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Properties

Label 15.5.f.a.13.2
Level $15$
Weight $5$
Character 15.13
Analytic conductor $1.551$
Analytic rank $0$
Dimension $8$
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] = [15,5,Mod(7,15)] 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("15.7"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 15.f (of order \(4\), degree \(2\), 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: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.2
Root \(-2.08045 + 2.08045i\) of defining polynomial
Character \(\chi\) \(=\) 15.13
Dual form 15.5.f.a.7.2

$q$-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+(-2.08045 + 2.08045i) q^{2} +(3.67423 + 3.67423i) q^{3} +7.34348i q^{4} +(-8.43390 + 23.5344i) q^{5} -15.2881 q^{6} +(65.1093 - 65.1093i) q^{7} +(-48.5649 - 48.5649i) q^{8} +27.0000i q^{9} +(-31.4158 - 66.5084i) q^{10} +56.3999 q^{11} +(-26.9817 + 26.9817i) q^{12} +(0.983822 + 0.983822i) q^{13} +270.913i q^{14} +(-117.459 + 55.4829i) q^{15} +84.5776 q^{16} +(159.144 - 159.144i) q^{17} +(-56.1721 - 56.1721i) q^{18} +265.552i q^{19} +(-172.825 - 61.9342i) q^{20} +478.454 q^{21} +(-117.337 + 117.337i) q^{22} +(-185.430 - 185.430i) q^{23} -356.877i q^{24} +(-482.739 - 396.974i) q^{25} -4.09358 q^{26} +(-99.2043 + 99.2043i) q^{27} +(478.129 + 478.129i) q^{28} -544.071i q^{29} +(128.938 - 359.797i) q^{30} -710.805 q^{31} +(601.079 - 601.079i) q^{32} +(207.227 + 207.227i) q^{33} +662.183i q^{34} +(983.184 + 2081.43i) q^{35} -198.274 q^{36} +(-639.026 + 639.026i) q^{37} +(-552.467 - 552.467i) q^{38} +7.22958i q^{39} +(1552.54 - 733.355i) q^{40} -1325.70 q^{41} +(-995.397 + 995.397i) q^{42} +(-22.3358 - 22.3358i) q^{43} +414.172i q^{44} +(-635.430 - 227.715i) q^{45} +771.553 q^{46} +(456.136 - 456.136i) q^{47} +(310.758 + 310.758i) q^{48} -6077.44i q^{49} +(1830.20 - 178.428i) q^{50} +1169.47 q^{51} +(-7.22468 + 7.22468i) q^{52} +(-424.212 - 424.212i) q^{53} -412.779i q^{54} +(-475.672 + 1327.34i) q^{55} -6324.05 q^{56} +(-975.701 + 975.701i) q^{57} +(1131.91 + 1131.91i) q^{58} +3460.30i q^{59} +(-407.437 - 862.559i) q^{60} +4937.82 q^{61} +(1478.79 - 1478.79i) q^{62} +(1757.95 + 1757.95i) q^{63} +3854.27i q^{64} +(-31.4511 + 14.8562i) q^{65} -862.248 q^{66} +(-3801.58 + 3801.58i) q^{67} +(1168.67 + 1168.67i) q^{68} -1362.62i q^{69} +(-6375.78 - 2284.85i) q^{70} +7950.20 q^{71} +(1311.25 - 1311.25i) q^{72} +(-1940.65 - 1940.65i) q^{73} -2658.92i q^{74} +(-315.118 - 3232.27i) q^{75} -1950.08 q^{76} +(3672.16 - 3672.16i) q^{77} +(-15.0408 - 15.0408i) q^{78} +4083.83i q^{79} +(-713.319 + 1990.48i) q^{80} -729.000 q^{81} +(2758.05 - 2758.05i) q^{82} +(-9103.10 - 9103.10i) q^{83} +3513.51i q^{84} +(2403.16 + 5087.58i) q^{85} +92.9369 q^{86} +(1999.04 - 1999.04i) q^{87} +(-2739.06 - 2739.06i) q^{88} +7016.23i q^{89} +(1795.73 - 848.228i) q^{90} +128.112 q^{91} +(1361.70 - 1361.70i) q^{92} +(-2611.66 - 2611.66i) q^{93} +1897.93i q^{94} +(-6249.62 - 2239.64i) q^{95} +4417.01 q^{96} +(2571.87 - 2571.87i) q^{97} +(12643.8 + 12643.8i) q^{98} +1522.80i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 84 q^{5} + 36 q^{6} + 20 q^{7} + 180 q^{8} + 104 q^{10} - 288 q^{11} - 360 q^{12} - 340 q^{13} + 144 q^{15} + 620 q^{16} + 900 q^{17} + 564 q^{20} + 792 q^{21} - 1100 q^{22} - 1560 q^{23} - 1204 q^{25}+ \cdots + 46440 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.08045 + 2.08045i −0.520112 + 0.520112i −0.917605 − 0.397493i \(-0.869880\pi\)
0.397493 + 0.917605i \(0.369880\pi\)
\(3\) 3.67423 + 3.67423i 0.408248 + 0.408248i
\(4\) 7.34348i 0.458968i
\(5\) −8.43390 + 23.5344i −0.337356 + 0.941377i
\(6\) −15.2881 −0.424669
\(7\) 65.1093 − 65.1093i 1.32876 − 1.32876i 0.422309 − 0.906452i \(-0.361220\pi\)
0.906452 − 0.422309i \(-0.138780\pi\)
\(8\) −48.5649 − 48.5649i −0.758826 − 0.758826i
\(9\) 27.0000i 0.333333i
\(10\) −31.4158 − 66.5084i −0.314158 − 0.665084i
\(11\) 56.3999 0.466115 0.233058 − 0.972463i \(-0.425127\pi\)
0.233058 + 0.972463i \(0.425127\pi\)
\(12\) −26.9817 + 26.9817i −0.187373 + 0.187373i
\(13\) 0.983822 + 0.983822i 0.00582143 + 0.00582143i 0.710012 − 0.704190i \(-0.248690\pi\)
−0.704190 + 0.710012i \(0.748690\pi\)
\(14\) 270.913i 1.38221i
\(15\) −117.459 + 55.4829i −0.522041 + 0.246591i
\(16\) 84.5776 0.330381
\(17\) 159.144 − 159.144i 0.550673 − 0.550673i −0.375962 − 0.926635i \(-0.622688\pi\)
0.926635 + 0.375962i \(0.122688\pi\)
\(18\) −56.1721 − 56.1721i −0.173371 − 0.173371i
\(19\) 265.552i 0.735602i 0.929905 + 0.367801i \(0.119889\pi\)
−0.929905 + 0.367801i \(0.880111\pi\)
\(20\) −172.825 − 61.9342i −0.432062 − 0.154836i
\(21\) 478.454 1.08493
\(22\) −117.337 + 117.337i −0.242432 + 0.242432i
\(23\) −185.430 − 185.430i −0.350529 − 0.350529i 0.509778 − 0.860306i \(-0.329728\pi\)
−0.860306 + 0.509778i \(0.829728\pi\)
\(24\) − 356.877i − 0.619579i
\(25\) −482.739 − 396.974i −0.772382 − 0.635159i
\(26\) −4.09358 −0.00605559
\(27\) −99.2043 + 99.2043i −0.136083 + 0.136083i
\(28\) 478.129 + 478.129i 0.609858 + 0.609858i
\(29\) − 544.071i − 0.646933i −0.946240 − 0.323467i \(-0.895152\pi\)
0.946240 − 0.323467i \(-0.104848\pi\)
\(30\) 128.938 − 359.797i 0.143265 − 0.399774i
\(31\) −710.805 −0.739651 −0.369826 − 0.929101i \(-0.620583\pi\)
−0.369826 + 0.929101i \(0.620583\pi\)
\(32\) 601.079 − 601.079i 0.586991 − 0.586991i
\(33\) 207.227 + 207.227i 0.190291 + 0.190291i
\(34\) 662.183i 0.572823i
\(35\) 983.184 + 2081.43i 0.802599 + 1.69913i
\(36\) −198.274 −0.152989
\(37\) −639.026 + 639.026i −0.466783 + 0.466783i −0.900871 − 0.434088i \(-0.857071\pi\)
0.434088 + 0.900871i \(0.357071\pi\)
\(38\) −552.467 − 552.467i −0.382595 − 0.382595i
\(39\) 7.22958i 0.00475318i
\(40\) 1552.54 − 733.355i 0.970336 − 0.458347i
\(41\) −1325.70 −0.788638 −0.394319 − 0.918974i \(-0.629019\pi\)
−0.394319 + 0.918974i \(0.629019\pi\)
\(42\) −995.397 + 995.397i −0.564284 + 0.564284i
\(43\) −22.3358 − 22.3358i −0.0120799 − 0.0120799i 0.701041 − 0.713121i \(-0.252719\pi\)
−0.713121 + 0.701041i \(0.752719\pi\)
\(44\) 414.172i 0.213932i
\(45\) −635.430 − 227.715i −0.313792 − 0.112452i
\(46\) 771.553 0.364628
\(47\) 456.136 − 456.136i 0.206490 − 0.206490i −0.596284 − 0.802774i \(-0.703357\pi\)
0.802774 + 0.596284i \(0.203357\pi\)
\(48\) 310.758 + 310.758i 0.134878 + 0.134878i
\(49\) − 6077.44i − 2.53121i
\(50\) 1830.20 − 178.428i 0.732078 − 0.0713713i
\(51\) 1169.47 0.449622
\(52\) −7.22468 + 7.22468i −0.00267185 + 0.00267185i
\(53\) −424.212 − 424.212i −0.151019 − 0.151019i 0.627554 − 0.778573i \(-0.284056\pi\)
−0.778573 + 0.627554i \(0.784056\pi\)
\(54\) − 412.779i − 0.141556i
\(55\) −475.672 + 1327.34i −0.157247 + 0.438790i
\(56\) −6324.05 −2.01660
\(57\) −975.701 + 975.701i −0.300308 + 0.300308i
\(58\) 1131.91 + 1131.91i 0.336478 + 0.336478i
\(59\) 3460.30i 0.994054i 0.867735 + 0.497027i \(0.165575\pi\)
−0.867735 + 0.497027i \(0.834425\pi\)
\(60\) −407.437 − 862.559i −0.113177 − 0.239600i
\(61\) 4937.82 1.32701 0.663507 − 0.748170i \(-0.269067\pi\)
0.663507 + 0.748170i \(0.269067\pi\)
\(62\) 1478.79 − 1478.79i 0.384701 − 0.384701i
\(63\) 1757.95 + 1757.95i 0.442920 + 0.442920i
\(64\) 3854.27i 0.940983i
\(65\) −31.4511 + 14.8562i −0.00744406 + 0.00351627i
\(66\) −862.248 −0.197945
\(67\) −3801.58 + 3801.58i −0.846866 + 0.846866i −0.989741 − 0.142874i \(-0.954366\pi\)
0.142874 + 0.989741i \(0.454366\pi\)
\(68\) 1168.67 + 1168.67i 0.252741 + 0.252741i
\(69\) − 1362.62i − 0.286206i
\(70\) −6375.78 − 2284.85i −1.30118 − 0.466296i
\(71\) 7950.20 1.57711 0.788554 − 0.614966i \(-0.210830\pi\)
0.788554 + 0.614966i \(0.210830\pi\)
\(72\) 1311.25 − 1311.25i 0.252942 − 0.252942i
\(73\) −1940.65 − 1940.65i −0.364168 − 0.364168i 0.501177 − 0.865345i \(-0.332901\pi\)
−0.865345 + 0.501177i \(0.832901\pi\)
\(74\) − 2658.92i − 0.485558i
\(75\) −315.118 − 3232.27i −0.0560211 − 0.574626i
\(76\) −1950.08 −0.337617
\(77\) 3672.16 − 3672.16i 0.619355 − 0.619355i
\(78\) −15.0408 − 15.0408i −0.00247218 − 0.00247218i
\(79\) 4083.83i 0.654355i 0.944963 + 0.327178i \(0.106098\pi\)
−0.944963 + 0.327178i \(0.893902\pi\)
\(80\) −713.319 + 1990.48i −0.111456 + 0.311013i
\(81\) −729.000 −0.111111
\(82\) 2758.05 − 2758.05i 0.410180 − 0.410180i
\(83\) −9103.10 − 9103.10i −1.32140 − 1.32140i −0.912646 − 0.408750i \(-0.865965\pi\)
−0.408750 − 0.912646i \(-0.634035\pi\)
\(84\) 3513.51i 0.497947i
\(85\) 2403.16 + 5087.58i 0.332618 + 0.704164i
\(86\) 92.9369 0.0125658
\(87\) 1999.04 − 1999.04i 0.264109 − 0.264109i
\(88\) −2739.06 − 2739.06i −0.353700 − 0.353700i
\(89\) 7016.23i 0.885775i 0.896577 + 0.442888i \(0.146046\pi\)
−0.896577 + 0.442888i \(0.853954\pi\)
\(90\) 1795.73 − 848.228i 0.221695 − 0.104719i
\(91\) 128.112 0.0154706
\(92\) 1361.70 − 1361.70i 0.160881 − 0.160881i
\(93\) −2611.66 − 2611.66i −0.301961 − 0.301961i
\(94\) 1897.93i 0.214796i
\(95\) −6249.62 − 2239.64i −0.692479 − 0.248160i
\(96\) 4417.01 0.479276
\(97\) 2571.87 − 2571.87i 0.273342 − 0.273342i −0.557102 − 0.830444i \(-0.688087\pi\)
0.830444 + 0.557102i \(0.188087\pi\)
\(98\) 12643.8 + 12643.8i 1.31651 + 1.31651i
\(99\) 1522.80i 0.155372i
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 15.5.f.a.13.2 yes 8
3.2 odd 2 45.5.g.e.28.3 8
4.3 odd 2 240.5.bg.c.193.2 8
5.2 odd 4 inner 15.5.f.a.7.2 ✓ 8
5.3 odd 4 75.5.f.e.7.3 8
5.4 even 2 75.5.f.e.43.3 8
15.2 even 4 45.5.g.e.37.3 8
15.8 even 4 225.5.g.m.82.2 8
15.14 odd 2 225.5.g.m.118.2 8
20.7 even 4 240.5.bg.c.97.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.5.f.a.7.2 ✓ 8 5.2 odd 4 inner
15.5.f.a.13.2 yes 8 1.1 even 1 trivial
45.5.g.e.28.3 8 3.2 odd 2
45.5.g.e.37.3 8 15.2 even 4
75.5.f.e.7.3 8 5.3 odd 4
75.5.f.e.43.3 8 5.4 even 2
225.5.g.m.82.2 8 15.8 even 4
225.5.g.m.118.2 8 15.14 odd 2
240.5.bg.c.97.2 8 20.7 even 4
240.5.bg.c.193.2 8 4.3 odd 2