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Properties

Label 615.4.f.b.286.37
Level $615$
Weight $4$
Character 615.286
Analytic conductor $36.286$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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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(286,615)] 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("615.286"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 615.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [42,4] 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: no
Analytic conductor: \(36.2861746535\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 286.37
Character \(\chi\) \(=\) 615.286
Dual form 615.4.f.b.286.38

$q$-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)
 
\(f(q)\) \(=\) \(q+4.57154 q^{2} +3.00000i q^{3} +12.8990 q^{4} +5.00000 q^{5} +13.7146i q^{6} +12.2406i q^{7} +22.3959 q^{8} -9.00000 q^{9} +22.8577 q^{10} +63.8798i q^{11} +38.6969i q^{12} +13.3238i q^{13} +55.9583i q^{14} +15.0000i q^{15} -0.808237 q^{16} -114.925i q^{17} -41.1439 q^{18} +85.3151i q^{19} +64.4949 q^{20} -36.7217 q^{21} +292.029i q^{22} -3.44876 q^{23} +67.1876i q^{24} +25.0000 q^{25} +60.9104i q^{26} -27.0000i q^{27} +157.891i q^{28} +188.030i q^{29} +68.5731i q^{30} +123.941 q^{31} -182.862 q^{32} -191.639 q^{33} -525.383i q^{34} +61.2029i q^{35} -116.091 q^{36} -302.519 q^{37} +390.021i q^{38} -39.9715 q^{39} +111.979 q^{40} +(259.436 - 40.1757i) q^{41} -167.875 q^{42} +261.636 q^{43} +823.984i q^{44} -45.0000 q^{45} -15.7661 q^{46} +232.164i q^{47} -2.42471i q^{48} +193.168 q^{49} +114.288 q^{50} +344.774 q^{51} +171.864i q^{52} -441.902i q^{53} -123.432i q^{54} +319.399i q^{55} +274.138i q^{56} -255.945 q^{57} +859.588i q^{58} +188.194 q^{59} +193.485i q^{60} +452.682 q^{61} +566.600 q^{62} -110.165i q^{63} -829.494 q^{64} +66.6191i q^{65} -876.087 q^{66} +435.626i q^{67} -1482.41i q^{68} -10.3463i q^{69} +279.792i q^{70} -479.138i q^{71} -201.563 q^{72} -644.292 q^{73} -1382.98 q^{74} +75.0000i q^{75} +1100.48i q^{76} -781.926 q^{77} -182.731 q^{78} -461.150i q^{79} -4.04118 q^{80} +81.0000 q^{81} +(1186.02 - 183.665i) q^{82} +1260.87 q^{83} -473.673 q^{84} -574.624i q^{85} +1196.08 q^{86} -564.091 q^{87} +1430.64i q^{88} +867.426i q^{89} -205.719 q^{90} -163.091 q^{91} -44.4854 q^{92} +371.822i q^{93} +1061.35i q^{94} +426.575i q^{95} -548.585i q^{96} -775.239i q^{97} +883.076 q^{98} -574.918i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 4 q^{2} + 168 q^{4} + 210 q^{5} + 48 q^{8} - 378 q^{9} + 20 q^{10} + 496 q^{16} - 36 q^{18} + 840 q^{20} - 12 q^{21} - 228 q^{23} + 1050 q^{25} + 156 q^{31} - 22 q^{32} - 1512 q^{36} - 744 q^{37}+ \cdots - 9412 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/615\mathbb{Z}\right)^\times\).

\(n\) \(206\) \(211\) \(247\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.57154 1.61628 0.808142 − 0.588988i \(-0.200474\pi\)
0.808142 + 0.588988i \(0.200474\pi\)
\(3\) 3.00000i 0.577350i
\(4\) 12.8990 1.61237
\(5\) 5.00000 0.447214
\(6\) 13.7146i 0.933162i
\(7\) 12.2406i 0.660929i 0.943818 + 0.330465i \(0.107205\pi\)
−0.943818 + 0.330465i \(0.892795\pi\)
\(8\) 22.3959 0.989767
\(9\) −9.00000 −0.333333
\(10\) 22.8577 0.722824
\(11\) 63.8798i 1.75095i 0.483260 + 0.875477i \(0.339453\pi\)
−0.483260 + 0.875477i \(0.660547\pi\)
\(12\) 38.6969i 0.930903i
\(13\) 13.3238i 0.284259i 0.989848 + 0.142129i \(0.0453949\pi\)
−0.989848 + 0.142129i \(0.954605\pi\)
\(14\) 55.9583i 1.06825i
\(15\) 15.0000i 0.258199i
\(16\) −0.808237 −0.0126287
\(17\) − 114.925i − 1.63961i −0.572644 − 0.819804i \(-0.694082\pi\)
0.572644 − 0.819804i \(-0.305918\pi\)
\(18\) −41.1439 −0.538761
\(19\) 85.3151i 1.03014i 0.857149 + 0.515069i \(0.172234\pi\)
−0.857149 + 0.515069i \(0.827766\pi\)
\(20\) 64.4949 0.721075
\(21\) −36.7217 −0.381588
\(22\) 292.029i 2.83004i
\(23\) −3.44876 −0.0312659 −0.0156330 − 0.999878i \(-0.504976\pi\)
−0.0156330 + 0.999878i \(0.504976\pi\)
\(24\) 67.1876i 0.571442i
\(25\) 25.0000 0.200000
\(26\) 60.9104i 0.459443i
\(27\) − 27.0000i − 0.192450i
\(28\) 157.891i 1.06566i
\(29\) 188.030i 1.20401i 0.798492 + 0.602006i \(0.205632\pi\)
−0.798492 + 0.602006i \(0.794368\pi\)
\(30\) 68.5731i 0.417323i
\(31\) 123.941 0.718078 0.359039 − 0.933323i \(-0.383105\pi\)
0.359039 + 0.933323i \(0.383105\pi\)
\(32\) −182.862 −1.01018
\(33\) −191.639 −1.01091
\(34\) − 525.383i − 2.65007i
\(35\) 61.2029i 0.295577i
\(36\) −116.091 −0.537457
\(37\) −302.519 −1.34416 −0.672078 − 0.740480i \(-0.734598\pi\)
−0.672078 + 0.740480i \(0.734598\pi\)
\(38\) 390.021i 1.66500i
\(39\) −39.9715 −0.164117
\(40\) 111.979 0.442637
\(41\) 259.436 − 40.1757i 0.988221 − 0.153034i
\(42\) −167.875 −0.616754
\(43\) 261.636 0.927885 0.463943 − 0.885865i \(-0.346434\pi\)
0.463943 + 0.885865i \(0.346434\pi\)
\(44\) 823.984i 2.82319i
\(45\) −45.0000 −0.149071
\(46\) −15.7661 −0.0505346
\(47\) 232.164i 0.720524i 0.932851 + 0.360262i \(0.117313\pi\)
−0.932851 + 0.360262i \(0.882687\pi\)
\(48\) − 2.42471i − 0.00729118i
\(49\) 193.168 0.563173
\(50\) 114.288 0.323257
\(51\) 344.774 0.946628
\(52\) 171.864i 0.458331i
\(53\) − 441.902i − 1.14528i −0.819806 − 0.572641i \(-0.805919\pi\)
0.819806 − 0.572641i \(-0.194081\pi\)
\(54\) − 123.432i − 0.311054i
\(55\) 319.399i 0.783050i
\(56\) 274.138i 0.654166i
\(57\) −255.945 −0.594751
\(58\) 859.588i 1.94602i
\(59\) 188.194 0.415268 0.207634 − 0.978207i \(-0.433424\pi\)
0.207634 + 0.978207i \(0.433424\pi\)
\(60\) 193.485i 0.416313i
\(61\) 452.682 0.950165 0.475082 − 0.879941i \(-0.342418\pi\)
0.475082 + 0.879941i \(0.342418\pi\)
\(62\) 566.600 1.16062
\(63\) − 110.165i − 0.220310i
\(64\) −829.494 −1.62011
\(65\) 66.6191i 0.127124i
\(66\) −876.087 −1.63392
\(67\) 435.626i 0.794330i 0.917747 + 0.397165i \(0.130006\pi\)
−0.917747 + 0.397165i \(0.869994\pi\)
\(68\) − 1482.41i − 2.64366i
\(69\) − 10.3463i − 0.0180514i
\(70\) 279.792i 0.477735i
\(71\) − 479.138i − 0.800890i −0.916321 − 0.400445i \(-0.868856\pi\)
0.916321 − 0.400445i \(-0.131144\pi\)
\(72\) −201.563 −0.329922
\(73\) −644.292 −1.03300 −0.516498 − 0.856289i \(-0.672765\pi\)
−0.516498 + 0.856289i \(0.672765\pi\)
\(74\) −1382.98 −2.17254
\(75\) 75.0000i 0.115470i
\(76\) 1100.48i 1.66097i
\(77\) −781.926 −1.15726
\(78\) −182.731 −0.265259
\(79\) − 461.150i − 0.656753i −0.944547 − 0.328376i \(-0.893499\pi\)
0.944547 − 0.328376i \(-0.106501\pi\)
\(80\) −4.04118 −0.00564773
\(81\) 81.0000 0.111111
\(82\) 1186.02 − 183.665i 1.59725 − 0.247346i
\(83\) 1260.87 1.66745 0.833724 − 0.552181i \(-0.186204\pi\)
0.833724 + 0.552181i \(0.186204\pi\)
\(84\) −473.673 −0.615261
\(85\) − 574.624i − 0.733255i
\(86\) 1196.08 1.49973
\(87\) −564.091 −0.695136
\(88\) 1430.64i 1.73304i
\(89\) 867.426i 1.03311i 0.856253 + 0.516556i \(0.172786\pi\)
−0.856253 + 0.516556i \(0.827214\pi\)
\(90\) −205.719 −0.240941
\(91\) −163.091 −0.187875
\(92\) −44.4854 −0.0504123
\(93\) 371.822i 0.414582i
\(94\) 1061.35i 1.16457i
\(95\) 426.575i 0.460692i
\(96\) − 548.585i − 0.583227i
\(97\) − 775.239i − 0.811480i −0.913988 − 0.405740i \(-0.867014\pi\)
0.913988 − 0.405740i \(-0.132986\pi\)
\(98\) 883.076 0.910246
\(99\) − 574.918i − 0.583651i
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.f.b.286.37 ✓ 42
41.40 even 2 inner 615.4.f.b.286.38 yes 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.4.f.b.286.37 ✓ 42 1.1 even 1 trivial
615.4.f.b.286.38 yes 42 41.40 even 2 inner