LMFDB - Embedded newform 4592.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4592,2,Mod(1,4592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4592 = 2^{4} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4592.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:	$36.6673046082$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{3} -0.618034 q^{5} +1.00000 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} -0.618034 q^{5} +1.00000 q^{7} -2.61803 q^{9} +1.00000 q^{11} -1.76393 q^{13} +0.381966 q^{15} +0.236068 q^{17} +3.85410 q^{19} -0.618034 q^{21} +1.38197 q^{23} -4.61803 q^{25} +3.47214 q^{27} +0.854102 q^{29} +3.09017 q^{31} -0.618034 q^{33} -0.618034 q^{35} -7.23607 q^{37} +1.09017 q^{39} -1.00000 q^{41} +1.00000 q^{43} +1.61803 q^{45} +3.70820 q^{47} +1.00000 q^{49} -0.145898 q^{51} +4.61803 q^{53} -0.618034 q^{55} -2.38197 q^{57} -10.6180 q^{59} +0.527864 q^{61} -2.61803 q^{63} +1.09017 q^{65} +0.909830 q^{67} -0.854102 q^{69} -11.9443 q^{71} +7.94427 q^{73} +2.85410 q^{75} +1.00000 q^{77} -1.70820 q^{79} +5.70820 q^{81} -3.47214 q^{83} -0.145898 q^{85} -0.527864 q^{87} +6.32624 q^{89} -1.76393 q^{91} -1.90983 q^{93} -2.38197 q^{95} -17.5623 q^{97} -2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9	$ 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 8 q^{13} + 3 q^{15} - 4 q^{17} + q^{19} + q^{21} + 5 q^{23} - 7 q^{25} - 2 q^{27} - 5 q^{29} - 5 q^{31} + q^{33} + q^{35} - 10 q^{37} - 9 q^{39} - 2 q^{41} + 2 q^{43} + q^{45} - 6 q^{47} + 2 q^{49} - 7 q^{51} + 7 q^{53} + q^{55} - 7 q^{57} - 19 q^{59} + 10 q^{61} - 3 q^{63} - 9 q^{65} + 13 q^{67} + 5 q^{69} - 6 q^{71} - 2 q^{73} - q^{75} + 2 q^{77} + 10 q^{79} - 2 q^{81} + 2 q^{83} - 7 q^{85} - 10 q^{87} - 3 q^{89} - 8 q^{91} - 15 q^{93} - 7 q^{95} - 15 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 8 * q^13 + 3 * q^15 - 4 * q^17 + q^19 + q^21 + 5 * q^23 - 7 * q^25 - 2 * q^27 - 5 * q^29 - 5 * q^31 + q^33 + q^35 - 10 * q^37 - 9 * q^39 - 2 * q^41 + 2 * q^43 + q^45 - 6 * q^47 + 2 * q^49 - 7 * q^51 + 7 * q^53 + q^55 - 7 * q^57 - 19 * q^59 + 10 * q^61 - 3 * q^63 - 9 * q^65 + 13 * q^67 + 5 * q^69 - 6 * q^71 - 2 * q^73 - q^75 + 2 * q^77 + 10 * q^79 - 2 * q^81 + 2 * q^83 - 7 * q^85 - 10 * q^87 - 3 * q^89 - 8 * q^91 - 15 * q^93 - 7 * q^95 - 15 * q^97 - 3 * 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	0
$2$	0	0
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−1.76393	−0.489227	−0.244613	−	0.969621i	$-0.578661\pi$
$13$	−1.76393	−0.489227	−0.244613	+	0.969621i	$0.578661\pi$
$14$	0	0
$15$	0.381966	0.0986232
$16$	0	0
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	0	0
$19$	3.85410	0.884192	0.442096	−	0.896968i	$-0.354235\pi$
$19$	3.85410	0.884192	0.442096	+	0.896968i	$0.354235\pi$
$20$	0	0
$21$	−0.618034	−0.134866
$22$	0	0
$23$	1.38197	0.288160	0.144080	−	0.989566i	$-0.453978\pi$
$23$	1.38197	0.288160	0.144080	+	0.989566i	$0.453978\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	0.854102	0.158603	0.0793014	−	0.996851i	$-0.474731\pi$
$29$	0.854102	0.158603	0.0793014	+	0.996851i	$0.474731\pi$
$30$	0	0
$31$	3.09017	0.555011	0.277505	−	0.960724i	$-0.410492\pi$
$31$	3.09017	0.555011	0.277505	+	0.960724i	$0.410492\pi$
$32$	0	0
$33$	−0.618034	−0.107586
$34$	0	0
$35$	−0.618034	−0.104467
$36$	0	0
$37$	−7.23607	−1.18960	−0.594801	−	0.803873i	$-0.702769\pi$
$37$	−7.23607	−1.18960	−0.594801	+	0.803873i	$0.702769\pi$
$38$	0	0
$39$	1.09017	0.174567
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	1.61803	0.241202
$46$	0	0
$47$	3.70820	0.540897	0.270449	−	0.962734i	$-0.412828\pi$
$47$	3.70820	0.540897	0.270449	+	0.962734i	$0.412828\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.145898	−0.0204298
$52$	0	0
$53$	4.61803	0.634336	0.317168	−	0.948369i	$-0.397268\pi$
$53$	4.61803	0.634336	0.317168	+	0.948369i	$0.397268\pi$
$54$	0	0
$55$	−0.618034	−0.0833357
$56$	0	0
$57$	−2.38197	−0.315499
$58$	0	0
$59$	−10.6180	−1.38235	−0.691175	−	0.722687i	$-0.742907\pi$
$59$	−10.6180	−1.38235	−0.691175	+	0.722687i	$0.742907\pi$
$60$	0	0
$61$	0.527864	0.0675861	0.0337930	−	0.999429i	$-0.489241\pi$
$61$	0.527864	0.0675861	0.0337930	+	0.999429i	$0.489241\pi$
$62$	0	0
$63$	−2.61803	−0.329841
$64$	0	0
$65$	1.09017	0.135219
$66$	0	0
$67$	0.909830	0.111153	0.0555767	−	0.998454i	$-0.482300\pi$
$67$	0.909830	0.111153	0.0555767	+	0.998454i	$0.482300\pi$
$68$	0	0
$69$	−0.854102	−0.102822
$70$	0	0
$71$	−11.9443	−1.41752	−0.708762	−	0.705448i	$-0.750746\pi$
$71$	−11.9443	−1.41752	−0.708762	+	0.705448i	$0.750746\pi$
$72$	0	0
$73$	7.94427	0.929807	0.464903	−	0.885361i	$-0.346089\pi$
$73$	7.94427	0.929807	0.464903	+	0.885361i	$0.346089\pi$
$74$	0	0
$75$	2.85410	0.329563
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−1.70820	−0.192188	−0.0960940	−	0.995372i	$-0.530635\pi$
$79$	−1.70820	−0.192188	−0.0960940	+	0.995372i	$0.530635\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	−3.47214	−0.381116	−0.190558	−	0.981676i	$-0.561030\pi$
$83$	−3.47214	−0.381116	−0.190558	+	0.981676i	$0.561030\pi$
$84$	0	0
$85$	−0.145898	−0.0158249
$86$	0	0
$87$	−0.527864	−0.0565930
$88$	0	0
$89$	6.32624	0.670580	0.335290	−	0.942115i	$-0.391166\pi$
$89$	6.32624	0.670580	0.335290	+	0.942115i	$0.391166\pi$
$90$	0	0
$91$	−1.76393	−0.184910
$92$	0	0
$93$	−1.90983	−0.198040
$94$	0	0
$95$	−2.38197	−0.244385
$96$	0	0
$97$	−17.5623	−1.78318	−0.891591	−	0.452842i	$-0.850410\pi$
$97$	−17.5623	−1.78318	−0.891591	+	0.452842i	$0.850410\pi$
$98$	0	0
$99$	−2.61803	−0.263122
$100$	0	0
$101$	−8.52786	−0.848554	−0.424277	−	0.905532i	$-0.639472\pi$
$101$	−8.52786	−0.848554	−0.424277	+	0.905532i	$0.639472\pi$
$102$	0	0
$103$	−10.0902	−0.994214	−0.497107	−	0.867689i	$-0.665604\pi$
$103$	−10.0902	−0.994214	−0.497107	+	0.867689i	$0.665604\pi$
$104$	0	0
$105$	0.381966	0.0372761
$106$	0	0
$107$	5.23607	0.506190	0.253095	−	0.967441i	$-0.418552\pi$
$107$	5.23607	0.506190	0.253095	+	0.967441i	$0.418552\pi$
$108$	0	0
$109$	−0.472136	−0.0452224	−0.0226112	−	0.999744i	$-0.507198\pi$
$109$	−0.472136	−0.0452224	−0.0226112	+	0.999744i	$0.507198\pi$
$110$	0	0
$111$	4.47214	0.424476
$112$	0	0
$113$	−12.4164	−1.16804	−0.584019	−	0.811740i	$-0.698521\pi$
$113$	−12.4164	−1.16804	−0.584019	+	0.811740i	$0.698521\pi$
$114$	0	0
$115$	−0.854102	−0.0796454
$116$	0	0
$117$	4.61803	0.426937
$118$	0	0
$119$	0.236068	0.0216403
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0.618034	0.0557262
$124$	0	0
$125$	5.94427	0.531672
$126$	0	0
$127$	16.7082	1.48261	0.741307	−	0.671166i	$-0.234206\pi$
$127$	16.7082	1.48261	0.741307	+	0.671166i	$0.234206\pi$
$128$	0	0
$129$	−0.618034	−0.0544149
$130$	0	0
$131$	−4.14590	−0.362229	−0.181114	−	0.983462i	$-0.557970\pi$
$131$	−4.14590	−0.362229	−0.181114	+	0.983462i	$0.557970\pi$
$132$	0	0
$133$	3.85410	0.334193
$134$	0	0
$135$	−2.14590	−0.184689
$136$	0	0
$137$	−8.79837	−0.751696	−0.375848	−	0.926681i	$-0.622648\pi$
$137$	−8.79837	−0.751696	−0.375848	+	0.926681i	$0.622648\pi$
$138$	0	0
$139$	−11.9443	−1.01310	−0.506550	−	0.862211i	$-0.669079\pi$
$139$	−11.9443	−1.01310	−0.506550	+	0.862211i	$0.669079\pi$
$140$	0	0
$141$	−2.29180	−0.193004
$142$	0	0
$143$	−1.76393	−0.147507
$144$	0	0
$145$	−0.527864	−0.0438367
$146$	0	0
$147$	−0.618034	−0.0509746
$148$	0	0
$149$	1.05573	0.0864886	0.0432443	−	0.999065i	$-0.486231\pi$
$149$	1.05573	0.0864886	0.0432443	+	0.999065i	$0.486231\pi$
$150$	0	0
$151$	2.05573	0.167293	0.0836464	−	0.996495i	$-0.473343\pi$
$151$	2.05573	0.167293	0.0836464	+	0.996495i	$0.473343\pi$
$152$	0	0
$153$	−0.618034	−0.0499651
$154$	0	0
$155$	−1.90983	−0.153401
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	0	0
$159$	−2.85410	−0.226345
$160$	0	0
$161$	1.38197	0.108914
$162$	0	0
$163$	−0.944272	−0.0739611	−0.0369805	−	0.999316i	$-0.511774\pi$
$163$	−0.944272	−0.0739611	−0.0369805	+	0.999316i	$0.511774\pi$
$164$	0	0
$165$	0.381966	0.0297360
$166$	0	0
$167$	−16.4164	−1.27034	−0.635170	−	0.772372i	$-0.719070\pi$
$167$	−16.4164	−1.27034	−0.635170	+	0.772372i	$0.719070\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	−10.0902	−0.771615
$172$	0	0
$173$	−8.70820	−0.662072	−0.331036	−	0.943618i	$-0.607398\pi$
$173$	−8.70820	−0.662072	−0.331036	+	0.943618i	$0.607398\pi$
$174$	0	0
$175$	−4.61803	−0.349091
$176$	0	0
$177$	6.56231	0.493253
$178$	0	0
$179$	26.0344	1.94591	0.972953	−	0.231004i	$-0.0742011\pi$
$179$	26.0344	1.94591	0.972953	+	0.231004i	$0.0742011\pi$
$180$	0	0
$181$	14.9443	1.11080	0.555399	−	0.831584i	$-0.312565\pi$
$181$	14.9443	1.11080	0.555399	+	0.831584i	$0.312565\pi$
$182$	0	0
$183$	−0.326238	−0.0241162
$184$	0	0
$185$	4.47214	0.328798
$186$	0	0
$187$	0.236068	0.0172630
$188$	0	0
$189$	3.47214	0.252561
$190$	0	0
$191$	−13.5279	−0.978842	−0.489421	−	0.872048i	$-0.662792\pi$
$191$	−13.5279	−0.978842	−0.489421	+	0.872048i	$0.662792\pi$
$192$	0	0
$193$	−9.29180	−0.668838	−0.334419	−	0.942424i	$-0.608540\pi$
$193$	−9.29180	−0.668838	−0.334419	+	0.942424i	$0.608540\pi$
$194$	0	0
$195$	−0.673762	−0.0482491
$196$	0	0
$197$	−7.38197	−0.525943	−0.262972	−	0.964804i	$-0.584703\pi$
$197$	−7.38197	−0.525943	−0.262972	+	0.964804i	$0.584703\pi$
$198$	0	0
$199$	−23.3607	−1.65599	−0.827997	−	0.560732i	$-0.810520\pi$
$199$	−23.3607	−1.65599	−0.827997	+	0.560732i	$0.810520\pi$
$200$	0	0
$201$	−0.562306	−0.0396620
$202$	0	0
$203$	0.854102	0.0599462
$204$	0	0
$205$	0.618034	0.0431654
$206$	0	0
$207$	−3.61803	−0.251471
$208$	0	0
$209$	3.85410	0.266594
$210$	0	0
$211$	15.0344	1.03501	0.517507	−	0.855679i	$-0.326860\pi$
$211$	15.0344	1.03501	0.517507	+	0.855679i	$0.326860\pi$
$212$	0	0
$213$	7.38197	0.505804
$214$	0	0
$215$	−0.618034	−0.0421496
$216$	0	0
$217$	3.09017	0.209774
$218$	0	0
$219$	−4.90983	−0.331776
$220$	0	0
$221$	−0.416408	−0.0280106
$222$	0	0
$223$	−14.3262	−0.959356	−0.479678	−	0.877445i	$-0.659246\pi$
$223$	−14.3262	−0.959356	−0.479678	+	0.877445i	$0.659246\pi$
$224$	0	0
$225$	12.0902	0.806011
$226$	0	0
$227$	25.6525	1.70261	0.851307	−	0.524667i	$-0.175810\pi$
$227$	25.6525	1.70261	0.851307	+	0.524667i	$0.175810\pi$
$228$	0	0
$229$	16.7639	1.10779	0.553896	−	0.832586i	$-0.313141\pi$
$229$	16.7639	1.10779	0.553896	+	0.832586i	$0.313141\pi$
$230$	0	0
$231$	−0.618034	−0.0406637
$232$	0	0
$233$	9.32624	0.610982	0.305491	−	0.952195i	$-0.401179\pi$
$233$	9.32624	0.610982	0.305491	+	0.952195i	$0.401179\pi$
$234$	0	0
$235$	−2.29180	−0.149500
$236$	0	0
$237$	1.05573	0.0685769
$238$	0	0
$239$	12.8541	0.831463	0.415731	−	0.909487i	$-0.363526\pi$
$239$	12.8541	0.831463	0.415731	+	0.909487i	$0.363526\pi$
$240$	0	0
$241$	−9.23607	−0.594947	−0.297474	−	0.954730i	$-0.596144\pi$
$241$	−9.23607	−0.594947	−0.297474	+	0.954730i	$0.596144\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	−0.618034	−0.0394847
$246$	0	0
$247$	−6.79837	−0.432570
$248$	0	0
$249$	2.14590	0.135991
$250$	0	0
$251$	−19.9443	−1.25887	−0.629436	−	0.777053i	$-0.716714\pi$
$251$	−19.9443	−1.25887	−0.629436	+	0.777053i	$0.716714\pi$
$252$	0	0
$253$	1.38197	0.0868835
$254$	0	0
$255$	0.0901699	0.00564666
$256$	0	0
$257$	−16.1246	−1.00583	−0.502913	−	0.864337i	$-0.667738\pi$
$257$	−16.1246	−1.00583	−0.502913	+	0.864337i	$0.667738\pi$
$258$	0	0
$259$	−7.23607	−0.449627
$260$	0	0
$261$	−2.23607	−0.138409
$262$	0	0
$263$	−0.708204	−0.0436697	−0.0218349	−	0.999762i	$-0.506951\pi$
$263$	−0.708204	−0.0436697	−0.0218349	+	0.999762i	$0.506951\pi$
$264$	0	0
$265$	−2.85410	−0.175326
$266$	0	0
$267$	−3.90983	−0.239278
$268$	0	0
$269$	−10.2705	−0.626204	−0.313102	−	0.949719i	$-0.601368\pi$
$269$	−10.2705	−0.626204	−0.313102	+	0.949719i	$0.601368\pi$
$270$	0	0
$271$	0.236068	0.0143401	0.00717005	−	0.999974i	$-0.497718\pi$
$271$	0.236068	0.0143401	0.00717005	+	0.999974i	$0.497718\pi$
$272$	0	0
$273$	1.09017	0.0659801
$274$	0	0
$275$	−4.61803	−0.278478
$276$	0	0
$277$	−23.1803	−1.39277	−0.696386	−	0.717668i	$-0.745210\pi$
$277$	−23.1803	−1.39277	−0.696386	+	0.717668i	$0.745210\pi$
$278$	0	0
$279$	−8.09017	−0.484346
$280$	0	0
$281$	−11.2918	−0.673612	−0.336806	−	0.941574i	$-0.609347\pi$
$281$	−11.2918	−0.673612	−0.336806	+	0.941574i	$0.609347\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	1.47214	0.0872018
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−16.9443	−0.996722
$290$	0	0
$291$	10.8541	0.636279
$292$	0	0
$293$	−7.76393	−0.453574	−0.226787	−	0.973944i	$-0.572822\pi$
$293$	−7.76393	−0.453574	−0.226787	+	0.973944i	$0.572822\pi$
$294$	0	0
$295$	6.56231	0.382072
$296$	0	0
$297$	3.47214	0.201474
$298$	0	0
$299$	−2.43769	−0.140975
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	5.27051	0.302783
$304$	0	0
$305$	−0.326238	−0.0186803
$306$	0	0
$307$	−32.4721	−1.85328	−0.926641	−	0.375947i	$-0.877318\pi$
$307$	−32.4721	−1.85328	−0.926641	+	0.375947i	$0.877318\pi$
$308$	0	0
$309$	6.23607	0.354758
$310$	0	0
$311$	1.61803	0.0917503	0.0458751	−	0.998947i	$-0.485392\pi$
$311$	1.61803	0.0917503	0.0458751	+	0.998947i	$0.485392\pi$
$312$	0	0
$313$	−20.5279	−1.16030	−0.580152	−	0.814508i	$-0.697007\pi$
$313$	−20.5279	−1.16030	−0.580152	+	0.814508i	$0.697007\pi$
$314$	0	0
$315$	1.61803	0.0911659
$316$	0	0
$317$	32.1246	1.80430	0.902149	−	0.431425i	$-0.141989\pi$
$317$	32.1246	1.80430	0.902149	+	0.431425i	$0.141989\pi$
$318$	0	0
$319$	0.854102	0.0478205
$320$	0	0
$321$	−3.23607	−0.180620
$322$	0	0
$323$	0.909830	0.0506243
$324$	0	0
$325$	8.14590	0.451853
$326$	0	0
$327$	0.291796	0.0161364
$328$	0	0
$329$	3.70820	0.204440
$330$	0	0
$331$	1.88854	0.103804	0.0519019	−	0.998652i	$-0.483472\pi$
$331$	1.88854	0.103804	0.0519019	+	0.998652i	$0.483472\pi$
$332$	0	0
$333$	18.9443	1.03814
$334$	0	0
$335$	−0.562306	−0.0307221
$336$	0	0
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	0	0
$339$	7.67376	0.416782
$340$	0	0
$341$	3.09017	0.167342
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.527864	0.0284192
$346$	0	0
$347$	17.7082	0.950626	0.475313	−	0.879817i	$-0.342335\pi$
$347$	17.7082	0.950626	0.475313	+	0.879817i	$0.342335\pi$
$348$	0	0
$349$	3.29180	0.176206	0.0881029	−	0.996111i	$-0.471920\pi$
$349$	3.29180	0.176206	0.0881029	+	0.996111i	$0.471920\pi$
$350$	0	0
$351$	−6.12461	−0.326908
$352$	0	0
$353$	−34.3607	−1.82883	−0.914417	−	0.404773i	$-0.867351\pi$
$353$	−34.3607	−1.82883	−0.914417	+	0.404773i	$0.867351\pi$
$354$	0	0
$355$	7.38197	0.391794
$356$	0	0
$357$	−0.145898	−0.00772174
$358$	0	0
$359$	0.180340	0.00951798	0.00475899	−	0.999989i	$-0.498485\pi$
$359$	0.180340	0.00951798	0.00475899	+	0.999989i	$0.498485\pi$
$360$	0	0
$361$	−4.14590	−0.218205
$362$	0	0
$363$	6.18034	0.324384
$364$	0	0
$365$	−4.90983	−0.256992
$366$	0	0
$367$	6.27051	0.327318	0.163659	−	0.986517i	$-0.447670\pi$
$367$	6.27051	0.327318	0.163659	+	0.986517i	$0.447670\pi$
$368$	0	0
$369$	2.61803	0.136289
$370$	0	0
$371$	4.61803	0.239756
$372$	0	0
$373$	−29.7426	−1.54002	−0.770008	−	0.638034i	$-0.779748\pi$
$373$	−29.7426	−1.54002	−0.770008	+	0.638034i	$0.779748\pi$
$374$	0	0
$375$	−3.67376	−0.189712
$376$	0	0
$377$	−1.50658	−0.0775927
$378$	0	0
$379$	−20.6525	−1.06085	−0.530423	−	0.847733i	$-0.677967\pi$
$379$	−20.6525	−1.06085	−0.530423	+	0.847733i	$0.677967\pi$
$380$	0	0
$381$	−10.3262	−0.529029
$382$	0	0
$383$	0.819660	0.0418827	0.0209413	−	0.999781i	$-0.493334\pi$
$383$	0.819660	0.0418827	0.0209413	+	0.999781i	$0.493334\pi$
$384$	0	0
$385$	−0.618034	−0.0314979
$386$	0	0
$387$	−2.61803	−0.133082
$388$	0	0
$389$	29.6869	1.50519	0.752593	−	0.658486i	$-0.228803\pi$
$389$	29.6869	1.50519	0.752593	+	0.658486i	$0.228803\pi$
$390$	0	0
$391$	0.326238	0.0164986
$392$	0	0
$393$	2.56231	0.129251
$394$	0	0
$395$	1.05573	0.0531194
$396$	0	0
$397$	21.4508	1.07659	0.538294	−	0.842757i	$-0.319069\pi$
$397$	21.4508	1.07659	0.538294	+	0.842757i	$0.319069\pi$
$398$	0	0
$399$	−2.38197	−0.119247
$400$	0	0
$401$	16.7639	0.837151	0.418575	−	0.908182i	$-0.362530\pi$
$401$	16.7639	0.837151	0.418575	+	0.908182i	$0.362530\pi$
$402$	0	0
$403$	−5.45085	−0.271526
$404$	0	0
$405$	−3.52786	−0.175301
$406$	0	0
$407$	−7.23607	−0.358679
$408$	0	0
$409$	5.65248	0.279497	0.139748	−	0.990187i	$-0.455371\pi$
$409$	5.65248	0.279497	0.139748	+	0.990187i	$0.455371\pi$
$410$	0	0
$411$	5.43769	0.268222
$412$	0	0
$413$	−10.6180	−0.522479
$414$	0	0
$415$	2.14590	0.105338
$416$	0	0
$417$	7.38197	0.361496
$418$	0	0
$419$	−24.2705	−1.18569	−0.592846	−	0.805316i	$-0.701996\pi$
$419$	−24.2705	−1.18569	−0.592846	+	0.805316i	$0.701996\pi$
$420$	0	0
$421$	−11.5279	−0.561834	−0.280917	−	0.959732i	$-0.590639\pi$
$421$	−11.5279	−0.561834	−0.280917	+	0.959732i	$0.590639\pi$
$422$	0	0
$423$	−9.70820	−0.472029
$424$	0	0
$425$	−1.09017	−0.0528810
$426$	0	0
$427$	0.527864	0.0255451
$428$	0	0
$429$	1.09017	0.0526339
$430$	0	0
$431$	0.527864	0.0254263	0.0127132	−	0.999919i	$-0.495953\pi$
$431$	0.527864	0.0254263	0.0127132	+	0.999919i	$0.495953\pi$
$432$	0	0
$433$	−30.0344	−1.44336	−0.721682	−	0.692225i	$-0.756631\pi$
$433$	−30.0344	−1.44336	−0.721682	+	0.692225i	$0.756631\pi$
$434$	0	0
$435$	0.326238	0.0156419
$436$	0	0
$437$	5.32624	0.254789
$438$	0	0
$439$	−3.94427	−0.188250	−0.0941249	−	0.995560i	$-0.530005\pi$
$439$	−3.94427	−0.188250	−0.0941249	+	0.995560i	$0.530005\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	0	0
$443$	−10.1459	−0.482046	−0.241023	−	0.970519i	$-0.577483\pi$
$443$	−10.1459	−0.482046	−0.241023	+	0.970519i	$0.577483\pi$
$444$	0	0
$445$	−3.90983	−0.185344
$446$	0	0
$447$	−0.652476	−0.0308610
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−1.00000	−0.0470882
$452$	0	0
$453$	−1.27051	−0.0596938
$454$	0	0
$455$	1.09017	0.0511080
$456$	0	0
$457$	10.4164	0.487259	0.243630	−	0.969868i	$-0.421662\pi$
$457$	10.4164	0.487259	0.243630	+	0.969868i	$0.421662\pi$
$458$	0	0
$459$	0.819660	0.0382585
$460$	0	0
$461$	−7.65248	−0.356411	−0.178206	−	0.983993i	$-0.557029\pi$
$461$	−7.65248	−0.356411	−0.178206	+	0.983993i	$0.557029\pi$
$462$	0	0
$463$	−33.6180	−1.56236	−0.781181	−	0.624304i	$-0.785383\pi$
$463$	−33.6180	−1.56236	−0.781181	+	0.624304i	$0.785383\pi$
$464$	0	0
$465$	1.18034	0.0547370
$466$	0	0
$467$	−0.236068	−0.0109239	−0.00546196	−	0.999985i	$-0.501739\pi$
$467$	−0.236068	−0.0109239	−0.00546196	+	0.999985i	$0.501739\pi$
$468$	0	0
$469$	0.909830	0.0420120
$470$	0	0
$471$	4.94427	0.227820
$472$	0	0
$473$	1.00000	0.0459800
$474$	0	0
$475$	−17.7984	−0.816645
$476$	0	0
$477$	−12.0902	−0.553571
$478$	0	0
$479$	19.5623	0.893825	0.446912	−	0.894578i	$-0.352524\pi$
$479$	19.5623	0.893825	0.446912	+	0.894578i	$0.352524\pi$
$480$	0	0
$481$	12.7639	0.581985
$482$	0	0
$483$	−0.854102	−0.0388630
$484$	0	0
$485$	10.8541	0.492859
$486$	0	0
$487$	15.2361	0.690412	0.345206	−	0.938527i	$-0.387809\pi$
$487$	15.2361	0.690412	0.345206	+	0.938527i	$0.387809\pi$
$488$	0	0
$489$	0.583592	0.0263909
$490$	0	0
$491$	12.2705	0.553760	0.276880	−	0.960904i	$-0.410699\pi$
$491$	12.2705	0.553760	0.276880	+	0.960904i	$0.410699\pi$
$492$	0	0
$493$	0.201626	0.00908078
$494$	0	0
$495$	1.61803	0.0727252
$496$	0	0
$497$	−11.9443	−0.535774
$498$	0	0
$499$	6.65248	0.297806	0.148903	−	0.988852i	$-0.452426\pi$
$499$	6.65248	0.297806	0.148903	+	0.988852i	$0.452426\pi$
$500$	0	0
$501$	10.1459	0.453285
$502$	0	0
$503$	35.6869	1.59120	0.795601	−	0.605822i	$-0.207155\pi$
$503$	35.6869	1.59120	0.795601	+	0.605822i	$0.207155\pi$
$504$	0	0
$505$	5.27051	0.234535
$506$	0	0
$507$	6.11146	0.271419
$508$	0	0
$509$	−13.0344	−0.577741	−0.288871	−	0.957368i	$-0.593280\pi$
$509$	−13.0344	−0.577741	−0.288871	+	0.957368i	$0.593280\pi$
$510$	0	0
$511$	7.94427	0.351434
$512$	0	0
$513$	13.3820	0.590828
$514$	0	0
$515$	6.23607	0.274794
$516$	0	0
$517$	3.70820	0.163087
$518$	0	0
$519$	5.38197	0.236242
$520$	0	0
$521$	−35.0902	−1.53733	−0.768664	−	0.639653i	$-0.779078\pi$
$521$	−35.0902	−1.53733	−0.768664	+	0.639653i	$0.779078\pi$
$522$	0	0
$523$	−8.52786	−0.372897	−0.186449	−	0.982465i	$-0.559698\pi$
$523$	−8.52786	−0.372897	−0.186449	+	0.982465i	$0.559698\pi$
$524$	0	0
$525$	2.85410	0.124563
$526$	0	0
$527$	0.729490	0.0317771
$528$	0	0
$529$	−21.0902	−0.916964
$530$	0	0
$531$	27.7984	1.20635
$532$	0	0
$533$	1.76393	0.0764044
$534$	0	0
$535$	−3.23607	−0.139907
$536$	0	0
$537$	−16.0902	−0.694342
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−1.74265	−0.0749222	−0.0374611	−	0.999298i	$-0.511927\pi$
$541$	−1.74265	−0.0749222	−0.0374611	+	0.999298i	$0.511927\pi$
$542$	0	0
$543$	−9.23607	−0.396358
$544$	0	0
$545$	0.291796	0.0124992
$546$	0	0
$547$	15.0000	0.641354	0.320677	−	0.947189i	$-0.396090\pi$
$547$	15.0000	0.641354	0.320677	+	0.947189i	$0.396090\pi$
$548$	0	0
$549$	−1.38197	−0.0589809
$550$	0	0
$551$	3.29180	0.140235
$552$	0	0
$553$	−1.70820	−0.0726402
$554$	0	0
$555$	−2.76393	−0.117322
$556$	0	0
$557$	23.2148	0.983642	0.491821	−	0.870696i	$-0.336331\pi$
$557$	23.2148	0.983642	0.491821	+	0.870696i	$0.336331\pi$
$558$	0	0
$559$	−1.76393	−0.0746064
$560$	0	0
$561$	−0.145898	−0.00615982
$562$	0	0
$563$	−0.527864	−0.0222468	−0.0111234	−	0.999938i	$-0.503541\pi$
$563$	−0.527864	−0.0222468	−0.0111234	+	0.999938i	$0.503541\pi$
$564$	0	0
$565$	7.67376	0.322838
$566$	0	0
$567$	5.70820	0.239722
$568$	0	0
$569$	2.61803	0.109754	0.0548768	−	0.998493i	$-0.482523\pi$
$569$	2.61803	0.109754	0.0548768	+	0.998493i	$0.482523\pi$
$570$	0	0
$571$	15.8885	0.664915	0.332457	−	0.943118i	$-0.392122\pi$
$571$	15.8885	0.664915	0.332457	+	0.943118i	$0.392122\pi$
$572$	0	0
$573$	8.36068	0.349272
$574$	0	0
$575$	−6.38197	−0.266146
$576$	0	0
$577$	1.18034	0.0491382	0.0245691	−	0.999698i	$-0.492179\pi$
$577$	1.18034	0.0491382	0.0245691	+	0.999698i	$0.492179\pi$
$578$	0	0
$579$	5.74265	0.238656
$580$	0	0
$581$	−3.47214	−0.144048
$582$	0	0
$583$	4.61803	0.191259
$584$	0	0
$585$	−2.85410	−0.118003
$586$	0	0
$587$	−21.0689	−0.869606	−0.434803	−	0.900526i	$-0.643182\pi$
$587$	−21.0689	−0.869606	−0.434803	+	0.900526i	$0.643182\pi$
$588$	0	0
$589$	11.9098	0.490736
$590$	0	0
$591$	4.56231	0.187668
$592$	0	0
$593$	18.8328	0.773371	0.386686	−	0.922212i	$-0.373620\pi$
$593$	18.8328	0.773371	0.386686	+	0.922212i	$0.373620\pi$
$594$	0	0
$595$	−0.145898	−0.00598124
$596$	0	0
$597$	14.4377	0.590895
$598$	0	0
$599$	4.65248	0.190095	0.0950475	−	0.995473i	$-0.469700\pi$
$599$	4.65248	0.190095	0.0950475	+	0.995473i	$0.469700\pi$
$600$	0	0
$601$	6.56231	0.267682	0.133841	−	0.991003i	$-0.457269\pi$
$601$	6.56231	0.267682	0.133841	+	0.991003i	$0.457269\pi$
$602$	0	0
$603$	−2.38197	−0.0970012
$604$	0	0
$605$	6.18034	0.251267
$606$	0	0
$607$	41.8328	1.69794	0.848971	−	0.528440i	$-0.177223\pi$
$607$	41.8328	1.69794	0.848971	+	0.528440i	$0.177223\pi$
$608$	0	0
$609$	−0.527864	−0.0213901
$610$	0	0
$611$	−6.54102	−0.264621
$612$	0	0
$613$	27.2705	1.10145	0.550723	−	0.834688i	$-0.314352\pi$
$613$	27.2705	1.10145	0.550723	+	0.834688i	$0.314352\pi$
$614$	0	0
$615$	−0.381966	−0.0154024
$616$	0	0
$617$	16.5967	0.668160	0.334080	−	0.942545i	$-0.391574\pi$
$617$	16.5967	0.668160	0.334080	+	0.942545i	$0.391574\pi$
$618$	0	0
$619$	21.3262	0.857174	0.428587	−	0.903501i	$-0.359012\pi$
$619$	21.3262	0.857174	0.428587	+	0.903501i	$0.359012\pi$
$620$	0	0
$621$	4.79837	0.192552
$622$	0	0
$623$	6.32624	0.253455
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	−2.38197	−0.0951266
$628$	0	0
$629$	−1.70820	−0.0681106
$630$	0	0
$631$	−44.4853	−1.77093	−0.885466	−	0.464705i	$-0.846161\pi$
$631$	−44.4853	−1.77093	−0.885466	+	0.464705i	$0.846161\pi$
$632$	0	0
$633$	−9.29180	−0.369316
$634$	0	0
$635$	−10.3262	−0.409784
$636$	0	0
$637$	−1.76393	−0.0698895
$638$	0	0
$639$	31.2705	1.23704
$640$	0	0
$641$	−5.18034	−0.204611	−0.102306	−	0.994753i	$-0.532622\pi$
$641$	−5.18034	−0.204611	−0.102306	+	0.994753i	$0.532622\pi$
$642$	0	0
$643$	36.9230	1.45610	0.728050	−	0.685524i	$-0.240427\pi$
$643$	36.9230	1.45610	0.728050	+	0.685524i	$0.240427\pi$
$644$	0	0
$645$	0.381966	0.0150399
$646$	0	0
$647$	−23.3607	−0.918403	−0.459202	−	0.888332i	$-0.651864\pi$
$647$	−23.3607	−0.918403	−0.459202	+	0.888332i	$0.651864\pi$
$648$	0	0
$649$	−10.6180	−0.416794
$650$	0	0
$651$	−1.90983	−0.0748521
$652$	0	0
$653$	25.3820	0.993273	0.496637	−	0.867959i	$-0.334568\pi$
$653$	25.3820	0.993273	0.496637	+	0.867959i	$0.334568\pi$
$654$	0	0
$655$	2.56231	0.100118
$656$	0	0
$657$	−20.7984	−0.811422
$658$	0	0
$659$	19.0689	0.742818	0.371409	−	0.928469i	$-0.378875\pi$
$659$	19.0689	0.742818	0.371409	+	0.928469i	$0.378875\pi$
$660$	0	0
$661$	35.2361	1.37052	0.685262	−	0.728297i	$-0.259688\pi$
$661$	35.2361	1.37052	0.685262	+	0.728297i	$0.259688\pi$
$662$	0	0
$663$	0.257354	0.00999481
$664$	0	0
$665$	−2.38197	−0.0923687
$666$	0	0
$667$	1.18034	0.0457029
$668$	0	0
$669$	8.85410	0.342319
$670$	0	0
$671$	0.527864	0.0203780
$672$	0	0
$673$	−33.8328	−1.30416	−0.652080	−	0.758151i	$-0.726103\pi$
$673$	−33.8328	−1.30416	−0.652080	+	0.758151i	$0.726103\pi$
$674$	0	0
$675$	−16.0344	−0.617166
$676$	0	0
$677$	43.0902	1.65609	0.828045	−	0.560662i	$-0.189453\pi$
$677$	43.0902	1.65609	0.828045	+	0.560662i	$0.189453\pi$
$678$	0	0
$679$	−17.5623	−0.673979
$680$	0	0
$681$	−15.8541	−0.607531
$682$	0	0
$683$	−35.9443	−1.37537	−0.687685	−	0.726009i	$-0.741373\pi$
$683$	−35.9443	−1.37537	−0.687685	+	0.726009i	$0.741373\pi$
$684$	0	0
$685$	5.43769	0.207764
$686$	0	0
$687$	−10.3607	−0.395285
$688$	0	0
$689$	−8.14590	−0.310334
$690$	0	0
$691$	−21.3262	−0.811288	−0.405644	−	0.914031i	$-0.632953\pi$
$691$	−21.3262	−0.811288	−0.405644	+	0.914031i	$0.632953\pi$
$692$	0	0
$693$	−2.61803	−0.0994509
$694$	0	0
$695$	7.38197	0.280014
$696$	0	0
$697$	−0.236068	−0.00894171
$698$	0	0
$699$	−5.76393	−0.218012
$700$	0	0
$701$	5.38197	0.203274	0.101637	−	0.994822i	$-0.467592\pi$
$701$	5.38197	0.203274	0.101637	+	0.994822i	$0.467592\pi$
$702$	0	0
$703$	−27.8885	−1.05184
$704$	0	0
$705$	1.41641	0.0533450
$706$	0	0
$707$	−8.52786	−0.320723
$708$	0	0
$709$	9.12461	0.342682	0.171341	−	0.985212i	$-0.445190\pi$
$709$	9.12461	0.342682	0.171341	+	0.985212i	$0.445190\pi$
$710$	0	0
$711$	4.47214	0.167718
$712$	0	0
$713$	4.27051	0.159932
$714$	0	0
$715$	1.09017	0.0407700
$716$	0	0
$717$	−7.94427	−0.296684
$718$	0	0
$719$	23.1803	0.864481	0.432240	−	0.901758i	$-0.357723\pi$
$719$	23.1803	0.864481	0.432240	+	0.901758i	$0.357723\pi$
$720$	0	0
$721$	−10.0902	−0.375778
$722$	0	0
$723$	5.70820	0.212290
$724$	0	0
$725$	−3.94427	−0.146487
$726$	0	0
$727$	28.1591	1.04436	0.522181	−	0.852835i	$-0.325119\pi$
$727$	28.1591	1.04436	0.522181	+	0.852835i	$0.325119\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	0.236068	0.00873129
$732$	0	0
$733$	−19.9443	−0.736658	−0.368329	−	0.929695i	$-0.620070\pi$
$733$	−19.9443	−0.736658	−0.368329	+	0.929695i	$0.620070\pi$
$734$	0	0
$735$	0.381966	0.0140890
$736$	0	0
$737$	0.909830	0.0335140
$738$	0	0
$739$	32.6738	1.20192	0.600962	−	0.799278i	$-0.294784\pi$
$739$	32.6738	1.20192	0.600962	+	0.799278i	$0.294784\pi$
$740$	0	0
$741$	4.20163	0.154351
$742$	0	0
$743$	9.68692	0.355379	0.177689	−	0.984087i	$-0.443138\pi$
$743$	9.68692	0.355379	0.177689	+	0.984087i	$0.443138\pi$
$744$	0	0
$745$	−0.652476	−0.0239049
$746$	0	0
$747$	9.09017	0.332592
$748$	0	0
$749$	5.23607	0.191322
$750$	0	0
$751$	31.5410	1.15095	0.575474	−	0.817820i	$-0.304818\pi$
$751$	31.5410	1.15095	0.575474	+	0.817820i	$0.304818\pi$
$752$	0	0
$753$	12.3262	0.449193
$754$	0	0
$755$	−1.27051	−0.0462386
$756$	0	0
$757$	−26.6525	−0.968701	−0.484350	−	0.874874i	$-0.660944\pi$
$757$	−26.6525	−0.968701	−0.484350	+	0.874874i	$0.660944\pi$
$758$	0	0
$759$	−0.854102	−0.0310019
$760$	0	0
$761$	39.7771	1.44192	0.720959	−	0.692978i	$-0.243702\pi$
$761$	39.7771	1.44192	0.720959	+	0.692978i	$0.243702\pi$
$762$	0	0
$763$	−0.472136	−0.0170925
$764$	0	0
$765$	0.381966	0.0138100
$766$	0	0
$767$	18.7295	0.676283
$768$	0	0
$769$	−23.8197	−0.858959	−0.429479	−	0.903077i	$-0.641303\pi$
$769$	−23.8197	−0.858959	−0.429479	+	0.903077i	$0.641303\pi$
$770$	0	0
$771$	9.96556	0.358901
$772$	0	0
$773$	−33.1803	−1.19341	−0.596707	−	0.802459i	$-0.703525\pi$
$773$	−33.1803	−1.19341	−0.596707	+	0.802459i	$0.703525\pi$
$774$	0	0
$775$	−14.2705	−0.512612
$776$	0	0
$777$	4.47214	0.160437
$778$	0	0
$779$	−3.85410	−0.138088
$780$	0	0
$781$	−11.9443	−0.427400
$782$	0	0
$783$	2.96556	0.105980
$784$	0	0
$785$	4.94427	0.176469
$786$	0	0
$787$	4.18034	0.149013	0.0745065	−	0.997221i	$-0.476262\pi$
$787$	4.18034	0.149013	0.0745065	+	0.997221i	$0.476262\pi$
$788$	0	0
$789$	0.437694	0.0155823
$790$	0	0
$791$	−12.4164	−0.441477
$792$	0	0
$793$	−0.931116	−0.0330649
$794$	0	0
$795$	1.76393	0.0625602
$796$	0	0
$797$	29.8328	1.05673	0.528366	−	0.849017i	$-0.322805\pi$
$797$	29.8328	1.05673	0.528366	+	0.849017i	$0.322805\pi$
$798$	0	0
$799$	0.875388	0.0309690
$800$	0	0
$801$	−16.5623	−0.585200
$802$	0	0
$803$	7.94427	0.280347
$804$	0	0
$805$	−0.854102	−0.0301031
$806$	0	0
$807$	6.34752	0.223443
$808$	0	0
$809$	−47.8115	−1.68096	−0.840482	−	0.541840i	$-0.817728\pi$
$809$	−47.8115	−1.68096	−0.840482	+	0.541840i	$0.817728\pi$
$810$	0	0
$811$	−49.0689	−1.72304	−0.861521	−	0.507722i	$-0.830488\pi$
$811$	−49.0689	−1.72304	−0.861521	+	0.507722i	$0.830488\pi$
$812$	0	0
$813$	−0.145898	−0.00511687
$814$	0	0
$815$	0.583592	0.0204423
$816$	0	0
$817$	3.85410	0.134838
$818$	0	0
$819$	4.61803	0.161367
$820$	0	0
$821$	50.7771	1.77213	0.886066	−	0.463559i	$-0.153428\pi$
$821$	50.7771	1.77213	0.886066	+	0.463559i	$0.153428\pi$
$822$	0	0
$823$	6.96556	0.242804	0.121402	−	0.992603i	$-0.461261\pi$
$823$	6.96556	0.242804	0.121402	+	0.992603i	$0.461261\pi$
$824$	0	0
$825$	2.85410	0.0993671
$826$	0	0
$827$	45.2492	1.57347	0.786735	−	0.617291i	$-0.211770\pi$
$827$	45.2492	1.57347	0.786735	+	0.617291i	$0.211770\pi$
$828$	0	0
$829$	37.3607	1.29759	0.648795	−	0.760963i	$-0.275273\pi$
$829$	37.3607	1.29759	0.648795	+	0.760963i	$0.275273\pi$
$830$	0	0
$831$	14.3262	0.496972
$832$	0	0
$833$	0.236068	0.00817927
$834$	0	0
$835$	10.1459	0.351113
$836$	0	0
$837$	10.7295	0.370865
$838$	0	0
$839$	−1.23607	−0.0426738	−0.0213369	−	0.999772i	$-0.506792\pi$
$839$	−1.23607	−0.0426738	−0.0213369	+	0.999772i	$0.506792\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	0	0
$843$	6.97871	0.240360
$844$	0	0
$845$	6.11146	0.210240
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.0000	−0.342796
$852$	0	0
$853$	−52.2492	−1.78898	−0.894490	−	0.447089i	$-0.852461\pi$
$853$	−52.2492	−1.78898	−0.894490	+	0.447089i	$0.852461\pi$
$854$	0	0
$855$	6.23607	0.213269
$856$	0	0
$857$	2.38197	0.0813664	0.0406832	−	0.999172i	$-0.487047\pi$
$857$	2.38197	0.0813664	0.0406832	+	0.999172i	$0.487047\pi$
$858$	0	0
$859$	19.9656	0.681216	0.340608	−	0.940205i	$-0.389367\pi$
$859$	19.9656	0.681216	0.340608	+	0.940205i	$0.389367\pi$
$860$	0	0
$861$	0.618034	0.0210625
$862$	0	0
$863$	29.7639	1.01318	0.506588	−	0.862188i	$-0.330907\pi$
$863$	29.7639	1.01318	0.506588	+	0.862188i	$0.330907\pi$
$864$	0	0
$865$	5.38197	0.182992
$866$	0	0
$867$	10.4721	0.355652
$868$	0	0
$869$	−1.70820	−0.0579468
$870$	0	0
$871$	−1.60488	−0.0543792
$872$	0	0
$873$	45.9787	1.55614
$874$	0	0
$875$	5.94427	0.200953
$876$	0	0
$877$	8.85410	0.298982	0.149491	−	0.988763i	$-0.452237\pi$
$877$	8.85410	0.298982	0.149491	+	0.988763i	$0.452237\pi$
$878$	0	0
$879$	4.79837	0.161845
$880$	0	0
$881$	29.4164	0.991064	0.495532	−	0.868590i	$-0.334973\pi$
$881$	29.4164	0.991064	0.495532	+	0.868590i	$0.334973\pi$
$882$	0	0
$883$	−3.38197	−0.113812	−0.0569061	−	0.998380i	$-0.518124\pi$
$883$	−3.38197	−0.113812	−0.0569061	+	0.998380i	$0.518124\pi$
$884$	0	0
$885$	−4.05573	−0.136332
$886$	0	0
$887$	−10.1115	−0.339509	−0.169755	−	0.985486i	$-0.554298\pi$
$887$	−10.1115	−0.339509	−0.169755	+	0.985486i	$0.554298\pi$
$888$	0	0
$889$	16.7082	0.560375
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	14.2918	0.478257
$894$	0	0
$895$	−16.0902	−0.537835
$896$	0	0
$897$	1.50658	0.0503032
$898$	0	0
$899$	2.63932	0.0880263
$900$	0	0
$901$	1.09017	0.0363188
$902$	0	0
$903$	−0.618034	−0.0205669
$904$	0	0
$905$	−9.23607	−0.307017
$906$	0	0
$907$	−12.1591	−0.403735	−0.201867	−	0.979413i	$-0.564701\pi$
$907$	−12.1591	−0.403735	−0.201867	+	0.979413i	$0.564701\pi$
$908$	0	0
$909$	22.3262	0.740515
$910$	0	0
$911$	56.3607	1.86731	0.933656	−	0.358170i	$-0.116599\pi$
$911$	56.3607	1.86731	0.933656	+	0.358170i	$0.116599\pi$
$912$	0	0
$913$	−3.47214	−0.114911
$914$	0	0
$915$	0.201626	0.00666555
$916$	0	0
$917$	−4.14590	−0.136910
$918$	0	0
$919$	56.4508	1.86214	0.931071	−	0.364838i	$-0.118876\pi$
$919$	56.4508	1.86214	0.931071	+	0.364838i	$0.118876\pi$
$920$	0	0
$921$	20.0689	0.661292
$922$	0	0
$923$	21.0689	0.693491
$924$	0	0
$925$	33.4164	1.09872
$926$	0	0
$927$	26.4164	0.867629
$928$	0	0
$929$	−24.5623	−0.805863	−0.402932	−	0.915230i	$-0.632009\pi$
$929$	−24.5623	−0.805863	−0.402932	+	0.915230i	$0.632009\pi$
$930$	0	0
$931$	3.85410	0.126313
$932$	0	0
$933$	−1.00000	−0.0327385
$934$	0	0
$935$	−0.145898	−0.00477138
$936$	0	0
$937$	−50.4164	−1.64703	−0.823516	−	0.567293i	$-0.807991\pi$
$937$	−50.4164	−1.64703	−0.823516	+	0.567293i	$0.807991\pi$
$938$	0	0
$939$	12.6869	0.414022
$940$	0	0
$941$	−36.8673	−1.20184	−0.600919	−	0.799310i	$-0.705199\pi$
$941$	−36.8673	−1.20184	−0.600919	+	0.799310i	$0.705199\pi$
$942$	0	0
$943$	−1.38197	−0.0450030
$944$	0	0
$945$	−2.14590	−0.0698061
$946$	0	0
$947$	−12.2705	−0.398738	−0.199369	−	0.979924i	$-0.563889\pi$
$947$	−12.2705	−0.398738	−0.199369	+	0.979924i	$0.563889\pi$
$948$	0	0
$949$	−14.0132	−0.454886
$950$	0	0
$951$	−19.8541	−0.643813
$952$	0	0
$953$	10.0902	0.326853	0.163426	−	0.986556i	$-0.447745\pi$
$953$	10.0902	0.326853	0.163426	+	0.986556i	$0.447745\pi$
$954$	0	0
$955$	8.36068	0.270545
$956$	0	0
$957$	−0.527864	−0.0170634
$958$	0	0
$959$	−8.79837	−0.284114
$960$	0	0
$961$	−21.4508	−0.691963
$962$	0	0
$963$	−13.7082	−0.441741
$964$	0	0
$965$	5.74265	0.184862
$966$	0	0
$967$	5.88854	0.189363	0.0946814	−	0.995508i	$-0.469817\pi$
$967$	5.88854	0.189363	0.0946814	+	0.995508i	$0.469817\pi$
$968$	0	0
$969$	−0.562306	−0.0180639
$970$	0	0
$971$	−4.59675	−0.147517	−0.0737583	−	0.997276i	$-0.523499\pi$
$971$	−4.59675	−0.147517	−0.0737583	+	0.997276i	$0.523499\pi$
$972$	0	0
$973$	−11.9443	−0.382916
$974$	0	0
$975$	−5.03444	−0.161231
$976$	0	0
$977$	51.7984	1.65718	0.828588	−	0.559858i	$-0.189144\pi$
$977$	51.7984	1.65718	0.828588	+	0.559858i	$0.189144\pi$
$978$	0	0
$979$	6.32624	0.202187
$980$	0	0
$981$	1.23607	0.0394646
$982$	0	0
$983$	−34.5279	−1.10127	−0.550634	−	0.834747i	$-0.685614\pi$
$983$	−34.5279	−1.10127	−0.550634	+	0.834747i	$0.685614\pi$
$984$	0	0
$985$	4.56231	0.145367
$986$	0	0
$987$	−2.29180	−0.0729487
$988$	0	0
$989$	1.38197	0.0439440
$990$	0	0
$991$	−16.8197	−0.534294	−0.267147	−	0.963656i	$-0.586081\pi$
$991$	−16.8197	−0.534294	−0.267147	+	0.963656i	$0.586081\pi$
$992$	0	0
$993$	−1.16718	−0.0370395
$994$	0	0
$995$	14.4377	0.457706
$996$	0	0
$997$	−55.3951	−1.75438	−0.877191	−	0.480142i	$-0.840585\pi$
$997$	−55.3951	−1.75438	−0.877191	+	0.480142i	$0.840585\pi$
$998$	0	0
$999$	−25.1246	−0.794908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4592.2.a.n.1.1		2
4.3	odd	2		287.2.a.b.1.1	✓	2
12.11	even	2		2583.2.a.g.1.2		2
20.19	odd	2		7175.2.a.g.1.2		2
28.27	even	2		2009.2.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.b.1.1	✓	2	4.3	odd	2
2009.2.a.a.1.1		2	28.27	even	2
2583.2.a.g.1.2		2	12.11	even	2
4592.2.a.n.1.1		2	1.1	even	1	trivial
7175.2.a.g.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4592.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} -0.618034 q^{5} +1.00000 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} -0.618034 q^{5} +1.00000 q^{7} -2.61803 q^{9} +1.00000 q^{11} -1.76393 q^{13} +0.381966 q^{15} +0.236068 q^{17} +3.85410 q^{19} -0.618034 q^{21} +1.38197 q^{23} -4.61803 q^{25} +3.47214 q^{27} +0.854102 q^{29} +3.09017 q^{31} -0.618034 q^{33} -0.618034 q^{35} -7.23607 q^{37} +1.09017 q^{39} -1.00000 q^{41} +1.00000 q^{43} +1.61803 q^{45} +3.70820 q^{47} +1.00000 q^{49} -0.145898 q^{51} +4.61803 q^{53} -0.618034 q^{55} -2.38197 q^{57} -10.6180 q^{59} +0.527864 q^{61} -2.61803 q^{63} +1.09017 q^{65} +0.909830 q^{67} -0.854102 q^{69} -11.9443 q^{71} +7.94427 q^{73} +2.85410 q^{75} +1.00000 q^{77} -1.70820 q^{79} +5.70820 q^{81} -3.47214 q^{83} -0.145898 q^{85} -0.527864 q^{87} +6.32624 q^{89} -1.76393 q^{91} -1.90983 q^{93} -2.38197 q^{95} -17.5623 q^{97} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9	\( 2 q + q^{3} + q^{5} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 8 q^{13} + 3 q^{15} - 4 q^{17} + q^{19} + q^{21} + 5 q^{23} - 7 q^{25} - 2 q^{27} - 5 q^{29} - 5 q^{31} + q^{33} + q^{35} - 10 q^{37} - 9 q^{39} - 2 q^{41} + 2 q^{43} + q^{45} - 6 q^{47} + 2 q^{49} - 7 q^{51} + 7 q^{53} + q^{55} - 7 q^{57} - 19 q^{59} + 10 q^{61} - 3 q^{63} - 9 q^{65} + 13 q^{67} + 5 q^{69} - 6 q^{71} - 2 q^{73} - q^{75} + 2 q^{77} + 10 q^{79} - 2 q^{81} + 2 q^{83} - 7 q^{85} - 10 q^{87} - 3 q^{89} - 8 q^{91} - 15 q^{93} - 7 q^{95} - 15 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 8 * q^13 + 3 * q^15 - 4 * q^17 + q^19 + q^21 + 5 * q^23 - 7 * q^25 - 2 * q^27 - 5 * q^29 - 5 * q^31 + q^33 + q^35 - 10 * q^37 - 9 * q^39 - 2 * q^41 + 2 * q^43 + q^45 - 6 * q^47 + 2 * q^49 - 7 * q^51 + 7 * q^53 + q^55 - 7 * q^57 - 19 * q^59 + 10 * q^61 - 3 * q^63 - 9 * q^65 + 13 * q^67 + 5 * q^69 - 6 * q^71 - 2 * q^73 - q^75 + 2 * q^77 + 10 * q^79 - 2 * q^81 + 2 * q^83 - 7 * q^85 - 10 * q^87 - 3 * q^89 - 8 * q^91 - 15 * q^93 - 7 * q^95 - 15 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more