LMFDB - Embedded newform 1100.2.b.e.749.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,2,Mod(749,1100)] 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("1100.749"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1100.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-10,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$8.78354422234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.1
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.2.b.e.749.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30278i q^{3} +4.30278i q^{7} -7.90833 q^{9} +1.00000 q^{11} +5.60555i q^{13} +0.697224i q^{17} +1.00000 q^{19} +14.2111 q^{21} +6.90833i q^{23} +16.2111i q^{27} +5.30278 q^{29} -5.60555 q^{31} -3.30278i q^{33} +0.394449i q^{37} +18.5139 q^{39} -6.21110 q^{41} +7.21110i q^{43} -1.60555i q^{47} -11.5139 q^{49} +2.30278 q^{51} -11.5139i q^{53} -3.30278i q^{57} -1.60555 q^{59} -0.302776 q^{61} -34.0278i q^{63} +8.00000i q^{67} +22.8167 q^{69} -4.60555 q^{71} -8.90833i q^{73} +4.30278i q^{77} -2.69722 q^{79} +29.8167 q^{81} +3.90833i q^{83} -17.5139i q^{87} +15.9083 q^{89} -24.1194 q^{91} +18.5139i q^{93} +18.1194i q^{97} -7.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{9} + 4 q^{11} + 4 q^{19} + 28 q^{21} + 14 q^{29} - 8 q^{31} + 38 q^{39} + 4 q^{41} - 10 q^{49} + 2 q^{51} + 8 q^{59} + 6 q^{61} + 48 q^{69} - 4 q^{71} - 18 q^{79} + 76 q^{81} + 42 q^{89} - 46 q^{91}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q - 10 * q^9 + 4 * q^11 + 4 * q^19 + 28 * q^21 + 14 * q^29 - 8 * q^31 + 38 * q^39 + 4 * q^41 - 10 * q^49 + 2 * q^51 + 8 * q^59 + 6 * q^61 + 48 * q^69 - 4 * q^71 - 18 * q^79 + 76 * q^81 + 42 * q^89 - 46 * q^91 - 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3.30278i	− 1.90686i	−0.301617	−	0.953429i	$-0.597526\pi$
$3$	− 3.30278i	− 1.90686i	0.301617	−	0.953429i	$-0.402474\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.30278i	1.62630i	0.582057	+	0.813148i	$0.302248\pi$
$7$	4.30278i	1.62630i	−0.582057	+	0.813148i	$0.697752\pi$
$8$	0	0
$9$	−7.90833	−2.63611
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.60555i	1.55470i	0.629068	+	0.777350i	$0.283437\pi$
$13$	5.60555i	1.55470i	−0.629068	+	0.777350i	$0.716563\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.697224i	0.169102i	0.996419	+	0.0845509i	$0.0269455\pi$
$17$	0.697224i	0.169102i	−0.996419	+	0.0845509i	$0.973054\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	14.2111	3.10112
$22$	0	0
$23$	6.90833i	1.44049i	0.693722	+	0.720243i	$0.255970\pi$
$23$	6.90833i	1.44049i	−0.693722	+	0.720243i	$0.744030\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.2111i	3.11983i
$28$	0	0
$29$	5.30278	0.984701	0.492350	−	0.870397i	$-0.336138\pi$
$29$	5.30278	0.984701	0.492350	+	0.870397i	$0.336138\pi$
$30$	0	0
$31$	−5.60555	−1.00679	−0.503393	−	0.864057i	$-0.667915\pi$
$31$	−5.60555	−1.00679	−0.503393	+	0.864057i	$0.667915\pi$
$32$	0	0
$33$	− 3.30278i	− 0.574939i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.394449i	0.0648470i	0.999474	+	0.0324235i	$0.0103225\pi$
$37$	0.394449i	0.0648470i	−0.999474	+	0.0324235i	$0.989677\pi$
$38$	0	0
$39$	18.5139	2.96459
$40$	0	0
$41$	−6.21110	−0.970011	−0.485006	−	0.874511i	$-0.661182\pi$
$41$	−6.21110	−0.970011	−0.485006	+	0.874511i	$0.661182\pi$
$42$	0	0
$43$	7.21110i	1.09968i	0.835269	+	0.549841i	$0.185312\pi$
$43$	7.21110i	1.09968i	−0.835269	+	0.549841i	$0.814688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.60555i	− 0.234194i	−0.993120	−	0.117097i	$-0.962641\pi$
$47$	− 1.60555i	− 0.234194i	0.993120	−	0.117097i	$-0.0373588\pi$
$48$	0	0
$49$	−11.5139	−1.64484
$50$	0	0
$51$	2.30278	0.322453
$52$	0	0
$53$	− 11.5139i	− 1.58155i	−0.612105	−	0.790776i	$-0.709677\pi$
$53$	− 11.5139i	− 1.58155i	0.612105	−	0.790776i	$-0.290323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.30278i	− 0.437463i
$58$	0	0
$59$	−1.60555	−0.209025	−0.104512	−	0.994524i	$-0.533328\pi$
$59$	−1.60555	−0.209025	−0.104512	+	0.994524i	$0.533328\pi$
$60$	0	0
$61$	−0.302776	−0.0387664	−0.0193832	−	0.999812i	$-0.506170\pi$
$61$	−0.302776	−0.0387664	−0.0193832	+	0.999812i	$0.506170\pi$
$62$	0	0
$63$	− 34.0278i	− 4.28709i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	22.8167	2.74680
$70$	0	0
$71$	−4.60555	−0.546578	−0.273289	−	0.961932i	$-0.588112\pi$
$71$	−4.60555	−0.546578	−0.273289	+	0.961932i	$0.588112\pi$
$72$	0	0
$73$	− 8.90833i	− 1.04264i	−0.853361	−	0.521320i	$-0.825440\pi$
$73$	− 8.90833i	− 1.04264i	0.853361	−	0.521320i	$-0.174560\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.30278i	0.490347i
$78$	0	0
$79$	−2.69722	−0.303461	−0.151731	−	0.988422i	$-0.548485\pi$
$79$	−2.69722	−0.303461	−0.151731	+	0.988422i	$0.548485\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	3.90833i	0.428995i	0.976725	+	0.214497i	$0.0688113\pi$
$83$	3.90833i	0.428995i	−0.976725	+	0.214497i	$0.931189\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 17.5139i	− 1.87768i
$88$	0	0
$89$	15.9083	1.68628	0.843140	−	0.537695i	$-0.180705\pi$
$89$	15.9083	1.68628	0.843140	+	0.537695i	$0.180705\pi$
$90$	0	0
$91$	−24.1194	−2.52840
$92$	0	0
$93$	18.5139i	1.91980i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.1194i	1.83975i	0.392212	+	0.919875i	$0.371710\pi$
$97$	18.1194i	1.83975i	−0.392212	+	0.919875i	$0.628290\pi$
$98$	0	0
$99$	−7.90833	−0.794817
$100$	0	0
$101$	11.5139	1.14567	0.572837	−	0.819669i	$-0.305843\pi$
$101$	11.5139	1.14567	0.572837	+	0.819669i	$0.305843\pi$
$102$	0	0
$103$	− 8.69722i	− 0.856963i	−0.903551	−	0.428481i	$-0.859049\pi$
$103$	− 8.69722i	− 0.856963i	0.903551	−	0.428481i	$-0.140951\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.2111i	− 1.18049i	−0.807223	−	0.590246i	$-0.799031\pi$
$107$	− 12.2111i	− 1.18049i	0.807223	−	0.590246i	$-0.200969\pi$
$108$	0	0
$109$	1.90833	0.182785	0.0913923	−	0.995815i	$-0.470868\pi$
$109$	1.90833	0.182785	0.0913923	+	0.995815i	$0.470868\pi$
$110$	0	0
$111$	1.30278	0.123654
$112$	0	0
$113$	− 1.60555i	− 0.151038i	−0.997144	−	0.0755188i	$-0.975939\pi$
$113$	− 1.60555i	− 0.151038i	0.997144	−	0.0755188i	$-0.0240613\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 44.3305i	− 4.09836i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.5139i	1.84967i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.09167i	0.363077i	0.983384	+	0.181539i	$0.0581077\pi$
$127$	4.09167i	0.363077i	−0.983384	+	0.181539i	$0.941892\pi$
$128$	0	0
$129$	23.8167	2.09694
$130$	0	0
$131$	−8.30278	−0.725417	−0.362708	−	0.931903i	$-0.618148\pi$
$131$	−8.30278	−0.725417	−0.362708	+	0.931903i	$0.618148\pi$
$132$	0	0
$133$	4.30278i	0.373098i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.90833i	0.846525i	0.906007	+	0.423263i	$0.139115\pi$
$137$	9.90833i	0.846525i	−0.906007	+	0.423263i	$0.860885\pi$
$138$	0	0
$139$	−1.78890	−0.151732	−0.0758662	−	0.997118i	$-0.524172\pi$
$139$	−1.78890	−0.151732	−0.0758662	+	0.997118i	$0.524172\pi$
$140$	0	0
$141$	−5.30278	−0.446574
$142$	0	0
$143$	5.60555i	0.468760i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	38.0278i	3.13648i
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	6.81665	0.554731	0.277366	−	0.960764i	$-0.410539\pi$
$151$	6.81665	0.554731	0.277366	+	0.960764i	$0.410539\pi$
$152$	0	0
$153$	− 5.51388i	− 0.445771i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 0.788897i	− 0.0629609i	−0.999504	−	0.0314804i	$-0.989978\pi$
$157$	− 0.788897i	− 0.0629609i	0.999504	−	0.0314804i	$-0.0100222\pi$
$158$	0	0
$159$	−38.0278	−3.01580
$160$	0	0
$161$	−29.7250	−2.34266
$162$	0	0
$163$	0.302776i	0.0237152i	0.999930	+	0.0118576i	$0.00377448\pi$
$163$	0.302776i	0.0237152i	−0.999930	+	0.0118576i	$0.996226\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.6333i	1.44189i	0.692993	+	0.720944i	$0.256292\pi$
$167$	18.6333i	1.44189i	−0.692993	+	0.720944i	$0.743708\pi$
$168$	0	0
$169$	−18.4222	−1.41709
$170$	0	0
$171$	−7.90833	−0.604765
$172$	0	0
$173$	24.2111i	1.84074i	0.391053	+	0.920368i	$0.372111\pi$
$173$	24.2111i	1.84074i	−0.391053	+	0.920368i	$0.627889\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.30278i	0.398581i
$178$	0	0
$179$	−18.9083	−1.41327	−0.706637	−	0.707576i	$-0.749789\pi$
$179$	−18.9083	−1.41327	−0.706637	+	0.707576i	$0.749789\pi$
$180$	0	0
$181$	2.90833	0.216174	0.108087	−	0.994141i	$-0.465527\pi$
$181$	2.90833	0.216174	0.108087	+	0.994141i	$0.465527\pi$
$182$	0	0
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.697224i	0.0509861i
$188$	0	0
$189$	−69.7527	−5.07377
$190$	0	0
$191$	2.09167	0.151348	0.0756741	−	0.997133i	$-0.475889\pi$
$191$	2.09167	0.151348	0.0756741	+	0.997133i	$0.475889\pi$
$192$	0	0
$193$	− 26.4222i	− 1.90191i	−0.309326	−	0.950956i	$-0.600104\pi$
$193$	− 26.4222i	− 1.90191i	0.309326	−	0.950956i	$-0.399896\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.0917i	− 0.790249i	−0.918628	−	0.395124i	$-0.870701\pi$
$197$	− 11.0917i	− 0.790249i	0.918628	−	0.395124i	$-0.129299\pi$
$198$	0	0
$199$	7.90833	0.560606	0.280303	−	0.959912i	$-0.409565\pi$
$199$	7.90833	0.560606	0.280303	+	0.959912i	$0.409565\pi$
$200$	0	0
$201$	26.4222	1.86368
$202$	0	0
$203$	22.8167i	1.60142i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 54.6333i	− 3.79728i
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−14.3944	−0.990955	−0.495477	−	0.868621i	$-0.665007\pi$
$211$	−14.3944	−0.990955	−0.495477	+	0.868621i	$0.665007\pi$
$212$	0	0
$213$	15.2111i	1.04225i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 24.1194i	− 1.63733i
$218$	0	0
$219$	−29.4222	−1.98817
$220$	0	0
$221$	−3.90833	−0.262903
$222$	0	0
$223$	− 6.39445i	− 0.428204i	−0.976811	−	0.214102i	$-0.931318\pi$
$223$	− 6.39445i	− 0.428204i	0.976811	−	0.214102i	$-0.0686825\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.9361i	1.38958i	0.719214	+	0.694788i	$0.244502\pi$
$227$	20.9361i	1.38958i	−0.719214	+	0.694788i	$0.755498\pi$
$228$	0	0
$229$	−0.880571	−0.0581897	−0.0290949	−	0.999577i	$-0.509262\pi$
$229$	−0.880571	−0.0581897	−0.0290949	+	0.999577i	$0.509262\pi$
$230$	0	0
$231$	14.2111	0.935022
$232$	0	0
$233$	24.9083i	1.63180i	0.578194	+	0.815899i	$0.303758\pi$
$233$	24.9083i	1.63180i	−0.578194	+	0.815899i	$0.696242\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.90833i	0.578658i
$238$	0	0
$239$	25.3305	1.63850	0.819248	−	0.573439i	$-0.194391\pi$
$239$	25.3305	1.63850	0.819248	+	0.573439i	$0.194391\pi$
$240$	0	0
$241$	−4.69722	−0.302575	−0.151287	−	0.988490i	$-0.548342\pi$
$241$	−4.69722	−0.302575	−0.151287	+	0.988490i	$0.548342\pi$
$242$	0	0
$243$	− 49.8444i	− 3.19752i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.60555i	0.356673i
$248$	0	0
$249$	12.9083	0.818032
$250$	0	0
$251$	16.3305	1.03077	0.515387	−	0.856958i	$-0.327648\pi$
$251$	16.3305	1.03077	0.515387	+	0.856958i	$0.327648\pi$
$252$	0	0
$253$	6.90833i	0.434323i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 0.422205i	− 0.0263364i	−0.999913	−	0.0131682i	$-0.995808\pi$
$257$	− 0.422205i	− 0.0263364i	0.999913	−	0.0131682i	$-0.00419169\pi$
$258$	0	0
$259$	−1.69722	−0.105460
$260$	0	0
$261$	−41.9361	−2.59578
$262$	0	0
$263$	− 10.8167i	− 0.666983i	−0.942753	−	0.333492i	$-0.891773\pi$
$263$	− 10.8167i	− 0.666983i	0.942753	−	0.333492i	$-0.108227\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 52.5416i	− 3.21550i
$268$	0	0
$269$	14.5139	0.884927	0.442463	−	0.896787i	$-0.354105\pi$
$269$	14.5139	0.884927	0.442463	+	0.896787i	$0.354105\pi$
$270$	0	0
$271$	−0.577795	−0.0350985	−0.0175493	−	0.999846i	$-0.505586\pi$
$271$	−0.577795	−0.0350985	−0.0175493	+	0.999846i	$0.505586\pi$
$272$	0	0
$273$	79.6611i	4.82131i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.7889i	− 0.948663i	−0.880346	−	0.474331i	$-0.842690\pi$
$277$	− 15.7889i	− 0.948663i	0.880346	−	0.474331i	$-0.157310\pi$
$278$	0	0
$279$	44.3305	2.65400
$280$	0	0
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	− 5.48612i	− 0.326116i	−0.986616	−	0.163058i	$-0.947864\pi$
$283$	− 5.48612i	− 0.326116i	0.986616	−	0.163058i	$-0.0521358\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 26.7250i	− 1.57753i
$288$	0	0
$289$	16.5139	0.971405
$290$	0	0
$291$	59.8444	3.50814
$292$	0	0
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.2111i	0.940664i
$298$	0	0
$299$	−38.7250	−2.23952
$300$	0	0
$301$	−31.0278	−1.78841
$302$	0	0
$303$	− 38.0278i	− 2.18464i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.3305i	1.38862i	0.719678	+	0.694308i	$0.244290\pi$
$307$	24.3305i	1.38862i	−0.719678	+	0.694308i	$0.755710\pi$
$308$	0	0
$309$	−28.7250	−1.63411
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	− 15.3944i	− 0.870146i	−0.900395	−	0.435073i	$-0.856723\pi$
$313$	− 15.3944i	− 0.870146i	0.900395	−	0.435073i	$-0.143277\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.3028i	− 1.30881i	−0.756142	−	0.654407i	$-0.772918\pi$
$317$	− 23.3028i	− 1.30881i	0.756142	−	0.654407i	$-0.227082\pi$
$318$	0	0
$319$	5.30278	0.296898
$320$	0	0
$321$	−40.3305	−2.25103
$322$	0	0
$323$	0.697224i	0.0387946i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6.30278i	− 0.348544i
$328$	0	0
$329$	6.90833	0.380868
$330$	0	0
$331$	−28.6333	−1.57383	−0.786914	−	0.617062i	$-0.788323\pi$
$331$	−28.6333	−1.57383	−0.786914	+	0.617062i	$0.788323\pi$
$332$	0	0
$333$	− 3.11943i	− 0.170944i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 19.2111i	− 1.04650i	−0.852181	−	0.523248i	$-0.824720\pi$
$337$	− 19.2111i	− 1.04650i	0.852181	−	0.523248i	$-0.175280\pi$
$338$	0	0
$339$	−5.30278	−0.288007
$340$	0	0
$341$	−5.60555	−0.303558
$342$	0	0
$343$	− 19.4222i	− 1.04870i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0917i	0.595432i	0.954654	+	0.297716i	$0.0962249\pi$
$347$	11.0917i	0.595432i	−0.954654	+	0.297716i	$0.903775\pi$
$348$	0	0
$349$	25.4222	1.36082	0.680410	−	0.732832i	$-0.261802\pi$
$349$	25.4222	1.36082	0.680410	+	0.732832i	$0.261802\pi$
$350$	0	0
$351$	−90.8722	−4.85040
$352$	0	0
$353$	− 12.2111i	− 0.649931i	−0.945726	−	0.324966i	$-0.894647\pi$
$353$	− 12.2111i	− 0.649931i	0.945726	−	0.324966i	$-0.105353\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.90833i	0.524404i
$358$	0	0
$359$	12.4222	0.655619	0.327809	−	0.944744i	$-0.393690\pi$
$359$	12.4222	0.655619	0.327809	+	0.944744i	$0.393690\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 3.30278i	− 0.173351i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	29.6972i	1.55018i	0.631849	+	0.775091i	$0.282296\pi$
$367$	29.6972i	1.55018i	−0.631849	+	0.775091i	$0.717704\pi$
$368$	0	0
$369$	49.1194	2.55706
$370$	0	0
$371$	49.5416	2.57207
$372$	0	0
$373$	− 22.0278i	− 1.14055i	−0.821452	−	0.570277i	$-0.806836\pi$
$373$	− 22.0278i	− 1.14055i	0.821452	−	0.570277i	$-0.193164\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	29.7250i	1.53091i
$378$	0	0
$379$	−14.2111	−0.729975	−0.364988	−	0.931012i	$-0.618927\pi$
$379$	−14.2111	−0.729975	−0.364988	+	0.931012i	$0.618927\pi$
$380$	0	0
$381$	13.5139	0.692337
$382$	0	0
$383$	− 15.2111i	− 0.777251i	−0.921396	−	0.388626i	$-0.872950\pi$
$383$	− 15.2111i	− 0.777251i	0.921396	−	0.388626i	$-0.127050\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 57.0278i	− 2.89888i
$388$	0	0
$389$	−15.6333	−0.792640	−0.396320	−	0.918112i	$-0.629713\pi$
$389$	−15.6333	−0.792640	−0.396320	+	0.918112i	$0.629713\pi$
$390$	0	0
$391$	−4.81665	−0.243589
$392$	0	0
$393$	27.4222i	1.38327i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.5139i	1.12994i	0.825112	+	0.564970i	$0.191112\pi$
$397$	22.5139i	1.12994i	−0.825112	+	0.564970i	$0.808888\pi$
$398$	0	0
$399$	14.2111	0.711445
$400$	0	0
$401$	−3.21110	−0.160355	−0.0801774	−	0.996781i	$-0.525549\pi$
$401$	−3.21110	−0.160355	−0.0801774	+	0.996781i	$0.525549\pi$
$402$	0	0
$403$	− 31.4222i	− 1.56525i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.394449i	0.0195521i
$408$	0	0
$409$	6.02776	0.298053	0.149027	−	0.988833i	$-0.452386\pi$
$409$	6.02776	0.298053	0.149027	+	0.988833i	$0.452386\pi$
$410$	0	0
$411$	32.7250	1.61420
$412$	0	0
$413$	− 6.90833i	− 0.339937i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.90833i	0.289332i
$418$	0	0
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	−16.9083	−0.824061	−0.412031	−	0.911170i	$-0.635180\pi$
$421$	−16.9083	−0.824061	−0.412031	+	0.911170i	$0.635180\pi$
$422$	0	0
$423$	12.6972i	0.617360i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.30278i	− 0.0630457i
$428$	0	0
$429$	18.5139	0.893858
$430$	0	0
$431$	18.2111	0.877198	0.438599	−	0.898683i	$-0.355475\pi$
$431$	18.2111	0.877198	0.438599	+	0.898683i	$0.355475\pi$
$432$	0	0
$433$	− 29.0000i	− 1.39365i	−0.717241	−	0.696826i	$-0.754595\pi$
$433$	− 29.0000i	− 1.39365i	0.717241	−	0.696826i	$-0.245405\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.90833i	0.330470i
$438$	0	0
$439$	−17.9083	−0.854718	−0.427359	−	0.904082i	$-0.640556\pi$
$439$	−17.9083	−0.854718	−0.427359	+	0.904082i	$0.640556\pi$
$440$	0	0
$441$	91.0555	4.33598
$442$	0	0
$443$	− 1.39445i	− 0.0662523i	−0.999451	−	0.0331261i	$-0.989454\pi$
$443$	− 1.39445i	− 0.0662523i	0.999451	−	0.0331261i	$-0.0105463\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 39.6333i	− 1.87459i
$448$	0	0
$449$	12.9083	0.609182	0.304591	−	0.952483i	$-0.401480\pi$
$449$	12.9083	0.609182	0.304591	+	0.952483i	$0.401480\pi$
$450$	0	0
$451$	−6.21110	−0.292469
$452$	0	0
$453$	− 22.5139i	− 1.05779i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19.1472i	− 0.895668i	−0.894117	−	0.447834i	$-0.852196\pi$
$457$	− 19.1472i	− 0.895668i	0.894117	−	0.447834i	$-0.147804\pi$
$458$	0	0
$459$	−11.3028	−0.527568
$460$	0	0
$461$	−1.18335	−0.0551139	−0.0275570	−	0.999620i	$-0.508773\pi$
$461$	−1.18335	−0.0551139	−0.0275570	+	0.999620i	$0.508773\pi$
$462$	0	0
$463$	39.6611i	1.84321i	0.388134	+	0.921603i	$0.373120\pi$
$463$	39.6611i	1.84321i	−0.388134	+	0.921603i	$0.626880\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.0278i	− 1.48207i	−0.671468	−	0.741034i	$-0.734336\pi$
$467$	− 32.0278i	− 1.48207i	0.671468	−	0.741034i	$-0.265664\pi$
$468$	0	0
$469$	−34.4222	−1.58947
$470$	0	0
$471$	−2.60555	−0.120057
$472$	0	0
$473$	7.21110i	0.331567i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	91.0555i	4.16915i
$478$	0	0
$479$	−38.4500	−1.75682	−0.878412	−	0.477905i	$-0.841396\pi$
$479$	−38.4500	−1.75682	−0.878412	+	0.477905i	$0.841396\pi$
$480$	0	0
$481$	−2.21110	−0.100818
$482$	0	0
$483$	98.1749i	4.46711i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0.394449i	0.0178742i	0.999960	+	0.00893709i	$0.00284480\pi$
$487$	0.394449i	0.0178742i	−0.999960	+	0.00893709i	$0.997155\pi$
$488$	0	0
$489$	1.00000	0.0452216
$490$	0	0
$491$	−30.6333	−1.38246	−0.691231	−	0.722634i	$-0.742931\pi$
$491$	−30.6333	−1.38246	−0.691231	+	0.722634i	$0.742931\pi$
$492$	0	0
$493$	3.69722i	0.166515i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 19.8167i	− 0.888898i
$498$	0	0
$499$	21.0917	0.944193	0.472096	−	0.881547i	$-0.343497\pi$
$499$	21.0917	0.944193	0.472096	+	0.881547i	$0.343497\pi$
$500$	0	0
$501$	61.5416	2.74948
$502$	0	0
$503$	− 13.8167i	− 0.616054i	−0.951378	−	0.308027i	$-0.900331\pi$
$503$	− 13.8167i	− 0.616054i	0.951378	−	0.308027i	$-0.0996687\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	60.8444i	2.70220i
$508$	0	0
$509$	−9.69722	−0.429822	−0.214911	−	0.976634i	$-0.568946\pi$
$509$	−9.69722	−0.429822	−0.214911	+	0.976634i	$0.568946\pi$
$510$	0	0
$511$	38.3305	1.69564
$512$	0	0
$513$	16.2111i	0.715738i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.60555i	− 0.0706121i
$518$	0	0
$519$	79.9638	3.51002
$520$	0	0
$521$	25.3944	1.11255	0.556275	−	0.830998i	$-0.312230\pi$
$521$	25.3944	1.11255	0.556275	+	0.830998i	$0.312230\pi$
$522$	0	0
$523$	1.63331i	0.0714196i	0.999362	+	0.0357098i	$0.0113692\pi$
$523$	1.63331i	0.0714196i	−0.999362	+	0.0357098i	$0.988631\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.90833i	− 0.170249i
$528$	0	0
$529$	−24.7250	−1.07500
$530$	0	0
$531$	12.6972	0.551013
$532$	0	0
$533$	− 34.8167i	− 1.50808i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	62.4500i	2.69491i
$538$	0	0
$539$	−11.5139	−0.495938
$540$	0	0
$541$	9.33053	0.401151	0.200575	−	0.979678i	$-0.435719\pi$
$541$	9.33053	0.401151	0.200575	+	0.979678i	$0.435719\pi$
$542$	0	0
$543$	− 9.60555i	− 0.412214i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.9083i	1.40706i	0.710666	+	0.703529i	$0.248394\pi$
$547$	32.9083i	1.40706i	−0.710666	+	0.703529i	$0.751606\pi$
$548$	0	0
$549$	2.39445	0.102193
$550$	0	0
$551$	5.30278	0.225906
$552$	0	0
$553$	− 11.6056i	− 0.493518i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.18335i	− 0.304368i	−0.988352	−	0.152184i	$-0.951369\pi$
$557$	− 7.18335i	− 0.304368i	0.988352	−	0.152184i	$-0.0486307\pi$
$558$	0	0
$559$	−40.4222	−1.70968
$560$	0	0
$561$	2.30278	0.0972233
$562$	0	0
$563$	29.0917i	1.22607i	0.790057	+	0.613034i	$0.210051\pi$
$563$	29.0917i	1.22607i	−0.790057	+	0.613034i	$0.789949\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	128.294i	5.38786i
$568$	0	0
$569$	−30.3583	−1.27269	−0.636343	−	0.771406i	$-0.719554\pi$
$569$	−30.3583	−1.27269	−0.636343	+	0.771406i	$0.719554\pi$
$570$	0	0
$571$	37.9361	1.58758	0.793788	−	0.608195i	$-0.208106\pi$
$571$	37.9361	1.58758	0.793788	+	0.608195i	$0.208106\pi$
$572$	0	0
$573$	− 6.90833i	− 0.288599i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.72498i	− 0.155073i	−0.996990	−	0.0775365i	$-0.975295\pi$
$577$	− 3.72498i	− 0.155073i	0.996990	−	0.0775365i	$-0.0247054\pi$
$578$	0	0
$579$	−87.2666	−3.62668
$580$	0	0
$581$	−16.8167	−0.697672
$582$	0	0
$583$	− 11.5139i	− 0.476856i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.275019i	0.0113513i	0.999984	+	0.00567563i	$0.00180662\pi$
$587$	0.275019i	0.0113513i	−0.999984	+	0.00567563i	$0.998193\pi$
$588$	0	0
$589$	−5.60555	−0.230973
$590$	0	0
$591$	−36.6333	−1.50689
$592$	0	0
$593$	− 11.7889i	− 0.484112i	−0.970262	−	0.242056i	$-0.922178\pi$
$593$	− 11.7889i	− 0.484112i	0.970262	−	0.242056i	$-0.0778218\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 26.1194i	− 1.06900i
$598$	0	0
$599$	12.2750	0.501544	0.250772	−	0.968046i	$-0.419316\pi$
$599$	12.2750	0.501544	0.250772	+	0.968046i	$0.419316\pi$
$600$	0	0
$601$	−27.5139	−1.12231	−0.561157	−	0.827709i	$-0.689644\pi$
$601$	−27.5139	−1.12231	−0.561157	+	0.827709i	$0.689644\pi$
$602$	0	0
$603$	− 63.2666i	− 2.57642i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.4222i	0.828912i	0.910069	+	0.414456i	$0.136028\pi$
$607$	20.4222i	0.828912i	−0.910069	+	0.414456i	$0.863972\pi$
$608$	0	0
$609$	75.3583	3.05367
$610$	0	0
$611$	9.00000	0.364101
$612$	0	0
$613$	− 24.3305i	− 0.982701i	−0.870962	−	0.491350i	$-0.836503\pi$
$613$	− 24.3305i	− 0.982701i	0.870962	−	0.491350i	$-0.163497\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.97224i	0.280692i	0.990103	+	0.140346i	$0.0448215\pi$
$617$	6.97224i	0.280692i	−0.990103	+	0.140346i	$0.955179\pi$
$618$	0	0
$619$	−6.81665	−0.273984	−0.136992	−	0.990572i	$-0.543744\pi$
$619$	−6.81665	−0.273984	−0.136992	+	0.990572i	$0.543744\pi$
$620$	0	0
$621$	−111.992	−4.49407
$622$	0	0
$623$	68.4500i	2.74239i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.30278i	− 0.131900i
$628$	0	0
$629$	−0.275019	−0.0109657
$630$	0	0
$631$	20.4861	0.815540	0.407770	−	0.913085i	$-0.366307\pi$
$631$	20.4861	0.815540	0.407770	+	0.913085i	$0.366307\pi$
$632$	0	0
$633$	47.5416i	1.88961i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 64.5416i	− 2.55723i
$638$	0	0
$639$	36.4222	1.44084
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	− 20.4222i	− 0.805373i	−0.915338	−	0.402687i	$-0.868076\pi$
$643$	− 20.4222i	− 0.805373i	0.915338	−	0.402687i	$-0.131924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.0000i	− 1.29736i	−0.761060	−	0.648682i	$-0.775321\pi$
$647$	− 33.0000i	− 1.29736i	0.761060	−	0.648682i	$-0.224679\pi$
$648$	0	0
$649$	−1.60555	−0.0630234
$650$	0	0
$651$	−79.6611	−3.12216
$652$	0	0
$653$	7.11943i	0.278605i	0.990250	+	0.139302i	$0.0444860\pi$
$653$	7.11943i	0.278605i	−0.990250	+	0.139302i	$0.955514\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	70.4500i	2.74851i
$658$	0	0
$659$	39.3583	1.53318	0.766591	−	0.642136i	$-0.221952\pi$
$659$	39.3583	1.53318	0.766591	+	0.642136i	$0.221952\pi$
$660$	0	0
$661$	0.816654	0.0317642	0.0158821	−	0.999874i	$-0.494944\pi$
$661$	0.816654	0.0317642	0.0158821	+	0.999874i	$0.494944\pi$
$662$	0	0
$663$	12.9083i	0.501318i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.6333i	1.41845i
$668$	0	0
$669$	−21.1194	−0.816524
$670$	0	0
$671$	−0.302776	−0.0116885
$672$	0	0
$673$	− 35.0000i	− 1.34915i	−0.738206	−	0.674575i	$-0.764327\pi$
$673$	− 35.0000i	− 1.34915i	0.738206	−	0.674575i	$-0.235673\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 26.2389i	− 1.00844i	−0.863575	−	0.504221i	$-0.831780\pi$
$677$	− 26.2389i	− 1.00844i	0.863575	−	0.504221i	$-0.168220\pi$
$678$	0	0
$679$	−77.9638	−2.99198
$680$	0	0
$681$	69.1472	2.64973
$682$	0	0
$683$	− 1.60555i	− 0.0614347i	−0.999528	−	0.0307174i	$-0.990221\pi$
$683$	− 1.60555i	− 0.0614347i	0.999528	−	0.0307174i	$-0.00977918\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.90833i	0.110960i
$688$	0	0
$689$	64.5416	2.45884
$690$	0	0
$691$	25.5139	0.970594	0.485297	−	0.874349i	$-0.338712\pi$
$691$	25.5139	0.970594	0.485297	+	0.874349i	$0.338712\pi$
$692$	0	0
$693$	− 34.0278i	− 1.29261i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.33053i	− 0.164031i
$698$	0	0
$699$	82.2666	3.11161
$700$	0	0
$701$	9.21110	0.347899	0.173949	−	0.984755i	$-0.444347\pi$
$701$	9.21110	0.347899	0.173949	+	0.984755i	$0.444347\pi$
$702$	0	0
$703$	0.394449i	0.0148769i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.5416i	1.86320i
$708$	0	0
$709$	7.63331	0.286675	0.143337	−	0.989674i	$-0.454217\pi$
$709$	7.63331	0.286675	0.143337	+	0.989674i	$0.454217\pi$
$710$	0	0
$711$	21.3305	0.799957
$712$	0	0
$713$	− 38.7250i	− 1.45026i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 83.6611i	− 3.12438i
$718$	0	0
$719$	45.4222	1.69396	0.846981	−	0.531623i	$-0.178418\pi$
$719$	45.4222	1.69396	0.846981	+	0.531623i	$0.178418\pi$
$720$	0	0
$721$	37.4222	1.39368
$722$	0	0
$723$	15.5139i	0.576967i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	16.0917i	0.596807i	0.954440	+	0.298404i	$0.0964541\pi$
$727$	16.0917i	0.596807i	−0.954440	+	0.298404i	$0.903546\pi$
$728$	0	0
$729$	−75.1749	−2.78426
$730$	0	0
$731$	−5.02776	−0.185958
$732$	0	0
$733$	19.4222i	0.717376i	0.933458	+	0.358688i	$0.116776\pi$
$733$	19.4222i	0.717376i	−0.933458	+	0.358688i	$0.883224\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000i	0.294684i
$738$	0	0
$739$	26.1194	0.960819	0.480409	−	0.877044i	$-0.340488\pi$
$739$	26.1194	0.960819	0.480409	+	0.877044i	$0.340488\pi$
$740$	0	0
$741$	18.5139	0.680124
$742$	0	0
$743$	0.908327i	0.0333233i	0.999861	+	0.0166616i	$0.00530381\pi$
$743$	0.908327i	0.0333233i	−0.999861	+	0.0166616i	$0.994696\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 30.9083i	− 1.13088i
$748$	0	0
$749$	52.5416	1.91983
$750$	0	0
$751$	53.3583	1.94707	0.973536	−	0.228535i	$-0.0733934\pi$
$751$	53.3583	1.94707	0.973536	+	0.228535i	$0.0733934\pi$
$752$	0	0
$753$	− 53.9361i	− 1.96554i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 25.0000i	− 0.908640i	−0.890838	−	0.454320i	$-0.849882\pi$
$757$	− 25.0000i	− 0.908640i	0.890838	−	0.454320i	$-0.150118\pi$
$758$	0	0
$759$	22.8167	0.828192
$760$	0	0
$761$	43.2666	1.56841	0.784207	−	0.620500i	$-0.213070\pi$
$761$	43.2666	1.56841	0.784207	+	0.620500i	$0.213070\pi$
$762$	0	0
$763$	8.21110i	0.297262i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 9.00000i	− 0.324971i
$768$	0	0
$769$	−44.6333	−1.60952	−0.804759	−	0.593602i	$-0.797706\pi$
$769$	−44.6333	−1.60952	−0.804759	+	0.593602i	$0.797706\pi$
$770$	0	0
$771$	−1.39445	−0.0502198
$772$	0	0
$773$	31.3305i	1.12688i	0.826157	+	0.563440i	$0.190523\pi$
$773$	31.3305i	1.12688i	−0.826157	+	0.563440i	$0.809477\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.60555i	0.201098i
$778$	0	0
$779$	−6.21110	−0.222536
$780$	0	0
$781$	−4.60555	−0.164800
$782$	0	0
$783$	85.9638i	3.07210i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31.8444i	− 1.13513i	−0.823328	−	0.567565i	$-0.807885\pi$
$787$	− 31.8444i	− 1.13513i	0.823328	−	0.567565i	$-0.192115\pi$
$788$	0	0
$789$	−35.7250	−1.27184
$790$	0	0
$791$	6.90833	0.245632
$792$	0	0
$793$	− 1.69722i	− 0.0602702i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.9083i	0.563502i	0.959488	+	0.281751i	$0.0909152\pi$
$797$	15.9083i	0.563502i	−0.959488	+	0.281751i	$0.909085\pi$
$798$	0	0
$799$	1.11943	0.0396026
$800$	0	0
$801$	−125.808	−4.44522
$802$	0	0
$803$	− 8.90833i	− 0.314368i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 47.9361i	− 1.68743i
$808$	0	0
$809$	−39.6333	−1.39343	−0.696716	−	0.717347i	$-0.745356\pi$
$809$	−39.6333	−1.39343	−0.696716	+	0.717347i	$0.745356\pi$
$810$	0	0
$811$	−2.39445	−0.0840805	−0.0420402	−	0.999116i	$-0.513386\pi$
$811$	−2.39445	−0.0840805	−0.0420402	+	0.999116i	$0.513386\pi$
$812$	0	0
$813$	1.90833i	0.0669279i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.21110i	0.252285i
$818$	0	0
$819$	190.744	6.66515
$820$	0	0
$821$	39.6333	1.38321	0.691606	−	0.722275i	$-0.256903\pi$
$821$	39.6333	1.38321	0.691606	+	0.722275i	$0.256903\pi$
$822$	0	0
$823$	− 11.6333i	− 0.405512i	−0.979229	−	0.202756i	$-0.935010\pi$
$823$	− 11.6333i	− 0.405512i	0.979229	−	0.202756i	$-0.0649898\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 37.3944i	− 1.30033i	−0.759792	−	0.650166i	$-0.774699\pi$
$827$	− 37.3944i	− 1.30033i	0.759792	−	0.650166i	$-0.225301\pi$
$828$	0	0
$829$	−0.880571	−0.0305835	−0.0152917	−	0.999883i	$-0.504868\pi$
$829$	−0.880571	−0.0305835	−0.0152917	+	0.999883i	$0.504868\pi$
$830$	0	0
$831$	−52.1472	−1.80897
$832$	0	0
$833$	− 8.02776i	− 0.278145i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 90.8722i	− 3.14100i
$838$	0	0
$839$	−36.9083	−1.27422	−0.637108	−	0.770774i	$-0.719870\pi$
$839$	−36.9083	−1.27422	−0.637108	+	0.770774i	$0.719870\pi$
$840$	0	0
$841$	−0.880571	−0.0303645
$842$	0	0
$843$	− 69.3583i	− 2.38883i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.30278i	0.147845i
$848$	0	0
$849$	−18.1194	−0.621857
$850$	0	0
$851$	−2.72498	−0.0934111
$852$	0	0
$853$	3.30278i	0.113085i	0.998400	+	0.0565424i	$0.0180076\pi$
$853$	3.30278i	0.113085i	−0.998400	+	0.0565424i	$0.981992\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.0278i	− 1.09405i	−0.837117	−	0.547024i	$-0.815761\pi$
$857$	− 32.0278i	− 1.09405i	0.837117	−	0.547024i	$-0.184239\pi$
$858$	0	0
$859$	−3.39445	−0.115817	−0.0579085	−	0.998322i	$-0.518443\pi$
$859$	−3.39445	−0.115817	−0.0579085	+	0.998322i	$0.518443\pi$
$860$	0	0
$861$	−88.2666	−3.00812
$862$	0	0
$863$	− 34.2666i	− 1.16645i	−0.812311	−	0.583225i	$-0.801791\pi$
$863$	− 34.2666i	− 1.16645i	0.812311	−	0.583225i	$-0.198209\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 54.5416i	− 1.85233i
$868$	0	0
$869$	−2.69722	−0.0914971
$870$	0	0
$871$	−44.8444	−1.51949
$872$	0	0
$873$	− 143.294i	− 4.84978i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.6333i	0.392829i	0.980521	+	0.196414i	$0.0629298\pi$
$877$	11.6333i	0.392829i	−0.980521	+	0.196414i	$0.937070\pi$
$878$	0	0
$879$	59.4500	2.00520
$880$	0	0
$881$	−55.7527	−1.87836	−0.939179	−	0.343429i	$-0.888412\pi$
$881$	−55.7527	−1.87836	−0.939179	+	0.343429i	$0.888412\pi$
$882$	0	0
$883$	3.78890i	0.127507i	0.997966	+	0.0637533i	$0.0203071\pi$
$883$	3.78890i	0.127507i	−0.997966	+	0.0637533i	$0.979693\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 4.39445i	− 0.147551i	−0.997275	−	0.0737756i	$-0.976495\pi$
$887$	− 4.39445i	− 0.147551i	0.997275	−	0.0737756i	$-0.0235048\pi$
$888$	0	0
$889$	−17.6056	−0.590471
$890$	0	0
$891$	29.8167	0.998895
$892$	0	0
$893$	− 1.60555i	− 0.0537277i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	127.900i	4.27045i
$898$	0	0
$899$	−29.7250	−0.991384
$900$	0	0
$901$	8.02776	0.267443
$902$	0	0
$903$	102.478i	3.41024i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 34.4222i	− 1.14297i	−0.820612	−	0.571485i	$-0.806367\pi$
$907$	− 34.4222i	− 1.14297i	0.820612	−	0.571485i	$-0.193633\pi$
$908$	0	0
$909$	−91.0555	−3.02012
$910$	0	0
$911$	9.63331	0.319166	0.159583	−	0.987185i	$-0.448985\pi$
$911$	9.63331	0.319166	0.159583	+	0.987185i	$0.448985\pi$
$912$	0	0
$913$	3.90833i	0.129347i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 35.7250i	− 1.17974i
$918$	0	0
$919$	−30.0555	−0.991440	−0.495720	−	0.868482i	$-0.665096\pi$
$919$	−30.0555	−0.991440	−0.495720	+	0.868482i	$0.665096\pi$
$920$	0	0
$921$	80.3583	2.64790
$922$	0	0
$923$	− 25.8167i	− 0.849766i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	68.7805i	2.25905i
$928$	0	0
$929$	47.6611	1.56371	0.781854	−	0.623461i	$-0.214274\pi$
$929$	47.6611	1.56371	0.781854	+	0.623461i	$0.214274\pi$
$930$	0	0
$931$	−11.5139	−0.377352
$932$	0	0
$933$	29.7250i	0.973152i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 16.0000i	− 0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$	− 16.0000i	− 0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$	0	0
$939$	−50.8444	−1.65924
$940$	0	0
$941$	33.4222	1.08953	0.544766	−	0.838588i	$-0.316618\pi$
$941$	33.4222	1.08953	0.544766	+	0.838588i	$0.316618\pi$
$942$	0	0
$943$	− 42.9083i	− 1.39729i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.4500i	0.859508i	0.902946	+	0.429754i	$0.141400\pi$
$947$	26.4500i	0.859508i	−0.902946	+	0.429754i	$0.858600\pi$
$948$	0	0
$949$	49.9361	1.62099
$950$	0	0
$951$	−76.9638	−2.49572
$952$	0	0
$953$	− 9.63331i	− 0.312053i	−0.987753	−	0.156027i	$-0.950131\pi$
$953$	− 9.63331i	− 0.312053i	0.987753	−	0.156027i	$-0.0498686\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 17.5139i	− 0.566143i
$958$	0	0
$959$	−42.6333	−1.37670
$960$	0	0
$961$	0.422205	0.0136195
$962$	0	0
$963$	96.5694i	3.11191i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.9361i	0.930522i	0.885174	+	0.465261i	$0.154039\pi$
$967$	28.9361i	0.930522i	−0.885174	+	0.465261i	$0.845961\pi$
$968$	0	0
$969$	2.30278	0.0739758
$970$	0	0
$971$	−41.0917	−1.31869	−0.659347	−	0.751839i	$-0.729167\pi$
$971$	−41.0917	−1.31869	−0.659347	+	0.751839i	$0.729167\pi$
$972$	0	0
$973$	− 7.69722i	− 0.246762i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.81665i	− 0.154098i	−0.997027	−	0.0770492i	$-0.975450\pi$
$977$	− 4.81665i	− 0.154098i	0.997027	−	0.0770492i	$-0.0245499\pi$
$978$	0	0
$979$	15.9083	0.508432
$980$	0	0
$981$	−15.0917	−0.481840
$982$	0	0
$983$	− 3.63331i	− 0.115885i	−0.998320	−	0.0579423i	$-0.981546\pi$
$983$	− 3.63331i	− 0.115885i	0.998320	−	0.0579423i	$-0.0184539\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 22.8167i	− 0.726262i
$988$	0	0
$989$	−49.8167	−1.58408
$990$	0	0
$991$	−32.7527	−1.04042	−0.520212	−	0.854037i	$-0.674147\pi$
$991$	−32.7527	−1.04042	−0.520212	+	0.854037i	$0.674147\pi$
$992$	0	0
$993$	94.5694i	3.00107i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	53.7805i	1.70325i	0.524155	+	0.851623i	$0.324381\pi$
$997$	53.7805i	1.70325i	−0.524155	+	0.851623i	$0.675619\pi$
$998$	0	0
$999$	−6.39445	−0.202311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.2.b.e.749.1		4
3.2	odd	2		9900.2.c.r.5149.4		4
4.3	odd	2		4400.2.b.r.4049.4		4
5.2	odd	4		1100.2.a.f.1.1	✓	2
5.3	odd	4		1100.2.a.i.1.2	yes	2
5.4	even	2	inner	1100.2.b.e.749.4		4
15.2	even	4		9900.2.a.bg.1.1		2
15.8	even	4		9900.2.a.by.1.2		2
15.14	odd	2		9900.2.c.r.5149.1		4
20.3	even	4		4400.2.a.bf.1.1		2
20.7	even	4		4400.2.a.bw.1.2		2
20.19	odd	2		4400.2.b.r.4049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.f.1.1	✓	2	5.2	odd	4
1100.2.a.i.1.2	yes	2	5.3	odd	4
1100.2.b.e.749.1		4	1.1	even	1	trivial
1100.2.b.e.749.4		4	5.4	even	2	inner
4400.2.a.bf.1.1		2	20.3	even	4
4400.2.a.bw.1.2		2	20.7	even	4
4400.2.b.r.4049.1		4	20.19	odd	2
4400.2.b.r.4049.4		4	4.3	odd	2
9900.2.a.bg.1.1		2	15.2	even	4
9900.2.a.by.1.2		2	15.8	even	4
9900.2.c.r.5149.1		4	15.14	odd	2
9900.2.c.r.5149.4		4	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.30278i q^{3} +4.30278i q^{7} -7.90833 q^{9} +1.00000 q^{11} +5.60555i q^{13} +0.697224i q^{17} +1.00000 q^{19} +14.2111 q^{21} +6.90833i q^{23} +16.2111i q^{27} +5.30278 q^{29} -5.60555 q^{31} -3.30278i q^{33} +0.394449i q^{37} +18.5139 q^{39} -6.21110 q^{41} +7.21110i q^{43} -1.60555i q^{47} -11.5139 q^{49} +2.30278 q^{51} -11.5139i q^{53} -3.30278i q^{57} -1.60555 q^{59} -0.302776 q^{61} -34.0278i q^{63} +8.00000i q^{67} +22.8167 q^{69} -4.60555 q^{71} -8.90833i q^{73} +4.30278i q^{77} -2.69722 q^{79} +29.8167 q^{81} +3.90833i q^{83} -17.5139i q^{87} +15.9083 q^{89} -24.1194 q^{91} +18.5139i q^{93} +18.1194i q^{97} -7.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{9} + 4 q^{11} + 4 q^{19} + 28 q^{21} + 14 q^{29} - 8 q^{31} + 38 q^{39} + 4 q^{41} - 10 q^{49} + 2 q^{51} + 8 q^{59} + 6 q^{61} + 48 q^{69} - 4 q^{71} - 18 q^{79} + 76 q^{81} + 42 q^{89} - 46 q^{91}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q - 10 * q^9 + 4 * q^11 + 4 * q^19 + 28 * q^21 + 14 * q^29 - 8 * q^31 + 38 * q^39 + 4 * q^41 - 10 * q^49 + 2 * q^51 + 8 * q^59 + 6 * q^61 + 48 * q^69 - 4 * q^71 - 18 * q^79 + 76 * q^81 + 42 * q^89 - 46 * q^91 - 10 * q^99

Introduction

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