LMFDB - Embedded newform 324.8.e.m.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,8,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	324.e (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:	$101.212748257$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3205 x^{14} + 7140274 x^{12} + 8220484645 x^{10} + 6820694102626 x^{8} + \cdots + 57\!\cdots\!00 $ x^16 + 3205x^14 + 7140274x^12 + 8220484645x^10 + 6820694102626x^8 + 2359067894603905x^6 + 587831501492949925x^4 + 68841856420333385500*x^2 + 5743303187209972810000
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{44} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.4
Root			$-19.2574 + 33.3549i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.8.e.m.217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.58929 + 11.4130i) q^{5} +(801.604 + 1388.42i) q^{7} +O(q^{10})$	$q+(-6.58929 + 11.4130i) q^{5} +(801.604 + 1388.42i) q^{7} +(3472.85 + 6015.16i) q^{11} +(-2098.03 + 3633.89i) q^{13} -31739.7 q^{17} -43478.0 q^{19} +(-56384.0 + 97659.9i) q^{23} +(38975.7 + 67507.8i) q^{25} +(5963.00 + 10328.2i) q^{29} +(126979. - 219934. i) q^{31} -21128.0 q^{35} +210306. q^{37} +(71570.1 - 123963. i) q^{41} +(-199683. - 345860. i) q^{43} +(-169785. - 294076. i) q^{47} +(-873367. + 1.51272e6i) q^{49} +719247. q^{53} -91534.5 q^{55} +(1.38205e6 - 2.39379e6i) q^{59} +(-186249. - 322592. i) q^{61} +(-27649.0 - 47889.5i) q^{65} +(1.18165e6 - 2.04668e6i) q^{67} -3.76399e6 q^{71} +2.80218e6 q^{73} +(-5.56771e6 + 9.64355e6i) q^{77} +(137299. + 237809. i) q^{79} +(-2.28487e6 - 3.95751e6i) q^{83} +(209142. - 362245. i) q^{85} -5.11950e6 q^{89} -6.72715e6 q^{91} +(286489. - 496213. i) q^{95} +(-2.30586e6 - 3.99386e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 560 q^{7}+O(q^{10}) $ 16 * q + 560 * q^7	$ 16 q + 560 q^{7} - 1480 q^{13} - 55264 q^{19} - 77936 q^{25} + 247424 q^{31} + 220400 q^{37} - 897040 q^{43} - 1329672 q^{49} + 1910880 q^{55} - 494968 q^{61} - 4698160 q^{67} + 21452240 q^{73} - 8887312 q^{79} - 15973992 q^{85} + 30743008 q^{91} - 33664240 q^{97}+O(q^{100}) $ 16 * q + 560 * q^7 - 1480 * q^13 - 55264 * q^19 - 77936 * q^25 + 247424 * q^31 + 220400 * q^37 - 897040 * q^43 - 1329672 * q^49 + 1910880 * q^55 - 494968 * q^61 - 4698160 * q^67 + 21452240 * q^73 - 8887312 * q^79 - 15973992 * q^85 + 30743008 * q^91 - 33664240 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−6.58929	+	11.4130i	−0.0235746	+	0.0408323i	−0.877572	−	0.479445i	$-0.840838\pi$
$5$	−6.58929	+	11.4130i	−0.0235746	+	0.0408323i	0.853997	+	0.520277i	$0.174171\pi$
$6$		0			0
$7$	801.604	+	1388.42i	0.883317	+	1.52995i	0.847630	+	0.530588i	$0.178029\pi$
$7$	801.604	+	1388.42i	0.883317	+	1.52995i	0.0356875	+	0.999363i	$0.488638\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3472.85	+	6015.16i	0.786705	+	1.36261i	0.927975	+	0.372642i	$0.121548\pi$
$11$	3472.85	+	6015.16i	0.786705	+	1.36261i	−0.141270	+	0.989971i	$0.545119\pi$
$12$		0			0
$13$	−2098.03	+	3633.89i	−0.264856	+	0.458743i	−0.967526	−	0.252772i	$-0.918658\pi$
$13$	−2098.03	+	3633.89i	−0.264856	+	0.458743i	0.702670	+	0.711516i	$0.251991\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−31739.7			−1.56687			−0.783433	−	0.621476i	$-0.786533\pi$
$17$	−31739.7			−1.56687			−0.783433	+	0.621476i	$0.786533\pi$
$18$		0			0
$19$	−43478.0			−1.45422			−0.727112	−	0.686518i	$-0.759138\pi$
$19$	−43478.0			−1.45422			−0.727112	+	0.686518i	$0.759138\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−56384.0	+	97659.9i	−0.966292	+	1.67367i	−0.260187	+	0.965558i	$0.583784\pi$
$23$	−56384.0	+	97659.9i	−0.966292	+	1.67367i	−0.706104	+	0.708108i	$0.749549\pi$
$24$		0			0
$25$	38975.7	+	67507.8i	0.498888	+	0.864100i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5963.00	+	10328.2i	0.0454017	+	0.0786380i	0.887833	−	0.460165i	$-0.152210\pi$
$29$	5963.00	+	10328.2i	0.0454017	+	0.0786380i	−0.842432	+	0.538803i	$0.818877\pi$
$30$		0			0
$31$	126979.	−	219934.i	0.765535	−	1.32595i	−0.174428	−	0.984670i	$-0.555808\pi$
$31$	126979.	−	219934.i	0.765535	−	1.32595i	0.939963	−	0.341276i	$-0.110859\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−21128.0			−0.0832953
$36$		0			0
$37$	210306.			0.682567			0.341283	−	0.939960i	$-0.389138\pi$
$37$	210306.			0.682567			0.341283	+	0.939960i	$0.389138\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	71570.1	−	123963.i	0.162177	−	0.280898i	−0.773472	−	0.633830i	$-0.781482\pi$
$41$	71570.1	−	123963.i	0.162177	−	0.280898i	0.935649	+	0.352932i	$0.114815\pi$
$42$		0			0
$43$	−199683.	−	345860.i	−0.383002	−	0.663378i	0.608488	−	0.793563i	$-0.291776\pi$
$43$	−199683.	−	345860.i	−0.383002	−	0.663378i	−0.991490	+	0.130185i	$0.958443\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−169785.	−	294076.i	−0.238537	−	0.413159i	0.721757	−	0.692146i	$-0.243335\pi$
$47$	−169785.	−	294076.i	−0.238537	−	0.413159i	−0.960295	+	0.278987i	$0.910001\pi$
$48$		0			0
$49$	−873367.	+	1.51272e6i	−1.06050	+	1.83684i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	719247.			0.663610			0.331805	−	0.943348i	$-0.392342\pi$
$53$	719247.			0.663610			0.331805	+	0.943348i	$0.392342\pi$
$54$		0			0
$55$	−91534.5			−0.0741849
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.38205e6	−	2.39379e6i	0.876079	−	1.51741i	0.0204695	−	0.999790i	$-0.493484\pi$
$59$	1.38205e6	−	2.39379e6i	0.876079	−	1.51741i	0.855609	−	0.517622i	$-0.173183\pi$
$60$		0			0
$61$	−186249.	−	322592.i	−0.105060	−	0.181970i	0.808703	−	0.588218i	$-0.200170\pi$
$61$	−186249.	−	322592.i	−0.105060	−	0.181970i	−0.913763	+	0.406248i	$0.866837\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−27649.0	−	47889.5i	−0.0124877	−	0.0216293i
$66$		0			0
$67$	1.18165e6	−	2.04668e6i	0.479986	−	0.831360i	−0.519750	−	0.854318i	$-0.673975\pi$
$67$	1.18165e6	−	2.04668e6i	0.479986	−	0.831360i	0.999736	+	0.0229580i	$0.00730840\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.76399e6			−1.24809			−0.624043	−	0.781390i	$-0.714511\pi$
$71$	−3.76399e6			−1.24809			−0.624043	+	0.781390i	$0.714511\pi$
$72$		0			0
$73$	2.80218e6			0.843073			0.421537	−	0.906811i	$-0.361491\pi$
$73$	2.80218e6			0.843073			0.421537	+	0.906811i	$0.361491\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.56771e6	+	9.64355e6i	−1.38982	+	2.40724i
$78$		0			0
$79$	137299.	+	237809.i	0.0313310	+	0.0542668i	0.881266	−	0.472621i	$-0.156692\pi$
$79$	137299.	+	237809.i	0.0313310	+	0.0542668i	−0.849935	+	0.526888i	$0.823359\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.28487e6	−	3.95751e6i	−0.438620	−	0.759712i	0.558963	−	0.829192i	$-0.311199\pi$
$83$	−2.28487e6	−	3.95751e6i	−0.438620	−	0.759712i	−0.997583	+	0.0694801i	$0.977866\pi$
$84$		0			0
$85$	209142.	−	362245.i	0.0369382	−	0.0639788i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.11950e6			−0.769773			−0.384886	−	0.922964i	$-0.625759\pi$
$89$	−5.11950e6			−0.769773			−0.384886	+	0.922964i	$0.625759\pi$
$90$		0			0
$91$	−6.72715e6			−0.935806
$92$		0			0
$93$		0			0
$94$		0			0
$95$	286489.	−	496213.i	0.0342827	−	0.0593794i
$96$		0			0
$97$	−2.30586e6	−	3.99386e6i	−0.256526	−	0.444316i	0.708783	−	0.705427i	$-0.249245\pi$
$97$	−2.30586e6	−	3.99386e6i	−0.256526	−	0.444316i	−0.965309	+	0.261110i	$0.915911\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.47532e6	+	7.75149e6i	0.432215	+	0.748618i	0.997064	−	0.0765763i	$-0.0243989\pi$
$101$	4.47532e6	+	7.75149e6i	0.432215	+	0.748618i	−0.564849	+	0.825194i	$0.691066\pi$
$102$		0			0
$103$	−6.13855e6	+	1.06323e7i	−0.553523	+	0.958730i	0.444494	+	0.895782i	$0.353384\pi$
$103$	−6.13855e6	+	1.06323e7i	−0.553523	+	0.958730i	−0.998017	+	0.0629479i	$0.979950\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.91165e7			−1.50857			−0.754285	−	0.656547i	$-0.772016\pi$
$107$	−1.91165e7			−1.50857			−0.754285	+	0.656547i	$0.772016\pi$
$108$		0			0
$109$	1.78953e7			1.32356			0.661782	−	0.749696i	$-0.269800\pi$
$109$	1.78953e7			1.32356			0.661782	+	0.749696i	$0.269800\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.26379e6	−	1.43133e7i	0.538772	−	0.933180i	−0.460199	−	0.887816i	$-0.652222\pi$
$113$	8.26379e6	−	1.43133e7i	0.538772	−	0.933180i	0.998971	−	0.0453644i	$-0.0144449\pi$
$114$		0			0
$115$	−743060.	−	1.28702e6i	−0.0455598	−	0.0789119i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.54427e7	−	4.40680e7i	−1.38404	−	2.39723i
$120$		0			0
$121$	−1.43778e7	+	2.49031e7i	−0.737809	+	1.27792i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.05686e6			−0.0941934
$126$		0			0
$127$	3.32673e7			1.44113			0.720567	−	0.693385i	$-0.243881\pi$
$127$	3.32673e7			1.44113			0.720567	+	0.693385i	$0.243881\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.69086e6	+	1.33210e7i	−0.298900	+	0.517709i	−0.975884	−	0.218288i	$-0.929953\pi$
$131$	−7.69086e6	+	1.33210e7i	−0.298900	+	0.517709i	0.676985	+	0.735997i	$0.263286\pi$
$132$		0			0
$133$	−3.48521e7	−	6.03656e7i	−1.28454	−	2.22489i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.54717e6	−	1.13400e7i	−0.217536	−	0.376784i	0.736518	−	0.676418i	$-0.236469\pi$
$137$	−6.54717e6	−	1.13400e7i	−0.217536	−	0.376784i	−0.954054	+	0.299634i	$0.903135\pi$
$138$		0			0
$139$	7.43895e6	−	1.28846e7i	0.234942	−	0.406931i	−0.724314	−	0.689470i	$-0.757843\pi$
$139$	7.43895e6	−	1.28846e7i	0.234942	−	0.406931i	0.959256	+	0.282539i	$0.0911768\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.91445e7			−0.833453
$144$		0			0
$145$	−157168.			−0.00428130
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.64037e7	+	4.57325e7i	−0.653902	+	1.13259i	0.328266	+	0.944585i	$0.393536\pi$
$149$	−2.64037e7	+	4.57325e7i	−0.653902	+	1.13259i	−0.982168	+	0.188006i	$0.939797\pi$
$150$		0			0
$151$	1.23026e7	+	2.13087e7i	0.290789	+	0.503661i	0.973996	−	0.226563i	$-0.0727490\pi$
$151$	1.23026e7	+	2.13087e7i	0.290789	+	0.503661i	−0.683208	+	0.730224i	$0.739416\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.67340e6	+	2.89841e6i	0.0360943	+	0.0625172i
$156$		0			0
$157$	−1.27011e7	+	2.19989e7i	−0.261934	+	0.453682i	−0.966756	−	0.255702i	$-0.917693\pi$
$157$	−1.27011e7	+	2.19989e7i	−0.261934	+	0.453682i	0.704822	+	0.709384i	$0.251027\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.80790e8			−3.41417
$162$		0			0
$163$	3.65162e7			0.660433			0.330216	−	0.943905i	$-0.392878\pi$
$163$	3.65162e7			0.660433			0.330216	+	0.943905i	$0.392878\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.92488e7	+	5.06604e7i	−0.485960	+	0.841707i	−0.999870	−	0.0161373i	$-0.994863\pi$
$167$	−2.92488e7	+	5.06604e7i	−0.485960	+	0.841707i	0.513910	+	0.857844i	$0.328196\pi$
$168$		0			0
$169$	2.25708e7	+	3.90938e7i	0.359703	+	0.623024i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.07947e7	−	8.79790e7i	−0.745859	−	1.29187i	−0.949792	−	0.312881i	$-0.898706\pi$
$173$	−5.07947e7	−	8.79790e7i	−0.745859	−	1.29187i	0.203933	−	0.978985i	$-0.434628\pi$
$174$		0			0
$175$	−6.24861e7	+	1.08229e8i	−0.881354	+	1.52655i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.18603e8			−1.54564			−0.772822	−	0.634623i	$-0.781156\pi$
$179$	−1.18603e8			−1.54564			−0.772822	+	0.634623i	$0.781156\pi$
$180$		0			0
$181$	−1.27758e7			−0.160145			−0.0800723	−	0.996789i	$-0.525515\pi$
$181$	−1.27758e7			−0.160145			−0.0800723	+	0.996789i	$0.525515\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.38577e6	+	2.40022e6i	−0.0160912	+	0.0278708i
$186$		0			0
$187$	−1.10227e8	−	1.90919e8i	−1.23266	−	2.13503i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.16382e6	+	1.24081e7i	0.0743923	+	0.128851i	0.900822	−	0.434189i	$-0.142965\pi$
$191$	7.16382e6	+	1.24081e7i	0.0743923	+	0.128851i	−0.826430	+	0.563040i	$0.809632\pi$
$192$		0			0
$193$	3.20070e7	−	5.54377e7i	0.320475	−	0.555079i	−0.660111	−	0.751168i	$-0.729491\pi$
$193$	3.20070e7	−	5.54377e7i	0.320475	−	0.555079i	0.980586	+	0.196089i	$0.0628242\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.78002e7			0.538639			0.269319	−	0.963051i	$-0.413201\pi$
$197$	5.78002e7			0.538639			0.269319	+	0.963051i	$0.413201\pi$
$198$		0			0
$199$	5.46950e7			0.491997			0.245998	−	0.969270i	$-0.420884\pi$
$199$	5.46950e7			0.491997			0.245998	+	0.969270i	$0.420884\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.55993e6	+	1.65583e7i	−0.0802082	+	0.138925i
$204$		0			0
$205$	943192.	+	1.63366e6i	0.00764648	+	0.0132441i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.50993e8	−	2.61527e8i	−1.14405	−	1.98155i
$210$		0			0
$211$	−4.78918e7	+	8.29510e7i	−0.350972	+	0.607901i	−0.986420	−	0.164243i	$-0.947482\pi$
$211$	−4.78918e7	+	8.29510e7i	−0.350972	+	0.607901i	0.635448	+	0.772144i	$0.280815\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.26306e6			0.0361164
$216$		0			0
$217$	4.07147e8			2.70484
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.65908e7	−	1.15339e8i	0.414993	−	0.718789i
$222$		0			0
$223$	−6.33462e7	−	1.09719e8i	−0.382520	−	0.662543i	0.608902	−	0.793245i	$-0.291610\pi$
$223$	−6.33462e7	−	1.09719e8i	−0.382520	−	0.662543i	−0.991422	+	0.130702i	$0.958277\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.57438e7	+	7.92305e7i	0.259562	+	0.449575i	0.966125	−	0.258076i	$-0.0830884\pi$
$227$	4.57438e7	+	7.92305e7i	0.259562	+	0.449575i	−0.706563	+	0.707651i	$0.749755\pi$
$228$		0			0
$229$	−1.23628e8	+	2.14131e8i	−0.680290	+	1.17830i	0.294603	+	0.955620i	$0.404813\pi$
$229$	−1.23628e8	+	2.14131e8i	−0.680290	+	1.17830i	−0.974892	+	0.222677i	$0.928521\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.92258e8			−0.995723			−0.497862	−	0.867256i	$-0.665881\pi$
$233$	−1.92258e8			−0.995723			−0.497862	+	0.867256i	$0.665881\pi$
$234$		0			0
$235$	4.47504e6			0.0224936
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.43028e7	+	5.94143e7i	−0.162531	+	0.281513i	−0.935776	−	0.352595i	$-0.885299\pi$
$239$	−3.43028e7	+	5.94143e7i	−0.162531	+	0.281513i	0.773244	+	0.634108i	$0.218633\pi$
$240$		0			0
$241$	−2.31631e7	−	4.01196e7i	−0.106595	−	0.184628i	0.807794	−	0.589465i	$-0.200661\pi$
$241$	−2.31631e7	−	4.01196e7i	−0.106595	−	0.184628i	−0.914389	+	0.404837i	$0.867328\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.15097e7	−	1.99354e7i	−0.0500016	−	0.0866053i
$246$		0			0
$247$	9.12179e7	−	1.57994e8i	0.385160	−	0.667116i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.28682e8			−1.31195			−0.655977	−	0.754781i	$-0.727743\pi$
$251$	−3.28682e8			−1.31195			−0.655977	+	0.754781i	$0.727743\pi$
$252$		0			0
$253$	−7.83253e8			−3.04075
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.92934e7	+	6.80581e7i	−0.144395	+	0.250100i	−0.929147	−	0.369710i	$-0.879457\pi$
$257$	−3.92934e7	+	6.80581e7i	−0.144395	+	0.250100i	0.784752	+	0.619810i	$0.212790\pi$
$258$		0			0
$259$	1.68582e8	+	2.91993e8i	0.602923	+	1.04429i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.45451e7	+	7.71544e7i	0.150992	+	0.261526i	0.931593	−	0.363504i	$-0.118420\pi$
$263$	4.45451e7	+	7.71544e7i	0.150992	+	0.261526i	−0.780600	+	0.625031i	$0.785086\pi$
$264$		0			0
$265$	−4.73933e6	+	8.20876e6i	−0.0156443	+	0.0270967i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3.08049e8			0.964909			0.482454	−	0.875921i	$-0.339745\pi$
$269$	3.08049e8			0.964909			0.482454	+	0.875921i	$0.339745\pi$
$270$		0			0
$271$	9.19656e7			0.280694			0.140347	−	0.990102i	$-0.455178\pi$
$271$	9.19656e7			0.280694			0.140347	+	0.990102i	$0.455178\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.70713e8	+	4.68889e8i	−0.784956	+	1.35958i
$276$		0			0
$277$	8.45194e7	+	1.46392e8i	0.238934	+	0.413845i	0.960409	−	0.278595i	$-0.0898688\pi$
$277$	8.45194e7	+	1.46392e8i	0.238934	+	0.413845i	−0.721475	+	0.692441i	$0.756535\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.37236e8	+	4.10905e8i	0.637836	+	1.10476i	0.985907	+	0.167295i	$0.0535034\pi$
$281$	2.37236e8	+	4.10905e8i	0.637836	+	1.10476i	−0.348071	+	0.937468i	$0.613163\pi$
$282$		0			0
$283$	1.20372e8	−	2.08490e8i	0.315698	−	0.546805i	−0.663888	−	0.747832i	$-0.731095\pi$
$283$	1.20372e8	−	2.08490e8i	0.315698	−	0.546805i	0.979586	+	0.201027i	$0.0644280\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.29484e8			0.573014
$288$		0			0
$289$	5.97071e8			1.45507
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.13504e8	−	1.96595e8i	0.263619	−	0.456601i	−0.703582	−	0.710614i	$-0.748417\pi$
$293$	1.13504e8	−	1.96595e8i	0.263619	−	0.456601i	0.967201	+	0.254013i	$0.0817507\pi$
$294$		0			0
$295$	1.82135e7	+	3.15467e7i	0.0413063	+	0.0715447i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.36590e8	−	4.09786e8i	−0.511855	−	0.886560i
$300$		0			0
$301$	3.20133e8	−	5.54486e8i	0.676624	−	1.17195i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.90898e6			0.00990699
$306$		0			0
$307$	−2.36817e8			−0.467119			−0.233560	−	0.972342i	$-0.575037\pi$
$307$	−2.36817e8			−0.467119			−0.233560	+	0.972342i	$0.575037\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.72111e8	−	6.44516e8i	0.701474	−	1.21499i	−0.266475	−	0.963842i	$-0.585859\pi$
$311$	3.72111e8	−	6.44516e8i	0.701474	−	1.21499i	0.967949	−	0.251147i	$-0.0808077\pi$
$312$		0			0
$313$	−3.31059e8	−	5.73411e8i	−0.610240	−	1.05697i	−0.991200	−	0.132375i	$-0.957740\pi$
$313$	−3.31059e8	−	5.73411e8i	−0.610240	−	1.05697i	0.380960	−	0.924591i	$-0.375594\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.53579e8	+	6.12417e8i	0.623418	+	1.07979i	0.988845	+	0.148951i	$0.0475897\pi$
$317$	3.53579e8	+	6.12417e8i	0.623418	+	1.07979i	−0.365427	+	0.930840i	$0.619077\pi$
$318$		0			0
$319$	−4.14172e7	+	7.17368e7i	−0.0714355	+	0.123730i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.37998e9			2.27858
$324$		0			0
$325$	−3.27088e8			−0.528534
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.72200e8	−	4.71465e8i	0.421408	−	0.729901i
$330$		0			0
$331$	4.99065e8	+	8.64405e8i	0.756413	+	1.31015i	0.944669	+	0.328025i	$0.106383\pi$
$331$	4.99065e8	+	8.64405e8i	0.756413	+	1.31015i	−0.188256	+	0.982120i	$0.560284\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.55725e7	+	2.69724e7i	0.0226309	+	0.0391979i
$336$		0			0
$337$	−3.58274e8	+	6.20549e8i	−0.509930	+	0.883225i	0.490003	+	0.871721i	$0.336996\pi$
$337$	−3.58274e8	+	6.20549e8i	−0.509930	+	0.883225i	−0.999934	+	0.0115049i	$0.996338\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.76391e9			2.40900
$342$		0			0
$343$	−1.48007e9			−1.98040
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.19735e8	−	9.00207e8i	0.667772	−	1.15662i	−0.310753	−	0.950491i	$-0.600581\pi$
$347$	5.19735e8	−	9.00207e8i	0.667772	−	1.15662i	0.978526	−	0.206125i	$-0.0660854\pi$
$348$		0			0
$349$	1.27845e8	+	2.21434e8i	0.160989	+	0.278841i	0.935224	−	0.354058i	$-0.115198\pi$
$349$	1.27845e8	+	2.21434e8i	0.160989	+	0.278841i	−0.774235	+	0.632898i	$0.781865\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.73385e8	+	4.73516e8i	0.330798	+	0.572959i	0.982669	−	0.185371i	$-0.0593488\pi$
$353$	2.73385e8	+	4.73516e8i	0.330798	+	0.572959i	−0.651871	+	0.758330i	$0.726015\pi$
$354$		0			0
$355$	2.48020e7	−	4.29584e7i	0.0294231	−	0.0509622i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.23140e8			−0.710813			−0.355407	−	0.934712i	$-0.615658\pi$
$359$	−6.23140e8			−0.710813			−0.355407	+	0.934712i	$0.615658\pi$
$360$		0			0
$361$	9.96461e8			1.11477
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.84643e7	+	3.19812e7i	−0.0198751	+	0.0344246i
$366$		0			0
$367$	4.99355e8	+	8.64908e8i	0.527325	+	0.913353i	0.999493	+	0.0318446i	$0.0101382\pi$
$367$	4.99355e8	+	8.64908e8i	0.527325	+	0.913353i	−0.472168	+	0.881508i	$0.656529\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.76552e8	+	9.98617e8i	0.586178	+	1.01529i
$372$		0			0
$373$	5.64259e8	−	9.77324e8i	0.562986	−	0.975120i	−0.434248	−	0.900793i	$-0.642986\pi$
$373$	5.64259e8	−	9.77324e8i	0.562986	−	0.975120i	0.997234	−	0.0743264i	$-0.0236807\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5.00421e7			−0.0480996
$378$		0			0
$379$	−9.94136e8			−0.938012			−0.469006	−	0.883195i	$-0.655388\pi$
$379$	−9.94136e8			−0.938012			−0.469006	+	0.883195i	$0.655388\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.26334e8	+	1.25805e9i	−0.660603	+	1.14420i	0.319855	+	0.947467i	$0.396366\pi$
$383$	−7.26334e8	+	1.25805e9i	−0.660603	+	1.14420i	−0.980457	+	0.196731i	$0.936967\pi$
$384$		0			0
$385$	−7.33744e7	−	1.27088e8i	−0.0655288	−	0.113499i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.48948e8	+	6.04396e8i	0.300564	+	0.520592i	0.976264	−	0.216585i	$-0.0694918\pi$
$389$	3.48948e8	+	6.04396e8i	0.300564	+	0.520592i	−0.675700	+	0.737177i	$0.736158\pi$
$390$		0			0
$391$	1.78961e9	−	3.09970e9i	1.51405	−	2.62241i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.61882e6			−0.00295445
$396$		0			0
$397$	−6.85058e8			−0.549491			−0.274746	−	0.961517i	$-0.588594\pi$
$397$	−6.85058e8			−0.549491			−0.274746	+	0.961517i	$0.588594\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.70211e8	+	8.14429e8i	−0.364156	+	0.630736i	−0.988640	−	0.150301i	$-0.951976\pi$
$401$	−4.70211e8	+	8.14429e8i	−0.364156	+	0.630736i	0.624485	+	0.781037i	$0.285309\pi$
$402$		0			0
$403$	5.32810e8	+	9.22853e8i	0.405513	+	0.702369i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.30361e8	+	1.26502e9i	0.536979	+	0.930074i
$408$		0			0
$409$	−8.68117e8	+	1.50362e9i	−0.627403	+	1.08669i	0.360668	+	0.932694i	$0.382549\pi$
$409$	−8.68117e8	+	1.50362e9i	−0.627403	+	1.08669i	−0.988071	+	0.153999i	$0.950785\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.43144e9			3.09542
$414$		0			0
$415$	6.02227e7			0.0413611
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.15321e8	+	1.06577e9i	−0.408651	+	0.707804i	−0.994739	−	0.102443i	$-0.967334\pi$
$419$	−6.15321e8	+	1.06577e9i	−0.408651	+	0.707804i	0.586088	+	0.810248i	$0.300667\pi$
$420$		0			0
$421$	6.82570e8	+	1.18225e9i	0.445820	+	0.772184i	0.998109	−	0.0614695i	$-0.0195787\pi$
$421$	6.82570e8	+	1.18225e9i	0.445820	+	0.772184i	−0.552289	+	0.833653i	$0.686245\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.23708e9	−	2.14268e9i	−0.781691	−	1.35393i
$426$		0			0
$427$	2.98595e8	−	5.17182e8i	0.185603	−	0.321474i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.26767e8			−0.437245			−0.218623	−	0.975809i	$-0.570156\pi$
$431$	−7.26767e8			−0.437245			−0.218623	+	0.975809i	$0.570156\pi$
$432$		0			0
$433$	−2.44713e9			−1.44860			−0.724301	−	0.689484i	$-0.757837\pi$
$433$	−2.44713e9			−1.44860			−0.724301	+	0.689484i	$0.757837\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.45146e9	−	4.24605e9i	1.40521	−	2.43389i
$438$		0			0
$439$	−9.82198e8	−	1.70122e9i	−0.554081	−	0.959696i	−0.997974	−	0.0636165i	$-0.979737\pi$
$439$	−9.82198e8	−	1.70122e9i	−0.554081	−	0.959696i	0.443894	−	0.896079i	$-0.353597\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.61317e8	+	7.99025e8i	0.252108	+	0.436664i	0.964106	−	0.265518i	$-0.0855429\pi$
$443$	4.61317e8	+	7.99025e8i	0.252108	+	0.436664i	−0.711998	+	0.702181i	$0.752210\pi$
$444$		0			0
$445$	3.37339e7	−	5.84288e7i	0.0181470	−	0.0314316i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.60127e9			−1.35620			−0.678099	−	0.734970i	$-0.737196\pi$
$449$	−2.60127e9			−1.35620			−0.678099	+	0.734970i	$0.737196\pi$
$450$		0			0
$451$	9.94210e8			0.510341
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.43271e7	−	7.67768e7i	0.0220612	−	0.0382111i
$456$		0			0
$457$	5.96931e8	+	1.03391e9i	0.292562	+	0.506732i	0.974415	−	0.224758i	$-0.0721590\pi$
$457$	5.96931e8	+	1.03391e9i	0.292562	+	0.506732i	−0.681853	+	0.731489i	$0.738826\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.16305e8	+	1.24068e9i	0.340522	+	0.589801i	0.984530	−	0.175218i	$-0.0560630\pi$
$461$	7.16305e8	+	1.24068e9i	0.340522	+	0.589801i	−0.644008	+	0.765019i	$0.722730\pi$
$462$		0			0
$463$	−1.95620e9	+	3.38824e9i	−0.915968	+	1.58650i	−0.110491	+	0.993877i	$0.535242\pi$
$463$	−1.95620e9	+	3.38824e9i	−0.915968	+	1.58650i	−0.805478	+	0.592626i	$0.798091\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.73537e8			0.351457			0.175728	−	0.984439i	$-0.443772\pi$
$467$	7.73537e8			0.351457			0.175728	+	0.984439i	$0.443772\pi$
$468$		0			0
$469$	3.78888e9			1.69592
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.38694e9	−	2.40224e9i	0.602618	−	1.04377i
$474$		0			0
$475$	−1.69458e9	−	2.93510e9i	−0.725496	−	1.25660i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.97145e8	+	5.14669e8i	0.123536	+	0.213971i	0.921160	−	0.389185i	$-0.127243\pi$
$479$	2.97145e8	+	5.14669e8i	0.123536	+	0.213971i	−0.797624	+	0.603155i	$0.793910\pi$
$480$		0			0
$481$	−4.41227e8	+	7.64227e8i	−0.180782	+	0.313123i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.07759e7			0.0241900
$486$		0			0
$487$	2.53444e8			0.0994331			0.0497165	−	0.998763i	$-0.484168\pi$
$487$	2.53444e8			0.0994331			0.0497165	+	0.998763i	$0.484168\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.01104e9	+	1.75117e9i	−0.385462	+	0.667640i	−0.991833	−	0.127542i	$-0.959291\pi$
$491$	−1.01104e9	+	1.75117e9i	−0.385462	+	0.667640i	0.606371	+	0.795182i	$0.292625\pi$
$492$		0			0
$493$	−1.89264e8	−	3.27815e8i	−0.0711384	−	0.123215i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.01723e9	−	5.22600e9i	−1.10246	−	1.90951i
$498$		0			0
$499$	5.63510e8	−	9.76029e8i	0.203025	−	0.351650i	−0.746477	−	0.665412i	$-0.768256\pi$
$499$	5.63510e8	−	9.76029e8i	0.203025	−	0.351650i	0.949502	+	0.313762i	$0.101589\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	2.33239e9			0.817171			0.408585	−	0.912720i	$-0.366022\pi$
$503$	2.33239e9			0.817171			0.408585	+	0.912720i	$0.366022\pi$
$504$		0			0
$505$	−1.17957e8			−0.0407571
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.79595e8	+	1.35030e9i	−0.262033	+	0.453855i	−0.966782	−	0.255602i	$-0.917726\pi$
$509$	−7.79595e8	+	1.35030e9i	−0.262033	+	0.453855i	0.704749	+	0.709457i	$0.251060\pi$
$510$		0			0
$511$	2.24624e9	+	3.89059e9i	0.744701	+	1.28986i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.08974e7	−	1.40118e8i	−0.0260981	−	0.0452033i
$516$		0			0
$517$	1.17928e9	−	2.04256e9i	0.375317	−	0.650068i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.32561e9			−0.410661			−0.205330	−	0.978693i	$-0.565827\pi$
$521$	−1.32561e9			−0.410661			−0.205330	+	0.978693i	$0.565827\pi$
$522$		0			0
$523$	−1.51546e9			−0.463221			−0.231611	−	0.972809i	$-0.574400\pi$
$523$	−1.51546e9			−0.463221			−0.231611	+	0.972809i	$0.574400\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.03027e9	+	6.98064e9i	−1.19949	+	2.07758i
$528$		0			0
$529$	−4.65589e9	−	8.06424e9i	−1.36744	−	2.36847i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.00312e8	+	5.20156e8i	0.0859068	+	0.148795i
$534$		0			0
$535$	1.25964e8	−	2.18176e8i	0.0355639	−	0.0615984i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1.21323e10			−3.33720
$540$		0			0
$541$	2.01912e9			0.548242			0.274121	−	0.961695i	$-0.411613\pi$
$541$	2.01912e9			0.548242			0.274121	+	0.961695i	$0.411613\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.17917e8	+	2.04238e8i	−0.0312025	+	0.0540442i
$546$		0			0
$547$	1.72818e9	+	2.99330e9i	0.451475	+	0.781977i	0.998478	−	0.0551535i	$-0.0175648\pi$
$547$	1.72818e9	+	2.99330e9i	0.451475	+	0.781977i	−0.547003	+	0.837130i	$0.684231\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.59259e8	−	4.49050e8i	−0.0660243	−	0.114357i
$552$		0			0
$553$	−2.20119e8	+	3.81258e8i	−0.0553504	+	0.0958697i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.42067e9			−1.32911			−0.664553	−	0.747241i	$-0.731378\pi$
$557$	−5.42067e9			−1.32911			−0.664553	+	0.747241i	$0.731378\pi$
$558$		0			0
$559$	1.67576e9			0.405760
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.69379e9	+	2.93373e9i	−0.400018	+	0.692852i	−0.993728	−	0.111828i	$-0.964330\pi$
$563$	−1.69379e9	+	2.93373e9i	−0.400018	+	0.692852i	0.593709	+	0.804680i	$0.297663\pi$
$564$		0			0
$565$	1.08905e8	+	1.88629e8i	0.0254026	+	0.0439986i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.93587e9	−	5.08508e9i	−0.668104	−	1.15719i	−0.978434	−	0.206561i	$-0.933773\pi$
$569$	−2.93587e9	−	5.08508e9i	−0.668104	−	1.15719i	0.310330	−	0.950629i	$-0.399561\pi$
$570$		0			0
$571$	−2.15789e9	+	3.73757e9i	−0.485068	+	0.840163i	−0.999853	−	0.0171568i	$-0.994539\pi$
$571$	−2.15789e9	+	3.73757e9i	−0.485068	+	0.840163i	0.514785	+	0.857320i	$0.327872\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.79041e9			−1.92829
$576$		0			0
$577$	1.82938e9			0.396450			0.198225	−	0.980157i	$-0.436482\pi$
$577$	1.82938e9			0.396450			0.198225	+	0.980157i	$0.436482\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.66313e9	−	6.34472e9i	0.774882	−	1.34213i
$582$		0			0
$583$	2.49784e9	+	4.32639e9i	0.522065	+	0.904243i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.23343e8	+	1.07966e9i	0.127202	+	0.220320i	0.922592	−	0.385778i	$-0.126067\pi$
$587$	6.23343e8	+	1.07966e9i	0.127202	+	0.220320i	−0.795390	+	0.606099i	$0.792734\pi$
$588$		0			0
$589$	−5.52078e9	+	9.56227e9i	−1.11326	+	1.92822i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.56586e9			1.68686			0.843431	−	0.537238i	$-0.180532\pi$
$593$	8.56586e9			1.68686			0.843431	+	0.537238i	$0.180532\pi$
$594$		0			0
$595$	6.70597e8			0.130513
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.74716e9	+	3.02617e9i	−0.332154	+	0.575307i	−0.982934	−	0.183959i	$-0.941109\pi$
$599$	−1.74716e9	+	3.02617e9i	−0.332154	+	0.575307i	0.650780	+	0.759266i	$0.274442\pi$
$600$		0			0
$601$	−3.78615e9	−	6.55781e9i	−0.711439	−	1.23225i	−0.964317	−	0.264750i	$-0.914710\pi$
$601$	−3.78615e9	−	6.55781e9i	−0.711439	−	1.23225i	0.252878	−	0.967498i	$-0.418623\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.89479e8	−	3.28188e8i	−0.0347870	−	0.0602529i
$606$		0			0
$607$	8.62910e8	−	1.49460e9i	0.156605	−	0.271248i	−0.777037	−	0.629454i	$-0.783278\pi$
$607$	8.62910e8	−	1.49460e9i	0.156605	−	0.271248i	0.933642	+	0.358207i	$0.116612\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.42485e9			0.252712
$612$		0			0
$613$	4.54370e9			0.796706			0.398353	−	0.917232i	$-0.369582\pi$
$613$	4.54370e9			0.796706			0.398353	+	0.917232i	$0.369582\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.02615e9	−	1.77735e9i	0.175879	−	0.304631i	−0.764586	−	0.644521i	$-0.777057\pi$
$617$	1.02615e9	−	1.77735e9i	0.175879	−	0.304631i	0.940465	+	0.339890i	$0.110390\pi$
$618$		0			0
$619$	−5.55637e9	−	9.62392e9i	−0.941617	−	1.63093i	−0.762387	−	0.647121i	$-0.775973\pi$
$619$	−5.55637e9	−	9.62392e9i	−0.941617	−	1.63093i	−0.179229	−	0.983807i	$-0.557360\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.10381e9	−	7.10801e9i	−0.679954	−	1.17771i
$624$		0			0
$625$	−3.03142e9	+	5.25057e9i	−0.496668	+	0.860254i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.67505e9			−1.06949
$630$		0			0
$631$	−4.39194e9			−0.695910			−0.347955	−	0.937511i	$-0.613124\pi$
$631$	−4.39194e9			−0.695910			−0.347955	+	0.937511i	$0.613124\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.19208e8	+	3.79679e8i	−0.0339741	+	0.0588449i
$636$		0			0
$637$	−3.66469e9	−	6.34743e9i	−0.561758	−	0.972994i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.20464e9	−	2.08649e9i	−0.180656	−	0.312906i	0.761448	−	0.648226i	$-0.224489\pi$
$641$	−1.20464e9	−	2.08649e9i	−0.180656	−	0.312906i	−0.942104	+	0.335320i	$0.891155\pi$
$642$		0			0
$643$	5.53259e9	−	9.58272e9i	0.820710	−	1.42151i	−0.0844440	−	0.996428i	$-0.526911\pi$
$643$	5.53259e9	−	9.58272e9i	0.820710	−	1.42151i	0.905154	−	0.425084i	$-0.139755\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.76306e9			−0.255919			−0.127960	−	0.991779i	$-0.540843\pi$
$647$	−1.76306e9			−0.255919			−0.127960	+	0.991779i	$0.540843\pi$
$648$		0			0
$649$	1.91987e10			2.75686
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.64242e8	−	6.30886e8i	0.0511911	−	0.0886655i	−0.839294	−	0.543677i	$-0.817032\pi$
$653$	3.64242e8	−	6.30886e8i	0.0511911	−	0.0886655i	0.890485	+	0.455012i	$0.150365\pi$
$654$		0			0
$655$	−1.01355e8	−	1.75551e8i	−0.0140929	−	0.0244095i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.92192e9	−	6.79296e9i	−0.533826	−	0.924614i	−0.999219	−	0.0395098i	$-0.987420\pi$
$659$	−3.92192e9	−	6.79296e9i	−0.533826	−	0.924614i	0.465393	−	0.885104i	$-0.345913\pi$
$660$		0			0
$661$	−4.39234e9	+	7.60776e9i	−0.591550	+	1.02459i	0.402474	+	0.915431i	$0.368150\pi$
$661$	−4.39234e9	+	7.60776e9i	−0.591550	+	1.02459i	−0.994024	+	0.109163i	$0.965183\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9.18602e8			0.121130
$666$		0			0
$667$	−1.34487e9			−0.175485
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.29363e9	−	2.24063e9i	0.165303	−	0.286313i
$672$		0			0
$673$	3.17483e9	+	5.49897e9i	0.401484	+	0.695390i	0.993905	−	0.110238i	$-0.0351613\pi$
$673$	3.17483e9	+	5.49897e9i	0.401484	+	0.695390i	−0.592422	+	0.805628i	$0.701828\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.97708e8	−	1.20847e9i	−0.0864199	−	0.149684i	0.819575	−	0.572971i	$-0.194209\pi$
$677$	−6.97708e8	−	1.20847e9i	−0.0864199	−	0.149684i	−0.905995	+	0.423288i	$0.860876\pi$
$678$		0			0
$679$	3.69677e9	−	6.40300e9i	0.453188	−	0.784945i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.46689e9			−1.13693			−0.568466	−	0.822707i	$-0.692463\pi$
$683$	−9.46689e9			−1.13693			−0.568466	+	0.822707i	$0.692463\pi$
$684$		0			0
$685$	1.72565e8			0.0205133
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.50900e9	+	2.61366e9i	−0.175761	+	0.304427i
$690$		0			0
$691$	6.39119e9	+	1.10699e10i	0.736900	+	1.27635i	0.953885	+	0.300174i	$0.0970447\pi$
$691$	6.39119e9	+	1.10699e10i	0.736900	+	1.27635i	−0.216984	+	0.976175i	$0.569622\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9.80348e7	+	1.69801e8i	0.0110773	+	0.0191864i
$696$		0			0
$697$	−2.27162e9	+	3.93455e9i	−0.254109	+	0.440130i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.73834e9			0.958111			0.479056	−	0.877785i	$-0.340979\pi$
$701$	8.73834e9			0.958111			0.479056	+	0.877785i	$0.340979\pi$
$702$		0			0
$703$	−9.14366e9			−0.992605
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.17488e9	+	1.24272e10i	−0.763566	+	1.32253i
$708$		0			0
$709$	2.81745e9	+	4.87997e9i	0.296889	+	0.514227i	0.975423	−	0.220342i	$-0.0707175\pi$
$709$	2.81745e9	+	4.87997e9i	0.296889	+	0.514227i	−0.678533	+	0.734570i	$0.737384\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.43191e10	+	2.48015e10i	1.47946	+	2.56250i
$714$		0			0
$715$	1.92042e8	−	3.32626e8i	0.0196483	−	0.0340318i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.93370e10			−1.94017			−0.970083	−	0.242775i	$-0.921942\pi$
$719$	−1.93370e10			−1.94017			−0.970083	+	0.242775i	$0.921942\pi$
$720$		0			0
$721$	−1.96828e10			−1.95575
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.64824e8	+	8.05099e8i	−0.0453008	+	0.0784632i
$726$		0			0
$727$	−6.45328e9	−	1.11774e10i	−0.622888	−	1.07887i	−0.988945	−	0.148281i	$-0.952626\pi$
$727$	−6.45328e9	−	1.11774e10i	−0.622888	−	1.07887i	0.366057	−	0.930592i	$-0.380707\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.33787e9	+	1.09775e10i	0.600112	+	1.03942i
$732$		0			0
$733$	4.31110e9	−	7.46704e9i	0.404319	−	0.700301i	−0.589923	−	0.807460i	$-0.700842\pi$
$733$	4.31110e9	−	7.46704e9i	0.404319	−	0.700301i	0.994242	+	0.107159i	$0.0341752\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.64148e10			1.51043
$738$		0			0
$739$	−1.86134e10			−1.69656			−0.848281	−	0.529546i	$-0.822362\pi$
$739$	−1.86134e10			−1.69656			−0.848281	+	0.529546i	$0.822362\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.18865e8	+	5.52290e8i	−0.0285197	+	0.0493976i	−0.879933	−	0.475098i	$-0.842413\pi$
$743$	−3.18865e8	+	5.52290e8i	−0.0285197	+	0.0493976i	0.851413	+	0.524495i	$0.175746\pi$
$744$		0			0
$745$	−3.47963e8	−	6.02690e8i	−0.0308309	−	0.0534007i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.53239e10	−	2.65417e10i	−1.33255	−	2.30804i
$750$		0			0
$751$	−2.06054e9	+	3.56896e9i	−0.177518	+	0.307470i	−0.941030	−	0.338324i	$-0.890140\pi$
$751$	−2.06054e9	+	3.56896e9i	−0.177518	+	0.307470i	0.763512	+	0.645794i	$0.223473\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.24261e8			−0.0274209
$756$		0			0
$757$	1.66290e10			1.39325			0.696626	−	0.717434i	$-0.254684\pi$
$757$	1.66290e10			1.39325			0.696626	+	0.717434i	$0.254684\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.93790e9	−	1.20168e10i	0.570665	−	0.988422i	−0.425832	−	0.904802i	$-0.640019\pi$
$761$	6.93790e9	−	1.20168e10i	0.570665	−	0.988422i	0.996498	−	0.0836194i	$-0.0266480\pi$
$762$		0			0
$763$	1.43449e10	+	2.48461e10i	1.16913	+	2.02499i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.79917e9	+	1.00445e10i	0.464069	+	0.803791i
$768$		0			0
$769$	1.45402e9	−	2.51843e9i	0.115299	−	0.199705i	−0.802600	−	0.596518i	$-0.796551\pi$
$769$	1.45402e9	−	2.51843e9i	0.115299	−	0.199705i	0.917899	+	0.396813i	$0.129884\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.15930e10			0.902747			0.451373	−	0.892335i	$-0.350934\pi$
$773$	1.15930e10			0.902747			0.451373	+	0.892335i	$0.350934\pi$
$774$		0			0
$775$	1.97963e10			1.52767
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.11172e9	+	5.38966e9i	−0.235841	+	0.408489i
$780$		0			0
$781$	−1.30718e10	−	2.26410e10i	−0.981875	−	1.70066i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.67382e8	−	2.89914e8i	−0.0123499	−	0.0213907i
$786$		0			0
$787$	3.67357e9	−	6.36281e9i	0.268644	−	0.465304i	−0.699868	−	0.714272i	$-0.746758\pi$
$787$	3.67357e9	−	6.36281e9i	0.268644	−	0.465304i	0.968512	+	0.248968i	$0.0800913\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.64972e10			1.90363
$792$		0			0
$793$	1.56302e9			0.111303
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.00220e10	−	1.73586e10i	0.701215	−	1.21454i	−0.266825	−	0.963745i	$-0.585975\pi$
$797$	1.00220e10	−	1.73586e10i	0.701215	−	1.21454i	0.968040	−	0.250795i	$-0.0806920\pi$
$798$		0			0
$799$	5.38892e9	+	9.33389e9i	0.373756	+	0.647364i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.73154e9	+	1.68555e10i	0.663250	+	1.14878i
$804$		0			0
$805$	1.19128e9	−	2.06336e9i	0.0804875	−	0.139408i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.67527e10			1.11241			0.556207	−	0.831044i	$-0.312256\pi$
$809$	1.67527e10			1.11241			0.556207	+	0.831044i	$0.312256\pi$
$810$		0			0
$811$	3.39682e9			0.223615			0.111807	−	0.993730i	$-0.464336\pi$
$811$	3.39682e9			0.223615			0.111807	+	0.993730i	$0.464336\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.40616e8	+	4.16758e8i	−0.0155694	+	0.0269670i
$816$		0			0
$817$	8.68179e9	+	1.50373e10i	0.556970	+	0.964701i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.51881e9	−	1.64871e10i	−0.600318	−	1.03978i	−0.992773	−	0.120010i	$-0.961707\pi$
$821$	−9.51881e9	−	1.64871e10i	−0.600318	−	1.03978i	0.392454	−	0.919771i	$-0.371626\pi$
$822$		0			0
$823$	1.15285e10	−	1.99680e10i	0.720898	−	1.24863i	−0.239742	−	0.970837i	$-0.577063\pi$
$823$	1.15285e10	−	1.99680e10i	0.720898	−	1.24863i	0.960640	−	0.277796i	$-0.0896039\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.00057e10			1.84474			0.922369	−	0.386309i	$-0.126250\pi$
$827$	3.00057e10			1.84474			0.922369	+	0.386309i	$0.126250\pi$
$828$		0			0
$829$	−1.43064e10			−0.872144			−0.436072	−	0.899912i	$-0.643631\pi$
$829$	−1.43064e10			−0.872144			−0.436072	+	0.899912i	$0.643631\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.77204e10	−	4.80132e10i	1.66166	−	2.87808i
$834$		0			0
$835$	−3.85457e8	−	6.67632e8i	−0.0229126	−	0.0396857i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.72280e9	+	1.51083e10i	0.509905	+	0.883181i	0.999934	+	0.0114748i	$0.00365263\pi$
$839$	8.72280e9	+	1.51083e10i	0.509905	+	0.883181i	−0.490030	+	0.871706i	$0.663014\pi$
$840$		0			0
$841$	8.55382e9	−	1.48157e10i	0.495877	−	0.858885i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.94903e8			−0.0339194
$846$		0			0
$847$	−4.61013e10			−2.60688
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.18579e10	+	2.05384e10i	−0.659558	+	1.14239i
$852$		0			0
$853$	−1.23762e10	−	2.14362e10i	−0.682756	−	1.18257i	−0.974136	−	0.225961i	$-0.927448\pi$
$853$	−1.23762e10	−	2.14362e10i	−0.682756	−	1.18257i	0.291380	−	0.956607i	$-0.405886\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.33437e10	+	2.31120e10i	0.724175	+	1.25431i	0.959313	+	0.282345i	$0.0911124\pi$
$857$	1.33437e10	+	2.31120e10i	0.724175	+	1.25431i	−0.235138	+	0.971962i	$0.575554\pi$
$858$		0			0
$859$	−1.61886e10	+	2.80395e10i	−0.871432	+	1.50936i	−0.0109167	+	0.999940i	$0.503475\pi$
$859$	−1.61886e10	+	2.80395e10i	−0.871432	+	1.50936i	−0.860515	+	0.509424i	$0.829858\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.52760e9			−0.0809042			−0.0404521	−	0.999181i	$-0.512880\pi$
$863$	−1.52760e9			−0.0809042			−0.0404521	+	0.999181i	$0.512880\pi$
$864$		0			0
$865$	1.33880e9			0.0703332
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.53641e8	+	1.65175e9i	−0.0492964	+	0.0853839i
$870$		0			0
$871$	4.95828e9	+	8.58800e9i	0.254254	+	0.440381i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.64879e9	−	2.85579e9i	−0.0832027	−	0.144111i
$876$		0			0
$877$	−9.07656e9	+	1.57211e10i	−0.454384	+	0.787015i	−0.998653	−	0.0518954i	$-0.983474\pi$
$877$	−9.07656e9	+	1.57211e10i	−0.454384	+	0.787015i	0.544269	+	0.838911i	$0.316807\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.17484e9			0.452046			0.226023	−	0.974122i	$-0.427428\pi$
$881$	9.17484e9			0.452046			0.226023	+	0.974122i	$0.427428\pi$
$882$		0			0
$883$	1.23071e10			0.601578			0.300789	−	0.953691i	$-0.402750\pi$
$883$	1.23071e10			0.601578			0.300789	+	0.953691i	$0.402750\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.55099e10	+	2.68639e10i	−0.746234	+	1.29252i	0.203381	+	0.979100i	$0.434807\pi$
$887$	−1.55099e10	+	2.68639e10i	−0.746234	+	1.29252i	−0.949616	+	0.313416i	$0.898526\pi$
$888$		0			0
$889$	2.66672e10	+	4.61890e10i	1.27298	+	2.20487i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.38190e9	+	1.27858e10i	0.346887	+	0.600826i
$894$		0			0
$895$	7.81508e8	−	1.35361e9i	0.0364379	−	0.0631122i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.02870e9			0.139026
$900$		0			0
$901$	−2.28287e10			−1.03979
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.41833e7	−	1.45810e8i	0.00377534	−	0.00653908i
$906$		0			0
$907$	−6.65417e7	−	1.15254e8i	−0.00296120	−	0.00512896i	0.864541	−	0.502562i	$-0.167609\pi$
$907$	−6.65417e7	−	1.15254e8i	−0.00296120	−	0.00512896i	−0.867502	+	0.497433i	$0.834276\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.33539e10	+	2.31297e10i	0.585187	+	1.01357i	0.994852	+	0.101338i	$0.0323124\pi$
$911$	1.33539e10	+	2.31297e10i	0.585187	+	1.01357i	−0.409665	+	0.912236i	$0.634354\pi$
$912$		0			0
$913$	1.58700e10	−	2.74877e10i	0.690129	−	1.19534i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.46601e10			−1.05609
$918$		0			0
$919$	8.37032e9			0.355744			0.177872	−	0.984054i	$-0.443079\pi$
$919$	8.37032e9			0.355744			0.177872	+	0.984054i	$0.443079\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	7.89695e9	−	1.36779e10i	0.330562	−	0.572551i
$924$		0			0
$925$	8.19681e9	+	1.41973e10i	0.340525	+	0.589806i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.66443e8	−	1.67393e9i	−0.0395477	−	0.0684986i	0.845574	−	0.533858i	$-0.179258\pi$
$929$	−9.66443e8	−	1.67393e9i	−0.0395477	−	0.0684986i	−0.885122	+	0.465360i	$0.845925\pi$
$930$		0			0
$931$	3.79722e10	−	6.57698e10i	1.54220	−	2.67118i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.90528e9			0.116238
$936$		0			0
$937$	−2.14070e10			−0.850093			−0.425047	−	0.905171i	$-0.639742\pi$
$937$	−2.14070e10			−0.850093			−0.425047	+	0.905171i	$0.639742\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.88033e10	+	3.25683e10i	−0.735650	+	1.27418i	0.218787	+	0.975773i	$0.429790\pi$
$941$	−1.88033e10	+	3.25683e10i	−0.735650	+	1.27418i	−0.954437	+	0.298411i	$0.903543\pi$
$942$		0			0
$943$	8.07081e9	+	1.39791e10i	0.313420	+	0.542859i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	8.87399e8	+	1.53702e9i	0.0339543	+	0.0588105i	0.882503	−	0.470306i	$-0.155857\pi$
$947$	8.87399e8	+	1.53702e9i	0.0339543	+	0.0588105i	−0.848549	+	0.529117i	$0.822523\pi$
$948$		0			0
$949$	−5.87904e9	+	1.01828e10i	−0.223293	+	0.386754i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	3.18953e10			1.19372			0.596859	−	0.802346i	$-0.296415\pi$
$953$	3.18953e10			1.19372			0.596859	+	0.802346i	$0.296415\pi$
$954$		0			0
$955$	−1.88818e8			−0.00701506
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.04965e10	−	1.81804e10i	0.384307	−	0.665639i
$960$		0			0
$961$	−1.84909e10	−	3.20272e10i	−0.672089	−	1.16409i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.21806e8	+	7.30590e8i	0.0151101	+	0.0261715i
$966$		0			0
$967$	2.43527e9	−	4.21802e9i	0.0866074	−	0.150008i	−0.819468	−	0.573125i	$-0.805731\pi$
$967$	2.43527e9	−	4.21802e9i	0.0866074	−	0.150008i	0.906075	+	0.423117i	$0.139064\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.51483e9			0.0531003			0.0265501	−	0.999647i	$-0.491548\pi$
$971$	1.51483e9			0.0531003			0.0265501	+	0.999647i	$0.491548\pi$
$972$		0			0
$973$	2.38524e10			0.830112
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.54996e9	−	1.65410e10i	0.327620	−	0.567454i	−0.654419	−	0.756132i	$-0.727087\pi$
$977$	9.54996e9	−	1.65410e10i	0.327620	−	0.567454i	0.982039	+	0.188678i	$0.0604201\pi$
$978$		0			0
$979$	−1.77793e10	−	3.07946e10i	−0.605584	−	1.04890i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.88595e10	−	3.26656e10i	−0.633275	−	1.09686i	−0.986878	−	0.161469i	$-0.948377\pi$
$983$	−1.88595e10	−	3.26656e10i	−0.633275	−	1.09686i	0.353602	−	0.935396i	$-0.384957\pi$
$984$		0			0
$985$	−3.80862e8	+	6.59673e8i	−0.0126982	+	0.0219939i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.50356e10			1.48036
$990$		0			0
$991$	−1.03952e10			−0.339292			−0.169646	−	0.985505i	$-0.554262\pi$
$991$	−1.03952e10			−0.339292			−0.169646	+	0.985505i	$0.554262\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.60401e8	+	6.24234e8i	−0.0115986	+	0.0200894i
$996$		0			0
$997$	2.60232e9	+	4.50734e9i	0.0831624	+	0.144042i	0.904607	−	0.426247i	$-0.140165\pi$
$997$	2.60232e9	+	4.50734e9i	0.0831624	+	0.144042i	−0.821444	+	0.570289i	$0.806831\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.8.e.m.109.4		16
3.2	odd	2	inner	324.8.e.m.109.5		16
9.2	odd	6	inner	324.8.e.m.217.5		16
9.4	even	3		324.8.a.e.1.5	yes	8
9.5	odd	6		324.8.a.e.1.4	✓	8
9.7	even	3	inner	324.8.e.m.217.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.8.a.e.1.4	✓	8	9.5	odd	6
324.8.a.e.1.5	yes	8	9.4	even	3
324.8.e.m.109.4		16	1.1	even	1	trivial
324.8.e.m.109.5		16	3.2	odd	2	inner
324.8.e.m.217.4		16	9.7	even	3	inner
324.8.e.m.217.5		16	9.2	odd	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.58929 + 11.4130i) q^{5} +(801.604 + 1388.42i) q^{7} +O(q^{10})\)	\(q+(-6.58929 + 11.4130i) q^{5} +(801.604 + 1388.42i) q^{7} +(3472.85 + 6015.16i) q^{11} +(-2098.03 + 3633.89i) q^{13} -31739.7 q^{17} -43478.0 q^{19} +(-56384.0 + 97659.9i) q^{23} +(38975.7 + 67507.8i) q^{25} +(5963.00 + 10328.2i) q^{29} +(126979. - 219934. i) q^{31} -21128.0 q^{35} +210306. q^{37} +(71570.1 - 123963. i) q^{41} +(-199683. - 345860. i) q^{43} +(-169785. - 294076. i) q^{47} +(-873367. + 1.51272e6i) q^{49} +719247. q^{53} -91534.5 q^{55} +(1.38205e6 - 2.39379e6i) q^{59} +(-186249. - 322592. i) q^{61} +(-27649.0 - 47889.5i) q^{65} +(1.18165e6 - 2.04668e6i) q^{67} -3.76399e6 q^{71} +2.80218e6 q^{73} +(-5.56771e6 + 9.64355e6i) q^{77} +(137299. + 237809. i) q^{79} +(-2.28487e6 - 3.95751e6i) q^{83} +(209142. - 362245. i) q^{85} -5.11950e6 q^{89} -6.72715e6 q^{91} +(286489. - 496213. i) q^{95} +(-2.30586e6 - 3.99386e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 560 q^{7}+O(q^{10}) \) 16 * q + 560 * q^7	\( 16 q + 560 q^{7} - 1480 q^{13} - 55264 q^{19} - 77936 q^{25} + 247424 q^{31} + 220400 q^{37} - 897040 q^{43} - 1329672 q^{49} + 1910880 q^{55} - 494968 q^{61} - 4698160 q^{67} + 21452240 q^{73} - 8887312 q^{79} - 15973992 q^{85} + 30743008 q^{91} - 33664240 q^{97}+O(q^{100}) \) 16 * q + 560 * q^7 - 1480 * q^13 - 55264 * q^19 - 77936 * q^25 + 247424 * q^31 + 220400 * q^37 - 897040 * q^43 - 1329672 * q^49 + 1910880 * q^55 - 494968 * q^61 - 4698160 * q^67 + 21452240 * q^73 - 8887312 * q^79 - 15973992 * q^85 + 30743008 * q^91 - 33664240 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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