LMFDB - Embedded newform 684.5.y.f.145.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$70.7050547493$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.4
Character	$\chi$	$=$	684.145
Dual form			684.5.y.f.217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-9.99470 - 17.3113i) q^{5} +36.5906 q^{7} +O(q^{10})$	$q+(-9.99470 - 17.3113i) q^{5} +36.5906 q^{7} -119.283 q^{11} +(-124.625 - 71.9523i) q^{13} +(262.361 + 454.423i) q^{17} +(341.976 + 115.645i) q^{19} +(-111.426 + 192.995i) q^{23} +(112.712 - 195.223i) q^{25} +(812.879 + 469.316i) q^{29} -715.372i q^{31} +(-365.712 - 633.432i) q^{35} -1845.18i q^{37} +(238.103 - 137.469i) q^{41} +(886.574 + 1535.59i) q^{43} +(335.666 - 581.391i) q^{47} -1062.13 q^{49} +(-604.760 - 349.158i) q^{53} +(1192.20 + 2064.95i) q^{55} +(80.5528 - 46.5072i) q^{59} +(-481.484 + 833.955i) q^{61} +2876.57i q^{65} +(-1332.48 - 769.310i) q^{67} +(-2559.51 + 1477.73i) q^{71} +(-1189.23 - 2059.81i) q^{73} -4364.65 q^{77} +(9515.54 - 5493.80i) q^{79} +9518.65 q^{83} +(5244.45 - 9083.65i) q^{85} +(-9881.74 - 5705.22i) q^{89} +(-4560.11 - 2632.78i) q^{91} +(-1415.98 - 7075.89i) q^{95} +(6999.64 - 4041.25i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 304 q^{7}+O(q^{10}) $ 24 * q + 304 * q^7	$ 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) $ 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−9.99470	−	17.3113i	−0.399788	−	0.692453i	0.593911	−	0.804530i	$-0.297583\pi$
$5$	−9.99470	−	17.3113i	−0.399788	−	0.692453i	−0.993700	+	0.112077i	$0.964250\pi$
$6$		0			0
$7$	36.5906			0.746747			0.373374	−	0.927681i	$-0.378201\pi$
$7$	36.5906			0.746747			0.373374	+	0.927681i	$0.378201\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−119.283			−0.985812			−0.492906	−	0.870083i	$-0.664065\pi$
$11$	−119.283			−0.985812			−0.492906	+	0.870083i	$0.664065\pi$
$12$		0			0
$13$	−124.625	−	71.9523i	−0.737426	−	0.425753i	0.0837067	−	0.996490i	$-0.473324\pi$
$13$	−124.625	−	71.9523i	−0.737426	−	0.425753i	−0.821133	+	0.570737i	$0.806657\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	262.361	+	454.423i	0.907824	+	1.57240i	0.817080	+	0.576524i	$0.195591\pi$
$17$	262.361	+	454.423i	0.907824	+	1.57240i	0.0907440	+	0.995874i	$0.471076\pi$
$18$		0			0
$19$	341.976	+	115.645i	0.947301	+	0.320345i
$19$	341.976	+	115.645i	0.947301	+	0.320345i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−111.426	+	192.995i	−0.210634	+	0.364829i	−0.951913	−	0.306368i	$-0.900886\pi$
$23$	−111.426	+	192.995i	−0.210634	+	0.364829i	0.741279	+	0.671197i	$0.234220\pi$
$24$		0			0
$25$	112.712	−	195.223i	0.180339	−	0.312356i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	812.879	+	469.316i	0.966562	+	0.558045i	0.898186	−	0.439615i	$-0.144885\pi$
$29$	812.879	+	469.316i	0.966562	+	0.558045i	0.0683756	+	0.997660i	$0.478218\pi$
$30$		0			0
$31$		−	715.372i		−	0.744404i	−0.928152	−	0.372202i	$-0.878603\pi$
$31$		−	715.372i		−	0.744404i	0.928152	−	0.372202i	$-0.121397\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−365.712	−	633.432i	−0.298541	−	0.517088i
$36$		0			0
$37$		−	1845.18i		−	1.34783i	−0.738809	−	0.673915i	$-0.764612\pi$
$37$		−	1845.18i		−	1.34783i	0.738809	−	0.673915i	$-0.235388\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	238.103	−	137.469i	0.141644	−	0.0817779i	−0.427504	−	0.904014i	$-0.640607\pi$
$41$	238.103	−	137.469i	0.141644	−	0.0817779i	0.569147	+	0.822236i	$0.307274\pi$
$42$		0			0
$43$	886.574	+	1535.59i	0.479488	+	0.830498i	0.999723	−	0.0235253i	$-0.00748901\pi$
$43$	886.574	+	1535.59i	0.479488	+	0.830498i	−0.520235	+	0.854023i	$0.674156\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	335.666	−	581.391i	0.151954	−	0.263192i	−0.779992	−	0.625790i	$-0.784777\pi$
$47$	335.666	−	581.391i	0.151954	−	0.263192i	0.931946	+	0.362598i	$0.118110\pi$
$48$		0			0
$49$	−1062.13			−0.442368
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−604.760	−	349.158i	−0.215294	−	0.124300i	0.388476	−	0.921459i	$-0.373002\pi$
$53$	−604.760	−	349.158i	−0.215294	−	0.124300i	−0.603769	+	0.797159i	$0.706335\pi$
$54$		0			0
$55$	1192.20	+	2064.95i	0.394116	+	0.682629i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	80.5528	−	46.5072i	0.0231407	−	0.0133603i	−0.488385	−	0.872628i	$-0.662414\pi$
$59$	80.5528	−	46.5072i	0.0231407	−	0.0133603i	0.511526	+	0.859268i	$0.329080\pi$
$60$		0			0
$61$	−481.484	+	833.955i	−0.129396	+	0.224121i	−0.923443	−	0.383736i	$-0.874637\pi$
$61$	−481.484	+	833.955i	−0.129396	+	0.224121i	0.794046	+	0.607857i	$0.207971\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			2876.57i			0.680844i
$66$		0			0
$67$	−1332.48	−	769.310i	−0.296833	−	0.171377i	0.344186	−	0.938901i	$-0.388155\pi$
$67$	−1332.48	−	769.310i	−0.296833	−	0.171377i	−0.641019	+	0.767525i	$0.721488\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2559.51	+	1477.73i	−0.507739	+	0.293143i	−0.731904	−	0.681408i	$-0.761368\pi$
$71$	−2559.51	+	1477.73i	−0.507739	+	0.293143i	0.224165	+	0.974551i	$0.428035\pi$
$72$		0			0
$73$	−1189.23	−	2059.81i	−0.223162	−	0.386528i	0.732605	−	0.680655i	$-0.238305\pi$
$73$	−1189.23	−	2059.81i	−0.223162	−	0.386528i	−0.955766	+	0.294127i	$0.904971\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4364.65			−0.736153
$78$		0			0
$79$	9515.54	−	5493.80i	1.52468	−	0.880276i	0.525110	−	0.851034i	$-0.324024\pi$
$79$	9515.54	−	5493.80i	1.52468	−	0.880276i	0.999572	−	0.0292415i	$-0.00930919\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9518.65			1.38172			0.690858	−	0.722990i	$-0.257233\pi$
$83$	9518.65			1.38172			0.690858	+	0.722990i	$0.257233\pi$
$84$		0			0
$85$	5244.45	−	9083.65i	0.725875	−	1.25725i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9881.74	−	5705.22i	−1.24754	−	0.720266i	−0.276919	−	0.960893i	$-0.589313\pi$
$89$	−9881.74	−	5705.22i	−1.24754	−	0.720266i	−0.970618	+	0.240627i	$0.922647\pi$
$90$		0			0
$91$	−4560.11	−	2632.78i	−0.550671	−	0.317930i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1415.98	−	7075.89i	−0.156896	−	0.784032i
$96$		0			0
$97$	6999.64	−	4041.25i	0.743931	−	0.429509i	−0.0795661	−	0.996830i	$-0.525353\pi$
$97$	6999.64	−	4041.25i	0.743931	−	0.429509i	0.823497	+	0.567321i	$0.192020\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6703.22	−	11610.3i	0.657114	−	1.13815i	−0.324246	−	0.945973i	$-0.605110\pi$
$101$	6703.22	−	11610.3i	0.657114	−	1.13815i	0.981359	−	0.192181i	$-0.0615562\pi$
$102$		0			0
$103$		−	15606.9i		−	1.47110i	−0.677472	−	0.735549i	$-0.736924\pi$
$103$		−	15606.9i		−	1.47110i	0.677472	−	0.735549i	$-0.263076\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17770.0i		−	1.55210i	−0.630671	−	0.776050i	$-0.717220\pi$
$107$		−	17770.0i		−	1.55210i	0.630671	−	0.776050i	$-0.282780\pi$
$108$		0			0
$109$	13035.6	−	7526.13i	1.09718	−	0.633459i	0.161704	−	0.986839i	$-0.448301\pi$
$109$	13035.6	−	7526.13i	1.09718	−	0.633459i	0.935480	+	0.353380i	$0.114968\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12964.3i			1.01529i	0.861565	+	0.507647i	$0.169485\pi$
$113$			12964.3i			1.01529i	−0.861565	+	0.507647i	$0.830515\pi$
$114$		0			0
$115$	4454.66			0.336836
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9599.96	+	16627.6i	0.677916	+	1.17418i
$120$		0			0
$121$	−412.502			−0.0281745
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−16999.5			−1.08797
$126$		0			0
$127$	4892.80	+	2824.86i	0.303354	+	0.175142i	0.643949	−	0.765069i	$-0.277295\pi$
$127$	4892.80	+	2824.86i	0.303354	+	0.175142i	−0.340594	+	0.940210i	$0.610628\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	16497.3	+	28574.1i	0.961324	+	1.66506i	0.719182	+	0.694822i	$0.244517\pi$
$131$	16497.3	+	28574.1i	0.961324	+	1.66506i	0.242143	+	0.970241i	$0.422150\pi$
$132$		0			0
$133$	12513.1	+	4231.51i	0.707395	+	0.239217i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3925.87	+	6799.81i	−0.209168	+	0.362289i	−0.951453	−	0.307795i	$-0.900409\pi$
$137$	−3925.87	+	6799.81i	−0.209168	+	0.362289i	0.742285	+	0.670085i	$0.233742\pi$
$138$		0			0
$139$	7866.10	−	13624.5i	0.407127	−	0.705164i	−0.587440	−	0.809268i	$-0.699864\pi$
$139$	7866.10	−	13624.5i	0.407127	−	0.705164i	0.994566	+	0.104104i	$0.0331974\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14865.7	+	8582.70i	0.726963	+	0.419713i
$144$		0			0
$145$		−	18762.7i		−	0.892399i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4489.87	−	7776.68i	−0.202237	−	0.350285i	0.747012	−	0.664811i	$-0.231488\pi$
$149$	−4489.87	−	7776.68i	−0.202237	−	0.350285i	−0.949249	+	0.314526i	$0.898155\pi$
$150$		0			0
$151$		−	18112.7i		−	0.794382i	−0.917736	−	0.397191i	$-0.869985\pi$
$151$		−	18112.7i		−	0.794382i	0.917736	−	0.397191i	$-0.130015\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12384.0	+	7149.93i	−0.515465	+	0.297604i
$156$		0			0
$157$	−15080.3	−	26119.9i	−0.611803	−	1.05967i	−0.990936	−	0.134332i	$-0.957111\pi$
$157$	−15080.3	−	26119.9i	−0.611803	−	1.05967i	0.379133	−	0.925342i	$-0.376222\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4077.13	+	7061.79i	−0.157291	+	0.272435i
$162$		0			0
$163$	34758.4			1.30823			0.654117	−	0.756394i	$-0.273041\pi$
$163$	34758.4			1.30823			0.654117	+	0.756394i	$0.273041\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18109.6	−	10455.6i	−0.649347	−	0.374901i	0.138859	−	0.990312i	$-0.455657\pi$
$167$	−18109.6	−	10455.6i	−0.649347	−	0.374901i	−0.788206	+	0.615411i	$0.788990\pi$
$168$		0			0
$169$	−3926.24	−	6800.45i	−0.137469	−	0.238103i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	40246.2	−	23236.1i	1.34472	−	0.776375i	0.357225	−	0.934018i	$-0.383723\pi$
$173$	40246.2	−	23236.1i	1.34472	−	0.776375i	0.987496	+	0.157643i	$0.0503896\pi$
$174$		0			0
$175$	4124.20	−	7143.32i	0.134668	−	0.233251i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12646.3i		−	0.394692i	−0.980334	−	0.197346i	$-0.936768\pi$
$179$		−	12646.3i		−	0.394692i	0.980334	−	0.197346i	$-0.0632323\pi$
$180$		0			0
$181$	51267.9	+	29599.6i	1.56491	+	0.903500i	0.996748	+	0.0805818i	$0.0256778\pi$
$181$	51267.9	+	29599.6i	1.56491	+	0.903500i	0.568160	+	0.822918i	$0.307656\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−31942.5	+	18442.0i	−0.933309	+	0.538846i
$186$		0			0
$187$	−31295.3	−	54205.1i	−0.894944	−	1.55009i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−32857.7			−0.900679			−0.450339	−	0.892858i	$-0.648697\pi$
$191$	−32857.7			−0.900679			−0.450339	+	0.892858i	$0.648697\pi$
$192$		0			0
$193$	580.118	−	334.931i	0.0155741	−	0.00899169i	−0.492193	−	0.870486i	$-0.663804\pi$
$193$	580.118	−	334.931i	0.0155741	−	0.00899169i	0.507767	+	0.861495i	$0.330471\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	55991.2			1.44274			0.721369	−	0.692551i	$-0.243513\pi$
$197$	55991.2			1.44274			0.721369	+	0.692551i	$0.243513\pi$
$198$		0			0
$199$	−4479.69	+	7759.04i	−0.113121	+	0.195930i	−0.917027	−	0.398825i	$-0.869418\pi$
$199$	−4479.69	+	7759.04i	−0.113121	+	0.195930i	0.803906	+	0.594756i	$0.202751\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	29743.7	+	17172.6i	0.721778	+	0.416719i
$204$		0			0
$205$	−4759.53	−	2747.92i	−0.113255	−	0.0653877i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−40792.0	−	13794.5i	−0.933861	−	0.315800i
$210$		0			0
$211$	40179.7	−	23197.7i	0.902488	−	0.521051i	0.0244812	−	0.999700i	$-0.492207\pi$
$211$	40179.7	−	23197.7i	0.902488	−	0.521051i	0.878006	+	0.478649i	$0.158873\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	17722.1	−	30695.5i	0.383387	−	0.664046i
$216$		0			0
$217$		−	26175.9i		−	0.555882i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	75509.9i		−	1.54604i
$222$		0			0
$223$	7415.30	−	4281.23i	0.149114	−	0.0860912i	−0.423586	−	0.905856i	$-0.639229\pi$
$223$	7415.30	−	4281.23i	0.149114	−	0.0860912i	0.572701	+	0.819764i	$0.305896\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			4156.24i			0.0806583i	0.999186	+	0.0403292i	$0.0128407\pi$
$227$			4156.24i			0.0806583i	−0.999186	+	0.0403292i	$0.987159\pi$
$228$		0			0
$229$	26506.8			0.505459			0.252729	−	0.967537i	$-0.418672\pi$
$229$	26506.8			0.505459			0.252729	+	0.967537i	$0.418672\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	44217.7	+	76587.2i	0.814487	+	1.41073i	0.909696	+	0.415275i	$0.136315\pi$
$233$	44217.7	+	76587.2i	0.814487	+	1.41073i	−0.0952093	+	0.995457i	$0.530352\pi$
$234$		0			0
$235$	−13419.5			−0.242997
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−81574.5			−1.42810			−0.714050	−	0.700095i	$-0.753141\pi$
$239$	−81574.5			−1.42810			−0.714050	+	0.700095i	$0.753141\pi$
$240$		0			0
$241$	64071.4	+	36991.6i	1.10314	+	0.636897i	0.937044	−	0.349213i	$-0.113551\pi$
$241$	64071.4	+	36991.6i	1.10314	+	0.636897i	0.166095	+	0.986110i	$0.446884\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10615.6	+	18386.8i	0.176854	+	0.306319i
$246$		0			0
$247$	−34297.8	−	39018.1i	−0.562176	−	0.639547i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	40291.0	−	69786.1i	0.639530	−	1.10770i	−0.346006	−	0.938232i	$-0.612462\pi$
$251$	40291.0	−	69786.1i	0.639530	−	1.10770i	0.985536	−	0.169466i	$-0.0542044\pi$
$252$		0			0
$253$	13291.2	−	23021.0i	0.207646	−	0.359653i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−40210.6	−	23215.6i	−0.608799	−	0.351490i	0.163696	−	0.986511i	$-0.447658\pi$
$257$	−40210.6	−	23215.6i	−0.608799	−	0.351490i	−0.772495	+	0.635020i	$0.780992\pi$
$258$		0			0
$259$		−	67516.2i		−	1.00649i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14902.2	−	25811.4i	−0.215447	−	0.373164i	0.737964	−	0.674840i	$-0.235787\pi$
$263$	−14902.2	−	25811.4i	−0.215447	−	0.373164i	−0.953411	+	0.301676i	$0.902454\pi$
$264$		0			0
$265$			13958.9i			0.198774i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−89929.2	+	51920.6i	−1.24278	+	0.717522i	−0.969660	−	0.244456i	$-0.921390\pi$
$269$	−89929.2	+	51920.6i	−1.24278	+	0.717522i	−0.273125	+	0.961979i	$0.588057\pi$
$270$		0			0
$271$	−29976.0	−	51920.0i	−0.408165	−	0.706962i	0.586519	−	0.809935i	$-0.300498\pi$
$271$	−29976.0	−	51920.0i	−0.408165	−	0.706962i	−0.994684	+	0.102973i	$0.967165\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13444.6	+	23286.8i	−0.177780	+	0.307925i
$276$		0			0
$277$	−13253.3			−0.172728			−0.0863641	−	0.996264i	$-0.527525\pi$
$277$	−13253.3			−0.172728			−0.0863641	+	0.996264i	$0.527525\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10099.2	+	5830.76i	0.127901	+	0.0738436i	0.562585	−	0.826739i	$-0.309807\pi$
$281$	10099.2	+	5830.76i	0.127901	+	0.0738436i	−0.434685	+	0.900583i	$0.643140\pi$
$282$		0			0
$283$	29554.6	+	51190.1i	0.369023	+	0.639166i	0.989413	−	0.145127i	$-0.0463592\pi$
$283$	29554.6	+	51190.1i	0.369023	+	0.639166i	−0.620390	+	0.784293i	$0.713026\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8712.33	−	5030.07i	0.105772	−	0.0610675i
$288$		0			0
$289$	−95906.4	+	166115.i	−1.14829	+	1.98890i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	29062.9i		−	0.338535i	−0.985570	−	0.169267i	$-0.945860\pi$
$293$		−	29062.9i		−	0.338535i	0.985570	−	0.169267i	$-0.0541401\pi$
$294$		0			0
$295$	−1610.20	−	929.651i	−0.0185028	−	0.0106826i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	27772.8	−	16034.6i	0.310654	−	0.179356i
$300$		0			0
$301$	32440.3	+	56188.2i	0.358057	+	0.620172i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19249.2			0.206925
$306$		0			0
$307$	3204.87	−	1850.33i	0.0340043	−	0.0196324i	−0.482901	−	0.875675i	$-0.660417\pi$
$307$	3204.87	−	1850.33i	0.0340043	−	0.0196324i	0.516906	+	0.856042i	$0.327084\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	24127.5			0.249454			0.124727	−	0.992191i	$-0.460194\pi$
$311$	24127.5			0.249454			0.124727	+	0.992191i	$0.460194\pi$
$312$		0			0
$313$	−26424.4	+	45768.4i	−0.269722	+	0.467173i	−0.968790	−	0.247883i	$-0.920265\pi$
$313$	−26424.4	+	45768.4i	−0.269722	+	0.467173i	0.699068	+	0.715055i	$0.253599\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	111008.	+	64090.3i	1.10468	+	0.637785i	0.937445	−	0.348133i	$-0.113184\pi$
$317$	111008.	+	64090.3i	1.10468	+	0.637785i	0.167230	+	0.985918i	$0.446518\pi$
$318$		0			0
$319$	−96962.8	−	55981.5i	−0.952849	−	0.550127i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	37169.6	+	185742.i	0.356273	+	1.78035i
$324$		0			0
$325$	−28093.4	+	16219.7i	−0.265973	+	0.153560i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12282.2	−	21273.5i	0.113471	−	0.196538i
$330$		0			0
$331$			137256.i			1.25278i	0.779509	+	0.626391i	$0.215469\pi$
$331$			137256.i			1.25278i	−0.779509	+	0.626391i	$0.784531\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			30756.1i			0.274057i
$336$		0			0
$337$	−30403.1	+	17553.2i	−0.267706	+	0.154560i	−0.627845	−	0.778339i	$-0.716063\pi$
$337$	−30403.1	+	17553.2i	−0.267706	+	0.154560i	0.360139	+	0.932899i	$0.382729\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			85331.9i			0.733842i
$342$		0			0
$343$	−126718.			−1.07708
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−54306.2	−	94061.1i	−0.451014	−	0.781180i	0.547435	−	0.836848i	$-0.315604\pi$
$347$	−54306.2	−	94061.1i	−0.451014	−	0.781180i	−0.998449	+	0.0556686i	$0.982271\pi$
$348$		0			0
$349$	−63086.9			−0.517950			−0.258975	−	0.965884i	$-0.583385\pi$
$349$	−63086.9			−0.517950			−0.258975	+	0.965884i	$0.583385\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	176520.			1.41659			0.708294	−	0.705917i	$-0.249465\pi$
$353$	176520.			1.41659			0.708294	+	0.705917i	$0.249465\pi$
$354$		0			0
$355$	51163.1	+	29539.0i	0.405976	+	0.234390i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−49723.8	−	86124.1i	−0.385811	−	0.668245i	0.606070	−	0.795411i	$-0.292745\pi$
$359$	−49723.8	−	86124.1i	−0.385811	−	0.668245i	−0.991881	+	0.127166i	$0.959412\pi$
$360$		0			0
$361$	103574.	+	79095.2i	0.794758	+	0.606926i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−23772.0	+	41174.3i	−0.178435	+	0.309058i
$366$		0			0
$367$	−130383.	+	225829.i	−0.968027	+	1.67667i	−0.266773	+	0.963759i	$0.585957\pi$
$367$	−130383.	+	225829.i	−0.968027	+	1.67667i	−0.701254	+	0.712912i	$0.747376\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−22128.5	−	12775.9i	−0.160770	−	0.0928206i
$372$		0			0
$373$			161961.i			1.16411i	0.813150	+	0.582054i	$0.197751\pi$
$373$			161961.i			1.16411i	−0.813150	+	0.582054i	$0.802249\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−67536.7	−	116977.i	−0.475179	−	0.823033i
$378$		0			0
$379$		−	154665.i		−	1.07675i	−0.842706	−	0.538374i	$-0.819039\pi$
$379$		−	154665.i		−	1.07675i	0.842706	−	0.538374i	$-0.180961\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−105912.	+	61148.2i	−0.722015	+	0.416856i	−0.815494	−	0.578766i	$-0.803535\pi$
$383$	−105912.	+	61148.2i	−0.722015	+	0.416856i	0.0934786	+	0.995621i	$0.470201\pi$
$384$		0			0
$385$	43623.4	+	75557.9i	0.294305	+	0.509751i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	77345.1	−	133966.i	0.511132	−	0.885307i	−0.488784	−	0.872405i	$-0.662560\pi$
$389$	77345.1	−	133966.i	0.511132	−	0.885307i	0.999917	−	0.0129027i	$-0.00410716\pi$
$390$		0			0
$391$	−116935.			−0.764876
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−190210.	−	109818.i	−1.21910	−	0.703848i
$396$		0			0
$397$	38452.9	+	66602.4i	0.243977	+	0.422580i	0.961843	−	0.273601i	$-0.0882146\pi$
$397$	38452.9	+	66602.4i	0.243977	+	0.422580i	−0.717867	+	0.696181i	$0.754881\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21466.0	−	12393.4i	0.133494	−	0.0770730i	−0.431766	−	0.901986i	$-0.642109\pi$
$401$	21466.0	−	12393.4i	0.133494	−	0.0770730i	0.565260	+	0.824913i	$0.308776\pi$
$402$		0			0
$403$	−51472.7	+	89153.2i	−0.316932	+	0.548943i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			220099.i			1.32871i
$408$		0			0
$409$	−58077.1	−	33530.8i	−0.347183	−	0.200446i	0.316261	−	0.948672i	$-0.397572\pi$
$409$	−58077.1	−	33530.8i	−0.347183	−	0.200446i	−0.663444	+	0.748226i	$0.730906\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2947.48	−	1701.73i	0.0172803	−	0.00997677i
$414$		0			0
$415$	−95136.1	−	164780.i	−0.552394	−	0.956774i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1229.90			0.00700556			0.00350278	−	0.999994i	$-0.498885\pi$
$419$	1229.90			0.00700556			0.00350278	+	0.999994i	$0.498885\pi$
$420$		0			0
$421$	−51036.9	+	29466.1i	−0.287952	+	0.166249i	−0.637018	−	0.770849i	$-0.719832\pi$
$421$	−51036.9	+	29466.1i	−0.287952	+	0.166249i	0.349066	+	0.937098i	$0.386499\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	118285.			0.654864
$426$		0			0
$427$	−17617.8	+	30514.9i	−0.0966264	+	0.167362i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18120.2	−	10461.7i	−0.0975456	−	0.0563180i	0.450434	−	0.892810i	$-0.351269\pi$
$431$	−18120.2	−	10461.7i	−0.0975456	−	0.0563180i	−0.547979	+	0.836492i	$0.684603\pi$
$432$		0			0
$433$	−102211.	−	59011.6i	−0.545157	−	0.314747i	0.202009	−	0.979384i	$-0.435253\pi$
$433$	−102211.	−	59011.6i	−0.545157	−	0.314747i	−0.747167	+	0.664637i	$0.768586\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−60423.6	+	53113.7i	−0.316405	+	0.278127i
$438$		0			0
$439$	31410.7	−	18135.0i	0.162985	−	0.0940996i	−0.416289	−	0.909232i	$-0.636669\pi$
$439$	31410.7	−	18135.0i	0.162985	−	0.0940996i	0.579274	+	0.815133i	$0.303336\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10697.1	+	18527.9i	−0.0545079	+	0.0944104i	−0.891992	−	0.452051i	$-0.850692\pi$
$443$	−10697.1	+	18527.9i	−0.0545079	+	0.0944104i	0.837484	+	0.546462i	$0.184026\pi$
$444$		0			0
$445$			228088.i			1.15181i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			277574.i			1.37685i	0.725307	+	0.688425i	$0.241698\pi$
$449$			277574.i			1.37685i	−0.725307	+	0.688425i	$0.758302\pi$
$450$		0			0
$451$	−28401.7	+	16397.7i	−0.139634	+	0.0806177i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			105255.i			0.508419i
$456$		0			0
$457$	159665.			0.764500			0.382250	−	0.924059i	$-0.375149\pi$
$457$	159665.			0.764500			0.382250	+	0.924059i	$0.375149\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−157382.	−	272593.i	−0.740546	−	1.28266i	−0.952247	−	0.305329i	$-0.901234\pi$
$461$	−157382.	−	272593.i	−0.740546	−	1.28266i	0.211701	−	0.977335i	$-0.432100\pi$
$462$		0			0
$463$	32233.8			0.150366			0.0751830	−	0.997170i	$-0.476046\pi$
$463$	32233.8			0.150366			0.0751830	+	0.997170i	$0.476046\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	326219.			1.49581			0.747904	−	0.663807i	$-0.231060\pi$
$467$	326219.			1.49581			0.747904	+	0.663807i	$0.231060\pi$
$468$		0			0
$469$	−48756.4	−	28149.5i	−0.221659	−	0.127975i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−105753.	−	183170.i	−0.472685	−	0.818715i
$474$		0			0
$475$	61121.1	−	53726.9i	0.270897	−	0.238125i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	151754.	−	262846.i	0.661407	−	1.14559i	−0.318839	−	0.947809i	$-0.603293\pi$
$479$	151754.	−	262846.i	0.661407	−	1.14559i	0.980246	−	0.197782i	$-0.0633739\pi$
$480$		0			0
$481$	−132765.	+	229955.i	−0.573843	+	0.993924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−139919.	−	80782.1i	−0.594829	−	0.343425i
$486$		0			0
$487$			135424.i			0.571003i	0.958378	+	0.285501i	$0.0921602\pi$
$487$			135424.i			0.571003i	−0.958378	+	0.285501i	$0.907840\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−58107.3	−	100645.i	−0.241028	−	0.417473i	0.719979	−	0.693996i	$-0.244151\pi$
$491$	−58107.3	−	100645.i	−0.241028	−	0.417473i	−0.961007	+	0.276523i	$0.910818\pi$
$492$		0			0
$493$			492521.i			2.02643i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−93654.1	+	54071.2i	−0.379152	+	0.218904i
$498$		0			0
$499$	−102031.	−	176724.i	−0.409763	−	0.709731i	0.585100	−	0.810961i	$-0.301055\pi$
$499$	−102031.	−	176724.i	−0.409763	−	0.709731i	−0.994863	+	0.101231i	$0.967722\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	79776.4	−	138177.i	0.315311	−	0.546134i	−0.664193	−	0.747561i	$-0.731225\pi$
$503$	79776.4	−	138177.i	0.315311	−	0.546134i	0.979503	+	0.201427i	$0.0645580\pi$
$504$		0			0
$505$	−267987.			−1.05083
$506$		0			0
$507$		0			0
$508$		0			0
$509$	65306.1	+	37704.5i	0.252068	+	0.145532i	0.620711	−	0.784040i	$-0.286844\pi$
$509$	65306.1	+	37704.5i	0.252068	+	0.145532i	−0.368643	+	0.929571i	$0.620177\pi$
$510$		0			0
$511$	−43514.7	−	75369.6i	−0.166646	−	0.288639i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−270176.	+	155986.i	−1.01867	+	0.588128i
$516$		0			0
$517$	−40039.4	+	69350.2i	−0.149798	+	0.259458i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	282223.i		−	1.03972i	−0.854251	−	0.519862i	$-0.825984\pi$
$521$		−	282223.i		−	1.03972i	0.854251	−	0.519862i	$-0.174016\pi$
$522$		0			0
$523$	−78675.5	−	45423.3i	−0.287631	−	0.166064i	0.349242	−	0.937033i	$-0.386439\pi$
$523$	−78675.5	−	45423.3i	−0.287631	−	0.166064i	−0.636873	+	0.770969i	$0.719772\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	325082.	−	187686.i	1.17050	−	0.675788i
$528$		0			0
$529$	115089.	+	199340.i	0.411266	+	0.712334i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−39564.7			−0.139269
$534$		0			0
$535$	−307622.	+	177606.i	−1.07476	+	0.620511i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	126694.			0.436092
$540$		0			0
$541$	−180295.	+	312279.i	−0.616011	+	1.06696i	0.374196	+	0.927350i	$0.377919\pi$
$541$	−180295.	+	312279.i	−0.616011	+	1.06696i	−0.990206	+	0.139612i	$0.955415\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−260575.	−	150443.i	−0.877282	−	0.506499i
$546$		0			0
$547$	−9456.44	−	5459.68i	−0.0316048	−	0.0182470i	0.484114	−	0.875005i	$-0.339142\pi$
$547$	−9456.44	−	5459.68i	−0.0316048	−	0.0182470i	−0.515719	+	0.856758i	$0.672475\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	223711.	+	254500.i	0.736858	+	0.838270i
$552$		0			0
$553$	348180.	−	201022.i	1.13855	−	0.657344i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−149408.	+	258783.i	−0.481576	+	0.834114i	−0.999776	−	0.0211453i	$-0.993269\pi$
$557$	−149408.	+	258783.i	−0.481576	+	0.834114i	0.518201	+	0.855259i	$0.326602\pi$
$558$		0			0
$559$		−	255164.i		−	0.816574i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			532306.i			1.67936i	0.543078	+	0.839682i	$0.317259\pi$
$563$			532306.i			1.67936i	−0.543078	+	0.839682i	$0.682741\pi$
$564$		0			0
$565$	224429.	−	129574.i	0.703044	−	0.405902i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			572245.i			1.76749i	0.467967	+	0.883746i	$0.344987\pi$
$569$			572245.i			1.76749i	−0.467967	+	0.883746i	$0.655013\pi$
$570$		0			0
$571$	−60094.6			−0.184316			−0.0921581	−	0.995744i	$-0.529377\pi$
$571$	−60094.6			−0.184316			−0.0921581	+	0.995744i	$0.529377\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	25117.9	+	43505.6i	0.0759711	+	0.131586i
$576$		0			0
$577$	320000.			0.961166			0.480583	−	0.876949i	$-0.340425\pi$
$577$	320000.			0.961166			0.480583	+	0.876949i	$0.340425\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	348293.			1.03179
$582$		0			0
$583$	72137.7	+	41648.7i	0.212239	+	0.122536i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−291869.	−	505533.i	−0.847056	−	1.46714i	−0.883823	−	0.467821i	$-0.845039\pi$
$587$	−291869.	−	505533.i	−0.847056	−	1.46714i	0.0367671	−	0.999324i	$-0.488294\pi$
$588$		0			0
$589$	82728.9	−	244640.i	0.238466	−	0.705175i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−209232.	+	362400.i	−0.595001	+	1.03057i	0.398546	+	0.917149i	$0.369515\pi$
$593$	−209232.	+	362400.i	−0.595001	+	1.03057i	−0.993547	+	0.113424i	$0.963818\pi$
$594$		0			0
$595$	191898.	−	332376.i	0.542045	−	0.938850i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	588305.	+	339658.i	1.63964	+	0.946648i	0.980956	+	0.194229i	$0.0622206\pi$
$599$	588305.	+	339658.i	1.63964	+	0.946648i	0.658686	+	0.752418i	$0.271113\pi$
$600$		0			0
$601$		−	101516.i		−	0.281051i	−0.990077	−	0.140526i	$-0.955121\pi$
$601$		−	101516.i		−	0.281051i	0.990077	−	0.140526i	$-0.0448793\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4122.84	+	7140.96i	0.0112638	+	0.0195095i
$606$		0			0
$607$		−	266838.i		−	0.724221i	−0.932135	−	0.362110i	$-0.882056\pi$
$607$		−	266838.i		−	0.724221i	0.932135	−	0.362110i	$-0.117944\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−83664.8	+	48303.9i	−0.224109	+	0.129390i
$612$		0			0
$613$	−139486.	−	241596.i	−0.371200	−	0.642938i	0.618550	−	0.785745i	$-0.287720\pi$
$613$	−139486.	−	241596.i	−0.371200	−	0.642938i	−0.989750	+	0.142807i	$0.954387\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−234275.	+	405776.i	−0.615397	+	1.06590i	0.374918	+	0.927058i	$0.377671\pi$
$617$	−234275.	+	405776.i	−0.615397	+	1.06590i	−0.990315	+	0.138841i	$0.955662\pi$
$618$		0			0
$619$	−280344.			−0.731661			−0.365830	−	0.930682i	$-0.619215\pi$
$619$	−280344.			−0.731661			−0.365830	+	0.930682i	$0.619215\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−361579.	−	208758.i	−0.931595	−	0.537857i
$624$		0			0
$625$	99459.7	+	172269.i	0.254617	+	0.441009i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	838492.	−	484103.i	2.11932	−	1.22359i
$630$		0			0
$631$	318112.	−	550986.i	0.798954	−	1.38383i	−0.121345	−	0.992610i	$-0.538721\pi$
$631$	318112.	−	550986.i	0.798954	−	1.38383i	0.920298	−	0.391218i	$-0.127946\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	112935.i		−	0.280078i
$636$		0			0
$637$	132367.	+	76422.4i	0.326214	+	0.188340i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−471255.	+	272079.i	−1.14694	+	0.662185i	−0.948139	−	0.317855i	$-0.897037\pi$
$641$	−471255.	+	272079.i	−1.14694	+	0.662185i	−0.198799	+	0.980040i	$0.563704\pi$
$642$		0			0
$643$	−260850.	−	451805.i	−0.630911	−	1.09277i	−0.987366	−	0.158458i	$-0.949348\pi$
$643$	−260850.	−	451805.i	−0.630911	−	1.09277i	0.356454	−	0.934313i	$-0.383985\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−339826.			−0.811798			−0.405899	−	0.913918i	$-0.633042\pi$
$647$	−339826.			−0.811798			−0.405899	+	0.913918i	$0.633042\pi$
$648$		0			0
$649$	−9608.60	+	5547.53i	−0.0228124	+	0.0131707i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	179308.			0.420508			0.210254	−	0.977647i	$-0.432571\pi$
$653$	179308.			0.420508			0.210254	+	0.977647i	$0.432571\pi$
$654$		0			0
$655$	329771.	−	571180.i	0.768652	−	1.33134i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	106078.	+	61244.1i	0.244261	+	0.141024i	0.617134	−	0.786858i	$-0.288294\pi$
$659$	106078.	+	61244.1i	0.244261	+	0.141024i	−0.372873	+	0.927883i	$0.621627\pi$
$660$		0			0
$661$	643594.	+	371579.i	1.47302	+	0.850450i	0.999539	−	0.0303528i	$-0.00966308\pi$
$661$	643594.	+	371579.i	1.47302	+	0.850450i	0.473483	+	0.880803i	$0.342996\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−51811.7	−	258911.i	−0.117161	−	0.585474i
$666$		0			0
$667$	−181151.	+	104587.i	−0.407182	+	0.235087i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	57433.0	−	99476.8i	0.127561	−	0.220941i
$672$		0			0
$673$			758095.i			1.67376i	0.547386	+	0.836880i	$0.315623\pi$
$673$			758095.i			1.67376i	−0.547386	+	0.836880i	$0.684377\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	632086.i		−	1.37911i	−0.724233	−	0.689555i	$-0.757806\pi$
$677$		−	632086.i		−	1.37911i	0.724233	−	0.689555i	$-0.242194\pi$
$678$		0			0
$679$	256121.	−	147872.i	0.555528	−	0.320734i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	673401.i		−	1.44355i	−0.692127	−	0.721776i	$-0.743326\pi$
$683$		−	673401.i		−	1.44355i	0.692127	−	0.721776i	$-0.256674\pi$
$684$		0			0
$685$	156952.			0.334491
$686$		0			0
$687$		0			0
$688$		0			0
$689$	50245.5	+	87027.7i	0.105842	+	0.183324i
$690$		0			0
$691$	−23708.7			−0.0496536			−0.0248268	−	0.999692i	$-0.507903\pi$
$691$	−23708.7			−0.0496536			−0.0248268	+	0.999692i	$0.507903\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−314477.			−0.651058
$696$		0			0
$697$	124938.	+	72132.9i	0.257175	+	0.148480i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19769.2	−	34241.3i	−0.0402304	−	0.0696811i	0.845209	−	0.534436i	$-0.179476\pi$
$701$	−19769.2	−	34241.3i	−0.0402304	−	0.0696811i	−0.885439	+	0.464755i	$0.846143\pi$
$702$		0			0
$703$	213385.	−	631006.i	0.431770	−	1.27680i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	245275.	−	424829.i	0.490698	−	0.849914i
$708$		0			0
$709$	88233.5	−	152825.i	0.175526	−	0.304020i	−0.764817	−	0.644247i	$-0.777171\pi$
$709$	88233.5	−	152825.i	0.175526	−	0.304020i	0.940343	+	0.340228i	$0.110504\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	138063.	+	79710.7i	0.271580	+	0.156797i
$714$		0			0
$715$		−	343126.i		−	0.671184i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	81799.7	+	141681.i	0.158232	+	0.274066i	0.934231	−	0.356668i	$-0.116087\pi$
$719$	81799.7	+	141681.i	0.158232	+	0.274066i	−0.775999	+	0.630734i	$0.782754\pi$
$720$		0			0
$721$		−	571065.i		−	1.09854i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	183242.	−	105795.i	0.348618	−	0.201274i
$726$		0			0
$727$	162054.	+	280686.i	0.306614	+	0.531071i	0.977619	−	0.210382i	$-0.0674706\pi$
$727$	162054.	+	280686.i	0.306614	+	0.531071i	−0.671005	+	0.741452i	$0.734137\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−465205.	+	805759.i	−0.870582	+	1.50789i
$732$		0			0
$733$	801693.			1.49211			0.746054	−	0.665885i	$-0.231946\pi$
$733$	801693.			1.49211			0.746054	+	0.665885i	$0.231946\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	158943.	+	91765.8i	0.292622	+	0.168945i
$738$		0			0
$739$	−489398.	−	847663.i	−0.896135	−	1.55215i	−0.832393	−	0.554185i	$-0.813030\pi$
$739$	−489398.	−	847663.i	−0.896135	−	1.55215i	−0.0637418	−	0.997966i	$-0.520303\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	284210.	−	164089.i	0.514828	−	0.297236i	−0.219988	−	0.975503i	$-0.570602\pi$
$743$	284210.	−	164089.i	0.514828	−	0.297236i	0.734816	+	0.678266i	$0.237268\pi$
$744$		0			0
$745$	−89749.8	+	155451.i	−0.161704	+	0.280080i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	650215.i		−	1.15903i
$750$		0			0
$751$	−617662.	−	356607.i	−1.09514	−	0.632281i	−0.160202	−	0.987084i	$-0.551215\pi$
$751$	−617662.	−	356607.i	−1.09514	−	0.632281i	−0.934941	+	0.354803i	$0.884548\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−313555.	+	181031.i	−0.550072	+	0.317584i
$756$		0			0
$757$	70575.8	+	122241.i	0.123158	+	0.213316i	0.921012	−	0.389535i	$-0.127364\pi$
$757$	70575.8	+	122241.i	0.123158	+	0.213316i	−0.797853	+	0.602852i	$0.794031\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	339525.			0.586276			0.293138	−	0.956070i	$-0.405300\pi$
$761$	339525.			0.586276			0.293138	+	0.956070i	$0.405300\pi$
$762$		0			0
$763$	476982.	−	275386.i	0.819319	−	0.473034i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13385.2			−0.0227527
$768$		0			0
$769$	−26731.7	+	46300.7i	−0.0452037	+	0.0782952i	−0.887742	−	0.460341i	$-0.847727\pi$
$769$	−26731.7	+	46300.7i	−0.0452037	+	0.0782952i	0.842538	+	0.538636i	$0.181060\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−635204.	−	366735.i	−1.06305	−	0.613753i	−0.136777	−	0.990602i	$-0.543674\pi$
$773$	−635204.	−	366735.i	−1.06305	−	0.613753i	−0.926275	+	0.376849i	$0.877008\pi$
$774$		0			0
$775$	−139657.	−	80630.9i	−0.232519	−	0.134245i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	97322.9	−	19475.7i	0.160376	−	0.0320935i
$780$		0			0
$781$	305307.	−	176269.i	0.500535	−	0.288984i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−301447.	+	522121.i	−0.489183	+	0.847290i
$786$		0			0
$787$		−	873749.i		−	1.41071i	−0.708855	−	0.705355i	$-0.750788\pi$
$787$		−	873749.i		−	1.41071i	0.708855	−	0.705355i	$-0.249212\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			474371.i			0.758168i
$792$		0			0
$793$	120010.	−	69287.7i	0.190840	−	0.110182i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.07688e6i			1.69532i	0.530543	+	0.847658i	$0.321988\pi$
$797$			1.07688e6i			1.69532i	−0.530543	+	0.847658i	$0.678012\pi$
$798$		0			0
$799$	352263.			0.551790
$800$		0			0
$801$		0			0
$802$		0			0
$803$	141855.	+	245700.i	0.219996	+	0.381044i
$804$		0			0
$805$	162999.			0.251532
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−970610.			−1.48302			−0.741511	−	0.670940i	$-0.765891\pi$
$809$	−970610.			−1.48302			−0.741511	+	0.670940i	$0.765891\pi$
$810$		0			0
$811$	6827.43	+	3941.82i	0.0103804	+	0.00599315i	0.505181	−	0.863013i	$-0.331426\pi$
$811$	6827.43	+	3941.82i	0.0103804	+	0.00599315i	−0.494801	+	0.869006i	$0.664759\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−347400.	−	601715.i	−0.523016	−	0.905890i
$816$		0			0
$817$	125604.	+	627662.i	0.188174	+	0.940333i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	432343.	−	748840.i	0.641419	−	1.11097i	−0.343697	−	0.939081i	$-0.611679\pi$
$821$	432343.	−	748840.i	0.641419	−	1.11097i	0.985116	−	0.171890i	$-0.0549873\pi$
$822$		0			0
$823$	36892.3	−	63899.3i	0.0544673	−	0.0943402i	−0.837506	−	0.546428i	$-0.815987\pi$
$823$	36892.3	−	63899.3i	0.0544673	−	0.0943402i	0.891973	+	0.452088i	$0.149321\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	507445.	+	292973.i	0.741956	+	0.428368i	0.822780	−	0.568360i	$-0.192422\pi$
$827$	507445.	+	292973.i	0.741956	+	0.428368i	−0.0808243	+	0.996728i	$0.525755\pi$
$828$		0			0
$829$		−	235392.i		−	0.342518i	−0.985226	−	0.171259i	$-0.945217\pi$
$829$		−	235392.i		−	0.342518i	0.985226	−	0.171259i	$-0.0547835\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−278661.	−	482655.i	−0.401593	−	0.695579i
$834$		0			0
$835$			418003.i			0.599523i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	470743.	−	271784.i	0.668744	−	0.386099i	−0.126857	−	0.991921i	$-0.540489\pi$
$839$	470743.	−	271784.i	0.668744	−	0.386099i	0.795601	+	0.605822i	$0.207155\pi$
$840$		0			0
$841$	86874.0	+	150470.i	0.122828	+	0.212745i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−78483.3	+	135937.i	−0.109917	+	0.190381i
$846$		0			0
$847$	−15093.7			−0.0210392
$848$		0			0
$849$		0			0
$850$		0			0
$851$	356110.	+	205600.i	0.491728	+	0.283899i
$852$		0			0
$853$	80195.4	+	138903.i	0.110218	+	0.190903i	0.915858	−	0.401502i	$-0.131512\pi$
$853$	80195.4	+	138903.i	0.110218	+	0.190903i	−0.805640	+	0.592405i	$0.798179\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	815016.	−	470550.i	1.10970	−	0.640684i	0.170946	−	0.985280i	$-0.445317\pi$
$857$	815016.	−	470550.i	1.10970	−	0.640684i	0.938751	+	0.344596i	$0.111984\pi$
$858$		0			0
$859$	−274856.	+	476064.i	−0.372493	+	0.645178i	−0.989948	−	0.141429i	$-0.954830\pi$
$859$	−274856.	+	476064.i	−0.372493	+	0.645178i	0.617455	+	0.786606i	$0.288164\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			32073.2i			0.0430646i	0.999768	+	0.0215323i	$0.00685448\pi$
$863$			32073.2i			0.0430646i	−0.999768	+	0.0215323i	$0.993146\pi$
$864$		0			0
$865$	−804497.	−	464476.i	−1.07521	−	0.620771i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.13505e6	+	655319.i	−1.50305	+	0.867787i
$870$		0			0
$871$	110707.	+	191750.i	0.145928	+	0.252755i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−622021.			−0.812436
$876$		0			0
$877$	1.12290e6	−	648306.i	1.45996	−	0.842909i	0.460953	−	0.887425i	$-0.347508\pi$
$877$	1.12290e6	−	648306.i	1.45996	−	0.842909i	0.999009	+	0.0445158i	$0.0141745\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	614294.			0.791451			0.395726	−	0.918369i	$-0.370493\pi$
$881$	614294.			0.791451			0.395726	+	0.918369i	$0.370493\pi$
$882$		0			0
$883$	118268.	−	204847.i	0.151686	−	0.262729i	−0.780161	−	0.625579i	$-0.784863\pi$
$883$	118268.	−	204847.i	0.151686	−	0.262729i	0.931848	+	0.362850i	$0.118196\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−82987.6	−	47912.9i	−0.105479	−	0.0608983i	0.446332	−	0.894867i	$-0.352730\pi$
$887$	−82987.6	−	47912.9i	−0.105479	−	0.0608983i	−0.551812	+	0.833969i	$0.686063\pi$
$888$		0			0
$889$	179031.	+	103363.i	0.226529	+	0.130787i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	182024.	−	160004.i	0.228258	−	0.200644i
$894$		0			0
$895$	−218925.	+	126396.i	−0.273306	+	0.157793i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	335735.	−	581511.i	0.415411	−	0.719513i
$900$		0			0
$901$		−	366422.i		−	0.451370i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.18336e6i		−	1.44483i
$906$		0			0
$907$	667240.	−	385231.i	0.811088	−	0.468282i	−0.0362459	−	0.999343i	$-0.511540\pi$
$907$	667240.	−	385231.i	0.811088	−	0.468282i	0.847333	+	0.531061i	$0.178207\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			397478.i			0.478935i	0.970904	+	0.239467i	$0.0769728\pi$
$911$			397478.i			0.478935i	−0.970904	+	0.239467i	$0.923027\pi$
$912$		0			0
$913$	−1.13542e6			−1.36211
$914$		0			0
$915$		0			0
$916$		0			0
$917$	603646.	+	1.04555e6i	0.717867	+	1.24338i
$918$		0			0
$919$	692743.			0.820241			0.410120	−	0.912031i	$-0.365487\pi$
$919$	692743.			0.820241			0.410120	+	0.912031i	$0.365487\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	425305.			0.499226
$924$		0			0
$925$	−360221.	−	207973.i	−0.421003	−	0.243066i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	620621.	+	1.07495e6i	0.719109	+	1.24553i	0.961353	+	0.275319i	$0.0887834\pi$
$929$	620621.	+	1.07495e6i	0.719109	+	1.24553i	−0.242244	+	0.970215i	$0.577883\pi$
$930$		0			0
$931$	−363221.	−	122829.i	−0.419056	−	0.141710i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−625575.	+	1.08353e6i	−0.715576	+	1.23941i
$936$		0			0
$937$	320016.	−	554283.i	0.364495	−	0.631324i	−0.624200	−	0.781265i	$-0.714575\pi$
$937$	320016.	−	554283.i	0.364495	−	0.631324i	0.988695	+	0.149940i	$0.0479082\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	414122.	+	239093.i	0.467680	+	0.270015i	0.715268	−	0.698850i	$-0.246305\pi$
$941$	414122.	+	239093.i	0.467680	+	0.270015i	−0.247588	+	0.968865i	$0.579638\pi$
$942$		0			0
$943$			61270.1i			0.0689009i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	70045.8	+	121323.i	0.0781055	+	0.135283i	0.902433	−	0.430831i	$-0.141780\pi$
$947$	70045.8	+	121323.i	0.0781055	+	0.135283i	−0.824327	+	0.566114i	$0.808446\pi$
$948$		0			0
$949$			342271.i			0.380047i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	290609.	−	167783.i	0.319980	−	0.184741i	−0.331403	−	0.943489i	$-0.607522\pi$
$953$	290609.	−	167783.i	0.319980	−	0.184741i	0.651384	+	0.758748i	$0.274189\pi$
$954$		0			0
$955$	328402.	+	568810.i	0.360081	+	0.623678i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−143650.	+	248809.i	−0.156196	+	0.270539i
$960$		0			0
$961$	411764.			0.445863
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11596.2	−	6695.08i	−0.0124526	−	0.00718954i
$966$		0			0
$967$	−216366.	−	374757.i	−0.231386	−	0.400772i	0.726830	−	0.686817i	$-0.240993\pi$
$967$	−216366.	−	374757.i	−0.231386	−	0.400772i	−0.958216	+	0.286045i	$0.907659\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−463756.	+	267750.i	−0.491871	+	0.283982i	−0.725350	−	0.688380i	$-0.758322\pi$
$971$	−463756.	+	267750.i	−0.491871	+	0.283982i	0.233479	+	0.972362i	$0.424989\pi$
$972$		0			0
$973$	287825.	−	498528.i	0.304021	−	0.526580i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	189635.i		−	0.198669i	−0.995054	−	0.0993345i	$-0.968329\pi$
$977$		−	189635.i		−	0.198669i	0.995054	−	0.0993345i	$-0.0316714\pi$
$978$		0			0
$979$	1.17873e6	+	680538.i	1.22984	+	0.710047i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−34717.5	+	20044.1i	−0.0359287	+	0.0207434i	−0.517857	−	0.855467i	$-0.673270\pi$
$983$	−34717.5	+	20044.1i	−0.0359287	+	0.0207434i	0.481928	+	0.876211i	$0.339937\pi$
$984$		0			0
$985$	−559615.	−	969282.i	−0.576789	−	0.999028i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−395148.			−0.403986
$990$		0			0
$991$	−182476.	+	105353.i	−0.185806	+	0.107275i	−0.590018	−	0.807390i	$-0.700879\pi$
$991$	−182476.	+	105353.i	−0.185806	+	0.107275i	0.404212	+	0.914665i	$0.367546\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	179093.			0.180897
$996$		0			0
$997$	−912098.	+	1.57980e6i	−0.917595	+	1.58932i	−0.114538	+	0.993419i	$0.536539\pi$
$997$	−912098.	+	1.57980e6i	−0.917595	+	1.58932i	−0.803057	+	0.595902i	$0.796795\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.y.f.145.4	✓	24
3.2	odd	2	inner	684.5.y.f.145.9	yes	24
19.8	odd	6	inner	684.5.y.f.217.4	yes	24
57.8	even	6	inner	684.5.y.f.217.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.5.y.f.145.4	✓	24	1.1	even	1	trivial
684.5.y.f.145.9	yes	24	3.2	odd	2	inner
684.5.y.f.217.4	yes	24	19.8	odd	6	inner
684.5.y.f.217.9	yes	24	57.8	even	6	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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-9.99470 - 17.3113i) q^{5} +36.5906 q^{7} +O(q^{10})\)	\(q+(-9.99470 - 17.3113i) q^{5} +36.5906 q^{7} -119.283 q^{11} +(-124.625 - 71.9523i) q^{13} +(262.361 + 454.423i) q^{17} +(341.976 + 115.645i) q^{19} +(-111.426 + 192.995i) q^{23} +(112.712 - 195.223i) q^{25} +(812.879 + 469.316i) q^{29} -715.372i q^{31} +(-365.712 - 633.432i) q^{35} -1845.18i q^{37} +(238.103 - 137.469i) q^{41} +(886.574 + 1535.59i) q^{43} +(335.666 - 581.391i) q^{47} -1062.13 q^{49} +(-604.760 - 349.158i) q^{53} +(1192.20 + 2064.95i) q^{55} +(80.5528 - 46.5072i) q^{59} +(-481.484 + 833.955i) q^{61} +2876.57i q^{65} +(-1332.48 - 769.310i) q^{67} +(-2559.51 + 1477.73i) q^{71} +(-1189.23 - 2059.81i) q^{73} -4364.65 q^{77} +(9515.54 - 5493.80i) q^{79} +9518.65 q^{83} +(5244.45 - 9083.65i) q^{85} +(-9881.74 - 5705.22i) q^{89} +(-4560.11 - 2632.78i) q^{91} +(-1415.98 - 7075.89i) q^{95} +(6999.64 - 4041.25i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 304 q^{7}+O(q^{10}) \) 24 * q + 304 * q^7	\( 24 q + 304 q^{7} + 516 q^{13} + 416 q^{19} - 5108 q^{25} - 720 q^{43} + 25440 q^{49} - 12536 q^{55} + 5020 q^{61} - 13080 q^{67} + 22572 q^{73} + 63096 q^{79} - 32624 q^{85} - 89568 q^{91} - 3888 q^{97}+O(q^{100}) \) 24 * q + 304 * q^7 + 516 * q^13 + 416 * q^19 - 5108 * q^25 - 720 * q^43 + 25440 * q^49 - 12536 * q^55 + 5020 * q^61 - 13080 * q^67 + 22572 * q^73 + 63096 * q^79 - 32624 * q^85 - 89568 * q^91 - 3888 * q^97

Introduction

L-functions

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