Properties

Label 605.2.j.h
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 7 x^{14} + 25 x^{12} - 57 x^{10} + 194 x^{8} - 303 x^{6} + 235 x^{4} - 33 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{14} + 25 x^{12} - 57 x^{10} + 194 x^{8} - 303 x^{6} + 235 x^{4} - 33 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 1829973 \nu^{15} - 24476532 \nu^{13} + 152757677 \nu^{11} - 557238290 \nu^{9} + 1500352252 \nu^{7} - 3810530967 \nu^{5} + 7929326490 \nu^{3} - 7498263355 \nu \)\()/ 2438578648 \)
\(\beta_{3}\)\(=\)\((\)\( -608673 \nu^{14} + 4779625 \nu^{12} - 9344761 \nu^{10} - 2560494 \nu^{8} - 9430964 \nu^{6} + 74558203 \nu^{4} + 726449607 \nu^{2} + 31775447 \)\()/ 609644662 \)
\(\beta_{4}\)\(=\)\((\)\( 2744937 \nu^{14} - 22433610 \nu^{12} + 99837091 \nu^{10} - 276654262 \nu^{8} + 814713604 \nu^{6} - 1632321387 \nu^{4} + 3415114020 \nu^{2} - 771926881 \)\()/ 2438578648 \)
\(\beta_{5}\)\(=\)\((\)\( 2744937 \nu^{15} - 22433610 \nu^{13} + 99837091 \nu^{11} - 276654262 \nu^{9} + 814713604 \nu^{7} - 1632321387 \nu^{5} + 3415114020 \nu^{3} - 771926881 \nu \)\()/ 2438578648 \)
\(\beta_{6}\)\(=\)\((\)\( -2924455 \nu^{14} + 43736988 \nu^{12} - 193036323 \nu^{10} + 488836470 \nu^{8} - 1053314964 \nu^{6} + 3520413317 \nu^{4} - 719547006 \nu^{2} - 236512683 \)\()/ 2438578648 \)
\(\beta_{7}\)\(=\)\((\)\( -1351905 \nu^{14} + 5558819 \nu^{12} - 7678521 \nu^{10} - 13223046 \nu^{8} - 42401020 \nu^{6} - 304413167 \nu^{4} + 664868085 \nu^{2} - 629543387 \)\()/ 609644662 \)
\(\beta_{8}\)\(=\)\((\)\( -6960689 \nu^{14} + 30587176 \nu^{12} - 61040185 \nu^{10} + 38446538 \nu^{8} - 665668084 \nu^{6} - 797037357 \nu^{4} + 1844303058 \nu^{2} - 1738523017 \)\()/ 2438578648 \)
\(\beta_{9}\)\(=\)\((\)\( -3962283 \nu^{14} + 31992860 \nu^{12} - 118526613 \nu^{10} + 271533274 \nu^{8} - 833575532 \nu^{6} + 1781437793 \nu^{4} - 742925482 \nu^{2} - 383811549 \)\()/ 1219289324 \)
\(\beta_{10}\)\(=\)\((\)\( 8234811 \nu^{15} - 67300830 \nu^{13} + 299511273 \nu^{11} - 829962786 \nu^{9} + 2444140812 \nu^{7} - 4896964161 \nu^{5} + 7806763412 \nu^{3} - 2315780643 \nu \)\()/ 2438578648 \)
\(\beta_{11}\)\(=\)\((\)\( 770947 \nu^{14} - 4704418 \nu^{12} + 13758165 \nu^{10} - 28071690 \nu^{8} + 126275508 \nu^{6} - 152673185 \nu^{4} - 21444456 \nu^{2} - 243303327 \)\()/ 221688968 \)
\(\beta_{12}\)\(=\)\((\)\( 10669503 \nu^{15} - 86419330 \nu^{13} + 336890317 \nu^{11} - 819720810 \nu^{9} + 2481864668 \nu^{7} - 5195196973 \nu^{5} + 4900964984 \nu^{3} - 4303783 \nu \)\()/ 2438578648 \)
\(\beta_{13}\)\(=\)\((\)\( -8815096 \nu^{15} + 58503253 \nu^{13} - 198965251 \nu^{11} + 429083674 \nu^{9} - 1573766376 \nu^{7} + 2199079808 \nu^{5} - 1528330963 \nu^{3} + 352179707 \nu \)\()/ 1219289324 \)
\(\beta_{14}\)\(=\)\((\)\(-19639683 \nu^{15} + 162786416 \nu^{13} - 643887411 \nu^{11} + 1627587846 \nu^{9} - 4886486364 \nu^{7} + 10096782513 \nu^{5} - 8290421402 \nu^{3} + 4755604557 \nu\)\()/ 2438578648 \)
\(\beta_{15}\)\(=\)\((\)\( -1496498 \nu^{15} + 9707543 \nu^{13} - 31032777 \nu^{11} + 60570826 \nu^{9} - 228914540 \nu^{7} + 268358314 \nu^{5} + 33071203 \nu^{3} - 172220447 \nu \)\()/ 110844484 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 2 \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{10} + 3 \beta_{5}\)
\(\nu^{4}\)\(=\)\(6 \beta_{9} - 4 \beta_{8} + \beta_{7} - 4 \beta_{6} + 4 \beta_{4} - \beta_{3} + 1\)
\(\nu^{5}\)\(=\)\(\beta_{15} - 5 \beta_{14} - \beta_{13} - 10 \beta_{12} - \beta_{10} + 15 \beta_{5} - 5 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-7 \beta_{11} - 22 \beta_{8} + 21 \beta_{7} + \beta_{4} - 7 \beta_{3} - 1\)
\(\nu^{7}\)\(=\)\(21 \beta_{15} - 21 \beta_{14} - 36 \beta_{13} - 36 \beta_{12} + 29 \beta_{5} - 28 \beta_{2} + 7 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-85 \beta_{11} - 78 \beta_{9} - 87 \beta_{8} + 85 \beta_{7} + 49 \beta_{6} - 7 \beta_{4} - 49 \beta_{3} - 87\)
\(\nu^{9}\)\(=\)\(85 \beta_{15} - 36 \beta_{14} - 136 \beta_{13} - 45 \beta_{12} + 36 \beta_{10} + 36 \beta_{5} - 121 \beta_{2} - 85 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-342 \beta_{11} - 396 \beta_{9} - 230 \beta_{8} + 166 \beta_{7} + 166 \beta_{6} - 166 \beta_{4} - 529\)
\(\nu^{11}\)\(=\)\(166 \beta_{15} - 220 \beta_{13} + 342 \beta_{10} - 220 \beta_{5} - 342 \beta_{2} - 475 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-728 \beta_{11} - 1653 \beta_{9} + 728 \beta_{6} - 1170 \beta_{4} + 651 \beta_{3} - 1653\)
\(\nu^{13}\)\(=\)\(728 \beta_{14} + 1002 \beta_{12} + 1379 \beta_{10} - 2823 \beta_{5} - 1002 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-4644 \beta_{9} + 3748 \beta_{8} - 3109 \beta_{7} + 2472 \beta_{6} - 3748 \beta_{4} + 3109 \beta_{3} - 3109\)
\(\nu^{15}\)\(=\)\(-3109 \beta_{15} + 5581 \beta_{14} + 4385 \beta_{13} + 8392 \beta_{12} + 3109 \beta_{10} - 13973 \beta_{5} + 5581 \beta_{2} - 1276 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{4} - \beta_{8} - \beta_{9}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.92464 + 0.625353i
−1.17360 + 0.381325i
1.17360 0.381325i
1.92464 0.625353i
−0.972539 + 1.33858i
−0.471815 + 0.649397i
0.471815 0.649397i
0.972539 1.33858i
−1.92464 0.625353i
−1.17360 0.381325i
1.17360 + 0.381325i
1.92464 + 0.625353i
−0.972539 1.33858i
−0.471815 0.649397i
0.471815 + 0.649397i
0.972539 + 1.33858i
−1.92464 0.625353i −1.54035 2.12011i 1.69513 + 1.23158i −2.19919 + 0.404418i 1.63880 + 5.04369i 0.567697 0.781367i −0.113351 0.156015i −1.19513 + 3.67823i 4.48555 + 0.596911i
9.2 −1.17360 0.381325i 0.213943 + 0.294468i −0.386111 0.280526i 2.13387 + 0.668267i −0.138796 0.427169i −1.51977 + 2.09178i 1.79681 + 2.47310i 0.886111 2.72717i −2.24948 1.59798i
9.3 1.17360 + 0.381325i −0.213943 0.294468i −0.386111 0.280526i −1.33354 + 1.79490i −0.138796 0.427169i 1.51977 2.09178i −1.79681 2.47310i 0.886111 2.72717i −2.24948 + 1.59798i
9.4 1.92464 + 0.625353i 1.54035 + 2.12011i 1.69513 + 1.23158i 2.01689 0.965471i 1.63880 + 5.04369i −0.567697 + 0.781367i 0.113351 + 0.156015i −1.19513 + 3.67823i 4.48555 0.596911i
124.1 −0.972539 1.33858i −1.87813 + 0.610243i −0.227943 + 0.701538i −1.51945 1.64051i 2.64342 + 1.92056i −2.13329 0.693148i −1.98645 + 0.645437i 0.727943 0.528882i −0.718246 + 3.62937i
124.2 −0.471815 0.649397i 1.67457 0.544099i 0.418926 1.28932i 2.07703 0.828231i −1.14342 0.830744i 0.563124 + 0.182970i −2.56176 + 0.832367i 0.0810736 0.0589034i −1.51782 0.958043i
124.3 0.471815 + 0.649397i −1.67457 + 0.544099i 0.418926 1.28932i −0.145858 + 2.23131i −1.14342 0.830744i −0.563124 0.182970i 2.56176 0.832367i 0.0810736 0.0589034i −1.51782 + 0.958043i
124.4 0.972539 + 1.33858i 1.87813 0.610243i −0.227943 + 0.701538i −2.02976 0.938132i 2.64342 + 1.92056i 2.13329 + 0.693148i 1.98645 0.645437i 0.727943 0.528882i −0.718246 3.62937i
269.1 −1.92464 + 0.625353i −1.54035 + 2.12011i 1.69513 1.23158i −2.19919 0.404418i 1.63880 5.04369i 0.567697 + 0.781367i −0.113351 + 0.156015i −1.19513 3.67823i 4.48555 0.596911i
269.2 −1.17360 + 0.381325i 0.213943 0.294468i −0.386111 + 0.280526i 2.13387 0.668267i −0.138796 + 0.427169i −1.51977 2.09178i 1.79681 2.47310i 0.886111 + 2.72717i −2.24948 + 1.59798i
269.3 1.17360 0.381325i −0.213943 + 0.294468i −0.386111 + 0.280526i −1.33354 1.79490i −0.138796 + 0.427169i 1.51977 + 2.09178i −1.79681 + 2.47310i 0.886111 + 2.72717i −2.24948 1.59798i
269.4 1.92464 0.625353i 1.54035 2.12011i 1.69513 1.23158i 2.01689 + 0.965471i 1.63880 5.04369i −0.567697 0.781367i 0.113351 0.156015i −1.19513 3.67823i 4.48555 + 0.596911i
444.1 −0.972539 + 1.33858i −1.87813 0.610243i −0.227943 0.701538i −1.51945 + 1.64051i 2.64342 1.92056i −2.13329 + 0.693148i −1.98645 0.645437i 0.727943 + 0.528882i −0.718246 3.62937i
444.2 −0.471815 + 0.649397i 1.67457 + 0.544099i 0.418926 + 1.28932i 2.07703 + 0.828231i −1.14342 + 0.830744i 0.563124 0.182970i −2.56176 0.832367i 0.0810736 + 0.0589034i −1.51782 + 0.958043i
444.3 0.471815 0.649397i −1.67457 0.544099i 0.418926 + 1.28932i −0.145858 2.23131i −1.14342 + 0.830744i −0.563124 + 0.182970i 2.56176 + 0.832367i 0.0810736 + 0.0589034i −1.51782 0.958043i
444.4 0.972539 1.33858i 1.87813 + 0.610243i −0.227943 0.701538i −2.02976 + 0.938132i 2.64342 1.92056i 2.13329 0.693148i 1.98645 + 0.645437i 0.727943 + 0.528882i −0.718246 + 3.62937i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 444.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.h 16
5.b even 2 1 inner 605.2.j.h 16
11.b odd 2 1 605.2.j.g 16
11.c even 5 2 55.2.j.a 16
11.c even 5 1 605.2.b.g 8
11.c even 5 1 inner 605.2.j.h 16
11.d odd 10 1 605.2.b.f 8
11.d odd 10 2 605.2.j.d 16
11.d odd 10 1 605.2.j.g 16
33.h odd 10 2 495.2.ba.a 16
44.h odd 10 2 880.2.cd.c 16
55.d odd 2 1 605.2.j.g 16
55.h odd 10 1 605.2.b.f 8
55.h odd 10 2 605.2.j.d 16
55.h odd 10 1 605.2.j.g 16
55.j even 10 2 55.2.j.a 16
55.j even 10 1 605.2.b.g 8
55.j even 10 1 inner 605.2.j.h 16
55.k odd 20 4 275.2.h.d 16
55.k odd 20 2 3025.2.a.bl 8
55.l even 20 2 3025.2.a.bk 8
165.o odd 10 2 495.2.ba.a 16
220.n odd 10 2 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 11.c even 5 2
55.2.j.a 16 55.j even 10 2
275.2.h.d 16 55.k odd 20 4
495.2.ba.a 16 33.h odd 10 2
495.2.ba.a 16 165.o odd 10 2
605.2.b.f 8 11.d odd 10 1
605.2.b.f 8 55.h odd 10 1
605.2.b.g 8 11.c even 5 1
605.2.b.g 8 55.j even 10 1
605.2.j.d 16 11.d odd 10 2
605.2.j.d 16 55.h odd 10 2
605.2.j.g 16 11.b odd 2 1
605.2.j.g 16 11.d odd 10 1
605.2.j.g 16 55.d odd 2 1
605.2.j.g 16 55.h odd 10 1
605.2.j.h 16 1.a even 1 1 trivial
605.2.j.h 16 5.b even 2 1 inner
605.2.j.h 16 11.c even 5 1 inner
605.2.j.h 16 55.j even 10 1 inner
880.2.cd.c 16 44.h odd 10 2
880.2.cd.c 16 220.n odd 10 2
3025.2.a.bk 8 55.l even 20 2
3025.2.a.bl 8 55.k odd 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\(T_{2}^{16} - \cdots\)
\(T_{19}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 121 - 33 T^{2} + 235 T^{4} - 303 T^{6} + 194 T^{8} - 57 T^{10} + 25 T^{12} - 7 T^{14} + T^{16} \)
$3$ \( 121 + 462 T^{2} + 6455 T^{4} - 5663 T^{6} + 2194 T^{8} - 427 T^{10} + 55 T^{12} - 7 T^{14} + T^{16} \)
$5$ \( 390625 + 156250 T - 156250 T^{2} - 118750 T^{3} + 11875 T^{4} + 37250 T^{5} + 10500 T^{6} - 3760 T^{7} - 3519 T^{8} - 752 T^{9} + 420 T^{10} + 298 T^{11} + 19 T^{12} - 38 T^{13} - 10 T^{14} + 2 T^{15} + T^{16} \)
$7$ \( 121 - 506 T^{2} + 867 T^{4} - 178 T^{6} + 1155 T^{8} - 262 T^{10} + 37 T^{12} - 4 T^{14} + T^{16} \)
$11$ \( T^{16} \)
$13$ \( 47265625 - 2578125 T^{2} + 3671875 T^{4} - 946875 T^{6} + 121250 T^{8} - 7125 T^{10} + 625 T^{12} - 35 T^{14} + T^{16} \)
$17$ \( 1675346761 - 196346007 T^{2} + 60857245 T^{4} - 13339737 T^{6} + 1538354 T^{8} - 82503 T^{10} + 1975 T^{12} - 8 T^{14} + T^{16} \)
$19$ \( ( 121 + 627 T + 1158 T^{2} - 771 T^{3} + 505 T^{4} - 141 T^{5} + 28 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$23$ \( ( 26411 + 16366 T^{2} + 2232 T^{4} + 99 T^{6} + T^{8} )^{2} \)
$29$ \( ( 121 + 319 T + 3180 T^{2} - 2109 T^{3} + 759 T^{4} - 209 T^{5} + 60 T^{6} - 11 T^{7} + T^{8} )^{2} \)
$31$ \( ( 1 + 3 T + 11 T^{2} + 19 T^{3} + 34 T^{4} + 47 T^{5} + 49 T^{6} + 11 T^{7} + T^{8} )^{2} \)
$37$ \( 15768841 + 31660783 T^{2} + 202738270 T^{4} - 53050607 T^{6} + 5512119 T^{8} - 96443 T^{10} + 3060 T^{12} - 73 T^{14} + T^{16} \)
$41$ \( ( 3876961 - 730499 T + 81580 T^{2} + 2859 T^{3} + 2069 T^{4} - 351 T^{5} + 130 T^{6} - 9 T^{7} + T^{8} )^{2} \)
$43$ \( ( 212531 + 79707 T^{2} + 8319 T^{4} + 173 T^{6} + T^{8} )^{2} \)
$47$ \( 59639012521 - 23039598373 T^{2} + 3345711493 T^{4} + 82983779 T^{6} + 18910630 T^{8} + 207559 T^{10} + 17003 T^{12} - 213 T^{14} + T^{16} \)
$53$ \( 239836452361 - 32889354498 T^{2} + 6788907085 T^{4} - 1292012988 T^{6} + 169780889 T^{8} - 2739312 T^{10} + 19495 T^{12} - 32 T^{14} + T^{16} \)
$59$ \( ( 1 - 18 T + 721 T^{2} - 1404 T^{3} + 5849 T^{4} - 312 T^{5} - 41 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$61$ \( ( 43681 + 29260 T + 10132 T^{2} - 650 T^{3} + 294 T^{4} - 260 T^{5} + 213 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$67$ \( ( 18491 + 46104 T^{2} + 5736 T^{4} + 149 T^{6} + T^{8} )^{2} \)
$71$ \( ( 6305121 + 2508489 T + 638604 T^{2} + 82863 T^{3} + 8019 T^{4} - 189 T^{5} + 36 T^{6} - 3 T^{7} + T^{8} )^{2} \)
$73$ \( 1636073786281 + 406010344311 T^{2} + 361779521217 T^{4} - 19377171147 T^{6} + 446327080 T^{8} - 4668843 T^{10} + 25397 T^{12} - 96 T^{14} + T^{16} \)
$79$ \( ( 101761 + 55506 T + 95555 T^{2} - 7511 T^{3} - 776 T^{4} + 269 T^{5} + 115 T^{6} + 11 T^{7} + T^{8} )^{2} \)
$83$ \( 45169425961 - 4511608068 T^{2} + 488428383 T^{4} - 50518596 T^{6} + 4579405 T^{8} - 79656 T^{10} + 3173 T^{12} - 78 T^{14} + T^{16} \)
$89$ \( ( 1871 + 486 T - 128 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$97$ \( 25937424601 - 1286153286 T^{2} + 752913425 T^{4} - 132559614 T^{6} + 11480359 T^{8} - 52734 T^{10} + 13605 T^{12} - 186 T^{14} + T^{16} \)
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