LMFDB - Embedded newform 496.6.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [496,6,Mod(1,496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("496.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 496 = 2^{4} \cdot 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	496.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,2,0,128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.5503746272$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 $ x^8 - x^7 - 199x^6 + 256x^5 + 12633x^4 - 18583x^3 - 260319x^2 + 410640x + 275908
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 5\cdot 13 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.513172$ of defining polynomial
Character	$\chi$	$=$	496.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-22.4697 q^{3} +77.4836 q^{5} -20.7014 q^{7} +261.888 q^{9} +13.6402 q^{11} +703.908 q^{13} -1741.03 q^{15} -1360.44 q^{17} +2516.75 q^{19} +465.155 q^{21} +3840.67 q^{23} +2878.70 q^{25} -424.401 q^{27} +1600.17 q^{29} -961.000 q^{31} -306.491 q^{33} -1604.02 q^{35} -7831.82 q^{37} -15816.6 q^{39} -1135.29 q^{41} -15289.3 q^{43} +20292.0 q^{45} -29909.8 q^{47} -16378.5 q^{49} +30568.7 q^{51} +17590.3 q^{53} +1056.89 q^{55} -56550.6 q^{57} +16403.3 q^{59} +23461.6 q^{61} -5421.45 q^{63} +54541.3 q^{65} -12225.2 q^{67} -86298.8 q^{69} +50123.9 q^{71} +63123.3 q^{73} -64683.6 q^{75} -282.371 q^{77} +34426.9 q^{79} -54102.5 q^{81} -34019.7 q^{83} -105412. q^{85} -35955.3 q^{87} +22185.6 q^{89} -14571.9 q^{91} +21593.4 q^{93} +195007. q^{95} -113665. q^{97} +3572.20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} + 128 q^{5} - 88 q^{7} + 1512 q^{9} - 574 q^{11} - 122 q^{13} + 524 q^{15} + 1932 q^{17} + 1796 q^{19} + 996 q^{21} + 4136 q^{23} + 3308 q^{25} + 2624 q^{27} + 5050 q^{29} - 7688 q^{31} + 27920 q^{33}+ \cdots + 180150 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 + 128 * q^5 - 88 * q^7 + 1512 * q^9 - 574 * q^11 - 122 * q^13 + 524 * q^15 + 1932 * q^17 + 1796 * q^19 + 996 * q^21 + 4136 * q^23 + 3308 * q^25 + 2624 * q^27 + 5050 * q^29 - 7688 * q^31 + 27920 * q^33 - 25700 * q^35 + 23674 * q^37 - 38652 * q^39 + 44828 * q^41 - 21058 * q^43 - 1344 * q^45 - 5348 * q^47 + 19356 * q^49 - 22300 * q^51 + 4926 * q^53 + 56892 * q^55 + 22072 * q^57 - 16944 * q^59 - 73682 * q^61 + 138784 * q^63 + 63316 * q^65 + 134768 * q^67 + 3992 * q^69 - 123724 * q^71 + 70792 * q^73 + 302902 * q^75 + 107932 * q^77 + 26036 * q^79 + 82528 * q^81 - 21882 * q^83 - 130464 * q^85 + 440244 * q^87 + 164392 * q^89 + 60028 * q^91 - 1922 * q^93 + 7404 * q^95 - 131948 * q^97 + 180150 * 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
$3$	−22.4697	−1.44143	−0.720716	−	0.693230i	$-0.756187\pi$
$3$	−22.4697	−1.44143	−0.720716	+	0.693230i	$0.756187\pi$
$4$	0	0
$5$	77.4836	1.38607	0.693034	−	0.720905i	$-0.256274\pi$
$5$	77.4836	1.38607	0.693034	+	0.720905i	$0.256274\pi$
$6$	0	0
$7$	−20.7014	−0.159682	−0.0798408	−	0.996808i	$-0.525441\pi$
$7$	−20.7014	−0.159682	−0.0798408	+	0.996808i	$0.525441\pi$
$8$	0	0
$9$	261.888	1.07773
$10$	0	0
$11$	13.6402	0.0339890	0.0169945	−	0.999856i	$-0.494590\pi$
$11$	13.6402	0.0339890	0.0169945	+	0.999856i	$0.494590\pi$
$12$	0	0
$13$	703.908	1.15520	0.577600	−	0.816320i	$-0.303989\pi$
$13$	703.908	1.15520	0.577600	+	0.816320i	$0.303989\pi$
$14$	0	0
$15$	−1741.03	−1.99792
$16$	0	0
$17$	−1360.44	−1.14171	−0.570857	−	0.821049i	$-0.693389\pi$
$17$	−1360.44	−1.14171	−0.570857	+	0.821049i	$0.693389\pi$
$18$	0	0
$19$	2516.75	1.59939	0.799697	−	0.600404i	$-0.204993\pi$
$19$	2516.75	1.59939	0.799697	+	0.600404i	$0.204993\pi$
$20$	0	0
$21$	465.155	0.230170
$22$	0	0
$23$	3840.67	1.51387	0.756934	−	0.653492i	$-0.226697\pi$
$23$	3840.67	1.51387	0.756934	+	0.653492i	$0.226697\pi$
$24$	0	0
$25$	2878.70	0.921185
$26$	0	0
$27$	−424.401	−0.112038
$28$	0	0
$29$	1600.17	0.353322	0.176661	−	0.984272i	$-0.443470\pi$
$29$	1600.17	0.353322	0.176661	+	0.984272i	$0.443470\pi$
$30$	0	0
$31$	−961.000	−0.179605
$31$	−961.000	−0.179605
$32$	0	0
$33$	−306.491	−0.0489929
$34$	0	0
$35$	−1604.02	−0.221330
$36$	0	0
$37$	−7831.82	−0.940500	−0.470250	−	0.882533i	$-0.655836\pi$
$37$	−7831.82	−0.940500	−0.470250	+	0.882533i	$0.655836\pi$
$38$	0	0
$39$	−15816.6	−1.66514
$40$	0	0
$41$	−1135.29	−0.105474	−0.0527371	−	0.998608i	$-0.516795\pi$
$41$	−1135.29	−0.105474	−0.0527371	+	0.998608i	$0.516795\pi$
$42$	0	0
$43$	−15289.3	−1.26100	−0.630501	−	0.776189i	$-0.717150\pi$
$43$	−15289.3	−1.26100	−0.630501	+	0.776189i	$0.717150\pi$
$44$	0	0
$45$	20292.0	1.49380
$46$	0	0
$47$	−29909.8	−1.97500	−0.987502	−	0.157604i	$-0.949623\pi$
$47$	−29909.8	−1.97500	−0.987502	+	0.157604i	$0.949623\pi$
$48$	0	0
$49$	−16378.5	−0.974502
$50$	0	0
$51$	30568.7	1.64570
$52$	0	0
$53$	17590.3	0.860169	0.430084	−	0.902789i	$-0.358484\pi$
$53$	17590.3	0.860169	0.430084	+	0.902789i	$0.358484\pi$
$54$	0	0
$55$	1056.89	0.0471111
$56$	0	0
$57$	−56550.6	−2.30542
$58$	0	0
$59$	16403.3	0.613481	0.306741	−	0.951793i	$-0.400762\pi$
$59$	16403.3	0.613481	0.306741	+	0.951793i	$0.400762\pi$
$60$	0	0
$61$	23461.6	0.807295	0.403648	−	0.914915i	$-0.367742\pi$
$61$	23461.6	0.807295	0.403648	+	0.914915i	$0.367742\pi$
$62$	0	0
$63$	−5421.45	−0.172093
$64$	0	0
$65$	54541.3	1.60119
$66$	0	0
$67$	−12225.2	−0.332712	−0.166356	−	0.986066i	$-0.553200\pi$
$67$	−12225.2	−0.332712	−0.166356	+	0.986066i	$0.553200\pi$
$68$	0	0
$69$	−86298.8	−2.18214
$70$	0	0
$71$	50123.9	1.18005	0.590023	−	0.807387i	$-0.299119\pi$
$71$	50123.9	1.18005	0.590023	+	0.807387i	$0.299119\pi$
$72$	0	0
$73$	63123.3	1.38638	0.693190	−	0.720755i	$-0.256205\pi$
$73$	63123.3	1.38638	0.693190	+	0.720755i	$0.256205\pi$
$74$	0	0
$75$	−64683.6	−1.32783
$76$	0	0
$77$	−282.371	−0.00542742
$78$	0	0
$79$	34426.9	0.620627	0.310313	−	0.950634i	$-0.399566\pi$
$79$	34426.9	0.620627	0.310313	+	0.950634i	$0.399566\pi$
$80$	0	0
$81$	−54102.5	−0.916231
$82$	0	0
$83$	−34019.7	−0.542044	−0.271022	−	0.962573i	$-0.587362\pi$
$83$	−34019.7	−0.542044	−0.271022	+	0.962573i	$0.587362\pi$
$84$	0	0
$85$	−105412.	−1.58249
$86$	0	0
$87$	−35955.3	−0.509290
$88$	0	0
$89$	22185.6	0.296890	0.148445	−	0.988921i	$-0.452573\pi$
$89$	22185.6	0.296890	0.148445	+	0.988921i	$0.452573\pi$
$90$	0	0
$91$	−14571.9	−0.184464
$92$	0	0
$93$	21593.4	0.258889
$94$	0	0
$95$	195007.	2.21687
$96$	0	0
$97$	−113665.	−1.22658	−0.613292	−	0.789857i	$-0.710155\pi$
$97$	−113665.	−1.22658	−0.613292	+	0.789857i	$0.710155\pi$
$98$	0	0
$99$	3572.20	0.0366309
$100$	0	0
$101$	136887.	1.33524	0.667620	−	0.744502i	$-0.267313\pi$
$101$	136887.	1.33524	0.667620	+	0.744502i	$0.267313\pi$
$102$	0	0
$103$	44100.8	0.409594	0.204797	−	0.978805i	$-0.434347\pi$
$103$	44100.8	0.409594	0.204797	+	0.978805i	$0.434347\pi$
$104$	0	0
$105$	36041.9	0.319032
$106$	0	0
$107$	−100756.	−0.850769	−0.425385	−	0.905013i	$-0.639861\pi$
$107$	−100756.	−0.850769	−0.425385	+	0.905013i	$0.639861\pi$
$108$	0	0
$109$	149149.	1.20241	0.601206	−	0.799094i	$-0.294687\pi$
$109$	149149.	1.20241	0.601206	+	0.799094i	$0.294687\pi$
$110$	0	0
$111$	175979.	1.35567
$112$	0	0
$113$	215378.	1.58674	0.793370	−	0.608740i	$-0.208325\pi$
$113$	215378.	1.58674	0.793370	+	0.608740i	$0.208325\pi$
$114$	0	0
$115$	297589.	2.09832
$116$	0	0
$117$	184345.	1.24499
$118$	0	0
$119$	28163.1	0.182311
$120$	0	0
$121$	−160865.	−0.998845
$122$	0	0
$123$	25509.6	0.152034
$124$	0	0
$125$	−19084.0	−0.109243
$126$	0	0
$127$	347582.	1.91226	0.956131	−	0.292939i	$-0.0946334\pi$
$127$	347582.	1.91226	0.956131	+	0.292939i	$0.0946334\pi$
$128$	0	0
$129$	343545.	1.81765
$130$	0	0
$131$	12743.3	0.0648790	0.0324395	−	0.999474i	$-0.489672\pi$
$131$	12743.3	0.0648790	0.0324395	+	0.999474i	$0.489672\pi$
$132$	0	0
$133$	−52100.2	−0.255394
$134$	0	0
$135$	−32884.1	−0.155293
$136$	0	0
$137$	54494.2	0.248056	0.124028	−	0.992279i	$-0.460419\pi$
$137$	54494.2	0.248056	0.124028	+	0.992279i	$0.460419\pi$
$138$	0	0
$139$	−292153.	−1.28255	−0.641273	−	0.767312i	$-0.721594\pi$
$139$	−292153.	−1.28255	−0.641273	+	0.767312i	$0.721594\pi$
$140$	0	0
$141$	672063.	2.84684
$142$	0	0
$143$	9601.43	0.0392641
$144$	0	0
$145$	123987.	0.489728
$146$	0	0
$147$	368019.	1.40468
$148$	0	0
$149$	−205340.	−0.757720	−0.378860	−	0.925454i	$-0.623684\pi$
$149$	−205340.	−0.757720	−0.378860	+	0.925454i	$0.623684\pi$
$150$	0	0
$151$	208992.	0.745911	0.372956	−	0.927849i	$-0.378344\pi$
$151$	208992.	0.745911	0.372956	+	0.927849i	$0.378344\pi$
$152$	0	0
$153$	−356283.	−1.23046
$154$	0	0
$155$	−74461.7	−0.248945
$156$	0	0
$157$	414617.	1.34245	0.671225	−	0.741254i	$-0.265769\pi$
$157$	414617.	1.34245	0.671225	+	0.741254i	$0.265769\pi$
$158$	0	0
$159$	−395249.	−1.23988
$160$	0	0
$161$	−79507.4	−0.241737
$162$	0	0
$163$	74081.6	0.218394	0.109197	−	0.994020i	$-0.465172\pi$
$163$	74081.6	0.218394	0.109197	+	0.994020i	$0.465172\pi$
$164$	0	0
$165$	−23748.0	−0.0679074
$166$	0	0
$167$	355478.	0.986329	0.493164	−	0.869936i	$-0.335840\pi$
$167$	355478.	0.986329	0.493164	+	0.869936i	$0.335840\pi$
$168$	0	0
$169$	124193.	0.334488
$170$	0	0
$171$	659105.	1.72371
$172$	0	0
$173$	511596.	1.29961	0.649803	−	0.760102i	$-0.274851\pi$
$173$	511596.	1.29961	0.649803	+	0.760102i	$0.274851\pi$
$174$	0	0
$175$	−59593.2	−0.147096
$176$	0	0
$177$	−368577.	−0.884291
$178$	0	0
$179$	−326430.	−0.761477	−0.380739	−	0.924683i	$-0.624330\pi$
$179$	−326430.	−0.761477	−0.380739	+	0.924683i	$0.624330\pi$
$180$	0	0
$181$	529461.	1.20126	0.600631	−	0.799527i	$-0.294916\pi$
$181$	529461.	1.20126	0.600631	+	0.799527i	$0.294916\pi$
$182$	0	0
$183$	−527174.	−1.16366
$184$	0	0
$185$	−606838.	−1.30360
$186$	0	0
$187$	−18556.7	−0.0388058
$188$	0	0
$189$	8785.71	0.0178905
$190$	0	0
$191$	227182.	0.450598	0.225299	−	0.974290i	$-0.427664\pi$
$191$	227182.	0.450598	0.225299	+	0.974290i	$0.427664\pi$
$192$	0	0
$193$	82697.5	0.159808	0.0799041	−	0.996803i	$-0.474539\pi$
$193$	82697.5	0.159808	0.0799041	+	0.996803i	$0.474539\pi$
$194$	0	0
$195$	−1.22553e6	−2.30800
$196$	0	0
$197$	−246100.	−0.451800	−0.225900	−	0.974150i	$-0.572532\pi$
$197$	−246100.	−0.451800	−0.225900	+	0.974150i	$0.572532\pi$
$198$	0	0
$199$	762519.	1.36495	0.682477	−	0.730907i	$-0.260903\pi$
$199$	762519.	1.36495	0.682477	+	0.730907i	$0.260903\pi$
$200$	0	0
$201$	274696.	0.479582
$202$	0	0
$203$	−33125.8	−0.0564191
$204$	0	0
$205$	−87966.1	−0.146194
$206$	0	0
$207$	1.00583e6	1.63154
$208$	0	0
$209$	34328.9	0.0543618
$210$	0	0
$211$	233035.	0.360343	0.180171	−	0.983635i	$-0.442335\pi$
$211$	233035.	0.360343	0.180171	+	0.983635i	$0.442335\pi$
$212$	0	0
$213$	−1.12627e6	−1.70096
$214$	0	0
$215$	−1.18467e6	−1.74783
$216$	0	0
$217$	19894.1	0.0286797
$218$	0	0
$219$	−1.41836e6	−1.99837
$220$	0	0
$221$	−957626.	−1.31891
$222$	0	0
$223$	861370.	1.15992	0.579959	−	0.814646i	$-0.303069\pi$
$223$	861370.	1.15992	0.579959	+	0.814646i	$0.303069\pi$
$224$	0	0
$225$	753897.	0.992786
$226$	0	0
$227$	−833699.	−1.07385	−0.536927	−	0.843629i	$-0.680415\pi$
$227$	−833699.	−1.07385	−0.536927	+	0.843629i	$0.680415\pi$
$228$	0	0
$229$	362589.	0.456906	0.228453	−	0.973555i	$-0.426633\pi$
$229$	362589.	0.456906	0.228453	+	0.973555i	$0.426633\pi$
$230$	0	0
$231$	6344.80	0.00782326
$232$	0	0
$233$	−273542.	−0.330092	−0.165046	−	0.986286i	$-0.552777\pi$
$233$	−273542.	−0.330092	−0.165046	+	0.986286i	$0.552777\pi$
$234$	0	0
$235$	−2.31751e6	−2.73749
$236$	0	0
$237$	−773563.	−0.894591
$238$	0	0
$239$	−1.26051e6	−1.42742	−0.713712	−	0.700439i	$-0.752987\pi$
$239$	−1.26051e6	−1.42742	−0.713712	+	0.700439i	$0.752987\pi$
$240$	0	0
$241$	−837791.	−0.929165	−0.464583	−	0.885530i	$-0.653796\pi$
$241$	−837791.	−0.929165	−0.464583	+	0.885530i	$0.653796\pi$
$242$	0	0
$243$	1.31880e6	1.43272
$244$	0	0
$245$	−1.26906e6	−1.35073
$246$	0	0
$247$	1.77156e6	1.84762
$248$	0	0
$249$	764412.	0.781320
$250$	0	0
$251$	1.42635e6	1.42903	0.714517	−	0.699619i	$-0.246647\pi$
$251$	1.42635e6	1.42903	0.714517	+	0.699619i	$0.246647\pi$
$252$	0	0
$253$	52387.5	0.0514549
$254$	0	0
$255$	2.36857e6	2.28106
$256$	0	0
$257$	763758.	0.721312	0.360656	−	0.932699i	$-0.382553\pi$
$257$	763758.	0.721312	0.360656	+	0.932699i	$0.382553\pi$
$258$	0	0
$259$	162130.	0.150181
$260$	0	0
$261$	419064.	0.380785
$262$	0	0
$263$	571402.	0.509393	0.254696	−	0.967021i	$-0.418025\pi$
$263$	571402.	0.509393	0.254696	+	0.967021i	$0.418025\pi$
$264$	0	0
$265$	1.36296e6	1.19225
$266$	0	0
$267$	−498504.	−0.427947
$268$	0	0
$269$	−222030.	−0.187081	−0.0935407	−	0.995615i	$-0.529819\pi$
$269$	−222030.	−0.187081	−0.0935407	+	0.995615i	$0.529819\pi$
$270$	0	0
$271$	1.47266e6	1.21809	0.609043	−	0.793137i	$-0.291553\pi$
$271$	1.47266e6	1.21809	0.609043	+	0.793137i	$0.291553\pi$
$272$	0	0
$273$	327426.	0.265893
$274$	0	0
$275$	39266.0	0.0313102
$276$	0	0
$277$	483155.	0.378344	0.189172	−	0.981944i	$-0.439420\pi$
$277$	483155.	0.378344	0.189172	+	0.981944i	$0.439420\pi$
$278$	0	0
$279$	−251674.	−0.193566
$280$	0	0
$281$	1.92887e6	1.45726	0.728631	−	0.684907i	$-0.240157\pi$
$281$	1.92887e6	1.45726	0.728631	+	0.684907i	$0.240157\pi$
$282$	0	0
$283$	−1.90674e6	−1.41523	−0.707614	−	0.706599i	$-0.750228\pi$
$283$	−1.90674e6	−1.41523	−0.707614	+	0.706599i	$0.750228\pi$
$284$	0	0
$285$	−4.38174e6	−3.19547
$286$	0	0
$287$	23502.1	0.0168423
$288$	0	0
$289$	430945.	0.303513
$290$	0	0
$291$	2.55402e6	1.76804
$292$	0	0
$293$	2.19302e6	1.49236	0.746179	−	0.665746i	$-0.231887\pi$
$293$	2.19302e6	1.49236	0.746179	+	0.665746i	$0.231887\pi$
$294$	0	0
$295$	1.27099e6	0.850327
$296$	0	0
$297$	−5788.91	−0.00380808
$298$	0	0
$299$	2.70348e6	1.74882
$300$	0	0
$301$	316510.	0.201359
$302$	0	0
$303$	−3.07582e6	−1.92466
$304$	0	0
$305$	1.81789e6	1.11897
$306$	0	0
$307$	−735651.	−0.445478	−0.222739	−	0.974878i	$-0.571500\pi$
$307$	−735651.	−0.445478	−0.222739	+	0.974878i	$0.571500\pi$
$308$	0	0
$309$	−990931.	−0.590401
$310$	0	0
$311$	−2.03955e6	−1.19573	−0.597865	−	0.801597i	$-0.703984\pi$
$311$	−2.03955e6	−1.19573	−0.597865	+	0.801597i	$0.703984\pi$
$312$	0	0
$313$	646526.	0.373014	0.186507	−	0.982454i	$-0.440283\pi$
$313$	646526.	0.373014	0.186507	+	0.982454i	$0.440283\pi$
$314$	0	0
$315$	−420073.	−0.238533
$316$	0	0
$317$	−2.74682e6	−1.53526	−0.767631	−	0.640892i	$-0.778565\pi$
$317$	−2.74682e6	−1.53526	−0.767631	+	0.640892i	$0.778565\pi$
$318$	0	0
$319$	21826.6	0.0120091
$320$	0	0
$321$	2.26396e6	1.22633
$322$	0	0
$323$	−3.42389e6	−1.82605
$324$	0	0
$325$	2.02634e6	1.06415
$326$	0	0
$327$	−3.35133e6	−1.73320
$328$	0	0
$329$	619174.	0.315372
$330$	0	0
$331$	1.07651e6	0.540068	0.270034	−	0.962851i	$-0.412965\pi$
$331$	1.07651e6	0.540068	0.270034	+	0.962851i	$0.412965\pi$
$332$	0	0
$333$	−2.05106e6	−1.01360
$334$	0	0
$335$	−947250.	−0.461161
$336$	0	0
$337$	−1.39604e6	−0.669613	−0.334806	−	0.942287i	$-0.608671\pi$
$337$	−1.39604e6	−0.669613	−0.334806	+	0.942287i	$0.608671\pi$
$338$	0	0
$339$	−4.83949e6	−2.28718
$340$	0	0
$341$	−13108.2	−0.00610461
$342$	0	0
$343$	686986.	0.315292
$344$	0	0
$345$	−6.68674e6	−3.02459
$346$	0	0
$347$	−192447.	−0.0858002	−0.0429001	−	0.999079i	$-0.513660\pi$
$347$	−192447.	−0.0858002	−0.0429001	+	0.999079i	$0.513660\pi$
$348$	0	0
$349$	−1.64570e6	−0.723247	−0.361623	−	0.932324i	$-0.617777\pi$
$349$	−1.64570e6	−0.723247	−0.361623	+	0.932324i	$0.617777\pi$
$350$	0	0
$351$	−298739.	−0.129427
$352$	0	0
$353$	1.39838e6	0.597292	0.298646	−	0.954364i	$-0.403465\pi$
$353$	1.39838e6	0.597292	0.298646	+	0.954364i	$0.403465\pi$
$354$	0	0
$355$	3.88378e6	1.63562
$356$	0	0
$357$	−632816.	−0.262789
$358$	0	0
$359$	−157908.	−0.0646650	−0.0323325	−	0.999477i	$-0.510294\pi$
$359$	−157908.	−0.0646650	−0.0323325	+	0.999477i	$0.510294\pi$
$360$	0	0
$361$	3.85792e6	1.55806
$362$	0	0
$363$	3.61459e6	1.43977
$364$	0	0
$365$	4.89102e6	1.92162
$366$	0	0
$367$	1.67243e6	0.648161	0.324081	−	0.946029i	$-0.394945\pi$
$367$	1.67243e6	0.648161	0.324081	+	0.946029i	$0.394945\pi$
$368$	0	0
$369$	−297318.	−0.113672
$370$	0	0
$371$	−364144.	−0.137353
$372$	0	0
$373$	−2.68053e6	−0.997583	−0.498791	−	0.866722i	$-0.666223\pi$
$373$	−2.68053e6	−0.997583	−0.498791	+	0.866722i	$0.666223\pi$
$374$	0	0
$375$	428812.	0.157467
$376$	0	0
$377$	1.12637e6	0.408158
$378$	0	0
$379$	949913.	0.339692	0.169846	−	0.985471i	$-0.445673\pi$
$379$	949913.	0.339692	0.169846	+	0.985471i	$0.445673\pi$
$380$	0	0
$381$	−7.81005e6	−2.75640
$382$	0	0
$383$	512479.	0.178517	0.0892584	−	0.996009i	$-0.471550\pi$
$383$	512479.	0.178517	0.0892584	+	0.996009i	$0.471550\pi$
$384$	0	0
$385$	−21879.1	−0.00752278
$386$	0	0
$387$	−4.00407e6	−1.35902
$388$	0	0
$389$	2.75649e6	0.923597	0.461798	−	0.886985i	$-0.347204\pi$
$389$	2.75649e6	0.923597	0.461798	+	0.886985i	$0.347204\pi$
$390$	0	0
$391$	−5.22501e6	−1.72840
$392$	0	0
$393$	−286339.	−0.0935187
$394$	0	0
$395$	2.66752e6	0.860231
$396$	0	0
$397$	−4.00659e6	−1.27585	−0.637923	−	0.770100i	$-0.720206\pi$
$397$	−4.00659e6	−1.27585	−0.637923	+	0.770100i	$0.720206\pi$
$398$	0	0
$399$	1.17068e6	0.368133
$400$	0	0
$401$	756703.	0.234998	0.117499	−	0.993073i	$-0.462512\pi$
$401$	756703.	0.234998	0.117499	+	0.993073i	$0.462512\pi$
$402$	0	0
$403$	−676455.	−0.207480
$404$	0	0
$405$	−4.19206e6	−1.26996
$406$	0	0
$407$	−106828.	−0.0319667
$408$	0	0
$409$	4.13510e6	1.22230	0.611150	−	0.791515i	$-0.290707\pi$
$409$	4.13510e6	1.22230	0.611150	+	0.791515i	$0.290707\pi$
$410$	0	0
$411$	−1.22447e6	−0.357555
$412$	0	0
$413$	−339572.	−0.0979617
$414$	0	0
$415$	−2.63596e6	−0.751310
$416$	0	0
$417$	6.56459e6	1.84870
$418$	0	0
$419$	−309961.	−0.0862526	−0.0431263	−	0.999070i	$-0.513732\pi$
$419$	−309961.	−0.0862526	−0.0431263	+	0.999070i	$0.513732\pi$
$420$	0	0
$421$	986987.	0.271398	0.135699	−	0.990750i	$-0.456672\pi$
$421$	986987.	0.271398	0.135699	+	0.990750i	$0.456672\pi$
$422$	0	0
$423$	−7.83300e6	−2.12852
$424$	0	0
$425$	−3.91631e6	−1.05173
$426$	0	0
$427$	−485688.	−0.128910
$428$	0	0
$429$	−215741.	−0.0565966
$430$	0	0
$431$	−1.55460e6	−0.403111	−0.201556	−	0.979477i	$-0.564600\pi$
$431$	−1.55460e6	−0.403111	−0.201556	+	0.979477i	$0.564600\pi$
$432$	0	0
$433$	−6.73484e6	−1.72626	−0.863132	−	0.504978i	$-0.831501\pi$
$433$	−6.73484e6	−1.72626	−0.863132	+	0.504978i	$0.831501\pi$
$434$	0	0
$435$	−2.78595e6	−0.705910
$436$	0	0
$437$	9.66600e6	2.42127
$438$	0	0
$439$	−692966.	−0.171613	−0.0858065	−	0.996312i	$-0.527347\pi$
$439$	−692966.	−0.171613	−0.0858065	+	0.996312i	$0.527347\pi$
$440$	0	0
$441$	−4.28931e6	−1.05025
$442$	0	0
$443$	583148.	0.141179	0.0705894	−	0.997505i	$-0.477512\pi$
$443$	583148.	0.141179	0.0705894	+	0.997505i	$0.477512\pi$
$444$	0	0
$445$	1.71902e6	0.411510
$446$	0	0
$447$	4.61394e6	1.09220
$448$	0	0
$449$	−4.44718e6	−1.04104	−0.520522	−	0.853848i	$-0.674263\pi$
$449$	−4.44718e6	−1.04104	−0.520522	+	0.853848i	$0.674263\pi$
$450$	0	0
$451$	−15485.5	−0.00358496
$452$	0	0
$453$	−4.69599e6	−1.07518
$454$	0	0
$455$	−1.12908e6	−0.255680
$456$	0	0
$457$	−7.39409e6	−1.65613	−0.828065	−	0.560633i	$-0.810558\pi$
$457$	−7.39409e6	−1.65613	−0.828065	+	0.560633i	$0.810558\pi$
$458$	0	0
$459$	577373.	0.127916
$460$	0	0
$461$	420559.	0.0921668	0.0460834	−	0.998938i	$-0.485326\pi$
$461$	420559.	0.0921668	0.0460834	+	0.998938i	$0.485326\pi$
$462$	0	0
$463$	8.68419e6	1.88268	0.941341	−	0.337458i	$-0.109567\pi$
$463$	8.68419e6	1.88268	0.941341	+	0.337458i	$0.109567\pi$
$464$	0	0
$465$	1.67313e6	0.358838
$466$	0	0
$467$	−5.35288e6	−1.13578	−0.567892	−	0.823103i	$-0.692241\pi$
$467$	−5.35288e6	−1.13578	−0.567892	+	0.823103i	$0.692241\pi$
$468$	0	0
$469$	253079.	0.0531280
$470$	0	0
$471$	−9.31632e6	−1.93505
$472$	0	0
$473$	−208548.	−0.0428602
$474$	0	0
$475$	7.24497e6	1.47334
$476$	0	0
$477$	4.60668e6	0.927027
$478$	0	0
$479$	1.50802e6	0.300310	0.150155	−	0.988662i	$-0.452023\pi$
$479$	1.50802e6	0.300310	0.150155	+	0.988662i	$0.452023\pi$
$480$	0	0
$481$	−5.51288e6	−1.08647
$482$	0	0
$483$	1.78651e6	0.348447
$484$	0	0
$485$	−8.80716e6	−1.70013
$486$	0	0
$487$	3.00208e6	0.573588	0.286794	−	0.957992i	$-0.407410\pi$
$487$	3.00208e6	0.573588	0.286794	+	0.957992i	$0.407410\pi$
$488$	0	0
$489$	−1.66459e6	−0.314801
$490$	0	0
$491$	6.33984e6	1.18679	0.593396	−	0.804911i	$-0.297787\pi$
$491$	6.33984e6	1.18679	0.593396	+	0.804911i	$0.297787\pi$
$492$	0	0
$493$	−2.17694e6	−0.403393
$494$	0	0
$495$	276787.	0.0507729
$496$	0	0
$497$	−1.03764e6	−0.188432
$498$	0	0
$499$	5.20836e6	0.936375	0.468188	−	0.883629i	$-0.344907\pi$
$499$	5.20836e6	0.936375	0.468188	+	0.883629i	$0.344907\pi$
$500$	0	0
$501$	−7.98749e6	−1.42173
$502$	0	0
$503$	4.63376e6	0.816608	0.408304	−	0.912846i	$-0.366120\pi$
$503$	4.63376e6	0.816608	0.408304	+	0.912846i	$0.366120\pi$
$504$	0	0
$505$	1.06065e7	1.85073
$506$	0	0
$507$	−2.79058e6	−0.482142
$508$	0	0
$509$	1.02585e6	0.175506	0.0877529	−	0.996142i	$-0.472031\pi$
$509$	1.02585e6	0.175506	0.0877529	+	0.996142i	$0.472031\pi$
$510$	0	0
$511$	−1.30674e6	−0.221380
$512$	0	0
$513$	−1.06811e6	−0.179194
$514$	0	0
$515$	3.41708e6	0.567725
$516$	0	0
$517$	−407975.	−0.0671285
$518$	0	0
$519$	−1.14954e7	−1.87330
$520$	0	0
$521$	−461300.	−0.0744542	−0.0372271	−	0.999307i	$-0.511852\pi$
$521$	−461300.	−0.0744542	−0.0372271	+	0.999307i	$0.511852\pi$
$522$	0	0
$523$	−1.69073e6	−0.270284	−0.135142	−	0.990826i	$-0.543149\pi$
$523$	−1.69073e6	−0.270284	−0.135142	+	0.990826i	$0.543149\pi$
$524$	0	0
$525$	1.33904e6	0.212029
$526$	0	0
$527$	1.30738e6	0.205058
$528$	0	0
$529$	8.31443e6	1.29179
$530$	0	0
$531$	4.29582e6	0.661165
$532$	0	0
$533$	−799138.	−0.121844
$534$	0	0
$535$	−7.80694e6	−1.17922
$536$	0	0
$537$	7.33478e6	1.09762
$538$	0	0
$539$	−223405.	−0.0331224
$540$	0	0
$541$	−1.30072e6	−0.191069	−0.0955344	−	0.995426i	$-0.530456\pi$
$541$	−1.30072e6	−0.191069	−0.0955344	+	0.995426i	$0.530456\pi$
$542$	0	0
$543$	−1.18968e7	−1.73154
$544$	0	0
$545$	1.15566e7	1.66662
$546$	0	0
$547$	−8.08359e6	−1.15514	−0.577572	−	0.816340i	$-0.696000\pi$
$547$	−8.08359e6	−1.15514	−0.577572	+	0.816340i	$0.696000\pi$
$548$	0	0
$549$	6.14429e6	0.870044
$550$	0	0
$551$	4.02722e6	0.565101
$552$	0	0
$553$	−712686.	−0.0991027
$554$	0	0
$555$	1.36355e7	1.87905
$556$	0	0
$557$	3.20915e6	0.438280	0.219140	−	0.975693i	$-0.429675\pi$
$557$	3.20915e6	0.438280	0.219140	+	0.975693i	$0.429675\pi$
$558$	0	0
$559$	−1.07622e7	−1.45671
$560$	0	0
$561$	416963.	0.0559359
$562$	0	0
$563$	8.74751e6	1.16309	0.581545	−	0.813514i	$-0.302448\pi$
$563$	8.74751e6	1.16309	0.581545	+	0.813514i	$0.302448\pi$
$564$	0	0
$565$	1.66883e7	2.19933
$566$	0	0
$567$	1.12000e6	0.146305
$568$	0	0
$569$	−7.46467e6	−0.966562	−0.483281	−	0.875465i	$-0.660555\pi$
$569$	−7.46467e6	−0.966562	−0.483281	+	0.875465i	$0.660555\pi$
$570$	0	0
$571$	4.98994e6	0.640479	0.320240	−	0.947337i	$-0.396237\pi$
$571$	4.98994e6	0.640479	0.320240	+	0.947337i	$0.396237\pi$
$572$	0	0
$573$	−5.10470e6	−0.649507
$574$	0	0
$575$	1.10562e7	1.39455
$576$	0	0
$577$	7.68798e6	0.961331	0.480666	−	0.876904i	$-0.340395\pi$
$577$	7.68798e6	0.961331	0.480666	+	0.876904i	$0.340395\pi$
$578$	0	0
$579$	−1.85819e6	−0.230353
$580$	0	0
$581$	704255.	0.0865545
$582$	0	0
$583$	239935.	0.0292363
$584$	0	0
$585$	1.42837e7	1.72564
$586$	0	0
$587$	−1.26155e7	−1.51116	−0.755580	−	0.655057i	$-0.772645\pi$
$587$	−1.26155e7	−1.51116	−0.755580	+	0.655057i	$0.772645\pi$
$588$	0	0
$589$	−2.41859e6	−0.287260
$590$	0	0
$591$	5.52980e6	0.651240
$592$	0	0
$593$	7.73692e6	0.903507	0.451754	−	0.892143i	$-0.350799\pi$
$593$	7.73692e6	0.903507	0.451754	+	0.892143i	$0.350799\pi$
$594$	0	0
$595$	2.18218e6	0.252695
$596$	0	0
$597$	−1.71336e7	−1.96749
$598$	0	0
$599$	1.42113e7	1.61833	0.809163	−	0.587584i	$-0.199921\pi$
$599$	1.42113e7	1.61833	0.809163	+	0.587584i	$0.199921\pi$
$600$	0	0
$601$	−448124.	−0.0506071	−0.0253036	−	0.999680i	$-0.508055\pi$
$601$	−448124.	−0.0506071	−0.0253036	+	0.999680i	$0.508055\pi$
$602$	0	0
$603$	−3.20162e6	−0.358573
$604$	0	0
$605$	−1.24644e7	−1.38447
$606$	0	0
$607$	−4.64948e6	−0.512192	−0.256096	−	0.966651i	$-0.582436\pi$
$607$	−4.64948e6	−0.512192	−0.256096	+	0.966651i	$0.582436\pi$
$608$	0	0
$609$	744326.	0.0813243
$610$	0	0
$611$	−2.10537e7	−2.28153
$612$	0	0
$613$	−2.36416e6	−0.254112	−0.127056	−	0.991896i	$-0.540553\pi$
$613$	−2.36416e6	−0.254112	−0.127056	+	0.991896i	$0.540553\pi$
$614$	0	0
$615$	1.97657e6	0.210729
$616$	0	0
$617$	5.59685e6	0.591876	0.295938	−	0.955207i	$-0.404368\pi$
$617$	5.59685e6	0.591876	0.295938	+	0.955207i	$0.404368\pi$
$618$	0	0
$619$	1.58165e7	1.65914	0.829569	−	0.558404i	$-0.188586\pi$
$619$	1.58165e7	1.65914	0.829569	+	0.558404i	$0.188586\pi$
$620$	0	0
$621$	−1.62999e6	−0.169611
$622$	0	0
$623$	−459273.	−0.0474079
$624$	0	0
$625$	−1.04746e7	−1.07260
$626$	0	0
$627$	−771360.	−0.0783589
$628$	0	0
$629$	1.06547e7	1.07378
$630$	0	0
$631$	−8.25092e6	−0.824952	−0.412476	−	0.910968i	$-0.635336\pi$
$631$	−8.25092e6	−0.824952	−0.412476	+	0.910968i	$0.635336\pi$
$632$	0	0
$633$	−5.23624e6	−0.519409
$634$	0	0
$635$	2.69319e7	2.65053
$636$	0	0
$637$	−1.15289e7	−1.12574
$638$	0	0
$639$	1.31268e7	1.27177
$640$	0	0
$641$	1.83243e7	1.76150	0.880751	−	0.473579i	$-0.157038\pi$
$641$	1.83243e7	1.76150	0.880751	+	0.473579i	$0.157038\pi$
$642$	0	0
$643$	−1.37837e7	−1.31474	−0.657369	−	0.753569i	$-0.728331\pi$
$643$	−1.37837e7	−1.31474	−0.657369	+	0.753569i	$0.728331\pi$
$644$	0	0
$645$	2.66191e7	2.51938
$646$	0	0
$647$	5.84932e6	0.549344	0.274672	−	0.961538i	$-0.411431\pi$
$647$	5.84932e6	0.549344	0.274672	+	0.961538i	$0.411431\pi$
$648$	0	0
$649$	223744.	0.0208516
$650$	0	0
$651$	−447014.	−0.0413398
$652$	0	0
$653$	−1.67161e7	−1.53410	−0.767048	−	0.641589i	$-0.778275\pi$
$653$	−1.67161e7	−1.53410	−0.767048	+	0.641589i	$0.778275\pi$
$654$	0	0
$655$	987397.	0.0899267
$656$	0	0
$657$	1.65312e7	1.49414
$658$	0	0
$659$	1.31524e7	1.17975	0.589877	−	0.807493i	$-0.299176\pi$
$659$	1.31524e7	1.17975	0.589877	+	0.807493i	$0.299176\pi$
$660$	0	0
$661$	1.26307e7	1.12441	0.562205	−	0.826998i	$-0.309953\pi$
$661$	1.26307e7	1.12441	0.562205	+	0.826998i	$0.309953\pi$
$662$	0	0
$663$	2.15176e7	1.90112
$664$	0	0
$665$	−4.03691e6	−0.353993
$666$	0	0
$667$	6.14572e6	0.534883
$668$	0	0
$669$	−1.93547e7	−1.67194
$670$	0	0
$671$	320020.	0.0274392
$672$	0	0
$673$	−8.15038e6	−0.693649	−0.346825	−	0.937930i	$-0.612740\pi$
$673$	−8.15038e6	−0.693649	−0.346825	+	0.937930i	$0.612740\pi$
$674$	0	0
$675$	−1.22172e6	−0.103208
$676$	0	0
$677$	−6.87475e6	−0.576481	−0.288241	−	0.957558i	$-0.593070\pi$
$677$	−6.87475e6	−0.576481	−0.288241	+	0.957558i	$0.593070\pi$
$678$	0	0
$679$	2.35303e6	0.195863
$680$	0	0
$681$	1.87330e7	1.54789
$682$	0	0
$683$	−4.71691e6	−0.386906	−0.193453	−	0.981110i	$-0.561969\pi$
$683$	−4.71691e6	−0.386906	−0.193453	+	0.981110i	$0.561969\pi$
$684$	0	0
$685$	4.22241e6	0.343822
$686$	0	0
$687$	−8.14728e6	−0.658598
$688$	0	0
$689$	1.23820e7	0.993668
$690$	0	0
$691$	−1.74137e6	−0.138738	−0.0693692	−	0.997591i	$-0.522099\pi$
$691$	−1.74137e6	−0.138738	−0.0693692	+	0.997591i	$0.522099\pi$
$692$	0	0
$693$	−73949.6	−0.00584928
$694$	0	0
$695$	−2.26371e7	−1.77770
$696$	0	0
$697$	1.54449e6	0.120421
$698$	0	0
$699$	6.14642e6	0.475805
$700$	0	0
$701$	1.57418e7	1.20993	0.604964	−	0.796253i	$-0.293188\pi$
$701$	1.57418e7	1.20993	0.604964	+	0.796253i	$0.293188\pi$
$702$	0	0
$703$	−1.97107e7	−1.50423
$704$	0	0
$705$	5.20739e7	3.94591
$706$	0	0
$707$	−2.83376e6	−0.213214
$708$	0	0
$709$	−2.30768e7	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$709$	−2.30768e7	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$710$	0	0
$711$	9.01599e6	0.668866
$712$	0	0
$713$	−3.69089e6	−0.271899
$714$	0	0
$715$	743953.	0.0544228
$716$	0	0
$717$	2.83234e7	2.05754
$718$	0	0
$719$	7.67945e6	0.553997	0.276999	−	0.960870i	$-0.410660\pi$
$719$	7.67945e6	0.553997	0.276999	+	0.960870i	$0.410660\pi$
$720$	0	0
$721$	−912949.	−0.0654046
$722$	0	0
$723$	1.88249e7	1.33933
$724$	0	0
$725$	4.60641e6	0.325475
$726$	0	0
$727$	1.11931e7	0.785445	0.392722	−	0.919657i	$-0.371533\pi$
$727$	1.11931e7	0.785445	0.392722	+	0.919657i	$0.371533\pi$
$728$	0	0
$729$	−1.64861e7	−1.14894
$730$	0	0
$731$	2.08002e7	1.43970
$732$	0	0
$733$	−5.04784e6	−0.347013	−0.173506	−	0.984833i	$-0.555510\pi$
$733$	−5.04784e6	−0.347013	−0.173506	+	0.984833i	$0.555510\pi$
$734$	0	0
$735$	2.85154e7	1.94698
$736$	0	0
$737$	−166754.	−0.0113085
$738$	0	0
$739$	−6.20023e6	−0.417635	−0.208817	−	0.977955i	$-0.566961\pi$
$739$	−6.20023e6	−0.417635	−0.208817	+	0.977955i	$0.566961\pi$
$740$	0	0
$741$	−3.98064e7	−2.66322
$742$	0	0
$743$	−2.87166e6	−0.190836	−0.0954181	−	0.995437i	$-0.530419\pi$
$743$	−2.87166e6	−0.190836	−0.0954181	+	0.995437i	$0.530419\pi$
$744$	0	0
$745$	−1.59105e7	−1.05025
$746$	0	0
$747$	−8.90933e6	−0.584176
$748$	0	0
$749$	2.08579e6	0.135852
$750$	0	0
$751$	−2.88744e7	−1.86816	−0.934079	−	0.357066i	$-0.883777\pi$
$751$	−2.88744e7	−1.86816	−0.934079	+	0.357066i	$0.883777\pi$
$752$	0	0
$753$	−3.20497e7	−2.05985
$754$	0	0
$755$	1.61934e7	1.03388
$756$	0	0
$757$	−2.22743e6	−0.141275	−0.0706375	−	0.997502i	$-0.522503\pi$
$757$	−2.22743e6	−0.141275	−0.0706375	+	0.997502i	$0.522503\pi$
$758$	0	0
$759$	−1.17713e6	−0.0741687
$760$	0	0
$761$	−1.04675e7	−0.655214	−0.327607	−	0.944814i	$-0.606242\pi$
$761$	−1.04675e7	−0.655214	−0.327607	+	0.944814i	$0.606242\pi$
$762$	0	0
$763$	−3.08759e6	−0.192003
$764$	0	0
$765$	−2.76061e7	−1.70550
$766$	0	0
$767$	1.15464e7	0.708694
$768$	0	0
$769$	1.71329e7	1.04476	0.522378	−	0.852714i	$-0.325045\pi$
$769$	1.71329e7	1.04476	0.522378	+	0.852714i	$0.325045\pi$
$770$	0	0
$771$	−1.71614e7	−1.03972
$772$	0	0
$773$	1.01987e7	0.613898	0.306949	−	0.951726i	$-0.400692\pi$
$773$	1.01987e7	0.613898	0.306949	+	0.951726i	$0.400692\pi$
$774$	0	0
$775$	−2.76643e6	−0.165450
$776$	0	0
$777$	−3.64301e6	−0.216475
$778$	0	0
$779$	−2.85723e6	−0.168695
$780$	0	0
$781$	683699.	0.0401086
$782$	0	0
$783$	−679113.	−0.0395857
$784$	0	0
$785$	3.21260e7	1.86073
$786$	0	0
$787$	−1.98443e7	−1.14208	−0.571042	−	0.820921i	$-0.693461\pi$
$787$	−1.98443e7	−1.14208	−0.571042	+	0.820921i	$0.693461\pi$
$788$	0	0
$789$	−1.28392e7	−0.734255
$790$	0	0
$791$	−4.45864e6	−0.253373
$792$	0	0
$793$	1.65148e7	0.932588
$794$	0	0
$795$	−3.06253e7	−1.71855
$796$	0	0
$797$	−260118.	−0.0145052	−0.00725261	−	0.999974i	$-0.502309\pi$
$797$	−260118.	−0.0145052	−0.00725261	+	0.999974i	$0.502309\pi$
$798$	0	0
$799$	4.06905e7	2.25489
$800$	0	0
$801$	5.81013e6	0.319967
$802$	0	0
$803$	861013.	0.0471217
$804$	0	0
$805$	−6.16052e6	−0.335064
$806$	0	0
$807$	4.98894e6	0.269665
$808$	0	0
$809$	−6.37528e6	−0.342474	−0.171237	−	0.985230i	$-0.554776\pi$
$809$	−6.37528e6	−0.342474	−0.171237	+	0.985230i	$0.554776\pi$
$810$	0	0
$811$	−3.33348e7	−1.77970	−0.889848	−	0.456257i	$-0.849190\pi$
$811$	−3.33348e7	−1.77970	−0.889848	+	0.456257i	$0.849190\pi$
$812$	0	0
$813$	−3.30902e7	−1.75579
$814$	0	0
$815$	5.74011e6	0.302709
$816$	0	0
$817$	−3.84792e7	−2.01684
$818$	0	0
$819$	−3.81620e6	−0.198802
$820$	0	0
$821$	1.71741e7	0.889232	0.444616	−	0.895721i	$-0.353340\pi$
$821$	1.71741e7	0.889232	0.444616	+	0.895721i	$0.353340\pi$
$822$	0	0
$823$	2.41387e7	1.24227	0.621134	−	0.783705i	$-0.286672\pi$
$823$	2.41387e7	1.24227	0.621134	+	0.783705i	$0.286672\pi$
$824$	0	0
$825$	−882296.	−0.0451315
$826$	0	0
$827$	−1.58197e7	−0.804332	−0.402166	−	0.915567i	$-0.631743\pi$
$827$	−1.58197e7	−0.804332	−0.402166	+	0.915567i	$0.631743\pi$
$828$	0	0
$829$	−7.34033e6	−0.370962	−0.185481	−	0.982648i	$-0.559384\pi$
$829$	−7.34033e6	−0.370962	−0.185481	+	0.982648i	$0.559384\pi$
$830$	0	0
$831$	−1.08563e7	−0.545357
$832$	0	0
$833$	2.22819e7	1.11260
$834$	0	0
$835$	2.75437e7	1.36712
$836$	0	0
$837$	407849.	0.0201227
$838$	0	0
$839$	−3.72549e6	−0.182717	−0.0913584	−	0.995818i	$-0.529121\pi$
$839$	−3.72549e6	−0.182717	−0.0913584	+	0.995818i	$0.529121\pi$
$840$	0	0
$841$	−1.79506e7	−0.875164
$842$	0	0
$843$	−4.33412e7	−2.10054
$844$	0	0
$845$	9.62293e6	0.463624
$846$	0	0
$847$	3.33013e6	0.159497
$848$	0	0
$849$	4.28440e7	2.03996
$850$	0	0
$851$	−3.00795e7	−1.42379
$852$	0	0
$853$	2.22483e7	1.04695	0.523473	−	0.852042i	$-0.324636\pi$
$853$	2.22483e7	1.04695	0.523473	+	0.852042i	$0.324636\pi$
$854$	0	0
$855$	5.10698e7	2.38918
$856$	0	0
$857$	−3.64158e7	−1.69370	−0.846852	−	0.531828i	$-0.821505\pi$
$857$	−3.64158e7	−1.69370	−0.846852	+	0.531828i	$0.821505\pi$
$858$	0	0
$859$	9.85579e6	0.455731	0.227865	−	0.973693i	$-0.426825\pi$
$859$	9.85579e6	0.455731	0.227865	+	0.973693i	$0.426825\pi$
$860$	0	0
$861$	−528085.	−0.0242770
$862$	0	0
$863$	−2.80815e7	−1.28349	−0.641745	−	0.766918i	$-0.721789\pi$
$863$	−2.80815e7	−1.28349	−0.641745	+	0.766918i	$0.721789\pi$
$864$	0	0
$865$	3.96403e7	1.80134
$866$	0	0
$867$	−9.68321e6	−0.437493
$868$	0	0
$869$	469590.	0.0210945
$870$	0	0
$871$	−8.60540e6	−0.384349
$872$	0	0
$873$	−2.97674e7	−1.32192
$874$	0	0
$875$	395067.	0.0174442
$876$	0	0
$877$	−3.47813e6	−0.152703	−0.0763514	−	0.997081i	$-0.524327\pi$
$877$	−3.47813e6	−0.152703	−0.0763514	+	0.997081i	$0.524327\pi$
$878$	0	0
$879$	−4.92764e7	−2.15113
$880$	0	0
$881$	2.91173e7	1.26390	0.631948	−	0.775011i	$-0.282256\pi$
$881$	2.91173e7	1.26390	0.631948	+	0.775011i	$0.282256\pi$
$882$	0	0
$883$	1.51390e7	0.653424	0.326712	−	0.945124i	$-0.394059\pi$
$883$	1.51390e7	0.653424	0.326712	+	0.945124i	$0.394059\pi$
$884$	0	0
$885$	−2.85587e7	−1.22569
$886$	0	0
$887$	−2.42213e6	−0.103369	−0.0516843	−	0.998663i	$-0.516459\pi$
$887$	−2.42213e6	−0.103369	−0.0516843	+	0.998663i	$0.516459\pi$
$888$	0	0
$889$	−7.19543e6	−0.305353
$890$	0	0
$891$	−737969.	−0.0311418
$892$	0	0
$893$	−7.52753e7	−3.15881
$894$	0	0
$895$	−2.52929e7	−1.05546
$896$	0	0
$897$	−6.07464e7	−2.52081
$898$	0	0
$899$	−1.53776e6	−0.0634585
$900$	0	0
$901$	−2.39306e7	−0.982068
$902$	0	0
$903$	−7.11188e6	−0.290245
$904$	0	0
$905$	4.10245e7	1.66503
$906$	0	0
$907$	−3.02210e6	−0.121981	−0.0609903	−	0.998138i	$-0.519426\pi$
$907$	−3.02210e6	−0.121981	−0.0609903	+	0.998138i	$0.519426\pi$
$908$	0	0
$909$	3.58491e7	1.43903
$910$	0	0
$911$	5.58822e6	0.223089	0.111544	−	0.993759i	$-0.464420\pi$
$911$	5.58822e6	0.223089	0.111544	+	0.993759i	$0.464420\pi$
$912$	0	0
$913$	−464034.	−0.0184235
$914$	0	0
$915$	−4.08474e7	−1.61291
$916$	0	0
$917$	−263805.	−0.0103600
$918$	0	0
$919$	−6.16660e6	−0.240856	−0.120428	−	0.992722i	$-0.538427\pi$
$919$	−6.16660e6	−0.240856	−0.120428	+	0.992722i	$0.538427\pi$
$920$	0	0
$921$	1.65299e7	0.642126
$922$	0	0
$923$	3.52826e7	1.36319
$924$	0	0
$925$	−2.25455e7	−0.866374
$926$	0	0
$927$	1.15494e7	0.441430
$928$	0	0
$929$	1.72089e7	0.654203	0.327102	−	0.944989i	$-0.393928\pi$
$929$	1.72089e7	0.654203	0.327102	+	0.944989i	$0.393928\pi$
$930$	0	0
$931$	−4.12204e7	−1.55861
$932$	0	0
$933$	4.58280e7	1.72356
$934$	0	0
$935$	−1.43784e6	−0.0537874
$936$	0	0
$937$	−4.03184e7	−1.50022	−0.750110	−	0.661313i	$-0.769999\pi$
$937$	−4.03184e7	−1.50022	−0.750110	+	0.661313i	$0.769999\pi$
$938$	0	0
$939$	−1.45272e7	−0.537674
$940$	0	0
$941$	−2.26902e7	−0.835343	−0.417671	−	0.908598i	$-0.637154\pi$
$941$	−2.26902e7	−0.835343	−0.417671	+	0.908598i	$0.637154\pi$
$942$	0	0
$943$	−4.36027e6	−0.159674
$944$	0	0
$945$	680748.	0.0247974
$946$	0	0
$947$	1.86682e7	0.676439	0.338219	−	0.941067i	$-0.390175\pi$
$947$	1.86682e7	0.676439	0.338219	+	0.941067i	$0.390175\pi$
$948$	0	0
$949$	4.44330e7	1.60155
$950$	0	0
$951$	6.17203e7	2.21298
$952$	0	0
$953$	−1.87730e7	−0.669580	−0.334790	−	0.942293i	$-0.608665\pi$
$953$	−1.87730e7	−0.669580	−0.334790	+	0.942293i	$0.608665\pi$
$954$	0	0
$955$	1.76028e7	0.624560
$956$	0	0
$957$	−490437.	−0.0173103
$958$	0	0
$959$	−1.12811e6	−0.0396099
$960$	0	0
$961$	923521.	0.0322581
$962$	0	0
$963$	−2.63868e7	−0.916897
$964$	0	0
$965$	6.40770e6	0.221505
$966$	0	0
$967$	352616.	0.0121265	0.00606325	−	0.999982i	$-0.498070\pi$
$967$	352616.	0.0121265	0.00606325	+	0.999982i	$0.498070\pi$
$968$	0	0
$969$	7.69338e7	2.63213
$970$	0	0
$971$	2.36047e6	0.0803433	0.0401716	−	0.999193i	$-0.487210\pi$
$971$	2.36047e6	0.0803433	0.0401716	+	0.999193i	$0.487210\pi$
$972$	0	0
$973$	6.04798e6	0.204799
$974$	0	0
$975$	−4.55313e7	−1.53390
$976$	0	0
$977$	−2.51190e7	−0.841910	−0.420955	−	0.907081i	$-0.638305\pi$
$977$	−2.51190e7	−0.841910	−0.420955	+	0.907081i	$0.638305\pi$
$978$	0	0
$979$	302616.	0.0100910
$980$	0	0
$981$	3.90602e7	1.29587
$982$	0	0
$983$	5.09828e7	1.68283	0.841414	−	0.540391i	$-0.181724\pi$
$983$	5.09828e7	1.68283	0.841414	+	0.540391i	$0.181724\pi$
$984$	0	0
$985$	−1.90687e7	−0.626226
$986$	0	0
$987$	−1.39127e7	−0.454588
$988$	0	0
$989$	−5.87211e7	−1.90899
$990$	0	0
$991$	−3.39948e7	−1.09958	−0.549792	−	0.835301i	$-0.685293\pi$
$991$	−3.39948e7	−1.09958	−0.549792	+	0.835301i	$0.685293\pi$
$992$	0	0
$993$	−2.41889e7	−0.778472
$994$	0	0
$995$	5.90827e7	1.89192
$996$	0	0
$997$	4.13726e7	1.31818	0.659090	−	0.752064i	$-0.270941\pi$
$997$	4.13726e7	1.31818	0.659090	+	0.752064i	$0.270941\pi$
$998$	0	0
$999$	3.32383e6	0.105372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	496.6.a.h.1.2		8
4.3	odd	2		31.6.a.b.1.5	✓	8
12.11	even	2		279.6.a.f.1.4		8
20.19	odd	2		775.6.a.b.1.4		8
124.123	even	2		961.6.a.c.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.6.a.b.1.5	✓	8	4.3	odd	2
279.6.a.f.1.4		8	12.11	even	2
496.6.a.h.1.2		8	1.1	even	1	trivial
775.6.a.b.1.4		8	20.19	odd	2
961.6.a.c.1.5		8	124.123	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:	\( N \)	\(=\)	\( 496 = 2^{4} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	496.a (trivial)

\(f(q)\)	\(=\)	\(q-22.4697 q^{3} +77.4836 q^{5} -20.7014 q^{7} +261.888 q^{9} +13.6402 q^{11} +703.908 q^{13} -1741.03 q^{15} -1360.44 q^{17} +2516.75 q^{19} +465.155 q^{21} +3840.67 q^{23} +2878.70 q^{25} -424.401 q^{27} +1600.17 q^{29} -961.000 q^{31} -306.491 q^{33} -1604.02 q^{35} -7831.82 q^{37} -15816.6 q^{39} -1135.29 q^{41} -15289.3 q^{43} +20292.0 q^{45} -29909.8 q^{47} -16378.5 q^{49} +30568.7 q^{51} +17590.3 q^{53} +1056.89 q^{55} -56550.6 q^{57} +16403.3 q^{59} +23461.6 q^{61} -5421.45 q^{63} +54541.3 q^{65} -12225.2 q^{67} -86298.8 q^{69} +50123.9 q^{71} +63123.3 q^{73} -64683.6 q^{75} -282.371 q^{77} +34426.9 q^{79} -54102.5 q^{81} -34019.7 q^{83} -105412. q^{85} -35955.3 q^{87} +22185.6 q^{89} -14571.9 q^{91} +21593.4 q^{93} +195007. q^{95} -113665. q^{97} +3572.20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} + 128 q^{5} - 88 q^{7} + 1512 q^{9} - 574 q^{11} - 122 q^{13} + 524 q^{15} + 1932 q^{17} + 1796 q^{19} + 996 q^{21} + 4136 q^{23} + 3308 q^{25} + 2624 q^{27} + 5050 q^{29} - 7688 q^{31} + 27920 q^{33}+ \cdots + 180150 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 + 128 * q^5 - 88 * q^7 + 1512 * q^9 - 574 * q^11 - 122 * q^13 + 524 * q^15 + 1932 * q^17 + 1796 * q^19 + 996 * q^21 + 4136 * q^23 + 3308 * q^25 + 2624 * q^27 + 5050 * q^29 - 7688 * q^31 + 27920 * q^33 - 25700 * q^35 + 23674 * q^37 - 38652 * q^39 + 44828 * q^41 - 21058 * q^43 - 1344 * q^45 - 5348 * q^47 + 19356 * q^49 - 22300 * q^51 + 4926 * q^53 + 56892 * q^55 + 22072 * q^57 - 16944 * q^59 - 73682 * q^61 + 138784 * q^63 + 63316 * q^65 + 134768 * q^67 + 3992 * q^69 - 123724 * q^71 + 70792 * q^73 + 302902 * q^75 + 107932 * q^77 + 26036 * q^79 + 82528 * q^81 - 21882 * q^83 - 130464 * q^85 + 440244 * q^87 + 164392 * q^89 + 60028 * q^91 - 1922 * q^93 + 7404 * q^95 - 131948 * q^97 + 180150 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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