Properties

Label 350.2.o.c
Level $350$
Weight $2$
Character orbit 350.o
Analytic conductor $2.795$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.o (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 171 \nu^{14} + 2802 \nu^{12} + 20266 \nu^{10} + 96110 \nu^{8} + 343988 \nu^{6} + 866714 \nu^{4} + 786719 \nu^{2} + 1302910 \)\()/1020740\)
\(\beta_{3}\)\(=\)\((\)\(-6279 \nu^{15} - 6023 \nu^{14} - 94829 \nu^{13} - 319853 \nu^{12} - 291088 \nu^{11} - 2335656 \nu^{10} - 1140202 \nu^{9} - 13637974 \nu^{8} - 2242730 \nu^{7} - 52276470 \nu^{6} - 2623012 \nu^{5} - 131660244 \nu^{4} - 4504619 \nu^{3} - 120098303 \nu^{2} - 163267041 \nu - 204554077\)\()/ 224562800 \)
\(\beta_{4}\)\(=\)\((\)\(6279 \nu^{15} - 6023 \nu^{14} + 94829 \nu^{13} - 319853 \nu^{12} + 291088 \nu^{11} - 2335656 \nu^{10} + 1140202 \nu^{9} - 13637974 \nu^{8} + 2242730 \nu^{7} - 52276470 \nu^{6} + 2623012 \nu^{5} - 131660244 \nu^{4} + 4504619 \nu^{3} - 120098303 \nu^{2} + 163267041 \nu - 204554077\)\()/ 224562800 \)
\(\beta_{5}\)\(=\)\((\)\( 657161 \nu^{14} + 6218761 \nu^{12} + 35853172 \nu^{10} + 144339958 \nu^{8} + 433352890 \nu^{6} + 516320528 \nu^{4} + 189804581 \nu^{2} + 1019822349 \)\()/ 785969800 \)
\(\beta_{6}\)\(=\)\((\)\( -623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} - 373946420 \nu^{6} - 445540384 \nu^{4} - 843429583 \nu^{2} - 1273004222 \)\()/ 392984900 \)
\(\beta_{7}\)\(=\)\((\)\(4174573 \nu^{15} + 307671 \nu^{14} + 41450603 \nu^{13} + 4646621 \nu^{12} + 238976156 \nu^{11} + 14263312 \nu^{10} + 995989274 \nu^{9} + 55869898 \nu^{8} + 2888475470 \nu^{7} + 109893770 \nu^{6} + 3441488944 \nu^{5} + 128527588 \nu^{4} + 1618097653 \nu^{3} + 220726331 \nu^{2} + 6797535927 \nu + 8000085009\)\()/ 11003577200 \)
\(\beta_{8}\)\(=\)\((\)\(-4174573 \nu^{15} + 307671 \nu^{14} - 41450603 \nu^{13} + 4646621 \nu^{12} - 238976156 \nu^{11} + 14263312 \nu^{10} - 995989274 \nu^{9} + 55869898 \nu^{8} - 2888475470 \nu^{7} + 109893770 \nu^{6} - 3441488944 \nu^{5} + 128527588 \nu^{4} - 1618097653 \nu^{3} + 220726331 \nu^{2} - 6797535927 \nu + 8000085009\)\()/ 11003577200 \)
\(\beta_{9}\)\(=\)\((\)\(-5413027 \nu^{15} - 6673919 \nu^{14} - 53207257 \nu^{13} - 67697189 \nu^{12} - 316254784 \nu^{11} - 390296228 \nu^{10} - 1397075246 \nu^{9} - 1650388502 \nu^{8} - 4444289310 \nu^{7} - 4717462610 \nu^{6} - 7454251516 \nu^{5} - 5620645072 \nu^{4} - 10077798427 \nu^{3} - 2889806479 \nu^{2} - 15782597033 \nu - 11101746201\)\()/ 11003577200 \)
\(\beta_{10}\)\(=\)\((\)\(-5413027 \nu^{15} + 6673919 \nu^{14} - 53207257 \nu^{13} + 67697189 \nu^{12} - 316254784 \nu^{11} + 390296228 \nu^{10} - 1397075246 \nu^{9} + 1650388502 \nu^{8} - 4444289310 \nu^{7} + 4717462610 \nu^{6} - 7454251516 \nu^{5} + 5620645072 \nu^{4} - 10077798427 \nu^{3} + 2889806479 \nu^{2} - 15782597033 \nu + 11101746201\)\()/ 11003577200 \)
\(\beta_{11}\)\(=\)\((\)\( 26590 \nu^{15} + 257521 \nu^{13} + 1484692 \nu^{11} + 6079906 \nu^{9} + 17945290 \nu^{7} + 21381008 \nu^{5} + 8929484 \nu^{3} + 42231189 \nu \)\()/50016260\)
\(\beta_{12}\)\(=\)\((\)\( -154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} - 133651610 \nu^{7} - 226254436 \nu^{5} - 303303017 \nu^{3} - 477500443 \nu \)\()/ 239208200 \)
\(\beta_{13}\)\(=\)\((\)\(7738601 \nu^{15} - 12843957 \nu^{14} + 58163261 \nu^{13} - 106723897 \nu^{12} + 313306372 \nu^{11} - 615297844 \nu^{10} + 1167320938 \nu^{9} - 2508526286 \nu^{8} + 3102633990 \nu^{7} - 7437029530 \nu^{6} + 1535163128 \nu^{5} - 8860887056 \nu^{4} + 6928535561 \nu^{3} - 16785607657 \nu^{2} + 9719272549 \nu - 17501784573\)\()/ 11003577200 \)
\(\beta_{14}\)\(=\)\((\)\(-5179651 \nu^{15} - 39396031 \nu^{13} - 212402552 \nu^{11} - 802913678 \nu^{9} - 2161262850 \nu^{7} - 1033713348 \nu^{5} - 4832044771 \nu^{3} - 6839548919 \nu\)\()/ 5501788600 \)
\(\beta_{15}\)\(=\)\((\)\(-850941 \nu^{15} - 939402 \nu^{14} - 7115276 \nu^{13} - 7955037 \nu^{12} - 39448752 \nu^{11} - 44968654 \nu^{10} - 154522158 \nu^{9} - 183171156 \nu^{8} - 427936390 \nu^{7} - 539065950 \nu^{6} - 355475148 \nu^{5} - 642101046 \nu^{4} - 610473801 \nu^{3} - 1214738142 \nu^{2} - 1179772034 \nu - 1821562113\)\()/ 785969800 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} + \beta_{13} + \beta_{7} - 2 \beta_{6} - \beta_{5} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{15} - 2 \beta_{14} - \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{15} - 5 \beta_{13} + 5 \beta_{8} + 8 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-14 \beta_{15} + 22 \beta_{14} + 14 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} - 3 \beta_{9} - 14 \beta_{7}\)
\(\nu^{6}\)\(=\)\(14 \beta_{15} + 14 \beta_{13} - 14 \beta_{10} + 14 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} - 22 \beta_{6} + 22 \beta_{5} - 14 \beta_{4} - 14 \beta_{3} - 22 \beta_{2} - 7\)
\(\nu^{7}\)\(=\)\(64 \beta_{15} - 94 \beta_{14} - 64 \beta_{13} + 94 \beta_{11} + 64 \beta_{8} + 2 \beta_{4} - 2 \beta_{3} - 7 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-75 \beta_{15} - 75 \beta_{13} + 62 \beta_{10} - 62 \beta_{9} - 75 \beta_{7} + 112 \beta_{6} - 87 \beta_{5} + 112\)
\(\nu^{9}\)\(=\)\(-75 \beta_{15} + 112 \beta_{14} + 75 \beta_{13} + 112 \beta_{12} - 385 \beta_{11} - 75 \beta_{10} - 75 \beta_{9} - 273 \beta_{8} + 198 \beta_{7} - 75 \beta_{4} + 75 \beta_{3} + 112 \beta_{1}\)
\(\nu^{10}\)\(=\)\(337 \beta_{15} + 337 \beta_{13} - 337 \beta_{8} - 486 \beta_{6} + 198 \beta_{4} + 198 \beta_{3} + 273 \beta_{2}\)
\(\nu^{11}\)\(=\)\(-332 \beta_{15} + 456 \beta_{14} + 332 \beta_{13} - 759 \beta_{12} + 535 \beta_{10} + 535 \beta_{9} - 332 \beta_{7}\)
\(\nu^{12}\)\(=\)\(332 \beta_{15} + 332 \beta_{13} - 332 \beta_{10} + 332 \beta_{9} + 2364 \beta_{8} + 2696 \beta_{7} - 456 \beta_{6} + 456 \beta_{5} - 332 \beta_{4} - 332 \beta_{3} - 456 \beta_{2} - 3803\)
\(\nu^{13}\)\(=\)\(1452 \beta_{15} - 2032 \beta_{14} - 1452 \beta_{13} + 2032 \beta_{11} + 1452 \beta_{8} + 2696 \beta_{4} - 2696 \beta_{3} - 3803 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-10647 \beta_{15} - 10647 \beta_{13} - 1244 \beta_{10} + 1244 \beta_{9} - 10647 \beta_{7} + 15030 \beta_{6} + 1771 \beta_{5} + 15030\)
\(\nu^{15}\)\(=\)\(-10647 \beta_{15} + 15030 \beta_{14} + 10647 \beta_{13} + 15030 \beta_{12} + 7801 \beta_{11} - 10647 \beta_{10} - 10647 \beta_{9} + 5503 \beta_{8} - 16150 \beta_{7} - 10647 \beta_{4} + 10647 \beta_{3} + 15030 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-\beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.45333 1.51725i
−0.587308 + 2.01725i
−0.144868 + 1.25092i
1.01089 0.750919i
0.144868 1.25092i
−1.01089 + 0.750919i
−1.45333 + 1.51725i
0.587308 2.01725i
0.144868 + 1.25092i
−1.01089 0.750919i
−1.45333 1.51725i
0.587308 + 2.01725i
1.45333 + 1.51725i
−0.587308 2.01725i
−0.144868 1.25092i
1.01089 + 0.750919i
−0.965926 0.258819i −0.304013 1.13459i 0.866025 + 0.500000i 0 1.17462i 2.55176 + 0.698943i −0.707107 0.707107i 1.40320 0.810140i 0
143.2 −0.965926 0.258819i 0.752300 + 2.80762i 0.866025 + 0.500000i 0 2.90667i −2.58583 + 0.559876i −0.707107 0.707107i −4.71872 + 2.72435i 0
143.3 0.965926 + 0.258819i −0.523277 1.95290i 0.866025 + 0.500000i 0 2.02179i −1.83959 1.90155i 0.707107 + 0.707107i −0.941911 + 0.543813i 0
143.4 0.965926 + 0.258819i 0.0749894 + 0.279864i 0.866025 + 0.500000i 0 0.289737i −0.126334 + 2.64273i 0.707107 + 0.707107i 2.52538 1.45803i 0
157.1 −0.258819 + 0.965926i −1.95290 + 0.523277i −0.866025 0.500000i 0 2.02179i 1.90155 1.83959i 0.707107 0.707107i 0.941911 0.543813i 0
157.2 −0.258819 + 0.965926i 0.279864 0.0749894i −0.866025 0.500000i 0 0.289737i −2.64273 0.126334i 0.707107 0.707107i −2.52538 + 1.45803i 0
157.3 0.258819 0.965926i −1.13459 + 0.304013i −0.866025 0.500000i 0 1.17462i −0.698943 + 2.55176i −0.707107 + 0.707107i −1.40320 + 0.810140i 0
157.4 0.258819 0.965926i 2.80762 0.752300i −0.866025 0.500000i 0 2.90667i −0.559876 2.58583i −0.707107 + 0.707107i 4.71872 2.72435i 0
243.1 −0.258819 0.965926i −1.95290 0.523277i −0.866025 + 0.500000i 0 2.02179i 1.90155 + 1.83959i 0.707107 + 0.707107i 0.941911 + 0.543813i 0
243.2 −0.258819 0.965926i 0.279864 + 0.0749894i −0.866025 + 0.500000i 0 0.289737i −2.64273 + 0.126334i 0.707107 + 0.707107i −2.52538 1.45803i 0
243.3 0.258819 + 0.965926i −1.13459 0.304013i −0.866025 + 0.500000i 0 1.17462i −0.698943 2.55176i −0.707107 0.707107i −1.40320 0.810140i 0
243.4 0.258819 + 0.965926i 2.80762 + 0.752300i −0.866025 + 0.500000i 0 2.90667i −0.559876 + 2.58583i −0.707107 0.707107i 4.71872 + 2.72435i 0
257.1 −0.965926 + 0.258819i −0.304013 + 1.13459i 0.866025 0.500000i 0 1.17462i 2.55176 0.698943i −0.707107 + 0.707107i 1.40320 + 0.810140i 0
257.2 −0.965926 + 0.258819i 0.752300 2.80762i 0.866025 0.500000i 0 2.90667i −2.58583 0.559876i −0.707107 + 0.707107i −4.71872 2.72435i 0
257.3 0.965926 0.258819i −0.523277 + 1.95290i 0.866025 0.500000i 0 2.02179i −1.83959 + 1.90155i 0.707107 0.707107i −0.941911 0.543813i 0
257.4 0.965926 0.258819i 0.0749894 0.279864i 0.866025 0.500000i 0 0.289737i −0.126334 2.64273i 0.707107 0.707107i 2.52538 + 1.45803i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.o.c 16
5.b even 2 1 70.2.k.a 16
5.c odd 4 1 70.2.k.a 16
5.c odd 4 1 inner 350.2.o.c 16
7.d odd 6 1 inner 350.2.o.c 16
15.d odd 2 1 630.2.bv.c 16
15.e even 4 1 630.2.bv.c 16
20.d odd 2 1 560.2.ci.c 16
20.e even 4 1 560.2.ci.c 16
35.c odd 2 1 490.2.l.c 16
35.f even 4 1 490.2.l.c 16
35.i odd 6 1 70.2.k.a 16
35.i odd 6 1 490.2.g.c 16
35.j even 6 1 490.2.g.c 16
35.j even 6 1 490.2.l.c 16
35.k even 12 1 70.2.k.a 16
35.k even 12 1 inner 350.2.o.c 16
35.k even 12 1 490.2.g.c 16
35.l odd 12 1 490.2.g.c 16
35.l odd 12 1 490.2.l.c 16
105.p even 6 1 630.2.bv.c 16
105.w odd 12 1 630.2.bv.c 16
140.s even 6 1 560.2.ci.c 16
140.x odd 12 1 560.2.ci.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.k.a 16 5.b even 2 1
70.2.k.a 16 5.c odd 4 1
70.2.k.a 16 35.i odd 6 1
70.2.k.a 16 35.k even 12 1
350.2.o.c 16 1.a even 1 1 trivial
350.2.o.c 16 5.c odd 4 1 inner
350.2.o.c 16 7.d odd 6 1 inner
350.2.o.c 16 35.k even 12 1 inner
490.2.g.c 16 35.i odd 6 1
490.2.g.c 16 35.j even 6 1
490.2.g.c 16 35.k even 12 1
490.2.g.c 16 35.l odd 12 1
490.2.l.c 16 35.c odd 2 1
490.2.l.c 16 35.f even 4 1
490.2.l.c 16 35.j even 6 1
490.2.l.c 16 35.l odd 12 1
560.2.ci.c 16 20.d odd 2 1
560.2.ci.c 16 20.e even 4 1
560.2.ci.c 16 140.s even 6 1
560.2.ci.c 16 140.x odd 12 1
630.2.bv.c 16 15.d odd 2 1
630.2.bv.c 16 15.e even 4 1
630.2.bv.c 16 105.p even 6 1
630.2.bv.c 16 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( 16 - 96 T + 288 T^{2} - 576 T^{3} - 276 T^{4} + 1968 T^{5} + 3528 T^{6} + 3780 T^{7} + 2357 T^{8} + 564 T^{9} - 84 T^{11} - 45 T^{12} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( 5764801 + 6588344 T + 3764768 T^{2} + 1344560 T^{3} + 175273 T^{4} - 54880 T^{5} - 20384 T^{6} + 4872 T^{7} + 4944 T^{8} + 696 T^{9} - 416 T^{10} - 160 T^{11} + 73 T^{12} + 80 T^{13} + 32 T^{14} + 8 T^{15} + T^{16} \)
$11$ \( ( 3844 + 5952 T + 8410 T^{2} + 1992 T^{3} + 807 T^{4} + 114 T^{5} + 49 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$13$ \( 16 + 2280 T^{4} + 1361 T^{8} + 90 T^{12} + T^{16} \)
$17$ \( 9834496 - 50577408 T + 130056192 T^{2} - 222953472 T^{3} + 273629952 T^{4} - 243823104 T^{5} + 162570240 T^{6} - 83276928 T^{7} + 33047888 T^{8} - 10173024 T^{9} + 2449440 T^{10} - 463968 T^{11} + 68652 T^{12} - 7776 T^{13} + 648 T^{14} - 36 T^{15} + T^{16} \)
$19$ \( 100000000 + 58000000 T^{2} + 23150000 T^{4} + 4844200 T^{6} + 730801 T^{8} + 53438 T^{10} + 2795 T^{12} + 62 T^{14} + T^{16} \)
$23$ \( 260144641 + 237612428 T + 108515912 T^{2} + 202471528 T^{3} + 80047982 T^{4} - 44050244 T^{5} + 5166336 T^{6} - 2575212 T^{7} + 443539 T^{8} + 48324 T^{9} - 5376 T^{10} + 3052 T^{11} - 658 T^{12} - 56 T^{13} + 8 T^{14} - 4 T^{15} + T^{16} \)
$29$ \( ( 329476 + 90948 T^{2} + 7325 T^{4} + 162 T^{6} + T^{8} )^{2} \)
$31$ \( ( 16 - 672 T + 9472 T^{2} - 2688 T^{3} - 412 T^{4} + 192 T^{5} + 32 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$37$ \( 65536 - 327680 T + 819200 T^{2} - 5005312 T^{3} + 12386048 T^{4} + 6952448 T^{5} + 1552512 T^{6} + 950256 T^{7} + 232225 T^{8} - 22860 T^{9} - 4536 T^{10} - 2296 T^{11} - 385 T^{12} + 56 T^{13} + 8 T^{14} + 4 T^{15} + T^{16} \)
$41$ \( ( 18769 + 31468 T^{2} + 5174 T^{4} + 140 T^{6} + T^{8} )^{2} \)
$43$ \( ( 784 - 896 T + 512 T^{2} + 976 T^{3} + 673 T^{4} + 140 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$47$ \( 9834496 - 71651328 T + 261015552 T^{2} - 633894912 T^{3} + 655934400 T^{4} + 400292352 T^{5} + 103800096 T^{6} + 18641688 T^{7} + 920753 T^{8} - 355332 T^{9} - 82584 T^{10} - 15288 T^{11} - 825 T^{12} + 288 T^{13} + 72 T^{14} + 12 T^{15} + T^{16} \)
$53$ \( 41740124416 + 16252791808 T + 3164260352 T^{2} + 14817500160 T^{3} - 96585232 T^{4} - 3470092096 T^{5} + 1286198656 T^{6} - 363698592 T^{7} + 99156337 T^{8} - 19525980 T^{9} + 2989576 T^{10} - 431240 T^{11} + 52271 T^{12} - 4776 T^{13} + 392 T^{14} - 28 T^{15} + T^{16} \)
$59$ \( 268435456 + 419430400 T^{2} + 581697536 T^{4} + 110116864 T^{6} + 16306432 T^{8} + 632192 T^{10} + 18608 T^{12} + 152 T^{14} + T^{16} \)
$61$ \( ( 148996 - 78744 T - 5042 T^{2} + 9996 T^{3} + 1607 T^{4} - 294 T^{5} - 37 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$67$ \( 3429742096 - 3117947360 T + 1417248800 T^{2} - 5674148832 T^{3} + 1483484684 T^{4} + 3138391520 T^{5} + 1227548872 T^{6} + 476714964 T^{7} + 180649957 T^{8} + 39085404 T^{9} + 6081184 T^{10} + 939268 T^{11} + 112235 T^{12} + 8184 T^{13} + 512 T^{14} + 32 T^{15} + T^{16} \)
$71$ \( ( -4424 + 1816 T - 190 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$73$ \( 1017603875209216 + 198908406202368 T + 19440056696832 T^{2} + 1266632589312 T^{3} - 616424309760 T^{4} - 46930397184 T^{5} + 3390958080 T^{6} - 1224624768 T^{7} + 171765008 T^{8} + 7027296 T^{9} - 295776 T^{10} + 125664 T^{11} - 16260 T^{12} - 288 T^{13} + 72 T^{14} - 12 T^{15} + T^{16} \)
$79$ \( 61465600000000 - 5720064000000 T^{2} + 349926400000 T^{4} - 12457574400 T^{6} + 323248896 T^{8} - 5240832 T^{10} + 59680 T^{12} - 288 T^{14} + T^{16} \)
$83$ \( 3812835757370896 + 5479636353768 T^{4} + 1373332049 T^{8} + 69978 T^{12} + T^{16} \)
$89$ \( 9971220736 + 6274951040 T^{2} + 3225808304 T^{4} + 417079160 T^{6} + 40392625 T^{8} + 1250110 T^{10} + 28859 T^{12} + 190 T^{14} + T^{16} \)
$97$ \( ( 3111696 + 8136 T^{4} + T^{8} )^{2} \)
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