Properties

Label 115.4.a.d.1.5
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $1$
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,-5] 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: \(1\)
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: \( x^{5} - 34x^{3} - 9x^{2} + 260x + 60 \) 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 1.5
Root \(4.98640\) 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)
 
\(f(q)\) \(=\) \(q+3.98640 q^{2} -7.39409 q^{3} +7.89136 q^{4} -5.00000 q^{5} -29.4758 q^{6} -3.57544 q^{7} -0.433099 q^{8} +27.6726 q^{9} -19.9320 q^{10} -43.8100 q^{11} -58.3494 q^{12} -65.2647 q^{13} -14.2531 q^{14} +36.9705 q^{15} -64.8574 q^{16} +25.0257 q^{17} +110.314 q^{18} +93.1311 q^{19} -39.4568 q^{20} +26.4371 q^{21} -174.644 q^{22} +23.0000 q^{23} +3.20238 q^{24} +25.0000 q^{25} -260.171 q^{26} -4.97335 q^{27} -28.2151 q^{28} -84.6447 q^{29} +147.379 q^{30} +275.951 q^{31} -255.082 q^{32} +323.935 q^{33} +99.7625 q^{34} +17.8772 q^{35} +218.374 q^{36} -270.136 q^{37} +371.257 q^{38} +482.574 q^{39} +2.16550 q^{40} +178.492 q^{41} +105.389 q^{42} -46.0544 q^{43} -345.720 q^{44} -138.363 q^{45} +91.6871 q^{46} -390.733 q^{47} +479.561 q^{48} -330.216 q^{49} +99.6599 q^{50} -185.043 q^{51} -515.027 q^{52} -460.760 q^{53} -19.8257 q^{54} +219.050 q^{55} +1.54852 q^{56} -688.620 q^{57} -337.427 q^{58} -171.714 q^{59} +291.747 q^{60} -432.337 q^{61} +1100.05 q^{62} -98.9417 q^{63} -498.000 q^{64} +326.324 q^{65} +1291.33 q^{66} +234.236 q^{67} +197.487 q^{68} -170.064 q^{69} +71.2656 q^{70} -465.620 q^{71} -11.9850 q^{72} +363.534 q^{73} -1076.87 q^{74} -184.852 q^{75} +734.930 q^{76} +156.640 q^{77} +1923.73 q^{78} +430.514 q^{79} +324.287 q^{80} -710.387 q^{81} +711.541 q^{82} +318.020 q^{83} +208.625 q^{84} -125.129 q^{85} -183.591 q^{86} +625.871 q^{87} +18.9741 q^{88} +1453.20 q^{89} -551.570 q^{90} +233.350 q^{91} +181.501 q^{92} -2040.41 q^{93} -1557.62 q^{94} -465.655 q^{95} +1886.10 q^{96} -919.515 q^{97} -1316.37 q^{98} -1212.34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} - 6 q^{3} + 33 q^{4} - 25 q^{5} - 36 q^{6} - 15 q^{7} - 102 q^{8} + 79 q^{9} + 25 q^{10} - 153 q^{11} - 30 q^{12} + 28 q^{13} + 69 q^{14} + 30 q^{15} + 145 q^{16} - 341 q^{17} - 409 q^{18}+ \cdots - 91 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.98640 1.40940 0.704702 − 0.709503i \(-0.251081\pi\)
0.704702 + 0.709503i \(0.251081\pi\)
\(3\) −7.39409 −1.42299 −0.711497 − 0.702689i \(-0.751982\pi\)
−0.711497 + 0.702689i \(0.751982\pi\)
\(4\) 7.89136 0.986419
\(5\) −5.00000 −0.447214
\(6\) −29.4758 −2.00557
\(7\) −3.57544 −0.193056 −0.0965278 − 0.995330i \(-0.530774\pi\)
−0.0965278 + 0.995330i \(0.530774\pi\)
\(8\) −0.433099 −0.0191405
\(9\) 27.6726 1.02491
\(10\) −19.9320 −0.630305
\(11\) −43.8100 −1.20084 −0.600419 − 0.799686i \(-0.705000\pi\)
−0.600419 + 0.799686i \(0.705000\pi\)
\(12\) −58.3494 −1.40367
\(13\) −65.2647 −1.39240 −0.696199 − 0.717848i \(-0.745127\pi\)
−0.696199 + 0.717848i \(0.745127\pi\)
\(14\) −14.2531 −0.272093
\(15\) 36.9705 0.636382
\(16\) −64.8574 −1.01340
\(17\) 25.0257 0.357037 0.178519 − 0.983937i \(-0.442870\pi\)
0.178519 + 0.983937i \(0.442870\pi\)
\(18\) 110.314 1.44451
\(19\) 93.1311 1.12451 0.562256 − 0.826963i \(-0.309934\pi\)
0.562256 + 0.826963i \(0.309934\pi\)
\(20\) −39.4568 −0.441140
\(21\) 26.4371 0.274717
\(22\) −174.644 −1.69247
\(23\) 23.0000 0.208514
\(24\) 3.20238 0.0272368
\(25\) 25.0000 0.200000
\(26\) −260.171 −1.96245
\(27\) −4.97335 −0.0354490
\(28\) −28.2151 −0.190434
\(29\) −84.6447 −0.542004 −0.271002 − 0.962579i \(-0.587355\pi\)
−0.271002 + 0.962579i \(0.587355\pi\)
\(30\) 147.379 0.896920
\(31\) 275.951 1.59879 0.799393 − 0.600809i \(-0.205155\pi\)
0.799393 + 0.600809i \(0.205155\pi\)
\(32\) −255.082 −1.40914
\(33\) 323.935 1.70879
\(34\) 99.7625 0.503209
\(35\) 17.8772 0.0863371
\(36\) 218.374 1.01099
\(37\) −270.136 −1.20027 −0.600137 − 0.799897i \(-0.704887\pi\)
−0.600137 + 0.799897i \(0.704887\pi\)
\(38\) 371.257 1.58489
\(39\) 482.574 1.98137
\(40\) 2.16550 0.00855988
\(41\) 178.492 0.679898 0.339949 − 0.940444i \(-0.389590\pi\)
0.339949 + 0.940444i \(0.389590\pi\)
\(42\) 105.389 0.387187
\(43\) −46.0544 −0.163331 −0.0816654 − 0.996660i \(-0.526024\pi\)
−0.0816654 + 0.996660i \(0.526024\pi\)
\(44\) −345.720 −1.18453
\(45\) −138.363 −0.458354
\(46\) 91.6871 0.293881
\(47\) −390.733 −1.21265 −0.606323 − 0.795219i \(-0.707356\pi\)
−0.606323 + 0.795219i \(0.707356\pi\)
\(48\) 479.561 1.44206
\(49\) −330.216 −0.962730
\(50\) 99.6599 0.281881
\(51\) −185.043 −0.508062
\(52\) −515.027 −1.37349
\(53\) −460.760 −1.19415 −0.597077 − 0.802184i \(-0.703671\pi\)
−0.597077 + 0.802184i \(0.703671\pi\)
\(54\) −19.8257 −0.0499619
\(55\) 219.050 0.537031
\(56\) 1.54852 0.00369517
\(57\) −688.620 −1.60017
\(58\) −337.427 −0.763903
\(59\) −171.714 −0.378902 −0.189451 − 0.981890i \(-0.560671\pi\)
−0.189451 + 0.981890i \(0.560671\pi\)
\(60\) 291.747 0.627740
\(61\) −432.337 −0.907460 −0.453730 − 0.891139i \(-0.649907\pi\)
−0.453730 + 0.891139i \(0.649907\pi\)
\(62\) 1100.05 2.25333
\(63\) −98.9417 −0.197865
\(64\) −498.000 −0.972657
\(65\) 326.324 0.622700
\(66\) 1291.33 2.40837
\(67\) 234.236 0.427112 0.213556 − 0.976931i \(-0.431495\pi\)
0.213556 + 0.976931i \(0.431495\pi\)
\(68\) 197.487 0.352188
\(69\) −170.064 −0.296715
\(70\) 71.2656 0.121684
\(71\) −465.620 −0.778295 −0.389147 − 0.921176i \(-0.627230\pi\)
−0.389147 + 0.921176i \(0.627230\pi\)
\(72\) −11.9850 −0.0196173
\(73\) 363.534 0.582856 0.291428 − 0.956593i \(-0.405870\pi\)
0.291428 + 0.956593i \(0.405870\pi\)
\(74\) −1076.87 −1.69167
\(75\) −184.852 −0.284599
\(76\) 734.930 1.10924
\(77\) 156.640 0.231828
\(78\) 1923.73 2.79256
\(79\) 430.514 0.613122 0.306561 − 0.951851i \(-0.400822\pi\)
0.306561 + 0.951851i \(0.400822\pi\)
\(80\) 324.287 0.453205
\(81\) −710.387 −0.974468
\(82\) 711.541 0.958250
\(83\) 318.020 0.420569 0.210285 − 0.977640i \(-0.432561\pi\)
0.210285 + 0.977640i \(0.432561\pi\)
\(84\) 208.625 0.270986
\(85\) −125.129 −0.159672
\(86\) −183.591 −0.230199
\(87\) 625.871 0.771269
\(88\) 18.9741 0.0229846
\(89\) 1453.20 1.73077 0.865385 − 0.501108i \(-0.167074\pi\)
0.865385 + 0.501108i \(0.167074\pi\)
\(90\) −551.570 −0.646006
\(91\) 233.350 0.268810
\(92\) 181.501 0.205683
\(93\) −2040.41 −2.27506
\(94\) −1557.62 −1.70911
\(95\) −465.655 −0.502897
\(96\) 1886.10 2.00520
\(97\) −919.515 −0.962501 −0.481250 − 0.876583i \(-0.659817\pi\)
−0.481250 + 0.876583i \(0.659817\pi\)
\(98\) −1316.37 −1.35687
\(99\) −1212.34 −1.23075
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.d.1.5 ✓ 5
3.2 odd 2 1035.4.a.m.1.1 5
4.3 odd 2 1840.4.a.p.1.4 5
5.2 odd 4 575.4.b.h.24.8 10
5.3 odd 4 575.4.b.h.24.3 10
5.4 even 2 575.4.a.k.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.d.1.5 ✓ 5 1.1 even 1 trivial
575.4.a.k.1.1 5 5.4 even 2
575.4.b.h.24.3 10 5.3 odd 4
575.4.b.h.24.8 10 5.2 odd 4
1035.4.a.m.1.1 5 3.2 odd 2
1840.4.a.p.1.4 5 4.3 odd 2