LMFDB - Embedded newform 3525.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2379008.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} + 23x - 4 $ x^5 - 10x^3 + 23x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.15072$ of defining polynomial
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15072 q^{2} -1.00000 q^{3} +2.62562 q^{4} +2.15072 q^{6} -2.17958 q^{7} -1.34553 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.15072 q^{2} -1.00000 q^{3} +2.62562 q^{4} +2.15072 q^{6} -2.17958 q^{7} -1.34553 q^{8} +1.00000 q^{9} -5.34213 q^{11} -2.62562 q^{12} -4.81703 q^{13} +4.68767 q^{14} -2.35737 q^{16} +4.15072 q^{17} -2.15072 q^{18} +2.38622 q^{19} +2.17958 q^{21} +11.4895 q^{22} +3.53694 q^{23} +1.34553 q^{24} +10.3601 q^{26} -1.00000 q^{27} -5.72273 q^{28} -5.96775 q^{29} +7.87908 q^{31} +7.76111 q^{32} +5.34213 q^{33} -8.92707 q^{34} +2.62562 q^{36} +4.95252 q^{37} -5.13210 q^{38} +4.81703 q^{39} +8.33643 q^{41} -4.68767 q^{42} +3.32690 q^{43} -14.0264 q^{44} -7.60699 q^{46} -1.00000 q^{47} +2.35737 q^{48} -2.24945 q^{49} -4.15072 q^{51} -12.6477 q^{52} +6.56630 q^{53} +2.15072 q^{54} +2.93269 q^{56} -2.38622 q^{57} +12.8350 q^{58} +2.30376 q^{59} -7.32512 q^{61} -16.9457 q^{62} -2.17958 q^{63} -11.9773 q^{64} -11.4895 q^{66} -8.32639 q^{67} +10.8982 q^{68} -3.53694 q^{69} +10.9067 q^{71} -1.34553 q^{72} -10.6650 q^{73} -10.6515 q^{74} +6.26529 q^{76} +11.6436 q^{77} -10.3601 q^{78} -5.92707 q^{79} +1.00000 q^{81} -17.9294 q^{82} +3.40936 q^{83} +5.72273 q^{84} -7.15525 q^{86} +5.96775 q^{87} +7.18801 q^{88} -0.138890 q^{89} +10.4991 q^{91} +9.28666 q^{92} -7.87908 q^{93} +2.15072 q^{94} -7.76111 q^{96} +18.2370 q^{97} +4.83795 q^{98} -5.34213 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9} - 2 q^{11} - 10 q^{12} - 7 q^{13} - 8 q^{14} + 8 q^{16} + 10 q^{17} + 2 q^{19} + 10 q^{21} + 4 q^{22} - 3 q^{23} - 16 q^{26} - 5 q^{27} - 12 q^{28} - 2 q^{29} - 6 q^{31} + 20 q^{32} + 2 q^{33} - 20 q^{34} + 10 q^{36} - 8 q^{37} + 8 q^{38} + 7 q^{39} + 8 q^{42} - 13 q^{43} + 4 q^{44} - 12 q^{46} - 5 q^{47} - 8 q^{48} + 13 q^{49} - 10 q^{51} - 34 q^{52} + 10 q^{53} - 28 q^{56} - 2 q^{57} + 4 q^{58} - 11 q^{59} + 11 q^{61} - 16 q^{62} - 10 q^{63} - 20 q^{64} - 4 q^{66} - 24 q^{67} + 20 q^{68} + 3 q^{69} - 11 q^{71} - 17 q^{73} + 12 q^{74} - 24 q^{76} + 12 q^{77} + 16 q^{78} - 5 q^{79} + 5 q^{81} - 64 q^{82} + 12 q^{84} + 12 q^{86} + 2 q^{87} + 4 q^{88} - 3 q^{89} + 22 q^{91} - 34 q^{92} + 6 q^{93} - 20 q^{96} + 14 q^{97} + 36 q^{98} - 2 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9 - 2 * q^11 - 10 * q^12 - 7 * q^13 - 8 * q^14 + 8 * q^16 + 10 * q^17 + 2 * q^19 + 10 * q^21 + 4 * q^22 - 3 * q^23 - 16 * q^26 - 5 * q^27 - 12 * q^28 - 2 * q^29 - 6 * q^31 + 20 * q^32 + 2 * q^33 - 20 * q^34 + 10 * q^36 - 8 * q^37 + 8 * q^38 + 7 * q^39 + 8 * q^42 - 13 * q^43 + 4 * q^44 - 12 * q^46 - 5 * q^47 - 8 * q^48 + 13 * q^49 - 10 * q^51 - 34 * q^52 + 10 * q^53 - 28 * q^56 - 2 * q^57 + 4 * q^58 - 11 * q^59 + 11 * q^61 - 16 * q^62 - 10 * q^63 - 20 * q^64 - 4 * q^66 - 24 * q^67 + 20 * q^68 + 3 * q^69 - 11 * q^71 - 17 * q^73 + 12 * q^74 - 24 * q^76 + 12 * q^77 + 16 * q^78 - 5 * q^79 + 5 * q^81 - 64 * q^82 + 12 * q^84 + 12 * q^86 + 2 * q^87 + 4 * q^88 - 3 * q^89 + 22 * q^91 - 34 * q^92 + 6 * q^93 - 20 * q^96 + 14 * q^97 + 36 * q^98 - 2 * 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 $

$2$	−2.15072	−1.52079	−0.760396	−	0.649460i	$-0.774995\pi$
$2$	−2.15072	−1.52079	−0.760396	+	0.649460i	$0.774995\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.62562	1.31281
$5$	0	0
$5$	0	0
$6$	2.15072	0.878030
$7$	−2.17958	−0.823802	−0.411901	−	0.911229i	$-0.635135\pi$
$7$	−2.17958	−0.823802	−0.411901	+	0.911229i	$0.635135\pi$
$8$	−1.34553	−0.475717
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.34213	−1.61071	−0.805357	−	0.592790i	$-0.798026\pi$
$11$	−5.34213	−1.61071	−0.805357	+	0.592790i	$0.798026\pi$
$12$	−2.62562	−0.757951
$13$	−4.81703	−1.33600	−0.668002	−	0.744160i	$-0.732850\pi$
$13$	−4.81703	−1.33600	−0.668002	+	0.744160i	$0.732850\pi$
$14$	4.68767	1.25283
$15$	0	0
$16$	−2.35737	−0.589342
$17$	4.15072	1.00670	0.503349	−	0.864083i	$-0.332101\pi$
$17$	4.15072	1.00670	0.503349	+	0.864083i	$0.332101\pi$
$18$	−2.15072	−0.506931
$19$	2.38622	0.547436	0.273718	−	0.961810i	$-0.411747\pi$
$19$	2.38622	0.547436	0.273718	+	0.961810i	$0.411747\pi$
$20$	0	0
$21$	2.17958	0.475622
$22$	11.4895	2.44956
$23$	3.53694	0.737503	0.368752	−	0.929528i	$-0.379785\pi$
$23$	3.53694	0.737503	0.368752	+	0.929528i	$0.379785\pi$
$24$	1.34553	0.274656
$25$	0	0
$26$	10.3601	2.03178
$27$	−1.00000	−0.192450
$28$	−5.72273	−1.08149
$29$	−5.96775	−1.10818	−0.554092	−	0.832455i	$-0.686934\pi$
$29$	−5.96775	−1.10818	−0.554092	+	0.832455i	$0.686934\pi$
$30$	0	0
$31$	7.87908	1.41512	0.707562	−	0.706651i	$-0.249795\pi$
$31$	7.87908	1.41512	0.707562	+	0.706651i	$0.249795\pi$
$32$	7.76111	1.37198
$33$	5.34213	0.929946
$34$	−8.92707	−1.53098
$35$	0	0
$36$	2.62562	0.437603
$37$	4.95252	0.814189	0.407095	−	0.913386i	$-0.366542\pi$
$37$	4.95252	0.814189	0.407095	+	0.913386i	$0.366542\pi$
$38$	−5.13210	−0.832536
$39$	4.81703	0.771342
$40$	0	0
$41$	8.33643	1.30193	0.650966	−	0.759107i	$-0.274364\pi$
$41$	8.33643	1.30193	0.650966	+	0.759107i	$0.274364\pi$
$42$	−4.68767	−0.723323
$43$	3.32690	0.507348	0.253674	−	0.967290i	$-0.418361\pi$
$43$	3.32690	0.507348	0.253674	+	0.967290i	$0.418361\pi$
$44$	−14.0264	−2.11456
$45$	0	0
$46$	−7.60699	−1.12159
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	2.35737	0.340257
$49$	−2.24945	−0.321350
$50$	0	0
$51$	−4.15072	−0.581218
$52$	−12.6477	−1.75392
$53$	6.56630	0.901951	0.450976	−	0.892536i	$-0.351076\pi$
$53$	6.56630	0.901951	0.450976	+	0.892536i	$0.351076\pi$
$54$	2.15072	0.292677
$55$	0	0
$56$	2.93269	0.391897
$57$	−2.38622	−0.316062
$58$	12.8350	1.68532
$59$	2.30376	0.299923	0.149962	−	0.988692i	$-0.452085\pi$
$59$	2.30376	0.299923	0.149962	+	0.988692i	$0.452085\pi$
$60$	0	0
$61$	−7.32512	−0.937885	−0.468943	−	0.883229i	$-0.655365\pi$
$61$	−7.32512	−0.937885	−0.468943	+	0.883229i	$0.655365\pi$
$62$	−16.9457	−2.15211
$63$	−2.17958	−0.274601
$64$	−11.9773	−1.49716
$65$	0	0
$66$	−11.4895	−1.41426
$67$	−8.32639	−1.01723	−0.508615	−	0.860994i	$-0.669842\pi$
$67$	−8.32639	−1.01723	−0.508615	+	0.860994i	$0.669842\pi$
$68$	10.8982	1.32160
$69$	−3.53694	−0.425798
$70$	0	0
$71$	10.9067	1.29438	0.647191	−	0.762328i	$-0.275944\pi$
$71$	10.9067	1.29438	0.647191	+	0.762328i	$0.275944\pi$
$72$	−1.34553	−0.158572
$73$	−10.6650	−1.24825	−0.624124	−	0.781326i	$-0.714544\pi$
$73$	−10.6650	−1.24825	−0.624124	+	0.781326i	$0.714544\pi$
$74$	−10.6515	−1.23821
$75$	0	0
$76$	6.26529	0.718678
$77$	11.6436	1.32691
$78$	−10.3601	−1.17305
$79$	−5.92707	−0.666847	−0.333424	−	0.942777i	$-0.608204\pi$
$79$	−5.92707	−0.666847	−0.333424	+	0.942777i	$0.608204\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−17.9294	−1.97997
$83$	3.40936	0.374226	0.187113	−	0.982338i	$-0.440087\pi$
$83$	3.40936	0.374226	0.187113	+	0.982338i	$0.440087\pi$
$84$	5.72273	0.624401
$85$	0	0
$86$	−7.15525	−0.771571
$87$	5.96775	0.639810
$88$	7.18801	0.766245
$89$	−0.138890	−0.0147223	−0.00736117	−	0.999973i	$-0.502343\pi$
$89$	−0.138890	−0.0147223	−0.00736117	+	0.999973i	$0.502343\pi$
$90$	0	0
$91$	10.4991	1.10060
$92$	9.28666	0.968201
$93$	−7.87908	−0.817022
$94$	2.15072	0.221830
$95$	0	0
$96$	−7.76111	−0.792115
$97$	18.2370	1.85168	0.925841	−	0.377913i	$-0.123358\pi$
$97$	18.2370	1.85168	0.925841	+	0.377913i	$0.123358\pi$
$98$	4.83795	0.488707
$99$	−5.34213	−0.536905
$100$	0	0
$101$	1.41380	0.140678	0.0703389	−	0.997523i	$-0.477592\pi$
$101$	1.41380	0.140678	0.0703389	+	0.997523i	$0.477592\pi$
$102$	8.92707	0.883911
$103$	1.22026	0.120236	0.0601179	−	0.998191i	$-0.480852\pi$
$103$	1.22026	0.120236	0.0601179	+	0.998191i	$0.480852\pi$
$104$	6.48146	0.635560
$105$	0	0
$106$	−14.1223	−1.37168
$107$	−3.03225	−0.293138	−0.146569	−	0.989200i	$-0.546823\pi$
$107$	−3.03225	−0.293138	−0.146569	+	0.989200i	$0.546823\pi$
$108$	−2.62562	−0.252650
$109$	−16.1501	−1.54690	−0.773448	−	0.633860i	$-0.781470\pi$
$109$	−16.1501	−1.54690	−0.773448	+	0.633860i	$0.781470\pi$
$110$	0	0
$111$	−4.95252	−0.470072
$112$	5.13806	0.485501
$113$	1.06545	0.100229	0.0501144	−	0.998743i	$-0.484041\pi$
$113$	1.06545	0.100229	0.0501144	+	0.998743i	$0.484041\pi$
$114$	5.13210	0.480665
$115$	0	0
$116$	−15.6690	−1.45483
$117$	−4.81703	−0.445334
$118$	−4.95474	−0.456121
$119$	−9.04682	−0.829320
$120$	0	0
$121$	17.5384	1.59440
$122$	15.7543	1.42633
$123$	−8.33643	−0.751671
$124$	20.6874	1.85779
$125$	0	0
$126$	4.68767	0.417611
$127$	−15.9355	−1.41405	−0.707024	−	0.707190i	$-0.749963\pi$
$127$	−15.9355	−1.41405	−0.707024	+	0.707190i	$0.749963\pi$
$128$	10.2376	0.904886
$129$	−3.32690	−0.292917
$130$	0	0
$131$	14.0472	1.22731	0.613653	−	0.789576i	$-0.289699\pi$
$131$	14.0472	1.22731	0.613653	+	0.789576i	$0.289699\pi$
$132$	14.0264	1.22084
$133$	−5.20094	−0.450979
$134$	17.9078	1.54700
$135$	0	0
$136$	−5.58493	−0.478904
$137$	0.696766	0.0595288	0.0297644	−	0.999557i	$-0.490524\pi$
$137$	0.696766	0.0595288	0.0297644	+	0.999557i	$0.490524\pi$
$138$	7.60699	0.647550
$139$	−7.72434	−0.655170	−0.327585	−	0.944822i	$-0.606235\pi$
$139$	−7.72434	−0.655170	−0.327585	+	0.944822i	$0.606235\pi$
$140$	0	0
$141$	1.00000	0.0842152
$142$	−23.4572	−1.96849
$143$	25.7332	2.15192
$144$	−2.35737	−0.196447
$145$	0	0
$146$	22.9375	1.89833
$147$	2.24945	0.185532
$148$	13.0034	1.06888
$149$	18.8728	1.54612	0.773060	−	0.634333i	$-0.218725\pi$
$149$	18.8728	1.54612	0.773060	+	0.634333i	$0.218725\pi$
$150$	0	0
$151$	−12.8819	−1.04831	−0.524157	−	0.851622i	$-0.675619\pi$
$151$	−12.8819	−1.04831	−0.524157	+	0.851622i	$0.675619\pi$
$152$	−3.21073	−0.260425
$153$	4.15072	0.335566
$154$	−25.0421	−2.01795
$155$	0	0
$156$	12.6477	1.01262
$157$	19.9050	1.58860	0.794298	−	0.607529i	$-0.207839\pi$
$157$	19.9050	1.58860	0.794298	+	0.607529i	$0.207839\pi$
$158$	12.7475	1.01414
$159$	−6.56630	−0.520742
$160$	0	0
$161$	−7.70903	−0.607557
$162$	−2.15072	−0.168977
$163$	−20.5327	−1.60825	−0.804123	−	0.594463i	$-0.797365\pi$
$163$	−20.5327	−1.60825	−0.804123	+	0.594463i	$0.797365\pi$
$164$	21.8883	1.70919
$165$	0	0
$166$	−7.33261	−0.569121
$167$	14.0107	1.08418	0.542089	−	0.840321i	$-0.317634\pi$
$167$	14.0107	1.08418	0.542089	+	0.840321i	$0.317634\pi$
$168$	−2.93269	−0.226262
$169$	10.2038	0.784904
$170$	0	0
$171$	2.38622	0.182479
$172$	8.73518	0.666051
$173$	−2.53354	−0.192622	−0.0963109	−	0.995351i	$-0.530704\pi$
$173$	−2.53354	−0.192622	−0.0963109	+	0.995351i	$0.530704\pi$
$174$	−12.8350	−0.973018
$175$	0	0
$176$	12.5934	0.949261
$177$	−2.30376	−0.173161
$178$	0.298715	0.0223896
$179$	−6.45958	−0.482812	−0.241406	−	0.970424i	$-0.577608\pi$
$179$	−6.45958	−0.482812	−0.241406	+	0.970424i	$0.577608\pi$
$180$	0	0
$181$	22.5712	1.67770	0.838851	−	0.544362i	$-0.183228\pi$
$181$	22.5712	1.67770	0.838851	+	0.544362i	$0.183228\pi$
$182$	−22.5806	−1.67379
$183$	7.32512	0.541488
$184$	−4.75907	−0.350843
$185$	0	0
$186$	16.9457	1.24252
$187$	−22.1737	−1.62150
$188$	−2.62562	−0.191493
$189$	2.17958	0.158541
$190$	0	0
$191$	1.87337	0.135553	0.0677763	−	0.997701i	$-0.478410\pi$
$191$	1.87337	0.135553	0.0677763	+	0.997701i	$0.478410\pi$
$192$	11.9773	0.864386
$193$	−19.1698	−1.37987	−0.689937	−	0.723870i	$-0.742362\pi$
$193$	−19.1698	−1.37987	−0.689937	+	0.723870i	$0.742362\pi$
$194$	−39.2227	−2.81602
$195$	0	0
$196$	−5.90620	−0.421872
$197$	−14.3877	−1.02508	−0.512540	−	0.858663i	$-0.671295\pi$
$197$	−14.3877	−1.02508	−0.512540	+	0.858663i	$0.671295\pi$
$198$	11.4895	0.816521
$199$	2.45174	0.173799	0.0868995	−	0.996217i	$-0.472304\pi$
$199$	2.45174	0.173799	0.0868995	+	0.996217i	$0.472304\pi$
$200$	0	0
$201$	8.32639	0.587298
$202$	−3.04068	−0.213942
$203$	13.0072	0.912924
$204$	−10.8982	−0.763028
$205$	0	0
$206$	−2.62444	−0.182854
$207$	3.53694	0.245834
$208$	11.3555	0.787362
$209$	−12.7475	−0.881762
$210$	0	0
$211$	−25.9964	−1.78966	−0.894832	−	0.446402i	$-0.852705\pi$
$211$	−25.9964	−1.78966	−0.894832	+	0.446402i	$0.852705\pi$
$212$	17.2406	1.18409
$213$	−10.9067	−0.747312
$214$	6.52153	0.445803
$215$	0	0
$216$	1.34553	0.0915518
$217$	−17.1730	−1.16578
$218$	34.7343	2.35251
$219$	10.6650	0.720676
$220$	0	0
$221$	−19.9942	−1.34495
$222$	10.6515	0.714883
$223$	−22.7959	−1.52653	−0.763264	−	0.646087i	$-0.776404\pi$
$223$	−22.7959	−1.52653	−0.763264	+	0.646087i	$0.776404\pi$
$224$	−16.9159	−1.13024
$225$	0	0
$226$	−2.29148	−0.152427
$227$	−18.4402	−1.22392	−0.611960	−	0.790889i	$-0.709618\pi$
$227$	−18.4402	−1.22392	−0.611960	+	0.790889i	$0.709618\pi$
$228$	−6.26529	−0.414929
$229$	13.4554	0.889160	0.444580	−	0.895739i	$-0.353353\pi$
$229$	13.4554	0.889160	0.444580	+	0.895739i	$0.353353\pi$
$230$	0	0
$231$	−11.6436	−0.766092
$232$	8.02980	0.527182
$233$	26.6513	1.74598	0.872991	−	0.487736i	$-0.162177\pi$
$233$	26.6513	1.74598	0.872991	+	0.487736i	$0.162177\pi$
$234$	10.3601	0.677261
$235$	0	0
$236$	6.04878	0.393742
$237$	5.92707	0.385004
$238$	19.4572	1.26122
$239$	−3.13031	−0.202483	−0.101242	−	0.994862i	$-0.532281\pi$
$239$	−3.13031	−0.202483	−0.101242	+	0.994862i	$0.532281\pi$
$240$	0	0
$241$	27.2266	1.75382	0.876910	−	0.480655i	$-0.159601\pi$
$241$	27.2266	1.75382	0.876910	+	0.480655i	$0.159601\pi$
$242$	−37.7203	−2.42475
$243$	−1.00000	−0.0641500
$244$	−19.2330	−1.23126
$245$	0	0
$246$	17.9294	1.14314
$247$	−11.4945	−0.731376
$248$	−10.6015	−0.673199
$249$	−3.40936	−0.216060
$250$	0	0
$251$	−1.06374	−0.0671430	−0.0335715	−	0.999436i	$-0.510688\pi$
$251$	−1.06374	−0.0671430	−0.0335715	+	0.999436i	$0.510688\pi$
$252$	−5.72273	−0.360498
$253$	−18.8948	−1.18791
$254$	34.2729	2.15047
$255$	0	0
$256$	1.93627	0.121017
$257$	−30.2577	−1.88742	−0.943710	−	0.330773i	$-0.892691\pi$
$257$	−30.2577	−1.88742	−0.943710	+	0.330773i	$0.892691\pi$
$258$	7.15525	0.445467
$259$	−10.7944	−0.670731
$260$	0	0
$261$	−5.96775	−0.369395
$262$	−30.2116	−1.86648
$263$	−14.6639	−0.904217	−0.452109	−	0.891963i	$-0.649328\pi$
$263$	−14.6639	−0.904217	−0.452109	+	0.891963i	$0.649328\pi$
$264$	−7.18801	−0.442392
$265$	0	0
$266$	11.1858	0.685845
$267$	0.138890	0.00849994
$268$	−21.8619	−1.33543
$269$	−27.8416	−1.69753	−0.848767	−	0.528766i	$-0.822655\pi$
$269$	−27.8416	−1.69753	−0.848767	+	0.528766i	$0.822655\pi$
$270$	0	0
$271$	0.590635	0.0358785	0.0179393	−	0.999839i	$-0.494289\pi$
$271$	0.590635	0.0358785	0.0179393	+	0.999839i	$0.494289\pi$
$272$	−9.78478	−0.593289
$273$	−10.4991	−0.635433
$274$	−1.49855	−0.0905309
$275$	0	0
$276$	−9.28666	−0.558991
$277$	29.5294	1.77425	0.887125	−	0.461529i	$-0.152699\pi$
$277$	29.5294	1.77425	0.887125	+	0.461529i	$0.152699\pi$
$278$	16.6129	0.996378
$279$	7.87908	0.471708
$280$	0	0
$281$	9.74306	0.581222	0.290611	−	0.956841i	$-0.406141\pi$
$281$	9.74306	0.581222	0.290611	+	0.956841i	$0.406141\pi$
$282$	−2.15072	−0.128074
$283$	−10.5670	−0.628141	−0.314070	−	0.949400i	$-0.601693\pi$
$283$	−10.5670	−0.628141	−0.314070	+	0.949400i	$0.601693\pi$
$284$	28.6367	1.69928
$285$	0	0
$286$	−55.3451	−3.27262
$287$	−18.1699	−1.07253
$288$	7.76111	0.457328
$289$	0.228518	0.0134422
$290$	0	0
$291$	−18.2370	−1.06907
$292$	−28.0023	−1.63871
$293$	−11.1033	−0.648659	−0.324330	−	0.945944i	$-0.605139\pi$
$293$	−11.1033	−0.648659	−0.324330	+	0.945944i	$0.605139\pi$
$294$	−4.83795	−0.282155
$295$	0	0
$296$	−6.66378	−0.387324
$297$	5.34213	0.309982
$298$	−40.5902	−2.35133
$299$	−17.0375	−0.985307
$300$	0	0
$301$	−7.25124	−0.417954
$302$	27.7054	1.59427
$303$	−1.41380	−0.0812204
$304$	−5.62519	−0.322627
$305$	0	0
$306$	−8.92707	−0.510326
$307$	−32.0060	−1.82668	−0.913339	−	0.407200i	$-0.866505\pi$
$307$	−32.0060	−1.82668	−0.913339	+	0.407200i	$0.866505\pi$
$308$	30.5716	1.74198
$309$	−1.22026	−0.0694182
$310$	0	0
$311$	−2.92528	−0.165878	−0.0829388	−	0.996555i	$-0.526431\pi$
$311$	−2.92528	−0.165878	−0.0829388	+	0.996555i	$0.526431\pi$
$312$	−6.48146	−0.366941
$313$	14.6678	0.829072	0.414536	−	0.910033i	$-0.363944\pi$
$313$	14.6678	0.829072	0.414536	+	0.910033i	$0.363944\pi$
$314$	−42.8103	−2.41592
$315$	0	0
$316$	−15.5622	−0.875443
$317$	−2.71685	−0.152593	−0.0762967	−	0.997085i	$-0.524310\pi$
$317$	−2.71685	−0.152593	−0.0762967	+	0.997085i	$0.524310\pi$
$318$	14.1223	0.791940
$319$	31.8805	1.78497
$320$	0	0
$321$	3.03225	0.169244
$322$	16.5800	0.923968
$323$	9.90453	0.551103
$324$	2.62562	0.145868
$325$	0	0
$326$	44.1602	2.44581
$327$	16.1501	0.893101
$328$	−11.2169	−0.619352
$329$	2.17958	0.120164
$330$	0	0
$331$	−0.313946	−0.0172561	−0.00862803	−	0.999963i	$-0.502746\pi$
$331$	−0.313946	−0.0172561	−0.00862803	+	0.999963i	$0.502746\pi$
$332$	8.95169	0.491288
$333$	4.95252	0.271396
$334$	−30.1331	−1.64881
$335$	0	0
$336$	−5.13806	−0.280304
$337$	15.4426	0.841210	0.420605	−	0.907244i	$-0.361818\pi$
$337$	15.4426	0.841210	0.420605	+	0.907244i	$0.361818\pi$
$338$	−21.9455	−1.19368
$339$	−1.06545	−0.0578671
$340$	0	0
$341$	−42.0911	−2.27936
$342$	−5.13210	−0.277512
$343$	20.1599	1.08853
$344$	−4.47645	−0.241354
$345$	0	0
$346$	5.44896	0.292938
$347$	0.478802	0.0257034	0.0128517	−	0.999917i	$-0.495909\pi$
$347$	0.478802	0.0257034	0.0128517	+	0.999917i	$0.495909\pi$
$348$	15.6690	0.839949
$349$	−2.10272	−0.112556	−0.0562781	−	0.998415i	$-0.517923\pi$
$349$	−2.10272	−0.112556	−0.0562781	+	0.998415i	$0.517923\pi$
$350$	0	0
$351$	4.81703	0.257114
$352$	−41.4609	−2.20987
$353$	8.83347	0.470158	0.235079	−	0.971976i	$-0.424465\pi$
$353$	8.83347	0.470158	0.235079	+	0.971976i	$0.424465\pi$
$354$	4.95474	0.263342
$355$	0	0
$356$	−0.364673	−0.0193276
$357$	9.04682	0.478808
$358$	13.8928	0.734256
$359$	−14.5452	−0.767666	−0.383833	−	0.923402i	$-0.625396\pi$
$359$	−14.5452	−0.767666	−0.383833	+	0.923402i	$0.625396\pi$
$360$	0	0
$361$	−13.3060	−0.700314
$362$	−48.5444	−2.55144
$363$	−17.5384	−0.920528
$364$	27.5666	1.44488
$365$	0	0
$366$	−15.7543	−0.823491
$367$	15.1631	0.791506	0.395753	−	0.918357i	$-0.370484\pi$
$367$	15.1631	0.791506	0.395753	+	0.918357i	$0.370484\pi$
$368$	−8.33787	−0.434641
$369$	8.33643	0.433977
$370$	0	0
$371$	−14.3118	−0.743029
$372$	−20.6874	−1.07259
$373$	−19.4329	−1.00620	−0.503098	−	0.864230i	$-0.667806\pi$
$373$	−19.4329	−1.00620	−0.503098	+	0.864230i	$0.667806\pi$
$374$	47.6896	2.46597
$375$	0	0
$376$	1.34553	0.0693905
$377$	28.7468	1.48054
$378$	−4.68767	−0.241108
$379$	12.9999	0.667758	0.333879	−	0.942616i	$-0.391642\pi$
$379$	12.9999	0.667758	0.333879	+	0.942616i	$0.391642\pi$
$380$	0	0
$381$	15.9355	0.816401
$382$	−4.02911	−0.206147
$383$	−35.9422	−1.83656	−0.918282	−	0.395928i	$-0.870423\pi$
$383$	−35.9422	−1.83656	−0.918282	+	0.395928i	$0.870423\pi$
$384$	−10.2376	−0.522436
$385$	0	0
$386$	41.2290	2.09850
$387$	3.32690	0.169116
$388$	47.8833	2.43091
$389$	6.74698	0.342085	0.171043	−	0.985264i	$-0.445286\pi$
$389$	6.74698	0.342085	0.171043	+	0.985264i	$0.445286\pi$
$390$	0	0
$391$	14.6809	0.742444
$392$	3.02671	0.152872
$393$	−14.0472	−0.708585
$394$	30.9439	1.55893
$395$	0	0
$396$	−14.0264	−0.704853
$397$	−10.6965	−0.536844	−0.268422	−	0.963301i	$-0.586502\pi$
$397$	−10.6965	−0.536844	−0.268422	+	0.963301i	$0.586502\pi$
$398$	−5.27301	−0.264312
$399$	5.20094	0.260373
$400$	0	0
$401$	5.12588	0.255974	0.127987	−	0.991776i	$-0.459148\pi$
$401$	5.12588	0.255974	0.127987	+	0.991776i	$0.459148\pi$
$402$	−17.9078	−0.893159
$403$	−37.9537	−1.89061
$404$	3.71209	0.184683
$405$	0	0
$406$	−27.9748	−1.38837
$407$	−26.4570	−1.31143
$408$	5.58493	0.276495
$409$	−16.9303	−0.837150	−0.418575	−	0.908182i	$-0.637470\pi$
$409$	−16.9303	−0.837150	−0.418575	+	0.908182i	$0.637470\pi$
$410$	0	0
$411$	−0.696766	−0.0343689
$412$	3.20394	0.157847
$413$	−5.02121	−0.247078
$414$	−7.60699	−0.373863
$415$	0	0
$416$	−37.3855	−1.83297
$417$	7.72434	0.378263
$418$	27.4164	1.34098
$419$	−17.5867	−0.859167	−0.429584	−	0.903027i	$-0.641340\pi$
$419$	−17.5867	−0.859167	−0.429584	+	0.903027i	$0.641340\pi$
$420$	0	0
$421$	−24.1556	−1.17727	−0.588636	−	0.808398i	$-0.700335\pi$
$421$	−24.1556	−1.17727	−0.588636	+	0.808398i	$0.700335\pi$
$422$	55.9111	2.72171
$423$	−1.00000	−0.0486217
$424$	−8.83517	−0.429074
$425$	0	0
$426$	23.4572	1.13651
$427$	15.9656	0.772632
$428$	−7.96152	−0.384835
$429$	−25.7332	−1.24241
$430$	0	0
$431$	38.2877	1.84425	0.922127	−	0.386888i	$-0.126450\pi$
$431$	38.2877	1.84425	0.922127	+	0.386888i	$0.126450\pi$
$432$	2.35737	0.113419
$433$	6.67225	0.320648	0.160324	−	0.987064i	$-0.448746\pi$
$433$	6.67225	0.320648	0.160324	+	0.987064i	$0.448746\pi$
$434$	36.9345	1.77291
$435$	0	0
$436$	−42.4039	−2.03078
$437$	8.43991	0.403736
$438$	−22.9375	−1.09600
$439$	−10.6917	−0.510286	−0.255143	−	0.966903i	$-0.582123\pi$
$439$	−10.6917	−0.510286	−0.255143	+	0.966903i	$0.582123\pi$
$440$	0	0
$441$	−2.24945	−0.107117
$442$	43.0019	2.04539
$443$	−36.4261	−1.73065	−0.865327	−	0.501207i	$-0.832889\pi$
$443$	−36.4261	−1.73065	−0.865327	+	0.501207i	$0.832889\pi$
$444$	−13.0034	−0.617115
$445$	0	0
$446$	49.0278	2.32153
$447$	−18.8728	−0.892653
$448$	26.1054	1.23336
$449$	−10.8510	−0.512089	−0.256045	−	0.966665i	$-0.582419\pi$
$449$	−10.8510	−0.512089	−0.256045	+	0.966665i	$0.582419\pi$
$450$	0	0
$451$	−44.5343	−2.09704
$452$	2.79745	0.131581
$453$	12.8819	0.605244
$454$	39.6598	1.86133
$455$	0	0
$456$	3.21073	0.150356
$457$	−8.13993	−0.380770	−0.190385	−	0.981710i	$-0.560974\pi$
$457$	−8.13993	−0.380770	−0.190385	+	0.981710i	$0.560974\pi$
$458$	−28.9389	−1.35223
$459$	−4.15072	−0.193739
$460$	0	0
$461$	15.5954	0.726351	0.363175	−	0.931721i	$-0.381693\pi$
$461$	15.5954	0.726351	0.363175	+	0.931721i	$0.381693\pi$
$462$	25.0421	1.16507
$463$	−21.9794	−1.02147	−0.510736	−	0.859738i	$-0.670627\pi$
$463$	−21.9794	−1.02147	−0.510736	+	0.859738i	$0.670627\pi$
$464$	14.0682	0.653099
$465$	0	0
$466$	−57.3195	−2.65528
$467$	18.4685	0.854622	0.427311	−	0.904105i	$-0.359461\pi$
$467$	18.4685	0.854622	0.427311	+	0.904105i	$0.359461\pi$
$468$	−12.6477	−0.584639
$469$	18.1480	0.837997
$470$	0	0
$471$	−19.9050	−0.917176
$472$	−3.09978	−0.142679
$473$	−17.7728	−0.817193
$474$	−12.7475	−0.585512
$475$	0	0
$476$	−23.7535	−1.08874
$477$	6.56630	0.300650
$478$	6.73244	0.307935
$479$	−20.7090	−0.946217	−0.473109	−	0.881004i	$-0.656868\pi$
$479$	−20.7090	−0.946217	−0.473109	+	0.881004i	$0.656868\pi$
$480$	0	0
$481$	−23.8564	−1.08776
$482$	−58.5569	−2.66719
$483$	7.70903	0.350773
$484$	46.0491	2.09314
$485$	0	0
$486$	2.15072	0.0975589
$487$	−8.75729	−0.396831	−0.198415	−	0.980118i	$-0.563579\pi$
$487$	−8.75729	−0.396831	−0.198415	+	0.980118i	$0.563579\pi$
$488$	9.85618	0.446168
$489$	20.5327	0.928521
$490$	0	0
$491$	−23.3878	−1.05548	−0.527738	−	0.849407i	$-0.676960\pi$
$491$	−23.3878	−1.05548	−0.527738	+	0.849407i	$0.676960\pi$
$492$	−21.8883	−0.986800
$493$	−24.7705	−1.11561
$494$	24.7215	1.11227
$495$	0	0
$496$	−18.5739	−0.833992
$497$	−23.7719	−1.06631
$498$	7.33261	0.328582
$499$	−6.91720	−0.309656	−0.154828	−	0.987941i	$-0.549482\pi$
$499$	−6.91720	−0.309656	−0.154828	+	0.987941i	$0.549482\pi$
$500$	0	0
$501$	−14.0107	−0.625950
$502$	2.28782	0.102110
$503$	−40.1341	−1.78949	−0.894746	−	0.446575i	$-0.852644\pi$
$503$	−40.1341	−1.78949	−0.894746	+	0.446575i	$0.852644\pi$
$504$	2.93269	0.130632
$505$	0	0
$506$	40.6376	1.80656
$507$	−10.2038	−0.453165
$508$	−41.8405	−1.85637
$509$	−26.7613	−1.18617	−0.593087	−	0.805138i	$-0.702091\pi$
$509$	−26.7613	−1.18617	−0.593087	+	0.805138i	$0.702091\pi$
$510$	0	0
$511$	23.2452	1.02831
$512$	−24.6396	−1.08893
$513$	−2.38622	−0.105354
$514$	65.0759	2.87037
$515$	0	0
$516$	−8.73518	−0.384545
$517$	5.34213	0.234947
$518$	23.2158	1.02004
$519$	2.53354	0.111210
$520$	0	0
$521$	11.7370	0.514206	0.257103	−	0.966384i	$-0.417232\pi$
$521$	11.7370	0.514206	0.257103	+	0.966384i	$0.417232\pi$
$522$	12.8350	0.561772
$523$	16.2155	0.709055	0.354527	−	0.935046i	$-0.384642\pi$
$523$	16.2155	0.709055	0.354527	+	0.935046i	$0.384642\pi$
$524$	36.8825	1.61122
$525$	0	0
$526$	31.5381	1.37513
$527$	32.7039	1.42460
$528$	−12.5934	−0.548056
$529$	−10.4900	−0.456089
$530$	0	0
$531$	2.30376	0.0999745
$532$	−13.6557	−0.592049
$533$	−40.1568	−1.73939
$534$	−0.298715	−0.0129266
$535$	0	0
$536$	11.2034	0.483914
$537$	6.45958	0.278751
$538$	59.8797	2.58160
$539$	12.0169	0.517603
$540$	0	0
$541$	−20.0303	−0.861171	−0.430585	−	0.902550i	$-0.641693\pi$
$541$	−20.0303	−0.861171	−0.430585	+	0.902550i	$0.641693\pi$
$542$	−1.27029	−0.0545638
$543$	−22.5712	−0.968621
$544$	32.2142	1.38117
$545$	0	0
$546$	22.5806	0.966361
$547$	8.91916	0.381356	0.190678	−	0.981653i	$-0.438931\pi$
$547$	8.91916	0.381356	0.190678	+	0.981653i	$0.438931\pi$
$548$	1.82944	0.0781499
$549$	−7.32512	−0.312628
$550$	0	0
$551$	−14.2404	−0.606659
$552$	4.75907	0.202559
$553$	12.9185	0.549350
$554$	−63.5096	−2.69827
$555$	0	0
$556$	−20.2812	−0.860114
$557$	32.5982	1.38123	0.690616	−	0.723222i	$-0.257340\pi$
$557$	32.5982	1.38123	0.690616	+	0.723222i	$0.257340\pi$
$558$	−16.9457	−0.717370
$559$	−16.0258	−0.677818
$560$	0	0
$561$	22.1737	0.936176
$562$	−20.9546	−0.883918
$563$	8.88638	0.374516	0.187258	−	0.982311i	$-0.440040\pi$
$563$	8.88638	0.374516	0.187258	+	0.982311i	$0.440040\pi$
$564$	2.62562	0.110558
$565$	0	0
$566$	22.7266	0.955272
$567$	−2.17958	−0.0915336
$568$	−14.6753	−0.615760
$569$	45.5793	1.91078	0.955391	−	0.295342i	$-0.0954337\pi$
$569$	45.5793	1.91078	0.955391	+	0.295342i	$0.0954337\pi$
$570$	0	0
$571$	−0.236696	−0.00990543	−0.00495271	−	0.999988i	$-0.501577\pi$
$571$	−0.236696	−0.00990543	−0.00495271	+	0.999988i	$0.501577\pi$
$572$	67.5656	2.82506
$573$	−1.87337	−0.0782613
$574$	39.0784	1.63110
$575$	0	0
$576$	−11.9773	−0.499053
$577$	25.8818	1.07747	0.538737	−	0.842474i	$-0.318902\pi$
$577$	25.8818	1.07747	0.538737	+	0.842474i	$0.318902\pi$
$578$	−0.491479	−0.0204428
$579$	19.1698	0.796670
$580$	0	0
$581$	−7.43097	−0.308288
$582$	39.2227	1.62583
$583$	−35.0781	−1.45279
$584$	14.3501	0.593813
$585$	0	0
$586$	23.8800	0.986476
$587$	−43.3613	−1.78971	−0.894857	−	0.446354i	$-0.852722\pi$
$587$	−43.3613	−1.78971	−0.894857	+	0.446354i	$0.852722\pi$
$588$	5.90620	0.243568
$589$	18.8012	0.774689
$590$	0	0
$591$	14.3877	0.591830
$592$	−11.6749	−0.479836
$593$	24.9952	1.02643	0.513216	−	0.858260i	$-0.328454\pi$
$593$	24.9952	1.02643	0.513216	+	0.858260i	$0.328454\pi$
$594$	−11.4895	−0.471418
$595$	0	0
$596$	49.5527	2.02976
$597$	−2.45174	−0.100343
$598$	36.6431	1.49845
$599$	−7.39323	−0.302079	−0.151040	−	0.988528i	$-0.548262\pi$
$599$	−7.39323	−0.302079	−0.151040	+	0.988528i	$0.548262\pi$
$600$	0	0
$601$	−11.2951	−0.460739	−0.230369	−	0.973103i	$-0.573993\pi$
$601$	−11.2951	−0.460739	−0.230369	+	0.973103i	$0.573993\pi$
$602$	15.5954	0.635622
$603$	−8.32639	−0.339077
$604$	−33.8229	−1.37624
$605$	0	0
$606$	3.04068	0.123519
$607$	12.2050	0.495387	0.247694	−	0.968838i	$-0.420327\pi$
$607$	12.2050	0.495387	0.247694	+	0.968838i	$0.420327\pi$
$608$	18.5197	0.751073
$609$	−13.0072	−0.527077
$610$	0	0
$611$	4.81703	0.194876
$612$	10.8982	0.440534
$613$	33.4594	1.35141	0.675707	−	0.737170i	$-0.263838\pi$
$613$	33.4594	1.35141	0.675707	+	0.737170i	$0.263838\pi$
$614$	68.8361	2.77800
$615$	0	0
$616$	−15.6668	−0.631234
$617$	11.9288	0.480234	0.240117	−	0.970744i	$-0.422814\pi$
$617$	11.9288	0.480234	0.240117	+	0.970744i	$0.422814\pi$
$618$	2.62444	0.105571
$619$	−39.8393	−1.60128	−0.800638	−	0.599148i	$-0.795506\pi$
$619$	−39.8393	−1.60128	−0.800638	+	0.599148i	$0.795506\pi$
$620$	0	0
$621$	−3.53694	−0.141933
$622$	6.29148	0.252265
$623$	0.302722	0.0121283
$624$	−11.3555	−0.454584
$625$	0	0
$626$	−31.5464	−1.26085
$627$	12.7475	0.509086
$628$	52.2630	2.08552
$629$	20.5566	0.819643
$630$	0	0
$631$	6.29270	0.250508	0.125254	−	0.992125i	$-0.460025\pi$
$631$	6.29270	0.250508	0.125254	+	0.992125i	$0.460025\pi$
$632$	7.97506	0.317231
$633$	25.9964	1.03326
$634$	5.84319	0.232063
$635$	0	0
$636$	−17.2406	−0.683634
$637$	10.8357	0.429325
$638$	−68.5663	−2.71456
$639$	10.9067	0.431461
$640$	0	0
$641$	−19.4250	−0.767242	−0.383621	−	0.923491i	$-0.625323\pi$
$641$	−19.4250	−0.767242	−0.383621	+	0.923491i	$0.625323\pi$
$642$	−6.52153	−0.257384
$643$	−18.8833	−0.744683	−0.372342	−	0.928096i	$-0.621445\pi$
$643$	−18.8833	−0.744683	−0.372342	+	0.928096i	$0.621445\pi$
$644$	−20.2410	−0.797606
$645$	0	0
$646$	−21.3019	−0.838113
$647$	−16.9011	−0.664452	−0.332226	−	0.943200i	$-0.607800\pi$
$647$	−16.9011	−0.664452	−0.332226	+	0.943200i	$0.607800\pi$
$648$	−1.34553	−0.0528575
$649$	−12.3070	−0.483091
$650$	0	0
$651$	17.1730	0.673065
$652$	−53.9110	−2.11132
$653$	20.0895	0.786161	0.393081	−	0.919504i	$-0.371409\pi$
$653$	20.0895	0.786161	0.393081	+	0.919504i	$0.371409\pi$
$654$	−34.7343	−1.35822
$655$	0	0
$656$	−19.6520	−0.767283
$657$	−10.6650	−0.416082
$658$	−4.68767	−0.182744
$659$	−7.11337	−0.277098	−0.138549	−	0.990356i	$-0.544244\pi$
$659$	−7.11337	−0.277098	−0.138549	+	0.990356i	$0.544244\pi$
$660$	0	0
$661$	−13.3991	−0.521166	−0.260583	−	0.965451i	$-0.583915\pi$
$661$	−13.3991	−0.521166	−0.260583	+	0.965451i	$0.583915\pi$
$662$	0.675212	0.0262429
$663$	19.9942	0.776509
$664$	−4.58741	−0.178026
$665$	0	0
$666$	−10.6515	−0.412738
$667$	−21.1076	−0.817289
$668$	36.7866	1.42332
$669$	22.7959	0.881341
$670$	0	0
$671$	39.1318	1.51067
$672$	16.9159	0.652546
$673$	−20.5062	−0.790457	−0.395229	−	0.918583i	$-0.629335\pi$
$673$	−20.5062	−0.790457	−0.395229	+	0.918583i	$0.629335\pi$
$674$	−33.2127	−1.27931
$675$	0	0
$676$	26.7912	1.03043
$677$	35.1460	1.35077	0.675385	−	0.737466i	$-0.263978\pi$
$677$	35.1460	1.35077	0.675385	+	0.737466i	$0.263978\pi$
$678$	2.29148	0.0880038
$679$	−39.7488	−1.52542
$680$	0	0
$681$	18.4402	0.706630
$682$	90.5264	3.46643
$683$	25.0104	0.956997	0.478498	−	0.878088i	$-0.341181\pi$
$683$	25.0104	0.956997	0.478498	+	0.878088i	$0.341181\pi$
$684$	6.26529	0.239559
$685$	0	0
$686$	−43.3583	−1.65543
$687$	−13.4554	−0.513357
$688$	−7.84273	−0.299001
$689$	−31.6301	−1.20501
$690$	0	0
$691$	−12.3442	−0.469594	−0.234797	−	0.972044i	$-0.575443\pi$
$691$	−12.3442	−0.469594	−0.234797	+	0.972044i	$0.575443\pi$
$692$	−6.65212	−0.252876
$693$	11.6436	0.442303
$694$	−1.02977	−0.0390896
$695$	0	0
$696$	−8.02980	−0.304369
$697$	34.6022	1.31065
$698$	4.52238	0.171175
$699$	−26.6513	−1.00804
$700$	0	0
$701$	−34.8408	−1.31592	−0.657960	−	0.753053i	$-0.728580\pi$
$701$	−34.8408	−1.31592	−0.657960	+	0.753053i	$0.728580\pi$
$702$	−10.3601	−0.391017
$703$	11.8178	0.445716
$704$	63.9843	2.41150
$705$	0	0
$706$	−18.9984	−0.715013
$707$	−3.08147	−0.115891
$708$	−6.04878	−0.227327
$709$	23.0456	0.865494	0.432747	−	0.901515i	$-0.357544\pi$
$709$	23.0456	0.865494	0.432747	+	0.901515i	$0.357544\pi$
$710$	0	0
$711$	−5.92707	−0.222282
$712$	0.186881	0.00700367
$713$	27.8678	1.04366
$714$	−19.4572	−0.728168
$715$	0	0
$716$	−16.9604	−0.633839
$717$	3.13031	0.116904
$718$	31.2827	1.16746
$719$	18.4572	0.688338	0.344169	−	0.938908i	$-0.388161\pi$
$719$	18.4572	0.688338	0.344169	+	0.938908i	$0.388161\pi$
$720$	0	0
$721$	−2.65965	−0.0990505
$722$	28.6175	1.06503
$723$	−27.2266	−1.01257
$724$	59.2633	2.20250
$725$	0	0
$726$	37.7203	1.39993
$727$	44.9031	1.66536	0.832682	−	0.553752i	$-0.186804\pi$
$727$	44.9031	1.66536	0.832682	+	0.553752i	$0.186804\pi$
$728$	−14.1268	−0.523576
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.8091	0.510747
$732$	19.2330	0.710871
$733$	24.6880	0.911873	0.455937	−	0.890012i	$-0.349304\pi$
$733$	24.6880	0.911873	0.455937	+	0.890012i	$0.349304\pi$
$734$	−32.6116	−1.20372
$735$	0	0
$736$	27.4506	1.01184
$737$	44.4807	1.63847
$738$	−17.9294	−0.659989
$739$	−50.1972	−1.84653	−0.923267	−	0.384158i	$-0.874492\pi$
$739$	−50.1972	−1.84653	−0.923267	+	0.384158i	$0.874492\pi$
$740$	0	0
$741$	11.4945	0.422260
$742$	30.7806	1.12999
$743$	3.88511	0.142531	0.0712654	−	0.997457i	$-0.477296\pi$
$743$	3.88511	0.142531	0.0712654	+	0.997457i	$0.477296\pi$
$744$	10.6015	0.388672
$745$	0	0
$746$	41.7947	1.53021
$747$	3.40936	0.124742
$748$	−58.2197	−2.12872
$749$	6.60901	0.241488
$750$	0	0
$751$	18.3816	0.670754	0.335377	−	0.942084i	$-0.391136\pi$
$751$	18.3816	0.670754	0.335377	+	0.942084i	$0.391136\pi$
$752$	2.35737	0.0859643
$753$	1.06374	0.0387650
$754$	−61.8265	−2.25159
$755$	0	0
$756$	5.72273	0.208134
$757$	−7.60705	−0.276483	−0.138241	−	0.990399i	$-0.544145\pi$
$757$	−7.60705	−0.276483	−0.138241	+	0.990399i	$0.544145\pi$
$758$	−27.9591	−1.01552
$759$	18.8948	0.685839
$760$	0	0
$761$	−15.4702	−0.560795	−0.280398	−	0.959884i	$-0.590466\pi$
$761$	−15.4702	−0.560795	−0.280398	+	0.959884i	$0.590466\pi$
$762$	−34.2729	−1.24158
$763$	35.2003	1.27434
$764$	4.91876	0.177955
$765$	0	0
$766$	77.3019	2.79303
$767$	−11.0973	−0.400699
$768$	−1.93627	−0.0698690
$769$	−27.8158	−1.00306	−0.501531	−	0.865139i	$-0.667230\pi$
$769$	−27.8158	−1.00306	−0.501531	+	0.865139i	$0.667230\pi$
$770$	0	0
$771$	30.2577	1.08970
$772$	−50.3326	−1.81151
$773$	34.2207	1.23083	0.615417	−	0.788202i	$-0.288988\pi$
$773$	34.2207	1.23083	0.615417	+	0.788202i	$0.288988\pi$
$774$	−7.15525	−0.257190
$775$	0	0
$776$	−24.5384	−0.880877
$777$	10.7944	0.387247
$778$	−14.5109	−0.520241
$779$	19.8925	0.712724
$780$	0	0
$781$	−58.2648	−2.08488
$782$	−31.5745	−1.12910
$783$	5.96775	0.213270
$784$	5.30278	0.189385
$785$	0	0
$786$	30.2116	1.07761
$787$	23.1787	0.826231	0.413116	−	0.910679i	$-0.364441\pi$
$787$	23.1787	0.826231	0.413116	+	0.910679i	$0.364441\pi$
$788$	−37.7766	−1.34573
$789$	14.6639	0.522050
$790$	0	0
$791$	−2.32222	−0.0825686
$792$	7.18801	0.255415
$793$	35.2853	1.25302
$794$	23.0053	0.816428
$795$	0	0
$796$	6.43732	0.228165
$797$	6.67370	0.236394	0.118197	−	0.992990i	$-0.462288\pi$
$797$	6.67370	0.236394	0.118197	+	0.992990i	$0.462288\pi$
$798$	−11.1858	−0.395973
$799$	−4.15072	−0.146842
$800$	0	0
$801$	−0.138890	−0.00490744
$802$	−11.0244	−0.389284
$803$	56.9740	2.01057
$804$	21.8619	0.771011
$805$	0	0
$806$	81.6280	2.87523
$807$	27.8416	0.980072
$808$	−1.90231	−0.0669229
$809$	−52.6329	−1.85047	−0.925237	−	0.379389i	$-0.876134\pi$
$809$	−52.6329	−1.85047	−0.925237	+	0.379389i	$0.876134\pi$
$810$	0	0
$811$	−13.0639	−0.458734	−0.229367	−	0.973340i	$-0.573666\pi$
$811$	−13.0639	−0.458734	−0.229367	+	0.973340i	$0.573666\pi$
$812$	34.1518	1.19849
$813$	−0.590635	−0.0207145
$814$	56.9018	1.99441
$815$	0	0
$816$	9.78478	0.342536
$817$	7.93871	0.277740
$818$	36.4125	1.27313
$819$	10.4991	0.366867
$820$	0	0
$821$	−30.8074	−1.07519	−0.537593	−	0.843204i	$-0.680666\pi$
$821$	−30.8074	−1.07519	−0.537593	+	0.843204i	$0.680666\pi$
$822$	1.49855	0.0522680
$823$	22.5779	0.787016	0.393508	−	0.919321i	$-0.371261\pi$
$823$	22.5779	0.787016	0.393508	+	0.919321i	$0.371261\pi$
$824$	−1.64190	−0.0571983
$825$	0	0
$826$	10.7992	0.375754
$827$	−5.52247	−0.192035	−0.0960175	−	0.995380i	$-0.530610\pi$
$827$	−5.52247	−0.192035	−0.0960175	+	0.995380i	$0.530610\pi$
$828$	9.28666	0.322734
$829$	47.5818	1.65259	0.826293	−	0.563241i	$-0.190446\pi$
$829$	47.5818	1.65259	0.826293	+	0.563241i	$0.190446\pi$
$830$	0	0
$831$	−29.5294	−1.02436
$832$	57.6949	2.00021
$833$	−9.33686	−0.323503
$834$	−16.6129	−0.575259
$835$	0	0
$836$	−33.4700	−1.15759
$837$	−7.87908	−0.272341
$838$	37.8242	1.30661
$839$	2.74622	0.0948101	0.0474050	−	0.998876i	$-0.484905\pi$
$839$	2.74622	0.0948101	0.0474050	+	0.998876i	$0.484905\pi$
$840$	0	0
$841$	6.61407	0.228071
$842$	51.9520	1.79038
$843$	−9.74306	−0.335569
$844$	−68.2566	−2.34949
$845$	0	0
$846$	2.15072	0.0739434
$847$	−38.2263	−1.31347
$848$	−15.4792	−0.531557
$849$	10.5670	0.362657
$850$	0	0
$851$	17.5168	0.600467
$852$	−28.6367	−0.981077
$853$	−39.9004	−1.36616	−0.683082	−	0.730341i	$-0.739361\pi$
$853$	−39.9004	−1.36616	−0.683082	+	0.730341i	$0.739361\pi$
$854$	−34.3377	−1.17501
$855$	0	0
$856$	4.07999	0.139451
$857$	10.3630	0.353993	0.176996	−	0.984212i	$-0.443362\pi$
$857$	10.3630	0.353993	0.176996	+	0.984212i	$0.443362\pi$
$858$	55.3451	1.88945
$859$	−8.39624	−0.286476	−0.143238	−	0.989688i	$-0.545751\pi$
$859$	−8.39624	−0.286476	−0.143238	+	0.989688i	$0.545751\pi$
$860$	0	0
$861$	18.1699	0.619228
$862$	−82.3463	−2.80473
$863$	1.64886	0.0561280	0.0280640	−	0.999606i	$-0.491066\pi$
$863$	1.64886	0.0561280	0.0280640	+	0.999606i	$0.491066\pi$
$864$	−7.76111	−0.264038
$865$	0	0
$866$	−14.3502	−0.487639
$867$	−0.228518	−0.00776087
$868$	−45.0898	−1.53045
$869$	31.6632	1.07410
$870$	0	0
$871$	40.1085	1.35902
$872$	21.7304	0.735885
$873$	18.2370	0.617227
$874$	−18.1519	−0.613998
$875$	0	0
$876$	28.0023	0.946110
$877$	−46.4032	−1.56693	−0.783463	−	0.621439i	$-0.786549\pi$
$877$	−46.4032	−1.56693	−0.783463	+	0.621439i	$0.786549\pi$
$878$	22.9949	0.776039
$879$	11.1033	0.374504
$880$	0	0
$881$	22.6990	0.764749	0.382374	−	0.924007i	$-0.375106\pi$
$881$	22.6990	0.764749	0.382374	+	0.924007i	$0.375106\pi$
$882$	4.83795	0.162902
$883$	41.9210	1.41076	0.705378	−	0.708832i	$-0.250777\pi$
$883$	41.9210	1.41076	0.705378	+	0.708832i	$0.250777\pi$
$884$	−52.4970	−1.76567
$885$	0	0
$886$	78.3424	2.63197
$887$	27.3212	0.917355	0.458677	−	0.888603i	$-0.348323\pi$
$887$	27.3212	0.917355	0.458677	+	0.888603i	$0.348323\pi$
$888$	6.66378	0.223622
$889$	34.7326	1.16490
$890$	0	0
$891$	−5.34213	−0.178968
$892$	−59.8534	−2.00404
$893$	−2.38622	−0.0798517
$894$	40.5902	1.35754
$895$	0	0
$896$	−22.3137	−0.745447
$897$	17.0375	0.568867
$898$	23.3375	0.778781
$899$	−47.0204	−1.56822
$900$	0	0
$901$	27.2549	0.907993
$902$	95.7811	3.18916
$903$	7.25124	0.241306
$904$	−1.43359	−0.0476805
$905$	0	0
$906$	−27.7054	−0.920451
$907$	9.78468	0.324895	0.162448	−	0.986717i	$-0.448061\pi$
$907$	9.78468	0.324895	0.162448	+	0.986717i	$0.448061\pi$
$908$	−48.4169	−1.60677
$909$	1.41380	0.0468926
$910$	0	0
$911$	−38.6073	−1.27912	−0.639558	−	0.768743i	$-0.720883\pi$
$911$	−38.6073	−1.27912	−0.639558	+	0.768743i	$0.720883\pi$
$912$	5.62519	0.186269
$913$	−18.2133	−0.602772
$914$	17.5068	0.579072
$915$	0	0
$916$	35.3288	1.16730
$917$	−30.6168	−1.01106
$918$	8.92707	0.294637
$919$	56.4215	1.86117	0.930587	−	0.366071i	$-0.119297\pi$
$919$	56.4215	1.86117	0.930587	+	0.366071i	$0.119297\pi$
$920$	0	0
$921$	32.0060	1.05463
$922$	−33.5414	−1.10463
$923$	−52.5377	−1.72930
$924$	−30.5716	−1.00573
$925$	0	0
$926$	47.2717	1.55345
$927$	1.22026	0.0400786
$928$	−46.3164	−1.52041
$929$	−37.7389	−1.23817	−0.619086	−	0.785323i	$-0.712497\pi$
$929$	−37.7389	−1.23817	−0.619086	+	0.785323i	$0.712497\pi$
$930$	0	0
$931$	−5.36768	−0.175919
$932$	69.9760	2.29214
$933$	2.92528	0.0957695
$934$	−39.7207	−1.29970
$935$	0	0
$936$	6.48146	0.211853
$937$	49.7218	1.62434	0.812170	−	0.583422i	$-0.198286\pi$
$937$	49.7218	1.62434	0.812170	+	0.583422i	$0.198286\pi$
$938$	−39.0313	−1.27442
$939$	−14.6678	−0.478665
$940$	0	0
$941$	16.9495	0.552538	0.276269	−	0.961080i	$-0.410902\pi$
$941$	16.9495	0.552538	0.276269	+	0.961080i	$0.410902\pi$
$942$	42.8103	1.39483
$943$	29.4855	0.960179
$944$	−5.43080	−0.176757
$945$	0	0
$946$	38.2243	1.24278
$947$	19.3787	0.629722	0.314861	−	0.949138i	$-0.398042\pi$
$947$	19.3787	0.629722	0.314861	+	0.949138i	$0.398042\pi$
$948$	15.5622	0.505437
$949$	51.3738	1.66766
$950$	0	0
$951$	2.71685	0.0880999
$952$	12.1728	0.394522
$953$	16.6549	0.539505	0.269753	−	0.962930i	$-0.413058\pi$
$953$	16.6549	0.539505	0.269753	+	0.962930i	$0.413058\pi$
$954$	−14.1223	−0.457227
$955$	0	0
$956$	−8.21900	−0.265822
$957$	−31.8805	−1.03055
$958$	44.5393	1.43900
$959$	−1.51865	−0.0490399
$960$	0	0
$961$	31.0799	1.00258
$962$	51.3086	1.65426
$963$	−3.03225	−0.0977128
$964$	71.4866	2.30243
$965$	0	0
$966$	−16.5800	−0.533453
$967$	−39.0713	−1.25645	−0.628225	−	0.778032i	$-0.716218\pi$
$967$	−39.0713	−1.25645	−0.628225	+	0.778032i	$0.716218\pi$
$968$	−23.5985	−0.758484
$969$	−9.90453	−0.318179
$970$	0	0
$971$	−11.2975	−0.362555	−0.181278	−	0.983432i	$-0.558023\pi$
$971$	−11.2975	−0.362555	−0.181278	+	0.983432i	$0.558023\pi$
$972$	−2.62562	−0.0842167
$973$	16.8358	0.539731
$974$	18.8345	0.603497
$975$	0	0
$976$	17.2680	0.552735
$977$	−37.9108	−1.21287	−0.606437	−	0.795132i	$-0.707402\pi$
$977$	−37.9108	−1.21287	−0.606437	+	0.795132i	$0.707402\pi$
$978$	−44.1602	−1.41209
$979$	0.741970	0.0237135
$980$	0	0
$981$	−16.1501	−0.515632
$982$	50.3007	1.60516
$983$	−21.2314	−0.677178	−0.338589	−	0.940934i	$-0.609950\pi$
$983$	−21.2314	−0.677178	−0.338589	+	0.940934i	$0.609950\pi$
$984$	11.2169	0.357583
$985$	0	0
$986$	53.2745	1.69661
$987$	−2.17958	−0.0693766
$988$	−30.1801	−0.960157
$989$	11.7671	0.374171
$990$	0	0
$991$	17.8023	0.565510	0.282755	−	0.959192i	$-0.408752\pi$
$991$	17.8023	0.565510	0.282755	+	0.959192i	$0.408752\pi$
$992$	61.1504	1.94153
$993$	0.313946	0.00996279
$994$	51.1268	1.62164
$995$	0	0
$996$	−8.95169	−0.283645
$997$	−50.1720	−1.58896	−0.794482	−	0.607287i	$-0.792258\pi$
$997$	−50.1720	−1.58896	−0.794482	+	0.607287i	$0.792258\pi$
$998$	14.8770	0.470923
$999$	−4.95252	−0.156691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.v.1.2		5
5.4	even	2		705.2.a.l.1.4	✓	5
15.14	odd	2		2115.2.a.q.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.l.1.4	✓	5	5.4	even	2
2115.2.a.q.1.2		5	15.14	odd	2
3525.2.a.v.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15072 q^{2} -1.00000 q^{3} +2.62562 q^{4} +2.15072 q^{6} -2.17958 q^{7} -1.34553 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.15072 q^{2} -1.00000 q^{3} +2.62562 q^{4} +2.15072 q^{6} -2.17958 q^{7} -1.34553 q^{8} +1.00000 q^{9} -5.34213 q^{11} -2.62562 q^{12} -4.81703 q^{13} +4.68767 q^{14} -2.35737 q^{16} +4.15072 q^{17} -2.15072 q^{18} +2.38622 q^{19} +2.17958 q^{21} +11.4895 q^{22} +3.53694 q^{23} +1.34553 q^{24} +10.3601 q^{26} -1.00000 q^{27} -5.72273 q^{28} -5.96775 q^{29} +7.87908 q^{31} +7.76111 q^{32} +5.34213 q^{33} -8.92707 q^{34} +2.62562 q^{36} +4.95252 q^{37} -5.13210 q^{38} +4.81703 q^{39} +8.33643 q^{41} -4.68767 q^{42} +3.32690 q^{43} -14.0264 q^{44} -7.60699 q^{46} -1.00000 q^{47} +2.35737 q^{48} -2.24945 q^{49} -4.15072 q^{51} -12.6477 q^{52} +6.56630 q^{53} +2.15072 q^{54} +2.93269 q^{56} -2.38622 q^{57} +12.8350 q^{58} +2.30376 q^{59} -7.32512 q^{61} -16.9457 q^{62} -2.17958 q^{63} -11.9773 q^{64} -11.4895 q^{66} -8.32639 q^{67} +10.8982 q^{68} -3.53694 q^{69} +10.9067 q^{71} -1.34553 q^{72} -10.6650 q^{73} -10.6515 q^{74} +6.26529 q^{76} +11.6436 q^{77} -10.3601 q^{78} -5.92707 q^{79} +1.00000 q^{81} -17.9294 q^{82} +3.40936 q^{83} +5.72273 q^{84} -7.15525 q^{86} +5.96775 q^{87} +7.18801 q^{88} -0.138890 q^{89} +10.4991 q^{91} +9.28666 q^{92} -7.87908 q^{93} +2.15072 q^{94} -7.76111 q^{96} +18.2370 q^{97} +4.83795 q^{98} -5.34213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 10 q^{4} - 10 q^{7} + 5 q^{9} - 2 q^{11} - 10 q^{12} - 7 q^{13} - 8 q^{14} + 8 q^{16} + 10 q^{17} + 2 q^{19} + 10 q^{21} + 4 q^{22} - 3 q^{23} - 16 q^{26} - 5 q^{27} - 12 q^{28} - 2 q^{29} - 6 q^{31} + 20 q^{32} + 2 q^{33} - 20 q^{34} + 10 q^{36} - 8 q^{37} + 8 q^{38} + 7 q^{39} + 8 q^{42} - 13 q^{43} + 4 q^{44} - 12 q^{46} - 5 q^{47} - 8 q^{48} + 13 q^{49} - 10 q^{51} - 34 q^{52} + 10 q^{53} - 28 q^{56} - 2 q^{57} + 4 q^{58} - 11 q^{59} + 11 q^{61} - 16 q^{62} - 10 q^{63} - 20 q^{64} - 4 q^{66} - 24 q^{67} + 20 q^{68} + 3 q^{69} - 11 q^{71} - 17 q^{73} + 12 q^{74} - 24 q^{76} + 12 q^{77} + 16 q^{78} - 5 q^{79} + 5 q^{81} - 64 q^{82} + 12 q^{84} + 12 q^{86} + 2 q^{87} + 4 q^{88} - 3 q^{89} + 22 q^{91} - 34 q^{92} + 6 q^{93} - 20 q^{96} + 14 q^{97} + 36 q^{98} - 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 10 * q^4 - 10 * q^7 + 5 * q^9 - 2 * q^11 - 10 * q^12 - 7 * q^13 - 8 * q^14 + 8 * q^16 + 10 * q^17 + 2 * q^19 + 10 * q^21 + 4 * q^22 - 3 * q^23 - 16 * q^26 - 5 * q^27 - 12 * q^28 - 2 * q^29 - 6 * q^31 + 20 * q^32 + 2 * q^33 - 20 * q^34 + 10 * q^36 - 8 * q^37 + 8 * q^38 + 7 * q^39 + 8 * q^42 - 13 * q^43 + 4 * q^44 - 12 * q^46 - 5 * q^47 - 8 * q^48 + 13 * q^49 - 10 * q^51 - 34 * q^52 + 10 * q^53 - 28 * q^56 - 2 * q^57 + 4 * q^58 - 11 * q^59 + 11 * q^61 - 16 * q^62 - 10 * q^63 - 20 * q^64 - 4 * q^66 - 24 * q^67 + 20 * q^68 + 3 * q^69 - 11 * q^71 - 17 * q^73 + 12 * q^74 - 24 * q^76 + 12 * q^77 + 16 * q^78 - 5 * q^79 + 5 * q^81 - 64 * q^82 + 12 * q^84 + 12 * q^86 + 2 * q^87 + 4 * q^88 - 3 * q^89 + 22 * q^91 - 34 * q^92 + 6 * q^93 - 20 * q^96 + 14 * q^97 + 36 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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