LMFDB - Embedded newform 5440.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5440,2,Mod(1,5440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5440 = 2^{6} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5440.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:	$43.4386186996$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.41421 q^{3} +1.00000 q^{5} -0.585786 q^{7} +8.65685 q^{9} +2.58579 q^{11} +2.82843 q^{13} +3.41421 q^{15} -1.00000 q^{17} +2.82843 q^{19} -2.00000 q^{21} -3.41421 q^{23} +1.00000 q^{25} +19.3137 q^{27} +4.82843 q^{29} +4.24264 q^{31} +8.82843 q^{33} -0.585786 q^{35} -6.48528 q^{37} +9.65685 q^{39} -6.48528 q^{41} -7.65685 q^{43} +8.65685 q^{45} -4.82843 q^{47} -6.65685 q^{49} -3.41421 q^{51} -0.343146 q^{53} +2.58579 q^{55} +9.65685 q^{57} +9.17157 q^{59} -7.65685 q^{61} -5.07107 q^{63} +2.82843 q^{65} +3.17157 q^{67} -11.6569 q^{69} -4.24264 q^{71} -4.82843 q^{73} +3.41421 q^{75} -1.51472 q^{77} +5.41421 q^{79} +39.9706 q^{81} -9.31371 q^{83} -1.00000 q^{85} +16.4853 q^{87} -2.34315 q^{89} -1.65685 q^{91} +14.4853 q^{93} +2.82843 q^{95} +3.65685 q^{97} +22.3848 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} + 8 q^{11} + 4 q^{15} - 2 q^{17} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 16 q^{27} + 4 q^{29} + 12 q^{33} - 4 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 4 q^{43} + 6 q^{45}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 + 8 * q^11 + 4 * q^15 - 2 * q^17 - 4 * q^21 - 4 * q^23 + 2 * q^25 + 16 * q^27 + 4 * q^29 + 12 * q^33 - 4 * q^35 + 4 * q^37 + 8 * q^39 + 4 * q^41 - 4 * q^43 + 6 * q^45 - 4 * q^47 - 2 * q^49 - 4 * q^51 - 12 * q^53 + 8 * q^55 + 8 * q^57 + 24 * q^59 - 4 * q^61 + 4 * q^63 + 12 * q^67 - 12 * q^69 - 4 * q^73 + 4 * q^75 - 20 * q^77 + 8 * q^79 + 46 * q^81 + 4 * q^83 - 2 * q^85 + 16 * q^87 - 16 * q^89 + 8 * q^91 + 12 * q^93 - 4 * q^97 + 8 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.41421	1.97120	0.985599	−	0.169102i	$-0.0540867\pi$
$3$	3.41421	1.97120	0.985599	+	0.169102i	$0.0540867\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	0	0
$9$	8.65685	2.88562
$10$	0	0
$11$	2.58579	0.779644	0.389822	−	0.920890i	$-0.372537\pi$
$11$	2.58579	0.779644	0.389822	+	0.920890i	$0.372537\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−3.41421	−0.711913	−0.355956	−	0.934503i	$-0.615845\pi$
$23$	−3.41421	−0.711913	−0.355956	+	0.934503i	$0.615845\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	19.3137	3.71692
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	0	0
$33$	8.82843	1.53683
$34$	0	0
$35$	−0.585786	−0.0990160
$36$	0	0
$37$	−6.48528	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$37$	−6.48528	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$38$	0	0
$39$	9.65685	1.54633
$40$	0	0
$41$	−6.48528	−1.01283	−0.506415	−	0.862290i	$-0.669030\pi$
$41$	−6.48528	−1.01283	−0.506415	+	0.862290i	$0.669030\pi$
$42$	0	0
$43$	−7.65685	−1.16766	−0.583830	−	0.811876i	$-0.698446\pi$
$43$	−7.65685	−1.16766	−0.583830	+	0.811876i	$0.698446\pi$
$44$	0	0
$45$	8.65685	1.29049
$46$	0	0
$47$	−4.82843	−0.704298	−0.352149	−	0.935944i	$-0.614549\pi$
$47$	−4.82843	−0.704298	−0.352149	+	0.935944i	$0.614549\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	−3.41421	−0.478086
$52$	0	0
$53$	−0.343146	−0.0471347	−0.0235673	−	0.999722i	$-0.507502\pi$
$53$	−0.343146	−0.0471347	−0.0235673	+	0.999722i	$0.507502\pi$
$54$	0	0
$55$	2.58579	0.348667
$56$	0	0
$57$	9.65685	1.27908
$58$	0	0
$59$	9.17157	1.19404	0.597019	−	0.802227i	$-0.296352\pi$
$59$	9.17157	1.19404	0.597019	+	0.802227i	$0.296352\pi$
$60$	0	0
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	0	0
$63$	−5.07107	−0.638894
$64$	0	0
$65$	2.82843	0.350823
$66$	0	0
$67$	3.17157	0.387469	0.193735	−	0.981054i	$-0.437940\pi$
$67$	3.17157	0.387469	0.193735	+	0.981054i	$0.437940\pi$
$68$	0	0
$69$	−11.6569	−1.40332
$70$	0	0
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	0	0
$75$	3.41421	0.394239
$76$	0	0
$77$	−1.51472	−0.172618
$78$	0	0
$79$	5.41421	0.609147	0.304573	−	0.952489i	$-0.401486\pi$
$79$	5.41421	0.609147	0.304573	+	0.952489i	$0.401486\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	0	0
$83$	−9.31371	−1.02231	−0.511156	−	0.859488i	$-0.670783\pi$
$83$	−9.31371	−1.02231	−0.511156	+	0.859488i	$0.670783\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	16.4853	1.76741
$88$	0	0
$89$	−2.34315	−0.248373	−0.124186	−	0.992259i	$-0.539632\pi$
$89$	−2.34315	−0.248373	−0.124186	+	0.992259i	$0.539632\pi$
$90$	0	0
$91$	−1.65685	−0.173686
$92$	0	0
$93$	14.4853	1.50205
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	0	0
$99$	22.3848	2.24975
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−0.828427	−0.0816274	−0.0408137	−	0.999167i	$-0.512995\pi$
$103$	−0.828427	−0.0816274	−0.0408137	+	0.999167i	$0.512995\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−11.8995	−1.15037	−0.575184	−	0.818024i	$-0.695069\pi$
$107$	−11.8995	−1.15037	−0.575184	+	0.818024i	$0.695069\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	−22.1421	−2.10164
$112$	0	0
$113$	−3.17157	−0.298356	−0.149178	−	0.988810i	$-0.547663\pi$
$113$	−3.17157	−0.298356	−0.149178	+	0.988810i	$0.547663\pi$
$114$	0	0
$115$	−3.41421	−0.318377
$116$	0	0
$117$	24.4853	2.26367
$118$	0	0
$119$	0.585786	0.0536990
$120$	0	0
$121$	−4.31371	−0.392155
$122$	0	0
$123$	−22.1421	−1.99649
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.3137	−1.53634	−0.768172	−	0.640244i	$-0.778833\pi$
$127$	−17.3137	−1.53634	−0.768172	+	0.640244i	$0.778833\pi$
$128$	0	0
$129$	−26.1421	−2.30169
$130$	0	0
$131$	13.8995	1.21440	0.607202	−	0.794547i	$-0.292292\pi$
$131$	13.8995	1.21440	0.607202	+	0.794547i	$0.292292\pi$
$132$	0	0
$133$	−1.65685	−0.143667
$134$	0	0
$135$	19.3137	1.66226
$136$	0	0
$137$	1.17157	0.100094	0.0500471	−	0.998747i	$-0.484063\pi$
$137$	1.17157	0.100094	0.0500471	+	0.998747i	$0.484063\pi$
$138$	0	0
$139$	−17.8995	−1.51822	−0.759108	−	0.650965i	$-0.774364\pi$
$139$	−17.8995	−1.51822	−0.759108	+	0.650965i	$0.774364\pi$
$140$	0	0
$141$	−16.4853	−1.38831
$142$	0	0
$143$	7.31371	0.611603
$144$	0	0
$145$	4.82843	0.400979
$146$	0	0
$147$	−22.7279	−1.87457
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	7.51472	0.611539	0.305770	−	0.952106i	$-0.401086\pi$
$151$	7.51472	0.611539	0.305770	+	0.952106i	$0.401086\pi$
$152$	0	0
$153$	−8.65685	−0.699865
$154$	0	0
$155$	4.24264	0.340777
$156$	0	0
$157$	21.3137	1.70102	0.850510	−	0.525960i	$-0.176294\pi$
$157$	21.3137	1.70102	0.850510	+	0.525960i	$0.176294\pi$
$158$	0	0
$159$	−1.17157	−0.0929118
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−0.585786	−0.0458823	−0.0229412	−	0.999737i	$-0.507303\pi$
$163$	−0.585786	−0.0458823	−0.0229412	+	0.999737i	$0.507303\pi$
$164$	0	0
$165$	8.82843	0.687292
$166$	0	0
$167$	2.24264	0.173541	0.0867704	−	0.996228i	$-0.472345\pi$
$167$	2.24264	0.173541	0.0867704	+	0.996228i	$0.472345\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	24.4853	1.87244
$172$	0	0
$173$	−12.8284	−0.975327	−0.487664	−	0.873032i	$-0.662151\pi$
$173$	−12.8284	−0.975327	−0.487664	+	0.873032i	$0.662151\pi$
$174$	0	0
$175$	−0.585786	−0.0442813
$176$	0	0
$177$	31.3137	2.35368
$178$	0	0
$179$	6.82843	0.510381	0.255190	−	0.966891i	$-0.417862\pi$
$179$	6.82843	0.510381	0.255190	+	0.966891i	$0.417862\pi$
$180$	0	0
$181$	−2.48528	−0.184730	−0.0923648	−	0.995725i	$-0.529443\pi$
$181$	−2.48528	−0.184730	−0.0923648	+	0.995725i	$0.529443\pi$
$182$	0	0
$183$	−26.1421	−1.93248
$184$	0	0
$185$	−6.48528	−0.476807
$186$	0	0
$187$	−2.58579	−0.189091
$188$	0	0
$189$	−11.3137	−0.822951
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−20.8284	−1.49926	−0.749631	−	0.661855i	$-0.769769\pi$
$193$	−20.8284	−1.49926	−0.749631	+	0.661855i	$0.769769\pi$
$194$	0	0
$195$	9.65685	0.691542
$196$	0	0
$197$	−12.8284	−0.913988	−0.456994	−	0.889470i	$-0.651074\pi$
$197$	−12.8284	−0.913988	−0.456994	+	0.889470i	$0.651074\pi$
$198$	0	0
$199$	24.2426	1.71852	0.859258	−	0.511543i	$-0.170926\pi$
$199$	24.2426	1.71852	0.859258	+	0.511543i	$0.170926\pi$
$200$	0	0
$201$	10.8284	0.763778
$202$	0	0
$203$	−2.82843	−0.198517
$204$	0	0
$205$	−6.48528	−0.452952
$206$	0	0
$207$	−29.5563	−2.05431
$208$	0	0
$209$	7.31371	0.505900
$210$	0	0
$211$	2.10051	0.144605	0.0723024	−	0.997383i	$-0.476965\pi$
$211$	2.10051	0.144605	0.0723024	+	0.997383i	$0.476965\pi$
$212$	0	0
$213$	−14.4853	−0.992515
$214$	0	0
$215$	−7.65685	−0.522193
$216$	0	0
$217$	−2.48528	−0.168712
$218$	0	0
$219$	−16.4853	−1.11397
$220$	0	0
$221$	−2.82843	−0.190261
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	8.65685	0.577124
$226$	0	0
$227$	22.7279	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$227$	22.7279	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$228$	0	0
$229$	0.686292	0.0453514	0.0226757	−	0.999743i	$-0.492781\pi$
$229$	0.686292	0.0453514	0.0226757	+	0.999743i	$0.492781\pi$
$230$	0	0
$231$	−5.17157	−0.340265
$232$	0	0
$233$	9.31371	0.610161	0.305081	−	0.952327i	$-0.401317\pi$
$233$	9.31371	0.610161	0.305081	+	0.952327i	$0.401317\pi$
$234$	0	0
$235$	−4.82843	−0.314972
$236$	0	0
$237$	18.4853	1.20075
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	12.8284	0.826352	0.413176	−	0.910651i	$-0.364419\pi$
$241$	12.8284	0.826352	0.413176	+	0.910651i	$0.364419\pi$
$242$	0	0
$243$	78.5269	5.03750
$244$	0	0
$245$	−6.65685	−0.425291
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−31.7990	−2.01518
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−8.82843	−0.555038
$254$	0	0
$255$	−3.41421	−0.213806
$256$	0	0
$257$	−2.82843	−0.176432	−0.0882162	−	0.996101i	$-0.528117\pi$
$257$	−2.82843	−0.176432	−0.0882162	+	0.996101i	$0.528117\pi$
$258$	0	0
$259$	3.79899	0.236058
$260$	0	0
$261$	41.7990	2.58729
$262$	0	0
$263$	9.31371	0.574308	0.287154	−	0.957884i	$-0.407291\pi$
$263$	9.31371	0.574308	0.287154	+	0.957884i	$0.407291\pi$
$264$	0	0
$265$	−0.343146	−0.0210793
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−18.9706	−1.15666	−0.578328	−	0.815804i	$-0.696295\pi$
$269$	−18.9706	−1.15666	−0.578328	+	0.815804i	$0.696295\pi$
$270$	0	0
$271$	−13.6569	−0.829595	−0.414797	−	0.909914i	$-0.636148\pi$
$271$	−13.6569	−0.829595	−0.414797	+	0.909914i	$0.636148\pi$
$272$	0	0
$273$	−5.65685	−0.342368
$274$	0	0
$275$	2.58579	0.155929
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	36.7279	2.19884
$280$	0	0
$281$	−4.34315	−0.259090	−0.129545	−	0.991574i	$-0.541352\pi$
$281$	−4.34315	−0.259090	−0.129545	+	0.991574i	$0.541352\pi$
$282$	0	0
$283$	−14.2426	−0.846637	−0.423319	−	0.905981i	$-0.639135\pi$
$283$	−14.2426	−0.846637	−0.423319	+	0.905981i	$0.639135\pi$
$284$	0	0
$285$	9.65685	0.572023
$286$	0	0
$287$	3.79899	0.224247
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.4853	0.731900
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	9.17157	0.533990
$296$	0	0
$297$	49.9411	2.89788
$298$	0	0
$299$	−9.65685	−0.558470
$300$	0	0
$301$	4.48528	0.258527
$302$	0	0
$303$	27.3137	1.56913
$304$	0	0
$305$	−7.65685	−0.438430
$306$	0	0
$307$	16.8284	0.960449	0.480225	−	0.877146i	$-0.340555\pi$
$307$	16.8284	0.960449	0.480225	+	0.877146i	$0.340555\pi$
$308$	0	0
$309$	−2.82843	−0.160904
$310$	0	0
$311$	15.0711	0.854602	0.427301	−	0.904109i	$-0.359464\pi$
$311$	15.0711	0.854602	0.427301	+	0.904109i	$0.359464\pi$
$312$	0	0
$313$	−5.79899	−0.327778	−0.163889	−	0.986479i	$-0.552404\pi$
$313$	−5.79899	−0.327778	−0.163889	+	0.986479i	$0.552404\pi$
$314$	0	0
$315$	−5.07107	−0.285722
$316$	0	0
$317$	21.3137	1.19710	0.598549	−	0.801087i	$-0.295744\pi$
$317$	21.3137	1.19710	0.598549	+	0.801087i	$0.295744\pi$
$318$	0	0
$319$	12.4853	0.699042
$320$	0	0
$321$	−40.6274	−2.26760
$322$	0	0
$323$	−2.82843	−0.157378
$324$	0	0
$325$	2.82843	0.156893
$326$	0	0
$327$	59.1127	3.26894
$328$	0	0
$329$	2.82843	0.155936
$330$	0	0
$331$	−26.8284	−1.47462	−0.737312	−	0.675553i	$-0.763905\pi$
$331$	−26.8284	−1.47462	−0.737312	+	0.675553i	$0.763905\pi$
$332$	0	0
$333$	−56.1421	−3.07657
$334$	0	0
$335$	3.17157	0.173282
$336$	0	0
$337$	−20.6274	−1.12365	−0.561824	−	0.827257i	$-0.689900\pi$
$337$	−20.6274	−1.12365	−0.561824	+	0.827257i	$0.689900\pi$
$338$	0	0
$339$	−10.8284	−0.588119
$340$	0	0
$341$	10.9706	0.594089
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−11.6569	−0.627584
$346$	0	0
$347$	23.6985	1.27220	0.636101	−	0.771606i	$-0.280546\pi$
$347$	23.6985	1.27220	0.636101	+	0.771606i	$0.280546\pi$
$348$	0	0
$349$	−31.6569	−1.69455	−0.847276	−	0.531152i	$-0.821759\pi$
$349$	−31.6569	−1.69455	−0.847276	+	0.531152i	$0.821759\pi$
$350$	0	0
$351$	54.6274	2.91580
$352$	0	0
$353$	−27.6569	−1.47203	−0.736013	−	0.676967i	$-0.763294\pi$
$353$	−27.6569	−1.47203	−0.736013	+	0.676967i	$0.763294\pi$
$354$	0	0
$355$	−4.24264	−0.225176
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	−31.7990	−1.67829	−0.839143	−	0.543910i	$-0.816943\pi$
$359$	−31.7990	−1.67829	−0.839143	+	0.543910i	$0.816943\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−14.7279	−0.773015
$364$	0	0
$365$	−4.82843	−0.252731
$366$	0	0
$367$	−14.2426	−0.743460	−0.371730	−	0.928341i	$-0.621235\pi$
$367$	−14.2426	−0.743460	−0.371730	+	0.928341i	$0.621235\pi$
$368$	0	0
$369$	−56.1421	−2.92264
$370$	0	0
$371$	0.201010	0.0104359
$372$	0	0
$373$	−11.7990	−0.610929	−0.305464	−	0.952204i	$-0.598812\pi$
$373$	−11.7990	−0.610929	−0.305464	+	0.952204i	$0.598812\pi$
$374$	0	0
$375$	3.41421	0.176309
$376$	0	0
$377$	13.6569	0.703364
$378$	0	0
$379$	26.8701	1.38022	0.690111	−	0.723703i	$-0.257562\pi$
$379$	26.8701	1.38022	0.690111	+	0.723703i	$0.257562\pi$
$380$	0	0
$381$	−59.1127	−3.02844
$382$	0	0
$383$	−34.2843	−1.75184	−0.875922	−	0.482452i	$-0.839746\pi$
$383$	−34.2843	−1.75184	−0.875922	+	0.482452i	$0.839746\pi$
$384$	0	0
$385$	−1.51472	−0.0771972
$386$	0	0
$387$	−66.2843	−3.36942
$388$	0	0
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	3.41421	0.172664
$392$	0	0
$393$	47.4558	2.39383
$394$	0	0
$395$	5.41421	0.272419
$396$	0	0
$397$	−13.3137	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$397$	−13.3137	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$398$	0	0
$399$	−5.65685	−0.283197
$400$	0	0
$401$	−16.3431	−0.816138	−0.408069	−	0.912951i	$-0.633798\pi$
$401$	−16.3431	−0.816138	−0.408069	+	0.912951i	$0.633798\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	0	0
$405$	39.9706	1.98615
$406$	0	0
$407$	−16.7696	−0.831236
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	−5.37258	−0.264368
$414$	0	0
$415$	−9.31371	−0.457192
$416$	0	0
$417$	−61.1127	−2.99270
$418$	0	0
$419$	−12.2426	−0.598092	−0.299046	−	0.954239i	$-0.596668\pi$
$419$	−12.2426	−0.598092	−0.299046	+	0.954239i	$0.596668\pi$
$420$	0	0
$421$	28.9706	1.41194	0.705969	−	0.708242i	$-0.250512\pi$
$421$	28.9706	1.41194	0.705969	+	0.708242i	$0.250512\pi$
$422$	0	0
$423$	−41.7990	−2.03234
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	4.48528	0.217058
$428$	0	0
$429$	24.9706	1.20559
$430$	0	0
$431$	1.41421	0.0681203	0.0340601	−	0.999420i	$-0.489156\pi$
$431$	1.41421	0.0681203	0.0340601	+	0.999420i	$0.489156\pi$
$432$	0	0
$433$	2.82843	0.135926	0.0679628	−	0.997688i	$-0.478350\pi$
$433$	2.82843	0.135926	0.0679628	+	0.997688i	$0.478350\pi$
$434$	0	0
$435$	16.4853	0.790409
$436$	0	0
$437$	−9.65685	−0.461950
$438$	0	0
$439$	5.41421	0.258406	0.129203	−	0.991618i	$-0.458758\pi$
$439$	5.41421	0.258406	0.129203	+	0.991618i	$0.458758\pi$
$440$	0	0
$441$	−57.6274	−2.74416
$442$	0	0
$443$	14.4853	0.688216	0.344108	−	0.938930i	$-0.388181\pi$
$443$	14.4853	0.688216	0.344108	+	0.938930i	$0.388181\pi$
$444$	0	0
$445$	−2.34315	−0.111076
$446$	0	0
$447$	−6.82843	−0.322974
$448$	0	0
$449$	−26.4853	−1.24992	−0.624959	−	0.780658i	$-0.714884\pi$
$449$	−26.4853	−1.24992	−0.624959	+	0.780658i	$0.714884\pi$
$450$	0	0
$451$	−16.7696	−0.789647
$452$	0	0
$453$	25.6569	1.20546
$454$	0	0
$455$	−1.65685	−0.0776745
$456$	0	0
$457$	39.1127	1.82961	0.914807	−	0.403890i	$-0.132342\pi$
$457$	39.1127	1.82961	0.914807	+	0.403890i	$0.132342\pi$
$458$	0	0
$459$	−19.3137	−0.901487
$460$	0	0
$461$	−41.5980	−1.93741	−0.968706	−	0.248213i	$-0.920157\pi$
$461$	−41.5980	−1.93741	−0.968706	+	0.248213i	$0.920157\pi$
$462$	0	0
$463$	−3.17157	−0.147395	−0.0736977	−	0.997281i	$-0.523480\pi$
$463$	−3.17157	−0.147395	−0.0736977	+	0.997281i	$0.523480\pi$
$464$	0	0
$465$	14.4853	0.671739
$466$	0	0
$467$	0.343146	0.0158789	0.00793945	−	0.999968i	$-0.497473\pi$
$467$	0.343146	0.0158789	0.00793945	+	0.999968i	$0.497473\pi$
$468$	0	0
$469$	−1.85786	−0.0857882
$470$	0	0
$471$	72.7696	3.35304
$472$	0	0
$473$	−19.7990	−0.910359
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	−2.97056	−0.136013
$478$	0	0
$479$	−15.7574	−0.719972	−0.359986	−	0.932958i	$-0.617219\pi$
$479$	−15.7574	−0.719972	−0.359986	+	0.932958i	$0.617219\pi$
$480$	0	0
$481$	−18.3431	−0.836375
$482$	0	0
$483$	6.82843	0.310704
$484$	0	0
$485$	3.65685	0.166049
$486$	0	0
$487$	−3.89949	−0.176703	−0.0883515	−	0.996089i	$-0.528160\pi$
$487$	−3.89949	−0.176703	−0.0883515	+	0.996089i	$0.528160\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	16.4853	0.743970	0.371985	−	0.928239i	$-0.378677\pi$
$491$	16.4853	0.743970	0.371985	+	0.928239i	$0.378677\pi$
$492$	0	0
$493$	−4.82843	−0.217461
$494$	0	0
$495$	22.3848	1.00612
$496$	0	0
$497$	2.48528	0.111480
$498$	0	0
$499$	−12.2426	−0.548056	−0.274028	−	0.961722i	$-0.588356\pi$
$499$	−12.2426	−0.548056	−0.274028	+	0.961722i	$0.588356\pi$
$500$	0	0
$501$	7.65685	0.342083
$502$	0	0
$503$	31.6985	1.41337	0.706683	−	0.707531i	$-0.250191\pi$
$503$	31.6985	1.41337	0.706683	+	0.707531i	$0.250191\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−17.0711	−0.758153
$508$	0	0
$509$	−20.6274	−0.914294	−0.457147	−	0.889391i	$-0.651129\pi$
$509$	−20.6274	−0.914294	−0.457147	+	0.889391i	$0.651129\pi$
$510$	0	0
$511$	2.82843	0.125122
$512$	0	0
$513$	54.6274	2.41186
$514$	0	0
$515$	−0.828427	−0.0365049
$516$	0	0
$517$	−12.4853	−0.549102
$518$	0	0
$519$	−43.7990	−1.92256
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	0.142136	0.00621516	0.00310758	−	0.999995i	$-0.499011\pi$
$523$	0.142136	0.00621516	0.00310758	+	0.999995i	$0.499011\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	−4.24264	−0.184812
$528$	0	0
$529$	−11.3431	−0.493180
$530$	0	0
$531$	79.3970	3.44554
$532$	0	0
$533$	−18.3431	−0.794530
$534$	0	0
$535$	−11.8995	−0.514460
$536$	0	0
$537$	23.3137	1.00606
$538$	0	0
$539$	−17.2132	−0.741425
$540$	0	0
$541$	42.7696	1.83881	0.919403	−	0.393316i	$-0.128672\pi$
$541$	42.7696	1.83881	0.919403	+	0.393316i	$0.128672\pi$
$542$	0	0
$543$	−8.48528	−0.364138
$544$	0	0
$545$	17.3137	0.741638
$546$	0	0
$547$	−7.21320	−0.308414	−0.154207	−	0.988039i	$-0.549282\pi$
$547$	−7.21320	−0.308414	−0.154207	+	0.988039i	$0.549282\pi$
$548$	0	0
$549$	−66.2843	−2.82894
$550$	0	0
$551$	13.6569	0.581802
$552$	0	0
$553$	−3.17157	−0.134869
$554$	0	0
$555$	−22.1421	−0.939881
$556$	0	0
$557$	−26.8284	−1.13676	−0.568378	−	0.822767i	$-0.692429\pi$
$557$	−26.8284	−1.13676	−0.568378	+	0.822767i	$0.692429\pi$
$558$	0	0
$559$	−21.6569	−0.915987
$560$	0	0
$561$	−8.82843	−0.372736
$562$	0	0
$563$	20.3431	0.857361	0.428681	−	0.903456i	$-0.358979\pi$
$563$	20.3431	0.857361	0.428681	+	0.903456i	$0.358979\pi$
$564$	0	0
$565$	−3.17157	−0.133429
$566$	0	0
$567$	−23.4142	−0.983305
$568$	0	0
$569$	−26.2843	−1.10189	−0.550947	−	0.834540i	$-0.685733\pi$
$569$	−26.2843	−1.10189	−0.550947	+	0.834540i	$0.685733\pi$
$570$	0	0
$571$	4.44365	0.185961	0.0929805	−	0.995668i	$-0.470361\pi$
$571$	4.44365	0.185961	0.0929805	+	0.995668i	$0.470361\pi$
$572$	0	0
$573$	40.9706	1.71157
$574$	0	0
$575$	−3.41421	−0.142383
$576$	0	0
$577$	22.8284	0.950360	0.475180	−	0.879889i	$-0.342383\pi$
$577$	22.8284	0.950360	0.475180	+	0.879889i	$0.342383\pi$
$578$	0	0
$579$	−71.1127	−2.95534
$580$	0	0
$581$	5.45584	0.226347
$582$	0	0
$583$	−0.887302	−0.0367483
$584$	0	0
$585$	24.4853	1.01234
$586$	0	0
$587$	−28.6274	−1.18158	−0.590790	−	0.806825i	$-0.701184\pi$
$587$	−28.6274	−1.18158	−0.590790	+	0.806825i	$0.701184\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	−43.7990	−1.80165
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0.585786	0.0240149
$596$	0	0
$597$	82.7696	3.38753
$598$	0	0
$599$	45.9411	1.87710	0.938552	−	0.345139i	$-0.112168\pi$
$599$	45.9411	1.87710	0.938552	+	0.345139i	$0.112168\pi$
$600$	0	0
$601$	−25.7990	−1.05236	−0.526181	−	0.850372i	$-0.676377\pi$
$601$	−25.7990	−1.05236	−0.526181	+	0.850372i	$0.676377\pi$
$602$	0	0
$603$	27.4558	1.11809
$604$	0	0
$605$	−4.31371	−0.175377
$606$	0	0
$607$	−4.58579	−0.186131	−0.0930657	−	0.995660i	$-0.529667\pi$
$607$	−4.58579	−0.186131	−0.0930657	+	0.995660i	$0.529667\pi$
$608$	0	0
$609$	−9.65685	−0.391315
$610$	0	0
$611$	−13.6569	−0.552497
$612$	0	0
$613$	46.9706	1.89712	0.948562	−	0.316593i	$-0.102539\pi$
$613$	46.9706	1.89712	0.948562	+	0.316593i	$0.102539\pi$
$614$	0	0
$615$	−22.1421	−0.892857
$616$	0	0
$617$	25.5147	1.02718	0.513592	−	0.858035i	$-0.328315\pi$
$617$	25.5147	1.02718	0.513592	+	0.858035i	$0.328315\pi$
$618$	0	0
$619$	16.9289	0.680431	0.340216	−	0.940347i	$-0.389500\pi$
$619$	16.9289	0.680431	0.340216	+	0.940347i	$0.389500\pi$
$620$	0	0
$621$	−65.9411	−2.64613
$622$	0	0
$623$	1.37258	0.0549914
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.9706	0.997228
$628$	0	0
$629$	6.48528	0.258585
$630$	0	0
$631$	−15.7990	−0.628948	−0.314474	−	0.949266i	$-0.601828\pi$
$631$	−15.7990	−0.628948	−0.314474	+	0.949266i	$0.601828\pi$
$632$	0	0
$633$	7.17157	0.285044
$634$	0	0
$635$	−17.3137	−0.687074
$636$	0	0
$637$	−18.8284	−0.746009
$638$	0	0
$639$	−36.7279	−1.45293
$640$	0	0
$641$	28.1421	1.11155	0.555774	−	0.831334i	$-0.312422\pi$
$641$	28.1421	1.11155	0.555774	+	0.831334i	$0.312422\pi$
$642$	0	0
$643$	47.6985	1.88104	0.940522	−	0.339732i	$-0.110336\pi$
$643$	47.6985	1.88104	0.940522	+	0.339732i	$0.110336\pi$
$644$	0	0
$645$	−26.1421	−1.02935
$646$	0	0
$647$	−20.8284	−0.818850	−0.409425	−	0.912344i	$-0.634271\pi$
$647$	−20.8284	−0.818850	−0.409425	+	0.912344i	$0.634271\pi$
$648$	0	0
$649$	23.7157	0.930924
$650$	0	0
$651$	−8.48528	−0.332564
$652$	0	0
$653$	−18.4853	−0.723385	−0.361692	−	0.932297i	$-0.617801\pi$
$653$	−18.4853	−0.723385	−0.361692	+	0.932297i	$0.617801\pi$
$654$	0	0
$655$	13.8995	0.543098
$656$	0	0
$657$	−41.7990	−1.63073
$658$	0	0
$659$	4.68629	0.182552	0.0912760	−	0.995826i	$-0.470905\pi$
$659$	4.68629	0.182552	0.0912760	+	0.995826i	$0.470905\pi$
$660$	0	0
$661$	−13.3137	−0.517843	−0.258922	−	0.965898i	$-0.583367\pi$
$661$	−13.3137	−0.517843	−0.258922	+	0.965898i	$0.583367\pi$
$662$	0	0
$663$	−9.65685	−0.375041
$664$	0	0
$665$	−1.65685	−0.0642501
$666$	0	0
$667$	−16.4853	−0.638313
$668$	0	0
$669$	20.4853	0.792007
$670$	0	0
$671$	−19.7990	−0.764332
$672$	0	0
$673$	−28.1421	−1.08480	−0.542400	−	0.840120i	$-0.682484\pi$
$673$	−28.1421	−1.08480	−0.542400	+	0.840120i	$0.682484\pi$
$674$	0	0
$675$	19.3137	0.743385
$676$	0	0
$677$	−4.62742	−0.177846	−0.0889230	−	0.996038i	$-0.528343\pi$
$677$	−4.62742	−0.177846	−0.0889230	+	0.996038i	$0.528343\pi$
$678$	0	0
$679$	−2.14214	−0.0822076
$680$	0	0
$681$	77.5980	2.97356
$682$	0	0
$683$	14.7279	0.563548	0.281774	−	0.959481i	$-0.409077\pi$
$683$	14.7279	0.563548	0.281774	+	0.959481i	$0.409077\pi$
$684$	0	0
$685$	1.17157	0.0447635
$686$	0	0
$687$	2.34315	0.0893966
$688$	0	0
$689$	−0.970563	−0.0369755
$690$	0	0
$691$	17.2132	0.654821	0.327411	−	0.944882i	$-0.393824\pi$
$691$	17.2132	0.654821	0.327411	+	0.944882i	$0.393824\pi$
$692$	0	0
$693$	−13.1127	−0.498110
$694$	0	0
$695$	−17.8995	−0.678967
$696$	0	0
$697$	6.48528	0.245648
$698$	0	0
$699$	31.7990	1.20275
$700$	0	0
$701$	−30.3431	−1.14604	−0.573022	−	0.819540i	$-0.694229\pi$
$701$	−30.3431	−1.14604	−0.573022	+	0.819540i	$0.694229\pi$
$702$	0	0
$703$	−18.3431	−0.691825
$704$	0	0
$705$	−16.4853	−0.620872
$706$	0	0
$707$	−4.68629	−0.176246
$708$	0	0
$709$	−25.7990	−0.968901	−0.484451	−	0.874819i	$-0.660981\pi$
$709$	−25.7990	−0.968901	−0.484451	+	0.874819i	$0.660981\pi$
$710$	0	0
$711$	46.8701	1.75776
$712$	0	0
$713$	−14.4853	−0.542478
$714$	0	0
$715$	7.31371	0.273517
$716$	0	0
$717$	68.2843	2.55012
$718$	0	0
$719$	−53.4975	−1.99512	−0.997560	−	0.0698205i	$-0.977757\pi$
$719$	−53.4975	−1.99512	−0.997560	+	0.0698205i	$0.977757\pi$
$720$	0	0
$721$	0.485281	0.0180728
$722$	0	0
$723$	43.7990	1.62890
$724$	0	0
$725$	4.82843	0.179323
$726$	0	0
$727$	11.4558	0.424874	0.212437	−	0.977175i	$-0.431860\pi$
$727$	11.4558	0.424874	0.212437	+	0.977175i	$0.431860\pi$
$728$	0	0
$729$	148.196	5.48874
$730$	0	0
$731$	7.65685	0.283199
$732$	0	0
$733$	−46.2843	−1.70955	−0.854774	−	0.519000i	$-0.826304\pi$
$733$	−46.2843	−1.70955	−0.854774	+	0.519000i	$0.826304\pi$
$734$	0	0
$735$	−22.7279	−0.838332
$736$	0	0
$737$	8.20101	0.302088
$738$	0	0
$739$	−27.7990	−1.02260	−0.511301	−	0.859402i	$-0.670836\pi$
$739$	−27.7990	−1.02260	−0.511301	+	0.859402i	$0.670836\pi$
$740$	0	0
$741$	27.3137	1.00339
$742$	0	0
$743$	14.0416	0.515137	0.257569	−	0.966260i	$-0.417079\pi$
$743$	14.0416	0.515137	0.257569	+	0.966260i	$0.417079\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	−80.6274	−2.95000
$748$	0	0
$749$	6.97056	0.254699
$750$	0	0
$751$	−46.1838	−1.68527	−0.842635	−	0.538485i	$-0.818997\pi$
$751$	−46.1838	−1.68527	−0.842635	+	0.538485i	$0.818997\pi$
$752$	0	0
$753$	40.9706	1.49305
$754$	0	0
$755$	7.51472	0.273489
$756$	0	0
$757$	−41.1716	−1.49641	−0.748203	−	0.663470i	$-0.769083\pi$
$757$	−41.1716	−1.49641	−0.748203	+	0.663470i	$0.769083\pi$
$758$	0	0
$759$	−30.1421	−1.09409
$760$	0	0
$761$	12.6863	0.459878	0.229939	−	0.973205i	$-0.426147\pi$
$761$	12.6863	0.459878	0.229939	+	0.973205i	$0.426147\pi$
$762$	0	0
$763$	−10.1421	−0.367170
$764$	0	0
$765$	−8.65685	−0.312989
$766$	0	0
$767$	25.9411	0.936680
$768$	0	0
$769$	−34.3431	−1.23845	−0.619223	−	0.785215i	$-0.712552\pi$
$769$	−34.3431	−1.23845	−0.619223	+	0.785215i	$0.712552\pi$
$770$	0	0
$771$	−9.65685	−0.347783
$772$	0	0
$773$	−27.1127	−0.975176	−0.487588	−	0.873074i	$-0.662123\pi$
$773$	−27.1127	−0.975176	−0.487588	+	0.873074i	$0.662123\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	0	0
$777$	12.9706	0.465316
$778$	0	0
$779$	−18.3431	−0.657211
$780$	0	0
$781$	−10.9706	−0.392558
$782$	0	0
$783$	93.2548	3.33266
$784$	0	0
$785$	21.3137	0.760719
$786$	0	0
$787$	31.2132	1.11263	0.556315	−	0.830971i	$-0.312215\pi$
$787$	31.2132	1.11263	0.556315	+	0.830971i	$0.312215\pi$
$788$	0	0
$789$	31.7990	1.13207
$790$	0	0
$791$	1.85786	0.0660581
$792$	0	0
$793$	−21.6569	−0.769057
$794$	0	0
$795$	−1.17157	−0.0415514
$796$	0	0
$797$	36.6274	1.29741	0.648705	−	0.761040i	$-0.275311\pi$
$797$	36.6274	1.29741	0.648705	+	0.761040i	$0.275311\pi$
$798$	0	0
$799$	4.82843	0.170817
$800$	0	0
$801$	−20.2843	−0.716709
$802$	0	0
$803$	−12.4853	−0.440596
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−64.7696	−2.28000
$808$	0	0
$809$	−31.6569	−1.11300	−0.556498	−	0.830849i	$-0.687855\pi$
$809$	−31.6569	−1.11300	−0.556498	+	0.830849i	$0.687855\pi$
$810$	0	0
$811$	−18.5858	−0.652635	−0.326318	−	0.945260i	$-0.605808\pi$
$811$	−18.5858	−0.652635	−0.326318	+	0.945260i	$0.605808\pi$
$812$	0	0
$813$	−46.6274	−1.63529
$814$	0	0
$815$	−0.585786	−0.0205192
$816$	0	0
$817$	−21.6569	−0.757677
$818$	0	0
$819$	−14.3431	−0.501190
$820$	0	0
$821$	−36.6274	−1.27831	−0.639153	−	0.769080i	$-0.720715\pi$
$821$	−36.6274	−1.27831	−0.639153	+	0.769080i	$0.720715\pi$
$822$	0	0
$823$	−25.0711	−0.873922	−0.436961	−	0.899480i	$-0.643945\pi$
$823$	−25.0711	−0.873922	−0.436961	+	0.899480i	$0.643945\pi$
$824$	0	0
$825$	8.82843	0.307366
$826$	0	0
$827$	29.5563	1.02777	0.513887	−	0.857858i	$-0.328205\pi$
$827$	29.5563	1.02777	0.513887	+	0.857858i	$0.328205\pi$
$828$	0	0
$829$	−39.9411	−1.38721	−0.693606	−	0.720354i	$-0.743979\pi$
$829$	−39.9411	−1.38721	−0.693606	+	0.720354i	$0.743979\pi$
$830$	0	0
$831$	34.1421	1.18438
$832$	0	0
$833$	6.65685	0.230646
$834$	0	0
$835$	2.24264	0.0776098
$836$	0	0
$837$	81.9411	2.83230
$838$	0	0
$839$	42.1838	1.45635	0.728173	−	0.685394i	$-0.240370\pi$
$839$	42.1838	1.45635	0.728173	+	0.685394i	$0.240370\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	−14.8284	−0.510718
$844$	0	0
$845$	−5.00000	−0.172005
$846$	0	0
$847$	2.52691	0.0868257
$848$	0	0
$849$	−48.6274	−1.66889
$850$	0	0
$851$	22.1421	0.759023
$852$	0	0
$853$	31.1716	1.06729	0.533647	−	0.845707i	$-0.320821\pi$
$853$	31.1716	1.06729	0.533647	+	0.845707i	$0.320821\pi$
$854$	0	0
$855$	24.4853	0.837379
$856$	0	0
$857$	13.5147	0.461654	0.230827	−	0.972995i	$-0.425857\pi$
$857$	13.5147	0.461654	0.230827	+	0.972995i	$0.425857\pi$
$858$	0	0
$859$	−36.7696	−1.25456	−0.627280	−	0.778793i	$-0.715832\pi$
$859$	−36.7696	−1.25456	−0.627280	+	0.778793i	$0.715832\pi$
$860$	0	0
$861$	12.9706	0.442036
$862$	0	0
$863$	6.48528	0.220762	0.110381	−	0.993889i	$-0.464793\pi$
$863$	6.48528	0.220762	0.110381	+	0.993889i	$0.464793\pi$
$864$	0	0
$865$	−12.8284	−0.436180
$866$	0	0
$867$	3.41421	0.115953
$868$	0	0
$869$	14.0000	0.474917
$870$	0	0
$871$	8.97056	0.303956
$872$	0	0
$873$	31.6569	1.07142
$874$	0	0
$875$	−0.585786	−0.0198032
$876$	0	0
$877$	2.28427	0.0771344	0.0385672	−	0.999256i	$-0.487721\pi$
$877$	2.28427	0.0771344	0.0385672	+	0.999256i	$0.487721\pi$
$878$	0	0
$879$	−61.4558	−2.07285
$880$	0	0
$881$	48.1421	1.62195	0.810975	−	0.585081i	$-0.198937\pi$
$881$	48.1421	1.62195	0.810975	+	0.585081i	$0.198937\pi$
$882$	0	0
$883$	−15.1716	−0.510564	−0.255282	−	0.966867i	$-0.582168\pi$
$883$	−15.1716	−0.510564	−0.255282	+	0.966867i	$0.582168\pi$
$884$	0	0
$885$	31.3137	1.05260
$886$	0	0
$887$	−14.9289	−0.501264	−0.250632	−	0.968082i	$-0.580638\pi$
$887$	−14.9289	−0.501264	−0.250632	+	0.968082i	$0.580638\pi$
$888$	0	0
$889$	10.1421	0.340156
$890$	0	0
$891$	103.355	3.46253
$892$	0	0
$893$	−13.6569	−0.457009
$894$	0	0
$895$	6.82843	0.228249
$896$	0	0
$897$	−32.9706	−1.10086
$898$	0	0
$899$	20.4853	0.683222
$900$	0	0
$901$	0.343146	0.0114318
$902$	0	0
$903$	15.3137	0.509608
$904$	0	0
$905$	−2.48528	−0.0826135
$906$	0	0
$907$	6.72792	0.223397	0.111698	−	0.993742i	$-0.464371\pi$
$907$	6.72792	0.223397	0.111698	+	0.993742i	$0.464371\pi$
$908$	0	0
$909$	69.2548	2.29704
$910$	0	0
$911$	12.2426	0.405617	0.202808	−	0.979218i	$-0.434993\pi$
$911$	12.2426	0.405617	0.202808	+	0.979218i	$0.434993\pi$
$912$	0	0
$913$	−24.0833	−0.797040
$914$	0	0
$915$	−26.1421	−0.864232
$916$	0	0
$917$	−8.14214	−0.268877
$918$	0	0
$919$	40.9706	1.35149	0.675747	−	0.737134i	$-0.263821\pi$
$919$	40.9706	1.35149	0.675747	+	0.737134i	$0.263821\pi$
$920$	0	0
$921$	57.4558	1.89323
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	−6.48528	−0.213235
$926$	0	0
$927$	−7.17157	−0.235545
$928$	0	0
$929$	35.4558	1.16327	0.581634	−	0.813450i	$-0.302414\pi$
$929$	35.4558	1.16327	0.581634	+	0.813450i	$0.302414\pi$
$930$	0	0
$931$	−18.8284	−0.617077
$932$	0	0
$933$	51.4558	1.68459
$934$	0	0
$935$	−2.58579	−0.0845643
$936$	0	0
$937$	34.2843	1.12002	0.560009	−	0.828486i	$-0.310798\pi$
$937$	34.2843	1.12002	0.560009	+	0.828486i	$0.310798\pi$
$938$	0	0
$939$	−19.7990	−0.646116
$940$	0	0
$941$	3.45584	0.112657	0.0563286	−	0.998412i	$-0.482061\pi$
$941$	3.45584	0.112657	0.0563286	+	0.998412i	$0.482061\pi$
$942$	0	0
$943$	22.1421	0.721047
$944$	0	0
$945$	−11.3137	−0.368035
$946$	0	0
$947$	35.8995	1.16658	0.583288	−	0.812265i	$-0.301766\pi$
$947$	35.8995	1.16658	0.583288	+	0.812265i	$0.301766\pi$
$948$	0	0
$949$	−13.6569	−0.443320
$950$	0	0
$951$	72.7696	2.35971
$952$	0	0
$953$	−21.8579	−0.708046	−0.354023	−	0.935237i	$-0.615186\pi$
$953$	−21.8579	−0.708046	−0.354023	+	0.935237i	$0.615186\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	42.6274	1.37795
$958$	0	0
$959$	−0.686292	−0.0221615
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	−103.012	−3.31952
$964$	0	0
$965$	−20.8284	−0.670491
$966$	0	0
$967$	−1.02944	−0.0331045	−0.0165522	−	0.999863i	$-0.505269\pi$
$967$	−1.02944	−0.0331045	−0.0165522	+	0.999863i	$0.505269\pi$
$968$	0	0
$969$	−9.65685	−0.310223
$970$	0	0
$971$	8.20101	0.263183	0.131591	−	0.991304i	$-0.457991\pi$
$971$	8.20101	0.263183	0.131591	+	0.991304i	$0.457991\pi$
$972$	0	0
$973$	10.4853	0.336143
$974$	0	0
$975$	9.65685	0.309267
$976$	0	0
$977$	59.2548	1.89573	0.947865	−	0.318672i	$-0.103237\pi$
$977$	59.2548	1.89573	0.947865	+	0.318672i	$0.103237\pi$
$978$	0	0
$979$	−6.05887	−0.193642
$980$	0	0
$981$	149.882	4.78537
$982$	0	0
$983$	−41.3553	−1.31903	−0.659515	−	0.751691i	$-0.729238\pi$
$983$	−41.3553	−1.31903	−0.659515	+	0.751691i	$0.729238\pi$
$984$	0	0
$985$	−12.8284	−0.408748
$986$	0	0
$987$	9.65685	0.307381
$988$	0	0
$989$	26.1421	0.831272
$990$	0	0
$991$	33.4142	1.06144	0.530719	−	0.847548i	$-0.321922\pi$
$991$	33.4142	1.06144	0.530719	+	0.847548i	$0.321922\pi$
$992$	0	0
$993$	−91.5980	−2.90677
$994$	0	0
$995$	24.2426	0.768543
$996$	0	0
$997$	−40.1421	−1.27131	−0.635657	−	0.771972i	$-0.719271\pi$
$997$	−40.1421	−1.27131	−0.635657	+	0.771972i	$0.719271\pi$
$998$	0	0
$999$	−125.255	−3.96289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5440.2.a.bm.1.2		2
4.3	odd	2		5440.2.a.ba.1.1		2
8.3	odd	2		1360.2.a.o.1.2		2
8.5	even	2		85.2.a.b.1.2	✓	2
24.5	odd	2		765.2.a.i.1.1		2
40.13	odd	4		425.2.b.e.324.2		4
40.19	odd	2		6800.2.a.ba.1.1		2
40.29	even	2		425.2.a.f.1.1		2
40.37	odd	4		425.2.b.e.324.3		4
56.13	odd	2		4165.2.a.q.1.2		2
120.29	odd	2		3825.2.a.p.1.2		2
136.13	even	4		1445.2.d.f.866.1		4
136.21	even	4		1445.2.d.f.866.2		4
136.101	even	2		1445.2.a.f.1.2		2
680.509	even	2		7225.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.b.1.2	✓	2	8.5	even	2
425.2.a.f.1.1		2	40.29	even	2
425.2.b.e.324.2		4	40.13	odd	4
425.2.b.e.324.3		4	40.37	odd	4
765.2.a.i.1.1		2	24.5	odd	2
1360.2.a.o.1.2		2	8.3	odd	2
1445.2.a.f.1.2		2	136.101	even	2
1445.2.d.f.866.1		4	136.13	even	4
1445.2.d.f.866.2		4	136.21	even	4
3825.2.a.p.1.2		2	120.29	odd	2
4165.2.a.q.1.2		2	56.13	odd	2
5440.2.a.ba.1.1		2	4.3	odd	2
5440.2.a.bm.1.2		2	1.1	even	1	trivial
6800.2.a.ba.1.1		2	40.19	odd	2
7225.2.a.o.1.1		2	680.509	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:	\( N \)	\(=\)	\( 5440 = 2^{6} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5440.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41421 q^{3} +1.00000 q^{5} -0.585786 q^{7} +8.65685 q^{9} +2.58579 q^{11} +2.82843 q^{13} +3.41421 q^{15} -1.00000 q^{17} +2.82843 q^{19} -2.00000 q^{21} -3.41421 q^{23} +1.00000 q^{25} +19.3137 q^{27} +4.82843 q^{29} +4.24264 q^{31} +8.82843 q^{33} -0.585786 q^{35} -6.48528 q^{37} +9.65685 q^{39} -6.48528 q^{41} -7.65685 q^{43} +8.65685 q^{45} -4.82843 q^{47} -6.65685 q^{49} -3.41421 q^{51} -0.343146 q^{53} +2.58579 q^{55} +9.65685 q^{57} +9.17157 q^{59} -7.65685 q^{61} -5.07107 q^{63} +2.82843 q^{65} +3.17157 q^{67} -11.6569 q^{69} -4.24264 q^{71} -4.82843 q^{73} +3.41421 q^{75} -1.51472 q^{77} +5.41421 q^{79} +39.9706 q^{81} -9.31371 q^{83} -1.00000 q^{85} +16.4853 q^{87} -2.34315 q^{89} -1.65685 q^{91} +14.4853 q^{93} +2.82843 q^{95} +3.65685 q^{97} +22.3848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{5} - 4 q^{7} + 6 q^{9} + 8 q^{11} + 4 q^{15} - 2 q^{17} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 16 q^{27} + 4 q^{29} + 12 q^{33} - 4 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 4 q^{43} + 6 q^{45}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^5 - 4 * q^7 + 6 * q^9 + 8 * q^11 + 4 * q^15 - 2 * q^17 - 4 * q^21 - 4 * q^23 + 2 * q^25 + 16 * q^27 + 4 * q^29 + 12 * q^33 - 4 * q^35 + 4 * q^37 + 8 * q^39 + 4 * q^41 - 4 * q^43 + 6 * q^45 - 4 * q^47 - 2 * q^49 - 4 * q^51 - 12 * q^53 + 8 * q^55 + 8 * q^57 + 24 * q^59 - 4 * q^61 + 4 * q^63 + 12 * q^67 - 12 * q^69 - 4 * q^73 + 4 * q^75 - 20 * q^77 + 8 * q^79 + 46 * q^81 + 4 * q^83 - 2 * q^85 + 16 * q^87 - 16 * q^89 + 8 * q^91 + 12 * q^93 - 4 * q^97 + 8 * q^99

Introduction

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