LMFDB - Embedded newform 3549.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3549,2,Mod(1,3549)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3549, 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("3549.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3549 = 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3549.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:	$28.3389076774$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	3549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.554958 q^{2} -1.00000 q^{3} -1.69202 q^{4} +0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.554958 q^{2} -1.00000 q^{3} -1.69202 q^{4} +0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} -0.445042 q^{11} +1.69202 q^{12} -0.554958 q^{14} +2.24698 q^{16} +3.60388 q^{17} -0.554958 q^{18} -4.09783 q^{19} -1.00000 q^{21} +0.246980 q^{22} -7.00969 q^{23} -2.04892 q^{24} -5.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +1.74094 q^{29} +6.49396 q^{31} -5.34481 q^{32} +0.445042 q^{33} -2.00000 q^{34} -1.69202 q^{36} -7.18598 q^{37} +2.27413 q^{38} -6.89008 q^{41} +0.554958 q^{42} +8.85086 q^{43} +0.753020 q^{44} +3.89008 q^{46} +9.48188 q^{47} -2.24698 q^{48} +1.00000 q^{49} +2.77479 q^{50} -3.60388 q^{51} +9.28382 q^{53} +0.554958 q^{54} +2.04892 q^{56} +4.09783 q^{57} -0.966148 q^{58} +4.98792 q^{59} -0.933624 q^{61} -3.60388 q^{62} +1.00000 q^{63} -1.52781 q^{64} -0.246980 q^{66} +1.70410 q^{67} -6.09783 q^{68} +7.00969 q^{69} -3.80194 q^{71} +2.04892 q^{72} +7.30559 q^{73} +3.98792 q^{74} +5.00000 q^{75} +6.93362 q^{76} -0.445042 q^{77} -0.621334 q^{79} +1.00000 q^{81} +3.82371 q^{82} -4.27413 q^{83} +1.69202 q^{84} -4.91185 q^{86} -1.74094 q^{87} -0.911854 q^{88} +8.81163 q^{89} +11.8605 q^{92} -6.49396 q^{93} -5.26205 q^{94} +5.34481 q^{96} -6.59179 q^{97} -0.554958 q^{98} -0.445042 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 3 q^{3} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 - 3 * q^3 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 2 q^{2} - 3 q^{3} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 6 q^{19} - 3 q^{21} - 4 q^{22} + q^{23} + 3 q^{24} - 15 q^{25} - 3 q^{27} - 9 q^{29} + 10 q^{31} + 7 q^{32} + q^{33} - 6 q^{34} - 7 q^{37} - 4 q^{38} - 20 q^{41} + 2 q^{42} + 13 q^{43} + 7 q^{44} + 11 q^{46} - 2 q^{48} + 3 q^{49} + 10 q^{50} - 2 q^{51} - 5 q^{53} + 2 q^{54} - 3 q^{56} - 6 q^{57} + 13 q^{58} - 4 q^{59} + 4 q^{61} - 2 q^{62} + 3 q^{63} - 11 q^{64} + 4 q^{66} + 19 q^{67} - q^{69} - 7 q^{71} - 3 q^{72} - 14 q^{73} - 7 q^{74} + 15 q^{75} + 14 q^{76} - q^{77} - 9 q^{79} + 3 q^{81} + 4 q^{82} - 2 q^{83} - 11 q^{86} + 9 q^{87} + q^{88} - 10 q^{93} + 14 q^{94} - 7 q^{96} + 8 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 - 3 * q^3 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - q^11 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 6 * q^19 - 3 * q^21 - 4 * q^22 + q^23 + 3 * q^24 - 15 * q^25 - 3 * q^27 - 9 * q^29 + 10 * q^31 + 7 * q^32 + q^33 - 6 * q^34 - 7 * q^37 - 4 * q^38 - 20 * q^41 + 2 * q^42 + 13 * q^43 + 7 * q^44 + 11 * q^46 - 2 * q^48 + 3 * q^49 + 10 * q^50 - 2 * q^51 - 5 * q^53 + 2 * q^54 - 3 * q^56 - 6 * q^57 + 13 * q^58 - 4 * q^59 + 4 * q^61 - 2 * q^62 + 3 * q^63 - 11 * q^64 + 4 * q^66 + 19 * q^67 - q^69 - 7 * q^71 - 3 * q^72 - 14 * q^73 - 7 * q^74 + 15 * q^75 + 14 * q^76 - q^77 - 9 * q^79 + 3 * q^81 + 4 * q^82 - 2 * q^83 - 11 * q^86 + 9 * q^87 + q^88 - 10 * q^93 + 14 * q^94 - 7 * q^96 + 8 * q^97 - 2 * q^98 - 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 $

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.69202	−0.846011
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0.554958	0.226561
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.04892	0.724402
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.445042	−0.134185	−0.0670926	−	0.997747i	$-0.521372\pi$
$11$	−0.445042	−0.134185	−0.0670926	+	0.997747i	$0.521372\pi$
$12$	1.69202	0.488445
$13$	0	0
$13$	0	0
$14$	−0.554958	−0.148319
$15$	0	0
$16$	2.24698	0.561745
$17$	3.60388	0.874068	0.437034	−	0.899445i	$-0.356029\pi$
$17$	3.60388	0.874068	0.437034	+	0.899445i	$0.356029\pi$
$18$	−0.554958	−0.130805
$19$	−4.09783	−0.940108	−0.470054	−	0.882638i	$-0.655766\pi$
$19$	−4.09783	−0.940108	−0.470054	+	0.882638i	$0.655766\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0.246980	0.0526562
$23$	−7.00969	−1.46162	−0.730811	−	0.682580i	$-0.760858\pi$
$23$	−7.00969	−1.46162	−0.730811	+	0.682580i	$0.760858\pi$
$24$	−2.04892	−0.418234
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−1.69202	−0.319762
$29$	1.74094	0.323284	0.161642	−	0.986849i	$-0.448321\pi$
$29$	1.74094	0.323284	0.161642	+	0.986849i	$0.448321\pi$
$30$	0	0
$31$	6.49396	1.16635	0.583175	−	0.812347i	$-0.301810\pi$
$31$	6.49396	1.16635	0.583175	+	0.812347i	$0.301810\pi$
$32$	−5.34481	−0.944839
$33$	0.445042	0.0774718
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−1.69202	−0.282004
$37$	−7.18598	−1.18137	−0.590684	−	0.806903i	$-0.701142\pi$
$37$	−7.18598	−1.18137	−0.590684	+	0.806903i	$0.701142\pi$
$38$	2.27413	0.368912
$39$	0	0
$40$	0	0
$41$	−6.89008	−1.07605	−0.538025	−	0.842929i	$-0.680829\pi$
$41$	−6.89008	−1.07605	−0.538025	+	0.842929i	$0.680829\pi$
$42$	0.554958	0.0856319
$43$	8.85086	1.34974	0.674871	−	0.737935i	$-0.264199\pi$
$43$	8.85086	1.34974	0.674871	+	0.737935i	$0.264199\pi$
$44$	0.753020	0.113522
$45$	0	0
$46$	3.89008	0.573562
$47$	9.48188	1.38307	0.691537	−	0.722341i	$-0.256934\pi$
$47$	9.48188	1.38307	0.691537	+	0.722341i	$0.256934\pi$
$48$	−2.24698	−0.324324
$49$	1.00000	0.142857
$50$	2.77479	0.392415
$51$	−3.60388	−0.504644
$52$	0	0
$53$	9.28382	1.27523	0.637615	−	0.770355i	$-0.279921\pi$
$53$	9.28382	1.27523	0.637615	+	0.770355i	$0.279921\pi$
$54$	0.554958	0.0755202
$55$	0	0
$56$	2.04892	0.273798
$57$	4.09783	0.542771
$58$	−0.966148	−0.126861
$59$	4.98792	0.649372	0.324686	−	0.945822i	$-0.394741\pi$
$59$	4.98792	0.649372	0.324686	+	0.945822i	$0.394741\pi$
$60$	0	0
$61$	−0.933624	−0.119538	−0.0597692	−	0.998212i	$-0.519036\pi$
$61$	−0.933624	−0.119538	−0.0597692	+	0.998212i	$0.519036\pi$
$62$	−3.60388	−0.457693
$63$	1.00000	0.125988
$64$	−1.52781	−0.190976
$65$	0	0
$66$	−0.246980	−0.0304011
$67$	1.70410	0.208189	0.104095	−	0.994567i	$-0.466806\pi$
$67$	1.70410	0.208189	0.104095	+	0.994567i	$0.466806\pi$
$68$	−6.09783	−0.739471
$69$	7.00969	0.843867
$70$	0	0
$71$	−3.80194	−0.451207	−0.225603	−	0.974219i	$-0.572435\pi$
$71$	−3.80194	−0.451207	−0.225603	+	0.974219i	$0.572435\pi$
$72$	2.04892	0.241467
$73$	7.30559	0.855054	0.427527	−	0.904003i	$-0.359385\pi$
$73$	7.30559	0.855054	0.427527	+	0.904003i	$0.359385\pi$
$74$	3.98792	0.463586
$75$	5.00000	0.577350
$76$	6.93362	0.795341
$77$	−0.445042	−0.0507172
$78$	0	0
$79$	−0.621334	−0.0699055	−0.0349528	−	0.999389i	$-0.511128\pi$
$79$	−0.621334	−0.0699055	−0.0349528	+	0.999389i	$0.511128\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.82371	0.422258
$83$	−4.27413	−0.469146	−0.234573	−	0.972098i	$-0.575369\pi$
$83$	−4.27413	−0.469146	−0.234573	+	0.972098i	$0.575369\pi$
$84$	1.69202	0.184615
$85$	0	0
$86$	−4.91185	−0.529659
$87$	−1.74094	−0.186648
$88$	−0.911854	−0.0972040
$89$	8.81163	0.934031	0.467015	−	0.884249i	$-0.345329\pi$
$89$	8.81163	0.934031	0.467015	+	0.884249i	$0.345329\pi$
$90$	0	0
$91$	0	0
$92$	11.8605	1.23655
$93$	−6.49396	−0.673392
$94$	−5.26205	−0.542739
$95$	0	0
$96$	5.34481	0.545503
$97$	−6.59179	−0.669295	−0.334648	−	0.942343i	$-0.608617\pi$
$97$	−6.59179	−0.669295	−0.334648	+	0.942343i	$0.608617\pi$
$98$	−0.554958	−0.0560592
$99$	−0.445042	−0.0447284
$100$	8.46011	0.846011
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	2.00000	0.198030
$103$	−15.3056	−1.50810	−0.754052	−	0.656815i	$-0.771903\pi$
$103$	−15.3056	−1.50810	−0.754052	+	0.656815i	$0.771903\pi$
$104$	0	0
$105$	0	0
$106$	−5.15213	−0.500419
$107$	−14.6649	−1.41771	−0.708853	−	0.705356i	$-0.750787\pi$
$107$	−14.6649	−1.41771	−0.708853	+	0.705356i	$0.750787\pi$
$108$	1.69202	0.162815
$109$	14.8605	1.42338	0.711691	−	0.702493i	$-0.247930\pi$
$109$	14.8605	1.42338	0.711691	+	0.702493i	$0.247930\pi$
$110$	0	0
$111$	7.18598	0.682063
$112$	2.24698	0.212320
$113$	−13.7071	−1.28945	−0.644727	−	0.764413i	$-0.723029\pi$
$113$	−13.7071	−1.28945	−0.644727	+	0.764413i	$0.723029\pi$
$114$	−2.27413	−0.212991
$115$	0	0
$116$	−2.94571	−0.273502
$117$	0	0
$118$	−2.76809	−0.254823
$119$	3.60388	0.330367
$120$	0	0
$121$	−10.8019	−0.981994
$122$	0.518122	0.0469086
$123$	6.89008	0.621258
$124$	−10.9879	−0.986744
$125$	0	0
$126$	−0.554958	−0.0494396
$127$	−16.1371	−1.43193	−0.715966	−	0.698135i	$-0.754014\pi$
$127$	−16.1371	−1.43193	−0.715966	+	0.698135i	$0.754014\pi$
$128$	11.5375	1.01978
$129$	−8.85086	−0.779274
$130$	0	0
$131$	−16.3177	−1.42568	−0.712841	−	0.701326i	$-0.752592\pi$
$131$	−16.3177	−1.42568	−0.712841	+	0.701326i	$0.752592\pi$
$132$	−0.753020	−0.0655420
$133$	−4.09783	−0.355327
$134$	−0.945706	−0.0816965
$135$	0	0
$136$	7.38404	0.633176
$137$	0.719169	0.0614427	0.0307214	−	0.999528i	$-0.490220\pi$
$137$	0.719169	0.0614427	0.0307214	+	0.999528i	$0.490220\pi$
$138$	−3.89008	−0.331146
$139$	2.12200	0.179986	0.0899928	−	0.995942i	$-0.471316\pi$
$139$	2.12200	0.179986	0.0899928	+	0.995942i	$0.471316\pi$
$140$	0	0
$141$	−9.48188	−0.798518
$142$	2.10992	0.177060
$143$	0	0
$144$	2.24698	0.187248
$145$	0	0
$146$	−4.05429	−0.335536
$147$	−1.00000	−0.0824786
$148$	12.1588	0.999450
$149$	−3.16852	−0.259575	−0.129788	−	0.991542i	$-0.541430\pi$
$149$	−3.16852	−0.259575	−0.129788	+	0.991542i	$0.541430\pi$
$150$	−2.77479	−0.226561
$151$	−0.929312	−0.0756264	−0.0378132	−	0.999285i	$-0.512039\pi$
$151$	−0.929312	−0.0756264	−0.0378132	+	0.999285i	$0.512039\pi$
$152$	−8.39612	−0.681016
$153$	3.60388	0.291356
$154$	0.246980	0.0199022
$155$	0	0
$156$	0	0
$157$	5.15346	0.411291	0.205645	−	0.978627i	$-0.434071\pi$
$157$	5.15346	0.411291	0.205645	+	0.978627i	$0.434071\pi$
$158$	0.344814	0.0274320
$159$	−9.28382	−0.736254
$160$	0	0
$161$	−7.00969	−0.552441
$162$	−0.554958	−0.0436016
$163$	15.0707	1.18043	0.590214	−	0.807247i	$-0.299043\pi$
$163$	15.0707	1.18043	0.590214	+	0.807247i	$0.299043\pi$
$164$	11.6582	0.910350
$165$	0	0
$166$	2.37196	0.184100
$167$	−11.9323	−0.923349	−0.461674	−	0.887049i	$-0.652751\pi$
$167$	−11.9323	−0.923349	−0.461674	+	0.887049i	$0.652751\pi$
$168$	−2.04892	−0.158077
$169$	0	0
$170$	0	0
$171$	−4.09783	−0.313369
$172$	−14.9758	−1.14190
$173$	3.03146	0.230478	0.115239	−	0.993338i	$-0.463237\pi$
$173$	3.03146	0.230478	0.115239	+	0.993338i	$0.463237\pi$
$174$	0.966148	0.0732435
$175$	−5.00000	−0.377964
$176$	−1.00000	−0.0753778
$177$	−4.98792	−0.374915
$178$	−4.89008	−0.366527
$179$	−9.87263	−0.737915	−0.368957	−	0.929446i	$-0.620285\pi$
$179$	−9.87263	−0.737915	−0.368957	+	0.929446i	$0.620285\pi$
$180$	0	0
$181$	−11.5362	−0.857477	−0.428738	−	0.903429i	$-0.641042\pi$
$181$	−11.5362	−0.857477	−0.428738	+	0.903429i	$0.641042\pi$
$182$	0	0
$183$	0.933624	0.0690155
$184$	−14.3623	−1.05880
$185$	0	0
$186$	3.60388	0.264249
$187$	−1.60388	−0.117287
$188$	−16.0435	−1.17010
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.12498	−0.226116	−0.113058	−	0.993588i	$-0.536065\pi$
$191$	−3.12498	−0.226116	−0.113058	+	0.993588i	$0.536065\pi$
$192$	1.52781	0.110260
$193$	21.0030	1.51183	0.755914	−	0.654671i	$-0.227193\pi$
$193$	21.0030	1.51183	0.755914	+	0.654671i	$0.227193\pi$
$194$	3.65817	0.262641
$195$	0	0
$196$	−1.69202	−0.120859
$197$	−11.6233	−0.828122	−0.414061	−	0.910249i	$-0.635890\pi$
$197$	−11.6233	−0.828122	−0.414061	+	0.910249i	$0.635890\pi$
$198$	0.246980	0.0175521
$199$	6.86592	0.486712	0.243356	−	0.969937i	$-0.421752\pi$
$199$	6.86592	0.486712	0.243356	+	0.969937i	$0.421752\pi$
$200$	−10.2446	−0.724402
$201$	−1.70410	−0.120198
$202$	3.32975	0.234280
$203$	1.74094	0.122190
$204$	6.09783	0.426934
$205$	0	0
$206$	8.49396	0.591802
$207$	−7.00969	−0.487207
$208$	0	0
$209$	1.82371	0.126149
$210$	0	0
$211$	6.88769	0.474168	0.237084	−	0.971489i	$-0.423808\pi$
$211$	6.88769	0.474168	0.237084	+	0.971489i	$0.423808\pi$
$212$	−15.7084	−1.07886
$213$	3.80194	0.260504
$214$	8.13839	0.556329
$215$	0	0
$216$	−2.04892	−0.139411
$217$	6.49396	0.440839
$218$	−8.24698	−0.558556
$219$	−7.30559	−0.493666
$220$	0	0
$221$	0	0
$222$	−3.98792	−0.267652
$223$	−17.7017	−1.18539	−0.592697	−	0.805425i	$-0.701937\pi$
$223$	−17.7017	−1.18539	−0.592697	+	0.805425i	$0.701937\pi$
$224$	−5.34481	−0.357115
$225$	−5.00000	−0.333333
$226$	7.60686	0.506001
$227$	−25.0315	−1.66140	−0.830698	−	0.556723i	$-0.812058\pi$
$227$	−25.0315	−1.66140	−0.830698	+	0.556723i	$0.812058\pi$
$228$	−6.93362	−0.459190
$229$	−24.4940	−1.61861	−0.809303	−	0.587391i	$-0.800155\pi$
$229$	−24.4940	−1.61861	−0.809303	+	0.587391i	$0.800155\pi$
$230$	0	0
$231$	0.445042	0.0292816
$232$	3.56704	0.234188
$233$	9.19269	0.602233	0.301116	−	0.953587i	$-0.402641\pi$
$233$	9.19269	0.602233	0.301116	+	0.953587i	$0.402641\pi$
$234$	0	0
$235$	0	0
$236$	−8.43967	−0.549375
$237$	0.621334	0.0403600
$238$	−2.00000	−0.129641
$239$	7.91723	0.512123	0.256062	−	0.966660i	$-0.417575\pi$
$239$	7.91723	0.512123	0.256062	+	0.966660i	$0.417575\pi$
$240$	0	0
$241$	1.51679	0.0977053	0.0488527	−	0.998806i	$-0.484444\pi$
$241$	1.51679	0.0977053	0.0488527	+	0.998806i	$0.484444\pi$
$242$	5.99462	0.385349
$243$	−1.00000	−0.0641500
$244$	1.57971	0.101131
$245$	0	0
$246$	−3.82371	−0.243791
$247$	0	0
$248$	13.3056	0.844906
$249$	4.27413	0.270862
$250$	0	0
$251$	−26.0978	−1.64728	−0.823640	−	0.567112i	$-0.808060\pi$
$251$	−26.0978	−1.64728	−0.823640	+	0.567112i	$0.808060\pi$
$252$	−1.69202	−0.106587
$253$	3.11960	0.196128
$254$	8.95539	0.561911
$255$	0	0
$256$	−3.34721	−0.209200
$257$	−5.42758	−0.338563	−0.169282	−	0.985568i	$-0.554145\pi$
$257$	−5.42758	−0.338563	−0.169282	+	0.985568i	$0.554145\pi$
$258$	4.91185	0.305799
$259$	−7.18598	−0.446515
$260$	0	0
$261$	1.74094	0.107761
$262$	9.05562	0.559458
$263$	−12.8955	−0.795168	−0.397584	−	0.917566i	$-0.630151\pi$
$263$	−12.8955	−0.795168	−0.397584	+	0.917566i	$0.630151\pi$
$264$	0.911854	0.0561207
$265$	0	0
$266$	2.27413	0.139436
$267$	−8.81163	−0.539263
$268$	−2.88338	−0.176130
$269$	0.141375	0.00861980	0.00430990	−	0.999991i	$-0.498628\pi$
$269$	0.141375	0.00861980	0.00430990	+	0.999991i	$0.498628\pi$
$270$	0	0
$271$	6.90946	0.419720	0.209860	−	0.977731i	$-0.432699\pi$
$271$	6.90946	0.419720	0.209860	+	0.977731i	$0.432699\pi$
$272$	8.09783	0.491003
$273$	0	0
$274$	−0.399108	−0.0241110
$275$	2.22521	0.134185
$276$	−11.8605	−0.713921
$277$	3.35929	0.201840	0.100920	−	0.994895i	$-0.467821\pi$
$277$	3.35929	0.201840	0.100920	+	0.994895i	$0.467821\pi$
$278$	−1.17762	−0.0706290
$279$	6.49396	0.388783
$280$	0	0
$281$	26.7439	1.59541	0.797704	−	0.603049i	$-0.206048\pi$
$281$	26.7439	1.59541	0.797704	+	0.603049i	$0.206048\pi$
$282$	5.26205	0.313350
$283$	14.6655	0.871771	0.435886	−	0.900002i	$-0.356435\pi$
$283$	14.6655	0.871771	0.435886	+	0.900002i	$0.356435\pi$
$284$	6.43296	0.381726
$285$	0	0
$286$	0	0
$287$	−6.89008	−0.406709
$288$	−5.34481	−0.314946
$289$	−4.01208	−0.236005
$290$	0	0
$291$	6.59179	0.386418
$292$	−12.3612	−0.723385
$293$	5.15346	0.301068	0.150534	−	0.988605i	$-0.451901\pi$
$293$	5.15346	0.301068	0.150534	+	0.988605i	$0.451901\pi$
$294$	0.554958	0.0323658
$295$	0	0
$296$	−14.7235	−0.855785
$297$	0.445042	0.0258239
$298$	1.75840	0.101861
$299$	0	0
$300$	−8.46011	−0.488445
$301$	8.85086	0.510155
$302$	0.515729	0.0296769
$303$	6.00000	0.344691
$304$	−9.20775	−0.528101
$305$	0	0
$306$	−2.00000	−0.114332
$307$	−15.4276	−0.880499	−0.440249	−	0.897876i	$-0.645110\pi$
$307$	−15.4276	−0.880499	−0.440249	+	0.897876i	$0.645110\pi$
$308$	0.753020	0.0429073
$309$	15.3056	0.870704
$310$	0	0
$311$	−34.9638	−1.98261	−0.991306	−	0.131574i	$-0.957997\pi$
$311$	−34.9638	−1.98261	−0.991306	+	0.131574i	$0.957997\pi$
$312$	0	0
$313$	−26.5569	−1.50108	−0.750542	−	0.660823i	$-0.770207\pi$
$313$	−26.5569	−1.50108	−0.750542	+	0.660823i	$0.770207\pi$
$314$	−2.85995	−0.161396
$315$	0	0
$316$	1.05131	0.0591408
$317$	−13.9511	−0.783571	−0.391785	−	0.920057i	$-0.628142\pi$
$317$	−13.9511	−0.783571	−0.391785	+	0.920057i	$0.628142\pi$
$318$	5.15213	0.288917
$319$	−0.774791	−0.0433799
$320$	0	0
$321$	14.6649	0.818513
$322$	3.89008	0.216786
$323$	−14.7681	−0.821718
$324$	−1.69202	−0.0940012
$325$	0	0
$326$	−8.36360	−0.463217
$327$	−14.8605	−0.821790
$328$	−14.1172	−0.779493
$329$	9.48188	0.522753
$330$	0	0
$331$	−5.46011	−0.300115	−0.150057	−	0.988677i	$-0.547946\pi$
$331$	−5.46011	−0.300115	−0.150057	+	0.988677i	$0.547946\pi$
$332$	7.23191	0.396903
$333$	−7.18598	−0.393789
$334$	6.62192	0.362336
$335$	0	0
$336$	−2.24698	−0.122583
$337$	−18.6262	−1.01464	−0.507318	−	0.861759i	$-0.669363\pi$
$337$	−18.6262	−1.01464	−0.507318	+	0.861759i	$0.669363\pi$
$338$	0	0
$339$	13.7071	0.744467
$340$	0	0
$341$	−2.89008	−0.156507
$342$	2.27413	0.122971
$343$	1.00000	0.0539949
$344$	18.1347	0.977756
$345$	0	0
$346$	−1.68233	−0.0904428
$347$	−36.4373	−1.95606	−0.978028	−	0.208475i	$-0.933150\pi$
$347$	−36.4373	−1.95606	−0.978028	+	0.208475i	$0.933150\pi$
$348$	2.94571	0.157906
$349$	−14.5810	−0.780505	−0.390253	−	0.920708i	$-0.627612\pi$
$349$	−14.5810	−0.780505	−0.390253	+	0.920708i	$0.627612\pi$
$350$	2.77479	0.148319
$351$	0	0
$352$	2.37867	0.126783
$353$	15.4276	0.821127	0.410564	−	0.911832i	$-0.365332\pi$
$353$	15.4276	0.821127	0.410564	+	0.911832i	$0.365332\pi$
$354$	2.76809	0.147122
$355$	0	0
$356$	−14.9095	−0.790200
$357$	−3.60388	−0.190737
$358$	5.47889	0.289569
$359$	−24.5894	−1.29778	−0.648889	−	0.760883i	$-0.724766\pi$
$359$	−24.5894	−1.29778	−0.648889	+	0.760883i	$0.724766\pi$
$360$	0	0
$361$	−2.20775	−0.116197
$362$	6.40209	0.336487
$363$	10.8019	0.566955
$364$	0	0
$365$	0	0
$366$	−0.518122	−0.0270827
$367$	28.3370	1.47918	0.739591	−	0.673057i	$-0.235019\pi$
$367$	28.3370	1.47918	0.739591	+	0.673057i	$0.235019\pi$
$368$	−15.7506	−0.821058
$369$	−6.89008	−0.358683
$370$	0	0
$371$	9.28382	0.481992
$372$	10.9879	0.569697
$373$	20.9661	1.08559	0.542793	−	0.839867i	$-0.317367\pi$
$373$	20.9661	1.08559	0.542793	+	0.839867i	$0.317367\pi$
$374$	0.890084	0.0460251
$375$	0	0
$376$	19.4276	1.00190
$377$	0	0
$378$	0.554958	0.0285440
$379$	−15.7845	−0.810794	−0.405397	−	0.914141i	$-0.632867\pi$
$379$	−15.7845	−0.810794	−0.405397	+	0.914141i	$0.632867\pi$
$380$	0	0
$381$	16.1371	0.826727
$382$	1.73423	0.0887311
$383$	−31.2814	−1.59841	−0.799203	−	0.601061i	$-0.794745\pi$
$383$	−31.2814	−1.59841	−0.799203	+	0.601061i	$0.794745\pi$
$384$	−11.5375	−0.588771
$385$	0	0
$386$	−11.6558	−0.593263
$387$	8.85086	0.449914
$388$	11.1535	0.566231
$389$	−8.07308	−0.409321	−0.204661	−	0.978833i	$-0.565609\pi$
$389$	−8.07308	−0.409321	−0.204661	+	0.978833i	$0.565609\pi$
$390$	0	0
$391$	−25.2620	−1.27756
$392$	2.04892	0.103486
$393$	16.3177	0.823117
$394$	6.45042	0.324967
$395$	0	0
$396$	0.753020	0.0378407
$397$	−17.5496	−0.880788	−0.440394	−	0.897805i	$-0.645161\pi$
$397$	−17.5496	−0.880788	−0.440394	+	0.897805i	$0.645161\pi$
$398$	−3.81030	−0.190993
$399$	4.09783	0.205148
$400$	−11.2349	−0.561745
$401$	−7.68664	−0.383853	−0.191926	−	0.981409i	$-0.561473\pi$
$401$	−7.68664	−0.383853	−0.191926	+	0.981409i	$0.561473\pi$
$402$	0.945706	0.0471675
$403$	0	0
$404$	10.1521	0.505087
$405$	0	0
$406$	−0.966148	−0.0479491
$407$	3.19806	0.158522
$408$	−7.38404	−0.365565
$409$	36.3129	1.79556	0.897778	−	0.440448i	$-0.145180\pi$
$409$	36.3129	1.79556	0.897778	+	0.440448i	$0.145180\pi$
$410$	0	0
$411$	−0.719169	−0.0354740
$412$	25.8974	1.27587
$413$	4.98792	0.245439
$414$	3.89008	0.191187
$415$	0	0
$416$	0	0
$417$	−2.12200	−0.103915
$418$	−1.01208	−0.0495025
$419$	2.75733	0.134704	0.0673522	−	0.997729i	$-0.478545\pi$
$419$	2.75733	0.134704	0.0673522	+	0.997729i	$0.478545\pi$
$420$	0	0
$421$	24.8552	1.21137	0.605683	−	0.795706i	$-0.292900\pi$
$421$	24.8552	1.21137	0.605683	+	0.795706i	$0.292900\pi$
$422$	−3.82238	−0.186071
$423$	9.48188	0.461025
$424$	19.0218	0.923779
$425$	−18.0194	−0.874068
$426$	−2.10992	−0.102226
$427$	−0.933624	−0.0451812
$428$	24.8133	1.19940
$429$	0	0
$430$	0	0
$431$	24.6515	1.18742	0.593710	−	0.804679i	$-0.297663\pi$
$431$	24.6515	1.18742	0.593710	+	0.804679i	$0.297663\pi$
$432$	−2.24698	−0.108108
$433$	29.6233	1.42360	0.711801	−	0.702381i	$-0.247880\pi$
$433$	29.6233	1.42360	0.711801	+	0.702381i	$0.247880\pi$
$434$	−3.60388	−0.172992
$435$	0	0
$436$	−25.1444	−1.20420
$437$	28.7245	1.37408
$438$	4.05429	0.193722
$439$	28.6219	1.36605	0.683025	−	0.730395i	$-0.260664\pi$
$439$	28.6219	1.36605	0.683025	+	0.730395i	$0.260664\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.3690	−1.11029	−0.555147	−	0.831752i	$-0.687338\pi$
$443$	−23.3690	−1.11029	−0.555147	+	0.831752i	$0.687338\pi$
$444$	−12.1588	−0.577033
$445$	0	0
$446$	9.82371	0.465166
$447$	3.16852	0.149866
$448$	−1.52781	−0.0721823
$449$	9.44073	0.445536	0.222768	−	0.974872i	$-0.428491\pi$
$449$	9.44073	0.445536	0.222768	+	0.974872i	$0.428491\pi$
$450$	2.77479	0.130805
$451$	3.06638	0.144390
$452$	23.1927	1.09089
$453$	0.929312	0.0436629
$454$	13.8914	0.651956
$455$	0	0
$456$	8.39612	0.393185
$457$	19.3110	0.903329	0.451664	−	0.892188i	$-0.350830\pi$
$457$	19.3110	0.903329	0.451664	+	0.892188i	$0.350830\pi$
$458$	13.5931	0.635165
$459$	−3.60388	−0.168215
$460$	0	0
$461$	−33.9215	−1.57988	−0.789942	−	0.613182i	$-0.789889\pi$
$461$	−33.9215	−1.57988	−0.789942	+	0.613182i	$0.789889\pi$
$462$	−0.246980	−0.0114905
$463$	20.3773	0.947015	0.473508	−	0.880790i	$-0.342988\pi$
$463$	20.3773	0.947015	0.473508	+	0.880790i	$0.342988\pi$
$464$	3.91185	0.181603
$465$	0	0
$466$	−5.10156	−0.236325
$467$	27.5448	1.27462	0.637311	−	0.770607i	$-0.280047\pi$
$467$	27.5448	1.27462	0.637311	+	0.770607i	$0.280047\pi$
$468$	0	0
$469$	1.70410	0.0786882
$470$	0	0
$471$	−5.15346	−0.237459
$472$	10.2198	0.470406
$473$	−3.93900	−0.181115
$474$	−0.344814	−0.0158378
$475$	20.4892	0.940108
$476$	−6.09783	−0.279494
$477$	9.28382	0.425077
$478$	−4.39373	−0.200965
$479$	2.05429	0.0938631	0.0469315	−	0.998898i	$-0.485056\pi$
$479$	2.05429	0.0938631	0.0469315	+	0.998898i	$0.485056\pi$
$480$	0	0
$481$	0	0
$482$	−0.841757	−0.0383410
$483$	7.00969	0.318952
$484$	18.2771	0.830778
$485$	0	0
$486$	0.554958	0.0251734
$487$	7.08516	0.321059	0.160530	−	0.987031i	$-0.448680\pi$
$487$	7.08516	0.321059	0.160530	+	0.987031i	$0.448680\pi$
$488$	−1.91292	−0.0865938
$489$	−15.0707	−0.681520
$490$	0	0
$491$	−36.8984	−1.66520	−0.832602	−	0.553872i	$-0.813150\pi$
$491$	−36.8984	−1.66520	−0.832602	+	0.553872i	$0.813150\pi$
$492$	−11.6582	−0.525591
$493$	6.27413	0.282572
$494$	0	0
$495$	0	0
$496$	14.5918	0.655191
$497$	−3.80194	−0.170540
$498$	−2.37196	−0.106290
$499$	−16.6069	−0.743425	−0.371713	−	0.928348i	$-0.621229\pi$
$499$	−16.6069	−0.743425	−0.371713	+	0.928348i	$0.621229\pi$
$500$	0	0
$501$	11.9323	0.533096
$502$	14.4832	0.646417
$503$	−4.39612	−0.196014	−0.0980068	−	0.995186i	$-0.531247\pi$
$503$	−4.39612	−0.196014	−0.0980068	+	0.995186i	$0.531247\pi$
$504$	2.04892	0.0912660
$505$	0	0
$506$	−1.73125	−0.0769635
$507$	0	0
$508$	27.3043	1.21143
$509$	28.9444	1.28294	0.641468	−	0.767149i	$-0.278326\pi$
$509$	28.9444	1.28294	0.641468	+	0.767149i	$0.278326\pi$
$510$	0	0
$511$	7.30559	0.323180
$512$	−21.2174	−0.937687
$513$	4.09783	0.180924
$514$	3.01208	0.132857
$515$	0	0
$516$	14.9758	0.659274
$517$	−4.21983	−0.185588
$518$	3.98792	0.175219
$519$	−3.03146	−0.133066
$520$	0	0
$521$	−30.6547	−1.34301	−0.671504	−	0.741001i	$-0.734351\pi$
$521$	−30.6547	−1.34301	−0.671504	+	0.741001i	$0.734351\pi$
$522$	−0.966148	−0.0422872
$523$	−15.8866	−0.694674	−0.347337	−	0.937740i	$-0.612914\pi$
$523$	−15.8866	−0.694674	−0.347337	+	0.937740i	$0.612914\pi$
$524$	27.6098	1.20614
$525$	5.00000	0.218218
$526$	7.15644	0.312036
$527$	23.4034	1.01947
$528$	1.00000	0.0435194
$529$	26.1357	1.13634
$530$	0	0
$531$	4.98792	0.216457
$532$	6.93362	0.300611
$533$	0	0
$534$	4.89008	0.211615
$535$	0	0
$536$	3.49157	0.150813
$537$	9.87263	0.426035
$538$	−0.0784573	−0.00338254
$539$	−0.445042	−0.0191693
$540$	0	0
$541$	−34.1062	−1.46634	−0.733170	−	0.680045i	$-0.761960\pi$
$541$	−34.1062	−1.46634	−0.733170	+	0.680045i	$0.761960\pi$
$542$	−3.83446	−0.164704
$543$	11.5362	0.495065
$544$	−19.2620	−0.825853
$545$	0	0
$546$	0	0
$547$	22.0597	0.943203	0.471602	−	0.881812i	$-0.343676\pi$
$547$	22.0597	0.943203	0.471602	+	0.881812i	$0.343676\pi$
$548$	−1.21685	−0.0519812
$549$	−0.933624	−0.0398461
$550$	−1.23490	−0.0526562
$551$	−7.13408	−0.303922
$552$	14.3623	0.611299
$553$	−0.621334	−0.0264218
$554$	−1.86426	−0.0792050
$555$	0	0
$556$	−3.59047	−0.152270
$557$	−43.6558	−1.84975	−0.924877	−	0.380266i	$-0.875833\pi$
$557$	−43.6558	−1.84975	−0.924877	+	0.380266i	$0.875833\pi$
$558$	−3.60388	−0.152564
$559$	0	0
$560$	0	0
$561$	1.60388	0.0677157
$562$	−14.8418	−0.626062
$563$	−34.4698	−1.45273	−0.726364	−	0.687310i	$-0.758791\pi$
$563$	−34.4698	−1.45273	−0.726364	+	0.687310i	$0.758791\pi$
$564$	16.0435	0.675555
$565$	0	0
$566$	−8.13872	−0.342096
$567$	1.00000	0.0419961
$568$	−7.78986	−0.326855
$569$	−5.69633	−0.238803	−0.119401	−	0.992846i	$-0.538098\pi$
$569$	−5.69633	−0.238803	−0.119401	+	0.992846i	$0.538098\pi$
$570$	0	0
$571$	−5.00777	−0.209569	−0.104784	−	0.994495i	$-0.533415\pi$
$571$	−5.00777	−0.209569	−0.104784	+	0.994495i	$0.533415\pi$
$572$	0	0
$573$	3.12498	0.130548
$574$	3.82371	0.159598
$575$	35.0484	1.46162
$576$	−1.52781	−0.0636588
$577$	−18.3370	−0.763381	−0.381691	−	0.924290i	$-0.624658\pi$
$577$	−18.3370	−0.763381	−0.381691	+	0.924290i	$0.624658\pi$
$578$	2.22654	0.0926117
$579$	−21.0030	−0.872854
$580$	0	0
$581$	−4.27413	−0.177321
$582$	−3.65817	−0.151636
$583$	−4.13169	−0.171117
$584$	14.9685	0.619403
$585$	0	0
$586$	−2.85995	−0.118144
$587$	38.8745	1.60452	0.802262	−	0.596972i	$-0.203630\pi$
$587$	38.8745	1.60452	0.802262	+	0.596972i	$0.203630\pi$
$588$	1.69202	0.0697778
$589$	−26.6112	−1.09649
$590$	0	0
$591$	11.6233	0.478117
$592$	−16.1468	−0.663627
$593$	−47.3551	−1.94464	−0.972320	−	0.233652i	$-0.924932\pi$
$593$	−47.3551	−1.94464	−0.972320	+	0.233652i	$0.924932\pi$
$594$	−0.246980	−0.0101337
$595$	0	0
$596$	5.36121	0.219604
$597$	−6.86592	−0.281003
$598$	0	0
$599$	25.4534	1.04000	0.519999	−	0.854167i	$-0.325932\pi$
$599$	25.4534	1.04000	0.519999	+	0.854167i	$0.325932\pi$
$600$	10.2446	0.418234
$601$	30.7982	1.25629	0.628143	−	0.778098i	$-0.283815\pi$
$601$	30.7982	1.25629	0.628143	+	0.778098i	$0.283815\pi$
$602$	−4.91185	−0.200192
$603$	1.70410	0.0693964
$604$	1.57242	0.0639807
$605$	0	0
$606$	−3.32975	−0.135262
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	21.9022	0.888250
$609$	−1.74094	−0.0705464
$610$	0	0
$611$	0	0
$612$	−6.09783	−0.246490
$613$	−17.5297	−0.708019	−0.354010	−	0.935242i	$-0.615182\pi$
$613$	−17.5297	−0.708019	−0.354010	+	0.935242i	$0.615182\pi$
$614$	8.56166	0.345521
$615$	0	0
$616$	−0.911854	−0.0367396
$617$	46.6741	1.87903	0.939514	−	0.342510i	$-0.111277\pi$
$617$	46.6741	1.87903	0.939514	+	0.342510i	$0.111277\pi$
$618$	−8.49396	−0.341677
$619$	14.1086	0.567072	0.283536	−	0.958962i	$-0.408492\pi$
$619$	14.1086	0.567072	0.283536	+	0.958962i	$0.408492\pi$
$620$	0	0
$621$	7.00969	0.281289
$622$	19.4034	0.778006
$623$	8.81163	0.353030
$624$	0	0
$625$	25.0000	1.00000
$626$	14.7380	0.589047
$627$	−1.82371	−0.0728319
$628$	−8.71976	−0.347956
$629$	−25.8974	−1.03260
$630$	0	0
$631$	−0.0140005	−0.000557353 0	−0.000278676	−	1.00000i	$-0.500089\pi$
$631$	−0.0140005	−0.000557353 0	−0.000278676		1.00000i	$0.500089\pi$
$632$	−1.27306	−0.0506397
$633$	−6.88769	−0.273761
$634$	7.74227	0.307485
$635$	0	0
$636$	15.7084	0.622879
$637$	0	0
$638$	0.429976	0.0170229
$639$	−3.80194	−0.150402
$640$	0	0
$641$	−30.1148	−1.18946	−0.594732	−	0.803924i	$-0.702742\pi$
$641$	−30.1148	−1.18946	−0.594732	+	0.803924i	$0.702742\pi$
$642$	−8.13839	−0.321197
$643$	−14.5617	−0.574256	−0.287128	−	0.957892i	$-0.592700\pi$
$643$	−14.5617	−0.574256	−0.287128	+	0.957892i	$0.592700\pi$
$644$	11.8605	0.467371
$645$	0	0
$646$	8.19567	0.322454
$647$	−47.7840	−1.87858	−0.939292	−	0.343120i	$-0.888516\pi$
$647$	−47.7840	−1.87858	−0.939292	+	0.343120i	$0.888516\pi$
$648$	2.04892	0.0804891
$649$	−2.21983	−0.0871360
$650$	0	0
$651$	−6.49396	−0.254518
$652$	−25.4999	−0.998654
$653$	34.0452	1.33229	0.666146	−	0.745822i	$-0.267943\pi$
$653$	34.0452	1.33229	0.666146	+	0.745822i	$0.267943\pi$
$654$	8.24698	0.322482
$655$	0	0
$656$	−15.4819	−0.604466
$657$	7.30559	0.285018
$658$	−5.26205	−0.205136
$659$	−46.4801	−1.81061	−0.905303	−	0.424766i	$-0.860356\pi$
$659$	−46.4801	−1.81061	−0.905303	+	0.424766i	$0.860356\pi$
$660$	0	0
$661$	−35.3551	−1.37515	−0.687577	−	0.726112i	$-0.741326\pi$
$661$	−35.3551	−1.37515	−0.687577	+	0.726112i	$0.741326\pi$
$662$	3.03013	0.117769
$663$	0	0
$664$	−8.75733	−0.339850
$665$	0	0
$666$	3.98792	0.154529
$667$	−12.2034	−0.472519
$668$	20.1897	0.781163
$669$	17.7017	0.684388
$670$	0	0
$671$	0.415502	0.0160403
$672$	5.34481	0.206181
$673$	43.0562	1.65970	0.829848	−	0.557990i	$-0.188427\pi$
$673$	43.0562	1.65970	0.829848	+	0.557990i	$0.188427\pi$
$674$	10.3368	0.398158
$675$	5.00000	0.192450
$676$	0	0
$677$	−24.1715	−0.928986	−0.464493	−	0.885577i	$-0.653763\pi$
$677$	−24.1715	−0.928986	−0.464493	+	0.885577i	$0.653763\pi$
$678$	−7.60686	−0.292140
$679$	−6.59179	−0.252970
$680$	0	0
$681$	25.0315	0.959208
$682$	1.60388	0.0614156
$683$	−10.1715	−0.389202	−0.194601	−	0.980882i	$-0.562341\pi$
$683$	−10.1715	−0.389202	−0.194601	+	0.980882i	$0.562341\pi$
$684$	6.93362	0.265114
$685$	0	0
$686$	−0.554958	−0.0211884
$687$	24.4940	0.934503
$688$	19.8877	0.758211
$689$	0	0
$690$	0	0
$691$	34.8176	1.32452	0.662262	−	0.749272i	$-0.269597\pi$
$691$	34.8176	1.32452	0.662262	+	0.749272i	$0.269597\pi$
$692$	−5.12929	−0.194986
$693$	−0.445042	−0.0169057
$694$	20.2212	0.767585
$695$	0	0
$696$	−3.56704	−0.135208
$697$	−24.8310	−0.940541
$698$	8.09187	0.306282
$699$	−9.19269	−0.347699
$700$	8.46011	0.319762
$701$	−12.7571	−0.481828	−0.240914	−	0.970546i	$-0.577447\pi$
$701$	−12.7571	−0.481828	−0.240914	+	0.970546i	$0.577447\pi$
$702$	0	0
$703$	29.4470	1.11061
$704$	0.679940	0.0256262
$705$	0	0
$706$	−8.56166	−0.322222
$707$	−6.00000	−0.225653
$708$	8.43967	0.317182
$709$	−33.9909	−1.27656	−0.638278	−	0.769806i	$-0.720353\pi$
$709$	−33.9909	−1.27656	−0.638278	+	0.769806i	$0.720353\pi$
$710$	0	0
$711$	−0.621334	−0.0233018
$712$	18.0543	0.676613
$713$	−45.5206	−1.70476
$714$	2.00000	0.0748481
$715$	0	0
$716$	16.7047	0.624284
$717$	−7.91723	−0.295674
$718$	13.6461	0.509267
$719$	−4.59658	−0.171423	−0.0857117	−	0.996320i	$-0.527316\pi$
$719$	−4.59658	−0.171423	−0.0857117	+	0.996320i	$0.527316\pi$
$720$	0	0
$721$	−15.3056	−0.570010
$722$	1.22521	0.0455976
$723$	−1.51679	−0.0564102
$724$	19.5195	0.725435
$725$	−8.70469	−0.323284
$726$	−5.99462	−0.222481
$727$	40.4349	1.49965	0.749823	−	0.661638i	$-0.230138\pi$
$727$	40.4349	1.49965	0.749823	+	0.661638i	$0.230138\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.8974	1.17977
$732$	−1.57971	−0.0583878
$733$	17.0207	0.628674	0.314337	−	0.949311i	$-0.398218\pi$
$733$	17.0207	0.628674	0.314337	+	0.949311i	$0.398218\pi$
$734$	−15.7259	−0.580453
$735$	0	0
$736$	37.4655	1.38100
$737$	−0.758397	−0.0279359
$738$	3.82371	0.140753
$739$	28.5483	1.05016	0.525082	−	0.851051i	$-0.324035\pi$
$739$	28.5483	1.05016	0.525082	+	0.851051i	$0.324035\pi$
$740$	0	0
$741$	0	0
$742$	−5.15213	−0.189141
$743$	−11.6732	−0.428249	−0.214125	−	0.976806i	$-0.568690\pi$
$743$	−11.6732	−0.428249	−0.214125	+	0.976806i	$0.568690\pi$
$744$	−13.3056	−0.487806
$745$	0	0
$746$	−11.6353	−0.426000
$747$	−4.27413	−0.156382
$748$	2.71379	0.0992261
$749$	−14.6649	−0.535843
$750$	0	0
$751$	41.9898	1.53223	0.766115	−	0.642703i	$-0.222187\pi$
$751$	41.9898	1.53223	0.766115	+	0.642703i	$0.222187\pi$
$752$	21.3056	0.776935
$753$	26.0978	0.951058
$754$	0	0
$755$	0	0
$756$	1.69202	0.0615382
$757$	44.9245	1.63281	0.816405	−	0.577480i	$-0.195964\pi$
$757$	44.9245	1.63281	0.816405	+	0.577480i	$0.195964\pi$
$758$	8.75973	0.318168
$759$	−3.11960	−0.113234
$760$	0	0
$761$	−23.1207	−0.838124	−0.419062	−	0.907958i	$-0.637641\pi$
$761$	−23.1207	−0.838124	−0.419062	+	0.907958i	$0.637641\pi$
$762$	−8.95539	−0.324420
$763$	14.8605	0.537988
$764$	5.28754	0.191296
$765$	0	0
$766$	17.3599	0.627238
$767$	0	0
$768$	3.34721	0.120782
$769$	10.4590	0.377163	0.188581	−	0.982058i	$-0.439611\pi$
$769$	10.4590	0.377163	0.188581	+	0.982058i	$0.439611\pi$
$770$	0	0
$771$	5.42758	0.195470
$772$	−35.5375	−1.27902
$773$	28.7525	1.03416	0.517079	−	0.855938i	$-0.327019\pi$
$773$	28.7525	1.03416	0.517079	+	0.855938i	$0.327019\pi$
$774$	−4.91185	−0.176553
$775$	−32.4698	−1.16635
$776$	−13.5060	−0.484839
$777$	7.18598	0.257796
$778$	4.48022	0.160624
$779$	28.2344	1.01160
$780$	0	0
$781$	1.69202	0.0605453
$782$	14.0194	0.501332
$783$	−1.74094	−0.0622161
$784$	2.24698	0.0802493
$785$	0	0
$786$	−9.05562	−0.323003
$787$	−34.2392	−1.22050	−0.610248	−	0.792210i	$-0.708930\pi$
$787$	−34.2392	−1.22050	−0.610248	+	0.792210i	$0.708930\pi$
$788$	19.6668	0.700601
$789$	12.8955	0.459091
$790$	0	0
$791$	−13.7071	−0.487368
$792$	−0.911854	−0.0324013
$793$	0	0
$794$	9.73928	0.345634
$795$	0	0
$796$	−11.6173	−0.411764
$797$	−19.1750	−0.679212	−0.339606	−	0.940568i	$-0.610294\pi$
$797$	−19.1750	−0.679212	−0.339606	+	0.940568i	$0.610294\pi$
$798$	−2.27413	−0.0805032
$799$	34.1715	1.20890
$800$	26.7241	0.944839
$801$	8.81163	0.311344
$802$	4.26577	0.150629
$803$	−3.25129	−0.114736
$804$	2.88338	0.101689
$805$	0	0
$806$	0	0
$807$	−0.141375	−0.00497664
$808$	−12.2935	−0.432484
$809$	−12.3394	−0.433832	−0.216916	−	0.976190i	$-0.569600\pi$
$809$	−12.3394	−0.433832	−0.216916	+	0.976190i	$0.569600\pi$
$810$	0	0
$811$	11.2078	0.393557	0.196779	−	0.980448i	$-0.436952\pi$
$811$	11.2078	0.393557	0.196779	+	0.980448i	$0.436952\pi$
$812$	−2.94571	−0.103374
$813$	−6.90946	−0.242325
$814$	−1.77479	−0.0622064
$815$	0	0
$816$	−8.09783	−0.283481
$817$	−36.2693	−1.26890
$818$	−20.1521	−0.704603
$819$	0	0
$820$	0	0
$821$	49.5666	1.72989	0.864943	−	0.501871i	$-0.167355\pi$
$821$	49.5666	1.72989	0.864943	+	0.501871i	$0.167355\pi$
$822$	0.399108	0.0139205
$823$	−20.0761	−0.699808	−0.349904	−	0.936786i	$-0.613786\pi$
$823$	−20.0761	−0.699808	−0.349904	+	0.936786i	$0.613786\pi$
$824$	−31.3599	−1.09247
$825$	−2.22521	−0.0774718
$826$	−2.76809	−0.0963140
$827$	50.8743	1.76907	0.884536	−	0.466472i	$-0.154475\pi$
$827$	50.8743	1.76907	0.884536	+	0.466472i	$0.154475\pi$
$828$	11.8605	0.412182
$829$	1.19700	0.0415734	0.0207867	−	0.999784i	$-0.493383\pi$
$829$	1.19700	0.0415734	0.0207867	+	0.999784i	$0.493383\pi$
$830$	0	0
$831$	−3.35929	−0.116532
$832$	0	0
$833$	3.60388	0.124867
$834$	1.17762	0.0407776
$835$	0	0
$836$	−3.08575	−0.106723
$837$	−6.49396	−0.224464
$838$	−1.53020	−0.0528600
$839$	−33.5905	−1.15967	−0.579836	−	0.814733i	$-0.696883\pi$
$839$	−33.5905	−1.15967	−0.579836	+	0.814733i	$0.696883\pi$
$840$	0	0
$841$	−25.9691	−0.895487
$842$	−13.7936	−0.475358
$843$	−26.7439	−0.921110
$844$	−11.6541	−0.401151
$845$	0	0
$846$	−5.26205	−0.180913
$847$	−10.8019	−0.371159
$848$	20.8605	0.716354
$849$	−14.6655	−0.503317
$850$	10.0000	0.342997
$851$	50.3715	1.72671
$852$	−6.43296	−0.220390
$853$	−38.7525	−1.32686	−0.663431	−	0.748238i	$-0.730900\pi$
$853$	−38.7525	−1.32686	−0.663431	+	0.748238i	$0.730900\pi$
$854$	0.518122	0.0177298
$855$	0	0
$856$	−30.0471	−1.02699
$857$	−7.48188	−0.255576	−0.127788	−	0.991802i	$-0.540788\pi$
$857$	−7.48188	−0.255576	−0.127788	+	0.991802i	$0.540788\pi$
$858$	0	0
$859$	−2.45042	−0.0836072	−0.0418036	−	0.999126i	$-0.513310\pi$
$859$	−2.45042	−0.0836072	−0.0418036	+	0.999126i	$0.513310\pi$
$860$	0	0
$861$	6.89008	0.234813
$862$	−13.6805	−0.465961
$863$	15.0175	0.511200	0.255600	−	0.966783i	$-0.417727\pi$
$863$	15.0175	0.511200	0.255600	+	0.966783i	$0.417727\pi$
$864$	5.34481	0.181834
$865$	0	0
$866$	−16.4397	−0.558643
$867$	4.01208	0.136257
$868$	−10.9879	−0.372954
$869$	0.276520	0.00938028
$870$	0	0
$871$	0	0
$872$	30.4480	1.03110
$873$	−6.59179	−0.223098
$874$	−15.9409	−0.539210
$875$	0	0
$876$	12.3612	0.417647
$877$	25.7372	0.869084	0.434542	−	0.900652i	$-0.356910\pi$
$877$	25.7372	0.869084	0.434542	+	0.900652i	$0.356910\pi$
$878$	−15.8840	−0.536058
$879$	−5.15346	−0.173822
$880$	0	0
$881$	36.8939	1.24299	0.621494	−	0.783419i	$-0.286526\pi$
$881$	36.8939	1.24299	0.621494	+	0.783419i	$0.286526\pi$
$882$	−0.554958	−0.0186864
$883$	−50.5273	−1.70038	−0.850190	−	0.526476i	$-0.823513\pi$
$883$	−50.5273	−1.70038	−0.850190	+	0.526476i	$0.823513\pi$
$884$	0	0
$885$	0	0
$886$	12.9688	0.435695
$887$	−28.1957	−0.946718	−0.473359	−	0.880870i	$-0.656959\pi$
$887$	−28.1957	−0.946718	−0.473359	+	0.880870i	$0.656959\pi$
$888$	14.7235	0.494088
$889$	−16.1371	−0.541220
$890$	0	0
$891$	−0.445042	−0.0149095
$892$	29.9517	1.00286
$893$	−38.8552	−1.30024
$894$	−1.75840	−0.0588096
$895$	0	0
$896$	11.5375	0.385441
$897$	0	0
$898$	−5.23921	−0.174835
$899$	11.3056	0.377062
$900$	8.46011	0.282004
$901$	33.4577	1.11464
$902$	−1.70171	−0.0566608
$903$	−8.85086	−0.294538
$904$	−28.0847	−0.934083
$905$	0	0
$906$	−0.515729	−0.0171340
$907$	−42.6389	−1.41580	−0.707901	−	0.706312i	$-0.750358\pi$
$907$	−42.6389	−1.41580	−0.707901	+	0.706312i	$0.750358\pi$
$908$	42.3538	1.40556
$909$	−6.00000	−0.199007
$910$	0	0
$911$	26.2825	0.870778	0.435389	−	0.900243i	$-0.356611\pi$
$911$	26.2825	0.870778	0.435389	+	0.900243i	$0.356611\pi$
$912$	9.20775	0.304899
$913$	1.90217	0.0629525
$914$	−10.7168	−0.354479
$915$	0	0
$916$	41.4443	1.36936
$917$	−16.3177	−0.538857
$918$	2.00000	0.0660098
$919$	−20.9071	−0.689661	−0.344830	−	0.938665i	$-0.612064\pi$
$919$	−20.9071	−0.689661	−0.344830	+	0.938665i	$0.612064\pi$
$920$	0	0
$921$	15.4276	0.508356
$922$	18.8250	0.619970
$923$	0	0
$924$	−0.753020	−0.0247726
$925$	35.9299	1.18137
$926$	−11.3086	−0.371623
$927$	−15.3056	−0.502701
$928$	−9.30499	−0.305451
$929$	24.3526	0.798982	0.399491	−	0.916737i	$-0.369187\pi$
$929$	24.3526	0.798982	0.399491	+	0.916737i	$0.369187\pi$
$930$	0	0
$931$	−4.09783	−0.134301
$932$	−15.5542	−0.509495
$933$	34.9638	1.14466
$934$	−15.2862	−0.500180
$935$	0	0
$936$	0	0
$937$	−25.7560	−0.841412	−0.420706	−	0.907197i	$-0.638218\pi$
$937$	−25.7560	−0.841412	−0.420706	+	0.907197i	$0.638218\pi$
$938$	−0.945706	−0.0308784
$939$	26.5569	0.866651
$940$	0	0
$941$	44.1258	1.43846	0.719231	−	0.694771i	$-0.244494\pi$
$941$	44.1258	1.43846	0.719231	+	0.694771i	$0.244494\pi$
$942$	2.85995	0.0931823
$943$	48.2973	1.57278
$944$	11.2078	0.364781
$945$	0	0
$946$	2.18598	0.0710724
$947$	29.8165	0.968907	0.484454	−	0.874817i	$-0.339018\pi$
$947$	29.8165	0.968907	0.484454	+	0.874817i	$0.339018\pi$
$948$	−1.05131	−0.0341450
$949$	0	0
$950$	−11.3706	−0.368912
$951$	13.9511	0.452395
$952$	7.38404	0.239318
$953$	3.46489	0.112239	0.0561194	−	0.998424i	$-0.482127\pi$
$953$	3.46489	0.112239	0.0561194	+	0.998424i	$0.482127\pi$
$954$	−5.15213	−0.166806
$955$	0	0
$956$	−13.3961	−0.433262
$957$	0.774791	0.0250454
$958$	−1.14005	−0.0368333
$959$	0.719169	0.0232232
$960$	0	0
$961$	11.1715	0.360371
$962$	0	0
$963$	−14.6649	−0.472569
$964$	−2.56645	−0.0826597
$965$	0	0
$966$	−3.89008	−0.125161
$967$	9.94810	0.319909	0.159955	−	0.987124i	$-0.448865\pi$
$967$	9.94810	0.319909	0.159955	+	0.987124i	$0.448865\pi$
$968$	−22.1323	−0.711358
$969$	14.7681	0.474419
$970$	0	0
$971$	15.7754	0.506256	0.253128	−	0.967433i	$-0.418541\pi$
$971$	15.7754	0.506256	0.253128	+	0.967433i	$0.418541\pi$
$972$	1.69202	0.0542716
$973$	2.12200	0.0680281
$974$	−3.93197	−0.125988
$975$	0	0
$976$	−2.09783	−0.0671501
$977$	−21.7178	−0.694815	−0.347408	−	0.937714i	$-0.612938\pi$
$977$	−21.7178	−0.694815	−0.347408	+	0.937714i	$0.612938\pi$
$978$	8.36360	0.267438
$979$	−3.92154	−0.125333
$980$	0	0
$981$	14.8605	0.474461
$982$	20.4771	0.653450
$983$	−4.99654	−0.159365	−0.0796825	−	0.996820i	$-0.525391\pi$
$983$	−4.99654	−0.159365	−0.0796825	+	0.996820i	$0.525391\pi$
$984$	14.1172	0.450040
$985$	0	0
$986$	−3.48188	−0.110886
$987$	−9.48188	−0.301811
$988$	0	0
$989$	−62.0417	−1.97281
$990$	0	0
$991$	7.98015	0.253498	0.126749	−	0.991935i	$-0.459546\pi$
$991$	7.98015	0.253498	0.126749	+	0.991935i	$0.459546\pi$
$992$	−34.7090	−1.10201
$993$	5.46011	0.173271
$994$	2.10992	0.0669225
$995$	0	0
$996$	−7.23191	−0.229152
$997$	−33.9952	−1.07664	−0.538320	−	0.842741i	$-0.680941\pi$
$997$	−33.9952	−1.07664	−0.538320	+	0.842741i	$0.680941\pi$
$998$	9.21611	0.291731
$999$	7.18598	0.227354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3549.2.a.g.1.2	✓	3
13.12	even	2		3549.2.a.r.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3549.2.a.g.1.2	✓	3	1.1	even	1	trivial
3549.2.a.r.1.2	yes	3	13.12	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:	\( N \)	\(=\)	\( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3549.a (trivial)

\(f(q)\)	\(=\)	\(q-0.554958 q^{2} -1.00000 q^{3} -1.69202 q^{4} +0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.554958 q^{2} -1.00000 q^{3} -1.69202 q^{4} +0.554958 q^{6} +1.00000 q^{7} +2.04892 q^{8} +1.00000 q^{9} -0.445042 q^{11} +1.69202 q^{12} -0.554958 q^{14} +2.24698 q^{16} +3.60388 q^{17} -0.554958 q^{18} -4.09783 q^{19} -1.00000 q^{21} +0.246980 q^{22} -7.00969 q^{23} -2.04892 q^{24} -5.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +1.74094 q^{29} +6.49396 q^{31} -5.34481 q^{32} +0.445042 q^{33} -2.00000 q^{34} -1.69202 q^{36} -7.18598 q^{37} +2.27413 q^{38} -6.89008 q^{41} +0.554958 q^{42} +8.85086 q^{43} +0.753020 q^{44} +3.89008 q^{46} +9.48188 q^{47} -2.24698 q^{48} +1.00000 q^{49} +2.77479 q^{50} -3.60388 q^{51} +9.28382 q^{53} +0.554958 q^{54} +2.04892 q^{56} +4.09783 q^{57} -0.966148 q^{58} +4.98792 q^{59} -0.933624 q^{61} -3.60388 q^{62} +1.00000 q^{63} -1.52781 q^{64} -0.246980 q^{66} +1.70410 q^{67} -6.09783 q^{68} +7.00969 q^{69} -3.80194 q^{71} +2.04892 q^{72} +7.30559 q^{73} +3.98792 q^{74} +5.00000 q^{75} +6.93362 q^{76} -0.445042 q^{77} -0.621334 q^{79} +1.00000 q^{81} +3.82371 q^{82} -4.27413 q^{83} +1.69202 q^{84} -4.91185 q^{86} -1.74094 q^{87} -0.911854 q^{88} +8.81163 q^{89} +11.8605 q^{92} -6.49396 q^{93} -5.26205 q^{94} +5.34481 q^{96} -6.59179 q^{97} -0.554958 q^{98} -0.445042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 3 q^{3} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 - 3 * q^3 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 2 q^{2} - 3 q^{3} + 2 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 6 q^{19} - 3 q^{21} - 4 q^{22} + q^{23} + 3 q^{24} - 15 q^{25} - 3 q^{27} - 9 q^{29} + 10 q^{31} + 7 q^{32} + q^{33} - 6 q^{34} - 7 q^{37} - 4 q^{38} - 20 q^{41} + 2 q^{42} + 13 q^{43} + 7 q^{44} + 11 q^{46} - 2 q^{48} + 3 q^{49} + 10 q^{50} - 2 q^{51} - 5 q^{53} + 2 q^{54} - 3 q^{56} - 6 q^{57} + 13 q^{58} - 4 q^{59} + 4 q^{61} - 2 q^{62} + 3 q^{63} - 11 q^{64} + 4 q^{66} + 19 q^{67} - q^{69} - 7 q^{71} - 3 q^{72} - 14 q^{73} - 7 q^{74} + 15 q^{75} + 14 q^{76} - q^{77} - 9 q^{79} + 3 q^{81} + 4 q^{82} - 2 q^{83} - 11 q^{86} + 9 q^{87} + q^{88} - 10 q^{93} + 14 q^{94} - 7 q^{96} + 8 q^{97} - 2 q^{98} - q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 - 3 * q^3 + 2 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - q^11 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 6 * q^19 - 3 * q^21 - 4 * q^22 + q^23 + 3 * q^24 - 15 * q^25 - 3 * q^27 - 9 * q^29 + 10 * q^31 + 7 * q^32 + q^33 - 6 * q^34 - 7 * q^37 - 4 * q^38 - 20 * q^41 + 2 * q^42 + 13 * q^43 + 7 * q^44 + 11 * q^46 - 2 * q^48 + 3 * q^49 + 10 * q^50 - 2 * q^51 - 5 * q^53 + 2 * q^54 - 3 * q^56 - 6 * q^57 + 13 * q^58 - 4 * q^59 + 4 * q^61 - 2 * q^62 + 3 * q^63 - 11 * q^64 + 4 * q^66 + 19 * q^67 - q^69 - 7 * q^71 - 3 * q^72 - 14 * q^73 - 7 * q^74 + 15 * q^75 + 14 * q^76 - q^77 - 9 * q^79 + 3 * q^81 + 4 * q^82 - 2 * q^83 - 11 * q^86 + 9 * q^87 + q^88 - 10 * q^93 + 14 * q^94 - 7 * q^96 + 8 * q^97 - 2 * q^98 - q^99

Introduction

L-functions

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Database

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