LMFDB - Embedded newform 2034.2.c.i.451.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,2,Mod(451,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.451"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2034.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,10,0,10,0,0,4,10,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2415717711$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 29x^{8} + 264x^{6} + 804x^{4} + 688x^{2} + 64 $ x^10 + 29x^8 + 264x^6 + 804x^4 + 688x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 678)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.9
Root			$3.19149i$ of defining polynomial
Character	$\chi$	$=$	2034.451
Dual form			2034.2.c.i.451.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.19149i q^{5} +4.56483 q^{7} +1.00000 q^{8} +3.19149i q^{10} +3.01570 q^{11} +3.43896 q^{13} +4.56483 q^{14} +1.00000 q^{16} +1.81816i q^{17} -5.01570i q^{19} +3.19149i q^{20} +3.01570 q^{22} +2.17579i q^{23} -5.18563 q^{25} +3.43896 q^{26} +4.56483 q^{28} -7.37713i q^{29} -9.20133 q^{31} +1.00000 q^{32} +1.81816i q^{34} +14.5686i q^{35} -3.26317i q^{37} -5.01570i q^{38} +3.19149i q^{40} -6.38299 q^{41} -8.75046i q^{43} +3.01570 q^{44} +2.17579i q^{46} +2.62667i q^{47} +13.8377 q^{49} -5.18563 q^{50} +3.43896 q^{52} -11.1020 q^{53} +9.62459i q^{55} +4.56483 q^{56} -7.37713i q^{58} +1.43896i q^{59} +8.19736 q^{61} -9.20133 q^{62} +1.00000 q^{64} +10.9754i q^{65} +8.75046i q^{67} +1.81816i q^{68} +14.5686i q^{70} +6.09703i q^{71} -3.26317i q^{74} -5.01570i q^{76} +13.7662 q^{77} -2.24747i q^{79} +3.19149i q^{80} -6.38299 q^{82} -10.7505 q^{83} -5.80264 q^{85} -8.75046i q^{86} +3.01570 q^{88} -12.8920i q^{89} +15.6983 q^{91} +2.17579i q^{92} +2.62667i q^{94} +16.0076 q^{95} -5.50885 q^{97} +13.8377 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} + 4 q^{7} + 10 q^{8} - 2 q^{11} + 6 q^{13} + 4 q^{14} + 10 q^{16} - 2 q^{22} - 8 q^{25} + 6 q^{26} + 4 q^{28} - 16 q^{31} + 10 q^{32} + 4 q^{41} - 2 q^{44} + 10 q^{49} - 8 q^{50}+ \cdots + 10 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 + 4 * q^7 + 10 * q^8 - 2 * q^11 + 6 * q^13 + 4 * q^14 + 10 * q^16 - 2 * q^22 - 8 * q^25 + 6 * q^26 + 4 * q^28 - 16 * q^31 + 10 * q^32 + 4 * q^41 - 2 * q^44 + 10 * q^49 - 8 * q^50 + 6 * q^52 - 22 * q^53 + 4 * q^56 + 58 * q^61 - 16 * q^62 + 10 * q^64 + 20 * q^77 + 4 * q^82 - 22 * q^83 - 82 * q^85 - 2 * q^88 - 38 * q^91 + 20 * q^95 + 26 * q^97 + 10 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times$.

$n$	$227$	$1585$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			3.19149i			1.42728i	0.700513	+	0.713640i	$0.252955\pi$
$5$			3.19149i			1.42728i	−0.700513	+	0.713640i	$0.747045\pi$
$6$		0			0
$7$	4.56483			1.72534			0.862671	−	0.505765i	$-0.168790\pi$
$7$	4.56483			1.72534			0.862671	+	0.505765i	$0.168790\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$			3.19149i			1.00924i
$11$	3.01570			0.909268			0.454634	−	0.890678i	$-0.349770\pi$
$11$	3.01570			0.909268			0.454634	+	0.890678i	$0.349770\pi$
$12$		0			0
$13$	3.43896			0.953797			0.476898	−	0.878958i	$-0.341761\pi$
$13$	3.43896			0.953797			0.476898	+	0.878958i	$0.341761\pi$
$14$	4.56483			1.22000
$15$		0			0
$16$	1.00000			0.250000
$17$			1.81816i			0.440968i	0.975391	+	0.220484i	$0.0707638\pi$
$17$			1.81816i			0.440968i	−0.975391	+	0.220484i	$0.929236\pi$
$18$		0			0
$19$		−	5.01570i		−	1.15068i	−0.817914	−	0.575341i	$-0.804870\pi$
$19$		−	5.01570i		−	1.15068i	0.817914	−	0.575341i	$-0.195130\pi$
$20$			3.19149i			0.713640i
$21$		0			0
$22$	3.01570			0.642950
$23$			2.17579i			0.453684i	0.973932	+	0.226842i	$0.0728401\pi$
$23$			2.17579i			0.453684i	−0.973932	+	0.226842i	$0.927160\pi$
$24$		0			0
$25$	−5.18563			−1.03713
$26$	3.43896			0.674436
$27$		0			0
$28$	4.56483			0.862671
$29$		−	7.37713i		−	1.36990i	−0.728591	−	0.684949i	$-0.759825\pi$
$29$		−	7.37713i		−	1.36990i	0.728591	−	0.684949i	$-0.240175\pi$
$30$		0			0
$31$	−9.20133			−1.65261			−0.826304	−	0.563224i	$-0.809561\pi$
$31$	−9.20133			−1.65261			−0.826304	+	0.563224i	$0.809561\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			1.81816i			0.311812i
$35$			14.5686i			2.46255i
$36$		0			0
$37$		−	3.26317i		−	0.536462i	−0.963355	−	0.268231i	$-0.913561\pi$
$37$		−	3.26317i		−	0.536462i	0.963355	−	0.268231i	$-0.0864391\pi$
$38$		−	5.01570i		−	0.813654i
$39$		0			0
$40$			3.19149i			0.504619i
$41$	−6.38299			−0.996855			−0.498428	−	0.866931i	$-0.666089\pi$
$41$	−6.38299			−0.996855			−0.498428	+	0.866931i	$0.666089\pi$
$42$		0			0
$43$		−	8.75046i		−	1.33443i	−0.744864	−	0.667216i	$-0.767486\pi$
$43$		−	8.75046i		−	1.33443i	0.744864	−	0.667216i	$-0.232514\pi$
$44$	3.01570			0.454634
$45$		0			0
$46$			2.17579i			0.320803i
$47$			2.62667i			0.383139i	0.981479	+	0.191569i	$0.0613577\pi$
$47$			2.62667i			0.383139i	−0.981479	+	0.191569i	$0.938642\pi$
$48$		0			0
$49$	13.8377			1.97681
$50$	−5.18563			−0.733359
$51$		0			0
$52$	3.43896			0.476898
$53$	−11.1020			−1.52498			−0.762491	−	0.646998i	$-0.776024\pi$
$53$	−11.1020			−1.52498			−0.762491	+	0.646998i	$0.776024\pi$
$54$		0			0
$55$			9.62459i			1.29778i
$56$	4.56483			0.610001
$57$		0			0
$58$		−	7.37713i		−	0.968664i
$59$			1.43896i			0.187337i	0.995603	+	0.0936685i	$0.0298594\pi$
$59$			1.43896i			0.187337i	−0.995603	+	0.0936685i	$0.970141\pi$
$60$		0			0
$61$	8.19736			1.04956			0.524782	−	0.851237i	$-0.324147\pi$
$61$	8.19736			1.04956			0.524782	+	0.851237i	$0.324147\pi$
$62$	−9.20133			−1.16857
$63$		0			0
$64$	1.00000			0.125000
$65$			10.9754i			1.36133i
$66$		0			0
$67$			8.75046i			1.06904i	0.845156	+	0.534519i	$0.179507\pi$
$67$			8.75046i			1.06904i	−0.845156	+	0.534519i	$0.820493\pi$
$68$			1.81816i			0.220484i
$69$		0			0
$70$			14.5686i			1.74128i
$71$			6.09703i			0.723585i	0.932259	+	0.361792i	$0.117835\pi$
$71$			6.09703i			0.723585i	−0.932259	+	0.361792i	$0.882165\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		−	3.26317i		−	0.379336i
$75$		0			0
$76$		−	5.01570i		−	0.575341i
$77$	13.7662			1.56880
$78$		0			0
$79$		−	2.24747i		−	0.252860i	−0.991976	−	0.126430i	$-0.959648\pi$
$79$		−	2.24747i		−	0.252860i	0.991976	−	0.126430i	$-0.0403519\pi$
$80$			3.19149i			0.356820i
$81$		0			0
$82$	−6.38299			−0.704883
$83$	−10.7505			−1.18002			−0.590008	−	0.807397i	$-0.700875\pi$
$83$	−10.7505			−1.18002			−0.590008	+	0.807397i	$0.700875\pi$
$84$		0			0
$85$	−5.80264			−0.629385
$86$		−	8.75046i		−	0.943586i
$87$		0			0
$88$	3.01570			0.321475
$89$		−	12.8920i		−	1.36655i	−0.730160	−	0.683276i	$-0.760555\pi$
$89$		−	12.8920i		−	1.36655i	0.730160	−	0.683276i	$-0.239445\pi$
$90$		0			0
$91$	15.6983			1.64563
$92$			2.17579i			0.226842i
$93$		0			0
$94$			2.62667i			0.270920i
$95$	16.0076			1.64234
$96$		0			0
$97$	−5.50885			−0.559339			−0.279670	−	0.960096i	$-0.590225\pi$
$97$	−5.50885			−0.559339			−0.279670	+	0.960096i	$0.590225\pi$
$98$	13.8377			1.39781
$99$		0			0
$100$	−5.18563			−0.518563
$101$			5.44322i			0.541621i	0.962633	+	0.270810i	$0.0872917\pi$
$101$			5.44322i			0.541621i	−0.962633	+	0.270810i	$0.912708\pi$
$102$		0			0
$103$			17.9035i			1.76408i	0.471173	+	0.882041i	$0.343831\pi$
$103$			17.9035i			1.76408i	−0.471173	+	0.882041i	$0.656169\pi$
$104$	3.43896			0.337218
$105$		0			0
$106$	−11.1020			−1.07833
$107$			8.53721i			0.825324i	0.910884	+	0.412662i	$0.135401\pi$
$107$			8.53721i			0.825324i	−0.910884	+	0.412662i	$0.864599\pi$
$108$		0			0
$109$	−2.59244			−0.248311			−0.124155	−	0.992263i	$-0.539622\pi$
$109$	−2.59244			−0.248311			−0.124155	+	0.992263i	$0.539622\pi$
$110$			9.62459i			0.917669i
$111$		0			0
$112$	4.56483			0.431336
$113$	7.83386	−	7.18544i	0.736948	−	0.675950i
$113$	7.83386	−	7.18544i	0.736948	−	0.675950i
$114$		0			0
$115$	−6.94402			−0.647534
$116$		−	7.37713i		−	0.684949i
$117$		0			0
$118$			1.43896i			0.132467i
$119$			8.29958i			0.760822i
$120$		0			0
$121$	−1.90554			−0.173231
$122$	8.19736			0.742154
$123$		0			0
$124$	−9.20133			−0.826304
$125$		−	0.592441i		−	0.0529896i
$126$		0			0
$127$	−13.0036			−1.15388			−0.576942	−	0.816785i	$-0.695754\pi$
$127$	−13.0036			−1.15388			−0.576942	+	0.816785i	$0.695754\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			10.9754i			0.962609i
$131$	−6.90554			−0.603340			−0.301670	−	0.953412i	$-0.597544\pi$
$131$	−6.90554			−0.603340			−0.301670	+	0.953412i	$0.597544\pi$
$132$		0			0
$133$		−	22.8958i		−	1.98532i
$134$			8.75046i			0.755925i
$135$		0			0
$136$			1.81816i			0.155906i
$137$		−	5.30771i		−	0.453468i	−0.973957	−	0.226734i	$-0.927195\pi$
$137$		−	5.30771i		−	0.453468i	0.973957	−	0.226734i	$-0.0728048\pi$
$138$		0			0
$139$	−0.548938			−0.0465604			−0.0232802	−	0.999729i	$-0.507411\pi$
$139$	−0.548938			−0.0465604			−0.0232802	+	0.999729i	$0.507411\pi$
$140$			14.5686i			1.23127i
$141$		0			0
$142$			6.09703i			0.511652i
$143$	10.3709			0.867257
$144$		0			0
$145$	23.5440			1.95523
$146$		0			0
$147$		0			0
$148$		−	3.26317i		−	0.268231i
$149$	16.8606			1.38128			0.690638	−	0.723201i	$-0.257330\pi$
$149$	16.8606			1.38128			0.690638	+	0.723201i	$0.257330\pi$
$150$		0			0
$151$		−	5.12540i		−	0.417099i	−0.978012	−	0.208549i	$-0.933126\pi$
$151$		−	5.12540i		−	0.417099i	0.978012	−	0.208549i	$-0.0668742\pi$
$152$		−	5.01570i		−	0.406827i
$153$		0			0
$154$	13.7662			1.10931
$155$		−	29.3660i		−	2.35873i
$156$		0			0
$157$	−0.506663			−0.0404361			−0.0202181	−	0.999796i	$-0.506436\pi$
$157$	−0.506663			−0.0404361			−0.0202181	+	0.999796i	$0.506436\pi$
$158$		−	2.24747i		−	0.178799i
$159$		0			0
$160$			3.19149i			0.252310i
$161$			9.93211i			0.782760i
$162$		0			0
$163$	−14.2049			−1.11262			−0.556308	−	0.830976i	$-0.687783\pi$
$163$	−14.2049			−1.11262			−0.556308	+	0.830976i	$0.687783\pi$
$164$	−6.38299			−0.498428
$165$		0			0
$166$	−10.7505			−0.834397
$167$		−	11.3348i		−	0.877117i	−0.898703	−	0.438559i	$-0.855489\pi$
$167$		−	11.3348i		−	0.877117i	0.898703	−	0.438559i	$-0.144511\pi$
$168$		0			0
$169$	−1.17353			−0.0902718
$170$	−5.80264			−0.445042
$171$		0			0
$172$		−	8.75046i		−	0.667216i
$173$	−15.4301			−1.17313			−0.586564	−	0.809903i	$-0.699520\pi$
$173$	−15.4301			−1.17313			−0.586564	+	0.809903i	$0.699520\pi$
$174$		0			0
$175$	−23.6715			−1.78940
$176$	3.01570			0.227317
$177$		0			0
$178$		−	12.8920i		−	0.966298i
$179$			24.4763i			1.82945i	0.404079	+	0.914724i	$0.367592\pi$
$179$			24.4763i			1.82945i	−0.404079	+	0.914724i	$0.632408\pi$
$180$		0			0
$181$			15.2183i			1.13116i	0.824692	+	0.565582i	$0.191348\pi$
$181$			15.2183i			1.13116i	−0.824692	+	0.565582i	$0.808652\pi$
$182$	15.6983			1.16363
$183$		0			0
$184$			2.17579i			0.160401i
$185$	10.4144			0.765681
$186$		0			0
$187$			5.48303i			0.400959i
$188$			2.62667i			0.191569i
$189$		0			0
$190$	16.0076			1.16131
$191$		−	25.6960i		−	1.85930i	−0.368446	−	0.929649i	$-0.620110\pi$
$191$		−	25.6960i		−	1.85930i	0.368446	−	0.929649i	$-0.379890\pi$
$192$		0			0
$193$			17.3161i			1.24644i	0.782045	+	0.623222i	$0.214177\pi$
$193$			17.3161i			1.24644i	−0.782045	+	0.623222i	$0.785823\pi$
$194$	−5.50885			−0.395513
$195$		0			0
$196$	13.8377			0.988404
$197$		−	10.6619i		−	0.759626i	−0.925063	−	0.379813i	$-0.875988\pi$
$197$		−	10.6619i		−	0.759626i	0.925063	−	0.379813i	$-0.124012\pi$
$198$		0			0
$199$			11.3087i			0.801651i	0.916154	+	0.400825i	$0.131277\pi$
$199$			11.3087i			0.801651i	−0.916154	+	0.400825i	$0.868723\pi$
$200$	−5.18563			−0.366680
$201$		0			0
$202$			5.44322i			0.382984i
$203$		−	33.6753i		−	2.36354i
$204$		0			0
$205$		−	20.3713i		−	1.42279i
$206$			17.9035i			1.24739i
$207$		0			0
$208$	3.43896			0.238449
$209$		−	15.1259i		−	1.04628i
$210$		0			0
$211$	17.8102			1.22611			0.613053	−	0.790042i	$-0.289941\pi$
$211$	17.8102			1.22611			0.613053	+	0.790042i	$0.289941\pi$
$212$	−11.1020			−0.762491
$213$		0			0
$214$			8.53721i			0.583592i
$215$	27.9270			1.90461
$216$		0			0
$217$	−42.0025			−2.85132
$218$	−2.59244			−0.175582
$219$		0			0
$220$			9.62459i			0.648890i
$221$			6.25258i			0.420594i
$222$		0			0
$223$			20.9136i			1.40048i	0.713909	+	0.700239i	$0.246923\pi$
$223$			20.9136i			1.40048i	−0.713909	+	0.700239i	$0.753077\pi$
$224$	4.56483			0.305000
$225$		0			0
$226$	7.83386	−	7.18544i	0.521101	−	0.477969i
$227$	7.60871			0.505008			0.252504	−	0.967596i	$-0.418746\pi$
$227$	7.60871			0.505008			0.252504	+	0.967596i	$0.418746\pi$
$228$		0			0
$229$		−	21.4085i		−	1.41472i	−0.706856	−	0.707358i	$-0.749887\pi$
$229$		−	21.4085i		−	1.41472i	0.706856	−	0.707358i	$-0.250113\pi$
$230$	−6.94402			−0.457875
$231$		0			0
$232$		−	7.37713i		−	0.484332i
$233$	−12.4144			−0.813294			−0.406647	−	0.913585i	$-0.633302\pi$
$233$	−12.4144			−0.813294			−0.406647	+	0.913585i	$0.633302\pi$
$234$		0			0
$235$	−8.38299			−0.546846
$236$			1.43896i			0.0936685i
$237$		0			0
$238$			8.29958i			0.537982i
$239$	9.97618			0.645305			0.322653	−	0.946517i	$-0.395425\pi$
$239$	9.97618			0.645305			0.322653	+	0.946517i	$0.395425\pi$
$240$		0			0
$241$	12.2766			0.790806			0.395403	−	0.918508i	$-0.370605\pi$
$241$	12.2766			0.790806			0.395403	+	0.918508i	$0.370605\pi$
$242$	−1.90554			−0.122493
$243$		0			0
$244$	8.19736			0.524782
$245$			44.1628i			2.82146i
$246$		0			0
$247$		−	17.2488i		−	1.09752i
$248$	−9.20133			−0.584285
$249$		0			0
$250$		−	0.592441i		−	0.0374693i
$251$	1.42251			0.0897882			0.0448941	−	0.998992i	$-0.485705\pi$
$251$	1.42251			0.0897882			0.0448941	+	0.998992i	$0.485705\pi$
$252$		0			0
$253$			6.56154i			0.412520i
$254$	−13.0036			−0.815919
$255$		0			0
$256$	1.00000			0.0625000
$257$	−23.5202			−1.46715			−0.733576	−	0.679608i	$-0.762150\pi$
$257$	−23.5202			−1.46715			−0.733576	+	0.679608i	$0.762150\pi$
$258$		0			0
$259$		−	14.8958i		−	0.925581i
$260$			10.9754i			0.680667i
$261$		0			0
$262$	−6.90554			−0.426626
$263$		−	29.8024i		−	1.83769i	−0.394614	−	0.918847i	$-0.629122\pi$
$263$		−	29.8024i		−	1.83769i	0.394614	−	0.918847i	$-0.370878\pi$
$264$		0			0
$265$		−	35.4321i		−	2.17658i
$266$		−	22.8958i		−	1.40383i
$267$		0			0
$268$			8.75046i			0.534519i
$269$		−	7.21117i		−	0.439673i	−0.975537	−	0.219837i	$-0.929448\pi$
$269$		−	7.21117i		−	0.439673i	0.975537	−	0.219837i	$-0.0705524\pi$
$270$		0			0
$271$		−	26.3396i		−	1.60002i	−0.599989	−	0.800008i	$-0.704828\pi$
$271$		−	26.3396i		−	1.60002i	0.599989	−	0.800008i	$-0.295172\pi$
$272$			1.81816i			0.110242i
$273$		0			0
$274$		−	5.30771i		−	0.320650i
$275$	−15.6383			−0.943026
$276$		0			0
$277$	29.5549			1.77578			0.887891	−	0.460053i	$-0.152170\pi$
$277$	29.5549			1.77578			0.887891	+	0.460053i	$0.152170\pi$
$278$	−0.548938			−0.0329231
$279$		0			0
$280$			14.5686i			0.870641i
$281$		−	21.5408i		−	1.28502i	−0.766278	−	0.642509i	$-0.777894\pi$
$281$		−	21.5408i		−	1.28502i	0.766278	−	0.642509i	$-0.222106\pi$
$282$		0			0
$283$	−21.8295			−1.29763			−0.648815	−	0.760946i	$-0.724735\pi$
$283$	−21.8295			−1.29763			−0.648815	+	0.760946i	$0.724735\pi$
$284$			6.09703i			0.361792i
$285$		0			0
$286$	10.3709			0.613244
$287$	−29.1372			−1.71992
$288$		0			0
$289$	13.6943			0.805547
$290$	23.5440			1.38255
$291$		0			0
$292$		0			0
$293$			22.9788i			1.34243i	0.741261	+	0.671216i	$0.234228\pi$
$293$			22.9788i			1.34243i	−0.741261	+	0.671216i	$0.765772\pi$
$294$		0			0
$295$	−4.59244			−0.267382
$296$		−	3.26317i		−	0.189668i
$297$		0			0
$298$	16.8606			0.976710
$299$			7.48247i			0.432722i
$300$		0			0
$301$		−	39.9443i		−	2.30235i
$302$		−	5.12540i		−	0.294933i
$303$		0			0
$304$		−	5.01570i		−	0.287670i
$305$			26.1618i			1.49802i
$306$		0			0
$307$	−6.08578			−0.347334			−0.173667	−	0.984804i	$-0.555562\pi$
$307$	−6.08578			−0.347334			−0.173667	+	0.984804i	$0.555562\pi$
$308$	13.7662			0.784400
$309$		0			0
$310$		−	29.3660i		−	1.66788i
$311$	−15.6439			−0.887084			−0.443542	−	0.896254i	$-0.646278\pi$
$311$	−15.6439			−0.887084			−0.443542	+	0.896254i	$0.646278\pi$
$312$		0			0
$313$	−7.81637			−0.441807			−0.220904	−	0.975296i	$-0.570901\pi$
$313$	−7.81637			−0.441807			−0.220904	+	0.975296i	$0.570901\pi$
$314$	−0.506663			−0.0285927
$315$		0			0
$316$		−	2.24747i		−	0.126430i
$317$	24.5287			1.37767			0.688835	−	0.724918i	$-0.258122\pi$
$317$	24.5287			1.37767			0.688835	+	0.724918i	$0.258122\pi$
$318$		0			0
$319$		−	22.2472i		−	1.24560i
$320$			3.19149i			0.178410i
$321$		0			0
$322$			9.93211i			0.553495i
$323$	9.11935			0.507414
$324$		0			0
$325$	−17.8332			−0.989208
$326$	−14.2049			−0.786739
$327$		0			0
$328$	−6.38299			−0.352442
$329$			11.9903i			0.661045i
$330$		0			0
$331$	−19.4586			−1.06954			−0.534772	−	0.844996i	$-0.679602\pi$
$331$	−19.4586			−1.06954			−0.534772	+	0.844996i	$0.679602\pi$
$332$	−10.7505			−0.590008
$333$		0			0
$334$		−	11.3348i		−	0.620215i
$335$	−27.9270			−1.52582
$336$		0			0
$337$	−29.7452			−1.62032			−0.810162	−	0.586206i	$-0.800621\pi$
$337$	−29.7452			−1.62032			−0.810162	+	0.586206i	$0.800621\pi$
$338$	−1.17353			−0.0638318
$339$		0			0
$340$	−5.80264			−0.314693
$341$	−27.7485			−1.50266
$342$		0			0
$343$	31.2127			1.68533
$344$		−	8.75046i		−	0.471793i
$345$		0			0
$346$	−15.4301			−0.829527
$347$	−22.1748			−1.19040			−0.595202	−	0.803576i	$-0.702928\pi$
$347$	−22.1748			−1.19040			−0.595202	+	0.803576i	$0.702928\pi$
$348$		0			0
$349$			32.9799i			1.76537i	0.469962	+	0.882687i	$0.344268\pi$
$349$			32.9799i			1.76537i	−0.469962	+	0.882687i	$0.655732\pi$
$350$	−23.6715			−1.26530
$351$		0			0
$352$	3.01570			0.160737
$353$	26.7853			1.42564			0.712818	−	0.701349i	$-0.247418\pi$
$353$	26.7853			1.42564			0.712818	+	0.701349i	$0.247418\pi$
$354$		0			0
$355$	−19.4586			−1.03276
$356$		−	12.8920i		−	0.683276i
$357$		0			0
$358$			24.4763i			1.29362i
$359$		−	7.12919i		−	0.376264i	−0.982144	−	0.188132i	$-0.939757\pi$
$359$		−	7.12919i		−	0.376264i	0.982144	−	0.188132i	$-0.0602433\pi$
$360$		0			0
$361$	−6.15727			−0.324067
$362$			15.2183i			0.799854i
$363$		0			0
$364$	15.6983			0.822813
$365$		0			0
$366$		0			0
$367$	14.3862			0.750954			0.375477	−	0.926832i	$-0.377479\pi$
$367$	14.3862			0.750954			0.375477	+	0.926832i	$0.377479\pi$
$368$			2.17579i			0.113421i
$369$		0			0
$370$	10.4144			0.541418
$371$	−50.6789			−2.63112
$372$		0			0
$373$			9.22204i			0.477499i	0.971081	+	0.238750i	$0.0767375\pi$
$373$			9.22204i			0.477499i	−0.971081	+	0.238750i	$0.923262\pi$
$374$			5.48303i			0.283521i
$375$		0			0
$376$			2.62667i			0.135460i
$377$		−	25.3697i		−	1.30660i
$378$		0			0
$379$			37.8014i			1.94173i	0.239637	+	0.970863i	$0.422972\pi$
$379$			37.8014i			1.94173i	−0.239637	+	0.970863i	$0.577028\pi$
$380$	16.0076			0.821172
$381$		0			0
$382$		−	25.6960i		−	1.31472i
$383$	22.1160			1.13007			0.565036	−	0.825066i	$-0.308862\pi$
$383$	22.1160			1.13007			0.565036	+	0.825066i	$0.308862\pi$
$384$		0			0
$385$			43.9346i			2.23912i
$386$			17.3161i			0.881368i
$387$		0			0
$388$	−5.50885			−0.279670
$389$	0.259311			0.0131476			0.00657380	−	0.999978i	$-0.497907\pi$
$389$	0.259311			0.0131476			0.00657380	+	0.999978i	$0.497907\pi$
$390$		0			0
$391$	−3.95594			−0.200060
$392$	13.8377			0.698907
$393$		0			0
$394$		−	10.6619i		−	0.537137i
$395$	7.17278			0.360902
$396$		0			0
$397$			19.2183i			0.964536i	0.876024	+	0.482268i	$0.160187\pi$
$397$			19.2183i			0.964536i	−0.876024	+	0.482268i	$0.839813\pi$
$398$			11.3087i			0.566853i
$399$		0			0
$400$	−5.18563			−0.259282
$401$	−7.66772			−0.382908			−0.191454	−	0.981502i	$-0.561320\pi$
$401$	−7.66772			−0.382908			−0.191454	+	0.981502i	$0.561320\pi$
$402$		0			0
$403$	−31.6430			−1.57625
$404$			5.44322i			0.270810i
$405$		0			0
$406$		−	33.6753i		−	1.67128i
$407$		−	9.84075i		−	0.487788i
$408$		0			0
$409$			28.7781i			1.42298i	0.702694	+	0.711492i	$0.251980\pi$
$409$			28.7781i			1.42298i	−0.702694	+	0.711492i	$0.748020\pi$
$410$		−	20.3713i		−	1.00606i
$411$		0			0
$412$			17.9035i			0.882041i
$413$			6.56862i			0.323221i
$414$		0			0
$415$		−	34.3100i		−	1.68421i
$416$	3.43896			0.168609
$417$		0			0
$418$		−	15.1259i		−	0.739830i
$419$		−	27.7414i		−	1.35526i	−0.735405	−	0.677628i	$-0.763008\pi$
$419$		−	27.7414i		−	1.35526i	0.735405	−	0.677628i	$-0.236992\pi$
$420$		0			0
$421$	33.8174			1.64816			0.824080	−	0.566473i	$-0.191692\pi$
$421$	33.8174			1.64816			0.824080	+	0.566473i	$0.191692\pi$
$422$	17.8102			0.866988
$423$		0			0
$424$	−11.1020			−0.539163
$425$		−	9.42830i		−	0.457340i
$426$		0			0
$427$	37.4195			1.81086
$428$			8.53721i			0.412662i
$429$		0			0
$430$	27.9270			1.34676
$431$		−	0.780893i		−	0.0376143i	−0.999823	−	0.0188071i	$-0.994013\pi$
$431$		−	0.780893i		−	0.0376143i	0.999823	−	0.0188071i	$-0.00598685\pi$
$432$		0			0
$433$		−	32.7127i		−	1.57207i	−0.618181	−	0.786036i	$-0.712130\pi$
$433$		−	32.7127i		−	1.57207i	0.618181	−	0.786036i	$-0.287870\pi$
$434$	−42.0025			−2.01618
$435$		0			0
$436$	−2.59244			−0.124155
$437$	10.9131			0.522045
$438$		0			0
$439$	10.5954			0.505690			0.252845	−	0.967507i	$-0.418634\pi$
$439$	10.5954			0.505690			0.252845	+	0.967507i	$0.418634\pi$
$440$			9.62459i			0.458835i
$441$		0			0
$442$			6.25258i			0.297405i
$443$	−38.0625			−1.80840			−0.904201	−	0.427107i	$-0.859533\pi$
$443$	−38.0625			−1.80840			−0.904201	+	0.427107i	$0.859533\pi$
$444$		0			0
$445$	41.1448			1.95045
$446$			20.9136i			0.990287i
$447$		0			0
$448$	4.56483			0.215668
$449$			11.8046i			0.557096i	0.960422	+	0.278548i	$0.0898531\pi$
$449$			11.8046i			0.557096i	−0.960422	+	0.278548i	$0.910147\pi$
$450$		0			0
$451$	−19.2492			−0.906409
$452$	7.83386	−	7.18544i	0.368474	−	0.337975i
$453$		0			0
$454$	7.60871			0.357094
$455$			50.1009i			2.34877i
$456$		0			0
$457$			1.59282i			0.0745088i	0.999306	+	0.0372544i	$0.0118612\pi$
$457$			1.59282i			0.0745088i	−0.999306	+	0.0372544i	$0.988139\pi$
$458$		−	21.4085i		−	1.00035i
$459$		0			0
$460$	−6.94402			−0.323767
$461$	9.26922			0.431711			0.215855	−	0.976425i	$-0.430746\pi$
$461$	9.26922			0.431711			0.215855	+	0.976425i	$0.430746\pi$
$462$		0			0
$463$	−38.4680			−1.78776			−0.893881	−	0.448305i	$-0.852028\pi$
$463$	−38.4680			−1.78776			−0.893881	+	0.448305i	$0.852028\pi$
$464$		−	7.37713i		−	0.342474i
$465$		0			0
$466$	−12.4144			−0.575086
$467$	26.7932			1.23984			0.619922	−	0.784664i	$-0.287164\pi$
$467$	26.7932			1.23984			0.619922	+	0.784664i	$0.287164\pi$
$468$		0			0
$469$			39.9443i			1.84446i
$470$	−8.38299			−0.386678
$471$		0			0
$472$			1.43896i			0.0662336i
$473$		−	26.3888i		−	1.21336i
$474$		0			0
$475$			26.0096i			1.19340i
$476$			8.29958i			0.380411i
$477$		0			0
$478$	9.97618			0.456300
$479$			1.46419i			0.0669006i	0.999440	+	0.0334503i	$0.0106495\pi$
$479$			1.46419i			0.0669006i	−0.999440	+	0.0334503i	$0.989350\pi$
$480$		0			0
$481$		−	11.2219i		−	0.511676i
$482$	12.2766			0.559185
$483$		0			0
$484$	−1.90554			−0.0866154
$485$		−	17.5815i		−	0.798333i
$486$		0			0
$487$		−	22.0357i		−	0.998531i	−0.866449	−	0.499266i	$-0.833603\pi$
$487$		−	22.0357i		−	0.998531i	0.866449	−	0.499266i	$-0.166397\pi$
$488$	8.19736			0.371077
$489$		0			0
$490$			44.1628i			1.99507i
$491$			34.3334i			1.54944i	0.632303	+	0.774722i	$0.282110\pi$
$491$			34.3334i			1.54944i	−0.632303	+	0.774722i	$0.717890\pi$
$492$		0			0
$493$	13.4128			0.604082
$494$		−	17.2488i		−	0.776061i
$495$		0			0
$496$	−9.20133			−0.413152
$497$			27.8319i			1.24843i
$498$		0			0
$499$		−	11.7445i		−	0.525755i	−0.964829	−	0.262878i	$-0.915328\pi$
$499$		−	11.7445i		−	0.525755i	0.964829	−	0.262878i	$-0.0846715\pi$
$500$		−	0.592441i		−	0.0264948i
$501$		0			0
$502$	1.42251			0.0634898
$503$	13.7212			0.611800			0.305900	−	0.952064i	$-0.401043\pi$
$503$	13.7212			0.611800			0.305900	+	0.952064i	$0.401043\pi$
$504$		0			0
$505$	−17.3720			−0.773044
$506$			6.56154i			0.291696i
$507$		0			0
$508$	−13.0036			−0.576942
$509$	−2.59100			−0.114844			−0.0574221	−	0.998350i	$-0.518288\pi$
$509$	−2.59100			−0.114844			−0.0574221	+	0.998350i	$0.518288\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$	−23.5202			−1.03743
$515$	−57.1388			−2.51784
$516$		0			0
$517$			7.92124i			0.348376i
$518$		−	14.8958i		−	0.654485i
$519$		0			0
$520$			10.9754i			0.481304i
$521$	−22.8095			−0.999301			−0.499651	−	0.866227i	$-0.666538\pi$
$521$	−22.8095			−0.999301			−0.499651	+	0.866227i	$0.666538\pi$
$522$		0			0
$523$			34.0527i			1.48902i	0.667611	+	0.744511i	$0.267317\pi$
$523$			34.0527i			1.48902i	−0.667611	+	0.744511i	$0.732683\pi$
$524$	−6.90554			−0.301670
$525$		0			0
$526$		−	29.8024i		−	1.29945i
$527$		−	16.7295i		−	0.728748i
$528$		0			0
$529$	18.2659			0.794171
$530$		−	35.4321i		−	1.53907i
$531$		0			0
$532$		−	22.8958i		−	0.992660i
$533$	−21.9509			−0.950797
$534$		0			0
$535$	−27.2465			−1.17797
$536$			8.75046i			0.377962i
$537$		0			0
$538$		−	7.21117i		−	0.310896i
$539$	41.7302			1.79745
$540$		0			0
$541$		−	36.7209i		−	1.57875i	−0.613909	−	0.789377i	$-0.710404\pi$
$541$		−	36.7209i		−	1.57875i	0.613909	−	0.789377i	$-0.289596\pi$
$542$		−	26.3396i		−	1.13138i
$543$		0			0
$544$			1.81816i			0.0779529i
$545$		−	8.27376i		−	0.354409i
$546$		0			0
$547$	24.1702			1.03345			0.516723	−	0.856153i	$-0.327152\pi$
$547$	24.1702			1.03345			0.516723	+	0.856153i	$0.327152\pi$
$548$		−	5.30771i		−	0.226734i
$549$		0			0
$550$	−15.6383			−0.666820
$551$	−37.0015			−1.57632
$552$		0			0
$553$		−	10.2593i		−	0.436270i
$554$	29.5549			1.25567
$555$		0			0
$556$	−0.548938			−0.0232802
$557$	22.1495			0.938504			0.469252	−	0.883064i	$-0.344524\pi$
$557$	22.1495			0.938504			0.469252	+	0.883064i	$0.344524\pi$
$558$		0			0
$559$		−	30.0925i		−	1.27278i
$560$			14.5686i			0.615637i
$561$		0			0
$562$		−	21.5408i		−	0.908645i
$563$	30.8223			1.29900			0.649502	−	0.760360i	$-0.274977\pi$
$563$	30.8223			1.29900			0.649502	+	0.760360i	$0.274977\pi$
$564$		0			0
$565$	22.9323	+	25.0017i	0.964769	+	1.05183i
$566$	−21.8295			−0.917564
$567$		0			0
$568$			6.09703i			0.255826i
$569$	25.9153			1.08643			0.543213	−	0.839595i	$-0.317208\pi$
$569$	25.9153			1.08643			0.543213	+	0.839595i	$0.317208\pi$
$570$		0			0
$571$		−	20.4731i		−	0.856773i	−0.903596	−	0.428387i	$-0.859082\pi$
$571$		−	20.4731i		−	0.856773i	0.903596	−	0.428387i	$-0.140918\pi$
$572$	10.3709			0.433629
$573$		0			0
$574$	−29.1372			−1.21616
$575$		−	11.2829i		−	0.470527i
$576$		0			0
$577$		−	1.21778i		−	0.0506970i	−0.999679	−	0.0253485i	$-0.991930\pi$
$577$		−	1.21778i		−	0.0506970i	0.999679	−	0.0253485i	$-0.00806955\pi$
$578$	13.6943			0.569608
$579$		0			0
$580$	23.5440			0.977613
$581$	−49.0740			−2.03593
$582$		0			0
$583$	−33.4805			−1.38662
$584$		0			0
$585$		0			0
$586$			22.9788i			0.949243i
$587$	41.4809			1.71210			0.856050	−	0.516894i	$-0.172912\pi$
$587$	41.4809			1.71210			0.856050	+	0.516894i	$0.172912\pi$
$588$		0			0
$589$			46.1512i			1.90163i
$590$	−4.59244			−0.189068
$591$		0			0
$592$		−	3.26317i		−	0.134116i
$593$	10.2103			0.419287			0.209644	−	0.977778i	$-0.432770\pi$
$593$	10.2103			0.419287			0.209644	+	0.977778i	$0.432770\pi$
$594$		0			0
$595$	−26.4881			−1.08591
$596$	16.8606			0.690638
$597$		0			0
$598$			7.48247i			0.305981i
$599$		−	27.3087i		−	1.11580i	−0.829907	−	0.557901i	$-0.811607\pi$
$599$		−	27.3087i		−	1.11580i	0.829907	−	0.557901i	$-0.188393\pi$
$600$		0			0
$601$	42.4904			1.73322			0.866611	−	0.498985i	$-0.166294\pi$
$601$	42.4904			1.73322			0.866611	+	0.498985i	$0.166294\pi$
$602$		−	39.9443i		−	1.62801i
$603$		0			0
$604$		−	5.12540i		−	0.208549i
$605$		−	6.08152i		−	0.247249i
$606$		0			0
$607$			8.43833i			0.342501i	0.985227	+	0.171251i	$0.0547808\pi$
$607$			8.43833i			0.342501i	−0.985227	+	0.171251i	$0.945219\pi$
$608$		−	5.01570i		−	0.203414i
$609$		0			0
$610$			26.1618i			1.05926i
$611$			9.03301i			0.365436i
$612$		0			0
$613$			17.8723i			0.721854i	0.932594	+	0.360927i	$0.117540\pi$
$613$			17.8723i			0.721854i	−0.932594	+	0.360927i	$0.882460\pi$
$614$	−6.08578			−0.245602
$615$		0			0
$616$	13.7662			0.554654
$617$	−0.747042			−0.0300748			−0.0150374	−	0.999887i	$-0.504787\pi$
$617$	−0.747042			−0.0300748			−0.0150374	+	0.999887i	$0.504787\pi$
$618$		0			0
$619$			5.33730i			0.214524i	0.994231	+	0.107262i	$0.0342084\pi$
$619$			5.33730i			0.214524i	−0.994231	+	0.107262i	$0.965792\pi$
$620$		−	29.3660i		−	1.17937i
$621$		0			0
$622$	−15.6439			−0.627263
$623$		−	58.8499i		−	2.35777i
$624$		0			0
$625$	−24.0374			−0.961495
$626$	−7.81637			−0.312405
$627$		0			0
$628$	−0.506663			−0.0202181
$629$	5.93297			0.236563
$630$		0			0
$631$		−	42.0940i		−	1.67573i	−0.545874	−	0.837867i	$-0.683802\pi$
$631$		−	42.0940i		−	1.67573i	0.545874	−	0.837867i	$-0.316198\pi$
$632$		−	2.24747i		−	0.0893995i
$633$		0			0
$634$	24.5287			0.974160
$635$		−	41.5009i		−	1.64691i
$636$		0			0
$637$	47.5872			1.88547
$638$		−	22.2472i		−	0.880776i
$639$		0			0
$640$			3.19149i			0.126155i
$641$			3.24869i			0.128315i	0.997940	+	0.0641577i	$0.0204361\pi$
$641$			3.24869i			0.128315i	−0.997940	+	0.0641577i	$0.979564\pi$
$642$		0			0
$643$		−	36.5162i		−	1.44006i	−0.693943	−	0.720030i	$-0.744128\pi$
$643$		−	36.5162i		−	1.44006i	0.693943	−	0.720030i	$-0.255872\pi$
$644$			9.93211i			0.391380i
$645$		0			0
$646$	9.11935			0.358796
$647$	−12.1549			−0.477860			−0.238930	−	0.971037i	$-0.576797\pi$
$647$	−12.1549			−0.477860			−0.238930	+	0.971037i	$0.576797\pi$
$648$		0			0
$649$			4.33948i			0.170340i
$650$	−17.8332			−0.699475
$651$		0			0
$652$	−14.2049			−0.556308
$653$	−2.22732			−0.0871618			−0.0435809	−	0.999050i	$-0.513877\pi$
$653$	−2.22732			−0.0871618			−0.0435809	+	0.999050i	$0.513877\pi$
$654$		0			0
$655$		−	22.0390i		−	0.861134i
$656$	−6.38299			−0.249214
$657$		0			0
$658$			11.9903i			0.467430i
$659$		−	18.2396i		−	0.710515i	−0.934768	−	0.355258i	$-0.884393\pi$
$659$		−	18.2396i		−	0.710515i	0.934768	−	0.355258i	$-0.115607\pi$
$660$		0			0
$661$		−	8.60864i		−	0.334837i	−0.985886	−	0.167419i	$-0.946457\pi$
$661$		−	8.60864i		−	0.334837i	0.985886	−	0.167419i	$-0.0535431\pi$
$662$	−19.4586			−0.756282
$663$		0			0
$664$	−10.7505			−0.417199
$665$	73.0719			2.83360
$666$		0			0
$667$	16.0511			0.621500
$668$		−	11.3348i		−	0.438559i
$669$		0			0
$670$	−27.9270			−1.07892
$671$	24.7208			0.954335
$672$		0			0
$673$		−	18.8002i		−	0.724694i	−0.932043	−	0.362347i	$-0.881976\pi$
$673$		−	18.8002i		−	0.724694i	0.932043	−	0.362347i	$-0.118024\pi$
$674$	−29.7452			−1.14574
$675$		0			0
$676$	−1.17353			−0.0451359
$677$	−4.11195			−0.158035			−0.0790176	−	0.996873i	$-0.525178\pi$
$677$	−4.11195			−0.158035			−0.0790176	+	0.996873i	$0.525178\pi$
$678$		0			0
$679$	−25.1470			−0.965052
$680$	−5.80264			−0.222521
$681$		0			0
$682$	−27.7485			−1.06254
$683$			3.84540i			0.147140i	0.997290	+	0.0735701i	$0.0234393\pi$
$683$			3.84540i			0.147140i	−0.997290	+	0.0735701i	$0.976561\pi$
$684$		0			0
$685$	16.9395			0.647225
$686$	31.2127			1.19171
$687$		0			0
$688$		−	8.75046i		−	0.333608i
$689$	−38.1795			−1.45452
$690$		0			0
$691$	11.2718			0.428799			0.214399	−	0.976746i	$-0.431221\pi$
$691$	11.2718			0.428799			0.214399	+	0.976746i	$0.431221\pi$
$692$	−15.4301			−0.586564
$693$		0			0
$694$	−22.1748			−0.841742
$695$		−	1.75193i		−	0.0664546i
$696$		0			0
$697$		−	11.6053i		−	0.439582i
$698$			32.9799i			1.24831i
$699$		0			0
$700$	−23.6715			−0.894699
$701$			25.1761i			0.950888i	0.879746	+	0.475444i	$0.157713\pi$
$701$			25.1761i			0.950888i	−0.879746	+	0.475444i	$0.842287\pi$
$702$		0			0
$703$	−16.3671			−0.617297
$704$	3.01570			0.113659
$705$		0			0
$706$	26.7853			1.00808
$707$			24.8474i			0.934482i
$708$		0			0
$709$	34.9471			1.31247			0.656233	−	0.754559i	$-0.272149\pi$
$709$	34.9471			1.31247			0.656233	+	0.754559i	$0.272149\pi$
$710$	−19.4586			−0.730270
$711$		0			0
$712$		−	12.8920i		−	0.483149i
$713$		−	20.0202i		−	0.749762i
$714$		0			0
$715$			33.0986i			1.23782i
$716$			24.4763i			0.914724i
$717$		0			0
$718$		−	7.12919i		−	0.266059i
$719$	25.1221			0.936895			0.468448	−	0.883491i	$-0.344813\pi$
$719$	25.1221			0.936895			0.468448	+	0.883491i	$0.344813\pi$
$720$		0			0
$721$			81.7263i			3.04364i
$722$	−6.15727			−0.229150
$723$		0			0
$724$			15.2183i			0.565582i
$725$			38.2551i			1.42076i
$726$		0			0
$727$	24.8550			0.921819			0.460910	−	0.887447i	$-0.347523\pi$
$727$	24.8550			0.921819			0.460910	+	0.887447i	$0.347523\pi$
$728$	15.6983			0.581817
$729$		0			0
$730$		0			0
$731$	15.9097			0.588443
$732$		0			0
$733$		−	21.3609i		−	0.788982i	−0.918900	−	0.394491i	$-0.870921\pi$
$733$		−	21.3609i		−	0.788982i	0.918900	−	0.394491i	$-0.129079\pi$
$734$	14.3862			0.531005
$735$		0			0
$736$			2.17579i			0.0802007i
$737$			26.3888i			0.972043i
$738$		0			0
$739$	0.905892			0.0333238			0.0166619	−	0.999861i	$-0.494696\pi$
$739$	0.905892			0.0333238			0.0166619	+	0.999861i	$0.494696\pi$
$740$	10.4144			0.382841
$741$		0			0
$742$	−50.6789			−1.86048
$743$			28.5438i			1.04717i	0.851974	+	0.523585i	$0.175406\pi$
$743$			28.5438i			1.04717i	−0.851974	+	0.523585i	$0.824594\pi$
$744$		0			0
$745$			53.8106i			1.97147i
$746$			9.22204i			0.337643i
$747$		0			0
$748$			5.48303i			0.200479i
$749$			38.9709i			1.42397i
$750$		0			0
$751$		−	52.7319i		−	1.92421i	−0.272673	−	0.962107i	$-0.587908\pi$
$751$		−	52.7319i		−	1.92421i	0.272673	−	0.962107i	$-0.412092\pi$
$752$			2.62667i			0.0957846i
$753$		0			0
$754$		−	25.3697i		−	0.923909i
$755$	16.3577			0.595316
$756$		0			0
$757$		−	27.4105i		−	0.996253i	−0.867104	−	0.498126i	$-0.834022\pi$
$757$		−	27.4105i		−	0.996253i	0.867104	−	0.498126i	$-0.165978\pi$
$758$			37.8014i			1.37301i
$759$		0			0
$760$	16.0076			0.580656
$761$	−28.8602			−1.04618			−0.523090	−	0.852277i	$-0.675221\pi$
$761$	−28.8602			−1.04618			−0.523090	+	0.852277i	$0.675221\pi$
$762$		0			0
$763$	−11.8340			−0.428421
$764$		−	25.6960i		−	0.929649i
$765$		0			0
$766$	22.1160			0.799082
$767$			4.94854i			0.178681i
$768$		0			0
$769$	6.64882			0.239762			0.119881	−	0.992788i	$-0.461749\pi$
$769$	6.64882			0.239762			0.119881	+	0.992788i	$0.461749\pi$
$770$			43.9346i			1.58329i
$771$		0			0
$772$			17.3161i			0.623222i
$773$	−28.2420			−1.01579			−0.507897	−	0.861418i	$-0.669577\pi$
$773$	−28.2420			−1.01579			−0.507897	+	0.861418i	$0.669577\pi$
$774$		0			0
$775$	47.7147			1.71396
$776$	−5.50885			−0.197756
$777$		0			0
$778$	0.259311			0.00929675
$779$			32.0152i			1.14706i
$780$		0			0
$781$			18.3868i			0.657933i
$782$	−3.95594			−0.141464
$783$		0			0
$784$	13.8377			0.494202
$785$		−	1.61701i		−	0.0577137i
$786$		0			0
$787$	2.44900			0.0872975			0.0436487	−	0.999047i	$-0.486102\pi$
$787$	2.44900			0.0872975			0.0436487	+	0.999047i	$0.486102\pi$
$788$		−	10.6619i		−	0.379813i
$789$		0			0
$790$	7.17278			0.255196
$791$	35.7602	−	32.8003i	1.27149	−	1.16624i
$792$		0			0
$793$	28.1904			1.00107
$794$			19.2183i			0.682030i
$795$		0			0
$796$			11.3087i			0.400825i
$797$			53.9703i			1.91173i	0.293814	+	0.955863i	$0.405075\pi$
$797$			53.9703i			1.91173i	−0.293814	+	0.955863i	$0.594925\pi$
$798$		0			0
$799$	−4.77570			−0.168952
$800$	−5.18563			−0.183340
$801$		0			0
$802$	−7.66772			−0.270757
$803$		0			0
$804$		0			0
$805$	−31.6983			−1.11722
$806$	−31.6430			−1.11458
$807$		0			0
$808$			5.44322i			0.191492i
$809$	33.8759			1.19101			0.595507	−	0.803350i	$-0.296951\pi$
$809$	33.8759			1.19101			0.595507	+	0.803350i	$0.296951\pi$
$810$		0			0
$811$			22.2590i			0.781618i	0.920472	+	0.390809i	$0.127805\pi$
$811$			22.2590i			0.781618i	−0.920472	+	0.390809i	$0.872195\pi$
$812$		−	33.6753i		−	1.18177i
$813$		0			0
$814$		−	9.84075i		−	0.344918i
$815$		−	45.3350i		−	1.58801i
$816$		0			0
$817$	−43.8897			−1.53551
$818$			28.7781i			1.00620i
$819$		0			0
$820$		−	20.3713i		−	0.711395i
$821$	−5.18221			−0.180861			−0.0904303	−	0.995903i	$-0.528824\pi$
$821$	−5.18221			−0.180861			−0.0904303	+	0.995903i	$0.528824\pi$
$822$		0			0
$823$	25.7350			0.897065			0.448532	−	0.893767i	$-0.351947\pi$
$823$	25.7350			0.897065			0.448532	+	0.893767i	$0.351947\pi$
$824$			17.9035i			0.623697i
$825$		0			0
$826$			6.56862i			0.228551i
$827$	−0.127467			−0.00443246			−0.00221623	−	0.999998i	$-0.500705\pi$
$827$	−0.127467			−0.00443246			−0.00221623	+	0.999998i	$0.500705\pi$
$828$		0			0
$829$			47.2109i			1.63970i	0.572576	+	0.819852i	$0.305944\pi$
$829$			47.2109i			1.63970i	−0.572576	+	0.819852i	$0.694056\pi$
$830$		−	34.3100i		−	1.19092i
$831$		0			0
$832$	3.43896			0.119225
$833$			25.1591i			0.871710i
$834$		0			0
$835$	36.1751			1.25189
$836$		−	15.1259i		−	0.523139i
$837$		0			0
$838$		−	27.7414i		−	0.958311i
$839$			18.1125i			0.625315i	0.949866	+	0.312657i	$0.101219\pi$
$839$			18.1125i			0.625315i	−0.949866	+	0.312657i	$0.898781\pi$
$840$		0			0
$841$	−25.4220			−0.876620
$842$	33.8174			1.16543
$843$		0			0
$844$	17.8102			0.613053
$845$		−	3.74532i		−	0.128843i
$846$		0			0
$847$	−8.69846			−0.298883
$848$	−11.1020			−0.381246
$849$		0			0
$850$		−	9.42830i		−	0.323388i
$851$	7.09998			0.243384
$852$		0			0
$853$	29.2726			1.00228			0.501138	−	0.865367i	$-0.332915\pi$
$853$	29.2726			1.00228			0.501138	+	0.865367i	$0.332915\pi$
$854$	37.4195			1.28047
$855$		0			0
$856$			8.53721i			0.291796i
$857$		−	12.7513i		−	0.435577i	−0.975996	−	0.217788i	$-0.930116\pi$
$857$		−	12.7513i		−	0.435577i	0.975996	−	0.217788i	$-0.0698842\pi$
$858$		0			0
$859$			30.0590i			1.02560i	0.858508	+	0.512800i	$0.171392\pi$
$859$			30.0590i			1.02560i	−0.858508	+	0.512800i	$0.828608\pi$
$860$	27.9270			0.952304
$861$		0			0
$862$		−	0.780893i		−	0.0265973i
$863$	−42.1546			−1.43496			−0.717479	−	0.696580i	$-0.754704\pi$
$863$	−42.1546			−1.43496			−0.717479	+	0.696580i	$0.754704\pi$
$864$		0			0
$865$		−	49.2450i		−	1.67438i
$866$		−	32.7127i		−	1.11162i
$867$		0			0
$868$	−42.0025			−1.42566
$869$		−	6.77770i		−	0.229918i
$870$		0			0
$871$			30.0925i			1.01965i
$872$	−2.59244			−0.0877911
$873$		0			0
$874$	10.9131			0.369142
$875$		−	2.70439i		−	0.0914252i
$876$		0			0
$877$			1.46904i			0.0496061i	0.999692	+	0.0248031i	$0.00789587\pi$
$877$			1.46904i			0.0496061i	−0.999692	+	0.0248031i	$0.992104\pi$
$878$	10.5954			0.357577
$879$		0			0
$880$			9.62459i			0.324445i
$881$			6.35716i			0.214178i	0.994249	+	0.107089i	$0.0341530\pi$
$881$			6.35716i			0.214178i	−0.994249	+	0.107089i	$0.965847\pi$
$882$		0			0
$883$			4.05873i			0.136587i	0.997665	+	0.0682935i	$0.0217554\pi$
$883$			4.05873i			0.136587i	−0.997665	+	0.0682935i	$0.978245\pi$
$884$			6.25258i			0.210297i
$885$		0			0
$886$	−38.0625			−1.27873
$887$		−	5.53349i		−	0.185797i	−0.995676	−	0.0928983i	$-0.970387\pi$
$887$		−	5.53349i		−	0.185797i	0.995676	−	0.0928983i	$-0.0296131\pi$
$888$		0			0
$889$	−59.3592			−1.99084
$890$	41.1448			1.37918
$891$		0			0
$892$			20.9136i			0.700239i
$893$	13.1746			0.440870
$894$		0			0
$895$	−78.1161			−2.61113
$896$	4.56483			0.152500
$897$		0			0
$898$			11.8046i			0.393926i
$899$			67.8794i			2.26390i
$900$		0			0
$901$		−	20.1853i		−	0.672469i
$902$	−19.2492			−0.640928
$903$		0			0
$904$	7.83386	−	7.18544i	0.260550	−	0.238984i
$905$	−48.5690			−1.61449
$906$		0			0
$907$		−	4.35610i		−	0.144642i	−0.997381	−	0.0723210i	$-0.976959\pi$
$907$		−	4.35610i		−	0.144642i	0.997381	−	0.0723210i	$-0.0230406\pi$
$908$	7.60871			0.252504
$909$		0			0
$910$			50.1009i			1.66083i
$911$	21.7212			0.719657			0.359828	−	0.933018i	$-0.382835\pi$
$911$	21.7212			0.719657			0.359828	+	0.933018i	$0.382835\pi$
$912$		0			0
$913$	−32.4202			−1.07295
$914$			1.59282i			0.0526857i
$915$		0			0
$916$		−	21.4085i		−	0.707358i
$917$	−31.5226			−1.04097
$918$		0			0
$919$	12.3436			0.407179			0.203590	−	0.979056i	$-0.434739\pi$
$919$	12.3436			0.407179			0.203590	+	0.979056i	$0.434739\pi$
$920$	−6.94402			−0.228938
$921$		0			0
$922$	9.26922			0.305266
$923$			20.9675i			0.690153i
$924$		0			0
$925$			16.9216i			0.556379i
$926$	−38.4680			−1.26414
$927$		0			0
$928$		−	7.37713i		−	0.242166i
$929$	50.3369			1.65150			0.825750	−	0.564036i	$-0.190752\pi$
$929$	50.3369			1.65150			0.825750	+	0.564036i	$0.190752\pi$
$930$		0			0
$931$		−	69.4055i		−	2.27467i
$932$	−12.4144			−0.406647
$933$		0			0
$934$	26.7932			0.876701
$935$	−17.4990			−0.572280
$936$		0			0
$937$			48.3471i			1.57943i	0.613474	+	0.789715i	$0.289772\pi$
$937$			48.3471i			1.57943i	−0.613474	+	0.789715i	$0.710228\pi$
$938$			39.9443i			1.30423i
$939$		0			0
$940$	−8.38299			−0.273423
$941$		−	3.20736i		−	0.104557i	−0.998633	−	0.0522785i	$-0.983352\pi$
$941$		−	3.20736i		−	0.104557i	0.998633	−	0.0522785i	$-0.0166483\pi$
$942$		0			0
$943$		−	13.8880i		−	0.452257i
$944$			1.43896i			0.0468343i
$945$		0			0
$946$		−	26.3888i		−	0.857973i
$947$			30.5339i			0.992219i	0.868260	+	0.496110i	$0.165239\pi$
$947$			30.5339i			0.992219i	−0.868260	+	0.496110i	$0.834761\pi$
$948$		0			0
$949$		0			0
$950$			26.0096i			0.843862i
$951$		0			0
$952$			8.29958i			0.268991i
$953$	15.0491			0.487488			0.243744	−	0.969840i	$-0.421624\pi$
$953$	15.0491			0.487488			0.243744	+	0.969840i	$0.421624\pi$
$954$		0			0
$955$	82.0087			2.65374
$956$	9.97618			0.322653
$957$		0			0
$958$			1.46419i			0.0473058i
$959$		−	24.2288i		−	0.782387i
$960$		0			0
$961$	53.6645			1.73111
$962$		−	11.2219i		−	0.361809i
$963$		0			0
$964$	12.2766			0.395403
$965$	−55.2644			−1.77902
$966$		0			0
$967$	21.0241			0.676088			0.338044	−	0.941130i	$-0.390235\pi$
$967$	21.0241			0.676088			0.338044	+	0.941130i	$0.390235\pi$
$968$	−1.90554			−0.0612464
$969$		0			0
$970$		−	17.5815i		−	0.564507i
$971$		−	33.4783i		−	1.07437i	−0.843464	−	0.537185i	$-0.819488\pi$
$971$		−	33.4783i		−	1.07437i	0.843464	−	0.537185i	$-0.180512\pi$
$972$		0			0
$973$	−2.50581			−0.0803326
$974$		−	22.0357i		−	0.706068i
$975$		0			0
$976$	8.19736			0.262391
$977$		−	26.9914i		−	0.863533i	−0.901985	−	0.431766i	$-0.857890\pi$
$977$		−	26.9914i		−	0.863533i	0.901985	−	0.431766i	$-0.142110\pi$
$978$		0			0
$979$		−	38.8785i		−	1.24256i
$980$			44.1628i			1.41073i
$981$		0			0
$982$			34.3334i			1.09562i
$983$		−	33.2301i		−	1.05988i	−0.848036	−	0.529938i	$-0.822215\pi$
$983$		−	33.2301i		−	1.05988i	0.848036	−	0.529938i	$-0.177785\pi$
$984$		0			0
$985$	34.0273			1.08420
$986$	13.4128			0.427150
$987$		0			0
$988$		−	17.2488i		−	0.548758i
$989$	19.0392			0.605410
$990$		0			0
$991$	−14.0675			−0.446867			−0.223434	−	0.974719i	$-0.571727\pi$
$991$	−14.0675			−0.446867			−0.223434	+	0.974719i	$0.571727\pi$
$992$	−9.20133			−0.292143
$993$		0			0
$994$			27.8319i			0.882775i
$995$	−36.0916			−1.14418
$996$		0			0
$997$			3.69070i			0.116886i	0.998291	+	0.0584429i	$0.0186135\pi$
$997$			3.69070i			0.116886i	−0.998291	+	0.0584429i	$0.981386\pi$
$998$		−	11.7445i		−	0.371765i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.2.c.i.451.9		10
3.2	odd	2		678.2.c.c.451.6	yes	10
113.112	even	2	inner	2034.2.c.i.451.2		10
339.338	odd	2		678.2.c.c.451.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
678.2.c.c.451.5	✓	10	339.338	odd	2
678.2.c.c.451.6	yes	10	3.2	odd	2
2034.2.c.i.451.2		10	113.112	even	2	inner
2034.2.c.i.451.9		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2034.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.19149i q^{5} +4.56483 q^{7} +1.00000 q^{8} +3.19149i q^{10} +3.01570 q^{11} +3.43896 q^{13} +4.56483 q^{14} +1.00000 q^{16} +1.81816i q^{17} -5.01570i q^{19} +3.19149i q^{20} +3.01570 q^{22} +2.17579i q^{23} -5.18563 q^{25} +3.43896 q^{26} +4.56483 q^{28} -7.37713i q^{29} -9.20133 q^{31} +1.00000 q^{32} +1.81816i q^{34} +14.5686i q^{35} -3.26317i q^{37} -5.01570i q^{38} +3.19149i q^{40} -6.38299 q^{41} -8.75046i q^{43} +3.01570 q^{44} +2.17579i q^{46} +2.62667i q^{47} +13.8377 q^{49} -5.18563 q^{50} +3.43896 q^{52} -11.1020 q^{53} +9.62459i q^{55} +4.56483 q^{56} -7.37713i q^{58} +1.43896i q^{59} +8.19736 q^{61} -9.20133 q^{62} +1.00000 q^{64} +10.9754i q^{65} +8.75046i q^{67} +1.81816i q^{68} +14.5686i q^{70} +6.09703i q^{71} -3.26317i q^{74} -5.01570i q^{76} +13.7662 q^{77} -2.24747i q^{79} +3.19149i q^{80} -6.38299 q^{82} -10.7505 q^{83} -5.80264 q^{85} -8.75046i q^{86} +3.01570 q^{88} -12.8920i q^{89} +15.6983 q^{91} +2.17579i q^{92} +2.62667i q^{94} +16.0076 q^{95} -5.50885 q^{97} +13.8377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} + 4 q^{7} + 10 q^{8} - 2 q^{11} + 6 q^{13} + 4 q^{14} + 10 q^{16} - 2 q^{22} - 8 q^{25} + 6 q^{26} + 4 q^{28} - 16 q^{31} + 10 q^{32} + 4 q^{41} - 2 q^{44} + 10 q^{49} - 8 q^{50}+ \cdots + 10 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 + 4 * q^7 + 10 * q^8 - 2 * q^11 + 6 * q^13 + 4 * q^14 + 10 * q^16 - 2 * q^22 - 8 * q^25 + 6 * q^26 + 4 * q^28 - 16 * q^31 + 10 * q^32 + 4 * q^41 - 2 * q^44 + 10 * q^49 - 8 * q^50 + 6 * q^52 - 22 * q^53 + 4 * q^56 + 58 * q^61 - 16 * q^62 + 10 * q^64 + 20 * q^77 + 4 * q^82 - 22 * q^83 - 82 * q^85 - 2 * q^88 - 38 * q^91 + 20 * q^95 + 26 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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