LMFDB - Embedded newform 1692.2.h.a.845.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1692,2,Mod(845,1692)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1692, 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("1692.845");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1692 = 2^{2} \cdot 3^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1692.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:	$13.5106880220$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26 x^{14} - 16 x^{13} + 269 x^{12} + 288 x^{11} + 850 x^{10} + 2032 x^{9} + 6628 x^{8} + \cdots + 253609 $ x^16 - 26x^14 - 16x^13 + 269x^12 + 288x^11 + 850x^10 + 2032x^9 + 6628x^8 + 7312x^7 + 53790x^6 + 18944x^5 + 201613x^4 + 27344x^3 + 355010x^2 + 13984x + 253609
Coefficient ring:	$\Z[a_1, \ldots, a_{47}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			845.13
Root			$-3.79828 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1692.845
Dual form			1692.2.h.a.845.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84774 q^{5} -3.53500 q^{7} +O(q^{10})$	$q+2.84774 q^{5} -3.53500 q^{7} +4.06156 q^{11} -4.24632i q^{13} -5.67895i q^{17} +0.496609i q^{19} -3.35925 q^{23} +3.10961 q^{25} -3.85978 q^{29} -0.715621i q^{31} -10.0668 q^{35} +10.1409 q^{37} +0.201778 q^{41} -9.49362i q^{43} +(3.15747 - 6.08526i) q^{47} +5.49625 q^{49} -3.66314i q^{53} +11.5663 q^{55} +9.26399i q^{59} +8.60586 q^{61} -12.0924i q^{65} +13.8063i q^{67} -4.39766i q^{71} -7.98827i q^{73} -14.3576 q^{77} +5.53500 q^{79} +2.85052i q^{83} -16.1722i q^{85} -7.77286i q^{89} +15.0108i q^{91} +1.41421i q^{95} -4.96039 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) $ 16 * q + 16 * q^25 + 8 * q^37 + 24 * q^49 + 8 * q^55 + 40 * q^61 + 32 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1692\mathbb{Z}\right)^\times$.

$n$	$757$	$847$	$1505$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.84774			1.27355			0.636774	−	0.771051i	$-0.280269\pi$
$5$	2.84774			1.27355			0.636774	+	0.771051i	$0.280269\pi$
$6$		0			0
$7$	−3.53500			−1.33611			−0.668053	−	0.744114i	$-0.732872\pi$
$7$	−3.53500			−1.33611			−0.668053	+	0.744114i	$0.732872\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.06156			1.22461			0.612303	−	0.790623i	$-0.290243\pi$
$11$	4.06156			1.22461			0.612303	+	0.790623i	$0.290243\pi$
$12$		0			0
$13$		−	4.24632i		−	1.17772i	−0.808236	−	0.588859i	$-0.799577\pi$
$13$		−	4.24632i		−	1.17772i	0.808236	−	0.588859i	$-0.200423\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.67895i		−	1.37735i	−0.725071	−	0.688674i	$-0.758193\pi$
$17$		−	5.67895i		−	1.37735i	0.725071	−	0.688674i	$-0.241807\pi$
$18$		0			0
$19$			0.496609i			0.113930i	0.998376	+	0.0569650i	$0.0181423\pi$
$19$			0.496609i			0.113930i	−0.998376	+	0.0569650i	$0.981858\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.35925			−0.700451			−0.350226	−	0.936665i	$-0.613895\pi$
$23$	−3.35925			−0.700451			−0.350226	+	0.936665i	$0.613895\pi$
$24$		0			0
$25$	3.10961			0.621923
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.85978			−0.716743			−0.358372	−	0.933579i	$-0.616668\pi$
$29$	−3.85978			−0.716743			−0.358372	+	0.933579i	$0.616668\pi$
$30$		0			0
$31$		−	0.715621i		−	0.128529i	−0.997933	−	0.0642647i	$-0.979530\pi$
$31$		−	0.715621i		−	0.128529i	0.997933	−	0.0642647i	$-0.0204702\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.0668			−1.70159
$36$		0			0
$37$	10.1409			1.66715			0.833574	−	0.552408i	$-0.186291\pi$
$37$	10.1409			1.66715			0.833574	+	0.552408i	$0.186291\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.201778			0.0315124			0.0157562	−	0.999876i	$-0.494984\pi$
$41$	0.201778			0.0315124			0.0157562	+	0.999876i	$0.494984\pi$
$42$		0			0
$43$		−	9.49362i		−	1.44776i	−0.689924	−	0.723882i	$-0.742356\pi$
$43$		−	9.49362i		−	1.44776i	0.689924	−	0.723882i	$-0.257644\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.15747	−	6.08526i	0.460564	−	0.887627i
$47$	3.15747	−	6.08526i	0.460564	−	0.887627i
$48$		0			0
$49$	5.49625			0.785178
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.66314i		−	0.503172i	−0.967835	−	0.251586i	$-0.919048\pi$
$53$		−	3.66314i		−	0.503172i	0.967835	−	0.251586i	$-0.0809521\pi$
$54$		0			0
$55$	11.5663			1.55959
$56$		0			0
$57$		0			0
$58$		0			0
$59$			9.26399i			1.20607i	0.797715	+	0.603034i	$0.206042\pi$
$59$			9.26399i			1.20607i	−0.797715	+	0.603034i	$0.793958\pi$
$60$		0			0
$61$	8.60586			1.10187			0.550934	−	0.834549i	$-0.314272\pi$
$61$	8.60586			1.10187			0.550934	+	0.834549i	$0.314272\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	12.0924i		−	1.49988i
$66$		0			0
$67$			13.8063i			1.68671i	0.537360	+	0.843353i	$0.319422\pi$
$67$			13.8063i			1.68671i	−0.537360	+	0.843353i	$0.680578\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.39766i		−	0.521906i	−0.965352	−	0.260953i	$-0.915963\pi$
$71$		−	4.39766i		−	0.521906i	0.965352	−	0.260953i	$-0.0840367\pi$
$72$		0			0
$73$		−	7.98827i		−	0.934957i	−0.884004	−	0.467478i	$-0.845163\pi$
$73$		−	7.98827i		−	0.934957i	0.884004	−	0.467478i	$-0.154837\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.3576			−1.63620
$78$		0			0
$79$	5.53500			0.622736			0.311368	−	0.950289i	$-0.399213\pi$
$79$	5.53500			0.622736			0.311368	+	0.950289i	$0.399213\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			2.85052i			0.312886i	0.987687	+	0.156443i	$0.0500027\pi$
$83$			2.85052i			0.312886i	−0.987687	+	0.156443i	$0.949997\pi$
$84$		0			0
$85$		−	16.1722i		−	1.75412i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	7.77286i		−	0.823922i	−0.911201	−	0.411961i	$-0.864844\pi$
$89$		−	7.77286i		−	0.823922i	0.911201	−	0.411961i	$-0.135156\pi$
$90$		0			0
$91$			15.0108i			1.57356i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1.41421i			0.145095i
$96$		0			0
$97$	−4.96039			−0.503652			−0.251826	−	0.967773i	$-0.581031\pi$
$97$	−4.96039			−0.503652			−0.251826	+	0.967773i	$0.581031\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	9.39691i		−	0.935027i	−0.883986	−	0.467514i	$-0.845150\pi$
$101$		−	9.39691i		−	0.935027i	0.883986	−	0.467514i	$-0.154850\pi$
$102$		0			0
$103$	2.49625			0.245962			0.122981	−	0.992409i	$-0.460755\pi$
$103$	2.49625			0.245962			0.122981	+	0.992409i	$0.460755\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.65117			−0.836340			−0.418170	−	0.908369i	$-0.637328\pi$
$107$	−8.65117			−0.836340			−0.418170	+	0.908369i	$0.637328\pi$
$108$		0			0
$109$		−	0.942392i		−	0.0902648i	−0.998981	−	0.0451324i	$-0.985629\pi$
$109$		−	0.942392i		−	0.0902648i	0.998981	−	0.0451324i	$-0.0143710\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.3191			1.34703			0.673515	−	0.739174i	$-0.264784\pi$
$113$	14.3191			1.34703			0.673515	+	0.739174i	$0.264784\pi$
$114$		0			0
$115$	−9.56625			−0.892058
$116$		0			0
$117$		0			0
$118$		0			0
$119$			20.0751i			1.84028i
$120$		0			0
$121$	5.49625			0.499659
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.38333			−0.481499
$126$		0			0
$127$		−	11.9844i		−	1.06345i	−0.846918	−	0.531723i	$-0.821545\pi$
$127$		−	11.9844i		−	1.06345i	0.846918	−	0.531723i	$-0.178455\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.56848i			0.573891i	0.957947	+	0.286945i	$0.0926398\pi$
$131$			6.56848i			0.573891i	−0.957947	+	0.286945i	$0.907360\pi$
$132$		0			0
$133$		−	1.75552i		−	0.152223i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.33693			−0.797708			−0.398854	−	0.917014i	$-0.630592\pi$
$137$	−9.33693			−0.797708			−0.398854	+	0.917014i	$0.630592\pi$
$138$		0			0
$139$			7.38640i			0.626507i	0.949670	+	0.313253i	$0.101419\pi$
$139$			7.38640i			0.626507i	−0.949670	+	0.313253i	$0.898581\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	17.2467i		−	1.44224i
$144$		0			0
$145$	−10.9916			−0.912806
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.58428i			0.703252i	0.936141	+	0.351626i	$0.114371\pi$
$149$			8.58428i			0.703252i	−0.936141	+	0.351626i	$0.885629\pi$
$150$		0			0
$151$		−	15.3273i		−	1.24732i	−0.781697	−	0.623658i	$-0.785646\pi$
$151$		−	15.3273i		−	1.24732i	0.781697	−	0.623658i	$-0.214354\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.03790i		−	0.163688i
$156$		0			0
$157$	21.5975			1.72367			0.861834	−	0.507190i	$-0.169316\pi$
$157$	21.5975			1.72367			0.861834	+	0.507190i	$0.169316\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	11.8749			0.935877
$162$		0			0
$163$			21.7946i			1.70708i	0.521026	+	0.853541i	$0.325550\pi$
$163$			21.7946i			1.70708i	−0.521026	+	0.853541i	$0.674450\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	10.3875			0.803806			0.401903	−	0.915682i	$-0.368349\pi$
$167$	10.3875			0.803806			0.401903	+	0.915682i	$0.368349\pi$
$168$		0			0
$169$	−5.03125			−0.387019
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.9757i			1.13858i	0.822138	+	0.569289i	$0.192781\pi$
$173$			14.9757i			1.13858i	−0.822138	+	0.569289i	$0.807219\pi$
$174$		0			0
$175$	−10.9925			−0.830954
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.78688			−0.432532			−0.216266	−	0.976334i	$-0.569388\pi$
$179$	−5.78688			−0.432532			−0.216266	+	0.976334i	$0.569388\pi$
$180$		0			0
$181$		−	7.98827i		−	0.593764i	−0.954914	−	0.296882i	$-0.904053\pi$
$181$		−	7.98827i		−	0.593764i	0.954914	−	0.296882i	$-0.0959468\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	28.8785			2.12319
$186$		0			0
$187$		−	23.0654i		−	1.68671i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.5780i		−	0.765397i	−0.923873	−	0.382698i	$-0.874995\pi$
$191$		−	10.5780i		−	0.765397i	0.923873	−	0.382698i	$-0.125005\pi$
$192$		0			0
$193$		−	4.96970i		−	0.357727i	−0.983874	−	0.178863i	$-0.942758\pi$
$193$		−	4.96970i		−	0.357727i	0.983874	−	0.178863i	$-0.0572420\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	27.1462i		−	1.93408i	−0.254616	−	0.967042i	$-0.581949\pi$
$197$		−	27.1462i		−	1.93408i	0.254616	−	0.967042i	$-0.418051\pi$
$198$		0			0
$199$			22.7370i			1.61178i	0.592065	+	0.805890i	$0.298313\pi$
$199$			22.7370i			1.61178i	−0.592065	+	0.805890i	$0.701687\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.6443			0.957644
$204$		0			0
$205$	0.574610			0.0401325
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.01701i			0.139519i
$210$		0			0
$211$			8.59015i			0.591370i	0.955285	+	0.295685i	$0.0955479\pi$
$211$			8.59015i			0.591370i	−0.955285	+	0.295685i	$0.904452\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	27.0353i		−	1.84380i
$216$		0			0
$217$			2.52972i			0.171729i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−24.1146			−1.62213
$222$		0			0
$223$		−	26.0683i		−	1.74566i	−0.488021	−	0.872832i	$-0.662281\pi$
$223$		−	26.0683i		−	1.74566i	0.488021	−	0.872832i	$-0.337719\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.3885			0.755883			0.377942	−	0.925829i	$-0.376632\pi$
$227$	11.3885			0.755883			0.377942	+	0.925829i	$0.376632\pi$
$228$		0			0
$229$			25.0165i			1.65314i	0.562836	+	0.826569i	$0.309710\pi$
$229$			25.0165i			1.65314i	−0.562836	+	0.826569i	$0.690290\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−27.2611			−1.78594			−0.892968	−	0.450119i	$-0.851381\pi$
$233$	−27.2611			−1.78594			−0.892968	+	0.450119i	$0.851381\pi$
$234$		0			0
$235$	8.99164	−	17.3292i	0.586550	−	1.13043i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			0.00120161i			7.77256e-5i	1.00000		3.88628e-5i	$1.23704e-5\pi$
$239$			0.00120161i			7.77256e-5i	−1.00000		3.88628e-5i	$0.999988\pi$
$240$		0			0
$241$	10.3858			0.669007			0.334504	−	0.942394i	$-0.391431\pi$
$241$	10.3858			0.669007			0.334504	+	0.942394i	$0.391431\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.6519			0.999961
$246$		0			0
$247$	2.10876			0.134177
$248$		0			0
$249$		0			0
$250$		0			0
$251$			14.9757i			0.945255i	0.881262	+	0.472627i	$0.156694\pi$
$251$			14.9757i			0.945255i	−0.881262	+	0.472627i	$0.843306\pi$
$252$		0			0
$253$	−13.6438			−0.857776
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.431074			0.0268896			0.0134448	−	0.999910i	$-0.495720\pi$
$257$	0.431074			0.0268896			0.0134448	+	0.999910i	$0.495720\pi$
$258$		0			0
$259$	−35.8480			−2.22748
$260$		0			0
$261$		0			0
$262$		0			0
$263$			26.1237i			1.61086i	0.592692	+	0.805429i	$0.298065\pi$
$263$			26.1237i			1.61086i	−0.592692	+	0.805429i	$0.701935\pi$
$264$		0			0
$265$		−	10.4317i		−	0.640813i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			26.4547i			1.61297i	0.591256	+	0.806484i	$0.298632\pi$
$269$			26.4547i			1.61297i	−0.591256	+	0.806484i	$0.701368\pi$
$270$		0			0
$271$	−19.5663			−1.18857			−0.594283	−	0.804256i	$-0.702564\pi$
$271$	−19.5663			−1.18857			−0.594283	+	0.804256i	$0.702564\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.6299			0.761610
$276$		0			0
$277$	32.8084			1.97126			0.985632	−	0.168908i	$-0.0540240\pi$
$277$	32.8084			1.97126			0.985632	+	0.168908i	$0.0540240\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.636004			0.0379408			0.0189704	−	0.999820i	$-0.493961\pi$
$281$	0.636004			0.0379408			0.0189704	+	0.999820i	$0.493961\pi$
$282$		0			0
$283$	−4.49625			−0.267274			−0.133637	−	0.991030i	$-0.542666\pi$
$283$	−4.49625			−0.267274			−0.133637	+	0.991030i	$0.542666\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.713285			−0.0421039
$288$		0			0
$289$	−15.2505			−0.897087
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−20.5646			−1.20140			−0.600698	−	0.799476i	$-0.705111\pi$
$293$	−20.5646			−1.20140			−0.600698	+	0.799476i	$0.705111\pi$
$294$		0			0
$295$			26.3814i			1.53598i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			14.2644i			0.824934i
$300$		0			0
$301$			33.5600i			1.93437i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	24.5072			1.40328
$306$		0			0
$307$	4.46330			0.254734			0.127367	−	0.991856i	$-0.459347\pi$
$307$	4.46330			0.254734			0.127367	+	0.991856i	$0.459347\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.88178			−0.390230			−0.195115	−	0.980780i	$-0.562508\pi$
$311$	−6.88178			−0.390230			−0.195115	+	0.980780i	$0.562508\pi$
$312$		0			0
$313$			8.36338i			0.472726i	0.971665	+	0.236363i	$0.0759554\pi$
$313$			8.36338i			0.472726i	−0.971665	+	0.236363i	$0.924045\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−35.0745			−1.96998			−0.984991	−	0.172606i	$-0.944781\pi$
$317$	−35.0745			−1.96998			−0.984991	+	0.172606i	$0.944781\pi$
$318$		0			0
$319$	−15.6767			−0.877727
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.82022			0.156921
$324$		0			0
$325$		−	13.2044i		−	0.732449i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.1617	+	21.5114i	−0.615362	+	1.18596i
$330$		0			0
$331$	−24.3905			−1.34062			−0.670311	−	0.742080i	$-0.733839\pi$
$331$	−24.3905			−1.34062			−0.670311	+	0.742080i	$0.733839\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			39.3167i			2.14810i
$336$		0			0
$337$	0.573760			0.0312547			0.0156274	−	0.999878i	$-0.495025\pi$
$337$	0.573760			0.0312547			0.0156274	+	0.999878i	$0.495025\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	2.90654i		−	0.157398i
$342$		0			0
$343$	5.31578			0.287025
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.20279i		−	0.386666i	−0.981133	−	0.193333i	$-0.938070\pi$
$347$		−	7.20279i		−	0.386666i	0.981133	−	0.193333i	$-0.0619297\pi$
$348$		0			0
$349$			32.7703i			1.75415i	0.480352	+	0.877076i	$0.340509\pi$
$349$			32.7703i			1.75415i	−0.480352	+	0.877076i	$0.659491\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			24.4400i			1.30081i	0.759587	+	0.650406i	$0.225401\pi$
$353$			24.4400i			1.30081i	−0.759587	+	0.650406i	$0.774599\pi$
$354$		0			0
$355$		−	12.5234i		−	0.664672i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.20931			0.116603			0.0583016	−	0.998299i	$-0.481431\pi$
$359$	2.20931			0.116603			0.0583016	+	0.998299i	$0.481431\pi$
$360$		0			0
$361$	18.7534			0.987020
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	22.7485i		−	1.19071i
$366$		0			0
$367$			29.7828i			1.55465i	0.629098	+	0.777326i	$0.283424\pi$
$367$			29.7828i			1.55465i	−0.629098	+	0.777326i	$0.716576\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			12.9492i			0.672290i
$372$		0			0
$373$			31.9097i			1.65222i	0.563507	+	0.826111i	$0.309452\pi$
$373$			31.9097i			1.65222i	−0.563507	+	0.826111i	$0.690548\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.3899i			0.844121i
$378$		0			0
$379$	−35.8167			−1.83978			−0.919891	−	0.392175i	$-0.871723\pi$
$379$	−35.8167			−1.83978			−0.919891	+	0.392175i	$0.871723\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.1796i		−	0.571250i	−0.958341	−	0.285625i	$-0.907799\pi$
$383$		−	11.1796i		−	0.571250i	0.958341	−	0.285625i	$-0.0922012\pi$
$384$		0			0
$385$	−40.8867			−2.08378
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.1703			−0.769165			−0.384582	−	0.923091i	$-0.625655\pi$
$389$	−15.1703			−0.769165			−0.384582	+	0.923091i	$0.625655\pi$
$390$		0			0
$391$			19.0770i			0.964765i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.7622			0.793084
$396$		0			0
$397$	−13.2100			−0.662992			−0.331496	−	0.943457i	$-0.607553\pi$
$397$	−13.2100			−0.662992			−0.331496	+	0.943457i	$0.607553\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.3509i			0.916398i	0.888850	+	0.458199i	$0.151505\pi$
$401$			18.3509i			0.916398i	−0.888850	+	0.458199i	$0.848495\pi$
$402$		0			0
$403$	−3.03876			−0.151371
$404$		0			0
$405$		0			0
$406$		0			0
$407$	41.1877			2.04160
$408$		0			0
$409$			9.87306i			0.488191i	0.969751	+	0.244096i	$0.0784911\pi$
$409$			9.87306i			0.488191i	−0.969751	+	0.244096i	$0.921509\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	32.7482i		−	1.61143i
$414$		0			0
$415$			8.11754i			0.398475i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	21.9943			1.07449			0.537246	−	0.843425i	$-0.319465\pi$
$419$	21.9943			1.07449			0.537246	+	0.843425i	$0.319465\pi$
$420$		0			0
$421$		−	27.2962i		−	1.33033i	−0.746694	−	0.665167i	$-0.768360\pi$
$421$		−	27.2962i		−	1.33033i	0.746694	−	0.665167i	$-0.231640\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	17.6593i		−	0.856604i
$426$		0			0
$427$	−30.4217			−1.47221
$428$		0			0
$429$		0			0
$430$		0			0
$431$			11.3894i			0.548609i	0.961643	+	0.274304i	$0.0884476\pi$
$431$			11.3894i			0.548609i	−0.961643	+	0.274304i	$0.911552\pi$
$432$		0			0
$433$			14.1468i			0.679852i	0.940452	+	0.339926i	$0.110402\pi$
$433$			14.1468i			0.679852i	−0.940452	+	0.339926i	$0.889598\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1.66823i		−	0.0798024i
$438$		0			0
$439$	−0.418730			−0.0199849			−0.00999245	−	0.999950i	$-0.503181\pi$
$439$	−0.418730			−0.0199849			−0.00999245	+	0.999950i	$0.503181\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.8222			1.46441			0.732203	−	0.681087i	$-0.238492\pi$
$443$	30.8222			1.46441			0.732203	+	0.681087i	$0.238492\pi$
$444$		0			0
$445$		−	22.1351i		−	1.04930i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7.94570			−0.374981			−0.187490	−	0.982266i	$-0.560035\pi$
$449$	−7.94570			−0.374981			−0.187490	+	0.982266i	$0.560035\pi$
$450$		0			0
$451$	0.819532			0.0385902
$452$		0			0
$453$		0			0
$454$		0			0
$455$			42.7467i			2.00400i
$456$		0			0
$457$	17.3509			0.811640			0.405820	−	0.913953i	$-0.366986\pi$
$457$	17.3509			0.811640			0.405820	+	0.913953i	$0.366986\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	32.2543			1.50223			0.751116	−	0.660170i	$-0.229516\pi$
$461$	32.2543			1.50223			0.751116	+	0.660170i	$0.229516\pi$
$462$		0			0
$463$		−	32.9496i		−	1.53130i	−0.643258	−	0.765649i	$-0.722418\pi$
$463$		−	32.9496i		−	1.53130i	0.643258	−	0.765649i	$-0.277582\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.6862			−0.494499			−0.247250	−	0.968952i	$-0.579527\pi$
$467$	−10.6862			−0.494499			−0.247250	+	0.968952i	$0.579527\pi$
$468$		0			0
$469$		−	48.8053i		−	2.25362i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	38.5589i		−	1.77294i
$474$		0			0
$475$			1.54426i			0.0708556i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	10.4230i		−	0.476238i	−0.971236	−	0.238119i	$-0.923469\pi$
$479$		−	10.4230i		−	0.476238i	0.971236	−	0.238119i	$-0.0765308\pi$
$480$		0			0
$481$		−	43.0614i		−	1.96343i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.1259			−0.641424
$486$		0			0
$487$	4.77326			0.216297			0.108149	−	0.994135i	$-0.465508\pi$
$487$	4.77326			0.216297			0.108149	+	0.994135i	$0.465508\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.70613i		−	0.167256i	−0.996497	−	0.0836278i	$-0.973349\pi$
$491$		−	3.70613i		−	0.167256i	0.996497	−	0.0836278i	$-0.0266507\pi$
$492$		0			0
$493$			21.9195i			0.987204i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			15.5457i			0.697321i
$498$		0			0
$499$			16.5784i			0.742152i	0.928602	+	0.371076i	$0.121011\pi$
$499$			16.5784i			0.742152i	−0.928602	+	0.371076i	$0.878989\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−36.0566			−1.60769			−0.803843	−	0.594842i	$-0.797215\pi$
$503$	−36.0566			−1.60769			−0.803843	+	0.594842i	$0.797215\pi$
$504$		0			0
$505$		−	26.7599i		−	1.19080i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	36.0281			1.59692			0.798458	−	0.602050i	$-0.205649\pi$
$509$	36.0281			1.59692			0.798458	+	0.602050i	$0.205649\pi$
$510$		0			0
$511$			28.2386i			1.24920i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.10865			0.313245
$516$		0			0
$517$	12.8242	−	24.7156i	0.564009	−	1.08699i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	16.5670i		−	0.725812i	−0.931826	−	0.362906i	$-0.881785\pi$
$521$		−	16.5670i		−	0.725812i	0.931826	−	0.362906i	$-0.118215\pi$
$522$		0			0
$523$	29.0435			1.26998			0.634991	−	0.772520i	$-0.281004\pi$
$523$	29.0435			1.26998			0.634991	+	0.772520i	$0.281004\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.06398			−0.177030
$528$		0			0
$529$	−11.7155			−0.509368
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.856813i		−	0.0371127i
$534$		0			0
$535$	−24.6363			−1.06512
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.3233			0.961533
$540$		0			0
$541$	36.6410			1.57532			0.787659	−	0.616111i	$-0.211293\pi$
$541$	36.6410			1.57532			0.787659	+	0.616111i	$0.211293\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	2.68369i		−	0.114956i
$546$		0			0
$547$			12.6570i			0.541173i	0.962696	+	0.270587i	$0.0872177\pi$
$547$			12.6570i			0.541173i	−0.962696	+	0.270587i	$0.912782\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.91680i		−	0.0816585i
$552$		0			0
$553$	−19.5663			−0.832042
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.64121			−0.281397			−0.140699	−	0.990052i	$-0.544935\pi$
$557$	−6.64121			−0.281397			−0.140699	+	0.990052i	$0.544935\pi$
$558$		0			0
$559$	−40.3130			−1.70506
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−34.4716			−1.45281			−0.726403	−	0.687269i	$-0.758809\pi$
$563$	−34.4716			−1.45281			−0.726403	+	0.687269i	$0.758809\pi$
$564$		0			0
$565$	40.7771			1.71551
$566$		0			0
$567$		0			0
$568$		0			0
$569$	34.2894			1.43748			0.718742	−	0.695276i	$-0.244718\pi$
$569$	34.2894			1.43748			0.718742	+	0.695276i	$0.244718\pi$
$570$		0			0
$571$	−8.74032			−0.365771			−0.182885	−	0.983134i	$-0.558544\pi$
$571$	−8.74032			−0.365771			−0.182885	+	0.983134i	$0.558544\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.4460			−0.435626
$576$		0			0
$577$		−	13.4658i		−	0.560587i	−0.959914	−	0.280294i	$-0.909568\pi$
$577$		−	13.4658i		−	0.560587i	0.959914	−	0.280294i	$-0.0904318\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	10.0766i		−	0.418048i
$582$		0			0
$583$		−	14.8781i		−	0.616187i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.2881			−0.548459			−0.274230	−	0.961664i	$-0.588423\pi$
$587$	−13.2881			−0.548459			−0.274230	+	0.961664i	$0.588423\pi$
$588$		0			0
$589$	0.355384			0.0146433
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.40881			0.386374			0.193187	−	0.981162i	$-0.438118\pi$
$593$	9.40881			0.386374			0.193187	+	0.981162i	$0.438118\pi$
$594$		0			0
$595$			57.1686i			2.34369i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−33.9655			−1.38779			−0.693897	−	0.720075i	$-0.744108\pi$
$599$	−33.9655			−1.38779			−0.693897	+	0.720075i	$0.744108\pi$
$600$		0			0
$601$	−7.11712			−0.290313			−0.145157	−	0.989409i	$-0.546369\pi$
$601$	−7.11712			−0.290313			−0.145157	+	0.989409i	$0.546369\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15.6519			0.636339
$606$		0			0
$607$		−	8.59015i		−	0.348663i	−0.984687	−	0.174332i	$-0.944224\pi$
$607$		−	8.59015i		−	0.348663i	0.984687	−	0.174332i	$-0.0557765\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−25.8400	−	13.4076i	−1.04537	−	0.542414i
$612$		0			0
$613$	29.8959			1.20749			0.603743	−	0.797179i	$-0.293675\pi$
$613$	29.8959			1.20749			0.603743	+	0.797179i	$0.293675\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.4587i			1.54829i	0.633009	+	0.774144i	$0.281819\pi$
$617$			38.4587i			1.54829i	−0.633009	+	0.774144i	$0.718181\pi$
$618$		0			0
$619$	27.5958			1.10917			0.554584	−	0.832127i	$-0.312877\pi$
$619$	27.5958			1.10917			0.554584	+	0.832127i	$0.312877\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			27.4771i			1.10085i
$624$		0			0
$625$	−30.8784			−1.23513
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	57.5894i		−	2.29624i
$630$		0			0
$631$			7.38983i			0.294184i	0.989123	+	0.147092i	$0.0469914\pi$
$631$			7.38983i			0.294184i	−0.989123	+	0.147092i	$0.953009\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	34.1285i		−	1.35435i
$636$		0			0
$637$		−	23.3388i		−	0.924718i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−29.4509			−1.16324			−0.581621	−	0.813460i	$-0.697581\pi$
$641$	−29.4509			−1.16324			−0.581621	+	0.813460i	$0.697581\pi$
$642$		0			0
$643$	36.9445			1.45695			0.728475	−	0.685072i	$-0.240229\pi$
$643$	36.9445			1.45695			0.728475	+	0.685072i	$0.240229\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	31.2129i		−	1.22711i	−0.789654	−	0.613553i	$-0.789740\pi$
$647$		−	31.2129i		−	1.22711i	0.789654	−	0.613553i	$-0.210260\pi$
$648$		0			0
$649$			37.6262i			1.47696i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			14.2841i			0.558981i	0.960148	+	0.279491i	$0.0901656\pi$
$653$			14.2841i			0.558981i	−0.960148	+	0.279491i	$0.909834\pi$
$654$		0			0
$655$			18.7053i			0.730877i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			29.1305i			1.13476i	0.823455	+	0.567381i	$0.192043\pi$
$659$			29.1305i			1.13476i	−0.823455	+	0.567381i	$0.807957\pi$
$660$		0			0
$661$	−18.7921			−0.730929			−0.365465	−	0.930825i	$-0.619090\pi$
$661$	−18.7921			−0.730929			−0.365465	+	0.930825i	$0.619090\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	4.99925i		−	0.193863i
$666$		0			0
$667$	12.9659			0.502043
$668$		0			0
$669$		0			0
$670$		0			0
$671$	34.9532			1.34935
$672$		0			0
$673$			37.9779i			1.46394i	0.681336	+	0.731970i	$0.261399\pi$
$673$			37.9779i			1.46394i	−0.681336	+	0.731970i	$0.738601\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.10902			−0.0426231			−0.0213115	−	0.999773i	$-0.506784\pi$
$677$	−1.10902			−0.0426231			−0.0213115	+	0.999773i	$0.506784\pi$
$678$		0			0
$679$	17.5350			0.672932
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1.91440i			0.0732525i	0.999329	+	0.0366262i	$0.0116611\pi$
$683$			1.91440i			0.0732525i	−0.999329	+	0.0366262i	$0.988339\pi$
$684$		0			0
$685$	−26.5891			−1.01592
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.5549			−0.592594
$690$		0			0
$691$		−	37.7632i		−	1.43658i	−0.695743	−	0.718291i	$-0.744925\pi$
$691$		−	37.7632i		−	1.43658i	0.695743	−	0.718291i	$-0.255075\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			21.0345i			0.797886i
$696$		0			0
$697$		−	1.14589i		−	0.0434035i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.73877			0.178981			0.0894904	−	0.995988i	$-0.471476\pi$
$701$	4.73877			0.178981			0.0894904	+	0.995988i	$0.471476\pi$
$702$		0			0
$703$			5.03605i			0.189938i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			33.2181i			1.24929i
$708$		0			0
$709$	28.9766			1.08824			0.544120	−	0.839007i	$-0.316864\pi$
$709$	28.9766			1.08824			0.544120	+	0.839007i	$0.316864\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.40395i			0.0900285i
$714$		0			0
$715$		−	49.1140i		−	1.83676i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1.92345i		−	0.0717327i	−0.999357	−	0.0358664i	$-0.988581\pi$
$719$		−	1.92345i		−	0.0717327i	0.999357	−	0.0358664i	$-0.0114191\pi$
$720$		0			0
$721$	−8.82423			−0.328632
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12.0024			−0.445759
$726$		0			0
$727$			26.3616i			0.977696i	0.872369	+	0.488848i	$0.162583\pi$
$727$			26.3616i			0.977696i	−0.872369	+	0.488848i	$0.837417\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−53.9138			−1.99407
$732$		0			0
$733$	41.9272			1.54862			0.774308	−	0.632809i	$-0.218098\pi$
$733$	41.9272			1.54862			0.774308	+	0.632809i	$0.218098\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			56.0750i			2.06555i
$738$		0			0
$739$	1.26951			0.0466997			0.0233498	−	0.999727i	$-0.492567\pi$
$739$	1.26951			0.0466997			0.0233498	+	0.999727i	$0.492567\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	8.22791			0.301853			0.150926	−	0.988545i	$-0.451774\pi$
$743$	8.22791			0.301853			0.150926	+	0.988545i	$0.451774\pi$
$744$		0			0
$745$			24.4458i			0.895625i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.5819			1.11744
$750$		0			0
$751$		−	12.8209i		−	0.467843i	−0.972255	−	0.233922i	$-0.924844\pi$
$751$		−	12.8209i		−	0.467843i	0.972255	−	0.233922i	$-0.0751559\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	43.6481i		−	1.58852i
$756$		0			0
$757$			7.35867i			0.267455i	0.991018	+	0.133728i	$0.0426947\pi$
$757$			7.35867i			0.267455i	−0.991018	+	0.133728i	$0.957305\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	29.7754i		−	1.07936i	−0.841871	−	0.539679i	$-0.818546\pi$
$761$		−	29.7754i		−	1.07936i	0.841871	−	0.539679i	$-0.181454\pi$
$762$		0			0
$763$			3.33136i			0.120603i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	39.3379			1.42041
$768$		0			0
$769$	−27.6296			−0.996349			−0.498174	−	0.867077i	$-0.665996\pi$
$769$	−27.6296			−0.996349			−0.498174	+	0.867077i	$0.665996\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	15.8767i		−	0.571044i	−0.958372	−	0.285522i	$-0.907833\pi$
$773$		−	15.8767i		−	0.571044i	0.958372	−	0.285522i	$-0.0921669\pi$
$774$		0			0
$775$		−	2.22530i		−	0.0799353i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			0.100205i			0.00359021i
$780$		0			0
$781$		−	17.8613i		−	0.639129i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	61.5040			2.19517
$786$		0			0
$787$		−	6.44401i		−	0.229704i	−0.993383	−	0.114852i	$-0.963361\pi$
$787$		−	6.44401i		−	0.229704i	0.993383	−	0.114852i	$-0.0366394\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−50.6181			−1.79977
$792$		0			0
$793$		−	36.5432i		−	1.29769i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−34.3832			−1.21792			−0.608958	−	0.793203i	$-0.708412\pi$
$797$	−34.3832			−1.21792			−0.608958	+	0.793203i	$0.708412\pi$
$798$		0			0
$799$	−34.5579	−	17.9311i	−1.22257	−	0.634357i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	32.4448i		−	1.14495i
$804$		0			0
$805$	33.8167			1.19188
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−46.6924			−1.64162			−0.820808	−	0.571204i	$-0.806476\pi$
$809$	−46.6924			−1.64162			−0.820808	+	0.571204i	$0.806476\pi$
$810$		0			0
$811$	27.1060			0.951819			0.475909	−	0.879494i	$-0.342119\pi$
$811$	27.1060			0.951819			0.475909	+	0.879494i	$0.342119\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			62.0652i			2.17405i
$816$		0			0
$817$	4.71462			0.164944
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−34.3257			−1.19798			−0.598988	−	0.800758i	$-0.704430\pi$
$821$	−34.3257			−1.19798			−0.598988	+	0.800758i	$0.704430\pi$
$822$		0			0
$823$	44.7629			1.56034			0.780169	−	0.625569i	$-0.215133\pi$
$823$	44.7629			1.56034			0.780169	+	0.625569i	$0.215133\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	4.93382i		−	0.171566i	−0.996314	−	0.0857828i	$-0.972661\pi$
$827$		−	4.93382i		−	0.171566i	0.996314	−	0.0857828i	$-0.0273391\pi$
$828$		0			0
$829$		−	44.7384i		−	1.55383i	−0.629605	−	0.776915i	$-0.716783\pi$
$829$		−	44.7384i		−	1.55383i	0.629605	−	0.776915i	$-0.283217\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	31.2129i		−	1.08146i
$834$		0			0
$835$	29.5808			1.02369
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.0834			−1.24574			−0.622869	−	0.782326i	$-0.714033\pi$
$839$	−36.0834			−1.24574			−0.622869	+	0.782326i	$0.714033\pi$
$840$		0			0
$841$	−14.1021			−0.486279
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.3277			−0.492887
$846$		0			0
$847$	−19.4292			−0.667597
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−34.0656			−1.16776
$852$		0			0
$853$	−6.99079			−0.239360			−0.119680	−	0.992813i	$-0.538187\pi$
$853$	−6.99079			−0.239360			−0.119680	+	0.992813i	$0.538187\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.94263			−0.134678			−0.0673389	−	0.997730i	$-0.521451\pi$
$857$	−3.94263			−0.134678			−0.0673389	+	0.997730i	$0.521451\pi$
$858$		0			0
$859$		−	19.6210i		−	0.669460i	−0.942314	−	0.334730i	$-0.891355\pi$
$859$		−	19.6210i		−	0.669460i	0.942314	−	0.334730i	$-0.108645\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	57.2873i		−	1.95008i	−0.222027	−	0.975041i	$-0.571267\pi$
$863$		−	57.2873i		−	1.95008i	0.222027	−	0.975041i	$-0.428733\pi$
$864$		0			0
$865$			42.6467i			1.45003i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	22.4807			0.762607
$870$		0			0
$871$	58.6259			1.98646
$872$		0			0
$873$		0			0
$874$		0			0
$875$	19.0301			0.643334
$876$		0			0
$877$		−	10.8155i		−	0.365212i	−0.983186	−	0.182606i	$-0.941547\pi$
$877$		−	10.8155i		−	0.365212i	0.983186	−	0.182606i	$-0.0584532\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	50.0989			1.68788			0.843938	−	0.536440i	$-0.180231\pi$
$881$	50.0989			1.68788			0.843938	+	0.536440i	$0.180231\pi$
$882$		0			0
$883$	25.6485			0.863140			0.431570	−	0.902080i	$-0.357960\pi$
$883$	25.6485			0.863140			0.431570	+	0.902080i	$0.357960\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.63164			0.289822			0.144911	−	0.989445i	$-0.453710\pi$
$887$	8.63164			0.289822			0.144911	+	0.989445i	$0.453710\pi$
$888$		0			0
$889$			42.3650i			1.42088i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.02200	+	1.56803i	0.101127	+	0.0524721i
$894$		0			0
$895$	−16.4795			−0.550850
$896$		0			0
$897$		0			0
$898$		0			0
$899$			2.76214i			0.0921225i
$900$		0			0
$901$	−20.8028			−0.693042
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	22.7485i		−	0.756186i
$906$		0			0
$907$	45.9584			1.52602			0.763012	−	0.646384i	$-0.223719\pi$
$907$	45.9584			1.52602			0.763012	+	0.646384i	$0.223719\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	30.5214i		−	1.01122i	−0.862763	−	0.505609i	$-0.831268\pi$
$911$		−	30.5214i		−	1.01122i	0.862763	−	0.505609i	$-0.168732\pi$
$912$		0			0
$913$			11.5776i			0.383161i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	23.2196i		−	0.766778i
$918$		0			0
$919$		−	33.1142i		−	1.09234i	−0.837675	−	0.546168i	$-0.816086\pi$
$919$		−	33.1142i		−	1.09234i	0.837675	−	0.546168i	$-0.183914\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18.6739			−0.614658
$924$		0			0
$925$	31.5342			1.03684
$926$		0			0
$927$		0			0
$928$		0			0
$929$			10.2210i			0.335340i	0.985843	+	0.167670i	$0.0536243\pi$
$929$			10.2210i			0.335340i	−0.985843	+	0.167670i	$0.946376\pi$
$930$		0			0
$931$			2.72949i			0.0894553i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	65.6842i		−	2.14810i
$936$		0			0
$937$			1.20374i			0.0393245i	0.999807	+	0.0196623i	$0.00625910\pi$
$937$			1.20374i			0.0393245i	−0.999807	+	0.0196623i	$0.993741\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	8.67387i		−	0.282760i	−0.989955	−	0.141380i	$-0.954846\pi$
$941$		−	8.67387i		−	0.282760i	0.989955	−	0.141380i	$-0.0451539\pi$
$942$		0			0
$943$	−0.677821			−0.0220729
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	35.5206i		−	1.15427i	−0.816650	−	0.577133i	$-0.804172\pi$
$947$		−	35.5206i		−	1.15427i	0.816650	−	0.577133i	$-0.195828\pi$
$948$		0			0
$949$	−33.9208			−1.10112
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−45.6033			−1.47724			−0.738618	−	0.674124i	$-0.764521\pi$
$953$	−45.6033			−1.47724			−0.738618	+	0.674124i	$0.764521\pi$
$954$		0			0
$955$		−	30.1234i		−	0.974769i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	33.0061			1.06582
$960$		0			0
$961$	30.4879			0.983480
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	14.1524i		−	0.455582i
$966$		0			0
$967$	−3.75423			−0.120728			−0.0603639	−	0.998176i	$-0.519226\pi$
$967$	−3.75423			−0.120728			−0.0603639	+	0.998176i	$0.519226\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.0369			0.675106			0.337553	−	0.941307i	$-0.390401\pi$
$971$	21.0369			0.675106			0.337553	+	0.941307i	$0.390401\pi$
$972$		0			0
$973$		−	26.1110i		−	0.837079i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	29.6204i		−	0.947640i	−0.880622	−	0.473820i	$-0.842875\pi$
$977$		−	29.6204i		−	0.947640i	0.880622	−	0.473820i	$-0.157125\pi$
$978$		0			0
$979$		−	31.5699i		−	1.00898i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−37.1371			−1.18449			−0.592245	−	0.805758i	$-0.701758\pi$
$983$	−37.1371			−1.18449			−0.592245	+	0.805758i	$0.701758\pi$
$984$		0			0
$985$		−	77.3052i		−	2.46315i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			31.8914i			1.01409i
$990$		0			0
$991$	46.9972			1.49292			0.746458	−	0.665433i	$-0.231753\pi$
$991$	46.9972			1.49292			0.746458	+	0.665433i	$0.231753\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			64.7489i			2.05268i
$996$		0			0
$997$		−	13.8296i		−	0.437987i	−0.975726	−	0.218993i	$-0.929723\pi$
$997$		−	13.8296i		−	0.437987i	0.975726	−	0.218993i	$-0.0702773\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1692.2.h.a.845.13	yes	16
3.2	odd	2	inner	1692.2.h.a.845.3	✓	16
4.3	odd	2		6768.2.o.d.5921.13		16
12.11	even	2		6768.2.o.d.5921.3		16
47.46	odd	2	inner	1692.2.h.a.845.4	yes	16
141.140	even	2	inner	1692.2.h.a.845.14	yes	16
188.187	even	2		6768.2.o.d.5921.4		16
564.563	odd	2		6768.2.o.d.5921.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1692.2.h.a.845.3	✓	16	3.2	odd	2	inner
1692.2.h.a.845.4	yes	16	47.46	odd	2	inner
1692.2.h.a.845.13	yes	16	1.1	even	1	trivial
1692.2.h.a.845.14	yes	16	141.140	even	2	inner
6768.2.o.d.5921.3		16	12.11	even	2
6768.2.o.d.5921.4		16	188.187	even	2
6768.2.o.d.5921.13		16	4.3	odd	2
6768.2.o.d.5921.14		16	564.563	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1692 = 2^{2} \cdot 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1692.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.84774 q^{5} -3.53500 q^{7} +O(q^{10})\)	\(q+2.84774 q^{5} -3.53500 q^{7} +4.06156 q^{11} -4.24632i q^{13} -5.67895i q^{17} +0.496609i q^{19} -3.35925 q^{23} +3.10961 q^{25} -3.85978 q^{29} -0.715621i q^{31} -10.0668 q^{35} +10.1409 q^{37} +0.201778 q^{41} -9.49362i q^{43} +(3.15747 - 6.08526i) q^{47} +5.49625 q^{49} -3.66314i q^{53} +11.5663 q^{55} +9.26399i q^{59} +8.60586 q^{61} -12.0924i q^{65} +13.8063i q^{67} -4.39766i q^{71} -7.98827i q^{73} -14.3576 q^{77} +5.53500 q^{79} +2.85052i q^{83} -16.1722i q^{85} -7.77286i q^{89} +15.0108i q^{91} +1.41421i q^{95} -4.96039 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) \) 16 * q + 16 * q^25 + 8 * q^37 + 24 * q^49 + 8 * q^55 + 40 * q^61 + 32 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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