LMFDB - Embedded newform 693.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$	\(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +2.30278 q^{10} -1.00000 q^{11} +3.60555 q^{13} +2.30278 q^{14} +0.302776 q^{16} -4.00000 q^{17} +3.00000 q^{19} -3.30278 q^{20} +2.30278 q^{22} -2.00000 q^{23} -4.00000 q^{25} -8.30278 q^{26} -3.30278 q^{28} +5.60555 q^{29} -2.00000 q^{31} +5.30278 q^{32} +9.21110 q^{34} +1.00000 q^{35} -8.21110 q^{37} -6.90833 q^{38} +3.00000 q^{40} +7.21110 q^{41} -5.21110 q^{43} -3.30278 q^{44} +4.60555 q^{46} -2.39445 q^{47} +1.00000 q^{49} +9.21110 q^{50} +11.9083 q^{52} +1.00000 q^{55} +3.00000 q^{56} -12.9083 q^{58} -7.60555 q^{59} -11.2111 q^{61} +4.60555 q^{62} -12.8167 q^{64} -3.60555 q^{65} +1.60555 q^{67} -13.2111 q^{68} -2.30278 q^{70} -11.2111 q^{71} -12.8167 q^{73} +18.9083 q^{74} +9.90833 q^{76} +1.00000 q^{77} -3.21110 q^{79} -0.302776 q^{80} -16.6056 q^{82} -12.0000 q^{83} +4.00000 q^{85} +12.0000 q^{86} +3.00000 q^{88} -6.00000 q^{89} -3.60555 q^{91} -6.60555 q^{92} +5.51388 q^{94} -3.00000 q^{95} +1.21110 q^{97} -2.30278 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + q^{10} - 2 q^{11} + q^{14} - 3 q^{16} - 8 q^{17} + 6 q^{19} - 3 q^{20} + q^{22} - 4 q^{23} - 8 q^{25} - 13 q^{26} - 3 q^{28} + 4 q^{29} - 4 q^{31} + 7 q^{32} + 4 q^{34} + 2 q^{35} - 2 q^{37} - 3 q^{38} + 6 q^{40} + 4 q^{43} - 3 q^{44} + 2 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} + 13 q^{52} + 2 q^{55} + 6 q^{56} - 15 q^{58} - 8 q^{59} - 8 q^{61} + 2 q^{62} - 4 q^{64} - 4 q^{67} - 12 q^{68} - q^{70} - 8 q^{71} - 4 q^{73} + 27 q^{74} + 9 q^{76} + 2 q^{77} + 8 q^{79} + 3 q^{80} - 26 q^{82} - 24 q^{83} + 8 q^{85} + 24 q^{86} + 6 q^{88} - 12 q^{89} - 6 q^{92} - 7 q^{94} - 6 q^{95} - 12 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + q^10 - 2 * q^11 + q^14 - 3 * q^16 - 8 * q^17 + 6 * q^19 - 3 * q^20 + q^22 - 4 * q^23 - 8 * q^25 - 13 * q^26 - 3 * q^28 + 4 * q^29 - 4 * q^31 + 7 * q^32 + 4 * q^34 + 2 * q^35 - 2 * q^37 - 3 * q^38 + 6 * q^40 + 4 * q^43 - 3 * q^44 + 2 * q^46 - 12 * q^47 + 2 * q^49 + 4 * q^50 + 13 * q^52 + 2 * q^55 + 6 * q^56 - 15 * q^58 - 8 * q^59 - 8 * q^61 + 2 * q^62 - 4 * q^64 - 4 * q^67 - 12 * q^68 - q^70 - 8 * q^71 - 4 * q^73 + 27 * q^74 + 9 * q^76 + 2 * q^77 + 8 * q^79 + 3 * q^80 - 26 * q^82 - 24 * q^83 + 8 * q^85 + 24 * q^86 + 6 * q^88 - 12 * q^89 - 6 * q^92 - 7 * q^94 - 6 * q^95 - 12 * q^97 - 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 $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	0	0
$10$	2.30278	0.728202
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.60555	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$13$	3.60555	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$14$	2.30278	0.615443
$15$	0	0
$16$	0.302776	0.0756939
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−3.30278	−0.738523
$21$	0	0
$22$	2.30278	0.490953
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−8.30278	−1.62831
$27$	0	0
$28$	−3.30278	−0.624166
$29$	5.60555	1.04092	0.520462	−	0.853885i	$-0.325760\pi$
$29$	5.60555	1.04092	0.520462	+	0.853885i	$0.325760\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.30278	0.937407
$33$	0	0
$34$	9.21110	1.57969
$35$	1.00000	0.169031
$36$	0	0
$37$	−8.21110	−1.34990	−0.674948	−	0.737865i	$-0.735834\pi$
$37$	−8.21110	−1.34990	−0.674948	+	0.737865i	$0.735834\pi$
$38$	−6.90833	−1.12068
$39$	0	0
$40$	3.00000	0.474342
$41$	7.21110	1.12619	0.563093	−	0.826394i	$-0.309611\pi$
$41$	7.21110	1.12619	0.563093	+	0.826394i	$0.309611\pi$
$42$	0	0
$43$	−5.21110	−0.794686	−0.397343	−	0.917670i	$-0.630068\pi$
$43$	−5.21110	−0.794686	−0.397343	+	0.917670i	$0.630068\pi$
$44$	−3.30278	−0.497912
$45$	0	0
$46$	4.60555	0.679051
$47$	−2.39445	−0.349266	−0.174633	−	0.984634i	$-0.555874\pi$
$47$	−2.39445	−0.349266	−0.174633	+	0.984634i	$0.555874\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.21110	1.30265
$51$	0	0
$52$	11.9083	1.65139
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	3.00000	0.400892
$57$	0	0
$58$	−12.9083	−1.69495
$59$	−7.60555	−0.990158	−0.495079	−	0.868848i	$-0.664861\pi$
$59$	−7.60555	−0.990158	−0.495079	+	0.868848i	$0.664861\pi$
$60$	0	0
$61$	−11.2111	−1.43543	−0.717717	−	0.696335i	$-0.754813\pi$
$61$	−11.2111	−1.43543	−0.717717	+	0.696335i	$0.754813\pi$
$62$	4.60555	0.584906
$63$	0	0
$64$	−12.8167	−1.60208
$65$	−3.60555	−0.447214
$66$	0	0
$67$	1.60555	0.196149	0.0980747	−	0.995179i	$-0.468732\pi$
$67$	1.60555	0.196149	0.0980747	+	0.995179i	$0.468732\pi$
$68$	−13.2111	−1.60208
$69$	0	0
$70$	−2.30278	−0.275234
$71$	−11.2111	−1.33051	−0.665257	−	0.746615i	$-0.731678\pi$
$71$	−11.2111	−1.33051	−0.665257	+	0.746615i	$0.731678\pi$
$72$	0	0
$73$	−12.8167	−1.50008	−0.750038	−	0.661395i	$-0.769965\pi$
$73$	−12.8167	−1.50008	−0.750038	+	0.661395i	$0.769965\pi$
$74$	18.9083	2.19805
$75$	0	0
$76$	9.90833	1.13656
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.21110	−0.361277	−0.180639	−	0.983550i	$-0.557816\pi$
$79$	−3.21110	−0.361277	−0.180639	+	0.983550i	$0.557816\pi$
$80$	−0.302776	−0.0338513
$81$	0	0
$82$	−16.6056	−1.83378
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	12.0000	1.29399
$87$	0	0
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−3.60555	−0.377964
$92$	−6.60555	−0.688676
$93$	0	0
$94$	5.51388	0.568713
$95$	−3.00000	−0.307794
$96$	0	0
$97$	1.21110	0.122969	0.0614844	−	0.998108i	$-0.480417\pi$
$97$	1.21110	0.122969	0.0614844	+	0.998108i	$0.480417\pi$
$98$	−2.30278	−0.232615
$99$	0	0
$100$	−13.2111	−1.32111
$101$	−12.4222	−1.23606	−0.618028	−	0.786156i	$-0.712068\pi$
$101$	−12.4222	−1.23606	−0.618028	+	0.786156i	$0.712068\pi$
$102$	0	0
$103$	−9.21110	−0.907597	−0.453798	−	0.891104i	$-0.649931\pi$
$103$	−9.21110	−0.907597	−0.453798	+	0.891104i	$0.649931\pi$
$104$	−10.8167	−1.06066
$105$	0	0
$106$	0	0
$107$	2.21110	0.213755	0.106878	−	0.994272i	$-0.465915\pi$
$107$	2.21110	0.213755	0.106878	+	0.994272i	$0.465915\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.30278	−0.219561
$111$	0	0
$112$	−0.302776	−0.0286096
$113$	5.21110	0.490219	0.245110	−	0.969495i	$-0.421176\pi$
$113$	5.21110	0.490219	0.245110	+	0.969495i	$0.421176\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	18.5139	1.71897
$117$	0	0
$118$	17.5139	1.61228
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	25.8167	2.33733
$123$	0	0
$124$	−6.60555	−0.593196
$125$	9.00000	0.804984
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	18.9083	1.67128
$129$	0	0
$130$	8.30278	0.728202
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	−3.69722	−0.319392
$135$	0	0
$136$	12.0000	1.02899
$137$	−6.42221	−0.548686	−0.274343	−	0.961632i	$-0.588460\pi$
$137$	−6.42221	−0.548686	−0.274343	+	0.961632i	$0.588460\pi$
$138$	0	0
$139$	14.4222	1.22328	0.611638	−	0.791138i	$-0.290511\pi$
$139$	14.4222	1.22328	0.611638	+	0.791138i	$0.290511\pi$
$140$	3.30278	0.279135
$141$	0	0
$142$	25.8167	2.16649
$143$	−3.60555	−0.301511
$144$	0	0
$145$	−5.60555	−0.465516
$146$	29.5139	2.44259
$147$	0	0
$148$	−27.1194	−2.22920
$149$	21.6056	1.77000	0.884998	−	0.465595i	$-0.154160\pi$
$149$	21.6056	1.77000	0.884998	+	0.465595i	$0.154160\pi$
$150$	0	0
$151$	18.4222	1.49918	0.749589	−	0.661904i	$-0.230251\pi$
$151$	18.4222	1.49918	0.749589	+	0.661904i	$0.230251\pi$
$152$	−9.00000	−0.729996
$153$	0	0
$154$	−2.30278	−0.185563
$155$	2.00000	0.160644
$156$	0	0
$157$	−4.78890	−0.382196	−0.191098	−	0.981571i	$-0.561205\pi$
$157$	−4.78890	−0.382196	−0.191098	+	0.981571i	$0.561205\pi$
$158$	7.39445	0.588271
$159$	0	0
$160$	−5.30278	−0.419221
$161$	2.00000	0.157622
$162$	0	0
$163$	−12.8167	−1.00388	−0.501939	−	0.864903i	$-0.667380\pi$
$163$	−12.8167	−1.00388	−0.501939	+	0.864903i	$0.667380\pi$
$164$	23.8167	1.85977
$165$	0	0
$166$	27.6333	2.14476
$167$	−23.2111	−1.79613	−0.898065	−	0.439864i	$-0.855027\pi$
$167$	−23.2111	−1.79613	−0.898065	+	0.439864i	$0.855027\pi$
$168$	0	0
$169$	0	0
$170$	−9.21110	−0.706459
$171$	0	0
$172$	−17.2111	−1.31233
$173$	16.4222	1.24856	0.624279	−	0.781202i	$-0.285393\pi$
$173$	16.4222	1.24856	0.624279	+	0.781202i	$0.285393\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−0.302776	−0.0228226
$177$	0	0
$178$	13.8167	1.03560
$179$	16.4222	1.22745	0.613727	−	0.789519i	$-0.289670\pi$
$179$	16.4222	1.22745	0.613727	+	0.789519i	$0.289670\pi$
$180$	0	0
$181$	8.42221	0.626018	0.313009	−	0.949750i	$-0.398663\pi$
$181$	8.42221	0.626018	0.313009	+	0.949750i	$0.398663\pi$
$182$	8.30278	0.615443
$183$	0	0
$184$	6.00000	0.442326
$185$	8.21110	0.603692
$186$	0	0
$187$	4.00000	0.292509
$188$	−7.90833	−0.576774
$189$	0	0
$190$	6.90833	0.501183
$191$	4.78890	0.346512	0.173256	−	0.984877i	$-0.444571\pi$
$191$	4.78890	0.346512	0.173256	+	0.984877i	$0.444571\pi$
$192$	0	0
$193$	26.4222	1.90191	0.950956	−	0.309326i	$-0.100104\pi$
$193$	26.4222	1.90191	0.950956	+	0.309326i	$0.100104\pi$
$194$	−2.78890	−0.200231
$195$	0	0
$196$	3.30278	0.235913
$197$	−15.2111	−1.08375	−0.541873	−	0.840460i	$-0.682285\pi$
$197$	−15.2111	−1.08375	−0.541873	+	0.840460i	$0.682285\pi$
$198$	0	0
$199$	−17.2111	−1.22006	−0.610031	−	0.792377i	$-0.708843\pi$
$199$	−17.2111	−1.22006	−0.610031	+	0.792377i	$0.708843\pi$
$200$	12.0000	0.848528
$201$	0	0
$202$	28.6056	2.01268
$203$	−5.60555	−0.393433
$204$	0	0
$205$	−7.21110	−0.503645
$206$	21.2111	1.47785
$207$	0	0
$208$	1.09167	0.0756939
$209$	−3.00000	−0.207514
$210$	0	0
$211$	10.4222	0.717494	0.358747	−	0.933435i	$-0.383204\pi$
$211$	10.4222	0.717494	0.358747	+	0.933435i	$0.383204\pi$
$212$	0	0
$213$	0	0
$214$	−5.09167	−0.348060
$215$	5.21110	0.355394
$216$	0	0
$217$	2.00000	0.135769
$218$	−23.0278	−1.55964
$219$	0	0
$220$	3.30278	0.222673
$221$	−14.4222	−0.970143
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−5.30278	−0.354307
$225$	0	0
$226$	−12.0000	−0.798228
$227$	16.4222	1.08998	0.544990	−	0.838443i	$-0.316533\pi$
$227$	16.4222	1.08998	0.544990	+	0.838443i	$0.316533\pi$
$228$	0	0
$229$	11.2111	0.740851	0.370425	−	0.928862i	$-0.379212\pi$
$229$	11.2111	0.740851	0.370425	+	0.928862i	$0.379212\pi$
$230$	−4.60555	−0.303681
$231$	0	0
$232$	−16.8167	−1.10407
$233$	−21.6333	−1.41725	−0.708623	−	0.705588i	$-0.750683\pi$
$233$	−21.6333	−1.41725	−0.708623	+	0.705588i	$0.750683\pi$
$234$	0	0
$235$	2.39445	0.156197
$236$	−25.1194	−1.63514
$237$	0	0
$238$	−9.21110	−0.597067
$239$	1.42221	0.0919948	0.0459974	−	0.998942i	$-0.485353\pi$
$239$	1.42221	0.0919948	0.0459974	+	0.998942i	$0.485353\pi$
$240$	0	0
$241$	10.3944	0.669565	0.334783	−	0.942295i	$-0.391337\pi$
$241$	10.3944	0.669565	0.334783	+	0.942295i	$0.391337\pi$
$242$	−2.30278	−0.148028
$243$	0	0
$244$	−37.0278	−2.37046
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	10.8167	0.688247
$248$	6.00000	0.381000
$249$	0	0
$250$	−20.7250	−1.31076
$251$	−16.3944	−1.03481	−0.517404	−	0.855741i	$-0.673102\pi$
$251$	−16.3944	−1.03481	−0.517404	+	0.855741i	$0.673102\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	23.0278	1.44489
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−18.2111	−1.13598	−0.567989	−	0.823036i	$-0.692278\pi$
$257$	−18.2111	−1.13598	−0.567989	+	0.823036i	$0.692278\pi$
$258$	0	0
$259$	8.21110	0.510213
$260$	−11.9083	−0.738523
$261$	0	0
$262$	13.8167	0.853596
$263$	−21.4222	−1.32095	−0.660475	−	0.750848i	$-0.729645\pi$
$263$	−21.4222	−1.32095	−0.660475	+	0.750848i	$0.729645\pi$
$264$	0	0
$265$	0	0
$266$	6.90833	0.423577
$267$	0	0
$268$	5.30278	0.323919
$269$	24.4222	1.48905	0.744524	−	0.667596i	$-0.232676\pi$
$269$	24.4222	1.48905	0.744524	+	0.667596i	$0.232676\pi$
$270$	0	0
$271$	28.2111	1.71370	0.856851	−	0.515564i	$-0.172417\pi$
$271$	28.2111	1.71370	0.856851	+	0.515564i	$0.172417\pi$
$272$	−1.21110	−0.0734339
$273$	0	0
$274$	14.7889	0.893430
$275$	4.00000	0.241209
$276$	0	0
$277$	−2.78890	−0.167569	−0.0837843	−	0.996484i	$-0.526701\pi$
$277$	−2.78890	−0.167569	−0.0837843	+	0.996484i	$0.526701\pi$
$278$	−33.2111	−1.99187
$279$	0	0
$280$	−3.00000	−0.179284
$281$	12.3944	0.739391	0.369695	−	0.929153i	$-0.379462\pi$
$281$	12.3944	0.739391	0.369695	+	0.929153i	$0.379462\pi$
$282$	0	0
$283$	−15.4222	−0.916755	−0.458377	−	0.888758i	$-0.651569\pi$
$283$	−15.4222	−0.916755	−0.458377	+	0.888758i	$0.651569\pi$
$284$	−37.0278	−2.19719
$285$	0	0
$286$	8.30278	0.490953
$287$	−7.21110	−0.425658
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	12.9083	0.758003
$291$	0	0
$292$	−42.3305	−2.47721
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	7.60555	0.442812
$296$	24.6333	1.43178
$297$	0	0
$298$	−49.7527	−2.88210
$299$	−7.21110	−0.417029
$300$	0	0
$301$	5.21110	0.300363
$302$	−42.4222	−2.44112
$303$	0	0
$304$	0.908327	0.0520961
$305$	11.2111	0.641946
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	3.30278	0.188193
$309$	0	0
$310$	−4.60555	−0.261578
$311$	1.57779	0.0894685	0.0447343	−	0.998999i	$-0.485756\pi$
$311$	1.57779	0.0894685	0.0447343	+	0.998999i	$0.485756\pi$
$312$	0	0
$313$	−26.4222	−1.49347	−0.746736	−	0.665121i	$-0.768380\pi$
$313$	−26.4222	−1.49347	−0.746736	+	0.665121i	$0.768380\pi$
$314$	11.0278	0.622332
$315$	0	0
$316$	−10.6056	−0.596609
$317$	24.4222	1.37169	0.685844	−	0.727749i	$-0.259433\pi$
$317$	24.4222	1.37169	0.685844	+	0.727749i	$0.259433\pi$
$318$	0	0
$319$	−5.60555	−0.313851
$320$	12.8167	0.716473
$321$	0	0
$322$	−4.60555	−0.256657
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−14.4222	−0.800000
$326$	29.5139	1.63462
$327$	0	0
$328$	−21.6333	−1.19450
$329$	2.39445	0.132010
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−39.6333	−2.17516
$333$	0	0
$334$	53.4500	2.92465
$335$	−1.60555	−0.0877206
$336$	0	0
$337$	36.4222	1.98404	0.992022	−	0.126065i	$-0.0402348\pi$
$337$	36.4222	1.98404	0.992022	+	0.126065i	$0.0402348\pi$
$338$	0	0
$339$	0	0
$340$	13.2111	0.716473
$341$	2.00000	0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	15.6333	0.842891
$345$	0	0
$346$	−37.8167	−2.03304
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	−33.2389	−1.77924	−0.889618	−	0.456706i	$-0.849029\pi$
$349$	−33.2389	−1.77924	−0.889618	+	0.456706i	$0.849029\pi$
$350$	−9.21110	−0.492354
$351$	0	0
$352$	−5.30278	−0.282639
$353$	12.6333	0.672403	0.336202	−	0.941790i	$-0.390858\pi$
$353$	12.6333	0.672403	0.336202	+	0.941790i	$0.390858\pi$
$354$	0	0
$355$	11.2111	0.595024
$356$	−19.8167	−1.05028
$357$	0	0
$358$	−37.8167	−1.99867
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−19.3944	−1.01935
$363$	0	0
$364$	−11.9083	−0.624166
$365$	12.8167	0.670854
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−0.605551	−0.0315665
$369$	0	0
$370$	−18.9083	−0.982997
$371$	0	0
$372$	0	0
$373$	4.78890	0.247960	0.123980	−	0.992285i	$-0.460434\pi$
$373$	4.78890	0.247960	0.123980	+	0.992285i	$0.460434\pi$
$374$	−9.21110	−0.476295
$375$	0	0
$376$	7.18335	0.370453
$377$	20.2111	1.04092
$378$	0	0
$379$	8.81665	0.452881	0.226441	−	0.974025i	$-0.427291\pi$
$379$	8.81665	0.452881	0.226441	+	0.974025i	$0.427291\pi$
$380$	−9.90833	−0.508286
$381$	0	0
$382$	−11.0278	−0.564229
$383$	26.4222	1.35011	0.675056	−	0.737767i	$-0.264120\pi$
$383$	26.4222	1.35011	0.675056	+	0.737767i	$0.264120\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−60.8444	−3.09690
$387$	0	0
$388$	4.00000	0.203069
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	−3.00000	−0.151523
$393$	0	0
$394$	35.0278	1.76467
$395$	3.21110	0.161568
$396$	0	0
$397$	36.8444	1.84917	0.924584	−	0.380978i	$-0.124413\pi$
$397$	36.8444	1.84917	0.924584	+	0.380978i	$0.124413\pi$
$398$	39.6333	1.98664
$399$	0	0
$400$	−1.21110	−0.0605551
$401$	9.21110	0.459981	0.229990	−	0.973193i	$-0.426131\pi$
$401$	9.21110	0.459981	0.229990	+	0.973193i	$0.426131\pi$
$402$	0	0
$403$	−7.21110	−0.359211
$404$	−41.0278	−2.04121
$405$	0	0
$406$	12.9083	0.640630
$407$	8.21110	0.407009
$408$	0	0
$409$	−23.2111	−1.14772	−0.573858	−	0.818955i	$-0.694554\pi$
$409$	−23.2111	−1.14772	−0.573858	+	0.818955i	$0.694554\pi$
$410$	16.6056	0.820090
$411$	0	0
$412$	−30.4222	−1.49879
$413$	7.60555	0.374245
$414$	0	0
$415$	12.0000	0.589057
$416$	19.1194	0.937407
$417$	0	0
$418$	6.90833	0.337897
$419$	−30.8167	−1.50549	−0.752746	−	0.658311i	$-0.771271\pi$
$419$	−30.8167	−1.50549	−0.752746	+	0.658311i	$0.771271\pi$
$420$	0	0
$421$	22.6333	1.10308	0.551540	−	0.834148i	$-0.314040\pi$
$421$	22.6333	1.10308	0.551540	+	0.834148i	$0.314040\pi$
$422$	−24.0000	−1.16830
$423$	0	0
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	11.2111	0.542543
$428$	7.30278	0.352993
$429$	0	0
$430$	−12.0000	−0.578691
$431$	19.4222	0.935535	0.467767	−	0.883852i	$-0.345059\pi$
$431$	19.4222	0.935535	0.467767	+	0.883852i	$0.345059\pi$
$432$	0	0
$433$	16.7889	0.806823	0.403411	−	0.915019i	$-0.367824\pi$
$433$	16.7889	0.806823	0.403411	+	0.915019i	$0.367824\pi$
$434$	−4.60555	−0.221074
$435$	0	0
$436$	33.0278	1.58174
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−20.6333	−0.984774	−0.492387	−	0.870376i	$-0.663876\pi$
$439$	−20.6333	−0.984774	−0.492387	+	0.870376i	$0.663876\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	33.2111	1.57969
$443$	−21.2111	−1.00777	−0.503885	−	0.863771i	$-0.668096\pi$
$443$	−21.2111	−1.00777	−0.503885	+	0.863771i	$0.668096\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−9.21110	−0.436158
$447$	0	0
$448$	12.8167	0.605530
$449$	33.2111	1.56733	0.783664	−	0.621184i	$-0.213348\pi$
$449$	33.2111	1.56733	0.783664	+	0.621184i	$0.213348\pi$
$450$	0	0
$451$	−7.21110	−0.339558
$452$	17.2111	0.809542
$453$	0	0
$454$	−37.8167	−1.77482
$455$	3.60555	0.169031
$456$	0	0
$457$	38.8444	1.81706	0.908532	−	0.417814i	$-0.137204\pi$
$457$	38.8444	1.81706	0.908532	+	0.417814i	$0.137204\pi$
$458$	−25.8167	−1.20633
$459$	0	0
$460$	6.60555	0.307985
$461$	−15.6333	−0.728116	−0.364058	−	0.931376i	$-0.618609\pi$
$461$	−15.6333	−0.728116	−0.364058	+	0.931376i	$0.618609\pi$
$462$	0	0
$463$	2.81665	0.130901	0.0654505	−	0.997856i	$-0.479152\pi$
$463$	2.81665	0.130901	0.0654505	+	0.997856i	$0.479152\pi$
$464$	1.69722	0.0787917
$465$	0	0
$466$	49.8167	2.30771
$467$	−13.1833	−0.610053	−0.305026	−	0.952344i	$-0.598665\pi$
$467$	−13.1833	−0.610053	−0.305026	+	0.952344i	$0.598665\pi$
$468$	0	0
$469$	−1.60555	−0.0741375
$470$	−5.51388	−0.254336
$471$	0	0
$472$	22.8167	1.05022
$473$	5.21110	0.239607
$474$	0	0
$475$	−12.0000	−0.550598
$476$	13.2111	0.605530
$477$	0	0
$478$	−3.27502	−0.149796
$479$	−34.8444	−1.59208	−0.796041	−	0.605243i	$-0.793076\pi$
$479$	−34.8444	−1.59208	−0.796041	+	0.605243i	$0.793076\pi$
$480$	0	0
$481$	−29.6056	−1.34990
$482$	−23.9361	−1.09026
$483$	0	0
$484$	3.30278	0.150126
$485$	−1.21110	−0.0549933
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	33.6333	1.52251
$489$	0	0
$490$	2.30278	0.104029
$491$	−12.2111	−0.551079	−0.275540	−	0.961290i	$-0.588857\pi$
$491$	−12.2111	−0.551079	−0.275540	+	0.961290i	$0.588857\pi$
$492$	0	0
$493$	−22.4222	−1.00985
$494$	−24.9083	−1.12068
$495$	0	0
$496$	−0.605551	−0.0271901
$497$	11.2111	0.502887
$498$	0	0
$499$	0.816654	0.0365584	0.0182792	−	0.999833i	$-0.494181\pi$
$499$	0.816654	0.0365584	0.0182792	+	0.999833i	$0.494181\pi$
$500$	29.7250	1.32934
$501$	0	0
$502$	37.7527	1.68499
$503$	24.4222	1.08893	0.544466	−	0.838783i	$-0.316732\pi$
$503$	24.4222	1.08893	0.544466	+	0.838783i	$0.316732\pi$
$504$	0	0
$505$	12.4222	0.552781
$506$	−4.60555	−0.204742
$507$	0	0
$508$	−33.0278	−1.46537
$509$	−4.42221	−0.196011	−0.0980054	−	0.995186i	$-0.531246\pi$
$509$	−4.42221	−0.196011	−0.0980054	+	0.995186i	$0.531246\pi$
$510$	0	0
$511$	12.8167	0.566975
$512$	3.42221	0.151242
$513$	0	0
$514$	41.9361	1.84972
$515$	9.21110	0.405890
$516$	0	0
$517$	2.39445	0.105308
$518$	−18.9083	−0.830784
$519$	0	0
$520$	10.8167	0.474342
$521$	−14.2111	−0.622600	−0.311300	−	0.950312i	$-0.600764\pi$
$521$	−14.2111	−0.622600	−0.311300	+	0.950312i	$0.600764\pi$
$522$	0	0
$523$	−43.0000	−1.88026	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−43.0000	−1.88026	−0.940129	+	0.340818i	$0.889296\pi$
$524$	−19.8167	−0.865695
$525$	0	0
$526$	49.3305	2.15091
$527$	8.00000	0.348485
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	−9.90833	−0.429580
$533$	26.0000	1.12619
$534$	0	0
$535$	−2.21110	−0.0955943
$536$	−4.81665	−0.208048
$537$	0	0
$538$	−56.2389	−2.42463
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−7.21110	−0.310030	−0.155015	−	0.987912i	$-0.549543\pi$
$541$	−7.21110	−0.310030	−0.155015	+	0.987912i	$0.549543\pi$
$542$	−64.9638	−2.79044
$543$	0	0
$544$	−21.2111	−0.909419
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−9.21110	−0.393838	−0.196919	−	0.980420i	$-0.563094\pi$
$547$	−9.21110	−0.393838	−0.196919	+	0.980420i	$0.563094\pi$
$548$	−21.2111	−0.906093
$549$	0	0
$550$	−9.21110	−0.392763
$551$	16.8167	0.716414
$552$	0	0
$553$	3.21110	0.136550
$554$	6.42221	0.272853
$555$	0	0
$556$	47.6333	2.02010
$557$	15.1833	0.643339	0.321670	−	0.946852i	$-0.395756\pi$
$557$	15.1833	0.643339	0.321670	+	0.946852i	$0.395756\pi$
$558$	0	0
$559$	−18.7889	−0.794686
$560$	0.302776	0.0127946
$561$	0	0
$562$	−28.5416	−1.20396
$563$	−41.2666	−1.73918	−0.869590	−	0.493774i	$-0.835617\pi$
$563$	−41.2666	−1.73918	−0.869590	+	0.493774i	$0.835617\pi$
$564$	0	0
$565$	−5.21110	−0.219233
$566$	35.5139	1.49276
$567$	0	0
$568$	33.6333	1.41122
$569$	−9.63331	−0.403849	−0.201925	−	0.979401i	$-0.564720\pi$
$569$	−9.63331	−0.403849	−0.201925	+	0.979401i	$0.564720\pi$
$570$	0	0
$571$	25.2111	1.05505	0.527526	−	0.849539i	$-0.323120\pi$
$571$	25.2111	1.05505	0.527526	+	0.849539i	$0.323120\pi$
$572$	−11.9083	−0.497912
$573$	0	0
$574$	16.6056	0.693102
$575$	8.00000	0.333623
$576$	0	0
$577$	23.2111	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$577$	23.2111	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$578$	2.30278	0.0957828
$579$	0	0
$580$	−18.5139	−0.768747
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	38.4500	1.59107
$585$	0	0
$586$	55.2666	2.28304
$587$	−4.39445	−0.181378	−0.0906892	−	0.995879i	$-0.528907\pi$
$587$	−4.39445	−0.181378	−0.0906892	+	0.995879i	$0.528907\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	−17.5139	−0.721035
$591$	0	0
$592$	−2.48612	−0.102179
$593$	33.2111	1.36382	0.681908	−	0.731438i	$-0.261150\pi$
$593$	33.2111	1.36382	0.681908	+	0.731438i	$0.261150\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	71.3583	2.92295
$597$	0	0
$598$	16.6056	0.679051
$599$	0.422205	0.0172508	0.00862542	−	0.999963i	$-0.497254\pi$
$599$	0.422205	0.0172508	0.00862542	+	0.999963i	$0.497254\pi$
$600$	0	0
$601$	3.18335	0.129851	0.0649257	−	0.997890i	$-0.479319\pi$
$601$	3.18335	0.129851	0.0649257	+	0.997890i	$0.479319\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	60.8444	2.47572
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	14.6333	0.593948	0.296974	−	0.954886i	$-0.404023\pi$
$607$	14.6333	0.593948	0.296974	+	0.954886i	$0.404023\pi$
$608$	15.9083	0.645168
$609$	0	0
$610$	−25.8167	−1.04529
$611$	−8.63331	−0.349266
$612$	0	0
$613$	25.2111	1.01827	0.509133	−	0.860688i	$-0.329966\pi$
$613$	25.2111	1.01827	0.509133	+	0.860688i	$0.329966\pi$
$614$	−18.4222	−0.743460
$615$	0	0
$616$	−3.00000	−0.120873
$617$	2.36669	0.0952794	0.0476397	−	0.998865i	$-0.484830\pi$
$617$	2.36669	0.0952794	0.0476397	+	0.998865i	$0.484830\pi$
$618$	0	0
$619$	−31.6333	−1.27145	−0.635725	−	0.771916i	$-0.719299\pi$
$619$	−31.6333	−1.27145	−0.635725	+	0.771916i	$0.719299\pi$
$620$	6.60555	0.265285
$621$	0	0
$622$	−3.63331	−0.145682
$623$	6.00000	0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	60.8444	2.43183
$627$	0	0
$628$	−15.8167	−0.631153
$629$	32.8444	1.30959
$630$	0	0
$631$	24.8444	0.989040	0.494520	−	0.869166i	$-0.335344\pi$
$631$	24.8444	0.989040	0.494520	+	0.869166i	$0.335344\pi$
$632$	9.63331	0.383192
$633$	0	0
$634$	−56.2389	−2.23353
$635$	10.0000	0.396838
$636$	0	0
$637$	3.60555	0.142857
$638$	12.9083	0.511046
$639$	0	0
$640$	−18.9083	−0.747417
$641$	21.6333	0.854464	0.427232	−	0.904142i	$-0.359489\pi$
$641$	21.6333	0.854464	0.427232	+	0.904142i	$0.359489\pi$
$642$	0	0
$643$	−30.0555	−1.18527	−0.592637	−	0.805470i	$-0.701913\pi$
$643$	−30.0555	−1.18527	−0.592637	+	0.805470i	$0.701913\pi$
$644$	6.60555	0.260295
$645$	0	0
$646$	27.6333	1.08722
$647$	41.6611	1.63787	0.818933	−	0.573890i	$-0.194566\pi$
$647$	41.6611	1.63787	0.818933	+	0.573890i	$0.194566\pi$
$648$	0	0
$649$	7.60555	0.298544
$650$	33.2111	1.30265
$651$	0	0
$652$	−42.3305	−1.65779
$653$	40.8444	1.59837	0.799183	−	0.601088i	$-0.205266\pi$
$653$	40.8444	1.59837	0.799183	+	0.601088i	$0.205266\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	2.18335	0.0852453
$657$	0	0
$658$	−5.51388	−0.214953
$659$	−33.0555	−1.28766	−0.643830	−	0.765169i	$-0.722656\pi$
$659$	−33.0555	−1.28766	−0.643830	+	0.765169i	$0.722656\pi$
$660$	0	0
$661$	−28.8444	−1.12192	−0.560959	−	0.827844i	$-0.689567\pi$
$661$	−28.8444	−1.12192	−0.560959	+	0.827844i	$0.689567\pi$
$662$	73.6888	2.86400
$663$	0	0
$664$	36.0000	1.39707
$665$	3.00000	0.116335
$666$	0	0
$667$	−11.2111	−0.434096
$668$	−76.6611	−2.96611
$669$	0	0
$670$	3.69722	0.142836
$671$	11.2111	0.432800
$672$	0	0
$673$	−24.0555	−0.927272	−0.463636	−	0.886026i	$-0.653455\pi$
$673$	−24.0555	−0.927272	−0.463636	+	0.886026i	$0.653455\pi$
$674$	−83.8722	−3.23064
$675$	0	0
$676$	0	0
$677$	−38.0555	−1.46259	−0.731296	−	0.682060i	$-0.761084\pi$
$677$	−38.0555	−1.46259	−0.731296	+	0.682060i	$0.761084\pi$
$678$	0	0
$679$	−1.21110	−0.0464779
$680$	−12.0000	−0.460179
$681$	0	0
$682$	−4.60555	−0.176356
$683$	−46.0555	−1.76227	−0.881133	−	0.472869i	$-0.843218\pi$
$683$	−46.0555	−1.76227	−0.881133	+	0.472869i	$0.843218\pi$
$684$	0	0
$685$	6.42221	0.245380
$686$	2.30278	0.0879204
$687$	0	0
$688$	−1.57779	−0.0601529
$689$	0	0
$690$	0	0
$691$	−42.8444	−1.62988	−0.814939	−	0.579547i	$-0.803230\pi$
$691$	−42.8444	−1.62988	−0.814939	+	0.579547i	$0.803230\pi$
$692$	54.2389	2.06185
$693$	0	0
$694$	64.4777	2.44754
$695$	−14.4222	−0.547065
$696$	0	0
$697$	−28.8444	−1.09256
$698$	76.5416	2.89714
$699$	0	0
$700$	13.2111	0.499333
$701$	39.2111	1.48098	0.740491	−	0.672066i	$-0.234593\pi$
$701$	39.2111	1.48098	0.740491	+	0.672066i	$0.234593\pi$
$702$	0	0
$703$	−24.6333	−0.929063
$704$	12.8167	0.483046
$705$	0	0
$706$	−29.0917	−1.09488
$707$	12.4222	0.467185
$708$	0	0
$709$	5.78890	0.217407	0.108703	−	0.994074i	$-0.465330\pi$
$709$	5.78890	0.217407	0.108703	+	0.994074i	$0.465330\pi$
$710$	−25.8167	−0.968882
$711$	0	0
$712$	18.0000	0.674579
$713$	4.00000	0.149801
$714$	0	0
$715$	3.60555	0.134840
$716$	54.2389	2.02700
$717$	0	0
$718$	−18.4222	−0.687511
$719$	7.97224	0.297315	0.148657	−	0.988889i	$-0.452505\pi$
$719$	7.97224	0.297315	0.148657	+	0.988889i	$0.452505\pi$
$720$	0	0
$721$	9.21110	0.343039
$722$	23.0278	0.857004
$723$	0	0
$724$	27.8167	1.03380
$725$	−22.4222	−0.832740
$726$	0	0
$727$	−4.42221	−0.164011	−0.0820053	−	0.996632i	$-0.526132\pi$
$727$	−4.42221	−0.164011	−0.0820053	+	0.996632i	$0.526132\pi$
$728$	10.8167	0.400892
$729$	0	0
$730$	−29.5139	−1.09236
$731$	20.8444	0.770958
$732$	0	0
$733$	35.2111	1.30055	0.650276	−	0.759698i	$-0.274653\pi$
$733$	35.2111	1.30055	0.650276	+	0.759698i	$0.274653\pi$
$734$	−23.0278	−0.849970
$735$	0	0
$736$	−10.6056	−0.390926
$737$	−1.60555	−0.0591412
$738$	0	0
$739$	−27.2111	−1.00098	−0.500488	−	0.865743i	$-0.666846\pi$
$739$	−27.2111	−1.00098	−0.500488	+	0.865743i	$0.666846\pi$
$740$	27.1194	0.996930
$741$	0	0
$742$	0	0
$743$	27.4222	1.00602	0.503012	−	0.864280i	$-0.332225\pi$
$743$	27.4222	1.00602	0.503012	+	0.864280i	$0.332225\pi$
$744$	0	0
$745$	−21.6056	−0.791566
$746$	−11.0278	−0.403755
$747$	0	0
$748$	13.2111	0.483046
$749$	−2.21110	−0.0807919
$750$	0	0
$751$	−40.4500	−1.47604	−0.738020	−	0.674779i	$-0.764239\pi$
$751$	−40.4500	−1.47604	−0.738020	+	0.674779i	$0.764239\pi$
$752$	−0.724981	−0.0264373
$753$	0	0
$754$	−46.5416	−1.69495
$755$	−18.4222	−0.670453
$756$	0	0
$757$	−46.6333	−1.69492	−0.847458	−	0.530862i	$-0.821868\pi$
$757$	−46.6333	−1.69492	−0.847458	+	0.530862i	$0.821868\pi$
$758$	−20.3028	−0.737430
$759$	0	0
$760$	9.00000	0.326464
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	15.8167	0.572226
$765$	0	0
$766$	−60.8444	−2.19840
$767$	−27.4222	−0.990158
$768$	0	0
$769$	−9.60555	−0.346385	−0.173193	−	0.984888i	$-0.555408\pi$
$769$	−9.60555	−0.346385	−0.173193	+	0.984888i	$0.555408\pi$
$770$	2.30278	0.0829863
$771$	0	0
$772$	87.2666	3.14079
$773$	−40.2666	−1.44829	−0.724145	−	0.689648i	$-0.757765\pi$
$773$	−40.2666	−1.44829	−0.724145	+	0.689648i	$0.757765\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−3.63331	−0.130428
$777$	0	0
$778$	−13.8167	−0.495351
$779$	21.6333	0.775094
$780$	0	0
$781$	11.2111	0.401165
$782$	−18.4222	−0.658777
$783$	0	0
$784$	0.302776	0.0108134
$785$	4.78890	0.170923
$786$	0	0
$787$	11.4222	0.407158	0.203579	−	0.979059i	$-0.434743\pi$
$787$	11.4222	0.407158	0.203579	+	0.979059i	$0.434743\pi$
$788$	−50.2389	−1.78969
$789$	0	0
$790$	−7.39445	−0.263083
$791$	−5.21110	−0.185285
$792$	0	0
$793$	−40.4222	−1.43543
$794$	−84.8444	−3.01102
$795$	0	0
$796$	−56.8444	−2.01480
$797$	−0.577795	−0.0204665	−0.0102333	−	0.999948i	$-0.503257\pi$
$797$	−0.577795	−0.0204665	−0.0102333	+	0.999948i	$0.503257\pi$
$798$	0	0
$799$	9.57779	0.338838
$800$	−21.2111	−0.749926
$801$	0	0
$802$	−21.2111	−0.748990
$803$	12.8167	0.452290
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	16.6056	0.584906
$807$	0	0
$808$	37.2666	1.31103
$809$	34.8167	1.22409	0.612044	−	0.790824i	$-0.290347\pi$
$809$	34.8167	1.22409	0.612044	+	0.790824i	$0.290347\pi$
$810$	0	0
$811$	23.4222	0.822465	0.411232	−	0.911531i	$-0.365098\pi$
$811$	23.4222	0.822465	0.411232	+	0.911531i	$0.365098\pi$
$812$	−18.5139	−0.649710
$813$	0	0
$814$	−18.9083	−0.662737
$815$	12.8167	0.448948
$816$	0	0
$817$	−15.6333	−0.546940
$818$	53.4500	1.86883
$819$	0	0
$820$	−23.8167	−0.831714
$821$	11.2389	0.392239	0.196119	−	0.980580i	$-0.437166\pi$
$821$	11.2389	0.392239	0.196119	+	0.980580i	$0.437166\pi$
$822$	0	0
$823$	−23.6056	−0.822838	−0.411419	−	0.911446i	$-0.634967\pi$
$823$	−23.6056	−0.822838	−0.411419	+	0.911446i	$0.634967\pi$
$824$	27.6333	0.962652
$825$	0	0
$826$	−17.5139	−0.609386
$827$	15.0555	0.523531	0.261766	−	0.965131i	$-0.415695\pi$
$827$	15.0555	0.523531	0.261766	+	0.965131i	$0.415695\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−27.6333	−0.959166
$831$	0	0
$832$	−46.2111	−1.60208
$833$	−4.00000	−0.138592
$834$	0	0
$835$	23.2111	0.803253
$836$	−9.90833	−0.342687
$837$	0	0
$838$	70.9638	2.45141
$839$	−22.3944	−0.773142	−0.386571	−	0.922260i	$-0.626341\pi$
$839$	−22.3944	−0.773142	−0.386571	+	0.922260i	$0.626341\pi$
$840$	0	0
$841$	2.42221	0.0835243
$842$	−52.1194	−1.79615
$843$	0	0
$844$	34.4222	1.18486
$845$	0	0
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	−36.8444	−1.26375
$851$	16.4222	0.562946
$852$	0	0
$853$	10.3667	0.354949	0.177474	−	0.984125i	$-0.443207\pi$
$853$	10.3667	0.354949	0.177474	+	0.984125i	$0.443207\pi$
$854$	−25.8167	−0.883428
$855$	0	0
$856$	−6.63331	−0.226722
$857$	−21.5778	−0.737083	−0.368542	−	0.929611i	$-0.620143\pi$
$857$	−21.5778	−0.737083	−0.368542	+	0.929611i	$0.620143\pi$
$858$	0	0
$859$	52.4222	1.78862	0.894311	−	0.447445i	$-0.147666\pi$
$859$	52.4222	1.78862	0.894311	+	0.447445i	$0.147666\pi$
$860$	17.2111	0.586894
$861$	0	0
$862$	−44.7250	−1.52334
$863$	56.0555	1.90815	0.954076	−	0.299565i	$-0.0968415\pi$
$863$	56.0555	1.90815	0.954076	+	0.299565i	$0.0968415\pi$
$864$	0	0
$865$	−16.4222	−0.558372
$866$	−38.6611	−1.31376
$867$	0	0
$868$	6.60555	0.224207
$869$	3.21110	0.108929
$870$	0	0
$871$	5.78890	0.196149
$872$	−30.0000	−1.01593
$873$	0	0
$874$	13.8167	0.467355
$875$	−9.00000	−0.304256
$876$	0	0
$877$	10.4222	0.351933	0.175966	−	0.984396i	$-0.443695\pi$
$877$	10.4222	0.351933	0.175966	+	0.984396i	$0.443695\pi$
$878$	47.5139	1.60352
$879$	0	0
$880$	0.302776	0.0102066
$881$	−31.0555	−1.04629	−0.523143	−	0.852245i	$-0.675241\pi$
$881$	−31.0555	−1.04629	−0.523143	+	0.852245i	$0.675241\pi$
$882$	0	0
$883$	44.8167	1.50820	0.754100	−	0.656759i	$-0.228073\pi$
$883$	44.8167	1.50820	0.754100	+	0.656759i	$0.228073\pi$
$884$	−47.6333	−1.60208
$885$	0	0
$886$	48.8444	1.64096
$887$	−11.6333	−0.390608	−0.195304	−	0.980743i	$-0.562569\pi$
$887$	−11.6333	−0.390608	−0.195304	+	0.980743i	$0.562569\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	−13.8167	−0.463135
$891$	0	0
$892$	13.2111	0.442340
$893$	−7.18335	−0.240382
$894$	0	0
$895$	−16.4222	−0.548934
$896$	−18.9083	−0.631683
$897$	0	0
$898$	−76.4777	−2.55209
$899$	−11.2111	−0.373911
$900$	0	0
$901$	0	0
$902$	16.6056	0.552904
$903$	0	0
$904$	−15.6333	−0.519956
$905$	−8.42221	−0.279964
$906$	0	0
$907$	44.8444	1.48903	0.744517	−	0.667603i	$-0.232680\pi$
$907$	44.8444	1.48903	0.744517	+	0.667603i	$0.232680\pi$
$908$	54.2389	1.79998
$909$	0	0
$910$	−8.30278	−0.275234
$911$	−27.6333	−0.915532	−0.457766	−	0.889073i	$-0.651350\pi$
$911$	−27.6333	−0.915532	−0.457766	+	0.889073i	$0.651350\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	−89.4500	−2.95874
$915$	0	0
$916$	37.0278	1.22343
$917$	6.00000	0.198137
$918$	0	0
$919$	−42.4222	−1.39938	−0.699690	−	0.714447i	$-0.746678\pi$
$919$	−42.4222	−1.39938	−0.699690	+	0.714447i	$0.746678\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	36.0000	1.18560
$923$	−40.4222	−1.33051
$924$	0	0
$925$	32.8444	1.07992
$926$	−6.48612	−0.213147
$927$	0	0
$928$	29.7250	0.975770
$929$	48.6333	1.59561	0.797804	−	0.602918i	$-0.205995\pi$
$929$	48.6333	1.59561	0.797804	+	0.602918i	$0.205995\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	−71.4500	−2.34042
$933$	0	0
$934$	30.3583	0.993354
$935$	−4.00000	−0.130814
$936$	0	0
$937$	16.0555	0.524511	0.262255	−	0.964999i	$-0.415534\pi$
$937$	16.0555	0.524511	0.262255	+	0.964999i	$0.415534\pi$
$938$	3.69722	0.120719
$939$	0	0
$940$	7.90833	0.257941
$941$	−26.8444	−0.875103	−0.437551	−	0.899193i	$-0.644154\pi$
$941$	−26.8444	−0.875103	−0.437551	+	0.899193i	$0.644154\pi$
$942$	0	0
$943$	−14.4222	−0.469652
$944$	−2.30278	−0.0749490
$945$	0	0
$946$	−12.0000	−0.390154
$947$	14.8444	0.482379	0.241189	−	0.970478i	$-0.422463\pi$
$947$	14.8444	0.482379	0.241189	+	0.970478i	$0.422463\pi$
$948$	0	0
$949$	−46.2111	−1.50008
$950$	27.6333	0.896543
$951$	0	0
$952$	−12.0000	−0.388922
$953$	−24.4500	−0.792012	−0.396006	−	0.918248i	$-0.629604\pi$
$953$	−24.4500	−0.792012	−0.396006	+	0.918248i	$0.629604\pi$
$954$	0	0
$955$	−4.78890	−0.154965
$956$	4.69722	0.151919
$957$	0	0
$958$	80.2389	2.59240
$959$	6.42221	0.207384
$960$	0	0
$961$	−27.0000	−0.870968
$962$	68.1749	2.19805
$963$	0	0
$964$	34.3305	1.10571
$965$	−26.4222	−0.850561
$966$	0	0
$967$	10.7889	0.346948	0.173474	−	0.984838i	$-0.444501\pi$
$967$	10.7889	0.346948	0.173474	+	0.984838i	$0.444501\pi$
$968$	−3.00000	−0.0964237
$969$	0	0
$970$	2.78890	0.0895461
$971$	−51.6611	−1.65788	−0.828941	−	0.559336i	$-0.811056\pi$
$971$	−51.6611	−1.65788	−0.828941	+	0.559336i	$0.811056\pi$
$972$	0	0
$973$	−14.4222	−0.462355
$974$	−18.4222	−0.590286
$975$	0	0
$976$	−3.39445	−0.108654
$977$	−12.7889	−0.409153	−0.204577	−	0.978851i	$-0.565582\pi$
$977$	−12.7889	−0.409153	−0.204577	+	0.978851i	$0.565582\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	−3.30278	−0.105503
$981$	0	0
$982$	28.1194	0.897327
$983$	43.2666	1.37999	0.689995	−	0.723814i	$-0.257613\pi$
$983$	43.2666	1.37999	0.689995	+	0.723814i	$0.257613\pi$
$984$	0	0
$985$	15.2111	0.484666
$986$	51.6333	1.64434
$987$	0	0
$988$	35.7250	1.13656
$989$	10.4222	0.331407
$990$	0	0
$991$	14.8167	0.470667	0.235333	−	0.971915i	$-0.424382\pi$
$991$	14.8167	0.470667	0.235333	+	0.971915i	$0.424382\pi$
$992$	−10.6056	−0.336727
$993$	0	0
$994$	−25.8167	−0.818855
$995$	17.2111	0.545629
$996$	0	0
$997$	16.0555	0.508483	0.254242	−	0.967141i	$-0.418174\pi$
$997$	16.0555	0.508483	0.254242	+	0.967141i	$0.418174\pi$
$998$	−1.88057	−0.0595284
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.g.1.1	✓	2
3.2	odd	2		693.2.a.i.1.2	yes	2
7.6	odd	2		4851.2.a.x.1.1		2
11.10	odd	2		7623.2.a.br.1.2		2
21.20	even	2		4851.2.a.z.1.2		2
33.32	even	2		7623.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.a.g.1.1	✓	2	1.1	even	1	trivial
693.2.a.i.1.2	yes	2	3.2	odd	2
4851.2.a.x.1.1		2	7.6	odd	2
4851.2.a.z.1.2		2	21.20	even	2
7623.2.a.bd.1.1		2	33.32	even	2
7623.2.a.br.1.2		2	11.10	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q-2.30278 q^{2} +3.30278 q^{4} -1.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +2.30278 q^{10} -1.00000 q^{11} +3.60555 q^{13} +2.30278 q^{14} +0.302776 q^{16} -4.00000 q^{17} +3.00000 q^{19} -3.30278 q^{20} +2.30278 q^{22} -2.00000 q^{23} -4.00000 q^{25} -8.30278 q^{26} -3.30278 q^{28} +5.60555 q^{29} -2.00000 q^{31} +5.30278 q^{32} +9.21110 q^{34} +1.00000 q^{35} -8.21110 q^{37} -6.90833 q^{38} +3.00000 q^{40} +7.21110 q^{41} -5.21110 q^{43} -3.30278 q^{44} +4.60555 q^{46} -2.39445 q^{47} +1.00000 q^{49} +9.21110 q^{50} +11.9083 q^{52} +1.00000 q^{55} +3.00000 q^{56} -12.9083 q^{58} -7.60555 q^{59} -11.2111 q^{61} +4.60555 q^{62} -12.8167 q^{64} -3.60555 q^{65} +1.60555 q^{67} -13.2111 q^{68} -2.30278 q^{70} -11.2111 q^{71} -12.8167 q^{73} +18.9083 q^{74} +9.90833 q^{76} +1.00000 q^{77} -3.21110 q^{79} -0.302776 q^{80} -16.6056 q^{82} -12.0000 q^{83} +4.00000 q^{85} +12.0000 q^{86} +3.00000 q^{88} -6.00000 q^{89} -3.60555 q^{91} -6.60555 q^{92} +5.51388 q^{94} -3.00000 q^{95} +1.21110 q^{97} -2.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{8} + q^{10} - 2 q^{11} + q^{14} - 3 q^{16} - 8 q^{17} + 6 q^{19} - 3 q^{20} + q^{22} - 4 q^{23} - 8 q^{25} - 13 q^{26} - 3 q^{28} + 4 q^{29} - 4 q^{31} + 7 q^{32} + 4 q^{34} + 2 q^{35} - 2 q^{37} - 3 q^{38} + 6 q^{40} + 4 q^{43} - 3 q^{44} + 2 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} + 13 q^{52} + 2 q^{55} + 6 q^{56} - 15 q^{58} - 8 q^{59} - 8 q^{61} + 2 q^{62} - 4 q^{64} - 4 q^{67} - 12 q^{68} - q^{70} - 8 q^{71} - 4 q^{73} + 27 q^{74} + 9 q^{76} + 2 q^{77} + 8 q^{79} + 3 q^{80} - 26 q^{82} - 24 q^{83} + 8 q^{85} + 24 q^{86} + 6 q^{88} - 12 q^{89} - 6 q^{92} - 7 q^{94} - 6 q^{95} - 12 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^8 + q^10 - 2 * q^11 + q^14 - 3 * q^16 - 8 * q^17 + 6 * q^19 - 3 * q^20 + q^22 - 4 * q^23 - 8 * q^25 - 13 * q^26 - 3 * q^28 + 4 * q^29 - 4 * q^31 + 7 * q^32 + 4 * q^34 + 2 * q^35 - 2 * q^37 - 3 * q^38 + 6 * q^40 + 4 * q^43 - 3 * q^44 + 2 * q^46 - 12 * q^47 + 2 * q^49 + 4 * q^50 + 13 * q^52 + 2 * q^55 + 6 * q^56 - 15 * q^58 - 8 * q^59 - 8 * q^61 + 2 * q^62 - 4 * q^64 - 4 * q^67 - 12 * q^68 - q^70 - 8 * q^71 - 4 * q^73 + 27 * q^74 + 9 * q^76 + 2 * q^77 + 8 * q^79 + 3 * q^80 - 26 * q^82 - 24 * q^83 + 8 * q^85 + 24 * q^86 + 6 * q^88 - 12 * q^89 - 6 * q^92 - 7 * q^94 - 6 * q^95 - 12 * q^97 - q^98

Introduction

L-functions

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