LMFDB - Embedded newform 648.4.i.u.433.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, 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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.4
Root			$-1.30421 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.4.i.u.217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.26031 - 9.11112i) q^{5} +(0.177444 + 0.307342i) q^{7} +O(q^{10})$	$q+(5.26031 - 9.11112i) q^{5} +(0.177444 + 0.307342i) q^{7} +(-4.30734 - 7.46054i) q^{11} +(17.3942 - 30.1277i) q^{13} +72.5080 q^{17} -78.3905 q^{19} +(24.6122 - 42.6295i) q^{23} +(7.15835 + 12.3986i) q^{25} +(-69.4036 - 120.211i) q^{29} +(31.2143 - 54.0648i) q^{31} +3.73364 q^{35} +36.0226 q^{37} +(-95.4936 + 165.400i) q^{41} +(-266.190 - 461.054i) q^{43} +(35.3418 + 61.2138i) q^{47} +(171.437 - 296.938i) q^{49} +245.898 q^{53} -90.6318 q^{55} +(-417.859 + 723.752i) q^{59} +(-240.379 - 416.348i) q^{61} +(-182.998 - 316.961i) q^{65} +(-55.2706 + 95.7315i) q^{67} -58.4968 q^{71} -313.784 q^{73} +(1.52863 - 2.64766i) q^{77} +(-497.611 - 861.887i) q^{79} +(-366.646 - 635.049i) q^{83} +(381.414 - 660.629i) q^{85} +994.244 q^{89} +12.3460 q^{91} +(-412.358 + 714.225i) q^{95} +(-529.088 - 916.408i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} - 8 q^{11} - 4 q^{13} + 32 q^{17} + 160 q^{19} - 200 q^{23} - 8 q^{25} - 216 q^{29} - 80 q^{31} + 816 q^{35} - 552 q^{37} - 384 q^{41} - 160 q^{43} - 768 q^{47} + 268 q^{49} + 1888 q^{53} + 608 q^{55} - 992 q^{59} + 548 q^{61} - 1328 q^{65} - 464 q^{67} + 3440 q^{71} - 1528 q^{73} - 1728 q^{77} - 688 q^{79} - 2128 q^{83} + 1324 q^{85} + 4224 q^{89} + 3552 q^{91} - 2056 q^{95} + 1816 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 8 * q^11 - 4 * q^13 + 32 * q^17 + 160 * q^19 - 200 * q^23 - 8 * q^25 - 216 * q^29 - 80 * q^31 + 816 * q^35 - 552 * q^37 - 384 * q^41 - 160 * q^43 - 768 * q^47 + 268 * q^49 + 1888 * q^53 + 608 * q^55 - 992 * q^59 + 548 * q^61 - 1328 * q^65 - 464 * q^67 + 3440 * q^71 - 1528 * q^73 - 1728 * q^77 - 688 * q^79 - 2128 * q^83 + 1324 * q^85 + 4224 * q^89 + 3552 * q^91 - 2056 * q^95 + 1816 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	5.26031	−	9.11112i	0.470496	−	0.814923i	−0.528935	−	0.848663i	$-0.677408\pi$
$5$	5.26031	−	9.11112i	0.470496	−	0.814923i	0.999431	+	0.0337394i	$0.0107416\pi$
$6$		0			0
$7$	0.177444	+	0.307342i	0.00958108	+	0.0165949i	0.870776	−	0.491680i	$-0.163617\pi$
$7$	0.177444	+	0.307342i	0.00958108	+	0.0165949i	−0.861195	+	0.508275i	$0.830284\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.30734	−	7.46054i	−0.118065	−	0.204494i	0.800936	−	0.598750i	$-0.204336\pi$
$11$	−4.30734	−	7.46054i	−0.118065	−	0.204494i	−0.919001	+	0.394256i	$0.871002\pi$
$12$		0			0
$13$	17.3942	−	30.1277i	0.371099	−	0.642762i	−0.618636	−	0.785678i	$-0.712314\pi$
$13$	17.3942	−	30.1277i	0.371099	−	0.642762i	0.989735	+	0.142916i	$0.0456477\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	72.5080			1.03446			0.517228	−	0.855847i	$-0.326964\pi$
$17$	72.5080			1.03446			0.517228	+	0.855847i	$0.326964\pi$
$18$		0			0
$19$	−78.3905			−0.946527			−0.473263	−	0.880921i	$-0.656924\pi$
$19$	−78.3905			−0.946527			−0.473263	+	0.880921i	$0.656924\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	24.6122	−	42.6295i	0.223130	−	0.386473i	−0.732627	−	0.680631i	$-0.761706\pi$
$23$	24.6122	−	42.6295i	0.223130	−	0.386473i	0.955757	+	0.294158i	$0.0950392\pi$
$24$		0			0
$25$	7.15835	+	12.3986i	0.0572668	+	0.0991890i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−69.4036	−	120.211i	−0.444411	−	0.769743i	0.553600	−	0.832783i	$-0.313254\pi$
$29$	−69.4036	−	120.211i	−0.444411	−	0.769743i	−0.998011	+	0.0630400i	$0.979920\pi$
$30$		0			0
$31$	31.2143	−	54.0648i	0.180847	−	0.313236i	−0.761322	−	0.648374i	$-0.775449\pi$
$31$	31.2143	−	54.0648i	0.180847	−	0.313236i	0.942169	+	0.335137i	$0.108783\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.73364			0.0180314
$36$		0			0
$37$	36.0226			0.160056			0.0800281	−	0.996793i	$-0.474499\pi$
$37$	36.0226			0.160056			0.0800281	+	0.996793i	$0.474499\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−95.4936	+	165.400i	−0.363746	+	0.630027i	−0.988574	−	0.150736i	$-0.951836\pi$
$41$	−95.4936	+	165.400i	−0.363746	+	0.630027i	0.624828	+	0.780763i	$0.285169\pi$
$42$		0			0
$43$	−266.190	−	461.054i	−0.944035	−	1.63512i	−0.757671	−	0.652636i	$-0.773663\pi$
$43$	−266.190	−	461.054i	−0.944035	−	1.63512i	−0.186364	−	0.982481i	$-0.559670\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	35.3418	+	61.2138i	0.109684	+	0.189978i	0.915642	−	0.401994i	$-0.131683\pi$
$47$	35.3418	+	61.2138i	0.109684	+	0.189978i	−0.805958	+	0.591972i	$0.798350\pi$
$48$		0			0
$49$	171.437	−	296.938i	0.499816	−	0.865707i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	245.898			0.637296			0.318648	−	0.947873i	$-0.396771\pi$
$53$	245.898			0.637296			0.318648	+	0.947873i	$0.396771\pi$
$54$		0			0
$55$	−90.6318			−0.222196
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−417.859	+	723.752i	−0.922043	+	1.59703i	−0.125794	+	0.992056i	$0.540148\pi$
$59$	−417.859	+	723.752i	−0.922043	+	1.59703i	−0.796249	+	0.604969i	$0.793185\pi$
$60$		0			0
$61$	−240.379	−	416.348i	−0.504547	−	0.873900i	−0.999986	−	0.00525798i	$-0.998326\pi$
$61$	−240.379	−	416.348i	−0.504547	−	0.873900i	0.495440	−	0.868642i	$-0.335007\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−182.998	−	316.961i	−0.349201	−	0.604834i
$66$		0			0
$67$	−55.2706	+	95.7315i	−0.100782	+	0.174559i	−0.912007	−	0.410175i	$-0.865468\pi$
$67$	−55.2706	+	95.7315i	−0.100782	+	0.174559i	0.811225	+	0.584734i	$0.198801\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−58.4968			−0.0977787			−0.0488894	−	0.998804i	$-0.515568\pi$
$71$	−58.4968			−0.0977787			−0.0488894	+	0.998804i	$0.515568\pi$
$72$		0			0
$73$	−313.784			−0.503090			−0.251545	−	0.967846i	$-0.580939\pi$
$73$	−313.784			−0.503090			−0.251545	+	0.967846i	$0.580939\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.52863	−	2.64766i	0.00226238	−	0.00391855i
$78$		0			0
$79$	−497.611	−	861.887i	−0.708679	−	1.22747i	−0.965347	−	0.260968i	$-0.915958\pi$
$79$	−497.611	−	861.887i	−0.708679	−	1.22747i	0.256669	−	0.966499i	$-0.417375\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−366.646	−	635.049i	−0.484875	−	0.839828i	0.514974	−	0.857206i	$-0.327802\pi$
$83$	−366.646	−	635.049i	−0.484875	−	0.839828i	−0.999849	+	0.0173777i	$0.994468\pi$
$84$		0			0
$85$	381.414	−	660.629i	0.486708	−	0.843003i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	994.244			1.18415			0.592077	−	0.805881i	$-0.298308\pi$
$89$	994.244			1.18415			0.592077	+	0.805881i	$0.298308\pi$
$90$		0			0
$91$	12.3460			0.0142221
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−412.358	+	714.225i	−0.445337	+	0.771347i
$96$		0			0
$97$	−529.088	−	916.408i	−0.553823	−	0.959249i	−0.997994	−	0.0633070i	$-0.979835\pi$
$97$	−529.088	−	916.408i	−0.553823	−	0.959249i	0.444172	−	0.895942i	$-0.353498\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−304.465	−	527.349i	−0.299955	−	0.519537i	0.676171	−	0.736745i	$-0.263638\pi$
$101$	−304.465	−	527.349i	−0.299955	−	0.519537i	−0.976125	+	0.217208i	$0.930305\pi$
$102$		0			0
$103$	906.306	−	1569.77i	0.867000	−	1.50169i	0.00195270	−	0.999998i	$-0.499378\pi$
$103$	906.306	−	1569.77i	0.867000	−	1.50169i	0.865047	−	0.501690i	$-0.167288\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	707.990			0.639664			0.319832	−	0.947474i	$-0.396374\pi$
$107$	707.990			0.639664			0.319832	+	0.947474i	$0.396374\pi$
$108$		0			0
$109$	−2101.35			−1.84654			−0.923270	−	0.384153i	$-0.874494\pi$
$109$	−2101.35			−1.84654			−0.923270	+	0.384153i	$0.874494\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	682.328	−	1181.83i	0.568035	−	0.983866i	−0.428725	−	0.903435i	$-0.641037\pi$
$113$	682.328	−	1181.83i	0.568035	−	0.983866i	0.996760	−	0.0804308i	$-0.0256296\pi$
$114$		0			0
$115$	−258.935	−	448.489i	−0.209964	−	0.363668i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.8661	+	22.2848i	0.00991122	+	0.0171667i
$120$		0			0
$121$	628.394	−	1088.41i	0.472121	−	0.817738i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1465.70			1.04877
$126$		0			0
$127$	1473.64			1.02964			0.514819	−	0.857299i	$-0.327859\pi$
$127$	1473.64			1.02964			0.514819	+	0.857299i	$0.327859\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−861.853	+	1492.77i	−0.574813	+	0.995605i	0.421249	+	0.906945i	$0.361592\pi$
$131$	−861.853	+	1492.77i	−0.574813	+	0.995605i	−0.996062	+	0.0886598i	$0.971742\pi$
$132$		0			0
$133$	−13.9099	−	24.0927i	−0.00906875	−	0.0157075i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−666.021	−	1153.58i	−0.415343	−	0.719395i	0.580122	−	0.814530i	$-0.303005\pi$
$137$	−666.021	−	1153.58i	−0.415343	−	0.719395i	−0.995464	+	0.0951350i	$0.969672\pi$
$138$		0			0
$139$	−1046.34	+	1812.32i	−0.638488	+	1.10589i	0.347277	+	0.937763i	$0.387106\pi$
$139$	−1046.34	+	1812.32i	−0.638488	+	1.10589i	−0.985765	+	0.168131i	$0.946227\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−299.691			−0.175255
$144$		0			0
$145$	−1460.34			−0.836375
$146$		0			0
$147$		0			0
$148$		0			0
$149$	708.617	−	1227.36i	0.389612	−	0.674828i	−0.602785	−	0.797903i	$-0.705942\pi$
$149$	708.617	−	1227.36i	0.389612	−	0.674828i	0.992397	+	0.123076i	$0.0392758\pi$
$150$		0			0
$151$	−1265.42	−	2191.77i	−0.681976	−	1.18122i	−0.974377	−	0.224921i	$-0.927788\pi$
$151$	−1265.42	−	2191.77i	−0.681976	−	1.18122i	0.292401	−	0.956296i	$-0.405546\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−328.394	−	568.795i	−0.170176	−	0.294753i
$156$		0			0
$157$	804.996	−	1394.29i	0.409208	−	0.708769i	−0.585593	−	0.810605i	$-0.699138\pi$
$157$	804.996	−	1394.29i	0.409208	−	0.708769i	0.994801	+	0.101836i	$0.0324717\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	17.4691			0.00855131
$162$		0			0
$163$	−162.622			−0.0781445			−0.0390722	−	0.999236i	$-0.512440\pi$
$163$	−162.622			−0.0781445			−0.0390722	+	0.999236i	$0.512440\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1879.83	+	3255.97i	−0.871053	+	1.50871i	−0.0101438	+	0.999949i	$0.503229\pi$
$167$	−1879.83	+	3255.97i	−0.871053	+	1.50871i	−0.860909	+	0.508759i	$0.830104\pi$
$168$		0			0
$169$	493.383	+	854.564i	0.224571	+	0.388969i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	119.704	+	207.334i	0.0526067	+	0.0911174i	0.891130	−	0.453749i	$-0.149914\pi$
$173$	119.704	+	207.334i	0.0526067	+	0.0911174i	−0.838523	+	0.544866i	$0.816580\pi$
$174$		0			0
$175$	−2.54041	+	4.40012i	−0.00109736	+	0.00190068i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2212.57			−0.923885			−0.461943	−	0.886910i	$-0.652847\pi$
$179$	−2212.57			−0.923885			−0.461943	+	0.886910i	$0.652847\pi$
$180$		0			0
$181$	1754.76			0.720608			0.360304	−	0.932835i	$-0.382673\pi$
$181$	1754.76			0.720608			0.360304	+	0.932835i	$0.382673\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	189.490	−	328.206i	0.0753058	−	0.130433i
$186$		0			0
$187$	−312.317	−	540.948i	−0.122133	−	0.211540i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2000.98	+	3465.80i	0.758040	+	1.31296i	0.943849	+	0.330378i	$0.107176\pi$
$191$	2000.98	+	3465.80i	0.758040	+	1.31296i	−0.185808	+	0.982586i	$0.559490\pi$
$192$		0			0
$193$	−317.205	+	549.415i	−0.118305	+	0.204911i	−0.919096	−	0.394033i	$-0.871079\pi$
$193$	−317.205	+	549.415i	−0.118305	+	0.204911i	0.800791	+	0.598944i	$0.204413\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	889.895			0.321839			0.160920	−	0.986967i	$-0.448554\pi$
$197$	889.895			0.321839			0.160920	+	0.986967i	$0.448554\pi$
$198$		0			0
$199$	1653.18			0.588900			0.294450	−	0.955667i	$-0.404864\pi$
$199$	1653.18			0.588900			0.294450	+	0.955667i	$0.404864\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.6305	−	42.6613i	0.00851588	−	0.0147499i
$204$		0			0
$205$	1004.65	+	1740.11i	0.342282	+	0.592851i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	337.655	+	584.835i	0.111751	+	0.193559i
$210$		0			0
$211$	−1685.31	+	2919.05i	−0.549866	+	0.952396i	0.448417	+	0.893824i	$0.351988\pi$
$211$	−1685.31	+	2919.05i	−0.549866	+	0.952396i	−0.998283	+	0.0585714i	$0.981345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5600.95			−1.77666
$216$		0			0
$217$	22.1552			0.00693084
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1261.22	−	2184.50i	0.383886	−	0.664910i
$222$		0			0
$223$	655.080	+	1134.63i	0.196715	+	0.340720i	0.947461	−	0.319870i	$-0.103639\pi$
$223$	655.080	+	1134.63i	0.196715	+	0.340720i	−0.750747	+	0.660590i	$0.770306\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−356.934	−	618.228i	−0.104364	−	0.180763i	0.809114	−	0.587651i	$-0.199947\pi$
$227$	−356.934	−	618.228i	−0.104364	−	0.180763i	−0.913478	+	0.406888i	$0.866614\pi$
$228$		0			0
$229$	−671.809	+	1163.61i	−0.193862	+	0.335779i	−0.946527	−	0.322625i	$-0.895435\pi$
$229$	−671.809	+	1163.61i	−0.193862	+	0.335779i	0.752665	+	0.658404i	$0.228768\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3886.80			1.09284			0.546422	−	0.837510i	$-0.315989\pi$
$233$	3886.80			1.09284			0.546422	+	0.837510i	$0.315989\pi$
$234$		0			0
$235$	743.635			0.206423
$236$		0			0
$237$		0			0
$238$		0			0
$239$	484.583	−	839.323i	0.131151	−	0.227160i	−0.792970	−	0.609261i	$-0.791466\pi$
$239$	484.583	−	839.323i	0.131151	−	0.227160i	0.924121	+	0.382101i	$0.124799\pi$
$240$		0			0
$241$	2985.89	+	5171.72i	0.798084	+	1.38232i	0.920862	+	0.389888i	$0.127486\pi$
$241$	2985.89	+	5171.72i	0.798084	+	1.38232i	−0.122779	+	0.992434i	$0.539181\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1803.62	−	3123.97i	−0.470323	−	0.814624i
$246$		0			0
$247$	−1363.54	+	2361.72i	−0.351255	+	0.608392i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−605.752			−0.152330			−0.0761648	−	0.997095i	$-0.524268\pi$
$251$	−605.752			−0.152330			−0.0761648	+	0.997095i	$0.524268\pi$
$252$		0			0
$253$	−424.052			−0.105375
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2988.71	−	5176.60i	0.725411	−	1.25645i	−0.233393	−	0.972382i	$-0.574983\pi$
$257$	2988.71	−	5176.60i	0.725411	−	1.25645i	0.958805	−	0.284067i	$-0.0916837\pi$
$258$		0			0
$259$	6.39200	+	11.0713i	0.00153351	+	0.00265612i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3429.70	+	5940.42i	0.804124	+	1.39278i	0.916881	+	0.399160i	$0.130698\pi$
$263$	3429.70	+	5940.42i	0.804124	+	1.39278i	−0.112758	+	0.993623i	$0.535968\pi$
$264$		0			0
$265$	1293.50	−	2240.41i	0.299845	−	0.519347i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1166.26			−0.264342			−0.132171	−	0.991227i	$-0.542195\pi$
$269$	−1166.26			−0.264342			−0.132171	+	0.991227i	$0.542195\pi$
$270$		0			0
$271$	−1286.99			−0.288483			−0.144241	−	0.989543i	$-0.546074\pi$
$271$	−1286.99			−0.288483			−0.144241	+	0.989543i	$0.546074\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	61.6669	−	106.810i	0.0135224	−	0.0234214i
$276$		0			0
$277$	3816.32	+	6610.06i	0.827800	+	1.43379i	0.899761	+	0.436384i	$0.143741\pi$
$277$	3816.32	+	6610.06i	0.827800	+	1.43379i	−0.0719611	+	0.997407i	$0.522926\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1532.19	+	2653.83i	0.325277	+	0.563397i	0.981568	−	0.191111i	$-0.0612091\pi$
$281$	1532.19	+	2653.83i	0.325277	+	0.563397i	−0.656291	+	0.754508i	$0.727876\pi$
$282$		0			0
$283$	−2119.31	+	3670.76i	−0.445159	+	0.771039i	−0.998063	−	0.0622063i	$-0.980186\pi$
$283$	−2119.31	+	3670.76i	−0.445159	+	0.771039i	0.552904	+	0.833245i	$0.313520\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−67.7791			−0.0139403
$288$		0			0
$289$	344.406			0.0701011
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2780.18	−	4815.41i	0.554333	−	0.960133i	−0.443622	−	0.896214i	$-0.646307\pi$
$293$	2780.18	−	4815.41i	0.554333	−	0.960133i	0.997955	−	0.0639191i	$-0.0203600\pi$
$294$		0			0
$295$	4396.13	+	7614.32i	0.867635	+	1.50279i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−856.219	−	1483.01i	−0.165607	−	0.286839i
$300$		0			0
$301$	94.4675	−	163.623i	0.0180898	−	0.0313324i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5057.86			−0.949549
$306$		0			0
$307$	6287.70			1.16892			0.584459	−	0.811423i	$-0.301307\pi$
$307$	6287.70			1.16892			0.584459	+	0.811423i	$0.301307\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1228.74	−	2128.25i	0.224038	−	0.388045i	−0.731993	−	0.681313i	$-0.761409\pi$
$311$	1228.74	−	2128.25i	0.224038	−	0.388045i	0.956030	+	0.293268i	$0.0947428\pi$
$312$		0			0
$313$	803.340	+	1391.42i	0.145072	+	0.251272i	0.929400	−	0.369075i	$-0.120325\pi$
$313$	803.340	+	1391.42i	0.145072	+	0.251272i	−0.784328	+	0.620346i	$0.786992\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2298.89	+	3981.80i	0.407314	+	0.705489i	0.994588	−	0.103900i	$-0.0331320\pi$
$317$	2298.89	+	3981.80i	0.407314	+	0.705489i	−0.587274	+	0.809389i	$0.699799\pi$
$318$		0			0
$319$	−597.890	+	1035.58i	−0.104939	+	0.181759i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5683.93			−0.979141
$324$		0			0
$325$	498.055			0.0850065
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.5424	+	21.7241i	−0.00210178	+	0.00364039i
$330$		0			0
$331$	−474.670	−	822.152i	−0.0788224	−	0.136524i	0.823920	−	0.566707i	$-0.191783\pi$
$331$	−474.670	−	822.152i	−0.0788224	−	0.136524i	−0.902742	+	0.430182i	$0.858449\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	581.481	+	1007.15i	0.0948349	+	0.164259i
$336$		0			0
$337$	−3750.27	+	6495.65i	−0.606202	+	1.04997i	0.385658	+	0.922642i	$0.373974\pi$
$337$	−3750.27	+	6495.65i	−0.606202	+	1.04997i	−0.991860	+	0.127331i	$0.959359\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−537.803			−0.0854067
$342$		0			0
$343$	243.409			0.0383173
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1134.36	−	1964.78i	0.175493	−	0.303962i	−0.764839	−	0.644221i	$-0.777182\pi$
$347$	1134.36	−	1964.78i	0.175493	−	0.303962i	0.940332	+	0.340259i	$0.110515\pi$
$348$		0			0
$349$	−1871.54	−	3241.60i	−0.287052	−	0.497189i	0.686053	−	0.727552i	$-0.259342\pi$
$349$	−1871.54	−	3241.60i	−0.287052	−	0.497189i	−0.973105	+	0.230363i	$0.926009\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1165.52	−	2018.74i	−0.175735	−	0.304381i	0.764681	−	0.644409i	$-0.222897\pi$
$353$	−1165.52	−	2018.74i	−0.175735	−	0.304381i	−0.940415	+	0.340028i	$0.889563\pi$
$354$		0			0
$355$	−307.711	+	532.971i	−0.0460045	+	0.0796821i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7364.86			1.08274			0.541368	−	0.840785i	$-0.317906\pi$
$359$	7364.86			1.08274			0.541368	+	0.840785i	$0.317906\pi$
$360$		0			0
$361$	−713.935			−0.104087
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1650.60	+	2858.92i	−0.236702	+	0.409980i
$366$		0			0
$367$	3191.08	+	5527.11i	0.453877	+	0.786138i	0.998623	−	0.0524634i	$-0.0167073\pi$
$367$	3191.08	+	5527.11i	0.453877	+	0.786138i	−0.544746	+	0.838601i	$0.683374\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	43.6331	+	75.5748i	0.00610598	+	0.0105759i
$372$		0			0
$373$	−4399.77	+	7620.62i	−0.610754	+	1.05786i	0.380359	+	0.924839i	$0.375800\pi$
$373$	−4399.77	+	7620.62i	−0.610754	+	1.05786i	−0.991114	+	0.133019i	$0.957533\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4828.88			−0.659682
$378$		0			0
$379$	8742.11			1.18483			0.592417	−	0.805631i	$-0.298174\pi$
$379$	8742.11			1.18483			0.592417	+	0.805631i	$0.298174\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7199.35	+	12469.6i	−0.960495	+	1.66363i	−0.239234	+	0.970962i	$0.576896\pi$
$383$	−7199.35	+	12469.6i	−0.960495	+	1.66363i	−0.721261	+	0.692664i	$0.756437\pi$
$384$		0			0
$385$	−16.0821	−	27.8550i	−0.00212888	−	0.00368733i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5122.02	+	8871.59i	0.667600	+	1.15632i	0.978573	+	0.205899i	$0.0660119\pi$
$389$	5122.02	+	8871.59i	0.667600	+	1.15632i	−0.310973	+	0.950419i	$0.600655\pi$
$390$		0			0
$391$	1784.58	−	3090.98i	0.230819	−	0.399789i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−10470.3			−1.33372
$396$		0			0
$397$	−3791.18			−0.479279			−0.239639	−	0.970862i	$-0.577029\pi$
$397$	−3791.18			−0.479279			−0.239639	+	0.970862i	$0.577029\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−684.394	+	1185.41i	−0.0852295	+	0.147622i	−0.905489	−	0.424370i	$-0.860496\pi$
$401$	−684.394	+	1185.41i	−0.0852295	+	0.147622i	0.820260	+	0.571992i	$0.193829\pi$
$402$		0			0
$403$	−1085.90	−	1880.83i	−0.134224	−	0.232483i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−155.162	−	268.748i	−0.0188970	−	0.0327306i
$408$		0			0
$409$	677.589	−	1173.62i	0.0819184	−	0.141887i	−0.822156	−	0.569263i	$-0.807229\pi$
$409$	677.589	−	1173.62i	0.0819184	−	0.141887i	0.904074	+	0.427376i	$0.140562\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−296.586			−0.0353367
$414$		0			0
$415$	−7714.68			−0.912527
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7056.01	−	12221.4i	0.822694	−	1.42495i	−0.0809752	−	0.996716i	$-0.525803\pi$
$419$	7056.01	−	12221.4i	0.822694	−	1.42495i	0.903669	−	0.428231i	$-0.140863\pi$
$420$		0			0
$421$	121.566	+	210.559i	0.0140731	+	0.0243753i	0.872976	−	0.487763i	$-0.162187\pi$
$421$	121.566	+	210.559i	0.0140731	+	0.0243753i	−0.858903	+	0.512138i	$0.828854\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	519.037	+	898.999i	0.0592400	+	0.102607i
$426$		0			0
$427$	85.3076	−	147.757i	0.00966821	−	0.0167458i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6214.92			0.694575			0.347288	−	0.937759i	$-0.387103\pi$
$431$	6214.92			0.694575			0.347288	+	0.937759i	$0.387103\pi$
$432$		0			0
$433$	7922.06			0.879238			0.439619	−	0.898184i	$-0.355114\pi$
$433$	7922.06			0.879238			0.439619	+	0.898184i	$0.355114\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1929.36	+	3341.75i	−0.211199	+	0.365807i
$438$		0			0
$439$	−5.05559	−	8.75654i	−0.000549636	−	0.000951997i	0.865750	−	0.500476i	$-0.166842\pi$
$439$	−5.05559	−	8.75654i	−0.000549636	−	0.000951997i	−0.866300	+	0.499524i	$0.833508\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4314.13	−	7472.29i	−0.462687	−	0.801398i	0.536406	−	0.843960i	$-0.319781\pi$
$443$	−4314.13	−	7472.29i	−0.462687	−	0.801398i	−0.999094	+	0.0425616i	$0.986448\pi$
$444$		0			0
$445$	5230.03	−	9058.68i	0.557140	−	0.964995i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3543.11			0.372405			0.186202	−	0.982511i	$-0.440382\pi$
$449$	3543.11			0.372405			0.186202	+	0.982511i	$0.440382\pi$
$450$		0			0
$451$	1645.29			0.171782
$452$		0			0
$453$		0			0
$454$		0			0
$455$	64.9438	−	112.486i	0.00669145	−	0.0115899i
$456$		0			0
$457$	−7333.85	−	12702.6i	−0.750685	−	1.30022i	−0.947491	−	0.319782i	$-0.896390\pi$
$457$	−7333.85	−	12702.6i	−0.750685	−	1.30022i	0.196806	−	0.980442i	$-0.436943\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7288.07	−	12623.3i	−0.736311	−	1.27533i	−0.954146	−	0.299342i	$-0.903233\pi$
$461$	−7288.07	−	12623.3i	−0.736311	−	1.27533i	0.217835	−	0.975986i	$-0.430101\pi$
$462$		0			0
$463$	2538.59	−	4396.97i	0.254813	−	0.441349i	−0.710032	−	0.704170i	$-0.751319\pi$
$463$	2538.59	−	4396.97i	0.254813	−	0.441349i	0.964845	+	0.262821i	$0.0846527\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12469.1			1.23555			0.617774	−	0.786356i	$-0.288035\pi$
$467$	12469.1			1.23555			0.617774	+	0.786356i	$0.288035\pi$
$468$		0			0
$469$	−39.2298			−0.00386240
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2293.14	+	3971.83i	−0.222915	+	0.386099i
$474$		0			0
$475$	−561.146	−	971.934i	−0.0542045	−	0.0938850i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1965.53	+	3404.39i	0.187489	+	0.324740i	0.944412	−	0.328763i	$-0.106632\pi$
$479$	1965.53	+	3404.39i	0.187489	+	0.324740i	−0.756923	+	0.653504i	$0.773298\pi$
$480$		0			0
$481$	626.584	−	1085.28i	0.0593967	−	0.102878i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11132.7			−1.04229
$486$		0			0
$487$	−20710.1			−1.92703			−0.963516	−	0.267652i	$-0.913752\pi$
$487$	−20710.1			−1.92703			−0.963516	+	0.267652i	$0.913752\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6248.38	−	10822.5i	0.574308	−	0.994731i	−0.421808	−	0.906685i	$-0.638604\pi$
$491$	6248.38	−	10822.5i	0.574308	−	0.994731i	0.996116	−	0.0880459i	$-0.0280622\pi$
$492$		0			0
$493$	−5032.32	−	8716.23i	−0.459724	−	0.796266i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.3799	−	17.9785i	−0.000936826	−	0.00162263i
$498$		0			0
$499$	5077.47	−	8794.44i	0.455509	−	0.788965i	−0.543208	−	0.839598i	$-0.682791\pi$
$499$	5077.47	−	8794.44i	0.455509	−	0.788965i	0.998717	+	0.0506334i	$0.0161240\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20309.2			1.80029			0.900143	−	0.435594i	$-0.143462\pi$
$503$	20309.2			1.80029			0.900143	+	0.435594i	$0.143462\pi$
$504$		0			0
$505$	−6406.32			−0.564510
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7097.29	+	12292.9i	−0.618039	+	1.07047i	0.371805	+	0.928311i	$0.378739\pi$
$509$	−7097.29	+	12292.9i	−0.618039	+	1.07047i	−0.989843	+	0.142163i	$0.954594\pi$
$510$		0			0
$511$	−55.6790	−	96.4389i	−0.00482015	−	0.00834874i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9534.90	−	16514.9i	−0.815840	−	1.41308i
$516$		0			0
$517$	304.459	−	527.338i	0.0258996	−	0.0448594i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10644.1			0.895057			0.447528	−	0.894270i	$-0.352304\pi$
$521$	10644.1			0.895057			0.447528	+	0.894270i	$0.352304\pi$
$522$		0			0
$523$	−2761.82			−0.230910			−0.115455	−	0.993313i	$-0.536833\pi$
$523$	−2761.82			−0.230910			−0.115455	+	0.993313i	$0.536833\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2263.29	−	3920.13i	0.187079	−	0.324029i
$528$		0			0
$529$	4871.98	+	8438.52i	0.400426	+	0.693558i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3322.07	+	5754.00i	0.269972	+	0.467605i
$534$		0			0
$535$	3724.25	−	6450.58i	0.300959	−	0.521277i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2953.75			−0.236043
$540$		0			0
$541$	13261.6			1.05390			0.526952	−	0.849895i	$-0.323335\pi$
$541$	13261.6			1.05390			0.526952	+	0.849895i	$0.323335\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11053.7	+	19145.6i	−0.868790	+	1.50479i
$546$		0			0
$547$	−10223.8	−	17708.1i	−0.799155	−	1.38418i	−0.920167	−	0.391526i	$-0.871947\pi$
$547$	−10223.8	−	17708.1i	−0.799155	−	1.38418i	0.121012	−	0.992651i	$-0.461386\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5440.58	+	9423.36i	0.420647	+	0.728582i
$552$		0			0
$553$	176.596	−	305.874i	0.0135798	−	0.0235209i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13888.2			1.05649			0.528243	−	0.849093i	$-0.322851\pi$
$557$	13888.2			1.05649			0.528243	+	0.849093i	$0.322851\pi$
$558$		0			0
$559$	−18520.6			−1.40132
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8439.11	−	14617.0i	0.631734	−	1.09419i	−0.355464	−	0.934690i	$-0.615677\pi$
$563$	8439.11	−	14617.0i	0.631734	−	1.09419i	0.987197	−	0.159505i	$-0.0509897\pi$
$564$		0			0
$565$	−7178.51	−	12433.5i	−0.534517	−	0.925810i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5926.87	−	10265.6i	−0.436674	−	0.756341i	0.560757	−	0.827981i	$-0.310510\pi$
$569$	−5926.87	−	10265.6i	−0.436674	−	0.756341i	−0.997431	+	0.0716394i	$0.977177\pi$
$570$		0			0
$571$	7006.38	−	12135.4i	0.513499	−	0.889406i	−0.486378	−	0.873748i	$-0.661682\pi$
$571$	7006.38	−	12135.4i	0.513499	−	0.889406i	0.999877	−	0.0156581i	$-0.00498433\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	704.730			0.0511118
$576$		0			0
$577$	2886.05			0.208228			0.104114	−	0.994565i	$-0.466799\pi$
$577$	2886.05			0.208228			0.104114	+	0.994565i	$0.466799\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	130.118	−	225.372i	0.00929125	−	0.0160929i
$582$		0			0
$583$	−1059.17	−	1834.53i	−0.0752422	−	0.130323i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7020.02	−	12159.0i	−0.493607	−	0.854952i	0.506366	−	0.862318i	$-0.330988\pi$
$587$	−7020.02	−	12159.0i	−0.493607	−	0.854952i	−0.999973	+	0.00736684i	$0.997655\pi$
$588$		0			0
$589$	−2446.91	+	4238.17i	−0.171177	+	0.296487i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18394.8			−1.27383			−0.636917	−	0.770932i	$-0.719791\pi$
$593$	−18394.8			−1.27383			−0.636917	+	0.770932i	$0.719791\pi$
$594$		0			0
$595$	270.719			0.0186528
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−14098.8	+	24419.8i	−0.961704	+	1.66572i	−0.243485	+	0.969905i	$0.578291\pi$
$599$	−14098.8	+	24419.8i	−0.961704	+	1.66572i	−0.718220	+	0.695816i	$0.755043\pi$
$600$		0			0
$601$	−13271.1	−	22986.2i	−0.900731	−	1.56011i	−0.826547	−	0.562868i	$-0.809698\pi$
$601$	−13271.1	−	22986.2i	−0.900731	−	1.56011i	−0.0741848	−	0.997245i	$-0.523635\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6611.09	−	11450.7i	−0.444263	−	0.769485i
$606$		0			0
$607$	5446.04	−	9432.82i	0.364165	−	0.630752i	−0.624477	−	0.781043i	$-0.714688\pi$
$607$	5446.04	−	9432.82i	0.364165	−	0.630752i	0.988642	+	0.150291i	$0.0480212\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2458.97			0.162814
$612$		0			0
$613$	6843.92			0.450935			0.225467	−	0.974251i	$-0.427609\pi$
$613$	6843.92			0.450935			0.225467	+	0.974251i	$0.427609\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−8209.71	+	14219.6i	−0.535673	+	0.927813i	0.463457	+	0.886119i	$0.346609\pi$
$617$	−8209.71	+	14219.6i	−0.535673	+	0.927813i	−0.999130	+	0.0416941i	$0.986725\pi$
$618$		0			0
$619$	−7033.72	−	12182.8i	−0.456719	−	0.791061i	0.542066	−	0.840336i	$-0.317642\pi$
$619$	−7033.72	−	12182.8i	−0.456719	−	0.791061i	−0.998785	+	0.0492751i	$0.984309\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	176.423	+	305.573i	0.0113455	+	0.0196509i
$624$		0			0
$625$	6815.22	−	11804.3i	0.436174	−	0.755476i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2611.92			0.165571
$630$		0			0
$631$	91.1166			0.00574848			0.00287424	−	0.999996i	$-0.499085\pi$
$631$	91.1166			0.00574848			0.00287424	+	0.999996i	$0.499085\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7751.78	−	13426.5i	0.484441	−	0.839077i
$636$		0			0
$637$	−5964.02	−	10330.0i	−0.370963	−	0.642526i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12314.9	+	21330.1i	0.758831	+	1.31433i	0.943447	+	0.331524i	$0.107563\pi$
$641$	12314.9	+	21330.1i	0.758831	+	1.31433i	−0.184616	+	0.982811i	$0.559104\pi$
$642$		0			0
$643$	−3033.79	+	5254.67i	−0.186067	+	0.322277i	−0.943935	−	0.330130i	$-0.892907\pi$
$643$	−3033.79	+	5254.67i	−0.186067	+	0.322277i	0.757869	+	0.652407i	$0.226241\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−890.884			−0.0541334			−0.0270667	−	0.999634i	$-0.508617\pi$
$647$	−890.884			−0.0541334			−0.0270667	+	0.999634i	$0.508617\pi$
$648$		0			0
$649$	7199.44			0.435443
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13850.5	+	23989.9i	−0.830036	+	1.43767i	0.0679720	+	0.997687i	$0.478347\pi$
$653$	−13850.5	+	23989.9i	−0.830036	+	1.43767i	−0.898008	+	0.439978i	$0.854986\pi$
$654$		0			0
$655$	9067.22	+	15704.9i	0.540894	+	0.936856i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9789.10	−	16955.2i	−0.578648	−	1.00225i	−0.995635	−	0.0933353i	$-0.970247\pi$
$659$	−9789.10	−	16955.2i	−0.578648	−	1.00225i	0.416987	−	0.908913i	$-0.363086\pi$
$660$		0			0
$661$	−5504.14	+	9533.45i	−0.323882	+	0.560981i	−0.981286	−	0.192558i	$-0.938321\pi$
$661$	−5504.14	+	9533.45i	−0.323882	+	0.560981i	0.657403	+	0.753539i	$0.271655\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−292.682			−0.0170672
$666$		0			0
$667$	−6832.70			−0.396646
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2070.79	+	3586.71i	−0.119138	+	0.206354i
$672$		0			0
$673$	−2246.03	−	3890.24i	−0.128645	−	0.222820i	0.794507	−	0.607255i	$-0.207729\pi$
$673$	−2246.03	−	3890.24i	−0.128645	−	0.222820i	−0.923152	+	0.384435i	$0.874396\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3036.02	+	5258.54i	0.172354	+	0.298526i	0.939242	−	0.343254i	$-0.111529\pi$
$677$	3036.02	+	5258.54i	0.172354	+	0.298526i	−0.766888	+	0.641781i	$0.778196\pi$
$678$		0			0
$679$	187.767	−	325.222i	0.0106124	−	0.0183813i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20709.0			1.16018			0.580092	−	0.814551i	$-0.303016\pi$
$683$	20709.0			1.16018			0.580092	+	0.814551i	$0.303016\pi$
$684$		0			0
$685$	−14013.9			−0.781669
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4277.20	−	7408.33i	0.236500	−	0.409630i
$690$		0			0
$691$	−3558.80	−	6164.03i	−0.195924	−	0.339350i	0.751279	−	0.659984i	$-0.229437\pi$
$691$	−3558.80	−	6164.03i	−0.195924	−	0.339350i	−0.947203	+	0.320635i	$0.896104\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11008.2	+	19066.7i	0.600812	+	1.04064i
$696$		0			0
$697$	−6924.05	+	11992.8i	−0.376280	+	0.651736i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−26295.0			−1.41676			−0.708379	−	0.705833i	$-0.750573\pi$
$701$	−26295.0			−1.41676			−0.708379	+	0.705833i	$0.750573\pi$
$702$		0			0
$703$	−2823.83			−0.151497
$704$		0			0
$705$		0			0
$706$		0			0
$707$	108.051	−	187.150i	0.00574778	−	0.00995545i
$708$		0			0
$709$	4364.83	+	7560.11i	0.231206	+	0.400460i	0.958163	−	0.286223i	$-0.0923997\pi$
$709$	4364.83	+	7560.11i	0.231206	+	0.400460i	−0.726958	+	0.686682i	$0.759066\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1536.51	−	2661.31i	−0.0807049	−	0.139785i
$714$		0			0
$715$	−1576.47	+	2730.52i	−0.0824567	+	0.142819i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32340.7			1.67747			0.838736	−	0.544538i	$-0.183295\pi$
$719$	32340.7			1.67747			0.838736	+	0.544538i	$0.183295\pi$
$720$		0			0
$721$	643.275			0.0332272
$722$		0			0
$723$		0			0
$724$		0			0
$725$	993.630	−	1721.02i	0.0509000	−	0.0881614i
$726$		0			0
$727$	13152.9	+	22781.5i	0.670997	+	1.16220i	0.977622	+	0.210371i	$0.0674671\pi$
$727$	13152.9	+	22781.5i	0.670997	+	1.16220i	−0.306624	+	0.951831i	$0.599200\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−19300.9	−	33430.1i	−0.976564	−	1.69146i
$732$		0			0
$733$	8186.73	−	14179.8i	0.412529	−	0.714521i	−0.582636	−	0.812733i	$-0.697979\pi$
$733$	8186.73	−	14179.8i	0.412529	−	0.714521i	0.995166	+	0.0982115i	$0.0313122\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	952.278			0.0475951
$738$		0			0
$739$	17567.9			0.874486			0.437243	−	0.899343i	$-0.355955\pi$
$739$	17567.9			0.874486			0.437243	+	0.899343i	$0.355955\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11410.7	−	19763.9i	0.563416	−	0.975865i	−0.433779	−	0.901019i	$-0.642820\pi$
$743$	11410.7	−	19763.9i	0.563416	−	0.975865i	0.997195	−	0.0748455i	$-0.0238464\pi$
$744$		0			0
$745$	−7455.09	−	12912.6i	−0.366622	−	0.635008i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	125.629	+	217.595i	0.00612867	+	0.0106152i
$750$		0			0
$751$	17748.5	−	30741.4i	0.862388	−	1.49370i	−0.00722941	−	0.999974i	$-0.502301\pi$
$751$	17748.5	−	30741.4i	0.862388	−	1.49370i	0.869617	−	0.493726i	$-0.164365\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−26626.0			−1.28347
$756$		0			0
$757$	−2676.83			−0.128522			−0.0642609	−	0.997933i	$-0.520469\pi$
$757$	−2676.83			−0.128522			−0.0642609	+	0.997933i	$0.520469\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1523.77	−	2639.24i	0.0725841	−	0.125719i	−0.827449	−	0.561541i	$-0.810209\pi$
$761$	1523.77	−	2639.24i	0.0725841	−	0.125719i	0.900033	+	0.435821i	$0.143542\pi$
$762$		0			0
$763$	−372.872	−	645.833i	−0.0176918	−	0.0306432i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14536.6	+	25178.2i	0.684338	+	1.18531i
$768$		0			0
$769$	−10296.2	+	17833.5i	−0.482823	+	0.836273i	−0.999805	−	0.0197225i	$-0.993722\pi$
$769$	−10296.2	+	17833.5i	−0.482823	+	0.836273i	0.516983	+	0.855996i	$0.327055\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3252.88			0.151356			0.0756779	−	0.997132i	$-0.475888\pi$
$773$	3252.88			0.151356			0.0756779	+	0.997132i	$0.475888\pi$
$774$		0			0
$775$	893.772			0.0414261
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7485.79	−	12965.8i	0.344296	−	0.596337i
$780$		0			0
$781$	251.966	+	436.417i	0.0115442	+	0.0199952i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8469.05	−	14668.8i	−0.385062	−	0.666946i
$786$		0			0
$787$	−19168.1	+	33200.2i	−0.868196	+	1.50376i	−0.00435673	+	0.999991i	$0.501387\pi$
$787$	−19168.1	+	33200.2i	−0.868196	+	1.50376i	−0.863839	+	0.503768i	$0.831947\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	484.300			0.0217696
$792$		0			0
$793$	−16724.8			−0.748947
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12578.9	−	21787.3i	0.559056	−	0.968313i	−0.438520	−	0.898721i	$-0.644497\pi$
$797$	12578.9	−	21787.3i	0.559056	−	0.968313i	0.997576	−	0.0695913i	$-0.0221695\pi$
$798$		0			0
$799$	2562.56	+	4438.49i	0.113463	+	0.196524i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1351.57	+	2340.99i	0.0593972	+	0.102879i
$804$		0			0
$805$	91.8931	−	159.163i	0.00402336	−	0.00696866i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−159.016			−0.00691064			−0.00345532	−	0.999994i	$-0.501100\pi$
$809$	−159.016			−0.00691064			−0.00345532	+	0.999994i	$0.501100\pi$
$810$		0			0
$811$	16146.0			0.699093			0.349547	−	0.936919i	$-0.386336\pi$
$811$	16146.0			0.699093			0.349547	+	0.936919i	$0.386336\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−855.443	+	1481.67i	−0.0367667	+	0.0636818i
$816$		0			0
$817$	20866.7	+	36142.2i	0.893555	+	1.54768i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16363.8	+	28343.0i	0.695616	+	1.20484i	0.969972	+	0.243215i	$0.0782020\pi$
$821$	16363.8	+	28343.0i	0.695616	+	1.20484i	−0.274356	+	0.961628i	$0.588465\pi$
$822$		0			0
$823$	16419.2	−	28438.9i	0.695428	−	1.20452i	−0.274608	−	0.961556i	$-0.588548\pi$
$823$	16419.2	−	28438.9i	0.695428	−	1.20452i	0.970036	−	0.242960i	$-0.0781185\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4725.60			−0.198700			−0.0993502	−	0.995053i	$-0.531676\pi$
$827$	−4725.60			−0.198700			−0.0993502	+	0.995053i	$0.531676\pi$
$828$		0			0
$829$	2080.19			0.0871510			0.0435755	−	0.999050i	$-0.486125\pi$
$829$	2080.19			0.0871510			0.0435755	+	0.999050i	$0.486125\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12430.6	−	21530.3i	0.517039	−	0.895537i
$834$		0			0
$835$	19777.0	+	34254.8i	0.819654	+	1.41968i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−164.372	−	284.701i	−0.00676371	−	0.0117151i	0.862624	−	0.505846i	$-0.168820\pi$
$839$	−164.372	−	284.701i	−0.00676371	−	0.0117151i	−0.869387	+	0.494131i	$0.835486\pi$
$840$		0			0
$841$	2560.78	−	4435.39i	0.104997	−	0.181860i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10381.4			0.422640
$846$		0			0
$847$	446.019			0.0180937
$848$		0			0
$849$		0			0
$850$		0			0
$851$	886.594	−	1535.63i	0.0357133	−	0.0618573i
$852$		0			0
$853$	15356.3	+	26597.9i	0.616400	+	1.06764i	0.990137	+	0.140101i	$0.0447428\pi$
$853$	15356.3	+	26597.9i	0.616400	+	1.06764i	−0.373738	+	0.927535i	$0.621924\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5484.54	+	9499.51i	0.218610	+	0.378643i	0.954383	−	0.298585i	$-0.0965146\pi$
$857$	5484.54	+	9499.51i	0.218610	+	0.378643i	−0.735774	+	0.677228i	$0.763181\pi$
$858$		0			0
$859$	−19632.9	+	34005.2i	−0.779821	+	1.35069i	0.152224	+	0.988346i	$0.451356\pi$
$859$	−19632.9	+	34005.2i	−0.779821	+	1.35069i	−0.932045	+	0.362343i	$0.881977\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−23715.3			−0.935433			−0.467716	−	0.883879i	$-0.654923\pi$
$863$	−23715.3			−0.935433			−0.467716	+	0.883879i	$0.654923\pi$
$864$		0			0
$865$	2518.73			0.0990050
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4286.76	+	7424.89i	−0.167340	+	0.289841i
$870$		0			0
$871$	1922.78	+	3330.35i	0.0748000	+	0.129557i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	260.079	+	450.471i	0.0100483	+	0.0174042i
$876$		0			0
$877$	−14343.5	+	24843.6i	−0.552275	+	0.956568i	0.445835	+	0.895115i	$0.352907\pi$
$877$	−14343.5	+	24843.6i	−0.552275	+	0.956568i	−0.998110	+	0.0614526i	$0.980427\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27975.8			−1.06984			−0.534920	−	0.844903i	$-0.679658\pi$
$881$	−27975.8			−1.06984			−0.534920	+	0.844903i	$0.679658\pi$
$882$		0			0
$883$	4610.87			0.175728			0.0878642	−	0.996132i	$-0.471996\pi$
$883$	4610.87			0.175728			0.0878642	+	0.996132i	$0.471996\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12831.9	+	22225.5i	−0.485741	+	0.841329i	−0.999866	−	0.0163868i	$-0.994784\pi$
$887$	−12831.9	+	22225.5i	−0.485741	+	0.841329i	0.514124	+	0.857716i	$0.328117\pi$
$888$		0			0
$889$	261.488	+	452.911i	0.00986506	+	0.0170868i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2770.46	−	4798.58i	−0.103819	−	0.179819i
$894$		0			0
$895$	−11638.8	+	20159.0i	−0.434685	+	0.752896i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8665.55			−0.321482
$900$		0			0
$901$	17829.6			0.659255
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9230.55	−	15987.8i	0.339043	−	0.587240i
$906$		0			0
$907$	−4666.86	−	8083.24i	−0.170850	−	0.295920i	0.767868	−	0.640609i	$-0.221318\pi$
$907$	−4666.86	−	8083.24i	−0.170850	−	0.295920i	−0.938717	+	0.344688i	$0.887985\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24069.0	+	41688.8i	0.875348	+	1.51615i	0.856392	+	0.516327i	$0.172701\pi$
$911$	24069.0	+	41688.8i	0.875348	+	1.51615i	0.0189566	+	0.999820i	$0.493966\pi$
$912$		0			0
$913$	−3158.54	+	5470.75i	−0.114493	+	0.198308i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−611.723			−0.0220293
$918$		0			0
$919$	−32300.1			−1.15939			−0.579696	−	0.814833i	$-0.696829\pi$
$919$	−32300.1			−1.15939			−0.579696	+	0.814833i	$0.696829\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1017.51	+	1762.37i	−0.0362856	+	0.0628485i
$924$		0			0
$925$	257.862	+	446.630i	0.00916590	+	0.0158758i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4506.77	+	7805.96i	0.159163	+	0.275678i	0.934567	−	0.355787i	$-0.115787\pi$
$929$	4506.77	+	7805.96i	0.159163	+	0.275678i	−0.775404	+	0.631465i	$0.782454\pi$
$930$		0			0
$931$	−13439.0	+	23277.1i	−0.473090	+	0.819415i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6571.53			−0.229852
$936$		0			0
$937$	47418.3			1.65324			0.826622	−	0.562758i	$-0.190260\pi$
$937$	47418.3			1.65324			0.826622	+	0.562758i	$0.190260\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10946.8	−	18960.4i	0.379229	−	0.656844i	−0.611721	−	0.791073i	$-0.709523\pi$
$941$	10946.8	−	18960.4i	0.379229	−	0.656844i	0.990950	+	0.134229i	$0.0428559\pi$
$942$		0			0
$943$	4700.61	+	8141.70i	0.162326	+	0.281156i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−22034.7	−	38165.3i	−0.756107	−	1.30962i	−0.944822	−	0.327583i	$-0.893766\pi$
$947$	−22034.7	−	38165.3i	−0.756107	−	1.30962i	0.188716	−	0.982032i	$-0.439568\pi$
$948$		0			0
$949$	−5458.02	+	9453.56i	−0.186696	+	0.323367i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	22429.1			0.762382			0.381191	−	0.924496i	$-0.375514\pi$
$953$	22429.1			0.762382			0.381191	+	0.924496i	$0.375514\pi$
$954$		0			0
$955$	42103.0			1.42662
$956$		0			0
$957$		0			0
$958$		0			0
$959$	236.363	−	409.393i	0.00795887	−	0.0137852i
$960$		0			0
$961$	12946.8	+	22424.6i	0.434589	+	0.752730i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3337.19	+	5780.18i	0.111324	+	0.192819i
$966$		0			0
$967$	2638.70	−	4570.36i	0.0877507	−	0.151989i	−0.818809	−	0.574066i	$-0.805365\pi$
$967$	2638.70	−	4570.36i	0.0877507	−	0.151989i	0.906560	+	0.422077i	$0.138699\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−53211.8			−1.75865			−0.879325	−	0.476223i	$-0.842005\pi$
$971$	−53211.8			−1.75865			−0.879325	+	0.476223i	$0.842005\pi$
$972$		0			0
$973$	−742.671			−0.0244696
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20046.8	+	34722.1i	−0.656453	+	1.13701i	0.325074	+	0.945689i	$0.394611\pi$
$977$	−20046.8	+	34722.1i	−0.656453	+	1.13701i	−0.981527	+	0.191322i	$0.938723\pi$
$978$		0			0
$979$	−4282.55	−	7417.60i	−0.139807	−	0.242153i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25123.7	+	43515.6i	0.815180	+	1.41193i	0.909198	+	0.416363i	$0.136696\pi$
$983$	25123.7	+	43515.6i	0.815180	+	1.41193i	−0.0940181	+	0.995570i	$0.529971\pi$
$984$		0			0
$985$	4681.12	−	8107.94i	0.151424	−	0.262274i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−26206.0			−0.842571
$990$		0			0
$991$	32333.6			1.03644			0.518219	−	0.855248i	$-0.326595\pi$
$991$	32333.6			1.03644			0.518219	+	0.855248i	$0.326595\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8696.26	−	15062.4i	0.277075	−	0.479909i
$996$		0			0
$997$	10619.5	+	18393.6i	0.337336	+	0.584283i	0.983931	−	0.178551i	$-0.0571408\pi$
$997$	10619.5	+	18393.6i	0.337336	+	0.584283i	−0.646595	+	0.762834i	$0.723808\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.u.433.4		8
3.2	odd	2		648.4.i.v.433.1		8
9.2	odd	6		648.4.i.v.217.1		8
9.4	even	3		648.4.a.j.1.1	yes	4
9.5	odd	6		648.4.a.g.1.4	✓	4
9.7	even	3	inner	648.4.i.u.217.4		8
36.23	even	6		1296.4.a.x.1.4		4
36.31	odd	6		1296.4.a.bb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.4.a.g.1.4	✓	4	9.5	odd	6
648.4.a.j.1.1	yes	4	9.4	even	3
648.4.i.u.217.4		8	9.7	even	3	inner
648.4.i.u.433.4		8	1.1	even	1	trivial
648.4.i.v.217.1		8	9.2	odd	6
648.4.i.v.433.1		8	3.2	odd	2
1296.4.a.x.1.4		4	36.23	even	6
1296.4.a.bb.1.1		4	36.31	odd	6

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.26031 - 9.11112i) q^{5} +(0.177444 + 0.307342i) q^{7} +O(q^{10})\)	\(q+(5.26031 - 9.11112i) q^{5} +(0.177444 + 0.307342i) q^{7} +(-4.30734 - 7.46054i) q^{11} +(17.3942 - 30.1277i) q^{13} +72.5080 q^{17} -78.3905 q^{19} +(24.6122 - 42.6295i) q^{23} +(7.15835 + 12.3986i) q^{25} +(-69.4036 - 120.211i) q^{29} +(31.2143 - 54.0648i) q^{31} +3.73364 q^{35} +36.0226 q^{37} +(-95.4936 + 165.400i) q^{41} +(-266.190 - 461.054i) q^{43} +(35.3418 + 61.2138i) q^{47} +(171.437 - 296.938i) q^{49} +245.898 q^{53} -90.6318 q^{55} +(-417.859 + 723.752i) q^{59} +(-240.379 - 416.348i) q^{61} +(-182.998 - 316.961i) q^{65} +(-55.2706 + 95.7315i) q^{67} -58.4968 q^{71} -313.784 q^{73} +(1.52863 - 2.64766i) q^{77} +(-497.611 - 861.887i) q^{79} +(-366.646 - 635.049i) q^{83} +(381.414 - 660.629i) q^{85} +994.244 q^{89} +12.3460 q^{91} +(-412.358 + 714.225i) q^{95} +(-529.088 - 916.408i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} - 8 q^{11} - 4 q^{13} + 32 q^{17} + 160 q^{19} - 200 q^{23} - 8 q^{25} - 216 q^{29} - 80 q^{31} + 816 q^{35} - 552 q^{37} - 384 q^{41} - 160 q^{43} - 768 q^{47} + 268 q^{49} + 1888 q^{53} + 608 q^{55} - 992 q^{59} + 548 q^{61} - 1328 q^{65} - 464 q^{67} + 3440 q^{71} - 1528 q^{73} - 1728 q^{77} - 688 q^{79} - 2128 q^{83} + 1324 q^{85} + 4224 q^{89} + 3552 q^{91} - 2056 q^{95} + 1816 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 8 * q^11 - 4 * q^13 + 32 * q^17 + 160 * q^19 - 200 * q^23 - 8 * q^25 - 216 * q^29 - 80 * q^31 + 816 * q^35 - 552 * q^37 - 384 * q^41 - 160 * q^43 - 768 * q^47 + 268 * q^49 + 1888 * q^53 + 608 * q^55 - 992 * q^59 + 548 * q^61 - 1328 * q^65 - 464 * q^67 + 3440 * q^71 - 1528 * q^73 - 1728 * q^77 - 688 * q^79 - 2128 * q^83 + 1324 * q^85 + 4224 * q^89 + 3552 * q^91 - 2056 * q^95 + 1816 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more