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Properties

Label 400.2.l.f.101.4
Level $400$
Weight $2$
Character 400.101
Analytic conductor $3.194$
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] = [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 = [12,-4,-2,2,0,6] 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.4767670494822400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 101.4
Root \(0.618969 + 1.27156i\) of defining polynomial
Character \(\chi\) \(=\) 400.101
Dual form 400.2.l.f.301.4

$q$-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+(-0.618969 - 1.27156i) q^{2} +(-2.16859 + 2.16859i) q^{3} +(-1.23375 + 1.57412i) q^{4} +(4.09979 + 1.41521i) q^{6} +3.30519i q^{7} +(2.76525 + 0.594467i) q^{8} -6.40553i q^{9} +(2.01163 + 2.01163i) q^{11} +(-0.738111 - 6.08911i) q^{12} +(-0.794042 + 0.794042i) q^{13} +(4.20276 - 2.04581i) q^{14} +(-0.955702 - 3.88415i) q^{16} -4.61575 q^{17} +(-8.14504 + 3.96483i) q^{18} +(-3.48786 + 3.48786i) q^{19} +(-7.16759 - 7.16759i) q^{21} +(1.31278 - 3.80306i) q^{22} -7.99801i q^{23} +(-7.28583 + 4.70753i) q^{24} +(1.50116 + 0.518188i) q^{26} +(7.38518 + 7.38518i) q^{27} +(-5.20276 - 4.07779i) q^{28} +(-1.95065 + 1.95065i) q^{29} -5.12695 q^{31} +(-4.34740 + 3.61941i) q^{32} -8.72480 q^{33} +(2.85701 + 5.86922i) q^{34} +(10.0831 + 7.90285i) q^{36} +(-0.448156 - 0.448156i) q^{37} +(6.59391 + 2.27616i) q^{38} -3.44390i q^{39} -4.02230i q^{41} +(-4.67754 + 13.5506i) q^{42} +(-4.97000 - 4.97000i) q^{43} +(-5.64841 + 0.684690i) q^{44} +(-10.1700 + 4.95052i) q^{46} +5.49112 q^{47} +(10.4956 + 6.35059i) q^{48} -3.92429 q^{49} +(10.0096 - 10.0096i) q^{51} +(-0.270264 - 2.22957i) q^{52} +(3.35125 + 3.35125i) q^{53} +(4.81954 - 13.9619i) q^{54} +(-1.96483 + 9.13968i) q^{56} -15.1274i q^{57} +(3.68777 + 1.27299i) q^{58} +(2.07673 + 2.07673i) q^{59} +(-0.557208 + 0.557208i) q^{61} +(3.17343 + 6.51925i) q^{62} +21.1715 q^{63} +(7.29322 + 3.28770i) q^{64} +(5.40038 + 11.0941i) q^{66} +(0.636094 - 0.636094i) q^{67} +(5.69470 - 7.26573i) q^{68} +(17.3444 + 17.3444i) q^{69} +6.85258i q^{71} +(3.80787 - 17.7129i) q^{72} -10.5177i q^{73} +(-0.292465 + 0.847255i) q^{74} +(-1.18714 - 9.79346i) q^{76} +(-6.64883 + 6.64883i) q^{77} +(-4.37914 + 2.13167i) q^{78} -17.3005 q^{79} -12.8142 q^{81} +(-5.11461 + 2.48968i) q^{82} +(-9.48015 + 9.48015i) q^{83} +(20.1257 - 2.43960i) q^{84} +(-3.24340 + 9.39596i) q^{86} -8.46030i q^{87} +(4.36682 + 6.75852i) q^{88} -7.62073i q^{89} +(-2.62446 - 2.62446i) q^{91} +(12.5898 + 9.86757i) q^{92} +(11.1182 - 11.1182i) q^{93} +(-3.39883 - 6.98231i) q^{94} +(1.57871 - 17.2767i) q^{96} -0.709082 q^{97} +(2.42901 + 4.98999i) q^{98} +(12.8856 - 12.8856i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 2 q^{3} + 2 q^{4} + 6 q^{6} + 8 q^{8} - 2 q^{11} - 8 q^{12} + 4 q^{13} + 14 q^{14} + 2 q^{16} + 8 q^{17} - 18 q^{18} - 14 q^{19} - 20 q^{21} - 2 q^{22} - 14 q^{24} - 16 q^{26} + 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618969 − 1.27156i −0.437677 − 0.899132i
\(3\) −2.16859 + 2.16859i −1.25203 + 1.25203i −0.297227 + 0.954807i \(0.596062\pi\)
−0.954807 + 0.297227i \(0.903938\pi\)
\(4\) −1.23375 + 1.57412i −0.616877 + 0.787060i
\(5\) 0 0
\(6\) 4.09979 + 1.41521i 1.67373 + 0.577757i
\(7\) 3.30519i 1.24924i 0.780927 + 0.624622i \(0.214747\pi\)
−0.780927 + 0.624622i \(0.785253\pi\)
\(8\) 2.76525 + 0.594467i 0.977664 + 0.210176i
\(9\) − 6.40553i − 2.13518i
\(10\) 0 0
\(11\) 2.01163 + 2.01163i 0.606530 + 0.606530i 0.942038 − 0.335507i \(-0.108908\pi\)
−0.335507 + 0.942038i \(0.608908\pi\)
\(12\) −0.738111 − 6.08911i −0.213074 − 1.75778i
\(13\) −0.794042 + 0.794042i −0.220228 + 0.220228i −0.808594 − 0.588367i \(-0.799771\pi\)
0.588367 + 0.808594i \(0.299771\pi\)
\(14\) 4.20276 − 2.04581i 1.12324 − 0.546766i
\(15\) 0 0
\(16\) −0.955702 − 3.88415i −0.238926 − 0.971038i
\(17\) −4.61575 −1.11948 −0.559741 − 0.828667i \(-0.689100\pi\)
−0.559741 + 0.828667i \(0.689100\pi\)
\(18\) −8.14504 + 3.96483i −1.91981 + 0.934518i
\(19\) −3.48786 + 3.48786i −0.800169 + 0.800169i −0.983122 − 0.182953i \(-0.941434\pi\)
0.182953 + 0.983122i \(0.441434\pi\)
\(20\) 0 0
\(21\) −7.16759 − 7.16759i −1.56410 − 1.56410i
\(22\) 1.31278 − 3.80306i 0.279886 − 0.810815i
\(23\) − 7.99801i − 1.66770i −0.551991 − 0.833850i \(-0.686132\pi\)
0.551991 − 0.833850i \(-0.313868\pi\)
\(24\) −7.28583 + 4.70753i −1.48721 + 0.960921i
\(25\) 0 0
\(26\) 1.50116 + 0.518188i 0.294402 + 0.101625i
\(27\) 7.38518 + 7.38518i 1.42128 + 1.42128i
\(28\) −5.20276 − 4.07779i −0.983230 − 0.770630i
\(29\) −1.95065 + 1.95065i −0.362227 + 0.362227i −0.864632 − 0.502406i \(-0.832448\pi\)
0.502406 + 0.864632i \(0.332448\pi\)
\(30\) 0 0
\(31\) −5.12695 −0.920828 −0.460414 − 0.887704i \(-0.652299\pi\)
−0.460414 + 0.887704i \(0.652299\pi\)
\(32\) −4.34740 + 3.61941i −0.768519 + 0.639827i
\(33\) −8.72480 −1.51879
\(34\) 2.85701 + 5.86922i 0.489972 + 1.00656i
\(35\) 0 0
\(36\) 10.0831 + 7.90285i 1.68051 + 1.31714i
\(37\) −0.448156 − 0.448156i −0.0736764 − 0.0736764i 0.669308 − 0.742985i \(-0.266591\pi\)
−0.742985 + 0.669308i \(0.766591\pi\)
\(38\) 6.59391 + 2.27616i 1.06967 + 0.369242i
\(39\) − 3.44390i − 0.551465i
\(40\) 0 0
\(41\) − 4.02230i − 0.628177i −0.949394 − 0.314089i \(-0.898301\pi\)
0.949394 − 0.314089i \(-0.101699\pi\)
\(42\) −4.67754 + 13.5506i −0.721760 + 2.09090i
\(43\) −4.97000 − 4.97000i −0.757918 − 0.757918i 0.218025 − 0.975943i \(-0.430039\pi\)
−0.975943 + 0.218025i \(0.930039\pi\)
\(44\) −5.64841 + 0.684690i −0.851530 + 0.103221i
\(45\) 0 0
\(46\) −10.1700 + 4.95052i −1.49948 + 0.729915i
\(47\) 5.49112 0.800962 0.400481 − 0.916305i \(-0.368843\pi\)
0.400481 + 0.916305i \(0.368843\pi\)
\(48\) 10.4956 + 6.35059i 1.51491 + 0.916629i
\(49\) −3.92429 −0.560612
\(50\) 0 0
\(51\) 10.0096 − 10.0096i 1.40163 − 1.40163i
\(52\) −0.270264 − 2.22957i −0.0374789 − 0.309186i
\(53\) 3.35125 + 3.35125i 0.460330 + 0.460330i 0.898763 − 0.438434i \(-0.144467\pi\)
−0.438434 + 0.898763i \(0.644467\pi\)
\(54\) 4.81954 − 13.9619i 0.655856 − 1.89998i
\(55\) 0 0
\(56\) −1.96483 + 9.13968i −0.262561 + 1.22134i
\(57\) − 15.1274i − 2.00368i
\(58\) 3.68777 + 1.27299i 0.484228 + 0.167151i
\(59\) 2.07673 + 2.07673i 0.270367 + 0.270367i 0.829248 − 0.558881i \(-0.188769\pi\)
−0.558881 + 0.829248i \(0.688769\pi\)
\(60\) 0 0
\(61\) −0.557208 + 0.557208i −0.0713432 + 0.0713432i −0.741878 − 0.670535i \(-0.766065\pi\)
0.670535 + 0.741878i \(0.266065\pi\)
\(62\) 3.17343 + 6.51925i 0.403026 + 0.827946i
\(63\) 21.1715 2.66736
\(64\) 7.29322 + 3.28770i 0.911652 + 0.410962i
\(65\) 0 0
\(66\) 5.40038 + 11.0941i 0.664741 + 1.36560i
\(67\) 0.636094 − 0.636094i 0.0777112 − 0.0777112i −0.667183 − 0.744894i \(-0.732500\pi\)
0.744894 + 0.667183i \(0.232500\pi\)
\(68\) 5.69470 − 7.26573i 0.690583 − 0.881100i
\(69\) 17.3444 + 17.3444i 2.08802 + 2.08802i
\(70\) 0 0
\(71\) 6.85258i 0.813252i 0.913595 + 0.406626i \(0.133295\pi\)
−0.913595 + 0.406626i \(0.866705\pi\)
\(72\) 3.80787 − 17.7129i 0.448762 − 2.08748i
\(73\) − 10.5177i − 1.23101i −0.788134 − 0.615504i \(-0.788953\pi\)
0.788134 − 0.615504i \(-0.211047\pi\)
\(74\) −0.292465 + 0.847255i −0.0339983 + 0.0984913i
\(75\) 0 0
\(76\) −1.18714 − 9.79346i −0.136175 − 1.12339i
\(77\) −6.64883 + 6.64883i −0.757705 + 0.757705i
\(78\) −4.37914 + 2.13167i −0.495840 + 0.241364i
\(79\) −17.3005 −1.94646 −0.973230 − 0.229833i \(-0.926182\pi\)
−0.973230 + 0.229833i \(0.926182\pi\)
\(80\) 0 0
\(81\) −12.8142 −1.42380
\(82\) −5.11461 + 2.48968i −0.564814 + 0.274939i
\(83\) −9.48015 + 9.48015i −1.04058 + 1.04058i −0.0414412 + 0.999141i \(0.513195\pi\)
−0.999141 + 0.0414412i \(0.986805\pi\)
\(84\) 20.1257 − 2.43960i 2.19589 − 0.266182i
\(85\) 0 0
\(86\) −3.24340 + 9.39596i −0.349745 + 1.01319i
\(87\) − 8.46030i − 0.907040i
\(88\) 4.36682 + 6.75852i 0.465505 + 0.720460i
\(89\) − 7.62073i − 0.807796i −0.914804 − 0.403898i \(-0.867655\pi\)
0.914804 − 0.403898i \(-0.132345\pi\)
\(90\) 0 0
\(91\) −2.62446 − 2.62446i −0.275118 − 0.275118i
\(92\) 12.5898 + 9.86757i 1.31258 + 1.02877i
\(93\) 11.1182 − 11.1182i 1.15291 − 1.15291i
\(94\) −3.39883 − 6.98231i −0.350563 − 0.720171i
\(95\) 0 0
\(96\) 1.57871 − 17.2767i 0.161127 − 1.76330i
\(97\) −0.709082 −0.0719964 −0.0359982 − 0.999352i \(-0.511461\pi\)
−0.0359982 + 0.999352i \(0.511461\pi\)
\(98\) 2.42901 + 4.98999i 0.245367 + 0.504065i
\(99\) 12.8856 − 12.8856i 1.29505 − 1.29505i
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.f.101.4 ✓ 12
4.3 odd 2 1600.2.l.g.1201.6 12
5.2 odd 4 400.2.q.e.149.6 12
5.3 odd 4 400.2.q.f.149.1 12
5.4 even 2 400.2.l.g.101.3 yes 12
16.3 odd 4 1600.2.l.g.401.6 12
16.13 even 4 inner 400.2.l.f.301.4 yes 12
20.3 even 4 1600.2.q.f.49.6 12
20.7 even 4 1600.2.q.e.49.1 12
20.19 odd 2 1600.2.l.f.1201.1 12
80.3 even 4 1600.2.q.e.849.1 12
80.13 odd 4 400.2.q.e.349.6 12
80.19 odd 4 1600.2.l.f.401.1 12
80.29 even 4 400.2.l.g.301.3 yes 12
80.67 even 4 1600.2.q.f.849.6 12
80.77 odd 4 400.2.q.f.349.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
400.2.l.f.101.4 ✓ 12 1.1 even 1 trivial
400.2.l.f.301.4 yes 12 16.13 even 4 inner
400.2.l.g.101.3 yes 12 5.4 even 2
400.2.l.g.301.3 yes 12 80.29 even 4
400.2.q.e.149.6 12 5.2 odd 4
400.2.q.e.349.6 12 80.13 odd 4
400.2.q.f.149.1 12 5.3 odd 4
400.2.q.f.349.1 12 80.77 odd 4
1600.2.l.f.401.1 12 80.19 odd 4
1600.2.l.f.1201.1 12 20.19 odd 2
1600.2.l.g.401.6 12 16.3 odd 4
1600.2.l.g.1201.6 12 4.3 odd 2
1600.2.q.e.49.1 12 20.7 even 4
1600.2.q.e.849.1 12 80.3 even 4
1600.2.q.f.49.6 12 20.3 even 4
1600.2.q.f.849.6 12 80.67 even 4