LMFDB - Embedded newform 525.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, 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("525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.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:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +6.47214 q^{11} +3.00000 q^{12} -4.47214 q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} +2.23607 q^{18} -2.47214 q^{19} -1.00000 q^{21} +14.4721 q^{22} -4.00000 q^{23} +2.23607 q^{24} -10.0000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -2.00000 q^{29} +1.52786 q^{31} -6.70820 q^{32} +6.47214 q^{33} +4.47214 q^{34} +3.00000 q^{36} +6.94427 q^{37} -5.52786 q^{38} -4.47214 q^{39} -2.00000 q^{41} -2.23607 q^{42} -8.94427 q^{43} +19.4164 q^{44} -8.94427 q^{46} -12.9443 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.00000 q^{51} -13.4164 q^{52} +3.52786 q^{53} +2.23607 q^{54} -2.23607 q^{56} -2.47214 q^{57} -4.47214 q^{58} -8.94427 q^{59} -2.00000 q^{61} +3.41641 q^{62} -1.00000 q^{63} -13.0000 q^{64} +14.4721 q^{66} +4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +5.52786 q^{71} +2.23607 q^{72} +12.4721 q^{73} +15.5279 q^{74} -7.41641 q^{76} -6.47214 q^{77} -10.0000 q^{78} +12.9443 q^{79} +1.00000 q^{81} -4.47214 q^{82} +16.9443 q^{83} -3.00000 q^{84} -20.0000 q^{86} -2.00000 q^{87} +14.4721 q^{88} -2.00000 q^{89} +4.47214 q^{91} -12.0000 q^{92} +1.52786 q^{93} -28.9443 q^{94} -6.70820 q^{96} -8.47214 q^{97} +2.23607 q^{98} +6.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{4} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^4 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 6 q^{4} - 2 q^{7} + 2 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 20 q^{22} - 8 q^{23} - 20 q^{26} + 2 q^{27} - 6 q^{28} - 4 q^{29} + 12 q^{31} + 4 q^{33} + 6 q^{36} - 4 q^{37} - 20 q^{38} - 4 q^{41} + 12 q^{44} - 8 q^{47} - 2 q^{48} + 2 q^{49} + 4 q^{51} + 16 q^{53} + 4 q^{57} - 4 q^{61} - 20 q^{62} - 2 q^{63} - 26 q^{64} + 20 q^{66} + 8 q^{67} + 12 q^{68} - 8 q^{69} + 20 q^{71} + 16 q^{73} + 40 q^{74} + 12 q^{76} - 4 q^{77} - 20 q^{78} + 8 q^{79} + 2 q^{81} + 16 q^{83} - 6 q^{84} - 40 q^{86} - 4 q^{87} + 20 q^{88} - 4 q^{89} - 24 q^{92} + 12 q^{93} - 40 q^{94} - 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^4 - 2 * q^7 + 2 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 20 * q^22 - 8 * q^23 - 20 * q^26 + 2 * q^27 - 6 * q^28 - 4 * q^29 + 12 * q^31 + 4 * q^33 + 6 * q^36 - 4 * q^37 - 20 * q^38 - 4 * q^41 + 12 * q^44 - 8 * q^47 - 2 * q^48 + 2 * q^49 + 4 * q^51 + 16 * q^53 + 4 * q^57 - 4 * q^61 - 20 * q^62 - 2 * q^63 - 26 * q^64 + 20 * q^66 + 8 * q^67 + 12 * q^68 - 8 * q^69 + 20 * q^71 + 16 * q^73 + 40 * q^74 + 12 * q^76 - 4 * q^77 - 20 * q^78 + 8 * q^79 + 2 * q^81 + 16 * q^83 - 6 * q^84 - 40 * q^86 - 4 * q^87 + 20 * q^88 - 4 * q^89 - 24 * q^92 + 12 * q^93 - 40 * q^94 - 8 * q^97 + 4 * 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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	2.23607	0.912871
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	6.47214	1.95142	0.975711	−	0.219061i	$-0.0702993\pi$
$11$	6.47214	1.95142	0.975711	+	0.219061i	$0.0702993\pi$
$12$	3.00000	0.866025
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	−2.23607	−0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	2.23607	0.527046
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	14.4721	3.08547
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−10.0000	−1.96116
$27$	1.00000	0.192450
$28$	−3.00000	−0.566947
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	−6.70820	−1.18585
$33$	6.47214	1.12665
$34$	4.47214	0.766965
$35$	0	0
$36$	3.00000	0.500000
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	−5.52786	−0.896738
$39$	−4.47214	−0.716115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	−2.23607	−0.345033
$43$	−8.94427	−1.36399	−0.681994	−	0.731357i	$-0.738887\pi$
$43$	−8.94427	−1.36399	−0.681994	+	0.731357i	$0.738887\pi$
$44$	19.4164	2.92713
$45$	0	0
$46$	−8.94427	−1.31876
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	−13.4164	−1.86052
$53$	3.52786	0.484589	0.242295	−	0.970203i	$-0.422100\pi$
$53$	3.52786	0.484589	0.242295	+	0.970203i	$0.422100\pi$
$54$	2.23607	0.304290
$55$	0	0
$56$	−2.23607	−0.298807
$57$	−2.47214	−0.327442
$58$	−4.47214	−0.587220
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.41641	0.433884
$63$	−1.00000	−0.125988
$64$	−13.0000	−1.62500
$65$	0	0
$66$	14.4721	1.78140
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	6.00000	0.727607
$69$	−4.00000	−0.481543
$70$	0	0
$71$	5.52786	0.656037	0.328018	−	0.944671i	$-0.393619\pi$
$71$	5.52786	0.656037	0.328018	+	0.944671i	$0.393619\pi$
$72$	2.23607	0.263523
$73$	12.4721	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$73$	12.4721	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$74$	15.5279	1.80508
$75$	0	0
$76$	−7.41641	−0.850720
$77$	−6.47214	−0.737568
$78$	−10.0000	−1.13228
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.47214	−0.493865
$83$	16.9443	1.85988	0.929938	−	0.367717i	$-0.119860\pi$
$83$	16.9443	1.85988	0.929938	+	0.367717i	$0.119860\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−20.0000	−2.15666
$87$	−2.00000	−0.214423
$88$	14.4721	1.54273
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	−12.0000	−1.25109
$93$	1.52786	0.158432
$94$	−28.9443	−2.98537
$95$	0	0
$96$	−6.70820	−0.684653
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	2.23607	0.225877
$99$	6.47214	0.650474
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	4.47214	0.442807
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−10.0000	−0.980581
$105$	0	0
$106$	7.88854	0.766203
$107$	12.9443	1.25137	0.625685	−	0.780076i	$-0.284820\pi$
$107$	12.9443	1.25137	0.625685	+	0.780076i	$0.284820\pi$
$108$	3.00000	0.288675
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	6.94427	0.659121
$112$	1.00000	0.0944911
$113$	−0.472136	−0.0444148	−0.0222074	−	0.999753i	$-0.507069\pi$
$113$	−0.472136	−0.0444148	−0.0222074	+	0.999753i	$0.507069\pi$
$114$	−5.52786	−0.517732
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−4.47214	−0.413449
$118$	−20.0000	−1.84115
$119$	−2.00000	−0.183340
$120$	0	0
$121$	30.8885	2.80805
$122$	−4.47214	−0.404888
$123$	−2.00000	−0.180334
$124$	4.58359	0.411619
$125$	0	0
$126$	−2.23607	−0.199205
$127$	4.94427	0.438733	0.219367	−	0.975643i	$-0.429601\pi$
$127$	4.94427	0.438733	0.219367	+	0.975643i	$0.429601\pi$
$128$	−15.6525	−1.38350
$129$	−8.94427	−0.787499
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	19.4164	1.68998
$133$	2.47214	0.214361
$134$	8.94427	0.772667
$135$	0	0
$136$	4.47214	0.383482
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	−8.94427	−0.761387
$139$	7.41641	0.629052	0.314526	−	0.949249i	$-0.398155\pi$
$139$	7.41641	0.629052	0.314526	+	0.949249i	$0.398155\pi$
$140$	0	0
$141$	−12.9443	−1.09010
$142$	12.3607	1.03729
$143$	−28.9443	−2.42044
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	27.8885	2.30807
$147$	1.00000	0.0824786
$148$	20.8328	1.71245
$149$	−14.9443	−1.22428	−0.612141	−	0.790748i	$-0.709692\pi$
$149$	−14.9443	−1.22428	−0.612141	+	0.790748i	$0.709692\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−5.52786	−0.448369
$153$	2.00000	0.161690
$154$	−14.4721	−1.16620
$155$	0	0
$156$	−13.4164	−1.07417
$157$	0.472136	0.0376806	0.0188403	−	0.999823i	$-0.494003\pi$
$157$	0.472136	0.0376806	0.0188403	+	0.999823i	$0.494003\pi$
$158$	28.9443	2.30268
$159$	3.52786	0.279778
$160$	0	0
$161$	4.00000	0.315244
$162$	2.23607	0.175682
$163$	16.9443	1.32718	0.663589	−	0.748097i	$-0.269032\pi$
$163$	16.9443	1.32718	0.663589	+	0.748097i	$0.269032\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	37.8885	2.94072
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−2.23607	−0.172516
$169$	7.00000	0.538462
$170$	0	0
$171$	−2.47214	−0.189049
$172$	−26.8328	−2.04598
$173$	2.94427	0.223849	0.111924	−	0.993717i	$-0.464299\pi$
$173$	2.94427	0.223849	0.111924	+	0.993717i	$0.464299\pi$
$174$	−4.47214	−0.339032
$175$	0	0
$176$	−6.47214	−0.487856
$177$	−8.94427	−0.672293
$178$	−4.47214	−0.335201
$179$	6.47214	0.483750	0.241875	−	0.970307i	$-0.422238\pi$
$179$	6.47214	0.483750	0.241875	+	0.970307i	$0.422238\pi$
$180$	0	0
$181$	1.05573	0.0784717	0.0392358	−	0.999230i	$-0.487508\pi$
$181$	1.05573	0.0784717	0.0392358	+	0.999230i	$0.487508\pi$
$182$	10.0000	0.741249
$183$	−2.00000	−0.147844
$184$	−8.94427	−0.659380
$185$	0	0
$186$	3.41641	0.250503
$187$	12.9443	0.946579
$188$	−38.8328	−2.83217
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	0.583592	0.0422272	0.0211136	−	0.999777i	$-0.493279\pi$
$191$	0.583592	0.0422272	0.0211136	+	0.999777i	$0.493279\pi$
$192$	−13.0000	−0.938194
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−18.9443	−1.36012
$195$	0	0
$196$	3.00000	0.214286
$197$	−15.5279	−1.10631	−0.553157	−	0.833077i	$-0.686577\pi$
$197$	−15.5279	−1.10631	−0.553157	+	0.833077i	$0.686577\pi$
$198$	14.4721	1.02849
$199$	27.4164	1.94350	0.971749	−	0.236017i	$-0.0758423\pi$
$199$	27.4164	1.94350	0.971749	+	0.236017i	$0.0758423\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	−31.3050	−2.20261
$203$	2.00000	0.140372
$204$	6.00000	0.420084
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	4.47214	0.310087
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	10.5836	0.726884
$213$	5.52786	0.378763
$214$	28.9443	1.97859
$215$	0	0
$216$	2.23607	0.152145
$217$	−1.52786	−0.103718
$218$	−4.47214	−0.302891
$219$	12.4721	0.842789
$220$	0	0
$221$	−8.94427	−0.601657
$222$	15.5279	1.04216
$223$	12.9443	0.866813	0.433406	−	0.901199i	$-0.357312\pi$
$223$	12.9443	0.866813	0.433406	+	0.901199i	$0.357312\pi$
$224$	6.70820	0.448211
$225$	0	0
$226$	−1.05573	−0.0702260
$227$	−0.944272	−0.0626735	−0.0313368	−	0.999509i	$-0.509976\pi$
$227$	−0.944272	−0.0626735	−0.0313368	+	0.999509i	$0.509976\pi$
$228$	−7.41641	−0.491164
$229$	23.8885	1.57860	0.789300	−	0.614008i	$-0.210444\pi$
$229$	23.8885	1.57860	0.789300	+	0.614008i	$0.210444\pi$
$230$	0	0
$231$	−6.47214	−0.425835
$232$	−4.47214	−0.293610
$233$	9.41641	0.616889	0.308445	−	0.951242i	$-0.400192\pi$
$233$	9.41641	0.616889	0.308445	+	0.951242i	$0.400192\pi$
$234$	−10.0000	−0.653720
$235$	0	0
$236$	−26.8328	−1.74667
$237$	12.9443	0.840821
$238$	−4.47214	−0.289886
$239$	−10.4721	−0.677386	−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721	−0.677386	−0.338693	+	0.940897i	$0.609985\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	69.0689	4.43992
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	0	0
$246$	−4.47214	−0.285133
$247$	11.0557	0.703459
$248$	3.41641	0.216942
$249$	16.9443	1.07380
$250$	0	0
$251$	16.9443	1.06951	0.534756	−	0.845006i	$-0.320403\pi$
$251$	16.9443	1.06951	0.534756	+	0.845006i	$0.320403\pi$
$252$	−3.00000	−0.188982
$253$	−25.8885	−1.62760
$254$	11.0557	0.693698
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−18.9443	−1.18171	−0.590856	−	0.806777i	$-0.701210\pi$
$257$	−18.9443	−1.18171	−0.590856	+	0.806777i	$0.701210\pi$
$258$	−20.0000	−1.24515
$259$	−6.94427	−0.431496
$260$	0	0
$261$	−2.00000	−0.123797
$262$	8.94427	0.552579
$263$	−7.05573	−0.435075	−0.217537	−	0.976052i	$-0.569802\pi$
$263$	−7.05573	−0.435075	−0.217537	+	0.976052i	$0.569802\pi$
$264$	14.4721	0.890698
$265$	0	0
$266$	5.52786	0.338935
$267$	−2.00000	−0.122398
$268$	12.0000	0.733017
$269$	11.8885	0.724857	0.362429	−	0.932012i	$-0.381948\pi$
$269$	11.8885	0.724857	0.362429	+	0.932012i	$0.381948\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	−2.00000	−0.121268
$273$	4.47214	0.270666
$274$	−7.88854	−0.476564
$275$	0	0
$276$	−12.0000	−0.722315
$277$	−18.9443	−1.13825	−0.569125	−	0.822251i	$-0.692718\pi$
$277$	−18.9443	−1.13825	−0.569125	+	0.822251i	$0.692718\pi$
$278$	16.5836	0.994618
$279$	1.52786	0.0914708
$280$	0	0
$281$	−10.9443	−0.652881	−0.326440	−	0.945218i	$-0.605849\pi$
$281$	−10.9443	−0.652881	−0.326440	+	0.945218i	$0.605849\pi$
$282$	−28.9443	−1.72361
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	16.5836	0.984055
$285$	0	0
$286$	−64.7214	−3.82705
$287$	2.00000	0.118056
$288$	−6.70820	−0.395285
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−8.47214	−0.496645
$292$	37.4164	2.18963
$293$	−5.05573	−0.295359	−0.147679	−	0.989035i	$-0.547180\pi$
$293$	−5.05573	−0.295359	−0.147679	+	0.989035i	$0.547180\pi$
$294$	2.23607	0.130410
$295$	0	0
$296$	15.5279	0.902539
$297$	6.47214	0.375551
$298$	−33.4164	−1.93576
$299$	17.8885	1.03452
$300$	0	0
$301$	8.94427	0.515539
$302$	−35.7771	−2.05874
$303$	−14.0000	−0.804279
$304$	2.47214	0.141787
$305$	0	0
$306$	4.47214	0.255655
$307$	15.0557	0.859276	0.429638	−	0.903001i	$-0.358641\pi$
$307$	15.0557	0.859276	0.429638	+	0.903001i	$0.358641\pi$
$308$	−19.4164	−1.10635
$309$	0	0
$310$	0	0
$311$	25.8885	1.46800	0.734002	−	0.679147i	$-0.237650\pi$
$311$	25.8885	1.46800	0.734002	+	0.679147i	$0.237650\pi$
$312$	−10.0000	−0.566139
$313$	17.4164	0.984434	0.492217	−	0.870473i	$-0.336187\pi$
$313$	17.4164	0.984434	0.492217	+	0.870473i	$0.336187\pi$
$314$	1.05573	0.0595782
$315$	0	0
$316$	38.8328	2.18452
$317$	−14.3607	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$317$	−14.3607	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$318$	7.88854	0.442368
$319$	−12.9443	−0.724740
$320$	0	0
$321$	12.9443	0.722479
$322$	8.94427	0.498445
$323$	−4.94427	−0.275107
$324$	3.00000	0.166667
$325$	0	0
$326$	37.8885	2.09845
$327$	−2.00000	−0.110600
$328$	−4.47214	−0.246932
$329$	12.9443	0.713641
$330$	0	0
$331$	0.944272	0.0519019	0.0259509	−	0.999663i	$-0.491739\pi$
$331$	0.944272	0.0519019	0.0259509	+	0.999663i	$0.491739\pi$
$332$	50.8328	2.78981
$333$	6.94427	0.380544
$334$	17.8885	0.978818
$335$	0	0
$336$	1.00000	0.0545545
$337$	23.8885	1.30129	0.650646	−	0.759381i	$-0.274498\pi$
$337$	23.8885	1.30129	0.650646	+	0.759381i	$0.274498\pi$
$338$	15.6525	0.851382
$339$	−0.472136	−0.0256429
$340$	0	0
$341$	9.88854	0.535495
$342$	−5.52786	−0.298913
$343$	−1.00000	−0.0539949
$344$	−20.0000	−1.07833
$345$	0	0
$346$	6.58359	0.353936
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	−6.00000	−0.321634
$349$	−11.8885	−0.636379	−0.318190	−	0.948027i	$-0.603075\pi$
$349$	−11.8885	−0.636379	−0.318190	+	0.948027i	$0.603075\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	−43.4164	−2.31410
$353$	−7.88854	−0.419865	−0.209932	−	0.977716i	$-0.567324\pi$
$353$	−7.88854	−0.419865	−0.209932	+	0.977716i	$0.567324\pi$
$354$	−20.0000	−1.06299
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−2.00000	−0.105851
$358$	14.4721	0.764876
$359$	18.4721	0.974922	0.487461	−	0.873145i	$-0.337923\pi$
$359$	18.4721	0.974922	0.487461	+	0.873145i	$0.337923\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	2.36068	0.124075
$363$	30.8885	1.62123
$364$	13.4164	0.703211
$365$	0	0
$366$	−4.47214	−0.233762
$367$	−3.05573	−0.159508	−0.0797539	−	0.996815i	$-0.525413\pi$
$367$	−3.05573	−0.159508	−0.0797539	+	0.996815i	$0.525413\pi$
$368$	4.00000	0.208514
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−3.52786	−0.183158
$372$	4.58359	0.237648
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	28.9443	1.49667
$375$	0	0
$376$	−28.9443	−1.49269
$377$	8.94427	0.460653
$378$	−2.23607	−0.115011
$379$	−37.8885	−1.94620	−0.973102	−	0.230375i	$-0.926005\pi$
$379$	−37.8885	−1.94620	−0.973102	+	0.230375i	$0.926005\pi$
$380$	0	0
$381$	4.94427	0.253303
$382$	1.30495	0.0667671
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	−15.6525	−0.798762
$385$	0	0
$386$	31.3050	1.59338
$387$	−8.94427	−0.454663
$388$	−25.4164	−1.29032
$389$	−6.94427	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$389$	−6.94427	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	2.23607	0.112938
$393$	4.00000	0.201773
$394$	−34.7214	−1.74924
$395$	0	0
$396$	19.4164	0.975711
$397$	13.4164	0.673350	0.336675	−	0.941621i	$-0.390698\pi$
$397$	13.4164	0.673350	0.336675	+	0.941621i	$0.390698\pi$
$398$	61.3050	3.07294
$399$	2.47214	0.123762
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	8.94427	0.446100
$403$	−6.83282	−0.340367
$404$	−42.0000	−2.08958
$405$	0	0
$406$	4.47214	0.221948
$407$	44.9443	2.22780
$408$	4.47214	0.221404
$409$	11.8885	0.587851	0.293925	−	0.955828i	$-0.405038\pi$
$409$	11.8885	0.587851	0.293925	+	0.955828i	$0.405038\pi$
$410$	0	0
$411$	−3.52786	−0.174017
$412$	0	0
$413$	8.94427	0.440119
$414$	−8.94427	−0.439587
$415$	0	0
$416$	30.0000	1.47087
$417$	7.41641	0.363183
$418$	−35.7771	−1.74991
$419$	−29.8885	−1.46015	−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−37.8885	−1.84439
$423$	−12.9443	−0.629372
$424$	7.88854	0.383102
$425$	0	0
$426$	12.3607	0.598877
$427$	2.00000	0.0967868
$428$	38.8328	1.87705
$429$	−28.9443	−1.39744
$430$	0	0
$431$	−18.4721	−0.889771	−0.444886	−	0.895587i	$-0.646756\pi$
$431$	−18.4721	−0.889771	−0.444886	+	0.895587i	$0.646756\pi$
$432$	−1.00000	−0.0481125
$433$	−16.4721	−0.791600	−0.395800	−	0.918337i	$-0.629533\pi$
$433$	−16.4721	−0.791600	−0.395800	+	0.918337i	$0.629533\pi$
$434$	−3.41641	−0.163993
$435$	0	0
$436$	−6.00000	−0.287348
$437$	9.88854	0.473033
$438$	27.8885	1.33257
$439$	1.52786	0.0729210	0.0364605	−	0.999335i	$-0.488392\pi$
$439$	1.52786	0.0729210	0.0364605	+	0.999335i	$0.488392\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−20.0000	−0.951303
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	20.8328	0.988682
$445$	0	0
$446$	28.9443	1.37055
$447$	−14.9443	−0.706840
$448$	13.0000	0.614192
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−12.9443	−0.609522
$452$	−1.41641	−0.0666222
$453$	−16.0000	−0.751746
$454$	−2.11146	−0.0990955
$455$	0	0
$456$	−5.52786	−0.258866
$457$	−6.94427	−0.324839	−0.162420	−	0.986722i	$-0.551930\pi$
$457$	−6.94427	−0.324839	−0.162420	+	0.986722i	$0.551930\pi$
$458$	53.4164	2.49598
$459$	2.00000	0.0933520
$460$	0	0
$461$	3.88854	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$461$	3.88854	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$462$	−14.4721	−0.673305
$463$	−20.9443	−0.973363	−0.486681	−	0.873580i	$-0.661793\pi$
$463$	−20.9443	−0.973363	−0.486681	+	0.873580i	$0.661793\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	21.0557	0.975388
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	−13.4164	−0.620174
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0.472136	0.0217549
$472$	−20.0000	−0.920575
$473$	−57.8885	−2.66172
$474$	28.9443	1.32945
$475$	0	0
$476$	−6.00000	−0.275010
$477$	3.52786	0.161530
$478$	−23.4164	−1.07104
$479$	−17.8885	−0.817348	−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885	−0.817348	−0.408674	+	0.912680i	$0.634009\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	−42.3607	−1.92948
$483$	4.00000	0.182006
$484$	92.6656	4.21207
$485$	0	0
$486$	2.23607	0.101430
$487$	20.9443	0.949076	0.474538	−	0.880235i	$-0.342615\pi$
$487$	20.9443	0.949076	0.474538	+	0.880235i	$0.342615\pi$
$488$	−4.47214	−0.202444
$489$	16.9443	0.766246
$490$	0	0
$491$	−21.3050	−0.961479	−0.480740	−	0.876863i	$-0.659632\pi$
$491$	−21.3050	−0.961479	−0.480740	+	0.876863i	$0.659632\pi$
$492$	−6.00000	−0.270501
$493$	−4.00000	−0.180151
$494$	24.7214	1.11227
$495$	0	0
$496$	−1.52786	−0.0686031
$497$	−5.52786	−0.247959
$498$	37.8885	1.69783
$499$	−13.8885	−0.621737	−0.310868	−	0.950453i	$-0.600620\pi$
$499$	−13.8885	−0.621737	−0.310868	+	0.950453i	$0.600620\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	37.8885	1.69105
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	−2.23607	−0.0996024
$505$	0	0
$506$	−57.8885	−2.57346
$507$	7.00000	0.310881
$508$	14.8328	0.658100
$509$	−23.8885	−1.05884	−0.529421	−	0.848360i	$-0.677591\pi$
$509$	−23.8885	−1.05884	−0.529421	+	0.848360i	$0.677591\pi$
$510$	0	0
$511$	−12.4721	−0.551735
$512$	11.1803	0.494106
$513$	−2.47214	−0.109147
$514$	−42.3607	−1.86845
$515$	0	0
$516$	−26.8328	−1.18125
$517$	−83.7771	−3.68451
$518$	−15.5279	−0.682255
$519$	2.94427	0.129239
$520$	0	0
$521$	−19.8885	−0.871333	−0.435666	−	0.900108i	$-0.643487\pi$
$521$	−19.8885	−0.871333	−0.435666	+	0.900108i	$0.643487\pi$
$522$	−4.47214	−0.195740
$523$	8.94427	0.391106	0.195553	−	0.980693i	$-0.437350\pi$
$523$	8.94427	0.391106	0.195553	+	0.980693i	$0.437350\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−15.7771	−0.687914
$527$	3.05573	0.133110
$528$	−6.47214	−0.281664
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−8.94427	−0.388148
$532$	7.41641	0.321542
$533$	8.94427	0.387419
$534$	−4.47214	−0.193528
$535$	0	0
$536$	8.94427	0.386334
$537$	6.47214	0.279293
$538$	26.5836	1.14610
$539$	6.47214	0.278775
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	−3.41641	−0.146747
$543$	1.05573	0.0453056
$544$	−13.4164	−0.575224
$545$	0	0
$546$	10.0000	0.427960
$547$	−5.88854	−0.251776	−0.125888	−	0.992044i	$-0.540178\pi$
$547$	−5.88854	−0.251776	−0.125888	+	0.992044i	$0.540178\pi$
$548$	−10.5836	−0.452109
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	4.94427	0.210633
$552$	−8.94427	−0.380693
$553$	−12.9443	−0.550446
$554$	−42.3607	−1.79973
$555$	0	0
$556$	22.2492	0.943577
$557$	−20.4721	−0.867432	−0.433716	−	0.901050i	$-0.642798\pi$
$557$	−20.4721	−0.867432	−0.433716	+	0.901050i	$0.642798\pi$
$558$	3.41641	0.144628
$559$	40.0000	1.69182
$560$	0	0
$561$	12.9443	0.546508
$562$	−24.4721	−1.03229
$563$	−13.8885	−0.585332	−0.292666	−	0.956215i	$-0.594542\pi$
$563$	−13.8885	−0.585332	−0.292666	+	0.956215i	$0.594542\pi$
$564$	−38.8328	−1.63516
$565$	0	0
$566$	−26.8328	−1.12787
$567$	−1.00000	−0.0419961
$568$	12.3607	0.518643
$569$	−39.8885	−1.67221	−0.836107	−	0.548566i	$-0.815174\pi$
$569$	−39.8885	−1.67221	−0.836107	+	0.548566i	$0.815174\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−86.8328	−3.63066
$573$	0.583592	0.0243799
$574$	4.47214	0.186663
$575$	0	0
$576$	−13.0000	−0.541667
$577$	−10.3607	−0.431321	−0.215660	−	0.976468i	$-0.569190\pi$
$577$	−10.3607	−0.431321	−0.215660	+	0.976468i	$0.569190\pi$
$578$	−29.0689	−1.20911
$579$	14.0000	0.581820
$580$	0	0
$581$	−16.9443	−0.702967
$582$	−18.9443	−0.785265
$583$	22.8328	0.945639
$584$	27.8885	1.15404
$585$	0	0
$586$	−11.3050	−0.467003
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	3.00000	0.123718
$589$	−3.77709	−0.155632
$590$	0	0
$591$	−15.5279	−0.638731
$592$	−6.94427	−0.285408
$593$	−23.8885	−0.980985	−0.490492	−	0.871445i	$-0.663183\pi$
$593$	−23.8885	−0.980985	−0.490492	+	0.871445i	$0.663183\pi$
$594$	14.4721	0.593799
$595$	0	0
$596$	−44.8328	−1.83642
$597$	27.4164	1.12208
$598$	40.0000	1.63572
$599$	12.3607	0.505044	0.252522	−	0.967591i	$-0.418740\pi$
$599$	12.3607	0.505044	0.252522	+	0.967591i	$0.418740\pi$
$600$	0	0
$601$	38.9443	1.58857	0.794285	−	0.607545i	$-0.207846\pi$
$601$	38.9443	1.58857	0.794285	+	0.607545i	$0.207846\pi$
$602$	20.0000	0.815139
$603$	4.00000	0.162893
$604$	−48.0000	−1.95309
$605$	0	0
$606$	−31.3050	−1.27168
$607$	−38.8328	−1.57618	−0.788088	−	0.615563i	$-0.788929\pi$
$607$	−38.8328	−1.57618	−0.788088	+	0.615563i	$0.788929\pi$
$608$	16.5836	0.672553
$609$	2.00000	0.0810441
$610$	0	0
$611$	57.8885	2.34192
$612$	6.00000	0.242536
$613$	6.94427	0.280477	0.140238	−	0.990118i	$-0.455213\pi$
$613$	6.94427	0.280477	0.140238	+	0.990118i	$0.455213\pi$
$614$	33.6656	1.35863
$615$	0	0
$616$	−14.4721	−0.583099
$617$	−16.4721	−0.663143	−0.331572	−	0.943430i	$-0.607579\pi$
$617$	−16.4721	−0.663143	−0.331572	+	0.943430i	$0.607579\pi$
$618$	0	0
$619$	39.4164	1.58428	0.792140	−	0.610340i	$-0.208967\pi$
$619$	39.4164	1.58428	0.792140	+	0.610340i	$0.208967\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	57.8885	2.32112
$623$	2.00000	0.0801283
$624$	4.47214	0.179029
$625$	0	0
$626$	38.9443	1.55653
$627$	−16.0000	−0.638978
$628$	1.41641	0.0565208
$629$	13.8885	0.553773
$630$	0	0
$631$	30.8328	1.22744	0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328	1.22744	0.613718	+	0.789526i	$0.289673\pi$
$632$	28.9443	1.15134
$633$	−16.9443	−0.673474
$634$	−32.1115	−1.27531
$635$	0	0
$636$	10.5836	0.419667
$637$	−4.47214	−0.177192
$638$	−28.9443	−1.14591
$639$	5.52786	0.218679
$640$	0	0
$641$	16.8328	0.664856	0.332428	−	0.943129i	$-0.392132\pi$
$641$	16.8328	0.664856	0.332428	+	0.943129i	$0.392132\pi$
$642$	28.9443	1.14234
$643$	15.0557	0.593740	0.296870	−	0.954918i	$-0.404057\pi$
$643$	15.0557	0.593740	0.296870	+	0.954918i	$0.404057\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	−11.0557	−0.434982
$647$	1.88854	0.0742463	0.0371232	−	0.999311i	$-0.488181\pi$
$647$	1.88854	0.0742463	0.0371232	+	0.999311i	$0.488181\pi$
$648$	2.23607	0.0878410
$649$	−57.8885	−2.27232
$650$	0	0
$651$	−1.52786	−0.0598817
$652$	50.8328	1.99077
$653$	22.5836	0.883764	0.441882	−	0.897073i	$-0.354311\pi$
$653$	22.5836	0.883764	0.441882	+	0.897073i	$0.354311\pi$
$654$	−4.47214	−0.174874
$655$	0	0
$656$	2.00000	0.0780869
$657$	12.4721	0.486584
$658$	28.9443	1.12837
$659$	21.3050	0.829923	0.414962	−	0.909839i	$-0.363795\pi$
$659$	21.3050	0.829923	0.414962	+	0.909839i	$0.363795\pi$
$660$	0	0
$661$	−35.8885	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$661$	−35.8885	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$662$	2.11146	0.0820641
$663$	−8.94427	−0.347367
$664$	37.8885	1.47036
$665$	0	0
$666$	15.5279	0.601693
$667$	8.00000	0.309761
$668$	24.0000	0.928588
$669$	12.9443	0.500454
$670$	0	0
$671$	−12.9443	−0.499708
$672$	6.70820	0.258775
$673$	−8.83282	−0.340480	−0.170240	−	0.985403i	$-0.554454\pi$
$673$	−8.83282	−0.340480	−0.170240	+	0.985403i	$0.554454\pi$
$674$	53.4164	2.05752
$675$	0	0
$676$	21.0000	0.807692
$677$	−21.0557	−0.809237	−0.404619	−	0.914485i	$-0.632596\pi$
$677$	−21.0557	−0.809237	−0.404619	+	0.914485i	$0.632596\pi$
$678$	−1.05573	−0.0405450
$679$	8.47214	0.325131
$680$	0	0
$681$	−0.944272	−0.0361846
$682$	22.1115	0.846691
$683$	1.88854	0.0722631	0.0361316	−	0.999347i	$-0.488496\pi$
$683$	1.88854	0.0722631	0.0361316	+	0.999347i	$0.488496\pi$
$684$	−7.41641	−0.283573
$685$	0	0
$686$	−2.23607	−0.0853735
$687$	23.8885	0.911405
$688$	8.94427	0.340997
$689$	−15.7771	−0.601059
$690$	0	0
$691$	44.3607	1.68756	0.843780	−	0.536689i	$-0.180325\pi$
$691$	44.3607	1.68756	0.843780	+	0.536689i	$0.180325\pi$
$692$	8.83282	0.335773
$693$	−6.47214	−0.245856
$694$	17.8885	0.679040
$695$	0	0
$696$	−4.47214	−0.169516
$697$	−4.00000	−0.151511
$698$	−26.5836	−1.00620
$699$	9.41641	0.356161
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	−10.0000	−0.377426
$703$	−17.1672	−0.647473
$704$	−84.1378	−3.17106
$705$	0	0
$706$	−17.6393	−0.663865
$707$	14.0000	0.526524
$708$	−26.8328	−1.00844
$709$	25.7771	0.968079	0.484039	−	0.875046i	$-0.339169\pi$
$709$	25.7771	0.968079	0.484039	+	0.875046i	$0.339169\pi$
$710$	0	0
$711$	12.9443	0.485448
$712$	−4.47214	−0.167600
$713$	−6.11146	−0.228876
$714$	−4.47214	−0.167365
$715$	0	0
$716$	19.4164	0.725625
$717$	−10.4721	−0.391089
$718$	41.3050	1.54149
$719$	−6.83282	−0.254821	−0.127411	−	0.991850i	$-0.540667\pi$
$719$	−6.83282	−0.254821	−0.127411	+	0.991850i	$0.540667\pi$
$720$	0	0
$721$	0	0
$722$	−28.8197	−1.07256
$723$	−18.9443	−0.704545
$724$	3.16718	0.117707
$725$	0	0
$726$	69.0689	2.56339
$727$	38.8328	1.44023	0.720115	−	0.693855i	$-0.244089\pi$
$727$	38.8328	1.44023	0.720115	+	0.693855i	$0.244089\pi$
$728$	10.0000	0.370625
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.8885	−0.661632
$732$	−6.00000	−0.221766
$733$	−10.5836	−0.390914	−0.195457	−	0.980712i	$-0.562619\pi$
$733$	−10.5836	−0.390914	−0.195457	+	0.980712i	$0.562619\pi$
$734$	−6.83282	−0.252204
$735$	0	0
$736$	26.8328	0.989071
$737$	25.8885	0.953617
$738$	−4.47214	−0.164622
$739$	−5.88854	−0.216614	−0.108307	−	0.994118i	$-0.534543\pi$
$739$	−5.88854	−0.216614	−0.108307	+	0.994118i	$0.534543\pi$
$740$	0	0
$741$	11.0557	0.406142
$742$	−7.88854	−0.289598
$743$	−34.8328	−1.27789	−0.638946	−	0.769252i	$-0.720629\pi$
$743$	−34.8328	−1.27789	−0.638946	+	0.769252i	$0.720629\pi$
$744$	3.41641	0.125252
$745$	0	0
$746$	−13.4164	−0.491210
$747$	16.9443	0.619958
$748$	38.8328	1.41987
$749$	−12.9443	−0.472973
$750$	0	0
$751$	−20.9443	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$752$	12.9443	0.472029
$753$	16.9443	0.617484
$754$	20.0000	0.728357
$755$	0	0
$756$	−3.00000	−0.109109
$757$	−31.8885	−1.15901	−0.579504	−	0.814969i	$-0.696754\pi$
$757$	−31.8885	−1.15901	−0.579504	+	0.814969i	$0.696754\pi$
$758$	−84.7214	−3.07722
$759$	−25.8885	−0.939695
$760$	0	0
$761$	−27.8885	−1.01096	−0.505479	−	0.862839i	$-0.668684\pi$
$761$	−27.8885	−1.01096	−0.505479	+	0.862839i	$0.668684\pi$
$762$	11.0557	0.400507
$763$	2.00000	0.0724049
$764$	1.75078	0.0633409
$765$	0	0
$766$	17.8885	0.646339
$767$	40.0000	1.44432
$768$	−9.00000	−0.324760
$769$	−52.8328	−1.90520	−0.952600	−	0.304226i	$-0.901602\pi$
$769$	−52.8328	−1.90520	−0.952600	+	0.304226i	$0.901602\pi$
$770$	0	0
$771$	−18.9443	−0.682261
$772$	42.0000	1.51161
$773$	42.9443	1.54460	0.772299	−	0.635259i	$-0.219107\pi$
$773$	42.9443	1.54460	0.772299	+	0.635259i	$0.219107\pi$
$774$	−20.0000	−0.718885
$775$	0	0
$776$	−18.9443	−0.680060
$777$	−6.94427	−0.249124
$778$	−15.5279	−0.556701
$779$	4.94427	0.177147
$780$	0	0
$781$	35.7771	1.28020
$782$	−17.8885	−0.639693
$783$	−2.00000	−0.0714742
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	8.94427	0.319032
$787$	−31.0557	−1.10702	−0.553509	−	0.832843i	$-0.686711\pi$
$787$	−31.0557	−1.10702	−0.553509	+	0.832843i	$0.686711\pi$
$788$	−46.5836	−1.65947
$789$	−7.05573	−0.251191
$790$	0	0
$791$	0.472136	0.0167872
$792$	14.4721	0.514245
$793$	8.94427	0.317620
$794$	30.0000	1.06466
$795$	0	0
$796$	82.2492	2.91525
$797$	18.9443	0.671041	0.335520	−	0.942033i	$-0.391088\pi$
$797$	18.9443	0.671041	0.335520	+	0.942033i	$0.391088\pi$
$798$	5.52786	0.195684
$799$	−25.8885	−0.915871
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	22.3607	0.789583
$803$	80.7214	2.84859
$804$	12.0000	0.423207
$805$	0	0
$806$	−15.2786	−0.538167
$807$	11.8885	0.418497
$808$	−31.3050	−1.10130
$809$	38.9443	1.36921	0.684604	−	0.728915i	$-0.259975\pi$
$809$	38.9443	1.36921	0.684604	+	0.728915i	$0.259975\pi$
$810$	0	0
$811$	55.4164	1.94593	0.972967	−	0.230946i	$-0.0741820\pi$
$811$	55.4164	1.94593	0.972967	+	0.230946i	$0.0741820\pi$
$812$	6.00000	0.210559
$813$	−1.52786	−0.0535845
$814$	100.498	3.52247
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	22.1115	0.773582
$818$	26.5836	0.929474
$819$	4.47214	0.156269
$820$	0	0
$821$	33.7771	1.17883	0.589414	−	0.807831i	$-0.299359\pi$
$821$	33.7771	1.17883	0.589414	+	0.807831i	$0.299359\pi$
$822$	−7.88854	−0.275145
$823$	−44.9443	−1.56666	−0.783329	−	0.621607i	$-0.786480\pi$
$823$	−44.9443	−1.56666	−0.783329	+	0.621607i	$0.786480\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	−12.9443	−0.450116	−0.225058	−	0.974345i	$-0.572257\pi$
$827$	−12.9443	−0.450116	−0.225058	+	0.974345i	$0.572257\pi$
$828$	−12.0000	−0.417029
$829$	−13.0557	−0.453444	−0.226722	−	0.973959i	$-0.572801\pi$
$829$	−13.0557	−0.453444	−0.226722	+	0.973959i	$0.572801\pi$
$830$	0	0
$831$	−18.9443	−0.657170
$832$	58.1378	2.01556
$833$	2.00000	0.0692959
$834$	16.5836	0.574243
$835$	0	0
$836$	−48.0000	−1.66011
$837$	1.52786	0.0528107
$838$	−66.8328	−2.30870
$839$	54.8328	1.89304	0.946520	−	0.322647i	$-0.104573\pi$
$839$	54.8328	1.89304	0.946520	+	0.322647i	$0.104573\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	49.1935	1.69532
$843$	−10.9443	−0.376941
$844$	−50.8328	−1.74974
$845$	0	0
$846$	−28.9443	−0.995125
$847$	−30.8885	−1.06134
$848$	−3.52786	−0.121147
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−27.7771	−0.952186
$852$	16.5836	0.568145
$853$	31.3050	1.07186	0.535931	−	0.844262i	$-0.319961\pi$
$853$	31.3050	1.07186	0.535931	+	0.844262i	$0.319961\pi$
$854$	4.47214	0.153033
$855$	0	0
$856$	28.9443	0.989295
$857$	−36.8328	−1.25819	−0.629093	−	0.777330i	$-0.716573\pi$
$857$	−36.8328	−1.25819	−0.629093	+	0.777330i	$0.716573\pi$
$858$	−64.7214	−2.20955
$859$	50.4721	1.72209	0.861044	−	0.508531i	$-0.169811\pi$
$859$	50.4721	1.72209	0.861044	+	0.508531i	$0.169811\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	−41.3050	−1.40685
$863$	−21.8885	−0.745095	−0.372547	−	0.928013i	$-0.621516\pi$
$863$	−21.8885	−0.745095	−0.372547	+	0.928013i	$0.621516\pi$
$864$	−6.70820	−0.228218
$865$	0	0
$866$	−36.8328	−1.25163
$867$	−13.0000	−0.441503
$868$	−4.58359	−0.155577
$869$	83.7771	2.84194
$870$	0	0
$871$	−17.8885	−0.606130
$872$	−4.47214	−0.151446
$873$	−8.47214	−0.286738
$874$	22.1115	0.747931
$875$	0	0
$876$	37.4164	1.26418
$877$	56.8328	1.91911	0.959554	−	0.281525i	$-0.0908402\pi$
$877$	56.8328	1.91911	0.959554	+	0.281525i	$0.0908402\pi$
$878$	3.41641	0.115298
$879$	−5.05573	−0.170525
$880$	0	0
$881$	−27.8885	−0.939589	−0.469794	−	0.882776i	$-0.655672\pi$
$881$	−27.8885	−0.939589	−0.469794	+	0.882776i	$0.655672\pi$
$882$	2.23607	0.0752923
$883$	37.8885	1.27505	0.637526	−	0.770429i	$-0.279958\pi$
$883$	37.8885	1.27505	0.637526	+	0.770429i	$0.279958\pi$
$884$	−26.8328	−0.902485
$885$	0	0
$886$	17.8885	0.600977
$887$	30.8328	1.03526	0.517632	−	0.855603i	$-0.326814\pi$
$887$	30.8328	1.03526	0.517632	+	0.855603i	$0.326814\pi$
$888$	15.5279	0.521081
$889$	−4.94427	−0.165826
$890$	0	0
$891$	6.47214	0.216825
$892$	38.8328	1.30022
$893$	32.0000	1.07084
$894$	−33.4164	−1.11761
$895$	0	0
$896$	15.6525	0.522913
$897$	17.8885	0.597281
$898$	−31.3050	−1.04466
$899$	−3.05573	−0.101914
$900$	0	0
$901$	7.05573	0.235060
$902$	−28.9443	−0.963739
$903$	8.94427	0.297647
$904$	−1.05573	−0.0351130
$905$	0	0
$906$	−35.7771	−1.18861
$907$	−53.8885	−1.78934	−0.894670	−	0.446728i	$-0.852589\pi$
$907$	−53.8885	−1.78934	−0.894670	+	0.446728i	$0.852589\pi$
$908$	−2.83282	−0.0940103
$909$	−14.0000	−0.464351
$910$	0	0
$911$	46.2492	1.53231	0.766153	−	0.642659i	$-0.222169\pi$
$911$	46.2492	1.53231	0.766153	+	0.642659i	$0.222169\pi$
$912$	2.47214	0.0818606
$913$	109.666	3.62940
$914$	−15.5279	−0.513616
$915$	0	0
$916$	71.6656	2.36790
$917$	−4.00000	−0.132092
$918$	4.47214	0.147602
$919$	−35.0557	−1.15638	−0.578191	−	0.815902i	$-0.696241\pi$
$919$	−35.0557	−1.15638	−0.578191	+	0.815902i	$0.696241\pi$
$920$	0	0
$921$	15.0557	0.496103
$922$	8.69505	0.286356
$923$	−24.7214	−0.813713
$924$	−19.4164	−0.638753
$925$	0	0
$926$	−46.8328	−1.53902
$927$	0	0
$928$	13.4164	0.440415
$929$	−16.1115	−0.528600	−0.264300	−	0.964441i	$-0.585141\pi$
$929$	−16.1115	−0.528600	−0.264300	+	0.964441i	$0.585141\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	28.2492	0.925334
$933$	25.8885	0.847553
$934$	20.0000	0.654420
$935$	0	0
$936$	−10.0000	−0.326860
$937$	52.4721	1.71419	0.857095	−	0.515158i	$-0.172267\pi$
$937$	52.4721	1.71419	0.857095	+	0.515158i	$0.172267\pi$
$938$	−8.94427	−0.292041
$939$	17.4164	0.568363
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	1.05573	0.0343975
$943$	8.00000	0.260516
$944$	8.94427	0.291111
$945$	0	0
$946$	−129.443	−4.20855
$947$	17.8885	0.581300	0.290650	−	0.956830i	$-0.406129\pi$
$947$	17.8885	0.581300	0.290650	+	0.956830i	$0.406129\pi$
$948$	38.8328	1.26123
$949$	−55.7771	−1.81060
$950$	0	0
$951$	−14.3607	−0.465677
$952$	−4.47214	−0.144943
$953$	33.4164	1.08246	0.541232	−	0.840873i	$-0.317958\pi$
$953$	33.4164	1.08246	0.541232	+	0.840873i	$0.317958\pi$
$954$	7.88854	0.255401
$955$	0	0
$956$	−31.4164	−1.01608
$957$	−12.9443	−0.418429
$958$	−40.0000	−1.29234
$959$	3.52786	0.113921
$960$	0	0
$961$	−28.6656	−0.924698
$962$	−69.4427	−2.23892
$963$	12.9443	0.417123
$964$	−56.8328	−1.83046
$965$	0	0
$966$	8.94427	0.287777
$967$	−25.8885	−0.832519	−0.416260	−	0.909246i	$-0.636659\pi$
$967$	−25.8885	−0.832519	−0.416260	+	0.909246i	$0.636659\pi$
$968$	69.0689	2.21996
$969$	−4.94427	−0.158833
$970$	0	0
$971$	40.9443	1.31396	0.656982	−	0.753906i	$-0.271833\pi$
$971$	40.9443	1.31396	0.656982	+	0.753906i	$0.271833\pi$
$972$	3.00000	0.0962250
$973$	−7.41641	−0.237759
$974$	46.8328	1.50062
$975$	0	0
$976$	2.00000	0.0640184
$977$	−30.5836	−0.978456	−0.489228	−	0.872156i	$-0.662721\pi$
$977$	−30.5836	−0.978456	−0.489228	+	0.872156i	$0.662721\pi$
$978$	37.8885	1.21154
$979$	−12.9443	−0.413701
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−47.6393	−1.52023
$983$	22.8328	0.728254	0.364127	−	0.931349i	$-0.381367\pi$
$983$	22.8328	0.728254	0.364127	+	0.931349i	$0.381367\pi$
$984$	−4.47214	−0.142566
$985$	0	0
$986$	−8.94427	−0.284844
$987$	12.9443	0.412021
$988$	33.1672	1.05519
$989$	35.7771	1.13765
$990$	0	0
$991$	4.94427	0.157060	0.0785300	−	0.996912i	$-0.474977\pi$
$991$	4.94427	0.157060	0.0785300	+	0.996912i	$0.474977\pi$
$992$	−10.2492	−0.325413
$993$	0.944272	0.0299656
$994$	−12.3607	−0.392057
$995$	0	0
$996$	50.8328	1.61070
$997$	5.41641	0.171539	0.0857697	−	0.996315i	$-0.472665\pi$
$997$	5.41641	0.171539	0.0857697	+	0.996315i	$0.472665\pi$
$998$	−31.0557	−0.983052
$999$	6.94427	0.219707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.a.g.1.2		2
3.2	odd	2		1575.2.a.r.1.1		2
4.3	odd	2		8400.2.a.cx.1.1		2
5.2	odd	4		525.2.d.c.274.3		4
5.3	odd	4		525.2.d.c.274.2		4
5.4	even	2		105.2.a.b.1.1	✓	2
7.6	odd	2		3675.2.a.y.1.2		2
15.2	even	4		1575.2.d.d.1324.1		4
15.8	even	4		1575.2.d.d.1324.4		4
15.14	odd	2		315.2.a.d.1.2		2
20.19	odd	2		1680.2.a.v.1.1		2
35.4	even	6		735.2.i.k.226.2		4
35.9	even	6		735.2.i.k.361.2		4
35.19	odd	6		735.2.i.i.361.2		4
35.24	odd	6		735.2.i.i.226.2		4
35.34	odd	2		735.2.a.k.1.1		2
40.19	odd	2		6720.2.a.cs.1.2		2
40.29	even	2		6720.2.a.cx.1.1		2
60.59	even	2		5040.2.a.bw.1.2		2
105.104	even	2		2205.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.1	✓	2	5.4	even	2
315.2.a.d.1.2		2	15.14	odd	2
525.2.a.g.1.2		2	1.1	even	1	trivial
525.2.d.c.274.2		4	5.3	odd	4
525.2.d.c.274.3		4	5.2	odd	4
735.2.a.k.1.1		2	35.34	odd	2
735.2.i.i.226.2		4	35.24	odd	6
735.2.i.i.361.2		4	35.19	odd	6
735.2.i.k.226.2		4	35.4	even	6
735.2.i.k.361.2		4	35.9	even	6
1575.2.a.r.1.1		2	3.2	odd	2
1575.2.d.d.1324.1		4	15.2	even	4
1575.2.d.d.1324.4		4	15.8	even	4
1680.2.a.v.1.1		2	20.19	odd	2
2205.2.a.w.1.2		2	105.104	even	2
3675.2.a.y.1.2		2	7.6	odd	2
5040.2.a.bw.1.2		2	60.59	even	2
6720.2.a.cs.1.2		2	40.19	odd	2
6720.2.a.cx.1.1		2	40.29	even	2
8400.2.a.cx.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +2.23607 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +6.47214 q^{11} +3.00000 q^{12} -4.47214 q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} +2.23607 q^{18} -2.47214 q^{19} -1.00000 q^{21} +14.4721 q^{22} -4.00000 q^{23} +2.23607 q^{24} -10.0000 q^{26} +1.00000 q^{27} -3.00000 q^{28} -2.00000 q^{29} +1.52786 q^{31} -6.70820 q^{32} +6.47214 q^{33} +4.47214 q^{34} +3.00000 q^{36} +6.94427 q^{37} -5.52786 q^{38} -4.47214 q^{39} -2.00000 q^{41} -2.23607 q^{42} -8.94427 q^{43} +19.4164 q^{44} -8.94427 q^{46} -12.9443 q^{47} -1.00000 q^{48} +1.00000 q^{49} +2.00000 q^{51} -13.4164 q^{52} +3.52786 q^{53} +2.23607 q^{54} -2.23607 q^{56} -2.47214 q^{57} -4.47214 q^{58} -8.94427 q^{59} -2.00000 q^{61} +3.41641 q^{62} -1.00000 q^{63} -13.0000 q^{64} +14.4721 q^{66} +4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +5.52786 q^{71} +2.23607 q^{72} +12.4721 q^{73} +15.5279 q^{74} -7.41641 q^{76} -6.47214 q^{77} -10.0000 q^{78} +12.9443 q^{79} +1.00000 q^{81} -4.47214 q^{82} +16.9443 q^{83} -3.00000 q^{84} -20.0000 q^{86} -2.00000 q^{87} +14.4721 q^{88} -2.00000 q^{89} +4.47214 q^{91} -12.0000 q^{92} +1.52786 q^{93} -28.9443 q^{94} -6.70820 q^{96} -8.47214 q^{97} +2.23607 q^{98} +6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^4 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 6 q^{4} - 2 q^{7} + 2 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 20 q^{22} - 8 q^{23} - 20 q^{26} + 2 q^{27} - 6 q^{28} - 4 q^{29} + 12 q^{31} + 4 q^{33} + 6 q^{36} - 4 q^{37} - 20 q^{38} - 4 q^{41} + 12 q^{44} - 8 q^{47} - 2 q^{48} + 2 q^{49} + 4 q^{51} + 16 q^{53} + 4 q^{57} - 4 q^{61} - 20 q^{62} - 2 q^{63} - 26 q^{64} + 20 q^{66} + 8 q^{67} + 12 q^{68} - 8 q^{69} + 20 q^{71} + 16 q^{73} + 40 q^{74} + 12 q^{76} - 4 q^{77} - 20 q^{78} + 8 q^{79} + 2 q^{81} + 16 q^{83} - 6 q^{84} - 40 q^{86} - 4 q^{87} + 20 q^{88} - 4 q^{89} - 24 q^{92} + 12 q^{93} - 40 q^{94} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^4 - 2 * q^7 + 2 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 20 * q^22 - 8 * q^23 - 20 * q^26 + 2 * q^27 - 6 * q^28 - 4 * q^29 + 12 * q^31 + 4 * q^33 + 6 * q^36 - 4 * q^37 - 20 * q^38 - 4 * q^41 + 12 * q^44 - 8 * q^47 - 2 * q^48 + 2 * q^49 + 4 * q^51 + 16 * q^53 + 4 * q^57 - 4 * q^61 - 20 * q^62 - 2 * q^63 - 26 * q^64 + 20 * q^66 + 8 * q^67 + 12 * q^68 - 8 * q^69 + 20 * q^71 + 16 * q^73 + 40 * q^74 + 12 * q^76 - 4 * q^77 - 20 * q^78 + 8 * q^79 + 2 * q^81 + 16 * q^83 - 6 * q^84 - 40 * q^86 - 4 * q^87 + 20 * q^88 - 4 * q^89 - 24 * q^92 + 12 * q^93 - 40 * q^94 - 8 * q^97 + 4 * q^99

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