LMFDB - Embedded newform 720.6.a.bh.1.2

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:	$0$
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			1.2
Root			$-28.1749$ 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)

$f(q)$	$=$	$q+25.0000 q^{5} +154.700 q^{7} +O(q^{10})$	$q+25.0000 q^{5} +154.700 q^{7} +254.700 q^{11} +928.198 q^{13} +1073.00 q^{17} -1123.00 q^{19} +3909.99 q^{23} +625.000 q^{25} -211.202 q^{29} +5513.99 q^{31} +3867.49 q^{35} -11707.8 q^{37} +10804.4 q^{41} +11974.0 q^{43} -25863.9 q^{47} +7124.97 q^{49} -22017.9 q^{53} +6367.49 q^{55} +33603.8 q^{59} +4019.95 q^{61} +23204.9 q^{65} -20401.3 q^{67} -7553.35 q^{71} -76116.7 q^{73} +39401.9 q^{77} +79357.9 q^{79} +18189.0 q^{83} +26824.9 q^{85} -47029.9 q^{89} +143592. q^{91} -28074.9 q^{95} -157218. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	154.700	1.19328	0.596642	−	0.802507i	$-0.296501\pi$
$7$	154.700	1.19328	0.596642	+	0.802507i	$0.296501\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	254.700	0.634668	0.317334	−	0.948314i	$-0.397212\pi$
$11$	254.700	0.634668	0.317334	+	0.948314i	$0.397212\pi$
$12$	0	0
$13$	928.198	1.52329	0.761644	−	0.647996i	$-0.224392\pi$
$13$	928.198	1.52329	0.761644	+	0.647996i	$0.224392\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1073.00	0.900484	0.450242	−	0.892907i	$-0.351338\pi$
$17$	1073.00	0.900484	0.450242	+	0.892907i	$0.351338\pi$
$18$	0	0
$19$	−1123.00	−0.713665	−0.356832	−	0.934168i	$-0.616143\pi$
$19$	−1123.00	−0.713665	−0.356832	+	0.934168i	$0.616143\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3909.99	1.54119	0.770595	−	0.637325i	$-0.219959\pi$
$23$	3909.99	1.54119	0.770595	+	0.637325i	$0.219959\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−211.202	−0.0466339	−0.0233170	−	0.999728i	$-0.507423\pi$
$29$	−211.202	−0.0466339	−0.0233170	+	0.999728i	$0.507423\pi$
$30$	0	0
$31$	5513.99	1.03053	0.515267	−	0.857030i	$-0.327693\pi$
$31$	5513.99	1.03053	0.515267	+	0.857030i	$0.327693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3867.49	0.533653
$36$	0	0
$37$	−11707.8	−1.40595	−0.702975	−	0.711215i	$-0.748145\pi$
$37$	−11707.8	−1.40595	−0.702975	+	0.711215i	$0.748145\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10804.4	1.00378	0.501892	−	0.864930i	$-0.332638\pi$
$41$	10804.4	1.00378	0.501892	+	0.864930i	$0.332638\pi$
$42$	0	0
$43$	11974.0	0.987570	0.493785	−	0.869584i	$-0.335613\pi$
$43$	11974.0	0.987570	0.493785	+	0.869584i	$0.335613\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−25863.9	−1.70785	−0.853926	−	0.520395i	$-0.825785\pi$
$47$	−25863.9	−1.70785	−0.853926	+	0.520395i	$0.825785\pi$
$48$	0	0
$49$	7124.97	0.423929
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−22017.9	−1.07668	−0.538340	−	0.842728i	$-0.680948\pi$
$53$	−22017.9	−1.07668	−0.538340	+	0.842728i	$0.680948\pi$
$54$	0	0
$55$	6367.49	0.283832
$56$	0	0
$57$	0	0
$58$	0	0
$59$	33603.8	1.25678	0.628389	−	0.777899i	$-0.283715\pi$
$59$	33603.8	1.25678	0.628389	+	0.777899i	$0.283715\pi$
$60$	0	0
$61$	4019.95	0.138323	0.0691617	−	0.997605i	$-0.477968\pi$
$61$	4019.95	0.138323	0.0691617	+	0.997605i	$0.477968\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23204.9	0.681235
$66$	0	0
$67$	−20401.3	−0.555226	−0.277613	−	0.960693i	$-0.589543\pi$
$67$	−20401.3	−0.555226	−0.277613	+	0.960693i	$0.589543\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7553.35	−0.177825	−0.0889127	−	0.996039i	$-0.528339\pi$
$71$	−7553.35	−0.177825	−0.0889127	+	0.996039i	$0.528339\pi$
$72$	0	0
$73$	−76116.7	−1.67176	−0.835878	−	0.548915i	$-0.815041\pi$
$73$	−76116.7	−1.67176	−0.835878	+	0.548915i	$0.815041\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	39401.9	0.757340
$78$	0	0
$79$	79357.9	1.43061	0.715307	−	0.698811i	$-0.246287\pi$
$79$	79357.9	1.43061	0.715307	+	0.698811i	$0.246287\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	18189.0	0.289811	0.144905	−	0.989446i	$-0.453712\pi$
$83$	18189.0	0.289811	0.144905	+	0.989446i	$0.453712\pi$
$84$	0	0
$85$	26824.9	0.402709
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−47029.9	−0.629360	−0.314680	−	0.949198i	$-0.601897\pi$
$89$	−47029.9	−0.629360	−0.314680	+	0.949198i	$0.601897\pi$
$90$	0	0
$91$	143592.	1.81772
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−28074.9	−0.319161
$96$	0	0
$97$	−157218.	−1.69658	−0.848289	−	0.529534i	$-0.822367\pi$
$97$	−157218.	−1.69658	−0.848289	+	0.529534i	$0.822367\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−131419.	−1.28190	−0.640952	−	0.767581i	$-0.721460\pi$
$101$	−131419.	−1.28190	−0.640952	+	0.767581i	$0.721460\pi$
$102$	0	0
$103$	−58471.0	−0.543060	−0.271530	−	0.962430i	$-0.587530\pi$
$103$	−58471.0	−0.543060	−0.271530	+	0.962430i	$0.587530\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16303.1	0.137661	0.0688305	−	0.997628i	$-0.478073\pi$
$107$	16303.1	0.137661	0.0688305	+	0.997628i	$0.478073\pi$
$108$	0	0
$109$	−132754.	−1.07024	−0.535120	−	0.844776i	$-0.679734\pi$
$109$	−132754.	−1.07024	−0.535120	+	0.844776i	$0.679734\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	262005.	1.93025	0.965123	−	0.261796i	$-0.0843147\pi$
$113$	262005.	1.93025	0.965123	+	0.261796i	$0.0843147\pi$
$114$	0	0
$115$	97749.8	0.689241
$116$	0	0
$117$	0	0
$118$	0	0
$119$	165992.	1.07453
$120$	0	0
$121$	−96179.1	−0.597197
$122$	0	0
$123$	0	0
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	118256.	0.650598	0.325299	−	0.945611i	$-0.394535\pi$
$127$	118256.	0.650598	0.325299	+	0.945611i	$0.394535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	78767.9	0.401025	0.200512	−	0.979691i	$-0.435739\pi$
$131$	78767.9	0.401025	0.200512	+	0.979691i	$0.435739\pi$
$132$	0	0
$133$	−173727.	−0.851605
$134$	0	0
$135$	0	0
$136$	0	0
$137$	222195.	1.01142	0.505711	−	0.862703i	$-0.331230\pi$
$137$	222195.	1.01142	0.505711	+	0.862703i	$0.331230\pi$
$138$	0	0
$139$	197493.	0.866991	0.433496	−	0.901156i	$-0.357280\pi$
$139$	197493.	0.866991	0.433496	+	0.901156i	$0.357280\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	236412.	0.966782
$144$	0	0
$145$	−5280.04	−0.0208553
$146$	0	0
$147$	0	0
$148$	0	0
$149$	271125.	1.00047	0.500236	−	0.865889i	$-0.333247\pi$
$149$	271125.	1.00047	0.500236	+	0.865889i	$0.333247\pi$
$150$	0	0
$151$	−285092.	−1.01752	−0.508760	−	0.860908i	$-0.669896\pi$
$151$	−285092.	−1.01752	−0.508760	+	0.860908i	$0.669896\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	137850.	0.460868
$156$	0	0
$157$	31343.6	0.101484	0.0507422	−	0.998712i	$-0.483841\pi$
$157$	31343.6	0.101484	0.0507422	+	0.998712i	$0.483841\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	604874.	1.83908
$162$	0	0
$163$	102629.	0.302552	0.151276	−	0.988492i	$-0.451662\pi$
$163$	102629.	0.302552	0.151276	+	0.988492i	$0.451662\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	519153.	1.44047	0.720235	−	0.693730i	$-0.244034\pi$
$167$	519153.	1.44047	0.720235	+	0.693730i	$0.244034\pi$
$168$	0	0
$169$	490258.	1.32041
$170$	0	0
$171$	0	0
$172$	0	0
$173$	55476.5	0.140927	0.0704634	−	0.997514i	$-0.477552\pi$
$173$	55476.5	0.140927	0.0704634	+	0.997514i	$0.477552\pi$
$174$	0	0
$175$	96687.3	0.238657
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−209934.	−0.489722	−0.244861	−	0.969558i	$-0.578742\pi$
$179$	−209934.	−0.489722	−0.244861	+	0.969558i	$0.578742\pi$
$180$	0	0
$181$	−544556.	−1.23551	−0.617755	−	0.786370i	$-0.711958\pi$
$181$	−544556.	−1.23551	−0.617755	+	0.786370i	$0.711958\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−292694.	−0.628760
$186$	0	0
$187$	273292.	0.571508
$188$	0	0
$189$	0	0
$190$	0	0
$191$	65044.9	0.129012	0.0645060	−	0.997917i	$-0.479453\pi$
$191$	65044.9	0.129012	0.0645060	+	0.997917i	$0.479453\pi$
$192$	0	0
$193$	−308570.	−0.596294	−0.298147	−	0.954520i	$-0.596369\pi$
$193$	−308570.	−0.596294	−0.298147	+	0.954520i	$0.596369\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−521030.	−0.956527	−0.478263	−	0.878216i	$-0.658734\pi$
$197$	−521030.	−0.956527	−0.478263	+	0.878216i	$0.658734\pi$
$198$	0	0
$199$	752130.	1.34636	0.673179	−	0.739480i	$-0.264928\pi$
$199$	752130.	1.34636	0.673179	+	0.739480i	$0.264928\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−32672.8	−0.0556476
$204$	0	0
$205$	270109.	0.448906
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−286027.	−0.452940
$210$	0	0
$211$	−322688.	−0.498973	−0.249487	−	0.968378i	$-0.580262\pi$
$211$	−322688.	−0.498973	−0.249487	+	0.968378i	$0.580262\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	299350.	0.441655
$216$	0	0
$217$	853012.	1.22972
$218$	0	0
$219$	0	0
$220$	0	0
$221$	995952.	1.37170
$222$	0	0
$223$	−967621.	−1.30300	−0.651498	−	0.758650i	$-0.725859\pi$
$223$	−967621.	−1.30300	−0.651498	+	0.758650i	$0.725859\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	44587.0	0.0574307	0.0287153	−	0.999588i	$-0.490858\pi$
$227$	44587.0	0.0574307	0.0287153	+	0.999588i	$0.490858\pi$
$228$	0	0
$229$	957827.	1.20698	0.603488	−	0.797372i	$-0.293777\pi$
$229$	957827.	1.20698	0.603488	+	0.797372i	$0.293777\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.02556e6	1.23758	0.618789	−	0.785557i	$-0.287624\pi$
$233$	1.02556e6	1.23758	0.618789	+	0.785557i	$0.287624\pi$
$234$	0	0
$235$	−646598.	−0.763774
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.18601e6	1.34306	0.671529	−	0.740978i	$-0.265638\pi$
$239$	1.18601e6	1.34306	0.671529	+	0.740978i	$0.265638\pi$
$240$	0	0
$241$	190575.	0.211360	0.105680	−	0.994400i	$-0.466298\pi$
$241$	190575.	0.211360	0.105680	+	0.994400i	$0.466298\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	178124.	0.189587
$246$	0	0
$247$	−1.04236e6	−1.08712
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.37627e6	1.37886	0.689429	−	0.724353i	$-0.257861\pi$
$251$	1.37627e6	1.37886	0.689429	+	0.724353i	$0.257861\pi$
$252$	0	0
$253$	995873.	0.978144
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.25590e6	1.18611	0.593053	−	0.805163i	$-0.297922\pi$
$257$	1.25590e6	1.18611	0.593053	+	0.805163i	$0.297922\pi$
$258$	0	0
$259$	−1.81119e6	−1.67770
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.18163e6	−1.05340	−0.526699	−	0.850052i	$-0.676570\pi$
$263$	−1.18163e6	−1.05340	−0.526699	+	0.850052i	$0.676570\pi$
$264$	0	0
$265$	−550448.	−0.481506
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.41078e6	1.18872	0.594358	−	0.804201i	$-0.297406\pi$
$269$	1.41078e6	1.18872	0.594358	+	0.804201i	$0.297406\pi$
$270$	0	0
$271$	−1.25536e6	−1.03835	−0.519177	−	0.854667i	$-0.673761\pi$
$271$	−1.25536e6	−1.03835	−0.519177	+	0.854667i	$0.673761\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	159187.	0.126934
$276$	0	0
$277$	−734415.	−0.575099	−0.287549	−	0.957766i	$-0.592841\pi$
$277$	−734415.	−0.575099	−0.287549	+	0.957766i	$0.592841\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−793701.	−0.599641	−0.299821	−	0.953996i	$-0.596927\pi$
$281$	−793701.	−0.599641	−0.299821	+	0.953996i	$0.596927\pi$
$282$	0	0
$283$	1.29354e6	0.960091	0.480045	−	0.877244i	$-0.340620\pi$
$283$	1.29354e6	0.960091	0.480045	+	0.877244i	$0.340620\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.67143e6	1.19780
$288$	0	0
$289$	−268536.	−0.189129
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.40112e6	0.953466	0.476733	−	0.879048i	$-0.341821\pi$
$293$	1.40112e6	0.953466	0.476733	+	0.879048i	$0.341821\pi$
$294$	0	0
$295$	840095.	0.562048
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.62925e6	2.34768
$300$	0	0
$301$	1.85237e6	1.17845
$302$	0	0
$303$	0	0
$304$	0	0
$305$	100499.	0.0618601
$306$	0	0
$307$	−422338.	−0.255749	−0.127875	−	0.991790i	$-0.540816\pi$
$307$	−422338.	−0.255749	−0.127875	+	0.991790i	$0.540816\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.08850e6	−0.638154	−0.319077	−	0.947729i	$-0.603373\pi$
$311$	−1.08850e6	−0.638154	−0.319077	+	0.947729i	$0.603373\pi$
$312$	0	0
$313$	1.81370e6	1.04642	0.523209	−	0.852204i	$-0.324735\pi$
$313$	1.81370e6	1.04642	0.523209	+	0.852204i	$0.324735\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.46607e6	1.37834	0.689172	−	0.724598i	$-0.257975\pi$
$317$	2.46607e6	1.37834	0.689172	+	0.724598i	$0.257975\pi$
$318$	0	0
$319$	−53793.0	−0.0295971
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.20497e6	−0.642643
$324$	0	0
$325$	580124.	0.304658
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00114e6	−2.03795
$330$	0	0
$331$	130507.	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$331$	130507.	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−510031.	−0.248305
$336$	0	0
$337$	1.13129e6	0.542623	0.271311	−	0.962492i	$-0.412543\pi$
$337$	1.13129e6	0.542623	0.271311	+	0.962492i	$0.412543\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.40441e6	0.654046
$342$	0	0
$343$	−1.49781e6	−0.687417
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.05924e6	−1.36392	−0.681962	−	0.731387i	$-0.738873\pi$
$347$	−3.05924e6	−1.36392	−0.681962	+	0.731387i	$0.738873\pi$
$348$	0	0
$349$	−2.63159e6	−1.15652	−0.578262	−	0.815851i	$-0.696269\pi$
$349$	−2.63159e6	−1.15652	−0.578262	+	0.815851i	$0.696269\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−623596.	−0.266358	−0.133179	−	0.991092i	$-0.542519\pi$
$353$	−623596.	−0.266358	−0.133179	+	0.991092i	$0.542519\pi$
$354$	0	0
$355$	−188834.	−0.0795259
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.99185e6	1.22519	0.612595	−	0.790397i	$-0.290126\pi$
$359$	2.99185e6	1.22519	0.612595	+	0.790397i	$0.290126\pi$
$360$	0	0
$361$	−1.21498e6	−0.490683
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.90292e6	−0.747632
$366$	0	0
$367$	−1.11600e6	−0.432513	−0.216257	−	0.976337i	$-0.569385\pi$
$367$	−1.11600e6	−0.432513	−0.216257	+	0.976337i	$0.569385\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.40616e6	−1.28479
$372$	0	0
$373$	1.85880e6	0.691767	0.345883	−	0.938277i	$-0.387579\pi$
$373$	1.85880e6	0.691767	0.345883	+	0.938277i	$0.387579\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−196037.	−0.0710369
$378$	0	0
$379$	−3.49696e6	−1.25053	−0.625263	−	0.780414i	$-0.715008\pi$
$379$	−3.49696e6	−1.25053	−0.625263	+	0.780414i	$0.715008\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.11608e6	1.08545	0.542727	−	0.839909i	$-0.317392\pi$
$383$	3.11608e6	1.08545	0.542727	+	0.839909i	$0.317392\pi$
$384$	0	0
$385$	985048.	0.338693
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.75310e6	−1.92765	−0.963825	−	0.266537i	$-0.914120\pi$
$389$	−5.75310e6	−1.92765	−0.963825	+	0.266537i	$0.914120\pi$
$390$	0	0
$391$	4.19541e6	1.38782
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.98395e6	0.639790
$396$	0	0
$397$	−3.74183e6	−1.19154	−0.595769	−	0.803156i	$-0.703153\pi$
$397$	−3.74183e6	−1.19154	−0.595769	+	0.803156i	$0.703153\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00047e6	0.931812	0.465906	−	0.884834i	$-0.345728\pi$
$401$	3.00047e6	0.931812	0.465906	+	0.884834i	$0.345728\pi$
$402$	0	0
$403$	5.11807e6	1.56980
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.98196e6	−0.892311
$408$	0	0
$409$	−3.54830e6	−1.04885	−0.524423	−	0.851458i	$-0.675719\pi$
$409$	−3.54830e6	−1.04885	−0.524423	+	0.851458i	$0.675719\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.19850e6	1.49969
$414$	0	0
$415$	454726.	0.129607
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.02437e6	−1.11986	−0.559929	−	0.828541i	$-0.689171\pi$
$419$	−4.02437e6	−1.11986	−0.559929	+	0.828541i	$0.689171\pi$
$420$	0	0
$421$	3.62675e6	0.997269	0.498634	−	0.866812i	$-0.333835\pi$
$421$	3.62675e6	0.997269	0.498634	+	0.866812i	$0.333835\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	670623.	0.180097
$426$	0	0
$427$	621884.	0.165059
$428$	0	0
$429$	0	0
$430$	0	0
$431$	400723.	0.103908	0.0519542	−	0.998649i	$-0.483455\pi$
$431$	400723.	0.103908	0.0519542	+	0.998649i	$0.483455\pi$
$432$	0	0
$433$	−2.07120e6	−0.530888	−0.265444	−	0.964126i	$-0.585519\pi$
$433$	−2.07120e6	−0.530888	−0.265444	+	0.964126i	$0.585519\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.39091e6	−1.09989
$438$	0	0
$439$	3.00868e6	0.745101	0.372551	−	0.928012i	$-0.378483\pi$
$439$	3.00868e6	0.745101	0.372551	+	0.928012i	$0.378483\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.04395e6	−1.94742	−0.973710	−	0.227789i	$-0.926850\pi$
$443$	−8.04395e6	−1.94742	−0.973710	+	0.227789i	$0.926850\pi$
$444$	0	0
$445$	−1.17575e6	−0.281458
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.27671e6	−0.532957	−0.266478	−	0.963841i	$-0.585860\pi$
$449$	−2.27671e6	−0.532957	−0.266478	+	0.963841i	$0.585860\pi$
$450$	0	0
$451$	2.75187e6	0.637069
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.58980e6	0.812908
$456$	0	0
$457$	1.49605e6	0.335087	0.167543	−	0.985865i	$-0.446417\pi$
$457$	1.49605e6	0.335087	0.167543	+	0.985865i	$0.446417\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.38157e6	−0.521928	−0.260964	−	0.965348i	$-0.584040\pi$
$461$	−2.38157e6	−0.521928	−0.260964	+	0.965348i	$0.584040\pi$
$462$	0	0
$463$	8.13436e6	1.76348	0.881740	−	0.471735i	$-0.156372\pi$
$463$	8.13436e6	1.76348	0.881740	+	0.471735i	$0.156372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−541215.	−0.114836	−0.0574180	−	0.998350i	$-0.518287\pi$
$467$	−541215.	−0.114836	−0.0574180	+	0.998350i	$0.518287\pi$
$468$	0	0
$469$	−3.15607e6	−0.662543
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.04977e6	0.626779
$474$	0	0
$475$	−701873.	−0.142733
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.71874e6	1.53712	0.768560	−	0.639778i	$-0.220974\pi$
$479$	7.71874e6	1.53712	0.768560	+	0.639778i	$0.220974\pi$
$480$	0	0
$481$	−1.08671e7	−2.14167
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.93046e6	−0.758732
$486$	0	0
$487$	7.29331e6	1.39348	0.696742	−	0.717322i	$-0.254632\pi$
$487$	7.29331e6	1.39348	0.696742	+	0.717322i	$0.254632\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.56353e6	−1.04147	−0.520735	−	0.853718i	$-0.674342\pi$
$491$	−5.56353e6	−1.04147	−0.520735	+	0.853718i	$0.674342\pi$
$492$	0	0
$493$	−226618.	−0.0419931
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.16850e6	−0.212196
$498$	0	0
$499$	−4.49967e6	−0.808964	−0.404482	−	0.914546i	$-0.632548\pi$
$499$	−4.49967e6	−0.808964	−0.404482	+	0.914546i	$0.632548\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.07774e7	−1.89931	−0.949655	−	0.313297i	$-0.898566\pi$
$503$	−1.07774e7	−1.89931	−0.949655	+	0.313297i	$0.898566\pi$
$504$	0	0
$505$	−3.28548e6	−0.573285
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.64953e6	−1.65087	−0.825433	−	0.564500i	$-0.809069\pi$
$509$	−9.64953e6	−1.65087	−0.825433	+	0.564500i	$0.809069\pi$
$510$	0	0
$511$	−1.17752e7	−1.99488
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.46177e6	−0.242864
$516$	0	0
$517$	−6.58753e6	−1.08392
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.14361e6	−0.345980	−0.172990	−	0.984924i	$-0.555343\pi$
$521$	−2.14361e6	−0.345980	−0.172990	+	0.984924i	$0.555343\pi$
$522$	0	0
$523$	2.93671e6	0.469470	0.234735	−	0.972059i	$-0.424578\pi$
$523$	2.93671e6	0.469470	0.234735	+	0.972059i	$0.424578\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.91649e6	0.927978
$528$	0	0
$529$	8.85170e6	1.37527
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.00286e7	1.52905
$534$	0	0
$535$	407578.	0.0615639
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.81473e6	0.269054
$540$	0	0
$541$	−6.50470e6	−0.955507	−0.477754	−	0.878494i	$-0.658549\pi$
$541$	−6.50470e6	−0.955507	−0.477754	+	0.878494i	$0.658549\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.31885e6	−0.478626
$546$	0	0
$547$	−3.23601e6	−0.462426	−0.231213	−	0.972903i	$-0.574269\pi$
$547$	−3.23601e6	−0.462426	−0.231213	+	0.972903i	$0.574269\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	237179.	0.0332810
$552$	0	0
$553$	1.22766e7	1.70713
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.14035e7	−1.55739	−0.778697	−	0.627400i	$-0.784119\pi$
$557$	−1.14035e7	−1.55739	−0.778697	+	0.627400i	$0.784119\pi$
$558$	0	0
$559$	1.11142e7	1.50435
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.17565e7	−1.56317	−0.781583	−	0.623801i	$-0.785588\pi$
$563$	−1.17565e7	−1.56317	−0.781583	+	0.623801i	$0.785588\pi$
$564$	0	0
$565$	6.55011e6	0.863233
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.38679e6	−0.309054	−0.154527	−	0.987989i	$-0.549385\pi$
$569$	−2.38679e6	−0.309054	−0.154527	+	0.987989i	$0.549385\pi$
$570$	0	0
$571$	−2.67918e6	−0.343884	−0.171942	−	0.985107i	$-0.555004\pi$
$571$	−2.67918e6	−0.343884	−0.171942	+	0.985107i	$0.555004\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.44375e6	0.308238
$576$	0	0
$577$	−342747.	−0.0428582	−0.0214291	−	0.999770i	$-0.506822\pi$
$577$	−342747.	−0.0428582	−0.0214291	+	0.999770i	$0.506822\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.81384e6	0.345827
$582$	0	0
$583$	−5.60796e6	−0.683334
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.61935e6	−0.193974	−0.0969872	−	0.995286i	$-0.530921\pi$
$587$	−1.61935e6	−0.193974	−0.0969872	+	0.995286i	$0.530921\pi$
$588$	0	0
$589$	−6.19219e6	−0.735455
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.55630e6	−0.298521	−0.149261	−	0.988798i	$-0.547689\pi$
$593$	−2.55630e6	−0.298521	−0.149261	+	0.988798i	$0.547689\pi$
$594$	0	0
$595$	4.14980e6	0.480546
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.68574e6	0.419718	0.209859	−	0.977732i	$-0.432700\pi$
$599$	3.68574e6	0.419718	0.209859	+	0.977732i	$0.432700\pi$
$600$	0	0
$601$	−1.50165e7	−1.69583	−0.847914	−	0.530133i	$-0.822142\pi$
$601$	−1.50165e7	−1.69583	−0.847914	+	0.530133i	$0.822142\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.40448e6	−0.267074
$606$	0	0
$607$	−7.63280e6	−0.840838	−0.420419	−	0.907330i	$-0.638117\pi$
$607$	−7.63280e6	−0.840838	−0.420419	+	0.907330i	$0.638117\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.40068e7	−2.60155
$612$	0	0
$613$	−7.60921e6	−0.817878	−0.408939	−	0.912562i	$-0.634101\pi$
$613$	−7.60921e6	−0.817878	−0.408939	+	0.912562i	$0.634101\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.01542e7	1.07382	0.536911	−	0.843639i	$-0.319591\pi$
$617$	1.01542e7	1.07382	0.536911	+	0.843639i	$0.319591\pi$
$618$	0	0
$619$	1.35012e7	1.41627	0.708134	−	0.706078i	$-0.249537\pi$
$619$	1.35012e7	1.41627	0.708134	+	0.706078i	$0.249537\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.27550e6	−0.751005
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.25624e7	−1.26604
$630$	0	0
$631$	−7.90006e6	−0.789873	−0.394936	−	0.918708i	$-0.629233\pi$
$631$	−7.90006e6	−0.789873	−0.394936	+	0.918708i	$0.629233\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.95639e6	0.290956
$636$	0	0
$637$	6.61338e6	0.645765
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.51910e6	−0.530546	−0.265273	−	0.964173i	$-0.585462\pi$
$641$	−5.51910e6	−0.530546	−0.265273	+	0.964173i	$0.585462\pi$
$642$	0	0
$643$	−560241.	−0.0534376	−0.0267188	−	0.999643i	$-0.508506\pi$
$643$	−560241.	−0.0534376	−0.0267188	+	0.999643i	$0.508506\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.50998e7	−1.41811	−0.709055	−	0.705153i	$-0.750878\pi$
$647$	−1.50998e7	−1.41811	−0.709055	+	0.705153i	$0.750878\pi$
$648$	0	0
$649$	8.55888e6	0.797636
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.97913e7	−1.81632	−0.908158	−	0.418629i	$-0.862511\pi$
$653$	−1.97913e7	−1.81632	−0.908158	+	0.418629i	$0.862511\pi$
$654$	0	0
$655$	1.96920e6	0.179344
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.53836e7	1.37989	0.689946	−	0.723861i	$-0.257634\pi$
$659$	1.53836e7	1.37989	0.689946	+	0.723861i	$0.257634\pi$
$660$	0	0
$661$	2.18766e7	1.94750	0.973748	−	0.227631i	$-0.0730979\pi$
$661$	2.18766e7	1.94750	0.973748	+	0.227631i	$0.0730979\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.34318e6	−0.380849
$666$	0	0
$667$	−825796.	−0.0718718
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.02388e6	0.0877894
$672$	0	0
$673$	1.18070e7	1.00485	0.502427	−	0.864620i	$-0.332441\pi$
$673$	1.18070e7	1.00485	0.502427	+	0.864620i	$0.332441\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.30369e7	1.09321	0.546604	−	0.837391i	$-0.315920\pi$
$677$	1.30369e7	1.09321	0.546604	+	0.837391i	$0.315920\pi$
$678$	0	0
$679$	−2.43216e7	−2.02450
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.00878e7	−0.827454	−0.413727	−	0.910401i	$-0.635773\pi$
$683$	−1.00878e7	−0.827454	−0.413727	+	0.910401i	$0.635773\pi$
$684$	0	0
$685$	5.55487e6	0.452322
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.04370e7	−1.64009
$690$	0	0
$691$	−2.37129e7	−1.88925	−0.944626	−	0.328148i	$-0.893575\pi$
$691$	−2.37129e7	−1.88925	−0.944626	+	0.328148i	$0.893575\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.93733e6	0.387730
$696$	0	0
$697$	1.15931e7	0.903891
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.44701e7	1.11218	0.556091	−	0.831122i	$-0.312301\pi$
$701$	1.44701e7	1.11218	0.556091	+	0.831122i	$0.312301\pi$
$702$	0	0
$703$	1.31478e7	1.00338
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.03305e7	−1.52968
$708$	0	0
$709$	1.83150e7	1.36833	0.684166	−	0.729326i	$-0.260166\pi$
$709$	1.83150e7	1.36833	0.684166	+	0.729326i	$0.260166\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.15597e7	1.58825
$714$	0	0
$715$	5.91029e6	0.432358
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.26298e7	1.63252	0.816260	−	0.577685i	$-0.196044\pi$
$719$	2.26298e7	1.63252	0.816260	+	0.577685i	$0.196044\pi$
$720$	0	0
$721$	−9.04544e6	−0.648025
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−132001.	−0.00932679
$726$	0	0
$727$	−1.80858e7	−1.26912	−0.634559	−	0.772875i	$-0.718818\pi$
$727$	−1.80858e7	−1.26912	−0.634559	+	0.772875i	$0.718818\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.28480e7	0.889291
$732$	0	0
$733$	1.35234e7	0.929667	0.464833	−	0.885398i	$-0.346114\pi$
$733$	1.35234e7	0.929667	0.464833	+	0.885398i	$0.346114\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.19619e6	−0.352384
$738$	0	0
$739$	6.04556e6	0.407217	0.203608	−	0.979052i	$-0.434733\pi$
$739$	6.04556e6	0.407217	0.203608	+	0.979052i	$0.434733\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.92433e7	1.27881	0.639407	−	0.768868i	$-0.279180\pi$
$743$	1.92433e7	1.27881	0.639407	+	0.768868i	$0.279180\pi$
$744$	0	0
$745$	6.77813e6	0.447424
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.52208e6	0.164269
$750$	0	0
$751$	1.58618e7	1.02625	0.513124	−	0.858314i	$-0.328488\pi$
$751$	1.58618e7	1.02625	0.513124	+	0.858314i	$0.328488\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.12730e6	−0.455049
$756$	0	0
$757$	−2.23189e7	−1.41558	−0.707789	−	0.706424i	$-0.750307\pi$
$757$	−2.23189e7	−1.41558	−0.707789	+	0.706424i	$0.750307\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.48599e7	1.55610	0.778050	−	0.628202i	$-0.216209\pi$
$761$	2.48599e7	1.55610	0.778050	+	0.628202i	$0.216209\pi$
$762$	0	0
$763$	−2.05370e7	−1.27710
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.11910e7	1.91443
$768$	0	0
$769$	−8.28627e6	−0.505293	−0.252646	−	0.967559i	$-0.581301\pi$
$769$	−8.28627e6	−0.505293	−0.252646	+	0.967559i	$0.581301\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.08819e7	1.25696	0.628478	−	0.777827i	$-0.283678\pi$
$773$	2.08819e7	1.25696	0.628478	+	0.777827i	$0.283678\pi$
$774$	0	0
$775$	3.44625e6	0.206107
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.21333e7	−0.716365
$780$	0	0
$781$	−1.92384e6	−0.112860
$782$	0	0
$783$	0	0
$784$	0	0
$785$	783590.	0.0453852
$786$	0	0
$787$	−1.35673e6	−0.0780828	−0.0390414	−	0.999238i	$-0.512430\pi$
$787$	−1.35673e6	−0.0780828	−0.0390414	+	0.999238i	$0.512430\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.05320e7	2.30333
$792$	0	0
$793$	3.73130e6	0.210706
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.60879e7	0.897125	0.448562	−	0.893752i	$-0.351936\pi$
$797$	1.60879e7	0.897125	0.448562	+	0.893752i	$0.351936\pi$
$798$	0	0
$799$	−2.77519e7	−1.53789
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.93869e7	−1.06101
$804$	0	0
$805$	1.51219e7	0.822461
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.05669e7	−1.64202	−0.821012	−	0.570911i	$-0.806590\pi$
$809$	−3.05669e7	−1.64202	−0.821012	+	0.570911i	$0.806590\pi$
$810$	0	0
$811$	−1.47697e7	−0.788531	−0.394265	−	0.918997i	$-0.629001\pi$
$811$	−1.47697e7	−0.788531	−0.394265	+	0.918997i	$0.629001\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.56572e6	0.135306
$816$	0	0
$817$	−1.34467e7	−0.704794
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.15008e7	−0.595484	−0.297742	−	0.954646i	$-0.596233\pi$
$821$	−1.15008e7	−0.595484	−0.297742	+	0.954646i	$0.596233\pi$
$822$	0	0
$823$	2.38471e7	1.22726	0.613629	−	0.789594i	$-0.289709\pi$
$823$	2.38471e7	1.22726	0.613629	+	0.789594i	$0.289709\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.47366e7	−0.749262	−0.374631	−	0.927174i	$-0.622231\pi$
$827$	−1.47366e7	−0.749262	−0.374631	+	0.927174i	$0.622231\pi$
$828$	0	0
$829$	−2.66517e6	−0.134691	−0.0673455	−	0.997730i	$-0.521453\pi$
$829$	−2.66517e6	−0.134691	−0.0673455	+	0.997730i	$0.521453\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.64506e6	0.381741
$834$	0	0
$835$	1.29788e7	0.644198
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.65882e6	0.473717	0.236859	−	0.971544i	$-0.423882\pi$
$839$	9.65882e6	0.473717	0.236859	+	0.971544i	$0.423882\pi$
$840$	0	0
$841$	−2.04665e7	−0.997825
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.22564e7	0.590504
$846$	0	0
$847$	−1.48789e7	−0.712626
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.57773e7	−2.16684
$852$	0	0
$853$	3.07432e7	1.44669	0.723346	−	0.690486i	$-0.242603\pi$
$853$	3.07432e7	1.44669	0.723346	+	0.690486i	$0.242603\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.16634e7	−1.00757	−0.503783	−	0.863830i	$-0.668059\pi$
$857$	−2.16634e7	−1.00757	−0.503783	+	0.863830i	$0.668059\pi$
$858$	0	0
$859$	1.47269e7	0.680972	0.340486	−	0.940250i	$-0.389408\pi$
$859$	1.47269e7	0.680972	0.340486	+	0.940250i	$0.389408\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.09448e7	0.500241	0.250120	−	0.968215i	$-0.419530\pi$
$863$	1.09448e7	0.500241	0.250120	+	0.968215i	$0.419530\pi$
$864$	0	0
$865$	1.38691e6	0.0630244
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.02124e7	0.907965
$870$	0	0
$871$	−1.89364e7	−0.845770
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.41718e6	0.106731
$876$	0	0
$877$	−3.16416e7	−1.38918	−0.694591	−	0.719405i	$-0.744415\pi$
$877$	−3.16416e7	−1.38918	−0.694591	+	0.719405i	$0.744415\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.69109e6	−0.420662	−0.210331	−	0.977630i	$-0.567454\pi$
$881$	−9.69109e6	−0.420662	−0.210331	+	0.977630i	$0.567454\pi$
$882$	0	0
$883$	4.26514e7	1.84091	0.920453	−	0.390852i	$-0.127820\pi$
$883$	4.26514e7	1.84091	0.920453	+	0.390852i	$0.127820\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.78141e7	0.760247	0.380123	−	0.924936i	$-0.375882\pi$
$887$	1.78141e7	0.760247	0.380123	+	0.924936i	$0.375882\pi$
$888$	0	0
$889$	1.82941e7	0.776349
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.90451e7	1.21883
$894$	0	0
$895$	−5.24834e6	−0.219010
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.16456e6	−0.0480578
$900$	0	0
$901$	−2.36251e7	−0.969533
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.36139e7	−0.552537
$906$	0	0
$907$	−3.18004e7	−1.28355	−0.641777	−	0.766892i	$-0.721802\pi$
$907$	−3.18004e7	−1.28355	−0.641777	+	0.766892i	$0.721802\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.82682e6	−0.312456	−0.156228	−	0.987721i	$-0.549934\pi$
$911$	−7.82682e6	−0.312456	−0.156228	+	0.987721i	$0.549934\pi$
$912$	0	0
$913$	4.63274e6	0.183934
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.21854e7	0.478537
$918$	0	0
$919$	2.33976e7	0.913865	0.456932	−	0.889501i	$-0.348948\pi$
$919$	2.33976e7	0.913865	0.456932	+	0.889501i	$0.348948\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.01100e6	−0.270879
$924$	0	0
$925$	−7.31736e6	−0.281190
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.79737e7	1.82375	0.911873	−	0.410473i	$-0.134636\pi$
$929$	4.79737e7	1.82375	0.911873	+	0.410473i	$0.134636\pi$
$930$	0	0
$931$	−8.00131e6	−0.302543
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.83229e6	0.255586
$936$	0	0
$937$	3.74198e7	1.39236	0.696182	−	0.717866i	$-0.254881\pi$
$937$	3.74198e7	1.39236	0.696182	+	0.717866i	$0.254881\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.41960e7	−0.890779	−0.445389	−	0.895337i	$-0.646935\pi$
$941$	−2.41960e7	−0.890779	−0.445389	+	0.895337i	$0.646935\pi$
$942$	0	0
$943$	4.22450e7	1.54702
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.96166e7	1.43550	0.717749	−	0.696302i	$-0.245173\pi$
$947$	3.96166e7	1.43550	0.717749	+	0.696302i	$0.245173\pi$
$948$	0	0
$949$	−7.06513e7	−2.54657
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.75028e6	0.240763	0.120381	−	0.992728i	$-0.461588\pi$
$953$	6.75028e6	0.240763	0.120381	+	0.992728i	$0.461588\pi$
$954$	0	0
$955$	1.62612e6	0.0576959
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.43734e7	1.20691
$960$	0	0
$961$	1.77496e6	0.0619983
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.71425e6	−0.266671
$966$	0	0
$967$	2.73183e7	0.939480	0.469740	−	0.882805i	$-0.344348\pi$
$967$	2.73183e7	0.939480	0.469740	+	0.882805i	$0.344348\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.55886e7	0.530591	0.265296	−	0.964167i	$-0.414530\pi$
$971$	1.55886e7	0.530591	0.265296	+	0.964167i	$0.414530\pi$
$972$	0	0
$973$	3.05521e7	1.03457
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.18561e6	0.173805	0.0869027	−	0.996217i	$-0.472303\pi$
$977$	5.18561e6	0.173805	0.0869027	+	0.996217i	$0.472303\pi$
$978$	0	0
$979$	−1.19785e7	−0.399434
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.33252e7	1.76015	0.880073	−	0.474838i	$-0.157493\pi$
$983$	5.33252e7	1.76015	0.880073	+	0.474838i	$0.157493\pi$
$984$	0	0
$985$	−1.30257e7	−0.427772
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.68182e7	1.52203
$990$	0	0
$991$	−1.14778e7	−0.371256	−0.185628	−	0.982620i	$-0.559432\pi$
$991$	−1.14778e7	−0.371256	−0.185628	+	0.982620i	$0.559432\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.88033e7	0.602109
$996$	0	0
$997$	−1.20931e7	−0.385301	−0.192651	−	0.981267i	$-0.561708\pi$
$997$	−1.20931e7	−0.385301	−0.192651	+	0.981267i	$0.561708\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.a (trivial)

\(f(q)\)	\(=\)	\(q+25.0000 q^{5} +154.700 q^{7} +O(q^{10})\)	\(q+25.0000 q^{5} +154.700 q^{7} +254.700 q^{11} +928.198 q^{13} +1073.00 q^{17} -1123.00 q^{19} +3909.99 q^{23} +625.000 q^{25} -211.202 q^{29} +5513.99 q^{31} +3867.49 q^{35} -11707.8 q^{37} +10804.4 q^{41} +11974.0 q^{43} -25863.9 q^{47} +7124.97 q^{49} -22017.9 q^{53} +6367.49 q^{55} +33603.8 q^{59} +4019.95 q^{61} +23204.9 q^{65} -20401.3 q^{67} -7553.35 q^{71} -76116.7 q^{73} +39401.9 q^{77} +79357.9 q^{79} +18189.0 q^{83} +26824.9 q^{85} -47029.9 q^{89} +143592. q^{91} -28074.9 q^{95} -157218. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{5} + 80 q^{7}+O(q^{10}) \) 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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