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Properties

Label 615.2.bo.a.353.3
Level $615$
Weight $2$
Character 615.353
Analytic conductor $4.911$
Analytic rank $0$
Dimension $640$
Inner twists $8$

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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] = [615,2,Mod(23,615)] 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("615.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(615, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 15, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 615 = 3 \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 615.bo (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.91079972431\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(80\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 353.3
Character \(\chi\) \(=\) 615.353
Dual form 615.2.bo.a.392.3

$q$-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+(-2.41646 + 1.23125i) q^{2} +(-1.43152 + 0.975068i) q^{3} +(3.14773 - 4.33248i) q^{4} +(-0.139262 + 2.23173i) q^{5} +(2.25865 - 4.11876i) q^{6} +(-0.133377 + 0.261766i) q^{7} +(-1.42349 + 8.98756i) q^{8} +(1.09849 - 2.79165i) q^{9} +(-2.41129 - 5.56434i) q^{10} +(0.691384 - 0.502320i) q^{11} +(-0.281572 + 9.27127i) q^{12} +(-0.209087 - 0.410357i) q^{13} -0.796766i q^{14} +(-1.97673 - 3.33055i) q^{15} +(-4.31638 - 13.2844i) q^{16} +(-0.545149 + 3.44194i) q^{17} +(0.782767 + 8.09842i) q^{18} +(2.49396 + 7.67562i) q^{19} +(9.23055 + 7.62822i) q^{20} +(-0.0643088 - 0.504774i) q^{21} +(-1.05222 + 2.06510i) q^{22} +(-1.95515 - 3.83720i) q^{23} +(-6.72572 - 14.2538i) q^{24} +(-4.96121 - 0.621588i) q^{25} +(1.01050 + 0.734172i) q^{26} +(1.14955 + 5.06740i) q^{27} +(0.714263 + 1.40182i) q^{28} +(-4.59308 + 6.32184i) q^{29} +(8.87741 + 5.61428i) q^{30} +(-2.48791 + 1.80757i) q^{31} +(13.9180 + 13.9180i) q^{32} +(-0.499933 + 1.39323i) q^{33} +(-2.92054 - 8.98851i) q^{34} +(-0.565616 - 0.334114i) q^{35} +(-8.63704 - 13.5465i) q^{36} +(-1.33540 + 0.211507i) q^{37} +(-15.4771 - 15.4771i) q^{38} +(0.699438 + 0.383559i) q^{39} +(-19.8595 - 4.42846i) q^{40} +(-6.22080 - 1.51713i) q^{41} +(0.776901 + 1.14059i) q^{42} +(2.50490 - 1.27631i) q^{43} -4.57657i q^{44} +(6.07723 + 2.84029i) q^{45} +(9.44908 + 6.86516i) q^{46} +(0.125989 + 0.247268i) q^{47} +(19.1322 + 14.8082i) q^{48} +(4.06376 + 5.59329i) q^{49} +(12.7539 - 4.60644i) q^{50} +(-2.57573 - 5.45875i) q^{51} +(-2.43601 - 0.385826i) q^{52} +(-1.02246 - 6.45554i) q^{53} +(-9.01705 - 10.8298i) q^{54} +(1.02476 + 1.61293i) q^{55} +(-2.16278 - 1.57135i) q^{56} +(-11.0544 - 8.55601i) q^{57} +(3.31525 - 20.9317i) q^{58} +(4.29829 - 13.2288i) q^{59} +(-20.6517 - 1.91952i) q^{60} +(0.477967 + 1.47103i) q^{61} +(3.78636 - 7.43116i) q^{62} +(0.584248 + 0.659888i) q^{63} +(-24.1999 - 7.86304i) q^{64} +(0.944922 - 0.409479i) q^{65} +(-0.507339 - 3.98221i) q^{66} +(-0.667460 - 4.21418i) q^{67} +(13.1961 + 13.1961i) q^{68} +(6.54036 + 3.58661i) q^{69} +(1.77817 + 0.110959i) q^{70} +(5.08655 - 3.69559i) q^{71} +(23.5265 + 13.8466i) q^{72} +(-6.24926 - 6.24926i) q^{73} +(2.96652 - 2.15530i) q^{74} +(7.70815 - 3.94770i) q^{75} +(41.1048 + 13.3557i) q^{76} +(0.0392760 + 0.247979i) q^{77} +(-2.16242 - 0.0656735i) q^{78} -14.9220 q^{79} +(30.2484 - 7.78297i) q^{80} +(-6.58665 - 6.13319i) q^{81} +(16.9003 - 3.99325i) q^{82} +(3.39317 - 3.39317i) q^{83} +(-2.38935 - 1.31028i) q^{84} +(-7.60555 - 1.69595i) q^{85} +(-4.48152 + 6.16829i) q^{86} +(0.410863 - 13.5284i) q^{87} +(3.53045 + 6.92890i) q^{88} +(-9.12371 + 2.96447i) q^{89} +(-18.1825 + 0.619124i) q^{90} +0.135305 q^{91} +(-22.7789 - 3.60782i) q^{92} +(1.79898 - 5.01345i) q^{93} +(-0.608897 - 0.442389i) q^{94} +(-17.4772 + 4.49692i) q^{95} +(-33.4949 - 6.35289i) q^{96} +(2.08326 + 13.1532i) q^{97} +(-16.7066 - 8.51246i) q^{98} +(-0.642827 - 2.48190i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 640 q - 20 q^{6} - 20 q^{7} + 12 q^{10} - 40 q^{12} - 20 q^{13} - 10 q^{15} + 112 q^{16} - 18 q^{18} + 20 q^{21} - 60 q^{22} - 28 q^{25} + 20 q^{28} - 10 q^{30} - 24 q^{31} + 6 q^{33} + 60 q^{36} - 100 q^{37}+ \cdots + 140 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(206\) \(211\) \(247\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41646 + 1.23125i −1.70869 + 0.870623i −0.725462 + 0.688262i \(0.758374\pi\)
−0.983232 + 0.182361i \(0.941626\pi\)
\(3\) −1.43152 + 0.975068i −0.826487 + 0.562956i
\(4\) 3.14773 − 4.33248i 1.57386 − 2.16624i
\(5\) −0.139262 + 2.23173i −0.0622797 + 0.998059i
\(6\) 2.25865 − 4.11876i 0.922092 − 1.68148i
\(7\) −0.133377 + 0.261766i −0.0504116 + 0.0989383i −0.914838 − 0.403821i \(-0.867682\pi\)
0.864426 + 0.502759i \(0.167682\pi\)
\(8\) −1.42349 + 8.98756i −0.503279 + 3.17758i
\(9\) 1.09849 − 2.79165i 0.366162 − 0.930551i
\(10\) −2.41129 − 5.56434i −0.762516 − 1.75960i
\(11\) 0.691384 − 0.502320i 0.208460 − 0.151455i −0.478657 − 0.878002i \(-0.658876\pi\)
0.687117 + 0.726547i \(0.258876\pi\)
\(12\) −0.281572 + 9.27127i −0.0812829 + 2.67638i
\(13\) −0.209087 − 0.410357i −0.0579904 − 0.113812i 0.860192 − 0.509970i \(-0.170343\pi\)
−0.918183 + 0.396157i \(0.870343\pi\)
\(14\) − 0.796766i − 0.212945i
\(15\) −1.97673 − 3.33055i −0.510389 − 0.859943i
\(16\) −4.31638 − 13.2844i −1.07909 − 3.32111i
\(17\) −0.545149 + 3.44194i −0.132218 + 0.834792i 0.829049 + 0.559175i \(0.188882\pi\)
−0.961268 + 0.275617i \(0.911118\pi\)
\(18\) 0.782767 + 8.09842i 0.184500 + 1.90882i
\(19\) 2.49396 + 7.67562i 0.572154 + 1.76091i 0.645673 + 0.763614i \(0.276577\pi\)
−0.0735193 + 0.997294i \(0.523423\pi\)
\(20\) 9.23055 + 7.62822i 2.06401 + 1.70572i
\(21\) −0.0643088 − 0.504774i −0.0140333 − 0.110151i
\(22\) −1.05222 + 2.06510i −0.224334 + 0.440281i
\(23\) −1.95515 − 3.83720i −0.407677 − 0.800111i 0.592307 − 0.805712i \(-0.298217\pi\)
−0.999984 + 0.00560089i \(0.998217\pi\)
\(24\) −6.72572 − 14.2538i −1.37288 − 2.90955i
\(25\) −4.96121 − 0.621588i −0.992242 − 0.124318i
\(26\) 1.01050 + 0.734172i 0.198176 + 0.143983i
\(27\) 1.14955 + 5.06740i 0.221230 + 0.975222i
\(28\) 0.714263 + 1.40182i 0.134983 + 0.264919i
\(29\) −4.59308 + 6.32184i −0.852914 + 1.17394i 0.130299 + 0.991475i \(0.458406\pi\)
−0.983213 + 0.182461i \(0.941594\pi\)
\(30\) 8.87741 + 5.61428i 1.62079 + 1.02502i
\(31\) −2.48791 + 1.80757i −0.446842 + 0.324650i −0.788348 − 0.615230i \(-0.789063\pi\)
0.341506 + 0.939880i \(0.389063\pi\)
\(32\) 13.9180 + 13.9180i 2.46038 + 2.46038i
\(33\) −0.499933 + 1.39323i −0.0870271 + 0.242530i
\(34\) −2.92054 − 8.98851i −0.500869 − 1.54152i
\(35\) −0.565616 − 0.334114i −0.0956066 − 0.0564756i
\(36\) −8.63704 − 13.5465i −1.43951 − 2.25776i
\(37\) −1.33540 + 0.211507i −0.219538 + 0.0347715i −0.265235 − 0.964184i \(-0.585449\pi\)
0.0456962 + 0.998955i \(0.485449\pi\)
\(38\) −15.4771 − 15.4771i −2.51072 − 2.51072i
\(39\) 0.699438 + 0.383559i 0.112000 + 0.0614186i
\(40\) −19.8595 − 4.42846i −3.14007 − 0.700201i
\(41\) −6.22080 − 1.51713i −0.971525 − 0.236936i
\(42\) 0.776901 + 1.14059i 0.119878 + 0.175996i
\(43\) 2.50490 − 1.27631i 0.381993 − 0.194635i −0.252441 − 0.967612i \(-0.581233\pi\)
0.634434 + 0.772977i \(0.281233\pi\)
\(44\) − 4.57657i − 0.689945i
\(45\) 6.07723 + 2.84029i 0.905940 + 0.423406i
\(46\) 9.44908 + 6.86516i 1.39319 + 1.01221i
\(47\) 0.125989 + 0.247268i 0.0183775 + 0.0360678i 0.900013 − 0.435864i \(-0.143557\pi\)
−0.881635 + 0.471931i \(0.843557\pi\)
\(48\) 19.1322 + 14.8082i 2.76150 + 2.13737i
\(49\) 4.06376 + 5.59329i 0.580538 + 0.799042i
\(50\) 12.7539 − 4.60644i 1.80367 − 0.651448i
\(51\) −2.57573 − 5.45875i −0.360674 − 0.764378i
\(52\) −2.43601 − 0.385826i −0.337814 − 0.0535045i
\(53\) −1.02246 − 6.45554i −0.140445 − 0.886737i −0.952806 − 0.303581i \(-0.901818\pi\)
0.812360 − 0.583156i \(-0.198182\pi\)
\(54\) −9.01705 − 10.8298i −1.22707 − 1.47375i
\(55\) 1.02476 + 1.61293i 0.138178 + 0.217488i
\(56\) −2.16278 − 1.57135i −0.289013 − 0.209980i
\(57\) −11.0544 − 8.55601i −1.46419 − 1.13327i
\(58\) 3.31525 − 20.9317i 0.435314 − 2.74846i
\(59\) 4.29829 − 13.2288i 0.559590 − 1.72224i −0.123913 − 0.992293i \(-0.539544\pi\)
0.683503 − 0.729948i \(-0.260456\pi\)
\(60\) −20.6517 − 1.91952i −2.66613 − 0.247810i
\(61\) 0.477967 + 1.47103i 0.0611975 + 0.188346i 0.976981 − 0.213325i \(-0.0684294\pi\)
−0.915784 + 0.401672i \(0.868429\pi\)
\(62\) 3.78636 − 7.43116i 0.480869 − 0.943758i
\(63\) 0.584248 + 0.659888i 0.0736083 + 0.0831380i
\(64\) −24.1999 − 7.86304i −3.02499 − 0.982880i
\(65\) 0.944922 − 0.409479i 0.117203 − 0.0507896i
\(66\) −0.507339 − 3.98221i −0.0624491 − 0.490177i
\(67\) −0.667460 − 4.21418i −0.0815432 − 0.514844i −0.994324 − 0.106394i \(-0.966070\pi\)
0.912781 − 0.408450i \(-0.133930\pi\)
\(68\) 13.1961 + 13.1961i 1.60027 + 1.60027i
\(69\) 6.54036 + 3.58661i 0.787367 + 0.431778i
\(70\) 1.77817 + 0.110959i 0.212531 + 0.0132621i
\(71\) 5.08655 − 3.69559i 0.603662 − 0.438586i −0.243515 − 0.969897i \(-0.578300\pi\)
0.847177 + 0.531311i \(0.178300\pi\)
\(72\) 23.5265 + 13.8466i 2.77262 + 1.63184i
\(73\) −6.24926 − 6.24926i −0.731421 − 0.731421i 0.239480 − 0.970901i \(-0.423023\pi\)
−0.970901 + 0.239480i \(0.923023\pi\)
\(74\) 2.96652 − 2.15530i 0.344851 − 0.250549i
\(75\) 7.70815 − 3.94770i 0.890061 − 0.455841i
\(76\) 41.1048 + 13.3557i 4.71504 + 1.53201i
\(77\) 0.0392760 + 0.247979i 0.00447591 + 0.0282598i
\(78\) −2.16242 − 0.0656735i −0.244846 − 0.00743606i
\(79\) −14.9220 −1.67885 −0.839425 − 0.543475i \(-0.817108\pi\)
−0.839425 + 0.543475i \(0.817108\pi\)
\(80\) 30.2484 − 7.78297i 3.38187 − 0.870162i
\(81\) −6.58665 − 6.13319i −0.731850 − 0.681465i
\(82\) 16.9003 − 3.99325i 1.86632 − 0.440981i
\(83\) 3.39317 − 3.39317i 0.372448 − 0.372448i −0.495920 − 0.868368i \(-0.665169\pi\)
0.868368 + 0.495920i \(0.165169\pi\)
\(84\) −2.38935 − 1.31028i −0.260699 − 0.142963i
\(85\) −7.60555 − 1.69595i −0.824937 − 0.183952i
\(86\) −4.48152 + 6.16829i −0.483255 + 0.665144i
\(87\) 0.410863 − 13.5284i 0.0440491 − 1.45040i
\(88\) 3.53045 + 6.92890i 0.376347 + 0.738623i
\(89\) −9.12371 + 2.96447i −0.967111 + 0.314234i −0.749649 − 0.661835i \(-0.769778\pi\)
−0.217462 + 0.976069i \(0.569778\pi\)
\(90\) −18.1825 + 0.619124i −1.91660 + 0.0652614i
\(91\) 0.135305 0.0141838
\(92\) −22.7789 − 3.60782i −2.37486 − 0.376141i
\(93\) 1.79898 − 5.01345i 0.186546 − 0.519871i
\(94\) −0.608897 − 0.442389i −0.0628029 − 0.0456290i
\(95\) −17.4772 + 4.49692i −1.79312 + 0.461374i
\(96\) −33.4949 − 6.35289i −3.41856 − 0.648389i
\(97\) 2.08326 + 13.1532i 0.211523 + 1.33550i 0.833523 + 0.552485i \(0.186320\pi\)
−0.622000 + 0.783017i \(0.713680\pi\)
\(98\) −16.7066 − 8.51246i −1.68763 − 0.859888i
\(99\) −0.642827 − 2.48190i −0.0646065 − 0.249440i
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 615.2.bo.a.353.3 yes 640
3.2 odd 2 inner 615.2.bo.a.353.78 yes 640
5.2 odd 4 inner 615.2.bo.a.107.3 yes 640
15.2 even 4 inner 615.2.bo.a.107.78 yes 640
41.23 even 10 inner 615.2.bo.a.23.78 yes 640
123.23 odd 10 inner 615.2.bo.a.23.3 ✓ 640
205.187 odd 20 inner 615.2.bo.a.392.78 yes 640
615.392 even 20 inner 615.2.bo.a.392.3 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.2.bo.a.23.3 ✓ 640 123.23 odd 10 inner
615.2.bo.a.23.78 yes 640 41.23 even 10 inner
615.2.bo.a.107.3 yes 640 5.2 odd 4 inner
615.2.bo.a.107.78 yes 640 15.2 even 4 inner
615.2.bo.a.353.3 yes 640 1.1 even 1 trivial
615.2.bo.a.353.78 yes 640 3.2 odd 2 inner
615.2.bo.a.392.3 yes 640 615.392 even 20 inner
615.2.bo.a.392.78 yes 640 205.187 odd 20 inner