LMFDB - Embedded newform 448.3.l.b.433.9

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			433.9
Character	$\chi$	$=$	448.433
Dual form			448.3.l.b.209.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.79667 - 1.79667i) q^{3} +(-3.85269 + 3.85269i) q^{5} +(6.95068 + 0.829499i) q^{7} -2.54394i q^{9} +O(q^{10})$	\(q+(-1.79667 - 1.79667i) q^{3} +(-3.85269 + 3.85269i) q^{5} +(6.95068 + 0.829499i) q^{7} -2.54394i q^{9} +(-14.1386 - 14.1386i) q^{11} +(5.62959 + 5.62959i) q^{13} +13.8440 q^{15} +23.7707i q^{17} +(6.44465 + 6.44465i) q^{19} +(-10.9977 - 13.9784i) q^{21} +15.1330i q^{23} -4.68642i q^{25} +(-20.7407 + 20.7407i) q^{27} +(-10.0604 + 10.0604i) q^{29} -22.6845i q^{31} +50.8047i q^{33} +(-29.9746 + 23.5830i) q^{35} +(31.9062 + 31.9062i) q^{37} -20.2290i q^{39} +49.4163 q^{41} +(34.1717 + 34.1717i) q^{43} +(9.80102 + 9.80102i) q^{45} +82.6364i q^{47} +(47.6239 + 11.5312i) q^{49} +(42.7081 - 42.7081i) q^{51} +(11.1883 + 11.1883i) q^{53} +108.943 q^{55} -23.1578i q^{57} +(-70.8011 + 70.8011i) q^{59} +(-24.6109 - 24.6109i) q^{61} +(2.11020 - 17.6821i) q^{63} -43.3781 q^{65} +(7.08521 - 7.08521i) q^{67} +(27.1891 - 27.1891i) q^{69} -43.8305i q^{71} -63.2501 q^{73} +(-8.41996 + 8.41996i) q^{75} +(-86.5447 - 110.001i) q^{77} -36.2088 q^{79} +51.6329 q^{81} +(22.7876 + 22.7876i) q^{83} +(-91.5811 - 91.5811i) q^{85} +36.1505 q^{87} +2.50810 q^{89} +(34.4597 + 43.7992i) q^{91} +(-40.7566 + 40.7566i) q^{93} -49.6585 q^{95} +135.308i q^{97} +(-35.9677 + 35.9677i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

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

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
$3$	−1.79667	−	1.79667i	−0.598890	−	0.598890i	0.341127	−	0.940017i	$-0.389191\pi$
$3$	−1.79667	−	1.79667i	−0.598890	−	0.598890i	−0.940017	+	0.341127i	$0.889191\pi$
$4$		0			0
$5$	−3.85269	+	3.85269i	−0.770538	+	0.770538i	−0.978200	−	0.207663i	$-0.933414\pi$
$5$	−3.85269	+	3.85269i	−0.770538	+	0.770538i	0.207663	+	0.978200i	$0.433414\pi$
$6$		0			0
$7$	6.95068	+	0.829499i	0.992954	+	0.118500i
$7$	6.95068	+	0.829499i	0.992954	+	0.118500i
$8$		0			0
$9$		−	2.54394i		−	0.282660i
$10$		0			0
$11$	−14.1386	−	14.1386i	−1.28532	−	1.28532i	−0.937595	−	0.347729i	$-0.886953\pi$
$11$	−14.1386	−	14.1386i	−1.28532	−	1.28532i	−0.347729	−	0.937595i	$-0.613047\pi$
$12$		0			0
$13$	5.62959	+	5.62959i	0.433045	+	0.433045i	0.889663	−	0.456618i	$-0.150939\pi$
$13$	5.62959	+	5.62959i	0.433045	+	0.433045i	−0.456618	+	0.889663i	$0.650939\pi$
$14$		0			0
$15$	13.8440			0.922935
$16$		0			0
$17$			23.7707i			1.39828i	0.714987	+	0.699138i	$0.246433\pi$
$17$			23.7707i			1.39828i	−0.714987	+	0.699138i	$0.753567\pi$
$18$		0			0
$19$	6.44465	+	6.44465i	0.339192	+	0.339192i	0.856063	−	0.516871i	$-0.172903\pi$
$19$	6.44465	+	6.44465i	0.339192	+	0.339192i	−0.516871	+	0.856063i	$0.672903\pi$
$20$		0			0
$21$	−10.9977	−	13.9784i	−0.523702	−	0.665639i
$22$		0			0
$23$			15.1330i			0.657958i	0.944337	+	0.328979i	$0.106705\pi$
$23$			15.1330i			0.657958i	−0.944337	+	0.328979i	$0.893295\pi$
$24$		0			0
$25$		−	4.68642i		−	0.187457i
$26$		0			0
$27$	−20.7407	+	20.7407i	−0.768173	+	0.768173i
$28$		0			0
$29$	−10.0604	+	10.0604i	−0.346911	+	0.346911i	−0.858958	−	0.512047i	$-0.828887\pi$
$29$	−10.0604	+	10.0604i	−0.346911	+	0.346911i	0.512047	+	0.858958i	$0.328887\pi$
$30$		0			0
$31$		−	22.6845i		−	0.731757i	−0.930663	−	0.365879i	$-0.880768\pi$
$31$		−	22.6845i		−	0.731757i	0.930663	−	0.365879i	$-0.119232\pi$
$32$		0			0
$33$			50.8047i			1.53954i
$34$		0			0
$35$	−29.9746	+	23.5830i	−0.856417	+	0.673800i
$36$		0			0
$37$	31.9062	+	31.9062i	0.862331	+	0.862331i	0.991608	−	0.129278i	$-0.0412659\pi$
$37$	31.9062	+	31.9062i	0.862331	+	0.862331i	−0.129278	+	0.991608i	$0.541266\pi$
$38$		0			0
$39$		−	20.2290i		−	0.518693i
$40$		0			0
$41$	49.4163			1.20528			0.602638	−	0.798015i	$-0.294116\pi$
$41$	49.4163			1.20528			0.602638	+	0.798015i	$0.294116\pi$
$42$		0			0
$43$	34.1717	+	34.1717i	0.794691	+	0.794691i	0.982253	−	0.187562i	$-0.0600584\pi$
$43$	34.1717	+	34.1717i	0.794691	+	0.794691i	−0.187562	+	0.982253i	$0.560058\pi$
$44$		0			0
$45$	9.80102	+	9.80102i	0.217800	+	0.217800i
$46$		0			0
$47$			82.6364i			1.75822i	0.476618	+	0.879110i	$0.341862\pi$
$47$			82.6364i			1.75822i	−0.476618	+	0.879110i	$0.658138\pi$
$48$		0			0
$49$	47.6239	+	11.5312i	0.971916	+	0.235330i
$50$		0			0
$51$	42.7081	−	42.7081i	0.837414	−	0.837414i
$52$		0			0
$53$	11.1883	+	11.1883i	0.211099	+	0.211099i	0.804734	−	0.593635i	$-0.202308\pi$
$53$	11.1883	+	11.1883i	0.211099	+	0.211099i	−0.593635	+	0.804734i	$0.702308\pi$
$54$		0			0
$55$	108.943			1.98078
$56$		0			0
$57$		−	23.1578i		−	0.406278i
$58$		0			0
$59$	−70.8011	+	70.8011i	−1.20002	+	1.20002i	−0.225857	+	0.974160i	$0.572518\pi$
$59$	−70.8011	+	70.8011i	−1.20002	+	1.20002i	−0.974160	+	0.225857i	$0.927482\pi$
$60$		0			0
$61$	−24.6109	−	24.6109i	−0.403457	−	0.403457i	0.475993	−	0.879449i	$-0.342089\pi$
$61$	−24.6109	−	24.6109i	−0.403457	−	0.403457i	−0.879449	+	0.475993i	$0.842089\pi$
$62$		0			0
$63$	2.11020	−	17.6821i	0.0334952	−	0.280669i
$64$		0			0
$65$	−43.3781			−0.667355
$66$		0			0
$67$	7.08521	−	7.08521i	0.105749	−	0.105749i	−0.652252	−	0.758002i	$-0.726176\pi$
$67$	7.08521	−	7.08521i	0.105749	−	0.105749i	0.758002	+	0.652252i	$0.226176\pi$
$68$		0			0
$69$	27.1891	−	27.1891i	0.394045	−	0.394045i
$70$		0			0
$71$		−	43.8305i		−	0.617330i	−0.951171	−	0.308665i	$-0.900118\pi$
$71$		−	43.8305i		−	0.617330i	0.951171	−	0.308665i	$-0.0998823\pi$
$72$		0			0
$73$	−63.2501			−0.866440			−0.433220	−	0.901288i	$-0.642623\pi$
$73$	−63.2501			−0.866440			−0.433220	+	0.901288i	$0.642623\pi$
$74$		0			0
$75$	−8.41996	+	8.41996i	−0.112266	+	0.112266i
$76$		0			0
$77$	−86.5447	−	110.001i	−1.12396	−	1.42858i
$78$		0			0
$79$	−36.2088			−0.458339			−0.229169	−	0.973387i	$-0.573601\pi$
$79$	−36.2088			−0.458339			−0.229169	+	0.973387i	$0.573601\pi$
$80$		0			0
$81$	51.6329			0.637443
$82$		0			0
$83$	22.7876	+	22.7876i	0.274550	+	0.274550i	0.830929	−	0.556379i	$-0.187810\pi$
$83$	22.7876	+	22.7876i	0.274550	+	0.274550i	−0.556379	+	0.830929i	$0.687810\pi$
$84$		0			0
$85$	−91.5811	−	91.5811i	−1.07742	−	1.07742i
$86$		0			0
$87$	36.1505			0.415523
$88$		0			0
$89$	2.50810			0.0281809			0.0140905	−	0.999901i	$-0.495515\pi$
$89$	2.50810			0.0281809			0.0140905	+	0.999901i	$0.495515\pi$
$90$		0			0
$91$	34.4597	+	43.7992i	0.378678	+	0.481310i
$92$		0			0
$93$	−40.7566	+	40.7566i	−0.438243	+	0.438243i
$94$		0			0
$95$	−49.6585			−0.522721
$96$		0			0
$97$			135.308i			1.39493i	0.716619	+	0.697465i	$0.245689\pi$
$97$			135.308i			1.39493i	−0.716619	+	0.697465i	$0.754311\pi$
$98$		0			0
$99$	−35.9677	+	35.9677i	−0.363310	+	0.363310i
$100$		0			0
$101$	54.4429	−	54.4429i	0.539039	−	0.539039i	−0.384208	−	0.923247i	$-0.625525\pi$
$101$	54.4429	−	54.4429i	0.539039	−	0.539039i	0.923247	+	0.384208i	$0.125525\pi$
$102$		0			0
$103$	29.9414			0.290693			0.145347	−	0.989381i	$-0.453570\pi$
$103$	29.9414			0.290693			0.145347	+	0.989381i	$0.453570\pi$
$104$		0			0
$105$	96.2254	+	11.4836i	0.916433	+	0.109368i
$106$		0			0
$107$	−14.5347	−	14.5347i	−0.135838	−	0.135838i	0.635918	−	0.771756i	$-0.280622\pi$
$107$	−14.5347	−	14.5347i	−0.135838	−	0.135838i	−0.771756	+	0.635918i	$0.780622\pi$
$108$		0			0
$109$	26.3830	−	26.3830i	0.242046	−	0.242046i	−0.575650	−	0.817696i	$-0.695251\pi$
$109$	26.3830	−	26.3830i	0.242046	−	0.242046i	0.817696	+	0.575650i	$0.195251\pi$
$110$		0			0
$111$		−	114.650i		−	1.03288i
$112$		0			0
$113$	−150.175			−1.32899			−0.664493	−	0.747295i	$-0.731352\pi$
$113$	−150.175			−1.32899			−0.664493	+	0.747295i	$0.731352\pi$
$114$		0			0
$115$	−58.3029	−	58.3029i	−0.506982	−	0.506982i
$116$		0			0
$117$	14.3214	−	14.3214i	0.122405	−	0.122405i
$118$		0			0
$119$	−19.7178	+	165.222i	−0.165696	+	1.38842i
$120$		0			0
$121$			278.798i			2.30412i
$122$		0			0
$123$	−88.7849	−	88.7849i	−0.721828	−	0.721828i
$124$		0			0
$125$	−78.2619	−	78.2619i	−0.626095	−	0.626095i
$126$		0			0
$127$	25.8584			0.203610			0.101805	−	0.994804i	$-0.467538\pi$
$127$	25.8584			0.203610			0.101805	+	0.994804i	$0.467538\pi$
$128$		0			0
$129$		−	122.791i		−	0.951866i
$130$		0			0
$131$	−116.316	−	116.316i	−0.887905	−	0.887905i	0.106417	−	0.994322i	$-0.466062\pi$
$131$	−116.316	−	116.316i	−0.887905	−	0.887905i	−0.994322	+	0.106417i	$0.966062\pi$
$132$		0			0
$133$	39.4489	+	50.1405i	0.296608	+	0.376996i
$134$		0			0
$135$		−	159.815i		−	1.18381i
$136$		0			0
$137$		−	187.219i		−	1.36656i	−0.730155	−	0.683281i	$-0.760552\pi$
$137$		−	187.219i		−	1.36656i	0.730155	−	0.683281i	$-0.239448\pi$
$138$		0			0
$139$	−36.2417	+	36.2417i	−0.260732	+	0.260732i	−0.825351	−	0.564620i	$-0.809023\pi$
$139$	−36.2417	+	36.2417i	−0.260732	+	0.260732i	0.564620	+	0.825351i	$0.309023\pi$
$140$		0			0
$141$	148.470	−	148.470i	1.05298	−	1.05298i
$142$		0			0
$143$		−	159.189i		−	1.11321i
$144$		0			0
$145$		−	77.5193i		−	0.534616i
$146$		0			0
$147$	−64.8467	−	106.282i	−0.441134	−	0.723008i
$148$		0			0
$149$	110.324	+	110.324i	0.740431	+	0.740431i	0.972661	−	0.232230i	$-0.0746021\pi$
$149$	110.324	+	110.324i	0.740431	+	0.740431i	−0.232230	+	0.972661i	$0.574602\pi$
$150$		0			0
$151$		−	79.7934i		−	0.528433i	−0.964463	−	0.264216i	$-0.914887\pi$
$151$		−	79.7934i		−	0.528433i	0.964463	−	0.264216i	$-0.0851133\pi$
$152$		0			0
$153$	60.4713			0.395237
$154$		0			0
$155$	87.3962	+	87.3962i	0.563847	+	0.563847i
$156$		0			0
$157$	−25.6416	−	25.6416i	−0.163322	−	0.163322i	0.620715	−	0.784037i	$-0.286843\pi$
$157$	−25.6416	−	25.6416i	−0.163322	−	0.163322i	−0.784037	+	0.620715i	$0.786843\pi$
$158$		0			0
$159$		−	40.2033i		−	0.252851i
$160$		0			0
$161$	−12.5528	+	105.185i	−0.0779680	+	0.653322i
$162$		0			0
$163$	−128.482	+	128.482i	−0.788236	+	0.788236i	−0.981205	−	0.192969i	$-0.938188\pi$
$163$	−128.482	+	128.482i	−0.788236	+	0.788236i	0.192969	+	0.981205i	$0.438188\pi$
$164$		0			0
$165$	−195.735	−	195.735i	−1.18627	−	1.18627i
$166$		0			0
$167$	−66.9309			−0.400784			−0.200392	−	0.979716i	$-0.564222\pi$
$167$	−66.9309			−0.400784			−0.200392	+	0.979716i	$0.564222\pi$
$168$		0			0
$169$		−	105.615i		−	0.624944i
$170$		0			0
$171$	16.3948	−	16.3948i	0.0958762	−	0.0958762i
$172$		0			0
$173$	104.632	+	104.632i	0.604808	+	0.604808i	0.941585	−	0.336777i	$-0.109337\pi$
$173$	104.632	+	104.632i	0.604808	+	0.604808i	−0.336777	+	0.941585i	$0.609337\pi$
$174$		0			0
$175$	3.88739	−	32.5738i	0.0222136	−	0.186136i
$176$		0			0
$177$	254.412			1.43736
$178$		0			0
$179$	134.155	−	134.155i	0.749470	−	0.749470i	−0.224910	−	0.974380i	$-0.572209\pi$
$179$	134.155	−	134.155i	0.749470	−	0.749470i	0.974380	+	0.224910i	$0.0722087\pi$
$180$		0			0
$181$	−17.0299	+	17.0299i	−0.0940877	+	0.0940877i	−0.752584	−	0.658496i	$-0.771193\pi$
$181$	−17.0299	+	17.0299i	−0.0940877	+	0.0940877i	0.658496	+	0.752584i	$0.271193\pi$
$182$		0			0
$183$			88.4352i			0.483253i
$184$		0			0
$185$	−245.850			−1.32892
$186$		0			0
$187$	336.083	−	336.083i	1.79724	−	1.79724i
$188$		0			0
$189$	−161.366	+	126.957i	−0.853789	+	0.671732i
$190$		0			0
$191$	173.782			0.909852			0.454926	−	0.890529i	$-0.349666\pi$
$191$	173.782			0.909852			0.454926	+	0.890529i	$0.349666\pi$
$192$		0			0
$193$	−31.4049			−0.162720			−0.0813598	−	0.996685i	$-0.525926\pi$
$193$	−31.4049			−0.162720			−0.0813598	+	0.996685i	$0.525926\pi$
$194$		0			0
$195$	77.9362	+	77.9362i	0.399673	+	0.399673i
$196$		0			0
$197$	212.604	+	212.604i	1.07921	+	1.07921i	0.996581	+	0.0826273i	$0.0263311\pi$
$197$	212.604	+	212.604i	1.07921	+	1.07921i	0.0826273	+	0.996581i	$0.473669\pi$
$198$		0			0
$199$	97.9436			0.492179			0.246089	−	0.969247i	$-0.420854\pi$
$199$	97.9436			0.492179			0.246089	+	0.969247i	$0.420854\pi$
$200$		0			0
$201$	−25.4596			−0.126665
$202$		0			0
$203$	−78.2719	+	61.5816i	−0.385576	+	0.303358i
$204$		0			0
$205$	−190.386	+	190.386i	−0.928711	+	0.928711i
$206$		0			0
$207$	38.4976			0.185979
$208$		0			0
$209$		−	182.236i		−	0.871943i
$210$		0			0
$211$	−23.3403	+	23.3403i	−0.110617	+	0.110617i	−0.760249	−	0.649632i	$-0.774923\pi$
$211$	−23.3403	+	23.3403i	−0.110617	+	0.110617i	0.649632	+	0.760249i	$0.274923\pi$
$212$		0			0
$213$	−78.7489	+	78.7489i	−0.369713	+	0.369713i
$214$		0			0
$215$	−263.306			−1.22468
$216$		0			0
$217$	18.8168	−	157.673i	0.0867132	−	0.726602i
$218$		0			0
$219$	113.640	+	113.640i	0.518903	+	0.518903i
$220$		0			0
$221$	−133.819	+	133.819i	−0.605517	+	0.605517i
$222$		0			0
$223$			57.9119i			0.259695i	0.991534	+	0.129847i	$0.0414487\pi$
$223$			57.9119i			0.259695i	−0.991534	+	0.129847i	$0.958551\pi$
$224$		0			0
$225$	−11.9220			−0.0529866
$226$		0			0
$227$	103.905	+	103.905i	0.457732	+	0.457732i	0.897910	−	0.440179i	$-0.145085\pi$
$227$	103.905	+	103.905i	0.457732	+	0.457732i	−0.440179	+	0.897910i	$0.645085\pi$
$228$		0			0
$229$	174.387	−	174.387i	0.761517	−	0.761517i	−0.215080	−	0.976596i	$-0.569001\pi$
$229$	174.387	−	174.387i	0.761517	−	0.761517i	0.976596	+	0.215080i	$0.0690011\pi$
$230$		0			0
$231$	−42.1425	+	353.127i	−0.182435	+	1.52869i
$232$		0			0
$233$			63.9575i			0.274496i	0.990537	+	0.137248i	$0.0438257\pi$
$233$			63.9575i			0.274496i	−0.990537	+	0.137248i	$0.956174\pi$
$234$		0			0
$235$	−318.372	−	318.372i	−1.35478	−	1.35478i
$236$		0			0
$237$	65.0552	+	65.0552i	0.274495	+	0.274495i
$238$		0			0
$239$	−316.396			−1.32383			−0.661917	−	0.749577i	$-0.730257\pi$
$239$	−316.396			−1.32383			−0.661917	+	0.749577i	$0.730257\pi$
$240$		0			0
$241$		−	110.426i		−	0.458199i	−0.973403	−	0.229100i	$-0.926422\pi$
$241$		−	110.426i		−	0.458199i	0.973403	−	0.229100i	$-0.0735781\pi$
$242$		0			0
$243$	93.8988	+	93.8988i	0.386415	+	0.386415i
$244$		0			0
$245$	−227.906	+	139.054i	−0.930228	+	0.567567i
$246$		0			0
$247$			72.5614i			0.293771i
$248$		0			0
$249$		−	81.8837i		−	0.328850i
$250$		0			0
$251$	−219.315	+	219.315i	−0.873767	+	0.873767i	−0.992881	−	0.119114i	$-0.961995\pi$
$251$	−219.315	+	219.315i	−0.873767	+	0.873767i	0.119114	+	0.992881i	$0.461995\pi$
$252$		0			0
$253$	213.959	−	213.959i	0.845689	−	0.845689i
$254$		0			0
$255$			329.082i			1.29052i
$256$		0			0
$257$		−	215.027i		−	0.836682i	−0.908290	−	0.418341i	$-0.862612\pi$
$257$		−	215.027i		−	0.836682i	0.908290	−	0.418341i	$-0.137388\pi$
$258$		0			0
$259$	195.304	+	248.236i	0.754069	+	0.958441i
$260$		0			0
$261$	25.5931	+	25.5931i	0.0980580	+	0.0980580i
$262$		0			0
$263$		−	207.720i		−	0.789810i	−0.918722	−	0.394905i	$-0.870778\pi$
$263$		−	207.720i		−	0.789810i	0.918722	−	0.394905i	$-0.129222\pi$
$264$		0			0
$265$	−86.2099			−0.325320
$266$		0			0
$267$	−4.50624	−	4.50624i	−0.0168773	−	0.0168773i
$268$		0			0
$269$	253.723	+	253.723i	0.943208	+	0.943208i	0.998472	−	0.0552633i	$-0.0175998\pi$
$269$	253.723	+	253.723i	0.943208	+	0.943208i	−0.0552633	+	0.998472i	$0.517600\pi$
$270$		0			0
$271$			420.547i			1.55183i	0.630835	+	0.775917i	$0.282712\pi$
$271$			420.547i			1.55183i	−0.630835	+	0.775917i	$0.717288\pi$
$272$		0			0
$273$	16.7800	−	140.606i	0.0614651	−	0.515039i
$274$		0			0
$275$	−66.2593	+	66.2593i	−0.240943	+	0.240943i
$276$		0			0
$277$	−170.534	−	170.534i	−0.615645	−	0.615645i	0.328766	−	0.944411i	$-0.393367\pi$
$277$	−170.534	−	170.534i	−0.615645	−	0.615645i	−0.944411	+	0.328766i	$0.893367\pi$
$278$		0			0
$279$	−57.7080			−0.206839
$280$		0			0
$281$		−	25.6535i		−	0.0912937i	−0.998958	−	0.0456469i	$-0.985465\pi$
$281$		−	25.6535i		−	0.0912937i	0.998958	−	0.0456469i	$-0.0145349\pi$
$282$		0			0
$283$	251.004	−	251.004i	0.886941	−	0.886941i	−0.107287	−	0.994228i	$-0.534216\pi$
$283$	251.004	−	251.004i	0.886941	−	0.886941i	0.994228	+	0.107287i	$0.0342164\pi$
$284$		0			0
$285$	89.2199	+	89.2199i	0.313052	+	0.313052i
$286$		0			0
$287$	343.477	+	40.9908i	1.19678	+	0.142825i
$288$		0			0
$289$	−276.046			−0.955176
$290$		0			0
$291$	243.104	−	243.104i	0.835411	−	0.835411i
$292$		0			0
$293$	−211.178	+	211.178i	−0.720743	+	0.720743i	−0.968756	−	0.248014i	$-0.920222\pi$
$293$	−211.178	+	211.178i	−0.720743	+	0.720743i	0.248014	+	0.968756i	$0.420222\pi$
$294$		0			0
$295$		−	545.549i		−	1.84932i
$296$		0			0
$297$	586.487			1.97470
$298$		0			0
$299$	−85.1928	+	85.1928i	−0.284926	+	0.284926i
$300$		0			0
$301$	209.171	+	265.862i	0.694921	+	0.883263i
$302$		0			0
$303$	−195.632			−0.645650
$304$		0			0
$305$	189.636			0.621757
$306$		0			0
$307$	−96.0938	−	96.0938i	−0.313009	−	0.313009i	0.533065	−	0.846074i	$-0.321040\pi$
$307$	−96.0938	−	96.0938i	−0.313009	−	0.313009i	−0.846074	+	0.533065i	$0.821040\pi$
$308$		0			0
$309$	−53.7949	−	53.7949i	−0.174094	−	0.174094i
$310$		0			0
$311$	−538.153			−1.73040			−0.865198	−	0.501431i	$-0.832807\pi$
$311$	−538.153			−1.73040			−0.865198	+	0.501431i	$0.832807\pi$
$312$		0			0
$313$	−435.372			−1.39097			−0.695483	−	0.718543i	$-0.744809\pi$
$313$	−435.372			−1.39097			−0.695483	+	0.718543i	$0.744809\pi$
$314$		0			0
$315$	59.9938	+	76.2537i	0.190457	+	0.242075i
$316$		0			0
$317$	430.328	−	430.328i	1.35750	−	1.35750i	0.480513	−	0.876988i	$-0.340450\pi$
$317$	430.328	−	430.328i	1.35750	−	1.35750i	0.876988	−	0.480513i	$-0.159550\pi$
$318$		0			0
$319$	284.480			0.891786
$320$		0			0
$321$			52.2281i			0.162704i
$322$		0			0
$323$	−153.194	+	153.194i	−0.474284	+	0.474284i
$324$		0			0
$325$	26.3826	−	26.3826i	0.0811773	−	0.0811773i
$326$		0			0
$327$	−94.8033			−0.289918
$328$		0			0
$329$	−68.5468	+	574.379i	−0.208349	+	1.74583i
$330$		0			0
$331$	160.024	+	160.024i	0.483455	+	0.483455i	0.906233	−	0.422778i	$-0.138945\pi$
$331$	160.024	+	160.024i	0.483455	+	0.483455i	−0.422778	+	0.906233i	$0.638945\pi$
$332$		0			0
$333$	81.1676	−	81.1676i	0.243747	−	0.243747i
$334$		0			0
$335$			54.5942i			0.162968i
$336$		0			0
$337$	−615.143			−1.82535			−0.912675	−	0.408686i	$-0.865987\pi$
$337$	−615.143			−1.82535			−0.912675	+	0.408686i	$0.865987\pi$
$338$		0			0
$339$	269.816	+	269.816i	0.795917	+	0.795917i
$340$		0			0
$341$	−320.726	+	320.726i	−0.940545	+	0.940545i
$342$		0			0
$343$	321.453	+	119.653i	0.937181	+	0.348844i
$344$		0			0
$345$			209.502i			0.607253i
$346$		0			0
$347$	−54.3434	−	54.3434i	−0.156609	−	0.156609i	0.624453	−	0.781062i	$-0.285322\pi$
$347$	−54.3434	−	54.3434i	−0.156609	−	0.156609i	−0.781062	+	0.624453i	$0.785322\pi$
$348$		0			0
$349$	−43.8104	−	43.8104i	−0.125531	−	0.125531i	0.641550	−	0.767081i	$-0.278292\pi$
$349$	−43.8104	−	43.8104i	−0.125531	−	0.125531i	−0.767081	+	0.641550i	$0.778292\pi$
$350$		0			0
$351$	−233.523			−0.665307
$352$		0			0
$353$			508.212i			1.43969i	0.694133	+	0.719847i	$0.255788\pi$
$353$			508.212i			1.43969i	−0.694133	+	0.719847i	$0.744212\pi$
$354$		0			0
$355$	168.865	+	168.865i	0.475676	+	0.475676i
$356$		0			0
$357$	332.277	−	261.424i	0.930747	−	0.732280i
$358$		0			0
$359$			637.838i			1.77671i	0.459159	+	0.888354i	$0.348151\pi$
$359$			637.838i			1.77671i	−0.459159	+	0.888354i	$0.651849\pi$
$360$		0			0
$361$		−	277.933i		−	0.769897i
$362$		0			0
$363$	500.908	−	500.908i	1.37991	−	1.37991i
$364$		0			0
$365$	243.683	−	243.683i	0.667625	−	0.667625i
$366$		0			0
$367$			306.129i			0.834138i	0.908875	+	0.417069i	$0.136943\pi$
$367$			306.129i			0.834138i	−0.908875	+	0.417069i	$0.863057\pi$
$368$		0			0
$369$		−	125.712i		−	0.340684i
$370$		0			0
$371$	68.4854	+	87.0467i	0.184597	+	0.234627i
$372$		0			0
$373$	487.497	+	487.497i	1.30696	+	1.30696i	0.923596	+	0.383366i	$0.125235\pi$
$373$	487.497	+	487.497i	1.30696	+	1.30696i	0.383366	+	0.923596i	$0.374765\pi$
$374$		0			0
$375$			281.222i			0.749925i
$376$		0			0
$377$	−113.272			−0.300456
$378$		0			0
$379$	260.138	+	260.138i	0.686379	+	0.686379i	0.961430	−	0.275050i	$-0.0886945\pi$
$379$	260.138	+	260.138i	0.686379	+	0.686379i	−0.275050	+	0.961430i	$0.588695\pi$
$380$		0			0
$381$	−46.4591	−	46.4591i	−0.121940	−	0.121940i
$382$		0			0
$383$		−	409.986i		−	1.07046i	−0.844706	−	0.535230i	$-0.820225\pi$
$383$		−	409.986i		−	1.07046i	0.844706	−	0.535230i	$-0.179775\pi$
$384$		0			0
$385$	757.228	+	90.3681i	1.96682	+	0.234722i
$386$		0			0
$387$	86.9309	−	86.9309i	0.224628	−	0.224628i
$388$		0			0
$389$	−294.541	−	294.541i	−0.757175	−	0.757175i	0.218632	−	0.975807i	$-0.429840\pi$
$389$	−294.541	−	294.541i	−0.757175	−	0.757175i	−0.975807	+	0.218632i	$0.929840\pi$
$390$		0			0
$391$	−359.723			−0.920007
$392$		0			0
$393$			417.962i			1.06352i
$394$		0			0
$395$	139.501	−	139.501i	0.353167	−	0.353167i
$396$		0			0
$397$	441.709	+	441.709i	1.11262	+	1.11262i	0.992796	+	0.119821i	$0.0382321\pi$
$397$	441.709	+	441.709i	1.11262	+	1.11262i	0.119821	+	0.992796i	$0.461768\pi$
$398$		0			0
$399$	19.2094	−	160.963i	0.0481439	−	0.403415i
$400$		0			0
$401$	−288.066			−0.718369			−0.359185	−	0.933266i	$-0.616945\pi$
$401$	−288.066			−0.718369			−0.359185	+	0.933266i	$0.616945\pi$
$402$		0			0
$403$	127.704	−	127.704i	0.316884	−	0.316884i
$404$		0			0
$405$	−198.925	+	198.925i	−0.491174	+	0.491174i
$406$		0			0
$407$		−	902.217i		−	2.21675i
$408$		0			0
$409$	−108.577			−0.265469			−0.132734	−	0.991152i	$-0.542376\pi$
$409$	−108.577			−0.265469			−0.132734	+	0.991152i	$0.542376\pi$
$410$		0			0
$411$	−336.371	+	336.371i	−0.818421	+	0.818421i
$412$		0			0
$413$	−550.845	+	433.386i	−1.33376	+	1.04936i
$414$		0			0
$415$	−175.587			−0.423102
$416$		0			0
$417$	130.229			0.312299
$418$		0			0
$419$	−217.069	−	217.069i	−0.518063	−	0.518063i	0.398922	−	0.916985i	$-0.369385\pi$
$419$	−217.069	−	217.069i	−0.518063	−	0.518063i	−0.916985	+	0.398922i	$0.869385\pi$
$420$		0			0
$421$	52.0878	+	52.0878i	0.123724	+	0.123724i	0.766258	−	0.642534i	$-0.222117\pi$
$421$	52.0878	+	52.0878i	0.123724	+	0.123724i	−0.642534	+	0.766258i	$0.722117\pi$
$422$		0			0
$423$	210.222			0.496979
$424$		0			0
$425$	111.400			0.262117
$426$		0			0
$427$	−150.647	−	191.477i	−0.352804	−	0.448423i
$428$		0			0
$429$	−286.010	+	286.010i	−0.666689	+	0.666689i
$430$		0			0
$431$	−456.883			−1.06005			−0.530027	−	0.847981i	$-0.677818\pi$
$431$	−456.883			−1.06005			−0.530027	+	0.847981i	$0.677818\pi$
$432$		0			0
$433$		−	315.767i		−	0.729254i	−0.931154	−	0.364627i	$-0.881196\pi$
$433$		−	315.767i		−	0.729254i	0.931154	−	0.364627i	$-0.118804\pi$
$434$		0			0
$435$	−139.277	+	139.277i	−0.320176	+	0.320176i
$436$		0			0
$437$	−97.5271	+	97.5271i	−0.223174	+	0.223174i
$438$		0			0
$439$	513.805			1.17040			0.585200	−	0.810889i	$-0.301016\pi$
$439$	513.805			1.17040			0.585200	+	0.810889i	$0.301016\pi$
$440$		0			0
$441$	29.3346	−	121.152i	0.0665184	−	0.274722i
$442$		0			0
$443$	215.639	+	215.639i	0.486770	+	0.486770i	0.907285	−	0.420516i	$-0.138151\pi$
$443$	215.639	+	215.639i	0.486770	+	0.486770i	−0.420516	+	0.907285i	$0.638151\pi$
$444$		0			0
$445$	−9.66294	+	9.66294i	−0.0217145	+	0.0217145i
$446$		0			0
$447$		−	396.433i		−	0.886874i
$448$		0			0
$449$	267.966			0.596805			0.298403	−	0.954440i	$-0.403546\pi$
$449$	267.966			0.596805			0.298403	+	0.954440i	$0.403546\pi$
$450$		0			0
$451$	−698.676	−	698.676i	−1.54917	−	1.54917i
$452$		0			0
$453$	−143.362	+	143.362i	−0.316473	+	0.316473i
$454$		0			0
$455$	−301.507	−	35.9821i	−0.662653	−	0.0790816i
$456$		0			0
$457$		−	549.921i		−	1.20333i	−0.798749	−	0.601664i	$-0.794504\pi$
$457$		−	549.921i		−	1.20333i	0.798749	−	0.601664i	$-0.205496\pi$
$458$		0			0
$459$	−493.020	−	493.020i	−1.07412	−	1.07412i
$460$		0			0
$461$	−159.019	−	159.019i	−0.344944	−	0.344944i	0.513279	−	0.858222i	$-0.328431\pi$
$461$	−159.019	−	159.019i	−0.344944	−	0.344944i	−0.858222	+	0.513279i	$0.828431\pi$
$462$		0			0
$463$	499.045			1.07785			0.538926	−	0.842353i	$-0.318830\pi$
$463$	499.045			1.07785			0.538926	+	0.842353i	$0.318830\pi$
$464$		0			0
$465$		−	314.045i		−	0.675365i
$466$		0			0
$467$	−198.453	−	198.453i	−0.424953	−	0.424953i	0.461952	−	0.886905i	$-0.347149\pi$
$467$	−198.453	−	198.453i	−0.424953	−	0.424953i	−0.886905	+	0.461952i	$0.847149\pi$
$468$		0			0
$469$	55.1242	−	43.3699i	0.117536	−	0.0924731i
$470$		0			0
$471$			92.1389i			0.195624i
$472$		0			0
$473$		−	966.278i		−	2.04287i
$474$		0			0
$475$	30.2024	−	30.2024i	0.0635839	−	0.0635839i
$476$		0			0
$477$	28.4623	−	28.4623i	0.0596694	−	0.0596694i
$478$		0			0
$479$			9.68687i			0.0202231i	0.999949	+	0.0101116i	$0.00321866\pi$
$479$			9.68687i			0.0202231i	−0.999949	+	0.0101116i	$0.996781\pi$
$480$		0			0
$481$			359.238i			0.746856i
$482$		0			0
$483$	211.536	−	166.429i	0.437963	−	0.344574i
$484$		0			0
$485$	−521.301	−	521.301i	−1.07485	−	1.07485i
$486$		0			0
$487$			489.167i			1.00445i	0.864737	+	0.502225i	$0.167485\pi$
$487$			489.167i			1.00445i	−0.864737	+	0.502225i	$0.832515\pi$
$488$		0			0
$489$	461.681			0.944134
$490$		0			0
$491$	367.418	+	367.418i	0.748305	+	0.748305i	0.974161	−	0.225856i	$-0.0725179\pi$
$491$	367.418	+	367.418i	0.748305	+	0.748305i	−0.225856	+	0.974161i	$0.572518\pi$
$492$		0			0
$493$	−239.143	−	239.143i	−0.485077	−	0.485077i
$494$		0			0
$495$		−	277.145i		−	0.559888i
$496$		0			0
$497$	36.3573	−	304.651i	0.0731536	−	0.612981i
$498$		0			0
$499$	−146.092	+	146.092i	−0.292770	+	0.292770i	−0.838173	−	0.545404i	$-0.816376\pi$
$499$	−146.092	+	146.092i	−0.292770	+	0.292770i	0.545404	+	0.838173i	$0.316376\pi$
$500$		0			0
$501$	120.253	+	120.253i	0.240026	+	0.240026i
$502$		0			0
$503$	74.6597			0.148429			0.0742144	−	0.997242i	$-0.476355\pi$
$503$	74.6597			0.148429			0.0742144	+	0.997242i	$0.476355\pi$
$504$		0			0
$505$			419.503i			0.830700i
$506$		0			0
$507$	−189.756	+	189.756i	−0.374273	+	0.374273i
$508$		0			0
$509$	−694.711	−	694.711i	−1.36486	−	1.36486i	−0.867610	−	0.497246i	$-0.834345\pi$
$509$	−694.711	−	694.711i	−1.36486	−	1.36486i	−0.497246	−	0.867610i	$-0.665655\pi$
$510$		0			0
$511$	−439.631	−	52.4659i	−0.860335	−	0.102673i
$512$		0			0
$513$	−267.333			−0.521116
$514$		0			0
$515$	−115.355	+	115.355i	−0.223990	+	0.223990i
$516$		0			0
$517$	1168.36	−	1168.36i	2.25988	−	2.25988i
$518$		0			0
$519$		−	375.978i		−	0.724427i
$520$		0			0
$521$	267.800			0.514012			0.257006	−	0.966410i	$-0.417264\pi$
$521$	267.800			0.514012			0.257006	+	0.966410i	$0.417264\pi$
$522$		0			0
$523$	−376.512	+	376.512i	−0.719907	+	0.719907i	−0.968586	−	0.248679i	$-0.920004\pi$
$523$	−376.512	+	376.512i	−0.719907	+	0.719907i	0.248679	+	0.968586i	$0.420004\pi$
$524$		0			0
$525$	−65.5088	+	51.5401i	−0.124779	+	0.0981716i
$526$		0			0
$527$	539.226			1.02320
$528$		0			0
$529$	299.991			0.567091
$530$		0			0
$531$	180.114	+	180.114i	0.339197	+	0.339197i
$532$		0			0
$533$	278.193	+	278.193i	0.521939	+	0.521939i
$534$		0			0
$535$	111.995			0.209337
$536$		0			0
$537$	−482.065			−0.897701
$538$		0			0
$539$	−510.299	−	836.367i	−0.946751	−	1.55170i
$540$		0			0
$541$	293.796	−	293.796i	0.543062	−	0.543062i	−0.381363	−	0.924425i	$-0.624545\pi$
$541$	293.796	−	293.796i	0.543062	−	0.543062i	0.924425	+	0.381363i	$0.124545\pi$
$542$		0			0
$543$	61.1942			0.112696
$544$		0			0
$545$			203.291i			0.373011i
$546$		0			0
$547$	−37.1542	+	37.1542i	−0.0679236	+	0.0679236i	−0.740253	−	0.672329i	$-0.765294\pi$
$547$	−37.1542	+	37.1542i	−0.0679236	+	0.0679236i	0.672329	+	0.740253i	$0.265294\pi$
$548$		0			0
$549$	−62.6086	+	62.6086i	−0.114041	+	0.114041i
$550$		0			0
$551$	−129.672			−0.235339
$552$		0			0
$553$	−251.675	−	30.0351i	−0.455109	−	0.0543131i
$554$		0			0
$555$	441.711	+	441.711i	0.795876	+	0.795876i
$556$		0			0
$557$	−550.381	+	550.381i	−0.988116	+	0.988116i	−0.999930	−	0.0118143i	$-0.996239\pi$
$557$	−550.381	+	550.381i	−0.988116	+	0.988116i	0.0118143	+	0.999930i	$0.496239\pi$
$558$		0			0
$559$			384.745i			0.688275i
$560$		0			0
$561$	−1207.66			−2.15270
$562$		0			0
$563$	382.090	+	382.090i	0.678668	+	0.678668i	0.959699	−	0.281031i	$-0.0906764\pi$
$563$	382.090	+	382.090i	0.678668	+	0.678668i	−0.281031	+	0.959699i	$0.590676\pi$
$564$		0			0
$565$	578.579	−	578.579i	1.02403	−	1.02403i
$566$		0			0
$567$	358.883	+	42.8294i	0.632951	+	0.0755369i
$568$		0			0
$569$			275.833i			0.484769i	0.970180	+	0.242384i	$0.0779295\pi$
$569$			275.833i			0.484769i	−0.970180	+	0.242384i	$0.922071\pi$
$570$		0			0
$571$	−277.271	−	277.271i	−0.485588	−	0.485588i	0.421323	−	0.906911i	$-0.361566\pi$
$571$	−277.271	−	277.271i	−0.485588	−	0.485588i	−0.906911	+	0.421323i	$0.861566\pi$
$572$		0			0
$573$	−312.229	−	312.229i	−0.544902	−	0.544902i
$574$		0			0
$575$	70.9198			0.123339
$576$		0			0
$577$			239.470i			0.415026i	0.978232	+	0.207513i	$0.0665369\pi$
$577$			239.470i			0.415026i	−0.978232	+	0.207513i	$0.933463\pi$
$578$		0			0
$579$	56.4242	+	56.4242i	0.0974512	+	0.0974512i
$580$		0			0
$581$	139.487	+	177.292i	0.240081	+	0.305149i
$582$		0			0
$583$		−	316.372i		−	0.542662i
$584$		0			0
$585$			110.351i			0.188635i
$586$		0			0
$587$	−404.308	+	404.308i	−0.688770	+	0.688770i	−0.961960	−	0.273190i	$-0.911921\pi$
$587$	−404.308	+	404.308i	−0.688770	+	0.688770i	0.273190	+	0.961960i	$0.411921\pi$
$588$		0			0
$589$	146.194	−	146.194i	0.248206	−	0.248206i
$590$		0			0
$591$		−	763.959i		−	1.29265i
$592$		0			0
$593$		−	56.8936i		−	0.0959419i	−0.998849	−	0.0479710i	$-0.984725\pi$
$593$		−	56.8936i		−	0.0959419i	0.998849	−	0.0479710i	$-0.0152755\pi$
$594$		0			0
$595$	−560.584	−	712.517i	−0.942158	−	1.19751i
$596$		0			0
$597$	−175.973	−	175.973i	−0.294761	−	0.294761i
$598$		0			0
$599$			376.943i			0.629287i	0.949210	+	0.314644i	$0.101885\pi$
$599$			376.943i			0.629287i	−0.949210	+	0.314644i	$0.898115\pi$
$600$		0			0
$601$	−840.880			−1.39913			−0.699567	−	0.714567i	$-0.746624\pi$
$601$	−840.880			−1.39913			−0.699567	+	0.714567i	$0.746624\pi$
$602$		0			0
$603$	−18.0244	−	18.0244i	−0.0298912	−	0.0298912i
$604$		0			0
$605$	−1074.12	−	1074.12i	−1.77541	−	1.77541i
$606$		0			0
$607$		−	375.286i		−	0.618263i	−0.951019	−	0.309132i	$-0.899962\pi$
$607$		−	375.286i		−	0.618263i	0.951019	−	0.309132i	$-0.100038\pi$
$608$		0			0
$609$	251.271	+	29.9868i	0.412596	+	0.0492395i
$610$		0			0
$611$	−465.209	+	465.209i	−0.761389	+	0.761389i
$612$		0			0
$613$	−243.106	−	243.106i	−0.396585	−	0.396585i	0.480442	−	0.877027i	$-0.340476\pi$
$613$	−243.106	−	243.106i	−0.396585	−	0.396585i	−0.877027	+	0.480442i	$0.840476\pi$
$614$		0			0
$615$	684.121			1.11239
$616$		0			0
$617$			384.257i			0.622783i	0.950282	+	0.311392i	$0.100795\pi$
$617$			384.257i			0.622783i	−0.950282	+	0.311392i	$0.899205\pi$
$618$		0			0
$619$	17.1513	−	17.1513i	0.0277081	−	0.0277081i	−0.693117	−	0.720825i	$-0.743763\pi$
$619$	17.1513	−	17.1513i	0.0277081	−	0.0277081i	0.720825	+	0.693117i	$0.243763\pi$
$620$		0			0
$621$	−313.869	−	313.869i	−0.505426	−	0.505426i
$622$		0			0
$623$	17.4330	+	2.08047i	0.0279824	+	0.00333944i
$624$		0			0
$625$	720.198			1.15232
$626$		0			0
$627$	−327.419	+	327.419i	−0.522199	+	0.522199i
$628$		0			0
$629$	−758.433	+	758.433i	−1.20578	+	1.20578i
$630$		0			0
$631$		−	161.350i		−	0.255706i	−0.991793	−	0.127853i	$-0.959191\pi$
$631$		−	161.350i		−	0.255706i	0.991793	−	0.127853i	$-0.0408086\pi$
$632$		0			0
$633$	83.8696			0.132495
$634$		0			0
$635$	−99.6245	+	99.6245i	−0.156889	+	0.156889i
$636$		0			0
$637$	203.187	+	333.018i	0.318975	+	0.522792i
$638$		0			0
$639$	−111.502			−0.174495
$640$		0			0
$641$	933.796			1.45678			0.728390	−	0.685163i	$-0.240269\pi$
$641$	933.796			1.45678			0.728390	+	0.685163i	$0.240269\pi$
$642$		0			0
$643$	−213.871	−	213.871i	−0.332614	−	0.332614i	0.520965	−	0.853578i	$-0.325572\pi$
$643$	−213.871	−	213.871i	−0.332614	−	0.332614i	−0.853578	+	0.520965i	$0.825572\pi$
$644$		0			0
$645$	473.074	+	473.074i	0.733449	+	0.733449i
$646$		0			0
$647$	957.177			1.47941			0.739704	−	0.672933i	$-0.234966\pi$
$647$	957.177			1.47941			0.739704	+	0.672933i	$0.234966\pi$
$648$		0			0
$649$	2002.05			3.08482
$650$		0			0
$651$	−317.093	+	249.478i	−0.487086	+	0.383223i
$652$		0			0
$653$	330.419	−	330.419i	0.506001	−	0.506001i	−0.407296	−	0.913296i	$-0.633528\pi$
$653$	330.419	−	330.419i	0.506001	−	0.506001i	0.913296	+	0.407296i	$0.133528\pi$
$654$		0			0
$655$	896.255			1.36833
$656$		0			0
$657$			160.905i			0.244908i
$658$		0			0
$659$	501.045	−	501.045i	0.760312	−	0.760312i	−0.216067	−	0.976379i	$-0.569323\pi$
$659$	501.045	−	501.045i	0.760312	−	0.760312i	0.976379	+	0.216067i	$0.0693229\pi$
$660$		0			0
$661$	174.444	−	174.444i	0.263909	−	0.263909i	−0.562731	−	0.826640i	$-0.690249\pi$
$661$	174.444	−	174.444i	0.263909	−	0.263909i	0.826640	+	0.562731i	$0.190249\pi$
$662$		0			0
$663$	480.858			0.725277
$664$		0			0
$665$	−345.160	−	41.1917i	−0.519038	−	0.0619423i
$666$		0			0
$667$	−152.245	−	152.245i	−0.228253	−	0.228253i
$668$		0			0
$669$	104.049	−	104.049i	0.155529	−	0.155529i
$670$		0			0
$671$			695.924i			1.03714i
$672$		0			0
$673$	−1132.80			−1.68321			−0.841607	−	0.540090i	$-0.818390\pi$
$673$	−1132.80			−1.68321			−0.841607	+	0.540090i	$0.818390\pi$
$674$		0			0
$675$	97.1996	+	97.1996i	0.143999	+	0.143999i
$676$		0			0
$677$	38.2381	−	38.2381i	0.0564817	−	0.0564817i	−0.678302	−	0.734783i	$-0.737284\pi$
$677$	38.2381	−	38.2381i	0.0564817	−	0.0564817i	0.734783	+	0.678302i	$0.237284\pi$
$678$		0			0
$679$	−112.238	+	940.484i	−0.165299	+	1.38510i
$680$		0			0
$681$		−	373.367i		−	0.548262i
$682$		0			0
$683$	306.215	+	306.215i	0.448339	+	0.448339i	0.894802	−	0.446463i	$-0.147317\pi$
$683$	306.215	+	306.215i	0.448339	+	0.448339i	−0.446463	+	0.894802i	$0.647317\pi$
$684$		0			0
$685$	721.297	+	721.297i	1.05299	+	1.05299i
$686$		0			0
$687$	−626.634			−0.912130
$688$		0			0
$689$			125.971i			0.182831i
$690$		0			0
$691$	−340.446	−	340.446i	−0.492686	−	0.492686i	0.416466	−	0.909152i	$-0.363269\pi$
$691$	−340.446	−	340.446i	−0.492686	−	0.492686i	−0.909152	+	0.416466i	$0.863269\pi$
$692$		0			0
$693$	−279.835	+	220.165i	−0.403802	+	0.317698i
$694$		0			0
$695$		−	279.256i		−	0.401807i
$696$		0			0
$697$			1174.66i			1.68531i
$698$		0			0
$699$	114.911	−	114.911i	0.164393	−	0.164393i
$700$		0			0
$701$	180.018	−	180.018i	0.256801	−	0.256801i	−0.566951	−	0.823752i	$-0.691877\pi$
$701$	180.018	−	180.018i	0.256801	−	0.256801i	0.823752	+	0.566951i	$0.191877\pi$
$702$		0			0
$703$			411.249i			0.584991i
$704$		0			0
$705$			1144.02i			1.62272i
$706$		0			0
$707$	423.576	−	333.255i	0.599117	−	0.471365i
$708$		0			0
$709$	−424.652	−	424.652i	−0.598945	−	0.598945i	0.341087	−	0.940032i	$-0.389205\pi$
$709$	−424.652	−	424.652i	−0.598945	−	0.598945i	−0.940032	+	0.341087i	$0.889205\pi$
$710$		0			0
$711$			92.1130i			0.129554i
$712$		0			0
$713$	343.285			0.481466
$714$		0			0
$715$	613.304	+	613.304i	0.857768	+	0.857768i
$716$		0			0
$717$	568.460	+	568.460i	0.792832	+	0.792832i
$718$		0			0
$719$		−	562.307i		−	0.782068i	−0.920376	−	0.391034i	$-0.872117\pi$
$719$		−	562.307i		−	0.782068i	0.920376	−	0.391034i	$-0.127883\pi$
$720$		0			0
$721$	208.113	+	24.8364i	0.288645	+	0.0344472i
$722$		0			0
$723$	−198.399	+	198.399i	−0.274411	+	0.274411i
$724$		0			0
$725$	47.1474	+	47.1474i	0.0650309	+	0.0650309i
$726$		0			0
$727$	97.7163			0.134410			0.0672052	−	0.997739i	$-0.478592\pi$
$727$	97.7163			0.134410			0.0672052	+	0.997739i	$0.478592\pi$
$728$		0			0
$729$		−	802.106i		−	1.10028i
$730$		0			0
$731$	−812.286	+	812.286i	−1.11120	+	1.11120i
$732$		0			0
$733$	−551.141	−	551.141i	−0.751897	−	0.751897i	0.222936	−	0.974833i	$-0.428436\pi$
$733$	−551.141	−	551.141i	−0.751897	−	0.751897i	−0.974833	+	0.222936i	$0.928436\pi$
$734$		0			0
$735$	659.306	+	159.638i	0.897015	+	0.217194i
$736$		0			0
$737$	−200.349			−0.271845
$738$		0			0
$739$	749.487	−	749.487i	1.01419	−	1.01419i	0.0142929	−	0.999898i	$-0.495450\pi$
$739$	749.487	−	749.487i	1.01419	−	1.01419i	0.999898	−	0.0142929i	$-0.00454971\pi$
$740$		0			0
$741$	130.369	−	130.369i	0.175937	−	0.175937i
$742$		0			0
$743$		−	848.086i		−	1.14143i	−0.821147	−	0.570717i	$-0.806665\pi$
$743$		−	848.086i		−	1.14143i	0.821147	−	0.570717i	$-0.193335\pi$
$744$		0			0
$745$	−850.090			−1.14106
$746$		0			0
$747$	57.9704	−	57.9704i	0.0776043	−	0.0776043i
$748$		0			0
$749$	−88.9694	−	113.082i	−0.118784	−	0.150978i
$750$		0			0
$751$	−153.228			−0.204032			−0.102016	−	0.994783i	$-0.532529\pi$
$751$	−153.228			−0.204032			−0.102016	+	0.994783i	$0.532529\pi$
$752$		0			0
$753$	788.076			1.04658
$754$		0			0
$755$	307.419	+	307.419i	0.407177	+	0.407177i
$756$		0			0
$757$	57.6855	+	57.6855i	0.0762028	+	0.0762028i	0.744181	−	0.667978i	$-0.232840\pi$
$757$	57.6855	+	57.6855i	0.0762028	+	0.0762028i	−0.667978	+	0.744181i	$0.732840\pi$
$758$		0			0
$759$	−768.830			−1.01295
$760$		0			0
$761$	293.973			0.386298			0.193149	−	0.981169i	$-0.438130\pi$
$761$	293.973			0.386298			0.193149	+	0.981169i	$0.438130\pi$
$762$		0			0
$763$	205.265	−	161.495i	0.269023	−	0.211658i
$764$		0			0
$765$	−232.977	+	232.977i	−0.304545	+	0.304545i
$766$		0			0
$767$	−797.161			−1.03932
$768$		0			0
$769$			1314.27i			1.70907i	0.519397	+	0.854533i	$0.326156\pi$
$769$			1314.27i			1.70907i	−0.519397	+	0.854533i	$0.673844\pi$
$770$		0			0
$771$	−386.333	+	386.333i	−0.501081	+	0.501081i
$772$		0			0
$773$	505.061	−	505.061i	0.653377	−	0.653377i	−0.300427	−	0.953805i	$-0.597129\pi$
$773$	505.061	−	505.061i	0.653377	−	0.653377i	0.953805	+	0.300427i	$0.0971293\pi$
$774$		0			0
$775$	−106.309			−0.137173
$776$		0			0
$777$	95.1021	−	796.896i	0.122397	−	1.02561i
$778$		0			0
$779$	318.471	+	318.471i	0.408820	+	0.408820i
$780$		0			0
$781$	−619.700	+	619.700i	−0.793470	+	0.793470i
$782$		0			0
$783$		−	417.320i		−	0.532975i
$784$		0			0
$785$	197.578			0.251692
$786$		0			0
$787$	−554.504	−	554.504i	−0.704579	−	0.704579i	0.260811	−	0.965390i	$-0.416010\pi$
$787$	−554.504	−	554.504i	−0.704579	−	0.704579i	−0.965390	+	0.260811i	$0.916010\pi$
$788$		0			0
$789$	−373.205	+	373.205i	−0.473010	+	0.473010i
$790$		0			0
$791$	−1043.82	−	124.570i	−1.31962	−	0.157485i
$792$		0			0
$793$		−	277.098i		−	0.349430i
$794$		0			0
$795$	154.891	+	154.891i	0.194831	+	0.194831i
$796$		0			0
$797$	1010.92	+	1010.92i	1.26841	+	1.26841i	0.946912	+	0.321494i	$0.104185\pi$
$797$	1010.92	+	1010.92i	1.26841	+	1.26841i	0.321494	+	0.946912i	$0.395815\pi$
$798$		0			0
$799$	−1964.32			−2.45848
$800$		0			0
$801$		−	6.38047i		−	0.00796563i
$802$		0			0
$803$	894.266	+	894.266i	1.11366	+	1.11366i
$804$		0			0
$805$	−356.882	−	453.607i	−0.443332	−	0.563487i
$806$		0			0
$807$		−	911.714i		−	1.12976i
$808$		0			0
$809$			422.423i			0.522154i	0.965318	+	0.261077i	$0.0840776\pi$
$809$			422.423i			0.522154i	−0.965318	+	0.261077i	$0.915922\pi$
$810$		0			0
$811$	121.240	−	121.240i	0.149494	−	0.149494i	−0.628398	−	0.777892i	$-0.716289\pi$
$811$	121.240	−	121.240i	0.149494	−	0.149494i	0.777892	+	0.628398i	$0.216289\pi$
$812$		0			0
$813$	755.584	−	755.584i	0.929378	−	0.929378i
$814$		0			0
$815$		−	990.005i		−	1.21473i
$816$		0			0
$817$			440.450i			0.539106i
$818$		0			0
$819$	111.423	−	87.6636i	0.136047	−	0.107037i
$820$		0			0
$821$	−192.305	−	192.305i	−0.234232	−	0.234232i	0.580224	−	0.814457i	$-0.302965\pi$
$821$	−192.305	−	192.305i	−0.234232	−	0.234232i	−0.814457	+	0.580224i	$0.802965\pi$
$822$		0			0
$823$			242.000i			0.294047i	0.989133	+	0.147023i	$0.0469692\pi$
$823$			242.000i			0.294047i	−0.989133	+	0.147023i	$0.953031\pi$
$824$		0			0
$825$	238.092			0.288597
$826$		0			0
$827$	7.18081	+	7.18081i	0.00868296	+	0.00868296i	0.711435	−	0.702752i	$-0.248046\pi$
$827$	7.18081	+	7.18081i	0.00868296	+	0.00868296i	−0.702752	+	0.711435i	$0.748046\pi$
$828$		0			0
$829$	559.201	+	559.201i	0.674549	+	0.674549i	0.958761	−	0.284212i	$-0.0917320\pi$
$829$	559.201	+	559.201i	0.674549	+	0.674549i	−0.284212	+	0.958761i	$0.591732\pi$
$830$		0			0
$831$			612.786i			0.737408i
$832$		0			0
$833$	−274.104	+	1132.05i	−0.329056	+	1.35901i
$834$		0			0
$835$	257.864	−	257.864i	0.308819	−	0.308819i
$836$		0			0
$837$	470.491	+	470.491i	0.562116	+	0.562116i
$838$		0			0
$839$	494.834			0.589790			0.294895	−	0.955530i	$-0.404715\pi$
$839$	494.834			0.589790			0.294895	+	0.955530i	$0.404715\pi$
$840$		0			0
$841$			638.576i			0.759305i
$842$		0			0
$843$	−46.0910	+	46.0910i	−0.0546750	+	0.0546750i
$844$		0			0
$845$	406.904	+	406.904i	0.481543	+	0.481543i
$846$		0			0
$847$	−231.263	+	1937.83i	−0.273037	+	2.28788i
$848$		0			0
$849$	−901.944			−1.06236
$850$		0			0
$851$	−482.838	+	482.838i	−0.567378	+	0.567378i
$852$		0			0
$853$	−460.356	+	460.356i	−0.539690	+	0.539690i	−0.923438	−	0.383748i	$-0.874633\pi$
$853$	−460.356	+	460.356i	−0.539690	+	0.539690i	0.383748	+	0.923438i	$0.374633\pi$
$854$		0			0
$855$			126.328i			0.147752i
$856$		0			0
$857$	−393.065			−0.458652			−0.229326	−	0.973350i	$-0.573652\pi$
$857$	−393.065			−0.458652			−0.229326	+	0.973350i	$0.573652\pi$
$858$		0			0
$859$	617.601	−	617.601i	0.718977	−	0.718977i	−0.249419	−	0.968396i	$-0.580240\pi$
$859$	617.601	−	617.601i	0.718977	−	0.718977i	0.968396	+	0.249419i	$0.0802395\pi$
$860$		0			0
$861$	−543.468	−	690.762i	−0.631206	−	0.802279i
$862$		0			0
$863$	−433.128			−0.501887			−0.250943	−	0.968002i	$-0.580741\pi$
$863$	−433.128			−0.501887			−0.250943	+	0.968002i	$0.580741\pi$
$864$		0			0
$865$	−806.227			−0.932055
$866$		0			0
$867$	495.964	+	495.964i	0.572046	+	0.572046i
$868$		0			0
$869$	511.940	+	511.940i	0.589114	+	0.589114i
$870$		0			0
$871$	79.7737			0.0915886
$872$		0			0
$873$	344.217			0.394292
$874$		0			0
$875$	−479.055	−	608.891i	−0.547491	−	0.695876i
$876$		0			0
$877$	527.680	−	527.680i	0.601687	−	0.601687i	−0.339073	−	0.940760i	$-0.610113\pi$
$877$	527.680	−	527.680i	0.601687	−	0.601687i	0.940760	+	0.339073i	$0.110113\pi$
$878$		0			0
$879$	758.833			0.863292
$880$		0			0
$881$		−	1266.15i		−	1.43717i	−0.695437	−	0.718587i	$-0.744789\pi$
$881$		−	1266.15i		−	1.43717i	0.695437	−	0.718587i	$-0.255211\pi$
$882$		0			0
$883$	−768.330	+	768.330i	−0.870136	+	0.870136i	−0.992487	−	0.122351i	$-0.960957\pi$
$883$	−768.330	+	768.330i	−0.870136	+	0.870136i	0.122351	+	0.992487i	$0.460957\pi$
$884$		0			0
$885$	−980.172	+	980.172i	−1.10754	+	1.10754i
$886$		0			0
$887$	1242.85			1.40118			0.700592	−	0.713562i	$-0.252919\pi$
$887$	1242.85			1.40118			0.700592	+	0.713562i	$0.252919\pi$
$888$		0			0
$889$	179.734	+	21.4496i	0.202175	+	0.0241277i
$890$		0			0
$891$	−730.014	−	730.014i	−0.819320	−	0.819320i
$892$		0			0
$893$	−532.562	+	532.562i	−0.596375	+	0.596375i
$894$		0			0
$895$			1033.72i			1.15499i
$896$		0			0
$897$	306.127			0.341279
$898$		0			0
$899$	228.215	+	228.215i	0.253855	+	0.253855i
$900$		0			0
$901$	−265.953	+	265.953i	−0.295175	+	0.295175i
$902$		0			0
$903$	101.855	−	853.479i	0.112796	−	0.945159i
$904$		0			0
$905$		−	131.222i		−	0.144996i
$906$		0			0
$907$	453.703	+	453.703i	0.500223	+	0.500223i	0.911507	−	0.411284i	$-0.134920\pi$
$907$	453.703	+	453.703i	0.500223	+	0.500223i	−0.411284	+	0.911507i	$0.634920\pi$
$908$		0			0
$909$	−138.500	−	138.500i	−0.152365	−	0.152365i
$910$		0			0
$911$	1772.87			1.94606			0.973032	−	0.230668i	$-0.0740912\pi$
$911$	1772.87			1.94606			0.973032	+	0.230668i	$0.0740912\pi$
$912$		0			0
$913$		−	644.368i		−	0.705770i
$914$		0			0
$915$	−340.713	−	340.713i	−0.372364	−	0.372364i
$916$		0			0
$917$	−711.988	−	904.956i	−0.776432	−	0.986866i
$918$		0			0
$919$		−	974.720i		−	1.06063i	−0.847800	−	0.530316i	$-0.822073\pi$
$919$		−	974.720i		−	1.06063i	0.847800	−	0.530316i	$-0.177927\pi$
$920$		0			0
$921$			345.298i			0.374916i
$922$		0			0
$923$	246.747	−	246.747i	0.267332	−	0.267332i
$924$		0			0
$925$	149.526	−	149.526i	0.161650	−	0.161650i
$926$		0			0
$927$		−	76.1693i		−	0.0821675i
$928$		0			0
$929$		−	1771.00i		−	1.90636i	−0.302410	−	0.953178i	$-0.597791\pi$
$929$		−	1771.00i		−	1.90636i	0.302410	−	0.953178i	$-0.402209\pi$
$930$		0			0
$931$	232.605	+	381.233i	0.249844	+	0.409488i
$932$		0			0
$933$	966.884	+	966.884i	1.03632	+	1.03632i
$934$		0			0
$935$			2589.65i			2.76968i
$936$		0			0
$937$	−887.636			−0.947317			−0.473659	−	0.880709i	$-0.657067\pi$
$937$	−887.636			−0.947317			−0.473659	+	0.880709i	$0.657067\pi$
$938$		0			0
$939$	782.221	+	782.221i	0.833036	+	0.833036i
$940$		0			0
$941$	776.267	+	776.267i	0.824939	+	0.824939i	0.986812	−	0.161873i	$-0.0517535\pi$
$941$	776.267	+	776.267i	0.824939	+	0.824939i	−0.161873	+	0.986812i	$0.551754\pi$
$942$		0			0
$943$			747.819i			0.793021i
$944$		0			0
$945$	132.566	−	1110.82i	0.140282	−	1.17547i
$946$		0			0
$947$	1029.00	−	1029.00i	1.08659	−	1.08659i	0.0907123	−	0.995877i	$-0.471086\pi$
$947$	1029.00	−	1029.00i	1.08659	−	1.08659i	0.995877	−	0.0907123i	$-0.0289144\pi$
$948$		0			0
$949$	−356.072	−	356.072i	−0.375208	−	0.375208i
$950$		0			0
$951$	−1546.32			−1.62599
$952$		0			0
$953$			531.911i			0.558144i	0.960270	+	0.279072i	$0.0900268\pi$
$953$			531.911i			0.558144i	−0.960270	+	0.279072i	$0.909973\pi$
$954$		0			0
$955$	−669.527	+	669.527i	−0.701075	+	0.701075i
$956$		0			0
$957$	−511.117	−	511.117i	−0.534082	−	0.534082i
$958$		0			0
$959$	155.298	−	1301.30i	0.161937	−	1.35693i
$960$		0			0
$961$	446.414			0.464531
$962$		0			0
$963$	−36.9754	+	36.9754i	−0.0383961	+	0.0383961i
$964$		0			0
$965$	120.993	−	120.993i	0.125382	−	0.125382i
$966$		0			0
$967$			1211.31i			1.25264i	0.779564	+	0.626322i	$0.215440\pi$
$967$			1211.31i			1.25264i	−0.779564	+	0.626322i	$0.784560\pi$
$968$		0			0
$969$	550.478			0.568089
$970$		0			0
$971$	−233.178	+	233.178i	−0.240142	+	0.240142i	−0.816909	−	0.576767i	$-0.804314\pi$
$971$	−233.178	+	233.178i	−0.240142	+	0.240142i	0.576767	+	0.816909i	$0.304314\pi$
$972$		0			0
$973$	−281.967	+	221.842i	−0.289791	+	0.227998i
$974$		0			0
$975$	−94.8018			−0.0972327
$976$		0			0
$977$	−522.328			−0.534624			−0.267312	−	0.963610i	$-0.586135\pi$
$977$	−522.328			−0.534624			−0.267312	+	0.963610i	$0.586135\pi$
$978$		0			0
$979$	−35.4610	−	35.4610i	−0.0362216	−	0.0362216i
$980$		0			0
$981$	−67.1169	−	67.1169i	−0.0684169	−	0.0684169i
$982$		0			0
$983$	−205.971			−0.209533			−0.104767	−	0.994497i	$-0.533410\pi$
$983$	−205.971			−0.209533			−0.104767	+	0.994497i	$0.533410\pi$
$984$		0			0
$985$	−1638.19			−1.66314
$986$		0			0
$987$	1155.13	−	908.814i	1.17034	−	0.920784i
$988$		0			0
$989$	−517.122	+	517.122i	−0.522874	+	0.522874i
$990$		0			0
$991$	1527.11			1.54098			0.770491	−	0.637451i	$-0.220011\pi$
$991$	1527.11			1.54098			0.770491	+	0.637451i	$0.220011\pi$
$992$		0			0
$993$		−	575.020i		−	0.579073i
$994$		0			0
$995$	−377.346	+	377.346i	−0.379243	+	0.379243i
$996$		0			0
$997$	871.463	−	871.463i	0.874085	−	0.874085i	−0.118830	−	0.992915i	$-0.537914\pi$
$997$	871.463	−	871.463i	0.874085	−	0.874085i	0.992915	+	0.118830i	$0.0379143\pi$
$998$		0			0
$999$	−1323.51			−1.32484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.433.9		56
4.3	odd	2		112.3.l.b.69.16	yes	56
7.6	odd	2	inner	448.3.l.b.433.20		56
16.3	odd	4		112.3.l.b.13.15	✓	56
16.13	even	4	inner	448.3.l.b.209.20		56
28.27	even	2		112.3.l.b.69.15	yes	56
112.13	odd	4	inner	448.3.l.b.209.9		56
112.83	even	4		112.3.l.b.13.16	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.15	✓	56	16.3	odd	4
112.3.l.b.13.16	yes	56	112.83	even	4
112.3.l.b.69.15	yes	56	28.27	even	2
112.3.l.b.69.16	yes	56	4.3	odd	2
448.3.l.b.209.9		56	112.13	odd	4	inner
448.3.l.b.209.20		56	16.13	even	4	inner
448.3.l.b.433.9		56	1.1	even	1	trivial
448.3.l.b.433.20		56	7.6	odd	2	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)

\(f(q)\)	\(=\)	\(q+(-1.79667 - 1.79667i) q^{3} +(-3.85269 + 3.85269i) q^{5} +(6.95068 + 0.829499i) q^{7} -2.54394i q^{9} +O(q^{10})\)	\(q+(-1.79667 - 1.79667i) q^{3} +(-3.85269 + 3.85269i) q^{5} +(6.95068 + 0.829499i) q^{7} -2.54394i q^{9} +(-14.1386 - 14.1386i) q^{11} +(5.62959 + 5.62959i) q^{13} +13.8440 q^{15} +23.7707i q^{17} +(6.44465 + 6.44465i) q^{19} +(-10.9977 - 13.9784i) q^{21} +15.1330i q^{23} -4.68642i q^{25} +(-20.7407 + 20.7407i) q^{27} +(-10.0604 + 10.0604i) q^{29} -22.6845i q^{31} +50.8047i q^{33} +(-29.9746 + 23.5830i) q^{35} +(31.9062 + 31.9062i) q^{37} -20.2290i q^{39} +49.4163 q^{41} +(34.1717 + 34.1717i) q^{43} +(9.80102 + 9.80102i) q^{45} +82.6364i q^{47} +(47.6239 + 11.5312i) q^{49} +(42.7081 - 42.7081i) q^{51} +(11.1883 + 11.1883i) q^{53} +108.943 q^{55} -23.1578i q^{57} +(-70.8011 + 70.8011i) q^{59} +(-24.6109 - 24.6109i) q^{61} +(2.11020 - 17.6821i) q^{63} -43.3781 q^{65} +(7.08521 - 7.08521i) q^{67} +(27.1891 - 27.1891i) q^{69} -43.8305i q^{71} -63.2501 q^{73} +(-8.41996 + 8.41996i) q^{75} +(-86.5447 - 110.001i) q^{77} -36.2088 q^{79} +51.6329 q^{81} +(22.7876 + 22.7876i) q^{83} +(-91.5811 - 91.5811i) q^{85} +36.1505 q^{87} +2.50810 q^{89} +(34.4597 + 43.7992i) q^{91} +(-40.7566 + 40.7566i) q^{93} -49.6585 q^{95} +135.308i q^{97} +(-35.9677 + 35.9677i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 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

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