Properties

Label 625.2.d.o
Level $625$
Weight $2$
Character orbit 625.d
Analytic conductor $4.991$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + x^{14} - 4 x^{12} - 49 x^{10} + 11 x^{8} + 395 x^{6} + 900 x^{4} + 1125 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + x^{14} - 4 x^{12} - 49 x^{10} + 11 x^{8} + 395 x^{6} + 900 x^{4} + 1125 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 22849 \nu^{15} - 2422021 \nu^{13} + 6573709 \nu^{11} - 1538146 \nu^{9} + 97097069 \nu^{7} - 420964990 \nu^{5} - 210825325 \nu^{3} - 35093875 \nu \)\()/ 858900625 \)
\(\beta_{2}\)\(=\)\((\)\( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + 77053830 \nu^{4} + 27131400 \nu^{2} + 181387875 \)\()/ 171780125 \)
\(\beta_{3}\)\(=\)\((\)\( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + 54634025 \nu^{4} + 77779875 \nu^{2} + 52412250 \)\()/15616375\)
\(\beta_{4}\)\(=\)\((\)\( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + 424939915 \nu^{4} + 771306350 \nu^{2} + 563878625 \)\()/ 171780125 \)
\(\beta_{5}\)\(=\)\((\)\( -281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} - 83450271 \nu^{4} - 152170830 \nu^{2} - 167096475 \)\()/34356025\)
\(\beta_{6}\)\(=\)\((\)\( -1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} - 137248467 \nu^{7} - 474206065 \nu^{5} + 222668250 \nu^{3} + 846395125 \nu \)\()/ 858900625 \)
\(\beta_{7}\)\(=\)\((\)\( -441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} - 135351515 \nu^{4} - 246968035 \nu^{2} - 215410900 \)\()/34356025\)
\(\beta_{8}\)\(=\)\((\)\( -2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} - 716617430 \nu^{4} - 884417625 \nu^{2} - 860602375 \)\()/ 171780125 \)
\(\beta_{9}\)\(=\)\((\)\( -626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} - 210462216 \nu^{4} - 281904075 \nu^{2} - 222107450 \)\()/34356025\)
\(\beta_{10}\)\(=\)\((\)\( -257689 \nu^{15} - 260409 \nu^{13} + 1184711 \nu^{11} + 13501191 \nu^{9} - 4440499 \nu^{7} - 113117275 \nu^{5} - 254856700 \nu^{3} - 189940875 \nu \)\()/78081875\)
\(\beta_{11}\)\(=\)\((\)\( 3711283 \nu^{15} + 2442853 \nu^{13} - 18411087 \nu^{11} - 175485672 \nu^{9} + 137417033 \nu^{7} + 1546922530 \nu^{5} + 2678832450 \nu^{3} + 1573606875 \nu \)\()/ 858900625 \)
\(\beta_{12}\)\(=\)\((\)\( 11296137 \nu^{15} - 2500148 \nu^{13} - 48798533 \nu^{11} - 491452273 \nu^{9} + 752782897 \nu^{7} + 3788808280 \nu^{5} + 5075395150 \nu^{3} + 4625990250 \nu \)\()/ 858900625 \)
\(\beta_{13}\)\(=\)\((\)\( 1033909 \nu^{15} - 651386 \nu^{13} - 3517956 \nu^{11} - 44316636 \nu^{9} + 84515054 \nu^{7} + 285294535 \nu^{5} + 432663800 \nu^{3} + 416073000 \nu \)\()/78081875\)
\(\beta_{14}\)\(=\)\((\)\( 1468861 \nu^{15} - 201274 \nu^{13} - 5426879 \nu^{11} - 66369199 \nu^{9} + 90901286 \nu^{7} + 466969935 \nu^{5} + 843562400 \nu^{3} + 722601875 \nu \)\()/78081875\)
\(\beta_{15}\)\(=\)\((\)\( 18299323 \nu^{15} - 8550017 \nu^{13} - 66698807 \nu^{11} - 789938892 \nu^{9} + 1385305413 \nu^{7} + 5454301045 \nu^{5} + 7862490225 \nu^{3} + 7417196625 \nu \)\()/ 858900625 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - 5 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - 5 \beta_{10} + 5 \beta_{6} + \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 2 \beta_{5} + 11 \beta_{4} - 4 \beta_{3} - \beta_{2} + 4\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-6 \beta_{15} + 14 \beta_{13} - 6 \beta_{12} + 5 \beta_{11} - 9 \beta_{1}\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{9} - 13 \beta_{8} + 3 \beta_{7} + 7 \beta_{5} + 6 \beta_{4} - 19 \beta_{3} - 26 \beta_{2} + 14\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(39 \beta_{15} - 20 \beta_{14} - 6 \beta_{13} - 36 \beta_{12} - 30 \beta_{10} - 29 \beta_{1}\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(23 \beta_{9} - 23 \beta_{8} + 63 \beta_{7} + 52 \beta_{5} + 156 \beta_{4} + 11 \beta_{3} - 11 \beta_{2} + 144\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(-26 \beta_{15} - 190 \beta_{14} + 184 \beta_{13} + 49 \beta_{12} + 190 \beta_{11} - 115 \beta_{10} + 115 \beta_{6} + 66 \beta_{1}\)\()/5\)
\(\nu^{8}\)\(=\)\((\)\(-57 \beta_{9} - 18 \beta_{8} + 363 \beta_{7} - 208 \beta_{5} + 381 \beta_{4} - 114 \beta_{3} - 96 \beta_{2} + 39\)\()/5\)
\(\nu^{9}\)\(=\)\((\)\(-\beta_{15} + 284 \beta_{13} - 286 \beta_{12} + 285 \beta_{11} + 285 \beta_{10} + 190 \beta_{6} - 479 \beta_{1}\)\()/5\)
\(\nu^{10}\)\(=\)\((\)\(208 \beta_{9} - 688 \beta_{8} + 208 \beta_{7} - 493 \beta_{5} - 149 \beta_{4} - 344 \beta_{3} - 896 \beta_{2} - 606\)\()/5\)
\(\nu^{11}\)\(=\)\((\)\(2344 \beta_{15} - 290 \beta_{14} - 1651 \beta_{13} - 1846 \beta_{12} + 195 \beta_{11} - 290 \beta_{10} + 95 \beta_{6} - 624 \beta_{1}\)\()/5\)
\(\nu^{12}\)\(=\)\((\)\(2208 \beta_{9} - 683 \beta_{8} + 2888 \beta_{7} - 683 \beta_{5} + 3956 \beta_{4} + 2891 \beta_{3} + 1104 \beta_{2} + 1784\)\()/5\)
\(\nu^{13}\)\(=\)\((\)\(-906 \beta_{15} - 4675 \beta_{14} + 2364 \beta_{13} + 3769 \beta_{12} + 7530 \beta_{11} + 4675 \beta_{6} + 5166 \beta_{1}\)\()/5\)
\(\nu^{14}\)\(=\)\((\)\(393 \beta_{9} + 1012 \beta_{8} + 10778 \beta_{7} - 18118 \beta_{5} - 619 \beta_{4} + 1631 \beta_{3} + 2024 \beta_{2} - 18511\)\()/5\)
\(\nu^{15}\)\(=\)\((\)\(1639 \beta_{15} + 19130 \beta_{14} - 12431 \beta_{13} - 9336 \beta_{12} + 30920 \beta_{10} - 13429 \beta_{1}\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{4} - \beta_{5} - \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−0.917186 + 1.66637i
−1.86824 + 0.357358i
1.86824 0.357358i
0.917186 1.66637i
−0.0566033 + 1.17421i
−0.644389 + 0.983224i
0.644389 0.983224i
0.0566033 1.17421i
−0.0566033 1.17421i
−0.644389 0.983224i
0.644389 + 0.983224i
0.0566033 + 1.17421i
−0.917186 1.66637i
−1.86824 0.357358i
1.86824 + 0.357358i
0.917186 + 1.66637i
−0.713605 2.19625i 0.384204 0.279141i −2.69625 + 1.95894i 0 −0.887234 0.644613i −3.03582 2.48990 + 1.80902i −0.857358 + 2.63868i 0
126.2 −0.350334 1.07822i −2.10569 + 1.52988i 0.578217 0.420099i 0 2.38723 + 1.73443i −0.407162 −2.48990 1.80902i 1.16637 3.58973i 0
126.3 0.350334 + 1.07822i 2.10569 1.52988i 0.578217 0.420099i 0 2.38723 + 1.73443i 0.407162 2.48990 + 1.80902i 1.16637 3.58973i 0
126.4 0.713605 + 2.19625i −0.384204 + 0.279141i −2.69625 + 1.95894i 0 −0.887234 0.644613i 3.03582 −2.48990 1.80902i −0.857358 + 2.63868i 0
251.1 −1.68703 + 1.22570i 0.679371 2.09089i 0.725700 2.23347i 0 1.41668 + 4.36010i −0.992398 0.224514 + 0.690983i −1.48322 1.07763i 0
251.2 −0.148189 + 0.107666i 0.454857 1.39991i −0.607666 + 1.87020i 0 0.0833172 + 0.256424i 3.26086 −0.224514 0.690983i 0.674207 + 0.489840i 0
251.3 0.148189 0.107666i −0.454857 + 1.39991i −0.607666 + 1.87020i 0 0.0833172 + 0.256424i −3.26086 0.224514 + 0.690983i 0.674207 + 0.489840i 0
251.4 1.68703 1.22570i −0.679371 + 2.09089i 0.725700 2.23347i 0 1.41668 + 4.36010i 0.992398 −0.224514 0.690983i −1.48322 1.07763i 0
376.1 −1.68703 1.22570i 0.679371 + 2.09089i 0.725700 + 2.23347i 0 1.41668 4.36010i −0.992398 0.224514 0.690983i −1.48322 + 1.07763i 0
376.2 −0.148189 0.107666i 0.454857 + 1.39991i −0.607666 1.87020i 0 0.0833172 0.256424i 3.26086 −0.224514 + 0.690983i 0.674207 0.489840i 0
376.3 0.148189 + 0.107666i −0.454857 1.39991i −0.607666 1.87020i 0 0.0833172 0.256424i −3.26086 0.224514 0.690983i 0.674207 0.489840i 0
376.4 1.68703 + 1.22570i −0.679371 2.09089i 0.725700 + 2.23347i 0 1.41668 4.36010i 0.992398 −0.224514 + 0.690983i −1.48322 + 1.07763i 0
501.1 −0.713605 + 2.19625i 0.384204 + 0.279141i −2.69625 1.95894i 0 −0.887234 + 0.644613i −3.03582 2.48990 1.80902i −0.857358 2.63868i 0
501.2 −0.350334 + 1.07822i −2.10569 1.52988i 0.578217 + 0.420099i 0 2.38723 1.73443i −0.407162 −2.48990 + 1.80902i 1.16637 + 3.58973i 0
501.3 0.350334 1.07822i 2.10569 + 1.52988i 0.578217 + 0.420099i 0 2.38723 1.73443i 0.407162 2.48990 1.80902i 1.16637 + 3.58973i 0
501.4 0.713605 2.19625i −0.384204 0.279141i −2.69625 1.95894i 0 −0.887234 + 0.644613i 3.03582 −2.48990 + 1.80902i −0.857358 2.63868i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.d.o 16
5.b even 2 1 inner 625.2.d.o 16
5.c odd 4 1 625.2.e.a 8
5.c odd 4 1 625.2.e.i 8
25.d even 5 2 125.2.d.b 16
25.d even 5 1 625.2.a.f 8
25.d even 5 1 inner 625.2.d.o 16
25.e even 10 2 125.2.d.b 16
25.e even 10 1 625.2.a.f 8
25.e even 10 1 inner 625.2.d.o 16
25.f odd 20 2 25.2.e.a 8
25.f odd 20 2 125.2.e.b 8
25.f odd 20 2 625.2.b.c 8
25.f odd 20 1 625.2.e.a 8
25.f odd 20 1 625.2.e.i 8
75.h odd 10 1 5625.2.a.x 8
75.j odd 10 1 5625.2.a.x 8
75.l even 20 2 225.2.m.a 8
100.h odd 10 1 10000.2.a.bj 8
100.j odd 10 1 10000.2.a.bj 8
100.l even 20 2 400.2.y.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.f odd 20 2
125.2.d.b 16 25.d even 5 2
125.2.d.b 16 25.e even 10 2
125.2.e.b 8 25.f odd 20 2
225.2.m.a 8 75.l even 20 2
400.2.y.c 8 100.l even 20 2
625.2.a.f 8 25.d even 5 1
625.2.a.f 8 25.e even 10 1
625.2.b.c 8 25.f odd 20 2
625.2.d.o 16 1.a even 1 1 trivial
625.2.d.o 16 5.b even 2 1 inner
625.2.d.o 16 25.d even 5 1 inner
625.2.d.o 16 25.e even 10 1 inner
625.2.e.a 8 5.c odd 4 1
625.2.e.a 8 25.f odd 20 1
625.2.e.i 8 5.c odd 4 1
625.2.e.i 8 25.f odd 20 1
5625.2.a.x 8 75.h odd 10 1
5625.2.a.x 8 75.j odd 10 1
10000.2.a.bj 8 100.h odd 10 1
10000.2.a.bj 8 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\(T_{2}^{16} + \cdots\)
\(T_{3}^{16} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 17 T^{2} + 863 T^{4} + 1246 T^{6} + 755 T^{8} + 146 T^{10} + 38 T^{12} + 8 T^{14} + T^{16} \)
$3$ \( 256 - 448 T^{2} + 4448 T^{4} + 4704 T^{6} + 2105 T^{8} + 399 T^{10} + 53 T^{12} + 7 T^{14} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 16 - 116 T^{2} + 121 T^{4} - 21 T^{6} + T^{8} )^{2} \)
$11$ \( ( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$13$ \( 1 - 3 T^{2} + 108 T^{4} + 479 T^{6} + 855 T^{8} + 479 T^{10} + 108 T^{12} - 3 T^{14} + T^{16} \)
$17$ \( 3748096 - 642752 T^{2} + 773408 T^{4} + 49696 T^{6} + 30905 T^{8} + 8816 T^{10} + 1178 T^{12} + 53 T^{14} + T^{16} \)
$19$ \( ( 400 + 800 T + 800 T^{2} + 450 T^{3} + 335 T^{4} + 135 T^{5} + 50 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$23$ \( 65536 + 28672 T^{2} + 85248 T^{4} + 107504 T^{6} + 64505 T^{8} + 6719 T^{10} + 333 T^{12} + 7 T^{14} + T^{16} \)
$29$ \( ( 483025 + 344025 T + 132750 T^{2} + 33775 T^{3} + 6885 T^{4} + 1105 T^{5} + 150 T^{6} + 15 T^{7} + T^{8} )^{2} \)
$31$ \( ( 1936 - 9064 T + 105252 T^{2} - 16322 T^{3} + 2255 T^{4} + 67 T^{5} - 3 T^{6} - T^{7} + T^{8} )^{2} \)
$37$ \( 13521270961 + 391634408 T^{2} + 495398478 T^{4} + 87235546 T^{6} + 6729555 T^{8} + 222206 T^{10} + 3463 T^{12} + 23 T^{14} + T^{16} \)
$41$ \( ( 13456 - 15544 T + 8332 T^{2} - 1222 T^{3} + 3805 T^{4} + 382 T^{5} + 62 T^{6} + 9 T^{7} + T^{8} )^{2} \)
$43$ \( ( 246016 - 56784 T^{2} + 4421 T^{4} - 129 T^{6} + T^{8} )^{2} \)
$47$ \( 4294967296 - 1291845632 T^{2} + 1518665728 T^{4} + 125924096 T^{6} + 5946105 T^{8} + 210016 T^{10} + 12298 T^{12} + 173 T^{14} + T^{16} \)
$53$ \( 76661949773761 + 1697131159592 T^{2} + 106782445478 T^{4} + 4408566354 T^{6} + 101911955 T^{8} + 1260894 T^{10} + 12863 T^{12} + 127 T^{14} + T^{16} \)
$59$ \( ( 4080400 + 2605800 T + 851400 T^{2} + 172200 T^{3} + 24985 T^{4} + 2610 T^{5} + 220 T^{6} + 15 T^{7} + T^{8} )^{2} \)
$61$ \( ( 116281 + 140151 T + 38437 T^{2} - 51672 T^{3} + 17305 T^{4} - 318 T^{5} + 252 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$67$ \( 60523872256 + 18295717888 T^{2} + 2484236288 T^{4} + 117354496 T^{6} + 31642880 T^{8} + 862976 T^{10} + 9968 T^{12} + 8 T^{14} + T^{16} \)
$71$ \( ( 24245776 + 9818456 T + 2542392 T^{2} + 427928 T^{3} + 54105 T^{4} + 5172 T^{5} + 462 T^{6} + 29 T^{7} + T^{8} )^{2} \)
$73$ \( 1 + 27 T^{2} + 303 T^{4} + 1054 T^{6} + 1755 T^{8} + 1354 T^{10} + 6078 T^{12} - 48 T^{14} + T^{16} \)
$79$ \( ( 33408400 + 4913000 T + 433500 T^{2} - 36550 T^{3} + 7935 T^{4} - 375 T^{5} + 270 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$83$ \( 9971220736 - 1312906688 T^{2} + 8731927008 T^{4} + 5084933504 T^{6} + 1144844105 T^{8} + 4438319 T^{10} + 37293 T^{12} + 247 T^{14} + T^{16} \)
$89$ \( ( 1392400 + 1014800 T + 629000 T^{2} + 206550 T^{3} + 48335 T^{4} + 7485 T^{5} + 740 T^{6} + 40 T^{7} + T^{8} )^{2} \)
$97$ \( 90802710507284161 + 2945611139430248 T^{2} + 42577340851068 T^{4} + 199383280546 T^{6} + 5403060030 T^{8} + 16977956 T^{10} + 81853 T^{12} + 338 T^{14} + T^{16} \)
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