LMFDB - Embedded newform 224.6.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [224,6,Mod(1,224)] 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("224.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	224.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.9259756381$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 $ x^5 - 229x^3 - 272x^2 + 7973*x - 13998
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.98949$ of defining polynomial
Character	$\chi$	$=$	224.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-19.9790 q^{3} +106.158 q^{5} +49.0000 q^{7} +156.159 q^{9} +452.252 q^{11} +886.820 q^{13} -2120.92 q^{15} +297.950 q^{17} -2286.25 q^{19} -978.970 q^{21} -555.550 q^{23} +8144.48 q^{25} +1734.99 q^{27} -8267.13 q^{29} -4247.14 q^{31} -9035.53 q^{33} +5201.73 q^{35} -758.789 q^{37} -17717.8 q^{39} +17202.5 q^{41} +5371.38 q^{43} +16577.5 q^{45} +25656.0 q^{47} +2401.00 q^{49} -5952.74 q^{51} +10834.4 q^{53} +48010.1 q^{55} +45677.0 q^{57} +2794.28 q^{59} +8464.18 q^{61} +7651.81 q^{63} +94142.9 q^{65} -14334.1 q^{67} +11099.3 q^{69} +61130.9 q^{71} -6008.30 q^{73} -162718. q^{75} +22160.3 q^{77} -25300.2 q^{79} -72610.0 q^{81} +5432.19 q^{83} +31629.7 q^{85} +165169. q^{87} +30312.7 q^{89} +43454.2 q^{91} +84853.5 q^{93} -242703. q^{95} -83046.9 q^{97} +70623.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) $ 5 * q - 10 * q^3 + 36 * q^5 + 245 * q^7 + 637 * q^9 + 116 * q^11 + 40 * q^13 + 16 * q^15 - 402 * q^17 - 3582 * q^19 - 490 * q^21 + 472 * q^23 + 9615 * q^25 + 356 * q^27 + 4754 * q^29 - 10500 * q^31 + 15864 * q^33 + 1764 * q^35 + 19642 * q^37 + 10872 * q^39 + 23398 * q^41 + 22044 * q^43 + 49476 * q^45 + 16004 * q^47 + 12005 * q^49 - 45676 * q^51 + 54246 * q^53 + 53456 * q^55 + 109556 * q^57 - 74366 * q^59 + 68316 * q^61 + 31213 * q^63 + 152568 * q^65 - 26560 * q^67 + 214720 * q^69 + 93072 * q^71 + 136098 * q^73 - 124510 * q^75 + 5684 * q^77 + 96080 * q^79 + 104801 * q^81 - 145894 * q^83 + 117352 * q^85 + 168876 * q^87 + 188554 * q^89 + 1960 * q^91 + 86296 * q^93 + 74736 * q^95 - 88146 * q^97 - 260236 * 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
$3$	−19.9790	−1.28165	−0.640826	−	0.767686i	$-0.721408\pi$
$3$	−19.9790	−1.28165	−0.640826	+	0.767686i	$0.721408\pi$
$4$	0	0
$5$	106.158	1.89901	0.949504	−	0.313754i	$-0.101587\pi$
$5$	106.158	1.89901	0.949504	+	0.313754i	$0.101587\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	156.159	0.642631
$10$	0	0
$11$	452.252	1.12693	0.563467	−	0.826138i	$-0.309467\pi$
$11$	452.252	1.12693	0.563467	+	0.826138i	$0.309467\pi$
$12$	0	0
$13$	886.820	1.45538	0.727691	−	0.685905i	$-0.240593\pi$
$13$	886.820	1.45538	0.727691	+	0.685905i	$0.240593\pi$
$14$	0	0
$15$	−2120.92	−2.43387
$16$	0	0
$17$	297.950	0.250047	0.125023	−	0.992154i	$-0.460099\pi$
$17$	297.950	0.250047	0.125023	+	0.992154i	$0.460099\pi$
$18$	0	0
$19$	−2286.25	−1.45291	−0.726457	−	0.687212i	$-0.758834\pi$
$19$	−2286.25	−1.45291	−0.726457	+	0.687212i	$0.758834\pi$
$20$	0	0
$21$	−978.970	−0.484419
$22$	0	0
$23$	−555.550	−0.218979	−0.109490	−	0.993988i	$-0.534922\pi$
$23$	−555.550	−0.218979	−0.109490	+	0.993988i	$0.534922\pi$
$24$	0	0
$25$	8144.48	2.60623
$26$	0	0
$27$	1734.99	0.458022
$28$	0	0
$29$	−8267.13	−1.82541	−0.912704	−	0.408621i	$-0.866010\pi$
$29$	−8267.13	−1.82541	−0.912704	+	0.408621i	$0.866010\pi$
$30$	0	0
$31$	−4247.14	−0.793765	−0.396883	−	0.917869i	$-0.629908\pi$
$31$	−4247.14	−0.793765	−0.396883	+	0.917869i	$0.629908\pi$
$32$	0	0
$33$	−9035.53	−1.44434
$34$	0	0
$35$	5201.73	0.717758
$36$	0	0
$37$	−758.789	−0.0911206	−0.0455603	−	0.998962i	$-0.514507\pi$
$37$	−758.789	−0.0911206	−0.0455603	+	0.998962i	$0.514507\pi$
$38$	0	0
$39$	−17717.8	−1.86529
$40$	0	0
$41$	17202.5	1.59820	0.799100	−	0.601198i	$-0.205310\pi$
$41$	17202.5	1.59820	0.799100	+	0.601198i	$0.205310\pi$
$42$	0	0
$43$	5371.38	0.443011	0.221505	−	0.975159i	$-0.428903\pi$
$43$	5371.38	0.443011	0.221505	+	0.975159i	$0.428903\pi$
$44$	0	0
$45$	16577.5	1.22036
$46$	0	0
$47$	25656.0	1.69412	0.847060	−	0.531497i	$-0.178370\pi$
$47$	25656.0	1.69412	0.847060	+	0.531497i	$0.178370\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−5952.74	−0.320473
$52$	0	0
$53$	10834.4	0.529806	0.264903	−	0.964275i	$-0.414660\pi$
$53$	10834.4	0.529806	0.264903	+	0.964275i	$0.414660\pi$
$54$	0	0
$55$	48010.1	2.14006
$56$	0	0
$57$	45677.0	1.86213
$58$	0	0
$59$	2794.28	0.104506	0.0522528	−	0.998634i	$-0.483360\pi$
$59$	2794.28	0.104506	0.0522528	+	0.998634i	$0.483360\pi$
$60$	0	0
$61$	8464.18	0.291246	0.145623	−	0.989340i	$-0.453481\pi$
$61$	8464.18	0.291246	0.145623	+	0.989340i	$0.453481\pi$
$62$	0	0
$63$	7651.81	0.242892
$64$	0	0
$65$	94142.9	2.76378
$66$	0	0
$67$	−14334.1	−0.390106	−0.195053	−	0.980793i	$-0.562488\pi$
$67$	−14334.1	−0.390106	−0.195053	+	0.980793i	$0.562488\pi$
$68$	0	0
$69$	11099.3	0.280655
$70$	0	0
$71$	61130.9	1.43918	0.719589	−	0.694400i	$-0.244330\pi$
$71$	61130.9	1.43918	0.719589	+	0.694400i	$0.244330\pi$
$72$	0	0
$73$	−6008.30	−0.131961	−0.0659803	−	0.997821i	$-0.521017\pi$
$73$	−6008.30	−0.131961	−0.0659803	+	0.997821i	$0.521017\pi$
$74$	0	0
$75$	−162718.	−3.34029
$76$	0	0
$77$	22160.3	0.425941
$78$	0	0
$79$	−25300.2	−0.456096	−0.228048	−	0.973650i	$-0.573234\pi$
$79$	−25300.2	−0.456096	−0.228048	+	0.973650i	$0.573234\pi$
$80$	0	0
$81$	−72610.0	−1.22966
$82$	0	0
$83$	5432.19	0.0865526	0.0432763	−	0.999063i	$-0.486220\pi$
$83$	5432.19	0.0865526	0.0432763	+	0.999063i	$0.486220\pi$
$84$	0	0
$85$	31629.7	0.474841
$86$	0	0
$87$	165169.	2.33954
$88$	0	0
$89$	30312.7	0.405649	0.202824	−	0.979215i	$-0.434988\pi$
$89$	30312.7	0.405649	0.202824	+	0.979215i	$0.434988\pi$
$90$	0	0
$91$	43454.2	0.550083
$92$	0	0
$93$	84853.5	1.01733
$94$	0	0
$95$	−242703.	−2.75910
$96$	0	0
$97$	−83046.9	−0.896177	−0.448089	−	0.893989i	$-0.647895\pi$
$97$	−83046.9	−0.896177	−0.448089	+	0.893989i	$0.647895\pi$
$98$	0	0
$99$	70623.4	0.724203
$100$	0	0
$101$	−136029.	−1.32687	−0.663433	−	0.748235i	$-0.730901\pi$
$101$	−136029.	−1.32687	−0.663433	+	0.748235i	$0.730901\pi$
$102$	0	0
$103$	−65195.4	−0.605514	−0.302757	−	0.953068i	$-0.597907\pi$
$103$	−65195.4	−0.605514	−0.302757	+	0.953068i	$0.597907\pi$
$104$	0	0
$105$	−103925.	−0.919916
$106$	0	0
$107$	−121743.	−1.02798	−0.513988	−	0.857797i	$-0.671833\pi$
$107$	−121743.	−1.02798	−0.513988	+	0.857797i	$0.671833\pi$
$108$	0	0
$109$	35552.4	0.286617	0.143309	−	0.989678i	$-0.454226\pi$
$109$	35552.4	0.286617	0.143309	+	0.989678i	$0.454226\pi$
$110$	0	0
$111$	15159.8	0.116785
$112$	0	0
$113$	186103.	1.37106	0.685530	−	0.728045i	$-0.259571\pi$
$113$	186103.	1.37106	0.685530	+	0.728045i	$0.259571\pi$
$114$	0	0
$115$	−58975.9	−0.415844
$116$	0	0
$117$	138485.	0.935274
$118$	0	0
$119$	14599.6	0.0945088
$120$	0	0
$121$	43480.8	0.269981
$122$	0	0
$123$	−343688.	−2.04834
$124$	0	0
$125$	532857.	3.05025
$126$	0	0
$127$	171332.	0.942601	0.471301	−	0.881973i	$-0.343785\pi$
$127$	171332.	0.942601	0.471301	+	0.881973i	$0.343785\pi$
$128$	0	0
$129$	−107315.	−0.567786
$130$	0	0
$131$	328559.	1.67277	0.836383	−	0.548145i	$-0.184666\pi$
$131$	328559.	1.67277	0.836383	+	0.548145i	$0.184666\pi$
$132$	0	0
$133$	−112026.	−0.549150
$134$	0	0
$135$	184182.	0.869788
$136$	0	0
$137$	32938.6	0.149935	0.0749675	−	0.997186i	$-0.476115\pi$
$137$	32938.6	0.149935	0.0749675	+	0.997186i	$0.476115\pi$
$138$	0	0
$139$	129231.	0.567324	0.283662	−	0.958924i	$-0.408451\pi$
$139$	129231.	0.567324	0.283662	+	0.958924i	$0.408451\pi$
$140$	0	0
$141$	−512580.	−2.17127
$142$	0	0
$143$	401066.	1.64012
$144$	0	0
$145$	−877621.	−3.46647
$146$	0	0
$147$	−47969.5	−0.183093
$148$	0	0
$149$	306271.	1.13016	0.565081	−	0.825035i	$-0.308845\pi$
$149$	306271.	1.13016	0.565081	+	0.825035i	$0.308845\pi$
$150$	0	0
$151$	263579.	0.940736	0.470368	−	0.882470i	$-0.344121\pi$
$151$	263579.	0.940736	0.470368	+	0.882470i	$0.344121\pi$
$152$	0	0
$153$	46527.7	0.160688
$154$	0	0
$155$	−450867.	−1.50737
$156$	0	0
$157$	−386226.	−1.25053	−0.625263	−	0.780414i	$-0.715008\pi$
$157$	−386226.	−1.25053	−0.625263	+	0.780414i	$0.715008\pi$
$158$	0	0
$159$	−216461.	−0.679027
$160$	0	0
$161$	−27221.9	−0.0827664
$162$	0	0
$163$	107342.	0.316448	0.158224	−	0.987403i	$-0.449423\pi$
$163$	107342.	0.316448	0.158224	+	0.987403i	$0.449423\pi$
$164$	0	0
$165$	−959192.	−2.74281
$166$	0	0
$167$	378368.	1.04984	0.524919	−	0.851152i	$-0.324095\pi$
$167$	378368.	1.04984	0.524919	+	0.851152i	$0.324095\pi$
$168$	0	0
$169$	415157.	1.11814
$170$	0	0
$171$	−357020.	−0.933688
$172$	0	0
$173$	−568280.	−1.44360	−0.721801	−	0.692101i	$-0.756685\pi$
$173$	−568280.	−1.44360	−0.721801	+	0.692101i	$0.756685\pi$
$174$	0	0
$175$	399080.	0.985064
$176$	0	0
$177$	−55826.8	−0.133940
$178$	0	0
$179$	102649.	0.239454	0.119727	−	0.992807i	$-0.461798\pi$
$179$	102649.	0.239454	0.119727	+	0.992807i	$0.461798\pi$
$180$	0	0
$181$	−214817.	−0.487386	−0.243693	−	0.969852i	$-0.578359\pi$
$181$	−214817.	−0.487386	−0.243693	+	0.969852i	$0.578359\pi$
$182$	0	0
$183$	−169106.	−0.373276
$184$	0	0
$185$	−80551.4	−0.173039
$186$	0	0
$187$	134748.	0.281786
$188$	0	0
$189$	85014.3	0.173116
$190$	0	0
$191$	693025.	1.37457	0.687283	−	0.726390i	$-0.258803\pi$
$191$	693025.	1.37457	0.687283	+	0.726390i	$0.258803\pi$
$192$	0	0
$193$	169011.	0.326604	0.163302	−	0.986576i	$-0.447786\pi$
$193$	169011.	0.326604	0.163302	+	0.986576i	$0.447786\pi$
$194$	0	0
$195$	−1.88088e6	−3.54221
$196$	0	0
$197$	−461835.	−0.847855	−0.423927	−	0.905696i	$-0.639349\pi$
$197$	−461835.	−0.847855	−0.423927	+	0.905696i	$0.639349\pi$
$198$	0	0
$199$	−844976.	−1.51256	−0.756278	−	0.654250i	$-0.772984\pi$
$199$	−844976.	−1.51256	−0.756278	+	0.654250i	$0.772984\pi$
$200$	0	0
$201$	286380.	0.499980
$202$	0	0
$203$	−405090.	−0.689939
$204$	0	0
$205$	1.82618e6	3.03500
$206$	0	0
$207$	−86754.3	−0.140723
$208$	0	0
$209$	−1.03396e6	−1.63734
$210$	0	0
$211$	497886.	0.769882	0.384941	−	0.922941i	$-0.374222\pi$
$211$	497886.	0.769882	0.384941	+	0.922941i	$0.374222\pi$
$212$	0	0
$213$	−1.22133e6	−1.84453
$214$	0	0
$215$	570214.	0.841282
$216$	0	0
$217$	−208110.	−0.300015
$218$	0	0
$219$	120040.	0.169127
$220$	0	0
$221$	264228.	0.363914
$222$	0	0
$223$	−976639.	−1.31514	−0.657570	−	0.753394i	$-0.728416\pi$
$223$	−976639.	−1.31514	−0.657570	+	0.753394i	$0.728416\pi$
$224$	0	0
$225$	1.27184e6	1.67485
$226$	0	0
$227$	−1.01893e6	−1.31244	−0.656222	−	0.754568i	$-0.727846\pi$
$227$	−1.01893e6	−1.31244	−0.656222	+	0.754568i	$0.727846\pi$
$228$	0	0
$229$	172096.	0.216862	0.108431	−	0.994104i	$-0.465417\pi$
$229$	172096.	0.216862	0.108431	+	0.994104i	$0.465417\pi$
$230$	0	0
$231$	−442741.	−0.545908
$232$	0	0
$233$	569270.	0.686955	0.343478	−	0.939161i	$-0.388395\pi$
$233$	569270.	0.686955	0.343478	+	0.939161i	$0.388395\pi$
$234$	0	0
$235$	2.72358e6	3.21715
$236$	0	0
$237$	505472.	0.584556
$238$	0	0
$239$	1.39868e6	1.58389	0.791943	−	0.610595i	$-0.209070\pi$
$239$	1.39868e6	1.58389	0.791943	+	0.610595i	$0.209070\pi$
$240$	0	0
$241$	1.18533e6	1.31461	0.657307	−	0.753623i	$-0.271695\pi$
$241$	1.18533e6	1.31461	0.657307	+	0.753623i	$0.271695\pi$
$242$	0	0
$243$	1.02907e6	1.11797
$244$	0	0
$245$	254885.	0.271287
$246$	0	0
$247$	−2.02749e6	−2.11455
$248$	0	0
$249$	−108530.	−0.110930
$250$	0	0
$251$	−781131.	−0.782599	−0.391300	−	0.920263i	$-0.627974\pi$
$251$	−781131.	−0.782599	−0.391300	+	0.920263i	$0.627974\pi$
$252$	0	0
$253$	−251248.	−0.246775
$254$	0	0
$255$	−631930.	−0.608581
$256$	0	0
$257$	−1.52738e6	−1.44249	−0.721247	−	0.692678i	$-0.756431\pi$
$257$	−1.52738e6	−1.44249	−0.721247	+	0.692678i	$0.756431\pi$
$258$	0	0
$259$	−37180.7	−0.0344404
$260$	0	0
$261$	−1.29099e6	−1.17306
$262$	0	0
$263$	−728038.	−0.649030	−0.324515	−	0.945881i	$-0.605201\pi$
$263$	−728038.	−0.649030	−0.324515	+	0.945881i	$0.605201\pi$
$264$	0	0
$265$	1.15016e6	1.00611
$266$	0	0
$267$	−605617.	−0.519901
$268$	0	0
$269$	−1.05070e6	−0.885313	−0.442656	−	0.896691i	$-0.645964\pi$
$269$	−1.05070e6	−0.885313	−0.442656	+	0.896691i	$0.645964\pi$
$270$	0	0
$271$	688732.	0.569675	0.284837	−	0.958576i	$-0.408060\pi$
$271$	688732.	0.569675	0.284837	+	0.958576i	$0.408060\pi$
$272$	0	0
$273$	−868170.	−0.705015
$274$	0	0
$275$	3.68336e6	2.93706
$276$	0	0
$277$	317611.	0.248711	0.124356	−	0.992238i	$-0.460314\pi$
$277$	317611.	0.248711	0.124356	+	0.992238i	$0.460314\pi$
$278$	0	0
$279$	−663231.	−0.510098
$280$	0	0
$281$	−1.01321e6	−0.765478	−0.382739	−	0.923856i	$-0.625019\pi$
$281$	−1.01321e6	−0.765478	−0.382739	+	0.923856i	$0.625019\pi$
$282$	0	0
$283$	1.28474e6	0.953561	0.476781	−	0.879022i	$-0.341804\pi$
$283$	1.28474e6	0.953561	0.476781	+	0.879022i	$0.341804\pi$
$284$	0	0
$285$	4.84897e6	3.53620
$286$	0	0
$287$	842921.	0.604063
$288$	0	0
$289$	−1.33108e6	−0.937477
$290$	0	0
$291$	1.65919e6	1.14859
$292$	0	0
$293$	2.37245e6	1.61446	0.807230	−	0.590237i	$-0.200966\pi$
$293$	2.37245e6	1.61446	0.807230	+	0.590237i	$0.200966\pi$
$294$	0	0
$295$	296634.	0.198457
$296$	0	0
$297$	784651.	0.516161
$298$	0	0
$299$	−492673.	−0.318699
$300$	0	0
$301$	263197.	0.167442
$302$	0	0
$303$	2.71772e6	1.70058
$304$	0	0
$305$	898539.	0.553079
$306$	0	0
$307$	−3.00154e6	−1.81760	−0.908800	−	0.417231i	$-0.863000\pi$
$307$	−3.00154e6	−1.81760	−0.908800	+	0.417231i	$0.863000\pi$
$308$	0	0
$309$	1.30254e6	0.776058
$310$	0	0
$311$	−378276.	−0.221773	−0.110886	−	0.993833i	$-0.535369\pi$
$311$	−378276.	−0.221773	−0.110886	+	0.993833i	$0.535369\pi$
$312$	0	0
$313$	−306373.	−0.176763	−0.0883813	−	0.996087i	$-0.528169\pi$
$313$	−306373.	−0.176763	−0.0883813	+	0.996087i	$0.528169\pi$
$314$	0	0
$315$	812299.	0.461254
$316$	0	0
$317$	−2.43596e6	−1.36152	−0.680758	−	0.732509i	$-0.738349\pi$
$317$	−2.43596e6	−1.36152	−0.680758	+	0.732509i	$0.738349\pi$
$318$	0	0
$319$	−3.73883e6	−2.05712
$320$	0	0
$321$	2.43229e6	1.31751
$322$	0	0
$323$	−681189.	−0.363297
$324$	0	0
$325$	7.22269e6	3.79307
$326$	0	0
$327$	−710299.	−0.367343
$328$	0	0
$329$	1.25714e6	0.640317
$330$	0	0
$331$	217384.	0.109058	0.0545291	−	0.998512i	$-0.482634\pi$
$331$	217384.	0.109058	0.0545291	+	0.998512i	$0.482634\pi$
$332$	0	0
$333$	−118492.	−0.0585569
$334$	0	0
$335$	−1.52167e6	−0.740815
$336$	0	0
$337$	1.34165e6	0.643522	0.321761	−	0.946821i	$-0.395725\pi$
$337$	1.34165e6	0.643522	0.321761	+	0.946821i	$0.395725\pi$
$338$	0	0
$339$	−3.71814e6	−1.75722
$340$	0	0
$341$	−1.92078e6	−0.894522
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	1.17828e6	0.532967
$346$	0	0
$347$	−1.47196e6	−0.656253	−0.328127	−	0.944634i	$-0.606417\pi$
$347$	−1.47196e6	−0.656253	−0.328127	+	0.944634i	$0.606417\pi$
$348$	0	0
$349$	−4.15482e6	−1.82595	−0.912975	−	0.408016i	$-0.866221\pi$
$349$	−4.15482e6	−1.82595	−0.912975	+	0.408016i	$0.866221\pi$
$350$	0	0
$351$	1.53862e6	0.666598
$352$	0	0
$353$	−2.41522e6	−1.03162	−0.515810	−	0.856703i	$-0.672509\pi$
$353$	−2.41522e6	−1.03162	−0.515810	+	0.856703i	$0.672509\pi$
$354$	0	0
$355$	6.48952e6	2.73301
$356$	0	0
$357$	−291684.	−0.121127
$358$	0	0
$359$	−803886.	−0.329199	−0.164599	−	0.986361i	$-0.552633\pi$
$359$	−803886.	−0.329199	−0.164599	+	0.986361i	$0.552633\pi$
$360$	0	0
$361$	2.75085e6	1.11096
$362$	0	0
$363$	−868701.	−0.346022
$364$	0	0
$365$	−637828.	−0.250594
$366$	0	0
$367$	5.03890e6	1.95286	0.976428	−	0.215842i	$-0.0692497\pi$
$367$	5.03890e6	1.95286	0.976428	+	0.215842i	$0.0692497\pi$
$368$	0	0
$369$	2.68633e6	1.02705
$370$	0	0
$371$	530888.	0.200248
$372$	0	0
$373$	−86405.3	−0.0321565	−0.0160782	−	0.999871i	$-0.505118\pi$
$373$	−86405.3	−0.0321565	−0.0160782	+	0.999871i	$0.505118\pi$
$374$	0	0
$375$	−1.06459e7	−3.90936
$376$	0	0
$377$	−7.33146e6	−2.65667
$378$	0	0
$379$	1.60986e6	0.575691	0.287845	−	0.957677i	$-0.407061\pi$
$379$	1.60986e6	0.575691	0.287845	+	0.957677i	$0.407061\pi$
$380$	0	0
$381$	−3.42303e6	−1.20809
$382$	0	0
$383$	−3.09358e6	−1.07762	−0.538808	−	0.842429i	$-0.681125\pi$
$383$	−3.09358e6	−1.07762	−0.538808	+	0.842429i	$0.681125\pi$
$384$	0	0
$385$	2.35249e6	0.808866
$386$	0	0
$387$	838791.	0.284693
$388$	0	0
$389$	331993.	0.111238	0.0556191	−	0.998452i	$-0.482287\pi$
$389$	331993.	0.111238	0.0556191	+	0.998452i	$0.482287\pi$
$390$	0	0
$391$	−165526.	−0.0547551
$392$	0	0
$393$	−6.56427e6	−2.14390
$394$	0	0
$395$	−2.68581e6	−0.866130
$396$	0	0
$397$	797824.	0.254057	0.127028	−	0.991899i	$-0.459456\pi$
$397$	797824.	0.254057	0.127028	+	0.991899i	$0.459456\pi$
$398$	0	0
$399$	2.23817e6	0.703819
$400$	0	0
$401$	−2.22782e6	−0.691862	−0.345931	−	0.938260i	$-0.612437\pi$
$401$	−2.22782e6	−0.691862	−0.345931	+	0.938260i	$0.612437\pi$
$402$	0	0
$403$	−3.76645e6	−1.15523
$404$	0	0
$405$	−7.70812e6	−2.33513
$406$	0	0
$407$	−343164.	−0.102687
$408$	0	0
$409$	1.00358e6	0.296650	0.148325	−	0.988939i	$-0.452612\pi$
$409$	1.00358e6	0.296650	0.148325	+	0.988939i	$0.452612\pi$
$410$	0	0
$411$	−658079.	−0.192165
$412$	0	0
$413$	136920.	0.0394994
$414$	0	0
$415$	576670.	0.164364
$416$	0	0
$417$	−2.58191e6	−0.727111
$418$	0	0
$419$	3.56464e6	0.991929	0.495965	−	0.868343i	$-0.334815\pi$
$419$	3.56464e6	0.991929	0.495965	+	0.868343i	$0.334815\pi$
$420$	0	0
$421$	2.39330e6	0.658100	0.329050	−	0.944312i	$-0.393271\pi$
$421$	2.39330e6	0.658100	0.329050	+	0.944312i	$0.393271\pi$
$422$	0	0
$423$	4.00642e6	1.08869
$424$	0	0
$425$	2.42665e6	0.651681
$426$	0	0
$427$	414745.	0.110081
$428$	0	0
$429$	−8.01289e6	−2.10206
$430$	0	0
$431$	130925.	0.0339491	0.0169745	−	0.999856i	$-0.494597\pi$
$431$	130925.	0.0339491	0.0169745	+	0.999856i	$0.494597\pi$
$432$	0	0
$433$	2.70732e6	0.693937	0.346969	−	0.937877i	$-0.387211\pi$
$433$	2.70732e6	0.693937	0.346969	+	0.937877i	$0.387211\pi$
$434$	0	0
$435$	1.75340e7	4.44280
$436$	0	0
$437$	1.27013e6	0.318158
$438$	0	0
$439$	−7.04646e6	−1.74506	−0.872529	−	0.488562i	$-0.837522\pi$
$439$	−7.04646e6	−1.74506	−0.872529	+	0.488562i	$0.837522\pi$
$440$	0	0
$441$	374939.	0.0918045
$442$	0	0
$443$	2.17797e6	0.527281	0.263641	−	0.964621i	$-0.415077\pi$
$443$	2.17797e6	0.527281	0.263641	+	0.964621i	$0.415077\pi$
$444$	0	0
$445$	3.21793e6	0.770331
$446$	0	0
$447$	−6.11899e6	−1.44847
$448$	0	0
$449$	−2.52327e6	−0.590674	−0.295337	−	0.955393i	$-0.595432\pi$
$449$	−2.52327e6	−0.590674	−0.295337	+	0.955393i	$0.595432\pi$
$450$	0	0
$451$	7.77985e6	1.80107
$452$	0	0
$453$	−5.26603e6	−1.20570
$454$	0	0
$455$	4.61300e6	1.04461
$456$	0	0
$457$	1.08980e6	0.244094	0.122047	−	0.992524i	$-0.461054\pi$
$457$	1.08980e6	0.244094	0.122047	+	0.992524i	$0.461054\pi$
$458$	0	0
$459$	516939.	0.114527
$460$	0	0
$461$	5.14703e6	1.12799	0.563994	−	0.825779i	$-0.309264\pi$
$461$	5.14703e6	1.12799	0.563994	+	0.825779i	$0.309264\pi$
$462$	0	0
$463$	5.03321e6	1.09117	0.545585	−	0.838055i	$-0.316307\pi$
$463$	5.03321e6	1.09117	0.545585	+	0.838055i	$0.316307\pi$
$464$	0	0
$465$	9.00786e6	1.93192
$466$	0	0
$467$	1.07086e6	0.227216	0.113608	−	0.993526i	$-0.463759\pi$
$467$	1.07086e6	0.227216	0.113608	+	0.993526i	$0.463759\pi$
$468$	0	0
$469$	−702370.	−0.147446
$470$	0	0
$471$	7.71640e6	1.60274
$472$	0	0
$473$	2.42921e6	0.499244
$474$	0	0
$475$	−1.86203e7	−3.78664
$476$	0	0
$477$	1.69190e6	0.340470
$478$	0	0
$479$	−5.30339e6	−1.05612	−0.528062	−	0.849206i	$-0.677081\pi$
$479$	−5.30339e6	−1.05612	−0.528062	+	0.849206i	$0.677081\pi$
$480$	0	0
$481$	−672909.	−0.132615
$482$	0	0
$483$	543866.	0.106078
$484$	0	0
$485$	−8.81607e6	−1.70185
$486$	0	0
$487$	−3.00137e6	−0.573453	−0.286726	−	0.958013i	$-0.592567\pi$
$487$	−3.00137e6	−0.573453	−0.286726	+	0.958013i	$0.592567\pi$
$488$	0	0
$489$	−2.14459e6	−0.405576
$490$	0	0
$491$	−7.37046e6	−1.37972	−0.689860	−	0.723943i	$-0.742328\pi$
$491$	−7.37046e6	−1.37972	−0.689860	+	0.723943i	$0.742328\pi$
$492$	0	0
$493$	−2.46319e6	−0.456437
$494$	0	0
$495$	7.49722e6	1.37527
$496$	0	0
$497$	2.99541e6	0.543958
$498$	0	0
$499$	−8624.58	−0.00155055	−0.000775276	−	1.00000i	$-0.500247\pi$
$499$	−8624.58	−0.00155055	−0.000775276		1.00000i	$0.500247\pi$
$500$	0	0
$501$	−7.55939e6	−1.34553
$502$	0	0
$503$	124603.	0.0219587	0.0109794	−	0.999940i	$-0.496505\pi$
$503$	124603.	0.0219587	0.0109794	+	0.999940i	$0.496505\pi$
$504$	0	0
$505$	−1.44405e7	−2.51973
$506$	0	0
$507$	−8.29441e6	−1.43306
$508$	0	0
$509$	−1.12727e6	−0.192856	−0.0964279	−	0.995340i	$-0.530742\pi$
$509$	−1.12727e6	−0.192856	−0.0964279	+	0.995340i	$0.530742\pi$
$510$	0	0
$511$	−294406.	−0.0498764
$512$	0	0
$513$	−3.96662e6	−0.665467
$514$	0	0
$515$	−6.92101e6	−1.14988
$516$	0	0
$517$	1.16030e7	1.90916
$518$	0	0
$519$	1.13537e7	1.85019
$520$	0	0
$521$	−7.96035e6	−1.28481	−0.642404	−	0.766367i	$-0.722063\pi$
$521$	−7.96035e6	−1.28481	−0.642404	+	0.766367i	$0.722063\pi$
$522$	0	0
$523$	−2.19765e6	−0.351321	−0.175661	−	0.984451i	$-0.556206\pi$
$523$	−2.19765e6	−0.351321	−0.175661	+	0.984451i	$0.556206\pi$
$524$	0	0
$525$	−7.97320e6	−1.26251
$526$	0	0
$527$	−1.26544e6	−0.198478
$528$	0	0
$529$	−6.12771e6	−0.952048
$530$	0	0
$531$	436352.	0.0671585
$532$	0	0
$533$	1.52555e7	2.32599
$534$	0	0
$535$	−1.29239e7	−1.95214
$536$	0	0
$537$	−2.05082e6	−0.306897
$538$	0	0
$539$	1.08586e6	0.160991
$540$	0	0
$541$	1.57555e6	0.231440	0.115720	−	0.993282i	$-0.463082\pi$
$541$	1.57555e6	0.231440	0.115720	+	0.993282i	$0.463082\pi$
$542$	0	0
$543$	4.29183e6	0.624659
$544$	0	0
$545$	3.77416e6	0.544288
$546$	0	0
$547$	−1.04857e7	−1.49840	−0.749201	−	0.662343i	$-0.769562\pi$
$547$	−1.04857e7	−1.49840	−0.749201	+	0.662343i	$0.769562\pi$
$548$	0	0
$549$	1.32176e6	0.187164
$550$	0	0
$551$	1.89007e7	2.65216
$552$	0	0
$553$	−1.23971e6	−0.172388
$554$	0	0
$555$	1.60933e6	0.221776
$556$	0	0
$557$	2.88209e6	0.393613	0.196807	−	0.980442i	$-0.436943\pi$
$557$	2.88209e6	0.393613	0.196807	+	0.980442i	$0.436943\pi$
$558$	0	0
$559$	4.76344e6	0.644750
$560$	0	0
$561$	−2.69214e6	−0.361152
$562$	0	0
$563$	−7.31106e6	−0.972096	−0.486048	−	0.873932i	$-0.661562\pi$
$563$	−7.31106e6	−0.972096	−0.486048	+	0.873932i	$0.661562\pi$
$564$	0	0
$565$	1.97562e7	2.60365
$566$	0	0
$567$	−3.55789e6	−0.464766
$568$	0	0
$569$	3.30620e6	0.428103	0.214052	−	0.976822i	$-0.431334\pi$
$569$	3.30620e6	0.428103	0.214052	+	0.976822i	$0.431334\pi$
$570$	0	0
$571$	−4.22306e6	−0.542047	−0.271024	−	0.962573i	$-0.587362\pi$
$571$	−4.22306e6	−0.542047	−0.271024	+	0.962573i	$0.587362\pi$
$572$	0	0
$573$	−1.38459e7	−1.76171
$574$	0	0
$575$	−4.52467e6	−0.570712
$576$	0	0
$577$	−4.28814e6	−0.536203	−0.268102	−	0.963391i	$-0.586396\pi$
$577$	−4.28814e6	−0.536203	−0.268102	+	0.963391i	$0.586396\pi$
$578$	0	0
$579$	−3.37666e6	−0.418592
$580$	0	0
$581$	266177.	0.0327138
$582$	0	0
$583$	4.89990e6	0.597057
$584$	0	0
$585$	1.47013e7	1.77609
$586$	0	0
$587$	−1.56309e7	−1.87236	−0.936180	−	0.351521i	$-0.885665\pi$
$587$	−1.56309e7	−1.87236	−0.936180	+	0.351521i	$0.885665\pi$
$588$	0	0
$589$	9.71003e6	1.15327
$590$	0	0
$591$	9.22699e6	1.08665
$592$	0	0
$593$	−6.78346e6	−0.792163	−0.396081	−	0.918215i	$-0.629630\pi$
$593$	−6.78346e6	−0.792163	−0.396081	+	0.918215i	$0.629630\pi$
$594$	0	0
$595$	1.54986e6	0.179473
$596$	0	0
$597$	1.68817e7	1.93857
$598$	0	0
$599$	−1.41543e6	−0.161184	−0.0805919	−	0.996747i	$-0.525681\pi$
$599$	−1.41543e6	−0.161184	−0.0805919	+	0.996747i	$0.525681\pi$
$600$	0	0
$601$	5.71218e6	0.645083	0.322542	−	0.946555i	$-0.395463\pi$
$601$	5.71218e6	0.645083	0.322542	+	0.946555i	$0.395463\pi$
$602$	0	0
$603$	−2.23840e6	−0.250694
$604$	0	0
$605$	4.61582e6	0.512697
$606$	0	0
$607$	7.41559e6	0.816910	0.408455	−	0.912779i	$-0.366068\pi$
$607$	7.41559e6	0.816910	0.408455	+	0.912779i	$0.366068\pi$
$608$	0	0
$609$	8.09327e6	0.884262
$610$	0	0
$611$	2.27522e7	2.46559
$612$	0	0
$613$	−1.00094e7	−1.07587	−0.537933	−	0.842988i	$-0.680795\pi$
$613$	−1.00094e7	−1.07587	−0.537933	+	0.842988i	$0.680795\pi$
$614$	0	0
$615$	−3.64851e7	−3.88981
$616$	0	0
$617$	1.32953e7	1.40600	0.702999	−	0.711190i	$-0.251844\pi$
$617$	1.32953e7	1.40600	0.702999	+	0.711190i	$0.251844\pi$
$618$	0	0
$619$	−785629.	−0.0824121	−0.0412060	−	0.999151i	$-0.513120\pi$
$619$	−785629.	−0.0824121	−0.0412060	+	0.999151i	$0.513120\pi$
$620$	0	0
$621$	−963871.	−0.100297
$622$	0	0
$623$	1.48532e6	0.153321
$624$	0	0
$625$	3.11155e7	3.18622
$626$	0	0
$627$	2.06575e7	2.09850
$628$	0	0
$629$	−226081.	−0.0227844
$630$	0	0
$631$	1.13220e7	1.13201	0.566005	−	0.824402i	$-0.308488\pi$
$631$	1.13220e7	1.13201	0.566005	+	0.824402i	$0.308488\pi$
$632$	0	0
$633$	−9.94726e6	−0.986720
$634$	0	0
$635$	1.81882e7	1.79001
$636$	0	0
$637$	2.12926e6	0.207912
$638$	0	0
$639$	9.54616e6	0.924861
$640$	0	0
$641$	−5.12306e6	−0.492475	−0.246237	−	0.969210i	$-0.579194\pi$
$641$	−5.12306e6	−0.492475	−0.246237	+	0.969210i	$0.579194\pi$
$642$	0	0
$643$	−6.74313e6	−0.643182	−0.321591	−	0.946879i	$-0.604218\pi$
$643$	−6.74313e6	−0.643182	−0.321591	+	0.946879i	$0.604218\pi$
$644$	0	0
$645$	−1.13923e7	−1.07823
$646$	0	0
$647$	−1.93239e7	−1.81482	−0.907410	−	0.420247i	$-0.861943\pi$
$647$	−1.93239e7	−1.81482	−0.907410	+	0.420247i	$0.861943\pi$
$648$	0	0
$649$	1.26372e6	0.117771
$650$	0	0
$651$	4.15782e6	0.384515
$652$	0	0
$653$	9.59779e6	0.880823	0.440411	−	0.897796i	$-0.354833\pi$
$653$	9.59779e6	0.880823	0.440411	+	0.897796i	$0.354833\pi$
$654$	0	0
$655$	3.48791e7	3.17660
$656$	0	0
$657$	−938252.	−0.0848020
$658$	0	0
$659$	−1.51712e7	−1.36084	−0.680419	−	0.732824i	$-0.738202\pi$
$659$	−1.51712e7	−1.36084	−0.680419	+	0.732824i	$0.738202\pi$
$660$	0	0
$661$	−1.17829e6	−0.104894	−0.0524468	−	0.998624i	$-0.516702\pi$
$661$	−1.17829e6	−0.104894	−0.0524468	+	0.998624i	$0.516702\pi$
$662$	0	0
$663$	−5.27901e6	−0.466411
$664$	0	0
$665$	−1.18925e7	−1.04284
$666$	0	0
$667$	4.59280e6	0.399727
$668$	0	0
$669$	1.95122e7	1.68555
$670$	0	0
$671$	3.82794e6	0.328215
$672$	0	0
$673$	1.34216e7	1.14226	0.571131	−	0.820859i	$-0.306505\pi$
$673$	1.34216e7	1.14226	0.571131	+	0.820859i	$0.306505\pi$
$674$	0	0
$675$	1.41306e7	1.19371
$676$	0	0
$677$	−3.24042e6	−0.271725	−0.135862	−	0.990728i	$-0.543380\pi$
$677$	−3.24042e6	−0.271725	−0.135862	+	0.990728i	$0.543380\pi$
$678$	0	0
$679$	−4.06930e6	−0.338723
$680$	0	0
$681$	2.03572e7	1.68210
$682$	0	0
$683$	−1.97746e7	−1.62202	−0.811011	−	0.585031i	$-0.801082\pi$
$683$	−1.97746e7	−1.62202	−0.811011	+	0.585031i	$0.801082\pi$
$684$	0	0
$685$	3.49669e6	0.284728
$686$	0	0
$687$	−3.43831e6	−0.277941
$688$	0	0
$689$	9.60821e6	0.771071
$690$	0	0
$691$	−1.59677e7	−1.27217	−0.636086	−	0.771618i	$-0.719448\pi$
$691$	−1.59677e7	−1.27217	−0.636086	+	0.771618i	$0.719448\pi$
$692$	0	0
$693$	3.46055e6	0.273723
$694$	0	0
$695$	1.37189e7	1.07735
$696$	0	0
$697$	5.12548e6	0.399625
$698$	0	0
$699$	−1.13734e7	−0.880438
$700$	0	0
$701$	−2.38418e6	−0.183250	−0.0916251	−	0.995794i	$-0.529206\pi$
$701$	−2.38418e6	−0.183250	−0.0916251	+	0.995794i	$0.529206\pi$
$702$	0	0
$703$	1.73478e6	0.132390
$704$	0	0
$705$	−5.44144e7	−4.12326
$706$	0	0
$707$	−6.66541e6	−0.501509
$708$	0	0
$709$	−2.13193e7	−1.59278	−0.796391	−	0.604782i	$-0.793260\pi$
$709$	−2.13193e7	−1.59278	−0.796391	+	0.604782i	$0.793260\pi$
$710$	0	0
$711$	−3.95086e6	−0.293102
$712$	0	0
$713$	2.35950e6	0.173818
$714$	0	0
$715$	4.25763e7	3.11460
$716$	0	0
$717$	−2.79442e7	−2.02999
$718$	0	0
$719$	2.57018e7	1.85413	0.927067	−	0.374897i	$-0.122322\pi$
$719$	2.57018e7	1.85413	0.927067	+	0.374897i	$0.122322\pi$
$720$	0	0
$721$	−3.19458e6	−0.228863
$722$	0	0
$723$	−2.36818e7	−1.68488
$724$	0	0
$725$	−6.73315e7	−4.75744
$726$	0	0
$727$	−1.23186e7	−0.864423	−0.432211	−	0.901772i	$-0.642267\pi$
$727$	−1.23186e7	−0.864423	−0.432211	+	0.901772i	$0.642267\pi$
$728$	0	0
$729$	−2.91556e6	−0.203190
$730$	0	0
$731$	1.60040e6	0.110773
$732$	0	0
$733$	1.74378e7	1.19876	0.599379	−	0.800466i	$-0.295414\pi$
$733$	1.74378e7	1.19876	0.599379	+	0.800466i	$0.295414\pi$
$734$	0	0
$735$	−5.09234e6	−0.347695
$736$	0	0
$737$	−6.48261e6	−0.439624
$738$	0	0
$739$	−2.50344e7	−1.68627	−0.843134	−	0.537704i	$-0.819292\pi$
$739$	−2.50344e7	−1.68627	−0.843134	+	0.537704i	$0.819292\pi$
$740$	0	0
$741$	4.05072e7	2.71011
$742$	0	0
$743$	−2.31077e7	−1.53562	−0.767811	−	0.640676i	$-0.778654\pi$
$743$	−2.31077e7	−1.53562	−0.767811	+	0.640676i	$0.778654\pi$
$744$	0	0
$745$	3.25131e7	2.14619
$746$	0	0
$747$	848288.	0.0556214
$748$	0	0
$749$	−5.96539e6	−0.388539
$750$	0	0
$751$	−1.74588e7	−1.12957	−0.564786	−	0.825238i	$-0.691041\pi$
$751$	−1.74588e7	−1.12957	−0.564786	+	0.825238i	$0.691041\pi$
$752$	0	0
$753$	1.56062e7	1.00302
$754$	0	0
$755$	2.79809e7	1.78647
$756$	0	0
$757$	6.63158e6	0.420608	0.210304	−	0.977636i	$-0.432555\pi$
$757$	6.63158e6	0.420608	0.210304	+	0.977636i	$0.432555\pi$
$758$	0	0
$759$	5.01968e6	0.316280
$760$	0	0
$761$	−1.74131e7	−1.08997	−0.544985	−	0.838446i	$-0.683465\pi$
$761$	−1.74131e7	−1.08997	−0.544985	+	0.838446i	$0.683465\pi$
$762$	0	0
$763$	1.74207e6	0.108331
$764$	0	0
$765$	4.93928e6	0.305148
$766$	0	0
$767$	2.47802e6	0.152095
$768$	0	0
$769$	2.83850e7	1.73090	0.865452	−	0.500991i	$-0.167031\pi$
$769$	2.83850e7	1.73090	0.865452	+	0.500991i	$0.167031\pi$
$770$	0	0
$771$	3.05154e7	1.84877
$772$	0	0
$773$	−2.59767e7	−1.56363	−0.781816	−	0.623509i	$-0.785706\pi$
$773$	−2.59767e7	−1.56363	−0.781816	+	0.623509i	$0.785706\pi$
$774$	0	0
$775$	−3.45907e7	−2.06874
$776$	0	0
$777$	742831.	0.0441405
$778$	0	0
$779$	−3.93292e7	−2.32205
$780$	0	0
$781$	2.76465e7	1.62186
$782$	0	0
$783$	−1.43434e7	−0.836078
$784$	0	0
$785$	−4.10009e7	−2.37476
$786$	0	0
$787$	−2.68085e7	−1.54290	−0.771448	−	0.636293i	$-0.780467\pi$
$787$	−2.68085e7	−1.54290	−0.771448	+	0.636293i	$0.780467\pi$
$788$	0	0
$789$	1.45455e7	0.831830
$790$	0	0
$791$	9.11903e6	0.518212
$792$	0	0
$793$	7.50621e6	0.423875
$794$	0	0
$795$	−2.29790e7	−1.28948
$796$	0	0
$797$	−2.93026e6	−0.163403	−0.0817014	−	0.996657i	$-0.526035\pi$
$797$	−2.93026e6	−0.163403	−0.0817014	+	0.996657i	$0.526035\pi$
$798$	0	0
$799$	7.64421e6	0.423609
$800$	0	0
$801$	4.73362e6	0.260683
$802$	0	0
$803$	−2.71726e6	−0.148711
$804$	0	0
$805$	−2.88982e6	−0.157174
$806$	0	0
$807$	2.09918e7	1.13466
$808$	0	0
$809$	2.98402e7	1.60299	0.801494	−	0.598003i	$-0.204039\pi$
$809$	2.98402e7	1.60299	0.801494	+	0.598003i	$0.204039\pi$
$810$	0	0
$811$	2.53235e7	1.35199	0.675993	−	0.736908i	$-0.263715\pi$
$811$	2.53235e7	1.35199	0.675993	+	0.736908i	$0.263715\pi$
$812$	0	0
$813$	−1.37602e7	−0.730124
$814$	0	0
$815$	1.13952e7	0.600937
$816$	0	0
$817$	−1.22803e7	−0.643657
$818$	0	0
$819$	6.78578e6	0.353500
$820$	0	0
$821$	−5.86859e6	−0.303862	−0.151931	−	0.988391i	$-0.548549\pi$
$821$	−5.86859e6	−0.303862	−0.151931	+	0.988391i	$0.548549\pi$
$822$	0	0
$823$	9.81684e6	0.505210	0.252605	−	0.967569i	$-0.418713\pi$
$823$	9.81684e6	0.505210	0.252605	+	0.967569i	$0.418713\pi$
$824$	0	0
$825$	−7.35897e7	−3.76428
$826$	0	0
$827$	−2.76584e7	−1.40625	−0.703127	−	0.711065i	$-0.748213\pi$
$827$	−2.76584e7	−1.40625	−0.703127	+	0.711065i	$0.748213\pi$
$828$	0	0
$829$	−2.06302e7	−1.04260	−0.521298	−	0.853374i	$-0.674552\pi$
$829$	−2.06302e7	−1.04260	−0.521298	+	0.853374i	$0.674552\pi$
$830$	0	0
$831$	−6.34553e6	−0.318761
$832$	0	0
$833$	715378.	0.0357210
$834$	0	0
$835$	4.01667e7	1.99365
$836$	0	0
$837$	−7.36873e6	−0.363562
$838$	0	0
$839$	1.55068e7	0.760533	0.380266	−	0.924877i	$-0.375832\pi$
$839$	1.55068e7	0.760533	0.380266	+	0.924877i	$0.375832\pi$
$840$	0	0
$841$	4.78343e7	2.33211
$842$	0	0
$843$	2.02429e7	0.981076
$844$	0	0
$845$	4.40721e7	2.12335
$846$	0	0
$847$	2.13056e6	0.102043
$848$	0	0
$849$	−2.56678e7	−1.22213
$850$	0	0
$851$	421545.	0.0199535
$852$	0	0
$853$	6.07567e6	0.285905	0.142952	−	0.989730i	$-0.454340\pi$
$853$	6.07567e6	0.285905	0.142952	+	0.989730i	$0.454340\pi$
$854$	0	0
$855$	−3.79004e7	−1.77308
$856$	0	0
$857$	−2.60713e7	−1.21258	−0.606291	−	0.795243i	$-0.707343\pi$
$857$	−2.60713e7	−1.21258	−0.606291	+	0.795243i	$0.707343\pi$
$858$	0	0
$859$	−3.89265e6	−0.179996	−0.0899980	−	0.995942i	$-0.528686\pi$
$859$	−3.89265e6	−0.179996	−0.0899980	+	0.995942i	$0.528686\pi$
$860$	0	0
$861$	−1.68407e7	−0.774198
$862$	0	0
$863$	1.73425e7	0.792657	0.396329	−	0.918109i	$-0.370284\pi$
$863$	1.73425e7	0.792657	0.396329	+	0.918109i	$0.370284\pi$
$864$	0	0
$865$	−6.03274e7	−2.74141
$866$	0	0
$867$	2.65937e7	1.20152
$868$	0	0
$869$	−1.14421e7	−0.513990
$870$	0	0
$871$	−1.27117e7	−0.567754
$872$	0	0
$873$	−1.29685e7	−0.575911
$874$	0	0
$875$	2.61100e7	1.15289
$876$	0	0
$877$	2.40336e6	0.105516	0.0527582	−	0.998607i	$-0.483199\pi$
$877$	2.40336e6	0.105516	0.0527582	+	0.998607i	$0.483199\pi$
$878$	0	0
$879$	−4.73990e7	−2.06918
$880$	0	0
$881$	−1.40789e7	−0.611125	−0.305563	−	0.952172i	$-0.598845\pi$
$881$	−1.40789e7	−0.611125	−0.305563	+	0.952172i	$0.598845\pi$
$882$	0	0
$883$	4.30002e7	1.85596	0.927979	−	0.372632i	$-0.121545\pi$
$883$	4.30002e7	1.85596	0.927979	+	0.372632i	$0.121545\pi$
$884$	0	0
$885$	−5.92645e6	−0.254353
$886$	0	0
$887$	−136209.	−0.00581297	−0.00290648	−	0.999996i	$-0.500925\pi$
$887$	−136209.	−0.00581297	−0.00290648	+	0.999996i	$0.500925\pi$
$888$	0	0
$889$	8.39524e6	0.356270
$890$	0	0
$891$	−3.28380e7	−1.38574
$892$	0	0
$893$	−5.86561e7	−2.46141
$894$	0	0
$895$	1.08970e7	0.454725
$896$	0	0
$897$	9.84309e6	0.408461
$898$	0	0
$899$	3.51117e7	1.44895
$900$	0	0
$901$	3.22813e6	0.132476
$902$	0	0
$903$	−5.25841e6	−0.214603
$904$	0	0
$905$	−2.28045e7	−0.925550
$906$	0	0
$907$	8.23331e6	0.332320	0.166160	−	0.986099i	$-0.446863\pi$
$907$	8.23331e6	0.332320	0.166160	+	0.986099i	$0.446863\pi$
$908$	0	0
$909$	−2.12422e7	−0.852686
$910$	0	0
$911$	−9.15065e6	−0.365306	−0.182653	−	0.983177i	$-0.558468\pi$
$911$	−9.15065e6	−0.365306	−0.182653	+	0.983177i	$0.558468\pi$
$912$	0	0
$913$	2.45672e6	0.0975391
$914$	0	0
$915$	−1.79519e7	−0.708855
$916$	0	0
$917$	1.60994e7	0.632246
$918$	0	0
$919$	−2.27063e7	−0.886865	−0.443433	−	0.896308i	$-0.646239\pi$
$919$	−2.27063e7	−0.886865	−0.443433	+	0.896308i	$0.646239\pi$
$920$	0	0
$921$	5.99677e7	2.32953
$922$	0	0
$923$	5.42121e7	2.09456
$924$	0	0
$925$	−6.17994e6	−0.237482
$926$	0	0
$927$	−1.01809e7	−0.389122
$928$	0	0
$929$	−1.95825e6	−0.0744439	−0.0372220	−	0.999307i	$-0.511851\pi$
$929$	−1.95825e6	−0.0744439	−0.0372220	+	0.999307i	$0.511851\pi$
$930$	0	0
$931$	−5.48929e6	−0.207559
$932$	0	0
$933$	7.55757e6	0.284235
$934$	0	0
$935$	1.43046e7	0.535115
$936$	0	0
$937$	−3.49113e7	−1.29902	−0.649512	−	0.760351i	$-0.725027\pi$
$937$	−3.49113e7	−1.29902	−0.649512	+	0.760351i	$0.725027\pi$
$938$	0	0
$939$	6.12103e6	0.226548
$940$	0	0
$941$	2.99605e7	1.10300	0.551500	−	0.834175i	$-0.314056\pi$
$941$	2.99605e7	1.10300	0.551500	+	0.834175i	$0.314056\pi$
$942$	0	0
$943$	−9.55683e6	−0.349973
$944$	0	0
$945$	9.02494e6	0.328749
$946$	0	0
$947$	1.74875e7	0.633655	0.316827	−	0.948483i	$-0.397382\pi$
$947$	1.74875e7	0.633655	0.316827	+	0.948483i	$0.397382\pi$
$948$	0	0
$949$	−5.32828e6	−0.192053
$950$	0	0
$951$	4.86681e7	1.74499
$952$	0	0
$953$	−1.25389e7	−0.447226	−0.223613	−	0.974678i	$-0.571785\pi$
$953$	−1.25389e7	−0.447226	−0.223613	+	0.974678i	$0.571785\pi$
$954$	0	0
$955$	7.35700e7	2.61031
$956$	0	0
$957$	7.46979e7	2.63651
$958$	0	0
$959$	1.61399e6	0.0566701
$960$	0	0
$961$	−1.05910e7	−0.369936
$962$	0	0
$963$	−1.90113e7	−0.660610
$964$	0	0
$965$	1.79418e7	0.620223
$966$	0	0
$967$	2.28111e7	0.784477	0.392238	−	0.919864i	$-0.371701\pi$
$967$	2.28111e7	0.784477	0.392238	+	0.919864i	$0.371701\pi$
$968$	0	0
$969$	1.36095e7	0.465620
$970$	0	0
$971$	−190729.	−0.00649186	−0.00324593	−	0.999995i	$-0.501033\pi$
$971$	−190729.	−0.00649186	−0.00324593	+	0.999995i	$0.501033\pi$
$972$	0	0
$973$	6.33234e6	0.214428
$974$	0	0
$975$	−1.44302e8	−4.86139
$976$	0	0
$977$	1.92970e7	0.646775	0.323388	−	0.946267i	$-0.395178\pi$
$977$	1.92970e7	0.646775	0.323388	+	0.946267i	$0.395178\pi$
$978$	0	0
$979$	1.37090e7	0.457140
$980$	0	0
$981$	5.55183e6	0.184189
$982$	0	0
$983$	−1.60890e7	−0.531062	−0.265531	−	0.964102i	$-0.585547\pi$
$983$	−1.60890e7	−0.531062	−0.265531	+	0.964102i	$0.585547\pi$
$984$	0	0
$985$	−4.90274e7	−1.61008
$986$	0	0
$987$	−2.51164e7	−0.820664
$988$	0	0
$989$	−2.98407e6	−0.0970103
$990$	0	0
$991$	−2.90847e7	−0.940762	−0.470381	−	0.882463i	$-0.655883\pi$
$991$	−2.90847e7	−0.940762	−0.470381	+	0.882463i	$0.655883\pi$
$992$	0	0
$993$	−4.34311e6	−0.139775
$994$	0	0
$995$	−8.97008e7	−2.87236
$996$	0	0
$997$	4.48593e7	1.42927	0.714636	−	0.699497i	$-0.246592\pi$
$997$	4.48593e7	1.42927	0.714636	+	0.699497i	$0.246592\pi$
$998$	0	0
$999$	−1.31649e6	−0.0417353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.6.a.i.1.2	✓	5
4.3	odd	2		224.6.a.j.1.4	yes	5
8.3	odd	2		448.6.a.be.1.2		5
8.5	even	2		448.6.a.bf.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.i.1.2	✓	5	1.1	even	1	trivial
224.6.a.j.1.4	yes	5	4.3	odd	2
448.6.a.be.1.2		5	8.3	odd	2
448.6.a.bf.1.4		5	8.5	even	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

\(f(q)\)	\(=\)	\(q-19.9790 q^{3} +106.158 q^{5} +49.0000 q^{7} +156.159 q^{9} +452.252 q^{11} +886.820 q^{13} -2120.92 q^{15} +297.950 q^{17} -2286.25 q^{19} -978.970 q^{21} -555.550 q^{23} +8144.48 q^{25} +1734.99 q^{27} -8267.13 q^{29} -4247.14 q^{31} -9035.53 q^{33} +5201.73 q^{35} -758.789 q^{37} -17717.8 q^{39} +17202.5 q^{41} +5371.38 q^{43} +16577.5 q^{45} +25656.0 q^{47} +2401.00 q^{49} -5952.74 q^{51} +10834.4 q^{53} +48010.1 q^{55} +45677.0 q^{57} +2794.28 q^{59} +8464.18 q^{61} +7651.81 q^{63} +94142.9 q^{65} -14334.1 q^{67} +11099.3 q^{69} +61130.9 q^{71} -6008.30 q^{73} -162718. q^{75} +22160.3 q^{77} -25300.2 q^{79} -72610.0 q^{81} +5432.19 q^{83} +31629.7 q^{85} +165169. q^{87} +30312.7 q^{89} +43454.2 q^{91} +84853.5 q^{93} -242703. q^{95} -83046.9 q^{97} +70623.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) \) 5 * q - 10 * q^3 + 36 * q^5 + 245 * q^7 + 637 * q^9 + 116 * q^11 + 40 * q^13 + 16 * q^15 - 402 * q^17 - 3582 * q^19 - 490 * q^21 + 472 * q^23 + 9615 * q^25 + 356 * q^27 + 4754 * q^29 - 10500 * q^31 + 15864 * q^33 + 1764 * q^35 + 19642 * q^37 + 10872 * q^39 + 23398 * q^41 + 22044 * q^43 + 49476 * q^45 + 16004 * q^47 + 12005 * q^49 - 45676 * q^51 + 54246 * q^53 + 53456 * q^55 + 109556 * q^57 - 74366 * q^59 + 68316 * q^61 + 31213 * q^63 + 152568 * q^65 - 26560 * q^67 + 214720 * q^69 + 93072 * q^71 + 136098 * q^73 - 124510 * q^75 + 5684 * q^77 + 96080 * q^79 + 104801 * q^81 - 145894 * q^83 + 117352 * q^85 + 168876 * q^87 + 188554 * q^89 + 1960 * q^91 + 86296 * q^93 + 74736 * q^95 - 88146 * q^97 - 260236 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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