LMFDB - Embedded newform 4410.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -0.585786 q^{11} +1.00000 q^{16} -1.41421 q^{17} -2.82843 q^{19} -1.00000 q^{20} +0.585786 q^{22} -4.82843 q^{23} +1.00000 q^{25} -8.24264 q^{29} +5.07107 q^{31} -1.00000 q^{32} +1.41421 q^{34} -1.41421 q^{37} +2.82843 q^{38} +1.00000 q^{40} +8.82843 q^{41} +4.58579 q^{43} -0.585786 q^{44} +4.82843 q^{46} +9.07107 q^{47} -1.00000 q^{50} +9.31371 q^{53} +0.585786 q^{55} +8.24264 q^{58} -2.48528 q^{59} -11.6569 q^{61} -5.07107 q^{62} +1.00000 q^{64} +7.89949 q^{67} -1.41421 q^{68} -10.8284 q^{71} +7.65685 q^{73} +1.41421 q^{74} -2.82843 q^{76} -11.3137 q^{79} -1.00000 q^{80} -8.82843 q^{82} +5.17157 q^{83} +1.41421 q^{85} -4.58579 q^{86} +0.585786 q^{88} +6.48528 q^{89} -4.82843 q^{92} -9.07107 q^{94} +2.82843 q^{95} +8.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} - 2 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} - 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{40} + 12 q^{41} + 12 q^{43} - 4 q^{44} + 4 q^{46} + 4 q^{47} - 2 q^{50} - 4 q^{53} + 4 q^{55} + 8 q^{58} + 12 q^{59} - 12 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{67} - 16 q^{71} + 4 q^{73} - 2 q^{80} - 12 q^{82} + 16 q^{83} - 12 q^{86} + 4 q^{88} - 4 q^{89} - 4 q^{92} - 4 q^{94} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^16 - 2 * q^20 + 4 * q^22 - 4 * q^23 + 2 * q^25 - 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^40 + 12 * q^41 + 12 * q^43 - 4 * q^44 + 4 * q^46 + 4 * q^47 - 2 * q^50 - 4 * q^53 + 4 * q^55 + 8 * q^58 + 12 * q^59 - 12 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^67 - 16 * q^71 + 4 * q^73 - 2 * q^80 - 12 * q^82 + 16 * q^83 - 12 * q^86 + 4 * q^88 - 4 * q^89 - 4 * q^92 - 4 * q^94 + 12 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0.585786	0.124890
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.24264	−1.53062	−0.765310	−	0.643662i	$-0.777414\pi$
$29$	−8.24264	−1.53062	−0.765310	+	0.643662i	$0.777414\pi$
$30$	0	0
$31$	5.07107	0.910791	0.455395	−	0.890289i	$-0.349498\pi$
$31$	5.07107	0.910791	0.455395	+	0.890289i	$0.349498\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.41421	0.242536
$35$	0	0
$36$	0	0
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	1.00000	0.158114
$41$	8.82843	1.37877	0.689384	−	0.724396i	$-0.257881\pi$
$41$	8.82843	1.37877	0.689384	+	0.724396i	$0.257881\pi$
$42$	0	0
$43$	4.58579	0.699326	0.349663	−	0.936876i	$-0.386296\pi$
$43$	4.58579	0.699326	0.349663	+	0.936876i	$0.386296\pi$
$44$	−0.585786	−0.0883106
$45$	0	0
$46$	4.82843	0.711913
$47$	9.07107	1.32315	0.661576	−	0.749878i	$-0.269888\pi$
$47$	9.07107	1.32315	0.661576	+	0.749878i	$0.269888\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0	0
$53$	9.31371	1.27934	0.639668	−	0.768651i	$-0.279072\pi$
$53$	9.31371	1.27934	0.639668	+	0.768651i	$0.279072\pi$
$54$	0	0
$55$	0.585786	0.0789874
$56$	0	0
$57$	0	0
$58$	8.24264	1.08231
$59$	−2.48528	−0.323556	−0.161778	−	0.986827i	$-0.551723\pi$
$59$	−2.48528	−0.323556	−0.161778	+	0.986827i	$0.551723\pi$
$60$	0	0
$61$	−11.6569	−1.49251	−0.746254	−	0.665662i	$-0.768149\pi$
$61$	−11.6569	−1.49251	−0.746254	+	0.665662i	$0.768149\pi$
$62$	−5.07107	−0.644026
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	7.89949	0.965077	0.482538	−	0.875875i	$-0.339715\pi$
$67$	7.89949	0.965077	0.482538	+	0.875875i	$0.339715\pi$
$68$	−1.41421	−0.171499
$69$	0	0
$70$	0	0
$71$	−10.8284	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$71$	−10.8284	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$72$	0	0
$73$	7.65685	0.896167	0.448084	−	0.893992i	$-0.352107\pi$
$73$	7.65685	0.896167	0.448084	+	0.893992i	$0.352107\pi$
$74$	1.41421	0.164399
$75$	0	0
$76$	−2.82843	−0.324443
$77$	0	0
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−8.82843	−0.974937
$83$	5.17157	0.567654	0.283827	−	0.958876i	$-0.408396\pi$
$83$	5.17157	0.567654	0.283827	+	0.958876i	$0.408396\pi$
$84$	0	0
$85$	1.41421	0.153393
$86$	−4.58579	−0.494498
$87$	0	0
$88$	0.585786	0.0624450
$89$	6.48528	0.687438	0.343719	−	0.939072i	$-0.388313\pi$
$89$	6.48528	0.687438	0.343719	+	0.939072i	$0.388313\pi$
$90$	0	0
$91$	0	0
$92$	−4.82843	−0.503398
$93$	0	0
$94$	−9.07107	−0.935609
$95$	2.82843	0.290191
$96$	0	0
$97$	8.82843	0.896391	0.448195	−	0.893936i	$-0.352067\pi$
$97$	8.82843	0.896391	0.448195	+	0.893936i	$0.352067\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	0	0
$103$	−10.1421	−0.999334	−0.499667	−	0.866217i	$-0.666544\pi$
$103$	−10.1421	−0.999334	−0.499667	+	0.866217i	$0.666544\pi$
$104$	0	0
$105$	0	0
$106$	−9.31371	−0.904627
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	−0.585786	−0.0558525
$111$	0	0
$112$	0	0
$113$	−9.31371	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$113$	−9.31371	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$114$	0	0
$115$	4.82843	0.450253
$116$	−8.24264	−0.765310
$117$	0	0
$118$	2.48528	0.228789
$119$	0	0
$120$	0	0
$121$	−10.6569	−0.968805
$122$	11.6569	1.05536
$123$	0	0
$124$	5.07107	0.455395
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.1421	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	16.8284	1.47031	0.735153	−	0.677901i	$-0.237110\pi$
$131$	16.8284	1.47031	0.735153	+	0.677901i	$0.237110\pi$
$132$	0	0
$133$	0	0
$134$	−7.89949	−0.682412
$135$	0	0
$136$	1.41421	0.121268
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	0	0
$139$	17.6569	1.49763	0.748817	−	0.662776i	$-0.230622\pi$
$139$	17.6569	1.49763	0.748817	+	0.662776i	$0.230622\pi$
$140$	0	0
$141$	0	0
$142$	10.8284	0.908701
$143$	0	0
$144$	0	0
$145$	8.24264	0.684514
$146$	−7.65685	−0.633686
$147$	0	0
$148$	−1.41421	−0.116248
$149$	13.8995	1.13869	0.569345	−	0.822098i	$-0.307197\pi$
$149$	13.8995	1.13869	0.569345	+	0.822098i	$0.307197\pi$
$150$	0	0
$151$	17.7990	1.44846	0.724231	−	0.689558i	$-0.242195\pi$
$151$	17.7990	1.44846	0.724231	+	0.689558i	$0.242195\pi$
$152$	2.82843	0.229416
$153$	0	0
$154$	0	0
$155$	−5.07107	−0.407318
$156$	0	0
$157$	0.343146	0.0273860	0.0136930	−	0.999906i	$-0.495641\pi$
$157$	0.343146	0.0273860	0.0136930	+	0.999906i	$0.495641\pi$
$158$	11.3137	0.900070
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	14.7279	1.15358	0.576790	−	0.816893i	$-0.304305\pi$
$163$	14.7279	1.15358	0.576790	+	0.816893i	$0.304305\pi$
$164$	8.82843	0.689384
$165$	0	0
$166$	−5.17157	−0.401392
$167$	−1.07107	−0.0828817	−0.0414409	−	0.999141i	$-0.513195\pi$
$167$	−1.07107	−0.0828817	−0.0414409	+	0.999141i	$0.513195\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−1.41421	−0.108465
$171$	0	0
$172$	4.58579	0.349663
$173$	−18.4853	−1.40541	−0.702705	−	0.711481i	$-0.748025\pi$
$173$	−18.4853	−1.40541	−0.702705	+	0.711481i	$0.748025\pi$
$174$	0	0
$175$	0	0
$176$	−0.585786	−0.0441553
$177$	0	0
$178$	−6.48528	−0.486092
$179$	−0.585786	−0.0437837	−0.0218919	−	0.999760i	$-0.506969\pi$
$179$	−0.585786	−0.0437837	−0.0218919	+	0.999760i	$0.506969\pi$
$180$	0	0
$181$	−8.82843	−0.656212	−0.328106	−	0.944641i	$-0.606410\pi$
$181$	−8.82843	−0.656212	−0.328106	+	0.944641i	$0.606410\pi$
$182$	0	0
$183$	0	0
$184$	4.82843	0.355956
$185$	1.41421	0.103975
$186$	0	0
$187$	0.828427	0.0605806
$188$	9.07107	0.661576
$189$	0	0
$190$	−2.82843	−0.205196
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	0	0
$193$	−21.7990	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$193$	−21.7990	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$194$	−8.82843	−0.633844
$195$	0	0
$196$	0	0
$197$	20.8284	1.48396	0.741982	−	0.670420i	$-0.233886\pi$
$197$	20.8284	1.48396	0.741982	+	0.670420i	$0.233886\pi$
$198$	0	0
$199$	2.92893	0.207626	0.103813	−	0.994597i	$-0.466896\pi$
$199$	2.92893	0.207626	0.103813	+	0.994597i	$0.466896\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−6.34315	−0.446302
$203$	0	0
$204$	0	0
$205$	−8.82843	−0.616604
$206$	10.1421	0.706636
$207$	0	0
$208$	0	0
$209$	1.65685	0.114607
$210$	0	0
$211$	−1.65685	−0.114063	−0.0570313	−	0.998372i	$-0.518163\pi$
$211$	−1.65685	−0.114063	−0.0570313	+	0.998372i	$0.518163\pi$
$212$	9.31371	0.639668
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−4.58579	−0.312748
$216$	0	0
$217$	0	0
$218$	−5.31371	−0.359890
$219$	0	0
$220$	0.585786	0.0394937
$221$	0	0
$222$	0	0
$223$	17.6569	1.18239	0.591195	−	0.806529i	$-0.298656\pi$
$223$	17.6569	1.18239	0.591195	+	0.806529i	$0.298656\pi$
$224$	0	0
$225$	0	0
$226$	9.31371	0.619539
$227$	13.6569	0.906437	0.453219	−	0.891399i	$-0.350276\pi$
$227$	13.6569	0.906437	0.453219	+	0.891399i	$0.350276\pi$
$228$	0	0
$229$	25.7990	1.70485	0.852423	−	0.522853i	$-0.175132\pi$
$229$	25.7990	1.70485	0.852423	+	0.522853i	$0.175132\pi$
$230$	−4.82843	−0.318377
$231$	0	0
$232$	8.24264	0.541156
$233$	−6.97056	−0.456657	−0.228328	−	0.973584i	$-0.573326\pi$
$233$	−6.97056	−0.456657	−0.228328	+	0.973584i	$0.573326\pi$
$234$	0	0
$235$	−9.07107	−0.591731
$236$	−2.48528	−0.161778
$237$	0	0
$238$	0	0
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	−16.2426	−1.04628	−0.523140	−	0.852247i	$-0.675240\pi$
$241$	−16.2426	−1.04628	−0.523140	+	0.852247i	$0.675240\pi$
$242$	10.6569	0.685049
$243$	0	0
$244$	−11.6569	−0.746254
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−5.07107	−0.322013
$249$	0	0
$250$	1.00000	0.0632456
$251$	17.6569	1.11449	0.557245	−	0.830348i	$-0.311858\pi$
$251$	17.6569	1.11449	0.557245	+	0.830348i	$0.311858\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	14.1421	0.887357
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.89949	−0.118487	−0.0592436	−	0.998244i	$-0.518869\pi$
$257$	−1.89949	−0.118487	−0.0592436	+	0.998244i	$0.518869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−16.8284	−1.03966
$263$	21.7990	1.34418	0.672092	−	0.740468i	$-0.265396\pi$
$263$	21.7990	1.34418	0.672092	+	0.740468i	$0.265396\pi$
$264$	0	0
$265$	−9.31371	−0.572137
$266$	0	0
$267$	0	0
$268$	7.89949	0.482538
$269$	−7.65685	−0.466847	−0.233423	−	0.972375i	$-0.574993\pi$
$269$	−7.65685	−0.466847	−0.233423	+	0.972375i	$0.574993\pi$
$270$	0	0
$271$	−14.2426	−0.865179	−0.432589	−	0.901591i	$-0.642400\pi$
$271$	−14.2426	−0.865179	−0.432589	+	0.901591i	$0.642400\pi$
$272$	−1.41421	−0.0857493
$273$	0	0
$274$	−16.0000	−0.966595
$275$	−0.585786	−0.0353243
$276$	0	0
$277$	21.8995	1.31581	0.657907	−	0.753100i	$-0.271442\pi$
$277$	21.8995	1.31581	0.657907	+	0.753100i	$0.271442\pi$
$278$	−17.6569	−1.05899
$279$	0	0
$280$	0	0
$281$	13.5147	0.806221	0.403110	−	0.915151i	$-0.367929\pi$
$281$	13.5147	0.806221	0.403110	+	0.915151i	$0.367929\pi$
$282$	0	0
$283$	30.4853	1.81216	0.906081	−	0.423104i	$-0.139060\pi$
$283$	30.4853	1.81216	0.906081	+	0.423104i	$0.139060\pi$
$284$	−10.8284	−0.642549
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	−8.24264	−0.484025
$291$	0	0
$292$	7.65685	0.448084
$293$	28.6274	1.67243	0.836216	−	0.548401i	$-0.184763\pi$
$293$	28.6274	1.67243	0.836216	+	0.548401i	$0.184763\pi$
$294$	0	0
$295$	2.48528	0.144699
$296$	1.41421	0.0821995
$297$	0	0
$298$	−13.8995	−0.805176
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−17.7990	−1.02422
$303$	0	0
$304$	−2.82843	−0.162221
$305$	11.6569	0.667470
$306$	0	0
$307$	−7.31371	−0.417415	−0.208708	−	0.977978i	$-0.566926\pi$
$307$	−7.31371	−0.417415	−0.208708	+	0.977978i	$0.566926\pi$
$308$	0	0
$309$	0	0
$310$	5.07107	0.288017
$311$	29.1716	1.65417	0.827084	−	0.562078i	$-0.189998\pi$
$311$	29.1716	1.65417	0.827084	+	0.562078i	$0.189998\pi$
$312$	0	0
$313$	28.8284	1.62948	0.814740	−	0.579827i	$-0.196880\pi$
$313$	28.8284	1.62948	0.814740	+	0.579827i	$0.196880\pi$
$314$	−0.343146	−0.0193648
$315$	0	0
$316$	−11.3137	−0.636446
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	4.82843	0.270340
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	0	0
$326$	−14.7279	−0.815704
$327$	0	0
$328$	−8.82843	−0.487468
$329$	0	0
$330$	0	0
$331$	−18.6274	−1.02386	−0.511928	−	0.859029i	$-0.671068\pi$
$331$	−18.6274	−1.02386	−0.511928	+	0.859029i	$0.671068\pi$
$332$	5.17157	0.283827
$333$	0	0
$334$	1.07107	0.0586062
$335$	−7.89949	−0.431596
$336$	0	0
$337$	−3.17157	−0.172767	−0.0863833	−	0.996262i	$-0.527531\pi$
$337$	−3.17157	−0.172767	−0.0863833	+	0.996262i	$0.527531\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	1.41421	0.0766965
$341$	−2.97056	−0.160865
$342$	0	0
$343$	0	0
$344$	−4.58579	−0.247249
$345$	0	0
$346$	18.4853	0.993775
$347$	−1.65685	−0.0889446	−0.0444723	−	0.999011i	$-0.514161\pi$
$347$	−1.65685	−0.0889446	−0.0444723	+	0.999011i	$0.514161\pi$
$348$	0	0
$349$	−5.79899	−0.310413	−0.155206	−	0.987882i	$-0.549604\pi$
$349$	−5.79899	−0.310413	−0.155206	+	0.987882i	$0.549604\pi$
$350$	0	0
$351$	0	0
$352$	0.585786	0.0312225
$353$	0.727922	0.0387434	0.0193717	−	0.999812i	$-0.493833\pi$
$353$	0.727922	0.0387434	0.0193717	+	0.999812i	$0.493833\pi$
$354$	0	0
$355$	10.8284	0.574713
$356$	6.48528	0.343719
$357$	0	0
$358$	0.585786	0.0309598
$359$	−18.8284	−0.993726	−0.496863	−	0.867829i	$-0.665515\pi$
$359$	−18.8284	−0.993726	−0.496863	+	0.867829i	$0.665515\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	8.82843	0.464012
$363$	0	0
$364$	0	0
$365$	−7.65685	−0.400778
$366$	0	0
$367$	16.4853	0.860525	0.430262	−	0.902704i	$-0.358421\pi$
$367$	16.4853	0.860525	0.430262	+	0.902704i	$0.358421\pi$
$368$	−4.82843	−0.251699
$369$	0	0
$370$	−1.41421	−0.0735215
$371$	0	0
$372$	0	0
$373$	11.0711	0.573238	0.286619	−	0.958045i	$-0.407469\pi$
$373$	11.0711	0.573238	0.286619	+	0.958045i	$0.407469\pi$
$374$	−0.828427	−0.0428369
$375$	0	0
$376$	−9.07107	−0.467805
$377$	0	0
$378$	0	0
$379$	−24.4853	−1.25772	−0.628862	−	0.777517i	$-0.716479\pi$
$379$	−24.4853	−1.25772	−0.628862	+	0.777517i	$0.716479\pi$
$380$	2.82843	0.145095
$381$	0	0
$382$	−11.3137	−0.578860
$383$	−15.8995	−0.812426	−0.406213	−	0.913778i	$-0.633151\pi$
$383$	−15.8995	−0.812426	−0.406213	+	0.913778i	$0.633151\pi$
$384$	0	0
$385$	0	0
$386$	21.7990	1.10954
$387$	0	0
$388$	8.82843	0.448195
$389$	−27.0711	−1.37256	−0.686279	−	0.727339i	$-0.740757\pi$
$389$	−27.0711	−1.37256	−0.686279	+	0.727339i	$0.740757\pi$
$390$	0	0
$391$	6.82843	0.345328
$392$	0	0
$393$	0	0
$394$	−20.8284	−1.04932
$395$	11.3137	0.569254
$396$	0	0
$397$	20.6274	1.03526	0.517630	−	0.855604i	$-0.326814\pi$
$397$	20.6274	1.03526	0.517630	+	0.855604i	$0.326814\pi$
$398$	−2.92893	−0.146814
$399$	0	0
$400$	1.00000	0.0500000
$401$	25.3137	1.26411	0.632053	−	0.774925i	$-0.282212\pi$
$401$	25.3137	1.26411	0.632053	+	0.774925i	$0.282212\pi$
$402$	0	0
$403$	0	0
$404$	6.34315	0.315583
$405$	0	0
$406$	0	0
$407$	0.828427	0.0410636
$408$	0	0
$409$	−18.3848	−0.909069	−0.454534	−	0.890729i	$-0.650194\pi$
$409$	−18.3848	−0.909069	−0.454534	+	0.890729i	$0.650194\pi$
$410$	8.82843	0.436005
$411$	0	0
$412$	−10.1421	−0.499667
$413$	0	0
$414$	0	0
$415$	−5.17157	−0.253863
$416$	0	0
$417$	0	0
$418$	−1.65685	−0.0810394
$419$	17.6569	0.862594	0.431297	−	0.902210i	$-0.358056\pi$
$419$	17.6569	0.862594	0.431297	+	0.902210i	$0.358056\pi$
$420$	0	0
$421$	16.6274	0.810371	0.405185	−	0.914235i	$-0.367207\pi$
$421$	16.6274	0.810371	0.405185	+	0.914235i	$0.367207\pi$
$422$	1.65685	0.0806544
$423$	0	0
$424$	−9.31371	−0.452314
$425$	−1.41421	−0.0685994
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	4.58579	0.221146
$431$	−14.1421	−0.681203	−0.340601	−	0.940208i	$-0.610631\pi$
$431$	−14.1421	−0.681203	−0.340601	+	0.940208i	$0.610631\pi$
$432$	0	0
$433$	8.82843	0.424267	0.212134	−	0.977241i	$-0.431959\pi$
$433$	8.82843	0.424267	0.212134	+	0.977241i	$0.431959\pi$
$434$	0	0
$435$	0	0
$436$	5.31371	0.254480
$437$	13.6569	0.653296
$438$	0	0
$439$	33.5563	1.60156	0.800779	−	0.598960i	$-0.204419\pi$
$439$	33.5563	1.60156	0.800779	+	0.598960i	$0.204419\pi$
$440$	−0.585786	−0.0279263
$441$	0	0
$442$	0	0
$443$	−13.6569	−0.648857	−0.324428	−	0.945910i	$-0.605172\pi$
$443$	−13.6569	−0.648857	−0.324428	+	0.945910i	$0.605172\pi$
$444$	0	0
$445$	−6.48528	−0.307432
$446$	−17.6569	−0.836076
$447$	0	0
$448$	0	0
$449$	25.1127	1.18514	0.592571	−	0.805518i	$-0.298113\pi$
$449$	25.1127	1.18514	0.592571	+	0.805518i	$0.298113\pi$
$450$	0	0
$451$	−5.17157	−0.243520
$452$	−9.31371	−0.438080
$453$	0	0
$454$	−13.6569	−0.640948
$455$	0	0
$456$	0	0
$457$	−9.79899	−0.458377	−0.229189	−	0.973382i	$-0.573607\pi$
$457$	−9.79899	−0.458377	−0.229189	+	0.973382i	$0.573607\pi$
$458$	−25.7990	−1.20551
$459$	0	0
$460$	4.82843	0.225127
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	33.6569	1.56417	0.782083	−	0.623174i	$-0.214157\pi$
$463$	33.6569	1.56417	0.782083	+	0.623174i	$0.214157\pi$
$464$	−8.24264	−0.382655
$465$	0	0
$466$	6.97056	0.322905
$467$	20.4853	0.947946	0.473973	−	0.880539i	$-0.342819\pi$
$467$	20.4853	0.947946	0.473973	+	0.880539i	$0.342819\pi$
$468$	0	0
$469$	0	0
$470$	9.07107	0.418417
$471$	0	0
$472$	2.48528	0.114394
$473$	−2.68629	−0.123516
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	0	0
$478$	23.3137	1.06634
$479$	−16.4853	−0.753232	−0.376616	−	0.926370i	$-0.622912\pi$
$479$	−16.4853	−0.753232	−0.376616	+	0.926370i	$0.622912\pi$
$480$	0	0
$481$	0	0
$482$	16.2426	0.739832
$483$	0	0
$484$	−10.6569	−0.484402
$485$	−8.82843	−0.400878
$486$	0	0
$487$	24.4853	1.10953	0.554767	−	0.832006i	$-0.312807\pi$
$487$	24.4853	1.10953	0.554767	+	0.832006i	$0.312807\pi$
$488$	11.6569	0.527681
$489$	0	0
$490$	0	0
$491$	−11.8995	−0.537017	−0.268508	−	0.963277i	$-0.586531\pi$
$491$	−11.8995	−0.537017	−0.268508	+	0.963277i	$0.586531\pi$
$492$	0	0
$493$	11.6569	0.524998
$494$	0	0
$495$	0	0
$496$	5.07107	0.227698
$497$	0	0
$498$	0	0
$499$	15.7990	0.707260	0.353630	−	0.935385i	$-0.384947\pi$
$499$	15.7990	0.707260	0.353630	+	0.935385i	$0.384947\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−17.6569	−0.788064
$503$	−23.8995	−1.06563	−0.532813	−	0.846233i	$-0.678865\pi$
$503$	−23.8995	−1.06563	−0.532813	+	0.846233i	$0.678865\pi$
$504$	0	0
$505$	−6.34315	−0.282266
$506$	−2.82843	−0.125739
$507$	0	0
$508$	−14.1421	−0.627456
$509$	18.3431	0.813046	0.406523	−	0.913641i	$-0.366741\pi$
$509$	18.3431	0.813046	0.406523	+	0.913641i	$0.366741\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	1.89949	0.0837831
$515$	10.1421	0.446916
$516$	0	0
$517$	−5.31371	−0.233697
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.1421	0.531957	0.265978	−	0.963979i	$-0.414305\pi$
$521$	12.1421	0.531957	0.265978	+	0.963979i	$0.414305\pi$
$522$	0	0
$523$	−21.1127	−0.923194	−0.461597	−	0.887090i	$-0.652723\pi$
$523$	−21.1127	−0.923194	−0.461597	+	0.887090i	$0.652723\pi$
$524$	16.8284	0.735153
$525$	0	0
$526$	−21.7990	−0.950481
$527$	−7.17157	−0.312399
$528$	0	0
$529$	0.313708	0.0136395
$530$	9.31371	0.404562
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−7.89949	−0.341206
$537$	0	0
$538$	7.65685	0.330110
$539$	0	0
$540$	0	0
$541$	−2.48528	−0.106851	−0.0534253	−	0.998572i	$-0.517014\pi$
$541$	−2.48528	−0.106851	−0.0534253	+	0.998572i	$0.517014\pi$
$542$	14.2426	0.611774
$543$	0	0
$544$	1.41421	0.0606339
$545$	−5.31371	−0.227614
$546$	0	0
$547$	27.4142	1.17215	0.586074	−	0.810258i	$-0.300673\pi$
$547$	27.4142	1.17215	0.586074	+	0.810258i	$0.300673\pi$
$548$	16.0000	0.683486
$549$	0	0
$550$	0.585786	0.0249780
$551$	23.3137	0.993197
$552$	0	0
$553$	0	0
$554$	−21.8995	−0.930420
$555$	0	0
$556$	17.6569	0.748817
$557$	−12.6274	−0.535041	−0.267520	−	0.963552i	$-0.586204\pi$
$557$	−12.6274	−0.535041	−0.267520	+	0.963552i	$0.586204\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−13.5147	−0.570084
$563$	26.1421	1.10176	0.550880	−	0.834585i	$-0.314292\pi$
$563$	26.1421	1.10176	0.550880	+	0.834585i	$0.314292\pi$
$564$	0	0
$565$	9.31371	0.391831
$566$	−30.4853	−1.28139
$567$	0	0
$568$	10.8284	0.454351
$569$	−34.9706	−1.46604	−0.733021	−	0.680206i	$-0.761890\pi$
$569$	−34.9706	−1.46604	−0.733021	+	0.680206i	$0.761890\pi$
$570$	0	0
$571$	4.68629	0.196115	0.0980576	−	0.995181i	$-0.468737\pi$
$571$	4.68629	0.196115	0.0980576	+	0.995181i	$0.468737\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.82843	−0.201359
$576$	0	0
$577$	−12.1421	−0.505484	−0.252742	−	0.967534i	$-0.581332\pi$
$577$	−12.1421	−0.505484	−0.252742	+	0.967534i	$0.581332\pi$
$578$	15.0000	0.623918
$579$	0	0
$580$	8.24264	0.342257
$581$	0	0
$582$	0	0
$583$	−5.45584	−0.225958
$584$	−7.65685	−0.316843
$585$	0	0
$586$	−28.6274	−1.18259
$587$	−19.7990	−0.817192	−0.408596	−	0.912715i	$-0.633981\pi$
$587$	−19.7990	−0.817192	−0.408596	+	0.912715i	$0.633981\pi$
$588$	0	0
$589$	−14.3431	−0.590999
$590$	−2.48528	−0.102317
$591$	0	0
$592$	−1.41421	−0.0581238
$593$	−37.2132	−1.52816	−0.764082	−	0.645120i	$-0.776808\pi$
$593$	−37.2132	−1.52816	−0.764082	+	0.645120i	$0.776808\pi$
$594$	0	0
$595$	0	0
$596$	13.8995	0.569345
$597$	0	0
$598$	0	0
$599$	−4.68629	−0.191477	−0.0957383	−	0.995407i	$-0.530521\pi$
$599$	−4.68629	−0.191477	−0.0957383	+	0.995407i	$0.530521\pi$
$600$	0	0
$601$	42.1838	1.72071	0.860356	−	0.509694i	$-0.170241\pi$
$601$	42.1838	1.72071	0.860356	+	0.509694i	$0.170241\pi$
$602$	0	0
$603$	0	0
$604$	17.7990	0.724231
$605$	10.6569	0.433263
$606$	0	0
$607$	−4.97056	−0.201749	−0.100874	−	0.994899i	$-0.532164\pi$
$607$	−4.97056	−0.201749	−0.100874	+	0.994899i	$0.532164\pi$
$608$	2.82843	0.114708
$609$	0	0
$610$	−11.6569	−0.471972
$611$	0	0
$612$	0	0
$613$	−5.21320	−0.210559	−0.105280	−	0.994443i	$-0.533574\pi$
$613$	−5.21320	−0.210559	−0.105280	+	0.994443i	$0.533574\pi$
$614$	7.31371	0.295157
$615$	0	0
$616$	0	0
$617$	44.2843	1.78282	0.891409	−	0.453200i	$-0.149718\pi$
$617$	44.2843	1.78282	0.891409	+	0.453200i	$0.149718\pi$
$618$	0	0
$619$	1.85786	0.0746739	0.0373369	−	0.999303i	$-0.488113\pi$
$619$	1.85786	0.0746739	0.0373369	+	0.999303i	$0.488113\pi$
$620$	−5.07107	−0.203659
$621$	0	0
$622$	−29.1716	−1.16967
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−28.8284	−1.15222
$627$	0	0
$628$	0.343146	0.0136930
$629$	2.00000	0.0797452
$630$	0	0
$631$	24.8284	0.988404	0.494202	−	0.869347i	$-0.335460\pi$
$631$	24.8284	0.988404	0.494202	+	0.869347i	$0.335460\pi$
$632$	11.3137	0.450035
$633$	0	0
$634$	−18.0000	−0.714871
$635$	14.1421	0.561214
$636$	0	0
$637$	0	0
$638$	−4.82843	−0.191159
$639$	0	0
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	48.4264	1.90975	0.954876	−	0.297006i	$-0.0959882\pi$
$643$	48.4264	1.90975	0.954876	+	0.297006i	$0.0959882\pi$
$644$	0	0
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−33.0711	−1.30016	−0.650079	−	0.759867i	$-0.725264\pi$
$647$	−33.0711	−1.30016	−0.650079	+	0.759867i	$0.725264\pi$
$648$	0	0
$649$	1.45584	0.0571469
$650$	0	0
$651$	0	0
$652$	14.7279	0.576790
$653$	−32.8284	−1.28468	−0.642338	−	0.766422i	$-0.722035\pi$
$653$	−32.8284	−1.28468	−0.642338	+	0.766422i	$0.722035\pi$
$654$	0	0
$655$	−16.8284	−0.657541
$656$	8.82843	0.344692
$657$	0	0
$658$	0	0
$659$	−20.1005	−0.783005	−0.391502	−	0.920177i	$-0.628044\pi$
$659$	−20.1005	−0.783005	−0.391502	+	0.920177i	$0.628044\pi$
$660$	0	0
$661$	−27.6569	−1.07573	−0.537863	−	0.843032i	$-0.680768\pi$
$661$	−27.6569	−1.07573	−0.537863	+	0.843032i	$0.680768\pi$
$662$	18.6274	0.723975
$663$	0	0
$664$	−5.17157	−0.200696
$665$	0	0
$666$	0	0
$667$	39.7990	1.54102
$668$	−1.07107	−0.0414409
$669$	0	0
$670$	7.89949	0.305184
$671$	6.82843	0.263609
$672$	0	0
$673$	41.5980	1.60348	0.801742	−	0.597670i	$-0.203907\pi$
$673$	41.5980	1.60348	0.801742	+	0.597670i	$0.203907\pi$
$674$	3.17157	0.122164
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−11.1716	−0.429358	−0.214679	−	0.976685i	$-0.568871\pi$
$677$	−11.1716	−0.429358	−0.214679	+	0.976685i	$0.568871\pi$
$678$	0	0
$679$	0	0
$680$	−1.41421	−0.0542326
$681$	0	0
$682$	2.97056	0.113749
$683$	−31.7990	−1.21675	−0.608377	−	0.793648i	$-0.708179\pi$
$683$	−31.7990	−1.21675	−0.608377	+	0.793648i	$0.708179\pi$
$684$	0	0
$685$	−16.0000	−0.611329
$686$	0	0
$687$	0	0
$688$	4.58579	0.174831
$689$	0	0
$690$	0	0
$691$	−12.2843	−0.467316	−0.233658	−	0.972319i	$-0.575070\pi$
$691$	−12.2843	−0.467316	−0.233658	+	0.972319i	$0.575070\pi$
$692$	−18.4853	−0.702705
$693$	0	0
$694$	1.65685	0.0628933
$695$	−17.6569	−0.669763
$696$	0	0
$697$	−12.4853	−0.472914
$698$	5.79899	0.219495
$699$	0	0
$700$	0	0
$701$	11.5563	0.436477	0.218239	−	0.975895i	$-0.429969\pi$
$701$	11.5563	0.436477	0.218239	+	0.975895i	$0.429969\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	−0.585786	−0.0220777
$705$	0	0
$706$	−0.727922	−0.0273957
$707$	0	0
$708$	0	0
$709$	−45.1127	−1.69424	−0.847121	−	0.531399i	$-0.821666\pi$
$709$	−45.1127	−1.69424	−0.847121	+	0.531399i	$0.821666\pi$
$710$	−10.8284	−0.406384
$711$	0	0
$712$	−6.48528	−0.243046
$713$	−24.4853	−0.916981
$714$	0	0
$715$	0	0
$716$	−0.585786	−0.0218919
$717$	0	0
$718$	18.8284	0.702671
$719$	12.2843	0.458126	0.229063	−	0.973412i	$-0.426434\pi$
$719$	12.2843	0.458126	0.229063	+	0.973412i	$0.426434\pi$
$720$	0	0
$721$	0	0
$722$	11.0000	0.409378
$723$	0	0
$724$	−8.82843	−0.328106
$725$	−8.24264	−0.306124
$726$	0	0
$727$	45.6569	1.69332	0.846659	−	0.532135i	$-0.178610\pi$
$727$	45.6569	1.69332	0.846659	+	0.532135i	$0.178610\pi$
$728$	0	0
$729$	0	0
$730$	7.65685	0.283393
$731$	−6.48528	−0.239867
$732$	0	0
$733$	−15.9411	−0.588799	−0.294399	−	0.955682i	$-0.595120\pi$
$733$	−15.9411	−0.588799	−0.294399	+	0.955682i	$0.595120\pi$
$734$	−16.4853	−0.608483
$735$	0	0
$736$	4.82843	0.177978
$737$	−4.62742	−0.170453
$738$	0	0
$739$	−43.1127	−1.58593	−0.792963	−	0.609270i	$-0.791463\pi$
$739$	−43.1127	−1.58593	−0.792963	+	0.609270i	$0.791463\pi$
$740$	1.41421	0.0519875
$741$	0	0
$742$	0	0
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	0	0
$745$	−13.8995	−0.509238
$746$	−11.0711	−0.405341
$747$	0	0
$748$	0.828427	0.0302903
$749$	0	0
$750$	0	0
$751$	25.5147	0.931045	0.465523	−	0.885036i	$-0.345866\pi$
$751$	25.5147	0.931045	0.465523	+	0.885036i	$0.345866\pi$
$752$	9.07107	0.330788
$753$	0	0
$754$	0	0
$755$	−17.7990	−0.647772
$756$	0	0
$757$	23.7574	0.863476	0.431738	−	0.901999i	$-0.357901\pi$
$757$	23.7574	0.863476	0.431738	+	0.901999i	$0.357901\pi$
$758$	24.4853	0.889345
$759$	0	0
$760$	−2.82843	−0.102598
$761$	18.9706	0.687682	0.343841	−	0.939028i	$-0.388272\pi$
$761$	18.9706	0.687682	0.343841	+	0.939028i	$0.388272\pi$
$762$	0	0
$763$	0	0
$764$	11.3137	0.409316
$765$	0	0
$766$	15.8995	0.574472
$767$	0	0
$768$	0	0
$769$	−30.5858	−1.10295	−0.551476	−	0.834191i	$-0.685935\pi$
$769$	−30.5858	−1.10295	−0.551476	+	0.834191i	$0.685935\pi$
$770$	0	0
$771$	0	0
$772$	−21.7990	−0.784563
$773$	−28.3431	−1.01943	−0.509716	−	0.860343i	$-0.670250\pi$
$773$	−28.3431	−1.01943	−0.509716	+	0.860343i	$0.670250\pi$
$774$	0	0
$775$	5.07107	0.182158
$776$	−8.82843	−0.316922
$777$	0	0
$778$	27.0711	0.970545
$779$	−24.9706	−0.894663
$780$	0	0
$781$	6.34315	0.226976
$782$	−6.82843	−0.244184
$783$	0	0
$784$	0	0
$785$	−0.343146	−0.0122474
$786$	0	0
$787$	28.9706	1.03269	0.516345	−	0.856381i	$-0.327292\pi$
$787$	28.9706	1.03269	0.516345	+	0.856381i	$0.327292\pi$
$788$	20.8284	0.741982
$789$	0	0
$790$	−11.3137	−0.402524
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−20.6274	−0.732040
$795$	0	0
$796$	2.92893	0.103813
$797$	−26.4853	−0.938157	−0.469078	−	0.883157i	$-0.655414\pi$
$797$	−26.4853	−0.938157	−0.469078	+	0.883157i	$0.655414\pi$
$798$	0	0
$799$	−12.8284	−0.453837
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−25.3137	−0.893858
$803$	−4.48528	−0.158282
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−6.34315	−0.223151
$809$	−25.1127	−0.882915	−0.441458	−	0.897282i	$-0.645538\pi$
$809$	−25.1127	−0.882915	−0.441458	+	0.897282i	$0.645538\pi$
$810$	0	0
$811$	−3.02944	−0.106378	−0.0531890	−	0.998584i	$-0.516939\pi$
$811$	−3.02944	−0.106378	−0.0531890	+	0.998584i	$0.516939\pi$
$812$	0	0
$813$	0	0
$814$	−0.828427	−0.0290364
$815$	−14.7279	−0.515897
$816$	0	0
$817$	−12.9706	−0.453783
$818$	18.3848	0.642809
$819$	0	0
$820$	−8.82843	−0.308302
$821$	4.44365	0.155084	0.0775422	−	0.996989i	$-0.475293\pi$
$821$	4.44365	0.155084	0.0775422	+	0.996989i	$0.475293\pi$
$822$	0	0
$823$	16.2843	0.567634	0.283817	−	0.958878i	$-0.408399\pi$
$823$	16.2843	0.567634	0.283817	+	0.958878i	$0.408399\pi$
$824$	10.1421	0.353318
$825$	0	0
$826$	0	0
$827$	11.1127	0.386426	0.193213	−	0.981157i	$-0.438109\pi$
$827$	11.1127	0.386426	0.193213	+	0.981157i	$0.438109\pi$
$828$	0	0
$829$	−45.3137	−1.57381	−0.786905	−	0.617074i	$-0.788318\pi$
$829$	−45.3137	−1.57381	−0.786905	+	0.617074i	$0.788318\pi$
$830$	5.17157	0.179508
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.07107	0.0370658
$836$	1.65685	0.0573035
$837$	0	0
$838$	−17.6569	−0.609946
$839$	−5.17157	−0.178543	−0.0892713	−	0.996007i	$-0.528454\pi$
$839$	−5.17157	−0.178543	−0.0892713	+	0.996007i	$0.528454\pi$
$840$	0	0
$841$	38.9411	1.34280
$842$	−16.6274	−0.573019
$843$	0	0
$844$	−1.65685	−0.0570313
$845$	13.0000	0.447214
$846$	0	0
$847$	0	0
$848$	9.31371	0.319834
$849$	0	0
$850$	1.41421	0.0485071
$851$	6.82843	0.234075
$852$	0	0
$853$	6.68629	0.228934	0.114467	−	0.993427i	$-0.463484\pi$
$853$	6.68629	0.228934	0.114467	+	0.993427i	$0.463484\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	−27.2721	−0.931596	−0.465798	−	0.884891i	$-0.654233\pi$
$857$	−27.2721	−0.931596	−0.465798	+	0.884891i	$0.654233\pi$
$858$	0	0
$859$	16.9706	0.579028	0.289514	−	0.957174i	$-0.406506\pi$
$859$	16.9706	0.579028	0.289514	+	0.957174i	$0.406506\pi$
$860$	−4.58579	−0.156374
$861$	0	0
$862$	14.1421	0.481683
$863$	0.970563	0.0330383	0.0165192	−	0.999864i	$-0.494742\pi$
$863$	0.970563	0.0330383	0.0165192	+	0.999864i	$0.494742\pi$
$864$	0	0
$865$	18.4853	0.628518
$866$	−8.82843	−0.300002
$867$	0	0
$868$	0	0
$869$	6.62742	0.224820
$870$	0	0
$871$	0	0
$872$	−5.31371	−0.179945
$873$	0	0
$874$	−13.6569	−0.461950
$875$	0	0
$876$	0	0
$877$	21.4142	0.723107	0.361553	−	0.932351i	$-0.382247\pi$
$877$	21.4142	0.723107	0.361553	+	0.932351i	$0.382247\pi$
$878$	−33.5563	−1.13247
$879$	0	0
$880$	0.585786	0.0197469
$881$	−31.6569	−1.06655	−0.533273	−	0.845943i	$-0.679038\pi$
$881$	−31.6569	−1.06655	−0.533273	+	0.845943i	$0.679038\pi$
$882$	0	0
$883$	−10.0416	−0.337928	−0.168964	−	0.985622i	$-0.554042\pi$
$883$	−10.0416	−0.337928	−0.168964	+	0.985622i	$0.554042\pi$
$884$	0	0
$885$	0	0
$886$	13.6569	0.458811
$887$	23.6985	0.795717	0.397859	−	0.917447i	$-0.369753\pi$
$887$	23.6985	0.795717	0.397859	+	0.917447i	$0.369753\pi$
$888$	0	0
$889$	0	0
$890$	6.48528	0.217387
$891$	0	0
$892$	17.6569	0.591195
$893$	−25.6569	−0.858574
$894$	0	0
$895$	0.585786	0.0195807
$896$	0	0
$897$	0	0
$898$	−25.1127	−0.838022
$899$	−41.7990	−1.39407
$900$	0	0
$901$	−13.1716	−0.438809
$902$	5.17157	0.172195
$903$	0	0
$904$	9.31371	0.309769
$905$	8.82843	0.293467
$906$	0	0
$907$	43.2132	1.43487	0.717435	−	0.696625i	$-0.245316\pi$
$907$	43.2132	1.43487	0.717435	+	0.696625i	$0.245316\pi$
$908$	13.6569	0.453219
$909$	0	0
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	−3.02944	−0.100260
$914$	9.79899	0.324122
$915$	0	0
$916$	25.7990	0.852423
$917$	0	0
$918$	0	0
$919$	26.7696	0.883046	0.441523	−	0.897250i	$-0.354438\pi$
$919$	26.7696	0.883046	0.441523	+	0.897250i	$0.354438\pi$
$920$	−4.82843	−0.159189
$921$	0	0
$922$	−14.0000	−0.461065
$923$	0	0
$924$	0	0
$925$	−1.41421	−0.0464991
$926$	−33.6569	−1.10603
$927$	0	0
$928$	8.24264	0.270578
$929$	11.6569	0.382449	0.191224	−	0.981546i	$-0.438754\pi$
$929$	11.6569	0.382449	0.191224	+	0.981546i	$0.438754\pi$
$930$	0	0
$931$	0	0
$932$	−6.97056	−0.228328
$933$	0	0
$934$	−20.4853	−0.670299
$935$	−0.828427	−0.0270925
$936$	0	0
$937$	−17.0294	−0.556327	−0.278164	−	0.960534i	$-0.589726\pi$
$937$	−17.0294	−0.556327	−0.278164	+	0.960534i	$0.589726\pi$
$938$	0	0
$939$	0	0
$940$	−9.07107	−0.295866
$941$	−36.2843	−1.18283	−0.591417	−	0.806366i	$-0.701431\pi$
$941$	−36.2843	−1.18283	−0.591417	+	0.806366i	$0.701431\pi$
$942$	0	0
$943$	−42.6274	−1.38814
$944$	−2.48528	−0.0808890
$945$	0	0
$946$	2.68629	0.0873389
$947$	−36.4853	−1.18561	−0.592806	−	0.805345i	$-0.701980\pi$
$947$	−36.4853	−1.18561	−0.592806	+	0.805345i	$0.701980\pi$
$948$	0	0
$949$	0	0
$950$	2.82843	0.0917663
$951$	0	0
$952$	0	0
$953$	23.5980	0.764414	0.382207	−	0.924077i	$-0.375164\pi$
$953$	23.5980	0.764414	0.382207	+	0.924077i	$0.375164\pi$
$954$	0	0
$955$	−11.3137	−0.366103
$956$	−23.3137	−0.754019
$957$	0	0
$958$	16.4853	0.532615
$959$	0	0
$960$	0	0
$961$	−5.28427	−0.170460
$962$	0	0
$963$	0	0
$964$	−16.2426	−0.523140
$965$	21.7990	0.701734
$966$	0	0
$967$	−44.2843	−1.42409	−0.712043	−	0.702136i	$-0.752230\pi$
$967$	−44.2843	−1.42409	−0.712043	+	0.702136i	$0.752230\pi$
$968$	10.6569	0.342524
$969$	0	0
$970$	8.82843	0.283464
$971$	−44.9706	−1.44317	−0.721587	−	0.692324i	$-0.756587\pi$
$971$	−44.9706	−1.44317	−0.721587	+	0.692324i	$0.756587\pi$
$972$	0	0
$973$	0	0
$974$	−24.4853	−0.784559
$975$	0	0
$976$	−11.6569	−0.373127
$977$	−5.31371	−0.170001	−0.0850003	−	0.996381i	$-0.527089\pi$
$977$	−5.31371	−0.170001	−0.0850003	+	0.996381i	$0.527089\pi$
$978$	0	0
$979$	−3.79899	−0.121416
$980$	0	0
$981$	0	0
$982$	11.8995	0.379728
$983$	16.8701	0.538071	0.269036	−	0.963130i	$-0.413295\pi$
$983$	16.8701	0.538071	0.269036	+	0.963130i	$0.413295\pi$
$984$	0	0
$985$	−20.8284	−0.663649
$986$	−11.6569	−0.371230
$987$	0	0
$988$	0	0
$989$	−22.1421	−0.704079
$990$	0	0
$991$	20.8284	0.661637	0.330818	−	0.943694i	$-0.392675\pi$
$991$	20.8284	0.661637	0.330818	+	0.943694i	$0.392675\pi$
$992$	−5.07107	−0.161007
$993$	0	0
$994$	0	0
$995$	−2.92893	−0.0928534
$996$	0	0
$997$	−2.68629	−0.0850757	−0.0425379	−	0.999095i	$-0.513544\pi$
$997$	−2.68629	−0.0850757	−0.0425379	+	0.999095i	$0.513544\pi$
$998$	−15.7990	−0.500108
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bn.1.2		2
3.2	odd	2		1470.2.a.v.1.1	yes	2
7.6	odd	2		4410.2.a.br.1.2		2
15.14	odd	2		7350.2.a.dd.1.1		2
21.2	odd	6		1470.2.i.u.361.2		4
21.5	even	6		1470.2.i.v.361.2		4
21.11	odd	6		1470.2.i.u.961.2		4
21.17	even	6		1470.2.i.v.961.2		4
21.20	even	2		1470.2.a.u.1.1	✓	2
105.104	even	2		7350.2.a.df.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.u.1.1	✓	2	21.20	even	2
1470.2.a.v.1.1	yes	2	3.2	odd	2
1470.2.i.u.361.2		4	21.2	odd	6
1470.2.i.u.961.2		4	21.11	odd	6
1470.2.i.v.361.2		4	21.5	even	6
1470.2.i.v.961.2		4	21.17	even	6
4410.2.a.bn.1.2		2	1.1	even	1	trivial
4410.2.a.br.1.2		2	7.6	odd	2
7350.2.a.dd.1.1		2	15.14	odd	2
7350.2.a.df.1.1		2	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -0.585786 q^{11} +1.00000 q^{16} -1.41421 q^{17} -2.82843 q^{19} -1.00000 q^{20} +0.585786 q^{22} -4.82843 q^{23} +1.00000 q^{25} -8.24264 q^{29} +5.07107 q^{31} -1.00000 q^{32} +1.41421 q^{34} -1.41421 q^{37} +2.82843 q^{38} +1.00000 q^{40} +8.82843 q^{41} +4.58579 q^{43} -0.585786 q^{44} +4.82843 q^{46} +9.07107 q^{47} -1.00000 q^{50} +9.31371 q^{53} +0.585786 q^{55} +8.24264 q^{58} -2.48528 q^{59} -11.6569 q^{61} -5.07107 q^{62} +1.00000 q^{64} +7.89949 q^{67} -1.41421 q^{68} -10.8284 q^{71} +7.65685 q^{73} +1.41421 q^{74} -2.82843 q^{76} -11.3137 q^{79} -1.00000 q^{80} -8.82843 q^{82} +5.17157 q^{83} +1.41421 q^{85} -4.58579 q^{86} +0.585786 q^{88} +6.48528 q^{89} -4.82843 q^{92} -9.07107 q^{94} +2.82843 q^{95} +8.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{16} - 2 q^{20} + 4 q^{22} - 4 q^{23} + 2 q^{25} - 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{40} + 12 q^{41} + 12 q^{43} - 4 q^{44} + 4 q^{46} + 4 q^{47} - 2 q^{50} - 4 q^{53} + 4 q^{55} + 8 q^{58} + 12 q^{59} - 12 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{67} - 16 q^{71} + 4 q^{73} - 2 q^{80} - 12 q^{82} + 16 q^{83} - 12 q^{86} + 4 q^{88} - 4 q^{89} - 4 q^{92} - 4 q^{94} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^16 - 2 * q^20 + 4 * q^22 - 4 * q^23 + 2 * q^25 - 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^40 + 12 * q^41 + 12 * q^43 - 4 * q^44 + 4 * q^46 + 4 * q^47 - 2 * q^50 - 4 * q^53 + 4 * q^55 + 8 * q^58 + 12 * q^59 - 12 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^67 - 16 * q^71 + 4 * q^73 - 2 * q^80 - 12 * q^82 + 16 * q^83 - 12 * q^86 + 4 * q^88 - 4 * q^89 - 4 * q^92 - 4 * q^94 + 12 * q^97

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