LMFDB - Embedded newform 9306.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9306,2,Mod(1,9306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9306, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9306.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9306 = 2 \cdot 3^{2} \cdot 11 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9306.a (trivial)

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:	yes
Analytic conductor:	$74.3087841210$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2949696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 16x^{2} + 10x - 16 $ x^5 - 2x^4 - 10x^3 + 16x^2 + 10x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3102)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.42308$ of defining polynomial
Character	$\chi$	$=$	9306.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.42308 q^{5} +2.58730 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.42308 q^{5} +2.58730 q^{7} -1.00000 q^{8} +3.42308 q^{10} -1.00000 q^{11} +3.70711 q^{13} -2.58730 q^{14} +1.00000 q^{16} -2.80457 q^{17} +4.11722 q^{19} -3.42308 q^{20} +1.00000 q^{22} +3.79312 q^{23} +6.71749 q^{25} -3.70711 q^{26} +2.58730 q^{28} +7.74870 q^{29} +2.00000 q^{31} -1.00000 q^{32} +2.80457 q^{34} -8.85655 q^{35} -2.42567 q^{37} -4.11722 q^{38} +3.42308 q^{40} +11.4439 q^{41} +4.70970 q^{43} -1.00000 q^{44} -3.79312 q^{46} -1.00000 q^{47} -0.305866 q^{49} -6.71749 q^{50} +3.70711 q^{52} +3.97918 q^{53} +3.42308 q^{55} -2.58730 q^{56} -7.74870 q^{58} -8.53093 q^{59} -6.05198 q^{61} -2.00000 q^{62} +1.00000 q^{64} -12.6897 q^{65} -8.33601 q^{67} -2.80457 q^{68} +8.85655 q^{70} +4.84099 q^{71} -13.0650 q^{73} +2.42567 q^{74} +4.11722 q^{76} -2.58730 q^{77} +6.37608 q^{79} -3.42308 q^{80} -11.4439 q^{82} +13.4542 q^{83} +9.60028 q^{85} -4.70970 q^{86} +1.00000 q^{88} -8.30480 q^{89} +9.59141 q^{91} +3.79312 q^{92} +1.00000 q^{94} -14.0936 q^{95} -7.22658 q^{97} +0.305866 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8} + 2 q^{10} - 5 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - q^{25} - 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} - 5 q^{32} + 6 q^{34} + 8 q^{35} - 12 q^{38} + 2 q^{40} - 2 q^{41} + 14 q^{43} - 5 q^{44} + 2 q^{46} - 5 q^{47} + 5 q^{49} + q^{50} + 6 q^{52} - 2 q^{53} + 2 q^{55} - 6 q^{56} - 4 q^{58} - 10 q^{59} + 14 q^{61} - 10 q^{62} + 5 q^{64} - 6 q^{68} - 8 q^{70} - 12 q^{71} - 2 q^{73} + 12 q^{76} - 6 q^{77} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 22 q^{85} - 14 q^{86} + 5 q^{88} + 12 q^{91} - 2 q^{92} + 5 q^{94} - 4 q^{95} + 22 q^{97} - 5 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8 + 2 * q^10 - 5 * q^11 + 6 * q^13 - 6 * q^14 + 5 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 + 5 * q^22 - 2 * q^23 - q^25 - 6 * q^26 + 6 * q^28 + 4 * q^29 + 10 * q^31 - 5 * q^32 + 6 * q^34 + 8 * q^35 - 12 * q^38 + 2 * q^40 - 2 * q^41 + 14 * q^43 - 5 * q^44 + 2 * q^46 - 5 * q^47 + 5 * q^49 + q^50 + 6 * q^52 - 2 * q^53 + 2 * q^55 - 6 * q^56 - 4 * q^58 - 10 * q^59 + 14 * q^61 - 10 * q^62 + 5 * q^64 - 6 * q^68 - 8 * q^70 - 12 * q^71 - 2 * q^73 + 12 * q^76 - 6 * q^77 - 2 * q^80 + 2 * q^82 - 14 * q^83 + 22 * q^85 - 14 * q^86 + 5 * q^88 + 12 * q^91 - 2 * q^92 + 5 * q^94 - 4 * q^95 + 22 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.42308	−1.53085	−0.765425	−	0.643526i	$-0.777471\pi$
$5$	−3.42308	−1.53085	−0.765425	+	0.643526i	$0.777471\pi$
$6$	0	0
$7$	2.58730	0.977908	0.488954	−	0.872309i	$-0.337379\pi$
$7$	2.58730	0.977908	0.488954	+	0.872309i	$0.337379\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.42308	1.08247
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.70711	1.02817	0.514084	−	0.857740i	$-0.328132\pi$
$13$	3.70711	1.02817	0.514084	+	0.857740i	$0.328132\pi$
$14$	−2.58730	−0.691486
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.80457	−0.680208	−0.340104	−	0.940388i	$-0.610462\pi$
$17$	−2.80457	−0.680208	−0.340104	+	0.940388i	$0.610462\pi$
$18$	0	0
$19$	4.11722	0.944554	0.472277	−	0.881450i	$-0.343432\pi$
$19$	4.11722	0.944554	0.472277	+	0.881450i	$0.343432\pi$
$20$	−3.42308	−0.765425
$21$	0	0
$22$	1.00000	0.213201
$23$	3.79312	0.790920	0.395460	−	0.918483i	$-0.370585\pi$
$23$	3.79312	0.790920	0.395460	+	0.918483i	$0.370585\pi$
$24$	0	0
$25$	6.71749	1.34350
$26$	−3.70711	−0.727024
$27$	0	0
$28$	2.58730	0.488954
$29$	7.74870	1.43890	0.719449	−	0.694545i	$-0.244394\pi$
$29$	7.74870	1.43890	0.719449	+	0.694545i	$0.244394\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.80457	0.480980
$35$	−8.85655	−1.49703
$36$	0	0
$37$	−2.42567	−0.398778	−0.199389	−	0.979920i	$-0.563896\pi$
$37$	−2.42567	−0.398778	−0.199389	+	0.979920i	$0.563896\pi$
$38$	−4.11722	−0.667901
$39$	0	0
$40$	3.42308	0.541237
$41$	11.4439	1.78723	0.893615	−	0.448835i	$-0.148161\pi$
$41$	11.4439	1.78723	0.893615	+	0.448835i	$0.148161\pi$
$42$	0	0
$43$	4.70970	0.718222	0.359111	−	0.933295i	$-0.383080\pi$
$43$	4.70970	0.718222	0.359111	+	0.933295i	$0.383080\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−3.79312	−0.559265
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−0.305866	−0.0436951
$50$	−6.71749	−0.949997
$51$	0	0
$52$	3.70711	0.514084
$53$	3.97918	0.546582	0.273291	−	0.961931i	$-0.411888\pi$
$53$	3.97918	0.546582	0.273291	+	0.961931i	$0.411888\pi$
$54$	0	0
$55$	3.42308	0.461568
$56$	−2.58730	−0.345743
$57$	0	0
$58$	−7.74870	−1.01745
$59$	−8.53093	−1.11063	−0.555316	−	0.831639i	$-0.687403\pi$
$59$	−8.53093	−1.11063	−0.555316	+	0.831639i	$0.687403\pi$
$60$	0	0
$61$	−6.05198	−0.774877	−0.387438	−	0.921896i	$-0.626640\pi$
$61$	−6.05198	−0.774877	−0.387438	+	0.921896i	$0.626640\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.6897	−1.57397
$66$	0	0
$67$	−8.33601	−1.01841	−0.509203	−	0.860647i	$-0.670060\pi$
$67$	−8.33601	−1.01841	−0.509203	+	0.860647i	$0.670060\pi$
$68$	−2.80457	−0.340104
$69$	0	0
$70$	8.85655	1.05856
$71$	4.84099	0.574519	0.287260	−	0.957853i	$-0.407256\pi$
$71$	4.84099	0.574519	0.287260	+	0.957853i	$0.407256\pi$
$72$	0	0
$73$	−13.0650	−1.52914	−0.764569	−	0.644542i	$-0.777048\pi$
$73$	−13.0650	−1.52914	−0.764569	+	0.644542i	$0.777048\pi$
$74$	2.42567	0.281979
$75$	0	0
$76$	4.11722	0.472277
$77$	−2.58730	−0.294850
$78$	0	0
$79$	6.37608	0.717365	0.358683	−	0.933460i	$-0.383226\pi$
$79$	6.37608	0.717365	0.358683	+	0.933460i	$0.383226\pi$
$80$	−3.42308	−0.382712
$81$	0	0
$82$	−11.4439	−1.26376
$83$	13.4542	1.47679	0.738397	−	0.674366i	$-0.235583\pi$
$83$	13.4542	1.47679	0.738397	+	0.674366i	$0.235583\pi$
$84$	0	0
$85$	9.60028	1.04130
$86$	−4.70970	−0.507860
$87$	0	0
$88$	1.00000	0.106600
$89$	−8.30480	−0.880307	−0.440153	−	0.897923i	$-0.645076\pi$
$89$	−8.30480	−0.880307	−0.440153	+	0.897923i	$0.645076\pi$
$90$	0	0
$91$	9.59141	1.00545
$92$	3.79312	0.395460
$93$	0	0
$94$	1.00000	0.103142
$95$	−14.0936	−1.44597
$96$	0	0
$97$	−7.22658	−0.733748	−0.366874	−	0.930271i	$-0.619572\pi$
$97$	−7.22658	−0.733748	−0.366874	+	0.930271i	$0.619572\pi$
$98$	0.305866	0.0308971
$99$	0	0
$100$	6.71749	0.671749
$101$	12.2068	1.21462	0.607312	−	0.794463i	$-0.292248\pi$
$101$	12.2068	1.21462	0.607312	+	0.794463i	$0.292248\pi$
$102$	0	0
$103$	−5.86592	−0.577986	−0.288993	−	0.957331i	$-0.593320\pi$
$103$	−5.86592	−0.577986	−0.288993	+	0.957331i	$0.593320\pi$
$104$	−3.70711	−0.363512
$105$	0	0
$106$	−3.97918	−0.386492
$107$	−10.2849	−0.994278	−0.497139	−	0.867671i	$-0.665616\pi$
$107$	−10.2849	−0.994278	−0.497139	+	0.867671i	$0.665616\pi$
$108$	0	0
$109$	−4.83730	−0.463329	−0.231665	−	0.972796i	$-0.574417\pi$
$109$	−4.83730	−0.463329	−0.231665	+	0.972796i	$0.574417\pi$
$110$	−3.42308	−0.326378
$111$	0	0
$112$	2.58730	0.244477
$113$	0.737812	0.0694075	0.0347038	−	0.999398i	$-0.488951\pi$
$113$	0.737812	0.0694075	0.0347038	+	0.999398i	$0.488951\pi$
$114$	0	0
$115$	−12.9842	−1.21078
$116$	7.74870	0.719449
$117$	0	0
$118$	8.53093	0.785336
$119$	−7.25627	−0.665181
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.05198	0.547920
$123$	0	0
$124$	2.00000	0.179605
$125$	−5.87912	−0.525845
$126$	0	0
$127$	18.8233	1.67029	0.835147	−	0.550027i	$-0.185383\pi$
$127$	18.8233	1.67029	0.835147	+	0.550027i	$0.185383\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.6897	1.11296
$131$	3.96079	0.346056	0.173028	−	0.984917i	$-0.444645\pi$
$131$	3.96079	0.346056	0.173028	+	0.984917i	$0.444645\pi$
$132$	0	0
$133$	10.6525	0.923688
$134$	8.33601	0.720121
$135$	0	0
$136$	2.80457	0.240490
$137$	−0.851425	−0.0727422	−0.0363711	−	0.999338i	$-0.511580\pi$
$137$	−0.851425	−0.0727422	−0.0363711	+	0.999338i	$0.511580\pi$
$138$	0	0
$139$	−10.3176	−0.875129	−0.437565	−	0.899187i	$-0.644159\pi$
$139$	−10.3176	−0.875129	−0.437565	+	0.899187i	$0.644159\pi$
$140$	−8.85655	−0.748515
$141$	0	0
$142$	−4.84099	−0.406247
$143$	−3.70711	−0.310004
$144$	0	0
$145$	−26.5245	−2.20274
$146$	13.0650	1.08126
$147$	0	0
$148$	−2.42567	−0.199389
$149$	3.48919	0.285845	0.142923	−	0.989734i	$-0.454350\pi$
$149$	3.48919	0.285845	0.142923	+	0.989734i	$0.454350\pi$
$150$	0	0
$151$	21.0257	1.71105	0.855526	−	0.517761i	$-0.173234\pi$
$151$	21.0257	1.71105	0.855526	+	0.517761i	$0.173234\pi$
$152$	−4.11722	−0.333950
$153$	0	0
$154$	2.58730	0.208491
$155$	−6.84617	−0.549897
$156$	0	0
$157$	−7.91912	−0.632014	−0.316007	−	0.948757i	$-0.602342\pi$
$157$	−7.91912	−0.632014	−0.316007	+	0.948757i	$0.602342\pi$
$158$	−6.37608	−0.507254
$159$	0	0
$160$	3.42308	0.270618
$161$	9.81394	0.773447
$162$	0	0
$163$	17.0182	1.33297	0.666483	−	0.745520i	$-0.267799\pi$
$163$	17.0182	1.33297	0.666483	+	0.745520i	$0.267799\pi$
$164$	11.4439	0.893615
$165$	0	0
$166$	−13.4542	−1.04425
$167$	2.88537	0.223277	0.111638	−	0.993749i	$-0.464390\pi$
$167$	2.88537	0.223277	0.111638	+	0.993749i	$0.464390\pi$
$168$	0	0
$169$	0.742658	0.0571276
$170$	−9.60028	−0.736308
$171$	0	0
$172$	4.70970	0.359111
$173$	1.94523	0.147893	0.0739465	−	0.997262i	$-0.476441\pi$
$173$	1.94523	0.147893	0.0739465	+	0.997262i	$0.476441\pi$
$174$	0	0
$175$	17.3802	1.31382
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	8.30480	0.622471
$179$	−6.70726	−0.501324	−0.250662	−	0.968075i	$-0.580648\pi$
$179$	−6.70726	−0.501324	−0.250662	+	0.968075i	$0.580648\pi$
$180$	0	0
$181$	16.4345	1.22157	0.610783	−	0.791798i	$-0.290855\pi$
$181$	16.4345	1.22157	0.610783	+	0.791798i	$0.290855\pi$
$182$	−9.59141	−0.710963
$183$	0	0
$184$	−3.79312	−0.279632
$185$	8.30328	0.610469
$186$	0	0
$187$	2.80457	0.205091
$188$	−1.00000	−0.0729325
$189$	0	0
$190$	14.0936	1.02246
$191$	−21.3054	−1.54161	−0.770804	−	0.637073i	$-0.780145\pi$
$191$	−21.3054	−1.54161	−0.770804	+	0.637073i	$0.780145\pi$
$192$	0	0
$193$	−0.172855	−0.0124424	−0.00622118	−	0.999981i	$-0.501980\pi$
$193$	−0.172855	−0.0124424	−0.00622118	+	0.999981i	$0.501980\pi$
$194$	7.22658	0.518839
$195$	0	0
$196$	−0.305866	−0.0218475
$197$	0.275162	0.0196045	0.00980225	−	0.999952i	$-0.496880\pi$
$197$	0.275162	0.0196045	0.00980225	+	0.999952i	$0.496880\pi$
$198$	0	0
$199$	−8.92691	−0.632812	−0.316406	−	0.948624i	$-0.602476\pi$
$199$	−8.92691	−0.632812	−0.316406	+	0.948624i	$0.602476\pi$
$200$	−6.71749	−0.474999
$201$	0	0
$202$	−12.2068	−0.858870
$203$	20.0482	1.40711
$204$	0	0
$205$	−39.1733	−2.73598
$206$	5.86592	0.408698
$207$	0	0
$208$	3.70711	0.257042
$209$	−4.11722	−0.284794
$210$	0	0
$211$	−11.9391	−0.821922	−0.410961	−	0.911653i	$-0.634807\pi$
$211$	−11.9391	−0.821922	−0.410961	+	0.911653i	$0.634807\pi$
$212$	3.97918	0.273291
$213$	0	0
$214$	10.2849	0.703061
$215$	−16.1217	−1.09949
$216$	0	0
$217$	5.17461	0.351275
$218$	4.83730	0.327623
$219$	0	0
$220$	3.42308	0.230784
$221$	−10.3968	−0.699368
$222$	0	0
$223$	−18.1395	−1.21471	−0.607356	−	0.794430i	$-0.707770\pi$
$223$	−18.1395	−1.21471	−0.607356	+	0.794430i	$0.707770\pi$
$224$	−2.58730	−0.172871
$225$	0	0
$226$	−0.737812	−0.0490785
$227$	−7.99322	−0.530529	−0.265264	−	0.964176i	$-0.585459\pi$
$227$	−7.99322	−0.530529	−0.265264	+	0.964176i	$0.585459\pi$
$228$	0	0
$229$	29.0901	1.92233	0.961163	−	0.275982i	$-0.0890031\pi$
$229$	29.0901	1.92233	0.961163	+	0.275982i	$0.0890031\pi$
$230$	12.9842	0.856150
$231$	0	0
$232$	−7.74870	−0.508727
$233$	10.2840	0.673729	0.336864	−	0.941553i	$-0.390634\pi$
$233$	10.2840	0.673729	0.336864	+	0.941553i	$0.390634\pi$
$234$	0	0
$235$	3.42308	0.223297
$236$	−8.53093	−0.555316
$237$	0	0
$238$	7.25627	0.470354
$239$	8.57778	0.554851	0.277425	−	0.960747i	$-0.410519\pi$
$239$	8.57778	0.554851	0.277425	+	0.960747i	$0.410519\pi$
$240$	0	0
$241$	28.5733	1.84057	0.920284	−	0.391250i	$-0.127957\pi$
$241$	28.5733	1.84057	0.920284	+	0.391250i	$0.127957\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−6.05198	−0.387438
$245$	1.04700	0.0668906
$246$	0	0
$247$	15.2630	0.971160
$248$	−2.00000	−0.127000
$249$	0	0
$250$	5.87912	0.371828
$251$	0.525723	0.0331834	0.0165917	−	0.999862i	$-0.494718\pi$
$251$	0.525723	0.0331834	0.0165917	+	0.999862i	$0.494718\pi$
$252$	0	0
$253$	−3.79312	−0.238471
$254$	−18.8233	−1.18108
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.9046	−0.680208	−0.340104	−	0.940388i	$-0.610462\pi$
$257$	−10.9046	−0.680208	−0.340104	+	0.940388i	$0.610462\pi$
$258$	0	0
$259$	−6.27595	−0.389968
$260$	−12.6897	−0.786984
$261$	0	0
$262$	−3.96079	−0.244699
$263$	−20.2597	−1.24927	−0.624634	−	0.780917i	$-0.714752\pi$
$263$	−20.2597	−1.24927	−0.624634	+	0.780917i	$0.714752\pi$
$264$	0	0
$265$	−13.6210	−0.836734
$266$	−10.6525	−0.653146
$267$	0	0
$268$	−8.33601	−0.509203
$269$	−22.6971	−1.38387	−0.691934	−	0.721961i	$-0.743241\pi$
$269$	−22.6971	−1.38387	−0.691934	+	0.721961i	$0.743241\pi$
$270$	0	0
$271$	−0.470086	−0.0285557	−0.0142778	−	0.999898i	$-0.504545\pi$
$271$	−0.470086	−0.0285557	−0.0142778	+	0.999898i	$0.504545\pi$
$272$	−2.80457	−0.170052
$273$	0	0
$274$	0.851425	0.0514365
$275$	−6.71749	−0.405080
$276$	0	0
$277$	13.3734	0.803532	0.401766	−	0.915742i	$-0.368397\pi$
$277$	13.3734	0.803532	0.401766	+	0.915742i	$0.368397\pi$
$278$	10.3176	0.618810
$279$	0	0
$280$	8.85655	0.529280
$281$	0.812569	0.0484738	0.0242369	−	0.999706i	$-0.492284\pi$
$281$	0.812569	0.0484738	0.0242369	+	0.999706i	$0.492284\pi$
$282$	0	0
$283$	−18.3632	−1.09158	−0.545789	−	0.837922i	$-0.683770\pi$
$283$	−18.3632	−1.09158	−0.545789	+	0.837922i	$0.683770\pi$
$284$	4.84099	0.287260
$285$	0	0
$286$	3.70711	0.219206
$287$	29.6087	1.74775
$288$	0	0
$289$	−9.13438	−0.537317
$290$	26.5245	1.55757
$291$	0	0
$292$	−13.0650	−0.764569
$293$	9.57880	0.559599	0.279800	−	0.960058i	$-0.409732\pi$
$293$	9.57880	0.559599	0.279800	+	0.960058i	$0.409732\pi$
$294$	0	0
$295$	29.2021	1.70021
$296$	2.42567	0.140989
$297$	0	0
$298$	−3.48919	−0.202123
$299$	14.0615	0.813197
$300$	0	0
$301$	12.1854	0.702356
$302$	−21.0257	−1.20990
$303$	0	0
$304$	4.11722	0.236139
$305$	20.7164	1.18622
$306$	0	0
$307$	11.8555	0.676631	0.338316	−	0.941033i	$-0.390143\pi$
$307$	11.8555	0.676631	0.338316	+	0.941033i	$0.390143\pi$
$308$	−2.58730	−0.147425
$309$	0	0
$310$	6.84617	0.388836
$311$	7.67674	0.435308	0.217654	−	0.976026i	$-0.430160\pi$
$311$	7.67674	0.435308	0.217654	+	0.976026i	$0.430160\pi$
$312$	0	0
$313$	−20.9945	−1.18668	−0.593341	−	0.804951i	$-0.702191\pi$
$313$	−20.9945	−1.18668	−0.593341	+	0.804951i	$0.702191\pi$
$314$	7.91912	0.446902
$315$	0	0
$316$	6.37608	0.358683
$317$	17.5100	0.983457	0.491729	−	0.870749i	$-0.336365\pi$
$317$	17.5100	0.983457	0.491729	+	0.870749i	$0.336365\pi$
$318$	0	0
$319$	−7.74870	−0.433844
$320$	−3.42308	−0.191356
$321$	0	0
$322$	−9.81394	−0.546910
$323$	−11.5470	−0.642494
$324$	0	0
$325$	24.9025	1.38134
$326$	−17.0182	−0.942550
$327$	0	0
$328$	−11.4439	−0.631881
$329$	−2.58730	−0.142643
$330$	0	0
$331$	8.00548	0.440021	0.220011	−	0.975497i	$-0.429391\pi$
$331$	8.00548	0.440021	0.220011	+	0.975497i	$0.429391\pi$
$332$	13.4542	0.738397
$333$	0	0
$334$	−2.88537	−0.157881
$335$	28.5348	1.55902
$336$	0	0
$337$	3.88235	0.211485	0.105743	−	0.994394i	$-0.466278\pi$
$337$	3.88235	0.211485	0.105743	+	0.994394i	$0.466278\pi$
$338$	−0.742658	−0.0403953
$339$	0	0
$340$	9.60028	0.520648
$341$	−2.00000	−0.108306
$342$	0	0
$343$	−18.9025	−1.02064
$344$	−4.70970	−0.253930
$345$	0	0
$346$	−1.94523	−0.104576
$347$	17.7511	0.952928	0.476464	−	0.879194i	$-0.341918\pi$
$347$	17.7511	0.952928	0.476464	+	0.879194i	$0.341918\pi$
$348$	0	0
$349$	0.0424606	0.00227286	0.00113643	−	0.999999i	$-0.499638\pi$
$349$	0.0424606	0.00227286	0.00113643	+	0.999999i	$0.499638\pi$
$350$	−17.3802	−0.929010
$351$	0	0
$352$	1.00000	0.0533002
$353$	−24.6000	−1.30933	−0.654663	−	0.755921i	$-0.727189\pi$
$353$	−24.6000	−1.30933	−0.654663	+	0.755921i	$0.727189\pi$
$354$	0	0
$355$	−16.5711	−0.879502
$356$	−8.30480	−0.440153
$357$	0	0
$358$	6.70726	0.354490
$359$	−8.50198	−0.448717	−0.224359	−	0.974507i	$-0.572029\pi$
$359$	−8.50198	−0.448717	−0.224359	+	0.974507i	$0.572029\pi$
$360$	0	0
$361$	−2.04852	−0.107817
$362$	−16.4345	−0.863777
$363$	0	0
$364$	9.59141	0.502727
$365$	44.7224	2.34088
$366$	0	0
$367$	4.57201	0.238657	0.119329	−	0.992855i	$-0.461926\pi$
$367$	4.57201	0.238657	0.119329	+	0.992855i	$0.461926\pi$
$368$	3.79312	0.197730
$369$	0	0
$370$	−8.30328	−0.431667
$371$	10.2953	0.534507
$372$	0	0
$373$	33.7598	1.74802	0.874009	−	0.485910i	$-0.161512\pi$
$373$	33.7598	1.74802	0.874009	+	0.485910i	$0.161512\pi$
$374$	−2.80457	−0.145021
$375$	0	0
$376$	1.00000	0.0515711
$377$	28.7253	1.47943
$378$	0	0
$379$	−5.99786	−0.308090	−0.154045	−	0.988064i	$-0.549230\pi$
$379$	−5.99786	−0.308090	−0.154045	+	0.988064i	$0.549230\pi$
$380$	−14.0936	−0.722985
$381$	0	0
$382$	21.3054	1.09008
$383$	−16.6120	−0.848835	−0.424418	−	0.905467i	$-0.639521\pi$
$383$	−16.6120	−0.848835	−0.424418	+	0.905467i	$0.639521\pi$
$384$	0	0
$385$	8.85655	0.451372
$386$	0.172855	0.00879808
$387$	0	0
$388$	−7.22658	−0.366874
$389$	12.2908	0.623166	0.311583	−	0.950219i	$-0.399141\pi$
$389$	12.2908	0.623166	0.311583	+	0.950219i	$0.399141\pi$
$390$	0	0
$391$	−10.6381	−0.537990
$392$	0.305866	0.0154485
$393$	0	0
$394$	−0.275162	−0.0138625
$395$	−21.8258	−1.09818
$396$	0	0
$397$	−6.86612	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$397$	−6.86612	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$398$	8.92691	0.447466
$399$	0	0
$400$	6.71749	0.335875
$401$	19.3735	0.967466	0.483733	−	0.875216i	$-0.339281\pi$
$401$	19.3735	0.967466	0.483733	+	0.875216i	$0.339281\pi$
$402$	0	0
$403$	7.41422	0.369329
$404$	12.2068	0.607312
$405$	0	0
$406$	−20.0482	−0.994977
$407$	2.42567	0.120236
$408$	0	0
$409$	40.1650	1.98603	0.993016	−	0.117977i	$-0.0376409\pi$
$409$	40.1650	1.98603	0.993016	+	0.117977i	$0.0376409\pi$
$410$	39.1733	1.93463
$411$	0	0
$412$	−5.86592	−0.288993
$413$	−22.0721	−1.08610
$414$	0	0
$415$	−46.0550	−2.26075
$416$	−3.70711	−0.181756
$417$	0	0
$418$	4.11722	0.201380
$419$	18.2873	0.893391	0.446695	−	0.894686i	$-0.352601\pi$
$419$	18.2873	0.893391	0.446695	+	0.894686i	$0.352601\pi$
$420$	0	0
$421$	−32.1520	−1.56699	−0.783496	−	0.621397i	$-0.786565\pi$
$421$	−32.1520	−1.56699	−0.783496	+	0.621397i	$0.786565\pi$
$422$	11.9391	0.581186
$423$	0	0
$424$	−3.97918	−0.193246
$425$	−18.8397	−0.913859
$426$	0	0
$427$	−15.6583	−0.757758
$428$	−10.2849	−0.497139
$429$	0	0
$430$	16.1217	0.777457
$431$	30.6748	1.47755	0.738777	−	0.673950i	$-0.235404\pi$
$431$	30.6748	1.47755	0.738777	+	0.673950i	$0.235404\pi$
$432$	0	0
$433$	−1.48522	−0.0713753	−0.0356877	−	0.999363i	$-0.511362\pi$
$433$	−1.48522	−0.0713753	−0.0356877	+	0.999363i	$0.511362\pi$
$434$	−5.17461	−0.248389
$435$	0	0
$436$	−4.83730	−0.231665
$437$	15.6171	0.747067
$438$	0	0
$439$	1.66242	0.0793429	0.0396715	−	0.999213i	$-0.487369\pi$
$439$	1.66242	0.0793429	0.0396715	+	0.999213i	$0.487369\pi$
$440$	−3.42308	−0.163189
$441$	0	0
$442$	10.3968	0.494528
$443$	−27.8186	−1.32170	−0.660850	−	0.750518i	$-0.729804\pi$
$443$	−27.8186	−1.32170	−0.660850	+	0.750518i	$0.729804\pi$
$444$	0	0
$445$	28.4280	1.34762
$446$	18.1395	0.858931
$447$	0	0
$448$	2.58730	0.122239
$449$	−3.62387	−0.171021	−0.0855104	−	0.996337i	$-0.527252\pi$
$449$	−3.62387	−0.171021	−0.0855104	+	0.996337i	$0.527252\pi$
$450$	0	0
$451$	−11.4439	−0.538870
$452$	0.737812	0.0347038
$453$	0	0
$454$	7.99322	0.375140
$455$	−32.8322	−1.53920
$456$	0	0
$457$	24.4088	1.14179	0.570897	−	0.821022i	$-0.306596\pi$
$457$	24.4088	1.14179	0.570897	+	0.821022i	$0.306596\pi$
$458$	−29.0901	−1.35929
$459$	0	0
$460$	−12.9842	−0.605389
$461$	13.7530	0.640541	0.320270	−	0.947326i	$-0.396226\pi$
$461$	13.7530	0.640541	0.320270	+	0.947326i	$0.396226\pi$
$462$	0	0
$463$	36.4095	1.69209	0.846047	−	0.533108i	$-0.178976\pi$
$463$	36.4095	1.69209	0.846047	+	0.533108i	$0.178976\pi$
$464$	7.74870	0.359725
$465$	0	0
$466$	−10.2840	−0.476398
$467$	17.8392	0.825498	0.412749	−	0.910845i	$-0.364569\pi$
$467$	17.8392	0.825498	0.412749	+	0.910845i	$0.364569\pi$
$468$	0	0
$469$	−21.5678	−0.995907
$470$	−3.42308	−0.157895
$471$	0	0
$472$	8.53093	0.392668
$473$	−4.70970	−0.216552
$474$	0	0
$475$	27.6574	1.26901
$476$	−7.25627	−0.332591
$477$	0	0
$478$	−8.57778	−0.392339
$479$	−28.3408	−1.29492	−0.647461	−	0.762099i	$-0.724169\pi$
$479$	−28.3408	−1.29492	−0.647461	+	0.762099i	$0.724169\pi$
$480$	0	0
$481$	−8.99223	−0.410010
$482$	−28.5733	−1.30148
$483$	0	0
$484$	1.00000	0.0454545
$485$	24.7372	1.12326
$486$	0	0
$487$	−33.2770	−1.50792	−0.753962	−	0.656918i	$-0.771860\pi$
$487$	−33.2770	−1.50792	−0.753962	+	0.656918i	$0.771860\pi$
$488$	6.05198	0.273960
$489$	0	0
$490$	−1.04700	−0.0472988
$491$	19.3015	0.871063	0.435532	−	0.900173i	$-0.356560\pi$
$491$	19.3015	0.871063	0.435532	+	0.900173i	$0.356560\pi$
$492$	0	0
$493$	−21.7318	−0.978750
$494$	−15.2630	−0.686714
$495$	0	0
$496$	2.00000	0.0898027
$497$	12.5251	0.561827
$498$	0	0
$499$	−19.9136	−0.891453	−0.445727	−	0.895169i	$-0.647055\pi$
$499$	−19.9136	−0.891453	−0.445727	+	0.895169i	$0.647055\pi$
$500$	−5.87912	−0.262922
$501$	0	0
$502$	−0.525723	−0.0234642
$503$	33.5932	1.49785	0.748924	−	0.662656i	$-0.230571\pi$
$503$	33.5932	1.49785	0.748924	+	0.662656i	$0.230571\pi$
$504$	0	0
$505$	−41.7850	−1.85941
$506$	3.79312	0.168625
$507$	0	0
$508$	18.8233	0.835147
$509$	−11.5018	−0.509810	−0.254905	−	0.966966i	$-0.582044\pi$
$509$	−11.5018	−0.509810	−0.254905	+	0.966966i	$0.582044\pi$
$510$	0	0
$511$	−33.8030	−1.49536
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.9046	0.480980
$515$	20.0795	0.884810
$516$	0	0
$517$	1.00000	0.0439799
$518$	6.27595	0.275749
$519$	0	0
$520$	12.6897	0.556482
$521$	41.8459	1.83330	0.916651	−	0.399689i	$-0.130882\pi$
$521$	41.8459	1.83330	0.916651	+	0.399689i	$0.130882\pi$
$522$	0	0
$523$	42.4532	1.85635	0.928174	−	0.372147i	$-0.121378\pi$
$523$	42.4532	1.85635	0.928174	+	0.372147i	$0.121378\pi$
$524$	3.96079	0.173028
$525$	0	0
$526$	20.2597	0.883366
$527$	−5.60914	−0.244338
$528$	0	0
$529$	−8.61226	−0.374446
$530$	13.6210	0.591660
$531$	0	0
$532$	10.6525	0.461844
$533$	42.4236	1.83757
$534$	0	0
$535$	35.2060	1.52209
$536$	8.33601	0.360061
$537$	0	0
$538$	22.6971	0.978542
$539$	0.305866	0.0131746
$540$	0	0
$541$	30.6489	1.31770	0.658849	−	0.752275i	$-0.271044\pi$
$541$	30.6489	1.31770	0.658849	+	0.752275i	$0.271044\pi$
$542$	0.470086	0.0201919
$543$	0	0
$544$	2.80457	0.120245
$545$	16.5585	0.709287
$546$	0	0
$547$	22.4418	0.959544	0.479772	−	0.877393i	$-0.340719\pi$
$547$	22.4418	0.959544	0.479772	+	0.877393i	$0.340719\pi$
$548$	−0.851425	−0.0363711
$549$	0	0
$550$	6.71749	0.286435
$551$	31.9031	1.35912
$552$	0	0
$553$	16.4968	0.701517
$554$	−13.3734	−0.568183
$555$	0	0
$556$	−10.3176	−0.437565
$557$	46.8219	1.98391	0.991954	−	0.126600i	$-0.0404063\pi$
$557$	46.8219	1.98391	0.991954	+	0.126600i	$0.0404063\pi$
$558$	0	0
$559$	17.4594	0.738453
$560$	−8.85655	−0.374258
$561$	0	0
$562$	−0.812569	−0.0342761
$563$	10.3702	0.437054	0.218527	−	0.975831i	$-0.429875\pi$
$563$	10.3702	0.437054	0.218527	+	0.975831i	$0.429875\pi$
$564$	0	0
$565$	−2.52559	−0.106252
$566$	18.3632	0.771863
$567$	0	0
$568$	−4.84099	−0.203123
$569$	34.2009	1.43378	0.716888	−	0.697188i	$-0.245566\pi$
$569$	34.2009	1.43378	0.716888	+	0.697188i	$0.245566\pi$
$570$	0	0
$571$	16.6377	0.696268	0.348134	−	0.937445i	$-0.386816\pi$
$571$	16.6377	0.696268	0.348134	+	0.937445i	$0.386816\pi$
$572$	−3.70711	−0.155002
$573$	0	0
$574$	−29.6087	−1.23584
$575$	25.4802	1.06260
$576$	0	0
$577$	13.5346	0.563453	0.281727	−	0.959495i	$-0.409093\pi$
$577$	13.5346	0.563453	0.281727	+	0.959495i	$0.409093\pi$
$578$	9.13438	0.379940
$579$	0	0
$580$	−26.5245	−1.10137
$581$	34.8102	1.44417
$582$	0	0
$583$	−3.97918	−0.164801
$584$	13.0650	0.540632
$585$	0	0
$586$	−9.57880	−0.395697
$587$	5.62270	0.232074	0.116037	−	0.993245i	$-0.462981\pi$
$587$	5.62270	0.232074	0.116037	+	0.993245i	$0.462981\pi$
$588$	0	0
$589$	8.23443	0.339294
$590$	−29.2021	−1.20223
$591$	0	0
$592$	−2.42567	−0.0996945
$593$	−7.70315	−0.316330	−0.158165	−	0.987413i	$-0.550558\pi$
$593$	−7.70315	−0.316330	−0.158165	+	0.987413i	$0.550558\pi$
$594$	0	0
$595$	24.8388	1.01829
$596$	3.48919	0.142923
$597$	0	0
$598$	−14.0615	−0.575017
$599$	19.8940	0.812848	0.406424	−	0.913685i	$-0.366776\pi$
$599$	19.8940	0.812848	0.406424	+	0.913685i	$0.366776\pi$
$600$	0	0
$601$	−5.77443	−0.235544	−0.117772	−	0.993041i	$-0.537575\pi$
$601$	−5.77443	−0.235544	−0.117772	+	0.993041i	$0.537575\pi$
$602$	−12.1854	−0.496640
$603$	0	0
$604$	21.0257	0.855526
$605$	−3.42308	−0.139168
$606$	0	0
$607$	−28.8224	−1.16987	−0.584933	−	0.811081i	$-0.698879\pi$
$607$	−28.8224	−1.16987	−0.584933	+	0.811081i	$0.698879\pi$
$608$	−4.11722	−0.166975
$609$	0	0
$610$	−20.7164	−0.838784
$611$	−3.70711	−0.149974
$612$	0	0
$613$	−29.7015	−1.19963	−0.599816	−	0.800138i	$-0.704760\pi$
$613$	−29.7015	−1.19963	−0.599816	+	0.800138i	$0.704760\pi$
$614$	−11.8555	−0.478450
$615$	0	0
$616$	2.58730	0.104245
$617$	35.3097	1.42151	0.710757	−	0.703437i	$-0.248352\pi$
$617$	35.3097	1.42151	0.710757	+	0.703437i	$0.248352\pi$
$618$	0	0
$619$	−17.4934	−0.703121	−0.351560	−	0.936165i	$-0.614349\pi$
$619$	−17.4934	−0.703121	−0.351560	+	0.936165i	$0.614349\pi$
$620$	−6.84617	−0.274949
$621$	0	0
$622$	−7.67674	−0.307809
$623$	−21.4870	−0.860859
$624$	0	0
$625$	−13.4627	−0.538510
$626$	20.9945	0.839110
$627$	0	0
$628$	−7.91912	−0.316007
$629$	6.80297	0.271252
$630$	0	0
$631$	−34.0825	−1.35680	−0.678401	−	0.734692i	$-0.737327\pi$
$631$	−34.0825	−1.35680	−0.678401	+	0.734692i	$0.737327\pi$
$632$	−6.37608	−0.253627
$633$	0	0
$634$	−17.5100	−0.695409
$635$	−64.4336	−2.55697
$636$	0	0
$637$	−1.13388	−0.0449258
$638$	7.74870	0.306774
$639$	0	0
$640$	3.42308	0.135309
$641$	2.52212	0.0996177	0.0498089	−	0.998759i	$-0.484139\pi$
$641$	2.52212	0.0996177	0.0498089	+	0.998759i	$0.484139\pi$
$642$	0	0
$643$	41.4677	1.63533	0.817663	−	0.575697i	$-0.195269\pi$
$643$	41.4677	1.63533	0.817663	+	0.575697i	$0.195269\pi$
$644$	9.81394	0.386723
$645$	0	0
$646$	11.5470	0.454312
$647$	−28.4779	−1.11958	−0.559791	−	0.828634i	$-0.689119\pi$
$647$	−28.4779	−1.11958	−0.559791	+	0.828634i	$0.689119\pi$
$648$	0	0
$649$	8.53093	0.334868
$650$	−24.9025	−0.976756
$651$	0	0
$652$	17.0182	0.666483
$653$	12.7890	0.500471	0.250236	−	0.968185i	$-0.419492\pi$
$653$	12.7890	0.500471	0.250236	+	0.968185i	$0.419492\pi$
$654$	0	0
$655$	−13.5581	−0.529760
$656$	11.4439	0.446807
$657$	0	0
$658$	2.58730	0.100864
$659$	−25.2522	−0.983684	−0.491842	−	0.870684i	$-0.663676\pi$
$659$	−25.2522	−0.983684	−0.491842	+	0.870684i	$0.663676\pi$
$660$	0	0
$661$	42.3456	1.64705	0.823527	−	0.567278i	$-0.192003\pi$
$661$	42.3456	1.64705	0.823527	+	0.567278i	$0.192003\pi$
$662$	−8.00548	−0.311142
$663$	0	0
$664$	−13.4542	−0.522126
$665$	−36.4643	−1.41403
$666$	0	0
$667$	29.3917	1.13805
$668$	2.88537	0.111638
$669$	0	0
$670$	−28.5348	−1.10240
$671$	6.05198	0.233634
$672$	0	0
$673$	13.2950	0.512486	0.256243	−	0.966612i	$-0.417515\pi$
$673$	13.2950	0.512486	0.256243	+	0.966612i	$0.417515\pi$
$674$	−3.88235	−0.149543
$675$	0	0
$676$	0.742658	0.0285638
$677$	16.3166	0.627100	0.313550	−	0.949572i	$-0.398482\pi$
$677$	16.3166	0.627100	0.313550	+	0.949572i	$0.398482\pi$
$678$	0	0
$679$	−18.6974	−0.717539
$680$	−9.60028	−0.368154
$681$	0	0
$682$	2.00000	0.0765840
$683$	37.5958	1.43856	0.719282	−	0.694718i	$-0.244471\pi$
$683$	37.5958	1.43856	0.719282	+	0.694718i	$0.244471\pi$
$684$	0	0
$685$	2.91450	0.111357
$686$	18.9025	0.721700
$687$	0	0
$688$	4.70970	0.179556
$689$	14.7512	0.561977
$690$	0	0
$691$	−30.2287	−1.14995	−0.574977	−	0.818170i	$-0.694989\pi$
$691$	−30.2287	−1.14995	−0.574977	+	0.818170i	$0.694989\pi$
$692$	1.94523	0.0739465
$693$	0	0
$694$	−17.7511	−0.673822
$695$	35.3181	1.33969
$696$	0	0
$697$	−32.0951	−1.21569
$698$	−0.0424606	−0.00160716
$699$	0	0
$700$	17.3802	0.656909
$701$	−27.7629	−1.04859	−0.524296	−	0.851536i	$-0.675671\pi$
$701$	−27.7629	−1.04859	−0.524296	+	0.851536i	$0.675671\pi$
$702$	0	0
$703$	−9.98702	−0.376668
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	24.6000	0.925833
$707$	31.5828	1.18779
$708$	0	0
$709$	−18.1589	−0.681972	−0.340986	−	0.940068i	$-0.610761\pi$
$709$	−18.1589	−0.681972	−0.340986	+	0.940068i	$0.610761\pi$
$710$	16.5711	0.621902
$711$	0	0
$712$	8.30480	0.311235
$713$	7.58623	0.284107
$714$	0	0
$715$	12.6897	0.474569
$716$	−6.70726	−0.250662
$717$	0	0
$718$	8.50198	0.317291
$719$	11.3848	0.424582	0.212291	−	0.977206i	$-0.431907\pi$
$719$	11.3848	0.424582	0.212291	+	0.977206i	$0.431907\pi$
$720$	0	0
$721$	−15.1769	−0.565218
$722$	2.04852	0.0762382
$723$	0	0
$724$	16.4345	0.610783
$725$	52.0519	1.93316
$726$	0	0
$727$	−20.6008	−0.764040	−0.382020	−	0.924154i	$-0.624771\pi$
$727$	−20.6008	−0.764040	−0.382020	+	0.924154i	$0.624771\pi$
$728$	−9.59141	−0.355481
$729$	0	0
$730$	−44.7224	−1.65525
$731$	−13.2087	−0.488541
$732$	0	0
$733$	40.2050	1.48501	0.742503	−	0.669843i	$-0.233639\pi$
$733$	40.2050	1.48501	0.742503	+	0.669843i	$0.233639\pi$
$734$	−4.57201	−0.168756
$735$	0	0
$736$	−3.79312	−0.139816
$737$	8.33601	0.307061
$738$	0	0
$739$	−15.7567	−0.579619	−0.289809	−	0.957084i	$-0.593592\pi$
$739$	−15.7567	−0.579619	−0.289809	+	0.957084i	$0.593592\pi$
$740$	8.30328	0.305234
$741$	0	0
$742$	−10.2953	−0.377954
$743$	−13.0968	−0.480475	−0.240238	−	0.970714i	$-0.577225\pi$
$743$	−13.0968	−0.480475	−0.240238	+	0.970714i	$0.577225\pi$
$744$	0	0
$745$	−11.9438	−0.437586
$746$	−33.7598	−1.23603
$747$	0	0
$748$	2.80457	0.102545
$749$	−26.6101	−0.972313
$750$	0	0
$751$	−21.8651	−0.797869	−0.398934	−	0.916979i	$-0.630620\pi$
$751$	−21.8651	−0.797869	−0.398934	+	0.916979i	$0.630620\pi$
$752$	−1.00000	−0.0364662
$753$	0	0
$754$	−28.7253	−1.04611
$755$	−71.9729	−2.61936
$756$	0	0
$757$	−9.21698	−0.334997	−0.167499	−	0.985872i	$-0.553569\pi$
$757$	−9.21698	−0.334997	−0.167499	+	0.985872i	$0.553569\pi$
$758$	5.99786	0.217852
$759$	0	0
$760$	14.0936	0.511228
$761$	11.7656	0.426503	0.213252	−	0.976997i	$-0.431595\pi$
$761$	11.7656	0.426503	0.213252	+	0.976997i	$0.431595\pi$
$762$	0	0
$763$	−12.5156	−0.453094
$764$	−21.3054	−0.770804
$765$	0	0
$766$	16.6120	0.600217
$767$	−31.6251	−1.14192
$768$	0	0
$769$	33.8580	1.22095	0.610475	−	0.792036i	$-0.290979\pi$
$769$	33.8580	1.22095	0.610475	+	0.792036i	$0.290979\pi$
$770$	−8.85655	−0.319168
$771$	0	0
$772$	−0.172855	−0.00622118
$773$	−8.45221	−0.304005	−0.152002	−	0.988380i	$-0.548572\pi$
$773$	−8.45221	−0.304005	−0.152002	+	0.988380i	$0.548572\pi$
$774$	0	0
$775$	13.4350	0.482599
$776$	7.22658	0.259419
$777$	0	0
$778$	−12.2908	−0.440645
$779$	47.1168	1.68814
$780$	0	0
$781$	−4.84099	−0.173224
$782$	10.6381	0.380416
$783$	0	0
$784$	−0.305866	−0.0109238
$785$	27.1078	0.967519
$786$	0	0
$787$	45.5121	1.62233	0.811165	−	0.584817i	$-0.198834\pi$
$787$	45.5121	1.62233	0.811165	+	0.584817i	$0.198834\pi$
$788$	0.275162	0.00980225
$789$	0	0
$790$	21.8258	0.776529
$791$	1.90894	0.0678742
$792$	0	0
$793$	−22.4353	−0.796703
$794$	6.86612	0.243670
$795$	0	0
$796$	−8.92691	−0.316406
$797$	0.830574	0.0294204	0.0147102	−	0.999892i	$-0.495317\pi$
$797$	0.830574	0.0294204	0.0147102	+	0.999892i	$0.495317\pi$
$798$	0	0
$799$	2.80457	0.0992186
$800$	−6.71749	−0.237499
$801$	0	0
$802$	−19.3735	−0.684102
$803$	13.0650	0.461052
$804$	0	0
$805$	−33.5939	−1.18403
$806$	−7.41422	−0.261155
$807$	0	0
$808$	−12.2068	−0.429435
$809$	−25.2088	−0.886293	−0.443147	−	0.896449i	$-0.646138\pi$
$809$	−25.2088	−0.886293	−0.443147	+	0.896449i	$0.646138\pi$
$810$	0	0
$811$	12.2428	0.429902	0.214951	−	0.976625i	$-0.431041\pi$
$811$	12.2428	0.429902	0.214951	+	0.976625i	$0.431041\pi$
$812$	20.0482	0.703555
$813$	0	0
$814$	−2.42567	−0.0850198
$815$	−58.2546	−2.04057
$816$	0	0
$817$	19.3909	0.678400
$818$	−40.1650	−1.40434
$819$	0	0
$820$	−39.1733	−1.36799
$821$	−19.4671	−0.679405	−0.339703	−	0.940533i	$-0.610326\pi$
$821$	−19.4671	−0.679405	−0.339703	+	0.940533i	$0.610326\pi$
$822$	0	0
$823$	−8.33465	−0.290528	−0.145264	−	0.989393i	$-0.546403\pi$
$823$	−8.33465	−0.290528	−0.145264	+	0.989393i	$0.546403\pi$
$824$	5.86592	0.204349
$825$	0	0
$826$	22.0721	0.767986
$827$	−21.5631	−0.749823	−0.374912	−	0.927061i	$-0.622327\pi$
$827$	−21.5631	−0.749823	−0.374912	+	0.927061i	$0.622327\pi$
$828$	0	0
$829$	7.55516	0.262402	0.131201	−	0.991356i	$-0.458117\pi$
$829$	7.55516	0.262402	0.131201	+	0.991356i	$0.458117\pi$
$830$	46.0550	1.59859
$831$	0	0
$832$	3.70711	0.128521
$833$	0.857821	0.0297218
$834$	0	0
$835$	−9.87687	−0.341803
$836$	−4.11722	−0.142397
$837$	0	0
$838$	−18.2873	−0.631723
$839$	−40.5087	−1.39851	−0.699257	−	0.714870i	$-0.746486\pi$
$839$	−40.5087	−1.39851	−0.699257	+	0.714870i	$0.746486\pi$
$840$	0	0
$841$	31.0424	1.07043
$842$	32.1520	1.10803
$843$	0	0
$844$	−11.9391	−0.410961
$845$	−2.54218	−0.0874537
$846$	0	0
$847$	2.58730	0.0889008
$848$	3.97918	0.136645
$849$	0	0
$850$	18.8397	0.646196
$851$	−9.20086	−0.315401
$852$	0	0
$853$	−29.5182	−1.01068	−0.505341	−	0.862920i	$-0.668633\pi$
$853$	−29.5182	−1.01068	−0.505341	+	0.862920i	$0.668633\pi$
$854$	15.6583	0.535816
$855$	0	0
$856$	10.2849	0.351530
$857$	−48.9998	−1.67380	−0.836901	−	0.547355i	$-0.815635\pi$
$857$	−48.9998	−1.67380	−0.836901	+	0.547355i	$0.815635\pi$
$858$	0	0
$859$	38.2805	1.30611	0.653057	−	0.757309i	$-0.273486\pi$
$859$	38.2805	1.30611	0.653057	+	0.757309i	$0.273486\pi$
$860$	−16.1217	−0.549745
$861$	0	0
$862$	−30.6748	−1.04479
$863$	31.0922	1.05839	0.529196	−	0.848500i	$-0.322494\pi$
$863$	31.0922	1.05839	0.529196	+	0.848500i	$0.322494\pi$
$864$	0	0
$865$	−6.65868	−0.226402
$866$	1.48522	0.0504700
$867$	0	0
$868$	5.17461	0.175638
$869$	−6.37608	−0.216294
$870$	0	0
$871$	−30.9025	−1.04709
$872$	4.83730	0.163812
$873$	0	0
$874$	−15.6171	−0.528256
$875$	−15.2111	−0.514228
$876$	0	0
$877$	9.08504	0.306780	0.153390	−	0.988166i	$-0.450981\pi$
$877$	9.08504	0.306780	0.153390	+	0.988166i	$0.450981\pi$
$878$	−1.66242	−0.0561039
$879$	0	0
$880$	3.42308	0.115392
$881$	11.1693	0.376305	0.188152	−	0.982140i	$-0.439750\pi$
$881$	11.1693	0.376305	0.188152	+	0.982140i	$0.439750\pi$
$882$	0	0
$883$	21.7751	0.732789	0.366395	−	0.930460i	$-0.380592\pi$
$883$	21.7751	0.732789	0.366395	+	0.930460i	$0.380592\pi$
$884$	−10.3968	−0.349684
$885$	0	0
$886$	27.8186	0.934583
$887$	19.0237	0.638752	0.319376	−	0.947628i	$-0.396527\pi$
$887$	19.0237	0.638752	0.319376	+	0.947628i	$0.396527\pi$
$888$	0	0
$889$	48.7015	1.63339
$890$	−28.4280	−0.952909
$891$	0	0
$892$	−18.1395	−0.607356
$893$	−4.11722	−0.137777
$894$	0	0
$895$	22.9595	0.767451
$896$	−2.58730	−0.0864357
$897$	0	0
$898$	3.62387	0.120930
$899$	15.4974	0.516867
$900$	0	0
$901$	−11.1599	−0.371790
$902$	11.4439	0.381039
$903$	0	0
$904$	−0.737812	−0.0245393
$905$	−56.2566	−1.87003
$906$	0	0
$907$	31.1826	1.03540	0.517701	−	0.855562i	$-0.326788\pi$
$907$	31.1826	1.03540	0.517701	+	0.855562i	$0.326788\pi$
$908$	−7.99322	−0.265264
$909$	0	0
$910$	32.8322	1.08838
$911$	24.5055	0.811903	0.405951	−	0.913895i	$-0.366940\pi$
$911$	24.5055	0.811903	0.405951	+	0.913895i	$0.366940\pi$
$912$	0	0
$913$	−13.4542	−0.445270
$914$	−24.4088	−0.807370
$915$	0	0
$916$	29.0901	0.961163
$917$	10.2478	0.338411
$918$	0	0
$919$	32.6564	1.07724	0.538618	−	0.842550i	$-0.318947\pi$
$919$	32.6564	1.07724	0.538618	+	0.842550i	$0.318947\pi$
$920$	12.9842	0.428075
$921$	0	0
$922$	−13.7530	−0.452931
$923$	17.9461	0.590702
$924$	0	0
$925$	−16.2944	−0.535758
$926$	−36.4095	−1.19649
$927$	0	0
$928$	−7.74870	−0.254364
$929$	−9.19187	−0.301576	−0.150788	−	0.988566i	$-0.548181\pi$
$929$	−9.19187	−0.301576	−0.150788	+	0.988566i	$0.548181\pi$
$930$	0	0
$931$	−1.25931	−0.0412724
$932$	10.2840	0.336864
$933$	0	0
$934$	−17.8392	−0.583715
$935$	−9.60028	−0.313963
$936$	0	0
$937$	−28.4805	−0.930419	−0.465209	−	0.885201i	$-0.654021\pi$
$937$	−28.4805	−0.930419	−0.465209	+	0.885201i	$0.654021\pi$
$938$	21.5678	0.704213
$939$	0	0
$940$	3.42308	0.111649
$941$	−29.7647	−0.970302	−0.485151	−	0.874431i	$-0.661235\pi$
$941$	−29.7647	−0.970302	−0.485151	+	0.874431i	$0.661235\pi$
$942$	0	0
$943$	43.4079	1.41355
$944$	−8.53093	−0.277658
$945$	0	0
$946$	4.70970	0.153126
$947$	0.845942	0.0274894	0.0137447	−	0.999906i	$-0.495625\pi$
$947$	0.845942	0.0274894	0.0137447	+	0.999906i	$0.495625\pi$
$948$	0	0
$949$	−48.4332	−1.57221
$950$	−27.6574	−0.897324
$951$	0	0
$952$	7.25627	0.235177
$953$	−58.3802	−1.89112	−0.945561	−	0.325446i	$-0.894486\pi$
$953$	−58.3802	−1.89112	−0.945561	+	0.325446i	$0.894486\pi$
$954$	0	0
$955$	72.9303	2.35997
$956$	8.57778	0.277425
$957$	0	0
$958$	28.3408	0.915648
$959$	−2.20289	−0.0711352
$960$	0	0
$961$	−27.0000	−0.870968
$962$	8.99223	0.289921
$963$	0	0
$964$	28.5733	0.920284
$965$	0.591696	0.0190474
$966$	0	0
$967$	1.58349	0.0509216	0.0254608	−	0.999676i	$-0.491895\pi$
$967$	1.58349	0.0509216	0.0254608	+	0.999676i	$0.491895\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−24.7372	−0.794263
$971$	−3.89533	−0.125007	−0.0625036	−	0.998045i	$-0.519908\pi$
$971$	−3.89533	−0.125007	−0.0625036	+	0.998045i	$0.519908\pi$
$972$	0	0
$973$	−26.6948	−0.855796
$974$	33.2770	1.06626
$975$	0	0
$976$	−6.05198	−0.193719
$977$	−41.1698	−1.31714	−0.658569	−	0.752520i	$-0.728838\pi$
$977$	−41.1698	−1.31714	−0.658569	+	0.752520i	$0.728838\pi$
$978$	0	0
$979$	8.30480	0.265422
$980$	1.04700	0.0334453
$981$	0	0
$982$	−19.3015	−0.615935
$983$	−33.4330	−1.06635	−0.533173	−	0.846006i	$-0.679000\pi$
$983$	−33.4330	−1.06635	−0.533173	+	0.846006i	$0.679000\pi$
$984$	0	0
$985$	−0.941903	−0.0300115
$986$	21.7318	0.692081
$987$	0	0
$988$	15.2630	0.485580
$989$	17.8644	0.568056
$990$	0	0
$991$	16.9621	0.538818	0.269409	−	0.963026i	$-0.413172\pi$
$991$	16.9621	0.538818	0.269409	+	0.963026i	$0.413172\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	−12.5251	−0.397272
$995$	30.5576	0.968740
$996$	0	0
$997$	−38.0048	−1.20362	−0.601812	−	0.798638i	$-0.705554\pi$
$997$	−38.0048	−1.20362	−0.601812	+	0.798638i	$0.705554\pi$
$998$	19.9136	0.630353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9306.2.a.be.1.1		5
3.2	odd	2		3102.2.a.v.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3102.2.a.v.1.5	✓	5	3.2	odd	2
9306.2.a.be.1.1		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9306 = 2 \cdot 3^{2} \cdot 11 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9306.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.42308 q^{5} +2.58730 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.42308 q^{5} +2.58730 q^{7} -1.00000 q^{8} +3.42308 q^{10} -1.00000 q^{11} +3.70711 q^{13} -2.58730 q^{14} +1.00000 q^{16} -2.80457 q^{17} +4.11722 q^{19} -3.42308 q^{20} +1.00000 q^{22} +3.79312 q^{23} +6.71749 q^{25} -3.70711 q^{26} +2.58730 q^{28} +7.74870 q^{29} +2.00000 q^{31} -1.00000 q^{32} +2.80457 q^{34} -8.85655 q^{35} -2.42567 q^{37} -4.11722 q^{38} +3.42308 q^{40} +11.4439 q^{41} +4.70970 q^{43} -1.00000 q^{44} -3.79312 q^{46} -1.00000 q^{47} -0.305866 q^{49} -6.71749 q^{50} +3.70711 q^{52} +3.97918 q^{53} +3.42308 q^{55} -2.58730 q^{56} -7.74870 q^{58} -8.53093 q^{59} -6.05198 q^{61} -2.00000 q^{62} +1.00000 q^{64} -12.6897 q^{65} -8.33601 q^{67} -2.80457 q^{68} +8.85655 q^{70} +4.84099 q^{71} -13.0650 q^{73} +2.42567 q^{74} +4.11722 q^{76} -2.58730 q^{77} +6.37608 q^{79} -3.42308 q^{80} -11.4439 q^{82} +13.4542 q^{83} +9.60028 q^{85} -4.70970 q^{86} +1.00000 q^{88} -8.30480 q^{89} +9.59141 q^{91} +3.79312 q^{92} +1.00000 q^{94} -14.0936 q^{95} -7.22658 q^{97} +0.305866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 2 q^{5} + 6 q^{7} - 5 q^{8} + 2 q^{10} - 5 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} - 6 q^{17} + 12 q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - q^{25} - 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} - 5 q^{32} + 6 q^{34} + 8 q^{35} - 12 q^{38} + 2 q^{40} - 2 q^{41} + 14 q^{43} - 5 q^{44} + 2 q^{46} - 5 q^{47} + 5 q^{49} + q^{50} + 6 q^{52} - 2 q^{53} + 2 q^{55} - 6 q^{56} - 4 q^{58} - 10 q^{59} + 14 q^{61} - 10 q^{62} + 5 q^{64} - 6 q^{68} - 8 q^{70} - 12 q^{71} - 2 q^{73} + 12 q^{76} - 6 q^{77} - 2 q^{80} + 2 q^{82} - 14 q^{83} + 22 q^{85} - 14 q^{86} + 5 q^{88} + 12 q^{91} - 2 q^{92} + 5 q^{94} - 4 q^{95} + 22 q^{97} - 5 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 2 * q^5 + 6 * q^7 - 5 * q^8 + 2 * q^10 - 5 * q^11 + 6 * q^13 - 6 * q^14 + 5 * q^16 - 6 * q^17 + 12 * q^19 - 2 * q^20 + 5 * q^22 - 2 * q^23 - q^25 - 6 * q^26 + 6 * q^28 + 4 * q^29 + 10 * q^31 - 5 * q^32 + 6 * q^34 + 8 * q^35 - 12 * q^38 + 2 * q^40 - 2 * q^41 + 14 * q^43 - 5 * q^44 + 2 * q^46 - 5 * q^47 + 5 * q^49 + q^50 + 6 * q^52 - 2 * q^53 + 2 * q^55 - 6 * q^56 - 4 * q^58 - 10 * q^59 + 14 * q^61 - 10 * q^62 + 5 * q^64 - 6 * q^68 - 8 * q^70 - 12 * q^71 - 2 * q^73 + 12 * q^76 - 6 * q^77 - 2 * q^80 + 2 * q^82 - 14 * q^83 + 22 * q^85 - 14 * q^86 + 5 * q^88 + 12 * q^91 - 2 * q^92 + 5 * q^94 - 4 * q^95 + 22 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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