LMFDB - Embedded newform 448.3.l.b.209.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(209,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("448.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			209.10
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.75534 + 1.75534i) q^{3} +(-3.25602 - 3.25602i) q^{5} +(-4.90539 + 4.99371i) q^{7} +2.83754i q^{9} +O(q^{10})$	\(q+(-1.75534 + 1.75534i) q^{3} +(-3.25602 - 3.25602i) q^{5} +(-4.90539 + 4.99371i) q^{7} +2.83754i q^{9} +(2.01178 - 2.01178i) q^{11} +(-6.07310 + 6.07310i) q^{13} +11.4309 q^{15} -9.55815i q^{17} +(25.2086 - 25.2086i) q^{19} +(-0.155043 - 17.3763i) q^{21} +14.0394i q^{23} -3.79671i q^{25} +(-20.7789 - 20.7789i) q^{27} +(-11.2161 - 11.2161i) q^{29} -16.5180i q^{31} +7.06271i q^{33} +(32.2316 - 0.287592i) q^{35} +(27.6654 - 27.6654i) q^{37} -21.3208i q^{39} -12.9650 q^{41} +(50.9760 - 50.9760i) q^{43} +(9.23909 - 9.23909i) q^{45} +66.5020i q^{47} +(-0.874351 - 48.9922i) q^{49} +(16.7778 + 16.7778i) q^{51} +(39.3660 - 39.3660i) q^{53} -13.1008 q^{55} +88.4994i q^{57} +(-30.4313 - 30.4313i) q^{59} +(-11.9950 + 11.9950i) q^{61} +(-14.1699 - 13.9192i) q^{63} +39.5482 q^{65} +(19.5294 + 19.5294i) q^{67} +(-24.6440 - 24.6440i) q^{69} -69.7628i q^{71} +143.326 q^{73} +(6.66453 + 6.66453i) q^{75} +(0.177693 + 19.9148i) q^{77} -66.7548 q^{79} +47.4105 q^{81} +(-60.7490 + 60.7490i) q^{83} +(-31.1215 + 31.1215i) q^{85} +39.3763 q^{87} +78.9394 q^{89} +(-0.536415 - 60.1182i) q^{91} +(28.9947 + 28.9947i) q^{93} -164.159 q^{95} +35.4878i q^{97} +(5.70850 + 5.70850i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−1.75534	+	1.75534i	−0.585114	+	0.585114i	−0.936304	−	0.351190i	$-0.885777\pi$
$3$	−1.75534	+	1.75534i	−0.585114	+	0.585114i	0.351190	+	0.936304i	$0.385777\pi$
$4$		0			0
$5$	−3.25602	−	3.25602i	−0.651203	−	0.651203i	0.302079	−	0.953283i	$-0.402319\pi$
$5$	−3.25602	−	3.25602i	−0.651203	−	0.651203i	−0.953283	+	0.302079i	$0.902319\pi$
$6$		0			0
$7$	−4.90539	+	4.99371i	−0.700770	+	0.713388i
$7$	−4.90539	+	4.99371i	−0.700770	+	0.713388i
$8$		0			0
$9$			2.83754i			0.315283i
$10$		0			0
$11$	2.01178	−	2.01178i	0.182889	−	0.182889i	−0.609725	−	0.792613i	$-0.708720\pi$
$11$	2.01178	−	2.01178i	0.182889	−	0.182889i	0.792613	+	0.609725i	$0.208720\pi$
$12$		0			0
$13$	−6.07310	+	6.07310i	−0.467162	+	0.467162i	−0.900994	−	0.433832i	$-0.857161\pi$
$13$	−6.07310	+	6.07310i	−0.467162	+	0.467162i	0.433832	+	0.900994i	$0.357161\pi$
$14$		0			0
$15$	11.4309			0.762057
$16$		0			0
$17$		−	9.55815i		−	0.562244i	−0.959672	−	0.281122i	$-0.909293\pi$
$17$		−	9.55815i		−	0.562244i	0.959672	−	0.281122i	$-0.0907065\pi$
$18$		0			0
$19$	25.2086	−	25.2086i	1.32677	−	1.32677i	0.418595	−	0.908173i	$-0.362523\pi$
$19$	25.2086	−	25.2086i	1.32677	−	1.32677i	0.908173	−	0.418595i	$-0.137477\pi$
$20$		0			0
$21$	−0.155043	−	17.3763i	−0.00738300	−	0.827444i
$22$		0			0
$23$			14.0394i			0.610408i	0.952287	+	0.305204i	$0.0987247\pi$
$23$			14.0394i			0.610408i	−0.952287	+	0.305204i	$0.901275\pi$
$24$		0			0
$25$		−	3.79671i		−	0.151869i
$26$		0			0
$27$	−20.7789	−	20.7789i	−0.769591	−	0.769591i
$28$		0			0
$29$	−11.2161	−	11.2161i	−0.386764	−	0.386764i	0.486768	−	0.873531i	$-0.338176\pi$
$29$	−11.2161	−	11.2161i	−0.386764	−	0.386764i	−0.873531	+	0.486768i	$0.838176\pi$
$30$		0			0
$31$		−	16.5180i		−	0.532837i	−0.963857	−	0.266419i	$-0.914160\pi$
$31$		−	16.5180i		−	0.532837i	0.963857	−	0.266419i	$-0.0858404\pi$
$32$		0			0
$33$			7.06271i			0.214022i
$34$		0			0
$35$	32.2316	−	0.287592i	0.920904	−	0.00821691i
$36$		0			0
$37$	27.6654	−	27.6654i	0.747714	−	0.747714i	−0.226335	−	0.974049i	$-0.572675\pi$
$37$	27.6654	−	27.6654i	0.747714	−	0.747714i	0.974049	+	0.226335i	$0.0726746\pi$
$38$		0			0
$39$		−	21.3208i		−	0.546686i
$40$		0			0
$41$	−12.9650			−0.316219			−0.158110	−	0.987422i	$-0.550540\pi$
$41$	−12.9650			−0.316219			−0.158110	+	0.987422i	$0.550540\pi$
$42$		0			0
$43$	50.9760	−	50.9760i	1.18549	−	1.18549i	0.207187	−	0.978301i	$-0.433569\pi$
$43$	50.9760	−	50.9760i	1.18549	−	1.18549i	0.978301	−	0.207187i	$-0.0664309\pi$
$44$		0			0
$45$	9.23909	−	9.23909i	0.205313	−	0.205313i
$46$		0			0
$47$			66.5020i			1.41494i	0.706745	+	0.707468i	$0.250163\pi$
$47$			66.5020i			1.41494i	−0.706745	+	0.707468i	$0.749837\pi$
$48$		0			0
$49$	−0.874351	−	48.9922i	−0.0178439	−	0.999841i
$50$		0			0
$51$	16.7778	+	16.7778i	0.328977	+	0.328977i
$52$		0			0
$53$	39.3660	−	39.3660i	0.742755	−	0.742755i	−0.230353	−	0.973107i	$-0.573988\pi$
$53$	39.3660	−	39.3660i	0.742755	−	0.742755i	0.973107	+	0.230353i	$0.0739880\pi$
$54$		0			0
$55$	−13.1008			−0.238196
$56$		0			0
$57$			88.4994i			1.55262i
$58$		0			0
$59$	−30.4313	−	30.4313i	−0.515785	−	0.515785i	0.400508	−	0.916293i	$-0.368834\pi$
$59$	−30.4313	−	30.4313i	−0.515785	−	0.515785i	−0.916293	+	0.400508i	$0.868834\pi$
$60$		0			0
$61$	−11.9950	+	11.9950i	−0.196639	+	0.196639i	−0.798558	−	0.601918i	$-0.794403\pi$
$61$	−11.9950	+	11.9950i	−0.196639	+	0.196639i	0.601918	+	0.798558i	$0.294403\pi$
$62$		0			0
$63$	−14.1699	−	13.9192i	−0.224919	−	0.220940i
$64$		0			0
$65$	39.5482			0.608434
$66$		0			0
$67$	19.5294	+	19.5294i	0.291483	+	0.291483i	0.837666	−	0.546183i	$-0.183920\pi$
$67$	19.5294	+	19.5294i	0.291483	+	0.291483i	−0.546183	+	0.837666i	$0.683920\pi$
$68$		0			0
$69$	−24.6440	−	24.6440i	−0.357159	−	0.357159i
$70$		0			0
$71$		−	69.7628i		−	0.982574i	−0.870998	−	0.491287i	$-0.836527\pi$
$71$		−	69.7628i		−	0.982574i	0.870998	−	0.491287i	$-0.163473\pi$
$72$		0			0
$73$	143.326			1.96337			0.981686	−	0.190507i	$-0.0610132\pi$
$73$	143.326			1.96337			0.981686	+	0.190507i	$0.0610132\pi$
$74$		0			0
$75$	6.66453	+	6.66453i	0.0888604	+	0.0888604i
$76$		0			0
$77$	0.177693	+	19.9148i	0.00230770	+	0.258633i
$78$		0			0
$79$	−66.7548			−0.844998			−0.422499	−	0.906363i	$-0.638847\pi$
$79$	−66.7548			−0.844998			−0.422499	+	0.906363i	$0.638847\pi$
$80$		0			0
$81$	47.4105			0.585314
$82$		0			0
$83$	−60.7490	+	60.7490i	−0.731916	+	0.731916i	−0.970999	−	0.239083i	$-0.923153\pi$
$83$	−60.7490	+	60.7490i	−0.731916	+	0.731916i	0.239083	+	0.970999i	$0.423153\pi$
$84$		0			0
$85$	−31.1215	+	31.1215i	−0.366135	+	0.366135i
$86$		0			0
$87$	39.3763			0.452602
$88$		0			0
$89$	78.9394			0.886960			0.443480	−	0.896284i	$-0.353744\pi$
$89$	78.9394			0.886960			0.443480	+	0.896284i	$0.353744\pi$
$90$		0			0
$91$	−0.536415	−	60.1182i	−0.00589467	−	0.660640i
$92$		0			0
$93$	28.9947	+	28.9947i	0.311771	+	0.311771i
$94$		0			0
$95$	−164.159			−1.72799
$96$		0			0
$97$			35.4878i			0.365854i	0.983126	+	0.182927i	$0.0585572\pi$
$97$			35.4878i			0.365854i	−0.983126	+	0.182927i	$0.941443\pi$
$98$		0			0
$99$	5.70850	+	5.70850i	0.0576616	+	0.0576616i
$100$		0			0
$101$	−135.312	−	135.312i	−1.33972	−	1.33972i	−0.896324	−	0.443400i	$-0.853772\pi$
$101$	−135.312	−	135.312i	−1.33972	−	1.33972i	−0.443400	−	0.896324i	$-0.646228\pi$
$102$		0			0
$103$	−76.3602			−0.741361			−0.370680	−	0.928760i	$-0.620875\pi$
$103$	−76.3602			−0.741361			−0.370680	+	0.928760i	$0.620875\pi$
$104$		0			0
$105$	−56.0727	+	57.0824i	−0.534026	+	0.543642i
$106$		0			0
$107$	−61.8636	+	61.8636i	−0.578164	+	0.578164i	−0.934397	−	0.356233i	$-0.884061\pi$
$107$	−61.8636	+	61.8636i	−0.578164	+	0.578164i	0.356233	+	0.934397i	$0.384061\pi$
$108$		0			0
$109$	−95.4954	−	95.4954i	−0.876105	−	0.876105i	0.117024	−	0.993129i	$-0.462664\pi$
$109$	−95.4954	−	95.4954i	−0.876105	−	0.876105i	−0.993129	+	0.117024i	$0.962664\pi$
$110$		0			0
$111$			97.1246i			0.874996i
$112$		0			0
$113$	−141.353			−1.25091			−0.625456	−	0.780260i	$-0.715087\pi$
$113$	−141.353			−1.25091			−0.625456	+	0.780260i	$0.715087\pi$
$114$		0			0
$115$	45.7125	−	45.7125i	0.397500	−	0.397500i
$116$		0			0
$117$	−17.2327	−	17.2327i	−0.147288	−	0.147288i
$118$		0			0
$119$	47.7306	+	46.8864i	0.401098	+	0.394003i
$120$		0			0
$121$			112.906i			0.933103i
$122$		0			0
$123$	22.7580	−	22.7580i	0.185024	−	0.185024i
$124$		0			0
$125$	−93.7626	+	93.7626i	−0.750101	+	0.750101i
$126$		0			0
$127$	80.1444			0.631058			0.315529	−	0.948916i	$-0.397818\pi$
$127$	80.1444			0.631058			0.315529	+	0.948916i	$0.397818\pi$
$128$		0			0
$129$			178.961i			1.38729i
$130$		0			0
$131$	90.0359	−	90.0359i	0.687297	−	0.687297i	−0.274337	−	0.961634i	$-0.588458\pi$
$131$	90.0359	−	90.0359i	0.687297	−	0.687297i	0.961634	+	0.274337i	$0.0884582\pi$
$132$		0			0
$133$	2.22658	+	249.542i	0.0167412	+	1.87626i
$134$		0			0
$135$			135.313i			1.00232i
$136$		0			0
$137$		−	253.079i		−	1.84729i	−0.383247	−	0.923646i	$-0.625194\pi$
$137$		−	253.079i		−	1.84729i	0.383247	−	0.923646i	$-0.374806\pi$
$138$		0			0
$139$	49.2592	+	49.2592i	0.354382	+	0.354382i	0.861737	−	0.507355i	$-0.169377\pi$
$139$	49.2592	+	49.2592i	0.354382	+	0.354382i	−0.507355	+	0.861737i	$0.669377\pi$
$140$		0			0
$141$	−116.734	−	116.734i	−0.827899	−	0.827899i
$142$		0			0
$143$			24.4354i			0.170877i
$144$		0			0
$145$			73.0399i			0.503723i
$146$		0			0
$147$	87.5329	+	84.4633i	0.595462	+	0.574580i
$148$		0			0
$149$	40.4528	−	40.4528i	0.271495	−	0.271495i	−0.558207	−	0.829702i	$-0.688510\pi$
$149$	40.4528	−	40.4528i	0.271495	−	0.271495i	0.829702	+	0.558207i	$0.188510\pi$
$150$		0			0
$151$		−	34.5523i		−	0.228823i	−0.993433	−	0.114412i	$-0.963502\pi$
$151$		−	34.5523i		−	0.228823i	0.993433	−	0.114412i	$-0.0364983\pi$
$152$		0			0
$153$	27.1217			0.177266
$154$		0			0
$155$	−53.7828	+	53.7828i	−0.346986	+	0.346986i
$156$		0			0
$157$	−17.2009	+	17.2009i	−0.109560	+	0.109560i	−0.759762	−	0.650202i	$-0.774684\pi$
$157$	−17.2009	+	17.2009i	−0.109560	+	0.109560i	0.650202	+	0.759762i	$0.274684\pi$
$158$		0			0
$159$			138.202i			0.869193i
$160$		0			0
$161$	−70.1087	−	68.8687i	−0.435458	−	0.427756i
$162$		0			0
$163$	38.9774	+	38.9774i	0.239125	+	0.239125i	0.816488	−	0.577363i	$-0.195918\pi$
$163$	38.9774	+	38.9774i	0.239125	+	0.239125i	−0.577363	+	0.816488i	$0.695918\pi$
$164$		0			0
$165$	22.9963	−	22.9963i	0.139372	−	0.139372i
$166$		0			0
$167$	137.053			0.820679			0.410339	−	0.911933i	$-0.365410\pi$
$167$	137.053			0.820679			0.410339	+	0.911933i	$0.365410\pi$
$168$		0			0
$169$			95.2349i			0.563520i
$170$		0			0
$171$	71.5305	+	71.5305i	0.418307	+	0.418307i
$172$		0			0
$173$	−59.0992	+	59.0992i	−0.341614	+	0.341614i	−0.856974	−	0.515360i	$-0.827658\pi$
$173$	−59.0992	+	59.0992i	−0.341614	+	0.341614i	0.515360	+	0.856974i	$0.327658\pi$
$174$		0			0
$175$	18.9597	+	18.6243i	0.108341	+	0.106425i
$176$		0			0
$177$	106.835			0.603586
$178$		0			0
$179$	−150.604	−	150.604i	−0.841364	−	0.841364i	0.147673	−	0.989036i	$-0.452822\pi$
$179$	−150.604	−	150.604i	−0.841364	−	0.841364i	−0.989036	+	0.147673i	$0.952822\pi$
$180$		0			0
$181$	195.403	+	195.403i	1.07958	+	1.07958i	0.996547	+	0.0830294i	$0.0264595\pi$
$181$	195.403	+	195.403i	1.07958	+	1.07958i	0.0830294	+	0.996547i	$0.473540\pi$
$182$		0			0
$183$		−	42.1106i		−	0.230113i
$184$		0			0
$185$	−180.158			−0.973828
$186$		0			0
$187$	−19.2288	−	19.2288i	−0.102828	−	0.102828i
$188$		0			0
$189$	205.693	−	1.83533i	1.08832	−	0.00971073i
$190$		0			0
$191$	−59.1682			−0.309781			−0.154891	−	0.987932i	$-0.549503\pi$
$191$	−59.1682			−0.309781			−0.154891	+	0.987932i	$0.549503\pi$
$192$		0			0
$193$	−190.929			−0.989269			−0.494634	−	0.869101i	$-0.664698\pi$
$193$	−190.929			−0.989269			−0.494634	+	0.869101i	$0.664698\pi$
$194$		0			0
$195$	−69.4207	+	69.4207i	−0.356004	+	0.356004i
$196$		0			0
$197$	142.902	−	142.902i	0.725390	−	0.725390i	−0.244308	−	0.969698i	$-0.578561\pi$
$197$	142.902	−	142.902i	0.725390	−	0.725390i	0.969698	+	0.244308i	$0.0785608\pi$
$198$		0			0
$199$	138.990			0.698440			0.349220	−	0.937041i	$-0.386447\pi$
$199$	138.990			0.698440			0.349220	+	0.937041i	$0.386447\pi$
$200$		0			0
$201$	−68.5614			−0.341102
$202$		0			0
$203$	111.030	−	0.990681i	0.546944	−	0.00488020i
$204$		0			0
$205$	42.2142	+	42.2142i	0.205923	+	0.205923i
$206$		0			0
$207$	−39.8374			−0.192451
$208$		0			0
$209$		−	101.428i		−	0.485302i
$210$		0			0
$211$	−212.666	−	212.666i	−1.00789	−	1.00789i	−0.999969	−	0.00792508i	$-0.997477\pi$
$211$	−212.666	−	212.666i	−1.00789	−	1.00789i	−0.00792508	−	0.999969i	$-0.502523\pi$
$212$		0			0
$213$	122.458	+	122.458i	0.574918	+	0.574918i
$214$		0			0
$215$	−331.957			−1.54399
$216$		0			0
$217$	82.4860	+	81.0270i	0.380120	+	0.373396i
$218$		0			0
$219$	−251.586	+	251.586i	−1.14880	+	1.14880i
$220$		0			0
$221$	58.0476	+	58.0476i	0.262659	+	0.262659i
$222$		0			0
$223$		−	346.003i		−	1.55158i	−0.630988	−	0.775792i	$-0.717350\pi$
$223$		−	346.003i		−	1.55158i	0.630988	−	0.775792i	$-0.282650\pi$
$224$		0			0
$225$	10.7733			0.0478815
$226$		0			0
$227$	−189.005	+	189.005i	−0.832622	+	0.832622i	−0.987875	−	0.155252i	$-0.950381\pi$
$227$	−189.005	+	189.005i	−0.832622	+	0.832622i	0.155252	+	0.987875i	$0.450381\pi$
$228$		0			0
$229$	−151.137	−	151.137i	−0.659987	−	0.659987i	0.295390	−	0.955377i	$-0.404550\pi$
$229$	−151.137	−	151.137i	−0.659987	−	0.659987i	−0.955377	+	0.295390i	$0.904550\pi$
$230$		0			0
$231$	−35.2692	−	34.6453i	−0.152680	−	0.149980i
$232$		0			0
$233$		−	10.5086i		−	0.0451015i	−0.999746	−	0.0225507i	$-0.992821\pi$
$233$		−	10.5086i		−	0.0451015i	0.999746	−	0.0225507i	$-0.00717873\pi$
$234$		0			0
$235$	216.532	−	216.532i	0.921411	−	0.921411i
$236$		0			0
$237$	117.178	−	117.178i	0.494420	−	0.494420i
$238$		0			0
$239$	362.334			1.51604			0.758021	−	0.652230i	$-0.226166\pi$
$239$	362.334			1.51604			0.758021	+	0.652230i	$0.226166\pi$
$240$		0			0
$241$		−	363.356i		−	1.50770i	−0.657046	−	0.753851i	$-0.728194\pi$
$241$		−	363.356i		−	1.50770i	0.657046	−	0.753851i	$-0.271806\pi$
$242$		0			0
$243$	103.789	−	103.789i	0.427115	−	0.427115i
$244$		0			0
$245$	−156.673	+	162.366i	−0.639480	+	0.662720i
$246$		0			0
$247$			306.189i			1.23963i
$248$		0			0
$249$		−	213.271i		−	0.856509i
$250$		0			0
$251$	54.1542	+	54.1542i	0.215754	+	0.215754i	0.806706	−	0.590952i	$-0.201248\pi$
$251$	54.1542	+	54.1542i	0.215754	+	0.215754i	−0.590952	+	0.806706i	$0.701248\pi$
$252$		0			0
$253$	28.2441	+	28.2441i	0.111637	+	0.111637i
$254$		0			0
$255$		−	109.258i		−	0.428462i
$256$		0			0
$257$			45.3701i			0.176537i	0.996097	+	0.0882687i	$0.0281334\pi$
$257$			45.3701i			0.176537i	−0.996097	+	0.0882687i	$0.971867\pi$
$258$		0			0
$259$	2.44358	+	273.863i	0.00943469	+	1.05739i
$260$		0			0
$261$	31.8263	−	31.8263i	0.121940	−	0.121940i
$262$		0			0
$263$		−	502.464i		−	1.91051i	−0.295787	−	0.955254i	$-0.595582\pi$
$263$		−	502.464i		−	1.91051i	0.295787	−	0.955254i	$-0.404418\pi$
$264$		0			0
$265$	−256.353			−0.967368
$266$		0			0
$267$	−138.566	+	138.566i	−0.518973	+	0.518973i
$268$		0			0
$269$	81.3588	−	81.3588i	0.302449	−	0.302449i	−0.539522	−	0.841971i	$-0.681395\pi$
$269$	81.3588	−	81.3588i	0.302449	−	0.302449i	0.841971	+	0.539522i	$0.181395\pi$
$270$		0			0
$271$		−	192.959i		−	0.712026i	−0.934481	−	0.356013i	$-0.884136\pi$
$271$		−	192.959i		−	0.712026i	0.934481	−	0.356013i	$-0.115864\pi$
$272$		0			0
$273$	106.470	+	104.587i	0.389999	+	0.383101i
$274$		0			0
$275$	−7.63814	−	7.63814i	−0.0277750	−	0.0277750i
$276$		0			0
$277$	−211.837	+	211.837i	−0.764756	+	0.764756i	−0.977178	−	0.212422i	$-0.931865\pi$
$277$	−211.837	+	211.837i	−0.764756	+	0.764756i	0.212422	+	0.977178i	$0.431865\pi$
$278$		0			0
$279$	46.8704			0.167994
$280$		0			0
$281$		−	85.3556i		−	0.303756i	−0.988399	−	0.151878i	$-0.951468\pi$
$281$		−	85.3556i		−	0.303756i	0.988399	−	0.151878i	$-0.0485322\pi$
$282$		0			0
$283$	−87.4756	−	87.4756i	−0.309101	−	0.309101i	0.535460	−	0.844561i	$-0.320138\pi$
$283$	−87.4756	−	87.4756i	−0.309101	−	0.309101i	−0.844561	+	0.535460i	$0.820138\pi$
$284$		0			0
$285$	288.156	−	288.156i	1.01107	−	1.01107i
$286$		0			0
$287$	63.5983	−	64.7435i	0.221597	−	0.225587i
$288$		0			0
$289$	197.642			0.683882
$290$		0			0
$291$	−62.2933	−	62.2933i	−0.214066	−	0.214066i
$292$		0			0
$293$	−30.7055	−	30.7055i	−0.104797	−	0.104797i	0.652764	−	0.757561i	$-0.273609\pi$
$293$	−30.7055	−	30.7055i	−0.104797	−	0.104797i	−0.757561	+	0.652764i	$0.773609\pi$
$294$		0			0
$295$			198.170i			0.671762i
$296$		0			0
$297$	−83.6052			−0.281499
$298$		0			0
$299$	−85.2627	−	85.2627i	−0.285159	−	0.285159i
$300$		0			0
$301$	4.50252	+	504.617i	0.0149585	+	1.67647i
$302$		0			0
$303$	475.038			1.56778
$304$		0			0
$305$	78.1117			0.256104
$306$		0			0
$307$	31.6433	−	31.6433i	0.103073	−	0.103073i	−0.653690	−	0.756763i	$-0.726780\pi$
$307$	31.6433	−	31.6433i	0.103073	−	0.103073i	0.756763	+	0.653690i	$0.226780\pi$
$308$		0			0
$309$	134.038	−	134.038i	0.433781	−	0.433781i
$310$		0			0
$311$	−499.800			−1.60708			−0.803538	−	0.595254i	$-0.797052\pi$
$311$	−499.800			−1.60708			−0.803538	+	0.595254i	$0.797052\pi$
$312$		0			0
$313$	52.7521			0.168537			0.0842685	−	0.996443i	$-0.473145\pi$
$313$	52.7521			0.168537			0.0842685	+	0.996443i	$0.473145\pi$
$314$		0			0
$315$	0.816055	+	91.4587i	0.00259065	+	0.290345i
$316$		0			0
$317$	218.091	+	218.091i	0.687983	+	0.687983i	0.961786	−	0.273803i	$-0.0882816\pi$
$317$	218.091	+	218.091i	0.687983	+	0.687983i	−0.273803	+	0.961786i	$0.588282\pi$
$318$		0			0
$319$	−45.1287			−0.141469
$320$		0			0
$321$		−	217.184i		−	0.676584i
$322$		0			0
$323$	−240.947	−	240.947i	−0.745967	−	0.745967i
$324$		0			0
$325$	23.0578	+	23.0578i	0.0709471	+	0.0709471i
$326$		0			0
$327$	335.254			1.02524
$328$		0			0
$329$	−332.092	−	326.218i	−1.00940	−	0.991544i
$330$		0			0
$331$	20.4515	−	20.4515i	0.0617871	−	0.0617871i	−0.675538	−	0.737325i	$-0.736089\pi$
$331$	20.4515	−	20.4515i	0.0617871	−	0.0617871i	0.737325	+	0.675538i	$0.236089\pi$
$332$		0			0
$333$	78.5018	+	78.5018i	0.235741	+	0.235741i
$334$		0			0
$335$		−	127.176i		−	0.379629i
$336$		0			0
$337$	451.093			1.33855			0.669277	−	0.743013i	$-0.266604\pi$
$337$	451.093			1.33855			0.669277	+	0.743013i	$0.266604\pi$
$338$		0			0
$339$	248.123	−	248.123i	0.731926	−	0.731926i
$340$		0			0
$341$	−33.2304	−	33.2304i	−0.0974500	−	0.0974500i
$342$		0			0
$343$	248.942	+	235.959i	0.725779	+	0.687928i
$344$		0			0
$345$			160.482i			0.465166i
$346$		0			0
$347$	313.926	−	313.926i	0.904687	−	0.904687i	−0.0911505	−	0.995837i	$-0.529054\pi$
$347$	313.926	−	313.926i	0.904687	−	0.904687i	0.995837	+	0.0911505i	$0.0290544\pi$
$348$		0			0
$349$	400.540	−	400.540i	1.14768	−	1.14768i	0.160670	−	0.987008i	$-0.448634\pi$
$349$	400.540	−	400.540i	1.14768	−	1.14768i	0.987008	−	0.160670i	$-0.0513655\pi$
$350$		0			0
$351$	252.385			0.719046
$352$		0			0
$353$			464.090i			1.31470i	0.753584	+	0.657352i	$0.228323\pi$
$353$			464.090i			1.31470i	−0.753584	+	0.657352i	$0.771677\pi$
$354$		0			0
$355$	−227.149	+	227.149i	−0.639855	+	0.639855i
$356$		0			0
$357$	−166.085	+	1.48192i	−0.465225	+	0.00415105i
$358$		0			0
$359$		−	126.355i		−	0.351963i	−0.984393	−	0.175982i	$-0.943690\pi$
$359$		−	126.355i		−	0.351963i	0.984393	−	0.175982i	$-0.0563099\pi$
$360$		0			0
$361$		−	909.946i		−	2.52063i
$362$		0			0
$363$	−198.188	−	198.188i	−0.545972	−	0.545972i
$364$		0			0
$365$	−466.672	−	466.672i	−1.27855	−	1.27855i
$366$		0			0
$367$			90.0661i			0.245412i	0.992443	+	0.122706i	$0.0391572\pi$
$367$			90.0661i			0.245412i	−0.992443	+	0.122706i	$0.960843\pi$
$368$		0			0
$369$		−	36.7887i		−	0.0996985i
$370$		0			0
$371$	3.47705	+	389.688i	0.00937211	+	1.05037i
$372$		0			0
$373$	−363.710	+	363.710i	−0.975094	+	0.975094i	−0.999697	−	0.0246037i	$-0.992168\pi$
$373$	−363.710	+	363.710i	−0.975094	+	0.975094i	0.0246037	+	0.999697i	$0.492168\pi$
$374$		0			0
$375$		−	329.171i		−	0.877789i
$376$		0			0
$377$	136.234			0.361362
$378$		0			0
$379$	−290.869	+	290.869i	−0.767465	+	0.767465i	−0.977660	−	0.210194i	$-0.932590\pi$
$379$	−290.869	+	290.869i	−0.767465	+	0.767465i	0.210194	+	0.977660i	$0.432590\pi$
$380$		0			0
$381$	−140.681	+	140.681i	−0.369241	+	0.369241i
$382$		0			0
$383$		−	234.051i		−	0.611099i	−0.952176	−	0.305550i	$-0.901160\pi$
$383$		−	234.051i		−	0.611099i	0.952176	−	0.305550i	$-0.0988402\pi$
$384$		0			0
$385$	64.2643	−	65.4214i	0.166920	−	0.169926i
$386$		0			0
$387$	144.647	+	144.647i	0.373764	+	0.373764i
$388$		0			0
$389$	83.4977	−	83.4977i	0.214647	−	0.214647i	−0.591591	−	0.806238i	$-0.701500\pi$
$389$	83.4977	−	83.4977i	0.214647	−	0.214647i	0.806238	+	0.591591i	$0.201500\pi$
$390$		0			0
$391$	134.191			0.343198
$392$		0			0
$393$			316.088i			0.804295i
$394$		0			0
$395$	217.355	+	217.355i	0.550265	+	0.550265i
$396$		0			0
$397$	−174.436	+	174.436i	−0.439386	+	0.439386i	−0.891805	−	0.452419i	$-0.850561\pi$
$397$	−174.436	+	174.436i	−0.439386	+	0.439386i	0.452419	+	0.891805i	$0.350561\pi$
$398$		0			0
$399$	−441.941	−	434.124i	−1.10762	−	1.08803i
$400$		0			0
$401$	−187.735			−0.468168			−0.234084	−	0.972216i	$-0.575209\pi$
$401$	−187.735			−0.468168			−0.234084	+	0.972216i	$0.575209\pi$
$402$		0			0
$403$	100.315	+	100.315i	0.248921	+	0.248921i
$404$		0			0
$405$	−154.369	−	154.369i	−0.381159	−	0.381159i
$406$		0			0
$407$		−	111.313i		−	0.273497i
$408$		0			0
$409$	144.940			0.354377			0.177188	−	0.984177i	$-0.443300\pi$
$409$	144.940			0.354377			0.177188	+	0.984177i	$0.443300\pi$
$410$		0			0
$411$	444.240	+	444.240i	1.08088	+	1.08088i
$412$		0			0
$413$	301.243	−	2.68789i	0.729401	−	0.00650820i
$414$		0			0
$415$	395.600			0.953253
$416$		0			0
$417$	−172.933			−0.414708
$418$		0			0
$419$	177.168	−	177.168i	0.422835	−	0.422835i	−0.463344	−	0.886179i	$-0.653350\pi$
$419$	177.168	−	177.168i	0.422835	−	0.422835i	0.886179	+	0.463344i	$0.153350\pi$
$420$		0			0
$421$	468.234	−	468.234i	1.11220	−	1.11220i	0.119342	−	0.992853i	$-0.461922\pi$
$421$	468.234	−	468.234i	1.11220	−	1.11220i	0.992853	−	0.119342i	$-0.0380784\pi$
$422$		0			0
$423$	−188.702			−0.446105
$424$		0			0
$425$	−36.2895			−0.0853871
$426$		0			0
$427$	−1.05947	−	118.740i	−0.00248120	−	0.278079i
$428$		0			0
$429$	−42.8926	−	42.8926i	−0.0999827	−	0.0999827i
$430$		0			0
$431$	−497.233			−1.15367			−0.576836	−	0.816860i	$-0.695713\pi$
$431$	−497.233			−1.15367			−0.576836	+	0.816860i	$0.695713\pi$
$432$		0			0
$433$		−	210.739i		−	0.486696i	−0.969939	−	0.243348i	$-0.921754\pi$
$433$		−	210.739i		−	0.486696i	0.969939	−	0.243348i	$-0.0782457\pi$
$434$		0			0
$435$	−128.210	−	128.210i	−0.294736	−	0.294736i
$436$		0			0
$437$	353.913	+	353.913i	0.809870	+	0.809870i
$438$		0			0
$439$	762.892			1.73779			0.868897	−	0.494993i	$-0.164829\pi$
$439$	762.892			1.73779			0.868897	+	0.494993i	$0.164829\pi$
$440$		0			0
$441$	139.017	−	2.48101i	0.315232	−	0.00562587i
$442$		0			0
$443$	140.252	−	140.252i	0.316596	−	0.316596i	−0.530862	−	0.847458i	$-0.678132\pi$
$443$	140.252	−	140.252i	0.316596	−	0.316596i	0.847458	+	0.530862i	$0.178132\pi$
$444$		0			0
$445$	−257.028	−	257.028i	−0.577591	−	0.577591i
$446$		0			0
$447$			142.017i			0.317711i
$448$		0			0
$449$	−599.904			−1.33609			−0.668044	−	0.744122i	$-0.732868\pi$
$449$	−599.904			−1.33609			−0.668044	+	0.744122i	$0.732868\pi$
$450$		0			0
$451$	−26.0827	+	26.0827i	−0.0578330	+	0.0578330i
$452$		0			0
$453$	60.6512	+	60.6512i	0.133888	+	0.133888i
$454$		0			0
$455$	−193.999	+	197.493i	−0.426372	+	0.434050i
$456$		0			0
$457$			557.580i			1.22009i	0.792368	+	0.610044i	$0.208848\pi$
$457$			557.580i			1.22009i	−0.792368	+	0.610044i	$0.791152\pi$
$458$		0			0
$459$	−198.608	+	198.608i	−0.432698	+	0.432698i
$460$		0			0
$461$	−482.593	+	482.593i	−1.04684	+	1.04684i	−0.0479917	+	0.998848i	$0.515282\pi$
$461$	−482.593	+	482.593i	−1.04684	+	1.04684i	−0.998848	+	0.0479917i	$0.984718\pi$
$462$		0			0
$463$	326.635			0.705475			0.352737	−	0.935722i	$-0.385251\pi$
$463$	326.635			0.705475			0.352737	+	0.935722i	$0.385251\pi$
$464$		0			0
$465$		−	188.814i		−	0.406052i
$466$		0			0
$467$	−213.948	+	213.948i	−0.458132	+	0.458132i	−0.898042	−	0.439910i	$-0.855010\pi$
$467$	−213.948	+	213.948i	−0.458132	+	0.458132i	0.439910	+	0.898042i	$0.355010\pi$
$468$		0			0
$469$	−193.323	+	1.72496i	−0.412203	+	0.00367794i
$470$		0			0
$471$		−	60.3870i		−	0.128210i
$472$		0			0
$473$		−	205.105i		−	0.433625i
$474$		0			0
$475$	−95.7098	−	95.7098i	−0.201494	−	0.201494i
$476$		0			0
$477$	111.703	+	111.703i	0.234178	+	0.234178i
$478$		0			0
$479$			437.605i			0.913581i	0.889574	+	0.456790i	$0.151001\pi$
$479$			437.605i			0.913581i	−0.889574	+	0.456790i	$0.848999\pi$
$480$		0			0
$481$			336.030i			0.698607i
$482$		0			0
$483$	243.953	−	2.17671i	0.505079	−	0.00450665i
$484$		0			0
$485$	115.549	−	115.549i	0.238245	−	0.238245i
$486$		0			0
$487$			681.542i			1.39947i	0.714402	+	0.699735i	$0.246699\pi$
$487$			681.542i			1.39947i	−0.714402	+	0.699735i	$0.753301\pi$
$488$		0			0
$489$	−136.837			−0.279831
$490$		0			0
$491$	−324.611	+	324.611i	−0.661122	+	0.661122i	−0.955644	−	0.294523i	$-0.904839\pi$
$491$	−324.611	+	324.611i	−0.661122	+	0.661122i	0.294523	+	0.955644i	$0.404839\pi$
$492$		0			0
$493$	−107.206	+	107.206i	−0.217455	+	0.217455i
$494$		0			0
$495$		−	37.1740i		−	0.0750989i
$496$		0			0
$497$	348.375	+	342.213i	0.700956	+	0.688558i
$498$		0			0
$499$	−23.3782	−	23.3782i	−0.0468501	−	0.0468501i	0.683294	−	0.730144i	$-0.260547\pi$
$499$	−23.3782	−	23.3782i	−0.0468501	−	0.0468501i	−0.730144	+	0.683294i	$0.760547\pi$
$500$		0			0
$501$	−240.576	+	240.576i	−0.480191	+	0.480191i
$502$		0			0
$503$	−693.554			−1.37884			−0.689418	−	0.724364i	$-0.742133\pi$
$503$	−693.554			−1.37884			−0.689418	+	0.724364i	$0.742133\pi$
$504$		0			0
$505$			881.157i			1.74487i
$506$		0			0
$507$	−167.170	−	167.170i	−0.329724	−	0.329724i
$508$		0			0
$509$	−175.357	+	175.357i	−0.344512	+	0.344512i	−0.858061	−	0.513548i	$-0.828331\pi$
$509$	−175.357	+	175.357i	−0.344512	+	0.344512i	0.513548	+	0.858061i	$0.328331\pi$
$510$		0			0
$511$	−703.070	+	715.730i	−1.37587	+	1.40065i
$512$		0			0
$513$	−1047.62			−2.04214
$514$		0			0
$515$	248.630	+	248.630i	0.482777	+	0.482777i
$516$		0			0
$517$	133.787	+	133.787i	0.258776	+	0.258776i
$518$		0			0
$519$		−	207.479i		−	0.399766i
$520$		0			0
$521$	−602.394			−1.15623			−0.578113	−	0.815957i	$-0.696211\pi$
$521$	−602.394			−1.15623			−0.578113	+	0.815957i	$0.696211\pi$
$522$		0			0
$523$	−296.480	−	296.480i	−0.566883	−	0.566883i	0.364371	−	0.931254i	$-0.381284\pi$
$523$	−296.480	−	296.480i	−0.566883	−	0.566883i	−0.931254	+	0.364371i	$0.881284\pi$
$524$		0			0
$525$	−65.9729	+	0.588654i	−0.125663	+	0.00112125i
$526$		0			0
$527$	−157.881			−0.299585
$528$		0			0
$529$	331.895			0.627401
$530$		0			0
$531$	86.3502	−	86.3502i	0.162618	−	0.162618i
$532$		0			0
$533$	78.7377	−	78.7377i	0.147726	−	0.147726i
$534$		0			0
$535$	402.858			0.753005
$536$		0			0
$537$	528.724			0.984588
$538$		0			0
$539$	−100.320	−	96.8023i	−0.186123	−	0.179596i
$540$		0			0
$541$	−221.579	−	221.579i	−0.409573	−	0.409573i	0.472017	−	0.881590i	$-0.343526\pi$
$541$	−221.579	−	221.579i	−0.409573	−	0.409573i	−0.881590	+	0.472017i	$0.843526\pi$
$542$		0			0
$543$	−686.000			−1.26335
$544$		0			0
$545$			621.869i			1.14104i
$546$		0			0
$547$	−545.973	−	545.973i	−0.998123	−	0.998123i	0.00187545	−	0.999998i	$-0.499403\pi$
$547$	−545.973	−	545.973i	−0.998123	−	0.998123i	−0.999998	+	0.00187545i	$0.999403\pi$
$548$		0			0
$549$	−34.0363	−	34.0363i	−0.0619969	−	0.0619969i
$550$		0			0
$551$	−565.486			−1.02629
$552$		0			0
$553$	327.458	−	333.355i	0.592149	−	0.602811i
$554$		0			0
$555$	316.239	−	316.239i	0.569800	−	0.569800i
$556$		0			0
$557$	87.7052	+	87.7052i	0.157460	+	0.157460i	0.781440	−	0.623980i	$-0.214485\pi$
$557$	87.7052	+	87.7052i	0.157460	+	0.157460i	−0.623980	+	0.781440i	$0.714485\pi$
$558$		0			0
$559$			619.165i			1.10763i
$560$		0			0
$561$	67.5064			0.120332
$562$		0			0
$563$	125.548	−	125.548i	0.222997	−	0.222997i	−0.586762	−	0.809759i	$-0.699598\pi$
$563$	125.548	−	125.548i	0.222997	−	0.222997i	0.809759	+	0.586762i	$0.199598\pi$
$564$		0			0
$565$	460.248	+	460.248i	0.814598	+	0.814598i
$566$		0			0
$567$	−232.567	+	236.754i	−0.410170	+	0.417556i
$568$		0			0
$569$			963.505i			1.69333i	0.532126	+	0.846665i	$0.321393\pi$
$569$			963.505i			1.69333i	−0.532126	+	0.846665i	$0.678607\pi$
$570$		0			0
$571$	254.297	−	254.297i	0.445355	−	0.445355i	−0.448452	−	0.893807i	$-0.648025\pi$
$571$	254.297	−	254.297i	0.445355	−	0.445355i	0.893807	+	0.448452i	$0.148025\pi$
$572$		0			0
$573$	103.860	−	103.860i	0.181257	−	0.181257i
$574$		0			0
$575$	53.3036			0.0927018
$576$		0			0
$577$		−	792.088i		−	1.37277i	−0.727239	−	0.686384i	$-0.759197\pi$
$577$		−	792.088i		−	1.37277i	0.727239	−	0.686384i	$-0.240803\pi$
$578$		0			0
$579$	335.146	−	335.146i	0.578835	−	0.578835i
$580$		0			0
$581$	−5.36574	−	601.361i	−0.00923535	−	1.03504i
$582$		0			0
$583$		−	158.391i		−	0.271683i
$584$		0			0
$585$			112.220i			0.191829i
$586$		0			0
$587$	436.848	+	436.848i	0.744204	+	0.744204i	0.973384	−	0.229180i	$-0.0736044\pi$
$587$	436.848	+	436.848i	0.744204	+	0.744204i	−0.229180	+	0.973384i	$0.573604\pi$
$588$		0			0
$589$	−416.394	−	416.394i	−0.706952	−	0.706952i
$590$		0			0
$591$			501.683i			0.848872i
$592$		0			0
$593$		−	668.916i		−	1.12802i	−0.825768	−	0.564010i	$-0.809258\pi$
$593$		−	668.916i		−	1.12802i	0.825768	−	0.564010i	$-0.190742\pi$
$594$		0			0
$595$	−2.74885	−	308.075i	−0.00461991	−	0.517773i
$596$		0			0
$597$	−243.974	+	243.974i	−0.408667	+	0.408667i
$598$		0			0
$599$		−	411.756i		−	0.687405i	−0.939079	−	0.343703i	$-0.888319\pi$
$599$		−	411.756i		−	0.687405i	0.939079	−	0.343703i	$-0.111681\pi$
$600$		0			0
$601$	688.982			1.14639			0.573196	−	0.819418i	$-0.305703\pi$
$601$	688.982			1.14639			0.573196	+	0.819418i	$0.305703\pi$
$602$		0			0
$603$	−55.4154	+	55.4154i	−0.0918995	+	0.0918995i
$604$		0			0
$605$	367.622	−	367.622i	0.607640	−	0.607640i
$606$		0			0
$607$			768.094i			1.26539i	0.774400	+	0.632697i	$0.218052\pi$
$607$			768.094i			1.26539i	−0.774400	+	0.632697i	$0.781948\pi$
$608$		0			0
$609$	−193.156	+	196.634i	−0.317170	+	0.322880i
$610$		0			0
$611$	−403.873	−	403.873i	−0.661004	−	0.661004i
$612$		0			0
$613$	−163.362	+	163.362i	−0.266495	+	0.266495i	−0.827686	−	0.561191i	$-0.810343\pi$
$613$	−163.362	+	163.362i	−0.266495	+	0.266495i	0.561191	+	0.827686i	$0.310343\pi$
$614$		0			0
$615$	−148.201			−0.240977
$616$		0			0
$617$			652.431i			1.05742i	0.848801	+	0.528712i	$0.177325\pi$
$617$			652.431i			1.05742i	−0.848801	+	0.528712i	$0.822675\pi$
$618$		0			0
$619$	78.1751	+	78.1751i	0.126293	+	0.126293i	0.767428	−	0.641135i	$-0.221536\pi$
$619$	78.1751	+	78.1751i	0.126293	+	0.126293i	−0.641135	+	0.767428i	$0.721536\pi$
$620$		0			0
$621$	291.724	−	291.724i	0.469765	−	0.469765i
$622$		0			0
$623$	−387.228	+	394.201i	−0.621554	+	0.632746i
$624$		0			0
$625$	515.667			0.825067
$626$		0			0
$627$	178.041	+	178.041i	0.283957	+	0.283957i
$628$		0			0
$629$	−264.430	−	264.430i	−0.420398	−	0.420398i
$630$		0			0
$631$			219.948i			0.348570i	0.984695	+	0.174285i	$0.0557614\pi$
$631$			219.948i			0.348570i	−0.984695	+	0.174285i	$0.944239\pi$
$632$		0			0
$633$	746.602			1.17947
$634$		0			0
$635$	−260.951	−	260.951i	−0.410947	−	0.410947i
$636$		0			0
$637$	302.845	+	292.225i	0.475423	+	0.458751i
$638$		0			0
$639$	197.955			0.309789
$640$		0			0
$641$	−186.577			−0.291072			−0.145536	−	0.989353i	$-0.546491\pi$
$641$	−186.577			−0.291072			−0.145536	+	0.989353i	$0.546491\pi$
$642$		0			0
$643$	215.904	−	215.904i	0.335776	−	0.335776i	−0.518999	−	0.854775i	$-0.673695\pi$
$643$	215.904	−	215.904i	0.335776	−	0.335776i	0.854775	+	0.518999i	$0.173695\pi$
$644$		0			0
$645$	582.699	−	582.699i	0.903409	−	0.903409i
$646$		0			0
$647$	663.677			1.02578			0.512888	−	0.858456i	$-0.328576\pi$
$647$	663.677			1.02578			0.512888	+	0.858456i	$0.328576\pi$
$648$		0			0
$649$	−122.442			−0.188663
$650$		0			0
$651$	−287.021	+	2.56099i	−0.440893	+	0.00393394i
$652$		0			0
$653$	−411.703	−	411.703i	−0.630479	−	0.630479i	0.317709	−	0.948188i	$-0.397086\pi$
$653$	−411.703	−	411.703i	−0.630479	−	0.630479i	−0.948188	+	0.317709i	$0.897086\pi$
$654$		0			0
$655$	−586.317			−0.895140
$656$		0			0
$657$			406.694i			0.619017i
$658$		0			0
$659$	149.906	+	149.906i	0.227475	+	0.227475i	0.811637	−	0.584162i	$-0.198577\pi$
$659$	149.906	+	149.906i	0.227475	+	0.227475i	−0.584162	+	0.811637i	$0.698577\pi$
$660$		0			0
$661$	−166.233	−	166.233i	−0.251487	−	0.251487i	0.570093	−	0.821580i	$-0.306907\pi$
$661$	−166.233	−	166.233i	−0.251487	−	0.251487i	−0.821580	+	0.570093i	$0.806907\pi$
$662$		0			0
$663$	−203.787			−0.307371
$664$		0			0
$665$	805.264	−	819.764i	1.21092	−	1.23273i
$666$		0			0
$667$	157.468	−	157.468i	0.236084	−	0.236084i
$668$		0			0
$669$	607.355	+	607.355i	0.907854	+	0.907854i
$670$		0			0
$671$			48.2625i			0.0719262i
$672$		0			0
$673$	315.076			0.468167			0.234083	−	0.972217i	$-0.424791\pi$
$673$	315.076			0.468167			0.234083	+	0.972217i	$0.424791\pi$
$674$		0			0
$675$	−78.8917	+	78.8917i	−0.116877	+	0.116877i
$676$		0			0
$677$	−496.264	−	496.264i	−0.733034	−	0.733034i	0.238186	−	0.971220i	$-0.423447\pi$
$677$	−496.264	−	496.264i	−0.733034	−	0.733034i	−0.971220	+	0.238186i	$0.923447\pi$
$678$		0			0
$679$	−177.216	−	174.082i	−0.260996	−	0.256379i
$680$		0			0
$681$		−	663.538i		−	0.974359i
$682$		0			0
$683$	387.884	−	387.884i	0.567913	−	0.567913i	−0.363631	−	0.931543i	$-0.618463\pi$
$683$	387.884	−	387.884i	0.567913	−	0.567913i	0.931543	+	0.363631i	$0.118463\pi$
$684$		0			0
$685$	−824.029	+	824.029i	−1.20296	+	1.20296i
$686$		0			0
$687$	530.594			0.772335
$688$		0			0
$689$			478.147i			0.693973i
$690$		0			0
$691$	−776.671	+	776.671i	−1.12398	+	1.12398i	−0.132844	+	0.991137i	$0.542411\pi$
$691$	−776.671	+	776.671i	−1.12398	+	1.12398i	−0.991137	+	0.132844i	$0.957589\pi$
$692$		0			0
$693$	−56.5090	+	0.504211i	−0.0815426	+	0.000727577i
$694$		0			0
$695$		−	320.777i		−	0.461550i
$696$		0			0
$697$			123.921i			0.177792i
$698$		0			0
$699$	18.4463	+	18.4463i	0.0263895	+	0.0263895i
$700$		0			0
$701$	−504.506	−	504.506i	−0.719695	−	0.719695i	0.248848	−	0.968543i	$-0.419948\pi$
$701$	−504.506	−	504.506i	−0.719695	−	0.719695i	−0.968543	+	0.248848i	$0.919948\pi$
$702$		0			0
$703$		−	1394.81i		−	1.98409i
$704$		0			0
$705$			760.174i			1.07826i
$706$		0			0
$707$	1339.47	−	11.9516i	1.89458	−	0.0169047i
$708$		0			0
$709$	329.265	−	329.265i	0.464408	−	0.464408i	−0.435689	−	0.900097i	$-0.643495\pi$
$709$	329.265	−	329.265i	0.464408	−	0.464408i	0.900097	+	0.435689i	$0.143495\pi$
$710$		0			0
$711$		−	189.420i		−	0.266413i
$712$		0			0
$713$	231.902			0.325249
$714$		0			0
$715$	79.5622	−	79.5622i	0.111276	−	0.111276i
$716$		0			0
$717$	−636.021	+	636.021i	−0.887058	+	0.887058i
$718$		0			0
$719$		−	612.721i		−	0.852185i	−0.904680	−	0.426093i	$-0.859890\pi$
$719$		−	612.721i		−	0.852185i	0.904680	−	0.426093i	$-0.140110\pi$
$720$		0			0
$721$	374.576	−	381.321i	0.519523	−	0.528878i
$722$		0			0
$723$	637.814	+	637.814i	0.882177	+	0.882177i
$724$		0			0
$725$	−42.5845	+	42.5845i	−0.0587372	+	0.0587372i
$726$		0			0
$727$	884.961			1.21728			0.608639	−	0.793447i	$-0.291716\pi$
$727$	884.961			1.21728			0.608639	+	0.793447i	$0.291716\pi$
$728$		0			0
$729$			791.064i			1.08514i
$730$		0			0
$731$	−487.236	−	487.236i	−0.666534	−	0.666534i
$732$		0			0
$733$	−630.619	+	630.619i	−0.860327	+	0.860327i	−0.991376	−	0.131049i	$-0.958165\pi$
$733$	−630.619	+	630.619i	−0.860327	+	0.860327i	0.131049	+	0.991376i	$0.458165\pi$
$734$		0			0
$735$	−9.99458	−	560.022i	−0.0135981	−	0.761935i
$736$		0			0
$737$	78.5774			0.106618
$738$		0			0
$739$	885.422	+	885.422i	1.19813	+	1.19813i	0.974725	+	0.223410i	$0.0717189\pi$
$739$	885.422	+	885.422i	1.19813	+	1.19813i	0.223410	+	0.974725i	$0.428281\pi$
$740$		0			0
$741$	−537.466	−	537.466i	−0.725325	−	0.725325i
$742$		0			0
$743$			454.580i			0.611817i	0.952061	+	0.305909i	$0.0989602\pi$
$743$			454.580i			0.611817i	−0.952061	+	0.305909i	$0.901040\pi$
$744$		0			0
$745$	−263.430			−0.353597
$746$		0			0
$747$	−172.378	−	172.378i	−0.230760	−	0.230760i
$748$		0			0
$749$	−5.46418	−	612.394i	−0.00729530	−	0.817615i
$750$		0			0
$751$	−220.761			−0.293956			−0.146978	−	0.989140i	$-0.546955\pi$
$751$	−220.761			−0.293956			−0.146978	+	0.989140i	$0.546955\pi$
$752$		0			0
$753$	−190.119			−0.252481
$754$		0			0
$755$	−112.503	+	112.503i	−0.149011	+	0.149011i
$756$		0			0
$757$	195.831	−	195.831i	0.258693	−	0.258693i	−0.565829	−	0.824522i	$-0.691444\pi$
$757$	195.831	−	195.831i	0.258693	−	0.258693i	0.824522	+	0.565829i	$0.191444\pi$
$758$		0			0
$759$	−99.1562			−0.130641
$760$		0			0
$761$	−1450.33			−1.90583			−0.952913	−	0.303244i	$-0.901930\pi$
$761$	−1450.33			−1.90583			−0.952913	+	0.303244i	$0.901930\pi$
$762$		0			0
$763$	945.319	−	8.43476i	1.23895	−	0.0110547i
$764$		0			0
$765$	−88.3086	−	88.3086i	−0.115436	−	0.115436i
$766$		0			0
$767$	369.625			0.481910
$768$		0			0
$769$			95.3790i			0.124030i	0.998075	+	0.0620150i	$0.0197527\pi$
$769$			95.3790i			0.124030i	−0.998075	+	0.0620150i	$0.980247\pi$
$770$		0			0
$771$	−79.6401	−	79.6401i	−0.103295	−	0.103295i
$772$		0			0
$773$	−811.244	−	811.244i	−1.04948	−	1.04948i	−0.998711	−	0.0507646i	$-0.983834\pi$
$773$	−811.244	−	811.244i	−1.04948	−	1.04948i	−0.0507646	−	0.998711i	$-0.516166\pi$
$774$		0			0
$775$	−62.7140			−0.0809212
$776$		0			0
$777$	−485.012	−	476.434i	−0.624211	−	0.613171i
$778$		0			0
$779$	−326.829	+	326.829i	−0.419550	+	0.419550i
$780$		0			0
$781$	−140.347	−	140.347i	−0.179702	−	0.179702i
$782$		0			0
$783$			466.119i			0.595299i
$784$		0			0
$785$	112.013			0.142692
$786$		0			0
$787$	96.6979	−	96.6979i	0.122869	−	0.122869i	−0.642998	−	0.765867i	$-0.722310\pi$
$787$	96.6979	−	96.6979i	0.122869	−	0.122869i	0.765867	+	0.642998i	$0.222310\pi$
$788$		0			0
$789$	881.996	+	881.996i	1.11787	+	1.11787i
$790$		0			0
$791$	693.391	−	705.876i	0.876601	−	0.892385i
$792$		0			0
$793$		−	145.694i		−	0.183725i
$794$		0			0
$795$	449.987	−	449.987i	0.566021	−	0.566021i
$796$		0			0
$797$	581.633	−	581.633i	0.729778	−	0.729778i	−0.240798	−	0.970575i	$-0.577409\pi$
$797$	581.633	−	581.633i	0.729778	−	0.729778i	0.970575	+	0.240798i	$0.0774090\pi$
$798$		0			0
$799$	635.636			0.795539
$800$		0			0
$801$			223.994i			0.279643i
$802$		0			0
$803$	288.340	−	288.340i	0.359079	−	0.359079i
$804$		0			0
$805$	4.03762	+	452.513i	0.00501567	+	0.562128i
$806$		0			0
$807$			285.625i			0.353935i
$808$		0			0
$809$			296.533i			0.366543i	0.983062	+	0.183272i	$0.0586688\pi$
$809$			296.533i			0.366543i	−0.983062	+	0.183272i	$0.941331\pi$
$810$		0			0
$811$	−605.219	−	605.219i	−0.746262	−	0.746262i	0.227513	−	0.973775i	$-0.426941\pi$
$811$	−605.219	−	605.219i	−0.746262	−	0.746262i	−0.973775	+	0.227513i	$0.926941\pi$
$812$		0			0
$813$	338.709	+	338.709i	0.416617	+	0.416617i
$814$		0			0
$815$		−	253.822i		−	0.311438i
$816$		0			0
$817$		−	2570.07i		−	3.14574i
$818$		0			0
$819$	170.588	−	1.52210i	0.208288	−	0.00185849i
$820$		0			0
$821$	460.505	−	460.505i	0.560907	−	0.560907i	−0.368658	−	0.929565i	$-0.620183\pi$
$821$	460.505	−	460.505i	0.560907	−	0.560907i	0.929565	+	0.368658i	$0.120183\pi$
$822$		0			0
$823$			1126.18i			1.36839i	0.729300	+	0.684194i	$0.239846\pi$
$823$			1126.18i			1.36839i	−0.729300	+	0.684194i	$0.760154\pi$
$824$		0			0
$825$	26.8151			0.0325031
$826$		0			0
$827$	−686.896	+	686.896i	−0.830587	+	0.830587i	−0.987597	−	0.157010i	$-0.949815\pi$
$827$	−686.896	+	686.896i	−0.830587	+	0.830587i	0.157010	+	0.987597i	$0.449815\pi$
$828$		0			0
$829$	732.088	−	732.088i	0.883098	−	0.883098i	−0.110750	−	0.993848i	$-0.535325\pi$
$829$	732.088	−	732.088i	0.883098	−	0.883098i	0.993848	+	0.110750i	$0.0353254\pi$
$830$		0			0
$831$		−	743.694i		−	0.894939i
$832$		0			0
$833$	−468.275	+	8.35718i	−0.562154	+	0.0100326i
$834$		0			0
$835$	−446.248	−	446.248i	−0.534429	−	0.534429i
$836$		0			0
$837$	−343.226	+	343.226i	−0.410067	+	0.410067i
$838$		0			0
$839$	−903.003			−1.07628			−0.538142	−	0.842854i	$-0.680874\pi$
$839$	−903.003			−1.07628			−0.538142	+	0.842854i	$0.680874\pi$
$840$		0			0
$841$		−	589.396i		−	0.700828i
$842$		0			0
$843$	149.828	+	149.828i	0.177732	+	0.177732i
$844$		0			0
$845$	310.086	−	310.086i	0.366966	−	0.366966i
$846$		0			0
$847$	−563.818	−	553.845i	−0.665664	−	0.653891i
$848$		0			0
$849$	307.099			0.361719
$850$		0			0
$851$	388.406	+	388.406i	0.456411	+	0.456411i
$852$		0			0
$853$	−38.7225	−	38.7225i	−0.0453957	−	0.0453957i	0.684045	−	0.729440i	$-0.260219\pi$
$853$	−38.7225	−	38.7225i	−0.0453957	−	0.0453957i	−0.729440	+	0.684045i	$0.760219\pi$
$854$		0			0
$855$		−	465.809i		−	0.544806i
$856$		0			0
$857$	653.919			0.763032			0.381516	−	0.924362i	$-0.375402\pi$
$857$	653.919			0.763032			0.381516	+	0.924362i	$0.375402\pi$
$858$		0			0
$859$	−340.187	−	340.187i	−0.396026	−	0.396026i	0.480802	−	0.876829i	$-0.340345\pi$
$859$	−340.187	−	340.187i	−0.396026	−	0.396026i	−0.876829	+	0.480802i	$0.840345\pi$
$860$		0			0
$861$	2.01013	+	225.284i	0.00233465	+	0.261654i
$862$		0			0
$863$	106.239			0.123104			0.0615519	−	0.998104i	$-0.480395\pi$
$863$	106.239			0.123104			0.0615519	+	0.998104i	$0.480395\pi$
$864$		0			0
$865$	384.856			0.444920
$866$		0			0
$867$	−346.929	+	346.929i	−0.400149	+	0.400149i
$868$		0			0
$869$	−134.296	+	134.296i	−0.154541	+	0.154541i
$870$		0			0
$871$	−237.207			−0.272339
$872$		0			0
$873$	−100.698			−0.115347
$874$		0			0
$875$	−8.28170	−	928.165i	−0.00946480	−	1.06076i
$876$		0			0
$877$	−535.021	−	535.021i	−0.610059	−	0.610059i	0.332903	−	0.942961i	$-0.391972\pi$
$877$	−535.021	−	535.021i	−0.610059	−	0.610059i	−0.942961	+	0.332903i	$0.891972\pi$
$878$		0			0
$879$	107.797			0.122636
$880$		0			0
$881$			370.517i			0.420564i	0.977641	+	0.210282i	$0.0674382\pi$
$881$			370.517i			0.420564i	−0.977641	+	0.210282i	$0.932562\pi$
$882$		0			0
$883$	−109.056	−	109.056i	−0.123507	−	0.123507i	0.642652	−	0.766158i	$-0.277834\pi$
$883$	−109.056	−	109.056i	−0.123507	−	0.123507i	−0.766158	+	0.642652i	$0.777834\pi$
$884$		0			0
$885$	−347.856	−	347.856i	−0.393057	−	0.393057i
$886$		0			0
$887$	1435.92			1.61884			0.809422	−	0.587227i	$-0.199780\pi$
$887$	1435.92			1.61884			0.809422	+	0.587227i	$0.199780\pi$
$888$		0			0
$889$	−393.139	+	400.218i	−0.442226	+	0.450189i
$890$		0			0
$891$	95.3792	−	95.3792i	0.107047	−	0.107047i
$892$		0			0
$893$	1676.42	+	1676.42i	1.87729	+	1.87729i
$894$		0			0
$895$			980.739i			1.09580i
$896$		0			0
$897$	299.330			0.333702
$898$		0			0
$899$	−185.268	+	185.268i	−0.206082	+	0.206082i
$900$		0			0
$901$	−376.266	−	376.266i	−0.417609	−	0.417609i
$902$		0			0
$903$	−893.679	−	877.872i	−0.989677	−	0.972172i
$904$		0			0
$905$		−	1272.47i		−	1.40605i
$906$		0			0
$907$	−726.879	+	726.879i	−0.801410	+	0.801410i	−0.983316	−	0.181906i	$-0.941773\pi$
$907$	−726.879	+	726.879i	−0.801410	+	0.801410i	0.181906	+	0.983316i	$0.441773\pi$
$908$		0			0
$909$	383.954	−	383.954i	0.422392	−	0.422392i
$910$		0			0
$911$	474.695			0.521070			0.260535	−	0.965464i	$-0.416101\pi$
$911$	474.695			0.521070			0.260535	+	0.965464i	$0.416101\pi$
$912$		0			0
$913$			244.427i			0.267718i
$914$		0			0
$915$	−137.113	+	137.113i	−0.149850	+	0.149850i
$916$		0			0
$917$	7.95254	+	891.275i	0.00867235	+	0.971946i
$918$		0			0
$919$		−	947.682i		−	1.03121i	−0.856827	−	0.515605i	$-0.827567\pi$
$919$		−	947.682i		−	1.03121i	0.856827	−	0.515605i	$-0.172433\pi$
$920$		0			0
$921$			111.090i			0.120619i
$922$		0			0
$923$	423.676	+	423.676i	0.459021	+	0.459021i
$924$		0			0
$925$	−105.038	−	105.038i	−0.113554	−	0.113554i
$926$		0			0
$927$		−	216.675i		−	0.233738i
$928$		0			0
$929$			448.957i			0.483269i	0.970367	+	0.241634i	$0.0776834\pi$
$929$			448.957i			0.483269i	−0.970367	+	0.241634i	$0.922317\pi$
$930$		0			0
$931$	−1257.07	−	1212.98i	−1.35023	−	1.30288i
$932$		0			0
$933$	877.321	−	877.321i	0.940323	−	0.940323i
$934$		0			0
$935$			125.219i			0.133924i
$936$		0			0
$937$	−857.680			−0.915347			−0.457674	−	0.889120i	$-0.651317\pi$
$937$	−857.680			−0.915347			−0.457674	+	0.889120i	$0.651317\pi$
$938$		0			0
$939$	−92.5980	+	92.5980i	−0.0986134	+	0.0986134i
$940$		0			0
$941$	151.657	−	151.657i	0.161166	−	0.161166i	−0.621917	−	0.783083i	$-0.713646\pi$
$941$	151.657	−	151.657i	0.161166	−	0.161166i	0.783083	+	0.621917i	$0.213646\pi$
$942$		0			0
$943$		−	182.021i		−	0.193023i
$944$		0			0
$945$	−675.715	−	663.764i	−0.715043	−	0.702395i
$946$		0			0
$947$	−908.719	−	908.719i	−0.959577	−	0.959577i	0.0396372	−	0.999214i	$-0.487380\pi$
$947$	−908.719	−	908.719i	−0.959577	−	0.959577i	−0.999214	+	0.0396372i	$0.987380\pi$
$948$		0			0
$949$	−870.434	+	870.434i	−0.917212	+	0.917212i
$950$		0			0
$951$	−765.648			−0.805097
$952$		0			0
$953$			844.182i			0.885815i	0.896567	+	0.442908i	$0.146053\pi$
$953$			844.182i			0.885815i	−0.896567	+	0.442908i	$0.853947\pi$
$954$		0			0
$955$	192.653	+	192.653i	0.201731	+	0.201731i
$956$		0			0
$957$	79.2164	−	79.2164i	0.0827758	−	0.0827758i
$958$		0			0
$959$	1263.80	+	1241.45i	1.31784	+	1.29453i
$960$		0			0
$961$	688.157			0.716084
$962$		0			0
$963$	−175.541	−	175.541i	−0.182285	−	0.182285i
$964$		0			0
$965$	621.668	+	621.668i	0.644215	+	0.644215i
$966$		0			0
$967$			1481.66i			1.53223i	0.642705	+	0.766114i	$0.277812\pi$
$967$			1481.66i			1.53223i	−0.642705	+	0.766114i	$0.722188\pi$
$968$		0			0
$969$	845.890			0.872952
$970$		0			0
$971$	736.360	+	736.360i	0.758352	+	0.758352i	0.976022	−	0.217671i	$-0.0698458\pi$
$971$	736.360	+	736.360i	0.758352	+	0.758352i	−0.217671	+	0.976022i	$0.569846\pi$
$972$		0			0
$973$	−487.621	+	4.35088i	−0.501152	+	0.00447161i
$974$		0			0
$975$	−80.9488			−0.0830244
$976$		0			0
$977$	736.267			0.753600			0.376800	−	0.926295i	$-0.377024\pi$
$977$	736.267			0.753600			0.376800	+	0.926295i	$0.377024\pi$
$978$		0			0
$979$	158.808	−	158.808i	0.162215	−	0.162215i
$980$		0			0
$981$	270.972	−	270.972i	0.276221	−	0.276221i
$982$		0			0
$983$	−564.432			−0.574194			−0.287097	−	0.957902i	$-0.592690\pi$
$983$	−564.432			−0.574194			−0.287097	+	0.957902i	$0.592690\pi$
$984$		0			0
$985$	−930.581			−0.944752
$986$		0			0
$987$	1155.56	−	10.3107i	1.17078	−	0.0104465i
$988$		0			0
$989$	715.672	+	715.672i	0.723632	+	0.723632i
$990$		0			0
$991$	715.833			0.722334			0.361167	−	0.932501i	$-0.382378\pi$
$991$	715.833			0.722334			0.361167	+	0.932501i	$0.382378\pi$
$992$		0			0
$993$			71.7989i			0.0723051i
$994$		0			0
$995$	−452.552	−	452.552i	−0.454826	−	0.454826i
$996$		0			0
$997$	892.876	+	892.876i	0.895562	+	0.895562i	0.995040	−	0.0994775i	$-0.0317171\pi$
$997$	892.876	+	892.876i	0.895562	+	0.895562i	−0.0994775	+	0.995040i	$0.531717\pi$
$998$		0			0
$999$	−1149.72			−1.15087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.10		56
4.3	odd	2		112.3.l.b.13.26	yes	56
7.6	odd	2	inner	448.3.l.b.209.19		56
16.5	even	4	inner	448.3.l.b.433.19		56
16.11	odd	4		112.3.l.b.69.25	yes	56
28.27	even	2		112.3.l.b.13.25	✓	56
112.27	even	4		112.3.l.b.69.26	yes	56
112.69	odd	4	inner	448.3.l.b.433.10		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.25	✓	56	28.27	even	2
112.3.l.b.13.26	yes	56	4.3	odd	2
112.3.l.b.69.25	yes	56	16.11	odd	4
112.3.l.b.69.26	yes	56	112.27	even	4
448.3.l.b.209.10		56	1.1	even	1	trivial
448.3.l.b.209.19		56	7.6	odd	2	inner
448.3.l.b.433.10		56	112.69	odd	4	inner
448.3.l.b.433.19		56	16.5	even	4	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.75534 + 1.75534i) q^{3} +(-3.25602 - 3.25602i) q^{5} +(-4.90539 + 4.99371i) q^{7} +2.83754i q^{9} +O(q^{10})\)	\(q+(-1.75534 + 1.75534i) q^{3} +(-3.25602 - 3.25602i) q^{5} +(-4.90539 + 4.99371i) q^{7} +2.83754i q^{9} +(2.01178 - 2.01178i) q^{11} +(-6.07310 + 6.07310i) q^{13} +11.4309 q^{15} -9.55815i q^{17} +(25.2086 - 25.2086i) q^{19} +(-0.155043 - 17.3763i) q^{21} +14.0394i q^{23} -3.79671i q^{25} +(-20.7789 - 20.7789i) q^{27} +(-11.2161 - 11.2161i) q^{29} -16.5180i q^{31} +7.06271i q^{33} +(32.2316 - 0.287592i) q^{35} +(27.6654 - 27.6654i) q^{37} -21.3208i q^{39} -12.9650 q^{41} +(50.9760 - 50.9760i) q^{43} +(9.23909 - 9.23909i) q^{45} +66.5020i q^{47} +(-0.874351 - 48.9922i) q^{49} +(16.7778 + 16.7778i) q^{51} +(39.3660 - 39.3660i) q^{53} -13.1008 q^{55} +88.4994i q^{57} +(-30.4313 - 30.4313i) q^{59} +(-11.9950 + 11.9950i) q^{61} +(-14.1699 - 13.9192i) q^{63} +39.5482 q^{65} +(19.5294 + 19.5294i) q^{67} +(-24.6440 - 24.6440i) q^{69} -69.7628i q^{71} +143.326 q^{73} +(6.66453 + 6.66453i) q^{75} +(0.177693 + 19.9148i) q^{77} -66.7548 q^{79} +47.4105 q^{81} +(-60.7490 + 60.7490i) q^{83} +(-31.1215 + 31.1215i) q^{85} +39.3763 q^{87} +78.9394 q^{89} +(-0.536415 - 60.1182i) q^{91} +(28.9947 + 28.9947i) q^{93} -164.159 q^{95} +35.4878i q^{97} +(5.70850 + 5.70850i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

L-functions

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