LMFDB - Embedded newform 4425.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4425,2,Mod(1,4425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4425, 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("4425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4425 = 3 \cdot 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4425.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:	$35.3338028944$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.22298624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 8x^{3} + 14x^{2} - 4x - 2 $ x^6 - 2x^5 - 7x^4 + 8x^3 + 14x^2 - 4*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 885)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.498703$ of defining polynomial
Character	$\chi$	$=$	4425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.501297 q^{2} +1.00000 q^{3} -1.74870 q^{4} +0.501297 q^{6} -2.82843 q^{7} -1.87921 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.501297 q^{2} +1.00000 q^{3} -1.74870 q^{4} +0.501297 q^{6} -2.82843 q^{7} -1.87921 q^{8} +1.00000 q^{9} -5.64222 q^{11} -1.74870 q^{12} -2.10485 q^{13} -1.41788 q^{14} +2.55536 q^{16} -3.64222 q^{17} +0.501297 q^{18} -1.55017 q^{19} -2.82843 q^{21} -2.82843 q^{22} +2.70894 q^{23} -1.87921 q^{24} -1.05515 q^{26} +1.00000 q^{27} +4.94607 q^{28} +2.09205 q^{29} -0.261025 q^{31} +5.03942 q^{32} -5.64222 q^{33} -1.82583 q^{34} -1.74870 q^{36} -2.10485 q^{37} -0.777095 q^{38} -2.10485 q^{39} -3.13345 q^{41} -1.41788 q^{42} +9.34122 q^{43} +9.86655 q^{44} +1.35798 q^{46} +12.5755 q^{47} +2.55536 q^{48} +1.00000 q^{49} -3.64222 q^{51} +3.68075 q^{52} +5.85519 q^{53} +0.501297 q^{54} +5.31522 q^{56} -1.55017 q^{57} +1.04874 q^{58} -1.00000 q^{59} -10.8544 q^{61} -0.130851 q^{62} -2.82843 q^{63} -2.58447 q^{64} -2.82843 q^{66} +2.89775 q^{67} +6.36915 q^{68} +2.70894 q^{69} -8.08471 q^{71} -1.87921 q^{72} +13.3893 q^{73} -1.05515 q^{74} +2.71078 q^{76} +15.9586 q^{77} -1.05515 q^{78} +9.72907 q^{79} +1.00000 q^{81} -1.57079 q^{82} +7.23099 q^{83} +4.94607 q^{84} +4.68273 q^{86} +2.09205 q^{87} +10.6029 q^{88} +14.1370 q^{89} +5.95341 q^{91} -4.73713 q^{92} -0.261025 q^{93} +6.30406 q^{94} +5.03942 q^{96} -6.78327 q^{97} +0.501297 q^{98} -5.64222 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9	$ 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9} - 2 q^{11} + 8 q^{12} - 2 q^{13} + 12 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 12 q^{23} + 12 q^{24} - 8 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 8 q^{31} + 28 q^{32} - 2 q^{33} + 8 q^{34} + 8 q^{36} - 2 q^{37} + 24 q^{38} - 2 q^{39} - 4 q^{41} + 18 q^{43} + 4 q^{44} + 16 q^{47} + 12 q^{48} + 6 q^{49} + 10 q^{51} - 12 q^{52} + 30 q^{53} + 4 q^{54} - 2 q^{57} + 16 q^{58} - 6 q^{59} + 4 q^{61} + 16 q^{62} + 20 q^{64} + 30 q^{67} + 20 q^{68} + 12 q^{69} - 24 q^{71} + 12 q^{72} + 6 q^{73} - 8 q^{74} - 4 q^{76} + 16 q^{77} - 8 q^{78} - 2 q^{79} + 6 q^{81} + 20 q^{83} + 16 q^{84} - 12 q^{87} + 16 q^{88} + 14 q^{89} - 48 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 28 q^{96} - 4 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9 - 2 * q^11 + 8 * q^12 - 2 * q^13 + 12 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 12 * q^23 + 12 * q^24 - 8 * q^26 + 6 * q^27 + 16 * q^28 - 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 8 * q^34 + 8 * q^36 - 2 * q^37 + 24 * q^38 - 2 * q^39 - 4 * q^41 + 18 * q^43 + 4 * q^44 + 16 * q^47 + 12 * q^48 + 6 * q^49 + 10 * q^51 - 12 * q^52 + 30 * q^53 + 4 * q^54 - 2 * q^57 + 16 * q^58 - 6 * q^59 + 4 * q^61 + 16 * q^62 + 20 * q^64 + 30 * q^67 + 20 * q^68 + 12 * q^69 - 24 * q^71 + 12 * q^72 + 6 * q^73 - 8 * q^74 - 4 * q^76 + 16 * q^77 - 8 * q^78 - 2 * q^79 + 6 * q^81 + 20 * q^83 + 16 * q^84 - 12 * q^87 + 16 * q^88 + 14 * q^89 - 48 * q^91 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 28 * q^96 - 4 * q^97 + 4 * 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 $

$2$	0.501297	0.354471	0.177235	−	0.984168i	$-0.443285\pi$
$2$	0.501297	0.354471	0.177235	+	0.984168i	$0.443285\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.74870	−0.874351
$5$	0	0
$5$	0	0
$6$	0.501297	0.204654
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	−1.87921	−0.664402
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.64222	−1.70119	−0.850596	−	0.525820i	$-0.823759\pi$
$11$	−5.64222	−1.70119	−0.850596	+	0.525820i	$0.823759\pi$
$12$	−1.74870	−0.504807
$13$	−2.10485	−0.583779	−0.291890	−	0.956452i	$-0.594284\pi$
$13$	−2.10485	−0.583779	−0.291890	+	0.956452i	$0.594284\pi$
$14$	−1.41788	−0.378945
$15$	0	0
$16$	2.55536	0.638839
$17$	−3.64222	−0.883367	−0.441683	−	0.897171i	$-0.645619\pi$
$17$	−3.64222	−0.883367	−0.441683	+	0.897171i	$0.645619\pi$
$18$	0.501297	0.118157
$19$	−1.55017	−0.355633	−0.177816	−	0.984064i	$-0.556903\pi$
$19$	−1.55017	−0.355633	−0.177816	+	0.984064i	$0.556903\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	−2.82843	−0.603023
$23$	2.70894	0.564853	0.282427	−	0.959289i	$-0.408861\pi$
$23$	2.70894	0.564853	0.282427	+	0.959289i	$0.408861\pi$
$24$	−1.87921	−0.383593
$25$	0	0
$26$	−1.05515	−0.206933
$27$	1.00000	0.192450
$28$	4.94607	0.934720
$29$	2.09205	0.388483	0.194242	−	0.980954i	$-0.437775\pi$
$29$	2.09205	0.388483	0.194242	+	0.980954i	$0.437775\pi$
$30$	0	0
$31$	−0.261025	−0.0468815	−0.0234408	−	0.999725i	$-0.507462\pi$
$31$	−0.261025	−0.0468815	−0.0234408	+	0.999725i	$0.507462\pi$
$32$	5.03942	0.890852
$33$	−5.64222	−0.982184
$34$	−1.82583	−0.313128
$35$	0	0
$36$	−1.74870	−0.291450
$37$	−2.10485	−0.346035	−0.173017	−	0.984919i	$-0.555352\pi$
$37$	−2.10485	−0.346035	−0.173017	+	0.984919i	$0.555352\pi$
$38$	−0.777095	−0.126061
$39$	−2.10485	−0.337045
$40$	0	0
$41$	−3.13345	−0.489362	−0.244681	−	0.969604i	$-0.578683\pi$
$41$	−3.13345	−0.489362	−0.244681	+	0.969604i	$0.578683\pi$
$42$	−1.41788	−0.218784
$43$	9.34122	1.42452	0.712262	−	0.701914i	$-0.247671\pi$
$43$	9.34122	1.42452	0.712262	+	0.701914i	$0.247671\pi$
$44$	9.86655	1.48744
$45$	0	0
$46$	1.35798	0.200224
$47$	12.5755	1.83432	0.917162	−	0.398515i	$-0.130474\pi$
$47$	12.5755	1.83432	0.917162	+	0.398515i	$0.130474\pi$
$48$	2.55536	0.368834
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.64222	−0.510012
$52$	3.68075	0.510428
$53$	5.85519	0.804272	0.402136	−	0.915580i	$-0.368268\pi$
$53$	5.85519	0.804272	0.402136	+	0.915580i	$0.368268\pi$
$54$	0.501297	0.0682179
$55$	0	0
$56$	5.31522	0.710276
$57$	−1.55017	−0.205325
$58$	1.04874	0.137706
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−10.8544	−1.38977	−0.694883	−	0.719122i	$-0.744544\pi$
$61$	−10.8544	−1.38977	−0.694883	+	0.719122i	$0.744544\pi$
$62$	−0.130851	−0.0166181
$63$	−2.82843	−0.356348
$64$	−2.58447	−0.323058
$65$	0	0
$66$	−2.82843	−0.348155
$67$	2.89775	0.354016	0.177008	−	0.984209i	$-0.443358\pi$
$67$	2.89775	0.354016	0.177008	+	0.984209i	$0.443358\pi$
$68$	6.36915	0.772372
$69$	2.70894	0.326118
$70$	0	0
$71$	−8.08471	−0.959478	−0.479739	−	0.877411i	$-0.659269\pi$
$71$	−8.08471	−0.959478	−0.479739	+	0.877411i	$0.659269\pi$
$72$	−1.87921	−0.221467
$73$	13.3893	1.56710	0.783548	−	0.621331i	$-0.213408\pi$
$73$	13.3893	1.56710	0.783548	+	0.621331i	$0.213408\pi$
$74$	−1.05515	−0.122659
$75$	0	0
$76$	2.71078	0.310948
$77$	15.9586	1.81865
$78$	−1.05515	−0.119473
$79$	9.72907	1.09461	0.547303	−	0.836934i	$-0.315654\pi$
$79$	9.72907	1.09461	0.547303	+	0.836934i	$0.315654\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.57079	−0.173465
$83$	7.23099	0.793704	0.396852	−	0.917883i	$-0.370103\pi$
$83$	7.23099	0.793704	0.396852	+	0.917883i	$0.370103\pi$
$84$	4.94607	0.539661
$85$	0	0
$86$	4.68273	0.504952
$87$	2.09205	0.224291
$88$	10.6029	1.13028
$89$	14.1370	1.49852	0.749261	−	0.662275i	$-0.230409\pi$
$89$	14.1370	1.49852	0.749261	+	0.662275i	$0.230409\pi$
$90$	0	0
$91$	5.95341	0.624086
$92$	−4.73713	−0.493880
$93$	−0.261025	−0.0270671
$94$	6.30406	0.650214
$95$	0	0
$96$	5.03942	0.514334
$97$	−6.78327	−0.688737	−0.344368	−	0.938835i	$-0.611907\pi$
$97$	−6.78327	−0.688737	−0.344368	+	0.938835i	$0.611907\pi$
$98$	0.501297	0.0506387
$99$	−5.64222	−0.567064
$100$	0	0
$101$	−13.2331	−1.31674	−0.658371	−	0.752693i	$-0.728754\pi$
$101$	−13.2331	−1.31674	−0.658371	+	0.752693i	$0.728754\pi$
$102$	−1.82583	−0.180784
$103$	−15.3892	−1.51634	−0.758172	−	0.652055i	$-0.773907\pi$
$103$	−15.3892	−1.51634	−0.758172	+	0.652055i	$0.773907\pi$
$104$	3.95546	0.387864
$105$	0	0
$106$	2.93519	0.285091
$107$	−9.87272	−0.954432	−0.477216	−	0.878786i	$-0.658354\pi$
$107$	−9.87272	−0.954432	−0.477216	+	0.878786i	$0.658354\pi$
$108$	−1.74870	−0.168269
$109$	12.4435	1.19187	0.595934	−	0.803033i	$-0.296782\pi$
$109$	12.4435	1.19187	0.595934	+	0.803033i	$0.296782\pi$
$110$	0	0
$111$	−2.10485	−0.199783
$112$	−7.22764	−0.682948
$113$	13.4260	1.26301	0.631504	−	0.775372i	$-0.282438\pi$
$113$	13.4260	1.26301	0.631504	+	0.775372i	$0.282438\pi$
$114$	−0.777095	−0.0727816
$115$	0	0
$116$	−3.65836	−0.339671
$117$	−2.10485	−0.194593
$118$	−0.501297	−0.0461482
$119$	10.3017	0.944359
$120$	0	0
$121$	20.8346	1.89405
$122$	−5.44129	−0.492632
$123$	−3.13345	−0.282533
$124$	0.456455	0.0409909
$125$	0	0
$126$	−1.41788	−0.126315
$127$	−2.23378	−0.198216	−0.0991081	−	0.995077i	$-0.531599\pi$
$127$	−2.23378	−0.198216	−0.0991081	+	0.995077i	$0.531599\pi$
$128$	−11.3744	−1.00537
$129$	9.34122	0.822449
$130$	0	0
$131$	−10.1708	−0.888628	−0.444314	−	0.895871i	$-0.646553\pi$
$131$	−10.1708	−0.888628	−0.444314	+	0.895871i	$0.646553\pi$
$132$	9.86655	0.858773
$133$	4.38454	0.380188
$134$	1.45263	0.125488
$135$	0	0
$136$	6.84450	0.586911
$137$	−0.374335	−0.0319816	−0.0159908	−	0.999872i	$-0.505090\pi$
$137$	−0.374335	−0.0319816	−0.0159908	+	0.999872i	$0.505090\pi$
$138$	1.35798	0.115599
$139$	9.30852	0.789538	0.394769	−	0.918780i	$-0.370825\pi$
$139$	9.30852	0.789538	0.394769	+	0.918780i	$0.370825\pi$
$140$	0	0
$141$	12.5755	1.05905
$142$	−4.05284	−0.340107
$143$	11.8760	0.993121
$144$	2.55536	0.212946
$145$	0	0
$146$	6.71201	0.555490
$147$	1.00000	0.0824786
$148$	3.68075	0.302556
$149$	−5.89805	−0.483187	−0.241594	−	0.970378i	$-0.577670\pi$
$149$	−5.89805	−0.483187	−0.241594	+	0.970378i	$0.577670\pi$
$150$	0	0
$151$	−8.11546	−0.660427	−0.330213	−	0.943906i	$-0.607121\pi$
$151$	−8.11546	−0.660427	−0.330213	+	0.943906i	$0.607121\pi$
$152$	2.91310	0.236283
$153$	−3.64222	−0.294456
$154$	8.00000	0.644658
$155$	0	0
$156$	3.68075	0.294696
$157$	13.5268	1.07955	0.539776	−	0.841809i	$-0.318509\pi$
$157$	13.5268	1.07955	0.539776	+	0.841809i	$0.318509\pi$
$158$	4.87716	0.388006
$159$	5.85519	0.464347
$160$	0	0
$161$	−7.66204	−0.603854
$162$	0.501297	0.0393856
$163$	−0.297680	−0.0233161	−0.0116581	−	0.999932i	$-0.503711\pi$
$163$	−0.297680	−0.0233161	−0.0116581	+	0.999932i	$0.503711\pi$
$164$	5.47946	0.427874
$165$	0	0
$166$	3.62488	0.281345
$167$	23.0662	1.78492	0.892459	−	0.451128i	$-0.148978\pi$
$167$	23.0662	1.78492	0.892459	+	0.451128i	$0.148978\pi$
$168$	5.31522	0.410078
$169$	−8.56962	−0.659202
$170$	0	0
$171$	−1.55017	−0.118544
$172$	−16.3350	−1.24553
$173$	−1.94915	−0.148191	−0.0740955	−	0.997251i	$-0.523607\pi$
$173$	−1.94915	−0.148191	−0.0740955	+	0.997251i	$0.523607\pi$
$174$	1.04874	0.0795046
$175$	0	0
$176$	−14.4179	−1.08679
$177$	−1.00000	−0.0751646
$178$	7.08685	0.531182
$179$	2.80645	0.209764	0.104882	−	0.994485i	$-0.466554\pi$
$179$	2.80645	0.209764	0.104882	+	0.994485i	$0.466554\pi$
$180$	0	0
$181$	17.9168	1.33174	0.665872	−	0.746066i	$-0.268060\pi$
$181$	17.9168	1.33174	0.665872	+	0.746066i	$0.268060\pi$
$182$	2.98443	0.221220
$183$	−10.8544	−0.802382
$184$	−5.09068	−0.375290
$185$	0	0
$186$	−0.130851	−0.00959448
$187$	20.5502	1.50278
$188$	−21.9908	−1.60384
$189$	−2.82843	−0.205738
$190$	0	0
$191$	−11.9347	−0.863565	−0.431783	−	0.901978i	$-0.642115\pi$
$191$	−11.9347	−0.863565	−0.431783	+	0.901978i	$0.642115\pi$
$192$	−2.58447	−0.186518
$193$	−1.55325	−0.111805	−0.0559025	−	0.998436i	$-0.517804\pi$
$193$	−1.55325	−0.111805	−0.0559025	+	0.998436i	$0.517804\pi$
$194$	−3.40044	−0.244137
$195$	0	0
$196$	−1.74870	−0.124907
$197$	−5.74160	−0.409072	−0.204536	−	0.978859i	$-0.565569\pi$
$197$	−5.74160	−0.409072	−0.204536	+	0.978859i	$0.565569\pi$
$198$	−2.82843	−0.201008
$199$	−17.5261	−1.24239	−0.621195	−	0.783656i	$-0.713352\pi$
$199$	−17.5261	−1.24239	−0.621195	+	0.783656i	$0.713352\pi$
$200$	0	0
$201$	2.89775	0.204391
$202$	−6.63372	−0.466747
$203$	−5.91720	−0.415306
$204$	6.36915	0.445929
$205$	0	0
$206$	−7.71456	−0.537499
$207$	2.70894	0.188284
$208$	−5.37864	−0.372941
$209$	8.74638	0.605000
$210$	0	0
$211$	−11.7563	−0.809336	−0.404668	−	0.914464i	$-0.632613\pi$
$211$	−11.7563	−0.809336	−0.404668	+	0.914464i	$0.632613\pi$
$212$	−10.2390	−0.703215
$213$	−8.08471	−0.553955
$214$	−4.94917	−0.338318
$215$	0	0
$216$	−1.87921	−0.127864
$217$	0.738291	0.0501184
$218$	6.23788	0.422483
$219$	13.3893	0.904763
$220$	0	0
$221$	7.66630	0.515691
$222$	−1.05515	−0.0708173
$223$	−1.27615	−0.0854572	−0.0427286	−	0.999087i	$-0.513605\pi$
$223$	−1.27615	−0.0854572	−0.0427286	+	0.999087i	$0.513605\pi$
$224$	−14.2536	−0.952361
$225$	0	0
$226$	6.73040	0.447700
$227$	0.327268	0.0217216	0.0108608	−	0.999941i	$-0.496543\pi$
$227$	0.327268	0.0217216	0.0108608	+	0.999941i	$0.496543\pi$
$228$	2.71078	0.179526
$229$	−14.4872	−0.957341	−0.478670	−	0.877995i	$-0.658881\pi$
$229$	−14.4872	−0.957341	−0.478670	+	0.877995i	$0.658881\pi$
$230$	0	0
$231$	15.9586	1.05000
$232$	−3.93140	−0.258109
$233$	−20.8956	−1.36892	−0.684459	−	0.729051i	$-0.739961\pi$
$233$	−20.8956	−1.36892	−0.684459	+	0.729051i	$0.739961\pi$
$234$	−1.05515	−0.0689776
$235$	0	0
$236$	1.74870	0.113831
$237$	9.72907	0.631971
$238$	5.16423	0.334748
$239$	21.2232	1.37281	0.686406	−	0.727219i	$-0.259187\pi$
$239$	21.2232	1.37281	0.686406	+	0.727219i	$0.259187\pi$
$240$	0	0
$241$	4.73427	0.304961	0.152481	−	0.988306i	$-0.451274\pi$
$241$	4.73427	0.304961	0.152481	+	0.988306i	$0.451274\pi$
$242$	10.4443	0.671387
$243$	1.00000	0.0641500
$244$	18.9811	1.21514
$245$	0	0
$246$	−1.57079	−0.100150
$247$	3.26287	0.207611
$248$	0.490522	0.0311482
$249$	7.23099	0.458245
$250$	0	0
$251$	−3.34219	−0.210957	−0.105479	−	0.994422i	$-0.533637\pi$
$251$	−3.34219	−0.210957	−0.105479	+	0.994422i	$0.533637\pi$
$252$	4.94607	0.311573
$253$	−15.2844	−0.960924
$254$	−1.11979	−0.0702618
$255$	0	0
$256$	−0.533037	−0.0333148
$257$	−13.1169	−0.818209	−0.409104	−	0.912488i	$-0.634159\pi$
$257$	−13.1169	−0.818209	−0.409104	+	0.912488i	$0.634159\pi$
$258$	4.68273	0.291534
$259$	5.95341	0.369927
$260$	0	0
$261$	2.09205	0.129494
$262$	−5.09860	−0.314993
$263$	17.5835	1.08425	0.542123	−	0.840299i	$-0.317621\pi$
$263$	17.5835	1.08425	0.542123	+	0.840299i	$0.317621\pi$
$264$	10.6029	0.652565
$265$	0	0
$266$	2.19796	0.134765
$267$	14.1370	0.865172
$268$	−5.06729	−0.309534
$269$	−19.7608	−1.20484	−0.602418	−	0.798181i	$-0.705796\pi$
$269$	−19.7608	−1.20484	−0.602418	+	0.798181i	$0.705796\pi$
$270$	0	0
$271$	30.4903	1.85215	0.926076	−	0.377336i	$-0.123160\pi$
$271$	30.4903	1.85215	0.926076	+	0.377336i	$0.123160\pi$
$272$	−9.30716	−0.564330
$273$	5.95341	0.360316
$274$	−0.187653	−0.0113365
$275$	0	0
$276$	−4.73713	−0.285142
$277$	29.8446	1.79319	0.896594	−	0.442854i	$-0.146034\pi$
$277$	29.8446	1.79319	0.896594	+	0.442854i	$0.146034\pi$
$278$	4.66634	0.279868
$279$	−0.261025	−0.0156272
$280$	0	0
$281$	30.0002	1.78966	0.894831	−	0.446404i	$-0.147296\pi$
$281$	30.0002	1.78966	0.894831	+	0.446404i	$0.147296\pi$
$282$	6.30406	0.375401
$283$	26.4639	1.57312	0.786559	−	0.617515i	$-0.211861\pi$
$283$	26.4639	1.57312	0.786559	+	0.617515i	$0.211861\pi$
$284$	14.1377	0.838920
$285$	0	0
$286$	5.95341	0.352032
$287$	8.86272	0.523150
$288$	5.03942	0.296951
$289$	−3.73427	−0.219663
$290$	0	0
$291$	−6.78327	−0.397642
$292$	−23.4138	−1.37019
$293$	19.9711	1.16673	0.583363	−	0.812212i	$-0.301737\pi$
$293$	19.9711	1.16673	0.583363	+	0.812212i	$0.301737\pi$
$294$	0.501297	0.0292363
$295$	0	0
$296$	3.95546	0.229906
$297$	−5.64222	−0.327395
$298$	−2.95668	−0.171276
$299$	−5.70191	−0.329750
$300$	0	0
$301$	−26.4210	−1.52288
$302$	−4.06826	−0.234102
$303$	−13.2331	−0.760222
$304$	−3.96123	−0.227192
$305$	0	0
$306$	−1.82583	−0.104376
$307$	−8.28808	−0.473026	−0.236513	−	0.971628i	$-0.576005\pi$
$307$	−8.28808	−0.473026	−0.236513	+	0.971628i	$0.576005\pi$
$308$	−27.9068	−1.59014
$309$	−15.3892	−0.875461
$310$	0	0
$311$	13.5542	0.768588	0.384294	−	0.923211i	$-0.374445\pi$
$311$	13.5542	0.768588	0.384294	+	0.923211i	$0.374445\pi$
$312$	3.95546	0.223934
$313$	1.54948	0.0875819	0.0437910	−	0.999041i	$-0.486056\pi$
$313$	1.54948	0.0875819	0.0437910	+	0.999041i	$0.486056\pi$
$314$	6.78092	0.382670
$315$	0	0
$316$	−17.0132	−0.957070
$317$	−12.4391	−0.698647	−0.349324	−	0.937002i	$-0.613589\pi$
$317$	−12.4391	−0.698647	−0.349324	+	0.937002i	$0.613589\pi$
$318$	2.93519	0.164597
$319$	−11.8038	−0.660885
$320$	0	0
$321$	−9.87272	−0.551041
$322$	−3.84096	−0.214048
$323$	5.64605	0.314154
$324$	−1.74870	−0.0971501
$325$	0	0
$326$	−0.149226	−0.00826488
$327$	12.4435	0.688126
$328$	5.88841	0.325133
$329$	−35.5689	−1.96097
$330$	0	0
$331$	−12.3722	−0.680039	−0.340019	−	0.940418i	$-0.610434\pi$
$331$	−12.3722	−0.680039	−0.340019	+	0.940418i	$0.610434\pi$
$332$	−12.6448	−0.693976
$333$	−2.10485	−0.115345
$334$	11.5630	0.632701
$335$	0	0
$336$	−7.22764	−0.394300
$337$	−17.9993	−0.980485	−0.490243	−	0.871586i	$-0.663092\pi$
$337$	−17.9993	−0.980485	−0.490243	+	0.871586i	$0.663092\pi$
$338$	−4.29593	−0.233668
$339$	13.4260	0.729199
$340$	0	0
$341$	1.47276	0.0797544
$342$	−0.777095	−0.0420205
$343$	16.9706	0.916324
$344$	−17.5542	−0.946457
$345$	0	0
$346$	−0.977103	−0.0525294
$347$	20.4234	1.09638	0.548192	−	0.836352i	$-0.315316\pi$
$347$	20.4234	1.09638	0.548192	+	0.836352i	$0.315316\pi$
$348$	−3.65836	−0.196109
$349$	18.7414	1.00320	0.501602	−	0.865099i	$-0.332744\pi$
$349$	18.7414	1.00320	0.501602	+	0.865099i	$0.332744\pi$
$350$	0	0
$351$	−2.10485	−0.112348
$352$	−28.4335	−1.51551
$353$	4.03369	0.214691	0.107346	−	0.994222i	$-0.465765\pi$
$353$	4.03369	0.214691	0.107346	+	0.994222i	$0.465765\pi$
$354$	−0.501297	−0.0266436
$355$	0	0
$356$	−24.7214	−1.31023
$357$	10.3017	0.545226
$358$	1.40687	0.0743552
$359$	15.5099	0.818583	0.409291	−	0.912404i	$-0.365776\pi$
$359$	15.5099	0.818583	0.409291	+	0.912404i	$0.365776\pi$
$360$	0	0
$361$	−16.5970	−0.873525
$362$	8.98164	0.472064
$363$	20.8346	1.09353
$364$	−10.4107	−0.545670
$365$	0	0
$366$	−5.44129	−0.284421
$367$	1.33468	0.0696695	0.0348348	−	0.999393i	$-0.488910\pi$
$367$	1.33468	0.0696695	0.0348348	+	0.999393i	$0.488910\pi$
$368$	6.92231	0.360851
$369$	−3.13345	−0.163121
$370$	0	0
$371$	−16.5610	−0.859803
$372$	0.456455	0.0236661
$373$	16.5481	0.856826	0.428413	−	0.903583i	$-0.359073\pi$
$373$	16.5481	0.856826	0.428413	+	0.903583i	$0.359073\pi$
$374$	10.3017	0.532690
$375$	0	0
$376$	−23.6320	−1.21873
$377$	−4.40344	−0.226789
$378$	−1.41788	−0.0729280
$379$	−1.44465	−0.0742067	−0.0371033	−	0.999311i	$-0.511813\pi$
$379$	−1.44465	−0.0742067	−0.0371033	+	0.999311i	$0.511813\pi$
$380$	0	0
$381$	−2.23378	−0.114440
$382$	−5.98284	−0.306109
$383$	−17.2683	−0.882368	−0.441184	−	0.897417i	$-0.645441\pi$
$383$	−17.2683	−0.882368	−0.441184	+	0.897417i	$0.645441\pi$
$384$	−11.3744	−0.580449
$385$	0	0
$386$	−0.778638	−0.0396316
$387$	9.34122	0.474841
$388$	11.8619	0.602197
$389$	−15.3713	−0.779355	−0.389678	−	0.920951i	$-0.627414\pi$
$389$	−15.3713	−0.779355	−0.389678	+	0.920951i	$0.627414\pi$
$390$	0	0
$391$	−9.86655	−0.498973
$392$	−1.87921	−0.0949146
$393$	−10.1708	−0.513050
$394$	−2.87825	−0.145004
$395$	0	0
$396$	9.86655	0.495813
$397$	−28.3065	−1.42066	−0.710331	−	0.703868i	$-0.751455\pi$
$397$	−28.3065	−1.42066	−0.710331	+	0.703868i	$0.751455\pi$
$398$	−8.78577	−0.440391
$399$	4.38454	0.219501
$400$	0	0
$401$	39.7916	1.98710	0.993548	−	0.113409i	$-0.0361771\pi$
$401$	39.7916	1.98710	0.993548	+	0.113409i	$0.0361771\pi$
$402$	1.45263	0.0724508
$403$	0.549418	0.0273685
$404$	23.1407	1.15129
$405$	0	0
$406$	−2.96628	−0.147214
$407$	11.8760	0.588671
$408$	6.84450	0.338853
$409$	22.7305	1.12395	0.561975	−	0.827155i	$-0.310042\pi$
$409$	22.7305	1.12395	0.561975	+	0.827155i	$0.310042\pi$
$410$	0	0
$411$	−0.374335	−0.0184646
$412$	26.9111	1.32582
$413$	2.82843	0.139178
$414$	1.35798	0.0667413
$415$	0	0
$416$	−10.6072	−0.520061
$417$	9.30852	0.455840
$418$	4.38454	0.214455
$419$	31.3738	1.53271	0.766355	−	0.642417i	$-0.222068\pi$
$419$	31.3738	1.53271	0.766355	+	0.642417i	$0.222068\pi$
$420$	0	0
$421$	−37.7198	−1.83835	−0.919176	−	0.393847i	$-0.871144\pi$
$421$	−37.7198	−1.83835	−0.919176	+	0.393847i	$0.871144\pi$
$422$	−5.89339	−0.286886
$423$	12.5755	0.611441
$424$	−11.0031	−0.534360
$425$	0	0
$426$	−4.05284	−0.196361
$427$	30.7010	1.48572
$428$	17.2644	0.834508
$429$	11.8760	0.573379
$430$	0	0
$431$	−9.77852	−0.471015	−0.235507	−	0.971873i	$-0.575675\pi$
$431$	−9.77852	−0.471015	−0.235507	+	0.971873i	$0.575675\pi$
$432$	2.55536	0.122945
$433$	16.5561	0.795635	0.397818	−	0.917465i	$-0.369768\pi$
$433$	16.5561	0.795635	0.397818	+	0.917465i	$0.369768\pi$
$434$	0.370103	0.0177655
$435$	0	0
$436$	−21.7599	−1.04211
$437$	−4.19931	−0.200880
$438$	6.71201	0.320712
$439$	−10.2511	−0.489259	−0.244630	−	0.969617i	$-0.578666\pi$
$439$	−10.2511	−0.489259	−0.244630	+	0.969617i	$0.578666\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	3.84310	0.182798
$443$	−4.63850	−0.220382	−0.110191	−	0.993910i	$-0.535146\pi$
$443$	−4.63850	−0.220382	−0.110191	+	0.993910i	$0.535146\pi$
$444$	3.68075	0.174681
$445$	0	0
$446$	−0.639729	−0.0302921
$447$	−5.89805	−0.278968
$448$	7.30998	0.345364
$449$	5.05224	0.238430	0.119215	−	0.992868i	$-0.461962\pi$
$449$	5.05224	0.238430	0.119215	+	0.992868i	$0.461962\pi$
$450$	0	0
$451$	17.6796	0.832499
$452$	−23.4780	−1.10431
$453$	−8.11546	−0.381298
$454$	0.164059	0.00769966
$455$	0	0
$456$	2.91310	0.136418
$457$	−23.0460	−1.07805	−0.539023	−	0.842291i	$-0.681206\pi$
$457$	−23.0460	−1.07805	−0.539023	+	0.842291i	$0.681206\pi$
$458$	−7.26239	−0.339349
$459$	−3.64222	−0.170004
$460$	0	0
$461$	19.4500	0.905878	0.452939	−	0.891542i	$-0.350376\pi$
$461$	19.4500	0.905878	0.452939	+	0.891542i	$0.350376\pi$
$462$	8.00000	0.372194
$463$	1.99925	0.0929131	0.0464565	−	0.998920i	$-0.485207\pi$
$463$	1.99925	0.0929131	0.0464565	+	0.998920i	$0.485207\pi$
$464$	5.34593	0.248178
$465$	0	0
$466$	−10.4749	−0.485241
$467$	7.48864	0.346533	0.173266	−	0.984875i	$-0.444568\pi$
$467$	7.48864	0.346533	0.173266	+	0.984875i	$0.444568\pi$
$468$	3.68075	0.170143
$469$	−8.19607	−0.378459
$470$	0	0
$471$	13.5268	0.623280
$472$	1.87921	0.0864978
$473$	−52.7052	−2.42339
$474$	4.87716	0.224015
$475$	0	0
$476$	−18.0147	−0.825701
$477$	5.85519	0.268091
$478$	10.6391	0.486622
$479$	6.37952	0.291488	0.145744	−	0.989322i	$-0.453442\pi$
$479$	6.37952	0.291488	0.145744	+	0.989322i	$0.453442\pi$
$480$	0	0
$481$	4.43038	0.202008
$482$	2.37328	0.108100
$483$	−7.66204	−0.348635
$484$	−36.4335	−1.65607
$485$	0	0
$486$	0.501297	0.0227393
$487$	15.6470	0.709035	0.354517	−	0.935049i	$-0.384645\pi$
$487$	15.6470	0.709035	0.354517	+	0.935049i	$0.384645\pi$
$488$	20.3978	0.923364
$489$	−0.297680	−0.0134616
$490$	0	0
$491$	25.9136	1.16946	0.584732	−	0.811226i	$-0.301200\pi$
$491$	25.9136	1.16946	0.584732	+	0.811226i	$0.301200\pi$
$492$	5.47946	0.247033
$493$	−7.61968	−0.343173
$494$	1.63567	0.0735921
$495$	0	0
$496$	−0.667013	−0.0299498
$497$	22.8670	1.02573
$498$	3.62488	0.162435
$499$	−40.8680	−1.82950	−0.914752	−	0.404015i	$-0.867614\pi$
$499$	−40.8680	−1.82950	−0.914752	+	0.404015i	$0.867614\pi$
$500$	0	0
$501$	23.0662	1.03052
$502$	−1.67543	−0.0747782
$503$	35.7640	1.59464	0.797320	−	0.603557i	$-0.206251\pi$
$503$	35.7640	1.59464	0.797320	+	0.603557i	$0.206251\pi$
$504$	5.31522	0.236759
$505$	0	0
$506$	−7.66204	−0.340619
$507$	−8.56962	−0.380590
$508$	3.90622	0.173310
$509$	20.6159	0.913785	0.456893	−	0.889522i	$-0.348962\pi$
$509$	20.6159	0.913785	0.456893	+	0.889522i	$0.348962\pi$
$510$	0	0
$511$	−37.8706	−1.67530
$512$	22.4816	0.993558
$513$	−1.55017	−0.0684416
$514$	−6.57546	−0.290031
$515$	0	0
$516$	−16.3350	−0.719109
$517$	−70.9536	−3.12054
$518$	2.98443	0.131128
$519$	−1.94915	−0.0855582
$520$	0	0
$521$	2.71963	0.119149	0.0595746	−	0.998224i	$-0.481026\pi$
$521$	2.71963	0.119149	0.0595746	+	0.998224i	$0.481026\pi$
$522$	1.04874	0.0459020
$523$	29.8932	1.30714	0.653568	−	0.756868i	$-0.273271\pi$
$523$	29.8932	1.30714	0.653568	+	0.756868i	$0.273271\pi$
$524$	17.7857	0.776972
$525$	0	0
$526$	8.81456	0.384333
$527$	0.950710	0.0414136
$528$	−14.4179	−0.627458
$529$	−15.6616	−0.680941
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	−7.66725	−0.332417
$533$	6.59542	0.285679
$534$	7.08685	0.306678
$535$	0	0
$536$	−5.44549	−0.235209
$537$	2.80645	0.121107
$538$	−9.90602	−0.427079
$539$	−5.64222	−0.243027
$540$	0	0
$541$	10.2324	0.439926	0.219963	−	0.975508i	$-0.429406\pi$
$541$	10.2324	0.439926	0.219963	+	0.975508i	$0.429406\pi$
$542$	15.2847	0.656534
$543$	17.9168	0.768883
$544$	−18.3547	−0.786949
$545$	0	0
$546$	2.98443	0.127722
$547$	−19.5084	−0.834118	−0.417059	−	0.908879i	$-0.636939\pi$
$547$	−19.5084	−0.834118	−0.417059	+	0.908879i	$0.636939\pi$
$548$	0.654599	0.0279631
$549$	−10.8544	−0.463256
$550$	0	0
$551$	−3.24302	−0.138157
$552$	−5.09068	−0.216674
$553$	−27.5180	−1.17018
$554$	14.9610	0.635633
$555$	0	0
$556$	−16.2778	−0.690333
$557$	3.90024	0.165258	0.0826292	−	0.996580i	$-0.473668\pi$
$557$	3.90024	0.165258	0.0826292	+	0.996580i	$0.473668\pi$
$558$	−0.130851	−0.00553937
$559$	−19.6618	−0.831607
$560$	0	0
$561$	20.5502	0.867628
$562$	15.0390	0.634383
$563$	−12.0190	−0.506539	−0.253269	−	0.967396i	$-0.581506\pi$
$563$	−12.0190	−0.506539	−0.253269	+	0.967396i	$0.581506\pi$
$564$	−21.9908	−0.925978
$565$	0	0
$566$	13.2663	0.557624
$567$	−2.82843	−0.118783
$568$	15.1929	0.637480
$569$	−41.4735	−1.73866	−0.869330	−	0.494233i	$-0.835449\pi$
$569$	−41.4735	−1.73866	−0.869330	+	0.494233i	$0.835449\pi$
$570$	0	0
$571$	−4.08499	−0.170951	−0.0854757	−	0.996340i	$-0.527241\pi$
$571$	−4.08499	−0.170951	−0.0854757	+	0.996340i	$0.527241\pi$
$572$	−20.7676	−0.868336
$573$	−11.9347	−0.498580
$574$	4.44286	0.185441
$575$	0	0
$576$	−2.58447	−0.107686
$577$	−31.2996	−1.30302	−0.651509	−	0.758641i	$-0.725864\pi$
$577$	−31.2996	−1.30302	−0.651509	+	0.758641i	$0.725864\pi$
$578$	−1.87198	−0.0778641
$579$	−1.55325	−0.0645507
$580$	0	0
$581$	−20.4523	−0.848506
$582$	−3.40044	−0.140953
$583$	−33.0362	−1.36822
$584$	−25.1613	−1.04118
$585$	0	0
$586$	10.0115	0.413570
$587$	2.92948	0.120913	0.0604563	−	0.998171i	$-0.480744\pi$
$587$	2.92948	0.120913	0.0604563	+	0.998171i	$0.480744\pi$
$588$	−1.74870	−0.0721152
$589$	0.404633	0.0166726
$590$	0	0
$591$	−5.74160	−0.236178
$592$	−5.37864	−0.221061
$593$	−11.0899	−0.455409	−0.227704	−	0.973730i	$-0.573122\pi$
$593$	−11.0899	−0.455409	−0.227704	+	0.973730i	$0.573122\pi$
$594$	−2.82843	−0.116052
$595$	0	0
$596$	10.3139	0.422475
$597$	−17.5261	−0.717294
$598$	−2.85835	−0.116887
$599$	18.6847	0.763438	0.381719	−	0.924278i	$-0.375332\pi$
$599$	18.6847	0.763438	0.381719	+	0.924278i	$0.375332\pi$
$600$	0	0
$601$	42.2644	1.72400	0.862000	−	0.506908i	$-0.169212\pi$
$601$	42.2644	1.72400	0.862000	+	0.506908i	$0.169212\pi$
$602$	−13.2448	−0.539816
$603$	2.89775	0.118005
$604$	14.1915	0.577444
$605$	0	0
$606$	−6.63372	−0.269476
$607$	−1.62103	−0.0657955	−0.0328978	−	0.999459i	$-0.510474\pi$
$607$	−1.62103	−0.0657955	−0.0328978	+	0.999459i	$0.510474\pi$
$608$	−7.81195	−0.316816
$609$	−5.91720	−0.239777
$610$	0	0
$611$	−26.4695	−1.07084
$612$	6.36915	0.257457
$613$	−40.4439	−1.63352	−0.816758	−	0.576981i	$-0.804231\pi$
$613$	−40.4439	−1.63352	−0.816758	+	0.576981i	$0.804231\pi$
$614$	−4.15479	−0.167674
$615$	0	0
$616$	−29.9896	−1.20832
$617$	48.5384	1.95408	0.977042	−	0.213047i	$-0.0683388\pi$
$617$	48.5384	1.95408	0.977042	+	0.213047i	$0.0683388\pi$
$618$	−7.71456	−0.310325
$619$	20.1317	0.809162	0.404581	−	0.914502i	$-0.367417\pi$
$619$	20.1317	0.809162	0.404581	+	0.914502i	$0.367417\pi$
$620$	0	0
$621$	2.70894	0.108706
$622$	6.79468	0.272442
$623$	−39.9855	−1.60199
$624$	−5.37864	−0.215318
$625$	0	0
$626$	0.776751	0.0310452
$627$	8.74638	0.349297
$628$	−23.6542	−0.943907
$629$	7.66630	0.305676
$630$	0	0
$631$	−10.0031	−0.398215	−0.199108	−	0.979978i	$-0.563804\pi$
$631$	−10.0031	−0.398215	−0.199108	+	0.979978i	$0.563804\pi$
$632$	−18.2830	−0.727259
$633$	−11.7563	−0.467270
$634$	−6.23567	−0.247650
$635$	0	0
$636$	−10.2390	−0.406002
$637$	−2.10485	−0.0833971
$638$	−5.91720	−0.234264
$639$	−8.08471	−0.319826
$640$	0	0
$641$	39.2361	1.54973	0.774866	−	0.632125i	$-0.217817\pi$
$641$	39.2361	1.54973	0.774866	+	0.632125i	$0.217817\pi$
$642$	−4.94917	−0.195328
$643$	32.9690	1.30017	0.650086	−	0.759861i	$-0.274733\pi$
$643$	32.9690	1.30017	0.650086	+	0.759861i	$0.274733\pi$
$644$	13.3986	0.527980
$645$	0	0
$646$	2.83035	0.111359
$647$	−31.1335	−1.22398	−0.611991	−	0.790865i	$-0.709631\pi$
$647$	−31.1335	−1.22398	−0.611991	+	0.790865i	$0.709631\pi$
$648$	−1.87921	−0.0738225
$649$	5.64222	0.221476
$650$	0	0
$651$	0.738291	0.0289359
$652$	0.520554	0.0203865
$653$	39.2971	1.53782	0.768908	−	0.639360i	$-0.220800\pi$
$653$	39.2971	1.53782	0.768908	+	0.639360i	$0.220800\pi$
$654$	6.23788	0.243920
$655$	0	0
$656$	−8.00707	−0.312624
$657$	13.3893	0.522365
$658$	−17.8306	−0.695108
$659$	−16.3227	−0.635842	−0.317921	−	0.948117i	$-0.602985\pi$
$659$	−16.3227	−0.635842	−0.317921	+	0.948117i	$0.602985\pi$
$660$	0	0
$661$	−39.5257	−1.53737	−0.768685	−	0.639628i	$-0.779089\pi$
$661$	−39.5257	−1.53737	−0.768685	+	0.639628i	$0.779089\pi$
$662$	−6.20216	−0.241054
$663$	7.66630	0.297735
$664$	−13.5886	−0.527339
$665$	0	0
$666$	−1.05515	−0.0408864
$667$	5.66723	0.219436
$668$	−40.3359	−1.56064
$669$	−1.27615	−0.0493387
$670$	0	0
$671$	61.2430	2.36426
$672$	−14.2536	−0.549846
$673$	−24.8712	−0.958716	−0.479358	−	0.877619i	$-0.659130\pi$
$673$	−24.8712	−0.958716	−0.479358	+	0.877619i	$0.659130\pi$
$674$	−9.02301	−0.347553
$675$	0	0
$676$	14.9857	0.576373
$677$	3.73774	0.143653	0.0718266	−	0.997417i	$-0.477117\pi$
$677$	3.73774	0.143653	0.0718266	+	0.997417i	$0.477117\pi$
$678$	6.73040	0.258480
$679$	19.1860	0.736291
$680$	0	0
$681$	0.327268	0.0125410
$682$	0.738291	0.0282706
$683$	6.51287	0.249208	0.124604	−	0.992207i	$-0.460234\pi$
$683$	6.51287	0.249208	0.124604	+	0.992207i	$0.460234\pi$
$684$	2.71078	0.103649
$685$	0	0
$686$	8.50730	0.324810
$687$	−14.4872	−0.552721
$688$	23.8702	0.910041
$689$	−12.3243	−0.469517
$690$	0	0
$691$	20.0701	0.763504	0.381752	−	0.924265i	$-0.375321\pi$
$691$	20.0701	0.763504	0.381752	+	0.924265i	$0.375321\pi$
$692$	3.40848	0.129571
$693$	15.9586	0.606217
$694$	10.2382	0.388636
$695$	0	0
$696$	−3.93140	−0.149019
$697$	11.4127	0.432286
$698$	9.39501	0.355606
$699$	−20.8956	−0.790345
$700$	0	0
$701$	40.1442	1.51622	0.758112	−	0.652124i	$-0.226122\pi$
$701$	40.1442	1.51622	0.758112	+	0.652124i	$0.226122\pi$
$702$	−1.05515	−0.0398242
$703$	3.26287	0.123061
$704$	14.5821	0.549584
$705$	0	0
$706$	2.02208	0.0761018
$707$	37.4289	1.40766
$708$	1.74870	0.0657202
$709$	−17.7064	−0.664978	−0.332489	−	0.943107i	$-0.607888\pi$
$709$	−17.7064	−0.664978	−0.332489	+	0.943107i	$0.607888\pi$
$710$	0	0
$711$	9.72907	0.364869
$712$	−26.5665	−0.995621
$713$	−0.707102	−0.0264812
$714$	5.16423	0.193267
$715$	0	0
$716$	−4.90764	−0.183407
$717$	21.2232	0.792593
$718$	7.77509	0.290164
$719$	−3.03887	−0.113331	−0.0566654	−	0.998393i	$-0.518047\pi$
$719$	−3.03887	−0.113331	−0.0566654	+	0.998393i	$0.518047\pi$
$720$	0	0
$721$	43.5272	1.62104
$722$	−8.32002	−0.309639
$723$	4.73427	0.176069
$724$	−31.3311	−1.16441
$725$	0	0
$726$	10.4443	0.387625
$727$	38.8352	1.44032	0.720159	−	0.693809i	$-0.244069\pi$
$727$	38.8352	1.44032	0.720159	+	0.693809i	$0.244069\pi$
$728$	−11.1877	−0.414645
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.0227	−1.25838
$732$	18.9811	0.701563
$733$	−18.6139	−0.687519	−0.343759	−	0.939058i	$-0.611700\pi$
$733$	−18.6139	−0.687519	−0.343759	+	0.939058i	$0.611700\pi$
$734$	0.669070	0.0246958
$735$	0	0
$736$	13.6515	0.503201
$737$	−16.3497	−0.602250
$738$	−1.57079	−0.0578215
$739$	18.8043	0.691727	0.345863	−	0.938285i	$-0.387586\pi$
$739$	18.8043	0.691727	0.345863	+	0.938285i	$0.387586\pi$
$740$	0	0
$741$	3.26287	0.119864
$742$	−8.30197	−0.304775
$743$	31.5390	1.15705	0.578527	−	0.815663i	$-0.303628\pi$
$743$	31.5390	1.15705	0.578527	+	0.815663i	$0.303628\pi$
$744$	0.490522	0.0179834
$745$	0	0
$746$	8.29549	0.303720
$747$	7.23099	0.264568
$748$	−35.9361	−1.31395
$749$	27.9243	1.02033
$750$	0	0
$751$	−30.0452	−1.09637	−0.548183	−	0.836358i	$-0.684680\pi$
$751$	−30.0452	−1.09637	−0.548183	+	0.836358i	$0.684680\pi$
$752$	32.1349	1.17184
$753$	−3.34219	−0.121796
$754$	−2.20743	−0.0803899
$755$	0	0
$756$	4.94607	0.179887
$757$	−35.5795	−1.29316	−0.646579	−	0.762847i	$-0.723801\pi$
$757$	−35.5795	−1.29316	−0.646579	+	0.762847i	$0.723801\pi$
$758$	−0.724199	−0.0263041
$759$	−15.2844	−0.554790
$760$	0	0
$761$	−28.0751	−1.01772	−0.508861	−	0.860849i	$-0.669933\pi$
$761$	−28.0751	−1.01772	−0.508861	+	0.860849i	$0.669933\pi$
$762$	−1.11979	−0.0405657
$763$	−35.1955	−1.27416
$764$	20.8702	0.755059
$765$	0	0
$766$	−8.65654	−0.312774
$767$	2.10485	0.0760016
$768$	−0.533037	−0.0192343
$769$	−39.6547	−1.42999	−0.714993	−	0.699131i	$-0.753570\pi$
$769$	−39.6547	−1.42999	−0.714993	+	0.699131i	$0.753570\pi$
$770$	0	0
$771$	−13.1169	−0.472393
$772$	2.71616	0.0977568
$773$	3.68323	0.132477	0.0662384	−	0.997804i	$-0.478900\pi$
$773$	3.68323	0.132477	0.0662384	+	0.997804i	$0.478900\pi$
$774$	4.68273	0.168317
$775$	0	0
$776$	12.7472	0.457598
$777$	5.95341	0.213577
$778$	−7.70558	−0.276259
$779$	4.85737	0.174033
$780$	0	0
$781$	45.6157	1.63226
$782$	−4.94607	−0.176871
$783$	2.09205	0.0747637
$784$	2.55536	0.0912628
$785$	0	0
$786$	−5.09860	−0.181861
$787$	−6.70495	−0.239006	−0.119503	−	0.992834i	$-0.538130\pi$
$787$	−6.70495	−0.239006	−0.119503	+	0.992834i	$0.538130\pi$
$788$	10.0403	0.357672
$789$	17.5835	0.625989
$790$	0	0
$791$	−37.9744	−1.35021
$792$	10.6029	0.376759
$793$	22.8469	0.811317
$794$	−14.1900	−0.503583
$795$	0	0
$796$	30.6478	1.08628
$797$	−52.6525	−1.86505	−0.932523	−	0.361111i	$-0.882398\pi$
$797$	−52.6525	−1.86505	−0.932523	+	0.361111i	$0.882398\pi$
$798$	2.19796	0.0778068
$799$	−45.8026	−1.62038
$800$	0	0
$801$	14.1370	0.499507
$802$	19.9474	0.704368
$803$	−75.5452	−2.66593
$804$	−5.06729	−0.178710
$805$	0	0
$806$	0.275422	0.00970132
$807$	−19.7608	−0.695612
$808$	24.8678	0.874847
$809$	21.4721	0.754918	0.377459	−	0.926026i	$-0.376798\pi$
$809$	21.4721	0.754918	0.377459	+	0.926026i	$0.376798\pi$
$810$	0	0
$811$	45.2520	1.58901	0.794506	−	0.607256i	$-0.207730\pi$
$811$	45.2520	1.58901	0.794506	+	0.607256i	$0.207730\pi$
$812$	10.3474	0.363123
$813$	30.4903	1.06934
$814$	5.95341	0.208667
$815$	0	0
$816$	−9.30716	−0.325816
$817$	−14.4805	−0.506607
$818$	11.3947	0.398407
$819$	5.95341	0.208029
$820$	0	0
$821$	22.2169	0.775376	0.387688	−	0.921791i	$-0.373274\pi$
$821$	22.2169	0.775376	0.387688	+	0.921791i	$0.373274\pi$
$822$	−0.187653	−0.00654515
$823$	5.59119	0.194897	0.0974484	−	0.995241i	$-0.468932\pi$
$823$	5.59119	0.194897	0.0974484	+	0.995241i	$0.468932\pi$
$824$	28.9196	1.00746
$825$	0	0
$826$	1.41788	0.0493345
$827$	−21.1945	−0.737006	−0.368503	−	0.929627i	$-0.620130\pi$
$827$	−21.1945	−0.737006	−0.368503	+	0.929627i	$0.620130\pi$
$828$	−4.73713	−0.164627
$829$	−22.6576	−0.786931	−0.393466	−	0.919339i	$-0.628724\pi$
$829$	−22.6576	−0.786931	−0.393466	+	0.919339i	$0.628724\pi$
$830$	0	0
$831$	29.8446	1.03530
$832$	5.43991	0.188595
$833$	−3.64222	−0.126195
$834$	4.66634	0.161582
$835$	0	0
$836$	−15.2948	−0.528982
$837$	−0.261025	−0.00902235
$838$	15.7276	0.543301
$839$	48.2991	1.66747	0.833734	−	0.552166i	$-0.186198\pi$
$839$	48.2991	1.66747	0.833734	+	0.552166i	$0.186198\pi$
$840$	0	0
$841$	−24.6233	−0.849081
$842$	−18.9089	−0.651642
$843$	30.0002	1.03326
$844$	20.5582	0.707643
$845$	0	0
$846$	6.30406	0.216738
$847$	−58.9291	−2.02483
$848$	14.9621	0.513800
$849$	26.4639	0.908240
$850$	0	0
$851$	−5.70191	−0.195459
$852$	14.1377	0.484351
$853$	−9.70733	−0.332373	−0.166186	−	0.986094i	$-0.553145\pi$
$853$	−9.70733	−0.332373	−0.166186	+	0.986094i	$0.553145\pi$
$854$	15.3903	0.526645
$855$	0	0
$856$	18.5529	0.634127
$857$	−23.2631	−0.794652	−0.397326	−	0.917678i	$-0.630062\pi$
$857$	−23.2631	−0.794652	−0.397326	+	0.917678i	$0.630062\pi$
$858$	5.95341	0.203246
$859$	−5.09966	−0.173998	−0.0869990	−	0.996208i	$-0.527728\pi$
$859$	−5.09966	−0.173998	−0.0869990	+	0.996208i	$0.527728\pi$
$860$	0	0
$861$	8.86272	0.302041
$862$	−4.90195	−0.166961
$863$	18.6706	0.635556	0.317778	−	0.948165i	$-0.397063\pi$
$863$	18.6706	0.635556	0.317778	+	0.948165i	$0.397063\pi$
$864$	5.03942	0.171445
$865$	0	0
$866$	8.29952	0.282029
$867$	−3.73427	−0.126822
$868$	−1.29105	−0.0438211
$869$	−54.8935	−1.86214
$870$	0	0
$871$	−6.09931	−0.206667
$872$	−23.3839	−0.791880
$873$	−6.78327	−0.229579
$874$	−2.10511	−0.0712062
$875$	0	0
$876$	−23.4138	−0.791080
$877$	−21.2540	−0.717697	−0.358849	−	0.933396i	$-0.616831\pi$
$877$	−21.2540	−0.717697	−0.358849	+	0.933396i	$0.616831\pi$
$878$	−5.13886	−0.173428
$879$	19.9711	0.673609
$880$	0	0
$881$	18.7433	0.631479	0.315740	−	0.948846i	$-0.397747\pi$
$881$	18.7433	0.631479	0.315740	+	0.948846i	$0.397747\pi$
$882$	0.501297	0.0168796
$883$	−3.70264	−0.124604	−0.0623019	−	0.998057i	$-0.519844\pi$
$883$	−3.70264	−0.124604	−0.0623019	+	0.998057i	$0.519844\pi$
$884$	−13.4061	−0.450895
$885$	0	0
$886$	−2.32527	−0.0781188
$887$	−20.0525	−0.673296	−0.336648	−	0.941631i	$-0.609293\pi$
$887$	−20.0525	−0.673296	−0.336648	+	0.941631i	$0.609293\pi$
$888$	3.95546	0.132736
$889$	6.31809	0.211902
$890$	0	0
$891$	−5.64222	−0.189021
$892$	2.23160	0.0747195
$893$	−19.4941	−0.652346
$894$	−2.95668	−0.0988861
$895$	0	0
$896$	32.1717	1.07478
$897$	−5.70191	−0.190381
$898$	2.53268	0.0845165
$899$	−0.546077	−0.0182127
$900$	0	0
$901$	−21.3259	−0.710467
$902$	8.86272	0.295096
$903$	−26.4210	−0.879235
$904$	−25.2303	−0.839146
$905$	0	0
$906$	−4.06826	−0.135159
$907$	37.8138	1.25559	0.627794	−	0.778380i	$-0.283958\pi$
$907$	37.8138	1.25559	0.627794	+	0.778380i	$0.283958\pi$
$908$	−0.572294	−0.0189923
$909$	−13.2331	−0.438914
$910$	0	0
$911$	3.63004	0.120269	0.0601343	−	0.998190i	$-0.480847\pi$
$911$	3.63004	0.120269	0.0601343	+	0.998190i	$0.480847\pi$
$912$	−3.96123	−0.131170
$913$	−40.7988	−1.35024
$914$	−11.5529	−0.382136
$915$	0	0
$916$	25.3338	0.837051
$917$	28.7674	0.949983
$918$	−1.82583	−0.0602615
$919$	19.5965	0.646429	0.323214	−	0.946326i	$-0.395236\pi$
$919$	19.5965	0.646429	0.323214	+	0.946326i	$0.395236\pi$
$920$	0	0
$921$	−8.28808	−0.273102
$922$	9.75024	0.321107
$923$	17.0171	0.560124
$924$	−27.9068	−0.918067
$925$	0	0
$926$	1.00222	0.0329350
$927$	−15.3892	−0.505448
$928$	10.5427	0.346081
$929$	−7.08332	−0.232396	−0.116198	−	0.993226i	$-0.537071\pi$
$929$	−7.08332	−0.232396	−0.116198	+	0.993226i	$0.537071\pi$
$930$	0	0
$931$	−1.55017	−0.0508047
$932$	36.5402	1.19691
$933$	13.5542	0.443744
$934$	3.75403	0.122836
$935$	0	0
$936$	3.95546	0.129288
$937$	49.3039	1.61069	0.805345	−	0.592807i	$-0.201980\pi$
$937$	49.3039	1.61069	0.805345	+	0.592807i	$0.201980\pi$
$938$	−4.10867	−0.134153
$939$	1.54948	0.0505655
$940$	0	0
$941$	−4.00877	−0.130682	−0.0653411	−	0.997863i	$-0.520814\pi$
$941$	−4.00877	−0.130682	−0.0653411	+	0.997863i	$0.520814\pi$
$942$	6.78092	0.220934
$943$	−8.48832	−0.276418
$944$	−2.55536	−0.0831698
$945$	0	0
$946$	−26.4210	−0.859020
$947$	19.7223	0.640888	0.320444	−	0.947267i	$-0.396168\pi$
$947$	19.7223	0.640888	0.320444	+	0.947267i	$0.396168\pi$
$948$	−17.0132	−0.552564
$949$	−28.1824	−0.914839
$950$	0	0
$951$	−12.4391	−0.403364
$952$	−19.3592	−0.627434
$953$	−1.68536	−0.0545941	−0.0272970	−	0.999627i	$-0.508690\pi$
$953$	−1.68536	−0.0545941	−0.0272970	+	0.999627i	$0.508690\pi$
$954$	2.93519	0.0950303
$955$	0	0
$956$	−37.1130	−1.20032
$957$	−11.8038	−0.381562
$958$	3.19804	0.103324
$959$	1.05878	0.0341897
$960$	0	0
$961$	−30.9319	−0.997802
$962$	2.22094	0.0716059
$963$	−9.87272	−0.318144
$964$	−8.27882	−0.266643
$965$	0	0
$966$	−3.84096	−0.123581
$967$	57.2181	1.84001	0.920006	−	0.391905i	$-0.128184\pi$
$967$	57.2181	1.84001	0.920006	+	0.391905i	$0.128184\pi$
$968$	−39.1526	−1.25841
$969$	5.64605	0.181377
$970$	0	0
$971$	20.9172	0.671264	0.335632	−	0.941993i	$-0.391050\pi$
$971$	20.9172	0.671264	0.335632	+	0.941993i	$0.391050\pi$
$972$	−1.74870	−0.0560896
$973$	−26.3285	−0.844052
$974$	7.84381	0.251332
$975$	0	0
$976$	−27.7369	−0.887838
$977$	7.14921	0.228723	0.114362	−	0.993439i	$-0.463518\pi$
$977$	7.14921	0.228723	0.114362	+	0.993439i	$0.463518\pi$
$978$	−0.149226	−0.00477173
$979$	−79.7641	−2.54927
$980$	0	0
$981$	12.4435	0.397290
$982$	12.9904	0.414541
$983$	−41.2064	−1.31428	−0.657140	−	0.753769i	$-0.728234\pi$
$983$	−41.2064	−1.31428	−0.657140	+	0.753769i	$0.728234\pi$
$984$	5.88841	0.187716
$985$	0	0
$986$	−3.81973	−0.121645
$987$	−35.5689	−1.13217
$988$	−5.70578	−0.181525
$989$	25.3048	0.804647
$990$	0	0
$991$	13.3978	0.425595	0.212797	−	0.977096i	$-0.431743\pi$
$991$	13.3978	0.425595	0.212797	+	0.977096i	$0.431743\pi$
$992$	−1.31542	−0.0417645
$993$	−12.3722	−0.392621
$994$	11.4632	0.363590
$995$	0	0
$996$	−12.6448	−0.400667
$997$	−52.4897	−1.66237	−0.831183	−	0.555999i	$-0.812336\pi$
$997$	−52.4897	−1.66237	−0.831183	+	0.555999i	$0.812336\pi$
$998$	−20.4870	−0.648506
$999$	−2.10485	−0.0665944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4425.2.a.bg.1.3		6
5.4	even	2		885.2.a.j.1.4	✓	6
15.14	odd	2		2655.2.a.v.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
885.2.a.j.1.4	✓	6	5.4	even	2
2655.2.a.v.1.3		6	15.14	odd	2
4425.2.a.bg.1.3		6	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:	\( N \)	\(=\)	\( 4425 = 3 \cdot 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4425.a (trivial)

\(f(q)\)	\(=\)	\(q+0.501297 q^{2} +1.00000 q^{3} -1.74870 q^{4} +0.501297 q^{6} -2.82843 q^{7} -1.87921 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.501297 q^{2} +1.00000 q^{3} -1.74870 q^{4} +0.501297 q^{6} -2.82843 q^{7} -1.87921 q^{8} +1.00000 q^{9} -5.64222 q^{11} -1.74870 q^{12} -2.10485 q^{13} -1.41788 q^{14} +2.55536 q^{16} -3.64222 q^{17} +0.501297 q^{18} -1.55017 q^{19} -2.82843 q^{21} -2.82843 q^{22} +2.70894 q^{23} -1.87921 q^{24} -1.05515 q^{26} +1.00000 q^{27} +4.94607 q^{28} +2.09205 q^{29} -0.261025 q^{31} +5.03942 q^{32} -5.64222 q^{33} -1.82583 q^{34} -1.74870 q^{36} -2.10485 q^{37} -0.777095 q^{38} -2.10485 q^{39} -3.13345 q^{41} -1.41788 q^{42} +9.34122 q^{43} +9.86655 q^{44} +1.35798 q^{46} +12.5755 q^{47} +2.55536 q^{48} +1.00000 q^{49} -3.64222 q^{51} +3.68075 q^{52} +5.85519 q^{53} +0.501297 q^{54} +5.31522 q^{56} -1.55017 q^{57} +1.04874 q^{58} -1.00000 q^{59} -10.8544 q^{61} -0.130851 q^{62} -2.82843 q^{63} -2.58447 q^{64} -2.82843 q^{66} +2.89775 q^{67} +6.36915 q^{68} +2.70894 q^{69} -8.08471 q^{71} -1.87921 q^{72} +13.3893 q^{73} -1.05515 q^{74} +2.71078 q^{76} +15.9586 q^{77} -1.05515 q^{78} +9.72907 q^{79} +1.00000 q^{81} -1.57079 q^{82} +7.23099 q^{83} +4.94607 q^{84} +4.68273 q^{86} +2.09205 q^{87} +10.6029 q^{88} +14.1370 q^{89} +5.95341 q^{91} -4.73713 q^{92} -0.261025 q^{93} +6.30406 q^{94} +5.03942 q^{96} -6.78327 q^{97} +0.501297 q^{98} -5.64222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9	\( 6 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 4 q^{6} + 12 q^{8} + 6 q^{9} - 2 q^{11} + 8 q^{12} - 2 q^{13} + 12 q^{16} + 10 q^{17} + 4 q^{18} - 2 q^{19} + 12 q^{23} + 12 q^{24} - 8 q^{26} + 6 q^{27} + 16 q^{28} - 12 q^{29} + 8 q^{31} + 28 q^{32} - 2 q^{33} + 8 q^{34} + 8 q^{36} - 2 q^{37} + 24 q^{38} - 2 q^{39} - 4 q^{41} + 18 q^{43} + 4 q^{44} + 16 q^{47} + 12 q^{48} + 6 q^{49} + 10 q^{51} - 12 q^{52} + 30 q^{53} + 4 q^{54} - 2 q^{57} + 16 q^{58} - 6 q^{59} + 4 q^{61} + 16 q^{62} + 20 q^{64} + 30 q^{67} + 20 q^{68} + 12 q^{69} - 24 q^{71} + 12 q^{72} + 6 q^{73} - 8 q^{74} - 4 q^{76} + 16 q^{77} - 8 q^{78} - 2 q^{79} + 6 q^{81} + 20 q^{83} + 16 q^{84} - 12 q^{87} + 16 q^{88} + 14 q^{89} - 48 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 28 q^{96} - 4 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 4 * q^6 + 12 * q^8 + 6 * q^9 - 2 * q^11 + 8 * q^12 - 2 * q^13 + 12 * q^16 + 10 * q^17 + 4 * q^18 - 2 * q^19 + 12 * q^23 + 12 * q^24 - 8 * q^26 + 6 * q^27 + 16 * q^28 - 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 8 * q^34 + 8 * q^36 - 2 * q^37 + 24 * q^38 - 2 * q^39 - 4 * q^41 + 18 * q^43 + 4 * q^44 + 16 * q^47 + 12 * q^48 + 6 * q^49 + 10 * q^51 - 12 * q^52 + 30 * q^53 + 4 * q^54 - 2 * q^57 + 16 * q^58 - 6 * q^59 + 4 * q^61 + 16 * q^62 + 20 * q^64 + 30 * q^67 + 20 * q^68 + 12 * q^69 - 24 * q^71 + 12 * q^72 + 6 * q^73 - 8 * q^74 - 4 * q^76 + 16 * q^77 - 8 * q^78 - 2 * q^79 + 6 * q^81 + 20 * q^83 + 16 * q^84 - 12 * q^87 + 16 * q^88 + 14 * q^89 - 48 * q^91 + 16 * q^92 + 8 * q^93 + 16 * q^94 + 28 * q^96 - 4 * q^97 + 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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