LMFDB - Embedded newform 224.3.v.a.69.1

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			69.1
Root			$-1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	224.69
Dual form			224.3.v.a.13.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.99607 - 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(-7.85905 - 1.49509i) q^{8} +(-6.36396 + 6.36396i) q^{9} +O(q^{10})$	\(q+(-1.99607 - 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(-7.85905 - 1.49509i) q^{8} +(-6.36396 + 6.36396i) q^{9} +(0.978370 - 2.36199i) q^{11} +(-9.26013 - 10.5000i) q^{14} +(15.5000 + 3.96863i) q^{16} +(13.5000 - 11.9059i) q^{18} +(-2.24873 + 4.59218i) q^{22} +(-30.1660 + 30.1660i) q^{23} +(17.6777 + 17.6777i) q^{25} +(17.1688 + 22.1186i) q^{28} +(15.9922 + 38.6086i) q^{29} +(-30.4421 - 9.86299i) q^{32} +(-28.4382 + 22.0742i) q^{36} +(40.7639 + 16.8850i) q^{37} +(-30.9438 + 74.7051i) q^{43} +(5.06378 - 8.88469i) q^{44} +(63.9918 - 56.4354i) q^{46} +49.0000i q^{49} +(-33.0719 - 37.5000i) q^{50} +(38.2953 - 92.4529i) q^{53} +(-31.5000 - 46.3006i) q^{56} +(-27.0860 - 79.0685i) q^{58} -63.0000 q^{63} +(59.5294 + 23.5000i) q^{64} +(5.16924 + 12.4797i) q^{67} +(-59.8665 - 59.8665i) q^{71} +(59.5294 - 40.5000i) q^{72} +(-79.2531 - 38.8092i) q^{74} +(16.5340 - 6.84859i) q^{77} -156.268i q^{79} -81.0000i q^{81} +(71.1227 - 145.241i) q^{86} +(-11.2205 + 17.1003i) q^{88} +(-134.801 + 104.635i) q^{92} +(6.13705 - 97.8077i) q^{98} +(8.80533 + 21.2579i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

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

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		−0.382683	−	0.923880i	$-0.625000\pi$
$3$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$4$	3.96863	+	0.500000i	0.992157	+	0.125000i
$5$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$5$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$6$		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
$11$	0.978370	−	2.36199i	0.0889427	−	0.214727i	−0.873149	−	0.487454i	$-0.837926\pi$
$11$	0.978370	−	2.36199i	0.0889427	−	0.214727i	0.962091	+	0.272727i	$0.0879257\pi$
$12$		0			0
$13$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$13$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$14$	−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		0.923880	−	0.382683i	$-0.125000\pi$
$19$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$20$		0			0
$21$		0			0
$22$	−2.24873	+	4.59218i	−0.102215	+	0.208735i
$23$	−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
$29$	15.9922	+	38.6086i	0.551455	+	1.33133i	0.916386	+	0.400295i	$0.131093\pi$
$29$	15.9922	+	38.6086i	0.551455	+	1.33133i	−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
$37$	40.7639	+	16.8850i	1.10173	+	0.456351i	0.858082	−	0.513514i	$-0.171656\pi$
$37$	40.7639	+	16.8850i	1.10173	+	0.456351i	0.243646	+	0.969864i	$0.421656\pi$
$38$		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
$43$	−30.9438	+	74.7051i	−0.719624	+	1.73733i	−0.0452058	+	0.998978i	$0.514394\pi$
$43$	−30.9438	+	74.7051i	−0.719624	+	1.73733i	−0.674419	+	0.738349i	$0.735606\pi$
$44$	5.06378	−	8.88469i	0.115086	−	0.201925i
$45$		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
$53$	38.2953	−	92.4529i	0.722552	−	1.74439i	0.0566038	−	0.998397i	$-0.481973\pi$
$53$	38.2953	−	92.4529i	0.722552	−	1.74439i	0.665948	−	0.745998i	$-0.268027\pi$
$54$		0			0
$55$		0			0
$56$	−31.5000	−	46.3006i	−0.562500	−	0.826797i
$57$		0			0
$58$	−27.0860	−	79.0685i	−0.467001	−	1.36325i
$59$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$59$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$60$		0			0
$61$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$61$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$62$		0			0
$63$	−63.0000			−1.00000
$64$	59.5294	+	23.5000i	0.930147	+	0.367188i
$65$		0			0
$66$		0			0
$67$	5.16924	+	12.4797i	0.0771529	+	0.186264i	0.957750	−	0.287602i	$-0.0928581\pi$
$67$	5.16924	+	12.4797i	0.0771529	+	0.186264i	−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	$-0.453334\pi$
$71$	−59.8665	−	59.8665i	−0.843190	−	0.843190i	−0.989272	+	0.146082i	$0.953334\pi$
$72$	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	$0.750000\pi$
$74$	−79.2531	−	38.8092i	−1.07099	−	0.524448i
$75$		0			0
$76$		0			0
$77$	16.5340	−	6.84859i	0.214727	−	0.0889427i
$78$		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		0.923880	−	0.382683i	$-0.125000\pi$
$83$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$		0			0
$86$	71.1227	−	145.241i	0.827008	−	1.68885i
$87$		0			0
$88$	−11.2205	+	17.1003i	−0.127505	+	0.194321i
$89$		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
$99$	8.80533	+	21.2579i	0.0889427	+	0.214727i
$100$	61.3172	+	78.9949i	0.613172	+	0.789949i
$101$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$101$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$102$		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
$106$	−88.0195	+	179.747i	−0.830373	+	1.69572i
$107$	−10.1775	+	24.5707i	−0.0951169	+	0.229632i	−0.964276	−	0.264901i	$-0.914661\pi$
$107$	−10.1775	+	24.5707i	−0.0951169	+	0.229632i	0.869159	+	0.494533i	$0.164661\pi$
$108$		0			0
$109$	157.826	−	65.3739i	1.44795	−	0.599760i	0.486239	−	0.873826i	$-0.338369\pi$
$109$	157.826	−	65.3739i	1.44795	−	0.599760i	0.961711	+	0.274066i	$0.0883686\pi$
$110$		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
$116$	44.1628	+	161.219i	0.380713	+	1.38982i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	80.9381	+	80.9381i	0.668910	+	0.668910i
$122$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$131$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$132$		0			0
$133$		0			0
$134$	−8.75517	−	25.5578i	−0.0653371	−	0.190730i
$135$		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		0.382683	−	0.923880i	$-0.375000\pi$
$139$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$140$		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
$148$	153.334	+	87.3921i	1.03604	+	0.590487i
$149$	109.556	−	264.490i	0.735272	−	1.77510i	0.111111	−	0.993808i	$-0.464559\pi$
$149$	109.556	−	264.490i	0.735272	−	1.77510i	0.624161	−	0.781296i	$-0.285441\pi$
$150$		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
$154$	−33.8608	+	11.5995i	−0.219875	+	0.0753214i
$155$		0			0
$156$		0			0
$157$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$157$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$158$	−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
$163$	20.3324	+	49.0868i	0.124739	+	0.301146i	0.973896	−	0.226994i	$-0.0728897\pi$
$163$	20.3324	+	49.0868i	0.124739	+	0.301146i	−0.849158	+	0.528140i	$0.822890\pi$
$164$		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
$172$	−160.157	+	281.005i	−0.931146	+	1.63375i
$173$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$173$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$174$		0			0
$175$			175.000i			1.00000i
$176$	24.5386	−	32.7281i	0.139424	−	0.185955i
$177$		0			0
$178$		0			0
$179$	211.247	−	87.5013i	1.18015	−	0.488834i	0.295615	−	0.955307i	$-0.404476\pi$
$179$	211.247	−	87.5013i	1.18015	−	0.488834i	0.884535	+	0.466473i	$0.154476\pi$
$180$		0			0
$181$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$181$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$182$		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	$-0.563069\pi$
$191$	−75.1937			−0.393684			−0.196842	+	0.980435i	$0.563069\pi$
$192$		0			0
$193$	−225.559			−1.16870			−0.584349	−	0.811502i	$-0.698650\pi$
$193$	−225.559			−1.16870			−0.584349	+	0.811502i	$0.698650\pi$
$194$		0			0
$195$		0			0
$196$	−24.5000	+	194.463i	−0.125000	+	0.992157i
$197$	135.160	+	55.9851i	0.686092	+	0.284189i	0.698371	−	0.715736i	$-0.253909\pi$
$197$	135.160	+	55.9851i	0.686092	+	0.284189i	−0.0122790	+	0.999925i	$0.503909\pi$
$198$	−14.9136	−	43.5353i	−0.0753214	−	0.219875i
$199$		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
$203$	−111.945	+	270.260i	−0.551455	+	1.33133i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	383.951i		−	1.85483i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	349.538	−	144.783i	1.65658	−	0.686176i	0.658768	−	0.752346i	$-0.271078\pi$
$211$	349.538	−	144.783i	1.65658	−	0.686176i	0.997808	+	0.0661700i	$0.0210780\pi$
$212$	198.206	−	347.764i	0.934934	−	1.64039i
$213$		0			0
$214$	23.3924	−	47.7702i	0.109310	−	0.223225i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−323.221	+	110.724i	−1.48267	+	0.507908i
$219$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$229$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$230$		0			0
$231$		0			0
$232$	−67.9601	−	327.337i	−0.292932	−	1.41093i
$233$	−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	$0.5\pi$
$242$	−151.421	−	171.696i	−0.625708	−	0.709486i
$243$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$251$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$252$	−250.023	−	31.5000i	−0.992157	−	0.125000i
$253$	41.7384	+	100.765i	0.164974	+	0.398282i
$254$	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
$259$	118.195	+	285.348i	0.456351	+	1.10173i
$260$		0			0
$261$	−347.477	−	143.930i	−1.33133	−	0.551455i
$262$		0			0
$263$	352.139	+	352.139i	1.33893	+	1.33893i	0.897095	+	0.441837i	$0.145673\pi$
$263$	352.139	+	352.139i	1.33893	+	1.33893i	0.441837	+	0.897095i	$0.354327\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	14.2750	+	52.1117i	0.0532648	+	0.194447i
$269$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$269$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$270$		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
$275$	59.0499	−	24.4593i	0.214727	−	0.0889427i
$276$		0			0
$277$	207.009	−	499.763i	0.747323	−	1.80420i	0.174237	−	0.984704i	$-0.444254\pi$
$277$	207.009	−	499.763i	0.747323	−	1.80420i	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	0.979210	+	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		−0.923880	−	0.382683i	$-0.875000\pi$
$283$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$284$	−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		−0.923880	−	0.382683i	$-0.875000\pi$
$293$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$294$		0			0
$295$		0			0
$296$	−295.121	−	193.646i	−0.997031	−	0.654208i
$297$		0			0
$298$	−251.807	+	514.221i	−0.844991	+	1.72557i
$299$		0			0
$300$		0			0
$301$	−522.935	+	216.607i	−1.73733	+	0.719624i
$302$	−155.921	+	137.510i	−0.516295	+	0.455330i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$307$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$308$	69.0414	−	18.9125i	0.224160	−	0.0614043i
$309$		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
$317$	237.250	+	572.771i	0.748421	+	1.80685i	0.567543	+	0.823344i	$0.307894\pi$
$317$	237.250	+	572.771i	0.748421	+	1.80685i	0.180879	+	0.983505i	$0.442106\pi$
$318$		0			0
$319$	106.840			0.334920
$320$		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
$326$	−34.4371	−	100.527i	−0.105635	−	0.308366i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−252.050	+	608.503i	−0.761481	+	1.83838i	−0.287755	+	0.957704i	$0.592909\pi$
$331$	−252.050	+	608.503i	−0.761481	+	1.83838i	−0.473725	+	0.880673i	$0.657091\pi$
$332$		0			0
$333$	−366.875	+	151.965i	−1.10173	+	0.456351i
$334$		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
$344$	354.880	−	540.847i	1.03163	−	1.57223i
$345$		0			0
$346$		0			0
$347$	631.092	+	261.407i	1.81871	+	0.753334i	0.976945	+	0.213490i	$0.0684831\pi$
$347$	631.092	+	261.407i	1.81871	+	0.753334i	0.841765	+	0.539844i	$0.181517\pi$
$348$		0			0
$349$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$349$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$350$	21.9180	−	349.313i	0.0626229	−	0.998037i
$351$		0			0
$352$	−53.0800	+	62.2544i	−0.150795	+	0.176859i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−432.624	+	148.201i	−1.20845	+	0.413970i
$359$	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
$371$	647.170	−	268.067i	1.74439	−	0.722552i
$372$		0			0
$373$	9.65880	−	23.3184i	0.0258949	−	0.0625158i	−0.910403	−	0.413722i	$-0.864228\pi$
$373$	9.65880	−	23.3184i	0.0258949	−	0.0625158i	0.936298	+	0.351206i	$0.114228\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−596.475	−	247.068i	−1.57381	−	0.651894i	−0.586393	−	0.810026i	$-0.699453\pi$
$379$	−596.475	−	247.068i	−1.57381	−	0.651894i	−0.987418	+	0.158132i	$0.949453\pi$
$380$		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
$387$	−278.495	−	672.346i	−0.719624	−	1.73733i
$388$		0			0
$389$	−578.727	−	239.716i	−1.48773	−	0.616238i	−0.516908	−	0.856041i	$-0.672917\pi$
$389$	−578.727	−	239.716i	−1.48773	−	0.616238i	−0.970821	+	0.239804i	$0.922917\pi$
$390$		0			0
$391$		0			0
$392$	73.2595	−	385.094i	0.186886	−	0.982382i
$393$		0			0
$394$	−262.778	−	128.679i	−0.666949	−	0.326596i
$395$		0			0
$396$	24.3161	+	88.7675i	0.0614043	+	0.224160i
$397$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$397$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$398$		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
$406$	257.300	−	525.438i	0.633744	−	1.29418i
$407$	79.7644	−	79.7644i	0.195981	−	0.195981i
$408$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$419$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$	433.539	+	179.578i	1.02978	+	0.426551i	0.832635	−	0.553823i	$-0.186832\pi$
$421$	433.539	+	179.578i	1.02978	+	0.426551i	0.197150	+	0.980373i	$0.436832\pi$
$422$	−715.837	+	245.220i	−1.69630	+	0.581090i
$423$		0			0
$424$	−439.190	+	669.337i	−1.03583	+	1.57863i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−52.6761	+	92.4231i	−0.123075	+	0.215942i
$429$		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
$436$	659.041	−	180.531i	1.51156	−	0.414063i
$437$		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
$443$	−460.494	−	190.743i	−1.03949	−	0.430571i	−0.203361	−	0.979104i	$-0.565187\pi$
$443$	−460.494	−	190.743i	−1.03949	−	0.430571i	−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	−0.482757	−	0.875754i	$-0.660365\pi$
$457$	179.600	−	179.600i	0.392997	−	0.392997i	0.875754	+	0.482757i	$0.160365\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$461$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$462$		0			0
$463$		−	27.5911i		−	0.0595920i	−0.999556	−	0.0297960i	$-0.990514\pi$
$463$		−	27.5911i		−	0.0595920i	0.999556	−	0.0297960i	$-0.00948576\pi$
$464$	94.6560	+	661.900i	0.204000	+	1.42651i
$465$		0			0
$466$	474.807	−	418.740i	1.01890	−	0.898584i
$467$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$		0			0
$469$	−36.1847	+	87.3576i	−0.0771529	+	0.186264i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	146.178	+	146.178i	0.309045	+	0.309045i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	344.657	+	832.076i	0.722552	+	1.74439i
$478$	−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
$484$	280.744	+	361.682i	0.580050	+	0.747277i
$485$		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
$491$	222.027	−	536.019i	0.452193	−	1.09169i	−0.519294	−	0.854596i	$-0.673805\pi$
$491$	222.027	−	536.019i	0.452193	−	1.09169i	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
$499$	−707.625	+	293.108i	−1.41809	+	0.587391i	−0.954379	−	0.298597i	$-0.903481\pi$
$499$	−707.625	+	293.108i	−1.41809	+	0.587391i	−0.463708	+	0.885988i	$0.653481\pi$
$500$		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
$506$	−70.6926	−	206.363i	−0.139709	−	0.407832i
$507$		0			0
$508$	−1008.00	−	126.996i	−1.98425	−	0.249992i
$509$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$509$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$510$		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
$518$	−200.187	−	584.378i	−0.386462	−	1.12814i
$519$		0			0
$520$		0			0
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$	675.564	+	330.815i	1.29418	+	0.633744i
$523$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$523$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$524$		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
$536$	−21.9671	−	105.807i	−0.0409835	−	0.197401i
$537$		0			0
$538$		0			0
$539$	115.738	+	47.9401i	0.214727	+	0.0889427i
$540$		0			0
$541$	−392.486	−	947.545i	−0.725483	−	1.75147i	−0.657091	−	0.753811i	$-0.728213\pi$
$541$	−392.486	−	947.545i	−0.725483	−	1.75147i	−0.0683919	−	0.997659i	$-0.521787\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−399.982	−	965.641i	−0.731228	−	1.76534i	−0.638463	−	0.769653i	$-0.720429\pi$
$547$	−399.982	−	965.641i	−0.731228	−	1.76534i	−0.0927652	−	0.995688i	$-0.529571\pi$
$548$	−84.1445	+	65.3144i	−0.153548	+	0.119187i
$549$		0			0
$550$	−120.931	+	41.4267i	−0.219875	+	0.0753214i
$551$		0			0
$552$		0			0
$553$	773.486	−	773.486i	1.39871	−	1.39871i
$554$	−475.798	+	971.637i	−0.858841	+	1.75386i
$555$		0			0
$556$		0			0
$557$	683.144	−	282.968i	1.22647	−	0.508021i	0.327009	−	0.945021i	$-0.393959\pi$
$557$	683.144	−	282.968i	1.22647	−	0.508021i	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		0.923880	−	0.382683i	$-0.125000\pi$
$563$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$564$		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	$0.0551217\pi$
$569$	658.532	+	658.532i	1.15735	+	1.15735i	0.172306	+	0.985044i	$0.444878\pi$
$570$		0			0
$571$	−893.751	−	370.204i	−1.56524	−	0.648343i	−0.579249	−	0.815151i	$-0.696654\pi$
$571$	−893.751	−	370.204i	−1.56524	−	0.648343i	−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
$583$	−180.906	−	180.906i	−0.310302	−	0.310302i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	564.831	+	423.494i	0.954106	+	0.715361i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	567.030	−	994.886i	0.951393	−	1.66927i
$597$		0			0
$598$		0			0
$599$	123.037	−	123.037i	0.205403	−	0.205403i	−0.596907	−	0.802310i	$-0.703604\pi$
$599$	123.037	−	123.037i	0.205403	−	0.205403i	0.802310	+	0.596907i	$0.203604\pi$
$600$		0			0
$601$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$601$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$602$	1070.95	−	366.868i	1.77898	−	0.609415i
$603$	−112.317	−	46.5232i	−0.186264	−	0.0771529i
$604$	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
$613$	−612.624	−	253.757i	−0.999386	−	0.413959i	−0.177814	−	0.984064i	$-0.556903\pi$
$613$	−612.624	−	253.757i	−0.999386	−	0.413959i	−0.821572	+	0.570105i	$0.806903\pi$
$614$		0			0
$615$		0			0
$616$	−140.181	+	29.1037i	−0.227566	+	0.0472462i
$617$	778.265	−	778.265i	1.26137	−	1.26137i	0.310939	−	0.950430i	$-0.399356\pi$
$617$	778.265	−	778.265i	1.26137	−	1.26137i	0.950430	−	0.310939i	$-0.100644\pi$
$618$		0			0
$619$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$619$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$620$		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	$-0.956341\pi$
$631$	−538.799	+	538.799i	−0.853881	+	0.853881i	0.136728	+	0.990609i	$0.456341\pi$
$632$	−233.635	+	1228.12i	−0.369675	+	1.94322i
$633$		0			0
$634$	−401.831	−	1173.01i	−0.633802	−	1.85017i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−213.260	−	13.3812i	−0.334263	−	0.0209737i
$639$	761.976			1.19245
$640$		0			0
$641$	98.7379			0.154037			0.0770186	−	0.997030i	$-0.475460\pi$
$641$	98.7379			0.154037			0.0770186	+	0.997030i	$0.475460\pi$
$642$		0			0
$643$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$643$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$644$	−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
$652$	56.1484	+	204.973i	0.0861171	+	0.314376i
$653$	−1206.23	+	499.635i	−1.84721	+	0.765138i	−0.914242	+	0.405169i	$0.867213\pi$
$653$	−1206.23	+	499.635i	−1.84721	+	0.765138i	−0.932964	+	0.359969i	$0.882787\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	115.914	−	48.0130i	0.175893	−	0.0728574i	−0.292999	−	0.956113i	$-0.594653\pi$
$659$	115.914	−	48.0130i	0.175893	−	0.0728574i	0.468892	+	0.883255i	$0.344653\pi$
$660$		0			0
$661$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$661$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$662$	579.323	−	1183.05i	0.875110	−	1.78708i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	751.344	−	257.383i	1.12814	−	0.386462i
$667$	−1647.09	−	682.246i	−2.46940	−	1.02286i
$668$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$677$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−98.6625	+	238.192i	−0.144455	+	0.348744i	−0.979502	−	0.201433i	$-0.935440\pi$
$683$	−98.6625	+	238.192i	−0.144455	+	0.348744i	0.835048	+	0.550178i	$0.185440\pi$
$684$		0			0
$685$		0			0
$686$	514.500	−	453.746i	0.750000	−	0.661438i
$687$		0			0
$688$	−776.106	+	1035.12i	−1.12806	+	1.50454i
$689$		0			0
$690$		0			0
$691$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$691$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$692$		0			0
$693$	−61.6373	+	148.806i	−0.0889427	+	0.214727i
$694$	−1226.97	−	600.830i	−1.76796	−	0.865749i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−87.5000	+	694.510i	−0.125000	+	0.992157i
$701$	242.149	+	584.599i	0.345434	+	0.833950i	0.997147	+	0.0754851i	$0.0240505\pi$
$701$	242.149	+	584.599i	0.345434	+	0.833950i	−0.651713	+	0.758465i	$0.725949\pi$
$702$		0			0
$703$		0			0
$704$	113.749	−	117.616i	0.161575	−	0.167069i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1291.86	+	535.106i	1.82209	+	0.754734i	0.974612	+	0.223900i	$0.0718789\pi$
$709$	1291.86	+	535.106i	1.82209	+	0.754734i	0.847476	+	0.530834i	$0.178121\pi$
$710$		0			0
$711$	994.482	+	994.482i	1.39871	+	1.39871i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	882.111	−	241.637i	1.23200	−	0.337481i
$717$		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
$725$	−399.805	+	965.214i	−0.551455	+	1.33133i
$726$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$		0			0
$736$	1215.84	−	620.790i	1.65196	−	0.843464i
$737$	34.5343			0.0468580
$738$		0			0
$739$	312.569	+	754.607i	0.422961	+	1.02112i	0.981469	+	0.191620i	$0.0613741\pi$
$739$	312.569	+	754.607i	0.422961	+	1.02112i	−0.558508	+	0.829499i	$0.688626\pi$
$740$		0			0
$741$		0			0
$742$	−1325.37	+	454.026i	−1.78622	+	0.611895i
$743$	−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
$746$	−22.2002	+	45.3355i	−0.0297590	+	0.0607715i
$747$		0			0
$748$		0			0
$749$	−171.995	+	71.2425i	−0.229632	+	0.0951169i
$750$		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
$757$	−407.368	+	983.472i	−0.538134	+	1.29917i	0.387890	+	0.921706i	$0.373204\pi$
$757$	−407.368	+	983.472i	−0.538134	+	1.29917i	−0.926024	+	0.377465i	$0.876796\pi$
$758$	1159.66	+	567.872i	1.52990	+	0.749171i
$759$		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
$763$	1104.79	+	457.617i	1.44795	+	0.599760i
$764$	−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		−0.923880	−	0.382683i	$-0.875000\pi$
$773$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$774$	471.688	+	1376.93i	0.609415	+	1.77898i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1125.16	+	550.975i	1.44622	+	0.708194i
$779$		0			0
$780$		0			0
$781$	−199.976	+	82.8328i	−0.256051	+	0.106060i
$782$		0			0
$783$		0			0
$784$	−194.463	+	759.500i	−0.248039	+	0.968750i
$785$		0			0
$786$		0			0
$787$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$787$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$788$	508.407	+	289.764i	0.645187	+	0.367721i
$789$		0			0
$790$		0			0
$791$	−628.838	+	628.838i	−0.794991	+	0.794991i
$792$	−37.4190	−	180.232i	−0.0472462	−	0.227566i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$797$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$798$		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	$0.715703\pi$
$809$	−1137.46	+	1137.46i	−1.40601	+	1.40601i	−0.779048	+	0.626964i	$0.784297\pi$
$810$		0			0
$811$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$811$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$812$	−579.399	+	1016.59i	−0.713546	+	1.25196i
$813$		0			0
$814$	−169.206	+	149.226i	−0.207870	+	0.183324i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	579.898	−	1400.00i	0.706331	−	1.70523i	−0.00264006	−	0.999997i	$-0.500840\pi$
$821$	579.898	−	1400.00i	0.706331	−	1.70523i	0.708971	−	0.705238i	$-0.249160\pi$
$822$		0			0
$823$	−439.820	+	439.820i	−0.534411	+	0.534411i	−0.921882	−	0.387471i	$-0.873349\pi$
$823$	−439.820	+	439.820i	−0.534411	+	0.534411i	0.387471	+	0.921882i	$0.373349\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1291.40	+	534.917i	1.56155	+	0.646817i	0.985358	−	0.170496i	$-0.0545369\pi$
$827$	1291.40	+	534.917i	1.56155	+	0.646817i	0.576195	+	0.817312i	$0.304537\pi$
$828$	191.975	−	1523.76i	0.231854	−	1.84029i
$829$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$829$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$830$		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
$841$	−640.195	+	640.195i	−0.761230	+	0.761230i
$842$	−842.885	−	412.750i	−1.00105	−	0.490201i
$843$		0			0
$844$	1459.58	−	399.822i	1.72936	−	0.473722i
$845$		0			0
$846$		0			0
$847$			801.246i			0.945982i
$848$	960.488	−	1281.04i	1.13265	−	1.51066i
$849$		0			0
$850$		0			0
$851$	−1739.04	+	720.333i	−2.04352	+	0.846455i
$852$		0			0
$853$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$853$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$854$		0			0
$855$		0			0
$856$	116.721	−	177.886i	0.136356	−	0.207811i
$857$		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		−0.923880	−	0.382683i	$-0.875000\pi$
$859$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$860$		0			0
$861$		0			0
$862$	20.2898	−	323.364i	0.0235381	−	0.375132i
$863$	1504.48			1.74331			0.871655	−	0.490119i	$-0.163047\pi$
$863$	1504.48			1.74331			0.871655	+	0.490119i	$0.163047\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−369.104	−	152.888i	−0.424745	−	0.175935i
$870$		0			0
$871$		0			0
$872$	−1338.11	+	277.812i	−1.53453	+	0.318591i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1618.72	−	670.498i	1.84575	−	0.764536i	0.905046	−	0.425314i	$-0.139836\pi$
$877$	1618.72	−	670.498i	1.84575	−	0.764536i	0.940706	−	0.339222i	$-0.110164\pi$
$878$		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
$883$	−1137.51	+	471.174i	−1.28824	+	0.533606i	−0.918460	−	0.395514i	$-0.870566\pi$
$883$	−1137.51	+	471.174i	−1.28824	+	0.533606i	−0.369778	+	0.929120i	$0.620566\pi$
$884$		0			0
$885$		0			0
$886$	895.291	+	438.412i	1.01049	+	0.494822i
$887$		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
$891$	−191.322	−	79.2480i	−0.214727	−	0.0889427i
$892$		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
$907$	584.951	−	1412.20i	0.644929	−	1.55700i	−0.175022	−	0.984564i	$-0.556000\pi$
$907$	584.951	−	1412.20i	0.644929	−	1.55700i	0.819952	−	0.572433i	$-0.194000\pi$
$908$		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	$-0.331594\pi$
$919$	1257.20	−	1257.20i	1.36801	−	1.36801i	0.863280	−	0.504725i	$-0.168406\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	422.124	+	1019.10i	0.456351	+	1.10173i
$926$	−3.45567	+	55.0739i	−0.00373182	+	0.0594750i
$927$		0			0
$928$	−106.040	−	1333.06i	−0.114267	−	1.43648i
$929$		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	$0.750000\pi$
$938$	83.1686	−	169.840i	0.0886658	−	0.181066i
$939$		0			0
$940$		0			0
$941$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$941$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−273.475	−	310.091i	−0.289085	−	0.327792i
$947$	1686.79	−	698.692i	1.78119	−	0.737795i	0.788807	−	0.614641i	$-0.210699\pi$
$947$	1686.79	−	698.692i	1.78119	−	0.737795i	0.992388	−	0.123154i	$-0.0393008\pi$
$948$		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	$-0.722762\pi$
$954$	−583.748	−	1704.05i	−0.611895	−	1.78622i
$955$		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
$963$	−91.5975	−	221.136i	−0.0951169	−	0.229632i
$964$		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	$0.944751\pi$
$968$	−515.087	−	757.107i	−0.532115	−	0.782135i
$969$		0			0
$970$		0			0
$971$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$971$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$972$		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
$981$	−588.365	+	1420.44i	−0.599760	+	1.44795i
$982$	−510.316	+	1042.13i	−0.519670	+	1.06123i
$983$		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
$989$	−1320.10	−	3187.01i	−1.33478	−	3.22245i
$990$		0			0
$991$	−6.39458			−0.00645266			−0.00322633	−	0.999995i	$-0.501027\pi$
$991$	−6.39458			−0.00645266			−0.00322633	+	0.999995i	$0.501027\pi$
$992$		0			0
$993$		0			0
$994$	−74.2267	+	1182.97i	−0.0746748	+	1.19011i
$995$		0			0
$996$		0			0
$997$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$997$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$998$	1449.18	−	496.438i	1.45209	−	0.497433i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.v.a.69.1	yes	8
7.6	odd	2	CM	224.3.v.a.69.1	yes	8
32.13	even	8	inner	224.3.v.a.13.1	✓	8
224.13	odd	8	inner	224.3.v.a.13.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.v.a.13.1	✓	8	32.13	even	8	inner
224.3.v.a.13.1	✓	8	224.13	odd	8	inner
224.3.v.a.69.1	yes	8	1.1	even	1	trivial
224.3.v.a.69.1	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)

\(f(q)\)	\(=\)	\(q+(-1.99607 - 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(-7.85905 - 1.49509i) q^{8} +(-6.36396 + 6.36396i) q^{9} +O(q^{10})\)	\(q+(-1.99607 - 0.125246i) q^{2} +(3.96863 + 0.500000i) q^{4} +(4.94975 + 4.94975i) q^{7} +(-7.85905 - 1.49509i) q^{8} +(-6.36396 + 6.36396i) q^{9} +(0.978370 - 2.36199i) q^{11} +(-9.26013 - 10.5000i) q^{14} +(15.5000 + 3.96863i) q^{16} +(13.5000 - 11.9059i) q^{18} +(-2.24873 + 4.59218i) q^{22} +(-30.1660 + 30.1660i) q^{23} +(17.6777 + 17.6777i) q^{25} +(17.1688 + 22.1186i) q^{28} +(15.9922 + 38.6086i) q^{29} +(-30.4421 - 9.86299i) q^{32} +(-28.4382 + 22.0742i) q^{36} +(40.7639 + 16.8850i) q^{37} +(-30.9438 + 74.7051i) q^{43} +(5.06378 - 8.88469i) q^{44} +(63.9918 - 56.4354i) q^{46} +49.0000i q^{49} +(-33.0719 - 37.5000i) q^{50} +(38.2953 - 92.4529i) q^{53} +(-31.5000 - 46.3006i) q^{56} +(-27.0860 - 79.0685i) q^{58} -63.0000 q^{63} +(59.5294 + 23.5000i) q^{64} +(5.16924 + 12.4797i) q^{67} +(-59.8665 - 59.8665i) q^{71} +(59.5294 - 40.5000i) q^{72} +(-79.2531 - 38.8092i) q^{74} +(16.5340 - 6.84859i) q^{77} -156.268i q^{79} -81.0000i q^{81} +(71.1227 - 145.241i) q^{86} +(-11.2205 + 17.1003i) q^{88} +(-134.801 + 104.635i) q^{92} +(6.13705 - 97.8077i) q^{98} +(8.80533 + 21.2579i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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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