Properties

Label 4016.2.a.j
Level 4016
Weight 2
Character orbit 4016.a
Self dual yes
Analytic conductor 32.068
Analytic rank 1
Dimension 14
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{4} q^{5} + ( -1 + \beta_{9} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{4} q^{5} + ( -1 + \beta_{9} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( -1 + \beta_{8} ) q^{11} + ( -\beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{13} + ( -\beta_{4} - \beta_{8} - \beta_{9} - \beta_{13} ) q^{15} + ( \beta_{1} + \beta_{6} ) q^{17} + ( -2 - \beta_{5} + \beta_{13} ) q^{19} + ( -\beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{11} ) q^{21} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{23} + ( 3 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{25} + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} + ( -1 - \beta_{2} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{29} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{10} - \beta_{12} ) q^{31} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{12} + \beta_{13} ) q^{33} + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{35} + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{10} + \beta_{12} ) q^{37} + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{39} + ( 2 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{13} ) q^{41} + ( -2 + \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{43} + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{45} + ( -1 + \beta_{1} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{47} + ( 2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{49} + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{51} + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{12} ) q^{53} + ( -4 - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{55} + ( 1 - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{57} + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{59} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{61} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{63} + ( 1 + \beta_{2} + \beta_{3} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{65} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{67} + ( -\beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{69} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{71} + ( -1 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{73} + ( -2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 6 \beta_{7} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} ) q^{75} + ( 1 - 2 \beta_{3} + 3 \beta_{5} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{77} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{79} + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + \beta_{12} ) q^{81} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{85} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{10} + 3 \beta_{12} - \beta_{13} ) q^{87} + ( -3 - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{13} ) q^{89} + ( -3 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{91} + ( -4 + 2 \beta_{1} - \beta_{3} + 3 \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{93} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{12} ) q^{95} + ( 2 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{97} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 3q^{3} - 2q^{5} - 8q^{7} + 21q^{9} + O(q^{10}) \) \( 14q - 3q^{3} - 2q^{5} - 8q^{7} + 21q^{9} - 9q^{11} - q^{13} - 14q^{15} - 27q^{19} - 3q^{21} - 13q^{23} + 26q^{25} - 15q^{27} - 25q^{31} + 16q^{33} - 21q^{35} - q^{37} - 33q^{39} + 10q^{41} - 35q^{43} - 4q^{45} - 6q^{47} + 36q^{49} - 48q^{51} - q^{53} - 41q^{55} + 14q^{57} - 30q^{59} + 3q^{61} - 31q^{63} + 7q^{65} - 22q^{67} - 17q^{69} - 6q^{71} + 5q^{73} - 4q^{75} - 14q^{77} - 56q^{79} + 26q^{81} + 28q^{83} - 23q^{85} - 11q^{87} - 24q^{89} - 38q^{91} - 55q^{93} + 4q^{95} + 6q^{97} - 12q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + 4161 x^{6} - 6861 x^{5} - 6676 x^{4} + 5599 x^{3} + 4627 x^{2} - 359 x - 196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-458652 \nu^{13} + 10207449 \nu^{12} + 8865462 \nu^{11} - 298701457 \nu^{10} - 45787181 \nu^{9} + 3246408968 \nu^{8} + 146603840 \nu^{7} - 16541854354 \nu^{6} - 1455748504 \nu^{5} + 39709795344 \nu^{4} + 7703220112 \nu^{3} - 36804146400 \nu^{2} - 13258305511 \nu + 1861877352\)\()/ 762886946 \)
\(\beta_{3}\)\(=\)\((\)\(458652 \nu^{13} - 10207449 \nu^{12} - 8865462 \nu^{11} + 298701457 \nu^{10} + 45787181 \nu^{9} - 3246408968 \nu^{8} - 146603840 \nu^{7} + 16541854354 \nu^{6} + 1455748504 \nu^{5} - 39709795344 \nu^{4} - 7703220112 \nu^{3} + 37567033346 \nu^{2} + 12495418565 \nu - 4913425136\)\()/ 762886946 \)
\(\beta_{4}\)\(=\)\((\)\(8519764 \nu^{13} + 100995904 \nu^{12} - 475048121 \nu^{11} - 2501409580 \nu^{10} + 7651912803 \nu^{9} + 22192164735 \nu^{8} - 47188811121 \nu^{7} - 94922312539 \nu^{6} + 114053323154 \nu^{5} + 203636821844 \nu^{4} - 66751900430 \nu^{3} - 178059623419 \nu^{2} - 50327477697 \nu + 9359448304\)\()/ 4577321676 \)
\(\beta_{5}\)\(=\)\((\)\(-4195687 \nu^{13} + 59312328 \nu^{12} - 28844005 \nu^{11} - 1424333823 \nu^{10} + 2228743711 \nu^{9} + 11739065967 \nu^{8} - 21011735390 \nu^{7} - 43345751602 \nu^{6} + 74926731662 \nu^{5} + 72932658519 \nu^{4} - 103565031713 \nu^{3} - 45673704976 \nu^{2} + 36469265332 \nu + 2862099808\)\()/ 1525773892 \)
\(\beta_{6}\)\(=\)\((\)\(-5336880 \nu^{13} - 865584 \nu^{12} + 175902437 \nu^{11} + 7885974 \nu^{10} - 2142375619 \nu^{9} - 43040057 \nu^{8} + 11998211113 \nu^{7} + 1393620799 \nu^{6} - 32452371932 \nu^{5} - 8827404356 \nu^{4} + 41134202654 \nu^{3} + 12822573733 \nu^{2} - 22034766939 \nu + 1054324184\)\()/ 1525773892 \)
\(\beta_{7}\)\(=\)\((\)\(34205344 \nu^{13} - 272353484 \nu^{12} - 487766789 \nu^{11} + 6852436190 \nu^{10} - 985278789 \nu^{9} - 60820846611 \nu^{8} + 33603498195 \nu^{7} + 248314486025 \nu^{6} - 134060247376 \nu^{5} - 476618765200 \nu^{4} + 150113282698 \nu^{3} + 352160580251 \nu^{2} + 5264197347 \nu - 18482722832\)\()/ 4577321676 \)
\(\beta_{8}\)\(=\)\((\)\(5967155 \nu^{13} - 22669783 \nu^{12} - 134567812 \nu^{11} + 542735784 \nu^{10} + 1026279357 \nu^{9} - 4439965553 \nu^{8} - 3819469690 \nu^{7} + 16339663095 \nu^{6} + 8758574070 \nu^{5} - 28676881891 \nu^{4} - 12289960304 \nu^{3} + 22212436924 \nu^{2} + 6353079607 \nu - 2832505686\)\()/ 762886946 \)
\(\beta_{9}\)\(=\)\((\)\(-13587249 \nu^{13} + 82033945 \nu^{12} + 220632596 \nu^{11} - 2025748304 \nu^{10} - 153782879 \nu^{9} + 17354603633 \nu^{8} - 10380694436 \nu^{7} - 66840155539 \nu^{6} + 50524361632 \nu^{5} + 117771424507 \nu^{4} - 72326314222 \nu^{3} - 79088806890 \nu^{2} + 14179372355 \nu + 6261708580\)\()/ 1525773892 \)
\(\beta_{10}\)\(=\)\((\)\(42935729 \nu^{13} - 93792808 \nu^{12} - 1159294432 \nu^{11} + 2336791369 \nu^{10} + 11502377406 \nu^{9} - 19984751130 \nu^{8} - 54780817125 \nu^{7} + 74365149889 \nu^{6} + 130314127684 \nu^{5} - 119198922599 \nu^{4} - 140109115921 \nu^{3} + 74681823445 \nu^{2} + 47947897389 \nu - 22751872012\)\()/ 4577321676 \)
\(\beta_{11}\)\(=\)\((\)\(20741705 \nu^{13} - 78046665 \nu^{12} - 457715319 \nu^{11} + 1856372484 \nu^{10} + 3173974206 \nu^{9} - 14783803588 \nu^{8} - 8133068789 \nu^{7} + 49588145194 \nu^{6} + 3889798330 \nu^{5} - 65520134901 \nu^{4} + 7319903428 \nu^{3} + 20909058893 \nu^{2} + 3171818988 \nu - 46956196\)\()/ 1525773892 \)
\(\beta_{12}\)\(=\)\((\)\(118858355 \nu^{13} - 427006354 \nu^{12} - 2693628022 \nu^{11} + 10377729199 \nu^{10} + 20013238272 \nu^{9} - 86552904492 \nu^{8} - 63314476149 \nu^{7} + 321067037137 \nu^{6} + 92843969962 \nu^{5} - 533348181419 \nu^{4} - 87018856909 \nu^{3} + 312683090797 \nu^{2} + 70791640479 \nu - 7920902056\)\()/ 4577321676 \)
\(\beta_{13}\)\(=\)\((\)\(40998083 \nu^{13} - 152031178 \nu^{12} - 851304577 \nu^{11} + 3546579085 \nu^{10} + 5069396045 \nu^{9} - 27563296343 \nu^{8} - 7226354746 \nu^{7} + 92002461912 \nu^{6} - 18695643274 \nu^{5} - 131587919995 \nu^{4} + 43902940955 \nu^{3} + 64100998940 \nu^{2} - 5970357810 \nu - 3159888728\)\()/ 1525773892 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{12} - 4 \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 11 \beta_{3} + 10 \beta_{2} + 9 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(\beta_{13} - 15 \beta_{11} + 13 \beta_{10} + 17 \beta_{9} + 13 \beta_{7} - 19 \beta_{6} - 18 \beta_{5} - 16 \beta_{3} - 5 \beta_{2} + 57 \beta_{1} + 5\)
\(\nu^{6}\)\(=\)\(\beta_{13} + 13 \beta_{12} - \beta_{11} - 65 \beta_{10} + 31 \beta_{9} + 22 \beta_{8} + 16 \beta_{7} - 35 \beta_{6} - 17 \beta_{5} + 34 \beta_{4} + 119 \beta_{3} + 101 \beta_{2} + 77 \beta_{1} + 227\)
\(\nu^{7}\)\(=\)\(13 \beta_{13} + 6 \beta_{12} - 186 \beta_{11} + 146 \beta_{10} + 220 \beta_{9} + 8 \beta_{8} + 135 \beta_{7} - 259 \beta_{6} - 253 \beta_{5} + 2 \beta_{4} - 210 \beta_{3} - 104 \beta_{2} + 510 \beta_{1} - 20\)
\(\nu^{8}\)\(=\)\(7 \beta_{13} + 158 \beta_{12} - 7 \beta_{11} - 849 \beta_{10} + 365 \beta_{9} + 326 \beta_{8} + 170 \beta_{7} - 462 \beta_{6} - 224 \beta_{5} + 438 \beta_{4} + 1316 \beta_{3} + 1065 \beta_{2} + 641 \beta_{1} + 2187\)
\(\nu^{9}\)\(=\)\(119 \beta_{13} + 143 \beta_{12} - 2190 \beta_{11} + 1644 \beta_{10} + 2590 \beta_{9} + 171 \beta_{8} + 1326 \beta_{7} - 3171 \beta_{6} - 3198 \beta_{5} + 2 \beta_{4} - 2623 \beta_{3} - 1572 \beta_{2} + 4876 \beta_{1} - 913\)
\(\nu^{10}\)\(=\)\(-50 \beta_{13} + 1940 \beta_{12} + 64 \beta_{11} - 10374 \beta_{10} + 3908 \beta_{9} + 4210 \beta_{8} + 1497 \beta_{7} - 5487 \beta_{6} - 2660 \beta_{5} + 5180 \beta_{4} + 14845 \beta_{3} + 11613 \beta_{2} + 5205 \beta_{1} + 22786\)
\(\nu^{11}\)\(=\)\(920 \beta_{13} + 2294 \beta_{12} - 25342 \beta_{11} + 18962 \beta_{10} + 29328 \beta_{9} + 2457 \beta_{8} + 12925 \beta_{7} - 37107 \beta_{6} - 38396 \beta_{5} - 564 \beta_{4} - 32149 \beta_{3} - 21153 \beta_{2} + 48906 \beta_{1} - 16315\)
\(\nu^{12}\)\(=\)\(-2101 \beta_{13} + 23661 \beta_{12} + 2860 \beta_{11} - 123400 \beta_{10} + 40080 \beta_{9} + 51131 \beta_{8} + 11385 \beta_{7} - 61898 \beta_{6} - 29604 \beta_{5} + 59402 \beta_{4} + 169642 \beta_{3} + 129551 \beta_{2} + 40862 \beta_{1} + 247897\)
\(\nu^{13}\)\(=\)\(6170 \beta_{13} + 31190 \beta_{12} - 291368 \beta_{11} + 223100 \beta_{10} + 326994 \beta_{9} + 29713 \beta_{8} + 127390 \beta_{7} - 425018 \beta_{6} - 448904 \beta_{5} - 14838 \beta_{4} - 390117 \beta_{3} - 269763 \beta_{2} + 507870 \beta_{1} - 238643\)

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
3.30568
2.87232
2.53364
2.37559
1.90340
1.46572
0.224635
−0.194346
−0.788858
−1.48367
−1.64615
−1.93078
−2.20631
−3.43087
0 −3.30568 0 0.595872 0 −0.878141 0 7.92752 0
1.2 0 −2.87232 0 −0.801100 0 4.32848 0 5.25020 0
1.3 0 −2.53364 0 2.70770 0 −1.66309 0 3.41934 0
1.4 0 −2.37559 0 −4.08952 0 −3.67423 0 2.64345 0
1.5 0 −1.90340 0 3.78264 0 0.441155 0 0.622940 0
1.6 0 −1.46572 0 0.817924 0 −4.79612 0 −0.851657 0
1.7 0 −0.224635 0 −2.42875 0 2.24830 0 −2.94954 0
1.8 0 0.194346 0 2.87489 0 −0.213356 0 −2.96223 0
1.9 0 0.788858 0 0.735420 0 −1.11520 0 −2.37770 0
1.10 0 1.48367 0 −3.74411 0 4.87233 0 −0.798715 0
1.11 0 1.64615 0 −4.00715 0 −2.94760 0 −0.290201 0
1.12 0 1.93078 0 3.14832 0 −3.83969 0 0.727926 0
1.13 0 2.20631 0 −0.868402 0 2.87616 0 1.86782 0
1.14 0 3.43087 0 −0.723735 0 −3.63900 0 8.77086 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 4016.2.a.j 14
4.b odd 2 1 1004.2.a.b 14
12.b even 2 1 9036.2.a.m 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1004.2.a.b 14 4.b odd 2 1
4016.2.a.j 14 1.a even 1 1 trivial
9036.2.a.m 14 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{14} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).