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Properties

Label 475.2.e.f.201.3
Level $475$
Weight $2$
Character 475.201
Analytic conductor $3.793$
Analytic rank $0$
Dimension $12$
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] = [475,2,Mod(26,475)] 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("475.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-2] 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.3
Root \(-0.0149173 - 0.0258375i\) of defining polynomial
Character \(\chi\) \(=\) 475.201
Dual form 475.2.e.f.26.3

$q$-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+(-0.155554 + 0.269427i) q^{2} +(-0.514917 + 0.891863i) q^{3} +(0.951606 + 1.64823i) q^{4} +(-0.160195 - 0.277466i) q^{6} -3.28038 q^{7} -1.21432 q^{8} +(0.969720 + 1.67960i) q^{9} -5.16792 q^{11} -1.95999 q^{12} +(-1.76735 - 3.06115i) q^{13} +(0.510276 - 0.883825i) q^{14} +(-1.71432 + 2.96929i) q^{16} +(0.504641 - 0.874064i) q^{17} -0.603375 q^{18} +(2.42346 - 3.62310i) q^{19} +(1.68913 - 2.92565i) q^{21} +(0.803890 - 1.39238i) q^{22} +(3.83176 + 6.63680i) q^{23} +(0.625274 - 1.08301i) q^{24} +1.09968 q^{26} -5.08681 q^{27} +(-3.12163 - 5.40683i) q^{28} +(2.01303 + 3.48667i) q^{29} -4.60077 q^{31} +(-1.74766 - 3.02703i) q^{32} +(2.66105 - 4.60908i) q^{33} +(0.156998 + 0.271928i) q^{34} +(-1.84558 + 3.19664i) q^{36} -6.48831 q^{37} +(0.599183 + 1.21653i) q^{38} +3.64017 q^{39} +(3.40277 - 5.89377i) q^{41} +(0.525500 + 0.910193i) q^{42} +(-3.15511 + 5.46481i) q^{43} +(-4.91782 - 8.51792i) q^{44} -2.38418 q^{46} +(1.92079 + 3.32690i) q^{47} +(-1.76547 - 3.05788i) q^{48} +3.76091 q^{49} +(0.519697 + 0.900141i) q^{51} +(3.36365 - 5.82601i) q^{52} +(3.55946 + 6.16516i) q^{53} +(0.791273 - 1.37052i) q^{54} +3.98343 q^{56} +(1.98343 + 4.02699i) q^{57} -1.25254 q^{58} +(-6.73649 + 11.6679i) q^{59} +(-3.06850 - 5.31480i) q^{61} +(0.715668 - 1.23957i) q^{62} +(-3.18105 - 5.50975i) q^{63} -5.76986 q^{64} +(0.827874 + 1.43392i) q^{66} +(5.59897 + 9.69770i) q^{67} +1.92088 q^{68} -7.89215 q^{69} +(0.227702 - 0.394391i) q^{71} +(-1.17755 - 2.03958i) q^{72} +(-2.06691 + 3.57999i) q^{73} +(1.00928 - 1.74813i) q^{74} +(8.27788 + 0.546653i) q^{76} +16.9528 q^{77} +(-0.566242 + 0.980760i) q^{78} +(-1.44414 + 2.50132i) q^{79} +(-0.289876 + 0.502080i) q^{81} +(1.05863 + 1.83360i) q^{82} -5.50061 q^{83} +6.42953 q^{84} +(-0.981579 - 1.70014i) q^{86} -4.14617 q^{87} +6.27551 q^{88} +(3.56433 + 6.17360i) q^{89} +(5.79760 + 10.0417i) q^{91} +(-7.29264 + 12.6312i) q^{92} +(2.36902 - 4.10326i) q^{93} -1.19514 q^{94} +3.59960 q^{96} +(5.41252 - 9.37476i) q^{97} +(-0.585025 + 1.01329i) q^{98} +(-5.01144 - 8.68006i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{6} + 4 q^{7} + 12 q^{8} - 7 q^{9} - 2 q^{11} + 14 q^{12} - 5 q^{13} + 6 q^{14} + 6 q^{16} + 3 q^{17} + 14 q^{18} - 6 q^{19} - 3 q^{21} - 9 q^{22} + 6 q^{23} - 11 q^{24}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.155554 + 0.269427i −0.109993 + 0.190514i −0.915767 − 0.401709i \(-0.868416\pi\)
0.805774 + 0.592223i \(0.201750\pi\)
\(3\) −0.514917 + 0.891863i −0.297288 + 0.514917i −0.975514 − 0.219935i \(-0.929415\pi\)
0.678227 + 0.734853i \(0.262749\pi\)
\(4\) 0.951606 + 1.64823i 0.475803 + 0.824115i
\(5\) 0 0
\(6\) −0.160195 − 0.277466i −0.0653993 − 0.113275i
\(7\) −3.28038 −1.23987 −0.619934 − 0.784654i \(-0.712841\pi\)
−0.619934 + 0.784654i \(0.712841\pi\)
\(8\) −1.21432 −0.429327
\(9\) 0.969720 + 1.67960i 0.323240 + 0.559868i
\(10\) 0 0
\(11\) −5.16792 −1.55819 −0.779093 − 0.626908i \(-0.784320\pi\)
−0.779093 + 0.626908i \(0.784320\pi\)
\(12\) −1.95999 −0.565801
\(13\) −1.76735 − 3.06115i −0.490176 − 0.849010i 0.509760 − 0.860317i \(-0.329734\pi\)
−0.999936 + 0.0113069i \(0.996401\pi\)
\(14\) 0.510276 − 0.883825i 0.136377 − 0.236212i
\(15\) 0 0
\(16\) −1.71432 + 2.96929i −0.428580 + 0.742322i
\(17\) 0.504641 − 0.874064i 0.122393 − 0.211992i −0.798318 − 0.602237i \(-0.794276\pi\)
0.920711 + 0.390245i \(0.127610\pi\)
\(18\) −0.603375 −0.142217
\(19\) 2.42346 − 3.62310i 0.555979 − 0.831196i
\(20\) 0 0
\(21\) 1.68913 − 2.92565i 0.368598 − 0.638430i
\(22\) 0.803890 − 1.39238i 0.171390 − 0.296856i
\(23\) 3.83176 + 6.63680i 0.798976 + 1.38387i 0.920283 + 0.391253i \(0.127958\pi\)
−0.121307 + 0.992615i \(0.538709\pi\)
\(24\) 0.625274 − 1.08301i 0.127634 − 0.221068i
\(25\) 0 0
\(26\) 1.09968 0.215664
\(27\) −5.08681 −0.978956
\(28\) −3.12163 − 5.40683i −0.589933 − 1.02179i
\(29\) 2.01303 + 3.48667i 0.373810 + 0.647458i 0.990148 − 0.140024i \(-0.0447179\pi\)
−0.616338 + 0.787482i \(0.711385\pi\)
\(30\) 0 0
\(31\) −4.60077 −0.826323 −0.413162 − 0.910658i \(-0.635576\pi\)
−0.413162 + 0.910658i \(0.635576\pi\)
\(32\) −1.74766 − 3.02703i −0.308945 − 0.535109i
\(33\) 2.66105 − 4.60908i 0.463230 − 0.802337i
\(34\) 0.156998 + 0.271928i 0.0269249 + 0.0466353i
\(35\) 0 0
\(36\) −1.84558 + 3.19664i −0.307597 + 0.532774i
\(37\) −6.48831 −1.06667 −0.533336 − 0.845904i \(-0.679062\pi\)
−0.533336 + 0.845904i \(0.679062\pi\)
\(38\) 0.599183 + 1.21653i 0.0972004 + 0.197348i
\(39\) 3.64017 0.582893
\(40\) 0 0
\(41\) 3.40277 − 5.89377i 0.531423 − 0.920451i −0.467904 − 0.883779i \(-0.654991\pi\)
0.999327 − 0.0366724i \(-0.0116758\pi\)
\(42\) 0.525500 + 0.910193i 0.0810865 + 0.140446i
\(43\) −3.15511 + 5.46481i −0.481150 + 0.833376i −0.999766 − 0.0216315i \(-0.993114\pi\)
0.518616 + 0.855007i \(0.326447\pi\)
\(44\) −4.91782 − 8.51792i −0.741390 − 1.28412i
\(45\) 0 0
\(46\) −2.38418 −0.351528
\(47\) 1.92079 + 3.32690i 0.280175 + 0.485278i 0.971428 − 0.237335i \(-0.0762740\pi\)
−0.691252 + 0.722613i \(0.742941\pi\)
\(48\) −1.76547 − 3.05788i −0.254823 − 0.441366i
\(49\) 3.76091 0.537273
\(50\) 0 0
\(51\) 0.519697 + 0.900141i 0.0727721 + 0.126045i
\(52\) 3.36365 − 5.82601i 0.466454 − 0.807923i
\(53\) 3.55946 + 6.16516i 0.488929 + 0.846850i 0.999919 − 0.0127367i \(-0.00405432\pi\)
−0.510990 + 0.859587i \(0.670721\pi\)
\(54\) 0.791273 − 1.37052i 0.107679 − 0.186505i
\(55\) 0 0
\(56\) 3.98343 0.532309
\(57\) 1.98343 + 4.02699i 0.262711 + 0.533388i
\(58\) −1.25254 −0.164466
\(59\) −6.73649 + 11.6679i −0.877016 + 1.51904i −0.0224174 + 0.999749i \(0.507136\pi\)
−0.854599 + 0.519288i \(0.826197\pi\)
\(60\) 0 0
\(61\) −3.06850 − 5.31480i −0.392881 − 0.680491i 0.599947 − 0.800040i \(-0.295188\pi\)
−0.992828 + 0.119549i \(0.961855\pi\)
\(62\) 0.715668 − 1.23957i 0.0908899 − 0.157426i
\(63\) −3.18105 − 5.50975i −0.400775 − 0.694163i
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) 0.827874 + 1.43392i 0.101904 + 0.176503i
\(67\) 5.59897 + 9.69770i 0.684023 + 1.18476i 0.973743 + 0.227652i \(0.0731047\pi\)
−0.289719 + 0.957112i \(0.593562\pi\)
\(68\) 1.92088 0.232941
\(69\) −7.89215 −0.950103
\(70\) 0 0
\(71\) 0.227702 − 0.394391i 0.0270233 − 0.0468056i −0.852198 − 0.523220i \(-0.824731\pi\)
0.879221 + 0.476415i \(0.158064\pi\)
\(72\) −1.17755 − 2.03958i −0.138776 − 0.240367i
\(73\) −2.06691 + 3.57999i −0.241914 + 0.419007i −0.961259 − 0.275646i \(-0.911108\pi\)
0.719346 + 0.694652i \(0.244442\pi\)
\(74\) 1.00928 − 1.74813i 0.117327 − 0.203216i
\(75\) 0 0
\(76\) 8.27788 + 0.546653i 0.949538 + 0.0627054i
\(77\) 16.9528 1.93195
\(78\) −0.566242 + 0.980760i −0.0641143 + 0.111049i
\(79\) −1.44414 + 2.50132i −0.162478 + 0.281421i −0.935757 − 0.352646i \(-0.885282\pi\)
0.773278 + 0.634067i \(0.218615\pi\)
\(80\) 0 0
\(81\) −0.289876 + 0.502080i −0.0322085 + 0.0557867i
\(82\) 1.05863 + 1.83360i 0.116906 + 0.202487i
\(83\) −5.50061 −0.603770 −0.301885 − 0.953344i \(-0.597616\pi\)
−0.301885 + 0.953344i \(0.597616\pi\)
\(84\) 6.42953 0.701519
\(85\) 0 0
\(86\) −0.981579 − 1.70014i −0.105846 − 0.183331i
\(87\) −4.14617 −0.444516
\(88\) 6.27551 0.668971
\(89\) 3.56433 + 6.17360i 0.377818 + 0.654400i 0.990745 − 0.135740i \(-0.0433411\pi\)
−0.612926 + 0.790140i \(0.710008\pi\)
\(90\) 0 0
\(91\) 5.79760 + 10.0417i 0.607754 + 1.05266i
\(92\) −7.29264 + 12.6312i −0.760311 + 1.31690i
\(93\) 2.36902 − 4.10326i 0.245656 − 0.425488i
\(94\) −1.19514 −0.123270
\(95\) 0 0
\(96\) 3.59960 0.367382
\(97\) 5.41252 − 9.37476i 0.549558 − 0.951862i −0.448747 − 0.893659i \(-0.648129\pi\)
0.998305 − 0.0582034i \(-0.0185372\pi\)
\(98\) −0.585025 + 1.01329i −0.0590964 + 0.102358i
\(99\) −5.01144 − 8.68006i −0.503668 − 0.872379i
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 475.2.e.f.201.3 yes 12
5.2 odd 4 475.2.j.d.49.6 24
5.3 odd 4 475.2.j.d.49.7 24
5.4 even 2 475.2.e.h.201.4 yes 12
19.7 even 3 inner 475.2.e.f.26.3 ✓ 12
19.8 odd 6 9025.2.a.bs.1.3 6
19.11 even 3 9025.2.a.bz.1.4 6
95.7 odd 12 475.2.j.d.349.7 24
95.49 even 6 9025.2.a.br.1.3 6
95.64 even 6 475.2.e.h.26.4 yes 12
95.83 odd 12 475.2.j.d.349.6 24
95.84 odd 6 9025.2.a.by.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.e.f.26.3 ✓ 12 19.7 even 3 inner
475.2.e.f.201.3 yes 12 1.1 even 1 trivial
475.2.e.h.26.4 yes 12 95.64 even 6
475.2.e.h.201.4 yes 12 5.4 even 2
475.2.j.d.49.6 24 5.2 odd 4
475.2.j.d.49.7 24 5.3 odd 4
475.2.j.d.349.6 24 95.83 odd 12
475.2.j.d.349.7 24 95.7 odd 12
9025.2.a.br.1.3 6 95.49 even 6
9025.2.a.bs.1.3 6 19.8 odd 6
9025.2.a.by.1.4 6 95.84 odd 6
9025.2.a.bz.1.4 6 19.11 even 3