LMFDB - Embedded newform 2100.3.e.c.449.32

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,3,Mod(449,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2100.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$57.2208555157$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.32
Character	$\chi$	$=$	2100.449
Dual form			2100.3.e.c.449.31

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.97031 + 0.421042i) q^{3} +2.64575i q^{7} +(8.64545 + 2.50125i) q^{9} +8.16468i q^{11} +5.77792i q^{13} -28.0655 q^{17} +4.50420 q^{19} +(-1.11397 + 7.85869i) q^{21} -13.6408 q^{23} +(24.6265 + 11.0696i) q^{27} +15.1126i q^{29} -57.3810 q^{31} +(-3.43768 + 24.2516i) q^{33} -9.23344i q^{37} +(-2.43275 + 17.1622i) q^{39} +23.0823i q^{41} +33.1640i q^{43} -4.35488 q^{47} -7.00000 q^{49} +(-83.3630 - 11.8167i) q^{51} -22.9301 q^{53} +(13.3789 + 1.89646i) q^{57} -36.3623i q^{59} -22.8074 q^{61} +(-6.61769 + 22.8737i) q^{63} -6.78336i q^{67} +(-40.5173 - 5.74334i) q^{69} +80.7397i q^{71} -120.569i q^{73} -21.6017 q^{77} +54.4972 q^{79} +(68.4875 + 43.2489i) q^{81} -57.2963 q^{83} +(-6.36306 + 44.8892i) q^{87} -150.774i q^{89} -15.2869 q^{91} +(-170.439 - 24.1598i) q^{93} +117.942i q^{97} +(-20.4219 + 70.5873i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 20 q^{9} - 96 q^{19} - 28 q^{21} + 48 q^{31} + 108 q^{39} - 224 q^{49} - 372 q^{51} - 304 q^{61} + 392 q^{69} - 424 q^{79} - 236 q^{81} + 392 q^{91} - 52 q^{99}+O(q^{100}) $ 32 * q + 20 * q^9 - 96 * q^19 - 28 * q^21 + 48 * q^31 + 108 * q^39 - 224 * q^49 - 372 * q^51 - 304 * q^61 + 392 * q^69 - 424 * q^79 - 236 * q^81 + 392 * q^91 - 52 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	2.97031	+	0.421042i	0.990102	+	0.140347i
$3$	2.97031	+	0.421042i	0.990102	+	0.140347i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			2.64575i			0.377964i
$7$			2.64575i			0.377964i
$8$		0			0
$9$	8.64545	+	2.50125i	0.960605	+	0.277917i
$10$		0			0
$11$			8.16468i			0.742243i	0.928584	+	0.371122i	$0.121027\pi$
$11$			8.16468i			0.742243i	−0.928584	+	0.371122i	$0.878973\pi$
$12$		0			0
$13$			5.77792i			0.444455i	0.974995	+	0.222228i	$0.0713328\pi$
$13$			5.77792i			0.444455i	−0.974995	+	0.222228i	$0.928667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−28.0655			−1.65091			−0.825454	−	0.564469i	$-0.809081\pi$
$17$	−28.0655			−1.65091			−0.825454	+	0.564469i	$0.809081\pi$
$18$		0			0
$19$	4.50420			0.237063			0.118532	−	0.992950i	$-0.462181\pi$
$19$	4.50420			0.237063			0.118532	+	0.992950i	$0.462181\pi$
$20$		0			0
$21$	−1.11397	+	7.85869i	−0.0530464	+	0.374223i
$22$		0			0
$23$	−13.6408			−0.593077			−0.296538	−	0.955021i	$-0.595832\pi$
$23$	−13.6408			−0.593077			−0.296538	+	0.955021i	$0.595832\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	24.6265	+	11.0696i	0.912092	+	0.409985i
$28$		0			0
$29$			15.1126i			0.521125i	0.965457	+	0.260563i	$0.0839080\pi$
$29$			15.1126i			0.521125i	−0.965457	+	0.260563i	$0.916092\pi$
$30$		0			0
$31$	−57.3810			−1.85100			−0.925500	−	0.378748i	$-0.876355\pi$
$31$	−57.3810			−1.85100			−0.925500	+	0.378748i	$0.876355\pi$
$32$		0			0
$33$	−3.43768	+	24.2516i	−0.104172	+	0.734897i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	9.23344i		−	0.249552i	−0.992185	−	0.124776i	$-0.960179\pi$
$37$		−	9.23344i		−	0.249552i	0.992185	−	0.124776i	$-0.0398213\pi$
$38$		0			0
$39$	−2.43275	+	17.1622i	−0.0623782	+	0.440056i
$40$		0			0
$41$			23.0823i			0.562983i	0.959564	+	0.281492i	$0.0908292\pi$
$41$			23.0823i			0.562983i	−0.959564	+	0.281492i	$0.909171\pi$
$42$		0			0
$43$			33.1640i			0.771255i	0.922654	+	0.385628i	$0.126015\pi$
$43$			33.1640i			0.771255i	−0.922654	+	0.385628i	$0.873985\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.35488			−0.0926571			−0.0463286	−	0.998926i	$-0.514752\pi$
$47$	−4.35488			−0.0926571			−0.0463286	+	0.998926i	$0.514752\pi$
$48$		0			0
$49$	−7.00000			−0.142857
$50$		0			0
$51$	−83.3630	−	11.8167i	−1.63457	−	0.231701i
$52$		0			0
$53$	−22.9301			−0.432643			−0.216322	−	0.976322i	$-0.569406\pi$
$53$	−22.9301			−0.432643			−0.216322	+	0.976322i	$0.569406\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	13.3789	+	1.89646i	0.234717	+	0.0332712i
$58$		0			0
$59$		−	36.3623i		−	0.616311i	−0.951336	−	0.308155i	$-0.900288\pi$
$59$		−	36.3623i		−	0.616311i	0.951336	−	0.308155i	$-0.0997117\pi$
$60$		0			0
$61$	−22.8074			−0.373892			−0.186946	−	0.982370i	$-0.559859\pi$
$61$	−22.8074			−0.373892			−0.186946	+	0.982370i	$0.559859\pi$
$62$		0			0
$63$	−6.61769	+	22.8737i	−0.105043	+	0.363075i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.78336i		−	0.101244i	−0.998718	−	0.0506221i	$-0.983880\pi$
$67$		−	6.78336i		−	0.101244i	0.998718	−	0.0506221i	$-0.0161204\pi$
$68$		0			0
$69$	−40.5173	−	5.74334i	−0.587207	−	0.0832368i
$70$		0			0
$71$			80.7397i			1.13718i	0.822622	+	0.568589i	$0.192511\pi$
$71$			80.7397i			1.13718i	−0.822622	+	0.568589i	$0.807489\pi$
$72$		0			0
$73$		−	120.569i		−	1.65163i	−0.563944	−	0.825813i	$-0.690717\pi$
$73$		−	120.569i		−	1.65163i	0.563944	−	0.825813i	$-0.309283\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−21.6017			−0.280542
$78$		0			0
$79$	54.4972			0.689838			0.344919	−	0.938633i	$-0.387906\pi$
$79$	54.4972			0.689838			0.344919	+	0.938633i	$0.387906\pi$
$80$		0			0
$81$	68.4875	+	43.2489i	0.845525	+	0.533937i
$82$		0			0
$83$	−57.2963			−0.690316			−0.345158	−	0.938545i	$-0.612175\pi$
$83$	−57.2963			−0.690316			−0.345158	+	0.938545i	$0.612175\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.36306	+	44.8892i	−0.0731386	+	0.515967i
$88$		0			0
$89$		−	150.774i		−	1.69409i	−0.531523	−	0.847044i	$-0.678380\pi$
$89$		−	150.774i		−	1.69409i	0.531523	−	0.847044i	$-0.321620\pi$
$90$		0			0
$91$	−15.2869			−0.167988
$92$		0			0
$93$	−170.439	−	24.1598i	−1.83268	−	0.259783i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			117.942i			1.21590i	0.793976	+	0.607949i	$0.208007\pi$
$97$			117.942i			1.21590i	−0.793976	+	0.607949i	$0.791993\pi$
$98$		0			0
$99$	−20.4219	+	70.5873i	−0.206282	+	0.713003i
$100$		0			0
$101$			140.433i			1.39043i	0.718804	+	0.695213i	$0.244690\pi$
$101$			140.433i			1.39043i	−0.718804	+	0.695213i	$0.755310\pi$
$102$		0			0
$103$		−	10.1147i		−	0.0982014i	−0.998794	−	0.0491007i	$-0.984364\pi$
$103$		−	10.1147i		−	0.0982014i	0.998794	−	0.0491007i	$-0.0156355\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	55.1068			0.515017			0.257509	−	0.966276i	$-0.417098\pi$
$107$	55.1068			0.515017			0.257509	+	0.966276i	$0.417098\pi$
$108$		0			0
$109$	−63.5989			−0.583477			−0.291738	−	0.956498i	$-0.594234\pi$
$109$	−63.5989			−0.583477			−0.291738	+	0.956498i	$0.594234\pi$
$110$		0			0
$111$	3.88767	−	27.4261i	0.0350241	−	0.247082i
$112$		0			0
$113$	25.6257			0.226776			0.113388	−	0.993551i	$-0.463830\pi$
$113$	25.6257			0.226776			0.113388	+	0.993551i	$0.463830\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−14.4520	+	49.9527i	−0.123522	+	0.426946i
$118$		0			0
$119$		−	74.2542i		−	0.623985i
$120$		0			0
$121$	54.3380			0.449075
$122$		0			0
$123$	−9.71863	+	68.5616i	−0.0790133	+	0.557411i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	69.5279i		−	0.547464i	−0.961806	−	0.273732i	$-0.911742\pi$
$127$		−	69.5279i		−	0.547464i	0.961806	−	0.273732i	$-0.0882581\pi$
$128$		0			0
$129$	−13.9634	+	98.5072i	−0.108244	+	0.763622i
$130$		0			0
$131$			114.353i			0.872927i	0.899722	+	0.436463i	$0.143769\pi$
$131$			114.353i			0.872927i	−0.899722	+	0.436463i	$0.856231\pi$
$132$		0			0
$133$			11.9170i			0.0896014i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−165.307			−1.20662			−0.603310	−	0.797507i	$-0.706152\pi$
$137$	−165.307			−1.20662			−0.603310	+	0.797507i	$0.706152\pi$
$138$		0			0
$139$	−207.941			−1.49598			−0.747989	−	0.663712i	$-0.768980\pi$
$139$	−207.941			−1.49598			−0.747989	+	0.663712i	$0.768980\pi$
$140$		0			0
$141$	−12.9353	−	1.83359i	−0.0917400	−	0.0130042i
$142$		0			0
$143$	−47.1749			−0.329894
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−20.7921	−	2.94730i	−0.141443	−	0.0200496i
$148$		0			0
$149$			127.831i			0.857927i	0.903322	+	0.428963i	$0.141121\pi$
$149$			127.831i			0.857927i	−0.903322	+	0.428963i	$0.858879\pi$
$150$		0			0
$151$	218.450			1.44669			0.723343	−	0.690489i	$-0.242604\pi$
$151$	218.450			1.44669			0.723343	+	0.690489i	$0.242604\pi$
$152$		0			0
$153$	−242.638	−	70.1987i	−1.58587	−	0.458815i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			274.083i			1.74575i	0.487942	+	0.872876i	$0.337748\pi$
$157$			274.083i			1.74575i	−0.487942	+	0.872876i	$0.662252\pi$
$158$		0			0
$159$	−68.1094	−	9.65454i	−0.428361	−	0.0607204i
$160$		0			0
$161$		−	36.0901i		−	0.224162i
$162$		0			0
$163$			283.820i			1.74123i	0.491968	+	0.870613i	$0.336278\pi$
$163$			283.820i			1.74123i	−0.491968	+	0.870613i	$0.663722\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−101.223			−0.606127			−0.303064	−	0.952970i	$-0.598009\pi$
$167$	−101.223			−0.606127			−0.303064	+	0.952970i	$0.598009\pi$
$168$		0			0
$169$	135.616			0.802459
$170$		0			0
$171$	38.9408	+	11.2661i	0.227724	+	0.0658838i
$172$		0			0
$173$	−332.624			−1.92268			−0.961340	−	0.275363i	$-0.911202\pi$
$173$	−332.624			−1.92268			−0.961340	+	0.275363i	$0.911202\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	15.3101	−	108.007i	0.0864977	−	0.610211i
$178$		0			0
$179$			15.1419i			0.0845918i	0.999105	+	0.0422959i	$0.0134672\pi$
$179$			15.1419i			0.0845918i	−0.999105	+	0.0422959i	$0.986533\pi$
$180$		0			0
$181$	244.178			1.34905			0.674525	−	0.738252i	$-0.264349\pi$
$181$	244.178			1.34905			0.674525	+	0.738252i	$0.264349\pi$
$182$		0			0
$183$	−67.7450	−	9.60288i	−0.370191	−	0.0524748i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	229.145i		−	1.22538i
$188$		0			0
$189$	−29.2874	+	65.1556i	−0.154960	+	0.344739i
$190$		0			0
$191$			328.924i			1.72211i	0.508508	+	0.861057i	$0.330197\pi$
$191$			328.924i			1.72211i	−0.508508	+	0.861057i	$0.669803\pi$
$192$		0			0
$193$		−	304.626i		−	1.57837i	−0.614154	−	0.789186i	$-0.710502\pi$
$193$		−	304.626i		−	1.57837i	0.614154	−	0.789186i	$-0.289498\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−235.048			−1.19314			−0.596568	−	0.802563i	$-0.703469\pi$
$197$	−235.048			−1.19314			−0.596568	+	0.802563i	$0.703469\pi$
$198$		0			0
$199$	298.075			1.49786			0.748931	−	0.662648i	$-0.230567\pi$
$199$	298.075			1.49786			0.748931	+	0.662648i	$0.230567\pi$
$200$		0			0
$201$	2.85608	−	20.1487i	0.0142094	−	0.100242i
$202$		0			0
$203$	−39.9843			−0.196967
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−117.931	−	34.1190i	−0.569713	−	0.164826i
$208$		0			0
$209$			36.7753i			0.175959i
$210$		0			0
$211$	−113.494			−0.537885			−0.268942	−	0.963156i	$-0.586674\pi$
$211$	−113.494			−0.537885			−0.268942	+	0.963156i	$0.586674\pi$
$212$		0			0
$213$	−33.9948	+	239.822i	−0.159600	+	1.12592i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	151.816i		−	0.699612i
$218$		0			0
$219$	50.7646	−	358.126i	0.231802	−	1.63528i
$220$		0			0
$221$		−	162.160i		−	0.733755i
$222$		0			0
$223$		−	337.861i		−	1.51507i	−0.652794	−	0.757536i	$-0.726403\pi$
$223$		−	337.861i		−	1.51507i	0.652794	−	0.757536i	$-0.273597\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	41.4984			0.182812			0.0914062	−	0.995814i	$-0.470864\pi$
$227$	41.4984			0.182812			0.0914062	+	0.995814i	$0.470864\pi$
$228$		0			0
$229$	341.169			1.48982			0.744911	−	0.667164i	$-0.232492\pi$
$229$	341.169			1.48982			0.744911	+	0.667164i	$0.232492\pi$
$230$		0			0
$231$	−64.1637	−	9.09524i	−0.277765	−	0.0393733i
$232$		0			0
$233$	113.530			0.487255			0.243627	−	0.969869i	$-0.421663\pi$
$233$	113.530			0.487255			0.243627	+	0.969869i	$0.421663\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	161.873	+	22.9456i	0.683010	+	0.0968170i
$238$		0			0
$239$			157.758i			0.660074i	0.943968	+	0.330037i	$0.107061\pi$
$239$			157.758i			0.660074i	−0.943968	+	0.330037i	$0.892939\pi$
$240$		0			0
$241$	247.947			1.02883			0.514413	−	0.857542i	$-0.328010\pi$
$241$	247.947			1.02883			0.514413	+	0.857542i	$0.328010\pi$
$242$		0			0
$243$	185.219	+	157.299i	0.762219	+	0.647319i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			26.0249i			0.105364i
$248$		0			0
$249$	−170.187	−	24.1242i	−0.683484	−	0.0968842i
$250$		0			0
$251$			276.737i			1.10254i	0.834328	+	0.551268i	$0.185856\pi$
$251$			276.737i			1.10254i	−0.834328	+	0.551268i	$0.814144\pi$
$252$		0			0
$253$		−	111.372i		−	0.440207i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−157.228			−0.611782			−0.305891	−	0.952067i	$-0.598954\pi$
$257$	−157.228			−0.611782			−0.305891	+	0.952067i	$0.598954\pi$
$258$		0			0
$259$	24.4294			0.0943219
$260$		0			0
$261$	−37.8005	+	130.655i	−0.144829	+	0.500596i
$262$		0			0
$263$	−306.093			−1.16385			−0.581926	−	0.813242i	$-0.697701\pi$
$263$	−306.093			−1.16385			−0.581926	+	0.813242i	$0.697701\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	63.4822	−	447.844i	0.237761	−	1.67732i
$268$		0			0
$269$		−	240.912i		−	0.895584i	−0.894138	−	0.447792i	$-0.852211\pi$
$269$		−	240.912i		−	0.895584i	0.894138	−	0.447792i	$-0.147789\pi$
$270$		0			0
$271$	0.693234			0.00255806			0.00127903	−	0.999999i	$-0.499593\pi$
$271$	0.693234			0.00255806			0.00127903	+	0.999999i	$0.499593\pi$
$272$		0			0
$273$	−45.4069	−	6.43645i	−0.166326	−	0.0235767i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	195.345i		−	0.705215i	−0.935771	−	0.352608i	$-0.885295\pi$
$277$		−	195.345i		−	0.705215i	0.935771	−	0.352608i	$-0.114705\pi$
$278$		0			0
$279$	−496.084	−	143.524i	−1.77808	−	0.514424i
$280$		0			0
$281$			407.985i			1.45191i	0.687745	+	0.725953i	$0.258601\pi$
$281$			407.985i			1.45191i	−0.687745	+	0.725953i	$0.741399\pi$
$282$		0			0
$283$			387.819i			1.37039i	0.728362	+	0.685193i	$0.240282\pi$
$283$			387.819i			1.37039i	−0.728362	+	0.685193i	$0.759718\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−61.0701			−0.212788
$288$		0			0
$289$	498.670			1.72550
$290$		0			0
$291$	−49.6586	+	350.324i	−0.170648	+	1.20386i
$292$		0			0
$293$	−98.4013			−0.335840			−0.167920	−	0.985801i	$-0.553705\pi$
$293$	−98.4013			−0.335840			−0.167920	+	0.985801i	$0.553705\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−90.3796	+	201.067i	−0.304308	+	0.676995i
$298$		0			0
$299$		−	78.8153i		−	0.263596i
$300$		0			0
$301$	−87.7437			−0.291507
$302$		0			0
$303$	−59.1283	+	417.129i	−0.195143	+	1.37666i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	172.156i		−	0.560769i	−0.959888	−	0.280384i	$-0.909538\pi$
$307$		−	172.156i		−	0.560769i	0.959888	−	0.280384i	$-0.0904620\pi$
$308$		0			0
$309$	4.25874	−	30.0439i	0.0137823	−	0.0972294i
$310$		0			0
$311$		−	463.864i		−	1.49152i	−0.666212	−	0.745762i	$-0.732086\pi$
$311$		−	463.864i		−	1.49152i	0.666212	−	0.745762i	$-0.267914\pi$
$312$		0			0
$313$			354.701i			1.13323i	0.823983	+	0.566615i	$0.191747\pi$
$313$			354.701i			1.13323i	−0.823983	+	0.566615i	$0.808253\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	44.9323			0.141742			0.0708711	−	0.997485i	$-0.477422\pi$
$317$	44.9323			0.141742			0.0708711	+	0.997485i	$0.477422\pi$
$318$		0			0
$319$	−123.390			−0.386802
$320$		0			0
$321$	163.684	+	23.2023i	0.509920	+	0.0722814i
$322$		0			0
$323$	−126.412			−0.391370
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−188.908	−	26.7779i	−0.577701	−	0.0818895i
$328$		0			0
$329$		−	11.5219i		−	0.0350211i
$330$		0			0
$331$	−31.4599			−0.0950451			−0.0475226	−	0.998870i	$-0.515133\pi$
$331$	−31.4599			−0.0950451			−0.0475226	+	0.998870i	$0.515133\pi$
$332$		0			0
$333$	23.0951	−	79.8272i	0.0693548	−	0.239721i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	392.031i		−	1.16330i	−0.813441	−	0.581648i	$-0.802408\pi$
$337$		−	392.031i		−	1.16330i	0.813441	−	0.581648i	$-0.197592\pi$
$338$		0			0
$339$	76.1163	+	10.7895i	0.224532	+	0.0318275i
$340$		0			0
$341$		−	468.497i		−	1.37389i
$342$		0			0
$343$		−	18.5203i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	649.989			1.87317			0.936584	−	0.350444i	$-0.113969\pi$
$347$	649.989			1.87317			0.936584	+	0.350444i	$0.113969\pi$
$348$		0			0
$349$	176.534			0.505827			0.252913	−	0.967489i	$-0.418611\pi$
$349$	176.534			0.505827			0.252913	+	0.967489i	$0.418611\pi$
$350$		0			0
$351$	−63.9592	+	142.290i	−0.182220	+	0.405384i
$352$		0			0
$353$	−84.4749			−0.239306			−0.119653	−	0.992816i	$-0.538178\pi$
$353$	−84.4749			−0.239306			−0.119653	+	0.992816i	$0.538178\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	31.2642	−	220.558i	0.0875747	−	0.617809i
$358$		0			0
$359$			175.365i			0.488481i	0.969715	+	0.244241i	$0.0785387\pi$
$359$			175.365i			0.488481i	−0.969715	+	0.244241i	$0.921461\pi$
$360$		0			0
$361$	−340.712			−0.943801
$362$		0			0
$363$	161.401	+	22.8786i	0.444630	+	0.0630265i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			643.483i			1.75336i	0.481075	+	0.876680i	$0.340247\pi$
$367$			643.483i			1.75336i	−0.481075	+	0.876680i	$0.659753\pi$
$368$		0			0
$369$	−57.7347	+	199.557i	−0.156462	+	0.540805i
$370$		0			0
$371$		−	60.6673i		−	0.163524i
$372$		0			0
$373$		−	312.351i		−	0.837402i	−0.908124	−	0.418701i	$-0.862486\pi$
$373$		−	312.351i		−	0.837402i	0.908124	−	0.418701i	$-0.137514\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−87.3196			−0.231617
$378$		0			0
$379$	−129.174			−0.340830			−0.170415	−	0.985372i	$-0.554511\pi$
$379$	−129.174			−0.340830			−0.170415	+	0.985372i	$0.554511\pi$
$380$		0			0
$381$	29.2742	−	206.519i	0.0768352	−	0.542045i
$382$		0			0
$383$	−371.018			−0.968716			−0.484358	−	0.874870i	$-0.660947\pi$
$383$	−371.018			−0.968716			−0.484358	+	0.874870i	$0.660947\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−82.9514	+	286.717i	−0.214345	+	0.740872i
$388$		0			0
$389$			239.364i			0.615332i	0.951494	+	0.307666i	$0.0995479\pi$
$389$			239.364i			0.615332i	−0.951494	+	0.307666i	$0.900452\pi$
$390$		0			0
$391$	382.834			0.979116
$392$		0			0
$393$	−48.1476	+	339.665i	−0.122513	+	0.864287i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			745.866i			1.87875i	0.342885	+	0.939377i	$0.388596\pi$
$397$			745.866i			1.87875i	−0.342885	+	0.939377i	$0.611404\pi$
$398$		0			0
$399$	−5.01756	+	35.3971i	−0.0125753	+	0.0887146i
$400$		0			0
$401$			338.979i			0.845335i	0.906285	+	0.422667i	$0.138906\pi$
$401$			338.979i			0.845335i	−0.906285	+	0.422667i	$0.861094\pi$
$402$		0			0
$403$		−	331.543i		−	0.822687i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	75.3881			0.185229
$408$		0			0
$409$	242.273			0.592355			0.296177	−	0.955133i	$-0.404288\pi$
$409$	242.273			0.592355			0.296177	+	0.955133i	$0.404288\pi$
$410$		0			0
$411$	−491.012	−	69.6012i	−1.19468	−	0.169346i
$412$		0			0
$413$	96.2057			0.232944
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−617.648	−	87.5519i	−1.48117	−	0.209957i
$418$		0			0
$419$			116.740i			0.278615i	0.990249	+	0.139308i	$0.0444877\pi$
$419$			116.740i			0.278615i	−0.990249	+	0.139308i	$0.955512\pi$
$420$		0			0
$421$	−57.9552			−0.137661			−0.0688304	−	0.997628i	$-0.521927\pi$
$421$	−57.9552			−0.137661			−0.0688304	+	0.997628i	$0.521927\pi$
$422$		0			0
$423$	−37.6499	−	10.8927i	−0.0890069	−	0.0257510i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	60.3427i		−	0.141318i
$428$		0			0
$429$	−140.124	−	19.8626i	−0.326629	−	0.0462998i
$430$		0			0
$431$		−	683.443i		−	1.58571i	−0.609408	−	0.792857i	$-0.708593\pi$
$431$		−	683.443i		−	1.58571i	0.609408	−	0.792857i	$-0.291407\pi$
$432$		0			0
$433$		−	710.818i		−	1.64161i	−0.571207	−	0.820806i	$-0.693525\pi$
$433$		−	710.818i		−	1.64161i	0.571207	−	0.820806i	$-0.306475\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−61.4407			−0.140597
$438$		0			0
$439$	113.873			0.259392			0.129696	−	0.991554i	$-0.458600\pi$
$439$	113.873			0.259392			0.129696	+	0.991554i	$0.458600\pi$
$440$		0			0
$441$	−60.5181	−	17.5088i	−0.137229	−	0.0397024i
$442$		0			0
$443$	285.321			0.644066			0.322033	−	0.946728i	$-0.395634\pi$
$443$	285.321			0.644066			0.322033	+	0.946728i	$0.395634\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−53.8223	+	379.698i	−0.120408	+	0.849435i
$448$		0			0
$449$		−	248.688i		−	0.553871i	−0.960888	−	0.276936i	$-0.910681\pi$
$449$		−	248.688i		−	0.553871i	0.960888	−	0.276936i	$-0.0893189\pi$
$450$		0			0
$451$	−188.460			−0.417871
$452$		0			0
$453$	648.863	+	91.9766i	1.43237	+	0.203039i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			179.703i			0.393224i	0.980481	+	0.196612i	$0.0629940\pi$
$457$			179.703i			0.393224i	−0.980481	+	0.196612i	$0.937006\pi$
$458$		0			0
$459$	−691.154	−	310.673i	−1.50578	−	0.676847i
$460$		0			0
$461$			496.992i			1.07807i	0.842282	+	0.539037i	$0.181212\pi$
$461$			496.992i			1.07807i	−0.842282	+	0.539037i	$0.818788\pi$
$462$		0			0
$463$		−	289.473i		−	0.625212i	−0.949883	−	0.312606i	$-0.898798\pi$
$463$		−	289.473i		−	0.625212i	0.949883	−	0.312606i	$-0.101202\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	200.191			0.428674			0.214337	−	0.976760i	$-0.431241\pi$
$467$	200.191			0.428674			0.214337	+	0.976760i	$0.431241\pi$
$468$		0			0
$469$	17.9471			0.0382667
$470$		0			0
$471$	−115.401	+	814.111i	−0.245012	+	1.72847i
$472$		0			0
$473$	−270.773			−0.572459
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−198.241	−	57.3539i	−0.415599	−	0.120239i
$478$		0			0
$479$		−	616.585i		−	1.28723i	−0.765348	−	0.643617i	$-0.777433\pi$
$479$		−	616.585i		−	1.28723i	0.765348	−	0.643617i	$-0.222567\pi$
$480$		0			0
$481$	53.3501			0.110915
$482$		0			0
$483$	15.1955	−	107.199i	0.0314606	−	0.221943i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			393.608i			0.808229i	0.914708	+	0.404115i	$0.132420\pi$
$487$			393.608i			0.808229i	−0.914708	+	0.404115i	$0.867580\pi$
$488$		0			0
$489$	−119.500	+	843.032i	−0.244377	+	1.72399i
$490$		0			0
$491$		−	602.139i		−	1.22635i	−0.789946	−	0.613176i	$-0.789892\pi$
$491$		−	602.139i		−	1.22635i	0.789946	−	0.613176i	$-0.210108\pi$
$492$		0			0
$493$		−	424.143i		−	0.860330i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−213.617			−0.429813
$498$		0			0
$499$	384.486			0.770513			0.385257	−	0.922810i	$-0.374113\pi$
$499$	384.486			0.770513			0.385257	+	0.922810i	$0.374113\pi$
$500$		0			0
$501$	−300.664	−	42.6193i	−0.600128	−	0.0850684i
$502$		0			0
$503$	458.304			0.911141			0.455571	−	0.890200i	$-0.349435\pi$
$503$	458.304			0.911141			0.455571	+	0.890200i	$0.349435\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	402.820	+	57.0999i	0.794517	+	0.112623i
$508$		0			0
$509$			715.661i			1.40601i	0.711183	+	0.703007i	$0.248160\pi$
$509$			715.661i			1.40601i	−0.711183	+	0.703007i	$0.751840\pi$
$510$		0			0
$511$	318.995			0.624256
$512$		0			0
$513$	110.923	+	49.8596i	0.216223	+	0.0971922i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	35.5562i		−	0.0687741i
$518$		0			0
$519$	−987.995	−	140.049i	−1.90365	−	0.269843i
$520$		0			0
$521$		−	58.2889i		−	0.111879i	−0.998434	−	0.0559394i	$-0.982185\pi$
$521$		−	58.2889i		−	0.111879i	0.998434	−	0.0559394i	$-0.0178154\pi$
$522$		0			0
$523$		−	676.567i		−	1.29363i	−0.762648	−	0.646814i	$-0.776101\pi$
$523$		−	676.567i		−	1.29363i	0.762648	−	0.646814i	$-0.223899\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1610.42			3.05583
$528$		0			0
$529$	−342.929			−0.648260
$530$		0			0
$531$	90.9513	−	314.369i	0.171283	−	0.592031i
$532$		0			0
$533$	−133.368			−0.250221
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.37540	+	44.9762i	−0.0118722	+	0.0837546i
$538$		0			0
$539$		−	57.1527i		−	0.106035i
$540$		0			0
$541$	−430.418			−0.795596			−0.397798	−	0.917473i	$-0.630226\pi$
$541$	−430.418			−0.795596			−0.397798	+	0.917473i	$0.630226\pi$
$542$		0			0
$543$	725.284	+	102.809i	1.33570	+	0.189336i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	110.245i		−	0.201546i	−0.994909	−	0.100773i	$-0.967868\pi$
$547$		−	110.245i		−	0.201546i	0.994909	−	0.100773i	$-0.0321315\pi$
$548$		0			0
$549$	−197.180	−	57.0470i	−0.359162	−	0.103911i
$550$		0			0
$551$			68.0703i			0.123540i
$552$		0			0
$553$			144.186i			0.260734i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−463.317			−0.831809			−0.415904	−	0.909408i	$-0.636535\pi$
$557$	−463.317			−0.831809			−0.415904	+	0.909408i	$0.636535\pi$
$558$		0			0
$559$	−191.619			−0.342789
$560$		0			0
$561$	96.4799	−	680.632i	0.171978	−	1.21325i
$562$		0			0
$563$	878.072			1.55963			0.779815	−	0.626010i	$-0.215313\pi$
$563$	878.072			1.55963			0.779815	+	0.626010i	$0.215313\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−114.426	+	181.201i	−0.201809	+	0.319578i
$568$		0			0
$569$			579.224i			1.01797i	0.860776	+	0.508984i	$0.169979\pi$
$569$			579.224i			1.01797i	−0.860776	+	0.508984i	$0.830021\pi$
$570$		0			0
$571$	471.669			0.826040			0.413020	−	0.910722i	$-0.364474\pi$
$571$	471.669			0.826040			0.413020	+	0.910722i	$0.364474\pi$
$572$		0			0
$573$	−138.491	+	977.005i	−0.241694	+	1.70507i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			291.140i			0.504576i	0.967652	+	0.252288i	$0.0811831\pi$
$577$			291.140i			0.504576i	−0.967652	+	0.252288i	$0.918817\pi$
$578$		0			0
$579$	128.260	−	904.833i	0.221521	−	1.56275i
$580$		0			0
$581$		−	151.592i		−	0.260915i
$582$		0			0
$583$		−	187.217i		−	0.321127i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	731.515			1.24619			0.623096	−	0.782145i	$-0.285874\pi$
$587$	731.515			1.24619			0.623096	+	0.782145i	$0.285874\pi$
$588$		0			0
$589$	−258.455			−0.438804
$590$		0			0
$591$	−698.164	−	98.9651i	−1.18133	−	0.167454i
$592$		0			0
$593$	683.187			1.15209			0.576043	−	0.817419i	$-0.304596\pi$
$593$	683.187			1.15209			0.576043	+	0.817419i	$0.304596\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	885.373	+	125.502i	1.48304	+	0.210221i
$598$		0			0
$599$			464.649i			0.775708i	0.921721	+	0.387854i	$0.126783\pi$
$599$			464.649i			0.775708i	−0.921721	+	0.387854i	$0.873217\pi$
$600$		0			0
$601$	−242.259			−0.403093			−0.201547	−	0.979479i	$-0.564597\pi$
$601$	−242.259			−0.403093			−0.201547	+	0.979479i	$0.564597\pi$
$602$		0			0
$603$	16.9669	−	58.6452i	0.0281374	−	0.0972557i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	143.456i		−	0.236337i	−0.992994	−	0.118168i	$-0.962298\pi$
$607$		−	143.456i		−	0.236337i	0.992994	−	0.118168i	$-0.0377022\pi$
$608$		0			0
$609$	−118.766	−	16.8351i	−0.195017	−	0.0276438i
$610$		0			0
$611$		−	25.1622i		−	0.0411820i
$612$		0			0
$613$			274.405i			0.447643i	0.974630	+	0.223821i	$0.0718533\pi$
$613$			274.405i			0.447643i	−0.974630	+	0.223821i	$0.928147\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	232.732			0.377199			0.188600	−	0.982054i	$-0.439605\pi$
$617$	232.732			0.377199			0.188600	+	0.982054i	$0.439605\pi$
$618$		0			0
$619$	1209.75			1.95437			0.977185	−	0.212392i	$-0.0681253\pi$
$619$	1209.75			1.95437			0.977185	+	0.212392i	$0.0681253\pi$
$620$		0			0
$621$	−335.924	−	150.998i	−0.540941	−	0.243152i
$622$		0			0
$623$	398.910			0.640305
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−15.4840	+	109.234i	−0.0246953	+	0.174217i
$628$		0			0
$629$			259.141i			0.411988i
$630$		0			0
$631$	581.467			0.921500			0.460750	−	0.887530i	$-0.347580\pi$
$631$	581.467			0.921500			0.460750	+	0.887530i	$0.347580\pi$
$632$		0			0
$633$	−337.111	−	47.7856i	−0.532561	−	0.0754907i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	40.4454i		−	0.0634936i
$638$		0			0
$639$	−201.950	+	698.031i	−0.316041	+	1.09238i
$640$		0			0
$641$		−	1052.94i		−	1.64265i	−0.570461	−	0.821325i	$-0.693235\pi$
$641$		−	1052.94i		−	1.64265i	0.570461	−	0.821325i	$-0.306765\pi$
$642$		0			0
$643$		−	307.328i		−	0.477959i	−0.971025	−	0.238980i	$-0.923187\pi$
$643$		−	307.328i		−	0.477959i	0.971025	−	0.238980i	$-0.0768129\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−765.017			−1.18241			−0.591203	−	0.806523i	$-0.701347\pi$
$647$	−765.017			−1.18241			−0.591203	+	0.806523i	$0.701347\pi$
$648$		0			0
$649$	296.887			0.457453
$650$		0			0
$651$	63.9209	−	450.940i	0.0981888	−	0.692688i
$652$		0			0
$653$	−455.317			−0.697270			−0.348635	−	0.937259i	$-0.613355\pi$
$653$	−455.317			−0.697270			−0.348635	+	0.937259i	$0.613355\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	301.573	−	1042.37i	0.459015	−	1.58656i
$658$		0			0
$659$		−	345.201i		−	0.523825i	−0.965092	−	0.261913i	$-0.915647\pi$
$659$		−	345.201i		−	0.523825i	0.965092	−	0.261913i	$-0.0843532\pi$
$660$		0			0
$661$	−487.944			−0.738191			−0.369096	−	0.929391i	$-0.620333\pi$
$661$	−487.944			−0.738191			−0.369096	+	0.929391i	$0.620333\pi$
$662$		0			0
$663$	68.2762	−	481.665i	0.102981	−	0.726493i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	206.148i		−	0.309067i
$668$		0			0
$669$	142.254	−	1003.55i	0.212636	−	1.50008i
$670$		0			0
$671$		−	186.215i		−	0.277519i
$672$		0			0
$673$			562.562i			0.835902i	0.908470	+	0.417951i	$0.137252\pi$
$673$			562.562i			0.835902i	−0.908470	+	0.417951i	$0.862748\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−944.244			−1.39475			−0.697373	−	0.716708i	$-0.745648\pi$
$677$	−944.244			−1.39475			−0.697373	+	0.716708i	$0.745648\pi$
$678$		0			0
$679$	−312.045			−0.459566
$680$		0			0
$681$	123.263	+	17.4726i	0.181003	+	0.0256573i
$682$		0			0
$683$	801.351			1.17328			0.586641	−	0.809847i	$-0.300450\pi$
$683$	801.351			1.17328			0.586641	+	0.809847i	$0.300450\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1013.38	+	143.647i	1.47508	+	0.209093i
$688$		0			0
$689$		−	132.488i		−	0.192291i
$690$		0			0
$691$	360.516			0.521731			0.260866	−	0.965375i	$-0.415992\pi$
$691$	360.516			0.521731			0.260866	+	0.965375i	$0.415992\pi$
$692$		0			0
$693$	−186.756	−	54.0313i	−0.269490	−	0.0779672i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	647.816i		−	0.929434i
$698$		0			0
$699$	337.220	+	47.8011i	0.482432	+	0.0683850i
$700$		0			0
$701$			818.208i			1.16720i	0.812041	+	0.583601i	$0.198357\pi$
$701$			818.208i			1.16720i	−0.812041	+	0.583601i	$0.801643\pi$
$702$		0			0
$703$		−	41.5892i		−	0.0591597i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−371.551			−0.525531
$708$		0			0
$709$	437.371			0.616884			0.308442	−	0.951243i	$-0.400192\pi$
$709$	437.371			0.616884			0.308442	+	0.951243i	$0.400192\pi$
$710$		0			0
$711$	471.152	+	136.311i	0.662662	+	0.191717i
$712$		0			0
$713$	782.721			1.09778
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−66.4227	+	468.589i	−0.0926397	+	0.653541i
$718$		0			0
$719$		−	527.093i		−	0.733091i	−0.930400	−	0.366546i	$-0.880540\pi$
$719$		−	527.093i		−	0.733091i	0.930400	−	0.366546i	$-0.119460\pi$
$720$		0			0
$721$	26.7611			0.0371166
$722$		0			0
$723$	736.479	+	104.396i	1.01864	+	0.144393i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	809.719i		−	1.11378i	−0.830586	−	0.556891i	$-0.811994\pi$
$727$		−	809.719i		−	1.11378i	0.830586	−	0.556891i	$-0.188006\pi$
$728$		0			0
$729$	483.929	+	545.210i	0.663825	+	0.747888i
$730$		0			0
$731$		−	930.762i		−	1.27327i
$732$		0			0
$733$		−	215.991i		−	0.294667i	−0.989087	−	0.147333i	$-0.952931\pi$
$733$		−	215.991i		−	0.294667i	0.989087	−	0.147333i	$-0.0470690\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	55.3839			0.0751478
$738$		0			0
$739$	1025.08			1.38712			0.693560	−	0.720398i	$-0.256041\pi$
$739$	1025.08			1.38712			0.693560	+	0.720398i	$0.256041\pi$
$740$		0			0
$741$	−10.9576	+	77.3020i	−0.0147876	+	0.104321i
$742$		0			0
$743$	168.132			0.226287			0.113144	−	0.993579i	$-0.463908\pi$
$743$	168.132			0.226287			0.113144	+	0.993579i	$0.463908\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−495.352	−	143.312i	−0.663121	−	0.191850i
$748$		0			0
$749$			145.799i			0.194658i
$750$		0			0
$751$	998.130			1.32907			0.664534	−	0.747258i	$-0.268630\pi$
$751$	998.130			1.32907			0.664534	+	0.747258i	$0.268630\pi$
$752$		0			0
$753$	−116.518	+	821.993i	−0.154738	+	1.09162i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	310.157i		−	0.409719i	−0.978791	−	0.204860i	$-0.934326\pi$
$757$		−	310.157i		−	0.409719i	0.978791	−	0.204860i	$-0.0656738\pi$
$758$		0			0
$759$	46.8925	−	330.810i	0.0617820	−	0.435850i
$760$		0			0
$761$		−	1386.50i		−	1.82194i	−0.412469	−	0.910972i	$-0.635333\pi$
$761$		−	1386.50i		−	1.82194i	0.412469	−	0.910972i	$-0.364667\pi$
$762$		0			0
$763$		−	168.267i		−	0.220533i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	210.099			0.273923
$768$		0			0
$769$	950.327			1.23580			0.617898	−	0.786258i	$-0.287985\pi$
$769$	950.327			1.23580			0.617898	+	0.786258i	$0.287985\pi$
$770$		0			0
$771$	−467.015	−	66.1996i	−0.605727	−	0.0858621i
$772$		0			0
$773$	898.658			1.16256			0.581279	−	0.813704i	$-0.302552\pi$
$773$	898.658			1.16256			0.581279	+	0.813704i	$0.302552\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	72.5628	+	10.2858i	0.0933884	+	0.0132378i
$778$		0			0
$779$			103.967i			0.133463i
$780$		0			0
$781$	−659.214			−0.844063
$782$		0			0
$783$	−167.291	+	372.171i	−0.213653	+	0.475314i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	409.324i		−	0.520107i	−0.965594	−	0.260054i	$-0.916260\pi$
$787$		−	409.324i		−	0.520107i	0.965594	−	0.260054i	$-0.0837402\pi$
$788$		0			0
$789$	−909.190	−	128.878i	−1.15233	−	0.163344i
$790$		0			0
$791$			67.7993i			0.0857134i
$792$		0			0
$793$		−	131.779i		−	0.166178i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−163.126			−0.204675			−0.102337	−	0.994750i	$-0.532632\pi$
$797$	−163.126			−0.204675			−0.102337	+	0.994750i	$0.532632\pi$
$798$		0			0
$799$	122.222			0.152968
$800$		0			0
$801$	377.123	−	1303.51i	0.470815	−	1.62735i
$802$		0			0
$803$	984.405			1.22591
$804$		0			0
$805$		0			0
$806$		0			0
$807$	101.434	−	715.583i	0.125693	−	0.886719i
$808$		0			0
$809$			226.377i			0.279823i	0.990164	+	0.139912i	$0.0446818\pi$
$809$			226.377i			0.279823i	−0.990164	+	0.139912i	$0.955318\pi$
$810$		0			0
$811$	−1232.29			−1.51948			−0.759738	−	0.650230i	$-0.774673\pi$
$811$	−1232.29			−1.51948			−0.759738	+	0.650230i	$0.774673\pi$
$812$		0			0
$813$	2.05912	+	0.291881i	0.00253274	+	0.000359017i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			149.377i			0.182836i
$818$		0			0
$819$	−132.162	−	38.2365i	−0.161370	−	0.0466868i
$820$		0			0
$821$			499.367i			0.608242i	0.952633	+	0.304121i	$0.0983627\pi$
$821$			499.367i			0.608242i	−0.952633	+	0.304121i	$0.901637\pi$
$822$		0			0
$823$		−	468.938i		−	0.569792i	−0.958558	−	0.284896i	$-0.908041\pi$
$823$		−	468.938i		−	0.569792i	0.958558	−	0.284896i	$-0.0919590\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.39906			0.00290092			0.00145046	−	0.999999i	$-0.499538\pi$
$827$	2.39906			0.00290092			0.00145046	+	0.999999i	$0.499538\pi$
$828$		0			0
$829$	−487.181			−0.587674			−0.293837	−	0.955856i	$-0.594932\pi$
$829$	−487.181			−0.587674			−0.293837	+	0.955856i	$0.594932\pi$
$830$		0			0
$831$	82.2484	−	580.234i	0.0989752	−	0.698235i
$832$		0			0
$833$	196.458			0.235844
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1413.09	−	635.184i	−1.68828	−	0.758881i
$838$		0			0
$839$		−	883.573i		−	1.05313i	−0.850136	−	0.526563i	$-0.823480\pi$
$839$		−	883.573i		−	1.05313i	0.850136	−	0.526563i	$-0.176520\pi$
$840$		0			0
$841$	612.608			0.728429
$842$		0			0
$843$	−171.779	+	1211.84i	−0.203771	+	1.43753i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			143.765i			0.169734i
$848$		0			0
$849$	−163.288	+	1151.94i	−0.192330	+	1.35682i
$850$		0			0
$851$			125.951i			0.148004i
$852$		0			0
$853$			835.699i			0.979717i	0.871802	+	0.489859i	$0.162952\pi$
$853$			835.699i			0.979717i	−0.871802	+	0.489859i	$0.837048\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1339.06			−1.56249			−0.781247	−	0.624222i	$-0.785416\pi$
$857$	−1339.06			−1.56249			−0.781247	+	0.624222i	$0.785416\pi$
$858$		0			0
$859$	334.667			0.389600			0.194800	−	0.980843i	$-0.437594\pi$
$859$	334.667			0.389600			0.194800	+	0.980843i	$0.437594\pi$
$860$		0			0
$861$	−181.397	−	25.7131i	−0.210682	−	0.0298642i
$862$		0			0
$863$	−995.095			−1.15306			−0.576532	−	0.817074i	$-0.695594\pi$
$863$	−995.095			−1.15306			−0.576532	+	0.817074i	$0.695594\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1481.20	+	209.961i	1.70842	+	0.242170i
$868$		0			0
$869$			444.952i			0.512027i
$870$		0			0
$871$	39.1937			0.0449985
$872$		0			0
$873$	−295.003	+	1019.66i	−0.337918	+	1.16800i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1095.18i			1.24878i	0.781112	+	0.624391i	$0.214653\pi$
$877$			1095.18i			1.24878i	−0.781112	+	0.624391i	$0.785347\pi$
$878$		0			0
$879$	−292.282	−	41.4311i	−0.332516	−	0.0471344i
$880$		0			0
$881$			1219.05i			1.38371i	0.722036	+	0.691855i	$0.243206\pi$
$881$			1219.05i			1.38371i	−0.722036	+	0.691855i	$0.756794\pi$
$882$		0			0
$883$		−	1339.04i		−	1.51646i	−0.651985	−	0.758232i	$-0.726064\pi$
$883$		−	1339.04i		−	1.51646i	0.651985	−	0.758232i	$-0.273936\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	938.587			1.05816			0.529079	−	0.848572i	$-0.322537\pi$
$887$	938.587			1.05816			0.529079	+	0.848572i	$0.322537\pi$
$888$		0			0
$889$	183.954			0.206922
$890$		0			0
$891$	−353.113	+	559.178i	−0.396311	+	0.627585i
$892$		0			0
$893$	−19.6153			−0.0219656
$894$		0			0
$895$		0			0
$896$		0			0
$897$	33.1846	−	234.106i	0.0369951	−	0.260987i
$898$		0			0
$899$		−	867.178i		−	0.964602i
$900$		0			0
$901$	643.543			0.714255
$902$		0			0
$903$	−260.626	−	36.9438i	−0.288622	−	0.0409123i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	1575.14i		−	1.73665i	−0.495994	−	0.868326i	$-0.665196\pi$
$907$		−	1575.14i		−	1.73665i	0.495994	−	0.868326i	$-0.334804\pi$
$908$		0			0
$909$	−351.258	+	1214.11i	−0.386423	+	1.33565i
$910$		0			0
$911$			566.329i			0.621656i	0.950466	+	0.310828i	$0.100606\pi$
$911$			566.329i			0.621656i	−0.950466	+	0.310828i	$0.899394\pi$
$912$		0			0
$913$		−	467.805i		−	0.512383i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−302.551			−0.329935
$918$		0			0
$919$	−276.281			−0.300632			−0.150316	−	0.988638i	$-0.548029\pi$
$919$	−276.281			−0.300632			−0.150316	+	0.988638i	$0.548029\pi$
$920$		0			0
$921$	72.4850	−	511.356i	0.0787025	−	0.555219i
$922$		0			0
$923$	−466.507			−0.505425
$924$		0			0
$925$		0			0
$926$		0			0
$927$	25.2995	−	87.4464i	0.0272918	−	0.0943327i
$928$		0			0
$929$		−	160.080i		−	0.172314i	−0.996282	−	0.0861571i	$-0.972541\pi$
$929$		−	160.080i		−	0.172314i	0.996282	−	0.0861571i	$-0.0274587\pi$
$930$		0			0
$931$	−31.5294			−0.0338662
$932$		0			0
$933$	195.306	−	1377.82i	0.209332	−	1.47676i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1084.01i		−	1.15689i	−0.815720	−	0.578446i	$-0.803659\pi$
$937$		−	1084.01i		−	1.15689i	0.815720	−	0.578446i	$-0.196341\pi$
$938$		0			0
$939$	−149.344	+	1053.57i	−0.159046	+	1.12201i
$940$		0			0
$941$		−	692.493i		−	0.735912i	−0.929843	−	0.367956i	$-0.880058\pi$
$941$		−	692.493i		−	0.735912i	0.929843	−	0.367956i	$-0.119942\pi$
$942$		0			0
$943$		−	314.860i		−	0.333892i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1795.10			1.89556			0.947781	−	0.318923i	$-0.103321\pi$
$947$	1795.10			1.89556			0.947781	+	0.318923i	$0.103321\pi$
$948$		0			0
$949$	696.636			0.734074
$950$		0			0
$951$	133.463	+	18.9184i	0.140339	+	0.0198932i
$952$		0			0
$953$	789.199			0.828121			0.414060	−	0.910249i	$-0.364110\pi$
$953$	789.199			0.828121			0.414060	+	0.910249i	$0.364110\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−366.505	−	51.9523i	−0.382973	−	0.0542867i
$958$		0			0
$959$		−	437.361i		−	0.456059i
$960$		0			0
$961$	2331.58			2.42620
$962$		0			0
$963$	476.423	+	137.836i	0.494728	+	0.143132i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1464.86i			1.51485i	0.652921	+	0.757426i	$0.273543\pi$
$967$			1464.86i			1.51485i	−0.652921	+	0.757426i	$0.726457\pi$
$968$		0			0
$969$	−375.484	−	53.2250i	−0.387496	−	0.0549277i
$970$		0			0
$971$		−	1278.47i		−	1.31665i	−0.752733	−	0.658326i	$-0.771265\pi$
$971$		−	1278.47i		−	1.31665i	0.752733	−	0.658326i	$-0.228735\pi$
$972$		0			0
$973$		−	550.160i		−	0.565426i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	403.422			0.412919			0.206460	−	0.978455i	$-0.433806\pi$
$977$	403.422			0.412919			0.206460	+	0.978455i	$0.433806\pi$
$978$		0			0
$979$	1231.02			1.25743
$980$		0			0
$981$	−549.841	−	159.077i	−0.560491	−	0.162158i
$982$		0			0
$983$	1086.01			1.10479			0.552394	−	0.833583i	$-0.313714\pi$
$983$	1086.01			1.10479			0.552394	+	0.833583i	$0.313714\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	4.85123	−	34.2237i	0.00491512	−	0.0346745i
$988$		0			0
$989$		−	452.382i		−	0.457414i
$990$		0			0
$991$	70.0427			0.0706788			0.0353394	−	0.999375i	$-0.488749\pi$
$991$	70.0427			0.0706788			0.0353394	+	0.999375i	$0.488749\pi$
$992$		0			0
$993$	−93.4457	−	13.2460i	−0.0941044	−	0.0133393i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			719.365i			0.721529i	0.932657	+	0.360765i	$0.117484\pi$
$997$			719.365i			0.721529i	−0.932657	+	0.360765i	$0.882516\pi$
$998$		0			0
$999$	102.210	−	227.387i	0.102313	−	0.227615i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.e.c.449.32		32
3.2	odd	2	inner	2100.3.e.c.449.2		32
5.2	odd	4		2100.3.g.d.701.7		16
5.3	odd	4		420.3.g.a.281.10	yes	16
5.4	even	2	inner	2100.3.e.c.449.1		32
15.2	even	4		2100.3.g.d.701.8		16
15.8	even	4		420.3.g.a.281.9	✓	16
15.14	odd	2	inner	2100.3.e.c.449.31		32
20.3	even	4		1680.3.l.b.1121.7		16
60.23	odd	4		1680.3.l.b.1121.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.g.a.281.9	✓	16	15.8	even	4
420.3.g.a.281.10	yes	16	5.3	odd	4
1680.3.l.b.1121.7		16	20.3	even	4
1680.3.l.b.1121.8		16	60.23	odd	4
2100.3.e.c.449.1		32	5.4	even	2	inner
2100.3.e.c.449.2		32	3.2	odd	2	inner
2100.3.e.c.449.31		32	15.14	odd	2	inner
2100.3.e.c.449.32		32	1.1	even	1	trivial
2100.3.g.d.701.7		16	5.2	odd	4
2100.3.g.d.701.8		16	15.2	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.97031 + 0.421042i) q^{3} +2.64575i q^{7} +(8.64545 + 2.50125i) q^{9} +8.16468i q^{11} +5.77792i q^{13} -28.0655 q^{17} +4.50420 q^{19} +(-1.11397 + 7.85869i) q^{21} -13.6408 q^{23} +(24.6265 + 11.0696i) q^{27} +15.1126i q^{29} -57.3810 q^{31} +(-3.43768 + 24.2516i) q^{33} -9.23344i q^{37} +(-2.43275 + 17.1622i) q^{39} +23.0823i q^{41} +33.1640i q^{43} -4.35488 q^{47} -7.00000 q^{49} +(-83.3630 - 11.8167i) q^{51} -22.9301 q^{53} +(13.3789 + 1.89646i) q^{57} -36.3623i q^{59} -22.8074 q^{61} +(-6.61769 + 22.8737i) q^{63} -6.78336i q^{67} +(-40.5173 - 5.74334i) q^{69} +80.7397i q^{71} -120.569i q^{73} -21.6017 q^{77} +54.4972 q^{79} +(68.4875 + 43.2489i) q^{81} -57.2963 q^{83} +(-6.36306 + 44.8892i) q^{87} -150.774i q^{89} -15.2869 q^{91} +(-170.439 - 24.1598i) q^{93} +117.942i q^{97} +(-20.4219 + 70.5873i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 20 q^{9} - 96 q^{19} - 28 q^{21} + 48 q^{31} + 108 q^{39} - 224 q^{49} - 372 q^{51} - 304 q^{61} + 392 q^{69} - 424 q^{79} - 236 q^{81} + 392 q^{91} - 52 q^{99}+O(q^{100}) \) 32 * q + 20 * q^9 - 96 * q^19 - 28 * q^21 + 48 * q^31 + 108 * q^39 - 224 * q^49 - 372 * q^51 - 304 * q^61 + 392 * q^69 - 424 * q^79 - 236 * q^81 + 392 * q^91 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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