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Properties

Label 615.4.a.k.1.5
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 1.5
Root \(-2.69918\) 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)
 
\(f(q)\) \(=\) \(q-2.69918 q^{2} +3.00000 q^{3} -0.714432 q^{4} +5.00000 q^{5} -8.09754 q^{6} +11.9424 q^{7} +23.5218 q^{8} +9.00000 q^{9} -13.4959 q^{10} +54.4129 q^{11} -2.14330 q^{12} +6.47213 q^{13} -32.2347 q^{14} +15.0000 q^{15} -57.7741 q^{16} +90.4792 q^{17} -24.2926 q^{18} +43.6737 q^{19} -3.57216 q^{20} +35.8272 q^{21} -146.870 q^{22} -67.8198 q^{23} +70.5654 q^{24} +25.0000 q^{25} -17.4694 q^{26} +27.0000 q^{27} -8.53204 q^{28} +5.95249 q^{29} -40.4877 q^{30} -168.690 q^{31} -32.2318 q^{32} +163.239 q^{33} -244.220 q^{34} +59.7120 q^{35} -6.42989 q^{36} +231.849 q^{37} -117.883 q^{38} +19.4164 q^{39} +117.609 q^{40} -41.0000 q^{41} -96.7041 q^{42} +64.5295 q^{43} -38.8743 q^{44} +45.0000 q^{45} +183.058 q^{46} -226.290 q^{47} -173.322 q^{48} -200.379 q^{49} -67.4795 q^{50} +271.438 q^{51} -4.62390 q^{52} +205.615 q^{53} -72.8778 q^{54} +272.064 q^{55} +280.907 q^{56} +131.021 q^{57} -16.0668 q^{58} -332.741 q^{59} -10.7165 q^{60} -11.9334 q^{61} +455.323 q^{62} +107.482 q^{63} +549.192 q^{64} +32.3606 q^{65} -440.610 q^{66} +272.333 q^{67} -64.6413 q^{68} -203.459 q^{69} -161.173 q^{70} +367.458 q^{71} +211.696 q^{72} +302.517 q^{73} -625.803 q^{74} +75.0000 q^{75} -31.2019 q^{76} +649.821 q^{77} -52.4083 q^{78} +225.989 q^{79} -288.871 q^{80} +81.0000 q^{81} +110.666 q^{82} -1054.68 q^{83} -25.5961 q^{84} +452.396 q^{85} -174.177 q^{86} +17.8575 q^{87} +1279.89 q^{88} -828.073 q^{89} -121.463 q^{90} +77.2928 q^{91} +48.4526 q^{92} -506.069 q^{93} +610.798 q^{94} +218.369 q^{95} -96.6954 q^{96} +813.944 q^{97} +540.859 q^{98} +489.716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{2} + 42 q^{3} + 91 q^{4} + 70 q^{5} + 15 q^{6} + 40 q^{7} + 81 q^{8} + 126 q^{9} + 25 q^{10} + 33 q^{11} + 273 q^{12} + 91 q^{13} + 84 q^{14} + 210 q^{15} + 331 q^{16} + 174 q^{17} + 45 q^{18}+ \cdots + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.69918 −0.954304 −0.477152 − 0.878821i \(-0.658331\pi\)
−0.477152 + 0.878821i \(0.658331\pi\)
\(3\) 3.00000 0.577350
\(4\) −0.714432 −0.0893040
\(5\) 5.00000 0.447214
\(6\) −8.09754 −0.550968
\(7\) 11.9424 0.644829 0.322415 − 0.946599i \(-0.395505\pi\)
0.322415 + 0.946599i \(0.395505\pi\)
\(8\) 23.5218 1.03953
\(9\) 9.00000 0.333333
\(10\) −13.4959 −0.426778
\(11\) 54.4129 1.49146 0.745732 − 0.666246i \(-0.232100\pi\)
0.745732 + 0.666246i \(0.232100\pi\)
\(12\) −2.14330 −0.0515597
\(13\) 6.47213 0.138080 0.0690402 − 0.997614i \(-0.478006\pi\)
0.0690402 + 0.997614i \(0.478006\pi\)
\(14\) −32.2347 −0.615363
\(15\) 15.0000 0.258199
\(16\) −57.7741 −0.902721
\(17\) 90.4792 1.29085 0.645424 − 0.763824i \(-0.276681\pi\)
0.645424 + 0.763824i \(0.276681\pi\)
\(18\) −24.2926 −0.318101
\(19\) 43.6737 0.527339 0.263670 − 0.964613i \(-0.415067\pi\)
0.263670 + 0.964613i \(0.415067\pi\)
\(20\) −3.57216 −0.0399380
\(21\) 35.8272 0.372292
\(22\) −146.870 −1.42331
\(23\) −67.8198 −0.614843 −0.307422 − 0.951573i \(-0.599466\pi\)
−0.307422 + 0.951573i \(0.599466\pi\)
\(24\) 70.5654 0.600171
\(25\) 25.0000 0.200000
\(26\) −17.4694 −0.131771
\(27\) 27.0000 0.192450
\(28\) −8.53204 −0.0575858
\(29\) 5.95249 0.0381155 0.0190578 − 0.999818i \(-0.493933\pi\)
0.0190578 + 0.999818i \(0.493933\pi\)
\(30\) −40.4877 −0.246400
\(31\) −168.690 −0.977340 −0.488670 − 0.872469i \(-0.662518\pi\)
−0.488670 + 0.872469i \(0.662518\pi\)
\(32\) −32.2318 −0.178057
\(33\) 163.239 0.861097
\(34\) −244.220 −1.23186
\(35\) 59.7120 0.288376
\(36\) −6.42989 −0.0297680
\(37\) 231.849 1.03016 0.515078 − 0.857143i \(-0.327763\pi\)
0.515078 + 0.857143i \(0.327763\pi\)
\(38\) −117.883 −0.503242
\(39\) 19.4164 0.0797207
\(40\) 117.609 0.464891
\(41\) −41.0000 −0.156174
\(42\) −96.7041 −0.355280
\(43\) 64.5295 0.228853 0.114426 − 0.993432i \(-0.463497\pi\)
0.114426 + 0.993432i \(0.463497\pi\)
\(44\) −38.8743 −0.133194
\(45\) 45.0000 0.149071
\(46\) 183.058 0.586748
\(47\) −226.290 −0.702294 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(48\) −173.322 −0.521186
\(49\) −200.379 −0.584195
\(50\) −67.4795 −0.190861
\(51\) 271.438 0.745272
\(52\) −4.62390 −0.0123311
\(53\) 205.615 0.532895 0.266448 − 0.963849i \(-0.414150\pi\)
0.266448 + 0.963849i \(0.414150\pi\)
\(54\) −72.8778 −0.183656
\(55\) 272.064 0.667003
\(56\) 280.907 0.670317
\(57\) 131.021 0.304459
\(58\) −16.0668 −0.0363738
\(59\) −332.741 −0.734223 −0.367112 − 0.930177i \(-0.619653\pi\)
−0.367112 + 0.930177i \(0.619653\pi\)
\(60\) −10.7165 −0.0230582
\(61\) −11.9334 −0.0250479 −0.0125239 − 0.999922i \(-0.503987\pi\)
−0.0125239 + 0.999922i \(0.503987\pi\)
\(62\) 455.323 0.932680
\(63\) 107.482 0.214943
\(64\) 549.192 1.07264
\(65\) 32.3606 0.0617514
\(66\) −440.610 −0.821748
\(67\) 272.333 0.496579 0.248289 − 0.968686i \(-0.420132\pi\)
0.248289 + 0.968686i \(0.420132\pi\)
\(68\) −64.6413 −0.115278
\(69\) −203.459 −0.354980
\(70\) −161.173 −0.275199
\(71\) 367.458 0.614214 0.307107 − 0.951675i \(-0.400639\pi\)
0.307107 + 0.951675i \(0.400639\pi\)
\(72\) 211.696 0.346509
\(73\) 302.517 0.485027 0.242514 − 0.970148i \(-0.422028\pi\)
0.242514 + 0.970148i \(0.422028\pi\)
\(74\) −625.803 −0.983083
\(75\) 75.0000 0.115470
\(76\) −31.2019 −0.0470935
\(77\) 649.821 0.961739
\(78\) −52.4083 −0.0760778
\(79\) 225.989 0.321845 0.160922 − 0.986967i \(-0.448553\pi\)
0.160922 + 0.986967i \(0.448553\pi\)
\(80\) −288.871 −0.403709
\(81\) 81.0000 0.111111
\(82\) 110.666 0.149037
\(83\) −1054.68 −1.39478 −0.697389 − 0.716693i \(-0.745655\pi\)
−0.697389 + 0.716693i \(0.745655\pi\)
\(84\) −25.5961 −0.0332472
\(85\) 452.396 0.577285
\(86\) −174.177 −0.218395
\(87\) 17.8575 0.0220060
\(88\) 1279.89 1.55042
\(89\) −828.073 −0.986242 −0.493121 − 0.869961i \(-0.664144\pi\)
−0.493121 + 0.869961i \(0.664144\pi\)
\(90\) −121.463 −0.142259
\(91\) 77.2928 0.0890383
\(92\) 48.4526 0.0549080
\(93\) −506.069 −0.564268
\(94\) 610.798 0.670202
\(95\) 218.369 0.235833
\(96\) −96.6954 −0.102801
\(97\) 813.944 0.851995 0.425997 − 0.904724i \(-0.359923\pi\)
0.425997 + 0.904724i \(0.359923\pi\)
\(98\) 540.859 0.557500
\(99\) 489.716 0.497155
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.1.5 ✓ 14
3.2 odd 2 1845.4.a.s.1.10 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.4.a.k.1.5 ✓ 14 1.1 even 1 trivial
1845.4.a.s.1.10 14 3.2 odd 2