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Properties

Label 400.2.l.h.301.2
Level $400$
Weight $2$
Character 400.301
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(101,400)] 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("400.101"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-4,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == 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 301.2
Root \(-0.296075 + 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 400.301
Dual form 400.2.l.h.101.2

$q$-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+(-1.09971 + 0.889181i) q^{2} +(0.120009 + 0.120009i) q^{3} +(0.418713 - 1.95568i) q^{4} +(-0.238684 - 0.0252650i) q^{6} -2.66881i q^{7} +(1.27849 + 2.52299i) q^{8} -2.97120i q^{9} +(-3.49714 + 3.49714i) q^{11} +(0.284948 - 0.184450i) q^{12} +(-2.94072 - 2.94072i) q^{13} +(2.37306 + 2.93491i) q^{14} +(-3.64936 - 1.63774i) q^{16} -1.85116 q^{17} +(2.64193 + 3.26745i) q^{18} +(-3.44856 - 3.44856i) q^{19} +(0.320281 - 0.320281i) q^{21} +(0.736240 - 6.95543i) q^{22} +0.707288i q^{23} +(-0.149351 + 0.456211i) q^{24} +(5.84877 + 0.619099i) q^{26} +(0.716597 - 0.716597i) q^{27} +(-5.21934 - 1.11747i) q^{28} +(-3.49909 - 3.49909i) q^{29} +6.84272 q^{31} +(5.46947 - 1.44391i) q^{32} -0.839377 q^{33} +(2.03573 - 1.64601i) q^{34} +(-5.81070 - 1.24408i) q^{36} +(0.0975060 - 0.0975060i) q^{37} +(6.85881 + 0.726013i) q^{38} -0.705826i q^{39} -10.2052i q^{41} +(-0.0674276 + 0.637004i) q^{42} +(-4.43844 + 4.43844i) q^{43} +(5.37499 + 8.30359i) q^{44} +(-0.628908 - 0.777810i) q^{46} +1.89428 q^{47} +(-0.241413 - 0.634498i) q^{48} -0.122561 q^{49} +(-0.222155 - 0.222155i) q^{51} +(-6.98243 + 4.51979i) q^{52} +(7.43897 - 7.43897i) q^{53} +(-0.150862 + 1.42523i) q^{54} +(6.73338 - 3.41205i) q^{56} -0.827717i q^{57} +(6.95931 + 0.736651i) q^{58} +(0.959574 - 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} +(-7.52499 + 6.08442i) q^{62} -7.92956 q^{63} +(-4.73092 + 6.45123i) q^{64} +(0.923069 - 0.746358i) q^{66} +(-3.49691 - 3.49691i) q^{67} +(-0.775103 + 3.62027i) q^{68} +(-0.0848809 + 0.0848809i) q^{69} +7.86777i q^{71} +(7.49629 - 3.79865i) q^{72} -15.6564i q^{73} +(-0.0205276 + 0.193929i) q^{74} +(-8.18824 + 5.30033i) q^{76} +(9.33322 + 9.33322i) q^{77} +(0.627607 + 0.776202i) q^{78} -6.70212 q^{79} -8.74159 q^{81} +(9.07431 + 11.2228i) q^{82} +(3.87327 + 3.87327i) q^{83} +(-0.492261 - 0.760473i) q^{84} +(0.934407 - 8.82755i) q^{86} -0.839845i q^{87} +(-13.2943 - 4.35218i) q^{88} +10.5055i q^{89} +(-7.84824 + 7.84824i) q^{91} +(1.38323 + 0.296151i) q^{92} +(0.821187 + 0.821187i) q^{93} +(-2.08316 + 1.68436i) q^{94} +(0.829667 + 0.483103i) q^{96} -4.79937 q^{97} +(0.134781 - 0.108979i) q^{98} +(10.3907 + 10.3907i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{4} - 12 q^{6} - 8 q^{11} + 12 q^{12} + 4 q^{14} + 16 q^{16} - 8 q^{19} + 20 q^{22} + 8 q^{24} - 16 q^{26} - 24 q^{27} + 4 q^{28} - 16 q^{29} + 16 q^{34} - 4 q^{36} + 16 q^{37} - 20 q^{38} - 60 q^{42}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.09971 + 0.889181i −0.777611 + 0.628746i
\(3\) 0.120009 + 0.120009i 0.0692872 + 0.0692872i 0.740901 − 0.671614i \(-0.234399\pi\)
−0.671614 + 0.740901i \(0.734399\pi\)
\(4\) 0.418713 − 1.95568i 0.209357 − 0.977839i
\(5\) 0 0
\(6\) −0.238684 − 0.0252650i −0.0974425 − 0.0103144i
\(7\) − 2.66881i − 1.00872i −0.863495 − 0.504358i \(-0.831729\pi\)
0.863495 − 0.504358i \(-0.168271\pi\)
\(8\) 1.27849 + 2.52299i 0.452015 + 0.892010i
\(9\) − 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 \(0.982166\pi\)
\(12\) 0.284948 − 0.184450i 0.0822574 − 0.0532460i
\(13\) −2.94072 − 2.94072i −0.815610 − 0.815610i 0.169858 − 0.985468i \(-0.445669\pi\)
−0.985468 + 0.169858i \(0.945669\pi\)
\(14\) 2.37306 + 2.93491i 0.634227 + 0.784389i
\(15\) 0 0
\(16\) −3.64936 − 1.63774i −0.912340 − 0.409434i
\(17\) −1.85116 −0.448971 −0.224486 − 0.974477i \(-0.572070\pi\)
−0.224486 + 0.974477i \(0.572070\pi\)
\(18\) 2.64193 + 3.26745i 0.622709 + 0.770144i
\(19\) −3.44856 − 3.44856i −0.791155 − 0.791155i 0.190527 − 0.981682i \(-0.438980\pi\)
−0.981682 + 0.190527i \(0.938980\pi\)
\(20\) 0 0
\(21\) 0.320281 − 0.320281i 0.0698911 − 0.0698911i
\(22\) 0.736240 − 6.95543i 0.156967 − 1.48290i
\(23\) 0.707288i 0.147480i 0.997278 + 0.0737399i \(0.0234935\pi\)
−0.997278 + 0.0737399i \(0.976507\pi\)
\(24\) −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
\(28\) −5.21934 − 1.11747i −0.986363 − 0.211181i
\(29\) −3.49909 − 3.49909i −0.649766 − 0.649766i 0.303171 − 0.952936i \(-0.401955\pi\)
−0.952936 + 0.303171i \(0.901955\pi\)
\(30\) 0 0
\(31\) 6.84272 1.22899 0.614494 − 0.788921i \(-0.289360\pi\)
0.614494 + 0.788921i \(0.289360\pi\)
\(32\) 5.46947 − 1.44391i 0.966875 − 0.255250i
\(33\) −0.839377 −0.146117
\(34\) 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 −0.699046 − 0.715076i \(-0.746392\pi\)
0.715076 + 0.699046i \(0.246392\pi\)
\(38\) 6.85881 + 0.726013i 1.11265 + 0.117775i
\(39\) − 0.705826i − 0.113023i
\(40\) 0 0
\(41\) − 10.2052i − 1.59379i −0.604117 − 0.796896i \(-0.706474\pi\)
0.604117 − 0.796896i \(-0.293526\pi\)
\(42\) −0.0674276 + 0.637004i −0.0104043 + 0.0982918i
\(43\) −4.43844 + 4.43844i −0.676855 + 0.676855i −0.959287 − 0.282432i \(-0.908859\pi\)
0.282432 + 0.959287i \(0.408859\pi\)
\(44\) 5.37499 + 8.30359i 0.810310 + 1.25181i
\(45\) 0 0
\(46\) −0.628908 − 0.777810i −0.0927274 − 0.114682i
\(47\) 1.89428 0.276310 0.138155 − 0.990411i \(-0.455883\pi\)
0.138155 + 0.990411i \(0.455883\pi\)
\(48\) −0.241413 − 0.634498i −0.0348449 − 0.0915820i
\(49\) −0.122561 −0.0175087
\(50\) 0 0
\(51\) −0.222155 − 0.222155i −0.0311079 − 0.0311079i
\(52\) −6.98243 + 4.51979i −0.968289 + 0.626782i
\(53\) 7.43897 − 7.43897i 1.02182 − 1.02182i 0.0220650 − 0.999757i \(-0.492976\pi\)
0.999757 − 0.0220650i \(-0.00702407\pi\)
\(54\) −0.150862 + 1.42523i −0.0205298 + 0.193949i
\(55\) 0 0
\(56\) 6.73338 − 3.41205i 0.899786 − 0.455955i
\(57\) − 0.827717i − 0.109634i
\(58\) 6.95931 + 0.736651i 0.913802 + 0.0967270i
\(59\) 0.959574 − 0.959574i 0.124926 − 0.124926i −0.641880 − 0.766805i \(-0.721845\pi\)
0.766805 + 0.641880i \(0.221845\pi\)
\(60\) 0 0
\(61\) 6.49825 + 6.49825i 0.832015 + 0.832015i 0.987792 − 0.155777i \(-0.0497881\pi\)
−0.155777 + 0.987792i \(0.549788\pi\)
\(62\) −7.52499 + 6.08442i −0.955674 + 0.772722i
\(63\) −7.92956 −0.999031
\(64\) −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 \(-0.347683\pi\)
−0.887679 + 0.460463i \(0.847683\pi\)
\(68\) −0.775103 + 3.62027i −0.0939951 + 0.439022i
\(69\) −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 \(0.845383\pi\)
\(72\) 7.49629 − 3.79865i 0.883446 − 0.447675i
\(73\) − 15.6564i − 1.83244i −0.400675 − 0.916220i \(-0.631224\pi\)
0.400675 − 0.916220i \(-0.368776\pi\)
\(74\) −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
\(78\) 0.627607 + 0.776202i 0.0710625 + 0.0878876i
\(79\) −6.70212 −0.754047 −0.377024 − 0.926204i \(-0.623052\pi\)
−0.377024 + 0.926204i \(0.623052\pi\)
\(80\) 0 0
\(81\) −8.74159 −0.971288
\(82\) 9.07431 + 11.2228i 1.00209 + 1.23935i
\(83\) 3.87327 + 3.87327i 0.425147 + 0.425147i 0.886971 − 0.461825i \(-0.152805\pi\)
−0.461825 + 0.886971i \(0.652805\pi\)
\(84\) −0.492261 − 0.760473i −0.0537101 − 0.0829744i
\(85\) 0 0
\(86\) 0.934407 − 8.82755i 0.100760 − 0.951900i
\(87\) − 0.839845i − 0.0900408i
\(88\) −13.2943 − 4.35218i −1.41718 − 0.463944i
\(89\) 10.5055i 1.11358i 0.830653 + 0.556790i \(0.187967\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(90\) 0 0
\(91\) −7.84824 + 7.84824i −0.822719 + 0.822719i
\(92\) 1.38323 + 0.296151i 0.144212 + 0.0308759i
\(93\) 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
\(97\) −4.79937 −0.487303 −0.243651 − 0.969863i \(-0.578345\pi\)
−0.243651 + 0.969863i \(0.578345\pi\)
\(98\) 0.134781 − 0.108979i 0.0136150 − 0.0110085i
\(99\) 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.l.h.301.2 16
4.3 odd 2 1600.2.l.i.401.4 16
5.2 odd 4 400.2.q.g.349.3 16
5.3 odd 4 400.2.q.h.349.6 16
5.4 even 2 80.2.l.a.61.7 yes 16
15.14 odd 2 720.2.t.c.541.2 16
16.5 even 4 inner 400.2.l.h.101.2 16
16.11 odd 4 1600.2.l.i.1201.4 16
20.3 even 4 1600.2.q.g.849.5 16
20.7 even 4 1600.2.q.h.849.4 16
20.19 odd 2 320.2.l.a.81.5 16
40.19 odd 2 640.2.l.a.161.4 16
40.29 even 2 640.2.l.b.161.5 16
60.59 even 2 2880.2.t.c.721.6 16
80.19 odd 4 640.2.l.a.481.4 16
80.27 even 4 1600.2.q.g.49.5 16
80.29 even 4 640.2.l.b.481.5 16
80.37 odd 4 400.2.q.h.149.6 16
80.43 even 4 1600.2.q.h.49.4 16
80.53 odd 4 400.2.q.g.149.3 16
80.59 odd 4 320.2.l.a.241.5 16
80.69 even 4 80.2.l.a.21.7 ✓ 16
160.59 odd 8 5120.2.a.u.1.4 8
160.69 even 8 5120.2.a.s.1.5 8
160.139 odd 8 5120.2.a.t.1.5 8
160.149 even 8 5120.2.a.v.1.4 8
240.59 even 4 2880.2.t.c.2161.7 16
240.149 odd 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 80.69 even 4
80.2.l.a.61.7 yes 16 5.4 even 2
320.2.l.a.81.5 16 20.19 odd 2
320.2.l.a.241.5 16 80.59 odd 4
400.2.l.h.101.2 16 16.5 even 4 inner
400.2.l.h.301.2 16 1.1 even 1 trivial
400.2.q.g.149.3 16 80.53 odd 4
400.2.q.g.349.3 16 5.2 odd 4
400.2.q.h.149.6 16 80.37 odd 4
400.2.q.h.349.6 16 5.3 odd 4
640.2.l.a.161.4 16 40.19 odd 2
640.2.l.a.481.4 16 80.19 odd 4
640.2.l.b.161.5 16 40.29 even 2
640.2.l.b.481.5 16 80.29 even 4
720.2.t.c.181.2 16 240.149 odd 4
720.2.t.c.541.2 16 15.14 odd 2
1600.2.l.i.401.4 16 4.3 odd 2
1600.2.l.i.1201.4 16 16.11 odd 4
1600.2.q.g.49.5 16 80.27 even 4
1600.2.q.g.849.5 16 20.3 even 4
1600.2.q.h.49.4 16 80.43 even 4
1600.2.q.h.849.4 16 20.7 even 4
2880.2.t.c.721.6 16 60.59 even 2
2880.2.t.c.2161.7 16 240.59 even 4
5120.2.a.s.1.5 8 160.69 even 8
5120.2.a.t.1.5 8 160.139 odd 8
5120.2.a.u.1.4 8 160.59 odd 8
5120.2.a.v.1.4 8 160.149 even 8