LMFDB - Embedded newform 440.2.b.d.89.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(89,440)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("440.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.51341768894$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.47985531136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 19x^{6} + 91x^{4} + 45x^{2} + 4 $ x^8 + 19x^6 + 91x^4 + 45*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.6
Root			$0.655762i$ of defining polynomial
Character	$\chi$	$=$	440.89
Dual form			440.2.b.d.89.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.655762i q^{3} +(-2.23285 + 0.119978i) q^{5} +0.415806i q^{7} +2.56998 q^{9} +O(q^{10})$	$q+0.655762i q^{3} +(-2.23285 + 0.119978i) q^{5} +0.415806i q^{7} +2.56998 q^{9} +1.00000 q^{11} +4.00000i q^{13} +(-0.0786769 - 1.46422i) q^{15} +6.51558i q^{17} -5.20406 q^{19} -0.272670 q^{21} +8.54830i q^{23} +(4.97121 - 0.535784i) q^{25} +3.65258i q^{27} -0.895717 q^{29} -6.73836 q^{31} +0.655762i q^{33} +(-0.0498875 - 0.928432i) q^{35} -8.96410i q^{37} -2.62305 q^{39} +10.0998 q^{41} -4.78825i q^{43} +(-5.73836 + 0.308340i) q^{45} -5.61986i q^{47} +6.82711 q^{49} -4.27267 q^{51} +10.0357i q^{53} +(-2.23285 + 0.119978i) q^{55} -3.41262i q^{57} +1.63408 q^{59} +7.10428 q^{61} +1.06861i q^{63} +(-0.479911 - 8.93139i) q^{65} -10.6914i q^{67} -5.60565 q^{69} -6.19302 q^{71} +3.16839i q^{73} +(0.351347 + 3.25993i) q^{75} +0.415806i q^{77} +11.2682 q^{79} +5.31471 q^{81} +16.2429i q^{83} +(-0.781725 - 14.5483i) q^{85} -0.587377i q^{87} -9.56998 q^{89} -1.66323 q^{91} -4.41876i q^{93} +(11.6199 - 0.624371i) q^{95} -0.591657i q^{97} +2.56998 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9}+O(q^{10}) $ 8 * q - 14 * q^9	$ 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/440\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$221$	$321$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			0.655762i			0.378604i	0.981919	+	0.189302i	$0.0606226\pi$
$3$			0.655762i			0.378604i	−0.981919	+	0.189302i	$0.939377\pi$
$4$		0			0
$5$	−2.23285	+	0.119978i	−0.998559	+	0.0536557i
$5$	−2.23285	+	0.119978i	−0.998559	+	0.0536557i
$6$		0			0
$7$			0.415806i			0.157160i	0.996908	+	0.0785800i	$0.0250386\pi$
$7$			0.415806i			0.157160i	−0.996908	+	0.0785800i	$0.974961\pi$
$8$		0			0
$9$	2.56998			0.856659
$10$		0			0
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	−0.0786769	−	1.46422i	−0.0203143	−	0.378059i
$16$		0			0
$17$			6.51558i			1.58026i	0.612939	+	0.790130i	$0.289987\pi$
$17$			6.51558i			1.58026i	−0.612939	+	0.790130i	$0.710013\pi$
$18$		0			0
$19$	−5.20406			−1.19389			−0.596946	−	0.802281i	$-0.703619\pi$
$19$	−5.20406			−1.19389			−0.596946	+	0.802281i	$0.703619\pi$
$20$		0			0
$21$	−0.272670			−0.0595015
$22$		0			0
$23$			8.54830i			1.78244i	0.453568	+	0.891221i	$0.350151\pi$
$23$			8.54830i			1.78244i	−0.453568	+	0.891221i	$0.649849\pi$
$24$		0			0
$25$	4.97121	−	0.535784i	0.994242	−	0.107157i
$26$		0			0
$27$			3.65258i			0.702939i
$28$		0			0
$29$	−0.895717			−0.166331			−0.0831653	−	0.996536i	$-0.526503\pi$
$29$	−0.895717			−0.166331			−0.0831653	+	0.996536i	$0.526503\pi$
$30$		0			0
$31$	−6.73836			−1.21025			−0.605123	−	0.796132i	$-0.706876\pi$
$31$	−6.73836			−1.21025			−0.605123	+	0.796132i	$0.706876\pi$
$32$		0			0
$33$			0.655762i			0.114153i
$34$		0			0
$35$	−0.0498875	−	0.928432i	−0.00843253	−	0.156934i
$36$		0			0
$37$		−	8.96410i		−	1.47369i	−0.676062	−	0.736845i	$-0.736315\pi$
$37$		−	8.96410i		−	1.47369i	0.676062	−	0.736845i	$-0.263685\pi$
$38$		0			0
$39$	−2.62305			−0.420024
$40$		0			0
$41$	10.0998			1.57732			0.788660	−	0.614830i	$-0.210775\pi$
$41$	10.0998			1.57732			0.788660	+	0.614830i	$0.210775\pi$
$42$		0			0
$43$		−	4.78825i		−	0.730201i	−0.930968	−	0.365101i	$-0.881035\pi$
$43$		−	4.78825i		−	0.730201i	0.930968	−	0.365101i	$-0.118965\pi$
$44$		0			0
$45$	−5.73836	+	0.308340i	−0.855425	+	0.0459646i
$46$		0			0
$47$		−	5.61986i		−	0.819741i	−0.912144	−	0.409871i	$-0.865574\pi$
$47$		−	5.61986i		−	0.819741i	0.912144	−	0.409871i	$-0.134426\pi$
$48$		0			0
$49$	6.82711			0.975301
$50$		0			0
$51$	−4.27267			−0.598293
$52$		0			0
$53$			10.0357i			1.37851i	0.724521	+	0.689253i	$0.242061\pi$
$53$			10.0357i			1.37851i	−0.724521	+	0.689253i	$0.757939\pi$
$54$		0			0
$55$	−2.23285	+	0.119978i	−0.301077	+	0.0161778i
$56$		0			0
$57$		−	3.41262i		−	0.452013i
$58$		0			0
$59$	1.63408			0.212739			0.106370	−	0.994327i	$-0.466077\pi$
$59$	1.63408			0.212739			0.106370	+	0.994327i	$0.466077\pi$
$60$		0			0
$61$	7.10428			0.909610			0.454805	−	0.890591i	$-0.349709\pi$
$61$	7.10428			0.909610			0.454805	+	0.890591i	$0.349709\pi$
$62$		0			0
$63$			1.06861i			0.134633i
$64$		0			0
$65$	−0.479911	−	8.93139i	−0.0595257	−	1.10780i
$66$		0			0
$67$		−	10.6914i		−	1.30617i	−0.757286	−	0.653083i	$-0.773475\pi$
$67$		−	10.6914i		−	1.30617i	0.757286	−	0.653083i	$-0.226525\pi$
$68$		0			0
$69$	−5.60565			−0.674841
$70$		0			0
$71$	−6.19302			−0.734977			−0.367488	−	0.930028i	$-0.619782\pi$
$71$	−6.19302			−0.734977			−0.367488	+	0.930028i	$0.619782\pi$
$72$		0			0
$73$			3.16839i			0.370832i	0.982660	+	0.185416i	$0.0593632\pi$
$73$			3.16839i			0.370832i	−0.982660	+	0.185416i	$0.940637\pi$
$74$		0			0
$75$	0.351347	+	3.25993i	0.0405700	+	0.376424i
$76$		0			0
$77$			0.415806i			0.0473855i
$78$		0			0
$79$	11.2682			1.26777			0.633884	−	0.773428i	$-0.281460\pi$
$79$	11.2682			1.26777			0.633884	+	0.773428i	$0.281460\pi$
$80$		0			0
$81$	5.31471			0.590523
$82$		0			0
$83$			16.2429i			1.78289i	0.453128	+	0.891446i	$0.350308\pi$
$83$			16.2429i			1.78289i	−0.453128	+	0.891446i	$0.649692\pi$
$84$		0			0
$85$	−0.781725	−	14.5483i	−0.0847900	−	1.57798i
$86$		0			0
$87$		−	0.587377i		−	0.0629735i
$88$		0			0
$89$	−9.56998			−1.01442			−0.507208	−	0.861824i	$-0.669322\pi$
$89$	−9.56998			−1.01442			−0.507208	+	0.861824i	$0.669322\pi$
$90$		0			0
$91$	−1.66323			−0.174353
$92$		0			0
$93$		−	4.41876i		−	0.458204i
$94$		0			0
$95$	11.6199	−	0.624371i	1.19217	−	0.0640591i
$96$		0			0
$97$		−	0.591657i		−	0.0600737i	−0.999549	−	0.0300368i	$-0.990438\pi$
$97$		−	0.591657i		−	0.0600737i	0.999549	−	0.0300368i	$-0.00956246\pi$
$98$		0			0
$99$	2.56998			0.258292
$100$		0			0
$101$	−4.62305			−0.460010			−0.230005	−	0.973189i	$-0.573874\pi$
$101$	−4.62305			−0.460010			−0.230005	+	0.973189i	$0.573874\pi$
$102$		0			0
$103$		−	8.24291i		−	0.812198i	−0.913829	−	0.406099i	$-0.866889\pi$
$103$		−	8.24291i		−	0.812198i	0.913829	−	0.406099i	$-0.133111\pi$
$104$		0			0
$105$	0.608830	−	0.0327143i	0.0594157	−	0.00319259i
$106$		0			0
$107$		−	17.6199i		−	1.70338i	−0.524049	−	0.851688i	$-0.675579\pi$
$107$		−	17.6199i		−	1.70338i	0.524049	−	0.851688i	$-0.324421\pi$
$108$		0			0
$109$	−3.66323			−0.350873			−0.175437	−	0.984491i	$-0.556134\pi$
$109$	−3.66323			−0.350873			−0.175437	+	0.984491i	$0.556134\pi$
$110$		0			0
$111$	5.87832			0.557945
$112$		0			0
$113$		−	11.4797i		−	1.07992i	−0.841691	−	0.539959i	$-0.818440\pi$
$113$		−	11.4797i		−	1.07992i	0.841691	−	0.539959i	$-0.181560\pi$
$114$		0			0
$115$	−1.02561	−	19.0870i	−0.0956382	−	1.77988i
$116$		0			0
$117$			10.2799i			0.950378i
$118$		0			0
$119$	−2.70922			−0.248354
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			6.62305i			0.597180i
$124$		0			0
$125$	−11.0357	+	1.79276i	−0.987060	+	0.160349i
$126$		0			0
$127$			9.13995i			0.811040i	0.914086	+	0.405520i	$0.132909\pi$
$127$			9.13995i			0.811040i	−0.914086	+	0.405520i	$0.867091\pi$
$128$		0			0
$129$	3.13995			0.276457
$130$		0			0
$131$	−2.48123			−0.216787			−0.108393	−	0.994108i	$-0.534571\pi$
$131$	−2.48123			−0.216787			−0.108393	+	0.994108i	$0.534571\pi$
$132$		0			0
$133$		−	2.16388i		−	0.187632i
$134$		0			0
$135$	−0.438228	−	8.15565i	−0.0377167	−	0.701926i
$136$		0			0
$137$		−	7.40834i		−	0.632937i	−0.948603	−	0.316469i	$-0.897503\pi$
$137$		−	7.40834i		−	0.632937i	0.948603	−	0.316469i	$-0.102497\pi$
$138$		0			0
$139$	1.69166			0.143485			0.0717424	−	0.997423i	$-0.477144\pi$
$139$	1.69166			0.143485			0.0717424	+	0.997423i	$0.477144\pi$
$140$		0			0
$141$	3.68529			0.310358
$142$		0			0
$143$			4.00000i			0.334497i
$144$		0			0
$145$	2.00000	−	0.107466i	0.166091	−	0.00892458i
$146$		0			0
$147$			4.47696i			0.369253i
$148$		0			0
$149$	−3.74940			−0.307163			−0.153581	−	0.988136i	$-0.549081\pi$
$149$	−3.74940			−0.307163			−0.153581	+	0.988136i	$0.549081\pi$
$150$		0			0
$151$	−15.1400			−1.23207			−0.616036	−	0.787718i	$-0.711262\pi$
$151$	−15.1400			−1.23207			−0.616036	+	0.787718i	$0.711262\pi$
$152$		0			0
$153$			16.7449i			1.35374i
$154$		0			0
$155$	15.0457	−	0.808454i	1.20850	−	0.0649366i
$156$		0			0
$157$		−	11.5871i		−	0.924755i	−0.886683	−	0.462378i	$-0.846996\pi$
$157$		−	11.5871i		−	0.924755i	0.886683	−	0.462378i	$-0.153004\pi$
$158$		0			0
$159$	−6.58101			−0.521908
$160$		0			0
$161$	−3.55444			−0.280129
$162$		0			0
$163$			6.82392i			0.534491i	0.963629	+	0.267245i	$0.0861134\pi$
$163$			6.82392i			0.534491i	−0.963629	+	0.267245i	$0.913887\pi$
$164$		0			0
$165$	−0.0786769	−	1.46422i	−0.00612499	−	0.113989i
$166$		0			0
$167$		−	7.10428i		−	0.549746i	−0.961481	−	0.274873i	$-0.911364\pi$
$167$		−	7.10428i		−	0.549746i	0.961481	−	0.274873i	$-0.0886358\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−13.3743			−1.02276
$172$		0			0
$173$			9.31152i			0.707942i	0.935256	+	0.353971i	$0.115169\pi$
$173$			9.31152i			0.707942i	−0.935256	+	0.353971i	$0.884831\pi$
$174$		0			0
$175$	0.222782	+	2.06706i	0.0168408	+	0.156255i
$176$		0			0
$177$			1.07157i			0.0805440i
$178$		0			0
$179$	26.6368			1.99093			0.995464	−	0.0951359i	$-0.0303286\pi$
$179$	26.6368			1.99093			0.995464	+	0.0951359i	$0.0303286\pi$
$180$		0			0
$181$	1.01103			0.0751495			0.0375748	−	0.999294i	$-0.488037\pi$
$181$	1.01103			0.0751495			0.0375748	+	0.999294i	$0.488037\pi$
$182$		0			0
$183$			4.65872i			0.344382i
$184$		0			0
$185$	1.07549	+	20.0155i	0.0790718	+	1.47157i
$186$		0			0
$187$			6.51558i			0.476466i
$188$		0			0
$189$	−1.51877			−0.110474
$190$		0			0
$191$	10.5655			0.764490			0.382245	−	0.924061i	$-0.375151\pi$
$191$	10.5655			0.764490			0.382245	+	0.924061i	$0.375151\pi$
$192$		0			0
$193$			11.8271i			0.851334i	0.904880	+	0.425667i	$0.139960\pi$
$193$			11.8271i			0.851334i	−0.904880	+	0.425667i	$0.860040\pi$
$194$		0			0
$195$	5.85686	−	0.314707i	0.419419	−	0.0225367i
$196$		0			0
$197$			1.31152i			0.0934422i	0.998908	+	0.0467211i	$0.0148772\pi$
$197$			1.31152i			0.0934422i	−0.998908	+	0.0467211i	$0.985123\pi$
$198$		0			0
$199$	11.8271			0.838401			0.419201	−	0.907894i	$-0.362310\pi$
$199$	11.8271			0.838401			0.419201	+	0.907894i	$0.362310\pi$
$200$		0			0
$201$	7.01103			0.494520
$202$		0			0
$203$		−	0.372445i		−	0.0261405i
$204$		0			0
$205$	−22.5513	+	1.21175i	−1.57505	+	0.0846322i
$206$		0			0
$207$			21.9689i			1.52695i
$208$		0			0
$209$	−5.20406			−0.359972
$210$		0			0
$211$	−8.37244			−0.576383			−0.288191	−	0.957573i	$-0.593054\pi$
$211$	−8.37244			−0.576383			−0.288191	+	0.957573i	$0.593054\pi$
$212$		0			0
$213$		−	4.06115i		−	0.278265i
$214$		0			0
$215$	0.574484	+	10.6914i	0.0391795	+	0.729150i
$216$		0			0
$217$		−	2.80185i		−	0.190202i
$218$		0			0
$219$	−2.07771			−0.140398
$220$		0			0
$221$	−26.0623			−1.75314
$222$		0			0
$223$		−	22.1461i		−	1.48301i	−0.670946	−	0.741506i	$-0.734112\pi$
$223$		−	22.1461i		−	1.48301i	0.670946	−	0.741506i	$-0.265888\pi$
$224$		0			0
$225$	12.7759	−	1.37695i	0.851726	−	0.0917968i
$226$		0			0
$227$			21.8194i			1.44821i	0.689692	+	0.724103i	$0.257746\pi$
$227$			21.8194i			1.44821i	−0.689692	+	0.724103i	$0.742254\pi$
$228$		0			0
$229$	−2.80247			−0.185192			−0.0925962	−	0.995704i	$-0.529517\pi$
$229$	−2.80247			−0.185192			−0.0925962	+	0.995704i	$0.529517\pi$
$230$		0			0
$231$	−0.272670			−0.0179404
$232$		0			0
$233$			8.24424i			0.540098i	0.962847	+	0.270049i	$0.0870399\pi$
$233$			8.24424i			0.540098i	−0.962847	+	0.270049i	$0.912960\pi$
$234$		0			0
$235$	0.674259	+	12.5483i	0.0439838	+	0.818561i
$236$		0			0
$237$			7.38923i			0.479982i
$238$		0			0
$239$	11.2682			0.728877			0.364438	−	0.931227i	$-0.381261\pi$
$239$	11.2682			0.728877			0.364438	+	0.931227i	$0.381261\pi$
$240$		0			0
$241$	22.3860			1.44201			0.721006	−	0.692929i	$-0.243680\pi$
$241$	22.3860			1.44201			0.721006	+	0.692929i	$0.243680\pi$
$242$		0			0
$243$			14.4429i			0.926514i
$244$		0			0
$245$	−15.2439	+	0.819101i	−0.973896	+	0.0523304i
$246$		0			0
$247$		−	20.8162i		−	1.32451i
$248$		0			0
$249$	−10.6515			−0.675010
$250$		0			0
$251$	3.84265			0.242546			0.121273	−	0.992619i	$-0.461302\pi$
$251$	3.84265			0.242546			0.121273	+	0.992619i	$0.461302\pi$
$252$		0			0
$253$			8.54830i			0.537427i
$254$		0			0
$255$	9.54022	−	0.512626i	0.597432	−	0.0321019i
$256$		0			0
$257$		−	10.6230i		−	0.662648i	−0.943517	−	0.331324i	$-0.892505\pi$
$257$		−	10.6230i		−	0.662648i	0.943517	−	0.331324i	$-0.107495\pi$
$258$		0			0
$259$	3.72733			0.231605
$260$		0			0
$261$	−2.30197			−0.142489
$262$		0			0
$263$			5.79276i			0.357197i	0.983922	+	0.178598i	$0.0571563\pi$
$263$			5.79276i			0.357197i	−0.983922	+	0.178598i	$0.942844\pi$
$264$		0			0
$265$	−1.20406	−	22.4081i	−0.0739647	−	1.37652i
$266$		0			0
$267$		−	6.27563i		−	0.384062i
$268$		0			0
$269$	8.20857			0.500485			0.250243	−	0.968183i	$-0.419490\pi$
$269$	8.20857			0.500485			0.250243	+	0.968183i	$0.419490\pi$
$270$		0			0
$271$	27.3111			1.65903			0.829515	−	0.558485i	$-0.188617\pi$
$271$	27.3111			1.65903			0.829515	+	0.558485i	$0.188617\pi$
$272$		0			0
$273$		−	1.09068i		−	0.0660109i
$274$		0			0
$275$	4.97121	−	0.535784i	0.299775	−	0.0323090i
$276$		0			0
$277$			14.4735i			0.869631i	0.900520	+	0.434815i	$0.143186\pi$
$277$			14.4735i			0.869631i	−0.900520	+	0.434815i	$0.856814\pi$
$278$		0			0
$279$	−17.3174			−1.03677
$280$		0			0
$281$	16.2799			0.971178			0.485589	−	0.874187i	$-0.338605\pi$
$281$	16.2799			0.971178			0.485589	+	0.874187i	$0.338605\pi$
$282$		0			0
$283$			10.5169i			0.625165i	0.949891	+	0.312583i	$0.101194\pi$
$283$			10.5169i			0.625165i	−0.949891	+	0.312583i	$0.898806\pi$
$284$		0			0
$285$	0.409439	+	7.61986i	0.0242531	+	0.451362i
$286$		0			0
$287$			4.19955i			0.247892i
$288$		0			0
$289$	−25.4528			−1.49722
$290$		0			0
$291$	0.387986			0.0227441
$292$		0			0
$293$		−	9.37695i		−	0.547807i	−0.961757	−	0.273904i	$-0.911685\pi$
$293$		−	9.37695i		−	0.547807i	0.961757	−	0.273904i	$-0.0883150\pi$
$294$		0			0
$295$	−3.64865	+	0.196053i	−0.212433	+	0.0114147i
$296$		0			0
$297$			3.65258i			0.211944i
$298$		0			0
$299$	−34.1932			−1.97744
$300$		0			0
$301$	1.99099			0.114758
$302$		0			0
$303$		−	3.03162i		−	0.174162i
$304$		0			0
$305$	−15.8628	+	0.852356i	−0.908300	+	0.0488058i
$306$		0			0
$307$		−	2.09978i		−	0.119840i	−0.998203	−	0.0599202i	$-0.980915\pi$
$307$		−	2.09978i		−	0.119840i	0.998203	−	0.0599202i	$-0.0190846\pi$
$308$		0			0
$309$	5.40539			0.307502
$310$		0			0
$311$	−20.3724			−1.15522			−0.577608	−	0.816314i	$-0.696014\pi$
$311$	−20.3724			−1.15522			−0.577608	+	0.816314i	$0.696014\pi$
$312$		0			0
$313$		−	23.8968i		−	1.35073i	−0.737485	−	0.675364i	$-0.763987\pi$
$313$		−	23.8968i		−	1.35073i	0.737485	−	0.675364i	$-0.236013\pi$
$314$		0			0
$315$	−0.128210	−	2.38605i	−0.00722380	−	0.134439i
$316$		0			0
$317$			5.86114i			0.329195i	0.986361	+	0.164597i	$0.0526325\pi$
$317$			5.86114i			0.329195i	−0.986361	+	0.164597i	$0.947368\pi$
$318$		0			0
$319$	−0.895717			−0.0501506
$320$		0			0
$321$	11.5544			0.644906
$322$		0			0
$323$		−	33.9075i		−	1.88666i
$324$		0			0
$325$	2.14314	+	19.8848i	0.118880	+	1.10301i
$326$		0			0
$327$		−	2.40220i		−	0.132842i
$328$		0			0
$329$	2.33677			0.128831
$330$		0			0
$331$	−0.574484			−0.0315765			−0.0157882	−	0.999875i	$-0.505026\pi$
$331$	−0.574484			−0.0315765			−0.0157882	+	0.999875i	$0.505026\pi$
$332$		0			0
$333$		−	23.0375i		−	1.26245i
$334$		0			0
$335$	1.28273	+	23.8723i	0.0700833	+	1.30428i
$336$		0			0
$337$			4.10747i			0.223748i	0.993722	+	0.111874i	$0.0356853\pi$
$337$			4.10747i			0.223748i	−0.993722	+	0.111874i	$0.964315\pi$
$338$		0			0
$339$	7.52794			0.408862
$340$		0			0
$341$	−6.73836			−0.364903
$342$		0			0
$343$			5.74940i			0.310438i
$344$		0			0
$345$	12.5166	−	0.672553i	0.673868	−	0.0362090i
$346$		0			0
$347$			8.30834i			0.446015i	0.974817	+	0.223008i	$0.0715875\pi$
$347$			8.30834i			0.446015i	−0.974817	+	0.223008i	$0.928413\pi$
$348$		0			0
$349$	18.4858			0.989523			0.494762	−	0.869029i	$-0.335255\pi$
$349$	18.4858			0.989523			0.494762	+	0.869029i	$0.335255\pi$
$350$		0			0
$351$	−14.6103			−0.779841
$352$		0			0
$353$			26.5199i			1.41151i	0.708456	+	0.705755i	$0.249392\pi$
$353$			26.5199i			1.41151i	−0.708456	+	0.705755i	$0.750608\pi$
$354$		0			0
$355$	13.8281	−	0.743025i	0.733918	−	0.0394357i
$356$		0			0
$357$		−	1.77660i		−	0.0940278i
$358$		0			0
$359$	18.8226			0.993419			0.496709	−	0.867917i	$-0.334541\pi$
$359$	18.8226			0.993419			0.496709	+	0.867917i	$0.334541\pi$
$360$		0			0
$361$	8.08222			0.425380
$362$		0			0
$363$			0.655762i			0.0344186i
$364$		0			0
$365$	−0.380136	−	7.07452i	−0.0198972	−	0.370298i
$366$		0			0
$367$			10.2115i			0.533037i	0.963830	+	0.266519i	$0.0858734\pi$
$367$			10.2115i			0.533037i	−0.963830	+	0.266519i	$0.914127\pi$
$368$		0			0
$369$	25.9562			1.35122
$370$		0			0
$371$	−4.17289			−0.216646
$372$		0			0
$373$		−	32.3363i		−	1.67431i	−0.546965	−	0.837156i	$-0.684217\pi$
$373$		−	32.3363i		−	1.67431i	0.546965	−	0.837156i	$-0.315783\pi$
$374$		0			0
$375$	−1.17562	−	7.23677i	−0.0607089	−	0.373705i
$376$		0			0
$377$		−	3.58287i		−	0.184527i
$378$		0			0
$379$	−15.3686			−0.789434			−0.394717	−	0.918803i	$-0.629157\pi$
$379$	−15.3686			−0.789434			−0.394717	+	0.918803i	$0.629157\pi$
$380$		0			0
$381$	−5.99363			−0.307063
$382$		0			0
$383$		−	13.6662i		−	0.698309i	−0.937065	−	0.349155i	$-0.886469\pi$
$383$		−	13.6662i		−	0.698309i	0.937065	−	0.349155i	$-0.113531\pi$
$384$		0			0
$385$	−0.0498875	−	0.928432i	−0.00254250	−	0.0473173i
$386$		0			0
$387$		−	12.3057i		−	0.625534i
$388$		0			0
$389$	9.01103			0.456878			0.228439	−	0.973558i	$-0.426638\pi$
$389$	9.01103			0.456878			0.228439	+	0.973558i	$0.426638\pi$
$390$		0			0
$391$	−55.6971			−2.81672
$392$		0			0
$393$		−	1.62710i		−	0.0820763i
$394$		0			0
$395$	−25.1601	+	1.35193i	−1.26594	+	0.0680229i
$396$		0			0
$397$			3.45466i			0.173384i	0.996235	+	0.0866922i	$0.0276297\pi$
$397$			3.45466i			0.173384i	−0.996235	+	0.0866922i	$0.972370\pi$
$398$		0			0
$399$	1.41899			0.0710384
$400$		0			0
$401$	−1.62756			−0.0812762			−0.0406381	−	0.999174i	$-0.512939\pi$
$401$	−1.62756			−0.0812762			−0.0406381	+	0.999174i	$0.512939\pi$
$402$		0			0
$403$		−	26.9535i		−	1.34265i
$404$		0			0
$405$	−11.8669	+	0.637647i	−0.589672	+	0.0316849i
$406$		0			0
$407$		−	8.96410i		−	0.444334i
$408$		0			0
$409$	−6.54534			−0.323646			−0.161823	−	0.986820i	$-0.551737\pi$
$409$	−6.54534			−0.323646			−0.161823	+	0.986820i	$0.551737\pi$
$410$		0			0
$411$	4.85811			0.239633
$412$		0			0
$413$			0.679461i			0.0334341i
$414$		0			0
$415$	−1.94879	−	36.2679i	−0.0956623	−	1.78032i
$416$		0			0
$417$			1.10933i			0.0543239i
$418$		0			0
$419$	34.4795			1.68443			0.842216	−	0.539141i	$-0.181251\pi$
$419$	34.4795			1.68443			0.842216	+	0.539141i	$0.181251\pi$
$420$		0			0
$421$	6.54534			0.319000			0.159500	−	0.987198i	$-0.449012\pi$
$421$	6.54534			0.319000			0.159500	+	0.987198i	$0.449012\pi$
$422$		0			0
$423$		−	14.4429i		−	0.702239i
$424$		0			0
$425$	3.49094	+	32.3903i	0.169336	+	1.57116i
$426$		0			0
$427$			2.95401i			0.142954i
$428$		0			0
$429$	−2.62305			−0.126642
$430$		0			0
$431$	36.1711			1.74230			0.871151	−	0.491016i	$-0.163374\pi$
$431$	36.1711			1.74230			0.871151	+	0.491016i	$0.163374\pi$
$432$		0			0
$433$		−	38.1027i		−	1.83110i	−0.402204	−	0.915550i	$-0.631756\pi$
$433$		−	38.1027i		−	1.83110i	0.402204	−	0.915550i	$-0.368244\pi$
$434$		0			0
$435$	0.0704722	+	1.31152i	0.00337889	+	0.0628828i
$436$		0			0
$437$		−	44.4858i		−	2.12805i
$438$		0			0
$439$	−12.1282			−0.578848			−0.289424	−	0.957201i	$-0.593464\pi$
$439$	−12.1282			−0.578848			−0.289424	+	0.957201i	$0.593464\pi$
$440$		0			0
$441$	17.5455			0.835500
$442$		0			0
$443$		−	16.5483i		−	0.786233i	−0.919489	−	0.393117i	$-0.871397\pi$
$443$		−	16.5483i		−	0.786233i	0.919489	−	0.393117i	$-0.128603\pi$
$444$		0			0
$445$	21.3683	−	1.14818i	1.01295	−	0.0544292i
$446$		0			0
$447$		−	2.45871i		−	0.116293i
$448$		0			0
$449$	−15.6056			−0.736476			−0.368238	−	0.929732i	$-0.620039\pi$
$449$	−15.6056			−0.736476			−0.368238	+	0.929732i	$0.620039\pi$
$450$		0			0
$451$	10.0998			0.475580
$452$		0			0
$453$		−	9.92820i		−	0.466468i
$454$		0			0
$455$	3.71373	−	0.199550i	0.174102	−	0.00935505i
$456$		0			0
$457$			5.87761i			0.274943i	0.990506	+	0.137471i	$0.0438975\pi$
$457$			5.87761i			0.274943i	−0.990506	+	0.137471i	$0.956102\pi$
$458$		0			0
$459$	−23.7987			−1.11083
$460$		0			0
$461$	36.7495			1.71159			0.855797	−	0.517312i	$-0.173067\pi$
$461$	36.7495			1.71159			0.855797	+	0.517312i	$0.173067\pi$
$462$		0			0
$463$		−	19.3081i		−	0.897324i	−0.893702	−	0.448662i	$-0.851901\pi$
$463$		−	19.3081i		−	0.897324i	0.893702	−	0.448662i	$-0.148099\pi$
$464$		0			0
$465$	0.530153	+	9.86642i	0.0245853	+	0.457544i
$466$		0			0
$467$			11.4129i			0.528127i	0.964505	+	0.264064i	$0.0850629\pi$
$467$			11.4129i			0.528127i	−0.964505	+	0.264064i	$0.914937\pi$
$468$		0			0
$469$	4.44556			0.205277
$470$		0			0
$471$	7.59841			0.350116
$472$		0			0
$473$		−	4.78825i		−	0.220164i
$474$		0			0
$475$	−25.8705	+	2.78825i	−1.18702	+	0.127934i
$476$		0			0
$477$			25.7914i			1.18091i
$478$		0			0
$479$	−29.2307			−1.33559			−0.667793	−	0.744347i	$-0.732761\pi$
$479$	−29.2307			−1.33559			−0.667793	+	0.744347i	$0.732761\pi$
$480$		0			0
$481$	35.8564			1.63491
$482$		0			0
$483$		−	2.33086i		−	0.106058i
$484$		0			0
$485$	0.0709857	+	1.32108i	0.00322329	+	0.0599871i
$486$		0			0
$487$			33.2932i			1.50866i	0.656496	+	0.754329i	$0.272038\pi$
$487$			33.2932i			1.50866i	−0.656496	+	0.754329i	$0.727962\pi$
$488$		0			0
$489$	−4.47487			−0.202361
$490$		0			0
$491$	−24.5151			−1.10635			−0.553176	−	0.833064i	$-0.686584\pi$
$491$	−24.5151			−1.10635			−0.553176	+	0.833064i	$0.686584\pi$
$492$		0			0
$493$		−	5.83612i		−	0.262846i
$494$		0			0
$495$	−5.73836	+	0.308340i	−0.257920	+	0.0138589i
$496$		0			0
$497$		−	2.57510i		−	0.115509i
$498$		0			0
$499$	18.9093			0.846497			0.423249	−	0.906014i	$-0.360890\pi$
$499$	18.9093			0.846497			0.423249	+	0.906014i	$0.360890\pi$
$500$		0			0
$501$	4.65872			0.208136
$502$		0			0
$503$			9.42031i			0.420031i	0.977698	+	0.210016i	$0.0673515\pi$
$503$			9.42031i			0.420031i	−0.977698	+	0.210016i	$0.932649\pi$
$504$		0			0
$505$	10.3226	−	0.554663i	0.459348	−	0.0246822i
$506$		0			0
$507$		−	1.96729i		−	0.0873702i
$508$		0			0
$509$	16.7804			0.743778			0.371889	−	0.928277i	$-0.378710\pi$
$509$	16.7804			0.743778			0.371889	+	0.928277i	$0.378710\pi$
$510$		0			0
$511$	−1.31744			−0.0582799
$512$		0			0
$513$		−	19.0082i		−	0.839234i
$514$		0			0
$515$	0.988966	+	18.4052i	0.0435791	+	0.811028i
$516$		0			0
$517$		−	5.61986i		−	0.247161i
$518$		0			0
$519$	−6.10614			−0.268030
$520$		0			0
$521$	−37.8273			−1.65724			−0.828621	−	0.559810i	$-0.810874\pi$
$521$	−37.8273			−1.65724			−0.828621	+	0.559810i	$0.810874\pi$
$522$		0			0
$523$		−	10.0998i		−	0.441632i	−0.975315	−	0.220816i	$-0.929128\pi$
$523$		−	10.0998i		−	0.441632i	0.975315	−	0.220816i	$-0.0708721\pi$
$524$		0			0
$525$	−1.35550	+	0.146092i	−0.0591589	+	0.00637599i
$526$		0			0
$527$		−	43.9044i		−	1.91250i
$528$		0			0
$529$	−50.0734			−2.17710
$530$		0			0
$531$	4.19955			0.182245
$532$		0			0
$533$			40.3991i			1.74988i
$534$		0			0
$535$	2.11399	+	39.3425i	0.0913959	+	1.70092i
$536$		0			0
$537$			17.4674i			0.753774i
$538$		0			0
$539$	6.82711			0.294064
$540$		0			0
$541$	−29.9269			−1.28666			−0.643329	−	0.765590i	$-0.722447\pi$
$541$	−29.9269			−1.28666			−0.643329	+	0.765590i	$0.722447\pi$
$542$		0			0
$543$			0.662997i			0.0284519i
$544$		0			0
$545$	8.17942	−	0.439506i	0.350368	−	0.0188264i
$546$		0			0
$547$			23.7630i			1.01603i	0.861347	+	0.508016i	$0.169621\pi$
$547$			23.7630i			1.01603i	−0.861347	+	0.508016i	$0.830379\pi$
$548$		0			0
$549$	18.2578			0.779226
$550$		0			0
$551$	4.66137			0.198581
$552$		0			0
$553$			4.68537i			0.199242i
$554$		0			0
$555$	−13.1254	+	0.705267i	−0.557141	+	0.0299369i
$556$		0			0
$557$		−	24.3363i		−	1.03116i	−0.856841	−	0.515581i	$-0.827576\pi$
$557$		−	24.3363i		−	1.03116i	0.856841	−	0.515581i	$-0.172424\pi$
$558$		0			0
$559$	19.1530			0.810086
$560$		0			0
$561$	−4.27267			−0.180392
$562$		0			0
$563$			8.85368i			0.373138i	0.982442	+	0.186569i	$0.0597368\pi$
$563$			8.85368i			0.373138i	−0.982442	+	0.186569i	$0.940263\pi$
$564$		0			0
$565$	1.37731	+	25.6324i	0.0579437	+	1.07836i
$566$		0			0
$567$			2.20989i			0.0928066i
$568$		0			0
$569$	−26.3860			−1.10616			−0.553080	−	0.833128i	$-0.686548\pi$
$569$	−26.3860			−1.10616			−0.553080	+	0.833128i	$0.686548\pi$
$570$		0			0
$571$	6.45280			0.270041			0.135021	−	0.990843i	$-0.456890\pi$
$571$	6.45280			0.270041			0.135021	+	0.990843i	$0.456890\pi$
$572$		0			0
$573$			6.92843i			0.289439i
$574$		0			0
$575$	4.58004	+	42.4954i	0.191001	+	1.77218i
$576$		0			0
$577$			12.1745i			0.506832i	0.967357	+	0.253416i	$0.0815542\pi$
$577$			12.1745i			0.506832i	−0.967357	+	0.253416i	$0.918446\pi$
$578$		0			0
$579$	−7.75576			−0.322319
$580$		0			0
$581$	−6.75390			−0.280199
$582$		0			0
$583$			10.0357i			0.415635i
$584$		0			0
$585$	−1.23336	−	22.9535i	−0.0509932	−	0.949009i
$586$		0			0
$587$		−	22.9521i		−	0.947336i	−0.880704	−	0.473668i	$-0.842930\pi$
$587$		−	22.9521i		−	0.947336i	0.880704	−	0.473668i	$-0.157070\pi$
$588$		0			0
$589$	35.0668			1.44490
$590$		0			0
$591$	−0.860047			−0.0353776
$592$		0			0
$593$		−	10.4081i		−	0.427410i	−0.976898	−	0.213705i	$-0.931447\pi$
$593$		−	10.4081i		−	0.427410i	0.976898	−	0.213705i	$-0.0685531\pi$
$594$		0			0
$595$	6.04927	−	0.325046i	0.247996	−	0.0133256i
$596$		0			0
$597$			7.75576i			0.317422i
$598$		0			0
$599$	8.98913			0.367286			0.183643	−	0.982993i	$-0.441211\pi$
$599$	8.98913			0.367286			0.183643	+	0.982993i	$0.441211\pi$
$600$		0			0
$601$	−3.57650			−0.145889			−0.0729443	−	0.997336i	$-0.523240\pi$
$601$	−3.57650			−0.145889			−0.0729443	+	0.997336i	$0.523240\pi$
$602$		0			0
$603$		−	27.4767i		−	1.11894i
$604$		0			0
$605$	−2.23285	+	0.119978i	−0.0907781	+	0.00487779i
$606$		0			0
$607$			27.0235i			1.09685i	0.836200	+	0.548424i	$0.184772\pi$
$607$			27.0235i			1.09685i	−0.836200	+	0.548424i	$0.815228\pi$
$608$		0			0
$609$	0.244235			0.00989691
$610$		0			0
$611$	22.4795			0.909421
$612$		0			0
$613$			44.5572i			1.79965i	0.436254	+	0.899823i	$0.356305\pi$
$613$			44.5572i			1.79965i	−0.436254	+	0.899823i	$0.643695\pi$
$614$		0			0
$615$	−0.794619	−	14.7883i	−0.0320421	−	0.596320i
$616$		0			0
$617$		−	15.2966i		−	0.615818i	−0.951416	−	0.307909i	$-0.900371\pi$
$617$		−	15.2966i		−	0.615818i	0.951416	−	0.307909i	$-0.0996292\pi$
$618$		0			0
$619$	−14.5655			−0.585436			−0.292718	−	0.956199i	$-0.594560\pi$
$619$	−14.5655			−0.585436			−0.292718	+	0.956199i	$0.594560\pi$
$620$		0			0
$621$	−31.2233			−1.25295
$622$		0			0
$623$		−	3.97926i		−	0.159426i
$624$		0			0
$625$	24.4259	−	5.32699i	0.977035	−	0.213080i
$626$		0			0
$627$		−	3.41262i		−	0.136287i
$628$		0			0
$629$	58.4063			2.32881
$630$		0			0
$631$	−22.0779			−0.878906			−0.439453	−	0.898266i	$-0.644828\pi$
$631$	−22.0779			−0.878906			−0.439453	+	0.898266i	$0.644828\pi$
$632$		0			0
$633$		−	5.49033i		−	0.218221i
$634$		0			0
$635$	−1.09659	−	20.4081i	−0.0435169	−	0.809871i
$636$		0			0
$637$			27.3084i			1.08200i
$638$		0			0
$639$	−15.9159			−0.629624
$640$		0			0
$641$	−31.2152			−1.23293			−0.616463	−	0.787384i	$-0.711435\pi$
$641$	−31.2152			−1.23293			−0.616463	+	0.787384i	$0.711435\pi$
$642$		0			0
$643$			10.4498i			0.412102i	0.978541	+	0.206051i	$0.0660612\pi$
$643$			10.4498i			0.412102i	−0.978541	+	0.206051i	$0.933939\pi$
$644$		0			0
$645$	−7.01103	+	0.376725i	−0.276059	+	0.0148335i
$646$		0			0
$647$			26.1307i			1.02730i	0.857999	+	0.513652i	$0.171708\pi$
$647$			26.1307i			1.02730i	−0.857999	+	0.513652i	$0.828292\pi$
$648$		0			0
$649$	1.63408			0.0641433
$650$		0			0
$651$	1.83735			0.0720114
$652$		0			0
$653$		−	44.2097i		−	1.73006i	−0.501719	−	0.865030i	$-0.667299\pi$
$653$		−	44.2097i		−	1.73006i	0.501719	−	0.865030i	$-0.332701\pi$
$654$		0			0
$655$	5.54022	−	0.297693i	0.216474	−	0.0116318i
$656$		0			0
$657$			8.14268i			0.317676i
$658$		0			0
$659$	−6.96706			−0.271398			−0.135699	−	0.990750i	$-0.543328\pi$
$659$	−6.96706			−0.271398			−0.135699	+	0.990750i	$0.543328\pi$
$660$		0			0
$661$	−9.19753			−0.357743			−0.178871	−	0.983872i	$-0.557245\pi$
$661$	−9.19753			−0.357743			−0.178871	+	0.983872i	$0.557245\pi$
$662$		0			0
$663$		−	17.0907i		−	0.663747i
$664$		0			0
$665$	0.259618	+	4.83161i	0.0100675	+	0.187362i
$666$		0			0
$667$		−	7.65686i		−	0.296475i
$668$		0			0
$669$	14.5226			0.561475
$670$		0			0
$671$	7.10428			0.274258
$672$		0			0
$673$			8.72415i			0.336291i	0.985762	+	0.168146i	$0.0537779\pi$
$673$			8.72415i			0.336291i	−0.985762	+	0.168146i	$0.946222\pi$
$674$		0			0
$675$	1.95699	+	18.1577i	0.0753247	+	0.698892i
$676$		0			0
$677$			8.76618i			0.336912i	0.985709	+	0.168456i	$0.0538781\pi$
$677$			8.76618i			0.336912i	−0.985709	+	0.168456i	$0.946122\pi$
$678$		0			0
$679$	0.246015			0.00944118
$680$		0			0
$681$	−14.3083			−0.548297
$682$		0			0
$683$			49.2320i			1.88381i	0.335877	+	0.941906i	$0.390967\pi$
$683$			49.2320i			1.88381i	−0.335877	+	0.941906i	$0.609033\pi$
$684$		0			0
$685$	0.888837	+	16.5417i	0.0339607	+	0.632026i
$686$		0			0
$687$		−	1.83775i		−	0.0701146i
$688$		0			0
$689$	−40.1427			−1.52931
$690$		0			0
$691$	−18.1483			−0.690395			−0.345198	−	0.938530i	$-0.612188\pi$
$691$	−18.1483			−0.690395			−0.345198	+	0.938530i	$0.612188\pi$
$692$		0			0
$693$			1.06861i			0.0405932i
$694$		0			0
$695$	−3.77722	+	0.202962i	−0.143278	+	0.00769877i
$696$		0			0
$697$			65.8059i			2.49258i
$698$		0			0
$699$	−5.40626			−0.204483
$700$		0			0
$701$	22.6587			0.855808			0.427904	−	0.903824i	$-0.359252\pi$
$701$	22.6587			0.855808			0.427904	+	0.903824i	$0.359252\pi$
$702$		0			0
$703$			46.6497i			1.75943i
$704$		0			0
$705$	−8.22869	+	0.442153i	−0.309911	+	0.0166525i
$706$		0			0
$707$		−	1.92229i		−	0.0722952i
$708$		0			0
$709$	36.7366			1.37967			0.689836	−	0.723966i	$-0.257683\pi$
$709$	36.7366			1.37967			0.689836	+	0.723966i	$0.257683\pi$
$710$		0			0
$711$	28.9589			1.08604
$712$		0			0
$713$		−	57.6015i		−	2.15719i
$714$		0			0
$715$	−0.479911	−	8.93139i	−0.0179477	−	0.334015i
$716$		0			0
$717$			7.38923i			0.275956i
$718$		0			0
$719$	−8.74738			−0.326222			−0.163111	−	0.986608i	$-0.552153\pi$
$719$	−8.74738			−0.326222			−0.163111	+	0.986608i	$0.552153\pi$
$720$		0			0
$721$	3.42745			0.127645
$722$		0			0
$723$			14.6799i			0.545952i
$724$		0			0
$725$	−4.45280	+	0.479911i	−0.165373	+	0.0178235i
$726$		0			0
$727$		−	2.20887i		−	0.0819226i	−0.999161	−	0.0409613i	$-0.986958\pi$
$727$		−	2.20887i		−	0.0819226i	0.999161	−	0.0409613i	$-0.0130420\pi$
$728$		0			0
$729$	6.47301			0.239741
$730$		0			0
$731$	31.1982			1.15391
$732$		0			0
$733$		−	3.58552i		−	0.132434i	−0.997805	−	0.0662171i	$-0.978907\pi$
$733$		−	3.58552i		−	0.132434i	0.997805	−	0.0662171i	$-0.0210930\pi$
$734$		0			0
$735$	−0.537135	−	9.99636i	−0.0198125	−	0.368721i
$736$		0			0
$737$		−	10.6914i		−	0.393824i
$738$		0			0
$739$	24.9314			0.917116			0.458558	−	0.888665i	$-0.348366\pi$
$739$	24.9314			0.917116			0.458558	+	0.888665i	$0.348366\pi$
$740$		0			0
$741$	13.6505			0.501463
$742$		0			0
$743$		−	21.4257i		−	0.786032i	−0.919532	−	0.393016i	$-0.871432\pi$
$743$		−	21.4257i		−	0.786032i	0.919532	−	0.393016i	$-0.128568\pi$
$744$		0			0
$745$	8.37183	−	0.449844i	0.306720	−	0.0164810i
$746$		0			0
$747$			41.7439i			1.52733i
$748$		0			0
$749$	7.32645			0.267703
$750$		0			0
$751$	28.1582			1.02751			0.513754	−	0.857938i	$-0.328254\pi$
$751$	28.1582			1.02751			0.513754	+	0.857938i	$0.328254\pi$
$752$		0			0
$753$			2.51986i			0.0918288i
$754$		0			0
$755$	33.8052	−	1.81646i	1.23030	−	0.0661077i
$756$		0			0
$757$		−	10.7513i		−	0.390761i	−0.980728	−	0.195381i	$-0.937406\pi$
$757$		−	10.7513i		−	0.390761i	0.980728	−	0.195381i	$-0.0625942\pi$
$758$		0			0
$759$	−5.60565			−0.203472
$760$		0			0
$761$	−38.0429			−1.37905			−0.689527	−	0.724260i	$-0.742182\pi$
$761$	−38.0429			−1.37905			−0.689527	+	0.724260i	$0.742182\pi$
$762$		0			0
$763$		−	1.52319i		−	0.0551433i
$764$		0			0
$765$	−2.00901	−	37.3888i	−0.0726361	−	1.35179i
$766$		0			0
$767$			6.53632i			0.236013i
$768$		0			0
$769$	−26.2280			−0.945805			−0.472903	−	0.881115i	$-0.656794\pi$
$769$	−26.2280			−0.945805			−0.472903	+	0.881115i	$0.656794\pi$
$770$		0			0
$771$	6.96619			0.250881
$772$		0			0
$773$			8.94499i			0.321729i	0.986977	+	0.160864i	$0.0514282\pi$
$773$			8.94499i			0.321729i	−0.986977	+	0.160864i	$0.948572\pi$
$774$		0			0
$775$	−33.4978	+	3.61031i	−1.20328	+	0.129686i
$776$		0			0
$777$			2.44424i			0.0876867i
$778$		0			0
$779$	−52.5598			−1.88315
$780$		0			0
$781$	−6.19302			−0.221604
$782$		0			0
$783$		−	3.27168i		−	0.116920i
$784$		0			0
$785$	1.39020	+	25.8723i	0.0496184	+	0.923423i
$786$		0			0
$787$		−	29.9716i		−	1.06837i	−0.845367	−	0.534185i	$-0.820618\pi$
$787$		−	29.9716i		−	1.06837i	0.845367	−	0.534185i	$-0.179382\pi$
$788$		0			0
$789$	−3.79867			−0.135236
$790$		0			0
$791$	4.77332			0.169720
$792$		0			0
$793$			28.4171i			1.00912i
$794$		0			0
$795$	14.6944	−	0.789575i	0.521156	−	0.0280033i
$796$		0			0
$797$		−	6.80022i		−	0.240876i	−0.992721	−	0.120438i	$-0.961570\pi$
$797$		−	6.80022i		−	0.240876i	0.992721	−	0.120438i	$-0.0384299\pi$
$798$		0			0
$799$	36.6167			1.29541
$800$		0			0
$801$	−24.5946			−0.869008
$802$		0			0
$803$			3.16839i			0.111810i
$804$		0			0
$805$	7.93651	−	0.426453i	0.279725	−	0.0150305i
$806$		0			0
$807$			5.38286i			0.189486i
$808$		0			0
$809$	−53.1246			−1.86776			−0.933880	−	0.357586i	$-0.883600\pi$
$809$	−53.1246			−1.86776			−0.933880	+	0.357586i	$0.883600\pi$
$810$		0			0
$811$	−30.0487			−1.05515			−0.527577	−	0.849507i	$-0.676899\pi$
$811$	−30.0487			−1.05515			−0.527577	+	0.849507i	$0.676899\pi$
$812$		0			0
$813$			17.9096i			0.628116i
$814$		0			0
$815$	−0.818719	−	15.2368i	−0.0286785	−	0.533721i
$816$		0			0
$817$			24.9183i			0.871782i
$818$		0			0
$819$	−4.27445			−0.149361
$820$		0			0
$821$	19.9069			0.694756			0.347378	−	0.937725i	$-0.387072\pi$
$821$	19.9069			0.694756			0.347378	+	0.937725i	$0.387072\pi$
$822$		0			0
$823$		−	33.2491i		−	1.15899i	−0.814976	−	0.579495i	$-0.803250\pi$
$823$		−	33.2491i		−	1.15899i	0.814976	−	0.579495i	$-0.196750\pi$
$824$		0			0
$825$	0.351347	+	3.25993i	0.0122323	+	0.113496i
$826$		0			0
$827$		−	28.3770i		−	0.986766i	−0.869812	−	0.493383i	$-0.835760\pi$
$827$		−	28.3770i		−	0.986766i	0.869812	−	0.493383i	$-0.164240\pi$
$828$		0			0
$829$	−3.63143			−0.126125			−0.0630625	−	0.998010i	$-0.520087\pi$
$829$	−3.63143			−0.126125			−0.0630625	+	0.998010i	$0.520087\pi$
$830$		0			0
$831$	−9.49120			−0.329246
$832$		0			0
$833$			44.4826i			1.54123i
$834$		0			0
$835$	0.852356	+	15.8628i	0.0294970	+	0.548954i
$836$		0			0
$837$		−	24.6124i		−	0.850729i
$838$		0			0
$839$	3.45660			0.119335			0.0596675	−	0.998218i	$-0.480996\pi$
$839$	3.45660			0.119335			0.0596675	+	0.998218i	$0.480996\pi$
$840$		0			0
$841$	−28.1977			−0.972334
$842$		0			0
$843$			10.6757i			0.367692i
$844$		0			0
$845$	6.69854	−	0.359933i	0.230437	−	0.0123821i
$846$		0			0
$847$			0.415806i			0.0142873i
$848$		0			0
$849$	−6.89659			−0.236690
$850$		0			0
$851$	76.6278			2.62677
$852$		0			0
$853$		−	47.4452i		−	1.62449i	−0.583315	−	0.812246i	$-0.698245\pi$
$853$		−	47.4452i		−	1.62449i	0.583315	−	0.812246i	$-0.301755\pi$
$854$		0			0
$855$	29.8628	−	1.60462i	1.02129	−	0.0548768i
$856$		0			0
$857$		−	35.8271i		−	1.22383i	−0.790923	−	0.611915i	$-0.790399\pi$
$857$		−	35.8271i		−	1.22383i	0.790923	−	0.611915i	$-0.209601\pi$
$858$		0			0
$859$	39.0539			1.33250			0.666252	−	0.745727i	$-0.267898\pi$
$859$	39.0539			1.33250			0.666252	+	0.745727i	$0.267898\pi$
$860$		0			0
$861$	−2.75390			−0.0938528
$862$		0			0
$863$		−	5.75072i		−	0.195757i	−0.995198	−	0.0978784i	$-0.968794\pi$
$863$		−	5.75072i		−	0.195757i	0.995198	−	0.0978784i	$-0.0312056\pi$
$864$		0			0
$865$	−1.11718	−	20.7912i	−0.0379851	−	0.706922i
$866$		0			0
$867$		−	16.6910i		−	0.566855i
$868$		0			0
$869$	11.2682			0.382246
$870$		0			0
$871$	42.7657			1.44906
$872$		0			0
$873$		−	1.52054i		−	0.0514626i
$874$		0			0
$875$	−0.745440	−	4.58870i	−0.0252005	−	0.155126i
$876$		0			0
$877$			8.32776i			0.281208i	0.990066	+	0.140604i	$0.0449045\pi$
$877$			8.32776i			0.281208i	−0.990066	+	0.140604i	$0.955095\pi$
$878$		0			0
$879$	6.14905			0.207402
$880$		0			0
$881$	−49.0954			−1.65407			−0.827033	−	0.562153i	$-0.809973\pi$
$881$	−49.0954			−1.65407			−0.827033	+	0.562153i	$0.809973\pi$
$882$		0			0
$883$			10.7526i			0.361853i	0.983497	+	0.180927i	$0.0579096\pi$
$883$			10.7526i			0.361853i	−0.983497	+	0.180927i	$0.942090\pi$
$884$		0			0
$885$	−0.128564	−	2.39265i	−0.00432164	−	0.0804280i
$886$		0			0
$887$		−	53.0591i		−	1.78155i	−0.454443	−	0.890776i	$-0.650162\pi$
$887$		−	53.0591i		−	1.78155i	0.454443	−	0.890776i	$-0.349838\pi$
$888$		0			0
$889$	−3.80045			−0.127463
$890$		0			0
$891$	5.31471			0.178049
$892$		0			0
$893$			29.2461i			0.978683i
$894$		0			0
$895$	−59.4759	+	3.19583i	−1.98806	+	0.106825i
$896$		0			0
$897$		−	22.4226i		−	0.748668i
$898$		0			0
$899$	6.03567			0.201301
$900$		0			0
$901$	−65.3882			−2.17840
$902$		0			0
$903$			1.30561i			0.0434481i
$904$		0			0
$905$	−2.25748	+	0.121302i	−0.0750413	+	0.00403220i
$906$		0			0
$907$			7.03885i			0.233721i	0.993148	+	0.116861i	$0.0372831\pi$
$907$			7.03885i			0.233721i	−0.993148	+	0.116861i	$0.962717\pi$
$908$		0			0
$909$	−11.8811			−0.394072
$910$		0			0
$911$	32.0980			1.06345			0.531727	−	0.846916i	$-0.321543\pi$
$911$	32.0980			1.06345			0.531727	+	0.846916i	$0.321543\pi$
$912$		0			0
$913$			16.2429i			0.537562i
$914$		0			0
$915$	−0.558943	−	10.4022i	−0.0184781	−	0.343886i
$916$		0			0
$917$		−	1.03171i		−	0.0340702i
$918$		0			0
$919$	1.05960			0.0349529			0.0174764	−	0.999847i	$-0.494437\pi$
$919$	1.05960			0.0349529			0.0174764	+	0.999847i	$0.494437\pi$
$920$		0			0
$921$	1.37695			0.0453721
$922$		0			0
$923$		−	24.7721i		−	0.815383i
$924$		0			0
$925$	−4.80282	−	44.5624i	−0.157916	−	1.46520i
$926$		0			0
$927$		−	21.1841i		−	0.695777i
$928$		0			0
$929$	12.9981			0.426455			0.213228	−	0.977003i	$-0.431602\pi$
$929$	12.9981			0.426455			0.213228	+	0.977003i	$0.431602\pi$
$930$		0			0
$931$	−35.5286			−1.16440
$932$		0			0
$933$		−	13.3595i		−	0.437370i
$934$		0			0
$935$	−0.781725	−	14.5483i	−0.0255651	−	0.475780i
$936$		0			0
$937$		−	39.9846i		−	1.30624i	−0.757254	−	0.653120i	$-0.773460\pi$
$937$		−	39.9846i		−	1.30624i	0.757254	−	0.653120i	$-0.226540\pi$
$938$		0			0
$939$	15.6706			0.511391
$940$		0			0
$941$	34.0267			1.10924			0.554619	−	0.832105i	$-0.312864\pi$
$941$	34.0267			1.10924			0.554619	+	0.832105i	$0.312864\pi$
$942$		0			0
$943$			86.3359i			2.81148i
$944$		0			0
$945$	3.39117	−	0.182218i	0.110315	−	0.00592755i
$946$		0			0
$947$		−	27.1506i		−	0.882276i	−0.897439	−	0.441138i	$-0.854575\pi$
$947$		−	27.1506i		−	0.882276i	0.897439	−	0.441138i	$-0.145425\pi$
$948$		0			0
$949$	−12.6735			−0.411401
$950$		0			0
$951$	−3.84351			−0.124634
$952$		0			0
$953$		−	43.1169i		−	1.39669i	−0.715760	−	0.698346i	$-0.753920\pi$
$953$		−	43.1169i		−	1.39669i	0.715760	−	0.698346i	$-0.246080\pi$
$954$		0			0
$955$	−23.5911	+	1.26762i	−0.763389	+	0.0410193i
$956$		0			0
$957$		−	0.587377i		−	0.0189872i
$958$		0			0
$959$	3.08044			0.0994725
$960$		0			0
$961$	14.4055			0.464695
$962$		0			0
$963$		−	45.2826i		−	1.45921i
$964$		0			0
$965$	−1.41899	−	26.4081i	−0.0456789	−	0.850107i
$966$		0			0
$967$			12.0641i			0.387955i	0.981006	+	0.193978i	$0.0621390\pi$
$967$			12.0641i			0.387955i	−0.981006	+	0.193978i	$0.937861\pi$
$968$		0			0
$969$	22.2352			0.714298
$970$		0			0
$971$	−3.81156			−0.122319			−0.0611595	−	0.998128i	$-0.519480\pi$
$971$	−3.81156			−0.122319			−0.0611595	+	0.998128i	$0.519480\pi$
$972$		0			0
$973$			0.703403i			0.0225501i
$974$		0			0
$975$	−13.0397	+	1.40539i	−0.417605	+	0.0450084i
$976$		0			0
$977$		−	39.4202i		−	1.26116i	−0.776123	−	0.630581i	$-0.782816\pi$
$977$		−	39.4202i		−	1.26116i	0.776123	−	0.630581i	$-0.217184\pi$
$978$		0			0
$979$	−9.56998			−0.305858
$980$		0			0
$981$	−9.41440			−0.300579
$982$		0			0
$983$		−	8.67651i		−	0.276738i	−0.990381	−	0.138369i	$-0.955814\pi$
$983$		−	8.67651i		−	0.276738i	0.990381	−	0.138369i	$-0.0441859\pi$
$984$		0			0
$985$	−0.157354	−	2.92843i	−0.00501371	−	0.0933076i
$986$		0			0
$987$			1.53237i			0.0487758i
$988$		0			0
$989$	40.9314			1.30154
$990$		0			0
$991$	35.7256			1.13486			0.567430	−	0.823422i	$-0.307938\pi$
$991$	35.7256			1.13486			0.567430	+	0.823422i	$0.307938\pi$
$992$		0			0
$993$		−	0.376725i		−	0.0119550i
$994$		0			0
$995$	−26.4081	+	1.41899i	−0.837194	+	0.0449850i
$996$		0			0
$997$			26.3453i			0.834365i	0.908823	+	0.417183i	$0.136982\pi$
$997$			26.3453i			0.834365i	−0.908823	+	0.417183i	$0.863018\pi$
$998$		0			0
$999$	32.7421			1.03591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.b.d.89.6	yes	8
3.2	odd	2		3960.2.d.f.3169.7		8
4.3	odd	2		880.2.b.j.529.3		8
5.2	odd	4		2200.2.a.y.1.3		4
5.3	odd	4		2200.2.a.x.1.2		4
5.4	even	2	inner	440.2.b.d.89.3	✓	8
15.14	odd	2		3960.2.d.f.3169.8		8
20.3	even	4		4400.2.a.ce.1.3		4
20.7	even	4		4400.2.a.cb.1.2		4
20.19	odd	2		880.2.b.j.529.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.3	✓	8	5.4	even	2	inner
440.2.b.d.89.6	yes	8	1.1	even	1	trivial
880.2.b.j.529.3		8	4.3	odd	2
880.2.b.j.529.6		8	20.19	odd	2
2200.2.a.x.1.2		4	5.3	odd	4
2200.2.a.y.1.3		4	5.2	odd	4
3960.2.d.f.3169.7		8	3.2	odd	2
3960.2.d.f.3169.8		8	15.14	odd	2
4400.2.a.cb.1.2		4	20.7	even	4
4400.2.a.ce.1.3		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.655762i q^{3} +(-2.23285 + 0.119978i) q^{5} +0.415806i q^{7} +2.56998 q^{9} +O(q^{10})\)	\(q+0.655762i q^{3} +(-2.23285 + 0.119978i) q^{5} +0.415806i q^{7} +2.56998 q^{9} +1.00000 q^{11} +4.00000i q^{13} +(-0.0786769 - 1.46422i) q^{15} +6.51558i q^{17} -5.20406 q^{19} -0.272670 q^{21} +8.54830i q^{23} +(4.97121 - 0.535784i) q^{25} +3.65258i q^{27} -0.895717 q^{29} -6.73836 q^{31} +0.655762i q^{33} +(-0.0498875 - 0.928432i) q^{35} -8.96410i q^{37} -2.62305 q^{39} +10.0998 q^{41} -4.78825i q^{43} +(-5.73836 + 0.308340i) q^{45} -5.61986i q^{47} +6.82711 q^{49} -4.27267 q^{51} +10.0357i q^{53} +(-2.23285 + 0.119978i) q^{55} -3.41262i q^{57} +1.63408 q^{59} +7.10428 q^{61} +1.06861i q^{63} +(-0.479911 - 8.93139i) q^{65} -10.6914i q^{67} -5.60565 q^{69} -6.19302 q^{71} +3.16839i q^{73} +(0.351347 + 3.25993i) q^{75} +0.415806i q^{77} +11.2682 q^{79} +5.31471 q^{81} +16.2429i q^{83} +(-0.781725 - 14.5483i) q^{85} -0.587377i q^{87} -9.56998 q^{89} -1.66323 q^{91} -4.41876i q^{93} +(11.6199 - 0.624371i) q^{95} -0.591657i q^{97} +2.56998 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9}+O(q^{10}) \) 8 * q - 14 * q^9	\( 8 q - 14 q^{9} + 8 q^{11} - 12 q^{15} + 18 q^{19} - 14 q^{21} - 2 q^{25} - 6 q^{29} - 30 q^{31} + 30 q^{35} - 8 q^{39} + 20 q^{41} - 22 q^{45} - 18 q^{49} - 46 q^{51} - 12 q^{59} + 58 q^{61} - 8 q^{65} + 60 q^{69} - 2 q^{71} + 26 q^{75} + 40 q^{79} + 88 q^{81} - 26 q^{85} - 42 q^{89} + 8 q^{91} + 28 q^{95} - 14 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 + 8 * q^11 - 12 * q^15 + 18 * q^19 - 14 * q^21 - 2 * q^25 - 6 * q^29 - 30 * q^31 + 30 * q^35 - 8 * q^39 + 20 * q^41 - 22 * q^45 - 18 * q^49 - 46 * q^51 - 12 * q^59 + 58 * q^61 - 8 * q^65 + 60 * q^69 - 2 * q^71 + 26 * q^75 + 40 * q^79 + 88 * q^81 - 26 * q^85 - 42 * q^89 + 8 * q^91 + 28 * q^95 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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