LMFDB - Embedded newform 220.3.f.a.21.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [220,3,Mod(21,220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(220, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("220.21");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	220.f (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:	$5.99456581593$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 67x^{6} + 1356x^{4} + 9065x^{2} + 17275 $ x^8 + 67x^6 + 1356x^4 + 9065*x^2 + 17275
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			21.4
Root			$1.79638i$ of defining polynomial
Character	$\chi$	$=$	220.21
Dual form			220.3.f.a.21.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41553 q^{3} +2.23607 q^{5} +0.558647i q^{7} +2.66584 q^{9} +O(q^{10})$	$q-3.41553 q^{3} +2.23607 q^{5} +0.558647i q^{7} +2.66584 q^{9} +(2.82054 - 10.6322i) q^{11} +18.1072i q^{13} -7.63736 q^{15} +32.0402i q^{17} +15.7408i q^{19} -1.90807i q^{21} +31.7468 q^{23} +5.00000 q^{25} +21.6345 q^{27} +48.3396i q^{29} -43.3557 q^{31} +(-9.63363 + 36.3147i) q^{33} +1.24917i q^{35} +21.4366 q^{37} -61.8458i q^{39} -38.9133i q^{41} +23.3044i q^{43} +5.96100 q^{45} -75.2975 q^{47} +48.6879 q^{49} -109.434i q^{51} -8.43533 q^{53} +(6.30692 - 23.7744i) q^{55} -53.7631i q^{57} -29.6094 q^{59} -64.1200i q^{61} +1.48926i q^{63} +40.4890i q^{65} +18.8211 q^{67} -108.432 q^{69} +94.8085 q^{71} -0.945835i q^{73} -17.0776 q^{75} +(5.93967 + 1.57568i) q^{77} -73.2825i q^{79} -97.8859 q^{81} +161.659i q^{83} +71.6440i q^{85} -165.105i q^{87} +91.5556 q^{89} -10.1155 q^{91} +148.083 q^{93} +35.1974i q^{95} -33.1491 q^{97} +(7.51910 - 28.3438i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 28 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 28 * q^9	$ 8 q - 8 q^{3} + 28 q^{9} + 24 q^{11} + 20 q^{15} + 56 q^{23} + 40 q^{25} - 80 q^{27} + 20 q^{31} + 88 q^{33} + 72 q^{37} - 184 q^{47} - 244 q^{49} + 136 q^{53} + 60 q^{55} - 16 q^{59} - 264 q^{67} - 56 q^{69} - 220 q^{71} - 40 q^{75} + 208 q^{77} - 224 q^{81} + 220 q^{89} - 72 q^{91} - 104 q^{93} - 8 q^{97} - 40 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 28 * q^9 + 24 * q^11 + 20 * q^15 + 56 * q^23 + 40 * q^25 - 80 * q^27 + 20 * q^31 + 88 * q^33 + 72 * q^37 - 184 * q^47 - 244 * q^49 + 136 * q^53 + 60 * q^55 - 16 * q^59 - 264 * q^67 - 56 * q^69 - 220 * q^71 - 40 * q^75 + 208 * q^77 - 224 * q^81 + 220 * q^89 - 72 * q^91 - 104 * q^93 - 8 * q^97 - 40 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.41553			−1.13851			−0.569255	−	0.822161i	$-0.692768\pi$
$3$	−3.41553			−1.13851			−0.569255	+	0.822161i	$0.692768\pi$
$4$		0			0
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$			0.558647i			0.0798067i	0.999204	+	0.0399033i	$0.0127050\pi$
$7$			0.558647i			0.0798067i	−0.999204	+	0.0399033i	$0.987295\pi$
$8$		0			0
$9$	2.66584			0.296204
$10$		0			0
$11$	2.82054	−	10.6322i	0.256413	−	0.966567i
$11$	2.82054	−	10.6322i	0.256413	−	0.966567i
$12$		0			0
$13$			18.1072i			1.39286i	0.717623	+	0.696432i	$0.245230\pi$
$13$			18.1072i			1.39286i	−0.717623	+	0.696432i	$0.754770\pi$
$14$		0			0
$15$	−7.63736			−0.509157
$16$		0			0
$17$			32.0402i			1.88472i	0.334605	+	0.942359i	$0.391397\pi$
$17$			32.0402i			1.88472i	−0.334605	+	0.942359i	$0.608603\pi$
$18$		0			0
$19$			15.7408i			0.828462i	0.910172	+	0.414231i	$0.135949\pi$
$19$			15.7408i			0.828462i	−0.910172	+	0.414231i	$0.864051\pi$
$20$		0			0
$21$		−	1.90807i		−	0.0908607i
$22$		0			0
$23$	31.7468			1.38030			0.690149	−	0.723667i	$-0.257545\pi$
$23$	31.7468			1.38030			0.690149	+	0.723667i	$0.257545\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$	21.6345			0.801278
$28$		0			0
$29$			48.3396i			1.66688i	0.552608	+	0.833442i	$0.313633\pi$
$29$			48.3396i			1.66688i	−0.552608	+	0.833442i	$0.686367\pi$
$30$		0			0
$31$	−43.3557			−1.39857			−0.699286	−	0.714842i	$-0.746498\pi$
$31$	−43.3557			−1.39857			−0.699286	+	0.714842i	$0.746498\pi$
$32$		0			0
$33$	−9.63363	+	36.3147i	−0.291928	+	1.10045i
$34$		0			0
$35$			1.24917i			0.0356906i
$36$		0			0
$37$	21.4366			0.579367			0.289683	−	0.957123i	$-0.406450\pi$
$37$	21.4366			0.579367			0.289683	+	0.957123i	$0.406450\pi$
$38$		0			0
$39$		−	61.8458i		−	1.58579i
$40$		0			0
$41$		−	38.9133i		−	0.949106i	−0.880227	−	0.474553i	$-0.842610\pi$
$41$		−	38.9133i		−	0.949106i	0.880227	−	0.474553i	$-0.157390\pi$
$42$		0			0
$43$			23.3044i			0.541964i	0.962584	+	0.270982i	$0.0873483\pi$
$43$			23.3044i			0.541964i	−0.962584	+	0.270982i	$0.912652\pi$
$44$		0			0
$45$	5.96100			0.132467
$46$		0			0
$47$	−75.2975			−1.60207			−0.801037	−	0.598614i	$-0.795718\pi$
$47$	−75.2975			−1.60207			−0.801037	+	0.598614i	$0.795718\pi$
$48$		0			0
$49$	48.6879			0.993631
$50$		0			0
$51$		−	109.434i		−	2.14577i
$52$		0			0
$53$	−8.43533			−0.159157			−0.0795786	−	0.996829i	$-0.525357\pi$
$53$	−8.43533			−0.159157			−0.0795786	+	0.996829i	$0.525357\pi$
$54$		0			0
$55$	6.30692	−	23.7744i	0.114671	−	0.432262i
$56$		0			0
$57$		−	53.7631i		−	0.943212i
$58$		0			0
$59$	−29.6094			−0.501854			−0.250927	−	0.968006i	$-0.580735\pi$
$59$	−29.6094			−0.501854			−0.250927	+	0.968006i	$0.580735\pi$
$60$		0			0
$61$		−	64.1200i		−	1.05115i	−0.850748	−	0.525573i	$-0.823851\pi$
$61$		−	64.1200i		−	1.05115i	0.850748	−	0.525573i	$-0.176149\pi$
$62$		0			0
$63$			1.48926i			0.0236391i
$64$		0			0
$65$			40.4890i			0.622908i
$66$		0			0
$67$	18.8211			0.280912			0.140456	−	0.990087i	$-0.455143\pi$
$67$	18.8211			0.280912			0.140456	+	0.990087i	$0.455143\pi$
$68$		0			0
$69$	−108.432			−1.57148
$70$		0			0
$71$	94.8085			1.33533			0.667666	−	0.744461i	$-0.267294\pi$
$71$	94.8085			1.33533			0.667666	+	0.744461i	$0.267294\pi$
$72$		0			0
$73$		−	0.945835i		−	0.0129566i	−0.999979	−	0.00647832i	$-0.997938\pi$
$73$		−	0.945835i		−	0.0129566i	0.999979	−	0.00647832i	$-0.00206213\pi$
$74$		0			0
$75$	−17.0776			−0.227702
$76$		0			0
$77$	5.93967	+	1.57568i	0.0771385	+	0.0204634i
$78$		0			0
$79$		−	73.2825i		−	0.927626i	−0.885933	−	0.463813i	$-0.846481\pi$
$79$		−	73.2825i		−	0.927626i	0.885933	−	0.463813i	$-0.153519\pi$
$80$		0			0
$81$	−97.8859			−1.20847
$82$		0			0
$83$			161.659i			1.94770i	0.227192	+	0.973850i	$0.427045\pi$
$83$			161.659i			1.94770i	−0.227192	+	0.973850i	$0.572955\pi$
$84$		0			0
$85$			71.6440i			0.842871i
$86$		0			0
$87$		−	165.105i		−	1.89776i
$88$		0			0
$89$	91.5556			1.02871			0.514357	−	0.857576i	$-0.328031\pi$
$89$	91.5556			1.02871			0.514357	+	0.857576i	$0.328031\pi$
$90$		0			0
$91$	−10.1155			−0.111160
$92$		0			0
$93$	148.083			1.59229
$94$		0			0
$95$			35.1974i			0.370499i
$96$		0			0
$97$	−33.1491			−0.341743			−0.170872	−	0.985293i	$-0.554658\pi$
$97$	−33.1491			−0.341743			−0.170872	+	0.985293i	$0.554658\pi$
$98$		0			0
$99$	7.51910	−	28.3438i	0.0759505	−	0.286301i
$100$		0			0
$101$			42.0879i			0.416712i	0.978053	+	0.208356i	$0.0668112\pi$
$101$			42.0879i			0.416712i	−0.978053	+	0.208356i	$0.933189\pi$
$102$		0			0
$103$	−75.9577			−0.737453			−0.368726	−	0.929538i	$-0.620206\pi$
$103$	−75.9577			−0.737453			−0.368726	+	0.929538i	$0.620206\pi$
$104$		0			0
$105$		−	4.26658i		−	0.0406341i
$106$		0			0
$107$		−	36.5864i		−	0.341929i	−0.985277	−	0.170965i	$-0.945312\pi$
$107$		−	36.5864i		−	0.341929i	0.985277	−	0.170965i	$-0.0546884\pi$
$108$		0			0
$109$		−	21.7920i		−	0.199927i	−0.994991	−	0.0999633i	$-0.968127\pi$
$109$		−	21.7920i		−	0.199927i	0.994991	−	0.0999633i	$-0.0318725\pi$
$110$		0			0
$111$	−73.2172			−0.659615
$112$		0			0
$113$	−47.1444			−0.417207			−0.208604	−	0.978000i	$-0.566892\pi$
$113$	−47.1444			−0.417207			−0.208604	+	0.978000i	$0.566892\pi$
$114$		0			0
$115$	70.9881			0.617288
$116$		0			0
$117$			48.2710i			0.412572i
$118$		0			0
$119$	−17.8991			−0.150413
$120$		0			0
$121$	−105.089	−	59.9773i	−0.868505	−	0.495680i
$122$		0			0
$123$			132.910i			1.08057i
$124$		0			0
$125$	11.1803			0.0894427
$126$		0			0
$127$		−	157.516i		−	1.24028i	−0.784490	−	0.620141i	$-0.787075\pi$
$127$		−	157.516i		−	1.24028i	0.784490	−	0.620141i	$-0.212925\pi$
$128$		0			0
$129$		−	79.5970i		−	0.617031i
$130$		0			0
$131$			106.106i			0.809966i	0.914324	+	0.404983i	$0.132723\pi$
$131$			106.106i			0.809966i	−0.914324	+	0.404983i	$0.867277\pi$
$132$		0			0
$133$	−8.79353			−0.0661168
$134$		0			0
$135$	48.3762			0.358343
$136$		0			0
$137$	87.9617			0.642057			0.321028	−	0.947070i	$-0.395972\pi$
$137$	87.9617			0.642057			0.321028	+	0.947070i	$0.395972\pi$
$138$		0			0
$139$			24.4159i			0.175654i	0.996136	+	0.0878268i	$0.0279922\pi$
$139$			24.4159i			0.175654i	−0.996136	+	0.0878268i	$0.972008\pi$
$140$		0			0
$141$	257.181			1.82398
$142$		0			0
$143$	192.521	+	51.0722i	1.34630	+	0.357148i
$144$		0			0
$145$			108.091i			0.745453i
$146$		0			0
$147$	−166.295			−1.13126
$148$		0			0
$149$			186.092i			1.24894i	0.781050	+	0.624469i	$0.214685\pi$
$149$			186.092i			1.24894i	−0.781050	+	0.624469i	$0.785315\pi$
$150$		0			0
$151$		−	5.26044i		−	0.0348374i	−0.999848	−	0.0174187i	$-0.994455\pi$
$151$		−	5.26044i		−	0.0348374i	0.999848	−	0.0174187i	$-0.00554482\pi$
$152$		0			0
$153$			85.4140i			0.558261i
$154$		0			0
$155$	−96.9463			−0.625460
$156$		0			0
$157$	−220.052			−1.40161			−0.700804	−	0.713354i	$-0.747175\pi$
$157$	−220.052			−1.40161			−0.700804	+	0.713354i	$0.747175\pi$
$158$		0			0
$159$	28.8111			0.181202
$160$		0			0
$161$			17.7353i			0.110157i
$162$		0			0
$163$	−11.9134			−0.0730881			−0.0365441	−	0.999332i	$-0.511635\pi$
$163$	−11.9134			−0.0730881			−0.0365441	+	0.999332i	$0.511635\pi$
$164$		0			0
$165$	−21.5415	+	81.2022i	−0.130554	+	0.492135i
$166$		0			0
$167$			10.5984i			0.0634633i	0.999496	+	0.0317317i	$0.0101022\pi$
$167$			10.5984i			0.0634633i	−0.999496	+	0.0317317i	$0.989898\pi$
$168$		0			0
$169$	−158.872			−0.940071
$170$		0			0
$171$			41.9624i			0.245394i
$172$		0			0
$173$			181.818i			1.05097i	0.850803	+	0.525485i	$0.176116\pi$
$173$			181.818i			1.05097i	−0.850803	+	0.525485i	$0.823884\pi$
$174$		0			0
$175$			2.79323i			0.0159613i
$176$		0			0
$177$	101.132			0.571366
$178$		0			0
$179$	72.6441			0.405833			0.202916	−	0.979196i	$-0.434958\pi$
$179$	72.6441			0.405833			0.202916	+	0.979196i	$0.434958\pi$
$180$		0			0
$181$	325.085			1.79605			0.898024	−	0.439946i	$-0.145003\pi$
$181$	325.085			1.79605			0.898024	+	0.439946i	$0.145003\pi$
$182$		0			0
$183$			219.004i			1.19674i
$184$		0			0
$185$	47.9336			0.259101
$186$		0			0
$187$	340.659	+	90.3706i	1.82171	+	0.483265i
$188$		0			0
$189$			12.0860i			0.0639473i
$190$		0			0
$191$	−199.153			−1.04268			−0.521342	−	0.853348i	$-0.674568\pi$
$191$	−199.153			−1.04268			−0.521342	+	0.853348i	$0.674568\pi$
$192$		0			0
$193$		−	284.097i		−	1.47201i	−0.676978	−	0.736003i	$-0.736711\pi$
$193$		−	284.097i		−	1.47201i	0.676978	−	0.736003i	$-0.263289\pi$
$194$		0			0
$195$		−	138.291i		−	0.709187i
$196$		0			0
$197$			230.008i			1.16755i	0.811914	+	0.583777i	$0.198426\pi$
$197$			230.008i			1.16755i	−0.811914	+	0.583777i	$0.801574\pi$
$198$		0			0
$199$	−123.328			−0.619740			−0.309870	−	0.950779i	$-0.600286\pi$
$199$	−123.328			−0.619740			−0.309870	+	0.950779i	$0.600286\pi$
$200$		0			0
$201$	−64.2841			−0.319822
$202$		0			0
$203$	−27.0048			−0.133028
$204$		0			0
$205$		−	87.0128i		−	0.424453i
$206$		0			0
$207$	84.6320			0.408850
$208$		0			0
$209$	167.360	+	44.3975i	0.800764	+	0.212428i
$210$		0			0
$211$		−	214.886i		−	1.01842i	−0.860643	−	0.509209i	$-0.829938\pi$
$211$		−	214.886i		−	1.01842i	0.860643	−	0.509209i	$-0.170062\pi$
$212$		0			0
$213$	−323.821			−1.52029
$214$		0			0
$215$			52.1103i			0.242373i
$216$		0			0
$217$		−	24.2205i		−	0.111615i
$218$		0			0
$219$			3.23053i			0.0147513i
$220$		0			0
$221$	−580.159			−2.62516
$222$		0			0
$223$	59.8548			0.268407			0.134204	−	0.990954i	$-0.457152\pi$
$223$	59.8548			0.268407			0.134204	+	0.990954i	$0.457152\pi$
$224$		0			0
$225$	13.3292			0.0592409
$226$		0			0
$227$		−	203.787i		−	0.897740i	−0.893597	−	0.448870i	$-0.851827\pi$
$227$		−	203.787i		−	0.897740i	0.893597	−	0.448870i	$-0.148173\pi$
$228$		0			0
$229$	265.375			1.15884			0.579421	−	0.815028i	$-0.303279\pi$
$229$	265.375			1.15884			0.579421	+	0.815028i	$0.303279\pi$
$230$		0			0
$231$	−20.2871	−	5.38180i	−0.0878230	−	0.0232978i
$232$		0			0
$233$		−	313.018i		−	1.34343i	−0.740812	−	0.671713i	$-0.765559\pi$
$233$		−	313.018i		−	1.34343i	0.740812	−	0.671713i	$-0.234441\pi$
$234$		0			0
$235$	−168.370			−0.716470
$236$		0			0
$237$			250.298i			1.05611i
$238$		0			0
$239$			159.796i			0.668604i	0.942466	+	0.334302i	$0.108501\pi$
$239$			159.796i			0.668604i	−0.942466	+	0.334302i	$0.891499\pi$
$240$		0			0
$241$		−	467.884i		−	1.94143i	−0.240242	−	0.970713i	$-0.577227\pi$
$241$		−	467.884i		−	1.94143i	0.240242	−	0.970713i	$-0.422773\pi$
$242$		0			0
$243$	139.621			0.574574
$244$		0			0
$245$	108.869			0.444365
$246$		0			0
$247$	−285.022			−1.15393
$248$		0			0
$249$		−	552.151i		−	2.21748i
$250$		0			0
$251$	312.122			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$251$	312.122			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$252$		0			0
$253$	89.5432	−	337.540i	0.353926	−	1.33415i
$254$		0			0
$255$		−	244.702i		−	0.959617i
$256$		0			0
$257$	139.735			0.543714			0.271857	−	0.962338i	$-0.412362\pi$
$257$	139.735			0.543714			0.271857	+	0.962338i	$0.412362\pi$
$258$		0			0
$259$			11.9755i			0.0462373i
$260$		0			0
$261$			128.866i			0.493738i
$262$		0			0
$263$		−	181.115i		−	0.688651i	−0.938850	−	0.344326i	$-0.888108\pi$
$263$		−	181.115i		−	0.688651i	0.938850	−	0.344326i	$-0.111892\pi$
$264$		0			0
$265$	−18.8620			−0.0711772
$266$		0			0
$267$	−312.711			−1.17120
$268$		0			0
$269$	−96.4486			−0.358545			−0.179273	−	0.983799i	$-0.557374\pi$
$269$	−96.4486			−0.358545			−0.179273	+	0.983799i	$0.557374\pi$
$270$		0			0
$271$			296.758i			1.09505i	0.836790	+	0.547525i	$0.184430\pi$
$271$			296.758i			1.09505i	−0.836790	+	0.547525i	$0.815570\pi$
$272$		0			0
$273$	34.5499			0.126557
$274$		0			0
$275$	14.1027	−	53.1612i	0.0512825	−	0.193313i
$276$		0			0
$277$			165.044i			0.595826i	0.954593	+	0.297913i	$0.0962905\pi$
$277$			165.044i			0.595826i	−0.954593	+	0.297913i	$0.903709\pi$
$278$		0			0
$279$	−115.579			−0.414263
$280$		0			0
$281$		−	229.230i		−	0.815765i	−0.913034	−	0.407883i	$-0.866267\pi$
$281$		−	229.230i		−	0.815765i	0.913034	−	0.407883i	$-0.133733\pi$
$282$		0			0
$283$		−	133.408i		−	0.471407i	−0.971825	−	0.235703i	$-0.924261\pi$
$283$		−	133.408i		−	0.471407i	0.971825	−	0.235703i	$-0.0757394\pi$
$284$		0			0
$285$		−	120.218i		−	0.421817i
$286$		0			0
$287$	21.7388			0.0757449
$288$		0			0
$289$	−737.574			−2.55216
$290$		0			0
$291$	113.222			0.389078
$292$		0			0
$293$		−	141.969i		−	0.484535i	−0.970209	−	0.242268i	$-0.922109\pi$
$293$		−	141.969i		−	0.484535i	0.970209	−	0.242268i	$-0.0778912\pi$
$294$		0			0
$295$	−66.2086			−0.224436
$296$		0			0
$297$	61.0210	−	230.023i	0.205458	−	0.774489i
$298$		0			0
$299$			574.848i			1.92257i
$300$		0			0
$301$	−13.0189			−0.0432523
$302$		0			0
$303$		−	143.752i		−	0.474430i
$304$		0			0
$305$		−	143.377i		−	0.470087i
$306$		0			0
$307$			405.167i			1.31976i	0.751370	+	0.659881i	$0.229393\pi$
$307$			405.167i			1.31976i	−0.751370	+	0.659881i	$0.770607\pi$
$308$		0			0
$309$	259.436			0.839597
$310$		0			0
$311$	518.227			1.66633			0.833163	−	0.553028i	$-0.186528\pi$
$311$	518.227			1.66633			0.833163	+	0.553028i	$0.186528\pi$
$312$		0			0
$313$	−177.327			−0.566539			−0.283270	−	0.959040i	$-0.591419\pi$
$313$	−177.327			−0.566539			−0.283270	+	0.959040i	$0.591419\pi$
$314$		0			0
$315$			3.33009i			0.0105717i
$316$		0			0
$317$	509.414			1.60698			0.803492	−	0.595315i	$-0.202973\pi$
$317$	509.414			1.60698			0.803492	+	0.595315i	$0.202973\pi$
$318$		0			0
$319$	513.958	+	136.344i	1.61115	+	0.427410i
$320$		0			0
$321$			124.962i			0.389290i
$322$		0			0
$323$	−504.337			−1.56142
$324$		0			0
$325$			90.5362i			0.278573i
$326$		0			0
$327$			74.4312i			0.227618i
$328$		0			0
$329$		−	42.0647i		−	0.127856i
$330$		0			0
$331$	−360.510			−1.08915			−0.544577	−	0.838711i	$-0.683310\pi$
$331$	−360.510			−1.08915			−0.544577	+	0.838711i	$0.683310\pi$
$332$		0			0
$333$	57.1464			0.171611
$334$		0			0
$335$	42.0853			0.125628
$336$		0			0
$337$			419.994i			1.24627i	0.782113	+	0.623136i	$0.214142\pi$
$337$			419.994i			1.24627i	−0.782113	+	0.623136i	$0.785858\pi$
$338$		0			0
$339$	161.023			0.474995
$340$		0			0
$341$	−122.286	+	460.968i	−0.358611	+	1.35181i
$342$		0			0
$343$			54.5730i			0.159105i
$344$		0			0
$345$	−242.462			−0.702788
$346$		0			0
$347$		−	278.399i		−	0.802303i	−0.916012	−	0.401151i	$-0.868610\pi$
$347$		−	278.399i		−	0.802303i	0.916012	−	0.401151i	$-0.131390\pi$
$348$		0			0
$349$		−	426.489i		−	1.22203i	−0.791618	−	0.611016i	$-0.790761\pi$
$349$		−	426.489i		−	1.22203i	0.791618	−	0.611016i	$-0.209239\pi$
$350$		0			0
$351$			391.741i			1.11607i
$352$		0			0
$353$	−114.688			−0.324894			−0.162447	−	0.986717i	$-0.551939\pi$
$353$	−114.688			−0.324894			−0.162447	+	0.986717i	$0.551939\pi$
$354$		0			0
$355$	211.998			0.597178
$356$		0			0
$357$	61.1351			0.171247
$358$		0			0
$359$			545.543i			1.51962i	0.650147	+	0.759809i	$0.274707\pi$
$359$			545.543i			1.51962i	−0.650147	+	0.759809i	$0.725293\pi$
$360$		0			0
$361$	113.228			0.313651
$362$		0			0
$363$	358.935	+	204.854i	0.988802	+	0.564337i
$364$		0			0
$365$		−	2.11495i		−	0.00579439i
$366$		0			0
$367$	265.619			0.723759			0.361879	−	0.932225i	$-0.382135\pi$
$367$	265.619			0.723759			0.361879	+	0.932225i	$0.382135\pi$
$368$		0			0
$369$		−	103.737i		−	0.281129i
$370$		0			0
$371$		−	4.71237i		−	0.0127018i
$372$		0			0
$373$		−	202.250i		−	0.542225i	−0.962548	−	0.271113i	$-0.912608\pi$
$373$		−	202.250i		−	0.542225i	0.962548	−	0.271113i	$-0.0873916\pi$
$374$		0			0
$375$	−38.1868			−0.101831
$376$		0			0
$377$	−875.297			−2.32174
$378$		0			0
$379$	114.650			0.302506			0.151253	−	0.988495i	$-0.451669\pi$
$379$	114.650			0.302506			0.151253	+	0.988495i	$0.451669\pi$
$380$		0			0
$381$			538.000i			1.41207i
$382$		0			0
$383$	−373.418			−0.974982			−0.487491	−	0.873128i	$-0.662088\pi$
$383$	−373.418			−0.974982			−0.487491	+	0.873128i	$0.662088\pi$
$384$		0			0
$385$	13.2815	+	3.52334i	0.0344974	+	0.00915153i
$386$		0			0
$387$			62.1259i			0.160532i
$388$		0			0
$389$	241.139			0.619896			0.309948	−	0.950754i	$-0.399688\pi$
$389$	241.139			0.619896			0.309948	+	0.950754i	$0.399688\pi$
$390$		0			0
$391$			1017.18i			2.60147i
$392$		0			0
$393$		−	362.406i		−	0.922154i
$394$		0			0
$395$		−	163.865i		−	0.414847i
$396$		0			0
$397$	467.321			1.17713			0.588566	−	0.808449i	$-0.299693\pi$
$397$	467.321			1.17713			0.588566	+	0.808449i	$0.299693\pi$
$398$		0			0
$399$	30.0346			0.0752746
$400$		0			0
$401$	680.945			1.69812			0.849059	−	0.528298i	$-0.177170\pi$
$401$	680.945			1.69812			0.849059	+	0.528298i	$0.177170\pi$
$402$		0			0
$403$		−	785.052i		−	1.94802i
$404$		0			0
$405$	−218.879			−0.540443
$406$		0			0
$407$	60.4627	−	227.919i	0.148557	−	0.559997i
$408$		0			0
$409$		−	599.500i		−	1.46577i	−0.680353	−	0.732885i	$-0.738173\pi$
$409$		−	599.500i		−	1.46577i	0.680353	−	0.732885i	$-0.261827\pi$
$410$		0			0
$411$	−300.436			−0.730988
$412$		0			0
$413$		−	16.5412i		−	0.0400513i
$414$		0			0
$415$			361.481i			0.871038i
$416$		0			0
$417$		−	83.3931i		−	0.199983i
$418$		0			0
$419$	−256.808			−0.612906			−0.306453	−	0.951886i	$-0.599142\pi$
$419$	−256.808			−0.612906			−0.306453	+	0.951886i	$0.599142\pi$
$420$		0			0
$421$	23.4654			0.0557373			0.0278686	−	0.999612i	$-0.491128\pi$
$421$	23.4654			0.0557373			0.0278686	+	0.999612i	$0.491128\pi$
$422$		0			0
$423$	−200.731			−0.474541
$424$		0			0
$425$			160.201i			0.376943i
$426$		0			0
$427$	35.8204			0.0838885
$428$		0			0
$429$	−657.559	−	174.438i	−1.53277	−	0.406616i
$430$		0			0
$431$			228.370i			0.529860i	0.964268	+	0.264930i	$0.0853488\pi$
$431$			228.370i			0.529860i	−0.964268	+	0.264930i	$0.914651\pi$
$432$		0			0
$433$	564.969			1.30478			0.652389	−	0.757884i	$-0.273767\pi$
$433$	564.969			1.30478			0.652389	+	0.757884i	$0.273767\pi$
$434$		0			0
$435$		−	369.187i		−	0.848705i
$436$		0			0
$437$			499.720i			1.14352i
$438$		0			0
$439$			500.617i			1.14036i	0.821521	+	0.570178i	$0.193126\pi$
$439$			500.617i			1.14036i	−0.821521	+	0.570178i	$0.806874\pi$
$440$		0			0
$441$	129.794			0.294318
$442$		0			0
$443$	365.948			0.826069			0.413034	−	0.910715i	$-0.364469\pi$
$443$	365.948			0.826069			0.413034	+	0.910715i	$0.364469\pi$
$444$		0			0
$445$	204.724			0.460055
$446$		0			0
$447$		−	635.602i		−	1.42193i
$448$		0			0
$449$	−316.528			−0.704963			−0.352481	−	0.935819i	$-0.614662\pi$
$449$	−316.528			−0.704963			−0.352481	+	0.935819i	$0.614662\pi$
$450$		0			0
$451$	−413.736	−	109.757i	−0.917375	−	0.243363i
$452$		0			0
$453$			17.9672i			0.0396627i
$454$		0			0
$455$	−22.6191			−0.0497122
$456$		0			0
$457$			291.964i			0.638870i	0.947608	+	0.319435i	$0.103493\pi$
$457$			291.964i			0.638870i	−0.947608	+	0.319435i	$0.896507\pi$
$458$		0			0
$459$			693.174i			1.51018i
$460$		0			0
$461$			431.260i			0.935489i	0.883864	+	0.467744i	$0.154933\pi$
$461$			431.260i			0.935489i	−0.883864	+	0.467744i	$0.845067\pi$
$462$		0			0
$463$	264.022			0.570243			0.285121	−	0.958491i	$-0.407966\pi$
$463$	264.022			0.570243			0.285121	+	0.958491i	$0.407966\pi$
$464$		0			0
$465$	331.123			0.712093
$466$		0			0
$467$	−477.602			−1.02270			−0.511351	−	0.859372i	$-0.670855\pi$
$467$	−477.602			−1.02270			−0.511351	+	0.859372i	$0.670855\pi$
$468$		0			0
$469$			10.5144i			0.0224187i
$470$		0			0
$471$	751.595			1.59574
$472$		0			0
$473$	247.778	+	65.7311i	0.523844	+	0.138966i
$474$		0			0
$475$			78.7039i			0.165692i
$476$		0			0
$477$	−22.4872			−0.0471430
$478$		0			0
$479$		−	279.214i		−	0.582909i	−0.956585	−	0.291455i	$-0.905861\pi$
$479$		−	279.214i		−	0.582909i	0.956585	−	0.291455i	$-0.0941393\pi$
$480$		0			0
$481$			388.157i			0.806979i
$482$		0			0
$483$		−	60.5753i		−	0.125415i
$484$		0			0
$485$	−74.1236			−0.152832
$486$		0			0
$487$	−279.655			−0.574241			−0.287121	−	0.957894i	$-0.592698\pi$
$487$	−279.655			−0.574241			−0.287121	+	0.957894i	$0.592698\pi$
$488$		0			0
$489$	40.6904			0.0832115
$490$		0			0
$491$		−	442.142i		−	0.900493i	−0.892904	−	0.450246i	$-0.851336\pi$
$491$		−	442.142i		−	0.900493i	0.892904	−	0.450246i	$-0.148664\pi$
$492$		0			0
$493$	−1548.81			−3.14160
$494$		0			0
$495$	16.8132	−	63.3788i	0.0339661	−	0.128038i
$496$		0			0
$497$			52.9645i			0.106568i
$498$		0			0
$499$	430.853			0.863432			0.431716	−	0.902010i	$-0.357908\pi$
$499$	430.853			0.863432			0.431716	+	0.902010i	$0.357908\pi$
$500$		0			0
$501$		−	36.1991i		−	0.0722536i
$502$		0			0
$503$		−	293.745i		−	0.583985i	−0.956421	−	0.291993i	$-0.905682\pi$
$503$		−	293.745i		−	0.583985i	0.956421	−	0.291993i	$-0.0943183\pi$
$504$		0			0
$505$			94.1113i			0.186359i
$506$		0			0
$507$	542.632			1.07028
$508$		0			0
$509$	276.608			0.543434			0.271717	−	0.962377i	$-0.412408\pi$
$509$	276.608			0.543434			0.271717	+	0.962377i	$0.412408\pi$
$510$		0			0
$511$	0.528388			0.00103403
$512$		0			0
$513$			340.544i			0.663828i
$514$		0			0
$515$	−169.846			−0.329799
$516$		0			0
$517$	−212.380	+	800.581i	−0.410792	+	1.54851i
$518$		0			0
$519$		−	621.004i		−	1.19654i
$520$		0			0
$521$	571.383			1.09670			0.548352	−	0.836248i	$-0.315255\pi$
$521$	571.383			1.09670			0.548352	+	0.836248i	$0.315255\pi$
$522$		0			0
$523$			255.631i			0.488778i	0.969677	+	0.244389i	$0.0785874\pi$
$523$			255.631i			0.488778i	−0.969677	+	0.244389i	$0.921413\pi$
$524$		0			0
$525$		−	9.54037i		−	0.0181721i
$526$		0			0
$527$		−	1389.13i		−	2.63591i
$528$		0			0
$529$	478.862			0.905222
$530$		0			0
$531$	−78.9339			−0.148651
$532$		0			0
$533$	704.613			1.32198
$534$		0			0
$535$		−	81.8098i		−	0.152915i
$536$		0			0
$537$	−248.118			−0.462045
$538$		0			0
$539$	137.326	−	517.662i	0.254780	−	0.960411i
$540$		0			0
$541$			214.227i			0.395983i	0.980204	+	0.197991i	$0.0634418\pi$
$541$			214.227i			0.395983i	−0.980204	+	0.197991i	$0.936558\pi$
$542$		0			0
$543$	−1110.34			−2.04482
$544$		0			0
$545$		−	48.7284i		−	0.0894099i
$546$		0			0
$547$		−	552.014i		−	1.00917i	−0.863363	−	0.504584i	$-0.831646\pi$
$547$		−	552.014i		−	1.00917i	0.863363	−	0.504584i	$-0.168354\pi$
$548$		0			0
$549$		−	170.933i		−	0.311354i
$550$		0			0
$551$	−760.903			−1.38095
$552$		0			0
$553$	40.9390			0.0740308
$554$		0			0
$555$	−163.719			−0.294989
$556$		0			0
$557$		−	159.593i		−	0.286523i	−0.989685	−	0.143261i	$-0.954241\pi$
$557$		−	159.593i		−	0.286523i	0.989685	−	0.143261i	$-0.0457590\pi$
$558$		0			0
$559$	−421.979			−0.754882
$560$		0			0
$561$	−1163.53	−	308.663i	−2.07403	−	0.550202i
$562$		0			0
$563$		−	603.391i		−	1.07174i	−0.844300	−	0.535871i	$-0.819983\pi$
$563$		−	603.391i		−	1.07174i	0.844300	−	0.535871i	$-0.180017\pi$
$564$		0			0
$565$	−105.418			−0.186581
$566$		0			0
$567$		−	54.6836i		−	0.0964437i
$568$		0			0
$569$		−	228.804i		−	0.402115i	−0.979579	−	0.201058i	$-0.935562\pi$
$569$		−	228.804i		−	0.402115i	0.979579	−	0.201058i	$-0.0644379\pi$
$570$		0			0
$571$		−	682.868i		−	1.19592i	−0.801527	−	0.597958i	$-0.795979\pi$
$571$		−	682.868i		−	1.19592i	0.801527	−	0.597958i	$-0.204021\pi$
$572$		0			0
$573$	680.212			1.18711
$574$		0			0
$575$	158.734			0.276060
$576$		0			0
$577$	486.375			0.842937			0.421468	−	0.906843i	$-0.361515\pi$
$577$	486.375			0.842937			0.421468	+	0.906843i	$0.361515\pi$
$578$		0			0
$579$			970.342i			1.67589i
$580$		0			0
$581$	−90.3103			−0.155439
$582$		0			0
$583$	−23.7922	+	89.6864i	−0.0408099	+	0.153836i
$584$		0			0
$585$			107.937i			0.184508i
$586$		0			0
$587$	411.640			0.701260			0.350630	−	0.936514i	$-0.385967\pi$
$587$	411.640			0.701260			0.350630	+	0.936514i	$0.385967\pi$
$588$		0			0
$589$		−	682.452i		−	1.15866i
$590$		0			0
$591$		−	785.599i		−	1.32927i
$592$		0			0
$593$			495.443i			0.835486i	0.908565	+	0.417743i	$0.137179\pi$
$593$			495.443i			0.835486i	−0.908565	+	0.417743i	$0.862821\pi$
$594$		0			0
$595$	−40.0237			−0.0672667
$596$		0			0
$597$	421.231			0.705580
$598$		0			0
$599$	−482.567			−0.805621			−0.402811	−	0.915283i	$-0.631967\pi$
$599$	−482.567			−0.805621			−0.402811	+	0.915283i	$0.631967\pi$
$600$		0			0
$601$		−	1171.94i		−	1.94999i	−0.222237	−	0.974993i	$-0.571336\pi$
$601$		−	1171.94i		−	1.94999i	0.222237	−	0.974993i	$-0.428664\pi$
$602$		0			0
$603$	50.1741			0.0832075
$604$		0			0
$605$	−234.986	−	134.113i	−0.388407	−	0.221675i
$606$		0			0
$607$		−	22.6334i		−	0.0372873i	−0.999826	−	0.0186437i	$-0.994065\pi$
$607$		−	22.6334i		−	0.0372873i	0.999826	−	0.0186437i	$-0.00593481\pi$
$608$		0			0
$609$	92.2355			0.151454
$610$		0			0
$611$		−	1363.43i		−	2.23147i
$612$		0			0
$613$		−	257.299i		−	0.419738i	−0.977730	−	0.209869i	$-0.932696\pi$
$613$		−	257.299i		−	0.419738i	0.977730	−	0.209869i	$-0.0673037\pi$
$614$		0			0
$615$			297.195i			0.483244i
$616$		0			0
$617$	−250.845			−0.406555			−0.203278	−	0.979121i	$-0.565159\pi$
$617$	−250.845			−0.406555			−0.203278	+	0.979121i	$0.565159\pi$
$618$		0			0
$619$	−454.630			−0.734459			−0.367230	−	0.930130i	$-0.619694\pi$
$619$	−454.630			−0.734459			−0.367230	+	0.930130i	$0.619694\pi$
$620$		0			0
$621$	686.828			1.10600
$622$		0			0
$623$			51.1472i			0.0820982i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−571.622	−	151.641i	−0.911678	−	0.241851i
$628$		0			0
$629$			686.832i			1.09194i
$630$		0			0
$631$	1097.79			1.73977			0.869883	−	0.493257i	$-0.164194\pi$
$631$	1097.79			1.73977			0.869883	+	0.493257i	$0.164194\pi$
$632$		0			0
$633$			733.950i			1.15948i
$634$		0			0
$635$		−	352.216i		−	0.554671i
$636$		0			0
$637$			881.604i			1.38399i
$638$		0			0
$639$	252.744			0.395531
$640$		0			0
$641$	−931.874			−1.45378			−0.726891	−	0.686753i	$-0.759035\pi$
$641$	−931.874			−1.45378			−0.726891	+	0.686753i	$0.759035\pi$
$642$		0			0
$643$	−462.937			−0.719963			−0.359982	−	0.932959i	$-0.617217\pi$
$643$	−462.937			−0.719963			−0.359982	+	0.932959i	$0.617217\pi$
$644$		0			0
$645$		−	177.984i		−	0.275945i
$646$		0			0
$647$	−337.698			−0.521944			−0.260972	−	0.965346i	$-0.584043\pi$
$647$	−337.698			−0.521944			−0.260972	+	0.965346i	$0.584043\pi$
$648$		0			0
$649$	−83.5144	+	314.814i	−0.128682	+	0.485076i
$650$		0			0
$651$			82.7259i			0.127075i
$652$		0			0
$653$	237.239			0.363307			0.181653	−	0.983363i	$-0.441855\pi$
$653$	237.239			0.363307			0.181653	+	0.983363i	$0.441855\pi$
$654$		0			0
$655$			237.259i			0.362228i
$656$		0			0
$657$		−	2.52144i		−	0.00383782i
$658$		0			0
$659$		−	482.840i		−	0.732686i	−0.930480	−	0.366343i	$-0.880610\pi$
$659$		−	482.840i		−	0.732686i	0.930480	−	0.366343i	$-0.119390\pi$
$660$		0			0
$661$	−649.514			−0.982624			−0.491312	−	0.870984i	$-0.663483\pi$
$661$	−649.514			−0.982624			−0.491312	+	0.870984i	$0.663483\pi$
$662$		0			0
$663$	1981.55			2.98876
$664$		0			0
$665$	−19.6629			−0.0295683
$666$		0			0
$667$			1534.63i			2.30079i
$668$		0			0
$669$	−204.436			−0.305584
$670$		0			0
$671$	−681.739	−	180.853i	−1.01600	−	0.269527i
$672$		0			0
$673$			1077.34i			1.60080i	0.599467	+	0.800400i	$0.295379\pi$
$673$			1077.34i			1.60080i	−0.599467	+	0.800400i	$0.704621\pi$
$674$		0			0
$675$	108.173			0.160256
$676$		0			0
$677$			82.8635i			0.122398i	0.998126	+	0.0611990i	$0.0194924\pi$
$677$			82.8635i			0.122398i	−0.998126	+	0.0611990i	$0.980508\pi$
$678$		0			0
$679$		−	18.5186i		−	0.0272734i
$680$		0			0
$681$			696.041i			1.02209i
$682$		0			0
$683$	388.525			0.568851			0.284426	−	0.958698i	$-0.408197\pi$
$683$	388.525			0.568851			0.284426	+	0.958698i	$0.408197\pi$
$684$		0			0
$685$	196.688			0.287136
$686$		0			0
$687$	−906.396			−1.31935
$688$		0			0
$689$		−	152.740i		−	0.221684i
$690$		0			0
$691$	764.841			1.10686			0.553430	−	0.832896i	$-0.313319\pi$
$691$	764.841			1.10686			0.553430	+	0.832896i	$0.313319\pi$
$692$		0			0
$693$	15.8342	+	4.20052i	0.0228488	+	0.00606136i
$694$		0			0
$695$			54.5955i			0.0785547i
$696$		0			0
$697$	1246.79			1.78880
$698$		0			0
$699$			1069.12i			1.52950i
$700$		0			0
$701$		−	229.490i		−	0.327375i	−0.986512	−	0.163688i	$-0.947661\pi$
$701$		−	229.490i		−	0.327375i	0.986512	−	0.163688i	$-0.0523389\pi$
$702$		0			0
$703$			337.428i			0.479983i
$704$		0			0
$705$	575.074			0.815708
$706$		0			0
$707$	−23.5122			−0.0332564
$708$		0			0
$709$	−197.960			−0.279210			−0.139605	−	0.990207i	$-0.544583\pi$
$709$	−197.960			−0.279210			−0.139605	+	0.990207i	$0.544583\pi$
$710$		0			0
$711$		−	195.359i		−	0.274767i
$712$		0			0
$713$	−1376.41			−1.93045
$714$		0			0
$715$	430.489	+	114.201i	0.602082	+	0.159721i
$716$		0			0
$717$		−	545.789i		−	0.761213i
$718$		0			0
$719$	955.392			1.32878			0.664390	−	0.747386i	$-0.268692\pi$
$719$	955.392			1.32878			0.664390	+	0.747386i	$0.268692\pi$
$720$		0			0
$721$		−	42.4335i		−	0.0588537i
$722$		0			0
$723$			1598.07i			2.21033i
$724$		0			0
$725$			241.698i			0.333377i
$726$		0			0
$727$	149.231			0.205270			0.102635	−	0.994719i	$-0.467273\pi$
$727$	149.231			0.205270			0.102635	+	0.994719i	$0.467273\pi$
$728$		0			0
$729$	404.092			0.554310
$730$		0			0
$731$	−746.679			−1.02145
$732$		0			0
$733$			884.031i			1.20604i	0.797724	+	0.603022i	$0.206037\pi$
$733$			884.031i			1.20604i	−0.797724	+	0.603022i	$0.793963\pi$
$734$		0			0
$735$	−371.847			−0.505914
$736$		0			0
$737$	53.0857	−	200.111i	0.0720295	−	0.271521i
$738$		0			0
$739$			185.448i			0.250944i	0.992097	+	0.125472i	$0.0400445\pi$
$739$			185.448i			0.250944i	−0.992097	+	0.125472i	$0.959955\pi$
$740$		0			0
$741$	973.501			1.31377
$742$		0			0
$743$			114.936i			0.154691i	0.997004	+	0.0773457i	$0.0246445\pi$
$743$			114.936i			0.154691i	−0.997004	+	0.0773457i	$0.975355\pi$
$744$		0			0
$745$			416.114i			0.558542i
$746$		0			0
$747$			430.957i			0.576917i
$748$		0			0
$749$	20.4389			0.0272882
$750$		0			0
$751$	−440.004			−0.585891			−0.292945	−	0.956129i	$-0.594635\pi$
$751$	−440.004			−0.585891			−0.292945	+	0.956129i	$0.594635\pi$
$752$		0			0
$753$	−1066.06			−1.41575
$754$		0			0
$755$		−	11.7627i		−	0.0155798i
$756$		0			0
$757$	−434.602			−0.574112			−0.287056	−	0.957914i	$-0.592677\pi$
$757$	−434.602			−0.574112			−0.287056	+	0.957914i	$0.592677\pi$
$758$		0			0
$759$	−305.837	+	1152.88i	−0.402948	+	1.51894i
$760$		0			0
$761$		−	1172.82i		−	1.54116i	−0.637346	−	0.770578i	$-0.719968\pi$
$761$		−	1172.82i		−	1.54116i	0.637346	−	0.770578i	$-0.280032\pi$
$762$		0			0
$763$	12.1740			0.0159555
$764$		0			0
$765$			190.991i			0.249662i
$766$		0			0
$767$		−	536.144i		−	0.699015i
$768$		0			0
$769$		−	947.915i		−	1.23266i	−0.787488	−	0.616330i	$-0.788619\pi$
$769$		−	947.915i		−	1.23266i	0.787488	−	0.616330i	$-0.211381\pi$
$770$		0			0
$771$	−477.268			−0.619024
$772$		0			0
$773$	1281.52			1.65785			0.828925	−	0.559359i	$-0.188953\pi$
$773$	1281.52			1.65785			0.828925	+	0.559359i	$0.188953\pi$
$774$		0			0
$775$	−216.779			−0.279714
$776$		0			0
$777$		−	40.9026i		−	0.0526416i
$778$		0			0
$779$	612.526			0.786298
$780$		0			0
$781$	267.411	−	1008.03i	0.342396	−	1.29069i
$782$		0			0
$783$			1045.80i			1.33564i
$784$		0			0
$785$	−492.052			−0.626818
$786$		0			0
$787$			707.698i			0.899235i	0.893221	+	0.449617i	$0.148440\pi$
$787$			707.698i			0.899235i	−0.893221	+	0.449617i	$0.851560\pi$
$788$		0			0
$789$			618.604i			0.784036i
$790$		0			0
$791$		−	26.3371i		−	0.0332959i
$792$		0			0
$793$	1161.04			1.46411
$794$		0			0
$795$	64.4236			0.0810360
$796$		0			0
$797$	241.596			0.303132			0.151566	−	0.988447i	$-0.451568\pi$
$797$	241.596			0.303132			0.151566	+	0.988447i	$0.451568\pi$
$798$		0			0
$799$		−	2412.55i		−	3.01946i
$800$		0			0
$801$	244.072			0.304710
$802$		0			0
$803$	−10.0563	−	2.66777i	−0.0125235	−	0.00332225i
$804$		0			0
$805$			39.6573i			0.0492637i
$806$		0			0
$807$	329.423			0.408207
$808$		0			0
$809$			1063.62i			1.31473i	0.753571	+	0.657366i	$0.228330\pi$
$809$			1063.62i			1.31473i	−0.753571	+	0.657366i	$0.771670\pi$
$810$		0			0
$811$			976.484i			1.20405i	0.798477	+	0.602025i	$0.205639\pi$
$811$			976.484i			1.20405i	−0.798477	+	0.602025i	$0.794361\pi$
$812$		0			0
$813$		−	1013.59i		−	1.24672i
$814$		0			0
$815$	−26.6391			−0.0326860
$816$		0			0
$817$	−366.830			−0.448996
$818$		0			0
$819$	−26.9664			−0.0329260
$820$		0			0
$821$			189.653i			0.231003i	0.993307	+	0.115501i	$0.0368475\pi$
$821$			189.653i			0.231003i	−0.993307	+	0.115501i	$0.963153\pi$
$822$		0			0
$823$	−155.966			−0.189510			−0.0947548	−	0.995501i	$-0.530207\pi$
$823$	−155.966			−0.189510			−0.0947548	+	0.995501i	$0.530207\pi$
$824$		0			0
$825$	−48.1682	+	181.574i	−0.0583857	+	0.220089i
$826$		0			0
$827$		−	1612.24i		−	1.94950i	−0.223292	−	0.974752i	$-0.571680\pi$
$827$		−	1612.24i		−	1.94950i	0.223292	−	0.974752i	$-0.428320\pi$
$828$		0			0
$829$	−555.833			−0.670486			−0.335243	−	0.942132i	$-0.608818\pi$
$829$	−555.833			−0.670486			−0.335243	+	0.942132i	$0.608818\pi$
$830$		0			0
$831$		−	563.712i		−	0.678354i
$832$		0			0
$833$			1559.97i			1.87271i
$834$		0			0
$835$			23.6987i			0.0283817i
$836$		0			0
$837$	−937.980			−1.12064
$838$		0			0
$839$	622.115			0.741496			0.370748	−	0.928733i	$-0.379101\pi$
$839$	622.115			0.741496			0.370748	+	0.928733i	$0.379101\pi$
$840$		0			0
$841$	−1495.72			−1.77850
$842$		0			0
$843$			782.942i			0.928757i
$844$		0			0
$845$	−355.249			−0.420413
$846$		0			0
$847$	33.5061	−	58.7077i	0.0395586	−	0.0693125i
$848$		0			0
$849$			455.659i			0.536701i
$850$		0			0
$851$	680.543			0.799698
$852$		0			0
$853$		−	1205.37i		−	1.41309i	−0.707668	−	0.706545i	$-0.750253\pi$
$853$		−	1205.37i		−	1.41309i	0.707668	−	0.706545i	$-0.249747\pi$
$854$		0			0
$855$			93.8307i			0.109744i
$856$		0			0
$857$		−	884.739i		−	1.03237i	−0.856478	−	0.516184i	$-0.827352\pi$
$857$		−	884.739i		−	1.03237i	0.856478	−	0.516184i	$-0.172648\pi$
$858$		0			0
$859$	−1362.42			−1.58605			−0.793024	−	0.609191i	$-0.791494\pi$
$859$	−1362.42			−1.58605			−0.793024	+	0.609191i	$0.791494\pi$
$860$		0			0
$861$	−74.2495			−0.0862364
$862$		0			0
$863$	−1254.23			−1.45333			−0.726667	−	0.686990i	$-0.758932\pi$
$863$	−1254.23			−1.45333			−0.726667	+	0.686990i	$0.758932\pi$
$864$		0			0
$865$			406.557i			0.470008i
$866$		0			0
$867$	2519.20			2.90566
$868$		0			0
$869$	−779.157	−	206.696i	−0.896613	−	0.237855i
$870$		0			0
$871$			340.799i			0.391273i
$872$		0			0
$873$	−88.3701			−0.101226
$874$		0			0
$875$			6.24586i			0.00713812i
$876$		0			0
$877$		−	86.1504i		−	0.0982331i	−0.998793	−	0.0491165i	$-0.984359\pi$
$877$		−	86.1504i		−	0.0982331i	0.998793	−	0.0491165i	$-0.0156406\pi$
$878$		0			0
$879$			484.899i			0.551648i
$880$		0			0
$881$	1011.19			1.14778			0.573888	−	0.818934i	$-0.305434\pi$
$881$	1011.19			1.14778			0.573888	+	0.818934i	$0.305434\pi$
$882$		0			0
$883$	442.707			0.501367			0.250683	−	0.968069i	$-0.419345\pi$
$883$	442.707			0.501367			0.250683	+	0.968069i	$0.419345\pi$
$884$		0			0
$885$	226.137			0.255522
$886$		0			0
$887$		−	706.419i		−	0.796414i	−0.917296	−	0.398207i	$-0.869633\pi$
$887$		−	706.419i		−	0.796414i	0.917296	−	0.398207i	$-0.130367\pi$
$888$		0			0
$889$	87.9957			0.0989828
$890$		0			0
$891$	−276.091	+	1040.75i	−0.309866	+	1.16807i
$892$		0			0
$893$		−	1185.24i		−	1.32726i
$894$		0			0
$895$	162.437			0.181494
$896$		0			0
$897$		−	1963.41i		−	2.18886i
$898$		0			0
$899$		−	2095.80i		−	2.33126i
$900$		0			0
$901$		−	270.270i		−	0.299966i
$902$		0			0
$903$	44.4666			0.0492432
$904$		0			0
$905$	726.912			0.803217
$906$		0			0
$907$	−963.841			−1.06267			−0.531335	−	0.847162i	$-0.678309\pi$
$907$	−963.841			−1.06267			−0.531335	+	0.847162i	$0.678309\pi$
$908$		0			0
$909$			112.199i			0.123432i
$910$		0			0
$911$	859.938			0.943949			0.471975	−	0.881612i	$-0.343541\pi$
$911$	859.938			0.943949			0.471975	+	0.881612i	$0.343541\pi$
$912$		0			0
$913$	1718.80	+	455.966i	1.88258	+	0.499415i
$914$		0			0
$915$			489.707i			0.535199i
$916$		0			0
$917$	−59.2755			−0.0646407
$918$		0			0
$919$		−	884.874i		−	0.962866i	−0.876483	−	0.481433i	$-0.840116\pi$
$919$		−	884.874i		−	0.962866i	0.876483	−	0.481433i	$-0.159884\pi$
$920$		0			0
$921$		−	1383.86i		−	1.50256i
$922$		0			0
$923$			1716.72i			1.85994i
$924$		0			0
$925$	107.183			0.115873
$926$		0			0
$927$	−202.491			−0.218437
$928$		0			0
$929$	−537.098			−0.578146			−0.289073	−	0.957307i	$-0.593347\pi$
$929$	−537.098			−0.578146			−0.289073	+	0.957307i	$0.593347\pi$
$930$		0			0
$931$			766.385i			0.823185i
$932$		0			0
$933$	−1770.02			−1.89713
$934$		0			0
$935$	761.737	+	202.075i	0.814692	+	0.216123i
$936$		0			0
$937$			1151.99i			1.22945i	0.788742	+	0.614724i	$0.210733\pi$
$937$			1151.99i			1.22945i	−0.788742	+	0.614724i	$0.789267\pi$
$938$		0			0
$939$	605.665			0.645010
$940$		0			0
$941$			1358.32i			1.44349i	0.692161	+	0.721743i	$0.256659\pi$
$941$			1358.32i			1.44349i	−0.692161	+	0.721743i	$0.743341\pi$
$942$		0			0
$943$		−	1235.38i		−	1.31005i
$944$		0			0
$945$			27.0252i			0.0285981i
$946$		0			0
$947$	−1834.97			−1.93767			−0.968834	−	0.247710i	$-0.920322\pi$
$947$	−1834.97			−1.93767			−0.968834	+	0.247710i	$0.920322\pi$
$948$		0			0
$949$	17.1265			0.0180469
$950$		0			0
$951$	−1739.92			−1.82957
$952$		0			0
$953$		−	249.548i		−	0.261856i	−0.991392	−	0.130928i	$-0.958204\pi$
$953$		−	249.548i		−	0.261856i	0.991392	−	0.130928i	$-0.0417956\pi$
$954$		0			0
$955$	−445.319			−0.466302
$956$		0			0
$957$	−1755.44	−	465.686i	−1.83432	−	0.486610i
$958$		0			0
$959$			49.1395i			0.0512404i
$960$		0			0
$961$	918.718			0.956002
$962$		0			0
$963$		−	97.5336i		−	0.101281i
$964$		0			0
$965$		−	635.261i		−	0.658301i
$966$		0			0
$967$			903.841i			0.934686i	0.884076	+	0.467343i	$0.154789\pi$
$967$			903.841i			0.934686i	−0.884076	+	0.467343i	$0.845211\pi$
$968$		0			0
$969$	1722.58			1.77769
$970$		0			0
$971$	−500.888			−0.515847			−0.257924	−	0.966165i	$-0.583038\pi$
$971$	−500.888			−0.515847			−0.257924	+	0.966165i	$0.583038\pi$
$972$		0			0
$973$	−13.6398			−0.0140183
$974$		0			0
$975$		−	309.229i		−	0.317158i
$976$		0			0
$977$	−549.099			−0.562026			−0.281013	−	0.959704i	$-0.590670\pi$
$977$	−549.099			−0.562026			−0.281013	+	0.959704i	$0.590670\pi$
$978$		0			0
$979$	258.236	−	973.441i	0.263775	−	0.994322i
$980$		0			0
$981$		−	58.0940i		−	0.0592191i
$982$		0			0
$983$	−245.607			−0.249855			−0.124927	−	0.992166i	$-0.539870\pi$
$983$	−245.607			−0.249855			−0.124927	+	0.992166i	$0.539870\pi$
$984$		0			0
$985$			514.314i			0.522146i
$986$		0			0
$987$			143.673i			0.145566i
$988$		0			0
$989$			739.842i			0.748071i
$990$		0			0
$991$	1154.02			1.16450			0.582251	−	0.813009i	$-0.302172\pi$
$991$	1154.02			1.16450			0.582251	+	0.813009i	$0.302172\pi$
$992$		0			0
$993$	1231.33			1.24001
$994$		0			0
$995$	−275.770			−0.277156
$996$		0			0
$997$		−	1220.05i		−	1.22372i	−0.790966	−	0.611861i	$-0.790421\pi$
$997$		−	1220.05i		−	1.22372i	0.790966	−	0.611861i	$-0.209579\pi$
$998$		0			0
$999$	463.770			0.464234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.3.f.a.21.4	yes	8
3.2	odd	2		1980.3.b.a.901.3		8
4.3	odd	2		880.3.j.b.241.5		8
5.2	odd	4		1100.3.e.b.549.11		16
5.3	odd	4		1100.3.e.b.549.6		16
5.4	even	2		1100.3.f.f.901.5		8
11.10	odd	2	inner	220.3.f.a.21.3	✓	8
33.32	even	2		1980.3.b.a.901.2		8
44.43	even	2		880.3.j.b.241.6		8
55.32	even	4		1100.3.e.b.549.12		16
55.43	even	4		1100.3.e.b.549.5		16
55.54	odd	2		1100.3.f.f.901.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.f.a.21.3	✓	8	11.10	odd	2	inner
220.3.f.a.21.4	yes	8	1.1	even	1	trivial
880.3.j.b.241.5		8	4.3	odd	2
880.3.j.b.241.6		8	44.43	even	2
1100.3.e.b.549.5		16	55.43	even	4
1100.3.e.b.549.6		16	5.3	odd	4
1100.3.e.b.549.11		16	5.2	odd	4
1100.3.e.b.549.12		16	55.32	even	4
1100.3.f.f.901.5		8	5.4	even	2
1100.3.f.f.901.6		8	55.54	odd	2
1980.3.b.a.901.2		8	33.32	even	2
1980.3.b.a.901.3		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	220.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.41553 q^{3} +2.23607 q^{5} +0.558647i q^{7} +2.66584 q^{9} +O(q^{10})\)	\(q-3.41553 q^{3} +2.23607 q^{5} +0.558647i q^{7} +2.66584 q^{9} +(2.82054 - 10.6322i) q^{11} +18.1072i q^{13} -7.63736 q^{15} +32.0402i q^{17} +15.7408i q^{19} -1.90807i q^{21} +31.7468 q^{23} +5.00000 q^{25} +21.6345 q^{27} +48.3396i q^{29} -43.3557 q^{31} +(-9.63363 + 36.3147i) q^{33} +1.24917i q^{35} +21.4366 q^{37} -61.8458i q^{39} -38.9133i q^{41} +23.3044i q^{43} +5.96100 q^{45} -75.2975 q^{47} +48.6879 q^{49} -109.434i q^{51} -8.43533 q^{53} +(6.30692 - 23.7744i) q^{55} -53.7631i q^{57} -29.6094 q^{59} -64.1200i q^{61} +1.48926i q^{63} +40.4890i q^{65} +18.8211 q^{67} -108.432 q^{69} +94.8085 q^{71} -0.945835i q^{73} -17.0776 q^{75} +(5.93967 + 1.57568i) q^{77} -73.2825i q^{79} -97.8859 q^{81} +161.659i q^{83} +71.6440i q^{85} -165.105i q^{87} +91.5556 q^{89} -10.1155 q^{91} +148.083 q^{93} +35.1974i q^{95} -33.1491 q^{97} +(7.51910 - 28.3438i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 28 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 28 * q^9	\( 8 q - 8 q^{3} + 28 q^{9} + 24 q^{11} + 20 q^{15} + 56 q^{23} + 40 q^{25} - 80 q^{27} + 20 q^{31} + 88 q^{33} + 72 q^{37} - 184 q^{47} - 244 q^{49} + 136 q^{53} + 60 q^{55} - 16 q^{59} - 264 q^{67} - 56 q^{69} - 220 q^{71} - 40 q^{75} + 208 q^{77} - 224 q^{81} + 220 q^{89} - 72 q^{91} - 104 q^{93} - 8 q^{97} - 40 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 28 * q^9 + 24 * q^11 + 20 * q^15 + 56 * q^23 + 40 * q^25 - 80 * q^27 + 20 * q^31 + 88 * q^33 + 72 * q^37 - 184 * q^47 - 244 * q^49 + 136 * q^53 + 60 * q^55 - 16 * q^59 - 264 * q^67 - 56 * q^69 - 220 * q^71 - 40 * q^75 + 208 * q^77 - 224 * q^81 + 220 * q^89 - 72 * q^91 - 104 * q^93 - 8 * q^97 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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