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Properties

Label 370.2.b.d.149.3
Level $370$
Weight $2$
Character 370.149
Analytic conductor $2.954$
Analytic rank $0$
Dimension $10$
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] = [370,2,Mod(149,370)] 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("370.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-10,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.12837029094400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.3
Root \(0.478560 - 0.478560i\) of defining polynomial
Character \(\chi\) \(=\) 370.149
Dual form 370.2.b.d.149.8

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -0.332924i q^{3} -1.00000 q^{4} +(-1.10390 + 1.94458i) q^{5} -0.332924 q^{6} -3.51336i q^{7} +1.00000i q^{8} +2.88916 q^{9} +(1.94458 + 1.10390i) q^{10} -0.290044 q^{11} +0.332924i q^{12} -7.12558i q^{13} -3.51336 q^{14} +(0.647398 + 0.367517i) q^{15} +1.00000 q^{16} -6.17921i q^{17} -2.88916i q^{18} -5.83553 q^{19} +(1.10390 - 1.94458i) q^{20} -1.16968 q^{21} +0.290044i q^{22} +6.45973i q^{23} +0.332924 q^{24} +(-2.56279 - 4.29326i) q^{25} -7.12558 q^{26} -1.96065i q^{27} +3.51336i q^{28} +1.18043 q^{29} +(0.367517 - 0.647398i) q^{30} +9.77838 q^{31} -1.00000i q^{32} +0.0965628i q^{33} -6.17921 q^{34} +(6.83201 + 3.87841i) q^{35} -2.88916 q^{36} -1.00000i q^{37} +5.83553i q^{38} -2.37228 q^{39} +(-1.94458 - 1.10390i) q^{40} +1.64077 q^{41} +1.16968i q^{42} +5.34889i q^{43} +0.290044 q^{44} +(-3.18936 + 5.61821i) q^{45} +6.45973 q^{46} +5.69256i q^{47} -0.332924i q^{48} -5.34367 q^{49} +(-4.29326 + 2.56279i) q^{50} -2.05721 q^{51} +7.12558i q^{52} -9.32034i q^{53} -1.96065 q^{54} +(0.320181 - 0.564014i) q^{55} +3.51336 q^{56} +1.94279i q^{57} -1.18043i q^{58} -5.94279 q^{59} +(-0.647398 - 0.367517i) q^{60} -1.27699 q^{61} -9.77838i q^{62} -10.1507i q^{63} -1.00000 q^{64} +(13.8563 + 7.86596i) q^{65} +0.0965628 q^{66} +6.04334i q^{67} +6.17921i q^{68} +2.15060 q^{69} +(3.87841 - 6.83201i) q^{70} +13.4995 q^{71} +2.88916i q^{72} -1.71348i q^{73} -1.00000 q^{74} +(-1.42933 + 0.853215i) q^{75} +5.83553 q^{76} +1.01903i q^{77} +2.37228i q^{78} -8.25422 q^{79} +(-1.10390 + 1.94458i) q^{80} +8.01474 q^{81} -1.64077i q^{82} -2.41807i q^{83} +1.16968 q^{84} +(12.0160 + 6.82126i) q^{85} +5.34889 q^{86} -0.392995i q^{87} -0.290044i q^{88} +10.2797 q^{89} +(5.61821 + 3.18936i) q^{90} -25.0347 q^{91} -6.45973i q^{92} -3.25546i q^{93} +5.69256 q^{94} +(6.44187 - 11.3477i) q^{95} -0.332924 q^{96} +15.1272i q^{97} +5.34367i q^{98} -0.837984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9} + 2 q^{10} + 6 q^{11} + 2 q^{14} + 10 q^{16} - 8 q^{19} - 6 q^{20} + 32 q^{21} + 4 q^{25} - 12 q^{26} - 22 q^{29} + 20 q^{30} + 46 q^{31} - 18 q^{34} + 32 q^{35} + 6 q^{36}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 0.332924i − 0.192214i −0.995371 − 0.0961070i \(-0.969361\pi\)
0.995371 − 0.0961070i \(-0.0306391\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.10390 + 1.94458i −0.493681 + 0.869643i
\(6\) −0.332924 −0.135916
\(7\) − 3.51336i − 1.32792i −0.747766 − 0.663962i \(-0.768874\pi\)
0.747766 − 0.663962i \(-0.231126\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.88916 0.963054
\(10\) 1.94458 + 1.10390i 0.614930 + 0.349085i
\(11\) −0.290044 −0.0874516 −0.0437258 − 0.999044i \(-0.513923\pi\)
−0.0437258 + 0.999044i \(0.513923\pi\)
\(12\) 0.332924i 0.0961070i
\(13\) − 7.12558i − 1.97628i −0.153559 − 0.988140i \(-0.549073\pi\)
0.153559 − 0.988140i \(-0.450927\pi\)
\(14\) −3.51336 −0.938984
\(15\) 0.647398 + 0.367517i 0.167158 + 0.0948925i
\(16\) 1.00000 0.250000
\(17\) − 6.17921i − 1.49868i −0.662187 − 0.749339i \(-0.730372\pi\)
0.662187 − 0.749339i \(-0.269628\pi\)
\(18\) − 2.88916i − 0.680982i
\(19\) −5.83553 −1.33876 −0.669381 − 0.742919i \(-0.733441\pi\)
−0.669381 + 0.742919i \(0.733441\pi\)
\(20\) 1.10390 − 1.94458i 0.246841 − 0.434821i
\(21\) −1.16968 −0.255246
\(22\) 0.290044i 0.0618376i
\(23\) 6.45973i 1.34695i 0.739212 + 0.673473i \(0.235198\pi\)
−0.739212 + 0.673473i \(0.764802\pi\)
\(24\) 0.332924 0.0679579
\(25\) −2.56279 − 4.29326i −0.512558 − 0.858653i
\(26\) −7.12558 −1.39744
\(27\) − 1.96065i − 0.377326i
\(28\) 3.51336i 0.663962i
\(29\) 1.18043 0.219201 0.109600 − 0.993976i \(-0.465043\pi\)
0.109600 + 0.993976i \(0.465043\pi\)
\(30\) 0.367517 − 0.647398i 0.0670991 − 0.118198i
\(31\) 9.77838 1.75625 0.878124 − 0.478433i \(-0.158795\pi\)
0.878124 + 0.478433i \(0.158795\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0.0965628i 0.0168094i
\(34\) −6.17921 −1.05972
\(35\) 6.83201 + 3.87841i 1.15482 + 0.655571i
\(36\) −2.88916 −0.481527
\(37\) − 1.00000i − 0.164399i
\(38\) 5.83553i 0.946648i
\(39\) −2.37228 −0.379869
\(40\) −1.94458 − 1.10390i −0.307465 − 0.174543i
\(41\) 1.64077 0.256245 0.128123 − 0.991758i \(-0.459105\pi\)
0.128123 + 0.991758i \(0.459105\pi\)
\(42\) 1.16968i 0.180486i
\(43\) 5.34889i 0.815698i 0.913049 + 0.407849i \(0.133721\pi\)
−0.913049 + 0.407849i \(0.866279\pi\)
\(44\) 0.290044 0.0437258
\(45\) −3.18936 + 5.61821i −0.475442 + 0.837513i
\(46\) 6.45973 0.952435
\(47\) 5.69256i 0.830346i 0.909743 + 0.415173i \(0.136279\pi\)
−0.909743 + 0.415173i \(0.863721\pi\)
\(48\) − 0.332924i − 0.0480535i
\(49\) −5.34367 −0.763382
\(50\) −4.29326 + 2.56279i −0.607159 + 0.362433i
\(51\) −2.05721 −0.288067
\(52\) 7.12558i 0.988140i
\(53\) − 9.32034i − 1.28025i −0.768272 − 0.640123i \(-0.778883\pi\)
0.768272 − 0.640123i \(-0.221117\pi\)
\(54\) −1.96065 −0.266810
\(55\) 0.320181 − 0.564014i 0.0431732 − 0.0760517i
\(56\) 3.51336 0.469492
\(57\) 1.94279i 0.257329i
\(58\) − 1.18043i − 0.154998i
\(59\) −5.94279 −0.773686 −0.386843 − 0.922146i \(-0.626434\pi\)
−0.386843 + 0.922146i \(0.626434\pi\)
\(60\) −0.647398 − 0.367517i −0.0835788 − 0.0474462i
\(61\) −1.27699 −0.163502 −0.0817512 − 0.996653i \(-0.526051\pi\)
−0.0817512 + 0.996653i \(0.526051\pi\)
\(62\) − 9.77838i − 1.24185i
\(63\) − 10.1507i − 1.27886i
\(64\) −1.00000 −0.125000
\(65\) 13.8563 + 7.86596i 1.71866 + 0.975652i
\(66\) 0.0965628 0.0118861
\(67\) 6.04334i 0.738312i 0.929368 + 0.369156i \(0.120353\pi\)
−0.929368 + 0.369156i \(0.879647\pi\)
\(68\) 6.17921i 0.749339i
\(69\) 2.15060 0.258902
\(70\) 3.87841 − 6.83201i 0.463559 − 0.816581i
\(71\) 13.4995 1.60210 0.801050 − 0.598597i \(-0.204275\pi\)
0.801050 + 0.598597i \(0.204275\pi\)
\(72\) 2.88916i 0.340491i
\(73\) − 1.71348i − 0.200548i −0.994960 − 0.100274i \(-0.968028\pi\)
0.994960 − 0.100274i \(-0.0319719\pi\)
\(74\) −1.00000 −0.116248
\(75\) −1.42933 + 0.853215i −0.165045 + 0.0985208i
\(76\) 5.83553 0.669381
\(77\) 1.01903i 0.116129i
\(78\) 2.37228i 0.268608i
\(79\) −8.25422 −0.928672 −0.464336 − 0.885659i \(-0.653707\pi\)
−0.464336 + 0.885659i \(0.653707\pi\)
\(80\) −1.10390 + 1.94458i −0.123420 + 0.217411i
\(81\) 8.01474 0.890526
\(82\) − 1.64077i − 0.181193i
\(83\) − 2.41807i − 0.265418i −0.991155 − 0.132709i \(-0.957632\pi\)
0.991155 − 0.132709i \(-0.0423676\pi\)
\(84\) 1.16968 0.127623
\(85\) 12.0160 + 6.82126i 1.30331 + 0.739869i
\(86\) 5.34889 0.576785
\(87\) − 0.392995i − 0.0421335i
\(88\) − 0.290044i − 0.0309188i
\(89\) 10.2797 1.08965 0.544823 − 0.838551i \(-0.316597\pi\)
0.544823 + 0.838551i \(0.316597\pi\)
\(90\) 5.61821 + 3.18936i 0.592211 + 0.336188i
\(91\) −25.0347 −2.62435
\(92\) − 6.45973i − 0.673473i
\(93\) − 3.25546i − 0.337576i
\(94\) 5.69256 0.587143
\(95\) 6.44187 − 11.3477i 0.660922 − 1.16425i
\(96\) −0.332924 −0.0339790
\(97\) 15.1272i 1.53594i 0.640489 + 0.767968i \(0.278732\pi\)
−0.640489 + 0.767968i \(0.721268\pi\)
\(98\) 5.34367i 0.539793i
\(99\) −0.837984 −0.0842206
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 370.2.b.d.149.3 ✓ 10
3.2 odd 2 3330.2.d.p.1999.9 10
5.2 odd 4 1850.2.a.be.1.3 5
5.3 odd 4 1850.2.a.bd.1.3 5
5.4 even 2 inner 370.2.b.d.149.8 yes 10
15.14 odd 2 3330.2.d.p.1999.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.3 ✓ 10 1.1 even 1 trivial
370.2.b.d.149.8 yes 10 5.4 even 2 inner
1850.2.a.bd.1.3 5 5.3 odd 4
1850.2.a.be.1.3 5 5.2 odd 4
3330.2.d.p.1999.4 10 15.14 odd 2
3330.2.d.p.1999.9 10 3.2 odd 2