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Properties

Label 615.4.a.k.1.8
Level $615$
Weight $4$
Character 615.1
Self dual yes
Analytic conductor $36.286$
Analytic rank $0$
Dimension $14$
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] = [615,4,Mod(1,615)] 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("615.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(615, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 615.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,5] 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: \(36.2861746535\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5 x^{13} - 89 x^{12} + 433 x^{11} + 3100 x^{10} - 14427 x^{9} - 53983 x^{8} + 233727 x^{7} + \cdots - 2084736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.15585\) of defining polynomial
Character \(\chi\) \(=\) 615.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.15585 q^{2} +3.00000 q^{3} -6.66400 q^{4} +5.00000 q^{5} +3.46756 q^{6} +8.87787 q^{7} -16.9494 q^{8} +9.00000 q^{9} +5.77927 q^{10} -34.6827 q^{11} -19.9920 q^{12} +32.1633 q^{13} +10.2615 q^{14} +15.0000 q^{15} +33.7210 q^{16} -43.2002 q^{17} +10.4027 q^{18} +142.462 q^{19} -33.3200 q^{20} +26.6336 q^{21} -40.0880 q^{22} -46.4479 q^{23} -50.8483 q^{24} +25.0000 q^{25} +37.1761 q^{26} +27.0000 q^{27} -59.1622 q^{28} +68.0397 q^{29} +17.3378 q^{30} +214.496 q^{31} +174.572 q^{32} -104.048 q^{33} -49.9331 q^{34} +44.3894 q^{35} -59.9760 q^{36} +331.767 q^{37} +164.665 q^{38} +96.4900 q^{39} -84.7472 q^{40} -41.0000 q^{41} +30.7845 q^{42} -178.226 q^{43} +231.125 q^{44} +45.0000 q^{45} -53.6870 q^{46} -127.463 q^{47} +101.163 q^{48} -264.183 q^{49} +28.8963 q^{50} -129.601 q^{51} -214.337 q^{52} +436.042 q^{53} +31.2080 q^{54} -173.413 q^{55} -150.475 q^{56} +427.386 q^{57} +78.6438 q^{58} -189.234 q^{59} -99.9601 q^{60} +327.524 q^{61} +247.925 q^{62} +79.9008 q^{63} -67.9883 q^{64} +160.817 q^{65} -120.264 q^{66} +235.822 q^{67} +287.887 q^{68} -139.344 q^{69} +51.3076 q^{70} +1024.20 q^{71} -152.545 q^{72} +684.701 q^{73} +383.474 q^{74} +75.0000 q^{75} -949.367 q^{76} -307.908 q^{77} +111.528 q^{78} +897.212 q^{79} +168.605 q^{80} +81.0000 q^{81} -47.3900 q^{82} +1310.14 q^{83} -177.487 q^{84} -216.001 q^{85} -206.003 q^{86} +204.119 q^{87} +587.851 q^{88} -1168.65 q^{89} +52.0134 q^{90} +285.542 q^{91} +309.529 q^{92} +643.487 q^{93} -147.329 q^{94} +712.310 q^{95} +523.716 q^{96} -1165.93 q^{97} -305.357 q^{98} -312.144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 42 q^{3} + 91 q^{4} + 70 q^{5} + 15 q^{6} + 40 q^{7} + 81 q^{8} + 126 q^{9} + 25 q^{10} + 33 q^{11} + 273 q^{12} + 91 q^{13} + 84 q^{14} + 210 q^{15} + 331 q^{16} + 174 q^{17} + 45 q^{18}+ \cdots + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.15585 0.408656 0.204328 − 0.978903i \(-0.434499\pi\)
0.204328 + 0.978903i \(0.434499\pi\)
\(3\) 3.00000 0.577350
\(4\) −6.66400 −0.833000
\(5\) 5.00000 0.447214
\(6\) 3.46756 0.235938
\(7\) 8.87787 0.479360 0.239680 − 0.970852i \(-0.422957\pi\)
0.239680 + 0.970852i \(0.422957\pi\)
\(8\) −16.9494 −0.749066
\(9\) 9.00000 0.333333
\(10\) 5.77927 0.182756
\(11\) −34.6827 −0.950656 −0.475328 − 0.879809i \(-0.657671\pi\)
−0.475328 + 0.879809i \(0.657671\pi\)
\(12\) −19.9920 −0.480933
\(13\) 32.1633 0.686193 0.343096 − 0.939300i \(-0.388524\pi\)
0.343096 + 0.939300i \(0.388524\pi\)
\(14\) 10.2615 0.195893
\(15\) 15.0000 0.258199
\(16\) 33.7210 0.526890
\(17\) −43.2002 −0.616329 −0.308165 − 0.951333i \(-0.599715\pi\)
−0.308165 + 0.951333i \(0.599715\pi\)
\(18\) 10.4027 0.136219
\(19\) 142.462 1.72016 0.860079 − 0.510161i \(-0.170414\pi\)
0.860079 + 0.510161i \(0.170414\pi\)
\(20\) −33.3200 −0.372529
\(21\) 26.6336 0.276759
\(22\) −40.0880 −0.388491
\(23\) −46.4479 −0.421089 −0.210545 − 0.977584i \(-0.567524\pi\)
−0.210545 + 0.977584i \(0.567524\pi\)
\(24\) −50.8483 −0.432474
\(25\) 25.0000 0.200000
\(26\) 37.1761 0.280417
\(27\) 27.0000 0.192450
\(28\) −59.1622 −0.399307
\(29\) 68.0397 0.435677 0.217839 − 0.975985i \(-0.430099\pi\)
0.217839 + 0.975985i \(0.430099\pi\)
\(30\) 17.3378 0.105514
\(31\) 214.496 1.24273 0.621364 − 0.783522i \(-0.286579\pi\)
0.621364 + 0.783522i \(0.286579\pi\)
\(32\) 174.572 0.964383
\(33\) −104.048 −0.548861
\(34\) −49.9331 −0.251866
\(35\) 44.3894 0.214376
\(36\) −59.9760 −0.277667
\(37\) 331.767 1.47411 0.737057 − 0.675830i \(-0.236215\pi\)
0.737057 + 0.675830i \(0.236215\pi\)
\(38\) 164.665 0.702953
\(39\) 96.4900 0.396173
\(40\) −84.7472 −0.334993
\(41\) −41.0000 −0.156174
\(42\) 30.7845 0.113099
\(43\) −178.226 −0.632076 −0.316038 − 0.948747i \(-0.602353\pi\)
−0.316038 + 0.948747i \(0.602353\pi\)
\(44\) 231.125 0.791897
\(45\) 45.0000 0.149071
\(46\) −53.6870 −0.172081
\(47\) −127.463 −0.395584 −0.197792 − 0.980244i \(-0.563377\pi\)
−0.197792 + 0.980244i \(0.563377\pi\)
\(48\) 101.163 0.304200
\(49\) −264.183 −0.770214
\(50\) 28.8963 0.0817312
\(51\) −129.601 −0.355838
\(52\) −214.337 −0.571599
\(53\) 436.042 1.13009 0.565046 − 0.825059i \(-0.308858\pi\)
0.565046 + 0.825059i \(0.308858\pi\)
\(54\) 31.2080 0.0786458
\(55\) −173.413 −0.425146
\(56\) −150.475 −0.359072
\(57\) 427.386 0.993134
\(58\) 78.6438 0.178042
\(59\) −189.234 −0.417562 −0.208781 − 0.977962i \(-0.566950\pi\)
−0.208781 + 0.977962i \(0.566950\pi\)
\(60\) −99.9601 −0.215080
\(61\) 327.524 0.687462 0.343731 − 0.939068i \(-0.388309\pi\)
0.343731 + 0.939068i \(0.388309\pi\)
\(62\) 247.925 0.507848
\(63\) 79.9008 0.159787
\(64\) −67.9883 −0.132790
\(65\) 160.817 0.306875
\(66\) −120.264 −0.224295
\(67\) 235.822 0.430003 0.215001 − 0.976614i \(-0.431024\pi\)
0.215001 + 0.976614i \(0.431024\pi\)
\(68\) 287.887 0.513403
\(69\) −139.344 −0.243116
\(70\) 51.3076 0.0876061
\(71\) 1024.20 1.71198 0.855991 − 0.516991i \(-0.172948\pi\)
0.855991 + 0.516991i \(0.172948\pi\)
\(72\) −152.545 −0.249689
\(73\) 684.701 1.09778 0.548891 − 0.835894i \(-0.315050\pi\)
0.548891 + 0.835894i \(0.315050\pi\)
\(74\) 383.474 0.602405
\(75\) 75.0000 0.115470
\(76\) −949.367 −1.43289
\(77\) −307.908 −0.455706
\(78\) 111.528 0.161899
\(79\) 897.212 1.27777 0.638887 − 0.769300i \(-0.279395\pi\)
0.638887 + 0.769300i \(0.279395\pi\)
\(80\) 168.605 0.235632
\(81\) 81.0000 0.111111
\(82\) −47.3900 −0.0638213
\(83\) 1310.14 1.73261 0.866305 − 0.499516i \(-0.166489\pi\)
0.866305 + 0.499516i \(0.166489\pi\)
\(84\) −177.487 −0.230540
\(85\) −216.001 −0.275631
\(86\) −206.003 −0.258301
\(87\) 204.119 0.251538
\(88\) 587.851 0.712104
\(89\) −1168.65 −1.39187 −0.695934 − 0.718106i \(-0.745009\pi\)
−0.695934 + 0.718106i \(0.745009\pi\)
\(90\) 52.0134 0.0609188
\(91\) 285.542 0.328933
\(92\) 309.529 0.350768
\(93\) 643.487 0.717489
\(94\) −147.329 −0.161658
\(95\) 712.310 0.769278
\(96\) 523.716 0.556787
\(97\) −1165.93 −1.22044 −0.610220 − 0.792232i \(-0.708919\pi\)
−0.610220 + 0.792232i \(0.708919\pi\)
\(98\) −305.357 −0.314752
\(99\) −312.144 −0.316885
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 615.4.a.k.1.8 ✓ 14
3.2 odd 2 1845.4.a.s.1.7 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.4.a.k.1.8 ✓ 14 1.1 even 1 trivial
1845.4.a.s.1.7 14 3.2 odd 2