LMFDB - Embedded newform 684.3.y.g.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 34x^{6} + 921x^{4} - 7990x^{2} + 55225 $ x^8 - 34x^6 + 921x^4 - 7990*x^2 + 55225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.2
Root			$2.69047 + 1.55335i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.3.y.g.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.19676 - 3.80490i) q^{5} +1.44949 q^{7} +O(q^{10})$	$q+(-2.19676 - 3.80490i) q^{5} +1.44949 q^{7} +15.1554 q^{11} +(-14.8485 - 8.57277i) q^{13} +(9.77447 + 16.9299i) q^{17} +(-3.34847 - 18.7026i) q^{19} +(-2.19676 + 3.80490i) q^{23} +(2.84847 - 4.93369i) q^{25} +(-29.3234 - 16.9299i) q^{29} -5.62475i q^{31} +(-3.18418 - 5.51517i) q^{35} -23.7238i q^{37} +(-32.2857 + 18.6401i) q^{41} +(-15.9722 - 27.6647i) q^{43} +(39.0979 - 67.7195i) q^{47} -46.8990 q^{49} +(-9.55255 - 5.51517i) q^{53} +(-33.2929 - 57.6649i) q^{55} +(-78.4177 + 45.2745i) q^{59} +(57.1413 - 98.9717i) q^{61} +75.3293i q^{65} +(-98.3105 - 56.7596i) q^{67} +(87.9702 - 50.7896i) q^{71} +(10.0505 + 17.4080i) q^{73} +21.9676 q^{77} +(-45.8258 + 26.4575i) q^{79} -29.2237 q^{83} +(42.9444 - 74.3819i) q^{85} +(126.846 + 73.2347i) q^{89} +(-21.5227 - 12.4261i) q^{91} +(-63.8059 + 53.8258i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 60 * q^13 + 32 * q^19 - 36 * q^25 - 20 * q^43 - 336 * q^49 + 8 * q^55 + 124 * q^61 - 228 * q^67 + 100 * q^73 - 396 * q^79 + 128 * q^85 - 84 * q^91 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$5$	−2.19676	−	3.80490i	−0.439352	−	0.760981i	0.558287	−	0.829648i	$-0.311459\pi$
$5$	−2.19676	−	3.80490i	−0.439352	−	0.760981i	−0.997640	+	0.0686670i	$0.978125\pi$
$6$		0			0
$7$	1.44949			0.207070			0.103535	−	0.994626i	$-0.466985\pi$
$7$	1.44949			0.207070			0.103535	+	0.994626i	$0.466985\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.1554			1.37777			0.688883	−	0.724873i	$-0.258102\pi$
$11$	15.1554			1.37777			0.688883	+	0.724873i	$0.258102\pi$
$12$		0			0
$13$	−14.8485	−	8.57277i	−1.14219	−	0.659444i	−0.195218	−	0.980760i	$-0.562541\pi$
$13$	−14.8485	−	8.57277i	−1.14219	−	0.659444i	−0.946972	+	0.321316i	$0.895875\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	9.77447	+	16.9299i	0.574969	+	0.995875i	0.996045	+	0.0888501i	$0.0283192\pi$
$17$	9.77447	+	16.9299i	0.574969	+	0.995875i	−0.421076	+	0.907025i	$0.638347\pi$
$18$		0			0
$19$	−3.34847	−	18.7026i	−0.176235	−	0.984348i
$19$	−3.34847	−	18.7026i	−0.176235	−	0.984348i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.19676	+	3.80490i	−0.0955114	+	0.165431i	−0.909822	−	0.414999i	$-0.863782\pi$
$23$	−2.19676	+	3.80490i	−0.0955114	+	0.165431i	0.814311	+	0.580429i	$0.197115\pi$
$24$		0			0
$25$	2.84847	−	4.93369i	0.113939	−	0.197348i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−29.3234	−	16.9299i	−1.01115	−	0.583789i	−0.0996235	−	0.995025i	$-0.531764\pi$
$29$	−29.3234	−	16.9299i	−1.01115	−	0.583789i	−0.911529	+	0.411236i	$0.865097\pi$
$30$		0			0
$31$		−	5.62475i		−	0.181443i	−0.995876	−	0.0907217i	$-0.971083\pi$
$31$		−	5.62475i		−	0.181443i	0.995876	−	0.0907217i	$-0.0289174\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.18418	−	5.51517i	−0.0909767	−	0.157576i
$36$		0			0
$37$		−	23.7238i		−	0.641184i	−0.947217	−	0.320592i	$-0.896118\pi$
$37$		−	23.7238i		−	0.641184i	0.947217	−	0.320592i	$-0.103882\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−32.2857	+	18.6401i	−0.787456	+	0.454638i	−0.839066	−	0.544030i	$-0.816898\pi$
$41$	−32.2857	+	18.6401i	−0.787456	+	0.454638i	0.0516104	+	0.998667i	$0.483565\pi$
$42$		0			0
$43$	−15.9722	−	27.6647i	−0.371446	−	0.643364i	0.618342	−	0.785909i	$-0.287805\pi$
$43$	−15.9722	−	27.6647i	−0.371446	−	0.643364i	−0.989788	+	0.142545i	$0.954471\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	39.0979	−	67.7195i	0.831870	−	1.44084i	−0.0646836	−	0.997906i	$-0.520604\pi$
$47$	39.0979	−	67.7195i	0.831870	−	1.44084i	0.896554	−	0.442935i	$-0.146063\pi$
$48$		0			0
$49$	−46.8990			−0.957122
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.55255	−	5.51517i	−0.180237	−	0.104060i	0.407167	−	0.913354i	$-0.366517\pi$
$53$	−9.55255	−	5.51517i	−0.180237	−	0.104060i	−0.587404	+	0.809294i	$0.699850\pi$
$54$		0			0
$55$	−33.2929	−	57.6649i	−0.605325	−	1.04845i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−78.4177	+	45.2745i	−1.32911	+	0.767364i	−0.985163	−	0.171624i	$-0.945099\pi$
$59$	−78.4177	+	45.2745i	−1.32911	+	0.767364i	−0.343951	+	0.938988i	$0.611765\pi$
$60$		0			0
$61$	57.1413	−	98.9717i	0.936743	−	1.62249i	0.165247	−	0.986252i	$-0.447158\pi$
$61$	57.1413	−	98.9717i	0.936743	−	1.62249i	0.771496	−	0.636234i	$-0.219509\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			75.3293i			1.15891i
$66$		0			0
$67$	−98.3105	−	56.7596i	−1.46732	−	0.847158i	−0.467989	−	0.883734i	$-0.655021\pi$
$67$	−98.3105	−	56.7596i	−1.46732	−	0.847158i	−0.999331	+	0.0365763i	$0.988355\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	87.9702	−	50.7896i	1.23902	−	0.715347i	0.270124	−	0.962825i	$-0.412935\pi$
$71$	87.9702	−	50.7896i	1.23902	−	0.715347i	0.968893	+	0.247478i	$0.0796019\pi$
$72$		0			0
$73$	10.0505	+	17.4080i	0.137678	+	0.238466i	0.926617	−	0.376006i	$-0.122703\pi$
$73$	10.0505	+	17.4080i	0.137678	+	0.238466i	−0.788939	+	0.614471i	$0.789369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	21.9676			0.285294
$78$		0			0
$79$	−45.8258	+	26.4575i	−0.580073	+	0.334905i	−0.761162	−	0.648561i	$-0.775371\pi$
$79$	−45.8258	+	26.4575i	−0.580073	+	0.334905i	0.181089	+	0.983467i	$0.442038\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−29.2237			−0.352092			−0.176046	−	0.984382i	$-0.556331\pi$
$83$	−29.2237			−0.352092			−0.176046	+	0.984382i	$0.556331\pi$
$84$		0			0
$85$	42.9444	−	74.3819i	0.505228	−	0.875081i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	126.846	+	73.2347i	1.42524	+	0.822862i	0.996740	−	0.0806807i	$-0.0257094\pi$
$89$	126.846	+	73.2347i	1.42524	+	0.822862i	0.428498	+	0.903543i	$0.359043\pi$
$90$		0			0
$91$	−21.5227	−	12.4261i	−0.236513	−	0.136551i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−63.8059	+	53.8258i	−0.671641	+	0.566587i
$96$		0			0
$97$	27.0000	−	15.5885i	0.278351	−	0.160706i	−0.354326	−	0.935122i	$-0.615290\pi$
$97$	27.0000	−	15.5885i	0.278351	−	0.160706i	0.632676	+	0.774416i	$0.281956\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	66.0026	−	114.320i	0.653491	−	1.13188i	−0.328778	−	0.944407i	$-0.606637\pi$
$101$	66.0026	−	114.320i	0.653491	−	1.13188i	0.982270	−	0.187473i	$-0.0600297\pi$
$102$		0			0
$103$			41.3156i			0.401122i	0.979681	+	0.200561i	$0.0642765\pi$
$103$			41.3156i			0.401122i	−0.979681	+	0.200561i	$0.935723\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	33.8598i		−	0.316446i	−0.987403	−	0.158223i	$-0.949423\pi$
$107$		−	33.8598i		−	0.316446i	0.987403	−	0.158223i	$-0.0505766\pi$
$108$		0			0
$109$	−30.3031	+	17.4955i	−0.278010	+	0.160509i	−0.632522	−	0.774542i	$-0.717980\pi$
$109$	−30.3031	+	17.4955i	−0.278010	+	0.160509i	0.354512	+	0.935051i	$0.384647\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	163.227i		−	1.44448i	−0.691641	−	0.722241i	$-0.743112\pi$
$113$		−	163.227i		−	1.44448i	0.691641	−	0.722241i	$-0.256888\pi$
$114$		0			0
$115$	19.3031			0.167853
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.1680	+	24.5397i	0.119059	+	0.206216i
$120$		0			0
$121$	108.687			0.898237
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−134.868			−1.07894
$126$		0			0
$127$	−135.576	−	78.2746i	−1.06752	−	0.616335i	−0.140020	−	0.990149i	$-0.544717\pi$
$127$	−135.576	−	78.2746i	−1.06752	−	0.616335i	−0.927504	+	0.373814i	$0.878050\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	107.963	+	186.997i	0.824145	+	1.42746i	0.902571	+	0.430541i	$0.141677\pi$
$131$	107.963	+	186.997i	0.824145	+	1.42746i	−0.0784255	+	0.996920i	$0.524989\pi$
$132$		0			0
$133$	−4.85357	−	27.1092i	−0.0364930	−	0.203829i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−29.7673	+	51.5584i	−0.217279	+	0.376339i	−0.953975	−	0.299885i	$-0.903052\pi$
$137$	−29.7673	+	51.5584i	−0.217279	+	0.376339i	0.736696	+	0.676224i	$0.236385\pi$
$138$		0			0
$139$	−1.82577	+	3.16232i	−0.0131350	+	0.0227505i	−0.872518	−	0.488582i	$-0.837514\pi$
$139$	−1.82577	+	3.16232i	−0.0131350	+	0.0227505i	0.859383	+	0.511332i	$0.170848\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−225.035	−	129.924i	−1.57367	−	0.908559i
$144$		0			0
$145$			148.764i			1.02596i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	29.0017	+	50.2325i	0.194643	+	0.337131i	0.946783	−	0.321872i	$-0.104312\pi$
$149$	29.0017	+	50.2325i	0.194643	+	0.337131i	−0.752141	+	0.659003i	$0.770979\pi$
$150$		0			0
$151$			8.64258i			0.0572356i	0.999590	+	0.0286178i	$0.00911058\pi$
$151$			8.64258i			0.0572356i	−0.999590	+	0.0286178i	$0.990889\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−21.4016	+	12.3562i	−0.138075	+	0.0797176i
$156$		0			0
$157$	60.0301	+	103.975i	0.382357	+	0.662262i	0.991399	−	0.130876i	$-0.0417790\pi$
$157$	60.0301	+	103.975i	0.382357	+	0.662262i	−0.609041	+	0.793138i	$0.708446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.18418	+	5.51517i	−0.0197775	+	0.0342557i
$162$		0			0
$163$	−51.8332			−0.317995			−0.158997	−	0.987279i	$-0.550826\pi$
$163$	−51.8332			−0.317995			−0.158997	+	0.987279i	$0.550826\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	125.215	+	72.2932i	0.749793	+	0.432893i	0.825619	−	0.564228i	$-0.190826\pi$
$167$	125.215	+	72.2932i	0.749793	+	0.432893i	−0.0758260	+	0.997121i	$0.524159\pi$
$168$		0			0
$169$	62.4847	+	108.227i	0.369732	+	0.640394i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−57.0161	+	32.9182i	−0.329573	+	0.190279i	−0.655651	−	0.755064i	$-0.727606\pi$
$173$	−57.0161	+	32.9182i	−0.329573	+	0.190279i	0.326079	+	0.945343i	$0.394273\pi$
$174$		0			0
$175$	4.12883	−	7.15134i	0.0235933	−	0.0408648i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	60.1097i		−	0.335808i	−0.985803	−	0.167904i	$-0.946300\pi$
$179$		−	60.1097i		−	0.335808i	0.985803	−	0.167904i	$-0.0536999\pi$
$180$		0			0
$181$	218.530	+	126.168i	1.20735	+	0.697063i	0.962179	−	0.272417i	$-0.0878231\pi$
$181$	218.530	+	126.168i	1.20735	+	0.697063i	0.245169	+	0.969480i	$0.421156\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−90.2668	+	52.1155i	−0.487928	+	0.281706i
$186$		0			0
$187$	148.136	+	256.579i	0.792172	+	1.37208i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	211.533			1.10750			0.553750	−	0.832683i	$-0.313196\pi$
$191$	211.533			1.10750			0.553750	+	0.832683i	$0.313196\pi$
$192$		0			0
$193$	48.4546	−	27.9753i	0.251060	−	0.144950i	−0.369189	−	0.929354i	$-0.620365\pi$
$193$	48.4546	−	27.9753i	0.251060	−	0.144950i	0.620250	+	0.784405i	$0.287031\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−95.1265			−0.482876			−0.241438	−	0.970416i	$-0.577619\pi$
$197$	−95.1265			−0.482876			−0.241438	+	0.970416i	$0.577619\pi$
$198$		0			0
$199$	−40.4115	+	69.9947i	−0.203073	+	0.351732i	−0.949517	−	0.313716i	$-0.898426\pi$
$199$	−40.4115	+	69.9947i	−0.203073	+	0.351732i	0.746444	+	0.665448i	$0.231759\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−42.5040	−	24.5397i	−0.209379	−	0.120885i
$204$		0			0
$205$	141.848	+	81.8960i	0.691941	+	0.399492i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−50.7475	−	283.446i	−0.242811	−	1.35620i
$210$		0			0
$211$	−46.0982	+	26.6148i	−0.218475	+	0.126137i	−0.605244	−	0.796040i	$-0.706924\pi$
$211$	−46.0982	+	26.6148i	−0.218475	+	0.126137i	0.386769	+	0.922177i	$0.373591\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−70.1742	+	121.545i	−0.326392	+	0.565327i
$216$		0			0
$217$		−	8.15301i		−	0.0375715i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	335.177i		−	1.51664i
$222$		0			0
$223$	−171.356	+	98.9324i	−0.768412	+	0.443643i	−0.832308	−	0.554314i	$-0.812981\pi$
$223$	−171.356	+	98.9324i	−0.768412	+	0.443643i	0.0638959	+	0.997957i	$0.479647\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	125.946i		−	0.554829i	−0.960750	−	0.277415i	$-0.910522\pi$
$227$		−	125.946i		−	0.554829i	0.960750	−	0.277415i	$-0.0894775\pi$
$228$		0			0
$229$	193.586			0.845352			0.422676	−	0.906281i	$-0.361091\pi$
$229$	193.586			0.845352			0.422676	+	0.906281i	$0.361091\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	148.592	+	257.369i	0.637734	+	1.10459i	0.985929	+	0.167165i	$0.0534613\pi$
$233$	148.592	+	257.369i	0.637734	+	1.10459i	−0.348195	+	0.937422i	$0.613205\pi$
$234$		0			0
$235$	−343.555			−1.46194
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−104.801			−0.438499			−0.219250	−	0.975669i	$-0.570361\pi$
$239$	−104.801			−0.438499			−0.219250	+	0.975669i	$0.570361\pi$
$240$		0			0
$241$	356.787	+	205.991i	1.48044	+	0.854735i	0.999755	−	0.0221551i	$-0.00705275\pi$
$241$	356.787	+	205.991i	1.48044	+	0.854735i	0.480690	+	0.876890i	$0.340386\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	103.026	+	178.446i	0.420514	+	0.728352i
$246$		0			0
$247$	−110.614	+	306.411i	−0.447828	+	1.24053i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−87.8705	+	152.196i	−0.350082	+	0.606359i	−0.986263	−	0.165180i	$-0.947180\pi$
$251$	−87.8705	+	152.196i	−0.350082	+	0.606359i	0.636182	+	0.771539i	$0.280513\pi$
$252$		0			0
$253$	−33.2929	+	57.6649i	−0.131592	+	0.227925i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	373.283	+	215.515i	1.45246	+	0.838579i	0.998621	−	0.0525037i	$-0.0167201\pi$
$257$	373.283	+	215.515i	1.45246	+	0.838579i	0.453841	+	0.891083i	$0.350053\pi$
$258$		0			0
$259$		−	34.3874i		−	0.132770i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	60.0781	+	104.058i	0.228434	+	0.395659i	0.957344	−	0.288950i	$-0.0933062\pi$
$263$	60.0781	+	104.058i	0.228434	+	0.395659i	−0.728910	+	0.684609i	$0.759973\pi$
$264$		0			0
$265$			48.4621i			0.182876i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	415.487	−	239.882i	1.54456	−	0.891754i	0.546021	−	0.837772i	$-0.316142\pi$
$269$	415.487	−	239.882i	1.54456	−	0.891754i	0.998542	−	0.0539820i	$-0.0171914\pi$
$270$		0			0
$271$	−63.0908	−	109.276i	−0.232807	−	0.403234i	0.725826	−	0.687879i	$-0.241458\pi$
$271$	−63.0908	−	109.276i	−0.232807	−	0.403234i	−0.958633	+	0.284644i	$0.908125\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	43.1697	−	74.7722i	0.156981	−	0.271899i
$276$		0			0
$277$	459.918			1.66036			0.830178	−	0.557499i	$-0.188239\pi$
$277$	459.918			1.66036			0.830178	+	0.557499i	$0.188239\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	65.2371	+	37.6647i	0.232161	+	0.134038i	0.611568	−	0.791192i	$-0.290539\pi$
$281$	65.2371	+	37.6647i	0.232161	+	0.134038i	−0.379408	+	0.925230i	$0.623872\pi$
$282$		0			0
$283$	178.753	+	309.609i	0.631634	+	1.09402i	0.987218	+	0.159378i	$0.0509490\pi$
$283$	178.753	+	309.609i	0.631634	+	1.09402i	−0.355583	+	0.934645i	$0.615718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−46.7978	+	27.0187i	−0.163058	+	0.0941418i
$288$		0			0
$289$	−46.5806	+	80.6800i	−0.161179	+	0.279169i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			228.294i			0.779161i	0.920992	+	0.389581i	$0.127380\pi$
$293$			228.294i			0.779161i	−0.920992	+	0.389581i	$0.872620\pi$
$294$		0			0
$295$	344.530	+	198.915i	1.16790	+	0.674287i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	65.2371	−	37.6647i	0.218184	−	0.125969i
$300$		0			0
$301$	−23.1515	−	40.0996i	−0.0769154	−	0.133221i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−502.104			−1.64624
$306$		0			0
$307$	−83.6969	+	48.3224i	−0.272628	+	0.157402i	−0.630082	−	0.776529i	$-0.716979\pi$
$307$	−83.6969	+	48.3224i	−0.272628	+	0.157402i	0.357453	+	0.933931i	$0.383645\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−414.722			−1.33351			−0.666755	−	0.745277i	$-0.732317\pi$
$311$	−414.722			−1.33351			−0.666755	+	0.745277i	$0.732317\pi$
$312$		0			0
$313$	164.000	−	284.056i	0.523962	−	0.907528i	−0.475649	−	0.879635i	$-0.657787\pi$
$313$	164.000	−	284.056i	0.523962	−	0.907528i	0.999611	−	0.0278932i	$-0.00887983\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−461.253	−	266.304i	−1.45506	−	0.840077i	−0.456295	−	0.889829i	$-0.650824\pi$
$317$	−461.253	−	266.304i	−1.45506	−	0.840077i	−0.998762	+	0.0497515i	$0.984157\pi$
$318$		0			0
$319$	−444.409	−	256.579i	−1.39313	−	0.804324i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	283.904	−	239.497i	0.878958	−	0.741478i
$324$		0			0
$325$	−84.5908	+	48.8385i	−0.260279	+	0.150272i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	56.6720	−	98.1588i	0.172255	−	0.298355i
$330$		0			0
$331$		−	138.310i		−	0.417856i	−0.977931	−	0.208928i	$-0.933003\pi$
$331$		−	138.310i		−	0.417856i	0.977931	−	0.208928i	$-0.0669975\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			498.749i			1.48880i
$336$		0			0
$337$	−200.409	+	115.706i	−0.594686	+	0.343342i	−0.766948	−	0.641709i	$-0.778226\pi$
$337$	−200.409	+	115.706i	−0.594686	+	0.343342i	0.172262	+	0.985051i	$0.444892\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	85.2454i		−	0.249986i
$342$		0			0
$343$	−139.005			−0.405261
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−291.137	−	504.264i	−0.839012	−	1.45321i	−0.890722	−	0.454549i	$-0.849800\pi$
$347$	−291.137	−	504.264i	−0.839012	−	1.45321i	0.0517097	−	0.998662i	$-0.483533\pi$
$348$		0			0
$349$	160.444			0.459725			0.229862	−	0.973223i	$-0.426172\pi$
$349$	160.444			0.459725			0.229862	+	0.973223i	$0.426172\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−162.959			−0.461641			−0.230821	−	0.972996i	$-0.574141\pi$
$353$	−162.959			−0.461641			−0.230821	+	0.972996i	$0.574141\pi$
$354$		0			0
$355$	−386.499	−	223.146i	−1.08873	−	0.628579i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−130.674	−	226.334i	−0.363994	−	0.630456i	0.624620	−	0.780928i	$-0.285254\pi$
$359$	−130.674	−	226.334i	−0.363994	−	0.630456i	−0.988614	+	0.150473i	$0.951920\pi$
$360$		0			0
$361$	−338.576	+	125.250i	−0.937882	+	0.346954i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	44.1572	−	76.4825i	0.120979	−	0.209541i
$366$		0			0
$367$	229.750	−	397.938i	0.626021	−	1.08430i	−0.362321	−	0.932053i	$-0.618016\pi$
$367$	229.750	−	397.938i	0.626021	−	1.08430i	0.988342	−	0.152247i	$-0.0486510\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.8463	−	7.99418i	−0.0373216	−	0.0215477i
$372$		0			0
$373$		−	48.6194i		−	0.130347i	−0.997874	−	0.0651734i	$-0.979240\pi$
$373$		−	48.6194i		−	0.130347i	0.997874	−	0.0651734i	$-0.0207601\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	290.272	+	502.766i	0.769952	+	1.33360i
$378$		0			0
$379$		−	523.962i		−	1.38249i	−0.722622	−	0.691243i	$-0.757063\pi$
$379$		−	523.962i		−	1.38249i	0.722622	−	0.691243i	$-0.242937\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	339.665	−	196.106i	0.886855	−	0.512026i	0.0139424	−	0.999903i	$-0.495562\pi$
$383$	339.665	−	196.106i	0.886855	−	0.512026i	0.872912	+	0.487877i	$0.162229\pi$
$384$		0			0
$385$	−48.2577	−	83.5847i	−0.125345	−	0.217103i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−266.108	+	460.912i	−0.684081	+	1.18486i	0.289644	+	0.957135i	$0.406463\pi$
$389$	−266.108	+	460.912i	−0.684081	+	1.18486i	−0.973725	+	0.227729i	$0.926870\pi$
$390$		0			0
$391$	−85.8888			−0.219664
$392$		0			0
$393$		0			0
$394$		0			0
$395$	201.337	+	116.242i	0.509713	+	0.294283i
$396$		0			0
$397$	−62.3332	−	107.964i	−0.157010	−	0.271950i	0.776779	−	0.629774i	$-0.216852\pi$
$397$	−62.3332	−	107.964i	−0.157010	−	0.271950i	−0.933789	+	0.357823i	$0.883519\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−174.975	+	101.022i	−0.436348	+	0.251926i	−0.702047	−	0.712130i	$-0.747730\pi$
$401$	−174.975	+	101.022i	−0.436348	+	0.251926i	0.265699	+	0.964056i	$0.414397\pi$
$402$		0			0
$403$	−48.2196	+	83.5189i	−0.119652	+	0.207243i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	359.544i		−	0.883401i
$408$		0			0
$409$	473.091	+	273.139i	1.15670	+	0.667822i	0.950511	−	0.310690i	$-0.100560\pi$
$409$	473.091	+	273.139i	1.15670	+	0.667822i	0.206190	+	0.978512i	$0.433893\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−113.666	+	65.6249i	−0.275219	+	0.158898i
$414$		0			0
$415$	64.1975	+	111.193i	0.154693	+	0.267936i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	529.552			1.26385			0.631924	−	0.775031i	$-0.282266\pi$
$419$	529.552			1.26385			0.631924	+	0.775031i	$0.282266\pi$
$420$		0			0
$421$	225.681	−	130.297i	0.536060	−	0.309494i	−0.207421	−	0.978252i	$-0.566507\pi$
$421$	225.681	−	130.297i	0.536060	−	0.309494i	0.743480	+	0.668758i	$0.233174\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	111.369			0.262045
$426$		0			0
$427$	82.8258	−	143.458i	0.193971	−	0.335968i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	208.226	+	120.219i	0.483123	+	0.278931i	0.721717	−	0.692188i	$-0.243353\pi$
$431$	208.226	+	120.219i	0.483123	+	0.278931i	−0.238594	+	0.971119i	$0.576686\pi$
$432$		0			0
$433$	−715.847	−	413.295i	−1.65323	−	0.954491i	−0.975733	−	0.218963i	$-0.929733\pi$
$433$	−715.847	−	413.295i	−1.65323	−	0.954491i	−0.677494	−	0.735528i	$-0.736934\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	78.5174	+	28.3446i	0.179674	+	0.0648618i
$438$		0			0
$439$	516.901	−	298.433i	1.17745	−	0.679801i	0.222027	−	0.975040i	$-0.428733\pi$
$439$	516.901	−	298.433i	1.17745	−	0.679801i	0.955423	+	0.295239i	$0.0953993\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−259.517	+	449.497i	−0.585818	+	1.01467i	0.408955	+	0.912554i	$0.365893\pi$
$443$	−259.517	+	449.497i	−0.585818	+	1.01467i	−0.994773	+	0.102112i	$0.967440\pi$
$444$		0			0
$445$		−	643.517i		−	1.44611i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	50.6169i		−	0.112732i	−0.998410	−	0.0563662i	$-0.982049\pi$
$449$		−	50.6169i		−	0.112732i	0.998410	−	0.0563662i	$-0.0179514\pi$
$450$		0			0
$451$	−489.303	+	282.499i	−1.08493	+	0.626384i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			109.189i			0.239976i
$456$		0			0
$457$	219.788			0.480936			0.240468	−	0.970657i	$-0.422699\pi$
$457$	219.788			0.480936			0.240468	+	0.970657i	$0.422699\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−23.1770	−	40.1437i	−0.0502754	−	0.0870796i	0.839793	−	0.542907i	$-0.182677\pi$
$461$	−23.1770	−	40.1437i	−0.0502754	−	0.0870796i	−0.890068	+	0.455828i	$0.849343\pi$
$462$		0			0
$463$	−678.308			−1.46503			−0.732514	−	0.680752i	$-0.761653\pi$
$463$	−678.308			−1.46503			−0.732514	+	0.680752i	$0.761653\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	687.475			1.47211			0.736054	−	0.676923i	$-0.236687\pi$
$467$	687.475			1.47211			0.736054	+	0.676923i	$0.236687\pi$
$468$		0			0
$469$	−142.500	−	82.2724i	−0.303838	−	0.175421i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−242.065	−	419.269i	−0.511766	−	0.886405i
$474$		0			0
$475$	−101.811	−	36.7535i	−0.214339	−	0.0773758i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−194.946	+	337.656i	−0.406985	+	0.704919i	−0.994550	−	0.104258i	$-0.966753\pi$
$479$	−194.946	+	337.656i	−0.406985	+	0.704919i	0.587565	+	0.809177i	$0.300087\pi$
$480$		0			0
$481$	−203.379	+	352.262i	−0.422824	+	0.732353i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−118.625	−	68.4883i	−0.244588	−	0.141213i
$486$		0			0
$487$		−	179.520i		−	0.368624i	−0.982868	−	0.184312i	$-0.940994\pi$
$487$		−	179.520i		−	0.368624i	0.982868	−	0.184312i	$-0.0590057\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−253.593	−	439.235i	−0.516482	−	0.894573i	−0.999817	−	0.0191376i	$-0.993908\pi$
$491$	−253.593	−	439.235i	−0.516482	−	0.894573i	0.483335	−	0.875436i	$-0.339425\pi$
$492$		0			0
$493$		−	661.923i		−	1.34264i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	127.512	−	73.6191i	0.256563	−	0.148127i
$498$		0			0
$499$	−359.729	−	623.069i	−0.720900	−	1.24864i	−0.960639	−	0.277799i	$-0.910395\pi$
$499$	−359.729	−	623.069i	−0.720900	−	1.24864i	0.239739	−	0.970837i	$-0.422938\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	315.102	−	545.773i	0.626445	−	1.08504i	−0.361814	−	0.932250i	$-0.617842\pi$
$503$	315.102	−	545.773i	0.626445	−	1.08504i	0.988259	−	0.152785i	$-0.0488242\pi$
$504$		0			0
$505$	−579.968			−1.14845
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−83.0780	−	47.9651i	−0.163218	−	0.0942340i	0.416166	−	0.909289i	$-0.363374\pi$
$509$	−83.0780	−	47.9651i	−0.163218	−	0.0942340i	−0.579384	+	0.815055i	$0.696707\pi$
$510$		0			0
$511$	14.5681	+	25.2327i	0.0285090	+	0.0493791i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	157.202	−	90.7606i	0.305246	−	0.176234i
$516$		0			0
$517$	592.545	−	1026.32i	1.14612	−	1.98514i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	253.352i		−	0.486281i	−0.969991	−	0.243140i	$-0.921822\pi$
$521$		−	253.352i		−	0.486281i	0.969991	−	0.243140i	$-0.0781776\pi$
$522$		0			0
$523$	−406.309	−	234.583i	−0.776882	−	0.448533i	0.0584420	−	0.998291i	$-0.481387\pi$
$523$	−406.309	−	234.583i	−0.776882	−	0.448533i	−0.835324	+	0.549758i	$0.814720\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	95.2263	−	54.9789i	0.180695	−	0.104324i
$528$		0			0
$529$	254.848	+	441.410i	0.481755	+	0.834424i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	639.191			1.19923
$534$		0			0
$535$	−128.833	+	74.3819i	−0.240810	+	0.139032i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−710.774			−1.31869
$540$		0			0
$541$	461.257	−	798.921i	0.852601	−	1.47675i	−0.0262519	−	0.999655i	$-0.508357\pi$
$541$	461.257	−	798.921i	0.852601	−	1.47675i	0.878853	−	0.477093i	$-0.158309\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	133.137	+	76.8668i	0.244289	+	0.141040i
$546$		0			0
$547$	159.901	+	92.3187i	0.292323	+	0.168773i	0.638989	−	0.769216i	$-0.279353\pi$
$547$	159.901	+	92.3187i	0.292323	+	0.168773i	−0.346666	+	0.937989i	$0.612686\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−218.444	+	605.114i	−0.396451	+	1.09821i
$552$		0			0
$553$	−66.4240	+	38.3499i	−0.120116	+	0.0693488i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−392.410	+	679.674i	−0.704507	+	1.22024i	0.262363	+	0.964969i	$0.415498\pi$
$557$	−392.410	+	679.674i	−0.704507	+	1.22024i	−0.966869	+	0.255272i	$0.917835\pi$
$558$		0			0
$559$			547.704i			0.979792i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			700.448i			1.24413i	0.782964	+	0.622067i	$0.213707\pi$
$563$			700.448i			1.24413i	−0.782964	+	0.622067i	$0.786293\pi$
$564$		0			0
$565$	−621.061	+	358.570i	−1.09922	+	0.634637i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	251.046i		−	0.441206i	−0.975364	−	0.220603i	$-0.929198\pi$
$569$		−	251.046i		−	0.441206i	0.975364	−	0.220603i	$-0.0708025\pi$
$570$		0			0
$571$	728.378			1.27562			0.637809	−	0.770194i	$-0.279841\pi$
$571$	728.378			1.27562			0.637809	+	0.770194i	$0.279841\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.5148	+	21.6763i	0.0217649	+	0.0376979i
$576$		0			0
$577$	−200.504			−0.347494			−0.173747	−	0.984790i	$-0.555588\pi$
$577$	−200.504			−0.347494			−0.173747	+	0.984790i	$0.555588\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−42.3594			−0.0729078
$582$		0			0
$583$	−144.773	−	83.5847i	−0.248324	−	0.143370i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−257.221	−	445.519i	−0.438195	−	0.758977i	0.559355	−	0.828928i	$-0.311049\pi$
$587$	−257.221	−	445.519i	−0.438195	−	0.758977i	−0.997550	+	0.0699515i	$0.977716\pi$
$588$		0			0
$589$	−105.197	+	18.8343i	−0.178603	+	0.0319767i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	399.444	−	691.858i	0.673599	−	1.16671i	−0.303277	−	0.952902i	$-0.598081\pi$
$593$	399.444	−	691.858i	0.673599	−	1.16671i	0.976876	−	0.213805i	$-0.0685859\pi$
$594$		0			0
$595$	62.2474	−	107.816i	0.104618	−	0.181203i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−147.881	−	85.3793i	−0.246880	−	0.142536i	0.371455	−	0.928451i	$-0.378859\pi$
$599$	−147.881	−	85.3793i	−0.246880	−	0.142536i	−0.618335	+	0.785915i	$0.712192\pi$
$600$		0			0
$601$			836.789i			1.39233i	0.717883	+	0.696164i	$0.245111\pi$
$601$			836.789i			1.39233i	−0.717883	+	0.696164i	$0.754889\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−238.759	−	413.543i	−0.394643	−	0.683542i
$606$		0			0
$607$		−	285.238i		−	0.469914i	−0.972006	−	0.234957i	$-0.924505\pi$
$607$		−	285.238i		−	0.469914i	0.972006	−	0.234957i	$-0.0754949\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1161.09	+	670.354i	−1.90031	+	1.09714i
$612$		0			0
$613$	−330.388	−	572.249i	−0.538969	−	0.933522i	−0.998960	−	0.0455986i	$-0.985480\pi$
$613$	−330.388	−	572.249i	−0.538969	−	0.933522i	0.459990	−	0.887924i	$-0.347853\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−372.340	+	644.912i	−0.603468	+	1.04524i	0.388823	+	0.921312i	$0.372882\pi$
$617$	−372.340	+	644.912i	−0.603468	+	1.04524i	−0.992292	+	0.123926i	$0.960452\pi$
$618$		0			0
$619$	94.1964			0.152175			0.0760876	−	0.997101i	$-0.475757\pi$
$619$	94.1964			0.152175			0.0760876	+	0.997101i	$0.475757\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	183.862	+	106.153i	0.295124	+	0.170390i
$624$		0			0
$625$	225.061	+	389.817i	0.360097	+	0.623707i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	401.641	−	231.888i	0.638539	−	0.368661i
$630$		0			0
$631$	284.195	−	492.240i	0.450388	−	0.780094i	−0.548022	−	0.836464i	$-0.684619\pi$
$631$	284.195	−	492.240i	0.450388	−	0.780094i	0.998410	+	0.0563693i	$0.0179524\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			687.802i			1.08315i
$636$		0			0
$637$	696.378	+	402.054i	1.09322	+	0.631168i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	233.256	−	134.670i	0.363894	−	0.210094i	−0.306894	−	0.951744i	$-0.599290\pi$
$641$	233.256	−	134.670i	0.363894	−	0.210094i	0.670787	+	0.741650i	$0.265956\pi$
$642$		0			0
$643$	577.725	+	1000.65i	0.898483	+	1.55622i	0.829434	+	0.558605i	$0.188663\pi$
$643$	577.725	+	1000.65i	0.898483	+	1.55622i	0.0690498	+	0.997613i	$0.478003\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1039.16			−1.60611			−0.803057	−	0.595902i	$-0.796795\pi$
$647$	−1039.16			−1.60611			−0.803057	+	0.595902i	$0.796795\pi$
$648$		0			0
$649$	−1188.45	+	686.154i	−1.83121	+	1.05725i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1115.67			1.70852			0.854262	−	0.519843i	$-0.174009\pi$
$653$	1115.67			1.70852			0.854262	+	0.519843i	$0.174009\pi$
$654$		0			0
$655$	474.338	−	821.578i	0.724181	−	1.25432i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−86.7060	−	50.0597i	−0.131572	−	0.0759632i	0.432769	−	0.901505i	$-0.357536\pi$
$659$	−86.7060	−	50.0597i	−0.131572	−	0.0759632i	−0.564341	+	0.825542i	$0.690870\pi$
$660$		0			0
$661$	−759.303	−	438.384i	−1.14872	−	0.663213i	−0.200144	−	0.979766i	$-0.564141\pi$
$661$	−759.303	−	438.384i	−1.14872	−	0.663213i	−0.948575	+	0.316553i	$0.897474\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−92.4859	+	78.0200i	−0.139077	+	0.117323i
$666$		0			0
$667$	128.833	−	74.3819i	0.193153	−	0.111517i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	866.001	−	1499.96i	1.29061	−	2.23541i
$672$		0			0
$673$			838.032i			1.24522i	0.782533	+	0.622609i	$0.213927\pi$
$673$			838.032i			1.24522i	−0.782533	+	0.622609i	$0.786073\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	572.195i		−	0.845193i	−0.906318	−	0.422596i	$-0.861119\pi$
$677$		−	572.195i		−	0.845193i	0.906318	−	0.422596i	$-0.138881\pi$
$678$		0			0
$679$	39.1362	−	22.5953i	0.0576380	−	0.0332773i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			191.437i			0.280289i	0.990131	+	0.140144i	$0.0447567\pi$
$683$			191.437i			0.280289i	−0.990131	+	0.140144i	$0.955243\pi$
$684$		0			0
$685$	261.566			0.381849
$686$		0			0
$687$		0			0
$688$		0			0
$689$	94.5605	+	163.784i	0.137243	+	0.237712i
$690$		0			0
$691$	1090.79			1.57856			0.789281	−	0.614033i	$-0.210454\pi$
$691$	1090.79			1.57856			0.789281	+	0.614033i	$0.210454\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.0431			0.0230836
$696$		0			0
$697$	−631.151	−	364.395i	−0.905525	−	0.522805i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	404.337	+	700.331i	0.576800	+	0.999046i	0.995844	+	0.0910803i	$0.0290320\pi$
$701$	404.337	+	700.331i	0.576800	+	0.999046i	−0.419044	+	0.907966i	$0.637635\pi$
$702$		0			0
$703$	−443.697	+	79.4384i	−0.631148	+	0.112999i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	95.6701	−	165.706i	0.135318	−	0.234378i
$708$		0			0
$709$	153.803	−	266.395i	0.216930	−	0.375733i	−0.736938	−	0.675960i	$-0.763729\pi$
$709$	153.803	−	266.395i	0.216930	−	0.375733i	0.953868	+	0.300227i	$0.0970625\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.4016	+	12.3562i	0.0300163	+	0.0173299i
$714$		0			0
$715$			1141.65i			1.59671i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	156.569	+	271.185i	0.217759	+	0.377169i	0.954122	−	0.299417i	$-0.0967920\pi$
$719$	156.569	+	271.185i	0.217759	+	0.377169i	−0.736364	+	0.676586i	$0.763459\pi$
$720$		0			0
$721$			59.8865i			0.0830604i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−167.054	+	96.4485i	−0.230419	+	0.133032i
$726$		0			0
$727$	295.082	+	511.098i	0.405890	+	0.703023i	0.994425	−	0.105449i	$-0.0336281\pi$
$727$	295.082	+	511.098i	0.405890	+	0.703023i	−0.588534	+	0.808472i	$0.700295\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	312.240	−	540.815i	0.427140	−	0.739829i
$732$		0			0
$733$	−775.514			−1.05800			−0.529000	−	0.848622i	$-0.677433\pi$
$733$	−775.514			−1.05800			−0.529000	+	0.848622i	$0.677433\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1489.94	−	860.215i	−2.02162	−	1.16718i
$738$		0			0
$739$	473.628	+	820.348i	0.640904	+	1.11008i	0.985231	+	0.171229i	$0.0547738\pi$
$739$	473.628	+	820.348i	0.640904	+	1.11008i	−0.344327	+	0.938850i	$0.611893\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−487.913	+	281.697i	−0.656680	+	0.379134i	−0.791011	−	0.611802i	$-0.790445\pi$
$743$	−487.913	+	281.697i	−0.656680	+	0.379134i	0.134331	+	0.990937i	$0.457112\pi$
$744$		0			0
$745$	127.420	−	220.698i	0.171033	−	0.296239i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	49.0794i		−	0.0655265i
$750$		0			0
$751$	863.689	+	498.651i	1.15005	+	0.663982i	0.948900	−	0.315578i	$-0.102198\pi$
$751$	863.689	+	498.651i	1.15005	+	0.663982i	0.201152	+	0.979560i	$0.435532\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	32.8842	−	18.9857i	0.0435552	−	0.0251466i
$756$		0			0
$757$	595.954	+	1032.22i	0.787258	+	1.36357i	0.927641	+	0.373473i	$0.121833\pi$
$757$	595.954	+	1032.22i	0.787258	+	1.36357i	−0.140383	+	0.990097i	$0.544833\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−516.905			−0.679244			−0.339622	−	0.940562i	$-0.610299\pi$
$761$	−516.905			−0.679244			−0.339622	+	0.940562i	$0.610299\pi$
$762$		0			0
$763$	−43.9240	+	25.3595i	−0.0575675	+	0.0332366i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1552.51			2.02413
$768$		0			0
$769$	−142.307	+	246.483i	−0.185055	+	0.320524i	−0.943595	−	0.331102i	$-0.892580\pi$
$769$	−142.307	+	246.483i	−0.185055	+	0.320524i	0.758540	+	0.651626i	$0.225913\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	792.031	+	457.280i	1.02462	+	0.591565i	0.915439	−	0.402457i	$-0.131844\pi$
$773$	792.031	+	457.280i	1.02462	+	0.591565i	0.109181	+	0.994022i	$0.465177\pi$
$774$		0			0
$775$	−27.7508	−	16.0219i	−0.0358074	−	0.0206734i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	456.727	+	541.411i	0.586299	+	0.695007i
$780$		0			0
$781$	1333.23	−	769.738i	1.70708	−	0.985581i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	263.744	−	456.818i	0.335979	−	0.581933i
$786$		0			0
$787$			199.823i			0.253905i	0.991909	+	0.126952i	$0.0405195\pi$
$787$			199.823i			0.253905i	−0.991909	+	0.126952i	$0.959480\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	236.595i		−	0.299109i
$792$		0			0
$793$	−1696.92	+	979.719i	−2.13988	+	1.23546i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1490.18i			1.86973i	0.355003	+	0.934865i	$0.384480\pi$
$797$			1490.18i			1.86973i	−0.355003	+	0.934865i	$0.615520\pi$
$798$		0			0
$799$	1528.64			1.91320
$800$		0			0
$801$		0			0
$802$		0			0
$803$	152.320	+	263.825i	0.189688	+	0.328550i
$804$		0			0
$805$	27.9796			0.0347573
$806$		0			0
$807$		0			0
$808$		0			0
$809$	235.121			0.290631			0.145316	−	0.989385i	$-0.453580\pi$
$809$	235.121			0.290631			0.145316	+	0.989385i	$0.453580\pi$
$810$		0			0
$811$	406.665	+	234.788i	0.501437	+	0.289505i	0.729307	−	0.684187i	$-0.239843\pi$
$811$	406.665	+	234.788i	0.501437	+	0.289505i	−0.227870	+	0.973692i	$0.573176\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	113.865	+	197.220i	0.139712	+	0.241988i
$816$		0			0
$817$	−463.919	+	391.356i	−0.567832	+	0.479016i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−370.642	+	641.971i	−0.451452	+	0.781938i	−0.998476	−	0.0551789i	$-0.982427\pi$
$821$	−370.642	+	641.971i	−0.451452	+	0.781938i	0.547025	+	0.837117i	$0.315760\pi$
$822$		0			0
$823$	−514.242	+	890.693i	−0.624838	+	1.08225i	0.363734	+	0.931503i	$0.381502\pi$
$823$	−514.242	+	890.693i	−0.624838	+	1.08225i	−0.988572	+	0.150749i	$0.951832\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	613.862	+	354.413i	0.742275	+	0.428553i	0.822896	−	0.568192i	$-0.192357\pi$
$827$	613.862	+	354.413i	0.742275	+	0.428553i	−0.0806207	+	0.996745i	$0.525690\pi$
$828$		0			0
$829$			695.252i			0.838664i	0.907833	+	0.419332i	$0.137736\pi$
$829$			695.252i			0.838664i	−0.907833	+	0.419332i	$0.862264\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−458.413	−	793.994i	−0.550315	−	0.953174i
$834$		0			0
$835$		−	635.244i		−	0.760771i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−747.665	+	431.664i	−0.891138	+	0.514499i	−0.874315	−	0.485360i	$-0.838689\pi$
$839$	−747.665	+	431.664i	−0.891138	+	0.514499i	−0.0168234	+	0.999858i	$0.505355\pi$
$840$		0			0
$841$	152.742	+	264.557i	0.181619	+	0.314574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	274.528	−	475.497i	0.324885	−	0.562718i
$846$		0			0
$847$	157.540			0.185998
$848$		0			0
$849$		0			0
$850$		0			0
$851$	90.2668	+	52.1155i	0.106071	+	0.0612404i
$852$		0			0
$853$	−92.1663	−	159.637i	−0.108050	−	0.187147i	0.806930	−	0.590646i	$-0.201127\pi$
$853$	−92.1663	−	159.637i	−0.108050	−	0.187147i	−0.914980	+	0.403499i	$0.867794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	684.058	−	394.941i	0.798201	−	0.460842i	−0.0446406	−	0.999003i	$-0.514214\pi$
$857$	684.058	−	394.941i	0.798201	−	0.460842i	0.842842	+	0.538161i	$0.180881\pi$
$858$		0			0
$859$	627.043	−	1086.07i	0.729968	−	1.26434i	−0.226928	−	0.973912i	$-0.572868\pi$
$859$	627.043	−	1086.07i	0.729968	−	1.26434i	0.956896	−	0.290430i	$-0.0937984\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1094.97i		−	1.26879i	−0.773009	−	0.634395i	$-0.781249\pi$
$863$		−	1094.97i		−	1.26879i	0.773009	−	0.634395i	$-0.218751\pi$
$864$		0			0
$865$	250.502	+	144.627i	0.289597	+	0.167199i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−694.509	+	400.975i	−0.799204	+	0.461421i
$870$		0			0
$871$	973.173	+	1685.59i	1.11731	+	1.93523i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−195.489			−0.223417
$876$		0			0
$877$	−15.6520	+	9.03671i	−0.0178473	+	0.0103041i	−0.508897	−	0.860827i	$-0.669947\pi$
$877$	−15.6520	+	9.03671i	−0.0178473	+	0.0103041i	0.491050	+	0.871131i	$0.336613\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	362.089			0.410998			0.205499	−	0.978657i	$-0.434118\pi$
$881$	362.089			0.410998			0.205499	+	0.978657i	$0.434118\pi$
$882$		0			0
$883$	517.836	−	896.918i	0.586451	−	1.01576i	−0.408242	−	0.912874i	$-0.633858\pi$
$883$	517.836	−	896.918i	0.586451	−	1.01576i	0.994693	−	0.102889i	$-0.0328086\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−374.913	−	216.456i	−0.422676	−	0.244032i	0.273546	−	0.961859i	$-0.411804\pi$
$887$	−374.913	−	216.456i	−0.422676	−	0.244032i	−0.696221	+	0.717827i	$0.745137\pi$
$888$		0			0
$889$	−196.515	−	113.458i	−0.221052	−	0.127624i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1397.45	−	504.476i	−1.56489	−	0.564923i
$894$		0			0
$895$	−228.712	+	132.047i	−0.255544	+	0.147538i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−95.2263	+	164.937i	−0.105925	+	0.183467i
$900$		0			0
$901$		−	215.631i		−	0.239325i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1108.65i		−	1.22503i
$906$		0			0
$907$	−1446.74	+	835.277i	−1.59509	+	0.920923i	−0.602670	+	0.797990i	$0.705897\pi$
$907$	−1446.74	+	835.277i	−1.59509	+	0.920923i	−0.992415	+	0.122933i	$0.960770\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1087.43i			1.19367i	0.802364	+	0.596835i	$0.203575\pi$
$911$			1087.43i			1.19367i	−0.802364	+	0.596835i	$0.796425\pi$
$912$		0			0
$913$	−442.897			−0.485101
$914$		0			0
$915$		0			0
$916$		0			0
$917$	156.491	+	271.051i	0.170656	+	0.295584i
$918$		0			0
$919$	−984.258			−1.07101			−0.535505	−	0.844532i	$-0.679879\pi$
$919$	−984.258			−1.07101			−0.535505	+	0.844532i	$0.679879\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1741.63			−1.88692
$924$		0			0
$925$	−117.046	−	67.5765i	−0.126536	−	0.0730557i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	223.803	+	387.638i	0.240907	+	0.417264i	0.960973	−	0.276642i	$-0.0892216\pi$
$929$	223.803	+	387.638i	0.240907	+	0.417264i	−0.720066	+	0.693906i	$0.755888\pi$
$930$		0			0
$931$	157.040	+	877.133i	0.168679	+	0.942141i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	650.840	−	1127.29i	0.696086	−	1.20566i
$936$		0			0
$937$	−153.045	+	265.082i	−0.163335	+	0.282905i	−0.936063	−	0.351833i	$-0.885558\pi$
$937$	−153.045	+	265.082i	−0.163335	+	0.282905i	0.772728	+	0.634738i	$0.218892\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−739.908	−	427.186i	−0.786299	−	0.453970i	0.0523588	−	0.998628i	$-0.483326\pi$
$941$	−739.908	−	427.186i	−0.786299	−	0.453970i	−0.838658	+	0.544658i	$0.816659\pi$
$942$		0			0
$943$		−	163.792i		−	0.173692i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−573.188	−	992.791i	−0.605267	−	1.04835i	−0.992009	−	0.126166i	$-0.959733\pi$
$947$	−573.188	−	992.791i	−0.605267	−	1.04835i	0.386742	−	0.922188i	$-0.373600\pi$
$948$		0			0
$949$		−	344.643i		−	0.363164i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−675.404	+	389.944i	−0.708713	+	0.409176i	−0.810584	−	0.585622i	$-0.800850\pi$
$953$	−675.404	+	389.944i	−0.708713	+	0.409176i	0.101871	+	0.994798i	$0.467517\pi$
$954$		0			0
$955$	−464.687	−	804.861i	−0.486583	−	0.842786i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−43.1473	+	74.7334i	−0.0449920	+	0.0779284i
$960$		0			0
$961$	929.362			0.967078
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−212.886	−	122.910i	−0.220608	−	0.127368i
$966$		0			0
$967$	687.674	+	1191.09i	0.711141	+	1.23173i	0.964429	+	0.264342i	$0.0851548\pi$
$967$	687.674	+	1191.09i	0.711141	+	1.23173i	−0.253288	+	0.967391i	$0.581512\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1484.68	−	857.179i	1.52902	−	0.882780i	0.529616	−	0.848238i	$-0.322336\pi$
$971$	1484.68	−	857.179i	1.52902	−	0.882780i	0.999403	−	0.0345418i	$-0.0109972\pi$
$972$		0			0
$973$	−2.64643	+	4.58375i	−0.00271986	+	0.00471094i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1792.68i			1.83489i	0.397866	+	0.917443i	$0.369751\pi$
$977$			1792.68i			1.83489i	−0.397866	+	0.917443i	$0.630249\pi$
$978$		0			0
$979$	1922.41	+	1109.90i	1.96364	+	1.13371i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−487.315	+	281.351i	−0.495742	+	0.286217i	−0.726954	−	0.686687i	$-0.759064\pi$
$983$	−487.315	+	281.351i	−0.495742	+	0.286217i	0.231211	+	0.972904i	$0.425731\pi$
$984$		0			0
$985$	208.970	+	361.947i	0.212153	+	0.367459i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	140.348			0.141909
$990$		0			0
$991$	1085.70	−	626.830i	1.09556	−	0.632523i	0.160510	−	0.987034i	$-0.448686\pi$
$991$	1085.70	−	626.830i	1.09556	−	0.632523i	0.935052	+	0.354511i	$0.115353\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	355.098			0.356882
$996$		0			0
$997$	502.212	−	869.856i	0.503723	−	0.872474i	−0.496268	−	0.868169i	$-0.665297\pi$
$997$	502.212	−	869.856i	0.503723	−	0.872474i	0.999991	−	0.00430418i	$-0.00137007\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.y.g.145.2	✓	8
3.2	odd	2	inner	684.3.y.g.145.3	yes	8
19.8	odd	6	inner	684.3.y.g.217.2	yes	8
57.8	even	6	inner	684.3.y.g.217.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.y.g.145.2	✓	8	1.1	even	1	trivial
684.3.y.g.145.3	yes	8	3.2	odd	2	inner
684.3.y.g.217.2	yes	8	19.8	odd	6	inner
684.3.y.g.217.3	yes	8	57.8	even	6	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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.19676 - 3.80490i) q^{5} +1.44949 q^{7} +O(q^{10})\)	\(q+(-2.19676 - 3.80490i) q^{5} +1.44949 q^{7} +15.1554 q^{11} +(-14.8485 - 8.57277i) q^{13} +(9.77447 + 16.9299i) q^{17} +(-3.34847 - 18.7026i) q^{19} +(-2.19676 + 3.80490i) q^{23} +(2.84847 - 4.93369i) q^{25} +(-29.3234 - 16.9299i) q^{29} -5.62475i q^{31} +(-3.18418 - 5.51517i) q^{35} -23.7238i q^{37} +(-32.2857 + 18.6401i) q^{41} +(-15.9722 - 27.6647i) q^{43} +(39.0979 - 67.7195i) q^{47} -46.8990 q^{49} +(-9.55255 - 5.51517i) q^{53} +(-33.2929 - 57.6649i) q^{55} +(-78.4177 + 45.2745i) q^{59} +(57.1413 - 98.9717i) q^{61} +75.3293i q^{65} +(-98.3105 - 56.7596i) q^{67} +(87.9702 - 50.7896i) q^{71} +(10.0505 + 17.4080i) q^{73} +21.9676 q^{77} +(-45.8258 + 26.4575i) q^{79} -29.2237 q^{83} +(42.9444 - 74.3819i) q^{85} +(126.846 + 73.2347i) q^{89} +(-21.5227 - 12.4261i) q^{91} +(-63.8059 + 53.8258i) q^{95} +(27.0000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} - 60 q^{13} + 32 q^{19} - 36 q^{25} - 20 q^{43} - 336 q^{49} + 8 q^{55} + 124 q^{61} - 228 q^{67} + 100 q^{73} - 396 q^{79} + 128 q^{85} - 84 q^{91} + 216 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 60 * q^13 + 32 * q^19 - 36 * q^25 - 20 * q^43 - 336 * q^49 + 8 * q^55 + 124 * q^61 - 228 * q^67 + 100 * q^73 - 396 * q^79 + 128 * q^85 - 84 * q^91 + 216 * q^97

Introduction

L-functions

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