LMFDB - Embedded newform 684.3.h.e.37.6

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			37.6
Root			$0.500000 - 2.79345i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.3.h.e.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.39180 q^{5} +3.28502 q^{7} +O(q^{10})$	$q+7.39180 q^{5} +3.28502 q^{7} +0.714984 q^{11} +25.4210i q^{13} +26.0686 q^{17} +(-15.7836 + 10.5773i) q^{19} -8.46325 q^{23} +29.6386 q^{25} -23.1292i q^{29} +18.8627i q^{31} +24.2822 q^{35} -48.5501i q^{37} +35.3808i q^{41} -24.5014 q^{43} +46.0990 q^{47} -38.2087 q^{49} +51.2119i q^{53} +5.28502 q^{55} -101.314i q^{59} +38.5721 q^{61} +187.907i q^{65} +44.2837i q^{67} +74.3410i q^{71} +112.200 q^{73} +2.34873 q^{77} -125.500i q^{79} +95.5672 q^{83} +192.694 q^{85} +28.0827i q^{89} +83.5083i q^{91} +(-116.669 + 78.1849i) q^{95} +55.7955i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	7.39180			1.47836			0.739180	−	0.673508i	$-0.235213\pi$
$5$	7.39180			1.47836			0.739180	+	0.673508i	$0.235213\pi$
$6$		0			0
$7$	3.28502			0.469288			0.234644	−	0.972081i	$-0.424608\pi$
$7$	3.28502			0.469288			0.234644	+	0.972081i	$0.424608\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.714984			0.0649986			0.0324993	−	0.999472i	$-0.489653\pi$
$11$	0.714984			0.0649986			0.0324993	+	0.999472i	$0.489653\pi$
$12$		0			0
$13$			25.4210i			1.95546i	0.209866	+	0.977730i	$0.432697\pi$
$13$			25.4210i			1.95546i	−0.209866	+	0.977730i	$0.567303\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.0686			1.53345			0.766724	−	0.641977i	$-0.221886\pi$
$17$	26.0686			1.53345			0.766724	+	0.641977i	$0.221886\pi$
$18$		0			0
$19$	−15.7836	+	10.5773i	−0.830715	+	0.556697i
$19$	−15.7836	+	10.5773i	−0.830715	+	0.556697i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.46325			−0.367967			−0.183984	−	0.982929i	$-0.558899\pi$
$23$	−8.46325			−0.367967			−0.183984	+	0.982929i	$0.558899\pi$
$24$		0			0
$25$	29.6386			1.18555
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	23.1292i		−	0.797557i	−0.917047	−	0.398778i	$-0.869434\pi$
$29$		−	23.1292i		−	0.797557i	0.917047	−	0.398778i	$-0.130566\pi$
$30$		0			0
$31$			18.8627i			0.608473i	0.952596	+	0.304237i	$0.0984014\pi$
$31$			18.8627i			0.608473i	−0.952596	+	0.304237i	$0.901599\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	24.2822			0.693776
$36$		0			0
$37$		−	48.5501i		−	1.31217i	−0.754689	−	0.656083i	$-0.772212\pi$
$37$		−	48.5501i		−	1.31217i	0.754689	−	0.656083i	$-0.227788\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			35.3808i			0.862946i	0.902126	+	0.431473i	$0.142006\pi$
$41$			35.3808i			0.862946i	−0.902126	+	0.431473i	$0.857994\pi$
$42$		0			0
$43$	−24.5014			−0.569801			−0.284900	−	0.958557i	$-0.591960\pi$
$43$	−24.5014			−0.569801			−0.284900	+	0.958557i	$0.591960\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	46.0990			0.980831			0.490415	−	0.871489i	$-0.336845\pi$
$47$	46.0990			0.980831			0.490415	+	0.871489i	$0.336845\pi$
$48$		0			0
$49$	−38.2087			−0.779769
$50$		0			0
$51$		0			0
$52$		0			0
$53$			51.2119i			0.966262i	0.875548	+	0.483131i	$0.160500\pi$
$53$			51.2119i			0.966262i	−0.875548	+	0.483131i	$0.839500\pi$
$54$		0			0
$55$	5.28502			0.0960912
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	101.314i		−	1.71719i	−0.512657	−	0.858594i	$-0.671339\pi$
$59$		−	101.314i		−	1.71719i	0.512657	−	0.858594i	$-0.328661\pi$
$60$		0			0
$61$	38.5721			0.632329			0.316165	−	0.948704i	$-0.397605\pi$
$61$	38.5721			0.632329			0.316165	+	0.948704i	$0.397605\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			187.907i			2.89087i
$66$		0			0
$67$			44.2837i			0.660950i	0.943815	+	0.330475i	$0.107209\pi$
$67$			44.2837i			0.660950i	−0.943815	+	0.330475i	$0.892791\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			74.3410i			1.04706i	0.852008	+	0.523528i	$0.175385\pi$
$71$			74.3410i			1.04706i	−0.852008	+	0.523528i	$0.824615\pi$
$72$		0			0
$73$	112.200			1.53699			0.768494	−	0.639857i	$-0.221006\pi$
$73$	112.200			1.53699			0.768494	+	0.639857i	$0.221006\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.34873			0.0305030
$78$		0			0
$79$		−	125.500i		−	1.58861i	−0.607519	−	0.794305i	$-0.707835\pi$
$79$		−	125.500i		−	1.58861i	0.607519	−	0.794305i	$-0.292165\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	95.5672			1.15141			0.575706	−	0.817657i	$-0.304727\pi$
$83$	95.5672			1.15141			0.575706	+	0.817657i	$0.304727\pi$
$84$		0			0
$85$	192.694			2.26699
$86$		0			0
$87$		0			0
$88$		0			0
$89$			28.0827i			0.315536i	0.987476	+	0.157768i	$0.0504298\pi$
$89$			28.0827i			0.315536i	−0.987476	+	0.157768i	$0.949570\pi$
$90$		0			0
$91$			83.5083i			0.917674i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−116.669	+	78.1849i	−1.22810	+	0.822999i
$96$		0			0
$97$			55.7955i			0.575212i	0.957749	+	0.287606i	$0.0928593\pi$
$97$			55.7955i			0.575212i	−0.957749	+	0.287606i	$0.907141\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	131.018			1.29721			0.648603	−	0.761127i	$-0.275354\pi$
$101$	131.018			1.29721			0.648603	+	0.761127i	$0.275354\pi$
$102$		0			0
$103$		−	74.6582i		−	0.724837i	−0.932015	−	0.362418i	$-0.881951\pi$
$103$		−	74.6582i		−	0.724837i	0.932015	−	0.362418i	$-0.118049\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	171.811i		−	1.60571i	−0.596173	−	0.802856i	$-0.703313\pi$
$107$		−	171.811i		−	1.60571i	0.596173	−	0.802856i	$-0.296687\pi$
$108$		0			0
$109$			34.9582i			0.320717i	0.987059	+	0.160359i	$0.0512651\pi$
$109$			34.9582i			0.320717i	−0.987059	+	0.160359i	$0.948735\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	42.8041i		−	0.378797i	−0.981900	−	0.189399i	$-0.939346\pi$
$113$		−	42.8041i		−	0.378797i	0.981900	−	0.189399i	$-0.0606538\pi$
$114$		0			0
$115$	−62.5586			−0.543988
$116$		0			0
$117$		0			0
$118$		0			0
$119$	85.6358			0.719628
$120$		0			0
$121$	−120.489			−0.995775
$122$		0			0
$123$		0			0
$124$		0			0
$125$	34.2879			0.274303
$126$		0			0
$127$			176.342i			1.38852i	0.719724	+	0.694260i	$0.244268\pi$
$127$			176.342i			1.38852i	−0.719724	+	0.694260i	$0.755732\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−202.114			−1.54285			−0.771427	−	0.636317i	$-0.780457\pi$
$131$	−202.114			−1.54285			−0.771427	+	0.636317i	$0.780457\pi$
$132$		0			0
$133$	−51.8493	+	34.7464i	−0.389845	+	0.261251i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.2891			0.0824021			0.0412011	−	0.999151i	$-0.486882\pi$
$137$	11.2891			0.0824021			0.0412011	+	0.999151i	$0.486882\pi$
$138$		0			0
$139$	−56.6427			−0.407502			−0.203751	−	0.979023i	$-0.565313\pi$
$139$	−56.6427			−0.407502			−0.203751	+	0.979023i	$0.565313\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			18.1756i			0.127102i
$144$		0			0
$145$		−	170.966i		−	1.17908i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.96590			0.0400396			0.0200198	−	0.999800i	$-0.493627\pi$
$149$	5.96590			0.0400396			0.0200198	+	0.999800i	$0.493627\pi$
$150$		0			0
$151$		−	28.0300i		−	0.185629i	−0.995683	−	0.0928146i	$-0.970414\pi$
$151$		−	28.0300i		−	0.185629i	0.995683	−	0.0928146i	$-0.0295864\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			139.429i			0.899542i
$156$		0			0
$157$	141.317			0.900108			0.450054	−	0.893001i	$-0.351405\pi$
$157$	141.317			0.900108			0.450054	+	0.893001i	$0.351405\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−27.8019			−0.172683
$162$		0			0
$163$	−181.550			−1.11380			−0.556902	−	0.830578i	$-0.688010\pi$
$163$	−181.550			−1.11380			−0.556902	+	0.830578i	$0.688010\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	301.737i		−	1.80681i	−0.428791	−	0.903404i	$-0.641060\pi$
$167$		−	301.737i		−	1.80681i	0.428791	−	0.903404i	$-0.358940\pi$
$168$		0			0
$169$	−477.226			−2.82382
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	195.469i		−	1.12988i	−0.825132	−	0.564940i	$-0.808899\pi$
$173$		−	195.469i		−	1.12988i	0.825132	−	0.564940i	$-0.191101\pi$
$174$		0			0
$175$	97.3634			0.556362
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	91.2818i		−	0.509954i	−0.966947	−	0.254977i	$-0.917932\pi$
$179$		−	91.2818i		−	0.509954i	0.966947	−	0.254977i	$-0.0820679\pi$
$180$		0			0
$181$		−	15.7783i		−	0.0871732i	−0.999050	−	0.0435866i	$-0.986122\pi$
$181$		−	15.7783i		−	0.0871732i	0.999050	−	0.0435866i	$-0.0138784\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	358.873i		−	1.93985i
$186$		0			0
$187$	18.6386			0.0996719
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−193.939			−1.01539			−0.507694	−	0.861538i	$-0.669502\pi$
$191$	−193.939			−1.01539			−0.507694	+	0.861538i	$0.669502\pi$
$192$		0			0
$193$		−	150.941i		−	0.782077i	−0.920374	−	0.391039i	$-0.872116\pi$
$193$		−	150.941i		−	0.782077i	0.920374	−	0.391039i	$-0.127884\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−68.0052			−0.345204			−0.172602	−	0.984992i	$-0.555217\pi$
$197$	−68.0052			−0.345204			−0.172602	+	0.984992i	$0.555217\pi$
$198$		0			0
$199$	29.4815			0.148148			0.0740741	−	0.997253i	$-0.476400\pi$
$199$	29.4815			0.148148			0.0740741	+	0.997253i	$0.476400\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	75.9796i		−	0.374284i
$204$		0			0
$205$			261.528i			1.27574i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.2850	+	7.56257i	−0.0539953	+	0.0361845i
$210$		0			0
$211$			9.92686i			0.0470467i	0.999723	+	0.0235234i	$0.00748841\pi$
$211$			9.92686i			0.0470467i	−0.999723	+	0.0235234i	$0.992512\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−181.110			−0.842370
$216$		0			0
$217$			61.9642i			0.285549i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			662.690i			2.99860i
$222$		0			0
$223$			177.702i			0.796871i	0.917196	+	0.398435i	$0.130447\pi$
$223$			177.702i			0.796871i	−0.917196	+	0.398435i	$0.869553\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	79.1694i		−	0.348764i	−0.984678	−	0.174382i	$-0.944207\pi$
$227$		−	79.1694i		−	0.348764i	0.984678	−	0.174382i	$-0.0557927\pi$
$228$		0			0
$229$	161.424			0.704908			0.352454	−	0.935829i	$-0.385347\pi$
$229$	161.424			0.704908			0.352454	+	0.935829i	$0.385347\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	196.958			0.845315			0.422658	−	0.906289i	$-0.361097\pi$
$233$	196.958			0.845315			0.422658	+	0.906289i	$0.361097\pi$
$234$		0			0
$235$	340.755			1.45002
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−179.160			−0.749621			−0.374811	−	0.927101i	$-0.622292\pi$
$239$	−179.160			−0.749621			−0.374811	+	0.927101i	$0.622292\pi$
$240$		0			0
$241$		−	464.116i		−	1.92579i	−0.269870	−	0.962897i	$-0.586981\pi$
$241$		−	464.116i		−	1.92579i	0.269870	−	0.962897i	$-0.413019\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−282.431			−1.15278
$246$		0			0
$247$	−268.884	−	401.234i	−1.08860	−	1.62443i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−140.760			−0.560798			−0.280399	−	0.959883i	$-0.590467\pi$
$251$	−140.760			−0.560798			−0.280399	+	0.959883i	$0.590467\pi$
$252$		0			0
$253$	−6.05109			−0.0239174
$254$		0			0
$255$		0			0
$256$		0			0
$257$			184.717i			0.718743i	0.933195	+	0.359371i	$0.117009\pi$
$257$			184.717i			0.718743i	−0.933195	+	0.359371i	$0.882991\pi$
$258$		0			0
$259$		−	159.488i		−	0.615784i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−254.011			−0.965822			−0.482911	−	0.875669i	$-0.660421\pi$
$263$	−254.011			−0.965822			−0.482911	+	0.875669i	$0.660421\pi$
$264$		0			0
$265$			378.548i			1.42848i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			54.2105i			0.201526i	0.994910	+	0.100763i	$0.0321284\pi$
$269$			54.2105i			0.201526i	−0.994910	+	0.100763i	$0.967872\pi$
$270$		0			0
$271$	397.966			1.46851			0.734254	−	0.678875i	$-0.237532\pi$
$271$	397.966			1.46851			0.734254	+	0.678875i	$0.237532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21.1912			0.0770587
$276$		0			0
$277$	−310.820			−1.12209			−0.561046	−	0.827784i	$-0.689601\pi$
$277$	−310.820			−1.12209			−0.561046	+	0.827784i	$0.689601\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	189.796i		−	0.675429i	−0.941249	−	0.337715i	$-0.890346\pi$
$281$		−	189.796i		−	0.675429i	0.941249	−	0.337715i	$-0.109654\pi$
$282$		0			0
$283$	−352.032			−1.24393			−0.621965	−	0.783045i	$-0.713665\pi$
$283$	−352.032			−1.24393			−0.621965	+	0.783045i	$0.713665\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			116.227i			0.404970i
$288$		0			0
$289$	390.572			1.35146
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	238.729i		−	0.814774i	−0.913256	−	0.407387i	$-0.866440\pi$
$293$		−	238.729i		−	0.814774i	0.913256	−	0.407387i	$-0.133560\pi$
$294$		0			0
$295$		−	748.893i		−	2.53862i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	215.144i		−	0.719546i
$300$		0			0
$301$	−80.4876			−0.267401
$302$		0			0
$303$		0			0
$304$		0			0
$305$	285.117			0.934809
$306$		0			0
$307$			78.3760i			0.255296i	0.991820	+	0.127648i	$0.0407428\pi$
$307$			78.3760i			0.255296i	−0.991820	+	0.127648i	$0.959257\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	257.012			0.826404			0.413202	−	0.910639i	$-0.364410\pi$
$311$	257.012			0.826404			0.413202	+	0.910639i	$0.364410\pi$
$312$		0			0
$313$	152.676			0.487784			0.243892	−	0.969802i	$-0.421576\pi$
$313$	152.676			0.487784			0.243892	+	0.969802i	$0.421576\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	293.190i		−	0.924889i	−0.886648	−	0.462444i	$-0.846972\pi$
$317$		−	293.190i		−	0.924889i	0.886648	−	0.462444i	$-0.153028\pi$
$318$		0			0
$319$		−	16.5370i		−	0.0518400i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−411.456	+	275.734i	−1.27386	+	0.853666i
$324$		0			0
$325$			753.443i			2.31829i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	151.436			0.460292
$330$		0			0
$331$		−	515.401i		−	1.55710i	−0.627580	−	0.778552i	$-0.715954\pi$
$331$		−	515.401i		−	1.55710i	0.627580	−	0.778552i	$-0.284046\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			327.336i			0.977122i
$336$		0			0
$337$			452.163i			1.34173i	0.741580	+	0.670865i	$0.234077\pi$
$337$			452.163i			1.34173i	−0.741580	+	0.670865i	$0.765923\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			13.4865i			0.0395499i
$342$		0			0
$343$	−286.482			−0.835224
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−378.380			−1.09043			−0.545217	−	0.838295i	$-0.683553\pi$
$347$	−378.380			−1.09043			−0.545217	+	0.838295i	$0.683553\pi$
$348$		0			0
$349$	−95.1859			−0.272739			−0.136369	−	0.990658i	$-0.543543\pi$
$349$	−95.1859			−0.272739			−0.136369	+	0.990658i	$0.543543\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−457.473			−1.29596			−0.647979	−	0.761659i	$-0.724385\pi$
$353$	−457.473			−1.29596			−0.647979	+	0.761659i	$0.724385\pi$
$354$		0			0
$355$			549.514i			1.54793i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−308.486			−0.859294			−0.429647	−	0.902997i	$-0.641362\pi$
$359$	−308.486			−0.859294			−0.429647	+	0.902997i	$0.641362\pi$
$360$		0			0
$361$	137.243	−	333.894i	0.380176	−	0.924914i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	829.360			2.27222
$366$		0			0
$367$	−432.928			−1.17964			−0.589820	−	0.807535i	$-0.700801\pi$
$367$	−432.928			−1.17964			−0.589820	+	0.807535i	$0.700801\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			168.232i			0.453455i
$372$		0			0
$373$		−	649.540i		−	1.74139i	−0.491820	−	0.870697i	$-0.663668\pi$
$373$		−	649.540i		−	1.74139i	0.491820	−	0.870697i	$-0.336332\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	587.966			1.55959
$378$		0			0
$379$			103.303i			0.272567i	0.990670	+	0.136283i	$0.0435158\pi$
$379$			103.303i			0.272567i	−0.990670	+	0.136283i	$0.956484\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			526.913i			1.37575i	0.725828	+	0.687876i	$0.241457\pi$
$383$			526.913i			1.37575i	−0.725828	+	0.687876i	$0.758543\pi$
$384$		0			0
$385$	17.3614			0.0450944
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−93.9846			−0.241606			−0.120803	−	0.992677i	$-0.538547\pi$
$389$	−93.9846			−0.241606			−0.120803	+	0.992677i	$0.538547\pi$
$390$		0			0
$391$	−220.625			−0.564259
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	927.672i		−	2.34854i
$396$		0			0
$397$	218.605			0.550643			0.275321	−	0.961352i	$-0.411216\pi$
$397$	218.605			0.550643			0.275321	+	0.961352i	$0.411216\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			0.984510i			0.00245514i	0.999999	+	0.00122757i	$0.000390747\pi$
$401$			0.984510i			0.00245514i	−0.999999	+	0.00122757i	$0.999609\pi$
$402$		0			0
$403$	−479.508			−1.18985
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	34.7126i		−	0.0852889i
$408$		0			0
$409$			711.378i			1.73931i	0.493659	+	0.869655i	$0.335659\pi$
$409$			711.378i			1.73931i	−0.493659	+	0.869655i	$0.664341\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	332.818i		−	0.805855i
$414$		0			0
$415$	706.413			1.70220
$416$		0			0
$417$		0			0
$418$		0			0
$419$	692.753			1.65335			0.826675	−	0.562680i	$-0.190230\pi$
$419$	692.753			1.65335			0.826675	+	0.562680i	$0.190230\pi$
$420$		0			0
$421$		−	205.304i		−	0.487658i	−0.969818	−	0.243829i	$-0.921596\pi$
$421$		−	205.304i		−	0.487658i	0.969818	−	0.243829i	$-0.0784035\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	772.638			1.81797
$426$		0			0
$427$	126.710			0.296744
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	534.388i		−	1.23988i	−0.784649	−	0.619940i	$-0.787157\pi$
$431$		−	534.388i		−	1.23988i	0.784649	−	0.619940i	$-0.212843\pi$
$432$		0			0
$433$			561.059i			1.29575i	0.761747	+	0.647874i	$0.224342\pi$
$433$			561.059i			1.29575i	−0.761747	+	0.647874i	$0.775658\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	133.581	−	89.5180i	0.305676	−	0.204847i
$438$		0			0
$439$			422.164i			0.961649i	0.876817	+	0.480825i	$0.159663\pi$
$439$			422.164i			0.961649i	−0.876817	+	0.480825i	$0.840337\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	780.825			1.76259			0.881293	−	0.472571i	$-0.156674\pi$
$443$	780.825			1.76259			0.881293	+	0.472571i	$0.156674\pi$
$444$		0			0
$445$			207.582i			0.466476i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			671.171i			1.49481i	0.664367	+	0.747406i	$0.268701\pi$
$449$			671.171i			1.49481i	−0.664367	+	0.747406i	$0.731299\pi$
$450$		0			0
$451$			25.2967i			0.0560903i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			617.277i			1.35665i
$456$		0			0
$457$	−371.441			−0.812781			−0.406390	−	0.913700i	$-0.633213\pi$
$457$	−371.441			−0.812781			−0.406390	+	0.913700i	$0.633213\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−301.718			−0.654487			−0.327243	−	0.944940i	$-0.606120\pi$
$461$	−301.718			−0.654487			−0.327243	+	0.944940i	$0.606120\pi$
$462$		0			0
$463$	665.736			1.43788			0.718938	−	0.695074i	$-0.244629\pi$
$463$	665.736			1.43788			0.718938	+	0.695074i	$0.244629\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	95.4717			0.204436			0.102218	−	0.994762i	$-0.467406\pi$
$467$	95.4717			0.204436			0.102218	+	0.994762i	$0.467406\pi$
$468$		0			0
$469$			145.473i			0.310176i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−17.5181			−0.0370362
$474$		0			0
$475$	−467.804	+	313.495i	−0.984851	+	0.659990i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	39.3541			0.0821590			0.0410795	−	0.999156i	$-0.486920\pi$
$479$	39.3541			0.0821590			0.0410795	+	0.999156i	$0.486920\pi$
$480$		0			0
$481$	1234.19			2.56589
$482$		0			0
$483$		0			0
$484$		0			0
$485$			412.429i			0.850369i
$486$		0			0
$487$			344.218i			0.706813i	0.935470	+	0.353407i	$0.114977\pi$
$487$			344.218i			0.706813i	−0.935470	+	0.353407i	$0.885023\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	582.955			1.18728			0.593641	−	0.804730i	$-0.297690\pi$
$491$	582.955			1.18728			0.593641	+	0.804730i	$0.297690\pi$
$492$		0			0
$493$		−	602.945i		−	1.22301i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			244.211i			0.491371i
$498$		0			0
$499$	−166.892			−0.334453			−0.167226	−	0.985919i	$-0.553481\pi$
$499$	−166.892			−0.334453			−0.167226	+	0.985919i	$0.553481\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−559.547			−1.11242			−0.556210	−	0.831042i	$-0.687745\pi$
$503$	−559.547			−1.11242			−0.556210	+	0.831042i	$0.687745\pi$
$504$		0			0
$505$	968.457			1.91774
$506$		0			0
$507$		0			0
$508$		0			0
$509$			371.176i			0.729226i	0.931159	+	0.364613i	$0.118799\pi$
$509$			371.176i			0.729226i	−0.931159	+	0.364613i	$0.881201\pi$
$510$		0			0
$511$	368.579			0.721290
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	551.858i		−	1.07157i
$516$		0			0
$517$	32.9601			0.0637526
$518$		0			0
$519$		0			0
$520$		0			0
$521$			967.227i			1.85648i	0.371981	+	0.928240i	$0.378679\pi$
$521$			967.227i			1.85648i	−0.371981	+	0.928240i	$0.621321\pi$
$522$		0			0
$523$			316.578i			0.605311i	0.953100	+	0.302656i	$0.0978732\pi$
$523$			316.578i			0.605311i	−0.953100	+	0.302656i	$0.902127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			491.724i			0.933062i
$528$		0			0
$529$	−457.373			−0.864600
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−899.415			−1.68746
$534$		0			0
$535$		−	1269.99i		−	2.37382i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−27.3186			−0.0506838
$540$		0			0
$541$	−700.118			−1.29412			−0.647059	−	0.762440i	$-0.724001\pi$
$541$	−700.118			−1.29412			−0.647059	+	0.762440i	$0.724001\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			258.404i			0.474135i
$546$		0			0
$547$			782.925i			1.43131i	0.698456	+	0.715653i	$0.253871\pi$
$547$			782.925i			1.43131i	−0.698456	+	0.715653i	$0.746129\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	244.643	+	365.061i	0.443998	+	0.662543i
$552$		0			0
$553$		−	412.270i		−	0.745515i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	358.027			0.642777			0.321388	−	0.946947i	$-0.395850\pi$
$557$	358.027			0.642777			0.321388	+	0.946947i	$0.395850\pi$
$558$		0			0
$559$		−	622.850i		−	1.11422i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	982.338i		−	1.74483i	−0.488769	−	0.872413i	$-0.662554\pi$
$563$		−	982.338i		−	1.74483i	0.488769	−	0.872413i	$-0.337446\pi$
$564$		0			0
$565$		−	316.399i		−	0.559998i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.6426i		−	0.0468236i	−0.999726	−	0.0234118i	$-0.992547\pi$
$569$		−	26.6426i		−	0.0468236i	0.999726	−	0.0234118i	$-0.00745289\pi$
$570$		0			0
$571$	309.230			0.541559			0.270779	−	0.962641i	$-0.412719\pi$
$571$	309.230			0.541559			0.270779	+	0.962641i	$0.412719\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−250.839			−0.436242
$576$		0			0
$577$	163.804			0.283889			0.141945	−	0.989875i	$-0.454665\pi$
$577$	163.804			0.283889			0.141945	+	0.989875i	$0.454665\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	313.940			0.540344
$582$		0			0
$583$			36.6157i			0.0628056i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	216.974			0.369632			0.184816	−	0.982773i	$-0.440831\pi$
$587$	216.974			0.369632			0.184816	+	0.982773i	$0.440831\pi$
$588$		0			0
$589$	−199.515	−	297.721i	−0.338736	−	0.505468i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	893.250			1.50632			0.753162	−	0.657835i	$-0.228528\pi$
$593$	893.250			1.50632			0.753162	+	0.657835i	$0.228528\pi$
$594$		0			0
$595$	633.002			1.06387
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1086.97i		−	1.81464i	−0.420446	−	0.907318i	$-0.638126\pi$
$599$		−	1086.97i		−	1.81464i	0.420446	−	0.907318i	$-0.361874\pi$
$600$		0			0
$601$		−	60.8027i		−	0.101169i	−0.998720	−	0.0505846i	$-0.983892\pi$
$601$		−	60.8027i		−	0.101169i	0.998720	−	0.0505846i	$-0.0161085\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−890.629			−1.47211
$606$		0			0
$607$			394.292i			0.649575i	0.945787	+	0.324788i	$0.105293\pi$
$607$			394.292i			0.649575i	−0.945787	+	0.324788i	$0.894707\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1171.88i			1.91798i
$612$		0			0
$613$	−899.074			−1.46668			−0.733340	−	0.679862i	$-0.762039\pi$
$613$	−899.074			−1.46668			−0.733340	+	0.679862i	$0.762039\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−593.943			−0.962630			−0.481315	−	0.876548i	$-0.659841\pi$
$617$	−593.943			−0.962630			−0.481315	+	0.876548i	$0.659841\pi$
$618$		0			0
$619$	−459.655			−0.742576			−0.371288	−	0.928518i	$-0.621084\pi$
$619$	−459.655			−0.742576			−0.371288	+	0.928518i	$0.621084\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			92.2521i			0.148077i
$624$		0			0
$625$	−487.517			−0.780027
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1265.63i		−	2.01214i
$630$		0			0
$631$	856.677			1.35765			0.678825	−	0.734300i	$-0.262490\pi$
$631$	856.677			1.35765			0.678825	+	0.734300i	$0.262490\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1303.48i			2.05273i
$636$		0			0
$637$		−	971.302i		−	1.52481i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	264.843i		−	0.413171i	−0.978429	−	0.206585i	$-0.933765\pi$
$641$		−	264.843i		−	0.413171i	0.978429	−	0.206585i	$-0.0662352\pi$
$642$		0			0
$643$	355.596			0.553026			0.276513	−	0.961010i	$-0.410821\pi$
$643$	355.596			0.553026			0.276513	+	0.961010i	$0.410821\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	589.871			0.911701			0.455851	−	0.890056i	$-0.349335\pi$
$647$	589.871			0.911701			0.455851	+	0.890056i	$0.349335\pi$
$648$		0			0
$649$		−	72.4379i		−	0.111615i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−891.917			−1.36588			−0.682938	−	0.730477i	$-0.739298\pi$
$653$	−891.917			−1.36588			−0.682938	+	0.730477i	$0.739298\pi$
$654$		0			0
$655$	−1493.99			−2.28089
$656$		0			0
$657$		0			0
$658$		0			0
$659$			828.284i			1.25688i	0.777858	+	0.628440i	$0.216306\pi$
$659$			828.284i			1.25688i	−0.777858	+	0.628440i	$0.783694\pi$
$660$		0			0
$661$		−	769.980i		−	1.16487i	−0.812877	−	0.582436i	$-0.802100\pi$
$661$		−	769.980i		−	1.16487i	0.812877	−	0.582436i	$-0.197900\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−383.260	+	256.839i	−0.576330	+	0.386223i
$666$		0			0
$667$			195.748i			0.293475i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	27.5784			0.0411005
$672$		0			0
$673$		−	372.195i		−	0.553038i	−0.961008	−	0.276519i	$-0.910819\pi$
$673$		−	372.195i		−	0.553038i	0.961008	−	0.276519i	$-0.0891809\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	484.617i		−	0.715830i	−0.933754	−	0.357915i	$-0.883488\pi$
$677$		−	484.617i		−	0.715830i	0.933754	−	0.357915i	$-0.116512\pi$
$678$		0			0
$679$			183.289i			0.269940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	779.041i		−	1.14062i	−0.821431	−	0.570308i	$-0.806824\pi$
$683$		−	779.041i		−	1.14062i	0.821431	−	0.570308i	$-0.193176\pi$
$684$		0			0
$685$	83.4467			0.121820
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1301.86			−1.88949
$690$		0			0
$691$	−76.9998			−0.111432			−0.0557162	−	0.998447i	$-0.517744\pi$
$691$	−76.9998			−0.111432			−0.0557162	+	0.998447i	$0.517744\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−418.691			−0.602434
$696$		0			0
$697$			922.328i			1.32328i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−754.168			−1.07585			−0.537923	−	0.842994i	$-0.680791\pi$
$701$	−754.168			−1.07585			−0.537923	+	0.842994i	$0.680791\pi$
$702$		0			0
$703$	513.527	+	766.295i	0.730479	+	1.09004i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	430.396			0.608763
$708$		0			0
$709$	48.7186			0.0687146			0.0343573	−	0.999410i	$-0.489062\pi$
$709$	48.7186			0.0687146			0.0343573	+	0.999410i	$0.489062\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	159.640i		−	0.223898i
$714$		0			0
$715$			134.350i			0.187903i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	583.791			0.811949			0.405974	−	0.913884i	$-0.366932\pi$
$719$	583.791			0.811949			0.405974	+	0.913884i	$0.366932\pi$
$720$		0			0
$721$		−	245.253i		−	0.340157i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	685.517i		−	0.945540i
$726$		0			0
$727$	−1005.23			−1.38271			−0.691357	−	0.722513i	$-0.742987\pi$
$727$	−1005.23			−1.38271			−0.691357	+	0.722513i	$0.742987\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−638.718			−0.873759
$732$		0			0
$733$	850.053			1.15969			0.579845	−	0.814727i	$-0.303113\pi$
$733$	850.053			1.15969			0.579845	+	0.814727i	$0.303113\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			31.6621i			0.0429608i
$738$		0			0
$739$	1215.10			1.64425			0.822127	−	0.569305i	$-0.192788\pi$
$739$	1215.10			1.64425			0.822127	+	0.569305i	$0.192788\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	661.859i		−	0.890792i	−0.895334	−	0.445396i	$-0.853063\pi$
$743$		−	661.859i		−	0.890792i	0.895334	−	0.445396i	$-0.146937\pi$
$744$		0			0
$745$	44.0987			0.0591929
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	564.402i		−	0.753541i
$750$		0			0
$751$			1354.03i			1.80297i	0.432812	+	0.901484i	$0.357521\pi$
$751$			1354.03i			1.80297i	−0.432812	+	0.901484i	$0.642479\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	207.192i		−	0.274426i
$756$		0			0
$757$	−972.577			−1.28478			−0.642389	−	0.766379i	$-0.722057\pi$
$757$	−972.577			−1.28478			−0.642389	+	0.766379i	$0.722057\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	644.913			0.847454			0.423727	−	0.905790i	$-0.360722\pi$
$761$	644.913			0.847454			0.423727	+	0.905790i	$0.360722\pi$
$762$		0			0
$763$			114.838i			0.150509i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2575.50			3.35789
$768$		0			0
$769$	−665.867			−0.865887			−0.432943	−	0.901421i	$-0.642525\pi$
$769$	−665.867			−0.865887			−0.432943	+	0.901421i	$0.642525\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	127.536i		−	0.164988i	−0.996592	−	0.0824942i	$-0.973711\pi$
$773$		−	127.536i		−	0.164988i	0.996592	−	0.0824942i	$-0.0262886\pi$
$774$		0			0
$775$			559.064i			0.721373i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−374.232	−	558.436i	−0.480400	−	0.716863i
$780$		0			0
$781$			53.1526i			0.0680572i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1044.59			1.33068
$786$		0			0
$787$			627.730i			0.797625i	0.917033	+	0.398812i	$0.130577\pi$
$787$			627.730i			0.797625i	−0.917033	+	0.398812i	$0.869423\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	140.612i		−	0.177765i
$792$		0			0
$793$			980.540i			1.23649i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1418.77i		−	1.78014i	−0.455827	−	0.890069i	$-0.650656\pi$
$797$		−	1418.77i		−	1.78014i	0.455827	−	0.890069i	$-0.349344\pi$
$798$		0			0
$799$	1201.74			1.50405
$800$		0			0
$801$		0			0
$802$		0			0
$803$	80.2213			0.0999020
$804$		0			0
$805$	−205.506			−0.255287
$806$		0			0
$807$		0			0
$808$		0			0
$809$	753.612			0.931535			0.465768	−	0.884907i	$-0.345778\pi$
$809$	753.612			0.931535			0.465768	+	0.884907i	$0.345778\pi$
$810$		0			0
$811$		−	225.670i		−	0.278261i	−0.990274	−	0.139131i	$-0.955569\pi$
$811$		−	225.670i		−	0.278261i	0.990274	−	0.139131i	$-0.0444308\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1341.98			−1.64660
$816$		0			0
$817$	386.720	−	259.158i	0.473342	−	0.317207i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−515.673			−0.628104			−0.314052	−	0.949406i	$-0.601687\pi$
$821$	−515.673			−0.628104			−0.314052	+	0.949406i	$0.601687\pi$
$822$		0			0
$823$	−1436.24			−1.74513			−0.872564	−	0.488500i	$-0.837544\pi$
$823$	−1436.24			−1.74513			−0.872564	+	0.488500i	$0.837544\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			134.792i			0.162989i	0.996674	+	0.0814944i	$0.0259693\pi$
$827$			134.792i			0.162989i	−0.996674	+	0.0814944i	$0.974031\pi$
$828$		0			0
$829$		−	376.958i		−	0.454714i	−0.973811	−	0.227357i	$-0.926992\pi$
$829$		−	376.958i		−	0.454714i	0.973811	−	0.227357i	$-0.0730084\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−996.047			−1.19573
$834$		0			0
$835$		−	2230.38i		−	2.67111i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	572.973i		−	0.682924i	−0.939896	−	0.341462i	$-0.889078\pi$
$839$		−	572.973i		−	0.682924i	0.939896	−	0.341462i	$-0.110922\pi$
$840$		0			0
$841$	306.042			0.363903
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3527.56			−4.17463
$846$		0			0
$847$	−395.808			−0.467305
$848$		0			0
$849$		0			0
$850$		0			0
$851$			410.892i			0.482834i
$852$		0			0
$853$	−1076.65			−1.26219			−0.631096	−	0.775705i	$-0.717395\pi$
$853$	−1076.65			−1.26219			−0.631096	+	0.775705i	$0.717395\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1506.83i			1.75826i	0.476578	+	0.879132i	$0.341877\pi$
$857$			1506.83i			1.75826i	−0.476578	+	0.879132i	$0.658123\pi$
$858$		0			0
$859$	−929.147			−1.08166			−0.540831	−	0.841131i	$-0.681890\pi$
$859$	−929.147			−1.08166			−0.540831	+	0.841131i	$0.681890\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	211.241i		−	0.244775i	−0.992482	−	0.122388i	$-0.960945\pi$
$863$		−	211.241i		−	0.244775i	0.992482	−	0.122388i	$-0.0390551\pi$
$864$		0			0
$865$		−	1444.87i		−	1.67037i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	89.7306i		−	0.103257i
$870$		0			0
$871$	−1125.73			−1.29246
$872$		0			0
$873$		0			0
$874$		0			0
$875$	112.636			0.128727
$876$		0			0
$877$		−	229.052i		−	0.261176i	−0.991437	−	0.130588i	$-0.958313\pi$
$877$		−	229.052i		−	0.261176i	0.991437	−	0.130588i	$-0.0416865\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1354.01			1.53690			0.768449	−	0.639911i	$-0.221029\pi$
$881$	1354.01			1.53690			0.768449	+	0.639911i	$0.221029\pi$
$882$		0			0
$883$	−391.523			−0.443401			−0.221701	−	0.975115i	$-0.571161\pi$
$883$	−391.523			−0.443401			−0.221701	+	0.975115i	$0.571161\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	697.322i		−	0.786158i	−0.919505	−	0.393079i	$-0.871410\pi$
$887$		−	697.322i		−	0.786158i	0.919505	−	0.393079i	$-0.128590\pi$
$888$		0			0
$889$			579.287i			0.651616i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−727.608	+	487.601i	−0.814791	+	0.546026i
$894$		0			0
$895$		−	674.736i		−	0.753895i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	436.278			0.485292
$900$		0			0
$901$			1335.02i			1.48171i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	116.630i		−	0.128873i
$906$		0			0
$907$		−	626.344i		−	0.690567i	−0.938498	−	0.345283i	$-0.887783\pi$
$907$		−	626.344i		−	0.690567i	0.938498	−	0.345283i	$-0.112217\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			517.549i			0.568111i	0.958808	+	0.284056i	$0.0916800\pi$
$911$			517.549i			0.568111i	−0.958808	+	0.284056i	$0.908320\pi$
$912$		0			0
$913$	68.3290			0.0748401
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−663.948			−0.724043
$918$		0			0
$919$	900.211			0.979555			0.489777	−	0.871848i	$-0.337078\pi$
$919$	900.211			0.979555			0.489777	+	0.871848i	$0.337078\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1889.82			−2.04748
$924$		0			0
$925$		−	1438.96i		−	1.55563i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	734.220			0.790333			0.395167	−	0.918609i	$-0.370687\pi$
$929$	734.220			0.790333			0.395167	+	0.918609i	$0.370687\pi$
$930$		0			0
$931$	603.070	−	404.143i	0.647766	−	0.434095i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	137.773			0.147351
$936$		0			0
$937$	832.710			0.888698			0.444349	−	0.895854i	$-0.353435\pi$
$937$	832.710			0.888698			0.444349	+	0.895854i	$0.353435\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1383.55i			1.47030i	0.677905	+	0.735149i	$0.262888\pi$
$941$			1383.55i			1.47030i	−0.677905	+	0.735149i	$0.737112\pi$
$942$		0			0
$943$		−	299.437i		−	0.317536i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−322.935			−0.341008			−0.170504	−	0.985357i	$-0.554540\pi$
$947$	−322.935			−0.341008			−0.170504	+	0.985357i	$0.554540\pi$
$948$		0			0
$949$			2852.24i			3.00552i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1815.82i		−	1.90537i	−0.303962	−	0.952684i	$-0.598309\pi$
$953$		−	1815.82i		−	1.90537i	0.303962	−	0.952684i	$-0.401691\pi$
$954$		0			0
$955$	−1433.56			−1.50111
$956$		0			0
$957$		0			0
$958$		0			0
$959$	37.0848			0.0386703
$960$		0			0
$961$	605.200			0.629760
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1115.72i		−	1.15619i
$966$		0			0
$967$	738.252			0.763446			0.381723	−	0.924277i	$-0.375331\pi$
$967$	738.252			0.763446			0.381723	+	0.924277i	$0.375331\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1140.17i			1.17422i	0.809507	+	0.587111i	$0.199735\pi$
$971$			1140.17i			1.17422i	−0.809507	+	0.587111i	$0.800265\pi$
$972$		0			0
$973$	−186.072			−0.191236
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1559.45i		−	1.59616i	−0.602550	−	0.798081i	$-0.705849\pi$
$977$		−	1559.45i		−	1.59616i	0.602550	−	0.798081i	$-0.294151\pi$
$978$		0			0
$979$			20.0787i			0.0205094i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			635.108i			0.646091i	0.946383	+	0.323046i	$0.104707\pi$
$983$			635.108i			0.646091i	−0.946383	+	0.323046i	$0.895293\pi$
$984$		0			0
$985$	−502.680			−0.510335
$986$		0			0
$987$		0			0
$988$		0			0
$989$	207.362			0.209668
$990$		0			0
$991$			700.746i			0.707110i	0.935414	+	0.353555i	$0.115027\pi$
$991$			700.746i			0.707110i	−0.935414	+	0.353555i	$0.884973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	217.921			0.219016
$996$		0			0
$997$	−1462.90			−1.46730			−0.733649	−	0.679529i	$-0.762184\pi$
$997$	−1462.90			−1.46730			−0.733649	+	0.679529i	$0.762184\pi$
$998$		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.37.6		6
3.2	odd	2		228.3.h.a.37.1	✓	6
4.3	odd	2		2736.3.o.m.721.6		6
12.11	even	2		912.3.o.c.721.4		6
19.18	odd	2	inner	684.3.h.e.37.5		6
57.56	even	2		228.3.h.a.37.4	yes	6
76.75	even	2		2736.3.o.m.721.5		6
228.227	odd	2		912.3.o.c.721.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.h.a.37.1	✓	6	3.2	odd	2
228.3.h.a.37.4	yes	6	57.56	even	2
684.3.h.e.37.5		6	19.18	odd	2	inner
684.3.h.e.37.6		6	1.1	even	1	trivial
912.3.o.c.721.1		6	228.227	odd	2
912.3.o.c.721.4		6	12.11	even	2
2736.3.o.m.721.5		6	76.75	even	2
2736.3.o.m.721.6		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+7.39180 q^{5} +3.28502 q^{7} +O(q^{10})\)	\(q+7.39180 q^{5} +3.28502 q^{7} +0.714984 q^{11} +25.4210i q^{13} +26.0686 q^{17} +(-15.7836 + 10.5773i) q^{19} -8.46325 q^{23} +29.6386 q^{25} -23.1292i q^{29} +18.8627i q^{31} +24.2822 q^{35} -48.5501i q^{37} +35.3808i q^{41} -24.5014 q^{43} +46.0990 q^{47} -38.2087 q^{49} +51.2119i q^{53} +5.28502 q^{55} -101.314i q^{59} +38.5721 q^{61} +187.907i q^{65} +44.2837i q^{67} +74.3410i q^{71} +112.200 q^{73} +2.34873 q^{77} -125.500i q^{79} +95.5672 q^{83} +192.694 q^{85} +28.0827i q^{89} +83.5083i q^{91} +(-116.669 + 78.1849i) q^{95} +55.7955i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 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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