LMFDB - Embedded newform 784.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 196)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-16.0000 q^{3} -16.0000 q^{5} +13.0000 q^{9} +O(q^{10})$

$q-16.0000 q^{3} -16.0000 q^{5} +13.0000 q^{9} +76.0000 q^{11} -880.000 q^{13} +256.000 q^{15} +1056.00 q^{17} +1936.00 q^{19} -936.000 q^{23} -2869.00 q^{25} +3680.00 q^{27} -3982.00 q^{29} +1568.00 q^{31} -1216.00 q^{33} +4938.00 q^{37} +14080.0 q^{39} +15840.0 q^{41} +16412.0 q^{43} -208.000 q^{45} -20768.0 q^{47} -16896.0 q^{51} -37402.0 q^{53} -1216.00 q^{55} -30976.0 q^{57} +21136.0 q^{59} +2992.00 q^{61} +14080.0 q^{65} +45836.0 q^{67} +14976.0 q^{69} +49840.0 q^{71} +56320.0 q^{73} +45904.0 q^{75} -40744.0 q^{79} -62039.0 q^{81} +112464. q^{83} -16896.0 q^{85} +63712.0 q^{87} -64256.0 q^{89} -25088.0 q^{93} -30976.0 q^{95} +2272.00 q^{97} +988.000 q^{99} +O(q^{100})$

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
$3$	−16.0000	−1.02640	−0.513200	−	0.858269i	$-0.671540\pi$
$3$	−16.0000	−1.02640	−0.513200	+	0.858269i	$0.671540\pi$
$4$	0	0
$5$	−16.0000	−0.286217	−0.143108	−	0.989707i	$-0.545710\pi$
$5$	−16.0000	−0.286217	−0.143108	+	0.989707i	$0.545710\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	13.0000	0.0534979
$10$	0	0
$11$	76.0000	0.189379	0.0946895	−	0.995507i	$-0.469814\pi$
$11$	76.0000	0.189379	0.0946895	+	0.995507i	$0.469814\pi$
$12$	0	0
$13$	−880.000	−1.44419	−0.722095	−	0.691794i	$-0.756821\pi$
$13$	−880.000	−1.44419	−0.722095	+	0.691794i	$0.756821\pi$
$14$	0	0
$15$	256.000	0.293773
$16$	0	0
$17$	1056.00	0.886220	0.443110	−	0.896467i	$-0.353875\pi$
$17$	1056.00	0.886220	0.443110	+	0.896467i	$0.353875\pi$
$18$	0	0
$19$	1936.00	1.23033	0.615165	−	0.788399i	$-0.289090\pi$
$19$	1936.00	1.23033	0.615165	+	0.788399i	$0.289090\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−936.000	−0.368940	−0.184470	−	0.982838i	$-0.559057\pi$
$23$	−936.000	−0.368940	−0.184470	+	0.982838i	$0.559057\pi$
$24$	0	0
$25$	−2869.00	−0.918080
$26$	0	0
$27$	3680.00	0.971490
$28$	0	0
$29$	−3982.00	−0.879238	−0.439619	−	0.898184i	$-0.644886\pi$
$29$	−3982.00	−0.879238	−0.439619	+	0.898184i	$0.644886\pi$
$30$	0	0
$31$	1568.00	0.293050	0.146525	−	0.989207i	$-0.453191\pi$
$31$	1568.00	0.293050	0.146525	+	0.989207i	$0.453191\pi$
$32$	0	0
$33$	−1216.00	−0.194379
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4938.00	0.592989	0.296495	−	0.955035i	$-0.404182\pi$
$37$	4938.00	0.592989	0.296495	+	0.955035i	$0.404182\pi$
$38$	0	0
$39$	14080.0	1.48232
$40$	0	0
$41$	15840.0	1.47162	0.735810	−	0.677188i	$-0.236802\pi$
$41$	15840.0	1.47162	0.735810	+	0.677188i	$0.236802\pi$
$42$	0	0
$43$	16412.0	1.35360	0.676800	−	0.736167i	$-0.263366\pi$
$43$	16412.0	1.35360	0.676800	+	0.736167i	$0.263366\pi$
$44$	0	0
$45$	−208.000	−0.0153120
$46$	0	0
$47$	−20768.0	−1.37136	−0.685678	−	0.727905i	$-0.740494\pi$
$47$	−20768.0	−1.37136	−0.685678	+	0.727905i	$0.740494\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−16896.0	−0.909617
$52$	0	0
$53$	−37402.0	−1.82896	−0.914482	−	0.404627i	$-0.867401\pi$
$53$	−37402.0	−1.82896	−0.914482	+	0.404627i	$0.867401\pi$
$54$	0	0
$55$	−1216.00	−0.0542034
$56$	0	0
$57$	−30976.0	−1.26281
$58$	0	0
$59$	21136.0	0.790483	0.395242	−	0.918577i	$-0.370661\pi$
$59$	21136.0	0.790483	0.395242	+	0.918577i	$0.370661\pi$
$60$	0	0
$61$	2992.00	0.102953	0.0514763	−	0.998674i	$-0.483607\pi$
$61$	2992.00	0.102953	0.0514763	+	0.998674i	$0.483607\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14080.0	0.413351
$66$	0	0
$67$	45836.0	1.24744	0.623720	−	0.781648i	$-0.285621\pi$
$67$	45836.0	1.24744	0.623720	+	0.781648i	$0.285621\pi$
$68$	0	0
$69$	14976.0	0.378681
$70$	0	0
$71$	49840.0	1.17336	0.586681	−	0.809818i	$-0.300434\pi$
$71$	49840.0	1.17336	0.586681	+	0.809818i	$0.300434\pi$
$72$	0	0
$73$	56320.0	1.23696	0.618480	−	0.785801i	$-0.287749\pi$
$73$	56320.0	1.23696	0.618480	+	0.785801i	$0.287749\pi$
$74$	0	0
$75$	45904.0	0.942318
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−40744.0	−0.734507	−0.367253	−	0.930121i	$-0.619702\pi$
$79$	−40744.0	−0.734507	−0.367253	+	0.930121i	$0.619702\pi$
$80$	0	0
$81$	−62039.0	−1.05064
$82$	0	0
$83$	112464.	1.79192	0.895959	−	0.444136i	$-0.146489\pi$
$83$	112464.	1.79192	0.895959	+	0.444136i	$0.146489\pi$
$84$	0	0
$85$	−16896.0	−0.253651
$86$	0	0
$87$	63712.0	0.902450
$88$	0	0
$89$	−64256.0	−0.859882	−0.429941	−	0.902857i	$-0.641466\pi$
$89$	−64256.0	−0.859882	−0.429941	+	0.902857i	$0.641466\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−25088.0	−0.300787
$94$	0	0
$95$	−30976.0	−0.352141
$96$	0	0
$97$	2272.00	0.0245177	0.0122588	−	0.999925i	$-0.496098\pi$
$97$	2272.00	0.0245177	0.0122588	+	0.999925i	$0.496098\pi$
$98$	0	0
$99$	988.000	0.0101314
$100$	0	0
$101$	110000.	1.07297	0.536487	−	0.843909i	$-0.319751\pi$
$101$	110000.	1.07297	0.536487	+	0.843909i	$0.319751\pi$
$102$	0	0
$103$	−84128.0	−0.781353	−0.390677	−	0.920528i	$-0.627759\pi$
$103$	−84128.0	−0.781353	−0.390677	+	0.920528i	$0.627759\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13948.0	−0.117775	−0.0588874	−	0.998265i	$-0.518755\pi$
$107$	−13948.0	−0.117775	−0.0588874	+	0.998265i	$0.518755\pi$
$108$	0	0
$109$	22594.0	0.182149	0.0910745	−	0.995844i	$-0.470970\pi$
$109$	22594.0	0.182149	0.0910745	+	0.995844i	$0.470970\pi$
$110$	0	0
$111$	−79008.0	−0.608644
$112$	0	0
$113$	94786.0	0.698310	0.349155	−	0.937065i	$-0.386469\pi$
$113$	94786.0	0.698310	0.349155	+	0.937065i	$0.386469\pi$
$114$	0	0
$115$	14976.0	0.105597
$116$	0	0
$117$	−11440.0	−0.0772612
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−155275.	−0.964136
$122$	0	0
$123$	−253440.	−1.51047
$124$	0	0
$125$	95904.0	0.548987
$126$	0	0
$127$	−140624.	−0.773660	−0.386830	−	0.922151i	$-0.626430\pi$
$127$	−140624.	−0.773660	−0.386830	+	0.922151i	$0.626430\pi$
$128$	0	0
$129$	−262592.	−1.38934
$130$	0	0
$131$	−242352.	−1.23387	−0.616934	−	0.787015i	$-0.711625\pi$
$131$	−242352.	−1.23387	−0.616934	+	0.787015i	$0.711625\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−58880.0	−0.278057
$136$	0	0
$137$	172886.	0.786970	0.393485	−	0.919331i	$-0.371269\pi$
$137$	172886.	0.786970	0.393485	+	0.919331i	$0.371269\pi$
$138$	0	0
$139$	−167376.	−0.734778	−0.367389	−	0.930067i	$-0.619748\pi$
$139$	−167376.	−0.734778	−0.367389	+	0.930067i	$0.619748\pi$
$140$	0	0
$141$	332288.	1.40756
$142$	0	0
$143$	−66880.0	−0.273499
$144$	0	0
$145$	63712.0	0.251652
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−404602.	−1.49301	−0.746504	−	0.665381i	$-0.768269\pi$
$149$	−404602.	−1.49301	−0.746504	+	0.665381i	$0.768269\pi$
$150$	0	0
$151$	−445192.	−1.58893	−0.794465	−	0.607309i	$-0.792249\pi$
$151$	−445192.	−1.58893	−0.794465	+	0.607309i	$0.792249\pi$
$152$	0	0
$153$	13728.0	0.0474110
$154$	0	0
$155$	−25088.0	−0.0838758
$156$	0	0
$157$	172496.	0.558509	0.279254	−	0.960217i	$-0.409913\pi$
$157$	172496.	0.558509	0.279254	+	0.960217i	$0.409913\pi$
$158$	0	0
$159$	598432.	1.87725
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−167484.	−0.493747	−0.246873	−	0.969048i	$-0.579403\pi$
$163$	−167484.	−0.493747	−0.246873	+	0.969048i	$0.579403\pi$
$164$	0	0
$165$	19456.0	0.0556344
$166$	0	0
$167$	206624.	0.573310	0.286655	−	0.958034i	$-0.407457\pi$
$167$	206624.	0.573310	0.286655	+	0.958034i	$0.407457\pi$
$168$	0	0
$169$	403107.	1.08568
$170$	0	0
$171$	25168.0	0.0658201
$172$	0	0
$173$	339856.	0.863336	0.431668	−	0.902033i	$-0.357925\pi$
$173$	339856.	0.863336	0.431668	+	0.902033i	$0.357925\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−338176.	−0.811353
$178$	0	0
$179$	−519540.	−1.21195	−0.605977	−	0.795482i	$-0.707218\pi$
$179$	−519540.	−1.21195	−0.605977	+	0.795482i	$0.707218\pi$
$180$	0	0
$181$	−830000.	−1.88314	−0.941568	−	0.336823i	$-0.890648\pi$
$181$	−830000.	−1.88314	−0.941568	+	0.336823i	$0.890648\pi$
$182$	0	0
$183$	−47872.0	−0.105671
$184$	0	0
$185$	−79008.0	−0.169723
$186$	0	0
$187$	80256.0	0.167832
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−301608.	−0.598218	−0.299109	−	0.954219i	$-0.596689\pi$
$191$	−301608.	−0.598218	−0.299109	+	0.954219i	$0.596689\pi$
$192$	0	0
$193$	−484462.	−0.936195	−0.468098	−	0.883677i	$-0.655060\pi$
$193$	−484462.	−0.936195	−0.468098	+	0.883677i	$0.655060\pi$
$194$	0	0
$195$	−225280.	−0.424264
$196$	0	0
$197$	−183018.	−0.335991	−0.167996	−	0.985788i	$-0.553729\pi$
$197$	−183018.	−0.335991	−0.167996	+	0.985788i	$0.553729\pi$
$198$	0	0
$199$	−904288.	−1.61873	−0.809364	−	0.587307i	$-0.800188\pi$
$199$	−904288.	−1.61873	−0.809364	+	0.587307i	$0.800188\pi$
$200$	0	0
$201$	−733376.	−1.28037
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−253440.	−0.421202
$206$	0	0
$207$	−12168.0	−0.0197376
$208$	0	0
$209$	147136.	0.232999
$210$	0	0
$211$	494428.	0.764534	0.382267	−	0.924052i	$-0.375143\pi$
$211$	494428.	0.764534	0.382267	+	0.924052i	$0.375143\pi$
$212$	0	0
$213$	−797440.	−1.20434
$214$	0	0
$215$	−262592.	−0.387423
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−901120.	−1.26962
$220$	0	0
$221$	−929280.	−1.27987
$222$	0	0
$223$	−389824.	−0.524936	−0.262468	−	0.964941i	$-0.584536\pi$
$223$	−389824.	−0.524936	−0.262468	+	0.964941i	$0.584536\pi$
$224$	0	0
$225$	−37297.0	−0.0491154
$226$	0	0
$227$	−79024.0	−0.101787	−0.0508937	−	0.998704i	$-0.516207\pi$
$227$	−79024.0	−0.101787	−0.0508937	+	0.998704i	$0.516207\pi$
$228$	0	0
$229$	1.25979e6	1.58749	0.793743	−	0.608253i	$-0.208129\pi$
$229$	1.25979e6	1.58749	0.793743	+	0.608253i	$0.208129\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−772618.	−0.932342	−0.466171	−	0.884695i	$-0.654367\pi$
$233$	−772618.	−0.932342	−0.466171	+	0.884695i	$0.654367\pi$
$234$	0	0
$235$	332288.	0.392505
$236$	0	0
$237$	651904.	0.753898
$238$	0	0
$239$	1.42507e6	1.61377	0.806886	−	0.590707i	$-0.201151\pi$
$239$	1.42507e6	1.61377	0.806886	+	0.590707i	$0.201151\pi$
$240$	0	0
$241$	−1.14858e6	−1.27385	−0.636923	−	0.770927i	$-0.719793\pi$
$241$	−1.14858e6	−1.27385	−0.636923	+	0.770927i	$0.719793\pi$
$242$	0	0
$243$	98384.0	0.106883
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.70368e6	−1.77683
$248$	0	0
$249$	−1.79942e6	−1.83923
$250$	0	0
$251$	−278096.	−0.278619	−0.139309	−	0.990249i	$-0.544488\pi$
$251$	−278096.	−0.278619	−0.139309	+	0.990249i	$0.544488\pi$
$252$	0	0
$253$	−71136.0	−0.0698696
$254$	0	0
$255$	270336.	0.260348
$256$	0	0
$257$	−357632.	−0.337756	−0.168878	−	0.985637i	$-0.554014\pi$
$257$	−357632.	−0.337756	−0.168878	+	0.985637i	$0.554014\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−51766.0	−0.0470374
$262$	0	0
$263$	84832.0	0.0756259	0.0378129	−	0.999285i	$-0.487961\pi$
$263$	84832.0	0.0756259	0.0378129	+	0.999285i	$0.487961\pi$
$264$	0	0
$265$	598432.	0.523480
$266$	0	0
$267$	1.02810e6	0.882583
$268$	0	0
$269$	2.19771e6	1.85178	0.925891	−	0.377790i	$-0.123316\pi$
$269$	2.19771e6	1.85178	0.925891	+	0.377790i	$0.123316\pi$
$270$	0	0
$271$	−475904.	−0.393637	−0.196819	−	0.980440i	$-0.563061\pi$
$271$	−475904.	−0.393637	−0.196819	+	0.980440i	$0.563061\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−218044.	−0.173865
$276$	0	0
$277$	−1.59401e6	−1.24822	−0.624111	−	0.781336i	$-0.714539\pi$
$277$	−1.59401e6	−1.24822	−0.624111	+	0.781336i	$0.714539\pi$
$278$	0	0
$279$	20384.0	0.0156776
$280$	0	0
$281$	−1.31558e6	−0.993919	−0.496959	−	0.867774i	$-0.665550\pi$
$281$	−1.31558e6	−0.993919	−0.496959	+	0.867774i	$0.665550\pi$
$282$	0	0
$283$	1.00866e6	0.748647	0.374323	−	0.927298i	$-0.377875\pi$
$283$	1.00866e6	0.748647	0.374323	+	0.927298i	$0.377875\pi$
$284$	0	0
$285$	495616.	0.361437
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−304721.	−0.214614
$290$	0	0
$291$	−36352.0	−0.0251649
$292$	0	0
$293$	−2.05762e6	−1.40022	−0.700108	−	0.714037i	$-0.746865\pi$
$293$	−2.05762e6	−1.40022	−0.700108	+	0.714037i	$0.746865\pi$
$294$	0	0
$295$	−338176.	−0.226250
$296$	0	0
$297$	279680.	0.183980
$298$	0	0
$299$	823680.	0.532820
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.76000e6	−1.10130
$304$	0	0
$305$	−47872.0	−0.0294667
$306$	0	0
$307$	2.07099e6	1.25410	0.627050	−	0.778979i	$-0.284262\pi$
$307$	2.07099e6	1.25410	0.627050	+	0.778979i	$0.284262\pi$
$308$	0	0
$309$	1.34605e6	0.801981
$310$	0	0
$311$	953216.	0.558844	0.279422	−	0.960168i	$-0.409857\pi$
$311$	953216.	0.558844	0.279422	+	0.960168i	$0.409857\pi$
$312$	0	0
$313$	−2.85840e6	−1.64916	−0.824579	−	0.565747i	$-0.808588\pi$
$313$	−2.85840e6	−1.64916	−0.824579	+	0.565747i	$0.808588\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.98760e6	−1.11092	−0.555458	−	0.831545i	$-0.687457\pi$
$317$	−1.98760e6	−1.11092	−0.555458	+	0.831545i	$0.687457\pi$
$318$	0	0
$319$	−302632.	−0.166509
$320$	0	0
$321$	223168.	0.120884
$322$	0	0
$323$	2.04442e6	1.09034
$324$	0	0
$325$	2.52472e6	1.32588
$326$	0	0
$327$	−361504.	−0.186958
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−394580.	−0.197954	−0.0989772	−	0.995090i	$-0.531557\pi$
$331$	−394580.	−0.197954	−0.0989772	+	0.995090i	$0.531557\pi$
$332$	0	0
$333$	64194.0	0.0317237
$334$	0	0
$335$	−733376.	−0.357038
$336$	0	0
$337$	2.19606e6	1.05334	0.526672	−	0.850069i	$-0.323440\pi$
$337$	2.19606e6	1.05334	0.526672	+	0.850069i	$0.323440\pi$
$338$	0	0
$339$	−1.51658e6	−0.716745
$340$	0	0
$341$	119168.	0.0554975
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−239616.	−0.108385
$346$	0	0
$347$	−4.02719e6	−1.79547	−0.897735	−	0.440536i	$-0.854789\pi$
$347$	−4.02719e6	−1.79547	−0.897735	+	0.440536i	$0.854789\pi$
$348$	0	0
$349$	1.96469e6	0.863436	0.431718	−	0.902009i	$-0.357908\pi$
$349$	1.96469e6	0.863436	0.431718	+	0.902009i	$0.357908\pi$
$350$	0	0
$351$	−3.23840e6	−1.40302
$352$	0	0
$353$	−3.34522e6	−1.42885	−0.714426	−	0.699711i	$-0.753312\pi$
$353$	−3.34522e6	−1.42885	−0.714426	+	0.699711i	$0.753312\pi$
$354$	0	0
$355$	−797440.	−0.335836
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.29513e6	0.939876	0.469938	−	0.882699i	$-0.344276\pi$
$359$	2.29513e6	0.939876	0.469938	+	0.882699i	$0.344276\pi$
$360$	0	0
$361$	1.27200e6	0.513710
$362$	0	0
$363$	2.48440e6	0.989589
$364$	0	0
$365$	−901120.	−0.354038
$366$	0	0
$367$	2.93869e6	1.13891	0.569454	−	0.822023i	$-0.307155\pi$
$367$	2.93869e6	1.13891	0.569454	+	0.822023i	$0.307155\pi$
$368$	0	0
$369$	205920.	0.0787286
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−414458.	−0.154244	−0.0771220	−	0.997022i	$-0.524573\pi$
$373$	−414458.	−0.154244	−0.0771220	+	0.997022i	$0.524573\pi$
$374$	0	0
$375$	−1.53446e6	−0.563480
$376$	0	0
$377$	3.50416e6	1.26979
$378$	0	0
$379$	−2.57111e6	−0.919438	−0.459719	−	0.888065i	$-0.652050\pi$
$379$	−2.57111e6	−0.919438	−0.459719	+	0.888065i	$0.652050\pi$
$380$	0	0
$381$	2.24998e6	0.794085
$382$	0	0
$383$	−2.56733e6	−0.894302	−0.447151	−	0.894458i	$-0.647561\pi$
$383$	−2.56733e6	−0.894302	−0.447151	+	0.894458i	$0.647561\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	213356.	0.0724148
$388$	0	0
$389$	3.37809e6	1.13187	0.565936	−	0.824449i	$-0.308515\pi$
$389$	3.37809e6	1.13187	0.565936	+	0.824449i	$0.308515\pi$
$390$	0	0
$391$	−988416.	−0.326962
$392$	0	0
$393$	3.87763e6	1.26644
$394$	0	0
$395$	651904.	0.210228
$396$	0	0
$397$	2.19771e6	0.699833	0.349917	−	0.936781i	$-0.386210\pi$
$397$	2.19771e6	0.699833	0.349917	+	0.936781i	$0.386210\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.47671e6	−1.07971	−0.539855	−	0.841758i	$-0.681521\pi$
$401$	−3.47671e6	−1.07971	−0.539855	+	0.841758i	$0.681521\pi$
$402$	0	0
$403$	−1.37984e6	−0.423220
$404$	0	0
$405$	992624.	0.300710
$406$	0	0
$407$	375288.	0.112300
$408$	0	0
$409$	1.68432e6	0.497870	0.248935	−	0.968520i	$-0.419919\pi$
$409$	1.68432e6	0.497870	0.248935	+	0.968520i	$0.419919\pi$
$410$	0	0
$411$	−2.76618e6	−0.807747
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.79942e6	−0.512877
$416$	0	0
$417$	2.67802e6	0.754177
$418$	0	0
$419$	−4.40475e6	−1.22571	−0.612853	−	0.790197i	$-0.709978\pi$
$419$	−4.40475e6	−1.22571	−0.612853	+	0.790197i	$0.709978\pi$
$420$	0	0
$421$	3.06601e6	0.843078	0.421539	−	0.906810i	$-0.361490\pi$
$421$	3.06601e6	0.843078	0.421539	+	0.906810i	$0.361490\pi$
$422$	0	0
$423$	−269984.	−0.0733647
$424$	0	0
$425$	−3.02966e6	−0.813621
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.07008e6	0.280720
$430$	0	0
$431$	−1.32035e6	−0.342371	−0.171185	−	0.985239i	$-0.554760\pi$
$431$	−1.32035e6	−0.342371	−0.171185	+	0.985239i	$0.554760\pi$
$432$	0	0
$433$	−2.91510e6	−0.747196	−0.373598	−	0.927591i	$-0.621876\pi$
$433$	−2.91510e6	−0.747196	−0.373598	+	0.927591i	$0.621876\pi$
$434$	0	0
$435$	−1.01939e6	−0.258296
$436$	0	0
$437$	−1.81210e6	−0.453918
$438$	0	0
$439$	7.06253e6	1.74904	0.874518	−	0.484993i	$-0.161178\pi$
$439$	7.06253e6	1.74904	0.874518	+	0.484993i	$0.161178\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.73812e6	1.87338	0.936690	−	0.350160i	$-0.113873\pi$
$443$	7.73812e6	1.87338	0.936690	+	0.350160i	$0.113873\pi$
$444$	0	0
$445$	1.02810e6	0.246112
$446$	0	0
$447$	6.47363e6	1.53242
$448$	0	0
$449$	−5.62457e6	−1.31666	−0.658330	−	0.752729i	$-0.728737\pi$
$449$	−5.62457e6	−1.31666	−0.658330	+	0.752729i	$0.728737\pi$
$450$	0	0
$451$	1.20384e6	0.278694
$452$	0	0
$453$	7.12307e6	1.63088
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.91516e6	1.32488	0.662439	−	0.749116i	$-0.269521\pi$
$457$	5.91516e6	1.32488	0.662439	+	0.749116i	$0.269521\pi$
$458$	0	0
$459$	3.88608e6	0.860954
$460$	0	0
$461$	−7.21195e6	−1.58052	−0.790261	−	0.612770i	$-0.790055\pi$
$461$	−7.21195e6	−1.58052	−0.790261	+	0.612770i	$0.790055\pi$
$462$	0	0
$463$	−5.22092e6	−1.13186	−0.565932	−	0.824452i	$-0.691484\pi$
$463$	−5.22092e6	−1.13186	−0.565932	+	0.824452i	$0.691484\pi$
$464$	0	0
$465$	401408.	0.0860902
$466$	0	0
$467$	5.81538e6	1.23392	0.616958	−	0.786996i	$-0.288365\pi$
$467$	5.81538e6	1.23392	0.616958	+	0.786996i	$0.288365\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.75994e6	−0.573254
$472$	0	0
$473$	1.24731e6	0.256343
$474$	0	0
$475$	−5.55438e6	−1.12954
$476$	0	0
$477$	−486226.	−0.0978458
$478$	0	0
$479$	−7.53878e6	−1.50128	−0.750641	−	0.660710i	$-0.770255\pi$
$479$	−7.53878e6	−1.50128	−0.750641	+	0.660710i	$0.770255\pi$
$480$	0	0
$481$	−4.34544e6	−0.856389
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−36352.0	−0.00701736
$486$	0	0
$487$	6.59660e6	1.26037	0.630185	−	0.776445i	$-0.282979\pi$
$487$	6.59660e6	1.26037	0.630185	+	0.776445i	$0.282979\pi$
$488$	0	0
$489$	2.67974e6	0.506782
$490$	0	0
$491$	9.96666e6	1.86572	0.932859	−	0.360242i	$-0.117306\pi$
$491$	9.96666e6	1.86572	0.932859	+	0.360242i	$0.117306\pi$
$492$	0	0
$493$	−4.20499e6	−0.779198
$494$	0	0
$495$	−15808.0	−0.00289977
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.94270e6	0.529047	0.264524	−	0.964379i	$-0.414785\pi$
$499$	2.94270e6	0.529047	0.264524	+	0.964379i	$0.414785\pi$
$500$	0	0
$501$	−3.30598e6	−0.588446
$502$	0	0
$503$	4.35142e6	0.766852	0.383426	−	0.923572i	$-0.374744\pi$
$503$	4.35142e6	0.766852	0.383426	+	0.923572i	$0.374744\pi$
$504$	0	0
$505$	−1.76000e6	−0.307103
$506$	0	0
$507$	−6.44971e6	−1.11435
$508$	0	0
$509$	−5.18494e6	−0.887053	−0.443527	−	0.896261i	$-0.646273\pi$
$509$	−5.18494e6	−0.887053	−0.443527	+	0.896261i	$0.646273\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.12448e6	1.19525
$514$	0	0
$515$	1.34605e6	0.223636
$516$	0	0
$517$	−1.57837e6	−0.259706
$518$	0	0
$519$	−5.43770e6	−0.886128
$520$	0	0
$521$	−7.41680e6	−1.19708	−0.598539	−	0.801094i	$-0.704252\pi$
$521$	−7.41680e6	−1.19708	−0.598539	+	0.801094i	$0.704252\pi$
$522$	0	0
$523$	−6.47346e6	−1.03486	−0.517430	−	0.855725i	$-0.673111\pi$
$523$	−6.47346e6	−1.03486	−0.517430	+	0.855725i	$0.673111\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.65581e6	0.259707
$528$	0	0
$529$	−5.56025e6	−0.863883
$530$	0	0
$531$	274768.	0.0422892
$532$	0	0
$533$	−1.39392e7	−2.12530
$534$	0	0
$535$	223168.	0.0337091
$536$	0	0
$537$	8.31264e6	1.24395
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.10987e6	1.33819	0.669097	−	0.743175i	$-0.266681\pi$
$541$	9.10987e6	1.33819	0.669097	+	0.743175i	$0.266681\pi$
$542$	0	0
$543$	1.32800e7	1.93285
$544$	0	0
$545$	−361504.	−0.0521341
$546$	0	0
$547$	5.63776e6	0.805635	0.402818	−	0.915280i	$-0.368031\pi$
$547$	5.63776e6	0.805635	0.402818	+	0.915280i	$0.368031\pi$
$548$	0	0
$549$	38896.0	0.00550775
$550$	0	0
$551$	−7.70915e6	−1.08175
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.26413e6	0.174204
$556$	0	0
$557$	3.71072e6	0.506781	0.253390	−	0.967364i	$-0.418454\pi$
$557$	3.71072e6	0.506781	0.253390	+	0.967364i	$0.418454\pi$
$558$	0	0
$559$	−1.44426e7	−1.95486
$560$	0	0
$561$	−1.28410e6	−0.172262
$562$	0	0
$563$	5.26117e6	0.699538	0.349769	−	0.936836i	$-0.386260\pi$
$563$	5.26117e6	0.699538	0.349769	+	0.936836i	$0.386260\pi$
$564$	0	0
$565$	−1.51658e6	−0.199868
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.80929e6	−0.363761	−0.181880	−	0.983321i	$-0.558218\pi$
$569$	−2.80929e6	−0.363761	−0.181880	+	0.983321i	$0.558218\pi$
$570$	0	0
$571$	8.19707e6	1.05213	0.526064	−	0.850445i	$-0.323667\pi$
$571$	8.19707e6	1.05213	0.526064	+	0.850445i	$0.323667\pi$
$572$	0	0
$573$	4.82573e6	0.614011
$574$	0	0
$575$	2.68538e6	0.338717
$576$	0	0
$577$	3.69722e6	0.462312	0.231156	−	0.972917i	$-0.425749\pi$
$577$	3.69722e6	0.462312	0.231156	+	0.972917i	$0.425749\pi$
$578$	0	0
$579$	7.75139e6	0.960911
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.84255e6	−0.346367
$584$	0	0
$585$	183040.	0.0221134
$586$	0	0
$587$	−8.25282e6	−0.988569	−0.494284	−	0.869300i	$-0.664570\pi$
$587$	−8.25282e6	−0.988569	−0.494284	+	0.869300i	$0.664570\pi$
$588$	0	0
$589$	3.03565e6	0.360548
$590$	0	0
$591$	2.92829e6	0.344862
$592$	0	0
$593$	6.31277e6	0.737196	0.368598	−	0.929589i	$-0.379838\pi$
$593$	6.31277e6	0.737196	0.368598	+	0.929589i	$0.379838\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.44686e7	1.66146
$598$	0	0
$599$	6.92976e6	0.789135	0.394567	−	0.918867i	$-0.370894\pi$
$599$	6.92976e6	0.789135	0.394567	+	0.918867i	$0.370894\pi$
$600$	0	0
$601$	−1.10092e7	−1.24328	−0.621638	−	0.783305i	$-0.713533\pi$
$601$	−1.10092e7	−1.24328	−0.621638	+	0.783305i	$0.713533\pi$
$602$	0	0
$603$	595868.	0.0667355
$604$	0	0
$605$	2.48440e6	0.275952
$606$	0	0
$607$	−333696.	−0.0367603	−0.0183802	−	0.999831i	$-0.505851\pi$
$607$	−333696.	−0.0367603	−0.0183802	+	0.999831i	$0.505851\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.82758e7	1.98050
$612$	0	0
$613$	1.75927e6	0.189096	0.0945480	−	0.995520i	$-0.469859\pi$
$613$	1.75927e6	0.189096	0.0945480	+	0.995520i	$0.469859\pi$
$614$	0	0
$615$	4.05504e6	0.432322
$616$	0	0
$617$	−1.46142e7	−1.54548	−0.772738	−	0.634725i	$-0.781113\pi$
$617$	−1.46142e7	−1.54548	−0.772738	+	0.634725i	$0.781113\pi$
$618$	0	0
$619$	−2.31544e6	−0.242888	−0.121444	−	0.992598i	$-0.538753\pi$
$619$	−2.31544e6	−0.242888	−0.121444	+	0.992598i	$0.538753\pi$
$620$	0	0
$621$	−3.44448e6	−0.358422
$622$	0	0
$623$	0	0
$624$	0	0
$625$	7.43116e6	0.760951
$626$	0	0
$627$	−2.35418e6	−0.239150
$628$	0	0
$629$	5.21453e6	0.525519
$630$	0	0
$631$	−1.47152e7	−1.47127	−0.735636	−	0.677377i	$-0.763117\pi$
$631$	−1.47152e7	−1.47127	−0.735636	+	0.677377i	$0.763117\pi$
$632$	0	0
$633$	−7.91085e6	−0.784718
$634$	0	0
$635$	2.24998e6	0.221434
$636$	0	0
$637$	0	0
$638$	0	0
$639$	647920.	0.0627725
$640$	0	0
$641$	1.07555e7	1.03392	0.516958	−	0.856011i	$-0.327064\pi$
$641$	1.07555e7	1.03392	0.516958	+	0.856011i	$0.327064\pi$
$642$	0	0
$643$	1.81097e7	1.72736	0.863681	−	0.504039i	$-0.168153\pi$
$643$	1.81097e7	1.72736	0.863681	+	0.504039i	$0.168153\pi$
$644$	0	0
$645$	4.20147e6	0.397651
$646$	0	0
$647$	3.87677e6	0.364090	0.182045	−	0.983290i	$-0.441728\pi$
$647$	3.87677e6	0.364090	0.182045	+	0.983290i	$0.441728\pi$
$648$	0	0
$649$	1.60634e6	0.149701
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.82557e6	−0.534632	−0.267316	−	0.963609i	$-0.586137\pi$
$653$	−5.82557e6	−0.534632	−0.267316	+	0.963609i	$0.586137\pi$
$654$	0	0
$655$	3.87763e6	0.353153
$656$	0	0
$657$	732160.	0.0661748
$658$	0	0
$659$	6.46989e6	0.580341	0.290171	−	0.956975i	$-0.406288\pi$
$659$	6.46989e6	0.580341	0.290171	+	0.956975i	$0.406288\pi$
$660$	0	0
$661$	4.18002e6	0.372113	0.186056	−	0.982539i	$-0.440429\pi$
$661$	4.18002e6	0.372113	0.186056	+	0.982539i	$0.440429\pi$
$662$	0	0
$663$	1.48685e7	1.31366
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.72715e6	0.324386
$668$	0	0
$669$	6.23718e6	0.538795
$670$	0	0
$671$	227392.	0.0194970
$672$	0	0
$673$	1.27159e7	1.08221	0.541104	−	0.840956i	$-0.318007\pi$
$673$	1.27159e7	1.08221	0.541104	+	0.840956i	$0.318007\pi$
$674$	0	0
$675$	−1.05579e7	−0.891906
$676$	0	0
$677$	−1.18710e7	−0.995443	−0.497722	−	0.867337i	$-0.665830\pi$
$677$	−1.18710e7	−0.995443	−0.497722	+	0.867337i	$0.665830\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.26438e6	0.104475
$682$	0	0
$683$	−1.39978e7	−1.14817	−0.574087	−	0.818794i	$-0.694643\pi$
$683$	−1.39978e7	−1.14817	−0.574087	+	0.818794i	$0.694643\pi$
$684$	0	0
$685$	−2.76618e6	−0.225244
$686$	0	0
$687$	−2.01567e7	−1.62940
$688$	0	0
$689$	3.29138e7	2.64137
$690$	0	0
$691$	−9.80037e6	−0.780813	−0.390407	−	0.920642i	$-0.627666\pi$
$691$	−9.80037e6	−0.780813	−0.390407	+	0.920642i	$0.627666\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.67802e6	0.210306
$696$	0	0
$697$	1.67270e7	1.30418
$698$	0	0
$699$	1.23619e7	0.956956
$700$	0	0
$701$	−1.80481e6	−0.138719	−0.0693597	−	0.997592i	$-0.522096\pi$
$701$	−1.80481e6	−0.138719	−0.0693597	+	0.997592i	$0.522096\pi$
$702$	0	0
$703$	9.55997e6	0.729572
$704$	0	0
$705$	−5.31661e6	−0.402867
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.47404e6	−0.483682	−0.241841	−	0.970316i	$-0.577751\pi$
$709$	−6.47404e6	−0.483682	−0.241841	+	0.970316i	$0.577751\pi$
$710$	0	0
$711$	−529672.	−0.0392946
$712$	0	0
$713$	−1.46765e6	−0.108118
$714$	0	0
$715$	1.07008e6	0.0782801
$716$	0	0
$717$	−2.28012e7	−1.65638
$718$	0	0
$719$	−2.34064e7	−1.68855	−0.844273	−	0.535913i	$-0.819967\pi$
$719$	−2.34064e7	−1.68855	−0.844273	+	0.535913i	$0.819967\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.83772e7	1.30748
$724$	0	0
$725$	1.14244e7	0.807210
$726$	0	0
$727$	−491936.	−0.0345201	−0.0172601	−	0.999851i	$-0.505494\pi$
$727$	−491936.	−0.0345201	−0.0172601	+	0.999851i	$0.505494\pi$
$728$	0	0
$729$	1.35013e7	0.940931
$730$	0	0
$731$	1.73311e7	1.19959
$732$	0	0
$733$	−7.86421e6	−0.540624	−0.270312	−	0.962773i	$-0.587127\pi$
$733$	−7.86421e6	−0.540624	−0.270312	+	0.962773i	$0.587127\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.48354e6	0.236239
$738$	0	0
$739$	−6.92547e6	−0.466485	−0.233243	−	0.972419i	$-0.574934\pi$
$739$	−6.92547e6	−0.466485	−0.233243	+	0.972419i	$0.574934\pi$
$740$	0	0
$741$	2.72589e7	1.82374
$742$	0	0
$743$	−1.59306e7	−1.05867	−0.529333	−	0.848414i	$-0.677558\pi$
$743$	−1.59306e7	−1.05867	−0.529333	+	0.848414i	$0.677558\pi$
$744$	0	0
$745$	6.47363e6	0.427324
$746$	0	0
$747$	1.46203e6	0.0958640
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.00152e7	−0.647980	−0.323990	−	0.946060i	$-0.605024\pi$
$751$	−1.00152e7	−0.647980	−0.323990	+	0.946060i	$0.605024\pi$
$752$	0	0
$753$	4.44954e6	0.285974
$754$	0	0
$755$	7.12307e6	0.454779
$756$	0	0
$757$	3.79619e6	0.240773	0.120387	−	0.992727i	$-0.461587\pi$
$757$	3.79619e6	0.240773	0.120387	+	0.992727i	$0.461587\pi$
$758$	0	0
$759$	1.13818e6	0.0717142
$760$	0	0
$761$	1.03484e7	0.647759	0.323880	−	0.946098i	$-0.395013\pi$
$761$	1.03484e7	0.647759	0.323880	+	0.946098i	$0.395013\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−219648.	−0.0135698
$766$	0	0
$767$	−1.85997e7	−1.14161
$768$	0	0
$769$	1.51131e7	0.921591	0.460796	−	0.887506i	$-0.347564\pi$
$769$	1.51131e7	0.921591	0.460796	+	0.887506i	$0.347564\pi$
$770$	0	0
$771$	5.72211e6	0.346673
$772$	0	0
$773$	5.66174e6	0.340801	0.170401	−	0.985375i	$-0.445494\pi$
$773$	5.66174e6	0.340801	0.170401	+	0.985375i	$0.445494\pi$
$774$	0	0
$775$	−4.49859e6	−0.269043
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.06662e7	1.81058
$780$	0	0
$781$	3.78784e6	0.222210
$782$	0	0
$783$	−1.46538e7	−0.854171
$784$	0	0
$785$	−2.75994e6	−0.159855
$786$	0	0
$787$	−1.73483e6	−0.0998437	−0.0499218	−	0.998753i	$-0.515897\pi$
$787$	−1.73483e6	−0.0998437	−0.0499218	+	0.998753i	$0.515897\pi$
$788$	0	0
$789$	−1.35731e6	−0.0776224
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.63296e6	−0.148683
$794$	0	0
$795$	−9.57491e6	−0.537300
$796$	0	0
$797$	−4.96709e6	−0.276985	−0.138492	−	0.990363i	$-0.544226\pi$
$797$	−4.96709e6	−0.276985	−0.138492	+	0.990363i	$0.544226\pi$
$798$	0	0
$799$	−2.19310e7	−1.21532
$800$	0	0
$801$	−835328.	−0.0460019
$802$	0	0
$803$	4.28032e6	0.234254
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.51634e7	−1.90067
$808$	0	0
$809$	−1.66238e7	−0.893013	−0.446506	−	0.894780i	$-0.647332\pi$
$809$	−1.66238e7	−0.893013	−0.446506	+	0.894780i	$0.647332\pi$
$810$	0	0
$811$	−1.67695e7	−0.895296	−0.447648	−	0.894210i	$-0.647738\pi$
$811$	−1.67695e7	−0.895296	−0.447648	+	0.894210i	$0.647738\pi$
$812$	0	0
$813$	7.61446e6	0.404029
$814$	0	0
$815$	2.67974e6	0.141319
$816$	0	0
$817$	3.17736e7	1.66537
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8.34150e6	−0.431903	−0.215951	−	0.976404i	$-0.569285\pi$
$821$	−8.34150e6	−0.431903	−0.215951	+	0.976404i	$0.569285\pi$
$822$	0	0
$823$	2.06990e7	1.06524	0.532622	−	0.846353i	$-0.321207\pi$
$823$	2.06990e7	1.06524	0.532622	+	0.846353i	$0.321207\pi$
$824$	0	0
$825$	3.48870e6	0.178455
$826$	0	0
$827$	3.25524e7	1.65508	0.827540	−	0.561407i	$-0.189740\pi$
$827$	3.25524e7	1.65508	0.827540	+	0.561407i	$0.189740\pi$
$828$	0	0
$829$	−1.64645e7	−0.832073	−0.416036	−	0.909348i	$-0.636581\pi$
$829$	−1.64645e7	−0.832073	−0.416036	+	0.909348i	$0.636581\pi$
$830$	0	0
$831$	2.55042e7	1.28118
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.30598e6	−0.164091
$836$	0	0
$837$	5.77024e6	0.284695
$838$	0	0
$839$	2.16446e7	1.06156	0.530781	−	0.847509i	$-0.321899\pi$
$839$	2.16446e7	1.06156	0.530781	+	0.847509i	$0.321899\pi$
$840$	0	0
$841$	−4.65482e6	−0.226941
$842$	0	0
$843$	2.10492e7	1.02016
$844$	0	0
$845$	−6.44971e6	−0.310741
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.61385e7	−0.768411
$850$	0	0
$851$	−4.62197e6	−0.218778
$852$	0	0
$853$	−2.02592e7	−0.953343	−0.476672	−	0.879081i	$-0.658157\pi$
$853$	−2.02592e7	−0.953343	−0.476672	+	0.879081i	$0.658157\pi$
$854$	0	0
$855$	−402688.	−0.0188388
$856$	0	0
$857$	1.94758e7	0.905823	0.452912	−	0.891555i	$-0.350385\pi$
$857$	1.94758e7	0.905823	0.452912	+	0.891555i	$0.350385\pi$
$858$	0	0
$859$	−8.66203e6	−0.400532	−0.200266	−	0.979742i	$-0.564181\pi$
$859$	−8.66203e6	−0.400532	−0.200266	+	0.979742i	$0.564181\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.55956e7	−0.712810	−0.356405	−	0.934332i	$-0.615998\pi$
$863$	−1.55956e7	−0.712810	−0.356405	+	0.934332i	$0.615998\pi$
$864$	0	0
$865$	−5.43770e6	−0.247101
$866$	0	0
$867$	4.87554e6	0.220280
$868$	0	0
$869$	−3.09654e6	−0.139100
$870$	0	0
$871$	−4.03357e7	−1.80154
$872$	0	0
$873$	29536.0	0.00131164
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.44599e6	0.414714	0.207357	−	0.978265i	$-0.433514\pi$
$877$	9.44599e6	0.414714	0.207357	+	0.978265i	$0.433514\pi$
$878$	0	0
$879$	3.29219e7	1.43718
$880$	0	0
$881$	−2.57866e7	−1.11932	−0.559661	−	0.828722i	$-0.689068\pi$
$881$	−2.57866e7	−1.11932	−0.559661	+	0.828722i	$0.689068\pi$
$882$	0	0
$883$	3.33298e7	1.43857	0.719284	−	0.694716i	$-0.244470\pi$
$883$	3.33298e7	1.43857	0.719284	+	0.694716i	$0.244470\pi$
$884$	0	0
$885$	5.41082e6	0.232223
$886$	0	0
$887$	9.27907e6	0.396000	0.198000	−	0.980202i	$-0.436555\pi$
$887$	9.27907e6	0.396000	0.198000	+	0.980202i	$0.436555\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.71496e6	−0.198968
$892$	0	0
$893$	−4.02068e7	−1.68722
$894$	0	0
$895$	8.31264e6	0.346882
$896$	0	0
$897$	−1.31789e7	−0.546887
$898$	0	0
$899$	−6.24378e6	−0.257661
$900$	0	0
$901$	−3.94965e7	−1.62086
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.32800e7	0.538985
$906$	0	0
$907$	−2.02398e7	−0.816934	−0.408467	−	0.912773i	$-0.633937\pi$
$907$	−2.02398e7	−0.816934	−0.408467	+	0.912773i	$0.633937\pi$
$908$	0	0
$909$	1.43000e6	0.0574019
$910$	0	0
$911$	−1.26979e7	−0.506917	−0.253459	−	0.967346i	$-0.581568\pi$
$911$	−1.26979e7	−0.506917	−0.253459	+	0.967346i	$0.581568\pi$
$912$	0	0
$913$	8.54726e6	0.339352
$914$	0	0
$915$	765952.	0.0302447
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.51393e7	−0.591314	−0.295657	−	0.955294i	$-0.595539\pi$
$919$	−1.51393e7	−0.591314	−0.295657	+	0.955294i	$0.595539\pi$
$920$	0	0
$921$	−3.31359e7	−1.28721
$922$	0	0
$923$	−4.38592e7	−1.69456
$924$	0	0
$925$	−1.41671e7	−0.544412
$926$	0	0
$927$	−1.09366e6	−0.0418008
$928$	0	0
$929$	−8.15971e6	−0.310196	−0.155098	−	0.987899i	$-0.549569\pi$
$929$	−8.15971e6	−0.310196	−0.155098	+	0.987899i	$0.549569\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.52515e7	−0.573597
$934$	0	0
$935$	−1.28410e6	−0.0480362
$936$	0	0
$937$	−2.65647e7	−0.988454	−0.494227	−	0.869333i	$-0.664549\pi$
$937$	−2.65647e7	−0.988454	−0.494227	+	0.869333i	$0.664549\pi$
$938$	0	0
$939$	4.57344e7	1.69270
$940$	0	0
$941$	−3.90535e7	−1.43776	−0.718880	−	0.695135i	$-0.755345\pi$
$941$	−3.90535e7	−1.43776	−0.718880	+	0.695135i	$0.755345\pi$
$942$	0	0
$943$	−1.48262e7	−0.542940
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.83971e7	−1.39131	−0.695655	−	0.718376i	$-0.744886\pi$
$947$	−3.83971e7	−1.39131	−0.695655	+	0.718376i	$0.744886\pi$
$948$	0	0
$949$	−4.95616e7	−1.78640
$950$	0	0
$951$	3.18016e7	1.14024
$952$	0	0
$953$	−2.78487e6	−0.0993282	−0.0496641	−	0.998766i	$-0.515815\pi$
$953$	−2.78487e6	−0.0993282	−0.0496641	+	0.998766i	$0.515815\pi$
$954$	0	0
$955$	4.82573e6	0.171220
$956$	0	0
$957$	4.84211e6	0.170905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.61705e7	−0.914122
$962$	0	0
$963$	−181324.	−0.00630071
$964$	0	0
$965$	7.75139e6	0.267955
$966$	0	0
$967$	7.71822e6	0.265430	0.132715	−	0.991154i	$-0.457630\pi$
$967$	7.71822e6	0.265430	0.132715	+	0.991154i	$0.457630\pi$
$968$	0	0
$969$	−3.27107e7	−1.11913
$970$	0	0
$971$	−2.19732e7	−0.747904	−0.373952	−	0.927448i	$-0.621998\pi$
$971$	−2.19732e7	−0.747904	−0.373952	+	0.927448i	$0.621998\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−4.03955e7	−1.36089
$976$	0	0
$977$	−1.08137e7	−0.362442	−0.181221	−	0.983442i	$-0.558005\pi$
$977$	−1.08137e7	−0.362442	−0.181221	+	0.983442i	$0.558005\pi$
$978$	0	0
$979$	−4.88346e6	−0.162844
$980$	0	0
$981$	293722.	0.00974460
$982$	0	0
$983$	−3.71136e7	−1.22504	−0.612519	−	0.790456i	$-0.709844\pi$
$983$	−3.71136e7	−1.22504	−0.612519	+	0.790456i	$0.709844\pi$
$984$	0	0
$985$	2.92829e6	0.0961664
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.53616e7	−0.499398
$990$	0	0
$991$	1.19036e7	0.385030	0.192515	−	0.981294i	$-0.438336\pi$
$991$	1.19036e7	0.385030	0.192515	+	0.981294i	$0.438336\pi$
$992$	0	0
$993$	6.31328e6	0.203180
$994$	0	0
$995$	1.44686e7	0.463307
$996$	0	0
$997$	164912.	0.00525429	0.00262715	−	0.999997i	$-0.499164\pi$
$997$	164912.	0.00525429	0.00262715	+	0.999997i	$0.499164\pi$
$998$	0	0
$999$	1.81718e7	0.576083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.b.1.1		1
4.3	odd	2		196.6.a.f.1.1	yes	1
7.6	odd	2		784.6.a.j.1.1		1
28.3	even	6		196.6.e.h.177.1		2
28.11	odd	6		196.6.e.c.177.1		2
28.19	even	6		196.6.e.h.165.1		2
28.23	odd	6		196.6.e.c.165.1		2
28.27	even	2		196.6.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.6.a.c.1.1	✓	1	28.27	even	2
196.6.a.f.1.1	yes	1	4.3	odd	2
196.6.e.c.165.1		2	28.23	odd	6
196.6.e.c.177.1		2	28.11	odd	6
196.6.e.h.165.1		2	28.19	even	6
196.6.e.h.177.1		2	28.3	even	6
784.6.a.b.1.1		1	1.1	even	1	trivial
784.6.a.j.1.1		1	7.6	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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