LMFDB - Embedded newform 300.5.b.e.149.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,5,Mod(149,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	300.b (of order $2$, degree $1$, not 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:	$31.0109889252$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2101x^{4} + 656100 $ x^8 + 2101*x^4 + 656100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.5
Root			$-3.12523 + 3.12523i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.5.b.e.149.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.97224 - 6.73293i) q^{3} -15.3976i q^{7} +(-9.66464 - 80.4214i) q^{9} +O(q^{10})$	$q+(5.97224 - 6.73293i) q^{3} -15.3976i q^{7} +(-9.66464 - 80.4214i) q^{9} -50.5601i q^{11} -14.6024i q^{13} -467.801 q^{17} +194.976 q^{19} +(-103.671 - 91.9580i) q^{21} +145.300 q^{23} +(-599.191 - 415.225i) q^{27} -891.423i q^{29} -950.939 q^{31} +(-340.418 - 301.957i) q^{33} -837.156i q^{37} +(-98.3171 - 87.2093i) q^{39} +1998.33i q^{41} -2096.53i q^{43} -493.802 q^{47} +2163.91 q^{49} +(-2793.82 + 3149.67i) q^{51} -4887.12 q^{53} +(1164.44 - 1312.76i) q^{57} +2224.65i q^{59} -5076.60 q^{61} +(-1238.29 + 148.812i) q^{63} -3403.17i q^{67} +(867.769 - 978.297i) q^{69} +9566.09i q^{71} -7982.30i q^{73} -778.503 q^{77} +1522.22 q^{79} +(-6374.19 + 1554.49i) q^{81} +1142.30 q^{83} +(-6001.89 - 5323.79i) q^{87} -10931.2i q^{89} -224.842 q^{91} +(-5679.24 + 6402.61i) q^{93} +30.9915i q^{97} +(-4066.12 + 488.646i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 142 q^{9}+O(q^{10}) $ 8 * q + 142 * q^9	$ 8 q + 142 q^{9} - 1072 q^{19} - 1268 q^{21} - 1028 q^{31} + 968 q^{39} + 8100 q^{49} - 3270 q^{51} + 2812 q^{61} - 18060 q^{69} + 35864 q^{79} - 35422 q^{81} + 15308 q^{91} - 45030 q^{99}+O(q^{100}) $ 8 * q + 142 * q^9 - 1072 * q^19 - 1268 * q^21 - 1028 * q^31 + 968 * q^39 + 8100 * q^49 - 3270 * q^51 + 2812 * q^61 - 18060 * q^69 + 35864 * q^79 - 35422 * q^81 + 15308 * q^91 - 45030 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$-1$

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
$2$		0			0
$3$	5.97224	−	6.73293i	0.663582	−	0.748103i
$3$	5.97224	−	6.73293i	0.663582	−	0.748103i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	15.3976i		−	0.314236i	−0.987580	−	0.157118i	$-0.949780\pi$
$7$		−	15.3976i		−	0.314236i	0.987580	−	0.157118i	$-0.0502203\pi$
$8$		0			0
$9$	−9.66464	−	80.4214i	−0.119317	−	0.992856i
$10$		0			0
$11$		−	50.5601i		−	0.417852i	−0.977931	−	0.208926i	$-0.933003\pi$
$11$		−	50.5601i		−	0.417852i	0.977931	−	0.208926i	$-0.0669969\pi$
$12$		0			0
$13$		−	14.6024i		−	0.0864049i	−0.999066	−	0.0432025i	$-0.986244\pi$
$13$		−	14.6024i		−	0.0864049i	0.999066	−	0.0432025i	$-0.0137561\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−467.801			−1.61869			−0.809345	−	0.587333i	$-0.800178\pi$
$17$	−467.801			−1.61869			−0.809345	+	0.587333i	$0.800178\pi$
$18$		0			0
$19$	194.976			0.540099			0.270049	−	0.962846i	$-0.412960\pi$
$19$	194.976			0.540099			0.270049	+	0.962846i	$0.412960\pi$
$20$		0			0
$21$	−103.671	−	91.9580i	−0.235081	−	0.208522i
$22$		0			0
$23$	145.300			0.274670			0.137335	−	0.990525i	$-0.456146\pi$
$23$	145.300			0.274670			0.137335	+	0.990525i	$0.456146\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−599.191	−	415.225i	−0.821935	−	0.569581i
$28$		0			0
$29$		−	891.423i		−	1.05996i	−0.848011	−	0.529978i	$-0.822200\pi$
$29$		−	891.423i		−	1.05996i	0.848011	−	0.529978i	$-0.177800\pi$
$30$		0			0
$31$	−950.939			−0.989531			−0.494765	−	0.869027i	$-0.664746\pi$
$31$	−950.939			−0.989531			−0.494765	+	0.869027i	$0.664746\pi$
$32$		0			0
$33$	−340.418	−	301.957i	−0.312597	−	0.277280i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	837.156i		−	0.611509i	−0.952110	−	0.305755i	$-0.901091\pi$
$37$		−	837.156i		−	0.611509i	0.952110	−	0.305755i	$-0.0989087\pi$
$38$		0			0
$39$	−98.3171	−	87.2093i	−0.0646398	−	0.0573368i
$40$		0			0
$41$			1998.33i			1.18877i	0.804180	+	0.594386i	$0.202605\pi$
$41$			1998.33i			1.18877i	−0.804180	+	0.594386i	$0.797395\pi$
$42$		0			0
$43$		−	2096.53i		−	1.13387i	−0.823762	−	0.566936i	$-0.808129\pi$
$43$		−	2096.53i		−	1.13387i	0.823762	−	0.566936i	$-0.191871\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−493.802			−0.223541			−0.111771	−	0.993734i	$-0.535652\pi$
$47$	−493.802			−0.223541			−0.111771	+	0.993734i	$0.535652\pi$
$48$		0			0
$49$	2163.91			0.901256
$50$		0			0
$51$	−2793.82	+	3149.67i	−1.07413	+	1.21095i
$52$		0			0
$53$	−4887.12			−1.73981			−0.869903	−	0.493223i	$-0.835819\pi$
$53$	−4887.12			−1.73981			−0.869903	+	0.493223i	$0.835819\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1164.44	−	1312.76i	0.358400	−	0.404050i
$58$		0			0
$59$			2224.65i			0.639083i	0.947572	+	0.319541i	$0.103529\pi$
$59$			2224.65i			0.639083i	−0.947572	+	0.319541i	$0.896471\pi$
$60$		0			0
$61$	−5076.60			−1.36431			−0.682155	−	0.731207i	$-0.738957\pi$
$61$	−5076.60			−1.36431			−0.682155	+	0.731207i	$0.738957\pi$
$62$		0			0
$63$	−1238.29	+	148.812i	−0.311991	+	0.0374936i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3403.17i		−	0.758112i	−0.925374	−	0.379056i	$-0.876249\pi$
$67$		−	3403.17i		−	0.758112i	0.925374	−	0.379056i	$-0.123751\pi$
$68$		0			0
$69$	867.769	−	978.297i	0.182266	−	0.205481i
$70$		0			0
$71$			9566.09i			1.89766i	0.315791	+	0.948829i	$0.397730\pi$
$71$			9566.09i			1.89766i	−0.315791	+	0.948829i	$0.602270\pi$
$72$		0			0
$73$		−	7982.30i		−	1.49790i	−0.662628	−	0.748949i	$-0.730559\pi$
$73$		−	7982.30i		−	1.49790i	0.662628	−	0.748949i	$-0.269441\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−778.503			−0.131304
$78$		0			0
$79$	1522.22			0.243906			0.121953	−	0.992536i	$-0.461084\pi$
$79$	1522.22			0.243906			0.121953	+	0.992536i	$0.461084\pi$
$80$		0			0
$81$	−6374.19	+	1554.49i	−0.971527	+	0.236928i
$82$		0			0
$83$	1142.30			0.165815			0.0829077	−	0.996557i	$-0.473579\pi$
$83$	1142.30			0.165815			0.0829077	+	0.996557i	$0.473579\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6001.89	−	5323.79i	−0.792956	−	0.703368i
$88$		0			0
$89$		−	10931.2i		−	1.38003i	−0.723795	−	0.690015i	$-0.757604\pi$
$89$		−	10931.2i		−	1.38003i	0.723795	−	0.690015i	$-0.242396\pi$
$90$		0			0
$91$	−224.842			−0.0271515
$92$		0			0
$93$	−5679.24	+	6402.61i	−0.656635	+	0.740271i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			30.9915i			0.00329381i	0.999999	+	0.00164691i	$0.000524227\pi$
$97$			30.9915i			0.00329381i	−0.999999	+	0.00164691i	$0.999476\pi$
$98$		0			0
$99$	−4066.12	+	488.646i	−0.414867	+	0.0498567i
$100$		0			0
$101$		−	5883.64i		−	0.576771i	−0.957514	−	0.288385i	$-0.906882\pi$
$101$		−	5883.64i		−	0.576771i	0.957514	−	0.288385i	$-0.0931184\pi$
$102$		0			0
$103$		−	10887.1i		−	1.02621i	−0.858326	−	0.513105i	$-0.828495\pi$
$103$		−	10887.1i		−	1.02621i	0.858326	−	0.513105i	$-0.171505\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18168.7			−1.58692			−0.793460	−	0.608622i	$-0.791723\pi$
$107$	−18168.7			−1.58692			−0.793460	+	0.608622i	$0.791723\pi$
$108$		0			0
$109$	15676.1			1.31943			0.659714	−	0.751517i	$-0.270678\pi$
$109$	15676.1			1.31943			0.659714	+	0.751517i	$0.270678\pi$
$110$		0			0
$111$	−5636.51	−	4999.70i	−0.457472	−	0.405787i
$112$		0			0
$113$	1310.11			0.102600			0.0513002	−	0.998683i	$-0.483663\pi$
$113$	1310.11			0.102600			0.0513002	+	0.998683i	$0.483663\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1174.35	+	141.127i	−0.0857877	+	0.0103095i
$118$		0			0
$119$			7203.00i			0.508651i
$120$		0			0
$121$	12084.7			0.825399
$122$		0			0
$123$	13454.6	+	11934.5i	0.889325	+	0.788849i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			7838.62i			0.485996i	0.970027	+	0.242998i	$0.0781308\pi$
$127$			7838.62i			0.485996i	−0.970027	+	0.242998i	$0.921869\pi$
$128$		0			0
$129$	−14115.8	−	12521.0i	−0.848253	−	0.752418i
$130$		0			0
$131$		−	12476.9i		−	0.727050i	−0.931584	−	0.363525i	$-0.881573\pi$
$131$		−	12476.9i		−	0.727050i	0.931584	−	0.363525i	$-0.118427\pi$
$132$		0			0
$133$		−	3002.15i		−	0.169719i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19266.2			1.02649			0.513244	−	0.858243i	$-0.328443\pi$
$137$	19266.2			1.02649			0.513244	+	0.858243i	$0.328443\pi$
$138$		0			0
$139$	−6673.02			−0.345376			−0.172688	−	0.984977i	$-0.555245\pi$
$139$	−6673.02			−0.345376			−0.172688	+	0.984977i	$0.555245\pi$
$140$		0			0
$141$	−2949.11	+	3324.73i	−0.148338	+	0.167232i
$142$		0			0
$143$	−738.301			−0.0361045
$144$		0			0
$145$		0			0
$146$		0			0
$147$	12923.4	−	14569.5i	0.598057	−	0.674232i
$148$		0			0
$149$			3613.85i			0.162779i	0.996682	+	0.0813893i	$0.0259357\pi$
$149$			3613.85i			0.162779i	−0.996682	+	0.0813893i	$0.974064\pi$
$150$		0			0
$151$	39087.7			1.71430			0.857149	−	0.515068i	$-0.172233\pi$
$151$	39087.7			1.71430			0.857149	+	0.515068i	$0.172233\pi$
$152$		0			0
$153$	4521.13	+	37621.2i	0.193137	+	1.60713i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			40667.8i			1.64988i	0.565222	+	0.824939i	$0.308790\pi$
$157$			40667.8i			1.64988i	−0.565222	+	0.824939i	$0.691210\pi$
$158$		0			0
$159$	−29187.0	+	32904.6i	−1.15450	+	1.30155i
$160$		0			0
$161$		−	2237.27i		−	0.0863112i
$162$		0			0
$163$		−	23364.7i		−	0.879396i	−0.898146	−	0.439698i	$-0.855086\pi$
$163$		−	23364.7i		−	0.879396i	0.898146	−	0.439698i	$-0.144914\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	41753.6			1.49714			0.748568	−	0.663058i	$-0.230742\pi$
$167$	41753.6			1.49714			0.748568	+	0.663058i	$0.230742\pi$
$168$		0			0
$169$	28347.8			0.992534
$170$		0			0
$171$	−1884.37	−	15680.2i	−0.0644427	−	0.536241i
$172$		0			0
$173$	30169.8			1.00805			0.504023	−	0.863690i	$-0.331853\pi$
$173$	30169.8			1.00805			0.504023	+	0.863690i	$0.331853\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	14978.4	+	13286.1i	0.478100	+	0.424084i
$178$		0			0
$179$			32585.4i			1.01699i	0.861065	+	0.508495i	$0.169798\pi$
$179$			32585.4i			1.01699i	−0.861065	+	0.508495i	$0.830202\pi$
$180$		0			0
$181$	43251.0			1.32020			0.660099	−	0.751179i	$-0.270514\pi$
$181$	43251.0			1.32020			0.660099	+	0.751179i	$0.270514\pi$
$182$		0			0
$183$	−30318.7	+	34180.4i	−0.905332	+	1.02064i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			23652.1i			0.676374i
$188$		0			0
$189$	−6393.45	+	9226.08i	−0.178983	+	0.258282i
$190$		0			0
$191$		−	34445.3i		−	0.944197i	−0.881546	−	0.472099i	$-0.843497\pi$
$191$		−	34445.3i		−	0.944197i	0.881546	−	0.472099i	$-0.156503\pi$
$192$		0			0
$193$			3562.54i			0.0956412i	0.998856	+	0.0478206i	$0.0152276\pi$
$193$			3562.54i			0.0956412i	−0.998856	+	0.0478206i	$0.984772\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	27839.1			0.717336			0.358668	−	0.933465i	$-0.383231\pi$
$197$	27839.1			0.717336			0.358668	+	0.933465i	$0.383231\pi$
$198$		0			0
$199$	−22102.2			−0.558121			−0.279061	−	0.960273i	$-0.590023\pi$
$199$	−22102.2			−0.558121			−0.279061	+	0.960273i	$0.590023\pi$
$200$		0			0
$201$	−22913.3	−	20324.5i	−0.567146	−	0.503070i
$202$		0			0
$203$	−13725.7			−0.333076
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1404.28	−	11685.3i	−0.0327727	−	0.272708i
$208$		0			0
$209$		−	9858.00i		−	0.225682i
$210$		0			0
$211$	71143.7			1.59798			0.798990	−	0.601345i	$-0.205368\pi$
$211$	71143.7			1.59798			0.798990	+	0.601345i	$0.205368\pi$
$212$		0			0
$213$	64407.8	+	57131.0i	1.41964	+	1.25925i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			14642.2i			0.310946i
$218$		0			0
$219$	−53744.2	−	47672.2i	−1.12058	−	0.993979i
$220$		0			0
$221$			6831.04i			0.139863i
$222$		0			0
$223$		−	76917.3i		−	1.54673i	−0.633961	−	0.773365i	$-0.718572\pi$
$223$		−	76917.3i		−	1.54673i	0.633961	−	0.773365i	$-0.281428\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−75766.1			−1.47036			−0.735180	−	0.677872i	$-0.762902\pi$
$227$	−75766.1			−1.47036			−0.735180	+	0.677872i	$0.762902\pi$
$228$		0			0
$229$	57716.4			1.10060			0.550298	−	0.834968i	$-0.314514\pi$
$229$	57716.4			1.10060			0.550298	+	0.834968i	$0.314514\pi$
$230$		0			0
$231$	−4649.41	+	5241.61i	−0.0871312	+	0.0982292i
$232$		0			0
$233$	65387.1			1.20443			0.602213	−	0.798335i	$-0.294286\pi$
$233$	65387.1			1.20443			0.602213	+	0.798335i	$0.294286\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	9091.06	−	10249.0i	0.161852	−	0.182467i
$238$		0			0
$239$			2699.52i			0.0472597i	0.999721	+	0.0236298i	$0.00752231\pi$
$239$			2699.52i			0.0472597i	−0.999721	+	0.0236298i	$0.992478\pi$
$240$		0			0
$241$	−3271.59			−0.0563281			−0.0281640	−	0.999603i	$-0.508966\pi$
$241$	−3271.59			−0.0563281			−0.0281640	+	0.999603i	$0.508966\pi$
$242$		0			0
$243$	−27602.0	+	52200.7i	−0.467442	+	0.884024i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	2847.12i		−	0.0466672i
$248$		0			0
$249$	6822.10	−	7691.04i	0.110032	−	0.124047i
$250$		0			0
$251$		−	91788.9i		−	1.45694i	−0.685076	−	0.728472i	$-0.740231\pi$
$251$		−	91788.9i		−	1.45694i	0.685076	−	0.728472i	$-0.259769\pi$
$252$		0			0
$253$		−	7346.41i		−	0.114771i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−43758.3			−0.662513			−0.331257	−	0.943541i	$-0.607473\pi$
$257$	−43758.3			−0.662513			−0.331257	+	0.943541i	$0.607473\pi$
$258$		0			0
$259$	−12890.2			−0.192158
$260$		0			0
$261$	−71689.4	+	8615.28i	−1.05238	+	0.126470i
$262$		0			0
$263$	−85928.0			−1.24229			−0.621145	−	0.783696i	$-0.713332\pi$
$263$	−85928.0			−1.24229			−0.621145	+	0.783696i	$0.713332\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−73599.1	−	65283.9i	−1.03240	−	0.915764i
$268$		0			0
$269$		−	78308.6i		−	1.08219i	−0.840961	−	0.541096i	$-0.818009\pi$
$269$		−	78308.6i		−	1.08219i	0.840961	−	0.541096i	$-0.181991\pi$
$270$		0			0
$271$	−36099.9			−0.491549			−0.245775	−	0.969327i	$-0.579042\pi$
$271$	−36099.9			−0.491549			−0.245775	+	0.969327i	$0.579042\pi$
$272$		0			0
$273$	−1342.81	+	1513.84i	−0.0180173	+	0.0203122i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			18413.4i			0.239979i	0.992775	+	0.119990i	$0.0382862\pi$
$277$			18413.4i			0.239979i	−0.992775	+	0.119990i	$0.961714\pi$
$278$		0			0
$279$	9190.49	+	76475.8i	0.118067	+	0.982462i
$280$		0			0
$281$			131788.i			1.66903i	0.550989	+	0.834513i	$0.314251\pi$
$281$			131788.i			1.66903i	−0.550989	+	0.834513i	$0.685749\pi$
$282$		0			0
$283$		−	108394.i		−	1.35342i	−0.736252	−	0.676708i	$-0.763406\pi$
$283$		−	108394.i		−	1.35342i	0.736252	−	0.676708i	$-0.236594\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	30769.4			0.373555
$288$		0			0
$289$	135317.			1.62016
$290$		0			0
$291$	208.664	+	185.089i	0.00246411	+	0.00218572i
$292$		0			0
$293$	6856.42			0.0798661			0.0399330	−	0.999202i	$-0.487286\pi$
$293$	6856.42			0.0798661			0.0399330	+	0.999202i	$0.487286\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−20993.8	+	30295.2i	−0.238001	+	0.343448i
$298$		0			0
$299$		−	2121.74i		−	0.0237328i
$300$		0			0
$301$	−32281.5			−0.356304
$302$		0			0
$303$	−39614.1	−	35138.5i	−0.431484	−	0.382735i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	88860.5i		−	0.942827i	−0.881912	−	0.471413i	$-0.843744\pi$
$307$		−	88860.5i		−	0.942827i	0.881912	−	0.471413i	$-0.156256\pi$
$308$		0			0
$309$	−73301.8	−	65020.2i	−0.767711	−	0.680975i
$310$		0			0
$311$			146598.i			1.51568i	0.652442	+	0.757839i	$0.273745\pi$
$311$			146598.i			1.51568i	−0.652442	+	0.757839i	$0.726255\pi$
$312$		0			0
$313$		−	31348.8i		−	0.319987i	−0.987118	−	0.159993i	$-0.948853\pi$
$313$		−	31348.8i		−	0.319987i	0.987118	−	0.159993i	$-0.0511473\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	61530.0			0.612306			0.306153	−	0.951982i	$-0.400958\pi$
$317$	61530.0			0.612306			0.306153	+	0.951982i	$0.400958\pi$
$318$		0			0
$319$	−45070.5			−0.442905
$320$		0			0
$321$	−108508.	+	122328.i	−1.05305	+	1.18718i
$322$		0			0
$323$	−91209.9			−0.874253
$324$		0			0
$325$		0			0
$326$		0			0
$327$	93621.5	−	105546.i	0.875549	−	0.987067i
$328$		0			0
$329$			7603.35i			0.0702447i
$330$		0			0
$331$	−127261.			−1.16156			−0.580779	−	0.814062i	$-0.697252\pi$
$331$	−127261.			−1.16156			−0.580779	+	0.814062i	$0.697252\pi$
$332$		0			0
$333$	−67325.2	+	8090.81i	−0.607141	+	0.0729632i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	23025.9i		−	0.202748i	−0.994848	−	0.101374i	$-0.967676\pi$
$337$		−	23025.9i		−	0.202748i	0.994848	−	0.101374i	$-0.0323238\pi$
$338$		0			0
$339$	7824.27	−	8820.85i	0.0680839	−	0.0767557i
$340$		0			0
$341$			48079.6i			0.413478i
$342$		0			0
$343$		−	70288.6i		−	0.597443i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−118367.			−0.983042			−0.491521	−	0.870866i	$-0.663559\pi$
$347$	−118367.			−0.983042			−0.491521	+	0.870866i	$0.663559\pi$
$348$		0			0
$349$	80175.7			0.658251			0.329126	−	0.944286i	$-0.393246\pi$
$349$	80175.7			0.658251			0.329126	+	0.944286i	$0.393246\pi$
$350$		0			0
$351$	−6063.29	+	8749.64i	−0.0492146	+	0.0710192i
$352$		0			0
$353$	−73798.3			−0.592239			−0.296120	−	0.955151i	$-0.595693\pi$
$353$	−73798.3			−0.592239			−0.296120	+	0.955151i	$0.595693\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	48497.3	+	43018.1i	0.380523	+	0.337532i
$358$		0			0
$359$			184883.i			1.43453i	0.696802	+	0.717264i	$0.254606\pi$
$359$			184883.i			1.43453i	−0.696802	+	0.717264i	$0.745394\pi$
$360$		0			0
$361$	−92305.5			−0.708293
$362$		0			0
$363$	72172.6	−	81365.2i	0.547721	−	0.617484i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	7132.72i		−	0.0529570i	−0.999649	−	0.0264785i	$-0.991571\pi$
$367$		−	7132.72i		−	0.0529570i	0.999649	−	0.0264785i	$-0.00842935\pi$
$368$		0			0
$369$	160708.	−	19313.1i	1.18028	−	0.141840i
$370$		0			0
$371$			75249.7i			0.546710i
$372$		0			0
$373$			90663.0i			0.651647i	0.945431	+	0.325823i	$0.105641\pi$
$373$			90663.0i			0.651647i	−0.945431	+	0.325823i	$0.894359\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13016.9			−0.0915854
$378$		0			0
$379$	−72641.4			−0.505715			−0.252857	−	0.967504i	$-0.581370\pi$
$379$	−72641.4			−0.505715			−0.252857	+	0.967504i	$0.581370\pi$
$380$		0			0
$381$	52776.9	+	46814.2i	0.363575	+	0.322498i
$382$		0			0
$383$	18040.8			0.122987			0.0614935	−	0.998107i	$-0.480414\pi$
$383$	18040.8			0.122987			0.0614935	+	0.998107i	$0.480414\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−168606.	+	20262.2i	−1.12577	+	0.135290i
$388$		0			0
$389$		−	151605.i		−	1.00188i	−0.865483	−	0.500939i	$-0.832988\pi$
$389$		−	151605.i		−	1.00188i	0.865483	−	0.500939i	$-0.167012\pi$
$390$		0			0
$391$	−67971.7			−0.444605
$392$		0			0
$393$	−84006.1	−	74515.1i	−0.543908	−	0.482458i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	105906.i		−	0.671951i	−0.941871	−	0.335975i	$-0.890934\pi$
$397$		−	105906.i		−	0.671951i	0.941871	−	0.335975i	$-0.109066\pi$
$398$		0			0
$399$	−20213.3	−	17929.6i	−0.126967	−	0.112622i
$400$		0			0
$401$		−	165722.i		−	1.03060i	−0.857010	−	0.515301i	$-0.827680\pi$
$401$		−	165722.i		−	1.03060i	0.857010	−	0.515301i	$-0.172320\pi$
$402$		0			0
$403$			13886.0i			0.0855003i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−42326.7			−0.255521
$408$		0			0
$409$	−25834.5			−0.154438			−0.0772189	−	0.997014i	$-0.524604\pi$
$409$	−25834.5			−0.154438			−0.0772189	+	0.997014i	$0.524604\pi$
$410$		0			0
$411$	115062.	−	129718.i	0.681160	−	0.767919i
$412$		0			0
$413$	34254.1			0.200823
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−39852.9	+	44928.9i	−0.229186	+	0.258377i
$418$		0			0
$419$		−	198893.i		−	1.13290i	−0.824097	−	0.566449i	$-0.808317\pi$
$419$		−	198893.i		−	1.13290i	0.824097	−	0.566449i	$-0.191683\pi$
$420$		0			0
$421$	106217.			0.599283			0.299641	−	0.954052i	$-0.403133\pi$
$421$	106217.			0.599283			0.299641	+	0.954052i	$0.403133\pi$
$422$		0			0
$423$	4772.42	+	39712.2i	0.0266721	+	0.221944i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			78167.3i			0.428716i
$428$		0			0
$429$	−4409.31	+	4970.93i	−0.0239583	+	0.0270099i
$430$		0			0
$431$			35487.9i			0.191040i	0.995427	+	0.0955202i	$0.0304515\pi$
$431$			35487.9i			0.191040i	−0.995427	+	0.0955202i	$0.969549\pi$
$432$		0			0
$433$			138725.i			0.739911i	0.929049	+	0.369955i	$0.120627\pi$
$433$			138725.i			0.739911i	−0.929049	+	0.369955i	$0.879373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	28330.0			0.148349
$438$		0			0
$439$	−235284.			−1.22086			−0.610428	−	0.792072i	$-0.709002\pi$
$439$	−235284.			−1.22086			−0.610428	+	0.792072i	$0.709002\pi$
$440$		0			0
$441$	−20913.5	−	174025.i	−0.107535	−	0.894817i
$442$		0			0
$443$	293963.			1.49791			0.748954	−	0.662622i	$-0.230557\pi$
$443$	293963.			1.49791			0.748954	+	0.662622i	$0.230557\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	24331.8	+	21582.8i	0.121775	+	0.108017i
$448$		0			0
$449$		−	150257.i		−	0.745318i	−0.927968	−	0.372659i	$-0.878446\pi$
$449$		−	150257.i		−	0.745318i	0.927968	−	0.372659i	$-0.121554\pi$
$450$		0			0
$451$	101036.			0.496732
$452$		0			0
$453$	233441.	−	263175.i	1.13758	−	1.28247i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	161306.i		−	0.772356i	−0.922424	−	0.386178i	$-0.873795\pi$
$457$		−	161306.i		−	0.772356i	0.922424	−	0.386178i	$-0.126205\pi$
$458$		0			0
$459$	280302.	+	194243.i	1.33046	+	0.921975i
$460$		0			0
$461$		−	339787.i		−	1.59884i	−0.600772	−	0.799421i	$-0.705140\pi$
$461$		−	339787.i		−	1.59884i	0.600772	−	0.799421i	$-0.294860\pi$
$462$		0			0
$463$			298343.i			1.39173i	0.718174	+	0.695863i	$0.244978\pi$
$463$			298343.i			1.39173i	−0.718174	+	0.695863i	$0.755022\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−70135.5			−0.321591			−0.160796	−	0.986988i	$-0.551406\pi$
$467$	−70135.5			−0.321591			−0.160796	+	0.986988i	$0.551406\pi$
$468$		0			0
$469$	−52400.5			−0.238226
$470$		0			0
$471$	273814.	+	242878.i	1.23428	+	1.09483i
$472$		0			0
$473$	−106001.			−0.473791
$474$		0			0
$475$		0			0
$476$		0			0
$477$	47232.2	+	393028.i	0.207588	+	1.72738i
$478$		0			0
$479$			388819.i			1.69464i	0.531085	+	0.847319i	$0.321785\pi$
$479$			388819.i			1.69464i	−0.531085	+	0.847319i	$0.678215\pi$
$480$		0			0
$481$	−12224.5			−0.0528374
$482$		0			0
$483$	−15063.4	−	13361.5i	−0.0645697	−	0.0572746i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			206792.i			0.871920i	0.899966	+	0.435960i	$0.143591\pi$
$487$			206792.i			0.871920i	−0.899966	+	0.435960i	$0.856409\pi$
$488$		0			0
$489$	−157313.	−	139539.i	−0.657879	−	0.583552i
$490$		0			0
$491$		−	276164.i		−	1.14552i	−0.819722	−	0.572762i	$-0.805872\pi$
$491$		−	276164.i		−	1.14552i	0.819722	−	0.572762i	$-0.194128\pi$
$492$		0			0
$493$			417009.i			1.71574i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	147295.			0.596312
$498$		0			0
$499$	−187937.			−0.754765			−0.377383	−	0.926057i	$-0.623176\pi$
$499$	−187937.			−0.754765			−0.377383	+	0.926057i	$0.623176\pi$
$500$		0			0
$501$	249363.	−	281124.i	0.993473	−	1.12001i
$502$		0			0
$503$	−18554.5			−0.0733355			−0.0366677	−	0.999328i	$-0.511674\pi$
$503$	−18554.5			−0.0733355			−0.0366677	+	0.999328i	$0.511674\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	169300.	−	190863.i	0.658628	−	0.742518i
$508$		0			0
$509$			412623.i			1.59264i	0.604874	+	0.796321i	$0.293223\pi$
$509$			412623.i			1.59264i	−0.604874	+	0.796321i	$0.706777\pi$
$510$		0			0
$511$	−122908.			−0.470694
$512$		0			0
$513$	−116828.	−	80958.7i	−0.443926	−	0.307630i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			24966.7i			0.0934072i
$518$		0			0
$519$	180181.	−	203131.i	0.668922	−	0.754122i
$520$		0			0
$521$			347649.i			1.28075i	0.768061	+	0.640377i	$0.221222\pi$
$521$			347649.i			1.28075i	−0.768061	+	0.640377i	$0.778778\pi$
$522$		0			0
$523$			288154.i			1.05347i	0.850030	+	0.526734i	$0.176584\pi$
$523$			288154.i			1.05347i	−0.850030	+	0.526734i	$0.823416\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	444851.			1.60174
$528$		0			0
$529$	−258729.			−0.924556
$530$		0			0
$531$	178909.	−	21500.4i	0.634517	−	0.0762531i
$532$		0			0
$533$	29180.4			0.102716
$534$		0			0
$535$		0			0
$536$		0			0
$537$	219395.	+	194608.i	0.760813	+	0.674857i
$538$		0			0
$539$		−	109408.i		−	0.376592i
$540$		0			0
$541$	397537.			1.35826			0.679131	−	0.734018i	$-0.262357\pi$
$541$	397537.			1.35826			0.679131	+	0.734018i	$0.262357\pi$
$542$		0			0
$543$	258305.	−	291206.i	0.876060	−	0.987644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			381872.i			1.27627i	0.769924	+	0.638136i	$0.220294\pi$
$547$			381872.i			1.27627i	−0.769924	+	0.638136i	$0.779706\pi$
$548$		0			0
$549$	49063.5	+	408267.i	0.162785	+	1.35456i
$550$		0			0
$551$		−	173806.i		−	0.572481i
$552$		0			0
$553$		−	23438.5i		−	0.0766441i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	162015.			0.522209			0.261105	−	0.965311i	$-0.415913\pi$
$557$	162015.			0.522209			0.261105	+	0.965311i	$0.415913\pi$
$558$		0			0
$559$	−30614.4			−0.0979721
$560$		0			0
$561$	159248.	+	141256.i	0.505997	+	0.448830i
$562$		0			0
$563$	306285.			0.966292			0.483146	−	0.875540i	$-0.339494\pi$
$563$	306285.			0.966292			0.483146	+	0.875540i	$0.339494\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	23935.3	+	98147.0i	0.0744514	+	0.305289i
$568$		0			0
$569$			2603.82i			0.00804241i	0.999992	+	0.00402121i	$0.00127999\pi$
$569$			2603.82i			0.00804241i	−0.999992	+	0.00402121i	$0.998720\pi$
$570$		0			0
$571$	−270421.			−0.829409			−0.414704	−	0.909956i	$-0.636115\pi$
$571$	−270421.			−0.829409			−0.414704	+	0.909956i	$0.636115\pi$
$572$		0			0
$573$	−231917.	−	205715.i	−0.706357	−	0.626553i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	569306.i		−	1.70999i	−0.518635	−	0.854996i	$-0.673560\pi$
$577$		−	569306.i		−	1.70999i	0.518635	−	0.854996i	$-0.326440\pi$
$578$		0			0
$579$	23986.3	+	21276.4i	0.0715495	+	0.0634658i
$580$		0			0
$581$		−	17588.7i		−	0.0521052i
$582$		0			0
$583$			247093.i			0.726982i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	232912.			0.675952			0.337976	−	0.941155i	$-0.390258\pi$
$587$	232912.			0.675952			0.337976	+	0.941155i	$0.390258\pi$
$588$		0			0
$589$	−185410.			−0.534445
$590$		0			0
$591$	166262.	−	187439.i	0.476012	−	0.536641i
$592$		0			0
$593$	−56505.6			−0.160688			−0.0803438	−	0.996767i	$-0.525602\pi$
$593$	−56505.6			−0.160688			−0.0803438	+	0.996767i	$0.525602\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−132000.	+	148812.i	−0.370360	+	0.417532i
$598$		0			0
$599$		−	408097.i		−	1.13739i	−0.822548	−	0.568696i	$-0.807448\pi$
$599$		−	408097.i		−	1.13739i	0.822548	−	0.568696i	$-0.192552\pi$
$600$		0			0
$601$	−227249.			−0.629147			−0.314574	−	0.949233i	$-0.601862\pi$
$601$	−227249.			−0.629147			−0.314574	+	0.949233i	$0.601862\pi$
$602$		0			0
$603$	−273687.	+	32890.4i	−0.752697	+	0.0904554i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			44956.0i			0.122014i	0.998137	+	0.0610071i	$0.0194312\pi$
$607$			44956.0i			0.122014i	−0.998137	+	0.0610071i	$0.980569\pi$
$608$		0			0
$609$	−81973.5	+	92414.4i	−0.221024	+	0.249175i
$610$		0			0
$611$			7210.71i			0.0193150i
$612$		0			0
$613$			512379.i			1.36355i	0.731563	+	0.681773i	$0.238791\pi$
$613$			512379.i			1.36355i	−0.731563	+	0.681773i	$0.761209\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−554076.			−1.45546			−0.727728	−	0.685866i	$-0.759424\pi$
$617$	−554076.			−1.45546			−0.727728	+	0.685866i	$0.759424\pi$
$618$		0			0
$619$	19229.5			0.0501865			0.0250932	−	0.999685i	$-0.492012\pi$
$619$	19229.5			0.0501865			0.0250932	+	0.999685i	$0.492012\pi$
$620$		0			0
$621$	−87062.6	−	60332.3i	−0.225761	−	0.156447i
$622$		0			0
$623$	−168314.			−0.433655
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−66373.2	−	58874.4i	−0.168833	−	0.149758i
$628$		0			0
$629$			391623.i			0.989844i
$630$		0			0
$631$	−242825.			−0.609867			−0.304934	−	0.952374i	$-0.598634\pi$
$631$	−242825.			−0.609867			−0.304934	+	0.952374i	$0.598634\pi$
$632$		0			0
$633$	424887.	−	479005.i	1.06039	−	1.19545i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	31598.4i		−	0.0778729i
$638$		0			0
$639$	769318.	−	92452.8i	1.88410	−	0.226422i
$640$		0			0
$641$			10523.4i			0.0256119i	0.999918	+	0.0128059i	$0.00407637\pi$
$641$			10523.4i			0.0256119i	−0.999918	+	0.0128059i	$0.995924\pi$
$642$		0			0
$643$			16119.5i			0.0389879i	0.999810	+	0.0194939i	$0.00620551\pi$
$643$			16119.5i			0.0389879i	−0.999810	+	0.0194939i	$0.993794\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−388292.			−0.927576			−0.463788	−	0.885946i	$-0.653510\pi$
$647$	−388292.			−0.927576			−0.463788	+	0.885946i	$0.653510\pi$
$648$		0			0
$649$	112478.			0.267042
$650$		0			0
$651$	98584.6	+	87446.5i	0.232620	+	0.206339i
$652$		0			0
$653$	93334.4			0.218885			0.109442	−	0.993993i	$-0.465093\pi$
$653$	93334.4			0.218885			0.109442	+	0.993993i	$0.465093\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−641947.	+	77146.0i	−1.48720	+	0.178724i
$658$		0			0
$659$		−	428636.i		−	0.987003i	−0.869745	−	0.493501i	$-0.835717\pi$
$659$		−	428636.i		−	0.987003i	0.869745	−	0.493501i	$-0.164283\pi$
$660$		0			0
$661$	40886.7			0.0935791			0.0467896	−	0.998905i	$-0.485101\pi$
$661$	40886.7			0.0935791			0.0467896	+	0.998905i	$0.485101\pi$
$662$		0			0
$663$	45992.9	+	40796.6i	0.104632	+	0.0928105i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	129524.i		−	0.291138i
$668$		0			0
$669$	−517879.	−	459369.i	−1.15711	−	1.02638i
$670$		0			0
$671$			256674.i			0.570080i
$672$		0			0
$673$		−	670267.i		−	1.47985i	−0.672689	−	0.739925i	$-0.734861\pi$
$673$		−	670267.i		−	1.47985i	0.672689	−	0.739925i	$-0.265139\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−404058.			−0.881590			−0.440795	−	0.897608i	$-0.645303\pi$
$677$	−404058.			−0.881590			−0.440795	+	0.897608i	$0.645303\pi$
$678$		0			0
$679$	477.194			0.00103504
$680$		0			0
$681$	−452494.	+	510128.i	−0.975705	+	1.09998i
$682$		0			0
$683$	−416295.			−0.892400			−0.446200	−	0.894933i	$-0.647223\pi$
$683$	−416295.			−0.892400			−0.446200	+	0.894933i	$0.647223\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	344696.	−	388600.i	0.730336	−	0.823359i
$688$		0			0
$689$			71363.8i			0.150328i
$690$		0			0
$691$	−562500.			−1.17806			−0.589029	−	0.808112i	$-0.700490\pi$
$691$	−562500.			−1.17806			−0.589029	+	0.808112i	$0.700490\pi$
$692$		0			0
$693$	7523.95	+	62608.3i	0.0156668	+	0.130366i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	934820.i		−	1.92425i
$698$		0			0
$699$	390508.	−	440247.i	0.799236	−	0.901035i
$700$		0			0
$701$		−	586463.i		−	1.19345i	−0.802445	−	0.596726i	$-0.796468\pi$
$701$		−	586463.i		−	1.19345i	0.802445	−	0.596726i	$-0.203532\pi$
$702$		0			0
$703$		−	163225.i		−	0.330275i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−90593.7			−0.181242
$708$		0			0
$709$	−253496.			−0.504288			−0.252144	−	0.967690i	$-0.581136\pi$
$709$	−253496.			−0.504288			−0.252144	+	0.967690i	$0.581136\pi$
$710$		0			0
$711$	−14711.7	−	122419.i	−0.0291021	−	0.242164i
$712$		0			0
$713$	−138172.			−0.271794
$714$		0			0
$715$		0			0
$716$		0			0
$717$	18175.7	+	16122.2i	0.0353551	+	0.0313607i
$718$		0			0
$719$		−	769020.i		−	1.48758i	−0.668414	−	0.743789i	$-0.733027\pi$
$719$		−	769020.i		−	1.48758i	0.668414	−	0.743789i	$-0.266973\pi$
$720$		0			0
$721$	−167634.			−0.322472
$722$		0			0
$723$	−19538.7	+	22027.4i	−0.0373783	+	0.0421392i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			679811.i			1.28623i	0.765769	+	0.643116i	$0.222359\pi$
$727$			679811.i			1.28623i	−0.765769	+	0.643116i	$0.777641\pi$
$728$		0			0
$729$	186618.	+	497597.i	0.351155	+	0.936317i
$730$		0			0
$731$			980760.i			1.83539i
$732$		0			0
$733$		−	311430.i		−	0.579633i	−0.957082	−	0.289817i	$-0.906406\pi$
$733$		−	311430.i		−	0.579633i	0.957082	−	0.289817i	$-0.0935943\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−172065.			−0.316779
$738$		0			0
$739$	425613.			0.779339			0.389669	−	0.920955i	$-0.372589\pi$
$739$	425613.			0.779339			0.389669	+	0.920955i	$0.372589\pi$
$740$		0			0
$741$	−19169.4	−	17003.7i	−0.0349119	−	0.0309675i
$742$		0			0
$743$	−240789.			−0.436173			−0.218087	−	0.975929i	$-0.569982\pi$
$743$	−240789.			−0.436173			−0.218087	+	0.975929i	$0.569982\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−11039.9	−	91865.5i	−0.0197845	−	0.164631i
$748$		0			0
$749$			279753.i			0.498668i
$750$		0			0
$751$	−767615.			−1.36102			−0.680508	−	0.732741i	$-0.738241\pi$
$751$	−767615.			−1.36102			−0.680508	+	0.732741i	$0.738241\pi$
$752$		0			0
$753$	−618008.	−	548186.i	−1.08994	−	0.966802i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	943107.i		−	1.64577i	−0.568208	−	0.822885i	$-0.692363\pi$
$757$		−	943107.i		−	1.64577i	0.568208	−	0.822885i	$-0.307637\pi$
$758$		0			0
$759$	−49462.8	−	43874.5i	−0.0858609	−	0.0761603i
$760$		0			0
$761$			1.04405e6i			1.80281i	0.432977	+	0.901405i	$0.357463\pi$
$761$			1.04405e6i			1.80281i	−0.432977	+	0.901405i	$0.642537\pi$
$762$		0			0
$763$		−	241374.i		−	0.414612i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	32485.2			0.0552199
$768$		0			0
$769$	428633.			0.724824			0.362412	−	0.932018i	$-0.381953\pi$
$769$	428633.			0.724824			0.362412	+	0.932018i	$0.381953\pi$
$770$		0			0
$771$	−261335.	+	294622.i	−0.439632	+	0.495628i
$772$		0			0
$773$	390545.			0.653600			0.326800	−	0.945094i	$-0.394030\pi$
$773$	390545.			0.653600			0.326800	+	0.945094i	$0.394030\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−76983.2	+	86788.6i	−0.127513	+	0.143754i
$778$		0			0
$779$			389625.i			0.642055i
$780$		0			0
$781$	483663.			0.792941
$782$		0			0
$783$	−370141.	+	534132.i	−0.603731	+	0.871215i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			936753.i			1.51243i	0.654323	+	0.756215i	$0.272954\pi$
$787$			936753.i			1.51243i	−0.654323	+	0.756215i	$0.727046\pi$
$788$		0			0
$789$	−513183.	+	578547.i	−0.824362	+	0.929361i
$790$		0			0
$791$		−	20172.4i		−	0.0322408i
$792$		0			0
$793$			74130.7i			0.117883i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	535580.			0.843156			0.421578	−	0.906792i	$-0.361476\pi$
$797$	535580.			0.843156			0.421578	+	0.906792i	$0.361476\pi$
$798$		0			0
$799$	231001.			0.361844
$800$		0			0
$801$	−879103.	+	105646.i	−1.37017	+	0.164660i
$802$		0			0
$803$	−403586.			−0.625900
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−527246.	−	467678.i	−0.809592	−	0.718124i
$808$		0			0
$809$			357054.i			0.545552i	0.962078	+	0.272776i	$0.0879419\pi$
$809$			357054.i			0.545552i	−0.962078	+	0.272776i	$0.912058\pi$
$810$		0			0
$811$	96582.5			0.146844			0.0734221	−	0.997301i	$-0.476608\pi$
$811$	96582.5			0.146844			0.0734221	+	0.997301i	$0.476608\pi$
$812$		0			0
$813$	−215597.	+	243058.i	−0.326184	+	0.367730i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	408772.i		−	0.612403i
$818$		0			0
$819$	2173.02	+	18082.1i	0.00323963	+	0.0269576i
$820$		0			0
$821$			183717.i			0.272560i	0.990670	+	0.136280i	$0.0435147\pi$
$821$			183717.i			0.272560i	−0.990670	+	0.136280i	$0.956485\pi$
$822$		0			0
$823$		−	715555.i		−	1.05644i	−0.849109	−	0.528218i	$-0.822860\pi$
$823$		−	715555.i		−	1.05644i	0.849109	−	0.528218i	$-0.177140\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.19158e6			1.74225			0.871126	−	0.491059i	$-0.163390\pi$
$827$	1.19158e6			1.74225			0.871126	+	0.491059i	$0.163390\pi$
$828$		0			0
$829$	146844.			0.213671			0.106836	−	0.994277i	$-0.465928\pi$
$829$	146844.			0.213671			0.106836	+	0.994277i	$0.465928\pi$
$830$		0			0
$831$	123976.	+	109969.i	0.179529	+	0.159246i
$832$		0			0
$833$	−1.01228e6			−1.45885
$834$		0			0
$835$		0			0
$836$		0			0
$837$	569794.	+	394853.i	0.813330	+	0.563618i
$838$		0			0
$839$		−	393256.i		−	0.558665i	−0.960194	−	0.279332i	$-0.909887\pi$
$839$		−	393256.i		−	0.558665i	0.960194	−	0.279332i	$-0.0901131\pi$
$840$		0			0
$841$	−87353.7			−0.123506
$842$		0			0
$843$	887318.	+	787069.i	1.24860	+	1.10754i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	186075.i		−	0.259370i
$848$		0			0
$849$	−729807.	−	647353.i	−1.01249	−	0.898103i
$850$		0			0
$851$		−	121639.i		−	0.167963i
$852$		0			0
$853$		−	369154.i		−	0.507352i	−0.967289	−	0.253676i	$-0.918360\pi$
$853$		−	369154.i		−	0.507352i	0.967289	−	0.253676i	$-0.0816397\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.02980e6			−1.40214			−0.701068	−	0.713095i	$-0.747293\pi$
$857$	−1.02980e6			−1.40214			−0.701068	+	0.713095i	$0.747293\pi$
$858$		0			0
$859$	33080.0			0.0448310			0.0224155	−	0.999749i	$-0.492864\pi$
$859$	33080.0			0.0448310			0.0224155	+	0.999749i	$0.492864\pi$
$860$		0			0
$861$	183762.	−	207168.i	0.247885	−	0.279458i
$862$		0			0
$863$	1.42924e6			1.91903			0.959516	−	0.281653i	$-0.0908825\pi$
$863$	1.42924e6			1.91903			0.959516	+	0.281653i	$0.0908825\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	808147.	−	911081.i	1.07511	−	1.21205i
$868$		0			0
$869$		−	76963.6i		−	0.101917i
$870$		0			0
$871$	−49694.5			−0.0655046
$872$		0			0
$873$	2492.38	−	299.522i	0.00327028	−	0.000393007i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			1.09234e6i			1.42023i	0.704085	+	0.710116i	$0.251358\pi$
$877$			1.09234e6i			1.42023i	−0.704085	+	0.710116i	$0.748642\pi$
$878$		0			0
$879$	40948.2	−	46163.8i	0.0529977	−	0.0597481i
$880$		0			0
$881$		−	431951.i		−	0.556522i	−0.960505	−	0.278261i	$-0.910242\pi$
$881$		−	431951.i		−	0.556522i	0.960505	−	0.278261i	$-0.0897581\pi$
$882$		0			0
$883$		−	337546.i		−	0.432924i	−0.976291	−	0.216462i	$-0.930548\pi$
$883$		−	337546.i		−	0.432924i	0.976291	−	0.216462i	$-0.0694518\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−297018.			−0.377516			−0.188758	−	0.982024i	$-0.560446\pi$
$887$	−297018.			−0.377516			−0.188758	+	0.982024i	$0.560446\pi$
$888$		0			0
$889$	120696.			0.152717
$890$		0			0
$891$	78595.1	+	322280.i	0.0990011	+	0.405955i
$892$		0			0
$893$	−96279.4			−0.120734
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−14285.5	−	12671.5i	−0.0177546	−	0.0157487i
$898$		0			0
$899$			847689.i			1.04886i
$900$		0			0
$901$	2.28620e6			2.81621
$902$		0			0
$903$	−192793.	+	217349.i	−0.236437	+	0.266552i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			900767.i			1.09496i	0.836819	+	0.547480i	$0.184413\pi$
$907$			900767.i			1.09496i	−0.836819	+	0.547480i	$0.815587\pi$
$908$		0			0
$909$	−473170.	+	56863.2i	−0.572650	+	0.0688183i
$910$		0			0
$911$		−	1.04199e6i		−	1.25553i	−0.778405	−	0.627763i	$-0.783971\pi$
$911$		−	1.04199e6i		−	1.25553i	0.778405	−	0.627763i	$-0.216029\pi$
$912$		0			0
$913$		−	57754.9i		−	0.0692863i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−192114.			−0.228465
$918$		0			0
$919$	644472.			0.763085			0.381542	−	0.924351i	$-0.375393\pi$
$919$	644472.			0.763085			0.381542	+	0.924351i	$0.375393\pi$
$920$		0			0
$921$	−598291.	−	530696.i	−0.705331	−	0.625643i
$922$		0			0
$923$	139688.			0.163967
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−875552.	+	105220.i	−1.01888	+	0.122444i
$928$		0			0
$929$		−	650980.i		−	0.754287i	−0.926155	−	0.377143i	$-0.876906\pi$
$929$		−	650980.i		−	0.754287i	0.926155	−	0.377143i	$-0.123094\pi$
$930$		0			0
$931$	421911.			0.486767
$932$		0			0
$933$	987033.	+	875518.i	1.13388	+	1.00578i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	274195.i		−	0.312306i	−0.987733	−	0.156153i	$-0.950091\pi$
$937$		−	274195.i		−	0.312306i	0.987733	−	0.156153i	$-0.0499093\pi$
$938$		0			0
$939$	−211069.	−	187223.i	−0.239383	−	0.212338i
$940$		0			0
$941$		−	1.37521e6i		−	1.55307i	−0.630076	−	0.776534i	$-0.716976\pi$
$941$		−	1.37521e6i		−	1.55307i	0.630076	−	0.776534i	$-0.283024\pi$
$942$		0			0
$943$			290358.i			0.326520i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5205.62			0.00580461			0.00290230	−	0.999996i	$-0.499076\pi$
$947$	5205.62			0.00580461			0.00290230	+	0.999996i	$0.499076\pi$
$948$		0			0
$949$	−116561.			−0.129426
$950$		0			0
$951$	367472.	−	414277.i	0.406315	−	0.458068i
$952$		0			0
$953$	−914858.			−1.00732			−0.503660	−	0.863902i	$-0.668014\pi$
$953$	−914858.			−1.00732			−0.503660	+	0.863902i	$0.668014\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−269172.	+	303456.i	−0.293904	+	0.331339i
$958$		0			0
$959$		−	296652.i		−	0.322560i
$960$		0			0
$961$	−19235.6			−0.0208286
$962$		0			0
$963$	175594.	+	1.46115e6i	0.189346	+	1.57558i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			529106.i			0.565835i	0.959144	+	0.282918i	$0.0913023\pi$
$967$			529106.i			0.565835i	−0.959144	+	0.282918i	$0.908698\pi$
$968$		0			0
$969$	−544728.	+	614110.i	−0.580139	+	0.654031i
$970$		0			0
$971$		−	130708.i		−	0.138632i	−0.997595	−	0.0693159i	$-0.977918\pi$
$971$		−	130708.i		−	0.138632i	0.997595	−	0.0693159i	$-0.0220816\pi$
$972$		0			0
$973$			102748.i			0.108530i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	704930.			0.738511			0.369255	−	0.929328i	$-0.379613\pi$
$977$	704930.			0.738511			0.369255	+	0.929328i	$0.379613\pi$
$978$		0			0
$979$	−552684.			−0.576649
$980$		0			0
$981$	−151504.	−	1.26069e6i	−0.157429	−	1.31000i
$982$		0			0
$983$	1.79145e6			1.85395			0.926975	−	0.375124i	$-0.122400\pi$
$983$	1.79145e6			1.85395			0.926975	+	0.375124i	$0.122400\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	51192.8	+	45409.1i	0.0525503	+	0.0466131i
$988$		0			0
$989$		−	304626.i		−	0.311440i
$990$		0			0
$991$	1.41235e6			1.43811			0.719057	−	0.694951i	$-0.244574\pi$
$991$	1.41235e6			1.43811			0.719057	+	0.694951i	$0.244574\pi$
$992$		0			0
$993$	−760036.	+	856842.i	−0.770789	+	0.868964i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			859976.i			0.865159i	0.901596	+	0.432579i	$0.142396\pi$
$997$			859976.i			0.865159i	−0.901596	+	0.432579i	$0.857604\pi$
$998$		0			0
$999$	−347608.	+	501616.i	−0.348304	+	0.502621i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.5.b.e.149.5		8
3.2	odd	2	inner	300.5.b.e.149.3		8
5.2	odd	4		300.5.g.e.101.1	✓	4
5.3	odd	4		300.5.g.f.101.4	yes	4
5.4	even	2	inner	300.5.b.e.149.4		8
15.2	even	4		300.5.g.e.101.2	yes	4
15.8	even	4		300.5.g.f.101.3	yes	4
15.14	odd	2	inner	300.5.b.e.149.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.5.b.e.149.3		8	3.2	odd	2	inner
300.5.b.e.149.4		8	5.4	even	2	inner
300.5.b.e.149.5		8	1.1	even	1	trivial
300.5.b.e.149.6		8	15.14	odd	2	inner
300.5.g.e.101.1	✓	4	5.2	odd	4
300.5.g.e.101.2	yes	4	15.2	even	4
300.5.g.f.101.3	yes	4	15.8	even	4
300.5.g.f.101.4	yes	4	5.3	odd	4

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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.97224 - 6.73293i) q^{3} -15.3976i q^{7} +(-9.66464 - 80.4214i) q^{9} +O(q^{10})\)	\(q+(5.97224 - 6.73293i) q^{3} -15.3976i q^{7} +(-9.66464 - 80.4214i) q^{9} -50.5601i q^{11} -14.6024i q^{13} -467.801 q^{17} +194.976 q^{19} +(-103.671 - 91.9580i) q^{21} +145.300 q^{23} +(-599.191 - 415.225i) q^{27} -891.423i q^{29} -950.939 q^{31} +(-340.418 - 301.957i) q^{33} -837.156i q^{37} +(-98.3171 - 87.2093i) q^{39} +1998.33i q^{41} -2096.53i q^{43} -493.802 q^{47} +2163.91 q^{49} +(-2793.82 + 3149.67i) q^{51} -4887.12 q^{53} +(1164.44 - 1312.76i) q^{57} +2224.65i q^{59} -5076.60 q^{61} +(-1238.29 + 148.812i) q^{63} -3403.17i q^{67} +(867.769 - 978.297i) q^{69} +9566.09i q^{71} -7982.30i q^{73} -778.503 q^{77} +1522.22 q^{79} +(-6374.19 + 1554.49i) q^{81} +1142.30 q^{83} +(-6001.89 - 5323.79i) q^{87} -10931.2i q^{89} -224.842 q^{91} +(-5679.24 + 6402.61i) q^{93} +30.9915i q^{97} +(-4066.12 + 488.646i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 142 q^{9}+O(q^{10}) \) 8 * q + 142 * q^9	\( 8 q + 142 q^{9} - 1072 q^{19} - 1268 q^{21} - 1028 q^{31} + 968 q^{39} + 8100 q^{49} - 3270 q^{51} + 2812 q^{61} - 18060 q^{69} + 35864 q^{79} - 35422 q^{81} + 15308 q^{91} - 45030 q^{99}+O(q^{100}) \) 8 * q + 142 * q^9 - 1072 * q^19 - 1268 * q^21 - 1028 * q^31 + 968 * q^39 + 8100 * q^49 - 3270 * q^51 + 2812 * q^61 - 18060 * q^69 + 35864 * q^79 - 35422 * q^81 + 15308 * q^91 - 45030 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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