LMFDB - Embedded newform 2070.3.c.a.91.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.13
Root			$-1.01877i$ of defining polynomial
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.a.91.12

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.61815i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.61815i q^{7} +2.82843 q^{8} +3.16228i q^{10} +12.3764i q^{11} -13.0302 q^{13} -10.7737i q^{14} +4.00000 q^{16} -9.13040i q^{17} -14.4549i q^{19} +4.47214i q^{20} +17.5029i q^{22} +(-22.5529 - 4.51289i) q^{23} -5.00000 q^{25} -18.4275 q^{26} -15.2363i q^{28} +21.2813 q^{29} +36.8428 q^{31} +5.65685 q^{32} -12.9123i q^{34} +17.0347 q^{35} -56.9603i q^{37} -20.4424i q^{38} +6.32456i q^{40} -70.7680 q^{41} -70.0086i q^{43} +24.7529i q^{44} +(-31.8946 - 6.38219i) q^{46} +66.2614 q^{47} -9.03623 q^{49} -7.07107 q^{50} -26.0604 q^{52} -77.4364i q^{53} -27.6746 q^{55} -21.5474i q^{56} +30.0963 q^{58} -82.7923 q^{59} -23.9941i q^{61} +52.1036 q^{62} +8.00000 q^{64} -29.1364i q^{65} +118.512i q^{67} -18.2608i q^{68} +24.0907 q^{70} -69.0263 q^{71} +25.9840 q^{73} -80.5540i q^{74} -28.9099i q^{76} +94.2857 q^{77} +28.8543i q^{79} +8.94427i q^{80} -100.081 q^{82} -69.3871i q^{83} +20.4162 q^{85} -99.0071i q^{86} +35.0059i q^{88} -45.4428i q^{89} +99.2661i q^{91} +(-45.1058 - 9.02579i) q^{92} +93.7077 q^{94} +32.3222 q^{95} +74.4458i q^{97} -12.7792 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	7.61815i		−	1.08831i	−0.838986	−	0.544154i	$-0.816851\pi$
$7$		−	7.61815i		−	1.08831i	0.838986	−	0.544154i	$-0.183149\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$			12.3764i			1.12513i	0.826752	+	0.562566i	$0.190186\pi$
$11$			12.3764i			1.12513i	−0.826752	+	0.562566i	$0.809814\pi$
$12$		0			0
$13$	−13.0302			−1.00232			−0.501162	−	0.865354i	$-0.667094\pi$
$13$	−13.0302			−1.00232			−0.501162	+	0.865354i	$0.667094\pi$
$14$		−	10.7737i		−	0.769549i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	9.13040i		−	0.537083i	−0.963268	−	0.268541i	$-0.913458\pi$
$17$		−	9.13040i		−	0.537083i	0.963268	−	0.268541i	$-0.0865416\pi$
$18$		0			0
$19$		−	14.4549i		−	0.760787i	−0.924825	−	0.380393i	$-0.875789\pi$
$19$		−	14.4549i		−	0.760787i	0.924825	−	0.380393i	$-0.124211\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$			17.5029i			0.795588i
$23$	−22.5529	−	4.51289i	−0.980561	−	0.196213i
$23$	−22.5529	−	4.51289i	−0.980561	−	0.196213i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−18.4275			−0.708750
$27$		0			0
$28$		−	15.2363i		−	0.544154i
$29$	21.2813			0.733839			0.366919	−	0.930253i	$-0.380412\pi$
$29$	21.2813			0.733839			0.366919	+	0.930253i	$0.380412\pi$
$30$		0			0
$31$	36.8428			1.18848			0.594239	−	0.804288i	$-0.297453\pi$
$31$	36.8428			1.18848			0.594239	+	0.804288i	$0.297453\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$		−	12.9123i		−	0.379775i
$35$	17.0347			0.486706
$36$		0			0
$37$		−	56.9603i		−	1.53947i	−0.638365	−	0.769734i	$-0.720389\pi$
$37$		−	56.9603i		−	1.53947i	0.638365	−	0.769734i	$-0.279611\pi$
$38$		−	20.4424i		−	0.537957i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	−70.7680			−1.72605			−0.863024	−	0.505163i	$-0.831433\pi$
$41$	−70.7680			−1.72605			−0.863024	+	0.505163i	$0.831433\pi$
$42$		0			0
$43$		−	70.0086i		−	1.62811i	−0.580790	−	0.814053i	$-0.697256\pi$
$43$		−	70.0086i		−	1.62811i	0.580790	−	0.814053i	$-0.302744\pi$
$44$			24.7529i			0.562566i
$45$		0			0
$46$	−31.8946	−	6.38219i	−0.693362	−	0.138743i
$47$	66.2614			1.40982			0.704908	−	0.709299i	$-0.250988\pi$
$47$	66.2614			1.40982			0.704908	+	0.709299i	$0.250988\pi$
$48$		0			0
$49$	−9.03623			−0.184413
$50$	−7.07107			−0.141421
$51$		0			0
$52$	−26.0604			−0.501162
$53$		−	77.4364i		−	1.46106i	−0.682879	−	0.730532i	$-0.739272\pi$
$53$		−	77.4364i		−	1.46106i	0.682879	−	0.730532i	$-0.260728\pi$
$54$		0			0
$55$	−27.6746			−0.503174
$56$		−	21.5474i		−	0.384775i
$57$		0			0
$58$	30.0963			0.518902
$59$	−82.7923			−1.40326			−0.701630	−	0.712541i	$-0.747544\pi$
$59$	−82.7923			−1.40326			−0.701630	+	0.712541i	$0.747544\pi$
$60$		0			0
$61$		−	23.9941i		−	0.393346i	−0.980469	−	0.196673i	$-0.936986\pi$
$61$		−	23.9941i		−	0.393346i	0.980469	−	0.196673i	$-0.0630137\pi$
$62$	52.1036			0.840381
$63$		0			0
$64$	8.00000			0.125000
$65$		−	29.1364i		−	0.448253i
$66$		0			0
$67$			118.512i			1.76884i	0.466695	+	0.884418i	$0.345445\pi$
$67$			118.512i			1.76884i	−0.466695	+	0.884418i	$0.654555\pi$
$68$		−	18.2608i		−	0.268541i
$69$		0			0
$70$	24.0907			0.344153
$71$	−69.0263			−0.972202			−0.486101	−	0.873903i	$-0.661581\pi$
$71$	−69.0263			−0.972202			−0.486101	+	0.873903i	$0.661581\pi$
$72$		0			0
$73$	25.9840			0.355945			0.177973	−	0.984035i	$-0.443046\pi$
$73$	25.9840			0.355945			0.177973	+	0.984035i	$0.443046\pi$
$74$		−	80.5540i		−	1.08857i
$75$		0			0
$76$		−	28.9099i		−	0.380393i
$77$	94.2857			1.22449
$78$		0			0
$79$			28.8543i			0.365244i	0.983183	+	0.182622i	$0.0584585\pi$
$79$			28.8543i			0.365244i	−0.983183	+	0.182622i	$0.941541\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	−100.081			−1.22050
$83$		−	69.3871i		−	0.835989i	−0.908449	−	0.417995i	$-0.862733\pi$
$83$		−	69.3871i		−	0.835989i	0.908449	−	0.417995i	$-0.137267\pi$
$84$		0			0
$85$	20.4162			0.240191
$86$		−	99.0071i		−	1.15125i
$87$		0			0
$88$			35.0059i			0.397794i
$89$		−	45.4428i		−	0.510594i	−0.966863	−	0.255297i	$-0.917827\pi$
$89$		−	45.4428i		−	0.510594i	0.966863	−	0.255297i	$-0.0821732\pi$
$90$		0			0
$91$			99.2661i			1.09084i
$92$	−45.1058	−	9.02579i	−0.490281	−	0.0981064i
$93$		0			0
$94$	93.7077			0.996891
$95$	32.3222			0.340234
$96$		0			0
$97$			74.4458i			0.767482i	0.923441	+	0.383741i	$0.125364\pi$
$97$			74.4458i			0.767482i	−0.923441	+	0.383741i	$0.874636\pi$
$98$	−12.7792			−0.130400
$99$		0			0
$100$	−10.0000			−0.100000
$101$	17.0563			0.168874			0.0844372	−	0.996429i	$-0.473091\pi$
$101$	17.0563			0.168874			0.0844372	+	0.996429i	$0.473091\pi$
$102$		0			0
$103$		−	153.952i		−	1.49468i	−0.664442	−	0.747340i	$-0.731331\pi$
$103$		−	153.952i		−	1.49468i	0.664442	−	0.747340i	$-0.268669\pi$
$104$	−36.8550			−0.354375
$105$		0			0
$106$		−	109.512i		−	1.03313i
$107$		−	112.821i		−	1.05441i	−0.849740	−	0.527203i	$-0.823241\pi$
$107$		−	112.821i		−	1.05441i	0.849740	−	0.527203i	$-0.176759\pi$
$108$		0			0
$109$		−	97.6061i		−	0.895468i	−0.894167	−	0.447734i	$-0.852231\pi$
$109$		−	97.6061i		−	0.895468i	0.894167	−	0.447734i	$-0.147769\pi$
$110$	−39.1378			−0.355798
$111$		0			0
$112$		−	30.4726i		−	0.272077i
$113$			57.0620i			0.504974i	0.967600	+	0.252487i	$0.0812485\pi$
$113$			57.0620i			0.504974i	−0.967600	+	0.252487i	$0.918752\pi$
$114$		0			0
$115$	10.0911	−	50.4298i	0.0877490	−	0.438520i
$116$	42.5627			0.366919
$117$		0			0
$118$	−117.086			−0.992255
$119$	−69.5568			−0.584511
$120$		0			0
$121$	−32.1765			−0.265921
$122$		−	33.9328i		−	0.278137i
$123$		0			0
$124$	73.6857			0.594239
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−151.376			−1.19194			−0.595969	−	0.803007i	$-0.703232\pi$
$127$	−151.376			−1.19194			−0.595969	+	0.803007i	$0.703232\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$		−	41.2051i		−	0.316962i
$131$	−46.7334			−0.356743			−0.178372	−	0.983963i	$-0.557083\pi$
$131$	−46.7334			−0.356743			−0.178372	+	0.983963i	$0.557083\pi$
$132$		0			0
$133$	−110.120			−0.827970
$134$			167.601i			1.25076i
$135$		0			0
$136$		−	25.8247i		−	0.189887i
$137$		−	31.1949i		−	0.227700i	−0.993498	−	0.113850i	$-0.963682\pi$
$137$		−	31.1949i		−	0.227700i	0.993498	−	0.113850i	$-0.0363183\pi$
$138$		0			0
$139$	36.5517			0.262962			0.131481	−	0.991319i	$-0.458027\pi$
$139$	36.5517			0.262962			0.131481	+	0.991319i	$0.458027\pi$
$140$	34.0694			0.243353
$141$		0			0
$142$	−97.6180			−0.687451
$143$		−	161.268i		−	1.12775i
$144$		0			0
$145$			47.5865i			0.328183i
$146$	36.7470			0.251691
$147$		0			0
$148$		−	113.921i		−	0.769734i
$149$			182.281i			1.22336i	0.791104	+	0.611681i	$0.209506\pi$
$149$			182.281i			1.22336i	−0.791104	+	0.611681i	$0.790494\pi$
$150$		0			0
$151$	40.9462			0.271167			0.135584	−	0.990766i	$-0.456709\pi$
$151$	40.9462			0.271167			0.135584	+	0.990766i	$0.456709\pi$
$152$		−	40.8848i		−	0.268979i
$153$		0			0
$154$	133.340			0.865845
$155$			82.3831i			0.531504i
$156$		0			0
$157$		−	252.396i		−	1.60762i	−0.594887	−	0.803810i	$-0.702803\pi$
$157$		−	252.396i		−	1.60762i	0.594887	−	0.803810i	$-0.297197\pi$
$158$			40.8062i			0.258267i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	−34.3799	+	171.811i	−0.213540	+	1.06715i
$162$		0			0
$163$	−46.9686			−0.288151			−0.144076	−	0.989567i	$-0.546021\pi$
$163$	−46.9686			−0.288151			−0.144076	+	0.989567i	$0.546021\pi$
$164$	−141.536			−0.863024
$165$		0			0
$166$		−	98.1282i		−	0.591134i
$167$	−78.4530			−0.469778			−0.234889	−	0.972022i	$-0.575473\pi$
$167$	−78.4530			−0.469778			−0.234889	+	0.972022i	$0.575473\pi$
$168$		0			0
$169$	0.786229			0.00465224
$170$	28.8729			0.169840
$171$		0			0
$172$		−	140.017i		−	0.814053i
$173$	−137.080			−0.792369			−0.396184	−	0.918171i	$-0.629666\pi$
$173$	−137.080			−0.792369			−0.396184	+	0.918171i	$0.629666\pi$
$174$		0			0
$175$			38.0908i			0.217661i
$176$			49.5058i			0.281283i
$177$		0			0
$178$		−	64.2659i		−	0.361044i
$179$	55.6699			0.311005			0.155503	−	0.987835i	$-0.450300\pi$
$179$	55.6699			0.311005			0.155503	+	0.987835i	$0.450300\pi$
$180$		0			0
$181$			23.7317i			0.131114i	0.997849	+	0.0655572i	$0.0208825\pi$
$181$			23.7317i			0.131114i	−0.997849	+	0.0655572i	$0.979118\pi$
$182$			140.383i			0.771337i
$183$		0			0
$184$	−63.7893	−	12.7644i	−0.346681	−	0.0693717i
$185$	127.367			0.688471
$186$		0			0
$187$	113.002			0.604289
$188$	132.523			0.704908
$189$		0			0
$190$	45.7105			0.240582
$191$		−	351.658i		−	1.84114i	−0.390576	−	0.920571i	$-0.627724\pi$
$191$		−	351.658i		−	1.84114i	0.390576	−	0.920571i	$-0.372276\pi$
$192$		0			0
$193$	−39.4808			−0.204564			−0.102282	−	0.994755i	$-0.532614\pi$
$193$	−39.4808			−0.204564			−0.102282	+	0.994755i	$0.532614\pi$
$194$			105.282i			0.542692i
$195$		0			0
$196$	−18.0725			−0.0922064
$197$	225.017			1.14222			0.571108	−	0.820875i	$-0.306514\pi$
$197$	225.017			1.14222			0.571108	+	0.820875i	$0.306514\pi$
$198$		0			0
$199$			37.0872i			0.186368i	0.995649	+	0.0931840i	$0.0297045\pi$
$199$			37.0872i			0.186368i	−0.995649	+	0.0931840i	$0.970296\pi$
$200$	−14.1421			−0.0707107
$201$		0			0
$202$	24.1213			0.119412
$203$		−	162.124i		−	0.798642i
$204$		0			0
$205$		−	158.242i		−	0.771912i
$206$		−	217.721i		−	1.05690i
$207$		0			0
$208$	−52.1208			−0.250581
$209$	178.901			0.855985
$210$		0			0
$211$	217.064			1.02874			0.514369	−	0.857569i	$-0.328026\pi$
$211$	217.064			1.02874			0.514369	+	0.857569i	$0.328026\pi$
$212$		−	154.873i		−	0.730532i
$213$		0			0
$214$		−	159.554i		−	0.745577i
$215$	156.544			0.728111
$216$		0			0
$217$		−	280.674i		−	1.29343i
$218$		−	138.036i		−	0.633192i
$219$		0			0
$220$	−55.3492			−0.251587
$221$			118.971i			0.538330i
$222$		0			0
$223$	269.398			1.20806			0.604030	−	0.796961i	$-0.293561\pi$
$223$	269.398			1.20806			0.604030	+	0.796961i	$0.293561\pi$
$224$		−	43.0948i		−	0.192387i
$225$		0			0
$226$			80.6979i			0.357070i
$227$		−	175.997i		−	0.775319i	−0.921803	−	0.387659i	$-0.873284\pi$
$227$		−	175.997i		−	0.775319i	0.921803	−	0.387659i	$-0.126716\pi$
$228$		0			0
$229$			63.0669i			0.275401i	0.990474	+	0.137701i	$0.0439712\pi$
$229$			63.0669i			0.275401i	−0.990474	+	0.137701i	$0.956029\pi$
$230$	14.2710	−	71.3186i	0.0620479	−	0.310081i
$231$		0			0
$232$	60.1927			0.259451
$233$	−4.48041			−0.0192292			−0.00961461	−	0.999954i	$-0.503060\pi$
$233$	−4.48041			−0.0192292			−0.00961461	+	0.999954i	$0.503060\pi$
$234$		0			0
$235$			148.165i			0.630489i
$236$	−165.585			−0.701630
$237$		0			0
$238$	−98.3682			−0.413312
$239$	−85.5205			−0.357826			−0.178913	−	0.983865i	$-0.557258\pi$
$239$	−85.5205			−0.357826			−0.178913	+	0.983865i	$0.557258\pi$
$240$		0			0
$241$		−	257.515i		−	1.06853i	−0.845318	−	0.534263i	$-0.820589\pi$
$241$		−	257.515i		−	1.06853i	0.845318	−	0.534263i	$-0.179411\pi$
$242$	−45.5044			−0.188035
$243$		0			0
$244$		−	47.9882i		−	0.196673i
$245$		−	20.2056i		−	0.0824719i
$246$		0			0
$247$			188.351i			0.762554i
$248$	104.207			0.420191
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$			121.438i			0.483816i	0.970299	+	0.241908i	$0.0777733\pi$
$251$			121.438i			0.483816i	−0.970299	+	0.241908i	$0.922227\pi$
$252$		0			0
$253$	55.8536	−	279.125i	0.220765	−	1.10326i
$254$	−214.078			−0.842827
$255$		0			0
$256$	16.0000			0.0625000
$257$	−192.295			−0.748230			−0.374115	−	0.927382i	$-0.622053\pi$
$257$	−192.295			−0.748230			−0.374115	+	0.927382i	$0.622053\pi$
$258$		0			0
$259$	−433.932			−1.67541
$260$		−	58.2728i		−	0.224126i
$261$		0			0
$262$	−66.0910			−0.252256
$263$			417.798i			1.58858i	0.607536	+	0.794292i	$0.292158\pi$
$263$			417.798i			1.58858i	−0.607536	+	0.794292i	$0.707842\pi$
$264$		0			0
$265$	173.153			0.653407
$266$	−155.733			−0.585463
$267$		0			0
$268$			237.024i			0.884418i
$269$	474.878			1.76534			0.882672	−	0.469989i	$-0.155742\pi$
$269$	474.878			1.76534			0.882672	+	0.469989i	$0.155742\pi$
$270$		0			0
$271$	−32.0762			−0.118362			−0.0591812	−	0.998247i	$-0.518849\pi$
$271$	−32.0762			−0.118362			−0.0591812	+	0.998247i	$0.518849\pi$
$272$		−	36.5216i		−	0.134271i
$273$		0			0
$274$		−	44.1162i		−	0.161008i
$275$		−	61.8822i		−	0.225026i
$276$		0			0
$277$	459.038			1.65718			0.828589	−	0.559858i	$-0.189144\pi$
$277$	459.038			1.65718			0.828589	+	0.559858i	$0.189144\pi$
$278$	51.6919			0.185942
$279$		0			0
$280$	48.1814			0.172076
$281$			321.543i			1.14428i	0.820156	+	0.572140i	$0.193887\pi$
$281$			321.543i			1.14428i	−0.820156	+	0.572140i	$0.806113\pi$
$282$		0			0
$283$			428.712i			1.51488i	0.652904	+	0.757441i	$0.273551\pi$
$283$			428.712i			1.51488i	−0.652904	+	0.757441i	$0.726449\pi$
$284$	−138.053			−0.486101
$285$		0			0
$286$		−	228.067i		−	0.797437i
$287$			539.121i			1.87847i
$288$		0			0
$289$	205.636			0.711542
$290$			67.2975i			0.232060i
$291$		0			0
$292$	51.9680			0.177973
$293$			359.840i			1.22812i	0.789258	+	0.614062i	$0.210466\pi$
$293$			359.840i			1.22812i	−0.789258	+	0.614062i	$0.789534\pi$
$294$		0			0
$295$		−	185.129i		−	0.627557i
$296$		−	161.108i		−	0.544284i
$297$		0			0
$298$			257.784i			0.865048i
$299$	293.869	+	58.8039i	0.982840	+	0.196669i
$300$		0			0
$301$	−533.336			−1.77188
$302$	57.9067			0.191744
$303$		0			0
$304$		−	57.8198i		−	0.190197i
$305$	53.6524			0.175910
$306$		0			0
$307$	−448.843			−1.46203			−0.731014	−	0.682362i	$-0.760953\pi$
$307$	−448.843			−1.46203			−0.731014	+	0.682362i	$0.760953\pi$
$308$	188.571			0.612245
$309$		0			0
$310$			116.507i			0.375830i
$311$	−187.562			−0.603094			−0.301547	−	0.953451i	$-0.597503\pi$
$311$	−187.562			−0.603094			−0.301547	+	0.953451i	$0.597503\pi$
$312$		0			0
$313$			93.4265i			0.298487i	0.988800	+	0.149244i	$0.0476839\pi$
$313$			93.4265i			0.298487i	−0.988800	+	0.149244i	$0.952316\pi$
$314$		−	356.942i		−	1.13676i
$315$		0			0
$316$			57.7086i			0.182622i
$317$	−453.069			−1.42924			−0.714620	−	0.699513i	$-0.753400\pi$
$317$	−453.069			−1.42924			−0.714620	+	0.699513i	$0.753400\pi$
$318$		0			0
$319$			263.387i			0.825665i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−48.6205	+	242.978i	−0.150995	+	0.754590i
$323$	−131.979			−0.408605
$324$		0			0
$325$	65.1510			0.200465
$326$	−66.4237			−0.203754
$327$		0			0
$328$	−200.162			−0.610250
$329$		−	504.789i		−	1.53431i
$330$		0			0
$331$	−178.326			−0.538749			−0.269374	−	0.963036i	$-0.586817\pi$
$331$	−178.326			−0.538749			−0.269374	+	0.963036i	$0.586817\pi$
$332$		−	138.774i		−	0.417995i
$333$		0			0
$334$	−110.949			−0.332183
$335$	−265.001			−0.791048
$336$		0			0
$337$		−	85.7271i		−	0.254383i	−0.991878	−	0.127192i	$-0.959404\pi$
$337$		−	85.7271i		−	0.254383i	0.991878	−	0.127192i	$-0.0405963\pi$
$338$	1.11190			0.00328963
$339$		0			0
$340$	40.8324			0.120095
$341$			455.984i			1.33720i
$342$		0			0
$343$		−	304.450i		−	0.887610i
$344$		−	198.014i		−	0.575623i
$345$		0			0
$346$	−193.860			−0.560289
$347$	−216.730			−0.624582			−0.312291	−	0.949986i	$-0.601096\pi$
$347$	−216.730			−0.624582			−0.312291	+	0.949986i	$0.601096\pi$
$348$		0			0
$349$	−528.822			−1.51525			−0.757624	−	0.652691i	$-0.773640\pi$
$349$	−528.822			−1.51525			−0.757624	+	0.652691i	$0.773640\pi$
$350$			53.8685i			0.153910i
$351$		0			0
$352$			70.0118i			0.198897i
$353$	28.7915			0.0815623			0.0407811	−	0.999168i	$-0.487015\pi$
$353$	28.7915			0.0815623			0.0407811	+	0.999168i	$0.487015\pi$
$354$		0			0
$355$		−	154.348i		−	0.434782i
$356$		−	90.8857i		−	0.255297i
$357$		0			0
$358$	78.7292			0.219914
$359$		−	55.5671i		−	0.154783i	−0.997001	−	0.0773914i	$-0.975341\pi$
$359$		−	55.5671i		−	0.154783i	0.997001	−	0.0773914i	$-0.0246591\pi$
$360$		0			0
$361$	152.055			0.421204
$362$			33.5617i			0.0927119i
$363$		0			0
$364$			198.532i			0.545418i
$365$			58.1020i			0.159184i
$366$		0			0
$367$		−	164.288i		−	0.447652i	−0.974629	−	0.223826i	$-0.928145\pi$
$367$		−	164.288i		−	0.447652i	0.974629	−	0.223826i	$-0.0718547\pi$
$368$	−90.2116	−	18.0516i	−0.245140	−	0.0490532i
$369$		0			0
$370$	180.124			0.486822
$371$	−589.922			−1.59009
$372$		0			0
$373$		−	25.1481i		−	0.0674212i	−0.999432	−	0.0337106i	$-0.989268\pi$
$373$		−	25.1481i		−	0.0674212i	0.999432	−	0.0337106i	$-0.0107325\pi$
$374$	159.809			0.427297
$375$		0			0
$376$	187.415			0.498445
$377$	−277.300			−0.735544
$378$		0			0
$379$		−	456.740i		−	1.20512i	−0.798074	−	0.602560i	$-0.794148\pi$
$379$		−	456.740i		−	1.20512i	0.798074	−	0.602560i	$-0.205852\pi$
$380$	64.6445			0.170117
$381$		0			0
$382$		−	497.319i		−	1.30188i
$383$		−	371.017i		−	0.968712i	−0.874871	−	0.484356i	$-0.839054\pi$
$383$		−	371.017i		−	0.968712i	0.874871	−	0.484356i	$-0.160946\pi$
$384$		0			0
$385$			210.829i			0.547608i
$386$	−55.8343			−0.144648
$387$		0			0
$388$			148.892i			0.383741i
$389$			400.013i			1.02831i	0.857697	+	0.514155i	$0.171895\pi$
$389$			400.013i			1.02831i	−0.857697	+	0.514155i	$0.828105\pi$
$390$		0			0
$391$	−41.2045	+	205.917i	−0.105382	+	0.526642i
$392$	−25.5583			−0.0651998
$393$		0			0
$394$	318.222			0.807669
$395$	−64.5202			−0.163342
$396$		0			0
$397$	264.267			0.665659			0.332830	−	0.942987i	$-0.391997\pi$
$397$	264.267			0.665659			0.332830	+	0.942987i	$0.391997\pi$
$398$			52.4493i			0.131782i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	425.319i		−	1.06065i	−0.847796	−	0.530323i	$-0.822071\pi$
$401$		−	425.319i		−	1.06065i	0.847796	−	0.530323i	$-0.177929\pi$
$402$		0			0
$403$	−480.070			−1.19124
$404$	34.1126			0.0844372
$405$		0			0
$406$		−	229.278i		−	0.564725i
$407$	704.966			1.73210
$408$		0			0
$409$	−241.558			−0.590607			−0.295304	−	0.955403i	$-0.595421\pi$
$409$	−241.558			−0.590607			−0.295304	+	0.955403i	$0.595421\pi$
$410$		−	223.788i		−	0.545824i
$411$		0			0
$412$		−	307.904i		−	0.747340i
$413$			630.725i			1.52718i
$414$		0			0
$415$	155.154			0.373866
$416$	−73.7100			−0.177187
$417$		0			0
$418$	253.004			0.605273
$419$			358.352i			0.855256i	0.903955	+	0.427628i	$0.140651\pi$
$419$			358.352i			0.855256i	−0.903955	+	0.427628i	$0.859349\pi$
$420$		0			0
$421$			672.051i			1.59632i	0.602446	+	0.798160i	$0.294193\pi$
$421$			672.051i			1.59632i	−0.602446	+	0.798160i	$0.705807\pi$
$422$	306.975			0.727428
$423$		0			0
$424$		−	219.023i		−	0.516564i
$425$			45.6520i			0.107417i
$426$		0			0
$427$	−182.791			−0.428081
$428$		−	225.643i		−	0.527203i
$429$		0			0
$430$	221.387			0.514852
$431$			819.926i			1.90238i	0.308604	+	0.951191i	$0.400138\pi$
$431$			819.926i			1.90238i	−0.308604	+	0.951191i	$0.599862\pi$
$432$		0			0
$433$			316.231i			0.730326i	0.930944	+	0.365163i	$0.118987\pi$
$433$			316.231i			0.730326i	−0.930944	+	0.365163i	$0.881013\pi$
$434$		−	396.933i		−	0.914593i
$435$		0			0
$436$		−	195.212i		−	0.447734i
$437$	−65.2336	+	326.001i	−0.149276	+	0.745998i
$438$		0			0
$439$	208.878			0.475805			0.237903	−	0.971289i	$-0.423540\pi$
$439$	208.878			0.475805			0.237903	+	0.971289i	$0.423540\pi$
$440$	−78.2755			−0.177899
$441$		0			0
$442$			168.250i			0.380657i
$443$	706.981			1.59589			0.797947	−	0.602728i	$-0.205919\pi$
$443$	706.981			1.59589			0.797947	+	0.602728i	$0.205919\pi$
$444$		0			0
$445$	101.613			0.228344
$446$	380.986			0.854228
$447$		0			0
$448$		−	60.9452i		−	0.136038i
$449$	524.552			1.16827			0.584134	−	0.811658i	$-0.301434\pi$
$449$	524.552			1.16827			0.584134	+	0.811658i	$0.301434\pi$
$450$		0			0
$451$		−	875.856i		−	1.94203i
$452$			114.124i			0.252487i
$453$		0			0
$454$		−	248.898i		−	0.548233i
$455$	−221.966			−0.487837
$456$		0			0
$457$			786.926i			1.72194i	0.508657	+	0.860969i	$0.330142\pi$
$457$			786.926i			1.72194i	−0.508657	+	0.860969i	$0.669858\pi$
$458$			89.1901i			0.194738i
$459$		0			0
$460$	20.1823	−	100.860i	0.0438745	−	0.219260i
$461$	38.2459			0.0829629			0.0414814	−	0.999139i	$-0.486792\pi$
$461$	38.2459			0.0829629			0.0414814	+	0.999139i	$0.486792\pi$
$462$		0			0
$463$	−525.945			−1.13595			−0.567976	−	0.823045i	$-0.692273\pi$
$463$	−525.945			−1.13595			−0.567976	+	0.823045i	$0.692273\pi$
$464$	85.1253			0.183460
$465$		0			0
$466$	−6.33625			−0.0135971
$467$		−	166.631i		−	0.356812i	−0.983957	−	0.178406i	$-0.942906\pi$
$467$		−	166.631i		−	0.356812i	0.983957	−	0.178406i	$-0.0570941\pi$
$468$		0			0
$469$	902.843			1.92504
$470$			209.537i			0.445823i
$471$		0			0
$472$	−234.172			−0.496127
$473$	866.458			1.83183
$474$		0			0
$475$			72.2747i			0.152157i
$476$	−139.114			−0.292255
$477$		0			0
$478$	−120.944			−0.253021
$479$		−	32.0653i		−	0.0669422i	−0.999440	−	0.0334711i	$-0.989344\pi$
$479$		−	32.0653i		−	0.0669422i	0.999440	−	0.0334711i	$-0.0106562\pi$
$480$		0			0
$481$			742.204i			1.54304i
$482$		−	364.181i		−	0.755562i
$483$		0			0
$484$	−64.3530			−0.132961
$485$	−166.466			−0.343228
$486$		0			0
$487$	685.808			1.40823			0.704115	−	0.710085i	$-0.251344\pi$
$487$	685.808			1.40823			0.704115	+	0.710085i	$0.251344\pi$
$488$		−	67.8655i		−	0.139069i
$489$		0			0
$490$		−	28.5751i		−	0.0583165i
$491$	−775.805			−1.58005			−0.790026	−	0.613074i	$-0.789933\pi$
$491$	−775.805			−1.58005			−0.790026	+	0.613074i	$0.789933\pi$
$492$		0			0
$493$		−	194.307i		−	0.394132i
$494$			266.368i			0.539207i
$495$		0			0
$496$	147.371			0.297120
$497$			525.853i			1.05805i
$498$		0			0
$499$	392.467			0.786507			0.393253	−	0.919430i	$-0.371350\pi$
$499$	392.467			0.786507			0.393253	+	0.919430i	$0.371350\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$			171.739i			0.342110i
$503$		−	633.449i		−	1.25934i	−0.776862	−	0.629671i	$-0.783190\pi$
$503$		−	633.449i		−	1.25934i	0.776862	−	0.629671i	$-0.216810\pi$
$504$		0			0
$505$			38.1391i			0.0755229i
$506$	78.9889	−	394.742i	0.156105	−	0.780123i
$507$		0			0
$508$	−302.752			−0.595969
$509$	−540.009			−1.06092			−0.530460	−	0.847710i	$-0.677981\pi$
$509$	−540.009			−1.06092			−0.530460	+	0.847710i	$0.677981\pi$
$510$		0			0
$511$		−	197.950i		−	0.387378i
$512$	22.6274			0.0441942
$513$		0			0
$514$	−271.946			−0.529078
$515$	344.247			0.668441
$516$		0			0
$517$			820.080i			1.58623i
$518$	−613.673			−1.18470
$519$		0			0
$520$		−	82.4102i		−	0.158481i
$521$		−	396.259i		−	0.760574i	−0.924869	−	0.380287i	$-0.875825\pi$
$521$		−	396.259i		−	0.760574i	0.924869	−	0.380287i	$-0.124175\pi$
$522$		0			0
$523$		−	27.4737i		−	0.0525309i	−0.999655	−	0.0262655i	$-0.991638\pi$
$523$		−	27.4737i		−	0.0525309i	0.999655	−	0.0262655i	$-0.00836152\pi$
$524$	−93.4668			−0.178372
$525$		0			0
$526$			590.855i			1.12330i
$527$		−	336.390i		−	0.638311i
$528$		0			0
$529$	488.268	+	203.558i	0.923001	+	0.384797i
$530$	244.875			0.462029
$531$		0			0
$532$	−220.240			−0.413985
$533$	922.121			1.73006
$534$		0			0
$535$	252.276			0.471544
$536$			335.203i			0.625378i
$537$		0			0
$538$	671.578			1.24829
$539$		−	111.836i		−	0.207489i
$540$		0			0
$541$	−802.725			−1.48378			−0.741890	−	0.670522i	$-0.766070\pi$
$541$	−802.725			−1.48378			−0.741890	+	0.670522i	$0.766070\pi$
$542$	−45.3626			−0.0836948
$543$		0			0
$544$		−	51.6494i		−	0.0949437i
$545$	218.254			0.400466
$546$		0			0
$547$	−439.126			−0.802790			−0.401395	−	0.915905i	$-0.631475\pi$
$547$	−439.126			−0.802790			−0.401395	+	0.915905i	$0.631475\pi$
$548$		−	62.3897i		−	0.113850i
$549$		0			0
$550$		−	87.5147i		−	0.159118i
$551$		−	307.620i		−	0.558295i
$552$		0			0
$553$	219.817			0.397498
$554$	649.178			1.17180
$555$		0			0
$556$	73.1034			0.131481
$557$			795.405i			1.42802i	0.700137	+	0.714008i	$0.253122\pi$
$557$			795.405i			1.42802i	−0.700137	+	0.714008i	$0.746878\pi$
$558$		0			0
$559$			912.226i			1.63189i
$560$	68.1388			0.121676
$561$		0			0
$562$			454.730i			0.809128i
$563$			214.573i			0.381124i	0.981675	+	0.190562i	$0.0610309\pi$
$563$			214.573i			0.381124i	−0.981675	+	0.190562i	$0.938969\pi$
$564$		0			0
$565$	−127.595			−0.225831
$566$			606.290i			1.07118i
$567$		0			0
$568$	−195.236			−0.343725
$569$		−	1101.54i		−	1.93592i	−0.251114	−	0.967958i	$-0.580797\pi$
$569$		−	1101.54i		−	1.93592i	0.251114	−	0.967958i	$-0.419203\pi$
$570$		0			0
$571$			560.543i			0.981686i	0.871248	+	0.490843i	$0.163311\pi$
$571$			560.543i			0.981686i	−0.871248	+	0.490843i	$0.836689\pi$
$572$		−	322.535i		−	0.563873i
$573$		0			0
$574$			762.433i			1.32828i
$575$	112.765	+	22.5645i	0.196112	+	0.0392425i
$576$		0			0
$577$	795.419			1.37854			0.689271	−	0.724504i	$-0.257931\pi$
$577$	795.419			1.37854			0.689271	+	0.724504i	$0.257931\pi$
$578$	290.813			0.503136
$579$		0			0
$580$			95.1730i			0.164091i
$581$	−528.602			−0.909813
$582$		0			0
$583$	958.387			1.64389
$584$	73.4939			0.125846
$585$		0			0
$586$			508.891i			0.868415i
$587$	−579.454			−0.987144			−0.493572	−	0.869705i	$-0.664309\pi$
$587$	−579.454			−0.987144			−0.493572	+	0.869705i	$0.664309\pi$
$588$		0			0
$589$		−	532.561i		−	0.904179i
$590$		−	261.812i		−	0.443750i
$591$		0			0
$592$		−	227.841i		−	0.384867i
$593$	175.139			0.295344			0.147672	−	0.989036i	$-0.452822\pi$
$593$	175.139			0.295344			0.147672	+	0.989036i	$0.452822\pi$
$594$		0			0
$595$		−	155.534i		−	0.261401i
$596$			364.562i			0.611681i
$597$		0			0
$598$	415.594	+	83.1613i	0.694973	+	0.139066i
$599$	512.672			0.855879			0.427940	−	0.903807i	$-0.359240\pi$
$599$	512.672			0.855879			0.427940	+	0.903807i	$0.359240\pi$
$600$		0			0
$601$	−175.570			−0.292130			−0.146065	−	0.989275i	$-0.546661\pi$
$601$	−175.570			−0.292130			−0.146065	+	0.989275i	$0.546661\pi$
$602$	−754.251			−1.25291
$603$		0			0
$604$	81.8925			0.135584
$605$		−	71.9488i		−	0.118924i
$606$		0			0
$607$	−173.873			−0.286447			−0.143223	−	0.989690i	$-0.545747\pi$
$607$	−173.873			−0.286447			−0.143223	+	0.989690i	$0.545747\pi$
$608$		−	81.7695i		−	0.134489i
$609$		0			0
$610$	75.8760			0.124387
$611$	−863.399			−1.41309
$612$		0			0
$613$			560.182i			0.913837i	0.889509	+	0.456919i	$0.151047\pi$
$613$			560.182i			0.913837i	−0.889509	+	0.456919i	$0.848953\pi$
$614$	−634.760			−1.03381
$615$		0			0
$616$	266.680			0.432922
$617$		−	308.240i		−	0.499578i	−0.968300	−	0.249789i	$-0.919639\pi$
$617$		−	308.240i		−	0.499578i	0.968300	−	0.249789i	$-0.0803613\pi$
$618$		0			0
$619$			424.114i			0.685161i	0.939489	+	0.342580i	$0.111301\pi$
$619$			424.114i			0.685161i	−0.939489	+	0.342580i	$0.888699\pi$
$620$			164.766i			0.265752i
$621$		0			0
$622$	−265.253			−0.426452
$623$	−346.190			−0.555683
$624$		0			0
$625$	25.0000			0.0400000
$626$			132.125i			0.211062i
$627$		0			0
$628$		−	504.792i		−	0.803810i
$629$	−520.070			−0.826821
$630$		0			0
$631$		−	172.251i		−	0.272981i	−0.990641	−	0.136491i	$-0.956418\pi$
$631$		−	172.251i		−	0.272981i	0.990641	−	0.136491i	$-0.0435823\pi$
$632$			81.6123i			0.129133i
$633$		0			0
$634$	−640.737			−1.01063
$635$		−	338.487i		−	0.533051i
$636$		0			0
$637$	117.744			0.184841
$638$			372.486i			0.583834i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$		−	365.954i		−	0.570912i	−0.958392	−	0.285456i	$-0.907855\pi$
$641$		−	365.954i		−	0.570912i	0.958392	−	0.285456i	$-0.0921450\pi$
$642$		0			0
$643$			823.317i			1.28043i	0.768195	+	0.640215i	$0.221155\pi$
$643$			823.317i			1.28043i	−0.768195	+	0.640215i	$0.778845\pi$
$644$	−68.7598	+	343.623i	−0.106770	+	0.533576i
$645$		0			0
$646$	−186.647			−0.288927
$647$	828.654			1.28076			0.640381	−	0.768057i	$-0.278776\pi$
$647$	828.654			1.28076			0.640381	+	0.768057i	$0.278776\pi$
$648$		0			0
$649$		−	1024.68i		−	1.57885i
$650$	92.1375			0.141750
$651$		0			0
$652$	−93.9372			−0.144076
$653$	912.811			1.39787			0.698937	−	0.715183i	$-0.253657\pi$
$653$	912.811			1.39787			0.698937	+	0.715183i	$0.253657\pi$
$654$		0			0
$655$		−	104.499i		−	0.159541i
$656$	−283.072			−0.431512
$657$		0			0
$658$		−	713.880i		−	1.08492i
$659$			247.106i			0.374971i	0.982267	+	0.187486i	$0.0600338\pi$
$659$			247.106i			0.374971i	−0.982267	+	0.187486i	$0.939966\pi$
$660$		0			0
$661$			937.016i			1.41757i	0.705423	+	0.708787i	$0.250757\pi$
$661$			937.016i			1.41757i	−0.705423	+	0.708787i	$0.749243\pi$
$662$	−252.191			−0.380953
$663$		0			0
$664$		−	196.256i		−	0.295567i
$665$		−	246.236i		−	0.370279i
$666$		0			0
$667$	−479.956	−	96.0404i	−0.719574	−	0.143989i
$668$	−156.906			−0.234889
$669$		0			0
$670$	−374.768			−0.559355
$671$	296.962			0.442566
$672$		0			0
$673$	1174.75			1.74554			0.872769	−	0.488134i	$-0.162322\pi$
$673$	1174.75			1.74554			0.872769	+	0.488134i	$0.162322\pi$
$674$		−	121.236i		−	0.179876i
$675$		0			0
$676$	1.57246			0.00232612
$677$			266.022i			0.392943i	0.980510	+	0.196471i	$0.0629483\pi$
$677$			266.022i			0.392943i	−0.980510	+	0.196471i	$0.937052\pi$
$678$		0			0
$679$	567.139			0.835256
$680$	57.7457			0.0849202
$681$		0			0
$682$			644.858i			0.945540i
$683$	214.808			0.314507			0.157254	−	0.987558i	$-0.449736\pi$
$683$	214.808			0.314507			0.157254	+	0.987558i	$0.449736\pi$
$684$		0			0
$685$	69.7539			0.101830
$686$		−	430.557i		−	0.627635i
$687$		0			0
$688$		−	280.034i		−	0.407027i
$689$			1009.01i			1.46446i
$690$		0			0
$691$	−998.531			−1.44505			−0.722526	−	0.691344i	$-0.757019\pi$
$691$	−998.531			−1.44505			−0.722526	+	0.691344i	$0.757019\pi$
$692$	−274.160			−0.396184
$693$		0			0
$694$	−306.502			−0.441646
$695$			81.7321i			0.117600i
$696$		0			0
$697$			646.140i			0.927030i
$698$	−747.867			−1.07144
$699$		0			0
$700$			76.1815i			0.108831i
$701$			1089.04i			1.55356i	0.629773	+	0.776779i	$0.283148\pi$
$701$			1089.04i			1.55356i	−0.629773	+	0.776779i	$0.716852\pi$
$702$		0			0
$703$	−823.358			−1.17121
$704$			99.0116i			0.140641i
$705$		0			0
$706$	40.7173			0.0576732
$707$		−	129.938i		−	0.183787i
$708$		0			0
$709$			779.102i			1.09887i	0.835535	+	0.549437i	$0.185158\pi$
$709$			779.102i			1.09887i	−0.835535	+	0.549437i	$0.814842\pi$
$710$		−	218.280i		−	0.307437i
$711$		0			0
$712$		−	128.532i		−	0.180522i
$713$	−830.913	−	166.268i	−1.16538	−	0.233195i
$714$		0			0
$715$	360.605			0.504343
$716$	111.340			0.155503
$717$		0			0
$718$		−	78.5837i		−	0.109448i
$719$	802.454			1.11607			0.558035	−	0.829817i	$-0.311555\pi$
$719$	802.454			1.11607			0.558035	+	0.829817i	$0.311555\pi$
$720$		0			0
$721$	−1172.83			−1.62667
$722$	215.038			0.297836
$723$		0			0
$724$			47.4634i			0.0655572i
$725$	−106.407			−0.146768
$726$		0			0
$727$		−	19.7852i		−	0.0272149i	−0.999907	−	0.0136075i	$-0.995668\pi$
$727$		−	19.7852i		−	0.0272149i	0.999907	−	0.0136075i	$-0.00433152\pi$
$728$			280.767i			0.385669i
$729$		0			0
$730$			82.1687i			0.112560i
$731$	−639.207			−0.874428
$732$		0			0
$733$		−	609.029i		−	0.830871i	−0.909623	−	0.415436i	$-0.863629\pi$
$733$		−	609.029i		−	0.830871i	0.909623	−	0.415436i	$-0.136371\pi$
$734$		−	232.339i		−	0.316538i
$735$		0			0
$736$	−127.579	−	25.5288i	−0.173340	−	0.0346858i
$737$	−1466.76			−1.99017
$738$		0			0
$739$	780.692			1.05642			0.528209	−	0.849115i	$-0.322864\pi$
$739$	780.692			1.05642			0.528209	+	0.849115i	$0.322864\pi$
$740$	254.734			0.344235
$741$		0			0
$742$	−834.276			−1.12436
$743$			46.4819i			0.0625598i	0.999511	+	0.0312799i	$0.00995832\pi$
$743$			46.4819i			0.0625598i	−0.999511	+	0.0312799i	$0.990042\pi$
$744$		0			0
$745$	−407.593			−0.547104
$746$		−	35.5648i		−	0.0476740i
$747$		0			0
$748$	226.004			0.302144
$749$	−859.490			−1.14752
$750$		0			0
$751$			1258.62i			1.67592i	0.545728	+	0.837962i	$0.316253\pi$
$751$			1258.62i			1.67592i	−0.545728	+	0.837962i	$0.683747\pi$
$752$	265.045			0.352454
$753$		0			0
$754$	−392.161			−0.520108
$755$			91.5586i			0.121270i
$756$		0			0
$757$		−	615.435i		−	0.812992i	−0.913653	−	0.406496i	$-0.866751\pi$
$757$		−	615.435i		−	0.812992i	0.913653	−	0.406496i	$-0.133249\pi$
$758$		−	645.928i		−	0.852148i
$759$		0			0
$760$	91.4211			0.120291
$761$	795.461			1.04528			0.522642	−	0.852552i	$-0.324947\pi$
$761$	795.461			1.04528			0.522642	+	0.852552i	$0.324947\pi$
$762$		0			0
$763$	−743.578			−0.974545
$764$		−	703.316i		−	0.920571i
$765$		0			0
$766$		−	524.697i		−	0.684983i
$767$	1078.80			1.40652
$768$		0			0
$769$		−	1031.00i		−	1.34070i	−0.742046	−	0.670349i	$-0.766145\pi$
$769$		−	1031.00i		−	1.34070i	0.742046	−	0.670349i	$-0.233855\pi$
$770$			298.157i			0.387217i
$771$		0			0
$772$	−78.9616			−0.102282
$773$			1281.23i			1.65747i	0.559638	+	0.828737i	$0.310940\pi$
$773$			1281.23i			1.65747i	−0.559638	+	0.828737i	$0.689060\pi$
$774$		0			0
$775$	−184.214			−0.237696
$776$			210.564i			0.271346i
$777$		0			0
$778$			565.704i			0.727126i
$779$			1022.95i			1.31315i
$780$		0			0
$781$		−	854.301i		−	1.09386i
$782$	−58.2720	+	291.211i	−0.0745166	+	0.372392i
$783$		0			0
$784$	−36.1449			−0.0461032
$785$	564.375			0.718949
$786$		0			0
$787$		−	1496.92i		−	1.90206i	−0.309096	−	0.951031i	$-0.600027\pi$
$787$		−	1496.92i		−	1.90206i	0.309096	−	0.951031i	$-0.399973\pi$
$788$	450.033			0.571108
$789$		0			0
$790$	−91.2453			−0.115500
$791$	434.707			0.549567
$792$		0			0
$793$			312.648i			0.394260i
$794$	373.730			0.470692
$795$		0			0
$796$			74.1745i			0.0931840i
$797$		−	87.1739i		−	0.109378i	−0.998503	−	0.0546888i	$-0.982583\pi$
$797$		−	87.1739i		−	0.109378i	0.998503	−	0.0546888i	$-0.0174167\pi$
$798$		0			0
$799$		−	604.993i		−	0.757188i
$800$	−28.2843			−0.0353553
$801$		0			0
$802$		−	601.492i		−	0.749990i
$803$			321.590i			0.400486i
$804$		0			0
$805$	−384.182	−	76.8758i	−0.477245	−	0.0954979i
$806$	−678.921			−0.842334
$807$		0			0
$808$	48.2425			0.0597061
$809$	−602.054			−0.744196			−0.372098	−	0.928194i	$-0.621361\pi$
$809$	−602.054			−0.744196			−0.372098	+	0.928194i	$0.621361\pi$
$810$		0			0
$811$	1198.57			1.47790			0.738948	−	0.673763i	$-0.235323\pi$
$811$	1198.57			1.47790			0.738948	+	0.673763i	$0.235323\pi$
$812$		−	324.249i		−	0.399321i
$813$		0			0
$814$	996.973			1.22478
$815$		−	105.025i		−	0.128865i
$816$		0			0
$817$	−1011.97			−1.23864
$818$	−341.615			−0.417622
$819$		0			0
$820$		−	316.484i		−	0.385956i
$821$	816.712			0.994777			0.497388	−	0.867528i	$-0.334292\pi$
$821$	816.712			0.994777			0.497388	+	0.867528i	$0.334292\pi$
$822$		0			0
$823$	143.774			0.174694			0.0873472	−	0.996178i	$-0.472161\pi$
$823$	143.774			0.174694			0.0873472	+	0.996178i	$0.472161\pi$
$824$		−	435.442i		−	0.528449i
$825$		0			0
$826$			891.979i			1.07988i
$827$		−	523.617i		−	0.633153i	−0.948567	−	0.316576i	$-0.897467\pi$
$827$		−	523.617i		−	0.633153i	0.948567	−	0.316576i	$-0.102533\pi$
$828$		0			0
$829$	735.461			0.887167			0.443583	−	0.896233i	$-0.353707\pi$
$829$	735.461			0.887167			0.443583	+	0.896233i	$0.353707\pi$
$830$	219.421			0.264363
$831$		0			0
$832$	−104.242			−0.125290
$833$			82.5044i			0.0990449i
$834$		0			0
$835$		−	175.426i		−	0.210091i
$836$	357.802			0.427993
$837$		0			0
$838$			506.786i			0.604757i
$839$		−	752.262i		−	0.896617i	−0.893879	−	0.448309i	$-0.852027\pi$
$839$		−	752.262i		−	0.896617i	0.893879	−	0.448309i	$-0.147973\pi$
$840$		0			0
$841$	−388.105			−0.461481
$842$			950.423i			1.12877i
$843$		0			0
$844$	434.128			0.514369
$845$			1.75806i			0.00208055i
$846$		0			0
$847$			245.125i			0.289404i
$848$		−	309.745i		−	0.365266i
$849$		0			0
$850$			64.5617i			0.0759549i
$851$	−257.056	+	1284.62i	−0.302063	+	1.50954i
$852$		0			0
$853$	−463.137			−0.542951			−0.271475	−	0.962445i	$-0.587512\pi$
$853$	−463.137			−0.542951			−0.271475	+	0.962445i	$0.587512\pi$
$854$	−258.505			−0.302699
$855$		0			0
$856$		−	319.107i		−	0.372789i
$857$	284.684			0.332186			0.166093	−	0.986110i	$-0.446885\pi$
$857$	284.684			0.332186			0.166093	+	0.986110i	$0.446885\pi$
$858$		0			0
$859$	625.746			0.728459			0.364230	−	0.931309i	$-0.381332\pi$
$859$	625.746			0.728459			0.364230	+	0.931309i	$0.381332\pi$
$860$	313.088			0.364056
$861$		0			0
$862$			1159.55i			1.34519i
$863$	−110.774			−0.128359			−0.0641793	−	0.997938i	$-0.520443\pi$
$863$	−110.774			−0.128359			−0.0641793	+	0.997938i	$0.520443\pi$
$864$		0			0
$865$		−	306.520i		−	0.354358i
$866$			447.219i			0.516419i
$867$		0			0
$868$		−	561.349i		−	0.646715i
$869$	−357.114			−0.410948
$870$		0			0
$871$		−	1544.24i		−	1.77295i
$872$		−	276.072i		−	0.316596i
$873$		0			0
$874$	−92.2543	+	461.035i	−0.105554	+	0.527500i
$875$	−85.1735			−0.0973412
$876$		0			0
$877$	−336.486			−0.383678			−0.191839	−	0.981426i	$-0.561445\pi$
$877$	−336.486			−0.383678			−0.191839	+	0.981426i	$0.561445\pi$
$878$	295.399			0.336445
$879$		0			0
$880$	−110.698			−0.125794
$881$			508.337i			0.577000i	0.957480	+	0.288500i	$0.0931565\pi$
$881$			508.337i			0.577000i	−0.957480	+	0.288500i	$0.906843\pi$
$882$		0			0
$883$	−334.663			−0.379006			−0.189503	−	0.981880i	$-0.560688\pi$
$883$	−334.663			−0.379006			−0.189503	+	0.981880i	$0.560688\pi$
$884$			237.942i			0.269165i
$885$		0			0
$886$	999.822			1.12847
$887$	278.810			0.314330			0.157165	−	0.987572i	$-0.449765\pi$
$887$	278.810			0.314330			0.157165	+	0.987572i	$0.449765\pi$
$888$		0			0
$889$			1153.21i			1.29719i
$890$	143.703			0.161464
$891$		0			0
$892$	538.795			0.604030
$893$		−	957.804i		−	1.07257i
$894$		0			0
$895$			124.482i			0.139086i
$896$		−	86.1895i		−	0.0961937i
$897$		0			0
$898$	741.828			0.826090
$899$	784.065			0.872152
$900$		0			0
$901$	−707.025			−0.784712
$902$		−	1238.65i		−	1.37322i
$903$		0			0
$904$			161.396i			0.178535i
$905$	−53.0657			−0.0586362
$906$		0			0
$907$			547.165i			0.603269i	0.953424	+	0.301634i	$0.0975322\pi$
$907$			547.165i			0.603269i	−0.953424	+	0.301634i	$0.902468\pi$
$908$		−	351.995i		−	0.387659i
$909$		0			0
$910$	−313.907			−0.344953
$911$		−	1702.27i		−	1.86857i	−0.356527	−	0.934285i	$-0.616039\pi$
$911$		−	1702.27i		−	1.86857i	0.356527	−	0.934285i	$-0.383961\pi$
$912$		0			0
$913$	858.766			0.940598
$914$			1112.88i			1.21759i
$915$		0			0
$916$			126.134i			0.137701i
$917$			356.022i			0.388247i
$918$		0			0
$919$			230.872i			0.251220i	0.992080	+	0.125610i	$0.0400889\pi$
$919$			230.872i			0.251220i	−0.992080	+	0.125610i	$0.959911\pi$
$920$	28.5420	−	142.637i	0.0310240	−	0.155040i
$921$		0			0
$922$	54.0879			0.0586636
$923$	899.427			0.974461
$924$		0			0
$925$			284.801i			0.307893i
$926$	−743.799			−0.803239
$927$		0			0
$928$	120.385			0.129726
$929$	−1312.19			−1.41247			−0.706237	−	0.707975i	$-0.749609\pi$
$929$	−1312.19			−1.41247			−0.706237	+	0.707975i	$0.749609\pi$
$930$		0			0
$931$			130.618i			0.140299i
$932$	−8.96082			−0.00961461
$933$		0			0
$934$		−	235.652i		−	0.252304i
$935$			252.680i			0.270246i
$936$		0			0
$937$			933.379i			0.996136i	0.867138	+	0.498068i	$0.165957\pi$
$937$			933.379i			0.996136i	−0.867138	+	0.498068i	$0.834043\pi$
$938$	1276.81			1.36121
$939$		0			0
$940$			296.330i			0.315244i
$941$			58.6896i			0.0623694i	0.999514	+	0.0311847i	$0.00992801\pi$
$941$			58.6896i			0.0623694i	−0.999514	+	0.0311847i	$0.990072\pi$
$942$		0			0
$943$	1596.02	+	319.368i	1.69250	+	0.338673i
$944$	−331.169			−0.350815
$945$		0			0
$946$	1225.36			1.29530
$947$	−1070.42			−1.13033			−0.565165	−	0.824978i	$-0.691188\pi$
$947$	−1070.42			−1.13033			−0.565165	+	0.824978i	$0.691188\pi$
$948$		0			0
$949$	−338.577			−0.356772
$950$			102.212i			0.107591i
$951$		0			0
$952$	−196.736			−0.206656
$953$		−	1370.22i		−	1.43780i	−0.695114	−	0.718899i	$-0.744646\pi$
$953$		−	1370.22i		−	1.43780i	0.695114	−	0.718899i	$-0.255354\pi$
$954$		0			0
$955$	786.331			0.823383
$956$	−171.041			−0.178913
$957$		0			0
$958$		−	45.3472i		−	0.0473353i
$959$	−237.647			−0.247807
$960$		0			0
$961$	396.395			0.412482
$962$			1049.64i			1.09110i
$963$		0			0
$964$		−	515.030i		−	0.534263i
$965$		−	88.2817i		−	0.0914837i
$966$		0			0
$967$	145.068			0.150019			0.0750094	−	0.997183i	$-0.476101\pi$
$967$	145.068			0.150019			0.0750094	+	0.997183i	$0.476101\pi$
$968$	−91.0088			−0.0940174
$969$		0			0
$970$	−235.418			−0.242699
$971$		−	996.133i		−	1.02588i	−0.858423	−	0.512942i	$-0.828556\pi$
$971$		−	996.133i		−	1.02588i	0.858423	−	0.512942i	$-0.171444\pi$
$972$		0			0
$973$		−	278.456i		−	0.286183i
$974$	969.880			0.995770
$975$		0			0
$976$		−	95.9764i		−	0.0983365i
$977$		−	161.683i		−	0.165489i	−0.996571	−	0.0827444i	$-0.973631\pi$
$977$		−	161.683i		−	0.165489i	0.996571	−	0.0827444i	$-0.0263685\pi$
$978$		0			0
$979$	562.421			0.574485
$980$		−	40.4112i		−	0.0412360i
$981$		0			0
$982$	−1097.15			−1.11727
$983$		−	1023.25i		−	1.04095i	−0.853878	−	0.520474i	$-0.825755\pi$
$983$		−	1023.25i		−	1.04095i	0.853878	−	0.520474i	$-0.174245\pi$
$984$		0			0
$985$			503.153i			0.510815i
$986$		−	274.792i		−	0.278693i
$987$		0			0
$988$			376.702i			0.381277i
$989$	−315.941	+	1578.90i	−0.319455	+	1.59646i
$990$		0			0
$991$	−518.210			−0.522917			−0.261458	−	0.965215i	$-0.584203\pi$
$991$	−518.210			−0.522917			−0.261458	+	0.965215i	$0.584203\pi$
$992$	208.415			0.210095
$993$		0			0
$994$			743.669i			0.748158i
$995$	−82.9296			−0.0833463
$996$		0			0
$997$	−1927.76			−1.93356			−0.966780	−	0.255611i	$-0.917723\pi$
$997$	−1927.76			−1.93356			−0.966780	+	0.255611i	$0.917723\pi$
$998$	555.032			0.556144
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.a.91.13		16
3.2	odd	2		230.3.d.a.91.5	✓	16
12.11	even	2		1840.3.k.d.321.5		16
15.2	even	4		1150.3.c.c.1149.11		32
15.8	even	4		1150.3.c.c.1149.22		32
15.14	odd	2		1150.3.d.b.551.12		16
23.22	odd	2	inner	2070.3.c.a.91.12		16
69.68	even	2		230.3.d.a.91.6	yes	16
276.275	odd	2		1840.3.k.d.321.6		16
345.68	odd	4		1150.3.c.c.1149.12		32
345.137	odd	4		1150.3.c.c.1149.21		32
345.344	even	2		1150.3.d.b.551.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.5	✓	16	3.2	odd	2
230.3.d.a.91.6	yes	16	69.68	even	2
1150.3.c.c.1149.11		32	15.2	even	4
1150.3.c.c.1149.12		32	345.68	odd	4
1150.3.c.c.1149.21		32	345.137	odd	4
1150.3.c.c.1149.22		32	15.8	even	4
1150.3.d.b.551.11		16	345.344	even	2
1150.3.d.b.551.12		16	15.14	odd	2
1840.3.k.d.321.5		16	12.11	even	2
1840.3.k.d.321.6		16	276.275	odd	2
2070.3.c.a.91.12		16	23.22	odd	2	inner
2070.3.c.a.91.13		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.61815i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -7.61815i q^{7} +2.82843 q^{8} +3.16228i q^{10} +12.3764i q^{11} -13.0302 q^{13} -10.7737i q^{14} +4.00000 q^{16} -9.13040i q^{17} -14.4549i q^{19} +4.47214i q^{20} +17.5029i q^{22} +(-22.5529 - 4.51289i) q^{23} -5.00000 q^{25} -18.4275 q^{26} -15.2363i q^{28} +21.2813 q^{29} +36.8428 q^{31} +5.65685 q^{32} -12.9123i q^{34} +17.0347 q^{35} -56.9603i q^{37} -20.4424i q^{38} +6.32456i q^{40} -70.7680 q^{41} -70.0086i q^{43} +24.7529i q^{44} +(-31.8946 - 6.38219i) q^{46} +66.2614 q^{47} -9.03623 q^{49} -7.07107 q^{50} -26.0604 q^{52} -77.4364i q^{53} -27.6746 q^{55} -21.5474i q^{56} +30.0963 q^{58} -82.7923 q^{59} -23.9941i q^{61} +52.1036 q^{62} +8.00000 q^{64} -29.1364i q^{65} +118.512i q^{67} -18.2608i q^{68} +24.0907 q^{70} -69.0263 q^{71} +25.9840 q^{73} -80.5540i q^{74} -28.9099i q^{76} +94.2857 q^{77} +28.8543i q^{79} +8.94427i q^{80} -100.081 q^{82} -69.3871i q^{83} +20.4162 q^{85} -99.0071i q^{86} +35.0059i q^{88} -45.4428i q^{89} +99.2661i q^{91} +(-45.1058 - 9.02579i) q^{92} +93.7077 q^{94} +32.3222 q^{95} +74.4458i q^{97} -12.7792 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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