Properties

Label 1856.4.a.bl
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + 7106592 x^{4} - 7979328 x^{3} - 38912400 x^{2} + 1083456 x + 14751072\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{3} + ( 1 - \beta_{4} ) q^{5} + ( -4 + \beta_{6} ) q^{7} + ( 11 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{3} + ( 1 - \beta_{4} ) q^{5} + ( -4 + \beta_{6} ) q^{7} + ( 11 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{9} + ( 3 + 2 \beta_{1} + \beta_{3} - \beta_{7} ) q^{11} + ( 3 + \beta_{6} + \beta_{8} + \beta_{11} ) q^{13} + ( 4 + 2 \beta_{1} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{15} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{17} + ( 12 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{19} + ( 9 - 12 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{10} - \beta_{11} ) q^{21} + ( -29 - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{23} + ( 41 + 12 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{25} + ( 24 + 14 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{27} -29 q^{29} + ( -33 + 11 \beta_{1} + \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + \beta_{5} - 4 \beta_{8} + \beta_{9} - \beta_{11} ) q^{31} + ( 59 + 16 \beta_{1} - 2 \beta_{2} + \beta_{3} - 8 \beta_{4} - \beta_{5} - 5 \beta_{6} + \beta_{7} + \beta_{9} - 4 \beta_{10} ) q^{33} + ( 16 - \beta_{2} - 5 \beta_{3} + 6 \beta_{4} - \beta_{5} + 8 \beta_{6} + 4 \beta_{7} - 3 \beta_{9} + 6 \beta_{10} - \beta_{11} ) q^{35} + ( 32 - 14 \beta_{1} - \beta_{2} - 8 \beta_{4} - 6 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} - \beta_{11} ) q^{37} + ( 12 + 5 \beta_{1} + \beta_{2} + 4 \beta_{3} - 7 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} + \beta_{7} - 3 \beta_{8} - 4 \beta_{9} + 3 \beta_{10} ) q^{39} + ( 20 \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 4 \beta_{10} - \beta_{11} ) q^{41} + ( 34 + 16 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} + \beta_{6} - 8 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{43} + ( 103 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 19 \beta_{4} - 3 \beta_{5} - 10 \beta_{6} - 5 \beta_{7} + 2 \beta_{8} - 7 \beta_{9} - 7 \beta_{10} + 3 \beta_{11} ) q^{45} + ( -40 + 16 \beta_{1} - 5 \beta_{2} - \beta_{3} - 6 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} + 6 \beta_{9} + 4 \beta_{10} ) q^{47} + ( 130 + 20 \beta_{1} + \beta_{2} + 4 \beta_{3} - 18 \beta_{4} - 5 \beta_{5} - \beta_{6} + 7 \beta_{7} - 11 \beta_{8} - 5 \beta_{9} + 7 \beta_{10} - 2 \beta_{11} ) q^{49} + ( 31 + 12 \beta_{1} - \beta_{2} - 4 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{51} + ( -52 - 8 \beta_{1} + \beta_{2} + 9 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} - \beta_{10} + \beta_{11} ) q^{53} + ( 1 + 39 \beta_{1} - 11 \beta_{4} - 7 \beta_{5} - 11 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} - 8 \beta_{9} - 5 \beta_{10} + 2 \beta_{11} ) q^{55} + ( -92 + 14 \beta_{1} - 7 \beta_{2} - 10 \beta_{3} + 11 \beta_{4} + 9 \beta_{6} + 2 \beta_{7} + 9 \beta_{8} + 10 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{57} + ( 100 + 10 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 10 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - \beta_{9} - 8 \beta_{10} - 5 \beta_{11} ) q^{59} + ( 115 - 18 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 8 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} - 6 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{61} + ( -375 + 24 \beta_{1} - 9 \beta_{3} + 42 \beta_{4} + 13 \beta_{5} + 22 \beta_{6} - 3 \beta_{7} + 10 \beta_{8} + 11 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} ) q^{63} + ( 62 + 2 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} - 17 \beta_{4} + 6 \beta_{5} + 13 \beta_{6} + 6 \beta_{8} - 10 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{65} + ( 133 + 6 \beta_{2} + 5 \beta_{3} + 4 \beta_{4} - \beta_{5} + 5 \beta_{7} - 2 \beta_{8} + 11 \beta_{9} - 2 \beta_{10} - 7 \beta_{11} ) q^{67} + ( 55 - 74 \beta_{1} + 2 \beta_{2} - \beta_{3} - 20 \beta_{4} - 2 \beta_{5} - \beta_{6} - 4 \beta_{7} + 8 \beta_{8} - 6 \beta_{9} - 4 \beta_{10} - 12 \beta_{11} ) q^{69} + ( -278 + 34 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + 22 \beta_{4} - 6 \beta_{6} - 4 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} - 8 \beta_{10} ) q^{71} + ( 34 + 26 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} + 18 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} + 7 \beta_{10} + 11 \beta_{11} ) q^{73} + ( 516 + 62 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 43 \beta_{4} - 4 \beta_{5} - 11 \beta_{6} - \beta_{8} - 17 \beta_{9} + 3 \beta_{10} + 11 \beta_{11} ) q^{75} + ( -59 - 6 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 20 \beta_{4} + 7 \beta_{6} + 14 \beta_{7} - 10 \beta_{8} + 12 \beta_{9} + 16 \beta_{10} - 4 \beta_{11} ) q^{77} + ( -155 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 12 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} - 4 \beta_{10} - 8 \beta_{11} ) q^{79} + ( 262 + 18 \beta_{1} + 10 \beta_{2} + 26 \beta_{3} - 33 \beta_{4} + 7 \beta_{5} - 10 \beta_{6} - 7 \beta_{7} - 20 \beta_{8} - 7 \beta_{9} - 8 \beta_{10} ) q^{81} + ( 309 + 34 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} - 14 \beta_{4} - 16 \beta_{5} - 14 \beta_{6} + 15 \beta_{7} - 6 \beta_{8} + 10 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} ) q^{83} + ( 75 - 28 \beta_{1} + 7 \beta_{2} - 11 \beta_{3} - 2 \beta_{4} + 14 \beta_{5} + 19 \beta_{6} + 4 \beta_{8} - 12 \beta_{9} + 3 \beta_{10} - \beta_{11} ) q^{85} + ( -29 - 29 \beta_{1} ) q^{87} + ( 54 - 40 \beta_{1} + 14 \beta_{2} - \beta_{3} + 8 \beta_{4} + \beta_{5} - 6 \beta_{6} - 11 \beta_{7} + \beta_{8} - \beta_{9} - 6 \beta_{10} + \beta_{11} ) q^{89} + ( 405 + 12 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} - 10 \beta_{4} + 14 \beta_{5} + 8 \beta_{6} - 11 \beta_{7} + 18 \beta_{8} - 12 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{91} + ( 260 + 24 \beta_{1} - \beta_{2} + 18 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} + \beta_{8} + 13 \beta_{9} - 11 \beta_{10} + 12 \beta_{11} ) q^{93} + ( -232 + 3 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} + 15 \beta_{6} + 7 \beta_{7} - 10 \beta_{8} - 17 \beta_{9} + 24 \beta_{10} - 3 \beta_{11} ) q^{95} + ( 72 - 46 \beta_{1} - 13 \beta_{2} - 11 \beta_{3} - 12 \beta_{4} - 13 \beta_{5} - 4 \beta_{6} - 11 \beta_{7} + 23 \beta_{8} + 7 \beta_{9} - 9 \beta_{10} - 2 \beta_{11} ) q^{97} + ( 567 + 117 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} - 33 \beta_{4} - 26 \beta_{6} - 10 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 23 \beta_{10} + 11 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9} + O(q^{10}) \) \( 12 q + 14 q^{3} + 10 q^{5} - 44 q^{7} + 130 q^{9} + 46 q^{11} + 34 q^{13} + 50 q^{15} + 36 q^{17} + 148 q^{19} + 92 q^{21} - 328 q^{23} + 486 q^{25} + 326 q^{27} - 348 q^{29} - 374 q^{31} + 710 q^{33} + 204 q^{35} + 340 q^{37} + 122 q^{39} + 32 q^{41} + 462 q^{43} + 1132 q^{45} - 434 q^{47} + 1508 q^{49} + 440 q^{51} - 610 q^{53} - 46 q^{55} - 932 q^{57} + 1240 q^{59} + 1228 q^{61} - 4240 q^{63} + 730 q^{65} + 1672 q^{67} + 528 q^{69} - 3220 q^{71} + 564 q^{73} + 6032 q^{75} - 644 q^{77} - 1862 q^{79} + 3040 q^{81} + 3736 q^{83} + 808 q^{85} - 406 q^{87} + 584 q^{89} + 4844 q^{91} + 3226 q^{93} - 2844 q^{95} + 904 q^{97} + 6832 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + 7106592 x^{4} - 7979328 x^{3} - 38912400 x^{2} + 1083456 x + 14751072\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-8262965802853 \nu^{11} + 902081932283552 \nu^{10} + 2346639969921367 \nu^{9} - 180796617966497194 \nu^{8} - 137831479565874128 \nu^{7} + 11676011560120460220 \nu^{6} + 2999195858668965360 \nu^{5} - 285345134208428931464 \nu^{4} - 48747936398114755248 \nu^{3} + 2144506383015197261040 \nu^{2} + 1372623293185654194000 \nu - 130880883178269802368\)\()/ 18803225825160531840 \)
\(\beta_{3}\)\(=\)\((\)\(-9923410960919 \nu^{11} - 37676872998341 \nu^{10} + 1932206317362965 \nu^{9} + 6266643552994801 \nu^{8} - 122407881172320082 \nu^{7} - 383841222389299668 \nu^{6} + 2864020741416137100 \nu^{5} + 9573935910987249320 \nu^{4} - 17382422610327413592 \nu^{3} - 84417525336207643200 \nu^{2} - 75826481215462680096 \nu - 24411272474786102064\)\()/ 1880322582516053184 \)
\(\beta_{4}\)\(=\)\((\)\(31658966007779 \nu^{11} + 107796402154664 \nu^{10} - 6304114806233345 \nu^{9} - 17621476139832082 \nu^{8} + 414949071090163264 \nu^{7} + 1060546087442001084 \nu^{6} - 10565197552015638384 \nu^{5} - 25927326058279232456 \nu^{4} + 88907411951978970000 \nu^{3} + 232114178944772669808 \nu^{2} - 35995190424411812400 \nu - 117083380965640227072\)\()/ 3760645165032106368 \)
\(\beta_{5}\)\(=\)\((\)\(79714148233592 \nu^{11} + 244066653715127 \nu^{10} - 15567651816457448 \nu^{9} - 40189407869822629 \nu^{8} + 978773849787088462 \nu^{7} + 2570196807407968200 \nu^{6} - 22288763754574032900 \nu^{5} - 69652022198234858144 \nu^{4} + 116862826758660754872 \nu^{3} + 708488448739440793200 \nu^{2} + 748881004783427192880 \nu - 370754794707176290608\)\()/ 9401612912580265920 \)
\(\beta_{6}\)\(=\)\((\)\(-17168595976539 \nu^{11} - 61050049383782 \nu^{10} + 3389509146986425 \nu^{9} + 10051587748607228 \nu^{8} - 219921611144934476 \nu^{7} - 609409510740200140 \nu^{6} + 5431079678282637528 \nu^{5} + 15025065960084577032 \nu^{4} - 41224085724211265728 \nu^{3} - 134903291594073354192 \nu^{2} - 38552590668837849264 \nu + 69134902374085319520\)\()/ 1253548388344035456 \)
\(\beta_{7}\)\(=\)\((\)\(29213196068751 \nu^{11} + 128904048786731 \nu^{10} - 5831092367414829 \nu^{9} - 22362014136328147 \nu^{8} + 388935303766329166 \nu^{7} + 1392617848807915940 \nu^{6} - 10179601450871767060 \nu^{5} - 34723972146274052712 \nu^{4} + 90319284922766926216 \nu^{3} + 301522966561976516960 \nu^{2} - 20376639934106823360 \nu - 138387215003650031664\)\()/ 1044623656953362880 \)
\(\beta_{8}\)\(=\)\((\)\(-558861445914547 \nu^{11} - 1443129095982772 \nu^{10} + 111817069614277393 \nu^{9} + 219758966810908694 \nu^{8} - 7377811995239846072 \nu^{7} - 12863573580235804380 \nu^{6} + 188854338533045876160 \nu^{5} + 313925490878338825864 \nu^{4} - 1612551093240424302192 \nu^{3} - 2884904342187036830640 \nu^{2} + 1115638517533144322160 \nu + 388242946812251435328\)\()/ 18803225825160531840 \)
\(\beta_{9}\)\(=\)\((\)\(-717031054281463 \nu^{11} - 2368265974363168 \nu^{10} + 143279267639749357 \nu^{9} + 387228589046785106 \nu^{8} - 9487533645764059328 \nu^{7} - 23562360702441730860 \nu^{6} + 244455104103922443120 \nu^{5} + 589736302397573718376 \nu^{4} - 2097932237598579735888 \nu^{3} - 5427714657682065409200 \nu^{2} + 722997879981424302960 \nu + 2643687924078247535232\)\()/ 18803225825160531840 \)
\(\beta_{10}\)\(=\)\((\)\(520026572887418 \nu^{11} + 1450712508774743 \nu^{10} - 104058321231612902 \nu^{9} - 226853177728184641 \nu^{8} + 6873952630563023158 \nu^{7} + 13506476427425555040 \nu^{6} - 175903275840468652020 \nu^{5} - 335334630008995508816 \nu^{4} + 1476510308933712790968 \nu^{3} + 3153902333494178895120 \nu^{2} - 329947139403080685360 \nu - 1250532375411231023472\)\()/ 9401612912580265920 \)
\(\beta_{11}\)\(=\)\((\)\(550701841585907 \nu^{11} + 1927048190173762 \nu^{10} - 110012287537477233 \nu^{9} - 318934469441125344 \nu^{8} + 7290092592720633012 \nu^{7} + 19449207468102971500 \nu^{6} - 187857599434843482120 \nu^{5} - 483056994237602318984 \nu^{4} + 1600970623934459622112 \nu^{3} + 4328285506266624032400 \nu^{2} - 276545945273205263760 \nu - 1718936033766509266848\)\()/ 6267741941720177280 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} - \beta_{4} + \beta_{3} + 37\)
\(\nu^{3}\)\(=\)\(3 \beta_{11} - 3 \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + 3 \beta_{6} + 6 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 65 \beta_{1} - 34\)
\(\nu^{4}\)\(=\)\(-12 \beta_{11} + 4 \beta_{10} - 11 \beta_{9} - 24 \beta_{8} + \beta_{7} - 97 \beta_{6} + 7 \beta_{5} - 132 \beta_{4} + 113 \beta_{3} + 18 \beta_{2} - 84 \beta_{1} + 2524\)
\(\nu^{5}\)\(=\)\(403 \beta_{11} - 335 \beta_{10} + 208 \beta_{9} + 215 \beta_{8} - 218 \beta_{7} + 447 \beta_{6} - 43 \beta_{5} + 819 \beta_{4} - 568 \beta_{3} - 304 \beta_{2} + 5316 \beta_{1} - 6939\)
\(\nu^{6}\)\(=\)\(-2178 \beta_{11} + 908 \beta_{10} - 2191 \beta_{9} - 3382 \beta_{8} + 199 \beta_{7} - 9563 \beta_{6} + 1041 \beta_{5} - 14650 \beta_{4} + 12069 \beta_{3} + 2886 \beta_{2} - 16672 \beta_{1} + 216826\)
\(\nu^{7}\)\(=\)\(45353 \beta_{11} - 34219 \beta_{10} + 27832 \beta_{9} + 30467 \beta_{8} - 20172 \beta_{7} + 58231 \beta_{6} - 8569 \beta_{5} + 102333 \beta_{4} - 81610 \beta_{3} - 38376 \beta_{2} + 492164 \beta_{1} - 1017961\)
\(\nu^{8}\)\(=\)\(-305376 \beta_{11} + 148482 \beta_{10} - 300869 \beta_{9} - 401240 \beta_{8} + 38209 \beta_{7} - 990991 \beta_{6} + 123045 \beta_{5} - 1595212 \beta_{4} + 1301983 \beta_{3} + 373750 \beta_{2} - 2469224 \beta_{1} + 20994312\)
\(\nu^{9}\)\(=\)\(4965951 \beta_{11} - 3534029 \beta_{10} + 3361826 \beta_{9} + 3819385 \beta_{8} - 1875142 \beta_{7} + 7243335 \beta_{6} - 1203425 \beta_{5} + 12496687 \beta_{4} - 10518512 \beta_{3} - 4554764 \beta_{2} + 49187864 \beta_{1} - 134215891\)
\(\nu^{10}\)\(=\)\(-38971046 \beta_{11} + 20764400 \beta_{10} - 36942863 \beta_{9} - 45988898 \beta_{8} + 6147163 \beta_{7} - 106364455 \beta_{6} + 13991589 \beta_{5} - 174987162 \beta_{4} + 142928305 \beta_{3} + 45318418 \beta_{2} - 326796968 \beta_{1} + 2172946166\)
\(\nu^{11}\)\(=\)\(546331693 \beta_{11} - 373781107 \beta_{10} + 393952724 \beta_{9} + 457049227 \beta_{8} - 181642712 \beta_{7} + 878676335 \beta_{6} - 149963201 \beta_{5} + 1501643573 \beta_{4} - 1287046938 \beta_{3} - 528029080 \beta_{2} + 5164547864 \beta_{1} - 16780997049\)

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
−10.7228
−6.97229
−6.04158
−3.34435
−2.71183
−0.676012
0.613335
4.35906
5.24549
5.28045
8.22268
8.74783
0 −9.72277 0 6.27199 0 −31.0130 0 67.5323 0
1.2 0 −5.97229 0 11.6005 0 −8.29658 0 8.66827 0
1.3 0 −5.04158 0 −10.9552 0 1.11655 0 −1.58244 0
1.4 0 −2.34435 0 −11.5573 0 −8.54075 0 −21.5040 0
1.5 0 −1.71183 0 −8.28780 0 9.40926 0 −24.0696 0
1.6 0 0.323988 0 5.78413 0 34.7780 0 −26.8950 0
1.7 0 1.61334 0 10.5322 0 −13.7418 0 −24.3972 0
1.8 0 5.35906 0 20.7045 0 26.2041 0 1.71955 0
1.9 0 6.24549 0 −5.77776 0 −10.3273 0 12.0061 0
1.10 0 6.28045 0 −18.9942 0 −32.3083 0 12.4440 0
1.11 0 9.22268 0 −9.91795 0 18.1458 0 58.0578 0
1.12 0 9.74783 0 20.5968 0 −29.4259 0 68.0201 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(29\) \(1\)

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 1856.4.a.bl 12
4.b odd 2 1 1856.4.a.bj 12
8.b even 2 1 928.4.a.h 12
8.d odd 2 1 928.4.a.j yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.h 12 8.b even 2 1
928.4.a.j yes 12 8.d odd 2 1
1856.4.a.bj 12 4.b odd 2 1
1856.4.a.bl 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\(T_{3}^{12} - \cdots\)
\(21\!\cdots\!00\)\( \)">\(T_{5}^{12} - \cdots\)
\(12\!\cdots\!28\)\( T_{7}^{2} + \)\(31\!\cdots\!96\)\( T_{7} - \)\(51\!\cdots\!32\)\( \)">\(T_{7}^{12} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( -11605016 + 34164648 T + 11333891 T^{2} - 18457158 T^{3} - 3164363 T^{4} + 2483536 T^{5} + 113150 T^{6} - 122236 T^{7} + 2402 T^{8} + 2360 T^{9} - 129 T^{10} - 14 T^{11} + T^{12} \)
$5$ \( 2158837575300 + 31718775060 T - 161375255195 T^{2} - 3397758554 T^{3} + 4379523877 T^{4} + 112129888 T^{5} - 54452022 T^{6} - 1372684 T^{7} + 331222 T^{8} + 6652 T^{9} - 943 T^{10} - 10 T^{11} + T^{12} \)
$7$ \( -51510514450432 + 31912018231296 T + 12388290966528 T^{2} + 520863366144 T^{3} - 169954399488 T^{4} - 15344836096 T^{5} + 376753216 T^{6} + 69463360 T^{7} + 566608 T^{8} - 100512 T^{9} - 1844 T^{10} + 44 T^{11} + T^{12} \)
$11$ \( -516490488078539224 + 62716505047302600 T + 4592861085597371 T^{2} - 640006085467158 T^{3} - 518511242587 T^{4} + 1492871753216 T^{5} - 18496706050 T^{6} - 1316717388 T^{7} + 22755250 T^{8} + 444072 T^{9} - 8761 T^{10} - 46 T^{11} + T^{12} \)
$13$ \( 4995316664264132 - 479720007227277036 T - 55790858317726395 T^{2} + 477279651509198 T^{3} + 177609138494565 T^{4} + 1092730661504 T^{5} - 184186318390 T^{6} - 1323523804 T^{7} + 80964694 T^{8} + 399772 T^{9} - 15247 T^{10} - 34 T^{11} + T^{12} \)
$17$ \( \)\(95\!\cdots\!48\)\( + 85873286390291456000 T - 3004330432318149632 T^{2} - 123097183801715712 T^{3} + 3101831266450688 T^{4} + 63189267350528 T^{5} - 1431659221312 T^{6} - 13929236928 T^{7} + 312352752 T^{8} + 1230368 T^{9} - 29740 T^{10} - 36 T^{11} + T^{12} \)
$19$ \( -\)\(91\!\cdots\!96\)\( + \)\(52\!\cdots\!56\)\( T + 4321526200801799168 T^{2} - 1191139605109548032 T^{3} + 9477606236754176 T^{4} + 520372558127104 T^{5} - 4697187804288 T^{6} - 90877248128 T^{7} + 715486080 T^{8} + 6399872 T^{9} - 45604 T^{10} - 148 T^{11} + T^{12} \)
$23$ \( -\)\(11\!\cdots\!16\)\( + \)\(11\!\cdots\!80\)\( T + 76555880381934954496 T^{2} - 1078761449437943808 T^{3} - 79476527286929152 T^{4} + 95192693214208 T^{5} + 25842793604352 T^{6} + 194741802368 T^{7} - 1248426896 T^{8} - 17563456 T^{9} - 20944 T^{10} + 328 T^{11} + T^{12} \)
$29$ \( ( 29 + T )^{12} \)
$31$ \( -\)\(25\!\cdots\!00\)\( + \)\(46\!\cdots\!00\)\( T + \)\(89\!\cdots\!95\)\( T^{2} + 33505525059912160894 T^{3} - 2437417651947684651 T^{4} - 20469268860656128 T^{5} + 124163171686270 T^{6} + 1639451996700 T^{7} - 200262974 T^{8} - 44066232 T^{9} - 79841 T^{10} + 374 T^{11} + T^{12} \)
$37$ \( \)\(54\!\cdots\!16\)\( + \)\(25\!\cdots\!96\)\( T + \)\(24\!\cdots\!24\)\( T^{2} - \)\(39\!\cdots\!76\)\( T^{3} + 16160457050850983936 T^{4} + 1536653994818404352 T^{5} - 3076717038281728 T^{6} - 22447507940352 T^{7} + 56900791552 T^{8} + 143712704 T^{9} - 401596 T^{10} - 340 T^{11} + T^{12} \)
$41$ \( \)\(67\!\cdots\!16\)\( + \)\(21\!\cdots\!84\)\( T - \)\(90\!\cdots\!12\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!12\)\( T^{4} + 340505162530304 T^{5} - 5222678753273600 T^{6} - 2025728971008 T^{7} + 75968898976 T^{8} + 16725216 T^{9} - 463816 T^{10} - 32 T^{11} + T^{12} \)
$43$ \( -\)\(12\!\cdots\!64\)\( + \)\(39\!\cdots\!56\)\( T - \)\(72\!\cdots\!41\)\( T^{2} - \)\(16\!\cdots\!94\)\( T^{3} + \)\(38\!\cdots\!69\)\( T^{4} + 5644338007435587520 T^{5} - 11643696529329890 T^{6} - 63274922588556 T^{7} + 132549467954 T^{8} + 287883976 T^{9} - 614857 T^{10} - 462 T^{11} + T^{12} \)
$47$ \( \)\(46\!\cdots\!32\)\( + \)\(72\!\cdots\!52\)\( T - \)\(13\!\cdots\!77\)\( T^{2} - \)\(99\!\cdots\!10\)\( T^{3} + \)\(15\!\cdots\!09\)\( T^{4} - 1442719608846441040 T^{5} - 25170746288200658 T^{6} + 39668008708900 T^{7} + 186468442770 T^{8} - 243958168 T^{9} - 698145 T^{10} + 434 T^{11} + T^{12} \)
$53$ \( \)\(10\!\cdots\!36\)\( + \)\(27\!\cdots\!96\)\( T + \)\(24\!\cdots\!77\)\( T^{2} + \)\(47\!\cdots\!22\)\( T^{3} - \)\(42\!\cdots\!11\)\( T^{4} - 20851488122583052624 T^{5} + 6867863031872378 T^{6} + 185271827174972 T^{7} + 170832479222 T^{8} - 590060308 T^{9} - 820807 T^{10} + 610 T^{11} + T^{12} \)
$59$ \( \)\(40\!\cdots\!72\)\( - \)\(11\!\cdots\!44\)\( T + \)\(89\!\cdots\!20\)\( T^{2} - \)\(20\!\cdots\!56\)\( T^{3} - \)\(12\!\cdots\!84\)\( T^{4} + 92263728210757243904 T^{5} + 38086838770582784 T^{6} - 631275557056640 T^{7} + 204506175856 T^{8} + 1533363776 T^{9} - 970544 T^{10} - 1240 T^{11} + T^{12} \)
$61$ \( \)\(33\!\cdots\!64\)\( + \)\(53\!\cdots\!40\)\( T - \)\(38\!\cdots\!92\)\( T^{2} - \)\(95\!\cdots\!64\)\( T^{3} - \)\(37\!\cdots\!36\)\( T^{4} + 3878058576863107584 T^{5} + 13292340597334976 T^{6} - 76396559791680 T^{7} - 73582246352 T^{8} + 545024000 T^{9} - 120060 T^{10} - 1228 T^{11} + T^{12} \)
$67$ \( \)\(31\!\cdots\!04\)\( - \)\(58\!\cdots\!68\)\( T + \)\(40\!\cdots\!64\)\( T^{2} + \)\(35\!\cdots\!00\)\( T^{3} - \)\(76\!\cdots\!60\)\( T^{4} + 93925447892353548288 T^{5} + 396453883356618752 T^{6} - 841887951278080 T^{7} - 520766044672 T^{8} + 2160883456 T^{9} - 544688 T^{10} - 1672 T^{11} + T^{12} \)
$71$ \( -\)\(50\!\cdots\!52\)\( - \)\(28\!\cdots\!00\)\( T - \)\(30\!\cdots\!56\)\( T^{2} - \)\(61\!\cdots\!20\)\( T^{3} + \)\(60\!\cdots\!68\)\( T^{4} + \)\(32\!\cdots\!80\)\( T^{5} + 337913146660160384 T^{6} - 1038646900741504 T^{7} - 2593843746912 T^{8} - 750778624 T^{9} + 2949212 T^{10} + 3220 T^{11} + T^{12} \)
$73$ \( -\)\(76\!\cdots\!00\)\( + \)\(13\!\cdots\!80\)\( T + \)\(87\!\cdots\!40\)\( T^{2} - \)\(49\!\cdots\!68\)\( T^{3} - \)\(81\!\cdots\!52\)\( T^{4} + \)\(49\!\cdots\!36\)\( T^{5} - 525861863156498432 T^{6} - 1431417292972032 T^{7} + 2085268635136 T^{8} + 1540375808 T^{9} - 2591580 T^{10} - 564 T^{11} + T^{12} \)
$79$ \( \)\(43\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T - \)\(23\!\cdots\!85\)\( T^{2} - \)\(14\!\cdots\!22\)\( T^{3} - \)\(47\!\cdots\!91\)\( T^{4} + 29296288484232441424 T^{5} + 165102425396794558 T^{6} + 72873547051532 T^{7} - 845849649166 T^{8} - 1414211176 T^{9} + 241975 T^{10} + 1862 T^{11} + T^{12} \)
$83$ \( -\)\(49\!\cdots\!00\)\( + \)\(29\!\cdots\!40\)\( T + \)\(20\!\cdots\!00\)\( T^{2} + \)\(22\!\cdots\!96\)\( T^{3} + \)\(44\!\cdots\!16\)\( T^{4} - \)\(48\!\cdots\!08\)\( T^{5} + 338154832515994880 T^{6} + 2969818568440704 T^{7} - 5930926098320 T^{8} + 2091806784 T^{9} + 3723312 T^{10} - 3736 T^{11} + T^{12} \)
$89$ \( \)\(74\!\cdots\!04\)\( - \)\(59\!\cdots\!04\)\( T - \)\(87\!\cdots\!44\)\( T^{2} + \)\(28\!\cdots\!04\)\( T^{3} + \)\(35\!\cdots\!96\)\( T^{4} + \)\(28\!\cdots\!24\)\( T^{5} - 1980499779154948864 T^{6} - 1375771433576320 T^{7} + 4161172740128 T^{8} + 1622143552 T^{9} - 3499592 T^{10} - 584 T^{11} + T^{12} \)
$97$ \( -\)\(53\!\cdots\!72\)\( + \)\(17\!\cdots\!28\)\( T - \)\(55\!\cdots\!88\)\( T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(43\!\cdots\!44\)\( T^{4} + \)\(43\!\cdots\!80\)\( T^{5} - 11343333354442469632 T^{6} - 6660503078015872 T^{7} + 12208195556128 T^{8} + 4230879232 T^{9} - 5779016 T^{10} - 904 T^{11} + T^{12} \)
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