LMFDB - Embedded newform 224.3.v.a.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(13,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 7, 4]))

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

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

chi := DirichletCharacter("224.13");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.v (of order $8$, degree $4$, 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:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{8}]$

Embedding invariants

Embedding label			181.2
Root			$1.28897 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	224.181
Dual form			224.3.v.a.125.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.99607 + 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(-4.94975 - 4.94975i) q^{7} +(7.85905 + 1.49509i) q^{8} +(6.36396 - 6.36396i) q^{9} +O(q^{10})$	\(q+(1.99607 + 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(-4.94975 - 4.94975i) q^{7} +(7.85905 + 1.49509i) q^{8} +(6.36396 - 6.36396i) q^{9} +(20.1876 + 8.36199i) q^{11} +(-9.26013 - 10.5000i) q^{14} +(15.5000 + 3.96863i) q^{16} +(13.5000 - 11.9059i) q^{18} +(39.2487 + 19.2196i) q^{22} +(-30.1660 + 30.1660i) q^{23} +(-17.6777 - 17.6777i) q^{25} +(-17.1688 - 22.1186i) q^{28} +(-37.1582 + 15.3914i) q^{29} +(30.4421 + 9.86299i) q^{32} +(28.4382 - 22.0742i) q^{36} +(22.7341 - 54.8850i) q^{37} +(-27.0562 - 11.2070i) q^{43} +(75.9362 + 43.2795i) q^{44} +(-63.9918 + 56.4354i) q^{46} +49.0000i q^{49} +(-33.0719 - 37.5000i) q^{50} +(-32.2953 - 13.3771i) q^{53} +(-31.5000 - 46.3006i) q^{56} +(-76.0983 + 26.0685i) q^{58} -63.0000 q^{63} +(59.5294 + 23.5000i) q^{64} +(-123.169 + 51.0184i) q^{67} +(59.8665 + 59.8665i) q^{71} +(59.5294 - 40.5000i) q^{72} +(52.2531 - 106.707i) q^{74} +(-58.5340 - 141.313i) q^{77} +156.268i q^{79} -81.0000i q^{81} +(-52.6025 - 25.7587i) q^{86} +(146.154 + 95.8997i) q^{88} +(-134.801 + 104.635i) q^{92} +(-6.13705 + 97.8077i) q^{98} +(181.689 - 75.2579i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 124 q^{16} + 108 q^{18} + 148 q^{22} - 72 q^{23} - 232 q^{43} + 324 q^{44} + 24 q^{53} - 252 q^{56} - 504 q^{63} - 472 q^{67} - 108 q^{74} - 168 q^{77} - 708 q^{92}+O(q^{100}) $ 8 * q + 124 * q^16 + 108 * q^18 + 148 * q^22 - 72 * q^23 - 232 * q^43 + 324 * q^44 + 24 * q^53 - 252 * q^56 - 504 * q^63 - 472 * q^67 - 108 * q^74 - 168 * q^77 - 708 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{8}\right)$

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 $

$2$	1.99607	+	0.125246i	0.998037	+	0.0626229i
$2$	1.99607	+	0.125246i	0.998037	+	0.0626229i
$3$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$3$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$4$	3.96863	+	0.500000i	0.992157	+	0.125000i
$5$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$5$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$6$		0			0
$7$	−4.94975	−	4.94975i	−0.707107	−	0.707107i
$7$	−4.94975	−	4.94975i	−0.707107	−	0.707107i
$8$	7.85905	+	1.49509i	0.982382	+	0.186886i
$9$	6.36396	−	6.36396i	0.707107	−	0.707107i
$10$		0			0
$11$	20.1876	+	8.36199i	1.83524	+	0.760181i	0.962091	+	0.272727i	$0.0879257\pi$
$11$	20.1876	+	8.36199i	1.83524	+	0.760181i	0.873149	+	0.487454i	$0.162074\pi$
$12$		0			0
$13$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$13$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$14$	−9.26013	−	10.5000i	−0.661438	−	0.750000i
$15$		0			0
$16$	15.5000	+	3.96863i	0.968750	+	0.248039i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	13.5000	−	11.9059i	0.750000	−	0.661438i
$19$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$19$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$20$		0			0
$21$		0			0
$22$	39.2487	+	19.2196i	1.78403	+	0.873617i
$23$	−30.1660	+	30.1660i	−1.31157	+	1.31157i	−0.391304	+	0.920261i	$0.627976\pi$
$23$	−30.1660	+	30.1660i	−1.31157	+	1.31157i	−0.920261	+	0.391304i	$0.872024\pi$
$24$		0			0
$25$	−17.6777	−	17.6777i	−0.707107	−	0.707107i
$26$		0			0
$27$		0			0
$28$	−17.1688	−	22.1186i	−0.613172	−	0.789949i
$29$	−37.1582	+	15.3914i	−1.28132	+	0.530739i	−0.916386	−	0.400295i	$-0.868907\pi$
$29$	−37.1582	+	15.3914i	−1.28132	+	0.530739i	−0.364931	+	0.931034i	$0.618907\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	30.4421	+	9.86299i	0.951316	+	0.308218i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	28.4382	−	22.0742i	0.789949	−	0.613172i
$37$	22.7341	−	54.8850i	0.614435	−	1.48338i	−0.243646	−	0.969864i	$-0.578344\pi$
$37$	22.7341	−	54.8850i	0.614435	−	1.48338i	0.858082	−	0.513514i	$-0.171656\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$	−27.0562	−	11.2070i	−0.629213	−	0.260628i	0.0452058	−	0.998978i	$-0.485606\pi$
$43$	−27.0562	−	11.2070i	−0.629213	−	0.260628i	−0.674419	+	0.738349i	$0.735606\pi$
$44$	75.9362	+	43.2795i	1.72582	+	0.983624i
$45$		0			0
$46$	−63.9918	+	56.4354i	−1.39113	+	1.22686i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$			49.0000i			1.00000i
$50$	−33.0719	−	37.5000i	−0.661438	−	0.750000i
$51$		0			0
$52$		0			0
$53$	−32.2953	−	13.3771i	−0.609344	−	0.252399i	0.0566038	−	0.998397i	$-0.481973\pi$
$53$	−32.2953	−	13.3771i	−0.609344	−	0.252399i	−0.665948	+	0.745998i	$0.731973\pi$
$54$		0			0
$55$		0			0
$56$	−31.5000	−	46.3006i	−0.562500	−	0.826797i
$57$		0			0
$58$	−76.0983	+	26.0685i	−1.31204	+	0.449457i
$59$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$59$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$60$		0			0
$61$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$61$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$62$		0			0
$63$	−63.0000			−1.00000
$64$	59.5294	+	23.5000i	0.930147	+	0.367188i
$65$		0			0
$66$		0			0
$67$	−123.169	+	51.0184i	−1.83835	+	0.761468i	−0.880597	+	0.473866i	$0.842858\pi$
$67$	−123.169	+	51.0184i	−1.83835	+	0.761468i	−0.957750	+	0.287602i	$0.907142\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	59.8665	+	59.8665i	0.843190	+	0.843190i	0.989272	−	0.146082i	$-0.0466664\pi$
$71$	59.8665	+	59.8665i	0.843190	+	0.843190i	−0.146082	+	0.989272i	$0.546666\pi$
$72$	59.5294	−	40.5000i	0.826797	−	0.562500i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	52.2531	−	106.707i	0.706123	−	1.44199i
$75$		0			0
$76$		0			0
$77$	−58.5340	−	141.313i	−0.760181	−	1.83524i
$78$		0			0
$79$			156.268i			1.97807i	0.147669	+	0.989037i	$0.452823\pi$
$79$			156.268i			1.97807i	−0.147669	+	0.989037i	$0.547177\pi$
$80$		0			0
$81$		−	81.0000i		−	1.00000i
$82$		0			0
$83$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$83$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$84$		0			0
$85$		0			0
$86$	−52.6025	−	25.7587i	−0.611657	−	0.299520i
$87$		0			0
$88$	146.154	+	95.8997i	1.66084	+	1.08977i
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$		0			0
$92$	−134.801	+	104.635i	−1.46522	+	1.13733i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−6.13705	+	97.8077i	−0.0626229	+	0.998037i
$99$	181.689	−	75.2579i	1.83524	−	0.760181i
$100$	−61.3172	−	78.9949i	−0.613172	−	0.789949i
$101$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$101$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	−62.7883	−	30.7466i	−0.592342	−	0.290062i
$107$	196.178	+	81.2594i	1.83343	+	0.759433i	0.964276	+	0.264901i	$0.0853391\pi$
$107$	196.178	+	81.2594i	1.83343	+	0.759433i	0.869159	+	0.494533i	$0.164661\pi$
$108$		0			0
$109$	−51.8265	−	125.120i	−0.475472	−	1.14789i	−0.961711	−	0.274066i	$-0.911631\pi$
$109$	−51.8265	−	125.120i	−0.475472	−	1.14789i	0.486239	−	0.873826i	$-0.338369\pi$
$110$		0			0
$111$		0			0
$112$	−57.0774	−	96.3648i	−0.509620	−	0.860400i
$113$		−	127.044i		−	1.12429i	−0.827040	−	0.562144i	$-0.809977\pi$
$113$		−	127.044i		−	1.12429i	0.827040	−	0.562144i	$-0.190023\pi$
$114$		0			0
$115$		0			0
$116$	−155.163	+	42.5037i	−1.33761	+	0.366412i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	252.058	+	252.058i	2.08312	+	2.08312i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−125.753	−	7.89049i	−0.998037	−	0.0626229i
$127$	−253.992			−1.99994			−0.999969	−	0.00787402i	$-0.997494\pi$
$127$	−253.992			−1.99994			−0.999969	+	0.00787402i	$0.997494\pi$
$128$	115.882	+	54.3636i	0.905327	+	0.424715i
$129$		0			0
$130$		0			0
$131$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$131$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$132$		0			0
$133$		0			0
$134$	−252.245	+	86.4100i	−1.88242	+	0.644851i
$135$		0			0
$136$		0			0
$137$	−18.8301	+	18.8301i	−0.137446	+	0.137446i	−0.772482	−	0.635036i	$-0.780985\pi$
$137$	−18.8301	+	18.8301i	−0.137446	+	0.137446i	0.635036	+	0.772482i	$0.280985\pi$
$138$		0			0
$139$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$139$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$140$		0			0
$141$		0			0
$142$	112.000	+	126.996i	0.788732	+	0.894338i
$143$		0			0
$144$	123.898	−	73.3852i	0.860400	−	0.509620i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	117.666	−	206.451i	0.795038	−	1.39494i
$149$	76.4445	+	31.6643i	0.513050	+	0.212512i	0.624161	−	0.781296i	$-0.285441\pi$
$149$	76.4445	+	31.6643i	0.513050	+	0.212512i	−0.111111	+	0.993808i	$0.535441\pi$
$150$		0			0
$151$	73.5020	−	73.5020i	0.486768	−	0.486768i	−0.420517	−	0.907285i	$-0.638151\pi$
$151$	73.5020	−	73.5020i	0.486768	−	0.486768i	0.907285	+	0.420517i	$0.138151\pi$
$152$		0			0
$153$		0			0
$154$	−99.1392	−	289.403i	−0.643761	−	1.87924i
$155$		0			0
$156$		0			0
$157$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$157$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$158$	−19.5719	+	311.922i	−0.123873	+	1.97419i
$159$		0			0
$160$		0			0
$161$	298.628			1.85483
$162$	10.1449	−	161.682i	0.0626229	−	0.998037i
$163$	297.158	−	123.087i	1.82305	−	0.755134i	0.849158	−	0.528140i	$-0.177110\pi$
$163$	297.158	−	123.087i	1.82305	−	0.755134i	0.973896	−	0.226994i	$-0.0728897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$	−119.501	+	119.501i	−0.707107	+	0.707107i
$170$		0			0
$171$		0			0
$172$	−101.772	−	58.0046i	−0.591699	−	0.337236i
$173$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$173$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$			175.000i			1.00000i
$176$	279.723	+	209.728i	1.58933	+	1.19164i
$177$		0			0
$178$		0			0
$179$	−105.417	−	254.499i	−0.588921	−	1.42178i	−0.884535	−	0.466473i	$-0.845524\pi$
$179$	−105.417	−	254.499i	−0.588921	−	1.42178i	0.295615	−	0.955307i	$-0.404476\pi$
$180$		0			0
$181$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$181$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$182$		0			0
$183$		0			0
$184$	−282.177	+	191.975i	−1.53357	+	1.04334i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	75.1937			0.393684			0.196842	−	0.980435i	$-0.436931\pi$
$191$	75.1937			0.393684			0.196842	+	0.980435i	$0.436931\pi$
$192$		0			0
$193$	225.559			1.16870			0.584349	−	0.811502i	$-0.301350\pi$
$193$	225.559			1.16870			0.584349	+	0.811502i	$0.301350\pi$
$194$		0			0
$195$		0			0
$196$	−24.5000	+	194.463i	−0.125000	+	0.992157i
$197$	139.998	−	337.985i	0.710650	−	1.71566i	0.0122790	−	0.999925i	$-0.496091\pi$
$197$	139.998	−	337.985i	0.710650	−	1.71566i	0.698371	−	0.715736i	$-0.253909\pi$
$198$	372.090	−	127.465i	1.87924	−	0.643761i
$199$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$199$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$200$	−112.500	−	165.359i	−0.562500	−	0.826797i
$201$		0			0
$202$		0			0
$203$	260.107	+	107.740i	1.28132	+	0.530739i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			383.951i			1.85483i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−71.5376	−	172.707i	−0.339041	−	0.818516i	−0.997808	−	0.0661700i	$-0.978922\pi$
$211$	−71.5376	−	172.707i	−0.339041	−	0.818516i	0.658768	−	0.752346i	$-0.271078\pi$
$212$	−121.479	−	69.2365i	−0.573015	−	0.326587i
$213$		0			0
$214$	381.408	+	186.770i	1.78228	+	0.872758i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−87.7788	−	256.240i	−0.402655	−	1.17541i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−101.861	−	199.500i	−0.454739	−	0.890625i
$225$	−225.000			−1.00000
$226$	15.9118	−	253.590i	0.0704061	−	1.12208i
$227$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$227$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$228$		0			0
$229$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$229$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$230$		0			0
$231$		0			0
$232$	−315.040	+	65.4072i	−1.35793	+	0.281927i
$233$	−223.826	+	223.826i	−0.960627	+	0.960627i	−0.999254	−	0.0386266i	$-0.987702\pi$
$233$	−223.826	+	223.826i	−0.960627	+	0.960627i	0.0386266	+	0.999254i	$0.487702\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	423.320i		−	1.77121i	−0.464435	−	0.885607i	$-0.653743\pi$
$239$		−	423.320i		−	1.77121i	0.464435	−	0.885607i	$-0.346257\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	471.557	+	534.696i	1.94858	+	2.20949i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$251$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$252$	−250.023	−	31.5000i	−0.992157	−	0.125000i
$253$	−861.229	+	356.733i	−3.40407	+	1.41001i
$254$	−506.987	−	31.8115i	−1.99601	−	0.125242i
$255$		0			0
$256$	224.500	+	123.027i	0.876953	+	0.480576i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−384.195	+	159.139i	−1.48338	+	0.614435i
$260$		0			0
$261$	−138.523	+	334.424i	−0.530739	+	1.28132i
$262$		0			0
$263$	−352.139	−	352.139i	−1.33893	−	1.33893i	−0.897095	−	0.441837i	$-0.854327\pi$
$263$	−352.139	−	352.139i	−1.33893	−	1.33893i	−0.441837	−	0.897095i	$-0.645673\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−514.322	+	140.888i	−1.91911	+	0.525702i
$269$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$269$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	−39.9446	+	35.2278i	−0.145783	+	0.128569i
$275$	−209.050	−	504.691i	−0.760181	−	1.83524i
$276$		0			0
$277$	110.482	+	45.7630i	0.398850	+	0.165209i	0.573087	−	0.819495i	$-0.305746\pi$
$277$	110.482	+	45.7630i	0.398850	+	0.165209i	−0.174237	+	0.984704i	$0.555746\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	332.158	+	332.158i	1.18206	+	1.18206i	0.979210	+	0.202847i	$0.0650194\pi$
$281$	332.158	+	332.158i	1.18206	+	1.18206i	0.202847	+	0.979210i	$0.434981\pi$
$282$		0			0
$283$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$283$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$284$	207.655	+	267.521i	0.731178	+	0.941976i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	256.500	−	130.965i	0.890625	−	0.454739i
$289$	−289.000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$293$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$294$		0			0
$295$		0			0
$296$	260.727	−	397.354i	0.880833	−	1.34241i
$297$		0			0
$298$	148.623	+	72.7787i	0.498735	+	0.244224i
$299$		0			0
$300$		0			0
$301$	78.4492	+	189.393i	0.260628	+	0.629213i
$302$	155.921	−	137.510i	0.516295	−	0.455330i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$307$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$308$	−161.643	−	590.087i	−0.524814	−	1.91587i
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	−78.1339	+	620.169i	−0.247259	+	1.96256i
$317$	122.573	−	50.7712i	0.386664	−	0.160162i	−0.180879	−	0.983505i	$-0.557894\pi$
$317$	122.573	−	50.7712i	0.386664	−	0.160162i	0.567543	+	0.823344i	$0.307894\pi$
$318$		0			0
$319$	−878.840			−2.75498
$320$		0			0
$321$		0			0
$322$	596.084	+	37.4019i	1.85119	+	0.116155i
$323$		0			0
$324$	40.5000	−	321.459i	0.125000	−	0.992157i
$325$		0			0
$326$	608.565	−	208.473i	1.86676	−	0.639486i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	61.5560	+	25.4973i	0.185970	+	0.0770312i	0.473725	−	0.880673i	$-0.342909\pi$
$331$	61.5560	+	25.4973i	0.185970	+	0.0770312i	−0.287755	+	0.957704i	$0.592909\pi$
$332$		0			0
$333$	−204.607	−	493.965i	−0.614435	−	1.48338i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			289.193i			0.858139i	0.903272	+	0.429069i	$0.141158\pi$
$337$			289.193i			0.858139i	−0.903272	+	0.429069i	$0.858842\pi$
$338$	−253.500	+	223.566i	−0.750000	+	0.661438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	242.538	−	242.538i	0.707107	−	0.707107i
$344$	−195.880	−	128.528i	−0.569419	−	0.373628i
$345$		0			0
$346$		0			0
$347$	46.9076	−	113.245i	0.135180	−	0.326354i	−0.841765	−	0.539844i	$-0.818483\pi$
$347$	46.9076	−	113.245i	0.135180	−	0.326354i	0.976945	+	0.213490i	$0.0684831\pi$
$348$		0			0
$349$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$349$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$350$	−21.9180	+	349.313i	−0.0626229	+	0.998037i
$351$		0			0
$352$	532.080	+	453.667i	1.51159	+	1.28883i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−178.545	−	521.201i	−0.498729	−	1.45587i
$359$	475.162	+	475.162i	1.32357	+	1.32357i	0.910864	+	0.412708i	$0.135417\pi$
$359$	475.162	+	475.162i	1.32357	+	1.32357i	0.412708	+	0.910864i	$0.364583\pi$
$360$		0			0
$361$	−255.266	+	255.266i	−0.707107	+	0.707107i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−587.291	+	347.856i	−1.59590	+	0.945260i
$369$		0			0
$370$		0			0
$371$	93.6399	+	226.067i	0.252399	+	0.609344i
$372$		0			0
$373$	688.820	+	285.318i	1.84670	+	0.764929i	0.936298	+	0.351206i	$0.114228\pi$
$373$	688.820	+	285.318i	1.84670	+	0.764929i	0.910403	+	0.413722i	$0.135772\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	151.988	−	366.932i	0.401024	−	0.968159i	−0.586393	−	0.810026i	$-0.699453\pi$
$379$	151.988	−	366.932i	0.401024	−	0.968159i	0.987418	−	0.158132i	$-0.0505473\pi$
$380$		0			0
$381$		0			0
$382$	150.092	+	9.41769i	0.392911	+	0.0246536i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	450.232	+	28.2503i	1.16640	+	0.0731873i
$387$	−243.505	+	100.863i	−0.629213	+	0.260628i
$388$		0			0
$389$	176.572	−	426.284i	0.453914	−	1.09584i	−0.516908	−	0.856041i	$-0.672917\pi$
$389$	176.572	−	426.284i	0.453914	−	1.09584i	0.970821	−	0.239804i	$-0.0770830\pi$
$390$		0			0
$391$		0			0
$392$	−73.2595	+	385.094i	−0.186886	+	0.982382i
$393$		0			0
$394$	321.778	−	657.109i	0.816695	−	1.66779i
$395$		0			0
$396$	758.684	−	207.826i	1.91587	−	0.524814i
$397$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$397$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$398$		0			0
$399$		0			0
$400$	−203.848	−	344.160i	−0.509620	−	0.860400i
$401$		−	759.181i		−	1.89322i	−0.322381	−	0.946610i	$-0.604483\pi$
$401$		−	759.181i		−	1.89322i	0.322381	−	0.946610i	$-0.395517\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	505.700	+	247.634i	1.24557	+	0.609937i
$407$	917.896	−	917.896i	2.25527	−	2.25527i
$408$		0			0
$409$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$409$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−48.0882	+	766.394i	−0.116155	+	1.85119i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$419$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$420$		0			0
$421$	−267.539	+	645.897i	−0.635485	+	1.53420i	0.197150	+	0.980373i	$0.436832\pi$
$421$	−267.539	+	645.897i	−0.635485	+	1.53420i	−0.832635	+	0.553823i	$0.813168\pi$
$422$	−121.163	−	353.696i	−0.287117	−	0.838142i
$423$		0			0
$424$	−233.810	−	153.416i	−0.551439	−	0.361830i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	737.926	+	420.577i	1.72413	+	0.982656i
$429$		0			0
$430$		0			0
$431$			162.000i			0.375870i	0.982181	+	0.187935i	$0.0601794\pi$
$431$			162.000i			0.375870i	−0.982181	+	0.187935i	$0.939821\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−143.120	−	522.469i	−0.328257	−	1.19832i
$437$		0			0
$438$		0			0
$439$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$439$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$	311.834	+	311.834i	0.707107	+	0.707107i
$442$		0			0
$443$	−280.316	+	676.743i	−0.632768	+	1.52764i	0.203361	+	0.979104i	$0.434813\pi$
$443$	−280.316	+	676.743i	−0.632768	+	1.52764i	−0.836129	+	0.548533i	$0.815187\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−178.336	−	410.975i	−0.398072	−	0.917354i
$449$	84.6640			0.188561			0.0942807	−	0.995546i	$-0.469945\pi$
$449$	84.6640			0.188561			0.0942807	+	0.995546i	$0.469945\pi$
$450$	−449.117	−	28.1803i	−0.998037	−	0.0626229i
$451$		0			0
$452$	63.5222	−	504.192i	0.140536	−	1.11547i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−179.600	+	179.600i	−0.392997	+	0.392997i	−0.875754	−	0.482757i	$-0.839635\pi$
$457$	−179.600	+	179.600i	−0.392997	+	0.392997i	0.482757	+	0.875754i	$0.339635\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$461$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$			27.5911i			0.0595920i	0.999556	+	0.0297960i	$0.00948576\pi$
$463$			27.5911i			0.0595920i	−0.999556	+	0.0297960i	$0.990514\pi$
$464$	−637.035	+	91.1001i	−1.37292	+	0.196337i
$465$		0			0
$466$	−474.807	+	418.740i	−1.01890	+	0.898584i
$467$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$467$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$468$		0			0
$469$	862.185	+	357.129i	1.83835	+	0.761468i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−452.487	−	452.487i	−0.956632	−	0.956632i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−290.657	+	120.394i	−0.609344	+	0.252399i
$478$	53.0191	−	844.979i	0.110919	−	1.76774i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	874.295	+	1126.35i	1.80639	+	2.32718i
$485$		0			0
$486$		0			0
$487$	245.486	+	245.486i	0.504078	+	0.504078i	0.912703	−	0.408624i	$-0.133991\pi$
$487$	245.486	+	245.486i	0.504078	+	0.504078i	−0.408624	+	0.912703i	$0.633991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	731.973	+	303.193i	1.49078	+	0.617502i	0.971487	−	0.237094i	$-0.0761949\pi$
$491$	731.973	+	303.193i	1.49078	+	0.617502i	0.519294	+	0.854596i	$0.326195\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	592.648i		−	1.19245i
$498$		0			0
$499$	−244.845	−	591.108i	−0.490671	−	1.18459i	−0.954379	−	0.298597i	$-0.903481\pi$
$499$	−244.845	−	591.108i	−0.490671	−	1.18459i	0.463708	−	0.885988i	$-0.346519\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	−495.120	−	94.1907i	−0.982382	−	0.186886i
$505$		0			0
$506$	−1763.76	+	604.199i	−3.48568	+	1.19407i
$507$		0			0
$508$	−1008.00	−	126.996i	−1.98425	−	0.249992i
$509$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$509$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$510$		0			0
$511$		0			0
$512$	432.710	+	273.690i	0.845137	+	0.534550i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−786.813	+	269.534i	−1.51894	+	0.520336i
$519$		0			0
$520$		0			0
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$	−318.387	+	650.185i	−0.609937	+	1.24557i
$523$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$523$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$524$		0			0
$525$		0			0
$526$	−658.792	−	747.000i	−1.25246	−	1.42015i
$527$		0			0
$528$		0			0
$529$		−	1290.98i		−	2.44041i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1044.27	+	216.807i	−1.94827	+	0.404490i
$537$		0			0
$538$		0			0
$539$	−409.738	+	989.194i	−0.760181	+	1.83524i
$540$		0			0
$541$	318.486	−	131.921i	0.588699	−	0.243847i	−0.0683919	−	0.997659i	$-0.521787\pi$
$541$	318.486	−	131.921i	0.588699	−	0.243847i	0.657091	+	0.753811i	$0.271787\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−298.497	+	123.641i	−0.545698	+	0.226035i	−0.638463	−	0.769653i	$-0.720429\pi$
$547$	−298.497	+	123.641i	−0.545698	+	0.226035i	0.0927652	+	0.995688i	$0.470429\pi$
$548$	−84.1445	+	65.3144i	−0.153548	+	0.119187i
$549$		0			0
$550$	−354.069	−	1033.58i	−0.643761	−	1.87924i
$551$		0			0
$552$		0			0
$553$	773.486	−	773.486i	1.39871	−	1.39871i
$554$	214.798	+	105.184i	0.387722	+	0.189862i
$555$		0			0
$556$		0			0
$557$	318.856	+	769.786i	0.572452	+	1.38202i	0.899461	+	0.437000i	$0.143959\pi$
$557$	318.856	+	769.786i	0.572452	+	1.38202i	−0.327009	+	0.945021i	$0.606041\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	621.411	+	704.614i	1.10571	+	1.25376i
$563$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$563$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−400.930	+	400.930i	−0.707107	+	0.707107i
$568$	380.988	+	560.000i	0.670754	+	0.985915i
$569$	−658.532	−	658.532i	−1.15735	−	1.15735i	−0.985044	−	0.172306i	$-0.944878\pi$
$569$	−658.532	−	658.532i	−1.15735	−	1.15735i	−0.172306	−	0.985044i	$-0.555122\pi$
$570$		0			0
$571$	−232.249	+	560.698i	−0.406740	+	0.981958i	0.579249	+	0.815151i	$0.303346\pi$
$571$	−232.249	+	560.698i	−0.406740	+	0.981958i	−0.985989	+	0.166807i	$0.946654\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1066.53			1.85483
$576$	528.396	−	229.290i	0.917354	−	0.398072i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−576.866	−	36.1960i	−0.998037	−	0.0626229i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−540.105	−	540.105i	−0.926424	−	0.926424i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$587$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	570.197	−	760.494i	0.963170	−	1.28462i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	287.547	+	163.886i	0.482462	+	0.274977i
$597$		0			0
$598$		0			0
$599$	−123.037	+	123.037i	−0.205403	+	0.205403i	−0.802310	−	0.596907i	$-0.796396\pi$
$599$	−123.037	+	123.037i	−0.205403	+	0.205403i	0.596907	+	0.802310i	$0.296396\pi$
$600$		0			0
$601$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$601$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$602$	132.870	+	387.868i	0.220714	+	0.644299i
$603$	−459.165	+	1108.52i	−0.761468	+	1.83835i
$604$	328.453	−	254.951i	0.543796	−	0.422104i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	394.624	−	952.706i	0.643758	−	1.55417i	−0.177814	−	0.984064i	$-0.556903\pi$
$613$	394.624	−	952.706i	0.643758	−	1.55417i	0.821572	−	0.570105i	$-0.193097\pi$
$614$		0			0
$615$		0			0
$616$	−248.745	−	1198.10i	−0.403807	−	1.94497i
$617$	−778.265	+	778.265i	−1.26137	+	1.26137i	−0.310939	+	0.950430i	$0.600644\pi$
$617$	−778.265	+	778.265i	−1.26137	+	1.26137i	−0.950430	+	0.310939i	$0.899356\pi$
$618$		0			0
$619$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$619$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$			625.000i			1.00000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	538.799	−	538.799i	0.853881	−	0.853881i	−0.136728	−	0.990609i	$-0.543659\pi$
$631$	538.799	−	538.799i	0.853881	−	0.853881i	0.990609	+	0.136728i	$0.0436586\pi$
$632$	−233.635	+	1228.12i	−0.369675	+	1.94322i
$633$		0			0
$634$	251.023	−	85.9914i	0.395935	−	0.135633i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−1754.23	−	110.071i	−2.74958	−	0.172525i
$639$	761.976			1.19245
$640$		0			0
$641$	−98.7379			−0.154037			−0.0770186	−	0.997030i	$-0.524540\pi$
$641$	−98.7379			−0.154037			−0.0770186	+	0.997030i	$0.524540\pi$
$642$		0			0
$643$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$643$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$644$	1185.14	+	149.314i	1.84029	+	0.231854i
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	121.102	−	636.583i	0.186886	−	0.982382i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1240.85	−	339.907i	1.90315	−	0.521329i
$653$	12.2256	+	29.5153i	0.0187222	+	0.0451995i	0.932964	−	0.359969i	$-0.117213\pi$
$653$	12.2256	+	29.5153i	0.0187222	+	0.0451995i	−0.914242	+	0.405169i	$0.867213\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	502.086	+	1212.14i	0.761891	+	1.83937i	0.468892	+	0.883255i	$0.344653\pi$
$659$	502.086	+	1212.14i	0.761891	+	1.83937i	0.292999	+	0.956113i	$0.405347\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$	119.677	+	58.6042i	0.180781	+	0.0885259i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−346.544	−	1011.62i	−0.520336	−	1.51894i
$667$	656.617	−	1585.21i	0.984433	−	2.37663i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1269.96			1.88701			0.943507	−	0.331352i	$-0.107505\pi$
$673$	1269.96			1.88701			0.943507	+	0.331352i	$0.107505\pi$
$674$	−36.2202	+	577.250i	−0.0537391	+	0.856454i
$675$		0			0
$676$	−534.006	+	414.505i	−0.789949	+	0.613172i
$677$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$677$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1239.34	−	513.350i	−1.81455	−	0.751611i	−0.979502	−	0.201433i	$-0.935440\pi$
$683$	−1239.34	−	513.350i	−1.81455	−	0.751611i	−0.835048	−	0.550178i	$-0.814560\pi$
$684$		0			0
$685$		0			0
$686$	514.500	−	453.746i	0.750000	−	0.661438i
$687$		0			0
$688$	−374.894	−	281.085i	−0.544904	−	0.408553i
$689$		0			0
$690$		0			0
$691$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$691$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$692$		0			0
$693$	−1271.82	−	526.806i	−1.83524	−	0.760181i
$694$	107.815	−	220.170i	0.155352	−	0.317248i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−87.5000	+	694.510i	−0.125000	+	0.992157i
$701$	1155.85	−	478.769i	1.64886	−	0.682980i	0.651713	−	0.758465i	$-0.274051\pi$
$701$	1155.85	−	478.769i	1.64886	−	0.682980i	0.997147	+	0.0754851i	$0.0240505\pi$
$702$		0			0
$703$		0			0
$704$	1005.25	+	972.194i	1.42791	+	1.38096i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	90.1395	−	217.616i	0.127136	−	0.306934i	−0.847476	−	0.530834i	$-0.821879\pi$
$709$	90.1395	−	217.616i	0.127136	−	0.306934i	0.974612	+	0.223900i	$0.0718789\pi$
$710$		0			0
$711$	994.482	+	994.482i	1.39871	+	1.39871i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−291.111	−	1062.72i	−0.406579	−	1.48424i
$717$		0			0
$718$	888.947	+	1007.97i	1.23809	+	1.40386i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−541.500	+	477.558i	−0.750000	+	0.661438i
$723$		0			0
$724$		0			0
$725$	928.955	+	384.786i	1.28132	+	0.530739i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−515.481	−	515.481i	−0.707107	−	0.707107i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$	−1215.84	+	620.790i	−1.65196	+	0.843464i
$737$	−2913.11			−3.95266
$738$		0			0
$739$	−1138.04	+	471.393i	−1.53998	+	0.637879i	−0.981469	−	0.191620i	$-0.938626\pi$
$739$	−1138.04	+	471.393i	−1.53998	+	0.637879i	−0.558508	+	0.829499i	$0.688626\pi$
$740$		0			0
$741$		0			0
$742$	158.598	+	462.974i	0.213744	+	0.623954i
$743$	−683.810	−	683.810i	−0.920337	−	0.920337i	0.0767160	−	0.997053i	$-0.475557\pi$
$743$	−683.810	−	683.810i	−0.920337	−	0.920337i	−0.997053	+	0.0767160i	$0.975557\pi$
$744$		0			0
$745$		0			0
$746$	1339.20	+	655.789i	1.79517	+	0.879073i
$747$		0			0
$748$		0			0
$749$	−568.816	−	1373.24i	−0.759433	−	1.83343i
$750$		0			0
$751$			1269.96i			1.69103i	0.533955	+	0.845513i	$0.320705\pi$
$751$			1269.96i			1.69103i	−0.533955	+	0.845513i	$0.679295\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−994.632	−	411.990i	−1.31391	−	0.544241i	−0.387890	−	0.921706i	$-0.626796\pi$
$757$	−994.632	−	411.990i	−1.31391	−	0.544241i	−0.926024	+	0.377465i	$0.876796\pi$
$758$	349.337	−	713.388i	0.460866	−	0.941145i
$759$		0			0
$760$		0			0
$761$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$761$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$	−362.785	+	875.842i	−0.475472	+	1.14789i
$764$	298.416	+	37.5968i	0.390596	+	0.0492105i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	895.158	+	112.779i	1.15953	+	0.146087i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$	−498.688	+	170.832i	−0.644299	+	0.220714i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	405.842	−	828.779i	0.521648	−	1.06527i
$779$		0			0
$780$		0			0
$781$	707.960	+	1709.17i	0.906479	+	2.18843i
$782$		0			0
$783$		0			0
$784$	−194.463	+	759.500i	−0.248039	+	0.968750i
$785$		0			0
$786$		0			0
$787$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$787$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$788$	724.593	−	1271.34i	0.919534	−	1.61337i
$789$		0			0
$790$		0			0
$791$	−628.838	+	628.838i	−0.794991	+	0.794991i
$792$	1540.42	−	319.815i	1.94497	−	0.403807i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$797$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$798$		0			0
$799$		0			0
$800$	−363.791	−	712.500i	−0.454739	−	0.890625i
$801$		0			0
$802$	95.0843	−	1515.38i	0.118559	−	1.88950i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1137.46	−	1137.46i	1.40601	−	1.40601i	0.626964	−	0.779048i	$-0.284297\pi$
$809$	1137.46	−	1137.46i	1.40601	−	1.40601i	0.779048	−	0.626964i	$-0.215703\pi$
$810$		0			0
$811$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$811$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$812$	978.399	+	557.634i	1.20493	+	0.686741i
$813$		0			0
$814$	1947.15	−	1717.23i	2.39208	−	2.10961i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	584.233	+	241.997i	0.711611	+	0.294759i	0.708971	−	0.705238i	$-0.249160\pi$
$821$	584.233	+	241.997i	0.711611	+	0.294759i	0.00264006	+	0.999997i	$0.499160\pi$
$822$		0			0
$823$	439.820	−	439.820i	0.534411	−	0.534411i	−0.387471	−	0.921882i	$-0.626651\pi$
$823$	439.820	−	439.820i	0.534411	−	0.534411i	0.921882	+	0.387471i	$0.126651\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	338.378	−	816.917i	0.409164	−	0.987808i	−0.576195	−	0.817312i	$-0.695463\pi$
$827$	338.378	−	816.917i	0.409164	−	0.987808i	0.985358	−	0.170496i	$-0.0545369\pi$
$828$	−191.975	+	1523.76i	−0.231854	+	1.84029i
$829$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$829$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	549.159	−	549.159i	0.652983	−	0.652983i
$842$	−614.924	+	1255.75i	−0.730313	+	1.49139i
$843$		0			0
$844$	−197.552	−	721.178i	−0.234067	−	0.854477i
$845$		0			0
$846$		0			0
$847$		−	2495.25i		−	2.94598i
$848$	−447.488	−	335.513i	−0.527698	−	0.395653i
$849$		0			0
$850$		0			0
$851$	969.864	+	2341.46i	1.13968	+	2.75142i
$852$		0			0
$853$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$853$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$854$		0			0
$855$		0			0
$856$	1420.28	+	931.925i	1.65920	+	1.08870i
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$859$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$860$		0			0
$861$		0			0
$862$	−20.2898	+	323.364i	−0.0235381	+	0.375132i
$863$	−1504.48			−1.74331			−0.871655	−	0.490119i	$-0.836953\pi$
$863$	−1504.48			−1.74331			−0.871655	+	0.490119i	$0.836953\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1306.71	+	3154.68i	−1.50369	+	3.63024i
$870$		0			0
$871$		0			0
$872$	−220.241	−	1060.81i	−0.252570	−	1.21653i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−31.2740	−	75.5022i	−0.0356603	−	0.0860915i	0.905046	−	0.425314i	$-0.139836\pi$
$877$	−31.2740	−	75.5022i	−0.0356603	−	0.0860915i	−0.940706	+	0.339222i	$0.889836\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	583.388	+	661.500i	0.661438	+	0.750000i
$883$	−484.486	−	1169.65i	−0.548682	−	1.32463i	−0.918460	−	0.395514i	$-0.870566\pi$
$883$	−484.486	−	1169.65i	−0.548682	−	1.32463i	0.369778	−	0.929120i	$-0.379434\pi$
$884$		0			0
$885$		0			0
$886$	−644.291	+	1315.72i	−0.727191	+	1.48501i
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$	1257.20	+	1257.20i	1.41417	+	1.41417i
$890$		0			0
$891$	677.322	−	1635.20i	0.760181	−	1.83524i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−304.500	−	842.672i	−0.339844	−	0.940482i
$897$		0			0
$898$	168.996	+	10.6038i	0.188191	+	0.0118083i
$899$		0			0
$900$	−892.941	−	112.500i	−0.992157	−	0.125000i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	189.943	−	998.449i	0.210114	−	1.10448i
$905$		0			0
$906$		0			0
$907$	−902.441	−	373.803i	−0.994974	−	0.412132i	−0.175022	−	0.984564i	$-0.556000\pi$
$907$	−902.441	−	373.803i	−0.994974	−	0.412132i	−0.819952	+	0.572433i	$0.806000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	931.304i		−	1.02229i	−0.859495	−	0.511144i	$-0.829222\pi$
$911$		−	931.304i		−	1.02229i	0.859495	−	0.511144i	$-0.170778\pi$
$912$		0			0
$913$		0			0
$914$	−380.988	+	336.000i	−0.416836	+	0.367615i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1257.20	+	1257.20i	−1.36801	+	1.36801i	−0.504725	+	0.863280i	$0.668406\pi$
$919$	−1257.20	+	1257.20i	−1.36801	+	1.36801i	−0.863280	+	0.504725i	$0.831594\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1372.12	+	568.353i	−1.48338	+	0.614435i
$926$	−3.45567	+	55.0739i	−0.00373182	+	0.0594750i
$927$		0			0
$928$	−1282.98	+	102.057i	−1.38252	+	0.109975i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−1000.20	+	776.369i	−1.07317	+	0.833014i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$	1676.26	+	820.840i	1.78705	+	0.875096i
$939$		0			0
$940$		0			0
$941$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$941$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−846.525	−	959.869i	−0.894847	−	1.01466i
$947$	−192.791	−	465.439i	−0.203581	−	0.491488i	0.788807	−	0.614641i	$-0.210699\pi$
$947$	−192.791	−	465.439i	−0.203581	−	0.491488i	−0.992388	+	0.123154i	$0.960699\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1342.81	−	1342.81i	−1.40904	−	1.40904i	−0.764953	−	0.644086i	$-0.777238\pi$
$953$	−1342.81	−	1342.81i	−1.40904	−	1.40904i	−0.644086	−	0.764953i	$-0.722762\pi$
$954$	−595.252	+	203.912i	−0.623954	+	0.213744i
$955$		0			0
$956$	211.660	−	1680.00i	0.221402	−	1.75732i
$957$		0			0
$958$		0			0
$959$	186.408			0.194377
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	1765.60	−	731.334i	1.83343	−	0.759433i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−785.470	−	785.470i	−0.812276	−	0.812276i	0.172699	−	0.984975i	$-0.444751\pi$
$967$	−785.470	−	785.470i	−0.812276	−	0.812276i	−0.984975	+	0.172699i	$0.944751\pi$
$968$	1604.09	+	2357.79i	1.65711	+	2.43573i
$969$		0			0
$970$		0			0
$971$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$971$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$972$		0			0
$973$		0			0
$974$	459.263	+	520.755i	0.471522	+	0.534656i
$975$		0			0
$976$		0			0
$977$			1947.27i			1.99311i	0.0829069	+	0.996557i	$0.473580\pi$
$977$			1947.27i			1.99311i	−0.0829069	+	0.996557i	$0.526420\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1126.08	−	466.438i	−1.14789	−	0.475472i
$982$	1423.10	+	696.873i	1.44919	+	0.709647i
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1154.25	−	478.105i	1.16709	−	0.483423i
$990$		0			0
$991$	6.39458			0.00645266			0.00322633	−	0.999995i	$-0.498973\pi$
$991$	6.39458			0.00645266			0.00322633	+	0.999995i	$0.498973\pi$
$992$		0			0
$993$		0			0
$994$	74.2267	−	1182.97i	0.0746748	−	1.19011i
$995$		0			0
$996$		0			0
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$	−414.695	−	1210.56i	−0.415526	−	1.21299i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.a.181.2	yes	8
7.6	odd	2	CM	224.3.v.a.181.2	yes	8
32.29	even	8	inner	224.3.v.a.125.2	✓	8
224.125	odd	8	inner	224.3.v.a.125.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.a.125.2	✓	8	32.29	even	8	inner
224.3.v.a.125.2	✓	8	224.125	odd	8	inner
224.3.v.a.181.2	yes	8	1.1	even	1	trivial
224.3.v.a.181.2	yes	8	7.6	odd	2	CM

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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.v (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.99607 + 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(-4.94975 - 4.94975i) q^{7} +(7.85905 + 1.49509i) q^{8} +(6.36396 - 6.36396i) q^{9} +O(q^{10})\)	\(q+(1.99607 + 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(-4.94975 - 4.94975i) q^{7} +(7.85905 + 1.49509i) q^{8} +(6.36396 - 6.36396i) q^{9} +(20.1876 + 8.36199i) q^{11} +(-9.26013 - 10.5000i) q^{14} +(15.5000 + 3.96863i) q^{16} +(13.5000 - 11.9059i) q^{18} +(39.2487 + 19.2196i) q^{22} +(-30.1660 + 30.1660i) q^{23} +(-17.6777 - 17.6777i) q^{25} +(-17.1688 - 22.1186i) q^{28} +(-37.1582 + 15.3914i) q^{29} +(30.4421 + 9.86299i) q^{32} +(28.4382 - 22.0742i) q^{36} +(22.7341 - 54.8850i) q^{37} +(-27.0562 - 11.2070i) q^{43} +(75.9362 + 43.2795i) q^{44} +(-63.9918 + 56.4354i) q^{46} +49.0000i q^{49} +(-33.0719 - 37.5000i) q^{50} +(-32.2953 - 13.3771i) q^{53} +(-31.5000 - 46.3006i) q^{56} +(-76.0983 + 26.0685i) q^{58} -63.0000 q^{63} +(59.5294 + 23.5000i) q^{64} +(-123.169 + 51.0184i) q^{67} +(59.8665 + 59.8665i) q^{71} +(59.5294 - 40.5000i) q^{72} +(52.2531 - 106.707i) q^{74} +(-58.5340 - 141.313i) q^{77} +156.268i q^{79} -81.0000i q^{81} +(-52.6025 - 25.7587i) q^{86} +(146.154 + 95.8997i) q^{88} +(-134.801 + 104.635i) q^{92} +(-6.13705 + 97.8077i) q^{98} +(181.689 - 75.2579i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 124 q^{16} + 108 q^{18} + 148 q^{22} - 72 q^{23} - 232 q^{43} + 324 q^{44} + 24 q^{53} - 252 q^{56} - 504 q^{63} - 472 q^{67} - 108 q^{74} - 168 q^{77} - 708 q^{92}+O(q^{100}) \) 8 * q + 124 * q^16 + 108 * q^18 + 148 * q^22 - 72 * q^23 - 232 * q^43 + 324 * q^44 + 24 * q^53 - 252 * q^56 - 504 * q^63 - 472 * q^67 - 108 * q^74 - 168 * q^77 - 708 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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