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Properties

Label 400.2.q.g.349.3
Level $400$
Weight $2$
Character 400.349
Analytic conductor $3.194$
Analytic rank $0$
Dimension $16$
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] = [400,2,Mod(149,400)] 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("400.149"); 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([0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.q (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-4] 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 349.3
Root \(-0.296075 - 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 400.349
Dual form 400.2.q.g.149.3

$q$-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+(-0.889181 - 1.09971i) q^{2} +(0.120009 - 0.120009i) q^{3} +(-0.418713 + 1.95568i) q^{4} +(-0.238684 - 0.0252650i) q^{6} +2.66881 q^{7} +(2.52299 - 1.27849i) q^{8} +2.97120i q^{9} +(-3.49714 + 3.49714i) q^{11} +(0.184450 + 0.284948i) q^{12} +(-2.94072 + 2.94072i) q^{13} +(-2.37306 - 2.93491i) q^{14} +(-3.64936 - 1.63774i) q^{16} -1.85116i q^{17} +(3.26745 - 2.64193i) q^{18} +(3.44856 + 3.44856i) q^{19} +(0.320281 - 0.320281i) q^{21} +(6.95543 + 0.736240i) q^{22} +0.707288 q^{23} +(0.149351 - 0.456211i) q^{24} +(5.84877 + 0.619099i) q^{26} +(0.716597 + 0.716597i) q^{27} +(-1.11747 + 5.21934i) q^{28} +(3.49909 + 3.49909i) q^{29} +6.84272 q^{31} +(1.44391 + 5.46947i) q^{32} +0.839377i q^{33} +(-2.03573 + 1.64601i) q^{34} +(-5.81070 - 1.24408i) q^{36} +(0.0975060 + 0.0975060i) q^{37} +(0.726013 - 6.85881i) q^{38} +0.705826i q^{39} -10.2052i q^{41} +(-0.637004 - 0.0674276i) q^{42} +(4.43844 + 4.43844i) q^{43} +(-5.37499 - 8.30359i) q^{44} +(-0.628908 - 0.777810i) q^{46} +1.89428i q^{47} +(-0.634498 + 0.241413i) q^{48} +0.122561 q^{49} +(-0.222155 - 0.222155i) q^{51} +(-4.51979 - 6.98243i) q^{52} +(-7.43897 - 7.43897i) q^{53} +(0.150862 - 1.42523i) q^{54} +(6.73338 - 3.41205i) q^{56} +0.827717 q^{57} +(0.736651 - 6.95931i) q^{58} +(-0.959574 + 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} +(-6.08442 - 7.52499i) q^{62} +7.92956i q^{63} +(4.73092 - 6.45123i) q^{64} +(0.923069 - 0.746358i) q^{66} +(3.49691 - 3.49691i) q^{67} +(3.62027 + 0.775103i) q^{68} +(0.0848809 - 0.0848809i) q^{69} +7.86777i q^{71} +(3.79865 + 7.49629i) q^{72} -15.6564 q^{73} +(0.0205276 - 0.193929i) q^{74} +(-8.18824 + 5.30033i) q^{76} +(-9.33322 + 9.33322i) q^{77} +(0.776202 - 0.627607i) q^{78} +6.70212 q^{79} -8.74159 q^{81} +(-11.2228 + 9.07431i) q^{82} +(3.87327 - 3.87327i) q^{83} +(0.492261 + 0.760473i) q^{84} +(0.934407 - 8.82755i) q^{86} +0.839845 q^{87} +(-4.35218 + 13.2943i) q^{88} -10.5055i q^{89} +(-7.84824 + 7.84824i) q^{91} +(-0.296151 + 1.38323i) q^{92} +(0.821187 - 0.821187i) q^{93} +(2.08316 - 1.68436i) q^{94} +(0.829667 + 0.483103i) q^{96} -4.79937i q^{97} +(-0.108979 - 0.134781i) q^{98} +(-10.3907 - 10.3907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 4 q^{4} - 12 q^{6} - 8 q^{7} + 8 q^{8} - 8 q^{11} - 20 q^{12} - 4 q^{14} + 16 q^{16} - 12 q^{18} + 8 q^{19} + 20 q^{22} - 24 q^{23} - 8 q^{24} - 16 q^{26} - 24 q^{27} - 20 q^{28} + 16 q^{29}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.889181 − 1.09971i −0.628746 − 0.777611i
\(3\) 0.120009 − 0.120009i 0.0692872 − 0.0692872i −0.671614 − 0.740901i \(-0.734399\pi\)
0.740901 + 0.671614i \(0.234399\pi\)
\(4\) −0.418713 + 1.95568i −0.209357 + 0.977839i
\(5\) 0 0
\(6\) −0.238684 − 0.0252650i −0.0974425 − 0.0103144i
\(7\) 2.66881 1.00872 0.504358 − 0.863495i \(-0.331729\pi\)
0.504358 + 0.863495i \(0.331729\pi\)
\(8\) 2.52299 − 1.27849i 0.892010 − 0.452015i
\(9\) 2.97120i 0.990399i
\(10\) 0 0
\(11\) −3.49714 + 3.49714i −1.05443 + 1.05443i −0.0559977 + 0.998431i \(0.517834\pi\)
−0.998431 + 0.0559977i \(0.982166\pi\)
\(12\) 0.184450 + 0.284948i 0.0532460 + 0.0822574i
\(13\) −2.94072 + 2.94072i −0.815610 + 0.815610i −0.985468 − 0.169858i \(-0.945669\pi\)
0.169858 + 0.985468i \(0.445669\pi\)
\(14\) −2.37306 − 2.93491i −0.634227 − 0.784389i
\(15\) 0 0
\(16\) −3.64936 − 1.63774i −0.912340 − 0.409434i
\(17\) − 1.85116i − 0.448971i −0.974477 − 0.224486i \(-0.927930\pi\)
0.974477 − 0.224486i \(-0.0720702\pi\)
\(18\) 3.26745 − 2.64193i 0.770144 − 0.622709i
\(19\) 3.44856 + 3.44856i 0.791155 + 0.791155i 0.981682 − 0.190527i \(-0.0610197\pi\)
−0.190527 + 0.981682i \(0.561020\pi\)
\(20\) 0 0
\(21\) 0.320281 − 0.320281i 0.0698911 − 0.0698911i
\(22\) 6.95543 + 0.736240i 1.48290 + 0.156967i
\(23\) 0.707288 0.147480 0.0737399 − 0.997278i \(-0.476507\pi\)
0.0737399 + 0.997278i \(0.476507\pi\)
\(24\) 0.149351 − 0.456211i 0.0304860 − 0.0931237i
\(25\) 0 0
\(26\) 5.84877 + 0.619099i 1.14704 + 0.121415i
\(27\) 0.716597 + 0.716597i 0.137909 + 0.137909i
\(28\) −1.11747 + 5.21934i −0.211181 + 0.986363i
\(29\) 3.49909 + 3.49909i 0.649766 + 0.649766i 0.952936 − 0.303171i \(-0.0980452\pi\)
−0.303171 + 0.952936i \(0.598045\pi\)
\(30\) 0 0
\(31\) 6.84272 1.22899 0.614494 − 0.788921i \(-0.289360\pi\)
0.614494 + 0.788921i \(0.289360\pi\)
\(32\) 1.44391 + 5.46947i 0.255250 + 0.966875i
\(33\) 0.839377i 0.146117i
\(34\) −2.03573 + 1.64601i −0.349125 + 0.282289i
\(35\) 0 0
\(36\) −5.81070 − 1.24408i −0.968451 − 0.207346i
\(37\) 0.0975060 + 0.0975060i 0.0160299 + 0.0160299i 0.715076 − 0.699046i \(-0.246392\pi\)
−0.699046 + 0.715076i \(0.746392\pi\)
\(38\) 0.726013 − 6.85881i 0.117775 − 1.11265i
\(39\) 0.705826i 0.113023i
\(40\) 0 0
\(41\) − 10.2052i − 1.59379i −0.604117 − 0.796896i \(-0.706474\pi\)
0.604117 − 0.796896i \(-0.293526\pi\)
\(42\) −0.637004 − 0.0674276i −0.0982918 − 0.0104043i
\(43\) 4.43844 + 4.43844i 0.676855 + 0.676855i 0.959287 − 0.282432i \(-0.0911412\pi\)
−0.282432 + 0.959287i \(0.591141\pi\)
\(44\) −5.37499 − 8.30359i −0.810310 − 1.25181i
\(45\) 0 0
\(46\) −0.628908 − 0.777810i −0.0927274 − 0.114682i
\(47\) 1.89428i 0.276310i 0.990411 + 0.138155i \(0.0441172\pi\)
−0.990411 + 0.138155i \(0.955883\pi\)
\(48\) −0.634498 + 0.241413i −0.0915820 + 0.0348449i
\(49\) 0.122561 0.0175087
\(50\) 0 0
\(51\) −0.222155 − 0.222155i −0.0311079 − 0.0311079i
\(52\) −4.51979 − 6.98243i −0.626782 − 0.968289i
\(53\) −7.43897 − 7.43897i −1.02182 − 1.02182i −0.999757 − 0.0220650i \(-0.992976\pi\)
−0.0220650 − 0.999757i \(-0.507024\pi\)
\(54\) 0.150862 − 1.42523i 0.0205298 − 0.193949i
\(55\) 0 0
\(56\) 6.73338 − 3.41205i 0.899786 − 0.455955i
\(57\) 0.827717 0.109634
\(58\) 0.736651 − 6.95931i 0.0967270 − 0.913802i
\(59\) −0.959574 + 0.959574i −0.124926 + 0.124926i −0.766805 − 0.641880i \(-0.778155\pi\)
0.641880 + 0.766805i \(0.278155\pi\)
\(60\) 0 0
\(61\) 6.49825 + 6.49825i 0.832015 + 0.832015i 0.987792 − 0.155777i \(-0.0497881\pi\)
−0.155777 + 0.987792i \(0.549788\pi\)
\(62\) −6.08442 − 7.52499i −0.772722 − 0.955674i
\(63\) 7.92956i 0.999031i
\(64\) 4.73092 − 6.45123i 0.591365 − 0.806404i
\(65\) 0 0
\(66\) 0.923069 − 0.746358i 0.113622 − 0.0918703i
\(67\) 3.49691 − 3.49691i 0.427216 − 0.427216i −0.460463 − 0.887679i \(-0.652317\pi\)
0.887679 + 0.460463i \(0.152317\pi\)
\(68\) 3.62027 + 0.775103i 0.439022 + 0.0939951i
\(69\) 0.0848809 − 0.0848809i 0.0102185 − 0.0102185i
\(70\) 0 0
\(71\) 7.86777i 0.933733i 0.884328 + 0.466866i \(0.154617\pi\)
−0.884328 + 0.466866i \(0.845383\pi\)
\(72\) 3.79865 + 7.49629i 0.447675 + 0.883446i
\(73\) −15.6564 −1.83244 −0.916220 − 0.400675i \(-0.868776\pi\)
−0.916220 + 0.400675i \(0.868776\pi\)
\(74\) 0.0205276 − 0.193929i 0.00238628 − 0.0225437i
\(75\) 0 0
\(76\) −8.18824 + 5.30033i −0.939256 + 0.607989i
\(77\) −9.33322 + 9.33322i −1.06362 + 1.06362i
\(78\) 0.776202 − 0.627607i 0.0878876 − 0.0710625i
\(79\) 6.70212 0.754047 0.377024 − 0.926204i \(-0.376948\pi\)
0.377024 + 0.926204i \(0.376948\pi\)
\(80\) 0 0
\(81\) −8.74159 −0.971288
\(82\) −11.2228 + 9.07431i −1.23935 + 1.00209i
\(83\) 3.87327 − 3.87327i 0.425147 − 0.425147i −0.461825 − 0.886971i \(-0.652805\pi\)
0.886971 + 0.461825i \(0.152805\pi\)
\(84\) 0.492261 + 0.760473i 0.0537101 + 0.0829744i
\(85\) 0 0
\(86\) 0.934407 − 8.82755i 0.100760 − 0.951900i
\(87\) 0.839845 0.0900408
\(88\) −4.35218 + 13.2943i −0.463944 + 1.41718i
\(89\) − 10.5055i − 1.11358i −0.830653 − 0.556790i \(-0.812033\pi\)
0.830653 − 0.556790i \(-0.187967\pi\)
\(90\) 0 0
\(91\) −7.84824 + 7.84824i −0.822719 + 0.822719i
\(92\) −0.296151 + 1.38323i −0.0308759 + 0.144212i
\(93\) 0.821187 − 0.821187i 0.0851531 − 0.0851531i
\(94\) 2.08316 − 1.68436i 0.214861 − 0.173729i
\(95\) 0 0
\(96\) 0.829667 + 0.483103i 0.0846776 + 0.0493065i
\(97\) − 4.79937i − 0.487303i −0.969863 − 0.243651i \(-0.921655\pi\)
0.969863 − 0.243651i \(-0.0783453\pi\)
\(98\) −0.108979 − 0.134781i −0.0110085 − 0.0136150i
\(99\) −10.3907 − 10.3907i −1.04430 − 1.04430i
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.q.g.349.3 16
4.3 odd 2 1600.2.q.h.849.4 16
5.2 odd 4 80.2.l.a.61.7 yes 16
5.3 odd 4 400.2.l.h.301.2 16
5.4 even 2 400.2.q.h.349.6 16
15.2 even 4 720.2.t.c.541.2 16
16.5 even 4 400.2.q.h.149.6 16
16.11 odd 4 1600.2.q.g.49.5 16
20.3 even 4 1600.2.l.i.401.4 16
20.7 even 4 320.2.l.a.81.5 16
20.19 odd 2 1600.2.q.g.849.5 16
40.27 even 4 640.2.l.a.161.4 16
40.37 odd 4 640.2.l.b.161.5 16
60.47 odd 4 2880.2.t.c.721.6 16
80.27 even 4 320.2.l.a.241.5 16
80.37 odd 4 80.2.l.a.21.7 ✓ 16
80.43 even 4 1600.2.l.i.1201.4 16
80.53 odd 4 400.2.l.h.101.2 16
80.59 odd 4 1600.2.q.h.49.4 16
80.67 even 4 640.2.l.a.481.4 16
80.69 even 4 inner 400.2.q.g.149.3 16
80.77 odd 4 640.2.l.b.481.5 16
160.27 even 8 5120.2.a.u.1.4 8
160.37 odd 8 5120.2.a.s.1.5 8
160.107 even 8 5120.2.a.t.1.5 8
160.117 odd 8 5120.2.a.v.1.4 8
240.107 odd 4 2880.2.t.c.2161.7 16
240.197 even 4 720.2.t.c.181.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 80.37 odd 4
80.2.l.a.61.7 yes 16 5.2 odd 4
320.2.l.a.81.5 16 20.7 even 4
320.2.l.a.241.5 16 80.27 even 4
400.2.l.h.101.2 16 80.53 odd 4
400.2.l.h.301.2 16 5.3 odd 4
400.2.q.g.149.3 16 80.69 even 4 inner
400.2.q.g.349.3 16 1.1 even 1 trivial
400.2.q.h.149.6 16 16.5 even 4
400.2.q.h.349.6 16 5.4 even 2
640.2.l.a.161.4 16 40.27 even 4
640.2.l.a.481.4 16 80.67 even 4
640.2.l.b.161.5 16 40.37 odd 4
640.2.l.b.481.5 16 80.77 odd 4
720.2.t.c.181.2 16 240.197 even 4
720.2.t.c.541.2 16 15.2 even 4
1600.2.l.i.401.4 16 20.3 even 4
1600.2.l.i.1201.4 16 80.43 even 4
1600.2.q.g.49.5 16 16.11 odd 4
1600.2.q.g.849.5 16 20.19 odd 2
1600.2.q.h.49.4 16 80.59 odd 4
1600.2.q.h.849.4 16 4.3 odd 2
2880.2.t.c.721.6 16 60.47 odd 4
2880.2.t.c.2161.7 16 240.107 odd 4
5120.2.a.s.1.5 8 160.37 odd 8
5120.2.a.t.1.5 8 160.107 even 8
5120.2.a.u.1.4 8 160.27 even 8
5120.2.a.v.1.4 8 160.117 odd 8