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Properties

Label 615.2.bo.a.23.4
Level $615$
Weight $2$
Character 615.23
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 23.4
Character \(\chi\) \(=\) 615.23
Dual form 615.2.bo.a.107.4

$q$-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+(-1.19491 + 2.34514i) q^{2} +(1.63566 - 0.569737i) q^{3} +(-2.89632 - 3.98644i) q^{4} +(1.74827 + 1.39411i) q^{5} +(-0.618358 + 4.51666i) q^{6} +(1.54409 - 0.786754i) q^{7} +(7.61041 - 1.20537i) q^{8} +(2.35080 - 1.86380i) q^{9} +(-5.35843 + 2.43410i) q^{10} +(-4.07910 - 2.96364i) q^{11} +(-7.00864 - 4.87034i) q^{12} +(3.81190 + 1.94226i) q^{13} +4.56122i q^{14} +(3.65386 + 1.28425i) q^{15} +(-3.22162 + 9.91512i) q^{16} +(0.618829 - 0.0980130i) q^{17} +(1.56188 + 7.74004i) q^{18} +(1.85850 - 5.71986i) q^{19} +(0.494008 - 11.0072i) q^{20} +(2.07737 - 2.16659i) q^{21} +(11.8243 - 6.02480i) q^{22} +(1.86389 + 0.949698i) q^{23} +(11.7613 - 6.30751i) q^{24} +(1.11289 + 4.87457i) q^{25} +(-9.10976 + 6.61863i) q^{26} +(2.78324 - 4.38789i) q^{27} +(-7.60854 - 3.87674i) q^{28} +(-2.68626 - 3.69732i) q^{29} +(-7.37779 + 7.03427i) q^{30} +(3.38900 + 2.46225i) q^{31} +(-8.50595 - 8.50595i) q^{32} +(-8.36055 - 2.52351i) q^{33} +(-0.509591 + 1.56836i) q^{34} +(3.79632 + 0.777182i) q^{35} +(-14.2386 - 3.97317i) q^{36} +(0.0564731 - 0.356557i) q^{37} +(11.1932 + 11.1932i) q^{38} +(7.34157 + 1.00511i) q^{39} +(14.9855 + 8.50246i) q^{40} +(2.92663 + 5.69516i) q^{41} +(2.59870 + 7.46063i) q^{42} +(-0.896904 + 1.76027i) q^{43} +24.8448i q^{44} +(6.70818 + 0.0188607i) q^{45} +(-4.45436 + 3.23628i) q^{46} +(7.90158 + 4.02605i) q^{47} +(0.379526 + 18.0533i) q^{48} +(-2.34926 + 3.23348i) q^{49} +(-12.7614 - 3.21479i) q^{50} +(0.956356 - 0.512886i) q^{51} +(-3.29778 - 20.8213i) q^{52} +(-9.57523 - 1.51657i) q^{53} +(6.96450 + 11.7702i) q^{54} +(-2.99972 - 10.8680i) q^{55} +(10.8028 - 7.84872i) q^{56} +(-0.218942 - 10.4146i) q^{57} +(11.8806 - 1.88170i) q^{58} +(-1.38192 - 4.25310i) q^{59} +(-5.46317 - 18.2855i) q^{60} +(-4.03068 + 12.4052i) q^{61} +(-9.82388 + 5.00552i) q^{62} +(2.16350 - 4.72738i) q^{63} +(10.2813 - 3.34059i) q^{64} +(3.95650 + 8.70982i) q^{65} +(15.9081 - 16.5913i) q^{66} +(-14.0045 - 2.21810i) q^{67} +(-2.18305 - 2.18305i) q^{68} +(3.58977 + 0.491462i) q^{69} +(-6.35886 + 7.97424i) q^{70} +(5.83094 + 4.23642i) q^{71} +(15.6440 - 17.0178i) q^{72} +(-7.45920 - 7.45920i) q^{73} +(0.768698 + 0.558492i) q^{74} +(4.59754 + 7.33911i) q^{75} +(-28.1847 + 9.15777i) q^{76} +(-8.63017 - 1.36689i) q^{77} +(-11.1296 + 16.0160i) q^{78} +12.8931 q^{79} +(-19.4551 + 12.8430i) q^{80} +(2.05251 - 8.76283i) q^{81} +(-16.8530 + 0.0581522i) q^{82} +(-3.14732 + 3.14732i) q^{83} +(-14.6537 - 2.00619i) q^{84} +(1.21852 + 0.691366i) q^{85} +(-3.05637 - 4.20674i) q^{86} +(-6.50032 - 4.51711i) q^{87} +(-34.6159 - 17.6377i) q^{88} +(-15.2669 - 4.96053i) q^{89} +(-8.05990 + 15.7091i) q^{90} +7.41401 q^{91} +(-1.61250 - 10.1809i) q^{92} +(6.94610 + 2.09658i) q^{93} +(-18.8834 + 13.7196i) q^{94} +(11.2233 - 7.40890i) q^{95} +(-18.7590 - 9.06672i) q^{96} +(12.9758 + 2.05516i) q^{97} +(-4.77581 - 9.37306i) q^{98} +(-15.1128 + 0.635699i) 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 \(\left(\mathbb{Z}/615\mathbb{Z}\right)^\times\).

\(n\) \(206\) \(211\) \(247\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.19491 + 2.34514i −0.844930 + 1.65827i −0.0962291 + 0.995359i \(0.530678\pi\)
−0.748700 + 0.662908i \(0.769322\pi\)
\(3\) 1.63566 − 0.569737i 0.944352 − 0.328938i
\(4\) −2.89632 − 3.98644i −1.44816 − 1.99322i
\(5\) 1.74827 + 1.39411i 0.781850 + 0.623467i
\(6\) −0.618358 + 4.51666i −0.252443 + 1.84392i
\(7\) 1.54409 − 0.786754i 0.583612 − 0.297365i −0.137147 − 0.990551i \(-0.543793\pi\)
0.720759 + 0.693185i \(0.243793\pi\)
\(8\) 7.61041 − 1.20537i 2.69068 − 0.426163i
\(9\) 2.35080 − 1.86380i 0.783600 − 0.621266i
\(10\) −5.35843 + 2.43410i −1.69448 + 0.769731i
\(11\) −4.07910 − 2.96364i −1.22990 − 0.893572i −0.233014 − 0.972473i \(-0.574859\pi\)
−0.996882 + 0.0789014i \(0.974859\pi\)
\(12\) −7.00864 − 4.87034i −2.02322 − 1.40595i
\(13\) 3.81190 + 1.94226i 1.05723 + 0.538686i 0.894076 − 0.447914i \(-0.147833\pi\)
0.163155 + 0.986601i \(0.447833\pi\)
\(14\) 4.56122i 1.21904i
\(15\) 3.65386 + 1.28425i 0.943423 + 0.331592i
\(16\) −3.22162 + 9.91512i −0.805404 + 2.47878i
\(17\) 0.618829 − 0.0980130i 0.150088 − 0.0237716i −0.0809382 − 0.996719i \(-0.525792\pi\)
0.231026 + 0.972947i \(0.425792\pi\)
\(18\) 1.56188 + 7.74004i 0.368139 + 1.82434i
\(19\) 1.85850 − 5.71986i 0.426368 − 1.31223i −0.475310 − 0.879818i \(-0.657664\pi\)
0.901678 − 0.432408i \(-0.142336\pi\)
\(20\) 0.494008 − 11.0072i 0.110464 − 2.46128i
\(21\) 2.07737 − 2.16659i 0.453320 − 0.472789i
\(22\) 11.8243 − 6.02480i 2.52096 − 1.28449i
\(23\) 1.86389 + 0.949698i 0.388648 + 0.198026i 0.637386 − 0.770545i \(-0.280016\pi\)
−0.248738 + 0.968571i \(0.580016\pi\)
\(24\) 11.7613 − 6.30751i 2.40077 − 1.28752i
\(25\) 1.11289 + 4.87457i 0.222578 + 0.974915i
\(26\) −9.10976 + 6.61863i −1.78657 + 1.29802i
\(27\) 2.78324 − 4.38789i 0.535636 − 0.844449i
\(28\) −7.60854 − 3.87674i −1.43788 − 0.732636i
\(29\) −2.68626 − 3.69732i −0.498826 − 0.686575i 0.483159 − 0.875532i \(-0.339489\pi\)
−0.981985 + 0.188957i \(0.939489\pi\)
\(30\) −7.37779 + 7.03427i −1.34699 + 1.28428i
\(31\) 3.38900 + 2.46225i 0.608682 + 0.442233i 0.848950 − 0.528474i \(-0.177235\pi\)
−0.240268 + 0.970707i \(0.577235\pi\)
\(32\) −8.50595 − 8.50595i −1.50365 − 1.50365i
\(33\) −8.36055 − 2.52351i −1.45538 − 0.439287i
\(34\) −0.509591 + 1.56836i −0.0873942 + 0.268972i
\(35\) 3.79632 + 0.777182i 0.641694 + 0.131368i
\(36\) −14.2386 − 3.97317i −2.37310 − 0.662195i
\(37\) 0.0564731 − 0.356557i 0.00928413 − 0.0586177i −0.982611 − 0.185674i \(-0.940553\pi\)
0.991895 + 0.127057i \(0.0405530\pi\)
\(38\) 11.1932 + 11.1932i 1.81577 + 1.81577i
\(39\) 7.34157 + 1.00511i 1.17559 + 0.160946i
\(40\) 14.9855 + 8.50246i 2.36941 + 1.34436i
\(41\) 2.92663 + 5.69516i 0.457062 + 0.889435i
\(42\) 2.59870 + 7.46063i 0.400988 + 1.15120i
\(43\) −0.896904 + 1.76027i −0.136777 + 0.268439i −0.949228 − 0.314590i \(-0.898133\pi\)
0.812451 + 0.583030i \(0.198133\pi\)
\(44\) 24.8448i 3.74549i
\(45\) 6.70818 + 0.0188607i 0.999996 + 0.00281158i
\(46\) −4.45436 + 3.23628i −0.656760 + 0.477164i
\(47\) 7.90158 + 4.02605i 1.15256 + 0.587260i 0.922531 − 0.385923i \(-0.126117\pi\)
0.230032 + 0.973183i \(0.426117\pi\)
\(48\) 0.379526 + 18.0533i 0.0547798 + 2.60577i
\(49\) −2.34926 + 3.23348i −0.335608 + 0.461925i
\(50\) −12.7614 − 3.21479i −1.80473 − 0.454640i
\(51\) 0.956356 − 0.512886i 0.133917 − 0.0718185i
\(52\) −3.29778 − 20.8213i −0.457319 − 2.88740i
\(53\) −9.57523 − 1.51657i −1.31526 − 0.208316i −0.540913 − 0.841079i \(-0.681921\pi\)
−0.774346 + 0.632762i \(0.781921\pi\)
\(54\) 6.96450 + 11.7702i 0.947748 + 1.60173i
\(55\) −2.99972 − 10.8680i −0.404482 − 1.46544i
\(56\) 10.8028 − 7.84872i 1.44359 − 1.04883i
\(57\) −0.218942 − 10.4146i −0.0289995 − 1.37945i
\(58\) 11.8806 − 1.88170i 1.56000 − 0.247079i
\(59\) −1.38192 − 4.25310i −0.179910 − 0.553707i 0.819913 − 0.572487i \(-0.194021\pi\)
−0.999824 + 0.0187809i \(0.994021\pi\)
\(60\) −5.46317 − 18.2855i −0.705292 − 2.36065i
\(61\) −4.03068 + 12.4052i −0.516076 + 1.58832i 0.265239 + 0.964183i \(0.414549\pi\)
−0.781315 + 0.624136i \(0.785451\pi\)
\(62\) −9.82388 + 5.00552i −1.24763 + 0.635701i
\(63\) 2.16350 − 4.72738i 0.272575 − 0.595594i
\(64\) 10.2813 − 3.34059i 1.28516 − 0.417574i
\(65\) 3.95650 + 8.70982i 0.490743 + 1.08032i
\(66\) 15.9081 − 16.5913i 1.95815 − 2.04225i
\(67\) −14.0045 − 2.21810i −1.71093 − 0.270984i −0.777274 − 0.629162i \(-0.783398\pi\)
−0.933653 + 0.358178i \(0.883398\pi\)
\(68\) −2.18305 − 2.18305i −0.264734 − 0.264734i
\(69\) 3.58977 + 0.491462i 0.432158 + 0.0591651i
\(70\) −6.35886 + 7.97424i −0.760030 + 0.953105i
\(71\) 5.83094 + 4.23642i 0.692005 + 0.502771i 0.877319 − 0.479908i \(-0.159330\pi\)
−0.185314 + 0.982679i \(0.559330\pi\)
\(72\) 15.6440 − 17.0178i 1.84366 − 2.00557i
\(73\) −7.45920 − 7.45920i −0.873033 − 0.873033i 0.119769 − 0.992802i \(-0.461785\pi\)
−0.992802 + 0.119769i \(0.961785\pi\)
\(74\) 0.768698 + 0.558492i 0.0893594 + 0.0649234i
\(75\) 4.59754 + 7.33911i 0.530879 + 0.847448i
\(76\) −28.1847 + 9.15777i −3.23301 + 1.05047i
\(77\) −8.63017 − 1.36689i −0.983500 − 0.155771i
\(78\) −11.1296 + 16.0160i −1.26018 + 1.81346i
\(79\) 12.8931 1.45058 0.725291 − 0.688442i \(-0.241705\pi\)
0.725291 + 0.688442i \(0.241705\pi\)
\(80\) −19.4551 + 12.8430i −2.17514 + 1.43589i
\(81\) 2.05251 − 8.76283i 0.228057 − 0.973648i
\(82\) −16.8530 + 0.0581522i −1.86111 + 0.00642183i
\(83\) −3.14732 + 3.14732i −0.345463 + 0.345463i −0.858417 − 0.512953i \(-0.828551\pi\)
0.512953 + 0.858417i \(0.328551\pi\)
\(84\) −14.6537 − 2.00619i −1.59885 − 0.218893i
\(85\) 1.21852 + 0.691366i 0.132167 + 0.0749891i
\(86\) −3.05637 − 4.20674i −0.329577 − 0.453624i
\(87\) −6.50032 − 4.51711i −0.696908 − 0.484285i
\(88\) −34.6159 − 17.6377i −3.69007 − 1.88018i
\(89\) −15.2669 − 4.96053i −1.61829 − 0.525815i −0.646754 − 0.762699i \(-0.723874\pi\)
−0.971538 + 0.236884i \(0.923874\pi\)
\(90\) −8.05990 + 15.7091i −0.849589 + 1.65589i
\(91\) 7.41401 0.777199
\(92\) −1.61250 − 10.1809i −0.168115 − 1.06143i
\(93\) 6.94610 + 2.09658i 0.720277 + 0.217405i
\(94\) −18.8834 + 13.7196i −1.94767 + 1.41506i
\(95\) 11.2233 − 7.40890i 1.15149 − 0.760138i
\(96\) −18.7590 − 9.06672i −1.91459 − 0.925369i
\(97\) 12.9758 + 2.05516i 1.31749 + 0.208670i 0.775303 − 0.631589i \(-0.217597\pi\)
0.542185 + 0.840259i \(0.317597\pi\)
\(98\) −4.77581 − 9.37306i −0.482430 − 0.946822i
\(99\) −15.1128 + 0.635699i −1.51889 + 0.0638901i
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.23.4 ✓ 640
3.2 odd 2 inner 615.2.bo.a.23.77 yes 640
5.2 odd 4 inner 615.2.bo.a.392.4 yes 640
15.2 even 4 inner 615.2.bo.a.392.77 yes 640
41.25 even 10 inner 615.2.bo.a.353.77 yes 640
123.107 odd 10 inner 615.2.bo.a.353.4 yes 640
205.107 odd 20 inner 615.2.bo.a.107.77 yes 640
615.107 even 20 inner 615.2.bo.a.107.4 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
615.2.bo.a.23.4 ✓ 640 1.1 even 1 trivial
615.2.bo.a.23.77 yes 640 3.2 odd 2 inner
615.2.bo.a.107.4 yes 640 615.107 even 20 inner
615.2.bo.a.107.77 yes 640 205.107 odd 20 inner
615.2.bo.a.353.4 yes 640 123.107 odd 10 inner
615.2.bo.a.353.77 yes 640 41.25 even 10 inner
615.2.bo.a.392.4 yes 640 5.2 odd 4 inner
615.2.bo.a.392.77 yes 640 15.2 even 4 inner