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Properties

Label 400.2.j.e.43.10
Level $400$
Weight $2$
Character 400.43
Analytic conductor $3.194$
Analytic rank $0$
Dimension $24$
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] = [400,2,Mod(43,400)] 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("400.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.10
Character \(\chi\) \(=\) 400.43
Dual form 400.2.j.e.307.10

$q$-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+(0.911865 - 1.08097i) q^{2} -0.619018i q^{3} +(-0.337006 - 1.97140i) q^{4} +(-0.669142 - 0.564461i) q^{6} +(1.82373 - 1.82373i) q^{7} +(-2.43834 - 1.43336i) q^{8} +2.61682 q^{9} +(-0.567849 + 0.567849i) q^{11} +(-1.22033 + 0.208613i) q^{12} -2.78771 q^{13} +(-0.308408 - 3.63440i) q^{14} +(-3.77285 + 1.32875i) q^{16} +(3.65193 - 3.65193i) q^{17} +(2.38618 - 2.82871i) q^{18} +(-4.51065 + 4.51065i) q^{19} +(-1.12892 - 1.12892i) q^{21} +(0.0960281 + 1.13163i) q^{22} +(-2.15520 - 2.15520i) q^{23} +(-0.887275 + 1.50937i) q^{24} +(-2.54201 + 3.01344i) q^{26} -3.47691i q^{27} +(-4.20991 - 2.98070i) q^{28} +(3.20259 + 3.20259i) q^{29} -3.54087i q^{31} +(-2.00399 + 5.28999i) q^{32} +(0.351509 + 0.351509i) q^{33} +(-0.617572 - 7.27770i) q^{34} +(-0.881883 - 5.15880i) q^{36} +5.22371 q^{37} +(0.762790 + 8.98900i) q^{38} +1.72564i q^{39} +8.76287i q^{41} +(-2.24976 + 0.190910i) q^{42} +10.8604 q^{43} +(1.31083 + 0.928090i) q^{44} +(-4.29497 + 0.364463i) q^{46} +(-3.22050 - 3.22050i) q^{47} +(0.822520 + 2.33547i) q^{48} +0.348024i q^{49} +(-2.26061 - 2.26061i) q^{51} +(0.939474 + 5.49569i) q^{52} +12.8658i q^{53} +(-3.75845 - 3.17047i) q^{54} +(-7.06092 + 1.83281i) q^{56} +(2.79218 + 2.79218i) q^{57} +(6.38224 - 0.541585i) q^{58} +(3.79319 + 3.79319i) q^{59} +(6.63395 - 6.63395i) q^{61} +(-3.82759 - 3.22879i) q^{62} +(4.77236 - 4.77236i) q^{63} +(3.89097 + 6.99002i) q^{64} +(0.700500 - 0.0594431i) q^{66} -7.78732 q^{67} +(-8.43014 - 5.96870i) q^{68} +(-1.33411 + 1.33411i) q^{69} +13.6650 q^{71} +(-6.38068 - 3.75083i) q^{72} +(-1.34382 + 1.34382i) q^{73} +(4.76332 - 5.64669i) q^{74} +(10.4124 + 7.37220i) q^{76} +2.07120i q^{77} +(1.86537 + 1.57355i) q^{78} -16.3528 q^{79} +5.69818 q^{81} +(9.47243 + 7.99055i) q^{82} -0.391056i q^{83} +(-1.84511 + 2.60601i) q^{84} +(9.90319 - 11.7398i) q^{86} +(1.98246 - 1.98246i) q^{87} +(2.19854 - 0.570676i) q^{88} +18.0317 q^{89} +(-5.08402 + 5.08402i) q^{91} +(-3.52246 + 4.97509i) q^{92} -2.19186 q^{93} +(-6.41793 + 0.544613i) q^{94} +(3.27460 + 1.24051i) q^{96} +(-6.43517 + 6.43517i) q^{97} +(0.376204 + 0.317350i) q^{98} +(-1.48596 + 1.48596i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{4} + 12 q^{6} - 40 q^{9} - 20 q^{11} + 44 q^{14} - 4 q^{16} - 12 q^{19} + 24 q^{24} + 32 q^{26} + 8 q^{29} + 32 q^{34} - 60 q^{36} - 44 q^{44} - 76 q^{46} + 20 q^{51} + 16 q^{54} - 28 q^{56}+ \cdots + 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.911865 − 1.08097i 0.644786 − 0.764363i
\(3\) − 0.619018i − 0.357390i −0.983904 − 0.178695i \(-0.942812\pi\)
0.983904 − 0.178695i \(-0.0571876\pi\)
\(4\) −0.337006 − 1.97140i −0.168503 − 0.985701i
\(5\) 0 0
\(6\) −0.669142 − 0.564461i −0.273176 − 0.230440i
\(7\) 1.82373 − 1.82373i 0.689305 − 0.689305i −0.272773 − 0.962078i \(-0.587941\pi\)
0.962078 + 0.272773i \(0.0879409\pi\)
\(8\) −2.43834 − 1.43336i −0.862082 − 0.506768i
\(9\) 2.61682 0.872272
\(10\) 0 0
\(11\) −0.567849 + 0.567849i −0.171213 + 0.171213i −0.787512 − 0.616299i \(-0.788631\pi\)
0.616299 + 0.787512i \(0.288631\pi\)
\(12\) −1.22033 + 0.208613i −0.352280 + 0.0602213i
\(13\) −2.78771 −0.773171 −0.386585 − 0.922254i \(-0.626346\pi\)
−0.386585 + 0.922254i \(0.626346\pi\)
\(14\) −0.308408 − 3.63440i −0.0824256 − 0.971333i
\(15\) 0 0
\(16\) −3.77285 + 1.32875i −0.943214 + 0.332187i
\(17\) 3.65193 − 3.65193i 0.885723 − 0.885723i −0.108386 − 0.994109i \(-0.534568\pi\)
0.994109 + 0.108386i \(0.0345683\pi\)
\(18\) 2.38618 − 2.82871i 0.562429 − 0.666733i
\(19\) −4.51065 + 4.51065i −1.03481 + 1.03481i −0.0354432 + 0.999372i \(0.511284\pi\)
−0.999372 + 0.0354432i \(0.988716\pi\)
\(20\) 0 0
\(21\) −1.12892 − 1.12892i −0.246351 − 0.246351i
\(22\) 0.0960281 + 1.13163i 0.0204733 + 0.241264i
\(23\) −2.15520 − 2.15520i −0.449391 − 0.449391i 0.445761 − 0.895152i \(-0.352933\pi\)
−0.895152 + 0.445761i \(0.852933\pi\)
\(24\) −0.887275 + 1.50937i −0.181114 + 0.308100i
\(25\) 0 0
\(26\) −2.54201 + 3.01344i −0.498529 + 0.590983i
\(27\) − 3.47691i − 0.669132i
\(28\) −4.20991 − 2.98070i −0.795598 − 0.563299i
\(29\) 3.20259 + 3.20259i 0.594705 + 0.594705i 0.938899 − 0.344193i \(-0.111848\pi\)
−0.344193 + 0.938899i \(0.611848\pi\)
\(30\) 0 0
\(31\) − 3.54087i − 0.635959i −0.948098 − 0.317980i \(-0.896996\pi\)
0.948098 − 0.317980i \(-0.103004\pi\)
\(32\) −2.00399 + 5.28999i −0.354259 + 0.935147i
\(33\) 0.351509 + 0.351509i 0.0611898 + 0.0611898i
\(34\) −0.617572 − 7.27770i −0.105913 − 1.24812i
\(35\) 0 0
\(36\) −0.881883 − 5.15880i −0.146980 − 0.859800i
\(37\) 5.22371 0.858773 0.429386 − 0.903121i \(-0.358730\pi\)
0.429386 + 0.903121i \(0.358730\pi\)
\(38\) 0.762790 + 8.98900i 0.123741 + 1.45821i
\(39\) 1.72564i 0.276324i
\(40\) 0 0
\(41\) 8.76287i 1.36853i 0.729233 + 0.684265i \(0.239877\pi\)
−0.729233 + 0.684265i \(0.760123\pi\)
\(42\) −2.24976 + 0.190910i −0.347145 + 0.0294581i
\(43\) 10.8604 1.65619 0.828096 − 0.560587i \(-0.189424\pi\)
0.828096 + 0.560587i \(0.189424\pi\)
\(44\) 1.31083 + 0.928090i 0.197615 + 0.139915i
\(45\) 0 0
\(46\) −4.29497 + 0.364463i −0.633259 + 0.0537372i
\(47\) −3.22050 − 3.22050i −0.469758 − 0.469758i 0.432078 − 0.901836i \(-0.357780\pi\)
−0.901836 + 0.432078i \(0.857780\pi\)
\(48\) 0.822520 + 2.33547i 0.118720 + 0.337095i
\(49\) 0.348024i 0.0497176i
\(50\) 0 0
\(51\) −2.26061 − 2.26061i −0.316549 − 0.316549i
\(52\) 0.939474 + 5.49569i 0.130282 + 0.762115i
\(53\) 12.8658i 1.76725i 0.468194 + 0.883626i \(0.344905\pi\)
−0.468194 + 0.883626i \(0.655095\pi\)
\(54\) −3.75845 − 3.17047i −0.511460 − 0.431447i
\(55\) 0 0
\(56\) −7.06092 + 1.83281i −0.943555 + 0.244919i
\(57\) 2.79218 + 2.79218i 0.369833 + 0.369833i
\(58\) 6.38224 − 0.541585i 0.838029 − 0.0711136i
\(59\) 3.79319 + 3.79319i 0.493832 + 0.493832i 0.909511 − 0.415679i \(-0.136456\pi\)
−0.415679 + 0.909511i \(0.636456\pi\)
\(60\) 0 0
\(61\) 6.63395 − 6.63395i 0.849390 − 0.849390i −0.140667 − 0.990057i \(-0.544925\pi\)
0.990057 + 0.140667i \(0.0449246\pi\)
\(62\) −3.82759 − 3.22879i −0.486104 − 0.410057i
\(63\) 4.77236 − 4.77236i 0.601261 − 0.601261i
\(64\) 3.89097 + 6.99002i 0.486371 + 0.873752i
\(65\) 0 0
\(66\) 0.700500 − 0.0594431i 0.0862256 − 0.00731694i
\(67\) −7.78732 −0.951373 −0.475686 − 0.879615i \(-0.657800\pi\)
−0.475686 + 0.879615i \(0.657800\pi\)
\(68\) −8.43014 − 5.96870i −1.02230 − 0.723811i
\(69\) −1.33411 + 1.33411i −0.160608 + 0.160608i
\(70\) 0 0
\(71\) 13.6650 1.62174 0.810868 − 0.585229i \(-0.198995\pi\)
0.810868 + 0.585229i \(0.198995\pi\)
\(72\) −6.38068 − 3.75083i −0.751970 − 0.442040i
\(73\) −1.34382 + 1.34382i −0.157282 + 0.157282i −0.781361 − 0.624079i \(-0.785474\pi\)
0.624079 + 0.781361i \(0.285474\pi\)
\(74\) 4.76332 − 5.64669i 0.553724 − 0.656415i
\(75\) 0 0
\(76\) 10.4124 + 7.37220i 1.19439 + 0.845649i
\(77\) 2.07120i 0.236036i
\(78\) 1.86537 + 1.57355i 0.211212 + 0.178170i
\(79\) −16.3528 −1.83984 −0.919918 − 0.392111i \(-0.871745\pi\)
−0.919918 + 0.392111i \(0.871745\pi\)
\(80\) 0 0
\(81\) 5.69818 0.633131
\(82\) 9.47243 + 7.99055i 1.04605 + 0.882409i
\(83\) − 0.391056i − 0.0429240i −0.999770 − 0.0214620i \(-0.993168\pi\)
0.999770 − 0.0214620i \(-0.00683209\pi\)
\(84\) −1.84511 + 2.60601i −0.201318 + 0.284339i
\(85\) 0 0
\(86\) 9.90319 − 11.7398i 1.06789 − 1.26593i
\(87\) 1.98246 − 1.98246i 0.212542 − 0.212542i
\(88\) 2.19854 − 0.570676i 0.234365 − 0.0608343i
\(89\) 18.0317 1.91135 0.955676 − 0.294419i \(-0.0951263\pi\)
0.955676 + 0.294419i \(0.0951263\pi\)
\(90\) 0 0
\(91\) −5.08402 + 5.08402i −0.532950 + 0.532950i
\(92\) −3.52246 + 4.97509i −0.367242 + 0.518689i
\(93\) −2.19186 −0.227286
\(94\) −6.41793 + 0.544613i −0.661959 + 0.0561726i
\(95\) 0 0
\(96\) 3.27460 + 1.24051i 0.334213 + 0.126609i
\(97\) −6.43517 + 6.43517i −0.653392 + 0.653392i −0.953808 − 0.300416i \(-0.902874\pi\)
0.300416 + 0.953808i \(0.402874\pi\)
\(98\) 0.376204 + 0.317350i 0.0380024 + 0.0320572i
\(99\) −1.48596 + 1.48596i −0.149344 + 0.149344i
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 400.2.j.e.43.10 yes 24
4.3 odd 2 1600.2.j.e.143.7 24
5.2 odd 4 400.2.s.e.107.9 yes 24
5.3 odd 4 400.2.s.e.107.4 yes 24
5.4 even 2 inner 400.2.j.e.43.3 ✓ 24
16.3 odd 4 400.2.s.e.243.9 yes 24
16.13 even 4 1600.2.s.e.943.7 24
20.3 even 4 1600.2.s.e.207.6 24
20.7 even 4 1600.2.s.e.207.7 24
20.19 odd 2 1600.2.j.e.143.6 24
80.3 even 4 inner 400.2.j.e.307.3 yes 24
80.13 odd 4 1600.2.j.e.1007.7 24
80.19 odd 4 400.2.s.e.243.4 yes 24
80.29 even 4 1600.2.s.e.943.6 24
80.67 even 4 inner 400.2.j.e.307.10 yes 24
80.77 odd 4 1600.2.j.e.1007.6 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.j.e.43.3 ✓ 24 5.4 even 2 inner
400.2.j.e.43.10 yes 24 1.1 even 1 trivial
400.2.j.e.307.3 yes 24 80.3 even 4 inner
400.2.j.e.307.10 yes 24 80.67 even 4 inner
400.2.s.e.107.4 yes 24 5.3 odd 4
400.2.s.e.107.9 yes 24 5.2 odd 4
400.2.s.e.243.4 yes 24 80.19 odd 4
400.2.s.e.243.9 yes 24 16.3 odd 4
1600.2.j.e.143.6 24 20.19 odd 2
1600.2.j.e.143.7 24 4.3 odd 2
1600.2.j.e.1007.6 24 80.77 odd 4
1600.2.j.e.1007.7 24 80.13 odd 4
1600.2.s.e.207.6 24 20.3 even 4
1600.2.s.e.207.7 24 20.7 even 4
1600.2.s.e.943.6 24 80.29 even 4
1600.2.s.e.943.7 24 16.13 even 4