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Properties

Label 735.4.a.o.1.2
Level $735$
Weight $4$
Character 735.1
Self dual yes
Analytic conductor $43.366$
Analytic rank $0$
Dimension $2$
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] = [735,4,Mod(1,735)] 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("735.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,6,2,-10,-6,0,-18,18,10,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(43.3664038542\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 735.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.82843 q^{2} +3.00000 q^{3} -4.65685 q^{4} -5.00000 q^{5} +5.48528 q^{6} -23.1421 q^{8} +9.00000 q^{9} -9.14214 q^{10} -64.5685 q^{11} -13.9706 q^{12} +32.3431 q^{13} -15.0000 q^{15} -5.05887 q^{16} +56.3431 q^{17} +16.4558 q^{18} +2.74517 q^{19} +23.2843 q^{20} -118.059 q^{22} +88.1665 q^{23} -69.4264 q^{24} +25.0000 q^{25} +59.1371 q^{26} +27.0000 q^{27} +246.735 q^{29} -27.4264 q^{30} +110.912 q^{31} +175.887 q^{32} -193.706 q^{33} +103.019 q^{34} -41.9117 q^{36} +120.676 q^{37} +5.01934 q^{38} +97.0294 q^{39} +115.711 q^{40} +176.274 q^{41} -443.362 q^{43} +300.686 q^{44} -45.0000 q^{45} +161.206 q^{46} +345.206 q^{47} -15.1766 q^{48} +45.7107 q^{50} +169.029 q^{51} -150.617 q^{52} +260.981 q^{53} +49.3675 q^{54} +322.843 q^{55} +8.23550 q^{57} +451.137 q^{58} -628.999 q^{59} +69.8528 q^{60} +115.206 q^{61} +202.794 q^{62} +362.068 q^{64} -161.716 q^{65} -354.177 q^{66} -951.480 q^{67} -262.382 q^{68} +264.500 q^{69} +356.264 q^{71} -208.279 q^{72} +656.754 q^{73} +220.648 q^{74} +75.0000 q^{75} -12.7838 q^{76} +177.411 q^{78} +440.195 q^{79} +25.2944 q^{80} +81.0000 q^{81} +322.304 q^{82} +54.4121 q^{83} -281.716 q^{85} -810.656 q^{86} +740.205 q^{87} +1494.25 q^{88} +1018.78 q^{89} -82.2792 q^{90} -410.579 q^{92} +332.735 q^{93} +631.184 q^{94} -13.7258 q^{95} +527.662 q^{96} +724.108 q^{97} -581.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 6 q^{6} - 18 q^{8} + 18 q^{9} + 10 q^{10} - 16 q^{11} + 6 q^{12} + 76 q^{13} - 30 q^{15} - 78 q^{16} + 124 q^{17} - 18 q^{18} + 96 q^{19} - 10 q^{20} - 304 q^{22}+ \cdots - 144 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82843 0.646447 0.323223 − 0.946323i \(-0.395234\pi\)
0.323223 + 0.946323i \(0.395234\pi\)
\(3\) 3.00000 0.577350
\(4\) −4.65685 −0.582107
\(5\) −5.00000 −0.447214
\(6\) 5.48528 0.373226
\(7\) 0 0
\(8\) −23.1421 −1.02275
\(9\) 9.00000 0.333333
\(10\) −9.14214 −0.289100
\(11\) −64.5685 −1.76983 −0.884916 − 0.465751i \(-0.845784\pi\)
−0.884916 + 0.465751i \(0.845784\pi\)
\(12\) −13.9706 −0.336080
\(13\) 32.3431 0.690029 0.345014 − 0.938597i \(-0.387874\pi\)
0.345014 + 0.938597i \(0.387874\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) −5.05887 −0.0790449
\(17\) 56.3431 0.803836 0.401918 − 0.915676i \(-0.368344\pi\)
0.401918 + 0.915676i \(0.368344\pi\)
\(18\) 16.4558 0.215482
\(19\) 2.74517 0.0331465 0.0165733 − 0.999863i \(-0.494724\pi\)
0.0165733 + 0.999863i \(0.494724\pi\)
\(20\) 23.2843 0.260326
\(21\) 0 0
\(22\) −118.059 −1.14410
\(23\) 88.1665 0.799304 0.399652 − 0.916667i \(-0.369131\pi\)
0.399652 + 0.916667i \(0.369131\pi\)
\(24\) −69.4264 −0.590484
\(25\) 25.0000 0.200000
\(26\) 59.1371 0.446067
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 246.735 1.57992 0.789958 − 0.613161i \(-0.210102\pi\)
0.789958 + 0.613161i \(0.210102\pi\)
\(30\) −27.4264 −0.166912
\(31\) 110.912 0.642591 0.321296 − 0.946979i \(-0.395882\pi\)
0.321296 + 0.946979i \(0.395882\pi\)
\(32\) 175.887 0.971649
\(33\) −193.706 −1.02181
\(34\) 103.019 0.519637
\(35\) 0 0
\(36\) −41.9117 −0.194036
\(37\) 120.676 0.536190 0.268095 − 0.963392i \(-0.413606\pi\)
0.268095 + 0.963392i \(0.413606\pi\)
\(38\) 5.01934 0.0214275
\(39\) 97.0294 0.398388
\(40\) 115.711 0.457387
\(41\) 176.274 0.671449 0.335724 − 0.941960i \(-0.391019\pi\)
0.335724 + 0.941960i \(0.391019\pi\)
\(42\) 0 0
\(43\) −443.362 −1.57238 −0.786188 − 0.617988i \(-0.787948\pi\)
−0.786188 + 0.617988i \(0.787948\pi\)
\(44\) 300.686 1.03023
\(45\) −45.0000 −0.149071
\(46\) 161.206 0.516707
\(47\) 345.206 1.07135 0.535675 − 0.844424i \(-0.320057\pi\)
0.535675 + 0.844424i \(0.320057\pi\)
\(48\) −15.1766 −0.0456366
\(49\) 0 0
\(50\) 45.7107 0.129289
\(51\) 169.029 0.464095
\(52\) −150.617 −0.401670
\(53\) 260.981 0.676386 0.338193 − 0.941077i \(-0.390184\pi\)
0.338193 + 0.941077i \(0.390184\pi\)
\(54\) 49.3675 0.124409
\(55\) 322.843 0.791493
\(56\) 0 0
\(57\) 8.23550 0.0191372
\(58\) 451.137 1.02133
\(59\) −628.999 −1.38794 −0.693972 − 0.720002i \(-0.744141\pi\)
−0.693972 + 0.720002i \(0.744141\pi\)
\(60\) 69.8528 0.150299
\(61\) 115.206 0.241814 0.120907 − 0.992664i \(-0.461420\pi\)
0.120907 + 0.992664i \(0.461420\pi\)
\(62\) 202.794 0.415401
\(63\) 0 0
\(64\) 362.068 0.707164
\(65\) −161.716 −0.308590
\(66\) −354.177 −0.660547
\(67\) −951.480 −1.73495 −0.867476 − 0.497479i \(-0.834259\pi\)
−0.867476 + 0.497479i \(0.834259\pi\)
\(68\) −262.382 −0.467919
\(69\) 264.500 0.461478
\(70\) 0 0
\(71\) 356.264 0.595504 0.297752 − 0.954643i \(-0.403763\pi\)
0.297752 + 0.954643i \(0.403763\pi\)
\(72\) −208.279 −0.340916
\(73\) 656.754 1.05298 0.526488 − 0.850183i \(-0.323509\pi\)
0.526488 + 0.850183i \(0.323509\pi\)
\(74\) 220.648 0.346618
\(75\) 75.0000 0.115470
\(76\) −12.7838 −0.0192948
\(77\) 0 0
\(78\) 177.411 0.257537
\(79\) 440.195 0.626909 0.313455 − 0.949603i \(-0.398514\pi\)
0.313455 + 0.949603i \(0.398514\pi\)
\(80\) 25.2944 0.0353500
\(81\) 81.0000 0.111111
\(82\) 322.304 0.434056
\(83\) 54.4121 0.0719579 0.0359790 − 0.999353i \(-0.488545\pi\)
0.0359790 + 0.999353i \(0.488545\pi\)
\(84\) 0 0
\(85\) −281.716 −0.359487
\(86\) −810.656 −1.01646
\(87\) 740.205 0.912165
\(88\) 1494.25 1.81009
\(89\) 1018.78 1.21338 0.606690 − 0.794938i \(-0.292497\pi\)
0.606690 + 0.794938i \(0.292497\pi\)
\(90\) −82.2792 −0.0963666
\(91\) 0 0
\(92\) −410.579 −0.465280
\(93\) 332.735 0.371000
\(94\) 631.184 0.692571
\(95\) −13.7258 −0.0148236
\(96\) 527.662 0.560982
\(97\) 724.108 0.757959 0.378979 − 0.925405i \(-0.376275\pi\)
0.378979 + 0.925405i \(0.376275\pi\)
\(98\) 0 0
\(99\) −581.117 −0.589944
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 735.4.a.o.1.2 2
3.2 odd 2 2205.4.a.bb.1.1 2
7.6 odd 2 105.4.a.e.1.2 ✓ 2
21.20 even 2 315.4.a.k.1.1 2
28.27 even 2 1680.4.a.bo.1.2 2
35.13 even 4 525.4.d.l.274.2 4
35.27 even 4 525.4.d.l.274.3 4
35.34 odd 2 525.4.a.l.1.1 2
105.104 even 2 1575.4.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.2 ✓ 2 7.6 odd 2
315.4.a.k.1.1 2 21.20 even 2
525.4.a.l.1.1 2 35.34 odd 2
525.4.d.l.274.2 4 35.13 even 4
525.4.d.l.274.3 4 35.27 even 4
735.4.a.o.1.2 2 1.1 even 1 trivial
1575.4.a.q.1.2 2 105.104 even 2
1680.4.a.bo.1.2 2 28.27 even 2
2205.4.a.bb.1.1 2 3.2 odd 2