LMFDB - Embedded newform 1225.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} -2.00000 q^{6} -2.23607 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} -2.00000 q^{6} -2.23607 q^{8} -1.47214 q^{9} +4.23607 q^{11} -0.763932 q^{12} +3.23607 q^{13} -4.85410 q^{16} +6.47214 q^{17} -2.38197 q^{18} -4.47214 q^{19} +6.85410 q^{22} +1.76393 q^{23} +2.76393 q^{24} +5.23607 q^{26} +5.52786 q^{27} +5.00000 q^{29} +9.70820 q^{31} -3.38197 q^{32} -5.23607 q^{33} +10.4721 q^{34} -0.909830 q^{36} +3.00000 q^{37} -7.23607 q^{38} -4.00000 q^{39} -9.23607 q^{41} +6.23607 q^{43} +2.61803 q^{44} +2.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} -8.00000 q^{51} +2.00000 q^{52} -0.472136 q^{53} +8.94427 q^{54} +5.52786 q^{57} +8.09017 q^{58} +1.70820 q^{59} -3.70820 q^{61} +15.7082 q^{62} +4.23607 q^{64} -8.47214 q^{66} +0.236068 q^{67} +4.00000 q^{68} -2.18034 q^{69} -4.70820 q^{71} +3.29180 q^{72} +13.2361 q^{73} +4.85410 q^{74} -2.76393 q^{76} -6.47214 q^{78} +11.1803 q^{79} -2.41641 q^{81} -14.9443 q^{82} -5.70820 q^{83} +10.0902 q^{86} -6.18034 q^{87} -9.47214 q^{88} -12.7639 q^{89} +1.09017 q^{92} -12.0000 q^{93} +3.23607 q^{94} +4.18034 q^{96} -0.763932 q^{97} -6.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{13} - 3 q^{16} + 4 q^{17} - 7 q^{18} + 7 q^{22} + 8 q^{23} + 10 q^{24} + 6 q^{26} + 20 q^{27} + 10 q^{29} + 6 q^{31} - 9 q^{32} - 6 q^{33} + 12 q^{34} - 13 q^{36} + 6 q^{37} - 10 q^{38} - 8 q^{39} - 14 q^{41} + 8 q^{43} + 3 q^{44} - q^{46} + 4 q^{47} + 12 q^{48} - 16 q^{51} + 4 q^{52} + 8 q^{53} + 20 q^{57} + 5 q^{58} - 10 q^{59} + 6 q^{61} + 18 q^{62} + 4 q^{64} - 8 q^{66} - 4 q^{67} + 8 q^{68} + 18 q^{69} + 4 q^{71} + 20 q^{72} + 22 q^{73} + 3 q^{74} - 10 q^{76} - 4 q^{78} + 22 q^{81} - 12 q^{82} + 2 q^{83} + 9 q^{86} + 10 q^{87} - 10 q^{88} - 30 q^{89} - 9 q^{92} - 24 q^{93} + 2 q^{94} - 14 q^{96} - 6 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9 + 4 * q^11 - 6 * q^12 + 2 * q^13 - 3 * q^16 + 4 * q^17 - 7 * q^18 + 7 * q^22 + 8 * q^23 + 10 * q^24 + 6 * q^26 + 20 * q^27 + 10 * q^29 + 6 * q^31 - 9 * q^32 - 6 * q^33 + 12 * q^34 - 13 * q^36 + 6 * q^37 - 10 * q^38 - 8 * q^39 - 14 * q^41 + 8 * q^43 + 3 * q^44 - q^46 + 4 * q^47 + 12 * q^48 - 16 * q^51 + 4 * q^52 + 8 * q^53 + 20 * q^57 + 5 * q^58 - 10 * q^59 + 6 * q^61 + 18 * q^62 + 4 * q^64 - 8 * q^66 - 4 * q^67 + 8 * q^68 + 18 * q^69 + 4 * q^71 + 20 * q^72 + 22 * q^73 + 3 * q^74 - 10 * q^76 - 4 * q^78 + 22 * q^81 - 12 * q^82 + 2 * q^83 + 9 * q^86 + 10 * q^87 - 10 * q^88 - 30 * q^89 - 9 * q^92 - 24 * q^93 + 2 * q^94 - 14 * q^96 - 6 * q^97 - 8 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	−1.47214	−0.490712
$10$	0	0
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	−0.763932	−0.220528
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	0	0
$15$	0	0
$16$	−4.85410	−1.21353
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	−2.38197	−0.561435
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	0	0
$22$	6.85410	1.46130
$23$	1.76393	0.367805	0.183903	−	0.982944i	$-0.441127\pi$
$23$	1.76393	0.367805	0.183903	+	0.982944i	$0.441127\pi$
$24$	2.76393	0.564185
$25$	0	0
$26$	5.23607	1.02688
$27$	5.52786	1.06384
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	9.70820	1.74364	0.871822	−	0.489822i	$-0.162938\pi$
$31$	9.70820	1.74364	0.871822	+	0.489822i	$0.162938\pi$
$32$	−3.38197	−0.597853
$33$	−5.23607	−0.911482
$34$	10.4721	1.79596
$35$	0	0
$36$	−0.909830	−0.151638
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−7.23607	−1.17385
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−9.23607	−1.44243	−0.721216	−	0.692711i	$-0.756416\pi$
$41$	−9.23607	−1.44243	−0.721216	+	0.692711i	$0.756416\pi$
$42$	0	0
$43$	6.23607	0.950991	0.475496	−	0.879718i	$-0.342269\pi$
$43$	6.23607	0.950991	0.475496	+	0.879718i	$0.342269\pi$
$44$	2.61803	0.394683
$45$	0	0
$46$	2.85410	0.420814
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	6.00000	0.866025
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	2.00000	0.277350
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	8.94427	1.21716
$55$	0	0
$56$	0	0
$57$	5.52786	0.732183
$58$	8.09017	1.06229
$59$	1.70820	0.222389	0.111195	−	0.993799i	$-0.464532\pi$
$59$	1.70820	0.222389	0.111195	+	0.993799i	$0.464532\pi$
$60$	0	0
$61$	−3.70820	−0.474787	−0.237393	−	0.971414i	$-0.576293\pi$
$61$	−3.70820	−0.474787	−0.237393	+	0.971414i	$0.576293\pi$
$62$	15.7082	1.99494
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	−8.47214	−1.04285
$67$	0.236068	0.0288403	0.0144201	−	0.999896i	$-0.495410\pi$
$67$	0.236068	0.0288403	0.0144201	+	0.999896i	$0.495410\pi$
$68$	4.00000	0.485071
$69$	−2.18034	−0.262482
$70$	0	0
$71$	−4.70820	−0.558761	−0.279381	−	0.960180i	$-0.590129\pi$
$71$	−4.70820	−0.558761	−0.279381	+	0.960180i	$0.590129\pi$
$72$	3.29180	0.387942
$73$	13.2361	1.54916	0.774582	−	0.632473i	$-0.217960\pi$
$73$	13.2361	1.54916	0.774582	+	0.632473i	$0.217960\pi$
$74$	4.85410	0.564278
$75$	0	0
$76$	−2.76393	−0.317045
$77$	0	0
$78$	−6.47214	−0.732825
$79$	11.1803	1.25789	0.628943	−	0.777451i	$-0.283488\pi$
$79$	11.1803	1.25789	0.628943	+	0.777451i	$0.283488\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	−14.9443	−1.65032
$83$	−5.70820	−0.626557	−0.313278	−	0.949661i	$-0.601427\pi$
$83$	−5.70820	−0.626557	−0.313278	+	0.949661i	$0.601427\pi$
$84$	0	0
$85$	0	0
$86$	10.0902	1.08805
$87$	−6.18034	−0.662602
$88$	−9.47214	−1.00973
$89$	−12.7639	−1.35297	−0.676487	−	0.736455i	$-0.736499\pi$
$89$	−12.7639	−1.35297	−0.676487	+	0.736455i	$0.736499\pi$
$90$	0	0
$91$	0	0
$92$	1.09017	0.113658
$93$	−12.0000	−1.24434
$94$	3.23607	0.333775
$95$	0	0
$96$	4.18034	0.426654
$97$	−0.763932	−0.0775655	−0.0387828	−	0.999248i	$-0.512348\pi$
$97$	−0.763932	−0.0775655	−0.0387828	+	0.999248i	$0.512348\pi$
$98$	0	0
$99$	−6.23607	−0.626748
$100$	0	0
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	−12.9443	−1.28167
$103$	0.472136	0.0465209	0.0232605	−	0.999729i	$-0.492595\pi$
$103$	0.472136	0.0465209	0.0232605	+	0.999729i	$0.492595\pi$
$104$	−7.23607	−0.709555
$105$	0	0
$106$	−0.763932	−0.0741996
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	3.41641	0.328744
$109$	−18.4164	−1.76397	−0.881986	−	0.471276i	$-0.843794\pi$
$109$	−18.4164	−1.76397	−0.881986	+	0.471276i	$0.843794\pi$
$110$	0	0
$111$	−3.70820	−0.351967
$112$	0	0
$113$	12.4164	1.16804	0.584019	−	0.811740i	$-0.301479\pi$
$113$	12.4164	1.16804	0.584019	+	0.811740i	$0.301479\pi$
$114$	8.94427	0.837708
$115$	0	0
$116$	3.09017	0.286915
$117$	−4.76393	−0.440426
$118$	2.76393	0.254441
$119$	0	0
$120$	0	0
$121$	6.94427	0.631297
$122$	−6.00000	−0.543214
$123$	11.4164	1.02938
$124$	6.00000	0.538816
$125$	0	0
$126$	0	0
$127$	−17.6525	−1.56640	−0.783202	−	0.621767i	$-0.786415\pi$
$127$	−17.6525	−1.56640	−0.783202	+	0.621767i	$0.786415\pi$
$128$	13.6180	1.20368
$129$	−7.70820	−0.678670
$130$	0	0
$131$	−0.944272	−0.0825014	−0.0412507	−	0.999149i	$-0.513134\pi$
$131$	−0.944272	−0.0825014	−0.0412507	+	0.999149i	$0.513134\pi$
$132$	−3.23607	−0.281664
$133$	0	0
$134$	0.381966	0.0329968
$135$	0	0
$136$	−14.4721	−1.24098
$137$	6.94427	0.593289	0.296645	−	0.954988i	$-0.404132\pi$
$137$	6.94427	0.593289	0.296645	+	0.954988i	$0.404132\pi$
$138$	−3.52786	−0.300312
$139$	20.6525	1.75172	0.875860	−	0.482565i	$-0.160295\pi$
$139$	20.6525	1.75172	0.875860	+	0.482565i	$0.160295\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	−7.61803	−0.639291
$143$	13.7082	1.14634
$144$	7.14590	0.595492
$145$	0	0
$146$	21.4164	1.77243
$147$	0	0
$148$	1.85410	0.152406
$149$	−13.9443	−1.14236	−0.571180	−	0.820825i	$-0.693514\pi$
$149$	−13.9443	−1.14236	−0.571180	+	0.820825i	$0.693514\pi$
$150$	0	0
$151$	−15.7639	−1.28285	−0.641425	−	0.767185i	$-0.721657\pi$
$151$	−15.7639	−1.28285	−0.641425	+	0.767185i	$0.721657\pi$
$152$	10.0000	0.811107
$153$	−9.52786	−0.770282
$154$	0	0
$155$	0	0
$156$	−2.47214	−0.197929
$157$	−5.23607	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$157$	−5.23607	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$158$	18.0902	1.43918
$159$	0.583592	0.0462819
$160$	0	0
$161$	0	0
$162$	−3.90983	−0.307185
$163$	−10.4721	−0.820241	−0.410120	−	0.912031i	$-0.634513\pi$
$163$	−10.4721	−0.820241	−0.410120	+	0.912031i	$0.634513\pi$
$164$	−5.70820	−0.445736
$165$	0	0
$166$	−9.23607	−0.716858
$167$	−0.763932	−0.0591148	−0.0295574	−	0.999563i	$-0.509410\pi$
$167$	−0.763932	−0.0591148	−0.0295574	+	0.999563i	$0.509410\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	6.58359	0.503460
$172$	3.85410	0.293873
$173$	20.4721	1.55647	0.778234	−	0.627975i	$-0.216116\pi$
$173$	20.4721	1.55647	0.778234	+	0.627975i	$0.216116\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	−20.5623	−1.54994
$177$	−2.11146	−0.158707
$178$	−20.6525	−1.54797
$179$	3.41641	0.255354	0.127677	−	0.991816i	$-0.459248\pi$
$179$	3.41641	0.255354	0.127677	+	0.991816i	$0.459248\pi$
$180$	0	0
$181$	14.1803	1.05402	0.527008	−	0.849860i	$-0.323314\pi$
$181$	14.1803	1.05402	0.527008	+	0.849860i	$0.323314\pi$
$182$	0	0
$183$	4.58359	0.338829
$184$	−3.94427	−0.290776
$185$	0	0
$186$	−19.4164	−1.42368
$187$	27.4164	2.00489
$188$	1.23607	0.0901495
$189$	0	0
$190$	0	0
$191$	−2.47214	−0.178877	−0.0894387	−	0.995992i	$-0.528507\pi$
$191$	−2.47214	−0.178877	−0.0894387	+	0.995992i	$0.528507\pi$
$192$	−5.23607	−0.377881
$193$	−14.4164	−1.03772	−0.518858	−	0.854861i	$-0.673643\pi$
$193$	−14.4164	−1.03772	−0.518858	+	0.854861i	$0.673643\pi$
$194$	−1.23607	−0.0887445
$195$	0	0
$196$	0	0
$197$	7.47214	0.532368	0.266184	−	0.963922i	$-0.414237\pi$
$197$	7.47214	0.532368	0.266184	+	0.963922i	$0.414237\pi$
$198$	−10.0902	−0.717077
$199$	−2.76393	−0.195930	−0.0979650	−	0.995190i	$-0.531233\pi$
$199$	−2.76393	−0.195930	−0.0979650	+	0.995190i	$0.531233\pi$
$200$	0	0
$201$	−0.291796	−0.0205817
$202$	−14.9443	−1.05148
$203$	0	0
$204$	−4.94427	−0.346168
$205$	0	0
$206$	0.763932	0.0532257
$207$	−2.59675	−0.180486
$208$	−15.7082	−1.08917
$209$	−18.9443	−1.31040
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−0.291796	−0.0200406
$213$	5.81966	0.398757
$214$	12.9443	0.884852
$215$	0	0
$216$	−12.3607	−0.841038
$217$	0	0
$218$	−29.7984	−2.01820
$219$	−16.3607	−1.10555
$220$	0	0
$221$	20.9443	1.40886
$222$	−6.00000	−0.402694
$223$	2.18034	0.146006	0.0730032	−	0.997332i	$-0.476742\pi$
$223$	2.18034	0.146006	0.0730032	+	0.997332i	$0.476742\pi$
$224$	0	0
$225$	0	0
$226$	20.0902	1.33638
$227$	5.41641	0.359500	0.179750	−	0.983712i	$-0.442471\pi$
$227$	5.41641	0.359500	0.179750	+	0.983712i	$0.442471\pi$
$228$	3.41641	0.226257
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	0	0
$231$	0	0
$232$	−11.1803	−0.734025
$233$	−9.94427	−0.651471	−0.325735	−	0.945461i	$-0.605612\pi$
$233$	−9.94427	−0.651471	−0.325735	+	0.945461i	$0.605612\pi$
$234$	−7.70820	−0.503901
$235$	0	0
$236$	1.05573	0.0687220
$237$	−13.8197	−0.897683
$238$	0	0
$239$	−14.4721	−0.936125	−0.468062	−	0.883695i	$-0.655048\pi$
$239$	−14.4721	−0.936125	−0.468062	+	0.883695i	$0.655048\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	11.2361	0.722282
$243$	−13.5967	−0.872232
$244$	−2.29180	−0.146717
$245$	0	0
$246$	18.4721	1.17774
$247$	−14.4721	−0.920840
$248$	−21.7082	−1.37847
$249$	7.05573	0.447139
$250$	0	0
$251$	2.47214	0.156040	0.0780199	−	0.996952i	$-0.475140\pi$
$251$	2.47214	0.156040	0.0780199	+	0.996952i	$0.475140\pi$
$252$	0	0
$253$	7.47214	0.469769
$254$	−28.5623	−1.79216
$255$	0	0
$256$	13.5623	0.847644
$257$	−18.6525	−1.16351	−0.581755	−	0.813364i	$-0.697634\pi$
$257$	−18.6525	−1.16351	−0.581755	+	0.813364i	$0.697634\pi$
$258$	−12.4721	−0.776481
$259$	0	0
$260$	0	0
$261$	−7.36068	−0.455615
$262$	−1.52786	−0.0943918
$263$	11.7639	0.725395	0.362698	−	0.931907i	$-0.381856\pi$
$263$	11.7639	0.725395	0.362698	+	0.931907i	$0.381856\pi$
$264$	11.7082	0.720590
$265$	0	0
$266$	0	0
$267$	15.7771	0.965542
$268$	0.145898	0.00891214
$269$	−1.70820	−0.104151	−0.0520755	−	0.998643i	$-0.516584\pi$
$269$	−1.70820	−0.104151	−0.0520755	+	0.998643i	$0.516584\pi$
$270$	0	0
$271$	−10.2918	−0.625182	−0.312591	−	0.949888i	$-0.601197\pi$
$271$	−10.2918	−0.625182	−0.312591	+	0.949888i	$0.601197\pi$
$272$	−31.4164	−1.90490
$273$	0	0
$274$	11.2361	0.678796
$275$	0	0
$276$	−1.34752	−0.0811114
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	33.4164	2.00418
$279$	−14.2918	−0.855627
$280$	0	0
$281$	29.3607	1.75151	0.875756	−	0.482755i	$-0.160364\pi$
$281$	29.3607	1.75151	0.875756	+	0.482755i	$0.160364\pi$
$282$	−4.00000	−0.238197
$283$	9.41641	0.559747	0.279874	−	0.960037i	$-0.409707\pi$
$283$	9.41641	0.559747	0.279874	+	0.960037i	$0.409707\pi$
$284$	−2.90983	−0.172667
$285$	0	0
$286$	22.1803	1.31155
$287$	0	0
$288$	4.97871	0.293374
$289$	24.8885	1.46403
$290$	0	0
$291$	0.944272	0.0553542
$292$	8.18034	0.478718
$293$	−9.12461	−0.533066	−0.266533	−	0.963826i	$-0.585878\pi$
$293$	−9.12461	−0.533066	−0.266533	+	0.963826i	$0.585878\pi$
$294$	0	0
$295$	0	0
$296$	−6.70820	−0.389906
$297$	23.4164	1.35876
$298$	−22.5623	−1.30700
$299$	5.70820	0.330114
$300$	0	0
$301$	0	0
$302$	−25.5066	−1.46774
$303$	11.4164	0.655855
$304$	21.7082	1.24505
$305$	0	0
$306$	−15.4164	−0.881297
$307$	−31.4164	−1.79303	−0.896515	−	0.443014i	$-0.853909\pi$
$307$	−31.4164	−1.79303	−0.896515	+	0.443014i	$0.853909\pi$
$308$	0	0
$309$	−0.583592	−0.0331994
$310$	0	0
$311$	20.3607	1.15455	0.577274	−	0.816550i	$-0.304116\pi$
$311$	20.3607	1.15455	0.577274	+	0.816550i	$0.304116\pi$
$312$	8.94427	0.506370
$313$	−28.4721	−1.60934	−0.804670	−	0.593722i	$-0.797658\pi$
$313$	−28.4721	−1.60934	−0.804670	+	0.593722i	$0.797658\pi$
$314$	−8.47214	−0.478110
$315$	0	0
$316$	6.90983	0.388708
$317$	−19.3607	−1.08740	−0.543702	−	0.839278i	$-0.682978\pi$
$317$	−19.3607	−1.08740	−0.543702	+	0.839278i	$0.682978\pi$
$318$	0.944272	0.0529521
$319$	21.1803	1.18587
$320$	0	0
$321$	−9.88854	−0.551925
$322$	0	0
$323$	−28.9443	−1.61050
$324$	−1.49342	−0.0829679
$325$	0	0
$326$	−16.9443	−0.938456
$327$	22.7639	1.25885
$328$	20.6525	1.14034
$329$	0	0
$330$	0	0
$331$	−11.2918	−0.620653	−0.310327	−	0.950630i	$-0.600438\pi$
$331$	−11.2918	−0.620653	−0.310327	+	0.950630i	$0.600438\pi$
$332$	−3.52786	−0.193617
$333$	−4.41641	−0.242018
$334$	−1.23607	−0.0676346
$335$	0	0
$336$	0	0
$337$	−7.52786	−0.410069	−0.205034	−	0.978755i	$-0.565731\pi$
$337$	−7.52786	−0.410069	−0.205034	+	0.978755i	$0.565731\pi$
$338$	−4.09017	−0.222476
$339$	−15.3475	−0.833563
$340$	0	0
$341$	41.1246	2.22702
$342$	10.6525	0.576020
$343$	0	0
$344$	−13.9443	−0.751825
$345$	0	0
$346$	33.1246	1.78079
$347$	15.7639	0.846252	0.423126	−	0.906071i	$-0.360933\pi$
$347$	15.7639	0.846252	0.423126	+	0.906071i	$0.360933\pi$
$348$	−3.81966	−0.204755
$349$	4.47214	0.239388	0.119694	−	0.992811i	$-0.461809\pi$
$349$	4.47214	0.239388	0.119694	+	0.992811i	$0.461809\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	−14.3262	−0.763591
$353$	−20.1803	−1.07409	−0.537046	−	0.843553i	$-0.680460\pi$
$353$	−20.1803	−1.07409	−0.537046	+	0.843553i	$0.680460\pi$
$354$	−3.41641	−0.181580
$355$	0	0
$356$	−7.88854	−0.418092
$357$	0	0
$358$	5.52786	0.292157
$359$	10.1246	0.534357	0.267178	−	0.963647i	$-0.413909\pi$
$359$	10.1246	0.534357	0.267178	+	0.963647i	$0.413909\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	22.9443	1.20592
$363$	−8.58359	−0.450522
$364$	0	0
$365$	0	0
$366$	7.41641	0.387662
$367$	−3.12461	−0.163103	−0.0815517	−	0.996669i	$-0.525988\pi$
$367$	−3.12461	−0.163103	−0.0815517	+	0.996669i	$0.525988\pi$
$368$	−8.56231	−0.446341
$369$	13.5967	0.707818
$370$	0	0
$371$	0	0
$372$	−7.41641	−0.384523
$373$	15.8328	0.819792	0.409896	−	0.912132i	$-0.365565\pi$
$373$	15.8328	0.819792	0.409896	+	0.912132i	$0.365565\pi$
$374$	44.3607	2.29384
$375$	0	0
$376$	−4.47214	−0.230633
$377$	16.1803	0.833330
$378$	0	0
$379$	11.1803	0.574295	0.287148	−	0.957886i	$-0.407293\pi$
$379$	11.1803	0.574295	0.287148	+	0.957886i	$0.407293\pi$
$380$	0	0
$381$	21.8197	1.11786
$382$	−4.00000	−0.204658
$383$	28.7639	1.46977	0.734884	−	0.678193i	$-0.237237\pi$
$383$	28.7639	1.46977	0.734884	+	0.678193i	$0.237237\pi$
$384$	−16.8328	−0.858996
$385$	0	0
$386$	−23.3262	−1.18727
$387$	−9.18034	−0.466663
$388$	−0.472136	−0.0239691
$389$	−32.8885	−1.66752	−0.833758	−	0.552131i	$-0.813815\pi$
$389$	−32.8885	−1.66752	−0.833758	+	0.552131i	$0.813815\pi$
$390$	0	0
$391$	11.4164	0.577353
$392$	0	0
$393$	1.16718	0.0588767
$394$	12.0902	0.609094
$395$	0	0
$396$	−3.85410	−0.193676
$397$	−26.9443	−1.35229	−0.676147	−	0.736767i	$-0.736352\pi$
$397$	−26.9443	−1.35229	−0.676147	+	0.736767i	$0.736352\pi$
$398$	−4.47214	−0.224168
$399$	0	0
$400$	0	0
$401$	11.4721	0.572891	0.286446	−	0.958097i	$-0.407526\pi$
$401$	11.4721	0.572891	0.286446	+	0.958097i	$0.407526\pi$
$402$	−0.472136	−0.0235480
$403$	31.4164	1.56496
$404$	−5.70820	−0.283994
$405$	0	0
$406$	0	0
$407$	12.7082	0.629922
$408$	17.8885	0.885615
$409$	−15.5279	−0.767803	−0.383902	−	0.923374i	$-0.625420\pi$
$409$	−15.5279	−0.767803	−0.383902	+	0.923374i	$0.625420\pi$
$410$	0	0
$411$	−8.58359	−0.423397
$412$	0.291796	0.0143758
$413$	0	0
$414$	−4.20163	−0.206499
$415$	0	0
$416$	−10.9443	−0.536587
$417$	−25.5279	−1.25010
$418$	−30.6525	−1.49926
$419$	3.81966	0.186603	0.0933013	−	0.995638i	$-0.470258\pi$
$419$	3.81966	0.186603	0.0933013	+	0.995638i	$0.470258\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	19.4164	0.945176
$423$	−2.94427	−0.143155
$424$	1.05573	0.0512707
$425$	0	0
$426$	9.41641	0.456226
$427$	0	0
$428$	4.94427	0.238990
$429$	−16.9443	−0.818077
$430$	0	0
$431$	26.4721	1.27512	0.637559	−	0.770402i	$-0.279944\pi$
$431$	26.4721	1.27512	0.637559	+	0.770402i	$0.279944\pi$
$432$	−26.8328	−1.29099
$433$	−16.3607	−0.786244	−0.393122	−	0.919486i	$-0.628605\pi$
$433$	−16.3607	−0.786244	−0.393122	+	0.919486i	$0.628605\pi$
$434$	0	0
$435$	0	0
$436$	−11.3820	−0.545097
$437$	−7.88854	−0.377360
$438$	−26.4721	−1.26489
$439$	21.7082	1.03608	0.518038	−	0.855358i	$-0.326663\pi$
$439$	21.7082	1.03608	0.518038	+	0.855358i	$0.326663\pi$
$440$	0	0
$441$	0	0
$442$	33.8885	1.61191
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	−2.29180	−0.108764
$445$	0	0
$446$	3.52786	0.167049
$447$	17.2361	0.815238
$448$	0	0
$449$	29.4721	1.39088	0.695438	−	0.718586i	$-0.255210\pi$
$449$	29.4721	1.39088	0.695438	+	0.718586i	$0.255210\pi$
$450$	0	0
$451$	−39.1246	−1.84231
$452$	7.67376	0.360943
$453$	19.4853	0.915499
$454$	8.76393	0.411312
$455$	0	0
$456$	−12.3607	−0.578842
$457$	−21.4721	−1.00442	−0.502212	−	0.864744i	$-0.667480\pi$
$457$	−21.4721	−1.00442	−0.502212	+	0.864744i	$0.667480\pi$
$458$	−7.23607	−0.338119
$459$	35.7771	1.66993
$460$	0	0
$461$	−8.18034	−0.380996	−0.190498	−	0.981688i	$-0.561010\pi$
$461$	−8.18034	−0.380996	−0.190498	+	0.981688i	$0.561010\pi$
$462$	0	0
$463$	21.8885	1.01725	0.508623	−	0.860989i	$-0.330155\pi$
$463$	21.8885	1.01725	0.508623	+	0.860989i	$0.330155\pi$
$464$	−24.2705	−1.12673
$465$	0	0
$466$	−16.0902	−0.745363
$467$	10.9443	0.506441	0.253220	−	0.967409i	$-0.418510\pi$
$467$	10.9443	0.506441	0.253220	+	0.967409i	$0.418510\pi$
$468$	−2.94427	−0.136099
$469$	0	0
$470$	0	0
$471$	6.47214	0.298220
$472$	−3.81966	−0.175814
$473$	26.4164	1.21463
$474$	−22.3607	−1.02706
$475$	0	0
$476$	0	0
$477$	0.695048	0.0318241
$478$	−23.4164	−1.07104
$479$	3.81966	0.174525	0.0872624	−	0.996185i	$-0.472188\pi$
$479$	3.81966	0.174525	0.0872624	+	0.996185i	$0.472188\pi$
$480$	0	0
$481$	9.70820	0.442656
$482$	20.1803	0.919189
$483$	0	0
$484$	4.29180	0.195082
$485$	0	0
$486$	−22.0000	−0.997940
$487$	10.2361	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$487$	10.2361	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$488$	8.29180	0.375352
$489$	12.9443	0.585360
$490$	0	0
$491$	−10.2361	−0.461947	−0.230974	−	0.972960i	$-0.574191\pi$
$491$	−10.2361	−0.461947	−0.230974	+	0.972960i	$0.574191\pi$
$492$	7.05573	0.318097
$493$	32.3607	1.45745
$494$	−23.4164	−1.05355
$495$	0	0
$496$	−47.1246	−2.11596
$497$	0	0
$498$	11.4164	0.511581
$499$	28.9443	1.29572	0.647862	−	0.761758i	$-0.275663\pi$
$499$	28.9443	1.29572	0.647862	+	0.761758i	$0.275663\pi$
$500$	0	0
$501$	0.944272	0.0421870
$502$	4.00000	0.178529
$503$	43.8885	1.95689	0.978447	−	0.206499i	$-0.0662072\pi$
$503$	43.8885	1.95689	0.978447	+	0.206499i	$0.0662072\pi$
$504$	0	0
$505$	0	0
$506$	12.0902	0.537474
$507$	3.12461	0.138769
$508$	−10.9098	−0.484045
$509$	−9.34752	−0.414322	−0.207161	−	0.978307i	$-0.566422\pi$
$509$	−9.34752	−0.414322	−0.207161	+	0.978307i	$0.566422\pi$
$510$	0	0
$511$	0	0
$512$	−5.29180	−0.233867
$513$	−24.7214	−1.09147
$514$	−30.1803	−1.33120
$515$	0	0
$516$	−4.76393	−0.209720
$517$	8.47214	0.372604
$518$	0	0
$519$	−25.3050	−1.11076
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	−11.9098	−0.521279
$523$	28.3607	1.24013	0.620063	−	0.784552i	$-0.287107\pi$
$523$	28.3607	1.24013	0.620063	+	0.784552i	$0.287107\pi$
$524$	−0.583592	−0.0254943
$525$	0	0
$526$	19.0344	0.829941
$527$	62.8328	2.73704
$528$	25.4164	1.10611
$529$	−19.8885	−0.864719
$530$	0	0
$531$	−2.51471	−0.109129
$532$	0	0
$533$	−29.8885	−1.29462
$534$	25.5279	1.10470
$535$	0	0
$536$	−0.527864	−0.0228003
$537$	−4.22291	−0.182232
$538$	−2.76393	−0.119162
$539$	0	0
$540$	0	0
$541$	−1.94427	−0.0835908	−0.0417954	−	0.999126i	$-0.513308\pi$
$541$	−1.94427	−0.0835908	−0.0417954	+	0.999126i	$0.513308\pi$
$542$	−16.6525	−0.715285
$543$	−17.5279	−0.752193
$544$	−21.8885	−0.938464
$545$	0	0
$546$	0	0
$547$	−14.2361	−0.608690	−0.304345	−	0.952562i	$-0.598438\pi$
$547$	−14.2361	−0.608690	−0.304345	+	0.952562i	$0.598438\pi$
$548$	4.29180	0.183336
$549$	5.45898	0.232984
$550$	0	0
$551$	−22.3607	−0.952597
$552$	4.87539	0.207510
$553$	0	0
$554$	25.7082	1.09224
$555$	0	0
$556$	12.7639	0.541311
$557$	−44.8885	−1.90199	−0.950994	−	0.309208i	$-0.899936\pi$
$557$	−44.8885	−1.90199	−0.950994	+	0.309208i	$0.899936\pi$
$558$	−23.1246	−0.978943
$559$	20.1803	0.853537
$560$	0	0
$561$	−33.8885	−1.43078
$562$	47.5066	2.00394
$563$	9.41641	0.396854	0.198427	−	0.980116i	$-0.436417\pi$
$563$	9.41641	0.396854	0.198427	+	0.980116i	$0.436417\pi$
$564$	−1.52786	−0.0643347
$565$	0	0
$566$	15.2361	0.640420
$567$	0	0
$568$	10.5279	0.441739
$569$	13.9443	0.584574	0.292287	−	0.956331i	$-0.405584\pi$
$569$	13.9443	0.584574	0.292287	+	0.956331i	$0.405584\pi$
$570$	0	0
$571$	−12.5967	−0.527157	−0.263579	−	0.964638i	$-0.584903\pi$
$571$	−12.5967	−0.527157	−0.263579	+	0.964638i	$0.584903\pi$
$572$	8.47214	0.354238
$573$	3.05573	0.127655
$574$	0	0
$575$	0	0
$576$	−6.23607	−0.259836
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	40.2705	1.67503
$579$	17.8197	0.740560
$580$	0	0
$581$	0	0
$582$	1.52786	0.0633320
$583$	−2.00000	−0.0828315
$584$	−29.5967	−1.22472
$585$	0	0
$586$	−14.7639	−0.609892
$587$	29.2361	1.20670	0.603351	−	0.797476i	$-0.293832\pi$
$587$	29.2361	1.20670	0.603351	+	0.797476i	$0.293832\pi$
$588$	0	0
$589$	−43.4164	−1.78894
$590$	0	0
$591$	−9.23607	−0.379921
$592$	−14.5623	−0.598507
$593$	−25.3050	−1.03915	−0.519575	−	0.854425i	$-0.673910\pi$
$593$	−25.3050	−1.03915	−0.519575	+	0.854425i	$0.673910\pi$
$594$	37.8885	1.55459
$595$	0	0
$596$	−8.61803	−0.353008
$597$	3.41641	0.139824
$598$	9.23607	0.377691
$599$	11.1803	0.456816	0.228408	−	0.973565i	$-0.426648\pi$
$599$	11.1803	0.456816	0.228408	+	0.973565i	$0.426648\pi$
$600$	0	0
$601$	19.0557	0.777299	0.388650	−	0.921386i	$-0.372942\pi$
$601$	19.0557	0.777299	0.388650	+	0.921386i	$0.372942\pi$
$602$	0	0
$603$	−0.347524	−0.0141523
$604$	−9.74265	−0.396423
$605$	0	0
$606$	18.4721	0.750379
$607$	−33.1246	−1.34449	−0.672243	−	0.740330i	$-0.734669\pi$
$607$	−33.1246	−1.34449	−0.672243	+	0.740330i	$0.734669\pi$
$608$	15.1246	0.613384
$609$	0	0
$610$	0	0
$611$	6.47214	0.261835
$612$	−5.88854	−0.238030
$613$	−17.5836	−0.710195	−0.355097	−	0.934829i	$-0.615552\pi$
$613$	−17.5836	−0.710195	−0.355097	+	0.934829i	$0.615552\pi$
$614$	−50.8328	−2.05145
$615$	0	0
$616$	0	0
$617$	11.9443	0.480858	0.240429	−	0.970667i	$-0.422712\pi$
$617$	11.9443	0.480858	0.240429	+	0.970667i	$0.422712\pi$
$618$	−0.944272	−0.0379842
$619$	−1.70820	−0.0686585	−0.0343293	−	0.999411i	$-0.510929\pi$
$619$	−1.70820	−0.0686585	−0.0343293	+	0.999411i	$0.510929\pi$
$620$	0	0
$621$	9.75078	0.391285
$622$	32.9443	1.32094
$623$	0	0
$624$	19.4164	0.777278
$625$	0	0
$626$	−46.0689	−1.84128
$627$	23.4164	0.935161
$628$	−3.23607	−0.129133
$629$	19.4164	0.774183
$630$	0	0
$631$	−3.65248	−0.145403	−0.0727014	−	0.997354i	$-0.523162\pi$
$631$	−3.65248	−0.145403	−0.0727014	+	0.997354i	$0.523162\pi$
$632$	−25.0000	−0.994447
$633$	−14.8328	−0.589551
$634$	−31.3262	−1.24412
$635$	0	0
$636$	0.360680	0.0143019
$637$	0	0
$638$	34.2705	1.35678
$639$	6.93112	0.274191
$640$	0	0
$641$	−9.83282	−0.388373	−0.194186	−	0.980965i	$-0.562207\pi$
$641$	−9.83282	−0.388373	−0.194186	+	0.980965i	$0.562207\pi$
$642$	−16.0000	−0.631470
$643$	−9.52786	−0.375742	−0.187871	−	0.982194i	$-0.560159\pi$
$643$	−9.52786	−0.375742	−0.187871	+	0.982194i	$0.560159\pi$
$644$	0	0
$645$	0	0
$646$	−46.8328	−1.84261
$647$	−15.8885	−0.624643	−0.312322	−	0.949976i	$-0.601107\pi$
$647$	−15.8885	−0.624643	−0.312322	+	0.949976i	$0.601107\pi$
$648$	5.40325	0.212260
$649$	7.23607	0.284041
$650$	0	0
$651$	0	0
$652$	−6.47214	−0.253468
$653$	42.9443	1.68054	0.840270	−	0.542169i	$-0.182397\pi$
$653$	42.9443	1.68054	0.840270	+	0.542169i	$0.182397\pi$
$654$	36.8328	1.44028
$655$	0	0
$656$	44.8328	1.75043
$657$	−19.4853	−0.760194
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−46.7214	−1.81725	−0.908625	−	0.417613i	$-0.862867\pi$
$661$	−46.7214	−1.81725	−0.908625	+	0.417613i	$0.862867\pi$
$662$	−18.2705	−0.710104
$663$	−25.8885	−1.00543
$664$	12.7639	0.495337
$665$	0	0
$666$	−7.14590	−0.276898
$667$	8.81966	0.341499
$668$	−0.472136	−0.0182675
$669$	−2.69505	−0.104197
$670$	0	0
$671$	−15.7082	−0.606408
$672$	0	0
$673$	28.4721	1.09752	0.548760	−	0.835980i	$-0.315100\pi$
$673$	28.4721	1.09752	0.548760	+	0.835980i	$0.315100\pi$
$674$	−12.1803	−0.469169
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	−30.3607	−1.16686	−0.583428	−	0.812165i	$-0.698289\pi$
$677$	−30.3607	−1.16686	−0.583428	+	0.812165i	$0.698289\pi$
$678$	−24.8328	−0.953699
$679$	0	0
$680$	0	0
$681$	−6.69505	−0.256555
$682$	66.5410	2.54799
$683$	−26.1246	−0.999630	−0.499815	−	0.866132i	$-0.666599\pi$
$683$	−26.1246	−0.999630	−0.499815	+	0.866132i	$0.666599\pi$
$684$	4.06888	0.155578
$685$	0	0
$686$	0	0
$687$	5.52786	0.210901
$688$	−30.2705	−1.15405
$689$	−1.52786	−0.0582070
$690$	0	0
$691$	−18.1803	−0.691613	−0.345806	−	0.938306i	$-0.612395\pi$
$691$	−18.1803	−0.691613	−0.345806	+	0.938306i	$0.612395\pi$
$692$	12.6525	0.480975
$693$	0	0
$694$	25.5066	0.968216
$695$	0	0
$696$	13.8197	0.523833
$697$	−59.7771	−2.26422
$698$	7.23607	0.273889
$699$	12.2918	0.464918
$700$	0	0
$701$	−46.9443	−1.77306	−0.886530	−	0.462670i	$-0.846891\pi$
$701$	−46.9443	−1.77306	−0.886530	+	0.462670i	$0.846891\pi$
$702$	28.9443	1.09243
$703$	−13.4164	−0.506009
$704$	17.9443	0.676300
$705$	0	0
$706$	−32.6525	−1.22889
$707$	0	0
$708$	−1.30495	−0.0490431
$709$	−47.8885	−1.79849	−0.899246	−	0.437443i	$-0.855884\pi$
$709$	−47.8885	−1.79849	−0.899246	+	0.437443i	$0.855884\pi$
$710$	0	0
$711$	−16.4590	−0.617260
$712$	28.5410	1.06962
$713$	17.1246	0.641322
$714$	0	0
$715$	0	0
$716$	2.11146	0.0789088
$717$	17.8885	0.668060
$718$	16.3820	0.611370
$719$	6.18034	0.230488	0.115244	−	0.993337i	$-0.463235\pi$
$719$	6.18034	0.230488	0.115244	+	0.993337i	$0.463235\pi$
$720$	0	0
$721$	0	0
$722$	1.61803	0.0602170
$723$	−15.4164	−0.573342
$724$	8.76393	0.325709
$725$	0	0
$726$	−13.8885	−0.515452
$727$	20.9443	0.776780	0.388390	−	0.921495i	$-0.373031\pi$
$727$	20.9443	0.776780	0.388390	+	0.921495i	$0.373031\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	40.3607	1.49279
$732$	2.83282	0.104704
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−5.05573	−0.186610
$735$	0	0
$736$	−5.96556	−0.219893
$737$	1.00000	0.0368355
$738$	22.0000	0.809831
$739$	−5.65248	−0.207930	−0.103965	−	0.994581i	$-0.533153\pi$
$739$	−5.65248	−0.207930	−0.103965	+	0.994581i	$0.533153\pi$
$740$	0	0
$741$	17.8885	0.657152
$742$	0	0
$743$	−1.52786	−0.0560519	−0.0280259	−	0.999607i	$-0.508922\pi$
$743$	−1.52786	−0.0560519	−0.0280259	+	0.999607i	$0.508922\pi$
$744$	26.8328	0.983739
$745$	0	0
$746$	25.6180	0.937943
$747$	8.40325	0.307459
$748$	16.9443	0.619544
$749$	0	0
$750$	0	0
$751$	20.9443	0.764267	0.382134	−	0.924107i	$-0.375189\pi$
$751$	20.9443	0.764267	0.382134	+	0.924107i	$0.375189\pi$
$752$	−9.70820	−0.354022
$753$	−3.05573	−0.111357
$754$	26.1803	0.953432
$755$	0	0
$756$	0	0
$757$	46.4164	1.68703	0.843517	−	0.537103i	$-0.180481\pi$
$757$	46.4164	1.68703	0.843517	+	0.537103i	$0.180481\pi$
$758$	18.0902	0.657065
$759$	−9.23607	−0.335248
$760$	0	0
$761$	43.7771	1.58692	0.793459	−	0.608624i	$-0.208278\pi$
$761$	43.7771	1.58692	0.793459	+	0.608624i	$0.208278\pi$
$762$	35.3050	1.27896
$763$	0	0
$764$	−1.52786	−0.0552762
$765$	0	0
$766$	46.5410	1.68160
$767$	5.52786	0.199600
$768$	−16.7639	−0.604916
$769$	−33.0132	−1.19048	−0.595242	−	0.803546i	$-0.702944\pi$
$769$	−33.0132	−1.19048	−0.595242	+	0.803546i	$0.702944\pi$
$770$	0	0
$771$	23.0557	0.830332
$772$	−8.90983	−0.320672
$773$	−27.8197	−1.00060	−0.500302	−	0.865851i	$-0.666778\pi$
$773$	−27.8197	−1.00060	−0.500302	+	0.865851i	$0.666778\pi$
$774$	−14.8541	−0.533920
$775$	0	0
$776$	1.70820	0.0613209
$777$	0	0
$778$	−53.2148	−1.90784
$779$	41.3050	1.47990
$780$	0	0
$781$	−19.9443	−0.713662
$782$	18.4721	0.660562
$783$	27.6393	0.987749
$784$	0	0
$785$	0	0
$786$	1.88854	0.0673621
$787$	−45.2361	−1.61249	−0.806246	−	0.591581i	$-0.798504\pi$
$787$	−45.2361	−1.61249	−0.806246	+	0.591581i	$0.798504\pi$
$788$	4.61803	0.164511
$789$	−14.5410	−0.517674
$790$	0	0
$791$	0	0
$792$	13.9443	0.495488
$793$	−12.0000	−0.426132
$794$	−43.5967	−1.54719
$795$	0	0
$796$	−1.70820	−0.0605457
$797$	8.58359	0.304046	0.152023	−	0.988377i	$-0.451421\pi$
$797$	8.58359	0.304046	0.152023	+	0.988377i	$0.451421\pi$
$798$	0	0
$799$	12.9443	0.457935
$800$	0	0
$801$	18.7902	0.663921
$802$	18.5623	0.655458
$803$	56.0689	1.97863
$804$	−0.180340	−0.00636010
$805$	0	0
$806$	50.8328	1.79051
$807$	2.11146	0.0743268
$808$	20.6525	0.726552
$809$	20.5279	0.721721	0.360861	−	0.932620i	$-0.382483\pi$
$809$	20.5279	0.721721	0.360861	+	0.932620i	$0.382483\pi$
$810$	0	0
$811$	−46.7214	−1.64061	−0.820304	−	0.571927i	$-0.806196\pi$
$811$	−46.7214	−1.64061	−0.820304	+	0.571927i	$0.806196\pi$
$812$	0	0
$813$	12.7214	0.446158
$814$	20.5623	0.720708
$815$	0	0
$816$	38.8328	1.35942
$817$	−27.8885	−0.975697
$818$	−25.1246	−0.878461
$819$	0	0
$820$	0	0
$821$	−24.8328	−0.866671	−0.433336	−	0.901233i	$-0.642664\pi$
$821$	−24.8328	−0.866671	−0.433336	+	0.901233i	$0.642664\pi$
$822$	−13.8885	−0.484419
$823$	−0.347524	−0.0121139	−0.00605697	−	0.999982i	$-0.501928\pi$
$823$	−0.347524	−0.0121139	−0.00605697	+	0.999982i	$0.501928\pi$
$824$	−1.05573	−0.0367780
$825$	0	0
$826$	0	0
$827$	−25.5410	−0.888148	−0.444074	−	0.895990i	$-0.646467\pi$
$827$	−25.5410	−0.888148	−0.444074	+	0.895990i	$0.646467\pi$
$828$	−1.60488	−0.0557734
$829$	52.3607	1.81856	0.909281	−	0.416183i	$-0.136632\pi$
$829$	52.3607	1.81856	0.909281	+	0.416183i	$0.136632\pi$
$830$	0	0
$831$	−19.6393	−0.681280
$832$	13.7082	0.475246
$833$	0	0
$834$	−41.3050	−1.43027
$835$	0	0
$836$	−11.7082	−0.404937
$837$	53.6656	1.85496
$838$	6.18034	0.213496
$839$	−0.652476	−0.0225260	−0.0112630	−	0.999937i	$-0.503585\pi$
$839$	−0.652476	−0.0225260	−0.0112630	+	0.999937i	$0.503585\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−21.0344	−0.724895
$843$	−36.2918	−1.24996
$844$	7.41641	0.255283
$845$	0	0
$846$	−4.76393	−0.163787
$847$	0	0
$848$	2.29180	0.0787006
$849$	−11.6393	−0.399460
$850$	0	0
$851$	5.29180	0.181400
$852$	3.59675	0.123223
$853$	−0.583592	−0.0199818	−0.00999091	−	0.999950i	$-0.503180\pi$
$853$	−0.583592	−0.0199818	−0.00999091	+	0.999950i	$0.503180\pi$
$854$	0	0
$855$	0	0
$856$	−17.8885	−0.611418
$857$	38.1803	1.30422	0.652108	−	0.758126i	$-0.273885\pi$
$857$	38.1803	1.30422	0.652108	+	0.758126i	$0.273885\pi$
$858$	−27.4164	−0.935981
$859$	22.3607	0.762937	0.381468	−	0.924382i	$-0.375419\pi$
$859$	22.3607	0.762937	0.381468	+	0.924382i	$0.375419\pi$
$860$	0	0
$861$	0	0
$862$	42.8328	1.45889
$863$	49.6525	1.69019	0.845095	−	0.534616i	$-0.179544\pi$
$863$	49.6525	1.69019	0.845095	+	0.534616i	$0.179544\pi$
$864$	−18.6950	−0.636018
$865$	0	0
$866$	−26.4721	−0.899560
$867$	−30.7639	−1.04480
$868$	0	0
$869$	47.3607	1.60660
$870$	0	0
$871$	0.763932	0.0258848
$872$	41.1803	1.39454
$873$	1.12461	0.0380623
$874$	−12.7639	−0.431746
$875$	0	0
$876$	−10.1115	−0.341634
$877$	−14.3607	−0.484926	−0.242463	−	0.970161i	$-0.577955\pi$
$877$	−14.3607	−0.484926	−0.242463	+	0.970161i	$0.577955\pi$
$878$	35.1246	1.18540
$879$	11.2786	0.380419
$880$	0	0
$881$	−28.1803	−0.949420	−0.474710	−	0.880142i	$-0.657447\pi$
$881$	−28.1803	−0.949420	−0.474710	+	0.880142i	$0.657447\pi$
$882$	0	0
$883$	−50.5967	−1.70272	−0.851358	−	0.524585i	$-0.824220\pi$
$883$	−50.5967	−1.70272	−0.851358	+	0.524585i	$0.824220\pi$
$884$	12.9443	0.435363
$885$	0	0
$886$	12.0000	0.403148
$887$	52.6525	1.76790	0.883949	−	0.467584i	$-0.154876\pi$
$887$	52.6525	1.76790	0.883949	+	0.467584i	$0.154876\pi$
$888$	8.29180	0.278254
$889$	0	0
$890$	0	0
$891$	−10.2361	−0.342921
$892$	1.34752	0.0451184
$893$	−8.94427	−0.299309
$894$	27.8885	0.932732
$895$	0	0
$896$	0	0
$897$	−7.05573	−0.235584
$898$	47.6869	1.59133
$899$	48.5410	1.61893
$900$	0	0
$901$	−3.05573	−0.101801
$902$	−63.3050	−2.10782
$903$	0	0
$904$	−27.7639	−0.923415
$905$	0	0
$906$	31.5279	1.04744
$907$	−18.8328	−0.625333	−0.312667	−	0.949863i	$-0.601222\pi$
$907$	−18.8328	−0.625333	−0.312667	+	0.949863i	$0.601222\pi$
$908$	3.34752	0.111091
$909$	13.5967	0.450976
$910$	0	0
$911$	23.1803	0.767999	0.383999	−	0.923333i	$-0.374546\pi$
$911$	23.1803	0.767999	0.383999	+	0.923333i	$0.374546\pi$
$912$	−26.8328	−0.888523
$913$	−24.1803	−0.800252
$914$	−34.7426	−1.14918
$915$	0	0
$916$	−2.76393	−0.0913229
$917$	0	0
$918$	57.8885	1.91061
$919$	−32.2361	−1.06337	−0.531685	−	0.846942i	$-0.678441\pi$
$919$	−32.2361	−1.06337	−0.531685	+	0.846942i	$0.678441\pi$
$920$	0	0
$921$	38.8328	1.27958
$922$	−13.2361	−0.435907
$923$	−15.2361	−0.501501
$924$	0	0
$925$	0	0
$926$	35.4164	1.16386
$927$	−0.695048	−0.0228284
$928$	−16.9098	−0.555092
$929$	−51.7082	−1.69649	−0.848246	−	0.529603i	$-0.822341\pi$
$929$	−51.7082	−1.69649	−0.848246	+	0.529603i	$0.822341\pi$
$930$	0	0
$931$	0	0
$932$	−6.14590	−0.201316
$933$	−25.1672	−0.823937
$934$	17.7082	0.579430
$935$	0	0
$936$	10.6525	0.348187
$937$	−30.7639	−1.00501	−0.502507	−	0.864573i	$-0.667589\pi$
$937$	−30.7639	−1.00501	−0.502507	+	0.864573i	$0.667589\pi$
$938$	0	0
$939$	35.1935	1.14850
$940$	0	0
$941$	0.763932	0.0249035	0.0124517	−	0.999922i	$-0.496036\pi$
$941$	0.763932	0.0249035	0.0124517	+	0.999922i	$0.496036\pi$
$942$	10.4721	0.341201
$943$	−16.2918	−0.530534
$944$	−8.29180	−0.269875
$945$	0	0
$946$	42.7426	1.38968
$947$	−18.8328	−0.611984	−0.305992	−	0.952034i	$-0.598988\pi$
$947$	−18.8328	−0.611984	−0.305992	+	0.952034i	$0.598988\pi$
$948$	−8.54102	−0.277399
$949$	42.8328	1.39041
$950$	0	0
$951$	23.9311	0.776020
$952$	0	0
$953$	−5.47214	−0.177260	−0.0886299	−	0.996065i	$-0.528249\pi$
$953$	−5.47214	−0.177260	−0.0886299	+	0.996065i	$0.528249\pi$
$954$	1.12461	0.0364107
$955$	0	0
$956$	−8.94427	−0.289278
$957$	−26.1803	−0.846290
$958$	6.18034	0.199678
$959$	0	0
$960$	0	0
$961$	63.2492	2.04030
$962$	15.7082	0.506453
$963$	−11.7771	−0.379511
$964$	7.70820	0.248265
$965$	0	0
$966$	0	0
$967$	−49.8885	−1.60431	−0.802154	−	0.597118i	$-0.796313\pi$
$967$	−49.8885	−1.60431	−0.802154	+	0.597118i	$0.796313\pi$
$968$	−15.5279	−0.499084
$969$	35.7771	1.14933
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	−8.40325	−0.269534
$973$	0	0
$974$	16.5623	0.530691
$975$	0	0
$976$	18.0000	0.576166
$977$	−2.52786	−0.0808735	−0.0404368	−	0.999182i	$-0.512875\pi$
$977$	−2.52786	−0.0808735	−0.0404368	+	0.999182i	$0.512875\pi$
$978$	20.9443	0.669724
$979$	−54.0689	−1.72805
$980$	0	0
$981$	27.1115	0.865602
$982$	−16.5623	−0.528524
$983$	−32.5410	−1.03790	−0.518949	−	0.854805i	$-0.673676\pi$
$983$	−32.5410	−1.03790	−0.518949	+	0.854805i	$0.673676\pi$
$984$	−25.5279	−0.813799
$985$	0	0
$986$	52.3607	1.66750
$987$	0	0
$988$	−8.94427	−0.284555
$989$	11.0000	0.349780
$990$	0	0
$991$	−9.18034	−0.291623	−0.145812	−	0.989312i	$-0.546579\pi$
$991$	−9.18034	−0.291623	−0.145812	+	0.989312i	$0.546579\pi$
$992$	−32.8328	−1.04244
$993$	13.9574	0.442926
$994$	0	0
$995$	0	0
$996$	4.36068	0.138173
$997$	18.5836	0.588548	0.294274	−	0.955721i	$-0.404922\pi$
$997$	18.5836	0.588548	0.294274	+	0.955721i	$0.404922\pi$
$998$	46.8328	1.48247
$999$	16.5836	0.524682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.u.1.2		2
5.2	odd	4		1225.2.b.k.99.4		4
5.3	odd	4		1225.2.b.k.99.1		4
5.4	even	2		1225.2.a.n.1.1		2
7.6	odd	2		175.2.a.e.1.2	yes	2
21.20	even	2		1575.2.a.n.1.1		2
28.27	even	2		2800.2.a.bp.1.1		2
35.13	even	4		175.2.b.c.99.1		4
35.27	even	4		175.2.b.c.99.4		4
35.34	odd	2		175.2.a.d.1.1	✓	2
105.62	odd	4		1575.2.d.k.1324.1		4
105.83	odd	4		1575.2.d.k.1324.4		4
105.104	even	2		1575.2.a.s.1.2		2
140.27	odd	4		2800.2.g.s.449.3		4
140.83	odd	4		2800.2.g.s.449.2		4
140.139	even	2		2800.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.1	✓	2	35.34	odd	2
175.2.a.e.1.2	yes	2	7.6	odd	2
175.2.b.c.99.1		4	35.13	even	4
175.2.b.c.99.4		4	35.27	even	4
1225.2.a.n.1.1		2	5.4	even	2
1225.2.a.u.1.2		2	1.1	even	1	trivial
1225.2.b.k.99.1		4	5.3	odd	4
1225.2.b.k.99.4		4	5.2	odd	4
1575.2.a.n.1.1		2	21.20	even	2
1575.2.a.s.1.2		2	105.104	even	2
1575.2.d.k.1324.1		4	105.62	odd	4
1575.2.d.k.1324.4		4	105.83	odd	4
2800.2.a.bh.1.2		2	140.139	even	2
2800.2.a.bp.1.1		2	28.27	even	2
2800.2.g.s.449.2		4	140.83	odd	4
2800.2.g.s.449.3		4	140.27	odd	4

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} -2.00000 q^{6} -2.23607 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} -2.00000 q^{6} -2.23607 q^{8} -1.47214 q^{9} +4.23607 q^{11} -0.763932 q^{12} +3.23607 q^{13} -4.85410 q^{16} +6.47214 q^{17} -2.38197 q^{18} -4.47214 q^{19} +6.85410 q^{22} +1.76393 q^{23} +2.76393 q^{24} +5.23607 q^{26} +5.52786 q^{27} +5.00000 q^{29} +9.70820 q^{31} -3.38197 q^{32} -5.23607 q^{33} +10.4721 q^{34} -0.909830 q^{36} +3.00000 q^{37} -7.23607 q^{38} -4.00000 q^{39} -9.23607 q^{41} +6.23607 q^{43} +2.61803 q^{44} +2.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} -8.00000 q^{51} +2.00000 q^{52} -0.472136 q^{53} +8.94427 q^{54} +5.52786 q^{57} +8.09017 q^{58} +1.70820 q^{59} -3.70820 q^{61} +15.7082 q^{62} +4.23607 q^{64} -8.47214 q^{66} +0.236068 q^{67} +4.00000 q^{68} -2.18034 q^{69} -4.70820 q^{71} +3.29180 q^{72} +13.2361 q^{73} +4.85410 q^{74} -2.76393 q^{76} -6.47214 q^{78} +11.1803 q^{79} -2.41641 q^{81} -14.9443 q^{82} -5.70820 q^{83} +10.0902 q^{86} -6.18034 q^{87} -9.47214 q^{88} -12.7639 q^{89} +1.09017 q^{92} -12.0000 q^{93} +3.23607 q^{94} +4.18034 q^{96} -0.763932 q^{97} -6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9} + 4 q^{11} - 6 q^{12} + 2 q^{13} - 3 q^{16} + 4 q^{17} - 7 q^{18} + 7 q^{22} + 8 q^{23} + 10 q^{24} + 6 q^{26} + 20 q^{27} + 10 q^{29} + 6 q^{31} - 9 q^{32} - 6 q^{33} + 12 q^{34} - 13 q^{36} + 6 q^{37} - 10 q^{38} - 8 q^{39} - 14 q^{41} + 8 q^{43} + 3 q^{44} - q^{46} + 4 q^{47} + 12 q^{48} - 16 q^{51} + 4 q^{52} + 8 q^{53} + 20 q^{57} + 5 q^{58} - 10 q^{59} + 6 q^{61} + 18 q^{62} + 4 q^{64} - 8 q^{66} - 4 q^{67} + 8 q^{68} + 18 q^{69} + 4 q^{71} + 20 q^{72} + 22 q^{73} + 3 q^{74} - 10 q^{76} - 4 q^{78} + 22 q^{81} - 12 q^{82} + 2 q^{83} + 9 q^{86} + 10 q^{87} - 10 q^{88} - 30 q^{89} - 9 q^{92} - 24 q^{93} + 2 q^{94} - 14 q^{96} - 6 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9 + 4 * q^11 - 6 * q^12 + 2 * q^13 - 3 * q^16 + 4 * q^17 - 7 * q^18 + 7 * q^22 + 8 * q^23 + 10 * q^24 + 6 * q^26 + 20 * q^27 + 10 * q^29 + 6 * q^31 - 9 * q^32 - 6 * q^33 + 12 * q^34 - 13 * q^36 + 6 * q^37 - 10 * q^38 - 8 * q^39 - 14 * q^41 + 8 * q^43 + 3 * q^44 - q^46 + 4 * q^47 + 12 * q^48 - 16 * q^51 + 4 * q^52 + 8 * q^53 + 20 * q^57 + 5 * q^58 - 10 * q^59 + 6 * q^61 + 18 * q^62 + 4 * q^64 - 8 * q^66 - 4 * q^67 + 8 * q^68 + 18 * q^69 + 4 * q^71 + 20 * q^72 + 22 * q^73 + 3 * q^74 - 10 * q^76 - 4 * q^78 + 22 * q^81 - 12 * q^82 + 2 * q^83 + 9 * q^86 + 10 * q^87 - 10 * q^88 - 30 * q^89 - 9 * q^92 - 24 * q^93 + 2 * q^94 - 14 * q^96 - 6 * q^97 - 8 * q^99

Introduction

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