LMFDB - Embedded newform 1680.3.bd.a.769.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,3,Mod(769,1680)]

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, 0, 1, 1]))

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

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

chi := DirichletCharacter("1680.769");

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.bd (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} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + \cdots + 33750 $ x^16 - 8x^15 + 96x^14 - 532x^13 + 3236x^12 - 12864x^11 + 49526x^10 - 141436x^9 + 362298x^8 - 722060x^7 + 1208164x^6 - 1570812x^5 + 1591101x^4 - 1183860x^3 + 619650x^2 - 202500*x + 33750
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.1
Root			$0.500000 + 2.68650i$ of defining polynomial
Character	$\chi$	$=$	1680.769
Dual form			1680.3.bd.a.769.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +(-4.59769 - 1.96501i) q^{5} +(6.81963 + 1.57881i) q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +(-4.59769 - 1.96501i) q^{5} +(6.81963 + 1.57881i) q^{7} +3.00000 q^{9} +8.15965 q^{11} -14.6064 q^{13} +(7.96343 + 3.40349i) q^{15} -5.81421 q^{17} +33.7736i q^{19} +(-11.8119 - 2.73459i) q^{21} -37.2576i q^{23} +(17.2775 + 18.0690i) q^{25} -5.19615 q^{27} -9.25305 q^{29} -19.2558i q^{31} -14.1329 q^{33} +(-28.2522 - 20.6595i) q^{35} -63.4350i q^{37} +25.2990 q^{39} -8.25880i q^{41} +42.0893i q^{43} +(-13.7931 - 5.89502i) q^{45} +23.3380 q^{47} +(44.0147 + 21.5339i) q^{49} +10.0705 q^{51} +71.3497i q^{53} +(-37.5155 - 16.0337i) q^{55} -58.4976i q^{57} -42.9350i q^{59} +34.2864i q^{61} +(20.4589 + 4.73644i) q^{63} +(67.1557 + 28.7017i) q^{65} +4.99889i q^{67} +64.5320i q^{69} +38.8120 q^{71} -124.629 q^{73} +(-29.9255 - 31.2964i) q^{75} +(55.6458 + 12.8826i) q^{77} +56.1842 q^{79} +9.00000 q^{81} -90.3980 q^{83} +(26.7319 + 11.4249i) q^{85} +16.0267 q^{87} -16.2289i q^{89} +(-99.6102 - 23.0608i) q^{91} +33.3520i q^{93} +(66.3653 - 155.280i) q^{95} -82.7605 q^{97} +24.4789 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9}+O(q^{10}) $ 16 * q + 48 * q^9	$ 16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 96 * q^11 + 24 * q^15 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 8 * q^35 + 144 * q^39 + 224 * q^49 + 48 * q^51 + 368 * q^65 + 384 * q^71 + 608 * q^79 + 144 * q^81 - 440 * q^85 - 224 * q^91 + 560 * q^95 - 288 * 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
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$		0			0
$5$	−4.59769	−	1.96501i	−0.919538	−	0.393001i
$5$	−4.59769	−	1.96501i	−0.919538	−	0.393001i
$6$		0			0
$7$	6.81963	+	1.57881i	0.974233	+	0.225545i
$7$	6.81963	+	1.57881i	0.974233	+	0.225545i
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	8.15965			0.741786			0.370893	−	0.928676i	$-0.379052\pi$
$11$	8.15965			0.741786			0.370893	+	0.928676i	$0.379052\pi$
$12$		0			0
$13$	−14.6064			−1.12357			−0.561785	−	0.827284i	$-0.689885\pi$
$13$	−14.6064			−1.12357			−0.561785	+	0.827284i	$0.689885\pi$
$14$		0			0
$15$	7.96343	+	3.40349i	0.530896	+	0.226899i
$16$		0			0
$17$	−5.81421			−0.342012			−0.171006	−	0.985270i	$-0.554702\pi$
$17$	−5.81421			−0.342012			−0.171006	+	0.985270i	$0.554702\pi$
$18$		0			0
$19$			33.7736i			1.77756i	0.458337	+	0.888778i	$0.348445\pi$
$19$			33.7736i			1.77756i	−0.458337	+	0.888778i	$0.651555\pi$
$20$		0			0
$21$	−11.8119	−	2.73459i	−0.562474	−	0.130218i
$22$		0			0
$23$		−	37.2576i		−	1.61989i	−0.586503	−	0.809947i	$-0.699496\pi$
$23$		−	37.2576i		−	1.61989i	0.586503	−	0.809947i	$-0.300504\pi$
$24$		0			0
$25$	17.2775	+	18.0690i	0.691100	+	0.722759i
$26$		0			0
$27$	−5.19615			−0.192450
$28$		0			0
$29$	−9.25305			−0.319071			−0.159535	−	0.987192i	$-0.551000\pi$
$29$	−9.25305			−0.319071			−0.159535	+	0.987192i	$0.551000\pi$
$30$		0			0
$31$		−	19.2558i		−	0.621154i	−0.950548	−	0.310577i	$-0.899478\pi$
$31$		−	19.2558i		−	0.621154i	0.950548	−	0.310577i	$-0.100522\pi$
$32$		0			0
$33$	−14.1329			−0.428270
$34$		0			0
$35$	−28.2522	−	20.6595i	−0.807205	−	0.590272i
$36$		0			0
$37$		−	63.4350i		−	1.71446i	−0.514933	−	0.857230i	$-0.672183\pi$
$37$		−	63.4350i		−	1.71446i	0.514933	−	0.857230i	$-0.327817\pi$
$38$		0			0
$39$	25.2990			0.648693
$40$		0			0
$41$		−	8.25880i		−	0.201434i	−0.994915	−	0.100717i	$-0.967886\pi$
$41$		−	8.25880i		−	0.201434i	0.994915	−	0.100717i	$-0.0321137\pi$
$42$		0			0
$43$			42.0893i			0.978822i	0.872053	+	0.489411i	$0.162788\pi$
$43$			42.0893i			0.978822i	−0.872053	+	0.489411i	$0.837212\pi$
$44$		0			0
$45$	−13.7931	−	5.89502i	−0.306513	−	0.131000i
$46$		0			0
$47$	23.3380			0.496552			0.248276	−	0.968689i	$-0.420136\pi$
$47$	23.3380			0.496552			0.248276	+	0.968689i	$0.420136\pi$
$48$		0			0
$49$	44.0147	+	21.5339i	0.898259	+	0.439466i
$50$		0			0
$51$	10.0705			0.197461
$52$		0			0
$53$			71.3497i			1.34622i	0.739542	+	0.673110i	$0.235042\pi$
$53$			71.3497i			1.34622i	−0.739542	+	0.673110i	$0.764958\pi$
$54$		0			0
$55$	−37.5155	−	16.0337i	−0.682100	−	0.291523i
$56$		0			0
$57$		−	58.4976i		−	1.02627i
$58$		0			0
$59$		−	42.9350i		−	0.727712i	−0.931455	−	0.363856i	$-0.881460\pi$
$59$		−	42.9350i		−	0.727712i	0.931455	−	0.363856i	$-0.118540\pi$
$60$		0			0
$61$			34.2864i			0.562072i	0.959697	+	0.281036i	$0.0906781\pi$
$61$			34.2864i			0.562072i	−0.959697	+	0.281036i	$0.909322\pi$
$62$		0			0
$63$	20.4589	+	4.73644i	0.324744	+	0.0751816i
$64$		0			0
$65$	67.1557	+	28.7017i	1.03316	+	0.441564i
$66$		0			0
$67$			4.99889i			0.0746102i	0.999304	+	0.0373051i	$0.0118773\pi$
$67$			4.99889i			0.0746102i	−0.999304	+	0.0373051i	$0.988123\pi$
$68$		0			0
$69$			64.5320i			0.935246i
$70$		0			0
$71$	38.8120			0.546648			0.273324	−	0.961922i	$-0.411877\pi$
$71$	38.8120			0.546648			0.273324	+	0.961922i	$0.411877\pi$
$72$		0			0
$73$	−124.629			−1.70725			−0.853626	−	0.520886i	$-0.825602\pi$
$73$	−124.629			−1.70725			−0.853626	+	0.520886i	$0.825602\pi$
$74$		0			0
$75$	−29.9255	−	31.2964i	−0.399007	−	0.417285i
$76$		0			0
$77$	55.6458	+	12.8826i	0.722672	+	0.167306i
$78$		0			0
$79$	56.1842			0.711192			0.355596	−	0.934640i	$-0.384278\pi$
$79$	56.1842			0.711192			0.355596	+	0.934640i	$0.384278\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−90.3980			−1.08913			−0.544566	−	0.838718i	$-0.683306\pi$
$83$	−90.3980			−1.08913			−0.544566	+	0.838718i	$0.683306\pi$
$84$		0			0
$85$	26.7319	+	11.4249i	0.314493	+	0.134411i
$86$		0			0
$87$	16.0267			0.184215
$88$		0			0
$89$		−	16.2289i		−	0.182347i	−0.995835	−	0.0911736i	$-0.970938\pi$
$89$		−	16.2289i		−	0.182347i	0.995835	−	0.0911736i	$-0.0290618\pi$
$90$		0			0
$91$	−99.6102	−	23.0608i	−1.09462	−	0.253415i
$92$		0			0
$93$			33.3520i			0.358624i
$94$		0			0
$95$	66.3653	−	155.280i	0.698582	−	1.63453i
$96$		0			0
$97$	−82.7605			−0.853201			−0.426601	−	0.904440i	$-0.640289\pi$
$97$	−82.7605			−0.853201			−0.426601	+	0.904440i	$0.640289\pi$
$98$		0			0
$99$	24.4789			0.247262
$100$		0			0
$101$			87.2055i			0.863420i	0.902012	+	0.431710i	$0.142090\pi$
$101$			87.2055i			0.863420i	−0.902012	+	0.431710i	$0.857910\pi$
$102$		0			0
$103$	−187.567			−1.82104			−0.910521	−	0.413462i	$-0.864319\pi$
$103$	−187.567			−1.82104			−0.910521	+	0.413462i	$0.864319\pi$
$104$		0			0
$105$	48.9342	+	35.7833i	0.466040	+	0.340794i
$106$		0			0
$107$			157.858i			1.47531i	0.675180	+	0.737653i	$0.264066\pi$
$107$			157.858i			1.47531i	−0.675180	+	0.737653i	$0.735934\pi$
$108$		0			0
$109$	−112.073			−1.02819			−0.514097	−	0.857732i	$-0.671873\pi$
$109$	−112.073			−1.02819			−0.514097	+	0.857732i	$0.671873\pi$
$110$		0			0
$111$			109.873i			0.989844i
$112$		0			0
$113$		−	141.987i		−	1.25653i	−0.778001	−	0.628263i	$-0.783766\pi$
$113$		−	141.987i		−	1.25653i	0.778001	−	0.628263i	$-0.216234\pi$
$114$		0			0
$115$	−73.2113	+	171.299i	−0.636620	+	1.48955i
$116$		0			0
$117$	−43.8192			−0.374523
$118$		0			0
$119$	−39.6507	−	9.17955i	−0.333199	−	0.0771391i
$120$		0			0
$121$	−54.4202			−0.449754
$122$		0			0
$123$			14.3047i			0.116298i
$124$		0			0
$125$	−43.9310	−	117.026i	−0.351448	−	0.936207i
$126$		0			0
$127$			202.414i			1.59381i	0.604104	+	0.796905i	$0.293531\pi$
$127$			202.414i			1.59381i	−0.604104	+	0.796905i	$0.706469\pi$
$128$		0			0
$129$		−	72.9008i		−	0.565123i
$130$		0			0
$131$			141.260i			1.07832i	0.842203	+	0.539160i	$0.181258\pi$
$131$			141.260i			1.07832i	−0.842203	+	0.539160i	$0.818742\pi$
$132$		0			0
$133$	−53.3222	+	230.323i	−0.400919	+	1.73175i
$134$		0			0
$135$	23.8903	+	10.2105i	0.176965	+	0.0756331i
$136$		0			0
$137$			50.6430i			0.369657i	0.982771	+	0.184829i	$0.0591730\pi$
$137$			50.6430i			0.369657i	−0.982771	+	0.184829i	$0.940827\pi$
$138$		0			0
$139$			74.0256i			0.532559i	0.963896	+	0.266279i	$0.0857943\pi$
$139$			74.0256i			0.532559i	−0.963896	+	0.266279i	$0.914206\pi$
$140$		0			0
$141$	−40.4225			−0.286685
$142$		0			0
$143$	−119.183			−0.833448
$144$		0			0
$145$	42.5426	+	18.1823i	0.293397	+	0.125395i
$146$		0			0
$147$	−76.2357	−	37.2977i	−0.518610	−	0.253726i
$148$		0			0
$149$	−226.979			−1.52335			−0.761676	−	0.647958i	$-0.775623\pi$
$149$	−226.979			−1.52335			−0.761676	+	0.647958i	$0.775623\pi$
$150$		0			0
$151$	−294.426			−1.94984			−0.974920	−	0.222558i	$-0.928559\pi$
$151$	−294.426			−1.94984			−0.974920	+	0.222558i	$0.928559\pi$
$152$		0			0
$153$	−17.4426			−0.114004
$154$		0			0
$155$	−37.8377	+	88.5321i	−0.244114	+	0.571175i
$156$		0			0
$157$	65.8596			0.419488			0.209744	−	0.977756i	$-0.432737\pi$
$157$	65.8596			0.419488			0.209744	+	0.977756i	$0.432737\pi$
$158$		0			0
$159$		−	123.581i		−	0.777241i
$160$		0			0
$161$	58.8227	−	254.083i	0.365359	−	1.57815i
$162$		0			0
$163$			28.2075i			0.173052i	0.996250	+	0.0865260i	$0.0275766\pi$
$163$			28.2075i			0.173052i	−0.996250	+	0.0865260i	$0.972423\pi$
$164$		0			0
$165$	64.9788	+	27.7713i	0.393811	+	0.168311i
$166$		0			0
$167$	128.461			0.769227			0.384614	−	0.923078i	$-0.374335\pi$
$167$	128.461			0.769227			0.384614	+	0.923078i	$0.374335\pi$
$168$		0			0
$169$	44.3469			0.262408
$170$		0			0
$171$			101.321i			0.592519i
$172$		0			0
$173$	−112.446			−0.649976			−0.324988	−	0.945718i	$-0.605360\pi$
$173$	−112.446			−0.649976			−0.324988	+	0.945718i	$0.605360\pi$
$174$		0			0
$175$	89.2986	+	150.502i	0.510278	+	0.860010i
$176$		0			0
$177$			74.3656i			0.420144i
$178$		0			0
$179$	68.9019			0.384927			0.192464	−	0.981304i	$-0.438352\pi$
$179$	68.9019			0.384927			0.192464	+	0.981304i	$0.438352\pi$
$180$		0			0
$181$			166.537i			0.920094i	0.887895	+	0.460047i	$0.152167\pi$
$181$			166.537i			0.920094i	−0.887895	+	0.460047i	$0.847833\pi$
$182$		0			0
$183$		−	59.3858i		−	0.324513i
$184$		0			0
$185$	−124.650	+	291.655i	−0.673785	+	1.57651i
$186$		0			0
$187$	−47.4419			−0.253700
$188$		0			0
$189$	−35.4358	−	8.20376i	−0.187491	−	0.0434061i
$190$		0			0
$191$	148.289			0.776382			0.388191	−	0.921579i	$-0.373100\pi$
$191$	148.289			0.776382			0.388191	+	0.921579i	$0.373100\pi$
$192$		0			0
$193$			35.1886i			0.182324i	0.995836	+	0.0911622i	$0.0290582\pi$
$193$			35.1886i			0.182324i	−0.995836	+	0.0911622i	$0.970942\pi$
$194$		0			0
$195$	−116.317	−	49.7127i	−0.596498	−	0.254937i
$196$		0			0
$197$		−	193.634i		−	0.982915i	−0.870902	−	0.491457i	$-0.836464\pi$
$197$		−	193.634i		−	0.982915i	0.870902	−	0.491457i	$-0.163536\pi$
$198$		0			0
$199$			181.838i			0.913759i	0.889529	+	0.456880i	$0.151033\pi$
$199$			181.838i			0.913759i	−0.889529	+	0.456880i	$0.848967\pi$
$200$		0			0
$201$		−	8.65832i		−	0.0430762i
$202$		0			0
$203$	−63.1023	−	14.6088i	−0.310849	−	0.0719647i
$204$		0			0
$205$	−16.2286	+	37.9714i	−0.0791638	+	0.185226i
$206$		0			0
$207$		−	111.773i		−	0.539965i
$208$		0			0
$209$			275.580i			1.31857i
$210$		0			0
$211$	−175.914			−0.833717			−0.416859	−	0.908971i	$-0.636869\pi$
$211$	−175.914			−0.833717			−0.416859	+	0.908971i	$0.636869\pi$
$212$		0			0
$213$	−67.2244			−0.315607
$214$		0			0
$215$	82.7058	−	193.514i	0.384678	−	0.900064i
$216$		0			0
$217$	30.4013	−	131.317i	0.140098	−	0.605149i
$218$		0			0
$219$	215.864			0.985683
$220$		0			0
$221$	84.9246			0.384274
$222$		0			0
$223$	20.5675			0.0922308			0.0461154	−	0.998936i	$-0.485316\pi$
$223$	20.5675			0.0922308			0.0461154	+	0.998936i	$0.485316\pi$
$224$		0			0
$225$	51.8325	+	54.2069i	0.230367	+	0.240920i
$226$		0			0
$227$	−414.087			−1.82417			−0.912085	−	0.410001i	$-0.865528\pi$
$227$	−414.087			−1.82417			−0.912085	+	0.410001i	$0.865528\pi$
$228$		0			0
$229$		−	195.520i		−	0.853799i	−0.904299	−	0.426899i	$-0.859606\pi$
$229$		−	195.520i		−	0.853799i	0.904299	−	0.426899i	$-0.140394\pi$
$230$		0			0
$231$	−96.3813	−	22.3133i	−0.417235	−	0.0965942i
$232$		0			0
$233$			81.3006i			0.348930i	0.984663	+	0.174465i	$0.0558195\pi$
$233$			81.3006i			0.348930i	−0.984663	+	0.174465i	$0.944180\pi$
$234$		0			0
$235$	−107.301	−	45.8592i	−0.456599	−	0.195146i
$236$		0			0
$237$	−97.3138			−0.410607
$238$		0			0
$239$	249.265			1.04295			0.521475	−	0.853267i	$-0.325382\pi$
$239$	249.265			1.04295			0.521475	+	0.853267i	$0.325382\pi$
$240$		0			0
$241$			262.343i			1.08856i	0.838904	+	0.544280i	$0.183197\pi$
$241$			262.343i			1.08856i	−0.838904	+	0.544280i	$0.816803\pi$
$242$		0			0
$243$	−15.5885			−0.0641500
$244$		0			0
$245$	−160.052	−	185.495i	−0.653272	−	0.757123i
$246$		0			0
$247$		−	493.310i		−	1.99721i
$248$		0			0
$249$	156.574			0.628811
$250$		0			0
$251$		−	141.917i		−	0.565406i	−0.959208	−	0.282703i	$-0.908769\pi$
$251$		−	141.917i		−	0.565406i	0.959208	−	0.282703i	$-0.0912310\pi$
$252$		0			0
$253$		−	304.008i		−	1.20161i
$254$		0			0
$255$	−46.3010	−	19.7886i	−0.181573	−	0.0776023i
$256$		0			0
$257$	−170.843			−0.664757			−0.332379	−	0.943146i	$-0.607851\pi$
$257$	−170.843			−0.664757			−0.332379	+	0.943146i	$0.607851\pi$
$258$		0			0
$259$	100.152	−	432.604i	0.386688	−	1.67028i
$260$		0			0
$261$	−27.7591			−0.106357
$262$		0			0
$263$			383.417i			1.45786i	0.684588	+	0.728930i	$0.259982\pi$
$263$			383.417i			1.45786i	−0.684588	+	0.728930i	$0.740018\pi$
$264$		0			0
$265$	140.203	−	328.044i	0.529066	−	1.23790i
$266$		0			0
$267$			28.1093i			0.105278i
$268$		0			0
$269$			154.512i			0.574393i	0.957872	+	0.287196i	$0.0927232\pi$
$269$			154.512i			0.574393i	−0.957872	+	0.287196i	$0.907277\pi$
$270$		0			0
$271$		−	320.328i		−	1.18202i	−0.806664	−	0.591010i	$-0.798729\pi$
$271$		−	320.328i		−	1.18202i	0.806664	−	0.591010i	$-0.201271\pi$
$272$		0			0
$273$	172.530	+	39.9425i	0.631978	+	0.146309i
$274$		0			0
$275$	140.978	+	147.436i	0.512648	+	0.536132i
$276$		0			0
$277$			222.854i			0.804526i	0.915524	+	0.402263i	$0.131776\pi$
$277$			222.854i			0.804526i	−0.915524	+	0.402263i	$0.868224\pi$
$278$		0			0
$279$		−	57.7673i		−	0.207051i
$280$		0			0
$281$	218.973			0.779263			0.389631	−	0.920971i	$-0.372602\pi$
$281$	218.973			0.779263			0.389631	+	0.920971i	$0.372602\pi$
$282$		0			0
$283$	36.3957			0.128607			0.0643034	−	0.997930i	$-0.479517\pi$
$283$	36.3957			0.128607			0.0643034	+	0.997930i	$0.479517\pi$
$284$		0			0
$285$	−114.948	+	268.954i	−0.403326	+	0.943697i
$286$		0			0
$287$	13.0391	−	56.3219i	0.0454324	−	0.196244i
$288$		0			0
$289$	−255.195			−0.883028
$290$		0			0
$291$	143.345			0.492596
$292$		0			0
$293$	168.440			0.574882			0.287441	−	0.957798i	$-0.407195\pi$
$293$	168.440			0.574882			0.287441	+	0.957798i	$0.407195\pi$
$294$		0			0
$295$	−84.3675	+	197.402i	−0.285991	+	0.669158i
$296$		0			0
$297$	−42.3988			−0.142757
$298$		0			0
$299$			544.199i			1.82006i
$300$		0			0
$301$	−66.4512	+	287.034i	−0.220768	+	0.953600i
$302$		0			0
$303$		−	151.044i		−	0.498496i
$304$		0			0
$305$	67.3730	−	157.638i	0.220895	−	0.516847i
$306$		0			0
$307$	223.400			0.727688			0.363844	−	0.931460i	$-0.381464\pi$
$307$	223.400			0.727688			0.363844	+	0.931460i	$0.381464\pi$
$308$		0			0
$309$	324.876			1.05138
$310$		0			0
$311$			46.0396i			0.148037i	0.997257	+	0.0740186i	$0.0235824\pi$
$311$			46.0396i			0.148037i	−0.997257	+	0.0740186i	$0.976418\pi$
$312$		0			0
$313$	80.0375			0.255711			0.127855	−	0.991793i	$-0.459191\pi$
$313$	80.0375			0.255711			0.127855	+	0.991793i	$0.459191\pi$
$314$		0			0
$315$	−84.7565	−	61.9785i	−0.269068	−	0.196757i
$316$		0			0
$317$			185.237i			0.584343i	0.956366	+	0.292171i	$0.0943777\pi$
$317$			185.237i			0.584343i	−0.956366	+	0.292171i	$0.905622\pi$
$318$		0			0
$319$	−75.5016			−0.236682
$320$		0			0
$321$		−	273.418i		−	0.851768i
$322$		0			0
$323$		−	196.367i		−	0.607946i
$324$		0			0
$325$	−252.362	−	263.923i	−0.776499	−	0.812070i
$326$		0			0
$327$	194.116			0.593628
$328$		0			0
$329$	159.156	+	36.8463i	0.483757	+	0.111995i
$330$		0			0
$331$	−61.4686			−0.185706			−0.0928529	−	0.995680i	$-0.529599\pi$
$331$	−61.4686			−0.185706			−0.0928529	+	0.995680i	$0.529599\pi$
$332$		0			0
$333$		−	190.305i		−	0.571487i
$334$		0			0
$335$	9.82284	−	22.9833i	0.0293219	−	0.0686069i
$336$		0			0
$337$			656.501i			1.94807i	0.226387	+	0.974037i	$0.427309\pi$
$337$			656.501i			1.94807i	−0.226387	+	0.974037i	$0.572691\pi$
$338$		0			0
$339$			245.929i			0.725456i
$340$		0			0
$341$		−	157.120i		−	0.460764i
$342$		0			0
$343$	266.166	+	216.344i	0.775994	+	0.630740i
$344$		0			0
$345$	126.806	−	296.698i	0.367553	−	0.859994i
$346$		0			0
$347$		−	321.323i		−	0.926002i	−0.886358	−	0.463001i	$-0.846773\pi$
$347$		−	321.323i		−	0.926002i	0.886358	−	0.463001i	$-0.153227\pi$
$348$		0			0
$349$			185.135i			0.530474i	0.964183	+	0.265237i	$0.0854501\pi$
$349$			185.135i			0.530474i	−0.964183	+	0.265237i	$0.914550\pi$
$350$		0			0
$351$	75.8971			0.216231
$352$		0			0
$353$	−597.338			−1.69218			−0.846088	−	0.533043i	$-0.821048\pi$
$353$	−597.338			−1.69218			−0.846088	+	0.533043i	$0.821048\pi$
$354$		0			0
$355$	−178.446	−	76.2658i	−0.502664	−	0.214833i
$356$		0			0
$357$	68.6771	+	15.8995i	0.192373	+	0.0445363i
$358$		0			0
$359$	−239.446			−0.666982			−0.333491	−	0.942753i	$-0.608227\pi$
$359$	−239.446			−0.666982			−0.333491	+	0.942753i	$0.608227\pi$
$360$		0			0
$361$	−779.655			−2.15971
$362$		0			0
$363$	94.2585			0.259665
$364$		0			0
$365$	573.007	+	244.898i	1.56988	+	0.670952i
$366$		0			0
$367$	398.743			1.08649			0.543246	−	0.839573i	$-0.317195\pi$
$367$	398.743			1.08649			0.543246	+	0.839573i	$0.317195\pi$
$368$		0			0
$369$		−	24.7764i		−	0.0671447i
$370$		0			0
$371$	−112.648	+	486.578i	−0.303633	+	1.31153i
$372$		0			0
$373$			34.7064i			0.0930466i	0.998917	+	0.0465233i	$0.0148142\pi$
$373$			34.7064i			0.0930466i	−0.998917	+	0.0465233i	$0.985186\pi$
$374$		0			0
$375$	76.0907	+	202.695i	0.202908	+	0.540520i
$376$		0			0
$377$	135.154			0.358498
$378$		0			0
$379$	109.217			0.288172			0.144086	−	0.989565i	$-0.453976\pi$
$379$	109.217			0.288172			0.144086	+	0.989565i	$0.453976\pi$
$380$		0			0
$381$		−	350.591i		−	0.920187i
$382$		0			0
$383$	336.420			0.878381			0.439191	−	0.898394i	$-0.355265\pi$
$383$	336.420			0.878381			0.439191	+	0.898394i	$0.355265\pi$
$384$		0			0
$385$	−230.528	−	168.574i	−0.598773	−	0.437855i
$386$		0			0
$387$			126.268i			0.326274i
$388$		0			0
$389$	−311.173			−0.799930			−0.399965	−	0.916530i	$-0.630978\pi$
$389$	−311.173			−0.799930			−0.399965	+	0.916530i	$0.630978\pi$
$390$		0			0
$391$			216.623i			0.554023i
$392$		0			0
$393$		−	244.669i		−	0.622568i
$394$		0			0
$395$	−258.317	−	110.402i	−0.653968	−	0.279499i
$396$		0			0
$397$	41.2679			0.103949			0.0519747	−	0.998648i	$-0.483448\pi$
$397$	41.2679			0.103949			0.0519747	+	0.998648i	$0.483448\pi$
$398$		0			0
$399$	92.3568	−	398.932i	0.231471	−	0.999829i
$400$		0			0
$401$	−495.642			−1.23601			−0.618007	−	0.786173i	$-0.712060\pi$
$401$	−495.642			−1.23601			−0.618007	+	0.786173i	$0.712060\pi$
$402$		0			0
$403$			281.258i			0.697910i
$404$		0			0
$405$	−41.3792	−	17.6851i	−0.102171	−	0.0436668i
$406$		0			0
$407$		−	517.608i		−	1.27176i
$408$		0			0
$409$		−	359.920i		−	0.880001i	−0.897998	−	0.440000i	$-0.854978\pi$
$409$		−	359.920i		−	0.880001i	0.897998	−	0.440000i	$-0.145022\pi$
$410$		0			0
$411$		−	87.7163i		−	0.213422i
$412$		0			0
$413$	67.7863	−	292.801i	0.164132	−	0.708960i
$414$		0			0
$415$	415.622	+	177.633i	1.00150	+	0.428030i
$416$		0			0
$417$		−	128.216i		−	0.307473i
$418$		0			0
$419$		−	482.910i		−	1.15253i	−0.817263	−	0.576265i	$-0.804510\pi$
$419$		−	482.910i		−	1.15253i	0.817263	−	0.576265i	$-0.195490\pi$
$420$		0			0
$421$	−523.520			−1.24351			−0.621757	−	0.783210i	$-0.713581\pi$
$421$	−523.520			−1.24351			−0.621757	+	0.783210i	$0.713581\pi$
$422$		0			0
$423$	70.0139			0.165517
$424$		0			0
$425$	−100.455	−	105.057i	−0.236365	−	0.247192i
$426$		0			0
$427$	−54.1319	+	233.821i	−0.126773	+	0.547589i
$428$		0			0
$429$	206.431			0.481191
$430$		0			0
$431$	−517.027			−1.19960			−0.599799	−	0.800150i	$-0.704753\pi$
$431$	−517.027			−1.19960			−0.599799	+	0.800150i	$0.704753\pi$
$432$		0			0
$433$	567.712			1.31111			0.655557	−	0.755146i	$-0.272434\pi$
$433$	567.712			1.31111			0.655557	+	0.755146i	$0.272434\pi$
$434$		0			0
$435$	−73.6860	−	31.4926i	−0.169393	−	0.0723969i
$436$		0			0
$437$	1258.32			2.87945
$438$		0			0
$439$			622.875i			1.41885i	0.704781	+	0.709425i	$0.251045\pi$
$439$			622.875i			1.41885i	−0.704781	+	0.709425i	$0.748955\pi$
$440$		0			0
$441$	132.044	+	64.6016i	0.299420	+	0.146489i
$442$		0			0
$443$		−	476.699i		−	1.07607i	−0.842923	−	0.538035i	$-0.819167\pi$
$443$		−	476.699i		−	1.07607i	0.842923	−	0.538035i	$-0.180833\pi$
$444$		0			0
$445$	−31.8899	+	74.6154i	−0.0716626	+	0.167675i
$446$		0			0
$447$	393.140			0.879507
$448$		0			0
$449$	861.931			1.91967			0.959834	−	0.280570i	$-0.0905233\pi$
$449$	861.931			1.91967			0.959834	+	0.280570i	$0.0905233\pi$
$450$		0			0
$451$		−	67.3889i		−	0.149421i
$452$		0			0
$453$	509.960			1.12574
$454$		0			0
$455$	412.662	+	301.761i	0.906950	+	0.663211i
$456$		0			0
$457$		−	27.5401i		−	0.0602629i	−0.999546	−	0.0301314i	$-0.990407\pi$
$457$		−	27.5401i		−	0.0602629i	0.999546	−	0.0301314i	$-0.00959258\pi$
$458$		0			0
$459$	30.2115			0.0658203
$460$		0			0
$461$		−	378.183i		−	0.820353i	−0.912006	−	0.410176i	$-0.865467\pi$
$461$		−	378.183i		−	0.820353i	0.912006	−	0.410176i	$-0.134533\pi$
$462$		0			0
$463$		−	724.994i		−	1.56586i	−0.622108	−	0.782931i	$-0.713724\pi$
$463$		−	724.994i		−	1.56586i	0.622108	−	0.782931i	$-0.286276\pi$
$464$		0			0
$465$	65.5369	−	153.342i	0.140939	−	0.329768i
$466$		0			0
$467$	−371.451			−0.795398			−0.397699	−	0.917516i	$-0.630191\pi$
$467$	−371.451			−0.795398			−0.397699	+	0.917516i	$0.630191\pi$
$468$		0			0
$469$	−7.89231	+	34.0905i	−0.0168280	+	0.0726877i
$470$		0			0
$471$	−114.072			−0.242191
$472$		0			0
$473$			343.434i			0.726076i
$474$		0			0
$475$	−610.254	+	583.523i	−1.28475	+	1.22847i
$476$		0			0
$477$			214.049i			0.448740i
$478$		0			0
$479$		−	860.650i		−	1.79677i	−0.439214	−	0.898383i	$-0.644743\pi$
$479$		−	860.650i		−	1.79677i	0.439214	−	0.898383i	$-0.355257\pi$
$480$		0			0
$481$			926.558i			1.92632i
$482$		0			0
$483$	−101.884	+	440.084i	−0.210940	+	0.911147i
$484$		0			0
$485$	380.507	+	162.625i	0.784551	+	0.335309i
$486$		0			0
$487$			887.173i			1.82171i	0.412726	+	0.910855i	$0.364577\pi$
$487$			887.173i			1.82171i	−0.412726	+	0.910855i	$0.635423\pi$
$488$		0			0
$489$		−	48.8568i		−	0.0999116i
$490$		0			0
$491$	50.1823			0.102204			0.0511021	−	0.998693i	$-0.483727\pi$
$491$	50.1823			0.102204			0.0511021	+	0.998693i	$0.483727\pi$
$492$		0			0
$493$	53.7991			0.109126
$494$		0			0
$495$	−112.547	−	48.1012i	−0.227367	−	0.0971742i
$496$		0			0
$497$	264.683	+	61.2769i	0.532562	+	0.123294i
$498$		0			0
$499$	−160.443			−0.321529			−0.160765	−	0.986993i	$-0.551396\pi$
$499$	−160.443			−0.321529			−0.160765	+	0.986993i	$0.551396\pi$
$500$		0			0
$501$	−222.501			−0.444113
$502$		0			0
$503$	742.042			1.47523			0.737616	−	0.675220i	$-0.235951\pi$
$503$	742.042			1.47523			0.737616	+	0.675220i	$0.235951\pi$
$504$		0			0
$505$	171.359	−	400.944i	0.339325	−	0.793948i
$506$		0			0
$507$	−76.8111			−0.151501
$508$		0			0
$509$		−	978.447i		−	1.92229i	−0.276038	−	0.961147i	$-0.589021\pi$
$509$		−	978.447i		−	1.92229i	0.276038	−	0.961147i	$-0.410979\pi$
$510$		0			0
$511$	−849.927	−	196.767i	−1.66326	−	0.385062i
$512$		0			0
$513$		−	175.493i		−	0.342091i
$514$		0			0
$515$	862.377	+	368.571i	1.67452	+	0.715672i
$516$		0			0
$517$	190.429			0.368335
$518$		0			0
$519$	194.762			0.375264
$520$		0			0
$521$			575.650i			1.10489i	0.833548	+	0.552447i	$0.186306\pi$
$521$			575.650i			1.10489i	−0.833548	+	0.552447i	$0.813694\pi$
$522$		0			0
$523$	−339.090			−0.648355			−0.324178	−	0.945996i	$-0.605088\pi$
$523$	−339.090			−0.648355			−0.324178	+	0.945996i	$0.605088\pi$
$524$		0			0
$525$	−154.670	−	260.677i	−0.294609	−	0.496527i
$526$		0			0
$527$			111.957i			0.212442i
$528$		0			0
$529$	−859.125			−1.62406
$530$		0			0
$531$		−	128.805i		−	0.242571i
$532$		0			0
$533$			120.631i			0.226325i
$534$		0			0
$535$	310.191	−	725.781i	0.579797	−	1.35660i
$536$		0			0
$537$	−119.342			−0.222238
$538$		0			0
$539$	359.144	+	175.709i	0.666316	+	0.325990i
$540$		0			0
$541$	35.8638			0.0662916			0.0331458	−	0.999451i	$-0.489447\pi$
$541$	35.8638			0.0662916			0.0331458	+	0.999451i	$0.489447\pi$
$542$		0			0
$543$		−	288.450i		−	0.531216i
$544$		0			0
$545$	515.277	+	220.224i	0.945463	+	0.404081i
$546$		0			0
$547$		−	700.845i		−	1.28125i	−0.767853	−	0.640626i	$-0.778675\pi$
$547$		−	700.845i		−	1.28125i	0.767853	−	0.640626i	$-0.221325\pi$
$548$		0			0
$549$			102.859i			0.187357i
$550$		0			0
$551$		−	312.508i		−	0.567166i
$552$		0			0
$553$	383.155	+	88.7044i	0.692867	+	0.160406i
$554$		0			0
$555$	215.901	−	505.161i	0.389010	−	0.910200i
$556$		0			0
$557$		−	639.349i		−	1.14784i	−0.818910	−	0.573922i	$-0.805421\pi$
$557$		−	639.349i		−	1.14784i	0.818910	−	0.573922i	$-0.194579\pi$
$558$		0			0
$559$		−	614.773i		−	1.09977i
$560$		0			0
$561$	82.1717			0.146474
$562$		0			0
$563$	−227.519			−0.404120			−0.202060	−	0.979373i	$-0.564764\pi$
$563$	−227.519			−0.404120			−0.202060	+	0.979373i	$0.564764\pi$
$564$		0			0
$565$	−279.006	+	652.814i	−0.493816	+	1.15542i
$566$		0			0
$567$	61.3767	+	14.2093i	0.108248	+	0.0250605i
$568$		0			0
$569$	613.901			1.07891			0.539456	−	0.842014i	$-0.318630\pi$
$569$	613.901			1.07891			0.539456	+	0.842014i	$0.318630\pi$
$570$		0			0
$571$	−366.674			−0.642161			−0.321081	−	0.947052i	$-0.604046\pi$
$571$	−366.674			−0.642161			−0.321081	+	0.947052i	$0.604046\pi$
$572$		0			0
$573$	−256.844			−0.448244
$574$		0			0
$575$	673.206	−	643.718i	1.17079	−	1.11951i
$576$		0			0
$577$	62.1951			0.107790			0.0538952	−	0.998547i	$-0.482836\pi$
$577$	62.1951			0.107790			0.0538952	+	0.998547i	$0.482836\pi$
$578$		0			0
$579$		−	60.9485i		−	0.105265i
$580$		0			0
$581$	−616.481	−	142.722i	−1.06107	−	0.245648i
$582$		0			0
$583$			582.188i			0.998607i
$584$		0			0
$585$	201.467	+	86.1050i	0.344388	+	0.147188i
$586$		0			0
$587$	−176.872			−0.301315			−0.150658	−	0.988586i	$-0.548139\pi$
$587$	−176.872			−0.301315			−0.150658	+	0.988586i	$0.548139\pi$
$588$		0			0
$589$	650.337			1.10414
$590$		0			0
$591$			335.384i			0.567486i
$592$		0			0
$593$	−154.878			−0.261176			−0.130588	−	0.991437i	$-0.541687\pi$
$593$	−154.878			−0.261176			−0.130588	+	0.991437i	$0.541687\pi$
$594$		0			0
$595$	164.264	+	120.119i	0.276074	+	0.201880i
$596$		0			0
$597$		−	314.953i		−	0.527559i
$598$		0			0
$599$	−900.825			−1.50388			−0.751940	−	0.659231i	$-0.770882\pi$
$599$	−900.825			−1.50388			−0.751940	+	0.659231i	$0.770882\pi$
$600$		0			0
$601$		−	682.387i		−	1.13542i	−0.823229	−	0.567710i	$-0.807830\pi$
$601$		−	682.387i		−	1.13542i	0.823229	−	0.567710i	$-0.192170\pi$
$602$		0			0
$603$			14.9967i			0.0248701i
$604$		0			0
$605$	250.207	+	106.936i	0.413566	+	0.176754i
$606$		0			0
$607$	367.814			0.605954			0.302977	−	0.952998i	$-0.402019\pi$
$607$	367.814			0.605954			0.302977	+	0.952998i	$0.402019\pi$
$608$		0			0
$609$	109.296	+	25.3032i	0.179469	+	0.0415488i
$610$		0			0
$611$	−340.883			−0.557911
$612$		0			0
$613$		−	117.271i		−	0.191306i	−0.995415	−	0.0956532i	$-0.969506\pi$
$613$		−	117.271i		−	0.191306i	0.995415	−	0.0956532i	$-0.0304940\pi$
$614$		0			0
$615$	28.1087	−	65.7684i	0.0457053	−	0.106940i
$616$		0			0
$617$		−	1070.18i		−	1.73449i	−0.497881	−	0.867245i	$-0.665888\pi$
$617$		−	1070.18i		−	1.73449i	0.497881	−	0.867245i	$-0.334112\pi$
$618$		0			0
$619$			532.192i			0.859761i	0.902886	+	0.429880i	$0.141444\pi$
$619$			532.192i			0.859761i	−0.902886	+	0.429880i	$0.858556\pi$
$620$		0			0
$621$			193.596i			0.311749i
$622$		0			0
$623$	25.6224	−	110.675i	0.0411275	−	0.177649i
$624$		0			0
$625$	−27.9756	+	624.374i	−0.0447610	+	0.998998i
$626$		0			0
$627$		−	477.319i		−	0.761275i
$628$		0			0
$629$			368.825i			0.586366i
$630$		0			0
$631$	−472.933			−0.749498			−0.374749	−	0.927126i	$-0.622271\pi$
$631$	−472.933			−0.749498			−0.374749	+	0.927126i	$0.622271\pi$
$632$		0			0
$633$	304.693			0.481347
$634$		0			0
$635$	397.745	−	930.636i	0.626369	−	1.46557i
$636$		0			0
$637$	−642.896	−	314.532i	−1.00926	−	0.493771i
$638$		0			0
$639$	116.436			0.182216
$640$		0			0
$641$	−486.990			−0.759734			−0.379867	−	0.925041i	$-0.624030\pi$
$641$	−486.990			−0.759734			−0.379867	+	0.925041i	$0.624030\pi$
$642$		0			0
$643$	−111.425			−0.173289			−0.0866447	−	0.996239i	$-0.527614\pi$
$643$	−111.425			−0.173289			−0.0866447	+	0.996239i	$0.527614\pi$
$644$		0			0
$645$	−143.251	+	335.176i	−0.222094	+	0.519652i
$646$		0			0
$647$	737.696			1.14018			0.570090	−	0.821583i	$-0.306908\pi$
$647$	737.696			1.14018			0.570090	+	0.821583i	$0.306908\pi$
$648$		0			0
$649$		−	350.334i		−	0.539806i
$650$		0			0
$651$	−52.6566	+	227.448i	−0.0808857	+	0.349383i
$652$		0			0
$653$		−	193.734i		−	0.296682i	−0.988936	−	0.148341i	$-0.952607\pi$
$653$		−	193.734i		−	0.296682i	0.988936	−	0.148341i	$-0.0473934\pi$
$654$		0			0
$655$	277.577	−	649.469i	0.423781	−	0.991556i
$656$		0			0
$657$	−373.888			−0.569084
$658$		0			0
$659$	823.339			1.24938			0.624688	−	0.780874i	$-0.285226\pi$
$659$	823.339			1.24938			0.624688	+	0.780874i	$0.285226\pi$
$660$		0			0
$661$		−	657.833i		−	0.995208i	−0.867404	−	0.497604i	$-0.834213\pi$
$661$		−	657.833i		−	0.995208i	0.867404	−	0.497604i	$-0.165787\pi$
$662$		0			0
$663$	−147.094			−0.221861
$664$		0			0
$665$	697.746	−	954.177i	1.04924	−	1.43485i
$666$		0			0
$667$			344.746i			0.516860i
$668$		0			0
$669$	−35.6239			−0.0532495
$670$		0			0
$671$			279.765i			0.416937i
$672$		0			0
$673$			727.300i			1.08068i	0.841446	+	0.540342i	$0.181705\pi$
$673$			727.300i			1.08068i	−0.841446	+	0.540342i	$0.818295\pi$
$674$		0			0
$675$	−89.7766	−	93.8891i	−0.133002	−	0.139095i
$676$		0			0
$677$	385.964			0.570110			0.285055	−	0.958511i	$-0.407988\pi$
$677$	385.964			0.570110			0.285055	+	0.958511i	$0.407988\pi$
$678$		0			0
$679$	−564.396	−	130.663i	−0.831217	−	0.192435i
$680$		0			0
$681$	717.219			1.05319
$682$		0			0
$683$			236.488i			0.346249i	0.984900	+	0.173124i	$0.0553863\pi$
$683$			236.488i			0.346249i	−0.984900	+	0.173124i	$0.944614\pi$
$684$		0			0
$685$	99.5138	−	232.841i	0.145276	−	0.339914i
$686$		0			0
$687$			338.650i			0.492941i
$688$		0			0
$689$		−	1042.16i		−	1.51257i
$690$		0			0
$691$			337.426i			0.488316i	0.969735	+	0.244158i	$0.0785116\pi$
$691$			337.426i			0.488316i	−0.969735	+	0.244158i	$0.921488\pi$
$692$		0			0
$693$	166.937	+	38.6477i	0.240891	+	0.0557687i
$694$		0			0
$695$	145.461	−	340.347i	0.209296	−	0.489708i
$696$		0			0
$697$			48.0184i			0.0688929i
$698$		0			0
$699$		−	140.817i		−	0.201455i
$700$		0			0
$701$	89.2192			0.127274			0.0636371	−	0.997973i	$-0.479730\pi$
$701$	89.2192			0.127274			0.0636371	+	0.997973i	$0.479730\pi$
$702$		0			0
$703$	2142.43			3.04755
$704$		0			0
$705$	185.850	+	79.4305i	0.263617	+	0.112667i
$706$		0			0
$707$	−137.681	+	594.709i	−0.194740	+	0.841172i
$708$		0			0
$709$	569.646			0.803450			0.401725	−	0.915760i	$-0.368411\pi$
$709$	569.646			0.803450			0.401725	+	0.915760i	$0.368411\pi$
$710$		0			0
$711$	168.552			0.237064
$712$		0			0
$713$	−717.423			−1.00620
$714$		0			0
$715$	547.967	+	234.195i	0.766387	+	0.327546i
$716$		0			0
$717$	−431.740			−0.602147
$718$		0			0
$719$			1261.33i			1.75428i	0.480233	+	0.877141i	$0.340552\pi$
$719$			1261.33i			1.75428i	−0.480233	+	0.877141i	$0.659448\pi$
$720$		0			0
$721$	−1279.14	−	296.134i	−1.77412	−	0.410727i
$722$		0			0
$723$		−	454.391i		−	0.628481i
$724$		0			0
$725$	−159.870	−	167.193i	−0.220510	−	0.230611i
$726$		0			0
$727$	−307.746			−0.423310			−0.211655	−	0.977344i	$-0.567885\pi$
$727$	−307.746			−0.423310			−0.211655	+	0.977344i	$0.567885\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	244.716i		−	0.334769i
$732$		0			0
$733$	633.490			0.864243			0.432121	−	0.901815i	$-0.357765\pi$
$733$	633.490			0.864243			0.432121	+	0.901815i	$0.357765\pi$
$734$		0			0
$735$	277.218	+	321.287i	0.377167	+	0.437125i
$736$		0			0
$737$			40.7891i			0.0553448i
$738$		0			0
$739$	−749.557			−1.01428			−0.507142	−	0.861862i	$-0.669298\pi$
$739$	−749.557			−1.01428			−0.507142	+	0.861862i	$0.669298\pi$
$740$		0			0
$741$			854.439i			1.15309i
$742$		0			0
$743$		−	145.699i		−	0.196095i	−0.995182	−	0.0980476i	$-0.968740\pi$
$743$		−	145.699i		−	0.196095i	0.995182	−	0.0980476i	$-0.0312597\pi$
$744$		0			0
$745$	1043.58	+	446.016i	1.40078	+	0.598679i
$746$		0			0
$747$	−271.194			−0.363044
$748$		0			0
$749$	−249.228	+	1076.53i	−0.332748	+	1.43729i
$750$		0			0
$751$	−156.811			−0.208803			−0.104402	−	0.994535i	$-0.533293\pi$
$751$	−156.811			−0.208803			−0.104402	+	0.994535i	$0.533293\pi$
$752$		0			0
$753$			245.807i			0.326437i
$754$		0			0
$755$	1353.68	+	578.548i	1.79295	+	0.766289i
$756$		0			0
$757$		−	296.377i		−	0.391515i	−0.980652	−	0.195758i	$-0.937283\pi$
$757$		−	296.377i		−	0.391515i	0.980652	−	0.195758i	$-0.0627166\pi$
$758$		0			0
$759$			526.558i			0.693752i
$760$		0			0
$761$			312.747i			0.410968i	0.978660	+	0.205484i	$0.0658769\pi$
$761$			312.747i			0.410968i	−0.978660	+	0.205484i	$0.934123\pi$
$762$		0			0
$763$	−764.297	−	176.942i	−1.00170	−	0.231904i
$764$		0			0
$765$	80.1958	+	34.2748i	0.104831	+	0.0448037i
$766$		0			0
$767$			627.125i			0.817634i
$768$		0			0
$769$			91.1460i			0.118525i	0.998242	+	0.0592627i	$0.0188750\pi$
$769$			91.1460i			0.118525i	−0.998242	+	0.0592627i	$0.981125\pi$
$770$		0			0
$771$	295.908			0.383798
$772$		0			0
$773$	−417.539			−0.540154			−0.270077	−	0.962839i	$-0.587049\pi$
$773$	−417.539			−0.540154			−0.270077	+	0.962839i	$0.587049\pi$
$774$		0			0
$775$	347.932	−	332.692i	0.448945	−	0.429280i
$776$		0			0
$777$	−173.469	+	749.291i	−0.223254	+	0.964339i
$778$		0			0
$779$	278.929			0.358061
$780$		0			0
$781$	316.692			0.405496
$782$		0			0
$783$	48.0802			0.0614052
$784$		0			0
$785$	−302.802	−	129.414i	−0.385735	−	0.164859i
$786$		0			0
$787$	318.111			0.404207			0.202104	−	0.979364i	$-0.435222\pi$
$787$	318.111			0.404207			0.202104	+	0.979364i	$0.435222\pi$
$788$		0			0
$789$		−	664.098i		−	0.841696i
$790$		0			0
$791$	224.172	−	968.302i	0.283403	−	1.22415i
$792$		0			0
$793$		−	500.801i		−	0.631527i
$794$		0			0
$795$	−242.838	+	568.188i	−0.305456	+	0.714702i
$796$		0			0
$797$	−1188.06			−1.49067			−0.745335	−	0.666690i	$-0.767711\pi$
$797$	−1188.06			−1.49067			−0.745335	+	0.666690i	$0.767711\pi$
$798$		0			0
$799$	−135.692			−0.169827
$800$		0			0
$801$		−	48.6867i		−	0.0607824i
$802$		0			0
$803$	−1016.93			−1.26642
$804$		0			0
$805$	−769.723	+	1052.61i	−0.956177	+	1.30759i
$806$		0			0
$807$		−	267.622i		−	0.331626i
$808$		0			0
$809$	−223.374			−0.276111			−0.138055	−	0.990425i	$-0.544085\pi$
$809$	−223.374			−0.276111			−0.138055	+	0.990425i	$0.544085\pi$
$810$		0			0
$811$			366.185i			0.451523i	0.974183	+	0.225761i	$0.0724870\pi$
$811$			366.185i			0.451523i	−0.974183	+	0.225761i	$0.927513\pi$
$812$		0			0
$813$			554.824i			0.682440i
$814$		0			0
$815$	55.4278	−	129.689i	0.0680096	−	0.159128i
$816$		0			0
$817$	−1421.51			−1.73991
$818$		0			0
$819$	−298.831	−	69.1824i	−0.364873	−	0.0844718i
$820$		0			0
$821$	−1547.39			−1.88476			−0.942381	−	0.334543i	$-0.891418\pi$
$821$	−1547.39			−1.88476			−0.942381	+	0.334543i	$0.891418\pi$
$822$		0			0
$823$		−	1284.25i		−	1.56045i	−0.625499	−	0.780225i	$-0.715105\pi$
$823$		−	1284.25i		−	1.56045i	0.625499	−	0.780225i	$-0.284895\pi$
$824$		0			0
$825$	−244.182	−	255.367i	−0.295978	−	0.309536i
$826$		0			0
$827$			807.319i			0.976202i	0.872787	+	0.488101i	$0.162310\pi$
$827$			807.319i			0.976202i	−0.872787	+	0.488101i	$0.837690\pi$
$828$		0			0
$829$			623.615i			0.752250i	0.926569	+	0.376125i	$0.122744\pi$
$829$			623.615i			0.752250i	−0.926569	+	0.376125i	$0.877256\pi$
$830$		0			0
$831$		−	385.994i		−	0.464493i
$832$		0			0
$833$	−255.911	−	125.202i	−0.307216	−	0.150303i
$834$		0			0
$835$	−590.623	−	252.426i	−0.707333	−	0.302307i
$836$		0			0
$837$			100.056i			0.119541i
$838$		0			0
$839$		−	1146.31i		−	1.36628i	−0.730286	−	0.683142i	$-0.760613\pi$
$839$		−	1146.31i		−	1.36628i	0.730286	−	0.683142i	$-0.239387\pi$
$840$		0			0
$841$	−755.381			−0.898194
$842$		0			0
$843$	−379.272			−0.449908
$844$		0			0
$845$	−203.893	−	87.1419i	−0.241294	−	0.103127i
$846$		0			0
$847$	−371.126	−	85.9194i	−0.438165	−	0.101440i
$848$		0			0
$849$	−63.0392			−0.0742511
$850$		0			0
$851$	−2363.43			−2.77724
$852$		0			0
$853$	948.920			1.11245			0.556225	−	0.831032i	$-0.312249\pi$
$853$	948.920			1.11245			0.556225	+	0.831032i	$0.312249\pi$
$854$		0			0
$855$	199.096	−	465.841i	0.232861	−	0.544844i
$856$		0			0
$857$	868.842			1.01382			0.506909	−	0.862000i	$-0.330788\pi$
$857$	868.842			1.01382			0.506909	+	0.862000i	$0.330788\pi$
$858$		0			0
$859$		−	119.981i		−	0.139675i	−0.997558	−	0.0698377i	$-0.977752\pi$
$859$		−	119.981i		−	0.139675i	0.997558	−	0.0698377i	$-0.0222482\pi$
$860$		0			0
$861$	−22.5844	+	97.5525i	−0.0262304	+	0.113301i
$862$		0			0
$863$			788.430i			0.913592i	0.889571	+	0.456796i	$0.151003\pi$
$863$			788.430i			0.913592i	−0.889571	+	0.456796i	$0.848997\pi$
$864$		0			0
$865$	516.991	+	220.957i	0.597678	+	0.255441i
$866$		0			0
$867$	442.011			0.509816
$868$		0			0
$869$	458.443			0.527552
$870$		0			0
$871$		−	73.0157i		−	0.0838298i
$872$		0			0
$873$	−248.282			−0.284400
$874$		0			0
$875$	−114.831	−	867.432i	−0.131235	−	0.991351i
$876$		0			0
$877$		−	234.663i		−	0.267574i	−0.991010	−	0.133787i	$-0.957286\pi$
$877$		−	234.663i		−	0.267574i	0.991010	−	0.133787i	$-0.0427139\pi$
$878$		0			0
$879$	−291.747			−0.331908
$880$		0			0
$881$			750.816i			0.852232i	0.904669	+	0.426116i	$0.140118\pi$
$881$			750.816i			0.852232i	−0.904669	+	0.426116i	$0.859882\pi$
$882$		0			0
$883$			1293.79i			1.46522i	0.680649	+	0.732610i	$0.261698\pi$
$883$			1293.79i			1.46522i	−0.680649	+	0.732610i	$0.738302\pi$
$884$		0			0
$885$	146.129	−	341.910i	0.165117	−	0.386339i
$886$		0			0
$887$	728.248			0.821024			0.410512	−	0.911855i	$-0.365350\pi$
$887$	728.248			0.821024			0.410512	+	0.911855i	$0.365350\pi$
$888$		0			0
$889$	−319.574	+	1380.39i	−0.359476	+	1.55274i
$890$		0			0
$891$	73.4368			0.0824207
$892$		0			0
$893$			788.206i			0.882650i
$894$		0			0
$895$	−316.790	−	135.393i	−0.353955	−	0.151277i
$896$		0			0
$897$		−	942.580i		−	1.05081i
$898$		0			0
$899$			178.175i			0.198192i
$900$		0			0
$901$		−	414.842i		−	0.460424i
$902$		0			0
$903$	115.097	−	497.157i	0.127461	−	0.550561i
$904$		0			0
$905$	327.246	−	765.685i	0.361598	−	0.846061i
$906$		0			0
$907$			913.713i			1.00740i	0.863878	+	0.503700i	$0.168028\pi$
$907$			913.713i			1.00740i	−0.863878	+	0.503700i	$0.831972\pi$
$908$		0			0
$909$			261.616i			0.287807i
$910$		0			0
$911$	1331.32			1.46138			0.730690	−	0.682709i	$-0.239198\pi$
$911$	1331.32			1.46138			0.730690	+	0.682709i	$0.239198\pi$
$912$		0			0
$913$	−737.615			−0.807903
$914$		0			0
$915$	−116.693	+	273.037i	−0.127534	+	0.298402i
$916$		0			0
$917$	−223.023	+	963.341i	−0.243210	+	1.05053i
$918$		0			0
$919$	111.140			0.120936			0.0604678	−	0.998170i	$-0.480741\pi$
$919$	111.140			0.120936			0.0604678	+	0.998170i	$0.480741\pi$
$920$		0			0
$921$	−386.941			−0.420131
$922$		0			0
$923$	−566.904			−0.614197
$924$		0			0
$925$	1146.21	−	1096.00i	1.23914	−	1.18486i
$926$		0			0
$927$	−562.702			−0.607014
$928$		0			0
$929$			571.383i			0.615051i	0.951540	+	0.307526i	$0.0995010\pi$
$929$			571.383i			0.615051i	−0.951540	+	0.307526i	$0.900499\pi$
$930$		0			0
$931$	−727.275	+	1486.53i	−0.781176	+	1.59671i
$932$		0			0
$933$		−	79.7428i		−	0.0854693i
$934$		0			0
$935$	218.123	+	93.2235i	0.233287	+	0.0997043i
$936$		0			0
$937$	1657.10			1.76852			0.884259	−	0.466997i	$-0.154664\pi$
$937$	1657.10			1.76852			0.884259	+	0.466997i	$0.154664\pi$
$938$		0			0
$939$	−138.629			−0.147635
$940$		0			0
$941$			519.083i			0.551629i	0.961211	+	0.275815i	$0.0889476\pi$
$941$			519.083i			0.551629i	−0.961211	+	0.275815i	$0.911052\pi$
$942$		0			0
$943$	−307.703			−0.326302
$944$		0			0
$945$	146.803	+	107.350i	0.155347	+	0.113598i
$946$		0			0
$947$			558.258i			0.589502i	0.955574	+	0.294751i	$0.0952366\pi$
$947$			558.258i			0.589502i	−0.955574	+	0.294751i	$0.904763\pi$
$948$		0			0
$949$	1820.39			1.91822
$950$		0			0
$951$		−	320.839i		−	0.337370i
$952$		0			0
$953$		−	291.413i		−	0.305784i	−0.988243	−	0.152892i	$-0.951141\pi$
$953$		−	291.413i		−	0.305784i	0.988243	−	0.152892i	$-0.0488587\pi$
$954$		0			0
$955$	−681.786	−	291.389i	−0.713912	−	0.305119i
$956$		0			0
$957$	130.773			0.136648
$958$		0			0
$959$	−79.9559	+	345.367i	−0.0833742	+	0.360132i
$960$		0			0
$961$	590.215			0.614167
$962$		0			0
$963$			473.573i			0.491768i
$964$		0			0
$965$	69.1458	−	161.786i	0.0716537	−	0.167654i
$966$		0			0
$967$		−	323.558i		−	0.334600i	−0.985906	−	0.167300i	$-0.946495\pi$
$967$		−	323.558i		−	0.334600i	0.985906	−	0.167300i	$-0.0535048\pi$
$968$		0			0
$969$			340.117i			0.350998i
$970$		0			0
$971$			219.659i			0.226219i	0.993583	+	0.113110i	$0.0360812\pi$
$971$			219.659i			0.226219i	−0.993583	+	0.113110i	$0.963919\pi$
$972$		0			0
$973$	−116.873	+	504.827i	−0.120116	+	0.518836i
$974$		0			0
$975$	437.104	+	457.127i	0.448312	+	0.468849i
$976$		0			0
$977$			1302.29i			1.33295i	0.745527	+	0.666475i	$0.232198\pi$
$977$			1302.29i			1.33295i	−0.745527	+	0.666475i	$0.767802\pi$
$978$		0			0
$979$		−	132.422i		−	0.135263i
$980$		0			0
$981$	−336.219			−0.342731
$982$		0			0
$983$	−649.905			−0.661144			−0.330572	−	0.943781i	$-0.607242\pi$
$983$	−649.905			−0.661144			−0.330572	+	0.943781i	$0.607242\pi$
$984$		0			0
$985$	−380.492	+	890.270i	−0.386287	+	0.903827i
$986$		0			0
$987$	−275.667	−	63.8196i	−0.279297	−	0.0646602i
$988$		0			0
$989$	1568.15			1.58559
$990$		0			0
$991$	575.568			0.580795			0.290397	−	0.956906i	$-0.406212\pi$
$991$	575.568			0.580795			0.290397	+	0.956906i	$0.406212\pi$
$992$		0			0
$993$	106.467			0.107217
$994$		0			0
$995$	357.313	−	836.035i	0.359108	−	0.840236i
$996$		0			0
$997$	−705.512			−0.707635			−0.353817	−	0.935315i	$-0.615117\pi$
$997$	−705.512			−0.707635			−0.353817	+	0.935315i	$0.615117\pi$
$998$		0			0
$999$			329.618i			0.329948i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.3.bd.a.769.1		16
4.3	odd	2		210.3.h.a.139.13	yes	16
5.4	even	2	inner	1680.3.bd.a.769.15		16
7.6	odd	2	inner	1680.3.bd.a.769.16		16
12.11	even	2		630.3.h.e.559.7		16
20.3	even	4		1050.3.f.e.601.14		16
20.7	even	4		1050.3.f.e.601.3		16
20.19	odd	2		210.3.h.a.139.4	✓	16
28.27	even	2		210.3.h.a.139.12	yes	16
35.34	odd	2	inner	1680.3.bd.a.769.2		16
60.59	even	2		630.3.h.e.559.10		16
84.83	odd	2		630.3.h.e.559.2		16
140.27	odd	4		1050.3.f.e.601.7		16
140.83	odd	4		1050.3.f.e.601.10		16
140.139	even	2		210.3.h.a.139.5	yes	16
420.419	odd	2		630.3.h.e.559.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.h.a.139.4	✓	16	20.19	odd	2
210.3.h.a.139.5	yes	16	140.139	even	2
210.3.h.a.139.12	yes	16	28.27	even	2
210.3.h.a.139.13	yes	16	4.3	odd	2
630.3.h.e.559.2		16	84.83	odd	2
630.3.h.e.559.7		16	12.11	even	2
630.3.h.e.559.10		16	60.59	even	2
630.3.h.e.559.15		16	420.419	odd	2
1050.3.f.e.601.3		16	20.7	even	4
1050.3.f.e.601.7		16	140.27	odd	4
1050.3.f.e.601.10		16	140.83	odd	4
1050.3.f.e.601.14		16	20.3	even	4
1680.3.bd.a.769.1		16	1.1	even	1	trivial
1680.3.bd.a.769.2		16	35.34	odd	2	inner
1680.3.bd.a.769.15		16	5.4	even	2	inner
1680.3.bd.a.769.16		16	7.6	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.bd (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(-4.59769 - 1.96501i) q^{5} +(6.81963 + 1.57881i) q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +(-4.59769 - 1.96501i) q^{5} +(6.81963 + 1.57881i) q^{7} +3.00000 q^{9} +8.15965 q^{11} -14.6064 q^{13} +(7.96343 + 3.40349i) q^{15} -5.81421 q^{17} +33.7736i q^{19} +(-11.8119 - 2.73459i) q^{21} -37.2576i q^{23} +(17.2775 + 18.0690i) q^{25} -5.19615 q^{27} -9.25305 q^{29} -19.2558i q^{31} -14.1329 q^{33} +(-28.2522 - 20.6595i) q^{35} -63.4350i q^{37} +25.2990 q^{39} -8.25880i q^{41} +42.0893i q^{43} +(-13.7931 - 5.89502i) q^{45} +23.3380 q^{47} +(44.0147 + 21.5339i) q^{49} +10.0705 q^{51} +71.3497i q^{53} +(-37.5155 - 16.0337i) q^{55} -58.4976i q^{57} -42.9350i q^{59} +34.2864i q^{61} +(20.4589 + 4.73644i) q^{63} +(67.1557 + 28.7017i) q^{65} +4.99889i q^{67} +64.5320i q^{69} +38.8120 q^{71} -124.629 q^{73} +(-29.9255 - 31.2964i) q^{75} +(55.6458 + 12.8826i) q^{77} +56.1842 q^{79} +9.00000 q^{81} -90.3980 q^{83} +(26.7319 + 11.4249i) q^{85} +16.0267 q^{87} -16.2289i q^{89} +(-99.6102 - 23.0608i) q^{91} +33.3520i q^{93} +(66.3653 - 155.280i) q^{95} -82.7605 q^{97} +24.4789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9}+O(q^{10}) \) 16 * q + 48 * q^9	\( 16 q + 48 q^{9} - 96 q^{11} + 24 q^{15} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 8 q^{35} + 144 q^{39} + 224 q^{49} + 48 q^{51} + 368 q^{65} + 384 q^{71} + 608 q^{79} + 144 q^{81} - 440 q^{85} - 224 q^{91} + 560 q^{95} - 288 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 96 * q^11 + 24 * q^15 + 24 * q^21 + 24 * q^25 + 64 * q^29 + 8 * q^35 + 144 * q^39 + 224 * q^49 + 48 * q^51 + 368 * q^65 + 384 * q^71 + 608 * q^79 + 144 * q^81 - 440 * q^85 - 224 * q^91 + 560 * q^95 - 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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