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Properties

Label 1936.4.a.bl.1.3
Level $1936$
Weight $4$
Character 1936.1
Self dual yes
Analytic conductor $114.228$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1936,4,Mod(1,1936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1936.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-6,0,-11,0,25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(114.227697771\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{37})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.15942\) of defining polynomial
Character \(\chi\) \(=\) 1936.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.54138 q^{3} -17.2669 q^{5} -9.56478 q^{7} -24.6241 q^{9} +6.48157 q^{13} -26.6148 q^{15} -94.5369 q^{17} -52.6159 q^{19} -14.7430 q^{21} -82.2579 q^{23} +173.145 q^{25} -79.5725 q^{27} -149.892 q^{29} -131.387 q^{31} +165.154 q^{35} -122.752 q^{37} +9.99057 q^{39} -190.098 q^{41} -320.348 q^{43} +425.182 q^{45} +441.172 q^{47} -251.515 q^{49} -145.717 q^{51} -123.572 q^{53} -81.1012 q^{57} +72.4923 q^{59} -248.668 q^{61} +235.525 q^{63} -111.916 q^{65} +128.926 q^{67} -126.791 q^{69} -992.858 q^{71} -516.964 q^{73} +266.883 q^{75} +574.689 q^{79} +542.200 q^{81} +188.076 q^{83} +1632.36 q^{85} -231.040 q^{87} -42.1254 q^{89} -61.9948 q^{91} -202.517 q^{93} +908.513 q^{95} -1285.62 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 11 q^{5} + 25 q^{7} - 62 q^{9} - 25 q^{13} - 2 q^{15} - 232 q^{17} + 154 q^{19} - 167 q^{21} + 6 q^{23} - 13 q^{25} + 144 q^{27} - 363 q^{29} - 37 q^{31} + 356 q^{35} + 93 q^{37} - 240 q^{39}+ \cdots - 1772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.54138 0.296639 0.148319 − 0.988939i \(-0.452614\pi\)
0.148319 + 0.988939i \(0.452614\pi\)
\(4\) 0 0
\(5\) −17.2669 −1.54440 −0.772198 − 0.635382i \(-0.780843\pi\)
−0.772198 + 0.635382i \(0.780843\pi\)
\(6\) 0 0
\(7\) −9.56478 −0.516450 −0.258225 − 0.966085i \(-0.583138\pi\)
−0.258225 + 0.966085i \(0.583138\pi\)
\(8\) 0 0
\(9\) −24.6241 −0.912005
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 6.48157 0.138282 0.0691409 − 0.997607i \(-0.477974\pi\)
0.0691409 + 0.997607i \(0.477974\pi\)
\(14\) 0 0
\(15\) −26.6148 −0.458128
\(16\) 0 0
\(17\) −94.5369 −1.34874 −0.674370 − 0.738394i \(-0.735585\pi\)
−0.674370 + 0.738394i \(0.735585\pi\)
\(18\) 0 0
\(19\) −52.6159 −0.635312 −0.317656 − 0.948206i \(-0.602896\pi\)
−0.317656 + 0.948206i \(0.602896\pi\)
\(20\) 0 0
\(21\) −14.7430 −0.153199
\(22\) 0 0
\(23\) −82.2579 −0.745737 −0.372868 − 0.927884i \(-0.621626\pi\)
−0.372868 + 0.927884i \(0.621626\pi\)
\(24\) 0 0
\(25\) 173.145 1.38516
\(26\) 0 0
\(27\) −79.5725 −0.567175
\(28\) 0 0
\(29\) −149.892 −0.959800 −0.479900 − 0.877323i \(-0.659327\pi\)
−0.479900 + 0.877323i \(0.659327\pi\)
\(30\) 0 0
\(31\) −131.387 −0.761218 −0.380609 − 0.924736i \(-0.624286\pi\)
−0.380609 + 0.924736i \(0.624286\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 165.154 0.797603
\(36\) 0 0
\(37\) −122.752 −0.545414 −0.272707 − 0.962097i \(-0.587919\pi\)
−0.272707 + 0.962097i \(0.587919\pi\)
\(38\) 0 0
\(39\) 9.99057 0.0410198
\(40\) 0 0
\(41\) −190.098 −0.724107 −0.362053 − 0.932157i \(-0.617924\pi\)
−0.362053 + 0.932157i \(0.617924\pi\)
\(42\) 0 0
\(43\) −320.348 −1.13611 −0.568054 − 0.822991i \(-0.692304\pi\)
−0.568054 + 0.822991i \(0.692304\pi\)
\(44\) 0 0
\(45\) 425.182 1.40850
\(46\) 0 0
\(47\) 441.172 1.36918 0.684591 − 0.728927i \(-0.259981\pi\)
0.684591 + 0.728927i \(0.259981\pi\)
\(48\) 0 0
\(49\) −251.515 −0.733280
\(50\) 0 0
\(51\) −145.717 −0.400089
\(52\) 0 0
\(53\) −123.572 −0.320262 −0.160131 − 0.987096i \(-0.551192\pi\)
−0.160131 + 0.987096i \(0.551192\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −81.1012 −0.188458
\(58\) 0 0
\(59\) 72.4923 0.159961 0.0799805 − 0.996796i \(-0.474514\pi\)
0.0799805 + 0.996796i \(0.474514\pi\)
\(60\) 0 0
\(61\) −248.668 −0.521945 −0.260972 − 0.965346i \(-0.584043\pi\)
−0.260972 + 0.965346i \(0.584043\pi\)
\(62\) 0 0
\(63\) 235.525 0.471005
\(64\) 0 0
\(65\) −111.916 −0.213562
\(66\) 0 0
\(67\) 128.926 0.235086 0.117543 − 0.993068i \(-0.462498\pi\)
0.117543 + 0.993068i \(0.462498\pi\)
\(68\) 0 0
\(69\) −126.791 −0.221215
\(70\) 0 0
\(71\) −992.858 −1.65959 −0.829793 − 0.558072i \(-0.811541\pi\)
−0.829793 + 0.558072i \(0.811541\pi\)
\(72\) 0 0
\(73\) −516.964 −0.828850 −0.414425 − 0.910083i \(-0.636017\pi\)
−0.414425 + 0.910083i \(0.636017\pi\)
\(74\) 0 0
\(75\) 266.883 0.410893
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 574.689 0.818450 0.409225 − 0.912433i \(-0.365799\pi\)
0.409225 + 0.912433i \(0.365799\pi\)
\(80\) 0 0
\(81\) 542.200 0.743759
\(82\) 0 0
\(83\) 188.076 0.248723 0.124362 − 0.992237i \(-0.460312\pi\)
0.124362 + 0.992237i \(0.460312\pi\)
\(84\) 0 0
\(85\) 1632.36 2.08299
\(86\) 0 0
\(87\) −231.040 −0.284714
\(88\) 0 0
\(89\) −42.1254 −0.0501717 −0.0250859 − 0.999685i \(-0.507986\pi\)
−0.0250859 + 0.999685i \(0.507986\pi\)
\(90\) 0 0
\(91\) −61.9948 −0.0714156
\(92\) 0 0
\(93\) −202.517 −0.225807
\(94\) 0 0
\(95\) 908.513 0.981173
\(96\) 0 0
\(97\) −1285.62 −1.34572 −0.672860 − 0.739770i \(-0.734934\pi\)
−0.672860 + 0.739770i \(0.734934\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1936.4.a.bl.1.3 4
4.3 odd 2 121.4.a.f.1.4 4
11.7 odd 10 176.4.m.c.49.1 8
11.8 odd 10 176.4.m.c.97.1 8
11.10 odd 2 1936.4.a.bk.1.3 4
12.11 even 2 1089.4.a.bh.1.1 4
44.3 odd 10 121.4.c.i.9.2 8
44.7 even 10 11.4.c.a.5.1 ✓ 8
44.15 odd 10 121.4.c.i.27.2 8
44.19 even 10 11.4.c.a.9.1 yes 8
44.27 odd 10 121.4.c.b.3.1 8
44.31 odd 10 121.4.c.b.81.1 8
44.35 even 10 121.4.c.h.81.2 8
44.39 even 10 121.4.c.h.3.2 8
44.43 even 2 121.4.a.g.1.1 4
132.95 odd 10 99.4.f.c.82.2 8
132.107 odd 10 99.4.f.c.64.2 8
132.131 odd 2 1089.4.a.y.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.c.a.5.1 ✓ 8 44.7 even 10
11.4.c.a.9.1 yes 8 44.19 even 10
99.4.f.c.64.2 8 132.107 odd 10
99.4.f.c.82.2 8 132.95 odd 10
121.4.a.f.1.4 4 4.3 odd 2
121.4.a.g.1.1 4 44.43 even 2
121.4.c.b.3.1 8 44.27 odd 10
121.4.c.b.81.1 8 44.31 odd 10
121.4.c.h.3.2 8 44.39 even 10
121.4.c.h.81.2 8 44.35 even 10
121.4.c.i.9.2 8 44.3 odd 10
121.4.c.i.27.2 8 44.15 odd 10
176.4.m.c.49.1 8 11.7 odd 10
176.4.m.c.97.1 8 11.8 odd 10
1089.4.a.y.1.4 4 132.131 odd 2
1089.4.a.bh.1.1 4 12.11 even 2
1936.4.a.bk.1.3 4 11.10 odd 2
1936.4.a.bl.1.3 4 1.1 even 1 trivial