LMFDB - Embedded newform 280.6.g.a.169.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,6,Mod(169,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("280.169");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.g (of order $2$, degree $1$, 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:	$44.9074695476$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 2997 x^{18} + 3735306 x^{16} + 2520827714 x^{14} + 1008202629141 x^{12} + 246520004342481 x^{10} + \cdots + 81\!\cdots\!00 $ x^20 + 2997x^18 + 3735306x^16 + 2520827714x^14 + 1008202629141x^12 + 246520004342481x^10 + 36629357739714496x^8 + 3153081134047078008x^6 + 139141465378172063856x^4 + 2257384477735837731600*x^2 + 810880935783229440000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{25}\cdot 5^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.20
Root			$26.0724i$ of defining polynomial
Character	$\chi$	$=$	280.169
Dual form			280.6.g.a.169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+26.0724i q^{3} +(-26.2314 - 49.3651i) q^{5} +49.0000i q^{7} -436.769 q^{9} +O(q^{10})$	$q+26.0724i q^{3} +(-26.2314 - 49.3651i) q^{5} +49.0000i q^{7} -436.769 q^{9} +711.472 q^{11} -1009.70i q^{13} +(1287.07 - 683.916i) q^{15} +1706.17i q^{17} +1910.64 q^{19} -1277.55 q^{21} +3817.51i q^{23} +(-1748.83 + 2589.83i) q^{25} -5052.03i q^{27} +3522.99 q^{29} -4042.45 q^{31} +18549.8i q^{33} +(2418.89 - 1285.34i) q^{35} -6343.71i q^{37} +26325.3 q^{39} +3368.40 q^{41} +7.74975i q^{43} +(11457.1 + 21561.2i) q^{45} +7471.06i q^{47} -2401.00 q^{49} -44483.9 q^{51} -7618.70i q^{53} +(-18662.9 - 35121.9i) q^{55} +49814.9i q^{57} +30756.5 q^{59} -34236.3 q^{61} -21401.7i q^{63} +(-49844.0 + 26485.9i) q^{65} +42305.4i q^{67} -99531.7 q^{69} -21086.2 q^{71} -5382.30i q^{73} +(-67523.1 - 45596.1i) q^{75} +34862.2i q^{77} -5475.96 q^{79} +25583.5 q^{81} +31229.0i q^{83} +(84225.2 - 44755.2i) q^{85} +91852.7i q^{87} +25325.5 q^{89} +49475.3 q^{91} -105396. i q^{93} +(-50118.8 - 94318.9i) q^{95} +60066.6i q^{97} -310749. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 14 q^{5} - 1134 q^{9}+O(q^{10}) $ 20 * q + 14 * q^5 - 1134 * q^9	$ 20 q + 14 q^{5} - 1134 q^{9} + 822 q^{11} + 1322 q^{15} + 3000 q^{19} - 882 q^{21} - 1944 q^{25} + 10406 q^{29} - 8532 q^{31} - 2058 q^{35} + 71066 q^{39} - 28880 q^{41} - 3922 q^{45} - 48020 q^{49} + 34770 q^{51} - 8844 q^{55} + 61196 q^{59} - 145052 q^{61} - 12722 q^{65} + 64980 q^{69} + 41464 q^{71} - 159032 q^{75} + 97174 q^{79} - 165852 q^{81} + 319566 q^{85} - 84604 q^{89} + 61642 q^{91} + 33416 q^{95} - 630244 q^{99}+O(q^{100}) $ 20 * q + 14 * q^5 - 1134 * q^9 + 822 * q^11 + 1322 * q^15 + 3000 * q^19 - 882 * q^21 - 1944 * q^25 + 10406 * q^29 - 8532 * q^31 - 2058 * q^35 + 71066 * q^39 - 28880 * q^41 - 3922 * q^45 - 48020 * q^49 + 34770 * q^51 - 8844 * q^55 + 61196 * q^59 - 145052 * q^61 - 12722 * q^65 + 64980 * q^69 + 41464 * q^71 - 159032 * q^75 + 97174 * q^79 - 165852 * q^81 + 319566 * q^85 - 84604 * q^89 + 61642 * q^91 + 33416 * q^95 - 630244 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			26.0724i			1.67254i	0.548315	+	0.836272i	$0.315270\pi$
$3$			26.0724i			1.67254i	−0.548315	+	0.836272i	$0.684730\pi$
$4$		0			0
$5$	−26.2314	−	49.3651i	−0.469242	−	0.883070i
$5$	−26.2314	−	49.3651i	−0.469242	−	0.883070i
$6$		0			0
$7$			49.0000i			0.377964i
$7$			49.0000i			0.377964i
$8$		0			0
$9$	−436.769			−1.79740
$10$		0			0
$11$	711.472			1.77287			0.886434	−	0.462855i	$-0.153175\pi$
$11$	711.472			1.77287			0.886434	+	0.462855i	$0.153175\pi$
$12$		0			0
$13$		−	1009.70i		−	1.65705i	−0.559956	−	0.828523i	$-0.689182\pi$
$13$		−	1009.70i		−	1.65705i	0.559956	−	0.828523i	$-0.310818\pi$
$14$		0			0
$15$	1287.07	−	683.916i	1.47697	−	0.784828i
$16$		0			0
$17$			1706.17i			1.43186i	0.698174	+	0.715928i	$0.253996\pi$
$17$			1706.17i			1.43186i	−0.698174	+	0.715928i	$0.746004\pi$
$18$		0			0
$19$	1910.64			1.21421			0.607106	−	0.794621i	$-0.292330\pi$
$19$	1910.64			1.21421			0.607106	+	0.794621i	$0.292330\pi$
$20$		0			0
$21$	−1277.55			−0.632162
$22$		0			0
$23$			3817.51i			1.50474i	0.658741	+	0.752369i	$0.271089\pi$
$23$			3817.51i			1.50474i	−0.658741	+	0.752369i	$0.728911\pi$
$24$		0			0
$25$	−1748.83	+	2589.83i	−0.559624	+	0.828746i
$26$		0			0
$27$		−	5052.03i		−	1.33369i
$28$		0			0
$29$	3522.99			0.777887			0.388943	−	0.921262i	$-0.372840\pi$
$29$	3522.99			0.777887			0.388943	+	0.921262i	$0.372840\pi$
$30$		0			0
$31$	−4042.45			−0.755510			−0.377755	−	0.925906i	$-0.623304\pi$
$31$	−4042.45			−0.755510			−0.377755	+	0.925906i	$0.623304\pi$
$32$		0			0
$33$			18549.8i			2.96520i
$34$		0			0
$35$	2418.89	−	1285.34i	0.333769	−	0.177357i
$36$		0			0
$37$		−	6343.71i		−	0.761796i	−0.924617	−	0.380898i	$-0.875615\pi$
$37$		−	6343.71i		−	0.761796i	0.924617	−	0.380898i	$-0.124385\pi$
$38$		0			0
$39$	26325.3			2.77148
$40$		0			0
$41$	3368.40			0.312942			0.156471	−	0.987683i	$-0.449988\pi$
$41$	3368.40			0.312942			0.156471	+	0.987683i	$0.449988\pi$
$42$		0			0
$43$			7.74975i			0.000639170i	1.00000		0.000319585i	$0.000101727\pi$
$43$			7.74975i			0.000639170i	−1.00000		0.000319585i	$0.999898\pi$
$44$		0			0
$45$	11457.1	+	21561.2i	0.843417	+	1.58723i
$46$		0			0
$47$			7471.06i			0.493330i	0.969101	+	0.246665i	$0.0793347\pi$
$47$			7471.06i			0.493330i	−0.969101	+	0.246665i	$0.920665\pi$
$48$		0			0
$49$	−2401.00			−0.142857
$50$		0			0
$51$	−44483.9			−2.39484
$52$		0			0
$53$		−	7618.70i		−	0.372556i	−0.982497	−	0.186278i	$-0.940358\pi$
$53$		−	7618.70i		−	0.372556i	0.982497	−	0.186278i	$-0.0596425\pi$
$54$		0			0
$55$	−18662.9	−	35121.9i	−0.831904	−	1.56557i
$56$		0			0
$57$			49814.9i			2.03082i
$58$		0			0
$59$	30756.5			1.15029			0.575144	−	0.818052i	$-0.304946\pi$
$59$	30756.5			1.15029			0.575144	+	0.818052i	$0.304946\pi$
$60$		0			0
$61$	−34236.3			−1.17805			−0.589024	−	0.808116i	$-0.700487\pi$
$61$	−34236.3			−1.17805			−0.589024	+	0.808116i	$0.700487\pi$
$62$		0			0
$63$		−	21401.7i		−	0.679355i
$64$		0			0
$65$	−49844.0	+	26485.9i	−1.46329	+	0.777555i
$66$		0			0
$67$			42305.4i			1.15135i	0.817677	+	0.575677i	$0.195261\pi$
$67$			42305.4i			1.15135i	−0.817677	+	0.575677i	$0.804739\pi$
$68$		0			0
$69$	−99531.7			−2.51674
$70$		0			0
$71$	−21086.2			−0.496424			−0.248212	−	0.968706i	$-0.579843\pi$
$71$	−21086.2			−0.496424			−0.248212	+	0.968706i	$0.579843\pi$
$72$		0			0
$73$		−	5382.30i		−	0.118212i	−0.998252	−	0.0591058i	$-0.981175\pi$
$73$		−	5382.30i		−	0.118212i	0.998252	−	0.0591058i	$-0.0188249\pi$
$74$		0			0
$75$	−67523.1	−	45596.1i	−1.38612	−	0.935996i
$76$		0			0
$77$			34862.2i			0.670081i
$78$		0			0
$79$	−5475.96			−0.0987172			−0.0493586	−	0.998781i	$-0.515718\pi$
$79$	−5475.96			−0.0987172			−0.0493586	+	0.998781i	$0.515718\pi$
$80$		0			0
$81$	25583.5			0.433259
$82$		0			0
$83$			31229.0i			0.497580i	0.968557	+	0.248790i	$0.0800329\pi$
$83$			31229.0i			0.497580i	−0.968557	+	0.248790i	$0.919967\pi$
$84$		0			0
$85$	84225.2	−	44755.2i	1.26443	−	0.671887i
$86$		0			0
$87$			91852.7i			1.30105i
$88$		0			0
$89$	25325.5			0.338909			0.169455	−	0.985538i	$-0.445799\pi$
$89$	25325.5			0.338909			0.169455	+	0.985538i	$0.445799\pi$
$90$		0			0
$91$	49475.3			0.626304
$92$		0			0
$93$		−	105396.i		−	1.26362i
$94$		0			0
$95$	−50118.8	−	94318.9i	−0.569759	−	1.07223i
$96$		0			0
$97$			60066.6i			0.648192i	0.946024	+	0.324096i	$0.105060\pi$
$97$			60066.6i			0.648192i	−0.946024	+	0.324096i	$0.894940\pi$
$98$		0			0
$99$	−310749.			−3.18656
$100$		0			0
$101$	−30614.9			−0.298627			−0.149314	−	0.988790i	$-0.547706\pi$
$101$	−30614.9			−0.298627			−0.149314	+	0.988790i	$0.547706\pi$
$102$		0			0
$103$			173209.i			1.60871i	0.594151	+	0.804354i	$0.297488\pi$
$103$			173209.i			1.60871i	−0.594151	+	0.804354i	$0.702512\pi$
$104$		0			0
$105$	33511.9	+	63066.2i	0.296637	+	0.558243i
$106$		0			0
$107$			94935.2i			0.801619i	0.916161	+	0.400809i	$0.131271\pi$
$107$			94935.2i			0.801619i	−0.916161	+	0.400809i	$0.868729\pi$
$108$		0			0
$109$	107592.			0.867389			0.433694	−	0.901060i	$-0.357210\pi$
$109$	107592.			0.867389			0.433694	+	0.901060i	$0.357210\pi$
$110$		0			0
$111$	165396.			1.27414
$112$		0			0
$113$			245229.i			1.80666i	0.428951	+	0.903328i	$0.358883\pi$
$113$			245229.i			1.80666i	−0.428951	+	0.903328i	$0.641117\pi$
$114$		0			0
$115$	188452.	−	100139.i	1.32879	−	0.706086i
$116$		0			0
$117$			441006.i			2.97838i
$118$		0			0
$119$	−83602.2			−0.541191
$120$		0			0
$121$	345142.			2.14306
$122$		0			0
$123$			87822.3i			0.523410i
$124$		0			0
$125$	173722.	+	18396.0i	0.994440	+	0.105305i
$126$		0			0
$127$			261908.i			1.44092i	0.693496	+	0.720460i	$0.256069\pi$
$127$			261908.i			1.44092i	−0.693496	+	0.720460i	$0.743931\pi$
$128$		0			0
$129$	−202.054			−0.00106904
$130$		0			0
$131$	−228038.			−1.16099			−0.580495	−	0.814263i	$-0.697141\pi$
$131$	−228038.			−1.16099			−0.580495	+	0.814263i	$0.697141\pi$
$132$		0			0
$133$			93621.3i			0.458929i
$134$		0			0
$135$	−249394.	+	132522.i	−1.17775	+	0.625825i
$136$		0			0
$137$		−	274421.i		−	1.24915i	−0.780963	−	0.624577i	$-0.785271\pi$
$137$		−	274421.i		−	1.24915i	0.780963	−	0.624577i	$-0.214729\pi$
$138$		0			0
$139$	235835.			1.03531			0.517657	−	0.855588i	$-0.326804\pi$
$139$	235835.			1.03531			0.517657	+	0.855588i	$0.326804\pi$
$140$		0			0
$141$	−194788.			−0.825116
$142$		0			0
$143$		−	718374.i		−	2.93772i
$144$		0			0
$145$	−92413.0	−	173913.i	−0.365017	−	0.686928i
$146$		0			0
$147$		−	62599.8i		−	0.238935i
$148$		0			0
$149$	304119.			1.12222			0.561109	−	0.827742i	$-0.310375\pi$
$149$	304119.			1.12222			0.561109	+	0.827742i	$0.310375\pi$
$150$		0			0
$151$	−307040.			−1.09585			−0.547927	−	0.836526i	$-0.684583\pi$
$151$	−307040.			−1.09585			−0.547927	+	0.836526i	$0.684583\pi$
$152$		0			0
$153$		−	745202.i		−	2.57363i
$154$		0			0
$155$	106039.	+	199556.i	0.354517	+	0.667168i
$156$		0			0
$157$		−	60977.5i		−	0.197433i	−0.995116	−	0.0987167i	$-0.968526\pi$
$157$		−	60977.5i		−	0.197433i	0.995116	−	0.0987167i	$-0.0314738\pi$
$158$		0			0
$159$	198638.			0.623116
$160$		0			0
$161$	−187058.			−0.568738
$162$		0			0
$163$			624676.i			1.84156i	0.390082	+	0.920780i	$0.372447\pi$
$163$			624676.i			1.84156i	−0.390082	+	0.920780i	$0.627553\pi$
$164$		0			0
$165$	915712.	−	486587.i	2.61848	−	1.39140i
$166$		0			0
$167$		−	439452.i		−	1.21933i	−0.792660	−	0.609663i	$-0.791305\pi$
$167$		−	439452.i		−	1.21933i	0.792660	−	0.609663i	$-0.208695\pi$
$168$		0			0
$169$	−648203.			−1.74580
$170$		0			0
$171$	−834509.			−2.18243
$172$		0			0
$173$		−	658578.i		−	1.67299i	−0.547978	−	0.836493i	$-0.684602\pi$
$173$		−	658578.i		−	1.67299i	0.547978	−	0.836493i	$-0.315398\pi$
$174$		0			0
$175$	−126902.	−	85692.5i	−0.313237	−	0.211518i
$176$		0			0
$177$			801894.i			1.92391i
$178$		0			0
$179$	−41141.9			−0.0959735			−0.0479868	−	0.998848i	$-0.515281\pi$
$179$	−41141.9			−0.0959735			−0.0479868	+	0.998848i	$0.515281\pi$
$180$		0			0
$181$	310451.			0.704363			0.352182	−	0.935932i	$-0.385440\pi$
$181$	310451.			0.704363			0.352182	+	0.935932i	$0.385440\pi$
$182$		0			0
$183$		−	892623.i		−	1.97034i
$184$		0			0
$185$	−313158.	+	166404.i	−0.672719	+	0.357467i
$186$		0			0
$187$			1.21389e6i			2.53849i
$188$		0			0
$189$	247549.			0.504089
$190$		0			0
$191$	−76802.8			−0.152333			−0.0761664	−	0.997095i	$-0.524268\pi$
$191$	−76802.8			−0.152333			−0.0761664	+	0.997095i	$0.524268\pi$
$192$		0			0
$193$			693759.i			1.34065i	0.742068	+	0.670325i	$0.233845\pi$
$193$			693759.i			1.34065i	−0.742068	+	0.670325i	$0.766155\pi$
$194$		0			0
$195$	−690550.	−	1.29955e6i	−1.30050	−	2.44741i
$196$		0			0
$197$			781250.i			1.43425i	0.696946	+	0.717124i	$0.254542\pi$
$197$			781250.i			1.43425i	−0.696946	+	0.717124i	$0.745458\pi$
$198$		0			0
$199$	458151.			0.820118			0.410059	−	0.912059i	$-0.365508\pi$
$199$	458151.			0.820118			0.410059	+	0.912059i	$0.365508\pi$
$200$		0			0
$201$	−1.10300e6			−1.92569
$202$		0			0
$203$			172626.i			0.294014i
$204$		0			0
$205$	−88358.0	−	166282.i	−0.146846	−	0.276350i
$206$		0			0
$207$		−	1.66737e6i		−	2.70462i
$208$		0			0
$209$	1.35937e6			2.15264
$210$		0			0
$211$	231114.			0.357372			0.178686	−	0.983906i	$-0.442815\pi$
$211$	231114.			0.357372			0.178686	+	0.983906i	$0.442815\pi$
$212$		0			0
$213$		−	549768.i		−	0.830292i
$214$		0			0
$215$	382.567	−	203.287i	0.000564432	−	0.000299925i
$216$		0			0
$217$		−	198080.i		−	0.285556i
$218$		0			0
$219$	140329.			0.197714
$220$		0			0
$221$	1.72272e6			2.37265
$222$		0			0
$223$		−	969513.i		−	1.30554i	−0.757554	−	0.652772i	$-0.773606\pi$
$223$		−	969513.i		−	1.30554i	0.757554	−	0.652772i	$-0.226394\pi$
$224$		0			0
$225$	763833.	−	1.13116e6i	1.00587	−	1.48959i
$226$		0			0
$227$			150908.i			0.194379i	0.995266	+	0.0971894i	$0.0309853\pi$
$227$			150908.i			0.194379i	−0.995266	+	0.0971894i	$0.969015\pi$
$228$		0			0
$229$	−125606.			−0.158278			−0.0791389	−	0.996864i	$-0.525217\pi$
$229$	−125606.			−0.158278			−0.0791389	+	0.996864i	$0.525217\pi$
$230$		0			0
$231$	−908939.			−1.12074
$232$		0			0
$233$			214202.i			0.258484i	0.991613	+	0.129242i	$0.0412545\pi$
$233$			214202.i			0.258484i	−0.991613	+	0.129242i	$0.958746\pi$
$234$		0			0
$235$	368810.	−	195976.i	0.435645	−	0.231491i
$236$		0			0
$237$		−	142771.i		−	0.165109i
$238$		0			0
$239$	862291.			0.976471			0.488235	−	0.872712i	$-0.337641\pi$
$239$	862291.			0.976471			0.488235	+	0.872712i	$0.337641\pi$
$240$		0			0
$241$	1.36118e6			1.50964			0.754821	−	0.655931i	$-0.227724\pi$
$241$	1.36118e6			1.50964			0.754821	+	0.655931i	$0.227724\pi$
$242$		0			0
$243$		−	560620.i		−	0.609050i
$244$		0			0
$245$	62981.6	+	118526.i	0.0670345	+	0.126153i
$246$		0			0
$247$		−	1.92917e6i		−	2.01201i
$248$		0			0
$249$	−814214.			−0.832224
$250$		0			0
$251$	−1.08922e6			−1.09127			−0.545633	−	0.838024i	$-0.683711\pi$
$251$	−1.08922e6			−1.09127			−0.545633	+	0.838024i	$0.683711\pi$
$252$		0			0
$253$			2.71606e6i			2.66770i
$254$		0			0
$255$	1.16687e6	+	2.19595e6i	1.12376	+	2.11481i
$256$		0			0
$257$		−	240538.i		−	0.227170i	−0.993528	−	0.113585i	$-0.963767\pi$
$257$		−	240538.i		−	0.227170i	0.993528	−	0.113585i	$-0.0362334\pi$
$258$		0			0
$259$	310842.			0.287932
$260$		0			0
$261$	−1.53873e6			−1.39818
$262$		0			0
$263$			268399.i			0.239272i	0.992818	+	0.119636i	$0.0381728\pi$
$263$			268399.i			0.239272i	−0.992818	+	0.119636i	$0.961827\pi$
$264$		0			0
$265$	−376098.	+	199849.i	−0.328993	+	0.174819i
$266$		0			0
$267$			660297.i			0.566841i
$268$		0			0
$269$	−719392.			−0.606156			−0.303078	−	0.952966i	$-0.598014\pi$
$269$	−719392.			−0.606156			−0.303078	+	0.952966i	$0.598014\pi$
$270$		0			0
$271$	1.57925e6			1.30625			0.653127	−	0.757248i	$-0.273457\pi$
$271$	1.57925e6			1.30625			0.653127	+	0.757248i	$0.273457\pi$
$272$		0			0
$273$			1.28994e6i			1.04752i
$274$		0			0
$275$	−1.24424e6	+	1.84259e6i	−0.992140	+	1.46926i
$276$		0			0
$277$			989731.i			0.775028i	0.921864	+	0.387514i	$0.126666\pi$
$277$			989731.i			0.775028i	−0.921864	+	0.387514i	$0.873334\pi$
$278$		0			0
$279$	1.76562e6			1.35796
$280$		0			0
$281$	1.16182e6			0.877758			0.438879	−	0.898546i	$-0.355376\pi$
$281$	1.16182e6			0.877758			0.438879	+	0.898546i	$0.355376\pi$
$282$		0			0
$283$		−	1.42804e6i		−	1.05992i	−0.848022	−	0.529962i	$-0.822206\pi$
$283$		−	1.42804e6i		−	1.05992i	0.848022	−	0.529962i	$-0.177794\pi$
$284$		0			0
$285$	2.45912e6	−	1.30672e6i	1.79336	−	0.952948i
$286$		0			0
$287$			165052.i			0.118281i
$288$		0			0
$289$	−1.49115e6			−1.05021
$290$		0			0
$291$	−1.56608e6			−1.08413
$292$		0			0
$293$		−	1.87283e6i		−	1.27447i	−0.770670	−	0.637235i	$-0.780078\pi$
$293$		−	1.87283e6i		−	1.27447i	0.770670	−	0.637235i	$-0.219922\pi$
$294$		0			0
$295$	−806786.	−	1.51830e6i	−0.539763	−	1.01578i
$296$		0			0
$297$		−	3.59438e6i		−	2.36446i
$298$		0			0
$299$	3.85455e6			2.49342
$300$		0			0
$301$	−379.738			−0.000241584
$302$		0			0
$303$		−	798204.i		−	0.499468i
$304$		0			0
$305$	898067.	+	1.69008e6i	0.552789	+	1.04030i
$306$		0			0
$307$		−	1.78870e6i		−	1.08316i	−0.840650	−	0.541579i	$-0.817827\pi$
$307$		−	1.78870e6i		−	1.08316i	0.840650	−	0.541579i	$-0.182173\pi$
$308$		0			0
$309$	−4.51597e6			−2.69063
$310$		0			0
$311$	−510314.			−0.299183			−0.149591	−	0.988748i	$-0.547796\pi$
$311$	−510314.			−0.299183			−0.149591	+	0.988748i	$0.547796\pi$
$312$		0			0
$313$			1.22593e6i			0.707304i	0.935377	+	0.353652i	$0.115060\pi$
$313$			1.22593e6i			0.707304i	−0.935377	+	0.353652i	$0.884940\pi$
$314$		0			0
$315$	−1.05650e6	+	561397.i	−0.599918	+	0.318782i
$316$		0			0
$317$			1.27697e6i			0.713725i	0.934157	+	0.356863i	$0.116154\pi$
$317$			1.27697e6i			0.713725i	−0.934157	+	0.356863i	$0.883846\pi$
$318$		0			0
$319$	2.50651e6			1.37909
$320$		0			0
$321$	−2.47519e6			−1.34074
$322$		0			0
$323$			3.25987e6i			1.73858i
$324$		0			0
$325$	2.61496e6	+	1.76579e6i	1.37327	+	0.927323i
$326$		0			0
$327$			2.80518e6i			1.45075i
$328$		0			0
$329$	−366082.			−0.186461
$330$		0			0
$331$	−2.06093e6			−1.03393			−0.516966	−	0.856006i	$-0.672939\pi$
$331$	−2.06093e6			−1.03393			−0.516966	+	0.856006i	$0.672939\pi$
$332$		0			0
$333$			2.77074e6i			1.36926i
$334$		0			0
$335$	2.08841e6	−	1.10973e6i	1.01673	−	0.540263i
$336$		0			0
$337$		−	3.36251e6i		−	1.61283i	−0.591350	−	0.806415i	$-0.701405\pi$
$337$		−	3.36251e6i		−	1.61283i	0.591350	−	0.806415i	$-0.298595\pi$
$338$		0			0
$339$	−6.39370e6			−3.02171
$340$		0			0
$341$	−2.87609e6			−1.33942
$342$		0			0
$343$		−	117649.i		−	0.0539949i
$344$		0			0
$345$	2.61086e6	+	4.91339e6i	1.18096	+	2.22246i
$346$		0			0
$347$		−	3.54426e6i		−	1.58016i	−0.613003	−	0.790080i	$-0.710039\pi$
$347$		−	3.54426e6i		−	1.58016i	0.613003	−	0.790080i	$-0.289961\pi$
$348$		0			0
$349$	−3.27698e6			−1.44016			−0.720079	−	0.693892i	$-0.755895\pi$
$349$	−3.27698e6			−1.44016			−0.720079	+	0.693892i	$0.755895\pi$
$350$		0			0
$351$	−5.10104e6			−2.20999
$352$		0			0
$353$		−	2.73905e6i		−	1.16994i	−0.811056	−	0.584969i	$-0.801107\pi$
$353$		−	2.73905e6i		−	1.16994i	0.811056	−	0.584969i	$-0.198893\pi$
$354$		0			0
$355$	553122.	+	1.04092e6i	0.232943	+	0.438377i
$356$		0			0
$357$		−	2.17971e6i		−	0.905166i
$358$		0			0
$359$	2.16903e6			0.888238			0.444119	−	0.895968i	$-0.353517\pi$
$359$	2.16903e6			0.888238			0.444119	+	0.895968i	$0.353517\pi$
$360$		0			0
$361$	1.17444e6			0.474312
$362$		0			0
$363$			8.99868e6i			3.58436i
$364$		0			0
$365$	−265698.	+	141185.i	−0.104389	+	0.0554699i
$366$		0			0
$367$			3.69984e6i			1.43390i	0.697125	+	0.716949i	$0.254462\pi$
$367$			3.69984e6i			1.43390i	−0.697125	+	0.716949i	$0.745538\pi$
$368$		0			0
$369$	−1.47122e6			−0.562484
$370$		0			0
$371$	373317.			0.140813
$372$		0			0
$373$		−	3.03604e6i		−	1.12989i	−0.825129	−	0.564945i	$-0.808897\pi$
$373$		−	3.03604e6i		−	1.12989i	0.825129	−	0.564945i	$-0.191103\pi$
$374$		0			0
$375$	−479627.	+	4.52933e6i	−0.176127	+	1.66325i
$376$		0			0
$377$		−	3.55717e6i		−	1.28899i
$378$		0			0
$379$	3.25231e6			1.16304			0.581520	−	0.813532i	$-0.302458\pi$
$379$	3.25231e6			1.16304			0.581520	+	0.813532i	$0.302458\pi$
$380$		0			0
$381$	−6.82857e6			−2.41000
$382$		0			0
$383$			340002.i			0.118436i	0.998245	+	0.0592182i	$0.0188608\pi$
$383$			340002.i			0.118436i	−0.998245	+	0.0592182i	$0.981139\pi$
$384$		0			0
$385$	1.72097e6	−	914484.i	0.591728	−	0.314430i
$386$		0			0
$387$		−	3384.85i		−	0.00114885i
$388$		0			0
$389$	4.00755e6			1.34278			0.671391	−	0.741103i	$-0.265697\pi$
$389$	4.00755e6			1.34278			0.671391	+	0.741103i	$0.265697\pi$
$390$		0			0
$391$	−6.51332e6			−2.15457
$392$		0			0
$393$		−	5.94549e6i		−	1.94181i
$394$		0			0
$395$	143642.	+	270321.i	0.0463222	+	0.0871741i
$396$		0			0
$397$			1.99548e6i			0.635435i	0.948185	+	0.317718i	$0.102916\pi$
$397$			1.99548e6i			0.635435i	−0.948185	+	0.317718i	$0.897084\pi$
$398$		0			0
$399$	−2.44093e6			−0.767579
$400$		0			0
$401$	−3.16688e6			−0.983492			−0.491746	−	0.870739i	$-0.663641\pi$
$401$	−3.16688e6			−0.983492			−0.491746	+	0.870739i	$0.663641\pi$
$402$		0			0
$403$			4.08167e6i			1.25191i
$404$		0			0
$405$	−671092.	−	1.26293e6i	−0.203303	−	0.382598i
$406$		0			0
$407$		−	4.51337e6i		−	1.35056i
$408$		0			0
$409$	−2.22231e6			−0.656897			−0.328448	−	0.944522i	$-0.606526\pi$
$409$	−2.22231e6			−0.656897			−0.328448	+	0.944522i	$0.606526\pi$
$410$		0			0
$411$	7.15481e6			2.08927
$412$		0			0
$413$			1.50707e6i			0.434768i
$414$		0			0
$415$	1.54162e6	−	819181.i	0.439398	−	0.233485i
$416$		0			0
$417$			6.14879e6i			1.73161i
$418$		0			0
$419$	3.86150e6			1.07454			0.537268	−	0.843412i	$-0.319456\pi$
$419$	3.86150e6			1.07454			0.537268	+	0.843412i	$0.319456\pi$
$420$		0			0
$421$	−3.35310e6			−0.922021			−0.461011	−	0.887395i	$-0.652513\pi$
$421$	−3.35310e6			−0.922021			−0.461011	+	0.887395i	$0.652513\pi$
$422$		0			0
$423$		−	3.26313e6i		−	0.886714i
$424$		0			0
$425$	−4.41869e6	−	2.98379e6i	−1.18665	−	0.801302i
$426$		0			0
$427$		−	1.67758e6i		−	0.445260i
$428$		0			0
$429$	1.87297e7			4.91347
$430$		0			0
$431$	−4.66490e6			−1.20962			−0.604810	−	0.796370i	$-0.706751\pi$
$431$	−4.66490e6			−1.20962			−0.604810	+	0.796370i	$0.706751\pi$
$432$		0			0
$433$			5.31445e6i			1.36219i	0.732193	+	0.681097i	$0.238497\pi$
$433$			5.31445e6i			1.36219i	−0.732193	+	0.681097i	$0.761503\pi$
$434$		0			0
$435$	4.53432e6	−	2.40943e6i	1.14892	−	0.610507i
$436$		0			0
$437$			7.29389e6i			1.82707i
$438$		0			0
$439$	1.40145e6			0.347069			0.173535	−	0.984828i	$-0.444481\pi$
$439$	1.40145e6			0.347069			0.173535	+	0.984828i	$0.444481\pi$
$440$		0			0
$441$	1.04868e6			0.256772
$442$		0			0
$443$			4.25926e6i			1.03116i	0.856842	+	0.515579i	$0.172423\pi$
$443$			4.25926e6i			1.03116i	−0.856842	+	0.515579i	$0.827577\pi$
$444$		0			0
$445$	−664325.	−	1.25020e6i	−0.159030	−	0.299281i
$446$		0			0
$447$			7.92910e6i			1.87696i
$448$		0			0
$449$	3.28668e6			0.769381			0.384690	−	0.923046i	$-0.374308\pi$
$449$	3.28668e6			0.769381			0.384690	+	0.923046i	$0.374308\pi$
$450$		0			0
$451$	2.39653e6			0.554806
$452$		0			0
$453$		−	8.00527e6i		−	1.83287i
$454$		0			0
$455$	−1.29781e6	−	2.44236e6i	−0.293888	−	0.553070i
$456$		0			0
$457$		−	4.72692e6i		−	1.05874i	−0.848392	−	0.529369i	$-0.822429\pi$
$457$		−	4.72692e6i		−	1.05874i	0.848392	−	0.529369i	$-0.177571\pi$
$458$		0			0
$459$	8.61961e6			1.90966
$460$		0			0
$461$	2.13612e6			0.468136			0.234068	−	0.972220i	$-0.424796\pi$
$461$	2.13612e6			0.468136			0.234068	+	0.972220i	$0.424796\pi$
$462$		0			0
$463$			6.57862e6i			1.42621i	0.701059	+	0.713104i	$0.252711\pi$
$463$			6.57862e6i			1.42621i	−0.701059	+	0.713104i	$0.747289\pi$
$464$		0			0
$465$	−5.20290e6	+	2.76469e6i	−1.11587	+	0.592946i
$466$		0			0
$467$		−	2.49778e6i		−	0.529983i	−0.964251	−	0.264991i	$-0.914631\pi$
$467$		−	2.49778e6i		−	0.529983i	0.964251	−	0.264991i	$-0.0853691\pi$
$468$		0			0
$469$	−2.07296e6			−0.435171
$470$		0			0
$471$	1.58983e6			0.330216
$472$		0			0
$473$			5513.73i			0.00113316i
$474$		0			0
$475$	−3.34138e6	+	4.94824e6i	−0.679503	+	1.00627i
$476$		0			0
$477$			3.32762e6i			0.669634i
$478$		0			0
$479$	−3.35165e6			−0.667452			−0.333726	−	0.942670i	$-0.608306\pi$
$479$	−3.35165e6			−0.667452			−0.333726	+	0.942670i	$0.608306\pi$
$480$		0			0
$481$	−6.40525e6			−1.26233
$482$		0			0
$483$		−	4.87705e6i		−	0.951239i
$484$		0			0
$485$	2.96519e6	−	1.57563e6i	0.572399	−	0.304159i
$486$		0			0
$487$		−	3.15469e6i		−	0.602746i	−0.953506	−	0.301373i	$-0.902555\pi$
$487$		−	3.15469e6i		−	0.602746i	0.953506	−	0.301373i	$-0.0974449\pi$
$488$		0			0
$489$	−1.62868e7			−3.08009
$490$		0			0
$491$	2.91678e6			0.546009			0.273004	−	0.962013i	$-0.411983\pi$
$491$	2.91678e6			0.546009			0.273004	+	0.962013i	$0.411983\pi$
$492$		0			0
$493$			6.01081e6i			1.11382i
$494$		0			0
$495$	8.15140e6	+	1.53402e7i	1.49527	+	2.81396i
$496$		0			0
$497$		−	1.03323e6i		−	0.187631i
$498$		0			0
$499$	−4.95680e6			−0.891148			−0.445574	−	0.895245i	$-0.647000\pi$
$499$	−4.95680e6			−0.891148			−0.445574	+	0.895245i	$0.647000\pi$
$500$		0			0
$501$	1.14576e7			2.03938
$502$		0			0
$503$		−	4.24710e6i		−	0.748467i	−0.927334	−	0.374234i	$-0.877906\pi$
$503$		−	4.24710e6i		−	0.748467i	0.927334	−	0.374234i	$-0.122094\pi$
$504$		0			0
$505$	803073.	+	1.51131e6i	0.140128	+	0.263709i
$506$		0			0
$507$		−	1.69002e7i		−	2.91993i
$508$		0			0
$509$	−7.44277e6			−1.27333			−0.636664	−	0.771142i	$-0.719686\pi$
$509$	−7.44277e6			−1.27333			−0.636664	+	0.771142i	$0.719686\pi$
$510$		0			0
$511$	263732.			0.0446798
$512$		0			0
$513$		−	9.65261e6i		−	1.61939i
$514$		0			0
$515$	8.55047e6	−	4.54351e6i	1.42060	−	0.754873i
$516$		0			0
$517$			5.31545e6i			0.874609i
$518$		0			0
$519$	1.71707e7			2.79814
$520$		0			0
$521$	2.10665e6			0.340015			0.170007	−	0.985443i	$-0.445621\pi$
$521$	2.10665e6			0.340015			0.170007	+	0.985443i	$0.445621\pi$
$522$		0			0
$523$		−	6.80764e6i		−	1.08828i	−0.838993	−	0.544142i	$-0.816855\pi$
$523$		−	6.80764e6i		−	1.08828i	0.838993	−	0.544142i	$-0.183145\pi$
$524$		0			0
$525$	2.23421e6	−	3.30863e6i	0.353773	−	0.523902i
$526$		0			0
$527$		−	6.89710e6i		−	1.08178i
$528$		0			0
$529$	−8.13707e6			−1.26424
$530$		0			0
$531$	−1.34335e7			−2.06753
$532$		0			0
$533$		−	3.40108e6i		−	0.518560i
$534$		0			0
$535$	4.68649e6	−	2.49029e6i	0.707885	−	0.376153i
$536$		0			0
$537$		−	1.07267e6i		−	0.160520i
$538$		0			0
$539$	−1.70825e6			−0.253267
$540$		0			0
$541$	−6.11948e6			−0.898921			−0.449460	−	0.893300i	$-0.648384\pi$
$541$	−6.11948e6			−0.898921			−0.449460	+	0.893300i	$0.648384\pi$
$542$		0			0
$543$			8.09420e6i			1.17808i
$544$		0			0
$545$	−2.82229e6	−	5.31129e6i	−0.407015	−	0.765965i
$546$		0			0
$547$		−	6.94424e6i		−	0.992331i	−0.868228	−	0.496165i	$-0.834741\pi$
$547$		−	6.94424e6i		−	0.992331i	0.868228	−	0.496165i	$-0.165259\pi$
$548$		0			0
$549$	1.49534e7			2.11743
$550$		0			0
$551$	6.73116e6			0.944520
$552$		0			0
$553$		−	268322.i		−	0.0373116i
$554$		0			0
$555$	−4.33856e6	−	8.16477e6i	−0.597879	−	1.12515i
$556$		0			0
$557$			1.02383e7i			1.39826i	0.714994	+	0.699131i	$0.246430\pi$
$557$			1.02383e7i			1.39826i	−0.714994	+	0.699131i	$0.753570\pi$
$558$		0			0
$559$	7824.93			0.00105913
$560$		0			0
$561$	−3.16491e7			−4.24574
$562$		0			0
$563$		−	1.29239e7i		−	1.71840i	−0.511640	−	0.859200i	$-0.670962\pi$
$563$		−	1.29239e7i		−	1.71840i	0.511640	−	0.859200i	$-0.329038\pi$
$564$		0			0
$565$	1.21057e7	−	6.43270e6i	1.59540	−	0.847758i
$566$		0			0
$567$			1.25359e6i			0.163757i
$568$		0			0
$569$	1.13337e7			1.46754			0.733772	−	0.679396i	$-0.237758\pi$
$569$	1.13337e7			1.46754			0.733772	+	0.679396i	$0.237758\pi$
$570$		0			0
$571$	9.81243e6			1.25946			0.629732	−	0.776812i	$-0.283165\pi$
$571$	9.81243e6			1.25946			0.629732	+	0.776812i	$0.283165\pi$
$572$		0			0
$573$		−	2.00243e6i		−	0.254783i
$574$		0			0
$575$	−9.88672e6	−	6.67617e6i	−1.24705	−	0.842088i
$576$		0			0
$577$			4.28759e6i			0.536134i	0.963400	+	0.268067i	$0.0863848\pi$
$577$			4.28759e6i			0.536134i	−0.963400	+	0.268067i	$0.913615\pi$
$578$		0			0
$579$	−1.80880e7			−2.24230
$580$		0			0
$581$	−1.53022e6			−0.188068
$582$		0			0
$583$		−	5.42050e6i		−	0.660492i
$584$		0			0
$585$	2.17703e7	−	1.15682e7i	2.63012	−	1.39758i
$586$		0			0
$587$		−	3.41305e6i		−	0.408834i	−0.978884	−	0.204417i	$-0.934470\pi$
$587$		−	3.41305e6i		−	0.408834i	0.978884	−	0.204417i	$-0.0655299\pi$
$588$		0			0
$589$	−7.72366e6			−0.917350
$590$		0			0
$591$	−2.03690e7			−2.39884
$592$		0			0
$593$			7.65845e6i			0.894343i	0.894448	+	0.447171i	$0.147569\pi$
$593$			7.65845e6i			0.894343i	−0.894448	+	0.447171i	$0.852431\pi$
$594$		0			0
$595$	2.19300e6	+	4.12703e6i	0.253949	+	0.477909i
$596$		0			0
$597$			1.19451e7i			1.37168i
$598$		0			0
$599$	−1.04255e7			−1.18722			−0.593610	−	0.804753i	$-0.702298\pi$
$599$	−1.04255e7			−1.18722			−0.593610	+	0.804753i	$0.702298\pi$
$600$		0			0
$601$	−1.42779e7			−1.61242			−0.806210	−	0.591629i	$-0.798485\pi$
$601$	−1.42779e7			−1.61242			−0.806210	+	0.591629i	$0.798485\pi$
$602$		0			0
$603$		−	1.84777e7i		−	2.06945i
$604$		0			0
$605$	−9.05357e6	−	1.70380e7i	−1.00561	−	1.89247i
$606$		0			0
$607$		−	2.07562e6i		−	0.228653i	−0.993443	−	0.114326i	$-0.963529\pi$
$607$		−	2.07562e6i		−	0.228653i	0.993443	−	0.114326i	$-0.0364709\pi$
$608$		0			0
$609$	−4.50078e6			−0.491751
$610$		0			0
$611$	7.54354e6			0.817470
$612$		0			0
$613$			9.82093e6i			1.05561i	0.849367	+	0.527803i	$0.176984\pi$
$613$			9.82093e6i			1.05561i	−0.849367	+	0.527803i	$0.823016\pi$
$614$		0			0
$615$	4.33536e6	−	2.30370e6i	0.462208	−	0.245606i
$616$		0			0
$617$		−	344649.i		−	0.0364472i	−0.999834	−	0.0182236i	$-0.994199\pi$
$617$		−	344649.i		−	0.0364472i	0.999834	−	0.0182236i	$-0.00580108\pi$
$618$		0			0
$619$	−1.47544e6			−0.154773			−0.0773865	−	0.997001i	$-0.524658\pi$
$619$	−1.47544e6			−0.154773			−0.0773865	+	0.997001i	$0.524658\pi$
$620$		0			0
$621$	1.92862e7			2.00686
$622$		0			0
$623$			1.24095e6i			0.128096i
$624$		0			0
$625$	−3.64884e6	−	9.05833e6i	−0.373641	−	0.927573i
$626$		0			0
$627$			3.54419e7i			3.60038i
$628$		0			0
$629$	1.08234e7			1.09078
$630$		0			0
$631$	1.00826e7			1.00809			0.504046	−	0.863677i	$-0.331844\pi$
$631$	1.00826e7			1.00809			0.504046	+	0.863677i	$0.331844\pi$
$632$		0			0
$633$			6.02570e6i			0.597721i
$634$		0			0
$635$	1.29291e7	−	6.87023e6i	1.27243	−	0.676140i
$636$		0			0
$637$			2.42429e6i			0.236721i
$638$		0			0
$639$	9.20982e6			0.892275
$640$		0			0
$641$	1.01775e7			0.978356			0.489178	−	0.872184i	$-0.337297\pi$
$641$	1.01775e7			0.978356			0.489178	+	0.872184i	$0.337297\pi$
$642$		0			0
$643$		−	1.03995e7i		−	0.991940i	−0.868340	−	0.495970i	$-0.834813\pi$
$643$		−	1.03995e7i		−	0.991940i	0.868340	−	0.495970i	$-0.165187\pi$
$644$		0			0
$645$	5300.17	+	9974.44i	0.000501639	+	0.000944037i
$646$		0			0
$647$			1.10090e7i			1.03392i	0.856011	+	0.516958i	$0.172936\pi$
$647$			1.10090e7i			1.03392i	−0.856011	+	0.516958i	$0.827064\pi$
$648$		0			0
$649$	2.18824e7			2.03931
$650$		0			0
$651$	5.16442e6			0.477605
$652$		0			0
$653$		−	1.09954e7i		−	1.00908i	−0.863388	−	0.504541i	$-0.831662\pi$
$653$		−	1.09954e7i		−	1.00908i	0.863388	−	0.504541i	$-0.168338\pi$
$654$		0			0
$655$	5.98176e6	+	1.12571e7i	0.544785	+	1.02524i
$656$		0			0
$657$			2.35082e6i			0.212474i
$658$		0			0
$659$	−7.09771e6			−0.636656			−0.318328	−	0.947981i	$-0.603121\pi$
$659$	−7.09771e6			−0.636656			−0.318328	+	0.947981i	$0.603121\pi$
$660$		0			0
$661$	−528198.			−0.0470211			−0.0235106	−	0.999724i	$-0.507484\pi$
$661$	−528198.			−0.0470211			−0.0235106	+	0.999724i	$0.507484\pi$
$662$		0			0
$663$			4.49154e7i			3.96836i
$664$		0			0
$665$	4.62163e6	−	2.45582e6i	0.405266	−	0.215349i
$666$		0			0
$667$			1.34491e7i			1.17052i
$668$		0			0
$669$	2.52775e7			2.18358
$670$		0			0
$671$	−2.43582e7			−2.08852
$672$		0			0
$673$		−	1.62860e6i		−	0.138604i	−0.997596	−	0.0693020i	$-0.977923\pi$
$673$		−	1.62860e6i		−	0.138604i	0.997596	−	0.0693020i	$-0.0220772\pi$
$674$		0			0
$675$	1.30839e7	+	8.83512e6i	1.10529	+	0.746368i
$676$		0			0
$677$		−	1.61732e7i		−	1.35620i	−0.734969	−	0.678100i	$-0.762804\pi$
$677$		−	1.61732e7i		−	1.35620i	0.734969	−	0.678100i	$-0.237196\pi$
$678$		0			0
$679$	−2.94326e6			−0.244993
$680$		0			0
$681$	−3.93454e6			−0.325107
$682$		0			0
$683$		−	1.78913e7i		−	1.46754i	−0.679397	−	0.733771i	$-0.737759\pi$
$683$		−	1.78913e7i		−	1.46754i	0.679397	−	0.733771i	$-0.262241\pi$
$684$		0			0
$685$	−1.35468e7	+	7.19845e6i	−1.10309	+	0.586155i
$686$		0			0
$687$		−	3.27484e6i		−	0.264727i
$688$		0			0
$689$	−7.69261e6			−0.617342
$690$		0			0
$691$	−996369.			−0.0793826			−0.0396913	−	0.999212i	$-0.512637\pi$
$691$	−996369.			−0.0793826			−0.0396913	+	0.999212i	$0.512637\pi$
$692$		0			0
$693$		−	1.52267e7i		−	1.20441i
$694$		0			0
$695$	−6.18630e6	−	1.16420e7i	−0.485813	−	0.914254i
$696$		0			0
$697$			5.74706e6i			0.448089i
$698$		0			0
$699$	−5.58476e6			−0.432326
$700$		0			0
$701$	1.11649e7			0.858141			0.429071	−	0.903271i	$-0.358841\pi$
$701$	1.11649e7			0.858141			0.429071	+	0.903271i	$0.358841\pi$
$702$		0			0
$703$		−	1.21205e7i		−	0.924983i
$704$		0			0
$705$	5.10957e6	+	9.61574e6i	0.387179	+	0.728635i
$706$		0			0
$707$		−	1.50013e6i		−	0.112871i
$708$		0			0
$709$	1.60104e7			1.19615			0.598076	−	0.801439i	$-0.295932\pi$
$709$	1.60104e7			1.19615			0.598076	+	0.801439i	$0.295932\pi$
$710$		0			0
$711$	2.39173e6			0.177435
$712$		0			0
$713$		−	1.54321e7i		−	1.13685i
$714$		0			0
$715$	−3.54626e7	+	1.88440e7i	−2.59421	+	1.37850i
$716$		0			0
$717$			2.24820e7i			1.63319i
$718$		0			0
$719$	−1.84103e6			−0.132812			−0.0664062	−	0.997793i	$-0.521153\pi$
$719$	−1.84103e6			−0.132812			−0.0664062	+	0.997793i	$0.521153\pi$
$720$		0			0
$721$	−8.48723e6			−0.608034
$722$		0			0
$723$			3.54893e7i			2.52494i
$724$		0			0
$725$	−6.16110e6	+	9.12395e6i	−0.435324	+	0.644671i
$726$		0			0
$727$			2.74414e6i			0.192562i	0.995354	+	0.0962810i	$0.0306947\pi$
$727$			2.74414e6i			0.192562i	−0.995354	+	0.0962810i	$0.969305\pi$
$728$		0			0
$729$	2.08335e7			1.45192
$730$		0			0
$731$	−13222.4			−0.000915200
$732$		0			0
$733$			8.91669e6i			0.612976i	0.951874	+	0.306488i	$0.0991540\pi$
$733$			8.91669e6i			0.612976i	−0.951874	+	0.306488i	$0.900846\pi$
$734$		0			0
$735$	−3.09025e6	+	1.64208e6i	−0.210996	+	0.112118i
$736$		0			0
$737$			3.00991e7i			2.04120i
$738$		0			0
$739$	8.73537e6			0.588397			0.294199	−	0.955744i	$-0.404947\pi$
$739$	8.73537e6			0.588397			0.294199	+	0.955744i	$0.404947\pi$
$740$		0			0
$741$	5.02982e7			3.36517
$742$		0			0
$743$		−	5.92592e6i		−	0.393808i	−0.980423	−	0.196904i	$-0.936911\pi$
$743$		−	5.92592e6i		−	0.393808i	0.980423	−	0.196904i	$-0.0630887\pi$
$744$		0			0
$745$	−7.97746e6	−	1.50128e7i	−0.526592	−	0.990997i
$746$		0			0
$747$		−	1.36399e7i		−	0.894352i
$748$		0			0
$749$	−4.65183e6			−0.302983
$750$		0			0
$751$	−1.32881e7			−0.859732			−0.429866	−	0.902893i	$-0.641439\pi$
$751$	−1.32881e7			−0.859732			−0.429866	+	0.902893i	$0.641439\pi$
$752$		0			0
$753$		−	2.83985e7i		−	1.82519i
$754$		0			0
$755$	8.05410e6	+	1.51571e7i	0.514221	+	0.967716i
$756$		0			0
$757$		−	416648.i		−	0.0264259i	−0.999913	−	0.0132129i	$-0.995794\pi$
$757$		−	416648.i		−	0.0264259i	0.999913	−	0.0132129i	$-0.00420593\pi$
$758$		0			0
$759$	−7.08141e7			−4.46185
$760$		0			0
$761$	−5.22709e6			−0.327189			−0.163594	−	0.986528i	$-0.552309\pi$
$761$	−5.22709e6			−0.327189			−0.163594	+	0.986528i	$0.552309\pi$
$762$		0			0
$763$			5.27201e6i			0.327842i
$764$		0			0
$765$	−3.67870e7	+	1.95477e7i	−2.27269	+	1.20765i
$766$		0			0
$767$		−	3.10548e7i		−	1.90608i
$768$		0			0
$769$	1.37199e7			0.836633			0.418317	−	0.908301i	$-0.362620\pi$
$769$	1.37199e7			0.836633			0.418317	+	0.908301i	$0.362620\pi$
$770$		0			0
$771$	6.27140e6			0.379952
$772$		0			0
$773$			1.47297e7i			0.886633i	0.896365	+	0.443316i	$0.146198\pi$
$773$			1.47297e7i			0.886633i	−0.896365	+	0.443316i	$0.853802\pi$
$774$		0			0
$775$	7.06954e6	−	1.04693e7i	0.422802	−	0.626127i
$776$		0			0
$777$			8.10438e6i			0.481579i
$778$		0			0
$779$	6.43580e6			0.379979
$780$		0			0
$781$	−1.50023e7			−0.880095
$782$		0			0
$783$		−	1.77982e7i		−	1.03746i
$784$		0			0
$785$	−3.01016e6	+	1.59953e6i	−0.174347	+	0.0926440i
$786$		0			0
$787$		−	7.49672e6i		−	0.431454i	−0.976454	−	0.215727i	$-0.930788\pi$
$787$		−	7.49672e6i		−	0.431454i	0.976454	−	0.215727i	$-0.0692121\pi$
$788$		0			0
$789$	−6.99781e6			−0.400193
$790$		0			0
$791$	−1.20162e7			−0.682852
$792$		0			0
$793$			3.45685e7i			1.95208i
$794$		0			0
$795$	−5.21055e6	−	9.80578e6i	−0.292392	−	0.550255i
$796$		0			0
$797$		−	1.77830e7i		−	0.991655i	−0.868421	−	0.495827i	$-0.834865\pi$
$797$		−	1.77830e7i		−	0.991655i	0.868421	−	0.495827i	$-0.165135\pi$
$798$		0			0
$799$	−1.27469e7			−0.706378
$800$		0			0
$801$	−1.10614e7			−0.609157
$802$		0			0
$803$		−	3.82936e6i		−	0.209574i
$804$		0			0
$805$	4.90680e6	+	9.23415e6i	0.266876	+	0.502235i
$806$		0			0
$807$		−	1.87563e7i		−	1.01382i
$808$		0			0
$809$	−2.63174e7			−1.41375			−0.706874	−	0.707340i	$-0.749895\pi$
$809$	−2.63174e7			−1.41375			−0.706874	+	0.707340i	$0.749895\pi$
$810$		0			0
$811$	6.87139e6			0.366853			0.183427	−	0.983033i	$-0.441281\pi$
$811$	6.87139e6			0.366853			0.183427	+	0.983033i	$0.441281\pi$
$812$		0			0
$813$			4.11748e7i			2.18477i
$814$		0			0
$815$	3.08372e7	−	1.63861e7i	1.62623	−	0.864137i
$816$		0			0
$817$			14807.0i			0.000776089i
$818$		0			0
$819$	−2.16093e7			−1.12572
$820$		0			0
$821$	−3.18061e6			−0.164684			−0.0823422	−	0.996604i	$-0.526240\pi$
$821$	−3.18061e6			−0.164684			−0.0823422	+	0.996604i	$0.526240\pi$
$822$		0			0
$823$			2.62296e7i			1.34987i	0.737876	+	0.674936i	$0.235829\pi$
$823$			2.62296e7i			1.34987i	−0.737876	+	0.674936i	$0.764171\pi$
$824$		0			0
$825$	−4.80408e7	−	3.24403e7i	−2.45740	−	1.65940i
$826$		0			0
$827$		−	3.05521e7i		−	1.55338i	−0.629883	−	0.776690i	$-0.716897\pi$
$827$		−	3.05521e7i		−	1.55338i	0.629883	−	0.776690i	$-0.283103\pi$
$828$		0			0
$829$	−1.90564e7			−0.963065			−0.481533	−	0.876428i	$-0.659920\pi$
$829$	−1.90564e7			−0.963065			−0.481533	+	0.876428i	$0.659920\pi$
$830$		0			0
$831$	−2.58046e7			−1.29627
$832$		0			0
$833$		−	4.09651e6i		−	0.204551i
$834$		0			0
$835$	−2.16936e7	+	1.15274e7i	−1.07675	+	0.572159i
$836$		0			0
$837$			2.04226e7i			1.00762i
$838$		0			0
$839$	1.27372e7			0.624698			0.312349	−	0.949967i	$-0.398884\pi$
$839$	1.27372e7			0.624698			0.312349	+	0.949967i	$0.398884\pi$
$840$		0			0
$841$	−8.09969e6			−0.394892
$842$		0			0
$843$			3.02915e7i			1.46809i
$844$		0			0
$845$	1.70033e7	+	3.19986e7i	0.819202	+	1.54166i
$846$		0			0
$847$			1.69120e7i			0.810001i
$848$		0			0
$849$	3.72324e7			1.77277
$850$		0			0
$851$	2.42172e7			1.14630
$852$		0			0
$853$		−	3.04694e7i		−	1.43381i	−0.697171	−	0.716905i	$-0.745558\pi$
$853$		−	3.04694e7i		−	1.43381i	0.697171	−	0.716905i	$-0.254442\pi$
$854$		0			0
$855$	2.18903e7	+	4.11956e7i	1.02409	+	1.92724i
$856$		0			0
$857$		−	3.79346e7i		−	1.76434i	−0.470929	−	0.882171i	$-0.656081\pi$
$857$		−	3.79346e7i		−	1.76434i	0.470929	−	0.882171i	$-0.343919\pi$
$858$		0			0
$859$	−3.23258e7			−1.49474			−0.747372	−	0.664406i	$-0.768685\pi$
$859$	−3.23258e7			−1.49474			−0.747372	+	0.664406i	$0.768685\pi$
$860$		0			0
$861$	−4.30329e6			−0.197830
$862$		0			0
$863$			1.81422e7i			0.829208i	0.910002	+	0.414604i	$0.136080\pi$
$863$			1.81422e7i			0.829208i	−0.910002	+	0.414604i	$0.863920\pi$
$864$		0			0
$865$	−3.25108e7	+	1.72754e7i	−1.47736	+	0.785035i
$866$		0			0
$867$		−	3.88779e7i		−	1.75653i
$868$		0			0
$869$	−3.89600e6			−0.175012
$870$		0			0
$871$	4.27158e7			1.90785
$872$		0			0
$873$		−	2.62352e7i		−	1.16506i
$874$		0			0
$875$	−901403.	+	8.51235e6i	−0.0398014	+	0.375863i
$876$		0			0
$877$		−	4.74849e6i		−	0.208476i	−0.994552	−	0.104238i	$-0.966760\pi$
$877$		−	4.74849e6i		−	0.208476i	0.994552	−	0.104238i	$-0.0332404\pi$
$878$		0			0
$879$	4.88292e7			2.13161
$880$		0			0
$881$	−9.55376e6			−0.414700			−0.207350	−	0.978267i	$-0.566484\pi$
$881$	−9.55376e6			−0.414700			−0.207350	+	0.978267i	$0.566484\pi$
$882$		0			0
$883$		−	1.96965e7i		−	0.850136i	−0.905162	−	0.425068i	$-0.860250\pi$
$883$		−	1.96965e7i		−	0.850136i	0.905162	−	0.425068i	$-0.139750\pi$
$884$		0			0
$885$	3.95856e7	−	2.10348e7i	1.69894	−	0.902777i
$886$		0			0
$887$		−	1.86968e7i		−	0.797919i	−0.916968	−	0.398960i	$-0.869371\pi$
$887$		−	1.86968e7i		−	0.797919i	0.916968	−	0.398960i	$-0.130629\pi$
$888$		0			0
$889$	−1.28335e7			−0.544617
$890$		0			0
$891$	1.82020e7			0.768111
$892$		0			0
$893$			1.42745e7i			0.599007i
$894$		0			0
$895$	1.07921e6	+	2.03097e6i	0.0450348	+	0.0847513i
$896$		0			0
$897$			1.00497e8i			4.17036i
$898$		0			0
$899$	−1.42415e7			−0.587702
$900$		0			0
$901$	1.29988e7			0.533447
$902$		0			0
$903$		−	9900.67i		−	0.000404059i
$904$		0			0
$905$	−8.14357e6	−	1.53255e7i	−0.330517	−	0.622002i
$906$		0			0
$907$			2.30074e7i			0.928643i	0.885667	+	0.464322i	$0.153702\pi$
$907$			2.30074e7i			0.928643i	−0.885667	+	0.464322i	$0.846298\pi$
$908$		0			0
$909$	1.33717e7			0.536754
$910$		0			0
$911$	1.72597e7			0.689027			0.344514	−	0.938781i	$-0.388044\pi$
$911$	1.72597e7			0.689027			0.344514	+	0.938781i	$0.388044\pi$
$912$		0			0
$913$			2.22186e7i			0.882143i
$914$		0			0
$915$	−4.40644e7	+	2.34148e7i	−1.73994	+	0.924564i
$916$		0			0
$917$		−	1.11739e7i		−	0.438813i
$918$		0			0
$919$	3.36268e7			1.31340			0.656699	−	0.754153i	$-0.271952\pi$
$919$	3.36268e7			1.31340			0.656699	+	0.754153i	$0.271952\pi$
$920$		0			0
$921$	4.66357e7			1.81163
$922$		0			0
$923$			2.12908e7i			0.822597i
$924$		0			0
$925$	1.64291e7	+	1.10940e7i	0.631336	+	0.426320i
$926$		0			0
$927$		−	7.56523e7i		−	2.89150i
$928$		0			0
$929$	9.20062e6			0.349766			0.174883	−	0.984589i	$-0.444045\pi$
$929$	9.20062e6			0.349766			0.174883	+	0.984589i	$0.444045\pi$
$930$		0			0
$931$	−4.58745e6			−0.173459
$932$		0			0
$933$		−	1.33051e7i		−	0.500396i
$934$		0			0
$935$	5.99239e7	−	3.18421e7i	2.24167	−	1.19117i
$936$		0			0
$937$		−	682322.i		−	0.0253887i	−0.999919	−	0.0126943i	$-0.995959\pi$
$937$		−	682322.i		−	0.0253887i	0.999919	−	0.0126943i	$-0.00404084\pi$
$938$		0			0
$939$	−3.19630e7			−1.18300
$940$		0			0
$941$	1.35362e7			0.498336			0.249168	−	0.968460i	$-0.419843\pi$
$941$	1.35362e7			0.498336			0.249168	+	0.968460i	$0.419843\pi$
$942$		0			0
$943$			1.28589e7i			0.470897i
$944$		0			0
$945$	−6.49357e6	−	1.22203e7i	−0.236540	−	0.445146i
$946$		0			0
$947$		−	1.25049e7i		−	0.453113i	−0.973998	−	0.226557i	$-0.927253\pi$
$947$		−	1.25049e7i		−	0.453113i	0.973998	−	0.226557i	$-0.0727468\pi$
$948$		0			0
$949$	−5.43451e6			−0.195882
$950$		0			0
$951$	−3.32935e7			−1.19374
$952$		0			0
$953$		−	3.75580e6i		−	0.133958i	−0.997754	−	0.0669792i	$-0.978664\pi$
$953$		−	3.75580e6i		−	0.133958i	0.997754	−	0.0669792i	$-0.0213361\pi$
$954$		0			0
$955$	2.01465e6	+	3.79138e6i	0.0714809	+	0.134520i
$956$		0			0
$957$			6.53507e7i			2.30659i
$958$		0			0
$959$	1.34466e7			0.472136
$960$		0			0
$961$	−1.22877e7			−0.429204
$962$		0			0
$963$		−	4.14648e7i		−	1.44083i
$964$		0			0
$965$	3.42475e7	−	1.81983e7i	1.18389	−	0.629089i
$966$		0			0
$967$		−	4.48649e7i		−	1.54291i	−0.636283	−	0.771455i	$-0.719529\pi$
$967$		−	4.48649e7i		−	1.54291i	0.636283	−	0.771455i	$-0.280471\pi$
$968$		0			0
$969$	−8.49926e7			−2.90785
$970$		0			0
$971$	5.66452e7			1.92804			0.964018	−	0.265835i	$-0.0856478\pi$
$971$	5.66452e7			1.92804			0.964018	+	0.265835i	$0.0856478\pi$
$972$		0			0
$973$			1.15559e7i			0.391312i
$974$		0			0
$975$	−4.60384e7	+	6.81782e7i	−1.55099	+	2.29686i
$976$		0			0
$977$			1.26701e7i			0.424661i	0.977198	+	0.212331i	$0.0681054\pi$
$977$			1.26701e7i			0.424661i	−0.977198	+	0.212331i	$0.931895\pi$
$978$		0			0
$979$	1.80184e7			0.600842
$980$		0			0
$981$	−4.69929e7			−1.55905
$982$		0			0
$983$			4.15709e7i			1.37216i	0.727525	+	0.686081i	$0.240671\pi$
$983$			4.15709e7i			1.37216i	−0.727525	+	0.686081i	$0.759329\pi$
$984$		0			0
$985$	3.85665e7	−	2.04933e7i	1.26654	−	0.673009i
$986$		0			0
$987$		−	9.54463e6i		−	0.311865i
$988$		0			0
$989$	−29584.8			−0.000961784
$990$		0			0
$991$	−1.35346e7			−0.437784			−0.218892	−	0.975749i	$-0.570244\pi$
$991$	−1.35346e7			−0.437784			−0.218892	+	0.975749i	$0.570244\pi$
$992$		0			0
$993$		−	5.37333e7i		−	1.72930i
$994$		0			0
$995$	−1.20180e7	−	2.26167e7i	−0.384833	−	0.724221i
$996$		0			0
$997$		−	2.19806e7i		−	0.700329i	−0.936688	−	0.350165i	$-0.886126\pi$
$997$		−	2.19806e7i		−	0.700329i	0.936688	−	0.350165i	$-0.113874\pi$
$998$		0			0
$999$	−3.20486e7			−1.01600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.g.a.169.20	yes	20
4.3	odd	2		560.6.g.g.449.1		20
5.4	even	2	inner	280.6.g.a.169.1	✓	20
20.19	odd	2		560.6.g.g.449.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.g.a.169.1	✓	20	5.4	even	2	inner
280.6.g.a.169.20	yes	20	1.1	even	1	trivial
560.6.g.g.449.1		20	4.3	odd	2
560.6.g.g.449.20		20	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+26.0724i q^{3} +(-26.2314 - 49.3651i) q^{5} +49.0000i q^{7} -436.769 q^{9} +O(q^{10})\)	\(q+26.0724i q^{3} +(-26.2314 - 49.3651i) q^{5} +49.0000i q^{7} -436.769 q^{9} +711.472 q^{11} -1009.70i q^{13} +(1287.07 - 683.916i) q^{15} +1706.17i q^{17} +1910.64 q^{19} -1277.55 q^{21} +3817.51i q^{23} +(-1748.83 + 2589.83i) q^{25} -5052.03i q^{27} +3522.99 q^{29} -4042.45 q^{31} +18549.8i q^{33} +(2418.89 - 1285.34i) q^{35} -6343.71i q^{37} +26325.3 q^{39} +3368.40 q^{41} +7.74975i q^{43} +(11457.1 + 21561.2i) q^{45} +7471.06i q^{47} -2401.00 q^{49} -44483.9 q^{51} -7618.70i q^{53} +(-18662.9 - 35121.9i) q^{55} +49814.9i q^{57} +30756.5 q^{59} -34236.3 q^{61} -21401.7i q^{63} +(-49844.0 + 26485.9i) q^{65} +42305.4i q^{67} -99531.7 q^{69} -21086.2 q^{71} -5382.30i q^{73} +(-67523.1 - 45596.1i) q^{75} +34862.2i q^{77} -5475.96 q^{79} +25583.5 q^{81} +31229.0i q^{83} +(84225.2 - 44755.2i) q^{85} +91852.7i q^{87} +25325.5 q^{89} +49475.3 q^{91} -105396. i q^{93} +(-50118.8 - 94318.9i) q^{95} +60066.6i q^{97} -310749. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{5} - 1134 q^{9}+O(q^{10}) \) 20 * q + 14 * q^5 - 1134 * q^9	\( 20 q + 14 q^{5} - 1134 q^{9} + 822 q^{11} + 1322 q^{15} + 3000 q^{19} - 882 q^{21} - 1944 q^{25} + 10406 q^{29} - 8532 q^{31} - 2058 q^{35} + 71066 q^{39} - 28880 q^{41} - 3922 q^{45} - 48020 q^{49} + 34770 q^{51} - 8844 q^{55} + 61196 q^{59} - 145052 q^{61} - 12722 q^{65} + 64980 q^{69} + 41464 q^{71} - 159032 q^{75} + 97174 q^{79} - 165852 q^{81} + 319566 q^{85} - 84604 q^{89} + 61642 q^{91} + 33416 q^{95} - 630244 q^{99}+O(q^{100}) \) 20 * q + 14 * q^5 - 1134 * q^9 + 822 * q^11 + 1322 * q^15 + 3000 * q^19 - 882 * q^21 - 1944 * q^25 + 10406 * q^29 - 8532 * q^31 - 2058 * q^35 + 71066 * q^39 - 28880 * q^41 - 3922 * q^45 - 48020 * q^49 + 34770 * q^51 - 8844 * q^55 + 61196 * q^59 - 145052 * q^61 - 12722 * q^65 + 64980 * q^69 + 41464 * q^71 - 159032 * q^75 + 97174 * q^79 - 165852 * q^81 + 319566 * q^85 - 84604 * q^89 + 61642 * q^91 + 33416 * q^95 - 630244 * q^99

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