LMFDB - Embedded newform 224.3.v.a.125.2

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			125.2
Root			$1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	224.125
Dual form			224.3.v.a.181.2

$q$-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 $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{8}\right)$

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	$-0.875000\pi$
$3$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$4$	3.96863	−	0.500000i	0.992157	−	0.125000i
$5$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$5$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$6$		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	0.873149	−	0.487454i	$-0.162074\pi$
$11$	20.1876	−	8.36199i	1.83524	−	0.760181i	0.962091	−	0.272727i	$-0.0879257\pi$
$12$		0			0
$13$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$13$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$14$	−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	$-0.375000\pi$
$19$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$20$		0			0
$21$		0			0
$22$	39.2487	−	19.2196i	1.78403	−	0.873617i
$23$	−30.1660	−	30.1660i	−1.31157	−	1.31157i	−0.920261	−	0.391304i	$-0.872024\pi$
$23$	−30.1660	−	30.1660i	−1.31157	−	1.31157i	−0.391304	−	0.920261i	$-0.627976\pi$
$24$		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	−0.364931	−	0.931034i	$-0.618907\pi$
$29$	−37.1582	−	15.3914i	−1.28132	−	0.530739i	−0.916386	+	0.400295i	$0.868907\pi$
$30$		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	0.858082	+	0.513514i	$0.171656\pi$
$37$	22.7341	+	54.8850i	0.614435	+	1.48338i	−0.243646	+	0.969864i	$0.578344\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$	−27.0562	+	11.2070i	−0.629213	+	0.260628i	−0.674419	−	0.738349i	$-0.735606\pi$
$43$	−27.0562	+	11.2070i	−0.629213	+	0.260628i	0.0452058	+	0.998978i	$0.485606\pi$
$44$	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	−0.665948	−	0.745998i	$-0.731973\pi$
$53$	−32.2953	+	13.3771i	−0.609344	+	0.252399i	0.0566038	+	0.998397i	$0.481973\pi$
$54$		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	$-0.625000\pi$
$59$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$60$		0			0
$61$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$61$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$62$		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	−0.957750	−	0.287602i	$-0.907142\pi$
$67$	−123.169	−	51.0184i	−1.83835	−	0.761468i	−0.880597	−	0.473866i	$-0.842858\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	59.8665	−	59.8665i	0.843190	−	0.843190i	−0.146082	−	0.989272i	$-0.546666\pi$
$71$	59.8665	−	59.8665i	0.843190	−	0.843190i	0.989272	+	0.146082i	$0.0466664\pi$
$72$	59.5294	+	40.5000i	0.826797	+	0.562500i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	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	$-0.547177\pi$
$79$		−	156.268i		−	1.97807i	0.147669	−	0.989037i	$-0.452823\pi$
$80$		0			0
$81$			81.0000i			1.00000i
$82$		0			0
$83$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$83$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$84$		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	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		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	$-0.625000\pi$
$101$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$102$		0			0
$103$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$103$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$104$		0			0
$105$		0			0
$106$	−62.7883	+	30.7466i	−0.592342	+	0.290062i
$107$	196.178	−	81.2594i	1.83343	−	0.759433i	0.869159	−	0.494533i	$-0.164661\pi$
$107$	196.178	−	81.2594i	1.83343	−	0.759433i	0.964276	−	0.264901i	$-0.0853391\pi$
$108$		0			0
$109$	−51.8265	+	125.120i	−0.475472	+	1.14789i	0.486239	+	0.873826i	$0.338369\pi$
$109$	−51.8265	+	125.120i	−0.475472	+	1.14789i	−0.961711	+	0.274066i	$0.911631\pi$
$110$		0			0
$111$		0			0
$112$	−57.0774	+	96.3648i	−0.509620	+	0.860400i
$113$			127.044i			1.12429i	0.827040	+	0.562144i	$0.190023\pi$
$113$			127.044i			1.12429i	−0.827040	+	0.562144i	$0.809977\pi$
$114$		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	$-0.875000\pi$
$131$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$132$		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	0.635036	−	0.772482i	$-0.280985\pi$
$137$	−18.8301	−	18.8301i	−0.137446	−	0.137446i	−0.772482	+	0.635036i	$0.780985\pi$
$138$		0			0
$139$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$139$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$140$		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	−0.111111	−	0.993808i	$-0.535441\pi$
$149$	76.4445	−	31.6643i	0.513050	−	0.212512i	0.624161	+	0.781296i	$0.285441\pi$
$150$		0			0
$151$	73.5020	+	73.5020i	0.486768	+	0.486768i	0.907285	−	0.420517i	$-0.138151\pi$
$151$	73.5020	+	73.5020i	0.486768	+	0.486768i	−0.420517	+	0.907285i	$0.638151\pi$
$152$		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	$-0.875000\pi$
$157$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$158$	−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	0.973896	+	0.226994i	$0.0728897\pi$
$163$	297.158	+	123.087i	1.82305	+	0.755134i	0.849158	+	0.528140i	$0.177110\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		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	$-0.375000\pi$
$173$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$174$		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	0.295615	+	0.955307i	$0.404476\pi$
$179$	−105.417	+	254.499i	−0.588921	+	1.42178i	−0.884535	+	0.466473i	$0.845524\pi$
$180$		0			0
$181$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$181$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$182$		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	0.698371	+	0.715736i	$0.253909\pi$
$197$	139.998	+	337.985i	0.710650	+	1.71566i	0.0122790	+	0.999925i	$0.496091\pi$
$198$	372.090	+	127.465i	1.87924	+	0.643761i
$199$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$199$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$200$	−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	0.658768	+	0.752346i	$0.271078\pi$
$211$	−71.5376	+	172.707i	−0.339041	+	0.818516i	−0.997808	+	0.0661700i	$0.978922\pi$
$212$	−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	$-0.875000\pi$
$227$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$228$		0			0
$229$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$229$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$230$		0			0
$231$		0			0
$232$	−315.040	−	65.4072i	−1.35793	−	0.281927i
$233$	−223.826	−	223.826i	−0.960627	−	0.960627i	0.0386266	−	0.999254i	$-0.487702\pi$
$233$	−223.826	−	223.826i	−0.960627	−	0.960627i	−0.999254	+	0.0386266i	$0.987702\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			423.320i			1.77121i	0.464435	+	0.885607i	$0.346257\pi$
$239$			423.320i			1.77121i	−0.464435	+	0.885607i	$0.653743\pi$
$240$		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	$-0.625000\pi$
$251$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$252$	−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	−0.441837	+	0.897095i	$0.645673\pi$
$263$	−352.139	+	352.139i	−1.33893	+	1.33893i	−0.897095	+	0.441837i	$0.854327\pi$
$264$		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	$-0.375000\pi$
$269$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$270$		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	−0.174237	−	0.984704i	$-0.555746\pi$
$277$	110.482	−	45.7630i	0.398850	−	0.165209i	0.573087	+	0.819495i	$0.305746\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	332.158	−	332.158i	1.18206	−	1.18206i	0.202847	−	0.979210i	$-0.434981\pi$
$281$	332.158	−	332.158i	1.18206	−	1.18206i	0.979210	−	0.202847i	$-0.0650194\pi$
$282$		0			0
$283$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$283$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$284$	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	$-0.625000\pi$
$293$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$294$		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	$-0.375000\pi$
$307$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$308$	−161.643	+	590.087i	−0.524814	+	1.91587i
$309$		0			0
$310$		0			0
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$		0			0
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$	−78.1339	−	620.169i	−0.247259	−	1.96256i
$317$	122.573	+	50.7712i	0.386664	+	0.160162i	0.567543	−	0.823344i	$-0.307894\pi$
$317$	122.573	+	50.7712i	0.386664	+	0.160162i	−0.180879	+	0.983505i	$0.557894\pi$
$318$		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	−0.287755	−	0.957704i	$-0.592909\pi$
$331$	61.5560	−	25.4973i	0.185970	−	0.0770312i	0.473725	+	0.880673i	$0.342909\pi$
$332$		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	$-0.858842\pi$
$337$		−	289.193i		−	0.858139i	0.903272	−	0.429069i	$-0.141158\pi$
$338$	−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	0.976945	−	0.213490i	$-0.0684831\pi$
$347$	46.9076	+	113.245i	0.135180	+	0.326354i	−0.841765	+	0.539844i	$0.818483\pi$
$348$		0			0
$349$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$349$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$350$	−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	0.412708	−	0.910864i	$-0.364583\pi$
$359$	475.162	−	475.162i	1.32357	−	1.32357i	0.910864	−	0.412708i	$-0.135417\pi$
$360$		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	0.910403	−	0.413722i	$-0.135772\pi$
$373$	688.820	−	285.318i	1.84670	−	0.764929i	0.936298	−	0.351206i	$-0.114228\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	151.988	+	366.932i	0.401024	+	0.968159i	0.987418	+	0.158132i	$0.0505473\pi$
$379$	151.988	+	366.932i	0.401024	+	0.968159i	−0.586393	+	0.810026i	$0.699453\pi$
$380$		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	0.970821	+	0.239804i	$0.0770830\pi$
$389$	176.572	+	426.284i	0.453914	+	1.09584i	−0.516908	+	0.856041i	$0.672917\pi$
$390$		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	$-0.375000\pi$
$397$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$398$		0			0
$399$		0			0
$400$	−203.848	+	344.160i	−0.509620	+	0.860400i
$401$			759.181i			1.89322i	0.322381	+	0.946610i	$0.395517\pi$
$401$			759.181i			1.89322i	−0.322381	+	0.946610i	$0.604483\pi$
$402$		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	$-0.250000\pi$
$409$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$410$		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	$-0.875000\pi$
$419$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$420$		0			0
$421$	−267.539	−	645.897i	−0.635485	−	1.53420i	−0.832635	−	0.553823i	$-0.813168\pi$
$421$	−267.539	−	645.897i	−0.635485	−	1.53420i	0.197150	−	0.980373i	$-0.436832\pi$
$422$	−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	$-0.939821\pi$
$431$		−	162.000i		−	0.375870i	0.982181	−	0.187935i	$-0.0601794\pi$
$432$		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	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$	311.834	−	311.834i	0.707107	−	0.707107i
$442$		0			0
$443$	−280.316	−	676.743i	−0.632768	−	1.52764i	−0.836129	−	0.548533i	$-0.815187\pi$
$443$	−280.316	−	676.743i	−0.632768	−	1.52764i	0.203361	−	0.979104i	$-0.434813\pi$
$444$		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	0.482757	−	0.875754i	$-0.339635\pi$
$457$	−179.600	−	179.600i	−0.392997	−	0.392997i	−0.875754	+	0.482757i	$0.839635\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$461$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$462$		0			0
$463$		−	27.5911i		−	0.0595920i	−0.999556	−	0.0297960i	$-0.990514\pi$
$463$		−	27.5911i		−	0.0595920i	0.999556	−	0.0297960i	$-0.00948576\pi$
$464$	−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	$-0.375000\pi$
$467$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$468$		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	−0.408624	−	0.912703i	$-0.633991\pi$
$487$	245.486	−	245.486i	0.504078	−	0.504078i	0.912703	+	0.408624i	$0.133991\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	731.973	−	303.193i	1.49078	−	0.617502i	0.519294	−	0.854596i	$-0.326195\pi$
$491$	731.973	−	303.193i	1.49078	−	0.617502i	0.971487	+	0.237094i	$0.0761949\pi$
$492$		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	0.463708	+	0.885988i	$0.346519\pi$
$499$	−244.845	+	591.108i	−0.490671	+	1.18459i	−0.954379	+	0.298597i	$0.903481\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$	−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	$-0.875000\pi$
$509$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$510$		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	$-0.750000\pi$
$521$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$522$	−318.387	−	650.185i	−0.609937	−	1.24557i
$523$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$523$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$524$		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	0.657091	−	0.753811i	$-0.271787\pi$
$541$	318.486	+	131.921i	0.588699	+	0.243847i	−0.0683919	+	0.997659i	$0.521787\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−298.497	−	123.641i	−0.545698	−	0.226035i	0.0927652	−	0.995688i	$-0.470429\pi$
$547$	−298.497	−	123.641i	−0.545698	−	0.226035i	−0.638463	+	0.769653i	$0.720429\pi$
$548$	−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	−0.327009	−	0.945021i	$-0.606041\pi$
$557$	318.856	−	769.786i	0.572452	−	1.38202i	0.899461	−	0.437000i	$-0.143959\pi$
$558$		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	$-0.375000\pi$
$563$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$564$		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	−0.172306	+	0.985044i	$0.555122\pi$
$569$	−658.532	+	658.532i	−1.15735	+	1.15735i	−0.985044	+	0.172306i	$0.944878\pi$
$570$		0			0
$571$	−232.249	−	560.698i	−0.406740	−	0.981958i	−0.985989	−	0.166807i	$-0.946654\pi$
$571$	−232.249	−	560.698i	−0.406740	−	0.981958i	0.579249	−	0.815151i	$-0.303346\pi$
$572$		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	$-0.125000\pi$
$587$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$588$		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	0.596907	−	0.802310i	$-0.296396\pi$
$599$	−123.037	−	123.037i	−0.205403	−	0.205403i	−0.802310	+	0.596907i	$0.796396\pi$
$600$		0			0
$601$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$601$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$602$	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	0.821572	+	0.570105i	$0.193097\pi$
$613$	394.624	+	952.706i	0.643758	+	1.55417i	−0.177814	+	0.984064i	$0.556903\pi$
$614$		0			0
$615$		0			0
$616$	−248.745	+	1198.10i	−0.403807	+	1.94497i
$617$	−778.265	−	778.265i	−1.26137	−	1.26137i	−0.950430	−	0.310939i	$-0.899356\pi$
$617$	−778.265	−	778.265i	−1.26137	−	1.26137i	−0.310939	−	0.950430i	$-0.600644\pi$
$618$		0			0
$619$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$619$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$620$		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	0.990609	−	0.136728i	$-0.0436586\pi$
$631$	538.799	+	538.799i	0.853881	+	0.853881i	−0.136728	+	0.990609i	$0.543659\pi$
$632$	−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	$-0.875000\pi$
$643$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$644$	1185.14	−	149.314i	1.84029	−	0.231854i
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	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	−0.914242	−	0.405169i	$-0.867213\pi$
$653$	12.2256	−	29.5153i	0.0187222	−	0.0451995i	0.932964	+	0.359969i	$0.117213\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	502.086	−	1212.14i	0.761891	−	1.83937i	0.292999	−	0.956113i	$-0.405347\pi$
$659$	502.086	−	1212.14i	0.761891	−	1.83937i	0.468892	−	0.883255i	$-0.344653\pi$
$660$		0			0
$661$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$661$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$662$	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	$-0.625000\pi$
$677$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1239.34	+	513.350i	−1.81455	+	0.751611i	−0.835048	+	0.550178i	$0.814560\pi$
$683$	−1239.34	+	513.350i	−1.81455	+	0.751611i	−0.979502	+	0.201433i	$0.935440\pi$
$684$		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	$-0.375000\pi$
$691$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$692$		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	0.997147	−	0.0754851i	$-0.0240505\pi$
$701$	1155.85	+	478.769i	1.64886	+	0.682980i	0.651713	+	0.758465i	$0.274051\pi$
$702$		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	0.974612	−	0.223900i	$-0.0718789\pi$
$709$	90.1395	+	217.616i	0.127136	+	0.306934i	−0.847476	+	0.530834i	$0.821879\pi$
$710$		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	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		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	$-0.875000\pi$
$733$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$734$		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	−0.558508	−	0.829499i	$-0.688626\pi$
$739$	−1138.04	−	471.393i	−1.53998	−	0.637879i	−0.981469	+	0.191620i	$0.938626\pi$
$740$		0			0
$741$		0			0
$742$	158.598	−	462.974i	0.213744	−	0.623954i
$743$	−683.810	+	683.810i	−0.920337	+	0.920337i	−0.997053	−	0.0767160i	$-0.975557\pi$
$743$	−683.810	+	683.810i	−0.920337	+	0.920337i	0.0767160	+	0.997053i	$0.475557\pi$
$744$		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	$-0.679295\pi$
$751$		−	1269.96i		−	1.69103i	0.533955	−	0.845513i	$-0.320705\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−994.632	+	411.990i	−1.31391	+	0.544241i	−0.926024	−	0.377465i	$-0.876796\pi$
$757$	−994.632	+	411.990i	−1.31391	+	0.544241i	−0.387890	+	0.921706i	$0.626796\pi$
$758$	349.337	+	713.388i	0.460866	+	0.941145i
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		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	$-0.625000\pi$
$773$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$774$	−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	$-0.375000\pi$
$787$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$788$	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	$-0.875000\pi$
$797$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$798$		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	0.779048	+	0.626964i	$0.215703\pi$
$809$	1137.46	+	1137.46i	1.40601	+	1.40601i	0.626964	+	0.779048i	$0.284297\pi$
$810$		0			0
$811$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$811$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$812$	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	0.00264006	−	0.999997i	$-0.499160\pi$
$821$	584.233	−	241.997i	0.711611	−	0.294759i	0.708971	+	0.705238i	$0.249160\pi$
$822$		0			0
$823$	439.820	+	439.820i	0.534411	+	0.534411i	0.921882	−	0.387471i	$-0.126651\pi$
$823$	439.820	+	439.820i	0.534411	+	0.534411i	−0.387471	+	0.921882i	$0.626651\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	338.378	+	816.917i	0.409164	+	0.987808i	0.985358	+	0.170496i	$0.0545369\pi$
$827$	338.378	+	816.917i	0.409164	+	0.987808i	−0.576195	+	0.817312i	$0.695463\pi$
$828$	−191.975	−	1523.76i	−0.231854	−	1.84029i
$829$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$829$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$830$		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	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		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	$-0.125000\pi$
$853$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$854$		0			0
$855$		0			0
$856$	1420.28	−	931.925i	1.65920	−	1.08870i
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$859$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$860$		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	−0.940706	−	0.339222i	$-0.889836\pi$
$877$	−31.2740	+	75.5022i	−0.0356603	+	0.0860915i	0.905046	+	0.425314i	$0.139836\pi$
$878$		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	0.369778	+	0.929120i	$0.379434\pi$
$883$	−484.486	+	1169.65i	−0.548682	+	1.32463i	−0.918460	+	0.395514i	$0.870566\pi$
$884$		0			0
$885$		0			0
$886$	−644.291	−	1315.72i	−0.727191	−	1.48501i
$887$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$887$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$888$		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	−0.819952	−	0.572433i	$-0.806000\pi$
$907$	−902.441	+	373.803i	−0.994974	+	0.412132i	−0.175022	+	0.984564i	$0.556000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			931.304i			1.02229i	0.859495	+	0.511144i	$0.170778\pi$
$911$			931.304i			1.02229i	−0.859495	+	0.511144i	$0.829222\pi$
$912$		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	−0.863280	−	0.504725i	$-0.831594\pi$
$919$	−1257.20	−	1257.20i	−1.36801	−	1.36801i	−0.504725	−	0.863280i	$-0.668406\pi$
$920$		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	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$	1676.26	−	820.840i	1.78705	−	0.875096i
$939$		0			0
$940$		0			0
$941$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$941$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$942$		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	−0.992388	−	0.123154i	$-0.960699\pi$
$947$	−192.791	+	465.439i	−0.203581	+	0.491488i	0.788807	+	0.614641i	$0.210699\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1342.81	+	1342.81i	−1.40904	+	1.40904i	−0.644086	+	0.764953i	$0.722762\pi$
$953$	−1342.81	+	1342.81i	−1.40904	+	1.40904i	−0.764953	+	0.644086i	$0.777238\pi$
$954$	−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	−0.984975	−	0.172699i	$-0.944751\pi$
$967$	−785.470	+	785.470i	−0.812276	+	0.812276i	0.172699	+	0.984975i	$0.444751\pi$
$968$	1604.09	−	2357.79i	1.65711	−	2.43573i
$969$		0			0
$970$		0			0
$971$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$971$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$972$		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	$-0.526420\pi$
$977$		−	1947.27i		−	1.99311i	0.0829069	−	0.996557i	$-0.473580\pi$
$978$		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	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		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	$-0.625000\pi$
$997$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$998$	−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.125.2	✓	8
7.6	odd	2	CM	224.3.v.a.125.2	✓	8
32.21	even	8	inner	224.3.v.a.181.2	yes	8
224.181	odd	8	inner	224.3.v.a.181.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.a.125.2	✓	8	1.1	even	1	trivial
224.3.v.a.125.2	✓	8	7.6	odd	2	CM
224.3.v.a.181.2	yes	8	32.21	even	8	inner
224.3.v.a.181.2	yes	8	224.181	odd	8	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
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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