Properties

Label 930.2.n.e
Level $930$
Weight $2$
Character orbit 930.n
Analytic conductor $7.426$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} + 16 x^{10} - 6 x^{9} + 161 x^{8} - 180 x^{7} + 1725 x^{6} - 2255 x^{5} + 17635 x^{4} - 18375 x^{3} + 60975 x^{2} + 2000 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} + 16 x^{10} - 6 x^{9} + 161 x^{8} - 180 x^{7} + 1725 x^{6} - 2255 x^{5} + 17635 x^{4} - 18375 x^{3} + 60975 x^{2} + 2000 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(3322281972019 \nu^{11} - 114656979031949 \nu^{10} - 42039345952763 \nu^{9} - 1446344359381892 \nu^{8} - 4528112783420713 \nu^{7} - 16341144970954958 \nu^{6} - 33087807003119337 \nu^{5} - 150045130062088020 \nu^{4} - 320325741801216475 \nu^{3} - 1212562781876652595 \nu^{2} - 4066488446345507205 \nu - 131773814276113450\)\()/ 4027168036955745330 \)
\(\beta_{2}\)\(=\)\((\)\(-112823041844241 \nu^{11} + 329247497650296 \nu^{10} - 1644860785336960 \nu^{9} + 3166436229279820 \nu^{8} - 12358310003335470 \nu^{7} + 45206178197892679 \nu^{6} - 200581418861056364 \nu^{5} + 475387905458896900 \nu^{4} - 2049502436858227190 \nu^{3} + 4170956410466929610 \nu^{2} - 6378771422167487115 \nu - 79418675560302400\)\()/ 4027168036955745330 \)
\(\beta_{3}\)\(=\)\((\)\(-145525184701524 \nu^{11} + 81175192550164 \nu^{10} - 2131397795696306 \nu^{9} + 351234336518491 \nu^{8} - 21413987347320571 \nu^{7} + 20741106213717582 \nu^{6} - 223168834519363377 \nu^{5} + 245469958776066705 \nu^{4} - 2712776923285910695 \nu^{3} + 1818406378451798780 \nu^{2} - 6455030843440640165 \nu - 130511659199335525\)\()/ 4027168036955745330 \)
\(\beta_{4}\)\(=\)\((\)\(187465168762881 \nu^{11} + 626582224648512 \nu^{10} + 2406078642231698 \nu^{9} + 8485330116947397 \nu^{8} + 25747569322466333 \nu^{7} + 68348535768531023 \nu^{6} + 179079646803900835 \nu^{5} + 669745210662038795 \nu^{4} + 1860639801988947325 \nu^{3} + 7190356009182501570 \nu^{2} + 233189167335082000 \nu + 15886129812897084325\)\()/ 4027168036955745330 \)
\(\beta_{5}\)\(=\)\((\)\(3176747022412096 \nu^{11} - 3289570064256337 \nu^{10} + 51157199856243832 \nu^{9} - 20705342919809536 \nu^{8} + 514622706837627276 \nu^{7} - 584172774037512750 \nu^{6} + 5525094791858758279 \nu^{5} - 7364145954400332844 \nu^{4} + 56497321645696209860 \nu^{3} - 60422228973680491190 \nu^{2} + 197873106102044483210 \nu - 25277377343295115\)\()/ 4027168036955745330 \)
\(\beta_{6}\)\(=\)\((\)\(-3176747022412096 \nu^{11} + 3289570064256337 \nu^{10} - 51157199856243832 \nu^{9} + 20705342919809536 \nu^{8} - 514622706837627276 \nu^{7} + 584172774037512750 \nu^{6} - 5525094791858758279 \nu^{5} + 7364145954400332844 \nu^{4} - 56497321645696209860 \nu^{3} + 60422228973680491190 \nu^{2} - 193845938065088737880 \nu + 25277377343295115\)\()/ 4027168036955745330 \)
\(\beta_{7}\)\(=\)\((\)\(-3227233225483213 \nu^{11} + 3191208800653911 \nu^{10} - 51769146933799591 \nu^{9} + 18918901688492695 \nu^{8} - 520953406836176730 \nu^{7} + 576374416026413952 \nu^{6} - 5575101240971762564 \nu^{5} + 7287911314809457639 \nu^{4} - 56992130748017203430 \nu^{3} + 58957461773627571845 \nu^{2} - 197920533173977390910 \nu - 6491405462849817175\)\()/ 4027168036955745330 \)
\(\beta_{8}\)\(=\)\((\)\(-5220466367973421 \nu^{11} + 5365991552674945 \nu^{10} - 83608637080124900 \nu^{9} + 33454196003536832 \nu^{8} - 840846319580239272 \nu^{7} + 961097933582536351 \nu^{6} - 9026045590967868807 \nu^{5} + 11995320494299427732 \nu^{4} - 92308394357987346040 \nu^{3} + 98638846434797521570 \nu^{2} - 320136343165631144255 \nu - 3985901892506201835\)\()/ 4027168036955745330 \)
\(\beta_{9}\)\(=\)\((\)\(-20423798056209645 \nu^{11} + 21085627500335359 \nu^{10} - 328048429442175024 \nu^{9} + 130184356242293759 \nu^{8} - 3300156092820715519 \nu^{7} + 3753906397499170609 \nu^{6} - 35428367573615196500 \nu^{5} + 46907509583354612425 \nu^{4} - 362128821612279045225 \nu^{3} + 384094338257193259840 \nu^{2} - 1262786767276716280850 \nu - 25538891027584600850\)\()/ 4027168036955745330 \)
\(\beta_{10}\)\(=\)\((\)\(20550414730983713 \nu^{11} - 20455879869589488 \nu^{10} + 329640044494412703 \nu^{9} - 124975585074945293 \nu^{8} + 3316491227925739768 \nu^{7} - 3716578247124392900 \nu^{6} + 35514900285316916445 \nu^{5} - 46540443917479469965 \nu^{4} + 363332754665948182650 \nu^{3} - 379708279637547204655 \nu^{2} + 1262928965137764119100 \nu + 15724276327979534350\)\()/ 4027168036955745330 \)
\(\beta_{11}\)\(=\)\((\)\(33316031350330734 \nu^{11} - 34370732068070704 \nu^{10} + 533491312627074709 \nu^{9} - 215260470336265129 \nu^{8} + 5364970830225498174 \nu^{7} - 6151311119051340505 \nu^{6} + 57607217895803319085 \nu^{5} - 76679332145286536485 \nu^{4} + 589166629757894218850 \nu^{3} - 627487295232999333815 \nu^{2} + 2043033868985440618875 \nu + 25437075821649341000\)\()/ 4027168036955745330 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{6} + \beta_{5}\)
\(\nu^{2}\)\(=\)\(\beta_{11} + 7 \beta_{8} - \beta_{7} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - 10 \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(-3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 7 \beta_{8} + 69 \beta_{7} + \beta_{6} - 6 \beta_{5} + 9 \beta_{4} + \beta_{2} + 3 \beta_{1}\)
\(\nu^{5}\)\(=\)\(18 \beta_{10} + 18 \beta_{9} - 32 \beta_{7} - 102 \beta_{6} - 134 \beta_{5} - \beta_{4} + 103 \beta_{3} - 103 \beta_{2} - 102 \beta_{1} - 6\)
\(\nu^{6}\)\(=\)\(53 \beta_{11} - 137 \beta_{10} + 53 \beta_{9} + 57 \beta_{8} + 63 \beta_{6} + 819 \beta_{5} + 137 \beta_{4} - 16 \beta_{3} + 16 \beta_{1} + 57\)
\(\nu^{7}\)\(=\)\(253 \beta_{11} - 227 \beta_{10} + 456 \beta_{8} + 392 \beta_{7} + 26 \beta_{6} + 26 \beta_{5} - 26 \beta_{3} + 1083 \beta_{2} + 392\)
\(\nu^{8}\)\(=\)\(-1537 \beta_{11} - 733 \beta_{9} - 7224 \beta_{8} - 7224 \beta_{7} - 7756 \beta_{5} - 1537 \beta_{4} - 797 \beta_{3} - 191 \beta_{2} - 191 \beta_{1} - 7756\)
\(\nu^{9}\)\(=\)\(-2812 \beta_{11} + 2812 \beta_{10} - 3258 \beta_{9} + 4347 \beta_{8} + 6796 \beta_{7} - 461 \beta_{6} + 3886 \beta_{5} + 446 \beta_{4} - 461 \beta_{2} + 11102 \beta_{1}\)
\(\nu^{10}\)\(=\)\(17172 \beta_{10} + 17172 \beta_{9} + 5462 \beta_{7} - 13758 \beta_{6} - 8296 \beta_{5} - 7829 \beta_{4} + 11777 \beta_{3} - 11777 \beta_{2} - 13758 \beta_{1} + 81271\)
\(\nu^{11}\)\(=\)\(33877 \beta_{11} - 40273 \beta_{10} + 33877 \beta_{9} - 45777 \beta_{8} + 117982 \beta_{6} + 165016 \beta_{5} + 40273 \beta_{4} + 6976 \beta_{3} - 6976 \beta_{1} - 45777\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
2.41709 + 1.75612i
−0.0163000 0.0118427i
−2.70981 1.96879i
1.05776 + 3.25545i
0.716623 + 2.20554i
−0.965367 2.97109i
1.05776 3.25545i
0.716623 2.20554i
−0.965367 + 2.97109i
2.41709 1.75612i
−0.0163000 + 0.0118427i
−2.70981 + 1.96879i
0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 −1.00000 −1.23226 + 3.79252i −0.309017 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
481.2 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 −1.00000 −0.302791 + 0.931895i −0.309017 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
481.3 0.809017 0.587785i −0.809017 0.587785i 0.309017 0.951057i 1.00000 −1.00000 0.726038 2.23451i −0.309017 0.951057i 0.309017 + 0.951057i 0.809017 0.587785i
721.1 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 −1.00000 −1.96024 1.42420i 0.809017 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.2 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 −1.00000 −1.06713 0.775312i 0.809017 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
721.3 −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 0.587785i 1.00000 −1.00000 3.33638 + 2.42402i 0.809017 0.587785i −0.809017 + 0.587785i −0.309017 + 0.951057i
841.1 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 −1.96024 + 1.42420i 0.809017 + 0.587785i −0.809017 0.587785i −0.309017 0.951057i
841.2 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 −1.06713 + 0.775312i 0.809017 + 0.587785i −0.809017 0.587785i −0.309017 0.951057i
841.3 −0.309017 0.951057i 0.309017 0.951057i −0.809017 + 0.587785i 1.00000 −1.00000 3.33638 2.42402i 0.809017 + 0.587785i −0.809017 0.587785i −0.309017 0.951057i
901.1 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 −1.23226 3.79252i −0.309017 + 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
901.2 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 −0.302791 0.931895i −0.309017 + 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
901.3 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 + 0.951057i 1.00000 −1.00000 0.726038 + 2.23451i −0.309017 + 0.951057i 0.309017 0.951057i 0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.e 12
31.d even 5 1 inner 930.2.n.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.e 12 1.a even 1 1 trivial
930.2.n.e 12 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{3} \)
$5$ \( ( -1 + T )^{12} \)
$7$ \( 14641 + 29645 T + 41603 T^{2} + 34332 T^{3} + 19194 T^{4} + 6877 T^{5} + 2921 T^{6} + 779 T^{7} + 132 T^{8} - 6 T^{9} + 11 T^{10} + T^{11} + T^{12} \)
$11$ \( 1296 + 2592 T + 4464 T^{2} + 7302 T^{3} + 10627 T^{4} + 5883 T^{5} + 2224 T^{6} + 351 T^{7} + 119 T^{8} - 21 T^{9} + 22 T^{10} - 5 T^{11} + T^{12} \)
$13$ \( 1442401 - 1468823 T + 1242839 T^{2} - 575388 T^{3} + 228312 T^{4} - 62997 T^{5} + 25699 T^{6} - 1949 T^{7} + 514 T^{8} - 76 T^{9} + 17 T^{10} + 5 T^{11} + T^{12} \)
$17$ \( 810000 - 1080000 T + 1008000 T^{2} - 555750 T^{3} + 222775 T^{4} - 54375 T^{5} + 14745 T^{6} - 1795 T^{7} + 606 T^{8} - 37 T^{9} + 9 T^{10} - 3 T^{11} + T^{12} \)
$19$ \( 164025 + 492075 T + 1986525 T^{2} + 1877175 T^{3} + 822960 T^{4} + 109755 T^{5} + 47070 T^{6} + 5205 T^{7} + 976 T^{8} + 3 T^{9} + 19 T^{10} + 2 T^{11} + T^{12} \)
$23$ \( 1296 + 6696 T + 15876 T^{2} + 18354 T^{3} + 14197 T^{4} + 4934 T^{5} + 301 T^{6} - 528 T^{7} + 659 T^{8} - 108 T^{9} + 43 T^{10} + T^{12} \)
$29$ \( 810000 + 1350000 T + 837000 T^{2} - 35250 T^{3} + 14275 T^{4} + 49075 T^{5} + 22580 T^{6} - 4725 T^{7} + 5291 T^{8} - 141 T^{9} + 116 T^{10} - T^{11} + T^{12} \)
$31$ \( 887503681 + 171774906 T + 112669562 T^{2} + 11529117 T^{3} + 5129818 T^{4} + 218302 T^{5} + 157243 T^{6} + 7042 T^{7} + 5338 T^{8} + 387 T^{9} + 122 T^{10} + 6 T^{11} + T^{12} \)
$37$ \( ( 3659 + 2067 T - 800 T^{2} - 565 T^{3} - 60 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$41$ \( 810000 - 1215000 T + 1224000 T^{2} - 1045500 T^{3} + 1008025 T^{4} - 447900 T^{5} + 207675 T^{6} - 84975 T^{7} + 25210 T^{8} - 2205 T^{9} + 185 T^{10} - 15 T^{11} + T^{12} \)
$43$ \( 149352841 - 86170271 T + 41576931 T^{2} - 10303524 T^{3} + 1954802 T^{4} - 300239 T^{5} + 82281 T^{6} - 13182 T^{7} + 2774 T^{8} - 502 T^{9} + 98 T^{10} - 10 T^{11} + T^{12} \)
$47$ \( 28772496 - 32473656 T + 22452912 T^{2} - 9396552 T^{3} + 2614993 T^{4} - 398957 T^{5} + 61793 T^{6} - 8937 T^{7} + 1598 T^{8} - 57 T^{9} + 47 T^{10} - T^{11} + T^{12} \)
$53$ \( 95179536 + 64857888 T + 25426656 T^{2} + 4538010 T^{3} + 9303715 T^{4} - 1310777 T^{5} + 709749 T^{6} - 24007 T^{7} + 890 T^{8} + 185 T^{9} + 141 T^{10} + 13 T^{11} + T^{12} \)
$59$ \( 216796176 + 132074280 T + 81675108 T^{2} + 47523738 T^{3} + 22442059 T^{4} + 4165203 T^{5} + 612056 T^{6} + 50941 T^{7} + 7287 T^{8} - 59 T^{9} + 96 T^{10} - 11 T^{11} + T^{12} \)
$61$ \( ( -275 + 4275 T + 5430 T^{2} + 2315 T^{3} + 430 T^{4} + 35 T^{5} + T^{6} )^{2} \)
$67$ \( ( 288101 - 170104 T + 840 T^{2} + 3925 T^{3} - 165 T^{4} - 19 T^{5} + T^{6} )^{2} \)
$71$ \( 75272976 - 156740616 T + 132517512 T^{2} - 15667332 T^{3} + 20824093 T^{4} - 3405837 T^{5} + 897093 T^{6} - 194017 T^{7} + 29898 T^{8} - 2737 T^{9} + 267 T^{10} - 21 T^{11} + T^{12} \)
$73$ \( 128845201 + 213954999 T + 126144337 T^{2} - 39501237 T^{3} + 30160288 T^{4} - 5242537 T^{5} + 1067138 T^{6} - 46387 T^{7} + 3288 T^{8} + 463 T^{9} + 187 T^{10} + 14 T^{11} + T^{12} \)
$79$ \( 59737441 - 119382134 T + 919874571 T^{2} + 492613039 T^{3} + 129564192 T^{4} + 17258989 T^{5} + 1488386 T^{6} + 6947 T^{7} + 3364 T^{8} + 1167 T^{9} + 173 T^{10} - 5 T^{11} + T^{12} \)
$83$ \( 10608176016 + 239774688 T + 1969880184 T^{2} + 590698758 T^{3} + 125870617 T^{4} + 19088592 T^{5} + 2990684 T^{6} + 289494 T^{7} + 21394 T^{8} + 306 T^{9} - 3 T^{10} + T^{12} \)
$89$ \( 26771504400 - 14946687000 T + 4543351560 T^{2} - 836256060 T^{3} + 110257021 T^{4} - 9320723 T^{5} + 820408 T^{6} - 88881 T^{7} + 16865 T^{8} - 1491 T^{9} + 178 T^{10} - 3 T^{11} + T^{12} \)
$97$ \( 7306041414961 + 2632959399931 T + 738427316631 T^{2} + 119243455564 T^{3} + 13763718512 T^{4} + 1077564529 T^{5} + 59412721 T^{6} + 2102877 T^{7} + 85794 T^{8} + 7082 T^{9} + 673 T^{10} + 35 T^{11} + T^{12} \)
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