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Properties

Label 465.2.t.d.119.5
Level $465$
Weight $2$
Character 465.119
Analytic conductor $3.713$
Analytic rank $0$
Dimension $104$
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] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 119.5
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.5

$q$-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-2.15798 q^{2} +(-0.783102 + 1.54491i) q^{3} +2.65688 q^{4} +(-0.0977344 - 2.23393i) q^{5} +(1.68992 - 3.33389i) q^{6} +(1.65711 + 0.956733i) q^{7} -1.41754 q^{8} +(-1.77350 - 2.41965i) q^{9} +(0.210909 + 4.82078i) q^{10} +(0.00428924 + 0.00742918i) q^{11} +(-2.08061 + 4.10465i) q^{12} +(-0.450786 - 0.780785i) q^{13} +(-3.57601 - 2.06461i) q^{14} +(3.52776 + 1.59840i) q^{15} -2.25474 q^{16} +(-3.97099 - 2.29265i) q^{17} +(3.82719 + 5.22155i) q^{18} +(-1.45167 + 2.51437i) q^{19} +(-0.259669 - 5.93529i) q^{20} +(-2.77575 + 1.81087i) q^{21} +(-0.00925610 - 0.0160320i) q^{22} +0.416389i q^{23} +(1.11008 - 2.18998i) q^{24} +(-4.98090 + 0.436664i) q^{25} +(0.972789 + 1.68492i) q^{26} +(5.12697 - 0.845077i) q^{27} +(4.40275 + 2.54193i) q^{28} -6.05444 q^{29} +(-7.61284 - 3.44933i) q^{30} +(2.11875 - 5.14887i) q^{31} +7.70077 q^{32} +(-0.0148363 + 0.000808691i) q^{33} +(8.56933 + 4.94751i) q^{34} +(1.97532 - 3.79537i) q^{35} +(-4.71199 - 6.42872i) q^{36} +(5.72431 - 9.91480i) q^{37} +(3.13268 - 5.42597i) q^{38} +(1.55926 - 0.0849911i) q^{39} +(0.138543 + 3.16669i) q^{40} +(4.57943 - 2.64394i) q^{41} +(5.99002 - 3.90782i) q^{42} +(2.58143 - 4.47117i) q^{43} +(0.0113960 + 0.0197385i) q^{44} +(-5.23199 + 4.19837i) q^{45} -0.898559i q^{46} -7.32357 q^{47} +(1.76569 - 3.48337i) q^{48} +(-1.66933 - 2.89136i) q^{49} +(10.7487 - 0.942313i) q^{50} +(6.65164 - 4.33945i) q^{51} +(-1.19769 - 2.07445i) q^{52} +(-4.41444 + 2.54868i) q^{53} +(-11.0639 + 1.82366i) q^{54} +(0.0161771 - 0.0103080i) q^{55} +(-2.34902 - 1.35621i) q^{56} +(-2.74768 - 4.21172i) q^{57} +13.0654 q^{58} +(-6.30537 - 3.64040i) q^{59} +(9.37285 + 4.24677i) q^{60} -4.80464i q^{61} +(-4.57223 + 11.1112i) q^{62} +(-0.623935 - 5.70639i) q^{63} -12.1086 q^{64} +(-1.70016 + 1.08334i) q^{65} +(0.0320165 - 0.00174514i) q^{66} +(-2.32351 + 1.34148i) q^{67} +(-10.5505 - 6.09132i) q^{68} +(-0.643284 - 0.326075i) q^{69} +(-4.26270 + 8.19035i) q^{70} +(13.2327 - 7.63991i) q^{71} +(2.51402 + 3.42995i) q^{72} +(0.452889 + 0.784427i) q^{73} +(-12.3530 + 21.3960i) q^{74} +(3.22594 - 8.03700i) q^{75} +(-3.85693 + 6.68040i) q^{76} +0.0164146i q^{77} +(-3.36484 + 0.183409i) q^{78} +(-7.83131 - 4.52141i) q^{79} +(0.220365 + 5.03693i) q^{80} +(-2.70937 + 8.58250i) q^{81} +(-9.88233 + 5.70557i) q^{82} +(8.54745 - 4.93487i) q^{83} +(-7.37485 + 4.81127i) q^{84} +(-4.73353 + 9.09500i) q^{85} +(-5.57068 + 9.64870i) q^{86} +(4.74125 - 9.35358i) q^{87} +(-0.00608018 - 0.0105312i) q^{88} -0.102395 q^{89} +(11.2905 - 9.06000i) q^{90} -1.72513i q^{91} +1.10630i q^{92} +(6.29535 + 7.30538i) q^{93} +15.8041 q^{94} +(5.75881 + 2.99720i) q^{95} +(-6.03048 + 11.8970i) q^{96} -3.26410i q^{97} +(3.60237 + 6.23949i) q^{98} +(0.0103690 - 0.0235541i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15798 −1.52592 −0.762962 − 0.646444i \(-0.776255\pi\)
−0.762962 + 0.646444i \(0.776255\pi\)
\(3\) −0.783102 + 1.54491i −0.452124 + 0.891955i
\(4\) 2.65688 1.32844
\(5\) −0.0977344 − 2.23393i −0.0437082 − 0.999044i
\(6\) 1.68992 − 3.33389i 0.689907 − 1.36106i
\(7\) 1.65711 + 0.956733i 0.626329 + 0.361611i 0.779329 − 0.626615i \(-0.215560\pi\)
−0.153000 + 0.988226i \(0.548894\pi\)
\(8\) −1.41754 −0.501177
\(9\) −1.77350 − 2.41965i −0.591168 − 0.806549i
\(10\) 0.210909 + 4.82078i 0.0666953 + 1.52446i
\(11\) 0.00428924 + 0.00742918i 0.00129325 + 0.00223998i 0.866671 − 0.498880i \(-0.166255\pi\)
−0.865378 + 0.501120i \(0.832922\pi\)
\(12\) −2.08061 + 4.10465i −0.600620 + 1.18491i
\(13\) −0.450786 − 0.780785i −0.125026 − 0.216551i 0.796717 − 0.604352i \(-0.206568\pi\)
−0.921743 + 0.387801i \(0.873235\pi\)
\(14\) −3.57601 − 2.06461i −0.955729 − 0.551791i
\(15\) 3.52776 + 1.59840i 0.910864 + 0.412706i
\(16\) −2.25474 −0.563684
\(17\) −3.97099 − 2.29265i −0.963108 − 0.556050i −0.0659795 − 0.997821i \(-0.521017\pi\)
−0.897128 + 0.441771i \(0.854351\pi\)
\(18\) 3.82719 + 5.22155i 0.902077 + 1.23073i
\(19\) −1.45167 + 2.51437i −0.333037 + 0.576837i −0.983106 − 0.183039i \(-0.941407\pi\)
0.650069 + 0.759875i \(0.274740\pi\)
\(20\) −0.259669 − 5.93529i −0.0580638 − 1.32717i
\(21\) −2.77575 + 1.81087i −0.605719 + 0.395164i
\(22\) −0.00925610 − 0.0160320i −0.00197341 − 0.00341804i
\(23\) 0.416389i 0.0868231i 0.999057 + 0.0434115i \(0.0138227\pi\)
−0.999057 + 0.0434115i \(0.986177\pi\)
\(24\) 1.11008 − 2.18998i 0.226594 − 0.447027i
\(25\) −4.98090 + 0.436664i −0.996179 + 0.0873328i
\(26\) 0.972789 + 1.68492i 0.190780 + 0.330440i
\(27\) 5.12697 − 0.845077i 0.986686 − 0.162635i
\(28\) 4.40275 + 2.54193i 0.832041 + 0.480379i
\(29\) −6.05444 −1.12428 −0.562141 − 0.827041i \(-0.690022\pi\)
−0.562141 + 0.827041i \(0.690022\pi\)
\(30\) −7.61284 − 3.44933i −1.38991 − 0.629758i
\(31\) 2.11875 − 5.14887i 0.380539 − 0.924765i
\(32\) 7.70077 1.36132
\(33\) −0.0148363 0.000808691i −0.00258268 0.000140775i
\(34\) 8.56933 + 4.94751i 1.46963 + 0.848490i
\(35\) 1.97532 − 3.79537i 0.333890 − 0.641535i
\(36\) −4.71199 − 6.42872i −0.785332 − 1.07145i
\(37\) 5.72431 − 9.91480i 0.941071 − 1.62998i 0.177638 − 0.984096i \(-0.443154\pi\)
0.763433 − 0.645887i \(-0.223512\pi\)
\(38\) 3.13268 − 5.42597i 0.508189 − 0.880208i
\(39\) 1.55926 − 0.0849911i 0.249681 − 0.0136095i
\(40\) 0.138543 + 3.16669i 0.0219055 + 0.500698i
\(41\) 4.57943 − 2.64394i 0.715187 − 0.412914i −0.0977914 − 0.995207i \(-0.531178\pi\)
0.812979 + 0.582293i \(0.197844\pi\)
\(42\) 5.99002 − 3.90782i 0.924281 − 0.602990i
\(43\) 2.58143 − 4.47117i 0.393664 − 0.681847i −0.599265 − 0.800550i \(-0.704541\pi\)
0.992930 + 0.118704i \(0.0378739\pi\)
\(44\) 0.0113960 + 0.0197385i 0.00171801 + 0.00297569i
\(45\) −5.23199 + 4.19837i −0.779939 + 0.625856i
\(46\) − 0.898559i − 0.132485i
\(47\) −7.32357 −1.06825 −0.534126 − 0.845405i \(-0.679359\pi\)
−0.534126 + 0.845405i \(0.679359\pi\)
\(48\) 1.76569 − 3.48337i 0.254855 − 0.502781i
\(49\) −1.66933 − 2.89136i −0.238475 − 0.413051i
\(50\) 10.7487 − 0.942313i 1.52009 − 0.133263i
\(51\) 6.65164 − 4.33945i 0.931416 − 0.607645i
\(52\) −1.19769 − 2.07445i −0.166089 − 0.287675i
\(53\) −4.41444 + 2.54868i −0.606370 + 0.350088i −0.771543 − 0.636177i \(-0.780515\pi\)
0.165173 + 0.986265i \(0.447182\pi\)
\(54\) −11.0639 + 1.82366i −1.50561 + 0.248169i
\(55\) 0.0161771 − 0.0103080i 0.00218132 − 0.00138992i
\(56\) −2.34902 − 1.35621i −0.313901 − 0.181231i
\(57\) −2.74768 − 4.21172i −0.363938 − 0.557856i
\(58\) 13.0654 1.71557
\(59\) −6.30537 − 3.64040i −0.820889 − 0.473940i 0.0298342 − 0.999555i \(-0.490502\pi\)
−0.850723 + 0.525615i \(0.823835\pi\)
\(60\) 9.37285 + 4.24677i 1.21003 + 0.548256i
\(61\) − 4.80464i − 0.615172i −0.951520 − 0.307586i \(-0.900479\pi\)
0.951520 − 0.307586i \(-0.0995211\pi\)
\(62\) −4.57223 + 11.1112i −0.580674 + 1.41112i
\(63\) −0.623935 − 5.70639i −0.0786084 − 0.718937i
\(64\) −12.1086 −1.51358
\(65\) −1.70016 + 1.08334i −0.210879 + 0.134371i
\(66\) 0.0320165 − 0.00174514i 0.00394096 − 0.000214812i
\(67\) −2.32351 + 1.34148i −0.283862 + 0.163888i −0.635171 − 0.772372i \(-0.719070\pi\)
0.351308 + 0.936260i \(0.385737\pi\)
\(68\) −10.5505 − 6.09132i −1.27943 − 0.738681i
\(69\) −0.643284 − 0.326075i −0.0774423 − 0.0392548i
\(70\) −4.26270 + 8.19035i −0.509490 + 0.978934i
\(71\) 13.2327 − 7.63991i 1.57043 − 0.906690i 0.574318 − 0.818633i \(-0.305267\pi\)
0.996115 − 0.0880574i \(-0.0280659\pi\)
\(72\) 2.51402 + 3.42995i 0.296280 + 0.404224i
\(73\) 0.452889 + 0.784427i 0.0530066 + 0.0918102i 0.891311 − 0.453392i \(-0.149786\pi\)
−0.838305 + 0.545202i \(0.816453\pi\)
\(74\) −12.3530 + 21.3960i −1.43600 + 2.48723i
\(75\) 3.22594 − 8.03700i 0.372500 − 0.928032i
\(76\) −3.85693 + 6.68040i −0.442420 + 0.766294i
\(77\) 0.0164146i 0.00187062i
\(78\) −3.36484 + 0.183409i −0.380994 + 0.0207670i
\(79\) −7.83131 − 4.52141i −0.881092 − 0.508699i −0.0100735 − 0.999949i \(-0.503207\pi\)
−0.871018 + 0.491251i \(0.836540\pi\)
\(80\) 0.220365 + 5.03693i 0.0246376 + 0.563146i
\(81\) −2.70937 + 8.58250i −0.301041 + 0.953611i
\(82\) −9.88233 + 5.70557i −1.09132 + 0.630075i
\(83\) 8.54745 − 4.93487i 0.938205 − 0.541673i 0.0488075 − 0.998808i \(-0.484458\pi\)
0.889397 + 0.457136i \(0.151125\pi\)
\(84\) −7.37485 + 4.81127i −0.804662 + 0.524952i
\(85\) −4.73353 + 9.09500i −0.513423 + 0.986491i
\(86\) −5.57068 + 9.64870i −0.600702 + 1.04045i
\(87\) 4.74125 − 9.35358i 0.508315 − 1.00281i
\(88\) −0.00608018 − 0.0105312i −0.000648149 − 0.00112263i
\(89\) −0.102395 −0.0108538 −0.00542692 − 0.999985i \(-0.501727\pi\)
−0.00542692 + 0.999985i \(0.501727\pi\)
\(90\) 11.2905 − 9.06000i 1.19013 − 0.955008i
\(91\) − 1.72513i − 0.180843i
\(92\) 1.10630i 0.115339i
\(93\) 6.29535 + 7.30538i 0.652798 + 0.757532i
\(94\) 15.8041 1.63007
\(95\) 5.75881 + 2.99720i 0.590842 + 0.307506i
\(96\) −6.03048 + 11.8970i −0.615484 + 1.21423i
\(97\) − 3.26410i − 0.331419i −0.986175 − 0.165710i \(-0.947009\pi\)
0.986175 − 0.165710i \(-0.0529915\pi\)
\(98\) 3.60237 + 6.23949i 0.363895 + 0.630284i
\(99\) 0.0103690 − 0.0235541i 0.00104212 − 0.00236728i
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.t.d.119.5 ✓ 104
3.2 odd 2 inner 465.2.t.d.119.47 yes 104
5.4 even 2 inner 465.2.t.d.119.48 yes 104
15.14 odd 2 inner 465.2.t.d.119.6 yes 104
31.6 odd 6 inner 465.2.t.d.254.6 yes 104
93.68 even 6 inner 465.2.t.d.254.48 yes 104
155.99 odd 6 inner 465.2.t.d.254.47 yes 104
465.254 even 6 inner 465.2.t.d.254.5 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.5 ✓ 104 1.1 even 1 trivial
465.2.t.d.119.6 yes 104 15.14 odd 2 inner
465.2.t.d.119.47 yes 104 3.2 odd 2 inner
465.2.t.d.119.48 yes 104 5.4 even 2 inner
465.2.t.d.254.5 yes 104 465.254 even 6 inner
465.2.t.d.254.6 yes 104 31.6 odd 6 inner
465.2.t.d.254.47 yes 104 155.99 odd 6 inner
465.2.t.d.254.48 yes 104 93.68 even 6 inner