LMFDB - Embedded newform 3300.2.x.d.1957.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3300,2,Mod(1693,3300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3300.1693"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$26.3506326670$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1957.13
Character	$\chi$	$=$	3300.1957
Dual form			3300.2.x.d.1693.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{3} +(-3.37614 + 3.37614i) q^{7} -1.00000i q^{9} +(-3.25700 + 0.626035i) q^{11} +(-1.72243 - 1.72243i) q^{13} +(-0.580573 + 0.580573i) q^{17} +4.50902 q^{19} +4.77458i q^{21} +(3.65309 - 3.65309i) q^{23} +(-0.707107 - 0.707107i) q^{27} -0.725530 q^{29} +6.16625 q^{31} +(-1.86038 + 2.74572i) q^{33} +(0.978843 + 0.978843i) q^{37} -2.43589 q^{39} +8.99367i q^{41} +(-0.936850 - 0.936850i) q^{43} +(-8.25919 - 8.25919i) q^{47} -15.7966i q^{49} +0.821055i q^{51} +(-0.742397 + 0.742397i) q^{53} +(3.18836 - 3.18836i) q^{57} -9.89830i q^{59} -14.4938i q^{61} +(3.37614 + 3.37614i) q^{63} +(-1.66012 - 1.66012i) q^{67} -5.16625i q^{69} +4.84839 q^{71} +(-8.52297 - 8.52297i) q^{73} +(8.88251 - 13.1097i) q^{77} +8.00257 q^{79} -1.00000 q^{81} +(8.89853 + 8.89853i) q^{83} +(-0.513027 + 0.513027i) q^{87} -12.7467i q^{89} +11.6304 q^{91} +(4.36020 - 4.36020i) q^{93} +(-1.29533 - 1.29533i) q^{97} +(0.626035 + 3.25700i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) $ 32 * q - 24 * q^11 + 16 * q^31 - 32 * q^71 - 32 * q^81 + 80 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$.

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.37614	+	3.37614i	−1.27606	+	1.27606i	−0.333206	+	0.942854i	$0.608130\pi$
$7$	−3.37614	+	3.37614i	−1.27606	+	1.27606i	−0.942854	+	0.333206i	$0.891870\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−3.25700	+	0.626035i	−0.982024	+	0.188757i
$11$	−3.25700	+	0.626035i	−0.982024	+	0.188757i
$12$		0			0
$13$	−1.72243	−	1.72243i	−0.477717	−	0.477717i	0.426684	−	0.904401i	$-0.359682\pi$
$13$	−1.72243	−	1.72243i	−0.477717	−	0.477717i	−0.904401	+	0.426684i	$0.859682\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.580573	+	0.580573i	−0.140810	+	0.140810i	−0.773998	−	0.633188i	$-0.781746\pi$
$17$	−0.580573	+	0.580573i	−0.140810	+	0.140810i	0.633188	+	0.773998i	$0.281746\pi$
$18$		0			0
$19$	4.50902			1.03444			0.517220	−	0.855853i	$-0.326967\pi$
$19$	4.50902			1.03444			0.517220	+	0.855853i	$0.326967\pi$
$20$		0			0
$21$			4.77458i			1.04190i
$22$		0			0
$23$	3.65309	−	3.65309i	0.761722	−	0.761722i	−0.214911	−	0.976634i	$-0.568946\pi$
$23$	3.65309	−	3.65309i	0.761722	−	0.761722i	0.976634	+	0.214911i	$0.0689463\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.707107	−	0.707107i	−0.136083	−	0.136083i
$28$		0			0
$29$	−0.725530			−0.134728			−0.0673638	−	0.997728i	$-0.521459\pi$
$29$	−0.725530			−0.134728			−0.0673638	+	0.997728i	$0.521459\pi$
$30$		0			0
$31$	6.16625			1.10749			0.553746	−	0.832686i	$-0.313198\pi$
$31$	6.16625			1.10749			0.553746	+	0.832686i	$0.313198\pi$
$32$		0			0
$33$	−1.86038	+	2.74572i	−0.323850	+	0.477969i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.978843	+	0.978843i	0.160921	+	0.160921i	0.782975	−	0.622054i	$-0.213701\pi$
$37$	0.978843	+	0.978843i	0.160921	+	0.160921i	−0.622054	+	0.782975i	$0.713701\pi$
$38$		0			0
$39$	−2.43589			−0.390055
$40$		0			0
$41$			8.99367i			1.40457i	0.711894	+	0.702287i	$0.247838\pi$
$41$			8.99367i			1.40457i	−0.711894	+	0.702287i	$0.752162\pi$
$42$		0			0
$43$	−0.936850	−	0.936850i	−0.142868	−	0.142868i	0.632055	−	0.774923i	$-0.282212\pi$
$43$	−0.936850	−	0.936850i	−0.142868	−	0.142868i	−0.774923	+	0.632055i	$0.782212\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.25919	−	8.25919i	−1.20473	−	1.20473i	−0.972712	−	0.232015i	$-0.925468\pi$
$47$	−8.25919	−	8.25919i	−1.20473	−	1.20473i	−0.232015	−	0.972712i	$-0.574532\pi$
$48$		0			0
$49$		−	15.7966i		−	2.25666i
$50$		0			0
$51$			0.821055i			0.114971i
$52$		0			0
$53$	−0.742397	+	0.742397i	−0.101976	+	0.101976i	−0.756254	−	0.654278i	$-0.772973\pi$
$53$	−0.742397	+	0.742397i	−0.101976	+	0.101976i	0.654278	+	0.756254i	$0.272973\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.18836	−	3.18836i	0.422308	−	0.422308i
$58$		0			0
$59$		−	9.89830i		−	1.28865i	−0.764752	−	0.644325i	$-0.777139\pi$
$59$		−	9.89830i		−	1.28865i	0.764752	−	0.644325i	$-0.222861\pi$
$60$		0			0
$61$		−	14.4938i		−	1.85574i	−0.372907	−	0.927869i	$-0.621639\pi$
$61$		−	14.4938i		−	1.85574i	0.372907	−	0.927869i	$-0.378361\pi$
$62$		0			0
$63$	3.37614	+	3.37614i	0.425353	+	0.425353i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.66012	−	1.66012i	−0.202815	−	0.202815i	0.598390	−	0.801205i	$-0.295807\pi$
$67$	−1.66012	−	1.66012i	−0.202815	−	0.202815i	−0.801205	+	0.598390i	$0.795807\pi$
$68$		0			0
$69$		−	5.16625i		−	0.621944i
$70$		0			0
$71$	4.84839			0.575399			0.287699	−	0.957721i	$-0.407110\pi$
$71$	4.84839			0.575399			0.287699	+	0.957721i	$0.407110\pi$
$72$		0			0
$73$	−8.52297	−	8.52297i	−0.997538	−	0.997538i	0.00245880	−	0.999997i	$-0.499217\pi$
$73$	−8.52297	−	8.52297i	−0.997538	−	0.997538i	−0.999997	+	0.00245880i	$0.999217\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.88251	−	13.1097i	1.01226	−	1.49399i
$78$		0			0
$79$	8.00257			0.900360			0.450180	−	0.892938i	$-0.351360\pi$
$79$	8.00257			0.900360			0.450180	+	0.892938i	$0.351360\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	8.89853	+	8.89853i	0.976741	+	0.976741i	0.999736	−	0.0229948i	$-0.00732012\pi$
$83$	8.89853	+	8.89853i	0.976741	+	0.976741i	−0.0229948	+	0.999736i	$0.507320\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.513027	+	0.513027i	−0.0550023	+	0.0550023i
$88$		0			0
$89$		−	12.7467i		−	1.35115i	−0.737293	−	0.675574i	$-0.763896\pi$
$89$		−	12.7467i		−	1.35115i	0.737293	−	0.675574i	$-0.236104\pi$
$90$		0			0
$91$	11.6304			1.21919
$92$		0			0
$93$	4.36020	−	4.36020i	0.452131	−	0.452131i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.29533	−	1.29533i	−0.131520	−	0.131520i	0.638282	−	0.769802i	$-0.279645\pi$
$97$	−1.29533	−	1.29533i	−0.131520	−	0.131520i	−0.769802	+	0.638282i	$0.779645\pi$
$98$		0			0
$99$	0.626035	+	3.25700i	0.0629189	+	0.327341i
$100$		0			0
$101$		−	14.0065i		−	1.39370i	−0.717216	−	0.696851i	$-0.754584\pi$
$101$		−	14.0065i		−	1.39370i	0.717216	−	0.696851i	$-0.245416\pi$
$102$		0			0
$103$	1.03528	−	1.03528i	0.102009	−	0.102009i	−0.654260	−	0.756269i	$-0.727020\pi$
$103$	1.03528	−	1.03528i	0.102009	−	0.102009i	0.756269	+	0.654260i	$0.227020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.43322	+	4.43322i	−0.428576	+	0.428576i	−0.888143	−	0.459567i	$-0.848005\pi$
$107$	−4.43322	+	4.43322i	−0.428576	+	0.428576i	0.459567	+	0.888143i	$0.348005\pi$
$108$		0			0
$109$	−8.89982			−0.852448			−0.426224	−	0.904618i	$-0.640156\pi$
$109$	−8.89982			−0.852448			−0.426224	+	0.904618i	$0.640156\pi$
$110$		0			0
$111$	1.38429			0.131391
$112$		0			0
$113$	−5.64138	+	5.64138i	−0.530696	+	0.530696i	−0.920780	−	0.390083i	$-0.872446\pi$
$113$	−5.64138	+	5.64138i	−0.530696	+	0.530696i	0.390083	+	0.920780i	$0.372446\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.72243	+	1.72243i	−0.159239	+	0.159239i
$118$		0			0
$119$		−	3.92019i		−	0.359363i
$120$		0			0
$121$	10.2162	−	4.07800i	0.928742	−	0.370727i
$122$		0			0
$123$	6.35948	+	6.35948i	0.573415	+	0.573415i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	12.3253	−	12.3253i	1.09369	−	1.09369i	0.0985630	−	0.995131i	$-0.468575\pi$
$127$	12.3253	−	12.3253i	1.09369	−	1.09369i	0.995131	−	0.0985630i	$-0.0314246\pi$
$128$		0			0
$129$	−1.32491			−0.116651
$130$		0			0
$131$		−	8.96807i		−	0.783544i	−0.920062	−	0.391772i	$-0.871862\pi$
$131$		−	8.96807i		−	0.783544i	0.920062	−	0.391772i	$-0.128138\pi$
$132$		0			0
$133$	−15.2231	+	15.2231i	−1.32001	+	1.32001i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−12.7124	−	12.7124i	−1.08610	−	1.08610i	−0.995926	−	0.0901711i	$-0.971259\pi$
$137$	−12.7124	−	12.7124i	−1.08610	−	1.08610i	−0.0901711	−	0.995926i	$-0.528741\pi$
$138$		0			0
$139$	9.54916			0.809949			0.404975	−	0.914328i	$-0.367280\pi$
$139$	9.54916			0.809949			0.404975	+	0.914328i	$0.367280\pi$
$140$		0			0
$141$	−11.6803			−0.983655
$142$		0			0
$143$	6.68828	+	4.53167i	0.559302	+	0.378958i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.1699	−	11.1699i	−0.921277	−	0.921277i
$148$		0			0
$149$	3.83513			0.314186			0.157093	−	0.987584i	$-0.449788\pi$
$149$	3.83513			0.314186			0.157093	+	0.987584i	$0.449788\pi$
$150$		0			0
$151$			13.4282i			1.09277i	0.837535	+	0.546384i	$0.183996\pi$
$151$			13.4282i			1.09277i	−0.837535	+	0.546384i	$0.816004\pi$
$152$		0			0
$153$	0.580573	+	0.580573i	0.0469366	+	0.0469366i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.43922	+	7.43922i	0.593714	+	0.593714i	0.938633	−	0.344918i	$-0.112093\pi$
$157$	7.43922	+	7.43922i	0.593714	+	0.593714i	−0.344918	+	0.938633i	$0.612093\pi$
$158$		0			0
$159$			1.04991i			0.0832631i
$160$		0			0
$161$			24.6667i			1.94401i
$162$		0			0
$163$	2.69539	−	2.69539i	0.211119	−	0.211119i	−0.593624	−	0.804743i	$-0.702303\pi$
$163$	2.69539	−	2.69539i	0.211119	−	0.211119i	0.804743	+	0.593624i	$0.202303\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.22436	−	7.22436i	0.559038	−	0.559038i	−0.369996	−	0.929033i	$-0.620641\pi$
$167$	7.22436	−	7.22436i	0.559038	−	0.559038i	0.929033	+	0.369996i	$0.120641\pi$
$168$		0			0
$169$		−	7.06644i		−	0.543572i
$170$		0			0
$171$		−	4.50902i		−	0.344813i
$172$		0			0
$173$	−6.40893	−	6.40893i	−0.487262	−	0.487262i	0.420179	−	0.907441i	$-0.361967\pi$
$173$	−6.40893	−	6.40893i	−0.487262	−	0.487262i	−0.907441	+	0.420179i	$0.861967\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.99916	−	6.99916i	−0.526089	−	0.526089i
$178$		0			0
$179$			9.54391i			0.713345i	0.934229	+	0.356673i	$0.116089\pi$
$179$			9.54391i			0.713345i	−0.934229	+	0.356673i	$0.883911\pi$
$180$		0			0
$181$	20.9562			1.55766			0.778832	−	0.627233i	$-0.215813\pi$
$181$	20.9562			1.55766			0.778832	+	0.627233i	$0.215813\pi$
$182$		0			0
$183$	−10.2486	−	10.2486i	−0.757602	−	0.757602i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.52747	−	2.25439i	0.111700	−	0.164857i
$188$		0			0
$189$	4.77458			0.347300
$190$		0			0
$191$	11.2028			0.810605			0.405303	−	0.914183i	$-0.367166\pi$
$191$	11.2028			0.810605			0.405303	+	0.914183i	$0.367166\pi$
$192$		0			0
$193$	6.90591	+	6.90591i	0.497098	+	0.497098i	0.910534	−	0.413435i	$-0.135671\pi$
$193$	6.90591	+	6.90591i	0.497098	+	0.497098i	−0.413435	+	0.910534i	$0.635671\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.01380	+	5.01380i	−0.357218	+	0.357218i	−0.862787	−	0.505568i	$-0.831283\pi$
$197$	−5.01380	+	5.01380i	−0.357218	+	0.357218i	0.505568	+	0.862787i	$0.331283\pi$
$198$		0			0
$199$		−	14.5586i		−	1.03203i	−0.856580	−	0.516014i	$-0.827415\pi$
$199$		−	14.5586i		−	1.03203i	0.856580	−	0.516014i	$-0.172585\pi$
$200$		0			0
$201$	−2.34776			−0.165598
$202$		0			0
$203$	2.44949	−	2.44949i	0.171920	−	0.171920i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.65309	−	3.65309i	−0.253907	−	0.253907i
$208$		0			0
$209$	−14.6859	+	2.82280i	−1.01584	+	0.195257i
$210$		0			0
$211$			6.72323i			0.462846i	0.972853	+	0.231423i	$0.0743382\pi$
$211$			6.72323i			0.462846i	−0.972853	+	0.231423i	$0.925662\pi$
$212$		0			0
$213$	3.42833	−	3.42833i	0.234905	−	0.234905i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−20.8181	+	20.8181i	−1.41323	+	1.41323i
$218$		0			0
$219$	−12.0533			−0.814487
$220$		0			0
$221$	2.00000			0.134535
$222$		0			0
$223$	−4.27414	+	4.27414i	−0.286218	+	0.286218i	−0.835583	−	0.549365i	$-0.814870\pi$
$223$	−4.27414	+	4.27414i	−0.286218	+	0.286218i	0.549365	+	0.835583i	$0.314870\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.3125	+	18.3125i	−1.21544	+	1.21544i	−0.246229	+	0.969212i	$0.579192\pi$
$227$	−18.3125	+	18.3125i	−1.21544	+	1.21544i	−0.969212	+	0.246229i	$0.920808\pi$
$228$		0			0
$229$		−	10.9402i		−	0.722949i	−0.932382	−	0.361474i	$-0.882274\pi$
$229$		−	10.9402i		−	0.722949i	0.932382	−	0.361474i	$-0.117726\pi$
$230$		0			0
$231$	−2.98906	−	15.5508i	−0.196665	−	1.02317i
$232$		0			0
$233$	−14.5572	−	14.5572i	−0.953674	−	0.953674i	0.0452992	−	0.998973i	$-0.485576\pi$
$233$	−14.5572	−	14.5572i	−0.953674	−	0.953674i	−0.998973	+	0.0452992i	$0.985576\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5.65867	−	5.65867i	0.367570	−	0.367570i
$238$		0			0
$239$	16.0097			1.03558			0.517792	−	0.855507i	$-0.326754\pi$
$239$	16.0097			1.03558			0.517792	+	0.855507i	$0.326754\pi$
$240$		0			0
$241$			23.4573i			1.51101i	0.655140	+	0.755507i	$0.272610\pi$
$241$			23.4573i			1.51101i	−0.655140	+	0.755507i	$0.727390\pi$
$242$		0			0
$243$	−0.707107	+	0.707107i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−7.76649	−	7.76649i	−0.494170	−	0.494170i
$248$		0			0
$249$	12.5844			0.797506
$250$		0			0
$251$	−5.10971			−0.322522			−0.161261	−	0.986912i	$-0.551556\pi$
$251$	−5.10971			−0.322522			−0.161261	+	0.986912i	$0.551556\pi$
$252$		0			0
$253$	−9.61117	+	14.1851i	−0.604249	+	0.891810i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.20035	−	4.20035i	−0.262011	−	0.262011i	0.563860	−	0.825871i	$-0.309316\pi$
$257$	−4.20035	−	4.20035i	−0.262011	−	0.262011i	−0.825871	+	0.563860i	$0.809316\pi$
$258$		0			0
$259$	−6.60942			−0.410689
$260$		0			0
$261$			0.725530i			0.0449092i
$262$		0			0
$263$	−19.5089	−	19.5089i	−1.20297	−	1.20297i	−0.973258	−	0.229715i	$-0.926221\pi$
$263$	−19.5089	−	19.5089i	−1.20297	−	1.20297i	−0.229715	−	0.973258i	$-0.573779\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−9.01328	−	9.01328i	−0.551604	−	0.551604i
$268$		0			0
$269$		−	8.29249i		−	0.505602i	−0.967518	−	0.252801i	$-0.918648\pi$
$269$		−	8.29249i		−	0.505602i	0.967518	−	0.252801i	$-0.0813518\pi$
$270$		0			0
$271$			25.8442i			1.56993i	0.619543	+	0.784963i	$0.287318\pi$
$271$			25.8442i			1.56993i	−0.619543	+	0.784963i	$0.712682\pi$
$272$		0			0
$273$	8.22390	−	8.22390i	0.497733	−	0.497733i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.69150	−	7.69150i	0.462137	−	0.462137i	−0.437218	−	0.899355i	$-0.644036\pi$
$277$	7.69150	−	7.69150i	0.462137	−	0.462137i	0.899355	+	0.437218i	$0.144036\pi$
$278$		0			0
$279$		−	6.16625i		−	0.369164i
$280$		0			0
$281$		−	15.1675i		−	0.904816i	−0.891811	−	0.452408i	$-0.850565\pi$
$281$		−	15.1675i		−	0.904816i	0.891811	−	0.452408i	$-0.149435\pi$
$282$		0			0
$283$	−5.32287	−	5.32287i	−0.316412	−	0.316412i	0.530975	−	0.847387i	$-0.321826\pi$
$283$	−5.32287	−	5.32287i	−0.316412	−	0.316412i	−0.847387	+	0.530975i	$0.821826\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.3638	−	30.3638i	−1.79232	−	1.79232i
$288$		0			0
$289$			16.3259i			0.960345i
$290$		0			0
$291$	−1.83187			−0.107386
$292$		0			0
$293$	4.80866	+	4.80866i	0.280925	+	0.280925i	0.833478	−	0.552553i	$-0.186346\pi$
$293$	4.80866	+	4.80866i	0.280925	+	0.280925i	−0.552553	+	0.833478i	$0.686346\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.74572	+	1.86038i	0.159323	+	0.107950i
$298$		0			0
$299$	−12.5844			−0.727776
$300$		0			0
$301$	6.32587			0.364617
$302$		0			0
$303$	−9.90411	−	9.90411i	−0.568977	−	0.568977i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7.92959	−	7.92959i	0.452566	−	0.452566i	−0.443639	−	0.896205i	$-0.646313\pi$
$307$	7.92959	−	7.92959i	0.452566	−	0.452566i	0.896205	+	0.443639i	$0.146313\pi$
$308$		0			0
$309$		−	1.46410i		−	0.0832898i
$310$		0			0
$311$	2.43608			0.138138			0.0690688	−	0.997612i	$-0.477997\pi$
$311$	2.43608			0.138138			0.0690688	+	0.997612i	$0.477997\pi$
$312$		0			0
$313$	−22.5292	+	22.5292i	−1.27343	+	1.27343i	−0.329151	+	0.944277i	$0.606762\pi$
$313$	−22.5292	+	22.5292i	−1.27343	+	1.27343i	−0.944277	+	0.329151i	$0.893238\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.0914	+	13.0914i	0.735285	+	0.735285i	0.971662	−	0.236376i	$-0.0759598\pi$
$317$	13.0914	+	13.0914i	0.735285	+	0.735285i	−0.236376	+	0.971662i	$0.575960\pi$
$318$		0			0
$319$	2.36306	−	0.454208i	0.132306	−	0.0254307i
$320$		0			0
$321$			6.26952i			0.349931i
$322$		0			0
$323$	−2.61782	+	2.61782i	−0.145659	+	0.145659i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−6.29312	+	6.29312i	−0.348010	+	0.348010i
$328$		0			0
$329$	55.7683			3.07461
$330$		0			0
$331$	33.7528			1.85522			0.927612	−	0.373546i	$-0.121858\pi$
$331$	33.7528			1.85522			0.927612	+	0.373546i	$0.121858\pi$
$332$		0			0
$333$	0.978843	−	0.978843i	0.0536403	−	0.0536403i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−13.2803	+	13.2803i	−0.723421	+	0.723421i	−0.969301	−	0.245879i	$-0.920923\pi$
$337$	−13.2803	+	13.2803i	−0.723421	+	0.723421i	0.245879	+	0.969301i	$0.420923\pi$
$338$		0			0
$339$			7.97811i			0.433312i
$340$		0			0
$341$	−20.0835	+	3.86029i	−1.08758	+	0.209047i
$342$		0			0
$343$	29.6985	+	29.6985i	1.60357	+	1.60357i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.9389	−	23.9389i	1.28511	−	1.28511i	0.347387	−	0.937722i	$-0.387069\pi$
$347$	23.9389	−	23.9389i	1.28511	−	1.28511i	0.937722	−	0.347387i	$-0.112931\pi$
$348$		0			0
$349$	−9.54916			−0.511155			−0.255577	−	0.966789i	$-0.582266\pi$
$349$	−9.54916			−0.511155			−0.255577	+	0.966789i	$0.582266\pi$
$350$		0			0
$351$			2.43589i			0.130018i
$352$		0			0
$353$	−14.4901	+	14.4901i	−0.771231	+	0.771231i	−0.978322	−	0.207091i	$-0.933600\pi$
$353$	−14.4901	+	14.4901i	−0.771231	+	0.771231i	0.207091	+	0.978322i	$0.433600\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.77199	−	2.77199i	−0.146709	−	0.146709i
$358$		0			0
$359$	−30.5172			−1.61064			−0.805318	−	0.592843i	$-0.798006\pi$
$359$	−30.5172			−1.61064			−0.805318	+	0.592843i	$0.798006\pi$
$360$		0			0
$361$	1.33123			0.0700647
$362$		0			0
$363$	4.34033	−	10.1075i	0.227808	−	0.530506i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	19.6444	+	19.6444i	1.02543	+	1.02543i	0.999668	+	0.0257602i	$0.00820063\pi$
$367$	19.6444	+	19.6444i	1.02543	+	1.02543i	0.0257602	+	0.999668i	$0.491799\pi$
$368$		0			0
$369$	8.99367			0.468192
$370$		0			0
$371$		−	5.01287i		−	0.260255i
$372$		0			0
$373$	13.3212	+	13.3212i	0.689745	+	0.689745i	0.962176	−	0.272430i	$-0.0878274\pi$
$373$	13.3212	+	13.3212i	0.689745	+	0.689745i	−0.272430	+	0.962176i	$0.587827\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.24968	+	1.24968i	0.0643617	+	0.0643617i
$378$		0			0
$379$		−	12.3857i		−	0.636209i	−0.948056	−	0.318105i	$-0.896954\pi$
$379$		−	12.3857i		−	0.636209i	0.948056	−	0.318105i	$-0.103046\pi$
$380$		0			0
$381$		−	17.4306i		−	0.892997i
$382$		0			0
$383$	25.5739	−	25.5739i	1.30677	−	1.30677i	0.383033	−	0.923735i	$-0.374880\pi$
$383$	25.5739	−	25.5739i	1.30677	−	1.30677i	0.923735	−	0.383033i	$-0.125120\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.936850	+	0.936850i	−0.0476228	+	0.0476228i
$388$		0			0
$389$		−	16.5140i		−	0.837294i	−0.908149	−	0.418647i	$-0.862505\pi$
$389$		−	16.5140i		−	0.837294i	0.908149	−	0.418647i	$-0.137495\pi$
$390$		0			0
$391$			4.24178i			0.214516i
$392$		0			0
$393$	−6.34138	−	6.34138i	−0.319880	−	0.319880i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	6.68134	+	6.68134i	0.335327	+	0.335327i	0.854605	−	0.519278i	$-0.173799\pi$
$397$	6.68134	+	6.68134i	0.335327	+	0.335327i	−0.519278	+	0.854605i	$0.673799\pi$
$398$		0			0
$399$			21.5287i			1.07778i
$400$		0			0
$401$	−22.4542			−1.12131			−0.560655	−	0.828050i	$-0.689451\pi$
$401$	−22.4542			−1.12131			−0.560655	+	0.828050i	$0.689451\pi$
$402$		0			0
$403$	−10.6210	−	10.6210i	−0.529068	−	0.529068i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.80089	−	2.57531i	−0.188403	−	0.127653i
$408$		0			0
$409$	−30.3079			−1.49863			−0.749314	−	0.662215i	$-0.769617\pi$
$409$	−30.3079			−1.49863			−0.749314	+	0.662215i	$0.769617\pi$
$410$		0			0
$411$	−17.9781			−0.886795
$412$		0			0
$413$	33.4180	+	33.4180i	1.64439	+	1.64439i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6.75227	−	6.75227i	0.330660	−	0.330660i
$418$		0			0
$419$			23.6570i			1.15572i	0.816136	+	0.577860i	$0.196112\pi$
$419$			23.6570i			1.15572i	−0.816136	+	0.577860i	$0.803888\pi$
$420$		0			0
$421$	−13.2991			−0.648160			−0.324080	−	0.946030i	$-0.605055\pi$
$421$	−13.2991			−0.648160			−0.324080	+	0.946030i	$0.605055\pi$
$422$		0			0
$423$	−8.25919	+	8.25919i	−0.401576	+	0.401576i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	48.9330	+	48.9330i	2.36803	+	2.36803i
$428$		0			0
$429$	7.93371	−	1.52495i	0.383043	−	0.0736255i
$430$		0			0
$431$		−	36.5579i		−	1.76093i	−0.474110	−	0.880466i	$-0.657230\pi$
$431$		−	36.5579i		−	1.76093i	0.474110	−	0.880466i	$-0.342770\pi$
$432$		0			0
$433$	−7.43922	+	7.43922i	−0.357506	+	0.357506i	−0.862893	−	0.505387i	$-0.831350\pi$
$433$	−7.43922	+	7.43922i	−0.357506	+	0.357506i	0.505387	+	0.862893i	$0.331350\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.4718	−	16.4718i	0.787955	−	0.787955i
$438$		0			0
$439$	−13.8638			−0.661682			−0.330841	−	0.943687i	$-0.607332\pi$
$439$	−13.8638			−0.661682			−0.330841	+	0.943687i	$0.607332\pi$
$440$		0			0
$441$	−15.7966			−0.752219
$442$		0			0
$443$	18.1870	−	18.1870i	0.864090	−	0.864090i	−0.127721	−	0.991810i	$-0.540766\pi$
$443$	18.1870	−	18.1870i	0.864090	−	0.864090i	0.991810	+	0.127721i	$0.0407661\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	2.71184	−	2.71184i	0.128266	−	0.128266i
$448$		0			0
$449$			18.2487i			0.861210i	0.902541	+	0.430605i	$0.141700\pi$
$449$			18.2487i			0.861210i	−0.902541	+	0.430605i	$0.858300\pi$
$450$		0			0
$451$	−5.63035	−	29.2924i	−0.265123	−	1.37933i
$452$		0			0
$453$	9.49515	+	9.49515i	0.446121	+	0.446121i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.18072	+	2.18072i	−0.102010	+	0.102010i	−0.756270	−	0.654260i	$-0.772980\pi$
$457$	−2.18072	+	2.18072i	−0.102010	+	0.102010i	0.654260	+	0.756270i	$0.272980\pi$
$458$		0			0
$459$	0.821055			0.0383236
$460$		0			0
$461$		−	39.3942i		−	1.83477i	−0.398002	−	0.917384i	$-0.630296\pi$
$461$		−	39.3942i		−	1.83477i	0.398002	−	0.917384i	$-0.369704\pi$
$462$		0			0
$463$	16.5184	−	16.5184i	0.767674	−	0.767674i	−0.210022	−	0.977697i	$-0.567354\pi$
$463$	16.5184	−	16.5184i	0.767674	−	0.767674i	0.977697	+	0.210022i	$0.0673536\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.2361	−	14.2361i	−0.658768	−	0.658768i	0.296321	−	0.955088i	$-0.404240\pi$
$467$	−14.2361	−	14.2361i	−0.658768	−	0.658768i	−0.955088	+	0.296321i	$0.904240\pi$
$468$		0			0
$469$	11.2096			0.517609
$470$		0			0
$471$	10.5206			0.484766
$472$		0			0
$473$	3.63783	+	2.46482i	0.167267	+	0.113333i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.742397	+	0.742397i	0.0339920	+	0.0339920i
$478$		0			0
$479$	−14.4709			−0.661192			−0.330596	−	0.943772i	$-0.607250\pi$
$479$	−14.4709			−0.661192			−0.330596	+	0.943772i	$0.607250\pi$
$480$		0			0
$481$		−	3.37199i		−	0.153749i
$482$		0			0
$483$	17.4420	+	17.4420i	0.793637	+	0.793637i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	11.6113	+	11.6113i	0.526157	+	0.526157i	0.919424	−	0.393267i	$-0.128655\pi$
$487$	11.6113	+	11.6113i	0.526157	+	0.526157i	−0.393267	+	0.919424i	$0.628655\pi$
$488$		0			0
$489$		−	3.81186i		−	0.172378i
$490$		0			0
$491$			35.8369i			1.61730i	0.588291	+	0.808649i	$0.299801\pi$
$491$			35.8369i			1.61730i	−0.588291	+	0.808649i	$0.700199\pi$
$492$		0			0
$493$	0.421224	−	0.421224i	0.0189710	−	0.0189710i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−16.3688	+	16.3688i	−0.734243	+	0.734243i
$498$		0			0
$499$			4.23932i			0.189778i	0.995488	+	0.0948890i	$0.0302496\pi$
$499$			4.23932i			0.189778i	−0.995488	+	0.0948890i	$0.969750\pi$
$500$		0			0
$501$		−	10.2168i		−	0.456452i
$502$		0			0
$503$	−14.7655	−	14.7655i	−0.658359	−	0.658359i	0.296632	−	0.954992i	$-0.404136\pi$
$503$	−14.7655	−	14.7655i	−0.658359	−	0.658359i	−0.954992	+	0.296632i	$0.904136\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−4.99672	−	4.99672i	−0.221912	−	0.221912i
$508$		0			0
$509$		−	0.613825i		−	0.0272073i	−0.999907	−	0.0136037i	$-0.995670\pi$
$509$		−	0.613825i		−	0.0272073i	0.999907	−	0.0136037i	$-0.00433031\pi$
$510$		0			0
$511$	57.5494			2.54584
$512$		0			0
$513$	−3.18836	−	3.18836i	−0.140769	−	0.140769i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	32.0708	+	21.7297i	1.41047	+	0.955670i
$518$		0			0
$519$	−9.06359			−0.397848
$520$		0			0
$521$	−11.9695			−0.524393			−0.262196	−	0.965015i	$-0.584447\pi$
$521$	−11.9695			−0.524393			−0.262196	+	0.965015i	$0.584447\pi$
$522$		0			0
$523$	−31.8513	−	31.8513i	−1.39276	−	1.39276i	−0.819080	−	0.573680i	$-0.805515\pi$
$523$	−31.8513	−	31.8513i	−1.39276	−	1.39276i	−0.573680	−	0.819080i	$-0.694485\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.57996	+	3.57996i	−0.155946	+	0.155946i
$528$		0			0
$529$		−	3.69015i		−	0.160442i
$530$		0			0
$531$	−9.89830			−0.429550
$532$		0			0
$533$	15.4910	−	15.4910i	0.670990	−	0.670990i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	6.74856	+	6.74856i	0.291222	+	0.291222i
$538$		0			0
$539$	9.88923	+	51.4496i	0.425959	+	2.21609i
$540$		0			0
$541$			12.0716i			0.518998i	0.965743	+	0.259499i	$0.0835574\pi$
$541$			12.0716i			0.518998i	−0.965743	+	0.259499i	$0.916443\pi$
$542$		0			0
$543$	14.8183	−	14.8183i	0.635914	−	0.635914i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	9.15947	−	9.15947i	0.391631	−	0.391631i	−0.483638	−	0.875268i	$-0.660685\pi$
$547$	9.15947	−	9.15947i	0.391631	−	0.391631i	0.875268	+	0.483638i	$0.160685\pi$
$548$		0			0
$549$	−14.4938			−0.618579
$550$		0			0
$551$	−3.27143			−0.139368
$552$		0			0
$553$	−27.0178	+	27.0178i	−1.14891	+	1.14891i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.54989	−	9.54989i	0.404642	−	0.404642i	−0.475223	−	0.879865i	$-0.657633\pi$
$557$	9.54989	−	9.54989i	0.404642	−	0.404642i	0.879865	+	0.475223i	$0.157633\pi$
$558$		0			0
$559$			3.22733i			0.136501i
$560$		0			0
$561$	−0.514009	−	2.67418i	−0.0217015	−	0.112904i
$562$		0			0
$563$	−0.279041	−	0.279041i	−0.0117602	−	0.0117602i	0.701202	−	0.712962i	$-0.252647\pi$
$563$	−0.279041	−	0.279041i	−0.0117602	−	0.0117602i	−0.712962	+	0.701202i	$0.752647\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.37614	−	3.37614i	0.141784	−	0.141784i
$568$		0			0
$569$	−12.9988			−0.544939			−0.272470	−	0.962164i	$-0.587840\pi$
$569$	−12.9988			−0.544939			−0.272470	+	0.962164i	$0.587840\pi$
$570$		0			0
$571$			10.9606i			0.458687i	0.973346	+	0.229343i	$0.0736579\pi$
$571$			10.9606i			0.458687i	−0.973346	+	0.229343i	$0.926342\pi$
$572$		0			0
$573$	7.92157	−	7.92157i	0.330928	−	0.330928i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	4.63627	+	4.63627i	0.193010	+	0.193010i	0.796996	−	0.603985i	$-0.206421\pi$
$577$	4.63627	+	4.63627i	0.193010	+	0.193010i	−0.603985	+	0.796996i	$0.706421\pi$
$578$		0			0
$579$	9.76644			0.405879
$580$		0			0
$581$	−60.0853			−2.49276
$582$		0			0
$583$	1.95322	−	2.88276i	0.0808943	−	0.119392i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.2005	+	13.2005i	0.544841	+	0.544841i	0.924944	−	0.380103i	$-0.124111\pi$
$587$	13.2005	+	13.2005i	0.544841	+	0.544841i	−0.380103	+	0.924944i	$0.624111\pi$
$588$		0			0
$589$	27.8037			1.14563
$590$		0			0
$591$			7.09058i			0.291667i
$592$		0			0
$593$	−14.5659	−	14.5659i	−0.598151	−	0.598151i	0.341669	−	0.939820i	$-0.389008\pi$
$593$	−14.5659	−	14.5659i	−0.598151	−	0.598151i	−0.939820	+	0.341669i	$0.889008\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−10.2945	−	10.2945i	−0.421324	−	0.421324i
$598$		0			0
$599$		−	38.2249i		−	1.56183i	−0.624638	−	0.780914i	$-0.714754\pi$
$599$		−	38.2249i		−	1.56183i	0.624638	−	0.780914i	$-0.285246\pi$
$600$		0			0
$601$		−	36.3360i		−	1.48218i	−0.671407	−	0.741089i	$-0.734310\pi$
$601$		−	36.3360i		−	1.48218i	0.671407	−	0.741089i	$-0.265690\pi$
$602$		0			0
$603$	−1.66012	+	1.66012i	−0.0676051	+	0.0676051i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−5.21319	+	5.21319i	−0.211597	+	0.211597i	−0.804946	−	0.593349i	$-0.797806\pi$
$607$	−5.21319	+	5.21319i	−0.211597	+	0.211597i	0.593349	+	0.804946i	$0.297806\pi$
$608$		0			0
$609$		−	3.46410i		−	0.140372i
$610$		0			0
$611$			28.4518i			1.15104i
$612$		0			0
$613$	−7.08689	−	7.08689i	−0.286237	−	0.286237i	0.549353	−	0.835590i	$-0.314874\pi$
$613$	−7.08689	−	7.08689i	−0.286237	−	0.286237i	−0.835590	+	0.549353i	$0.814874\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.3159	+	19.3159i	0.777627	+	0.777627i	0.979427	−	0.201800i	$-0.0646790\pi$
$617$	19.3159	+	19.3159i	0.777627	+	0.777627i	−0.201800	+	0.979427i	$0.564679\pi$
$618$		0			0
$619$			23.6634i			0.951113i	0.879685	+	0.475556i	$0.157753\pi$
$619$			23.6634i			0.951113i	−0.879685	+	0.475556i	$0.842247\pi$
$620$		0			0
$621$	−5.16625			−0.207315
$622$		0			0
$623$	43.0346	+	43.0346i	1.72414	+	1.72414i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−8.38847	+	12.3805i	−0.335003	+	0.494430i
$628$		0			0
$629$	−1.13658			−0.0453184
$630$		0			0
$631$	3.73017			0.148496			0.0742478	−	0.997240i	$-0.476344\pi$
$631$	3.73017			0.148496			0.0742478	+	0.997240i	$0.476344\pi$
$632$		0			0
$633$	4.75404	+	4.75404i	0.188956	+	0.188956i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−27.2086	+	27.2086i	−1.07804	+	1.07804i
$638$		0			0
$639$		−	4.84839i		−	0.191800i
$640$		0			0
$641$	−19.1988			−0.758307			−0.379154	−	0.925334i	$-0.623785\pi$
$641$	−19.1988			−0.758307			−0.379154	+	0.925334i	$0.623785\pi$
$642$		0			0
$643$	23.5162	−	23.5162i	0.927389	−	0.927389i	−0.0701478	−	0.997537i	$-0.522347\pi$
$643$	23.5162	−	23.5162i	0.927389	−	0.927389i	0.997537	+	0.0701478i	$0.0223471\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.53754	−	5.53754i	−0.217703	−	0.217703i	0.589827	−	0.807530i	$-0.299196\pi$
$647$	−5.53754	−	5.53754i	−0.217703	−	0.217703i	−0.807530	+	0.589827i	$0.799196\pi$
$648$		0			0
$649$	6.19669	+	32.2388i	0.243241	+	1.26548i
$650$		0			0
$651$			29.4413i			1.15389i
$652$		0			0
$653$	19.0350	−	19.0350i	0.744898	−	0.744898i	−0.228618	−	0.973516i	$-0.573421\pi$
$653$	19.0350	−	19.0350i	0.744898	−	0.744898i	0.973516	+	0.228618i	$0.0734208\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−8.52297	+	8.52297i	−0.332513	+	0.332513i
$658$		0			0
$659$	−15.3342			−0.597334			−0.298667	−	0.954357i	$-0.596542\pi$
$659$	−15.3342			−0.597334			−0.298667	+	0.954357i	$0.596542\pi$
$660$		0			0
$661$	−1.77522			−0.0690481			−0.0345240	−	0.999404i	$-0.510992\pi$
$661$	−1.77522			−0.0690481			−0.0345240	+	0.999404i	$0.510992\pi$
$662$		0			0
$663$	1.41421	−	1.41421i	0.0549235	−	0.0549235i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.65043	+	2.65043i	−0.102625	+	0.102625i
$668$		0			0
$669$			6.04455i			0.233696i
$670$		0			0
$671$	9.07362	+	47.2063i	0.350283	+	1.82238i
$672$		0			0
$673$	−24.4078	−	24.4078i	−0.940850	−	0.940850i	0.0574956	−	0.998346i	$-0.481688\pi$
$673$	−24.4078	−	24.4078i	−0.940850	−	0.940850i	−0.998346	+	0.0574956i	$0.981688\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−24.8887	+	24.8887i	−0.956551	+	0.956551i	−0.999095	−	0.0425440i	$-0.986454\pi$
$677$	−24.8887	+	24.8887i	−0.956551	+	0.956551i	0.0425440	+	0.999095i	$0.486454\pi$
$678$		0			0
$679$	8.74639			0.335656
$680$		0			0
$681$			25.8977i			0.992403i
$682$		0			0
$683$	−15.6593	+	15.6593i	−0.599188	+	0.599188i	−0.940096	−	0.340909i	$-0.889265\pi$
$683$	−15.6593	+	15.6593i	−0.599188	+	0.599188i	0.340909	+	0.940096i	$0.389265\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−7.73589	−	7.73589i	−0.295143	−	0.295143i
$688$		0			0
$689$	2.55746			0.0974315
$690$		0			0
$691$	−30.5812			−1.16336			−0.581682	−	0.813416i	$-0.697605\pi$
$691$	−30.5812			−1.16336			−0.581682	+	0.813416i	$0.697605\pi$
$692$		0			0
$693$	−13.1097	−	8.88251i	−0.497995	−	0.337419i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−5.22148	−	5.22148i	−0.197778	−	0.197778i
$698$		0			0
$699$	−20.5870			−0.778672
$700$		0			0
$701$			1.76677i			0.0667299i	0.999443	+	0.0333650i	$0.0106224\pi$
$701$			1.76677i			0.0667299i	−0.999443	+	0.0333650i	$0.989378\pi$
$702$		0			0
$703$	4.41362	+	4.41362i	0.166463	+	0.166463i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	47.2880	+	47.2880i	1.77845	+	1.77845i
$708$		0			0
$709$		−	13.8444i		−	0.519938i	−0.965617	−	0.259969i	$-0.916288\pi$
$709$		−	13.8444i		−	0.519938i	0.965617	−	0.259969i	$-0.0837123\pi$
$710$		0			0
$711$		−	8.00257i		−	0.300120i
$712$		0			0
$713$	22.5259	−	22.5259i	0.843601	−	0.843601i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	11.3206	−	11.3206i	0.422775	−	0.422775i
$718$		0			0
$719$			39.2569i			1.46404i	0.681285	+	0.732018i	$0.261422\pi$
$719$			39.2569i			1.46404i	−0.681285	+	0.732018i	$0.738578\pi$
$720$		0			0
$721$			6.99047i			0.260339i
$722$		0			0
$723$	16.5868	+	16.5868i	0.616869	+	0.616869i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.319144	−	0.319144i	−0.0118364	−	0.0118364i	0.701164	−	0.713000i	$-0.252664\pi$
$727$	−0.319144	−	0.319144i	−0.0118364	−	0.0118364i	−0.713000	+	0.701164i	$0.752664\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	1.08782			0.0402345
$732$		0			0
$733$	−9.07457	−	9.07457i	−0.335177	−	0.335177i	0.519372	−	0.854548i	$-0.326166\pi$
$733$	−9.07457	−	9.07457i	−0.335177	−	0.335177i	−0.854548	+	0.519372i	$0.826166\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.44630	+	4.36771i	0.237452	+	0.160887i
$738$		0			0
$739$	40.3688			1.48499			0.742496	−	0.669851i	$-0.233642\pi$
$739$	40.3688			1.48499			0.742496	+	0.669851i	$0.233642\pi$
$740$		0			0
$741$	−10.9835			−0.403488
$742$		0			0
$743$	−28.6094	−	28.6094i	−1.04958	−	1.04958i	−0.998705	−	0.0508708i	$-0.983800\pi$
$743$	−28.6094	−	28.6094i	−1.04958	−	1.04958i	−0.0508708	−	0.998705i	$-0.516200\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	8.89853	−	8.89853i	0.325580	−	0.325580i
$748$		0			0
$749$		−	29.9343i		−	1.09378i
$750$		0			0
$751$	16.3325			0.595982			0.297991	−	0.954569i	$-0.403684\pi$
$751$	16.3325			0.595982			0.297991	+	0.954569i	$0.403684\pi$
$752$		0			0
$753$	−3.61311	+	3.61311i	−0.131669	+	0.131669i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	35.1913	+	35.1913i	1.27905	+	1.27905i	0.941199	+	0.337852i	$0.109700\pi$
$757$	35.1913	+	35.1913i	1.27905	+	1.27905i	0.337852	+	0.941199i	$0.390300\pi$
$758$		0			0
$759$	3.23426	+	16.8265i	0.117396	+	0.610763i
$760$		0			0
$761$			0.642348i			0.0232851i	0.999932	+	0.0116425i	$0.00370602\pi$
$761$			0.642348i			0.0232851i	−0.999932	+	0.0116425i	$0.996294\pi$
$762$		0			0
$763$	30.0470	−	30.0470i	1.08777	−	1.08777i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−17.0492	+	17.0492i	−0.615610	+	0.615610i
$768$		0			0
$769$	20.3400			0.733481			0.366740	−	0.930323i	$-0.380474\pi$
$769$	20.3400			0.733481			0.366740	+	0.930323i	$0.380474\pi$
$770$		0			0
$771$	−5.94020			−0.213931
$772$		0			0
$773$	−36.5793	+	36.5793i	−1.31567	+	1.31567i	−0.398496	+	0.917170i	$0.630468\pi$
$773$	−36.5793	+	36.5793i	−1.31567	+	1.31567i	−0.917170	+	0.398496i	$0.869532\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−4.67356	+	4.67356i	−0.167663	+	0.167663i
$778$		0			0
$779$			40.5526i			1.45295i
$780$		0			0
$781$	−15.7912	+	3.03527i	−0.565055	+	0.108610i
$782$		0			0
$783$	0.513027	+	0.513027i	0.0183341	+	0.0183341i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−33.9266	+	33.9266i	−1.20935	+	1.20935i	−0.238117	+	0.971236i	$0.576530\pi$
$787$	−33.9266	+	33.9266i	−1.20935	+	1.20935i	−0.971236	+	0.238117i	$0.923470\pi$
$788$		0			0
$789$	−27.5898			−0.982223
$790$		0			0
$791$		−	38.0921i		−	1.35440i
$792$		0			0
$793$	−24.9646	+	24.9646i	−0.886518	+	0.886518i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.4607	−	33.4607i	−1.18524	−	1.18524i	−0.978369	−	0.206868i	$-0.933673\pi$
$797$	−33.4607	−	33.4607i	−1.18524	−	1.18524i	−0.206868	−	0.978369i	$-0.566327\pi$
$798$		0			0
$799$	9.59014			0.339275
$800$		0			0
$801$	−12.7467			−0.450382
$802$		0			0
$803$	33.0950	+	22.4237i	1.16790	+	0.791314i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.86368	−	5.86368i	−0.206411	−	0.206411i
$808$		0			0
$809$	−31.0715			−1.09241			−0.546207	−	0.837650i	$-0.683929\pi$
$809$	−31.0715			−1.09241			−0.546207	+	0.837650i	$0.683929\pi$
$810$		0			0
$811$		−	29.8391i		−	1.04779i	−0.851783	−	0.523895i	$-0.824478\pi$
$811$		−	29.8391i		−	1.04779i	0.851783	−	0.523895i	$-0.175522\pi$
$812$		0			0
$813$	18.2746	+	18.2746i	0.640919	+	0.640919i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.22427	−	4.22427i	−0.147789	−	0.147789i
$818$		0			0
$819$		−	11.6304i		−	0.406397i
$820$		0			0
$821$			1.73200i			0.0604474i	0.999543	+	0.0302237i	$0.00962196\pi$
$821$			1.73200i			0.0604474i	−0.999543	+	0.0302237i	$0.990378\pi$
$822$		0			0
$823$	−5.48153	+	5.48153i	−0.191074	+	0.191074i	−0.796160	−	0.605086i	$-0.793139\pi$
$823$	−5.48153	+	5.48153i	−0.191074	+	0.191074i	0.605086	+	0.796160i	$0.293139\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11.7909	+	11.7909i	−0.410011	+	0.410011i	−0.881742	−	0.471731i	$-0.843629\pi$
$827$	−11.7909	+	11.7909i	−0.410011	+	0.410011i	0.471731	+	0.881742i	$0.343629\pi$
$828$		0			0
$829$		−	12.4323i		−	0.431792i	−0.976416	−	0.215896i	$-0.930733\pi$
$829$		−	12.4323i		−	0.431792i	0.976416	−	0.215896i	$-0.0692673\pi$
$830$		0			0
$831$		−	10.8774i		−	0.377334i
$832$		0			0
$833$	9.17109	+	9.17109i	0.317759	+	0.317759i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−4.36020	−	4.36020i	−0.150710	−	0.150710i
$838$		0			0
$839$			51.0135i			1.76118i	0.473876	+	0.880591i	$0.342854\pi$
$839$			51.0135i			1.76118i	−0.473876	+	0.880591i	$0.657146\pi$
$840$		0			0
$841$	−28.4736			−0.981848
$842$		0			0
$843$	−10.7250	−	10.7250i	−0.369390	−	0.369390i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−20.7233	+	48.2590i	−0.712060	+	1.65820i
$848$		0			0
$849$	−7.52767			−0.258349
$850$		0			0
$851$	7.15161			0.245154
$852$		0			0
$853$	−5.71242	−	5.71242i	−0.195590	−	0.195590i	0.602517	−	0.798106i	$-0.294165\pi$
$853$	−5.71242	−	5.71242i	−0.195590	−	0.195590i	−0.798106	+	0.602517i	$0.794165\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.401296	+	0.401296i	−0.0137080	+	0.0137080i	−0.713928	−	0.700220i	$-0.753085\pi$
$857$	−0.401296	+	0.401296i	−0.0137080	+	0.0137080i	0.700220	+	0.713928i	$0.253085\pi$
$858$		0			0
$859$			0.216664i			0.00739247i	0.999993	+	0.00369623i	$0.00117655\pi$
$859$			0.216664i			0.00739247i	−0.999993	+	0.00369623i	$0.998823\pi$
$860$		0			0
$861$	−42.9410			−1.46342
$862$		0			0
$863$	−6.01688	+	6.01688i	−0.204817	+	0.204817i	−0.802060	−	0.597243i	$-0.796263\pi$
$863$	−6.01688	+	6.01688i	−0.204817	+	0.204817i	0.597243	+	0.802060i	$0.296263\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	11.5441	+	11.5441i	0.392059	+	0.392059i
$868$		0			0
$869$	−26.0644	+	5.00989i	−0.884175	+	0.169949i
$870$		0			0
$871$			5.71888i			0.193777i
$872$		0			0
$873$	−1.29533	+	1.29533i	−0.0438401	+	0.0438401i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	29.5816	−	29.5816i	0.998898	−	0.998898i	−0.00110139	−	0.999999i	$-0.500351\pi$
$877$	29.5816	−	29.5816i	0.998898	−	0.998898i	0.999999	+	0.00110139i	$0.000350584\pi$
$878$		0			0
$879$	6.80047			0.229374
$880$		0			0
$881$	24.9063			0.839115			0.419558	−	0.907729i	$-0.362185\pi$
$881$	24.9063			0.839115			0.419558	+	0.907729i	$0.362185\pi$
$882$		0			0
$883$	14.6232	−	14.6232i	0.492108	−	0.492108i	−0.416862	−	0.908970i	$-0.636870\pi$
$883$	14.6232	−	14.6232i	0.492108	−	0.492108i	0.908970	+	0.416862i	$0.136870\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	36.4810	−	36.4810i	1.22491	−	1.22491i	0.259047	−	0.965865i	$-0.416592\pi$
$887$	36.4810	−	36.4810i	1.22491	−	1.22491i	0.965865	−	0.259047i	$-0.0834083\pi$
$888$		0			0
$889$			83.2238i			2.79124i
$890$		0			0
$891$	3.25700	−	0.626035i	0.109114	−	0.0209730i
$892$		0			0
$893$	−37.2408	−	37.2408i	−1.24622	−	1.24622i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−8.89853	+	8.89853i	−0.297113	+	0.297113i
$898$		0			0
$899$	−4.47380			−0.149210
$900$		0			0
$901$		−	0.862032i		−	0.0287184i
$902$		0			0
$903$	4.47306	−	4.47306i	0.148854	−	0.148854i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	33.0998	+	33.0998i	1.09906	+	1.09906i	0.994521	+	0.104539i	$0.0333367\pi$
$907$	33.0998	+	33.0998i	1.09906	+	1.09906i	0.104539	+	0.994521i	$0.466663\pi$
$908$		0			0
$909$	−14.0065			−0.464567
$910$		0			0
$911$	−37.0971			−1.22908			−0.614541	−	0.788885i	$-0.710659\pi$
$911$	−37.0971			−1.22908			−0.614541	+	0.788885i	$0.710659\pi$
$912$		0			0
$913$	−34.5534	−	23.4118i	−1.14355	−	0.774816i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	30.2774	+	30.2774i	0.999849	+	0.999849i
$918$		0			0
$919$	−32.9167			−1.08582			−0.542911	−	0.839790i	$-0.682678\pi$
$919$	−32.9167			−1.08582			−0.542911	+	0.839790i	$0.682678\pi$
$920$		0			0
$921$		−	11.2141i		−	0.369518i
$922$		0			0
$923$	−8.35104	−	8.35104i	−0.274878	−	0.274878i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.03528	−	1.03528i	−0.0340029	−	0.0340029i
$928$		0			0
$929$		−	19.8943i		−	0.652711i	−0.945247	−	0.326356i	$-0.894179\pi$
$929$		−	19.8943i		−	0.652711i	0.945247	−	0.326356i	$-0.105821\pi$
$930$		0			0
$931$		−	71.2272i		−	2.33438i
$932$		0			0
$933$	1.72257	−	1.72257i	0.0563944	−	0.0563944i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	29.9868	−	29.9868i	0.979627	−	0.979627i	−0.0201696	−	0.999797i	$-0.506421\pi$
$937$	29.9868	−	29.9868i	0.979627	−	0.979627i	0.999797	+	0.0201696i	$0.00642062\pi$
$938$		0			0
$939$			31.8612i			1.03975i
$940$		0			0
$941$			39.6233i			1.29168i	0.763471	+	0.645842i	$0.223493\pi$
$941$			39.6233i			1.29168i	−0.763471	+	0.645842i	$0.776507\pi$
$942$		0			0
$943$	32.8547	+	32.8547i	1.06990	+	1.06990i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.8680	−	12.8680i	−0.418152	−	0.418152i	0.466414	−	0.884566i	$-0.345546\pi$
$947$	−12.8680	−	12.8680i	−0.418152	−	0.418152i	−0.884566	+	0.466414i	$0.845546\pi$
$948$		0			0
$949$			29.3605i			0.953083i
$950$		0			0
$951$	18.5140			0.600358
$952$		0			0
$953$	25.7516	+	25.7516i	0.834175	+	0.834175i	0.988085	−	0.153910i	$-0.0491866\pi$
$953$	25.7516	+	25.7516i	0.834175	+	0.834175i	−0.153910	+	0.988085i	$0.549187\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.34976	−	1.99211i	0.0436315	−	0.0643956i
$958$		0			0
$959$	85.8379			2.77185
$960$		0			0
$961$	7.02266			0.226537
$962$		0			0
$963$	4.43322	+	4.43322i	0.142859	+	0.142859i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	10.3322	−	10.3322i	0.332262	−	0.332262i	−0.521183	−	0.853445i	$-0.674509\pi$
$967$	10.3322	−	10.3322i	0.332262	−	0.332262i	0.853445	+	0.521183i	$0.174509\pi$
$968$		0			0
$969$			3.70215i			0.118930i
$970$		0			0
$971$	36.9482			1.18572			0.592862	−	0.805304i	$-0.297998\pi$
$971$	36.9482			1.18572			0.592862	+	0.805304i	$0.297998\pi$
$972$		0			0
$973$	−32.2393	+	32.2393i	−1.03354	+	1.03354i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.5851	+	22.5851i	0.722562	+	0.722562i	0.969126	−	0.246564i	$-0.0793016\pi$
$977$	22.5851	+	22.5851i	0.722562	+	0.722562i	−0.246564	+	0.969126i	$0.579302\pi$
$978$		0			0
$979$	7.97988	+	41.5161i	0.255038	+	1.32686i
$980$		0			0
$981$			8.89982i			0.284149i
$982$		0			0
$983$	−19.0373	+	19.0373i	−0.607196	+	0.607196i	−0.942212	−	0.335016i	$-0.891258\pi$
$983$	−19.0373	+	19.0373i	−0.607196	+	0.607196i	0.335016	+	0.942212i	$0.391258\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	39.4342	−	39.4342i	1.25520	−	1.25520i
$988$		0			0
$989$	−6.84480			−0.217652
$990$		0			0
$991$	25.8791			0.822076			0.411038	−	0.911618i	$-0.365166\pi$
$991$	25.8791			0.822076			0.411038	+	0.911618i	$0.365166\pi$
$992$		0			0
$993$	23.8669	−	23.8669i	0.757392	−	0.757392i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	33.6498	−	33.6498i	1.06570	−	1.06570i	0.0680146	−	0.997684i	$-0.478334\pi$
$997$	33.6498	−	33.6498i	1.06570	−	1.06570i	0.997684	−	0.0680146i	$-0.0216664\pi$
$998$		0			0
$999$		−	1.38429i		−	0.0437971i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.x.d.1957.13	yes	32
5.2	odd	4	inner	3300.2.x.d.1693.1	✓	32
5.3	odd	4	inner	3300.2.x.d.1693.14	yes	32
5.4	even	2	inner	3300.2.x.d.1957.2	yes	32
11.10	odd	2	inner	3300.2.x.d.1957.14	yes	32
55.32	even	4	inner	3300.2.x.d.1693.2	yes	32
55.43	even	4	inner	3300.2.x.d.1693.13	yes	32
55.54	odd	2	inner	3300.2.x.d.1957.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3300.2.x.d.1693.1	✓	32	5.2	odd	4	inner
3300.2.x.d.1693.2	yes	32	55.32	even	4	inner
3300.2.x.d.1693.13	yes	32	55.43	even	4	inner
3300.2.x.d.1693.14	yes	32	5.3	odd	4	inner
3300.2.x.d.1957.1	yes	32	55.54	odd	2	inner
3300.2.x.d.1957.2	yes	32	5.4	even	2	inner
3300.2.x.d.1957.13	yes	32	1.1	even	1	trivial
3300.2.x.d.1957.14	yes	32	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{3} +(-3.37614 + 3.37614i) q^{7} -1.00000i q^{9} +(-3.25700 + 0.626035i) q^{11} +(-1.72243 - 1.72243i) q^{13} +(-0.580573 + 0.580573i) q^{17} +4.50902 q^{19} +4.77458i q^{21} +(3.65309 - 3.65309i) q^{23} +(-0.707107 - 0.707107i) q^{27} -0.725530 q^{29} +6.16625 q^{31} +(-1.86038 + 2.74572i) q^{33} +(0.978843 + 0.978843i) q^{37} -2.43589 q^{39} +8.99367i q^{41} +(-0.936850 - 0.936850i) q^{43} +(-8.25919 - 8.25919i) q^{47} -15.7966i q^{49} +0.821055i q^{51} +(-0.742397 + 0.742397i) q^{53} +(3.18836 - 3.18836i) q^{57} -9.89830i q^{59} -14.4938i q^{61} +(3.37614 + 3.37614i) q^{63} +(-1.66012 - 1.66012i) q^{67} -5.16625i q^{69} +4.84839 q^{71} +(-8.52297 - 8.52297i) q^{73} +(8.88251 - 13.1097i) q^{77} +8.00257 q^{79} -1.00000 q^{81} +(8.89853 + 8.89853i) q^{83} +(-0.513027 + 0.513027i) q^{87} -12.7467i q^{89} +11.6304 q^{91} +(4.36020 - 4.36020i) q^{93} +(-1.29533 - 1.29533i) q^{97} +(0.626035 + 3.25700i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) \) 32 * q - 24 * q^11 + 16 * q^31 - 32 * q^71 - 32 * q^81 + 80 * q^91

Introduction

L-functions

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