LMFDB - Embedded newform 4005.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.23607 q^{7} -3.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.23607 q^{7} -3.00000 q^{8} +1.00000 q^{10} +6.47214 q^{11} +6.47214 q^{13} +3.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.76393 q^{19} -1.00000 q^{20} +6.47214 q^{22} +0.763932 q^{23} +1.00000 q^{25} +6.47214 q^{26} -3.23607 q^{28} -4.47214 q^{29} -1.23607 q^{31} +5.00000 q^{32} +2.00000 q^{34} +3.23607 q^{35} -6.47214 q^{37} -2.76393 q^{38} -3.00000 q^{40} +10.0000 q^{41} +0.763932 q^{43} -6.47214 q^{44} +0.763932 q^{46} -8.00000 q^{47} +3.47214 q^{49} +1.00000 q^{50} -6.47214 q^{52} -2.00000 q^{53} +6.47214 q^{55} -9.70820 q^{56} -4.47214 q^{58} -1.23607 q^{59} -8.47214 q^{61} -1.23607 q^{62} +7.00000 q^{64} +6.47214 q^{65} -2.47214 q^{67} -2.00000 q^{68} +3.23607 q^{70} +5.52786 q^{71} +2.00000 q^{73} -6.47214 q^{74} +2.76393 q^{76} +20.9443 q^{77} -4.94427 q^{79} -1.00000 q^{80} +10.0000 q^{82} +15.2361 q^{83} +2.00000 q^{85} +0.763932 q^{86} -19.4164 q^{88} +1.00000 q^{89} +20.9443 q^{91} -0.763932 q^{92} -8.00000 q^{94} -2.76393 q^{95} -6.00000 q^{97} +3.47214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 4 q^{11} + 4 q^{13} + 2 q^{14} - 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 2 q^{31} + 10 q^{32} + 4 q^{34} + 2 q^{35} - 4 q^{37} - 10 q^{38} - 6 q^{40} + 20 q^{41} + 6 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} - 2 q^{49} + 2 q^{50} - 4 q^{52} - 4 q^{53} + 4 q^{55} - 6 q^{56} + 2 q^{59} - 8 q^{61} + 2 q^{62} + 14 q^{64} + 4 q^{65} + 4 q^{67} - 4 q^{68} + 2 q^{70} + 20 q^{71} + 4 q^{73} - 4 q^{74} + 10 q^{76} + 24 q^{77} + 8 q^{79} - 2 q^{80} + 20 q^{82} + 26 q^{83} + 4 q^{85} + 6 q^{86} - 12 q^{88} + 2 q^{89} + 24 q^{91} - 6 q^{92} - 16 q^{94} - 10 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 + 2 * q^10 + 4 * q^11 + 4 * q^13 + 2 * q^14 - 2 * q^16 + 4 * q^17 - 10 * q^19 - 2 * q^20 + 4 * q^22 + 6 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 2 * q^31 + 10 * q^32 + 4 * q^34 + 2 * q^35 - 4 * q^37 - 10 * q^38 - 6 * q^40 + 20 * q^41 + 6 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 - 2 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 + 4 * q^55 - 6 * q^56 + 2 * q^59 - 8 * q^61 + 2 * q^62 + 14 * q^64 + 4 * q^65 + 4 * q^67 - 4 * q^68 + 2 * q^70 + 20 * q^71 + 4 * q^73 - 4 * q^74 + 10 * q^76 + 24 * q^77 + 8 * q^79 - 2 * q^80 + 20 * q^82 + 26 * q^83 + 4 * q^85 + 6 * q^86 - 12 * q^88 + 2 * q^89 + 24 * q^91 - 6 * q^92 - 16 * q^94 - 10 * q^95 - 12 * q^97 - 2 * q^98

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	6.47214	1.95142	0.975711	−	0.219061i	$-0.0702993\pi$
$11$	6.47214	1.95142	0.975711	+	0.219061i	$0.0702993\pi$
$12$	0	0
$13$	6.47214	1.79505	0.897524	−	0.440966i	$-0.145364\pi$
$13$	6.47214	1.79505	0.897524	+	0.440966i	$0.145364\pi$
$14$	3.23607	0.864876
$15$	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	$0.422021\pi$
$18$	0	0
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	6.47214	1.37986
$23$	0.763932	0.159291	0.0796454	−	0.996823i	$-0.474621\pi$
$23$	0.763932	0.159291	0.0796454	+	0.996823i	$0.474621\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.47214	1.26929
$27$	0	0
$28$	−3.23607	−0.611559
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	2.00000	0.342997
$35$	3.23607	0.546995
$36$	0	0
$37$	−6.47214	−1.06401	−0.532006	−	0.846740i	$-0.678562\pi$
$37$	−6.47214	−1.06401	−0.532006	+	0.846740i	$0.678562\pi$
$38$	−2.76393	−0.448369
$39$	0	0
$40$	−3.00000	−0.474342
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	0.763932	0.116499	0.0582493	−	0.998302i	$-0.481448\pi$
$43$	0.763932	0.116499	0.0582493	+	0.998302i	$0.481448\pi$
$44$	−6.47214	−0.975711
$45$	0	0
$46$	0.763932	0.112636
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	1.00000	0.141421
$51$	0	0
$52$	−6.47214	−0.897524
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	−9.70820	−1.29731
$57$	0	0
$58$	−4.47214	−0.587220
$59$	−1.23607	−0.160922	−0.0804612	−	0.996758i	$-0.525639\pi$
$59$	−1.23607	−0.160922	−0.0804612	+	0.996758i	$0.525639\pi$
$60$	0	0
$61$	−8.47214	−1.08475	−0.542373	−	0.840138i	$-0.682474\pi$
$61$	−8.47214	−1.08475	−0.542373	+	0.840138i	$0.682474\pi$
$62$	−1.23607	−0.156981
$63$	0	0
$64$	7.00000	0.875000
$65$	6.47214	0.802770
$66$	0	0
$67$	−2.47214	−0.302019	−0.151010	−	0.988532i	$-0.548252\pi$
$67$	−2.47214	−0.302019	−0.151010	+	0.988532i	$0.548252\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	3.23607	0.386784
$71$	5.52786	0.656037	0.328018	−	0.944671i	$-0.393619\pi$
$71$	5.52786	0.656037	0.328018	+	0.944671i	$0.393619\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−6.47214	−0.752371
$75$	0	0
$76$	2.76393	0.317045
$77$	20.9443	2.38682
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	10.0000	1.10432
$83$	15.2361	1.67238	0.836188	−	0.548443i	$-0.184779\pi$
$83$	15.2361	1.67238	0.836188	+	0.548443i	$0.184779\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0.763932	0.0823769
$87$	0	0
$88$	−19.4164	−2.06980
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	20.9443	2.19556
$92$	−0.763932	−0.0796454
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−2.76393	−0.283573
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	3.47214	0.350739
$99$	0	0
$100$	−1.00000	−0.100000
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	−14.6525	−1.44375	−0.721876	−	0.692023i	$-0.756720\pi$
$103$	−14.6525	−1.44375	−0.721876	+	0.692023i	$0.756720\pi$
$104$	−19.4164	−1.90394
$105$	0	0
$106$	−2.00000	−0.194257
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	−8.47214	−0.811483	−0.405742	−	0.913988i	$-0.632987\pi$
$109$	−8.47214	−0.811483	−0.405742	+	0.913988i	$0.632987\pi$
$110$	6.47214	0.617094
$111$	0	0
$112$	−3.23607	−0.305780
$113$	−3.05573	−0.287459	−0.143729	−	0.989617i	$-0.545909\pi$
$113$	−3.05573	−0.287459	−0.143729	+	0.989617i	$0.545909\pi$
$114$	0	0
$115$	0.763932	0.0712370
$116$	4.47214	0.415227
$117$	0	0
$118$	−1.23607	−0.113789
$119$	6.47214	0.593300
$120$	0	0
$121$	30.8885	2.80805
$122$	−8.47214	−0.767031
$123$	0	0
$124$	1.23607	0.111002
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.763932	0.0677880	0.0338940	−	0.999425i	$-0.489209\pi$
$127$	0.763932	0.0677880	0.0338940	+	0.999425i	$0.489209\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	6.47214	0.567644
$131$	9.52786	0.832453	0.416227	−	0.909261i	$-0.363352\pi$
$131$	9.52786	0.832453	0.416227	+	0.909261i	$0.363352\pi$
$132$	0	0
$133$	−8.94427	−0.775567
$134$	−2.47214	−0.213560
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	6.47214	0.548959	0.274480	−	0.961593i	$-0.411494\pi$
$139$	6.47214	0.548959	0.274480	+	0.961593i	$0.411494\pi$
$140$	−3.23607	−0.273498
$141$	0	0
$142$	5.52786	0.463888
$143$	41.8885	3.50290
$144$	0	0
$145$	−4.47214	−0.371391
$146$	2.00000	0.165521
$147$	0	0
$148$	6.47214	0.532006
$149$	9.41641	0.771422	0.385711	−	0.922620i	$-0.373956\pi$
$149$	9.41641	0.771422	0.385711	+	0.922620i	$0.373956\pi$
$150$	0	0
$151$	−5.23607	−0.426105	−0.213053	−	0.977041i	$-0.568341\pi$
$151$	−5.23607	−0.426105	−0.213053	+	0.977041i	$0.568341\pi$
$152$	8.29180	0.672553
$153$	0	0
$154$	20.9443	1.68774
$155$	−1.23607	−0.0992834
$156$	0	0
$157$	−10.9443	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$158$	−4.94427	−0.393345
$159$	0	0
$160$	5.00000	0.395285
$161$	2.47214	0.194832
$162$	0	0
$163$	16.1803	1.26734	0.633671	−	0.773603i	$-0.281547\pi$
$163$	16.1803	1.26734	0.633671	+	0.773603i	$0.281547\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	15.2361	1.18255
$167$	−11.4164	−0.883428	−0.441714	−	0.897156i	$-0.645629\pi$
$167$	−11.4164	−0.883428	−0.441714	+	0.897156i	$0.645629\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	2.00000	0.153393
$171$	0	0
$172$	−0.763932	−0.0582493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	3.23607	0.244624
$176$	−6.47214	−0.487856
$177$	0	0
$178$	1.00000	0.0749532
$179$	1.52786	0.114198	0.0570990	−	0.998369i	$-0.481815\pi$
$179$	1.52786	0.114198	0.0570990	+	0.998369i	$0.481815\pi$
$180$	0	0
$181$	−12.4721	−0.927047	−0.463523	−	0.886085i	$-0.653415\pi$
$181$	−12.4721	−0.927047	−0.463523	+	0.886085i	$0.653415\pi$
$182$	20.9443	1.55249
$183$	0	0
$184$	−2.29180	−0.168953
$185$	−6.47214	−0.475841
$186$	0	0
$187$	12.9443	0.946579
$188$	8.00000	0.583460
$189$	0	0
$190$	−2.76393	−0.200517
$191$	8.29180	0.599973	0.299987	−	0.953943i	$-0.403018\pi$
$191$	8.29180	0.599973	0.299987	+	0.953943i	$0.403018\pi$
$192$	0	0
$193$	4.94427	0.355896	0.177948	−	0.984040i	$-0.443054\pi$
$193$	4.94427	0.355896	0.177948	+	0.984040i	$0.443054\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−3.47214	−0.248010
$197$	6.47214	0.461121	0.230560	−	0.973058i	$-0.425944\pi$
$197$	6.47214	0.461121	0.230560	+	0.973058i	$0.425944\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	4.47214	0.314658
$203$	−14.4721	−1.01574
$204$	0	0
$205$	10.0000	0.698430
$206$	−14.6525	−1.02089
$207$	0	0
$208$	−6.47214	−0.448762
$209$	−17.8885	−1.23738
$210$	0	0
$211$	−18.1803	−1.25159	−0.625793	−	0.779989i	$-0.715225\pi$
$211$	−18.1803	−1.25159	−0.625793	+	0.779989i	$0.715225\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	8.94427	0.611418
$215$	0.763932	0.0520997
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−8.47214	−0.573805
$219$	0	0
$220$	−6.47214	−0.436351
$221$	12.9443	0.870726
$222$	0	0
$223$	12.9443	0.866813	0.433406	−	0.901199i	$-0.357312\pi$
$223$	12.9443	0.866813	0.433406	+	0.901199i	$0.357312\pi$
$224$	16.1803	1.08109
$225$	0	0
$226$	−3.05573	−0.203264
$227$	−7.41641	−0.492244	−0.246122	−	0.969239i	$-0.579156\pi$
$227$	−7.41641	−0.492244	−0.246122	+	0.969239i	$0.579156\pi$
$228$	0	0
$229$	7.52786	0.497455	0.248728	−	0.968573i	$-0.419988\pi$
$229$	7.52786	0.497455	0.248728	+	0.968573i	$0.419988\pi$
$230$	0.763932	0.0503722
$231$	0	0
$232$	13.4164	0.880830
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	1.23607	0.0804612
$237$	0	0
$238$	6.47214	0.419526
$239$	−8.29180	−0.536352	−0.268176	−	0.963370i	$-0.586421\pi$
$239$	−8.29180	−0.536352	−0.268176	+	0.963370i	$0.586421\pi$
$240$	0	0
$241$	18.9443	1.22031	0.610154	−	0.792283i	$-0.291108\pi$
$241$	18.9443	1.22031	0.610154	+	0.792283i	$0.291108\pi$
$242$	30.8885	1.98559
$243$	0	0
$244$	8.47214	0.542373
$245$	3.47214	0.221827
$246$	0	0
$247$	−17.8885	−1.13822
$248$	3.70820	0.235471
$249$	0	0
$250$	1.00000	0.0632456
$251$	17.5279	1.10635	0.553174	−	0.833065i	$-0.313416\pi$
$251$	17.5279	1.10635	0.553174	+	0.833065i	$0.313416\pi$
$252$	0	0
$253$	4.94427	0.310844
$254$	0.763932	0.0479334
$255$	0	0
$256$	−17.0000	−1.06250
$257$	22.9443	1.43122	0.715612	−	0.698498i	$-0.246148\pi$
$257$	22.9443	1.43122	0.715612	+	0.698498i	$0.246148\pi$
$258$	0	0
$259$	−20.9443	−1.30141
$260$	−6.47214	−0.401385
$261$	0	0
$262$	9.52786	0.588633
$263$	−30.8328	−1.90123	−0.950616	−	0.310368i	$-0.899548\pi$
$263$	−30.8328	−1.90123	−0.950616	+	0.310368i	$0.899548\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	−8.94427	−0.548408
$267$	0	0
$268$	2.47214	0.151010
$269$	−25.4164	−1.54967	−0.774833	−	0.632166i	$-0.782166\pi$
$269$	−25.4164	−1.54967	−0.774833	+	0.632166i	$0.782166\pi$
$270$	0	0
$271$	−26.4721	−1.60807	−0.804034	−	0.594583i	$-0.797317\pi$
$271$	−26.4721	−1.60807	−0.804034	+	0.594583i	$0.797317\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−8.00000	−0.483298
$275$	6.47214	0.390284
$276$	0	0
$277$	−2.94427	−0.176904	−0.0884521	−	0.996080i	$-0.528192\pi$
$277$	−2.94427	−0.176904	−0.0884521	+	0.996080i	$0.528192\pi$
$278$	6.47214	0.388173
$279$	0	0
$280$	−9.70820	−0.580176
$281$	−1.05573	−0.0629795	−0.0314897	−	0.999504i	$-0.510025\pi$
$281$	−1.05573	−0.0629795	−0.0314897	+	0.999504i	$0.510025\pi$
$282$	0	0
$283$	−8.94427	−0.531682	−0.265841	−	0.964017i	$-0.585650\pi$
$283$	−8.94427	−0.531682	−0.265841	+	0.964017i	$0.585650\pi$
$284$	−5.52786	−0.328018
$285$	0	0
$286$	41.8885	2.47692
$287$	32.3607	1.91019
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−4.47214	−0.262613
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−15.4164	−0.900636	−0.450318	−	0.892868i	$-0.648689\pi$
$293$	−15.4164	−0.900636	−0.450318	+	0.892868i	$0.648689\pi$
$294$	0	0
$295$	−1.23607	−0.0719667
$296$	19.4164	1.12856
$297$	0	0
$298$	9.41641	0.545478
$299$	4.94427	0.285935
$300$	0	0
$301$	2.47214	0.142492
$302$	−5.23607	−0.301302
$303$	0	0
$304$	2.76393	0.158522
$305$	−8.47214	−0.485113
$306$	0	0
$307$	−15.4164	−0.879861	−0.439930	−	0.898032i	$-0.644997\pi$
$307$	−15.4164	−0.879861	−0.439930	+	0.898032i	$0.644997\pi$
$308$	−20.9443	−1.19341
$309$	0	0
$310$	−1.23607	−0.0702039
$311$	15.4164	0.874184	0.437092	−	0.899417i	$-0.356008\pi$
$311$	15.4164	0.874184	0.437092	+	0.899417i	$0.356008\pi$
$312$	0	0
$313$	−17.8885	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$313$	−17.8885	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$314$	−10.9443	−0.617621
$315$	0	0
$316$	4.94427	0.278137
$317$	−10.9443	−0.614692	−0.307346	−	0.951598i	$-0.599441\pi$
$317$	−10.9443	−0.614692	−0.307346	+	0.951598i	$0.599441\pi$
$318$	0	0
$319$	−28.9443	−1.62057
$320$	7.00000	0.391312
$321$	0	0
$322$	2.47214	0.137767
$323$	−5.52786	−0.307579
$324$	0	0
$325$	6.47214	0.359010
$326$	16.1803	0.896146
$327$	0	0
$328$	−30.0000	−1.65647
$329$	−25.8885	−1.42728
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−15.2361	−0.836188
$333$	0	0
$334$	−11.4164	−0.624678
$335$	−2.47214	−0.135067
$336$	0	0
$337$	16.9443	0.923013	0.461507	−	0.887137i	$-0.347309\pi$
$337$	16.9443	0.923013	0.461507	+	0.887137i	$0.347309\pi$
$338$	28.8885	1.57133
$339$	0	0
$340$	−2.00000	−0.108465
$341$	−8.00000	−0.433224
$342$	0	0
$343$	−11.4164	−0.616428
$344$	−2.29180	−0.123565
$345$	0	0
$346$	6.00000	0.322562
$347$	−34.8328	−1.86992	−0.934962	−	0.354749i	$-0.884566\pi$
$347$	−34.8328	−1.86992	−0.934962	+	0.354749i	$0.884566\pi$
$348$	0	0
$349$	−10.3607	−0.554594	−0.277297	−	0.960784i	$-0.589439\pi$
$349$	−10.3607	−0.554594	−0.277297	+	0.960784i	$0.589439\pi$
$350$	3.23607	0.172975
$351$	0	0
$352$	32.3607	1.72483
$353$	−15.0557	−0.801336	−0.400668	−	0.916223i	$-0.631222\pi$
$353$	−15.0557	−0.801336	−0.400668	+	0.916223i	$0.631222\pi$
$354$	0	0
$355$	5.52786	0.293389
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	1.52786	0.0807501
$359$	11.7082	0.617935	0.308968	−	0.951073i	$-0.400017\pi$
$359$	11.7082	0.617935	0.308968	+	0.951073i	$0.400017\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	−12.4721	−0.655521
$363$	0	0
$364$	−20.9443	−1.09778
$365$	2.00000	0.104685
$366$	0	0
$367$	9.88854	0.516178	0.258089	−	0.966121i	$-0.416907\pi$
$367$	9.88854	0.516178	0.258089	+	0.966121i	$0.416907\pi$
$368$	−0.763932	−0.0398227
$369$	0	0
$370$	−6.47214	−0.336470
$371$	−6.47214	−0.336017
$372$	0	0
$373$	24.8328	1.28579	0.642897	−	0.765952i	$-0.277732\pi$
$373$	24.8328	1.28579	0.642897	+	0.765952i	$0.277732\pi$
$374$	12.9443	0.669332
$375$	0	0
$376$	24.0000	1.23771
$377$	−28.9443	−1.49071
$378$	0	0
$379$	30.5410	1.56879	0.784393	−	0.620264i	$-0.212974\pi$
$379$	30.5410	1.56879	0.784393	+	0.620264i	$0.212974\pi$
$380$	2.76393	0.141787
$381$	0	0
$382$	8.29180	0.424245
$383$	−0.763932	−0.0390351	−0.0195176	−	0.999810i	$-0.506213\pi$
$383$	−0.763932	−0.0390351	−0.0195176	+	0.999810i	$0.506213\pi$
$384$	0	0
$385$	20.9443	1.06742
$386$	4.94427	0.251657
$387$	0	0
$388$	6.00000	0.304604
$389$	−5.41641	−0.274623	−0.137311	−	0.990528i	$-0.543846\pi$
$389$	−5.41641	−0.274623	−0.137311	+	0.990528i	$0.543846\pi$
$390$	0	0
$391$	1.52786	0.0772674
$392$	−10.4164	−0.526108
$393$	0	0
$394$	6.47214	0.326061
$395$	−4.94427	−0.248773
$396$	0	0
$397$	−10.4721	−0.525581	−0.262791	−	0.964853i	$-0.584643\pi$
$397$	−10.4721	−0.525581	−0.262791	+	0.964853i	$0.584643\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−16.4721	−0.822579	−0.411290	−	0.911505i	$-0.634922\pi$
$401$	−16.4721	−0.822579	−0.411290	+	0.911505i	$0.634922\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−4.47214	−0.222497
$405$	0	0
$406$	−14.4721	−0.718240
$407$	−41.8885	−2.07634
$408$	0	0
$409$	5.41641	0.267824	0.133912	−	0.990993i	$-0.457246\pi$
$409$	5.41641	0.267824	0.133912	+	0.990993i	$0.457246\pi$
$410$	10.0000	0.493865
$411$	0	0
$412$	14.6525	0.721876
$413$	−4.00000	−0.196827
$414$	0	0
$415$	15.2361	0.747909
$416$	32.3607	1.58661
$417$	0	0
$418$	−17.8885	−0.874957
$419$	31.7082	1.54905	0.774524	−	0.632545i	$-0.217990\pi$
$419$	31.7082	1.54905	0.774524	+	0.632545i	$0.217990\pi$
$420$	0	0
$421$	−31.3050	−1.52571	−0.762855	−	0.646570i	$-0.776203\pi$
$421$	−31.3050	−1.52571	−0.762855	+	0.646570i	$0.776203\pi$
$422$	−18.1803	−0.885005
$423$	0	0
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	−27.4164	−1.32677
$428$	−8.94427	−0.432338
$429$	0	0
$430$	0.763932	0.0368401
$431$	−27.1246	−1.30655	−0.653273	−	0.757122i	$-0.726605\pi$
$431$	−27.1246	−1.30655	−0.653273	+	0.757122i	$0.726605\pi$
$432$	0	0
$433$	−11.0557	−0.531304	−0.265652	−	0.964069i	$-0.585587\pi$
$433$	−11.0557	−0.531304	−0.265652	+	0.964069i	$0.585587\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	8.47214	0.405742
$437$	−2.11146	−0.101005
$438$	0	0
$439$	−9.81966	−0.468667	−0.234333	−	0.972156i	$-0.575291\pi$
$439$	−9.81966	−0.468667	−0.234333	+	0.972156i	$0.575291\pi$
$440$	−19.4164	−0.925641
$441$	0	0
$442$	12.9443	0.615696
$443$	−13.8885	−0.659865	−0.329932	−	0.944005i	$-0.607026\pi$
$443$	−13.8885	−0.659865	−0.329932	+	0.944005i	$0.607026\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	12.9443	0.612929
$447$	0	0
$448$	22.6525	1.07023
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	64.7214	3.04761
$452$	3.05573	0.143729
$453$	0	0
$454$	−7.41641	−0.348069
$455$	20.9443	0.981883
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	7.52786	0.351754
$459$	0	0
$460$	−0.763932	−0.0356185
$461$	22.3607	1.04144	0.520720	−	0.853727i	$-0.325663\pi$
$461$	22.3607	1.04144	0.520720	+	0.853727i	$0.325663\pi$
$462$	0	0
$463$	17.8885	0.831351	0.415676	−	0.909513i	$-0.363545\pi$
$463$	17.8885	0.831351	0.415676	+	0.909513i	$0.363545\pi$
$464$	4.47214	0.207614
$465$	0	0
$466$	26.0000	1.20443
$467$	20.3607	0.942180	0.471090	−	0.882085i	$-0.343861\pi$
$467$	20.3607	0.942180	0.471090	+	0.882085i	$0.343861\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	−8.00000	−0.369012
$471$	0	0
$472$	3.70820	0.170684
$473$	4.94427	0.227338
$474$	0	0
$475$	−2.76393	−0.126818
$476$	−6.47214	−0.296650
$477$	0	0
$478$	−8.29180	−0.379258
$479$	4.94427	0.225910	0.112955	−	0.993600i	$-0.463968\pi$
$479$	4.94427	0.225910	0.112955	+	0.993600i	$0.463968\pi$
$480$	0	0
$481$	−41.8885	−1.90995
$482$	18.9443	0.862888
$483$	0	0
$484$	−30.8885	−1.40402
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−21.3050	−0.965420	−0.482710	−	0.875780i	$-0.660347\pi$
$487$	−21.3050	−0.965420	−0.482710	+	0.875780i	$0.660347\pi$
$488$	25.4164	1.15055
$489$	0	0
$490$	3.47214	0.156855
$491$	−29.5967	−1.33568	−0.667841	−	0.744304i	$-0.732782\pi$
$491$	−29.5967	−1.33568	−0.667841	+	0.744304i	$0.732782\pi$
$492$	0	0
$493$	−8.94427	−0.402830
$494$	−17.8885	−0.804844
$495$	0	0
$496$	1.23607	0.0555011
$497$	17.8885	0.802411
$498$	0	0
$499$	27.1246	1.21426	0.607132	−	0.794601i	$-0.292320\pi$
$499$	27.1246	1.21426	0.607132	+	0.794601i	$0.292320\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	17.5279	0.782307
$503$	41.1246	1.83366	0.916828	−	0.399283i	$-0.130741\pi$
$503$	41.1246	1.83366	0.916828	+	0.399283i	$0.130741\pi$
$504$	0	0
$505$	4.47214	0.199007
$506$	4.94427	0.219800
$507$	0	0
$508$	−0.763932	−0.0338940
$509$	7.52786	0.333667	0.166833	−	0.985985i	$-0.446646\pi$
$509$	7.52786	0.333667	0.166833	+	0.985985i	$0.446646\pi$
$510$	0	0
$511$	6.47214	0.286310
$512$	−11.0000	−0.486136
$513$	0	0
$514$	22.9443	1.01203
$515$	−14.6525	−0.645665
$516$	0	0
$517$	−51.7771	−2.27715
$518$	−20.9443	−0.920238
$519$	0	0
$520$	−19.4164	−0.851466
$521$	18.9443	0.829964	0.414982	−	0.909830i	$-0.363788\pi$
$521$	18.9443	0.829964	0.414982	+	0.909830i	$0.363788\pi$
$522$	0	0
$523$	−26.4721	−1.15755	−0.578773	−	0.815489i	$-0.696468\pi$
$523$	−26.4721	−1.15755	−0.578773	+	0.815489i	$0.696468\pi$
$524$	−9.52786	−0.416227
$525$	0	0
$526$	−30.8328	−1.34437
$527$	−2.47214	−0.107688
$528$	0	0
$529$	−22.4164	−0.974626
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	8.94427	0.387783
$533$	64.7214	2.80339
$534$	0	0
$535$	8.94427	0.386695
$536$	7.41641	0.320340
$537$	0	0
$538$	−25.4164	−1.09578
$539$	22.4721	0.967943
$540$	0	0
$541$	−1.41641	−0.0608961	−0.0304481	−	0.999536i	$-0.509693\pi$
$541$	−1.41641	−0.0608961	−0.0304481	+	0.999536i	$0.509693\pi$
$542$	−26.4721	−1.13708
$543$	0	0
$544$	10.0000	0.428746
$545$	−8.47214	−0.362906
$546$	0	0
$547$	46.0689	1.96976	0.984882	−	0.173229i	$-0.0554200\pi$
$547$	46.0689	1.96976	0.984882	+	0.173229i	$0.0554200\pi$
$548$	8.00000	0.341743
$549$	0	0
$550$	6.47214	0.275973
$551$	12.3607	0.526583
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−2.94427	−0.125090
$555$	0	0
$556$	−6.47214	−0.274480
$557$	22.4721	0.952175	0.476087	−	0.879398i	$-0.342055\pi$
$557$	22.4721	0.952175	0.476087	+	0.879398i	$0.342055\pi$
$558$	0	0
$559$	4.94427	0.209120
$560$	−3.23607	−0.136749
$561$	0	0
$562$	−1.05573	−0.0445332
$563$	−45.1246	−1.90178	−0.950888	−	0.309536i	$-0.899826\pi$
$563$	−45.1246	−1.90178	−0.950888	+	0.309536i	$0.899826\pi$
$564$	0	0
$565$	−3.05573	−0.128555
$566$	−8.94427	−0.375956
$567$	0	0
$568$	−16.5836	−0.695832
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−31.7082	−1.32695	−0.663474	−	0.748200i	$-0.730918\pi$
$571$	−31.7082	−1.32695	−0.663474	+	0.748200i	$0.730918\pi$
$572$	−41.8885	−1.75145
$573$	0	0
$574$	32.3607	1.35071
$575$	0.763932	0.0318582
$576$	0	0
$577$	−18.8328	−0.784020	−0.392010	−	0.919961i	$-0.628220\pi$
$577$	−18.8328	−0.784020	−0.392010	+	0.919961i	$0.628220\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	4.47214	0.185695
$581$	49.3050	2.04551
$582$	0	0
$583$	−12.9443	−0.536097
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−15.4164	−0.636846
$587$	−12.3607	−0.510180	−0.255090	−	0.966917i	$-0.582105\pi$
$587$	−12.3607	−0.510180	−0.255090	+	0.966917i	$0.582105\pi$
$588$	0	0
$589$	3.41641	0.140771
$590$	−1.23607	−0.0508881
$591$	0	0
$592$	6.47214	0.266003
$593$	12.9443	0.531558	0.265779	−	0.964034i	$-0.414371\pi$
$593$	12.9443	0.531558	0.265779	+	0.964034i	$0.414371\pi$
$594$	0	0
$595$	6.47214	0.265332
$596$	−9.41641	−0.385711
$597$	0	0
$598$	4.94427	0.202186
$599$	25.5967	1.04585	0.522927	−	0.852377i	$-0.324840\pi$
$599$	25.5967	1.04585	0.522927	+	0.852377i	$0.324840\pi$
$600$	0	0
$601$	31.8885	1.30076	0.650380	−	0.759609i	$-0.274610\pi$
$601$	31.8885	1.30076	0.650380	+	0.759609i	$0.274610\pi$
$602$	2.47214	0.100757
$603$	0	0
$604$	5.23607	0.213053
$605$	30.8885	1.25580
$606$	0	0
$607$	−14.4721	−0.587406	−0.293703	−	0.955897i	$-0.594888\pi$
$607$	−14.4721	−0.587406	−0.293703	+	0.955897i	$0.594888\pi$
$608$	−13.8197	−0.560461
$609$	0	0
$610$	−8.47214	−0.343027
$611$	−51.7771	−2.09468
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−15.4164	−0.622156
$615$	0	0
$616$	−62.8328	−2.53161
$617$	43.7771	1.76240	0.881200	−	0.472744i	$-0.156737\pi$
$617$	43.7771	1.76240	0.881200	+	0.472744i	$0.156737\pi$
$618$	0	0
$619$	−5.88854	−0.236681	−0.118340	−	0.992973i	$-0.537757\pi$
$619$	−5.88854	−0.236681	−0.118340	+	0.992973i	$0.537757\pi$
$620$	1.23607	0.0496417
$621$	0	0
$622$	15.4164	0.618142
$623$	3.23607	0.129650
$624$	0	0
$625$	1.00000	0.0400000
$626$	−17.8885	−0.714970
$627$	0	0
$628$	10.9443	0.436724
$629$	−12.9443	−0.516122
$630$	0	0
$631$	−34.4721	−1.37231	−0.686157	−	0.727453i	$-0.740704\pi$
$631$	−34.4721	−1.37231	−0.686157	+	0.727453i	$0.740704\pi$
$632$	14.8328	0.590018
$633$	0	0
$634$	−10.9443	−0.434653
$635$	0.763932	0.0303157
$636$	0	0
$637$	22.4721	0.890378
$638$	−28.9443	−1.14591
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−20.3607	−0.802947	−0.401473	−	0.915871i	$-0.631502\pi$
$643$	−20.3607	−0.802947	−0.401473	+	0.915871i	$0.631502\pi$
$644$	−2.47214	−0.0974158
$645$	0	0
$646$	−5.52786	−0.217491
$647$	13.7082	0.538925	0.269463	−	0.963011i	$-0.413154\pi$
$647$	13.7082	0.538925	0.269463	+	0.963011i	$0.413154\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	6.47214	0.253858
$651$	0	0
$652$	−16.1803	−0.633671
$653$	−4.58359	−0.179370	−0.0896849	−	0.995970i	$-0.528586\pi$
$653$	−4.58359	−0.179370	−0.0896849	+	0.995970i	$0.528586\pi$
$654$	0	0
$655$	9.52786	0.372284
$656$	−10.0000	−0.390434
$657$	0	0
$658$	−25.8885	−1.00924
$659$	9.52786	0.371153	0.185576	−	0.982630i	$-0.440585\pi$
$659$	9.52786	0.371153	0.185576	+	0.982630i	$0.440585\pi$
$660$	0	0
$661$	10.5836	0.411654	0.205827	−	0.978588i	$-0.434012\pi$
$661$	10.5836	0.411654	0.205827	+	0.978588i	$0.434012\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−45.7082	−1.77382
$665$	−8.94427	−0.346844
$666$	0	0
$667$	−3.41641	−0.132284
$668$	11.4164	0.441714
$669$	0	0
$670$	−2.47214	−0.0955069
$671$	−54.8328	−2.11680
$672$	0	0
$673$	−31.8885	−1.22921	−0.614607	−	0.788834i	$-0.710685\pi$
$673$	−31.8885	−1.22921	−0.614607	+	0.788834i	$0.710685\pi$
$674$	16.9443	0.652669
$675$	0	0
$676$	−28.8885	−1.11110
$677$	−49.3050	−1.89494	−0.947472	−	0.319840i	$-0.896371\pi$
$677$	−49.3050	−1.89494	−0.947472	+	0.319840i	$0.896371\pi$
$678$	0	0
$679$	−19.4164	−0.745133
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−20.1803	−0.772179	−0.386090	−	0.922461i	$-0.626174\pi$
$683$	−20.1803	−0.772179	−0.386090	+	0.922461i	$0.626174\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	−11.4164	−0.435880
$687$	0	0
$688$	−0.763932	−0.0291246
$689$	−12.9443	−0.493137
$690$	0	0
$691$	22.4721	0.854880	0.427440	−	0.904044i	$-0.359415\pi$
$691$	22.4721	0.854880	0.427440	+	0.904044i	$0.359415\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−34.8328	−1.32224
$695$	6.47214	0.245502
$696$	0	0
$697$	20.0000	0.757554
$698$	−10.3607	−0.392158
$699$	0	0
$700$	−3.23607	−0.122312
$701$	−33.7771	−1.27574	−0.637871	−	0.770143i	$-0.720185\pi$
$701$	−33.7771	−1.27574	−0.637871	+	0.770143i	$0.720185\pi$
$702$	0	0
$703$	17.8885	0.674679
$704$	45.3050	1.70749
$705$	0	0
$706$	−15.0557	−0.566630
$707$	14.4721	0.544281
$708$	0	0
$709$	22.3607	0.839773	0.419886	−	0.907577i	$-0.362070\pi$
$709$	22.3607	0.839773	0.419886	+	0.907577i	$0.362070\pi$
$710$	5.52786	0.207457
$711$	0	0
$712$	−3.00000	−0.112430
$713$	−0.944272	−0.0353633
$714$	0	0
$715$	41.8885	1.56654
$716$	−1.52786	−0.0570990
$717$	0	0
$718$	11.7082	0.436946
$719$	−51.1246	−1.90663	−0.953313	−	0.301984i	$-0.902351\pi$
$719$	−51.1246	−1.90663	−0.953313	+	0.301984i	$0.902351\pi$
$720$	0	0
$721$	−47.4164	−1.76588
$722$	−11.3607	−0.422801
$723$	0	0
$724$	12.4721	0.463523
$725$	−4.47214	−0.166091
$726$	0	0
$727$	9.70820	0.360057	0.180029	−	0.983661i	$-0.442381\pi$
$727$	9.70820	0.360057	0.180029	+	0.983661i	$0.442381\pi$
$728$	−62.8328	−2.32874
$729$	0	0
$730$	2.00000	0.0740233
$731$	1.52786	0.0565101
$732$	0	0
$733$	−25.0557	−0.925454	−0.462727	−	0.886501i	$-0.653129\pi$
$733$	−25.0557	−0.925454	−0.462727	+	0.886501i	$0.653129\pi$
$734$	9.88854	0.364993
$735$	0	0
$736$	3.81966	0.140795
$737$	−16.0000	−0.589368
$738$	0	0
$739$	16.0689	0.591103	0.295552	−	0.955327i	$-0.404497\pi$
$739$	16.0689	0.591103	0.295552	+	0.955327i	$0.404497\pi$
$740$	6.47214	0.237920
$741$	0	0
$742$	−6.47214	−0.237600
$743$	12.1803	0.446853	0.223427	−	0.974721i	$-0.428276\pi$
$743$	12.1803	0.446853	0.223427	+	0.974721i	$0.428276\pi$
$744$	0	0
$745$	9.41641	0.344990
$746$	24.8328	0.909194
$747$	0	0
$748$	−12.9443	−0.473289
$749$	28.9443	1.05760
$750$	0	0
$751$	5.52786	0.201715	0.100857	−	0.994901i	$-0.467841\pi$
$751$	5.52786	0.201715	0.100857	+	0.994901i	$0.467841\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−28.9443	−1.05409
$755$	−5.23607	−0.190560
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	30.5410	1.10930
$759$	0	0
$760$	8.29180	0.300775
$761$	4.47214	0.162115	0.0810574	−	0.996709i	$-0.474170\pi$
$761$	4.47214	0.162115	0.0810574	+	0.996709i	$0.474170\pi$
$762$	0	0
$763$	−27.4164	−0.992541
$764$	−8.29180	−0.299987
$765$	0	0
$766$	−0.763932	−0.0276020
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−35.5279	−1.28117	−0.640584	−	0.767888i	$-0.721308\pi$
$769$	−35.5279	−1.28117	−0.640584	+	0.767888i	$0.721308\pi$
$770$	20.9443	0.754779
$771$	0	0
$772$	−4.94427	−0.177948
$773$	−8.36068	−0.300713	−0.150356	−	0.988632i	$-0.548042\pi$
$773$	−8.36068	−0.300713	−0.150356	+	0.988632i	$0.548042\pi$
$774$	0	0
$775$	−1.23607	−0.0444009
$776$	18.0000	0.646162
$777$	0	0
$778$	−5.41641	−0.194188
$779$	−27.6393	−0.990281
$780$	0	0
$781$	35.7771	1.28020
$782$	1.52786	0.0546363
$783$	0	0
$784$	−3.47214	−0.124005
$785$	−10.9443	−0.390618
$786$	0	0
$787$	32.5410	1.15996	0.579981	−	0.814630i	$-0.303060\pi$
$787$	32.5410	1.15996	0.579981	+	0.814630i	$0.303060\pi$
$788$	−6.47214	−0.230560
$789$	0	0
$790$	−4.94427	−0.175909
$791$	−9.88854	−0.351596
$792$	0	0
$793$	−54.8328	−1.94717
$794$	−10.4721	−0.371642
$795$	0	0
$796$	8.00000	0.283552
$797$	35.8885	1.27124	0.635619	−	0.772003i	$-0.280745\pi$
$797$	35.8885	1.27124	0.635619	+	0.772003i	$0.280745\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	5.00000	0.176777
$801$	0	0
$802$	−16.4721	−0.581651
$803$	12.9443	0.456793
$804$	0	0
$805$	2.47214	0.0871313
$806$	−8.00000	−0.281788
$807$	0	0
$808$	−13.4164	−0.471988
$809$	29.7771	1.04691	0.523453	−	0.852054i	$-0.324643\pi$
$809$	29.7771	1.04691	0.523453	+	0.852054i	$0.324643\pi$
$810$	0	0
$811$	−2.11146	−0.0741433	−0.0370716	−	0.999313i	$-0.511803\pi$
$811$	−2.11146	−0.0741433	−0.0370716	+	0.999313i	$0.511803\pi$
$812$	14.4721	0.507872
$813$	0	0
$814$	−41.8885	−1.46819
$815$	16.1803	0.566773
$816$	0	0
$817$	−2.11146	−0.0738705
$818$	5.41641	0.189380
$819$	0	0
$820$	−10.0000	−0.349215
$821$	−4.47214	−0.156079	−0.0780393	−	0.996950i	$-0.524866\pi$
$821$	−4.47214	−0.156079	−0.0780393	+	0.996950i	$0.524866\pi$
$822$	0	0
$823$	33.5279	1.16871	0.584354	−	0.811499i	$-0.301348\pi$
$823$	33.5279	1.16871	0.584354	+	0.811499i	$0.301348\pi$
$824$	43.9574	1.53133
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−17.1246	−0.595481	−0.297741	−	0.954647i	$-0.596233\pi$
$827$	−17.1246	−0.595481	−0.297741	+	0.954647i	$0.596233\pi$
$828$	0	0
$829$	−40.2492	−1.39791	−0.698957	−	0.715164i	$-0.746352\pi$
$829$	−40.2492	−1.39791	−0.698957	+	0.715164i	$0.746352\pi$
$830$	15.2361	0.528852
$831$	0	0
$832$	45.3050	1.57067
$833$	6.94427	0.240605
$834$	0	0
$835$	−11.4164	−0.395081
$836$	17.8885	0.618688
$837$	0	0
$838$	31.7082	1.09534
$839$	4.29180	0.148169	0.0740846	−	0.997252i	$-0.476397\pi$
$839$	4.29180	0.148169	0.0740846	+	0.997252i	$0.476397\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−31.3050	−1.07884
$843$	0	0
$844$	18.1803	0.625793
$845$	28.8885	0.993796
$846$	0	0
$847$	99.9574	3.43458
$848$	2.00000	0.0686803
$849$	0	0
$850$	2.00000	0.0685994
$851$	−4.94427	−0.169487
$852$	0	0
$853$	17.5279	0.600143	0.300071	−	0.953917i	$-0.402990\pi$
$853$	17.5279	0.600143	0.300071	+	0.953917i	$0.402990\pi$
$854$	−27.4164	−0.938170
$855$	0	0
$856$	−26.8328	−0.917127
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	35.1246	1.19844	0.599218	−	0.800586i	$-0.295478\pi$
$859$	35.1246	1.19844	0.599218	+	0.800586i	$0.295478\pi$
$860$	−0.763932	−0.0260499
$861$	0	0
$862$	−27.1246	−0.923868
$863$	−45.4853	−1.54834	−0.774169	−	0.632979i	$-0.781832\pi$
$863$	−45.4853	−1.54834	−0.774169	+	0.632979i	$0.781832\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	−11.0557	−0.375689
$867$	0	0
$868$	4.00000	0.135769
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−16.0000	−0.542139
$872$	25.4164	0.860708
$873$	0	0
$874$	−2.11146	−0.0714211
$875$	3.23607	0.109399
$876$	0	0
$877$	−41.5279	−1.40230	−0.701148	−	0.713016i	$-0.747329\pi$
$877$	−41.5279	−1.40230	−0.701148	+	0.713016i	$0.747329\pi$
$878$	−9.81966	−0.331397
$879$	0	0
$880$	−6.47214	−0.218176
$881$	7.52786	0.253620	0.126810	−	0.991927i	$-0.459526\pi$
$881$	7.52786	0.253620	0.126810	+	0.991927i	$0.459526\pi$
$882$	0	0
$883$	−6.06888	−0.204234	−0.102117	−	0.994772i	$-0.532562\pi$
$883$	−6.06888	−0.204234	−0.102117	+	0.994772i	$0.532562\pi$
$884$	−12.9443	−0.435363
$885$	0	0
$886$	−13.8885	−0.466595
$887$	28.5410	0.958314	0.479157	−	0.877729i	$-0.340943\pi$
$887$	28.5410	0.958314	0.479157	+	0.877729i	$0.340943\pi$
$888$	0	0
$889$	2.47214	0.0829128
$890$	1.00000	0.0335201
$891$	0	0
$892$	−12.9443	−0.433406
$893$	22.1115	0.739932
$894$	0	0
$895$	1.52786	0.0510709
$896$	−9.70820	−0.324328
$897$	0	0
$898$	14.0000	0.467186
$899$	5.52786	0.184365
$900$	0	0
$901$	−4.00000	−0.133259
$902$	64.7214	2.15499
$903$	0	0
$904$	9.16718	0.304896
$905$	−12.4721	−0.414588
$906$	0	0
$907$	37.5279	1.24609	0.623046	−	0.782185i	$-0.285895\pi$
$907$	37.5279	1.24609	0.623046	+	0.782185i	$0.285895\pi$
$908$	7.41641	0.246122
$909$	0	0
$910$	20.9443	0.694296
$911$	34.4721	1.14211	0.571056	−	0.820911i	$-0.306534\pi$
$911$	34.4721	1.14211	0.571056	+	0.820911i	$0.306534\pi$
$912$	0	0
$913$	98.6099	3.26351
$914$	28.0000	0.926158
$915$	0	0
$916$	−7.52786	−0.248728
$917$	30.8328	1.01819
$918$	0	0
$919$	38.5410	1.27135	0.635675	−	0.771956i	$-0.280722\pi$
$919$	38.5410	1.27135	0.635675	+	0.771956i	$0.280722\pi$
$920$	−2.29180	−0.0755583
$921$	0	0
$922$	22.3607	0.736410
$923$	35.7771	1.17762
$924$	0	0
$925$	−6.47214	−0.212803
$926$	17.8885	0.587854
$927$	0	0
$928$	−22.3607	−0.734025
$929$	−24.1115	−0.791071	−0.395536	−	0.918451i	$-0.629441\pi$
$929$	−24.1115	−0.791071	−0.395536	+	0.918451i	$0.629441\pi$
$930$	0	0
$931$	−9.59675	−0.314521
$932$	−26.0000	−0.851658
$933$	0	0
$934$	20.3607	0.666222
$935$	12.9443	0.423323
$936$	0	0
$937$	−0.111456	−0.00364111	−0.00182056	−	0.999998i	$-0.500580\pi$
$937$	−0.111456	−0.00364111	−0.00182056	+	0.999998i	$0.500580\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	8.00000	0.260931
$941$	−52.2492	−1.70328	−0.851638	−	0.524130i	$-0.824390\pi$
$941$	−52.2492	−1.70328	−0.851638	+	0.524130i	$0.824390\pi$
$942$	0	0
$943$	7.63932	0.248770
$944$	1.23607	0.0402306
$945$	0	0
$946$	4.94427	0.160752
$947$	20.3607	0.661633	0.330817	−	0.943695i	$-0.392676\pi$
$947$	20.3607	0.661633	0.330817	+	0.943695i	$0.392676\pi$
$948$	0	0
$949$	12.9443	0.420189
$950$	−2.76393	−0.0896738
$951$	0	0
$952$	−19.4164	−0.629289
$953$	49.8885	1.61605	0.808024	−	0.589149i	$-0.200537\pi$
$953$	49.8885	1.61605	0.808024	+	0.589149i	$0.200537\pi$
$954$	0	0
$955$	8.29180	0.268316
$956$	8.29180	0.268176
$957$	0	0
$958$	4.94427	0.159742
$959$	−25.8885	−0.835985
$960$	0	0
$961$	−29.4721	−0.950714
$962$	−41.8885	−1.35054
$963$	0	0
$964$	−18.9443	−0.610154
$965$	4.94427	0.159162
$966$	0	0
$967$	23.2361	0.747222	0.373611	−	0.927586i	$-0.378120\pi$
$967$	23.2361	0.747222	0.373611	+	0.927586i	$0.378120\pi$
$968$	−92.6656	−2.97839
$969$	0	0
$970$	−6.00000	−0.192648
$971$	−53.8885	−1.72937	−0.864683	−	0.502318i	$-0.832481\pi$
$971$	−53.8885	−1.72937	−0.864683	+	0.502318i	$0.832481\pi$
$972$	0	0
$973$	20.9443	0.671443
$974$	−21.3050	−0.682655
$975$	0	0
$976$	8.47214	0.271186
$977$	14.0000	0.447900	0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000	0.447900	0.223950	+	0.974601i	$0.428105\pi$
$978$	0	0
$979$	6.47214	0.206850
$980$	−3.47214	−0.110913
$981$	0	0
$982$	−29.5967	−0.944470
$983$	−28.9443	−0.923179	−0.461589	−	0.887094i	$-0.652721\pi$
$983$	−28.9443	−0.923179	−0.461589	+	0.887094i	$0.652721\pi$
$984$	0	0
$985$	6.47214	0.206219
$986$	−8.94427	−0.284844
$987$	0	0
$988$	17.8885	0.569110
$989$	0.583592	0.0185572
$990$	0	0
$991$	−60.6525	−1.92669	−0.963345	−	0.268267i	$-0.913549\pi$
$991$	−60.6525	−1.92669	−0.963345	+	0.268267i	$0.913549\pi$
$992$	−6.18034	−0.196226
$993$	0	0
$994$	17.8885	0.567390
$995$	−8.00000	−0.253617
$996$	0	0
$997$	38.9443	1.23338	0.616689	−	0.787207i	$-0.288474\pi$
$997$	38.9443	1.23338	0.616689	+	0.787207i	$0.288474\pi$
$998$	27.1246	0.858615
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.h.1.2		2
3.2	odd	2		445.2.a.a.1.1	✓	2
12.11	even	2		7120.2.a.w.1.2		2
15.14	odd	2		2225.2.a.e.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.a.1.1	✓	2	3.2	odd	2
2225.2.a.e.1.2		2	15.14	odd	2
4005.2.a.h.1.2		2	1.1	even	1	trivial
7120.2.a.w.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.23607 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +3.23607 q^{7} -3.00000 q^{8} +1.00000 q^{10} +6.47214 q^{11} +6.47214 q^{13} +3.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.76393 q^{19} -1.00000 q^{20} +6.47214 q^{22} +0.763932 q^{23} +1.00000 q^{25} +6.47214 q^{26} -3.23607 q^{28} -4.47214 q^{29} -1.23607 q^{31} +5.00000 q^{32} +2.00000 q^{34} +3.23607 q^{35} -6.47214 q^{37} -2.76393 q^{38} -3.00000 q^{40} +10.0000 q^{41} +0.763932 q^{43} -6.47214 q^{44} +0.763932 q^{46} -8.00000 q^{47} +3.47214 q^{49} +1.00000 q^{50} -6.47214 q^{52} -2.00000 q^{53} +6.47214 q^{55} -9.70820 q^{56} -4.47214 q^{58} -1.23607 q^{59} -8.47214 q^{61} -1.23607 q^{62} +7.00000 q^{64} +6.47214 q^{65} -2.47214 q^{67} -2.00000 q^{68} +3.23607 q^{70} +5.52786 q^{71} +2.00000 q^{73} -6.47214 q^{74} +2.76393 q^{76} +20.9443 q^{77} -4.94427 q^{79} -1.00000 q^{80} +10.0000 q^{82} +15.2361 q^{83} +2.00000 q^{85} +0.763932 q^{86} -19.4164 q^{88} +1.00000 q^{89} +20.9443 q^{91} -0.763932 q^{92} -8.00000 q^{94} -2.76393 q^{95} -6.00000 q^{97} +3.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 4 q^{11} + 4 q^{13} + 2 q^{14} - 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 2 q^{31} + 10 q^{32} + 4 q^{34} + 2 q^{35} - 4 q^{37} - 10 q^{38} - 6 q^{40} + 20 q^{41} + 6 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} - 2 q^{49} + 2 q^{50} - 4 q^{52} - 4 q^{53} + 4 q^{55} - 6 q^{56} + 2 q^{59} - 8 q^{61} + 2 q^{62} + 14 q^{64} + 4 q^{65} + 4 q^{67} - 4 q^{68} + 2 q^{70} + 20 q^{71} + 4 q^{73} - 4 q^{74} + 10 q^{76} + 24 q^{77} + 8 q^{79} - 2 q^{80} + 20 q^{82} + 26 q^{83} + 4 q^{85} + 6 q^{86} - 12 q^{88} + 2 q^{89} + 24 q^{91} - 6 q^{92} - 16 q^{94} - 10 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 + 2 * q^10 + 4 * q^11 + 4 * q^13 + 2 * q^14 - 2 * q^16 + 4 * q^17 - 10 * q^19 - 2 * q^20 + 4 * q^22 + 6 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 2 * q^31 + 10 * q^32 + 4 * q^34 + 2 * q^35 - 4 * q^37 - 10 * q^38 - 6 * q^40 + 20 * q^41 + 6 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 - 2 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 + 4 * q^55 - 6 * q^56 + 2 * q^59 - 8 * q^61 + 2 * q^62 + 14 * q^64 + 4 * q^65 + 4 * q^67 - 4 * q^68 + 2 * q^70 + 20 * q^71 + 4 * q^73 - 4 * q^74 + 10 * q^76 + 24 * q^77 + 8 * q^79 - 2 * q^80 + 20 * q^82 + 26 * q^83 + 4 * q^85 + 6 * q^86 - 12 * q^88 + 2 * q^89 + 24 * q^91 - 6 * q^92 - 16 * q^94 - 10 * q^95 - 12 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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