LMFDB - Embedded newform 324.8.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,8,Mod(1,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("324.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	324.a (trivial)

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:	yes
Analytic conductor:	$101.212748257$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 1289x^{5} + 4994x^{4} + 496633x^{3} - 2291461x^{2} - 56851263x + 373225328 $ x^7 - 3x^6 - 1289x^5 + 4994x^4 + 496633x^3 - 2291461x^2 - 56851263x + 373225328
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{15} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.55063$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-309.722 q^{5} -84.7846 q^{7} +O(q^{10})$	$q-309.722 q^{5} -84.7846 q^{7} -7751.16 q^{11} +11836.0 q^{13} +8281.86 q^{17} +47232.8 q^{19} +52759.9 q^{23} +17802.8 q^{25} -57297.6 q^{29} -96490.8 q^{31} +26259.7 q^{35} +178612. q^{37} +127229. q^{41} +708886. q^{43} -872717. q^{47} -816355. q^{49} -364628. q^{53} +2.40071e6 q^{55} +2.07234e6 q^{59} -1.10181e6 q^{61} -3.66588e6 q^{65} -3.26107e6 q^{67} +2.68245e6 q^{71} -2.96061e6 q^{73} +657179. q^{77} -5.00187e6 q^{79} -1.73719e6 q^{83} -2.56508e6 q^{85} -9.55486e6 q^{89} -1.00351e6 q^{91} -1.46290e7 q^{95} +1.78520e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) $ 7 * q - 321 * q^5 + 83 * q^7	$ 7 q - 321 q^{5} + 83 q^{7} + 111 q^{11} + 1847 q^{13} - 48 q^{17} + 10124 q^{19} - 19119 q^{23} + 73378 q^{25} - 6045 q^{29} + 153089 q^{31} + 13713 q^{35} + 69674 q^{37} - 446631 q^{41} + 384347 q^{43} - 298413 q^{47} + 351876 q^{49} - 454038 q^{53} - 1263483 q^{55} - 2619543 q^{59} + 146231 q^{61} - 2535735 q^{65} - 1637419 q^{67} - 4353492 q^{71} - 2132260 q^{73} - 9785451 q^{77} - 2402185 q^{79} - 12936357 q^{83} + 1015002 q^{85} - 19684830 q^{89} - 492203 q^{91} - 22685196 q^{95} + 2853257 q^{97}+O(q^{100}) $ 7 * q - 321 * q^5 + 83 * q^7 + 111 * q^11 + 1847 * q^13 - 48 * q^17 + 10124 * q^19 - 19119 * q^23 + 73378 * q^25 - 6045 * q^29 + 153089 * q^31 + 13713 * q^35 + 69674 * q^37 - 446631 * q^41 + 384347 * q^43 - 298413 * q^47 + 351876 * q^49 - 454038 * q^53 - 1263483 * q^55 - 2619543 * q^59 + 146231 * q^61 - 2535735 * q^65 - 1637419 * q^67 - 4353492 * q^71 - 2132260 * q^73 - 9785451 * q^77 - 2402185 * q^79 - 12936357 * q^83 + 1015002 * q^85 - 19684830 * q^89 - 492203 * q^91 - 22685196 * q^95 + 2853257 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−309.722	−1.10810	−0.554048	−	0.832485i	$-0.686917\pi$
$5$	−309.722	−1.10810	−0.554048	+	0.832485i	$0.686917\pi$
$6$	0	0
$7$	−84.7846	−0.0934273	−0.0467136	−	0.998908i	$-0.514875\pi$
$7$	−84.7846	−0.0934273	−0.0467136	+	0.998908i	$0.514875\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−7751.16	−1.75587	−0.877935	−	0.478780i	$-0.841079\pi$
$11$	−7751.16	−1.75587	−0.877935	+	0.478780i	$0.841079\pi$
$12$	0	0
$13$	11836.0	1.49419	0.747093	−	0.664719i	$-0.231449\pi$
$13$	11836.0	1.49419	0.747093	+	0.664719i	$0.231449\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8281.86	0.408843	0.204422	−	0.978883i	$-0.434469\pi$
$17$	8281.86	0.408843	0.204422	+	0.978883i	$0.434469\pi$
$18$	0	0
$19$	47232.8	1.57981	0.789907	−	0.613227i	$-0.210129\pi$
$19$	47232.8	1.57981	0.789907	+	0.613227i	$0.210129\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	52759.9	0.904183	0.452092	−	0.891972i	$-0.350678\pi$
$23$	52759.9	0.904183	0.452092	+	0.891972i	$0.350678\pi$
$24$	0	0
$25$	17802.8	0.227875
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−57297.6	−0.436258	−0.218129	−	0.975920i	$-0.569995\pi$
$29$	−57297.6	−0.436258	−0.218129	+	0.975920i	$0.569995\pi$
$30$	0	0
$31$	−96490.8	−0.581728	−0.290864	−	0.956764i	$-0.593943\pi$
$31$	−96490.8	−0.581728	−0.290864	+	0.956764i	$0.593943\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	26259.7	0.103526
$36$	0	0
$37$	178612.	0.579703	0.289851	−	0.957072i	$-0.406394\pi$
$37$	178612.	0.579703	0.289851	+	0.957072i	$0.406394\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	127229.	0.288299	0.144150	−	0.989556i	$-0.453955\pi$
$41$	127229.	0.288299	0.144150	+	0.989556i	$0.453955\pi$
$42$	0	0
$43$	708886.	1.35968	0.679840	−	0.733360i	$-0.262049\pi$
$43$	708886.	1.35968	0.679840	+	0.733360i	$0.262049\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−872717.	−1.22611	−0.613057	−	0.790038i	$-0.710061\pi$
$47$	−872717.	−1.22611	−0.613057	+	0.790038i	$0.710061\pi$
$48$	0	0
$49$	−816355.	−0.991271
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−364628.	−0.336422	−0.168211	−	0.985751i	$-0.553799\pi$
$53$	−364628.	−0.336422	−0.168211	+	0.985751i	$0.553799\pi$
$54$	0	0
$55$	2.40071e6	1.94567
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.07234e6	1.31365	0.656824	−	0.754044i	$-0.271900\pi$
$59$	2.07234e6	1.31365	0.656824	+	0.754044i	$0.271900\pi$
$60$	0	0
$61$	−1.10181e6	−0.621516	−0.310758	−	0.950489i	$-0.600583\pi$
$61$	−1.10181e6	−0.621516	−0.310758	+	0.950489i	$0.600583\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.66588e6	−1.65570
$66$	0	0
$67$	−3.26107e6	−1.32464	−0.662321	−	0.749220i	$-0.730428\pi$
$67$	−3.26107e6	−1.32464	−0.662321	+	0.749220i	$0.730428\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.68245e6	0.889463	0.444732	−	0.895664i	$-0.353299\pi$
$71$	2.68245e6	0.889463	0.444732	+	0.895664i	$0.353299\pi$
$72$	0	0
$73$	−2.96061e6	−0.890741	−0.445370	−	0.895346i	$-0.646928\pi$
$73$	−2.96061e6	−0.890741	−0.445370	+	0.895346i	$0.646928\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	657179.	0.164046
$78$	0	0
$79$	−5.00187e6	−1.14140	−0.570700	−	0.821159i	$-0.693328\pi$
$79$	−5.00187e6	−1.14140	−0.570700	+	0.821159i	$0.693328\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.73719e6	−0.333483	−0.166742	−	0.986001i	$-0.553325\pi$
$83$	−1.73719e6	−0.333483	−0.166742	+	0.986001i	$0.553325\pi$
$84$	0	0
$85$	−2.56508e6	−0.453037
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.55486e6	−1.43668	−0.718339	−	0.695693i	$-0.755097\pi$
$89$	−9.55486e6	−1.43668	−0.718339	+	0.695693i	$0.755097\pi$
$90$	0	0
$91$	−1.00351e6	−0.139598
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.46290e7	−1.75058
$96$	0	0
$97$	1.78520e7	1.98603	0.993016	−	0.117982i	$-0.0376425\pi$
$97$	1.78520e7	1.98603	0.993016	+	0.117982i	$0.0376425\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.65452e6	0.932408	0.466204	−	0.884677i	$-0.345621\pi$
$101$	9.65452e6	0.932408	0.466204	+	0.884677i	$0.345621\pi$
$102$	0	0
$103$	1.16497e7	1.05047	0.525235	−	0.850957i	$-0.323977\pi$
$103$	1.16497e7	1.05047	0.525235	+	0.850957i	$0.323977\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.05708e7	−0.834187	−0.417094	−	0.908864i	$-0.636951\pi$
$107$	−1.05708e7	−0.834187	−0.417094	+	0.908864i	$0.636951\pi$
$108$	0	0
$109$	5.34351e6	0.395215	0.197608	−	0.980281i	$-0.436683\pi$
$109$	5.34351e6	0.395215	0.197608	+	0.980281i	$0.436683\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.68727e6	−0.501184	−0.250592	−	0.968093i	$-0.580625\pi$
$113$	−7.68727e6	−0.501184	−0.250592	+	0.968093i	$0.580625\pi$
$114$	0	0
$115$	−1.63409e7	−1.00192
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−702174.	−0.0381971
$120$	0	0
$121$	4.05933e7	2.08308
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.86831e7	0.855588
$126$	0	0
$127$	−3.18274e7	−1.37876	−0.689380	−	0.724400i	$-0.742117\pi$
$127$	−3.18274e7	−1.37876	−0.689380	+	0.724400i	$0.742117\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.82651e7	−1.09850	−0.549252	−	0.835657i	$-0.685087\pi$
$131$	−2.82651e7	−1.09850	−0.549252	+	0.835657i	$0.685087\pi$
$132$	0	0
$133$	−4.00461e6	−0.147598
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.20847e7	−1.39831	−0.699153	−	0.714972i	$-0.746440\pi$
$137$	−4.20847e7	−1.39831	−0.699153	+	0.714972i	$0.746440\pi$
$138$	0	0
$139$	−6.04498e6	−0.190916	−0.0954582	−	0.995433i	$-0.530432\pi$
$139$	−6.04498e6	−0.190916	−0.0954582	+	0.995433i	$0.530432\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.17431e7	−2.62360
$144$	0	0
$145$	1.77463e7	0.483415
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.55413e7	−1.87082	−0.935411	−	0.353563i	$-0.884970\pi$
$149$	−7.55413e7	−1.87082	−0.935411	+	0.353563i	$0.884970\pi$
$150$	0	0
$151$	−1.87099e7	−0.442234	−0.221117	−	0.975247i	$-0.570970\pi$
$151$	−1.87099e7	−0.442234	−0.221117	+	0.975247i	$0.570970\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.98853e7	0.644610
$156$	0	0
$157$	−5.59729e7	−1.15433	−0.577164	−	0.816629i	$-0.695841\pi$
$157$	−5.59729e7	−1.15433	−0.577164	+	0.816629i	$0.695841\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.47322e6	−0.0844754
$162$	0	0
$163$	−3.27218e7	−0.591808	−0.295904	−	0.955218i	$-0.595621\pi$
$163$	−3.27218e7	−0.591808	−0.295904	+	0.955218i	$0.595621\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.01016e7	0.998569	0.499285	−	0.866438i	$-0.333596\pi$
$167$	6.01016e7	0.998569	0.499285	+	0.866438i	$0.333596\pi$
$168$	0	0
$169$	7.73434e7	1.23259
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.65282e7	−1.41740	−0.708701	−	0.705509i	$-0.750718\pi$
$173$	−9.65282e7	−1.41740	−0.708701	+	0.705509i	$0.750718\pi$
$174$	0	0
$175$	−1.50940e6	−0.0212898
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.51149e6	−0.0327300	−0.0163650	−	0.999866i	$-0.505209\pi$
$179$	−2.51149e6	−0.0327300	−0.0163650	+	0.999866i	$0.505209\pi$
$180$	0	0
$181$	4.80546e7	0.602365	0.301183	−	0.953566i	$-0.402619\pi$
$181$	4.80546e7	0.602365	0.301183	+	0.953566i	$0.402619\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.53202e7	−0.642366
$186$	0	0
$187$	−6.41940e7	−0.717875
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.18692e8	−1.23255	−0.616275	−	0.787531i	$-0.711359\pi$
$191$	−1.18692e8	−1.23255	−0.616275	+	0.787531i	$0.711359\pi$
$192$	0	0
$193$	−1.24351e8	−1.24508	−0.622542	−	0.782586i	$-0.713900\pi$
$193$	−1.24351e8	−1.24508	−0.622542	+	0.782586i	$0.713900\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.69395e7	−0.251048	−0.125524	−	0.992091i	$-0.540061\pi$
$197$	−2.69395e7	−0.251048	−0.125524	+	0.992091i	$0.540061\pi$
$198$	0	0
$199$	9.55484e7	0.859484	0.429742	−	0.902952i	$-0.358605\pi$
$199$	9.55484e7	0.859484	0.429742	+	0.902952i	$0.358605\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.85795e6	0.0407584
$204$	0	0
$205$	−3.94057e7	−0.319463
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.66109e8	−2.77395
$210$	0	0
$211$	1.71821e7	0.125918	0.0629590	−	0.998016i	$-0.479946\pi$
$211$	1.71821e7	0.125918	0.0629590	+	0.998016i	$0.479946\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.19558e8	−1.50666
$216$	0	0
$217$	8.18093e6	0.0543492
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.80245e7	0.610888
$222$	0	0
$223$	−5.68895e7	−0.343530	−0.171765	−	0.985138i	$-0.554947\pi$
$223$	−5.68895e7	−0.343530	−0.171765	+	0.985138i	$0.554947\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.69031e8	0.959129	0.479564	−	0.877507i	$-0.340795\pi$
$227$	1.69031e8	0.959129	0.479564	+	0.877507i	$0.340795\pi$
$228$	0	0
$229$	−1.65468e8	−0.910521	−0.455261	−	0.890358i	$-0.650454\pi$
$229$	−1.65468e8	−0.910521	−0.455261	+	0.890358i	$0.650454\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.64114e8	0.849964	0.424982	−	0.905202i	$-0.360280\pi$
$233$	1.64114e8	0.849964	0.424982	+	0.905202i	$0.360280\pi$
$234$	0	0
$235$	2.70300e8	1.35865
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.60895e8	−1.23616	−0.618078	−	0.786117i	$-0.712088\pi$
$239$	−2.60895e8	−1.23616	−0.618078	+	0.786117i	$0.712088\pi$
$240$	0	0
$241$	4.20299e8	1.93419	0.967094	−	0.254420i	$-0.0818847\pi$
$241$	4.20299e8	1.93419	0.967094	+	0.254420i	$0.0818847\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.52843e8	1.09842
$246$	0	0
$247$	5.59049e8	2.36054
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−153195.	−0.000611485 0	−0.000305742	−	1.00000i	$-0.500097\pi$
$251$	−153195.	−0.000611485 0	−0.000305742		1.00000i	$0.500097\pi$
$252$	0	0
$253$	−4.08950e8	−1.58763
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.46390e6	0.00537955	0.00268978	−	0.999996i	$-0.499144\pi$
$257$	1.46390e6	0.00537955	0.00268978	+	0.999996i	$0.499144\pi$
$258$	0	0
$259$	−1.51436e7	−0.0541600
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.31558e7	−0.146283	−0.0731416	−	0.997322i	$-0.523302\pi$
$263$	−4.31558e7	−0.146283	−0.0731416	+	0.997322i	$0.523302\pi$
$264$	0	0
$265$	1.12933e8	0.372788
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.63006e8	−0.510588	−0.255294	−	0.966863i	$-0.582172\pi$
$269$	−1.63006e8	−0.510588	−0.255294	+	0.966863i	$0.582172\pi$
$270$	0	0
$271$	−1.42292e8	−0.434297	−0.217149	−	0.976139i	$-0.569676\pi$
$271$	−1.42292e8	−0.434297	−0.217149	+	0.976139i	$0.569676\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.37992e8	−0.400119
$276$	0	0
$277$	−2.27537e8	−0.643239	−0.321619	−	0.946869i	$-0.604227\pi$
$277$	−2.27537e8	−0.643239	−0.321619	+	0.946869i	$0.604227\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.23676e8	1.40796	0.703980	−	0.710220i	$-0.251405\pi$
$281$	5.23676e8	1.40796	0.703980	+	0.710220i	$0.251405\pi$
$282$	0	0
$283$	−2.93209e8	−0.768997	−0.384499	−	0.923126i	$-0.625626\pi$
$283$	−2.93209e8	−0.768997	−0.384499	+	0.923126i	$0.625626\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.07871e7	−0.0269350
$288$	0	0
$289$	−3.41749e8	−0.832847
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.62533e8	−1.07425	−0.537126	−	0.843502i	$-0.680490\pi$
$293$	−4.62533e8	−1.07425	−0.537126	+	0.843502i	$0.680490\pi$
$294$	0	0
$295$	−6.41849e8	−1.45565
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.24468e8	1.35102
$300$	0	0
$301$	−6.01026e7	−0.127031
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.41255e8	0.688699
$306$	0	0
$307$	−3.00609e8	−0.592949	−0.296475	−	0.955041i	$-0.595811\pi$
$307$	−3.00609e8	−0.592949	−0.296475	+	0.955041i	$0.595811\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.03408e9	1.94936	0.974681	−	0.223600i	$-0.0717809\pi$
$311$	1.03408e9	1.94936	0.974681	+	0.223600i	$0.0717809\pi$
$312$	0	0
$313$	−2.89476e8	−0.533589	−0.266795	−	0.963753i	$-0.585965\pi$
$313$	−2.89476e8	−0.533589	−0.266795	+	0.963753i	$0.585965\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.94824e8	−1.75404	−0.877018	−	0.480457i	$-0.840471\pi$
$317$	−9.94824e8	−1.75404	−0.877018	+	0.480457i	$0.840471\pi$
$318$	0	0
$319$	4.44123e8	0.766012
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.91175e8	0.645896
$324$	0	0
$325$	2.10714e8	0.340488
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.39930e7	0.114553
$330$	0	0
$331$	4.41910e8	0.669785	0.334893	−	0.942256i	$-0.391300\pi$
$331$	4.41910e8	0.669785	0.334893	+	0.942256i	$0.391300\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.01003e9	1.46783
$336$	0	0
$337$	5.00442e8	0.712277	0.356139	−	0.934433i	$-0.384093\pi$
$337$	5.00442e8	0.712277	0.356139	+	0.934433i	$0.384093\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.47916e8	1.02144
$342$	0	0
$343$	1.39038e8	0.186039
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.40700e8	0.309260	0.154630	−	0.987972i	$-0.450581\pi$
$347$	2.40700e8	0.309260	0.154630	+	0.987972i	$0.450581\pi$
$348$	0	0
$349$	6.05261e8	0.762173	0.381086	−	0.924539i	$-0.375550\pi$
$349$	6.05261e8	0.762173	0.381086	+	0.924539i	$0.375550\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.87421e8	0.589784	0.294892	−	0.955531i	$-0.404716\pi$
$353$	4.87421e8	0.589784	0.294892	+	0.955531i	$0.404716\pi$
$354$	0	0
$355$	−8.30815e8	−0.985610
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.62775e8	0.299746	0.149873	−	0.988705i	$-0.452114\pi$
$359$	2.62775e8	0.299746	0.149873	+	0.988705i	$0.452114\pi$
$360$	0	0
$361$	1.33706e9	1.49581
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.16967e8	0.987026
$366$	0	0
$367$	1.35121e9	1.42690	0.713448	−	0.700708i	$-0.247132\pi$
$367$	1.35121e9	1.42690	0.713448	+	0.700708i	$0.247132\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.09148e7	0.0314310
$372$	0	0
$373$	−1.42991e9	−1.42669	−0.713343	−	0.700815i	$-0.752820\pi$
$373$	−1.42991e9	−1.42669	−0.713343	+	0.700815i	$0.752820\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.78177e8	−0.651851
$378$	0	0
$379$	3.75136e8	0.353957	0.176979	−	0.984215i	$-0.443368\pi$
$379$	3.75136e8	0.353957	0.176979	+	0.984215i	$0.443368\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.79667e9	1.63408	0.817039	−	0.576583i	$-0.195614\pi$
$383$	1.79667e9	1.63408	0.817039	+	0.576583i	$0.195614\pi$
$384$	0	0
$385$	−2.03543e8	−0.181779
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.94959e8	−0.254061	−0.127030	−	0.991899i	$-0.540545\pi$
$389$	−2.94959e8	−0.254061	−0.127030	+	0.991899i	$0.540545\pi$
$390$	0	0
$391$	4.36950e8	0.369669
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.54919e9	1.26478
$396$	0	0
$397$	1.63577e8	0.131207	0.0656034	−	0.997846i	$-0.479103\pi$
$397$	1.63577e8	0.131207	0.0656034	+	0.997846i	$0.479103\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.03826e7	0.0622525	0.0311262	−	0.999515i	$-0.490091\pi$
$401$	8.03826e7	0.0622525	0.0311262	+	0.999515i	$0.490091\pi$
$402$	0	0
$403$	−1.14207e9	−0.869210
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.38445e9	−1.01788
$408$	0	0
$409$	−1.08157e9	−0.781669	−0.390835	−	0.920461i	$-0.627814\pi$
$409$	−1.08157e9	−0.781669	−0.390835	+	0.920461i	$0.627814\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.75702e8	−0.122730
$414$	0	0
$415$	5.38046e8	0.369531
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.08423e8	−0.271244	−0.135622	−	0.990761i	$-0.543303\pi$
$419$	−4.08423e8	−0.271244	−0.135622	+	0.990761i	$0.543303\pi$
$420$	0	0
$421$	−1.43219e9	−0.935437	−0.467718	−	0.883878i	$-0.654924\pi$
$421$	−1.43219e9	−0.935437	−0.467718	+	0.883878i	$0.654924\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.47440e8	0.0931652
$426$	0	0
$427$	9.34165e7	0.0580665
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.35879e9	1.41912	0.709560	−	0.704645i	$-0.248894\pi$
$431$	2.35879e9	1.41912	0.709560	+	0.704645i	$0.248894\pi$
$432$	0	0
$433$	−8.58258e8	−0.508055	−0.254027	−	0.967197i	$-0.581755\pi$
$433$	−8.58258e8	−0.508055	−0.254027	+	0.967197i	$0.581755\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.49200e9	1.42844
$438$	0	0
$439$	1.19564e9	0.674487	0.337244	−	0.941417i	$-0.390505\pi$
$439$	1.19564e9	0.674487	0.337244	+	0.941417i	$0.390505\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.00748e8	0.109708	0.0548540	−	0.998494i	$-0.482531\pi$
$443$	2.00748e8	0.109708	0.0548540	+	0.998494i	$0.482531\pi$
$444$	0	0
$445$	2.95935e9	1.59198
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.40652e8	0.177602	0.0888011	−	0.996049i	$-0.471696\pi$
$449$	3.40652e8	0.177602	0.0888011	+	0.996049i	$0.471696\pi$
$450$	0	0
$451$	−9.86174e8	−0.506216
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.10810e8	0.154688
$456$	0	0
$457$	−6.83492e8	−0.334986	−0.167493	−	0.985873i	$-0.553567\pi$
$457$	−6.83492e8	−0.334986	−0.167493	+	0.985873i	$0.553567\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.18702e9	1.03968	0.519839	−	0.854264i	$-0.325992\pi$
$461$	2.18702e9	1.03968	0.519839	+	0.854264i	$0.325992\pi$
$462$	0	0
$463$	−2.07608e9	−0.972099	−0.486049	−	0.873931i	$-0.661562\pi$
$463$	−2.07608e9	−0.972099	−0.486049	+	0.873931i	$0.661562\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.75732e9	−0.798440	−0.399220	−	0.916855i	$-0.630719\pi$
$467$	−1.75732e9	−0.798440	−0.399220	+	0.916855i	$0.630719\pi$
$468$	0	0
$469$	2.76488e8	0.123758
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.49469e9	−2.38742
$474$	0	0
$475$	8.40874e8	0.360001
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.76969e9	1.56722	0.783612	−	0.621250i	$-0.213375\pi$
$479$	3.76969e9	1.56722	0.783612	+	0.621250i	$0.213375\pi$
$480$	0	0
$481$	2.11406e9	0.866184
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.52916e9	−2.20071
$486$	0	0
$487$	−2.09613e9	−0.822371	−0.411185	−	0.911552i	$-0.634885\pi$
$487$	−2.09613e9	−0.822371	−0.411185	+	0.911552i	$0.634885\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.07566e9	−0.791355	−0.395678	−	0.918389i	$-0.629490\pi$
$491$	−2.07566e9	−0.791355	−0.395678	+	0.918389i	$0.629490\pi$
$492$	0	0
$493$	−4.74531e8	−0.178361
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.27431e8	−0.0831001
$498$	0	0
$499$	−4.28277e9	−1.54302	−0.771511	−	0.636215i	$-0.780499\pi$
$499$	−4.28277e9	−1.54302	−0.771511	+	0.636215i	$0.780499\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.90648e9	0.667949	0.333975	−	0.942582i	$-0.391610\pi$
$503$	1.90648e9	0.667949	0.333975	+	0.942582i	$0.391610\pi$
$504$	0	0
$505$	−2.99022e9	−1.03320
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.22835e9	−0.748981	−0.374491	−	0.927231i	$-0.622182\pi$
$509$	−2.22835e9	−0.748981	−0.374491	+	0.927231i	$0.622182\pi$
$510$	0	0
$511$	2.51014e8	0.0832195
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.60816e9	−1.16402
$516$	0	0
$517$	6.76457e9	2.15290
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.82365e9	−0.874738	−0.437369	−	0.899282i	$-0.644090\pi$
$521$	−2.82365e9	−0.874738	−0.437369	+	0.899282i	$0.644090\pi$
$522$	0	0
$523$	−4.68151e9	−1.43097	−0.715484	−	0.698630i	$-0.753794\pi$
$523$	−4.68151e9	−1.43097	−0.715484	+	0.698630i	$0.753794\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.99123e8	−0.237835
$528$	0	0
$529$	−6.21221e8	−0.182453
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.50589e9	0.430773
$534$	0	0
$535$	3.27400e9	0.924359
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.32770e9	1.74054
$540$	0	0
$541$	5.31805e9	1.44398	0.721992	−	0.691901i	$-0.243227\pi$
$541$	5.31805e9	1.44398	0.721992	+	0.691901i	$0.243227\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.65500e9	−0.437936
$546$	0	0
$547$	−2.41392e9	−0.630619	−0.315310	−	0.948989i	$-0.602108\pi$
$547$	−2.41392e9	−0.630619	−0.315310	+	0.948989i	$0.602108\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.70632e9	−0.689206
$552$	0	0
$553$	4.24082e8	0.106638
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.13472e9	−1.50419	−0.752093	−	0.659057i	$-0.770956\pi$
$557$	−6.13472e9	−1.50419	−0.752093	+	0.659057i	$0.770956\pi$
$558$	0	0
$559$	8.39041e9	2.03162
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.18283e9	1.22402	0.612009	−	0.790851i	$-0.290362\pi$
$563$	5.18283e9	1.22402	0.612009	+	0.790851i	$0.290362\pi$
$564$	0	0
$565$	2.38092e9	0.555360
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.37825e9	−0.996340	−0.498170	−	0.867079i	$-0.665994\pi$
$569$	−4.37825e9	−0.996340	−0.498170	+	0.867079i	$0.665994\pi$
$570$	0	0
$571$	−1.10717e9	−0.248879	−0.124440	−	0.992227i	$-0.539713\pi$
$571$	−1.10717e9	−0.248879	−0.124440	+	0.992227i	$0.539713\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9.39271e8	0.206041
$576$	0	0
$577$	−1.82372e9	−0.395223	−0.197611	−	0.980280i	$-0.563318\pi$
$577$	−1.82372e9	−0.395223	−0.197611	+	0.980280i	$0.563318\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.47287e8	0.0311564
$582$	0	0
$583$	2.82629e9	0.590713
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.89279e8	−0.0998442	−0.0499221	−	0.998753i	$-0.515897\pi$
$587$	−4.89279e8	−0.0998442	−0.0499221	+	0.998753i	$0.515897\pi$
$588$	0	0
$589$	−4.55753e9	−0.919022
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.90482e9	−0.768969	−0.384485	−	0.923131i	$-0.625621\pi$
$593$	−3.90482e9	−0.768969	−0.384485	+	0.923131i	$0.625621\pi$
$594$	0	0
$595$	2.17479e8	0.0423260
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.83210e9	−1.67907	−0.839537	−	0.543302i	$-0.817174\pi$
$599$	−8.83210e9	−1.67907	−0.839537	+	0.543302i	$0.817174\pi$
$600$	0	0
$601$	3.85382e9	0.724153	0.362077	−	0.932148i	$-0.382068\pi$
$601$	3.85382e9	0.724153	0.362077	+	0.932148i	$0.382068\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.25727e10	−2.30825
$606$	0	0
$607$	−2.33672e9	−0.424079	−0.212040	−	0.977261i	$-0.568011\pi$
$607$	−2.33672e9	−0.424079	−0.212040	+	0.977261i	$0.568011\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.03295e10	−1.83204
$612$	0	0
$613$	7.33773e9	1.28662	0.643310	−	0.765606i	$-0.277561\pi$
$613$	7.33773e9	1.28662	0.643310	+	0.765606i	$0.277561\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.08980e9	−0.529581	−0.264790	−	0.964306i	$-0.585303\pi$
$617$	−3.08980e9	−0.529581	−0.264790	+	0.964306i	$0.585303\pi$
$618$	0	0
$619$	−8.91809e8	−0.151131	−0.0755657	−	0.997141i	$-0.524076\pi$
$619$	−8.91809e8	−0.151131	−0.0755657	+	0.997141i	$0.524076\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.10105e8	0.134225
$624$	0	0
$625$	−7.17742e9	−1.17595
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.47924e9	0.237007
$630$	0	0
$631$	−4.89308e8	−0.0775317	−0.0387659	−	0.999248i	$-0.512343\pi$
$631$	−4.89308e8	−0.0775317	−0.0387659	+	0.999248i	$0.512343\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.85765e9	1.52780
$636$	0	0
$637$	−9.66241e9	−1.48114
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.43888e9	−1.11559	−0.557795	−	0.829979i	$-0.688352\pi$
$641$	−7.43888e9	−1.11559	−0.557795	+	0.829979i	$0.688352\pi$
$642$	0	0
$643$	5.29076e8	0.0784838	0.0392419	−	0.999230i	$-0.487506\pi$
$643$	5.29076e8	0.0784838	0.0392419	+	0.999230i	$0.487506\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.96481e9	1.01098	0.505492	−	0.862831i	$-0.331311\pi$
$647$	6.96481e9	1.01098	0.505492	+	0.862831i	$0.331311\pi$
$648$	0	0
$649$	−1.60630e10	−2.30659
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.41113e9	−0.760487	−0.380243	−	0.924886i	$-0.624160\pi$
$653$	−5.41113e9	−0.760487	−0.380243	+	0.924886i	$0.624160\pi$
$654$	0	0
$655$	8.75433e9	1.21725
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.35800e9	0.729296	0.364648	−	0.931145i	$-0.381189\pi$
$659$	5.35800e9	0.729296	0.364648	+	0.931145i	$0.381189\pi$
$660$	0	0
$661$	−1.25783e10	−1.69401	−0.847004	−	0.531586i	$-0.821596\pi$
$661$	−1.25783e10	−1.69401	−0.847004	+	0.531586i	$0.821596\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.24032e9	0.163552
$666$	0	0
$667$	−3.02301e9	−0.394457
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.54031e9	1.09130
$672$	0	0
$673$	−3.65795e9	−0.462578	−0.231289	−	0.972885i	$-0.574294\pi$
$673$	−3.65795e9	−0.462578	−0.231289	+	0.972885i	$0.574294\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.21689e8	−0.0522314	−0.0261157	−	0.999659i	$-0.508314\pi$
$677$	−4.21689e8	−0.0522314	−0.0261157	+	0.999659i	$0.508314\pi$
$678$	0	0
$679$	−1.51358e9	−0.185550
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.64344e9	0.437561	0.218781	−	0.975774i	$-0.429792\pi$
$683$	3.64344e9	0.437561	0.218781	+	0.975774i	$0.429792\pi$
$684$	0	0
$685$	1.30346e10	1.54946
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.31575e9	−0.502677
$690$	0	0
$691$	−8.03103e9	−0.925972	−0.462986	−	0.886365i	$-0.653222\pi$
$691$	−8.03103e9	−0.925972	−0.462986	+	0.886365i	$0.653222\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.87226e9	0.211554
$696$	0	0
$697$	1.05369e9	0.117869
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.28577e9	−1.01813	−0.509067	−	0.860727i	$-0.670009\pi$
$701$	−9.28577e9	−1.01813	−0.509067	+	0.860727i	$0.670009\pi$
$702$	0	0
$703$	8.43636e9	0.915822
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.18554e8	−0.0871123
$708$	0	0
$709$	−3.06841e9	−0.323334	−0.161667	−	0.986845i	$-0.551687\pi$
$709$	−3.06841e9	−0.323334	−0.161667	+	0.986845i	$0.551687\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.09084e9	−0.525988
$714$	0	0
$715$	2.84149e10	2.90720
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.57787e10	−1.58314	−0.791572	−	0.611076i	$-0.790737\pi$
$719$	−1.57787e10	−1.58314	−0.791572	+	0.611076i	$0.790737\pi$
$720$	0	0
$721$	−9.87713e8	−0.0981426
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.02005e9	−0.0994124
$726$	0	0
$727$	2.32317e9	0.224238	0.112119	−	0.993695i	$-0.464236\pi$
$727$	2.32317e9	0.224238	0.112119	+	0.993695i	$0.464236\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.87090e9	0.555896
$732$	0	0
$733$	8.22553e8	0.0771436	0.0385718	−	0.999256i	$-0.487719\pi$
$733$	8.22553e8	0.0771436	0.0385718	+	0.999256i	$0.487719\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.52771e10	2.32590
$738$	0	0
$739$	−2.27700e9	−0.207543	−0.103771	−	0.994601i	$-0.533091\pi$
$739$	−2.27700e9	−0.207543	−0.103771	+	0.994601i	$0.533091\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.59879e9	−0.142998	−0.0714991	−	0.997441i	$-0.522778\pi$
$743$	−1.59879e9	−0.142998	−0.0714991	+	0.997441i	$0.522778\pi$
$744$	0	0
$745$	2.33968e10	2.07305
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.96238e8	0.0779358
$750$	0	0
$751$	6.87936e9	0.592663	0.296332	−	0.955085i	$-0.404237\pi$
$751$	6.87936e9	0.592663	0.296332	+	0.955085i	$0.404237\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.79487e9	0.490037
$756$	0	0
$757$	−1.58272e10	−1.32607	−0.663037	−	0.748586i	$-0.730733\pi$
$757$	−1.58272e10	−1.32607	−0.663037	+	0.748586i	$0.730733\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.48160e10	−1.21866	−0.609332	−	0.792915i	$-0.708562\pi$
$761$	−1.48160e10	−1.21866	−0.609332	+	0.792915i	$0.708562\pi$
$762$	0	0
$763$	−4.53047e8	−0.0369239
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.45283e10	1.96283
$768$	0	0
$769$	1.77521e10	1.40769	0.703845	−	0.710353i	$-0.251465\pi$
$769$	1.77521e10	1.40769	0.703845	+	0.710353i	$0.251465\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.11167e9	−0.320177	−0.160089	−	0.987103i	$-0.551178\pi$
$773$	−4.11167e9	−0.320177	−0.160089	+	0.987103i	$0.551178\pi$
$774$	0	0
$775$	−1.71780e9	−0.132561
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00939e9	0.455459
$780$	0	0
$781$	−2.07921e10	−1.56178
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.73360e10	1.27910
$786$	0	0
$787$	−1.73472e10	−1.26858	−0.634289	−	0.773096i	$-0.718707\pi$
$787$	−1.73472e10	−1.26858	−0.634289	+	0.773096i	$0.718707\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.51762e8	0.0468243
$792$	0	0
$793$	−1.30411e10	−0.928661
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.25465e9	0.507589	0.253794	−	0.967258i	$-0.418321\pi$
$797$	7.25465e9	0.507589	0.253794	+	0.967258i	$0.418321\pi$
$798$	0	0
$799$	−7.22772e9	−0.501289
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.29482e10	1.56403
$804$	0	0
$805$	1.38546e9	0.0936068
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.10660e9	0.206284	0.103142	−	0.994667i	$-0.467110\pi$
$809$	3.10660e9	0.206284	0.103142	+	0.994667i	$0.467110\pi$
$810$	0	0
$811$	−1.31378e9	−0.0864868	−0.0432434	−	0.999065i	$-0.513769\pi$
$811$	−1.31378e9	−0.0864868	−0.0432434	+	0.999065i	$0.513769\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.01347e10	0.655780
$816$	0	0
$817$	3.34827e10	2.14804
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.14000e9	−0.576428	−0.288214	−	0.957566i	$-0.593061\pi$
$821$	−9.14000e9	−0.576428	−0.288214	+	0.957566i	$0.593061\pi$
$822$	0	0
$823$	8.29555e8	0.0518736	0.0259368	−	0.999664i	$-0.491743\pi$
$823$	8.29555e8	0.0518736	0.0259368	+	0.999664i	$0.491743\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.66542e10	1.02389	0.511946	−	0.859018i	$-0.328925\pi$
$827$	1.66542e10	1.02389	0.511946	+	0.859018i	$0.328925\pi$
$828$	0	0
$829$	5.12080e9	0.312174	0.156087	−	0.987743i	$-0.450112\pi$
$829$	5.12080e9	0.312174	0.156087	+	0.987743i	$0.450112\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.76094e9	−0.405274
$834$	0	0
$835$	−1.86148e10	−1.10651
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.19094e9	0.128075	0.0640374	−	0.997947i	$-0.479602\pi$
$839$	2.19094e9	0.128075	0.0640374	+	0.997947i	$0.479602\pi$
$840$	0	0
$841$	−1.39669e10	−0.809679
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.39550e10	−1.36583
$846$	0	0
$847$	−3.44169e9	−0.194617
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.42356e9	0.524157
$852$	0	0
$853$	1.55327e10	0.856892	0.428446	−	0.903567i	$-0.359061\pi$
$853$	1.55327e10	0.856892	0.428446	+	0.903567i	$0.359061\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.19374e8	0.0444682	0.0222341	−	0.999753i	$-0.492922\pi$
$857$	8.19374e8	0.0444682	0.0222341	+	0.999753i	$0.492922\pi$
$858$	0	0
$859$	4.05717e9	0.218397	0.109199	−	0.994020i	$-0.465172\pi$
$859$	4.05717e9	0.218397	0.109199	+	0.994020i	$0.465172\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.83217e10	1.49997	0.749984	−	0.661456i	$-0.230061\pi$
$863$	2.83217e10	1.49997	0.749984	+	0.661456i	$0.230061\pi$
$864$	0	0
$865$	2.98969e10	1.57062
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.87703e10	2.00415
$870$	0	0
$871$	−3.85982e10	−1.97926
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.58404e9	−0.0799352
$876$	0	0
$877$	−1.31183e10	−0.656718	−0.328359	−	0.944553i	$-0.606495\pi$
$877$	−1.31183e10	−0.656718	−0.328359	+	0.944553i	$0.606495\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.39271e10	1.17889	0.589447	−	0.807807i	$-0.299346\pi$
$881$	2.39271e10	1.17889	0.589447	+	0.807807i	$0.299346\pi$
$882$	0	0
$883$	1.13578e10	0.555177	0.277589	−	0.960700i	$-0.410465\pi$
$883$	1.13578e10	0.555177	0.277589	+	0.960700i	$0.410465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.60096e9	0.413823	0.206911	−	0.978360i	$-0.433659\pi$
$887$	8.60096e9	0.413823	0.206911	+	0.978360i	$0.433659\pi$
$888$	0	0
$889$	2.69847e9	0.128814
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.12209e10	−1.93703
$894$	0	0
$895$	7.77864e8	0.0362680
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.52869e9	0.253783
$900$	0	0
$901$	−3.01980e9	−0.137544
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.48836e10	−0.667478
$906$	0	0
$907$	2.45218e9	0.109126	0.0545628	−	0.998510i	$-0.482623\pi$
$907$	2.45218e9	0.109126	0.0545628	+	0.998510i	$0.482623\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.78405e9	−0.297286	−0.148643	−	0.988891i	$-0.547491\pi$
$911$	−6.78405e9	−0.297286	−0.148643	+	0.988891i	$0.547491\pi$
$912$	0	0
$913$	1.34652e10	0.585553
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.39645e9	0.102630
$918$	0	0
$919$	−2.90815e10	−1.23598	−0.617991	−	0.786185i	$-0.712053\pi$
$919$	−2.90815e10	−1.23598	−0.617991	+	0.786185i	$0.712053\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.17496e10	1.32902
$924$	0	0
$925$	3.17979e9	0.132100
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.35380e9	−0.260003	−0.130002	−	0.991514i	$-0.541498\pi$
$929$	−6.35380e9	−0.260003	−0.130002	+	0.991514i	$0.541498\pi$
$930$	0	0
$931$	−3.85587e10	−1.56602
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.98823e10	0.795474
$936$	0	0
$937$	6.75315e9	0.268175	0.134087	−	0.990970i	$-0.457190\pi$
$937$	6.75315e9	0.268175	0.134087	+	0.990970i	$0.457190\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.30223e10	1.29194	0.645972	−	0.763361i	$-0.276452\pi$
$941$	3.30223e10	1.29194	0.645972	+	0.763361i	$0.276452\pi$
$942$	0	0
$943$	6.71260e9	0.260675
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.09234e10	0.417960	0.208980	−	0.977920i	$-0.432986\pi$
$947$	1.09234e10	0.417960	0.208980	+	0.977920i	$0.432986\pi$
$948$	0	0
$949$	−3.50419e10	−1.33093
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.00332e10	−0.749765	−0.374883	−	0.927072i	$-0.622317\pi$
$953$	−2.00332e10	−0.749765	−0.374883	+	0.927072i	$0.622317\pi$
$954$	0	0
$955$	3.67615e10	1.36578
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.56813e9	0.130640
$960$	0	0
$961$	−1.82021e10	−0.661593
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.85142e10	1.37967
$966$	0	0
$967$	−1.80472e10	−0.641826	−0.320913	−	0.947109i	$-0.603990\pi$
$967$	−1.80472e10	−0.641826	−0.320913	+	0.947109i	$0.603990\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.28443e10	−1.50185	−0.750924	−	0.660388i	$-0.770392\pi$
$971$	−4.28443e10	−1.50185	−0.750924	+	0.660388i	$0.770392\pi$
$972$	0	0
$973$	5.12521e8	0.0178368
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.80560e10	−1.64860	−0.824302	−	0.566150i	$-0.808432\pi$
$977$	−4.80560e10	−1.64860	−0.824302	+	0.566150i	$0.808432\pi$
$978$	0	0
$979$	7.40613e10	2.52262
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.86180e10	0.625167	0.312584	−	0.949890i	$-0.398806\pi$
$983$	1.86180e10	0.625167	0.312584	+	0.949890i	$0.398806\pi$
$984$	0	0
$985$	8.34375e9	0.278185
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.74007e10	1.22940
$990$	0	0
$991$	3.26103e10	1.06438	0.532191	−	0.846624i	$-0.321369\pi$
$991$	3.26103e10	1.06438	0.532191	+	0.846624i	$0.321369\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.95935e10	−0.952390
$996$	0	0
$997$	−1.83146e10	−0.585281	−0.292641	−	0.956222i	$-0.594534\pi$
$997$	−1.83146e10	−0.585281	−0.292641	+	0.956222i	$0.594534\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.8.a.c.1.2		7
3.2	odd	2		324.8.a.d.1.6		7
9.2	odd	6		108.8.e.a.37.2		14
9.4	even	3		36.8.e.a.25.2	yes	14
9.5	odd	6		108.8.e.a.73.2		14
9.7	even	3		36.8.e.a.13.2	✓	14
36.7	odd	6		144.8.i.d.49.6		14
36.11	even	6		432.8.i.d.145.2		14
36.23	even	6		432.8.i.d.289.2		14
36.31	odd	6		144.8.i.d.97.6		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.8.e.a.13.2	✓	14	9.7	even	3
36.8.e.a.25.2	yes	14	9.4	even	3
108.8.e.a.37.2		14	9.2	odd	6
108.8.e.a.73.2		14	9.5	odd	6
144.8.i.d.49.6		14	36.7	odd	6
144.8.i.d.97.6		14	36.31	odd	6
324.8.a.c.1.2		7	1.1	even	1	trivial
324.8.a.d.1.6		7	3.2	odd	2
432.8.i.d.145.2		14	36.11	even	6
432.8.i.d.289.2		14	36.23	even	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q-309.722 q^{5} -84.7846 q^{7} +O(q^{10})\)	\(q-309.722 q^{5} -84.7846 q^{7} -7751.16 q^{11} +11836.0 q^{13} +8281.86 q^{17} +47232.8 q^{19} +52759.9 q^{23} +17802.8 q^{25} -57297.6 q^{29} -96490.8 q^{31} +26259.7 q^{35} +178612. q^{37} +127229. q^{41} +708886. q^{43} -872717. q^{47} -816355. q^{49} -364628. q^{53} +2.40071e6 q^{55} +2.07234e6 q^{59} -1.10181e6 q^{61} -3.66588e6 q^{65} -3.26107e6 q^{67} +2.68245e6 q^{71} -2.96061e6 q^{73} +657179. q^{77} -5.00187e6 q^{79} -1.73719e6 q^{83} -2.56508e6 q^{85} -9.55486e6 q^{89} -1.00351e6 q^{91} -1.46290e7 q^{95} +1.78520e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 321 q^{5} + 83 q^{7}+O(q^{10}) \) 7 * q - 321 * q^5 + 83 * q^7	\( 7 q - 321 q^{5} + 83 q^{7} + 111 q^{11} + 1847 q^{13} - 48 q^{17} + 10124 q^{19} - 19119 q^{23} + 73378 q^{25} - 6045 q^{29} + 153089 q^{31} + 13713 q^{35} + 69674 q^{37} - 446631 q^{41} + 384347 q^{43} - 298413 q^{47} + 351876 q^{49} - 454038 q^{53} - 1263483 q^{55} - 2619543 q^{59} + 146231 q^{61} - 2535735 q^{65} - 1637419 q^{67} - 4353492 q^{71} - 2132260 q^{73} - 9785451 q^{77} - 2402185 q^{79} - 12936357 q^{83} + 1015002 q^{85} - 19684830 q^{89} - 492203 q^{91} - 22685196 q^{95} + 2853257 q^{97}+O(q^{100}) \) 7 * q - 321 * q^5 + 83 * q^7 + 111 * q^11 + 1847 * q^13 - 48 * q^17 + 10124 * q^19 - 19119 * q^23 + 73378 * q^25 - 6045 * q^29 + 153089 * q^31 + 13713 * q^35 + 69674 * q^37 - 446631 * q^41 + 384347 * q^43 - 298413 * q^47 + 351876 * q^49 - 454038 * q^53 - 1263483 * q^55 - 2619543 * q^59 + 146231 * q^61 - 2535735 * q^65 - 1637419 * q^67 - 4353492 * q^71 - 2132260 * q^73 - 9785451 * q^77 - 2402185 * q^79 - 12936357 * q^83 + 1015002 * q^85 - 19684830 * q^89 - 492203 * q^91 - 22685196 * q^95 + 2853257 * q^97

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