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Properties

Label 16.50.a
Level $16$
Weight $50$
Character orbit 16.a
Rep. character $\chi_{16}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $6$
Sturm bound $100$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(100\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{50}(\Gamma_0(16))\).

Total New Old
Modular forms 101 25 76
Cusp forms 95 24 71
Eisenstein series 6 1 5

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim
\(+\)\(12\)
\(-\)\(12\)

Trace form

\( 24 q - 282429536480 q^{3} - 49765646857595760 q^{5} + 427551211265309052480 q^{7} + 1735383646546148183660344 q^{9} + O(q^{10}) \) \( 24 q - 282429536480 q^{3} - 49765646857595760 q^{5} + 427551211265309052480 q^{7} + 1735383646546148183660344 q^{9} - 5965094907798124174492320 q^{11} - 501253751084719188271650480 q^{13} + 20225530713250057332492499648 q^{15} - 900121517787726004390281925200 q^{17} - 41915945386446592589572282432608 q^{19} - 47567727177571645469136768177408 q^{21} + 2766640949271615676032435942025920 q^{23} + 60636935669043827108065668197043432 q^{25} - 88174877579641089173530708008655040 q^{27} - 47191916222304823315349937419838768 q^{29} - 1683203093341346965905552451853461248 q^{31} + 1917940087203527494563148236565266560 q^{33} - 118885527518577851494756181045537737344 q^{35} - 249177375413409342305630594967600509040 q^{37} + 400396872805334194976598975897740223424 q^{39} - 838650874744058404954556014177505595408 q^{41} - 20532807888317128723239974842284365216160 q^{43} - 46427221828995385816441379020023557910704 q^{45} + 224356340095433299777163722168038919689600 q^{47} + 564591431112221840524524727300126664999256 q^{49} + 691612958110646942007532278024172764024896 q^{51} + 545602615383253865861927196168593942468880 q^{53} - 4364714165285415916424717202159680403939264 q^{55} + 10568017961999516249593329151154386458037120 q^{57} + 29610164707137046069458355541835150857966304 q^{59} - 69495322972920651733058159941133978266937904 q^{61} - 1508971501658105460282302005402688969062080 q^{63} + 129244990271800956557083890869657384326565856 q^{65} - 471104668828923699791739939555781834745100000 q^{67} + 570845924559537635194686025932025733420695808 q^{69} + 422072076980405661322360577961914693630664768 q^{71} - 3584298389346694422349008713029944696296378640 q^{73} + 4335191886442514241896657831847540661673530592 q^{75} - 7879665243880756808836177155385413667527786240 q^{77} - 22059504135983791220262623661115557268745834880 q^{79} + 16008191081704580760365889791852065450486205272 q^{81} - 75163084390108773702138483906018911490660063840 q^{83} + 167768442509238389129648845677775700378519435808 q^{85} + 869189954217035911876668042671957506547742528960 q^{87} + 520337022736253157420047249484103985963577325680 q^{89} - 697835174304315737733514939539473537953683650688 q^{91} - 2231016057393575998042653196283028291618398458880 q^{93} - 2099932286238058228434915885679080872755995236928 q^{95} - 2898362097159276004644781181897349369214268314320 q^{97} - 7815754146023241126994189137050168117232873867808 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{50}^{\mathrm{new}}(\Gamma_0(16))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
16.50.a.a 16.a 1.a $2$ $243.306$ \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(0\) \(-281051075592\) \(83\!\cdots\!00\) \(43\!\cdots\!36\) $-$ $\mathrm{SU}(2)$ \(q+(-140525537796-\beta )q^{3}+\cdots\)
16.50.a.b 16.a 1.a $3$ $243.306$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(16203614388\) \(-10\!\cdots\!30\) \(12\!\cdots\!96\) $-$ $\mathrm{SU}(2)$ \(q+(5401204796+\beta _{1})q^{3}+(-33937672133946810+\cdots)q^{5}+\cdots\)
16.50.a.c 16.a 1.a $3$ $243.306$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(326954692404\) \(63\!\cdots\!50\) \(-50\!\cdots\!92\) $-$ $\mathrm{SU}(2)$ \(q+(108984897468+\beta _{1}-\beta _{2})q^{3}+\cdots\)
16.50.a.d 16.a 1.a $4$ $243.306$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-168739618320\) \(-50\!\cdots\!16\) \(-77\!\cdots\!60\) $-$ $\mathrm{SU}(2)$ \(q+(-42184904580+\beta _{1})q^{3}+\cdots\)
16.50.a.e 16.a 1.a $6$ $243.306$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-122200046808\) \(-11\!\cdots\!24\) \(18\!\cdots\!80\) $+$ $\mathrm{SU}(2)$ \(q+(-20366674468+\beta _{1})q^{3}+\cdots\)
16.50.a.f 16.a 1.a $6$ $243.306$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-53597102552\) \(24\!\cdots\!60\) \(45\!\cdots\!20\) $+$ $\mathrm{SU}(2)$ \(q+(-8932850425-\beta _{1})q^{3}+(4132476630741815+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{50}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{50}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{50}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{50}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{50}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{50}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)