LMFDB - Embedded newform 684.3.h.e.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.219615408.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 $ x^6 - 3x^5 + 22x^4 - 39x^3 + 112x^2 - 93*x + 39
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.1
Root			$0.500000 - 0.460304i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.3.h.e.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83539 q^{5} +5.24085 q^{7} +O(q^{10})$	$q-5.83539 q^{5} +5.24085 q^{7} -1.24085 q^{11} -5.89843i q^{13} +1.57007 q^{17} +(10.6708 + 15.7205i) q^{19} -27.5579 q^{23} +9.05177 q^{25} -15.9159i q^{29} +53.2554i q^{31} -30.5824 q^{35} -10.0175i q^{37} +69.8102i q^{41} -52.9116 q^{43} -12.2135 q^{47} -21.5335 q^{49} +40.4288i q^{53} +7.24085 q^{55} +75.8194i q^{59} -28.0275 q^{61} +34.4197i q^{65} +47.3570i q^{67} +56.3447i q^{71} -74.9363 q^{73} -6.50312 q^{77} -38.2585i q^{79} +42.6584 q^{83} -9.16198 q^{85} +24.5128i q^{89} -30.9128i q^{91} +(-62.2682 - 91.7353i) q^{95} -3.19999i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 6 * q + 2 * q^5 - 2 * q^7	$ 6 q + 2 q^{5} - 2 q^{7} + 26 q^{11} + 50 q^{17} - 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 210 q^{43} - 22 q^{47} - 36 q^{49} + 10 q^{55} + 214 q^{61} + 102 q^{73} - 266 q^{77} + 404 q^{83} + 370 q^{85} - 358 q^{95}+O(q^{100}) $ 6 * q + 2 * q^5 - 2 * q^7 + 26 * q^11 + 50 * q^17 - 10 * q^19 - 28 * q^23 + 28 * q^25 - 2 * q^35 - 210 * q^43 - 22 * q^47 - 36 * q^49 + 10 * q^55 + 214 * q^61 + 102 * q^73 - 266 * q^77 + 404 * q^83 + 370 * q^85 - 358 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−5.83539			−1.16708			−0.583539	−	0.812085i	$-0.698332\pi$
$5$	−5.83539			−1.16708			−0.583539	+	0.812085i	$0.698332\pi$
$6$		0			0
$7$	5.24085			0.748693			0.374347	−	0.927289i	$-0.377867\pi$
$7$	5.24085			0.748693			0.374347	+	0.927289i	$0.377867\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.24085			−0.112805			−0.0564023	−	0.998408i	$-0.517963\pi$
$11$	−1.24085			−0.112805			−0.0564023	+	0.998408i	$0.517963\pi$
$12$		0			0
$13$		−	5.89843i		−	0.453726i	−0.973927	−	0.226863i	$-0.927153\pi$
$13$		−	5.89843i		−	0.453726i	0.973927	−	0.226863i	$-0.0728469\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.57007			0.0923572			0.0461786	−	0.998933i	$-0.485296\pi$
$17$	1.57007			0.0923572			0.0461786	+	0.998933i	$0.485296\pi$
$18$		0			0
$19$	10.6708	+	15.7205i	0.561620	+	0.827395i
$19$	10.6708	+	15.7205i	0.561620	+	0.827395i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−27.5579			−1.19817			−0.599086	−	0.800685i	$-0.704469\pi$
$23$	−27.5579			−1.19817			−0.599086	+	0.800685i	$0.704469\pi$
$24$		0			0
$25$	9.05177			0.362071
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	15.9159i		−	0.548826i	−0.961612	−	0.274413i	$-0.911517\pi$
$29$		−	15.9159i		−	0.548826i	0.961612	−	0.274413i	$-0.0884835\pi$
$30$		0			0
$31$			53.2554i			1.71792i	0.512046	+	0.858958i	$0.328888\pi$
$31$			53.2554i			1.71792i	−0.512046	+	0.858958i	$0.671112\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−30.5824			−0.873783
$36$		0			0
$37$		−	10.0175i		−	0.270744i	−0.990795	−	0.135372i	$-0.956777\pi$
$37$		−	10.0175i		−	0.270744i	0.990795	−	0.135372i	$-0.0432229\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			69.8102i			1.70269i	0.524607	+	0.851344i	$0.324212\pi$
$41$			69.8102i			1.70269i	−0.524607	+	0.851344i	$0.675788\pi$
$42$		0			0
$43$	−52.9116			−1.23050			−0.615252	−	0.788331i	$-0.710946\pi$
$43$	−52.9116			−1.23050			−0.615252	+	0.788331i	$0.710946\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.2135			−0.259863			−0.129931	−	0.991523i	$-0.541476\pi$
$47$	−12.2135			−0.259863			−0.129931	+	0.991523i	$0.541476\pi$
$48$		0			0
$49$	−21.5335			−0.439459
$50$		0			0
$51$		0			0
$52$		0			0
$53$			40.4288i			0.762807i	0.924409	+	0.381403i	$0.124559\pi$
$53$			40.4288i			0.762807i	−0.924409	+	0.381403i	$0.875441\pi$
$54$		0			0
$55$	7.24085			0.131652
$56$		0			0
$57$		0			0
$58$		0			0
$59$			75.8194i			1.28507i	0.766255	+	0.642537i	$0.222118\pi$
$59$			75.8194i			1.28507i	−0.766255	+	0.642537i	$0.777882\pi$
$60$		0			0
$61$	−28.0275			−0.459468			−0.229734	−	0.973254i	$-0.573786\pi$
$61$	−28.0275			−0.459468			−0.229734	+	0.973254i	$0.573786\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			34.4197i			0.529533i
$66$		0			0
$67$			47.3570i			0.706820i	0.935468	+	0.353410i	$0.114978\pi$
$67$			47.3570i			0.706820i	−0.935468	+	0.353410i	$0.885022\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			56.3447i			0.793588i	0.917908	+	0.396794i	$0.129877\pi$
$71$			56.3447i			0.793588i	−0.917908	+	0.396794i	$0.870123\pi$
$72$		0			0
$73$	−74.9363			−1.02652			−0.513262	−	0.858232i	$-0.671563\pi$
$73$	−74.9363			−1.02652			−0.513262	+	0.858232i	$0.671563\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.50312			−0.0844561
$78$		0			0
$79$		−	38.2585i		−	0.484285i	−0.970241	−	0.242143i	$-0.922150\pi$
$79$		−	38.2585i		−	0.484285i	0.970241	−	0.242143i	$-0.0778502\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	42.6584			0.513957			0.256979	−	0.966417i	$-0.417273\pi$
$83$	42.6584			0.513957			0.256979	+	0.966417i	$0.417273\pi$
$84$		0			0
$85$	−9.16198			−0.107788
$86$		0			0
$87$		0			0
$88$		0			0
$89$			24.5128i			0.275425i	0.990472	+	0.137712i	$0.0439750\pi$
$89$			24.5128i			0.275425i	−0.990472	+	0.137712i	$0.956025\pi$
$90$		0			0
$91$		−	30.9128i		−	0.339701i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−62.2682	−	91.7353i	−0.655454	−	0.965635i
$96$		0			0
$97$		−	3.19999i		−	0.0329896i	−0.999864	−	0.0164948i	$-0.994749\pi$
$97$		−	3.19999i		−	0.0329896i	0.999864	−	0.0164948i	$-0.00525070\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	67.7650			0.670941			0.335471	−	0.942051i	$-0.391105\pi$
$101$	67.7650			0.670941			0.335471	+	0.942051i	$0.391105\pi$
$102$		0			0
$103$		−	50.0554i		−	0.485975i	−0.970029	−	0.242987i	$-0.921873\pi$
$103$		−	50.0554i		−	0.485975i	0.970029	−	0.242987i	$-0.0781274\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	128.605i		−	1.20192i	−0.799279	−	0.600960i	$-0.794785\pi$
$107$		−	128.605i		−	1.20192i	0.799279	−	0.600960i	$-0.205215\pi$
$108$		0			0
$109$		−	40.9303i		−	0.375508i	−0.982216	−	0.187754i	$-0.939879\pi$
$109$		−	40.9303i		−	0.375508i	0.982216	−	0.187754i	$-0.0601207\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			161.546i			1.42961i	0.699326	+	0.714803i	$0.253484\pi$
$113$			161.546i			1.42961i	−0.699326	+	0.714803i	$0.746516\pi$
$114$		0			0
$115$	160.811			1.39836
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.22851			0.0691472
$120$		0			0
$121$	−119.460			−0.987275
$122$		0			0
$123$		0			0
$124$		0			0
$125$	93.0641			0.744513
$126$		0			0
$127$			26.4617i			0.208360i	0.994558	+	0.104180i	$0.0332218\pi$
$127$			26.4617i			0.208360i	−0.994558	+	0.104180i	$0.966778\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−23.7282			−0.181132			−0.0905658	−	0.995890i	$-0.528868\pi$
$131$	−23.7282			−0.181132			−0.0905658	+	0.995890i	$0.528868\pi$
$132$		0			0
$133$	55.9240	+	82.3889i	0.420481	+	0.619465i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−127.778			−0.932683			−0.466342	−	0.884605i	$-0.654428\pi$
$137$	−127.778			−0.932683			−0.466342	+	0.884605i	$0.654428\pi$
$138$		0			0
$139$	104.967			0.755156			0.377578	−	0.925978i	$-0.376757\pi$
$139$	104.967			0.755156			0.377578	+	0.925978i	$0.376757\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7.31908i			0.0511824i
$144$		0			0
$145$			92.8758i			0.640522i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−144.372			−0.968940			−0.484470	−	0.874808i	$-0.660988\pi$
$149$	−144.372			−0.968940			−0.484470	+	0.874808i	$0.660988\pi$
$150$		0			0
$151$			34.0021i			0.225180i	0.993642	+	0.112590i	$0.0359146\pi$
$151$			34.0021i			0.225180i	−0.993642	+	0.112590i	$0.964085\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	310.766i		−	2.00494i
$156$		0			0
$157$	−167.385			−1.06615			−0.533073	−	0.846069i	$-0.678963\pi$
$157$	−167.385			−1.06615			−0.533073	+	0.846069i	$0.678963\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−144.427			−0.897063
$162$		0			0
$163$	212.281			1.30234			0.651169	−	0.758933i	$-0.274279\pi$
$163$	212.281			1.30234			0.651169	+	0.758933i	$0.274279\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			52.3097i			0.313231i	0.987660	+	0.156616i	$0.0500584\pi$
$167$			52.3097i			0.313231i	−0.987660	+	0.156616i	$0.949942\pi$
$168$		0			0
$169$	134.208			0.794133
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	14.9128i		−	0.0862014i	−0.999071	−	0.0431007i	$-0.986276\pi$
$173$		−	14.9128i		−	0.0862014i	0.999071	−	0.0431007i	$-0.0137236\pi$
$174$		0			0
$175$	47.4390			0.271080
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	183.640i		−	1.02592i	−0.858412	−	0.512961i	$-0.828548\pi$
$179$		−	183.640i		−	1.02592i	0.858412	−	0.512961i	$-0.171452\pi$
$180$		0			0
$181$			87.8964i			0.485616i	0.970074	+	0.242808i	$0.0780684\pi$
$181$			87.8964i			0.485616i	−0.970074	+	0.242808i	$0.921932\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			58.4561i			0.315979i
$186$		0			0
$187$	−1.94823			−0.0104183
$188$		0			0
$189$		0			0
$190$		0			0
$191$	269.940			1.41330			0.706649	−	0.707565i	$-0.250206\pi$
$191$	269.940			1.41330			0.706649	+	0.707565i	$0.250206\pi$
$192$		0			0
$193$			361.323i			1.87214i	0.351814	+	0.936070i	$0.385565\pi$
$193$			361.323i			1.87214i	−0.351814	+	0.936070i	$0.614435\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	24.6863			0.125311			0.0626556	−	0.998035i	$-0.480043\pi$
$197$	24.6863			0.125311			0.0626556	+	0.998035i	$0.480043\pi$
$198$		0			0
$199$	−339.851			−1.70779			−0.853897	−	0.520441i	$-0.825767\pi$
$199$	−339.851			−1.70779			−0.853897	+	0.520441i	$0.825767\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	83.4131i		−	0.410902i
$204$		0			0
$205$		−	407.370i		−	1.98717i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−13.2409	−	19.5068i	−0.0633534	−	0.0933341i
$210$		0			0
$211$		−	376.489i		−	1.78431i	−0.451730	−	0.892155i	$-0.649193\pi$
$211$		−	376.489i		−	1.78431i	0.451730	−	0.892155i	$-0.350807\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	308.760			1.43609
$216$		0			0
$217$			279.104i			1.28619i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	9.26096i		−	0.0419048i
$222$		0			0
$223$		−	370.168i		−	1.65995i	−0.557804	−	0.829973i	$-0.688356\pi$
$223$		−	370.168i		−	1.65995i	0.557804	−	0.829973i	$-0.311644\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	341.595i		−	1.50482i	−0.658693	−	0.752411i	$-0.728891\pi$
$227$		−	341.595i		−	1.50482i	0.658693	−	0.752411i	$-0.271109\pi$
$228$		0			0
$229$	369.046			1.61155			0.805777	−	0.592219i	$-0.201748\pi$
$229$	369.046			1.61155			0.805777	+	0.592219i	$0.201748\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−75.5098			−0.324076			−0.162038	−	0.986784i	$-0.551807\pi$
$233$	−75.5098			−0.324076			−0.162038	+	0.986784i	$0.551807\pi$
$234$		0			0
$235$	71.2708			0.303280
$236$		0			0
$237$		0			0
$238$		0			0
$239$	399.287			1.67066			0.835329	−	0.549750i	$-0.185277\pi$
$239$	399.287			1.67066			0.835329	+	0.549750i	$0.185277\pi$
$240$		0			0
$241$		−	228.357i		−	0.947541i	−0.880648	−	0.473771i	$-0.842893\pi$
$241$		−	228.357i		−	0.947541i	0.880648	−	0.473771i	$-0.157107\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	125.656			0.512883
$246$		0			0
$247$	92.7264	−	62.9409i	0.375410	−	0.254821i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.0827			0.0600904			0.0300452	−	0.999549i	$-0.490435\pi$
$251$	15.0827			0.0600904			0.0300452	+	0.999549i	$0.490435\pi$
$252$		0			0
$253$	34.1953			0.135159
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	223.762i		−	0.870669i	−0.900269	−	0.435335i	$-0.856630\pi$
$257$		−	223.762i		−	0.870669i	0.900269	−	0.435335i	$-0.143370\pi$
$258$		0			0
$259$		−	52.5003i		−	0.202704i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	50.2140			0.190928			0.0954638	−	0.995433i	$-0.469567\pi$
$263$	50.2140			0.190928			0.0954638	+	0.995433i	$0.469567\pi$
$264$		0			0
$265$		−	235.918i		−	0.890255i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	329.132i		−	1.22354i	−0.791035	−	0.611770i	$-0.790458\pi$
$269$		−	329.132i		−	1.22354i	0.791035	−	0.611770i	$-0.209542\pi$
$270$		0			0
$271$	−283.879			−1.04752			−0.523762	−	0.851864i	$-0.675472\pi$
$271$	−283.879			−1.04752			−0.523762	+	0.851864i	$0.675472\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.2319			−0.0408433
$276$		0			0
$277$	167.315			0.604025			0.302012	−	0.953304i	$-0.402342\pi$
$277$	167.315			0.604025			0.302012	+	0.953304i	$0.402342\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			491.818i			1.75024i	0.483903	+	0.875122i	$0.339219\pi$
$281$			491.818i			1.75024i	−0.483903	+	0.875122i	$0.660781\pi$
$282$		0			0
$283$	−41.3743			−0.146199			−0.0730995	−	0.997325i	$-0.523289\pi$
$283$	−41.3743			−0.146199			−0.0730995	+	0.997325i	$0.523289\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			365.865i			1.27479i
$288$		0			0
$289$	−286.535			−0.991470
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	416.306i		−	1.42084i	−0.703778	−	0.710420i	$-0.748505\pi$
$293$		−	416.306i		−	1.42084i	0.703778	−	0.710420i	$-0.251495\pi$
$294$		0			0
$295$		−	442.436i		−	1.49978i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			162.549i			0.543641i
$300$		0			0
$301$	−277.302			−0.921269
$302$		0			0
$303$		0			0
$304$		0			0
$305$	163.551			0.536234
$306$		0			0
$307$			536.008i			1.74595i	0.487762	+	0.872976i	$0.337813\pi$
$307$			536.008i			1.74595i	−0.487762	+	0.872976i	$0.662187\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	405.293			1.30319			0.651596	−	0.758566i	$-0.274100\pi$
$311$	405.293			1.30319			0.651596	+	0.758566i	$0.274100\pi$
$312$		0			0
$313$	−64.9274			−0.207436			−0.103718	−	0.994607i	$-0.533074\pi$
$313$	−64.9274			−0.207436			−0.103718	+	0.994607i	$0.533074\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			466.133i			1.47045i	0.677823	+	0.735225i	$0.262924\pi$
$317$			466.133i			1.47045i	−0.677823	+	0.735225i	$0.737076\pi$
$318$		0			0
$319$			19.7493i			0.0619101i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.7539	+	24.6823i	0.0518696	+	0.0764159i
$324$		0			0
$325$		−	53.3913i		−	0.164281i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−64.0094			−0.194557
$330$		0			0
$331$			423.761i			1.28024i	0.768273	+	0.640122i	$0.221116\pi$
$331$			423.761i			1.28024i	−0.768273	+	0.640122i	$0.778884\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	276.346i		−	0.824915i
$336$		0			0
$337$			204.478i			0.606760i	0.952870	+	0.303380i	$0.0981151\pi$
$337$			204.478i			0.606760i	−0.952870	+	0.303380i	$0.901885\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	66.0820i		−	0.193789i
$342$		0			0
$343$	−369.655			−1.07771
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−384.045			−1.10676			−0.553380	−	0.832929i	$-0.686662\pi$
$347$	−384.045			−1.10676			−0.553380	+	0.832929i	$0.686662\pi$
$348$		0			0
$349$	376.053			1.07751			0.538757	−	0.842461i	$-0.318894\pi$
$349$	376.053			1.07751			0.538757	+	0.842461i	$0.318894\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−695.360			−1.96986			−0.984929	−	0.172959i	$-0.944667\pi$
$353$	−695.360			−1.96986			−0.984929	+	0.172959i	$0.944667\pi$
$354$		0			0
$355$		−	328.793i		−	0.926179i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−456.970			−1.27290			−0.636448	−	0.771319i	$-0.719597\pi$
$359$	−456.970			−1.27290			−0.636448	+	0.771319i	$0.719597\pi$
$360$		0			0
$361$	−133.269	+	335.500i	−0.369166	+	0.929363i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	437.283			1.19803
$366$		0			0
$367$	337.233			0.918891			0.459446	−	0.888206i	$-0.348048\pi$
$367$	337.233			0.918891			0.459446	+	0.888206i	$0.348048\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			211.881i			0.571108i
$372$		0			0
$373$			343.151i			0.919977i	0.887925	+	0.459988i	$0.152146\pi$
$373$			343.151i			0.919977i	−0.887925	+	0.459988i	$0.847854\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−93.8791			−0.249016
$378$		0			0
$379$			397.158i			1.04791i	0.851746	+	0.523955i	$0.175544\pi$
$379$			397.158i			1.04791i	−0.851746	+	0.523955i	$0.824456\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	474.318i		−	1.23843i	−0.785222	−	0.619214i	$-0.787451\pi$
$383$		−	474.318i		−	1.23843i	0.785222	−	0.619214i	$-0.212549\pi$
$384$		0			0
$385$	37.9482			0.0985668
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−539.232			−1.38620			−0.693101	−	0.720841i	$-0.743756\pi$
$389$	−539.232			−1.38620			−0.693101	+	0.720841i	$0.743756\pi$
$390$		0			0
$391$	−43.2679			−0.110660
$392$		0			0
$393$		0			0
$394$		0			0
$395$			223.254i			0.565199i
$396$		0			0
$397$	421.186			1.06092			0.530461	−	0.847709i	$-0.322019\pi$
$397$	421.186			1.06092			0.530461	+	0.847709i	$0.322019\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			433.330i			1.08062i	0.841465	+	0.540312i	$0.181694\pi$
$401$			433.330i			1.08062i	−0.841465	+	0.540312i	$0.818306\pi$
$402$		0			0
$403$	314.123			0.779463
$404$		0			0
$405$		0			0
$406$		0			0
$407$			12.4302i			0.0305411i
$408$		0			0
$409$			569.984i			1.39361i	0.717263	+	0.696803i	$0.245395\pi$
$409$			569.984i			1.39361i	−0.717263	+	0.696803i	$0.754605\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			397.358i			0.962126i
$414$		0			0
$415$	−248.929			−0.599828
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−156.568			−0.373672			−0.186836	−	0.982391i	$-0.559823\pi$
$419$	−156.568			−0.373672			−0.186836	+	0.982391i	$0.559823\pi$
$420$		0			0
$421$		−	367.275i		−	0.872387i	−0.899853	−	0.436193i	$-0.856326\pi$
$421$		−	367.275i		−	0.872387i	0.899853	−	0.436193i	$-0.143674\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.2119			0.0334398
$426$		0			0
$427$	−146.888			−0.344000
$428$		0			0
$429$		0			0
$430$		0			0
$431$			289.780i			0.672343i	0.941801	+	0.336171i	$0.109132\pi$
$431$			289.780i			0.672343i	−0.941801	+	0.336171i	$0.890868\pi$
$432$		0			0
$433$		−	284.531i		−	0.657116i	−0.944484	−	0.328558i	$-0.893437\pi$
$433$		−	284.531i		−	0.657116i	0.944484	−	0.328558i	$-0.106563\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−294.065	−	433.225i	−0.672917	−	0.991361i
$438$		0			0
$439$		−	628.180i		−	1.43093i	−0.698646	−	0.715467i	$-0.746214\pi$
$439$		−	628.180i		−	1.43093i	0.698646	−	0.715467i	$-0.253786\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	416.332			0.939801			0.469900	−	0.882720i	$-0.344290\pi$
$443$	416.332			0.939801			0.469900	+	0.882720i	$0.344290\pi$
$444$		0			0
$445$		−	143.042i		−	0.321442i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	499.834i		−	1.11322i	−0.830775	−	0.556608i	$-0.812103\pi$
$449$		−	499.834i		−	1.11322i	0.830775	−	0.556608i	$-0.187897\pi$
$450$		0			0
$451$		−	86.6241i		−	0.192071i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			180.388i			0.396458i
$456$		0			0
$457$	143.028			0.312973			0.156486	−	0.987680i	$-0.449983\pi$
$457$	143.028			0.312973			0.156486	+	0.987680i	$0.449983\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−562.915			−1.22107			−0.610537	−	0.791988i	$-0.709046\pi$
$461$	−562.915			−1.22107			−0.610537	+	0.791988i	$0.709046\pi$
$462$		0			0
$463$	253.173			0.546810			0.273405	−	0.961899i	$-0.411850\pi$
$463$	253.173			0.546810			0.273405	+	0.961899i	$0.411850\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−246.479			−0.527793			−0.263896	−	0.964551i	$-0.585008\pi$
$467$	−246.479			−0.527793			−0.263896	+	0.964551i	$0.585008\pi$
$468$		0			0
$469$			248.191i			0.529192i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	65.6555			0.138806
$474$		0			0
$475$	96.5895	+	142.299i	0.203346	+	0.299576i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	223.144			0.465853			0.232927	−	0.972494i	$-0.425170\pi$
$479$	223.144			0.465853			0.232927	+	0.972494i	$0.425170\pi$
$480$		0			0
$481$	−59.0876			−0.122843
$482$		0			0
$483$		0			0
$484$		0			0
$485$			18.6732i			0.0385015i
$486$		0			0
$487$			674.619i			1.38526i	0.721295	+	0.692628i	$0.243547\pi$
$487$			674.619i			1.38526i	−0.721295	+	0.692628i	$0.756453\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−31.6662			−0.0644933			−0.0322466	−	0.999480i	$-0.510266\pi$
$491$	−31.6662			−0.0644933			−0.0322466	+	0.999480i	$0.510266\pi$
$492$		0			0
$493$		−	24.9892i		−	0.0506880i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			295.294i			0.594153i
$498$		0			0
$499$	151.589			0.303786			0.151893	−	0.988397i	$-0.451463\pi$
$499$	151.589			0.303786			0.151893	+	0.988397i	$0.451463\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−355.069			−0.705903			−0.352952	−	0.935642i	$-0.614822\pi$
$503$	−355.069			−0.705903			−0.352952	+	0.935642i	$0.614822\pi$
$504$		0			0
$505$	−395.435			−0.783040
$506$		0			0
$507$		0			0
$508$		0			0
$509$			411.332i			0.808118i	0.914733	+	0.404059i	$0.132401\pi$
$509$			411.332i			0.808118i	−0.914733	+	0.404059i	$0.867599\pi$
$510$		0			0
$511$	−392.730			−0.768552
$512$		0			0
$513$		0			0
$514$		0			0
$515$			292.093i			0.567171i
$516$		0			0
$517$	15.1552			0.0293137
$518$		0			0
$519$		0			0
$520$		0			0
$521$			538.069i			1.03276i	0.856359	+	0.516381i	$0.172721\pi$
$521$			538.069i			1.03276i	−0.856359	+	0.516381i	$0.827279\pi$
$522$		0			0
$523$			367.164i			0.702035i	0.936369	+	0.351017i	$0.114164\pi$
$523$			367.164i			0.702035i	−0.936369	+	0.351017i	$0.885836\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			83.6148i			0.158662i
$528$		0			0
$529$	230.440			0.435615
$530$		0			0
$531$		0			0
$532$		0			0
$533$	411.771			0.772554
$534$		0			0
$535$			750.463i			1.40273i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.7198			0.0495730
$540$		0			0
$541$	−873.057			−1.61378			−0.806892	−	0.590699i	$-0.798852\pi$
$541$	−873.057			−1.61378			−0.806892	+	0.590699i	$0.798852\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			238.844i			0.438247i
$546$		0			0
$547$		−	855.495i		−	1.56398i	−0.623293	−	0.781988i	$-0.714206\pi$
$547$		−	855.495i		−	1.56398i	0.623293	−	0.781988i	$-0.285794\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	250.207	−	169.836i	0.454096	−	0.308232i
$552$		0			0
$553$		−	200.507i		−	0.362581i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	140.061			0.251455			0.125728	−	0.992065i	$-0.459873\pi$
$557$	140.061			0.251455			0.125728	+	0.992065i	$0.459873\pi$
$558$		0			0
$559$			312.096i			0.558311i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	10.5527i		−	0.0187437i	−0.999956	−	0.00937185i	$-0.997017\pi$
$563$		−	10.5527i		−	0.0187437i	0.999956	−	0.00937185i	$-0.00298320\pi$
$564$		0			0
$565$		−	942.681i		−	1.66846i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			971.756i			1.70783i	0.520412	+	0.853916i	$0.325779\pi$
$569$			971.756i			1.70783i	−0.520412	+	0.853916i	$0.674221\pi$
$570$		0			0
$571$	−895.734			−1.56871			−0.784355	−	0.620312i	$-0.787006\pi$
$571$	−895.734			−1.56871			−0.784355	+	0.620312i	$0.787006\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−249.448			−0.433823
$576$		0			0
$577$	209.918			0.363809			0.181905	−	0.983316i	$-0.441774\pi$
$577$	209.918			0.363809			0.181905	+	0.983316i	$0.441774\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	223.567			0.384796
$582$		0			0
$583$		−	50.1661i		−	0.0860482i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1032.25			−1.75852			−0.879258	−	0.476345i	$-0.841961\pi$
$587$	−1032.25			−1.75852			−0.879258	+	0.476345i	$0.841961\pi$
$588$		0			0
$589$	−837.202	+	568.277i	−1.42140	+	0.964816i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1030.88			1.73842			0.869210	−	0.494442i	$-0.164628\pi$
$593$	1030.88			1.73842			0.869210	+	0.494442i	$0.164628\pi$
$594$		0			0
$595$	−48.0166			−0.0807001
$596$		0			0
$597$		0			0
$598$		0			0
$599$			83.7382i			0.139797i	0.997554	+	0.0698983i	$0.0222675\pi$
$599$			83.7382i			0.139797i	−0.997554	+	0.0698983i	$0.977733\pi$
$600$		0			0
$601$			293.467i			0.488298i	0.969738	+	0.244149i	$0.0785086\pi$
$601$			293.467i			0.488298i	−0.969738	+	0.244149i	$0.921491\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	697.097			1.15223
$606$		0			0
$607$		−	418.633i		−	0.689676i	−0.938662	−	0.344838i	$-0.887934\pi$
$607$		−	418.633i		−	0.689676i	0.938662	−	0.344838i	$-0.112066\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			72.0408i			0.117906i
$612$		0			0
$613$	−377.709			−0.616165			−0.308083	−	0.951360i	$-0.599687\pi$
$613$	−377.709			−0.616165			−0.308083	+	0.951360i	$0.599687\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−845.723			−1.37070			−0.685351	−	0.728213i	$-0.740351\pi$
$617$	−845.723			−1.37070			−0.685351	+	0.728213i	$0.740351\pi$
$618$		0			0
$619$	410.355			0.662932			0.331466	−	0.943467i	$-0.392457\pi$
$619$	410.355			0.662932			0.331466	+	0.943467i	$0.392457\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			128.468i			0.206209i
$624$		0			0
$625$	−769.360			−1.23098
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	15.7282i		−	0.0250051i
$630$		0			0
$631$	313.898			0.497462			0.248731	−	0.968573i	$-0.419987\pi$
$631$	313.898			0.497462			0.248731	+	0.968573i	$0.419987\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	154.414i		−	0.243172i
$636$		0			0
$637$			127.014i			0.199394i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			425.842i			0.664339i	0.943220	+	0.332170i	$0.107781\pi$
$641$			425.842i			0.664339i	−0.943220	+	0.332170i	$0.892219\pi$
$642$		0			0
$643$	1038.06			1.61441			0.807204	−	0.590272i	$-0.200980\pi$
$643$	1038.06			1.61441			0.807204	+	0.590272i	$0.200980\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−174.684			−0.269991			−0.134995	−	0.990846i	$-0.543102\pi$
$647$	−174.684			−0.269991			−0.134995	+	0.990846i	$0.543102\pi$
$648$		0			0
$649$		−	94.0806i		−	0.144962i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1247.54			1.91048			0.955238	−	0.295839i	$-0.0955992\pi$
$653$	1247.54			1.91048			0.955238	+	0.295839i	$0.0955992\pi$
$654$		0			0
$655$	138.464			0.211395
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	746.281i		−	1.13244i	−0.824253	−	0.566222i	$-0.808405\pi$
$659$		−	746.281i		−	1.13244i	0.824253	−	0.566222i	$-0.191595\pi$
$660$		0			0
$661$			73.1406i			0.110651i	0.998468	+	0.0553257i	$0.0176197\pi$
$661$			73.1406i			0.110651i	−0.998468	+	0.0553257i	$0.982380\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−326.338	−	480.771i	−0.490734	−	0.722964i
$666$		0			0
$667$			438.611i			0.657587i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	34.7780			0.0518301
$672$		0			0
$673$		−	939.481i		−	1.39596i	−0.716117	−	0.697980i	$-0.754082\pi$
$673$		−	939.481i		−	1.39596i	0.716117	−	0.697980i	$-0.245918\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	666.060i		−	0.983841i	−0.870640	−	0.491920i	$-0.836295\pi$
$677$		−	666.060i		−	0.983841i	0.870640	−	0.491920i	$-0.163705\pi$
$678$		0			0
$679$		−	16.7707i		−	0.0246991i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	79.9509i		−	0.117058i	−0.998286	−	0.0585292i	$-0.981359\pi$
$683$		−	79.9509i		−	0.117058i	0.998286	−	0.0585292i	$-0.0186411\pi$
$684$		0			0
$685$	745.632			1.08851
$686$		0			0
$687$		0			0
$688$		0			0
$689$	238.466			0.346105
$690$		0			0
$691$	208.333			0.301495			0.150748	−	0.988572i	$-0.451832\pi$
$691$	208.333			0.301495			0.150748	+	0.988572i	$0.451832\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−612.521			−0.881326
$696$		0			0
$697$			109.607i			0.157256i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.0082			−0.0299688			−0.0149844	−	0.999888i	$-0.504770\pi$
$701$	−21.0082			−0.0299688			−0.0149844	+	0.999888i	$0.504770\pi$
$702$		0			0
$703$	157.480	−	106.895i	0.224012	−	0.152055i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	355.147			0.502329
$708$		0			0
$709$	230.915			0.325691			0.162845	−	0.986652i	$-0.447933\pi$
$709$	230.915			0.325691			0.162845	+	0.986652i	$0.447933\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1467.61i		−	2.05836i
$714$		0			0
$715$		−	42.7097i		−	0.0597338i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	846.640			1.17752			0.588762	−	0.808306i	$-0.299615\pi$
$719$	846.640			1.17752			0.588762	+	0.808306i	$0.299615\pi$
$720$		0			0
$721$		−	262.333i		−	0.363846i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	144.068i		−	0.198714i
$726$		0			0
$727$	1078.59			1.48362			0.741810	−	0.670610i	$-0.233968\pi$
$727$	1078.59			1.48362			0.741810	+	0.670610i	$0.233968\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−83.0751			−0.113646
$732$		0			0
$733$	−648.893			−0.885256			−0.442628	−	0.896705i	$-0.645954\pi$
$733$	−648.893			−0.885256			−0.442628	+	0.896705i	$0.645954\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	58.7630i		−	0.0797326i
$738$		0			0
$739$	−1022.09			−1.38307			−0.691534	−	0.722344i	$-0.743065\pi$
$739$	−1022.09			−1.38307			−0.691534	+	0.722344i	$0.743065\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	257.760i		−	0.346918i	−0.984841	−	0.173459i	$-0.944506\pi$
$743$		−	257.760i		−	0.346918i	0.984841	−	0.173459i	$-0.0554944\pi$
$744$		0			0
$745$	842.468			1.13083
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	674.002i		−	0.899869i
$750$		0			0
$751$		−	553.605i		−	0.737157i	−0.929597	−	0.368578i	$-0.879845\pi$
$751$		−	553.605i		−	0.737157i	0.929597	−	0.368578i	$-0.120155\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	198.416i		−	0.262802i
$756$		0			0
$757$	456.061			0.602458			0.301229	−	0.953552i	$-0.402603\pi$
$757$	456.061			0.602458			0.301229	+	0.953552i	$0.402603\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−536.682			−0.705233			−0.352616	−	0.935768i	$-0.614708\pi$
$761$	−536.682			−0.705233			−0.352616	+	0.935768i	$0.614708\pi$
$762$		0			0
$763$		−	214.510i		−	0.281140i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	447.215			0.583071
$768$		0			0
$769$	814.347			1.05897			0.529484	−	0.848320i	$-0.322385\pi$
$769$	814.347			1.05897			0.529484	+	0.848320i	$0.322385\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1451.94i		−	1.87831i	−0.343490	−	0.939156i	$-0.611609\pi$
$773$		−	1451.94i		−	1.87831i	0.343490	−	0.939156i	$-0.388391\pi$
$774$		0			0
$775$			482.056i			0.622008i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1097.45	+	744.930i	−1.40880	+	0.956264i
$780$		0			0
$781$		−	69.9154i		−	0.0895204i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	976.756			1.24427
$786$		0			0
$787$			389.048i			0.494343i	0.968972	+	0.247172i	$0.0795012\pi$
$787$			389.048i			0.494343i	−0.968972	+	0.247172i	$0.920499\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			846.636i			1.07034i
$792$		0			0
$793$			165.318i			0.208472i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1243.63i		−	1.56039i	−0.625539	−	0.780193i	$-0.715121\pi$
$797$		−	1243.63i		−	1.56039i	0.625539	−	0.780193i	$-0.284879\pi$
$798$		0			0
$799$	−19.1761			−0.0240002
$800$		0			0
$801$		0			0
$802$		0			0
$803$	92.9848			0.115797
$804$		0			0
$805$	842.788			1.04694
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−683.371			−0.844711			−0.422355	−	0.906430i	$-0.638797\pi$
$809$	−683.371			−0.844711			−0.422355	+	0.906430i	$0.638797\pi$
$810$		0			0
$811$		−	68.3221i		−	0.0842443i	−0.999112	−	0.0421221i	$-0.986588\pi$
$811$		−	68.3221i		−	0.0842443i	0.999112	−	0.0421221i	$-0.0134119\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1238.74			−1.51993
$816$		0			0
$817$	−564.608	−	831.798i	−0.691075	−	1.01811i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1353.27			1.64832			0.824161	−	0.566356i	$-0.191647\pi$
$821$	1353.27			1.64832			0.824161	+	0.566356i	$0.191647\pi$
$822$		0			0
$823$	239.439			0.290935			0.145467	−	0.989363i	$-0.453531\pi$
$823$	239.439			0.290935			0.145467	+	0.989363i	$0.453531\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			31.7267i			0.0383636i	0.999816	+	0.0191818i	$0.00610614\pi$
$827$			31.7267i			0.0383636i	−0.999816	+	0.0191818i	$0.993894\pi$
$828$		0			0
$829$			37.0436i			0.0446847i	0.999750	+	0.0223424i	$0.00711239\pi$
$829$			37.0436i			0.0446847i	−0.999750	+	0.0223424i	$0.992888\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−33.8091			−0.0405872
$834$		0			0
$835$		−	305.247i		−	0.365566i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	526.714i		−	0.627788i	−0.949458	−	0.313894i	$-0.898366\pi$
$839$		−	526.714i		−	0.627788i	0.949458	−	0.313894i	$-0.101634\pi$
$840$		0			0
$841$	587.683			0.698790
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−783.159			−0.926815
$846$		0			0
$847$	−626.074			−0.739166
$848$		0			0
$849$		0			0
$850$		0			0
$851$			276.062i			0.324397i
$852$		0			0
$853$	104.843			0.122911			0.0614557	−	0.998110i	$-0.480426\pi$
$853$	104.843			0.122911			0.0614557	+	0.998110i	$0.480426\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			192.081i			0.224132i	0.993701	+	0.112066i	$0.0357469\pi$
$857$			192.081i			0.224132i	−0.993701	+	0.112066i	$0.964253\pi$
$858$		0			0
$859$	653.580			0.760861			0.380431	−	0.924809i	$-0.375776\pi$
$859$	653.580			0.760861			0.380431	+	0.924809i	$0.375776\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1166.58i		−	1.35177i	−0.737005	−	0.675887i	$-0.763761\pi$
$863$		−	1166.58i		−	1.35177i	0.737005	−	0.675887i	$-0.236239\pi$
$864$		0			0
$865$			87.0222i			0.100604i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			47.4732i			0.0546296i
$870$		0			0
$871$	279.332			0.320703
$872$		0			0
$873$		0			0
$874$		0			0
$875$	487.735			0.557412
$876$		0			0
$877$		−	596.868i		−	0.680579i	−0.940321	−	0.340289i	$-0.889475\pi$
$877$		−	596.868i		−	0.680579i	0.940321	−	0.340289i	$-0.110525\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1397.14			1.58585			0.792927	−	0.609317i	$-0.208556\pi$
$881$	1397.14			1.58585			0.792927	+	0.609317i	$0.208556\pi$
$882$		0			0
$883$	148.675			0.168375			0.0841876	−	0.996450i	$-0.473171\pi$
$883$	148.675			0.168375			0.0841876	+	0.996450i	$0.473171\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1414.59i			1.59480i	0.603450	+	0.797401i	$0.293792\pi$
$887$			1414.59i			1.59480i	−0.603450	+	0.797401i	$0.706208\pi$
$888$		0			0
$889$			138.682i			0.155997i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−130.328	−	192.003i	−0.145944	−	0.215009i
$894$		0			0
$895$			1071.61i			1.19733i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	847.610			0.942837
$900$		0			0
$901$			63.4761i			0.0704507i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	512.910i		−	0.566751i
$906$		0			0
$907$			906.084i			0.998991i	0.866317	+	0.499495i	$0.166481\pi$
$907$			906.084i			0.998991i	−0.866317	+	0.499495i	$0.833519\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	132.618i		−	0.145574i	−0.997348	−	0.0727870i	$-0.976811\pi$
$911$		−	132.618i		−	0.145574i	0.997348	−	0.0727870i	$-0.0231893\pi$
$912$		0			0
$913$	−52.9328			−0.0579768
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−124.356			−0.135612
$918$		0			0
$919$	−860.497			−0.936341			−0.468170	−	0.883638i	$-0.655087\pi$
$919$	−860.497			−0.936341			−0.468170	+	0.883638i	$0.655087\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	332.345			0.360071
$924$		0			0
$925$		−	90.6763i		−	0.0980284i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−403.521			−0.434360			−0.217180	−	0.976132i	$-0.569686\pi$
$929$	−403.521			−0.434360			−0.217180	+	0.976132i	$0.569686\pi$
$930$		0			0
$931$	−229.779	−	338.517i	−0.246809	−	0.363606i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.3687			0.0121590
$936$		0			0
$937$	559.112			0.596704			0.298352	−	0.954456i	$-0.403563\pi$
$937$	559.112			0.596704			0.298352	+	0.954456i	$0.403563\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			824.556i			0.876255i	0.898913	+	0.438127i	$0.144358\pi$
$941$			824.556i			0.876255i	−0.898913	+	0.438127i	$0.855642\pi$
$942$		0			0
$943$		−	1923.83i		−	2.04011i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−79.0790			−0.0835048			−0.0417524	−	0.999128i	$-0.513294\pi$
$947$	−79.0790			−0.0835048			−0.0417524	+	0.999128i	$0.513294\pi$
$948$		0			0
$949$			442.007i			0.465761i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			230.825i			0.242208i	0.992640	+	0.121104i	$0.0386435\pi$
$953$			230.825i			0.242208i	−0.992640	+	0.121104i	$0.961356\pi$
$954$		0			0
$955$	−1575.20			−1.64943
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−669.663			−0.698293
$960$		0			0
$961$	−1875.14			−1.95124
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	2108.46i		−	2.18493i
$966$		0			0
$967$	−1527.67			−1.57980			−0.789901	−	0.613235i	$-0.789868\pi$
$967$	−1527.67			−1.57980			−0.789901	+	0.613235i	$0.789868\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	285.090i		−	0.293605i	−0.989166	−	0.146802i	$-0.953102\pi$
$971$		−	285.090i		−	0.293605i	0.989166	−	0.146802i	$-0.0468981\pi$
$972$		0			0
$973$	550.115			0.565380
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1833.67i			1.87684i	0.345499	+	0.938419i	$0.387710\pi$
$977$			1833.67i			1.87684i	−0.345499	+	0.938419i	$0.612290\pi$
$978$		0			0
$979$		−	30.4168i		−	0.0310692i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1140.39i		−	1.16011i	−0.814578	−	0.580054i	$-0.803031\pi$
$983$		−	1140.39i		−	1.16011i	0.814578	−	0.580054i	$-0.196969\pi$
$984$		0			0
$985$	−144.054			−0.146248
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1458.14			1.47435
$990$		0			0
$991$			579.463i			0.584725i	0.956308	+	0.292363i	$0.0944414\pi$
$991$			579.463i			0.584725i	−0.956308	+	0.292363i	$0.905559\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1983.16			1.99313
$996$		0			0
$997$	59.6215			0.0598009			0.0299005	−	0.999553i	$-0.490481\pi$
$997$	59.6215			0.0598009			0.0299005	+	0.999553i	$0.490481\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.h.e.37.1		6
3.2	odd	2		228.3.h.a.37.6	yes	6
4.3	odd	2		2736.3.o.m.721.1		6
12.11	even	2		912.3.o.c.721.3		6
19.18	odd	2	inner	684.3.h.e.37.2		6
57.56	even	2		228.3.h.a.37.3	✓	6
76.75	even	2		2736.3.o.m.721.2		6
228.227	odd	2		912.3.o.c.721.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.h.a.37.3	✓	6	57.56	even	2
228.3.h.a.37.6	yes	6	3.2	odd	2
684.3.h.e.37.1		6	1.1	even	1	trivial
684.3.h.e.37.2		6	19.18	odd	2	inner
912.3.o.c.721.3		6	12.11	even	2
912.3.o.c.721.6		6	228.227	odd	2
2736.3.o.m.721.1		6	4.3	odd	2
2736.3.o.m.721.2		6	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.83539 q^{5} +5.24085 q^{7} +O(q^{10})\)	\(q-5.83539 q^{5} +5.24085 q^{7} -1.24085 q^{11} -5.89843i q^{13} +1.57007 q^{17} +(10.6708 + 15.7205i) q^{19} -27.5579 q^{23} +9.05177 q^{25} -15.9159i q^{29} +53.2554i q^{31} -30.5824 q^{35} -10.0175i q^{37} +69.8102i q^{41} -52.9116 q^{43} -12.2135 q^{47} -21.5335 q^{49} +40.4288i q^{53} +7.24085 q^{55} +75.8194i q^{59} -28.0275 q^{61} +34.4197i q^{65} +47.3570i q^{67} +56.3447i q^{71} -74.9363 q^{73} -6.50312 q^{77} -38.2585i q^{79} +42.6584 q^{83} -9.16198 q^{85} +24.5128i q^{89} -30.9128i q^{91} +(-62.2682 - 91.7353i) q^{95} -3.19999i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 6 * q + 2 * q^5 - 2 * q^7	\( 6 q + 2 q^{5} - 2 q^{7} + 26 q^{11} + 50 q^{17} - 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 210 q^{43} - 22 q^{47} - 36 q^{49} + 10 q^{55} + 214 q^{61} + 102 q^{73} - 266 q^{77} + 404 q^{83} + 370 q^{85} - 358 q^{95}+O(q^{100}) \) 6 * q + 2 * q^5 - 2 * q^7 + 26 * q^11 + 50 * q^17 - 10 * q^19 - 28 * q^23 + 28 * q^25 - 2 * q^35 - 210 * q^43 - 22 * q^47 - 36 * q^49 + 10 * q^55 + 214 * q^61 + 102 * q^73 - 266 * q^77 + 404 * q^83 + 370 * q^85 - 358 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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