LMFDB - Embedded newform 1680.3.l.a.1121.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,3,Mod(1121,1680)]

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

lf = mfeigenbasis(mf)

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")

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

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} + 46x^{14} + 823x^{12} + 7252x^{10} + 32831x^{8} + 71486x^{6} + 60809x^{4} + 15680x^{2} + 576 $ x^16 + 46x^14 + 823x^12 + 7252x^10 + 32831x^8 + 71486x^6 + 60809x^4 + 15680*x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.3
Root			$-2.02253i$ of defining polynomial
Character	$\chi$	$=$	1680.1121
Dual form			1680.3.l.a.1121.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.91626 - 0.703870i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(8.00914 + 4.10533i) q^{9} +O(q^{10})$	\(q+(-2.91626 - 0.703870i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(8.00914 + 4.10533i) q^{9} +11.6583i q^{11} -20.3660 q^{13} +(-1.57390 + 6.52095i) q^{15} +9.92200i q^{17} +4.91979 q^{19} +(7.71570 + 1.86226i) q^{21} +11.4646i q^{23} -5.00000 q^{25} +(-20.4671 - 17.6096i) q^{27} -37.5101i q^{29} +21.2282 q^{31} +(8.20592 - 33.9986i) q^{33} +5.91608i q^{35} +57.1939 q^{37} +(59.3924 + 14.3350i) q^{39} -20.2428i q^{41} -5.88954 q^{43} +(9.17980 - 17.9090i) q^{45} +74.5051i q^{47} +7.00000 q^{49} +(6.98379 - 28.9351i) q^{51} +88.2001i q^{53} +26.0687 q^{55} +(-14.3474 - 3.46289i) q^{57} -71.6589i q^{59} -26.0462 q^{61} +(-21.1902 - 10.8617i) q^{63} +45.5397i q^{65} -76.4691 q^{67} +(8.06960 - 33.4338i) q^{69} -47.1551i q^{71} -135.859 q^{73} +(14.5813 + 3.51935i) q^{75} -30.8449i q^{77} +98.2712 q^{79} +(47.2925 + 65.7603i) q^{81} -55.8693i q^{83} +22.1863 q^{85} +(-26.4022 + 109.389i) q^{87} -105.666i q^{89} +53.8833 q^{91} +(-61.9069 - 14.9419i) q^{93} -11.0010i q^{95} +61.8996 q^{97} +(-47.8612 + 93.3728i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{3} + 22 q^{9}+O(q^{10}) $ 16 * q - 8 * q^3 + 22 * q^9	$ 16 q - 8 q^{3} + 22 q^{9} - 10 q^{15} + 16 q^{19} - 14 q^{21} - 80 q^{25} + 148 q^{27} + 72 q^{31} - 4 q^{33} - 40 q^{37} - 90 q^{39} - 280 q^{43} + 40 q^{45} + 112 q^{49} - 38 q^{51} - 80 q^{55} - 36 q^{57} - 56 q^{61} + 56 q^{63} + 120 q^{67} + 60 q^{69} - 208 q^{73} + 40 q^{75} + 204 q^{79} + 458 q^{81} + 100 q^{85} + 324 q^{87} + 28 q^{91} + 108 q^{93} + 728 q^{97} + 166 q^{99}+O(q^{100}) $ 16 * q - 8 * q^3 + 22 * q^9 - 10 * q^15 + 16 * q^19 - 14 * q^21 - 80 * q^25 + 148 * q^27 + 72 * q^31 - 4 * q^33 - 40 * q^37 - 90 * q^39 - 280 * q^43 + 40 * q^45 + 112 * q^49 - 38 * q^51 - 80 * q^55 - 36 * q^57 - 56 * q^61 + 56 * q^63 + 120 * q^67 + 60 * q^69 - 208 * q^73 + 40 * q^75 + 204 * q^79 + 458 * q^81 + 100 * q^85 + 324 * q^87 + 28 * q^91 + 108 * q^93 + 728 * q^97 + 166 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.91626	−	0.703870i	−0.972086	−	0.234623i
$3$	−2.91626	−	0.703870i	−0.972086	−	0.234623i
$4$		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
$9$	8.00914	+	4.10533i	0.889904	+	0.456148i
$10$		0			0
$11$			11.6583i			1.05984i	0.848046	+	0.529922i	$0.177779\pi$
$11$			11.6583i			1.05984i	−0.848046	+	0.529922i	$0.822221\pi$
$12$		0			0
$13$	−20.3660			−1.56661			−0.783306	−	0.621636i	$-0.786468\pi$
$13$	−20.3660			−1.56661			−0.783306	+	0.621636i	$0.786468\pi$
$14$		0			0
$15$	−1.57390	+	6.52095i	−0.104927	+	0.434730i
$16$		0			0
$17$			9.92200i			0.583647i	0.956472	+	0.291824i	$0.0942620\pi$
$17$			9.92200i			0.583647i	−0.956472	+	0.291824i	$0.905738\pi$
$18$		0			0
$19$	4.91979			0.258936			0.129468	−	0.991584i	$-0.458673\pi$
$19$	4.91979			0.258936			0.129468	+	0.991584i	$0.458673\pi$
$20$		0			0
$21$	7.71570	+	1.86226i	0.367414	+	0.0886792i
$22$		0			0
$23$			11.4646i			0.498462i	0.968444	+	0.249231i	$0.0801779\pi$
$23$			11.4646i			0.498462i	−0.968444	+	0.249231i	$0.919822\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−20.4671	−	17.6096i	−0.758041	−	0.652207i
$28$		0			0
$29$		−	37.5101i		−	1.29345i	−0.762722	−	0.646726i	$-0.776138\pi$
$29$		−	37.5101i		−	1.29345i	0.762722	−	0.646726i	$-0.223862\pi$
$30$		0			0
$31$	21.2282			0.684780			0.342390	−	0.939558i	$-0.388764\pi$
$31$	21.2282			0.684780			0.342390	+	0.939558i	$0.388764\pi$
$32$		0			0
$33$	8.20592	−	33.9986i	0.248664	−	1.03026i
$34$		0			0
$35$			5.91608i			0.169031i
$36$		0			0
$37$	57.1939			1.54578			0.772890	−	0.634540i	$-0.218810\pi$
$37$	57.1939			1.54578			0.772890	+	0.634540i	$0.218810\pi$
$38$		0			0
$39$	59.3924	+	14.3350i	1.52288	+	0.367564i
$40$		0			0
$41$		−	20.2428i		−	0.493728i	−0.969050	−	0.246864i	$-0.920600\pi$
$41$		−	20.2428i		−	0.493728i	0.969050	−	0.246864i	$-0.0794001\pi$
$42$		0			0
$43$	−5.88954			−0.136966			−0.0684830	−	0.997652i	$-0.521816\pi$
$43$	−5.88954			−0.136966			−0.0684830	+	0.997652i	$0.521816\pi$
$44$		0			0
$45$	9.17980	−	17.9090i	0.203996	−	0.397977i
$46$		0			0
$47$			74.5051i			1.58521i	0.609733	+	0.792607i	$0.291277\pi$
$47$			74.5051i			1.58521i	−0.609733	+	0.792607i	$0.708723\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	6.98379	−	28.9351i	0.136937	−	0.567355i
$52$		0			0
$53$			88.2001i			1.66415i	0.554661	+	0.832077i	$0.312848\pi$
$53$			88.2001i			1.66415i	−0.554661	+	0.832077i	$0.687152\pi$
$54$		0			0
$55$	26.0687			0.473977
$56$		0			0
$57$	−14.3474	−	3.46289i	−0.251708	−	0.0607524i
$58$		0			0
$59$		−	71.6589i		−	1.21456i	−0.794489	−	0.607279i	$-0.792261\pi$
$59$		−	71.6589i		−	1.21456i	0.794489	−	0.607279i	$-0.207739\pi$
$60$		0			0
$61$	−26.0462			−0.426987			−0.213494	−	0.976944i	$-0.568484\pi$
$61$	−26.0462			−0.426987			−0.213494	+	0.976944i	$0.568484\pi$
$62$		0			0
$63$	−21.1902	−	10.8617i	−0.336352	−	0.172408i
$64$		0			0
$65$			45.5397i			0.700610i
$66$		0			0
$67$	−76.4691			−1.14133			−0.570665	−	0.821183i	$-0.693315\pi$
$67$	−76.4691			−1.14133			−0.570665	+	0.821183i	$0.693315\pi$
$68$		0			0
$69$	8.06960	−	33.4338i	0.116951	−	0.484548i
$70$		0			0
$71$		−	47.1551i		−	0.664156i	−0.943252	−	0.332078i	$-0.892250\pi$
$71$		−	47.1551i		−	0.664156i	0.943252	−	0.332078i	$-0.107750\pi$
$72$		0			0
$73$	−135.859			−1.86109			−0.930544	−	0.366179i	$-0.880666\pi$
$73$	−135.859			−1.86109			−0.930544	+	0.366179i	$0.880666\pi$
$74$		0			0
$75$	14.5813	+	3.51935i	0.194417	+	0.0469246i
$76$		0			0
$77$		−	30.8449i		−	0.400584i
$78$		0			0
$79$	98.2712			1.24394			0.621969	−	0.783041i	$-0.286333\pi$
$79$	98.2712			1.24394			0.621969	+	0.783041i	$0.286333\pi$
$80$		0			0
$81$	47.2925	+	65.7603i	0.583858	+	0.811856i
$82$		0			0
$83$		−	55.8693i		−	0.673124i	−0.941661	−	0.336562i	$-0.890736\pi$
$83$		−	55.8693i		−	0.673124i	0.941661	−	0.336562i	$-0.109264\pi$
$84$		0			0
$85$	22.1863			0.261015
$86$		0			0
$87$	−26.4022	+	109.389i	−0.303474	+	1.25735i
$88$		0			0
$89$		−	105.666i		−	1.18726i	−0.804740	−	0.593628i	$-0.797695\pi$
$89$		−	105.666i		−	1.18726i	0.804740	−	0.593628i	$-0.202305\pi$
$90$		0			0
$91$	53.8833			0.592124
$92$		0			0
$93$	−61.9069	−	14.9419i	−0.665666	−	0.160665i
$94$		0			0
$95$		−	11.0010i		−	0.115800i
$96$		0			0
$97$	61.8996			0.638141			0.319070	−	0.947731i	$-0.396629\pi$
$97$	61.8996			0.638141			0.319070	+	0.947731i	$0.396629\pi$
$98$		0			0
$99$	−47.8612	+	93.3728i	−0.483446	+	0.943160i
$100$		0			0
$101$		−	11.1708i		−	0.110602i	−0.998470	−	0.0553011i	$-0.982388\pi$
$101$		−	11.1708i		−	0.110602i	0.998470	−	0.0553011i	$-0.0176119\pi$
$102$		0			0
$103$	−38.6092			−0.374846			−0.187423	−	0.982279i	$-0.560014\pi$
$103$	−38.6092			−0.374846			−0.187423	+	0.982279i	$0.560014\pi$
$104$		0			0
$105$	4.16415	−	17.2528i	0.0396586	−	0.164313i
$106$		0			0
$107$		−	196.812i		−	1.83936i	−0.392668	−	0.919680i	$-0.628448\pi$
$107$		−	196.812i		−	1.83936i	0.392668	−	0.919680i	$-0.371552\pi$
$108$		0			0
$109$	47.3856			0.434730			0.217365	−	0.976090i	$-0.430254\pi$
$109$	47.3856			0.434730			0.217365	+	0.976090i	$0.430254\pi$
$110$		0			0
$111$	−166.792	−	40.2570i	−1.50263	−	0.362676i
$112$		0			0
$113$			64.2642i			0.568710i	0.958719	+	0.284355i	$0.0917794\pi$
$113$			64.2642i			0.568710i	−0.958719	+	0.284355i	$0.908221\pi$
$114$		0			0
$115$	25.6357			0.222919
$116$		0			0
$117$	−163.114	−	83.6090i	−1.39413	−	0.714607i
$118$		0			0
$119$		−	26.2511i		−	0.220598i
$120$		0			0
$121$	−14.9157			−0.123271
$122$		0			0
$123$	−14.2483	+	59.0334i	−0.115840	+	0.479946i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	88.6515			0.698044			0.349022	−	0.937115i	$-0.386514\pi$
$127$	88.6515			0.698044			0.349022	+	0.937115i	$0.386514\pi$
$128$		0			0
$129$	17.1754	+	4.14547i	0.133143	+	0.0321354i
$130$		0			0
$131$		−	116.606i		−	0.890121i	−0.895501	−	0.445060i	$-0.853182\pi$
$131$		−	116.606i		−	0.890121i	0.895501	−	0.445060i	$-0.146818\pi$
$132$		0			0
$133$	−13.0165			−0.0978686
$134$		0			0
$135$	−39.3763	+	45.7658i	−0.291676	+	0.339006i
$136$		0			0
$137$		−	80.7032i		−	0.589075i	−0.955640	−	0.294537i	$-0.904834\pi$
$137$		−	80.7032i		−	0.589075i	0.955640	−	0.294537i	$-0.0951655\pi$
$138$		0			0
$139$	−194.771			−1.40123			−0.700616	−	0.713538i	$-0.747091\pi$
$139$	−194.771			−1.40123			−0.700616	+	0.713538i	$0.747091\pi$
$140$		0			0
$141$	52.4418	−	217.276i	0.371928	−	1.54097i
$142$		0			0
$143$		−	237.432i		−	1.66037i
$144$		0			0
$145$	−83.8752			−0.578450
$146$		0			0
$147$	−20.4138	−	4.92709i	−0.138869	−	0.0335176i
$148$		0			0
$149$		−	251.636i		−	1.68883i	−0.535690	−	0.844415i	$-0.679948\pi$
$149$		−	251.636i		−	1.68883i	0.535690	−	0.844415i	$-0.320052\pi$
$150$		0			0
$151$	−138.915			−0.919969			−0.459984	−	0.887927i	$-0.652145\pi$
$151$	−138.915			−0.919969			−0.459984	+	0.887927i	$0.652145\pi$
$152$		0			0
$153$	−40.7331	+	79.4667i	−0.266229	+	0.519390i
$154$		0			0
$155$		−	47.4677i		−	0.306243i
$156$		0			0
$157$	−25.8481			−0.164638			−0.0823189	−	0.996606i	$-0.526233\pi$
$157$	−25.8481			−0.164638			−0.0823189	+	0.996606i	$0.526233\pi$
$158$		0			0
$159$	62.0814	−	257.214i	0.390449	−	1.61770i
$160$		0			0
$161$		−	30.3326i		−	0.188401i
$162$		0			0
$163$	−94.9187			−0.582323			−0.291162	−	0.956674i	$-0.594042\pi$
$163$	−94.9187			−0.582323			−0.291162	+	0.956674i	$0.594042\pi$
$164$		0			0
$165$	−76.0232	−	18.3490i	−0.460747	−	0.111206i
$166$		0			0
$167$		−	325.029i		−	1.94628i	−0.230206	−	0.973142i	$-0.573940\pi$
$167$		−	325.029i		−	1.94628i	0.230206	−	0.973142i	$-0.426060\pi$
$168$		0			0
$169$	245.772			1.45427
$170$		0			0
$171$	39.4032	+	20.1974i	0.230428	+	0.118113i
$172$		0			0
$173$		−	97.5515i		−	0.563881i	−0.959432	−	0.281941i	$-0.909022\pi$
$173$		−	97.5515i		−	0.563881i	0.959432	−	0.281941i	$-0.0909781\pi$
$174$		0			0
$175$	13.2288			0.0755929
$176$		0			0
$177$	−50.4385	+	208.976i	−0.284963	+	1.18065i
$178$		0			0
$179$		−	172.813i		−	0.965435i	−0.875776	−	0.482718i	$-0.839650\pi$
$179$		−	172.813i		−	0.965435i	0.875776	−	0.482718i	$-0.160350\pi$
$180$		0			0
$181$	−8.15596			−0.0450606			−0.0225303	−	0.999746i	$-0.507172\pi$
$181$	−8.15596			−0.0450606			−0.0225303	+	0.999746i	$0.507172\pi$
$182$		0			0
$183$	75.9575	+	18.3331i	0.415069	+	0.100181i
$184$		0			0
$185$		−	127.889i		−	0.691294i
$186$		0			0
$187$	−115.674			−0.618575
$188$		0			0
$189$	54.1508	+	46.5906i	0.286512	+	0.246511i
$190$		0			0
$191$			119.244i			0.624313i	0.950031	+	0.312156i	$0.101051\pi$
$191$			119.244i			0.624313i	−0.950031	+	0.312156i	$0.898949\pi$
$192$		0			0
$193$	137.385			0.711838			0.355919	−	0.934517i	$-0.384168\pi$
$193$	137.385			0.711838			0.355919	+	0.934517i	$0.384168\pi$
$194$		0			0
$195$	32.0540	−	132.805i	0.164379	−	0.681054i
$196$		0			0
$197$			202.161i			1.02620i	0.858330	+	0.513098i	$0.171502\pi$
$197$			202.161i			1.02620i	−0.858330	+	0.513098i	$0.828498\pi$
$198$		0			0
$199$	−59.0561			−0.296764			−0.148382	−	0.988930i	$-0.547407\pi$
$199$	−59.0561			−0.296764			−0.148382	+	0.988930i	$0.547407\pi$
$200$		0			0
$201$	223.004	+	53.8243i	1.10947	+	0.267783i
$202$		0			0
$203$			99.2425i			0.488879i
$204$		0			0
$205$	−45.2644			−0.220802
$206$		0			0
$207$	−47.0661	+	91.8218i	−0.227373	+	0.443583i
$208$		0			0
$209$			57.3563i			0.274432i
$210$		0			0
$211$	−13.5306			−0.0641261			−0.0320630	−	0.999486i	$-0.510208\pi$
$211$	−13.5306			−0.0641261			−0.0320630	+	0.999486i	$0.510208\pi$
$212$		0			0
$213$	−33.1910	+	137.516i	−0.155826	+	0.645617i
$214$		0			0
$215$			13.1694i			0.0612530i
$216$		0			0
$217$	−56.1645			−0.258823
$218$		0			0
$219$	396.201	+	95.6274i	1.80914	+	0.436655i
$220$		0			0
$221$		−	202.071i		−	0.914349i
$222$		0			0
$223$	342.401			1.53543			0.767715	−	0.640792i	$-0.221394\pi$
$223$	342.401			1.53543			0.767715	+	0.640792i	$0.221394\pi$
$224$		0			0
$225$	−40.0457	−	20.5267i	−0.177981	−	0.0912296i
$226$		0			0
$227$			15.4878i			0.0682283i	0.999418	+	0.0341141i	$0.0108610\pi$
$227$			15.4878i			0.0682283i	−0.999418	+	0.0341141i	$0.989139\pi$
$228$		0			0
$229$	−178.803			−0.780798			−0.390399	−	0.920646i	$-0.627663\pi$
$229$	−178.803			−0.780798			−0.390399	+	0.920646i	$0.627663\pi$
$230$		0			0
$231$	−21.7108	+	89.9518i	−0.0939862	+	0.389402i
$232$		0			0
$233$		−	415.344i		−	1.78259i	−0.453422	−	0.891296i	$-0.649797\pi$
$233$		−	415.344i		−	1.78259i	0.453422	−	0.891296i	$-0.350203\pi$
$234$		0			0
$235$	166.598			0.708929
$236$		0			0
$237$	−286.584	−	69.1701i	−1.20922	−	0.291857i
$238$		0			0
$239$			204.961i			0.857577i	0.903405	+	0.428788i	$0.141059\pi$
$239$			204.961i			0.857577i	−0.903405	+	0.428788i	$0.858941\pi$
$240$		0			0
$241$	−75.9054			−0.314960			−0.157480	−	0.987522i	$-0.550337\pi$
$241$	−75.9054			−0.314960			−0.157480	+	0.987522i	$0.550337\pi$
$242$		0			0
$243$	−91.6305	−	225.062i	−0.377080	−	0.926181i
$244$		0			0
$245$		−	15.6525i		−	0.0638877i
$246$		0			0
$247$	−100.196			−0.405653
$248$		0			0
$249$	−39.3247	+	162.929i	−0.157930	+	0.654335i
$250$		0			0
$251$		−	25.9960i		−	0.103570i	−0.998658	−	0.0517849i	$-0.983509\pi$
$251$		−	25.9960i		−	0.103570i	0.998658	−	0.0517849i	$-0.0164910\pi$
$252$		0			0
$253$	−133.658			−0.528292
$254$		0			0
$255$	−64.7009	−	15.6162i	−0.253729	−	0.0612402i
$256$		0			0
$257$		−	244.092i		−	0.949775i	−0.880047	−	0.474887i	$-0.842489\pi$
$257$		−	244.092i		−	0.949775i	0.880047	−	0.474887i	$-0.157511\pi$
$258$		0			0
$259$	−151.321			−0.584250
$260$		0			0
$261$	153.992	−	300.424i	0.590006	−	1.15105i
$262$		0			0
$263$			98.5645i			0.374770i	0.982287	+	0.187385i	$0.0600012\pi$
$263$			98.5645i			0.374770i	−0.982287	+	0.187385i	$0.939999\pi$
$264$		0			0
$265$	197.221			0.744232
$266$		0			0
$267$	−74.3749	+	308.149i	−0.278558	+	1.15412i
$268$		0			0
$269$			32.3831i			0.120383i	0.998187	+	0.0601917i	$0.0191712\pi$
$269$			32.3831i			0.120383i	−0.998187	+	0.0601917i	$0.980829\pi$
$270$		0			0
$271$	−251.624			−0.928501			−0.464251	−	0.885704i	$-0.653676\pi$
$271$	−251.624			−0.928501			−0.464251	+	0.885704i	$0.653676\pi$
$272$		0			0
$273$	−157.138	−	37.9268i	−0.575596	−	0.138926i
$274$		0			0
$275$		−	58.2915i		−	0.211969i
$276$		0			0
$277$	−319.412			−1.15311			−0.576556	−	0.817058i	$-0.695604\pi$
$277$	−319.412			−1.15311			−0.576556	+	0.817058i	$0.695604\pi$
$278$		0			0
$279$	170.019	+	87.1488i	0.609389	+	0.312361i
$280$		0			0
$281$			234.200i			0.833450i	0.909033	+	0.416725i	$0.136822\pi$
$281$			234.200i			0.833450i	−0.909033	+	0.416725i	$0.863178\pi$
$282$		0			0
$283$	430.416			1.52091			0.760453	−	0.649393i	$-0.224977\pi$
$283$	430.416			1.52091			0.760453	+	0.649393i	$0.224977\pi$
$284$		0			0
$285$	−7.74325	+	32.0817i	−0.0271693	+	0.112567i
$286$		0			0
$287$			53.5575i			0.186612i
$288$		0			0
$289$	190.554			0.659356
$290$		0			0
$291$	−180.515	−	43.5693i	−0.620328	−	0.149723i
$292$		0			0
$293$		−	359.848i		−	1.22815i	−0.789247	−	0.614075i	$-0.789529\pi$
$293$		−	359.848i		−	1.22815i	0.789247	−	0.614075i	$-0.210471\pi$
$294$		0			0
$295$	−160.234			−0.543166
$296$		0			0
$297$	205.298	−	238.611i	0.691238	−	0.803405i
$298$		0			0
$299$		−	233.488i		−	0.780897i
$300$		0			0
$301$	15.5822			0.0517683
$302$		0			0
$303$	−7.86280	+	32.5770i	−0.0259498	+	0.107515i
$304$		0			0
$305$			58.2411i			0.190955i
$306$		0			0
$307$	−146.927			−0.478590			−0.239295	−	0.970947i	$-0.576916\pi$
$307$	−146.927			−0.478590			−0.239295	+	0.970947i	$0.576916\pi$
$308$		0			0
$309$	112.594	+	27.1758i	0.364383	+	0.0879476i
$310$		0			0
$311$		−	18.9887i		−	0.0610570i	−0.999534	−	0.0305285i	$-0.990281\pi$
$311$		−	18.9887i		−	0.0610570i	0.999534	−	0.0305285i	$-0.00971903\pi$
$312$		0			0
$313$	−253.927			−0.811268			−0.405634	−	0.914036i	$-0.632949\pi$
$313$	−253.927			−0.811268			−0.405634	+	0.914036i	$0.632949\pi$
$314$		0			0
$315$	−24.2875	+	47.3827i	−0.0771031	+	0.150421i
$316$		0			0
$317$		−	576.396i		−	1.81829i	−0.416485	−	0.909143i	$-0.636738\pi$
$317$		−	576.396i		−	1.81829i	0.416485	−	0.909143i	$-0.363262\pi$
$318$		0			0
$319$	437.304			1.37086
$320$		0			0
$321$	−138.530	+	573.954i	−0.431557	+	1.78802i
$322$		0			0
$323$			48.8141i			0.151127i
$324$		0			0
$325$	101.830			0.313322
$326$		0			0
$327$	−138.189	−	33.3533i	−0.422595	−	0.101998i
$328$		0			0
$329$		−	197.122i		−	0.599155i
$330$		0			0
$331$	−246.809			−0.745646			−0.372823	−	0.927903i	$-0.621610\pi$
$331$	−246.809			−0.745646			−0.372823	+	0.927903i	$0.621610\pi$
$332$		0			0
$333$	458.073	+	234.800i	1.37560	+	0.705104i
$334$		0			0
$335$			170.990i			0.510418i
$336$		0			0
$337$	−137.732			−0.408701			−0.204351	−	0.978898i	$-0.565508\pi$
$337$	−137.732			−0.408701			−0.204351	+	0.978898i	$0.565508\pi$
$338$		0			0
$339$	45.2336	−	187.411i	0.133432	−	0.552835i
$340$		0			0
$341$			247.484i			0.725761i
$342$		0			0
$343$	−18.5203			−0.0539949
$344$		0			0
$345$	−74.7603	−	18.0442i	−0.216697	−	0.0523020i
$346$		0			0
$347$		−	445.192i		−	1.28298i	−0.767133	−	0.641488i	$-0.778318\pi$
$347$		−	445.192i		−	1.28298i	0.767133	−	0.641488i	$-0.221682\pi$
$348$		0			0
$349$	483.226			1.38460			0.692301	−	0.721609i	$-0.256597\pi$
$349$	483.226			1.38460			0.692301	+	0.721609i	$0.256597\pi$
$350$		0			0
$351$	416.832	+	358.636i	1.18756	+	1.02176i
$352$		0			0
$353$			229.392i			0.649835i	0.945742	+	0.324917i	$0.105336\pi$
$353$			229.392i			0.649835i	−0.945742	+	0.324917i	$0.894664\pi$
$354$		0			0
$355$	−105.442			−0.297020
$356$		0			0
$357$	−18.4774	+	76.5552i	−0.0517574	+	0.214440i
$358$		0			0
$359$		−	33.5036i		−	0.0933247i	−0.998911	−	0.0466624i	$-0.985142\pi$
$359$		−	33.5036i		−	0.0933247i	0.998911	−	0.0466624i	$-0.0148585\pi$
$360$		0			0
$361$	−336.796			−0.932952
$362$		0			0
$363$	43.4982	+	10.4987i	0.119830	+	0.0289221i
$364$		0			0
$365$			303.791i			0.832304i
$366$		0			0
$367$	−466.167			−1.27021			−0.635105	−	0.772426i	$-0.719043\pi$
$367$	−466.167			−1.27021			−0.635105	+	0.772426i	$0.719043\pi$
$368$		0			0
$369$	83.1036	−	162.128i	0.225213	−	0.439370i
$370$		0			0
$371$		−	233.356i		−	0.628991i
$372$		0			0
$373$	−583.312			−1.56384			−0.781920	−	0.623379i	$-0.785759\pi$
$373$	−583.312			−1.56384			−0.781920	+	0.623379i	$0.785759\pi$
$374$		0			0
$375$	7.86950	−	32.6048i	0.0209853	−	0.0869461i
$376$		0			0
$377$			763.930i			2.02634i
$378$		0			0
$379$	320.826			0.846507			0.423253	−	0.906011i	$-0.360888\pi$
$379$	320.826			0.846507			0.423253	+	0.906011i	$0.360888\pi$
$380$		0			0
$381$	−258.531	−	62.3991i	−0.678559	−	0.163777i
$382$		0			0
$383$			636.582i			1.66209i	0.556203	+	0.831046i	$0.312258\pi$
$383$			636.582i			1.66209i	−0.556203	+	0.831046i	$0.687742\pi$
$384$		0			0
$385$	−68.9714			−0.179146
$386$		0			0
$387$	−47.1701	−	24.1785i	−0.121887	−	0.0624767i
$388$		0			0
$389$		−	67.4908i		−	0.173498i	−0.996230	−	0.0867491i	$-0.972352\pi$
$389$		−	67.4908i		−	0.173498i	0.996230	−	0.0867491i	$-0.0276478\pi$
$390$		0			0
$391$	−113.752			−0.290926
$392$		0			0
$393$	−82.0753	+	340.053i	−0.208843	+	0.865274i
$394$		0			0
$395$		−	219.741i		−	0.556306i
$396$		0			0
$397$	567.020			1.42826			0.714131	−	0.700012i	$-0.246822\pi$
$397$	567.020			1.42826			0.714131	+	0.700012i	$0.246822\pi$
$398$		0			0
$399$	37.9596	+	9.16194i	0.0951368	+	0.0229623i
$400$		0			0
$401$		−	151.329i		−	0.377378i	−0.982037	−	0.188689i	$-0.939576\pi$
$401$		−	151.329i		−	0.377378i	0.982037	−	0.188689i	$-0.0604238\pi$
$402$		0			0
$403$	−432.333			−1.07279
$404$		0			0
$405$	147.045	−	105.749i	0.363073	−	0.261109i
$406$		0			0
$407$			666.783i			1.63829i
$408$		0			0
$409$	557.799			1.36381			0.681906	−	0.731440i	$-0.261151\pi$
$409$	557.799			1.36381			0.681906	+	0.731440i	$0.261151\pi$
$410$		0			0
$411$	−56.8045	+	235.352i	−0.138211	+	0.572631i
$412$		0			0
$413$			189.592i			0.459059i
$414$		0			0
$415$	−124.928			−0.301030
$416$		0			0
$417$	568.004	+	137.094i	1.36212	+	0.328762i
$418$		0			0
$419$		−	217.458i		−	0.518992i	−0.965744	−	0.259496i	$-0.916444\pi$
$419$		−	217.458i		−	0.518992i	0.965744	−	0.259496i	$-0.0835564\pi$
$420$		0			0
$421$	344.111			0.817366			0.408683	−	0.912676i	$-0.365988\pi$
$421$	344.111			0.817366			0.408683	+	0.912676i	$0.365988\pi$
$422$		0			0
$423$	−305.868	+	596.721i	−0.723092	+	1.41069i
$424$		0			0
$425$		−	49.6100i		−	0.116729i
$426$		0			0
$427$	68.9118			0.161386
$428$		0			0
$429$	−167.121	+	692.414i	−0.389560	+	1.61402i
$430$		0			0
$431$			733.518i			1.70190i	0.525248	+	0.850949i	$0.323973\pi$
$431$			733.518i			1.70190i	−0.525248	+	0.850949i	$0.676027\pi$
$432$		0			0
$433$	489.197			1.12978			0.564892	−	0.825165i	$-0.308918\pi$
$433$	489.197			1.12978			0.564892	+	0.825165i	$0.308918\pi$
$434$		0			0
$435$	244.602	+	59.0372i	0.562303	+	0.135718i
$436$		0			0
$437$			56.4035i			0.129070i
$438$		0			0
$439$	787.369			1.79355			0.896775	−	0.442486i	$-0.145903\pi$
$439$	787.369			1.79355			0.896775	+	0.442486i	$0.145903\pi$
$440$		0			0
$441$	56.0639	+	28.7373i	0.127129	+	0.0651640i
$442$		0			0
$443$		−	351.199i		−	0.792773i	−0.918084	−	0.396387i	$-0.870264\pi$
$443$		−	351.199i		−	0.792773i	0.918084	−	0.396387i	$-0.129736\pi$
$444$		0			0
$445$	−236.276			−0.530957
$446$		0			0
$447$	−177.119	+	733.835i	−0.396239	+	1.64169i
$448$		0			0
$449$			85.6820i			0.190828i	0.995438	+	0.0954142i	$0.0304176\pi$
$449$			85.6820i			0.190828i	−0.995438	+	0.0954142i	$0.969582\pi$
$450$		0			0
$451$	235.997			0.523275
$452$		0			0
$453$	405.113	+	97.7782i	0.894289	+	0.215846i
$454$		0			0
$455$		−	120.487i		−	0.264806i
$456$		0			0
$457$	−442.093			−0.967381			−0.483691	−	0.875239i	$-0.660704\pi$
$457$	−442.093			−0.967381			−0.483691	+	0.875239i	$0.660704\pi$
$458$		0			0
$459$	174.722	−	203.075i	0.380659	−	0.442428i
$460$		0			0
$461$		−	386.251i		−	0.837854i	−0.908020	−	0.418927i	$-0.862406\pi$
$461$		−	386.251i		−	0.837854i	0.908020	−	0.418927i	$-0.137594\pi$
$462$		0			0
$463$	300.426			0.648868			0.324434	−	0.945908i	$-0.394826\pi$
$463$	300.426			0.648868			0.324434	+	0.945908i	$0.394826\pi$
$464$		0			0
$465$	−33.4111	+	138.428i	−0.0718517	+	0.297695i
$466$		0			0
$467$			29.5201i			0.0632123i	0.999500	+	0.0316062i	$0.0100622\pi$
$467$			29.5201i			0.0632123i	−0.999500	+	0.0316062i	$0.989938\pi$
$468$		0			0
$469$	202.318			0.431382
$470$		0			0
$471$	75.3798	+	18.1937i	0.160042	+	0.0386278i
$472$		0			0
$473$		−	68.6619i		−	0.145163i
$474$		0			0
$475$	−24.5989			−0.0517872
$476$		0			0
$477$	−362.091	+	706.407i	−0.759100	+	1.48094i
$478$		0			0
$479$		−	183.487i		−	0.383063i	−0.981486	−	0.191531i	$-0.938655\pi$
$479$		−	183.487i		−	0.383063i	0.981486	−	0.191531i	$-0.0613454\pi$
$480$		0			0
$481$	−1164.81			−2.42164
$482$		0			0
$483$	−21.3502	+	88.4576i	−0.0442032	+	0.183142i
$484$		0			0
$485$		−	138.412i		−	0.285385i
$486$		0			0
$487$	529.580			1.08743			0.543717	−	0.839269i	$-0.317016\pi$
$487$	529.580			1.08743			0.543717	+	0.839269i	$0.317016\pi$
$488$		0			0
$489$	276.807	+	66.8104i	0.566068	+	0.136627i
$490$		0			0
$491$			640.468i			1.30442i	0.758040	+	0.652208i	$0.226157\pi$
$491$			640.468i			1.30442i	−0.758040	+	0.652208i	$0.773843\pi$
$492$		0			0
$493$	372.176			0.754920
$494$		0			0
$495$	208.788	+	107.021i	0.421794	+	0.216204i
$496$		0			0
$497$			124.761i			0.251027i
$498$		0			0
$499$	−121.650			−0.243788			−0.121894	−	0.992543i	$-0.538897\pi$
$499$	−121.650			−0.243788			−0.121894	+	0.992543i	$0.538897\pi$
$500$		0			0
$501$	−228.778	+	947.870i	−0.456643	+	1.89196i
$502$		0			0
$503$		−	524.536i		−	1.04281i	−0.853308	−	0.521407i	$-0.825407\pi$
$503$		−	524.536i		−	1.04281i	0.853308	−	0.521407i	$-0.174593\pi$
$504$		0			0
$505$	−24.9787			−0.0494628
$506$		0			0
$507$	−716.736	−	172.992i	−1.41368	−	0.341206i
$508$		0			0
$509$		−	572.981i		−	1.12570i	−0.826559	−	0.562849i	$-0.809705\pi$
$509$		−	572.981i		−	1.12570i	0.826559	−	0.562849i	$-0.190295\pi$
$510$		0			0
$511$	359.450			0.703425
$512$		0			0
$513$	−100.694	−	86.6355i	−0.196284	−	0.168880i
$514$		0			0
$515$			86.3327i			0.167636i
$516$		0			0
$517$	−868.602			−1.68008
$518$		0			0
$519$	−68.6635	+	284.485i	−0.132300	+	0.548141i
$520$		0			0
$521$		−	247.906i		−	0.475827i	−0.971286	−	0.237913i	$-0.923537\pi$
$521$		−	247.906i		−	0.475827i	0.971286	−	0.237913i	$-0.0764634\pi$
$522$		0			0
$523$	431.945			0.825898			0.412949	−	0.910754i	$-0.364499\pi$
$523$	431.945			0.825898			0.412949	+	0.910754i	$0.364499\pi$
$524$		0			0
$525$	−38.5785	−	9.31132i	−0.0734828	−	0.0177358i
$526$		0			0
$527$			210.626i			0.399670i
$528$		0			0
$529$	397.562			0.751535
$530$		0			0
$531$	294.183	−	573.926i	0.554018	−	1.08084i
$532$		0			0
$533$			412.265i			0.773480i
$534$		0			0
$535$	−440.084			−0.822587
$536$		0			0
$537$	−121.638	+	503.967i	−0.226513	+	0.938486i
$538$		0			0
$539$			81.6080i			0.151406i
$540$		0			0
$541$	961.822			1.77786			0.888930	−	0.458044i	$-0.151450\pi$
$541$	961.822			1.77786			0.888930	+	0.458044i	$0.151450\pi$
$542$		0			0
$543$	23.7849	+	5.74073i	0.0438028	+	0.0105723i
$544$		0			0
$545$		−	105.957i		−	0.194417i
$546$		0			0
$547$	82.6144			0.151032			0.0755159	−	0.997145i	$-0.475940\pi$
$547$	82.6144			0.151032			0.0755159	+	0.997145i	$0.475940\pi$
$548$		0			0
$549$	−208.608	−	106.928i	−0.379978	−	0.194769i
$550$		0			0
$551$		−	184.542i		−	0.334922i
$552$		0			0
$553$	−260.001			−0.470165
$554$		0			0
$555$	−90.0174	+	372.959i	−0.162194	+	0.671997i
$556$		0			0
$557$			576.766i			1.03549i	0.855536	+	0.517743i	$0.173228\pi$
$557$			576.766i			1.03549i	−0.855536	+	0.517743i	$0.826772\pi$
$558$		0			0
$559$	119.946			0.214573
$560$		0			0
$561$	337.334	+	81.4191i	0.601309	+	0.145132i
$562$		0			0
$563$		−	486.496i		−	0.864115i	−0.901846	−	0.432057i	$-0.857788\pi$
$563$		−	486.496i		−	0.864115i	0.901846	−	0.432057i	$-0.142212\pi$
$564$		0			0
$565$	143.699			0.254335
$566$		0			0
$567$	−125.124	−	173.985i	−0.220678	−	0.306853i
$568$		0			0
$569$			355.587i			0.624934i	0.949929	+	0.312467i	$0.101155\pi$
$569$			355.587i			0.624934i	−0.949929	+	0.312467i	$0.898845\pi$
$570$		0			0
$571$	−104.407			−0.182849			−0.0914244	−	0.995812i	$-0.529142\pi$
$571$	−104.407			−0.182849			−0.0914244	+	0.995812i	$0.529142\pi$
$572$		0			0
$573$	83.9321	−	347.746i	0.146478	−	0.606886i
$574$		0			0
$575$		−	57.3231i		−	0.0996924i
$576$		0			0
$577$	−502.560			−0.870987			−0.435494	−	0.900192i	$-0.643426\pi$
$577$	−502.560			−0.870987			−0.435494	+	0.900192i	$0.643426\pi$
$578$		0			0
$579$	−400.649	−	96.7009i	−0.691968	−	0.167014i
$580$		0			0
$581$			147.816i			0.254417i
$582$		0			0
$583$	−1028.26			−1.76374
$584$		0			0
$585$	−186.955	+	364.733i	−0.319582	+	0.623476i
$586$		0			0
$587$			361.958i			0.616623i	0.951285	+	0.308312i	$0.0997639\pi$
$587$			361.958i			0.616623i	−0.951285	+	0.308312i	$0.900236\pi$
$588$		0			0
$589$	104.438			0.177314
$590$		0			0
$591$	142.295	−	589.553i	0.240769	−	0.997551i
$592$		0			0
$593$		−	761.082i		−	1.28344i	−0.766938	−	0.641722i	$-0.778220\pi$
$593$		−	761.082i		−	1.28344i	0.766938	−	0.641722i	$-0.221780\pi$
$594$		0			0
$595$	−58.6994			−0.0986544
$596$		0			0
$597$	172.223	+	41.5678i	0.288481	+	0.0696278i
$598$		0			0
$599$			534.644i			0.892560i	0.894893	+	0.446280i	$0.147252\pi$
$599$			534.644i			0.892560i	−0.894893	+	0.446280i	$0.852748\pi$
$600$		0			0
$601$	−237.242			−0.394746			−0.197373	−	0.980328i	$-0.563241\pi$
$601$	−237.242			−0.394746			−0.197373	+	0.980328i	$0.563241\pi$
$602$		0			0
$603$	−612.452	−	313.931i	−1.01567	−	0.520615i
$604$		0			0
$605$			33.3526i			0.0551283i
$606$		0			0
$607$	−530.470			−0.873921			−0.436961	−	0.899481i	$-0.643945\pi$
$607$	−530.470			−0.873921			−0.436961	+	0.899481i	$0.643945\pi$
$608$		0			0
$609$	69.8537	−	289.417i	0.114702	−	0.475233i
$610$		0			0
$611$		−	1517.37i		−	2.48342i
$612$		0			0
$613$	313.472			0.511374			0.255687	−	0.966760i	$-0.417698\pi$
$613$	313.472			0.511374			0.255687	+	0.966760i	$0.417698\pi$
$614$		0			0
$615$	132.003	+	31.8602i	0.214638	+	0.0518052i
$616$		0			0
$617$		−	752.431i		−	1.21950i	−0.792594	−	0.609750i	$-0.791270\pi$
$617$		−	752.431i		−	1.21950i	0.792594	−	0.609750i	$-0.208730\pi$
$618$		0			0
$619$	157.272			0.254074			0.127037	−	0.991898i	$-0.459453\pi$
$619$	157.272			0.254074			0.127037	+	0.991898i	$0.459453\pi$
$620$		0			0
$621$	201.888	−	234.648i	0.325101	−	0.377855i
$622$		0			0
$623$			279.565i			0.448741i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	40.3713	−	167.266i	0.0643881	−	0.266772i
$628$		0			0
$629$			567.478i			0.902190i
$630$		0			0
$631$	481.330			0.762806			0.381403	−	0.924409i	$-0.375441\pi$
$631$	481.330			0.762806			0.381403	+	0.924409i	$0.375441\pi$
$632$		0			0
$633$	39.4588	+	9.52378i	0.0623361	+	0.0150455i
$634$		0			0
$635$		−	198.231i		−	0.312175i
$636$		0			0
$637$	−142.562			−0.223802
$638$		0			0
$639$	193.587	−	377.671i	0.302953	−	0.591035i
$640$		0			0
$641$			91.6756i			0.143020i	0.997440	+	0.0715098i	$0.0227817\pi$
$641$			91.6756i			0.143020i	−0.997440	+	0.0715098i	$0.977218\pi$
$642$		0			0
$643$	983.203			1.52909			0.764544	−	0.644572i	$-0.222964\pi$
$643$	983.203			1.52909			0.764544	+	0.644572i	$0.222964\pi$
$644$		0			0
$645$	9.26954	−	38.4054i	0.0143714	−	0.0595432i
$646$		0			0
$647$		−	146.732i		−	0.226788i	−0.993550	−	0.113394i	$-0.963828\pi$
$647$		−	146.732i		−	0.226788i	0.993550	−	0.113394i	$-0.0361722\pi$
$648$		0			0
$649$	835.420			1.28724
$650$		0			0
$651$	163.790	+	39.5325i	0.251598	+	0.0607258i
$652$		0			0
$653$			589.430i			0.902649i	0.892360	+	0.451324i	$0.149048\pi$
$653$			589.430i			0.902649i	−0.892360	+	0.451324i	$0.850952\pi$
$654$		0			0
$655$	−260.738			−0.398074
$656$		0			0
$657$	−1088.12	−	557.748i	−1.65619	−	0.848932i
$658$		0			0
$659$		−	27.8483i		−	0.0422584i	−0.999777	−	0.0211292i	$-0.993274\pi$
$659$		−	27.8483i		−	0.0422584i	0.999777	−	0.0211292i	$-0.00672613\pi$
$660$		0			0
$661$	60.5008			0.0915292			0.0457646	−	0.998952i	$-0.485428\pi$
$661$	60.5008			0.0915292			0.0457646	+	0.998952i	$0.485428\pi$
$662$		0			0
$663$	−142.232	+	589.292i	−0.214527	+	0.888826i
$664$		0			0
$665$			29.1058i			0.0437682i
$666$		0			0
$667$	430.040			0.644737
$668$		0			0
$669$	−998.529	−	241.005i	−1.49257	−	0.360247i
$670$		0			0
$671$		−	303.654i		−	0.452540i
$672$		0			0
$673$	1111.43			1.65145			0.825727	−	0.564070i	$-0.190765\pi$
$673$	1111.43			1.65145			0.825727	+	0.564070i	$0.190765\pi$
$674$		0			0
$675$	102.335	+	88.0480i	0.151608	+	0.130441i
$676$		0			0
$677$			703.912i			1.03975i	0.854242	+	0.519876i	$0.174022\pi$
$677$			703.912i			1.03975i	−0.854242	+	0.519876i	$0.825978\pi$
$678$		0			0
$679$	−163.771			−0.241194
$680$		0			0
$681$	10.9014	−	45.1665i	0.0160079	−	0.0663238i
$682$		0			0
$683$			828.212i			1.21261i	0.795233	+	0.606305i	$0.207349\pi$
$683$			828.212i			1.21261i	−0.795233	+	0.606305i	$0.792651\pi$
$684$		0			0
$685$	−180.458			−0.263442
$686$		0			0
$687$	521.435	+	125.854i	0.759003	+	0.183193i
$688$		0			0
$689$		−	1796.28i		−	2.60708i
$690$		0			0
$691$	−287.201			−0.415631			−0.207816	−	0.978168i	$-0.566635\pi$
$691$	−287.201			−0.415631			−0.207816	+	0.978168i	$0.566635\pi$
$692$		0			0
$693$	126.629	−	247.041i	0.182725	−	0.356481i
$694$		0			0
$695$			435.522i			0.626650i
$696$		0			0
$697$	200.849			0.288163
$698$		0			0
$699$	−292.348	+	1211.25i	−0.418237	+	1.73283i
$700$		0			0
$701$		−	285.350i		−	0.407061i	−0.979069	−	0.203531i	$-0.934758\pi$
$701$		−	285.350i		−	0.407061i	0.979069	−	0.203531i	$-0.0652416\pi$
$702$		0			0
$703$	281.382			0.400258
$704$		0			0
$705$	−485.844	−	117.264i	−0.689141	−	0.166331i
$706$		0			0
$707$			29.5552i			0.0418037i
$708$		0			0
$709$	522.639			0.737149			0.368575	−	0.929598i	$-0.379846\pi$
$709$	522.639			0.737149			0.368575	+	0.929598i	$0.379846\pi$
$710$		0			0
$711$	787.067	+	403.436i	1.10699	+	0.567420i
$712$		0			0
$713$			243.373i			0.341337i
$714$		0			0
$715$	−530.915			−0.742538
$716$		0			0
$717$	144.266	−	597.719i	0.201207	−	0.833639i
$718$		0			0
$719$			640.717i			0.891123i	0.895251	+	0.445561i	$0.146996\pi$
$719$			640.717i			0.891123i	−0.895251	+	0.445561i	$0.853004\pi$
$720$		0			0
$721$	102.150			0.141679
$722$		0			0
$723$	221.360	+	53.4275i	0.306169	+	0.0738970i
$724$		0			0
$725$			187.551i			0.258691i
$726$		0			0
$727$	−466.941			−0.642285			−0.321142	−	0.947031i	$-0.604067\pi$
$727$	−466.941			−0.642285			−0.321142	+	0.947031i	$0.604067\pi$
$728$		0			0
$729$	108.804	+	720.835i	0.149251	+	0.988799i
$730$		0			0
$731$		−	58.4360i		−	0.0799398i
$732$		0			0
$733$	418.565			0.571030			0.285515	−	0.958374i	$-0.407835\pi$
$733$	418.565			0.571030			0.285515	+	0.958374i	$0.407835\pi$
$734$		0			0
$735$	−11.0173	+	45.6467i	−0.0149895	+	0.0621043i
$736$		0			0
$737$		−	891.499i		−	1.20963i
$738$		0			0
$739$	−112.745			−0.152565			−0.0762824	−	0.997086i	$-0.524305\pi$
$739$	−112.745			−0.152565			−0.0762824	+	0.997086i	$0.524305\pi$
$740$		0			0
$741$	292.198	+	70.5250i	0.394329	+	0.0951755i
$742$		0			0
$743$		−	195.638i		−	0.263308i	−0.991296	−	0.131654i	$-0.957971\pi$
$743$		−	195.638i		−	0.263308i	0.991296	−	0.131654i	$-0.0420288\pi$
$744$		0			0
$745$	−562.674			−0.755268
$746$		0			0
$747$	229.362	−	447.465i	0.307044	−	0.599016i
$748$		0			0
$749$			520.714i			0.695213i
$750$		0			0
$751$	216.850			0.288748			0.144374	−	0.989523i	$-0.453883\pi$
$751$	216.850			0.288748			0.144374	+	0.989523i	$0.453883\pi$
$752$		0			0
$753$	−18.2978	+	75.8111i	−0.0242999	+	0.100679i
$754$		0			0
$755$			310.624i			0.411422i
$756$		0			0
$757$	−944.974			−1.24831			−0.624157	−	0.781299i	$-0.714558\pi$
$757$	−944.974			−1.24831			−0.624157	+	0.781299i	$0.714558\pi$
$758$		0			0
$759$	389.781	+	94.0778i	0.513546	+	0.123950i
$760$		0			0
$761$			438.658i			0.576424i	0.957567	+	0.288212i	$0.0930607\pi$
$761$			438.658i			0.576424i	−0.957567	+	0.288212i	$0.906939\pi$
$762$		0			0
$763$	−125.370			−0.164312
$764$		0			0
$765$	177.693	+	91.0820i	0.232278	+	0.119061i
$766$		0			0
$767$			1459.40i			1.90274i
$768$		0			0
$769$	−265.037			−0.344651			−0.172325	−	0.985040i	$-0.555128\pi$
$769$	−265.037			−0.344651			−0.172325	+	0.985040i	$0.555128\pi$
$770$		0			0
$771$	−171.809	+	711.836i	−0.222839	+	0.923263i
$772$		0			0
$773$		−	500.020i		−	0.646856i	−0.946253	−	0.323428i	$-0.895165\pi$
$773$		−	500.020i		−	0.646856i	0.946253	−	0.323428i	$-0.104835\pi$
$774$		0			0
$775$	−106.141			−0.136956
$776$		0			0
$777$	441.290	+	106.510i	0.567941	+	0.137079i
$778$		0			0
$779$		−	99.5904i		−	0.127844i
$780$		0			0
$781$	549.748			0.703902
$782$		0			0
$783$	−660.538	+	767.723i	−0.843599	+	0.980490i
$784$		0			0
$785$			57.7982i			0.0736282i
$786$		0			0
$787$	−466.854			−0.593207			−0.296603	−	0.955001i	$-0.595854\pi$
$787$	−466.854			−0.593207			−0.296603	+	0.955001i	$0.595854\pi$
$788$		0			0
$789$	69.3765	−	287.440i	0.0879297	−	0.364309i
$790$		0			0
$791$		−	170.027i		−	0.214952i
$792$		0			0
$793$	530.456			0.668924
$794$		0			0
$795$	−575.149	−	138.818i	−0.723458	−	0.174614i
$796$		0			0
$797$			1227.75i			1.54046i	0.637763	+	0.770232i	$0.279860\pi$
$797$			1227.75i			1.54046i	−0.637763	+	0.770232i	$0.720140\pi$
$798$		0			0
$799$	−739.239			−0.925206
$800$		0			0
$801$	433.793	−	846.292i	0.541564	−	1.05654i
$802$		0			0
$803$		−	1583.89i		−	1.97246i
$804$		0			0
$805$	−67.8257			−0.0842555
$806$		0			0
$807$	22.7935	−	94.4376i	0.0282447	−	0.117023i
$808$		0			0
$809$		−	327.028i		−	0.404237i	−0.979361	−	0.202119i	$-0.935217\pi$
$809$		−	327.028i		−	0.404237i	0.979361	−	0.202119i	$-0.0647826\pi$
$810$		0			0
$811$	−1069.52			−1.31877			−0.659384	−	0.751807i	$-0.729183\pi$
$811$	−1069.52			−1.31877			−0.659384	+	0.751807i	$0.729183\pi$
$812$		0			0
$813$	733.800	+	177.110i	0.902584	+	0.217848i
$814$		0			0
$815$			212.245i			0.260423i
$816$		0			0
$817$	−28.9753			−0.0354654
$818$		0			0
$819$	431.558	+	221.209i	0.526933	+	0.270096i
$820$		0			0
$821$			184.485i			0.224707i	0.993668	+	0.112354i	$0.0358389\pi$
$821$			184.485i			0.224707i	−0.993668	+	0.112354i	$0.964161\pi$
$822$		0			0
$823$	739.393			0.898412			0.449206	−	0.893428i	$-0.351707\pi$
$823$	739.393			0.898412			0.449206	+	0.893428i	$0.351707\pi$
$824$		0			0
$825$	−41.0296	+	169.993i	−0.0497328	+	0.206052i
$826$		0			0
$827$		−	1253.54i		−	1.51577i	−0.652388	−	0.757885i	$-0.726233\pi$
$827$		−	1253.54i		−	1.51577i	0.652388	−	0.757885i	$-0.273767\pi$
$828$		0			0
$829$	−1566.71			−1.88988			−0.944941	−	0.327240i	$-0.893881\pi$
$829$	−1566.71			−1.88988			−0.944941	+	0.327240i	$0.893881\pi$
$830$		0			0
$831$	931.488	+	224.824i	1.12092	+	0.270547i
$832$		0			0
$833$			69.4540i			0.0833782i
$834$		0			0
$835$	−726.788			−0.870405
$836$		0			0
$837$	−434.479	−	373.820i	−0.519091	−	0.446619i
$838$		0			0
$839$		−	836.567i		−	0.997100i	−0.866861	−	0.498550i	$-0.833866\pi$
$839$		−	836.567i		−	0.997100i	0.866861	−	0.498550i	$-0.166134\pi$
$840$		0			0
$841$	−566.010			−0.673020
$842$		0			0
$843$	164.846	−	682.987i	0.195547	−	0.810186i
$844$		0			0
$845$		−	549.564i		−	0.650371i
$846$		0			0
$847$	39.4633			0.0465919
$848$		0			0
$849$	−1255.21	−	302.957i	−1.47845	−	0.356840i
$850$		0			0
$851$			655.706i			0.770513i
$852$		0			0
$853$	−361.497			−0.423794			−0.211897	−	0.977292i	$-0.567964\pi$
$853$	−361.497			−0.423794			−0.211897	+	0.977292i	$0.567964\pi$
$854$		0			0
$855$	45.1627	−	88.1083i	0.0528218	−	0.103051i
$856$		0			0
$857$			106.923i			0.124764i	0.998052	+	0.0623820i	$0.0198697\pi$
$857$			106.923i			0.124764i	−0.998052	+	0.0623820i	$0.980130\pi$
$858$		0			0
$859$	290.602			0.338302			0.169151	−	0.985590i	$-0.445897\pi$
$859$	290.602			0.338302			0.169151	+	0.985590i	$0.445897\pi$
$860$		0			0
$861$	37.6975	−	156.188i	0.0437834	−	0.181403i
$862$		0			0
$863$		−	74.3624i		−	0.0861673i	−0.999071	−	0.0430837i	$-0.986282\pi$
$863$		−	74.3624i		−	0.0861673i	0.999071	−	0.0430837i	$-0.0137182\pi$
$864$		0			0
$865$	−218.132			−0.252175
$866$		0			0
$867$	−555.705	−	134.125i	−0.640951	−	0.154700i
$868$		0			0
$869$			1145.67i			1.31838i
$870$		0			0
$871$	1557.37			1.78802
$872$		0			0
$873$	495.763	+	254.119i	0.567884	+	0.291087i
$874$		0			0
$875$		−	29.5804i		−	0.0338062i
$876$		0			0
$877$	−1221.03			−1.39228			−0.696138	−	0.717908i	$-0.745100\pi$
$877$	−1221.03			−1.39228			−0.696138	+	0.717908i	$0.745100\pi$
$878$		0			0
$879$	−253.286	+	1049.41i	−0.288153	+	1.19387i
$880$		0			0
$881$		−	946.113i		−	1.07391i	−0.843611	−	0.536954i	$-0.819575\pi$
$881$		−	946.113i		−	1.07391i	0.843611	−	0.536954i	$-0.180425\pi$
$882$		0			0
$883$	866.198			0.980971			0.490486	−	0.871449i	$-0.336819\pi$
$883$	866.198			0.980971			0.490486	+	0.871449i	$0.336819\pi$
$884$		0			0
$885$	467.284	+	112.784i	0.528005	+	0.127439i
$886$		0			0
$887$		−	814.659i		−	0.918443i	−0.888322	−	0.459222i	$-0.848128\pi$
$887$		−	814.659i		−	0.918443i	0.888322	−	0.459222i	$-0.151872\pi$
$888$		0			0
$889$	−234.550			−0.263836
$890$		0			0
$891$	−766.653	+	551.350i	−0.860441	+	0.618799i
$892$		0			0
$893$			366.549i			0.410469i
$894$		0			0
$895$	−386.421			−0.431756
$896$		0			0
$897$	−164.345	+	680.912i	−0.183217	+	0.759099i
$898$		0			0
$899$		−	796.272i		−	0.885731i
$900$		0			0
$901$	−875.122			−0.971278
$902$		0			0
$903$	−45.4419	−	10.9679i	−0.0503232	−	0.0121460i
$904$		0			0
$905$			18.2373i			0.0201517i
$906$		0			0
$907$	−374.381			−0.412768			−0.206384	−	0.978471i	$-0.566170\pi$
$907$	−374.381			−0.412768			−0.206384	+	0.978471i	$0.566170\pi$
$908$		0			0
$909$	45.8599	−	89.4686i	0.0504509	−	0.0984252i
$910$		0			0
$911$		−	586.070i		−	0.643326i	−0.946854	−	0.321663i	$-0.895758\pi$
$911$		−	586.070i		−	0.643326i	0.946854	−	0.321663i	$-0.104242\pi$
$912$		0			0
$913$	651.340			0.713407
$914$		0			0
$915$	40.9942	−	169.846i	0.0448024	−	0.185624i
$916$		0			0
$917$			308.510i			0.336434i
$918$		0			0
$919$	903.038			0.982632			0.491316	−	0.870982i	$-0.336516\pi$
$919$	903.038			0.982632			0.491316	+	0.870982i	$0.336516\pi$
$920$		0			0
$921$	428.478	+	103.418i	0.465231	+	0.112288i
$922$		0			0
$923$			960.358i			1.04047i
$924$		0			0
$925$	−285.969			−0.309156
$926$		0			0
$927$	−309.226	−	158.503i	−0.333577	−	0.170985i
$928$		0			0
$929$			1184.88i			1.27543i	0.770271	+	0.637717i	$0.220121\pi$
$929$			1184.88i			1.27543i	−0.770271	+	0.637717i	$0.779879\pi$
$930$		0			0
$931$	34.4385			0.0369909
$932$		0			0
$933$	−13.3656	+	55.3760i	−0.0143254	+	0.0593526i
$934$		0			0
$935$			258.654i			0.276635i
$936$		0			0
$937$	802.075			0.856003			0.428002	−	0.903778i	$-0.359218\pi$
$937$	802.075			0.856003			0.428002	+	0.903778i	$0.359218\pi$
$938$		0			0
$939$	740.517	+	178.731i	0.788623	+	0.190342i
$940$		0			0
$941$		−	379.438i		−	0.403229i	−0.979465	−	0.201614i	$-0.935381\pi$
$941$		−	379.438i		−	0.403229i	0.979465	−	0.201614i	$-0.0646188\pi$
$942$		0			0
$943$	232.077			0.246105
$944$		0			0
$945$	104.180	−	121.085i	0.110243	−	0.128132i
$946$		0			0
$947$		−	611.809i		−	0.646050i	−0.946391	−	0.323025i	$-0.895300\pi$
$947$		−	611.809i		−	0.646050i	0.946391	−	0.323025i	$-0.104700\pi$
$948$		0			0
$949$	2766.91			2.91560
$950$		0			0
$951$	−405.708	+	1680.92i	−0.426612	+	1.76753i
$952$		0			0
$953$			410.950i			0.431217i	0.976480	+	0.215609i	$0.0691735\pi$
$953$			410.950i			0.431217i	−0.976480	+	0.215609i	$0.930826\pi$
$954$		0			0
$955$	266.637			0.279201
$956$		0			0
$957$	−1275.29	−	307.805i	−1.33259	−	0.321635i
$958$		0			0
$959$			213.521i			0.222649i
$960$		0			0
$961$	−510.364			−0.531076
$962$		0			0
$963$	807.977	−	1576.29i	0.839021	−	1.63685i
$964$		0			0
$965$		−	307.201i		−	0.318343i
$966$		0			0
$967$	−1368.83			−1.41555			−0.707773	−	0.706440i	$-0.750300\pi$
$967$	−1368.83			−1.41555			−0.707773	+	0.706440i	$0.750300\pi$
$968$		0			0
$969$	34.3588	−	142.355i	0.0354580	−	0.146909i
$970$		0			0
$971$		−	918.521i		−	0.945954i	−0.881075	−	0.472977i	$-0.843179\pi$
$971$		−	918.521i		−	0.945954i	0.881075	−	0.472977i	$-0.156821\pi$
$972$		0			0
$973$	515.316			0.529616
$974$		0			0
$975$	−296.962	−	71.6749i	−0.304577	−	0.0735127i
$976$		0			0
$977$		−	1327.47i		−	1.35872i	−0.733807	−	0.679358i	$-0.762258\pi$
$977$		−	1327.47i		−	1.35872i	0.733807	−	0.679358i	$-0.237742\pi$
$978$		0			0
$979$	1231.88			1.25831
$980$		0			0
$981$	379.517	+	194.533i	0.386868	+	0.198301i
$982$		0			0
$983$		−	1045.07i		−	1.06314i	−0.847015	−	0.531570i	$-0.821602\pi$
$983$		−	1045.07i		−	1.06314i	0.847015	−	0.531570i	$-0.178398\pi$
$984$		0			0
$985$	452.045			0.458929
$986$		0			0
$987$	−138.748	+	574.858i	−0.140576	+	0.582430i
$988$		0			0
$989$		−	67.5213i		−	0.0682723i
$990$		0			0
$991$	636.280			0.642058			0.321029	−	0.947069i	$-0.395971\pi$
$991$	636.280			0.642058			0.321029	+	0.947069i	$0.395971\pi$
$992$		0			0
$993$	719.758	+	173.721i	0.724832	+	0.174946i
$994$		0			0
$995$			132.053i			0.132717i
$996$		0			0
$997$	607.263			0.609090			0.304545	−	0.952498i	$-0.401496\pi$
$997$	607.263			0.609090			0.304545	+	0.952498i	$0.401496\pi$
$998$		0			0
$999$	−1170.59	−	1007.16i	−1.17176	−	1.00817i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.l.a.1121.3		16
3.2	odd	2	inner	1680.3.l.a.1121.4		16
4.3	odd	2		105.3.c.a.71.12	yes	16
12.11	even	2		105.3.c.a.71.5	✓	16
20.3	even	4		525.3.f.b.449.23		32
20.7	even	4		525.3.f.b.449.9		32
20.19	odd	2		525.3.c.b.176.5		16
60.23	odd	4		525.3.f.b.449.10		32
60.47	odd	4		525.3.f.b.449.24		32
60.59	even	2		525.3.c.b.176.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.3.c.a.71.5	✓	16	12.11	even	2
105.3.c.a.71.12	yes	16	4.3	odd	2
525.3.c.b.176.5		16	20.19	odd	2
525.3.c.b.176.12		16	60.59	even	2
525.3.f.b.449.9		32	20.7	even	4
525.3.f.b.449.10		32	60.23	odd	4
525.3.f.b.449.23		32	20.3	even	4
525.3.f.b.449.24		32	60.47	odd	4
1680.3.l.a.1121.3		16	1.1	even	1	trivial
1680.3.l.a.1121.4		16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-2.91626 - 0.703870i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(8.00914 + 4.10533i) q^{9} +O(q^{10})\)	\(q+(-2.91626 - 0.703870i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(8.00914 + 4.10533i) q^{9} +11.6583i q^{11} -20.3660 q^{13} +(-1.57390 + 6.52095i) q^{15} +9.92200i q^{17} +4.91979 q^{19} +(7.71570 + 1.86226i) q^{21} +11.4646i q^{23} -5.00000 q^{25} +(-20.4671 - 17.6096i) q^{27} -37.5101i q^{29} +21.2282 q^{31} +(8.20592 - 33.9986i) q^{33} +5.91608i q^{35} +57.1939 q^{37} +(59.3924 + 14.3350i) q^{39} -20.2428i q^{41} -5.88954 q^{43} +(9.17980 - 17.9090i) q^{45} +74.5051i q^{47} +7.00000 q^{49} +(6.98379 - 28.9351i) q^{51} +88.2001i q^{53} +26.0687 q^{55} +(-14.3474 - 3.46289i) q^{57} -71.6589i q^{59} -26.0462 q^{61} +(-21.1902 - 10.8617i) q^{63} +45.5397i q^{65} -76.4691 q^{67} +(8.06960 - 33.4338i) q^{69} -47.1551i q^{71} -135.859 q^{73} +(14.5813 + 3.51935i) q^{75} -30.8449i q^{77} +98.2712 q^{79} +(47.2925 + 65.7603i) q^{81} -55.8693i q^{83} +22.1863 q^{85} +(-26.4022 + 109.389i) q^{87} -105.666i q^{89} +53.8833 q^{91} +(-61.9069 - 14.9419i) q^{93} -11.0010i q^{95} +61.8996 q^{97} +(-47.8612 + 93.3728i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{3} + 22 q^{9}+O(q^{10}) \) 16 * q - 8 * q^3 + 22 * q^9	\( 16 q - 8 q^{3} + 22 q^{9} - 10 q^{15} + 16 q^{19} - 14 q^{21} - 80 q^{25} + 148 q^{27} + 72 q^{31} - 4 q^{33} - 40 q^{37} - 90 q^{39} - 280 q^{43} + 40 q^{45} + 112 q^{49} - 38 q^{51} - 80 q^{55} - 36 q^{57} - 56 q^{61} + 56 q^{63} + 120 q^{67} + 60 q^{69} - 208 q^{73} + 40 q^{75} + 204 q^{79} + 458 q^{81} + 100 q^{85} + 324 q^{87} + 28 q^{91} + 108 q^{93} + 728 q^{97} + 166 q^{99}+O(q^{100}) \) 16 * q - 8 * q^3 + 22 * q^9 - 10 * q^15 + 16 * q^19 - 14 * q^21 - 80 * q^25 + 148 * q^27 + 72 * q^31 - 4 * q^33 - 40 * q^37 - 90 * q^39 - 280 * q^43 + 40 * q^45 + 112 * q^49 - 38 * q^51 - 80 * q^55 - 36 * q^57 - 56 * q^61 + 56 * q^63 + 120 * q^67 + 60 * q^69 - 208 * q^73 + 40 * q^75 + 204 * q^79 + 458 * q^81 + 100 * q^85 + 324 * q^87 + 28 * q^91 + 108 * q^93 + 728 * q^97 + 166 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more