LMFDB - Embedded newform 177.4.d.a.176.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(176,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, 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("177.176");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.d (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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-59}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 15 $ x^2 - x + 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			176.2
Root			$0.500000 - 3.84057i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.4.d.a.176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.50000 + 3.84057i) q^{3} -8.00000 q^{4} -7.68115i q^{5} +29.0000 q^{7} +(-2.50000 + 26.8840i) q^{9} +O(q^{10})$	$q+(3.50000 + 3.84057i) q^{3} -8.00000 q^{4} -7.68115i q^{5} +29.0000 q^{7} +(-2.50000 + 26.8840i) q^{9} +(-28.0000 - 30.7246i) q^{12} +(29.5000 - 26.8840i) q^{15} +64.0000 q^{16} +61.4492i q^{17} +119.000 q^{19} +61.4492i q^{20} +(101.500 + 111.377i) q^{21} +66.0000 q^{25} +(-112.000 + 84.4926i) q^{27} -232.000 q^{28} +268.840i q^{29} -222.753i q^{35} +(20.0000 - 215.072i) q^{36} -7.68115i q^{41} +(206.500 + 19.2029i) q^{45} +(224.000 + 245.797i) q^{48} +498.000 q^{49} +(-236.000 + 215.072i) q^{51} -698.984i q^{53} +(416.500 + 457.028i) q^{57} -453.188i q^{59} +(-236.000 + 215.072i) q^{60} +(-72.5000 + 779.636i) q^{63} -512.000 q^{64} -491.593i q^{68} -1182.90i q^{71} +(231.000 + 253.478i) q^{75} -952.000 q^{76} -835.000 q^{79} -491.593i q^{80} +(-716.500 - 134.420i) q^{81} +(-812.000 - 891.013i) q^{84} +472.000 q^{85} +(-1032.50 + 940.940i) q^{87} -914.056i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{3} - 16 q^{4} + 58 q^{7} - 5 q^{9}+O(q^{10}) $ 2 * q + 7 * q^3 - 16 * q^4 + 58 * q^7 - 5 * q^9	$ 2 q + 7 q^{3} - 16 q^{4} + 58 q^{7} - 5 q^{9} - 56 q^{12} + 59 q^{15} + 128 q^{16} + 238 q^{19} + 203 q^{21} + 132 q^{25} - 224 q^{27} - 464 q^{28} + 40 q^{36} + 413 q^{45} + 448 q^{48} + 996 q^{49} - 472 q^{51} + 833 q^{57} - 472 q^{60} - 145 q^{63} - 1024 q^{64} + 462 q^{75} - 1904 q^{76} - 1670 q^{79} - 1433 q^{81} - 1624 q^{84} + 944 q^{85} - 2065 q^{87}+O(q^{100}) $ 2 * q + 7 * q^3 - 16 * q^4 + 58 * q^7 - 5 * q^9 - 56 * q^12 + 59 * q^15 + 128 * q^16 + 238 * q^19 + 203 * q^21 + 132 * q^25 - 224 * q^27 - 464 * q^28 + 40 * q^36 + 413 * q^45 + 448 * q^48 + 996 * q^49 - 472 * q^51 + 833 * q^57 - 472 * q^60 - 145 * q^63 - 1024 * q^64 + 462 * q^75 - 1904 * q^76 - 1670 * q^79 - 1433 * q^81 - 1624 * q^84 + 944 * q^85 - 2065 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$-1$

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 $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	3.50000	+	3.84057i	0.673575	+	0.739119i
$3$	3.50000	+	3.84057i	0.673575	+	0.739119i
$4$	−8.00000			−1.00000
$5$		−	7.68115i		−	0.687023i	−0.939149	−	0.343511i	$-0.888384\pi$
$5$		−	7.68115i		−	0.687023i	0.939149	−	0.343511i	$-0.111616\pi$
$6$		0			0
$7$	29.0000			1.56585			0.782926	−	0.622114i	$-0.213726\pi$
$7$	29.0000			1.56585			0.782926	+	0.622114i	$0.213726\pi$
$8$		0			0
$9$	−2.50000	+	26.8840i	−0.0925926	+	0.995704i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−28.0000	−	30.7246i	−0.673575	−	0.739119i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	29.5000	−	26.8840i	0.507791	−	0.462761i
$16$	64.0000			1.00000
$17$			61.4492i			0.876683i	0.898808	+	0.438342i	$0.144434\pi$
$17$			61.4492i			0.876683i	−0.898808	+	0.438342i	$0.855566\pi$
$18$		0			0
$19$	119.000			1.43687			0.718433	−	0.695596i	$-0.244859\pi$
$19$	119.000			1.43687			0.718433	+	0.695596i	$0.244859\pi$
$20$			61.4492i			0.687023i
$21$	101.500	+	111.377i	1.05472	+	1.15735i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	66.0000			0.528000
$26$		0			0
$27$	−112.000	+	84.4926i	−0.798311	+	0.602245i
$28$	−232.000			−1.56585
$29$			268.840i			1.72146i	0.509061	+	0.860730i	$0.329993\pi$
$29$			268.840i			1.72146i	−0.509061	+	0.860730i	$0.670007\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	222.753i		−	1.07578i
$36$	20.0000	−	215.072i	0.0925926	−	0.995704i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.68115i		−	0.0292584i	−0.999893	−	0.0146292i	$-0.995343\pi$
$41$		−	7.68115i		−	0.0292584i	0.999893	−	0.0146292i	$-0.00465678\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	206.500	+	19.2029i	0.684071	+	0.0636132i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	224.000	+	245.797i	0.673575	+	0.739119i
$49$	498.000			1.45190
$50$		0			0
$51$	−236.000	+	215.072i	−0.647973	+	0.590512i
$52$		0			0
$53$		−	698.984i		−	1.81156i	−0.423744	−	0.905782i	$-0.639285\pi$
$53$		−	698.984i		−	1.81156i	0.423744	−	0.905782i	$-0.360715\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	416.500	+	457.028i	0.967838	+	1.06202i
$58$		0			0
$59$		−	453.188i		−	1.00000i
$59$		−	453.188i		−	1.00000i
$60$	−236.000	+	215.072i	−0.507791	+	0.462761i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−72.5000	+	779.636i	−0.144986	+	1.55913i
$64$	−512.000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		−	491.593i		−	0.876683i
$69$		0			0
$70$		0			0
$71$		−	1182.90i		−	1.97724i	−0.150437	−	0.988620i	$-0.548068\pi$
$71$		−	1182.90i		−	1.97724i	0.150437	−	0.988620i	$-0.451932\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	231.000	+	253.478i	0.355648	+	0.390255i
$76$	−952.000			−1.43687
$77$		0			0
$78$		0			0
$79$	−835.000			−1.18918			−0.594588	−	0.804031i	$-0.702685\pi$
$79$	−835.000			−1.18918			−0.594588	+	0.804031i	$0.702685\pi$
$80$		−	491.593i		−	0.687023i
$81$	−716.500	−	134.420i	−0.982853	−	0.184390i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−812.000	−	891.013i	−1.05472	−	1.15735i
$85$	472.000			0.602301
$86$		0			0
$87$	−1032.50	+	940.940i	−1.27236	+	1.15953i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	914.056i		−	0.987160i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−528.000			−0.528000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	855.500	−	779.636i	0.795126	−	0.724616i
$106$		0			0
$107$			2204.49i			1.99174i	0.0908010	+	0.995869i	$0.471057\pi$
$107$			2204.49i			1.99174i	−0.0908010	+	0.995869i	$0.528943\pi$
$108$	896.000	−	675.941i	0.798311	−	0.602245i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	1856.00			1.56585
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	2150.72i		−	1.72146i
$117$		0			0
$118$		0			0
$119$			1782.03i			1.37276i
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$	29.5000	−	26.8840i	0.0216254	−	0.0197077i
$124$		0			0
$125$		−	1467.10i		−	1.04977i
$126$		0			0
$127$	821.000			0.573638			0.286819	−	0.957985i	$-0.407402\pi$
$127$	821.000			0.573638			0.286819	+	0.957985i	$0.407402\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	3451.00			2.24992
$134$		0			0
$135$	649.000	+	860.288i	0.413756	+	0.548458i
$136$		0			0
$137$			3172.31i			1.97831i	0.146862	+	0.989157i	$0.453083\pi$
$137$			3172.31i			1.97831i	−0.146862	+	0.989157i	$0.546917\pi$
$138$		0			0
$139$	−1960.00			−1.19601			−0.598004	−	0.801493i	$-0.704039\pi$
$139$	−1960.00			−1.19601			−0.598004	+	0.801493i	$0.704039\pi$
$140$			1782.03i			1.07578i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−160.000	+	1720.58i	−0.0925926	+	0.995704i
$145$	2065.00			1.18268
$146$		0			0
$147$	1743.00	+	1912.61i	0.977961	+	1.07312i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−1652.00	−	153.623i	−0.872917	−	0.0811744i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	2684.50	−	2446.44i	1.33896	−	1.22022i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4048.00			−1.94518			−0.972588	−	0.232533i	$-0.925299\pi$
$163$	−4048.00			−1.94518			−0.972588	+	0.232533i	$0.925299\pi$
$164$			61.4492i			0.0292584i
$165$		0			0
$166$		0			0
$167$		−	3325.94i		−	1.54113i	−0.637362	−	0.770565i	$-0.719974\pi$
$167$		−	3325.94i		−	1.54113i	0.637362	−	0.770565i	$-0.280026\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$	−297.500	+	3199.20i	−0.133043	+	1.43069i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	1914.00			0.826770
$176$		0			0
$177$	1740.50	−	1586.16i	0.739119	−	0.673575i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−1652.00	−	153.623i	−0.684071	−	0.0636132i
$181$	−4795.00			−1.96911			−0.984557	−	0.175066i	$-0.943986\pi$
$181$	−4795.00			−1.96911			−0.984557	+	0.175066i	$0.943986\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−3248.00	+	2450.29i	−1.25004	+	0.943027i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−1792.00	−	1966.37i	−0.673575	−	0.739119i
$193$	1937.00			0.722426			0.361213	−	0.932483i	$-0.382363\pi$
$193$	1937.00			0.722426			0.361213	+	0.932483i	$0.382363\pi$
$194$		0			0
$195$		0			0
$196$	−3984.00			−1.45190
$197$		−	4086.37i		−	1.47788i	−0.673773	−	0.738939i	$-0.735327\pi$
$197$		−	4086.37i		−	1.47788i	0.673773	−	0.738939i	$-0.264673\pi$
$198$		0			0
$199$	5411.00			1.92752			0.963758	−	0.266779i	$-0.0859592\pi$
$199$	5411.00			1.92752			0.963758	+	0.266779i	$0.0859592\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			7796.36i			2.69555i
$204$	1888.00	−	1720.58i	0.647973	−	0.590512i
$205$	−59.0000			−0.0201012
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			5591.87i			1.81156i
$213$	4543.00	−	4140.14i	1.46141	−	1.33182i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	5852.00			1.75730			0.878652	−	0.477462i	$-0.158443\pi$
$223$	5852.00			1.75730			0.878652	+	0.477462i	$0.158443\pi$
$224$		0			0
$225$	−165.000	+	1774.34i	−0.0488889	+	0.525732i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−3332.00	−	3656.23i	−0.967838	−	1.06202i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$			3625.50i			1.00000i
$237$	−2922.50	−	3206.88i	−0.800999	−	0.878941i
$238$		0			0
$239$			7043.61i			1.90633i	0.302448	+	0.953166i	$0.402196\pi$
$239$			7043.61i			1.90633i	−0.302448	+	0.953166i	$0.597804\pi$
$240$	1888.00	−	1720.58i	0.507791	−	0.462761i
$241$	−5803.00			−1.55105			−0.775527	−	0.631314i	$-0.782516\pi$
$241$	−5803.00			−1.55105			−0.775527	+	0.631314i	$0.782516\pi$
$242$		0			0
$243$	−1991.50	−	3222.24i	−0.525740	−	0.850645i
$244$		0			0
$245$		−	3825.21i		−	0.997485i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7750.28i		−	1.94898i	−0.224439	−	0.974488i	$-0.572055\pi$
$251$		−	7750.28i		−	1.94898i	0.224439	−	0.974488i	$-0.427945\pi$
$252$	580.000	−	6237.09i	0.144986	−	1.55913i
$253$		0			0
$254$		0			0
$255$	1652.00	+	1812.75i	0.405695	+	0.445172i
$256$	4096.00			1.00000
$257$		−	3325.94i		−	0.807261i	−0.914922	−	0.403631i	$-0.867748\pi$
$257$		−	3325.94i		−	0.807261i	0.914922	−	0.403631i	$-0.132252\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−7227.50	−	672.100i	−1.71407	−	0.159394i
$262$		0			0
$263$		−	8441.58i		−	1.97920i	−0.143840	−	0.989601i	$-0.545945\pi$
$263$		−	8441.58i		−	1.97920i	0.143840	−	0.989601i	$-0.454055\pi$
$264$		0			0
$265$	−5369.00			−1.24459
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−8575.00			−1.92212			−0.961059	−	0.276342i	$-0.910878\pi$
$271$	−8575.00			−1.92212			−0.961059	+	0.276342i	$0.910878\pi$
$272$			3932.75i			0.876683i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9151.00			−1.98495			−0.992473	−	0.122460i	$-0.960922\pi$
$277$	−9151.00			−1.98495			−0.992473	+	0.122460i	$0.960922\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			268.840i			0.0570735i	0.999593	+	0.0285368i	$0.00908476\pi$
$281$			268.840i			0.0570735i	−0.999593	+	0.0285368i	$0.990915\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			9463.17i			1.97724i
$285$	3510.50	−	3199.20i	0.729628	−	0.664927i
$286$		0			0
$287$		−	222.753i		−	0.0458143i
$288$		0			0
$289$	1137.00			0.231427
$290$		0			0
$291$		0			0
$292$		0			0
$293$			5384.48i			1.07360i	0.843709	+	0.536800i	$0.180367\pi$
$293$			5384.48i			1.07360i	−0.843709	+	0.536800i	$0.819633\pi$
$294$		0			0
$295$	−3481.00			−0.687023
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1848.00	−	2027.82i	−0.355648	−	0.390255i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	7616.00			1.43687
$305$		0			0
$306$		0			0
$307$	−3661.00			−0.680600			−0.340300	−	0.940317i	$-0.610529\pi$
$307$	−3661.00			−0.680600			−0.340300	+	0.940317i	$0.610529\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8718.10i		−	1.58958i	−0.606887	−	0.794788i	$-0.707582\pi$
$311$		−	8718.10i		−	1.58958i	0.606887	−	0.794788i	$-0.292418\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	5988.50	+	556.883i	1.07115	+	0.0996089i
$316$	6680.00			1.18918
$317$		−	215.072i		−	0.0381062i	−0.999818	−	0.0190531i	$-0.993935\pi$
$317$		−	215.072i		−	0.0381062i	0.999818	−	0.0190531i	$-0.00606515\pi$
$318$		0			0
$319$		0			0
$320$			3932.75i			0.687023i
$321$	−8466.50	+	7715.71i	−1.47213	+	1.34159i
$322$		0			0
$323$			7312.45i			1.25968i
$324$	5732.00	+	1075.36i	0.982853	+	0.184390i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	12017.0			1.99551			0.997755	−	0.0669643i	$-0.0213314\pi$
$331$	12017.0			1.99551			0.997755	+	0.0669643i	$0.0213314\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	6496.00	+	7128.10i	1.05472	+	1.15735i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	−3776.00			−0.602301
$341$		0			0
$342$		0			0
$343$	4495.00			0.707601
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	8260.00	−	7527.52i	1.27236	−	1.15953i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−9086.00			−1.35841
$356$		0			0
$357$	−6844.00	+	6237.09i	−1.01463	+	0.924655i
$358$		0			0
$359$			8011.44i			1.17779i	0.808209	+	0.588896i	$0.200437\pi$
$359$			8011.44i			1.17779i	−0.808209	+	0.588896i	$0.799563\pi$
$360$		0			0
$361$	7302.00			1.06459
$362$		0			0
$363$	−4658.50	−	5111.80i	−0.673575	−	0.739119i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	206.500	+	19.2029i	0.0291327	+	0.00270911i
$370$		0			0
$371$		−	20270.5i		−	2.83664i
$372$		0			0
$373$	4898.00			0.679916			0.339958	−	0.940441i	$-0.389587\pi$
$373$	4898.00			0.679916			0.339958	+	0.940441i	$0.389587\pi$
$374$		0			0
$375$	5634.50	−	5134.85i	0.775905	−	0.707099i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	13025.0			1.76530			0.882651	−	0.470029i	$-0.155757\pi$
$379$	13025.0			1.76530			0.882651	+	0.470029i	$0.155757\pi$
$380$			7312.45i			0.987160i
$381$	2873.50	+	3153.11i	0.386388	+	0.423986i
$382$		0			0
$383$			14025.8i			1.87124i	0.353014	+	0.935618i	$0.385157\pi$
$383$			14025.8i			1.87124i	−0.353014	+	0.935618i	$0.614843\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	11829.0i		−	1.54178i	−0.636968	−	0.770890i	$-0.719812\pi$
$389$		−	11829.0i		−	1.54178i	0.636968	−	0.770890i	$-0.280188\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			6413.76i			0.816990i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	12078.5	+	13253.8i	1.51549	+	1.66296i
$400$	4224.00			0.528000
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1032.50	+	5503.54i	−0.126680	+	0.675242i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−12183.5	+	11103.1i	−1.46221	+	1.33254i
$412$		0			0
$413$		−	13142.4i		−	1.56585i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−6860.00	−	7527.52i	−0.805601	−	0.883991i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−6844.00	+	6237.09i	−0.795126	+	0.724616i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			4055.64i			0.462889i
$426$		0			0
$427$		0			0
$428$		−	17635.9i		−	1.99174i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−7168.00	+	5407.53i	−0.798311	+	0.602245i
$433$	−14623.0			−1.62295			−0.811474	−	0.584388i	$-0.801334\pi$
$433$	−14623.0			−1.62295			−0.811474	+	0.584388i	$0.801334\pi$
$434$		0			0
$435$	7227.50	+	7930.78i	0.796626	+	0.874142i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−3220.00			−0.350073			−0.175037	−	0.984562i	$-0.556004\pi$
$439$	−3220.00			−0.350073			−0.175037	+	0.984562i	$0.556004\pi$
$440$		0			0
$441$	−1245.00	+	13388.2i	−0.134435	+	1.44566i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−14848.0			−1.56585
$449$		−	9409.40i		−	0.988992i	−0.869180	−	0.494496i	$-0.835353\pi$
$449$		−	9409.40i		−	0.988992i	0.869180	−	0.494496i	$-0.164647\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	−5192.00	−	6882.31i	−0.527978	−	0.699866i
$460$		0			0
$461$		−	13488.1i		−	1.36270i	−0.731959	−	0.681348i	$-0.761394\pi$
$461$		−	13488.1i		−	1.36270i	0.731959	−	0.681348i	$-0.238606\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			17205.8i			1.72146i
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	7854.00			0.758666
$476$		−	14256.2i		−	1.37276i
$477$	18791.5	+	1747.46i	1.80378	+	0.167737i
$478$		0			0
$479$		−	1459.42i		−	0.139212i	−0.997575	−	0.0696059i	$-0.977826\pi$
$479$		−	1459.42i		−	0.139212i	0.997575	−	0.0696059i	$-0.0221742\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	10648.0			1.00000
$485$		0			0
$486$		0			0
$487$	19721.0			1.83500			0.917499	−	0.397739i	$-0.130205\pi$
$487$	19721.0			1.83500			0.917499	+	0.397739i	$0.130205\pi$
$488$		0			0
$489$	−14168.0	−	15546.6i	−1.31022	−	1.43772i
$490$		0			0
$491$			9947.08i			0.914268i	0.889398	+	0.457134i	$0.151124\pi$
$491$			9947.08i			0.914268i	−0.889398	+	0.457134i	$0.848876\pi$
$492$	−236.000	+	215.072i	−0.0216254	+	0.0197077i
$493$	−16520.0			−1.50918
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	34304.0i		−	3.09607i
$498$		0			0
$499$	−9511.00			−0.853248			−0.426624	−	0.904429i	$-0.640297\pi$
$499$	−9511.00			−0.853248			−0.426624	+	0.904429i	$0.640297\pi$
$500$			11736.8i			1.04977i
$501$	12773.5	−	11640.8i	1.13908	−	1.03807i
$502$		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
$507$	7689.50	+	8437.74i	0.673575	+	0.739119i
$508$	−6568.00			−0.573638
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−13328.0	+	10054.6i	−1.14707	+	0.865346i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	11552.4i		−	0.971443i	−0.874114	−	0.485721i	$-0.838557\pi$
$521$		−	11552.4i		−	0.971443i	0.874114	−	0.485721i	$-0.161443\pi$
$522$		0			0
$523$	6167.00			0.515610			0.257805	−	0.966197i	$-0.417001\pi$
$523$	6167.00			0.515610			0.257805	+	0.966197i	$0.417001\pi$
$524$		0			0
$525$	6699.00	+	7350.86i	0.556892	+	0.611081i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	12183.5	+	1132.97i	0.995704	+	0.0925926i
$532$	−27608.0			−2.24992
$533$		0			0
$534$		0			0
$535$	16933.0			1.36837
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−5192.00	−	6882.31i	−0.413756	−	0.548458i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−16782.5	−	18415.5i	−1.32635	−	1.45541i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12616.0			−0.986145			−0.493072	−	0.869988i	$-0.664126\pi$
$547$	−12616.0			−0.986145			−0.493072	+	0.869988i	$0.664126\pi$
$548$		−	25378.5i		−	1.97831i
$549$		0			0
$550$		0			0
$551$			31992.0i			2.47351i
$552$		0			0
$553$	−24215.0			−1.86207
$554$		0			0
$555$		0			0
$556$	15680.0			1.19601
$557$			8011.44i			0.609435i	0.952443	+	0.304718i	$0.0985621\pi$
$557$			8011.44i			0.609435i	−0.952443	+	0.304718i	$0.901438\pi$
$558$		0			0
$559$		0			0
$560$		−	14256.2i		−	1.07578i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−20778.5	−	3898.18i	−1.53900	−	0.288727i
$568$		0			0
$569$		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
$575$		0			0
$576$	1280.00	−	13764.6i	0.0925926	−	0.995704i
$577$	9569.00			0.690403			0.345202	−	0.938529i	$-0.387811\pi$
$577$	9569.00			0.690403			0.345202	+	0.938529i	$0.387811\pi$
$578$		0			0
$579$	6779.50	+	7439.19i	0.486609	+	0.533959i
$580$	−16520.0			−1.18268
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−13944.0	−	15300.8i	−0.977961	−	1.07312i
$589$		0			0
$590$		0			0
$591$	15694.0	−	14302.3i	1.09233	−	0.995461i
$592$		0			0
$593$			6767.09i			0.468619i	0.972162	+	0.234309i	$0.0752829\pi$
$593$			6767.09i			0.468619i	−0.972162	+	0.234309i	$0.924717\pi$
$594$		0			0
$595$	13688.0			0.943115
$596$		0			0
$597$	18938.5	+	20781.3i	1.29833	+	1.42466i
$598$		0			0
$599$			24464.4i			1.66877i	0.551186	+	0.834383i	$0.314176\pi$
$599$			24464.4i			1.66877i	−0.551186	+	0.834383i	$0.685824\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			10223.6i			0.687023i
$606$		0			0
$607$	25319.0			1.69303			0.846513	−	0.532368i	$-0.178698\pi$
$607$	25319.0			1.69303			0.846513	+	0.532368i	$0.178698\pi$
$608$		0			0
$609$	−29942.5	+	27287.3i	−1.99233	+	1.81566i
$610$		0			0
$611$		0			0
$612$	13216.0	+	1228.98i	0.872917	+	0.0811744i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	−206.500	−	226.594i	−0.0135396	−	0.0148571i
$616$		0			0
$617$		−	16184.2i		−	1.05600i	−0.849245	−	0.527999i	$-0.822943\pi$
$617$		−	16184.2i		−	1.05600i	0.849245	−	0.527999i	$-0.177057\pi$
$618$		0			0
$619$	30485.0			1.97948			0.989738	−	0.142894i	$-0.0456409\pi$
$619$	30485.0			1.97948			0.989738	+	0.142894i	$0.0456409\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−3019.00			−0.193216
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−29860.0			−1.88385			−0.941924	−	0.335827i	$-0.890984\pi$
$631$	−29860.0			−1.88385			−0.941924	+	0.335827i	$0.890984\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	6306.22i		−	0.394102i
$636$	−21476.0	+	19571.6i	−1.33896	+	1.22022i
$637$		0			0
$638$		0			0
$639$	31801.0	+	2957.24i	1.96875	+	0.183078i
$640$		0			0
$641$			17205.8i			1.06020i	0.847936	+	0.530099i	$0.177845\pi$
$641$			17205.8i			1.06020i	−0.847936	+	0.530099i	$0.822155\pi$
$642$		0			0
$643$	−18907.0			−1.15959			−0.579797	−	0.814761i	$-0.696868\pi$
$643$	−18907.0			−1.15959			−0.579797	+	0.814761i	$0.696868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32898.3i			1.99902i	0.0312629	+	0.999511i	$0.490047\pi$
$647$			32898.3i			1.99902i	−0.0312629	+	0.999511i	$0.509953\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	32384.0			1.94518
$653$		−	24894.6i		−	1.49188i	−0.666011	−	0.745942i	$-0.731999\pi$
$653$		−	24894.6i		−	1.49188i	0.666011	−	0.745942i	$-0.268001\pi$
$654$		0			0
$655$		0			0
$656$		−	491.593i		−	0.0292584i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	31745.0			1.86798			0.933992	−	0.357294i	$-0.116301\pi$
$661$	31745.0			1.86798			0.933992	+	0.357294i	$0.116301\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	26507.6i		−	1.54575i
$666$		0			0
$667$		0			0
$668$			26607.5i			1.54113i
$669$	20482.0	+	22475.0i	1.18368	+	1.29886i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−7392.00	+	5576.51i	−0.421508	+	0.317985i
$676$	−17576.0			−1.00000
$677$		−	33397.6i		−	1.89597i	−0.318308	−	0.947987i	$-0.603115\pi$
$677$		−	33397.6i		−	1.89597i	0.318308	−	0.947987i	$-0.396885\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	2380.00	−	25593.6i	0.133043	−	1.43069i
$685$	24367.0			1.35915
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			15055.0i			0.821684i
$696$		0			0
$697$	472.000			0.0256503
$698$		0			0
$699$		0			0
$700$	−15312.0			−0.826770
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−13924.0	+	12689.3i	−0.739119	+	0.673575i
$709$	−14929.0			−0.790790			−0.395395	−	0.918511i	$-0.629392\pi$
$709$	−14929.0			−0.790790			−0.395395	+	0.918511i	$0.629392\pi$
$710$		0			0
$711$	2087.50	−	22448.1i	0.110109	−	1.18407i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−27051.5	+	24652.6i	−1.40901	+	1.28406i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	13216.0	+	1228.98i	0.684071	+	0.0636132i
$721$		0			0
$722$		0			0
$723$	−20310.5	−	22286.8i	−1.04475	−	1.14641i
$724$	38360.0			1.96911
$725$			17743.4i			0.908931i
$726$		0			0
$727$	39116.0			1.99551			0.997753	−	0.0670071i	$-0.0213450\pi$
$727$	39116.0			1.99551			0.997753	+	0.0670071i	$0.0213450\pi$
$728$		0			0
$729$	5405.00	−	18926.3i	0.274602	−	0.961558i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−9898.00			−0.498760			−0.249380	−	0.968406i	$-0.580227\pi$
$733$	−9898.00			−0.498760			−0.249380	+	0.968406i	$0.580227\pi$
$734$		0			0
$735$	14691.0	−	13388.2i	0.737259	−	0.671881i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			35594.4i			1.75751i	0.477269	+	0.878757i	$0.341627\pi$
$743$			35594.4i			1.75751i	−0.477269	+	0.878757i	$0.658373\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			63930.2i			3.11877i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	29765.5	−	27126.0i	1.44052	−	1.31278i
$754$		0			0
$755$		0			0
$756$	25984.0	−	19602.3i	1.25004	−	0.943027i
$757$	−19861.0			−0.953580			−0.476790	−	0.879017i	$-0.658200\pi$
$757$	−19861.0			−0.953580			−0.476790	+	0.879017i	$0.658200\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			34419.2i			1.63955i	0.572688	+	0.819774i	$0.305901\pi$
$761$			34419.2i			1.63955i	−0.572688	+	0.819774i	$0.694099\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1180.00	+	12689.3i	−0.0557686	+	0.599714i
$766$		0			0
$767$		0			0
$768$	14336.0	+	15731.0i	0.673575	+	0.739119i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	12773.5	−	11640.8i	0.596662	−	0.543751i
$772$	−15496.0			−0.722426
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	914.056i		−	0.0420404i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−22715.0	−	30110.1i	−1.03674	−	1.37426i
$784$	31872.0			1.45190
$785$		0			0
$786$		0			0
$787$	−42784.0			−1.93785			−0.968923	−	0.247362i	$-0.920436\pi$
$787$	−42784.0			−1.93785			−0.968923	+	0.247362i	$0.920436\pi$
$788$			32691.0i			1.47788i
$789$	32420.5	−	29545.5i	1.46286	−	1.33314i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−18791.5	−	20620.0i	−0.838322	−	0.919896i
$796$	−43288.0			−1.92752
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		−	62370.9i		−	2.69555i
$813$	−30012.5	−	32932.9i	−1.29469	−	1.42067i
$814$		0			0
$815$			31093.3i			1.33638i
$816$	−15104.0	+	13764.6i	−0.647973	+	0.590512i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	472.000			0.0201012
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	18603.7i		−	0.782243i	−0.920339	−	0.391122i	$-0.872087\pi$
$827$		−	18603.7i		−	0.782243i	0.920339	−	0.391122i	$-0.127913\pi$
$828$		0			0
$829$	28091.0			1.17689			0.588444	−	0.808538i	$-0.299741\pi$
$829$	28091.0			1.17689			0.588444	+	0.808538i	$0.299741\pi$
$830$		0			0
$831$	−32028.5	−	35145.1i	−1.33701	−	1.46711i
$832$		0			0
$833$			30601.7i			1.27285i
$834$		0			0
$835$	−25547.0			−1.05879
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−47886.0			−1.96343
$842$		0			0
$843$	−1032.50	+	940.940i	−0.0421841	+	0.0384433i
$844$		0			0
$845$		−	16875.5i		−	0.687023i
$846$		0			0
$847$	−38599.0			−1.56585
$848$		−	44735.0i		−	1.81156i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−36344.0	+	33121.1i	−1.46141	+	1.33182i
$853$	13853.0			0.556058			0.278029	−	0.960573i	$-0.410319\pi$
$853$	13853.0			0.556058			0.278029	+	0.960573i	$0.410319\pi$
$854$		0			0
$855$	24573.5	+	2285.14i	0.982919	+	0.0914037i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$	855.500	−	779.636i	0.0338622	−	0.0308594i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	3979.50	+	4366.73i	0.155883	+	0.171052i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	42545.9i		−	1.64379i
$876$		0			0
$877$	50591.0			1.94793			0.973966	−	0.226693i	$-0.0727915\pi$
$877$	50591.0			1.94793			0.973966	+	0.226693i	$0.0727915\pi$
$878$		0			0
$879$	−20679.5	+	18845.7i	−0.793518	+	0.723151i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−52477.0			−1.99999			−0.999995	−	0.00307379i	$-0.999022\pi$
$883$	−52477.0			−1.99999			−0.999995	+	0.00307379i	$0.999022\pi$
$884$		0			0
$885$	−12183.5	−	13369.0i	−0.462761	−	0.507791i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	23809.0			0.898232
$890$		0			0
$891$		0			0
$892$	−46816.0			−1.75730
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1320.00	−	14194.8i	0.0488889	−	0.525732i
$901$	42952.0			1.58817
$902$		0			0
$903$		0			0
$904$		0			0
$905$			36831.1i			1.35283i
$906$		0			0
$907$	−20869.0			−0.763995			−0.381998	−	0.924163i	$-0.624764\pi$
$907$	−20869.0			−0.763995			−0.381998	+	0.924163i	$0.624764\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			8979.26i			0.326560i	0.986580	+	0.163280i	$0.0522074\pi$
$911$			8979.26i			0.326560i	−0.986580	+	0.163280i	$0.947793\pi$
$912$	26656.0	+	29249.8i	0.967838	+	1.06202i
$913$		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
$921$	−12813.5	−	14060.3i	−0.458436	−	0.503044i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	59262.0			2.08618
$932$		0			0
$933$	33482.5	−	30513.4i	1.17489	−	1.07070i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		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
$943$		0			0
$944$		−	29004.0i		−	1.00000i
$945$	18821.0	+	24948.4i	0.647881	+	0.858805i
$946$		0			0
$947$		−	54897.1i		−	1.88376i	−0.335954	−	0.941878i	$-0.609059\pi$
$947$		−	54897.1i		−	1.88376i	0.335954	−	0.941878i	$-0.390941\pi$
$948$	23380.0	+	25655.0i	0.800999	+	0.878941i
$949$		0			0
$950$		0			0
$951$	826.000	−	752.752i	0.0281650	−	0.0256674i
$952$		0			0
$953$		−	56348.9i		−	1.91534i	−0.287868	−	0.957670i	$-0.592946\pi$
$953$		−	56348.9i		−	1.91534i	0.287868	−	0.957670i	$-0.407054\pi$
$954$		0			0
$955$		0			0
$956$		−	56348.9i		−	1.90633i
$957$		0			0
$958$		0			0
$959$			91997.1i			3.09775i
$960$	−15104.0	+	13764.6i	−0.507791	+	0.462761i
$961$	29791.0			1.00000
$962$		0			0
$963$	−59265.5	−	5511.22i	−1.98318	−	0.184420i
$964$	46424.0			1.55105
$965$		−	14878.4i		−	0.496323i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−28084.0	+	25593.6i	−0.931051	+	0.848487i
$970$		0			0
$971$		−	57109.3i		−	1.88746i	−0.330715	−	0.943731i	$-0.607290\pi$
$971$		−	57109.3i		−	1.88746i	0.330715	−	0.943731i	$-0.392710\pi$
$972$	15932.0	+	25777.9i	0.525740	+	0.850645i
$973$	−56840.0			−1.87277
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$			30601.7i			0.997485i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−31388.0			−1.01533
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	42059.5	+	46152.2i	1.34413	+	1.47492i
$994$		0			0
$995$		−	41562.7i		−	1.32425i
$996$		0			0
$997$	61859.0			1.96499			0.982495	−	0.186291i	$-0.0596468\pi$
$997$	61859.0			1.96499			0.982495	+	0.186291i	$0.0596468\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.d.a.176.2	yes	2
3.2	odd	2	inner	177.4.d.a.176.1	✓	2
59.58	odd	2	CM	177.4.d.a.176.2	yes	2
177.176	even	2	inner	177.4.d.a.176.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.d.a.176.1	✓	2	3.2	odd	2	inner
177.4.d.a.176.1	✓	2	177.176	even	2	inner
177.4.d.a.176.2	yes	2	1.1	even	1	trivial
177.4.d.a.176.2	yes	2	59.58	odd	2	CM

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(3.50000 + 3.84057i) q^{3} -8.00000 q^{4} -7.68115i q^{5} +29.0000 q^{7} +(-2.50000 + 26.8840i) q^{9} +O(q^{10})\)	\(q+(3.50000 + 3.84057i) q^{3} -8.00000 q^{4} -7.68115i q^{5} +29.0000 q^{7} +(-2.50000 + 26.8840i) q^{9} +(-28.0000 - 30.7246i) q^{12} +(29.5000 - 26.8840i) q^{15} +64.0000 q^{16} +61.4492i q^{17} +119.000 q^{19} +61.4492i q^{20} +(101.500 + 111.377i) q^{21} +66.0000 q^{25} +(-112.000 + 84.4926i) q^{27} -232.000 q^{28} +268.840i q^{29} -222.753i q^{35} +(20.0000 - 215.072i) q^{36} -7.68115i q^{41} +(206.500 + 19.2029i) q^{45} +(224.000 + 245.797i) q^{48} +498.000 q^{49} +(-236.000 + 215.072i) q^{51} -698.984i q^{53} +(416.500 + 457.028i) q^{57} -453.188i q^{59} +(-236.000 + 215.072i) q^{60} +(-72.5000 + 779.636i) q^{63} -512.000 q^{64} -491.593i q^{68} -1182.90i q^{71} +(231.000 + 253.478i) q^{75} -952.000 q^{76} -835.000 q^{79} -491.593i q^{80} +(-716.500 - 134.420i) q^{81} +(-812.000 - 891.013i) q^{84} +472.000 q^{85} +(-1032.50 + 940.940i) q^{87} -914.056i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{3} - 16 q^{4} + 58 q^{7} - 5 q^{9}+O(q^{10}) \) 2 * q + 7 * q^3 - 16 * q^4 + 58 * q^7 - 5 * q^9	\( 2 q + 7 q^{3} - 16 q^{4} + 58 q^{7} - 5 q^{9} - 56 q^{12} + 59 q^{15} + 128 q^{16} + 238 q^{19} + 203 q^{21} + 132 q^{25} - 224 q^{27} - 464 q^{28} + 40 q^{36} + 413 q^{45} + 448 q^{48} + 996 q^{49} - 472 q^{51} + 833 q^{57} - 472 q^{60} - 145 q^{63} - 1024 q^{64} + 462 q^{75} - 1904 q^{76} - 1670 q^{79} - 1433 q^{81} - 1624 q^{84} + 944 q^{85} - 2065 q^{87}+O(q^{100}) \) 2 * q + 7 * q^3 - 16 * q^4 + 58 * q^7 - 5 * q^9 - 56 * q^12 + 59 * q^15 + 128 * q^16 + 238 * q^19 + 203 * q^21 + 132 * q^25 - 224 * q^27 - 464 * q^28 + 40 * q^36 + 413 * q^45 + 448 * q^48 + 996 * q^49 - 472 * q^51 + 833 * q^57 - 472 * q^60 - 145 * q^63 - 1024 * q^64 + 462 * q^75 - 1904 * q^76 - 1670 * q^79 - 1433 * q^81 - 1624 * q^84 + 944 * q^85 - 2065 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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