Properties

Label 115.4.a.e.1.2
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $5$
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] = [115,4,Mod(1,115)] 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("115.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.49214\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.49214 q^{2} +9.02447 q^{3} -5.77352 q^{4} -5.00000 q^{5} -13.4658 q^{6} +4.33445 q^{7} +20.5520 q^{8} +54.4411 q^{9} +7.46070 q^{10} +48.1836 q^{11} -52.1029 q^{12} +53.0174 q^{13} -6.46761 q^{14} -45.1224 q^{15} +15.5216 q^{16} -61.0120 q^{17} -81.2338 q^{18} +12.0115 q^{19} +28.8676 q^{20} +39.1161 q^{21} -71.8968 q^{22} -23.0000 q^{23} +185.471 q^{24} +25.0000 q^{25} -79.1094 q^{26} +247.642 q^{27} -25.0250 q^{28} +198.213 q^{29} +67.3289 q^{30} -11.9661 q^{31} -187.577 q^{32} +434.832 q^{33} +91.0386 q^{34} -21.6723 q^{35} -314.317 q^{36} -230.018 q^{37} -17.9229 q^{38} +478.454 q^{39} -102.760 q^{40} -393.486 q^{41} -58.3668 q^{42} -167.496 q^{43} -278.189 q^{44} -272.206 q^{45} +34.3192 q^{46} +531.002 q^{47} +140.074 q^{48} -324.213 q^{49} -37.3035 q^{50} -550.602 q^{51} -306.097 q^{52} +373.570 q^{53} -369.516 q^{54} -240.918 q^{55} +89.0818 q^{56} +108.398 q^{57} -295.762 q^{58} -825.880 q^{59} +260.515 q^{60} -632.806 q^{61} +17.8551 q^{62} +235.972 q^{63} +155.718 q^{64} -265.087 q^{65} -648.830 q^{66} -38.7503 q^{67} +352.254 q^{68} -207.563 q^{69} +32.3381 q^{70} +1109.37 q^{71} +1118.88 q^{72} -446.171 q^{73} +343.219 q^{74} +225.612 q^{75} -69.3488 q^{76} +208.850 q^{77} -713.921 q^{78} +642.608 q^{79} -77.6081 q^{80} +764.926 q^{81} +587.137 q^{82} -292.575 q^{83} -225.838 q^{84} +305.060 q^{85} +249.927 q^{86} +1788.77 q^{87} +990.271 q^{88} -273.272 q^{89} +406.169 q^{90} +229.801 q^{91} +132.791 q^{92} -107.988 q^{93} -792.329 q^{94} -60.0577 q^{95} -1692.78 q^{96} -959.139 q^{97} +483.771 q^{98} +2623.17 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19}+ \cdots + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.49214 −0.527551 −0.263776 − 0.964584i \(-0.584968\pi\)
−0.263776 + 0.964584i \(0.584968\pi\)
\(3\) 9.02447 1.73676 0.868380 − 0.495899i \(-0.165161\pi\)
0.868380 + 0.495899i \(0.165161\pi\)
\(4\) −5.77352 −0.721689
\(5\) −5.00000 −0.447214
\(6\) −13.4658 −0.916231
\(7\) 4.33445 0.234038 0.117019 − 0.993130i \(-0.462666\pi\)
0.117019 + 0.993130i \(0.462666\pi\)
\(8\) 20.5520 0.908280
\(9\) 54.4411 2.01634
\(10\) 7.46070 0.235928
\(11\) 48.1836 1.32072 0.660360 − 0.750949i \(-0.270404\pi\)
0.660360 + 0.750949i \(0.270404\pi\)
\(12\) −52.1029 −1.25340
\(13\) 53.0174 1.13111 0.565553 − 0.824712i \(-0.308663\pi\)
0.565553 + 0.824712i \(0.308663\pi\)
\(14\) −6.46761 −0.123467
\(15\) −45.1224 −0.776703
\(16\) 15.5216 0.242525
\(17\) −61.0120 −0.870447 −0.435223 − 0.900323i \(-0.643331\pi\)
−0.435223 + 0.900323i \(0.643331\pi\)
\(18\) −81.2338 −1.06372
\(19\) 12.0115 0.145034 0.0725168 − 0.997367i \(-0.476897\pi\)
0.0725168 + 0.997367i \(0.476897\pi\)
\(20\) 28.8676 0.322749
\(21\) 39.1161 0.406469
\(22\) −71.8968 −0.696747
\(23\) −23.0000 −0.208514
\(24\) 185.471 1.57746
\(25\) 25.0000 0.200000
\(26\) −79.1094 −0.596716
\(27\) 247.642 1.76514
\(28\) −25.0250 −0.168903
\(29\) 198.213 1.26922 0.634609 − 0.772834i \(-0.281161\pi\)
0.634609 + 0.772834i \(0.281161\pi\)
\(30\) 67.3289 0.409751
\(31\) −11.9661 −0.0693284 −0.0346642 − 0.999399i \(-0.511036\pi\)
−0.0346642 + 0.999399i \(0.511036\pi\)
\(32\) −187.577 −1.03622
\(33\) 434.832 2.29377
\(34\) 91.0386 0.459205
\(35\) −21.6723 −0.104665
\(36\) −314.317 −1.45517
\(37\) −230.018 −1.02202 −0.511010 − 0.859575i \(-0.670728\pi\)
−0.511010 + 0.859575i \(0.670728\pi\)
\(38\) −17.9229 −0.0765126
\(39\) 478.454 1.96446
\(40\) −102.760 −0.406195
\(41\) −393.486 −1.49883 −0.749417 − 0.662098i \(-0.769666\pi\)
−0.749417 + 0.662098i \(0.769666\pi\)
\(42\) −58.3668 −0.214433
\(43\) −167.496 −0.594019 −0.297010 − 0.954874i \(-0.595989\pi\)
−0.297010 + 0.954874i \(0.595989\pi\)
\(44\) −278.189 −0.953149
\(45\) −272.206 −0.901734
\(46\) 34.3192 0.110002
\(47\) 531.002 1.64797 0.823984 − 0.566612i \(-0.191746\pi\)
0.823984 + 0.566612i \(0.191746\pi\)
\(48\) 140.074 0.421208
\(49\) −324.213 −0.945226
\(50\) −37.3035 −0.105510
\(51\) −550.602 −1.51176
\(52\) −306.097 −0.816307
\(53\) 373.570 0.968186 0.484093 − 0.875017i \(-0.339150\pi\)
0.484093 + 0.875017i \(0.339150\pi\)
\(54\) −369.516 −0.931200
\(55\) −240.918 −0.590644
\(56\) 89.0818 0.212572
\(57\) 108.398 0.251889
\(58\) −295.762 −0.669578
\(59\) −825.880 −1.82238 −0.911190 − 0.411987i \(-0.864835\pi\)
−0.911190 + 0.411987i \(0.864835\pi\)
\(60\) 260.515 0.560538
\(61\) −632.806 −1.32824 −0.664119 − 0.747627i \(-0.731193\pi\)
−0.664119 + 0.747627i \(0.731193\pi\)
\(62\) 17.8551 0.0365743
\(63\) 235.972 0.471900
\(64\) 155.718 0.304136
\(65\) −265.087 −0.505846
\(66\) −648.830 −1.21008
\(67\) −38.7503 −0.0706583 −0.0353292 − 0.999376i \(-0.511248\pi\)
−0.0353292 + 0.999376i \(0.511248\pi\)
\(68\) 352.254 0.628192
\(69\) −207.563 −0.362140
\(70\) 32.3381 0.0552163
\(71\) 1109.37 1.85434 0.927168 − 0.374645i \(-0.122236\pi\)
0.927168 + 0.374645i \(0.122236\pi\)
\(72\) 1118.88 1.83140
\(73\) −446.171 −0.715348 −0.357674 − 0.933847i \(-0.616430\pi\)
−0.357674 + 0.933847i \(0.616430\pi\)
\(74\) 343.219 0.539168
\(75\) 225.612 0.347352
\(76\) −69.3488 −0.104669
\(77\) 208.850 0.309099
\(78\) −713.921 −1.03635
\(79\) 642.608 0.915179 0.457589 − 0.889164i \(-0.348713\pi\)
0.457589 + 0.889164i \(0.348713\pi\)
\(80\) −77.6081 −0.108461
\(81\) 764.926 1.04928
\(82\) 587.137 0.790712
\(83\) −292.575 −0.386919 −0.193460 − 0.981108i \(-0.561971\pi\)
−0.193460 + 0.981108i \(0.561971\pi\)
\(84\) −225.838 −0.293344
\(85\) 305.060 0.389276
\(86\) 249.927 0.313376
\(87\) 1788.77 2.20433
\(88\) 990.271 1.19958
\(89\) −273.272 −0.325469 −0.162735 − 0.986670i \(-0.552031\pi\)
−0.162735 + 0.986670i \(0.552031\pi\)
\(90\) 406.169 0.475711
\(91\) 229.801 0.264722
\(92\) 132.791 0.150483
\(93\) −107.988 −0.120407
\(94\) −792.329 −0.869388
\(95\) −60.0577 −0.0648610
\(96\) −1692.78 −1.79967
\(97\) −959.139 −1.00398 −0.501989 − 0.864874i \(-0.667398\pi\)
−0.501989 + 0.864874i \(0.667398\pi\)
\(98\) 483.771 0.498655
\(99\) 2623.17 2.66302
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 115.4.a.e.1.2 ✓ 5
3.2 odd 2 1035.4.a.k.1.4 5
4.3 odd 2 1840.4.a.n.1.1 5
5.2 odd 4 575.4.b.i.24.4 10
5.3 odd 4 575.4.b.i.24.7 10
5.4 even 2 575.4.a.j.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.2 ✓ 5 1.1 even 1 trivial
575.4.a.j.1.4 5 5.4 even 2
575.4.b.i.24.4 10 5.2 odd 4
575.4.b.i.24.7 10 5.3 odd 4
1035.4.a.k.1.4 5 3.2 odd 2
1840.4.a.n.1.1 5 4.3 odd 2