LMFDB - Embedded newform 1680.2.f.g.881.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(881,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.881"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [4,0,-1,0,4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 2x^{2} - 3x + 9$ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.3
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.g.881.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.18614 - 1.26217i) q^{3} +1.00000 q^{5} +(2.00000 - 1.73205i) q^{7} +(-0.186141 - 2.99422i) q^{9} -2.52434i q^{11} -4.10891i q^{13} +(1.18614 - 1.26217i) q^{15} +4.37228 q^{17} +3.46410i q^{19} +(0.186141 - 4.57879i) q^{21} +8.51278i q^{23} +1.00000 q^{25} +(-4.00000 - 3.31662i) q^{27} -0.939764i q^{29} -3.46410i q^{31} +(-3.18614 - 2.99422i) q^{33} +(2.00000 - 1.73205i) q^{35} -6.74456 q^{37} +(-5.18614 - 4.87375i) q^{39} -6.00000 q^{41} -4.74456 q^{43} +(-0.186141 - 2.99422i) q^{45} +1.62772 q^{47} +(1.00000 - 6.92820i) q^{49} +(5.18614 - 5.51856i) q^{51} +1.87953i q^{53} -2.52434i q^{55} +(4.37228 + 4.10891i) q^{57} +8.74456 q^{59} +6.92820i q^{61} +(-5.55842 - 5.66603i) q^{63} -4.10891i q^{65} -4.74456 q^{67} +(10.7446 + 10.0974i) q^{69} +0.294954i q^{71} +6.92820i q^{73} +(1.18614 - 1.26217i) q^{75} +(-4.37228 - 5.04868i) q^{77} +2.37228 q^{79} +(-8.93070 + 1.11469i) q^{81} +17.4891 q^{83} +4.37228 q^{85} +(-1.18614 - 1.11469i) q^{87} +14.7446 q^{89} +(-7.11684 - 8.21782i) q^{91} +(-4.37228 - 4.10891i) q^{93} +3.46410i q^{95} -11.0371i q^{97} +(-7.55842 + 0.469882i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - q^{3} + 4 q^{5} + 8 q^{7} + 5 q^{9} - q^{15} + 6 q^{17} - 5 q^{21} + 4 q^{25} - 16 q^{27} - 7 q^{33} + 8 q^{35} - 4 q^{37} - 15 q^{39} - 24 q^{41} + 4 q^{43} + 5 q^{45} + 18 q^{47} + 4 q^{49} + 15 q^{51}+ \cdots - 13 q^{99}+O(q^{100})$ 4 * q - q^3 + 4 * q^5 + 8 * q^7 + 5 * q^9 - q^15 + 6 * q^17 - 5 * q^21 + 4 * q^25 - 16 * q^27 - 7 * q^33 + 8 * q^35 - 4 * q^37 - 15 * q^39 - 24 * q^41 + 4 * q^43 + 5 * q^45 + 18 * q^47 + 4 * q^49 + 15 * q^51 + 6 * q^57 + 12 * q^59 - 5 * q^63 + 4 * q^67 + 20 * q^69 - q^75 - 6 * q^77 - 2 * q^79 - 7 * q^81 + 24 * q^83 + 6 * q^85 + q^87 + 36 * q^89 + 6 * q^91 - 6 * q^93 - 13 * 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
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$3$	1.18614	−	1.26217i	0.684819	−	0.728714i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	−0.186141	−	2.99422i	−0.0620469	−	0.998073i
$10$		0			0
$11$		−	2.52434i		−	0.761116i	−0.924757	−	0.380558i	$-0.875732\pi$
$11$		−	2.52434i		−	0.761116i	0.924757	−	0.380558i	$-0.124268\pi$
$12$		0			0
$13$		−	4.10891i		−	1.13961i	−0.821781	−	0.569804i	$-0.807019\pi$
$13$		−	4.10891i		−	1.13961i	0.821781	−	0.569804i	$-0.192981\pi$
$14$		0			0
$15$	1.18614	−	1.26217i	0.306260	−	0.325891i
$16$		0			0
$17$	4.37228			1.06043			0.530217	−	0.847862i	$-0.322110\pi$
$17$	4.37228			1.06043			0.530217	+	0.847862i	$0.322110\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	0.186141	−	4.57879i	0.0406192	−	0.999175i
$22$		0			0
$23$			8.51278i			1.77504i	0.460772	+	0.887518i	$0.347572\pi$
$23$			8.51278i			1.77504i	−0.460772	+	0.887518i	$0.652428\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−4.00000	−	3.31662i	−0.769800	−	0.638285i
$28$		0			0
$29$		−	0.939764i		−	0.174510i	−0.996186	−	0.0872549i	$-0.972191\pi$
$29$		−	0.939764i		−	0.174510i	0.996186	−	0.0872549i	$-0.0278095\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$	−3.18614	−	2.99422i	−0.554636	−	0.521227i
$34$		0			0
$35$	2.00000	−	1.73205i	0.338062	−	0.292770i
$36$		0			0
$37$	−6.74456			−1.10880			−0.554400	−	0.832251i	$-0.687052\pi$
$37$	−6.74456			−1.10880			−0.554400	+	0.832251i	$0.687052\pi$
$38$		0			0
$39$	−5.18614	−	4.87375i	−0.830447	−	0.780424i
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	−4.74456			−0.723539			−0.361770	−	0.932268i	$-0.617827\pi$
$43$	−4.74456			−0.723539			−0.361770	+	0.932268i	$0.617827\pi$
$44$		0			0
$45$	−0.186141	−	2.99422i	−0.0277482	−	0.446352i
$46$		0			0
$47$	1.62772			0.237427			0.118714	−	0.992929i	$-0.462123\pi$
$47$	1.62772			0.237427			0.118714	+	0.992929i	$0.462123\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	5.18614	−	5.51856i	0.726205	−	0.772753i
$52$		0			0
$53$			1.87953i			0.258173i	0.991633	+	0.129086i	$0.0412045\pi$
$53$			1.87953i			0.258173i	−0.991633	+	0.129086i	$0.958796\pi$
$54$		0			0
$55$		−	2.52434i		−	0.340382i
$56$		0			0
$57$	4.37228	+	4.10891i	0.579123	+	0.544239i
$58$		0			0
$59$	8.74456			1.13845			0.569223	−	0.822183i	$-0.307244\pi$
$59$	8.74456			1.13845			0.569223	+	0.822183i	$0.307244\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−5.55842	−	5.66603i	−0.700295	−	0.713853i
$64$		0			0
$65$		−	4.10891i		−	0.509648i
$66$		0			0
$67$	−4.74456			−0.579641			−0.289820	−	0.957081i	$-0.593596\pi$
$67$	−4.74456			−0.579641			−0.289820	+	0.957081i	$0.593596\pi$
$68$		0			0
$69$	10.7446	+	10.0974i	1.29349	+	1.21558i
$70$		0			0
$71$			0.294954i			0.0350046i	0.999847	+	0.0175023i	$0.00557143\pi$
$71$			0.294954i			0.0350046i	−0.999847	+	0.0175023i	$0.994429\pi$
$72$		0			0
$73$			6.92820i			0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$			6.92820i			0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$		0			0
$75$	1.18614	−	1.26217i	0.136964	−	0.145743i
$76$		0			0
$77$	−4.37228	−	5.04868i	−0.498268	−	0.575350i
$78$		0			0
$79$	2.37228			0.266903			0.133451	−	0.991055i	$-0.457394\pi$
$79$	2.37228			0.266903			0.133451	+	0.991055i	$0.457394\pi$
$80$		0			0
$81$	−8.93070	+	1.11469i	−0.992300	+	0.123855i
$82$		0			0
$83$	17.4891			1.91968			0.959840	−	0.280546i	$-0.0905157\pi$
$83$	17.4891			1.91968			0.959840	+	0.280546i	$0.0905157\pi$
$84$		0			0
$85$	4.37228			0.474240
$86$		0			0
$87$	−1.18614	−	1.11469i	−0.127168	−	0.119508i
$88$		0			0
$89$	14.7446			1.56292			0.781460	−	0.623955i	$-0.214475\pi$
$89$	14.7446			1.56292			0.781460	+	0.623955i	$0.214475\pi$
$90$		0			0
$91$	−7.11684	−	8.21782i	−0.746048	−	0.861462i
$92$		0			0
$93$	−4.37228	−	4.10891i	−0.453384	−	0.426074i
$94$		0			0
$95$			3.46410i			0.355409i
$96$		0			0
$97$		−	11.0371i		−	1.12065i	−0.828273	−	0.560325i	$-0.810676\pi$
$97$		−	11.0371i		−	1.12065i	0.828273	−	0.560325i	$-0.189324\pi$
$98$		0			0
$99$	−7.55842	+	0.469882i	−0.759650	+	0.0472249i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$			6.28339i			0.619121i	0.950880	+	0.309561i	$0.100182\pi$
$103$			6.28339i			0.619121i	−0.950880	+	0.309561i	$0.899818\pi$
$104$		0			0
$105$	0.186141	−	4.57879i	0.0181655	−	0.446845i
$106$		0			0
$107$		−	6.63325i		−	0.641260i	−0.947204	−	0.320630i	$-0.896105\pi$
$107$		−	6.63325i		−	0.641260i	0.947204	−	0.320630i	$-0.103895\pi$
$108$		0			0
$109$	−17.1168			−1.63950			−0.819748	−	0.572724i	$-0.805887\pi$
$109$	−17.1168			−1.63950			−0.819748	+	0.572724i	$0.805887\pi$
$110$		0			0
$111$	−8.00000	+	8.51278i	−0.759326	+	0.807997i
$112$		0			0
$113$			3.16915i			0.298128i	0.988828	+	0.149064i	$0.0476261\pi$
$113$			3.16915i			0.298128i	−0.988828	+	0.149064i	$0.952374\pi$
$114$		0			0
$115$			8.51278i			0.793821i
$116$		0			0
$117$	−12.3030	+	0.764836i	−1.13741	+	0.0707091i
$118$		0			0
$119$	8.74456	−	7.57301i	0.801613	−	0.694217i
$120$		0			0
$121$	4.62772			0.420702
$122$		0			0
$123$	−7.11684	+	7.57301i	−0.641704	+	0.682836i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	0.744563			0.0660693			0.0330346	−	0.999454i	$-0.489483\pi$
$127$	0.744563			0.0660693			0.0330346	+	0.999454i	$0.489483\pi$
$128$		0			0
$129$	−5.62772	+	5.98844i	−0.495493	+	0.527253i
$130$		0			0
$131$	−5.48913			−0.479587			−0.239794	−	0.970824i	$-0.577080\pi$
$131$	−5.48913			−0.479587			−0.239794	+	0.970824i	$0.577080\pi$
$132$		0			0
$133$	6.00000	+	6.92820i	0.520266	+	0.600751i
$134$		0			0
$135$	−4.00000	−	3.31662i	−0.344265	−	0.285450i
$136$		0			0
$137$		−	13.2665i		−	1.13343i	−0.823913	−	0.566717i	$-0.808213\pi$
$137$		−	13.2665i		−	1.13343i	0.823913	−	0.566717i	$-0.191787\pi$
$138$		0			0
$139$		−	18.6101i		−	1.57849i	−0.614078	−	0.789245i	$-0.710472\pi$
$139$		−	18.6101i		−	1.57849i	0.614078	−	0.789245i	$-0.289528\pi$
$140$		0			0
$141$	1.93070	−	2.05446i	0.162595	−	0.173016i
$142$		0			0
$143$	−10.3723			−0.867374
$144$		0			0
$145$		−	0.939764i		−	0.0780431i
$146$		0			0
$147$	−7.55842	−	9.47999i	−0.623408	−	0.781897i
$148$		0			0
$149$			3.16915i			0.259627i	0.991538	+	0.129813i	$0.0414378\pi$
$149$			3.16915i			0.259627i	−0.991538	+	0.129813i	$0.958562\pi$
$150$		0			0
$151$	2.37228			0.193054			0.0965268	−	0.995330i	$-0.469227\pi$
$151$	2.37228			0.193054			0.0965268	+	0.995330i	$0.469227\pi$
$152$		0			0
$153$	−0.813859	−	13.0916i	−0.0657966	−	1.05839i
$154$		0			0
$155$		−	3.46410i		−	0.278243i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	2.37228	+	2.22938i	0.188134	+	0.176802i
$160$		0			0
$161$	14.7446	+	17.0256i	1.16203	+	1.34180i
$162$		0			0
$163$	−8.00000			−0.626608			−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000			−0.626608			−0.313304	+	0.949653i	$0.601436\pi$
$164$		0			0
$165$	−3.18614	−	2.99422i	−0.248041	−	0.233100i
$166$		0			0
$167$	4.88316			0.377870			0.188935	−	0.981990i	$-0.439496\pi$
$167$	4.88316			0.377870			0.188935	+	0.981990i	$0.439496\pi$
$168$		0			0
$169$	−3.88316			−0.298704
$170$		0			0
$171$	10.3723	−	0.644810i	0.793188	−	0.0493099i
$172$		0			0
$173$	−1.11684			−0.0849121			−0.0424560	−	0.999098i	$-0.513518\pi$
$173$	−1.11684			−0.0849121			−0.0424560	+	0.999098i	$0.513518\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$	10.3723	−	11.0371i	0.779628	−	0.829600i
$178$		0			0
$179$		−	6.63325i		−	0.495792i	−0.968787	−	0.247896i	$-0.920261\pi$
$179$		−	6.63325i		−	0.495792i	0.968787	−	0.247896i	$-0.0797392\pi$
$180$		0			0
$181$			1.28962i			0.0958567i	0.998851	+	0.0479284i	$0.0152619\pi$
$181$			1.28962i			0.0958567i	−0.998851	+	0.0479284i	$0.984738\pi$
$182$		0			0
$183$	8.74456	+	8.21782i	0.646417	+	0.607479i
$184$		0			0
$185$	−6.74456			−0.495870
$186$		0			0
$187$		−	11.0371i		−	0.807114i
$188$		0			0
$189$	−13.7446	+	0.294954i	−0.999770	+	0.0214547i
$190$		0			0
$191$			20.8395i			1.50789i	0.656935	+	0.753947i	$0.271853\pi$
$191$			20.8395i			1.50789i	−0.656935	+	0.753947i	$0.728147\pi$
$192$		0			0
$193$	−12.2337			−0.880600			−0.440300	−	0.897851i	$-0.645128\pi$
$193$	−12.2337			−0.880600			−0.440300	+	0.897851i	$0.645128\pi$
$194$		0			0
$195$	−5.18614	−	4.87375i	−0.371387	−	0.349016i
$196$		0			0
$197$			15.7359i			1.12114i	0.828107	+	0.560569i	$0.189418\pi$
$197$			15.7359i			1.12114i	−0.828107	+	0.560569i	$0.810582\pi$
$198$		0			0
$199$			26.8280i			1.90178i	0.309524	+	0.950892i	$0.399830\pi$
$199$			26.8280i			1.90178i	−0.309524	+	0.950892i	$0.600170\pi$
$200$		0			0
$201$	−5.62772	+	5.98844i	−0.396949	+	0.422392i
$202$		0			0
$203$	−1.62772	−	1.87953i	−0.114243	−	0.131917i
$204$		0			0
$205$	−6.00000			−0.419058
$206$		0			0
$207$	25.4891	−	1.58457i	1.77162	−	0.110136i
$208$		0			0
$209$	8.74456			0.604874
$210$		0			0
$211$	11.1168			0.765315			0.382658	−	0.923890i	$-0.375009\pi$
$211$	11.1168			0.765315			0.382658	+	0.923890i	$0.375009\pi$
$212$		0			0
$213$	0.372281	+	0.349857i	0.0255083	+	0.0239718i
$214$		0			0
$215$	−4.74456			−0.323576
$216$		0			0
$217$	−6.00000	−	6.92820i	−0.407307	−	0.470317i
$218$		0			0
$219$	8.74456	+	8.21782i	0.590903	+	0.555309i
$220$		0			0
$221$		−	17.9653i		−	1.20848i
$222$		0			0
$223$			8.86263i			0.593486i	0.954957	+	0.296743i	$0.0959004\pi$
$223$			8.86263i			0.593486i	−0.954957	+	0.296743i	$0.904100\pi$
$224$		0			0
$225$	−0.186141	−	2.99422i	−0.0124094	−	0.199615i
$226$		0			0
$227$	−24.6060			−1.63316			−0.816578	−	0.577235i	$-0.804131\pi$
$227$	−24.6060			−1.63316			−0.816578	+	0.577235i	$0.804131\pi$
$228$		0			0
$229$			15.1460i			1.00088i	0.865772	+	0.500439i	$0.166828\pi$
$229$			15.1460i			1.00088i	−0.865772	+	0.500439i	$0.833172\pi$
$230$		0			0
$231$	−11.5584	−	0.469882i	−0.760488	−	0.0309160i
$232$		0			0
$233$			17.0256i			1.11538i	0.830049	+	0.557691i	$0.188312\pi$
$233$			17.0256i			1.11538i	−0.830049	+	0.557691i	$0.811688\pi$
$234$		0			0
$235$	1.62772			0.106181
$236$		0			0
$237$	2.81386	−	2.99422i	0.182780	−	0.194495i
$238$		0			0
$239$			20.8395i			1.34800i	0.738733	+	0.673998i	$0.235424\pi$
$239$			20.8395i			1.34800i	−0.738733	+	0.673998i	$0.764576\pi$
$240$		0			0
$241$		−	16.4356i		−	1.05871i	−0.848399	−	0.529357i	$-0.822433\pi$
$241$		−	16.4356i		−	1.05871i	0.848399	−	0.529357i	$-0.177567\pi$
$242$		0			0
$243$	−9.18614	+	12.5942i	−0.589291	+	0.807921i
$244$		0			0
$245$	1.00000	−	6.92820i	0.0638877	−	0.442627i
$246$		0			0
$247$	14.2337			0.905668
$248$		0			0
$249$	20.7446	−	22.0742i	1.31463	−	1.39890i
$250$		0			0
$251$	5.48913			0.346471			0.173235	−	0.984880i	$-0.444578\pi$
$251$	5.48913			0.346471			0.173235	+	0.984880i	$0.444578\pi$
$252$		0			0
$253$	21.4891			1.35101
$254$		0			0
$255$	5.18614	−	5.51856i	0.324769	−	0.345585i
$256$		0			0
$257$	0.510875			0.0318675			0.0159337	−	0.999873i	$-0.494928\pi$
$257$	0.510875			0.0318675			0.0159337	+	0.999873i	$0.494928\pi$
$258$		0			0
$259$	−13.4891	+	11.6819i	−0.838173	+	0.725880i
$260$		0			0
$261$	−2.81386	+	0.174928i	−0.174174	+	0.0108278i
$262$		0			0
$263$			0.294954i			0.0181876i	0.999959	+	0.00909381i	$0.00289469\pi$
$263$			0.294954i			0.0181876i	−0.999959	+	0.00909381i	$0.997105\pi$
$264$		0			0
$265$			1.87953i			0.115458i
$266$		0			0
$267$	17.4891	−	18.6101i	1.07032	−	1.13892i
$268$		0			0
$269$	2.74456			0.167339			0.0836695	−	0.996494i	$-0.473336\pi$
$269$	2.74456			0.167339			0.0836695	+	0.996494i	$0.473336\pi$
$270$		0			0
$271$			4.75372i			0.288768i	0.989522	+	0.144384i	$0.0461201\pi$
$271$			4.75372i			0.288768i	−0.989522	+	0.144384i	$0.953880\pi$
$272$		0			0
$273$	−18.8139	−	0.764836i	−1.13867	−	0.0462900i
$274$		0			0
$275$		−	2.52434i		−	0.152223i
$276$		0			0
$277$	28.2337			1.69640			0.848199	−	0.529678i	$-0.177687\pi$
$277$	28.2337			1.69640			0.848199	+	0.529678i	$0.177687\pi$
$278$		0			0
$279$	−10.3723	+	0.644810i	−0.620972	+	0.0386038i
$280$		0			0
$281$			28.0627i			1.67408i	0.547143	+	0.837039i	$0.315715\pi$
$281$			28.0627i			1.67408i	−0.547143	+	0.837039i	$0.684285\pi$
$282$		0			0
$283$		−	0.644810i		−	0.0383300i	−0.999816	−	0.0191650i	$-0.993899\pi$
$283$		−	0.644810i		−	0.0383300i	0.999816	−	0.0191650i	$-0.00610078\pi$
$284$		0			0
$285$	4.37228	+	4.10891i	0.258992	+	0.243391i
$286$		0			0
$287$	−12.0000	+	10.3923i	−0.708338	+	0.613438i
$288$		0			0
$289$	2.11684			0.124520
$290$		0			0
$291$	−13.9307	−	13.0916i	−0.816632	−	0.767441i
$292$		0			0
$293$	−10.8832			−0.635801			−0.317900	−	0.948124i	$-0.602978\pi$
$293$	−10.8832			−0.635801			−0.317900	+	0.948124i	$0.602978\pi$
$294$		0			0
$295$	8.74456			0.509128
$296$		0			0
$297$	−8.37228	+	10.0974i	−0.485809	+	0.585908i
$298$		0			0
$299$	34.9783			2.02284
$300$		0			0
$301$	−9.48913	+	8.21782i	−0.546944	+	0.473667i
$302$		0			0
$303$	7.11684	−	7.57301i	0.408852	−	0.435058i
$304$		0			0
$305$			6.92820i			0.396708i
$306$		0			0
$307$		−	22.7190i		−	1.29664i	−0.761366	−	0.648322i	$-0.775471\pi$
$307$		−	22.7190i		−	1.29664i	0.761366	−	0.648322i	$-0.224529\pi$
$308$		0			0
$309$	7.93070	+	7.45299i	0.451162	+	0.423986i
$310$		0			0
$311$	14.2337			0.807118			0.403559	−	0.914954i	$-0.367773\pi$
$311$	14.2337			0.807118			0.403559	+	0.914954i	$0.367773\pi$
$312$		0			0
$313$			5.39853i			0.305143i	0.988292	+	0.152572i	$0.0487554\pi$
$313$			5.39853i			0.305143i	−0.988292	+	0.152572i	$0.951245\pi$
$314$		0			0
$315$	−5.55842	−	5.66603i	−0.313182	−	0.319245i
$316$		0			0
$317$			1.87953i			0.105565i	0.998606	+	0.0527824i	$0.0168090\pi$
$317$			1.87953i			0.105565i	−0.998606	+	0.0527824i	$0.983191\pi$
$318$		0			0
$319$	−2.37228			−0.132822
$320$		0			0
$321$	−8.37228	−	7.86797i	−0.467295	−	0.439147i
$322$		0			0
$323$			15.1460i			0.842747i
$324$		0			0
$325$		−	4.10891i		−	0.227921i
$326$		0			0
$327$	−20.3030	+	21.6043i	−1.12276	+	1.19472i
$328$		0			0
$329$	3.25544	−	2.81929i	0.179478	−	0.155433i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$	1.25544	+	20.1947i	0.0687975	+	1.10666i
$334$		0			0
$335$	−4.74456			−0.259223
$336$		0			0
$337$	16.2337			0.884305			0.442153	−	0.896940i	$-0.354215\pi$
$337$	16.2337			0.884305			0.442153	+	0.896940i	$0.354215\pi$
$338$		0			0
$339$	4.00000	+	3.75906i	0.217250	+	0.204164i
$340$		0			0
$341$	−8.74456			−0.473545
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	10.7446	+	10.0974i	0.578468	+	0.543623i
$346$		0			0
$347$		−	14.8511i		−	0.797247i	−0.917115	−	0.398624i	$-0.869488\pi$
$347$		−	14.8511i		−	0.797247i	0.917115	−	0.398624i	$-0.130512\pi$
$348$		0			0
$349$		−	15.1460i		−	0.810748i	−0.914151	−	0.405374i	$-0.867141\pi$
$349$		−	15.1460i		−	0.810748i	0.914151	−	0.405374i	$-0.132859\pi$
$350$		0			0
$351$	−13.6277	+	16.4356i	−0.727394	+	0.877270i
$352$		0			0
$353$	25.1168			1.33683			0.668417	−	0.743786i	$-0.266972\pi$
$353$	25.1168			1.33683			0.668417	+	0.743786i	$0.266972\pi$
$354$		0			0
$355$			0.294954i			0.0156545i
$356$		0			0
$357$	0.813859	−	20.0198i	0.0430740	−	1.05956i
$358$		0			0
$359$			16.7306i			0.883007i	0.897259	+	0.441504i	$0.145555\pi$
$359$			16.7306i			0.883007i	−0.897259	+	0.441504i	$0.854445\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	5.48913	−	5.84096i	0.288104	−	0.306571i
$364$		0			0
$365$			6.92820i			0.362639i
$366$		0			0
$367$		−	35.2858i		−	1.84191i	−0.389675	−	0.920953i	$-0.627413\pi$
$367$		−	35.2858i		−	1.84191i	0.389675	−	0.920953i	$-0.372587\pi$
$368$		0			0
$369$	1.11684	+	17.9653i	0.0581406	+	0.935237i
$370$		0			0
$371$	3.25544	+	3.75906i	0.169014	+	0.195160i
$372$		0			0
$373$	−13.2554			−0.686341			−0.343170	−	0.939273i	$-0.611501\pi$
$373$	−13.2554			−0.686341			−0.343170	+	0.939273i	$0.611501\pi$
$374$		0			0
$375$	1.18614	−	1.26217i	0.0612520	−	0.0651781i
$376$		0			0
$377$	−3.86141			−0.198873
$378$		0			0
$379$	21.4891			1.10382			0.551911	−	0.833903i	$-0.313899\pi$
$379$	21.4891			1.10382			0.551911	+	0.833903i	$0.313899\pi$
$380$		0			0
$381$	0.883156	−	0.939764i	0.0452455	−	0.0481456i
$382$		0			0
$383$	−5.48913			−0.280481			−0.140241	−	0.990117i	$-0.544788\pi$
$383$	−5.48913			−0.280481			−0.140241	+	0.990117i	$0.544788\pi$
$384$		0			0
$385$	−4.37228	−	5.04868i	−0.222832	−	0.257304i
$386$		0			0
$387$	0.883156	+	14.2063i	0.0448933	+	0.722145i
$388$		0			0
$389$			29.3523i			1.48822i	0.668057	+	0.744110i	$0.267126\pi$
$389$			29.3523i			1.48822i	−0.668057	+	0.744110i	$0.732874\pi$
$390$		0			0
$391$			37.2203i			1.88231i
$392$		0			0
$393$	−6.51087	+	6.92820i	−0.328430	+	0.349482i
$394$		0			0
$395$	2.37228			0.119362
$396$		0			0
$397$		−	26.1831i		−	1.31409i	−0.753850	−	0.657047i	$-0.771805\pi$
$397$		−	26.1831i		−	1.31409i	0.753850	−	0.657047i	$-0.228195\pi$
$398$		0			0
$399$	15.8614	+	0.644810i	0.794064	+	0.0322809i
$400$		0			0
$401$			5.98844i			0.299048i	0.988758	+	0.149524i	$0.0477742\pi$
$401$			5.98844i			0.299048i	−0.988758	+	0.149524i	$0.952226\pi$
$402$		0			0
$403$	−14.2337			−0.709030
$404$		0			0
$405$	−8.93070	+	1.11469i	−0.443770	+	0.0553895i
$406$		0			0
$407$			17.0256i			0.843925i
$408$		0			0
$409$			24.6535i			1.21904i	0.792772	+	0.609518i	$0.208637\pi$
$409$			24.6535i			1.21904i	−0.792772	+	0.609518i	$0.791363\pi$
$410$		0			0
$411$	−16.7446	−	15.7359i	−0.825948	−	0.776196i
$412$		0			0
$413$	17.4891	−	15.1460i	0.860584	−	0.745287i
$414$		0			0
$415$	17.4891			0.858507
$416$		0			0
$417$	−23.4891	−	22.0742i	−1.15027	−	1.08098i
$418$		0			0
$419$	−8.74456			−0.427200			−0.213600	−	0.976921i	$-0.568519\pi$
$419$	−8.74456			−0.427200			−0.213600	+	0.976921i	$0.568519\pi$
$420$		0			0
$421$	14.6060			0.711851			0.355926	−	0.934514i	$-0.384166\pi$
$421$	14.6060			0.711851			0.355926	+	0.934514i	$0.384166\pi$
$422$		0			0
$423$	−0.302985	−	4.87375i	−0.0147316	−	0.236970i
$424$		0			0
$425$	4.37228			0.212087
$426$		0			0
$427$	12.0000	+	13.8564i	0.580721	+	0.670559i
$428$		0			0
$429$	−12.3030	+	13.0916i	−0.593994	+	0.632067i
$430$		0			0
$431$		−	15.0911i		−	0.726914i	−0.931611	−	0.363457i	$-0.881596\pi$
$431$		−	15.0911i		−	0.726914i	0.931611	−	0.363457i	$-0.118404\pi$
$432$		0			0
$433$			37.2203i			1.78869i	0.447377	+	0.894346i	$0.352358\pi$
$433$			37.2203i			1.78869i	−0.447377	+	0.894346i	$0.647642\pi$
$434$		0			0
$435$	−1.18614	−	1.11469i	−0.0568711	−	0.0534454i
$436$		0			0
$437$	−29.4891			−1.41066
$438$		0			0
$439$			18.6101i			0.888213i	0.895974	+	0.444106i	$0.146479\pi$
$439$			18.6101i			0.888213i	−0.895974	+	0.444106i	$0.853521\pi$
$440$		0			0
$441$	−20.9307	−	1.70460i	−0.996700	−	0.0811714i
$442$		0			0
$443$		−	6.63325i		−	0.315155i	−0.987507	−	0.157578i	$-0.949632\pi$
$443$		−	6.63325i		−	0.315155i	0.987507	−	0.157578i	$-0.0503684\pi$
$444$		0			0
$445$	14.7446			0.698959
$446$		0			0
$447$	4.00000	+	3.75906i	0.189194	+	0.177797i
$448$		0			0
$449$			11.6270i			0.548713i	0.961628	+	0.274357i	$0.0884648\pi$
$449$			11.6270i			0.548713i	−0.961628	+	0.274357i	$0.911535\pi$
$450$		0			0
$451$			15.1460i			0.713199i
$452$		0			0
$453$	2.81386	−	2.99422i	0.132207	−	0.140681i
$454$		0			0
$455$	−7.11684	−	8.21782i	−0.333643	−	0.385258i
$456$		0			0
$457$	12.9783			0.607097			0.303548	−	0.952816i	$-0.401829\pi$
$457$	12.9783			0.607097			0.303548	+	0.952816i	$0.401829\pi$
$458$		0			0
$459$	−17.4891	−	14.5012i	−0.816322	−	0.676859i
$460$		0			0
$461$	2.74456			0.127827			0.0639135	−	0.997955i	$-0.479642\pi$
$461$	2.74456			0.127827			0.0639135	+	0.997955i	$0.479642\pi$
$462$		0			0
$463$	32.4674			1.50889			0.754443	−	0.656365i	$-0.227907\pi$
$463$	32.4674			1.50889			0.754443	+	0.656365i	$0.227907\pi$
$464$		0			0
$465$	−4.37228	−	4.10891i	−0.202760	−	0.190546i
$466$		0			0
$467$	31.1168			1.43992			0.719958	−	0.694018i	$-0.244161\pi$
$467$	31.1168			1.43992			0.719958	+	0.694018i	$0.244161\pi$
$468$		0			0
$469$	−9.48913	+	8.21782i	−0.438167	+	0.379464i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			11.9769i			0.550697i
$474$		0			0
$475$			3.46410i			0.158944i
$476$		0			0
$477$	5.62772	−	0.349857i	0.257676	−	0.0160188i
$478$		0			0
$479$	−41.4891			−1.89569			−0.947843	−	0.318737i	$-0.896741\pi$
$479$	−41.4891			−1.89569			−0.947843	+	0.318737i	$0.896741\pi$
$480$		0			0
$481$			27.7128i			1.26360i
$482$		0			0
$483$	38.9783	+	1.58457i	1.77357	+	0.0721006i
$484$		0			0
$485$		−	11.0371i		−	0.501170i
$486$		0			0
$487$	−37.4891			−1.69879			−0.849397	−	0.527754i	$-0.823034\pi$
$487$	−37.4891			−1.69879			−0.849397	+	0.527754i	$0.823034\pi$
$488$		0			0
$489$	−9.48913	+	10.0974i	−0.429113	+	0.456618i
$490$		0			0
$491$			5.69349i			0.256943i	0.991713	+	0.128472i	$0.0410072\pi$
$491$			5.69349i			0.256943i	−0.991713	+	0.128472i	$0.958993\pi$
$492$		0			0
$493$		−	4.10891i		−	0.185056i
$494$		0			0
$495$	−7.55842	+	0.469882i	−0.339726	+	0.0211196i
$496$		0			0
$497$	0.510875	+	0.589907i	0.0229159	+	0.0264610i
$498$		0			0
$499$	−41.3505			−1.85110			−0.925552	−	0.378620i	$-0.876399\pi$
$499$	−41.3505			−1.85110			−0.925552	+	0.378620i	$0.876399\pi$
$500$		0			0
$501$	5.79211	−	6.16337i	0.258772	−	0.275359i
$502$		0			0
$503$	−12.6060			−0.562072			−0.281036	−	0.959697i	$-0.590678\pi$
$503$	−12.6060			−0.562072			−0.281036	+	0.959697i	$0.590678\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$	−4.60597	+	4.90120i	−0.204558	+	0.217670i
$508$		0			0
$509$	−3.76631			−0.166939			−0.0834694	−	0.996510i	$-0.526600\pi$
$509$	−3.76631			−0.166939			−0.0834694	+	0.996510i	$0.526600\pi$
$510$		0			0
$511$	12.0000	+	13.8564i	0.530849	+	0.612971i
$512$		0			0
$513$	11.4891	−	13.8564i	0.507257	−	0.611775i
$514$		0			0
$515$			6.28339i			0.276879i
$516$		0			0
$517$		−	4.10891i		−	0.180710i
$518$		0			0
$519$	−1.32473	+	1.40965i	−0.0581494	+	0.0618766i
$520$		0			0
$521$	−34.4674			−1.51004			−0.755022	−	0.655700i	$-0.772374\pi$
$521$	−34.4674			−1.51004			−0.755022	+	0.655700i	$0.772374\pi$
$522$		0			0
$523$		−	10.3923i		−	0.454424i	−0.973845	−	0.227212i	$-0.927039\pi$
$523$		−	10.3923i		−	0.454424i	0.973845	−	0.227212i	$-0.0729610\pi$
$524$		0			0
$525$	0.186141	−	4.57879i	0.00812384	−	0.199835i
$526$		0			0
$527$		−	15.1460i		−	0.659771i
$528$		0			0
$529$	−49.4674			−2.15076
$530$		0			0
$531$	−1.62772	−	26.1831i	−0.0706370	−	1.13625i
$532$		0			0
$533$			24.6535i			1.06786i
$534$		0			0
$535$		−	6.63325i		−	0.286780i
$536$		0			0
$537$	−8.37228	−	7.86797i	−0.361291	−	0.339528i
$538$		0			0
$539$	−17.4891	−	2.52434i	−0.753310	−	0.108731i
$540$		0			0
$541$	24.3723			1.04785			0.523923	−	0.851766i	$-0.324468\pi$
$541$	24.3723			1.04785			0.523923	+	0.851766i	$0.324468\pi$
$542$		0			0
$543$	1.62772	+	1.52967i	0.0698521	+	0.0656445i
$544$		0			0
$545$	−17.1168			−0.733205
$546$		0			0
$547$	2.97825			0.127341			0.0636704	−	0.997971i	$-0.479719\pi$
$547$	2.97825			0.127341			0.0636704	+	0.997971i	$0.479719\pi$
$548$		0			0
$549$	20.7446	−	1.28962i	0.885356	−	0.0550397i
$550$		0			0
$551$	3.25544			0.138686
$552$		0			0
$553$	4.74456	−	4.10891i	0.201759	−	0.174729i
$554$		0			0
$555$	−8.00000	+	8.51278i	−0.339581	+	0.361347i
$556$		0			0
$557$		−	17.6155i		−	0.746391i	−0.927753	−	0.373196i	$-0.878262\pi$
$557$		−	17.6155i		−	0.746391i	0.927753	−	0.373196i	$-0.121738\pi$
$558$		0			0
$559$			19.4950i			0.824550i
$560$		0			0
$561$	−13.9307	−	13.0916i	−0.588155	−	0.552727i
$562$		0			0
$563$	17.4891			0.737079			0.368539	−	0.929612i	$-0.379858\pi$
$563$	17.4891			0.737079			0.368539	+	0.929612i	$0.379858\pi$
$564$		0			0
$565$			3.16915i			0.133327i
$566$		0			0
$567$	−15.9307	+	17.6978i	−0.669027	+	0.743238i
$568$		0			0
$569$			23.9538i			1.00419i	0.864811	+	0.502097i	$0.167438\pi$
$569$			23.9538i			1.00419i	−0.864811	+	0.502097i	$0.832562\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	26.3030	+	24.7186i	1.09882	+	1.03263i
$574$		0			0
$575$			8.51278i			0.355007i
$576$		0			0
$577$		−	2.81929i		−	0.117369i	−0.998277	−	0.0586843i	$-0.981309\pi$
$577$		−	2.81929i		−	0.117369i	0.998277	−	0.0586843i	$-0.0186905\pi$
$578$		0			0
$579$	−14.5109	+	15.4410i	−0.603051	+	0.641705i
$580$		0			0
$581$	34.9783	−	30.2921i	1.45114	−	1.25673i
$582$		0			0
$583$	4.74456			0.196500
$584$		0			0
$585$	−12.3030	+	0.764836i	−0.508666	+	0.0316221i
$586$		0			0
$587$	−17.4891			−0.721853			−0.360927	−	0.932594i	$-0.617540\pi$
$587$	−17.4891			−0.721853			−0.360927	+	0.932594i	$0.617540\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$	19.8614	+	18.6650i	0.816989	+	0.767777i
$592$		0			0
$593$	4.37228			0.179548			0.0897740	−	0.995962i	$-0.471386\pi$
$593$	4.37228			0.179548			0.0897740	+	0.995962i	$0.471386\pi$
$594$		0			0
$595$	8.74456	−	7.57301i	0.358492	−	0.310463i
$596$		0			0
$597$	33.8614	+	31.8217i	1.38586	+	1.30238i
$598$		0			0
$599$			34.6959i			1.41764i	0.705391	+	0.708818i	$0.250771\pi$
$599$			34.6959i			1.41764i	−0.705391	+	0.708818i	$0.749229\pi$
$600$		0			0
$601$			22.0742i			0.900427i	0.892921	+	0.450213i	$0.148652\pi$
$601$			22.0742i			0.900427i	−0.892921	+	0.450213i	$0.851348\pi$
$602$		0			0
$603$	0.883156	+	14.2063i	0.0359649	+	0.578524i
$604$		0			0
$605$	4.62772			0.188144
$606$		0			0
$607$			8.86263i			0.359723i	0.983692	+	0.179862i	$0.0575650\pi$
$607$			8.86263i			0.359723i	−0.983692	+	0.179862i	$0.942435\pi$
$608$		0			0
$609$	−4.30298	−	0.174928i	−0.174366	−	0.00708845i
$610$		0			0
$611$		−	6.68815i		−	0.270574i
$612$		0			0
$613$	−27.4891			−1.11028			−0.555138	−	0.831758i	$-0.687334\pi$
$613$	−27.4891			−1.11028			−0.555138	+	0.831758i	$0.687334\pi$
$614$		0			0
$615$	−7.11684	+	7.57301i	−0.286979	+	0.305373i
$616$		0			0
$617$		−	24.5437i		−	0.988091i	−0.869436	−	0.494045i	$-0.835518\pi$
$617$		−	24.5437i		−	0.988091i	0.869436	−	0.494045i	$-0.164482\pi$
$618$		0			0
$619$		−	18.6101i		−	0.748004i	−0.927428	−	0.374002i	$-0.877985\pi$
$619$		−	18.6101i		−	0.748004i	0.927428	−	0.374002i	$-0.122015\pi$
$620$		0			0
$621$	28.2337	−	34.0511i	1.13298	−	1.36642i
$622$		0			0
$623$	29.4891	−	25.5383i	1.18146	−	1.02317i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	10.3723	−	11.0371i	0.414229	−	0.440780i
$628$		0			0
$629$	−29.4891			−1.17581
$630$		0			0
$631$	23.1168			0.920267			0.460134	−	0.887850i	$-0.347802\pi$
$631$	23.1168			0.920267			0.460134	+	0.887850i	$0.347802\pi$
$632$		0			0
$633$	13.1861	−	14.0313i	0.524102	−	0.557695i
$634$		0			0
$635$	0.744563			0.0295471
$636$		0			0
$637$	−28.4674	−	4.10891i	−1.12792	−	0.162801i
$638$		0			0
$639$	0.883156	−	0.0549029i	0.0349371	−	0.00217192i
$640$		0			0
$641$		−	36.6303i		−	1.44681i	−0.690423	−	0.723406i	$-0.742576\pi$
$641$		−	36.6303i		−	1.44681i	0.690423	−	0.723406i	$-0.257424\pi$
$642$		0			0
$643$			10.1523i			0.400366i	0.979759	+	0.200183i	$0.0641536\pi$
$643$			10.1523i			0.400366i	−0.979759	+	0.200183i	$0.935846\pi$
$644$		0			0
$645$	−5.62772	+	5.98844i	−0.221591	+	0.235795i
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$		−	22.0742i		−	0.866489i
$650$		0			0
$651$	−15.8614	−	0.644810i	−0.621658	−	0.0252721i
$652$		0			0
$653$		−	39.6897i		−	1.55318i	−0.630008	−	0.776589i	$-0.716948\pi$
$653$		−	39.6897i		−	1.55318i	0.630008	−	0.776589i	$-0.283052\pi$
$654$		0			0
$655$	−5.48913			−0.214478
$656$		0			0
$657$	20.7446	−	1.28962i	0.809322	−	0.0503129i
$658$		0			0
$659$			35.9855i			1.40180i	0.713261	+	0.700899i	$0.247218\pi$
$659$			35.9855i			1.40180i	−0.713261	+	0.700899i	$0.752782\pi$
$660$		0			0
$661$		−	6.92820i		−	0.269476i	−0.990881	−	0.134738i	$-0.956981\pi$
$661$		−	6.92820i		−	0.269476i	0.990881	−	0.134738i	$-0.0430193\pi$
$662$		0			0
$663$	−22.6753	−	21.3094i	−0.880634	−	0.827588i
$664$		0			0
$665$	6.00000	+	6.92820i	0.232670	+	0.268664i
$666$		0			0
$667$	8.00000			0.309761
$668$		0			0
$669$	11.1861	+	10.5123i	0.432481	+	0.406430i
$670$		0			0
$671$	17.4891			0.675160
$672$		0			0
$673$	−12.2337			−0.471574			−0.235787	−	0.971805i	$-0.575767\pi$
$673$	−12.2337			−0.471574			−0.235787	+	0.971805i	$0.575767\pi$
$674$		0			0
$675$	−4.00000	−	3.31662i	−0.153960	−	0.127657i
$676$		0			0
$677$	−18.6060			−0.715085			−0.357543	−	0.933897i	$-0.616385\pi$
$677$	−18.6060			−0.715085			−0.357543	+	0.933897i	$0.616385\pi$
$678$		0			0
$679$	−19.1168	−	22.0742i	−0.733637	−	0.847131i
$680$		0			0
$681$	−29.1861	+	31.0569i	−1.11842	+	1.19010i
$682$		0			0
$683$			34.4559i			1.31842i	0.751960	+	0.659209i	$0.229109\pi$
$683$			34.4559i			1.31842i	−0.751960	+	0.659209i	$0.770891\pi$
$684$		0			0
$685$		−	13.2665i		−	0.506887i
$686$		0			0
$687$	19.1168	+	17.9653i	0.729353	+	0.685420i
$688$		0			0
$689$	7.72281			0.294216
$690$		0			0
$691$		−	21.1894i		−	0.806082i	−0.915182	−	0.403041i	$-0.867953\pi$
$691$		−	21.1894i		−	0.806082i	0.915182	−	0.403041i	$-0.132047\pi$
$692$		0			0
$693$	−14.3030	+	14.0313i	−0.543325	+	0.533006i
$694$		0			0
$695$		−	18.6101i		−	0.705922i
$696$		0			0
$697$	−26.2337			−0.993672
$698$		0			0
$699$	21.4891	+	20.1947i	0.812793	+	0.763834i
$700$		0			0
$701$		−	42.5090i		−	1.60554i	−0.596287	−	0.802771i	$-0.703358\pi$
$701$		−	42.5090i		−	1.60554i	0.596287	−	0.802771i	$-0.296642\pi$
$702$		0			0
$703$		−	23.3639i		−	0.881184i
$704$		0			0
$705$	1.93070	−	2.05446i	0.0727145	−	0.0773753i
$706$		0			0
$707$	12.0000	−	10.3923i	0.451306	−	0.390843i
$708$		0			0
$709$	6.88316			0.258502			0.129251	−	0.991612i	$-0.458743\pi$
$709$	6.88316			0.258502			0.129251	+	0.991612i	$0.458743\pi$
$710$		0			0
$711$	−0.441578	−	7.10313i	−0.0165605	−	0.266388i
$712$		0			0
$713$	29.4891			1.10438
$714$		0			0
$715$	−10.3723			−0.387901
$716$		0			0
$717$	26.3030	+	24.7186i	0.982303	+	0.923133i
$718$		0			0
$719$	28.4674			1.06165			0.530827	−	0.847480i	$-0.321881\pi$
$719$	28.4674			1.06165			0.530827	+	0.847480i	$0.321881\pi$
$720$		0			0
$721$	10.8832	+	12.5668i	0.405310	+	0.468012i
$722$		0			0
$723$	−20.7446	−	19.4950i	−0.771499	−	0.725026i
$724$		0			0
$725$		−	0.939764i		−	0.0349020i
$726$		0			0
$727$		−	3.46410i		−	0.128476i	−0.997935	−	0.0642382i	$-0.979538\pi$
$727$		−	3.46410i		−	0.128476i	0.997935	−	0.0642382i	$-0.0204617\pi$
$728$		0			0
$729$	5.00000	+	26.5330i	0.185185	+	0.982704i
$730$		0			0
$731$	−20.7446			−0.767265
$732$		0			0
$733$		−	40.0395i		−	1.47889i	−0.673215	−	0.739447i	$-0.735087\pi$
$733$		−	40.0395i		−	1.47889i	0.673215	−	0.739447i	$-0.264913\pi$
$734$		0			0
$735$	−7.55842	−	9.47999i	−0.278797	−	0.349675i
$736$		0			0
$737$			11.9769i			0.441174i
$738$		0			0
$739$	14.3723			0.528693			0.264346	−	0.964428i	$-0.414844\pi$
$739$	14.3723			0.528693			0.264346	+	0.964428i	$0.414844\pi$
$740$		0			0
$741$	16.8832	−	17.9653i	0.620218	−	0.659972i
$742$		0			0
$743$		−	16.1407i		−	0.592145i	−0.955166	−	0.296072i	$-0.904323\pi$
$743$		−	16.1407i		−	0.592145i	0.955166	−	0.296072i	$-0.0956769\pi$
$744$		0			0
$745$			3.16915i			0.116109i
$746$		0			0
$747$	−3.25544	−	52.3663i	−0.119110	−	1.91598i
$748$		0			0
$749$	−11.4891	−	13.2665i	−0.419804	−	0.484747i
$750$		0			0
$751$	−29.3505			−1.07102			−0.535508	−	0.844530i	$-0.679880\pi$
$751$	−29.3505			−1.07102			−0.535508	+	0.844530i	$0.679880\pi$
$752$		0			0
$753$	6.51087	−	6.92820i	0.237269	−	0.252478i
$754$		0			0
$755$	2.37228			0.0863362
$756$		0			0
$757$	−19.7663			−0.718419			−0.359209	−	0.933257i	$-0.616954\pi$
$757$	−19.7663			−0.718419			−0.359209	+	0.933257i	$0.616954\pi$
$758$		0			0
$759$	25.4891	−	27.1229i	0.925197	−	0.984499i
$760$		0			0
$761$	8.23369			0.298471			0.149235	−	0.988802i	$-0.452319\pi$
$761$	8.23369			0.298471			0.149235	+	0.988802i	$0.452319\pi$
$762$		0			0
$763$	−34.2337	+	29.6472i	−1.23934	+	1.07330i
$764$		0			0
$765$	−0.813859	−	13.0916i	−0.0294251	−	0.473327i
$766$		0			0
$767$		−	35.9306i		−	1.29738i
$768$		0			0
$769$		−	38.5099i		−	1.38870i	−0.719637	−	0.694351i	$-0.755692\pi$
$769$		−	38.5099i		−	1.38870i	0.719637	−	0.694351i	$-0.244308\pi$
$770$		0			0
$771$	0.605969	−	0.644810i	0.0218234	−	0.0232223i
$772$		0			0
$773$	−52.3723			−1.88370			−0.941850	−	0.336034i	$-0.890914\pi$
$773$	−52.3723			−1.88370			−0.941850	+	0.336034i	$0.890914\pi$
$774$		0			0
$775$		−	3.46410i		−	0.124434i
$776$		0			0
$777$	−1.25544	+	30.8820i	−0.0450386	+	1.10788i
$778$		0			0
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	0.744563			0.0266425
$782$		0			0
$783$	−3.11684	+	3.75906i	−0.111387	+	0.134338i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	3.22405i		−	0.114925i	−0.998348	−	0.0574625i	$-0.981699\pi$
$787$		−	3.22405i		−	0.114925i	0.998348	−	0.0574625i	$-0.0183010\pi$
$788$		0			0
$789$	0.372281	+	0.349857i	0.0132536	+	0.0124552i
$790$		0			0
$791$	5.48913	+	6.33830i	0.195171	+	0.225364i
$792$		0			0
$793$	28.4674			1.01091
$794$		0			0
$795$	2.37228	+	2.22938i	0.0841361	+	0.0790681i
$796$		0			0
$797$	−14.1386			−0.500815			−0.250407	−	0.968141i	$-0.580565\pi$
$797$	−14.1386			−0.500815			−0.250407	+	0.968141i	$0.580565\pi$
$798$		0			0
$799$	7.11684			0.251776
$800$		0			0
$801$	−2.74456	−	44.1485i	−0.0969744	−	1.55991i
$802$		0			0
$803$	17.4891			0.617178
$804$		0			0
$805$	14.7446	+	17.0256i	0.519678	+	0.600072i
$806$		0			0
$807$	3.25544	−	3.46410i	0.114597	−	0.121942i
$808$		0			0
$809$		−	13.5065i		−	0.474865i	−0.971404	−	0.237433i	$-0.923694\pi$
$809$		−	13.5065i		−	0.474865i	0.971404	−	0.237433i	$-0.0763059\pi$
$810$		0			0
$811$		−	18.6101i		−	0.653490i	−0.945113	−	0.326745i	$-0.894048\pi$
$811$		−	18.6101i		−	0.653490i	0.945113	−	0.326745i	$-0.105952\pi$
$812$		0			0
$813$	6.00000	+	5.63858i	0.210429	+	0.197754i
$814$		0			0
$815$	−8.00000			−0.280228
$816$		0			0
$817$		−	16.4356i		−	0.575011i
$818$		0			0
$819$	−23.2812	+	22.8391i	−0.813512	+	0.798062i
$820$		0			0
$821$			34.9909i			1.22119i	0.791943	+	0.610595i	$0.209070\pi$
$821$			34.9909i			1.22119i	−0.791943	+	0.610595i	$0.790930\pi$
$822$		0			0
$823$	13.7663			0.479863			0.239932	−	0.970790i	$-0.422875\pi$
$823$	13.7663			0.479863			0.239932	+	0.970790i	$0.422875\pi$
$824$		0			0
$825$	−3.18614	−	2.99422i	−0.110927	−	0.104245i
$826$		0			0
$827$		−	14.8511i		−	0.516422i	−0.966088	−	0.258211i	$-0.916867\pi$
$827$		−	14.8511i		−	0.516422i	0.966088	−	0.258211i	$-0.0831330\pi$
$828$		0			0
$829$			12.5668i			0.436463i	0.975897	+	0.218231i	$0.0700287\pi$
$829$			12.5668i			0.436463i	−0.975897	+	0.218231i	$0.929971\pi$
$830$		0			0
$831$	33.4891	−	35.6357i	1.16172	−	1.23619i
$832$		0			0
$833$	4.37228	−	30.2921i	0.151491	−	1.04956i
$834$		0			0
$835$	4.88316			0.168989
$836$		0			0
$837$	−11.4891	+	13.8564i	−0.397122	+	0.478947i
$838$		0			0
$839$	30.5109			1.05335			0.526676	−	0.850066i	$-0.323438\pi$
$839$	30.5109			1.05335			0.526676	+	0.850066i	$0.323438\pi$
$840$		0			0
$841$	28.1168			0.969546
$842$		0			0
$843$	35.4198	+	33.2863i	1.21992	+	1.14644i
$844$		0			0
$845$	−3.88316			−0.133585
$846$		0			0
$847$	9.25544	−	8.01544i	0.318021	−	0.275414i
$848$		0			0
$849$	−0.813859	−	0.764836i	−0.0279316	−	0.0262491i
$850$		0			0
$851$		−	57.4150i		−	1.96816i
$852$		0			0
$853$			32.8713i			1.12549i	0.826630	+	0.562746i	$0.190255\pi$
$853$			32.8713i			1.12549i	−0.826630	+	0.562746i	$0.809745\pi$
$854$		0			0
$855$	10.3723	−	0.644810i	0.354725	−	0.0220520i
$856$		0			0
$857$	46.4674			1.58730			0.793648	−	0.608378i	$-0.208179\pi$
$857$	46.4674			1.58730			0.793648	+	0.608378i	$0.208179\pi$
$858$		0			0
$859$			14.2612i			0.486585i	0.969953	+	0.243292i	$0.0782274\pi$
$859$			14.2612i			0.486585i	−0.969953	+	0.243292i	$0.921773\pi$
$860$		0			0
$861$	−1.11684	+	27.4728i	−0.0380619	+	0.936269i
$862$		0			0
$863$		−	41.2743i		−	1.40499i	−0.711688	−	0.702496i	$-0.752069\pi$
$863$		−	41.2743i		−	1.40499i	0.711688	−	0.702496i	$-0.247931\pi$
$864$		0			0
$865$	−1.11684			−0.0379738
$866$		0			0
$867$	2.51087	−	2.67181i	0.0852738	−	0.0907396i
$868$		0			0
$869$		−	5.98844i		−	0.203144i
$870$		0			0
$871$			19.4950i			0.660563i
$872$		0			0
$873$	−33.0475	+	2.05446i	−1.11849	+	0.0695328i
$874$		0			0
$875$	2.00000	−	1.73205i	0.0676123	−	0.0585540i
$876$		0			0
$877$	−38.4674			−1.29895			−0.649475	−	0.760383i	$-0.725011\pi$
$877$	−38.4674			−1.29895			−0.649475	+	0.760383i	$0.725011\pi$
$878$		0			0
$879$	−12.9090	+	13.7364i	−0.435408	+	0.463317i
$880$		0			0
$881$	32.2337			1.08598			0.542990	−	0.839739i	$-0.317292\pi$
$881$	32.2337			1.08598			0.542990	+	0.839739i	$0.317292\pi$
$882$		0			0
$883$	−49.4891			−1.66544			−0.832721	−	0.553693i	$-0.813218\pi$
$883$	−49.4891			−1.66544			−0.832721	+	0.553693i	$0.813218\pi$
$884$		0			0
$885$	10.3723	−	11.0371i	0.348660	−	0.371009i
$886$		0			0
$887$	−18.5109			−0.621534			−0.310767	−	0.950486i	$-0.600586\pi$
$887$	−18.5109			−0.621534			−0.310767	+	0.950486i	$0.600586\pi$
$888$		0			0
$889$	1.48913	−	1.28962i	0.0499437	−	0.0432525i
$890$		0			0
$891$	2.81386	+	22.5441i	0.0942678	+	0.755256i
$892$		0			0
$893$			5.63858i			0.188688i
$894$		0			0
$895$		−	6.63325i		−	0.221725i
$896$		0			0
$897$	41.4891	−	44.1485i	1.38528	−	1.47407i
$898$		0			0
$899$	−3.25544			−0.108575
$900$		0			0
$901$			8.21782i			0.273775i
$902$		0			0
$903$	−0.883156	+	21.7244i	−0.0293896	+	0.722942i
$904$		0			0
$905$			1.28962i			0.0428684i
$906$		0			0
$907$	−8.00000			−0.265636			−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000			−0.265636			−0.132818	+	0.991140i	$0.542403\pi$
$908$		0			0
$909$	−1.11684	−	17.9653i	−0.0370434	−	0.595872i
$910$		0			0
$911$			2.87419i			0.0952263i	0.998866	+	0.0476132i	$0.0151615\pi$
$911$			2.87419i			0.0952263i	−0.998866	+	0.0476132i	$0.984839\pi$
$912$		0			0
$913$		−	44.1485i		−	1.46110i
$914$		0			0
$915$	8.74456	+	8.21782i	0.289086	+	0.271673i
$916$		0			0
$917$	−10.9783	+	9.50744i	−0.362534	+	0.313963i
$918$		0			0
$919$	5.62772			0.185641			0.0928207	−	0.995683i	$-0.470412\pi$
$919$	5.62772			0.185641			0.0928207	+	0.995683i	$0.470412\pi$
$920$		0			0
$921$	−28.6753	−	26.9480i	−0.944882	−	0.887966i
$922$		0			0
$923$	1.21194			0.0398914
$924$		0			0
$925$	−6.74456			−0.221760
$926$		0			0
$927$	18.8139	−	1.16959i	0.617928	−	0.0384145i
$928$		0			0
$929$	52.9783			1.73816			0.869080	−	0.494672i	$-0.164712\pi$
$929$	52.9783			1.73816			0.869080	+	0.494672i	$0.164712\pi$
$930$		0			0
$931$	24.0000	+	3.46410i	0.786568	+	0.113531i
$932$		0			0
$933$	16.8832	−	17.9653i	0.552730	−	0.588158i
$934$		0			0
$935$		−	11.0371i		−	0.360952i
$936$		0			0
$937$		−	0.240051i		−	0.00784212i	−0.999992	−	0.00392106i	$-0.998752\pi$
$937$		−	0.240051i		−	0.00784212i	0.999992	−	0.00392106i	$-0.00124811\pi$
$938$		0			0
$939$	6.81386	+	6.40342i	0.222362	+	0.208968i
$940$		0			0
$941$	−38.7446			−1.26304			−0.631518	−	0.775361i	$-0.717568\pi$
$941$	−38.7446			−1.26304			−0.631518	+	0.775361i	$0.717568\pi$
$942$		0			0
$943$		−	51.0767i		−	1.66329i
$944$		0			0
$945$	−13.7446	+	0.294954i	−0.447111	+	0.00959484i
$946$		0			0
$947$			34.9360i			1.13527i	0.823282	+	0.567633i	$0.192141\pi$
$947$			34.9360i			1.13527i	−0.823282	+	0.567633i	$0.807859\pi$
$948$		0			0
$949$	28.4674			0.924090
$950$		0			0
$951$	2.37228	+	2.22938i	0.0769265	+	0.0722927i
$952$		0			0
$953$		−	7.62792i		−	0.247092i	−0.992339	−	0.123546i	$-0.960573\pi$
$953$		−	7.62792i		−	0.247092i	0.992339	−	0.123546i	$-0.0394267\pi$
$954$		0			0
$955$			20.8395i			0.674351i
$956$		0			0
$957$	−2.81386	+	2.99422i	−0.0909592	+	0.0967894i
$958$		0			0
$959$	−22.9783	−	26.5330i	−0.742006	−	0.856795i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	−19.8614	+	1.23472i	−0.640025	+	0.0397882i
$964$		0			0
$965$	−12.2337			−0.393816
$966$		0			0
$967$	−10.2337			−0.329093			−0.164547	−	0.986369i	$-0.552616\pi$
$967$	−10.2337			−0.329093			−0.164547	+	0.986369i	$0.552616\pi$
$968$		0			0
$969$	19.1168	+	17.9653i	0.614122	+	0.577129i
$970$		0			0
$971$	−37.2119			−1.19419			−0.597094	−	0.802171i	$-0.703678\pi$
$971$	−37.2119			−1.19419			−0.597094	+	0.802171i	$0.703678\pi$
$972$		0			0
$973$	−32.2337	−	37.2203i	−1.03336	−	1.19323i
$974$		0			0
$975$	−5.18614	−	4.87375i	−0.166089	−	0.156085i
$976$		0			0
$977$		−	7.62792i		−	0.244039i	−0.992528	−	0.122019i	$-0.961063\pi$
$977$		−	7.62792i		−	0.244039i	0.992528	−	0.122019i	$-0.0389370\pi$
$978$		0			0
$979$		−	37.2203i		−	1.18956i
$980$		0			0
$981$	3.18614	+	51.2516i	0.101726	+	1.63634i
$982$		0			0
$983$	−39.8614			−1.27138			−0.635691	−	0.771944i	$-0.719285\pi$
$983$	−39.8614			−1.27138			−0.635691	+	0.771944i	$0.719285\pi$
$984$		0			0
$985$			15.7359i			0.501389i
$986$		0			0
$987$	0.302985	−	7.45299i	0.00964411	−	0.237231i
$988$		0			0
$989$		−	40.3894i		−	1.28431i
$990$		0			0
$991$	−25.4891			−0.809689			−0.404844	−	0.914386i	$-0.632674\pi$
$991$	−25.4891			−0.809689			−0.404844	+	0.914386i	$0.632674\pi$
$992$		0			0
$993$	4.74456	−	5.04868i	0.150564	−	0.160215i
$994$		0			0
$995$			26.8280i			0.850503i
$996$		0			0
$997$		−	31.8217i		−	1.00780i	−0.863761	−	0.503902i	$-0.831897\pi$
$997$		−	31.8217i		−	1.00780i	0.863761	−	0.503902i	$-0.168103\pi$
$998$		0			0
$999$	26.9783	+	22.3692i	0.853554	+	0.707730i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.g.881.3		4
3.2	odd	2		1680.2.f.h.881.1		4
4.3	odd	2		105.2.b.d.41.3	yes	4
7.6	odd	2		1680.2.f.h.881.2		4
12.11	even	2		105.2.b.c.41.2	✓	4
20.3	even	4		525.2.g.d.524.5		8
20.7	even	4		525.2.g.d.524.4		8
20.19	odd	2		525.2.b.e.251.2		4
21.20	even	2	inner	1680.2.f.g.881.4		4
28.3	even	6		735.2.s.i.656.1		4
28.11	odd	6		735.2.s.j.656.1		4
28.19	even	6		735.2.s.h.521.2		4
28.23	odd	6		735.2.s.g.521.2		4
28.27	even	2		105.2.b.c.41.3	yes	4
60.23	odd	4		525.2.g.e.524.4		8
60.47	odd	4		525.2.g.e.524.5		8
60.59	even	2		525.2.b.g.251.3		4
84.11	even	6		735.2.s.h.656.2		4
84.23	even	6		735.2.s.i.521.1		4
84.47	odd	6		735.2.s.j.521.1		4
84.59	odd	6		735.2.s.g.656.2		4
84.83	odd	2		105.2.b.d.41.2	yes	4
140.27	odd	4		525.2.g.e.524.3		8
140.83	odd	4		525.2.g.e.524.6		8
140.139	even	2		525.2.b.g.251.2		4
420.83	even	4		525.2.g.d.524.3		8
420.167	even	4		525.2.g.d.524.6		8
420.419	odd	2		525.2.b.e.251.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.b.c.41.2	✓	4	12.11	even	2
105.2.b.c.41.3	yes	4	28.27	even	2
105.2.b.d.41.2	yes	4	84.83	odd	2
105.2.b.d.41.3	yes	4	4.3	odd	2
525.2.b.e.251.2		4	20.19	odd	2
525.2.b.e.251.3		4	420.419	odd	2
525.2.b.g.251.2		4	140.139	even	2
525.2.b.g.251.3		4	60.59	even	2
525.2.g.d.524.3		8	420.83	even	4
525.2.g.d.524.4		8	20.7	even	4
525.2.g.d.524.5		8	20.3	even	4
525.2.g.d.524.6		8	420.167	even	4
525.2.g.e.524.3		8	140.27	odd	4
525.2.g.e.524.4		8	60.23	odd	4
525.2.g.e.524.5		8	60.47	odd	4
525.2.g.e.524.6		8	140.83	odd	4
735.2.s.g.521.2		4	28.23	odd	6
735.2.s.g.656.2		4	84.59	odd	6
735.2.s.h.521.2		4	28.19	even	6
735.2.s.h.656.2		4	84.11	even	6
735.2.s.i.521.1		4	84.23	even	6
735.2.s.i.656.1		4	28.3	even	6
735.2.s.j.521.1		4	84.47	odd	6
735.2.s.j.656.1		4	28.11	odd	6
1680.2.f.g.881.3		4	1.1	even	1	trivial
1680.2.f.g.881.4		4	21.20	even	2	inner
1680.2.f.h.881.1		4	3.2	odd	2
1680.2.f.h.881.2		4	7.6	odd	2

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

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1680.2.f.g.881.3

Level	$1680$
Weight	$2$
Character	1680.881
Analytic conductor	$13.415$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$