LMFDB - Embedded newform 784.6.a.r.1.1

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,-14,0,42,0,0,0,652] 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{130}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 130 $ x^2 - 130
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-11.4018$ 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)

$f(q)$	$=$	$q-29.8035 q^{3} +21.0000 q^{5} +645.249 q^{9} -331.874 q^{11} -66.8211 q^{13} -625.874 q^{15} -240.537 q^{17} +441.979 q^{19} +1071.37 q^{23} -2684.00 q^{25} -11988.4 q^{27} +1791.75 q^{29} -5688.74 q^{31} +9891.00 q^{33} +11210.7 q^{37} +1991.50 q^{39} -12077.9 q^{41} +9921.98 q^{43} +13550.2 q^{45} -16869.2 q^{47} +7168.84 q^{51} +5298.77 q^{53} -6969.35 q^{55} -13172.5 q^{57} +41384.8 q^{59} +21523.3 q^{61} -1403.24 q^{65} +26618.8 q^{67} -31930.5 q^{69} +58096.5 q^{71} +39987.5 q^{73} +79992.6 q^{75} +43949.8 q^{79} +200502. q^{81} +22421.4 q^{83} -5051.27 q^{85} -53400.4 q^{87} -24062.0 q^{89} +169544. q^{93} +9281.56 q^{95} -71896.4 q^{97} -214141. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} + 42 q^{5} + 652 q^{9} + 294 q^{11} + 140 q^{13} - 294 q^{15} - 1302 q^{17} - 1442 q^{19} - 2646 q^{23} - 5368 q^{25} - 15722 q^{27} + 1668 q^{29} - 14798 q^{31} + 19782 q^{33} + 5182 q^{37}+ \cdots - 209916 q^{99}+O(q^{100}) $ 2 * q - 14 * q^3 + 42 * q^5 + 652 * q^9 + 294 * q^11 + 140 * q^13 - 294 * q^15 - 1302 * q^17 - 1442 * q^19 - 2646 * q^23 - 5368 * q^25 - 15722 * q^27 + 1668 * q^29 - 14798 * q^31 + 19782 * q^33 + 5182 * q^37 + 5260 * q^39 + 5124 * q^41 + 4520 * q^43 + 13692 * q^45 - 14994 * q^47 - 9606 * q^51 + 24006 * q^53 + 6174 * q^55 - 42946 * q^57 + 38850 * q^59 + 23618 * q^61 + 2940 * q^65 - 32002 * q^67 - 90678 * q^69 + 89376 * q^71 + 47138 * q^73 + 37576 * q^75 + 40970 * q^79 + 139858 * q^81 + 68376 * q^83 - 27342 * q^85 - 55356 * q^87 - 123102 * q^89 + 25586 * q^93 - 30282 * q^95 + 43652 * q^97 - 209916 * 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
$3$	−29.8035	−1.91190	−0.955948	−	0.293536i	$-0.905168\pi$
$3$	−29.8035	−1.91190	−0.955948	+	0.293536i	$0.905168\pi$
$4$	0	0
$5$	21.0000	0.375659	0.187830	−	0.982202i	$-0.439855\pi$
$5$	21.0000	0.375659	0.187830	+	0.982202i	$0.439855\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	645.249	2.65535
$10$	0	0
$11$	−331.874	−0.826973	−0.413486	−	0.910510i	$-0.635689\pi$
$11$	−331.874	−0.826973	−0.413486	+	0.910510i	$0.635689\pi$
$12$	0	0
$13$	−66.8211	−0.109662	−0.0548308	−	0.998496i	$-0.517462\pi$
$13$	−66.8211	−0.109662	−0.0548308	+	0.998496i	$0.517462\pi$
$14$	0	0
$15$	−625.874	−0.718222
$16$	0	0
$17$	−240.537	−0.201864	−0.100932	−	0.994893i	$-0.532182\pi$
$17$	−240.537	−0.201864	−0.100932	+	0.994893i	$0.532182\pi$
$18$	0	0
$19$	441.979	0.280878	0.140439	−	0.990089i	$-0.455149\pi$
$19$	441.979	0.280878	0.140439	+	0.990089i	$0.455149\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1071.37	0.422298	0.211149	−	0.977454i	$-0.432279\pi$
$23$	1071.37	0.422298	0.211149	+	0.977454i	$0.432279\pi$
$24$	0	0
$25$	−2684.00	−0.858880
$26$	0	0
$27$	−11988.4	−3.16485
$28$	0	0
$29$	1791.75	0.395623	0.197812	−	0.980240i	$-0.436617\pi$
$29$	1791.75	0.395623	0.197812	+	0.980240i	$0.436617\pi$
$30$	0	0
$31$	−5688.74	−1.06319	−0.531596	−	0.846998i	$-0.678407\pi$
$31$	−5688.74	−1.06319	−0.531596	+	0.846998i	$0.678407\pi$
$32$	0	0
$33$	9891.00	1.58109
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11210.7	1.34626	0.673131	−	0.739523i	$-0.264949\pi$
$37$	11210.7	1.34626	0.673131	+	0.739523i	$0.264949\pi$
$38$	0	0
$39$	1991.50	0.209662
$40$	0	0
$41$	−12077.9	−1.12210	−0.561048	−	0.827783i	$-0.689602\pi$
$41$	−12077.9	−1.12210	−0.561048	+	0.827783i	$0.689602\pi$
$42$	0	0
$43$	9921.98	0.818328	0.409164	−	0.912461i	$-0.365820\pi$
$43$	9921.98	0.818328	0.409164	+	0.912461i	$0.365820\pi$
$44$	0	0
$45$	13550.2	0.997506
$46$	0	0
$47$	−16869.2	−1.11391	−0.556956	−	0.830542i	$-0.688031\pi$
$47$	−16869.2	−1.11391	−0.556956	+	0.830542i	$0.688031\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7168.84	0.385943
$52$	0	0
$53$	5298.77	0.259111	0.129555	−	0.991572i	$-0.458645\pi$
$53$	5298.77	0.259111	0.129555	+	0.991572i	$0.458645\pi$
$54$	0	0
$55$	−6969.35	−0.310660
$56$	0	0
$57$	−13172.5	−0.537009
$58$	0	0
$59$	41384.8	1.54778	0.773892	−	0.633317i	$-0.218307\pi$
$59$	41384.8	1.54778	0.773892	+	0.633317i	$0.218307\pi$
$60$	0	0
$61$	21523.3	0.740601	0.370300	−	0.928912i	$-0.379255\pi$
$61$	21523.3	0.740601	0.370300	+	0.928912i	$0.379255\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1403.24	−0.0411954
$66$	0	0
$67$	26618.8	0.724437	0.362219	−	0.932093i	$-0.382019\pi$
$67$	26618.8	0.724437	0.362219	+	0.932093i	$0.382019\pi$
$68$	0	0
$69$	−31930.5	−0.807390
$70$	0	0
$71$	58096.5	1.36774	0.683870	−	0.729603i	$-0.260295\pi$
$71$	58096.5	1.36774	0.683870	+	0.729603i	$0.260295\pi$
$72$	0	0
$73$	39987.5	0.878248	0.439124	−	0.898426i	$-0.355289\pi$
$73$	39987.5	0.878248	0.439124	+	0.898426i	$0.355289\pi$
$74$	0	0
$75$	79992.6	1.64209
$76$	0	0
$77$	0	0
$78$	0	0
$79$	43949.8	0.792299	0.396150	−	0.918186i	$-0.370346\pi$
$79$	43949.8	0.792299	0.396150	+	0.918186i	$0.370346\pi$
$80$	0	0
$81$	200502.	3.39552
$82$	0	0
$83$	22421.4	0.357246	0.178623	−	0.983918i	$-0.442836\pi$
$83$	22421.4	0.357246	0.178623	+	0.983918i	$0.442836\pi$
$84$	0	0
$85$	−5051.27	−0.0758322
$86$	0	0
$87$	−53400.4	−0.756390
$88$	0	0
$89$	−24062.0	−0.322001	−0.161001	−	0.986954i	$-0.551472\pi$
$89$	−24062.0	−0.322001	−0.161001	+	0.986954i	$0.551472\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	169544.	2.03271
$94$	0	0
$95$	9281.56	0.105514
$96$	0	0
$97$	−71896.4	−0.775850	−0.387925	−	0.921691i	$-0.626808\pi$
$97$	−71896.4	−0.775850	−0.387925	+	0.921691i	$0.626808\pi$
$98$	0	0
$99$	−214141.	−2.19590
$100$	0	0
$101$	15865.5	0.154757	0.0773783	−	0.997002i	$-0.475345\pi$
$101$	15865.5	0.154757	0.0773783	+	0.997002i	$0.475345\pi$
$102$	0	0
$103$	−160574.	−1.49135	−0.745677	−	0.666307i	$-0.767874\pi$
$103$	−160574.	−1.49135	−0.745677	+	0.666307i	$0.767874\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	205047.	1.73139	0.865693	−	0.500575i	$-0.166878\pi$
$107$	205047.	1.73139	0.865693	+	0.500575i	$0.166878\pi$
$108$	0	0
$109$	−112544.	−0.907313	−0.453657	−	0.891177i	$-0.649881\pi$
$109$	−112544.	−0.907313	−0.453657	+	0.891177i	$0.649881\pi$
$110$	0	0
$111$	−334119.	−2.57391
$112$	0	0
$113$	−118332.	−0.871782	−0.435891	−	0.900000i	$-0.643567\pi$
$113$	−118332.	−0.871782	−0.435891	+	0.900000i	$0.643567\pi$
$114$	0	0
$115$	22498.7	0.158640
$116$	0	0
$117$	−43116.2	−0.291190
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−50910.9	−0.316116
$122$	0	0
$123$	359962.	2.14533
$124$	0	0
$125$	−121989.	−0.698306
$126$	0	0
$127$	245195.	1.34897	0.674485	−	0.738289i	$-0.264366\pi$
$127$	245195.	1.34897	0.674485	+	0.738289i	$0.264366\pi$
$128$	0	0
$129$	−295710.	−1.56456
$130$	0	0
$131$	69993.7	0.356353	0.178177	−	0.983999i	$-0.442980\pi$
$131$	69993.7	0.356353	0.178177	+	0.983999i	$0.442980\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−251757.	−1.18891
$136$	0	0
$137$	−187664.	−0.854239	−0.427119	−	0.904195i	$-0.640472\pi$
$137$	−187664.	−0.854239	−0.427119	+	0.904195i	$0.640472\pi$
$138$	0	0
$139$	−78272.7	−0.343616	−0.171808	−	0.985130i	$-0.554961\pi$
$139$	−78272.7	−0.343616	−0.171808	+	0.985130i	$0.554961\pi$
$140$	0	0
$141$	502763.	2.12968
$142$	0	0
$143$	22176.1	0.0906872
$144$	0	0
$145$	37626.7	0.148620
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−92166.4	−0.340100	−0.170050	−	0.985435i	$-0.554393\pi$
$149$	−92166.4	−0.340100	−0.170050	+	0.985435i	$0.554393\pi$
$150$	0	0
$151$	−53416.0	−0.190646	−0.0953232	−	0.995446i	$-0.530388\pi$
$151$	−53416.0	−0.190646	−0.0953232	+	0.995446i	$0.530388\pi$
$152$	0	0
$153$	−155206.	−0.536019
$154$	0	0
$155$	−119463.	−0.399398
$156$	0	0
$157$	280501.	0.908206	0.454103	−	0.890949i	$-0.349960\pi$
$157$	280501.	0.908206	0.454103	+	0.890949i	$0.349960\pi$
$158$	0	0
$159$	−157922.	−0.495393
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−626810.	−1.84785	−0.923925	−	0.382574i	$-0.875038\pi$
$163$	−626810.	−1.84785	−0.923925	+	0.382574i	$0.875038\pi$
$164$	0	0
$165$	207711.	0.593950
$166$	0	0
$167$	293997.	0.815741	0.407870	−	0.913040i	$-0.366272\pi$
$167$	293997.	0.815741	0.407870	+	0.913040i	$0.366272\pi$
$168$	0	0
$169$	−366828.	−0.987974
$170$	0	0
$171$	285187.	0.745828
$172$	0	0
$173$	−714187.	−1.81425	−0.907124	−	0.420864i	$-0.861727\pi$
$173$	−714187.	−1.81425	−0.907124	+	0.420864i	$0.861727\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.23341e6	−2.95920
$178$	0	0
$179$	−580050.	−1.35311	−0.676554	−	0.736393i	$-0.736528\pi$
$179$	−580050.	−1.35311	−0.676554	+	0.736393i	$0.736528\pi$
$180$	0	0
$181$	308046.	0.698907	0.349454	−	0.936954i	$-0.386367\pi$
$181$	308046.	0.698907	0.349454	+	0.936954i	$0.386367\pi$
$182$	0	0
$183$	−641470.	−1.41595
$184$	0	0
$185$	235425.	0.505736
$186$	0	0
$187$	79827.8	0.166936
$188$	0	0
$189$	0	0
$190$	0	0
$191$	202789.	0.402218	0.201109	−	0.979569i	$-0.435545\pi$
$191$	202789.	0.402218	0.201109	+	0.979569i	$0.435545\pi$
$192$	0	0
$193$	605988.	1.17104	0.585519	−	0.810659i	$-0.300891\pi$
$193$	605988.	1.17104	0.585519	+	0.810659i	$0.300891\pi$
$194$	0	0
$195$	41821.5	0.0787614
$196$	0	0
$197$	−310.819	−0.000570614 0	−0.000285307	−	1.00000i	$-0.500091\pi$
$197$	−310.819	−0.000570614 0	−0.000285307		1.00000i	$0.500091\pi$
$198$	0	0
$199$	−409666.	−0.733327	−0.366664	−	0.930354i	$-0.619500\pi$
$199$	−409666.	−0.733327	−0.366664	+	0.930354i	$0.619500\pi$
$200$	0	0
$201$	−793332.	−1.38505
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−253635.	−0.421526
$206$	0	0
$207$	691300.	1.12135
$208$	0	0
$209$	−146681.	−0.232278
$210$	0	0
$211$	441339.	0.682442	0.341221	−	0.939983i	$-0.389160\pi$
$211$	441339.	0.682442	0.341221	+	0.939983i	$0.389160\pi$
$212$	0	0
$213$	−1.73148e6	−2.61498
$214$	0	0
$215$	208362.	0.307412
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.19177e6	−1.67912
$220$	0	0
$221$	16072.9	0.0221368
$222$	0	0
$223$	265133.	0.357028	0.178514	−	0.983937i	$-0.442871\pi$
$223$	265133.	0.357028	0.178514	+	0.983937i	$0.442871\pi$
$224$	0	0
$225$	−1.73185e6	−2.28062
$226$	0	0
$227$	−1.43044e6	−1.84249	−0.921244	−	0.388985i	$-0.872826\pi$
$227$	−1.43044e6	−1.84249	−0.921244	+	0.388985i	$0.872826\pi$
$228$	0	0
$229$	1.25805e6	1.58529	0.792646	−	0.609682i	$-0.208703\pi$
$229$	1.25805e6	1.58529	0.792646	+	0.609682i	$0.208703\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	224723.	0.271180	0.135590	−	0.990765i	$-0.456707\pi$
$233$	224723.	0.271180	0.135590	+	0.990765i	$0.456707\pi$
$234$	0	0
$235$	−354254.	−0.418452
$236$	0	0
$237$	−1.30986e6	−1.51479
$238$	0	0
$239$	−1.28193e6	−1.45167	−0.725837	−	0.687867i	$-0.758547\pi$
$239$	−1.28193e6	−1.45167	−0.725837	+	0.687867i	$0.758547\pi$
$240$	0	0
$241$	576125.	0.638961	0.319480	−	0.947593i	$-0.396492\pi$
$241$	576125.	0.638961	0.319480	+	0.947593i	$0.396492\pi$
$242$	0	0
$243$	−3.06247e6	−3.32703
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−29533.5	−0.0308015
$248$	0	0
$249$	−668236.	−0.683017
$250$	0	0
$251$	−609040.	−0.610185	−0.305092	−	0.952323i	$-0.598687\pi$
$251$	−609040.	−0.610185	−0.305092	+	0.952323i	$0.598687\pi$
$252$	0	0
$253$	−355559.	−0.349229
$254$	0	0
$255$	150546.	0.144983
$256$	0	0
$257$	−1.03067e6	−0.973389	−0.486695	−	0.873572i	$-0.661798\pi$
$257$	−1.03067e6	−0.973389	−0.486695	+	0.873572i	$0.661798\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.15612e6	1.05052
$262$	0	0
$263$	910585.	0.811767	0.405883	−	0.913925i	$-0.366964\pi$
$263$	910585.	0.811767	0.405883	+	0.913925i	$0.366964\pi$
$264$	0	0
$265$	111274.	0.0973374
$266$	0	0
$267$	717133.	0.615633
$268$	0	0
$269$	−1.42100e6	−1.19733	−0.598663	−	0.801001i	$-0.704301\pi$
$269$	−1.42100e6	−1.19733	−0.598663	+	0.801001i	$0.704301\pi$
$270$	0	0
$271$	−297530.	−0.246097	−0.123049	−	0.992401i	$-0.539267\pi$
$271$	−297530.	−0.246097	−0.123049	+	0.992401i	$0.539267\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	890749.	0.710270
$276$	0	0
$277$	849612.	0.665305	0.332653	−	0.943049i	$-0.392056\pi$
$277$	849612.	0.665305	0.332653	+	0.943049i	$0.392056\pi$
$278$	0	0
$279$	−3.67065e6	−2.82314
$280$	0	0
$281$	−7680.49	−0.00580261	−0.00290130	−	0.999996i	$-0.500924\pi$
$281$	−7680.49	−0.00580261	−0.00290130	+	0.999996i	$0.500924\pi$
$282$	0	0
$283$	−985368.	−0.731362	−0.365681	−	0.930740i	$-0.619164\pi$
$283$	−985368.	−0.731362	−0.365681	+	0.930740i	$0.619164\pi$
$284$	0	0
$285$	−276623.	−0.201733
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.36200e6	−0.959251
$290$	0	0
$291$	2.14277e6	1.48335
$292$	0	0
$293$	2.16934e6	1.47625	0.738124	−	0.674665i	$-0.235712\pi$
$293$	2.16934e6	1.47625	0.738124	+	0.674665i	$0.235712\pi$
$294$	0	0
$295$	869080.	0.581440
$296$	0	0
$297$	3.97865e6	2.61724
$298$	0	0
$299$	−71590.0	−0.0463099
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−472846.	−0.295879
$304$	0	0
$305$	451989.	0.278214
$306$	0	0
$307$	−1.39093e6	−0.842287	−0.421143	−	0.906994i	$-0.638371\pi$
$307$	−1.39093e6	−0.842287	−0.421143	+	0.906994i	$0.638371\pi$
$308$	0	0
$309$	4.78566e6	2.85132
$310$	0	0
$311$	2.03794e6	1.19479	0.597394	−	0.801948i	$-0.296203\pi$
$311$	2.03794e6	1.19479	0.597394	+	0.801948i	$0.296203\pi$
$312$	0	0
$313$	1.16958e6	0.674789	0.337395	−	0.941363i	$-0.390454\pi$
$313$	1.16958e6	0.674789	0.337395	+	0.941363i	$0.390454\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	801137.	0.447774	0.223887	−	0.974615i	$-0.428125\pi$
$317$	801137.	0.447774	0.223887	+	0.974615i	$0.428125\pi$
$318$	0	0
$319$	−594634.	−0.327170
$320$	0	0
$321$	−6.11112e6	−3.31023
$322$	0	0
$323$	−106312.	−0.0566992
$324$	0	0
$325$	179348.	0.0941862
$326$	0	0
$327$	3.35421e6	1.73469
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.64951e6	−1.32922	−0.664608	−	0.747192i	$-0.731402\pi$
$331$	−2.64951e6	−1.32922	−0.664608	+	0.747192i	$0.731402\pi$
$332$	0	0
$333$	7.23371e6	3.57479
$334$	0	0
$335$	558994.	0.272142
$336$	0	0
$337$	−3.40056e6	−1.63108	−0.815541	−	0.578699i	$-0.803560\pi$
$337$	−3.40056e6	−1.63108	−0.815541	+	0.578699i	$0.803560\pi$
$338$	0	0
$339$	3.52672e6	1.66676
$340$	0	0
$341$	1.88794e6	0.879230
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−670541.	−0.303304
$346$	0	0
$347$	617709.	0.275398	0.137699	−	0.990474i	$-0.456029\pi$
$347$	617709.	0.275398	0.137699	+	0.990474i	$0.456029\pi$
$348$	0	0
$349$	−2.70539e6	−1.18896	−0.594479	−	0.804111i	$-0.702642\pi$
$349$	−2.70539e6	−1.18896	−0.594479	+	0.804111i	$0.702642\pi$
$350$	0	0
$351$	801080.	0.347063
$352$	0	0
$353$	3.18064e6	1.35856	0.679279	−	0.733880i	$-0.262293\pi$
$353$	3.18064e6	1.35856	0.679279	+	0.733880i	$0.262293\pi$
$354$	0	0
$355$	1.22003e6	0.513805
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.45970e6	1.82629	0.913145	−	0.407635i	$-0.133646\pi$
$359$	4.45970e6	1.82629	0.913145	+	0.407635i	$0.133646\pi$
$360$	0	0
$361$	−2.28075e6	−0.921108
$362$	0	0
$363$	1.51732e6	0.604382
$364$	0	0
$365$	839738.	0.329922
$366$	0	0
$367$	461247.	0.178759	0.0893795	−	0.995998i	$-0.471512\pi$
$367$	461247.	0.178759	0.0893795	+	0.995998i	$0.471512\pi$
$368$	0	0
$369$	−7.79322e6	−2.97955
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.41954e6	1.27261	0.636305	−	0.771437i	$-0.280462\pi$
$373$	3.41954e6	1.27261	0.636305	+	0.771437i	$0.280462\pi$
$374$	0	0
$375$	3.63570e6	1.33509
$376$	0	0
$377$	−119726.	−0.0433847
$378$	0	0
$379$	−16355.6	−0.00584882	−0.00292441	−	0.999996i	$-0.500931\pi$
$379$	−16355.6	−0.00584882	−0.00292441	+	0.999996i	$0.500931\pi$
$380$	0	0
$381$	−7.30767e6	−2.57909
$382$	0	0
$383$	−3.43643e6	−1.19705	−0.598523	−	0.801105i	$-0.704246\pi$
$383$	−3.43643e6	−1.19705	−0.598523	+	0.801105i	$0.704246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.40215e6	2.17294
$388$	0	0
$389$	−4.81052e6	−1.61183	−0.805914	−	0.592033i	$-0.798325\pi$
$389$	−4.81052e6	−1.61183	−0.805914	+	0.592033i	$0.798325\pi$
$390$	0	0
$391$	−257704.	−0.0852469
$392$	0	0
$393$	−2.08606e6	−0.681310
$394$	0	0
$395$	922946.	0.297635
$396$	0	0
$397$	3.29848e6	1.05036	0.525180	−	0.850991i	$-0.323998\pi$
$397$	3.29848e6	1.05036	0.525180	+	0.850991i	$0.323998\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	136164.	0.0422865	0.0211432	−	0.999776i	$-0.493269\pi$
$401$	136164.	0.0422865	0.0211432	+	0.999776i	$0.493269\pi$
$402$	0	0
$403$	380127.	0.116591
$404$	0	0
$405$	4.21054e6	1.27556
$406$	0	0
$407$	−3.72054e6	−1.11332
$408$	0	0
$409$	−2.74869e6	−0.812490	−0.406245	−	0.913764i	$-0.633162\pi$
$409$	−2.74869e6	−0.812490	−0.406245	+	0.913764i	$0.633162\pi$
$410$	0	0
$411$	5.59304e6	1.63322
$412$	0	0
$413$	0	0
$414$	0	0
$415$	470849.	0.134203
$416$	0	0
$417$	2.33280e6	0.656958
$418$	0	0
$419$	−1.87219e6	−0.520973	−0.260486	−	0.965478i	$-0.583883\pi$
$419$	−1.87219e6	−0.520973	−0.260486	+	0.965478i	$0.583883\pi$
$420$	0	0
$421$	655225.	0.180171	0.0900856	−	0.995934i	$-0.471286\pi$
$421$	655225.	0.180171	0.0900856	+	0.995934i	$0.471286\pi$
$422$	0	0
$423$	−1.08849e7	−2.95782
$424$	0	0
$425$	645601.	0.173377
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−660927.	−0.173385
$430$	0	0
$431$	−6.37690e6	−1.65355	−0.826773	−	0.562535i	$-0.809826\pi$
$431$	−6.37690e6	−1.65355	−0.826773	+	0.562535i	$0.809826\pi$
$432$	0	0
$433$	−4.80003e6	−1.23034	−0.615168	−	0.788396i	$-0.710912\pi$
$433$	−4.80003e6	−1.23034	−0.615168	+	0.788396i	$0.710912\pi$
$434$	0	0
$435$	−1.12141e6	−0.284145
$436$	0	0
$437$	473522.	0.118614
$438$	0	0
$439$	−5.59508e6	−1.38562	−0.692811	−	0.721119i	$-0.743628\pi$
$439$	−5.59508e6	−1.38562	−0.692811	+	0.721119i	$0.743628\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.14010e6	−1.00231	−0.501154	−	0.865358i	$-0.667091\pi$
$443$	−4.14010e6	−1.00231	−0.501154	+	0.865358i	$0.667091\pi$
$444$	0	0
$445$	−505303.	−0.120963
$446$	0	0
$447$	2.74688e6	0.650237
$448$	0	0
$449$	−305966.	−0.0716239	−0.0358119	−	0.999359i	$-0.511402\pi$
$449$	−305966.	−0.0716239	−0.0358119	+	0.999359i	$0.511402\pi$
$450$	0	0
$451$	4.00832e6	0.927943
$452$	0	0
$453$	1.59198e6	0.364496
$454$	0	0
$455$	0	0
$456$	0	0
$457$	892608.	0.199926	0.0999632	−	0.994991i	$-0.468127\pi$
$457$	892608.	0.199926	0.0999632	+	0.994991i	$0.468127\pi$
$458$	0	0
$459$	2.88366e6	0.638870
$460$	0	0
$461$	3.01465e6	0.660670	0.330335	−	0.943864i	$-0.392838\pi$
$461$	3.01465e6	0.660670	0.330335	+	0.943864i	$0.392838\pi$
$462$	0	0
$463$	−3.45497e6	−0.749017	−0.374508	−	0.927224i	$-0.622188\pi$
$463$	−3.45497e6	−0.749017	−0.374508	+	0.927224i	$0.622188\pi$
$464$	0	0
$465$	3.56043e6	0.763607
$466$	0	0
$467$	2.83776e6	0.602120	0.301060	−	0.953605i	$-0.402660\pi$
$467$	2.83776e6	0.602120	0.301060	+	0.953605i	$0.402660\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−8.35990e6	−1.73640
$472$	0	0
$473$	−3.29284e6	−0.676734
$474$	0	0
$475$	−1.18627e6	−0.241240
$476$	0	0
$477$	3.41903e6	0.688028
$478$	0	0
$479$	−4.70442e6	−0.936845	−0.468422	−	0.883505i	$-0.655177\pi$
$479$	−4.70442e6	−0.936845	−0.468422	+	0.883505i	$0.655177\pi$
$480$	0	0
$481$	−749113.	−0.147633
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.50982e6	−0.291455
$486$	0	0
$487$	−4.16634e6	−0.796036	−0.398018	−	0.917378i	$-0.630302\pi$
$487$	−4.16634e6	−0.796036	−0.398018	+	0.917378i	$0.630302\pi$
$488$	0	0
$489$	1.86811e7	3.53290
$490$	0	0
$491$	−1.57876e6	−0.295537	−0.147768	−	0.989022i	$-0.547209\pi$
$491$	−1.57876e6	−0.295537	−0.147768	+	0.989022i	$0.547209\pi$
$492$	0	0
$493$	−430981.	−0.0798622
$494$	0	0
$495$	−4.49697e6	−0.824910
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.55712e6	−0.279943	−0.139972	−	0.990156i	$-0.544701\pi$
$499$	−1.55712e6	−0.279943	−0.139972	+	0.990156i	$0.544701\pi$
$500$	0	0
$501$	−8.76215e6	−1.55961
$502$	0	0
$503$	−1.19459e6	−0.210523	−0.105261	−	0.994445i	$-0.533568\pi$
$503$	−1.19459e6	−0.210523	−0.105261	+	0.994445i	$0.533568\pi$
$504$	0	0
$505$	333175.	0.0581358
$506$	0	0
$507$	1.09328e7	1.88890
$508$	0	0
$509$	−3.16857e6	−0.542087	−0.271043	−	0.962567i	$-0.587369\pi$
$509$	−3.16857e6	−0.542087	−0.271043	+	0.962567i	$0.587369\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.29864e6	−0.888936
$514$	0	0
$515$	−3.37204e6	−0.560241
$516$	0	0
$517$	5.59846e6	0.921175
$518$	0	0
$519$	2.12853e7	3.46865
$520$	0	0
$521$	1.00658e6	0.162462	0.0812312	−	0.996695i	$-0.474115\pi$
$521$	1.00658e6	0.162462	0.0812312	+	0.996695i	$0.474115\pi$
$522$	0	0
$523$	8.78985e6	1.40517	0.702583	−	0.711602i	$-0.252030\pi$
$523$	8.78985e6	1.40517	0.702583	+	0.711602i	$0.252030\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.36835e6	0.214620
$528$	0	0
$529$	−5.28851e6	−0.821664
$530$	0	0
$531$	2.67035e7	4.10990
$532$	0	0
$533$	807055.	0.123051
$534$	0	0
$535$	4.30599e6	0.650412
$536$	0	0
$537$	1.72875e7	2.58700
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.54864e6	−0.521277	−0.260639	−	0.965436i	$-0.583933\pi$
$541$	−3.54864e6	−0.521277	−0.260639	+	0.965436i	$0.583933\pi$
$542$	0	0
$543$	−9.18086e6	−1.33624
$544$	0	0
$545$	−2.36343e6	−0.340841
$546$	0	0
$547$	−4.68179e6	−0.669027	−0.334513	−	0.942391i	$-0.608572\pi$
$547$	−4.68179e6	−0.669027	−0.334513	+	0.942391i	$0.608572\pi$
$548$	0	0
$549$	1.38879e7	1.96655
$550$	0	0
$551$	791915.	0.111122
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.01650e6	−0.966914
$556$	0	0
$557$	5.79507e6	0.791445	0.395723	−	0.918370i	$-0.370494\pi$
$557$	5.79507e6	0.791445	0.395723	+	0.918370i	$0.370494\pi$
$558$	0	0
$559$	−662997.	−0.0897392
$560$	0	0
$561$	−2.37915e6	−0.319165
$562$	0	0
$563$	−7.59703e6	−1.01012	−0.505060	−	0.863084i	$-0.668530\pi$
$563$	−7.59703e6	−1.01012	−0.505060	+	0.863084i	$0.668530\pi$
$564$	0	0
$565$	−2.48498e6	−0.327493
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.81699e6	−1.27115	−0.635576	−	0.772038i	$-0.719237\pi$
$569$	−9.81699e6	−1.27115	−0.635576	+	0.772038i	$0.719237\pi$
$570$	0	0
$571$	−5.25888e6	−0.674999	−0.337499	−	0.941326i	$-0.609581\pi$
$571$	−5.25888e6	−0.674999	−0.337499	+	0.941326i	$0.609581\pi$
$572$	0	0
$573$	−6.04384e6	−0.769000
$574$	0	0
$575$	−2.87555e6	−0.362703
$576$	0	0
$577$	−8.63468e6	−1.07971	−0.539855	−	0.841758i	$-0.681521\pi$
$577$	−8.63468e6	−1.07971	−0.539855	+	0.841758i	$0.681521\pi$
$578$	0	0
$579$	−1.80606e7	−2.23890
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.75852e6	−0.214277
$584$	0	0
$585$	−905441.	−0.109388
$586$	0	0
$587$	−3.32014e6	−0.397705	−0.198852	−	0.980029i	$-0.563721\pi$
$587$	−3.32014e6	−0.397705	−0.198852	+	0.980029i	$0.563721\pi$
$588$	0	0
$589$	−2.51430e6	−0.298627
$590$	0	0
$591$	9263.50	0.00109095
$592$	0	0
$593$	6.40001e6	0.747384	0.373692	−	0.927553i	$-0.378092\pi$
$593$	6.40001e6	0.747384	0.373692	+	0.927553i	$0.378092\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.22095e7	1.40204
$598$	0	0
$599$	−7.49862e6	−0.853915	−0.426957	−	0.904272i	$-0.640415\pi$
$599$	−7.49862e6	−0.853915	−0.426957	+	0.904272i	$0.640415\pi$
$600$	0	0
$601$	227052.	0.0256412	0.0128206	−	0.999918i	$-0.495919\pi$
$601$	227052.	0.0256412	0.0128206	+	0.999918i	$0.495919\pi$
$602$	0	0
$603$	1.71757e7	1.92363
$604$	0	0
$605$	−1.06913e6	−0.118752
$606$	0	0
$607$	1.63037e7	1.79603	0.898015	−	0.439965i	$-0.145009\pi$
$607$	1.63037e7	1.79603	0.898015	+	0.439965i	$0.145009\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.12722e6	0.122153
$612$	0	0
$613$	−1.04062e7	−1.11852	−0.559259	−	0.828993i	$-0.688914\pi$
$613$	−1.04062e7	−1.11852	−0.559259	+	0.828993i	$0.688914\pi$
$614$	0	0
$615$	7.55921e6	0.805914
$616$	0	0
$617$	4.74140e6	0.501411	0.250705	−	0.968063i	$-0.419337\pi$
$617$	4.74140e6	0.501411	0.250705	+	0.968063i	$0.419337\pi$
$618$	0	0
$619$	−1.08534e7	−1.13851	−0.569256	−	0.822160i	$-0.692769\pi$
$619$	−1.08534e7	−1.13851	−0.569256	+	0.822160i	$0.692769\pi$
$620$	0	0
$621$	−1.28440e7	−1.33651
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.82573e6	0.596555
$626$	0	0
$627$	4.37161e6	0.444092
$628$	0	0
$629$	−2.69659e6	−0.271762
$630$	0	0
$631$	−1.58978e7	−1.58951	−0.794757	−	0.606928i	$-0.792402\pi$
$631$	−1.58978e7	−1.58951	−0.794757	+	0.606928i	$0.792402\pi$
$632$	0	0
$633$	−1.31534e7	−1.30476
$634$	0	0
$635$	5.14909e6	0.506753
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.74867e7	3.63183
$640$	0	0
$641$	1.11828e7	1.07499	0.537496	−	0.843267i	$-0.319371\pi$
$641$	1.11828e7	1.07499	0.537496	+	0.843267i	$0.319371\pi$
$642$	0	0
$643$	1.55983e6	0.148782	0.0743909	−	0.997229i	$-0.476299\pi$
$643$	1.55983e6	0.148782	0.0743909	+	0.997229i	$0.476299\pi$
$644$	0	0
$645$	−6.20991e6	−0.587741
$646$	0	0
$647$	−1.54824e7	−1.45404	−0.727020	−	0.686616i	$-0.759095\pi$
$647$	−1.54824e7	−1.45404	−0.727020	+	0.686616i	$0.759095\pi$
$648$	0	0
$649$	−1.37345e7	−1.27998
$650$	0	0
$651$	0	0
$652$	0	0
$653$	614752.	0.0564179	0.0282090	−	0.999602i	$-0.491020\pi$
$653$	614752.	0.0564179	0.0282090	+	0.999602i	$0.491020\pi$
$654$	0	0
$655$	1.46987e6	0.133867
$656$	0	0
$657$	2.58019e7	2.33205
$658$	0	0
$659$	1.32697e7	1.19028	0.595139	−	0.803623i	$-0.297097\pi$
$659$	1.32697e7	1.19028	0.595139	+	0.803623i	$0.297097\pi$
$660$	0	0
$661$	−5.03584e6	−0.448299	−0.224150	−	0.974555i	$-0.571960\pi$
$661$	−5.03584e6	−0.448299	−0.224150	+	0.974555i	$0.571960\pi$
$662$	0	0
$663$	−479030.	−0.0423232
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.91962e6	0.167071
$668$	0	0
$669$	−7.90190e6	−0.682600
$670$	0	0
$671$	−7.14301e6	−0.612457
$672$	0	0
$673$	8.28068e6	0.704739	0.352369	−	0.935861i	$-0.385376\pi$
$673$	8.28068e6	0.704739	0.352369	+	0.935861i	$0.385376\pi$
$674$	0	0
$675$	3.21770e7	2.71823
$676$	0	0
$677$	1.52513e7	1.27889	0.639446	−	0.768836i	$-0.279164\pi$
$677$	1.52513e7	1.27889	0.639446	+	0.768836i	$0.279164\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.26321e7	3.52265
$682$	0	0
$683$	−7.88673e6	−0.646912	−0.323456	−	0.946243i	$-0.604845\pi$
$683$	−7.88673e6	−0.646912	−0.323456	+	0.946243i	$0.604845\pi$
$684$	0	0
$685$	−3.94094e6	−0.320903
$686$	0	0
$687$	−3.74943e7	−3.03091
$688$	0	0
$689$	−354069.	−0.0284145
$690$	0	0
$691$	−1.66190e6	−0.132407	−0.0662033	−	0.997806i	$-0.521089\pi$
$691$	−1.66190e6	−0.132407	−0.0662033	+	0.997806i	$0.521089\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.64373e6	−0.129083
$696$	0	0
$697$	2.90517e6	0.226511
$698$	0	0
$699$	−6.69754e6	−0.518469
$700$	0	0
$701$	−1.39364e7	−1.07117	−0.535583	−	0.844483i	$-0.679908\pi$
$701$	−1.39364e7	−1.07117	−0.535583	+	0.844483i	$0.679908\pi$
$702$	0	0
$703$	4.95490e6	0.378135
$704$	0	0
$705$	1.05580e7	0.800036
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.01874e7	−0.761108	−0.380554	−	0.924759i	$-0.624267\pi$
$709$	−1.01874e7	−0.761108	−0.380554	+	0.924759i	$0.624267\pi$
$710$	0	0
$711$	2.83586e7	2.10383
$712$	0	0
$713$	−6.09473e6	−0.448984
$714$	0	0
$715$	465699.	0.0340675
$716$	0	0
$717$	3.82060e7	2.77545
$718$	0	0
$719$	1.39169e7	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$719$	1.39169e7	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.71705e7	−1.22163
$724$	0	0
$725$	−4.80905e6	−0.339793
$726$	0	0
$727$	3.42063e6	0.240033	0.120016	−	0.992772i	$-0.461705\pi$
$727$	3.42063e6	0.240033	0.120016	+	0.992772i	$0.461705\pi$
$728$	0	0
$729$	4.25504e7	2.96541
$730$	0	0
$731$	−2.38660e6	−0.165191
$732$	0	0
$733$	1.03162e7	0.709186	0.354593	−	0.935021i	$-0.384619\pi$
$733$	1.03162e7	0.709186	0.354593	+	0.935021i	$0.384619\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.83406e6	−0.599090
$738$	0	0
$739$	1.03890e7	0.699779	0.349889	−	0.936791i	$-0.386219\pi$
$739$	1.03890e7	0.699779	0.349889	+	0.936791i	$0.386219\pi$
$740$	0	0
$741$	880202.	0.0588893
$742$	0	0
$743$	1.04738e7	0.696035	0.348018	−	0.937488i	$-0.386855\pi$
$743$	1.04738e7	0.696035	0.348018	+	0.937488i	$0.386855\pi$
$744$	0	0
$745$	−1.93550e6	−0.127762
$746$	0	0
$747$	1.44674e7	0.948612
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.40445e6	0.608462	0.304231	−	0.952598i	$-0.401600\pi$
$751$	9.40445e6	0.608462	0.304231	+	0.952598i	$0.401600\pi$
$752$	0	0
$753$	1.81515e7	1.16661
$754$	0	0
$755$	−1.12174e6	−0.0716181
$756$	0	0
$757$	−1.33677e7	−0.847848	−0.423924	−	0.905698i	$-0.639348\pi$
$757$	−1.33677e7	−0.847848	−0.423924	+	0.905698i	$0.639348\pi$
$758$	0	0
$759$	1.05969e7	0.667690
$760$	0	0
$761$	2.22623e7	1.39350	0.696752	−	0.717312i	$-0.254628\pi$
$761$	2.22623e7	1.39350	0.696752	+	0.717312i	$0.254628\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.25933e6	−0.201361
$766$	0	0
$767$	−2.76537e6	−0.169733
$768$	0	0
$769$	−9.65833e6	−0.588961	−0.294480	−	0.955658i	$-0.595147\pi$
$769$	−9.65833e6	−0.588961	−0.294480	+	0.955658i	$0.595147\pi$
$770$	0	0
$771$	3.07176e7	1.86102
$772$	0	0
$773$	5.10951e6	0.307561	0.153780	−	0.988105i	$-0.450855\pi$
$773$	5.10951e6	0.307561	0.153780	+	0.988105i	$0.450855\pi$
$774$	0	0
$775$	1.52686e7	0.913154
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.33816e6	−0.315172
$780$	0	0
$781$	−1.92807e7	−1.13108
$782$	0	0
$783$	−2.14802e7	−1.25209
$784$	0	0
$785$	5.89051e6	0.341176
$786$	0	0
$787$	2.51592e7	1.44797	0.723984	−	0.689816i	$-0.242309\pi$
$787$	2.51592e7	1.44797	0.723984	+	0.689816i	$0.242309\pi$
$788$	0	0
$789$	−2.71386e7	−1.55201
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.43821e6	−0.0812155
$794$	0	0
$795$	−3.31636e6	−0.186099
$796$	0	0
$797$	3.35976e6	0.187354	0.0936768	−	0.995603i	$-0.470138\pi$
$797$	3.35976e6	0.187354	0.0936768	+	0.995603i	$0.470138\pi$
$798$	0	0
$799$	4.05767e6	0.224859
$800$	0	0
$801$	−1.55260e7	−0.855024
$802$	0	0
$803$	−1.32708e7	−0.726287
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.23507e7	2.28916
$808$	0	0
$809$	−2.97050e7	−1.59572	−0.797862	−	0.602840i	$-0.794036\pi$
$809$	−2.97050e7	−1.59572	−0.797862	+	0.602840i	$0.794036\pi$
$810$	0	0
$811$	1.26386e7	0.674757	0.337379	−	0.941369i	$-0.390460\pi$
$811$	1.26386e7	0.674757	0.337379	+	0.941369i	$0.390460\pi$
$812$	0	0
$813$	8.86743e6	0.470513
$814$	0	0
$815$	−1.31630e7	−0.694162
$816$	0	0
$817$	4.38531e6	0.229850
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.61760e7	−1.35533	−0.677666	−	0.735370i	$-0.737008\pi$
$821$	−2.61760e7	−1.35533	−0.677666	+	0.735370i	$0.737008\pi$
$822$	0	0
$823$	7.29843e6	0.375604	0.187802	−	0.982207i	$-0.439864\pi$
$823$	7.29843e6	0.375604	0.187802	+	0.982207i	$0.439864\pi$
$824$	0	0
$825$	−2.65474e7	−1.35796
$826$	0	0
$827$	−1.18681e7	−0.603418	−0.301709	−	0.953400i	$-0.597557\pi$
$827$	−1.18681e7	−0.603418	−0.301709	+	0.953400i	$0.597557\pi$
$828$	0	0
$829$	1.04338e7	0.527298	0.263649	−	0.964619i	$-0.415074\pi$
$829$	1.04338e7	0.527298	0.263649	+	0.964619i	$0.415074\pi$
$830$	0	0
$831$	−2.53214e7	−1.27199
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.17394e6	0.306441
$836$	0	0
$837$	6.81991e7	3.36484
$838$	0	0
$839$	−2.91444e7	−1.42939	−0.714695	−	0.699436i	$-0.753435\pi$
$839$	−2.91444e7	−1.42939	−0.714695	+	0.699436i	$0.753435\pi$
$840$	0	0
$841$	−1.73008e7	−0.843482
$842$	0	0
$843$	228906.	0.0110940
$844$	0	0
$845$	−7.70339e6	−0.371142
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.93674e7	1.39829
$850$	0	0
$851$	1.20108e7	0.568524
$852$	0	0
$853$	−2.63032e7	−1.23776	−0.618879	−	0.785487i	$-0.712413\pi$
$853$	−2.63032e7	−1.23776	−0.618879	+	0.785487i	$0.712413\pi$
$854$	0	0
$855$	5.98892e6	0.280177
$856$	0	0
$857$	−3.43598e6	−0.159808	−0.0799040	−	0.996803i	$-0.525461\pi$
$857$	−3.43598e6	−0.159808	−0.0799040	+	0.996803i	$0.525461\pi$
$858$	0	0
$859$	1.09139e7	0.504659	0.252329	−	0.967641i	$-0.418803\pi$
$859$	1.09139e7	0.504659	0.252329	+	0.967641i	$0.418803\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.19058e7	−1.00123	−0.500614	−	0.865671i	$-0.666892\pi$
$863$	−2.19058e7	−1.00123	−0.500614	+	0.865671i	$0.666892\pi$
$864$	0	0
$865$	−1.49979e7	−0.681539
$866$	0	0
$867$	4.05923e7	1.83399
$868$	0	0
$869$	−1.45858e7	−0.655210
$870$	0	0
$871$	−1.77869e6	−0.0794430
$872$	0	0
$873$	−4.63911e7	−2.06015
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.46927e6	0.284025	0.142012	−	0.989865i	$-0.454643\pi$
$877$	6.46927e6	0.284025	0.142012	+	0.989865i	$0.454643\pi$
$878$	0	0
$879$	−6.46540e7	−2.82243
$880$	0	0
$881$	−2.03983e7	−0.885430	−0.442715	−	0.896662i	$-0.645985\pi$
$881$	−2.03983e7	−0.885430	−0.442715	+	0.896662i	$0.645985\pi$
$882$	0	0
$883$	−1.62381e7	−0.700862	−0.350431	−	0.936589i	$-0.613965\pi$
$883$	−1.62381e7	−0.700862	−0.350431	+	0.936589i	$0.613965\pi$
$884$	0	0
$885$	−2.59016e7	−1.11165
$886$	0	0
$887$	−6.79027e6	−0.289786	−0.144893	−	0.989447i	$-0.546284\pi$
$887$	−6.79027e6	−0.289786	−0.144893	+	0.989447i	$0.546284\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.65413e7	−2.80800
$892$	0	0
$893$	−7.45585e6	−0.312873
$894$	0	0
$895$	−1.21810e7	−0.508308
$896$	0	0
$897$	2.13363e6	0.0885398
$898$	0	0
$899$	−1.01928e7	−0.420623
$900$	0	0
$901$	−1.27455e6	−0.0523052
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.46897e6	0.262551
$906$	0	0
$907$	−3.42142e7	−1.38098	−0.690492	−	0.723340i	$-0.742606\pi$
$907$	−3.42142e7	−1.38098	−0.690492	+	0.723340i	$0.742606\pi$
$908$	0	0
$909$	1.02372e7	0.410932
$910$	0	0
$911$	4.47390e6	0.178604	0.0893019	−	0.996005i	$-0.471536\pi$
$911$	4.47390e6	0.178604	0.0893019	+	0.996005i	$0.471536\pi$
$912$	0	0
$913$	−7.44107e6	−0.295433
$914$	0	0
$915$	−1.34709e7	−0.531916
$916$	0	0
$917$	0	0
$918$	0	0
$919$	537900.	0.0210094	0.0105047	−	0.999945i	$-0.496656\pi$
$919$	537900.	0.0210094	0.0105047	+	0.999945i	$0.496656\pi$
$920$	0	0
$921$	4.14547e7	1.61036
$922$	0	0
$923$	−3.88207e6	−0.149989
$924$	0	0
$925$	−3.00896e7	−1.15628
$926$	0	0
$927$	−1.03610e8	−3.96006
$928$	0	0
$929$	1.77241e7	0.673792	0.336896	−	0.941542i	$-0.390623\pi$
$929$	1.77241e7	0.673792	0.336896	+	0.941542i	$0.390623\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.07379e7	−2.28431
$934$	0	0
$935$	1.67638e6	0.0627111
$936$	0	0
$937$	3.32444e7	1.23700	0.618499	−	0.785786i	$-0.287741\pi$
$937$	3.32444e7	1.23700	0.618499	+	0.785786i	$0.287741\pi$
$938$	0	0
$939$	−3.48575e7	−1.29013
$940$	0	0
$941$	−2.78480e7	−1.02523	−0.512613	−	0.858620i	$-0.671322\pi$
$941$	−2.78480e7	−1.02523	−0.512613	+	0.858620i	$0.671322\pi$
$942$	0	0
$943$	−1.29398e7	−0.473859
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.30569e7	0.835459	0.417729	−	0.908571i	$-0.362826\pi$
$947$	2.30569e7	0.835459	0.417729	+	0.908571i	$0.362826\pi$
$948$	0	0
$949$	−2.67201e6	−0.0963102
$950$	0	0
$951$	−2.38767e7	−0.856097
$952$	0	0
$953$	2.85536e7	1.01843	0.509213	−	0.860641i	$-0.329937\pi$
$953$	2.85536e7	1.01843	0.509213	+	0.860641i	$0.329937\pi$
$954$	0	0
$955$	4.25858e6	0.151097
$956$	0	0
$957$	1.77222e7	0.625514
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.73258e6	0.130377
$962$	0	0
$963$	1.32306e8	4.59743
$964$	0	0
$965$	1.27257e7	0.439911
$966$	0	0
$967$	3.45310e7	1.18753	0.593763	−	0.804640i	$-0.297642\pi$
$967$	3.45310e7	1.18753	0.593763	+	0.804640i	$0.297642\pi$
$968$	0	0
$969$	3.16848e6	0.108403
$970$	0	0
$971$	1.24933e7	0.425236	0.212618	−	0.977135i	$-0.431801\pi$
$971$	1.24933e7	0.425236	0.212618	+	0.977135i	$0.431801\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.34519e6	−0.180074
$976$	0	0
$977$	−3.89440e7	−1.30528	−0.652641	−	0.757667i	$-0.726339\pi$
$977$	−3.89440e7	−1.30528	−0.652641	+	0.757667i	$0.726339\pi$
$978$	0	0
$979$	7.98556e6	0.266286
$980$	0	0
$981$	−7.26191e7	−2.40923
$982$	0	0
$983$	279901.	0.00923889	0.00461945	−	0.999989i	$-0.498530\pi$
$983$	279901.	0.00923889	0.00461945	+	0.999989i	$0.498530\pi$
$984$	0	0
$985$	−6527.20	−0.000214356 0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.06301e7	0.345578
$990$	0	0
$991$	−1.96343e7	−0.635084	−0.317542	−	0.948244i	$-0.602858\pi$
$991$	−1.96343e7	−0.635084	−0.317542	+	0.948244i	$0.602858\pi$
$992$	0	0
$993$	7.89647e7	2.54132
$994$	0	0
$995$	−8.60299e6	−0.275481
$996$	0	0
$997$	3.73374e7	1.18961	0.594807	−	0.803868i	$-0.297228\pi$
$997$	3.73374e7	1.18961	0.594807	+	0.803868i	$0.297228\pi$
$998$	0	0
$999$	−1.34399e8	−4.26072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.r.1.1		2
4.3	odd	2		98.6.a.f.1.2		2
7.3	odd	6		112.6.i.b.65.1		4
7.5	odd	6		112.6.i.b.81.1		4
7.6	odd	2		784.6.a.bc.1.2		2
12.11	even	2		882.6.a.bl.1.1		2
28.3	even	6		14.6.c.b.9.2	✓	4
28.11	odd	6		98.6.c.f.79.1		4
28.19	even	6		14.6.c.b.11.2	yes	4
28.23	odd	6		98.6.c.f.67.1		4
28.27	even	2		98.6.a.c.1.1		2
84.47	odd	6		126.6.g.e.109.1		4
84.59	odd	6		126.6.g.e.37.1		4
84.83	odd	2		882.6.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.2	✓	4	28.3	even	6
14.6.c.b.11.2	yes	4	28.19	even	6
98.6.a.c.1.1		2	28.27	even	2
98.6.a.f.1.2		2	4.3	odd	2
98.6.c.f.67.1		4	28.23	odd	6
98.6.c.f.79.1		4	28.11	odd	6
112.6.i.b.65.1		4	7.3	odd	6
112.6.i.b.81.1		4	7.5	odd	6
126.6.g.e.37.1		4	84.59	odd	6
126.6.g.e.109.1		4	84.47	odd	6
784.6.a.r.1.1		2	1.1	even	1	trivial
784.6.a.bc.1.2		2	7.6	odd	2
882.6.a.bl.1.1		2	12.11	even	2
882.6.a.bt.1.1		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)

\(f(q)\)	\(=\)	\(q-29.8035 q^{3} +21.0000 q^{5} +645.249 q^{9} -331.874 q^{11} -66.8211 q^{13} -625.874 q^{15} -240.537 q^{17} +441.979 q^{19} +1071.37 q^{23} -2684.00 q^{25} -11988.4 q^{27} +1791.75 q^{29} -5688.74 q^{31} +9891.00 q^{33} +11210.7 q^{37} +1991.50 q^{39} -12077.9 q^{41} +9921.98 q^{43} +13550.2 q^{45} -16869.2 q^{47} +7168.84 q^{51} +5298.77 q^{53} -6969.35 q^{55} -13172.5 q^{57} +41384.8 q^{59} +21523.3 q^{61} -1403.24 q^{65} +26618.8 q^{67} -31930.5 q^{69} +58096.5 q^{71} +39987.5 q^{73} +79992.6 q^{75} +43949.8 q^{79} +200502. q^{81} +22421.4 q^{83} -5051.27 q^{85} -53400.4 q^{87} -24062.0 q^{89} +169544. q^{93} +9281.56 q^{95} -71896.4 q^{97} -214141. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} + 42 q^{5} + 652 q^{9} + 294 q^{11} + 140 q^{13} - 294 q^{15} - 1302 q^{17} - 1442 q^{19} - 2646 q^{23} - 5368 q^{25} - 15722 q^{27} + 1668 q^{29} - 14798 q^{31} + 19782 q^{33} + 5182 q^{37}+ \cdots - 209916 q^{99}+O(q^{100}) \) 2 * q - 14 * q^3 + 42 * q^5 + 652 * q^9 + 294 * q^11 + 140 * q^13 - 294 * q^15 - 1302 * q^17 - 1442 * q^19 - 2646 * q^23 - 5368 * q^25 - 15722 * q^27 + 1668 * q^29 - 14798 * q^31 + 19782 * q^33 + 5182 * q^37 + 5260 * q^39 + 5124 * q^41 + 4520 * q^43 + 13692 * q^45 - 14994 * q^47 - 9606 * q^51 + 24006 * q^53 + 6174 * q^55 - 42946 * q^57 + 38850 * q^59 + 23618 * q^61 + 2940 * q^65 - 32002 * q^67 - 90678 * q^69 + 89376 * q^71 + 47138 * q^73 + 37576 * q^75 + 40970 * q^79 + 139858 * q^81 + 68376 * q^83 - 27342 * q^85 - 55356 * q^87 - 123102 * q^89 + 25586 * q^93 - 30282 * q^95 + 43652 * q^97 - 209916 * q^99

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