LMFDB - Embedded newform 882.3.b.i.197.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(197,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.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:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.3.b.i.197.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +2.00000i q^{5} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} +2.00000i q^{5} -2.82843i q^{8} -2.82843 q^{10} -2.82843i q^{11} +12.7279 q^{13} +4.00000 q^{16} -22.0000i q^{17} -5.65685 q^{19} -4.00000i q^{20} +4.00000 q^{22} +2.82843i q^{23} +21.0000 q^{25} +18.0000i q^{26} -35.3553i q^{29} -33.9411 q^{31} +5.65685i q^{32} +31.1127 q^{34} +64.0000 q^{37} -8.00000i q^{38} +5.65685 q^{40} +20.0000i q^{41} +44.0000 q^{43} +5.65685i q^{44} -4.00000 q^{46} +68.0000i q^{47} +29.6985i q^{50} -25.4558 q^{52} +18.3848i q^{53} +5.65685 q^{55} +50.0000 q^{58} -100.000i q^{59} +52.3259 q^{61} -48.0000i q^{62} -8.00000 q^{64} +25.4558i q^{65} +120.000 q^{67} +44.0000i q^{68} +8.48528i q^{71} -74.9533 q^{73} +90.5097i q^{74} +11.3137 q^{76} +92.0000 q^{79} +8.00000i q^{80} -28.2843 q^{82} +112.000i q^{83} +44.0000 q^{85} +62.2254i q^{86} -8.00000 q^{88} -20.0000i q^{89} -5.65685i q^{92} -96.1665 q^{94} -11.3137i q^{95} -26.8701 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 84 q^{25} + 256 q^{37} + 176 q^{43} - 16 q^{46} + 200 q^{58} - 32 q^{64} + 480 q^{67} + 368 q^{79} + 176 q^{85} - 32 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 16 * q^16 + 16 * q^22 + 84 * q^25 + 256 * q^37 + 176 * q^43 - 16 * q^46 + 200 * q^58 - 32 * q^64 + 480 * q^67 + 368 * q^79 + 176 * q^85 - 32 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	2.00000i	0.400000i	0.979796	+	0.200000i	$0.0640942\pi$
$5$	2.00000i	0.400000i	−0.979796	+	0.200000i	$0.935906\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−2.82843	−0.282843
$11$	− 2.82843i	− 0.257130i	−0.991701	−	0.128565i	$-0.958963\pi$
$11$	− 2.82843i	− 0.257130i	0.991701	−	0.128565i	$-0.0410371\pi$
$12$	0	0
$13$	12.7279	0.979071	0.489535	−	0.871983i	$-0.337166\pi$
$13$	12.7279	0.979071	0.489535	+	0.871983i	$0.337166\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	− 22.0000i	− 1.29412i	−0.762440	−	0.647059i	$-0.775999\pi$
$17$	− 22.0000i	− 1.29412i	0.762440	−	0.647059i	$-0.224001\pi$
$18$	0	0
$19$	−5.65685	−0.297729	−0.148865	−	0.988858i	$-0.547562\pi$
$19$	−5.65685	−0.297729	−0.148865	+	0.988858i	$0.547562\pi$
$20$	− 4.00000i	− 0.200000i
$21$	0	0
$22$	4.00000	0.181818
$23$	2.82843i	0.122975i	0.998108	+	0.0614875i	$0.0195844\pi$
$23$	2.82843i	0.122975i	−0.998108	+	0.0614875i	$0.980416\pi$
$24$	0	0
$25$	21.0000	0.840000
$26$	18.0000i	0.692308i
$27$	0	0
$28$	0	0
$29$	− 35.3553i	− 1.21915i	−0.792729	−	0.609575i	$-0.791340\pi$
$29$	− 35.3553i	− 1.21915i	0.792729	−	0.609575i	$-0.208660\pi$
$30$	0	0
$31$	−33.9411	−1.09488	−0.547438	−	0.836847i	$-0.684397\pi$
$31$	−33.9411	−1.09488	−0.547438	+	0.836847i	$0.684397\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	31.1127	0.915079
$35$	0	0
$36$	0	0
$37$	64.0000	1.72973	0.864865	−	0.502005i	$-0.167404\pi$
$37$	64.0000	1.72973	0.864865	+	0.502005i	$0.167404\pi$
$38$	− 8.00000i	− 0.210526i
$39$	0	0
$40$	5.65685	0.141421
$41$	20.0000i	0.487805i	0.969800	+	0.243902i	$0.0784277\pi$
$41$	20.0000i	0.487805i	−0.969800	+	0.243902i	$0.921572\pi$
$42$	0	0
$43$	44.0000	1.02326	0.511628	−	0.859207i	$-0.329043\pi$
$43$	44.0000	1.02326	0.511628	+	0.859207i	$0.329043\pi$
$44$	5.65685i	0.128565i
$45$	0	0
$46$	−4.00000	−0.0869565
$47$	68.0000i	1.44681i	0.690425	+	0.723404i	$0.257424\pi$
$47$	68.0000i	1.44681i	−0.690425	+	0.723404i	$0.742576\pi$
$48$	0	0
$49$	0	0
$50$	29.6985i	0.593970i
$51$	0	0
$52$	−25.4558	−0.489535
$53$	18.3848i	0.346883i	0.984844	+	0.173441i	$0.0554887\pi$
$53$	18.3848i	0.346883i	−0.984844	+	0.173441i	$0.944511\pi$
$54$	0	0
$55$	5.65685	0.102852
$56$	0	0
$57$	0	0
$58$	50.0000	0.862069
$59$	− 100.000i	− 1.69492i	−0.530863	−	0.847458i	$-0.678132\pi$
$59$	− 100.000i	− 1.69492i	0.530863	−	0.847458i	$-0.321868\pi$
$60$	0	0
$61$	52.3259	0.857802	0.428901	−	0.903352i	$-0.358901\pi$
$61$	52.3259	0.857802	0.428901	+	0.903352i	$0.358901\pi$
$62$	− 48.0000i	− 0.774194i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	25.4558i	0.391628i
$66$	0	0
$67$	120.000	1.79104	0.895522	−	0.445016i	$-0.146802\pi$
$67$	120.000	1.79104	0.895522	+	0.445016i	$0.146802\pi$
$68$	44.0000i	0.647059i
$69$	0	0
$70$	0	0
$71$	8.48528i	0.119511i	0.998213	+	0.0597555i	$0.0190321\pi$
$71$	8.48528i	0.119511i	−0.998213	+	0.0597555i	$0.980968\pi$
$72$	0	0
$73$	−74.9533	−1.02676	−0.513379	−	0.858162i	$-0.671606\pi$
$73$	−74.9533	−1.02676	−0.513379	+	0.858162i	$0.671606\pi$
$74$	90.5097i	1.22310i
$75$	0	0
$76$	11.3137	0.148865
$77$	0	0
$78$	0	0
$79$	92.0000	1.16456	0.582278	−	0.812989i	$-0.302161\pi$
$79$	92.0000	1.16456	0.582278	+	0.812989i	$0.302161\pi$
$80$	8.00000i	0.100000i
$81$	0	0
$82$	−28.2843	−0.344930
$83$	112.000i	1.34940i	0.738093	+	0.674699i	$0.235726\pi$
$83$	112.000i	1.34940i	−0.738093	+	0.674699i	$0.764274\pi$
$84$	0	0
$85$	44.0000	0.517647
$86$	62.2254i	0.723551i
$87$	0	0
$88$	−8.00000	−0.0909091
$89$	− 20.0000i	− 0.224719i	−0.993668	−	0.112360i	$-0.964159\pi$
$89$	− 20.0000i	− 0.224719i	0.993668	−	0.112360i	$-0.0358408\pi$
$90$	0	0
$91$	0	0
$92$	− 5.65685i	− 0.0614875i
$93$	0	0
$94$	−96.1665	−1.02305
$95$	− 11.3137i	− 0.119092i
$96$	0	0
$97$	−26.8701	−0.277011	−0.138505	−	0.990362i	$-0.544230\pi$
$97$	−26.8701	−0.277011	−0.138505	+	0.990362i	$0.544230\pi$
$98$	0	0
$99$	0	0
$100$	−42.0000	−0.420000
$101$	− 84.0000i	− 0.831683i	−0.909437	−	0.415842i	$-0.863487\pi$
$101$	− 84.0000i	− 0.831683i	0.909437	−	0.415842i	$-0.136513\pi$
$102$	0	0
$103$	158.392	1.53779	0.768893	−	0.639378i	$-0.220808\pi$
$103$	158.392	1.53779	0.768893	+	0.639378i	$0.220808\pi$
$104$	− 36.0000i	− 0.346154i
$105$	0	0
$106$	−26.0000	−0.245283
$107$	138.593i	1.29526i	0.761955	+	0.647631i	$0.224240\pi$
$107$	138.593i	1.29526i	−0.761955	+	0.647631i	$0.775760\pi$
$108$	0	0
$109$	70.0000	0.642202	0.321101	−	0.947045i	$-0.395947\pi$
$109$	70.0000	0.642202	0.321101	+	0.947045i	$0.395947\pi$
$110$	8.00000i	0.0727273i
$111$	0	0
$112$	0	0
$113$	21.2132i	0.187727i	0.995585	+	0.0938637i	$0.0299218\pi$
$113$	21.2132i	0.187727i	−0.995585	+	0.0938637i	$0.970078\pi$
$114$	0	0
$115$	−5.65685	−0.0491900
$116$	70.7107i	0.609575i
$117$	0	0
$118$	141.421	1.19849
$119$	0	0
$120$	0	0
$121$	113.000	0.933884
$122$	74.0000i	0.606557i
$123$	0	0
$124$	67.8823	0.547438
$125$	92.0000i	0.736000i
$126$	0	0
$127$	−20.0000	−0.157480	−0.0787402	−	0.996895i	$-0.525090\pi$
$127$	−20.0000	−0.157480	−0.0787402	+	0.996895i	$0.525090\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−36.0000	−0.276923
$131$	− 144.000i	− 1.09924i	−0.835416	−	0.549618i	$-0.814773\pi$
$131$	− 144.000i	− 1.09924i	0.835416	−	0.549618i	$-0.185227\pi$
$132$	0	0
$133$	0	0
$134$	169.706i	1.26646i
$135$	0	0
$136$	−62.2254	−0.457540
$137$	− 199.404i	− 1.45550i	−0.685840	−	0.727752i	$-0.740565\pi$
$137$	− 199.404i	− 1.45550i	0.685840	−	0.727752i	$-0.259435\pi$
$138$	0	0
$139$	−5.65685	−0.0406968	−0.0203484	−	0.999793i	$-0.506478\pi$
$139$	−5.65685	−0.0406968	−0.0203484	+	0.999793i	$0.506478\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−0.0845070
$143$	− 36.0000i	− 0.251748i
$144$	0	0
$145$	70.7107	0.487660
$146$	− 106.000i	− 0.726027i
$147$	0	0
$148$	−128.000	−0.864865
$149$	18.3848i	0.123388i	0.998095	+	0.0616939i	$0.0196503\pi$
$149$	18.3848i	0.123388i	−0.998095	+	0.0616939i	$0.980350\pi$
$150$	0	0
$151$	−64.0000	−0.423841	−0.211921	−	0.977287i	$-0.567972\pi$
$151$	−64.0000	−0.423841	−0.211921	+	0.977287i	$0.567972\pi$
$152$	16.0000i	0.105263i
$153$	0	0
$154$	0	0
$155$	− 67.8823i	− 0.437950i
$156$	0	0
$157$	−162.635	−1.03589	−0.517944	−	0.855414i	$-0.673303\pi$
$157$	−162.635	−1.03589	−0.517944	+	0.855414i	$0.673303\pi$
$158$	130.108i	0.823466i
$159$	0	0
$160$	−11.3137	−0.0707107
$161$	0	0
$162$	0	0
$163$	−224.000	−1.37423	−0.687117	−	0.726547i	$-0.741124\pi$
$163$	−224.000	−1.37423	−0.687117	+	0.726547i	$0.741124\pi$
$164$	− 40.0000i	− 0.243902i
$165$	0	0
$166$	−158.392	−0.954168
$167$	292.000i	1.74850i	0.485473	+	0.874251i	$0.338647\pi$
$167$	292.000i	1.74850i	−0.485473	+	0.874251i	$0.661353\pi$
$168$	0	0
$169$	−7.00000	−0.0414201
$170$	62.2254i	0.366032i
$171$	0	0
$172$	−88.0000	−0.511628
$173$	− 156.000i	− 0.901734i	−0.892591	−	0.450867i	$-0.851115\pi$
$173$	− 156.000i	− 0.901734i	0.892591	−	0.450867i	$-0.148885\pi$
$174$	0	0
$175$	0	0
$176$	− 11.3137i	− 0.0642824i
$177$	0	0
$178$	28.2843	0.158900
$179$	93.3381i	0.521442i	0.965414	+	0.260721i	$0.0839602\pi$
$179$	93.3381i	0.521442i	−0.965414	+	0.260721i	$0.916040\pi$
$180$	0	0
$181$	−190.919	−1.05480	−0.527400	−	0.849617i	$-0.676833\pi$
$181$	−190.919	−1.05480	−0.527400	+	0.849617i	$0.676833\pi$
$182$	0	0
$183$	0	0
$184$	8.00000	0.0434783
$185$	128.000i	0.691892i
$186$	0	0
$187$	−62.2254	−0.332756
$188$	− 136.000i	− 0.723404i
$189$	0	0
$190$	16.0000	0.0842105
$191$	− 132.936i	− 0.696000i	−0.937494	−	0.348000i	$-0.886861\pi$
$191$	− 132.936i	− 0.696000i	0.937494	−	0.348000i	$-0.113139\pi$
$192$	0	0
$193$	122.000	0.632124	0.316062	−	0.948738i	$-0.397639\pi$
$193$	122.000	0.632124	0.316062	+	0.948738i	$0.397639\pi$
$194$	− 38.0000i	− 0.195876i
$195$	0	0
$196$	0	0
$197$	7.07107i	0.0358937i	0.999839	+	0.0179469i	$0.00571297\pi$
$197$	7.07107i	0.0358937i	−0.999839	+	0.0179469i	$0.994287\pi$
$198$	0	0
$199$	373.352	1.87614	0.938071	−	0.346442i	$-0.112610\pi$
$199$	373.352	1.87614	0.938071	+	0.346442i	$0.112610\pi$
$200$	− 59.3970i	− 0.296985i
$201$	0	0
$202$	118.794	0.588089
$203$	0	0
$204$	0	0
$205$	−40.0000	−0.195122
$206$	224.000i	1.08738i
$207$	0	0
$208$	50.9117	0.244768
$209$	16.0000i	0.0765550i
$210$	0	0
$211$	−12.0000	−0.0568720	−0.0284360	−	0.999596i	$-0.509053\pi$
$211$	−12.0000	−0.0568720	−0.0284360	+	0.999596i	$0.509053\pi$
$212$	− 36.7696i	− 0.173441i
$213$	0	0
$214$	−196.000	−0.915888
$215$	88.0000i	0.409302i
$216$	0	0
$217$	0	0
$218$	98.9949i	0.454105i
$219$	0	0
$220$	−11.3137	−0.0514259
$221$	− 280.014i	− 1.26703i
$222$	0	0
$223$	−203.647	−0.913214	−0.456607	−	0.889668i	$-0.650935\pi$
$223$	−203.647	−0.913214	−0.456607	+	0.889668i	$0.650935\pi$
$224$	0	0
$225$	0	0
$226$	−30.0000	−0.132743
$227$	− 204.000i	− 0.898678i	−0.893361	−	0.449339i	$-0.851659\pi$
$227$	− 204.000i	− 0.898678i	0.893361	−	0.449339i	$-0.148341\pi$
$228$	0	0
$229$	−46.6690	−0.203795	−0.101897	−	0.994795i	$-0.532491\pi$
$229$	−46.6690	−0.203795	−0.101897	+	0.994795i	$0.532491\pi$
$230$	− 8.00000i	− 0.0347826i
$231$	0	0
$232$	−100.000	−0.431034
$233$	− 309.713i	− 1.32924i	−0.747182	−	0.664620i	$-0.768594\pi$
$233$	− 309.713i	− 1.32924i	0.747182	−	0.664620i	$-0.231406\pi$
$234$	0	0
$235$	−136.000	−0.578723
$236$	200.000i	0.847458i
$237$	0	0
$238$	0	0
$239$	− 42.4264i	− 0.177516i	−0.996053	−	0.0887582i	$-0.971710\pi$
$239$	− 42.4264i	− 0.177516i	0.996053	−	0.0887582i	$-0.0282898\pi$
$240$	0	0
$241$	377.595	1.56678	0.783392	−	0.621528i	$-0.213488\pi$
$241$	377.595	1.56678	0.783392	+	0.621528i	$0.213488\pi$
$242$	159.806i	0.660356i
$243$	0	0
$244$	−104.652	−0.428901
$245$	0	0
$246$	0	0
$247$	−72.0000	−0.291498
$248$	96.0000i	0.387097i
$249$	0	0
$250$	−130.108	−0.520431
$251$	− 332.000i	− 1.32271i	−0.750073	−	0.661355i	$-0.769982\pi$
$251$	− 332.000i	− 1.32271i	0.750073	−	0.661355i	$-0.230018\pi$
$252$	0	0
$253$	8.00000	0.0316206
$254$	− 28.2843i	− 0.111355i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 180.000i	− 0.700389i	−0.936677	−	0.350195i	$-0.886115\pi$
$257$	− 180.000i	− 0.700389i	0.936677	−	0.350195i	$-0.113885\pi$
$258$	0	0
$259$	0	0
$260$	− 50.9117i	− 0.195814i
$261$	0	0
$262$	203.647	0.777278
$263$	− 161.220i	− 0.613005i	−0.951870	−	0.306503i	$-0.900841\pi$
$263$	− 161.220i	− 0.613005i	0.951870	−	0.306503i	$-0.0991588\pi$
$264$	0	0
$265$	−36.7696	−0.138753
$266$	0	0
$267$	0	0
$268$	−240.000	−0.895522
$269$	526.000i	1.95539i	0.210029	+	0.977695i	$0.432644\pi$
$269$	526.000i	1.95539i	−0.210029	+	0.977695i	$0.567356\pi$
$270$	0	0
$271$	−429.921	−1.58642	−0.793212	−	0.608946i	$-0.791593\pi$
$271$	−429.921	−1.58642	−0.793212	+	0.608946i	$0.791593\pi$
$272$	− 88.0000i	− 0.323529i
$273$	0	0
$274$	282.000	1.02920
$275$	− 59.3970i	− 0.215989i
$276$	0	0
$277$	64.0000	0.231047	0.115523	−	0.993305i	$-0.463145\pi$
$277$	64.0000	0.231047	0.115523	+	0.993305i	$0.463145\pi$
$278$	− 8.00000i	− 0.0287770i
$279$	0	0
$280$	0	0
$281$	− 448.306i	− 1.59539i	−0.603058	−	0.797697i	$-0.706051\pi$
$281$	− 448.306i	− 1.59539i	0.603058	−	0.797697i	$-0.293949\pi$
$282$	0	0
$283$	124.451	0.439755	0.219878	−	0.975527i	$-0.429434\pi$
$283$	124.451	0.439755	0.219878	+	0.975527i	$0.429434\pi$
$284$	− 16.9706i	− 0.0597555i
$285$	0	0
$286$	50.9117	0.178013
$287$	0	0
$288$	0	0
$289$	−195.000	−0.674740
$290$	100.000i	0.344828i
$291$	0	0
$292$	149.907	0.513379
$293$	− 284.000i	− 0.969283i	−0.874713	−	0.484642i	$-0.838950\pi$
$293$	− 284.000i	− 0.969283i	0.874713	−	0.484642i	$-0.161050\pi$
$294$	0	0
$295$	200.000	0.677966
$296$	− 181.019i	− 0.611552i
$297$	0	0
$298$	−26.0000	−0.0872483
$299$	36.0000i	0.120401i
$300$	0	0
$301$	0	0
$302$	− 90.5097i	− 0.299701i
$303$	0	0
$304$	−22.6274	−0.0744323
$305$	104.652i	0.343121i
$306$	0	0
$307$	−282.843	−0.921312	−0.460656	−	0.887579i	$-0.652386\pi$
$307$	−282.843	−0.921312	−0.460656	+	0.887579i	$0.652386\pi$
$308$	0	0
$309$	0	0
$310$	96.0000	0.309677
$311$	− 188.000i	− 0.604502i	−0.953228	−	0.302251i	$-0.902262\pi$
$311$	− 188.000i	− 0.604502i	0.953228	−	0.302251i	$-0.0977380\pi$
$312$	0	0
$313$	−216.375	−0.691293	−0.345646	−	0.938365i	$-0.612340\pi$
$313$	−216.375	−0.691293	−0.345646	+	0.938365i	$0.612340\pi$
$314$	− 230.000i	− 0.732484i
$315$	0	0
$316$	−184.000	−0.582278
$317$	− 456.791i	− 1.44098i	−0.693465	−	0.720491i	$-0.743917\pi$
$317$	− 456.791i	− 1.44098i	0.693465	−	0.720491i	$-0.256083\pi$
$318$	0	0
$319$	−100.000	−0.313480
$320$	− 16.0000i	− 0.0500000i
$321$	0	0
$322$	0	0
$323$	124.451i	0.385297i
$324$	0	0
$325$	267.286	0.822420
$326$	− 316.784i	− 0.971730i
$327$	0	0
$328$	56.5685	0.172465
$329$	0	0
$330$	0	0
$331$	−500.000	−1.51057	−0.755287	−	0.655394i	$-0.772503\pi$
$331$	−500.000	−1.51057	−0.755287	+	0.655394i	$0.772503\pi$
$332$	− 224.000i	− 0.674699i
$333$	0	0
$334$	−412.950	−1.23638
$335$	240.000i	0.716418i
$336$	0	0
$337$	86.0000	0.255193	0.127596	−	0.991826i	$-0.459274\pi$
$337$	86.0000	0.255193	0.127596	+	0.991826i	$0.459274\pi$
$338$	− 9.89949i	− 0.0292884i
$339$	0	0
$340$	−88.0000	−0.258824
$341$	96.0000i	0.281525i
$342$	0	0
$343$	0	0
$344$	− 124.451i	− 0.361776i
$345$	0	0
$346$	220.617	0.637622
$347$	− 246.073i	− 0.709145i	−0.935029	−	0.354572i	$-0.884626\pi$
$347$	− 246.073i	− 0.709145i	0.935029	−	0.354572i	$-0.115374\pi$
$348$	0	0
$349$	445.477	1.27644	0.638220	−	0.769854i	$-0.279671\pi$
$349$	445.477	1.27644	0.638220	+	0.769854i	$0.279671\pi$
$350$	0	0
$351$	0	0
$352$	16.0000	0.0454545
$353$	218.000i	0.617564i	0.951133	+	0.308782i	$0.0999213\pi$
$353$	218.000i	0.617564i	−0.951133	+	0.308782i	$0.900079\pi$
$354$	0	0
$355$	−16.9706	−0.0478044
$356$	40.0000i	0.112360i
$357$	0	0
$358$	−132.000	−0.368715
$359$	370.524i	1.03210i	0.856558	+	0.516050i	$0.172598\pi$
$359$	370.524i	1.03210i	−0.856558	+	0.516050i	$0.827402\pi$
$360$	0	0
$361$	−329.000	−0.911357
$362$	− 270.000i	− 0.745856i
$363$	0	0
$364$	0	0
$365$	− 149.907i	− 0.410703i
$366$	0	0
$367$	446.891	1.21769	0.608844	−	0.793290i	$-0.291634\pi$
$367$	446.891	1.21769	0.608844	+	0.793290i	$0.291634\pi$
$368$	11.3137i	0.0307438i
$369$	0	0
$370$	−181.019	−0.489241
$371$	0	0
$372$	0	0
$373$	286.000	0.766756	0.383378	−	0.923592i	$-0.374761\pi$
$373$	286.000	0.766756	0.383378	+	0.923592i	$0.374761\pi$
$374$	− 88.0000i	− 0.235294i
$375$	0	0
$376$	192.333	0.511524
$377$	− 450.000i	− 1.19363i
$378$	0	0
$379$	−188.000	−0.496042	−0.248021	−	0.968755i	$-0.579780\pi$
$379$	−188.000	−0.496042	−0.248021	+	0.968755i	$0.579780\pi$
$380$	22.6274i	0.0595458i
$381$	0	0
$382$	188.000	0.492147
$383$	352.000i	0.919060i	0.888162	+	0.459530i	$0.151982\pi$
$383$	352.000i	0.919060i	−0.888162	+	0.459530i	$0.848018\pi$
$384$	0	0
$385$	0	0
$386$	172.534i	0.446979i
$387$	0	0
$388$	53.7401	0.138505
$389$	− 205.061i	− 0.527149i	−0.964639	−	0.263575i	$-0.915099\pi$
$389$	− 205.061i	− 0.527149i	0.964639	−	0.263575i	$-0.0849015\pi$
$390$	0	0
$391$	62.2254	0.159144
$392$	0	0
$393$	0	0
$394$	−10.0000	−0.0253807
$395$	184.000i	0.465823i
$396$	0	0
$397$	244.659	0.616269	0.308135	−	0.951343i	$-0.400295\pi$
$397$	244.659	0.616269	0.308135	+	0.951343i	$0.400295\pi$
$398$	528.000i	1.32663i
$399$	0	0
$400$	84.0000	0.210000
$401$	− 445.477i	− 1.11092i	−0.831545	−	0.555458i	$-0.812543\pi$
$401$	− 445.477i	− 1.11092i	0.831545	−	0.555458i	$-0.187457\pi$
$402$	0	0
$403$	−432.000	−1.07196
$404$	168.000i	0.415842i
$405$	0	0
$406$	0	0
$407$	− 181.019i	− 0.444765i
$408$	0	0
$409$	−750.947	−1.83606	−0.918029	−	0.396514i	$-0.870220\pi$
$409$	−750.947	−1.83606	−0.918029	+	0.396514i	$0.870220\pi$
$410$	− 56.5685i	− 0.137972i
$411$	0	0
$412$	−316.784	−0.768893
$413$	0	0
$414$	0	0
$415$	−224.000	−0.539759
$416$	72.0000i	0.173077i
$417$	0	0
$418$	−22.6274	−0.0541326
$419$	236.000i	0.563246i	0.959525	+	0.281623i	$0.0908727\pi$
$419$	236.000i	0.563246i	−0.959525	+	0.281623i	$0.909127\pi$
$420$	0	0
$421$	−96.0000	−0.228029	−0.114014	−	0.993479i	$-0.536371\pi$
$421$	−96.0000	−0.228029	−0.114014	+	0.993479i	$0.536371\pi$
$422$	− 16.9706i	− 0.0402146i
$423$	0	0
$424$	52.0000	0.122642
$425$	− 462.000i	− 1.08706i
$426$	0	0
$427$	0	0
$428$	− 277.186i	− 0.647631i
$429$	0	0
$430$	−124.451	−0.289420
$431$	857.013i	1.98843i	0.107407	+	0.994215i	$0.465745\pi$
$431$	857.013i	1.98843i	−0.107407	+	0.994215i	$0.534255\pi$
$432$	0	0
$433$	156.978	0.362535	0.181268	−	0.983434i	$-0.441980\pi$
$433$	156.978	0.362535	0.181268	+	0.983434i	$0.441980\pi$
$434$	0	0
$435$	0	0
$436$	−140.000	−0.321101
$437$	− 16.0000i	− 0.0366133i
$438$	0	0
$439$	−644.881	−1.46898	−0.734489	−	0.678621i	$-0.762578\pi$
$439$	−644.881	−1.46898	−0.734489	+	0.678621i	$0.762578\pi$
$440$	− 16.0000i	− 0.0363636i
$441$	0	0
$442$	396.000	0.895928
$443$	579.828i	1.30887i	0.756120	+	0.654433i	$0.227093\pi$
$443$	579.828i	1.30887i	−0.756120	+	0.654433i	$0.772907\pi$
$444$	0	0
$445$	40.0000	0.0898876
$446$	− 288.000i	− 0.645740i
$447$	0	0
$448$	0	0
$449$	869.741i	1.93706i	0.248891	+	0.968532i	$0.419934\pi$
$449$	869.741i	1.93706i	−0.248891	+	0.968532i	$0.580066\pi$
$450$	0	0
$451$	56.5685	0.125429
$452$	− 42.4264i	− 0.0938637i
$453$	0	0
$454$	288.500	0.635462
$455$	0	0
$456$	0	0
$457$	384.000	0.840263	0.420131	−	0.907463i	$-0.361984\pi$
$457$	384.000	0.840263	0.420131	+	0.907463i	$0.361984\pi$
$458$	− 66.0000i	− 0.144105i
$459$	0	0
$460$	11.3137	0.0245950
$461$	268.000i	0.581345i	0.956823	+	0.290672i	$0.0938790\pi$
$461$	268.000i	0.581345i	−0.956823	+	0.290672i	$0.906121\pi$
$462$	0	0
$463$	−716.000	−1.54644	−0.773218	−	0.634140i	$-0.781354\pi$
$463$	−716.000	−1.54644	−0.773218	+	0.634140i	$0.781354\pi$
$464$	− 141.421i	− 0.304787i
$465$	0	0
$466$	438.000	0.939914
$467$	308.000i	0.659529i	0.944063	+	0.329764i	$0.106969\pi$
$467$	308.000i	0.659529i	−0.944063	+	0.329764i	$0.893031\pi$
$468$	0	0
$469$	0	0
$470$	− 192.333i	− 0.409219i
$471$	0	0
$472$	−282.843	−0.599243
$473$	− 124.451i	− 0.263109i
$474$	0	0
$475$	−118.794	−0.250093
$476$	0	0
$477$	0	0
$478$	60.0000	0.125523
$479$	− 388.000i	− 0.810021i	−0.914312	−	0.405010i	$-0.867268\pi$
$479$	− 388.000i	− 0.810021i	0.914312	−	0.405010i	$-0.132732\pi$
$480$	0	0
$481$	814.587	1.69353
$482$	534.000i	1.10788i
$483$	0	0
$484$	−226.000	−0.466942
$485$	− 53.7401i	− 0.110804i
$486$	0	0
$487$	120.000	0.246407	0.123203	−	0.992381i	$-0.460683\pi$
$487$	120.000	0.246407	0.123203	+	0.992381i	$0.460683\pi$
$488$	− 148.000i	− 0.303279i
$489$	0	0
$490$	0	0
$491$	619.426i	1.26156i	0.775962	+	0.630780i	$0.217265\pi$
$491$	619.426i	1.26156i	−0.775962	+	0.630780i	$0.782735\pi$
$492$	0	0
$493$	−777.817	−1.57772
$494$	− 101.823i	− 0.206120i
$495$	0	0
$496$	−135.765	−0.273719
$497$	0	0
$498$	0	0
$499$	476.000	0.953908	0.476954	−	0.878928i	$-0.341741\pi$
$499$	476.000	0.953908	0.476954	+	0.878928i	$0.341741\pi$
$500$	− 184.000i	− 0.368000i
$501$	0	0
$502$	469.519	0.935297
$503$	− 968.000i	− 1.92445i	−0.272250	−	0.962227i	$-0.587768\pi$
$503$	− 968.000i	− 1.92445i	0.272250	−	0.962227i	$-0.412232\pi$
$504$	0	0
$505$	168.000	0.332673
$506$	11.3137i	0.0223591i
$507$	0	0
$508$	40.0000	0.0787402
$509$	− 254.000i	− 0.499018i	−0.968373	−	0.249509i	$-0.919731\pi$
$509$	− 254.000i	− 0.499018i	0.968373	−	0.249509i	$-0.0802692\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	254.558	0.495250
$515$	316.784i	0.615114i
$516$	0	0
$517$	192.333	0.372017
$518$	0	0
$519$	0	0
$520$	72.0000	0.138462
$521$	806.000i	1.54702i	0.633781	+	0.773512i	$0.281502\pi$
$521$	806.000i	1.54702i	−0.633781	+	0.773512i	$0.718498\pi$
$522$	0	0
$523$	610.940	1.16815	0.584073	−	0.811701i	$-0.301458\pi$
$523$	610.940	1.16815	0.584073	+	0.811701i	$0.301458\pi$
$524$	288.000i	0.549618i
$525$	0	0
$526$	228.000	0.433460
$527$	746.705i	1.41690i
$528$	0	0
$529$	521.000	0.984877
$530$	− 52.0000i	− 0.0981132i
$531$	0	0
$532$	0	0
$533$	254.558i	0.477596i
$534$	0	0
$535$	−277.186	−0.518104
$536$	− 339.411i	− 0.633230i
$537$	0	0
$538$	−743.876	−1.38267
$539$	0	0
$540$	0	0
$541$	−80.0000	−0.147874	−0.0739372	−	0.997263i	$-0.523556\pi$
$541$	−80.0000	−0.147874	−0.0739372	+	0.997263i	$0.523556\pi$
$542$	− 608.000i	− 1.12177i
$543$	0	0
$544$	124.451	0.228770
$545$	140.000i	0.256881i
$546$	0	0
$547$	256.000	0.468007	0.234004	−	0.972236i	$-0.424817\pi$
$547$	256.000	0.468007	0.234004	+	0.972236i	$0.424817\pi$
$548$	398.808i	0.727752i
$549$	0	0
$550$	84.0000	0.152727
$551$	200.000i	0.362976i
$552$	0	0
$553$	0	0
$554$	90.5097i	0.163375i
$555$	0	0
$556$	11.3137	0.0203484
$557$	784.889i	1.40914i	0.709637	+	0.704568i	$0.248859\pi$
$557$	784.889i	1.40914i	−0.709637	+	0.704568i	$0.751141\pi$
$558$	0	0
$559$	560.029	1.00184
$560$	0	0
$561$	0	0
$562$	634.000	1.12811
$563$	− 604.000i	− 1.07282i	−0.843956	−	0.536412i	$-0.819779\pi$
$563$	− 604.000i	− 1.07282i	0.843956	−	0.536412i	$-0.180221\pi$
$564$	0	0
$565$	−42.4264	−0.0750910
$566$	176.000i	0.310954i
$567$	0	0
$568$	24.0000	0.0422535
$569$	434.164i	0.763029i	0.924363	+	0.381515i	$0.124597\pi$
$569$	434.164i	0.763029i	−0.924363	+	0.381515i	$0.875403\pi$
$570$	0	0
$571$	−248.000	−0.434326	−0.217163	−	0.976135i	$-0.569680\pi$
$571$	−248.000	−0.434326	−0.217163	+	0.976135i	$0.569680\pi$
$572$	72.0000i	0.125874i
$573$	0	0
$574$	0	0
$575$	59.3970i	0.103299i
$576$	0	0
$577$	−767.918	−1.33088	−0.665440	−	0.746451i	$-0.731756\pi$
$577$	−767.918	−1.33088	−0.665440	+	0.746451i	$0.731756\pi$
$578$	− 275.772i	− 0.477114i
$579$	0	0
$580$	−141.421	−0.243830
$581$	0	0
$582$	0	0
$583$	52.0000	0.0891938
$584$	212.000i	0.363014i
$585$	0	0
$586$	401.637	0.685387
$587$	1140.00i	1.94208i	0.238920	+	0.971039i	$0.423207\pi$
$587$	1140.00i	1.94208i	−0.238920	+	0.971039i	$0.576793\pi$
$588$	0	0
$589$	192.000	0.325976
$590$	282.843i	0.479394i
$591$	0	0
$592$	256.000	0.432432
$593$	682.000i	1.15008i	0.818124	+	0.575042i	$0.195014\pi$
$593$	682.000i	1.15008i	−0.818124	+	0.575042i	$0.804986\pi$
$594$	0	0
$595$	0	0
$596$	− 36.7696i	− 0.0616939i
$597$	0	0
$598$	−50.9117	−0.0851366
$599$	1055.00i	1.76127i	0.473791	+	0.880637i	$0.342885\pi$
$599$	1055.00i	1.76127i	−0.473791	+	0.880637i	$0.657115\pi$
$600$	0	0
$601$	555.786	0.924769	0.462384	−	0.886680i	$-0.346994\pi$
$601$	555.786	0.924769	0.462384	+	0.886680i	$0.346994\pi$
$602$	0	0
$603$	0	0
$604$	128.000	0.211921
$605$	226.000i	0.373554i
$606$	0	0
$607$	424.264	0.698952	0.349476	−	0.936945i	$-0.386360\pi$
$607$	424.264	0.698952	0.349476	+	0.936945i	$0.386360\pi$
$608$	− 32.0000i	− 0.0526316i
$609$	0	0
$610$	−148.000	−0.242623
$611$	865.499i	1.41653i
$612$	0	0
$613$	312.000	0.508972	0.254486	−	0.967076i	$-0.418094\pi$
$613$	312.000	0.508972	0.254486	+	0.967076i	$0.418094\pi$
$614$	− 400.000i	− 0.651466i
$615$	0	0
$616$	0	0
$617$	− 490.732i	− 0.795352i	−0.917526	−	0.397676i	$-0.869817\pi$
$617$	− 490.732i	− 0.795352i	0.917526	−	0.397676i	$-0.130183\pi$
$618$	0	0
$619$	−277.186	−0.447796	−0.223898	−	0.974613i	$-0.571878\pi$
$619$	−277.186	−0.447796	−0.223898	+	0.974613i	$0.571878\pi$
$620$	135.765i	0.218975i
$621$	0	0
$622$	265.872	0.427447
$623$	0	0
$624$	0	0
$625$	341.000	0.545600
$626$	− 306.000i	− 0.488818i
$627$	0	0
$628$	325.269	0.517944
$629$	− 1408.00i	− 2.23847i
$630$	0	0
$631$	−316.000	−0.500792	−0.250396	−	0.968143i	$-0.580561\pi$
$631$	−316.000	−0.500792	−0.250396	+	0.968143i	$0.580561\pi$
$632$	− 260.215i	− 0.411733i
$633$	0	0
$634$	646.000	1.01893
$635$	− 40.0000i	− 0.0629921i
$636$	0	0
$637$	0	0
$638$	− 141.421i	− 0.221664i
$639$	0	0
$640$	22.6274	0.0353553
$641$	− 434.164i	− 0.677322i	−0.940908	−	0.338661i	$-0.890026\pi$
$641$	− 434.164i	− 0.677322i	0.940908	−	0.338661i	$-0.109974\pi$
$642$	0	0
$643$	−158.392	−0.246333	−0.123166	−	0.992386i	$-0.539305\pi$
$643$	−158.392	−0.246333	−0.123166	+	0.992386i	$0.539305\pi$
$644$	0	0
$645$	0	0
$646$	−176.000	−0.272446
$647$	− 876.000i	− 1.35394i	−0.736010	−	0.676971i	$-0.763292\pi$
$647$	− 876.000i	− 1.35394i	0.736010	−	0.676971i	$-0.236708\pi$
$648$	0	0
$649$	−282.843	−0.435813
$650$	378.000i	0.581538i
$651$	0	0
$652$	448.000	0.687117
$653$	− 292.742i	− 0.448304i	−0.974554	−	0.224152i	$-0.928039\pi$
$653$	− 292.742i	− 0.448304i	0.974554	−	0.224152i	$-0.0719611\pi$
$654$	0	0
$655$	288.000	0.439695
$656$	80.0000i	0.121951i
$657$	0	0
$658$	0	0
$659$	907.925i	1.37773i	0.724889	+	0.688866i	$0.241891\pi$
$659$	907.925i	1.37773i	−0.724889	+	0.688866i	$0.758109\pi$
$660$	0	0
$661$	−784.889	−1.18743	−0.593713	−	0.804677i	$-0.702339\pi$
$661$	−784.889	−1.18743	−0.593713	+	0.804677i	$0.702339\pi$
$662$	− 707.107i	− 1.06814i
$663$	0	0
$664$	316.784	0.477084
$665$	0	0
$666$	0	0
$667$	100.000	0.149925
$668$	− 584.000i	− 0.874251i
$669$	0	0
$670$	−339.411	−0.506584
$671$	− 148.000i	− 0.220566i
$672$	0	0
$673$	−264.000	−0.392273	−0.196137	−	0.980577i	$-0.562840\pi$
$673$	−264.000	−0.392273	−0.196137	+	0.980577i	$0.562840\pi$
$674$	121.622i	0.180449i
$675$	0	0
$676$	14.0000	0.0207101
$677$	− 300.000i	− 0.443131i	−0.975145	−	0.221566i	$-0.928883\pi$
$677$	− 300.000i	− 0.443131i	0.975145	−	0.221566i	$-0.0711167\pi$
$678$	0	0
$679$	0	0
$680$	− 124.451i	− 0.183016i
$681$	0	0
$682$	−135.765	−0.199068
$683$	1162.48i	1.70203i	0.525145	+	0.851013i	$0.324011\pi$
$683$	1162.48i	1.70203i	−0.525145	+	0.851013i	$0.675989\pi$
$684$	0	0
$685$	398.808	0.582202
$686$	0	0
$687$	0	0
$688$	176.000	0.255814
$689$	234.000i	0.339623i
$690$	0	0
$691$	−509.117	−0.736783	−0.368391	−	0.929671i	$-0.620091\pi$
$691$	−509.117	−0.736783	−0.368391	+	0.929671i	$0.620091\pi$
$692$	312.000i	0.450867i
$693$	0	0
$694$	348.000	0.501441
$695$	− 11.3137i	− 0.0162787i
$696$	0	0
$697$	440.000	0.631277
$698$	630.000i	0.902579i
$699$	0	0
$700$	0	0
$701$	182.434i	0.260248i	0.991498	+	0.130124i	$0.0415375\pi$
$701$	182.434i	0.260248i	−0.991498	+	0.130124i	$0.958463\pi$
$702$	0	0
$703$	−362.039	−0.514991
$704$	22.6274i	0.0321412i
$705$	0	0
$706$	−308.299	−0.436684
$707$	0	0
$708$	0	0
$709$	1318.00	1.85896	0.929478	−	0.368877i	$-0.120258\pi$
$709$	1318.00	1.85896	0.929478	+	0.368877i	$0.120258\pi$
$710$	− 24.0000i	− 0.0338028i
$711$	0	0
$712$	−56.5685	−0.0794502
$713$	− 96.0000i	− 0.134642i
$714$	0	0
$715$	72.0000	0.100699
$716$	− 186.676i	− 0.260721i
$717$	0	0
$718$	−524.000	−0.729805
$719$	1016.00i	1.41307i	0.707676	+	0.706537i	$0.249744\pi$
$719$	1016.00i	1.41307i	−0.707676	+	0.706537i	$0.750256\pi$
$720$	0	0
$721$	0	0
$722$	− 465.276i	− 0.644427i
$723$	0	0
$724$	381.838	0.527400
$725$	− 742.462i	− 1.02409i
$726$	0	0
$727$	384.666	0.529114	0.264557	−	0.964370i	$-0.414774\pi$
$727$	384.666	0.529114	0.264557	+	0.964370i	$0.414774\pi$
$728$	0	0
$729$	0	0
$730$	212.000	0.290411
$731$	− 968.000i	− 1.32421i
$732$	0	0
$733$	555.786	0.758235	0.379117	−	0.925349i	$-0.376228\pi$
$733$	555.786	0.758235	0.379117	+	0.925349i	$0.376228\pi$
$734$	632.000i	0.861035i
$735$	0	0
$736$	−16.0000	−0.0217391
$737$	− 339.411i	− 0.460531i
$738$	0	0
$739$	−1200.00	−1.62382	−0.811908	−	0.583785i	$-0.801571\pi$
$739$	−1200.00	−1.62382	−0.811908	+	0.583785i	$0.801571\pi$
$740$	− 256.000i	− 0.345946i
$741$	0	0
$742$	0	0
$743$	− 1309.56i	− 1.76253i	−0.472620	−	0.881266i	$-0.656692\pi$
$743$	− 1309.56i	− 1.76253i	0.472620	−	0.881266i	$-0.343308\pi$
$744$	0	0
$745$	−36.7696	−0.0493551
$746$	404.465i	0.542178i
$747$	0	0
$748$	124.451	0.166378
$749$	0	0
$750$	0	0
$751$	−568.000	−0.756325	−0.378162	−	0.925739i	$-0.623444\pi$
$751$	−568.000	−0.756325	−0.378162	+	0.925739i	$0.623444\pi$
$752$	272.000i	0.361702i
$753$	0	0
$754$	636.396	0.844027
$755$	− 128.000i	− 0.169536i
$756$	0	0
$757$	−358.000	−0.472919	−0.236460	−	0.971641i	$-0.575987\pi$
$757$	−358.000	−0.472919	−0.236460	+	0.971641i	$0.575987\pi$
$758$	− 265.872i	− 0.350755i
$759$	0	0
$760$	−32.0000	−0.0421053
$761$	508.000i	0.667543i	0.942654	+	0.333771i	$0.108321\pi$
$761$	508.000i	0.667543i	−0.942654	+	0.333771i	$0.891679\pi$
$762$	0	0
$763$	0	0
$764$	265.872i	0.348000i
$765$	0	0
$766$	−497.803	−0.649874
$767$	− 1272.79i	− 1.65944i
$768$	0	0
$769$	861.256	1.11997	0.559984	−	0.828503i	$-0.310807\pi$
$769$	861.256	1.11997	0.559984	+	0.828503i	$0.310807\pi$
$770$	0	0
$771$	0	0
$772$	−244.000	−0.316062
$773$	892.000i	1.15395i	0.816763	+	0.576973i	$0.195766\pi$
$773$	892.000i	1.15395i	−0.816763	+	0.576973i	$0.804234\pi$
$774$	0	0
$775$	−712.764	−0.919695
$776$	76.0000i	0.0979381i
$777$	0	0
$778$	290.000	0.372751
$779$	− 113.137i	− 0.145234i
$780$	0	0
$781$	24.0000	0.0307298
$782$	88.0000i	0.112532i
$783$	0	0
$784$	0	0
$785$	− 325.269i	− 0.414356i
$786$	0	0
$787$	429.921	0.546278	0.273139	−	0.961975i	$-0.411938\pi$
$787$	429.921	0.546278	0.273139	+	0.961975i	$0.411938\pi$
$788$	− 14.1421i	− 0.0179469i
$789$	0	0
$790$	−260.215	−0.329386
$791$	0	0
$792$	0	0
$793$	666.000	0.839849
$794$	346.000i	0.435768i
$795$	0	0
$796$	−746.705	−0.938071
$797$	540.000i	0.677541i	0.940869	+	0.338770i	$0.110011\pi$
$797$	540.000i	0.677541i	−0.940869	+	0.338770i	$0.889989\pi$
$798$	0	0
$799$	1496.00	1.87234
$800$	118.794i	0.148492i
$801$	0	0
$802$	630.000	0.785536
$803$	212.000i	0.264010i
$804$	0	0
$805$	0	0
$806$	− 610.940i	− 0.757990i
$807$	0	0
$808$	−237.588	−0.294044
$809$	872.570i	1.07858i	0.842121	+	0.539289i	$0.181307\pi$
$809$	872.570i	1.07858i	−0.842121	+	0.539289i	$0.818693\pi$
$810$	0	0
$811$	−492.146	−0.606839	−0.303419	−	0.952857i	$-0.598128\pi$
$811$	−492.146	−0.606839	−0.303419	+	0.952857i	$0.598128\pi$
$812$	0	0
$813$	0	0
$814$	256.000	0.314496
$815$	− 448.000i	− 0.549693i
$816$	0	0
$817$	−248.902	−0.304653
$818$	− 1062.00i	− 1.29829i
$819$	0	0
$820$	80.0000	0.0975610
$821$	− 750.947i	− 0.914674i	−0.889293	−	0.457337i	$-0.848803\pi$
$821$	− 750.947i	− 0.914674i	0.889293	−	0.457337i	$-0.151197\pi$
$822$	0	0
$823$	−116.000	−0.140948	−0.0704739	−	0.997514i	$-0.522451\pi$
$823$	−116.000	−0.140948	−0.0704739	+	0.997514i	$0.522451\pi$
$824$	− 448.000i	− 0.543689i
$825$	0	0
$826$	0	0
$827$	873.984i	1.05681i	0.848992	+	0.528406i	$0.177210\pi$
$827$	873.984i	1.05681i	−0.848992	+	0.528406i	$0.822790\pi$
$828$	0	0
$829$	272.943	0.329244	0.164622	−	0.986357i	$-0.447360\pi$
$829$	272.943	0.329244	0.164622	+	0.986357i	$0.447360\pi$
$830$	− 316.784i	− 0.381667i
$831$	0	0
$832$	−101.823	−0.122384
$833$	0	0
$834$	0	0
$835$	−584.000	−0.699401
$836$	− 32.0000i	− 0.0382775i
$837$	0	0
$838$	−333.754	−0.398275
$839$	188.000i	0.224076i	0.993704	+	0.112038i	$0.0357379\pi$
$839$	188.000i	0.224076i	−0.993704	+	0.112038i	$0.964262\pi$
$840$	0	0
$841$	−409.000	−0.486326
$842$	− 135.765i	− 0.161241i
$843$	0	0
$844$	24.0000	0.0284360
$845$	− 14.0000i	− 0.0165680i
$846$	0	0
$847$	0	0
$848$	73.5391i	0.0867206i
$849$	0	0
$850$	653.367	0.768667
$851$	181.019i	0.212714i
$852$	0	0
$853$	63.6396	0.0746068	0.0373034	−	0.999304i	$-0.488123\pi$
$853$	63.6396	0.0746068	0.0373034	+	0.999304i	$0.488123\pi$
$854$	0	0
$855$	0	0
$856$	392.000	0.457944
$857$	− 260.000i	− 0.303384i	−0.988428	−	0.151692i	$-0.951528\pi$
$857$	− 260.000i	− 0.303384i	0.988428	−	0.151692i	$-0.0484722\pi$
$858$	0	0
$859$	−791.960	−0.921955	−0.460978	−	0.887412i	$-0.652501\pi$
$859$	−791.960	−0.921955	−0.460978	+	0.887412i	$0.652501\pi$
$860$	− 176.000i	− 0.204651i
$861$	0	0
$862$	−1212.00	−1.40603
$863$	− 421.436i	− 0.488338i	−0.969733	−	0.244169i	$-0.921485\pi$
$863$	− 421.436i	− 0.488338i	0.969733	−	0.244169i	$-0.0785152\pi$
$864$	0	0
$865$	312.000	0.360694
$866$	222.000i	0.256351i
$867$	0	0
$868$	0	0
$869$	− 260.215i	− 0.299442i
$870$	0	0
$871$	1527.35	1.75356
$872$	− 197.990i	− 0.227053i
$873$	0	0
$874$	22.6274	0.0258895
$875$	0	0
$876$	0	0
$877$	−952.000	−1.08552	−0.542759	−	0.839888i	$-0.682620\pi$
$877$	−952.000	−1.08552	−0.542759	+	0.839888i	$0.682620\pi$
$878$	− 912.000i	− 1.03872i
$879$	0	0
$880$	22.6274	0.0257130
$881$	− 1158.00i	− 1.31442i	−0.753710	−	0.657208i	$-0.771737\pi$
$881$	− 1158.00i	− 1.31442i	0.753710	−	0.657208i	$-0.228263\pi$
$882$	0	0
$883$	−1036.00	−1.17327	−0.586636	−	0.809850i	$-0.699548\pi$
$883$	−1036.00	−1.17327	−0.586636	+	0.809850i	$0.699548\pi$
$884$	560.029i	0.633516i
$885$	0	0
$886$	−820.000	−0.925508
$887$	460.000i	0.518602i	0.965797	+	0.259301i	$0.0834922\pi$
$887$	460.000i	0.518602i	−0.965797	+	0.259301i	$0.916508\pi$
$888$	0	0
$889$	0	0
$890$	56.5685i	0.0635602i
$891$	0	0
$892$	407.294	0.456607
$893$	− 384.666i	− 0.430757i
$894$	0	0
$895$	−186.676	−0.208577
$896$	0	0
$897$	0	0
$898$	−1230.00	−1.36971
$899$	1200.00i	1.33482i
$900$	0	0
$901$	404.465	0.448907
$902$	80.0000i	0.0886918i
$903$	0	0
$904$	60.0000	0.0663717
$905$	− 381.838i	− 0.421920i
$906$	0	0
$907$	220.000	0.242558	0.121279	−	0.992618i	$-0.461300\pi$
$907$	220.000	0.242558	0.121279	+	0.992618i	$0.461300\pi$
$908$	408.000i	0.449339i
$909$	0	0
$910$	0	0
$911$	− 1049.35i	− 1.15186i	−0.817498	−	0.575931i	$-0.804640\pi$
$911$	− 1049.35i	− 1.15186i	0.817498	−	0.575931i	$-0.195360\pi$
$912$	0	0
$913$	316.784	0.346970
$914$	543.058i	0.594155i
$915$	0	0
$916$	93.3381	0.101897
$917$	0	0
$918$	0	0
$919$	−320.000	−0.348205	−0.174102	−	0.984728i	$-0.555702\pi$
$919$	−320.000	−0.348205	−0.174102	+	0.984728i	$0.555702\pi$
$920$	16.0000i	0.0173913i
$921$	0	0
$922$	−379.009	−0.411073
$923$	108.000i	0.117010i
$924$	0	0
$925$	1344.00	1.45297
$926$	− 1012.58i	− 1.09350i
$927$	0	0
$928$	200.000	0.215517
$929$	− 1756.00i	− 1.89020i	−0.326775	−	0.945102i	$-0.605962\pi$
$929$	− 1756.00i	− 1.89020i	0.326775	−	0.945102i	$-0.394038\pi$
$930$	0	0
$931$	0	0
$932$	619.426i	0.664620i
$933$	0	0
$934$	−435.578	−0.466357
$935$	− 124.451i	− 0.133102i
$936$	0	0
$937$	−1740.90	−1.85795	−0.928974	−	0.370145i	$-0.879308\pi$
$937$	−1740.90	−1.85795	−0.928974	+	0.370145i	$0.879308\pi$
$938$	0	0
$939$	0	0
$940$	272.000	0.289362
$941$	626.000i	0.665250i	0.943059	+	0.332625i	$0.107934\pi$
$941$	626.000i	0.665250i	−0.943059	+	0.332625i	$0.892066\pi$
$942$	0	0
$943$	−56.5685	−0.0599878
$944$	− 400.000i	− 0.423729i
$945$	0	0
$946$	176.000	0.186047
$947$	789.131i	0.833296i	0.909068	+	0.416648i	$0.136795\pi$
$947$	789.131i	0.833296i	−0.909068	+	0.416648i	$0.863205\pi$
$948$	0	0
$949$	−954.000	−1.00527
$950$	− 168.000i	− 0.176842i
$951$	0	0
$952$	0	0
$953$	− 193.747i	− 0.203302i	−0.994820	−	0.101651i	$-0.967587\pi$
$953$	− 193.747i	− 0.203302i	0.994820	−	0.101651i	$-0.0324126\pi$
$954$	0	0
$955$	265.872	0.278400
$956$	84.8528i	0.0887582i
$957$	0	0
$958$	548.715	0.572771
$959$	0	0
$960$	0	0
$961$	191.000	0.198751
$962$	1152.00i	1.19751i
$963$	0	0
$964$	−755.190	−0.783392
$965$	244.000i	0.252850i
$966$	0	0
$967$	132.000	0.136505	0.0682523	−	0.997668i	$-0.478258\pi$
$967$	132.000	0.136505	0.0682523	+	0.997668i	$0.478258\pi$
$968$	− 319.612i	− 0.330178i
$969$	0	0
$970$	76.0000	0.0783505
$971$	− 160.000i	− 0.164779i	−0.996600	−	0.0823893i	$-0.973745\pi$
$971$	− 160.000i	− 0.164779i	0.996600	−	0.0823893i	$-0.0262551\pi$
$972$	0	0
$973$	0	0
$974$	169.706i	0.174236i
$975$	0	0
$976$	209.304	0.214450
$977$	− 1497.65i	− 1.53291i	−0.642299	−	0.766455i	$-0.722019\pi$
$977$	− 1497.65i	− 1.53291i	0.642299	−	0.766455i	$-0.277981\pi$
$978$	0	0
$979$	−56.5685	−0.0577820
$980$	0	0
$981$	0	0
$982$	−876.000	−0.892057
$983$	328.000i	0.333672i	0.985985	+	0.166836i	$0.0533551\pi$
$983$	328.000i	0.333672i	−0.985985	+	0.166836i	$0.946645\pi$
$984$	0	0
$985$	−14.1421	−0.0143575
$986$	− 1100.00i	− 1.11562i
$987$	0	0
$988$	144.000	0.145749
$989$	124.451i	0.125835i
$990$	0	0
$991$	−16.0000	−0.0161453	−0.00807265	−	0.999967i	$-0.502570\pi$
$991$	−16.0000	−0.0161453	−0.00807265	+	0.999967i	$0.502570\pi$
$992$	− 192.000i	− 0.193548i
$993$	0	0
$994$	0	0
$995$	746.705i	0.750457i
$996$	0	0
$997$	875.398	0.878032	0.439016	−	0.898479i	$-0.355327\pi$
$997$	875.398	0.878032	0.439016	+	0.898479i	$0.355327\pi$
$998$	673.166i	0.674515i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.b.i.197.4	yes	4
3.2	odd	2	inner	882.3.b.i.197.1	✓	4
7.2	even	3		882.3.s.f.557.4		8
7.3	odd	6		882.3.s.f.863.2		8
7.4	even	3		882.3.s.f.863.1		8
7.5	odd	6		882.3.s.f.557.3		8
7.6	odd	2	inner	882.3.b.i.197.3	yes	4
21.2	odd	6		882.3.s.f.557.1		8
21.5	even	6		882.3.s.f.557.2		8
21.11	odd	6		882.3.s.f.863.4		8
21.17	even	6		882.3.s.f.863.3		8
21.20	even	2	inner	882.3.b.i.197.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.3.b.i.197.1	✓	4	3.2	odd	2	inner
882.3.b.i.197.2	yes	4	21.20	even	2	inner
882.3.b.i.197.3	yes	4	7.6	odd	2	inner
882.3.b.i.197.4	yes	4	1.1	even	1	trivial
882.3.s.f.557.1		8	21.2	odd	6
882.3.s.f.557.2		8	21.5	even	6
882.3.s.f.557.3		8	7.5	odd	6
882.3.s.f.557.4		8	7.2	even	3
882.3.s.f.863.1		8	7.4	even	3
882.3.s.f.863.2		8	7.3	odd	6
882.3.s.f.863.3		8	21.17	even	6
882.3.s.f.863.4		8	21.11	odd	6

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.00000i q^{5} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.00000i q^{5} -2.82843i q^{8} -2.82843 q^{10} -2.82843i q^{11} +12.7279 q^{13} +4.00000 q^{16} -22.0000i q^{17} -5.65685 q^{19} -4.00000i q^{20} +4.00000 q^{22} +2.82843i q^{23} +21.0000 q^{25} +18.0000i q^{26} -35.3553i q^{29} -33.9411 q^{31} +5.65685i q^{32} +31.1127 q^{34} +64.0000 q^{37} -8.00000i q^{38} +5.65685 q^{40} +20.0000i q^{41} +44.0000 q^{43} +5.65685i q^{44} -4.00000 q^{46} +68.0000i q^{47} +29.6985i q^{50} -25.4558 q^{52} +18.3848i q^{53} +5.65685 q^{55} +50.0000 q^{58} -100.000i q^{59} +52.3259 q^{61} -48.0000i q^{62} -8.00000 q^{64} +25.4558i q^{65} +120.000 q^{67} +44.0000i q^{68} +8.48528i q^{71} -74.9533 q^{73} +90.5097i q^{74} +11.3137 q^{76} +92.0000 q^{79} +8.00000i q^{80} -28.2843 q^{82} +112.000i q^{83} +44.0000 q^{85} +62.2254i q^{86} -8.00000 q^{88} -20.0000i q^{89} -5.65685i q^{92} -96.1665 q^{94} -11.3137i q^{95} -26.8701 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 84 q^{25} + 256 q^{37} + 176 q^{43} - 16 q^{46} + 200 q^{58} - 32 q^{64} + 480 q^{67} + 368 q^{79} + 176 q^{85} - 32 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 16 * q^16 + 16 * q^22 + 84 * q^25 + 256 * q^37 + 176 * q^43 - 16 * q^46 + 200 * q^58 - 32 * q^64 + 480 * q^67 + 368 * q^79 + 176 * q^85 - 32 * q^88

Introduction

L-functions

Modular forms

Varieties

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Database

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