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Properties

Label 465.2.ba.a.4.1
Level $465$
Weight $2$
Character 465.4
Analytic conductor $3.713$
Analytic rank $0$
Dimension $128$
Inner twists $4$

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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] = [465,2,Mod(4,465)] 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("465.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.ba (of order \(10\), degree \(4\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 4.1
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.1

$q$-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+(-2.49874 + 0.811888i) q^{2} +(0.951057 + 0.309017i) q^{3} +(3.96648 - 2.88182i) q^{4} +(2.23400 - 0.0962096i) q^{5} -2.62733 q^{6} +(2.20823 + 3.03937i) q^{7} +(-4.48287 + 6.17014i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-5.50406 + 2.05416i) q^{10} +(-5.20715 + 3.78322i) q^{11} +(4.66288 - 1.51506i) q^{12} +(-2.59397 - 0.842832i) q^{13} +(-7.98541 - 5.80174i) q^{14} +(2.15439 + 0.598842i) q^{15} +(3.16191 - 9.73137i) q^{16} +(0.349529 - 0.481085i) q^{17} +(-2.49874 - 0.811888i) q^{18} +(1.89752 + 5.83996i) q^{19} +(8.58385 - 6.81959i) q^{20} +(1.16093 + 3.57299i) q^{21} +(9.93975 - 13.6809i) q^{22} +(-1.75277 + 2.41248i) q^{23} +(-6.17014 + 4.48287i) q^{24} +(4.98149 - 0.429864i) q^{25} +7.16593 q^{26} +(0.587785 + 0.809017i) q^{27} +(17.5178 + 5.69188i) q^{28} +(0.428967 + 1.32022i) q^{29} +(-5.86944 + 0.252774i) q^{30} +(-1.85161 - 5.25086i) q^{31} +11.6298i q^{32} +(-6.12138 + 1.98896i) q^{33} +(-0.482793 + 1.48588i) q^{34} +(5.22559 + 6.57748i) q^{35} +4.90284 q^{36} -5.83719i q^{37} +(-9.48280 - 13.0519i) q^{38} +(-2.20656 - 1.60316i) q^{39} +(-9.42109 + 14.2154i) q^{40} +(-0.820258 - 2.52450i) q^{41} +(-5.80174 - 7.98541i) q^{42} +(-4.39688 + 1.42863i) q^{43} +(-9.75153 + 30.0121i) q^{44} +(1.86389 + 1.23528i) q^{45} +(2.42104 - 7.45121i) q^{46} +(4.96408 + 1.61293i) q^{47} +(6.01432 - 8.27800i) q^{48} +(-2.19836 + 6.76584i) q^{49} +(-12.0984 + 5.11853i) q^{50} +(0.481085 - 0.349529i) q^{51} +(-12.7178 + 4.13227i) q^{52} +(6.31718 - 8.69485i) q^{53} +(-2.12555 - 1.54430i) q^{54} +(-11.2688 + 8.95268i) q^{55} -28.6525 q^{56} +6.14050i q^{57} +(-2.14375 - 2.95062i) q^{58} +(-0.869637 + 2.67647i) q^{59} +(10.2711 - 3.83326i) q^{60} +2.55465 q^{61} +(8.88980 + 11.6172i) q^{62} +3.75686i q^{63} +(-3.11829 - 9.59709i) q^{64} +(-5.87601 - 1.63332i) q^{65} +(13.6809 - 9.93975i) q^{66} +0.653667i q^{67} -2.91550i q^{68} +(-2.41248 + 1.75277i) q^{69} +(-18.3976 - 12.1928i) q^{70} +(5.87286 + 4.26688i) q^{71} +(-7.25343 + 2.35678i) q^{72} +(9.38870 + 12.9224i) q^{73} +(4.73915 + 14.5856i) q^{74} +(4.87051 + 1.13054i) q^{75} +(24.3562 + 17.6958i) q^{76} +(-22.9972 - 7.47224i) q^{77} +(6.81521 + 2.21440i) q^{78} +(-2.94357 - 2.13863i) q^{79} +(6.12745 - 22.0441i) q^{80} +(0.309017 + 0.951057i) q^{81} +(4.09922 + 5.64209i) q^{82} +(6.87002 - 2.23221i) q^{83} +(14.9015 + 10.8266i) q^{84} +(0.734562 - 1.10837i) q^{85} +(9.82675 - 7.13955i) q^{86} +1.38817i q^{87} -49.0885i q^{88} +(8.78902 - 6.38560i) q^{89} +(-5.66028 - 1.57335i) q^{90} +(-3.16641 - 9.74520i) q^{91} +14.6202i q^{92} +(-0.138380 - 5.56604i) q^{93} -13.7134 q^{94} +(4.80091 + 12.8639i) q^{95} +(-3.59381 + 11.0606i) q^{96} +(-0.605490 - 0.833385i) q^{97} -18.6909i q^{98} -6.43640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 36 q^{4} - 20 q^{6} + 32 q^{9} + 2 q^{10} + 4 q^{14} - 4 q^{15} - 40 q^{16} - 4 q^{19} + 36 q^{20} + 20 q^{21} - 20 q^{24} + 12 q^{25} + 24 q^{26} - 28 q^{29} - 8 q^{30} - 12 q^{31} - 44 q^{34}+ \cdots - 20 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.49874 + 0.811888i −1.76687 + 0.574092i −0.997875 − 0.0651597i \(-0.979244\pi\)
−0.768998 + 0.639251i \(0.779244\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) 3.96648 − 2.88182i 1.98324 − 1.44091i
\(5\) 2.23400 − 0.0962096i 0.999074 − 0.0430262i
\(6\) −2.62733 −1.07260
\(7\) 2.20823 + 3.03937i 0.834632 + 1.14877i 0.987043 + 0.160455i \(0.0512963\pi\)
−0.152411 + 0.988317i \(0.548704\pi\)
\(8\) −4.48287 + 6.17014i −1.58493 + 2.18147i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −5.50406 + 2.05416i −1.74054 + 0.649582i
\(11\) −5.20715 + 3.78322i −1.57002 + 1.14068i −0.642875 + 0.765971i \(0.722259\pi\)
−0.927141 + 0.374712i \(0.877741\pi\)
\(12\) 4.66288 − 1.51506i 1.34606 − 0.437360i
\(13\) −2.59397 − 0.842832i −0.719438 − 0.233760i −0.0736587 − 0.997284i \(-0.523468\pi\)
−0.645780 + 0.763524i \(0.723468\pi\)
\(14\) −7.98541 − 5.80174i −2.13419 − 1.55058i
\(15\) 2.15439 + 0.598842i 0.556261 + 0.154620i
\(16\) 3.16191 − 9.73137i 0.790478 − 2.43284i
\(17\) 0.349529 − 0.481085i 0.0847732 − 0.116680i −0.764524 − 0.644595i \(-0.777026\pi\)
0.849297 + 0.527915i \(0.177026\pi\)
\(18\) −2.49874 − 0.811888i −0.588958 − 0.191364i
\(19\) 1.89752 + 5.83996i 0.435321 + 1.33978i 0.892758 + 0.450537i \(0.148768\pi\)
−0.457437 + 0.889242i \(0.651232\pi\)
\(20\) 8.58385 − 6.81959i 1.91941 − 1.52491i
\(21\) 1.16093 + 3.57299i 0.253337 + 0.779690i
\(22\) 9.93975 − 13.6809i 2.11916 − 2.91678i
\(23\) −1.75277 + 2.41248i −0.365478 + 0.503037i −0.951665 − 0.307139i \(-0.900628\pi\)
0.586187 + 0.810176i \(0.300628\pi\)
\(24\) −6.17014 + 4.48287i −1.25947 + 0.915061i
\(25\) 4.98149 − 0.429864i 0.996297 − 0.0859728i
\(26\) 7.16593 1.40536
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) 17.5178 + 5.69188i 3.31055 + 1.07566i
\(29\) 0.428967 + 1.32022i 0.0796571 + 0.245159i 0.982952 − 0.183860i \(-0.0588592\pi\)
−0.903295 + 0.429019i \(0.858859\pi\)
\(30\) −5.86944 + 0.252774i −1.07161 + 0.0461500i
\(31\) −1.85161 − 5.25086i −0.332559 − 0.943082i
\(32\) 11.6298i 2.05588i
\(33\) −6.12138 + 1.98896i −1.06559 + 0.346233i
\(34\) −0.482793 + 1.48588i −0.0827983 + 0.254827i
\(35\) 5.22559 + 6.57748i 0.883287 + 1.11180i
\(36\) 4.90284 0.817140
\(37\) − 5.83719i − 0.959629i −0.877370 − 0.479814i \(-0.840704\pi\)
0.877370 − 0.479814i \(-0.159296\pi\)
\(38\) −9.48280 − 13.0519i −1.53831 − 2.11731i
\(39\) −2.20656 − 1.60316i −0.353333 − 0.256711i
\(40\) −9.42109 + 14.2154i −1.48960 + 2.24765i
\(41\) −0.820258 − 2.52450i −0.128103 − 0.394260i 0.866351 − 0.499436i \(-0.166459\pi\)
−0.994454 + 0.105176i \(0.966459\pi\)
\(42\) −5.80174 − 7.98541i −0.895228 − 1.23217i
\(43\) −4.39688 + 1.42863i −0.670518 + 0.217865i −0.624439 − 0.781073i \(-0.714673\pi\)
−0.0460788 + 0.998938i \(0.514673\pi\)
\(44\) −9.75153 + 30.0121i −1.47010 + 4.52450i
\(45\) 1.86389 + 1.23528i 0.277853 + 0.184144i
\(46\) 2.42104 − 7.45121i 0.356963 − 1.09862i
\(47\) 4.96408 + 1.61293i 0.724086 + 0.235270i 0.647794 − 0.761816i \(-0.275692\pi\)
0.0762920 + 0.997086i \(0.475692\pi\)
\(48\) 6.01432 − 8.27800i 0.868092 − 1.19483i
\(49\) −2.19836 + 6.76584i −0.314051 + 0.966549i
\(50\) −12.0984 + 5.11853i −1.71097 + 0.723869i
\(51\) 0.481085 − 0.349529i 0.0673654 − 0.0489439i
\(52\) −12.7178 + 4.13227i −1.76365 + 0.573043i
\(53\) 6.31718 − 8.69485i 0.867731 − 1.19433i −0.111939 − 0.993715i \(-0.535706\pi\)
0.979670 − 0.200615i \(-0.0642939\pi\)
\(54\) −2.12555 − 1.54430i −0.289251 − 0.210153i
\(55\) −11.2688 + 8.95268i −1.51948 + 1.20718i
\(56\) −28.6525 −3.82885
\(57\) 6.14050i 0.813329i
\(58\) −2.14375 − 2.95062i −0.281488 − 0.387435i
\(59\) −0.869637 + 2.67647i −0.113217 + 0.348447i −0.991571 − 0.129564i \(-0.958642\pi\)
0.878354 + 0.478011i \(0.158642\pi\)
\(60\) 10.2711 − 3.83326i 1.32599 − 0.494871i
\(61\) 2.55465 0.327089 0.163545 − 0.986536i \(-0.447707\pi\)
0.163545 + 0.986536i \(0.447707\pi\)
\(62\) 8.88980 + 11.6172i 1.12901 + 1.47539i
\(63\) 3.75686i 0.473320i
\(64\) −3.11829 − 9.59709i −0.389786 − 1.19964i
\(65\) −5.87601 − 1.63332i −0.728830 − 0.202588i
\(66\) 13.6809 − 9.93975i 1.68400 − 1.22350i
\(67\) 0.653667i 0.0798581i 0.999203 + 0.0399291i \(0.0127132\pi\)
−0.999203 + 0.0399291i \(0.987287\pi\)
\(68\) − 2.91550i − 0.353556i
\(69\) −2.41248 + 1.75277i −0.290429 + 0.211009i
\(70\) −18.3976 − 12.1928i −2.19893 − 1.45732i
\(71\) 5.87286 + 4.26688i 0.696980 + 0.506385i 0.878947 − 0.476919i \(-0.158247\pi\)
−0.181967 + 0.983305i \(0.558247\pi\)
\(72\) −7.25343 + 2.35678i −0.854825 + 0.277749i
\(73\) 9.38870 + 12.9224i 1.09886 + 1.51246i 0.836881 + 0.547385i \(0.184376\pi\)
0.261983 + 0.965072i \(0.415624\pi\)
\(74\) 4.73915 + 14.5856i 0.550915 + 1.69554i
\(75\) 4.87051 + 1.13054i 0.562398 + 0.130543i
\(76\) 24.3562 + 17.6958i 2.79385 + 2.02985i
\(77\) −22.9972 − 7.47224i −2.62077 − 0.851540i
\(78\) 6.81521 + 2.21440i 0.771670 + 0.250731i
\(79\) −2.94357 − 2.13863i −0.331178 − 0.240615i 0.409753 − 0.912197i \(-0.365615\pi\)
−0.740930 + 0.671582i \(0.765615\pi\)
\(80\) 6.12745 − 22.0441i 0.685070 − 2.46460i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 4.09922 + 5.64209i 0.452683 + 0.623064i
\(83\) 6.87002 − 2.23221i 0.754083 − 0.245016i 0.0933458 − 0.995634i \(-0.470244\pi\)
0.660737 + 0.750617i \(0.270244\pi\)
\(84\) 14.9015 + 10.8266i 1.62589 + 1.18128i
\(85\) 0.734562 − 1.10837i 0.0796744 − 0.120220i
\(86\) 9.82675 − 7.13955i 1.05965 − 0.769878i
\(87\) 1.38817i 0.148827i
\(88\) − 49.0885i − 5.23285i
\(89\) 8.78902 − 6.38560i 0.931634 − 0.676872i −0.0147582 − 0.999891i \(-0.504698\pi\)
0.946392 + 0.323019i \(0.104698\pi\)
\(90\) −5.66028 − 1.57335i −0.596646 − 0.165846i
\(91\) −3.16641 − 9.74520i −0.331930 − 1.02157i
\(92\) 14.6202i 1.52426i
\(93\) −0.138380 − 5.56604i −0.0143494 − 0.577172i
\(94\) −13.7134 −1.41443
\(95\) 4.80091 + 12.8639i 0.492563 + 1.31981i
\(96\) −3.59381 + 11.0606i −0.366792 + 1.12887i
\(97\) −0.605490 − 0.833385i −0.0614782 − 0.0846175i 0.777171 − 0.629290i \(-0.216654\pi\)
−0.838649 + 0.544672i \(0.816654\pi\)
\(98\) − 18.6909i − 1.88806i
\(99\) −6.43640 −0.646882
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 465.2.ba.a.4.1 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.32 yes 128
31.8 even 5 inner 465.2.ba.a.349.32 yes 128
155.39 even 10 inner 465.2.ba.a.349.1 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.1 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.32 yes 128 5.4 even 2 inner
465.2.ba.a.349.1 yes 128 155.39 even 10 inner
465.2.ba.a.349.32 yes 128 31.8 even 5 inner