LMFDB - Embedded newform 3360.2.g.c.1681.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1681,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1681"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.3058043990573056.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{7} - 16x^{5} + 16x^{2} + 64 $ x^12 + x^10 - 8x^7 - 16x^5 + 16*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.8
Root			$-1.23149 + 0.695292i$ of defining polynomial
Character	$\chi$	$=$	3360.1681
Dual form			3360.2.g.c.1681.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -3.45686i q^{11} +3.45686i q^{13} +1.00000 q^{15} +1.15712 q^{17} -5.88796i q^{19} +1.00000i q^{21} -2.37595 q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.63769i q^{29} +5.40739 q^{31} +3.45686 q^{33} -1.00000i q^{35} -4.15908i q^{37} -3.45686 q^{39} +3.26174 q^{41} -4.97544i q^{43} +1.00000i q^{45} -13.0778 q^{47} +1.00000 q^{49} +1.15712i q^{51} +5.92814i q^{53} -3.45686 q^{55} +5.88796 q^{57} -12.5571i q^{59} -13.8161i q^{61} -1.00000 q^{63} +3.45686 q^{65} -3.21227i q^{67} -2.37595i q^{69} -2.33991 q^{71} +5.95172 q^{73} -1.00000i q^{75} -3.45686i q^{77} +14.2934 q^{79} +1.00000 q^{81} +8.18770i q^{83} -1.15712i q^{85} -3.63769 q^{87} -2.61172 q^{89} +3.45686i q^{91} +5.40739i q^{93} -5.88796 q^{95} +0.962000 q^{97} +3.45686i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 12 q^{9} + 12 q^{15} - 40 q^{23} - 12 q^{25} + 8 q^{31} + 32 q^{47} + 12 q^{49} - 12 q^{63} + 8 q^{71} + 80 q^{79} + 12 q^{81} - 16 q^{87}+O(q^{100}) $ 12 * q + 12 * q^7 - 12 * q^9 + 12 * q^15 - 40 * q^23 - 12 * q^25 + 8 * q^31 + 32 * q^47 + 12 * q^49 - 12 * q^63 + 8 * q^71 + 80 * q^79 + 12 * q^81 - 16 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 3.45686i	− 1.04228i	−0.853470	−	0.521141i	$-0.825506\pi$
$11$	− 3.45686i	− 1.04228i	0.853470	−	0.521141i	$-0.174494\pi$
$12$	0	0
$13$	3.45686i	0.958761i	0.877607	+	0.479380i	$0.159139\pi$
$13$	3.45686i	0.958761i	−0.877607	+	0.479380i	$0.840861\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.15712	0.280643	0.140321	−	0.990106i	$-0.455186\pi$
$17$	1.15712	0.280643	0.140321	+	0.990106i	$0.455186\pi$
$18$	0	0
$19$	− 5.88796i	− 1.35079i	−0.737456	−	0.675396i	$-0.763973\pi$
$19$	− 5.88796i	− 1.35079i	0.737456	−	0.675396i	$-0.236027\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	−2.37595	−0.495420	−0.247710	−	0.968834i	$-0.579678\pi$
$23$	−2.37595	−0.495420	−0.247710	+	0.968834i	$0.579678\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	3.63769i	0.675503i	0.941235	+	0.337751i	$0.109666\pi$
$29$	3.63769i	0.675503i	−0.941235	+	0.337751i	$0.890334\pi$
$30$	0	0
$31$	5.40739	0.971196	0.485598	−	0.874182i	$-0.338602\pi$
$31$	5.40739	0.971196	0.485598	+	0.874182i	$0.338602\pi$
$32$	0	0
$33$	3.45686	0.601762
$34$	0	0
$35$	− 1.00000i	− 0.169031i
$36$	0	0
$37$	− 4.15908i	− 0.683749i	−0.939746	−	0.341874i	$-0.888938\pi$
$37$	− 4.15908i	− 0.683749i	0.939746	−	0.341874i	$-0.111062\pi$
$38$	0	0
$39$	−3.45686	−0.553541
$40$	0	0
$41$	3.26174	0.509398	0.254699	−	0.967020i	$-0.418024\pi$
$41$	3.26174	0.509398	0.254699	+	0.967020i	$0.418024\pi$
$42$	0	0
$43$	− 4.97544i	− 0.758747i	−0.925244	−	0.379373i	$-0.876140\pi$
$43$	− 4.97544i	− 0.758747i	0.925244	−	0.379373i	$-0.123860\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	−13.0778	−1.90760	−0.953800	−	0.300444i	$-0.902865\pi$
$47$	−13.0778	−1.90760	−0.953800	+	0.300444i	$0.902865\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.15712i	0.162029i
$52$	0	0
$53$	5.92814i	0.814292i	0.913363	+	0.407146i	$0.133476\pi$
$53$	5.92814i	0.814292i	−0.913363	+	0.407146i	$0.866524\pi$
$54$	0	0
$55$	−3.45686	−0.466123
$56$	0	0
$57$	5.88796	0.779880
$58$	0	0
$59$	− 12.5571i	− 1.63479i	−0.576075	−	0.817397i	$-0.695416\pi$
$59$	− 12.5571i	− 1.63479i	0.576075	−	0.817397i	$-0.304584\pi$
$60$	0	0
$61$	− 13.8161i	− 1.76897i	−0.466568	−	0.884485i	$-0.654510\pi$
$61$	− 13.8161i	− 1.76897i	0.466568	−	0.884485i	$-0.345490\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.45686	0.428771
$66$	0	0
$67$	− 3.21227i	− 0.392441i	−0.980560	−	0.196221i	$-0.937133\pi$
$67$	− 3.21227i	− 0.392441i	0.980560	−	0.196221i	$-0.0628669\pi$
$68$	0	0
$69$	− 2.37595i	− 0.286031i
$70$	0	0
$71$	−2.33991	−0.277697	−0.138848	−	0.990314i	$-0.544340\pi$
$71$	−2.33991	−0.277697	−0.138848	+	0.990314i	$0.544340\pi$
$72$	0	0
$73$	5.95172	0.696596	0.348298	−	0.937384i	$-0.386760\pi$
$73$	5.95172	0.696596	0.348298	+	0.937384i	$0.386760\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	− 3.45686i	− 0.393946i
$78$	0	0
$79$	14.2934	1.60813	0.804066	−	0.594540i	$-0.202666\pi$
$79$	14.2934	1.60813	0.804066	+	0.594540i	$0.202666\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.18770i	0.898717i	0.893351	+	0.449359i	$0.148347\pi$
$83$	8.18770i	0.898717i	−0.893351	+	0.449359i	$0.851653\pi$
$84$	0	0
$85$	− 1.15712i	− 0.125507i
$86$	0	0
$87$	−3.63769	−0.390002
$88$	0	0
$89$	−2.61172	−0.276842	−0.138421	−	0.990373i	$-0.544203\pi$
$89$	−2.61172	−0.276842	−0.138421	+	0.990373i	$0.544203\pi$
$90$	0	0
$91$	3.45686i	0.362377i
$92$	0	0
$93$	5.40739i	0.560720i
$94$	0	0
$95$	−5.88796	−0.604092
$96$	0	0
$97$	0.962000	0.0976763	0.0488382	−	0.998807i	$-0.484448\pi$
$97$	0.962000	0.0976763	0.0488382	+	0.998807i	$0.484448\pi$
$98$	0	0
$99$	3.45686i	0.347428i
$100$	0	0
$101$	− 8.65394i	− 0.861099i	−0.902567	−	0.430550i	$-0.858320\pi$
$101$	− 8.65394i	− 0.861099i	0.902567	−	0.430550i	$-0.141680\pi$
$102$	0	0
$103$	−12.5278	−1.23440	−0.617202	−	0.786805i	$-0.711734\pi$
$103$	−12.5278	−1.23440	−0.617202	+	0.786805i	$0.711734\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	− 13.7265i	− 1.32699i	−0.748182	−	0.663493i	$-0.769073\pi$
$107$	− 13.7265i	− 1.32699i	0.748182	−	0.663493i	$-0.230927\pi$
$108$	0	0
$109$	− 0.912317i	− 0.0873841i	−0.999045	−	0.0436921i	$-0.986088\pi$
$109$	− 0.912317i	− 0.0873841i	0.999045	−	0.0436921i	$-0.0139120\pi$
$110$	0	0
$111$	4.15908	0.394763
$112$	0	0
$113$	−1.85435	−0.174443	−0.0872215	−	0.996189i	$-0.527799\pi$
$113$	−1.85435	−0.174443	−0.0872215	+	0.996189i	$0.527799\pi$
$114$	0	0
$115$	2.37595i	0.221559i
$116$	0	0
$117$	− 3.45686i	− 0.319587i
$118$	0	0
$119$	1.15712	0.106073
$120$	0	0
$121$	−0.949887	−0.0863534
$122$	0	0
$123$	3.26174i	0.294101i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	9.75004	0.865176	0.432588	−	0.901592i	$-0.357600\pi$
$127$	9.75004	0.865176	0.432588	+	0.901592i	$0.357600\pi$
$128$	0	0
$129$	4.97544	0.438063
$130$	0	0
$131$	− 7.32913i	− 0.640349i	−0.947359	−	0.320175i	$-0.896258\pi$
$131$	− 7.32913i	− 0.640349i	0.947359	−	0.320175i	$-0.103742\pi$
$132$	0	0
$133$	− 5.88796i	− 0.510551i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−4.09652	−0.349989	−0.174995	−	0.984569i	$-0.555991\pi$
$137$	−4.09652	−0.349989	−0.174995	+	0.984569i	$0.555991\pi$
$138$	0	0
$139$	− 18.1531i	− 1.53972i	−0.638211	−	0.769862i	$-0.720325\pi$
$139$	− 18.1531i	− 1.53972i	0.638211	−	0.769862i	$-0.279675\pi$
$140$	0	0
$141$	− 13.0778i	− 1.10135i
$142$	0	0
$143$	11.9499	0.999300
$144$	0	0
$145$	3.63769	0.302094
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	− 4.96179i	− 0.406486i	−0.979128	−	0.203243i	$-0.934852\pi$
$149$	− 4.96179i	− 0.406486i	0.979128	−	0.203243i	$-0.0651481\pi$
$150$	0	0
$151$	19.8109	1.61219	0.806096	−	0.591785i	$-0.201577\pi$
$151$	19.8109	1.61219	0.806096	+	0.591785i	$0.201577\pi$
$152$	0	0
$153$	−1.15712	−0.0935475
$154$	0	0
$155$	− 5.40739i	− 0.434332i
$156$	0	0
$157$	22.7995i	1.81960i	0.415052	+	0.909798i	$0.363763\pi$
$157$	22.7995i	1.81960i	−0.415052	+	0.909798i	$0.636237\pi$
$158$	0	0
$159$	−5.92814	−0.470132
$160$	0	0
$161$	−2.37595	−0.187251
$162$	0	0
$163$	− 7.57926i	− 0.593654i	−0.954931	−	0.296827i	$-0.904072\pi$
$163$	− 7.57926i	− 0.593654i	0.954931	−	0.296827i	$-0.0959284\pi$
$164$	0	0
$165$	− 3.45686i	− 0.269116i
$166$	0	0
$167$	−7.14851	−0.553168	−0.276584	−	0.960990i	$-0.589202\pi$
$167$	−7.14851	−0.553168	−0.276584	+	0.960990i	$0.589202\pi$
$168$	0	0
$169$	1.05011	0.0807779
$170$	0	0
$171$	5.88796i	0.450264i
$172$	0	0
$173$	1.60976i	0.122388i	0.998126	+	0.0611940i	$0.0194908\pi$
$173$	1.60976i	0.122388i	−0.998126	+	0.0611940i	$0.980509\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	12.5571	0.943849
$178$	0	0
$179$	− 17.9373i	− 1.34069i	−0.742048	−	0.670347i	$-0.766145\pi$
$179$	− 17.9373i	− 1.34069i	0.742048	−	0.670347i	$-0.233855\pi$
$180$	0	0
$181$	12.5922i	0.935973i	0.883736	+	0.467986i	$0.155020\pi$
$181$	12.5922i	0.935973i	−0.883736	+	0.467986i	$0.844980\pi$
$182$	0	0
$183$	13.8161	1.02132
$184$	0	0
$185$	−4.15908	−0.305782
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	−0.362860	−0.0262556	−0.0131278	−	0.999914i	$-0.504179\pi$
$191$	−0.362860	−0.0262556	−0.0131278	+	0.999914i	$0.504179\pi$
$192$	0	0
$193$	−21.4416	−1.54340	−0.771698	−	0.635989i	$-0.780592\pi$
$193$	−21.4416	−1.54340	−0.771698	+	0.635989i	$0.780592\pi$
$194$	0	0
$195$	3.45686i	0.247551i
$196$	0	0
$197$	− 19.6997i	− 1.40355i	−0.712400	−	0.701773i	$-0.752392\pi$
$197$	− 19.6997i	− 1.40355i	0.712400	−	0.701773i	$-0.247608\pi$
$198$	0	0
$199$	−8.33424	−0.590798	−0.295399	−	0.955374i	$-0.595453\pi$
$199$	−8.33424	−0.590798	−0.295399	+	0.955374i	$0.595453\pi$
$200$	0	0
$201$	3.21227	0.226576
$202$	0	0
$203$	3.63769i	0.255316i
$204$	0	0
$205$	− 3.26174i	− 0.227810i
$206$	0	0
$207$	2.37595	0.165140
$208$	0	0
$209$	−20.3539	−1.40791
$210$	0	0
$211$	15.0108i	1.03339i	0.856171	+	0.516693i	$0.172837\pi$
$211$	15.0108i	1.03339i	−0.856171	+	0.516693i	$0.827163\pi$
$212$	0	0
$213$	− 2.33991i	− 0.160328i
$214$	0	0
$215$	−4.97544	−0.339322
$216$	0	0
$217$	5.40739	0.367077
$218$	0	0
$219$	5.95172i	0.402180i
$220$	0	0
$221$	4.00000i	0.269069i
$222$	0	0
$223$	20.0655	1.34369	0.671843	−	0.740693i	$-0.265503\pi$
$223$	20.0655	1.34369	0.671843	+	0.740693i	$0.265503\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 21.6784i	− 1.43884i	−0.694573	−	0.719422i	$-0.744407\pi$
$227$	− 21.6784i	− 1.43884i	0.694573	−	0.719422i	$-0.255593\pi$
$228$	0	0
$229$	7.83904i	0.518019i	0.965875	+	0.259009i	$0.0833960\pi$
$229$	7.83904i	0.518019i	−0.965875	+	0.259009i	$0.916604\pi$
$230$	0	0
$231$	3.45686	0.227445
$232$	0	0
$233$	24.9587	1.63510	0.817550	−	0.575857i	$-0.195332\pi$
$233$	24.9587	1.63510	0.817550	+	0.575857i	$0.195332\pi$
$234$	0	0
$235$	13.0778i	0.853104i
$236$	0	0
$237$	14.2934i	0.928455i
$238$	0	0
$239$	2.64981	0.171402	0.0857009	−	0.996321i	$-0.472687\pi$
$239$	2.64981	0.171402	0.0857009	+	0.996321i	$0.472687\pi$
$240$	0	0
$241$	−5.92835	−0.381878	−0.190939	−	0.981602i	$-0.561153\pi$
$241$	−5.92835	−0.381878	−0.190939	+	0.981602i	$0.561153\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	20.3539	1.29509
$248$	0	0
$249$	−8.18770	−0.518875
$250$	0	0
$251$	13.3090i	0.840056i	0.907511	+	0.420028i	$0.137980\pi$
$251$	13.3090i	0.840056i	−0.907511	+	0.420028i	$0.862020\pi$
$252$	0	0
$253$	8.21334i	0.516368i
$254$	0	0
$255$	1.15712	0.0724616
$256$	0	0
$257$	21.0189	1.31112	0.655560	−	0.755143i	$-0.272432\pi$
$257$	21.0189	1.31112	0.655560	+	0.755143i	$0.272432\pi$
$258$	0	0
$259$	− 4.15908i	− 0.258433i
$260$	0	0
$261$	− 3.63769i	− 0.225168i
$262$	0	0
$263$	21.3307	1.31531	0.657654	−	0.753320i	$-0.271549\pi$
$263$	21.3307	1.31531	0.657654	+	0.753320i	$0.271549\pi$
$264$	0	0
$265$	5.92814	0.364162
$266$	0	0
$267$	− 2.61172i	− 0.159835i
$268$	0	0
$269$	10.3670i	0.632087i	0.948745	+	0.316043i	$0.102355\pi$
$269$	10.3670i	0.632087i	−0.948745	+	0.316043i	$0.897645\pi$
$270$	0	0
$271$	18.7588	1.13951	0.569757	−	0.821813i	$-0.307037\pi$
$271$	18.7588	1.13951	0.569757	+	0.821813i	$0.307037\pi$
$272$	0	0
$273$	−3.45686	−0.209219
$274$	0	0
$275$	3.45686i	0.208457i
$276$	0	0
$277$	− 25.1344i	− 1.51018i	−0.655621	−	0.755090i	$-0.727593\pi$
$277$	− 25.1344i	− 1.51018i	0.655621	−	0.755090i	$-0.272407\pi$
$278$	0	0
$279$	−5.40739	−0.323732
$280$	0	0
$281$	8.31424	0.495986	0.247993	−	0.968762i	$-0.420229\pi$
$281$	8.31424	0.495986	0.247993	+	0.968762i	$0.420229\pi$
$282$	0	0
$283$	− 10.9655i	− 0.651831i	−0.945399	−	0.325916i	$-0.894327\pi$
$283$	− 10.9655i	− 0.651831i	0.945399	−	0.325916i	$-0.105673\pi$
$284$	0	0
$285$	− 5.88796i	− 0.348773i
$286$	0	0
$287$	3.26174	0.192535
$288$	0	0
$289$	−15.6611	−0.921240
$290$	0	0
$291$	0.962000i	0.0563934i
$292$	0	0
$293$	− 17.0513i	− 0.996148i	−0.867134	−	0.498074i	$-0.834041\pi$
$293$	− 17.0513i	− 0.996148i	0.867134	−	0.498074i	$-0.165959\pi$
$294$	0	0
$295$	−12.5571	−0.731102
$296$	0	0
$297$	−3.45686	−0.200587
$298$	0	0
$299$	− 8.21334i	− 0.474989i
$300$	0	0
$301$	− 4.97544i	− 0.286779i
$302$	0	0
$303$	8.65394	0.497156
$304$	0	0
$305$	−13.8161	−0.791108
$306$	0	0
$307$	22.1618i	1.26484i	0.774625	+	0.632421i	$0.217939\pi$
$307$	22.1618i	1.26484i	−0.774625	+	0.632421i	$0.782061\pi$
$308$	0	0
$309$	− 12.5278i	− 0.712683i
$310$	0	0
$311$	−15.5650	−0.882609	−0.441304	−	0.897357i	$-0.645484\pi$
$311$	−15.5650	−0.882609	−0.441304	+	0.897357i	$0.645484\pi$
$312$	0	0
$313$	25.6560	1.45016	0.725081	−	0.688663i	$-0.241802\pi$
$313$	25.6560	1.45016	0.725081	+	0.688663i	$0.241802\pi$
$314$	0	0
$315$	1.00000i	0.0563436i
$316$	0	0
$317$	− 21.9692i	− 1.23391i	−0.786998	−	0.616956i	$-0.788366\pi$
$317$	− 21.9692i	− 1.23391i	0.786998	−	0.616956i	$-0.211634\pi$
$318$	0	0
$319$	12.5750	0.704065
$320$	0	0
$321$	13.7265	0.766136
$322$	0	0
$323$	− 6.81307i	− 0.379090i
$324$	0	0
$325$	− 3.45686i	− 0.191752i
$326$	0	0
$327$	0.912317	0.0504512
$328$	0	0
$329$	−13.0778	−0.721005
$330$	0	0
$331$	− 17.1940i	− 0.945068i	−0.881312	−	0.472534i	$-0.843339\pi$
$331$	− 17.1940i	− 0.945068i	0.881312	−	0.472534i	$-0.156661\pi$
$332$	0	0
$333$	4.15908i	0.227916i
$334$	0	0
$335$	−3.21227	−0.175505
$336$	0	0
$337$	2.89078	0.157471	0.0787353	−	0.996896i	$-0.474912\pi$
$337$	2.89078	0.157471	0.0787353	+	0.996896i	$0.474912\pi$
$338$	0	0
$339$	− 1.85435i	− 0.100715i
$340$	0	0
$341$	− 18.6926i	− 1.01226i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.37595	−0.127917
$346$	0	0
$347$	35.6670i	1.91471i	0.288922	+	0.957353i	$0.406703\pi$
$347$	35.6670i	1.91471i	−0.288922	+	0.957353i	$0.593297\pi$
$348$	0	0
$349$	− 4.17512i	− 0.223489i	−0.993737	−	0.111745i	$-0.964356\pi$
$349$	− 4.17512i	− 0.223489i	0.993737	−	0.111745i	$-0.0356438\pi$
$350$	0	0
$351$	3.45686	0.184514
$352$	0	0
$353$	−25.4091	−1.35239	−0.676195	−	0.736723i	$-0.736372\pi$
$353$	−25.4091	−1.35239	−0.676195	+	0.736723i	$0.736372\pi$
$354$	0	0
$355$	2.33991i	0.124190i
$356$	0	0
$357$	1.15712i	0.0612412i
$358$	0	0
$359$	−7.66401	−0.404491	−0.202245	−	0.979335i	$-0.564824\pi$
$359$	−7.66401	−0.404491	−0.202245	+	0.979335i	$0.564824\pi$
$360$	0	0
$361$	−15.6681	−0.824637
$362$	0	0
$363$	− 0.949887i	− 0.0498561i
$364$	0	0
$365$	− 5.95172i	− 0.311527i
$366$	0	0
$367$	−18.5662	−0.969149	−0.484574	−	0.874750i	$-0.661026\pi$
$367$	−18.5662	−0.969149	−0.484574	+	0.874750i	$0.661026\pi$
$368$	0	0
$369$	−3.26174	−0.169799
$370$	0	0
$371$	5.92814i	0.307773i
$372$	0	0
$373$	26.1444i	1.35371i	0.736118	+	0.676853i	$0.236657\pi$
$373$	26.1444i	1.35371i	−0.736118	+	0.676853i	$0.763343\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−12.5750	−0.647646
$378$	0	0
$379$	12.3622i	0.635003i	0.948258	+	0.317501i	$0.102844\pi$
$379$	12.3622i	0.635003i	−0.948258	+	0.317501i	$0.897156\pi$
$380$	0	0
$381$	9.75004i	0.499510i
$382$	0	0
$383$	−17.1552	−0.876592	−0.438296	−	0.898831i	$-0.644418\pi$
$383$	−17.1552	−0.876592	−0.438296	+	0.898831i	$0.644418\pi$
$384$	0	0
$385$	−3.45686	−0.176178
$386$	0	0
$387$	4.97544i	0.252916i
$388$	0	0
$389$	− 10.3584i	− 0.525191i	−0.964906	−	0.262595i	$-0.915422\pi$
$389$	− 10.3584i	− 0.525191i	0.964906	−	0.262595i	$-0.0845784\pi$
$390$	0	0
$391$	−2.74926	−0.139036
$392$	0	0
$393$	7.32913	0.369706
$394$	0	0
$395$	− 14.2934i	− 0.719178i
$396$	0	0
$397$	2.12996i	0.106899i	0.998571	+	0.0534497i	$0.0170217\pi$
$397$	2.12996i	0.106899i	−0.998571	+	0.0534497i	$0.982978\pi$
$398$	0	0
$399$	5.88796	0.294767
$400$	0	0
$401$	30.7553	1.53585	0.767924	−	0.640541i	$-0.221290\pi$
$401$	30.7553	1.53585	0.767924	+	0.640541i	$0.221290\pi$
$402$	0	0
$403$	18.6926i	0.931144i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	−14.3774	−0.712660
$408$	0	0
$409$	2.50335	0.123783	0.0618913	−	0.998083i	$-0.480287\pi$
$409$	2.50335	0.123783	0.0618913	+	0.998083i	$0.480287\pi$
$410$	0	0
$411$	− 4.09652i	− 0.202066i
$412$	0	0
$413$	− 12.5571i	− 0.617894i
$414$	0	0
$415$	8.18770	0.401919
$416$	0	0
$417$	18.1531	0.888960
$418$	0	0
$419$	36.8405i	1.79978i	0.436118	+	0.899889i	$0.356353\pi$
$419$	36.8405i	1.79978i	−0.436118	+	0.899889i	$0.643647\pi$
$420$	0	0
$421$	− 15.2623i	− 0.743837i	−0.928265	−	0.371919i	$-0.878700\pi$
$421$	− 15.2623i	− 0.743837i	0.928265	−	0.371919i	$-0.121300\pi$
$422$	0	0
$423$	13.0778	0.635866
$424$	0	0
$425$	−1.15712	−0.0561285
$426$	0	0
$427$	− 13.8161i	− 0.668608i
$428$	0	0
$429$	11.9499i	0.576946i
$430$	0	0
$431$	−13.1302	−0.632460	−0.316230	−	0.948683i	$-0.602417\pi$
$431$	−13.1302	−0.632460	−0.316230	+	0.948683i	$0.602417\pi$
$432$	0	0
$433$	29.3975	1.41275	0.706376	−	0.707837i	$-0.250329\pi$
$433$	29.3975	1.41275	0.706376	+	0.707837i	$0.250329\pi$
$434$	0	0
$435$	3.63769i	0.174414i
$436$	0	0
$437$	13.9895i	0.669209i
$438$	0	0
$439$	−22.6867	−1.08278	−0.541389	−	0.840773i	$-0.682101\pi$
$439$	−22.6867	−1.08278	−0.541389	+	0.840773i	$0.682101\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	35.7201i	1.69711i	0.529106	+	0.848556i	$0.322527\pi$
$443$	35.7201i	1.69711i	−0.529106	+	0.848556i	$0.677473\pi$
$444$	0	0
$445$	2.61172i	0.123808i
$446$	0	0
$447$	4.96179	0.234685
$448$	0	0
$449$	−23.1828	−1.09406	−0.547031	−	0.837112i	$-0.684242\pi$
$449$	−23.1828	−1.09406	−0.547031	+	0.837112i	$0.684242\pi$
$450$	0	0
$451$	− 11.2754i	− 0.530937i
$452$	0	0
$453$	19.8109i	0.930799i
$454$	0	0
$455$	3.45686	0.162060
$456$	0	0
$457$	−2.26681	−0.106037	−0.0530185	−	0.998594i	$-0.516884\pi$
$457$	−2.26681	−0.106037	−0.0530185	+	0.998594i	$0.516884\pi$
$458$	0	0
$459$	− 1.15712i	− 0.0540097i
$460$	0	0
$461$	1.56217i	0.0727575i	0.999338	+	0.0363787i	$0.0115823\pi$
$461$	1.56217i	0.0727575i	−0.999338	+	0.0363787i	$0.988418\pi$
$462$	0	0
$463$	32.5631	1.51334	0.756668	−	0.653799i	$-0.226826\pi$
$463$	32.5631	1.51334	0.756668	+	0.653799i	$0.226826\pi$
$464$	0	0
$465$	5.40739	0.250762
$466$	0	0
$467$	23.9641i	1.10892i	0.832209	+	0.554462i	$0.187076\pi$
$467$	23.9641i	1.10892i	−0.832209	+	0.554462i	$0.812924\pi$
$468$	0	0
$469$	− 3.21227i	− 0.148329i
$470$	0	0
$471$	−22.7995	−1.05054
$472$	0	0
$473$	−17.1994	−0.790829
$474$	0	0
$475$	5.88796i	0.270158i
$476$	0	0
$477$	− 5.92814i	− 0.271431i
$478$	0	0
$479$	−20.5143	−0.937321	−0.468661	−	0.883378i	$-0.655263\pi$
$479$	−20.5143	−0.937321	−0.468661	+	0.883378i	$0.655263\pi$
$480$	0	0
$481$	14.3774	0.655551
$482$	0	0
$483$	− 2.37595i	− 0.108110i
$484$	0	0
$485$	− 0.962000i	− 0.0436822i
$486$	0	0
$487$	−35.7867	−1.62165	−0.810826	−	0.585288i	$-0.800982\pi$
$487$	−35.7867	−1.62165	−0.810826	+	0.585288i	$0.800982\pi$
$488$	0	0
$489$	7.57926	0.342746
$490$	0	0
$491$	− 23.8613i	− 1.07684i	−0.842675	−	0.538422i	$-0.819021\pi$
$491$	− 23.8613i	− 1.07684i	0.842675	−	0.538422i	$-0.180979\pi$
$492$	0	0
$493$	4.20925i	0.189575i
$494$	0	0
$495$	3.45686	0.155374
$496$	0	0
$497$	−2.33991	−0.104960
$498$	0	0
$499$	− 34.5650i	− 1.54734i	−0.633587	−	0.773672i	$-0.718418\pi$
$499$	− 34.5650i	− 1.54734i	0.633587	−	0.773672i	$-0.281582\pi$
$500$	0	0
$501$	− 7.14851i	− 0.319372i
$502$	0	0
$503$	−15.1957	−0.677542	−0.338771	−	0.940869i	$-0.610011\pi$
$503$	−15.1957	−0.677542	−0.338771	+	0.940869i	$0.610011\pi$
$504$	0	0
$505$	−8.65394	−0.385095
$506$	0	0
$507$	1.05011i	0.0466372i
$508$	0	0
$509$	6.28118i	0.278408i	0.990264	+	0.139204i	$0.0444544\pi$
$509$	6.28118i	0.278408i	−0.990264	+	0.139204i	$0.955546\pi$
$510$	0	0
$511$	5.95172	0.263289
$512$	0	0
$513$	−5.88796	−0.259960
$514$	0	0
$515$	12.5278i	0.552042i
$516$	0	0
$517$	45.2083i	1.98826i
$518$	0	0
$519$	−1.60976	−0.0706607
$520$	0	0
$521$	5.88279	0.257730	0.128865	−	0.991662i	$-0.458867\pi$
$521$	5.88279	0.257730	0.128865	+	0.991662i	$0.458867\pi$
$522$	0	0
$523$	1.16662i	0.0510129i	0.999675	+	0.0255065i	$0.00811984\pi$
$523$	1.16662i	0.0510129i	−0.999675	+	0.0255065i	$0.991880\pi$
$524$	0	0
$525$	− 1.00000i	− 0.0436436i
$526$	0	0
$527$	6.25699	0.272559
$528$	0	0
$529$	−17.3549	−0.754559
$530$	0	0
$531$	12.5571i	0.544931i
$532$	0	0
$533$	11.2754i	0.488391i
$534$	0	0
$535$	−13.7265	−0.593446
$536$	0	0
$537$	17.9373	0.774050
$538$	0	0
$539$	− 3.45686i	− 0.148898i
$540$	0	0
$541$	13.7558i	0.591408i	0.955280	+	0.295704i	$0.0955541\pi$
$541$	13.7558i	0.591408i	−0.955280	+	0.295704i	$0.904446\pi$
$542$	0	0
$543$	−12.5922	−0.540384
$544$	0	0
$545$	−0.912317	−0.0390794
$546$	0	0
$547$	15.0151i	0.641998i	0.947080	+	0.320999i	$0.104019\pi$
$547$	15.0151i	0.641998i	−0.947080	+	0.320999i	$0.895981\pi$
$548$	0	0
$549$	13.8161i	0.589657i
$550$	0	0
$551$	21.4186	0.912463
$552$	0	0
$553$	14.2934	0.607817
$554$	0	0
$555$	− 4.15908i	− 0.176543i
$556$	0	0
$557$	− 16.4658i	− 0.697678i	−0.937183	−	0.348839i	$-0.886576\pi$
$557$	− 16.4658i	− 0.697678i	0.937183	−	0.348839i	$-0.113424\pi$
$558$	0	0
$559$	17.1994	0.727457
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	− 5.43891i	− 0.229223i	−0.993410	−	0.114611i	$-0.963438\pi$
$563$	− 5.43891i	− 0.229223i	0.993410	−	0.114611i	$-0.0365623\pi$
$564$	0	0
$565$	1.85435i	0.0780132i
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−46.1886	−1.93633	−0.968163	−	0.250319i	$-0.919464\pi$
$569$	−46.1886	−1.93633	−0.968163	+	0.250319i	$0.919464\pi$
$570$	0	0
$571$	30.7029i	1.28488i	0.766338	+	0.642438i	$0.222077\pi$
$571$	30.7029i	1.28488i	−0.766338	+	0.642438i	$0.777923\pi$
$572$	0	0
$573$	− 0.362860i	− 0.0151587i
$574$	0	0
$575$	2.37595	0.0990840
$576$	0	0
$577$	42.3096	1.76137	0.880687	−	0.473699i	$-0.157082\pi$
$577$	42.3096	1.76137	0.880687	+	0.473699i	$0.157082\pi$
$578$	0	0
$579$	− 21.4416i	− 0.891080i
$580$	0	0
$581$	8.18770i	0.339683i
$582$	0	0
$583$	20.4927	0.848723
$584$	0	0
$585$	−3.45686	−0.142924
$586$	0	0
$587$	33.7492i	1.39298i	0.717567	+	0.696490i	$0.245256\pi$
$587$	33.7492i	1.39298i	−0.717567	+	0.696490i	$0.754744\pi$
$588$	0	0
$589$	− 31.8385i	− 1.31188i
$590$	0	0
$591$	19.6997	0.810338
$592$	0	0
$593$	−18.7826	−0.771308	−0.385654	−	0.922643i	$-0.626024\pi$
$593$	−18.7826	−0.771308	−0.385654	+	0.922643i	$0.626024\pi$
$594$	0	0
$595$	− 1.15712i	− 0.0474373i
$596$	0	0
$597$	− 8.33424i	− 0.341098i
$598$	0	0
$599$	−36.0553	−1.47318	−0.736590	−	0.676340i	$-0.763565\pi$
$599$	−36.0553	−1.47318	−0.736590	+	0.676340i	$0.763565\pi$
$600$	0	0
$601$	−40.1760	−1.63881	−0.819406	−	0.573213i	$-0.805697\pi$
$601$	−40.1760	−1.63881	−0.819406	+	0.573213i	$0.805697\pi$
$602$	0	0
$603$	3.21227i	0.130814i
$604$	0	0
$605$	0.949887i	0.0386184i
$606$	0	0
$607$	1.19349	0.0484425	0.0242212	−	0.999707i	$-0.492289\pi$
$607$	1.19349	0.0484425	0.0242212	+	0.999707i	$0.492289\pi$
$608$	0	0
$609$	−3.63769	−0.147407
$610$	0	0
$611$	− 45.2083i	− 1.82893i
$612$	0	0
$613$	− 5.58254i	− 0.225477i	−0.993625	−	0.112738i	$-0.964038\pi$
$613$	− 5.58254i	− 0.225477i	0.993625	−	0.112738i	$-0.0359622\pi$
$614$	0	0
$615$	3.26174	0.131526
$616$	0	0
$617$	1.04871	0.0422194	0.0211097	−	0.999777i	$-0.493280\pi$
$617$	1.04871	0.0422194	0.0211097	+	0.999777i	$0.493280\pi$
$618$	0	0
$619$	20.7700i	0.834818i	0.908719	+	0.417409i	$0.137062\pi$
$619$	20.7700i	0.834818i	−0.908719	+	0.417409i	$0.862938\pi$
$620$	0	0
$621$	2.37595i	0.0953437i
$622$	0	0
$623$	−2.61172	−0.104637
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 20.3539i	− 0.812855i
$628$	0	0
$629$	− 4.81255i	− 0.191889i
$630$	0	0
$631$	13.4246	0.534426	0.267213	−	0.963637i	$-0.413897\pi$
$631$	13.4246	0.534426	0.267213	+	0.963637i	$0.413897\pi$
$632$	0	0
$633$	−15.0108	−0.596626
$634$	0	0
$635$	− 9.75004i	− 0.386918i
$636$	0	0
$637$	3.45686i	0.136966i
$638$	0	0
$639$	2.33991	0.0925656
$640$	0	0
$641$	−5.50664	−0.217499	−0.108750	−	0.994069i	$-0.534685\pi$
$641$	−5.50664	−0.217499	−0.108750	+	0.994069i	$0.534685\pi$
$642$	0	0
$643$	44.9187i	1.77142i	0.464239	+	0.885710i	$0.346328\pi$
$643$	44.9187i	1.77142i	−0.464239	+	0.885710i	$0.653672\pi$
$644$	0	0
$645$	− 4.97544i	− 0.195908i
$646$	0	0
$647$	−33.1838	−1.30459	−0.652295	−	0.757965i	$-0.726194\pi$
$647$	−33.1838	−1.30459	−0.652295	+	0.757965i	$0.726194\pi$
$648$	0	0
$649$	−43.4081	−1.70392
$650$	0	0
$651$	5.40739i	0.211932i
$652$	0	0
$653$	− 13.1201i	− 0.513429i	−0.966487	−	0.256714i	$-0.917360\pi$
$653$	− 13.1201i	− 0.513429i	0.966487	−	0.256714i	$-0.0826400\pi$
$654$	0	0
$655$	−7.32913	−0.286373
$656$	0	0
$657$	−5.95172	−0.232199
$658$	0	0
$659$	− 13.6449i	− 0.531528i	−0.964038	−	0.265764i	$-0.914376\pi$
$659$	− 13.6449i	− 0.531528i	0.964038	−	0.265764i	$-0.0856243\pi$
$660$	0	0
$661$	35.1275i	1.36630i	0.730278	+	0.683150i	$0.239391\pi$
$661$	35.1275i	1.36630i	−0.730278	+	0.683150i	$0.760609\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−5.88796	−0.228325
$666$	0	0
$667$	− 8.64299i	− 0.334658i
$668$	0	0
$669$	20.0655i	0.775778i
$670$	0	0
$671$	−47.7603	−1.84377
$672$	0	0
$673$	−7.01462	−0.270394	−0.135197	−	0.990819i	$-0.543167\pi$
$673$	−7.01462	−0.270394	−0.135197	+	0.990819i	$0.543167\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	− 24.1419i	− 0.927850i	−0.885874	−	0.463925i	$-0.846441\pi$
$677$	− 24.1419i	− 0.927850i	0.885874	−	0.463925i	$-0.153559\pi$
$678$	0	0
$679$	0.962000	0.0369182
$680$	0	0
$681$	21.6784	0.830718
$682$	0	0
$683$	13.3089i	0.509252i	0.967040	+	0.254626i	$0.0819523\pi$
$683$	13.3089i	0.509252i	−0.967040	+	0.254626i	$0.918048\pi$
$684$	0	0
$685$	4.09652i	0.156520i
$686$	0	0
$687$	−7.83904	−0.299078
$688$	0	0
$689$	−20.4927	−0.780711
$690$	0	0
$691$	1.62524i	0.0618271i	0.999522	+	0.0309136i	$0.00984166\pi$
$691$	1.62524i	0.0618271i	−0.999522	+	0.0309136i	$0.990158\pi$
$692$	0	0
$693$	3.45686i	0.131315i
$694$	0	0
$695$	−18.1531	−0.688585
$696$	0	0
$697$	3.77422	0.142959
$698$	0	0
$699$	24.9587i	0.944026i
$700$	0	0
$701$	35.1489i	1.32755i	0.747931	+	0.663777i	$0.231048\pi$
$701$	35.1489i	1.32755i	−0.747931	+	0.663777i	$0.768952\pi$
$702$	0	0
$703$	−24.4885	−0.923602
$704$	0	0
$705$	−13.0778	−0.492540
$706$	0	0
$707$	− 8.65394i	− 0.325465i
$708$	0	0
$709$	− 20.0114i	− 0.751542i	−0.926713	−	0.375771i	$-0.877378\pi$
$709$	− 20.0114i	− 0.751542i	0.926713	−	0.375771i	$-0.122622\pi$
$710$	0	0
$711$	−14.2934	−0.536044
$712$	0	0
$713$	−12.8477	−0.481150
$714$	0	0
$715$	− 11.9499i	− 0.446900i
$716$	0	0
$717$	2.64981i	0.0989589i
$718$	0	0
$719$	31.7901	1.18557	0.592785	−	0.805361i	$-0.298028\pi$
$719$	31.7901	1.18557	0.592785	+	0.805361i	$0.298028\pi$
$720$	0	0
$721$	−12.5278	−0.466561
$722$	0	0
$723$	− 5.92835i	− 0.220478i
$724$	0	0
$725$	− 3.63769i	− 0.135101i
$726$	0	0
$727$	−25.9394	−0.962041	−0.481020	−	0.876709i	$-0.659734\pi$
$727$	−25.9394	−0.962041	−0.481020	+	0.876709i	$0.659734\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 5.75717i	− 0.212937i
$732$	0	0
$733$	16.8200i	0.621262i	0.950531	+	0.310631i	$0.100540\pi$
$733$	16.8200i	0.621262i	−0.950531	+	0.310631i	$0.899460\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−11.1044	−0.409035
$738$	0	0
$739$	− 43.9284i	− 1.61593i	−0.589228	−	0.807967i	$-0.700568\pi$
$739$	− 43.9284i	− 1.61593i	0.589228	−	0.807967i	$-0.299432\pi$
$740$	0	0
$741$	20.3539i	0.747718i
$742$	0	0
$743$	24.5519	0.900723	0.450362	−	0.892846i	$-0.351295\pi$
$743$	24.5519	0.900723	0.450362	+	0.892846i	$0.351295\pi$
$744$	0	0
$745$	−4.96179	−0.181786
$746$	0	0
$747$	− 8.18770i	− 0.299572i
$748$	0	0
$749$	− 13.7265i	− 0.501554i
$750$	0	0
$751$	10.0855	0.368026	0.184013	−	0.982924i	$-0.441091\pi$
$751$	10.0855	0.368026	0.184013	+	0.982924i	$0.441091\pi$
$752$	0	0
$753$	−13.3090	−0.485007
$754$	0	0
$755$	− 19.8109i	− 0.720994i
$756$	0	0
$757$	20.2698i	0.736719i	0.929684	+	0.368359i	$0.120080\pi$
$757$	20.2698i	0.736719i	−0.929684	+	0.368359i	$0.879920\pi$
$758$	0	0
$759$	−8.21334	−0.298125
$760$	0	0
$761$	19.4494	0.705042	0.352521	−	0.935804i	$-0.385325\pi$
$761$	19.4494	0.705042	0.352521	+	0.935804i	$0.385325\pi$
$762$	0	0
$763$	− 0.912317i	− 0.0330281i
$764$	0	0
$765$	1.15712i	0.0418357i
$766$	0	0
$767$	43.4081	1.56738
$768$	0	0
$769$	−22.6523	−0.816863	−0.408431	−	0.912789i	$-0.633924\pi$
$769$	−22.6523	−0.816863	−0.408431	+	0.912789i	$0.633924\pi$
$770$	0	0
$771$	21.0189i	0.756976i
$772$	0	0
$773$	32.3487i	1.16350i	0.813367	+	0.581751i	$0.197632\pi$
$773$	32.3487i	1.16350i	−0.813367	+	0.581751i	$0.802368\pi$
$774$	0	0
$775$	−5.40739	−0.194239
$776$	0	0
$777$	4.15908	0.149206
$778$	0	0
$779$	− 19.2050i	− 0.688091i
$780$	0	0
$781$	8.08876i	0.289439i
$782$	0	0
$783$	3.63769	0.130001
$784$	0	0
$785$	22.7995	0.813748
$786$	0	0
$787$	31.4411i	1.12076i	0.828237	+	0.560378i	$0.189344\pi$
$787$	31.4411i	1.12076i	−0.828237	+	0.560378i	$0.810656\pi$
$788$	0	0
$789$	21.3307i	0.759393i
$790$	0	0
$791$	−1.85435	−0.0659332
$792$	0	0
$793$	47.7603	1.69602
$794$	0	0
$795$	5.92814i	0.210249i
$796$	0	0
$797$	− 28.1515i	− 0.997177i	−0.866839	−	0.498589i	$-0.833852\pi$
$797$	− 28.1515i	− 0.997177i	0.866839	−	0.498589i	$-0.166148\pi$
$798$	0	0
$799$	−15.1326	−0.535354
$800$	0	0
$801$	2.61172	0.0922807
$802$	0	0
$803$	− 20.5743i	− 0.726050i
$804$	0	0
$805$	2.37595i	0.0837413i
$806$	0	0
$807$	−10.3670	−0.364936
$808$	0	0
$809$	14.9984	0.527316	0.263658	−	0.964616i	$-0.415071\pi$
$809$	14.9984	0.527316	0.263658	+	0.964616i	$0.415071\pi$
$810$	0	0
$811$	18.8356i	0.661406i	0.943735	+	0.330703i	$0.107286\pi$
$811$	18.8356i	0.661406i	−0.943735	+	0.330703i	$0.892714\pi$
$812$	0	0
$813$	18.7588i	0.657899i
$814$	0	0
$815$	−7.57926	−0.265490
$816$	0	0
$817$	−29.2952	−1.02491
$818$	0	0
$819$	− 3.45686i	− 0.120792i
$820$	0	0
$821$	35.1179i	1.22562i	0.790229	+	0.612811i	$0.209961\pi$
$821$	35.1179i	1.22562i	−0.790229	+	0.612811i	$0.790039\pi$
$822$	0	0
$823$	−10.8181	−0.377097	−0.188548	−	0.982064i	$-0.560378\pi$
$823$	−10.8181	−0.377097	−0.188548	+	0.982064i	$0.560378\pi$
$824$	0	0
$825$	−3.45686	−0.120352
$826$	0	0
$827$	− 38.7858i	− 1.34871i	−0.738406	−	0.674357i	$-0.764421\pi$
$827$	− 38.7858i	− 1.34871i	0.738406	−	0.674357i	$-0.235579\pi$
$828$	0	0
$829$	− 38.0268i	− 1.32072i	−0.750947	−	0.660362i	$-0.770403\pi$
$829$	− 38.0268i	− 1.32072i	0.750947	−	0.660362i	$-0.229597\pi$
$830$	0	0
$831$	25.1344	0.871903
$832$	0	0
$833$	1.15712	0.0400918
$834$	0	0
$835$	7.14851i	0.247384i
$836$	0	0
$837$	− 5.40739i	− 0.186907i
$838$	0	0
$839$	−33.7281	−1.16442	−0.582211	−	0.813038i	$-0.697812\pi$
$839$	−33.7281	−1.16442	−0.582211	+	0.813038i	$0.697812\pi$
$840$	0	0
$841$	15.7672	0.543696
$842$	0	0
$843$	8.31424i	0.286358i
$844$	0	0
$845$	− 1.05011i	− 0.0361250i
$846$	0	0
$847$	−0.949887	−0.0326385
$848$	0	0
$849$	10.9655	0.376335
$850$	0	0
$851$	9.88178i	0.338743i
$852$	0	0
$853$	43.8436i	1.50118i	0.660770	+	0.750589i	$0.270230\pi$
$853$	43.8436i	1.50118i	−0.660770	+	0.750589i	$0.729770\pi$
$854$	0	0
$855$	5.88796	0.201364
$856$	0	0
$857$	−20.7396	−0.708451	−0.354225	−	0.935160i	$-0.615255\pi$
$857$	−20.7396	−0.708451	−0.354225	+	0.935160i	$0.615255\pi$
$858$	0	0
$859$	− 24.7972i	− 0.846071i	−0.906113	−	0.423036i	$-0.860965\pi$
$859$	− 24.7972i	− 0.846071i	0.906113	−	0.423036i	$-0.139035\pi$
$860$	0	0
$861$	3.26174i	0.111160i
$862$	0	0
$863$	31.3753	1.06803	0.534014	−	0.845475i	$-0.320683\pi$
$863$	31.3753	1.06803	0.534014	+	0.845475i	$0.320683\pi$
$864$	0	0
$865$	1.60976	0.0547335
$866$	0	0
$867$	− 15.6611i	− 0.531878i
$868$	0	0
$869$	− 49.4103i	− 1.67613i
$870$	0	0
$871$	11.1044	0.376257
$872$	0	0
$873$	−0.962000	−0.0325588
$874$	0	0
$875$	1.00000i	0.0338062i
$876$	0	0
$877$	22.7957i	0.769755i	0.922968	+	0.384878i	$0.125756\pi$
$877$	22.7957i	0.769755i	−0.922968	+	0.384878i	$0.874244\pi$
$878$	0	0
$879$	17.0513	0.575127
$880$	0	0
$881$	−14.2101	−0.478749	−0.239375	−	0.970927i	$-0.576942\pi$
$881$	−14.2101	−0.478749	−0.239375	+	0.970927i	$0.576942\pi$
$882$	0	0
$883$	11.3850i	0.383135i	0.981479	+	0.191568i	$0.0613572\pi$
$883$	11.3850i	0.383135i	−0.981479	+	0.191568i	$0.938643\pi$
$884$	0	0
$885$	− 12.5571i	− 0.422102i
$886$	0	0
$887$	11.5024	0.386212	0.193106	−	0.981178i	$-0.438144\pi$
$887$	11.5024	0.386212	0.193106	+	0.981178i	$0.438144\pi$
$888$	0	0
$889$	9.75004	0.327006
$890$	0	0
$891$	− 3.45686i	− 0.115809i
$892$	0	0
$893$	77.0018i	2.57677i
$894$	0	0
$895$	−17.9373	−0.599577
$896$	0	0
$897$	8.21334	0.274235
$898$	0	0
$899$	19.6704i	0.656045i
$900$	0	0
$901$	6.85956i	0.228525i
$902$	0	0
$903$	4.97544	0.165572
$904$	0	0
$905$	12.5922	0.418580
$906$	0	0
$907$	34.8718i	1.15790i	0.815364	+	0.578949i	$0.196537\pi$
$907$	34.8718i	1.15790i	−0.815364	+	0.578949i	$0.803463\pi$
$908$	0	0
$909$	8.65394i	0.287033i
$910$	0	0
$911$	−8.14049	−0.269706	−0.134853	−	0.990866i	$-0.543056\pi$
$911$	−8.14049	−0.269706	−0.134853	+	0.990866i	$0.543056\pi$
$912$	0	0
$913$	28.3038	0.936718
$914$	0	0
$915$	− 13.8161i	− 0.456746i
$916$	0	0
$917$	− 7.32913i	− 0.242029i
$918$	0	0
$919$	16.9557	0.559318	0.279659	−	0.960099i	$-0.409779\pi$
$919$	16.9557	0.559318	0.279659	+	0.960099i	$0.409779\pi$
$920$	0	0
$921$	−22.1618	−0.730257
$922$	0	0
$923$	− 8.08876i	− 0.266245i
$924$	0	0
$925$	4.15908i	0.136750i
$926$	0	0
$927$	12.5278	0.411468
$928$	0	0
$929$	−11.1454	−0.365670	−0.182835	−	0.983144i	$-0.558527\pi$
$929$	−11.1454	−0.365670	−0.182835	+	0.983144i	$0.558527\pi$
$930$	0	0
$931$	− 5.88796i	− 0.192970i
$932$	0	0
$933$	− 15.5650i	− 0.509575i
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−18.5833	−0.607089	−0.303545	−	0.952817i	$-0.598170\pi$
$937$	−18.5833	−0.607089	−0.303545	+	0.952817i	$0.598170\pi$
$938$	0	0
$939$	25.6560i	0.837252i
$940$	0	0
$941$	41.3754i	1.34880i	0.738367	+	0.674399i	$0.235597\pi$
$941$	41.3754i	1.34880i	−0.738367	+	0.674399i	$0.764403\pi$
$942$	0	0
$943$	−7.74974	−0.252366
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	48.0763i	1.56227i	0.624363	+	0.781134i	$0.285359\pi$
$947$	48.0763i	1.56227i	−0.624363	+	0.781134i	$0.714641\pi$
$948$	0	0
$949$	20.5743i	0.667869i
$950$	0	0
$951$	21.9692	0.712399
$952$	0	0
$953$	17.6285	0.571042	0.285521	−	0.958372i	$-0.407833\pi$
$953$	17.6285	0.571042	0.285521	+	0.958372i	$0.407833\pi$
$954$	0	0
$955$	0.362860i	0.0117419i
$956$	0	0
$957$	12.5750i	0.406492i
$958$	0	0
$959$	−4.09652	−0.132283
$960$	0	0
$961$	−1.76015	−0.0567791
$962$	0	0
$963$	13.7265i	0.442329i
$964$	0	0
$965$	21.4416i	0.690228i
$966$	0	0
$967$	16.9567	0.545291	0.272646	−	0.962115i	$-0.412101\pi$
$967$	16.9567	0.545291	0.272646	+	0.962115i	$0.412101\pi$
$968$	0	0
$969$	6.81307	0.218867
$970$	0	0
$971$	− 15.0779i	− 0.483873i	−0.970292	−	0.241937i	$-0.922217\pi$
$971$	− 15.0779i	− 0.483873i	0.970292	−	0.241937i	$-0.0777826\pi$
$972$	0	0
$973$	− 18.1531i	− 0.581961i
$974$	0	0
$975$	3.45686	0.110708
$976$	0	0
$977$	39.9107	1.27686	0.638428	−	0.769682i	$-0.279585\pi$
$977$	39.9107	1.27686	0.638428	+	0.769682i	$0.279585\pi$
$978$	0	0
$979$	9.02837i	0.288548i
$980$	0	0
$981$	0.912317i	0.0291280i
$982$	0	0
$983$	−24.0536	−0.767190	−0.383595	−	0.923502i	$-0.625314\pi$
$983$	−24.0536	−0.767190	−0.383595	+	0.923502i	$0.625314\pi$
$984$	0	0
$985$	−19.6997	−0.627685
$986$	0	0
$987$	− 13.0778i	− 0.416272i
$988$	0	0
$989$	11.8214i	0.375899i
$990$	0	0
$991$	47.3455	1.50398	0.751990	−	0.659174i	$-0.229094\pi$
$991$	47.3455	1.50398	0.751990	+	0.659174i	$0.229094\pi$
$992$	0	0
$993$	17.1940	0.545636
$994$	0	0
$995$	8.33424i	0.264213i
$996$	0	0
$997$	25.2053i	0.798261i	0.916894	+	0.399130i	$0.130688\pi$
$997$	25.2053i	0.798261i	−0.916894	+	0.399130i	$0.869312\pi$
$998$	0	0
$999$	−4.15908	−0.131588

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.g.c.1681.8		12
4.3	odd	2		840.2.g.b.421.5	✓	12
8.3	odd	2		840.2.g.b.421.6	yes	12
8.5	even	2	inner	3360.2.g.c.1681.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.g.b.421.5	✓	12	4.3	odd	2
840.2.g.b.421.6	yes	12	8.3	odd	2
3360.2.g.c.1681.5		12	8.5	even	2	inner
3360.2.g.c.1681.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{5} +1.00000 q^{7} -1.00000 q^{9} -3.45686i q^{11} +3.45686i q^{13} +1.00000 q^{15} +1.15712 q^{17} -5.88796i q^{19} +1.00000i q^{21} -2.37595 q^{23} -1.00000 q^{25} -1.00000i q^{27} +3.63769i q^{29} +5.40739 q^{31} +3.45686 q^{33} -1.00000i q^{35} -4.15908i q^{37} -3.45686 q^{39} +3.26174 q^{41} -4.97544i q^{43} +1.00000i q^{45} -13.0778 q^{47} +1.00000 q^{49} +1.15712i q^{51} +5.92814i q^{53} -3.45686 q^{55} +5.88796 q^{57} -12.5571i q^{59} -13.8161i q^{61} -1.00000 q^{63} +3.45686 q^{65} -3.21227i q^{67} -2.37595i q^{69} -2.33991 q^{71} +5.95172 q^{73} -1.00000i q^{75} -3.45686i q^{77} +14.2934 q^{79} +1.00000 q^{81} +8.18770i q^{83} -1.15712i q^{85} -3.63769 q^{87} -2.61172 q^{89} +3.45686i q^{91} +5.40739i q^{93} -5.88796 q^{95} +0.962000 q^{97} +3.45686i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 12 q^{9} + 12 q^{15} - 40 q^{23} - 12 q^{25} + 8 q^{31} + 32 q^{47} + 12 q^{49} - 12 q^{63} + 8 q^{71} + 80 q^{79} + 12 q^{81} - 16 q^{87}+O(q^{100}) \) 12 * q + 12 * q^7 - 12 * q^9 + 12 * q^15 - 40 * q^23 - 12 * q^25 + 8 * q^31 + 32 * q^47 + 12 * q^49 - 12 * q^63 + 8 * q^71 + 80 * q^79 + 12 * q^81 - 16 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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