LMFDB - Embedded newform 1840.3.k.d.321.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not 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:	$50.1363686423$
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^{9} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.6
Root			$-1.01877i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.d.321.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34854 q^{3} +2.23607i q^{5} -7.61815i q^{7} -3.48436 q^{9} +O(q^{10})$	$q-2.34854 q^{3} +2.23607i q^{5} -7.61815i q^{7} -3.48436 q^{9} -12.3764i q^{11} -13.0302 q^{13} -5.25149i q^{15} -9.13040i q^{17} -14.4549i q^{19} +17.8915i q^{21} +(-22.5529 + 4.51289i) q^{23} -5.00000 q^{25} +29.3200 q^{27} -21.2813 q^{29} -36.8428 q^{31} +29.0666i q^{33} +17.0347 q^{35} +56.9603i q^{37} +30.6019 q^{39} +70.7680 q^{41} -70.0086i q^{43} -7.79128i q^{45} +66.2614 q^{47} -9.03623 q^{49} +21.4431i q^{51} -77.4364i q^{53} +27.6746 q^{55} +33.9480i q^{57} -82.7923 q^{59} +23.9941i q^{61} +26.5444i q^{63} -29.1364i q^{65} +118.512i q^{67} +(52.9664 - 10.5987i) q^{69} -69.0263 q^{71} +25.9840 q^{73} +11.7427 q^{75} -94.2857 q^{77} +28.8543i q^{79} -37.4999 q^{81} +69.3871i q^{83} +20.4162 q^{85} +49.9800 q^{87} -45.4428i q^{89} +99.2661i q^{91} +86.5269 q^{93} +32.3222 q^{95} -74.4458i q^{97} +43.1241i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−2.34854			−0.782846			−0.391423	−	0.920211i	$-0.628017\pi$
$3$	−2.34854			−0.782846			−0.391423	+	0.920211i	$0.628017\pi$
$4$		0			0
$5$			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	$-0.183149\pi$
$8$		0			0
$9$	−3.48436			−0.387152
$10$		0			0
$11$		−	12.3764i		−	1.12513i	−0.826752	−	0.562566i	$-0.809814\pi$
$11$		−	12.3764i		−	1.12513i	0.826752	−	0.562566i	$-0.190186\pi$
$12$		0			0
$13$	−13.0302			−1.00232			−0.501162	−	0.865354i	$-0.667094\pi$
$13$	−13.0302			−1.00232			−0.501162	+	0.865354i	$0.667094\pi$
$14$		0			0
$15$		−	5.25149i		−	0.350100i
$16$		0			0
$17$		−	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	$-0.124211\pi$
$20$		0			0
$21$			17.8915i			0.851977i
$22$		0			0
$23$	−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
$26$		0			0
$27$	29.3200			1.08593
$28$		0			0
$29$	−21.2813			−0.733839			−0.366919	−	0.930253i	$-0.619588\pi$
$29$	−21.2813			−0.733839			−0.366919	+	0.930253i	$0.619588\pi$
$30$		0			0
$31$	−36.8428			−1.18848			−0.594239	−	0.804288i	$-0.702547\pi$
$31$	−36.8428			−1.18848			−0.594239	+	0.804288i	$0.702547\pi$
$32$		0			0
$33$			29.0666i			0.880805i
$34$		0			0
$35$	17.0347			0.486706
$36$		0			0
$37$			56.9603i			1.53947i	0.638365	+	0.769734i	$0.279611\pi$
$37$			56.9603i			1.53947i	−0.638365	+	0.769734i	$0.720389\pi$
$38$		0			0
$39$	30.6019			0.784665
$40$		0			0
$41$	70.7680			1.72605			0.863024	−	0.505163i	$-0.168567\pi$
$41$	70.7680			1.72605			0.863024	+	0.505163i	$0.168567\pi$
$42$		0			0
$43$		−	70.0086i		−	1.62811i	−0.580790	−	0.814053i	$-0.697256\pi$
$43$		−	70.0086i		−	1.62811i	0.580790	−	0.814053i	$-0.302744\pi$
$44$		0			0
$45$		−	7.79128i		−	0.173139i
$46$		0			0
$47$	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
$50$		0			0
$51$			21.4431i			0.420453i
$52$		0			0
$53$		−	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
$56$		0			0
$57$			33.9480i			0.595579i
$58$		0			0
$59$	−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	$0.0630137\pi$
$61$			23.9941i			0.393346i	−0.980469	+	0.196673i	$0.936986\pi$
$62$		0			0
$63$			26.5444i			0.421340i
$64$		0			0
$65$		−	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	$0.654555\pi$
$68$		0			0
$69$	52.9664	−	10.5987i	0.767629	−	0.153604i
$70$		0			0
$71$	−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	$0.443046\pi$
$74$		0			0
$75$	11.7427			0.156569
$76$		0			0
$77$	−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	$0.941541\pi$
$80$		0			0
$81$	−37.4999			−0.462962
$82$		0			0
$83$			69.3871i			0.835989i	0.908449	+	0.417995i	$0.137267\pi$
$83$			69.3871i			0.835989i	−0.908449	+	0.417995i	$0.862733\pi$
$84$		0			0
$85$	20.4162			0.240191
$86$		0			0
$87$	49.9800			0.574483
$88$		0			0
$89$		−	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
$92$		0			0
$93$	86.5269			0.930396
$94$		0			0
$95$	32.3222			0.340234
$96$		0			0
$97$		−	74.4458i		−	0.767482i	−0.923441	−	0.383741i	$-0.874636\pi$
$97$		−	74.4458i		−	0.767482i	0.923441	−	0.383741i	$-0.125364\pi$
$98$		0			0
$99$			43.1241i			0.435597i
$100$		0			0
$101$	−17.0563			−0.168874			−0.0844372	−	0.996429i	$-0.526909\pi$
$101$	−17.0563			−0.168874			−0.0844372	+	0.996429i	$0.526909\pi$
$102$		0			0
$103$		−	153.952i		−	1.49468i	−0.664442	−	0.747340i	$-0.731331\pi$
$103$		−	153.952i		−	1.49468i	0.664442	−	0.747340i	$-0.268669\pi$
$104$		0			0
$105$	−40.0067			−0.381016
$106$		0			0
$107$			112.821i			1.05441i	0.849740	+	0.527203i	$0.176759\pi$
$107$			112.821i			1.05441i	−0.849740	+	0.527203i	$0.823241\pi$
$108$		0			0
$109$			97.6061i			0.895468i	0.894167	+	0.447734i	$0.147769\pi$
$109$			97.6061i			0.895468i	−0.894167	+	0.447734i	$0.852231\pi$
$110$		0			0
$111$		−	133.773i		−	1.20517i
$112$		0			0
$113$			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
$116$		0			0
$117$	45.4020			0.388051
$118$		0			0
$119$	−69.5568			−0.584511
$120$		0			0
$121$	−32.1765			−0.265921
$122$		0			0
$123$	−166.201			−1.35123
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	151.376			1.19194			0.595969	−	0.803007i	$-0.296768\pi$
$127$	151.376			1.19194			0.595969	+	0.803007i	$0.296768\pi$
$128$		0			0
$129$			164.418i			1.27456i
$130$		0			0
$131$	−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
$134$		0			0
$135$			65.5616i			0.485641i
$136$		0			0
$137$		−	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	$-0.541973\pi$
$139$	−36.5517			−0.262962			−0.131481	+	0.991319i	$0.541973\pi$
$140$		0			0
$141$	−155.617			−1.10367
$142$		0			0
$143$			161.268i			1.12775i
$144$		0			0
$145$		−	47.5865i		−	0.328183i
$146$		0			0
$147$	21.2219			0.144367
$148$		0			0
$149$			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	$-0.543291\pi$
$151$	−40.9462			−0.271167			−0.135584	+	0.990766i	$0.543291\pi$
$152$		0			0
$153$			31.8136i			0.207932i
$154$		0			0
$155$		−	82.3831i		−	0.531504i
$156$		0			0
$157$			252.396i			1.60762i	0.594887	+	0.803810i	$0.297197\pi$
$157$			252.396i			1.60762i	−0.594887	+	0.803810i	$0.702803\pi$
$158$		0			0
$159$			181.862i			1.14379i
$160$		0			0
$161$	34.3799	+	171.811i	0.213540	+	1.06715i
$162$		0			0
$163$	46.9686			0.288151			0.144076	−	0.989567i	$-0.453979\pi$
$163$	46.9686			0.288151			0.144076	+	0.989567i	$0.453979\pi$
$164$		0			0
$165$	−64.9948			−0.393908
$166$		0			0
$167$	−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
$170$		0			0
$171$			50.3663i			0.294540i
$172$		0			0
$173$	137.080			0.792369			0.396184	−	0.918171i	$-0.370334\pi$
$173$	137.080			0.792369			0.396184	+	0.918171i	$0.370334\pi$
$174$		0			0
$175$			38.0908i			0.217661i
$176$		0			0
$177$	194.441			1.09854
$178$		0			0
$179$	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	$-0.979118\pi$
$181$		−	23.7317i		−	0.131114i	0.997849	−	0.0655572i	$-0.0208825\pi$
$182$		0			0
$183$		−	56.3511i		−	0.307929i
$184$		0			0
$185$	−127.367			−0.688471
$186$		0			0
$187$	−113.002			−0.604289
$188$		0			0
$189$		−	223.364i		−	1.18182i
$190$		0			0
$191$			351.658i			1.84114i	0.390576	+	0.920571i	$0.372276\pi$
$191$			351.658i			1.84114i	−0.390576	+	0.920571i	$0.627724\pi$
$192$		0			0
$193$	−39.4808			−0.204564			−0.102282	−	0.994755i	$-0.532614\pi$
$193$	−39.4808			−0.204564			−0.102282	+	0.994755i	$0.532614\pi$
$194$		0			0
$195$			68.4280i			0.350913i
$196$		0			0
$197$	−225.017			−1.14222			−0.571108	−	0.820875i	$-0.693486\pi$
$197$	−225.017			−1.14222			−0.571108	+	0.820875i	$0.693486\pi$
$198$		0			0
$199$			37.0872i			0.186368i	0.995649	+	0.0931840i	$0.0297045\pi$
$199$			37.0872i			0.186368i	−0.995649	+	0.0931840i	$0.970296\pi$
$200$		0			0
$201$		−	278.330i		−	1.38473i
$202$		0			0
$203$			162.124i			0.798642i
$204$		0			0
$205$			158.242i			0.771912i
$206$		0			0
$207$	78.5826	−	15.7246i	0.379626	−	0.0759641i
$208$		0			0
$209$	−178.901			−0.855985
$210$		0			0
$211$	−217.064			−1.02874			−0.514369	−	0.857569i	$-0.671974\pi$
$211$	−217.064			−1.02874			−0.514369	+	0.857569i	$0.671974\pi$
$212$		0			0
$213$	162.111			0.761085
$214$		0			0
$215$	156.544			0.728111
$216$		0			0
$217$			280.674i			1.29343i
$218$		0			0
$219$	−61.0245			−0.278651
$220$		0			0
$221$			118.971i			0.538330i
$222$		0			0
$223$	−269.398			−1.20806			−0.604030	−	0.796961i	$-0.706439\pi$
$223$	−269.398			−1.20806			−0.604030	+	0.796961i	$0.706439\pi$
$224$		0			0
$225$	17.4218			0.0774303
$226$		0			0
$227$			175.997i			0.775319i	0.921803	+	0.387659i	$0.126716\pi$
$227$			175.997i			0.775319i	−0.921803	+	0.387659i	$0.873284\pi$
$228$		0			0
$229$		−	63.0669i		−	0.275401i	−0.990474	−	0.137701i	$-0.956029\pi$
$229$		−	63.0669i		−	0.275401i	0.990474	−	0.137701i	$-0.0439712\pi$
$230$		0			0
$231$	221.434			0.958587
$232$		0			0
$233$	4.48041			0.0192292			0.00961461	−	0.999954i	$-0.496940\pi$
$233$	4.48041			0.0192292			0.00961461	+	0.999954i	$0.496940\pi$
$234$		0			0
$235$			148.165i			0.630489i
$236$		0			0
$237$		−	67.7655i		−	0.285930i
$238$		0			0
$239$	−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	$0.179411\pi$
$241$			257.515i			1.06853i	−0.845318	+	0.534263i	$0.820589\pi$
$242$		0			0
$243$	−175.810			−0.723498
$244$		0			0
$245$		−	20.2056i		−	0.0824719i
$246$		0			0
$247$			188.351i			0.762554i
$248$		0			0
$249$		−	162.958i		−	0.654451i
$250$		0			0
$251$		−	121.438i		−	0.483816i	−0.970299	−	0.241908i	$-0.922227\pi$
$251$		−	121.438i		−	0.483816i	0.970299	−	0.241908i	$-0.0777733\pi$
$252$		0			0
$253$	55.8536	+	279.125i	0.220765	+	1.10326i
$254$		0			0
$255$	−47.9482			−0.188032
$256$		0			0
$257$	192.295			0.748230			0.374115	−	0.927382i	$-0.377947\pi$
$257$	192.295			0.748230			0.374115	+	0.927382i	$0.377947\pi$
$258$		0			0
$259$	433.932			1.67541
$260$		0			0
$261$	74.1519			0.284107
$262$		0			0
$263$		−	417.798i		−	1.58858i	−0.607536	−	0.794292i	$-0.707842\pi$
$263$		−	417.798i		−	1.58858i	0.607536	−	0.794292i	$-0.292158\pi$
$264$		0			0
$265$	173.153			0.653407
$266$		0			0
$267$			106.724i			0.399716i
$268$		0			0
$269$	−474.878			−1.76534			−0.882672	−	0.469989i	$-0.844258\pi$
$269$	−474.878			−1.76534			−0.882672	+	0.469989i	$0.844258\pi$
$270$		0			0
$271$	32.0762			0.118362			0.0591812	−	0.998247i	$-0.481151\pi$
$271$	32.0762			0.118362			0.0591812	+	0.998247i	$0.481151\pi$
$272$		0			0
$273$		−	233.130i		−	0.853957i
$274$		0			0
$275$			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	$0.189144\pi$
$278$		0			0
$279$	128.374			0.460121
$280$		0			0
$281$			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	$0.726449\pi$
$284$		0			0
$285$	−75.9100			−0.266351
$286$		0			0
$287$		−	539.121i		−	1.87847i
$288$		0			0
$289$	205.636			0.711542
$290$		0			0
$291$			174.839i			0.600821i
$292$		0			0
$293$			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
$296$		0			0
$297$		−	362.878i		−	1.22181i
$298$		0			0
$299$	293.869	−	58.8039i	0.982840	−	0.196669i
$300$		0			0
$301$	−533.336			−1.77188
$302$		0			0
$303$	40.0574			0.132203
$304$		0			0
$305$	−53.6524			−0.175910
$306$		0			0
$307$	448.843			1.46203			0.731014	−	0.682362i	$-0.239047\pi$
$307$	448.843			1.46203			0.731014	+	0.682362i	$0.239047\pi$
$308$		0			0
$309$			361.562i			1.17010i
$310$		0			0
$311$	−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	$-0.952316\pi$
$313$		−	93.4265i		−	0.298487i	0.988800	−	0.149244i	$-0.0476839\pi$
$314$		0			0
$315$	−59.3551			−0.188429
$316$		0			0
$317$	453.069			1.42924			0.714620	−	0.699513i	$-0.246600\pi$
$317$	453.069			1.42924			0.714620	+	0.699513i	$0.246600\pi$
$318$		0			0
$319$			263.387i			0.825665i
$320$		0			0
$321$		−	264.965i		−	0.825437i
$322$		0			0
$323$	−131.979			−0.408605
$324$		0			0
$325$	65.1510			0.200465
$326$		0			0
$327$		−	229.232i		−	0.701014i
$328$		0			0
$329$		−	504.789i		−	1.53431i
$330$		0			0
$331$	178.326			0.538749			0.269374	−	0.963036i	$-0.413183\pi$
$331$	178.326			0.538749			0.269374	+	0.963036i	$0.413183\pi$
$332$		0			0
$333$		−	198.470i		−	0.596007i
$334$		0			0
$335$	−265.001			−0.791048
$336$		0			0
$337$			85.7271i			0.254383i	0.991878	+	0.127192i	$0.0405963\pi$
$337$			85.7271i			0.254383i	−0.991878	+	0.127192i	$0.959404\pi$
$338$		0			0
$339$		−	134.012i		−	0.395317i
$340$		0			0
$341$			455.984i			1.33720i
$342$		0			0
$343$		−	304.450i		−	0.887610i
$344$		0			0
$345$	23.6994	+	118.436i	0.0686940	+	0.343294i
$346$		0			0
$347$	−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	$0.773640\pi$
$350$		0			0
$351$	−382.046			−1.08845
$352$		0			0
$353$	−28.7915			−0.0815623			−0.0407811	−	0.999168i	$-0.512985\pi$
$353$	−28.7915			−0.0815623			−0.0407811	+	0.999168i	$0.512985\pi$
$354$		0			0
$355$		−	154.348i		−	0.434782i
$356$		0			0
$357$	163.357			0.457582
$358$		0			0
$359$			55.5671i			0.154783i	0.997001	+	0.0773914i	$0.0246591\pi$
$359$			55.5671i			0.154783i	−0.997001	+	0.0773914i	$0.975341\pi$
$360$		0			0
$361$	152.055			0.421204
$362$		0			0
$363$	75.5677			0.208176
$364$		0			0
$365$			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	$-0.0718547\pi$
$368$		0			0
$369$	−246.581			−0.668242
$370$		0			0
$371$	−589.922			−1.59009
$372$		0			0
$373$			25.1481i			0.0674212i	0.999432	+	0.0337106i	$0.0107325\pi$
$373$			25.1481i			0.0674212i	−0.999432	+	0.0337106i	$0.989268\pi$
$374$		0			0
$375$			26.2575i			0.0700199i
$376$		0			0
$377$	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	$-0.205852\pi$
$380$		0			0
$381$	−355.513			−0.933104
$382$		0			0
$383$			371.017i			0.968712i	0.874871	+	0.484356i	$0.160946\pi$
$383$			371.017i			0.968712i	−0.874871	+	0.484356i	$0.839054\pi$
$384$		0			0
$385$		−	210.829i		−	0.547608i
$386$		0			0
$387$			243.935i			0.630324i
$388$		0			0
$389$			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
$392$		0			0
$393$	109.755			0.279275
$394$		0			0
$395$	−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	$0.391997\pi$
$398$		0			0
$399$	258.621			0.648173
$400$		0			0
$401$		−	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
$404$		0			0
$405$		−	83.8524i		−	0.207043i
$406$		0			0
$407$	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	$0.595421\pi$
$410$		0			0
$411$			73.2624i			0.178254i
$412$		0			0
$413$			630.725i			1.52718i
$414$		0			0
$415$	−155.154			−0.373866
$416$		0			0
$417$	85.8431			0.205859
$418$		0			0
$419$		−	358.352i		−	0.855256i	−0.903955	−	0.427628i	$-0.859349\pi$
$419$		−	358.352i		−	0.855256i	0.903955	−	0.427628i	$-0.140651\pi$
$420$		0			0
$421$		−	672.051i		−	1.59632i	−0.602446	−	0.798160i	$-0.705807\pi$
$421$		−	672.051i		−	1.59632i	0.602446	−	0.798160i	$-0.294193\pi$
$422$		0			0
$423$	−230.879			−0.545813
$424$		0			0
$425$			45.6520i			0.107417i
$426$		0			0
$427$	182.791			0.428081
$428$		0			0
$429$		−	378.743i		−	0.882852i
$430$		0			0
$431$		−	819.926i		−	1.90238i	−0.308604	−	0.951191i	$-0.599862\pi$
$431$		−	819.926i		−	1.90238i	0.308604	−	0.951191i	$-0.400138\pi$
$432$		0			0
$433$		−	316.231i		−	0.730326i	−0.930944	−	0.365163i	$-0.881013\pi$
$433$		−	316.231i		−	0.730326i	0.930944	−	0.365163i	$-0.118987\pi$
$434$		0			0
$435$			111.759i			0.256917i
$436$		0			0
$437$	65.2336	+	326.001i	0.149276	+	0.745998i
$438$		0			0
$439$	−208.878			−0.475805			−0.237903	−	0.971289i	$-0.576460\pi$
$439$	−208.878			−0.475805			−0.237903	+	0.971289i	$0.576460\pi$
$440$		0			0
$441$	31.4855			0.0713957
$442$		0			0
$443$	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
$446$		0			0
$447$		−	428.094i		−	0.957705i
$448$		0			0
$449$	−524.552			−1.16827			−0.584134	−	0.811658i	$-0.698566\pi$
$449$	−524.552			−1.16827			−0.584134	+	0.811658i	$0.698566\pi$
$450$		0			0
$451$		−	875.856i		−	1.94203i
$452$		0			0
$453$	96.1639			0.212282
$454$		0			0
$455$	−221.966			−0.487837
$456$		0			0
$457$		−	786.926i		−	1.72194i	−0.508657	−	0.860969i	$-0.669858\pi$
$457$		−	786.926i		−	1.72194i	0.508657	−	0.860969i	$-0.330142\pi$
$458$		0			0
$459$		−	267.704i		−	0.583232i
$460$		0			0
$461$	−38.2459			−0.0829629			−0.0414814	−	0.999139i	$-0.513208\pi$
$461$	−38.2459			−0.0829629			−0.0414814	+	0.999139i	$0.513208\pi$
$462$		0			0
$463$	525.945			1.13595			0.567976	−	0.823045i	$-0.307727\pi$
$463$	525.945			1.13595			0.567976	+	0.823045i	$0.307727\pi$
$464$		0			0
$465$			193.480i			0.416086i
$466$		0			0
$467$			166.631i			0.356812i	0.983957	+	0.178406i	$0.0570941\pi$
$467$			166.631i			0.356812i	−0.983957	+	0.178406i	$0.942906\pi$
$468$		0			0
$469$	902.843			1.92504
$470$		0			0
$471$		−	592.762i		−	1.25852i
$472$		0			0
$473$	−866.458			−1.83183
$474$		0			0
$475$			72.2747i			0.152157i
$476$		0			0
$477$			269.817i			0.565653i
$478$		0			0
$479$			32.0653i			0.0669422i	0.999440	+	0.0334711i	$0.0106562\pi$
$479$			32.0653i			0.0669422i	−0.999440	+	0.0334711i	$0.989344\pi$
$480$		0			0
$481$		−	742.204i		−	1.54304i
$482$		0			0
$483$	−80.7425	−	403.506i	−0.167169	−	0.835416i
$484$		0			0
$485$	166.466			0.343228
$486$		0			0
$487$	−685.808			−1.40823			−0.704115	−	0.710085i	$-0.748656\pi$
$487$	−685.808			−1.40823			−0.704115	+	0.710085i	$0.748656\pi$
$488$		0			0
$489$	−110.308			−0.225578
$490$		0			0
$491$	−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
$494$		0			0
$495$	−96.4283			−0.194805
$496$		0			0
$497$			525.853i			1.05805i
$498$		0			0
$499$	−392.467			−0.786507			−0.393253	−	0.919430i	$-0.628650\pi$
$499$	−392.467			−0.786507			−0.393253	+	0.919430i	$0.628650\pi$
$500$		0			0
$501$	184.250			0.367764
$502$		0			0
$503$			633.449i			1.25934i	0.776862	+	0.629671i	$0.216810\pi$
$503$			633.449i			1.25934i	−0.776862	+	0.629671i	$0.783190\pi$
$504$		0			0
$505$		−	38.1391i		−	0.0755229i
$506$		0			0
$507$	−1.84649			−0.00364199
$508$		0			0
$509$	540.009			1.06092			0.530460	−	0.847710i	$-0.322019\pi$
$509$	540.009			1.06092			0.530460	+	0.847710i	$0.322019\pi$
$510$		0			0
$511$		−	197.950i		−	0.387378i
$512$		0			0
$513$		−	423.819i		−	0.826158i
$514$		0			0
$515$	344.247			0.668441
$516$		0			0
$517$		−	820.080i		−	1.58623i
$518$		0			0
$519$	−321.937			−0.620303
$520$		0			0
$521$		−	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	$-0.00836152\pi$
$524$		0			0
$525$		−	89.4576i		−	0.170395i
$526$		0			0
$527$			336.390i			0.638311i
$528$		0			0
$529$	488.268	−	203.558i	0.923001	−	0.384797i
$530$		0			0
$531$	288.479			0.543274
$532$		0			0
$533$	−922.121			−1.73006
$534$		0			0
$535$	−252.276			−0.471544
$536$		0			0
$537$	−130.743			−0.243469
$538$		0			0
$539$			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	$0.766070\pi$
$542$		0			0
$543$			55.7349i			0.102643i
$544$		0			0
$545$	−218.254			−0.400466
$546$		0			0
$547$	439.126			0.802790			0.401395	−	0.915905i	$-0.368525\pi$
$547$	439.126			0.802790			0.401395	+	0.915905i	$0.368525\pi$
$548$		0			0
$549$		−	83.6042i		−	0.152284i
$550$		0			0
$551$			307.620i			0.558295i
$552$		0			0
$553$	219.817			0.397498
$554$		0			0
$555$	299.127			0.538967
$556$		0			0
$557$			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
$560$		0			0
$561$	265.390			0.473065
$562$		0			0
$563$		−	214.573i		−	0.381124i	−0.981675	−	0.190562i	$-0.938969\pi$
$563$		−	214.573i		−	0.381124i	0.981675	−	0.190562i	$-0.0610309\pi$
$564$		0			0
$565$	−127.595			−0.225831
$566$		0			0
$567$			285.680i			0.503845i
$568$		0			0
$569$		−	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	$0.836689\pi$
$572$		0			0
$573$		−	825.882i		−	1.44133i
$574$		0			0
$575$	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	$0.257931\pi$
$578$		0			0
$579$	92.7222			0.160142
$580$		0			0
$581$	528.602			0.909813
$582$		0			0
$583$	−958.387			−1.64389
$584$		0			0
$585$			101.522i			0.173542i
$586$		0			0
$587$	−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
$590$		0			0
$591$	528.460			0.894180
$592$		0			0
$593$	−175.139			−0.295344			−0.147672	−	0.989036i	$-0.547178\pi$
$593$	−175.139			−0.295344			−0.147672	+	0.989036i	$0.547178\pi$
$594$		0			0
$595$		−	155.534i		−	0.261401i
$596$		0			0
$597$		−	87.1009i		−	0.145898i
$598$		0			0
$599$	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	$0.546661\pi$
$602$		0			0
$603$		−	412.939i		−	0.684808i
$604$		0			0
$605$		−	71.9488i		−	0.118924i
$606$		0			0
$607$	173.873			0.286447			0.143223	−	0.989690i	$-0.454253\pi$
$607$	173.873			0.286447			0.143223	+	0.989690i	$0.454253\pi$
$608$		0			0
$609$		−	380.755i		−	0.625214i
$610$		0			0
$611$	−863.399			−1.41309
$612$		0			0
$613$		−	560.182i		−	0.913837i	−0.889509	−	0.456919i	$-0.848953\pi$
$613$		−	560.182i		−	0.913837i	0.889509	−	0.456919i	$-0.151047\pi$
$614$		0			0
$615$		−	371.638i		−	0.604289i
$616$		0			0
$617$		−	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	$0.888699\pi$
$620$		0			0
$621$	−661.252	+	132.318i	−1.06482	+	0.213073i
$622$		0			0
$623$	−346.190			−0.555683
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	420.156			0.670105
$628$		0			0
$629$	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	$-0.0435823\pi$
$632$		0			0
$633$	509.783			0.805344
$634$		0			0
$635$			338.487i			0.533051i
$636$		0			0
$637$	117.744			0.184841
$638$		0			0
$639$	240.513			0.376390
$640$		0			0
$641$		−	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	$0.778845\pi$
$644$		0			0
$645$	−367.650			−0.569999
$646$		0			0
$647$	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
$650$		0			0
$651$		−	659.175i		−	1.01256i
$652$		0			0
$653$	−912.811			−1.39787			−0.698937	−	0.715183i	$-0.746343\pi$
$653$	−912.811			−1.39787			−0.698937	+	0.715183i	$0.746343\pi$
$654$		0			0
$655$		−	104.499i		−	0.159541i
$656$		0			0
$657$	−90.5378			−0.137805
$658$		0			0
$659$		−	247.106i		−	0.374971i	−0.982267	−	0.187486i	$-0.939966\pi$
$659$		−	247.106i		−	0.374971i	0.982267	−	0.187486i	$-0.0600338\pi$
$660$		0			0
$661$		−	937.016i		−	1.41757i	−0.705423	−	0.708787i	$-0.749243\pi$
$661$		−	937.016i		−	1.41757i	0.705423	−	0.708787i	$-0.250757\pi$
$662$		0			0
$663$		−	279.408i		−	0.421430i
$664$		0			0
$665$		−	246.236i		−	0.370279i
$666$		0			0
$667$	479.956	−	96.0404i	0.719574	−	0.143989i
$668$		0			0
$669$	632.691			0.945726
$670$		0			0
$671$	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	$0.162322\pi$
$674$		0			0
$675$	−146.600			−0.217185
$676$		0			0
$677$			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
$680$		0			0
$681$		−	413.337i		−	0.606955i
$682$		0			0
$683$	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
$686$		0			0
$687$			148.115i			0.215597i
$688$		0			0
$689$			1009.01i			1.46446i
$690$		0			0
$691$	998.531			1.44505			0.722526	−	0.691344i	$-0.242981\pi$
$691$	998.531			1.44505			0.722526	+	0.691344i	$0.242981\pi$
$692$		0			0
$693$	328.526			0.474063
$694$		0			0
$695$		−	81.7321i		−	0.117600i
$696$		0			0
$697$		−	646.140i		−	0.927030i
$698$		0			0
$699$	−10.5224			−0.0150535
$700$		0			0
$701$			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
$704$		0			0
$705$		−	347.971i		−	0.493576i
$706$		0			0
$707$			129.938i			0.183787i
$708$		0			0
$709$		−	779.102i		−	1.09887i	−0.835535	−	0.549437i	$-0.814842\pi$
$709$		−	779.102i		−	1.09887i	0.835535	−	0.549437i	$-0.185158\pi$
$710$		0			0
$711$		−	100.539i		−	0.141405i
$712$		0			0
$713$	830.913	−	166.268i	1.16538	−	0.233195i
$714$		0			0
$715$	−360.605			−0.504343
$716$		0			0
$717$	200.848			0.280123
$718$		0			0
$719$	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
$722$		0			0
$723$		−	604.784i		−	0.836492i
$724$		0			0
$725$	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	$-0.00433152\pi$
$728$		0			0
$729$	750.396			1.02935
$730$		0			0
$731$	−639.207			−0.874428
$732$		0			0
$733$			609.029i			0.830871i	0.909623	+	0.415436i	$0.136371\pi$
$733$			609.029i			0.830871i	−0.909623	+	0.415436i	$0.863629\pi$
$734$		0			0
$735$			47.4537i			0.0645628i
$736$		0			0
$737$	1466.76			1.99017
$738$		0			0
$739$	−780.692			−1.05642			−0.528209	−	0.849115i	$-0.677136\pi$
$739$	−780.692			−1.05642			−0.528209	+	0.849115i	$0.677136\pi$
$740$		0			0
$741$		−	442.349i		−	0.596963i
$742$		0			0
$743$		−	46.4819i		−	0.0625598i	−0.999511	−	0.0312799i	$-0.990042\pi$
$743$		−	46.4819i		−	0.0625598i	0.999511	−	0.0312799i	$-0.00995832\pi$
$744$		0			0
$745$	−407.593			−0.547104
$746$		0			0
$747$		−	241.770i		−	0.323655i
$748$		0			0
$749$	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	$0.683747\pi$
$752$		0			0
$753$			285.202i			0.378754i
$754$		0			0
$755$		−	91.5586i		−	0.121270i
$756$		0			0
$757$			615.435i			0.812992i	0.913653	+	0.406496i	$0.133249\pi$
$757$			615.435i			0.812992i	−0.913653	+	0.406496i	$0.866751\pi$
$758$		0			0
$759$	−131.174	−	655.536i	−0.172825	−	0.863684i
$760$		0			0
$761$	−795.461			−1.04528			−0.522642	−	0.852552i	$-0.675053\pi$
$761$	−795.461			−1.04528			−0.522642	+	0.852552i	$0.675053\pi$
$762$		0			0
$763$	743.578			0.974545
$764$		0			0
$765$	−71.1375			−0.0929902
$766$		0			0
$767$	1078.80			1.40652
$768$		0			0
$769$			1031.00i			1.34070i	0.742046	+	0.670349i	$0.233855\pi$
$769$			1031.00i			1.34070i	−0.742046	+	0.670349i	$0.766145\pi$
$770$		0			0
$771$	−451.613			−0.585749
$772$		0			0
$773$			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
$776$		0			0
$777$	−1019.11			−1.31159
$778$		0			0
$779$		−	1022.95i		−	1.31315i
$780$		0			0
$781$			854.301i			1.09386i
$782$		0			0
$783$	−623.969			−0.796895
$784$		0			0
$785$	−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	$-0.399973\pi$
$788$		0			0
$789$			981.215i			1.24362i
$790$		0			0
$791$	434.707			0.549567
$792$		0			0
$793$		−	312.648i		−	0.394260i
$794$		0			0
$795$	−406.657			−0.511518
$796$		0			0
$797$		−	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
$800$		0			0
$801$			158.339i			0.197677i
$802$		0			0
$803$		−	321.590i		−	0.400486i
$804$		0			0
$805$	−384.182	+	76.8758i	−0.477245	+	0.0954979i
$806$		0			0
$807$	1115.27			1.38199
$808$		0			0
$809$	602.054			0.744196			0.372098	−	0.928194i	$-0.378639\pi$
$809$	602.054			0.744196			0.372098	+	0.928194i	$0.378639\pi$
$810$		0			0
$811$	−1198.57			−1.47790			−0.738948	−	0.673763i	$-0.764677\pi$
$811$	−1198.57			−1.47790			−0.738948	+	0.673763i	$0.764677\pi$
$812$		0			0
$813$	−75.3322			−0.0926595
$814$		0			0
$815$			105.025i			0.128865i
$816$		0			0
$817$	−1011.97			−1.23864
$818$		0			0
$819$		−	345.879i		−	0.422319i
$820$		0			0
$821$	−816.712			−0.994777			−0.497388	−	0.867528i	$-0.665708\pi$
$821$	−816.712			−0.994777			−0.497388	+	0.867528i	$0.665708\pi$
$822$		0			0
$823$	−143.774			−0.174694			−0.0873472	−	0.996178i	$-0.527839\pi$
$823$	−143.774			−0.174694			−0.0873472	+	0.996178i	$0.527839\pi$
$824$		0			0
$825$		−	145.333i		−	0.176161i
$826$		0			0
$827$			523.617i			0.633153i	0.948567	+	0.316576i	$0.102533\pi$
$827$			523.617i			0.633153i	−0.948567	+	0.316576i	$0.897467\pi$
$828$		0			0
$829$	735.461			0.887167			0.443583	−	0.896233i	$-0.353707\pi$
$829$	735.461			0.887167			0.443583	+	0.896233i	$0.353707\pi$
$830$		0			0
$831$	−1078.07			−1.29732
$832$		0			0
$833$			82.5044i			0.0990449i
$834$		0			0
$835$		−	175.426i		−	0.210091i
$836$		0			0
$837$	−1080.23			−1.29060
$838$		0			0
$839$			752.262i			0.896617i	0.893879	+	0.448309i	$0.147973\pi$
$839$			752.262i			0.896617i	−0.893879	+	0.448309i	$0.852027\pi$
$840$		0			0
$841$	−388.105			−0.461481
$842$		0			0
$843$		−	755.156i		−	0.895796i
$844$		0			0
$845$			1.75806i			0.00208055i
$846$		0			0
$847$			245.125i			0.289404i
$848$		0			0
$849$		−	1006.85i		−	1.18592i
$850$		0			0
$851$	−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	$0.587512\pi$
$854$		0			0
$855$	−112.622			−0.131722
$856$		0			0
$857$	−284.684			−0.332186			−0.166093	−	0.986110i	$-0.553115\pi$
$857$	−284.684			−0.332186			−0.166093	+	0.986110i	$0.553115\pi$
$858$		0			0
$859$	−625.746			−0.728459			−0.364230	−	0.931309i	$-0.618668\pi$
$859$	−625.746			−0.728459			−0.364230	+	0.931309i	$0.618668\pi$
$860$		0			0
$861$			1266.15i			1.47055i
$862$		0			0
$863$	−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
$866$		0			0
$867$	−482.944			−0.557028
$868$		0			0
$869$	357.114			0.410948
$870$		0			0
$871$		−	1544.24i		−	1.77295i
$872$		0			0
$873$			259.396i			0.297132i
$874$		0			0
$875$	−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	$0.561445\pi$
$878$		0			0
$879$		−	845.099i		−	0.961433i
$880$		0			0
$881$			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	$-0.439312\pi$
$883$	334.663			0.379006			0.189503	+	0.981880i	$0.439312\pi$
$884$		0			0
$885$			434.783i			0.491281i
$886$		0			0
$887$	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
$890$		0			0
$891$			464.116i			0.520893i
$892$		0			0
$893$		−	957.804i		−	1.07257i
$894$		0			0
$895$			124.482i			0.139086i
$896$		0			0
$897$	−690.163	+	138.103i	−0.769412	+	0.153961i
$898$		0			0
$899$	784.065			0.872152
$900$		0			0
$901$	−707.025			−0.784712
$902$		0			0
$903$	1252.56			1.38711
$904$		0			0
$905$	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	$0.902468\pi$
$908$		0			0
$909$	59.4304			0.0653800
$910$		0			0
$911$			1702.27i			1.86857i	0.356527	+	0.934285i	$0.383961\pi$
$911$			1702.27i			1.86857i	−0.356527	+	0.934285i	$0.616039\pi$
$912$		0			0
$913$	858.766			0.940598
$914$		0			0
$915$	126.005			0.137710
$916$		0			0
$917$			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	$0.959911\pi$
$920$		0			0
$921$	−1054.12			−1.14454
$922$		0			0
$923$	899.427			0.974461
$924$		0			0
$925$		−	284.801i		−	0.307893i
$926$		0			0
$927$			536.425i			0.578668i
$928$		0			0
$929$	1312.19			1.41247			0.706237	−	0.707975i	$-0.250391\pi$
$929$	1312.19			1.41247			0.706237	+	0.707975i	$0.250391\pi$
$930$		0			0
$931$			130.618i			0.140299i
$932$		0			0
$933$	440.497			0.472130
$934$		0			0
$935$		−	252.680i		−	0.270246i
$936$		0			0
$937$		−	933.379i		−	0.996136i	−0.867138	−	0.498068i	$-0.834043\pi$
$937$		−	933.379i		−	0.996136i	0.867138	−	0.498068i	$-0.165957\pi$
$938$		0			0
$939$			219.416i			0.233670i
$940$		0			0
$941$			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
$944$		0			0
$945$	499.458			0.528527
$946$		0			0
$947$	−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
$950$		0			0
$951$	−1064.05			−1.11888
$952$		0			0
$953$		−	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
$956$		0			0
$957$		−	618.575i		−	0.646369i
$958$		0			0
$959$	−237.647			−0.247807
$960$		0			0
$961$	396.395			0.412482
$962$		0			0
$963$		−	393.111i		−	0.408215i
$964$		0			0
$965$		−	88.2817i		−	0.0914837i
$966$		0			0
$967$	−145.068			−0.150019			−0.0750094	−	0.997183i	$-0.523899\pi$
$967$	−145.068			−0.150019			−0.0750094	+	0.997183i	$0.523899\pi$
$968$		0			0
$969$	309.959			0.319875
$970$		0			0
$971$			996.133i			1.02588i	0.858423	+	0.512942i	$0.171444\pi$
$971$			996.133i			1.02588i	−0.858423	+	0.512942i	$0.828556\pi$
$972$		0			0
$973$			278.456i			0.286183i
$974$		0			0
$975$	−153.010			−0.156933
$976$		0			0
$977$		−	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
$980$		0			0
$981$		−	340.095i		−	0.346682i
$982$		0			0
$983$			1023.25i			1.04095i	0.853878	+	0.520474i	$0.174245\pi$
$983$			1023.25i			1.04095i	−0.853878	+	0.520474i	$0.825755\pi$
$984$		0			0
$985$		−	503.153i		−	0.510815i
$986$		0			0
$987$			1185.52i			1.20113i
$988$		0			0
$989$	315.941	+	1578.90i	0.319455	+	1.59646i
$990$		0			0
$991$	518.210			0.522917			0.261458	−	0.965215i	$-0.415797\pi$
$991$	518.210			0.522917			0.261458	+	0.965215i	$0.415797\pi$
$992$		0			0
$993$	−418.805			−0.421757
$994$		0			0
$995$	−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	$0.917723\pi$
$998$		0			0
$999$			1670.08i			1.67175i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.d.321.6		16
4.3	odd	2		230.3.d.a.91.6	yes	16
12.11	even	2		2070.3.c.a.91.12		16
20.3	even	4		1150.3.c.c.1149.12		32
20.7	even	4		1150.3.c.c.1149.21		32
20.19	odd	2		1150.3.d.b.551.11		16
23.22	odd	2	inner	1840.3.k.d.321.5		16
92.91	even	2		230.3.d.a.91.5	✓	16
276.275	odd	2		2070.3.c.a.91.13		16
460.183	odd	4		1150.3.c.c.1149.22		32
460.367	odd	4		1150.3.c.c.1149.11		32
460.459	even	2		1150.3.d.b.551.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.5	✓	16	92.91	even	2
230.3.d.a.91.6	yes	16	4.3	odd	2
1150.3.c.c.1149.11		32	460.367	odd	4
1150.3.c.c.1149.12		32	20.3	even	4
1150.3.c.c.1149.21		32	20.7	even	4
1150.3.c.c.1149.22		32	460.183	odd	4
1150.3.d.b.551.11		16	20.19	odd	2
1150.3.d.b.551.12		16	460.459	even	2
1840.3.k.d.321.5		16	23.22	odd	2	inner
1840.3.k.d.321.6		16	1.1	even	1	trivial
2070.3.c.a.91.12		16	12.11	even	2
2070.3.c.a.91.13		16	276.275	odd	2

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.34854 q^{3} +2.23607i q^{5} -7.61815i q^{7} -3.48436 q^{9} +O(q^{10})\)	\(q-2.34854 q^{3} +2.23607i q^{5} -7.61815i q^{7} -3.48436 q^{9} -12.3764i q^{11} -13.0302 q^{13} -5.25149i q^{15} -9.13040i q^{17} -14.4549i q^{19} +17.8915i q^{21} +(-22.5529 + 4.51289i) q^{23} -5.00000 q^{25} +29.3200 q^{27} -21.2813 q^{29} -36.8428 q^{31} +29.0666i q^{33} +17.0347 q^{35} +56.9603i q^{37} +30.6019 q^{39} +70.7680 q^{41} -70.0086i q^{43} -7.79128i q^{45} +66.2614 q^{47} -9.03623 q^{49} +21.4431i q^{51} -77.4364i q^{53} +27.6746 q^{55} +33.9480i q^{57} -82.7923 q^{59} +23.9941i q^{61} +26.5444i q^{63} -29.1364i q^{65} +118.512i q^{67} +(52.9664 - 10.5987i) q^{69} -69.0263 q^{71} +25.9840 q^{73} +11.7427 q^{75} -94.2857 q^{77} +28.8543i q^{79} -37.4999 q^{81} +69.3871i q^{83} +20.4162 q^{85} +49.9800 q^{87} -45.4428i q^{89} +99.2661i q^{91} +86.5269 q^{93} +32.3222 q^{95} -74.4458i q^{97} +43.1241i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} + 24 q^{13} - 4 q^{23} - 80 q^{25} + 96 q^{27} - 108 q^{29} + 116 q^{31} - 60 q^{35} - 248 q^{39} - 156 q^{41} + 128 q^{47} - 28 q^{49} - 204 q^{59} - 268 q^{69} - 236 q^{71} - 112 q^{73} - 936 q^{77} - 136 q^{81} + 60 q^{85} + 152 q^{87} + 856 q^{93} + 160 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 + 24 * q^13 - 4 * q^23 - 80 * q^25 + 96 * q^27 - 108 * q^29 + 116 * q^31 - 60 * q^35 - 248 * q^39 - 156 * q^41 + 128 * q^47 - 28 * q^49 - 204 * q^59 - 268 * q^69 - 236 * q^71 - 112 * q^73 - 936 * q^77 - 136 * q^81 + 60 * q^85 + 152 * q^87 + 856 * q^93 + 160 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more