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Properties

Label 700.2.bj.a.323.2
Level $700$
Weight $2$
Character 700.323
Analytic conductor $5.590$
Analytic rank $0$
Dimension $720$
Inner twists $4$

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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] = [700,2,Mod(127,700)] 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("700.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(90\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 323.2
Character \(\chi\) \(=\) 700.323
Dual form 700.2.bj.a.687.2

$q$-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+(-1.41279 - 0.0633751i) q^{2} +(0.834190 + 1.63719i) q^{3} +(1.99197 + 0.179072i) q^{4} +(-1.91219 - 1.15911i) q^{5} +(-1.07478 - 2.36588i) q^{6} +(0.707107 + 0.707107i) q^{7} +(-2.80289 - 0.379233i) q^{8} +(-0.221165 + 0.304407i) q^{9} +(2.62807 + 1.75877i) q^{10} +(0.917897 + 1.26338i) q^{11} +(1.36851 + 3.41061i) q^{12} +(-0.136798 - 0.863706i) q^{13} +(-0.954182 - 1.04381i) q^{14} +(0.302556 - 4.09754i) q^{15} +(3.93587 + 0.713411i) q^{16} +(-1.39517 - 0.710875i) q^{17} +(0.331752 - 0.416048i) q^{18} +(2.27371 + 6.99775i) q^{19} +(-3.60145 - 2.65133i) q^{20} +(-0.567807 + 1.74753i) q^{21} +(-1.21673 - 1.84306i) q^{22} +(-1.02865 + 6.49463i) q^{23} +(-1.71727 - 4.90522i) q^{24} +(2.31293 + 4.43287i) q^{25} +(0.138529 + 1.22891i) q^{26} +(4.76165 + 0.754172i) q^{27} +(1.28191 + 1.53516i) q^{28} +(-0.567208 - 0.184297i) q^{29} +(-0.687131 + 5.76979i) q^{30} +(6.39327 - 2.07730i) q^{31} +(-5.51535 - 1.25734i) q^{32} +(-1.30269 + 2.55667i) q^{33} +(1.92603 + 1.09274i) q^{34} +(-0.532506 - 2.17174i) q^{35} +(-0.495063 + 0.566764i) q^{36} +(-5.42145 + 0.858673i) q^{37} +(-2.76879 - 10.0305i) q^{38} +(1.29994 - 0.944460i) q^{39} +(4.92008 + 3.97402i) q^{40} +(7.10317 + 5.16075i) q^{41} +(0.912944 - 2.43291i) q^{42} +(-4.06563 + 4.06563i) q^{43} +(1.60219 + 2.68097i) q^{44} +(0.775750 - 0.325729i) q^{45} +(1.86487 - 9.11038i) q^{46} +(-5.50151 + 2.80316i) q^{47} +(2.11527 + 7.03889i) q^{48} +1.00000i q^{49} +(-2.98675 - 6.40932i) q^{50} -2.87716i q^{51} +(-0.117831 - 1.74497i) q^{52} +(2.25678 - 1.14989i) q^{53} +(-6.67944 - 1.36726i) q^{54} +(-0.290798 - 3.47976i) q^{55} +(-1.71378 - 2.25010i) q^{56} +(-9.55995 + 9.55995i) q^{57} +(0.789668 + 0.296320i) q^{58} +(-10.8163 - 7.85848i) q^{59} +(1.33644 - 8.10798i) q^{60} +(-0.470316 + 0.341705i) q^{61} +(-9.16402 + 2.52962i) q^{62} +(-0.371635 + 0.0588612i) q^{63} +(7.71237 + 2.12589i) q^{64} +(-0.739548 + 1.81013i) q^{65} +(2.00246 - 3.52949i) q^{66} +(-3.37846 + 6.63059i) q^{67} +(-2.65184 - 1.66587i) q^{68} +(-11.4910 + 3.73367i) q^{69} +(0.614687 + 3.10196i) q^{70} +(-6.49492 - 2.11033i) q^{71} +(0.735341 - 0.769346i) q^{72} +(13.5462 + 2.14550i) q^{73} +(7.71380 - 0.869542i) q^{74} +(-5.32804 + 7.48456i) q^{75} +(3.27605 + 14.3464i) q^{76} +(-0.244291 + 1.54239i) q^{77} +(-1.89640 + 1.25194i) q^{78} +(0.112435 - 0.346038i) q^{79} +(-6.69919 - 5.92628i) q^{80} +(3.08622 + 9.49842i) q^{81} +(-9.70824 - 7.74124i) q^{82} +(8.12296 + 4.13885i) q^{83} +(-1.44399 + 3.37935i) q^{84} +(1.84385 + 2.97648i) q^{85} +(6.00155 - 5.48623i) q^{86} +(-0.171430 - 1.08237i) q^{87} +(-2.09365 - 3.88920i) q^{88} +(6.83491 + 9.40744i) q^{89} +(-1.11662 + 0.411024i) q^{90} +(0.514002 - 0.707463i) q^{91} +(-3.21204 + 12.7529i) q^{92} +(8.73414 + 8.73414i) q^{93} +(7.95014 - 3.61162i) q^{94} +(3.76341 - 16.0165i) q^{95} +(-2.54235 - 10.0785i) q^{96} +(-6.10824 - 11.9881i) q^{97} +(0.0633751 - 1.41279i) q^{98} -0.587587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 720 q + 16 q^{10} + 16 q^{12} + 4 q^{13} + 20 q^{17} - 28 q^{18} - 20 q^{20} - 4 q^{22} + 20 q^{25} + 4 q^{30} - 20 q^{37} - 64 q^{40} - 80 q^{42} - 140 q^{44} - 20 q^{45} - 236 q^{48} - 40 q^{50} - 16 q^{52}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41279 − 0.0633751i −0.998995 − 0.0448130i
\(3\) 0.834190 + 1.63719i 0.481620 + 0.945233i 0.996142 + 0.0877554i \(0.0279694\pi\)
−0.514522 + 0.857477i \(0.672031\pi\)
\(4\) 1.99197 + 0.179072i 0.995984 + 0.0895359i
\(5\) −1.91219 − 1.15911i −0.855156 − 0.518370i
\(6\) −1.07478 − 2.36588i −0.438778 − 0.965866i
\(7\) 0.707107 + 0.707107i 0.267261 + 0.267261i
\(8\) −2.80289 − 0.379233i −0.990971 − 0.134079i
\(9\) −0.221165 + 0.304407i −0.0737215 + 0.101469i
\(10\) 2.62807 + 1.75877i 0.831068 + 0.556171i
\(11\) 0.917897 + 1.26338i 0.276756 + 0.380922i 0.924656 − 0.380803i \(-0.124352\pi\)
−0.647900 + 0.761726i \(0.724352\pi\)
\(12\) 1.36851 + 3.41061i 0.395053 + 0.984558i
\(13\) −0.136798 − 0.863706i −0.0379408 − 0.239549i 0.961429 − 0.275054i \(-0.0886959\pi\)
−0.999369 + 0.0355054i \(0.988696\pi\)
\(14\) −0.954182 − 1.04381i −0.255016 − 0.278970i
\(15\) 0.302556 − 4.09754i 0.0781197 − 1.05798i
\(16\) 3.93587 + 0.713411i 0.983967 + 0.178353i
\(17\) −1.39517 − 0.710875i −0.338378 − 0.172412i 0.276541 − 0.961002i \(-0.410812\pi\)
−0.614920 + 0.788590i \(0.710812\pi\)
\(18\) 0.331752 − 0.416048i 0.0781946 − 0.0980634i
\(19\) 2.27371 + 6.99775i 0.521624 + 1.60539i 0.770896 + 0.636961i \(0.219809\pi\)
−0.249272 + 0.968433i \(0.580191\pi\)
\(20\) −3.60145 − 2.65133i −0.805309 − 0.592855i
\(21\) −0.567807 + 1.74753i −0.123906 + 0.381342i
\(22\) −1.21673 − 1.84306i −0.259408 − 0.392942i
\(23\) −1.02865 + 6.49463i −0.214488 + 1.35422i 0.611816 + 0.791000i \(0.290439\pi\)
−0.826304 + 0.563224i \(0.809561\pi\)
\(24\) −1.71727 − 4.90522i −0.350536 − 1.00127i
\(25\) 2.31293 + 4.43287i 0.462585 + 0.886575i
\(26\) 0.138529 + 1.22891i 0.0271678 + 0.241009i
\(27\) 4.76165 + 0.754172i 0.916381 + 0.145140i
\(28\) 1.28191 + 1.53516i 0.242258 + 0.290117i
\(29\) −0.567208 − 0.184297i −0.105328 − 0.0342231i 0.255879 − 0.966709i \(-0.417635\pi\)
−0.361207 + 0.932486i \(0.617635\pi\)
\(30\) −0.687131 + 5.76979i −0.125452 + 1.05342i
\(31\) 6.39327 − 2.07730i 1.14827 − 0.373094i 0.327776 − 0.944756i \(-0.393701\pi\)
0.820490 + 0.571662i \(0.193701\pi\)
\(32\) −5.51535 − 1.25734i −0.974986 − 0.222268i
\(33\) −1.30269 + 2.55667i −0.226769 + 0.445059i
\(34\) 1.92603 + 1.09274i 0.330312 + 0.187403i
\(35\) −0.532506 − 2.17174i −0.0900100 − 0.367090i
\(36\) −0.495063 + 0.566764i −0.0825106 + 0.0944607i
\(37\) −5.42145 + 0.858673i −0.891281 + 0.141165i −0.585246 − 0.810856i \(-0.699002\pi\)
−0.306034 + 0.952020i \(0.599002\pi\)
\(38\) −2.76879 − 10.0305i −0.449158 − 1.62716i
\(39\) 1.29994 − 0.944460i 0.208157 − 0.151235i
\(40\) 4.92008 + 3.97402i 0.777932 + 0.628348i
\(41\) 7.10317 + 5.16075i 1.10933 + 0.805974i 0.982558 − 0.185958i \(-0.0595387\pi\)
0.126771 + 0.991932i \(0.459539\pi\)
\(42\) 0.912944 − 2.43291i 0.140870 − 0.375407i
\(43\) −4.06563 + 4.06563i −0.620002 + 0.620002i −0.945532 − 0.325529i \(-0.894457\pi\)
0.325529 + 0.945532i \(0.394457\pi\)
\(44\) 1.60219 + 2.68097i 0.241538 + 0.404172i
\(45\) 0.775750 − 0.325729i 0.115642 − 0.0485568i
\(46\) 1.86487 − 9.11038i 0.274959 − 1.34325i
\(47\) −5.50151 + 2.80316i −0.802477 + 0.408883i −0.806592 − 0.591109i \(-0.798690\pi\)
0.00411439 + 0.999992i \(0.498690\pi\)
\(48\) 2.11527 + 7.03889i 0.305313 + 1.01598i
\(49\) 1.00000i 0.142857i
\(50\) −2.98675 − 6.40932i −0.422390 − 0.906414i
\(51\) − 2.87716i − 0.402884i
\(52\) −0.117831 − 1.74497i −0.0163402 − 0.241984i
\(53\) 2.25678 − 1.14989i 0.309992 − 0.157949i −0.292073 − 0.956396i \(-0.594345\pi\)
0.602065 + 0.798447i \(0.294345\pi\)
\(54\) −6.67944 − 1.36726i −0.908956 − 0.186060i
\(55\) −0.290798 − 3.47976i −0.0392112 − 0.469210i
\(56\) −1.71378 − 2.25010i −0.229014 − 0.300682i
\(57\) −9.55995 + 9.55995i −1.26625 + 1.26625i
\(58\) 0.789668 + 0.296320i 0.103688 + 0.0389088i
\(59\) −10.8163 − 7.85848i −1.40816 − 1.02309i −0.993587 − 0.113070i \(-0.963931\pi\)
−0.414572 − 0.910017i \(-0.636069\pi\)
\(60\) 1.33644 − 8.10798i 0.172533 − 1.04674i
\(61\) −0.470316 + 0.341705i −0.0602179 + 0.0437508i −0.617487 − 0.786581i \(-0.711849\pi\)
0.557269 + 0.830332i \(0.311849\pi\)
\(62\) −9.16402 + 2.52962i −1.16383 + 0.321262i
\(63\) −0.371635 + 0.0588612i −0.0468216 + 0.00741582i
\(64\) 7.71237 + 2.12589i 0.964046 + 0.265737i
\(65\) −0.739548 + 1.81013i −0.0917297 + 0.224519i
\(66\) 2.00246 − 3.52949i 0.246485 − 0.434450i
\(67\) −3.37846 + 6.63059i −0.412744 + 0.810056i 0.587256 + 0.809402i \(0.300208\pi\)
−1.00000 0.000654498i \(0.999792\pi\)
\(68\) −2.65184 − 1.66587i −0.321582 − 0.202017i
\(69\) −11.4910 + 3.73367i −1.38336 + 0.449481i
\(70\) 0.614687 + 3.10196i 0.0734691 + 0.370755i
\(71\) −6.49492 − 2.11033i −0.770805 − 0.250450i −0.102896 − 0.994692i \(-0.532811\pi\)
−0.667910 + 0.744242i \(0.732811\pi\)
\(72\) 0.735341 − 0.769346i 0.0866607 − 0.0906683i
\(73\) 13.5462 + 2.14550i 1.58546 + 0.251112i 0.886044 − 0.463601i \(-0.153443\pi\)
0.699417 + 0.714714i \(0.253443\pi\)
\(74\) 7.71380 − 0.869542i 0.896711 − 0.101082i
\(75\) −5.32804 + 7.48456i −0.615229 + 0.864243i
\(76\) 3.27605 + 14.3464i 0.375789 + 1.64565i
\(77\) −0.244291 + 1.54239i −0.0278396 + 0.175772i
\(78\) −1.89640 + 1.25194i −0.214725 + 0.141755i
\(79\) 0.112435 − 0.346038i 0.0126499 − 0.0389323i −0.944532 − 0.328418i \(-0.893484\pi\)
0.957182 + 0.289486i \(0.0934844\pi\)
\(80\) −6.69919 − 5.92628i −0.748993 − 0.662578i
\(81\) 3.08622 + 9.49842i 0.342914 + 1.05538i
\(82\) −9.70824 − 7.74124i −1.07210 − 0.854877i
\(83\) 8.12296 + 4.13885i 0.891610 + 0.454298i 0.838884 − 0.544311i \(-0.183209\pi\)
0.0527268 + 0.998609i \(0.483209\pi\)
\(84\) −1.44399 + 3.37935i −0.157552 + 0.368717i
\(85\) 1.84385 + 2.97648i 0.199993 + 0.322845i
\(86\) 6.00155 − 5.48623i 0.647164 − 0.591595i
\(87\) −0.171430 − 1.08237i −0.0183792 − 0.116042i
\(88\) −2.09365 − 3.88920i −0.223184 − 0.414590i
\(89\) 6.83491 + 9.40744i 0.724499 + 0.997187i 0.999362 + 0.0357038i \(0.0113673\pi\)
−0.274864 + 0.961483i \(0.588633\pi\)
\(90\) −1.11662 + 0.411024i −0.117702 + 0.0433258i
\(91\) 0.514002 − 0.707463i 0.0538821 − 0.0741623i
\(92\) −3.21204 + 12.7529i −0.334878 + 1.32958i
\(93\) 8.73414 + 8.73414i 0.905688 + 0.905688i
\(94\) 7.95014 − 3.61162i 0.819994 − 0.372510i
\(95\) 3.76341 − 16.0165i 0.386118 − 1.64326i
\(96\) −2.54235 − 10.0785i −0.259478 − 1.02864i
\(97\) −6.10824 − 11.9881i −0.620198 − 1.21721i −0.960864 − 0.277022i \(-0.910652\pi\)
0.340666 − 0.940184i \(-0.389348\pi\)
\(98\) 0.0633751 − 1.41279i 0.00640186 − 0.142714i
\(99\) −0.587587 −0.0590547
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 700.2.bj.a.323.2 ✓ 720
4.3 odd 2 inner 700.2.bj.a.323.83 yes 720
25.12 odd 20 inner 700.2.bj.a.687.83 yes 720
100.87 even 20 inner 700.2.bj.a.687.2 yes 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.bj.a.323.2 ✓ 720 1.1 even 1 trivial
700.2.bj.a.323.83 yes 720 4.3 odd 2 inner
700.2.bj.a.687.2 yes 720 100.87 even 20 inner
700.2.bj.a.687.83 yes 720 25.12 odd 20 inner