Properties

Label 115.4.a.e.1.5
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 1.5
Root \(4.60878\) 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)
 
\(f(q)\) \(=\) \(q+5.60878 q^{2} -1.89520 q^{3} +23.4584 q^{4} -5.00000 q^{5} -10.6297 q^{6} +11.4426 q^{7} +86.7031 q^{8} -23.4082 q^{9} -28.0439 q^{10} +37.7245 q^{11} -44.4584 q^{12} -8.69346 q^{13} +64.1788 q^{14} +9.47598 q^{15} +298.631 q^{16} -105.687 q^{17} -131.292 q^{18} -128.279 q^{19} -117.292 q^{20} -21.6859 q^{21} +211.588 q^{22} -23.0000 q^{23} -164.319 q^{24} +25.0000 q^{25} -48.7597 q^{26} +95.5335 q^{27} +268.425 q^{28} -133.383 q^{29} +53.1487 q^{30} +106.008 q^{31} +981.333 q^{32} -71.4953 q^{33} -592.773 q^{34} -57.2128 q^{35} -549.121 q^{36} -248.835 q^{37} -719.491 q^{38} +16.4758 q^{39} -433.515 q^{40} +134.233 q^{41} -121.631 q^{42} +108.684 q^{43} +884.957 q^{44} +117.041 q^{45} -129.002 q^{46} -76.2000 q^{47} -565.965 q^{48} -212.068 q^{49} +140.220 q^{50} +200.297 q^{51} -203.935 q^{52} +476.207 q^{53} +535.827 q^{54} -188.622 q^{55} +992.105 q^{56} +243.114 q^{57} -748.118 q^{58} +608.000 q^{59} +222.292 q^{60} -366.273 q^{61} +594.575 q^{62} -267.850 q^{63} +3115.03 q^{64} +43.4673 q^{65} -401.001 q^{66} +136.041 q^{67} -2479.24 q^{68} +43.5895 q^{69} -320.894 q^{70} -152.874 q^{71} -2029.57 q^{72} +1228.16 q^{73} -1395.66 q^{74} -47.3799 q^{75} -3009.23 q^{76} +431.664 q^{77} +92.4092 q^{78} -364.637 q^{79} -1493.16 q^{80} +450.968 q^{81} +752.882 q^{82} -762.744 q^{83} -508.717 q^{84} +528.433 q^{85} +609.583 q^{86} +252.788 q^{87} +3270.83 q^{88} +271.222 q^{89} +656.459 q^{90} -99.4754 q^{91} -539.544 q^{92} -200.906 q^{93} -427.389 q^{94} +641.396 q^{95} -1859.82 q^{96} +574.510 q^{97} -1189.44 q^{98} -883.063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19}+ \cdots + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.60878 1.98300 0.991502 − 0.130090i \(-0.0415267\pi\)
0.991502 + 0.130090i \(0.0415267\pi\)
\(3\) −1.89520 −0.364731 −0.182365 − 0.983231i \(-0.558375\pi\)
−0.182365 + 0.983231i \(0.558375\pi\)
\(4\) 23.4584 2.93231
\(5\) −5.00000 −0.447214
\(6\) −10.6297 −0.723262
\(7\) 11.4426 0.617840 0.308920 − 0.951088i \(-0.400032\pi\)
0.308920 + 0.951088i \(0.400032\pi\)
\(8\) 86.7031 3.83177
\(9\) −23.4082 −0.866972
\(10\) −28.0439 −0.886826
\(11\) 37.7245 1.03403 0.517016 − 0.855976i \(-0.327043\pi\)
0.517016 + 0.855976i \(0.327043\pi\)
\(12\) −44.4584 −1.06950
\(13\) −8.69346 −0.185472 −0.0927358 − 0.995691i \(-0.529561\pi\)
−0.0927358 + 0.995691i \(0.529561\pi\)
\(14\) 64.1788 1.22518
\(15\) 9.47598 0.163112
\(16\) 298.631 4.66611
\(17\) −105.687 −1.50781 −0.753905 − 0.656983i \(-0.771832\pi\)
−0.753905 + 0.656983i \(0.771832\pi\)
\(18\) −131.292 −1.71921
\(19\) −128.279 −1.54891 −0.774455 − 0.632629i \(-0.781976\pi\)
−0.774455 + 0.632629i \(0.781976\pi\)
\(20\) −117.292 −1.31137
\(21\) −21.6859 −0.225345
\(22\) 211.588 2.05049
\(23\) −23.0000 −0.208514
\(24\) −164.319 −1.39756
\(25\) 25.0000 0.200000
\(26\) −48.7597 −0.367791
\(27\) 95.5335 0.680942
\(28\) 268.425 1.81170
\(29\) −133.383 −0.854092 −0.427046 − 0.904230i \(-0.640446\pi\)
−0.427046 + 0.904230i \(0.640446\pi\)
\(30\) 53.1487 0.323453
\(31\) 106.008 0.614179 0.307090 − 0.951681i \(-0.400645\pi\)
0.307090 + 0.951681i \(0.400645\pi\)
\(32\) 981.333 5.42115
\(33\) −71.4953 −0.377143
\(34\) −592.773 −2.98999
\(35\) −57.2128 −0.276306
\(36\) −549.121 −2.54223
\(37\) −248.835 −1.10563 −0.552814 − 0.833305i \(-0.686446\pi\)
−0.552814 + 0.833305i \(0.686446\pi\)
\(38\) −719.491 −3.07150
\(39\) 16.4758 0.0676472
\(40\) −433.515 −1.71362
\(41\) 134.233 0.511308 0.255654 − 0.966768i \(-0.417709\pi\)
0.255654 + 0.966768i \(0.417709\pi\)
\(42\) −121.631 −0.446860
\(43\) 108.684 0.385444 0.192722 − 0.981253i \(-0.438268\pi\)
0.192722 + 0.981253i \(0.438268\pi\)
\(44\) 884.957 3.03210
\(45\) 117.041 0.387721
\(46\) −129.002 −0.413485
\(47\) −76.2000 −0.236488 −0.118244 − 0.992985i \(-0.537726\pi\)
−0.118244 + 0.992985i \(0.537726\pi\)
\(48\) −565.965 −1.70187
\(49\) −212.068 −0.618274
\(50\) 140.220 0.396601
\(51\) 200.297 0.549945
\(52\) −203.935 −0.543859
\(53\) 476.207 1.23419 0.617095 − 0.786889i \(-0.288309\pi\)
0.617095 + 0.786889i \(0.288309\pi\)
\(54\) 535.827 1.35031
\(55\) −188.622 −0.462433
\(56\) 992.105 2.36742
\(57\) 243.114 0.564935
\(58\) −748.118 −1.69367
\(59\) 608.000 1.34161 0.670804 − 0.741635i \(-0.265949\pi\)
0.670804 + 0.741635i \(0.265949\pi\)
\(60\) 222.292 0.478296
\(61\) −366.273 −0.768794 −0.384397 − 0.923168i \(-0.625591\pi\)
−0.384397 + 0.923168i \(0.625591\pi\)
\(62\) 594.575 1.21792
\(63\) −267.850 −0.535650
\(64\) 3115.03 6.08405
\(65\) 43.4673 0.0829454
\(66\) −401.001 −0.747877
\(67\) 136.041 0.248060 0.124030 − 0.992278i \(-0.460418\pi\)
0.124030 + 0.992278i \(0.460418\pi\)
\(68\) −2479.24 −4.42136
\(69\) 43.5895 0.0760516
\(70\) −320.894 −0.547917
\(71\) −152.874 −0.255533 −0.127766 − 0.991804i \(-0.540781\pi\)
−0.127766 + 0.991804i \(0.540781\pi\)
\(72\) −2029.57 −3.32204
\(73\) 1228.16 1.96911 0.984556 − 0.175070i \(-0.0560151\pi\)
0.984556 + 0.175070i \(0.0560151\pi\)
\(74\) −1395.66 −2.19246
\(75\) −47.3799 −0.0729461
\(76\) −3009.23 −4.54188
\(77\) 431.664 0.638867
\(78\) 92.4092 0.134145
\(79\) −364.637 −0.519302 −0.259651 − 0.965702i \(-0.583608\pi\)
−0.259651 + 0.965702i \(0.583608\pi\)
\(80\) −1493.16 −2.08675
\(81\) 450.968 0.618611
\(82\) 752.882 1.01393
\(83\) −762.744 −1.00870 −0.504350 − 0.863499i \(-0.668268\pi\)
−0.504350 + 0.863499i \(0.668268\pi\)
\(84\) −508.717 −0.660781
\(85\) 528.433 0.674313
\(86\) 609.583 0.764338
\(87\) 252.788 0.311514
\(88\) 3270.83 3.96217
\(89\) 271.222 0.323028 0.161514 − 0.986870i \(-0.448362\pi\)
0.161514 + 0.986870i \(0.448362\pi\)
\(90\) 656.459 0.768853
\(91\) −99.4754 −0.114592
\(92\) −539.544 −0.611428
\(93\) −200.906 −0.224010
\(94\) −427.389 −0.468956
\(95\) 641.396 0.692694
\(96\) −1859.82 −1.97726
\(97\) 574.510 0.601367 0.300684 − 0.953724i \(-0.402785\pi\)
0.300684 + 0.953724i \(0.402785\pi\)
\(98\) −1189.44 −1.22604
\(99\) −883.063 −0.896477
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.1.5 ✓ 5
3.2 odd 2 1035.4.a.k.1.1 5
4.3 odd 2 1840.4.a.n.1.3 5
5.2 odd 4 575.4.b.i.24.10 10
5.3 odd 4 575.4.b.i.24.1 10
5.4 even 2 575.4.a.j.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.5 ✓ 5 1.1 even 1 trivial
575.4.a.j.1.1 5 5.4 even 2
575.4.b.i.24.1 10 5.3 odd 4
575.4.b.i.24.10 10 5.2 odd 4
1035.4.a.k.1.1 5 3.2 odd 2
1840.4.a.n.1.3 5 4.3 odd 2