LMFDB - Embedded newform 69.4.c.a.68.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,4,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$4.07113179040$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-23}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 6 $ x^2 - x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.1
Root			$0.500000 + 2.39792i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.4.c.a.68.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.79583i q^{2} +(2.00000 - 4.79583i) q^{3} -15.0000 q^{4} +(-23.0000 - 9.59166i) q^{6} +33.5708i q^{8} +(-19.0000 - 19.1833i) q^{9} +O(q^{10})$	$q-4.79583i q^{2} +(2.00000 - 4.79583i) q^{3} -15.0000 q^{4} +(-23.0000 - 9.59166i) q^{6} +33.5708i q^{8} +(-19.0000 - 19.1833i) q^{9} +(-30.0000 + 71.9375i) q^{12} +74.0000 q^{13} +41.0000 q^{16} +(-92.0000 + 91.1208i) q^{18} -110.304i q^{23} +(161.000 + 67.1416i) q^{24} -125.000 q^{25} -354.892i q^{26} +(-130.000 + 52.7541i) q^{27} -134.283i q^{29} +344.000 q^{31} +71.9375i q^{32} +(285.000 + 287.750i) q^{36} +(148.000 - 354.892i) q^{39} -306.933i q^{41} -529.000 q^{46} +642.641i q^{47} +(82.0000 - 196.629i) q^{48} +343.000 q^{49} +599.479i q^{50} -1110.00 q^{52} +(253.000 + 623.458i) q^{54} -644.000 q^{58} +815.291i q^{59} -1649.77i q^{62} +673.000 q^{64} +(-529.000 - 220.608i) q^{69} -220.608i q^{71} +(644.000 - 637.846i) q^{72} +1226.00 q^{73} +(-250.000 + 599.479i) q^{75} +(-1702.00 - 709.783i) q^{78} +(-7.00000 + 728.966i) q^{81} -1472.00 q^{82} +(-644.000 - 268.567i) q^{87} +1654.56i q^{92} +(688.000 - 1649.77i) q^{93} +3082.00 q^{94} +(345.000 + 143.875i) q^{96} -1644.97i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 30 q^{4} - 46 q^{6} - 38 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 30 * q^4 - 46 * q^6 - 38 * q^9	$ 2 q + 4 q^{3} - 30 q^{4} - 46 q^{6} - 38 q^{9} - 60 q^{12} + 148 q^{13} + 82 q^{16} - 184 q^{18} + 322 q^{24} - 250 q^{25} - 260 q^{27} + 688 q^{31} + 570 q^{36} + 296 q^{39} - 1058 q^{46} + 164 q^{48} + 686 q^{49} - 2220 q^{52} + 506 q^{54} - 1288 q^{58} + 1346 q^{64} - 1058 q^{69} + 1288 q^{72} + 2452 q^{73} - 500 q^{75} - 3404 q^{78} - 14 q^{81} - 2944 q^{82} - 1288 q^{87} + 1376 q^{93} + 6164 q^{94} + 690 q^{96}+O(q^{100}) $ 2 * q + 4 * q^3 - 30 * q^4 - 46 * q^6 - 38 * q^9 - 60 * q^12 + 148 * q^13 + 82 * q^16 - 184 * q^18 + 322 * q^24 - 250 * q^25 - 260 * q^27 + 688 * q^31 + 570 * q^36 + 296 * q^39 - 1058 * q^46 + 164 * q^48 + 686 * q^49 - 2220 * q^52 + 506 * q^54 - 1288 * q^58 + 1346 * q^64 - 1058 * q^69 + 1288 * q^72 + 2452 * q^73 - 500 * q^75 - 3404 * q^78 - 14 * q^81 - 2944 * q^82 - 1288 * q^87 + 1376 * q^93 + 6164 * q^94 + 690 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	4.79583i		−	1.69558i	−0.530330	−	0.847791i	$-0.677932\pi$
$2$		−	4.79583i		−	1.69558i	0.530330	−	0.847791i	$-0.322068\pi$
$3$	2.00000	−	4.79583i	0.384900	−	0.922958i
$3$	2.00000	−	4.79583i	0.384900	−	0.922958i
$4$	−15.0000			−1.87500
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−23.0000	−	9.59166i	−1.56495	−	0.652630i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			33.5708i			1.48363i
$9$	−19.0000	−	19.1833i	−0.703704	−	0.710494i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−30.0000	+	71.9375i	−0.721688	+	1.73055i
$13$	74.0000			1.57876			0.789381	−	0.613904i	$-0.210402\pi$
$13$	74.0000			1.57876			0.789381	+	0.613904i	$0.210402\pi$
$14$		0			0
$15$		0			0
$16$	41.0000			0.640625
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−92.0000	+	91.1208i	−1.20470	+	1.19319i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	110.304i		−	1.00000i
$23$		−	110.304i		−	1.00000i
$24$	161.000	+	67.1416i	1.36933	+	0.571051i
$25$	−125.000			−1.00000
$26$		−	354.892i		−	2.67692i
$27$	−130.000	+	52.7541i	−0.926612	+	0.376020i
$28$		0			0
$29$		−	134.283i		−	0.859854i	−0.902864	−	0.429927i	$-0.858539\pi$
$29$		−	134.283i		−	0.859854i	0.902864	−	0.429927i	$-0.141461\pi$
$30$		0			0
$31$	344.000			1.99304			0.996520	−	0.0833571i	$-0.0265642\pi$
$31$	344.000			1.99304			0.996520	+	0.0833571i	$0.0265642\pi$
$32$			71.9375i			0.397402i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	285.000	+	287.750i	1.31944	+	1.33218i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	148.000	−	354.892i	0.607666	−	1.45713i
$40$		0			0
$41$		−	306.933i		−	1.16914i	−0.811342	−	0.584572i	$-0.801262\pi$
$41$		−	306.933i		−	1.16914i	0.811342	−	0.584572i	$-0.198738\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−529.000			−1.69558
$47$			642.641i			1.99444i	0.0744843	+	0.997222i	$0.476269\pi$
$47$			642.641i			1.99444i	−0.0744843	+	0.997222i	$0.523731\pi$
$48$	82.0000	−	196.629i	0.246577	−	0.591270i
$49$	343.000			1.00000
$50$			599.479i			1.69558i
$51$		0			0
$52$	−1110.00			−2.96018
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	253.000	+	623.458i	0.637573	+	1.57115i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−644.000			−1.45795
$59$			815.291i			1.79902i	0.436905	+	0.899508i	$0.356075\pi$
$59$			815.291i			1.79902i	−0.436905	+	0.899508i	$0.643925\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		−	1649.77i		−	3.37936i
$63$		0			0
$64$	673.000			1.31445
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−529.000	−	220.608i	−0.922958	−	0.384900i
$70$		0			0
$71$		−	220.608i		−	0.368752i	−0.982856	−	0.184376i	$-0.940974\pi$
$71$		−	220.608i		−	0.368752i	0.982856	−	0.184376i	$-0.0590264\pi$
$72$	644.000	−	637.846i	1.05411	−	1.04404i
$73$	1226.00			1.96565			0.982825	−	0.184540i	$-0.0590796\pi$
$73$	1226.00			1.96565			0.982825	+	0.184540i	$0.0590796\pi$
$74$		0			0
$75$	−250.000	+	599.479i	−0.384900	+	0.922958i
$76$		0			0
$77$		0			0
$78$	−1702.00	−	709.783i	−2.47069	−	1.03035i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−7.00000	+	728.966i	−0.00960219	+	0.999954i
$82$	−1472.00			−1.98238
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−644.000	−	268.567i	−0.793610	−	0.330958i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$			1654.56i			1.87500i
$93$	688.000	−	1649.77i	0.767121	−	1.83949i
$94$	3082.00			3.38174
$95$		0			0
$96$	345.000	+	143.875i	0.366786	+	0.152960i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	1644.97i		−	1.69558i
$99$		0			0
$100$	1875.00			1.87500
$101$			1246.92i			1.22844i	0.789133	+	0.614222i	$0.210530\pi$
$101$			1246.92i			1.22844i	−0.789133	+	0.614222i	$0.789470\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			2484.24i			2.34231i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	1950.00	−	791.312i	1.73740	−	0.705038i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			2014.25i			1.61223i
$117$	−1406.00	−	1419.57i	−1.11098	−	1.12170i
$118$	3910.00			3.05038
$119$		0			0
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$	−1472.00	−	613.866i	−1.07907	−	0.450004i
$124$	−5160.00			−3.73695
$125$		0			0
$126$		0			0
$127$	−2608.00			−1.82223			−0.911113	−	0.412158i	$-0.864775\pi$
$127$	−2608.00			−1.82223			−0.911113	+	0.412158i	$0.864775\pi$
$128$		−	2652.09i		−	1.83136i
$129$		0			0
$130$		0			0
$131$		−	1083.86i		−	0.722879i	−0.932396	−	0.361439i	$-0.882285\pi$
$131$		−	1083.86i		−	0.722879i	0.932396	−	0.361439i	$-0.117715\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−1058.00	+	2536.99i	−0.652630	+	1.56495i
$139$	−412.000			−0.251406			−0.125703	−	0.992068i	$-0.540119\pi$
$139$	−412.000			−0.251406			−0.125703	+	0.992068i	$0.540119\pi$
$140$		0			0
$141$	3082.00	+	1285.28i	1.84079	+	0.767662i
$142$	−1058.00			−0.625249
$143$		0			0
$144$	−779.000	−	786.516i	−0.450810	−	0.455160i
$145$		0			0
$146$		−	5879.69i		−	3.33292i
$147$	686.000	−	1644.97i	0.384900	−	0.922958i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	2875.00	+	1198.96i	1.56495	+	0.652630i
$151$	−3616.00			−1.94878			−0.974390	−	0.224863i	$-0.927807\pi$
$151$	−3616.00			−1.94878			−0.974390	+	0.224863i	$0.927807\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2220.00	+	5323.37i	−1.13937	+	2.73212i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	3496.00	+	33.5708i	1.69550	+	0.0162813i
$163$	−1636.00			−0.786144			−0.393072	−	0.919508i	$-0.628588\pi$
$163$	−1636.00			−0.786144			−0.393072	+	0.919508i	$0.628588\pi$
$164$			4604.00i			2.19215i
$165$		0			0
$166$		0			0
$167$			3922.99i			1.81778i	0.417030	+	0.908892i	$0.363071\pi$
$167$			3922.99i			1.81778i	−0.417030	+	0.908892i	$0.636929\pi$
$168$		0			0
$169$	3279.00			1.49249
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2896.68i		−	1.27301i	−0.771273	−	0.636505i	$-0.780380\pi$
$173$		−	2896.68i		−	1.27301i	0.771273	−	0.636505i	$-0.219620\pi$
$174$	−1288.00	+	3088.52i	−0.561167	+	1.34563i
$175$		0			0
$176$		0			0
$177$	3910.00	+	1630.58i	1.66042	+	0.692441i
$178$		0			0
$179$			3922.99i			1.63809i	0.573730	+	0.819045i	$0.305496\pi$
$179$			3922.99i			1.63809i	−0.573730	+	0.819045i	$0.694504\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	3703.00			1.48363
$185$		0			0
$186$	−7912.00	−	3299.53i	−3.11901	−	1.30072i
$187$		0			0
$188$		−	9639.62i		−	3.73958i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	1346.00	−	3227.59i	0.505933	−	1.21319i
$193$	4286.00			1.59851			0.799257	−	0.600990i	$-0.205227\pi$
$193$	4286.00			1.59851			0.799257	+	0.600990i	$0.205227\pi$
$194$		0			0
$195$		0			0
$196$	−5145.00			−1.87500
$197$			556.316i			0.201197i	0.994927	+	0.100599i	$0.0320758\pi$
$197$			556.316i			0.201197i	−0.994927	+	0.100599i	$0.967924\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	4196.35i		−	1.48363i
$201$		0			0
$202$	5980.00			2.08293
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2116.00	+	2095.78i	−0.710494	+	0.703704i
$208$	3034.00			1.01139
$209$		0			0
$210$		0			0
$211$	2468.00			0.805233			0.402616	−	0.915369i	$-0.368101\pi$
$211$	2468.00			0.805233			0.402616	+	0.915369i	$0.368101\pi$
$212$		0			0
$213$	−1058.00	−	441.217i	−0.340343	−	0.141933i
$214$		0			0
$215$		0			0
$216$	−1771.00	−	4364.21i	−0.557876	−	1.37475i
$217$		0			0
$218$		0			0
$219$	2452.00	−	5879.69i	0.756579	−	1.81421i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4840.00			−1.45341			−0.726705	−	0.686950i	$-0.758949\pi$
$223$	−4840.00			−1.45341			−0.726705	+	0.686950i	$0.758949\pi$
$224$		0			0
$225$	2375.00	+	2397.92i	0.703704	+	0.710494i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	4508.00			1.27571
$233$			4181.97i			1.17584i	0.808921	+	0.587918i	$0.200052\pi$
$233$			4181.97i			1.17584i	−0.808921	+	0.587918i	$0.799948\pi$
$234$	−6808.00	+	6742.94i	−1.90194	+	1.88376i
$235$		0			0
$236$		−	12229.4i		−	3.37315i
$237$		0			0
$238$		0			0
$239$		−	565.908i		−	0.153161i	−0.997063	−	0.0765807i	$-0.975600\pi$
$239$		−	565.908i		−	0.153161i	0.997063	−	0.0765807i	$-0.0244003\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			6383.25i			1.69558i
$243$	3482.00	+	1491.50i	0.919220	+	0.393745i
$244$		0			0
$245$		0			0
$246$	−2944.00	+	7059.46i	−0.763019	+	1.82965i
$247$		0			0
$248$			11548.4i			2.95694i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$			12507.5i			3.08973i
$255$		0			0
$256$	−7335.00			−1.79077
$257$		−	7558.23i		−	1.83451i	−0.398299	−	0.917256i	$-0.630399\pi$
$257$		−	7558.23i		−	1.83451i	0.398299	−	0.917256i	$-0.369601\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2576.00	+	2551.38i	−0.610921	+	0.605083i
$262$	−5198.00			−1.22570
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	8766.78i		−	1.98706i	−0.113556	−	0.993532i	$-0.536224\pi$
$269$		−	8766.78i		−	1.98706i	0.113556	−	0.993532i	$-0.463776\pi$
$270$		0			0
$271$	8912.00			1.99766			0.998829	−	0.0483752i	$-0.0154043\pi$
$271$	8912.00			1.99766			0.998829	+	0.0483752i	$0.0154043\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	7935.00	+	3309.12i	1.73055	+	0.721688i
$277$	1838.00			0.398681			0.199341	−	0.979930i	$-0.436120\pi$
$277$	1838.00			0.398681			0.199341	+	0.979930i	$0.436120\pi$
$278$			1975.88i			0.426279i
$279$	−6536.00	−	6599.06i	−1.40251	−	1.41604i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	6164.00	−	14780.8i	1.30163	−	3.12121i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			3309.12i			0.691410i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1380.00	−	1366.81i	0.282352	−	0.279653i
$289$	−4913.00			−1.00000
$290$		0			0
$291$		0			0
$292$	−18390.0			−3.68559
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−7889.00	−	3289.94i	−1.56495	−	0.652630i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	8162.51i		−	1.57876i
$300$	3750.00	−	8992.18i	0.721688	−	1.73055i
$301$		0			0
$302$			17341.7i			3.30432i
$303$	5980.00	+	2493.83i	1.13380	+	0.472828i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10420.0			−1.93714			−0.968568	−	0.248749i	$-0.919981\pi$
$307$	−10420.0			−1.93714			−0.968568	+	0.248749i	$0.919981\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			9275.14i			1.69114i	0.533864	+	0.845570i	$0.320739\pi$
$311$			9275.14i			1.69114i	−0.533864	+	0.845570i	$0.679261\pi$
$312$	11914.0	+	4968.48i	2.16185	+	0.901554i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11183.9i		−	1.98154i	−0.135542	−	0.990772i	$-0.543277\pi$
$317$		−	11183.9i		−	1.98154i	0.135542	−	0.990772i	$-0.456723\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	105.000	−	10934.5i	0.0180041	−	1.87491i
$325$	−9250.00			−1.57876
$326$			7845.98i			1.33297i
$327$		0			0
$328$	10304.0			1.73458
$329$		0			0
$330$		0			0
$331$	12044.0			1.99999			0.999997	−	0.00238915i	$-0.000760490\pi$
$331$	12044.0			1.99999			0.999997	+	0.00238915i	$0.000760490\pi$
$332$		0			0
$333$		0			0
$334$	18814.0			3.08220
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	15725.5i		−	2.53064i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−13892.0			−2.15849
$347$			9102.49i			1.40821i	0.710098	+	0.704103i	$0.248650\pi$
$347$			9102.49i			1.40821i	−0.710098	+	0.704103i	$0.751350\pi$
$348$	9660.00	+	4028.50i	1.48802	+	0.620547i
$349$	9722.00			1.49114			0.745568	−	0.666429i	$-0.232178\pi$
$349$	9722.00			1.49114			0.745568	+	0.666429i	$0.232178\pi$
$350$		0			0
$351$	−9620.00	+	3903.81i	−1.46290	+	0.593646i
$352$		0			0
$353$			12814.5i			1.93214i	0.258283	+	0.966069i	$0.416843\pi$
$353$			12814.5i			1.93214i	−0.258283	+	0.966069i	$0.583157\pi$
$354$	7820.00	−	18751.7i	1.17409	−	2.81537i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	18814.0			2.77752
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6859.00			1.00000
$362$		0			0
$363$	−2662.00	+	6383.25i	−0.384900	+	0.922958i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	4522.47i		−	0.640625i
$369$	−5888.00	+	5831.73i	−0.830669	+	0.822731i
$370$		0			0
$371$		0			0
$372$	−10320.0	+	24746.5i	−1.43835	+	3.44905i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−21574.0			−2.95903
$377$		−	9936.96i		−	1.35751i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−5216.00	+	12507.5i	−0.701375	+	1.68184i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−12719.0	−	5304.19i	−1.69027	−	0.704891i
$385$		0			0
$386$		−	20554.9i		−	2.71041i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$			11514.8i			1.48363i
$393$	−5198.00	−	2167.72i	−0.667187	−	0.278236i
$394$	2668.00			0.341147
$395$		0			0
$396$		0			0
$397$	−15082.0			−1.90666			−0.953330	−	0.301931i	$-0.902369\pi$
$397$	−15082.0			−1.90666			−0.953330	+	0.301931i	$0.902369\pi$
$398$		0			0
$399$		0			0
$400$	−5125.00			−0.640625
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	25456.0			3.14654
$404$		−	18703.7i		−	2.30333i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5002.00			−0.604726			−0.302363	−	0.953193i	$-0.597775\pi$
$409$	−5002.00			−0.604726			−0.302363	+	0.953193i	$0.597775\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	10051.0	+	10148.0i	1.19319	+	1.20470i
$415$		0			0
$416$			5323.37i			0.627403i
$417$	−824.000	+	1975.88i	−0.0967661	+	0.232037i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		−	11836.1i		−	1.36534i
$423$	12328.0	−	12210.2i	1.41704	−	1.40350i
$424$		0			0
$425$		0			0
$426$	−2116.00	+	5073.99i	−0.240659	+	0.577079i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−5330.00	+	2162.92i	−0.593611	+	0.240888i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−28198.0	−	11759.4i	−3.07615	−	1.28284i
$439$	−7288.00			−0.792340			−0.396170	−	0.918177i	$-0.629661\pi$
$439$	−7288.00			−0.792340			−0.396170	+	0.918177i	$0.629661\pi$
$440$		0			0
$441$	−6517.00	−	6579.88i	−0.703704	−	0.710494i
$442$		0			0
$443$		−	5572.76i		−	0.597674i	−0.954304	−	0.298837i	$-0.903401\pi$
$443$		−	5572.76i		−	0.597674i	0.954304	−	0.298837i	$-0.0965987\pi$
$444$		0			0
$445$		0			0
$446$			23211.8i			2.46438i
$447$		0			0
$448$		0			0
$449$		−	4795.83i		−	0.504074i	−0.967718	−	0.252037i	$-0.918900\pi$
$449$		−	4795.83i		−	0.504074i	0.967718	−	0.252037i	$-0.0811005\pi$
$450$	11500.0	−	11390.1i	1.20470	−	1.19319i
$451$		0			0
$452$		0			0
$453$	−7232.00	+	17341.7i	−0.750086	+	1.79864i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			19547.8i			1.97491i	0.157909	+	0.987454i	$0.449525\pi$
$461$			19547.8i			1.97491i	−0.157909	+	0.987454i	$0.550475\pi$
$462$		0			0
$463$	−11680.0			−1.17239			−0.586194	−	0.810171i	$-0.699374\pi$
$463$	−11680.0			−1.17239			−0.586194	+	0.810171i	$0.699374\pi$
$464$		−	5505.61i		−	0.550844i
$465$		0			0
$466$	20056.0			1.99373
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	21090.0	+	21293.5i	2.08309	+	2.10319i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−27370.0			−2.66908
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−2714.00			−0.259698
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	19965.0			1.87500
$485$		0			0
$486$	7153.00	−	16699.1i	0.667627	−	1.55861i
$487$	−19672.0			−1.83044			−0.915219	−	0.402957i	$-0.867983\pi$
$487$	−19672.0			−1.83044			−0.915219	+	0.402957i	$0.867983\pi$
$488$		0			0
$489$	−3272.00	+	7845.98i	−0.302587	+	0.725578i
$490$		0			0
$491$		−	6781.31i		−	0.623291i	−0.950198	−	0.311646i	$-0.899120\pi$
$491$		−	6781.31i		−	0.623291i	0.950198	−	0.311646i	$-0.100880\pi$
$492$	22080.0	+	9208.00i	2.02326	+	0.843757i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	14104.0			1.27679
$497$		0			0
$498$		0			0
$499$	15788.0			1.41637			0.708184	−	0.706028i	$-0.249515\pi$
$499$	15788.0			1.41637			0.708184	+	0.706028i	$0.249515\pi$
$500$		0			0
$501$	18814.0	+	7845.98i	1.67774	+	0.699666i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6558.00	−	15725.5i	0.574460	−	1.37751i
$508$	39120.0			3.41667
$509$		−	21542.9i		−	1.87597i	−0.346669	−	0.937987i	$-0.612687\pi$
$509$		−	21542.9i		−	1.87597i	0.346669	−	0.937987i	$-0.387313\pi$
$510$		0			0
$511$		0			0
$512$			13960.7i			1.20504i
$513$		0			0
$514$	−36248.0			−3.11057
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−13892.0	−	5793.36i	−1.17493	−	0.489982i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	12236.0	+	12354.1i	1.02597	+	1.03587i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$			16257.9i			1.35540i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	15640.0	−	15490.5i	1.27819	−	1.26597i
$532$		0			0
$533$		−	22713.1i		−	1.84580i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	18814.0	+	7845.98i	1.51189	+	0.630501i
$538$	−42044.0			−3.36923
$539$		0			0
$540$		0			0
$541$	−934.000			−0.0742251			−0.0371126	−	0.999311i	$-0.511816\pi$
$541$	−934.000			−0.0742251			−0.0371126	+	0.999311i	$0.511816\pi$
$542$		−	42740.5i		−	3.38719i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	21332.0			1.66744			0.833721	−	0.552186i	$-0.186206\pi$
$547$	21332.0			1.66744			0.833721	+	0.552186i	$0.186206\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	7406.00	−	17759.0i	0.571051	−	1.36933i
$553$		0			0
$554$		−	8814.74i		−	0.675997i
$555$		0			0
$556$	6180.00			0.471386
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−31648.0	+	31345.6i	−2.40102	+	2.37807i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−46230.0	−	19279.2i	−3.45148	−	1.43937i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	7406.00			0.547093
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			13788.0i			1.00000i
$576$	−12787.0	−	12910.4i	−0.924986	−	0.933910i
$577$	−21778.0			−1.57128			−0.785641	−	0.618682i	$-0.787667\pi$
$577$	−21778.0			−1.57128			−0.785641	+	0.618682i	$0.787667\pi$
$578$			23561.9i			1.69558i
$579$	8572.00	−	20554.9i	0.615268	−	1.47536i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$			41157.8i			2.91631i
$585$		0			0
$586$		0			0
$587$		−	28362.5i		−	1.99429i	−0.0755174	−	0.997144i	$-0.524061\pi$
$587$		−	28362.5i		−	1.99429i	0.0755174	−	0.997144i	$-0.475939\pi$
$588$	−10290.0	+	24674.6i	−0.721688	+	1.73055i
$589$		0			0
$590$		0			0
$591$	2668.00	+	1112.63i	0.185697	+	0.0774410i
$592$		0			0
$593$			11778.6i			0.815662i	0.913057	+	0.407831i	$0.133715\pi$
$593$			11778.6i			0.815662i	−0.913057	+	0.407831i	$0.866285\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−39146.0			−2.67692
$599$			16353.8i			1.11552i	0.830002	+	0.557761i	$0.188339\pi$
$599$			16353.8i			1.11552i	−0.830002	+	0.557761i	$0.811661\pi$
$600$	−20125.0	−	8392.71i	−1.36933	−	0.571051i
$601$	326.000			0.0221262			0.0110631	−	0.999939i	$-0.496478\pi$
$601$	326.000			0.0221262			0.0110631	+	0.999939i	$0.496478\pi$
$602$		0			0
$603$		0			0
$604$	54240.0			3.65396
$605$		0			0
$606$	11960.0	−	28679.1i	0.801719	−	1.92245i
$607$	8840.00			0.591111			0.295556	−	0.955326i	$-0.404495\pi$
$607$	8840.00			0.591111			0.295556	+	0.955326i	$0.404495\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			47555.5i			3.14875i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			49972.6i			3.28457i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	5819.00	+	14339.5i	0.376020	+	0.926612i
$622$	44482.0			2.86747
$623$		0			0
$624$	6068.00	−	14550.6i	0.389286	−	0.933475i
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	4936.00	−	11836.1i	0.309934	−	0.743196i
$634$	−53636.0			−3.35987
$635$		0			0
$636$		0			0
$637$	25382.0			1.57876
$638$		0			0
$639$	−4232.00	+	4191.56i	−0.261996	+	0.259492i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	16277.1i		−	0.989052i	−0.869163	−	0.494526i	$-0.835342\pi$
$647$		−	16277.1i		−	0.989052i	0.869163	−	0.494526i	$-0.164658\pi$
$648$	−24472.0	−	234.996i	−1.48357	−	0.0142461i
$649$		0			0
$650$			44361.4i			2.67692i
$651$		0			0
$652$	24540.0			1.47402
$653$			25072.6i			1.50255i	0.659988	+	0.751276i	$0.270561\pi$
$653$			25072.6i			1.50255i	−0.659988	+	0.751276i	$0.729439\pi$
$654$		0			0
$655$		0			0
$656$		−	12584.3i		−	0.748983i
$657$	−23294.0	−	23518.8i	−1.38324	−	1.39658i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	57761.0i		−	3.39116i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−14812.0			−0.859854
$668$		−	58844.9i		−	3.40835i
$669$	−9680.00	+	23211.8i	−0.559418	+	1.34144i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33586.0			−1.92369			−0.961846	−	0.273590i	$-0.911789\pi$
$673$	−33586.0			−1.92369			−0.961846	+	0.273590i	$0.911789\pi$
$674$		0			0
$675$	16250.0	−	6594.27i	0.926612	−	0.376020i
$676$	−49185.0			−2.79842
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	4882.16i		−	0.273515i	−0.990605	−	0.136757i	$-0.956332\pi$
$683$		−	4882.16i		−	0.273515i	0.990605	−	0.136757i	$-0.0436681\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−6028.00			−0.331861			−0.165930	−	0.986137i	$-0.553063\pi$
$691$	−6028.00			−0.331861			−0.165930	+	0.986137i	$0.553063\pi$
$692$			43450.2i			2.38689i
$693$		0			0
$694$	43654.0			2.38773
$695$		0			0
$696$	9016.00	−	21619.6i	0.491021	−	1.17743i
$697$		0			0
$698$		−	46625.1i		−	2.52835i
$699$	20056.0	+	8363.93i	1.08525	+	0.452579i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	18722.0	+	46135.9i	1.00658	+	2.48047i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	61456.0			3.27610
$707$		0			0
$708$	−58650.0	−	24458.7i	−3.11328	−	1.29833i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	37944.6i		−	1.99304i
$714$		0			0
$715$		0			0
$716$		−	58844.9i		−	3.07142i
$717$	−2714.00	−	1131.82i	−0.141361	−	0.0589518i
$718$		0			0
$719$			6858.04i			0.355719i	0.984056	+	0.177859i	$0.0569172\pi$
$719$			6858.04i			0.355719i	−0.984056	+	0.177859i	$0.943083\pi$
$720$		0			0
$721$		0			0
$722$		−	32894.6i		−	1.69558i
$723$		0			0
$724$		0			0
$725$			16785.4i			0.859854i
$726$	30613.0	+	12766.5i	1.56495	+	0.652630i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	14117.0	−	13716.1i	0.717218	−	0.696849i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	7935.00			0.397402
$737$		0			0
$738$	27968.0	+	28237.9i	1.39501	+	1.40847i
$739$	30548.0			1.52060			0.760302	−	0.649570i	$-0.225051\pi$
$739$	30548.0			1.52060			0.760302	+	0.649570i	$0.225051\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	55384.0	+	23096.7i	2.72913	+	1.13813i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			26348.3i			1.27769i
$753$		0			0
$754$	−47656.0			−2.30176
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			15231.6i			0.725550i	0.931877	+	0.362775i	$0.118171\pi$
$761$			15231.6i			0.725550i	−0.931877	+	0.362775i	$0.881829\pi$
$762$	59984.0	+	25015.1i	2.85169	+	1.18924i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			60331.6i			2.84022i
$768$	−14670.0	+	35177.4i	−0.689268	+	1.65281i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−36248.0	−	15116.5i	−1.69318	−	0.706104i
$772$	−64290.0			−2.99721
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−43000.0			−1.99304
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	7084.00	+	17456.8i	0.323322	+	0.796751i
$784$	14063.0			0.640625
$785$		0			0
$786$	−10396.0	+	24928.7i	−0.471772	+	1.13127i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	8344.75i		−	0.377245i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$			72330.7i			3.23290i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	8992.18i		−	0.397402i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	122083.i		−	5.33521i
$807$	−42044.0	−	17533.6i	−1.83398	−	0.764821i
$808$	−41860.0			−1.82256
$809$			36640.2i			1.59234i	0.605076	+	0.796168i	$0.293143\pi$
$809$			36640.2i			1.59234i	−0.605076	+	0.796168i	$0.706857\pi$
$810$		0			0
$811$	−3364.00			−0.145655			−0.0728274	−	0.997345i	$-0.523202\pi$
$811$	−3364.00			−0.145655			−0.0728274	+	0.997345i	$0.523202\pi$
$812$		0			0
$813$	17824.0	−	42740.5i	0.768899	−	1.84376i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$			23988.7i			1.02536i
$819$		0			0
$820$		0			0
$821$		−	40189.1i		−	1.70841i	−0.519933	−	0.854207i	$-0.674043\pi$
$821$		−	40189.1i		−	1.70841i	0.519933	−	0.854207i	$-0.325957\pi$
$822$		0			0
$823$	−17584.0			−0.744763			−0.372381	−	0.928080i	$-0.621459\pi$
$823$	−17584.0			−0.744763			−0.372381	+	0.928080i	$0.621459\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	31740.0	−	31436.7i	1.33218	−	1.31944i
$829$	−4966.00			−0.208053			−0.104027	−	0.994575i	$-0.533173\pi$
$829$	−4966.00			−0.208053			−0.104027	+	0.994575i	$0.533173\pi$
$830$		0			0
$831$	3676.00	−	8814.74i	0.153452	−	0.367966i
$832$	49802.0			2.07521
$833$		0			0
$834$	9476.00	+	3951.77i	0.393438	+	0.164075i
$835$		0			0
$836$		0			0
$837$	−44720.0	+	18147.4i	−1.84677	+	0.749423i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	6357.00			0.260650
$842$		0			0
$843$		0			0
$844$	−37020.0			−1.50981
$845$		0			0
$846$	−58558.0	−	59123.0i	−2.37975	−	2.40271i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	15870.0	+	6618.25i	0.638142	+	0.266124i
$853$	24590.0			0.987041			0.493520	−	0.869734i	$-0.335710\pi$
$853$	24590.0			0.987041			0.493520	+	0.869734i	$0.335710\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	42088.2i		−	1.67760i	−0.544437	−	0.838802i	$-0.683257\pi$
$857$		−	42088.2i		−	1.67760i	0.544437	−	0.838802i	$-0.316743\pi$
$858$		0			0
$859$	50348.0			1.99983			0.999914	−	0.0131439i	$-0.00418394\pi$
$859$	50348.0			1.99983			0.999914	+	0.0131439i	$0.00418394\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	29743.7i		−	1.17322i	−0.809870	−	0.586610i	$-0.800462\pi$
$863$		−	29743.7i		−	1.17322i	0.809870	−	0.586610i	$-0.199538\pi$
$864$	−3795.00	−	9351.87i	−0.149431	−	0.368237i
$865$		0			0
$866$		0			0
$867$	−9826.00	+	23561.9i	−0.384900	+	0.922958i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−36780.0	+	88195.3i	−1.41859	+	3.40165i
$877$	−34090.0			−1.31259			−0.656293	−	0.754506i	$-0.727876\pi$
$877$	−34090.0			−1.31259			−0.656293	+	0.754506i	$0.727876\pi$
$878$			34952.0i			1.34348i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−31556.0	+	31254.4i	−1.20470	+	1.19319i
$883$	−2860.00			−0.109000			−0.0544998	−	0.998514i	$-0.517356\pi$
$883$	−2860.00			−0.109000			−0.0544998	+	0.998514i	$0.517356\pi$
$884$		0			0
$885$		0			0
$886$	−26726.0			−1.01341
$887$			52782.9i			1.99806i	0.0440623	+	0.999029i	$0.485970\pi$
$887$			52782.9i			1.99806i	−0.0440623	+	0.999029i	$0.514030\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	72600.0			2.72514
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−39146.0	−	16325.0i	−1.45713	−	0.607666i
$898$	−23000.0			−0.854699
$899$		−	46193.4i		−	1.71372i
$900$	−35625.0	−	35968.7i	−1.31944	−	1.33218i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	83168.0	+	34683.5i	3.04975	+	1.27183i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	23920.0	−	23691.4i	0.872801	−	0.864460i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−20840.0	+	49972.6i	−0.745604	+	1.78790i
$922$	93748.0			3.34862
$923$		−	16325.0i		−	0.582171i
$924$		0			0
$925$		0			0
$926$			56015.3i			1.98788i
$927$		0			0
$928$	9660.00			0.341708
$929$		−	27585.6i		−	0.974225i	−0.873339	−	0.487112i	$-0.838050\pi$
$929$		−	27585.6i		−	0.974225i	0.873339	−	0.487112i	$-0.161950\pi$
$930$		0			0
$931$		0			0
$932$		−	62729.5i		−	2.20469i
$933$	44482.0	+	18550.3i	1.56085	+	0.650920i
$934$		0			0
$935$		0			0
$936$	47656.0	−	47200.6i	1.66419	−	1.64829i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	−33856.0			−1.16914
$944$			33426.9i			1.15249i
$945$		0			0
$946$		0			0
$947$			55027.4i			1.88823i	0.329623	+	0.944113i	$0.393078\pi$
$947$			55027.4i			1.88823i	−0.329623	+	0.944113i	$0.606922\pi$
$948$		0			0
$949$	90724.0			3.10329
$950$		0			0
$951$	−53636.0	−	22367.8i	−1.82888	−	0.762696i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			8488.62i			0.287177i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	88545.0			2.97221
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−8512.00			−0.283069			−0.141534	−	0.989933i	$-0.545204\pi$
$967$	−8512.00			−0.283069			−0.141534	+	0.989933i	$0.545204\pi$
$968$		−	44682.8i		−	1.48363i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−52230.0	−	22372.6i	−1.72354	−	0.738272i
$973$		0			0
$974$			94343.6i			3.10366i
$975$	−18500.0	+	44361.4i	−0.607666	+	1.45713i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	37628.0	+	15692.0i	1.23028	+	0.513061i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−32522.0			−1.05684
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	20608.0	−	49416.2i	0.667641	−	1.60095i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	9128.00			0.292594			0.146297	−	0.989241i	$-0.453265\pi$
$991$	9128.00			0.292594			0.146297	+	0.989241i	$0.453265\pi$
$992$			24746.5i			0.792038i
$993$	24088.0	−	57761.0i	0.769798	−	1.84591i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−60190.0			−1.91197			−0.955986	−	0.293412i	$-0.905209\pi$
$997$	−60190.0			−1.91197			−0.955986	+	0.293412i	$0.905209\pi$
$998$		−	75716.6i		−	2.40157i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.c.a.68.1	✓	2
3.2	odd	2	inner	69.4.c.a.68.2	yes	2
23.22	odd	2	CM	69.4.c.a.68.1	✓	2
69.68	even	2	inner	69.4.c.a.68.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.c.a.68.1	✓	2	1.1	even	1	trivial
69.4.c.a.68.1	✓	2	23.22	odd	2	CM
69.4.c.a.68.2	yes	2	3.2	odd	2	inner
69.4.c.a.68.2	yes	2	69.68	even	2	inner

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.79583i q^{2} +(2.00000 - 4.79583i) q^{3} -15.0000 q^{4} +(-23.0000 - 9.59166i) q^{6} +33.5708i q^{8} +(-19.0000 - 19.1833i) q^{9} +O(q^{10})\)	\(q-4.79583i q^{2} +(2.00000 - 4.79583i) q^{3} -15.0000 q^{4} +(-23.0000 - 9.59166i) q^{6} +33.5708i q^{8} +(-19.0000 - 19.1833i) q^{9} +(-30.0000 + 71.9375i) q^{12} +74.0000 q^{13} +41.0000 q^{16} +(-92.0000 + 91.1208i) q^{18} -110.304i q^{23} +(161.000 + 67.1416i) q^{24} -125.000 q^{25} -354.892i q^{26} +(-130.000 + 52.7541i) q^{27} -134.283i q^{29} +344.000 q^{31} +71.9375i q^{32} +(285.000 + 287.750i) q^{36} +(148.000 - 354.892i) q^{39} -306.933i q^{41} -529.000 q^{46} +642.641i q^{47} +(82.0000 - 196.629i) q^{48} +343.000 q^{49} +599.479i q^{50} -1110.00 q^{52} +(253.000 + 623.458i) q^{54} -644.000 q^{58} +815.291i q^{59} -1649.77i q^{62} +673.000 q^{64} +(-529.000 - 220.608i) q^{69} -220.608i q^{71} +(644.000 - 637.846i) q^{72} +1226.00 q^{73} +(-250.000 + 599.479i) q^{75} +(-1702.00 - 709.783i) q^{78} +(-7.00000 + 728.966i) q^{81} -1472.00 q^{82} +(-644.000 - 268.567i) q^{87} +1654.56i q^{92} +(688.000 - 1649.77i) q^{93} +3082.00 q^{94} +(345.000 + 143.875i) q^{96} -1644.97i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 30 q^{4} - 46 q^{6} - 38 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 30 * q^4 - 46 * q^6 - 38 * q^9	\( 2 q + 4 q^{3} - 30 q^{4} - 46 q^{6} - 38 q^{9} - 60 q^{12} + 148 q^{13} + 82 q^{16} - 184 q^{18} + 322 q^{24} - 250 q^{25} - 260 q^{27} + 688 q^{31} + 570 q^{36} + 296 q^{39} - 1058 q^{46} + 164 q^{48} + 686 q^{49} - 2220 q^{52} + 506 q^{54} - 1288 q^{58} + 1346 q^{64} - 1058 q^{69} + 1288 q^{72} + 2452 q^{73} - 500 q^{75} - 3404 q^{78} - 14 q^{81} - 2944 q^{82} - 1288 q^{87} + 1376 q^{93} + 6164 q^{94} + 690 q^{96}+O(q^{100}) \) 2 * q + 4 * q^3 - 30 * q^4 - 46 * q^6 - 38 * q^9 - 60 * q^12 + 148 * q^13 + 82 * q^16 - 184 * q^18 + 322 * q^24 - 250 * q^25 - 260 * q^27 + 688 * q^31 + 570 * q^36 + 296 * q^39 - 1058 * q^46 + 164 * q^48 + 686 * q^49 - 2220 * q^52 + 506 * q^54 - 1288 * q^58 + 1346 * q^64 - 1058 * q^69 + 1288 * q^72 + 2452 * q^73 - 500 * q^75 - 3404 * q^78 - 14 * q^81 - 2944 * q^82 - 1288 * q^87 + 1376 * q^93 + 6164 * q^94 + 690 * q^96

Introduction

L-functions

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