LMFDB - Embedded newform 448.3.c.g.321.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,3,Mod(321,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.321"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4694952902656.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 $ x^8 + 20x^6 + 56x^4 + 20*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.1
Root			$4.08932i$ of defining polynomial
Character	$\chi$	$=$	448.321
Dual form			448.3.c.g.321.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.43734i q^{3} -6.12900i q^{5} +(6.21005 + 3.23038i) q^{7} -20.5647 q^{9} -15.2485 q^{11} -3.00789i q^{13} -33.3255 q^{15} -21.6213i q^{17} +11.8981i q^{19} +(17.5647 - 33.7661i) q^{21} +1.72205 q^{23} -12.5647 q^{25} +62.8810i q^{27} +41.1293 q^{29} +8.50759i q^{31} +82.9114i q^{33} +(19.7990 - 38.0614i) q^{35} -53.1293 q^{37} -16.3549 q^{39} -43.0162i q^{41} -36.6446 q^{43} +126.041i q^{45} +30.2569i q^{47} +(28.1293 + 40.1216i) q^{49} -117.563 q^{51} +30.0000 q^{53} +93.4582i q^{55} +64.6940 q^{57} -73.0525i q^{59} +5.67609i q^{61} +(-127.707 - 66.4316i) q^{63} -18.4353 q^{65} +28.7750 q^{67} -9.36335i q^{69} +5.53191 q^{71} -94.9429i q^{73} +68.3183i q^{75} +(-94.6940 - 49.2585i) q^{77} +81.5337 q^{79} +156.823 q^{81} +86.2943i q^{83} -132.517 q^{85} -223.634i q^{87} +27.6371i q^{89} +(9.71661 - 18.6791i) q^{91} +46.2586 q^{93} +72.9234 q^{95} -100.959i q^{97} +313.581 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 56 q^{9} + 32 q^{21} + 8 q^{25} + 112 q^{29} - 208 q^{37} + 8 q^{49} + 240 q^{53} + 192 q^{57} - 256 q^{65} - 432 q^{77} + 712 q^{81} - 192 q^{85} - 64 q^{93}+O(q^{100}) $ 8 * q - 56 * q^9 + 32 * q^21 + 8 * q^25 + 112 * q^29 - 208 * q^37 + 8 * q^49 + 240 * q^53 + 192 * q^57 - 256 * q^65 - 432 * q^77 + 712 * q^81 - 192 * q^85 - 64 * q^93

Character values

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

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	5.43734i		−	1.81245i	−0.422800	−	0.906223i	$-0.638953\pi$
$3$		−	5.43734i		−	1.81245i	0.422800	−	0.906223i	$-0.361047\pi$
$4$		0			0
$5$		−	6.12900i		−	1.22580i	−0.790160	−	0.612900i	$-0.790003\pi$
$5$		−	6.12900i		−	1.22580i	0.790160	−	0.612900i	$-0.209997\pi$
$6$		0			0
$7$	6.21005	+	3.23038i	0.887149	+	0.461483i
$7$	6.21005	+	3.23038i	0.887149	+	0.461483i
$8$		0			0
$9$	−20.5647			−2.28496
$10$		0			0
$11$	−15.2485			−1.38623			−0.693114	−	0.720828i	$-0.743762\pi$
$11$	−15.2485			−1.38623			−0.693114	+	0.720828i	$0.743762\pi$
$12$		0			0
$13$		−	3.00789i		−	0.231376i	−0.993286	−	0.115688i	$-0.963093\pi$
$13$		−	3.00789i		−	0.231376i	0.993286	−	0.115688i	$-0.0369073\pi$
$14$		0			0
$15$	−33.3255			−2.22170
$16$		0			0
$17$		−	21.6213i		−	1.27184i	−0.771753	−	0.635922i	$-0.780620\pi$
$17$		−	21.6213i		−	1.27184i	0.771753	−	0.635922i	$-0.219380\pi$
$18$		0			0
$19$			11.8981i			0.626216i	0.949718	+	0.313108i	$0.101370\pi$
$19$			11.8981i			0.626216i	−0.949718	+	0.313108i	$0.898630\pi$
$20$		0			0
$21$	17.5647	−	33.7661i	0.836412	−	1.60791i
$22$		0			0
$23$	1.72205			0.0748715			0.0374358	−	0.999299i	$-0.488081\pi$
$23$	1.72205			0.0748715			0.0374358	+	0.999299i	$0.488081\pi$
$24$		0			0
$25$	−12.5647			−0.502586
$26$		0			0
$27$			62.8810i			2.32893i
$28$		0			0
$29$	41.1293			1.41825			0.709126	−	0.705082i	$-0.249090\pi$
$29$	41.1293			1.41825			0.709126	+	0.705082i	$0.249090\pi$
$30$		0			0
$31$			8.50759i			0.274438i	0.990541	+	0.137219i	$0.0438165\pi$
$31$			8.50759i			0.274438i	−0.990541	+	0.137219i	$0.956184\pi$
$32$		0			0
$33$			82.9114i			2.51247i
$34$		0			0
$35$	19.7990	−	38.0614i	0.565685	−	1.08747i
$36$		0			0
$37$	−53.1293			−1.43593			−0.717964	−	0.696080i	$-0.754926\pi$
$37$	−53.1293			−1.43593			−0.717964	+	0.696080i	$0.754926\pi$
$38$		0			0
$39$	−16.3549			−0.419356
$40$		0			0
$41$		−	43.0162i		−	1.04918i	−0.851356	−	0.524588i	$-0.824219\pi$
$41$		−	43.0162i		−	1.04918i	0.851356	−	0.524588i	$-0.175781\pi$
$42$		0			0
$43$	−36.6446			−0.852200			−0.426100	−	0.904676i	$-0.640113\pi$
$43$	−36.6446			−0.852200			−0.426100	+	0.904676i	$0.640113\pi$
$44$		0			0
$45$			126.041i			2.80091i
$46$		0			0
$47$			30.2569i			0.643765i	0.946780	+	0.321882i	$0.104316\pi$
$47$			30.2569i			0.643765i	−0.946780	+	0.321882i	$0.895684\pi$
$48$		0			0
$49$	28.1293	+	40.1216i	0.574068	+	0.818808i
$50$		0			0
$51$	−117.563			−2.30515
$52$		0			0
$53$	30.0000			0.566038			0.283019	−	0.959114i	$-0.408664\pi$
$53$	30.0000			0.566038			0.283019	+	0.959114i	$0.408664\pi$
$54$		0			0
$55$			93.4582i			1.69924i
$56$		0			0
$57$	64.6940			1.13498
$58$		0			0
$59$		−	73.0525i		−	1.23818i	−0.785321	−	0.619089i	$-0.787502\pi$
$59$		−	73.0525i		−	1.23818i	0.785321	−	0.619089i	$-0.212498\pi$
$60$		0			0
$61$			5.67609i			0.0930506i	0.998917	+	0.0465253i	$0.0148148\pi$
$61$			5.67609i			0.0930506i	−0.998917	+	0.0465253i	$0.985185\pi$
$62$		0			0
$63$	−127.707	−	66.4316i	−2.02710	−	1.05447i
$64$		0			0
$65$	−18.4353			−0.283621
$66$		0			0
$67$	28.7750			0.429477			0.214739	−	0.976672i	$-0.431110\pi$
$67$	28.7750			0.429477			0.214739	+	0.976672i	$0.431110\pi$
$68$		0			0
$69$		−	9.36335i		−	0.135701i
$70$		0			0
$71$	5.53191			0.0779142			0.0389571	−	0.999241i	$-0.487596\pi$
$71$	5.53191			0.0779142			0.0389571	+	0.999241i	$0.487596\pi$
$72$		0			0
$73$		−	94.9429i		−	1.30059i	−0.759683	−	0.650294i	$-0.774646\pi$
$73$		−	94.9429i		−	1.30059i	0.759683	−	0.650294i	$-0.225354\pi$
$74$		0			0
$75$			68.3183i			0.910911i
$76$		0			0
$77$	−94.6940	−	49.2585i	−1.22979	−	0.639720i
$78$		0			0
$79$	81.5337			1.03207			0.516036	−	0.856567i	$-0.327407\pi$
$79$	81.5337			1.03207			0.516036	+	0.856567i	$0.327407\pi$
$80$		0			0
$81$	156.823			1.93609
$82$		0			0
$83$			86.2943i			1.03969i	0.854261	+	0.519845i	$0.174010\pi$
$83$			86.2943i			1.03969i	−0.854261	+	0.519845i	$0.825990\pi$
$84$		0			0
$85$	−132.517			−1.55903
$86$		0			0
$87$		−	223.634i		−	2.57051i
$88$		0			0
$89$			27.6371i			0.310529i	0.987873	+	0.155265i	$0.0496231\pi$
$89$			27.6371i			0.310529i	−0.987873	+	0.155265i	$0.950377\pi$
$90$		0			0
$91$	9.71661	−	18.6791i	0.106776	−	0.205265i
$92$		0			0
$93$	46.2586			0.497405
$94$		0			0
$95$	72.9234			0.767615
$96$		0			0
$97$		−	100.959i		−	1.04081i	−0.853919	−	0.520406i	$-0.825781\pi$
$97$		−	100.959i		−	1.04081i	0.853919	−	0.520406i	$-0.174219\pi$
$98$		0			0
$99$	313.581			3.16748
$100$		0			0
$101$		−	107.541i		−	1.06476i	−0.846506	−	0.532379i	$-0.821298\pi$
$101$		−	107.541i		−	1.06476i	0.846506	−	0.532379i	$-0.178702\pi$
$102$		0			0
$103$			59.1076i			0.573860i	0.957952	+	0.286930i	$0.0926347\pi$
$103$			59.1076i			0.573860i	−0.957952	+	0.286930i	$0.907365\pi$
$104$		0			0
$105$	−206.953	−	107.654i	−1.97098	−	1.02527i
$106$		0			0
$107$	−45.9954			−0.429864			−0.214932	−	0.976629i	$-0.568953\pi$
$107$	−45.9954			−0.429864			−0.214932	+	0.976629i	$0.568953\pi$
$108$		0			0
$109$	23.7414			0.217811			0.108905	−	0.994052i	$-0.465265\pi$
$109$	23.7414			0.217811			0.108905	+	0.994052i	$0.465265\pi$
$110$		0			0
$111$			288.882i			2.60254i
$112$		0			0
$113$	−187.211			−1.65674			−0.828368	−	0.560184i	$-0.810731\pi$
$113$	−187.211			−1.65674			−0.828368	+	0.560184i	$0.810731\pi$
$114$		0			0
$115$		−	10.5544i		−	0.0917776i
$116$		0			0
$117$			61.8562i			0.528685i
$118$		0			0
$119$	69.8451	−	134.270i	0.586934	−	1.12832i
$120$		0			0
$121$	111.517			0.921630
$122$		0			0
$123$	−233.894			−1.90158
$124$		0			0
$125$		−	76.2162i		−	0.609730i
$126$		0			0
$127$	−206.350			−1.62480			−0.812402	−	0.583097i	$-0.801841\pi$
$127$	−206.350			−1.62480			−0.812402	+	0.583097i	$0.801841\pi$
$128$		0			0
$129$			199.249i			1.54457i
$130$		0			0
$131$		−	186.979i		−	1.42732i	−0.700492	−	0.713660i	$-0.747036\pi$
$131$		−	186.979i		−	1.42732i	0.700492	−	0.713660i	$-0.252964\pi$
$132$		0			0
$133$	−38.4353	+	73.8877i	−0.288988	+	0.555547i
$134$		0			0
$135$	385.398			2.85480
$136$		0			0
$137$	−2.51728			−0.0183743			−0.00918715	−	0.999958i	$-0.502924\pi$
$137$	−2.51728			−0.0183743			−0.00918715	+	0.999958i	$0.502924\pi$
$138$		0			0
$139$			18.6791i			0.134382i	0.997740	+	0.0671910i	$0.0214037\pi$
$139$			18.6791i			0.134382i	−0.997740	+	0.0671910i	$0.978596\pi$
$140$		0			0
$141$	164.517			1.16679
$142$		0			0
$143$			45.8658i			0.320740i
$144$		0			0
$145$		−	252.082i		−	1.73849i
$146$		0			0
$147$	218.155	−	152.949i	1.48405	−	1.04047i
$148$		0			0
$149$	2.51728			0.0168945			0.00844725	−	0.999964i	$-0.497311\pi$
$149$	2.51728			0.0168945			0.00844725	+	0.999964i	$0.497311\pi$
$150$		0			0
$151$	78.4553			0.519572			0.259786	−	0.965666i	$-0.416348\pi$
$151$	78.4553			0.519572			0.259786	+	0.965666i	$0.416348\pi$
$152$		0			0
$153$			444.636i			2.90612i
$154$		0			0
$155$	52.1430			0.336407
$156$		0			0
$157$		−	298.559i		−	1.90165i	−0.309732	−	0.950824i	$-0.600239\pi$
$157$		−	298.559i		−	1.90165i	0.309732	−	0.950824i	$-0.399761\pi$
$158$		0			0
$159$		−	163.120i		−	1.02591i
$160$		0			0
$161$	10.6940	+	5.56286i	0.0664222	+	0.0345519i
$162$		0			0
$163$	−8.86012			−0.0543566			−0.0271783	−	0.999631i	$-0.508652\pi$
$163$	−8.86012			−0.0543566			−0.0271783	+	0.999631i	$0.508652\pi$
$164$		0			0
$165$	508.164			3.07978
$166$		0			0
$167$		−	252.930i		−	1.51455i	−0.653095	−	0.757276i	$-0.726530\pi$
$167$		−	252.930i		−	1.51455i	0.653095	−	0.757276i	$-0.273470\pi$
$168$		0			0
$169$	159.953			0.946465
$170$		0			0
$171$		−	244.680i		−	1.43088i
$172$		0			0
$173$		−	7.89137i		−	0.0456149i	−0.999740	−	0.0228074i	$-0.992740\pi$
$173$		−	7.89137i		−	0.0456149i	0.999740	−	0.0228074i	$-0.00726046\pi$
$174$		0			0
$175$	−78.0271	−	40.5886i	−0.445869	−	0.231935i
$176$		0			0
$177$	−397.211			−2.24413
$178$		0			0
$179$	298.573			1.66800			0.834002	−	0.551761i	$-0.186044\pi$
$179$	298.573			1.66800			0.834002	+	0.551761i	$0.186044\pi$
$180$		0			0
$181$			281.821i			1.55702i	0.627631	+	0.778511i	$0.284025\pi$
$181$			281.821i			1.55702i	−0.627631	+	0.778511i	$0.715975\pi$
$182$		0			0
$183$	30.8628			0.168649
$184$		0			0
$185$			325.630i			1.76016i
$186$		0			0
$187$			329.694i			1.76307i
$188$		0			0
$189$	−203.129	+	390.494i	−1.07476	+	2.06610i
$190$		0			0
$191$	20.5396			0.107537			0.0537686	−	0.998553i	$-0.482877\pi$
$191$	20.5396			0.107537			0.0537686	+	0.998553i	$0.482877\pi$
$192$		0			0
$193$	191.565			0.992563			0.496282	−	0.868162i	$-0.334698\pi$
$193$	191.565			0.992563			0.496282	+	0.868162i	$0.334698\pi$
$194$		0			0
$195$			100.239i			0.514047i
$196$		0			0
$197$	113.129			0.574261			0.287130	−	0.957892i	$-0.407299\pi$
$197$	113.129			0.574261			0.287130	+	0.957892i	$0.407299\pi$
$198$		0			0
$199$		−	132.543i		−	0.666045i	−0.942919	−	0.333023i	$-0.891931\pi$
$199$		−	132.543i		−	0.666045i	0.942919	−	0.333023i	$-0.108069\pi$
$200$		0			0
$201$		−	156.459i		−	0.778405i
$202$		0			0
$203$	255.415	+	132.863i	1.25820	+	0.654499i
$204$		0			0
$205$	−263.647			−1.28608
$206$		0			0
$207$	−35.4133			−0.171079
$208$		0			0
$209$		−	181.428i		−	0.868078i
$210$		0			0
$211$	−235.366			−1.11548			−0.557739	−	0.830016i	$-0.688331\pi$
$211$	−235.366			−1.11548			−0.557739	+	0.830016i	$0.688331\pi$
$212$		0			0
$213$		−	30.0789i		−	0.141215i
$214$		0			0
$215$			224.595i			1.04463i
$216$		0			0
$217$	−27.4827	+	52.8325i	−0.126648	+	0.243468i
$218$		0			0
$219$	−516.237			−2.35725
$220$		0			0
$221$	−65.0346			−0.294274
$222$		0			0
$223$		−	209.306i		−	0.938593i	−0.883041	−	0.469297i	$-0.844508\pi$
$223$		−	209.306i		−	0.938593i	0.883041	−	0.469297i	$-0.155492\pi$
$224$		0			0
$225$	258.388			1.14839
$226$		0			0
$227$			217.751i			0.959256i	0.877472	+	0.479628i	$0.159228\pi$
$227$			217.751i			0.959256i	−0.877472	+	0.479628i	$0.840772\pi$
$228$		0			0
$229$			51.1341i			0.223293i	0.993748	+	0.111646i	$0.0356124\pi$
$229$			51.1341i			0.223293i	−0.993748	+	0.111646i	$0.964388\pi$
$230$		0			0
$231$	−267.835	+	514.883i	−1.15946	+	2.22893i
$232$		0			0
$233$	−242.517			−1.04085			−0.520423	−	0.853908i	$-0.674226\pi$
$233$	−242.517			−1.04085			−0.520423	+	0.853908i	$0.674226\pi$
$234$		0			0
$235$	185.445			0.789127
$236$		0			0
$237$		−	443.326i		−	1.87057i
$238$		0			0
$239$	17.4613			0.0730598			0.0365299	−	0.999333i	$-0.488370\pi$
$239$	17.4613			0.0730598			0.0365299	+	0.999333i	$0.488370\pi$
$240$		0			0
$241$		−	68.8417i		−	0.285650i	−0.989748	−	0.142825i	$-0.954381\pi$
$241$		−	68.8417i		−	0.285650i	0.989748	−	0.142825i	$-0.0456186\pi$
$242$		0			0
$243$		−	286.773i		−	1.18013i
$244$		0			0
$245$	245.905	−	172.405i	1.00369	−	0.703692i
$246$		0			0
$247$	35.7881			0.144891
$248$		0			0
$249$	469.211			1.88438
$250$		0			0
$251$		−	122.246i		−	0.487037i	−0.969896	−	0.243518i	$-0.921698\pi$
$251$		−	122.246i		−	0.487037i	0.969896	−	0.243518i	$-0.0783016\pi$
$252$		0			0
$253$	−26.2586			−0.103789
$254$		0			0
$255$			720.541i			2.82565i
$256$		0			0
$257$		−	312.466i		−	1.21582i	−0.794006	−	0.607910i	$-0.792008\pi$
$257$		−	312.466i		−	1.21582i	0.794006	−	0.607910i	$-0.207992\pi$
$258$		0			0
$259$	−329.935	−	171.628i	−1.27388	−	0.662655i
$260$		0			0
$261$	−845.810			−3.24065
$262$		0			0
$263$	373.959			1.42190			0.710949	−	0.703244i	$-0.248266\pi$
$263$	373.959			1.42190			0.710949	+	0.703244i	$0.248266\pi$
$264$		0			0
$265$		−	183.870i		−	0.693849i
$266$		0			0
$267$	150.272			0.562818
$268$		0			0
$269$			180.636i			0.671508i	0.941950	+	0.335754i	$0.108991\pi$
$269$			180.636i			0.671508i	−0.941950	+	0.335754i	$0.891009\pi$
$270$		0			0
$271$		−	275.195i		−	1.01548i	−0.861511	−	0.507740i	$-0.830481\pi$
$271$		−	275.195i		−	1.01548i	0.861511	−	0.507740i	$-0.169519\pi$
$272$		0			0
$273$	−101.565	−	52.8325i	−0.372032	−	0.193526i
$274$		0			0
$275$	191.592			0.696700
$276$		0			0
$277$	−376.940			−1.36079			−0.680397	−	0.732844i	$-0.738193\pi$
$277$	−376.940			−1.36079			−0.680397	+	0.732844i	$0.738193\pi$
$278$		0			0
$279$		−	174.956i		−	0.627081i
$280$		0			0
$281$	62.5173			0.222481			0.111241	−	0.993793i	$-0.464518\pi$
$281$	62.5173			0.222481			0.111241	+	0.993793i	$0.464518\pi$
$282$		0			0
$283$			103.435i			0.365494i	0.983160	+	0.182747i	$0.0584989\pi$
$283$			103.435i			0.365494i	−0.983160	+	0.182747i	$0.941501\pi$
$284$		0			0
$285$		−	396.510i		−	1.39126i
$286$		0			0
$287$	138.959	−	267.133i	0.484177	−	0.930776i
$288$		0			0
$289$	−178.483			−0.617587
$290$		0			0
$291$	−548.947			−1.88641
$292$		0			0
$293$			507.624i			1.73251i	0.499606	+	0.866253i	$0.333478\pi$
$293$			507.624i			1.73251i	−0.499606	+	0.866253i	$0.666522\pi$
$294$		0			0
$295$	−447.739			−1.51776
$296$		0			0
$297$		−	958.842i		−	3.22842i
$298$		0			0
$299$		−	5.17972i		−	0.0173235i
$300$		0			0
$301$	−227.565	−	118.376i	−0.756029	−	0.393275i
$302$		0			0
$303$	−584.735			−1.92982
$304$		0			0
$305$	34.7887			0.114061
$306$		0			0
$307$		−	419.316i		−	1.36585i	−0.730489	−	0.682925i	$-0.760708\pi$
$307$		−	419.316i		−	1.36585i	0.730489	−	0.682925i	$-0.239292\pi$
$308$		0			0
$309$	321.388			1.04009
$310$		0			0
$311$		−	171.628i		−	0.551858i	−0.961178	−	0.275929i	$-0.911015\pi$
$311$		−	171.628i		−	0.551858i	0.961178	−	0.275929i	$-0.0889854\pi$
$312$		0			0
$313$			503.484i			1.60857i	0.594240	+	0.804287i	$0.297453\pi$
$313$			503.484i			1.60857i	−0.594240	+	0.804287i	$0.702547\pi$
$314$		0			0
$315$	−407.159	+	782.719i	−1.29257	+	2.48482i
$316$		0			0
$317$	205.647			0.648727			0.324364	−	0.945932i	$-0.394850\pi$
$317$	205.647			0.648727			0.324364	+	0.945932i	$0.394850\pi$
$318$		0			0
$319$	−627.161			−1.96602
$320$		0			0
$321$			250.093i			0.779105i
$322$		0			0
$323$	257.253			0.796448
$324$		0			0
$325$			37.7931i			0.116286i
$326$		0			0
$327$		−	129.090i		−	0.394770i
$328$		0			0
$329$	−97.7414	+	187.897i	−0.297086	+	0.571115i
$330$		0			0
$331$	−19.6740			−0.0594382			−0.0297191	−	0.999558i	$-0.509461\pi$
$331$	−19.6740			−0.0594382			−0.0297191	+	0.999558i	$0.509461\pi$
$332$		0			0
$333$	1092.59			3.28104
$334$		0			0
$335$		−	176.362i		−	0.526454i
$336$		0			0
$337$	317.823			0.943096			0.471548	−	0.881840i	$-0.343695\pi$
$337$	317.823			0.943096			0.471548	+	0.881840i	$0.343695\pi$
$338$		0			0
$339$			1017.93i			3.00275i
$340$		0			0
$341$		−	129.728i		−	0.380434i
$342$		0			0
$343$	45.0765	+	340.025i	0.131418	+	0.991327i
$344$		0			0
$345$	−57.3880			−0.166342
$346$		0			0
$347$	−416.385			−1.19996			−0.599979	−	0.800016i	$-0.704824\pi$
$347$	−416.385			−1.19996			−0.599979	+	0.800016i	$0.704824\pi$
$348$		0			0
$349$			333.344i			0.955140i	0.878594	+	0.477570i	$0.158482\pi$
$349$			333.344i			0.955140i	−0.878594	+	0.477570i	$0.841518\pi$
$350$		0			0
$351$	189.139			0.538857
$352$		0			0
$353$			230.864i			0.654005i	0.945023	+	0.327003i	$0.106039\pi$
$353$			230.864i			0.654005i	−0.945023	+	0.327003i	$0.893961\pi$
$354$		0			0
$355$		−	33.9051i		−	0.0955073i
$356$		0			0
$357$	−730.069	−	379.772i	−2.04501	−	1.06379i
$358$		0			0
$359$	−16.9796			−0.0472970			−0.0236485	−	0.999720i	$-0.507528\pi$
$359$	−16.9796			−0.0472970			−0.0236485	+	0.999720i	$0.507528\pi$
$360$		0			0
$361$	219.435			0.607854
$362$		0			0
$363$		−	606.357i		−	1.67041i
$364$		0			0
$365$	−581.905			−1.59426
$366$		0			0
$367$			457.377i			1.24626i	0.782119	+	0.623129i	$0.214139\pi$
$367$			457.377i			1.24626i	−0.782119	+	0.623129i	$0.785861\pi$
$368$		0			0
$369$			884.614i			2.39733i
$370$		0			0
$371$	186.301	+	96.9113i	0.502160	+	0.261217i
$372$		0			0
$373$	541.974			1.45301			0.726507	−	0.687159i	$-0.241142\pi$
$373$	541.974			1.45301			0.726507	+	0.687159i	$0.241142\pi$
$374$		0			0
$375$	−414.413			−1.10510
$376$		0			0
$377$		−	123.712i		−	0.328149i
$378$		0			0
$379$	−254.799			−0.672294			−0.336147	−	0.941810i	$-0.609124\pi$
$379$	−254.799			−0.672294			−0.336147	+	0.941810i	$0.609124\pi$
$380$		0			0
$381$			1122.00i			2.94487i
$382$		0			0
$383$			295.983i			0.772803i	0.922331	+	0.386401i	$0.126282\pi$
$383$			295.983i			0.772803i	−0.922331	+	0.386401i	$0.873718\pi$
$384$		0			0
$385$	−301.905	+	580.380i	−0.784169	+	1.50748i
$386$		0			0
$387$	753.584			1.94725
$388$		0			0
$389$	538.164			1.38345			0.691727	−	0.722159i	$-0.256850\pi$
$389$	538.164			1.38345			0.691727	+	0.722159i	$0.256850\pi$
$390$		0			0
$391$		−	37.2329i		−	0.0952249i
$392$		0			0
$393$	−1016.67			−2.58694
$394$		0			0
$395$		−	499.720i		−	1.26511i
$396$		0			0
$397$			547.293i			1.37857i	0.724490	+	0.689286i	$0.242075\pi$
$397$			547.293i			1.37857i	−0.724490	+	0.689286i	$0.757925\pi$
$398$		0			0
$399$	401.753	+	208.986i	1.00690	+	0.523774i
$400$		0			0
$401$	203.375			0.507170			0.253585	−	0.967313i	$-0.418390\pi$
$401$	203.375			0.507170			0.253585	+	0.967313i	$0.418390\pi$
$402$		0			0
$403$	25.5899			0.0634984
$404$		0			0
$405$		−	961.170i		−	2.37326i
$406$		0			0
$407$	810.143			1.99052
$408$		0			0
$409$			103.627i			0.253366i	0.991943	+	0.126683i	$0.0404332\pi$
$409$			103.627i			0.253366i	−0.991943	+	0.126683i	$0.959567\pi$
$410$		0			0
$411$			13.6873i			0.0333024i
$412$		0			0
$413$	235.987	−	453.659i	0.571398	−	1.09845i
$414$		0			0
$415$	528.898			1.27445
$416$		0			0
$417$	101.565			0.243560
$418$		0			0
$419$		−	438.503i		−	1.04655i	−0.852165	−	0.523273i	$-0.824711\pi$
$419$		−	438.503i		−	1.04655i	0.852165	−	0.523273i	$-0.175289\pi$
$420$		0			0
$421$	741.293			1.76079			0.880396	−	0.474240i	$-0.157277\pi$
$421$	741.293			1.76079			0.880396	+	0.474240i	$0.157277\pi$
$422$		0			0
$423$		−	622.224i		−	1.47098i
$424$		0			0
$425$			271.665i			0.639212i
$426$		0			0
$427$	−18.3359	+	35.2488i	−0.0429412	+	0.0825498i
$428$		0			0
$429$	249.388			0.581324
$430$		0			0
$431$	82.8809			0.192299			0.0961495	−	0.995367i	$-0.469347\pi$
$431$	82.8809			0.192299			0.0961495	+	0.995367i	$0.469347\pi$
$432$		0			0
$433$		−	607.840i		−	1.40379i	−0.712282	−	0.701893i	$-0.752338\pi$
$433$		−	607.840i		−	1.40379i	0.712282	−	0.701893i	$-0.247662\pi$
$434$		0			0
$435$	−1370.65			−3.15093
$436$		0			0
$437$			20.4891i			0.0468857i
$438$		0			0
$439$			654.207i			1.49022i	0.666941	+	0.745111i	$0.267603\pi$
$439$			654.207i			1.49022i	−0.666941	+	0.745111i	$0.732397\pi$
$440$		0			0
$441$	−578.470	−	825.087i	−1.31172	−	1.87094i
$442$		0			0
$443$	566.176			1.27805			0.639025	−	0.769186i	$-0.279338\pi$
$443$	566.176			1.27805			0.639025	+	0.769186i	$0.279338\pi$
$444$		0			0
$445$	169.388			0.380647
$446$		0			0
$447$		−	13.6873i		−	0.0306204i
$448$		0			0
$449$	348.776			0.776784			0.388392	−	0.921494i	$-0.373031\pi$
$449$	348.776			0.776784			0.388392	+	0.921494i	$0.373031\pi$
$450$		0			0
$451$			655.934i			1.45440i
$452$		0			0
$453$		−	426.588i		−	0.941696i
$454$		0			0
$455$	−114.484	−	59.5531i	−0.251614	−	0.130886i
$456$		0			0
$457$	354.694			0.776136			0.388068	−	0.921631i	$-0.373143\pi$
$457$	354.694			0.776136			0.388068	+	0.921631i	$0.373143\pi$
$458$		0			0
$459$	1359.57			2.96203
$460$		0			0
$461$			220.531i			0.478375i	0.970973	+	0.239187i	$0.0768810\pi$
$461$			220.531i			0.478375i	−0.970973	+	0.239187i	$0.923119\pi$
$462$		0			0
$463$	497.178			1.07382			0.536910	−	0.843640i	$-0.319592\pi$
$463$	497.178			1.07382			0.536910	+	0.843640i	$0.319592\pi$
$464$		0			0
$465$		−	283.519i		−	0.609719i
$466$		0			0
$467$		−	299.499i		−	0.641326i	−0.947193	−	0.320663i	$-0.896094\pi$
$467$		−	299.499i		−	0.641326i	0.947193	−	0.320663i	$-0.103906\pi$
$468$		0			0
$469$	178.694	+	92.9541i	0.381011	+	0.198196i
$470$		0			0
$471$	−1623.37			−3.44663
$472$		0			0
$473$	558.776			1.18134
$474$		0			0
$475$		−	149.496i		−	0.314727i
$476$		0			0
$477$	−616.940			−1.29337
$478$		0			0
$479$			524.867i			1.09576i	0.836558	+	0.547878i	$0.184564\pi$
$479$			524.867i			1.09576i	−0.836558	+	0.547878i	$0.815436\pi$
$480$		0			0
$481$			159.807i			0.332239i
$482$		0			0
$483$	30.2471	−	58.1468i	0.0626235	−	0.120387i
$484$		0			0
$485$	−618.776			−1.27583
$486$		0			0
$487$	−62.2345			−0.127792			−0.0638958	−	0.997957i	$-0.520353\pi$
$487$	−62.2345			−0.127792			−0.0638958	+	0.997957i	$0.520353\pi$
$488$		0			0
$489$			48.1755i			0.0985184i
$490$		0			0
$491$	127.136			0.258933			0.129467	−	0.991584i	$-0.458674\pi$
$491$	127.136			0.258933			0.129467	+	0.991584i	$0.458674\pi$
$492$		0			0
$493$		−	889.271i		−	1.80380i
$494$		0			0
$495$		−	1921.94i		−	3.88270i
$496$		0			0
$497$	34.3534	+	17.8702i	0.0691215	+	0.0359560i
$498$		0			0
$499$	−699.460			−1.40172			−0.700862	−	0.713297i	$-0.747201\pi$
$499$	−699.460			−1.40172			−0.700862	+	0.713297i	$0.747201\pi$
$500$		0			0
$501$	−1375.27			−2.74505
$502$		0			0
$503$		−	856.217i		−	1.70222i	−0.524986	−	0.851111i	$-0.675930\pi$
$503$		−	856.217i		−	1.70222i	0.524986	−	0.851111i	$-0.324070\pi$
$504$		0			0
$505$	−659.116			−1.30518
$506$		0			0
$507$		−	869.717i		−	1.71542i
$508$		0			0
$509$			83.6547i			0.164351i	0.996618	+	0.0821755i	$0.0261868\pi$
$509$			83.6547i			0.164351i	−0.996618	+	0.0821755i	$0.973813\pi$
$510$		0			0
$511$	306.701	−	589.600i	0.600199	−	1.15382i
$512$		0			0
$513$	−748.164			−1.45841
$514$		0			0
$515$	362.270			0.703438
$516$		0			0
$517$		−	461.374i		−	0.892405i
$518$		0			0
$519$	−42.9081			−0.0826745
$520$		0			0
$521$			563.287i			1.08117i	0.841291	+	0.540583i	$0.181796\pi$
$521$			563.287i			1.08117i	−0.841291	+	0.540583i	$0.818204\pi$
$522$		0			0
$523$			414.011i			0.791607i	0.918335	+	0.395804i	$0.129534\pi$
$523$			414.011i			0.791607i	−0.918335	+	0.395804i	$0.870466\pi$
$524$		0			0
$525$	−220.694	+	424.260i	−0.420369	+	0.808114i
$526$		0			0
$527$	183.946			0.349043
$528$		0			0
$529$	−526.035			−0.994394
$530$		0			0
$531$			1502.30i			2.82919i
$532$		0			0
$533$	−129.388			−0.242754
$534$		0			0
$535$			281.906i			0.526927i
$536$		0			0
$537$		−	1623.44i		−	3.02317i
$538$		0			0
$539$	−428.930	−	611.795i	−0.795789	−	1.13505i
$540$		0			0
$541$	86.7068			0.160271			0.0801357	−	0.996784i	$-0.474465\pi$
$541$	86.7068			0.160271			0.0801357	+	0.996784i	$0.474465\pi$
$542$		0			0
$543$	1532.36			2.82202
$544$		0			0
$545$		−	145.511i		−	0.266992i
$546$		0			0
$547$	827.355			1.51253			0.756266	−	0.654265i	$-0.227022\pi$
$547$	827.355			1.51253			0.756266	+	0.654265i	$0.227022\pi$
$548$		0			0
$549$		−	116.727i		−	0.212617i
$550$		0			0
$551$			489.361i			0.888132i
$552$		0			0
$553$	506.328	+	263.385i	0.915602	+	0.476283i
$554$		0			0
$555$	1770.56			3.19020
$556$		0			0
$557$	343.741			0.617130			0.308565	−	0.951203i	$-0.400151\pi$
$557$	343.741			0.617130			0.308565	+	0.951203i	$0.400151\pi$
$558$		0			0
$559$			110.223i			0.197179i
$560$		0			0
$561$	1792.66			3.19546
$562$		0			0
$563$		−	308.968i		−	0.548788i	−0.961617	−	0.274394i	$-0.911523\pi$
$563$		−	308.968i		−	0.548788i	0.961617	−	0.274394i	$-0.0884772\pi$
$564$		0			0
$565$			1147.42i			2.03083i
$566$		0			0
$567$	973.880	+	506.598i	1.71760	+	0.893472i
$568$		0			0
$569$	182.177			0.320170			0.160085	−	0.987103i	$-0.448823\pi$
$569$	182.177			0.320170			0.160085	+	0.987103i	$0.448823\pi$
$570$		0			0
$571$	378.037			0.662061			0.331031	−	0.943620i	$-0.392604\pi$
$571$	378.037			0.662061			0.331031	+	0.943620i	$0.392604\pi$
$572$		0			0
$573$		−	111.681i		−	0.194905i
$574$		0			0
$575$	−21.6369			−0.0376294
$576$		0			0
$577$			81.6019i			0.141424i	0.997497	+	0.0707122i	$0.0225272\pi$
$577$			81.6019i			0.141424i	−0.997497	+	0.0707122i	$0.977473\pi$
$578$		0			0
$579$		−	1041.60i		−	1.79897i
$580$		0			0
$581$	−278.763	+	535.891i	−0.479799	+	0.922360i
$582$		0			0
$583$	−457.456			−0.784658
$584$		0			0
$585$	379.116			0.648062
$586$		0			0
$587$		−	37.9917i		−	0.0647217i	−0.999476	−	0.0323609i	$-0.989697\pi$
$587$		−	37.9917i		−	0.0647217i	0.999476	−	0.0323609i	$-0.0103026\pi$
$588$		0			0
$589$	−101.224			−0.171858
$590$		0			0
$591$		−	615.123i		−	1.04082i
$592$		0			0
$593$		−	406.099i		−	0.684822i	−0.939550	−	0.342411i	$-0.888757\pi$
$593$		−	406.099i		−	0.684822i	0.939550	−	0.342411i	$-0.111243\pi$
$594$		0			0
$595$	−822.938	−	428.081i	−1.38309	−	0.719464i
$596$		0			0
$597$	−720.681			−1.20717
$598$		0			0
$599$	292.800			0.488815			0.244407	−	0.969673i	$-0.421407\pi$
$599$	292.800			0.488815			0.244407	+	0.969673i	$0.421407\pi$
$600$		0			0
$601$		−	14.6997i		−	0.0244588i	−0.999925	−	0.0122294i	$-0.996107\pi$
$601$		−	14.6997i		−	0.0244588i	0.999925	−	0.0122294i	$-0.00389284\pi$
$602$		0			0
$603$	−591.748			−0.981340
$604$		0			0
$605$		−	683.490i		−	1.12973i
$606$		0			0
$607$			959.534i			1.58078i	0.612604	+	0.790390i	$0.290122\pi$
$607$			959.534i			1.58078i	−0.612604	+	0.790390i	$0.709878\pi$
$608$		0			0
$609$	722.423	−	1388.78i	1.18624	−	2.28042i
$610$		0			0
$611$	91.0095			0.148952
$612$		0			0
$613$	896.397			1.46231			0.731156	−	0.682211i	$-0.238981\pi$
$613$	896.397			1.46231			0.731156	+	0.682211i	$0.238981\pi$
$614$		0			0
$615$			1433.54i			2.33095i
$616$		0			0
$617$	137.280			0.222497			0.111248	−	0.993793i	$-0.464515\pi$
$617$	137.280			0.222497			0.111248	+	0.993793i	$0.464515\pi$
$618$		0			0
$619$		−	703.714i		−	1.13686i	−0.822733	−	0.568428i	$-0.807552\pi$
$619$		−	703.714i		−	1.13686i	0.822733	−	0.568428i	$-0.192448\pi$
$620$		0			0
$621$			108.284i			0.174370i
$622$		0			0
$623$	−89.2783	+	171.628i	−0.143304	+	0.275486i
$624$		0			0
$625$	−781.246			−1.24999
$626$		0			0
$627$	−986.487			−1.57334
$628$		0			0
$629$			1148.73i			1.82628i
$630$		0			0
$631$	87.6903			0.138970			0.0694852	−	0.997583i	$-0.477864\pi$
$631$	87.6903			0.138970			0.0694852	+	0.997583i	$0.477864\pi$
$632$		0			0
$633$			1279.77i			2.02175i
$634$		0			0
$635$			1264.72i			1.99169i
$636$		0			0
$637$	120.681	−	84.6098i	0.189452	−	0.132825i
$638$		0			0
$639$	−113.762			−0.178031
$640$		0			0
$641$	−380.953			−0.594310			−0.297155	−	0.954829i	$-0.596038\pi$
$641$	−380.953			−0.594310			−0.297155	+	0.954829i	$0.596038\pi$
$642$		0			0
$643$			441.511i			0.686642i	0.939218	+	0.343321i	$0.111552\pi$
$643$			441.511i			0.686642i	−0.939218	+	0.343321i	$0.888448\pi$
$644$		0			0
$645$	1221.20			1.89333
$646$		0			0
$647$		−	844.897i		−	1.30587i	−0.757414	−	0.652934i	$-0.773538\pi$
$647$		−	844.897i		−	1.30587i	0.757414	−	0.652934i	$-0.226462\pi$
$648$		0			0
$649$			1113.94i			1.71640i
$650$		0			0
$651$	287.268	+	149.433i	0.441272	+	0.229544i
$652$		0			0
$653$	−478.845			−0.733300			−0.366650	−	0.930359i	$-0.619495\pi$
$653$	−478.845			−0.733300			−0.366650	+	0.930359i	$0.619495\pi$
$654$		0			0
$655$	−1145.99			−1.74961
$656$		0			0
$657$			1952.47i			2.97179i
$658$		0			0
$659$	257.012			0.390003			0.195002	−	0.980803i	$-0.437529\pi$
$659$	257.012			0.390003			0.195002	+	0.980803i	$0.437529\pi$
$660$		0			0
$661$		−	788.652i		−	1.19312i	−0.802568	−	0.596560i	$-0.796534\pi$
$661$		−	788.652i		−	1.19312i	0.802568	−	0.596560i	$-0.203466\pi$
$662$		0			0
$663$			353.615i			0.533356i
$664$		0			0
$665$	452.858	+	235.570i	0.680989	+	0.354241i
$666$		0			0
$667$	70.8266			0.106187
$668$		0			0
$669$	−1138.07			−1.70115
$670$		0			0
$671$		−	86.5519i		−	0.128989i
$672$		0			0
$673$	−511.362			−0.759825			−0.379913	−	0.925022i	$-0.624046\pi$
$673$	−511.362			−0.759825			−0.379913	+	0.925022i	$0.624046\pi$
$674$		0			0
$675$		−	790.078i		−	1.17049i
$676$		0			0
$677$			543.542i			0.802868i	0.915888	+	0.401434i	$0.131488\pi$
$677$			543.542i			0.802868i	−0.915888	+	0.401434i	$0.868512\pi$
$678$		0			0
$679$	326.135	−	626.958i	0.480316	−	0.923355i
$680$		0			0
$681$	1183.99			1.73860
$682$		0			0
$683$	−785.312			−1.14980			−0.574899	−	0.818224i	$-0.694959\pi$
$683$	−785.312			−1.14980			−0.574899	+	0.818224i	$0.694959\pi$
$684$		0			0
$685$			15.4284i			0.0225232i
$686$		0			0
$687$	278.033			0.404706
$688$		0			0
$689$		−	90.2366i		−	0.130967i
$690$		0			0
$691$			1063.98i			1.53978i	0.638179	+	0.769888i	$0.279688\pi$
$691$			1063.98i			1.53978i	−0.638179	+	0.769888i	$0.720312\pi$
$692$		0			0
$693$	1947.35	+	1012.98i	2.81003	+	1.46174i
$694$		0			0
$695$	114.484			0.164726
$696$		0			0
$697$	−930.069			−1.33439
$698$		0			0
$699$			1318.65i			1.88648i
$700$		0			0
$701$	113.810			0.162354			0.0811772	−	0.996700i	$-0.474132\pi$
$701$	113.810			0.162354			0.0811772	+	0.996700i	$0.474132\pi$
$702$		0			0
$703$		−	632.138i		−	0.899200i
$704$		0			0
$705$		−	1008.33i		−	1.43025i
$706$		0			0
$707$	347.397	−	667.832i	0.491367	−	0.944600i
$708$		0			0
$709$	−418.000			−0.589563			−0.294781	−	0.955565i	$-0.595247\pi$
$709$	−418.000			−0.589563			−0.294781	+	0.955565i	$0.595247\pi$
$710$		0			0
$711$	−1676.71			−2.35824
$712$		0			0
$713$			14.6505i			0.0205476i
$714$		0			0
$715$	281.112			0.393163
$716$		0			0
$717$		−	94.9429i		−	0.132417i
$718$		0			0
$719$		−	752.720i		−	1.04690i	−0.852057	−	0.523449i	$-0.824645\pi$
$719$		−	752.720i		−	1.04690i	0.852057	−	0.523449i	$-0.175355\pi$
$720$		0			0
$721$	−190.940	+	367.061i	−0.264826	+	0.509099i
$722$		0			0
$723$	−374.316			−0.517726
$724$		0			0
$725$	−516.776			−0.712794
$726$		0			0
$727$			269.945i			0.371314i	0.982615	+	0.185657i	$0.0594414\pi$
$727$			269.945i			0.371314i	−0.982615	+	0.185657i	$0.940559\pi$
$728$		0			0
$729$	−147.871			−0.202840
$730$		0			0
$731$			792.306i			1.08387i
$732$		0			0
$733$		−	805.971i		−	1.09955i	−0.835312	−	0.549776i	$-0.814713\pi$
$733$		−	805.971i		−	1.09955i	0.835312	−	0.549776i	$-0.185287\pi$
$734$		0			0
$735$	−937.423	−	1337.07i	−1.27540	−	1.81914i
$736$		0			0
$737$	−438.776			−0.595354
$738$		0			0
$739$	672.425			0.909912			0.454956	−	0.890514i	$-0.349655\pi$
$739$	672.425			0.909912			0.454956	+	0.890514i	$0.349655\pi$
$740$		0			0
$741$		−	194.592i		−	0.262607i
$742$		0			0
$743$	367.686			0.494867			0.247434	−	0.968905i	$-0.420413\pi$
$743$	367.686			0.494867			0.247434	+	0.968905i	$0.420413\pi$
$744$		0			0
$745$		−	15.4284i		−	0.0207093i
$746$		0			0
$747$		−	1774.61i		−	2.37565i
$748$		0			0
$749$	−285.634	−	148.583i	−0.381354	−	0.198375i
$750$		0			0
$751$	181.778			0.242048			0.121024	−	0.992650i	$-0.461382\pi$
$751$	181.778			0.242048			0.121024	+	0.992650i	$0.461382\pi$
$752$		0			0
$753$	−664.694			−0.882728
$754$		0			0
$755$		−	480.853i		−	0.636891i
$756$		0			0
$757$	63.7414			0.0842026			0.0421013	−	0.999113i	$-0.486595\pi$
$757$	63.7414			0.0842026			0.0421013	+	0.999113i	$0.486595\pi$
$758$		0			0
$759$			142.777i			0.188112i
$760$		0			0
$761$		−	1279.94i		−	1.68192i	−0.541096	−	0.840961i	$-0.681990\pi$
$761$		−	1279.94i		−	1.68192i	0.541096	−	0.840961i	$-0.318010\pi$
$762$		0			0
$763$	147.435	+	76.6936i	0.193231	+	0.100516i
$764$		0			0
$765$	2725.17			3.56232
$766$		0			0
$767$	−219.734			−0.286485
$768$		0			0
$769$		−	321.701i		−	0.418337i	−0.977880	−	0.209169i	$-0.932924\pi$
$769$		−	321.701i		−	0.418337i	0.977880	−	0.209169i	$-0.0670758\pi$
$770$		0			0
$771$	−1698.98			−2.20361
$772$		0			0
$773$			697.333i			0.902112i	0.892496	+	0.451056i	$0.148953\pi$
$773$			697.333i			0.902112i	−0.892496	+	0.451056i	$0.851047\pi$
$774$		0			0
$775$		−	106.895i		−	0.137929i
$776$		0			0
$777$	−933.198	+	1793.97i	−1.20103	+	2.30884i
$778$		0			0
$779$	511.811			0.657011
$780$		0			0
$781$	−84.3534			−0.108007
$782$		0			0
$783$			2586.25i			3.30300i
$784$		0			0
$785$	−1829.87			−2.33104
$786$		0			0
$787$			330.202i			0.419570i	0.977748	+	0.209785i	$0.0672764\pi$
$787$			330.202i			0.419570i	−0.977748	+	0.209785i	$0.932724\pi$
$788$		0			0
$789$		−	2033.34i		−	2.57711i
$790$		0			0
$791$	−1162.59	−	604.763i	−1.46977	−	0.764555i
$792$		0			0
$793$	17.0730			0.0215297
$794$		0			0
$795$	−999.764			−1.25756
$796$		0			0
$797$		−	456.879i		−	0.573248i	−0.958043	−	0.286624i	$-0.907467\pi$
$797$		−	456.879i		−	0.573248i	0.958043	−	0.286624i	$-0.0925331\pi$
$798$		0			0
$799$	654.196			0.818768
$800$		0			0
$801$		−	568.348i		−	0.709548i
$802$		0			0
$803$			1447.74i			1.80291i
$804$		0			0
$805$	34.0948	−	65.5434i	0.0423537	−	0.0814204i
$806$		0			0
$807$	982.178			1.21707
$808$		0			0
$809$	175.211			0.216578			0.108289	−	0.994119i	$-0.465463\pi$
$809$	175.211			0.216578			0.108289	+	0.994119i	$0.465463\pi$
$810$		0			0
$811$		−	149.621i		−	0.184489i	−0.995736	−	0.0922446i	$-0.970596\pi$
$811$		−	149.621i		−	0.184489i	0.995736	−	0.0922446i	$-0.0294042\pi$
$812$		0			0
$813$	−1496.33			−1.84050
$814$		0			0
$815$			54.3037i			0.0666303i
$816$		0			0
$817$		−	436.001i		−	0.533661i
$818$		0			0
$819$	−199.819	+	384.130i	−0.243979	+	0.469023i
$820$		0			0
$821$	880.750			1.07278			0.536389	−	0.843971i	$-0.319788\pi$
$821$	880.750			1.07278			0.536389	+	0.843971i	$0.319788\pi$
$822$		0			0
$823$	−1149.93			−1.39724			−0.698621	−	0.715492i	$-0.746202\pi$
$823$	−1149.93			−1.39724			−0.698621	+	0.715492i	$0.746202\pi$
$824$		0			0
$825$		−	1041.75i		−	1.26273i
$826$		0			0
$827$	−633.041			−0.765466			−0.382733	−	0.923859i	$-0.625017\pi$
$827$	−633.041			−0.765466			−0.382733	+	0.923859i	$0.625017\pi$
$828$		0			0
$829$		−	900.963i		−	1.08681i	−0.839472	−	0.543404i	$-0.817135\pi$
$829$		−	900.963i		−	1.08681i	0.839472	−	0.543404i	$-0.182865\pi$
$830$		0			0
$831$			2049.55i			2.46637i
$832$		0			0
$833$	867.483	−	608.194i	1.04140	−	0.730125i
$834$		0			0
$835$	−1550.21			−1.85654
$836$		0			0
$837$	−534.965			−0.639146
$838$		0			0
$839$			1587.19i			1.89176i	0.324514	+	0.945881i	$0.394799\pi$
$839$			1587.19i			1.89176i	−0.324514	+	0.945881i	$0.605201\pi$
$840$		0			0
$841$	850.621			1.01144
$842$		0			0
$843$		−	339.928i		−	0.403236i
$844$		0			0
$845$		−	980.350i		−	1.16018i
$846$		0			0
$847$	692.527	+	360.243i	0.817624	+	0.425316i
$848$		0			0
$849$	562.410			0.662438
$850$		0			0
$851$	−91.4911			−0.107510
$852$		0			0
$853$		−	614.776i		−	0.720722i	−0.932813	−	0.360361i	$-0.882654\pi$
$853$		−	614.776i		−	0.720722i	0.932813	−	0.360361i	$-0.117346\pi$
$854$		0			0
$855$	−1499.65			−1.75397
$856$		0			0
$857$		−	741.142i		−	0.864809i	−0.901680	−	0.432405i	$-0.857665\pi$
$857$		−	741.142i		−	0.864809i	0.901680	−	0.432405i	$-0.142335\pi$
$858$		0			0
$859$		−	686.448i		−	0.799125i	−0.916706	−	0.399563i	$-0.869162\pi$
$859$		−	686.448i		−	0.799125i	0.916706	−	0.399563i	$-0.130838\pi$
$860$		0			0
$861$	−1452.49	−	755.566i	−1.68698	−	0.877544i
$862$		0			0
$863$	−1375.44			−1.59378			−0.796892	−	0.604122i	$-0.793524\pi$
$863$	−1375.44			−1.59378			−0.796892	+	0.604122i	$0.793524\pi$
$864$		0			0
$865$	−48.3662			−0.0559147
$866$		0			0
$867$			970.471i			1.11934i
$868$		0			0
$869$	−1243.27			−1.43069
$870$		0			0
$871$		−	86.5519i		−	0.0993707i
$872$		0			0
$873$			2076.18i			2.37821i
$874$		0			0
$875$	246.207	−	473.306i	0.281380	−	0.540921i
$876$		0			0
$877$	1319.39			1.50443			0.752217	−	0.658916i	$-0.228985\pi$
$877$	1319.39			1.50443			0.752217	+	0.658916i	$0.228985\pi$
$878$		0			0
$879$	2760.12			3.14007
$880$		0			0
$881$		−	270.582i		−	0.307130i	−0.988139	−	0.153565i	$-0.950924\pi$
$881$		−	270.582i		−	0.307130i	0.988139	−	0.153565i	$-0.0490755\pi$
$882$		0			0
$883$	1491.67			1.68932			0.844660	−	0.535303i	$-0.179803\pi$
$883$	1491.67			1.68932			0.844660	+	0.535303i	$0.179803\pi$
$884$		0			0
$885$			2434.51i			2.75086i
$886$		0			0
$887$		−	79.8264i		−	0.0899960i	−0.998987	−	0.0449980i	$-0.985672\pi$
$887$		−	79.8264i		−	0.0899960i	0.998987	−	0.0449980i	$-0.0143281\pi$
$888$		0			0
$889$	−1281.44	−	666.589i	−1.44144	−	0.749819i
$890$		0			0
$891$	−2391.32			−2.68386
$892$		0			0
$893$	−360.000			−0.403135
$894$		0			0
$895$		−	1829.95i		−	2.04464i
$896$		0			0
$897$	−28.1639			−0.0313979
$898$		0			0
$899$			349.911i			0.389223i
$900$		0			0
$901$		−	648.640i		−	0.719912i
$902$		0			0
$903$	−643.650	+	1237.35i	−0.712791	+	1.37026i
$904$		0			0
$905$	1727.28			1.90860
$906$		0			0
$907$	845.557			0.932256			0.466128	−	0.884717i	$-0.345649\pi$
$907$	845.557			0.932256			0.466128	+	0.884717i	$0.345649\pi$
$908$		0			0
$909$			2211.54i			2.43293i
$910$		0			0
$911$	−1377.51			−1.51209			−0.756045	−	0.654519i	$-0.772871\pi$
$911$	−1377.51			−1.51209			−0.756045	+	0.654519i	$0.772871\pi$
$912$		0			0
$913$		−	1315.86i		−	1.44125i
$914$		0			0
$915$		−	189.158i		−	0.206730i
$916$		0			0
$917$	604.013	−	1161.15i	0.658684	−	1.26625i
$918$		0			0
$919$	648.950			0.706148			0.353074	−	0.935595i	$-0.385136\pi$
$919$	648.950			0.706148			0.353074	+	0.935595i	$0.385136\pi$
$920$		0			0
$921$	−2279.96			−2.47553
$922$		0			0
$923$		−	16.6394i		−	0.0180275i
$924$		0			0
$925$	667.552			0.721678
$926$		0			0
$927$		−	1215.53i		−	1.31125i
$928$		0			0
$929$			1394.62i			1.50120i	0.660756	+	0.750601i	$0.270236\pi$
$929$			1394.62i			1.50120i	−0.660756	+	0.750601i	$0.729764\pi$
$930$		0			0
$931$	−477.370	+	334.685i	−0.512750	+	0.359490i
$932$		0			0
$933$	−933.198			−1.00021
$934$		0			0
$935$	2020.69			2.16117
$936$		0			0
$937$			433.382i			0.462521i	0.972892	+	0.231260i	$0.0742850\pi$
$937$			433.382i			0.462521i	−0.972892	+	0.231260i	$0.925715\pi$
$938$		0			0
$939$	2737.61			2.91546
$940$		0			0
$941$			891.147i			0.947021i	0.880788	+	0.473511i	$0.157013\pi$
$941$			891.147i			0.947021i	−0.880788	+	0.473511i	$0.842987\pi$
$942$		0			0
$943$		−	74.0759i		−	0.0785535i
$944$		0			0
$945$	2393.34	+	1244.98i	2.53263	+	1.31744i
$946$		0			0
$947$	1021.88			1.07907			0.539537	−	0.841962i	$-0.318599\pi$
$947$	1021.88			1.07907			0.539537	+	0.841962i	$0.318599\pi$
$948$		0			0
$949$	−285.577			−0.300925
$950$		0			0
$951$		−	1118.17i		−	1.17578i
$952$		0			0
$953$	−1322.52			−1.38774			−0.693871	−	0.720100i	$-0.744096\pi$
$953$	−1322.52			−1.38774			−0.693871	+	0.720100i	$0.744096\pi$
$954$		0			0
$955$		−	125.887i		−	0.131819i
$956$		0			0
$957$			3410.09i			3.56331i
$958$		0			0
$959$	−15.6324	−	8.13176i	−0.0163008	−	0.00847942i
$960$		0			0
$961$	888.621			0.924684
$962$		0			0
$963$	945.881			0.982223
$964$		0			0
$965$		−	1174.10i		−	1.21668i
$966$		0			0
$967$	−1681.56			−1.73894			−0.869471	−	0.493983i	$-0.835540\pi$
$967$	−1681.56			−1.73894			−0.869471	+	0.493983i	$0.835540\pi$
$968$		0			0
$969$		−	1398.77i		−	1.44352i
$970$		0			0
$971$			1177.91i			1.21309i	0.795049	+	0.606546i	$0.207445\pi$
$971$			1177.91i			1.21309i	−0.795049	+	0.606546i	$0.792555\pi$
$972$		0			0
$973$	−60.3406	+	115.998i	−0.0620150	+	0.119217i
$974$		0			0
$975$	205.494			0.210763
$976$		0			0
$977$	1440.21			1.47411			0.737056	−	0.675832i	$-0.236215\pi$
$977$	1440.21			1.47411			0.737056	+	0.675832i	$0.236215\pi$
$978$		0			0
$979$		−	421.425i		−	0.430465i
$980$		0			0
$981$	−488.233			−0.497689
$982$		0			0
$983$			283.257i			0.288156i	0.989566	+	0.144078i	$0.0460215\pi$
$983$			283.257i			0.288156i	−0.989566	+	0.144078i	$0.953978\pi$
$984$		0			0
$985$		−	693.370i		−	0.703929i
$986$		0			0
$987$	1021.66	+	531.453i	1.03512	+	0.538453i
$988$		0			0
$989$	−63.1037			−0.0638055
$990$		0			0
$991$	−1079.85			−1.08966			−0.544830	−	0.838547i	$-0.683406\pi$
$991$	−1079.85			−1.08966			−0.544830	+	0.838547i	$0.683406\pi$
$992$		0			0
$993$			106.974i			0.107729i
$994$		0			0
$995$	−812.356			−0.816438
$996$		0			0
$997$			1572.31i			1.57704i	0.615010	+	0.788520i	$0.289152\pi$
$997$			1572.31i			1.57704i	−0.615010	+	0.788520i	$0.710848\pi$
$998$		0			0
$999$		−	3340.82i		−	3.34417i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.c.g.321.1		8
4.3	odd	2	inner	448.3.c.g.321.7		8
7.6	odd	2	inner	448.3.c.g.321.8		8
8.3	odd	2		224.3.c.a.97.2	yes	8
8.5	even	2		224.3.c.a.97.8	yes	8
24.5	odd	2		2016.3.f.c.1441.2		8
24.11	even	2		2016.3.f.c.1441.1		8
28.27	even	2	inner	448.3.c.g.321.2		8
56.13	odd	2		224.3.c.a.97.1	✓	8
56.27	even	2		224.3.c.a.97.7	yes	8
168.83	odd	2		2016.3.f.c.1441.7		8
168.125	even	2		2016.3.f.c.1441.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.c.a.97.1	✓	8	56.13	odd	2
224.3.c.a.97.2	yes	8	8.3	odd	2
224.3.c.a.97.7	yes	8	56.27	even	2
224.3.c.a.97.8	yes	8	8.5	even	2
448.3.c.g.321.1		8	1.1	even	1	trivial
448.3.c.g.321.2		8	28.27	even	2	inner
448.3.c.g.321.7		8	4.3	odd	2	inner
448.3.c.g.321.8		8	7.6	odd	2	inner
2016.3.f.c.1441.1		8	24.11	even	2
2016.3.f.c.1441.2		8	24.5	odd	2
2016.3.f.c.1441.7		8	168.83	odd	2
2016.3.f.c.1441.8		8	168.125	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.43734i q^{3} -6.12900i q^{5} +(6.21005 + 3.23038i) q^{7} -20.5647 q^{9} -15.2485 q^{11} -3.00789i q^{13} -33.3255 q^{15} -21.6213i q^{17} +11.8981i q^{19} +(17.5647 - 33.7661i) q^{21} +1.72205 q^{23} -12.5647 q^{25} +62.8810i q^{27} +41.1293 q^{29} +8.50759i q^{31} +82.9114i q^{33} +(19.7990 - 38.0614i) q^{35} -53.1293 q^{37} -16.3549 q^{39} -43.0162i q^{41} -36.6446 q^{43} +126.041i q^{45} +30.2569i q^{47} +(28.1293 + 40.1216i) q^{49} -117.563 q^{51} +30.0000 q^{53} +93.4582i q^{55} +64.6940 q^{57} -73.0525i q^{59} +5.67609i q^{61} +(-127.707 - 66.4316i) q^{63} -18.4353 q^{65} +28.7750 q^{67} -9.36335i q^{69} +5.53191 q^{71} -94.9429i q^{73} +68.3183i q^{75} +(-94.6940 - 49.2585i) q^{77} +81.5337 q^{79} +156.823 q^{81} +86.2943i q^{83} -132.517 q^{85} -223.634i q^{87} +27.6371i q^{89} +(9.71661 - 18.6791i) q^{91} +46.2586 q^{93} +72.9234 q^{95} -100.959i q^{97} +313.581 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 56 q^{9} + 32 q^{21} + 8 q^{25} + 112 q^{29} - 208 q^{37} + 8 q^{49} + 240 q^{53} + 192 q^{57} - 256 q^{65} - 432 q^{77} + 712 q^{81} - 192 q^{85} - 64 q^{93}+O(q^{100}) \) 8 * q - 56 * q^9 + 32 * q^21 + 8 * q^25 + 112 * q^29 - 208 * q^37 + 8 * q^49 + 240 * q^53 + 192 * q^57 - 256 * q^65 - 432 * q^77 + 712 * q^81 - 192 * q^85 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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