LMFDB - Embedded newform 220.4.b.a.89.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [220,4,Mod(89,220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("220.89"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	220.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$12.9804202013$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.2
Root			$0.500000 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	220.89
Dual form			220.4.b.a.89.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.35890i q^{3} +(-2.50000 - 10.8972i) q^{5} +8.71780i q^{7} +8.00000 q^{9} -11.0000 q^{11} +69.7424i q^{13} +(47.5000 - 10.8972i) q^{15} +26.1534i q^{17} +68.0000 q^{19} -38.0000 q^{21} +117.690i q^{23} +(-112.500 + 54.4862i) q^{25} +152.561i q^{27} -260.000 q^{29} +175.000 q^{31} -47.9479i q^{33} +(95.0000 - 21.7945i) q^{35} +169.997i q^{37} -304.000 q^{39} -380.000 q^{41} +305.123i q^{43} +(-20.0000 - 87.1780i) q^{45} -305.123i q^{47} +267.000 q^{49} -114.000 q^{51} +453.325i q^{53} +(27.5000 + 119.870i) q^{55} +296.405i q^{57} +143.000 q^{59} +676.000 q^{61} +69.7424i q^{63} +(760.000 - 174.356i) q^{65} -527.427i q^{67} -513.000 q^{69} +1035.00 q^{71} +331.276i q^{73} +(-237.500 - 490.376i) q^{75} -95.8958i q^{77} -218.000 q^{79} -449.000 q^{81} -758.448i q^{83} +(285.000 - 65.3835i) q^{85} -1133.31i q^{87} -1279.00 q^{89} -608.000 q^{91} +762.807i q^{93} +(-170.000 - 741.013i) q^{95} -771.525i q^{97} -88.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{5} + 16 q^{9} - 22 q^{11} + 95 q^{15} + 136 q^{19} - 76 q^{21} - 225 q^{25} - 520 q^{29} + 350 q^{31} + 190 q^{35} - 608 q^{39} - 760 q^{41} - 40 q^{45} + 534 q^{49} - 228 q^{51} + 55 q^{55}+ \cdots - 176 q^{99}+O(q^{100}) $ 2 * q - 5 * q^5 + 16 * q^9 - 22 * q^11 + 95 * q^15 + 136 * q^19 - 76 * q^21 - 225 * q^25 - 520 * q^29 + 350 * q^31 + 190 * q^35 - 608 * q^39 - 760 * q^41 - 40 * q^45 + 534 * q^49 - 228 * q^51 + 55 * q^55 + 286 * q^59 + 1352 * q^61 + 1520 * q^65 - 1026 * q^69 + 2070 * q^71 - 475 * q^75 - 436 * q^79 - 898 * q^81 + 570 * q^85 - 2558 * q^89 - 1216 * q^91 - 340 * q^95 - 176 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/220\mathbb{Z}\right)^\times$.

$n$	$101$	$111$	$177$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			4.35890i			0.838870i	0.907785	+	0.419435i	$0.137772\pi$
$3$			4.35890i			0.838870i	−0.907785	+	0.419435i	$0.862228\pi$
$4$		0			0
$5$	−2.50000	−	10.8972i	−0.223607	−	0.974679i
$5$	−2.50000	−	10.8972i	−0.223607	−	0.974679i
$6$		0			0
$7$			8.71780i			0.470717i	0.971909	+	0.235358i	$0.0756264\pi$
$7$			8.71780i			0.470717i	−0.971909	+	0.235358i	$0.924374\pi$
$8$		0			0
$9$	8.00000			0.296296
$10$		0			0
$11$	−11.0000			−0.301511
$11$	−11.0000			−0.301511
$12$		0			0
$13$			69.7424i			1.48793i	0.668220	+	0.743964i	$0.267056\pi$
$13$			69.7424i			1.48793i	−0.668220	+	0.743964i	$0.732944\pi$
$14$		0			0
$15$	47.5000	−	10.8972i	0.817630	−	0.187577i
$16$		0			0
$17$			26.1534i			0.373125i	0.982443	+	0.186563i	$0.0597347\pi$
$17$			26.1534i			0.373125i	−0.982443	+	0.186563i	$0.940265\pi$
$18$		0			0
$19$	68.0000			0.821067			0.410533	−	0.911846i	$-0.365343\pi$
$19$	68.0000			0.821067			0.410533	+	0.911846i	$0.365343\pi$
$20$		0			0
$21$	−38.0000			−0.394870
$22$		0			0
$23$			117.690i			1.06696i	0.845812	+	0.533481i	$0.179116\pi$
$23$			117.690i			1.06696i	−0.845812	+	0.533481i	$0.820884\pi$
$24$		0			0
$25$	−112.500	+	54.4862i	−0.900000	+	0.435890i
$26$		0			0
$27$			152.561i			1.08742i
$28$		0			0
$29$	−260.000			−1.66485			−0.832427	−	0.554134i	$-0.813049\pi$
$29$	−260.000			−1.66485			−0.832427	+	0.554134i	$0.813049\pi$
$30$		0			0
$31$	175.000			1.01390			0.506950	−	0.861975i	$-0.330773\pi$
$31$	175.000			1.01390			0.506950	+	0.861975i	$0.330773\pi$
$32$		0			0
$33$		−	47.9479i		−	0.252929i
$34$		0			0
$35$	95.0000	−	21.7945i	0.458798	−	0.105255i
$36$		0			0
$37$			169.997i			0.755334i	0.925942	+	0.377667i	$0.123274\pi$
$37$			169.997i			0.755334i	−0.925942	+	0.377667i	$0.876726\pi$
$38$		0			0
$39$	−304.000			−1.24818
$40$		0			0
$41$	−380.000			−1.44746			−0.723732	−	0.690081i	$-0.757575\pi$
$41$	−380.000			−1.44746			−0.723732	+	0.690081i	$0.757575\pi$
$42$		0			0
$43$			305.123i			1.08211i	0.840987	+	0.541056i	$0.181975\pi$
$43$			305.123i			1.08211i	−0.840987	+	0.541056i	$0.818025\pi$
$44$		0			0
$45$	−20.0000	−	87.1780i	−0.0662539	−	0.288794i
$46$		0			0
$47$		−	305.123i		−	0.946952i	−0.880807	−	0.473476i	$-0.842999\pi$
$47$		−	305.123i		−	0.946952i	0.880807	−	0.473476i	$-0.157001\pi$
$48$		0			0
$49$	267.000			0.778426
$50$		0			0
$51$	−114.000			−0.313004
$52$		0			0
$53$			453.325i			1.17489i	0.809265	+	0.587444i	$0.199866\pi$
$53$			453.325i			1.17489i	−0.809265	+	0.587444i	$0.800134\pi$
$54$		0			0
$55$	27.5000	+	119.870i	0.0674200	+	0.293877i
$56$		0			0
$57$			296.405i			0.688769i
$58$		0			0
$59$	143.000			0.315543			0.157771	−	0.987476i	$-0.449569\pi$
$59$	143.000			0.315543			0.157771	+	0.987476i	$0.449569\pi$
$60$		0			0
$61$	676.000			1.41890			0.709450	−	0.704756i	$-0.248943\pi$
$61$	676.000			1.41890			0.709450	+	0.704756i	$0.248943\pi$
$62$		0			0
$63$			69.7424i			0.139472i
$64$		0			0
$65$	760.000	−	174.356i	1.45025	−	0.332711i
$66$		0			0
$67$		−	527.427i		−	0.961723i	−0.876797	−	0.480861i	$-0.840324\pi$
$67$		−	527.427i		−	0.961723i	0.876797	−	0.480861i	$-0.159676\pi$
$68$		0			0
$69$	−513.000			−0.895043
$70$		0			0
$71$	1035.00			1.73003			0.865013	−	0.501749i	$-0.167310\pi$
$71$	1035.00			1.73003			0.865013	+	0.501749i	$0.167310\pi$
$72$		0			0
$73$			331.276i			0.531136i	0.964092	+	0.265568i	$0.0855596\pi$
$73$			331.276i			0.531136i	−0.964092	+	0.265568i	$0.914440\pi$
$74$		0			0
$75$	−237.500	−	490.376i	−0.365655	−	0.754983i
$76$		0			0
$77$		−	95.8958i		−	0.141926i
$78$		0			0
$79$	−218.000			−0.310467			−0.155234	−	0.987878i	$-0.549613\pi$
$79$	−218.000			−0.310467			−0.155234	+	0.987878i	$0.549613\pi$
$80$		0			0
$81$	−449.000			−0.615912
$82$		0			0
$83$		−	758.448i		−	1.00302i	−0.865152	−	0.501509i	$-0.832778\pi$
$83$		−	758.448i		−	1.00302i	0.865152	−	0.501509i	$-0.167222\pi$
$84$		0			0
$85$	285.000	−	65.3835i	0.363678	−	0.0834333i
$86$		0			0
$87$		−	1133.31i		−	1.39660i
$88$		0			0
$89$	−1279.00			−1.52330			−0.761650	−	0.647988i	$-0.775610\pi$
$89$	−1279.00			−1.52330			−0.761650	+	0.647988i	$0.775610\pi$
$90$		0			0
$91$	−608.000			−0.700393
$92$		0			0
$93$			762.807i			0.850532i
$94$		0			0
$95$	−170.000	−	741.013i	−0.183596	−	0.800277i
$96$		0			0
$97$		−	771.525i		−	0.807593i	−0.914849	−	0.403796i	$-0.867690\pi$
$97$		−	771.525i		−	0.807593i	0.914849	−	0.403796i	$-0.132310\pi$
$98$		0			0
$99$	−88.0000			−0.0893367
$100$		0			0
$101$	638.000			0.628548			0.314274	−	0.949332i	$-0.398239\pi$
$101$	638.000			0.628548			0.314274	+	0.949332i	$0.398239\pi$
$102$		0			0
$103$		−	531.786i		−	0.508722i	−0.967109	−	0.254361i	$-0.918135\pi$
$103$		−	531.786i		−	0.508722i	0.967109	−	0.254361i	$-0.0818652\pi$
$104$		0			0
$105$	95.0000	+	414.095i	0.0882957	+	0.384872i
$106$		0			0
$107$		−	61.0246i		−	0.0551352i	−0.999620	−	0.0275676i	$-0.991224\pi$
$107$		−	61.0246i		−	0.0551352i	0.999620	−	0.0275676i	$-0.00877616\pi$
$108$		0			0
$109$	142.000			0.124781			0.0623905	−	0.998052i	$-0.480128\pi$
$109$	142.000			0.124781			0.0623905	+	0.998052i	$0.480128\pi$
$110$		0			0
$111$	−741.000			−0.633627
$112$		0			0
$113$		−	2332.01i		−	1.94139i	−0.240314	−	0.970695i	$-0.577250\pi$
$113$		−	2332.01i		−	1.94139i	0.240314	−	0.970695i	$-0.422750\pi$
$114$		0			0
$115$	1282.50	−	294.226i	1.03995	−	0.238580i
$116$		0			0
$117$			557.939i			0.440867i
$118$		0			0
$119$	−228.000			−0.175636
$120$		0			0
$121$	121.000			0.0909091
$122$		0			0
$123$		−	1656.38i		−	1.21423i
$124$		0			0
$125$	875.000	+	1089.72i	0.626099	+	0.779744i
$126$		0			0
$127$			976.393i			0.682212i	0.940025	+	0.341106i	$0.110801\pi$
$127$			976.393i			0.682212i	−0.940025	+	0.341106i	$0.889199\pi$
$128$		0			0
$129$	−1330.00			−0.907752
$130$		0			0
$131$	774.000			0.516219			0.258110	−	0.966116i	$-0.416900\pi$
$131$	774.000			0.516219			0.258110	+	0.966116i	$0.416900\pi$
$132$		0			0
$133$			592.810i			0.386490i
$134$		0			0
$135$	1662.50	−	381.404i	1.05989	−	0.243156i
$136$		0			0
$137$			39.2301i			0.0244646i	0.999925	+	0.0122323i	$0.00389376\pi$
$137$			39.2301i			0.0244646i	−0.999925	+	0.0122323i	$0.996106\pi$
$138$		0			0
$139$	−2986.00			−1.82208			−0.911040	−	0.412317i	$-0.864720\pi$
$139$	−2986.00			−1.82208			−0.911040	+	0.412317i	$0.864720\pi$
$140$		0			0
$141$	1330.00			0.794370
$142$		0			0
$143$		−	767.166i		−	0.448627i
$144$		0			0
$145$	650.000	+	2833.28i	0.372273	+	1.62270i
$146$		0			0
$147$			1163.83i			0.652998i
$148$		0			0
$149$	1546.00			0.850022			0.425011	−	0.905188i	$-0.360270\pi$
$149$	1546.00			0.850022			0.425011	+	0.905188i	$0.360270\pi$
$150$		0			0
$151$	3150.00			1.69764			0.848819	−	0.528683i	$-0.177314\pi$
$151$	3150.00			1.69764			0.848819	+	0.528683i	$0.177314\pi$
$152$		0			0
$153$			209.227i			0.110556i
$154$		0			0
$155$	−437.500	−	1907.02i	−0.226715	−	0.988228i
$156$		0			0
$157$			501.273i			0.254815i	0.991850	+	0.127408i	$0.0406656\pi$
$157$			501.273i			0.254815i	−0.991850	+	0.127408i	$0.959334\pi$
$158$		0			0
$159$	−1976.00			−0.985579
$160$		0			0
$161$	−1026.00			−0.502237
$162$		0			0
$163$		−	932.804i		−	0.448239i	−0.974562	−	0.224119i	$-0.928049\pi$
$163$		−	932.804i		−	0.448239i	0.974562	−	0.224119i	$-0.0719505\pi$
$164$		0			0
$165$	−522.500	+	119.870i	−0.246525	+	0.0565566i
$166$		0			0
$167$			1952.79i			0.904857i	0.891801	+	0.452429i	$0.149442\pi$
$167$			1952.79i			0.904857i	−0.891801	+	0.452429i	$0.850558\pi$
$168$		0			0
$169$	−2667.00			−1.21393
$170$		0			0
$171$	544.000			0.243279
$172$		0			0
$173$			2345.09i			1.03060i	0.857010	+	0.515300i	$0.172319\pi$
$173$			2345.09i			1.03060i	−0.857010	+	0.515300i	$0.827681\pi$
$174$		0			0
$175$	−475.000	−	980.752i	−0.205181	−	0.423645i
$176$		0			0
$177$			623.323i			0.264699i
$178$		0			0
$179$	−699.000			−0.291875			−0.145938	−	0.989294i	$-0.546620\pi$
$179$	−699.000			−0.291875			−0.145938	+	0.989294i	$0.546620\pi$
$180$		0			0
$181$	−2603.00			−1.06895			−0.534474	−	0.845185i	$-0.679490\pi$
$181$	−2603.00			−1.06895			−0.534474	+	0.845185i	$0.679490\pi$
$182$		0			0
$183$			2946.62i			1.19027i
$184$		0			0
$185$	1852.50	−	424.993i	0.736208	−	0.168898i
$186$		0			0
$187$		−	287.687i		−	0.112502i
$188$		0			0
$189$	−1330.00			−0.511869
$190$		0			0
$191$	1329.00			0.503472			0.251736	−	0.967796i	$-0.418999\pi$
$191$	1329.00			0.503472			0.251736	+	0.967796i	$0.418999\pi$
$192$		0			0
$193$			1394.85i			0.520225i	0.965578	+	0.260112i	$0.0837596\pi$
$193$			1394.85i			0.520225i	−0.965578	+	0.260112i	$0.916240\pi$
$194$		0			0
$195$	760.000	+	3312.76i	0.279101	+	1.21657i
$196$		0			0
$197$		−	2327.65i		−	0.841819i	−0.907103	−	0.420909i	$-0.861711\pi$
$197$		−	2327.65i		−	0.841819i	0.907103	−	0.420909i	$-0.138289\pi$
$198$		0			0
$199$	8.00000			0.00284977			0.00142489	−	0.999999i	$-0.499546\pi$
$199$	8.00000			0.00284977			0.00142489	+	0.999999i	$0.499546\pi$
$200$		0			0
$201$	2299.00			0.806761
$202$		0			0
$203$		−	2266.63i		−	0.783675i
$204$		0			0
$205$	950.000	+	4140.95i	0.323663	+	1.41081i
$206$		0			0
$207$			941.522i			0.316137i
$208$		0			0
$209$	−748.000			−0.247561
$210$		0			0
$211$	−2840.00			−0.926605			−0.463303	−	0.886200i	$-0.653336\pi$
$211$	−2840.00			−0.926605			−0.463303	+	0.886200i	$0.653336\pi$
$212$		0			0
$213$			4511.46i			1.45127i
$214$		0			0
$215$	3325.00	−	762.807i	1.05471	−	0.241968i
$216$		0			0
$217$			1525.61i			0.477260i
$218$		0			0
$219$	−1444.00			−0.445555
$220$		0			0
$221$	−1824.00			−0.555183
$222$		0			0
$223$		−	4154.03i		−	1.24742i	−0.781656	−	0.623710i	$-0.785625\pi$
$223$		−	4154.03i		−	1.24742i	0.781656	−	0.623710i	$-0.214375\pi$
$224$		0			0
$225$	−900.000	+	435.890i	−0.266667	+	0.129153i
$226$		0			0
$227$			2467.14i			0.721364i	0.932689	+	0.360682i	$0.117456\pi$
$227$			2467.14i			0.721364i	−0.932689	+	0.360682i	$0.882544\pi$
$228$		0			0
$229$	−5813.00			−1.67744			−0.838720	−	0.544563i	$-0.816696\pi$
$229$	−5813.00			−1.67744			−0.838720	+	0.544563i	$0.816696\pi$
$230$		0			0
$231$	418.000			0.119058
$232$		0			0
$233$		−	2022.53i		−	0.568671i	−0.958725	−	0.284335i	$-0.908227\pi$
$233$		−	2022.53i		−	0.568671i	0.958725	−	0.284335i	$-0.0917729\pi$
$234$		0			0
$235$	−3325.00	+	762.807i	−0.922975	+	0.211745i
$236$		0			0
$237$		−	950.240i		−	0.260442i
$238$		0			0
$239$	246.000			0.0665792			0.0332896	−	0.999446i	$-0.489402\pi$
$239$	246.000			0.0665792			0.0332896	+	0.999446i	$0.489402\pi$
$240$		0			0
$241$	3388.00			0.905561			0.452781	−	0.891622i	$-0.350432\pi$
$241$	3388.00			0.905561			0.452781	+	0.891622i	$0.350432\pi$
$242$		0			0
$243$			2162.01i			0.570754i
$244$		0			0
$245$	−667.500	−	2909.57i	−0.174061	−	0.758715i
$246$		0			0
$247$			4742.48i			1.22169i
$248$		0			0
$249$	3306.00			0.841403
$250$		0			0
$251$	1091.00			0.274356			0.137178	−	0.990546i	$-0.456197\pi$
$251$	1091.00			0.274356			0.137178	+	0.990546i	$0.456197\pi$
$252$		0			0
$253$		−	1294.59i		−	0.321701i
$254$		0			0
$255$	285.000	+	1242.29i	0.0699898	+	0.305078i
$256$		0			0
$257$		−	4132.24i		−	1.00296i	−0.865168	−	0.501482i	$-0.832788\pi$
$257$		−	4132.24i		−	1.00296i	0.865168	−	0.501482i	$-0.167212\pi$
$258$		0			0
$259$	−1482.00			−0.355548
$260$		0			0
$261$	−2080.00			−0.493290
$262$		0			0
$263$		−	6982.96i		−	1.63721i	−0.574353	−	0.818607i	$-0.694746\pi$
$263$		−	6982.96i		−	1.63721i	0.574353	−	0.818607i	$-0.305254\pi$
$264$		0			0
$265$	4940.00	−	1133.31i	1.14514	−	0.262713i
$266$		0			0
$267$		−	5575.03i		−	1.27785i
$268$		0			0
$269$	314.000			0.0711707			0.0355853	−	0.999367i	$-0.488670\pi$
$269$	314.000			0.0711707			0.0355853	+	0.999367i	$0.488670\pi$
$270$		0			0
$271$	3180.00			0.712809			0.356405	−	0.934332i	$-0.384003\pi$
$271$	3180.00			0.712809			0.356405	+	0.934332i	$0.384003\pi$
$272$		0			0
$273$		−	2650.21i		−	0.587539i
$274$		0			0
$275$	1237.50	−	599.349i	0.271360	−	0.131426i
$276$		0			0
$277$			3879.42i			0.841487i	0.907180	+	0.420743i	$0.138231\pi$
$277$			3879.42i			0.841487i	−0.907180	+	0.420743i	$0.861769\pi$
$278$		0			0
$279$	1400.00			0.300415
$280$		0			0
$281$	−4218.00			−0.895462			−0.447731	−	0.894168i	$-0.647768\pi$
$281$	−4218.00			−0.895462			−0.447731	+	0.894168i	$0.647768\pi$
$282$		0			0
$283$			8351.65i			1.75425i	0.480258	+	0.877127i	$0.340543\pi$
$283$			8351.65i			1.75425i	−0.480258	+	0.877127i	$0.659457\pi$
$284$		0			0
$285$	3230.00	−	741.013i	0.671329	−	0.154013i
$286$		0			0
$287$		−	3312.76i		−	0.681346i
$288$		0			0
$289$	4229.00			0.860778
$290$		0			0
$291$	3363.00			0.677466
$292$		0			0
$293$			6363.99i			1.26890i	0.772963	+	0.634451i	$0.218774\pi$
$293$			6363.99i			1.26890i	−0.772963	+	0.634451i	$0.781226\pi$
$294$		0			0
$295$	−357.500	−	1558.31i	−0.0705575	−	0.307553i
$296$		0			0
$297$		−	1678.18i		−	0.327871i
$298$		0			0
$299$	−8208.00			−1.58756
$300$		0			0
$301$	−2660.00			−0.509368
$302$		0			0
$303$			2780.98i			0.527271i
$304$		0			0
$305$	−1690.00	−	7366.54i	−0.317276	−	1.38297i
$306$		0			0
$307$		−	6608.09i		−	1.22848i	−0.789119	−	0.614240i	$-0.789462\pi$
$307$		−	6608.09i		−	1.22848i	0.789119	−	0.614240i	$-0.210538\pi$
$308$		0			0
$309$	2318.00			0.426752
$310$		0			0
$311$	5812.00			1.05971			0.529853	−	0.848090i	$-0.322247\pi$
$311$	5812.00			1.05971			0.529853	+	0.848090i	$0.322247\pi$
$312$		0			0
$313$		−	5888.87i		−	1.06345i	−0.846918	−	0.531723i	$-0.821545\pi$
$313$		−	5888.87i		−	1.06345i	0.846918	−	0.531723i	$-0.178455\pi$
$314$		0			0
$315$	760.000	−	174.356i	0.135940	−	0.0311868i
$316$		0			0
$317$		−	8966.26i		−	1.58863i	−0.607507	−	0.794314i	$-0.707831\pi$
$317$		−	8966.26i		−	1.58863i	0.607507	−	0.794314i	$-0.292169\pi$
$318$		0			0
$319$	2860.00			0.501973
$320$		0			0
$321$	266.000			0.0462513
$322$		0			0
$323$			1778.43i			0.306361i
$324$		0			0
$325$	−3800.00	−	7846.02i	−0.648573	−	1.33913i
$326$		0			0
$327$			618.964i			0.104675i
$328$		0			0
$329$	2660.00			0.445746
$330$		0			0
$331$	8683.00			1.44188			0.720938	−	0.693000i	$-0.243711\pi$
$331$	8683.00			1.44188			0.720938	+	0.693000i	$0.243711\pi$
$332$		0			0
$333$			1359.98i			0.223803i
$334$		0			0
$335$	−5747.50	+	1318.57i	−0.937372	+	0.215048i
$336$		0			0
$337$			5152.22i			0.832817i	0.909178	+	0.416408i	$0.136711\pi$
$337$			5152.22i			0.832817i	−0.909178	+	0.416408i	$0.863289\pi$
$338$		0			0
$339$	10165.0			1.62858
$340$		0			0
$341$	−1925.00			−0.305703
$342$		0			0
$343$			5317.86i			0.837135i
$344$		0			0
$345$	1282.50	+	5590.29i	0.200138	+	0.872380i
$346$		0			0
$347$			52.3068i			0.00809215i	0.999992	+	0.00404607i	$0.00128791\pi$
$347$			52.3068i			0.00809215i	−0.999992	+	0.00404607i	$0.998712\pi$
$348$		0			0
$349$	−2126.00			−0.326081			−0.163040	−	0.986619i	$-0.552130\pi$
$349$	−2126.00			−0.326081			−0.163040	+	0.986619i	$0.552130\pi$
$350$		0			0
$351$	−10640.0			−1.61801
$352$		0			0
$353$		−	7588.84i		−	1.14423i	−0.820173	−	0.572115i	$-0.806123\pi$
$353$		−	7588.84i		−	1.14423i	0.820173	−	0.572115i	$-0.193877\pi$
$354$		0			0
$355$	−2587.50	−	11278.7i	−0.386846	−	1.68622i
$356$		0			0
$357$		−	993.829i		−	0.147336i
$358$		0			0
$359$	−4156.00			−0.610990			−0.305495	−	0.952194i	$-0.598822\pi$
$359$	−4156.00			−0.610990			−0.305495	+	0.952194i	$0.598822\pi$
$360$		0			0
$361$	−2235.00			−0.325849
$362$		0			0
$363$			527.427i			0.0762610i
$364$		0			0
$365$	3610.00	−	828.191i	0.517688	−	0.118766i
$366$		0			0
$367$			13299.0i			1.89156i	0.324809	+	0.945780i	$0.394700\pi$
$367$			13299.0i			1.89156i	−0.324809	+	0.945780i	$0.605300\pi$
$368$		0			0
$369$	−3040.00			−0.428878
$370$		0			0
$371$	−3952.00			−0.553039
$372$		0			0
$373$		−	5622.98i		−	0.780555i	−0.920697	−	0.390277i	$-0.872379\pi$
$373$		−	5622.98i		−	0.780555i	0.920697	−	0.390277i	$-0.127621\pi$
$374$		0			0
$375$	−4750.00	+	3814.04i	−0.654104	+	0.525216i
$376$		0			0
$377$		−	18133.0i		−	2.47718i
$378$		0			0
$379$	−631.000			−0.0855206			−0.0427603	−	0.999085i	$-0.513615\pi$
$379$	−631.000			−0.0855206			−0.0427603	+	0.999085i	$0.513615\pi$
$380$		0			0
$381$	−4256.00			−0.572287
$382$		0			0
$383$		−	7091.93i		−	0.946164i	−0.881019	−	0.473082i	$-0.843142\pi$
$383$		−	7091.93i		−	0.946164i	0.881019	−	0.473082i	$-0.156858\pi$
$384$		0			0
$385$	−1045.00	+	239.739i	−0.138333	+	0.0317357i
$386$		0			0
$387$			2440.98i			0.320626i
$388$		0			0
$389$	5613.00			0.731595			0.365797	−	0.930694i	$-0.380796\pi$
$389$	5613.00			0.731595			0.365797	+	0.930694i	$0.380796\pi$
$390$		0			0
$391$	−3078.00			−0.398110
$392$		0			0
$393$			3373.79i			0.433041i
$394$		0			0
$395$	545.000	+	2375.60i	0.0694226	+	0.302606i
$396$		0			0
$397$			14018.2i			1.77218i	0.463516	+	0.886088i	$0.346588\pi$
$397$			14018.2i			1.77218i	−0.463516	+	0.886088i	$0.653412\pi$
$398$		0			0
$399$	−2584.00			−0.324215
$400$		0			0
$401$	−162.000			−0.0201743			−0.0100871	−	0.999949i	$-0.503211\pi$
$401$	−162.000			−0.0201743			−0.0100871	+	0.999949i	$0.503211\pi$
$402$		0			0
$403$			12204.9i			1.50861i
$404$		0			0
$405$	1122.50	+	4892.86i	0.137722	+	0.600317i
$406$		0			0
$407$		−	1869.97i		−	0.227742i
$408$		0			0
$409$	14142.0			1.70972			0.854862	−	0.518856i	$-0.173642\pi$
$409$	14142.0			1.70972			0.854862	+	0.518856i	$0.173642\pi$
$410$		0			0
$411$	−171.000			−0.0205226
$412$		0			0
$413$			1246.65i			0.148531i
$414$		0			0
$415$	−8265.00	+	1896.12i	−0.977621	+	0.224282i
$416$		0			0
$417$		−	13015.7i		−	1.52849i
$418$		0			0
$419$	11532.0			1.34457			0.672285	−	0.740292i	$-0.265313\pi$
$419$	11532.0			1.34457			0.672285	+	0.740292i	$0.265313\pi$
$420$		0			0
$421$	−3430.00			−0.397074			−0.198537	−	0.980093i	$-0.563619\pi$
$421$	−3430.00			−0.397074			−0.198537	+	0.980093i	$0.563619\pi$
$422$		0			0
$423$		−	2440.98i		−	0.280578i
$424$		0			0
$425$	−1425.00	−	2942.26i	−0.162642	−	0.335813i
$426$		0			0
$427$			5893.23i			0.667900i
$428$		0			0
$429$	3344.00			0.376340
$430$		0			0
$431$	8658.00			0.967613			0.483806	−	0.875175i	$-0.339254\pi$
$431$	8658.00			0.967613			0.483806	+	0.875175i	$0.339254\pi$
$432$		0			0
$433$		−	745.372i		−	0.0827258i	−0.999144	−	0.0413629i	$-0.986830\pi$
$433$		−	745.372i		−	0.0827258i	0.999144	−	0.0413629i	$-0.0131700\pi$
$434$		0			0
$435$	−12350.0	+	2833.28i	−1.36123	+	0.312289i
$436$		0			0
$437$			8002.94i			0.876047i
$438$		0			0
$439$	−4532.00			−0.492712			−0.246356	−	0.969179i	$-0.579233\pi$
$439$	−4532.00			−0.492712			−0.246356	+	0.969179i	$0.579233\pi$
$440$		0			0
$441$	2136.00			0.230645
$442$		0			0
$443$			4310.95i			0.462346i	0.972913	+	0.231173i	$0.0742564\pi$
$443$			4310.95i			0.462346i	−0.972913	+	0.231173i	$0.925744\pi$
$444$		0			0
$445$	3197.50	+	13937.6i	0.340620	+	1.48473i
$446$		0			0
$447$			6738.86i			0.713058i
$448$		0			0
$449$	2333.00			0.245214			0.122607	−	0.992455i	$-0.460875\pi$
$449$	2333.00			0.245214			0.122607	+	0.992455i	$0.460875\pi$
$450$		0			0
$451$	4180.00			0.436427
$452$		0			0
$453$			13730.5i			1.42410i
$454$		0			0
$455$	1520.00	+	6625.53i	0.156613	+	0.682658i
$456$		0			0
$457$		−	6921.93i		−	0.708521i	−0.935147	−	0.354261i	$-0.884733\pi$
$457$		−	6921.93i		−	0.708521i	0.935147	−	0.354261i	$-0.115267\pi$
$458$		0			0
$459$	−3990.00			−0.405746
$460$		0			0
$461$	5332.00			0.538690			0.269345	−	0.963044i	$-0.413193\pi$
$461$	5332.00			0.538690			0.269345	+	0.963044i	$0.413193\pi$
$462$		0			0
$463$		−	9314.97i		−	0.934996i	−0.883994	−	0.467498i	$-0.845155\pi$
$463$		−	9314.97i		−	0.934996i	0.883994	−	0.467498i	$-0.154845\pi$
$464$		0			0
$465$	8312.50	−	1907.02i	0.828996	−	0.190185i
$466$		0			0
$467$			17623.0i			1.74625i	0.487501	+	0.873123i	$0.337909\pi$
$467$			17623.0i			1.74625i	−0.487501	+	0.873123i	$0.662091\pi$
$468$		0			0
$469$	4598.00			0.452699
$470$		0			0
$471$	−2185.00			−0.213757
$472$		0			0
$473$		−	3356.35i		−	0.326269i
$474$		0			0
$475$	−7650.00	+	3705.06i	−0.738960	+	0.357895i
$476$		0			0
$477$			3626.60i			0.348115i
$478$		0			0
$479$	13748.0			1.31140			0.655702	−	0.755020i	$-0.272373\pi$
$479$	13748.0			1.31140			0.655702	+	0.755020i	$0.272373\pi$
$480$		0			0
$481$	−11856.0			−1.12388
$482$		0			0
$483$		−	4472.23i		−	0.421312i
$484$		0			0
$485$	−8407.50	+	1928.81i	−0.787144	+	0.180583i
$486$		0			0
$487$		−	4101.72i		−	0.381657i	−0.981623	−	0.190828i	$-0.938883\pi$
$487$		−	4101.72i		−	0.381657i	0.981623	−	0.190828i	$-0.0611174\pi$
$488$		0			0
$489$	4066.00			0.376014
$490$		0			0
$491$	−4016.00			−0.369123			−0.184562	−	0.982821i	$-0.559087\pi$
$491$	−4016.00			−0.369123			−0.184562	+	0.982821i	$0.559087\pi$
$492$		0			0
$493$		−	6799.88i		−	0.621199i
$494$		0			0
$495$	220.000	+	958.958i	0.0199763	+	0.0870746i
$496$		0			0
$497$			9022.92i			0.814353i
$498$		0			0
$499$	14236.0			1.27714			0.638568	−	0.769565i	$-0.279527\pi$
$499$	14236.0			1.27714			0.638568	+	0.769565i	$0.279527\pi$
$500$		0			0
$501$	−8512.00			−0.759058
$502$		0			0
$503$			18089.4i			1.60351i	0.597650	+	0.801757i	$0.296101\pi$
$503$			18089.4i			1.60351i	−0.597650	+	0.801757i	$0.703899\pi$
$504$		0			0
$505$	−1595.00	−	6952.44i	−0.140548	−	0.612633i
$506$		0			0
$507$		−	11625.2i		−	1.01833i
$508$		0			0
$509$	8379.00			0.729652			0.364826	−	0.931076i	$-0.381129\pi$
$509$	8379.00			0.729652			0.364826	+	0.931076i	$0.381129\pi$
$510$		0			0
$511$	−2888.00			−0.250015
$512$		0			0
$513$			10374.2i			0.892848i
$514$		0			0
$515$	−5795.00	+	1329.46i	−0.495841	+	0.113754i
$516$		0			0
$517$			3356.35i			0.285517i
$518$		0			0
$519$	−10222.0			−0.864539
$520$		0			0
$521$	−20277.0			−1.70509			−0.852545	−	0.522654i	$-0.824942\pi$
$521$	−20277.0			−1.70509			−0.852545	+	0.522654i	$0.824942\pi$
$522$		0			0
$523$			12152.6i			1.01605i	0.861341	+	0.508027i	$0.169625\pi$
$523$			12152.6i			1.01605i	−0.861341	+	0.508027i	$0.830375\pi$
$524$		0			0
$525$	4275.00	−	2070.48i	0.355383	−	0.172120i
$526$		0			0
$527$			4576.84i			0.378312i
$528$		0			0
$529$	−1684.00			−0.138407
$530$		0			0
$531$	1144.00			0.0934941
$532$		0			0
$533$		−	26502.1i		−	2.15372i
$534$		0			0
$535$	−665.000	+	152.561i	−0.0537392	+	0.0123286i
$536$		0			0
$537$		−	3046.87i		−	0.244846i
$538$		0			0
$539$	−2937.00			−0.234704
$540$		0			0
$541$	−4796.00			−0.381139			−0.190569	−	0.981674i	$-0.561033\pi$
$541$	−4796.00			−0.381139			−0.190569	+	0.981674i	$0.561033\pi$
$542$		0			0
$543$		−	11346.2i		−	0.896708i
$544$		0			0
$545$	−355.000	−	1547.41i	−0.0279019	−	0.121622i
$546$		0			0
$547$		−	16790.5i		−	1.31245i	−0.754566	−	0.656224i	$-0.772153\pi$
$547$		−	16790.5i		−	1.31245i	0.754566	−	0.656224i	$-0.227847\pi$
$548$		0			0
$549$	5408.00			0.420415
$550$		0			0
$551$	−17680.0			−1.36696
$552$		0			0
$553$		−	1900.48i		−	0.146142i
$554$		0			0
$555$	1852.50	+	8074.86i	0.141683	+	0.617583i
$556$		0			0
$557$			9859.83i			0.750044i	0.927016	+	0.375022i	$0.122365\pi$
$557$			9859.83i			0.750044i	−0.927016	+	0.375022i	$0.877635\pi$
$558$		0			0
$559$	−21280.0			−1.61010
$560$		0			0
$561$	1254.00			0.0943742
$562$		0			0
$563$		−	10095.2i		−	0.755706i	−0.925866	−	0.377853i	$-0.876662\pi$
$563$		−	10095.2i		−	0.755706i	0.925866	−	0.377853i	$-0.123338\pi$
$564$		0			0
$565$	−25412.5	+	5830.03i	−1.89223	+	0.434108i
$566$		0			0
$567$		−	3914.29i		−	0.289920i
$568$		0			0
$569$	12240.0			0.901806			0.450903	−	0.892573i	$-0.351102\pi$
$569$	12240.0			0.901806			0.450903	+	0.892573i	$0.351102\pi$
$570$		0			0
$571$	21224.0			1.55551			0.777755	−	0.628567i	$-0.216358\pi$
$571$	21224.0			1.55551			0.777755	+	0.628567i	$0.216358\pi$
$572$		0			0
$573$			5792.98i			0.422347i
$574$		0			0
$575$	−6412.50	−	13240.2i	−0.465078	−	0.960265i
$576$		0			0
$577$		−	972.034i		−	0.0701323i	−0.999385	−	0.0350661i	$-0.988836\pi$
$577$		−	972.034i		−	0.0701323i	0.999385	−	0.0350661i	$-0.0111642\pi$
$578$		0			0
$579$	−6080.00			−0.436401
$580$		0			0
$581$	6612.00			0.472138
$582$		0			0
$583$		−	4986.58i		−	0.354242i
$584$		0			0
$585$	6080.00	−	1394.85i	0.429704	−	0.0985809i
$586$		0			0
$587$			7662.94i			0.538814i	0.963026	+	0.269407i	$0.0868276\pi$
$587$			7662.94i			0.538814i	−0.963026	+	0.269407i	$0.913172\pi$
$588$		0			0
$589$	11900.0			0.832480
$590$		0			0
$591$	10146.0			0.706177
$592$		0			0
$593$		−	14541.3i		−	1.00698i	−0.864001	−	0.503490i	$-0.832049\pi$
$593$		−	14541.3i		−	1.00698i	0.864001	−	0.503490i	$-0.167951\pi$
$594$		0			0
$595$	570.000	+	2484.57i	0.0392735	+	0.171189i
$596$		0			0
$597$			34.8712i			0.00239059i
$598$		0			0
$599$	20520.0			1.39971			0.699853	−	0.714286i	$-0.253249\pi$
$599$	20520.0			1.39971			0.699853	+	0.714286i	$0.253249\pi$
$600$		0			0
$601$	12726.0			0.863734			0.431867	−	0.901937i	$-0.357855\pi$
$601$	12726.0			0.863734			0.431867	+	0.901937i	$0.357855\pi$
$602$		0			0
$603$		−	4219.41i		−	0.284955i
$604$		0			0
$605$	−302.500	−	1318.57i	−0.0203279	−	0.0886072i
$606$		0			0
$607$		−	3338.92i		−	0.223266i	−0.993750	−	0.111633i	$-0.964392\pi$
$607$		−	3338.92i		−	0.223266i	0.993750	−	0.111633i	$-0.0356081\pi$
$608$		0			0
$609$	9880.00			0.657402
$610$		0			0
$611$	21280.0			1.40900
$612$		0			0
$613$			5457.34i			0.359576i	0.983705	+	0.179788i	$0.0575411\pi$
$613$			5457.34i			0.359576i	−0.983705	+	0.179788i	$0.942459\pi$
$614$		0			0
$615$	−18050.0	+	4140.95i	−1.18349	+	0.271511i
$616$		0			0
$617$		−	1272.80i		−	0.0830485i	−0.999137	−	0.0415243i	$-0.986779\pi$
$617$		−	1272.80i		−	0.0830485i	0.999137	−	0.0415243i	$-0.0132214\pi$
$618$		0			0
$619$	17307.0			1.12379			0.561896	−	0.827208i	$-0.310072\pi$
$619$	17307.0			1.12379			0.561896	+	0.827208i	$0.310072\pi$
$620$		0			0
$621$	−17955.0			−1.16024
$622$		0			0
$623$		−	11150.1i		−	0.717043i
$624$		0			0
$625$	9687.50	−	12259.4i	0.620000	−	0.784602i
$626$		0			0
$627$		−	3260.46i		−	0.207672i
$628$		0			0
$629$	−4446.00			−0.281834
$630$		0			0
$631$	24977.0			1.57578			0.787891	−	0.615814i	$-0.211173\pi$
$631$	24977.0			1.57578			0.787891	+	0.615814i	$0.211173\pi$
$632$		0			0
$633$		−	12379.3i		−	0.777302i
$634$		0			0
$635$	10640.0	−	2440.98i	0.664938	−	0.152547i
$636$		0			0
$637$			18621.2i			1.15824i
$638$		0			0
$639$	8280.00			0.512601
$640$		0			0
$641$	−23151.0			−1.42654			−0.713268	−	0.700891i	$-0.752786\pi$
$641$	−23151.0			−1.42654			−0.713268	+	0.700891i	$0.752786\pi$
$642$		0			0
$643$		−	710.501i		−	0.0435761i	−0.999763	−	0.0217880i	$-0.993064\pi$
$643$		−	710.501i		−	0.0435761i	0.999763	−	0.0217880i	$-0.00693589\pi$
$644$		0			0
$645$	3325.00	+	14493.3i	0.202979	+	0.884767i
$646$		0			0
$647$		−	3081.74i		−	0.187258i	−0.995607	−	0.0936289i	$-0.970153\pi$
$647$		−	3081.74i		−	0.187258i	0.995607	−	0.0936289i	$-0.0298467\pi$
$648$		0			0
$649$	−1573.00			−0.0951397
$650$		0			0
$651$	−6650.00			−0.400360
$652$		0			0
$653$			579.734i			0.0347423i	0.999849	+	0.0173712i	$0.00552969\pi$
$653$			579.734i			0.0347423i	−0.999849	+	0.0173712i	$0.994470\pi$
$654$		0			0
$655$	−1935.00	−	8434.47i	−0.115430	−	0.503148i
$656$		0			0
$657$			2650.21i			0.157374i
$658$		0			0
$659$	3458.00			0.204408			0.102204	−	0.994763i	$-0.467411\pi$
$659$	3458.00			0.204408			0.102204	+	0.994763i	$0.467411\pi$
$660$		0			0
$661$	−12983.0			−0.763964			−0.381982	−	0.924170i	$-0.624758\pi$
$661$	−12983.0			−0.763964			−0.381982	+	0.924170i	$0.624758\pi$
$662$		0			0
$663$		−	7950.63i		−	0.465727i
$664$		0			0
$665$	6460.00	−	1482.03i	0.376704	−	0.0864218i
$666$		0			0
$667$		−	30599.5i		−	1.77634i
$668$		0			0
$669$	18107.0			1.04642
$670$		0			0
$671$	−7436.00			−0.427815
$672$		0			0
$673$		−	31357.9i		−	1.79608i	−0.439918	−	0.898038i	$-0.644993\pi$
$673$		−	31357.9i		−	1.79608i	0.439918	−	0.898038i	$-0.355007\pi$
$674$		0			0
$675$	−8312.50	−	17163.2i	−0.473997	−	0.978682i
$676$		0			0
$677$			8796.26i			0.499361i	0.968328	+	0.249681i	$0.0803257\pi$
$677$			8796.26i			0.499361i	−0.968328	+	0.249681i	$0.919674\pi$
$678$		0			0
$679$	6726.00			0.380148
$680$		0			0
$681$	−10754.0			−0.605131
$682$		0			0
$683$			6355.27i			0.356044i	0.984027	+	0.178022i	$0.0569698\pi$
$683$			6355.27i			0.356044i	−0.984027	+	0.178022i	$0.943030\pi$
$684$		0			0
$685$	427.500	−	98.0752i	0.0238452	−	0.00547046i
$686$		0			0
$687$		−	25338.3i		−	1.40716i
$688$		0			0
$689$	−31616.0			−1.74815
$690$		0			0
$691$	11819.0			0.650674			0.325337	−	0.945598i	$-0.394522\pi$
$691$	11819.0			0.650674			0.325337	+	0.945598i	$0.394522\pi$
$692$		0			0
$693$		−	767.166i		−	0.0420523i
$694$		0			0
$695$	7465.00	+	32539.2i	0.407430	+	1.77594i
$696$		0			0
$697$		−	9938.29i		−	0.540085i
$698$		0			0
$699$	8816.00			0.477041
$700$		0			0
$701$	6978.00			0.375971			0.187985	−	0.982172i	$-0.439804\pi$
$701$	6978.00			0.375971			0.187985	+	0.982172i	$0.439804\pi$
$702$		0			0
$703$			11559.8i			0.620179i
$704$		0			0
$705$	−3325.00	−	14493.3i	−0.177627	−	0.774256i
$706$		0			0
$707$			5561.96i			0.295868i
$708$		0			0
$709$	−17947.0			−0.950654			−0.475327	−	0.879809i	$-0.657670\pi$
$709$	−17947.0			−0.950654			−0.475327	+	0.879809i	$0.657670\pi$
$710$		0			0
$711$	−1744.00			−0.0919903
$712$		0			0
$713$			20595.8i			1.08179i
$714$		0			0
$715$	−8360.00	+	1917.92i	−0.437268	+	0.100316i
$716$		0			0
$717$			1072.29i			0.0558513i
$718$		0			0
$719$	−905.000			−0.0469413			−0.0234707	−	0.999725i	$-0.507472\pi$
$719$	−905.000			−0.0469413			−0.0234707	+	0.999725i	$0.507472\pi$
$720$		0			0
$721$	4636.00			0.239464
$722$		0			0
$723$			14767.9i			0.759649i
$724$		0			0
$725$	29250.0	−	14166.4i	1.49837	−	0.725693i
$726$		0			0
$727$		−	30961.3i		−	1.57949i	−0.613435	−	0.789745i	$-0.710213\pi$
$727$		−	30961.3i		−	1.57949i	0.613435	−	0.789745i	$-0.289787\pi$
$728$		0			0
$729$	−21547.0			−1.09470
$730$		0			0
$731$	−7980.00			−0.403763
$732$		0			0
$733$			23520.6i			1.18520i	0.805496	+	0.592602i	$0.201899\pi$
$733$			23520.6i			1.18520i	−0.805496	+	0.592602i	$0.798101\pi$
$734$		0			0
$735$	12682.5	−	2909.57i	0.636464	−	0.146015i
$736$		0			0
$737$			5801.69i			0.289970i
$738$		0			0
$739$	−30654.0			−1.52588			−0.762940	−	0.646469i	$-0.776245\pi$
$739$	−30654.0			−1.52588			−0.762940	+	0.646469i	$0.776245\pi$
$740$		0			0
$741$	−20672.0			−1.02484
$742$		0			0
$743$			40154.2i			1.98266i	0.131408	+	0.991328i	$0.458050\pi$
$743$			40154.2i			1.98266i	−0.131408	+	0.991328i	$0.541950\pi$
$744$		0			0
$745$	−3865.00	−	16847.1i	−0.190071	−	0.828499i
$746$		0			0
$747$		−	6067.59i		−	0.297191i
$748$		0			0
$749$	532.000			0.0259531
$750$		0			0
$751$	19735.0			0.958909			0.479454	−	0.877567i	$-0.340835\pi$
$751$	19735.0			0.958909			0.479454	+	0.877567i	$0.340835\pi$
$752$		0			0
$753$			4755.56i			0.230149i
$754$		0			0
$755$	−7875.00	−	34326.3i	−0.379603	−	1.65465i
$756$		0			0
$757$			10583.4i			0.508138i	0.967186	+	0.254069i	$0.0817690\pi$
$757$			10583.4i			0.508138i	−0.967186	+	0.254069i	$0.918231\pi$
$758$		0			0
$759$	5643.00			0.269866
$760$		0			0
$761$	6876.00			0.327536			0.163768	−	0.986499i	$-0.447635\pi$
$761$	6876.00			0.327536			0.163768	+	0.986499i	$0.447635\pi$
$762$		0			0
$763$			1237.93i			0.0587365i
$764$		0			0
$765$	2280.00	−	523.068i	0.107756	−	0.0247210i
$766$		0			0
$767$			9973.16i			0.469505i
$768$		0			0
$769$	−16956.0			−0.795122			−0.397561	−	0.917576i	$-0.630143\pi$
$769$	−16956.0			−0.795122			−0.397561	+	0.917576i	$0.630143\pi$
$770$		0			0
$771$	18012.0			0.841357
$772$		0			0
$773$			29954.4i			1.39377i	0.717184	+	0.696884i	$0.245431\pi$
$773$			29954.4i			1.39377i	−0.717184	+	0.696884i	$0.754569\pi$
$774$		0			0
$775$	−19687.5	+	9535.09i	−0.912511	+	0.441949i
$776$		0			0
$777$		−	6459.89i		−	0.298259i
$778$		0			0
$779$	−25840.0			−1.18846
$780$		0			0
$781$	−11385.0			−0.521623
$782$		0			0
$783$		−	39666.0i		−	1.81040i
$784$		0			0
$785$	5462.50	−	1253.18i	0.248363	−	0.0569784i
$786$		0			0
$787$			1098.44i			0.0497525i	0.999691	+	0.0248763i	$0.00791918\pi$
$787$			1098.44i			0.0497525i	−0.999691	+	0.0248763i	$0.992081\pi$
$788$		0			0
$789$	30438.0			1.37341
$790$		0			0
$791$	20330.0			0.913845
$792$		0			0
$793$			47145.9i			2.11122i
$794$		0			0
$795$	4940.00	+	21533.0i	0.220382	+	0.960623i
$796$		0			0
$797$			14118.5i			0.627481i	0.949509	+	0.313740i	$0.101582\pi$
$797$			14118.5i			0.627481i	−0.949509	+	0.313740i	$0.898418\pi$
$798$		0			0
$799$	7980.00			0.353332
$800$		0			0
$801$	−10232.0			−0.451348
$802$		0			0
$803$		−	3644.04i		−	0.160144i
$804$		0			0
$805$	2565.00	+	11180.6i	0.112304	+	0.489520i
$806$		0			0
$807$			1368.69i			0.0597030i
$808$		0			0
$809$	30076.0			1.30707			0.653533	−	0.756898i	$-0.273286\pi$
$809$	30076.0			1.30707			0.653533	+	0.756898i	$0.273286\pi$
$810$		0			0
$811$	−7062.00			−0.305771			−0.152886	−	0.988244i	$-0.548857\pi$
$811$	−7062.00			−0.305771			−0.152886	+	0.988244i	$0.548857\pi$
$812$		0			0
$813$			13861.3i			0.597954i
$814$		0			0
$815$	−10165.0	+	2332.01i	−0.436889	+	0.100229i
$816$		0			0
$817$			20748.4i			0.888486i
$818$		0			0
$819$	−4864.00			−0.207524
$820$		0			0
$821$	−14090.0			−0.598958			−0.299479	−	0.954103i	$-0.596813\pi$
$821$	−14090.0			−0.598958			−0.299479	+	0.954103i	$0.596813\pi$
$822$		0			0
$823$		−	10300.1i		−	0.436255i	−0.975920	−	0.218128i	$-0.930005\pi$
$823$		−	10300.1i		−	0.436255i	0.975920	−	0.218128i	$-0.0699949\pi$
$824$		0			0
$825$	2612.50	+	5394.14i	0.110249	+	0.227636i
$826$		0			0
$827$		−	16110.5i		−	0.677408i	−0.940893	−	0.338704i	$-0.890011\pi$
$827$		−	16110.5i		−	0.677408i	0.940893	−	0.338704i	$-0.109989\pi$
$828$		0			0
$829$	14611.0			0.612136			0.306068	−	0.952010i	$-0.400986\pi$
$829$	14611.0			0.612136			0.306068	+	0.952010i	$0.400986\pi$
$830$		0			0
$831$	−16910.0			−0.705898
$832$		0			0
$833$			6982.96i			0.290450i
$834$		0			0
$835$	21280.0	−	4881.97i	0.881946	−	0.202332i
$836$		0			0
$837$			26698.3i			1.10254i
$838$		0			0
$839$	−37259.0			−1.53316			−0.766581	−	0.642147i	$-0.778044\pi$
$839$	−37259.0			−1.53316			−0.766581	+	0.642147i	$0.778044\pi$
$840$		0			0
$841$	43211.0			1.77174
$842$		0			0
$843$		−	18385.8i		−	0.751177i
$844$		0			0
$845$	6667.50	+	29063.0i	0.271443	+	1.18319i
$846$		0			0
$847$			1054.85i			0.0427924i
$848$		0			0
$849$	−36404.0			−1.47159
$850$		0			0
$851$	−20007.0			−0.805912
$852$		0			0
$853$		−	5239.40i		−	0.210309i	−0.994456	−	0.105154i	$-0.966466\pi$
$853$		−	5239.40i		−	0.210309i	0.994456	−	0.105154i	$-0.0335337\pi$
$854$		0			0
$855$	−1360.00	−	5928.10i	−0.0543989	−	0.237119i
$856$		0			0
$857$			26781.1i			1.06747i	0.845651	+	0.533736i	$0.179213\pi$
$857$			26781.1i			1.06747i	−0.845651	+	0.533736i	$0.820787\pi$
$858$		0			0
$859$	−29955.0			−1.18982			−0.594908	−	0.803794i	$-0.702811\pi$
$859$	−29955.0			−1.18982			−0.594908	+	0.803794i	$0.702811\pi$
$860$		0			0
$861$	14440.0			0.571561
$862$		0			0
$863$			14462.8i			0.570475i	0.958457	+	0.285238i	$0.0920726\pi$
$863$			14462.8i			0.570475i	−0.958457	+	0.285238i	$0.907927\pi$
$864$		0			0
$865$	25555.0	−	5862.72i	1.00450	−	0.230449i
$866$		0			0
$867$			18433.8i			0.722081i
$868$		0			0
$869$	2398.00			0.0936094
$870$		0			0
$871$	36784.0			1.43097
$872$		0			0
$873$		−	6172.20i		−	0.239287i
$874$		0			0
$875$	−9500.00	+	7628.07i	−0.367038	+	0.294715i
$876$		0			0
$877$		−	24898.0i		−	0.958662i	−0.877634	−	0.479331i	$-0.840879\pi$
$877$		−	24898.0i		−	0.958662i	0.877634	−	0.479331i	$-0.159121\pi$
$878$		0			0
$879$	−27740.0			−1.06444
$880$		0			0
$881$	−19987.0			−0.764335			−0.382168	−	0.924093i	$-0.624822\pi$
$881$	−19987.0			−0.764335			−0.382168	+	0.924093i	$0.624822\pi$
$882$		0			0
$883$		−	5466.06i		−	0.208321i	−0.994560	−	0.104161i	$-0.966784\pi$
$883$		−	5466.06i		−	0.208321i	0.994560	−	0.104161i	$-0.0332156\pi$
$884$		0			0
$885$	6792.50	−	1558.31i	0.257997	−	0.0591886i
$886$		0			0
$887$			25586.7i			0.968567i	0.874911	+	0.484283i	$0.160920\pi$
$887$			25586.7i			0.968567i	−0.874911	+	0.484283i	$0.839080\pi$
$888$		0			0
$889$	−8512.00			−0.321129
$890$		0			0
$891$	4939.00			0.185705
$892$		0			0
$893$		−	20748.4i		−	0.777511i
$894$		0			0
$895$	1747.50	+	7617.18i	0.0652653	+	0.284485i
$896$		0			0
$897$		−	35777.8i		−	1.33176i
$898$		0			0
$899$	−45500.0			−1.68800
$900$		0			0
$901$	−11856.0			−0.438380
$902$		0			0
$903$		−	11594.7i		−	0.427294i
$904$		0			0
$905$	6507.50	+	28365.5i	0.239024	+	1.04188i
$906$		0			0
$907$		−	1368.69i		−	0.0501067i	−0.999686	−	0.0250533i	$-0.992024\pi$
$907$		−	1368.69i		−	0.0501067i	0.999686	−	0.0250533i	$-0.00797556\pi$
$908$		0			0
$909$	5104.00			0.186237
$910$		0			0
$911$	−20068.0			−0.729838			−0.364919	−	0.931039i	$-0.618903\pi$
$911$	−20068.0			−0.729838			−0.364919	+	0.931039i	$0.618903\pi$
$912$		0			0
$913$			8342.93i			0.302421i
$914$		0			0
$915$	32110.0	−	7366.54i	1.16014	−	0.266153i
$916$		0			0
$917$			6747.58i			0.242993i
$918$		0			0
$919$	10946.0			0.392900			0.196450	−	0.980514i	$-0.437059\pi$
$919$	10946.0			0.392900			0.196450	+	0.980514i	$0.437059\pi$
$920$		0			0
$921$	28804.0			1.03054
$922$		0			0
$923$			72183.4i			2.57415i
$924$		0			0
$925$	−9262.50	−	19124.7i	−0.329242	−	0.679800i
$926$		0			0
$927$		−	4254.29i		−	0.150733i
$928$		0			0
$929$	33338.0			1.17738			0.588689	−	0.808360i	$-0.299644\pi$
$929$	33338.0			1.17738			0.588689	+	0.808360i	$0.299644\pi$
$930$		0			0
$931$	18156.0			0.639139
$932$		0			0
$933$			25333.9i			0.888955i
$934$		0			0
$935$	−3135.00	+	719.218i	−0.109653	+	0.0251561i
$936$		0			0
$937$			51487.3i			1.79511i	0.440903	+	0.897555i	$0.354658\pi$
$937$			51487.3i			1.79511i	−0.440903	+	0.897555i	$0.645342\pi$
$938$		0			0
$939$	25669.0			0.892094
$940$		0			0
$941$	−21590.0			−0.747942			−0.373971	−	0.927440i	$-0.622004\pi$
$941$	−21590.0			−0.747942			−0.373971	+	0.927440i	$0.622004\pi$
$942$		0			0
$943$		−	44722.3i		−	1.54439i
$944$		0			0
$945$	3325.00	+	14493.3i	0.114457	+	0.498908i
$946$		0			0
$947$		−	23420.4i		−	0.803653i	−0.915716	−	0.401827i	$-0.868375\pi$
$947$		−	23420.4i		−	0.803653i	0.915716	−	0.401827i	$-0.131625\pi$
$948$		0			0
$949$	−23104.0			−0.790292
$950$		0			0
$951$	39083.0			1.33265
$952$		0			0
$953$		−	22221.7i		−	0.755331i	−0.925942	−	0.377665i	$-0.876727\pi$
$953$		−	22221.7i		−	0.755331i	0.925942	−	0.377665i	$-0.123273\pi$
$954$		0			0
$955$	−3322.50	−	14482.4i	−0.112580	−	0.490723i
$956$		0			0
$957$			12466.5i			0.421090i
$958$		0			0
$959$	−342.000			−0.0115159
$960$		0			0
$961$	834.000			0.0279950
$962$		0			0
$963$		−	488.197i		−	0.0163364i
$964$		0			0
$965$	15200.0	−	3487.12i	0.507052	−	0.116326i
$966$		0			0
$967$			15883.8i			0.528221i	0.964492	+	0.264110i	$0.0850783\pi$
$967$			15883.8i			0.528221i	−0.964492	+	0.264110i	$0.914922\pi$
$968$		0			0
$969$	−7752.00			−0.256997
$970$		0			0
$971$	−13965.0			−0.461543			−0.230771	−	0.973008i	$-0.574125\pi$
$971$	−13965.0			−0.461543			−0.230771	+	0.973008i	$0.574125\pi$
$972$		0			0
$973$		−	26031.3i		−	0.857684i
$974$		0			0
$975$	34200.0	−	16563.8i	1.12336	−	0.544068i
$976$		0			0
$977$		−	36418.6i		−	1.19256i	−0.802775	−	0.596282i	$-0.796644\pi$
$977$		−	36418.6i		−	1.19256i	0.802775	−	0.596282i	$-0.203356\pi$
$978$		0			0
$979$	14069.0			0.459292
$980$		0			0
$981$	1136.00			0.0369722
$982$		0			0
$983$		−	49844.0i		−	1.61727i	−0.588310	−	0.808635i	$-0.700207\pi$
$983$		−	49844.0i		−	1.61727i	0.588310	−	0.808635i	$-0.299793\pi$
$984$		0			0
$985$	−25365.0	+	5819.13i	−0.820504	+	0.188236i
$986$		0			0
$987$			11594.7i			0.373923i
$988$		0			0
$989$	−35910.0			−1.15457
$990$		0			0
$991$	55024.0			1.76377			0.881884	−	0.471466i	$-0.156275\pi$
$991$	55024.0			1.76377			0.881884	+	0.471466i	$0.156275\pi$
$992$		0			0
$993$			37848.3i			1.20955i
$994$		0			0
$995$	−20.0000	−	87.1780i	−0.000637229	−	0.00277762i
$996$		0			0
$997$		−	51740.1i		−	1.64356i	−0.569807	−	0.821779i	$-0.692982\pi$
$997$		−	51740.1i		−	1.64356i	0.569807	−	0.821779i	$-0.307018\pi$
$998$		0			0
$999$	−25935.0			−0.821368

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.4.b.a.89.2	yes	2
3.2	odd	2		1980.4.c.a.1189.2		2
4.3	odd	2		880.4.b.b.529.1		2
5.2	odd	4		1100.4.a.f.1.2		2
5.3	odd	4		1100.4.a.f.1.1		2
5.4	even	2	inner	220.4.b.a.89.1	✓	2
15.14	odd	2		1980.4.c.a.1189.1		2
20.19	odd	2		880.4.b.b.529.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.b.a.89.1	✓	2	5.4	even	2	inner
220.4.b.a.89.2	yes	2	1.1	even	1	trivial
880.4.b.b.529.1		2	4.3	odd	2
880.4.b.b.529.2		2	20.19	odd	2
1100.4.a.f.1.1		2	5.3	odd	4
1100.4.a.f.1.2		2	5.2	odd	4
1980.4.c.a.1189.1		2	15.14	odd	2
1980.4.c.a.1189.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	220.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.35890i q^{3} +(-2.50000 - 10.8972i) q^{5} +8.71780i q^{7} +8.00000 q^{9} -11.0000 q^{11} +69.7424i q^{13} +(47.5000 - 10.8972i) q^{15} +26.1534i q^{17} +68.0000 q^{19} -38.0000 q^{21} +117.690i q^{23} +(-112.500 + 54.4862i) q^{25} +152.561i q^{27} -260.000 q^{29} +175.000 q^{31} -47.9479i q^{33} +(95.0000 - 21.7945i) q^{35} +169.997i q^{37} -304.000 q^{39} -380.000 q^{41} +305.123i q^{43} +(-20.0000 - 87.1780i) q^{45} -305.123i q^{47} +267.000 q^{49} -114.000 q^{51} +453.325i q^{53} +(27.5000 + 119.870i) q^{55} +296.405i q^{57} +143.000 q^{59} +676.000 q^{61} +69.7424i q^{63} +(760.000 - 174.356i) q^{65} -527.427i q^{67} -513.000 q^{69} +1035.00 q^{71} +331.276i q^{73} +(-237.500 - 490.376i) q^{75} -95.8958i q^{77} -218.000 q^{79} -449.000 q^{81} -758.448i q^{83} +(285.000 - 65.3835i) q^{85} -1133.31i q^{87} -1279.00 q^{89} -608.000 q^{91} +762.807i q^{93} +(-170.000 - 741.013i) q^{95} -771.525i q^{97} -88.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{5} + 16 q^{9} - 22 q^{11} + 95 q^{15} + 136 q^{19} - 76 q^{21} - 225 q^{25} - 520 q^{29} + 350 q^{31} + 190 q^{35} - 608 q^{39} - 760 q^{41} - 40 q^{45} + 534 q^{49} - 228 q^{51} + 55 q^{55}+ \cdots - 176 q^{99}+O(q^{100}) \) 2 * q - 5 * q^5 + 16 * q^9 - 22 * q^11 + 95 * q^15 + 136 * q^19 - 76 * q^21 - 225 * q^25 - 520 * q^29 + 350 * q^31 + 190 * q^35 - 608 * q^39 - 760 * q^41 - 40 * q^45 + 534 * q^49 - 228 * q^51 + 55 * q^55 + 286 * q^59 + 1352 * q^61 + 1520 * q^65 - 1026 * q^69 + 2070 * q^71 - 475 * q^75 - 436 * q^79 - 898 * q^81 + 570 * q^85 - 2558 * q^89 - 1216 * q^91 - 340 * q^95 - 176 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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