↑

Properties

Label 370.2.b.d.149.1
Level $370$
Weight $2$
Character 370.149
Analytic conductor $2.954$
Analytic rank $0$
Dimension $10$
Inner twists $2$

Related objects

Downloads

Learn more

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 149.1
Root \(1.51933 - 1.51933i\) of defining polynomial
Character \(\chi\) \(=\) 370.149
Dual form 370.2.b.d.149.10

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.72987i q^{3} -1.00000 q^{4} +(-1.42149 - 1.72608i) q^{5} -2.72987 q^{6} +4.14336i q^{7} +1.00000i q^{8} -4.45216 q^{9} +(-1.72608 + 1.42149i) q^{10} -4.76853 q^{11} +2.72987i q^{12} -3.91744i q^{13} +4.14336 q^{14} +(-4.71197 + 3.88048i) q^{15} +1.00000 q^{16} -3.31637i q^{17} +4.45216i q^{18} +1.85109 q^{19} +(1.42149 + 1.72608i) q^{20} +11.3108 q^{21} +4.76853i q^{22} -1.54229i q^{23} +2.72987 q^{24} +(-0.958719 + 4.90723i) q^{25} -3.91744 q^{26} +3.96421i q^{27} -4.14336i q^{28} -8.87323 q^{29} +(3.88048 + 4.71197i) q^{30} +9.75286 q^{31} -1.00000i q^{32} +13.0174i q^{33} -3.31637 q^{34} +(7.15179 - 5.88976i) q^{35} +4.45216 q^{36} -1.00000i q^{37} -1.85109i q^{38} -10.6941 q^{39} +(1.72608 - 1.42149i) q^{40} -5.06977 q^{41} -11.3108i q^{42} -9.99446i q^{43} +4.76853 q^{44} +(6.32872 + 7.68480i) q^{45} -1.54229 q^{46} -4.82700i q^{47} -2.72987i q^{48} -10.1675 q^{49} +(4.90723 + 0.958719i) q^{50} -9.05324 q^{51} +3.91744i q^{52} -5.13611i q^{53} +3.96421 q^{54} +(6.77843 + 8.23088i) q^{55} -4.14336 q^{56} -5.05324i q^{57} +8.87323i q^{58} +1.05324 q^{59} +(4.71197 - 3.88048i) q^{60} -4.14422 q^{61} -9.75286i q^{62} -18.4469i q^{63} -1.00000 q^{64} +(-6.76182 + 5.56861i) q^{65} +13.0174 q^{66} -1.00811i q^{67} +3.31637i q^{68} -4.21025 q^{69} +(-5.88976 - 7.15179i) q^{70} +6.45248 q^{71} -4.45216i q^{72} +10.7714i q^{73} -1.00000 q^{74} +(13.3961 + 2.61717i) q^{75} -1.85109 q^{76} -19.7578i q^{77} +10.6941i q^{78} +1.19856 q^{79} +(-1.42149 - 1.72608i) q^{80} -2.53473 q^{81} +5.06977i q^{82} +10.6932i q^{83} -11.3108 q^{84} +(-5.72432 + 4.71419i) q^{85} -9.99446 q^{86} +24.2227i q^{87} -4.76853i q^{88} -7.29569 q^{89} +(7.68480 - 6.32872i) q^{90} +16.2314 q^{91} +1.54229i q^{92} -26.6240i q^{93} -4.82700 q^{94} +(-2.63131 - 3.19514i) q^{95} -2.72987 q^{96} -14.8988i q^{97} +10.1675i q^{98} +21.2303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9} + 2 q^{10} + 6 q^{11} + 2 q^{14} + 10 q^{16} - 8 q^{19} - 6 q^{20} + 32 q^{21} + 4 q^{25} - 12 q^{26} - 22 q^{29} + 20 q^{30} + 46 q^{31} - 18 q^{34} + 32 q^{35} + 6 q^{36}+ \cdots + 22 q^{99}+O(q^{100}) \) 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
\(3\) − 2.72987i − 1.57609i −0.615619 − 0.788044i \(-0.711094\pi\)
0.615619 − 0.788044i \(-0.288906\pi\)
\(4\) −1.00000 −0.500000
\(5\) −1.42149 − 1.72608i −0.635711 − 0.771927i
\(6\) −2.72987 −1.11446
\(7\) 4.14336i 1.56604i 0.621994 + 0.783022i \(0.286323\pi\)
−0.621994 + 0.783022i \(0.713677\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −4.45216 −1.48405
\(10\) −1.72608 + 1.42149i −0.545835 + 0.449515i
\(11\) −4.76853 −1.43777 −0.718883 − 0.695131i \(-0.755346\pi\)
−0.718883 + 0.695131i \(0.755346\pi\)
\(12\) 2.72987i 0.788044i
\(13\) − 3.91744i − 1.08650i −0.839570 − 0.543251i \(-0.817193\pi\)
0.839570 − 0.543251i \(-0.182807\pi\)
\(14\) 4.14336 1.10736
\(15\) −4.71197 + 3.88048i −1.21663 + 1.00194i
\(16\) 1.00000 0.250000
\(17\) − 3.31637i − 0.804337i −0.915566 − 0.402169i \(-0.868257\pi\)
0.915566 − 0.402169i \(-0.131743\pi\)
\(18\) 4.45216i 1.04939i
\(19\) 1.85109 0.424670 0.212335 − 0.977197i \(-0.431893\pi\)
0.212335 + 0.977197i \(0.431893\pi\)
\(20\) 1.42149 + 1.72608i 0.317855 + 0.385964i
\(21\) 11.3108 2.46822
\(22\) 4.76853i 1.01665i
\(23\) − 1.54229i − 0.321590i −0.986988 − 0.160795i \(-0.948594\pi\)
0.986988 − 0.160795i \(-0.0514058\pi\)
\(24\) 2.72987 0.557231
\(25\) −0.958719 + 4.90723i −0.191744 + 0.981445i
\(26\) −3.91744 −0.768273
\(27\) 3.96421i 0.762913i
\(28\) − 4.14336i − 0.783022i
\(29\) −8.87323 −1.64772 −0.823859 − 0.566795i \(-0.808183\pi\)
−0.823859 + 0.566795i \(0.808183\pi\)
\(30\) 3.88048 + 4.71197i 0.708476 + 0.860284i
\(31\) 9.75286 1.75166 0.875832 − 0.482615i \(-0.160313\pi\)
0.875832 + 0.482615i \(0.160313\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 13.0174i 2.26605i
\(34\) −3.31637 −0.568752
\(35\) 7.15179 − 5.88976i 1.20887 − 0.995551i
\(36\) 4.45216 0.742027
\(37\) − 1.00000i − 0.164399i
\(38\) − 1.85109i − 0.300287i
\(39\) −10.6941 −1.71242
\(40\) 1.72608 − 1.42149i 0.272918 − 0.224758i
\(41\) −5.06977 −0.791764 −0.395882 − 0.918301i \(-0.629561\pi\)
−0.395882 + 0.918301i \(0.629561\pi\)
\(42\) − 11.3108i − 1.74530i
\(43\) − 9.99446i − 1.52414i −0.647494 − 0.762070i \(-0.724183\pi\)
0.647494 − 0.762070i \(-0.275817\pi\)
\(44\) 4.76853 0.718883
\(45\) 6.32872 + 7.68480i 0.943429 + 1.14558i
\(46\) −1.54229 −0.227399
\(47\) − 4.82700i − 0.704090i −0.935983 − 0.352045i \(-0.885486\pi\)
0.935983 − 0.352045i \(-0.114514\pi\)
\(48\) − 2.72987i − 0.394022i
\(49\) −10.1675 −1.45249
\(50\) 4.90723 + 0.958719i 0.693986 + 0.135583i
\(51\) −9.05324 −1.26771
\(52\) 3.91744i 0.543251i
\(53\) − 5.13611i − 0.705499i −0.935718 − 0.352750i \(-0.885247\pi\)
0.935718 − 0.352750i \(-0.114753\pi\)
\(54\) 3.96421 0.539461
\(55\) 6.77843 + 8.23088i 0.914003 + 1.10985i
\(56\) −4.14336 −0.553680
\(57\) − 5.05324i − 0.669317i
\(58\) 8.87323i 1.16511i
\(59\) 1.05324 0.137120 0.0685598 − 0.997647i \(-0.478160\pi\)
0.0685598 + 0.997647i \(0.478160\pi\)
\(60\) 4.71197 − 3.88048i 0.608313 − 0.500968i
\(61\) −4.14422 −0.530613 −0.265306 − 0.964164i \(-0.585473\pi\)
−0.265306 + 0.964164i \(0.585473\pi\)
\(62\) − 9.75286i − 1.23861i
\(63\) − 18.4469i − 2.32410i
\(64\) −1.00000 −0.125000
\(65\) −6.76182 + 5.56861i −0.838701 + 0.690701i
\(66\) 13.0174 1.60234
\(67\) − 1.00811i − 0.123160i −0.998102 − 0.0615800i \(-0.980386\pi\)
0.998102 − 0.0615800i \(-0.0196139\pi\)
\(68\) 3.31637i 0.402169i
\(69\) −4.21025 −0.506855
\(70\) −5.88976 − 7.15179i −0.703961 − 0.854802i
\(71\) 6.45248 0.765768 0.382884 − 0.923796i \(-0.374931\pi\)
0.382884 + 0.923796i \(0.374931\pi\)
\(72\) − 4.45216i − 0.524693i
\(73\) 10.7714i 1.26070i 0.776312 + 0.630349i \(0.217088\pi\)
−0.776312 + 0.630349i \(0.782912\pi\)
\(74\) −1.00000 −0.116248
\(75\) 13.3961 + 2.61717i 1.54684 + 0.302205i
\(76\) −1.85109 −0.212335
\(77\) − 19.7578i − 2.25161i
\(78\) 10.6941i 1.21087i
\(79\) 1.19856 0.134849 0.0674243 − 0.997724i \(-0.478522\pi\)
0.0674243 + 0.997724i \(0.478522\pi\)
\(80\) −1.42149 − 1.72608i −0.158928 − 0.192982i
\(81\) −2.53473 −0.281636
\(82\) 5.06977i 0.559862i
\(83\) 10.6932i 1.17373i 0.809683 + 0.586867i \(0.199639\pi\)
−0.809683 + 0.586867i \(0.800361\pi\)
\(84\) −11.3108 −1.23411
\(85\) −5.72432 + 4.71419i −0.620890 + 0.511326i
\(86\) −9.99446 −1.07773
\(87\) 24.2227i 2.59695i
\(88\) − 4.76853i − 0.508327i
\(89\) −7.29569 −0.773342 −0.386671 − 0.922218i \(-0.626375\pi\)
−0.386671 + 0.922218i \(0.626375\pi\)
\(90\) 7.68480 − 6.32872i 0.810049 − 0.667105i
\(91\) 16.2314 1.70151
\(92\) 1.54229i 0.160795i
\(93\) − 26.6240i − 2.76078i
\(94\) −4.82700 −0.497867
\(95\) −2.63131 − 3.19514i −0.269967 − 0.327814i
\(96\) −2.72987 −0.278616
\(97\) − 14.8988i − 1.51274i −0.654143 − 0.756371i \(-0.726970\pi\)
0.654143 − 0.756371i \(-0.273030\pi\)
\(98\) 10.1675i 1.02707i
\(99\) 21.2303 2.13372
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.149.1 ✓ 10
3.2 odd 2 3330.2.d.p.1999.10 10
5.2 odd 4 1850.2.a.be.1.1 5
5.3 odd 4 1850.2.a.bd.1.5 5
5.4 even 2 inner 370.2.b.d.149.10 yes 10
15.14 odd 2 3330.2.d.p.1999.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.1 ✓ 10 1.1 even 1 trivial
370.2.b.d.149.10 yes 10 5.4 even 2 inner
1850.2.a.bd.1.5 5 5.3 odd 4
1850.2.a.be.1.1 5 5.2 odd 4
3330.2.d.p.1999.5 10 15.14 odd 2
3330.2.d.p.1999.10 10 3.2 odd 2