LMFDB - Embedded newform 2004.2.h.a.1001.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2004,2,Mod(1001,2004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2004.1001");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2004 = 2^{2} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2004.h (of order $2$, degree $1$, 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:	$16.0020205651$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1001.13
Character	$\chi$	$=$	2004.1001
Dual form			2004.2.h.a.1001.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.34740 - 1.08836i) q^{3} -3.01275 q^{5} -3.17416 q^{7} +(0.630964 + 2.93290i) q^{9} +O(q^{10})$	\(q+(-1.34740 - 1.08836i) q^{3} -3.01275 q^{5} -3.17416 q^{7} +(0.630964 + 2.93290i) q^{9} +4.05776i q^{11} +2.81029i q^{13} +(4.05938 + 3.27895i) q^{15} -2.19734 q^{17} -4.15012 q^{19} +(4.27686 + 3.45462i) q^{21} +4.69008 q^{23} +4.07667 q^{25} +(2.34187 - 4.63849i) q^{27} -2.09874i q^{29} -5.12816 q^{31} +(4.41629 - 5.46742i) q^{33} +9.56296 q^{35} +7.67475i q^{37} +(3.05860 - 3.78658i) q^{39} +0.449127 q^{41} +8.78091i q^{43} +(-1.90094 - 8.83609i) q^{45} -8.78834i q^{47} +3.07531 q^{49} +(2.96069 + 2.39148i) q^{51} +10.9061 q^{53} -12.2250i q^{55} +(5.59187 + 4.51681i) q^{57} -7.31611 q^{59} -13.8352 q^{61} +(-2.00278 - 9.30949i) q^{63} -8.46671i q^{65} -10.3620i q^{67} +(-6.31940 - 5.10447i) q^{69} -5.85050 q^{71} +5.54493i q^{73} +(-5.49290 - 4.43687i) q^{75} -12.8800i q^{77} +2.67815i q^{79} +(-8.20377 + 3.70111i) q^{81} -5.70992 q^{83} +6.62003 q^{85} +(-2.28418 + 2.82784i) q^{87} -7.55160i q^{89} -8.92033i q^{91} +(6.90968 + 5.58126i) q^{93} +12.5033 q^{95} -0.448156 q^{97} +(-11.9010 + 2.56030i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 8 q^{9}+O(q^{10}) $ 56 * q - 8 * q^9	$ 56 q - 8 q^{9} - 4 q^{19} + 4 q^{21} + 52 q^{25} + 12 q^{27} + 4 q^{31} - 8 q^{33} + 32 q^{49} + 24 q^{57} + 28 q^{61} - 26 q^{63} - 54 q^{75} - 24 q^{81} + 8 q^{85} + 14 q^{87} - 20 q^{93} + 36 q^{97} + 32 q^{99}+O(q^{100}) $ 56 * q - 8 * q^9 - 4 * q^19 + 4 * q^21 + 52 * q^25 + 12 * q^27 + 4 * q^31 - 8 * q^33 + 32 * q^49 + 24 * q^57 + 28 * q^61 - 26 * q^63 - 54 * q^75 - 24 * q^81 + 8 * q^85 + 14 * q^87 - 20 * q^93 + 36 * q^97 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2004\mathbb{Z}\right)^\times$.

$n$	$673$	$1003$	$1337$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.34740	−	1.08836i	−0.777921	−	0.628362i
$3$	−1.34740	−	1.08836i	−0.777921	−	0.628362i
$4$		0			0
$5$	−3.01275			−1.34734			−0.673672	−	0.739031i	$-0.735284\pi$
$5$	−3.01275			−1.34734			−0.673672	+	0.739031i	$0.735284\pi$
$6$		0			0
$7$	−3.17416			−1.19972			−0.599860	−	0.800105i	$-0.704777\pi$
$7$	−3.17416			−1.19972			−0.599860	+	0.800105i	$0.704777\pi$
$8$		0			0
$9$	0.630964	+	2.93290i	0.210321	+	0.977632i
$10$		0			0
$11$			4.05776i			1.22346i	0.791066	+	0.611731i	$0.209526\pi$
$11$			4.05776i			1.22346i	−0.791066	+	0.611731i	$0.790474\pi$
$12$		0			0
$13$			2.81029i			0.779435i	0.920934	+	0.389717i	$0.127427\pi$
$13$			2.81029i			0.779435i	−0.920934	+	0.389717i	$0.872573\pi$
$14$		0			0
$15$	4.05938	+	3.27895i	1.04813	+	0.846620i
$16$		0			0
$17$	−2.19734			−0.532933			−0.266466	−	0.963844i	$-0.585856\pi$
$17$	−2.19734			−0.532933			−0.266466	+	0.963844i	$0.585856\pi$
$18$		0			0
$19$	−4.15012			−0.952104			−0.476052	−	0.879417i	$-0.657933\pi$
$19$	−4.15012			−0.952104			−0.476052	+	0.879417i	$0.657933\pi$
$20$		0			0
$21$	4.27686	+	3.45462i	0.933288	+	0.753859i
$22$		0			0
$23$	4.69008			0.977949			0.488974	−	0.872298i	$-0.337371\pi$
$23$	4.69008			0.977949			0.488974	+	0.872298i	$0.337371\pi$
$24$		0			0
$25$	4.07667			0.815335
$26$		0			0
$27$	2.34187	−	4.63849i	0.450694	−	0.892679i
$28$		0			0
$29$		−	2.09874i		−	0.389727i	−0.980830	−	0.194863i	$-0.937574\pi$
$29$		−	2.09874i		−	0.389727i	0.980830	−	0.194863i	$-0.0624263\pi$
$30$		0			0
$31$	−5.12816			−0.921045			−0.460523	−	0.887648i	$-0.652338\pi$
$31$	−5.12816			−0.921045			−0.460523	+	0.887648i	$0.652338\pi$
$32$		0			0
$33$	4.41629	−	5.46742i	0.768777	−	0.951756i
$34$		0			0
$35$	9.56296			1.61644
$36$		0			0
$37$			7.67475i			1.26172i	0.775896	+	0.630861i	$0.217298\pi$
$37$			7.67475i			1.26172i	−0.775896	+	0.630861i	$0.782702\pi$
$38$		0			0
$39$	3.05860	−	3.78658i	0.489768	−	0.606339i
$40$		0			0
$41$	0.449127			0.0701419			0.0350710	−	0.999385i	$-0.488834\pi$
$41$	0.449127			0.0701419			0.0350710	+	0.999385i	$0.488834\pi$
$42$		0			0
$43$			8.78091i			1.33908i	0.742778	+	0.669538i	$0.233508\pi$
$43$			8.78091i			1.33908i	−0.742778	+	0.669538i	$0.766492\pi$
$44$		0			0
$45$	−1.90094	−	8.83609i	−0.283375	−	1.31721i
$46$		0			0
$47$		−	8.78834i		−	1.28191i	−0.767578	−	0.640956i	$-0.778538\pi$
$47$		−	8.78834i		−	1.28191i	0.767578	−	0.640956i	$-0.221462\pi$
$48$		0			0
$49$	3.07531			0.439330
$50$		0			0
$51$	2.96069	+	2.39148i	0.414579	+	0.334875i
$52$		0			0
$53$	10.9061			1.49807			0.749034	−	0.662532i	$-0.230518\pi$
$53$	10.9061			1.49807			0.749034	+	0.662532i	$0.230518\pi$
$54$		0			0
$55$		−	12.2250i		−	1.64842i
$56$		0			0
$57$	5.59187	+	4.51681i	0.740661	+	0.598266i
$58$		0			0
$59$	−7.31611			−0.952477			−0.476238	−	0.879316i	$-0.658000\pi$
$59$	−7.31611			−0.952477			−0.476238	+	0.879316i	$0.658000\pi$
$60$		0			0
$61$	−13.8352			−1.77141			−0.885707	−	0.464244i	$-0.846326\pi$
$61$	−13.8352			−1.77141			−0.885707	+	0.464244i	$0.846326\pi$
$62$		0			0
$63$	−2.00278	−	9.30949i	−0.252327	−	1.17289i
$64$		0			0
$65$		−	8.46671i		−	1.05017i
$66$		0			0
$67$		−	10.3620i		−	1.26592i	−0.774185	−	0.632960i	$-0.781840\pi$
$67$		−	10.3620i		−	1.26592i	0.774185	−	0.632960i	$-0.218160\pi$
$68$		0			0
$69$	−6.31940	−	5.10447i	−0.760767	−	0.614506i
$70$		0			0
$71$	−5.85050			−0.694327			−0.347163	−	0.937805i	$-0.612855\pi$
$71$	−5.85050			−0.694327			−0.347163	+	0.937805i	$0.612855\pi$
$72$		0			0
$73$			5.54493i			0.648985i	0.945888	+	0.324492i	$0.105193\pi$
$73$			5.54493i			0.648985i	−0.945888	+	0.324492i	$0.894807\pi$
$74$		0			0
$75$	−5.49290	−	4.43687i	−0.634266	−	0.512326i
$76$		0			0
$77$		−	12.8800i		−	1.46781i
$78$		0			0
$79$			2.67815i			0.301315i	0.988586	+	0.150658i	$0.0481391\pi$
$79$			2.67815i			0.301315i	−0.988586	+	0.150658i	$0.951861\pi$
$80$		0			0
$81$	−8.20377	+	3.70111i	−0.911530	+	0.411234i
$82$		0			0
$83$	−5.70992			−0.626745			−0.313373	−	0.949630i	$-0.601459\pi$
$83$	−5.70992			−0.626745			−0.313373	+	0.949630i	$0.601459\pi$
$84$		0			0
$85$	6.62003			0.718043
$86$		0			0
$87$	−2.28418	+	2.82784i	−0.244889	+	0.303176i
$88$		0			0
$89$		−	7.55160i		−	0.800468i	−0.916413	−	0.400234i	$-0.868929\pi$
$89$		−	7.55160i		−	0.800468i	0.916413	−	0.400234i	$-0.131071\pi$
$90$		0			0
$91$		−	8.92033i		−	0.935104i
$92$		0			0
$93$	6.90968	+	5.58126i	0.716500	+	0.578750i
$94$		0			0
$95$	12.5033			1.28281
$96$		0			0
$97$	−0.448156			−0.0455033			−0.0227517	−	0.999741i	$-0.507243\pi$
$97$	−0.448156			−0.0455033			−0.0227517	+	0.999741i	$0.507243\pi$
$98$		0			0
$99$	−11.9010	+	2.56030i	−1.19610	+	0.257320i
$100$		0			0
$101$	7.98275			0.794313			0.397156	−	0.917751i	$-0.369997\pi$
$101$	7.98275			0.794313			0.397156	+	0.917751i	$0.369997\pi$
$102$		0			0
$103$		−	14.9968i		−	1.47768i	−0.673881	−	0.738840i	$-0.735374\pi$
$103$		−	14.9968i		−	1.47768i	0.673881	−	0.738840i	$-0.264626\pi$
$104$		0			0
$105$	−12.8851	−	10.4079i	−1.25746	−	1.01571i
$106$		0			0
$107$		−	12.7718i		−	1.23470i	−0.786688	−	0.617350i	$-0.788206\pi$
$107$		−	12.7718i		−	1.23470i	0.786688	−	0.617350i	$-0.211794\pi$
$108$		0			0
$109$		−	5.04339i		−	0.483069i	−0.970392	−	0.241534i	$-0.922349\pi$
$109$		−	5.04339i		−	0.483069i	0.970392	−	0.241534i	$-0.0776507\pi$
$110$		0			0
$111$	8.35286	−	10.3409i	0.792818	−	0.981519i
$112$		0			0
$113$	−5.51829			−0.519117			−0.259558	−	0.965727i	$-0.583577\pi$
$113$	−5.51829			−0.519117			−0.259558	+	0.965727i	$0.583577\pi$
$114$		0			0
$115$	−14.1300			−1.31763
$116$		0			0
$117$	−8.24230	+	1.77319i	−0.762001	+	0.163932i
$118$		0			0
$119$	6.97471			0.639370
$120$		0			0
$121$	−5.46544			−0.496858
$122$		0			0
$123$	−0.605153	−	0.488810i	−0.0545649	−	0.0440745i
$124$		0			0
$125$	2.78175			0.248808
$126$		0			0
$127$	−11.5203			−1.02226			−0.511132	−	0.859503i	$-0.670774\pi$
$127$	−11.5203			−1.02226			−0.511132	+	0.859503i	$0.670774\pi$
$128$		0			0
$129$	9.55675	−	11.8314i	0.841425	−	1.04169i
$130$		0			0
$131$	18.5402			1.61986			0.809931	−	0.586525i	$-0.199505\pi$
$131$	18.5402			1.61986			0.809931	+	0.586525i	$0.199505\pi$
$132$		0			0
$133$	13.1732			1.14226
$134$		0			0
$135$	−7.05549	+	13.9746i	−0.607239	+	1.20274i
$136$		0			0
$137$			8.54840i			0.730339i	0.930941	+	0.365170i	$0.118989\pi$
$137$			8.54840i			0.730339i	−0.930941	+	0.365170i	$0.881011\pi$
$138$		0			0
$139$			2.46191i			0.208816i	0.994535	+	0.104408i	$0.0332948\pi$
$139$			2.46191i			0.208816i	−0.994535	+	0.104408i	$0.966705\pi$
$140$		0			0
$141$	−9.56484	+	11.8414i	−0.805505	+	0.997226i
$142$		0			0
$143$	−11.4035			−0.953609
$144$		0			0
$145$			6.32299i			0.525096i
$146$		0			0
$147$	−4.14367	−	3.34703i	−0.341764	−	0.276058i
$148$		0			0
$149$	14.6113			1.19700			0.598501	−	0.801122i	$-0.295763\pi$
$149$	14.6113			1.19700			0.598501	+	0.801122i	$0.295763\pi$
$150$		0			0
$151$		−	19.8977i		−	1.61926i	−0.586944	−	0.809628i	$-0.699669\pi$
$151$		−	19.8977i		−	1.61926i	0.586944	−	0.809628i	$-0.300331\pi$
$152$		0			0
$153$	−1.38644	−	6.44456i	−0.112087	−	0.521012i
$154$		0			0
$155$	15.4499			1.24096
$156$		0			0
$157$	0.540423			0.0431305			0.0215652	−	0.999767i	$-0.493135\pi$
$157$	0.540423			0.0431305			0.0215652	+	0.999767i	$0.493135\pi$
$158$		0			0
$159$	−14.6949	−	11.8697i	−1.16538	−	0.941329i
$160$		0			0
$161$	−14.8871			−1.17327
$162$		0			0
$163$			19.8121i			1.55180i	0.630855	+	0.775901i	$0.282704\pi$
$163$			19.8121i			1.55180i	−0.630855	+	0.775901i	$0.717296\pi$
$164$		0			0
$165$	−13.3052	+	16.4720i	−1.03581	+	1.28234i
$166$		0			0
$167$	8.27036	+	9.92981i	0.639980	+	0.768392i
$167$	8.27036	+	9.92981i	0.639980	+	0.768392i
$168$		0			0
$169$	5.10226			0.392481
$170$		0			0
$171$	−2.61858	−	12.1719i	−0.200248	−	0.930807i
$172$		0			0
$173$			10.9955i			0.835971i	0.908454	+	0.417986i	$0.137264\pi$
$173$			10.9955i			0.835971i	−0.908454	+	0.417986i	$0.862736\pi$
$174$		0			0
$175$	−12.9400			−0.978174
$176$		0			0
$177$	9.85772	+	7.96253i	0.740952	+	0.598501i
$178$		0			0
$179$		−	10.5715i		−	0.790153i	−0.918648	−	0.395076i	$-0.870718\pi$
$179$		−	10.5715i		−	0.790153i	0.918648	−	0.395076i	$-0.129282\pi$
$180$		0			0
$181$	9.81665			0.729666			0.364833	−	0.931073i	$-0.381126\pi$
$181$	9.81665			0.729666			0.364833	+	0.931073i	$0.381126\pi$
$182$		0			0
$183$	18.6415	+	15.0576i	1.37802	+	1.11309i
$184$		0			0
$185$		−	23.1221i		−	1.69997i
$186$		0			0
$187$		−	8.91628i		−	0.652023i
$188$		0			0
$189$	−7.43349	+	14.7233i	−0.540707	+	1.07097i
$190$		0			0
$191$		−	15.8558i		−	1.14728i	−0.819107	−	0.573641i	$-0.805530\pi$
$191$		−	15.8558i		−	1.14728i	0.819107	−	0.573641i	$-0.194470\pi$
$192$		0			0
$193$			18.2463i			1.31340i	0.754153	+	0.656699i	$0.228048\pi$
$193$			18.2463i			1.31340i	−0.754153	+	0.656699i	$0.771952\pi$
$194$		0			0
$195$	−9.21479	+	11.4080i	−0.659885	+	0.816946i
$196$		0			0
$197$	25.5950			1.82357			0.911784	−	0.410670i	$-0.134705\pi$
$197$	25.5950			1.82357			0.911784	+	0.410670i	$0.134705\pi$
$198$		0			0
$199$	7.79815			0.552796			0.276398	−	0.961043i	$-0.410859\pi$
$199$	7.79815			0.552796			0.276398	+	0.961043i	$0.410859\pi$
$200$		0			0
$201$	−11.2775	+	13.9617i	−0.795456	+	0.984785i
$202$		0			0
$203$			6.66175i			0.467563i
$204$		0			0
$205$	−1.35311			−0.0945053
$206$		0			0
$207$	2.95927	+	13.7555i	0.205684	+	0.956074i
$208$		0			0
$209$		−	16.8402i		−	1.16486i
$210$		0			0
$211$	5.43732			0.374320			0.187160	−	0.982329i	$-0.440072\pi$
$211$	5.43732			0.374320			0.187160	+	0.982329i	$0.440072\pi$
$212$		0			0
$213$	7.88296	+	6.36743i	0.540131	+	0.436289i
$214$		0			0
$215$		−	26.4547i		−	1.80420i
$216$		0			0
$217$	16.2776			1.10500
$218$		0			0
$219$	6.03485	−	7.47123i	0.407798	−	0.504859i
$220$		0			0
$221$		−	6.17516i		−	0.415386i
$222$		0			0
$223$	−25.0465			−1.67724			−0.838619	−	0.544719i	$-0.816636\pi$
$223$	−25.0465			−1.67724			−0.838619	+	0.544719i	$0.816636\pi$
$224$		0			0
$225$	2.57224	+	11.9565i	0.171482	+	0.797098i
$226$		0			0
$227$	2.55743			0.169743			0.0848714	−	0.996392i	$-0.472952\pi$
$227$	2.55743			0.169743			0.0848714	+	0.996392i	$0.472952\pi$
$228$		0			0
$229$	−7.03173			−0.464670			−0.232335	−	0.972636i	$-0.574637\pi$
$229$	−7.03173			−0.464670			−0.232335	+	0.972636i	$0.574637\pi$
$230$		0			0
$231$	−14.0180	+	17.3545i	−0.922318	+	1.14184i
$232$		0			0
$233$			3.24931i			0.212869i	0.994320	+	0.106435i	$0.0339435\pi$
$233$			3.24931i			0.212869i	−0.994320	+	0.106435i	$0.966056\pi$
$234$		0			0
$235$			26.4771i			1.72718i
$236$		0			0
$237$	2.91478	−	3.60853i	0.189335	−	0.234399i
$238$		0			0
$239$		−	3.84970i		−	0.249016i	−0.992219	−	0.124508i	$-0.960265\pi$
$239$		−	3.84970i		−	0.249016i	0.992219	−	0.124508i	$-0.0397353\pi$
$240$		0			0
$241$		−	16.2181i		−	1.04470i	−0.852732	−	0.522349i	$-0.825056\pi$
$241$		−	16.2181i		−	1.04470i	0.852732	−	0.522349i	$-0.174944\pi$
$242$		0			0
$243$	15.0819	+	3.94175i	0.967502	+	0.252863i
$244$		0			0
$245$	−9.26514			−0.591928
$246$		0			0
$247$		−	11.6631i		−	0.742103i
$248$		0			0
$249$	7.69354	+	6.21442i	0.487558	+	0.393823i
$250$		0			0
$251$			2.87277i			0.181328i	0.995882	+	0.0906638i	$0.0288989\pi$
$251$			2.87277i			0.181328i	−0.995882	+	0.0906638i	$0.971101\pi$
$252$		0			0
$253$			19.0312i			1.19648i
$254$		0			0
$255$	−8.91982	−	7.20495i	−0.558581	−	0.451191i
$256$		0			0
$257$	−8.44081			−0.526523			−0.263262	−	0.964724i	$-0.584798\pi$
$257$	−8.44081			−0.526523			−0.263262	+	0.964724i	$0.584798\pi$
$258$		0			0
$259$		−	24.3609i		−	1.51371i
$260$		0			0
$261$	6.15539	−	1.32423i	0.381009	−	0.0819679i
$262$		0			0
$263$		−	9.84125i		−	0.606838i	−0.952857	−	0.303419i	$-0.901872\pi$
$263$		−	9.84125i		−	0.606838i	0.952857	−	0.303419i	$-0.0981281\pi$
$264$		0			0
$265$	−32.8574			−2.01841
$266$		0			0
$267$	−8.21883	+	10.1750i	−0.502984	+	0.622701i
$268$		0			0
$269$	−15.3257			−0.934425			−0.467212	−	0.884145i	$-0.654742\pi$
$269$	−15.3257			−0.934425			−0.467212	+	0.884145i	$0.654742\pi$
$270$		0			0
$271$		−	26.9286i		−	1.63580i	−0.575361	−	0.817899i	$-0.695139\pi$
$271$		−	26.9286i		−	1.63580i	0.575361	−	0.817899i	$-0.304861\pi$
$272$		0			0
$273$	−9.70849	+	12.0192i	−0.587584	+	0.727437i
$274$		0			0
$275$			16.5422i			0.997531i
$276$		0			0
$277$		−	10.4453i		−	0.627596i	−0.949490	−	0.313798i	$-0.898398\pi$
$277$		−	10.4453i		−	0.627596i	0.949490	−	0.313798i	$-0.101602\pi$
$278$		0			0
$279$	−3.23569	−	15.0404i	−0.193716	−	0.900443i
$280$		0			0
$281$			2.06253i			0.123040i	0.998106	+	0.0615200i	$0.0195948\pi$
$281$			2.06253i			0.123040i	−0.998106	+	0.0615200i	$0.980405\pi$
$282$		0			0
$283$	32.9818			1.96056			0.980282	−	0.197603i	$-0.0633157\pi$
$283$	32.9818			1.96056			0.980282	+	0.197603i	$0.0633157\pi$
$284$		0			0
$285$	−16.8469	−	13.6080i	−0.997925	−	0.806070i
$286$		0			0
$287$	−1.42560			−0.0841507
$288$		0			0
$289$	−12.1717			−0.715983
$290$		0			0
$291$	0.603844	+	0.487753i	0.0353980	+	0.0285926i
$292$		0			0
$293$			14.5875i			0.852209i	0.904674	+	0.426104i	$0.140114\pi$
$293$			14.5875i			0.852209i	−0.904674	+	0.426104i	$0.859886\pi$
$294$		0			0
$295$	22.0416			1.28331
$296$		0			0
$297$	18.8219	+	9.50277i	1.09216	+	0.551407i
$298$		0			0
$299$			13.1805i			0.762247i
$300$		0			0
$301$		−	27.8720i		−	1.60652i
$302$		0			0
$303$	−10.7559	−	8.68807i	−0.617913	−	0.499116i
$304$		0			0
$305$	41.6820			2.38670
$306$		0			0
$307$			22.0182i			1.25664i	0.777954	+	0.628322i	$0.216258\pi$
$307$			22.0182i			1.25664i	−0.777954	+	0.628322i	$0.783742\pi$
$308$		0			0
$309$	−16.3219	+	20.2067i	−0.928519	+	1.14952i
$310$		0			0
$311$		−	9.12088i		−	0.517198i	−0.965985	−	0.258599i	$-0.916739\pi$
$311$		−	9.12088i		−	0.517198i	0.965985	−	0.258599i	$-0.0832608\pi$
$312$		0			0
$313$		−	10.4767i		−	0.592179i	−0.955160	−	0.296089i	$-0.904317\pi$
$313$		−	10.4767i		−	0.592179i	0.955160	−	0.296089i	$-0.0956826\pi$
$314$		0			0
$315$	6.03389	+	28.0472i	0.339971	+	1.58028i
$316$		0			0
$317$		−	4.88166i		−	0.274181i	−0.990558	−	0.137091i	$-0.956225\pi$
$317$		−	4.88166i		−	0.274181i	0.990558	−	0.137091i	$-0.0437752\pi$
$318$		0			0
$319$	8.51620			0.476815
$320$		0			0
$321$	−13.9003	+	17.2088i	−0.775840	+	0.960500i
$322$		0			0
$323$	9.11922			0.507407
$324$		0			0
$325$			11.4566i			0.635500i
$326$		0			0
$327$	−5.48900	+	6.79545i	−0.303542	+	0.375789i
$328$		0			0
$329$			27.8956i			1.53794i
$330$		0			0
$331$			20.1793i			1.10916i	0.832132	+	0.554578i	$0.187120\pi$
$331$			20.1793i			1.10916i	−0.832132	+	0.554578i	$0.812880\pi$
$332$		0			0
$333$	−22.5093	+	4.84250i	−1.23350	+	0.265367i
$334$		0			0
$335$			31.2181i			1.70563i
$336$		0			0
$337$	26.1788			1.42605			0.713024	−	0.701140i	$-0.247325\pi$
$337$	26.1788			1.42605			0.713024	+	0.701140i	$0.247325\pi$
$338$		0			0
$339$	7.43533	+	6.00586i	0.403832	+	0.326193i
$340$		0			0
$341$		−	20.8089i		−	1.12686i
$342$		0			0
$343$	12.4576			0.672648
$344$		0			0
$345$	19.0388	+	15.3785i	1.02501	+	0.827951i
$346$		0			0
$347$	−12.6340			−0.678227			−0.339114	−	0.940745i	$-0.610127\pi$
$347$	−12.6340			−0.678227			−0.339114	+	0.940745i	$0.610127\pi$
$348$		0			0
$349$			12.5612i			0.672386i	0.941793	+	0.336193i	$0.109139\pi$
$349$			12.5612i			0.672386i	−0.941793	+	0.336193i	$0.890861\pi$
$350$		0			0
$351$	13.0355	+	6.58135i	0.695785	+	0.351287i
$352$		0			0
$353$			30.8243i			1.64061i	0.571923	+	0.820307i	$0.306198\pi$
$353$			30.8243i			1.64061i	−0.571923	+	0.820307i	$0.693802\pi$
$354$		0			0
$355$	17.6261			0.935497
$356$		0			0
$357$	−9.39771	−	7.59096i	−0.497379	−	0.401756i
$358$		0			0
$359$		−	29.1337i		−	1.53762i	−0.639477	−	0.768810i	$-0.720849\pi$
$359$		−	29.1337i		−	1.53762i	0.639477	−	0.768810i	$-0.279151\pi$
$360$		0			0
$361$	−1.77647			−0.0934987
$362$		0			0
$363$	7.36413	+	5.94835i	0.386517	+	0.312207i
$364$		0			0
$365$		−	16.7055i		−	0.874405i
$366$		0			0
$367$	18.9558			0.989485			0.494742	−	0.869040i	$-0.335262\pi$
$367$	18.9558			0.989485			0.494742	+	0.869040i	$0.335262\pi$
$368$		0			0
$369$	0.283383	+	1.31724i	0.0147524	+	0.0685730i
$370$		0			0
$371$	−34.6177			−1.79726
$372$		0			0
$373$		−	24.1417i		−	1.25001i	−0.780621	−	0.625005i	$-0.785097\pi$
$373$		−	24.1417i		−	1.25001i	0.780621	−	0.625005i	$-0.214903\pi$
$374$		0			0
$375$	−3.74813	−	3.02754i	−0.193553	−	0.156341i
$376$		0			0
$377$	5.89808			0.303766
$378$		0			0
$379$			2.87437i			0.147647i	0.997271	+	0.0738233i	$0.0235201\pi$
$379$			2.87437i			0.147647i	−0.997271	+	0.0738233i	$0.976480\pi$
$380$		0			0
$381$	15.5225	+	12.5382i	0.795240	+	0.642352i
$382$		0			0
$383$		−	10.2016i		−	0.521280i	−0.965436	−	0.260640i	$-0.916067\pi$
$383$		−	10.2016i		−	0.521280i	0.965436	−	0.260640i	$-0.0839335\pi$
$384$		0			0
$385$			38.8042i			1.97765i
$386$		0			0
$387$	−25.7535	+	5.54044i	−1.30912	+	0.281636i
$388$		0			0
$389$	−35.8123			−1.81575			−0.907877	−	0.419237i	$-0.862298\pi$
$389$	−35.8123			−1.81575			−0.907877	+	0.419237i	$0.862298\pi$
$390$		0			0
$391$	−10.3057			−0.521181
$392$		0			0
$393$	−24.9810	−	20.1783i	−1.26012	−	1.01786i
$394$		0			0
$395$		−	8.06859i		−	0.405975i
$396$		0			0
$397$	12.9019			0.647528			0.323764	−	0.946138i	$-0.395052\pi$
$397$	12.9019			0.647528			0.323764	+	0.946138i	$0.395052\pi$
$398$		0			0
$399$	−17.7495	−	14.3371i	−0.888587	−	0.717752i
$400$		0			0
$401$	24.3427			1.21562			0.607809	−	0.794084i	$-0.292049\pi$
$401$	24.3427			1.21562			0.607809	+	0.794084i	$0.292049\pi$
$402$		0			0
$403$		−	14.4116i		−	0.717895i
$404$		0			0
$405$	24.7159	−	11.1505i	1.22814	−	0.554074i
$406$		0			0
$407$	−31.1423			−1.54367
$408$		0			0
$409$	7.12084			0.352103			0.176051	−	0.984381i	$-0.443668\pi$
$409$	7.12084			0.352103			0.176051	+	0.984381i	$0.443668\pi$
$410$		0			0
$411$	9.30370	−	11.5181i	0.458918	−	0.568146i
$412$		0			0
$413$	23.2225			1.14271
$414$		0			0
$415$	17.2026			0.844441
$416$		0			0
$417$	2.67943	−	3.31717i	0.131212	−	0.162443i
$418$		0			0
$419$		−	3.60372i		−	0.176053i	−0.996118	−	0.0880267i	$-0.971944\pi$
$419$		−	3.60372i		−	0.176053i	0.996118	−	0.0880267i	$-0.0280561\pi$
$420$		0			0
$421$	3.88137			0.189166			0.0945832	−	0.995517i	$-0.469848\pi$
$421$	3.88137			0.189166			0.0945832	+	0.995517i	$0.469848\pi$
$422$		0			0
$423$	25.7753	−	5.54513i	1.25324	−	0.269614i
$424$		0			0
$425$	−8.95783			−0.434518
$426$		0			0
$427$	43.9152			2.12520
$428$		0			0
$429$	15.3651	+	12.4111i	0.741832	+	0.599212i
$430$		0			0
$431$		−	28.7404i		−	1.38438i	−0.721718	−	0.692188i	$-0.756647\pi$
$431$		−	28.7404i		−	1.38438i	0.721718	−	0.692188i	$-0.243353\pi$
$432$		0			0
$433$	13.5388			0.650634			0.325317	−	0.945605i	$-0.394529\pi$
$433$	13.5388			0.650634			0.325317	+	0.945605i	$0.394529\pi$
$434$		0			0
$435$	6.88166	−	8.51958i	0.329950	−	0.408483i
$436$		0			0
$437$	−19.4644			−0.931109
$438$		0			0
$439$			7.32987i			0.349835i	0.984583	+	0.174918i	$0.0559659\pi$
$439$			7.32987i			0.349835i	−0.984583	+	0.174918i	$0.944034\pi$
$440$		0			0
$441$	1.94041	+	9.01956i	0.0924005	+	0.429503i
$442$		0			0
$443$	−22.8926			−1.08766			−0.543829	−	0.839196i	$-0.683026\pi$
$443$	−22.8926			−1.08766			−0.543829	+	0.839196i	$0.683026\pi$
$444$		0			0
$445$			22.7511i			1.07851i
$446$		0			0
$447$	−19.6872	−	15.9022i	−0.931172	−	0.752151i
$448$		0			0
$449$			15.0367i			0.709625i	0.934937	+	0.354813i	$0.115455\pi$
$449$			15.0367i			0.709625i	−0.934937	+	0.354813i	$0.884545\pi$
$450$		0			0
$451$			1.82245i			0.0858159i
$452$		0			0
$453$	−21.6558	+	26.8102i	−1.01748	+	1.25965i
$454$		0			0
$455$			26.8747i			1.25991i
$456$		0			0
$457$			36.6485i			1.71434i	0.515032	+	0.857171i	$0.327780\pi$
$457$			36.6485i			1.71434i	−0.515032	+	0.857171i	$0.672220\pi$
$458$		0			0
$459$	−5.14589	+	10.1923i	−0.240189	+	0.475738i
$460$		0			0
$461$			24.3754i			1.13528i	0.823278	+	0.567639i	$0.192143\pi$
$461$			24.3754i			1.13528i	−0.823278	+	0.567639i	$0.807857\pi$
$462$		0			0
$463$		−	31.8499i		−	1.48019i	−0.672503	−	0.740095i	$-0.734781\pi$
$463$		−	31.8499i		−	1.48019i	0.672503	−	0.740095i	$-0.265219\pi$
$464$		0			0
$465$	−20.8171	−	16.8150i	−0.965372	−	0.779775i
$466$		0			0
$467$			7.82188i			0.361953i	0.983487	+	0.180977i	$0.0579258\pi$
$467$			7.82188i			0.361953i	−0.983487	+	0.180977i	$0.942074\pi$
$468$		0			0
$469$			32.8907i			1.51875i
$470$		0			0
$471$	−0.728165	−	0.588172i	−0.0335521	−	0.0271016i
$472$		0			0
$473$	−35.6308			−1.63831
$474$		0			0
$475$	−16.9187			−0.776283
$476$		0			0
$477$	6.88136	+	31.9864i	0.315076	+	1.46456i
$478$		0			0
$479$	8.41249			0.384377			0.192188	−	0.981358i	$-0.438442\pi$
$479$	8.41249			0.384377			0.192188	+	0.981358i	$0.438442\pi$
$480$		0			0
$481$	−21.5683			−0.983430
$482$		0			0
$483$	20.0588	+	16.2024i	0.912708	+	0.737236i
$484$		0			0
$485$	1.35018			0.0613086
$486$		0			0
$487$			19.3033i			0.874718i	0.899287	+	0.437359i	$0.144086\pi$
$487$			19.3033i			0.874718i	−0.899287	+	0.437359i	$0.855914\pi$
$488$		0			0
$489$	21.5626	−	26.6948i	0.975094	−	1.20718i
$490$		0			0
$491$			15.6151i			0.704701i	0.935868	+	0.352350i	$0.114617\pi$
$491$			15.6151i			0.704701i	−0.935868	+	0.352350i	$0.885383\pi$
$492$		0			0
$493$			4.61164i			0.207698i
$494$		0			0
$495$	35.8548	−	7.71356i	1.61155	−	0.346699i
$496$		0			0
$497$	18.5704			0.832998
$498$		0			0
$499$		−	32.0615i		−	1.43527i	−0.696419	−	0.717635i	$-0.745224\pi$
$499$		−	32.0615i		−	1.43527i	0.696419	−	0.717635i	$-0.254776\pi$
$500$		0			0
$501$	−0.336311	−	22.3805i	−0.0150253	−	0.999887i
$502$		0			0
$503$			20.3177i			0.905920i	0.891531	+	0.452960i	$0.149632\pi$
$503$			20.3177i			0.905920i	−0.891531	+	0.452960i	$0.850368\pi$
$504$		0			0
$505$	−24.0500			−1.07021
$506$		0			0
$507$	−6.87477	−	5.55307i	−0.305319	−	0.246620i
$508$		0			0
$509$			18.7036i			0.829020i	0.910045	+	0.414510i	$0.136047\pi$
$509$			18.7036i			0.829020i	−0.910045	+	0.414510i	$0.863953\pi$
$510$		0			0
$511$		−	17.6005i		−	0.778600i
$512$		0			0
$513$	−9.71907	+	19.2503i	−0.429107	+	0.849923i
$514$		0			0
$515$			45.1817i			1.99094i
$516$		0			0
$517$	35.6610			1.56837
$518$		0			0
$519$	11.9670	−	14.8153i	0.525293	−	0.650319i
$520$		0			0
$521$	−8.04946			−0.352653			−0.176327	−	0.984332i	$-0.556421\pi$
$521$	−8.04946			−0.352653			−0.176327	+	0.984332i	$0.556421\pi$
$522$		0			0
$523$	−36.7013			−1.60483			−0.802417	−	0.596764i	$-0.796453\pi$
$523$	−36.7013			−1.60483			−0.802417	+	0.596764i	$0.796453\pi$
$524$		0			0
$525$	17.4354	+	14.0833i	0.760942	+	0.614648i
$526$		0			0
$527$	11.2683			0.490855
$528$		0			0
$529$	−1.00317			−0.0436161
$530$		0			0
$531$	−4.61621	−	21.4574i	−0.200326	−	0.931172i
$532$		0			0
$533$			1.26218i			0.0546711i
$534$		0			0
$535$			38.4784i			1.66357i
$536$		0			0
$537$	−11.5056	+	14.2441i	−0.496502	+	0.614676i
$538$		0			0
$539$			12.4789i			0.537503i
$540$		0			0
$541$			12.6561i			0.544127i	0.962279	+	0.272063i	$0.0877061\pi$
$541$			12.6561i			0.544127i	−0.962279	+	0.272063i	$0.912294\pi$
$542$		0			0
$543$	−13.2269	−	10.6840i	−0.567622	−	0.458495i
$544$		0			0
$545$			15.1945i			0.650860i
$546$		0			0
$547$			26.1871i			1.11968i	0.828601	+	0.559840i	$0.189137\pi$
$547$			26.1871i			1.11968i	−0.828601	+	0.559840i	$0.810863\pi$
$548$		0			0
$549$	−8.72952	−	40.5772i	−0.372567	−	1.73179i
$550$		0			0
$551$			8.71004i			0.371060i
$552$		0			0
$553$		−	8.50088i		−	0.361494i
$554$		0			0
$555$	−25.1651	+	31.1547i	−1.06820	+	1.32244i
$556$		0			0
$557$		−	17.4856i		−	0.740887i	−0.928855	−	0.370443i	$-0.879206\pi$
$557$		−	17.4856i		−	0.740887i	0.928855	−	0.370443i	$-0.120794\pi$
$558$		0			0
$559$	−24.6769			−1.04372
$560$		0			0
$561$	−9.70408	+	12.0138i	−0.409707	+	0.507222i
$562$		0			0
$563$		−	33.4515i		−	1.40981i	−0.709300	−	0.704907i	$-0.750989\pi$
$563$		−	33.4515i		−	1.40981i	0.709300	−	0.704907i	$-0.249011\pi$
$564$		0			0
$565$	16.6252			0.699428
$566$		0			0
$567$	26.0401	−	11.7479i	1.09358	−	0.493366i
$568$		0			0
$569$	8.72419			0.365737			0.182868	−	0.983137i	$-0.441462\pi$
$569$	8.72419			0.365737			0.182868	+	0.983137i	$0.441462\pi$
$570$		0			0
$571$		−	3.97889i		−	0.166511i	−0.996528	−	0.0832556i	$-0.973468\pi$
$571$		−	3.97889i		−	0.166511i	0.996528	−	0.0832556i	$-0.0265318\pi$
$572$		0			0
$573$	−17.2567	+	21.3640i	−0.720909	+	0.892495i
$574$		0			0
$575$	19.1199			0.797356
$576$		0			0
$577$	−33.1199			−1.37880			−0.689401	−	0.724380i	$-0.742126\pi$
$577$	−33.1199			−1.37880			−0.689401	+	0.724380i	$0.742126\pi$
$578$		0			0
$579$	19.8585	−	24.5850i	0.825290	−	1.02172i
$580$		0			0
$581$	18.1242			0.751919
$582$		0			0
$583$			44.2543i			1.83283i
$584$		0			0
$585$	24.8320	−	5.34220i	1.02668	−	0.220873i
$586$		0			0
$587$	−19.8588			−0.819659			−0.409829	−	0.912162i	$-0.634412\pi$
$587$	−19.8588			−0.819659			−0.409829	+	0.912162i	$0.634412\pi$
$588$		0			0
$589$	21.2825			0.876930
$590$		0			0
$591$	−34.4867	−	27.8565i	−1.41859	−	1.14586i
$592$		0			0
$593$	−5.73548			−0.235528			−0.117764	−	0.993042i	$-0.537573\pi$
$593$	−5.73548			−0.235528			−0.117764	+	0.993042i	$0.537573\pi$
$594$		0			0
$595$	−21.0131			−0.861452
$596$		0			0
$597$	−10.5072	−	8.48716i	−0.430032	−	0.347356i
$598$		0			0
$599$		−	26.7361i		−	1.09241i	−0.837653	−	0.546203i	$-0.816073\pi$
$599$		−	26.7361i		−	1.09241i	0.837653	−	0.546203i	$-0.183927\pi$
$600$		0			0
$601$	−25.7890			−1.05196			−0.525978	−	0.850498i	$-0.676301\pi$
$601$	−25.7890			−1.05196			−0.525978	+	0.850498i	$0.676301\pi$
$602$		0			0
$603$	30.3907	−	6.53805i	1.23760	−	0.266250i
$604$		0			0
$605$	16.4660			0.669439
$606$		0			0
$607$		−	32.0071i		−	1.29913i	−0.760307	−	0.649564i	$-0.774952\pi$
$607$		−	32.0071i		−	1.29913i	0.760307	−	0.649564i	$-0.225048\pi$
$608$		0			0
$609$	7.25035	−	8.97603i	0.293799	−	0.363727i
$610$		0			0
$611$	24.6978			0.999167
$612$		0			0
$613$	18.7394			0.756879			0.378439	−	0.925626i	$-0.376461\pi$
$613$	18.7394			0.756879			0.378439	+	0.925626i	$0.376461\pi$
$614$		0			0
$615$	1.82318	+	1.47266i	0.0735176	+	0.0593835i
$616$		0			0
$617$		−	4.39733i		−	0.177030i	−0.996075	−	0.0885150i	$-0.971788\pi$
$617$		−	4.39733i		−	0.177030i	0.996075	−	0.0885150i	$-0.0282121\pi$
$618$		0			0
$619$		−	6.39703i		−	0.257118i	−0.991702	−	0.128559i	$-0.958965\pi$
$619$		−	6.39703i		−	0.257118i	0.991702	−	0.128559i	$-0.0410352\pi$
$620$		0			0
$621$	10.9836	−	21.7549i	0.440756	−	0.872994i
$622$		0			0
$623$			23.9700i			0.960338i
$624$		0			0
$625$	−28.7641			−1.15056
$626$		0			0
$627$	−18.3281	+	22.6905i	−0.731956	+	0.906171i
$628$		0			0
$629$		−	16.8640i		−	0.672413i
$630$		0			0
$631$	−22.5355			−0.897124			−0.448562	−	0.893752i	$-0.648064\pi$
$631$	−22.5355			−0.897124			−0.448562	+	0.893752i	$0.648064\pi$
$632$		0			0
$633$	−7.32623	−	5.91773i	−0.291192	−	0.235209i
$634$		0			0
$635$	34.7079			1.37734
$636$		0			0
$637$			8.64252i			0.342429i
$638$		0			0
$639$	−3.69146	−	17.1589i	−0.146032	−	0.678796i
$640$		0			0
$641$	−11.1224			−0.439306			−0.219653	−	0.975578i	$-0.570493\pi$
$641$	−11.1224			−0.439306			−0.219653	+	0.975578i	$0.570493\pi$
$642$		0			0
$643$			36.8683i			1.45394i	0.686667	+	0.726972i	$0.259073\pi$
$643$			36.8683i			1.45394i	−0.686667	+	0.726972i	$0.740927\pi$
$644$		0			0
$645$	−28.7921	+	35.6450i	−1.13369	+	1.40352i
$646$		0			0
$647$	43.9225			1.72677			0.863387	−	0.504543i	$-0.168339\pi$
$647$	43.9225			1.72677			0.863387	+	0.504543i	$0.168339\pi$
$648$		0			0
$649$		−	29.6871i		−	1.16532i
$650$		0			0
$651$	−21.9324	−	17.7158i	−0.859600	−	0.694338i
$652$		0			0
$653$		−	4.48149i		−	0.175374i	−0.996148	−	0.0876872i	$-0.972052\pi$
$653$		−	4.48149i		−	0.175374i	0.996148	−	0.0876872i	$-0.0279476\pi$
$654$		0			0
$655$	−55.8569			−2.18251
$656$		0			0
$657$	−16.2627	+	3.49865i	−0.634468	+	0.136495i
$658$		0			0
$659$	−22.8073			−0.888447			−0.444224	−	0.895916i	$-0.646520\pi$
$659$	−22.8073			−0.888447			−0.444224	+	0.895916i	$0.646520\pi$
$660$		0			0
$661$		−	32.7339i		−	1.27320i	−0.771194	−	0.636601i	$-0.780340\pi$
$661$		−	32.7339i		−	1.27320i	0.771194	−	0.636601i	$-0.219660\pi$
$662$		0			0
$663$	−6.72077	+	8.32040i	−0.261013	+	0.323138i
$664$		0			0
$665$	−39.6875			−1.53901
$666$		0			0
$667$		−	9.84326i		−	0.381133i
$668$		0			0
$669$	33.7476	+	27.2595i	1.30476	+	1.05391i
$670$		0			0
$671$		−	56.1399i		−	2.16726i
$672$		0			0
$673$		−	28.6438i		−	1.10414i	−0.833798	−	0.552069i	$-0.813838\pi$
$673$		−	28.6438i		−	1.10414i	0.833798	−	0.552069i	$-0.186162\pi$
$674$		0			0
$675$	9.54706	−	18.9096i	0.367466	−	0.727832i
$676$		0			0
$677$		−	8.77741i		−	0.337343i	−0.985672	−	0.168672i	$-0.946052\pi$
$677$		−	8.77741i		−	0.337343i	0.985672	−	0.168672i	$-0.0539477\pi$
$678$		0			0
$679$	1.42252			0.0545913
$680$		0			0
$681$	−3.44588	−	2.78340i	−0.132046	−	0.106660i
$682$		0			0
$683$	−12.9621			−0.495979			−0.247990	−	0.968763i	$-0.579770\pi$
$683$	−12.9621			−0.495979			−0.247990	+	0.968763i	$0.579770\pi$
$684$		0			0
$685$		−	25.7542i		−	0.984018i
$686$		0			0
$687$	9.47455	+	7.65303i	0.361477	+	0.291981i
$688$		0			0
$689$			30.6493i			1.16765i
$690$		0			0
$691$			27.1220i			1.03177i	0.856658	+	0.515885i	$0.172537\pi$
$691$			27.1220i			1.03177i	−0.856658	+	0.515885i	$0.827463\pi$
$692$		0			0
$693$	37.7757	−	8.12682i	1.43498	−	0.308712i
$694$		0			0
$695$		−	7.41712i		−	0.281347i
$696$		0			0
$697$	−0.986884			−0.0373809
$698$		0			0
$699$	3.53640	−	4.37811i	0.133759	−	0.165596i
$700$		0			0
$701$		−	49.6669i		−	1.87589i	−0.346779	−	0.937947i	$-0.612725\pi$
$701$		−	49.6669i		−	1.87589i	0.346779	−	0.937947i	$-0.387275\pi$
$702$		0			0
$703$		−	31.8512i		−	1.20129i
$704$		0			0
$705$	28.8165	−	35.6752i	1.08529	−	1.34361i
$706$		0			0
$707$	−25.3385			−0.952954
$708$		0			0
$709$			1.93531i			0.0726821i	0.999339	+	0.0363410i	$0.0115703\pi$
$709$			1.93531i			0.0726821i	−0.999339	+	0.0363410i	$0.988430\pi$
$710$		0			0
$711$	−7.85473	+	1.68982i	−0.294575	+	0.0633730i
$712$		0			0
$713$	−24.0515			−0.900735
$714$		0			0
$715$	34.3559			1.28484
$716$		0			0
$717$	−4.18984	+	5.18707i	−0.156472	+	0.193715i
$718$		0			0
$719$	15.0805			0.562408			0.281204	−	0.959648i	$-0.409266\pi$
$719$	15.0805			0.562408			0.281204	+	0.959648i	$0.409266\pi$
$720$		0			0
$721$			47.6023i			1.77280i
$722$		0			0
$723$	−17.6510	+	21.8522i	−0.656448	+	0.812692i
$724$		0			0
$725$		−	8.55588i		−	0.317758i
$726$		0			0
$727$			44.3277i			1.64402i	0.569471	+	0.822011i	$0.307148\pi$
$727$			44.3277i			1.64402i	−0.569471	+	0.822011i	$0.692852\pi$
$728$		0			0
$729$	−16.0313	−	21.7255i	−0.593750	−	0.804650i
$730$		0			0
$731$		−	19.2946i		−	0.713637i
$732$		0			0
$733$	−16.5739			−0.612172			−0.306086	−	0.952004i	$-0.599020\pi$
$733$	−16.5739			−0.612172			−0.306086	+	0.952004i	$0.599020\pi$
$734$		0			0
$735$	12.4838	+	10.0838i	0.460473	+	0.371945i
$736$		0			0
$737$	42.0465			1.54880
$738$		0			0
$739$			32.7848i			1.20601i	0.797739	+	0.603003i	$0.206029\pi$
$739$			32.7848i			1.20601i	−0.797739	+	0.603003i	$0.793971\pi$
$740$		0			0
$741$	−12.6936	+	15.7148i	−0.466309	+	0.577297i
$742$		0			0
$743$		−	12.9115i		−	0.473676i	−0.971549	−	0.236838i	$-0.923889\pi$
$743$		−	12.9115i		−	0.473676i	0.971549	−	0.236838i	$-0.0761111\pi$
$744$		0			0
$745$	−44.0201			−1.61277
$746$		0			0
$747$	−3.60276	−	16.7466i	−0.131818	−	0.612726i
$748$		0			0
$749$			40.5399i			1.48130i
$750$		0			0
$751$		−	1.63472i		−	0.0596516i	−0.999555	−	0.0298258i	$-0.990505\pi$
$751$		−	1.63472i		−	0.0596516i	0.999555	−	0.0298258i	$-0.00949526\pi$
$752$		0			0
$753$	3.12659	−	3.87076i	0.113939	−	0.141058i
$754$		0			0
$755$			59.9470i			2.18169i
$756$		0			0
$757$	0.159682			0.00580372			0.00290186	−	0.999996i	$-0.499076\pi$
$757$	0.159682			0.00580372			0.00290186	+	0.999996i	$0.499076\pi$
$758$		0			0
$759$	20.7127	−	25.6426i	0.751825	−	0.930769i
$760$		0			0
$761$		−	44.6281i		−	1.61777i	−0.587970	−	0.808883i	$-0.700073\pi$
$761$		−	44.6281i		−	1.61777i	0.587970	−	0.808883i	$-0.299927\pi$
$762$		0			0
$763$			16.0085i			0.579548i
$764$		0			0
$765$	4.17701	+	19.4159i	0.151020	+	0.701982i
$766$		0			0
$767$		−	20.5604i		−	0.742394i
$768$		0			0
$769$		−	14.1447i		−	0.510071i	−0.966932	−	0.255036i	$-0.917913\pi$
$769$		−	14.1447i		−	0.510071i	0.966932	−	0.255036i	$-0.0820872\pi$
$770$		0			0
$771$	11.3731	+	9.18660i	0.409593	+	0.330847i
$772$		0			0
$773$	−10.6410			−0.382730			−0.191365	−	0.981519i	$-0.561291\pi$
$773$	−10.6410			−0.382730			−0.191365	+	0.981519i	$0.561291\pi$
$774$		0			0
$775$	−20.9058			−0.750960
$776$		0			0
$777$	−26.5133	+	32.8238i	−0.951160	+	1.17755i
$778$		0			0
$779$	−1.86393			−0.0667824
$780$		0			0
$781$		−	23.7400i		−	0.849482i
$782$		0			0
$783$	−9.73500	−	4.91499i	−0.347901	−	0.175647i
$784$		0			0
$785$	−1.62816			−0.0581115
$786$		0			0
$787$		−	48.0013i		−	1.71106i	−0.517753	−	0.855530i	$-0.673231\pi$
$787$		−	48.0013i		−	1.71106i	0.517753	−	0.855530i	$-0.326769\pi$
$788$		0			0
$789$	−10.7108	+	13.2601i	−0.381314	+	0.472072i
$790$		0			0
$791$	17.5159			0.622795
$792$		0			0
$793$		−	38.8809i		−	1.38070i
$794$		0			0
$795$	44.2719	+	35.7605i	1.57016	+	1.26829i
$796$		0			0
$797$	25.3959			0.899569			0.449784	−	0.893137i	$-0.351501\pi$
$797$	25.3959			0.899569			0.449784	+	0.893137i	$0.351501\pi$
$798$		0			0
$799$			19.3110i			0.683173i
$800$		0			0
$801$	22.1481	−	4.76479i	0.782563	−	0.168356i
$802$		0			0
$803$	−22.5000			−0.794008
$804$		0			0
$805$	44.8510			1.58079
$806$		0			0
$807$	20.6498	+	16.6798i	0.726908	+	0.587157i
$808$		0			0
$809$		−	24.3601i		−	0.856454i	−0.903671	−	0.428227i	$-0.859138\pi$
$809$		−	24.3601i		−	0.856454i	0.903671	−	0.428227i	$-0.140862\pi$
$810$		0			0
$811$		−	24.7712i		−	0.869835i	−0.900470	−	0.434918i	$-0.856777\pi$
$811$		−	24.7712i		−	0.869835i	0.900470	−	0.434918i	$-0.143223\pi$
$812$		0			0
$813$	−29.3079	+	36.2836i	−1.02787	+	1.27252i
$814$		0			0
$815$		−	59.6889i		−	2.09081i
$816$		0			0
$817$		−	36.4418i		−	1.27494i
$818$		0			0
$819$	26.1624	−	5.62841i	0.914188	−	0.196673i
$820$		0			0
$821$	16.7159			0.583389			0.291694	−	0.956512i	$-0.405781\pi$
$821$	16.7159			0.583389			0.291694	+	0.956512i	$0.405781\pi$
$822$		0			0
$823$		−	5.32039i		−	0.185457i	−0.995691	−	0.0927286i	$-0.970441\pi$
$823$		−	5.32039i		−	0.185457i	0.995691	−	0.0927286i	$-0.0295589\pi$
$824$		0			0
$825$	18.0038	−	22.2889i	0.626811	−	0.776000i
$826$		0			0
$827$	13.1772			0.458217			0.229108	−	0.973401i	$-0.426419\pi$
$827$	13.1772			0.458217			0.229108	+	0.973401i	$0.426419\pi$
$828$		0			0
$829$			29.3203i			1.01833i	0.860667	+	0.509167i	$0.170047\pi$
$829$			29.3203i			1.01833i	−0.860667	+	0.509167i	$0.829953\pi$
$830$		0			0
$831$	−11.3682	+	14.0740i	−0.394358	+	0.488220i
$832$		0			0
$833$	−6.75749			−0.234133
$834$		0			0
$835$	−24.9166	−	29.9160i	−0.862273	−	1.03529i
$836$		0			0
$837$	−12.0095	+	23.7869i	−0.415109	+	0.822197i
$838$		0			0
$839$			54.2464i			1.87280i	0.350941	+	0.936398i	$0.385862\pi$
$839$			54.2464i			1.87280i	−0.350941	+	0.936398i	$0.614138\pi$
$840$		0			0
$841$	24.5953			0.848113
$842$		0			0
$843$	2.24476	−	2.77905i	0.0773137	−	0.0957154i
$844$		0			0
$845$	−15.3718			−0.528807
$846$		0			0
$847$	17.3482			0.596091
$848$		0			0
$849$	−44.4396	−	35.8959i	−1.52516	−	1.23194i
$850$		0			0
$851$			35.9952i			1.23390i
$852$		0			0
$853$	16.0438			0.549328			0.274664	−	0.961540i	$-0.411433\pi$
$853$	16.0438			0.549328			0.274664	+	0.961540i	$0.411433\pi$
$854$		0			0
$855$	7.88913	+	36.6709i	0.269803	+	1.25412i
$856$		0			0
$857$		−	40.5609i		−	1.38553i	−0.721162	−	0.692767i	$-0.756391\pi$
$857$		−	40.5609i		−	1.38553i	0.721162	−	0.692767i	$-0.243609\pi$
$858$		0			0
$859$	48.1289			1.64214			0.821069	−	0.570829i	$-0.193378\pi$
$859$	48.1289			1.64214			0.821069	+	0.570829i	$0.193378\pi$
$860$		0			0
$861$	1.92086	+	1.55156i	0.0654626	+	0.0528771i
$862$		0			0
$863$			6.26818i			0.213371i	0.994293	+	0.106686i	$0.0340238\pi$
$863$			6.26818i			0.213371i	−0.994293	+	0.106686i	$0.965976\pi$
$864$		0			0
$865$		−	33.1267i		−	1.12634i
$866$		0			0
$867$	16.4001	+	13.2471i	0.556978	+	0.449897i
$868$		0			0
$869$	−10.8673			−0.368647
$870$		0			0
$871$	29.1202			0.986702
$872$		0			0
$873$	−0.282770	−	1.31439i	−0.00957033	−	0.0444855i
$874$		0			0
$875$	−8.82974			−0.298500
$876$		0			0
$877$	−2.90778			−0.0981888			−0.0490944	−	0.998794i	$-0.515634\pi$
$877$	−2.90778			−0.0981888			−0.0490944	+	0.998794i	$0.515634\pi$
$878$		0			0
$879$	15.8763	−	19.6551i	0.535496	−	0.662951i
$880$		0			0
$881$	−6.84456			−0.230599			−0.115300	−	0.993331i	$-0.536783\pi$
$881$	−6.84456			−0.230599			−0.115300	+	0.993331i	$0.536783\pi$
$882$		0			0
$883$	34.9179			1.17508			0.587541	−	0.809194i	$-0.300096\pi$
$883$	34.9179			1.17508			0.587541	+	0.809194i	$0.300096\pi$
$884$		0			0
$885$	−29.6989	−	23.9891i	−0.998316	−	0.806386i
$886$		0			0
$887$	27.7480			0.931686			0.465843	−	0.884867i	$-0.345751\pi$
$887$	27.7480			0.931686			0.465843	+	0.884867i	$0.345751\pi$
$888$		0			0
$889$	36.5674			1.22643
$890$		0			0
$891$	−15.0182	−	33.2889i	−0.503129	−	1.11522i
$892$		0			0
$893$			36.4727i			1.22051i
$894$		0			0
$895$			31.8494i			1.06461i
$896$		0			0
$897$	14.3451	−	17.7594i	0.478968	−	0.592968i
$898$		0			0
$899$			10.7627i			0.358956i
$900$		0			0
$901$	−23.9644			−0.798369
$902$		0			0
$903$	−30.3347	+	37.5547i	−1.00947	+	1.24974i
$904$		0			0
$905$	−29.5751			−0.983111
$906$		0			0
$907$	−49.1492			−1.63197			−0.815986	−	0.578071i	$-0.803806\pi$
$907$	−49.1492			−1.63197			−0.815986	+	0.578071i	$0.803806\pi$
$908$		0			0
$909$	5.03683	+	23.4126i	0.167061	+	0.776546i
$910$		0			0
$911$			37.7917i			1.25210i	0.779785	+	0.626048i	$0.215328\pi$
$911$			37.7917i			1.25210i	−0.779785	+	0.626048i	$0.784672\pi$
$912$		0			0
$913$		−	23.1695i		−	0.766799i
$914$		0			0
$915$	−56.1623	−	45.3648i	−1.85667	−	1.49972i
$916$		0			0
$917$	−58.8495			−1.94338
$918$		0			0
$919$	−2.81977			−0.0930157			−0.0465078	−	0.998918i	$-0.514809\pi$
$919$	−2.81977			−0.0930157			−0.0465078	+	0.998918i	$0.514809\pi$
$920$		0			0
$921$	23.9636	−	29.6673i	0.789628	−	0.977569i
$922$		0			0
$923$		−	16.4416i		−	0.541183i
$924$		0			0
$925$			31.2875i			1.02873i
$926$		0			0
$927$	43.9841	−	9.46246i	1.44463	−	0.310788i
$928$		0			0
$929$		−	11.0753i		−	0.363370i	−0.983357	−	0.181685i	$-0.941845\pi$
$929$		−	11.0753i		−	0.363370i	0.983357	−	0.181685i	$-0.0581552\pi$
$930$		0			0
$931$	−12.7629			−0.418288
$932$		0			0
$933$	−9.92676	+	12.2895i	−0.324988	+	0.402339i
$934$		0			0
$935$			26.8625i			0.878499i
$936$		0			0
$937$		−	52.1524i		−	1.70374i	−0.523749	−	0.851872i	$-0.675467\pi$
$937$		−	52.1524i		−	1.70374i	0.523749	−	0.851872i	$-0.324533\pi$
$938$		0			0
$939$	−11.4024	+	14.1163i	−0.372103	+	0.460668i
$940$		0			0
$941$	30.7815			1.00345			0.501724	−	0.865028i	$-0.332699\pi$
$941$	30.7815			1.00345			0.501724	+	0.865028i	$0.332699\pi$
$942$		0			0
$943$	2.10644			0.0685952
$944$		0			0
$945$	22.3953	−	44.3578i	0.728518	−	1.44296i
$946$		0			0
$947$			48.2547i			1.56807i	0.620719	+	0.784033i	$0.286841\pi$
$947$			48.2547i			1.56807i	−0.620719	+	0.784033i	$0.713159\pi$
$948$		0			0
$949$	−15.5829			−0.505841
$950$		0			0
$951$	−5.31298	+	6.57754i	−0.172285	+	0.213291i
$952$		0			0
$953$	30.5920			0.990973			0.495487	−	0.868616i	$-0.334990\pi$
$953$	30.5920			0.990973			0.495487	+	0.868616i	$0.334990\pi$
$954$		0			0
$955$			47.7695i			1.54578i
$956$		0			0
$957$	−11.4747	−	9.26865i	−0.370925	−	0.299613i
$958$		0			0
$959$		−	27.1340i		−	0.876203i
$960$		0			0
$961$	−4.70196			−0.151676
$962$		0			0
$963$	37.4585	−	8.05858i	1.20708	−	0.259684i
$964$		0			0
$965$		−	54.9716i		−	1.76960i
$966$		0			0
$967$	−26.4597			−0.850887			−0.425443	−	0.904985i	$-0.639882\pi$
$967$	−26.4597			−0.850887			−0.425443	+	0.904985i	$0.639882\pi$
$968$		0			0
$969$	−12.2872	−	9.92496i	−0.394723	−	0.318836i
$970$		0			0
$971$	−29.5318			−0.947722			−0.473861	−	0.880600i	$-0.657140\pi$
$971$	−29.5318			−0.947722			−0.473861	+	0.880600i	$0.657140\pi$
$972$		0			0
$973$		−	7.81450i		−	0.250521i
$974$		0			0
$975$	12.4689	−	15.4367i	0.399324	−	0.494369i
$976$		0			0
$977$	26.1580			0.836868			0.418434	−	0.908247i	$-0.362579\pi$
$977$	26.1580			0.836868			0.418434	+	0.908247i	$0.362579\pi$
$978$		0			0
$979$	30.6426			0.979342
$980$		0			0
$981$	14.7917	−	3.18220i	0.472264	−	0.101600i
$982$		0			0
$983$	12.6040			0.402005			0.201003	−	0.979591i	$-0.435580\pi$
$983$	12.6040			0.402005			0.201003	+	0.979591i	$0.435580\pi$
$984$		0			0
$985$	−77.1114			−2.45697
$986$		0			0
$987$	30.3604	−	37.5865i	0.966381	−	1.19639i
$988$		0			0
$989$			41.1831i			1.30955i
$990$		0			0
$991$		−	52.9747i		−	1.68280i	−0.540415	−	0.841399i	$-0.681733\pi$
$991$		−	52.9747i		−	1.68280i	0.540415	−	0.841399i	$-0.318267\pi$
$992$		0			0
$993$	21.9623	−	27.1896i	0.696952	−	0.862836i
$994$		0			0
$995$	−23.4939			−0.744807
$996$		0			0
$997$	59.3828			1.88067			0.940336	−	0.340248i	$-0.110511\pi$
$997$	59.3828			1.88067			0.940336	+	0.340248i	$0.110511\pi$
$998$		0			0
$999$	35.5993	+	17.9733i	1.12631	+	0.568650i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2004.2.h.a.1001.13	✓	56
3.2	odd	2	inner	2004.2.h.a.1001.16	yes	56
167.166	odd	2	inner	2004.2.h.a.1001.14	yes	56
501.500	even	2	inner	2004.2.h.a.1001.15	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.h.a.1001.13	✓	56	1.1	even	1	trivial
2004.2.h.a.1001.14	yes	56	167.166	odd	2	inner
2004.2.h.a.1001.15	yes	56	501.500	even	2	inner
2004.2.h.a.1001.16	yes	56	3.2	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:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2004.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.34740 - 1.08836i) q^{3} -3.01275 q^{5} -3.17416 q^{7} +(0.630964 + 2.93290i) q^{9} +O(q^{10})\)	\(q+(-1.34740 - 1.08836i) q^{3} -3.01275 q^{5} -3.17416 q^{7} +(0.630964 + 2.93290i) q^{9} +4.05776i q^{11} +2.81029i q^{13} +(4.05938 + 3.27895i) q^{15} -2.19734 q^{17} -4.15012 q^{19} +(4.27686 + 3.45462i) q^{21} +4.69008 q^{23} +4.07667 q^{25} +(2.34187 - 4.63849i) q^{27} -2.09874i q^{29} -5.12816 q^{31} +(4.41629 - 5.46742i) q^{33} +9.56296 q^{35} +7.67475i q^{37} +(3.05860 - 3.78658i) q^{39} +0.449127 q^{41} +8.78091i q^{43} +(-1.90094 - 8.83609i) q^{45} -8.78834i q^{47} +3.07531 q^{49} +(2.96069 + 2.39148i) q^{51} +10.9061 q^{53} -12.2250i q^{55} +(5.59187 + 4.51681i) q^{57} -7.31611 q^{59} -13.8352 q^{61} +(-2.00278 - 9.30949i) q^{63} -8.46671i q^{65} -10.3620i q^{67} +(-6.31940 - 5.10447i) q^{69} -5.85050 q^{71} +5.54493i q^{73} +(-5.49290 - 4.43687i) q^{75} -12.8800i q^{77} +2.67815i q^{79} +(-8.20377 + 3.70111i) q^{81} -5.70992 q^{83} +6.62003 q^{85} +(-2.28418 + 2.82784i) q^{87} -7.55160i q^{89} -8.92033i q^{91} +(6.90968 + 5.58126i) q^{93} +12.5033 q^{95} -0.448156 q^{97} +(-11.9010 + 2.56030i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 8 q^{9}+O(q^{10}) \) 56 * q - 8 * q^9	\( 56 q - 8 q^{9} - 4 q^{19} + 4 q^{21} + 52 q^{25} + 12 q^{27} + 4 q^{31} - 8 q^{33} + 32 q^{49} + 24 q^{57} + 28 q^{61} - 26 q^{63} - 54 q^{75} - 24 q^{81} + 8 q^{85} + 14 q^{87} - 20 q^{93} + 36 q^{97} + 32 q^{99}+O(q^{100}) \) 56 * q - 8 * q^9 - 4 * q^19 + 4 * q^21 + 52 * q^25 + 12 * q^27 + 4 * q^31 - 8 * q^33 + 32 * q^49 + 24 * q^57 + 28 * q^61 - 26 * q^63 - 54 * q^75 - 24 * q^81 + 8 * q^85 + 14 * q^87 - 20 * q^93 + 36 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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