LMFDB - Embedded newform 4025.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.1397868136$
Analytic rank:	$1$
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} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 $ x^8 - x^7 - 10x^6 + 9x^5 + 28x^4 - 22x^3 - 16x^2 + 7x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.09801$ of defining polynomial
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09801 q^{2} -0.493654 q^{3} -0.794374 q^{4} +0.542037 q^{6} +1.00000 q^{7} +3.06825 q^{8} -2.75631 q^{9} +O(q^{10})$	\(q-1.09801 q^{2} -0.493654 q^{3} -0.794374 q^{4} +0.542037 q^{6} +1.00000 q^{7} +3.06825 q^{8} -2.75631 q^{9} -1.70792 q^{11} +0.392146 q^{12} -4.14713 q^{13} -1.09801 q^{14} -1.78022 q^{16} +3.30617 q^{17} +3.02645 q^{18} +2.28046 q^{19} -0.493654 q^{21} +1.87532 q^{22} -1.00000 q^{23} -1.51465 q^{24} +4.55360 q^{26} +2.84162 q^{27} -0.794374 q^{28} +3.22858 q^{29} +5.89075 q^{31} -4.18180 q^{32} +0.843123 q^{33} -3.63021 q^{34} +2.18954 q^{36} +1.92372 q^{37} -2.50397 q^{38} +2.04725 q^{39} -1.92865 q^{41} +0.542037 q^{42} -6.45568 q^{43} +1.35673 q^{44} +1.09801 q^{46} +5.31771 q^{47} +0.878814 q^{48} +1.00000 q^{49} -1.63210 q^{51} +3.29438 q^{52} -3.23409 q^{53} -3.12013 q^{54} +3.06825 q^{56} -1.12576 q^{57} -3.54501 q^{58} -9.01538 q^{59} +4.38020 q^{61} -6.46810 q^{62} -2.75631 q^{63} +8.15210 q^{64} -0.925758 q^{66} +1.05619 q^{67} -2.62634 q^{68} +0.493654 q^{69} +11.7201 q^{71} -8.45704 q^{72} -13.6891 q^{73} -2.11226 q^{74} -1.81154 q^{76} -1.70792 q^{77} -2.24790 q^{78} +13.7225 q^{79} +6.86614 q^{81} +2.11768 q^{82} -8.57893 q^{83} +0.392146 q^{84} +7.08840 q^{86} -1.59380 q^{87} -5.24033 q^{88} -3.30414 q^{89} -4.14713 q^{91} +0.794374 q^{92} -2.90799 q^{93} -5.83890 q^{94} +2.06436 q^{96} -10.4592 q^{97} -1.09801 q^{98} +4.70756 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7}+O(q^{10}) $ 8 * q - q^2 - 4 * q^3 + 5 * q^4 - q^6 + 8 * q^7	$ 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} + 3 q^{11} - 9 q^{12} - 5 q^{13} - q^{14} - q^{16} - 5 q^{17} - 2 q^{18} - 2 q^{19} - 4 q^{21} - 21 q^{22} - 8 q^{23} - 6 q^{24} + 18 q^{26} - 7 q^{27} + 5 q^{28} - 9 q^{29} - 3 q^{31} - 6 q^{32} - 4 q^{33} - 10 q^{34} + 16 q^{36} - 6 q^{37} - 4 q^{38} - 2 q^{39} - 7 q^{41} - q^{42} - 8 q^{43} + 4 q^{44} + q^{46} - 22 q^{47} - 9 q^{48} + 8 q^{49} - 12 q^{51} - 11 q^{52} - 21 q^{53} - 15 q^{54} - 8 q^{57} - 16 q^{58} + 14 q^{59} + 8 q^{61} - 12 q^{62} - 40 q^{64} + 55 q^{66} - 21 q^{67} - 3 q^{68} + 4 q^{69} + 11 q^{71} + q^{72} - 26 q^{73} - 41 q^{74} + 21 q^{76} + 3 q^{77} - 17 q^{78} - 16 q^{79} - 20 q^{81} + q^{82} - 20 q^{83} - 9 q^{84} + 14 q^{86} + 29 q^{87} - 32 q^{88} + 15 q^{89} - 5 q^{91} - 5 q^{92} - 19 q^{93} + 21 q^{94} + 52 q^{96} - q^{97} - q^{98} + 15 q^{99}+O(q^{100}) $ 8 * q - q^2 - 4 * q^3 + 5 * q^4 - q^6 + 8 * q^7 + 3 * q^11 - 9 * q^12 - 5 * q^13 - q^14 - q^16 - 5 * q^17 - 2 * q^18 - 2 * q^19 - 4 * q^21 - 21 * q^22 - 8 * q^23 - 6 * q^24 + 18 * q^26 - 7 * q^27 + 5 * q^28 - 9 * q^29 - 3 * q^31 - 6 * q^32 - 4 * q^33 - 10 * q^34 + 16 * q^36 - 6 * q^37 - 4 * q^38 - 2 * q^39 - 7 * q^41 - q^42 - 8 * q^43 + 4 * q^44 + q^46 - 22 * q^47 - 9 * q^48 + 8 * q^49 - 12 * q^51 - 11 * q^52 - 21 * q^53 - 15 * q^54 - 8 * q^57 - 16 * q^58 + 14 * q^59 + 8 * q^61 - 12 * q^62 - 40 * q^64 + 55 * q^66 - 21 * q^67 - 3 * q^68 + 4 * q^69 + 11 * q^71 + q^72 - 26 * q^73 - 41 * q^74 + 21 * q^76 + 3 * q^77 - 17 * q^78 - 16 * q^79 - 20 * q^81 + q^82 - 20 * q^83 - 9 * q^84 + 14 * q^86 + 29 * q^87 - 32 * q^88 + 15 * q^89 - 5 * q^91 - 5 * q^92 - 19 * q^93 + 21 * q^94 + 52 * q^96 - q^97 - q^98 + 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.09801	−0.776410	−0.388205	−	0.921573i	$-0.626905\pi$
$2$	−1.09801	−0.776410	−0.388205	+	0.921573i	$0.626905\pi$
$3$	−0.493654	−0.285011	−0.142506	−	0.989794i	$-0.545516\pi$
$3$	−0.493654	−0.285011	−0.142506	+	0.989794i	$0.545516\pi$
$4$	−0.794374	−0.397187
$5$	0	0
$5$	0	0
$6$	0.542037	0.221286
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.06825	1.08479
$9$	−2.75631	−0.918769
$10$	0	0
$11$	−1.70792	−0.514958	−0.257479	−	0.966284i	$-0.582892\pi$
$11$	−1.70792	−0.514958	−0.257479	+	0.966284i	$0.582892\pi$
$12$	0.392146	0.113203
$13$	−4.14713	−1.15021	−0.575104	−	0.818080i	$-0.695038\pi$
$13$	−4.14713	−1.15021	−0.575104	+	0.818080i	$0.695038\pi$
$14$	−1.09801	−0.293456
$15$	0	0
$16$	−1.78022	−0.445056
$17$	3.30617	0.801864	0.400932	−	0.916108i	$-0.368686\pi$
$17$	3.30617	0.801864	0.400932	+	0.916108i	$0.368686\pi$
$18$	3.02645	0.713341
$19$	2.28046	0.523174	0.261587	−	0.965180i	$-0.415754\pi$
$19$	2.28046	0.523174	0.261587	+	0.965180i	$0.415754\pi$
$20$	0	0
$21$	−0.493654	−0.107724
$22$	1.87532	0.399819
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.51465	−0.309178
$25$	0	0
$26$	4.55360	0.893034
$27$	2.84162	0.546871
$28$	−0.794374	−0.150123
$29$	3.22858	0.599532	0.299766	−	0.954013i	$-0.403091\pi$
$29$	3.22858	0.599532	0.299766	+	0.954013i	$0.403091\pi$
$30$	0	0
$31$	5.89075	1.05801	0.529005	−	0.848619i	$-0.322565\pi$
$31$	5.89075	1.05801	0.529005	+	0.848619i	$0.322565\pi$
$32$	−4.18180	−0.739245
$33$	0.843123	0.146769
$34$	−3.63021	−0.622576
$35$	0	0
$36$	2.18954	0.364923
$37$	1.92372	0.316258	0.158129	−	0.987418i	$-0.449454\pi$
$37$	1.92372	0.316258	0.158129	+	0.987418i	$0.449454\pi$
$38$	−2.50397	−0.406197
$39$	2.04725	0.327822
$40$	0	0
$41$	−1.92865	−0.301205	−0.150603	−	0.988594i	$-0.548121\pi$
$41$	−1.92865	−0.301205	−0.150603	+	0.988594i	$0.548121\pi$
$42$	0.542037	0.0836382
$43$	−6.45568	−0.984482	−0.492241	−	0.870459i	$-0.663822\pi$
$43$	−6.45568	−0.984482	−0.492241	+	0.870459i	$0.663822\pi$
$44$	1.35673	0.204535
$45$	0	0
$46$	1.09801	0.161893
$47$	5.31771	0.775667	0.387834	−	0.921729i	$-0.373223\pi$
$47$	5.31771	0.775667	0.387834	+	0.921729i	$0.373223\pi$
$48$	0.878814	0.126846
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.63210	−0.228540
$52$	3.29438	0.456848
$53$	−3.23409	−0.444236	−0.222118	−	0.975020i	$-0.571297\pi$
$53$	−3.23409	−0.444236	−0.222118	+	0.975020i	$0.571297\pi$
$54$	−3.12013	−0.424596
$55$	0	0
$56$	3.06825	0.410012
$57$	−1.12576	−0.149110
$58$	−3.54501	−0.465483
$59$	−9.01538	−1.17370	−0.586851	−	0.809695i	$-0.699633\pi$
$59$	−9.01538	−1.17370	−0.586851	+	0.809695i	$0.699633\pi$
$60$	0	0
$61$	4.38020	0.560827	0.280413	−	0.959879i	$-0.409528\pi$
$61$	4.38020	0.560827	0.280413	+	0.959879i	$0.409528\pi$
$62$	−6.46810	−0.821450
$63$	−2.75631	−0.347262
$64$	8.15210	1.01901
$65$	0	0
$66$	−0.925758	−0.113953
$67$	1.05619	0.129034	0.0645170	−	0.997917i	$-0.479449\pi$
$67$	1.05619	0.129034	0.0645170	+	0.997917i	$0.479449\pi$
$68$	−2.62634	−0.318490
$69$	0.493654	0.0594290
$70$	0	0
$71$	11.7201	1.39092	0.695461	−	0.718564i	$-0.255200\pi$
$71$	11.7201	1.39092	0.695461	+	0.718564i	$0.255200\pi$
$72$	−8.45704	−0.996671
$73$	−13.6891	−1.60219	−0.801095	−	0.598537i	$-0.795749\pi$
$73$	−13.6891	−1.60219	−0.801095	+	0.598537i	$0.795749\pi$
$74$	−2.11226	−0.245546
$75$	0	0
$76$	−1.81154	−0.207798
$77$	−1.70792	−0.194636
$78$	−2.24790	−0.254525
$79$	13.7225	1.54391	0.771953	−	0.635680i	$-0.219280\pi$
$79$	13.7225	1.54391	0.771953	+	0.635680i	$0.219280\pi$
$80$	0	0
$81$	6.86614	0.762904
$82$	2.11768	0.233859
$83$	−8.57893	−0.941659	−0.470830	−	0.882224i	$-0.656045\pi$
$83$	−8.57893	−0.941659	−0.470830	+	0.882224i	$0.656045\pi$
$84$	0.392146	0.0427866
$85$	0	0
$86$	7.08840	0.764362
$87$	−1.59380	−0.170873
$88$	−5.24033	−0.558622
$89$	−3.30414	−0.350238	−0.175119	−	0.984547i	$-0.556031\pi$
$89$	−3.30414	−0.350238	−0.175119	+	0.984547i	$0.556031\pi$
$90$	0	0
$91$	−4.14713	−0.434738
$92$	0.794374	0.0828192
$93$	−2.90799	−0.301545
$94$	−5.83890	−0.602236
$95$	0	0
$96$	2.06436	0.210693
$97$	−10.4592	−1.06197	−0.530985	−	0.847381i	$-0.678178\pi$
$97$	−10.4592	−1.06197	−0.530985	+	0.847381i	$0.678178\pi$
$98$	−1.09801	−0.110916
$99$	4.70756	0.473127
$100$	0	0
$101$	−5.27615	−0.524996	−0.262498	−	0.964933i	$-0.584546\pi$
$101$	−5.27615	−0.524996	−0.262498	+	0.964933i	$0.584546\pi$
$102$	1.79207	0.177441
$103$	3.19221	0.314538	0.157269	−	0.987556i	$-0.449731\pi$
$103$	3.19221	0.314538	0.157269	+	0.987556i	$0.449731\pi$
$104$	−12.7244	−1.24773
$105$	0	0
$106$	3.55106	0.344910
$107$	1.39458	0.134819	0.0674095	−	0.997725i	$-0.478527\pi$
$107$	1.39458	0.134819	0.0674095	+	0.997725i	$0.478527\pi$
$108$	−2.25731	−0.217210
$109$	10.6980	1.02469	0.512343	−	0.858781i	$-0.328777\pi$
$109$	10.6980	1.02469	0.512343	+	0.858781i	$0.328777\pi$
$110$	0	0
$111$	−0.949653	−0.0901370
$112$	−1.78022	−0.168215
$113$	3.36774	0.316810	0.158405	−	0.987374i	$-0.449365\pi$
$113$	3.36774	0.316810	0.158405	+	0.987374i	$0.449365\pi$
$114$	1.23609	0.115771
$115$	0	0
$116$	−2.56470	−0.238126
$117$	11.4308	1.05678
$118$	9.89898	0.911275
$119$	3.30617	0.303076
$120$	0	0
$121$	−8.08300	−0.734818
$122$	−4.80950	−0.435432
$123$	0.952088	0.0858469
$124$	−4.67946	−0.420228
$125$	0	0
$126$	3.02645	0.269618
$127$	2.24197	0.198943	0.0994715	−	0.995040i	$-0.468285\pi$
$127$	2.24197	0.198943	0.0994715	+	0.995040i	$0.468285\pi$
$128$	−0.587494	−0.0519276
$129$	3.18687	0.280589
$130$	0	0
$131$	9.88736	0.863862	0.431931	−	0.901907i	$-0.357832\pi$
$131$	9.88736	0.863862	0.431931	+	0.901907i	$0.357832\pi$
$132$	−0.669755	−0.0582947
$133$	2.28046	0.197741
$134$	−1.15971	−0.100183
$135$	0	0
$136$	10.1442	0.869855
$137$	−18.1991	−1.55485	−0.777426	−	0.628975i	$-0.783475\pi$
$137$	−18.1991	−1.55485	−0.777426	+	0.628975i	$0.783475\pi$
$138$	−0.542037	−0.0461413
$139$	−0.0928647	−0.00787668	−0.00393834	−	0.999992i	$-0.501254\pi$
$139$	−0.0928647	−0.00787668	−0.00393834	+	0.999992i	$0.501254\pi$
$140$	0	0
$141$	−2.62511	−0.221074
$142$	−12.8688	−1.07993
$143$	7.08299	0.592309
$144$	4.90684	0.408903
$145$	0	0
$146$	15.0308	1.24396
$147$	−0.493654	−0.0407159
$148$	−1.52815	−0.125613
$149$	9.51665	0.779634	0.389817	−	0.920892i	$-0.372538\pi$
$149$	9.51665	0.779634	0.389817	+	0.920892i	$0.372538\pi$
$150$	0	0
$151$	−8.52256	−0.693556	−0.346778	−	0.937947i	$-0.612724\pi$
$151$	−8.52256	−0.693556	−0.346778	+	0.937947i	$0.612724\pi$
$152$	6.99703	0.567534
$153$	−9.11282	−0.736728
$154$	1.87532	0.151117
$155$	0	0
$156$	−1.62628	−0.130207
$157$	11.5417	0.921128	0.460564	−	0.887627i	$-0.347647\pi$
$157$	11.5417	0.921128	0.460564	+	0.887627i	$0.347647\pi$
$158$	−15.0675	−1.19870
$159$	1.59652	0.126612
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	−7.53909	−0.592327
$163$	−10.8232	−0.847738	−0.423869	−	0.905723i	$-0.639328\pi$
$163$	−10.8232	−0.847738	−0.423869	+	0.905723i	$0.639328\pi$
$164$	1.53207	0.119635
$165$	0	0
$166$	9.41975	0.731114
$167$	−5.93162	−0.459002	−0.229501	−	0.973308i	$-0.573709\pi$
$167$	−5.93162	−0.459002	−0.229501	+	0.973308i	$0.573709\pi$
$168$	−1.51465	−0.116858
$169$	4.19873	0.322979
$170$	0	0
$171$	−6.28565	−0.480675
$172$	5.12822	0.391023
$173$	15.7589	1.19813	0.599063	−	0.800702i	$-0.295540\pi$
$173$	15.7589	1.19813	0.599063	+	0.800702i	$0.295540\pi$
$174$	1.75001	0.132668
$175$	0	0
$176$	3.04048	0.229185
$177$	4.45048	0.334519
$178$	3.62798	0.271929
$179$	−21.6794	−1.62039	−0.810197	−	0.586157i	$-0.800640\pi$
$179$	−21.6794	−1.62039	−0.810197	+	0.586157i	$0.800640\pi$
$180$	0	0
$181$	22.6425	1.68300	0.841502	−	0.540254i	$-0.181672\pi$
$181$	22.6425	1.68300	0.841502	+	0.540254i	$0.181672\pi$
$182$	4.55360	0.337535
$183$	−2.16230	−0.159842
$184$	−3.06825	−0.226194
$185$	0	0
$186$	3.19301	0.234122
$187$	−5.64668	−0.412926
$188$	−4.22425	−0.308085
$189$	2.84162	0.206698
$190$	0	0
$191$	−8.84286	−0.639847	−0.319924	−	0.947443i	$-0.603657\pi$
$191$	−8.84286	−0.639847	−0.319924	+	0.947443i	$0.603657\pi$
$192$	−4.02432	−0.290430
$193$	−25.8391	−1.85994	−0.929968	−	0.367640i	$-0.880166\pi$
$193$	−25.8391	−1.85994	−0.929968	+	0.367640i	$0.880166\pi$
$194$	11.4843	0.824525
$195$	0	0
$196$	−0.794374	−0.0567410
$197$	9.81220	0.699090	0.349545	−	0.936920i	$-0.386336\pi$
$197$	9.81220	0.699090	0.349545	+	0.936920i	$0.386336\pi$
$198$	−5.16894	−0.367341
$199$	−8.49219	−0.601995	−0.300998	−	0.953625i	$-0.597320\pi$
$199$	−8.49219	−0.601995	−0.300998	+	0.953625i	$0.597320\pi$
$200$	0	0
$201$	−0.521392	−0.0367762
$202$	5.79326	0.407612
$203$	3.22858	0.226602
$204$	1.29650	0.0907732
$205$	0	0
$206$	−3.50508	−0.244210
$207$	2.75631	0.191576
$208$	7.38282	0.511907
$209$	−3.89485	−0.269412
$210$	0	0
$211$	23.8973	1.64516	0.822579	−	0.568651i	$-0.192535\pi$
$211$	23.8973	1.64516	0.822579	+	0.568651i	$0.192535\pi$
$212$	2.56907	0.176445
$213$	−5.78569	−0.396429
$214$	−1.53126	−0.104675
$215$	0	0
$216$	8.71881	0.593240
$217$	5.89075	0.399890
$218$	−11.7466	−0.795578
$219$	6.75769	0.456642
$220$	0	0
$221$	−13.7111	−0.922311
$222$	1.04273	0.0699833
$223$	−14.1866	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$223$	−14.1866	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$224$	−4.18180	−0.279408
$225$	0	0
$226$	−3.69781	−0.245975
$227$	−7.58503	−0.503436	−0.251718	−	0.967801i	$-0.580996\pi$
$227$	−7.58503	−0.503436	−0.251718	+	0.967801i	$0.580996\pi$
$228$	0.894273	0.0592247
$229$	−22.4398	−1.48287	−0.741433	−	0.671027i	$-0.765853\pi$
$229$	−22.4398	−1.48287	−0.741433	+	0.671027i	$0.765853\pi$
$230$	0	0
$231$	0.843123	0.0554734
$232$	9.90609	0.650367
$233$	1.14889	0.0752662	0.0376331	−	0.999292i	$-0.488018\pi$
$233$	1.14889	0.0752662	0.0376331	+	0.999292i	$0.488018\pi$
$234$	−12.5511	−0.820491
$235$	0	0
$236$	7.16158	0.466179
$237$	−6.77418	−0.440031
$238$	−3.63021	−0.235311
$239$	−23.0049	−1.48806	−0.744030	−	0.668146i	$-0.767088\pi$
$239$	−23.0049	−1.48806	−0.744030	+	0.668146i	$0.767088\pi$
$240$	0	0
$241$	−21.3119	−1.37282	−0.686409	−	0.727215i	$-0.740814\pi$
$241$	−21.3119	−1.37282	−0.686409	+	0.727215i	$0.740814\pi$
$242$	8.87522	0.570521
$243$	−11.9144	−0.764307
$244$	−3.47951	−0.222753
$245$	0	0
$246$	−1.04540	−0.0666524
$247$	−9.45738	−0.601759
$248$	18.0743	1.14772
$249$	4.23502	0.268384
$250$	0	0
$251$	10.5858	0.668172	0.334086	−	0.942543i	$-0.391572\pi$
$251$	10.5858	0.668172	0.334086	+	0.942543i	$0.391572\pi$
$252$	2.18954	0.137928
$253$	1.70792	0.107376
$254$	−2.46171	−0.154461
$255$	0	0
$256$	−15.6591	−0.978696
$257$	−19.7890	−1.23440	−0.617202	−	0.786804i	$-0.711734\pi$
$257$	−19.7890	−1.23440	−0.617202	+	0.786804i	$0.711734\pi$
$258$	−3.49922	−0.217852
$259$	1.92372	0.119534
$260$	0	0
$261$	−8.89895	−0.550831
$262$	−10.8564	−0.670712
$263$	5.15660	0.317969	0.158985	−	0.987281i	$-0.449178\pi$
$263$	5.15660	0.317969	0.158985	+	0.987281i	$0.449178\pi$
$264$	2.58691	0.159213
$265$	0	0
$266$	−2.50397	−0.153528
$267$	1.63110	0.0998219
$268$	−0.839009	−0.0512506
$269$	−4.92544	−0.300309	−0.150155	−	0.988663i	$-0.547977\pi$
$269$	−4.92544	−0.300309	−0.150155	+	0.988663i	$0.547977\pi$
$270$	0	0
$271$	−8.15010	−0.495083	−0.247542	−	0.968877i	$-0.579623\pi$
$271$	−8.15010	−0.495083	−0.247542	+	0.968877i	$0.579623\pi$
$272$	−5.88572	−0.356874
$273$	2.04725	0.123905
$274$	19.9828	1.20720
$275$	0	0
$276$	−0.392146	−0.0236044
$277$	−13.4936	−0.810750	−0.405375	−	0.914151i	$-0.632859\pi$
$277$	−13.4936	−0.810750	−0.405375	+	0.914151i	$0.632859\pi$
$278$	0.101966	0.00611554
$279$	−16.2367	−0.972066
$280$	0	0
$281$	−17.2634	−1.02985	−0.514925	−	0.857236i	$-0.672180\pi$
$281$	−17.2634	−1.02985	−0.514925	+	0.857236i	$0.672180\pi$
$282$	2.88240	0.171644
$283$	−27.4389	−1.63107	−0.815537	−	0.578705i	$-0.803558\pi$
$283$	−27.4389	−1.63107	−0.815537	+	0.578705i	$0.803558\pi$
$284$	−9.31016	−0.552456
$285$	0	0
$286$	−7.77719	−0.459875
$287$	−1.92865	−0.113845
$288$	11.5263	0.679195
$289$	−6.06924	−0.357014
$290$	0	0
$291$	5.16322	0.302674
$292$	10.8743	0.636369
$293$	−3.95396	−0.230993	−0.115496	−	0.993308i	$-0.536846\pi$
$293$	−3.95396	−0.230993	−0.115496	+	0.993308i	$0.536846\pi$
$294$	0.542037	0.0316123
$295$	0	0
$296$	5.90246	0.343073
$297$	−4.85327	−0.281615
$298$	−10.4494	−0.605316
$299$	4.14713	0.239835
$300$	0	0
$301$	−6.45568	−0.372099
$302$	9.35786	0.538484
$303$	2.60459	0.149630
$304$	−4.05973	−0.232841
$305$	0	0
$306$	10.0060	0.572003
$307$	12.2263	0.697790	0.348895	−	0.937162i	$-0.386557\pi$
$307$	12.2263	0.697790	0.348895	+	0.937162i	$0.386557\pi$
$308$	1.35673	0.0773068
$309$	−1.57585	−0.0896468
$310$	0	0
$311$	−6.45372	−0.365957	−0.182978	−	0.983117i	$-0.558574\pi$
$311$	−6.45372	−0.365957	−0.182978	+	0.983117i	$0.558574\pi$
$312$	6.28148	0.355619
$313$	−12.1777	−0.688322	−0.344161	−	0.938911i	$-0.611837\pi$
$313$	−12.1777	−0.688322	−0.344161	+	0.938911i	$0.611837\pi$
$314$	−12.6729	−0.715173
$315$	0	0
$316$	−10.9008	−0.613219
$317$	20.8022	1.16837	0.584184	−	0.811622i	$-0.301415\pi$
$317$	20.8022	1.16837	0.584184	+	0.811622i	$0.301415\pi$
$318$	−1.75300	−0.0983032
$319$	−5.51416	−0.308734
$320$	0	0
$321$	−0.688440	−0.0384250
$322$	1.09801	0.0611897
$323$	7.53959	0.419514
$324$	−5.45428	−0.303016
$325$	0	0
$326$	11.8840	0.658193
$327$	−5.28113	−0.292047
$328$	−5.91760	−0.326745
$329$	5.31771	0.293175
$330$	0	0
$331$	2.61980	0.143997	0.0719987	−	0.997405i	$-0.477062\pi$
$331$	2.61980	0.143997	0.0719987	+	0.997405i	$0.477062\pi$
$332$	6.81487	0.374015
$333$	−5.30236	−0.290568
$334$	6.51298	0.356374
$335$	0	0
$336$	0.878814	0.0479433
$337$	−26.0537	−1.41924	−0.709618	−	0.704587i	$-0.751132\pi$
$337$	−26.0537	−1.41924	−0.709618	+	0.704587i	$0.751132\pi$
$338$	−4.61025	−0.250764
$339$	−1.66250	−0.0902945
$340$	0	0
$341$	−10.0609	−0.544831
$342$	6.90170	0.373201
$343$	1.00000	0.0539949
$344$	−19.8077	−1.06796
$345$	0	0
$346$	−17.3034	−0.930238
$347$	−13.9463	−0.748676	−0.374338	−	0.927292i	$-0.622130\pi$
$347$	−13.9463	−0.748676	−0.374338	+	0.927292i	$0.622130\pi$
$348$	1.26607	0.0678687
$349$	−8.08333	−0.432691	−0.216345	−	0.976317i	$-0.569414\pi$
$349$	−8.08333	−0.432691	−0.216345	+	0.976317i	$0.569414\pi$
$350$	0	0
$351$	−11.7846	−0.629015
$352$	7.14219	0.380680
$353$	−9.69120	−0.515811	−0.257905	−	0.966170i	$-0.583032\pi$
$353$	−9.69120	−0.515811	−0.257905	+	0.966170i	$0.583032\pi$
$354$	−4.88667	−0.259724
$355$	0	0
$356$	2.62472	0.139110
$357$	−1.63210	−0.0863801
$358$	23.8042	1.25809
$359$	−12.4408	−0.656600	−0.328300	−	0.944573i	$-0.606476\pi$
$359$	−12.4408	−0.656600	−0.328300	+	0.944573i	$0.606476\pi$
$360$	0	0
$361$	−13.7995	−0.726289
$362$	−24.8617	−1.30670
$363$	3.99021	0.209432
$364$	3.29438	0.172672
$365$	0	0
$366$	2.37423	0.124103
$367$	−7.33266	−0.382762	−0.191381	−	0.981516i	$-0.561297\pi$
$367$	−7.33266	−0.382762	−0.191381	+	0.981516i	$0.561297\pi$
$368$	1.78022	0.0928005
$369$	5.31596	0.276738
$370$	0	0
$371$	−3.23409	−0.167906
$372$	2.31003	0.119770
$373$	−1.86686	−0.0966625	−0.0483313	−	0.998831i	$-0.515390\pi$
$373$	−1.86686	−0.0966625	−0.0483313	+	0.998831i	$0.515390\pi$
$374$	6.20012	0.320600
$375$	0	0
$376$	16.3161	0.841437
$377$	−13.3894	−0.689587
$378$	−3.12013	−0.160482
$379$	−5.94785	−0.305520	−0.152760	−	0.988263i	$-0.548816\pi$
$379$	−5.94785	−0.305520	−0.152760	+	0.988263i	$0.548816\pi$
$380$	0	0
$381$	−1.10676	−0.0567010
$382$	9.70955	0.496784
$383$	2.04405	0.104446	0.0522230	−	0.998635i	$-0.483369\pi$
$383$	2.04405	0.104446	0.0522230	+	0.998635i	$0.483369\pi$
$384$	0.290019	0.0148000
$385$	0	0
$386$	28.3715	1.44407
$387$	17.7938	0.904511
$388$	8.30851	0.421801
$389$	−0.774203	−0.0392536	−0.0196268	−	0.999807i	$-0.506248\pi$
$389$	−0.774203	−0.0392536	−0.0196268	+	0.999807i	$0.506248\pi$
$390$	0	0
$391$	−3.30617	−0.167200
$392$	3.06825	0.154970
$393$	−4.88093	−0.246211
$394$	−10.7739	−0.542781
$395$	0	0
$396$	−3.73956	−0.187920
$397$	−24.7805	−1.24370	−0.621848	−	0.783138i	$-0.713618\pi$
$397$	−24.7805	−1.24370	−0.621848	+	0.783138i	$0.713618\pi$
$398$	9.32451	0.467395
$399$	−1.12576	−0.0563584
$400$	0	0
$401$	−10.8322	−0.540936	−0.270468	−	0.962729i	$-0.587178\pi$
$401$	−10.8322	−0.540936	−0.270468	+	0.962729i	$0.587178\pi$
$402$	0.572494	0.0285534
$403$	−24.4297	−1.21693
$404$	4.19123	0.208522
$405$	0	0
$406$	−3.54501	−0.175936
$407$	−3.28557	−0.162859
$408$	−5.00771	−0.247918
$409$	8.11331	0.401177	0.200589	−	0.979676i	$-0.435715\pi$
$409$	8.11331	0.401177	0.200589	+	0.979676i	$0.435715\pi$
$410$	0	0
$411$	8.98404	0.443150
$412$	−2.53581	−0.124930
$413$	−9.01538	−0.443618
$414$	−3.02645	−0.148742
$415$	0	0
$416$	17.3425	0.850285
$417$	0.0458430	0.00224494
$418$	4.27659	0.209175
$419$	4.19628	0.205002	0.102501	−	0.994733i	$-0.467316\pi$
$419$	4.19628	0.205002	0.102501	+	0.994733i	$0.467316\pi$
$420$	0	0
$421$	−19.4817	−0.949482	−0.474741	−	0.880126i	$-0.657458\pi$
$421$	−19.4817	−0.949482	−0.474741	+	0.880126i	$0.657458\pi$
$422$	−26.2395	−1.27732
$423$	−14.6572	−0.712659
$424$	−9.92299	−0.481903
$425$	0	0
$426$	6.35274	0.307791
$427$	4.38020	0.211973
$428$	−1.10782	−0.0535484
$429$	−3.49654	−0.168815
$430$	0	0
$431$	22.2321	1.07088	0.535441	−	0.844573i	$-0.320146\pi$
$431$	22.2321	1.07088	0.535441	+	0.844573i	$0.320146\pi$
$432$	−5.05872	−0.243388
$433$	36.1430	1.73692	0.868461	−	0.495757i	$-0.165109\pi$
$433$	36.1430	1.73692	0.868461	+	0.495757i	$0.165109\pi$
$434$	−6.46810	−0.310479
$435$	0	0
$436$	−8.49825	−0.406992
$437$	−2.28046	−0.109089
$438$	−7.42001	−0.354542
$439$	14.2271	0.679024	0.339512	−	0.940602i	$-0.389738\pi$
$439$	14.2271	0.679024	0.339512	+	0.940602i	$0.389738\pi$
$440$	0	0
$441$	−2.75631	−0.131253
$442$	15.0550	0.716092
$443$	27.0868	1.28693	0.643467	−	0.765474i	$-0.277495\pi$
$443$	27.0868	1.28693	0.643467	+	0.765474i	$0.277495\pi$
$444$	0.754379	0.0358012
$445$	0	0
$446$	15.5770	0.737593
$447$	−4.69793	−0.222205
$448$	8.15210	0.385151
$449$	10.9937	0.518823	0.259411	−	0.965767i	$-0.416471\pi$
$449$	10.9937	0.518823	0.259411	+	0.965767i	$0.416471\pi$
$450$	0	0
$451$	3.29399	0.155108
$452$	−2.67524	−0.125833
$453$	4.20720	0.197671
$454$	8.32844	0.390873
$455$	0	0
$456$	−3.45411	−0.161754
$457$	−12.4143	−0.580715	−0.290358	−	0.956918i	$-0.593774\pi$
$457$	−12.4143	−0.580715	−0.290358	+	0.956918i	$0.593774\pi$
$458$	24.6392	1.15131
$459$	9.39489	0.438516
$460$	0	0
$461$	24.1837	1.12635	0.563175	−	0.826338i	$-0.309580\pi$
$461$	24.1837	1.12635	0.563175	+	0.826338i	$0.309580\pi$
$462$	−0.925758	−0.0430701
$463$	19.4383	0.903376	0.451688	−	0.892176i	$-0.350822\pi$
$463$	19.4383	0.903376	0.451688	+	0.892176i	$0.350822\pi$
$464$	−5.74759	−0.266825
$465$	0	0
$466$	−1.26149	−0.0584375
$467$	−22.1898	−1.02682	−0.513411	−	0.858143i	$-0.671618\pi$
$467$	−22.1898	−1.02682	−0.513411	+	0.858143i	$0.671618\pi$
$468$	−9.08030	−0.419737
$469$	1.05619	0.0487703
$470$	0	0
$471$	−5.69761	−0.262532
$472$	−27.6615	−1.27322
$473$	11.0258	0.506967
$474$	7.43812	0.341644
$475$	0	0
$476$	−2.62634	−0.120378
$477$	8.91414	0.408150
$478$	25.2596	1.15535
$479$	−26.5405	−1.21267	−0.606334	−	0.795210i	$-0.707360\pi$
$479$	−26.5405	−1.21267	−0.606334	+	0.795210i	$0.707360\pi$
$480$	0	0
$481$	−7.97793	−0.363762
$482$	23.4007	1.06587
$483$	0.493654	0.0224620
$484$	6.42092	0.291860
$485$	0	0
$486$	13.0821	0.593416
$487$	−16.7043	−0.756945	−0.378472	−	0.925613i	$-0.623550\pi$
$487$	−16.7043	−0.756945	−0.378472	+	0.925613i	$0.623550\pi$
$488$	13.4395	0.608379
$489$	5.34291	0.241615
$490$	0	0
$491$	35.1859	1.58792	0.793960	−	0.607970i	$-0.208016\pi$
$491$	35.1859	1.58792	0.793960	+	0.607970i	$0.208016\pi$
$492$	−0.756314	−0.0340973
$493$	10.6742	0.480743
$494$	10.3843	0.467212
$495$	0	0
$496$	−10.4868	−0.470873
$497$	11.7201	0.525719
$498$	−4.65010	−0.208376
$499$	−13.9565	−0.624777	−0.312389	−	0.949954i	$-0.601129\pi$
$499$	−13.9565	−0.624777	−0.312389	+	0.949954i	$0.601129\pi$
$500$	0	0
$501$	2.92817	0.130821
$502$	−11.6234	−0.518776
$503$	−34.5791	−1.54181	−0.770904	−	0.636952i	$-0.780195\pi$
$503$	−34.5791	−1.54181	−0.770904	+	0.636952i	$0.780195\pi$
$504$	−8.45704	−0.376706
$505$	0	0
$506$	−1.87532	−0.0833680
$507$	−2.07272	−0.0920527
$508$	−1.78096	−0.0790175
$509$	24.3286	1.07834	0.539172	−	0.842195i	$-0.318737\pi$
$509$	24.3286	1.07834	0.539172	+	0.842195i	$0.318737\pi$
$510$	0	0
$511$	−13.6891	−0.605571
$512$	18.3689	0.811797
$513$	6.48021	0.286108
$514$	21.7285	0.958405
$515$	0	0
$516$	−2.53157	−0.111446
$517$	−9.08223	−0.399436
$518$	−2.11226	−0.0928076
$519$	−7.77944	−0.341480
$520$	0	0
$521$	3.43542	0.150509	0.0752543	−	0.997164i	$-0.476023\pi$
$521$	3.43542	0.150509	0.0752543	+	0.997164i	$0.476023\pi$
$522$	9.77114	0.427671
$523$	18.1301	0.792776	0.396388	−	0.918083i	$-0.370264\pi$
$523$	18.1301	0.792776	0.396388	+	0.918083i	$0.370264\pi$
$524$	−7.85426	−0.343115
$525$	0	0
$526$	−5.66199	−0.246875
$527$	19.4758	0.848380
$528$	−1.50095	−0.0653203
$529$	1.00000	0.0434783
$530$	0	0
$531$	24.8492	1.07836
$532$	−1.81154	−0.0785401
$533$	7.99839	0.346449
$534$	−1.79097	−0.0775028
$535$	0	0
$536$	3.24065	0.139975
$537$	10.7021	0.461831
$538$	5.40818	0.233163
$539$	−1.70792	−0.0735654
$540$	0	0
$541$	−2.46086	−0.105801	−0.0529003	−	0.998600i	$-0.516847\pi$
$541$	−2.46086	−0.105801	−0.0529003	+	0.998600i	$0.516847\pi$
$542$	8.94889	0.384388
$543$	−11.1776	−0.479675
$544$	−13.8257	−0.592774
$545$	0	0
$546$	−2.24790	−0.0962013
$547$	−34.6349	−1.48088	−0.740442	−	0.672121i	$-0.765384\pi$
$547$	−34.6349	−1.48088	−0.740442	+	0.672121i	$0.765384\pi$
$548$	14.4569	0.617567
$549$	−12.0732	−0.515270
$550$	0	0
$551$	7.36265	0.313659
$552$	1.51465	0.0644680
$553$	13.7225	0.583541
$554$	14.8161	0.629474
$555$	0	0
$556$	0.0737693	0.00312851
$557$	14.6447	0.620514	0.310257	−	0.950653i	$-0.399585\pi$
$557$	14.6447	0.620514	0.310257	+	0.950653i	$0.399585\pi$
$558$	17.8281	0.754722
$559$	26.7726	1.13236
$560$	0	0
$561$	2.78751	0.117689
$562$	18.9554	0.799586
$563$	−18.0454	−0.760524	−0.380262	−	0.924879i	$-0.624166\pi$
$563$	−18.0454	−0.760524	−0.380262	+	0.924879i	$0.624166\pi$
$564$	2.08532	0.0878077
$565$	0	0
$566$	30.1282	1.26638
$567$	6.86614	0.288351
$568$	35.9603	1.50886
$569$	11.1181	0.466093	0.233046	−	0.972466i	$-0.425131\pi$
$569$	11.1181	0.466093	0.233046	+	0.972466i	$0.425131\pi$
$570$	0	0
$571$	−3.11104	−0.130193	−0.0650966	−	0.997879i	$-0.520736\pi$
$571$	−3.11104	−0.130193	−0.0650966	+	0.997879i	$0.520736\pi$
$572$	−5.62654	−0.235257
$573$	4.36532	0.182364
$574$	2.11768	0.0883904
$575$	0	0
$576$	−22.4697	−0.936237
$577$	32.5581	1.35541	0.677707	−	0.735333i	$-0.262974\pi$
$577$	32.5581	1.35541	0.677707	+	0.735333i	$0.262974\pi$
$578$	6.66408	0.277189
$579$	12.7556	0.530103
$580$	0	0
$581$	−8.57893	−0.355914
$582$	−5.66927	−0.234999
$583$	5.52357	0.228763
$584$	−42.0016	−1.73804
$585$	0	0
$586$	4.34149	0.179345
$587$	4.52127	0.186613	0.0933065	−	0.995637i	$-0.470256\pi$
$587$	4.52127	0.186613	0.0933065	+	0.995637i	$0.470256\pi$
$588$	0.392146	0.0161718
$589$	13.4336	0.553523
$590$	0	0
$591$	−4.84383	−0.199249
$592$	−3.42465	−0.140752
$593$	40.0455	1.64447	0.822236	−	0.569147i	$-0.192726\pi$
$593$	40.0455	1.64447	0.822236	+	0.569147i	$0.192726\pi$
$594$	5.32894	0.218649
$595$	0	0
$596$	−7.55977	−0.309660
$597$	4.19220	0.171575
$598$	−4.55360	−0.186210
$599$	1.72966	0.0706721	0.0353361	−	0.999375i	$-0.488750\pi$
$599$	1.72966	0.0706721	0.0353361	+	0.999375i	$0.488750\pi$
$600$	0	0
$601$	−28.8314	−1.17606	−0.588029	−	0.808840i	$-0.700096\pi$
$601$	−28.8314	−1.17606	−0.588029	+	0.808840i	$0.700096\pi$
$602$	7.08840	0.288902
$603$	−2.91118	−0.118552
$604$	6.77010	0.275471
$605$	0	0
$606$	−2.85987	−0.116174
$607$	2.16493	0.0878719	0.0439359	−	0.999034i	$-0.486010\pi$
$607$	2.16493	0.0878719	0.0439359	+	0.999034i	$0.486010\pi$
$608$	−9.53643	−0.386753
$609$	−1.59380	−0.0645841
$610$	0	0
$611$	−22.0533	−0.892179
$612$	7.23898	0.292619
$613$	−33.8646	−1.36778	−0.683890	−	0.729585i	$-0.739713\pi$
$613$	−33.8646	−1.36778	−0.683890	+	0.729585i	$0.739713\pi$
$614$	−13.4246	−0.541771
$615$	0	0
$616$	−5.24033	−0.211139
$617$	−18.7033	−0.752965	−0.376482	−	0.926424i	$-0.622866\pi$
$617$	−18.7033	−0.752965	−0.376482	+	0.926424i	$0.622866\pi$
$618$	1.73030	0.0696027
$619$	13.4203	0.539407	0.269703	−	0.962943i	$-0.413074\pi$
$619$	13.4203	0.539407	0.269703	+	0.962943i	$0.413074\pi$
$620$	0	0
$621$	−2.84162	−0.114030
$622$	7.08624	0.284133
$623$	−3.30414	−0.132378
$624$	−3.64456	−0.145899
$625$	0	0
$626$	13.3712	0.534420
$627$	1.92271	0.0767856
$628$	−9.16842	−0.365860
$629$	6.36015	0.253596
$630$	0	0
$631$	−28.6674	−1.14123	−0.570615	−	0.821218i	$-0.693295\pi$
$631$	−28.6674	−1.14123	−0.570615	+	0.821218i	$0.693295\pi$
$632$	42.1042	1.67481
$633$	−11.7970	−0.468889
$634$	−22.8410	−0.907132
$635$	0	0
$636$	−1.26823	−0.0502888
$637$	−4.14713	−0.164315
$638$	6.05461	0.239704
$639$	−32.3042	−1.27794
$640$	0	0
$641$	−32.2452	−1.27361	−0.636805	−	0.771025i	$-0.719744\pi$
$641$	−32.2452	−1.27361	−0.636805	+	0.771025i	$0.719744\pi$
$642$	0.755914	0.0298335
$643$	−45.0160	−1.77526	−0.887629	−	0.460559i	$-0.847649\pi$
$643$	−45.0160	−1.77526	−0.887629	+	0.460559i	$0.847649\pi$
$644$	0.794374	0.0313027
$645$	0	0
$646$	−8.27855	−0.325715
$647$	−21.3550	−0.839553	−0.419777	−	0.907627i	$-0.637892\pi$
$647$	−21.3550	−0.839553	−0.419777	+	0.907627i	$0.637892\pi$
$648$	21.0670	0.827591
$649$	15.3976	0.604408
$650$	0	0
$651$	−2.90799	−0.113973
$652$	8.59766	0.336710
$653$	10.5192	0.411648	0.205824	−	0.978589i	$-0.434013\pi$
$653$	10.5192	0.411648	0.205824	+	0.978589i	$0.434013\pi$
$654$	5.79874	0.226749
$655$	0	0
$656$	3.43344	0.134053
$657$	37.7314	1.47204
$658$	−5.83890	−0.227624
$659$	37.7075	1.46888	0.734438	−	0.678675i	$-0.237446\pi$
$659$	37.7075	1.46888	0.734438	+	0.678675i	$0.237446\pi$
$660$	0	0
$661$	−9.71879	−0.378017	−0.189009	−	0.981975i	$-0.560527\pi$
$661$	−9.71879	−0.378017	−0.189009	+	0.981975i	$0.560527\pi$
$662$	−2.87657	−0.111801
$663$	6.76856	0.262869
$664$	−26.3223	−1.02150
$665$	0	0
$666$	5.82205	0.225600
$667$	−3.22858	−0.125011
$668$	4.71192	0.182310
$669$	7.00327	0.270762
$670$	0	0
$671$	−7.48103	−0.288802
$672$	2.06436	0.0796345
$673$	32.4376	1.25038	0.625189	−	0.780473i	$-0.285022\pi$
$673$	32.4376	1.25038	0.625189	+	0.780473i	$0.285022\pi$
$674$	28.6072	1.10191
$675$	0	0
$676$	−3.33536	−0.128283
$677$	−44.7583	−1.72020	−0.860101	−	0.510124i	$-0.829599\pi$
$677$	−44.7583	−1.72020	−0.860101	+	0.510124i	$0.829599\pi$
$678$	1.82544	0.0701056
$679$	−10.4592	−0.401387
$680$	0	0
$681$	3.74438	0.143485
$682$	11.0470	0.423012
$683$	−26.3174	−1.00701	−0.503503	−	0.863993i	$-0.667956\pi$
$683$	−26.3174	−1.00701	−0.503503	+	0.863993i	$0.667956\pi$
$684$	4.99315	0.190918
$685$	0	0
$686$	−1.09801	−0.0419222
$687$	11.0775	0.422633
$688$	11.4926	0.438149
$689$	13.4122	0.510964
$690$	0	0
$691$	−20.2738	−0.771250	−0.385625	−	0.922656i	$-0.626014\pi$
$691$	−20.2738	−0.771250	−0.385625	+	0.922656i	$0.626014\pi$
$692$	−12.5185	−0.475880
$693$	4.70756	0.178825
$694$	15.3132	0.581280
$695$	0	0
$696$	−4.89018	−0.185362
$697$	−6.37646	−0.241526
$698$	8.87558	0.335946
$699$	−0.567154	−0.0214517
$700$	0	0
$701$	22.3016	0.842318	0.421159	−	0.906987i	$-0.361623\pi$
$701$	22.3016	0.842318	0.421159	+	0.906987i	$0.361623\pi$
$702$	12.9396	0.488374
$703$	4.38697	0.165458
$704$	−13.9232	−0.524749
$705$	0	0
$706$	10.6410	0.400481
$707$	−5.27615	−0.198430
$708$	−3.53535	−0.132866
$709$	−33.5683	−1.26068	−0.630342	−	0.776317i	$-0.717085\pi$
$709$	−33.5683	−1.26068	−0.630342	+	0.776317i	$0.717085\pi$
$710$	0	0
$711$	−37.8235	−1.41849
$712$	−10.1379	−0.379935
$713$	−5.89075	−0.220610
$714$	1.79207	0.0670664
$715$	0	0
$716$	17.2215	0.643599
$717$	11.3564	0.424114
$718$	13.6601	0.509791
$719$	31.1917	1.16326	0.581628	−	0.813455i	$-0.302416\pi$
$719$	31.1917	1.16326	0.581628	+	0.813455i	$0.302416\pi$
$720$	0	0
$721$	3.19221	0.118884
$722$	15.1520	0.563899
$723$	10.5207	0.391269
$724$	−17.9866	−0.668467
$725$	0	0
$726$	−4.38129	−0.162605
$727$	4.59454	0.170402	0.0852010	−	0.996364i	$-0.472847\pi$
$727$	4.59454	0.170402	0.0852010	+	0.996364i	$0.472847\pi$
$728$	−12.7244	−0.471599
$729$	−14.7168	−0.545068
$730$	0	0
$731$	−21.3436	−0.789421
$732$	1.71768	0.0634871
$733$	21.0857	0.778818	0.389409	−	0.921065i	$-0.372679\pi$
$733$	21.0857	0.778818	0.389409	+	0.921065i	$0.372679\pi$
$734$	8.05133	0.297180
$735$	0	0
$736$	4.18180	0.154143
$737$	−1.80389	−0.0664471
$738$	−5.83698	−0.214862
$739$	9.13608	0.336076	0.168038	−	0.985780i	$-0.446257\pi$
$739$	9.13608	0.336076	0.168038	+	0.985780i	$0.446257\pi$
$740$	0	0
$741$	4.66867	0.171508
$742$	3.55106	0.130364
$743$	−22.1049	−0.810951	−0.405476	−	0.914106i	$-0.632894\pi$
$743$	−22.1049	−0.810951	−0.405476	+	0.914106i	$0.632894\pi$
$744$	−8.92245	−0.327113
$745$	0	0
$746$	2.04984	0.0750498
$747$	23.6461	0.865167
$748$	4.48558	0.164009
$749$	1.39458	0.0509568
$750$	0	0
$751$	−34.5623	−1.26119	−0.630597	−	0.776110i	$-0.717190\pi$
$751$	−34.5623	−1.26119	−0.630597	+	0.776110i	$0.717190\pi$
$752$	−9.46671	−0.345215
$753$	−5.22574	−0.190437
$754$	14.7016	0.535402
$755$	0	0
$756$	−2.25731	−0.0820976
$757$	−7.87248	−0.286130	−0.143065	−	0.989713i	$-0.545696\pi$
$757$	−7.87248	−0.286130	−0.143065	+	0.989713i	$0.545696\pi$
$758$	6.53079	0.237209
$759$	−0.843123	−0.0306034
$760$	0	0
$761$	13.5062	0.489601	0.244801	−	0.969573i	$-0.421278\pi$
$761$	13.5062	0.489601	0.244801	+	0.969573i	$0.421278\pi$
$762$	1.21523	0.0440232
$763$	10.6980	0.387295
$764$	7.02454	0.254139
$765$	0	0
$766$	−2.24439	−0.0810930
$767$	37.3880	1.35000
$768$	7.73019	0.278939
$769$	−15.0405	−0.542374	−0.271187	−	0.962527i	$-0.587416\pi$
$769$	−15.0405	−0.542374	−0.271187	+	0.962527i	$0.587416\pi$
$770$	0	0
$771$	9.76893	0.351819
$772$	20.5259	0.738742
$773$	47.1901	1.69731	0.848654	−	0.528948i	$-0.177413\pi$
$773$	47.1901	1.69731	0.848654	+	0.528948i	$0.177413\pi$
$774$	−19.5378	−0.702272
$775$	0	0
$776$	−32.0914	−1.15202
$777$	−0.949653	−0.0340686
$778$	0.850082	0.0304769
$779$	−4.39822	−0.157583
$780$	0	0
$781$	−20.0171	−0.716267
$782$	3.63021	0.129816
$783$	9.17441	0.327867
$784$	−1.78022	−0.0635794
$785$	0	0
$786$	5.35931	0.191160
$787$	32.7302	1.16670	0.583352	−	0.812219i	$-0.301741\pi$
$787$	32.7302	1.16670	0.583352	+	0.812219i	$0.301741\pi$
$788$	−7.79455	−0.277670
$789$	−2.54557	−0.0906249
$790$	0	0
$791$	3.36774	0.119743
$792$	14.4440	0.513244
$793$	−18.1653	−0.645067
$794$	27.2092	0.965619
$795$	0	0
$796$	6.74597	0.239105
$797$	−42.9287	−1.52061	−0.760306	−	0.649565i	$-0.774951\pi$
$797$	−42.9287	−1.52061	−0.760306	+	0.649565i	$0.774951\pi$
$798$	1.23609	0.0437573
$799$	17.5813	0.621980
$800$	0	0
$801$	9.10723	0.321788
$802$	11.8939	0.419988
$803$	23.3799	0.825060
$804$	0.414180	0.0146070
$805$	0	0
$806$	26.8241	0.944838
$807$	2.43146	0.0855915
$808$	−16.1885	−0.569511
$809$	0.599782	0.0210872	0.0105436	−	0.999944i	$-0.496644\pi$
$809$	0.599782	0.0210872	0.0105436	+	0.999944i	$0.496644\pi$
$810$	0	0
$811$	4.78801	0.168130	0.0840649	−	0.996460i	$-0.473210\pi$
$811$	4.78801	0.168130	0.0840649	+	0.996460i	$0.473210\pi$
$812$	−2.56470	−0.0900033
$813$	4.02333	0.141104
$814$	3.60758	0.126446
$815$	0	0
$816$	2.90551	0.101713
$817$	−14.7219	−0.515055
$818$	−8.90850	−0.311478
$819$	11.4308	0.399423
$820$	0	0
$821$	4.17656	0.145763	0.0728814	−	0.997341i	$-0.476781\pi$
$821$	4.17656	0.145763	0.0728814	+	0.997341i	$0.476781\pi$
$822$	−9.86457	−0.344066
$823$	27.6949	0.965383	0.482692	−	0.875790i	$-0.339659\pi$
$823$	27.6949	0.965383	0.482692	+	0.875790i	$0.339659\pi$
$824$	9.79450	0.341208
$825$	0	0
$826$	9.89898	0.344430
$827$	−41.1730	−1.43173	−0.715863	−	0.698241i	$-0.753966\pi$
$827$	−41.1730	−1.43173	−0.715863	+	0.698241i	$0.753966\pi$
$828$	−2.18954	−0.0760917
$829$	15.3651	0.533651	0.266826	−	0.963745i	$-0.414025\pi$
$829$	15.3651	0.533651	0.266826	+	0.963745i	$0.414025\pi$
$830$	0	0
$831$	6.66115	0.231073
$832$	−33.8079	−1.17208
$833$	3.30617	0.114552
$834$	−0.0503361	−0.00174300
$835$	0	0
$836$	3.09397	0.107007
$837$	16.7393	0.578595
$838$	−4.60756	−0.159165
$839$	27.5235	0.950216	0.475108	−	0.879927i	$-0.342409\pi$
$839$	27.5235	0.950216	0.475108	+	0.879927i	$0.342409\pi$
$840$	0	0
$841$	−18.5763	−0.640561
$842$	21.3911	0.737188
$843$	8.52216	0.293519
$844$	−18.9834	−0.653435
$845$	0	0
$846$	16.0938	0.553316
$847$	−8.08300	−0.277735
$848$	5.75740	0.197710
$849$	13.5453	0.464875
$850$	0	0
$851$	−1.92372	−0.0659443
$852$	4.59600	0.157456
$853$	−24.0506	−0.823476	−0.411738	−	0.911302i	$-0.635078\pi$
$853$	−24.0506	−0.823476	−0.411738	+	0.911302i	$0.635078\pi$
$854$	−4.80950	−0.164578
$855$	0	0
$856$	4.27892	0.146250
$857$	32.4249	1.10761	0.553807	−	0.832645i	$-0.313175\pi$
$857$	32.4249	1.10761	0.553807	+	0.832645i	$0.313175\pi$
$858$	3.83924	0.131070
$859$	−39.6874	−1.35412	−0.677059	−	0.735929i	$-0.736746\pi$
$859$	−39.6874	−1.35412	−0.677059	+	0.735929i	$0.736746\pi$
$860$	0	0
$861$	0.952088	0.0324471
$862$	−24.4110	−0.831443
$863$	28.0426	0.954581	0.477290	−	0.878746i	$-0.341619\pi$
$863$	28.0426	0.954581	0.477290	+	0.878746i	$0.341619\pi$
$864$	−11.8831	−0.404271
$865$	0	0
$866$	−39.6854	−1.34857
$867$	2.99610	0.101753
$868$	−4.67946	−0.158831
$869$	−23.4370	−0.795046
$870$	0	0
$871$	−4.38016	−0.148416
$872$	32.8243	1.11157
$873$	28.8287	0.975705
$874$	2.50397	0.0846980
$875$	0	0
$876$	−5.36813	−0.181372
$877$	−19.8953	−0.671816	−0.335908	−	0.941895i	$-0.609043\pi$
$877$	−19.8953	−0.671816	−0.335908	+	0.941895i	$0.609043\pi$
$878$	−15.6215	−0.527202
$879$	1.95189	0.0658355
$880$	0	0
$881$	44.8609	1.51140	0.755701	−	0.654917i	$-0.227296\pi$
$881$	44.8609	1.51140	0.755701	+	0.654917i	$0.227296\pi$
$882$	3.02645	0.101906
$883$	−2.28574	−0.0769212	−0.0384606	−	0.999260i	$-0.512245\pi$
$883$	−2.28574	−0.0769212	−0.0384606	+	0.999260i	$0.512245\pi$
$884$	10.8918	0.366330
$885$	0	0
$886$	−29.7416	−0.999189
$887$	−9.22524	−0.309753	−0.154877	−	0.987934i	$-0.549498\pi$
$887$	−9.22524	−0.309753	−0.154877	+	0.987934i	$0.549498\pi$
$888$	−2.91377	−0.0977798
$889$	2.24197	0.0751934
$890$	0	0
$891$	−11.7268	−0.392864
$892$	11.2695	0.377329
$893$	12.1268	0.405809
$894$	5.15838	0.172522
$895$	0	0
$896$	−0.587494	−0.0196268
$897$	−2.04725	−0.0683557
$898$	−12.0711	−0.402819
$899$	19.0188	0.634311
$900$	0	0
$901$	−10.6924	−0.356217
$902$	−3.61684	−0.120428
$903$	3.18687	0.106053
$904$	10.3331	0.343673
$905$	0	0
$906$	−4.61955	−0.153474
$907$	31.9544	1.06103	0.530515	−	0.847676i	$-0.321999\pi$
$907$	31.9544	1.06103	0.530515	+	0.847676i	$0.321999\pi$
$908$	6.02535	0.199958
$909$	14.5427	0.482350
$910$	0	0
$911$	−18.4386	−0.610897	−0.305448	−	0.952209i	$-0.598806\pi$
$911$	−18.4386	−0.610897	−0.305448	+	0.952209i	$0.598806\pi$
$912$	2.00410	0.0663624
$913$	14.6521	0.484915
$914$	13.6310	0.450873
$915$	0	0
$916$	17.8256	0.588975
$917$	9.88736	0.326509
$918$	−10.3157	−0.340468
$919$	47.8031	1.57688	0.788440	−	0.615112i	$-0.210889\pi$
$919$	47.8031	1.57688	0.788440	+	0.615112i	$0.210889\pi$
$920$	0	0
$921$	−6.03555	−0.198878
$922$	−26.5540	−0.874509
$923$	−48.6049	−1.59985
$924$	−0.669755	−0.0220333
$925$	0	0
$926$	−21.3435	−0.701390
$927$	−8.79871	−0.288987
$928$	−13.5013	−0.443201
$929$	49.5998	1.62732	0.813658	−	0.581344i	$-0.197473\pi$
$929$	49.5998	1.62732	0.813658	+	0.581344i	$0.197473\pi$
$930$	0	0
$931$	2.28046	0.0747391
$932$	−0.912647	−0.0298948
$933$	3.18590	0.104302
$934$	24.3646	0.797236
$935$	0	0
$936$	35.0725	1.14638
$937$	−32.5744	−1.06416	−0.532079	−	0.846695i	$-0.678589\pi$
$937$	−32.5744	−1.06416	−0.532079	+	0.846695i	$0.678589\pi$
$938$	−1.15971	−0.0378658
$939$	6.01155	0.196180
$940$	0	0
$941$	42.1235	1.37319	0.686593	−	0.727042i	$-0.259105\pi$
$941$	42.1235	1.37319	0.686593	+	0.727042i	$0.259105\pi$
$942$	6.25603	0.203832
$943$	1.92865	0.0628056
$944$	16.0494	0.522363
$945$	0	0
$946$	−12.1064	−0.393615
$947$	36.1706	1.17539	0.587694	−	0.809084i	$-0.300036\pi$
$947$	36.1706	1.17539	0.587694	+	0.809084i	$0.300036\pi$
$948$	5.38123	0.174774
$949$	56.7706	1.84285
$950$	0	0
$951$	−10.2691	−0.332998
$952$	10.1442	0.328774
$953$	43.0942	1.39596	0.697979	−	0.716118i	$-0.254083\pi$
$953$	43.0942	1.39596	0.697979	+	0.716118i	$0.254083\pi$
$954$	−9.78781	−0.316892
$955$	0	0
$956$	18.2745	0.591038
$957$	2.72209	0.0879927
$958$	29.1418	0.941528
$959$	−18.1991	−0.587679
$960$	0	0
$961$	3.70092	0.119385
$962$	8.75985	0.282429
$963$	−3.84389	−0.123867
$964$	16.9296	0.545266
$965$	0	0
$966$	−0.542037	−0.0174398
$967$	−36.9112	−1.18698	−0.593492	−	0.804840i	$-0.702251\pi$
$967$	−36.9112	−1.18698	−0.593492	+	0.804840i	$0.702251\pi$
$968$	−24.8007	−0.797124
$969$	−3.72195	−0.119566
$970$	0	0
$971$	25.8982	0.831112	0.415556	−	0.909568i	$-0.363587\pi$
$971$	25.8982	0.831112	0.415556	+	0.909568i	$0.363587\pi$
$972$	9.46446	0.303573
$973$	−0.0928647	−0.00297711
$974$	18.3415	0.587700
$975$	0	0
$976$	−7.79772	−0.249599
$977$	14.4726	0.463018	0.231509	−	0.972833i	$-0.425634\pi$
$977$	14.4726	0.463018	0.231509	+	0.972833i	$0.425634\pi$
$978$	−5.86657	−0.187592
$979$	5.64322	0.180358
$980$	0	0
$981$	−29.4871	−0.941450
$982$	−38.6345	−1.23288
$983$	−1.36113	−0.0434132	−0.0217066	−	0.999764i	$-0.506910\pi$
$983$	−1.36113	−0.0434132	−0.0217066	+	0.999764i	$0.506910\pi$
$984$	2.92125	0.0931259
$985$	0	0
$986$	−11.7204	−0.373254
$987$	−2.62511	−0.0835581
$988$	7.51269	0.239011
$989$	6.45568	0.205279
$990$	0	0
$991$	59.3459	1.88519	0.942593	−	0.333945i	$-0.108380\pi$
$991$	59.3459	1.88519	0.942593	+	0.333945i	$0.108380\pi$
$992$	−24.6339	−0.782128
$993$	−1.29328	−0.0410409
$994$	−12.8688	−0.408174
$995$	0	0
$996$	−3.36419	−0.106598
$997$	29.2437	0.926159	0.463079	−	0.886317i	$-0.346745\pi$
$997$	29.2437	0.926159	0.463079	+	0.886317i	$0.346745\pi$
$998$	15.3243	0.485084
$999$	5.46649	0.172952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.u.1.3	✓	8
5.4	even	2		4025.2.a.v.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4025.2.a.u.1.3	✓	8	1.1	even	1	trivial
4025.2.a.v.1.6	yes	8	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.09801 q^{2} -0.493654 q^{3} -0.794374 q^{4} +0.542037 q^{6} +1.00000 q^{7} +3.06825 q^{8} -2.75631 q^{9} +O(q^{10})\)	\(q-1.09801 q^{2} -0.493654 q^{3} -0.794374 q^{4} +0.542037 q^{6} +1.00000 q^{7} +3.06825 q^{8} -2.75631 q^{9} -1.70792 q^{11} +0.392146 q^{12} -4.14713 q^{13} -1.09801 q^{14} -1.78022 q^{16} +3.30617 q^{17} +3.02645 q^{18} +2.28046 q^{19} -0.493654 q^{21} +1.87532 q^{22} -1.00000 q^{23} -1.51465 q^{24} +4.55360 q^{26} +2.84162 q^{27} -0.794374 q^{28} +3.22858 q^{29} +5.89075 q^{31} -4.18180 q^{32} +0.843123 q^{33} -3.63021 q^{34} +2.18954 q^{36} +1.92372 q^{37} -2.50397 q^{38} +2.04725 q^{39} -1.92865 q^{41} +0.542037 q^{42} -6.45568 q^{43} +1.35673 q^{44} +1.09801 q^{46} +5.31771 q^{47} +0.878814 q^{48} +1.00000 q^{49} -1.63210 q^{51} +3.29438 q^{52} -3.23409 q^{53} -3.12013 q^{54} +3.06825 q^{56} -1.12576 q^{57} -3.54501 q^{58} -9.01538 q^{59} +4.38020 q^{61} -6.46810 q^{62} -2.75631 q^{63} +8.15210 q^{64} -0.925758 q^{66} +1.05619 q^{67} -2.62634 q^{68} +0.493654 q^{69} +11.7201 q^{71} -8.45704 q^{72} -13.6891 q^{73} -2.11226 q^{74} -1.81154 q^{76} -1.70792 q^{77} -2.24790 q^{78} +13.7225 q^{79} +6.86614 q^{81} +2.11768 q^{82} -8.57893 q^{83} +0.392146 q^{84} +7.08840 q^{86} -1.59380 q^{87} -5.24033 q^{88} -3.30414 q^{89} -4.14713 q^{91} +0.794374 q^{92} -2.90799 q^{93} -5.83890 q^{94} +2.06436 q^{96} -10.4592 q^{97} -1.09801 q^{98} +4.70756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7}+O(q^{10}) \) 8 * q - q^2 - 4 * q^3 + 5 * q^4 - q^6 + 8 * q^7	\( 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7} + 3 q^{11} - 9 q^{12} - 5 q^{13} - q^{14} - q^{16} - 5 q^{17} - 2 q^{18} - 2 q^{19} - 4 q^{21} - 21 q^{22} - 8 q^{23} - 6 q^{24} + 18 q^{26} - 7 q^{27} + 5 q^{28} - 9 q^{29} - 3 q^{31} - 6 q^{32} - 4 q^{33} - 10 q^{34} + 16 q^{36} - 6 q^{37} - 4 q^{38} - 2 q^{39} - 7 q^{41} - q^{42} - 8 q^{43} + 4 q^{44} + q^{46} - 22 q^{47} - 9 q^{48} + 8 q^{49} - 12 q^{51} - 11 q^{52} - 21 q^{53} - 15 q^{54} - 8 q^{57} - 16 q^{58} + 14 q^{59} + 8 q^{61} - 12 q^{62} - 40 q^{64} + 55 q^{66} - 21 q^{67} - 3 q^{68} + 4 q^{69} + 11 q^{71} + q^{72} - 26 q^{73} - 41 q^{74} + 21 q^{76} + 3 q^{77} - 17 q^{78} - 16 q^{79} - 20 q^{81} + q^{82} - 20 q^{83} - 9 q^{84} + 14 q^{86} + 29 q^{87} - 32 q^{88} + 15 q^{89} - 5 q^{91} - 5 q^{92} - 19 q^{93} + 21 q^{94} + 52 q^{96} - q^{97} - q^{98} + 15 q^{99}+O(q^{100}) \) 8 * q - q^2 - 4 * q^3 + 5 * q^4 - q^6 + 8 * q^7 + 3 * q^11 - 9 * q^12 - 5 * q^13 - q^14 - q^16 - 5 * q^17 - 2 * q^18 - 2 * q^19 - 4 * q^21 - 21 * q^22 - 8 * q^23 - 6 * q^24 + 18 * q^26 - 7 * q^27 + 5 * q^28 - 9 * q^29 - 3 * q^31 - 6 * q^32 - 4 * q^33 - 10 * q^34 + 16 * q^36 - 6 * q^37 - 4 * q^38 - 2 * q^39 - 7 * q^41 - q^42 - 8 * q^43 + 4 * q^44 + q^46 - 22 * q^47 - 9 * q^48 + 8 * q^49 - 12 * q^51 - 11 * q^52 - 21 * q^53 - 15 * q^54 - 8 * q^57 - 16 * q^58 + 14 * q^59 + 8 * q^61 - 12 * q^62 - 40 * q^64 + 55 * q^66 - 21 * q^67 - 3 * q^68 + 4 * q^69 + 11 * q^71 + q^72 - 26 * q^73 - 41 * q^74 + 21 * q^76 + 3 * q^77 - 17 * q^78 - 16 * q^79 - 20 * q^81 + q^82 - 20 * q^83 - 9 * q^84 + 14 * q^86 + 29 * q^87 - 32 * q^88 + 15 * q^89 - 5 * q^91 - 5 * q^92 - 19 * q^93 + 21 * q^94 + 52 * q^96 - q^97 - q^98 + 15 * q^99

Introduction

L-functions

Modular forms

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Database

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