LMFDB - Embedded newform 784.6.a.y.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-118] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+13.5647 q^{3} +94.9526 q^{5} -59.0000 q^{9} -476.000 q^{11} -963.091 q^{13} +1288.00 q^{15} +895.268 q^{17} +637.539 q^{19} -3696.00 q^{23} +5891.00 q^{25} -4096.53 q^{27} +1394.00 q^{29} +1926.18 q^{31} -6456.78 q^{33} +12090.0 q^{37} -13064.0 q^{39} -15219.5 q^{41} -9724.00 q^{43} -5602.20 q^{45} -29272.5 q^{47} +12144.0 q^{51} +4310.00 q^{53} -45197.4 q^{55} +8648.00 q^{57} -20848.9 q^{59} -9291.79 q^{61} -91448.0 q^{65} -20236.0 q^{67} -50135.0 q^{69} -29792.0 q^{71} -11285.8 q^{73} +79909.4 q^{75} +33176.0 q^{79} -41231.0 q^{81} -3540.38 q^{83} +85008.0 q^{85} +18909.1 q^{87} +70753.3 q^{89} +26128.0 q^{93} +60536.0 q^{95} +17769.7 q^{97} +28084.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 118 q^{9} - 952 q^{11} + 2576 q^{15} - 7392 q^{23} + 11782 q^{25} + 2788 q^{29} + 24180 q^{37} - 26128 q^{39} - 19448 q^{43} + 24288 q^{51} + 8620 q^{53} + 17296 q^{57} - 182896 q^{65} - 40472 q^{67}+ \cdots + 56168 q^{99}+O(q^{100}) $ 2 * q - 118 * q^9 - 952 * q^11 + 2576 * q^15 - 7392 * q^23 + 11782 * q^25 + 2788 * q^29 + 24180 * q^37 - 26128 * q^39 - 19448 * q^43 + 24288 * q^51 + 8620 * q^53 + 17296 * q^57 - 182896 * q^65 - 40472 * q^67 - 59584 * q^71 + 66352 * q^79 - 82462 * q^81 + 170016 * q^85 + 52256 * q^93 + 121072 * q^95 + 56168 * 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
$3$	13.5647	0.870173	0.435087	−	0.900389i	$-0.356718\pi$
$3$	13.5647	0.870173	0.435087	+	0.900389i	$0.356718\pi$
$4$	0	0
$5$	94.9526	1.69856	0.849282	−	0.527939i	$-0.177035\pi$
$5$	94.9526	1.69856	0.849282	+	0.527939i	$0.177035\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−59.0000	−0.242798
$10$	0	0
$11$	−476.000	−1.18611	−0.593055	−	0.805162i	$-0.702078\pi$
$11$	−476.000	−1.18611	−0.593055	+	0.805162i	$0.702078\pi$
$12$	0	0
$13$	−963.091	−1.58055	−0.790276	−	0.612751i	$-0.790063\pi$
$13$	−963.091	−1.58055	−0.790276	+	0.612751i	$0.790063\pi$
$14$	0	0
$15$	1288.00	1.47805
$16$	0	0
$17$	895.268	0.751330	0.375665	−	0.926756i	$-0.377414\pi$
$17$	895.268	0.751330	0.375665	+	0.926756i	$0.377414\pi$
$18$	0	0
$19$	637.539	0.405156	0.202578	−	0.979266i	$-0.435068\pi$
$19$	637.539	0.405156	0.202578	+	0.979266i	$0.435068\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3696.00	−1.45684	−0.728421	−	0.685130i	$-0.759745\pi$
$23$	−3696.00	−1.45684	−0.728421	+	0.685130i	$0.759745\pi$
$24$	0	0
$25$	5891.00	1.88512
$26$	0	0
$27$	−4096.53	−1.08145
$28$	0	0
$29$	1394.00	0.307799	0.153900	−	0.988086i	$-0.450817\pi$
$29$	1394.00	0.307799	0.153900	+	0.988086i	$0.450817\pi$
$30$	0	0
$31$	1926.18	0.359992	0.179996	−	0.983667i	$-0.442391\pi$
$31$	1926.18	0.359992	0.179996	+	0.983667i	$0.442391\pi$
$32$	0	0
$33$	−6456.78	−1.03212
$34$	0	0
$35$	0	0
$36$	0	0
$37$	12090.0	1.45185	0.725925	−	0.687773i	$-0.241412\pi$
$37$	12090.0	1.45185	0.725925	+	0.687773i	$0.241412\pi$
$38$	0	0
$39$	−13064.0	−1.37535
$40$	0	0
$41$	−15219.5	−1.41398	−0.706988	−	0.707225i	$-0.749947\pi$
$41$	−15219.5	−1.41398	−0.706988	+	0.707225i	$0.749947\pi$
$42$	0	0
$43$	−9724.00	−0.801999	−0.400999	−	0.916078i	$-0.631337\pi$
$43$	−9724.00	−0.801999	−0.400999	+	0.916078i	$0.631337\pi$
$44$	0	0
$45$	−5602.20	−0.412409
$46$	0	0
$47$	−29272.5	−1.93293	−0.966464	−	0.256802i	$-0.917331\pi$
$47$	−29272.5	−1.93293	−0.966464	+	0.256802i	$0.917331\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12144.0	0.653787
$52$	0	0
$53$	4310.00	0.210760	0.105380	−	0.994432i	$-0.466394\pi$
$53$	4310.00	0.210760	0.105380	+	0.994432i	$0.466394\pi$
$54$	0	0
$55$	−45197.4	−2.01469
$56$	0	0
$57$	8648.00	0.352556
$58$	0	0
$59$	−20848.9	−0.779745	−0.389873	−	0.920869i	$-0.627481\pi$
$59$	−20848.9	−0.779745	−0.389873	+	0.920869i	$0.627481\pi$
$60$	0	0
$61$	−9291.79	−0.319724	−0.159862	−	0.987139i	$-0.551105\pi$
$61$	−9291.79	−0.319724	−0.159862	+	0.987139i	$0.551105\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−91448.0	−2.68467
$66$	0	0
$67$	−20236.0	−0.550729	−0.275364	−	0.961340i	$-0.588798\pi$
$67$	−20236.0	−0.550729	−0.275364	+	0.961340i	$0.588798\pi$
$68$	0	0
$69$	−50135.0	−1.26770
$70$	0	0
$71$	−29792.0	−0.701381	−0.350690	−	0.936491i	$-0.614053\pi$
$71$	−29792.0	−0.701381	−0.350690	+	0.936491i	$0.614053\pi$
$72$	0	0
$73$	−11285.8	−0.247871	−0.123935	−	0.992290i	$-0.539552\pi$
$73$	−11285.8	−0.247871	−0.123935	+	0.992290i	$0.539552\pi$
$74$	0	0
$75$	79909.4	1.64038
$76$	0	0
$77$	0	0
$78$	0	0
$79$	33176.0	0.598076	0.299038	−	0.954241i	$-0.403334\pi$
$79$	33176.0	0.598076	0.299038	+	0.954241i	$0.403334\pi$
$80$	0	0
$81$	−41231.0	−0.698251
$82$	0	0
$83$	−3540.38	−0.0564098	−0.0282049	−	0.999602i	$-0.508979\pi$
$83$	−3540.38	−0.0564098	−0.0282049	+	0.999602i	$0.508979\pi$
$84$	0	0
$85$	85008.0	1.27618
$86$	0	0
$87$	18909.1	0.267839
$88$	0	0
$89$	70753.3	0.946829	0.473414	−	0.880840i	$-0.343021\pi$
$89$	70753.3	0.946829	0.473414	+	0.880840i	$0.343021\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	26128.0	0.313256
$94$	0	0
$95$	60536.0	0.688184
$96$	0	0
$97$	17769.7	0.191757	0.0958784	−	0.995393i	$-0.469434\pi$
$97$	17769.7	0.191757	0.0958784	+	0.995393i	$0.469434\pi$
$98$	0	0
$99$	28084.0	0.287986
$100$	0	0
$101$	−89920.1	−0.877109	−0.438554	−	0.898705i	$-0.644509\pi$
$101$	−89920.1	−0.877109	−0.438554	+	0.898705i	$0.644509\pi$
$102$	0	0
$103$	−8111.67	−0.0753385	−0.0376693	−	0.999290i	$-0.511993\pi$
$103$	−8111.67	−0.0753385	−0.0376693	+	0.999290i	$0.511993\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−125908.	−1.06315	−0.531574	−	0.847012i	$-0.678399\pi$
$107$	−125908.	−1.06315	−0.531574	+	0.847012i	$0.678399\pi$
$108$	0	0
$109$	89170.0	0.718874	0.359437	−	0.933169i	$-0.382969\pi$
$109$	89170.0	0.718874	0.359437	+	0.933169i	$0.382969\pi$
$110$	0	0
$111$	163997.	1.26336
$112$	0	0
$113$	−29702.0	−0.218821	−0.109411	−	0.993997i	$-0.534896\pi$
$113$	−29702.0	−0.218821	−0.109411	+	0.993997i	$0.534896\pi$
$114$	0	0
$115$	−350945.	−2.47454
$116$	0	0
$117$	56822.4	0.383756
$118$	0	0
$119$	0	0
$120$	0	0
$121$	65525.0	0.406859
$122$	0	0
$123$	−206448.	−1.23040
$124$	0	0
$125$	262639.	1.50343
$126$	0	0
$127$	243112.	1.33751	0.668755	−	0.743483i	$-0.266827\pi$
$127$	243112.	1.33751	0.668755	+	0.743483i	$0.266827\pi$
$128$	0	0
$129$	−131903.	−0.697878
$130$	0	0
$131$	107147.	0.545510	0.272755	−	0.962084i	$-0.412065\pi$
$131$	107147.	0.545510	0.272755	+	0.962084i	$0.412065\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−388976.	−1.83691
$136$	0	0
$137$	−332842.	−1.51508	−0.757542	−	0.652786i	$-0.773600\pi$
$137$	−332842.	−1.51508	−0.757542	+	0.652786i	$0.773600\pi$
$138$	0	0
$139$	−186582.	−0.819092	−0.409546	−	0.912290i	$-0.634313\pi$
$139$	−186582.	−0.819092	−0.409546	+	0.912290i	$0.634313\pi$
$140$	0	0
$141$	−397072.	−1.68198
$142$	0	0
$143$	458431.	1.87471
$144$	0	0
$145$	132364.	0.522817
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−202330.	−0.746611	−0.373306	−	0.927708i	$-0.621776\pi$
$149$	−202330.	−0.746611	−0.373306	+	0.927708i	$0.621776\pi$
$150$	0	0
$151$	−345760.	−1.23405	−0.617024	−	0.786944i	$-0.711662\pi$
$151$	−345760.	−1.23405	−0.617024	+	0.786944i	$0.711662\pi$
$152$	0	0
$153$	−52820.8	−0.182422
$154$	0	0
$155$	182896.	0.611470
$156$	0	0
$157$	82866.5	0.268306	0.134153	−	0.990961i	$-0.457169\pi$
$157$	82866.5	0.268306	0.134153	+	0.990961i	$0.457169\pi$
$158$	0	0
$159$	58463.7	0.183397
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−127476.	−0.375802	−0.187901	−	0.982188i	$-0.560168\pi$
$163$	−127476.	−0.375802	−0.187901	+	0.982188i	$0.560168\pi$
$164$	0	0
$165$	−613088.	−1.75313
$166$	0	0
$167$	671261.	1.86252	0.931258	−	0.364360i	$-0.118712\pi$
$167$	671261.	1.86252	0.931258	+	0.364360i	$0.118712\pi$
$168$	0	0
$169$	556251.	1.49815
$170$	0	0
$171$	−37614.8	−0.0983713
$172$	0	0
$173$	−347784.	−0.883476	−0.441738	−	0.897144i	$-0.645638\pi$
$173$	−347784.	−0.883476	−0.441738	+	0.897144i	$0.645638\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−282808.	−0.678514
$178$	0	0
$179$	625236.	1.45852	0.729258	−	0.684238i	$-0.239865\pi$
$179$	625236.	1.45852	0.729258	+	0.684238i	$0.239865\pi$
$180$	0	0
$181$	−241492.	−0.547906	−0.273953	−	0.961743i	$-0.588331\pi$
$181$	−241492.	−0.547906	−0.273953	+	0.961743i	$0.588331\pi$
$182$	0	0
$183$	−126040.	−0.278215
$184$	0	0
$185$	1.14798e6	2.46606
$186$	0	0
$187$	−426147.	−0.891160
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−212952.	−0.422375	−0.211188	−	0.977446i	$-0.567733\pi$
$191$	−212952.	−0.422375	−0.211188	+	0.977446i	$0.567733\pi$
$192$	0	0
$193$	135002.	0.260884	0.130442	−	0.991456i	$-0.458360\pi$
$193$	135002.	0.260884	0.130442	+	0.991456i	$0.458360\pi$
$194$	0	0
$195$	−1.24046e6	−2.33613
$196$	0	0
$197$	548838.	1.00758	0.503789	−	0.863827i	$-0.331939\pi$
$197$	548838.	1.00758	0.503789	+	0.863827i	$0.331939\pi$
$198$	0	0
$199$	631869.	1.13108	0.565541	−	0.824720i	$-0.308667\pi$
$199$	631869.	1.13108	0.565541	+	0.824720i	$0.308667\pi$
$200$	0	0
$201$	−274494.	−0.479229
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.44514e6	−2.40173
$206$	0	0
$207$	218064.	0.353719
$208$	0	0
$209$	−303469.	−0.480560
$210$	0	0
$211$	159940.	0.247315	0.123658	−	0.992325i	$-0.460538\pi$
$211$	159940.	0.247315	0.123658	+	0.992325i	$0.460538\pi$
$212$	0	0
$213$	−404118.	−0.610323
$214$	0	0
$215$	−923319.	−1.36225
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−153088.	−0.215690
$220$	0	0
$221$	−862224.	−1.18752
$222$	0	0
$223$	876765.	1.18065	0.590325	−	0.807166i	$-0.299000\pi$
$223$	876765.	1.18065	0.590325	+	0.807166i	$0.299000\pi$
$224$	0	0
$225$	−347569.	−0.457704
$226$	0	0
$227$	−218513.	−0.281458	−0.140729	−	0.990048i	$-0.544945\pi$
$227$	−218513.	−0.281458	−0.140729	+	0.990048i	$0.544945\pi$
$228$	0	0
$229$	−1.42786e6	−1.79927	−0.899634	−	0.436644i	$-0.856167\pi$
$229$	−1.42786e6	−1.79927	−0.899634	+	0.436644i	$0.856167\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−429418.	−0.518192	−0.259096	−	0.965852i	$-0.583425\pi$
$233$	−429418.	−0.518192	−0.259096	+	0.965852i	$0.583425\pi$
$234$	0	0
$235$	−2.77950e6	−3.28320
$236$	0	0
$237$	450021.	0.520430
$238$	0	0
$239$	338328.	0.383127	0.191564	−	0.981480i	$-0.438644\pi$
$239$	338328.	0.383127	0.191564	+	0.981480i	$0.438644\pi$
$240$	0	0
$241$	−536645.	−0.595175	−0.297587	−	0.954695i	$-0.596182\pi$
$241$	−536645.	−0.595175	−0.297587	+	0.954695i	$0.596182\pi$
$242$	0	0
$243$	436172.	0.473851
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−614008.	−0.640371
$248$	0	0
$249$	−48024.0	−0.0490863
$250$	0	0
$251$	−548813.	−0.549844	−0.274922	−	0.961466i	$-0.588652\pi$
$251$	−548813.	−0.549844	−0.274922	+	0.961466i	$0.588652\pi$
$252$	0	0
$253$	1.75930e6	1.72798
$254$	0	0
$255$	1.15310e6	1.11050
$256$	0	0
$257$	1.69591e6	1.60166	0.800828	−	0.598894i	$-0.204393\pi$
$257$	1.69591e6	1.60166	0.800828	+	0.598894i	$0.204393\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−82246.0	−0.0747332
$262$	0	0
$263$	−1.96616e6	−1.75279	−0.876394	−	0.481594i	$-0.840058\pi$
$263$	−1.96616e6	−1.75279	−0.876394	+	0.481594i	$0.840058\pi$
$264$	0	0
$265$	409246.	0.357989
$266$	0	0
$267$	959744.	0.823905
$268$	0	0
$269$	431858.	0.363882	0.181941	−	0.983309i	$-0.441762\pi$
$269$	431858.	0.363882	0.181941	+	0.983309i	$0.441762\pi$
$270$	0	0
$271$	−590768.	−0.488645	−0.244323	−	0.969694i	$-0.578566\pi$
$271$	−590768.	−0.488645	−0.244323	+	0.969694i	$0.578566\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.80412e6	−2.23596
$276$	0	0
$277$	828262.	0.648587	0.324294	−	0.945956i	$-0.394873\pi$
$277$	828262.	0.648587	0.324294	+	0.945956i	$0.394873\pi$
$278$	0	0
$279$	−113645.	−0.0874055
$280$	0	0
$281$	−719130.	−0.543302	−0.271651	−	0.962396i	$-0.587570\pi$
$281$	−719130.	−0.543302	−0.271651	+	0.962396i	$0.587570\pi$
$282$	0	0
$283$	−2.12413e6	−1.57658	−0.788288	−	0.615306i	$-0.789033\pi$
$283$	−2.12413e6	−1.57658	−0.788288	+	0.615306i	$0.789033\pi$
$284$	0	0
$285$	821150.	0.598840
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−618353.	−0.435504
$290$	0	0
$291$	241040.	0.166862
$292$	0	0
$293$	313710.	0.213481	0.106740	−	0.994287i	$-0.465959\pi$
$293$	313710.	0.213481	0.106740	+	0.994287i	$0.465959\pi$
$294$	0	0
$295$	−1.97966e6	−1.32445
$296$	0	0
$297$	1.94995e6	1.28272
$298$	0	0
$299$	3.55958e6	2.30261
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.21974e6	−0.763237
$304$	0	0
$305$	−882280.	−0.543071
$306$	0	0
$307$	2.82279e6	1.70936	0.854679	−	0.519157i	$-0.173754\pi$
$307$	2.82279e6	1.70936	0.854679	+	0.519157i	$0.173754\pi$
$308$	0	0
$309$	−110032.	−0.0655576
$310$	0	0
$311$	−441557.	−0.258872	−0.129436	−	0.991588i	$-0.541317\pi$
$311$	−441557.	−0.258872	−0.129436	+	0.991588i	$0.541317\pi$
$312$	0	0
$313$	1.42339e6	0.821229	0.410615	−	0.911809i	$-0.365314\pi$
$313$	1.42339e6	0.821229	0.410615	+	0.911809i	$0.365314\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.25360e6	0.700665	0.350332	−	0.936625i	$-0.386069\pi$
$317$	1.25360e6	0.700665	0.350332	+	0.936625i	$0.386069\pi$
$318$	0	0
$319$	−663544.	−0.365084
$320$	0	0
$321$	−1.70790e6	−0.925123
$322$	0	0
$323$	570768.	0.304406
$324$	0	0
$325$	−5.67357e6	−2.97953
$326$	0	0
$327$	1.20956e6	0.625545
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3.38223e6	1.69681	0.848404	−	0.529349i	$-0.177564\pi$
$331$	3.38223e6	1.69681	0.848404	+	0.529349i	$0.177564\pi$
$332$	0	0
$333$	−713310.	−0.352507
$334$	0	0
$335$	−1.92146e6	−0.935448
$336$	0	0
$337$	1.94529e6	0.933058	0.466529	−	0.884506i	$-0.345504\pi$
$337$	1.94529e6	0.933058	0.466529	+	0.884506i	$0.345504\pi$
$338$	0	0
$339$	−402898.	−0.190412
$340$	0	0
$341$	−916862.	−0.426991
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.76045e6	−2.15328
$346$	0	0
$347$	2.08232e6	0.928378	0.464189	−	0.885736i	$-0.346346\pi$
$347$	2.08232e6	0.928378	0.464189	+	0.885736i	$0.346346\pi$
$348$	0	0
$349$	3.15765e6	1.38772	0.693858	−	0.720112i	$-0.255910\pi$
$349$	3.15765e6	1.38772	0.693858	+	0.720112i	$0.255910\pi$
$350$	0	0
$351$	3.94533e6	1.70929
$352$	0	0
$353$	−2.55021e6	−1.08928	−0.544640	−	0.838670i	$-0.683334\pi$
$353$	−2.55021e6	−1.08928	−0.544640	+	0.838670i	$0.683334\pi$
$354$	0	0
$355$	−2.82883e6	−1.19134
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.43467e6	−0.997021	−0.498511	−	0.866884i	$-0.666119\pi$
$359$	−2.43467e6	−0.997021	−0.498511	+	0.866884i	$0.666119\pi$
$360$	0	0
$361$	−2.06964e6	−0.835848
$362$	0	0
$363$	888824.	0.354038
$364$	0	0
$365$	−1.07162e6	−0.421024
$366$	0	0
$367$	−4.33141e6	−1.67867	−0.839333	−	0.543617i	$-0.817054\pi$
$367$	−4.33141e6	−1.67867	−0.839333	+	0.543617i	$0.817054\pi$
$368$	0	0
$369$	897953.	0.343311
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.66435e6	−1.36372	−0.681859	−	0.731484i	$-0.738828\pi$
$373$	−3.66435e6	−1.36372	−0.681859	+	0.731484i	$0.738828\pi$
$374$	0	0
$375$	3.56261e6	1.30825
$376$	0	0
$377$	−1.34255e6	−0.486493
$378$	0	0
$379$	1.24352e6	0.444689	0.222344	−	0.974968i	$-0.428629\pi$
$379$	1.24352e6	0.444689	0.222344	+	0.974968i	$0.428629\pi$
$380$	0	0
$381$	3.29773e6	1.16387
$382$	0	0
$383$	1.50527e6	0.524346	0.262173	−	0.965021i	$-0.415561\pi$
$383$	1.50527e6	0.524346	0.262173	+	0.965021i	$0.415561\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	573716.	0.194724
$388$	0	0
$389$	−3.85930e6	−1.29311	−0.646554	−	0.762868i	$-0.723790\pi$
$389$	−3.85930e6	−1.29311	−0.646554	+	0.762868i	$0.723790\pi$
$390$	0	0
$391$	−3.30891e6	−1.09457
$392$	0	0
$393$	1.45342e6	0.474688
$394$	0	0
$395$	3.15015e6	1.01587
$396$	0	0
$397$	−3.32013e6	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$397$	−3.32013e6	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.69337e6	1.14699	0.573497	−	0.819207i	$-0.305586\pi$
$401$	3.69337e6	1.14699	0.573497	+	0.819207i	$0.305586\pi$
$402$	0	0
$403$	−1.85509e6	−0.568986
$404$	0	0
$405$	−3.91499e6	−1.18602
$406$	0	0
$407$	−5.75484e6	−1.72206
$408$	0	0
$409$	4.04213e6	1.19482	0.597410	−	0.801936i	$-0.296197\pi$
$409$	4.04213e6	1.19482	0.597410	+	0.801936i	$0.296197\pi$
$410$	0	0
$411$	−4.51489e6	−1.31839
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−336168.	−0.0958156
$416$	0	0
$417$	−2.53092e6	−0.712752
$418$	0	0
$419$	−2.42383e6	−0.674476	−0.337238	−	0.941419i	$-0.609493\pi$
$419$	−2.42383e6	−0.674476	−0.337238	+	0.941419i	$0.609493\pi$
$420$	0	0
$421$	−6.65639e6	−1.83035	−0.915174	−	0.403058i	$-0.867947\pi$
$421$	−6.65639e6	−1.83035	−0.915174	+	0.403058i	$0.867947\pi$
$422$	0	0
$423$	1.72708e6	0.469312
$424$	0	0
$425$	5.27402e6	1.41635
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.21846e6	1.63132
$430$	0	0
$431$	610520.	0.158309	0.0791547	−	0.996862i	$-0.474778\pi$
$431$	610520.	0.158309	0.0791547	+	0.996862i	$0.474778\pi$
$432$	0	0
$433$	−29977.9	−0.00768390	−0.00384195	−	0.999993i	$-0.501223\pi$
$433$	−29977.9	−0.00768390	−0.00384195	+	0.999993i	$0.501223\pi$
$434$	0	0
$435$	1.79547e6	0.454941
$436$	0	0
$437$	−2.35634e6	−0.590249
$438$	0	0
$439$	369827.	0.0915877	0.0457939	−	0.998951i	$-0.485418\pi$
$439$	369827.	0.0915877	0.0457939	+	0.998951i	$0.485418\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	639036.	0.154709	0.0773546	−	0.997004i	$-0.475353\pi$
$443$	639036.	0.154709	0.0773546	+	0.997004i	$0.475353\pi$
$444$	0	0
$445$	6.71821e6	1.60825
$446$	0	0
$447$	−2.74454e6	−0.649681
$448$	0	0
$449$	1.90682e6	0.446368	0.223184	−	0.974776i	$-0.428355\pi$
$449$	1.90682e6	0.446368	0.223184	+	0.974776i	$0.428355\pi$
$450$	0	0
$451$	7.24451e6	1.67713
$452$	0	0
$453$	−4.69012e6	−1.07384
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.60039e6	1.47836	0.739178	−	0.673510i	$-0.235214\pi$
$457$	6.60039e6	1.47836	0.739178	+	0.673510i	$0.235214\pi$
$458$	0	0
$459$	−3.66749e6	−0.812525
$460$	0	0
$461$	−4.93085e6	−1.08061	−0.540305	−	0.841469i	$-0.681691\pi$
$461$	−4.93085e6	−1.08061	−0.540305	+	0.841469i	$0.681691\pi$
$462$	0	0
$463$	−257576.	−0.0558410	−0.0279205	−	0.999610i	$-0.508889\pi$
$463$	−257576.	−0.0558410	−0.0279205	+	0.999610i	$0.508889\pi$
$464$	0	0
$465$	2.48092e6	0.532085
$466$	0	0
$467$	541976.	0.114997	0.0574987	−	0.998346i	$-0.481688\pi$
$467$	541976.	0.114997	0.0574987	+	0.998346i	$0.481688\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.12406e6	0.233472
$472$	0	0
$473$	4.62862e6	0.951260
$474$	0	0
$475$	3.75574e6	0.763769
$476$	0	0
$477$	−254290.	−0.0511721
$478$	0	0
$479$	−6.28828e6	−1.25226	−0.626128	−	0.779720i	$-0.715361\pi$
$479$	−6.28828e6	−1.25226	−0.626128	+	0.779720i	$0.715361\pi$
$480$	0	0
$481$	−1.16438e7	−2.29473
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.68728e6	0.325711
$486$	0	0
$487$	6.18478e6	1.18169	0.590843	−	0.806787i	$-0.298795\pi$
$487$	6.18478e6	1.18169	0.590843	+	0.806787i	$0.298795\pi$
$488$	0	0
$489$	−1.72917e6	−0.327013
$490$	0	0
$491$	4.93753e6	0.924286	0.462143	−	0.886806i	$-0.347081\pi$
$491$	4.93753e6	0.924286	0.462143	+	0.886806i	$0.347081\pi$
$492$	0	0
$493$	1.24800e6	0.231259
$494$	0	0
$495$	2.66665e6	0.489162
$496$	0	0
$497$	0	0
$498$	0	0
$499$	742212.	0.133437	0.0667186	−	0.997772i	$-0.478747\pi$
$499$	742212.	0.133437	0.0667186	+	0.997772i	$0.478747\pi$
$500$	0	0
$501$	9.10542e6	1.62071
$502$	0	0
$503$	2.92872e6	0.516128	0.258064	−	0.966128i	$-0.416915\pi$
$503$	2.92872e6	0.516128	0.258064	+	0.966128i	$0.416915\pi$
$504$	0	0
$505$	−8.53815e6	−1.48983
$506$	0	0
$507$	7.54536e6	1.30365
$508$	0	0
$509$	6.05761e6	1.03635	0.518176	−	0.855274i	$-0.326611\pi$
$509$	6.05761e6	1.03635	0.518176	+	0.855274i	$0.326611\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.61170e6	−0.438156
$514$	0	0
$515$	−770224.	−0.127967
$516$	0	0
$517$	1.39337e7	2.29267
$518$	0	0
$519$	−4.71758e6	−0.768777
$520$	0	0
$521$	4.99920e6	0.806875	0.403438	−	0.915007i	$-0.367815\pi$
$521$	4.99920e6	0.806875	0.403438	+	0.915007i	$0.367815\pi$
$522$	0	0
$523$	−9.19002e6	−1.46914	−0.734568	−	0.678535i	$-0.762615\pi$
$523$	−9.19002e6	−1.46914	−0.734568	+	0.678535i	$0.762615\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.72445e6	0.270473
$528$	0	0
$529$	7.22407e6	1.12239
$530$	0	0
$531$	1.23008e6	0.189321
$532$	0	0
$533$	1.46578e7	2.23486
$534$	0	0
$535$	−1.19553e7	−1.80583
$536$	0	0
$537$	8.48111e6	1.26916
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.53883e6	−1.10742	−0.553708	−	0.832711i	$-0.686788\pi$
$541$	−7.53883e6	−1.10742	−0.553708	+	0.832711i	$0.686788\pi$
$542$	0	0
$543$	−3.27575e6	−0.476773
$544$	0	0
$545$	8.46693e6	1.22105
$546$	0	0
$547$	−3.73311e6	−0.533460	−0.266730	−	0.963771i	$-0.585943\pi$
$547$	−3.73311e6	−0.533460	−0.266730	+	0.963771i	$0.585943\pi$
$548$	0	0
$549$	548216.	0.0776284
$550$	0	0
$551$	888729.	0.124707
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.55719e7	2.14590
$556$	0	0
$557$	−7.95391e6	−1.08628	−0.543141	−	0.839642i	$-0.682765\pi$
$557$	−7.95391e6	−1.08628	−0.543141	+	0.839642i	$0.682765\pi$
$558$	0	0
$559$	9.36510e6	1.26760
$560$	0	0
$561$	−5.78054e6	−0.775464
$562$	0	0
$563$	8.85242e6	1.17704	0.588520	−	0.808483i	$-0.299711\pi$
$563$	8.85242e6	1.17704	0.588520	+	0.808483i	$0.299711\pi$
$564$	0	0
$565$	−2.82028e6	−0.371682
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.43774e6	0.833591	0.416795	−	0.909000i	$-0.363153\pi$
$569$	6.43774e6	0.833591	0.416795	+	0.909000i	$0.363153\pi$
$570$	0	0
$571$	1.06947e7	1.37271	0.686353	−	0.727269i	$-0.259211\pi$
$571$	1.06947e7	1.37271	0.686353	+	0.727269i	$0.259211\pi$
$572$	0	0
$573$	−2.88862e6	−0.367540
$574$	0	0
$575$	−2.17731e7	−2.74632
$576$	0	0
$577$	−1.30088e7	−1.62666	−0.813330	−	0.581802i	$-0.802348\pi$
$577$	−1.30088e7	−1.62666	−0.813330	+	0.581802i	$0.802348\pi$
$578$	0	0
$579$	1.83126e6	0.227014
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.05156e6	−0.249984
$584$	0	0
$585$	5.39543e6	0.651833
$586$	0	0
$587$	1.27410e7	1.52619	0.763096	−	0.646285i	$-0.223678\pi$
$587$	1.27410e7	1.52619	0.763096	+	0.646285i	$0.223678\pi$
$588$	0	0
$589$	1.22802e6	0.145853
$590$	0	0
$591$	7.44480e6	0.876767
$592$	0	0
$593$	−2.71592e6	−0.317161	−0.158580	−	0.987346i	$-0.550692\pi$
$593$	−2.71592e6	−0.317161	−0.158580	+	0.987346i	$0.550692\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.57109e6	0.984238
$598$	0	0
$599$	1.04748e7	1.19284	0.596418	−	0.802674i	$-0.296590\pi$
$599$	1.04748e7	1.19284	0.596418	+	0.802674i	$0.296590\pi$
$600$	0	0
$601$	−754738.	−0.0852334	−0.0426167	−	0.999091i	$-0.513569\pi$
$601$	−754738.	−0.0852334	−0.0426167	+	0.999091i	$0.513569\pi$
$602$	0	0
$603$	1.19392e6	0.133716
$604$	0	0
$605$	6.22177e6	0.691076
$606$	0	0
$607$	9.17503e6	1.01073	0.505366	−	0.862905i	$-0.331358\pi$
$607$	9.17503e6	1.01073	0.505366	+	0.862905i	$0.331358\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.81921e7	3.05509
$612$	0	0
$613$	−8.85554e6	−0.951840	−0.475920	−	0.879489i	$-0.657885\pi$
$613$	−8.85554e6	−0.951840	−0.475920	+	0.879489i	$0.657885\pi$
$614$	0	0
$615$	−1.96028e7	−2.08992
$616$	0	0
$617$	7.50753e6	0.793934	0.396967	−	0.917833i	$-0.370063\pi$
$617$	7.50753e6	0.793934	0.396967	+	0.917833i	$0.370063\pi$
$618$	0	0
$619$	−1.01449e7	−1.06420	−0.532098	−	0.846683i	$-0.678596\pi$
$619$	−1.01449e7	−1.06420	−0.532098	+	0.846683i	$0.678596\pi$
$620$	0	0
$621$	1.51408e7	1.57550
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.52888e6	0.668557
$626$	0	0
$627$	−4.11645e6	−0.418171
$628$	0	0
$629$	1.08238e7	1.09082
$630$	0	0
$631$	9.28258e6	0.928101	0.464050	−	0.885809i	$-0.346396\pi$
$631$	9.28258e6	0.928101	0.464050	+	0.885809i	$0.346396\pi$
$632$	0	0
$633$	2.16953e6	0.215207
$634$	0	0
$635$	2.30841e7	2.27185
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.75773e6	0.170294
$640$	0	0
$641$	1.70740e7	1.64130	0.820652	−	0.571428i	$-0.193610\pi$
$641$	1.70740e7	1.64130	0.820652	+	0.571428i	$0.193610\pi$
$642$	0	0
$643$	2.73150e6	0.260540	0.130270	−	0.991479i	$-0.458416\pi$
$643$	2.73150e6	0.260540	0.130270	+	0.991479i	$0.458416\pi$
$644$	0	0
$645$	−1.25245e7	−1.18539
$646$	0	0
$647$	5.66533e6	0.532065	0.266033	−	0.963964i	$-0.414287\pi$
$647$	5.66533e6	0.532065	0.266033	+	0.963964i	$0.414287\pi$
$648$	0	0
$649$	9.92407e6	0.924864
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.13187e6	0.746290	0.373145	−	0.927773i	$-0.378279\pi$
$653$	8.13187e6	0.746290	0.373145	+	0.927773i	$0.378279\pi$
$654$	0	0
$655$	1.01739e7	0.926584
$656$	0	0
$657$	665862.	0.0601826
$658$	0	0
$659$	1.99012e6	0.178512	0.0892558	−	0.996009i	$-0.471551\pi$
$659$	1.99012e6	0.178512	0.0892558	+	0.996009i	$0.471551\pi$
$660$	0	0
$661$	−3.18755e6	−0.283761	−0.141881	−	0.989884i	$-0.545315\pi$
$661$	−3.18755e6	−0.283761	−0.141881	+	0.989884i	$0.545315\pi$
$662$	0	0
$663$	−1.16958e7	−1.03334
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.15222e6	−0.448415
$668$	0	0
$669$	1.18930e7	1.02737
$670$	0	0
$671$	4.42289e6	0.379228
$672$	0	0
$673$	−1.72276e7	−1.46618	−0.733090	−	0.680131i	$-0.761923\pi$
$673$	−1.72276e7	−1.46618	−0.733090	+	0.680131i	$0.761923\pi$
$674$	0	0
$675$	−2.41326e7	−2.03866
$676$	0	0
$677$	−1.17235e7	−0.983072	−0.491536	−	0.870857i	$-0.663564\pi$
$677$	−1.17235e7	−0.983072	−0.491536	+	0.870857i	$0.663564\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.96406e6	−0.244917
$682$	0	0
$683$	−6.40383e6	−0.525276	−0.262638	−	0.964894i	$-0.584593\pi$
$683$	−6.40383e6	−0.525276	−0.262638	+	0.964894i	$0.584593\pi$
$684$	0	0
$685$	−3.16042e7	−2.57347
$686$	0	0
$687$	−1.93684e7	−1.56568
$688$	0	0
$689$	−4.15092e6	−0.333117
$690$	0	0
$691$	−1.14960e7	−0.915904	−0.457952	−	0.888977i	$-0.651417\pi$
$691$	−1.14960e7	−0.915904	−0.457952	+	0.888977i	$0.651417\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.77164e7	−1.39128
$696$	0	0
$697$	−1.36256e7	−1.06236
$698$	0	0
$699$	−5.82491e6	−0.450917
$700$	0	0
$701$	−1.71167e7	−1.31560	−0.657802	−	0.753191i	$-0.728514\pi$
$701$	−1.71167e7	−1.31560	−0.657802	+	0.753191i	$0.728514\pi$
$702$	0	0
$703$	7.70785e6	0.588227
$704$	0	0
$705$	−3.77030e7	−2.85696
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.69689e7	1.26776	0.633882	−	0.773429i	$-0.281460\pi$
$709$	1.69689e7	1.26776	0.633882	+	0.773429i	$0.281460\pi$
$710$	0	0
$711$	−1.95738e6	−0.145212
$712$	0	0
$713$	−7.11917e6	−0.524452
$714$	0	0
$715$	4.35292e7	3.18432
$716$	0	0
$717$	4.58930e6	0.333387
$718$	0	0
$719$	−2.40621e7	−1.73585	−0.867923	−	0.496698i	$-0.834546\pi$
$719$	−2.40621e7	−1.73585	−0.867923	+	0.496698i	$0.834546\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−7.27941e6	−0.517905
$724$	0	0
$725$	8.21205e6	0.580239
$726$	0	0
$727$	−3.24345e6	−0.227599	−0.113800	−	0.993504i	$-0.536302\pi$
$727$	−3.24345e6	−0.227599	−0.113800	+	0.993504i	$0.536302\pi$
$728$	0	0
$729$	1.59357e7	1.11058
$730$	0	0
$731$	−8.70558e6	−0.602566
$732$	0	0
$733$	2.59315e7	1.78266	0.891328	−	0.453359i	$-0.149774\pi$
$733$	2.59315e7	1.78266	0.891328	+	0.453359i	$0.149774\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.63234e6	0.653225
$738$	0	0
$739$	5.53387e6	0.372750	0.186375	−	0.982479i	$-0.440326\pi$
$739$	5.53387e6	0.372750	0.186375	+	0.982479i	$0.440326\pi$
$740$	0	0
$741$	−8.32881e6	−0.557234
$742$	0	0
$743$	−1.09491e7	−0.727622	−0.363811	−	0.931473i	$-0.618525\pi$
$743$	−1.09491e7	−0.727622	−0.363811	+	0.931473i	$0.618525\pi$
$744$	0	0
$745$	−1.92118e7	−1.26817
$746$	0	0
$747$	208882.	0.0136962
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.93260e6	−0.319136	−0.159568	−	0.987187i	$-0.551010\pi$
$751$	−4.93260e6	−0.319136	−0.159568	+	0.987187i	$0.551010\pi$
$752$	0	0
$753$	−7.44446e6	−0.478460
$754$	0	0
$755$	−3.28308e7	−2.09611
$756$	0	0
$757$	7.69782e6	0.488234	0.244117	−	0.969746i	$-0.421502\pi$
$757$	7.69782e6	0.488234	0.244117	+	0.969746i	$0.421502\pi$
$758$	0	0
$759$	2.38643e7	1.50364
$760$	0	0
$761$	7.27427e6	0.455331	0.227666	−	0.973739i	$-0.426891\pi$
$761$	7.27427e6	0.455331	0.227666	+	0.973739i	$0.426891\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5.01547e6	−0.309855
$766$	0	0
$767$	2.00794e7	1.23243
$768$	0	0
$769$	1.72394e7	1.05125	0.525626	−	0.850716i	$-0.323831\pi$
$769$	1.72394e7	1.05125	0.525626	+	0.850716i	$0.323831\pi$
$770$	0	0
$771$	2.30044e7	1.39372
$772$	0	0
$773$	3.96168e6	0.238468	0.119234	−	0.992866i	$-0.461956\pi$
$773$	3.96168e6	0.238468	0.119234	+	0.992866i	$0.461956\pi$
$774$	0	0
$775$	1.13471e7	0.678628
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.70306e6	−0.572882
$780$	0	0
$781$	1.41810e7	0.831915
$782$	0	0
$783$	−5.71056e6	−0.332870
$784$	0	0
$785$	7.86839e6	0.455734
$786$	0	0
$787$	−3.41283e6	−0.196416	−0.0982082	−	0.995166i	$-0.531311\pi$
$787$	−3.41283e6	−0.196416	−0.0982082	+	0.995166i	$0.531311\pi$
$788$	0	0
$789$	−2.66703e7	−1.52523
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.94884e6	0.505340
$794$	0	0
$795$	5.55128e6	0.311512
$796$	0	0
$797$	−5.38044e6	−0.300035	−0.150017	−	0.988683i	$-0.547933\pi$
$797$	−5.38044e6	−0.300035	−0.150017	+	0.988683i	$0.547933\pi$
$798$	0	0
$799$	−2.62068e7	−1.45227
$800$	0	0
$801$	−4.17444e6	−0.229888
$802$	0	0
$803$	5.37204e6	0.294002
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.85801e6	0.316640
$808$	0	0
$809$	−1.99148e7	−1.06980	−0.534901	−	0.844915i	$-0.679651\pi$
$809$	−1.99148e7	−1.06980	−0.534901	+	0.844915i	$0.679651\pi$
$810$	0	0
$811$	−2.24825e7	−1.20031	−0.600153	−	0.799886i	$-0.704893\pi$
$811$	−2.24825e7	−1.20031	−0.600153	+	0.799886i	$0.704893\pi$
$812$	0	0
$813$	−8.01357e6	−0.425206
$814$	0	0
$815$	−1.21042e7	−0.638324
$816$	0	0
$817$	−6.19943e6	−0.324935
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.88281e7	−0.974872	−0.487436	−	0.873159i	$-0.662068\pi$
$821$	−1.88281e7	−0.974872	−0.487436	+	0.873159i	$0.662068\pi$
$822$	0	0
$823$	9.23000e6	0.475009	0.237505	−	0.971386i	$-0.423671\pi$
$823$	9.23000e6	0.475009	0.237505	+	0.971386i	$0.423671\pi$
$824$	0	0
$825$	−3.80369e7	−1.94567
$826$	0	0
$827$	−3.44089e7	−1.74947	−0.874736	−	0.484600i	$-0.838965\pi$
$827$	−3.44089e7	−1.74947	−0.874736	+	0.484600i	$0.838965\pi$
$828$	0	0
$829$	2.51500e7	1.27102	0.635509	−	0.772093i	$-0.280790\pi$
$829$	2.51500e7	1.27102	0.635509	+	0.772093i	$0.280790\pi$
$830$	0	0
$831$	1.12351e7	0.564383
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.37380e7	3.16360
$836$	0	0
$837$	−7.89066e6	−0.389314
$838$	0	0
$839$	3.27543e7	1.60644	0.803219	−	0.595684i	$-0.203119\pi$
$839$	3.27543e7	1.60644	0.803219	+	0.595684i	$0.203119\pi$
$840$	0	0
$841$	−1.85679e7	−0.905260
$842$	0	0
$843$	−9.75475e6	−0.472767
$844$	0	0
$845$	5.28175e7	2.54470
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.88131e7	−1.37190
$850$	0	0
$851$	−4.46846e7	−2.11512
$852$	0	0
$853$	1.70918e7	0.804295	0.402148	−	0.915575i	$-0.368264\pi$
$853$	1.70918e7	0.804295	0.402148	+	0.915575i	$0.368264\pi$
$854$	0	0
$855$	−3.57162e6	−0.167090
$856$	0	0
$857$	7.19282e6	0.334539	0.167270	−	0.985911i	$-0.446505\pi$
$857$	7.19282e6	0.334539	0.167270	+	0.985911i	$0.446505\pi$
$858$	0	0
$859$	1.49120e7	0.689531	0.344765	−	0.938689i	$-0.387958\pi$
$859$	1.49120e7	0.689531	0.344765	+	0.938689i	$0.387958\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.24943e7	−1.48518	−0.742592	−	0.669744i	$-0.766404\pi$
$863$	−3.24943e7	−1.48518	−0.742592	+	0.669744i	$0.766404\pi$
$864$	0	0
$865$	−3.30230e7	−1.50064
$866$	0	0
$867$	−8.38775e6	−0.378964
$868$	0	0
$869$	−1.57918e7	−0.709384
$870$	0	0
$871$	1.94891e7	0.870455
$872$	0	0
$873$	−1.04841e6	−0.0465582
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.72512e7	−0.757391	−0.378695	−	0.925521i	$-0.623627\pi$
$877$	−1.72512e7	−0.757391	−0.378695	+	0.925521i	$0.623627\pi$
$878$	0	0
$879$	4.25537e6	0.185765
$880$	0	0
$881$	3.45237e7	1.49857	0.749286	−	0.662247i	$-0.230397\pi$
$881$	3.45237e7	1.49857	0.749286	+	0.662247i	$0.230397\pi$
$882$	0	0
$883$	3.99893e7	1.72600	0.863002	−	0.505200i	$-0.168581\pi$
$883$	3.99893e7	1.72600	0.863002	+	0.505200i	$0.168581\pi$
$884$	0	0
$885$	−2.68534e7	−1.15250
$886$	0	0
$887$	−3.71547e7	−1.58564	−0.792819	−	0.609457i	$-0.791388\pi$
$887$	−3.71547e7	−1.58564	−0.792819	+	0.609457i	$0.791388\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.96260e7	0.828203
$892$	0	0
$893$	−1.86624e7	−0.783138
$894$	0	0
$895$	5.93678e7	2.47738
$896$	0	0
$897$	4.82845e7	2.00367
$898$	0	0
$899$	2.68510e6	0.110805
$900$	0	0
$901$	3.85860e6	0.158350
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.29303e7	−0.930653
$906$	0	0
$907$	−1.78366e7	−0.719934	−0.359967	−	0.932965i	$-0.617212\pi$
$907$	−1.78366e7	−0.719934	−0.359967	+	0.932965i	$0.617212\pi$
$908$	0	0
$909$	5.30529e6	0.212961
$910$	0	0
$911$	600440.	0.0239703	0.0119852	−	0.999928i	$-0.496185\pi$
$911$	600440.	0.0239703	0.0119852	+	0.999928i	$0.496185\pi$
$912$	0	0
$913$	1.68522e6	0.0669082
$914$	0	0
$915$	−1.19678e7	−0.472566
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.14543e7	−1.61913	−0.809564	−	0.587032i	$-0.800296\pi$
$919$	−4.14543e7	−1.61913	−0.809564	+	0.587032i	$0.800296\pi$
$920$	0	0
$921$	3.82902e7	1.48744
$922$	0	0
$923$	2.86924e7	1.10857
$924$	0	0
$925$	7.12222e7	2.73691
$926$	0	0
$927$	478588.	0.0182921
$928$	0	0
$929$	−4.87049e7	−1.85154	−0.925771	−	0.378085i	$-0.876583\pi$
$929$	−4.87049e7	−1.85154	−0.925771	+	0.378085i	$0.876583\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.98957e6	−0.225264
$934$	0	0
$935$	−4.04638e7	−1.51369
$936$	0	0
$937$	2.45567e7	0.913737	0.456868	−	0.889534i	$-0.348971\pi$
$937$	2.45567e7	0.913737	0.456868	+	0.889534i	$0.348971\pi$
$938$	0	0
$939$	1.93079e7	0.714612
$940$	0	0
$941$	8.38235e6	0.308597	0.154299	−	0.988024i	$-0.450688\pi$
$941$	8.38235e6	0.308597	0.154299	+	0.988024i	$0.450688\pi$
$942$	0	0
$943$	5.62515e7	2.05994
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.11502e7	−0.404026	−0.202013	−	0.979383i	$-0.564748\pi$
$947$	−1.11502e7	−0.404026	−0.202013	+	0.979383i	$0.564748\pi$
$948$	0	0
$949$	1.08692e7	0.391773
$950$	0	0
$951$	1.70046e7	0.609700
$952$	0	0
$953$	−1.57312e7	−0.561088	−0.280544	−	0.959841i	$-0.590515\pi$
$953$	−1.57312e7	−0.561088	−0.280544	+	0.959841i	$0.590515\pi$
$954$	0	0
$955$	−2.02204e7	−0.717431
$956$	0	0
$957$	−9.00075e6	−0.317687
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.49190e7	−0.870406
$962$	0	0
$963$	7.42857e6	0.258131
$964$	0	0
$965$	1.28188e7	0.443128
$966$	0	0
$967$	−5.38684e7	−1.85254	−0.926271	−	0.376857i	$-0.877005\pi$
$967$	−5.38684e7	−1.85254	−0.926271	+	0.376857i	$0.877005\pi$
$968$	0	0
$969$	7.74227e6	0.264886
$970$	0	0
$971$	3.06393e7	1.04287	0.521436	−	0.853291i	$-0.325397\pi$
$971$	3.06393e7	1.04287	0.521436	+	0.853291i	$0.325397\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.69600e7	−2.59271
$976$	0	0
$977$	1.15906e7	0.388480	0.194240	−	0.980954i	$-0.437776\pi$
$977$	1.15906e7	0.388480	0.194240	+	0.980954i	$0.437776\pi$
$978$	0	0
$979$	−3.36786e7	−1.12304
$980$	0	0
$981$	−5.26103e6	−0.174541
$982$	0	0
$983$	609460.	0.0201169	0.0100585	−	0.999949i	$-0.496798\pi$
$983$	609460.	0.0201169	0.0100585	+	0.999949i	$0.496798\pi$
$984$	0	0
$985$	5.21136e7	1.71144
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.59399e7	1.16839
$990$	0	0
$991$	−2.18372e7	−0.706339	−0.353170	−	0.935559i	$-0.614896\pi$
$991$	−2.18372e7	−0.706339	−0.353170	+	0.935559i	$0.614896\pi$
$992$	0	0
$993$	4.58788e7	1.47652
$994$	0	0
$995$	5.99976e7	1.92122
$996$	0	0
$997$	3.46465e7	1.10388	0.551939	−	0.833885i	$-0.313888\pi$
$997$	3.46465e7	1.10388	0.551939	+	0.833885i	$0.313888\pi$
$998$	0	0
$999$	−4.95270e7	−1.57010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.y.1.2		2
4.3	odd	2		98.6.a.e.1.1	✓	2
7.6	odd	2	inner	784.6.a.y.1.1		2
12.11	even	2		882.6.a.bo.1.1		2
28.3	even	6		98.6.c.g.79.1		4
28.11	odd	6		98.6.c.g.79.2		4
28.19	even	6		98.6.c.g.67.1		4
28.23	odd	6		98.6.c.g.67.2		4
28.27	even	2		98.6.a.e.1.2	yes	2
84.83	odd	2		882.6.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.6.a.e.1.1	✓	2	4.3	odd	2
98.6.a.e.1.2	yes	2	28.27	even	2
98.6.c.g.67.1		4	28.19	even	6
98.6.c.g.67.2		4	28.23	odd	6
98.6.c.g.79.1		4	28.3	even	6
98.6.c.g.79.2		4	28.11	odd	6
784.6.a.y.1.1		2	7.6	odd	2	inner
784.6.a.y.1.2		2	1.1	even	1	trivial
882.6.a.bo.1.1		2	12.11	even	2
882.6.a.bo.1.2		2	84.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+13.5647 q^{3} +94.9526 q^{5} -59.0000 q^{9} -476.000 q^{11} -963.091 q^{13} +1288.00 q^{15} +895.268 q^{17} +637.539 q^{19} -3696.00 q^{23} +5891.00 q^{25} -4096.53 q^{27} +1394.00 q^{29} +1926.18 q^{31} -6456.78 q^{33} +12090.0 q^{37} -13064.0 q^{39} -15219.5 q^{41} -9724.00 q^{43} -5602.20 q^{45} -29272.5 q^{47} +12144.0 q^{51} +4310.00 q^{53} -45197.4 q^{55} +8648.00 q^{57} -20848.9 q^{59} -9291.79 q^{61} -91448.0 q^{65} -20236.0 q^{67} -50135.0 q^{69} -29792.0 q^{71} -11285.8 q^{73} +79909.4 q^{75} +33176.0 q^{79} -41231.0 q^{81} -3540.38 q^{83} +85008.0 q^{85} +18909.1 q^{87} +70753.3 q^{89} +26128.0 q^{93} +60536.0 q^{95} +17769.7 q^{97} +28084.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 118 q^{9} - 952 q^{11} + 2576 q^{15} - 7392 q^{23} + 11782 q^{25} + 2788 q^{29} + 24180 q^{37} - 26128 q^{39} - 19448 q^{43} + 24288 q^{51} + 8620 q^{53} + 17296 q^{57} - 182896 q^{65} - 40472 q^{67}+ \cdots + 56168 q^{99}+O(q^{100}) \) 2 * q - 118 * q^9 - 952 * q^11 + 2576 * q^15 - 7392 * q^23 + 11782 * q^25 + 2788 * q^29 + 24180 * q^37 - 26128 * q^39 - 19448 * q^43 + 24288 * q^51 + 8620 * q^53 + 17296 * q^57 - 182896 * q^65 - 40472 * q^67 - 59584 * q^71 + 66352 * q^79 - 82462 * q^81 + 170016 * q^85 + 52256 * q^93 + 121072 * q^95 + 56168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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