LMFDB - Embedded newform 2100.3.e.c.449.4

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			449.4
Character	$\chi$	$=$	2100.449
Dual form			2100.3.e.c.449.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.86332 + 0.895194i) q^{3} +2.64575i q^{7} +(7.39726 - 5.12646i) q^{9} -3.00338i q^{11} -24.3241i q^{13} +3.60390 q^{17} +16.8805 q^{19} +(-2.36846 - 7.57565i) q^{21} -12.4015 q^{23} +(-16.5916 + 21.3007i) q^{27} -13.5807i q^{29} +5.19503 q^{31} +(2.68860 + 8.59964i) q^{33} +17.7081i q^{37} +(21.7747 + 69.6477i) q^{39} +19.6909i q^{41} -39.7227i q^{43} -72.1833 q^{47} -7.00000 q^{49} +(-10.3191 + 3.22619i) q^{51} -14.4407 q^{53} +(-48.3343 + 15.1113i) q^{57} +74.4914i q^{59} +66.1551 q^{61} +(13.5633 + 19.5713i) q^{63} +47.8138i q^{67} +(35.5094 - 11.1017i) q^{69} +46.8183i q^{71} -42.2852i q^{73} +7.94619 q^{77} -3.12717 q^{79} +(28.4388 - 75.8435i) q^{81} -98.3606 q^{83} +(12.1574 + 38.8860i) q^{87} -154.840i q^{89} +64.3554 q^{91} +(-14.8751 + 4.65056i) q^{93} +92.3775i q^{97} +(-15.3967 - 22.2168i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.86332	+	0.895194i	−0.954442	+	0.298398i
$3$	−2.86332	+	0.895194i	−0.954442	+	0.298398i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			2.64575i			0.377964i
$7$			2.64575i			0.377964i
$8$		0			0
$9$	7.39726	−	5.12646i	0.821917	−	0.569607i
$10$		0			0
$11$		−	3.00338i		−	0.273034i	−0.990638	−	0.136517i	$-0.956409\pi$
$11$		−	3.00338i		−	0.273034i	0.990638	−	0.136517i	$-0.0435909\pi$
$12$		0			0
$13$		−	24.3241i		−	1.87108i	−0.353220	−	0.935540i	$-0.614913\pi$
$13$		−	24.3241i		−	1.87108i	0.353220	−	0.935540i	$-0.385087\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.60390			0.211994			0.105997	−	0.994366i	$-0.466197\pi$
$17$	3.60390			0.211994			0.105997	+	0.994366i	$0.466197\pi$
$18$		0			0
$19$	16.8805			0.888447			0.444223	−	0.895916i	$-0.353480\pi$
$19$	16.8805			0.888447			0.444223	+	0.895916i	$0.353480\pi$
$20$		0			0
$21$	−2.36846	−	7.57565i	−0.112784	−	0.360745i
$22$		0			0
$23$	−12.4015			−0.539194			−0.269597	−	0.962973i	$-0.586890\pi$
$23$	−12.4015			−0.539194			−0.269597	+	0.962973i	$0.586890\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−16.5916	+	21.3007i	−0.614503	+	0.788915i
$28$		0			0
$29$		−	13.5807i		−	0.468301i	−0.972200	−	0.234150i	$-0.924769\pi$
$29$		−	13.5807i		−	0.468301i	0.972200	−	0.234150i	$-0.0752308\pi$
$30$		0			0
$31$	5.19503			0.167582			0.0837908	−	0.996483i	$-0.473297\pi$
$31$	5.19503			0.167582			0.0837908	+	0.996483i	$0.473297\pi$
$32$		0			0
$33$	2.68860	+	8.59964i	0.0814729	+	0.260595i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			17.7081i			0.478598i	0.970946	+	0.239299i	$0.0769177\pi$
$37$			17.7081i			0.478598i	−0.970946	+	0.239299i	$0.923082\pi$
$38$		0			0
$39$	21.7747	+	69.6477i	0.558327	+	1.78584i
$40$		0			0
$41$			19.6909i			0.480267i	0.970740	+	0.240133i	$0.0771912\pi$
$41$			19.6909i			0.480267i	−0.970740	+	0.240133i	$0.922809\pi$
$42$		0			0
$43$		−	39.7227i		−	0.923783i	−0.886937	−	0.461891i	$-0.847171\pi$
$43$		−	39.7227i		−	0.923783i	0.886937	−	0.461891i	$-0.152829\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−72.1833			−1.53582			−0.767908	−	0.640561i	$-0.778702\pi$
$47$	−72.1833			−1.53582			−0.767908	+	0.640561i	$0.778702\pi$
$48$		0			0
$49$	−7.00000			−0.142857
$50$		0			0
$51$	−10.3191	+	3.22619i	−0.202336	+	0.0632587i
$52$		0			0
$53$	−14.4407			−0.272467			−0.136233	−	0.990677i	$-0.543500\pi$
$53$	−14.4407			−0.272467			−0.136233	+	0.990677i	$0.543500\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−48.3343	+	15.1113i	−0.847971	+	0.265111i
$58$		0			0
$59$			74.4914i			1.26257i	0.775552	+	0.631283i	$0.217471\pi$
$59$			74.4914i			1.26257i	−0.775552	+	0.631283i	$0.782529\pi$
$60$		0			0
$61$	66.1551			1.08451			0.542255	−	0.840214i	$-0.317571\pi$
$61$	66.1551			1.08451			0.542255	+	0.840214i	$0.317571\pi$
$62$		0			0
$63$	13.5633	+	19.5713i	0.215291	+	0.310656i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			47.8138i			0.713639i	0.934173	+	0.356820i	$0.116139\pi$
$67$			47.8138i			0.713639i	−0.934173	+	0.356820i	$0.883861\pi$
$68$		0			0
$69$	35.5094	−	11.1017i	0.514629	−	0.160894i
$70$		0			0
$71$			46.8183i			0.659413i	0.944083	+	0.329707i	$0.106950\pi$
$71$			46.8183i			0.659413i	−0.944083	+	0.329707i	$0.893050\pi$
$72$		0			0
$73$		−	42.2852i		−	0.579249i	−0.957140	−	0.289624i	$-0.906470\pi$
$73$		−	42.2852i		−	0.579249i	0.957140	−	0.289624i	$-0.0935304\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.94619			0.103197
$78$		0			0
$79$	−3.12717			−0.0395844			−0.0197922	−	0.999804i	$-0.506300\pi$
$79$	−3.12717			−0.0395844			−0.0197922	+	0.999804i	$0.506300\pi$
$80$		0			0
$81$	28.4388	−	75.8435i	0.351097	−	0.936339i
$82$		0			0
$83$	−98.3606			−1.18507			−0.592534	−	0.805545i	$-0.701872\pi$
$83$	−98.3606			−1.18507			−0.592534	+	0.805545i	$0.701872\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	12.1574	+	38.8860i	0.139740	+	0.446966i
$88$		0			0
$89$		−	154.840i		−	1.73977i	−0.493255	−	0.869885i	$-0.664193\pi$
$89$		−	154.840i		−	1.73977i	0.493255	−	0.869885i	$-0.335807\pi$
$90$		0			0
$91$	64.3554			0.707202
$92$		0			0
$93$	−14.8751	+	4.65056i	−0.159947	+	0.0500060i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			92.3775i			0.952346i	0.879352	+	0.476173i	$0.157976\pi$
$97$			92.3775i			0.952346i	−0.879352	+	0.476173i	$0.842024\pi$
$98$		0			0
$99$	−15.3967	−	22.2168i	−0.155522	−	0.224412i
$100$		0			0
$101$		−	94.4792i		−	0.935438i	−0.883877	−	0.467719i	$-0.845076\pi$
$101$		−	94.4792i		−	0.935438i	0.883877	−	0.467719i	$-0.154924\pi$
$102$		0			0
$103$			23.8293i			0.231352i	0.993287	+	0.115676i	$0.0369034\pi$
$103$			23.8293i			0.231352i	−0.993287	+	0.115676i	$0.963097\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	150.037			1.40221			0.701107	−	0.713056i	$-0.252690\pi$
$107$	150.037			1.40221			0.701107	+	0.713056i	$0.252690\pi$
$108$		0			0
$109$	−161.310			−1.47991			−0.739953	−	0.672658i	$-0.765152\pi$
$109$	−161.310			−1.47991			−0.739953	+	0.672658i	$0.765152\pi$
$110$		0			0
$111$	−15.8522	−	50.7042i	−0.142813	−	0.456794i
$112$		0			0
$113$	−200.957			−1.77838			−0.889190	−	0.457538i	$-0.848731\pi$
$113$	−200.957			−1.77838			−0.889190	+	0.457538i	$0.848731\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−124.696	−	179.931i	−1.06578	−	1.53787i
$118$		0			0
$119$			9.53503i			0.0801263i
$120$		0			0
$121$	111.980			0.925452
$122$		0			0
$123$	−17.6272	−	56.3815i	−0.143311	−	0.458387i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	89.2829i		−	0.703015i	−0.936185	−	0.351507i	$-0.885669\pi$
$127$		−	89.2829i		−	0.703015i	0.936185	−	0.351507i	$-0.114331\pi$
$128$		0			0
$129$	35.5595	+	113.739i	0.275655	+	0.881697i
$130$		0			0
$131$		−	133.670i		−	1.02038i	−0.860061	−	0.510191i	$-0.829575\pi$
$131$		−	133.670i		−	1.02038i	0.860061	−	0.510191i	$-0.170425\pi$
$132$		0			0
$133$			44.6616i			0.335801i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	51.8272			0.378301			0.189150	−	0.981948i	$-0.439427\pi$
$137$	51.8272			0.378301			0.189150	+	0.981948i	$0.439427\pi$
$138$		0			0
$139$	190.393			1.36973			0.684866	−	0.728669i	$-0.259861\pi$
$139$	190.393			1.36973			0.684866	+	0.728669i	$0.259861\pi$
$140$		0			0
$141$	206.684	−	64.6180i	1.46585	−	0.458284i
$142$		0			0
$143$	−73.0543			−0.510869
$144$		0			0
$145$		0			0
$146$		0			0
$147$	20.0433	−	6.26636i	0.136349	−	0.0426283i
$148$		0			0
$149$			7.21233i			0.0484049i	0.999707	+	0.0242024i	$0.00770463\pi$
$149$			7.21233i			0.0484049i	−0.999707	+	0.0242024i	$0.992295\pi$
$150$		0			0
$151$	−113.626			−0.752491			−0.376245	−	0.926520i	$-0.622785\pi$
$151$	−113.626			−0.752491			−0.376245	+	0.926520i	$0.622785\pi$
$152$		0			0
$153$	26.6590	−	18.4753i	0.174242	−	0.120753i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	204.211i		−	1.30071i	−0.759630	−	0.650355i	$-0.774620\pi$
$157$		−	204.211i		−	1.30071i	0.759630	−	0.650355i	$-0.225380\pi$
$158$		0			0
$159$	41.3485	−	12.9272i	0.260053	−	0.0813034i
$160$		0			0
$161$		−	32.8112i		−	0.203796i
$162$		0			0
$163$			77.2206i			0.473746i	0.971541	+	0.236873i	$0.0761225\pi$
$163$			77.2206i			0.473746i	−0.971541	+	0.236873i	$0.923877\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−284.381			−1.70288			−0.851439	−	0.524454i	$-0.824269\pi$
$167$	−284.381			−1.70288			−0.851439	+	0.524454i	$0.824269\pi$
$168$		0			0
$169$	−422.659			−2.50094
$170$		0			0
$171$	124.869	−	86.5372i	0.730230	−	0.506065i
$172$		0			0
$173$	−267.316			−1.54518			−0.772590	−	0.634905i	$-0.781039\pi$
$173$	−267.316			−1.54518			−0.772590	+	0.634905i	$0.781039\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−66.6842	−	213.293i	−0.376747	−	1.20505i
$178$		0			0
$179$		−	23.3604i		−	0.130505i	−0.997869	−	0.0652525i	$-0.979215\pi$
$179$		−	23.3604i		−	0.130505i	0.997869	−	0.0652525i	$-0.0207853\pi$
$180$		0			0
$181$	−119.639			−0.660987			−0.330494	−	0.943808i	$-0.607215\pi$
$181$	−119.639			−0.660987			−0.330494	+	0.943808i	$0.607215\pi$
$182$		0			0
$183$	−189.424	+	59.2217i	−1.03510	+	0.323616i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	10.8239i		−	0.0578817i
$188$		0			0
$189$	−56.3563	−	43.8972i	−0.298182	−	0.232260i
$190$		0			0
$191$			76.1540i			0.398712i	0.979927	+	0.199356i	$0.0638850\pi$
$191$			76.1540i			0.398712i	−0.979927	+	0.199356i	$0.936115\pi$
$192$		0			0
$193$		−	116.189i		−	0.602015i	−0.953622	−	0.301007i	$-0.902677\pi$
$193$		−	116.189i		−	0.602015i	0.953622	−	0.301007i	$-0.0973229\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−247.485			−1.25627			−0.628134	−	0.778105i	$-0.716181\pi$
$197$	−247.485			−1.25627			−0.628134	+	0.778105i	$0.716181\pi$
$198$		0			0
$199$	43.5013			0.218599			0.109300	−	0.994009i	$-0.465139\pi$
$199$	43.5013			0.218599			0.109300	+	0.994009i	$0.465139\pi$
$200$		0			0
$201$	−42.8026	−	136.906i	−0.212948	−	0.681127i
$202$		0			0
$203$	35.9312			0.177001
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−91.7368	+	63.5756i	−0.443173	+	0.307128i
$208$		0			0
$209$		−	50.6985i		−	0.242576i
$210$		0			0
$211$	−283.206			−1.34221			−0.671104	−	0.741363i	$-0.734180\pi$
$211$	−283.206			−1.34221			−0.671104	+	0.741363i	$0.734180\pi$
$212$		0			0
$213$	−41.9115	−	134.056i	−0.196767	−	0.629371i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			13.7448i			0.0633399i
$218$		0			0
$219$	37.8534	+	121.076i	0.172847	+	0.552859i
$220$		0			0
$221$		−	87.6615i		−	0.396659i
$222$		0			0
$223$		−	288.852i		−	1.29530i	−0.761937	−	0.647651i	$-0.775751\pi$
$223$		−	288.852i		−	1.29530i	0.761937	−	0.647651i	$-0.224249\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	333.944			1.47112			0.735559	−	0.677460i	$-0.236920\pi$
$227$	333.944			1.47112			0.735559	+	0.677460i	$0.236920\pi$
$228$		0			0
$229$	11.1825			0.0488318			0.0244159	−	0.999702i	$-0.492227\pi$
$229$	11.1825			0.0488318			0.0244159	+	0.999702i	$0.492227\pi$
$230$		0			0
$231$	−22.7525	+	7.11338i	−0.0984958	+	0.0307938i
$232$		0			0
$233$	−249.339			−1.07012			−0.535062	−	0.844813i	$-0.679712\pi$
$233$	−249.339			−1.07012			−0.535062	+	0.844813i	$0.679712\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.95410	−	2.79942i	0.0377810	−	0.0118119i
$238$		0			0
$239$			144.066i			0.602786i	0.953500	+	0.301393i	$0.0974516\pi$
$239$			144.066i			0.602786i	−0.953500	+	0.301393i	$0.902548\pi$
$240$		0			0
$241$	−347.745			−1.44292			−0.721462	−	0.692454i	$-0.756530\pi$
$241$	−347.745			−1.44292			−0.721462	+	0.692454i	$0.756530\pi$
$242$		0			0
$243$	−13.5350	+	242.623i	−0.0556995	+	0.998448i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	410.602i		−	1.66236i
$248$		0			0
$249$	281.638	−	88.0518i	1.13108	−	0.353622i
$250$		0			0
$251$		−	453.378i		−	1.80629i	−0.429339	−	0.903143i	$-0.641253\pi$
$251$		−	453.378i		−	1.80629i	0.429339	−	0.903143i	$-0.358747\pi$
$252$		0			0
$253$			37.2463i			0.147218i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	422.789			1.64509			0.822547	−	0.568697i	$-0.192552\pi$
$257$	422.789			1.64509			0.822547	+	0.568697i	$0.192552\pi$
$258$		0			0
$259$	−46.8513			−0.180893
$260$		0			0
$261$	−69.6211	−	100.460i	−0.266747	−	0.384905i
$262$		0			0
$263$	−375.524			−1.42785			−0.713924	−	0.700223i	$-0.753084\pi$
$263$	−375.524			−1.42785			−0.713924	+	0.700223i	$0.753084\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	138.611	+	443.356i	0.519144	+	1.66051i
$268$		0			0
$269$			1.41316i			0.00525338i	0.999997	+	0.00262669i	$0.000836103\pi$
$269$			1.41316i			0.00525338i	−0.999997	+	0.00262669i	$0.999164\pi$
$270$		0			0
$271$	226.054			0.834146			0.417073	−	0.908873i	$-0.363056\pi$
$271$	226.054			0.834146			0.417073	+	0.908873i	$0.363056\pi$
$272$		0			0
$273$	−184.270	+	57.6105i	−0.674983	+	0.211028i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			355.341i			1.28282i	0.767198	+	0.641410i	$0.221650\pi$
$277$			355.341i			1.28282i	−0.767198	+	0.641410i	$0.778350\pi$
$278$		0			0
$279$	38.4290	−	26.6321i	0.137738	−	0.0954556i
$280$		0			0
$281$		−	195.986i		−	0.697458i	−0.937224	−	0.348729i	$-0.886613\pi$
$281$		−	195.986i		−	0.697458i	0.937224	−	0.348729i	$-0.113387\pi$
$282$		0			0
$283$		−	485.087i		−	1.71409i	−0.515244	−	0.857044i	$-0.672299\pi$
$283$		−	485.087i		−	1.71409i	0.515244	−	0.857044i	$-0.327701\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−52.0973			−0.181524
$288$		0			0
$289$	−276.012			−0.955058
$290$		0			0
$291$	−82.6958	−	264.507i	−0.284178	−	0.908958i
$292$		0			0
$293$	378.468			1.29170			0.645849	−	0.763465i	$-0.276503\pi$
$293$	378.468			1.29170			0.645849	+	0.763465i	$0.276503\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	63.9740	+	49.8308i	0.215401	+	0.167780i
$298$		0			0
$299$			301.654i			1.00888i
$300$		0			0
$301$	105.096			0.349157
$302$		0			0
$303$	84.5772	+	270.525i	0.279133	+	0.892821i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			573.684i			1.86868i	0.356385	+	0.934339i	$0.384009\pi$
$307$			573.684i			1.86868i	−0.356385	+	0.934339i	$0.615991\pi$
$308$		0			0
$309$	−21.3318	−	68.2309i	−0.0690349	−	0.220812i
$310$		0			0
$311$		−	441.918i		−	1.42096i	−0.703718	−	0.710479i	$-0.748478\pi$
$311$		−	441.918i		−	1.42096i	0.703718	−	0.710479i	$-0.251522\pi$
$312$		0			0
$313$		−	378.973i		−	1.21078i	−0.795930	−	0.605388i	$-0.793018\pi$
$313$		−	378.973i		−	1.21078i	0.795930	−	0.605388i	$-0.206982\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−588.177			−1.85545			−0.927724	−	0.373266i	$-0.878238\pi$
$317$	−588.177			−1.85545			−0.927724	+	0.373266i	$0.878238\pi$
$318$		0			0
$319$	−40.7881			−0.127862
$320$		0			0
$321$	−429.604	+	134.312i	−1.33833	+	0.418418i
$322$		0			0
$323$	60.8357			0.188346
$324$		0			0
$325$		0			0
$326$		0			0
$327$	461.882	−	144.404i	1.41248	−	0.441601i
$328$		0			0
$329$		−	190.979i		−	0.580484i
$330$		0			0
$331$	205.546			0.620984			0.310492	−	0.950576i	$-0.399506\pi$
$331$	205.546			0.620984			0.310492	+	0.950576i	$0.399506\pi$
$332$		0			0
$333$	90.7801	+	130.992i	0.272613	+	0.393368i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	456.465i		−	1.35450i	−0.735755	−	0.677248i	$-0.763172\pi$
$337$		−	456.465i		−	1.35450i	0.735755	−	0.677248i	$-0.236828\pi$
$338$		0			0
$339$	575.405	−	179.895i	1.69736	−	0.530665i
$340$		0			0
$341$		−	15.6026i		−	0.0457555i
$342$		0			0
$343$		−	18.5203i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−302.625			−0.872118			−0.436059	−	0.899918i	$-0.643626\pi$
$347$	−302.625			−0.872118			−0.436059	+	0.899918i	$0.643626\pi$
$348$		0			0
$349$	−19.9121			−0.0570549			−0.0285274	−	0.999593i	$-0.509082\pi$
$349$	−19.9121			−0.0570549			−0.0285274	+	0.999593i	$0.509082\pi$
$350$		0			0
$351$	518.119	+	403.574i	1.47612	+	1.14978i
$352$		0			0
$353$	219.478			0.621750			0.310875	−	0.950451i	$-0.399378\pi$
$353$	219.478			0.621750			0.310875	+	0.950451i	$0.399378\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−8.53570	−	27.3019i	−0.0239095	−	0.0764759i
$358$		0			0
$359$			309.375i			0.861770i	0.902407	+	0.430885i	$0.141799\pi$
$359$			309.375i			0.861770i	−0.902407	+	0.430885i	$0.858201\pi$
$360$		0			0
$361$	−76.0491			−0.210662
$362$		0			0
$363$	−320.634	+	100.244i	−0.883290	+	0.276153i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	516.947i		−	1.40857i	−0.709915	−	0.704287i	$-0.751267\pi$
$367$		−	516.947i		−	1.40857i	0.709915	−	0.704287i	$-0.248733\pi$
$368$		0			0
$369$	100.945	+	145.659i	0.273563	+	0.394740i
$370$		0			0
$371$		−	38.2066i		−	0.102983i
$372$		0			0
$373$		−	417.787i		−	1.12007i	−0.828468	−	0.560037i	$-0.810787\pi$
$373$		−	417.787i		−	1.12007i	0.828468	−	0.560037i	$-0.189213\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−330.338			−0.876229
$378$		0			0
$379$	−468.822			−1.23700			−0.618499	−	0.785786i	$-0.712259\pi$
$379$	−468.822			−1.23700			−0.618499	+	0.785786i	$0.712259\pi$
$380$		0			0
$381$	79.9255	+	255.646i	0.209778	+	0.670987i
$382$		0			0
$383$	260.516			0.680198			0.340099	−	0.940390i	$-0.389539\pi$
$383$	260.516			0.680198			0.340099	+	0.940390i	$0.389539\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−203.637	−	293.839i	−0.526193	−	0.759273i
$388$		0			0
$389$		−	724.894i		−	1.86348i	−0.363125	−	0.931740i	$-0.618290\pi$
$389$		−	724.894i		−	1.86348i	0.363125	−	0.931740i	$-0.381710\pi$
$390$		0			0
$391$	−44.6937			−0.114306
$392$		0			0
$393$	119.661	+	382.741i	0.304480	+	0.973895i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			394.446i			0.993567i	0.867874	+	0.496784i	$0.165486\pi$
$397$			394.446i			0.993567i	−0.867874	+	0.496784i	$0.834514\pi$
$398$		0			0
$399$	−39.9808	−	127.881i	−0.100202	−	0.320503i
$400$		0			0
$401$			340.136i			0.848219i	0.905611	+	0.424110i	$0.139413\pi$
$401$			340.136i			0.848219i	−0.905611	+	0.424110i	$0.860587\pi$
$402$		0			0
$403$		−	126.364i		−	0.313559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	53.1842			0.130674
$408$		0			0
$409$	646.306			1.58021			0.790105	−	0.612971i	$-0.210026\pi$
$409$	646.306			1.58021			0.790105	+	0.612971i	$0.210026\pi$
$410$		0			0
$411$	−148.398	+	46.3954i	−0.361066	+	0.112884i
$412$		0			0
$413$	−197.086			−0.477205
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−545.157	+	170.438i	−1.30733	+	0.408725i
$418$		0			0
$419$			774.244i			1.84784i	0.382590	+	0.923918i	$0.375032\pi$
$419$			774.244i			1.84784i	−0.382590	+	0.923918i	$0.624968\pi$
$420$		0			0
$421$	77.6930			0.184544			0.0922719	−	0.995734i	$-0.470587\pi$
$421$	77.6930			0.184544			0.0922719	+	0.995734i	$0.470587\pi$
$422$		0			0
$423$	−533.959	+	370.045i	−1.26231	+	0.874811i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			175.030i			0.409906i
$428$		0			0
$429$	209.178	−	65.3977i	0.487595	−	0.152442i
$430$		0			0
$431$			301.177i			0.698787i	0.936976	+	0.349394i	$0.113612\pi$
$431$			301.177i			0.698787i	−0.936976	+	0.349394i	$0.886388\pi$
$432$		0			0
$433$		−	98.3440i		−	0.227123i	−0.993531	−	0.113561i	$-0.963774\pi$
$433$		−	98.3440i		−	0.227123i	0.993531	−	0.113561i	$-0.0362258\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−209.343			−0.479045
$438$		0			0
$439$	−525.993			−1.19816			−0.599081	−	0.800688i	$-0.704467\pi$
$439$	−525.993			−1.19816			−0.599081	+	0.800688i	$0.704467\pi$
$440$		0			0
$441$	−51.7808	+	35.8852i	−0.117417	+	0.0813724i
$442$		0			0
$443$	547.933			1.23687			0.618434	−	0.785837i	$-0.287767\pi$
$443$	547.933			1.23687			0.618434	+	0.785837i	$0.287767\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−6.45643	−	20.6512i	−0.0144439	−	0.0461996i
$448$		0			0
$449$		−	302.192i		−	0.673033i	−0.941678	−	0.336517i	$-0.890751\pi$
$449$		−	302.192i		−	0.673033i	0.941678	−	0.336517i	$-0.109249\pi$
$450$		0			0
$451$	59.1393			0.131129
$452$		0			0
$453$	325.348	−	101.717i	0.718208	−	0.224542i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	202.530i		−	0.443172i	−0.975141	−	0.221586i	$-0.928877\pi$
$457$		−	202.530i		−	0.443172i	0.975141	−	0.221586i	$-0.0711234\pi$
$458$		0			0
$459$	−59.7944	+	76.7657i	−0.130271	+	0.167245i
$460$		0			0
$461$			850.540i			1.84499i	0.386010	+	0.922495i	$0.373853\pi$
$461$			850.540i			1.84499i	−0.386010	+	0.922495i	$0.626147\pi$
$462$		0			0
$463$			371.502i			0.802380i	0.915995	+	0.401190i	$0.131403\pi$
$463$			371.502i			0.802380i	−0.915995	+	0.401190i	$0.868597\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	571.243			1.22322			0.611609	−	0.791160i	$-0.290523\pi$
$467$	571.243			1.22322			0.611609	+	0.791160i	$0.290523\pi$
$468$		0			0
$469$	−126.503			−0.269730
$470$		0			0
$471$	182.809	+	584.724i	0.388129	+	1.24145i
$472$		0			0
$473$	−119.302			−0.252224
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−106.822	+	74.0298i	−0.223945	+	0.155199i
$478$		0			0
$479$			433.086i			0.904145i	0.891981	+	0.452073i	$0.149315\pi$
$479$			433.086i			0.904145i	−0.891981	+	0.452073i	$0.850685\pi$
$480$		0			0
$481$	430.734			0.895496
$482$		0			0
$483$	29.3723	+	93.9490i	0.0608123	+	0.194511i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			319.390i			0.655831i	0.944707	+	0.327916i	$0.106346\pi$
$487$			319.390i			0.655831i	−0.944707	+	0.327916i	$0.893654\pi$
$488$		0			0
$489$	−69.1274	−	221.108i	−0.141365	−	0.452163i
$490$		0			0
$491$			106.542i			0.216989i	0.994097	+	0.108495i	$0.0346031\pi$
$491$			106.542i			0.216989i	−0.994097	+	0.108495i	$0.965397\pi$
$492$		0			0
$493$		−	48.9436i		−	0.0992772i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−123.870			−0.249235
$498$		0			0
$499$	−189.829			−0.380420			−0.190210	−	0.981743i	$-0.560917\pi$
$499$	−189.829			−0.380420			−0.190210	+	0.981743i	$0.560917\pi$
$500$		0			0
$501$	814.274	−	254.576i	1.62530	−	0.508135i
$502$		0			0
$503$	18.3298			0.0364410			0.0182205	−	0.999834i	$-0.494200\pi$
$503$	18.3298			0.0364410			0.0182205	+	0.999834i	$0.494200\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1210.21	−	378.362i	2.38700	−	0.746276i
$508$		0			0
$509$		−	102.736i		−	0.201839i	−0.994895	−	0.100920i	$-0.967821\pi$
$509$		−	102.736i		−	0.201839i	0.994895	−	0.100920i	$-0.0321785\pi$
$510$		0			0
$511$	111.876			0.218936
$512$		0			0
$513$	−280.074	+	359.566i	−0.545953	+	0.700909i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			216.794i			0.419330i
$518$		0			0
$519$	765.413	−	239.300i	1.47478	−	0.461079i
$520$		0			0
$521$		−	341.510i		−	0.655489i	−0.944766	−	0.327744i	$-0.893711\pi$
$521$		−	341.510i		−	0.655489i	0.944766	−	0.327744i	$-0.106289\pi$
$522$		0			0
$523$			400.488i			0.765751i	0.923800	+	0.382875i	$0.125066\pi$
$523$			400.488i			0.765751i	−0.923800	+	0.382875i	$0.874934\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.7224			0.0355264
$528$		0			0
$529$	−375.204			−0.709270
$530$		0			0
$531$	381.877	+	551.032i	0.719166	+	1.03773i
$532$		0			0
$533$	478.963			0.898618
$534$		0			0
$535$		0			0
$536$		0			0
$537$	20.9121	+	66.8884i	0.0389424	+	0.124559i
$538$		0			0
$539$			21.0236i			0.0390049i
$540$		0			0
$541$	−516.907			−0.955465			−0.477733	−	0.878505i	$-0.658541\pi$
$541$	−516.907			−0.955465			−0.477733	+	0.878505i	$0.658541\pi$
$542$		0			0
$543$	342.564	−	107.100i	0.630874	−	0.197237i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			725.077i			1.32555i	0.748818	+	0.662776i	$0.230622\pi$
$547$			725.077i			1.32555i	−0.748818	+	0.662776i	$0.769378\pi$
$548$		0			0
$549$	489.367	−	339.142i	0.891378	−	0.617744i
$550$		0			0
$551$		−	229.249i		−	0.416060i
$552$		0			0
$553$		−	8.27371i		−	0.0149615i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	741.121			1.33056			0.665279	−	0.746595i	$-0.268313\pi$
$557$	741.121			1.33056			0.665279	+	0.746595i	$0.268313\pi$
$558$		0			0
$559$	−966.216			−1.72847
$560$		0			0
$561$	9.68947	+	30.9923i	0.0172718	+	0.0552447i
$562$		0			0
$563$	926.643			1.64590			0.822951	−	0.568112i	$-0.192326\pi$
$563$	926.643			1.64590			0.822951	+	0.568112i	$0.192326\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	200.663	+	75.2420i	0.353903	+	0.132702i
$568$		0			0
$569$			116.739i			0.205166i	0.994724	+	0.102583i	$0.0327107\pi$
$569$			116.739i			0.205166i	−0.994724	+	0.102583i	$0.967289\pi$
$570$		0			0
$571$	−202.431			−0.354520			−0.177260	−	0.984164i	$-0.556723\pi$
$571$	−202.431			−0.354520			−0.177260	+	0.984164i	$0.556723\pi$
$572$		0			0
$573$	−68.1726	−	218.054i	−0.118975	−	0.380547i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			82.6440i			0.143230i	0.997432	+	0.0716152i	$0.0228154\pi$
$577$			82.6440i			0.143230i	−0.997432	+	0.0716152i	$0.977185\pi$
$578$		0			0
$579$	104.012	+	332.686i	0.179640	+	0.574588i
$580$		0			0
$581$		−	260.238i		−	0.447914i
$582$		0			0
$583$			43.3710i			0.0743927i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−108.578			−0.184970			−0.0924851	−	0.995714i	$-0.529481\pi$
$587$	−108.578			−0.184970			−0.0924851	+	0.995714i	$0.529481\pi$
$588$		0			0
$589$	87.6947			0.148887
$590$		0			0
$591$	708.630	−	221.547i	1.19903	−	0.374868i
$592$		0			0
$593$	−395.830			−0.667504			−0.333752	−	0.942661i	$-0.608315\pi$
$593$	−395.830			−0.667504			−0.333752	+	0.942661i	$0.608315\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−124.558	+	38.9420i	−0.208640	+	0.0652296i
$598$		0			0
$599$		−	59.2409i		−	0.0988996i	−0.998777	−	0.0494498i	$-0.984253\pi$
$599$		−	59.2409i		−	0.0988996i	0.998777	−	0.0494498i	$-0.0157468\pi$
$600$		0			0
$601$	−350.659			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$601$	−350.659			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$602$		0			0
$603$	245.116	+	353.691i	0.406494	+	0.586552i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	319.382i		−	0.526164i	−0.964773	−	0.263082i	$-0.915261\pi$
$607$		−	319.382i		−	0.526164i	0.964773	−	0.263082i	$-0.0847390\pi$
$608$		0			0
$609$	−102.883	+	32.1654i	−0.168937	+	0.0528168i
$610$		0			0
$611$			1755.79i			2.87363i
$612$		0			0
$613$			423.614i			0.691051i	0.938409	+	0.345525i	$0.112299\pi$
$613$			423.614i			0.691051i	−0.938409	+	0.345525i	$0.887701\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−986.842			−1.59942			−0.799710	−	0.600386i	$-0.795014\pi$
$617$	−986.842			−1.59942			−0.799710	+	0.600386i	$0.795014\pi$
$618$		0			0
$619$	−857.133			−1.38471			−0.692353	−	0.721559i	$-0.743426\pi$
$619$	−857.133			−1.38471			−0.692353	+	0.721559i	$0.743426\pi$
$620$		0			0
$621$	205.760	−	264.160i	0.331336	−	0.425378i
$622$		0			0
$623$	409.667			0.657571
$624$		0			0
$625$		0			0
$626$		0			0
$627$	45.3850	+	145.166i	0.0723843	+	0.231525i
$628$		0			0
$629$			63.8184i			0.101460i
$630$		0			0
$631$	−320.357			−0.507697			−0.253848	−	0.967244i	$-0.581696\pi$
$631$	−320.357			−0.507697			−0.253848	+	0.967244i	$0.581696\pi$
$632$		0			0
$633$	810.911	−	253.524i	1.28106	−	0.400512i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			170.268i			0.267297i
$638$		0			0
$639$	240.012	+	346.327i	0.375606	+	0.541983i
$640$		0			0
$641$			280.940i			0.438284i	0.975693	+	0.219142i	$0.0703259\pi$
$641$			280.940i			0.438284i	−0.975693	+	0.219142i	$0.929674\pi$
$642$		0			0
$643$			206.903i			0.321777i	0.986973	+	0.160889i	$0.0514360\pi$
$643$			206.903i			0.321777i	−0.986973	+	0.160889i	$0.948564\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	907.218			1.40219			0.701096	−	0.713067i	$-0.252694\pi$
$647$	907.218			1.40219			0.701096	+	0.713067i	$0.252694\pi$
$648$		0			0
$649$	223.726			0.344724
$650$		0			0
$651$	−12.3042	−	39.3557i	−0.0189005	−	0.0604542i
$652$		0			0
$653$	93.6270			0.143380			0.0716899	−	0.997427i	$-0.477161\pi$
$653$	93.6270			0.143380			0.0716899	+	0.997427i	$0.477161\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−216.773	−	312.794i	−0.329944	−	0.476095i
$658$		0			0
$659$		−	223.298i		−	0.338844i	−0.985544	−	0.169422i	$-0.945810\pi$
$659$		−	223.298i		−	0.338844i	0.985544	−	0.169422i	$-0.0541900\pi$
$660$		0			0
$661$	−409.255			−0.619145			−0.309572	−	0.950876i	$-0.600186\pi$
$661$	−409.255			−0.619145			−0.309572	+	0.950876i	$0.600186\pi$
$662$		0			0
$663$	78.4741	+	251.003i	0.118362	+	0.378587i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			168.421i			0.252505i
$668$		0			0
$669$	258.579	+	827.078i	0.386515	+	1.23629i
$670$		0			0
$671$		−	198.689i		−	0.296109i
$672$		0			0
$673$		−	256.438i		−	0.381037i	−0.981684	−	0.190519i	$-0.938983\pi$
$673$		−	256.438i		−	0.381037i	0.981684	−	0.190519i	$-0.0610170\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−186.843			−0.275986			−0.137993	−	0.990433i	$-0.544065\pi$
$677$	−186.843			−0.275986			−0.137993	+	0.990433i	$0.544065\pi$
$678$		0			0
$679$	−244.408			−0.359953
$680$		0			0
$681$	−956.190	+	298.944i	−1.40410	+	0.438979i
$682$		0			0
$683$	639.899			0.936895			0.468447	−	0.883491i	$-0.344814\pi$
$683$	639.899			0.936895			0.468447	+	0.883491i	$0.344814\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−32.0191	+	10.0105i	−0.0466071	+	0.0145713i
$688$		0			0
$689$			351.257i			0.509807i
$690$		0			0
$691$	−784.721			−1.13563			−0.567815	−	0.823156i	$-0.692211\pi$
$691$	−784.721			−1.13563			−0.567815	+	0.823156i	$0.692211\pi$
$692$		0			0
$693$	58.7800	−	40.7358i	0.0848196	−	0.0587819i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			70.9642i			0.101814i
$698$		0			0
$699$	713.938	−	223.206i	1.02137	−	0.319322i
$700$		0			0
$701$			743.171i			1.06016i	0.847948	+	0.530079i	$0.177838\pi$
$701$			743.171i			1.06016i	−0.847948	+	0.530079i	$0.822162\pi$
$702$		0			0
$703$			298.922i			0.425209i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	249.968			0.353562
$708$		0			0
$709$	742.471			1.04721			0.523605	−	0.851961i	$-0.324587\pi$
$709$	742.471			1.04721			0.523605	+	0.851961i	$0.324587\pi$
$710$		0			0
$711$	−23.1325	+	16.0313i	−0.0325351	+	0.0225475i
$712$		0			0
$713$	−64.4259			−0.0903590
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−128.967	−	412.507i	−0.179870	−	0.575324i
$718$		0			0
$719$		−	1075.69i		−	1.49609i	−0.663648	−	0.748045i	$-0.730993\pi$
$719$		−	1075.69i		−	1.49609i	0.663648	−	0.748045i	$-0.269007\pi$
$720$		0			0
$721$	−63.0463			−0.0874428
$722$		0			0
$723$	995.706	−	311.299i	1.37719	−	0.430565i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	276.007i		−	0.379652i	−0.981818	−	0.189826i	$-0.939208\pi$
$727$		−	276.007i		−	0.379652i	0.981818	−	0.189826i	$-0.0607924\pi$
$728$		0			0
$729$	−178.439	−	706.824i	−0.244773	−	0.969580i
$730$		0			0
$731$		−	143.157i		−	0.195837i
$732$		0			0
$733$		−	1323.23i		−	1.80523i	−0.430448	−	0.902616i	$-0.641644\pi$
$733$		−	1323.23i		−	1.80523i	0.430448	−	0.902616i	$-0.358356\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	143.603			0.194848
$738$		0			0
$739$	−11.1722			−0.0151181			−0.00755903	−	0.999971i	$-0.502406\pi$
$739$	−11.1722			−0.0151181			−0.00755903	+	0.999971i	$0.502406\pi$
$740$		0			0
$741$	367.568	+	1175.69i	0.496043	+	1.58662i
$742$		0			0
$743$	532.660			0.716904			0.358452	−	0.933548i	$-0.383305\pi$
$743$	532.660			0.716904			0.358452	+	0.933548i	$0.383305\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−727.599	+	504.242i	−0.974028	+	0.675023i
$748$		0			0
$749$			396.960i			0.529987i
$750$		0			0
$751$	−843.951			−1.12377			−0.561885	−	0.827215i	$-0.689924\pi$
$751$	−843.951			−1.12377			−0.561885	+	0.827215i	$0.689924\pi$
$752$		0			0
$753$	405.861	+	1298.17i	0.538992	+	1.72400i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			303.221i			0.400556i	0.979739	+	0.200278i	$0.0641846\pi$
$757$			303.221i			0.400556i	−0.979739	+	0.200278i	$0.935815\pi$
$758$		0			0
$759$	−33.3426	−	106.648i	−0.0439297	−	0.140511i
$760$		0			0
$761$		−	115.096i		−	0.151243i	−0.997137	−	0.0756215i	$-0.975906\pi$
$761$		−	115.096i		−	0.151243i	0.997137	−	0.0756215i	$-0.0240941\pi$
$762$		0			0
$763$		−	426.786i		−	0.559352i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1811.93			2.36236
$768$		0			0
$769$	−801.344			−1.04206			−0.521030	−	0.853538i	$-0.674452\pi$
$769$	−801.344			−1.04206			−0.521030	+	0.853538i	$0.674452\pi$
$770$		0			0
$771$	−1210.58	+	378.478i	−1.57015	+	0.490893i
$772$		0			0
$773$	94.0269			0.121639			0.0608194	−	0.998149i	$-0.480629\pi$
$773$	94.0269			0.121639			0.0608194	+	0.998149i	$0.480629\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	134.151	−	41.9410i	0.172652	−	0.0539781i
$778$		0			0
$779$			332.393i			0.426691i
$780$		0			0
$781$	140.613			0.180042
$782$		0			0
$783$	289.279	+	225.326i	0.369450	+	0.287772i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			114.020i			0.144880i	0.997373	+	0.0724398i	$0.0230785\pi$
$787$			114.020i			0.144880i	−0.997373	+	0.0724398i	$0.976921\pi$
$788$		0			0
$789$	1075.25	−	336.167i	1.36280	−	0.426067i
$790$		0			0
$791$		−	531.682i		−	0.672165i
$792$		0			0
$793$		−	1609.16i		−	2.02921i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−879.789			−1.10388			−0.551938	−	0.833885i	$-0.686111\pi$
$797$	−879.789			−1.10388			−0.551938	+	0.833885i	$0.686111\pi$
$798$		0			0
$799$	−260.142			−0.325584
$800$		0			0
$801$	−793.779	−	1145.39i	−0.990984	−	1.42995i
$802$		0			0
$803$	−126.998			−0.158155
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.26505	−	4.04634i	−0.00156760	−	0.00501405i
$808$		0			0
$809$		−	1429.94i		−	1.76753i	−0.467927	−	0.883767i	$-0.654999\pi$
$809$		−	1429.94i		−	1.76753i	0.467927	−	0.883767i	$-0.345001\pi$
$810$		0			0
$811$	−1369.69			−1.68889			−0.844445	−	0.535643i	$-0.820069\pi$
$811$	−1369.69			−1.68889			−0.844445	+	0.535643i	$0.820069\pi$
$812$		0			0
$813$	−647.265	+	202.362i	−0.796144	+	0.248907i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	670.538i		−	0.820732i
$818$		0			0
$819$	476.053	−	329.915i	0.581262	−	0.402827i
$820$		0			0
$821$			1221.52i			1.48785i	0.668265	+	0.743923i	$0.267037\pi$
$821$			1221.52i			1.48785i	−0.668265	+	0.743923i	$0.732963\pi$
$822$		0			0
$823$		−	223.454i		−	0.271512i	−0.990742	−	0.135756i	$-0.956654\pi$
$823$		−	223.454i		−	0.271512i	0.990742	−	0.135756i	$-0.0433463\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−150.303			−0.181744			−0.0908722	−	0.995863i	$-0.528965\pi$
$827$	−150.303			−0.181744			−0.0908722	+	0.995863i	$0.528965\pi$
$828$		0			0
$829$	994.512			1.19965			0.599826	−	0.800130i	$-0.295236\pi$
$829$	994.512			1.19965			0.599826	+	0.800130i	$0.295236\pi$
$830$		0			0
$831$	−318.099	−	1017.46i	−0.382791	−	1.22438i
$832$		0			0
$833$	−25.2273			−0.0302849
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−86.1937	+	110.658i	−0.102979	+	0.132208i
$838$		0			0
$839$		−	582.305i		−	0.694047i	−0.937857	−	0.347023i	$-0.887192\pi$
$839$		−	582.305i		−	0.694047i	0.937857	−	0.347023i	$-0.112808\pi$
$840$		0			0
$841$	656.564			0.780694
$842$		0			0
$843$	175.445	+	561.171i	0.208120	+	0.665683i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			296.271i			0.349788i
$848$		0			0
$849$	434.246	+	1388.96i	0.511480	+	1.63600i
$850$		0			0
$851$		−	219.607i		−	0.258057i
$852$		0			0
$853$			1513.46i			1.77428i	0.461503	+	0.887139i	$0.347310\pi$
$853$			1513.46i			1.77428i	−0.461503	+	0.887139i	$0.652690\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1275.06			1.48782			0.743911	−	0.668278i	$-0.232969\pi$
$857$	1275.06			1.48782			0.743911	+	0.668278i	$0.232969\pi$
$858$		0			0
$859$	330.107			0.384292			0.192146	−	0.981366i	$-0.438455\pi$
$859$	330.107			0.384292			0.192146	+	0.981366i	$0.438455\pi$
$860$		0			0
$861$	149.172	−	46.6372i	0.173254	−	0.0541663i
$862$		0			0
$863$	51.0607			0.0591665			0.0295832	−	0.999562i	$-0.490582\pi$
$863$	51.0607			0.0591665			0.0295832	+	0.999562i	$0.490582\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	790.312	−	247.084i	0.911547	−	0.284987i
$868$		0			0
$869$			9.39207i			0.0108079i
$870$		0			0
$871$	1163.03			1.33528
$872$		0			0
$873$	473.570	+	683.340i	0.542463	+	0.782750i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	19.8915i		−	0.0226813i	−0.999936	−	0.0113406i	$-0.996390\pi$
$877$		−	19.8915i		−	0.0226813i	0.999936	−	0.0113406i	$-0.00360991\pi$
$878$		0			0
$879$	−1083.68	+	338.802i	−1.23285	+	0.385440i
$880$		0			0
$881$			946.321i			1.07414i	0.843536	+	0.537072i	$0.180470\pi$
$881$			946.321i			1.07414i	−0.843536	+	0.537072i	$0.819530\pi$
$882$		0			0
$883$		−	1500.11i		−	1.69888i	−0.527684	−	0.849441i	$-0.676940\pi$
$883$		−	1500.11i		−	1.69888i	0.527684	−	0.849441i	$-0.323060\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	515.237			0.580876			0.290438	−	0.956894i	$-0.406199\pi$
$887$	515.237			0.580876			0.290438	+	0.956894i	$0.406199\pi$
$888$		0			0
$889$	236.220			0.265715
$890$		0			0
$891$	−227.787	−	85.4125i	−0.255653	−	0.0958614i
$892$		0			0
$893$	−1218.49			−1.36449
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−270.038	−	863.732i	−0.301046	−	0.962912i
$898$		0			0
$899$		−	70.5523i		−	0.0784786i
$900$		0			0
$901$	−52.0430			−0.0577614
$902$		0			0
$903$	−300.925	+	94.0815i	−0.333250	+	0.104188i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	841.383i		−	0.927655i	−0.885926	−	0.463827i	$-0.846476\pi$
$907$		−	841.383i		−	0.927655i	0.885926	−	0.463827i	$-0.153524\pi$
$908$		0			0
$909$	−484.344	−	698.887i	−0.532831	−	0.768852i
$910$		0			0
$911$		−	564.445i		−	0.619589i	−0.950804	−	0.309794i	$-0.899740\pi$
$911$		−	564.445i		−	0.619589i	0.950804	−	0.309794i	$-0.100260\pi$
$912$		0			0
$913$			295.414i			0.323564i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	353.658			0.385668
$918$		0			0
$919$	−1171.81			−1.27509			−0.637547	−	0.770411i	$-0.720051\pi$
$919$	−1171.81			−1.27509			−0.637547	+	0.770411i	$0.720051\pi$
$920$		0			0
$921$	−513.558	−	1642.64i	−0.557610	−	1.78354i
$922$		0			0
$923$	1138.81			1.23382
$924$		0			0
$925$		0			0
$926$		0			0
$927$	122.160	+	176.271i	0.131780	+	0.190152i
$928$		0			0
$929$		−	812.845i		−	0.874967i	−0.899226	−	0.437484i	$-0.855870\pi$
$929$		−	812.845i		−	0.874967i	0.899226	−	0.437484i	$-0.144130\pi$
$930$		0			0
$931$	−118.163			−0.126921
$932$		0			0
$933$	395.602	+	1265.36i	0.424011	+	1.35622i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1673.57i		−	1.78610i	−0.449962	−	0.893048i	$-0.648563\pi$
$937$		−	1673.57i		−	1.78610i	0.449962	−	0.893048i	$-0.351437\pi$
$938$		0			0
$939$	339.254	+	1085.12i	0.361293	+	1.15562i
$940$		0			0
$941$		−	974.517i		−	1.03562i	−0.855496	−	0.517809i	$-0.826748\pi$
$941$		−	974.517i		−	1.03562i	0.855496	−	0.517809i	$-0.173252\pi$
$942$		0			0
$943$		−	244.196i		−	0.258957i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−876.037			−0.925066			−0.462533	−	0.886602i	$-0.653059\pi$
$947$	−876.037			−0.925066			−0.462533	+	0.886602i	$0.653059\pi$
$948$		0			0
$949$	−1028.55			−1.08382
$950$		0			0
$951$	1684.14	−	526.533i	1.77092	−	0.553662i
$952$		0			0
$953$	6.64325			0.00697088			0.00348544	−	0.999994i	$-0.498891\pi$
$953$	6.64325			0.00697088			0.00348544	+	0.999994i	$0.498891\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	116.789	−	36.5132i	0.122037	−	0.0381538i
$958$		0			0
$959$			137.122i			0.142984i
$960$		0			0
$961$	−934.012			−0.971916
$962$		0			0
$963$	1109.86	−	769.158i	1.15250	−	0.798710i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1147.22i			1.18637i	0.805067	+	0.593184i	$0.202129\pi$
$967$			1147.22i			1.18637i	−0.805067	+	0.593184i	$0.797871\pi$
$968$		0			0
$969$	−174.192	+	54.4597i	−0.179765	+	0.0562020i
$970$		0			0
$971$			48.8617i			0.0503210i	0.999683	+	0.0251605i	$0.00800968\pi$
$971$			48.8617i			0.0503210i	−0.999683	+	0.0251605i	$0.991990\pi$
$972$		0			0
$973$			503.732i			0.517710i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	454.682			0.465386			0.232693	−	0.972550i	$-0.425246\pi$
$977$	454.682			0.465386			0.232693	+	0.972550i	$0.425246\pi$
$978$		0			0
$979$	−465.042			−0.475017
$980$		0			0
$981$	−1193.25	+	826.948i	−1.21636	+	0.842965i
$982$		0			0
$983$	−493.543			−0.502078			−0.251039	−	0.967977i	$-0.580772\pi$
$983$	−493.543			−0.502078			−0.251039	+	0.967977i	$0.580772\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	170.963	+	546.835i	0.173215	+	0.554038i
$988$		0			0
$989$			492.619i			0.498098i
$990$		0			0
$991$	321.693			0.324614			0.162307	−	0.986740i	$-0.448107\pi$
$991$	321.693			0.324614			0.162307	+	0.986740i	$0.448107\pi$
$992$		0			0
$993$	−588.544	+	184.003i	−0.592693	+	0.185300i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1254.19i			1.25796i	0.777421	+	0.628980i	$0.216527\pi$
$997$			1254.19i			1.25796i	−0.777421	+	0.628980i	$0.783473\pi$
$998$		0			0
$999$	−377.196	−	293.806i	−0.377573	−	0.294100i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.3.e.c.449.4		32
3.2	odd	2	inner	2100.3.e.c.449.30		32
5.2	odd	4		420.3.g.a.281.12	yes	16
5.3	odd	4		2100.3.g.d.701.5		16
5.4	even	2	inner	2100.3.e.c.449.29		32
15.2	even	4		420.3.g.a.281.11	✓	16
15.8	even	4		2100.3.g.d.701.6		16
15.14	odd	2	inner	2100.3.e.c.449.3		32
20.7	even	4		1680.3.l.b.1121.5		16
60.47	odd	4		1680.3.l.b.1121.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.3.g.a.281.11	✓	16	15.2	even	4
420.3.g.a.281.12	yes	16	5.2	odd	4
1680.3.l.b.1121.5		16	20.7	even	4
1680.3.l.b.1121.6		16	60.47	odd	4
2100.3.e.c.449.3		32	15.14	odd	2	inner
2100.3.e.c.449.4		32	1.1	even	1	trivial
2100.3.e.c.449.29		32	5.4	even	2	inner
2100.3.e.c.449.30		32	3.2	odd	2	inner
2100.3.g.d.701.5		16	5.3	odd	4
2100.3.g.d.701.6		16	15.8	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)

\(f(q)\)	\(=\)	\(q+(-2.86332 + 0.895194i) q^{3} +2.64575i q^{7} +(7.39726 - 5.12646i) q^{9} -3.00338i q^{11} -24.3241i q^{13} +3.60390 q^{17} +16.8805 q^{19} +(-2.36846 - 7.57565i) q^{21} -12.4015 q^{23} +(-16.5916 + 21.3007i) q^{27} -13.5807i q^{29} +5.19503 q^{31} +(2.68860 + 8.59964i) q^{33} +17.7081i q^{37} +(21.7747 + 69.6477i) q^{39} +19.6909i q^{41} -39.7227i q^{43} -72.1833 q^{47} -7.00000 q^{49} +(-10.3191 + 3.22619i) q^{51} -14.4407 q^{53} +(-48.3343 + 15.1113i) q^{57} +74.4914i q^{59} +66.1551 q^{61} +(13.5633 + 19.5713i) q^{63} +47.8138i q^{67} +(35.5094 - 11.1017i) q^{69} +46.8183i q^{71} -42.2852i q^{73} +7.94619 q^{77} -3.12717 q^{79} +(28.4388 - 75.8435i) q^{81} -98.3606 q^{83} +(12.1574 + 38.8860i) q^{87} -154.840i q^{89} +64.3554 q^{91} +(-14.8751 + 4.65056i) q^{93} +92.3775i q^{97} +(-15.3967 - 22.2168i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 20 q^{9} - 96 q^{19} - 28 q^{21} + 48 q^{31} + 108 q^{39} - 224 q^{49} - 372 q^{51} - 304 q^{61} + 392 q^{69} - 424 q^{79} - 236 q^{81} + 392 q^{91} - 52 q^{99}+O(q^{100}) \) 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

Related objects

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