LMFDB - Embedded newform 425.6.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,6,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.1631234205$
Analytic rank:	$1$
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} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.99434$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.99434 q^{2} -7.11254 q^{3} -16.0453 q^{4} +28.4099 q^{6} +173.307 q^{7} +191.909 q^{8} -192.412 q^{9} -87.6841 q^{11} +114.123 q^{12} -230.405 q^{13} -692.246 q^{14} -253.100 q^{16} -289.000 q^{17} +768.557 q^{18} -1762.14 q^{19} -1232.65 q^{21} +350.240 q^{22} +1157.50 q^{23} -1364.96 q^{24} +920.314 q^{26} +3096.89 q^{27} -2780.76 q^{28} +987.385 q^{29} +1687.64 q^{31} -5130.12 q^{32} +623.657 q^{33} +1154.36 q^{34} +3087.30 q^{36} +13054.0 q^{37} +7038.59 q^{38} +1638.76 q^{39} +10518.3 q^{41} +4923.63 q^{42} +8230.48 q^{43} +1406.92 q^{44} -4623.45 q^{46} -3626.35 q^{47} +1800.19 q^{48} +13228.3 q^{49} +2055.53 q^{51} +3696.91 q^{52} -23579.8 q^{53} -12370.0 q^{54} +33259.2 q^{56} +12533.3 q^{57} -3943.95 q^{58} +24480.0 q^{59} -12032.1 q^{61} -6740.99 q^{62} -33346.3 q^{63} +28590.6 q^{64} -2491.10 q^{66} +37268.9 q^{67} +4637.08 q^{68} -8232.78 q^{69} -20741.1 q^{71} -36925.5 q^{72} -65497.7 q^{73} -52142.1 q^{74} +28274.1 q^{76} -15196.3 q^{77} -6545.78 q^{78} -59371.4 q^{79} +24729.3 q^{81} -42013.8 q^{82} -15877.2 q^{83} +19778.3 q^{84} -32875.3 q^{86} -7022.82 q^{87} -16827.4 q^{88} -37021.7 q^{89} -39930.8 q^{91} -18572.4 q^{92} -12003.4 q^{93} +14484.9 q^{94} +36488.2 q^{96} +46409.1 q^{97} -52838.3 q^{98} +16871.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) $ 5 * q + 7 * q^2 + 36 * q^3 + 43 * q^4 - 105 * q^6 + 204 * q^7 + 63 * q^8 + 531 * q^9 - 792 * q^11 - 785 * q^12 - 88 * q^13 + 860 * q^14 - 2365 * q^16 - 1445 * q^17 + 2052 * q^18 - 5160 * q^19 - 6428 * q^21 + 3058 * q^22 + 6140 * q^23 - 11751 * q^24 - 9643 * q^26 + 13380 * q^27 + 4568 * q^28 - 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 13072 * q^33 - 2023 * q^34 + 2048 * q^36 - 9834 * q^37 - 10625 * q^38 + 10108 * q^39 - 5302 * q^41 - 16222 * q^42 + 4916 * q^43 - 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 45557 * q^48 - 26483 * q^49 - 10404 * q^51 + 11525 * q^52 - 56492 * q^53 + 64635 * q^54 + 73228 * q^56 - 67554 * q^57 - 54213 * q^58 - 96200 * q^59 - 7284 * q^61 - 122335 * q^62 - 64108 * q^63 + 28483 * q^64 + 115232 * q^66 + 60308 * q^67 - 12427 * q^68 - 89076 * q^69 - 25304 * q^71 - 118722 * q^72 - 163328 * q^73 - 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 134895 * q^78 - 50940 * q^79 + 92445 * q^81 - 191390 * q^82 + 179060 * q^83 - 128054 * q^84 + 87086 * q^86 - 55764 * q^87 - 167634 * q^88 - 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 202558 * q^93 + 100515 * q^94 + 428113 * q^96 + 106076 * q^97 - 49329 * q^98 - 535112 * 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 $

$2$	−3.99434	−0.706106	−0.353053	−	0.935603i	$-0.614856\pi$
$2$	−3.99434	−0.706106	−0.353053	+	0.935603i	$0.614856\pi$
$3$	−7.11254	−0.456270	−0.228135	−	0.973630i	$-0.573263\pi$
$3$	−7.11254	−0.456270	−0.228135	+	0.973630i	$0.573263\pi$
$4$	−16.0453	−0.501415
$5$	0	0
$5$	0	0
$6$	28.4099	0.322175
$7$	173.307	1.33681	0.668407	−	0.743796i	$-0.266977\pi$
$7$	173.307	1.33681	0.668407	+	0.743796i	$0.266977\pi$
$8$	191.909	1.06016
$9$	−192.412	−0.791818
$10$	0	0
$11$	−87.6841	−0.218494	−0.109247	−	0.994015i	$-0.534844\pi$
$11$	−87.6841	−0.218494	−0.109247	+	0.994015i	$0.534844\pi$
$12$	114.123	0.228781
$13$	−230.405	−0.378123	−0.189062	−	0.981965i	$-0.560545\pi$
$13$	−230.405	−0.378123	−0.189062	+	0.981965i	$0.560545\pi$
$14$	−692.246	−0.943932
$15$	0	0
$16$	−253.100	−0.247168
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	768.557	0.559107
$19$	−1762.14	−1.11984	−0.559921	−	0.828546i	$-0.689169\pi$
$19$	−1762.14	−1.11984	−0.559921	+	0.828546i	$0.689169\pi$
$20$	0	0
$21$	−1232.65	−0.609948
$22$	350.240	0.154280
$23$	1157.50	0.456249	0.228124	−	0.973632i	$-0.426741\pi$
$23$	1157.50	0.456249	0.228124	+	0.973632i	$0.426741\pi$
$24$	−1364.96	−0.483718
$25$	0	0
$26$	920.314	0.266995
$27$	3096.89	0.817553
$28$	−2780.76	−0.670298
$29$	987.385	0.218017	0.109009	−	0.994041i	$-0.465232\pi$
$29$	987.385	0.218017	0.109009	+	0.994041i	$0.465232\pi$
$30$	0	0
$31$	1687.64	0.315409	0.157705	−	0.987486i	$-0.449591\pi$
$31$	1687.64	0.315409	0.157705	+	0.987486i	$0.449591\pi$
$32$	−5130.12	−0.885631
$33$	623.657	0.0996922
$34$	1154.36	0.171256
$35$	0	0
$36$	3087.30	0.397029
$37$	13054.0	1.56762	0.783808	−	0.621003i	$-0.213275\pi$
$37$	13054.0	1.56762	0.783808	+	0.621003i	$0.213275\pi$
$38$	7038.59	0.790727
$39$	1638.76	0.172526
$40$	0	0
$41$	10518.3	0.977209	0.488605	−	0.872505i	$-0.337506\pi$
$41$	10518.3	0.977209	0.488605	+	0.872505i	$0.337506\pi$
$42$	4923.63	0.430688
$43$	8230.48	0.678819	0.339410	−	0.940639i	$-0.389773\pi$
$43$	8230.48	0.678819	0.339410	+	0.940639i	$0.389773\pi$
$44$	1406.92	0.109556
$45$	0	0
$46$	−4623.45	−0.322160
$47$	−3626.35	−0.239456	−0.119728	−	0.992807i	$-0.538202\pi$
$47$	−3626.35	−0.239456	−0.119728	+	0.992807i	$0.538202\pi$
$48$	1800.19	0.112775
$49$	13228.3	0.787071
$50$	0	0
$51$	2055.53	0.110662
$52$	3696.91	0.189597
$53$	−23579.8	−1.15306	−0.576529	−	0.817077i	$-0.695593\pi$
$53$	−23579.8	−1.15306	−0.576529	+	0.817077i	$0.695593\pi$
$54$	−12370.0	−0.577278
$55$	0	0
$56$	33259.2	1.41723
$57$	12533.3	0.510951
$58$	−3943.95	−0.153943
$59$	24480.0	0.915550	0.457775	−	0.889068i	$-0.348646\pi$
$59$	24480.0	0.915550	0.457775	+	0.889068i	$0.348646\pi$
$60$	0	0
$61$	−12032.1	−0.414015	−0.207007	−	0.978339i	$-0.566372\pi$
$61$	−12032.1	−0.414015	−0.207007	+	0.978339i	$0.566372\pi$
$62$	−6740.99	−0.222712
$63$	−33346.3	−1.05851
$64$	28590.6	0.872517
$65$	0	0
$66$	−2491.10	−0.0703932
$67$	37268.9	1.01428	0.507142	−	0.861862i	$-0.330702\pi$
$67$	37268.9	1.01428	0.507142	+	0.861862i	$0.330702\pi$
$68$	4637.08	0.121611
$69$	−8232.78	−0.208173
$70$	0	0
$71$	−20741.1	−0.488298	−0.244149	−	0.969738i	$-0.578509\pi$
$71$	−20741.1	−0.488298	−0.244149	+	0.969738i	$0.578509\pi$
$72$	−36925.5	−0.839451
$73$	−65497.7	−1.43853	−0.719265	−	0.694735i	$-0.755521\pi$
$73$	−65497.7	−1.43853	−0.719265	+	0.694735i	$0.755521\pi$
$74$	−52142.1	−1.10690
$75$	0	0
$76$	28274.1	0.561506
$77$	−15196.3	−0.292086
$78$	−6545.78	−0.121822
$79$	−59371.4	−1.07031	−0.535155	−	0.844754i	$-0.679747\pi$
$79$	−59371.4	−1.07031	−0.535155	+	0.844754i	$0.679747\pi$
$80$	0	0
$81$	24729.3	0.418793
$82$	−42013.8	−0.690013
$83$	−15877.2	−0.252975	−0.126488	−	0.991968i	$-0.540370\pi$
$83$	−15877.2	−0.252975	−0.126488	+	0.991968i	$0.540370\pi$
$84$	19778.3	0.305837
$85$	0	0
$86$	−32875.3	−0.479318
$87$	−7022.82	−0.0994748
$88$	−16827.4	−0.231638
$89$	−37021.7	−0.495429	−0.247714	−	0.968833i	$-0.579679\pi$
$89$	−37021.7	−0.495429	−0.247714	+	0.968833i	$0.579679\pi$
$90$	0	0
$91$	−39930.8	−0.505480
$92$	−18572.4	−0.228770
$93$	−12003.4	−0.143912
$94$	14484.9	0.169081
$95$	0	0
$96$	36488.2	0.404087
$97$	46409.1	0.500811	0.250405	−	0.968141i	$-0.419436\pi$
$97$	46409.1	0.500811	0.250405	+	0.968141i	$0.419436\pi$
$98$	−52838.3	−0.555755
$99$	16871.5	0.173007
$100$	0	0
$101$	6117.21	0.0596691	0.0298346	−	0.999555i	$-0.490502\pi$
$101$	6117.21	0.0596691	0.0298346	+	0.999555i	$0.490502\pi$
$102$	−8210.46	−0.0781389
$103$	−8455.69	−0.0785337	−0.0392668	−	0.999229i	$-0.512502\pi$
$103$	−8455.69	−0.0785337	−0.0392668	+	0.999229i	$0.512502\pi$
$104$	−44216.8	−0.400870
$105$	0	0
$106$	94185.8	0.814180
$107$	68521.4	0.578584	0.289292	−	0.957241i	$-0.406580\pi$
$107$	68521.4	0.578584	0.289292	+	0.957241i	$0.406580\pi$
$108$	−49690.4	−0.409933
$109$	−217169.	−1.75078	−0.875389	−	0.483419i	$-0.839395\pi$
$109$	−217169.	−1.75078	−0.875389	+	0.483419i	$0.839395\pi$
$110$	0	0
$111$	−92847.2	−0.715256
$112$	−43864.1	−0.330418
$113$	122902.	0.905449	0.452725	−	0.891650i	$-0.350452\pi$
$113$	122902.	0.905449	0.452725	+	0.891650i	$0.350452\pi$
$114$	−50062.3	−0.360785
$115$	0	0
$116$	−15842.9	−0.109317
$117$	44332.6	0.299405
$118$	−97781.5	−0.646475
$119$	−50085.7	−0.324225
$120$	0	0
$121$	−153362.	−0.952260
$122$	48060.1	0.292338
$123$	−74812.2	−0.445871
$124$	−27078.6	−0.158151
$125$	0	0
$126$	133196.	0.747422
$127$	315079.	1.73345	0.866723	−	0.498789i	$-0.166222\pi$
$127$	315079.	1.73345	0.866723	+	0.498789i	$0.166222\pi$
$128$	49963.2	0.269541
$129$	−58539.7	−0.309725
$130$	0	0
$131$	101214.	0.515301	0.257651	−	0.966238i	$-0.417052\pi$
$131$	101214.	0.515301	0.257651	+	0.966238i	$0.417052\pi$
$132$	−10006.8	−0.0499871
$133$	−305392.	−1.49702
$134$	−148865.	−0.716192
$135$	0	0
$136$	−55461.7	−0.257126
$137$	−16395.5	−0.0746317	−0.0373158	−	0.999304i	$-0.511881\pi$
$137$	−16395.5	−0.0746317	−0.0373158	+	0.999304i	$0.511881\pi$
$138$	32884.5	0.146992
$139$	−74588.7	−0.327443	−0.163722	−	0.986507i	$-0.552350\pi$
$139$	−74588.7	−0.327443	−0.163722	+	0.986507i	$0.552350\pi$
$140$	0	0
$141$	25792.6	0.109256
$142$	82846.8	0.344790
$143$	20202.8	0.0826176
$144$	48699.5	0.195712
$145$	0	0
$146$	261620.	1.01575
$147$	−94086.9	−0.359117
$148$	−209455.	−0.786026
$149$	−520566.	−1.92092	−0.960462	−	0.278411i	$-0.910192\pi$
$149$	−520566.	−1.92092	−0.960462	+	0.278411i	$0.910192\pi$
$150$	0	0
$151$	118996.	0.424709	0.212355	−	0.977193i	$-0.431887\pi$
$151$	118996.	0.424709	0.212355	+	0.977193i	$0.431887\pi$
$152$	−338171.	−1.18721
$153$	55607.0	0.192044
$154$	60699.0	0.206243
$155$	0	0
$156$	−26294.4	−0.0865072
$157$	−443199.	−1.43499	−0.717496	−	0.696563i	$-0.754712\pi$
$157$	−443199.	−1.43499	−0.717496	+	0.696563i	$0.754712\pi$
$158$	237149.	0.755752
$159$	167713.	0.526106
$160$	0	0
$161$	200603.	0.609920
$162$	−98777.2	−0.295712
$163$	−477473.	−1.40760	−0.703801	−	0.710397i	$-0.748515\pi$
$163$	−477473.	−1.40760	−0.703801	+	0.710397i	$0.748515\pi$
$164$	−168770.	−0.489987
$165$	0	0
$166$	63418.8	0.178627
$167$	−571338.	−1.58527	−0.792633	−	0.609699i	$-0.791290\pi$
$167$	−571338.	−1.58527	−0.792633	+	0.609699i	$0.791290\pi$
$168$	−236557.	−0.646641
$169$	−318207.	−0.857023
$170$	0	0
$171$	339057.	0.886711
$172$	−132060.	−0.340370
$173$	−139939.	−0.355486	−0.177743	−	0.984077i	$-0.556880\pi$
$173$	−139939.	−0.355486	−0.177743	+	0.984077i	$0.556880\pi$
$174$	28051.5	0.0702397
$175$	0	0
$176$	22192.9	0.0540048
$177$	−174115.	−0.417738
$178$	147877.	0.349825
$179$	−519689.	−1.21230	−0.606151	−	0.795349i	$-0.707287\pi$
$179$	−519689.	−1.21230	−0.606151	+	0.795349i	$0.707287\pi$
$180$	0	0
$181$	−403570.	−0.915635	−0.457817	−	0.889046i	$-0.651369\pi$
$181$	−403570.	−0.915635	−0.457817	+	0.889046i	$0.651369\pi$
$182$	159497.	0.356922
$183$	85578.6	0.188902
$184$	222135.	0.483696
$185$	0	0
$186$	47945.6	0.101617
$187$	25340.7	0.0529926
$188$	58185.8	0.120067
$189$	536712.	1.09292
$190$	0	0
$191$	−241070.	−0.478146	−0.239073	−	0.971002i	$-0.576844\pi$
$191$	−241070.	−0.478146	−0.239073	+	0.971002i	$0.576844\pi$
$192$	−203352.	−0.398103
$193$	394284.	0.761932	0.380966	−	0.924589i	$-0.375591\pi$
$193$	394284.	0.761932	0.380966	+	0.924589i	$0.375591\pi$
$194$	−185373.	−0.353625
$195$	0	0
$196$	−212252.	−0.394649
$197$	−291003.	−0.534234	−0.267117	−	0.963664i	$-0.586071\pi$
$197$	−291003.	−0.534234	−0.267117	+	0.963664i	$0.586071\pi$
$198$	−67390.3	−0.122161
$199$	−442108.	−0.791400	−0.395700	−	0.918380i	$-0.629498\pi$
$199$	−442108.	−0.791400	−0.395700	+	0.918380i	$0.629498\pi$
$200$	0	0
$201$	−265077.	−0.462787
$202$	−24434.2	−0.0421327
$203$	171121.	0.291449
$204$	−32981.5	−0.0554874
$205$	0	0
$206$	33774.9	0.0554531
$207$	−222717.	−0.361266
$208$	58315.5	0.0934600
$209$	154512.	0.244679
$210$	0	0
$211$	832091.	1.28666	0.643331	−	0.765588i	$-0.277552\pi$
$211$	832091.	1.28666	0.643331	+	0.765588i	$0.277552\pi$
$212$	378345.	0.578160
$213$	147522.	0.222796
$214$	−273697.	−0.408542
$215$	0	0
$216$	594320.	0.866735
$217$	292479.	0.421644
$218$	867445.	1.23623
$219$	465856.	0.656358
$220$	0	0
$221$	66587.0	0.0917083
$222$	370863.	0.505046
$223$	−98752.3	−0.132980	−0.0664898	−	0.997787i	$-0.521180\pi$
$223$	−98752.3	−0.132980	−0.0664898	+	0.997787i	$0.521180\pi$
$224$	−889085.	−1.18392
$225$	0	0
$226$	−490913.	−0.639343
$227$	1.33977e6	1.72570	0.862851	−	0.505458i	$-0.168676\pi$
$227$	1.33977e6	1.72570	0.862851	+	0.505458i	$0.168676\pi$
$228$	−201101.	−0.256198
$229$	1.44996e6	1.82712	0.913558	−	0.406709i	$-0.133324\pi$
$229$	1.44996e6	1.82712	0.913558	+	0.406709i	$0.133324\pi$
$230$	0	0
$231$	108084.	0.133270
$232$	189488.	0.231133
$233$	−977654.	−1.17976	−0.589882	−	0.807489i	$-0.700826\pi$
$233$	−977654.	−1.17976	−0.589882	+	0.807489i	$0.700826\pi$
$234$	−177079.	−0.211411
$235$	0	0
$236$	−392789.	−0.459071
$237$	422282.	0.488351
$238$	200059.	0.228937
$239$	−11746.4	−0.0133017	−0.00665087	−	0.999978i	$-0.502117\pi$
$239$	−11746.4	−0.0133017	−0.00665087	+	0.999978i	$0.502117\pi$
$240$	0	0
$241$	−61628.4	−0.0683500	−0.0341750	−	0.999416i	$-0.510880\pi$
$241$	−61628.4	−0.0683500	−0.0341750	+	0.999416i	$0.510880\pi$
$242$	612581.	0.672396
$243$	−928431.	−1.00864
$244$	193058.	0.207593
$245$	0	0
$246$	298825.	0.314832
$247$	406006.	0.423438
$248$	323873.	0.334384
$249$	112927.	0.115425
$250$	0	0
$251$	−1.60993e6	−1.61296	−0.806479	−	0.591262i	$-0.798630\pi$
$251$	−1.60993e6	−1.61296	−0.806479	+	0.591262i	$0.798630\pi$
$252$	535050.	0.530754
$253$	−101494.	−0.0996876
$254$	−1.25853e6	−1.22400
$255$	0	0
$256$	−1.11447e6	−1.06284
$257$	−1.20312e6	−1.13626	−0.568128	−	0.822940i	$-0.692332\pi$
$257$	−1.20312e6	−1.13626	−0.568128	+	0.822940i	$0.692332\pi$
$258$	233827.	0.218698
$259$	2.26235e6	2.09561
$260$	0	0
$261$	−189984.	−0.172630
$262$	−404282.	−0.363857
$263$	1.43642e6	1.28054	0.640269	−	0.768151i	$-0.278823\pi$
$263$	1.43642e6	1.28054	0.640269	+	0.768151i	$0.278823\pi$
$264$	119685.	0.105689
$265$	0	0
$266$	1.21984e6	1.05706
$267$	263318.	0.226049
$268$	−597990.	−0.508577
$269$	2.18824e6	1.84380	0.921900	−	0.387429i	$-0.126637\pi$
$269$	2.18824e6	1.84380	0.921900	+	0.387429i	$0.126637\pi$
$270$	0	0
$271$	−796919.	−0.659160	−0.329580	−	0.944128i	$-0.606907\pi$
$271$	−796919.	−0.659160	−0.329580	+	0.944128i	$0.606907\pi$
$272$	73146.0	0.0599471
$273$	284009.	0.230635
$274$	65489.1	0.0526978
$275$	0	0
$276$	132097.	0.104381
$277$	−2.39021e6	−1.87170	−0.935850	−	0.352399i	$-0.885366\pi$
$277$	−2.39021e6	−1.87170	−0.935850	+	0.352399i	$0.885366\pi$
$278$	297932.	0.231210
$279$	−324721.	−0.249747
$280$	0	0
$281$	−2.37607e6	−1.79512	−0.897561	−	0.440890i	$-0.854663\pi$
$281$	−2.37607e6	−1.79512	−0.897561	+	0.440890i	$0.854663\pi$
$282$	−103024.	−0.0771466
$283$	−1.62108e6	−1.20320	−0.601600	−	0.798797i	$-0.705470\pi$
$283$	−1.62108e6	−1.20320	−0.601600	+	0.798797i	$0.705470\pi$
$284$	332796.	0.244840
$285$	0	0
$286$	−80697.0	−0.0583367
$287$	1.82290e6	1.30635
$288$	987095.	0.701258
$289$	83521.0	0.0588235
$290$	0	0
$291$	−330087.	−0.228505
$292$	1.05093e6	0.721301
$293$	390563.	0.265780	0.132890	−	0.991131i	$-0.457574\pi$
$293$	390563.	0.265780	0.132890	+	0.991131i	$0.457574\pi$
$294$	375815.	0.253574
$295$	0	0
$296$	2.50518e6	1.66192
$297$	−271548.	−0.178630
$298$	2.07932e6	1.35638
$299$	−266694.	−0.172518
$300$	0	0
$301$	1.42640e6	0.907455
$302$	−475312.	−0.299890
$303$	−43508.9	−0.0272252
$304$	445999.	0.276790
$305$	0	0
$306$	−222113.	−0.135603
$307$	2.54435e6	1.54074	0.770371	−	0.637595i	$-0.220071\pi$
$307$	2.54435e6	1.54074	0.770371	+	0.637595i	$0.220071\pi$
$308$	243828.	0.146456
$309$	60141.5	0.0358326
$310$	0	0
$311$	−227988.	−0.133663	−0.0668315	−	0.997764i	$-0.521289\pi$
$311$	−227988.	−0.133663	−0.0668315	+	0.997764i	$0.521289\pi$
$312$	314494.	0.182905
$313$	−208383.	−0.120227	−0.0601133	−	0.998192i	$-0.519146\pi$
$313$	−208383.	−0.120227	−0.0601133	+	0.998192i	$0.519146\pi$
$314$	1.77029e6	1.01326
$315$	0	0
$316$	952631.	0.536670
$317$	−643310.	−0.359560	−0.179780	−	0.983707i	$-0.557539\pi$
$317$	−643310.	−0.359560	−0.179780	+	0.983707i	$0.557539\pi$
$318$	−669901.	−0.371486
$319$	−86578.0	−0.0476355
$320$	0	0
$321$	−487361.	−0.263991
$322$	−801276.	−0.430668
$323$	509259.	0.271602
$324$	−396789.	−0.209989
$325$	0	0
$326$	1.90719e6	0.993916
$327$	1.54462e6	0.798827
$328$	2.01856e6	1.03600
$329$	−628472.	−0.320108
$330$	0	0
$331$	−727410.	−0.364930	−0.182465	−	0.983212i	$-0.558408\pi$
$331$	−727410.	−0.364930	−0.182465	+	0.983212i	$0.558408\pi$
$332$	254754.	0.126846
$333$	−2.51174e6	−1.24127
$334$	2.28212e6	1.11936
$335$	0	0
$336$	311985.	0.150760
$337$	−3.44274e6	−1.65131	−0.825655	−	0.564175i	$-0.809194\pi$
$337$	−3.44274e6	−1.65131	−0.825655	+	0.564175i	$0.809194\pi$
$338$	1.27102e6	0.605149
$339$	−874149.	−0.413129
$340$	0	0
$341$	−147979.	−0.0689150
$342$	−1.35431e6	−0.626112
$343$	−620214.	−0.284647
$344$	1.57950e6	0.719655
$345$	0	0
$346$	558962.	0.251011
$347$	2.39988e6	1.06995	0.534977	−	0.844867i	$-0.320320\pi$
$347$	2.39988e6	1.06995	0.534977	+	0.844867i	$0.320320\pi$
$348$	112683.	0.0498782
$349$	2.73418e6	1.20161	0.600805	−	0.799395i	$-0.294847\pi$
$349$	2.73418e6	1.20161	0.600805	+	0.799395i	$0.294847\pi$
$350$	0	0
$351$	−713537.	−0.309136
$352$	449830.	0.193505
$353$	−565662.	−0.241613	−0.120806	−	0.992676i	$-0.538548\pi$
$353$	−565662.	−0.241613	−0.120806	+	0.992676i	$0.538548\pi$
$354$	695476.	0.294967
$355$	0	0
$356$	594023.	0.248415
$357$	356237.	0.147934
$358$	2.07581e6	0.856013
$359$	4.64822e6	1.90349	0.951744	−	0.306892i	$-0.0992891\pi$
$359$	4.64822e6	1.90349	0.951744	+	0.306892i	$0.0992891\pi$
$360$	0	0
$361$	629048.	0.254048
$362$	1.61199e6	0.646535
$363$	1.09080e6	0.434488
$364$	640700.	0.253455
$365$	0	0
$366$	−341830.	−0.133385
$367$	−2.72804e6	−1.05727	−0.528635	−	0.848849i	$-0.677296\pi$
$367$	−2.72804e6	−1.05727	−0.528635	+	0.848849i	$0.677296\pi$
$368$	−292964.	−0.112770
$369$	−2.02385e6	−0.773772
$370$	0	0
$371$	−4.08655e6	−1.54142
$372$	192598.	0.0721595
$373$	1.22410e6	0.455560	0.227780	−	0.973713i	$-0.426853\pi$
$373$	1.22410e6	0.455560	0.227780	+	0.973713i	$0.426853\pi$
$374$	−101219.	−0.0374183
$375$	0	0
$376$	−695930.	−0.253861
$377$	−227498.	−0.0824375
$378$	−2.14381e6	−0.771714
$379$	−2.84850e6	−1.01864	−0.509318	−	0.860579i	$-0.670102\pi$
$379$	−2.84850e6	−1.01864	−0.509318	+	0.860579i	$0.670102\pi$
$380$	0	0
$381$	−2.24102e6	−0.790920
$382$	962916.	0.337621
$383$	2.70280e6	0.941493	0.470747	−	0.882268i	$-0.343985\pi$
$383$	2.70280e6	0.941493	0.470747	+	0.882268i	$0.343985\pi$
$384$	−355366.	−0.122984
$385$	0	0
$386$	−1.57490e6	−0.538005
$387$	−1.58364e6	−0.537501
$388$	−744646.	−0.251114
$389$	−3.25126e6	−1.08937	−0.544687	−	0.838639i	$-0.683352\pi$
$389$	−3.25126e6	−1.08937	−0.544687	+	0.838639i	$0.683352\pi$
$390$	0	0
$391$	−334518.	−0.110657
$392$	2.53863e6	0.834419
$393$	−719887.	−0.235116
$394$	1.16236e6	0.377225
$395$	0	0
$396$	−270707.	−0.0867484
$397$	−1.24083e6	−0.395128	−0.197564	−	0.980290i	$-0.563303\pi$
$397$	−1.24083e6	−0.395128	−0.197564	+	0.980290i	$0.563303\pi$
$398$	1.76593e6	0.558812
$399$	2.17211e6	0.683046
$400$	0	0
$401$	3.42036e6	1.06221	0.531106	−	0.847305i	$-0.321776\pi$
$401$	3.42036e6	1.06221	0.531106	+	0.847305i	$0.321776\pi$
$402$	1.05881e6	0.326777
$403$	−388840.	−0.119264
$404$	−98152.3	−0.0299190
$405$	0	0
$406$	−683513.	−0.205794
$407$	−1.14463e6	−0.342514
$408$	394474.	0.117319
$409$	−1.33834e6	−0.395603	−0.197802	−	0.980242i	$-0.563380\pi$
$409$	−1.33834e6	−0.395603	−0.197802	+	0.980242i	$0.563380\pi$
$410$	0	0
$411$	116614.	0.0340522
$412$	135674.	0.0393779
$413$	4.24256e6	1.22392
$414$	889606.	0.255092
$415$	0	0
$416$	1.18200e6	0.334877
$417$	530515.	0.149403
$418$	−617173.	−0.172769
$419$	−863879.	−0.240391	−0.120195	−	0.992750i	$-0.538352\pi$
$419$	−863879.	−0.240391	−0.120195	+	0.992750i	$0.538352\pi$
$420$	0	0
$421$	−3.33617e6	−0.917366	−0.458683	−	0.888600i	$-0.651679\pi$
$421$	−3.33617e6	−0.917366	−0.458683	+	0.888600i	$0.651679\pi$
$422$	−3.32365e6	−0.908520
$423$	697753.	0.189605
$424$	−4.52518e6	−1.22242
$425$	0	0
$426$	−589251.	−0.157317
$427$	−2.08524e6	−0.553460
$428$	−1.09944e6	−0.290111
$429$	−143694.	−0.0376959
$430$	0	0
$431$	−7.28648e6	−1.88940	−0.944702	−	0.327931i	$-0.893649\pi$
$431$	−7.28648e6	−1.88940	−0.944702	+	0.327931i	$0.893649\pi$
$432$	−783823.	−0.202073
$433$	5.56653e6	1.42681	0.713403	−	0.700754i	$-0.247153\pi$
$433$	5.56653e6	1.42681	0.713403	+	0.700754i	$0.247153\pi$
$434$	−1.16826e6	−0.297725
$435$	0	0
$436$	3.48453e6	0.877866
$437$	−2.03968e6	−0.510927
$438$	−1.86078e6	−0.463458
$439$	−2.68452e6	−0.664821	−0.332411	−	0.943135i	$-0.607862\pi$
$439$	−2.68452e6	−0.664821	−0.332411	+	0.943135i	$0.607862\pi$
$440$	0	0
$441$	−2.54528e6	−0.623217
$442$	−265971.	−0.0647558
$443$	3.23851e6	0.784037	0.392018	−	0.919957i	$-0.371777\pi$
$443$	3.23851e6	0.784037	0.392018	+	0.919957i	$0.371777\pi$
$444$	1.48976e6	0.358640
$445$	0	0
$446$	394450.	0.0938976
$447$	3.70255e6	0.876460
$448$	4.95496e6	1.16639
$449$	−3.08073e6	−0.721171	−0.360585	−	0.932726i	$-0.617423\pi$
$449$	−3.08073e6	−0.721171	−0.360585	+	0.932726i	$0.617423\pi$
$450$	0	0
$451$	−922292.	−0.213514
$452$	−1.97200e6	−0.454006
$453$	−846368.	−0.193782
$454$	−5.35150e6	−1.21853
$455$	0	0
$456$	2.40526e6	0.541688
$457$	−7.77565e6	−1.74159	−0.870795	−	0.491646i	$-0.836396\pi$
$457$	−7.77565e6	−1.74159	−0.870795	+	0.491646i	$0.836396\pi$
$458$	−5.79161e6	−1.29014
$459$	−895000.	−0.198286
$460$	0	0
$461$	6.52169e6	1.42925	0.714625	−	0.699508i	$-0.246597\pi$
$461$	6.52169e6	1.42925	0.714625	+	0.699508i	$0.246597\pi$
$462$	−431724.	−0.0941026
$463$	−200647.	−0.0434992	−0.0217496	−	0.999763i	$-0.506924\pi$
$463$	−200647.	−0.0434992	−0.0217496	+	0.999763i	$0.506924\pi$
$464$	−249907.	−0.0538870
$465$	0	0
$466$	3.90508e6	0.833038
$467$	−6.15254e6	−1.30546	−0.652728	−	0.757592i	$-0.726375\pi$
$467$	−6.15254e6	−1.30546	−0.652728	+	0.757592i	$0.726375\pi$
$468$	−711329.	−0.150126
$469$	6.45896e6	1.35591
$470$	0	0
$471$	3.15227e6	0.654744
$472$	4.69794e6	0.970627
$473$	−721683.	−0.148318
$474$	−1.68674e6	−0.344827
$475$	0	0
$476$	803639.	0.162571
$477$	4.53704e6	0.913011
$478$	46918.9	0.00939244
$479$	8101.82	0.00161341	0.000806703	−	1.00000i	$-0.499743\pi$
$479$	8101.82	0.00161341	0.000806703		1.00000i	$0.499743\pi$
$480$	0	0
$481$	−3.00771e6	−0.592752
$482$	246165.	0.0482623
$483$	−1.42680e6	−0.278288
$484$	2.46074e6	0.477478
$485$	0	0
$486$	3.70847e6	0.712203
$487$	−3.69236e6	−0.705475	−0.352738	−	0.935722i	$-0.614749\pi$
$487$	−3.69236e6	−0.705475	−0.352738	+	0.935722i	$0.614749\pi$
$488$	−2.30906e6	−0.438921
$489$	3.39605e6	0.642246
$490$	0	0
$491$	7.60793e6	1.42417	0.712087	−	0.702091i	$-0.247750\pi$
$491$	7.60793e6	1.42417	0.712087	+	0.702091i	$0.247750\pi$
$492$	1.20038e6	0.223566
$493$	−285354.	−0.0528770
$494$	−1.62173e6	−0.298992
$495$	0	0
$496$	−427141.	−0.0779592
$497$	−3.59457e6	−0.652764
$498$	−451069.	−0.0815023
$499$	−6.92842e6	−1.24561	−0.622806	−	0.782376i	$-0.714007\pi$
$499$	−6.92842e6	−1.24561	−0.622806	+	0.782376i	$0.714007\pi$
$500$	0	0
$501$	4.06367e6	0.723309
$502$	6.43061e6	1.13892
$503$	−7.03374e6	−1.23956	−0.619778	−	0.784777i	$-0.712777\pi$
$503$	−7.03374e6	−1.23956	−0.619778	+	0.784777i	$0.712777\pi$
$504$	−6.39945e6	−1.12219
$505$	0	0
$506$	405403.	0.0703900
$507$	2.26326e6	0.391034
$508$	−5.05553e6	−0.869176
$509$	−600480.	−0.102732	−0.0513658	−	0.998680i	$-0.516357\pi$
$509$	−600480.	−0.102732	−0.0513658	+	0.998680i	$0.516357\pi$
$510$	0	0
$511$	−1.13512e7	−1.92305
$512$	2.85275e6	0.480937
$513$	−5.45715e6	−0.915530
$514$	4.80567e6	0.802317
$515$	0	0
$516$	939285.	0.155301
$517$	317974.	0.0523196
$518$	−9.03659e6	−1.47972
$519$	995320.	0.162198
$520$	0	0
$521$	9.66108e6	1.55931	0.779653	−	0.626212i	$-0.215395\pi$
$521$	9.66108e6	1.55931	0.779653	+	0.626212i	$0.215395\pi$
$522$	758861.	0.121895
$523$	−183520.	−0.0293379	−0.0146689	−	0.999892i	$-0.504669\pi$
$523$	−183520.	−0.0293379	−0.0146689	+	0.999892i	$0.504669\pi$
$524$	−1.62400e6	−0.258380
$525$	0	0
$526$	−5.73755e6	−0.904195
$527$	−487727.	−0.0764980
$528$	−157848.	−0.0246408
$529$	−5.09653e6	−0.791837
$530$	0	0
$531$	−4.71025e6	−0.724949
$532$	4.90009e6	0.750629
$533$	−2.42348e6	−0.369505
$534$	−1.05178e6	−0.159615
$535$	0	0
$536$	7.15224e6	1.07530
$537$	3.69631e6	0.553137
$538$	−8.74055e6	−1.30192
$539$	−1.15991e6	−0.171970
$540$	0	0
$541$	2.98842e6	0.438984	0.219492	−	0.975614i	$-0.429560\pi$
$541$	2.98842e6	0.438984	0.219492	+	0.975614i	$0.429560\pi$
$542$	3.18316e6	0.465437
$543$	2.87041e6	0.417777
$544$	1.48260e6	0.214797
$545$	0	0
$546$	−1.13443e6	−0.162853
$547$	−3.29405e6	−0.470719	−0.235359	−	0.971908i	$-0.575627\pi$
$547$	−3.29405e6	−0.470719	−0.235359	+	0.971908i	$0.575627\pi$
$548$	263070.	0.0374214
$549$	2.31511e6	0.327824
$550$	0	0
$551$	−1.73991e6	−0.244145
$552$	−1.57994e6	−0.220696
$553$	−1.02895e7	−1.43081
$554$	9.54729e6	1.32162
$555$	0	0
$556$	1.19680e6	0.164185
$557$	−7.56776e6	−1.03354	−0.516772	−	0.856123i	$-0.672867\pi$
$557$	−7.56776e6	−1.03354	−0.516772	+	0.856123i	$0.672867\pi$
$558$	1.29704e6	0.176348
$559$	−1.89634e6	−0.256677
$560$	0	0
$561$	−180237.	−0.0241789
$562$	9.49084e6	1.26755
$563$	744130.	0.0989413	0.0494706	−	0.998776i	$-0.484247\pi$
$563$	744130.	0.0989413	0.0494706	+	0.998776i	$0.484247\pi$
$564$	−413849.	−0.0547828
$565$	0	0
$566$	6.47513e6	0.849586
$567$	4.28576e6	0.559848
$568$	−3.98040e6	−0.517673
$569$	1.24094e7	1.60684	0.803418	−	0.595416i	$-0.203013\pi$
$569$	1.24094e7	1.60684	0.803418	+	0.595416i	$0.203013\pi$
$570$	0	0
$571$	−1.37776e7	−1.76841	−0.884205	−	0.467099i	$-0.845299\pi$
$571$	−1.37776e7	−1.76841	−0.884205	+	0.467099i	$0.845299\pi$
$572$	−324160.	−0.0414257
$573$	1.71462e6	0.218164
$574$	−7.28128e6	−0.922419
$575$	0	0
$576$	−5.50117e6	−0.690874
$577$	−778896.	−0.0973957	−0.0486979	−	0.998814i	$-0.515507\pi$
$577$	−778896.	−0.0973957	−0.0486979	+	0.998814i	$0.515507\pi$
$578$	−333611.	−0.0415356
$579$	−2.80437e6	−0.347647
$580$	0	0
$581$	−2.75163e6	−0.338181
$582$	1.31848e6	0.161349
$583$	2.06758e6	0.251936
$584$	−1.25696e7	−1.52507
$585$	0	0
$586$	−1.56004e6	−0.187669
$587$	4.35917e6	0.522165	0.261083	−	0.965316i	$-0.415920\pi$
$587$	4.35917e6	0.522165	0.261083	+	0.965316i	$0.415920\pi$
$588$	1.50965e6	0.180066
$589$	−2.97386e6	−0.353209
$590$	0	0
$591$	2.06977e6	0.243755
$592$	−3.30397e6	−0.387465
$593$	5.91260e6	0.690465	0.345232	−	0.938517i	$-0.387800\pi$
$593$	5.91260e6	0.690465	0.345232	+	0.938517i	$0.387800\pi$
$594$	1.08465e6	0.126132
$595$	0	0
$596$	8.35263e6	0.963180
$597$	3.14451e6	0.361092
$598$	1.06527e6	0.121816
$599$	−1.26400e7	−1.43939	−0.719697	−	0.694289i	$-0.755719\pi$
$599$	−1.26400e7	−1.43939	−0.719697	+	0.694289i	$0.755719\pi$
$600$	0	0
$601$	−4.28012e6	−0.483359	−0.241680	−	0.970356i	$-0.577698\pi$
$601$	−4.28012e6	−0.483359	−0.241680	+	0.970356i	$0.577698\pi$
$602$	−5.69752e6	−0.640759
$603$	−7.17097e6	−0.803128
$604$	−1.90933e6	−0.212955
$605$	0	0
$606$	173789.	0.0192239
$607$	−7.57648e6	−0.834633	−0.417317	−	0.908761i	$-0.637029\pi$
$607$	−7.57648e6	−0.834633	−0.417317	+	0.908761i	$0.637029\pi$
$608$	9.04000e6	0.991767
$609$	−1.21710e6	−0.132979
$610$	0	0
$611$	835529.	0.0905438
$612$	−892229.	−0.0962937
$613$	1.01912e7	1.09540	0.547700	−	0.836675i	$-0.315504\pi$
$613$	1.01912e7	1.09540	0.547700	+	0.836675i	$0.315504\pi$
$614$	−1.01630e7	−1.08793
$615$	0	0
$616$	−2.91630e6	−0.309657
$617$	1.03958e7	1.09938	0.549688	−	0.835370i	$-0.314747\pi$
$617$	1.03958e7	1.09938	0.549688	+	0.835370i	$0.314747\pi$
$618$	−240225.	−0.0253016
$619$	4.99485e6	0.523957	0.261979	−	0.965074i	$-0.415625\pi$
$619$	4.99485e6	0.523957	0.261979	+	0.965074i	$0.415625\pi$
$620$	0	0
$621$	3.58465e6	0.373007
$622$	910661.	0.0943802
$623$	−6.41612e6	−0.662296
$624$	−414772.	−0.0426430
$625$	0	0
$626$	832350.	0.0848926
$627$	−1.09897e6	−0.111640
$628$	7.11125e6	0.719526
$629$	−3.77261e6	−0.380203
$630$	0	0
$631$	3.61063e6	0.361002	0.180501	−	0.983575i	$-0.442228\pi$
$631$	3.61063e6	0.361002	0.180501	+	0.983575i	$0.442228\pi$
$632$	−1.13939e7	−1.13470
$633$	−5.91828e6	−0.587065
$634$	2.56959e6	0.253888
$635$	0	0
$636$	−2.69099e6	−0.263797
$637$	−3.04786e6	−0.297610
$638$	345822.	0.0336357
$639$	3.99082e6	0.386643
$640$	0	0
$641$	4.92622e6	0.473553	0.236777	−	0.971564i	$-0.423909\pi$
$641$	4.92622e6	0.473553	0.236777	+	0.971564i	$0.423909\pi$
$642$	1.94669e6	0.186405
$643$	−6.90346e6	−0.658475	−0.329238	−	0.944247i	$-0.606792\pi$
$643$	−6.90346e6	−0.658475	−0.329238	+	0.944247i	$0.606792\pi$
$644$	−3.21873e6	−0.305823
$645$	0	0
$646$	−2.03415e6	−0.191780
$647$	−9.62754e6	−0.904180	−0.452090	−	0.891972i	$-0.649321\pi$
$647$	−9.62754e6	−0.904180	−0.452090	+	0.891972i	$0.649321\pi$
$648$	4.74578e6	0.443987
$649$	−2.14651e6	−0.200042
$650$	0	0
$651$	−2.08027e6	−0.192383
$652$	7.66119e6	0.705793
$653$	2.23052e6	0.204703	0.102351	−	0.994748i	$-0.467363\pi$
$653$	2.23052e6	0.204703	0.102351	+	0.994748i	$0.467363\pi$
$654$	−6.16974e6	−0.564057
$655$	0	0
$656$	−2.66220e6	−0.241535
$657$	1.26025e7	1.13905
$658$	2.51033e6	0.226030
$659$	9.19034e6	0.824362	0.412181	−	0.911102i	$-0.364767\pi$
$659$	9.19034e6	0.824362	0.412181	+	0.911102i	$0.364767\pi$
$660$	0	0
$661$	−5.98312e6	−0.532628	−0.266314	−	0.963886i	$-0.585806\pi$
$661$	−5.98312e6	−0.532628	−0.266314	+	0.963886i	$0.585806\pi$
$662$	2.90552e6	0.257679
$663$	−473603.	−0.0418438
$664$	−3.04697e6	−0.268194
$665$	0	0
$666$	1.00328e7	0.876465
$667$	1.14290e6	0.0994702
$668$	9.16728e6	0.794876
$669$	702380.	0.0606746
$670$	0	0
$671$	1.05502e6	0.0904597
$672$	6.32366e6	0.540189
$673$	−1.17789e7	−1.00246	−0.501230	−	0.865314i	$-0.667119\pi$
$673$	−1.17789e7	−1.00246	−0.501230	+	0.865314i	$0.667119\pi$
$674$	1.37514e7	1.16600
$675$	0	0
$676$	5.10571e6	0.429724
$677$	−7.90617e6	−0.662971	−0.331485	−	0.943460i	$-0.607550\pi$
$677$	−7.90617e6	−0.662971	−0.331485	+	0.943460i	$0.607550\pi$
$678$	3.49164e6	0.291713
$679$	8.04301e6	0.669490
$680$	0	0
$681$	−9.52918e6	−0.787386
$682$	591078.	0.0486613
$683$	1.92817e7	1.58159	0.790796	−	0.612080i	$-0.209667\pi$
$683$	1.92817e7	1.58159	0.790796	+	0.612080i	$0.209667\pi$
$684$	−5.44026e6	−0.444610
$685$	0	0
$686$	2.47734e6	0.200991
$687$	−1.03129e7	−0.833658
$688$	−2.08314e6	−0.167783
$689$	5.43291e6	0.435998
$690$	0	0
$691$	1.39881e7	1.11446	0.557228	−	0.830360i	$-0.311865\pi$
$691$	1.39881e7	1.11446	0.557228	+	0.830360i	$0.311865\pi$
$692$	2.24535e6	0.178246
$693$	2.92394e6	0.231279
$694$	−9.58591e6	−0.755501
$695$	0	0
$696$	−1.34774e6	−0.105459
$697$	−3.03980e6	−0.237008
$698$	−1.09212e7	−0.848464
$699$	6.95361e6	0.538291
$700$	0	0
$701$	8.03578e6	0.617636	0.308818	−	0.951121i	$-0.400067\pi$
$701$	8.03578e6	0.617636	0.308818	+	0.951121i	$0.400067\pi$
$702$	2.85011e6	0.218282
$703$	−2.30030e7	−1.75548
$704$	−2.50695e6	−0.190640
$705$	0	0
$706$	2.25944e6	0.170604
$707$	1.06015e6	0.0797665
$708$	2.79373e6	0.209460
$709$	2.03760e7	1.52231	0.761156	−	0.648569i	$-0.224632\pi$
$709$	2.03760e7	1.52231	0.761156	+	0.648569i	$0.224632\pi$
$710$	0	0
$711$	1.14238e7	0.847491
$712$	−7.10480e6	−0.525233
$713$	1.95344e6	0.143905
$714$	−1.42293e6	−0.104457
$715$	0	0
$716$	8.33855e6	0.607866
$717$	83546.5	0.00606919
$718$	−1.85665e7	−1.34406
$719$	−4.67132e6	−0.336990	−0.168495	−	0.985702i	$-0.553891\pi$
$719$	−4.67132e6	−0.336990	−0.168495	+	0.985702i	$0.553891\pi$
$720$	0	0
$721$	−1.46543e6	−0.104985
$722$	−2.51263e6	−0.179385
$723$	438335.	0.0311860
$724$	6.47539e6	0.459113
$725$	0	0
$726$	−4.35701e6	−0.306794
$727$	−2.43338e6	−0.170755	−0.0853777	−	0.996349i	$-0.527210\pi$
$727$	−2.43338e6	−0.170755	−0.0853777	+	0.996349i	$0.527210\pi$
$728$	−7.66307e6	−0.535889
$729$	594288.	0.0414170
$730$	0	0
$731$	−2.37861e6	−0.164638
$732$	−1.37313e6	−0.0947185
$733$	−3.71293e6	−0.255245	−0.127622	−	0.991823i	$-0.540735\pi$
$733$	−3.71293e6	−0.255245	−0.127622	+	0.991823i	$0.540735\pi$
$734$	1.08967e7	0.746544
$735$	0	0
$736$	−5.93812e6	−0.404068
$737$	−3.26789e6	−0.221615
$738$	8.08395e6	0.546365
$739$	−9.12940e6	−0.614937	−0.307469	−	0.951558i	$-0.599482\pi$
$739$	−9.12940e6	−0.614937	−0.307469	+	0.951558i	$0.599482\pi$
$740$	0	0
$741$	−2.88774e6	−0.193202
$742$	1.63230e7	1.08841
$743$	1.91143e7	1.27024	0.635121	−	0.772413i	$-0.280950\pi$
$743$	1.91143e7	1.27024	0.635121	+	0.772413i	$0.280950\pi$
$744$	−2.30356e6	−0.152569
$745$	0	0
$746$	−4.88947e6	−0.321673
$747$	3.05496e6	0.200310
$748$	−406599.	−0.0265713
$749$	1.18752e7	0.773459
$750$	0	0
$751$	−2.65439e7	−1.71738	−0.858688	−	0.512499i	$-0.828720\pi$
$751$	−2.65439e7	−1.71738	−0.858688	+	0.512499i	$0.828720\pi$
$752$	917831.	0.0591859
$753$	1.14507e7	0.735945
$754$	908704.	0.0582095
$755$	0	0
$756$	−8.61169e6	−0.548004
$757$	−6.73189e6	−0.426970	−0.213485	−	0.976946i	$-0.568481\pi$
$757$	−6.73189e6	−0.426970	−0.213485	+	0.976946i	$0.568481\pi$
$758$	1.13779e7	0.719264
$759$	721884.	0.0454845
$760$	0	0
$761$	−1.85804e7	−1.16304	−0.581520	−	0.813532i	$-0.697542\pi$
$761$	−1.85804e7	−1.16304	−0.581520	+	0.813532i	$0.697542\pi$
$762$	8.95137e6	0.558473
$763$	−3.76369e7	−2.34046
$764$	3.86804e6	0.239749
$765$	0	0
$766$	−1.07959e7	−0.664794
$767$	−5.64032e6	−0.346191
$768$	7.92672e6	0.484943
$769$	−1.10439e7	−0.673454	−0.336727	−	0.941602i	$-0.609320\pi$
$769$	−1.10439e7	−0.673454	−0.336727	+	0.941602i	$0.609320\pi$
$770$	0	0
$771$	8.55725e6	0.518439
$772$	−6.32640e6	−0.382044
$773$	2.59749e7	1.56353	0.781764	−	0.623575i	$-0.214320\pi$
$773$	2.59749e7	1.56353	0.781764	+	0.623575i	$0.214320\pi$
$774$	6.32560e6	0.379533
$775$	0	0
$776$	8.90632e6	0.530938
$777$	−1.60911e7	−0.956164
$778$	1.29866e7	0.769213
$779$	−1.85348e7	−1.09432
$780$	0	0
$781$	1.81866e6	0.106690
$782$	1.33618e6	0.0781353
$783$	3.05782e6	0.178241
$784$	−3.34809e6	−0.194539
$785$	0	0
$786$	2.87547e6	0.166017
$787$	4.08254e6	0.234960	0.117480	−	0.993075i	$-0.462518\pi$
$787$	4.08254e6	0.234960	0.117480	+	0.993075i	$0.462518\pi$
$788$	4.66922e6	0.267873
$789$	−1.02166e7	−0.584271
$790$	0	0
$791$	2.12998e7	1.21042
$792$	3.23778e6	0.183415
$793$	2.77225e6	0.156549
$794$	4.95631e6	0.279002
$795$	0	0
$796$	7.09375e6	0.396820
$797$	1.74482e7	0.972982	0.486491	−	0.873686i	$-0.338277\pi$
$797$	1.74482e7	0.972982	0.486491	+	0.873686i	$0.338277\pi$
$798$	−8.67614e6	−0.482303
$799$	1.04802e6	0.0580766
$800$	0	0
$801$	7.12341e6	0.392289
$802$	−1.36621e7	−0.750034
$803$	5.74311e6	0.314310
$804$	4.25323e6	0.232048
$805$	0	0
$806$	1.55316e6	0.0842127
$807$	−1.55639e7	−0.841270
$808$	1.17395e6	0.0632587
$809$	2.43120e7	1.30602	0.653010	−	0.757349i	$-0.273506\pi$
$809$	2.43120e7	1.30602	0.653010	+	0.757349i	$0.273506\pi$
$810$	0	0
$811$	−1.99198e7	−1.06349	−0.531745	−	0.846905i	$-0.678463\pi$
$811$	−1.99198e7	−1.06349	−0.531745	+	0.846905i	$0.678463\pi$
$812$	−2.74568e6	−0.146137
$813$	5.66812e6	0.300755
$814$	4.57204e6	0.241851
$815$	0	0
$816$	−520254.	−0.0273521
$817$	−1.45033e7	−0.760171
$818$	5.34580e6	0.279338
$819$	7.68315e6	0.400248
$820$	0	0
$821$	2.06289e7	1.06811	0.534057	−	0.845449i	$-0.320667\pi$
$821$	2.06289e7	1.06811	0.534057	+	0.845449i	$0.320667\pi$
$822$	−465794.	−0.0240444
$823$	−381768.	−0.0196472	−0.00982358	−	0.999952i	$-0.503127\pi$
$823$	−381768.	−0.0196472	−0.00982358	+	0.999952i	$0.503127\pi$
$824$	−1.62272e6	−0.0832581
$825$	0	0
$826$	−1.69462e7	−0.864217
$827$	770156.	0.0391575	0.0195788	−	0.999808i	$-0.493767\pi$
$827$	770156.	0.0391575	0.0195788	+	0.999808i	$0.493767\pi$
$828$	3.57355e6	0.181144
$829$	1.26773e7	0.640678	0.320339	−	0.947303i	$-0.396203\pi$
$829$	1.26773e7	0.640678	0.320339	+	0.947303i	$0.396203\pi$
$830$	0	0
$831$	1.70005e7	0.854000
$832$	−6.58742e6	−0.329919
$833$	−3.82298e6	−0.190893
$834$	−2.11906e6	−0.105494
$835$	0	0
$836$	−2.47919e6	−0.122686
$837$	5.22642e6	0.257864
$838$	3.45062e6	0.169741
$839$	−7.96879e6	−0.390830	−0.195415	−	0.980721i	$-0.562605\pi$
$839$	−7.96879e6	−0.390830	−0.195415	+	0.980721i	$0.562605\pi$
$840$	0	0
$841$	−1.95362e7	−0.952468
$842$	1.33258e7	0.647757
$843$	1.68999e7	0.819060
$844$	−1.33511e7	−0.645152
$845$	0	0
$846$	−2.78706e6	−0.133881
$847$	−2.65788e7	−1.27299
$848$	5.96806e6	0.284999
$849$	1.15300e7	0.548984
$850$	0	0
$851$	1.51100e7	0.715223
$852$	−2.36703e6	−0.111713
$853$	2.65563e7	1.24967	0.624834	−	0.780758i	$-0.285167\pi$
$853$	2.65563e7	1.24967	0.624834	+	0.780758i	$0.285167\pi$
$854$	8.32916e6	0.390802
$855$	0	0
$856$	1.31499e7	0.613390
$857$	3.20080e7	1.48870	0.744349	−	0.667791i	$-0.232760\pi$
$857$	3.20080e7	1.48870	0.744349	+	0.667791i	$0.232760\pi$
$858$	573961.	0.0266173
$859$	−3.73003e7	−1.72476	−0.862380	−	0.506261i	$-0.831027\pi$
$859$	−3.73003e7	−1.72476	−0.862380	+	0.506261i	$0.831027\pi$
$860$	0	0
$861$	−1.29655e7	−0.596047
$862$	2.91047e7	1.33412
$863$	2.78130e7	1.27122	0.635610	−	0.772010i	$-0.280749\pi$
$863$	2.78130e7	1.27122	0.635610	+	0.772010i	$0.280749\pi$
$864$	−1.58874e7	−0.724050
$865$	0	0
$866$	−2.22346e7	−1.00748
$867$	−594047.	−0.0268394
$868$	−4.69291e6	−0.211418
$869$	5.20593e6	0.233856
$870$	0	0
$871$	−8.58694e6	−0.383524
$872$	−4.16766e7	−1.85610
$873$	−8.92965e6	−0.396551
$874$	8.14718e6	0.360768
$875$	0	0
$876$	−7.47478e6	−0.329108
$877$	−1.05484e7	−0.463112	−0.231556	−	0.972822i	$-0.574382\pi$
$877$	−1.05484e7	−0.463112	−0.231556	+	0.972822i	$0.574382\pi$
$878$	1.07229e7	0.469434
$879$	−2.77790e6	−0.121267
$880$	0	0
$881$	−1.26816e7	−0.550472	−0.275236	−	0.961377i	$-0.588756\pi$
$881$	−1.26816e7	−0.550472	−0.275236	+	0.961377i	$0.588756\pi$
$882$	1.01667e7	0.440057
$883$	2.21555e7	0.956267	0.478134	−	0.878287i	$-0.341313\pi$
$883$	2.21555e7	0.956267	0.478134	+	0.878287i	$0.341313\pi$
$884$	−1.06841e6	−0.0459839
$885$	0	0
$886$	−1.29357e7	−0.553613
$887$	3.61505e7	1.54279	0.771393	−	0.636359i	$-0.219560\pi$
$887$	3.61505e7	1.54279	0.771393	+	0.636359i	$0.219560\pi$
$888$	−1.78182e7	−0.758284
$889$	5.46054e7	2.31730
$890$	0	0
$891$	−2.16837e6	−0.0915037
$892$	1.58451e6	0.0666779
$893$	6.39015e6	0.268153
$894$	−1.47892e7	−0.618873
$895$	0	0
$896$	8.65897e6	0.360327
$897$	1.89687e6	0.0787149
$898$	1.23055e7	0.509223
$899$	1.66635e6	0.0687648
$900$	0	0
$901$	6.81457e6	0.279658
$902$	3.68394e6	0.150764
$903$	−1.01453e7	−0.414044
$904$	2.35861e7	0.959919
$905$	0	0
$906$	3.38068e6	0.136831
$907$	−1.61778e7	−0.652982	−0.326491	−	0.945200i	$-0.605866\pi$
$907$	−1.61778e7	−0.652982	−0.326491	+	0.945200i	$0.605866\pi$
$908$	−2.14970e7	−0.865293
$909$	−1.17702e6	−0.0472471
$910$	0	0
$911$	−3.23865e7	−1.29291	−0.646454	−	0.762953i	$-0.723749\pi$
$911$	−3.23865e7	−1.29291	−0.646454	+	0.762953i	$0.723749\pi$
$912$	−3.17219e6	−0.126291
$913$	1.39218e6	0.0552736
$914$	3.10586e7	1.22975
$915$	0	0
$916$	−2.32649e7	−0.916143
$917$	1.75410e7	0.688861
$918$	3.57493e6	0.140011
$919$	3.65233e7	1.42653	0.713265	−	0.700894i	$-0.247215\pi$
$919$	3.65233e7	1.42653	0.713265	+	0.700894i	$0.247215\pi$
$920$	0	0
$921$	−1.80968e7	−0.702995
$922$	−2.60498e7	−1.00920
$923$	4.77884e6	0.184637
$924$	−1.73424e6	−0.0668235
$925$	0	0
$926$	801454.	0.0307150
$927$	1.62697e6	0.0621843
$928$	−5.06540e6	−0.193083
$929$	−4.58804e7	−1.74416	−0.872082	−	0.489359i	$-0.837231\pi$
$929$	−4.58804e7	−1.74416	−0.872082	+	0.489359i	$0.837231\pi$
$930$	0	0
$931$	−2.33102e7	−0.881396
$932$	1.56867e7	0.591551
$933$	1.62158e6	0.0609864
$934$	2.45753e7	0.921790
$935$	0	0
$936$	8.50782e6	0.317416
$937$	3.82986e7	1.42506	0.712532	−	0.701640i	$-0.247548\pi$
$937$	3.82986e7	1.42506	0.712532	+	0.701640i	$0.247548\pi$
$938$	−2.57993e7	−0.957415
$939$	1.48213e6	0.0548558
$940$	0	0
$941$	−5.66114e6	−0.208416	−0.104208	−	0.994556i	$-0.533231\pi$
$941$	−5.66114e6	−0.208416	−0.104208	+	0.994556i	$0.533231\pi$
$942$	−1.25912e7	−0.462318
$943$	1.21750e7	0.445851
$944$	−6.19591e6	−0.226295
$945$	0	0
$946$	2.88264e6	0.104728
$947$	1.06045e7	0.384252	0.192126	−	0.981370i	$-0.438462\pi$
$947$	1.06045e7	0.384252	0.192126	+	0.981370i	$0.438462\pi$
$948$	−6.77563e6	−0.244866
$949$	1.50910e7	0.543942
$950$	0	0
$951$	4.57557e6	0.164057
$952$	−9.61190e6	−0.343729
$953$	3.61608e6	0.128975	0.0644875	−	0.997919i	$-0.479459\pi$
$953$	3.61608e6	0.128975	0.0644875	+	0.997919i	$0.479459\pi$
$954$	−1.81224e7	−0.644682
$955$	0	0
$956$	188474.	0.00666969
$957$	615790.	0.0217346
$958$	−32361.4	−0.00113924
$959$	−2.84145e6	−0.0997686
$960$	0	0
$961$	−2.57810e7	−0.900517
$962$	1.20138e7	0.418545
$963$	−1.31843e7	−0.458133
$964$	988845.	0.0342717
$965$	0	0
$966$	5.69911e6	0.196501
$967$	−1.47146e7	−0.506039	−0.253019	−	0.967461i	$-0.581424\pi$
$967$	−1.47146e7	−0.506039	−0.253019	+	0.967461i	$0.581424\pi$
$968$	−2.94316e7	−1.00955
$969$	−3.62213e6	−0.123924
$970$	0	0
$971$	−2.36857e7	−0.806191	−0.403095	−	0.915158i	$-0.632066\pi$
$971$	−2.36857e7	−0.806191	−0.403095	+	0.915158i	$0.632066\pi$
$972$	1.48969e7	0.505745
$973$	−1.29267e7	−0.437731
$974$	1.47485e7	0.498140
$975$	0	0
$976$	3.04532e6	0.102331
$977$	4.30799e7	1.44390	0.721952	−	0.691944i	$-0.243245\pi$
$977$	4.30799e7	1.44390	0.721952	+	0.691944i	$0.243245\pi$
$978$	−1.35650e7	−0.453494
$979$	3.24622e6	0.108248
$980$	0	0
$981$	4.17858e7	1.38630
$982$	−3.03886e7	−1.00562
$983$	−2.70662e7	−0.893396	−0.446698	−	0.894685i	$-0.647400\pi$
$983$	−2.70662e7	−0.893396	−0.446698	+	0.894685i	$0.647400\pi$
$984$	−1.43571e7	−0.472694
$985$	0	0
$986$	1.13980e6	0.0373368
$987$	4.47004e6	0.146056
$988$	−6.51448e6	−0.212318
$989$	9.52679e6	0.309711
$990$	0	0
$991$	4.43526e7	1.43461	0.717306	−	0.696758i	$-0.245375\pi$
$991$	4.43526e7	1.43461	0.717306	+	0.696758i	$0.245375\pi$
$992$	−8.65778e6	−0.279336
$993$	5.17373e6	0.166506
$994$	1.43579e7	0.460920
$995$	0	0
$996$	−1.81195e6	−0.0578758
$997$	−2.28040e7	−0.726564	−0.363282	−	0.931679i	$-0.618344\pi$
$997$	−2.28040e7	−0.726564	−0.363282	+	0.931679i	$0.618344\pi$
$998$	2.76744e7	0.879534
$999$	4.04268e7	1.28161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.d.1.2		5
5.4	even	2		85.6.a.a.1.4	✓	5
15.14	odd	2		765.6.a.g.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.4	✓	5	5.4	even	2
425.6.a.d.1.2		5	1.1	even	1	trivial
765.6.a.g.1.2		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q-3.99434 q^{2} -7.11254 q^{3} -16.0453 q^{4} +28.4099 q^{6} +173.307 q^{7} +191.909 q^{8} -192.412 q^{9} -87.6841 q^{11} +114.123 q^{12} -230.405 q^{13} -692.246 q^{14} -253.100 q^{16} -289.000 q^{17} +768.557 q^{18} -1762.14 q^{19} -1232.65 q^{21} +350.240 q^{22} +1157.50 q^{23} -1364.96 q^{24} +920.314 q^{26} +3096.89 q^{27} -2780.76 q^{28} +987.385 q^{29} +1687.64 q^{31} -5130.12 q^{32} +623.657 q^{33} +1154.36 q^{34} +3087.30 q^{36} +13054.0 q^{37} +7038.59 q^{38} +1638.76 q^{39} +10518.3 q^{41} +4923.63 q^{42} +8230.48 q^{43} +1406.92 q^{44} -4623.45 q^{46} -3626.35 q^{47} +1800.19 q^{48} +13228.3 q^{49} +2055.53 q^{51} +3696.91 q^{52} -23579.8 q^{53} -12370.0 q^{54} +33259.2 q^{56} +12533.3 q^{57} -3943.95 q^{58} +24480.0 q^{59} -12032.1 q^{61} -6740.99 q^{62} -33346.3 q^{63} +28590.6 q^{64} -2491.10 q^{66} +37268.9 q^{67} +4637.08 q^{68} -8232.78 q^{69} -20741.1 q^{71} -36925.5 q^{72} -65497.7 q^{73} -52142.1 q^{74} +28274.1 q^{76} -15196.3 q^{77} -6545.78 q^{78} -59371.4 q^{79} +24729.3 q^{81} -42013.8 q^{82} -15877.2 q^{83} +19778.3 q^{84} -32875.3 q^{86} -7022.82 q^{87} -16827.4 q^{88} -37021.7 q^{89} -39930.8 q^{91} -18572.4 q^{92} -12003.4 q^{93} +14484.9 q^{94} +36488.2 q^{96} +46409.1 q^{97} -52838.3 q^{98} +16871.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) \) 5 * q + 7 * q^2 + 36 * q^3 + 43 * q^4 - 105 * q^6 + 204 * q^7 + 63 * q^8 + 531 * q^9 - 792 * q^11 - 785 * q^12 - 88 * q^13 + 860 * q^14 - 2365 * q^16 - 1445 * q^17 + 2052 * q^18 - 5160 * q^19 - 6428 * q^21 + 3058 * q^22 + 6140 * q^23 - 11751 * q^24 - 9643 * q^26 + 13380 * q^27 + 4568 * q^28 - 7328 * q^29 - 8968 * q^31 + 4703 * q^32 - 13072 * q^33 - 2023 * q^34 + 2048 * q^36 - 9834 * q^37 - 10625 * q^38 + 10108 * q^39 - 5302 * q^41 - 16222 * q^42 + 4916 * q^43 - 31514 * q^44 - 16768 * q^46 + 20364 * q^47 - 45557 * q^48 - 26483 * q^49 - 10404 * q^51 + 11525 * q^52 - 56492 * q^53 + 64635 * q^54 + 73228 * q^56 - 67554 * q^57 - 54213 * q^58 - 96200 * q^59 - 7284 * q^61 - 122335 * q^62 - 64108 * q^63 + 28483 * q^64 + 115232 * q^66 + 60308 * q^67 - 12427 * q^68 - 89076 * q^69 - 25304 * q^71 - 118722 * q^72 - 163328 * q^73 - 82768 * q^74 + 92027 * q^76 - 19912 * q^77 - 134895 * q^78 - 50940 * q^79 + 92445 * q^81 - 191390 * q^82 + 179060 * q^83 - 128054 * q^84 + 87086 * q^86 - 55764 * q^87 - 167634 * q^88 - 387356 * q^89 - 115200 * q^91 + 293228 * q^92 + 202558 * q^93 + 100515 * q^94 + 428113 * q^96 + 106076 * q^97 - 49329 * q^98 - 535112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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