4889762954980130 T^{18} + 16269393339596440 T^{19} + 48216250513245985 T^{20} + 119176204191763835 T^{21} + 244430902547416781 T^{22} + 353814783205469041 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 6 T + 202 T^{2} - 1066 T^{3} + 18815 T^{4} - 88824 T^{5} + 1079326 T^{6} - 4570047 T^{7} + 43249037 T^{8} - 166059267 T^{9} + 1369526848 T^{10} - 5039552536 T^{11} + 41133646998 T^{12} - 156226128616 T^{13} + 1316115300928 T^{14} - 4947071623197 T^{15} + 39941393899277 T^{16} - 130836565640097 T^{17} + 957905797999006 T^{18} - 2443780435795464 T^{19} + 16047144869452415 T^{20} - 28184637223275286 T^{21} + 165564913970121802 T^{22} - 152450861378428986 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 7 T + 266 T^{2} - 1606 T^{3} + 35104 T^{4} - 190635 T^{5} + 3077220 T^{6} - 15279486 T^{7} + 200225468 T^{8} - 912382020 T^{9} + 10212857752 T^{10} - 42495240726 T^{11} + 419463956610 T^{12} - 1572323906862 T^{13} + 13981402262488 T^{14} - 46214886459060 T^{15} + 375254763332348 T^{16} - 1059540020166102 T^{17} + 7895304620302980 T^{18} - 18097338397249455 T^{19} + 123302078750442784 T^{20} - 208718554110893662 T^{21} + 1279083443063147834 T^{22} - 1245423352456222891 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 8 T + 210 T^{2} + 1836 T^{3} + 25968 T^{4} + 214790 T^{5} + 2323316 T^{6} + 17480343 T^{7} + 160141110 T^{8} + 1094530654 T^{9} + 8822694258 T^{10} + 55173358345 T^{11} + 397594665394 T^{12} + 2262107692145 T^{13} + 14830949047698 T^{14} + 75436147204334 T^{15} + 452520503134710 T^{16} + 2025206132156943 T^{17} + 11035993184783156 T^{18} + 41831270486899990 T^{19} + 207352538349814128 T^{20} + 601073231547312396 T^{21} + 2818758455132004210 T^{22} + 4402632253729987528 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + 8 T + 243 T^{2} + 1570 T^{3} + 28840 T^{4} + 150146 T^{5} + 2269784 T^{6} + 9907625 T^{7} + 140394002 T^{8} + 545817614 T^{9} + 7474610385 T^{10} + 26919387673 T^{11} + 346179346594 T^{12} + 1157533669939 T^{13} + 13820554601865 T^{14} + 43396321036298 T^{15} + 479979154431602 T^{16} + 1456504525077875 T^{17} + 14348128706811416 T^{18} + 40812477183271622 T^{19} + 337087696006012840 T^{20} + 789070400740843510 T^{21} + 5251590202128072507 T^{22} + 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 16 T + 357 T^{2} + 3968 T^{3} + 54911 T^{4} + 497508 T^{5} + 5443282 T^{6} + 42837179 T^{7} + 403460171 T^{8} + 2847775608 T^{9} + 24127276587 T^{10} + 156134470657 T^{11} + 1220882385238 T^{12} + 7338320120879 T^{13} + 53297153980683 T^{14} + 295664606949384 T^{15} + 1968756930685451 T^{16} + 9824493117615253 T^{17} + 58674308774469778 T^{18} + 252049055415306204 T^{19} + 1307501561883958271 T^{20} + 4440709717271779456 T^{21} + 18777890208191327493 T^{22} + 39554547441344196848 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 9 T + 364 T^{2} + 2648 T^{3} + 63946 T^{4} + 380636 T^{5} + 7277843 T^{6} + 35772976 T^{7} + 611076458 T^{8} + 2519222669 T^{9} + 41087609373 T^{10} + 147982600782 T^{11} + 2340259069438 T^{12} + 7843077841446 T^{13} + 115415094728757 T^{14} + 375054313292713 T^{15} + 4821687181396298 T^{16} + 14960097334397168 T^{17} + 161308740492164747 T^{18} + 447137349422996332 T^{19} + 3981258163044890506 T^{20} + 8737773991092048184 T^{21} + 63659039213046749836 T^{22} + 83421323364349724373 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 29 T + 751 T^{2} + 13806 T^{3} + 226698 T^{4} + 3155964 T^{5} + 40064037 T^{6} + 456699304 T^{7} + 4815443937 T^{8} + 46780459281 T^{9} + 424380475788 T^{10} + 3590840392348 T^{11} + 28503939680408 T^{12} + 211859583148532 T^{13} + 1477268436218028 T^{14} + 9607723946672499 T^{15} + 58350472559890257 T^{16} + 326505429765987896 T^{17} + 1689922460472768717 T^{18} + 7854094494635310516 T^{19} + 33286166544024362058 T^{20} + \)\(11\!\cdots\!34\)\( T^{21} + \)\(38\!\cdots\!51\)\( T^{22} + \)\(87\!\cdots\!11\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 26 T + 811 T^{2} + 15010 T^{3} + 278860 T^{4} + 4057732 T^{5} + 56493224 T^{6} + 678244599 T^{7} + 7664308564 T^{8} + 77893196608 T^{9} + 740665052183 T^{10} + 6454026745289 T^{11} + 52513812625194 T^{12} + 393695631462629 T^{13} + 2756014659172943 T^{14} + 17680275659280448 T^{15} + 106118797752082324 T^{16} + 572842879488628299 T^{17} + 2910552049339829864 T^{18} + 12752408173493164372 T^{19} + 53459501302421779660 T^{20} + \)\(17\!\cdots\!10\)\( T^{21} + \)\(57\!\cdots\!11\)\( T^{22} + \)\(11\!\cdots\!86\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 12 T + 541 T^{2} - 6152 T^{3} + 146173 T^{4} - 1560061 T^{5} + 25832558 T^{6} - 255792465 T^{7} + 3306058168 T^{8} - 29991143507 T^{9} + 321676689949 T^{10} - 2633052193083 T^{11} + 24350241505332 T^{12} - 176414496936561 T^{13} + 1444006661181061 T^{14} - 9020226294595841 T^{15} + 66620778176406328 T^{16} - 345351829177918755 T^{17} + 2336771403966858302 T^{18} - 9455079807711804703 T^{19} + 59356130631486884893 T^{20} - \)\(16\!\cdots\!44\)\( T^{21} + \)\(98\!\cdots\!09\)\( T^{22} - \)\(14\!\cdots\!96\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 35 T + 1135 T^{2} + 24696 T^{3} + 491537 T^{4} + 8021287 T^{5} + 120877529 T^{6} + 1593235761 T^{7} + 19548155567 T^{8} + 215684766884 T^{9} + 2226478765392 T^{10} + 20926580550085 T^{11} + 184460124370606 T^{12} + 1485787219056035 T^{13} + 11223679456341072 T^{14} + 77195950600219324 T^{15} + 496751493406978127 T^{16} + 2874562723059021111 T^{17} + 15484445784568911209 T^{18} + 72954569089939669217 T^{19} + \)\(31\!\cdots\!57\)\( T^{20} + \)\(11\!\cdots\!76\)\( T^{21} + \)\(36\!\cdots\!35\)\( T^{22} + \)\(80\!\cdots\!85\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 18 T + 354 T^{2} - 5804 T^{3} + 82357 T^{4} - 1064849 T^{5} + 13002100 T^{6} - 144999477 T^{7} + 1559317972 T^{8} - 15575597371 T^{9} + 150648276022 T^{10} - 1373951341057 T^{11} + 12006180010132 T^{12} - 100298447897161 T^{13} + 802804662921238 T^{14} - 6059172162474307 T^{15} + 44281887564487252 T^{16} - 300594296768556861 T^{17} + 1967662743632206900 T^{18} - 11763811265661921353 T^{19} + 66417633788120828917 T^{20} - \)\(34\!\cdots\!52\)\( T^{21} + \)\(15\!\cdots\!46\)\( T^{22} - \)\(56\!\cdots\!86\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 37 T + 1105 T^{2} + 25037 T^{3} + 491127 T^{4} + 8319158 T^{5} + 127058602 T^{6} + 1747643419 T^{7} + 22062467511 T^{8} + 255455090796 T^{9} + 2741158508173 T^{10} + 27197618826911 T^{11} + 251063744352914 T^{12} + 2148611887325969 T^{13} + 17107570249507693 T^{14} + 125949322510969044 T^{15} + 859334896613318391 T^{16} + 5377597365604188181 T^{17} + 30886352262235441642 T^{18} + \)\(15\!\cdots\!22\)\( T^{19} + \)\(74\!\cdots\!47\)\( T^{20} + \)\(30\!\cdots\!03\)\( T^{21} + \)\(10\!\cdots\!05\)\( T^{22} + \)\(27\!\cdots\!23\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + 24 T + 753 T^{2} + 12791 T^{3} + 240301 T^{4} + 3213851 T^{5} + 45719477 T^{6} + 510509135 T^{7} + 6050773221 T^{8} + 59078147485 T^{9} + 622265694970 T^{10} + 5563156908874 T^{11} + 54682684150922 T^{12} + 461742023436542 T^{13} + 4286788372648330 T^{14} + 33780116716005695 T^{15} + 287159537820421941 T^{16} + 2010916231387773805 T^{17} + 14947542880615408013 T^{18} + 87211224609063723577 T^{19} + \)\(54\!\cdots\!41\)\( T^{20} + \)\(23\!\cdots\!73\)\( T^{21} + \)\(11\!\cdots\!97\)\( T^{22} + \)\(30\!\cdots\!08\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 15 T + 841 T^{2} - 11662 T^{3} + 344802 T^{4} - 4319083 T^{5} + 89943638 T^{6} - 1006107027 T^{7} + 16494253885 T^{8} - 163662185312 T^{9} + 2230819592745 T^{10} - 19500674759813 T^{11} + 227730176149968 T^{12} - 1735560053623357 T^{13} + 17670321994133145 T^{14} - 115376767117215328 T^{15} + 1034886452367856285 T^{16} - 5618161450824648123 T^{17} + 44700305326968856118 T^{18} - \)\(19\!\cdots\!07\)\( T^{19} + \)\(13\!\cdots\!62\)\( T^{20} - \)\(40\!\cdots\!58\)\( T^{21} + \)\(26\!\cdots\!41\)\( T^{22} - \)\(41\!\cdots\!35\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 11 T + 1026 T^{2} + 10059 T^{3} + 492409 T^{4} + 4310715 T^{5} + 146403265 T^{6} + 1142816468 T^{7} + 30068791037 T^{8} + 208306392171 T^{9} + 4498037411981 T^{10} + 27417640536048 T^{11} + 502791327262770 T^{12} + 2659511131996656 T^{13} + 42322034009329229 T^{14} + 190115619861883083 T^{15} + 2661968451044854397 T^{16} + 9813753862018952276 T^{17} + \)\(12\!\cdots\!85\)\( T^{18} + \)\(34\!\cdots\!95\)\( T^{19} + \)\(38\!\cdots\!49\)\( T^{20} + \)\(76\!\cdots\!03\)\( T^{21} + \)\(75\!\cdots\!74\)\( T^{22} + \)\(78\!\cdots\!83\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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