LMFDB - Embedded newform 684.3.bj.a.125.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(125,684)]

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, 3, 4]))

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

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

chi := DirichletCharacter("684.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.bj (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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.4
Character	$\chi$	$=$	684.125
Dual form			684.3.bj.a.197.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.59674 - 2.07658i) q^{5} +4.23001 q^{7} +O(q^{10})$	$q+(-3.59674 - 2.07658i) q^{5} +4.23001 q^{7} +14.0399i q^{11} +(-6.49853 - 11.2558i) q^{13} +(-8.58068 - 4.95406i) q^{17} +(6.13661 - 17.9817i) q^{19} +(-1.94069 + 1.12046i) q^{23} +(-3.87565 - 6.71283i) q^{25} +(-3.25040 + 1.87662i) q^{29} +15.9509 q^{31} +(-15.2142 - 8.78394i) q^{35} -21.7181 q^{37} +(-53.3655 - 30.8106i) q^{41} +(-18.1483 + 31.4337i) q^{43} +(-48.8612 + 28.2101i) q^{47} -31.1070 q^{49} +(-21.9351 + 12.6642i) q^{53} +(29.1549 - 50.4978i) q^{55} +(-41.1184 - 23.7397i) q^{59} +(-22.8300 - 39.5427i) q^{61} +53.9788i q^{65} +(-37.6376 - 65.1901i) q^{67} +(-97.0949 - 56.0578i) q^{71} +(62.8386 - 108.840i) q^{73} +59.3888i q^{77} +(4.87284 - 8.44001i) q^{79} +40.9269i q^{83} +(20.5750 + 35.6369i) q^{85} +(95.2647 - 55.0011i) q^{89} +(-27.4888 - 47.6121i) q^{91} +(-59.4122 + 51.9324i) q^{95} +(-5.36484 + 9.29218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 16 q^{7}+O(q^{10}) $ 24 * q - 16 * q^7	$ 24 q - 16 q^{7} + 4 q^{13} - 60 q^{19} + 24 q^{25} + 40 q^{31} + 224 q^{37} + 52 q^{43} + 144 q^{49} + 156 q^{55} - 72 q^{61} + 124 q^{67} - 88 q^{73} - 28 q^{79} - 192 q^{85} + 236 q^{91}+O(q^{100}) $ 24 * q - 16 * q^7 + 4 * q^13 - 60 * q^19 + 24 * q^25 + 40 * q^31 + 224 * q^37 + 52 * q^43 + 144 * q^49 + 156 * q^55 - 72 * q^61 + 124 * q^67 - 88 * q^73 - 28 * q^79 - 192 * q^85 + 236 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

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
$5$	−3.59674	−	2.07658i	−0.719348	−	0.415316i	0.0951649	−	0.995462i	$-0.469662\pi$
$5$	−3.59674	−	2.07658i	−0.719348	−	0.415316i	−0.814512	+	0.580146i	$0.802995\pi$
$6$		0			0
$7$	4.23001			0.604287			0.302143	−	0.953262i	$-0.402298\pi$
$7$	4.23001			0.604287			0.302143	+	0.953262i	$0.402298\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			14.0399i			1.27635i	0.769890	+	0.638176i	$0.220311\pi$
$11$			14.0399i			1.27635i	−0.769890	+	0.638176i	$0.779689\pi$
$12$		0			0
$13$	−6.49853	−	11.2558i	−0.499887	−	0.865830i	0.500113	−	0.865960i	$-0.333292\pi$
$13$	−6.49853	−	11.2558i	−0.499887	−	0.865830i	−1.00000		0.000130349i	$0.999959\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8.58068	−	4.95406i	−0.504746	−	0.291415i	0.225925	−	0.974145i	$-0.427459\pi$
$17$	−8.58068	−	4.95406i	−0.504746	−	0.291415i	−0.730671	+	0.682729i	$0.760793\pi$
$18$		0			0
$19$	6.13661	−	17.9817i	0.322980	−	0.946406i
$19$	6.13661	−	17.9817i	0.322980	−	0.946406i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.94069	+	1.12046i	−0.0843780	+	0.0487157i	−0.541595	−	0.840639i	$-0.682179\pi$
$23$	−1.94069	+	1.12046i	−0.0843780	+	0.0487157i	0.457217	+	0.889355i	$0.348846\pi$
$24$		0			0
$25$	−3.87565	−	6.71283i	−0.155026	−	0.268513i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.25040	+	1.87662i	−0.112083	+	0.0647110i	−0.554993	−	0.831855i	$-0.687279\pi$
$29$	−3.25040	+	1.87662i	−0.112083	+	0.0647110i	0.442911	+	0.896566i	$0.353946\pi$
$30$		0			0
$31$	15.9509			0.514546			0.257273	−	0.966339i	$-0.417176\pi$
$31$	15.9509			0.514546			0.257273	+	0.966339i	$0.417176\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−15.2142	−	8.78394i	−0.434692	−	0.250970i
$36$		0			0
$37$	−21.7181			−0.586976			−0.293488	−	0.955963i	$-0.594816\pi$
$37$	−21.7181			−0.586976			−0.293488	+	0.955963i	$0.594816\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−53.3655	−	30.8106i	−1.30160	−	0.751477i	−0.320919	−	0.947107i	$-0.603992\pi$
$41$	−53.3655	−	30.8106i	−1.30160	−	0.751477i	−0.980678	+	0.195629i	$0.937325\pi$
$42$		0			0
$43$	−18.1483	+	31.4337i	−0.422052	+	0.731016i	−0.996140	−	0.0877777i	$-0.972023\pi$
$43$	−18.1483	+	31.4337i	−0.422052	+	0.731016i	0.574088	+	0.818794i	$0.305357\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−48.8612	+	28.2101i	−1.03960	+	0.600214i	−0.919720	−	0.392574i	$-0.871585\pi$
$47$	−48.8612	+	28.2101i	−1.03960	+	0.600214i	−0.119881	+	0.992788i	$0.538251\pi$
$48$		0			0
$49$	−31.1070			−0.634837
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−21.9351	+	12.6642i	−0.413869	+	0.238947i	−0.692451	−	0.721465i	$-0.743469\pi$
$53$	−21.9351	+	12.6642i	−0.413869	+	0.238947i	0.278582	+	0.960413i	$0.410136\pi$
$54$		0			0
$55$	29.1549	−	50.4978i	0.530089	−	0.918141i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−41.1184	−	23.7397i	−0.696922	−	0.402368i	0.109278	−	0.994011i	$-0.465146\pi$
$59$	−41.1184	−	23.7397i	−0.696922	−	0.402368i	−0.806200	+	0.591643i	$0.798479\pi$
$60$		0			0
$61$	−22.8300	−	39.5427i	−0.374262	−	0.648241i	0.615954	−	0.787782i	$-0.288771\pi$
$61$	−22.8300	−	39.5427i	−0.374262	−	0.648241i	−0.990216	+	0.139541i	$0.955437\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			53.9788i			0.830443i
$66$		0			0
$67$	−37.6376	−	65.1901i	−0.561754	−	0.972987i	−0.997344	−	0.0728417i	$-0.976793\pi$
$67$	−37.6376	−	65.1901i	−0.561754	−	0.972987i	0.435589	−	0.900146i	$-0.356540\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−97.0949	−	56.0578i	−1.36753	−	0.789546i	−0.376921	−	0.926245i	$-0.623017\pi$
$71$	−97.0949	−	56.0578i	−1.36753	−	0.789546i	−0.990613	+	0.136699i	$0.956351\pi$
$72$		0			0
$73$	62.8386	−	108.840i	0.860803	−	1.49095i	−0.0103529	−	0.999946i	$-0.503295\pi$
$73$	62.8386	−	108.840i	0.860803	−	1.49095i	0.871155	−	0.491007i	$-0.163371\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			59.3888i			0.771283i
$78$		0			0
$79$	4.87284	−	8.44001i	0.0616816	−	0.106836i	−0.833536	−	0.552466i	$-0.813687\pi$
$79$	4.87284	−	8.44001i	0.0616816	−	0.106836i	0.895217	+	0.445630i	$0.147020\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			40.9269i			0.493096i	0.969131	+	0.246548i	$0.0792963\pi$
$83$			40.9269i			0.493096i	−0.969131	+	0.246548i	$0.920704\pi$
$84$		0			0
$85$	20.5750	+	35.6369i	0.242058	+	0.419258i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	95.2647	−	55.0011i	1.07039	−	0.617990i	0.142102	−	0.989852i	$-0.454614\pi$
$89$	95.2647	−	55.0011i	1.07039	−	0.617990i	0.928288	+	0.371862i	$0.121281\pi$
$90$		0			0
$91$	−27.4888	−	47.6121i	−0.302075	−	0.523210i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−59.4122	+	51.9324i	−0.625392	+	0.546656i
$96$		0			0
$97$	−5.36484	+	9.29218i	−0.0553076	+	0.0957956i	−0.892354	−	0.451337i	$-0.850947\pi$
$97$	−5.36484	+	9.29218i	−0.0553076	+	0.0957956i	0.837046	+	0.547132i	$0.184281\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−25.5587	+	14.7564i	−0.253057	+	0.146102i	−0.621163	−	0.783681i	$-0.713340\pi$
$101$	−25.5587	+	14.7564i	−0.253057	+	0.146102i	0.368106	+	0.929784i	$0.380006\pi$
$102$		0			0
$103$	157.217			1.52638			0.763189	−	0.646176i	$-0.223633\pi$
$103$	157.217			1.52638			0.763189	+	0.646176i	$0.223633\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	119.707i		−	1.11876i	−0.828911	−	0.559381i	$-0.811039\pi$
$107$		−	119.707i		−	1.11876i	0.828911	−	0.559381i	$-0.188961\pi$
$108$		0			0
$109$	−67.0499	+	116.134i	−0.615137	+	1.06545i	0.375224	+	0.926934i	$0.377566\pi$
$109$	−67.0499	+	116.134i	−0.615137	+	1.06545i	−0.990361	+	0.138514i	$0.955767\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			50.2999i			0.445132i	0.974918	+	0.222566i	$0.0714432\pi$
$113$			50.2999i			0.445132i	−0.974918	+	0.222566i	$0.928557\pi$
$114$		0			0
$115$	9.30689			0.0809295
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−36.2963	−	20.9557i	−0.305011	−	0.176098i
$120$		0			0
$121$	−76.1182			−0.629076
$122$		0			0
$123$		0			0
$124$		0			0
$125$			136.021i			1.08817i
$126$		0			0
$127$	34.8685	+	60.3941i	0.274555	+	0.475544i	0.970023	−	0.243014i	$-0.0781360\pi$
$127$	34.8685	+	60.3941i	0.274555	+	0.475544i	−0.695467	+	0.718558i	$0.744803\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−44.0389	−	25.4259i	−0.336175	−	0.194091i	0.322404	−	0.946602i	$-0.395509\pi$
$131$	−44.0389	−	25.4259i	−0.336175	−	0.194091i	−0.658579	+	0.752511i	$0.728842\pi$
$132$		0			0
$133$	25.9579	−	76.0628i	0.195172	−	0.571901i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	26.9818	−	15.5780i	0.196948	−	0.113708i	−0.398283	−	0.917262i	$-0.630394\pi$
$137$	26.9818	−	15.5780i	0.196948	−	0.113708i	0.595231	+	0.803555i	$0.297061\pi$
$138$		0			0
$139$	−37.2037	−	64.4387i	−0.267653	−	0.463588i	0.700602	−	0.713552i	$-0.252915\pi$
$139$	−37.2037	−	64.4387i	−0.267653	−	0.463588i	−0.968255	+	0.249964i	$0.919581\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	158.030	−	91.2386i	1.10510	−	0.638032i
$144$		0			0
$145$	15.5878			0.107502
$146$		0			0
$147$		0			0
$148$		0			0
$149$	115.708	+	66.8042i	0.776566	+	0.448350i	0.835212	−	0.549928i	$-0.185345\pi$
$149$	115.708	+	66.8042i	0.776566	+	0.448350i	−0.0586460	+	0.998279i	$0.518678\pi$
$150$		0			0
$151$	−88.9340			−0.588967			−0.294484	−	0.955657i	$-0.595148\pi$
$151$	−88.9340			−0.588967			−0.294484	+	0.955657i	$0.595148\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−57.3713	−	33.1233i	−0.370137	−	0.213699i
$156$		0			0
$157$	−142.896	+	247.503i	−0.910165	+	1.57645i	−0.0963345	+	0.995349i	$0.530712\pi$
$157$	−142.896	+	247.503i	−0.910165	+	1.57645i	−0.813830	+	0.581103i	$0.802621\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.20915	+	4.73956i	−0.0509885	+	0.0294382i
$162$		0			0
$163$	64.5898			0.396257			0.198128	−	0.980176i	$-0.436514\pi$
$163$	64.5898			0.396257			0.198128	+	0.980176i	$0.436514\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	174.687	−	100.856i	1.04603	−	0.603926i	0.124495	−	0.992220i	$-0.460269\pi$
$167$	174.687	−	100.856i	1.04603	−	0.603926i	0.921535	+	0.388294i	$0.126936\pi$
$168$		0			0
$169$	0.0381526	−	0.0660822i	0.000225755	−	0.000391019i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−62.3889	−	36.0202i	−0.360629	−	0.208210i	0.308727	−	0.951151i	$-0.400097\pi$
$173$	−62.3889	−	36.0202i	−0.360629	−	0.208210i	−0.669357	+	0.742941i	$0.733430\pi$
$174$		0			0
$175$	−16.3940	−	28.3953i	−0.0936802	−	0.162259i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	24.0617i		−	0.134423i	−0.997739	−	0.0672115i	$-0.978590\pi$
$179$		−	24.0617i		−	0.134423i	0.997739	−	0.0672115i	$-0.0214102\pi$
$180$		0			0
$181$	8.23206	+	14.2583i	0.0454810	+	0.0787754i	0.887870	−	0.460095i	$-0.152185\pi$
$181$	8.23206	+	14.2583i	0.0454810	+	0.0787754i	−0.842389	+	0.538870i	$0.818851\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	78.1144	+	45.0994i	0.422240	+	0.243780i
$186$		0			0
$187$	69.5544	−	120.472i	0.371948	−	0.644234i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	237.594i		−	1.24395i	−0.783039	−	0.621973i	$-0.786331\pi$
$191$		−	237.594i		−	1.24395i	0.783039	−	0.621973i	$-0.213669\pi$
$192$		0			0
$193$	−106.769	+	184.928i	−0.553205	+	0.958179i	0.444836	+	0.895612i	$0.353262\pi$
$193$	−106.769	+	184.928i	−0.553205	+	0.958179i	−0.998041	+	0.0625667i	$0.980071\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			49.6500i			0.252031i	0.992028	+	0.126015i	$0.0402188\pi$
$197$			49.6500i			0.252031i	−0.992028	+	0.126015i	$0.959781\pi$
$198$		0			0
$199$	129.987	+	225.144i	0.653200	+	1.13137i	0.982342	+	0.187095i	$0.0599071\pi$
$199$	129.987	+	225.144i	0.653200	+	1.13137i	−0.329142	+	0.944280i	$0.606760\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.7492	+	7.93812i	−0.0677302	+	0.0391040i
$204$		0			0
$205$	127.961	+	221.635i	0.624200	+	1.08115i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	252.461	+	86.1573i	1.20795	+	0.412236i
$210$		0			0
$211$	−60.1860	+	104.245i	−0.285242	+	0.494053i	−0.972668	−	0.232201i	$-0.925407\pi$
$211$	−60.1860	+	104.245i	−0.285242	+	0.494053i	0.687426	+	0.726254i	$0.258741\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	130.549	−	75.3725i	0.607205	−	0.350570i
$216$		0			0
$217$	67.4725			0.310933
$218$		0			0
$219$		0			0
$220$		0			0
$221$			128.776i			0.582699i
$222$		0			0
$223$	94.7814	−	164.166i	0.425029	−	0.736171i	−0.571394	−	0.820676i	$-0.693597\pi$
$223$	94.7814	−	164.166i	0.425029	−	0.736171i	0.996423	+	0.0845044i	$0.0269307\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	0.299330i		−	0.00131863i	−1.00000		0.000659317i	$-0.999790\pi$
$227$		−	0.299330i		−	0.00131863i	1.00000		0.000659317i	$-0.000209867\pi$
$228$		0			0
$229$	−228.265			−0.996790			−0.498395	−	0.866950i	$-0.666077\pi$
$229$	−228.265			−0.996790			−0.498395	+	0.866950i	$0.666077\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	287.053	+	165.730i	1.23199	+	0.711287i	0.967444	−	0.253087i	$-0.0814458\pi$
$233$	287.053	+	165.730i	1.23199	+	0.711287i	0.264542	+	0.964374i	$0.414779\pi$
$234$		0			0
$235$	234.321			0.997113
$236$		0			0
$237$		0			0
$238$		0			0
$239$			205.711i			0.860714i	0.902659	+	0.430357i	$0.141612\pi$
$239$			205.711i			0.860714i	−0.902659	+	0.430357i	$0.858388\pi$
$240$		0			0
$241$	123.153	+	213.307i	0.511007	+	0.885090i	0.999919	+	0.0127570i	$0.00406080\pi$
$241$	123.153	+	213.307i	0.511007	+	0.885090i	−0.488911	+	0.872333i	$0.662606\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	111.884	+	64.5962i	0.456669	+	0.263658i
$246$		0			0
$247$	−242.277	+	47.7823i	−0.980880	+	0.193451i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	45.4865	−	26.2617i	0.181221	−	0.104628i	−0.406645	−	0.913586i	$-0.633301\pi$
$251$	45.4865	−	26.2617i	0.181221	−	0.104628i	0.587866	+	0.808958i	$0.299968\pi$
$252$		0			0
$253$	−15.7311	−	27.2471i	−0.0621784	−	0.107696i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	383.442	−	221.380i	1.49199	−	0.861402i	0.492034	−	0.870576i	$-0.336254\pi$
$257$	383.442	−	221.380i	1.49199	−	0.861402i	0.999958	+	0.00917428i	$0.00292030\pi$
$258$		0			0
$259$	−91.8678			−0.354702
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.76163	+	2.17178i	0.0143028	+	0.00825771i	0.507134	−	0.861867i	$-0.330705\pi$
$263$	3.76163	+	2.17178i	0.0143028	+	0.00825771i	−0.492832	+	0.870125i	$0.664038\pi$
$264$		0			0
$265$	105.193			0.396954
$266$		0			0
$267$		0			0
$268$		0			0
$269$	77.6431	+	44.8272i	0.288636	+	0.166644i	0.637327	−	0.770594i	$-0.280040\pi$
$269$	77.6431	+	44.8272i	0.288636	+	0.166644i	−0.348691	+	0.937238i	$0.613374\pi$
$270$		0			0
$271$	144.056	−	249.512i	0.531570	−	0.920707i	−0.467751	−	0.883861i	$-0.654935\pi$
$271$	144.056	−	249.512i	0.531570	−	0.920707i	0.999321	−	0.0368464i	$-0.0117312\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	94.2473	−	54.4137i	0.342717	−	0.197868i
$276$		0			0
$277$	313.089			1.13029			0.565143	−	0.824993i	$-0.308821\pi$
$277$	313.089			1.13029			0.565143	+	0.824993i	$0.308821\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	264.402	−	152.652i	0.940931	−	0.543247i	0.0506790	−	0.998715i	$-0.483861\pi$
$281$	264.402	−	152.652i	0.940931	−	0.543247i	0.890252	+	0.455468i	$0.150528\pi$
$282$		0			0
$283$	−172.287	+	298.410i	−0.608788	+	1.05445i	0.382652	+	0.923892i	$0.375011\pi$
$283$	−172.287	+	298.410i	−0.608788	+	1.05445i	−0.991440	+	0.130560i	$0.958323\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−225.736	−	130.329i	−0.786538	−	0.454108i
$288$		0			0
$289$	−95.4146	−	165.263i	−0.330154	−	0.571844i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	139.349i		−	0.475596i	−0.971315	−	0.237798i	$-0.923574\pi$
$293$		−	139.349i		−	0.475596i	0.971315	−	0.237798i	$-0.0764255\pi$
$294$		0			0
$295$	98.5947	+	170.771i	0.334219	+	0.578885i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	25.2233	+	14.5627i	0.0843590	+	0.0487047i
$300$		0			0
$301$	−76.7673	+	132.965i	−0.255041	+	0.441743i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			189.633i			0.621748i
$306$		0			0
$307$	77.7041	−	134.587i	0.253108	−	0.438395i	−0.711272	−	0.702917i	$-0.751881\pi$
$307$	77.7041	−	134.587i	0.253108	−	0.438395i	0.964380	+	0.264521i	$0.0852139\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	505.981i		−	1.62695i	−0.581600	−	0.813475i	$-0.697573\pi$
$311$		−	505.981i		−	1.62695i	0.581600	−	0.813475i	$-0.302427\pi$
$312$		0			0
$313$	−125.126	−	216.724i	−0.399762	−	0.692409i	0.593934	−	0.804514i	$-0.297574\pi$
$313$	−125.126	−	216.724i	−0.399762	−	0.692409i	−0.993696	+	0.112105i	$0.964241\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−276.479	+	159.625i	−0.872175	+	0.503550i	−0.868070	−	0.496441i	$-0.834640\pi$
$317$	−276.479	+	159.625i	−0.872175	+	0.503550i	−0.00410458	+	0.999992i	$0.501307\pi$
$318$		0			0
$319$	−26.3475	−	45.6352i	−0.0825941	−	0.143057i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−141.739	+	123.894i	−0.438820	+	0.383573i
$324$		0			0
$325$	−50.3721	+	87.2470i	−0.154991	+	0.268452i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−206.683	+	119.329i	−0.628217	+	0.362701i
$330$		0			0
$331$	309.260			0.934319			0.467160	−	0.884173i	$-0.345277\pi$
$331$	309.260			0.934319			0.467160	+	0.884173i	$0.345277\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			312.629i			0.933221i
$336$		0			0
$337$	−175.382	+	303.770i	−0.520421	+	0.901396i	0.479297	+	0.877653i	$0.340892\pi$
$337$	−175.382	+	303.770i	−0.520421	+	0.901396i	−0.999718	+	0.0237432i	$0.992442\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			223.949i			0.656742i
$342$		0			0
$343$	−338.853			−0.987911
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−191.199	−	110.389i	−0.551005	−	0.318123i	0.198522	−	0.980096i	$-0.436386\pi$
$347$	−191.199	−	110.389i	−0.551005	−	0.318123i	−0.749527	+	0.661974i	$0.769719\pi$
$348$		0			0
$349$	−242.093			−0.693677			−0.346838	−	0.937925i	$-0.612745\pi$
$349$	−242.093			−0.693677			−0.346838	+	0.937925i	$0.612745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	223.397i		−	0.632851i	−0.948617	−	0.316426i	$-0.897517\pi$
$353$		−	223.397i		−	0.632851i	0.948617	−	0.316426i	$-0.102483\pi$
$354$		0			0
$355$	232.817	+	403.250i	0.655822	+	1.13592i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−449.396	−	259.459i	−1.25180	−	0.722726i	−0.280332	−	0.959903i	$-0.590445\pi$
$359$	−449.396	−	259.459i	−1.25180	−	0.722726i	−0.971466	+	0.237177i	$0.923778\pi$
$360$		0			0
$361$	−285.684	−	220.694i	−0.791368	−	0.611340i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−452.028	+	260.978i	−1.23843	+	0.715009i
$366$		0			0
$367$	−90.2493	−	156.316i	−0.245911	−	0.425930i	0.716476	−	0.697611i	$-0.245754\pi$
$367$	−90.2493	−	156.316i	−0.245911	−	0.425930i	−0.962387	+	0.271681i	$0.912420\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−92.7855	+	53.5697i	−0.250096	+	0.144393i
$372$		0			0
$373$	−653.204			−1.75122			−0.875609	−	0.483021i	$-0.839539\pi$
$373$	−653.204			−1.75122			−0.875609	+	0.483021i	$0.839539\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	42.2457	+	24.3905i	0.112057	+	0.0646964i
$378$		0			0
$379$	507.851			1.33998			0.669988	−	0.742372i	$-0.266299\pi$
$379$	507.851			1.33998			0.669988	+	0.742372i	$0.266299\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	338.570	+	195.473i	0.883994	+	0.510374i	0.871973	−	0.489554i	$-0.162840\pi$
$383$	338.570	+	195.473i	0.883994	+	0.510374i	0.0120208	+	0.999928i	$0.496174\pi$
$384$		0			0
$385$	123.325	−	213.606i	0.320326	−	0.554821i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	283.906	−	163.913i	0.729835	−	0.421371i	−0.0885266	−	0.996074i	$-0.528216\pi$
$389$	283.906	−	163.913i	0.729835	−	0.421371i	0.818362	+	0.574703i	$0.194883\pi$
$390$		0			0
$391$	22.2033			0.0567859
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−35.0527	+	20.2377i	−0.0887410	+	0.0512346i
$396$		0			0
$397$	342.484	−	593.200i	0.862681	−	1.49421i	−0.00665080	−	0.999978i	$-0.502117\pi$
$397$	342.484	−	593.200i	0.862681	−	1.49421i	0.869332	−	0.494229i	$-0.164550\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−314.647	−	181.662i	−0.784657	−	0.453022i	0.0534212	−	0.998572i	$-0.482987\pi$
$401$	−314.647	−	181.662i	−0.784657	−	0.453022i	−0.838078	+	0.545550i	$0.816321\pi$
$402$		0			0
$403$	−103.658	−	179.540i	−0.257215	−	0.445509i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	304.920i		−	0.749189i
$408$		0			0
$409$	43.4155	+	75.1978i	0.106150	+	0.183858i	0.914208	−	0.405246i	$-0.132814\pi$
$409$	43.4155	+	75.1978i	0.106150	+	0.183858i	−0.808057	+	0.589104i	$0.799481\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−173.931	−	100.419i	−0.421141	−	0.243146i
$414$		0			0
$415$	84.9880	−	147.204i	0.204790	−	0.354707i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	244.411i		−	0.583320i	−0.956522	−	0.291660i	$-0.905793\pi$
$419$		−	244.411i		−	0.583320i	0.956522	−	0.291660i	$-0.0942075\pi$
$420$		0			0
$421$	−0.718794	+	1.24499i	−0.00170735	+	0.00295721i	−0.866878	−	0.498521i	$-0.833877\pi$
$421$	−0.718794	+	1.24499i	−0.00170735	+	0.00295721i	0.865170	+	0.501478i	$0.167210\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			76.8008i			0.180708i
$426$		0			0
$427$	−96.5711	−	167.266i	−0.226162	−	0.391724i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−352.626	+	203.589i	−0.818158	+	0.472364i	−0.849781	−	0.527136i	$-0.823266\pi$
$431$	−352.626	+	203.589i	−0.818158	+	0.472364i	0.0316230	+	0.999500i	$0.489932\pi$
$432$		0			0
$433$	224.924	+	389.579i	0.519454	+	0.899721i	0.999744	+	0.0226110i	$0.00719792\pi$
$433$	224.924	+	389.579i	0.519454	+	0.899721i	−0.480290	+	0.877109i	$0.659469\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.23851	+	41.7728i	0.0188524	+	0.0955900i
$438$		0			0
$439$	262.683	−	454.980i	0.598366	−	1.03640i	−0.394696	−	0.918812i	$-0.629150\pi$
$439$	262.683	−	454.980i	0.598366	−	1.03640i	0.993062	−	0.117589i	$-0.0375166\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	207.177	−	119.613i	0.467667	−	0.270008i	−0.247595	−	0.968864i	$-0.579640\pi$
$443$	207.177	−	119.613i	0.467667	−	0.270008i	0.715263	+	0.698856i	$0.246307\pi$
$444$		0			0
$445$	−456.856			−1.02664
$446$		0			0
$447$		0			0
$448$		0			0
$449$			323.795i			0.721147i	0.932731	+	0.360574i	$0.117419\pi$
$449$			323.795i			0.721147i	−0.932731	+	0.360574i	$0.882581\pi$
$450$		0			0
$451$	432.577	−	749.245i	0.959150	−	1.66130i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			228.331i			0.501826i
$456$		0			0
$457$	324.399			0.709844			0.354922	−	0.934896i	$-0.384507\pi$
$457$	324.399			0.709844			0.354922	+	0.934896i	$0.384507\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	522.958	+	301.930i	1.13440	+	0.654946i	0.945038	−	0.326962i	$-0.106025\pi$
$461$	522.958	+	301.930i	1.13440	+	0.654946i	0.189362	+	0.981907i	$0.439358\pi$
$462$		0			0
$463$	−237.035			−0.511954			−0.255977	−	0.966683i	$-0.582397\pi$
$463$	−237.035			−0.511954			−0.255977	+	0.966683i	$0.582397\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			412.245i			0.882751i	0.897322	+	0.441376i	$0.145509\pi$
$467$			412.245i			0.882751i	−0.897322	+	0.441376i	$0.854491\pi$
$468$		0			0
$469$	−159.207	−	275.755i	−0.339461	−	0.587964i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−441.325	−	254.799i	−0.933034	−	0.538688i
$474$		0			0
$475$	−144.491	+	28.4968i	−0.304193	+	0.0599934i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−684.912	+	395.434i	−1.42988	+	0.825542i	−0.997111	−	0.0759632i	$-0.975797\pi$
$479$	−684.912	+	395.434i	−1.42988	+	0.825542i	−0.432769	+	0.901505i	$0.642464\pi$
$480$		0			0
$481$	141.136	+	244.455i	0.293422	+	0.508221i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	38.5918	−	22.2810i	0.0795708	−	0.0459402i
$486$		0			0
$487$	−61.6406			−0.126572			−0.0632860	−	0.997995i	$-0.520158\pi$
$487$	−61.6406			−0.126572			−0.0632860	+	0.997995i	$0.520158\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.13392	−	5.27347i	−0.0186027	−	0.0107403i	0.490670	−	0.871346i	$-0.336752\pi$
$491$	−9.13392	−	5.27347i	−0.0186027	−	0.0107403i	−0.509273	+	0.860605i	$0.670085\pi$
$492$		0			0
$493$	37.1875			0.0754311
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−410.712	−	237.125i	−0.826383	−	0.477112i
$498$		0			0
$499$	−48.7012	+	84.3529i	−0.0975976	+	0.169044i	−0.910690	−	0.413091i	$-0.864449\pi$
$499$	−48.7012	+	84.3529i	−0.0975976	+	0.169044i	0.813092	+	0.582135i	$0.197782\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−444.712	+	256.754i	−0.884119	+	0.510446i	−0.872014	−	0.489481i	$-0.837186\pi$
$503$	−444.712	+	256.754i	−0.884119	+	0.510446i	−0.0121045	+	0.999927i	$0.503853\pi$
$504$		0			0
$505$	122.571			0.242715
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−746.779	+	431.153i	−1.46715	+	0.847059i	−0.999324	−	0.0367617i	$-0.988296\pi$
$509$	−746.779	+	431.153i	−1.46715	+	0.847059i	−0.467825	+	0.883821i	$0.654962\pi$
$510$		0			0
$511$	265.808	−	460.393i	0.520172	−	0.900964i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−565.468	−	326.473i	−1.09800	−	0.633928i
$516$		0			0
$517$	−396.066	−	686.006i	−0.766085	−	1.32690i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	788.311i		−	1.51307i	−0.653951	−	0.756537i	$-0.726890\pi$
$521$		−	788.311i		−	1.51307i	0.653951	−	0.756537i	$-0.273110\pi$
$522$		0			0
$523$	135.871	+	235.335i	0.259791	+	0.449971i	0.966186	−	0.257847i	$-0.0830130\pi$
$523$	135.871	+	235.335i	0.259791	+	0.449971i	−0.706395	+	0.707818i	$0.749680\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−136.870	−	79.0217i	−0.259715	−	0.149946i
$528$		0			0
$529$	−261.989	+	453.778i	−0.495254	+	0.857804i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			800.894i			1.50262i
$534$		0			0
$535$	−248.582	+	430.556i	−0.464639	+	0.804778i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	436.739i		−	0.810276i
$540$		0			0
$541$	−169.858	−	294.202i	−0.313970	−	0.543811i	0.665248	−	0.746622i	$-0.268326\pi$
$541$	−169.858	−	294.202i	−0.313970	−	0.543811i	−0.979218	+	0.202811i	$0.934992\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	482.322	−	278.469i	0.884994	−	0.510952i
$546$		0			0
$547$	−205.911	−	356.648i	−0.376436	−	0.652007i	0.614105	−	0.789225i	$-0.289517\pi$
$547$	−205.911	−	356.648i	−0.376436	−	0.652007i	−0.990541	+	0.137218i	$0.956184\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.7984	+	69.9639i	0.0250424	+	0.126976i
$552$		0			0
$553$	20.6122	−	35.7013i	0.0372734	−	0.0645594i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−224.969	+	129.886i	−0.403894	+	0.233188i	−0.688163	−	0.725556i	$-0.741582\pi$
$557$	−224.969	+	129.886i	−0.403894	+	0.233188i	0.284269	+	0.958745i	$0.408249\pi$
$558$		0			0
$559$	471.748			0.843914
$560$		0			0
$561$		0			0
$562$		0			0
$563$			543.269i			0.964954i	0.875909	+	0.482477i	$0.160263\pi$
$563$			543.269i			0.964954i	−0.875909	+	0.482477i	$0.839737\pi$
$564$		0			0
$565$	104.452	−	180.915i	0.184870	−	0.320204i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	638.654i		−	1.12241i	−0.827675	−	0.561207i	$-0.810337\pi$
$569$		−	638.654i		−	1.12241i	0.827675	−	0.561207i	$-0.189663\pi$
$570$		0			0
$571$	−462.570			−0.810106			−0.405053	−	0.914293i	$-0.632747\pi$
$571$	−462.570			−0.810106			−0.405053	+	0.914293i	$0.632747\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	15.0429	+	8.68503i	0.0261616	+	0.0151044i
$576$		0			0
$577$	666.137			1.15448			0.577242	−	0.816573i	$-0.304129\pi$
$577$	666.137			1.15448			0.577242	+	0.816573i	$0.304129\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			173.121i			0.297971i
$582$		0			0
$583$	−177.804	−	307.965i	−0.304981	−	0.528243i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−739.307	−	426.839i	−1.25947	−	0.727153i	−0.286495	−	0.958082i	$-0.592490\pi$
$587$	−739.307	−	426.839i	−1.25947	−	0.727153i	−0.972971	+	0.230929i	$0.925824\pi$
$588$		0			0
$589$	97.8846	−	286.825i	0.166188	−	0.486969i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	51.2238	−	29.5741i	0.0863809	−	0.0498720i	−0.456187	−	0.889884i	$-0.650785\pi$
$593$	51.2238	−	29.5741i	0.0863809	−	0.0498720i	0.542568	+	0.840012i	$0.317452\pi$
$594$		0			0
$595$	87.0323	+	150.744i	0.146273	+	0.253352i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	660.176	−	381.153i	1.10213	−	0.636315i	0.165350	−	0.986235i	$-0.447125\pi$
$599$	660.176	−	381.153i	1.10213	−	0.636315i	0.936780	+	0.349920i	$0.113791\pi$
$600$		0			0
$601$	423.724			0.705032			0.352516	−	0.935806i	$-0.385326\pi$
$601$	423.724			0.705032			0.352516	+	0.935806i	$0.385326\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	273.777	+	158.065i	0.452524	+	0.261265i
$606$		0			0
$607$	−664.575			−1.09485			−0.547426	−	0.836854i	$-0.684392\pi$
$607$	−664.575			−1.09485			−0.547426	+	0.836854i	$0.684392\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	635.053	+	366.648i	1.03937	+	0.600078i
$612$		0			0
$613$	−314.973	+	545.549i	−0.513822	+	0.889967i	0.486049	+	0.873932i	$0.338438\pi$
$613$	−314.973	+	545.549i	−0.513822	+	0.889967i	−0.999871	+	0.0160350i	$0.994896\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	73.4504	−	42.4066i	0.119044	−	0.0687303i	−0.439295	−	0.898343i	$-0.644772\pi$
$617$	73.4504	−	42.4066i	0.119044	−	0.0687303i	0.558340	+	0.829612i	$0.311439\pi$
$618$		0			0
$619$	−1087.39			−1.75669			−0.878343	−	0.478030i	$-0.841351\pi$
$619$	−1087.39			−1.75669			−0.878343	+	0.478030i	$0.841351\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	402.971	−	232.655i	0.646823	−	0.373443i
$624$		0			0
$625$	185.567	−	321.412i	0.296908	−	0.514259i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	186.356	+	107.593i	0.296274	+	0.171054i
$630$		0			0
$631$	215.282	+	372.879i	0.341176	+	0.590934i	0.984651	−	0.174533i	$-0.0558415\pi$
$631$	215.282	+	372.879i	0.341176	+	0.590934i	−0.643476	+	0.765467i	$0.722508\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	289.629i		−	0.456108i
$636$		0			0
$637$	202.150	+	350.134i	0.317347	+	0.549661i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1049.92	−	606.171i	−1.63794	−	0.945665i	−0.981541	−	0.191252i	$-0.938745\pi$
$641$	−1049.92	−	606.171i	−1.63794	−	0.945665i	−0.656399	−	0.754414i	$-0.727921\pi$
$642$		0			0
$643$	−570.835	+	988.715i	−0.887768	+	1.53766i	−0.0452608	+	0.998975i	$0.514412\pi$
$643$	−570.835	+	988.715i	−0.887768	+	1.53766i	−0.842507	+	0.538685i	$0.818921\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			765.970i			1.18388i	0.805982	+	0.591940i	$0.201638\pi$
$647$			765.970i			1.18388i	−0.805982	+	0.591940i	$0.798362\pi$
$648$		0			0
$649$	333.303	−	577.297i	0.513564	−	0.889518i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			57.7200i			0.0883920i	0.999023	+	0.0441960i	$0.0140726\pi$
$653$			57.7200i			0.0883920i	−0.999023	+	0.0441960i	$0.985927\pi$
$654$		0			0
$655$	105.598	+	182.900i	0.161218	+	0.279237i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	967.814	−	558.768i	1.46861	−	0.847903i	0.469229	−	0.883076i	$-0.344532\pi$
$659$	967.814	−	558.768i	1.46861	−	0.847903i	0.999381	+	0.0351737i	$0.0111985\pi$
$660$		0			0
$661$	64.5440	+	111.793i	0.0976459	+	0.169128i	0.910710	−	0.413047i	$-0.135535\pi$
$661$	64.5440	+	111.793i	0.0976459	+	0.169128i	−0.813064	+	0.582174i	$0.802202\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−251.314	+	219.674i	−0.377916	+	0.330337i
$666$		0			0
$667$	4.20536	−	7.28389i	0.00630488	−	0.0109204i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	555.175	−	320.530i	0.827384	−	0.477691i
$672$		0			0
$673$	525.024			0.780126			0.390063	−	0.920788i	$-0.372453\pi$
$673$	525.024			0.780126			0.390063	+	0.920788i	$0.372453\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			747.884i			1.10470i	0.833611	+	0.552351i	$0.186269\pi$
$677$			747.884i			1.10470i	−0.833611	+	0.552351i	$0.813731\pi$
$678$		0			0
$679$	−22.6933	+	39.3060i	−0.0334217	+	0.0578880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	17.5209i		−	0.0256529i	−0.999918	−	0.0128264i	$-0.995917\pi$
$683$		−	17.5209i		−	0.0256529i	0.999918	−	0.0128264i	$-0.00408289\pi$
$684$		0			0
$685$	−129.396			−0.188899
$686$		0			0
$687$		0			0
$688$		0			0
$689$	285.091	+	164.598i	0.413775	+	0.238893i
$690$		0			0
$691$	439.543			0.636098			0.318049	−	0.948074i	$-0.396972\pi$
$691$	439.543			0.636098			0.318049	+	0.948074i	$0.396972\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			309.026i			0.444641i
$696$		0			0
$697$	305.275	+	528.751i	0.437984	+	0.758610i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−777.026	−	448.616i	−1.10845	−	0.639966i	−0.170026	−	0.985440i	$-0.554385\pi$
$701$	−777.026	−	448.616i	−1.10845	−	0.639966i	−0.938429	+	0.345473i	$0.887718\pi$
$702$		0			0
$703$	−133.276	+	390.529i	−0.189581	+	0.555518i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−108.114	+	62.4195i	−0.152919	+	0.0882878i
$708$		0			0
$709$	375.521	+	650.422i	0.529649	+	0.917379i	0.999402	+	0.0345811i	$0.0110097\pi$
$709$	375.521	+	650.422i	0.529649	+	0.917379i	−0.469753	+	0.882798i	$0.655657\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−30.9558	+	17.8724i	−0.0434163	+	0.0250664i
$714$		0			0
$715$	−757.856			−1.05994
$716$		0			0
$717$		0			0
$718$		0			0
$719$	449.042	+	259.254i	0.624537	+	0.360576i	0.778633	−	0.627479i	$-0.215913\pi$
$719$	449.042	+	259.254i	0.624537	+	0.360576i	−0.154096	+	0.988056i	$0.549247\pi$
$720$		0			0
$721$	665.029			0.922370
$722$		0			0
$723$		0			0
$724$		0			0
$725$	25.1948	+	14.5462i	0.0347515	+	0.0200638i
$726$		0			0
$727$	−475.501	+	823.592i	−0.654060	+	1.13286i	0.328069	+	0.944654i	$0.393602\pi$
$727$	−475.501	+	823.592i	−0.654060	+	1.13286i	−0.982129	+	0.188211i	$0.939731\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	311.449	−	179.815i	0.426058	−	0.245985i
$732$		0			0
$733$	−894.995			−1.22100			−0.610501	−	0.792015i	$-0.709032\pi$
$733$	−894.995			−1.22100			−0.610501	+	0.792015i	$0.709032\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	915.262	−	528.427i	1.24187	−	0.716997i
$738$		0			0
$739$	568.726	−	985.062i	0.769588	−	1.33297i	−0.168198	−	0.985753i	$-0.553795\pi$
$739$	568.726	−	985.062i	0.769588	−	1.33297i	0.937786	−	0.347213i	$-0.112872\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1062.56	+	613.467i	1.43009	+	0.825663i	0.997127	−	0.0757489i	$-0.0241347\pi$
$743$	1062.56	+	613.467i	1.43009	+	0.825663i	0.432963	+	0.901412i	$0.357468\pi$
$744$		0			0
$745$	−277.448	−	480.555i	−0.372414	−	0.645040i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	506.364i		−	0.676053i
$750$		0			0
$751$	490.683	+	849.888i	0.653373	+	1.13168i	0.982299	+	0.187320i	$0.0599800\pi$
$751$	490.683	+	849.888i	0.653373	+	1.13168i	−0.328926	+	0.944356i	$0.606687\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	319.872	+	184.678i	0.423672	+	0.244607i
$756$		0			0
$757$	517.674	−	896.637i	0.683849	−	1.18446i	−0.289948	−	0.957042i	$-0.593638\pi$
$757$	517.674	−	896.637i	0.683849	−	1.18446i	0.973797	−	0.227419i	$-0.0730286\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			494.493i			0.649794i	0.945749	+	0.324897i	$0.105330\pi$
$761$			494.493i			0.649794i	−0.945749	+	0.324897i	$0.894670\pi$
$762$		0			0
$763$	−283.622	+	491.247i	−0.371719	+	0.643836i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			617.093i			0.804555i
$768$		0			0
$769$	457.121	+	791.757i	0.594436	+	1.02959i	0.993626	+	0.112725i	$0.0359578\pi$
$769$	457.121	+	791.757i	0.594436	+	1.02959i	−0.399191	+	0.916868i	$0.630709\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	931.860	−	538.010i	1.20551	−	0.696002i	0.243736	−	0.969842i	$-0.421627\pi$
$773$	931.860	−	538.010i	1.20551	−	0.696002i	0.961775	+	0.273840i	$0.0882938\pi$
$774$		0			0
$775$	−61.8202	−	107.076i	−0.0797680	−	0.138162i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−881.510	+	770.530i	−1.13159	+	0.989127i
$780$		0			0
$781$	787.044	−	1363.20i	1.00774	−	1.74546i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1027.92	−	593.469i	1.30945	−	0.756011i
$786$		0			0
$787$	771.646			0.980490			0.490245	−	0.871585i	$-0.336907\pi$
$787$	771.646			0.980490			0.490245	+	0.871585i	$0.336907\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			212.769i			0.268987i
$792$		0			0
$793$	−296.723	+	513.939i	−0.374178	+	0.648095i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	702.863i		−	0.881886i	−0.897535	−	0.440943i	$-0.854644\pi$
$797$		−	702.863i		−	0.881886i	0.897535	−	0.440943i	$-0.145356\pi$
$798$		0			0
$799$	559.017			0.699646
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1528.10	+	882.246i	1.90298	+	1.09869i
$804$		0			0
$805$	39.3682			0.0489046
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1566.47i		−	1.93630i	−0.250370	−	0.968150i	$-0.580552\pi$
$809$		−	1566.47i		−	1.93630i	0.250370	−	0.968150i	$-0.419448\pi$
$810$		0			0
$811$	−357.593	−	619.369i	−0.440928	−	0.763710i	0.556830	−	0.830626i	$-0.312017\pi$
$811$	−357.593	−	619.369i	−0.440928	−	0.763710i	−0.997759	+	0.0669160i	$0.978684\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−232.313	−	134.126i	−0.285046	−	0.164572i
$816$		0			0
$817$	453.863	+	519.233i	0.555524	+	0.635536i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1265.69	+	730.745i	−1.54164	+	0.890067i	−0.542907	+	0.839793i	$0.682676\pi$
$821$	−1265.69	+	730.745i	−1.54164	+	0.890067i	−0.998735	+	0.0502744i	$0.983990\pi$
$822$		0			0
$823$	−38.9668	−	67.4924i	−0.0473472	−	0.0820078i	0.841381	−	0.540443i	$-0.181743\pi$
$823$	−38.9668	−	67.4924i	−0.0473472	−	0.0820078i	−0.888728	+	0.458435i	$0.848410\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1332.74	−	769.459i	1.61154	−	0.930422i	0.622525	−	0.782600i	$-0.286107\pi$
$827$	1332.74	−	769.459i	1.61154	−	0.930422i	0.989014	−	0.147822i	$-0.0472264\pi$
$828$		0			0
$829$	−928.465			−1.11998			−0.559991	−	0.828499i	$-0.689195\pi$
$829$	−928.465			−1.11998			−0.559991	+	0.828499i	$0.689195\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	266.919	+	154.106i	0.320431	+	0.185001i
$834$		0			0
$835$	−837.738			−1.00328
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−116.948	−	67.5200i	−0.139390	−	0.0804768i	0.428683	−	0.903455i	$-0.358978\pi$
$839$	−116.948	−	67.5200i	−0.139390	−	0.0804768i	−0.568073	+	0.822978i	$0.692311\pi$
$840$		0			0
$841$	−413.457	+	716.128i	−0.491625	+	0.851519i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.274450	+	0.158454i	−0.000324793	+	0.000187519i
$846$		0			0
$847$	−321.981			−0.380142
$848$		0			0
$849$		0			0
$850$		0			0
$851$	42.1482	−	24.3343i	0.0495279	−	0.0285949i
$852$		0			0
$853$	184.342	−	319.289i	0.216110	−	0.374313i	−0.737506	−	0.675341i	$-0.763996\pi$
$853$	184.342	−	319.289i	0.216110	−	0.374313i	0.953615	+	0.301028i	$0.0973297\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−394.797	−	227.936i	−0.460673	−	0.265970i	0.251654	−	0.967817i	$-0.419025\pi$
$857$	−394.797	−	227.936i	−0.460673	−	0.265970i	−0.712327	+	0.701847i	$0.752359\pi$
$858$		0			0
$859$	275.072	+	476.439i	0.320224	+	0.554643i	0.980534	−	0.196349i	$-0.0629086\pi$
$859$	275.072	+	476.439i	0.320224	+	0.554643i	−0.660310	+	0.750993i	$0.729575\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	11.5243i		−	0.0133538i	−0.999978	−	0.00667688i	$-0.997875\pi$
$863$		−	11.5243i		−	0.0133538i	0.999978	−	0.00667688i	$-0.00212533\pi$
$864$		0			0
$865$	149.598	+	259.111i	0.172945	+	0.299550i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	118.497	+	68.4141i	0.136360	+	0.0787274i
$870$		0			0
$871$	−489.178	+	847.281i	−0.561628	+	0.972768i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			575.371i			0.657567i
$876$		0			0
$877$	−410.441	+	710.905i	−0.468006	+	0.810610i	−0.999332	−	0.0365580i	$-0.988361\pi$
$877$	−410.441	+	710.905i	−0.468006	+	0.810610i	0.531326	+	0.847168i	$0.321694\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	811.431i		−	0.921034i	−0.887651	−	0.460517i	$-0.847664\pi$
$881$		−	811.431i		−	0.921034i	0.887651	−	0.460517i	$-0.152336\pi$
$882$		0			0
$883$	153.564	+	265.980i	0.173911	+	0.301223i	0.939784	−	0.341769i	$-0.111026\pi$
$883$	153.564	+	265.980i	0.173911	+	0.301223i	−0.765873	+	0.642992i	$0.777693\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	552.484	−	318.977i	0.622868	−	0.359613i	−0.155117	−	0.987896i	$-0.549575\pi$
$887$	552.484	−	318.977i	0.622868	−	0.359613i	0.777985	+	0.628283i	$0.216242\pi$
$888$		0			0
$889$	147.494	+	255.467i	0.165910	+	0.287365i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	207.423	+	1051.72i	0.232276	+	1.17774i
$894$		0			0
$895$	−49.9660	+	86.5437i	−0.0558279	+	0.0966968i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−51.8469	+	29.9338i	−0.0576717	+	0.0332968i
$900$		0			0
$901$	250.957			0.278531
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	68.3780i		−	0.0755558i
$906$		0			0
$907$	249.819	−	432.699i	0.275434	−	0.477066i	−0.694811	−	0.719193i	$-0.744512\pi$
$907$	249.819	−	432.699i	0.275434	−	0.477066i	0.970245	+	0.242127i	$0.0778451\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	479.613i		−	0.526469i	−0.964732	−	0.263235i	$-0.915211\pi$
$911$		−	479.613i		−	0.526469i	0.964732	−	0.263235i	$-0.0847893\pi$
$912$		0			0
$913$	−574.609			−0.629364
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−186.285	−	107.552i	−0.203146	−	0.117286i
$918$		0			0
$919$	591.828			0.643991			0.321996	−	0.946741i	$-0.395646\pi$
$919$	591.828			0.643991			0.321996	+	0.946741i	$0.395646\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1457.17i			1.57874i
$924$		0			0
$925$	84.1719	+	145.790i	0.0909966	+	0.157611i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1276.23	−	736.834i	−1.37377	−	0.793148i	−0.382371	−	0.924009i	$-0.624892\pi$
$929$	−1276.23	−	736.834i	−1.37377	−	0.793148i	−0.991401	+	0.130861i	$0.958226\pi$
$930$		0			0
$931$	−190.892	+	559.358i	−0.205040	+	0.600814i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−500.338	+	288.870i	−0.535120	+	0.308952i
$936$		0			0
$937$	590.021	+	1021.95i	0.629691	+	1.09066i	0.987614	+	0.156906i	$0.0501519\pi$
$937$	590.021	+	1021.95i	0.629691	+	1.09066i	−0.357923	+	0.933751i	$0.616515\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	585.870	−	338.252i	0.622603	−	0.359460i	−0.155279	−	0.987871i	$-0.549628\pi$
$941$	585.870	−	338.252i	0.622603	−	0.359460i	0.777882	+	0.628411i	$0.216294\pi$
$942$		0			0
$943$	138.088			0.146435
$944$		0			0
$945$		0			0
$946$		0			0
$947$	533.197	+	307.842i	0.563039	+	0.325070i	0.754364	−	0.656456i	$-0.227945\pi$
$947$	533.197	+	307.842i	0.563039	+	0.325070i	−0.191326	+	0.981527i	$0.561279\pi$
$948$		0			0
$949$	−1633.43			−1.72122
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1563.42	+	902.643i	1.64053	+	0.947160i	0.980645	+	0.195792i	$0.0627277\pi$
$953$	1563.42	+	902.643i	1.64053	+	0.947160i	0.659883	+	0.751368i	$0.270606\pi$
$954$		0			0
$955$	−493.381	+	854.562i	−0.516630	+	0.894829i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	114.133	−	65.8950i	0.119013	−	0.0687122i
$960$		0			0
$961$	−706.568			−0.735243
$962$		0			0
$963$		0			0
$964$		0			0
$965$	768.037	−	443.426i	0.795893	−	0.459509i
$966$		0			0
$967$	121.852	−	211.053i	0.126010	−	0.218256i	−0.796117	−	0.605142i	$-0.793116\pi$
$967$	121.852	−	211.053i	0.126010	−	0.218256i	0.922127	+	0.386887i	$0.126450\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1521.31	−	878.327i	−1.56674	−	0.904559i	−0.996545	−	0.0830512i	$-0.973534\pi$
$971$	−1521.31	−	878.327i	−1.56674	−	0.904559i	−0.570197	−	0.821508i	$-0.693133\pi$
$972$		0			0
$973$	−157.372	−	272.576i	−0.161739	−	0.280140i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1881.50i		−	1.92579i	−0.269875	−	0.962895i	$-0.586982\pi$
$977$		−	1881.50i		−	1.92579i	0.269875	−	0.962895i	$-0.413018\pi$
$978$		0			0
$979$	772.209	+	1337.50i	0.788773	+	1.36620i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	383.562	+	221.450i	0.390195	+	0.225279i	0.682245	−	0.731124i	$-0.261004\pi$
$983$	383.562	+	221.450i	0.390195	+	0.225279i	−0.292050	+	0.956403i	$0.594337\pi$
$984$		0			0
$985$	103.102	−	178.578i	0.104672	−	0.181298i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	81.3376i		−	0.0822422i
$990$		0			0
$991$	359.853	−	623.283i	0.363121	−	0.628943i	−0.625352	−	0.780343i	$-0.715045\pi$
$991$	359.853	−	623.283i	0.363121	−	0.628943i	0.988473	+	0.151399i	$0.0483779\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	1079.71i		−	1.08514i
$996$		0			0
$997$	−59.8849	−	103.724i	−0.0600650	−	0.104036i	0.834429	−	0.551115i	$-0.185797\pi$
$997$	−59.8849	−	103.724i	−0.0600650	−	0.104036i	−0.894494	+	0.447079i	$0.852464\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.bj.a.125.4	✓	24
3.2	odd	2	inner	684.3.bj.a.125.9	yes	24
19.7	even	3	inner	684.3.bj.a.197.9	yes	24
57.26	odd	6	inner	684.3.bj.a.197.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.bj.a.125.4	✓	24	1.1	even	1	trivial
684.3.bj.a.125.9	yes	24	3.2	odd	2	inner
684.3.bj.a.197.4	yes	24	57.26	odd	6	inner
684.3.bj.a.197.9	yes	24	19.7	even	3	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.bj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.59674 - 2.07658i) q^{5} +4.23001 q^{7} +O(q^{10})\)	\(q+(-3.59674 - 2.07658i) q^{5} +4.23001 q^{7} +14.0399i q^{11} +(-6.49853 - 11.2558i) q^{13} +(-8.58068 - 4.95406i) q^{17} +(6.13661 - 17.9817i) q^{19} +(-1.94069 + 1.12046i) q^{23} +(-3.87565 - 6.71283i) q^{25} +(-3.25040 + 1.87662i) q^{29} +15.9509 q^{31} +(-15.2142 - 8.78394i) q^{35} -21.7181 q^{37} +(-53.3655 - 30.8106i) q^{41} +(-18.1483 + 31.4337i) q^{43} +(-48.8612 + 28.2101i) q^{47} -31.1070 q^{49} +(-21.9351 + 12.6642i) q^{53} +(29.1549 - 50.4978i) q^{55} +(-41.1184 - 23.7397i) q^{59} +(-22.8300 - 39.5427i) q^{61} +53.9788i q^{65} +(-37.6376 - 65.1901i) q^{67} +(-97.0949 - 56.0578i) q^{71} +(62.8386 - 108.840i) q^{73} +59.3888i q^{77} +(4.87284 - 8.44001i) q^{79} +40.9269i q^{83} +(20.5750 + 35.6369i) q^{85} +(95.2647 - 55.0011i) q^{89} +(-27.4888 - 47.6121i) q^{91} +(-59.4122 + 51.9324i) q^{95} +(-5.36484 + 9.29218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 16 q^{7}+O(q^{10}) \) 24 * q - 16 * q^7	\( 24 q - 16 q^{7} + 4 q^{13} - 60 q^{19} + 24 q^{25} + 40 q^{31} + 224 q^{37} + 52 q^{43} + 144 q^{49} + 156 q^{55} - 72 q^{61} + 124 q^{67} - 88 q^{73} - 28 q^{79} - 192 q^{85} + 236 q^{91}+O(q^{100}) \) 24 * q - 16 * q^7 + 4 * q^13 - 60 * q^19 + 24 * q^25 + 40 * q^31 + 224 * q^37 + 52 * q^43 + 144 * q^49 + 156 * q^55 - 72 * q^61 + 124 * q^67 - 88 * q^73 - 28 * q^79 - 192 * q^85 + 236 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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