LMFDB - Embedded newform 720.6.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("720.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.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:	$115.476350265$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3289}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 822 $ x^2 - x - 822
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$29.1749$ of defining polynomial
Character	$\chi$	$=$	720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-25.0000 q^{5} -74.6996 q^{7} +O(q^{10})$	$q-25.0000 q^{5} -74.6996 q^{7} -25.3004 q^{11} -448.198 q^{13} +1221.00 q^{17} +1171.00 q^{19} +677.992 q^{23} +625.000 q^{25} +1128.80 q^{29} +926.008 q^{31} +1867.49 q^{35} +5267.77 q^{37} +1124.38 q^{41} +7386.01 q^{43} -10839.9 q^{47} -11227.0 q^{49} -23861.9 q^{53} +632.510 q^{55} +20763.8 q^{59} -28095.9 q^{61} +11204.9 q^{65} +62641.3 q^{67} -20433.4 q^{71} -20143.3 q^{73} +1889.93 q^{77} +10538.1 q^{79} -38835.0 q^{83} -30524.9 q^{85} -21789.9 q^{89} +33480.2 q^{91} -29274.9 q^{95} -71881.7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{5} + 80 q^{7}+O(q^{10}) $ 2 * q - 50 * q^5 + 80 * q^7	$ 2 q - 50 q^{5} + 80 q^{7} - 280 q^{11} + 480 q^{13} + 148 q^{17} + 48 q^{19} - 3232 q^{23} + 1250 q^{25} + 1340 q^{29} + 6440 q^{31} - 2000 q^{35} - 6440 q^{37} - 9680 q^{41} + 19360 q^{43} + 15024 q^{47} - 4102 q^{49} - 1844 q^{53} + 7000 q^{55} - 12840 q^{59} - 24076 q^{61} - 12000 q^{65} + 42240 q^{67} - 12880 q^{71} - 96260 q^{73} - 37512 q^{77} + 89896 q^{79} - 57024 q^{83} - 3700 q^{85} + 25240 q^{89} + 177072 q^{91} - 1200 q^{95} - 229100 q^{97}+O(q^{100}) $ 2 * q - 50 * q^5 + 80 * q^7 - 280 * q^11 + 480 * q^13 + 148 * q^17 + 48 * q^19 - 3232 * q^23 + 1250 * q^25 + 1340 * q^29 + 6440 * q^31 - 2000 * q^35 - 6440 * q^37 - 9680 * q^41 + 19360 * q^43 + 15024 * q^47 - 4102 * q^49 - 1844 * q^53 + 7000 * q^55 - 12840 * q^59 - 24076 * q^61 - 12000 * q^65 + 42240 * q^67 - 12880 * q^71 - 96260 * q^73 - 37512 * q^77 + 89896 * q^79 - 57024 * q^83 - 3700 * q^85 + 25240 * q^89 + 177072 * q^91 - 1200 * q^95 - 229100 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−74.6996	−0.576200	−0.288100	−	0.957600i	$-0.593024\pi$
$7$	−74.6996	−0.576200	−0.288100	+	0.957600i	$0.593024\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−25.3004	−0.0630443	−0.0315221	−	0.999503i	$-0.510035\pi$
$11$	−25.3004	−0.0630443	−0.0315221	+	0.999503i	$0.510035\pi$
$12$	0	0
$13$	−448.198	−0.735548	−0.367774	−	0.929915i	$-0.619880\pi$
$13$	−448.198	−0.735548	−0.367774	+	0.929915i	$0.619880\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1221.00	1.02469	0.512344	−	0.858780i	$-0.328777\pi$
$17$	1221.00	1.02469	0.512344	+	0.858780i	$0.328777\pi$
$18$	0	0
$19$	1171.00	0.744169	0.372084	−	0.928199i	$-0.378643\pi$
$19$	1171.00	0.744169	0.372084	+	0.928199i	$0.378643\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	677.992	0.267242	0.133621	−	0.991032i	$-0.457339\pi$
$23$	677.992	0.267242	0.133621	+	0.991032i	$0.457339\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1128.80	0.249242	0.124621	−	0.992204i	$-0.460228\pi$
$29$	1128.80	0.249242	0.124621	+	0.992204i	$0.460228\pi$
$30$	0	0
$31$	926.008	0.173065	0.0865327	−	0.996249i	$-0.472421\pi$
$31$	926.008	0.173065	0.0865327	+	0.996249i	$0.472421\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1867.49	0.257684
$36$	0	0
$37$	5267.77	0.632590	0.316295	−	0.948661i	$-0.397561\pi$
$37$	5267.77	0.632590	0.316295	+	0.948661i	$0.397561\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1124.38	0.104461	0.0522304	−	0.998635i	$-0.483367\pi$
$41$	1124.38	0.104461	0.0522304	+	0.998635i	$0.483367\pi$
$42$	0	0
$43$	7386.01	0.609170	0.304585	−	0.952485i	$-0.401482\pi$
$43$	7386.01	0.609170	0.304585	+	0.952485i	$0.401482\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10839.9	−0.715784	−0.357892	−	0.933763i	$-0.616504\pi$
$47$	−10839.9	−0.715784	−0.357892	+	0.933763i	$0.616504\pi$
$48$	0	0
$49$	−11227.0	−0.667994
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−23861.9	−1.16685	−0.583426	−	0.812166i	$-0.698288\pi$
$53$	−23861.9	−1.16685	−0.583426	+	0.812166i	$0.698288\pi$
$54$	0	0
$55$	632.510	0.0281942
$56$	0	0
$57$	0	0
$58$	0	0
$59$	20763.8	0.776563	0.388282	−	0.921541i	$-0.373069\pi$
$59$	20763.8	0.776563	0.388282	+	0.921541i	$0.373069\pi$
$60$	0	0
$61$	−28095.9	−0.966761	−0.483380	−	0.875410i	$-0.660591\pi$
$61$	−28095.9	−0.966761	−0.483380	+	0.875410i	$0.660591\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11204.9	0.328947
$66$	0	0
$67$	62641.3	1.70480	0.852400	−	0.522890i	$-0.175146\pi$
$67$	62641.3	1.70480	0.852400	+	0.522890i	$0.175146\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−20433.4	−0.481054	−0.240527	−	0.970642i	$-0.577320\pi$
$71$	−20433.4	−0.481054	−0.240527	+	0.970642i	$0.577320\pi$
$72$	0	0
$73$	−20143.3	−0.442408	−0.221204	−	0.975228i	$-0.570999\pi$
$73$	−20143.3	−0.442408	−0.221204	+	0.975228i	$0.570999\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1889.93	0.0363261
$78$	0	0
$79$	10538.1	0.189975	0.0949873	−	0.995478i	$-0.469719\pi$
$79$	10538.1	0.189975	0.0949873	+	0.995478i	$0.469719\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−38835.0	−0.618768	−0.309384	−	0.950937i	$-0.600123\pi$
$83$	−38835.0	−0.618768	−0.309384	+	0.950937i	$0.600123\pi$
$84$	0	0
$85$	−30524.9	−0.458255
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−21789.9	−0.291595	−0.145797	−	0.989314i	$-0.546575\pi$
$89$	−21789.9	−0.291595	−0.145797	+	0.989314i	$0.546575\pi$
$90$	0	0
$91$	33480.2	0.423823
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−29274.9	−0.332802
$96$	0	0
$97$	−71881.7	−0.775692	−0.387846	−	0.921724i	$-0.626781\pi$
$97$	−71881.7	−0.775692	−0.387846	+	0.921724i	$0.626781\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−90639.2	−0.884123	−0.442062	−	0.896985i	$-0.645753\pi$
$101$	−90639.2	−0.884123	−0.442062	+	0.896985i	$0.645753\pi$
$102$	0	0
$103$	120231.	1.11667	0.558333	−	0.829617i	$-0.311441\pi$
$103$	120231.	1.11667	0.558333	+	0.829617i	$0.311441\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−78240.9	−0.660654	−0.330327	−	0.943866i	$-0.607159\pi$
$107$	−78240.9	−0.660654	−0.330327	+	0.943866i	$0.607159\pi$
$108$	0	0
$109$	−141930.	−1.14422	−0.572108	−	0.820178i	$-0.693874\pi$
$109$	−141930.	−1.14422	−0.572108	+	0.820178i	$0.693874\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−21135.4	−0.155709	−0.0778547	−	0.996965i	$-0.524807\pi$
$113$	−21135.4	−0.155709	−0.0778547	+	0.996965i	$0.524807\pi$
$114$	0	0
$115$	−16949.8	−0.119514
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−91207.9	−0.590426
$120$	0	0
$121$	−160411.	−0.996025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	27184.3	0.149557	0.0747787	−	0.997200i	$-0.476175\pi$
$127$	27184.3	0.149557	0.0747787	+	0.997200i	$0.476175\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	41207.9	0.209798	0.104899	−	0.994483i	$-0.466548\pi$
$131$	41207.9	0.209798	0.104899	+	0.994483i	$0.466548\pi$
$132$	0	0
$133$	−87472.9	−0.428790
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−86849.2	−0.395334	−0.197667	−	0.980269i	$-0.563337\pi$
$137$	−86849.2	−0.395334	−0.197667	+	0.980269i	$0.563337\pi$
$138$	0	0
$139$	199787.	0.877062	0.438531	−	0.898716i	$-0.355499\pi$
$139$	199787.	0.877062	0.438531	+	0.898716i	$0.355499\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11339.6	0.0463721
$144$	0	0
$145$	−28220.0	−0.111464
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−145415.	−0.536590	−0.268295	−	0.963337i	$-0.586460\pi$
$149$	−145415.	−0.536590	−0.268295	+	0.963337i	$0.586460\pi$
$150$	0	0
$151$	198940.	0.710036	0.355018	−	0.934860i	$-0.384475\pi$
$151$	198940.	0.710036	0.355018	+	0.934860i	$0.384475\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−23150.2	−0.0773972
$156$	0	0
$157$	−95743.6	−0.309999	−0.155000	−	0.987915i	$-0.549538\pi$
$157$	−95743.6	−0.309999	−0.155000	+	0.987915i	$0.549538\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−50645.7	−0.153985
$162$	0	0
$163$	254491.	0.750246	0.375123	−	0.926975i	$-0.377600\pi$
$163$	254491.	0.750246	0.375123	+	0.926975i	$0.377600\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	26817.1	0.0744081	0.0372040	−	0.999308i	$-0.488155\pi$
$167$	26817.1	0.0744081	0.0372040	+	0.999308i	$0.488155\pi$
$168$	0	0
$169$	−170412.	−0.458969
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−353695.	−0.898492	−0.449246	−	0.893408i	$-0.648307\pi$
$173$	−353695.	−0.898492	−0.449246	+	0.893408i	$0.648307\pi$
$174$	0	0
$175$	−46687.3	−0.115240
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−822134.	−1.91783	−0.958914	−	0.283696i	$-0.908439\pi$
$179$	−822134.	−1.91783	−0.958914	+	0.283696i	$0.908439\pi$
$180$	0	0
$181$	336336.	0.763093	0.381547	−	0.924350i	$-0.375392\pi$
$181$	336336.	0.763093	0.381547	+	0.924350i	$0.375392\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−131694.	−0.282903
$186$	0	0
$187$	−30891.7	−0.0646007
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−717915.	−1.42393	−0.711967	−	0.702213i	$-0.752195\pi$
$191$	−717915.	−1.42393	−0.711967	+	0.702213i	$0.752195\pi$
$192$	0	0
$193$	546630.	1.05633	0.528166	−	0.849141i	$-0.322880\pi$
$193$	546630.	1.05633	0.528166	+	0.849141i	$0.322880\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−676434.	−1.24182	−0.620912	−	0.783881i	$-0.713237\pi$
$197$	−676434.	−1.24182	−0.620912	+	0.783881i	$0.713237\pi$
$198$	0	0
$199$	−796314.	−1.42545	−0.712725	−	0.701444i	$-0.752539\pi$
$199$	−796314.	−1.42545	−0.712725	+	0.701444i	$0.752539\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−84320.8	−0.143613
$204$	0	0
$205$	−28109.5	−0.0467163
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−29626.7	−0.0469156
$210$	0	0
$211$	14528.4	0.0224653	0.0112327	−	0.999937i	$-0.496424\pi$
$211$	14528.4	0.0224653	0.0112327	+	0.999937i	$0.496424\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−184650.	−0.272429
$216$	0	0
$217$	−69172.4	−0.0997203
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−547248.	−0.753708
$222$	0	0
$223$	−246619.	−0.332097	−0.166048	−	0.986118i	$-0.553101\pi$
$223$	−246619.	−0.332097	−0.166048	+	0.986118i	$0.553101\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−654789.	−0.843406	−0.421703	−	0.906734i	$-0.638568\pi$
$227$	−654789.	−0.843406	−0.421703	+	0.906734i	$0.638568\pi$
$228$	0	0
$229$	347625.	0.438049	0.219024	−	0.975719i	$-0.429713\pi$
$229$	347625.	0.438049	0.219024	+	0.975719i	$0.429713\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−789281.	−0.952450	−0.476225	−	0.879324i	$-0.657995\pi$
$233$	−789281.	−0.952450	−0.476225	+	0.879324i	$0.657995\pi$
$234$	0	0
$235$	270998.	0.320108
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−700146.	−0.792855	−0.396428	−	0.918066i	$-0.629750\pi$
$239$	−700146.	−0.792855	−0.396428	+	0.918066i	$0.629750\pi$
$240$	0	0
$241$	612669.	0.679491	0.339745	−	0.940517i	$-0.389659\pi$
$241$	612669.	0.679491	0.339745	+	0.940517i	$0.389659\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	280674.	0.298736
$246$	0	0
$247$	−524838.	−0.547372
$248$	0	0
$249$	0	0
$250$	0	0
$251$	802792.	0.804301	0.402150	−	0.915574i	$-0.368263\pi$
$251$	802792.	0.804301	0.402150	+	0.915574i	$0.368263\pi$
$252$	0	0
$253$	−17153.5	−0.0168481
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−675524.	−0.637981	−0.318991	−	0.947758i	$-0.603344\pi$
$257$	−675524.	−0.637981	−0.318991	+	0.947758i	$0.603344\pi$
$258$	0	0
$259$	−393500.	−0.364499
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−648975.	−0.578547	−0.289273	−	0.957247i	$-0.593414\pi$
$263$	−648975.	−0.578547	−0.289273	+	0.957247i	$0.593414\pi$
$264$	0	0
$265$	596548.	0.521832
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.04196e6	0.877950	0.438975	−	0.898499i	$-0.355342\pi$
$269$	1.04196e6	0.877950	0.438975	+	0.898499i	$0.355342\pi$
$270$	0	0
$271$	−1.07414e6	−0.888456	−0.444228	−	0.895914i	$-0.646522\pi$
$271$	−1.07414e6	−0.888456	−0.444228	+	0.895914i	$0.646522\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15812.7	−0.0126089
$276$	0	0
$277$	1.62886e6	1.27551	0.637754	−	0.770240i	$-0.279864\pi$
$277$	1.62886e6	1.27551	0.637754	+	0.770240i	$0.279864\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.90862e6	−1.44196	−0.720981	−	0.692955i	$-0.756309\pi$
$281$	−1.90862e6	−1.44196	−0.720981	+	0.692955i	$0.756309\pi$
$282$	0	0
$283$	−1.45146e6	−1.07730	−0.538651	−	0.842529i	$-0.681066\pi$
$283$	−1.45146e6	−1.07730	−0.538651	+	0.842529i	$0.681066\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−83990.7	−0.0601903
$288$	0	0
$289$	70974.4	0.0499870
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.50664e6	−1.02528	−0.512638	−	0.858605i	$-0.671332\pi$
$293$	−1.50664e6	−1.02528	−0.512638	+	0.858605i	$0.671332\pi$
$294$	0	0
$295$	−519095.	−0.347290
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−303874.	−0.196570
$300$	0	0
$301$	−551732.	−0.351004
$302$	0	0
$303$	0	0
$304$	0	0
$305$	702399.	0.432349
$306$	0	0
$307$	−1.26102e6	−0.763619	−0.381809	−	0.924241i	$-0.624699\pi$
$307$	−1.26102e6	−0.763619	−0.381809	+	0.924241i	$0.624699\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.30506e6	−0.765117	−0.382559	−	0.923931i	$-0.624957\pi$
$311$	−1.30506e6	−0.765117	−0.382559	+	0.923931i	$0.624957\pi$
$312$	0	0
$313$	1.13652e6	0.655715	0.327857	−	0.944727i	$-0.393673\pi$
$313$	1.13652e6	0.655715	0.327857	+	0.944727i	$0.393673\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	805162.	0.450024	0.225012	−	0.974356i	$-0.427758\pi$
$317$	805162.	0.450024	0.225012	+	0.974356i	$0.427758\pi$
$318$	0	0
$319$	−28559.0	−0.0157133
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.42978e6	0.762541
$324$	0	0
$325$	−280124.	−0.147110
$326$	0	0
$327$	0	0
$328$	0	0
$329$	809739.	0.412435
$330$	0	0
$331$	−1.91803e6	−0.962243	−0.481122	−	0.876654i	$-0.659770\pi$
$331$	−1.91803e6	−0.962243	−0.481122	+	0.876654i	$0.659770\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.56603e6	−0.762410
$336$	0	0
$337$	−581867.	−0.279093	−0.139546	−	0.990216i	$-0.544564\pi$
$337$	−581867.	−0.279093	−0.139546	+	0.990216i	$0.544564\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−23428.4	−0.0109108
$342$	0	0
$343$	2.09413e6	0.961098
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−533148.	−0.237697	−0.118849	−	0.992912i	$-0.537920\pi$
$347$	−533148.	−0.237697	−0.118849	+	0.992912i	$0.537920\pi$
$348$	0	0
$349$	1.94263e6	0.853743	0.426871	−	0.904312i	$-0.359616\pi$
$349$	1.94263e6	0.853743	0.426871	+	0.904312i	$0.359616\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.54441e6	−1.08680	−0.543400	−	0.839474i	$-0.682863\pi$
$353$	−2.54441e6	−1.08680	−0.543400	+	0.839474i	$0.682863\pi$
$354$	0	0
$355$	510834.	0.215134
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.44239e6	−1.40969	−0.704846	−	0.709361i	$-0.748984\pi$
$359$	−3.44239e6	−1.40969	−0.704846	+	0.709361i	$0.748984\pi$
$360$	0	0
$361$	−1.10487e6	−0.446213
$362$	0	0
$363$	0	0
$364$	0	0
$365$	503582.	0.197851
$366$	0	0
$367$	1.00984e6	0.391370	0.195685	−	0.980667i	$-0.437307\pi$
$367$	1.00984e6	0.391370	0.195685	+	0.980667i	$0.437307\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.78248e6	0.672340
$372$	0	0
$373$	581043.	0.216240	0.108120	−	0.994138i	$-0.465517\pi$
$373$	581043.	0.216240	0.108120	+	0.994138i	$0.465517\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−505925.	−0.183330
$378$	0	0
$379$	2.78629e6	0.996386	0.498193	−	0.867066i	$-0.333997\pi$
$379$	2.78629e6	0.996386	0.498193	+	0.867066i	$0.333997\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.03659e6	1.40610	0.703052	−	0.711138i	$-0.251820\pi$
$383$	4.03659e6	1.40610	0.703052	+	0.711138i	$0.251820\pi$
$384$	0	0
$385$	−47248.2	−0.0162455
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.33212e6	0.446343	0.223172	−	0.974779i	$-0.428359\pi$
$389$	1.33212e6	0.446343	0.223172	+	0.974779i	$0.428359\pi$
$390$	0	0
$391$	827826.	0.273840
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−263453.	−0.0849592
$396$	0	0
$397$	−2.44389e6	−0.778225	−0.389113	−	0.921190i	$-0.627218\pi$
$397$	−2.44389e6	−0.778225	−0.389113	+	0.921190i	$0.627218\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.45171e6	0.450836	0.225418	−	0.974262i	$-0.427625\pi$
$401$	1.45171e6	0.450836	0.225418	+	0.974262i	$0.427625\pi$
$402$	0	0
$403$	−415035.	−0.127298
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−133277.	−0.0398812
$408$	0	0
$409$	1.01216e6	0.299185	0.149593	−	0.988748i	$-0.452204\pi$
$409$	1.01216e6	0.299185	0.149593	+	0.988748i	$0.452204\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.55105e6	−0.447456
$414$	0	0
$415$	970874.	0.276721
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.49465e6	−0.972453	−0.486226	−	0.873833i	$-0.661627\pi$
$419$	−3.49465e6	−0.972453	−0.486226	+	0.873833i	$0.661627\pi$
$420$	0	0
$421$	−3.06712e6	−0.843384	−0.421692	−	0.906739i	$-0.638564\pi$
$421$	−3.06712e6	−0.843384	−0.421692	+	0.906739i	$0.638564\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	763123.	0.204938
$426$	0	0
$427$	2.09876e6	0.557048
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.62936e6	0.941103	0.470552	−	0.882373i	$-0.344055\pi$
$431$	3.62936e6	0.941103	0.470552	+	0.882373i	$0.344055\pi$
$432$	0	0
$433$	−2.81354e6	−0.721162	−0.360581	−	0.932728i	$-0.617422\pi$
$433$	−2.81354e6	−0.721162	−0.360581	+	0.932728i	$0.617422\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	793926.	0.198873
$438$	0	0
$439$	730749.	0.180970	0.0904851	−	0.995898i	$-0.471158\pi$
$439$	730749.	0.180970	0.0904851	+	0.995898i	$0.471158\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.95201e6	1.68306	0.841532	−	0.540207i	$-0.181654\pi$
$443$	6.95201e6	1.68306	0.841532	+	0.540207i	$0.181654\pi$
$444$	0	0
$445$	544747.	0.130405
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.62997e6	1.78610	0.893052	−	0.449953i	$-0.148559\pi$
$449$	7.62997e6	1.78610	0.893052	+	0.449953i	$0.148559\pi$
$450$	0	0
$451$	−28447.2	−0.00658565
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−837005.	−0.189539
$456$	0	0
$457$	−2.02843e6	−0.454329	−0.227165	−	0.973856i	$-0.572946\pi$
$457$	−2.02843e6	−0.454329	−0.227165	+	0.973856i	$0.572946\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.97995e6	−0.653065	−0.326533	−	0.945186i	$-0.605880\pi$
$461$	−2.97995e6	−0.653065	−0.326533	+	0.945186i	$0.605880\pi$
$462$	0	0
$463$	4.58348e6	0.993673	0.496836	−	0.867844i	$-0.334495\pi$
$463$	4.58348e6	0.993673	0.496836	+	0.867844i	$0.334495\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.56294e6	1.39254	0.696268	−	0.717782i	$-0.254842\pi$
$467$	6.56294e6	1.39254	0.696268	+	0.717782i	$0.254842\pi$
$468$	0	0
$469$	−4.67928e6	−0.982306
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−186869.	−0.0384047
$474$	0	0
$475$	731873.	0.148834
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.34262e6	−1.06394	−0.531968	−	0.846764i	$-0.678547\pi$
$479$	−5.34262e6	−1.06394	−0.531968	+	0.846764i	$0.678547\pi$
$480$	0	0
$481$	−2.36100e6	−0.465301
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.79704e6	0.346900
$486$	0	0
$487$	70213.5	0.0134152	0.00670761	−	0.999978i	$-0.497865\pi$
$487$	70213.5	0.0134152	0.00670761	+	0.999978i	$0.497865\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.82867e6	1.27830	0.639149	−	0.769083i	$-0.279287\pi$
$491$	6.82867e6	1.27830	0.639149	+	0.769083i	$0.279287\pi$
$492$	0	0
$493$	1.37826e6	0.255396
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.52636e6	0.277183
$498$	0	0
$499$	3.83899e6	0.690186	0.345093	−	0.938568i	$-0.387847\pi$
$499$	3.83899e6	0.690186	0.345093	+	0.938568i	$0.387847\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	353546.	0.0623054	0.0311527	−	0.999515i	$-0.490082\pi$
$503$	353546.	0.0623054	0.0311527	+	0.999515i	$0.490082\pi$
$504$	0	0
$505$	2.26598e6	0.395392
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.12553e7	1.92559	0.962795	−	0.270234i	$-0.0871011\pi$
$509$	1.12553e7	1.92559	0.962795	+	0.270234i	$0.0871011\pi$
$510$	0	0
$511$	1.50470e6	0.254916
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.00577e6	−0.499388
$516$	0	0
$517$	274255.	0.0451261
$518$	0	0
$519$	0	0
$520$	0	0
$521$	914030.	0.147525	0.0737626	−	0.997276i	$-0.476499\pi$
$521$	914030.	0.147525	0.0737626	+	0.997276i	$0.476499\pi$
$522$	0	0
$523$	5.94873e6	0.950976	0.475488	−	0.879722i	$-0.342271\pi$
$523$	5.94873e6	0.950976	0.475488	+	0.879722i	$0.342271\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.13065e6	0.177338
$528$	0	0
$529$	−5.97667e6	−0.928582
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−503944.	−0.0768360
$534$	0	0
$535$	1.95602e6	0.295454
$536$	0	0
$537$	0	0
$538$	0	0
$539$	284047.	0.0421132
$540$	0	0
$541$	−9.84475e6	−1.44614	−0.723072	−	0.690773i	$-0.757271\pi$
$541$	−9.84475e6	−1.44614	−0.723072	+	0.690773i	$0.757271\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.54825e6	0.511709
$546$	0	0
$547$	−3.96367e6	−0.566407	−0.283204	−	0.959060i	$-0.591397\pi$
$547$	−3.96367e6	−0.566407	−0.283204	+	0.959060i	$0.591397\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.32182e6	0.185478
$552$	0	0
$553$	−787193.	−0.109463
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.63137e6	−0.495943	−0.247972	−	0.968767i	$-0.579764\pi$
$557$	−3.63137e6	−0.495943	−0.247972	+	0.968767i	$0.579764\pi$
$558$	0	0
$559$	−3.31039e6	−0.448074
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.35366e7	1.79986	0.899929	−	0.436036i	$-0.143618\pi$
$563$	1.35366e7	1.79986	0.899929	+	0.436036i	$0.143618\pi$
$564$	0	0
$565$	528385.	0.0696353
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.29336e7	1.67471	0.837357	−	0.546657i	$-0.184100\pi$
$569$	1.29336e7	1.67471	0.837357	+	0.546657i	$0.184100\pi$
$570$	0	0
$571$	−3.72754e6	−0.478445	−0.239222	−	0.970965i	$-0.576892\pi$
$571$	−3.72754e6	−0.478445	−0.239222	+	0.970965i	$0.576892\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	423745.	0.0534484
$576$	0	0
$577$	−8.75619e6	−1.09490	−0.547452	−	0.836837i	$-0.684402\pi$
$577$	−8.75619e6	−1.09490	−0.547452	+	0.836837i	$0.684402\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.90096e6	0.356534
$582$	0	0
$583$	603716.	0.0735633
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.43946e7	1.72427	0.862133	−	0.506683i	$-0.169128\pi$
$587$	1.43946e7	1.72427	0.862133	+	0.506683i	$0.169128\pi$
$588$	0	0
$589$	1.08435e6	0.128790
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.92954e6	−0.926000	−0.463000	−	0.886358i	$-0.653227\pi$
$593$	−7.92954e6	−0.926000	−0.463000	+	0.886358i	$0.653227\pi$
$594$	0	0
$595$	2.28020e6	0.264046
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.06787e7	−1.21605	−0.608027	−	0.793916i	$-0.708039\pi$
$599$	−1.06787e7	−1.21605	−0.608027	+	0.793916i	$0.708039\pi$
$600$	0	0
$601$	1.53468e7	1.73313	0.866566	−	0.499063i	$-0.166322\pi$
$601$	1.53468e7	1.73313	0.866566	+	0.499063i	$0.166322\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.01027e6	0.445436
$606$	0	0
$607$	−2.00312e6	−0.220666	−0.110333	−	0.993895i	$-0.535192\pi$
$607$	−2.00312e6	−0.220666	−0.110333	+	0.993895i	$0.535192\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.85843e6	0.526494
$612$	0	0
$613$	1.47512e7	1.58554	0.792771	−	0.609520i	$-0.208638\pi$
$613$	1.47512e7	1.58554	0.792771	+	0.609520i	$0.208638\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.52506e6	0.478533	0.239266	−	0.970954i	$-0.423093\pi$
$617$	4.52506e6	0.478533	0.239266	+	0.970954i	$0.423093\pi$
$618$	0	0
$619$	−1.68139e7	−1.76377	−0.881886	−	0.471463i	$-0.843726\pi$
$619$	−1.68139e7	−1.76377	−0.881886	+	0.471463i	$0.843726\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.62770e6	0.168017
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.43193e6	0.648208
$630$	0	0
$631$	−3.37631e6	−0.337574	−0.168787	−	0.985653i	$-0.553985\pi$
$631$	−3.37631e6	−0.337574	−0.168787	+	0.985653i	$0.553985\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−679606.	−0.0668841
$636$	0	0
$637$	5.03190e6	0.491342
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.01525e7	−0.975954	−0.487977	−	0.872856i	$-0.662265\pi$
$641$	−1.01525e7	−0.975954	−0.487977	+	0.872856i	$0.662265\pi$
$642$	0	0
$643$	7.28200e6	0.694582	0.347291	−	0.937757i	$-0.387102\pi$
$643$	7.28200e6	0.694582	0.347291	+	0.937757i	$0.387102\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.68876e7	−1.58602	−0.793009	−	0.609209i	$-0.791487\pi$
$647$	−1.68876e7	−1.58602	−0.793009	+	0.609209i	$0.791487\pi$
$648$	0	0
$649$	−525332.	−0.0489579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.61969e7	−1.48644	−0.743221	−	0.669046i	$-0.766703\pi$
$653$	−1.61969e7	−1.48644	−0.743221	+	0.669046i	$0.766703\pi$
$654$	0	0
$655$	−1.03020e6	−0.0938247
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.40541e6	0.574558	0.287279	−	0.957847i	$-0.407249\pi$
$659$	6.40541e6	0.574558	0.287279	+	0.957847i	$0.407249\pi$
$660$	0	0
$661$	−1.00053e7	−0.890689	−0.445345	−	0.895359i	$-0.646919\pi$
$661$	−1.00053e7	−0.890689	−0.445345	+	0.895359i	$0.646919\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.18682e6	0.191761
$666$	0	0
$667$	765316.	0.0666080
$668$	0	0
$669$	0	0
$670$	0	0
$671$	710838.	0.0609487
$672$	0	0
$673$	2.05563e7	1.74947	0.874737	−	0.484598i	$-0.161034\pi$
$673$	2.05563e7	1.74947	0.874737	+	0.484598i	$0.161034\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.73748e6	0.648825	0.324413	−	0.945916i	$-0.394833\pi$
$677$	7.73748e6	0.648825	0.324413	+	0.945916i	$0.394833\pi$
$678$	0	0
$679$	5.36954e6	0.446954
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.71602e7	1.40757	0.703784	−	0.710414i	$-0.251492\pi$
$683$	1.71602e7	1.40757	0.703784	+	0.710414i	$0.251492\pi$
$684$	0	0
$685$	2.17123e6	0.176799
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.06949e7	0.858276
$690$	0	0
$691$	−1.17727e7	−0.937952	−0.468976	−	0.883211i	$-0.655377\pi$
$691$	−1.17727e7	−0.937952	−0.468976	+	0.883211i	$0.655377\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.99467e6	−0.392234
$696$	0	0
$697$	1.37286e6	0.107040
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−862109.	−0.0662624	−0.0331312	−	0.999451i	$-0.510548\pi$
$701$	−862109.	−0.0662624	−0.0331312	+	0.999451i	$0.510548\pi$
$702$	0	0
$703$	6.16854e6	0.470754
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.77071e6	0.509432
$708$	0	0
$709$	1.51814e7	1.13422	0.567109	−	0.823643i	$-0.308062\pi$
$709$	1.51814e7	1.13422	0.567109	+	0.823643i	$0.308062\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	627826.	0.0462504
$714$	0	0
$715$	−283489.	−0.0207382
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.21811e7	−1.60015	−0.800075	−	0.599900i	$-0.795207\pi$
$719$	−2.21811e7	−1.60015	−0.800075	+	0.599900i	$0.795207\pi$
$720$	0	0
$721$	−8.98121e6	−0.643423
$722$	0	0
$723$	0	0
$724$	0	0
$725$	705499.	0.0498484
$726$	0	0
$727$	−2.24588e7	−1.57598	−0.787991	−	0.615687i	$-0.788879\pi$
$727$	−2.24588e7	−1.57598	−0.787991	+	0.615687i	$0.788879\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.01829e6	0.624210
$732$	0	0
$733$	9.75716e6	0.670754	0.335377	−	0.942084i	$-0.391136\pi$
$733$	9.75716e6	0.670754	0.335377	+	0.942084i	$0.391136\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.58485e6	−0.107478
$738$	0	0
$739$	2.55181e6	0.171885	0.0859425	−	0.996300i	$-0.472610\pi$
$739$	2.55181e6	0.171885	0.0859425	+	0.996300i	$0.472610\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.80305e6	0.585007	0.292503	−	0.956264i	$-0.405512\pi$
$743$	8.80305e6	0.585007	0.292503	+	0.956264i	$0.405512\pi$
$744$	0	0
$745$	3.63537e6	0.239970
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.84456e6	0.380669
$750$	0	0
$751$	−3.05319e7	−1.97540	−0.987698	−	0.156376i	$-0.950019\pi$
$751$	−3.05319e7	−1.97540	−0.987698	+	0.156376i	$0.950019\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.97350e6	−0.317538
$756$	0	0
$757$	4.08905e6	0.259348	0.129674	−	0.991557i	$-0.458607\pi$
$757$	4.08905e6	0.259348	0.129674	+	0.991557i	$0.458607\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.22130e6	0.514611	0.257305	−	0.966330i	$-0.417165\pi$
$761$	8.22130e6	0.514611	0.257305	+	0.966330i	$0.417165\pi$
$762$	0	0
$763$	1.06021e7	0.659297
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.30629e6	−0.571200
$768$	0	0
$769$	1.34850e6	0.0822310	0.0411155	−	0.999154i	$-0.486909\pi$
$769$	1.34850e6	0.0822310	0.0411155	+	0.999154i	$0.486909\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.03585e6	0.363320	0.181660	−	0.983361i	$-0.441853\pi$
$773$	6.03585e6	0.363320	0.181660	+	0.983361i	$0.441853\pi$
$774$	0	0
$775$	578755.	0.0346131
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.31664e6	0.0777365
$780$	0	0
$781$	516972.	0.0303277
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.39359e6	0.138636
$786$	0	0
$787$	−8.62455e6	−0.496363	−0.248182	−	0.968713i	$-0.579833\pi$
$787$	−8.62455e6	−0.496363	−0.248182	+	0.968713i	$0.579833\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.57881e6	0.0897197
$792$	0	0
$793$	1.25925e7	0.711099
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.81538e6	−0.324289	−0.162144	−	0.986767i	$-0.551841\pi$
$797$	−5.81538e6	−0.324289	−0.162144	+	0.986767i	$0.551841\pi$
$798$	0	0
$799$	−1.32355e7	−0.733456
$800$	0	0
$801$	0	0
$802$	0	0
$803$	509633.	0.0278913
$804$	0	0
$805$	1.26614e6	0.0688642
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.79657e7	−1.50229	−0.751147	−	0.660135i	$-0.770499\pi$
$809$	−2.79657e7	−1.50229	−0.751147	+	0.660135i	$0.770499\pi$
$810$	0	0
$811$	7.59446e6	0.405457	0.202728	−	0.979235i	$-0.435019\pi$
$811$	7.59446e6	0.405457	0.202728	+	0.979235i	$0.435019\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.36228e6	−0.335520
$816$	0	0
$817$	8.64899e6	0.453325
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.46608e7	−0.759102	−0.379551	−	0.925171i	$-0.623922\pi$
$821$	−1.46608e7	−0.759102	−0.379551	+	0.925171i	$0.623922\pi$
$822$	0	0
$823$	9.79434e6	0.504052	0.252026	−	0.967720i	$-0.418903\pi$
$823$	9.79434e6	0.504052	0.252026	+	0.967720i	$0.418903\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.98732e6	0.304417	0.152209	−	0.988348i	$-0.451361\pi$
$827$	5.98732e6	0.304417	0.152209	+	0.988348i	$0.451361\pi$
$828$	0	0
$829$	−2.20356e7	−1.11363	−0.556813	−	0.830638i	$-0.687976\pi$
$829$	−2.20356e7	−1.11363	−0.556813	+	0.830638i	$0.687976\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.37081e7	−0.684486
$834$	0	0
$835$	−670427.	−0.0332763
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.12407e7	−1.53220	−0.766101	−	0.642721i	$-0.777806\pi$
$839$	−3.12407e7	−1.53220	−0.766101	+	0.642721i	$0.777806\pi$
$840$	0	0
$841$	−1.92370e7	−0.937878
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.26030e6	0.205257
$846$	0	0
$847$	1.19826e7	0.573910
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.57151e6	0.169055
$852$	0	0
$853$	2.24467e7	1.05628	0.528141	−	0.849156i	$-0.322889\pi$
$853$	2.24467e7	1.05628	0.528141	+	0.849156i	$0.322889\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.59301e7	−1.20601	−0.603007	−	0.797736i	$-0.706031\pi$
$857$	−2.59301e7	−1.20601	−0.603007	+	0.797736i	$0.706031\pi$
$858$	0	0
$859$	−1.38585e7	−0.640817	−0.320408	−	0.947280i	$-0.603820\pi$
$859$	−1.38585e7	−0.640817	−0.320408	+	0.947280i	$0.603820\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.88792e7	−1.31995	−0.659976	−	0.751287i	$-0.729434\pi$
$863$	−2.88792e7	−1.31995	−0.659976	+	0.751287i	$0.729434\pi$
$864$	0	0
$865$	8.84239e6	0.401818
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−266619.	−0.0119768
$870$	0	0
$871$	−2.80757e7	−1.25396
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.16718e6	0.0515369
$876$	0	0
$877$	−1.65549e7	−0.726821	−0.363411	−	0.931629i	$-0.618388\pi$
$877$	−1.65549e7	−0.726821	−0.363411	+	0.931629i	$0.618388\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.79460e7	−0.778983	−0.389492	−	0.921030i	$-0.627349\pi$
$881$	−1.79460e7	−0.778983	−0.389492	+	0.921030i	$0.627349\pi$
$882$	0	0
$883$	3.15783e7	1.36297	0.681487	−	0.731830i	$-0.261334\pi$
$883$	3.15783e7	1.36297	0.681487	+	0.731830i	$0.261334\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.86752e6	−0.378437	−0.189218	−	0.981935i	$-0.560595\pi$
$887$	−8.86752e6	−0.378437	−0.189218	+	0.981935i	$0.560595\pi$
$888$	0	0
$889$	−2.03065e6	−0.0861750
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.26935e7	−0.532664
$894$	0	0
$895$	2.05533e7	0.857679
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.04528e6	0.0431352
$900$	0	0
$901$	−2.91353e7	−1.19566
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.40841e6	−0.341266
$906$	0	0
$907$	5.10629e6	0.206104	0.103052	−	0.994676i	$-0.467139\pi$
$907$	5.10629e6	0.206104	0.103052	+	0.994676i	$0.467139\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.61646e7	−1.84295	−0.921474	−	0.388441i	$-0.873014\pi$
$911$	−4.61646e7	−1.84295	−0.921474	+	0.388441i	$0.873014\pi$
$912$	0	0
$913$	982540.	0.0390098
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.07821e6	−0.120886
$918$	0	0
$919$	2.19340e7	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$919$	2.19340e7	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.15818e6	0.353838
$924$	0	0
$925$	3.29236e6	0.126518
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.00716e7	0.382877	0.191439	−	0.981505i	$-0.438685\pi$
$929$	1.00716e7	0.382877	0.191439	+	0.981505i	$0.438685\pi$
$930$	0	0
$931$	−1.31467e7	−0.497100
$932$	0	0
$933$	0	0
$934$	0	0
$935$	772292.	0.0288903
$936$	0	0
$937$	−5.19945e7	−1.93468	−0.967338	−	0.253488i	$-0.918422\pi$
$937$	−5.19945e7	−1.93468	−0.967338	+	0.253488i	$0.918422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.85215e6	−0.325893	−0.162946	−	0.986635i	$-0.552100\pi$
$941$	−8.85215e6	−0.325893	−0.162946	+	0.986635i	$0.552100\pi$
$942$	0	0
$943$	762321.	0.0279163
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.94610e6	0.106751	0.0533756	−	0.998575i	$-0.483002\pi$
$947$	2.94610e6	0.106751	0.0533756	+	0.998575i	$0.483002\pi$
$948$	0	0
$949$	9.02818e6	0.325413
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.23774e7	−0.441467	−0.220734	−	0.975334i	$-0.570845\pi$
$953$	−1.23774e7	−0.441467	−0.220734	+	0.975334i	$0.570845\pi$
$954$	0	0
$955$	1.79479e7	0.636802
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.48760e6	0.227792
$960$	0	0
$961$	−2.77717e7	−0.970048
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.36658e7	−0.472406
$966$	0	0
$967$	1.02250e6	0.0351639	0.0175819	−	0.999845i	$-0.494403\pi$
$967$	1.02250e6	0.0351639	0.0175819	+	0.999845i	$0.494403\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.32659e7	0.451531	0.225766	−	0.974182i	$-0.427512\pi$
$971$	1.32659e7	0.451531	0.225766	+	0.974182i	$0.427512\pi$
$972$	0	0
$973$	−1.49240e7	−0.505363
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.04067e7	0.348799	0.174399	−	0.984675i	$-0.444202\pi$
$977$	1.04067e7	0.348799	0.174399	+	0.984675i	$0.444202\pi$
$978$	0	0
$979$	551293.	0.0183834
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.30270e7	−1.75030	−0.875152	−	0.483849i	$-0.839238\pi$
$983$	−5.30270e7	−1.75030	−0.875152	+	0.483849i	$0.839238\pi$
$984$	0	0
$985$	1.69108e7	0.555360
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.00766e6	0.162796
$990$	0	0
$991$	3.23719e7	1.04709	0.523545	−	0.851998i	$-0.324609\pi$
$991$	3.23719e7	1.04709	0.523545	+	0.851998i	$0.324609\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.99079e7	0.637480
$996$	0	0
$997$	8.68632e6	0.276757	0.138378	−	0.990379i	$-0.455811\pi$
$997$	8.68632e6	0.276757	0.138378	+	0.990379i	$0.455811\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.a.bb.1.1		2
3.2	odd	2		720.6.a.bh.1.1		2
4.3	odd	2		360.6.a.j.1.2	✓	2
12.11	even	2		360.6.a.m.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.6.a.j.1.2	✓	2	4.3	odd	2
360.6.a.m.1.2	yes	2	12.11	even	2
720.6.a.bb.1.1		2	1.1	even	1	trivial
720.6.a.bh.1.1		2	3.2	odd	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} -74.6996 q^{7} +O(q^{10})\)	\(q-25.0000 q^{5} -74.6996 q^{7} -25.3004 q^{11} -448.198 q^{13} +1221.00 q^{17} +1171.00 q^{19} +677.992 q^{23} +625.000 q^{25} +1128.80 q^{29} +926.008 q^{31} +1867.49 q^{35} +5267.77 q^{37} +1124.38 q^{41} +7386.01 q^{43} -10839.9 q^{47} -11227.0 q^{49} -23861.9 q^{53} +632.510 q^{55} +20763.8 q^{59} -28095.9 q^{61} +11204.9 q^{65} +62641.3 q^{67} -20433.4 q^{71} -20143.3 q^{73} +1889.93 q^{77} +10538.1 q^{79} -38835.0 q^{83} -30524.9 q^{85} -21789.9 q^{89} +33480.2 q^{91} -29274.9 q^{95} -71881.7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{5} + 80 q^{7}+O(q^{10}) \) 2 * q - 50 * q^5 + 80 * q^7	\( 2 q - 50 q^{5} + 80 q^{7} - 280 q^{11} + 480 q^{13} + 148 q^{17} + 48 q^{19} - 3232 q^{23} + 1250 q^{25} + 1340 q^{29} + 6440 q^{31} - 2000 q^{35} - 6440 q^{37} - 9680 q^{41} + 19360 q^{43} + 15024 q^{47} - 4102 q^{49} - 1844 q^{53} + 7000 q^{55} - 12840 q^{59} - 24076 q^{61} - 12000 q^{65} + 42240 q^{67} - 12880 q^{71} - 96260 q^{73} - 37512 q^{77} + 89896 q^{79} - 57024 q^{83} - 3700 q^{85} + 25240 q^{89} + 177072 q^{91} - 1200 q^{95} - 229100 q^{97}+O(q^{100}) \) 2 * q - 50 * q^5 + 80 * q^7 - 280 * q^11 + 480 * q^13 + 148 * q^17 + 48 * q^19 - 3232 * q^23 + 1250 * q^25 + 1340 * q^29 + 6440 * q^31 - 2000 * q^35 - 6440 * q^37 - 9680 * q^41 + 19360 * q^43 + 15024 * q^47 - 4102 * q^49 - 1844 * q^53 + 7000 * q^55 - 12840 * q^59 - 24076 * q^61 - 12000 * q^65 + 42240 * q^67 - 12880 * q^71 - 96260 * q^73 - 37512 * q^77 + 89896 * q^79 - 57024 * q^83 - 3700 * q^85 + 25240 * q^89 + 177072 * q^91 - 1200 * q^95 - 229100 * q^97

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