LMFDB - Embedded newform 448.6.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.367637.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 107x + 282 $ x^3 - x^2 - 107*x + 282
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.76097$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.52194 q^{3} -72.6328 q^{5} -49.0000 q^{7} -186.420 q^{9} +O(q^{10})$	$q-7.52194 q^{3} -72.6328 q^{5} -49.0000 q^{7} -186.420 q^{9} -30.4503 q^{11} +1145.83 q^{13} +546.340 q^{15} -514.240 q^{17} +2319.63 q^{19} +368.575 q^{21} -409.434 q^{23} +2150.53 q^{25} +3230.07 q^{27} +1693.66 q^{29} +7030.24 q^{31} +229.045 q^{33} +3559.01 q^{35} -3935.80 q^{37} -8618.86 q^{39} -9236.29 q^{41} -21356.2 q^{43} +13540.2 q^{45} -9948.52 q^{47} +2401.00 q^{49} +3868.09 q^{51} +34576.4 q^{53} +2211.69 q^{55} -17448.1 q^{57} -46307.9 q^{59} +33817.6 q^{61} +9134.60 q^{63} -83224.8 q^{65} +49266.7 q^{67} +3079.74 q^{69} -49543.4 q^{71} +33822.3 q^{73} -16176.1 q^{75} +1492.07 q^{77} +622.967 q^{79} +21003.7 q^{81} -97253.4 q^{83} +37350.7 q^{85} -12739.6 q^{87} -16975.2 q^{89} -56145.6 q^{91} -52881.0 q^{93} -168481. q^{95} -65462.3 q^{97} +5676.56 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{3} + 14 q^{5} - 147 q^{7} + 151 q^{9}+O(q^{10}) $ 3 * q - 8 * q^3 + 14 * q^5 - 147 * q^7 + 151 * q^9	$ 3 q - 8 q^{3} + 14 q^{5} - 147 q^{7} + 151 q^{9} - 600 q^{11} + 974 q^{13} + 120 q^{15} + 718 q^{17} + 1056 q^{19} + 392 q^{21} - 3760 q^{23} - 147 q^{25} + 2872 q^{27} + 9134 q^{29} + 1448 q^{31} - 14872 q^{33} - 686 q^{35} + 4998 q^{37} + 824 q^{39} - 23186 q^{41} - 29880 q^{43} + 28350 q^{45} - 10840 q^{47} + 7203 q^{49} - 3776 q^{51} + 28006 q^{53} - 14952 q^{55} - 53504 q^{57} + 17456 q^{59} + 92294 q^{61} - 7399 q^{63} - 95300 q^{65} - 56024 q^{67} + 63480 q^{69} - 77064 q^{71} - 46346 q^{73} - 50768 q^{75} + 29400 q^{77} + 4376 q^{79} - 121973 q^{81} - 107128 q^{83} + 94348 q^{85} - 82720 q^{87} + 29814 q^{89} - 47726 q^{91} + 108432 q^{93} - 205304 q^{95} - 156482 q^{97} - 83128 q^{99}+O(q^{100}) $ 3 * q - 8 * q^3 + 14 * q^5 - 147 * q^7 + 151 * q^9 - 600 * q^11 + 974 * q^13 + 120 * q^15 + 718 * q^17 + 1056 * q^19 + 392 * q^21 - 3760 * q^23 - 147 * q^25 + 2872 * q^27 + 9134 * q^29 + 1448 * q^31 - 14872 * q^33 - 686 * q^35 + 4998 * q^37 + 824 * q^39 - 23186 * q^41 - 29880 * q^43 + 28350 * q^45 - 10840 * q^47 + 7203 * q^49 - 3776 * q^51 + 28006 * q^53 - 14952 * q^55 - 53504 * q^57 + 17456 * q^59 + 92294 * q^61 - 7399 * q^63 - 95300 * q^65 - 56024 * q^67 + 63480 * q^69 - 77064 * q^71 - 46346 * q^73 - 50768 * q^75 + 29400 * q^77 + 4376 * q^79 - 121973 * q^81 - 107128 * q^83 + 94348 * q^85 - 82720 * q^87 + 29814 * q^89 - 47726 * q^91 + 108432 * q^93 - 205304 * q^95 - 156482 * q^97 - 83128 * q^99

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
$3$	−7.52194	−0.482533	−0.241266	−	0.970459i	$-0.577563\pi$
$3$	−7.52194	−0.482533	−0.241266	+	0.970459i	$0.577563\pi$
$4$	0	0
$5$	−72.6328	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$5$	−72.6328	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−186.420	−0.767162
$10$	0	0
$11$	−30.4503	−0.0758770	−0.0379385	−	0.999280i	$-0.512079\pi$
$11$	−30.4503	−0.0758770	−0.0379385	+	0.999280i	$0.512079\pi$
$12$	0	0
$13$	1145.83	1.88045	0.940225	−	0.340555i	$-0.110615\pi$
$13$	1145.83	1.88045	0.940225	+	0.340555i	$0.110615\pi$
$14$	0	0
$15$	546.340	0.626953
$16$	0	0
$17$	−514.240	−0.431563	−0.215781	−	0.976442i	$-0.569230\pi$
$17$	−514.240	−0.431563	−0.215781	+	0.976442i	$0.569230\pi$
$18$	0	0
$19$	2319.63	1.47413	0.737063	−	0.675824i	$-0.236212\pi$
$19$	2319.63	1.47413	0.737063	+	0.675824i	$0.236212\pi$
$20$	0	0
$21$	368.575	0.182380
$22$	0	0
$23$	−409.434	−0.161385	−0.0806927	−	0.996739i	$-0.525713\pi$
$23$	−409.434	−0.161385	−0.0806927	+	0.996739i	$0.525713\pi$
$24$	0	0
$25$	2150.53	0.688169
$26$	0	0
$27$	3230.07	0.852714
$28$	0	0
$29$	1693.66	0.373965	0.186982	−	0.982363i	$-0.440129\pi$
$29$	1693.66	0.373965	0.186982	+	0.982363i	$0.440129\pi$
$30$	0	0
$31$	7030.24	1.31391	0.656955	−	0.753930i	$-0.271844\pi$
$31$	7030.24	1.31391	0.656955	+	0.753930i	$0.271844\pi$
$32$	0	0
$33$	229.045	0.0366131
$34$	0	0
$35$	3559.01	0.491088
$36$	0	0
$37$	−3935.80	−0.472638	−0.236319	−	0.971676i	$-0.575941\pi$
$37$	−3935.80	−0.472638	−0.236319	+	0.971676i	$0.575941\pi$
$38$	0	0
$39$	−8618.86	−0.907378
$40$	0	0
$41$	−9236.29	−0.858100	−0.429050	−	0.903281i	$-0.641152\pi$
$41$	−9236.29	−0.858100	−0.429050	+	0.903281i	$0.641152\pi$
$42$	0	0
$43$	−21356.2	−1.76138	−0.880691	−	0.473691i	$-0.842921\pi$
$43$	−21356.2	−1.76138	−0.880691	+	0.473691i	$0.842921\pi$
$44$	0	0
$45$	13540.2	0.996770
$46$	0	0
$47$	−9948.52	−0.656922	−0.328461	−	0.944518i	$-0.606530\pi$
$47$	−9948.52	−0.656922	−0.328461	+	0.944518i	$0.606530\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	3868.09	0.208243
$52$	0	0
$53$	34576.4	1.69079	0.845396	−	0.534140i	$-0.179364\pi$
$53$	34576.4	1.69079	0.845396	+	0.534140i	$0.179364\pi$
$54$	0	0
$55$	2211.69	0.0985866
$56$	0	0
$57$	−17448.1	−0.711314
$58$	0	0
$59$	−46307.9	−1.73191	−0.865954	−	0.500123i	$-0.833288\pi$
$59$	−46307.9	−1.73191	−0.865954	+	0.500123i	$0.833288\pi$
$60$	0	0
$61$	33817.6	1.16364	0.581819	−	0.813318i	$-0.302341\pi$
$61$	33817.6	1.16364	0.581819	+	0.813318i	$0.302341\pi$
$62$	0	0
$63$	9134.60	0.289960
$64$	0	0
$65$	−83224.8	−2.44326
$66$	0	0
$67$	49266.7	1.34081	0.670404	−	0.741996i	$-0.266121\pi$
$67$	49266.7	1.34081	0.670404	+	0.741996i	$0.266121\pi$
$68$	0	0
$69$	3079.74	0.0778737
$70$	0	0
$71$	−49543.4	−1.16638	−0.583190	−	0.812336i	$-0.698196\pi$
$71$	−49543.4	−1.16638	−0.583190	+	0.812336i	$0.698196\pi$
$72$	0	0
$73$	33822.3	0.742841	0.371420	−	0.928465i	$-0.378871\pi$
$73$	33822.3	0.742841	0.371420	+	0.928465i	$0.378871\pi$
$74$	0	0
$75$	−16176.1	−0.332064
$76$	0	0
$77$	1492.07	0.0286788
$78$	0	0
$79$	622.967	0.0112305	0.00561523	−	0.999984i	$-0.498213\pi$
$79$	622.967	0.0112305	0.00561523	+	0.999984i	$0.498213\pi$
$80$	0	0
$81$	21003.7	0.355700
$82$	0	0
$83$	−97253.4	−1.54956	−0.774782	−	0.632228i	$-0.782140\pi$
$83$	−97253.4	−1.54956	−0.774782	+	0.632228i	$0.782140\pi$
$84$	0	0
$85$	37350.7	0.560728
$86$	0	0
$87$	−12739.6	−0.180450
$88$	0	0
$89$	−16975.2	−0.227164	−0.113582	−	0.993529i	$-0.536232\pi$
$89$	−16975.2	−0.227164	−0.113582	+	0.993529i	$0.536232\pi$
$90$	0	0
$91$	−56145.6	−0.710743
$92$	0	0
$93$	−52881.0	−0.634005
$94$	0	0
$95$	−168481.	−1.91532
$96$	0	0
$97$	−65462.3	−0.706419	−0.353209	−	0.935544i	$-0.614910\pi$
$97$	−65462.3	−0.706419	−0.353209	+	0.935544i	$0.614910\pi$
$98$	0	0
$99$	5676.56	0.0582100
$100$	0	0
$101$	121485.	1.18500	0.592500	−	0.805570i	$-0.298141\pi$
$101$	121485.	1.18500	0.592500	+	0.805570i	$0.298141\pi$
$102$	0	0
$103$	−83208.1	−0.772810	−0.386405	−	0.922329i	$-0.626283\pi$
$103$	−83208.1	−0.772810	−0.386405	+	0.922329i	$0.626283\pi$
$104$	0	0
$105$	−26770.6	−0.236966
$106$	0	0
$107$	−33139.0	−0.279821	−0.139910	−	0.990164i	$-0.544681\pi$
$107$	−33139.0	−0.279821	−0.139910	+	0.990164i	$0.544681\pi$
$108$	0	0
$109$	−12724.1	−0.102580	−0.0512899	−	0.998684i	$-0.516333\pi$
$109$	−12724.1	−0.102580	−0.0512899	+	0.998684i	$0.516333\pi$
$110$	0	0
$111$	29604.9	0.228063
$112$	0	0
$113$	−20385.5	−0.150184	−0.0750922	−	0.997177i	$-0.523925\pi$
$113$	−20385.5	−0.150184	−0.0750922	+	0.997177i	$0.523925\pi$
$114$	0	0
$115$	29738.3	0.209687
$116$	0	0
$117$	−213606.	−1.44261
$118$	0	0
$119$	25197.8	0.163115
$120$	0	0
$121$	−160124.	−0.994243
$122$	0	0
$123$	69474.8	0.414061
$124$	0	0
$125$	70778.7	0.405161
$126$	0	0
$127$	235783.	1.29719	0.648594	−	0.761135i	$-0.275357\pi$
$127$	235783.	1.29719	0.648594	+	0.761135i	$0.275357\pi$
$128$	0	0
$129$	160640.	0.849925
$130$	0	0
$131$	−15276.4	−0.0777755	−0.0388878	−	0.999244i	$-0.512381\pi$
$131$	−15276.4	−0.0777755	−0.0388878	+	0.999244i	$0.512381\pi$
$132$	0	0
$133$	−113662.	−0.557167
$134$	0	0
$135$	−234609.	−1.10793
$136$	0	0
$137$	148016.	0.673763	0.336881	−	0.941547i	$-0.390628\pi$
$137$	148016.	0.673763	0.336881	+	0.941547i	$0.390628\pi$
$138$	0	0
$139$	−143481.	−0.629879	−0.314939	−	0.949112i	$-0.601984\pi$
$139$	−143481.	−0.629879	−0.314939	+	0.949112i	$0.601984\pi$
$140$	0	0
$141$	74832.1	0.316986
$142$	0	0
$143$	−34890.9	−0.142683
$144$	0	0
$145$	−123015.	−0.485891
$146$	0	0
$147$	−18060.2	−0.0689332
$148$	0	0
$149$	35178.8	0.129812	0.0649061	−	0.997891i	$-0.479325\pi$
$149$	35178.8	0.129812	0.0649061	+	0.997891i	$0.479325\pi$
$150$	0	0
$151$	37656.6	0.134400	0.0671998	−	0.997740i	$-0.478593\pi$
$151$	37656.6	0.134400	0.0671998	+	0.997740i	$0.478593\pi$
$152$	0	0
$153$	95864.9	0.331079
$154$	0	0
$155$	−510626.	−1.70716
$156$	0	0
$157$	−440876.	−1.42747	−0.713736	−	0.700415i	$-0.752998\pi$
$157$	−440876.	−1.42747	−0.713736	+	0.700415i	$0.752998\pi$
$158$	0	0
$159$	−260082.	−0.815863
$160$	0	0
$161$	20062.3	0.0609979
$162$	0	0
$163$	−273443.	−0.806118	−0.403059	−	0.915174i	$-0.632053\pi$
$163$	−273443.	−0.806118	−0.403059	+	0.915174i	$0.632053\pi$
$164$	0	0
$165$	−16636.2	−0.0475713
$166$	0	0
$167$	−657930.	−1.82553	−0.912764	−	0.408487i	$-0.866056\pi$
$167$	−657930.	−1.82553	−0.912764	+	0.408487i	$0.866056\pi$
$168$	0	0
$169$	941632.	2.53609
$170$	0	0
$171$	−432426.	−1.13089
$172$	0	0
$173$	−605134.	−1.53722	−0.768610	−	0.639717i	$-0.779051\pi$
$173$	−605134.	−1.53722	−0.768610	+	0.639717i	$0.779051\pi$
$174$	0	0
$175$	−105376.	−0.260103
$176$	0	0
$177$	348325.	0.835703
$178$	0	0
$179$	−370201.	−0.863585	−0.431793	−	0.901973i	$-0.642119\pi$
$179$	−370201.	−0.863585	−0.431793	+	0.901973i	$0.642119\pi$
$180$	0	0
$181$	−330624.	−0.750132	−0.375066	−	0.926998i	$-0.622380\pi$
$181$	−330624.	−0.750132	−0.375066	+	0.926998i	$0.622380\pi$
$182$	0	0
$183$	−254374.	−0.561494
$184$	0	0
$185$	285868.	0.614097
$186$	0	0
$187$	15658.8	0.0327457
$188$	0	0
$189$	−158274.	−0.322295
$190$	0	0
$191$	20402.0	0.0404660	0.0202330	−	0.999795i	$-0.493559\pi$
$191$	20402.0	0.0404660	0.0202330	+	0.999795i	$0.493559\pi$
$192$	0	0
$193$	97967.0	0.189316	0.0946578	−	0.995510i	$-0.469824\pi$
$193$	97967.0	0.189316	0.0946578	+	0.995510i	$0.469824\pi$
$194$	0	0
$195$	626012.	1.17895
$196$	0	0
$197$	950513.	1.74499	0.872494	−	0.488625i	$-0.162501\pi$
$197$	950513.	1.74499	0.872494	+	0.488625i	$0.162501\pi$
$198$	0	0
$199$	−780360.	−1.39689	−0.698445	−	0.715664i	$-0.746124\pi$
$199$	−780360.	−1.39689	−0.698445	+	0.715664i	$0.746124\pi$
$200$	0	0
$201$	−370581.	−0.646984
$202$	0	0
$203$	−82989.2	−0.141345
$204$	0	0
$205$	670858.	1.11493
$206$	0	0
$207$	76326.8	0.123809
$208$	0	0
$209$	−70633.4	−0.111852
$210$	0	0
$211$	262454.	0.405833	0.202917	−	0.979196i	$-0.434958\pi$
$211$	262454.	0.405833	0.202917	+	0.979196i	$0.434958\pi$
$212$	0	0
$213$	372663.	0.562817
$214$	0	0
$215$	1.55116e6	2.28856
$216$	0	0
$217$	−344482.	−0.496612
$218$	0	0
$219$	−254409.	−0.358445
$220$	0	0
$221$	−589232.	−0.811532
$222$	0	0
$223$	−404645.	−0.544894	−0.272447	−	0.962171i	$-0.587833\pi$
$223$	−404645.	−0.544894	−0.272447	+	0.962171i	$0.587833\pi$
$224$	0	0
$225$	−400902.	−0.527937
$226$	0	0
$227$	418192.	0.538656	0.269328	−	0.963048i	$-0.413198\pi$
$227$	418192.	0.538656	0.269328	+	0.963048i	$0.413198\pi$
$228$	0	0
$229$	−434849.	−0.547961	−0.273981	−	0.961735i	$-0.588340\pi$
$229$	−434849.	−0.547961	−0.273981	+	0.961735i	$0.588340\pi$
$230$	0	0
$231$	−11223.2	−0.0138385
$232$	0	0
$233$	−188015.	−0.226884	−0.113442	−	0.993545i	$-0.536188\pi$
$233$	−188015.	−0.226884	−0.113442	+	0.993545i	$0.536188\pi$
$234$	0	0
$235$	722589.	0.853536
$236$	0	0
$237$	−4685.92	−0.00541906
$238$	0	0
$239$	−1.49827e6	−1.69666	−0.848331	−	0.529466i	$-0.822392\pi$
$239$	−1.49827e6	−1.69666	−0.848331	+	0.529466i	$0.822392\pi$
$240$	0	0
$241$	34638.8	0.0384167	0.0192084	−	0.999816i	$-0.493885\pi$
$241$	34638.8	0.0384167	0.0192084	+	0.999816i	$0.493885\pi$
$242$	0	0
$243$	−942897.	−1.02435
$244$	0	0
$245$	−174391.	−0.185614
$246$	0	0
$247$	2.65790e6	2.77202
$248$	0	0
$249$	731534.	0.747715
$250$	0	0
$251$	880721.	0.882376	0.441188	−	0.897415i	$-0.354557\pi$
$251$	880721.	0.882376	0.441188	+	0.897415i	$0.354557\pi$
$252$	0	0
$253$	12467.4	0.0122454
$254$	0	0
$255$	−280950.	−0.270569
$256$	0	0
$257$	245881.	0.232216	0.116108	−	0.993237i	$-0.462958\pi$
$257$	245881.	0.232216	0.116108	+	0.993237i	$0.462958\pi$
$258$	0	0
$259$	192854.	0.178640
$260$	0	0
$261$	−315732.	−0.286892
$262$	0	0
$263$	1.44452e6	1.28776	0.643878	−	0.765128i	$-0.277325\pi$
$263$	1.44452e6	1.28776	0.643878	+	0.765128i	$0.277325\pi$
$264$	0	0
$265$	−2.51138e6	−2.19684
$266$	0	0
$267$	127686.	0.109614
$268$	0	0
$269$	428084.	0.360702	0.180351	−	0.983602i	$-0.442277\pi$
$269$	428084.	0.360702	0.180351	+	0.983602i	$0.442277\pi$
$270$	0	0
$271$	−37799.8	−0.0312655	−0.0156328	−	0.999878i	$-0.504976\pi$
$271$	−37799.8	−0.0312655	−0.0156328	+	0.999878i	$0.504976\pi$
$272$	0	0
$273$	422324.	0.342957
$274$	0	0
$275$	−65484.2	−0.0522162
$276$	0	0
$277$	2.06008e6	1.61319	0.806593	−	0.591107i	$-0.201309\pi$
$277$	2.06008e6	1.61319	0.806593	+	0.591107i	$0.201309\pi$
$278$	0	0
$279$	−1.31058e6	−1.00798
$280$	0	0
$281$	1.21937e6	0.921235	0.460617	−	0.887599i	$-0.347628\pi$
$281$	1.21937e6	0.921235	0.460617	+	0.887599i	$0.347628\pi$
$282$	0	0
$283$	1.27677e6	0.947645	0.473823	−	0.880620i	$-0.342874\pi$
$283$	1.27677e6	0.947645	0.473823	+	0.880620i	$0.342874\pi$
$284$	0	0
$285$	1.26731e6	0.924207
$286$	0	0
$287$	452578.	0.324331
$288$	0	0
$289$	−1.15541e6	−0.813754
$290$	0	0
$291$	492404.	0.340870
$292$	0	0
$293$	−2.11406e6	−1.43863	−0.719313	−	0.694686i	$-0.755543\pi$
$293$	−2.11406e6	−1.43863	−0.719313	+	0.694686i	$0.755543\pi$
$294$	0	0
$295$	3.36347e6	2.25026
$296$	0	0
$297$	−98356.8	−0.0647013
$298$	0	0
$299$	−469141.	−0.303477
$300$	0	0
$301$	1.04646e6	0.665740
$302$	0	0
$303$	−913801.	−0.571801
$304$	0	0
$305$	−2.45627e6	−1.51191
$306$	0	0
$307$	1.75588e6	1.06329	0.531643	−	0.846969i	$-0.321575\pi$
$307$	1.75588e6	1.06329	0.531643	+	0.846969i	$0.321575\pi$
$308$	0	0
$309$	625886.	0.372906
$310$	0	0
$311$	792917.	0.464865	0.232432	−	0.972613i	$-0.425332\pi$
$311$	792917.	0.464865	0.232432	+	0.972613i	$0.425332\pi$
$312$	0	0
$313$	−2.19228e6	−1.26484	−0.632420	−	0.774626i	$-0.717938\pi$
$313$	−2.19228e6	−1.26484	−0.632420	+	0.774626i	$0.717938\pi$
$314$	0	0
$315$	−663472.	−0.376744
$316$	0	0
$317$	−1.08545e6	−0.606685	−0.303342	−	0.952882i	$-0.598103\pi$
$317$	−1.08545e6	−0.606685	−0.303342	+	0.952882i	$0.598103\pi$
$318$	0	0
$319$	−51572.4	−0.0283753
$320$	0	0
$321$	249270.	0.135023
$322$	0	0
$323$	−1.19285e6	−0.636178
$324$	0	0
$325$	2.46414e6	1.29407
$326$	0	0
$327$	95710.2	0.0494981
$328$	0	0
$329$	487477.	0.248293
$330$	0	0
$331$	−1.77389e6	−0.889933	−0.444967	−	0.895547i	$-0.646784\pi$
$331$	−1.77389e6	−0.889933	−0.444967	+	0.895547i	$0.646784\pi$
$332$	0	0
$333$	733713.	0.362590
$334$	0	0
$335$	−3.57838e6	−1.74211
$336$	0	0
$337$	−3.08310e6	−1.47881	−0.739407	−	0.673259i	$-0.764894\pi$
$337$	−3.08310e6	−1.47881	−0.739407	+	0.673259i	$0.764894\pi$
$338$	0	0
$339$	153338.	0.0724689
$340$	0	0
$341$	−214073.	−0.0996956
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−223690.	−0.101181
$346$	0	0
$347$	3.14121e6	1.40047	0.700234	−	0.713914i	$-0.253079\pi$
$347$	3.14121e6	1.40047	0.700234	+	0.713914i	$0.253079\pi$
$348$	0	0
$349$	2.21743e6	0.974511	0.487256	−	0.873259i	$-0.337998\pi$
$349$	2.21743e6	0.974511	0.487256	+	0.873259i	$0.337998\pi$
$350$	0	0
$351$	3.70111e6	1.60348
$352$	0	0
$353$	923780.	0.394577	0.197288	−	0.980345i	$-0.436786\pi$
$353$	923780.	0.394577	0.197288	+	0.980345i	$0.436786\pi$
$354$	0	0
$355$	3.59848e6	1.51547
$356$	0	0
$357$	−189536.	−0.0787085
$358$	0	0
$359$	311845.	0.127703	0.0638517	−	0.997959i	$-0.479662\pi$
$359$	311845.	0.127703	0.0638517	+	0.997959i	$0.479662\pi$
$360$	0	0
$361$	2.90458e6	1.17305
$362$	0	0
$363$	1.20444e6	0.479755
$364$	0	0
$365$	−2.45661e6	−0.965170
$366$	0	0
$367$	−1.27344e6	−0.493531	−0.246765	−	0.969075i	$-0.579368\pi$
$367$	−1.27344e6	−0.493531	−0.246765	+	0.969075i	$0.579368\pi$
$368$	0	0
$369$	1.72183e6	0.658302
$370$	0	0
$371$	−1.69424e6	−0.639059
$372$	0	0
$373$	−406594.	−0.151317	−0.0756587	−	0.997134i	$-0.524106\pi$
$373$	−406594.	−0.151317	−0.0756587	+	0.997134i	$0.524106\pi$
$374$	0	0
$375$	−532393.	−0.195503
$376$	0	0
$377$	1.94064e6	0.703222
$378$	0	0
$379$	−4.05172e6	−1.44891	−0.724455	−	0.689322i	$-0.757908\pi$
$379$	−4.05172e6	−1.44891	−0.724455	+	0.689322i	$0.757908\pi$
$380$	0	0
$381$	−1.77354e6	−0.625935
$382$	0	0
$383$	2.95208e6	1.02833	0.514163	−	0.857692i	$-0.328103\pi$
$383$	2.95208e6	1.02833	0.514163	+	0.857692i	$0.328103\pi$
$384$	0	0
$385$	−108373.	−0.0372622
$386$	0	0
$387$	3.98124e6	1.35127
$388$	0	0
$389$	−2.34441e6	−0.785523	−0.392761	−	0.919640i	$-0.628480\pi$
$389$	−2.34441e6	−0.785523	−0.392761	+	0.919640i	$0.628480\pi$
$390$	0	0
$391$	210547.	0.0696479
$392$	0	0
$393$	114908.	0.0375292
$394$	0	0
$395$	−45247.8	−0.0145917
$396$	0	0
$397$	−799866.	−0.254707	−0.127354	−	0.991857i	$-0.540648\pi$
$397$	−799866.	−0.254707	−0.127354	+	0.991857i	$0.540648\pi$
$398$	0	0
$399$	854957.	0.268851
$400$	0	0
$401$	4.66435e6	1.44854	0.724269	−	0.689517i	$-0.242177\pi$
$401$	4.66435e6	1.44854	0.724269	+	0.689517i	$0.242177\pi$
$402$	0	0
$403$	8.05545e6	2.47074
$404$	0	0
$405$	−1.52556e6	−0.462159
$406$	0	0
$407$	119846.	0.0358623
$408$	0	0
$409$	−3.45860e6	−1.02233	−0.511165	−	0.859482i	$-0.670786\pi$
$409$	−3.45860e6	−1.02233	−0.511165	+	0.859482i	$0.670786\pi$
$410$	0	0
$411$	−1.11337e6	−0.325113
$412$	0	0
$413$	2.26909e6	0.654600
$414$	0	0
$415$	7.06379e6	2.01334
$416$	0	0
$417$	1.07925e6	0.303937
$418$	0	0
$419$	2.36121e6	0.657052	0.328526	−	0.944495i	$-0.393448\pi$
$419$	2.36121e6	0.657052	0.328526	+	0.944495i	$0.393448\pi$
$420$	0	0
$421$	−3.57530e6	−0.983123	−0.491561	−	0.870843i	$-0.663574\pi$
$421$	−3.57530e6	−0.983123	−0.491561	+	0.870843i	$0.663574\pi$
$422$	0	0
$423$	1.85461e6	0.503966
$424$	0	0
$425$	−1.10589e6	−0.296988
$426$	0	0
$427$	−1.65706e6	−0.439814
$428$	0	0
$429$	262447.	0.0688491
$430$	0	0
$431$	−5.65850e6	−1.46726	−0.733631	−	0.679548i	$-0.762176\pi$
$431$	−5.65850e6	−1.46726	−0.733631	+	0.679548i	$0.762176\pi$
$432$	0	0
$433$	6.09905e6	1.56330	0.781650	−	0.623717i	$-0.214378\pi$
$433$	6.09905e6	1.56330	0.781650	+	0.623717i	$0.214378\pi$
$434$	0	0
$435$	925312.	0.234458
$436$	0	0
$437$	−949734.	−0.237902
$438$	0	0
$439$	2.81345e6	0.696753	0.348376	−	0.937355i	$-0.386733\pi$
$439$	2.81345e6	0.696753	0.348376	+	0.937355i	$0.386733\pi$
$440$	0	0
$441$	−447595.	−0.109595
$442$	0	0
$443$	−7.64719e6	−1.85137	−0.925683	−	0.378300i	$-0.876509\pi$
$443$	−7.64719e6	−1.85137	−0.925683	+	0.378300i	$0.876509\pi$
$444$	0	0
$445$	1.23296e6	0.295153
$446$	0	0
$447$	−264613.	−0.0626386
$448$	0	0
$449$	−1.32460e6	−0.310076	−0.155038	−	0.987909i	$-0.549550\pi$
$449$	−1.32460e6	−0.310076	−0.155038	+	0.987909i	$0.549550\pi$
$450$	0	0
$451$	281248.	0.0651100
$452$	0	0
$453$	−283250.	−0.0648522
$454$	0	0
$455$	4.07802e6	0.923465
$456$	0	0
$457$	5.79243e6	1.29739	0.648694	−	0.761049i	$-0.275315\pi$
$457$	5.79243e6	1.29739	0.648694	+	0.761049i	$0.275315\pi$
$458$	0	0
$459$	−1.66104e6	−0.367999
$460$	0	0
$461$	4.41695e6	0.967988	0.483994	−	0.875071i	$-0.339186\pi$
$461$	4.41695e6	0.967988	0.483994	+	0.875071i	$0.339186\pi$
$462$	0	0
$463$	−2.22867e6	−0.483163	−0.241582	−	0.970380i	$-0.577666\pi$
$463$	−2.22867e6	−0.483163	−0.241582	+	0.970380i	$0.577666\pi$
$464$	0	0
$465$	3.84090e6	0.823760
$466$	0	0
$467$	−1.80111e6	−0.382164	−0.191082	−	0.981574i	$-0.561200\pi$
$467$	−1.80111e6	−0.382164	−0.191082	+	0.981574i	$0.561200\pi$
$468$	0	0
$469$	−2.41407e6	−0.506778
$470$	0	0
$471$	3.31624e6	0.688802
$472$	0	0
$473$	650304.	0.133648
$474$	0	0
$475$	4.98843e6	1.01445
$476$	0	0
$477$	−6.44575e6	−1.29711
$478$	0	0
$479$	−6.08283e6	−1.21134	−0.605672	−	0.795715i	$-0.707095\pi$
$479$	−6.08283e6	−1.21134	−0.605672	+	0.795715i	$0.707095\pi$
$480$	0	0
$481$	−4.50976e6	−0.888772
$482$	0	0
$483$	−150907.	−0.0294335
$484$	0	0
$485$	4.75471e6	0.917847
$486$	0	0
$487$	−8.08489e6	−1.54473	−0.772364	−	0.635180i	$-0.780926\pi$
$487$	−8.08489e6	−1.54473	−0.772364	+	0.635180i	$0.780926\pi$
$488$	0	0
$489$	2.05682e6	0.388978
$490$	0	0
$491$	−2.16092e6	−0.404516	−0.202258	−	0.979332i	$-0.564828\pi$
$491$	−2.16092e6	−0.404516	−0.202258	+	0.979332i	$0.564828\pi$
$492$	0	0
$493$	−870947.	−0.161389
$494$	0	0
$495$	−412305.	−0.0756319
$496$	0	0
$497$	2.42763e6	0.440850
$498$	0	0
$499$	−4.38369e6	−0.788113	−0.394056	−	0.919086i	$-0.628929\pi$
$499$	−4.38369e6	−0.788113	−0.394056	+	0.919086i	$0.628929\pi$
$500$	0	0
$501$	4.94891e6	0.880877
$502$	0	0
$503$	−6.50358e6	−1.14613	−0.573063	−	0.819511i	$-0.694245\pi$
$503$	−6.50358e6	−1.14613	−0.573063	+	0.819511i	$0.694245\pi$
$504$	0	0
$505$	−8.82378e6	−1.53967
$506$	0	0
$507$	−7.08290e6	−1.22375
$508$	0	0
$509$	−8.90877e6	−1.52413	−0.762067	−	0.647498i	$-0.775815\pi$
$509$	−8.90877e6	−1.52413	−0.762067	+	0.647498i	$0.775815\pi$
$510$	0	0
$511$	−1.65729e6	−0.280767
$512$	0	0
$513$	7.49257e6	1.25701
$514$	0	0
$515$	6.04364e6	1.00411
$516$	0	0
$517$	302935.	0.0498452
$518$	0	0
$519$	4.55178e6	0.741759
$520$	0	0
$521$	−2.87271e6	−0.463658	−0.231829	−	0.972756i	$-0.574471\pi$
$521$	−2.87271e6	−0.463658	−0.231829	+	0.972756i	$0.574471\pi$
$522$	0	0
$523$	−6.21693e6	−0.993852	−0.496926	−	0.867793i	$-0.665538\pi$
$523$	−6.21693e6	−0.993852	−0.496926	+	0.867793i	$0.665538\pi$
$524$	0	0
$525$	792631.	0.125508
$526$	0	0
$527$	−3.61523e6	−0.567035
$528$	0	0
$529$	−6.26871e6	−0.973955
$530$	0	0
$531$	8.63274e6	1.32865
$532$	0	0
$533$	−1.05832e7	−1.61361
$534$	0	0
$535$	2.40698e6	0.363570
$536$	0	0
$537$	2.78463e6	0.416708
$538$	0	0
$539$	−73111.2	−0.0108396
$540$	0	0
$541$	6.30994e6	0.926898	0.463449	−	0.886124i	$-0.346612\pi$
$541$	6.30994e6	0.926898	0.463449	+	0.886124i	$0.346612\pi$
$542$	0	0
$543$	2.48693e6	0.361963
$544$	0	0
$545$	924190.	0.133282
$546$	0	0
$547$	−1.07052e7	−1.52976	−0.764882	−	0.644170i	$-0.777203\pi$
$547$	−1.07052e7	−1.52976	−0.764882	+	0.644170i	$0.777203\pi$
$548$	0	0
$549$	−6.30429e6	−0.892700
$550$	0	0
$551$	3.92866e6	0.551271
$552$	0	0
$553$	−30525.4	−0.00424471
$554$	0	0
$555$	−2.15028e6	−0.296322
$556$	0	0
$557$	−7.92588e6	−1.08245	−0.541227	−	0.840877i	$-0.682040\pi$
$557$	−7.92588e6	−1.08245	−0.541227	+	0.840877i	$0.682040\pi$
$558$	0	0
$559$	−2.44706e7	−3.31219
$560$	0	0
$561$	−117784.	−0.0158009
$562$	0	0
$563$	8.39442e6	1.11614	0.558072	−	0.829793i	$-0.311541\pi$
$563$	8.39442e6	1.11614	0.558072	+	0.829793i	$0.311541\pi$
$564$	0	0
$565$	1.48065e6	0.195134
$566$	0	0
$567$	−1.02918e6	−0.134442
$568$	0	0
$569$	−3.87124e6	−0.501267	−0.250634	−	0.968082i	$-0.580639\pi$
$569$	−3.87124e6	−0.501267	−0.250634	+	0.968082i	$0.580639\pi$
$570$	0	0
$571$	−1.27091e7	−1.63126	−0.815631	−	0.578572i	$-0.803610\pi$
$571$	−1.27091e7	−1.63126	−0.815631	+	0.578572i	$0.803610\pi$
$572$	0	0
$573$	−153463.	−0.0195261
$574$	0	0
$575$	−880498.	−0.111060
$576$	0	0
$577$	−4.77969e6	−0.597669	−0.298834	−	0.954305i	$-0.596598\pi$
$577$	−4.77969e6	−0.597669	−0.298834	+	0.954305i	$0.596598\pi$
$578$	0	0
$579$	−736902.	−0.0913510
$580$	0	0
$581$	4.76542e6	0.585680
$582$	0	0
$583$	−1.05286e6	−0.128292
$584$	0	0
$585$	1.55148e7	1.87438
$586$	0	0
$587$	−2.61275e6	−0.312970	−0.156485	−	0.987680i	$-0.550016\pi$
$587$	−2.61275e6	−0.312970	−0.156485	+	0.987680i	$0.550016\pi$
$588$	0	0
$589$	1.63075e7	1.93687
$590$	0	0
$591$	−7.14970e6	−0.842014
$592$	0	0
$593$	−1.43447e7	−1.67516	−0.837579	−	0.546316i	$-0.816030\pi$
$593$	−1.43447e7	−1.67516	−0.837579	+	0.546316i	$0.816030\pi$
$594$	0	0
$595$	−1.83019e6	−0.211935
$596$	0	0
$597$	5.86982e6	0.674045
$598$	0	0
$599$	7.08290e6	0.806574	0.403287	−	0.915074i	$-0.367868\pi$
$599$	7.08290e6	0.806574	0.403287	+	0.915074i	$0.367868\pi$
$600$	0	0
$601$	−1.48490e7	−1.67691	−0.838457	−	0.544967i	$-0.816542\pi$
$601$	−1.48490e7	−1.67691	−0.838457	+	0.544967i	$0.816542\pi$
$602$	0	0
$603$	−9.18432e6	−1.02862
$604$	0	0
$605$	1.16302e7	1.29182
$606$	0	0
$607$	−1.20896e7	−1.33181	−0.665903	−	0.746038i	$-0.731954\pi$
$607$	−1.20896e7	−1.33181	−0.665903	+	0.746038i	$0.731954\pi$
$608$	0	0
$609$	624240.	0.0682038
$610$	0	0
$611$	−1.13993e7	−1.23531
$612$	0	0
$613$	1.66773e7	1.79257	0.896284	−	0.443480i	$-0.146256\pi$
$613$	1.66773e7	1.79257	0.896284	+	0.443480i	$0.146256\pi$
$614$	0	0
$615$	−5.04615e6	−0.537988
$616$	0	0
$617$	−1.01868e7	−1.07727	−0.538634	−	0.842540i	$-0.681059\pi$
$617$	−1.01868e7	−1.07727	−0.538634	+	0.842540i	$0.681059\pi$
$618$	0	0
$619$	5.88545e6	0.617381	0.308690	−	0.951163i	$-0.400109\pi$
$619$	5.88545e6	0.617381	0.308690	+	0.951163i	$0.400109\pi$
$620$	0	0
$621$	−1.32250e6	−0.137615
$622$	0	0
$623$	831784.	0.0858599
$624$	0	0
$625$	−1.18613e7	−1.21459
$626$	0	0
$627$	531300.	0.0539723
$628$	0	0
$629$	2.02395e6	0.203973
$630$	0	0
$631$	1.09945e7	1.09926	0.549632	−	0.835407i	$-0.314768\pi$
$631$	1.09945e7	1.09926	0.549632	+	0.835407i	$0.314768\pi$
$632$	0	0
$633$	−1.97417e6	−0.195828
$634$	0	0
$635$	−1.71256e7	−1.68543
$636$	0	0
$637$	2.75114e6	0.268636
$638$	0	0
$639$	9.23591e6	0.894803
$640$	0	0
$641$	−453144.	−0.0435604	−0.0217802	−	0.999763i	$-0.506933\pi$
$641$	−453144.	−0.0435604	−0.0217802	+	0.999763i	$0.506933\pi$
$642$	0	0
$643$	1.72050e7	1.64107	0.820537	−	0.571593i	$-0.193674\pi$
$643$	1.72050e7	1.64107	0.820537	+	0.571593i	$0.193674\pi$
$644$	0	0
$645$	−1.16678e7	−1.10430
$646$	0	0
$647$	−4.68036e6	−0.439560	−0.219780	−	0.975549i	$-0.570534\pi$
$647$	−4.68036e6	−0.439560	−0.219780	+	0.975549i	$0.570534\pi$
$648$	0	0
$649$	1.41009e6	0.131412
$650$	0	0
$651$	2.59117e6	0.239631
$652$	0	0
$653$	−9.29058e6	−0.852629	−0.426314	−	0.904575i	$-0.640188\pi$
$653$	−9.29058e6	−0.852629	−0.426314	+	0.904575i	$0.640188\pi$
$654$	0	0
$655$	1.10957e6	0.101053
$656$	0	0
$657$	−6.30516e6	−0.569879
$658$	0	0
$659$	1.60390e7	1.43868	0.719339	−	0.694659i	$-0.244445\pi$
$659$	1.60390e7	1.43868	0.719339	+	0.694659i	$0.244445\pi$
$660$	0	0
$661$	6.02455e6	0.536316	0.268158	−	0.963375i	$-0.413585\pi$
$661$	6.02455e6	0.536316	0.268158	+	0.963375i	$0.413585\pi$
$662$	0	0
$663$	4.43217e6	0.391591
$664$	0	0
$665$	8.25558e6	0.723925
$666$	0	0
$667$	−693441.	−0.0603524
$668$	0	0
$669$	3.04372e6	0.262929
$670$	0	0
$671$	−1.02976e6	−0.0882934
$672$	0	0
$673$	−1.58601e6	−0.134980	−0.0674898	−	0.997720i	$-0.521499\pi$
$673$	−1.58601e6	−0.134980	−0.0674898	+	0.997720i	$0.521499\pi$
$674$	0	0
$675$	6.94636e6	0.586811
$676$	0	0
$677$	−5.60343e6	−0.469875	−0.234938	−	0.972010i	$-0.575489\pi$
$677$	−5.60343e6	−0.469875	−0.234938	+	0.972010i	$0.575489\pi$
$678$	0	0
$679$	3.20765e6	0.267001
$680$	0	0
$681$	−3.14562e6	−0.259919
$682$	0	0
$683$	−2.55974e6	−0.209963	−0.104982	−	0.994474i	$-0.533478\pi$
$683$	−2.55974e6	−0.209963	−0.104982	+	0.994474i	$0.533478\pi$
$684$	0	0
$685$	−1.07508e7	−0.875417
$686$	0	0
$687$	3.27091e6	0.264409
$688$	0	0
$689$	3.96187e7	3.17945
$690$	0	0
$691$	−1.02347e7	−0.815415	−0.407708	−	0.913113i	$-0.633672\pi$
$691$	−1.02347e7	−0.815415	−0.407708	+	0.913113i	$0.633672\pi$
$692$	0	0
$693$	−278151.	−0.0220013
$694$	0	0
$695$	1.04214e7	0.818399
$696$	0	0
$697$	4.74967e6	0.370324
$698$	0	0
$699$	1.41424e6	0.109479
$700$	0	0
$701$	1.25770e7	0.966677	0.483339	−	0.875434i	$-0.339424\pi$
$701$	1.25770e7	0.966677	0.483339	+	0.875434i	$0.339424\pi$
$702$	0	0
$703$	−9.12960e6	−0.696728
$704$	0	0
$705$	−5.43527e6	−0.411859
$706$	0	0
$707$	−5.95275e6	−0.447888
$708$	0	0
$709$	2.54796e7	1.90361	0.951804	−	0.306706i	$-0.0992269\pi$
$709$	2.54796e7	1.90361	0.951804	+	0.306706i	$0.0992269\pi$
$710$	0	0
$711$	−116134.	−0.00861558
$712$	0	0
$713$	−2.87842e6	−0.212046
$714$	0	0
$715$	2.53422e6	0.185387
$716$	0	0
$717$	1.12699e7	0.818695
$718$	0	0
$719$	9.61567e6	0.693677	0.346839	−	0.937925i	$-0.387255\pi$
$719$	9.61567e6	0.693677	0.346839	+	0.937925i	$0.387255\pi$
$720$	0	0
$721$	4.07720e6	0.292095
$722$	0	0
$723$	−260551.	−0.0185373
$724$	0	0
$725$	3.64226e6	0.257351
$726$	0	0
$727$	1.37720e7	0.966409	0.483204	−	0.875508i	$-0.339473\pi$
$727$	1.37720e7	0.966409	0.483204	+	0.875508i	$0.339473\pi$
$728$	0	0
$729$	1.98851e6	0.138582
$730$	0	0
$731$	1.09822e7	0.760147
$732$	0	0
$733$	−1.99018e7	−1.36814	−0.684072	−	0.729414i	$-0.739793\pi$
$733$	−1.99018e7	−1.36814	−0.684072	+	0.729414i	$0.739793\pi$
$734$	0	0
$735$	1.31176e6	0.0895646
$736$	0	0
$737$	−1.50019e6	−0.101736
$738$	0	0
$739$	−1.83447e7	−1.23566	−0.617832	−	0.786310i	$-0.711989\pi$
$739$	−1.83447e7	−1.23566	−0.617832	+	0.786310i	$0.711989\pi$
$740$	0	0
$741$	−1.99926e7	−1.33759
$742$	0	0
$743$	−2.37153e7	−1.57600	−0.788000	−	0.615675i	$-0.788884\pi$
$743$	−2.37153e7	−1.57600	−0.788000	+	0.615675i	$0.788884\pi$
$744$	0	0
$745$	−2.55514e6	−0.168664
$746$	0	0
$747$	1.81300e7	1.18877
$748$	0	0
$749$	1.62381e6	0.105762
$750$	0	0
$751$	−1.63612e7	−1.05856	−0.529279	−	0.848448i	$-0.677538\pi$
$751$	−1.63612e7	−1.05856	−0.529279	+	0.848448i	$0.677538\pi$
$752$	0	0
$753$	−6.62473e6	−0.425775
$754$	0	0
$755$	−2.73510e6	−0.174625
$756$	0	0
$757$	−5.73349e6	−0.363646	−0.181823	−	0.983331i	$-0.558200\pi$
$757$	−5.73349e6	−0.363646	−0.181823	+	0.983331i	$0.558200\pi$
$758$	0	0
$759$	−93778.9	−0.00590882
$760$	0	0
$761$	−7.71654e6	−0.483015	−0.241508	−	0.970399i	$-0.577642\pi$
$761$	−7.71654e6	−0.483015	−0.241508	+	0.970399i	$0.577642\pi$
$762$	0	0
$763$	623483.	0.0387715
$764$	0	0
$765$	−6.96294e6	−0.430169
$766$	0	0
$767$	−5.30610e7	−3.25677
$768$	0	0
$769$	−2.68723e7	−1.63866	−0.819330	−	0.573322i	$-0.805654\pi$
$769$	−2.68723e7	−1.63866	−0.819330	+	0.573322i	$0.805654\pi$
$770$	0	0
$771$	−1.84950e6	−0.112052
$772$	0	0
$773$	−2.92190e6	−0.175880	−0.0879401	−	0.996126i	$-0.528028\pi$
$773$	−2.92190e6	−0.175880	−0.0879401	+	0.996126i	$0.528028\pi$
$774$	0	0
$775$	1.51187e7	0.904192
$776$	0	0
$777$	−1.45064e6	−0.0861998
$778$	0	0
$779$	−2.14248e7	−1.26495
$780$	0	0
$781$	1.50861e6	0.0885014
$782$	0	0
$783$	5.47064e6	0.318885
$784$	0	0
$785$	3.20221e7	1.85471
$786$	0	0
$787$	−2.40079e7	−1.38171	−0.690855	−	0.722994i	$-0.742766\pi$
$787$	−2.40079e7	−1.38171	−0.690855	+	0.722994i	$0.742766\pi$
$788$	0	0
$789$	−1.08656e7	−0.621384
$790$	0	0
$791$	998888.	0.0567644
$792$	0	0
$793$	3.87492e7	2.18816
$794$	0	0
$795$	1.88905e7	1.06005
$796$	0	0
$797$	1.26503e6	0.0705433	0.0352717	−	0.999378i	$-0.488770\pi$
$797$	1.26503e6	0.0705433	0.0352717	+	0.999378i	$0.488770\pi$
$798$	0	0
$799$	5.11593e6	0.283503
$800$	0	0
$801$	3.16452e6	0.174272
$802$	0	0
$803$	−1.02990e6	−0.0563645
$804$	0	0
$805$	−1.45718e6	−0.0792543
$806$	0	0
$807$	−3.22002e6	−0.174050
$808$	0	0
$809$	3.50966e7	1.88536	0.942679	−	0.333702i	$-0.108298\pi$
$809$	3.50966e7	1.88536	0.942679	+	0.333702i	$0.108298\pi$
$810$	0	0
$811$	1.24676e7	0.665629	0.332814	−	0.942992i	$-0.392002\pi$
$811$	1.24676e7	0.665629	0.332814	+	0.942992i	$0.392002\pi$
$812$	0	0
$813$	284328.	0.0150866
$814$	0	0
$815$	1.98610e7	1.04738
$816$	0	0
$817$	−4.95386e7	−2.59650
$818$	0	0
$819$	1.04667e7	0.545255
$820$	0	0
$821$	−5.42888e6	−0.281094	−0.140547	−	0.990074i	$-0.544886\pi$
$821$	−5.42888e6	−0.281094	−0.140547	+	0.990074i	$0.544886\pi$
$822$	0	0
$823$	1.65454e7	0.851487	0.425744	−	0.904844i	$-0.360013\pi$
$823$	1.65454e7	0.851487	0.425744	+	0.904844i	$0.360013\pi$
$824$	0	0
$825$	492568.	0.0251960
$826$	0	0
$827$	−1.05506e7	−0.536429	−0.268214	−	0.963359i	$-0.586434\pi$
$827$	−1.05506e7	−0.536429	−0.268214	+	0.963359i	$0.586434\pi$
$828$	0	0
$829$	8.49412e6	0.429271	0.214636	−	0.976694i	$-0.431144\pi$
$829$	8.49412e6	0.429271	0.214636	+	0.976694i	$0.431144\pi$
$830$	0	0
$831$	−1.54958e7	−0.778415
$832$	0	0
$833$	−1.23469e6	−0.0616518
$834$	0	0
$835$	4.77873e7	2.37190
$836$	0	0
$837$	2.27082e7	1.12039
$838$	0	0
$839$	6.01104e6	0.294812	0.147406	−	0.989076i	$-0.452908\pi$
$839$	6.01104e6	0.294812	0.147406	+	0.989076i	$0.452908\pi$
$840$	0	0
$841$	−1.76427e7	−0.860150
$842$	0	0
$843$	−9.17204e6	−0.444526
$844$	0	0
$845$	−6.83934e7	−3.29513
$846$	0	0
$847$	7.84607e6	0.375788
$848$	0	0
$849$	−9.60377e6	−0.457270
$850$	0	0
$851$	1.61145e6	0.0762768
$852$	0	0
$853$	1.12183e6	0.0527903	0.0263951	−	0.999652i	$-0.491597\pi$
$853$	1.12183e6	0.0527903	0.0263951	+	0.999652i	$0.491597\pi$
$854$	0	0
$855$	3.14083e7	1.46936
$856$	0	0
$857$	6.05078e6	0.281423	0.140711	−	0.990051i	$-0.455061\pi$
$857$	6.05078e6	0.281423	0.140711	+	0.990051i	$0.455061\pi$
$858$	0	0
$859$	3.36344e7	1.55525	0.777625	−	0.628728i	$-0.216424\pi$
$859$	3.36344e7	1.55525	0.777625	+	0.628728i	$0.216424\pi$
$860$	0	0
$861$	−3.40427e6	−0.156500
$862$	0	0
$863$	1.90095e7	0.868846	0.434423	−	0.900709i	$-0.356952\pi$
$863$	1.90095e7	0.868846	0.434423	+	0.900709i	$0.356952\pi$
$864$	0	0
$865$	4.39526e7	1.99730
$866$	0	0
$867$	8.69095e6	0.392663
$868$	0	0
$869$	−18969.5	−0.000852133 0
$870$	0	0
$871$	5.64513e7	2.52132
$872$	0	0
$873$	1.22035e7	0.541938
$874$	0	0
$875$	−3.46816e6	−0.153136
$876$	0	0
$877$	2.94632e7	1.29354	0.646772	−	0.762684i	$-0.276119\pi$
$877$	2.94632e7	1.29354	0.646772	+	0.762684i	$0.276119\pi$
$878$	0	0
$879$	1.59018e7	0.694184
$880$	0	0
$881$	3.76458e7	1.63409	0.817047	−	0.576571i	$-0.195610\pi$
$881$	3.76458e7	1.63409	0.817047	+	0.576571i	$0.195610\pi$
$882$	0	0
$883$	−2.89106e6	−0.124783	−0.0623915	−	0.998052i	$-0.519873\pi$
$883$	−2.89106e6	−0.124783	−0.0623915	+	0.998052i	$0.519873\pi$
$884$	0	0
$885$	−2.52998e7	−1.08582
$886$	0	0
$887$	−2.21452e7	−0.945085	−0.472542	−	0.881308i	$-0.656664\pi$
$887$	−2.21452e7	−0.945085	−0.472542	+	0.881308i	$0.656664\pi$
$888$	0	0
$889$	−1.15534e7	−0.490291
$890$	0	0
$891$	−639570.	−0.0269894
$892$	0	0
$893$	−2.30769e7	−0.968385
$894$	0	0
$895$	2.68888e7	1.12205
$896$	0	0
$897$	3.52885e6	0.146438
$898$	0	0
$899$	1.19068e7	0.491356
$900$	0	0
$901$	−1.77806e7	−0.729683
$902$	0	0
$903$	−7.87138e6	−0.321241
$904$	0	0
$905$	2.40141e7	0.974643
$906$	0	0
$907$	3.93856e7	1.58972	0.794858	−	0.606795i	$-0.207545\pi$
$907$	3.93856e7	1.58972	0.794858	+	0.606795i	$0.207545\pi$
$908$	0	0
$909$	−2.26472e7	−0.909087
$910$	0	0
$911$	1.88553e6	0.0752726	0.0376363	−	0.999292i	$-0.488017\pi$
$911$	1.88553e6	0.0752726	0.0376363	+	0.999292i	$0.488017\pi$
$912$	0	0
$913$	2.96140e6	0.117576
$914$	0	0
$915$	1.84759e7	0.729546
$916$	0	0
$917$	748544.	0.0293964
$918$	0	0
$919$	−1.46417e7	−0.571877	−0.285939	−	0.958248i	$-0.592305\pi$
$919$	−1.46417e7	−0.571877	−0.285939	+	0.958248i	$0.592305\pi$
$920$	0	0
$921$	−1.32077e7	−0.513070
$922$	0	0
$923$	−5.67683e7	−2.19332
$924$	0	0
$925$	−8.46405e6	−0.325255
$926$	0	0
$927$	1.55117e7	0.592870
$928$	0	0
$929$	−3.94772e7	−1.50075	−0.750373	−	0.661014i	$-0.770126\pi$
$929$	−3.94772e7	−1.50075	−0.750373	+	0.661014i	$0.770126\pi$
$930$	0	0
$931$	5.56943e6	0.210589
$932$	0	0
$933$	−5.96427e6	−0.224312
$934$	0	0
$935$	−1.13734e6	−0.0425463
$936$	0	0
$937$	−1.93619e7	−0.720441	−0.360221	−	0.932867i	$-0.617299\pi$
$937$	−1.93619e7	−0.720441	−0.360221	+	0.932867i	$0.617299\pi$
$938$	0	0
$939$	1.64902e7	0.610327
$940$	0	0
$941$	3.57379e7	1.31569	0.657847	−	0.753151i	$-0.271467\pi$
$941$	3.57379e7	1.31569	0.657847	+	0.753151i	$0.271467\pi$
$942$	0	0
$943$	3.78165e6	0.138485
$944$	0	0
$945$	1.14959e7	0.418757
$946$	0	0
$947$	9.81761e6	0.355739	0.177869	−	0.984054i	$-0.443080\pi$
$947$	9.81761e6	0.355739	0.177869	+	0.984054i	$0.443080\pi$
$948$	0	0
$949$	3.87546e7	1.39687
$950$	0	0
$951$	8.16472e6	0.292745
$952$	0	0
$953$	−1.44558e7	−0.515596	−0.257798	−	0.966199i	$-0.582997\pi$
$953$	−1.44558e7	−0.515596	−0.257798	+	0.966199i	$0.582997\pi$
$954$	0	0
$955$	−1.48186e6	−0.0525772
$956$	0	0
$957$	387925.	0.0136920
$958$	0	0
$959$	−7.25278e6	−0.254658
$960$	0	0
$961$	2.07951e7	0.726361
$962$	0	0
$963$	6.17779e6	0.214668
$964$	0	0
$965$	−7.11562e6	−0.245977
$966$	0	0
$967$	1.66324e7	0.571989	0.285995	−	0.958231i	$-0.407676\pi$
$967$	1.66324e7	0.571989	0.285995	+	0.958231i	$0.407676\pi$
$968$	0	0
$969$	8.97252e6	0.306977
$970$	0	0
$971$	−2.74729e7	−0.935098	−0.467549	−	0.883967i	$-0.654863\pi$
$971$	−2.74729e7	−0.935098	−0.467549	+	0.883967i	$0.654863\pi$
$972$	0	0
$973$	7.03056e6	0.238072
$974$	0	0
$975$	−1.85351e7	−0.624429
$976$	0	0
$977$	−4.45569e7	−1.49341	−0.746705	−	0.665156i	$-0.768365\pi$
$977$	−4.45569e7	−1.49341	−0.746705	+	0.665156i	$0.768365\pi$
$978$	0	0
$979$	516900.	0.0172365
$980$	0	0
$981$	2.37204e6	0.0786954
$982$	0	0
$983$	−5.82935e7	−1.92414	−0.962070	−	0.272803i	$-0.912049\pi$
$983$	−5.82935e7	−1.92414	−0.962070	+	0.272803i	$0.912049\pi$
$984$	0	0
$985$	−6.90384e7	−2.26725
$986$	0	0
$987$	−3.66678e6	−0.119810
$988$	0	0
$989$	8.74397e6	0.284261
$990$	0	0
$991$	1.41083e7	0.456343	0.228172	−	0.973621i	$-0.426725\pi$
$991$	1.41083e7	0.456343	0.228172	+	0.973621i	$0.426725\pi$
$992$	0	0
$993$	1.33431e7	0.429422
$994$	0	0
$995$	5.66797e7	1.81497
$996$	0	0
$997$	4.20913e6	0.134108	0.0670540	−	0.997749i	$-0.478640\pi$
$997$	4.20913e6	0.134108	0.0670540	+	0.997749i	$0.478640\pi$
$998$	0	0
$999$	−1.27129e7	−0.403025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.ba.1.2		3
4.3	odd	2		448.6.a.bb.1.2		3
8.3	odd	2		224.6.a.e.1.2	✓	3
8.5	even	2		224.6.a.f.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.e.1.2	✓	3	8.3	odd	2
224.6.a.f.1.2	yes	3	8.5	even	2
448.6.a.ba.1.2		3	1.1	even	1	trivial
448.6.a.bb.1.2		3	4.3	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q-7.52194 q^{3} -72.6328 q^{5} -49.0000 q^{7} -186.420 q^{9} +O(q^{10})\)	\(q-7.52194 q^{3} -72.6328 q^{5} -49.0000 q^{7} -186.420 q^{9} -30.4503 q^{11} +1145.83 q^{13} +546.340 q^{15} -514.240 q^{17} +2319.63 q^{19} +368.575 q^{21} -409.434 q^{23} +2150.53 q^{25} +3230.07 q^{27} +1693.66 q^{29} +7030.24 q^{31} +229.045 q^{33} +3559.01 q^{35} -3935.80 q^{37} -8618.86 q^{39} -9236.29 q^{41} -21356.2 q^{43} +13540.2 q^{45} -9948.52 q^{47} +2401.00 q^{49} +3868.09 q^{51} +34576.4 q^{53} +2211.69 q^{55} -17448.1 q^{57} -46307.9 q^{59} +33817.6 q^{61} +9134.60 q^{63} -83224.8 q^{65} +49266.7 q^{67} +3079.74 q^{69} -49543.4 q^{71} +33822.3 q^{73} -16176.1 q^{75} +1492.07 q^{77} +622.967 q^{79} +21003.7 q^{81} -97253.4 q^{83} +37350.7 q^{85} -12739.6 q^{87} -16975.2 q^{89} -56145.6 q^{91} -52881.0 q^{93} -168481. q^{95} -65462.3 q^{97} +5676.56 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{3} + 14 q^{5} - 147 q^{7} + 151 q^{9}+O(q^{10}) \) 3 * q - 8 * q^3 + 14 * q^5 - 147 * q^7 + 151 * q^9	\( 3 q - 8 q^{3} + 14 q^{5} - 147 q^{7} + 151 q^{9} - 600 q^{11} + 974 q^{13} + 120 q^{15} + 718 q^{17} + 1056 q^{19} + 392 q^{21} - 3760 q^{23} - 147 q^{25} + 2872 q^{27} + 9134 q^{29} + 1448 q^{31} - 14872 q^{33} - 686 q^{35} + 4998 q^{37} + 824 q^{39} - 23186 q^{41} - 29880 q^{43} + 28350 q^{45} - 10840 q^{47} + 7203 q^{49} - 3776 q^{51} + 28006 q^{53} - 14952 q^{55} - 53504 q^{57} + 17456 q^{59} + 92294 q^{61} - 7399 q^{63} - 95300 q^{65} - 56024 q^{67} + 63480 q^{69} - 77064 q^{71} - 46346 q^{73} - 50768 q^{75} + 29400 q^{77} + 4376 q^{79} - 121973 q^{81} - 107128 q^{83} + 94348 q^{85} - 82720 q^{87} + 29814 q^{89} - 47726 q^{91} + 108432 q^{93} - 205304 q^{95} - 156482 q^{97} - 83128 q^{99}+O(q^{100}) \) 3 * q - 8 * q^3 + 14 * q^5 - 147 * q^7 + 151 * q^9 - 600 * q^11 + 974 * q^13 + 120 * q^15 + 718 * q^17 + 1056 * q^19 + 392 * q^21 - 3760 * q^23 - 147 * q^25 + 2872 * q^27 + 9134 * q^29 + 1448 * q^31 - 14872 * q^33 - 686 * q^35 + 4998 * q^37 + 824 * q^39 - 23186 * q^41 - 29880 * q^43 + 28350 * q^45 - 10840 * q^47 + 7203 * q^49 - 3776 * q^51 + 28006 * q^53 - 14952 * q^55 - 53504 * q^57 + 17456 * q^59 + 92294 * q^61 - 7399 * q^63 - 95300 * q^65 - 56024 * q^67 + 63480 * q^69 - 77064 * q^71 - 46346 * q^73 - 50768 * q^75 + 29400 * q^77 + 4376 * q^79 - 121973 * q^81 - 107128 * q^83 + 94348 * q^85 - 82720 * q^87 + 29814 * q^89 - 47726 * q^91 + 108432 * q^93 - 205304 * q^95 - 156482 * q^97 - 83128 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more