LMFDB - Embedded newform 1680.3.l.b.1121.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,3,Mod(1121,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.1121"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1680.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,8,0,0,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$45.7766844125$
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} - 8 x^{15} + 37 x^{14} - 116 x^{13} + 298 x^{12} - 548 x^{11} + 267 x^{10} + 3264 x^{9} + \cdots + 43046721 $ x^16 - 8x^15 + 37x^14 - 116x^13 + 298x^12 - 548x^11 + 267x^10 + 3264x^9 - 13590x^8 + 29376x^7 + 21627x^6 - 399492x^5 + 1955178x^4 - 6849684x^3 + 19663317x^2 - 38263752*x + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.11
Root			$1.83120 - 2.37628i$ of defining polynomial
Character	$\chi$	$=$	1680.1121
Dual form			1680.3.l.b.1121.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.83120 - 2.37628i) q^{3} +2.23607i q^{5} +2.64575 q^{7} +(-2.29343 - 8.70288i) q^{9} +19.2815i q^{11} -5.14247 q^{13} +(5.31353 + 4.09468i) q^{15} +2.27339i q^{17} -19.4150 q^{19} +(4.84489 - 6.28705i) q^{21} +4.35527i q^{23} -5.00000 q^{25} +(-24.8802 - 10.4869i) q^{27} -0.296771i q^{29} -49.1895 q^{31} +(45.8183 + 35.3083i) q^{33} +5.91608i q^{35} -36.4996 q^{37} +(-9.41688 + 12.2200i) q^{39} -66.2231i q^{41} -41.1470 q^{43} +(19.4602 - 5.12826i) q^{45} +39.7444i q^{47} +7.00000 q^{49} +(5.40223 + 4.16304i) q^{51} -47.0397i q^{53} -43.1148 q^{55} +(-35.5526 + 46.1354i) q^{57} -35.5445i q^{59} -64.1701 q^{61} +(-6.06784 - 23.0257i) q^{63} -11.4989i q^{65} -116.159 q^{67} +(10.3493 + 7.97536i) q^{69} +102.295i q^{71} +82.3841 q^{73} +(-9.15599 + 11.8814i) q^{75} +51.0141i q^{77} +76.4237 q^{79} +(-70.4804 + 39.9189i) q^{81} +7.12740i q^{83} -5.08346 q^{85} +(-0.705211 - 0.543446i) q^{87} -12.7853i q^{89} -13.6057 q^{91} +(-90.0757 + 116.888i) q^{93} -43.4132i q^{95} -164.329 q^{97} +(167.805 - 44.2208i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) $ 16 * q + 8 * q^3 - 10 * q^9 - 10 * q^15 - 48 * q^19 - 14 * q^21 - 80 * q^25 - 28 * q^27 - 24 * q^31 + 92 * q^33 - 40 * q^37 + 54 * q^39 + 168 * q^43 + 40 * q^45 + 112 * q^49 + 186 * q^51 + 80 * q^55 + 252 * q^57 - 152 * q^61 + 56 * q^63 - 264 * q^67 - 196 * q^69 + 240 * q^73 - 40 * q^75 - 212 * q^79 - 118 * q^81 - 60 * q^85 - 60 * q^87 - 196 * q^91 - 244 * q^93 - 360 * q^97 - 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	1.83120	−	2.37628i	0.610399	−	0.792094i
$3$	1.83120	−	2.37628i	0.610399	−	0.792094i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		0			0
$9$	−2.29343	−	8.70288i	−0.254825	−	0.966987i
$10$		0			0
$11$			19.2815i			1.75287i	0.481524	+	0.876433i	$0.340083\pi$
$11$			19.2815i			1.75287i	−0.481524	+	0.876433i	$0.659917\pi$
$12$		0			0
$13$	−5.14247			−0.395574			−0.197787	−	0.980245i	$-0.563376\pi$
$13$	−5.14247			−0.395574			−0.197787	+	0.980245i	$0.563376\pi$
$14$		0			0
$15$	5.31353	+	4.09468i	0.354235	+	0.272979i
$16$		0			0
$17$			2.27339i			0.133729i	0.997762	+	0.0668645i	$0.0212995\pi$
$17$			2.27339i			0.133729i	−0.997762	+	0.0668645i	$0.978700\pi$
$18$		0			0
$19$	−19.4150			−1.02184			−0.510920	−	0.859628i	$-0.670695\pi$
$19$	−19.4150			−1.02184			−0.510920	+	0.859628i	$0.670695\pi$
$20$		0			0
$21$	4.84489	−	6.28705i	0.230709	−	0.299383i
$22$		0			0
$23$			4.35527i			0.189359i	0.995508	+	0.0946797i	$0.0301827\pi$
$23$			4.35527i			0.189359i	−0.995508	+	0.0946797i	$0.969817\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−24.8802	−	10.4869i	−0.921490	−	0.388403i
$28$		0			0
$29$		−	0.296771i		−	0.0102335i	−0.999987	−	0.00511674i	$-0.998371\pi$
$29$		−	0.296771i		−	0.0102335i	0.999987	−	0.00511674i	$-0.00162871\pi$
$30$		0			0
$31$	−49.1895			−1.58676			−0.793379	−	0.608728i	$-0.791680\pi$
$31$	−49.1895			−1.58676			−0.793379	+	0.608728i	$0.791680\pi$
$32$		0			0
$33$	45.8183	+	35.3083i	1.38843	+	1.06995i
$34$		0			0
$35$			5.91608i			0.169031i
$36$		0			0
$37$	−36.4996			−0.986476			−0.493238	−	0.869894i	$-0.664187\pi$
$37$	−36.4996			−0.986476			−0.493238	+	0.869894i	$0.664187\pi$
$38$		0			0
$39$	−9.41688	+	12.2200i	−0.241458	+	0.313332i
$40$		0			0
$41$		−	66.2231i		−	1.61520i	−0.589733	−	0.807598i	$-0.700767\pi$
$41$		−	66.2231i		−	1.61520i	0.589733	−	0.807598i	$-0.299233\pi$
$42$		0			0
$43$	−41.1470			−0.956906			−0.478453	−	0.878113i	$-0.658802\pi$
$43$	−41.1470			−0.956906			−0.478453	+	0.878113i	$0.658802\pi$
$44$		0			0
$45$	19.4602	−	5.12826i	0.432450	−	0.113961i
$46$		0			0
$47$			39.7444i			0.845626i	0.906217	+	0.422813i	$0.138957\pi$
$47$			39.7444i			0.845626i	−0.906217	+	0.422813i	$0.861043\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	5.40223	+	4.16304i	0.105926	+	0.0816282i
$52$		0			0
$53$		−	47.0397i		−	0.887541i	−0.896140	−	0.443771i	$-0.853641\pi$
$53$		−	47.0397i		−	0.887541i	0.896140	−	0.443771i	$-0.146359\pi$
$54$		0			0
$55$	−43.1148			−0.783905
$56$		0			0
$57$	−35.5526	+	46.1354i	−0.623730	+	0.809393i
$58$		0			0
$59$		−	35.5445i		−	0.602449i	−0.953553	−	0.301224i	$-0.902605\pi$
$59$		−	35.5445i		−	0.602449i	0.953553	−	0.301224i	$-0.0973953\pi$
$60$		0			0
$61$	−64.1701			−1.05197			−0.525985	−	0.850494i	$-0.676303\pi$
$61$	−64.1701			−1.05197			−0.525985	+	0.850494i	$0.676303\pi$
$62$		0			0
$63$	−6.06784	−	23.0257i	−0.0963149	−	0.365487i
$64$		0			0
$65$		−	11.4989i		−	0.176906i
$66$		0			0
$67$	−116.159			−1.73372			−0.866861	−	0.498549i	$-0.833866\pi$
$67$	−116.159			−1.73372			−0.866861	+	0.498549i	$0.833866\pi$
$68$		0			0
$69$	10.3493	+	7.97536i	0.149990	+	0.115585i
$70$		0			0
$71$			102.295i			1.44077i	0.693574	+	0.720385i	$0.256035\pi$
$71$			102.295i			1.44077i	−0.693574	+	0.720385i	$0.743965\pi$
$72$		0			0
$73$	82.3841			1.12855			0.564275	−	0.825587i	$-0.309156\pi$
$73$	82.3841			1.12855			0.564275	+	0.825587i	$0.309156\pi$
$74$		0			0
$75$	−9.15599	+	11.8814i	−0.122080	+	0.158419i
$76$		0			0
$77$			51.0141i			0.662521i
$78$		0			0
$79$	76.4237			0.967389			0.483694	−	0.875237i	$-0.339295\pi$
$79$	76.4237			0.967389			0.483694	+	0.875237i	$0.339295\pi$
$80$		0			0
$81$	−70.4804	+	39.9189i	−0.870128	+	0.492826i
$82$		0			0
$83$			7.12740i			0.0858723i	0.999078	+	0.0429361i	$0.0136712\pi$
$83$			7.12740i			0.0858723i	−0.999078	+	0.0429361i	$0.986329\pi$
$84$		0			0
$85$	−5.08346			−0.0598055
$86$		0			0
$87$	−0.705211	−	0.543446i	−0.00810587	−	0.00624650i
$88$		0			0
$89$		−	12.7853i		−	0.143655i	−0.997417	−	0.0718274i	$-0.977117\pi$
$89$		−	12.7853i		−	0.143655i	0.997417	−	0.0718274i	$-0.0228831\pi$
$90$		0			0
$91$	−13.6057			−0.149513
$92$		0			0
$93$	−90.0757	+	116.888i	−0.968556	+	1.25686i
$94$		0			0
$95$		−	43.4132i		−	0.456981i
$96$		0			0
$97$	−164.329			−1.69412			−0.847059	−	0.531499i	$-0.821629\pi$
$97$	−164.329			−1.69412			−0.847059	+	0.531499i	$0.821629\pi$
$98$		0			0
$99$	167.805	−	44.2208i	1.69500	−	0.446675i
$100$		0			0
$101$		−	128.611i		−	1.27337i	−0.771123	−	0.636686i	$-0.780305\pi$
$101$		−	128.611i		−	1.27337i	0.771123	−	0.636686i	$-0.219695\pi$
$102$		0			0
$103$	85.6904			0.831946			0.415973	−	0.909377i	$-0.363441\pi$
$103$	85.6904			0.831946			0.415973	+	0.909377i	$0.363441\pi$
$104$		0			0
$105$	14.0583	+	10.8335i	0.133888	+	0.103176i
$106$		0			0
$107$			56.4021i			0.527123i	0.964643	+	0.263561i	$0.0848971\pi$
$107$			56.4021i			0.527123i	−0.964643	+	0.263561i	$0.915103\pi$
$108$		0			0
$109$	95.6789			0.877788			0.438894	−	0.898539i	$-0.355370\pi$
$109$	95.6789			0.877788			0.438894	+	0.898539i	$0.355370\pi$
$110$		0			0
$111$	−66.8380	+	86.7334i	−0.602145	+	0.781382i
$112$		0			0
$113$			145.741i			1.28975i	0.764289	+	0.644874i	$0.223090\pi$
$113$			145.741i			1.28975i	−0.764289	+	0.644874i	$0.776910\pi$
$114$		0			0
$115$	−9.73868			−0.0846841
$116$		0			0
$117$	11.7939	+	44.7543i	0.100802	+	0.382515i
$118$		0			0
$119$			6.01484i			0.0505448i
$120$		0			0
$121$	−250.777			−2.07254
$122$		0			0
$123$	−157.365	−	121.268i	−1.27939	−	0.985915i
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	147.325			1.16004			0.580021	−	0.814601i	$-0.303044\pi$
$127$	147.325			1.16004			0.580021	+	0.814601i	$0.303044\pi$
$128$		0			0
$129$	−75.3482	+	97.7768i	−0.584095	+	0.757960i
$130$		0			0
$131$			64.8142i			0.494765i	0.968918	+	0.247383i	$0.0795704\pi$
$131$			64.8142i			0.494765i	−0.968918	+	0.247383i	$0.920430\pi$
$132$		0			0
$133$	−51.3671			−0.386219
$134$		0			0
$135$	23.4494	−	55.6339i	0.173699	−	0.412103i
$136$		0			0
$137$		−	112.339i		−	0.819995i	−0.912087	−	0.409997i	$-0.865530\pi$
$137$		−	112.339i		−	0.819995i	0.912087	−	0.409997i	$-0.134470\pi$
$138$		0			0
$139$	−191.737			−1.37940			−0.689700	−	0.724095i	$-0.742258\pi$
$139$	−191.737			−1.37940			−0.689700	+	0.724095i	$0.742258\pi$
$140$		0			0
$141$	94.4440	+	72.7799i	0.669815	+	0.516170i
$142$		0			0
$143$		−	99.1546i		−	0.693389i
$144$		0			0
$145$	0.663599			0.00457655
$146$		0			0
$147$	12.8184	−	16.6340i	0.0871999	−	0.113156i
$148$		0			0
$149$		−	36.0538i		−	0.241972i	−0.992654	−	0.120986i	$-0.961394\pi$
$149$		−	36.0538i		−	0.241972i	0.992654	−	0.120986i	$-0.0386056\pi$
$150$		0			0
$151$	142.221			0.941862			0.470931	−	0.882170i	$-0.343918\pi$
$151$	142.221			0.941862			0.470931	+	0.882170i	$0.343918\pi$
$152$		0			0
$153$	19.7851	−	5.21387i	0.129314	−	0.0340776i
$154$		0			0
$155$		−	109.991i		−	0.709620i
$156$		0			0
$157$	31.7422			0.202180			0.101090	−	0.994877i	$-0.467767\pi$
$157$	31.7422			0.202180			0.101090	+	0.994877i	$0.467767\pi$
$158$		0			0
$159$	−111.780	−	86.1390i	−0.703016	−	0.541754i
$160$		0			0
$161$			11.5230i			0.0715712i
$162$		0			0
$163$	6.49009			0.0398165			0.0199083	−	0.999802i	$-0.493663\pi$
$163$	6.49009			0.0398165			0.0199083	+	0.999802i	$0.493663\pi$
$164$		0			0
$165$	−78.9517	+	102.453i	−0.478495	+	0.620927i
$166$		0			0
$167$			60.8968i			0.364652i	0.983238	+	0.182326i	$0.0583626\pi$
$167$			60.8968i			0.364652i	−0.983238	+	0.182326i	$0.941637\pi$
$168$		0			0
$169$	−142.555			−0.843521
$170$		0			0
$171$	44.5268	+	168.966i	0.260391	+	0.988106i
$172$		0			0
$173$			330.679i			1.91144i	0.294275	+	0.955721i	$0.404922\pi$
$173$			330.679i			1.91144i	−0.294275	+	0.955721i	$0.595078\pi$
$174$		0			0
$175$	−13.2288			−0.0755929
$176$		0			0
$177$	−84.4637	−	65.0890i	−0.477196	−	0.367734i
$178$		0			0
$179$			281.089i			1.57033i	0.619288	+	0.785164i	$0.287421\pi$
$179$			281.089i			1.57033i	−0.619288	+	0.785164i	$0.712579\pi$
$180$		0			0
$181$	157.168			0.868332			0.434166	−	0.900833i	$-0.357043\pi$
$181$	157.168			0.868332			0.434166	+	0.900833i	$0.357043\pi$
$182$		0			0
$183$	−117.508	+	152.486i	−0.642121	+	0.833258i
$184$		0			0
$185$		−	81.6156i		−	0.441166i
$186$		0			0
$187$	−43.8345			−0.234409
$188$		0			0
$189$	−65.8269	−	27.7457i	−0.348290	−	0.146802i
$190$		0			0
$191$		−	197.219i		−	1.03256i	−0.856420	−	0.516280i	$-0.827317\pi$
$191$		−	197.219i		−	1.03256i	0.856420	−	0.516280i	$-0.172683\pi$
$192$		0			0
$193$	165.188			0.855894			0.427947	−	0.903804i	$-0.359237\pi$
$193$	165.188			0.855894			0.427947	+	0.903804i	$0.359237\pi$
$194$		0			0
$195$	−27.3246	−	21.0568i	−0.140126	−	0.107983i
$196$		0			0
$197$			90.3439i			0.458598i	0.973356	+	0.229299i	$0.0736434\pi$
$197$			90.3439i			0.458598i	−0.973356	+	0.229299i	$0.926357\pi$
$198$		0			0
$199$	9.39271			0.0471996			0.0235998	−	0.999721i	$-0.492487\pi$
$199$	9.39271			0.0471996			0.0235998	+	0.999721i	$0.492487\pi$
$200$		0			0
$201$	−212.711	+	276.028i	−1.05826	+	1.37327i
$202$		0			0
$203$		−	0.785181i		−	0.00386789i
$204$		0			0
$205$	148.079			0.722338
$206$		0			0
$207$	37.9034	−	9.98849i	0.183108	−	0.0482536i
$208$		0			0
$209$		−	374.350i		−	1.79115i
$210$		0			0
$211$	220.196			1.04358			0.521792	−	0.853073i	$-0.325264\pi$
$211$	220.196			1.04358			0.521792	+	0.853073i	$0.325264\pi$
$212$		0			0
$213$	243.081	+	187.322i	1.14123	+	0.879445i
$214$		0			0
$215$		−	92.0074i		−	0.427941i
$216$		0			0
$217$	−130.143			−0.599738
$218$		0			0
$219$	150.862	−	195.768i	0.688866	−	0.893917i
$220$		0			0
$221$		−	11.6909i		−	0.0528998i
$222$		0			0
$223$	−349.896			−1.56904			−0.784519	−	0.620104i	$-0.787090\pi$
$223$	−349.896			−1.56904			−0.784519	+	0.620104i	$0.787090\pi$
$224$		0			0
$225$	11.4671	+	43.5144i	0.0509651	+	0.193397i
$226$		0			0
$227$			337.232i			1.48560i	0.669512	+	0.742801i	$0.266503\pi$
$227$			337.232i			1.48560i	−0.669512	+	0.742801i	$0.733497\pi$
$228$		0			0
$229$	−3.13719			−0.0136995			−0.00684977	−	0.999977i	$-0.502180\pi$
$229$	−3.13719			−0.0136995			−0.00684977	+	0.999977i	$0.502180\pi$
$230$		0			0
$231$	121.224	+	93.4170i	0.524779	+	0.404402i
$232$		0			0
$233$			165.629i			0.710854i	0.934704	+	0.355427i	$0.115665\pi$
$233$			165.629i			0.710854i	−0.934704	+	0.355427i	$0.884335\pi$
$234$		0			0
$235$	−88.8713			−0.378176
$236$		0			0
$237$	139.947	−	181.604i	0.590493	−	0.766263i
$238$		0			0
$239$			20.4912i			0.0857372i	0.999081	+	0.0428686i	$0.0136497\pi$
$239$			20.4912i			0.0857372i	−0.999081	+	0.0428686i	$0.986350\pi$
$240$		0			0
$241$	−88.8198			−0.368547			−0.184273	−	0.982875i	$-0.558993\pi$
$241$	−88.8198			−0.368547			−0.184273	+	0.982875i	$0.558993\pi$
$242$		0			0
$243$	−34.2051	+	240.581i	−0.140762	+	0.990044i
$244$		0			0
$245$			15.6525i			0.0638877i
$246$		0			0
$247$	99.8408			0.404214
$248$		0			0
$249$	16.9367	+	13.0517i	0.0680189	+	0.0524164i
$250$		0			0
$251$			163.714i			0.652247i	0.945327	+	0.326123i	$0.105742\pi$
$251$			163.714i			0.652247i	−0.945327	+	0.326123i	$0.894258\pi$
$252$		0			0
$253$	−83.9762			−0.331922
$254$		0			0
$255$	−9.30883	+	12.0797i	−0.0365052	+	0.0473715i
$256$		0			0
$257$		−	308.577i		−	1.20069i	−0.799741	−	0.600345i	$-0.795030\pi$
$257$		−	308.577i		−	1.20069i	0.799741	−	0.600345i	$-0.204970\pi$
$258$		0			0
$259$	−96.5689			−0.372853
$260$		0			0
$261$	−2.58276	+	0.680622i	−0.00989564	+	0.00260775i
$262$		0			0
$263$			305.110i			1.16012i	0.814575	+	0.580058i	$0.196970\pi$
$263$			305.110i			1.16012i	−0.814575	+	0.580058i	$0.803030\pi$
$264$		0			0
$265$	105.184			0.396920
$266$		0			0
$267$	−30.3814	−	23.4124i	−0.113788	−	0.0876868i
$268$		0			0
$269$			371.149i			1.37974i	0.723935	+	0.689868i	$0.242331\pi$
$269$			371.149i			1.37974i	−0.723935	+	0.689868i	$0.757669\pi$
$270$		0			0
$271$	282.816			1.04360			0.521801	−	0.853067i	$-0.325260\pi$
$271$	282.816			1.04360			0.521801	+	0.853067i	$0.325260\pi$
$272$		0			0
$273$	−24.9147	+	32.3309i	−0.0912627	+	0.118428i
$274$		0			0
$275$		−	96.4076i		−	0.350573i
$276$		0			0
$277$	177.234			0.639833			0.319916	−	0.947446i	$-0.396345\pi$
$277$	177.234			0.639833			0.319916	+	0.947446i	$0.396345\pi$
$278$		0			0
$279$	112.813	+	428.090i	0.404346	+	1.53437i
$280$		0			0
$281$			54.9268i			0.195469i	0.995213	+	0.0977346i	$0.0311596\pi$
$281$			54.9268i			0.195469i	−0.995213	+	0.0977346i	$0.968840\pi$
$282$		0			0
$283$	173.185			0.611962			0.305981	−	0.952038i	$-0.401016\pi$
$283$	173.185			0.611962			0.305981	+	0.952038i	$0.401016\pi$
$284$		0			0
$285$	−103.162	−	79.4981i	−0.361971	−	0.278941i
$286$		0			0
$287$		−	175.210i		−	0.610487i
$288$		0			0
$289$	283.832			0.982117
$290$		0			0
$291$	−300.920	+	390.493i	−1.03409	+	1.34190i
$292$		0			0
$293$			212.515i			0.725306i	0.931924	+	0.362653i	$0.118129\pi$
$293$			212.515i			0.725306i	−0.931924	+	0.362653i	$0.881871\pi$
$294$		0			0
$295$	79.4799			0.269423
$296$		0			0
$297$	202.203	−	479.729i	0.680818	−	1.61525i
$298$		0			0
$299$		−	22.3968i		−	0.0749058i
$300$		0			0
$301$	−108.865			−0.361677
$302$		0			0
$303$	−305.615	−	235.512i	−1.00863	−	0.777266i
$304$		0			0
$305$		−	143.489i		−	0.470455i
$306$		0			0
$307$	−545.093			−1.77555			−0.887774	−	0.460279i	$-0.847749\pi$
$307$	−545.093			−1.77555			−0.887774	+	0.460279i	$0.847749\pi$
$308$		0			0
$309$	156.916	−	203.625i	0.507819	−	0.658979i
$310$		0			0
$311$		−	362.627i		−	1.16600i	−0.812471	−	0.583001i	$-0.801878\pi$
$311$		−	362.627i		−	1.16600i	0.812471	−	0.583001i	$-0.198122\pi$
$312$		0			0
$313$	−164.567			−0.525774			−0.262887	−	0.964827i	$-0.584675\pi$
$313$	−164.567			−0.525774			−0.262887	+	0.964827i	$0.584675\pi$
$314$		0			0
$315$	51.4870	−	13.5681i	0.163451	−	0.0430733i
$316$		0			0
$317$		−	394.214i		−	1.24358i	−0.783185	−	0.621789i	$-0.786406\pi$
$317$		−	394.214i		−	1.24358i	0.783185	−	0.621789i	$-0.213594\pi$
$318$		0			0
$319$	5.72219			0.0179379
$320$		0			0
$321$	134.027	+	103.283i	0.417531	+	0.321755i
$322$		0			0
$323$		−	44.1378i		−	0.136650i
$324$		0			0
$325$	25.7123			0.0791149
$326$		0			0
$327$	175.207	−	227.360i	0.535801	−	0.695291i
$328$		0			0
$329$			105.154i			0.319617i
$330$		0			0
$331$	−628.823			−1.89977			−0.949884	−	0.312604i	$-0.898799\pi$
$331$	−628.823			−1.89977			−0.949884	+	0.312604i	$0.898799\pi$
$332$		0			0
$333$	83.7093	+	317.652i	0.251379	+	0.953910i
$334$		0			0
$335$		−	259.740i		−	0.775344i
$336$		0			0
$337$	−179.118			−0.531506			−0.265753	−	0.964041i	$-0.585621\pi$
$337$	−179.118			−0.531506			−0.265753	+	0.964041i	$0.585621\pi$
$338$		0			0
$339$	346.323	+	266.882i	1.02160	+	0.787261i
$340$		0			0
$341$		−	948.448i		−	2.78137i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$	−17.8334	+	23.1418i	−0.0516911	+	0.0670778i
$346$		0			0
$347$		−	478.907i		−	1.38013i	−0.723745	−	0.690067i	$-0.757581\pi$
$347$		−	478.907i		−	1.38013i	0.723745	−	0.690067i	$-0.242419\pi$
$348$		0			0
$349$	468.111			1.34129			0.670646	−	0.741777i	$-0.266017\pi$
$349$	468.111			1.34129			0.670646	+	0.741777i	$0.266017\pi$
$350$		0			0
$351$	127.946	+	53.9284i	0.364518	+	0.153642i
$352$		0			0
$353$		−	69.9260i		−	0.198091i	−0.995083	−	0.0990453i	$-0.968421\pi$
$353$		−	69.9260i		−	0.198091i	0.995083	−	0.0990453i	$-0.0315789\pi$
$354$		0			0
$355$	−228.738			−0.644332
$356$		0			0
$357$	14.2929	+	11.0144i	0.0400363	+	0.0308525i
$358$		0			0
$359$		−	454.508i		−	1.26604i	−0.774136	−	0.633020i	$-0.781815\pi$
$359$		−	454.508i		−	1.26604i	0.774136	−	0.633020i	$-0.218185\pi$
$360$		0			0
$361$	15.9403			0.0441560
$362$		0			0
$363$	−459.223	+	595.917i	−1.26508	+	1.64165i
$364$		0			0
$365$			184.216i			0.504703i
$366$		0			0
$367$	−190.322			−0.518587			−0.259294	−	0.965799i	$-0.583490\pi$
$367$	−190.322			−0.518587			−0.259294	+	0.965799i	$0.583490\pi$
$368$		0			0
$369$	−576.332	+	151.878i	−1.56187	+	0.411593i
$370$		0			0
$371$		−	124.455i		−	0.335459i
$372$		0			0
$373$	−492.934			−1.32154			−0.660770	−	0.750589i	$-0.729770\pi$
$373$	−492.934			−1.32154			−0.660770	+	0.750589i	$0.729770\pi$
$374$		0			0
$375$	−26.5676	−	20.4734i	−0.0708470	−	0.0545958i
$376$		0			0
$377$			1.52613i			0.00404810i
$378$		0			0
$379$	−411.232			−1.08504			−0.542522	−	0.840041i	$-0.682531\pi$
$379$	−411.232			−1.08504			−0.542522	+	0.840041i	$0.682531\pi$
$380$		0			0
$381$	269.782	−	350.087i	0.708089	−	0.918862i
$382$		0			0
$383$		−	506.545i		−	1.32257i	−0.750133	−	0.661286i	$-0.770011\pi$
$383$		−	506.545i		−	1.32257i	0.750133	−	0.661286i	$-0.229989\pi$
$384$		0			0
$385$	−114.071			−0.296288
$386$		0			0
$387$	94.3676	+	358.097i	0.243844	+	0.925316i
$388$		0			0
$389$		−	168.573i		−	0.433349i	−0.976244	−	0.216675i	$-0.930479\pi$
$389$		−	168.573i		−	0.433349i	0.976244	−	0.216675i	$-0.0695211\pi$
$390$		0			0
$391$	−9.90124			−0.0253229
$392$		0			0
$393$	154.017	+	118.688i	0.391900	+	0.302004i
$394$		0			0
$395$			170.889i			0.432629i
$396$		0			0
$397$	−309.834			−0.780439			−0.390220	−	0.920722i	$-0.627601\pi$
$397$	−309.834			−0.780439			−0.390220	+	0.920722i	$0.627601\pi$
$398$		0			0
$399$	−94.0634	+	122.063i	−0.235748	+	0.305922i
$400$		0			0
$401$			699.211i			1.74367i	0.489802	+	0.871834i	$0.337069\pi$
$401$			699.211i			1.74367i	−0.489802	+	0.871834i	$0.662931\pi$
$402$		0			0
$403$	252.955			0.627681
$404$		0			0
$405$	−89.2613	−	157.599i	−0.220398	−	0.389133i
$406$		0			0
$407$		−	703.768i		−	1.72916i
$408$		0			0
$409$	589.473			1.44125			0.720627	−	0.693323i	$-0.243854\pi$
$409$	589.473			1.44125			0.720627	+	0.693323i	$0.243854\pi$
$410$		0			0
$411$	−266.950	−	205.715i	−0.649513	−	0.500524i
$412$		0			0
$413$		−	94.0419i		−	0.227704i
$414$		0			0
$415$	−15.9373			−0.0384032
$416$		0			0
$417$	−351.108	+	455.620i	−0.841985	+	1.09261i
$418$		0			0
$419$		−	305.382i		−	0.728835i	−0.931236	−	0.364418i	$-0.881268\pi$
$419$		−	305.382i		−	0.728835i	0.931236	−	0.364418i	$-0.118732\pi$
$420$		0			0
$421$	−83.6282			−0.198642			−0.0993209	−	0.995055i	$-0.531667\pi$
$421$	−83.6282			−0.198642			−0.0993209	+	0.995055i	$0.531667\pi$
$422$		0			0
$423$	345.891	−	91.1510i	0.817710	−	0.215487i
$424$		0			0
$425$		−	11.3670i		−	0.0267458i
$426$		0			0
$427$	−169.778			−0.397607
$428$		0			0
$429$	−235.619	−	181.572i	−0.549229	−	0.423244i
$430$		0			0
$431$			802.649i			1.86229i	0.364643	+	0.931147i	$0.381191\pi$
$431$			802.649i			1.86229i	−0.364643	+	0.931147i	$0.618809\pi$
$432$		0			0
$433$	−37.4318			−0.0864476			−0.0432238	−	0.999065i	$-0.513763\pi$
$433$	−37.4318			−0.0864476			−0.0432238	+	0.999065i	$0.513763\pi$
$434$		0			0
$435$	1.21518	−	1.57690i	0.00279352	−	0.00362506i
$436$		0			0
$437$		−	84.5573i		−	0.193495i
$438$		0			0
$439$	119.327			0.271815			0.135907	−	0.990722i	$-0.456605\pi$
$439$	119.327			0.271815			0.135907	+	0.990722i	$0.456605\pi$
$440$		0			0
$441$	−16.0540	−	60.9202i	−0.0364036	−	0.138141i
$442$		0			0
$443$			326.408i			0.736813i	0.929665	+	0.368407i	$0.120097\pi$
$443$			326.408i			0.736813i	−0.929665	+	0.368407i	$0.879903\pi$
$444$		0			0
$445$	28.5888			0.0642444
$446$		0			0
$447$	−85.6739	−	66.0216i	−0.191664	−	0.147699i
$448$		0			0
$449$		−	94.0364i		−	0.209435i	−0.994502	−	0.104718i	$-0.966606\pi$
$449$		−	94.0364i		−	0.209435i	0.994502	−	0.104718i	$-0.0333939\pi$
$450$		0			0
$451$	1276.88			2.83122
$452$		0			0
$453$	260.435	−	337.958i	0.574912	−	0.746043i
$454$		0			0
$455$		−	30.4232i		−	0.0668643i
$456$		0			0
$457$	−445.185			−0.974146			−0.487073	−	0.873361i	$-0.661935\pi$
$457$	−445.185			−0.974146			−0.487073	+	0.873361i	$0.661935\pi$
$458$		0			0
$459$	23.8408	−	56.5626i	0.0519407	−	0.123230i
$460$		0			0
$461$			437.152i			0.948269i	0.880452	+	0.474135i	$0.157239\pi$
$461$			437.152i			0.948269i	−0.880452	+	0.474135i	$0.842761\pi$
$462$		0			0
$463$	518.312			1.11946			0.559732	−	0.828674i	$-0.310904\pi$
$463$	518.312			1.11946			0.559732	+	0.828674i	$0.310904\pi$
$464$		0			0
$465$	−261.370	−	201.415i	−0.562085	−	0.433151i
$466$		0			0
$467$		−	697.335i		−	1.49322i	−0.665260	−	0.746612i	$-0.731679\pi$
$467$		−	697.335i		−	1.49322i	0.665260	−	0.746612i	$-0.268321\pi$
$468$		0			0
$469$	−307.329			−0.655286
$470$		0			0
$471$	58.1262	−	75.4284i	0.123410	−	0.160145i
$472$		0			0
$473$		−	793.376i		−	1.67733i
$474$		0			0
$475$	97.0748			0.204368
$476$		0			0
$477$	−409.381	+	107.882i	−0.858241	+	0.226168i
$478$		0			0
$479$		−	921.591i		−	1.92399i	−0.273067	−	0.961995i	$-0.588038\pi$
$479$		−	921.591i		−	1.92399i	0.273067	−	0.961995i	$-0.411962\pi$
$480$		0			0
$481$	187.698			0.390225
$482$		0			0
$483$	27.3818	+	21.1008i	0.0566911	+	0.0436870i
$484$		0			0
$485$		−	367.452i		−	0.757632i
$486$		0			0
$487$	839.254			1.72331			0.861657	−	0.507490i	$-0.169427\pi$
$487$	839.254			1.72331			0.861657	+	0.507490i	$0.169427\pi$
$488$		0			0
$489$	11.8846	−	15.4223i	0.0243040	−	0.0315384i
$490$		0			0
$491$		−	610.574i		−	1.24353i	−0.783204	−	0.621765i	$-0.786416\pi$
$491$		−	610.574i		−	1.24353i	0.783204	−	0.621765i	$-0.213584\pi$
$492$		0			0
$493$	0.674677			0.00136851
$494$		0			0
$495$	98.8807	+	375.223i	0.199759	+	0.758026i
$496$		0			0
$497$			270.646i			0.544560i
$498$		0			0
$499$	947.765			1.89933			0.949664	−	0.313269i	$-0.101424\pi$
$499$	947.765			1.89933			0.949664	+	0.313269i	$0.101424\pi$
$500$		0			0
$501$	144.708	+	111.514i	0.288838	+	0.222583i
$502$		0			0
$503$			215.893i			0.429210i	0.976701	+	0.214605i	$0.0688464\pi$
$503$			215.893i			0.429210i	−0.976701	+	0.214605i	$0.931154\pi$
$504$		0			0
$505$	287.582			0.569470
$506$		0			0
$507$	−261.046	+	338.751i	−0.514885	+	0.668148i
$508$		0			0
$509$			494.254i			0.971030i	0.874228	+	0.485515i	$0.161368\pi$
$509$			494.254i			0.971030i	−0.874228	+	0.485515i	$0.838632\pi$
$510$		0			0
$511$	217.968			0.426552
$512$		0			0
$513$	483.048	+	203.602i	0.941615	+	0.396885i
$514$		0			0
$515$			191.610i			0.372057i
$516$		0			0
$517$	−766.333			−1.48227
$518$		0			0
$519$	785.787	+	605.539i	1.51404	+	1.16674i
$520$		0			0
$521$		−	98.2752i		−	0.188628i	−0.995543	−	0.0943140i	$-0.969934\pi$
$521$		−	98.2752i		−	0.188628i	0.995543	−	0.0943140i	$-0.0300658\pi$
$522$		0			0
$523$	227.467			0.434927			0.217463	−	0.976068i	$-0.430222\pi$
$523$	227.467			0.434927			0.217463	+	0.976068i	$0.430222\pi$
$524$		0			0
$525$	−24.2245	+	31.4352i	−0.0461419	+	0.0598767i
$526$		0			0
$527$		−	111.827i		−	0.212196i
$528$		0			0
$529$	510.032			0.964143
$530$		0			0
$531$	−309.340	+	81.5187i	−0.582560	+	0.153519i
$532$		0			0
$533$			340.550i			0.638931i
$534$		0			0
$535$	−126.119			−0.235736
$536$		0			0
$537$	667.946	+	514.729i	1.24385	+	0.958527i
$538$		0			0
$539$			134.971i			0.250409i
$540$		0			0
$541$	510.563			0.943740			0.471870	−	0.881668i	$-0.343579\pi$
$541$	510.563			0.943740			0.471870	+	0.881668i	$0.343579\pi$
$542$		0			0
$543$	287.806	−	373.476i	0.530029	−	0.687800i
$544$		0			0
$545$			213.945i			0.392559i
$546$		0			0
$547$	−485.854			−0.888215			−0.444108	−	0.895974i	$-0.646479\pi$
$547$	−485.854			−0.888215			−0.444108	+	0.895974i	$0.646479\pi$
$548$		0			0
$549$	147.170	+	558.465i	0.268068	+	1.01724i
$550$		0			0
$551$			5.76179i			0.0104570i
$552$		0			0
$553$	202.198			0.365639
$554$		0			0
$555$	−193.942	−	149.454i	−0.349445	−	0.269287i
$556$		0			0
$557$			383.384i			0.688302i	0.938914	+	0.344151i	$0.111833\pi$
$557$			383.384i			0.688302i	−0.938914	+	0.344151i	$0.888167\pi$
$558$		0			0
$559$	211.597			0.378528
$560$		0			0
$561$	−80.2697	+	104.163i	−0.143083	+	0.185674i
$562$		0			0
$563$		−	854.637i		−	1.51800i	−0.651088	−	0.759002i	$-0.725687\pi$
$563$		−	854.637i		−	1.51800i	0.651088	−	0.759002i	$-0.274313\pi$
$564$		0			0
$565$	−325.888			−0.576793
$566$		0			0
$567$	−186.474	+	105.615i	−0.328878	+	0.186271i
$568$		0			0
$569$		−	197.946i		−	0.347883i	−0.984756	−	0.173942i	$-0.944350\pi$
$569$		−	197.946i		−	0.347883i	0.984756	−	0.173942i	$-0.0556504\pi$
$570$		0			0
$571$	−582.578			−1.02028			−0.510139	−	0.860092i	$-0.670406\pi$
$571$	−582.578			−1.02028			−0.510139	+	0.860092i	$0.670406\pi$
$572$		0			0
$573$	−468.647	−	361.147i	−0.817884	−	0.630273i
$574$		0			0
$575$		−	21.7763i		−	0.0378719i
$576$		0			0
$577$	828.323			1.43557			0.717784	−	0.696266i	$-0.245156\pi$
$577$	828.323			1.43557			0.717784	+	0.696266i	$0.245156\pi$
$578$		0			0
$579$	302.491	−	392.532i	0.522437	−	0.677949i
$580$		0			0
$581$			18.8573i			0.0324567i
$582$		0			0
$583$	906.997			1.55574
$584$		0			0
$585$	−100.074	+	26.3719i	−0.171066	+	0.0450802i
$586$		0			0
$587$		−	20.1899i		−	0.0343950i	−0.999852	−	0.0171975i	$-0.994526\pi$
$587$		−	20.1899i		−	0.0343950i	0.999852	−	0.0171975i	$-0.00547441\pi$
$588$		0			0
$589$	955.011			1.62141
$590$		0			0
$591$	214.682	+	165.438i	0.363253	+	0.279928i
$592$		0			0
$593$		−	297.750i		−	0.502108i	−0.967973	−	0.251054i	$-0.919223\pi$
$593$		−	297.750i		−	0.502108i	0.967973	−	0.251054i	$-0.0807772\pi$
$594$		0			0
$595$	−13.4496			−0.0226043
$596$		0			0
$597$	17.1999	−	22.3197i	0.0288106	−	0.0373865i
$598$		0			0
$599$			229.232i			0.382692i	0.981523	+	0.191346i	$0.0612852\pi$
$599$			229.232i			0.382692i	−0.981523	+	0.191346i	$0.938715\pi$
$600$		0			0
$601$	15.1940			0.0252812			0.0126406	−	0.999920i	$-0.495976\pi$
$601$	15.1940			0.0252812			0.0126406	+	0.999920i	$0.495976\pi$
$602$		0			0
$603$	266.403	+	1010.92i	0.441796	+	1.67649i
$604$		0			0
$605$		−	560.755i		−	0.926868i
$606$		0			0
$607$	−714.266			−1.17672			−0.588358	−	0.808601i	$-0.700225\pi$
$607$	−714.266			−1.17672			−0.588358	+	0.808601i	$0.700225\pi$
$608$		0			0
$609$	−1.86581	−	1.43782i	−0.00306373	−	0.00236096i
$610$		0			0
$611$		−	204.384i		−	0.334508i
$612$		0			0
$613$	−126.711			−0.206707			−0.103354	−	0.994645i	$-0.532957\pi$
$613$	−126.711			−0.206707			−0.103354	+	0.994645i	$0.532957\pi$
$614$		0			0
$615$	271.163	−	351.878i	0.440915	−	0.572159i
$616$		0			0
$617$		−	363.016i		−	0.588357i	−0.955751	−	0.294178i	$-0.904954\pi$
$617$		−	363.016i		−	0.588357i	0.955751	−	0.294178i	$-0.0950460\pi$
$618$		0			0
$619$	−374.347			−0.604761			−0.302380	−	0.953187i	$-0.597781\pi$
$619$	−374.347			−0.604761			−0.302380	+	0.953187i	$0.597781\pi$
$620$		0			0
$621$	45.6732	−	108.360i	0.0735477	−	0.174493i
$622$		0			0
$623$		−	33.8267i		−	0.0542964i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−889.561	−	685.509i	−1.41876	−	1.09332i
$628$		0			0
$629$		−	82.9781i		−	0.131921i
$630$		0			0
$631$	−454.126			−0.719692			−0.359846	−	0.933012i	$-0.617171\pi$
$631$	−454.126			−0.719692			−0.359846	+	0.933012i	$0.617171\pi$
$632$		0			0
$633$	403.223	−	523.248i	0.637003	−	0.826617i
$634$		0			0
$635$			329.430i			0.518787i
$636$		0			0
$637$	−35.9973			−0.0565106
$638$		0			0
$639$	890.259	−	234.605i	1.39321	−	0.367145i
$640$		0			0
$641$		−	42.7418i		−	0.0666799i	−0.999444	−	0.0333400i	$-0.989386\pi$
$641$		−	42.7418i		−	0.0666799i	0.999444	−	0.0333400i	$-0.0106144\pi$
$642$		0			0
$643$	809.610			1.25911			0.629557	−	0.776955i	$-0.283237\pi$
$643$	809.610			1.25911			0.629557	+	0.776955i	$0.283237\pi$
$644$		0			0
$645$	−218.636	−	168.484i	−0.338970	−	0.261215i
$646$		0			0
$647$		−	1007.94i		−	1.55787i	−0.627102	−	0.778937i	$-0.715759\pi$
$647$		−	1007.94i		−	1.55787i	0.627102	−	0.778937i	$-0.284241\pi$
$648$		0			0
$649$	685.352			1.05601
$650$		0			0
$651$	−238.318	+	309.257i	−0.366080	+	0.475049i
$652$		0			0
$653$		−	474.829i		−	0.727151i	−0.931565	−	0.363575i	$-0.881556\pi$
$653$		−	474.829i		−	0.727151i	0.931565	−	0.363575i	$-0.118444\pi$
$654$		0			0
$655$	−144.929			−0.221266
$656$		0			0
$657$	−188.942	−	716.979i	−0.287583	−	1.09129i
$658$		0			0
$659$			1304.52i			1.97955i	0.142633	+	0.989776i	$0.454443\pi$
$659$			1304.52i			1.97955i	−0.142633	+	0.989776i	$0.545557\pi$
$660$		0			0
$661$	928.326			1.40443			0.702213	−	0.711967i	$-0.252195\pi$
$661$	928.326			1.40443			0.702213	+	0.711967i	$0.252195\pi$
$662$		0			0
$663$	−27.7808	−	21.4083i	−0.0419016	−	0.0322900i
$664$		0			0
$665$		−	114.860i		−	0.172722i
$666$		0			0
$667$	1.29252			0.00193781
$668$		0			0
$669$	−640.728	+	831.450i	−0.957740	+	1.24283i
$670$		0			0
$671$		−	1237.30i		−	1.84396i
$672$		0			0
$673$	−825.850			−1.22712			−0.613558	−	0.789649i	$-0.710262\pi$
$673$	−825.850			−1.22712			−0.613558	+	0.789649i	$0.710262\pi$
$674$		0			0
$675$	124.401	+	52.4344i	0.184298	+	0.0776806i
$676$		0			0
$677$		−	1161.82i		−	1.71613i	−0.513544	−	0.858064i	$-0.671668\pi$
$677$		−	1161.82i		−	1.71613i	0.513544	−	0.858064i	$-0.328332\pi$
$678$		0			0
$679$	−434.775			−0.640316
$680$		0			0
$681$	801.358	+	617.538i	1.17674	+	0.906811i
$682$		0			0
$683$			325.686i			0.476846i	0.971161	+	0.238423i	$0.0766305\pi$
$683$			325.686i			0.476846i	−0.971161	+	0.238423i	$0.923370\pi$
$684$		0			0
$685$	251.198			0.366713
$686$		0			0
$687$	−5.74482	+	7.45485i	−0.00836218	+	0.0108513i
$688$		0			0
$689$			241.900i			0.351089i
$690$		0			0
$691$	36.1475			0.0523118			0.0261559	−	0.999658i	$-0.491673\pi$
$691$	36.1475			0.0523118			0.0261559	+	0.999658i	$0.491673\pi$
$692$		0			0
$693$	443.970	−	116.997i	0.640649	−	0.168827i
$694$		0			0
$695$		−	428.736i		−	0.616886i
$696$		0			0
$697$	150.551			0.215999
$698$		0			0
$699$	393.581	+	303.299i	0.563063	+	0.433905i
$700$		0			0
$701$			121.382i			0.173156i	0.996245	+	0.0865778i	$0.0275931\pi$
$701$			121.382i			0.173156i	−0.996245	+	0.0865778i	$0.972407\pi$
$702$		0			0
$703$	708.638			1.00802
$704$		0			0
$705$	−162.741	+	211.183i	−0.230838	+	0.299551i
$706$		0			0
$707$		−	340.272i		−	0.481290i
$708$		0			0
$709$	860.497			1.21368			0.606839	−	0.794825i	$-0.292437\pi$
$709$	860.497			1.21368			0.606839	+	0.794825i	$0.292437\pi$
$710$		0			0
$711$	−175.272	−	665.107i	−0.246515	−	0.935452i
$712$		0			0
$713$		−	214.233i		−	0.300468i
$714$		0			0
$715$	221.716			0.310093
$716$		0			0
$717$	48.6928	+	37.5234i	0.0679119	+	0.0523339i
$718$		0			0
$719$			583.081i			0.810961i	0.914104	+	0.405480i	$0.132896\pi$
$719$			583.081i			0.810961i	−0.914104	+	0.405480i	$0.867104\pi$
$720$		0			0
$721$	226.716			0.314446
$722$		0			0
$723$	−162.647	+	211.061i	−0.224961	+	0.291924i
$724$		0			0
$725$			1.48385i			0.00204669i
$726$		0			0
$727$	337.474			0.464200			0.232100	−	0.972692i	$-0.425440\pi$
$727$	337.474			0.464200			0.232100	+	0.972692i	$0.425440\pi$
$728$		0			0
$729$	509.051	+	521.832i	0.698287	+	0.715818i
$730$		0			0
$731$		−	93.5433i		−	0.127966i
$732$		0			0
$733$	403.486			0.550459			0.275229	−	0.961379i	$-0.411246\pi$
$733$	403.486			0.550459			0.275229	+	0.961379i	$0.411246\pi$
$734$		0			0
$735$	37.1947	+	28.6628i	0.0506050	+	0.0389970i
$736$		0			0
$737$		−	2239.73i		−	3.03898i
$738$		0			0
$739$	−1149.86			−1.55597			−0.777987	−	0.628281i	$-0.783759\pi$
$739$	−1149.86			−1.55597			−0.777987	+	0.628281i	$0.783759\pi$
$740$		0			0
$741$	182.828	−	237.250i	0.246732	−	0.320175i
$742$		0			0
$743$			408.388i			0.549647i	0.961495	+	0.274824i	$0.0886195\pi$
$743$			408.388i			0.549647i	−0.961495	+	0.274824i	$0.911381\pi$
$744$		0			0
$745$	80.6187			0.108213
$746$		0			0
$747$	62.0289	−	16.3462i	0.0830374	−	0.0218824i
$748$		0			0
$749$			149.226i			0.199234i
$750$		0			0
$751$	−103.547			−0.137878			−0.0689391	−	0.997621i	$-0.521961\pi$
$751$	−103.547			−0.137878			−0.0689391	+	0.997621i	$0.521961\pi$
$752$		0			0
$753$	389.030	+	299.793i	0.516641	+	0.398131i
$754$		0			0
$755$			318.016i			0.421214i
$756$		0			0
$757$	−505.410			−0.667648			−0.333824	−	0.942635i	$-0.608339\pi$
$757$	−505.410			−0.667648			−0.333824	+	0.942635i	$0.608339\pi$
$758$		0			0
$759$	−153.777	+	199.551i	−0.202605	+	0.262913i
$760$		0			0
$761$		−	246.614i		−	0.324066i	−0.986785	−	0.162033i	$-0.948195\pi$
$761$		−	246.614i		−	0.324066i	0.986785	−	0.162033i	$-0.0518050\pi$
$762$		0			0
$763$	253.143			0.331773
$764$		0			0
$765$	11.6586	+	44.2408i	0.0152399	+	0.0578311i
$766$		0			0
$767$			182.786i			0.238313i
$768$		0			0
$769$	−944.766			−1.22856			−0.614282	−	0.789086i	$-0.710554\pi$
$769$	−944.766			−1.22856			−0.614282	+	0.789086i	$0.710554\pi$
$770$		0			0
$771$	−733.267	−	565.066i	−0.951059	−	0.732900i
$772$		0			0
$773$		−	792.538i		−	1.02528i	−0.858605	−	0.512638i	$-0.828668\pi$
$773$		−	792.538i		−	1.02528i	0.858605	−	0.512638i	$-0.171332\pi$
$774$		0			0
$775$	245.947			0.317352
$776$		0			0
$777$	−176.837	+	229.475i	−0.227589	+	0.295335i
$778$		0			0
$779$			1285.72i			1.65047i
$780$		0			0
$781$	−1972.40			−2.52548
$782$		0			0
$783$	−3.11220	+	7.38372i	−0.00397471	+	0.00943004i
$784$		0			0
$785$			70.9777i			0.0904174i
$786$		0			0
$787$	−65.5806			−0.0833298			−0.0416649	−	0.999132i	$-0.513266\pi$
$787$	−65.5806			−0.0833298			−0.0416649	+	0.999132i	$0.513266\pi$
$788$		0			0
$789$	725.028	+	558.717i	0.918920	+	0.708134i
$790$		0			0
$791$			385.596i			0.487479i
$792$		0			0
$793$	329.993			0.416132
$794$		0			0
$795$	192.613	−	249.947i	0.242280	−	0.314398i
$796$		0			0
$797$			1262.46i			1.58401i	0.610512	+	0.792007i	$0.290964\pi$
$797$			1262.46i			1.58401i	−0.610512	+	0.792007i	$0.709036\pi$
$798$		0			0
$799$	−90.3548			−0.113085
$800$		0			0
$801$	−111.269	+	29.3221i	−0.138912	+	0.0366069i
$802$		0			0
$803$			1588.49i			1.97820i
$804$		0			0
$805$	−25.7661			−0.0320076
$806$		0			0
$807$	881.955	+	679.647i	1.09288	+	0.842190i
$808$		0			0
$809$		−	591.358i		−	0.730974i	−0.930816	−	0.365487i	$-0.880902\pi$
$809$		−	591.358i		−	0.730974i	0.930816	−	0.365487i	$-0.119098\pi$
$810$		0			0
$811$	249.297			0.307394			0.153697	−	0.988118i	$-0.450882\pi$
$811$	249.297			0.307394			0.153697	+	0.988118i	$0.450882\pi$
$812$		0			0
$813$	517.893	−	672.051i	0.637014	−	0.826631i
$814$		0			0
$815$			14.5123i			0.0178065i
$816$		0			0
$817$	798.866			0.977805
$818$		0			0
$819$	31.2037	+	118.409i	0.0380997	+	0.144577i
$820$		0			0
$821$			1412.27i			1.72019i	0.510137	+	0.860093i	$0.329595\pi$
$821$			1412.27i			1.72019i	−0.510137	+	0.860093i	$0.670405\pi$
$822$		0			0
$823$	12.3889			0.0150533			0.00752664	−	0.999972i	$-0.497604\pi$
$823$	12.3889			0.0150533			0.00752664	+	0.999972i	$0.497604\pi$
$824$		0			0
$825$	−229.092	−	176.541i	−0.277687	−	0.213990i
$826$		0			0
$827$			200.645i			0.242618i	0.992615	+	0.121309i	$0.0387091\pi$
$827$			200.645i			0.242618i	−0.992615	+	0.121309i	$0.961291\pi$
$828$		0			0
$829$	−762.542			−0.919833			−0.459917	−	0.887962i	$-0.652121\pi$
$829$	−762.542			−0.919833			−0.459917	+	0.887962i	$0.652121\pi$
$830$		0			0
$831$	324.550	−	421.157i	0.390553	−	0.506808i
$832$		0			0
$833$			15.9138i			0.0191042i
$834$		0			0
$835$	−136.169			−0.163077
$836$		0			0
$837$	1223.85	+	515.844i	1.46218	+	0.616301i
$838$		0			0
$839$			32.4426i			0.0386682i	0.999813	+	0.0193341i	$0.00615461\pi$
$839$			32.4426i			0.0386682i	−0.999813	+	0.0193341i	$0.993845\pi$
$840$		0			0
$841$	840.912			0.999895
$842$		0			0
$843$	130.522	+	100.582i	0.154830	+	0.119314i
$844$		0			0
$845$		−	318.763i		−	0.377234i
$846$		0			0
$847$	−663.494			−0.783346
$848$		0			0
$849$	317.136	−	411.537i	0.373541	−	0.484731i
$850$		0			0
$851$		−	158.966i		−	0.186799i
$852$		0			0
$853$	−156.786			−0.183806			−0.0919029	−	0.995768i	$-0.529295\pi$
$853$	−156.786			−0.183806			−0.0919029	+	0.995768i	$0.529295\pi$
$854$		0			0
$855$	−377.820	+	99.5649i	−0.441894	+	0.116450i
$856$		0			0
$857$		−	862.268i		−	1.00615i	−0.864244	−	0.503073i	$-0.832203\pi$
$857$		−	862.268i		−	1.00615i	0.864244	−	0.503073i	$-0.167797\pi$
$858$		0			0
$859$	−1326.28			−1.54399			−0.771993	−	0.635632i	$-0.780740\pi$
$859$	−1326.28			−1.54399			−0.771993	+	0.635632i	$0.780740\pi$
$860$		0			0
$861$	−416.348	−	320.844i	−0.483563	−	0.372641i
$862$		0			0
$863$			1214.13i			1.40687i	0.710759	+	0.703436i	$0.248352\pi$
$863$			1214.13i			1.40687i	−0.710759	+	0.703436i	$0.751648\pi$
$864$		0			0
$865$	−739.422			−0.854823
$866$		0			0
$867$	519.752	−	674.464i	0.599483	−	0.777928i
$868$		0			0
$869$			1473.57i			1.69570i
$870$		0			0
$871$	597.346			0.685816
$872$		0			0
$873$	376.878	+	1430.14i	0.431704	+	1.63819i
$874$		0			0
$875$		−	29.5804i		−	0.0338062i
$876$		0			0
$877$	−830.750			−0.947263			−0.473632	−	0.880723i	$-0.657057\pi$
$877$	−830.750			−0.947263			−0.473632	+	0.880723i	$0.657057\pi$
$878$		0			0
$879$	504.995	+	389.157i	0.574511	+	0.442727i
$880$		0			0
$881$			974.644i			1.10629i	0.833084	+	0.553146i	$0.186573\pi$
$881$			974.644i			1.10629i	−0.833084	+	0.553146i	$0.813427\pi$
$882$		0			0
$883$	−583.358			−0.660654			−0.330327	−	0.943866i	$-0.607159\pi$
$883$	−583.358			−0.660654			−0.330327	+	0.943866i	$0.607159\pi$
$884$		0			0
$885$	145.543	−	188.867i	0.164456	−	0.213409i
$886$		0			0
$887$		−	897.766i		−	1.01214i	−0.862493	−	0.506069i	$-0.831098\pi$
$887$		−	897.766i		−	1.01214i	0.862493	−	0.506069i	$-0.168902\pi$
$888$		0			0
$889$	389.786			0.438455
$890$		0			0
$891$	−769.697	−	1358.97i	−0.863857	−	1.52522i
$892$		0			0
$893$		−	771.636i		−	0.864094i
$894$		0			0
$895$	−628.533			−0.702272
$896$		0			0
$897$	−53.2212	−	41.0130i	−0.0593324	−	0.0457224i
$898$		0			0
$899$			14.5980i			0.0162380i
$900$		0			0
$901$	106.940			0.118690
$902$		0			0
$903$	−199.353	+	258.693i	−0.220767	+	0.286482i
$904$		0			0
$905$			351.438i			0.388330i
$906$		0			0
$907$	−1418.76			−1.56423			−0.782115	−	0.623134i	$-0.785859\pi$
$907$	−1418.76			−1.56423			−0.782115	+	0.623134i	$0.785859\pi$
$908$		0			0
$909$	−1119.28	+	294.959i	−1.23134	+	0.324488i
$910$		0			0
$911$			620.721i			0.681362i	0.940179	+	0.340681i	$0.110658\pi$
$911$			620.721i			0.681362i	−0.940179	+	0.340681i	$0.889342\pi$
$912$		0			0
$913$	−137.427			−0.150523
$914$		0			0
$915$	−340.970	−	262.756i	−0.372645	−	0.287165i
$916$		0			0
$917$			171.482i			0.187004i
$918$		0			0
$919$	−662.040			−0.720392			−0.360196	−	0.932877i	$-0.617290\pi$
$919$	−662.040			−0.720392			−0.360196	+	0.932877i	$0.617290\pi$
$920$		0			0
$921$	−998.174	+	1295.30i	−1.08379	+	1.40640i
$922$		0			0
$923$		−	526.047i		−	0.569932i
$924$		0			0
$925$	182.498			0.197295
$926$		0			0
$927$	−196.525	−	745.754i	−0.212001	−	0.804481i
$928$		0			0
$929$			3.51829i			0.00378717i	0.999998	+	0.00189359i	$0.000602748\pi$
$929$			3.51829i			0.00378717i	−0.999998	+	0.00189359i	$0.999397\pi$
$930$		0			0
$931$	−135.905			−0.145977
$932$		0			0
$933$	−861.703	−	664.041i	−0.923583	−	0.711727i
$934$		0			0
$935$		−	98.0170i		−	0.104831i
$936$		0			0
$937$	−893.777			−0.953871			−0.476935	−	0.878938i	$-0.658252\pi$
$937$	−893.777			−0.953871			−0.476935	+	0.878938i	$0.658252\pi$
$938$		0			0
$939$	−301.355	+	391.058i	−0.320932	+	0.416462i
$940$		0			0
$941$		−	824.629i		−	0.876332i	−0.898894	−	0.438166i	$-0.855628\pi$
$941$		−	824.629i		−	0.876332i	0.898894	−	0.438166i	$-0.144372\pi$
$942$		0			0
$943$	288.419			0.305853
$944$		0			0
$945$	62.0412	−	147.193i	0.0656520	−	0.155760i
$946$		0			0
$947$			974.692i			1.02924i	0.857418	+	0.514621i	$0.172067\pi$
$947$			974.692i			1.02924i	−0.857418	+	0.514621i	$0.827933\pi$
$948$		0			0
$949$	−423.658			−0.446425
$950$		0			0
$951$	−936.764	−	721.884i	−0.985031	−	0.759079i
$952$		0			0
$953$		−	264.699i		−	0.277753i	−0.990310	−	0.138877i	$-0.955651\pi$
$953$		−	264.699i		−	0.277753i	0.990310	−	0.138877i	$-0.0443492\pi$
$954$		0			0
$955$	440.995			0.461774
$956$		0			0
$957$	10.4785	−	13.5975i	0.0109493	−	0.0142085i
$958$		0			0
$959$		−	297.222i		−	0.309929i
$960$		0			0
$961$	1458.61			1.51780
$962$		0			0
$963$	490.861	−	129.354i	0.509721	−	0.134324i
$964$		0			0
$965$			369.371i			0.382768i
$966$		0			0
$967$	−365.750			−0.378232			−0.189116	−	0.981955i	$-0.560562\pi$
$967$	−365.750			−0.378232			−0.189116	+	0.981955i	$0.560562\pi$
$968$		0			0
$969$	−104.884	−	80.8251i	−0.108239	−	0.0834109i
$970$		0			0
$971$			401.640i			0.413635i	0.978380	+	0.206818i	$0.0663107\pi$
$971$			401.640i			0.413635i	−0.978380	+	0.206818i	$0.933689\pi$
$972$		0			0
$973$	−507.287			−0.521364
$974$		0			0
$975$	47.0844	−	61.0998i	0.0482917	−	0.0626664i
$976$		0			0
$977$		−	1202.29i		−	1.23060i	−0.788294	−	0.615299i	$-0.789035\pi$
$977$		−	1202.29i		−	1.23060i	0.788294	−	0.615299i	$-0.210965\pi$
$978$		0			0
$979$	246.520			0.251808
$980$		0			0
$981$	−219.433	−	832.683i	−0.223683	−	0.848810i
$982$		0			0
$983$		−	1503.26i		−	1.52926i	−0.644469	−	0.764630i	$-0.722922\pi$
$983$		−	1503.26i		−	1.52926i	0.644469	−	0.764630i	$-0.277078\pi$
$984$		0			0
$985$	−202.015			−0.205091
$986$		0			0
$987$	249.875	+	192.558i	0.253166	+	0.195094i
$988$		0			0
$989$		−	179.206i		−	0.181199i
$990$		0			0
$991$	1172.96			1.18361			0.591805	−	0.806081i	$-0.298415\pi$
$991$	1172.96			1.18361			0.591805	+	0.806081i	$0.298415\pi$
$992$		0			0
$993$	−1151.50	+	1494.26i	−1.15962	+	1.50479i
$994$		0			0
$995$			21.0027i			0.0211083i
$996$		0			0
$997$	−376.288			−0.377421			−0.188710	−	0.982033i	$-0.560431\pi$
$997$	−376.288			−0.377421			−0.188710	+	0.982033i	$0.560431\pi$
$998$		0			0
$999$	908.119	+	382.767i	0.909028	+	0.383150i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.l.b.1121.11		16
3.2	odd	2	inner	1680.3.l.b.1121.12		16
4.3	odd	2		420.3.g.a.281.6	yes	16
12.11	even	2		420.3.g.a.281.5	✓	16
20.3	even	4		2100.3.e.c.449.9		32
20.7	even	4		2100.3.e.c.449.24		32
20.19	odd	2		2100.3.g.d.701.11		16
60.23	odd	4		2100.3.e.c.449.23		32
60.47	odd	4		2100.3.e.c.449.10		32
60.59	even	2		2100.3.g.d.701.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.g.a.281.5	✓	16	12.11	even	2
420.3.g.a.281.6	yes	16	4.3	odd	2
1680.3.l.b.1121.11		16	1.1	even	1	trivial
1680.3.l.b.1121.12		16	3.2	odd	2	inner
2100.3.e.c.449.9		32	20.3	even	4
2100.3.e.c.449.10		32	60.47	odd	4
2100.3.e.c.449.23		32	60.23	odd	4
2100.3.e.c.449.24		32	20.7	even	4
2100.3.g.d.701.11		16	20.19	odd	2
2100.3.g.d.701.12		16	60.59	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1680.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.83120 - 2.37628i) q^{3} +2.23607i q^{5} +2.64575 q^{7} +(-2.29343 - 8.70288i) q^{9} +19.2815i q^{11} -5.14247 q^{13} +(5.31353 + 4.09468i) q^{15} +2.27339i q^{17} -19.4150 q^{19} +(4.84489 - 6.28705i) q^{21} +4.35527i q^{23} -5.00000 q^{25} +(-24.8802 - 10.4869i) q^{27} -0.296771i q^{29} -49.1895 q^{31} +(45.8183 + 35.3083i) q^{33} +5.91608i q^{35} -36.4996 q^{37} +(-9.41688 + 12.2200i) q^{39} -66.2231i q^{41} -41.1470 q^{43} +(19.4602 - 5.12826i) q^{45} +39.7444i q^{47} +7.00000 q^{49} +(5.40223 + 4.16304i) q^{51} -47.0397i q^{53} -43.1148 q^{55} +(-35.5526 + 46.1354i) q^{57} -35.5445i q^{59} -64.1701 q^{61} +(-6.06784 - 23.0257i) q^{63} -11.4989i q^{65} -116.159 q^{67} +(10.3493 + 7.97536i) q^{69} +102.295i q^{71} +82.3841 q^{73} +(-9.15599 + 11.8814i) q^{75} +51.0141i q^{77} +76.4237 q^{79} +(-70.4804 + 39.9189i) q^{81} +7.12740i q^{83} -5.08346 q^{85} +(-0.705211 - 0.543446i) q^{87} -12.7853i q^{89} -13.6057 q^{91} +(-90.0757 + 116.888i) q^{93} -43.4132i q^{95} -164.329 q^{97} +(167.805 - 44.2208i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 10 q^{9} - 10 q^{15} - 48 q^{19} - 14 q^{21} - 80 q^{25} - 28 q^{27} - 24 q^{31} + 92 q^{33} - 40 q^{37} + 54 q^{39} + 168 q^{43} + 40 q^{45} + 112 q^{49} + 186 q^{51} + 80 q^{55} + 252 q^{57}+ \cdots - 26 q^{99}+O(q^{100}) \) 16 * q + 8 * q^3 - 10 * q^9 - 10 * q^15 - 48 * q^19 - 14 * q^21 - 80 * q^25 - 28 * q^27 - 24 * q^31 + 92 * q^33 - 40 * q^37 + 54 * q^39 + 168 * q^43 + 40 * q^45 + 112 * q^49 + 186 * q^51 + 80 * q^55 + 252 * q^57 - 152 * q^61 + 56 * q^63 - 264 * q^67 - 196 * q^69 + 240 * q^73 - 40 * q^75 - 212 * q^79 - 118 * q^81 - 60 * q^85 - 60 * q^87 - 196 * q^91 - 244 * q^93 - 360 * q^97 - 26 * q^99

Introduction

L-functions

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