LMFDB - Embedded newform 84.8.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,8,Mod(1,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("84.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	84.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:	$26.2403421407$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 912 $ x^2 - x - 912
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-29.7035$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +494.442 q^{5} +343.000 q^{7} +729.000 q^{9} +47.0920 q^{11} +3624.60 q^{13} -13349.9 q^{15} -14642.7 q^{17} -3395.95 q^{19} -9261.00 q^{21} -17982.5 q^{23} +166348. q^{25} -19683.0 q^{27} +139916. q^{29} +228888. q^{31} -1271.48 q^{33} +169594. q^{35} +438263. q^{37} -97864.2 q^{39} +312110. q^{41} -556478. q^{43} +360448. q^{45} -794234. q^{47} +117649. q^{49} +395353. q^{51} +2.04720e6 q^{53} +23284.3 q^{55} +91690.7 q^{57} +2.56746e6 q^{59} -2.46108e6 q^{61} +250047. q^{63} +1.79215e6 q^{65} +2.15480e6 q^{67} +485527. q^{69} -2.38723e6 q^{71} -1.97256e6 q^{73} -4.49139e6 q^{75} +16152.6 q^{77} +117694. q^{79} +531441. q^{81} -509356. q^{83} -7.23997e6 q^{85} -3.77773e6 q^{87} -4.16031e6 q^{89} +1.24324e6 q^{91} -6.17998e6 q^{93} -1.67910e6 q^{95} +8.40834e6 q^{97} +34330.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} + 264 q^{5} + 686 q^{7} + 1458 q^{9} - 4980 q^{11} - 10148 q^{13} - 7128 q^{15} + 17832 q^{17} + 6256 q^{19} - 18522 q^{21} + 14052 q^{23} + 141326 q^{25} - 39366 q^{27} + 243588 q^{29} + 470824 q^{31}+ \cdots - 3630420 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 + 264 * q^5 + 686 * q^7 + 1458 * q^9 - 4980 * q^11 - 10148 * q^13 - 7128 * q^15 + 17832 * q^17 + 6256 * q^19 - 18522 * q^21 + 14052 * q^23 + 141326 * q^25 - 39366 * q^27 + 243588 * q^29 + 470824 * q^31 + 134460 * q^33 + 90552 * q^35 + 311116 * q^37 + 273996 * q^39 + 919248 * q^41 - 112616 * q^43 + 192456 * q^45 - 102456 * q^47 + 235298 * q^49 - 481464 * q^51 + 2720028 * q^53 + 1181736 * q^55 - 168912 * q^57 - 19008 * q^59 - 925148 * q^61 + 500094 * q^63 + 4965936 * q^65 - 2053424 * q^67 - 379404 * q^69 - 869508 * q^71 + 3505228 * q^73 - 3815802 * q^75 - 1708140 * q^77 - 6640856 * q^79 + 1062882 * q^81 + 7856760 * q^83 - 14723496 * q^85 - 6576876 * q^87 - 9330384 * q^89 - 3480764 * q^91 - 12712248 * q^93 - 3903312 * q^95 - 2220212 * q^97 - 3630420 * q^99

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	494.442	1.76897	0.884484	−	0.466570i	$-0.154510\pi$
$5$	494.442	1.76897	0.884484	+	0.466570i	$0.154510\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	47.0920	0.0106678	0.00533388	−	0.999986i	$-0.498302\pi$
$11$	47.0920	0.0106678	0.00533388	+	0.999986i	$0.498302\pi$
$12$	0	0
$13$	3624.60	0.457571	0.228786	−	0.973477i	$-0.426525\pi$
$13$	3624.60	0.457571	0.228786	+	0.973477i	$0.426525\pi$
$14$	0	0
$15$	−13349.9	−1.02131
$16$	0	0
$17$	−14642.7	−0.722854	−0.361427	−	0.932401i	$-0.617710\pi$
$17$	−14642.7	−0.722854	−0.361427	+	0.932401i	$0.617710\pi$
$18$	0	0
$19$	−3395.95	−0.113586	−0.0567929	−	0.998386i	$-0.518087\pi$
$19$	−3395.95	−0.113586	−0.0567929	+	0.998386i	$0.518087\pi$
$20$	0	0
$21$	−9261.00	−0.218218
$22$	0	0
$23$	−17982.5	−0.308178	−0.154089	−	0.988057i	$-0.549244\pi$
$23$	−17982.5	−0.308178	−0.154089	+	0.988057i	$0.549244\pi$
$24$	0	0
$25$	166348.	2.12925
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	139916.	1.06531	0.532653	−	0.846334i	$-0.321195\pi$
$29$	139916.	1.06531	0.532653	+	0.846334i	$0.321195\pi$
$30$	0	0
$31$	228888.	1.37993	0.689965	−	0.723843i	$-0.257626\pi$
$31$	228888.	1.37993	0.689965	+	0.723843i	$0.257626\pi$
$32$	0	0
$33$	−1271.48	−0.00615903
$34$	0	0
$35$	169594.	0.668607
$36$	0	0
$37$	438263.	1.42242	0.711211	−	0.702979i	$-0.248147\pi$
$37$	438263.	1.42242	0.711211	+	0.702979i	$0.248147\pi$
$38$	0	0
$39$	−97864.2	−0.264179
$40$	0	0
$41$	312110.	0.707236	0.353618	−	0.935390i	$-0.384951\pi$
$41$	312110.	0.707236	0.353618	+	0.935390i	$0.384951\pi$
$42$	0	0
$43$	−556478.	−1.06735	−0.533676	−	0.845689i	$-0.679190\pi$
$43$	−556478.	−1.06735	−0.533676	+	0.845689i	$0.679190\pi$
$44$	0	0
$45$	360448.	0.589656
$46$	0	0
$47$	−794234.	−1.11585	−0.557925	−	0.829891i	$-0.688402\pi$
$47$	−794234.	−1.11585	−0.557925	+	0.829891i	$0.688402\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	395353.	0.417340
$52$	0	0
$53$	2.04720e6	1.88884	0.944421	−	0.328739i	$-0.106624\pi$
$53$	2.04720e6	1.88884	0.944421	+	0.328739i	$0.106624\pi$
$54$	0	0
$55$	23284.3	0.0188709
$56$	0	0
$57$	91690.7	0.0655788
$58$	0	0
$59$	2.56746e6	1.62750	0.813750	−	0.581215i	$-0.197422\pi$
$59$	2.56746e6	1.62750	0.813750	+	0.581215i	$0.197422\pi$
$60$	0	0
$61$	−2.46108e6	−1.38826	−0.694130	−	0.719850i	$-0.744211\pi$
$61$	−2.46108e6	−1.38826	−0.694130	+	0.719850i	$0.744211\pi$
$62$	0	0
$63$	250047.	0.125988
$64$	0	0
$65$	1.79215e6	0.809429
$66$	0	0
$67$	2.15480e6	0.875277	0.437638	−	0.899151i	$-0.355815\pi$
$67$	2.15480e6	0.875277	0.437638	+	0.899151i	$0.355815\pi$
$68$	0	0
$69$	485527.	0.177927
$70$	0	0
$71$	−2.38723e6	−0.791570	−0.395785	−	0.918343i	$-0.629528\pi$
$71$	−2.38723e6	−0.791570	−0.395785	+	0.918343i	$0.629528\pi$
$72$	0	0
$73$	−1.97256e6	−0.593473	−0.296736	−	0.954959i	$-0.595898\pi$
$73$	−1.97256e6	−0.593473	−0.296736	+	0.954959i	$0.595898\pi$
$74$	0	0
$75$	−4.49139e6	−1.22932
$76$	0	0
$77$	16152.6	0.00403203
$78$	0	0
$79$	117694.	0.0268572	0.0134286	−	0.999910i	$-0.495725\pi$
$79$	117694.	0.0268572	0.0134286	+	0.999910i	$0.495725\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−509356.	−0.0977796	−0.0488898	−	0.998804i	$-0.515568\pi$
$83$	−509356.	−0.0977796	−0.0488898	+	0.998804i	$0.515568\pi$
$84$	0	0
$85$	−7.23997e6	−1.27871
$86$	0	0
$87$	−3.77773e6	−0.615055
$88$	0	0
$89$	−4.16031e6	−0.625548	−0.312774	−	0.949828i	$-0.601258\pi$
$89$	−4.16031e6	−0.625548	−0.312774	+	0.949828i	$0.601258\pi$
$90$	0	0
$91$	1.24324e6	0.172946
$92$	0	0
$93$	−6.17998e6	−0.796703
$94$	0	0
$95$	−1.67910e6	−0.200930
$96$	0	0
$97$	8.40834e6	0.935425	0.467713	−	0.883881i	$-0.345078\pi$
$97$	8.40834e6	0.935425	0.467713	+	0.883881i	$0.345078\pi$
$98$	0	0
$99$	34330.1	0.00355592
$100$	0	0
$101$	8.43612e6	0.814738	0.407369	−	0.913264i	$-0.366446\pi$
$101$	8.43612e6	0.814738	0.407369	+	0.913264i	$0.366446\pi$
$102$	0	0
$103$	−220795.	−0.0199094	−0.00995471	−	0.999950i	$-0.503169\pi$
$103$	−220795.	−0.0199094	−0.00995471	+	0.999950i	$0.503169\pi$
$104$	0	0
$105$	−4.57902e6	−0.386021
$106$	0	0
$107$	−1.33020e7	−1.04972	−0.524861	−	0.851188i	$-0.675883\pi$
$107$	−1.33020e7	−1.04972	−0.524861	+	0.851188i	$0.675883\pi$
$108$	0	0
$109$	−2.11624e7	−1.56521	−0.782605	−	0.622518i	$-0.786110\pi$
$109$	−2.11624e7	−1.56521	−0.782605	+	0.622518i	$0.786110\pi$
$110$	0	0
$111$	−1.18331e7	−0.821235
$112$	0	0
$113$	2.23896e7	1.45972	0.729862	−	0.683594i	$-0.239584\pi$
$113$	2.23896e7	1.45972	0.729862	+	0.683594i	$0.239584\pi$
$114$	0	0
$115$	−8.89129e6	−0.545158
$116$	0	0
$117$	2.64233e6	0.152524
$118$	0	0
$119$	−5.02245e6	−0.273213
$120$	0	0
$121$	−1.94850e7	−0.999886
$122$	0	0
$123$	−8.42698e6	−0.408323
$124$	0	0
$125$	4.36209e7	1.99761
$126$	0	0
$127$	−3.45607e7	−1.49716	−0.748582	−	0.663042i	$-0.769265\pi$
$127$	−3.45607e7	−1.49716	−0.748582	+	0.663042i	$0.769265\pi$
$128$	0	0
$129$	1.50249e7	0.616237
$130$	0	0
$131$	4.06358e7	1.57928	0.789641	−	0.613569i	$-0.210267\pi$
$131$	4.06358e7	1.57928	0.789641	+	0.613569i	$0.210267\pi$
$132$	0	0
$133$	−1.16481e6	−0.0429314
$134$	0	0
$135$	−9.73210e6	−0.340438
$136$	0	0
$137$	3.35277e7	1.11399	0.556995	−	0.830516i	$-0.311954\pi$
$137$	3.35277e7	1.11399	0.556995	+	0.830516i	$0.311954\pi$
$138$	0	0
$139$	3.42048e7	1.08028	0.540139	−	0.841576i	$-0.318372\pi$
$139$	3.42048e7	1.08028	0.540139	+	0.841576i	$0.318372\pi$
$140$	0	0
$141$	2.14443e7	0.644236
$142$	0	0
$143$	170690.	0.00488126
$144$	0	0
$145$	6.91804e7	1.88449
$146$	0	0
$147$	−3.17652e6	−0.0824786
$148$	0	0
$149$	2.21749e7	0.549174	0.274587	−	0.961562i	$-0.411459\pi$
$149$	2.21749e7	0.549174	0.274587	+	0.961562i	$0.411459\pi$
$150$	0	0
$151$	−6.18743e7	−1.46248	−0.731241	−	0.682119i	$-0.761059\pi$
$151$	−6.18743e7	−1.46248	−0.731241	+	0.682119i	$0.761059\pi$
$152$	0	0
$153$	−1.06745e7	−0.240951
$154$	0	0
$155$	1.13172e8	2.44105
$156$	0	0
$157$	−2.00578e7	−0.413652	−0.206826	−	0.978378i	$-0.566313\pi$
$157$	−2.00578e7	−0.413652	−0.206826	+	0.978378i	$0.566313\pi$
$158$	0	0
$159$	−5.52745e7	−1.09052
$160$	0	0
$161$	−6.16799e6	−0.116480
$162$	0	0
$163$	2.15447e7	0.389658	0.194829	−	0.980837i	$-0.437585\pi$
$163$	2.15447e7	0.389658	0.194829	+	0.980837i	$0.437585\pi$
$164$	0	0
$165$	−628675.	−0.0108951
$166$	0	0
$167$	−3.48173e7	−0.578479	−0.289239	−	0.957257i	$-0.593402\pi$
$167$	−3.48173e7	−0.578479	−0.289239	+	0.957257i	$0.593402\pi$
$168$	0	0
$169$	−4.96108e7	−0.790629
$170$	0	0
$171$	−2.47565e6	−0.0378619
$172$	0	0
$173$	−8.84532e7	−1.29883	−0.649415	−	0.760434i	$-0.724986\pi$
$173$	−8.84532e7	−1.29883	−0.649415	+	0.760434i	$0.724986\pi$
$174$	0	0
$175$	5.70572e7	0.804781
$176$	0	0
$177$	−6.93213e7	−0.939638
$178$	0	0
$179$	−1.10608e8	−1.44146	−0.720729	−	0.693216i	$-0.756193\pi$
$179$	−1.10608e8	−1.44146	−0.720729	+	0.693216i	$0.756193\pi$
$180$	0	0
$181$	1.31624e8	1.64991	0.824955	−	0.565198i	$-0.191200\pi$
$181$	1.31624e8	1.64991	0.824955	+	0.565198i	$0.191200\pi$
$182$	0	0
$183$	6.64491e7	0.801512
$184$	0	0
$185$	2.16695e8	2.51622
$186$	0	0
$187$	−689555.	−0.00771122
$188$	0	0
$189$	−6.75127e6	−0.0727393
$190$	0	0
$191$	4.48491e7	0.465733	0.232866	−	0.972509i	$-0.425190\pi$
$191$	4.48491e7	0.465733	0.232866	+	0.972509i	$0.425190\pi$
$192$	0	0
$193$	1.21202e7	0.121356	0.0606778	−	0.998157i	$-0.480674\pi$
$193$	1.21202e7	0.121356	0.0606778	+	0.998157i	$0.480674\pi$
$194$	0	0
$195$	−4.83882e7	−0.467324
$196$	0	0
$197$	−5.60365e7	−0.522202	−0.261101	−	0.965311i	$-0.584086\pi$
$197$	−5.60365e7	−0.522202	−0.261101	+	0.965311i	$0.584086\pi$
$198$	0	0
$199$	−1.07524e8	−0.967211	−0.483606	−	0.875286i	$-0.660673\pi$
$199$	−1.07524e8	−0.967211	−0.483606	+	0.875286i	$0.660673\pi$
$200$	0	0
$201$	−5.81796e7	−0.505341
$202$	0	0
$203$	4.79912e7	0.402648
$204$	0	0
$205$	1.54320e8	1.25108
$206$	0	0
$207$	−1.31092e7	−0.102726
$208$	0	0
$209$	−159922.	−0.00121170
$210$	0	0
$211$	−2.45877e8	−1.80190	−0.900948	−	0.433928i	$-0.857127\pi$
$211$	−2.45877e8	−1.80190	−0.900948	+	0.433928i	$0.857127\pi$
$212$	0	0
$213$	6.44551e7	0.457013
$214$	0	0
$215$	−2.75146e8	−1.88811
$216$	0	0
$217$	7.85086e7	0.521565
$218$	0	0
$219$	5.32592e7	0.342642
$220$	0	0
$221$	−5.30740e7	−0.330757
$222$	0	0
$223$	−2.69676e8	−1.62845	−0.814226	−	0.580547i	$-0.802839\pi$
$223$	−2.69676e8	−1.62845	−0.814226	+	0.580547i	$0.802839\pi$
$224$	0	0
$225$	1.21267e8	0.709750
$226$	0	0
$227$	−6.95068e7	−0.394400	−0.197200	−	0.980363i	$-0.563185\pi$
$227$	−6.95068e7	−0.394400	−0.197200	+	0.980363i	$0.563185\pi$
$228$	0	0
$229$	−2.38154e8	−1.31049	−0.655245	−	0.755416i	$-0.727435\pi$
$229$	−2.38154e8	−1.31049	−0.655245	+	0.755416i	$0.727435\pi$
$230$	0	0
$231$	−436119.	−0.00232790
$232$	0	0
$233$	−2.85310e8	−1.47765	−0.738825	−	0.673898i	$-0.764619\pi$
$233$	−2.85310e8	−1.47765	−0.738825	+	0.673898i	$0.764619\pi$
$234$	0	0
$235$	−3.92702e8	−1.97390
$236$	0	0
$237$	−3.17774e6	−0.0155060
$238$	0	0
$239$	2.74585e8	1.30102	0.650511	−	0.759497i	$-0.274555\pi$
$239$	2.74585e8	1.30102	0.650511	+	0.759497i	$0.274555\pi$
$240$	0	0
$241$	2.63747e8	1.21375	0.606874	−	0.794798i	$-0.292423\pi$
$241$	2.63747e8	1.21375	0.606874	+	0.794798i	$0.292423\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	5.81706e7	0.252710
$246$	0	0
$247$	−1.23090e7	−0.0519735
$248$	0	0
$249$	1.37526e7	0.0564531
$250$	0	0
$251$	−2.33340e8	−0.931391	−0.465695	−	0.884945i	$-0.654196\pi$
$251$	−2.33340e8	−0.931391	−0.465695	+	0.884945i	$0.654196\pi$
$252$	0	0
$253$	−846832.	−0.00328757
$254$	0	0
$255$	1.95479e8	0.738261
$256$	0	0
$257$	−1.30277e8	−0.478742	−0.239371	−	0.970928i	$-0.576941\pi$
$257$	−1.30277e8	−0.478742	−0.239371	+	0.970928i	$0.576941\pi$
$258$	0	0
$259$	1.50324e8	0.537625
$260$	0	0
$261$	1.01999e8	0.355102
$262$	0	0
$263$	−2.21134e8	−0.749566	−0.374783	−	0.927113i	$-0.622283\pi$
$263$	−2.21134e8	−0.749566	−0.374783	+	0.927113i	$0.622283\pi$
$264$	0	0
$265$	1.01222e9	3.34130
$266$	0	0
$267$	1.12328e8	0.361160
$268$	0	0
$269$	−1.56343e8	−0.489717	−0.244859	−	0.969559i	$-0.578742\pi$
$269$	−1.56343e8	−0.489717	−0.244859	+	0.969559i	$0.578742\pi$
$270$	0	0
$271$	4.78673e8	1.46099	0.730494	−	0.682919i	$-0.239290\pi$
$271$	4.78673e8	1.46099	0.730494	+	0.682919i	$0.239290\pi$
$272$	0	0
$273$	−3.35674e7	−0.0998502
$274$	0	0
$275$	7.83365e6	0.0227143
$276$	0	0
$277$	−4.92210e8	−1.39146	−0.695730	−	0.718303i	$-0.744919\pi$
$277$	−4.92210e8	−1.39146	−0.695730	+	0.718303i	$0.744919\pi$
$278$	0	0
$279$	1.66859e8	0.459977
$280$	0	0
$281$	−1.40370e8	−0.377401	−0.188700	−	0.982035i	$-0.560428\pi$
$281$	−1.40370e8	−0.377401	−0.188700	+	0.982035i	$0.560428\pi$
$282$	0	0
$283$	1.67907e8	0.440369	0.220184	−	0.975458i	$-0.429334\pi$
$283$	1.67907e8	0.440369	0.220184	+	0.975458i	$0.429334\pi$
$284$	0	0
$285$	4.53357e7	0.116007
$286$	0	0
$287$	1.07054e8	0.267310
$288$	0	0
$289$	−1.95930e8	−0.477483
$290$	0	0
$291$	−2.27025e8	−0.540068
$292$	0	0
$293$	4.28971e8	0.996302	0.498151	−	0.867090i	$-0.334013\pi$
$293$	4.28971e8	0.996302	0.498151	+	0.867090i	$0.334013\pi$
$294$	0	0
$295$	1.26946e9	2.87900
$296$	0	0
$297$	−926913.	−0.00205301
$298$	0	0
$299$	−6.51793e7	−0.141013
$300$	0	0
$301$	−1.90872e8	−0.403422
$302$	0	0
$303$	−2.27775e8	−0.470389
$304$	0	0
$305$	−1.21686e9	−2.45579
$306$	0	0
$307$	3.34691e8	0.660176	0.330088	−	0.943950i	$-0.392921\pi$
$307$	3.34691e8	0.660176	0.330088	+	0.943950i	$0.392921\pi$
$308$	0	0
$309$	5.96146e6	0.0114947
$310$	0	0
$311$	7.62982e8	1.43831	0.719155	−	0.694849i	$-0.244529\pi$
$311$	7.62982e8	1.43831	0.719155	+	0.694849i	$0.244529\pi$
$312$	0	0
$313$	−4.65273e6	−0.00857636	−0.00428818	−	0.999991i	$-0.501365\pi$
$313$	−4.65273e6	−0.00857636	−0.00428818	+	0.999991i	$0.501365\pi$
$314$	0	0
$315$	1.23634e8	0.222869
$316$	0	0
$317$	−4.99816e8	−0.881258	−0.440629	−	0.897689i	$-0.645245\pi$
$317$	−4.99816e8	−0.881258	−0.440629	+	0.897689i	$0.645245\pi$
$318$	0	0
$319$	6.58893e6	0.0113644
$320$	0	0
$321$	3.59155e8	0.606057
$322$	0	0
$323$	4.97259e7	0.0821058
$324$	0	0
$325$	6.02944e8	0.974283
$326$	0	0
$327$	5.71386e8	0.903675
$328$	0	0
$329$	−2.72422e8	−0.421752
$330$	0	0
$331$	−3.89513e8	−0.590369	−0.295185	−	0.955440i	$-0.595381\pi$
$331$	−3.89513e8	−0.590369	−0.295185	+	0.955440i	$0.595381\pi$
$332$	0	0
$333$	3.19493e8	0.474141
$334$	0	0
$335$	1.06542e9	1.54834
$336$	0	0
$337$	1.70288e8	0.242371	0.121185	−	0.992630i	$-0.461330\pi$
$337$	1.70288e8	0.242371	0.121185	+	0.992630i	$0.461330\pi$
$338$	0	0
$339$	−6.04518e8	−0.842772
$340$	0	0
$341$	1.07788e7	0.0147208
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	2.40065e8	0.314747
$346$	0	0
$347$	−6.84193e8	−0.879074	−0.439537	−	0.898224i	$-0.644858\pi$
$347$	−6.84193e8	−0.879074	−0.439537	+	0.898224i	$0.644858\pi$
$348$	0	0
$349$	−7.67796e8	−0.966845	−0.483423	−	0.875387i	$-0.660607\pi$
$349$	−7.67796e8	−0.966845	−0.483423	+	0.875387i	$0.660607\pi$
$350$	0	0
$351$	−7.13430e7	−0.0880596
$352$	0	0
$353$	−1.27279e8	−0.154009	−0.0770046	−	0.997031i	$-0.524536\pi$
$353$	−1.27279e8	−0.154009	−0.0770046	+	0.997031i	$0.524536\pi$
$354$	0	0
$355$	−1.18034e9	−1.40026
$356$	0	0
$357$	1.35606e8	0.157740
$358$	0	0
$359$	1.33959e8	0.152806	0.0764032	−	0.997077i	$-0.475656\pi$
$359$	1.33959e8	0.152806	0.0764032	+	0.997077i	$0.475656\pi$
$360$	0	0
$361$	−8.82339e8	−0.987098
$362$	0	0
$363$	5.26094e8	0.577285
$364$	0	0
$365$	−9.75317e8	−1.04983
$366$	0	0
$367$	−1.18654e7	−0.0125300	−0.00626499	−	0.999980i	$-0.501994\pi$
$367$	−1.18654e7	−0.0125300	−0.00626499	+	0.999980i	$0.501994\pi$
$368$	0	0
$369$	2.27528e8	0.235745
$370$	0	0
$371$	7.02191e8	0.713915
$372$	0	0
$373$	−1.58982e9	−1.58623	−0.793117	−	0.609069i	$-0.791543\pi$
$373$	−1.58982e9	−1.58623	−0.793117	+	0.609069i	$0.791543\pi$
$374$	0	0
$375$	−1.17777e9	−1.15332
$376$	0	0
$377$	5.07140e8	0.487453
$378$	0	0
$379$	4.80180e8	0.453072	0.226536	−	0.974003i	$-0.427260\pi$
$379$	4.80180e8	0.453072	0.226536	+	0.974003i	$0.427260\pi$
$380$	0	0
$381$	9.33139e8	0.864388
$382$	0	0
$383$	−1.34744e9	−1.22550	−0.612752	−	0.790275i	$-0.709938\pi$
$383$	−1.34744e9	−1.22550	−0.612752	+	0.790275i	$0.709938\pi$
$384$	0	0
$385$	7.98650e6	0.00713254
$386$	0	0
$387$	−4.05672e8	−0.355784
$388$	0	0
$389$	1.31503e8	0.113270	0.0566348	−	0.998395i	$-0.481963\pi$
$389$	1.31503e8	0.113270	0.0566348	+	0.998395i	$0.481963\pi$
$390$	0	0
$391$	2.63312e8	0.222768
$392$	0	0
$393$	−1.09717e9	−0.911799
$394$	0	0
$395$	5.81929e7	0.0475095
$396$	0	0
$397$	−2.42834e9	−1.94779	−0.973894	−	0.227001i	$-0.927108\pi$
$397$	−2.42834e9	−1.94779	−0.973894	+	0.227001i	$0.927108\pi$
$398$	0	0
$399$	3.14499e7	0.0247864
$400$	0	0
$401$	8.32155e8	0.644464	0.322232	−	0.946661i	$-0.395567\pi$
$401$	8.32155e8	0.644464	0.322232	+	0.946661i	$0.395567\pi$
$402$	0	0
$403$	8.29628e8	0.631416
$404$	0	0
$405$	2.62767e8	0.196552
$406$	0	0
$407$	2.06387e7	0.0151740
$408$	0	0
$409$	2.50268e9	1.80873	0.904364	−	0.426762i	$-0.140346\pi$
$409$	2.50268e9	1.80873	0.904364	+	0.426762i	$0.140346\pi$
$410$	0	0
$411$	−9.05247e8	−0.643162
$412$	0	0
$413$	8.80638e8	0.615137
$414$	0	0
$415$	−2.51847e8	−0.172969
$416$	0	0
$417$	−9.23529e8	−0.623698
$418$	0	0
$419$	8.38811e8	0.557076	0.278538	−	0.960425i	$-0.410150\pi$
$419$	8.38811e8	0.557076	0.278538	+	0.960425i	$0.410150\pi$
$420$	0	0
$421$	2.01998e9	1.31935	0.659673	−	0.751553i	$-0.270695\pi$
$421$	2.01998e9	1.31935	0.659673	+	0.751553i	$0.270695\pi$
$422$	0	0
$423$	−5.78996e8	−0.371950
$424$	0	0
$425$	−2.43578e9	−1.53914
$426$	0	0
$427$	−8.44150e8	−0.524713
$428$	0	0
$429$	−4.60863e6	−0.00281819
$430$	0	0
$431$	2.61116e9	1.57095	0.785476	−	0.618893i	$-0.212418\pi$
$431$	2.61116e9	1.57095	0.785476	+	0.618893i	$0.212418\pi$
$432$	0	0
$433$	−1.05965e9	−0.627270	−0.313635	−	0.949544i	$-0.601547\pi$
$433$	−1.05965e9	−0.627270	−0.313635	+	0.949544i	$0.601547\pi$
$434$	0	0
$435$	−1.86787e9	−1.08801
$436$	0	0
$437$	6.10676e7	0.0350047
$438$	0	0
$439$	−7.50659e8	−0.423465	−0.211732	−	0.977328i	$-0.567911\pi$
$439$	−7.50659e8	−0.423465	−0.211732	+	0.977328i	$0.567911\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	0	0
$443$	1.40587e9	0.768300	0.384150	−	0.923271i	$-0.374495\pi$
$443$	1.40587e9	0.768300	0.384150	+	0.923271i	$0.374495\pi$
$444$	0	0
$445$	−2.05703e9	−1.10657
$446$	0	0
$447$	−5.98722e8	−0.317066
$448$	0	0
$449$	−1.95998e9	−1.02186	−0.510928	−	0.859623i	$-0.670698\pi$
$449$	−1.95998e9	−1.02186	−0.510928	+	0.859623i	$0.670698\pi$
$450$	0	0
$451$	1.46979e7	0.00754462
$452$	0	0
$453$	1.67060e9	0.844365
$454$	0	0
$455$	6.14709e8	0.305935
$456$	0	0
$457$	1.85623e9	0.909755	0.454878	−	0.890554i	$-0.349683\pi$
$457$	1.85623e9	0.909755	0.454878	+	0.890554i	$0.349683\pi$
$458$	0	0
$459$	2.88212e8	0.139113
$460$	0	0
$461$	−1.66716e9	−0.792546	−0.396273	−	0.918133i	$-0.629697\pi$
$461$	−1.66716e9	−0.792546	−0.396273	+	0.918133i	$0.629697\pi$
$462$	0	0
$463$	−1.83551e7	−0.00859457	−0.00429728	−	0.999991i	$-0.501368\pi$
$463$	−1.83551e7	−0.00859457	−0.00429728	+	0.999991i	$0.501368\pi$
$464$	0	0
$465$	−3.05564e9	−1.40934
$466$	0	0
$467$	3.89247e9	1.76855	0.884273	−	0.466970i	$-0.154655\pi$
$467$	3.89247e9	1.76855	0.884273	+	0.466970i	$0.154655\pi$
$468$	0	0
$469$	7.39097e8	0.330824
$470$	0	0
$471$	5.41561e8	0.238822
$472$	0	0
$473$	−2.62057e7	−0.0113863
$474$	0	0
$475$	−5.64908e8	−0.241852
$476$	0	0
$477$	1.49241e9	0.629614
$478$	0	0
$479$	4.13175e9	1.71775	0.858875	−	0.512185i	$-0.171164\pi$
$479$	4.13175e9	1.71775	0.858875	+	0.512185i	$0.171164\pi$
$480$	0	0
$481$	1.58853e9	0.650859
$482$	0	0
$483$	1.66536e8	0.0672500
$484$	0	0
$485$	4.15743e9	1.65474
$486$	0	0
$487$	1.58468e9	0.621714	0.310857	−	0.950457i	$-0.399384\pi$
$487$	1.58468e9	0.621714	0.310857	+	0.950457i	$0.399384\pi$
$488$	0	0
$489$	−5.81707e8	−0.224969
$490$	0	0
$491$	2.49037e9	0.949464	0.474732	−	0.880130i	$-0.342545\pi$
$491$	2.49037e9	0.949464	0.474732	+	0.880130i	$0.342545\pi$
$492$	0	0
$493$	−2.04875e9	−0.770061
$494$	0	0
$495$	1.69742e7	0.00629031
$496$	0	0
$497$	−8.18819e8	−0.299186
$498$	0	0
$499$	−3.17754e9	−1.14483	−0.572413	−	0.819966i	$-0.693992\pi$
$499$	−3.17754e9	−1.14483	−0.572413	+	0.819966i	$0.693992\pi$
$500$	0	0
$501$	9.40067e8	0.333985
$502$	0	0
$503$	4.91322e9	1.72139	0.860693	−	0.509124i	$-0.170031\pi$
$503$	4.91322e9	1.72139	0.860693	+	0.509124i	$0.170031\pi$
$504$	0	0
$505$	4.17117e9	1.44125
$506$	0	0
$507$	1.33949e9	0.456470
$508$	0	0
$509$	9.09545e8	0.305711	0.152856	−	0.988249i	$-0.451153\pi$
$509$	9.09545e8	0.305711	0.152856	+	0.988249i	$0.451153\pi$
$510$	0	0
$511$	−6.76589e8	−0.224312
$512$	0	0
$513$	6.68425e7	0.0218596
$514$	0	0
$515$	−1.09170e8	−0.0352191
$516$	0	0
$517$	−3.74021e7	−0.0119036
$518$	0	0
$519$	2.38824e9	0.749880
$520$	0	0
$521$	1.16113e9	0.359707	0.179853	−	0.983693i	$-0.442438\pi$
$521$	1.16113e9	0.359707	0.179853	+	0.983693i	$0.442438\pi$
$522$	0	0
$523$	1.80210e9	0.550837	0.275418	−	0.961324i	$-0.411184\pi$
$523$	1.80210e9	0.550837	0.275418	+	0.961324i	$0.411184\pi$
$524$	0	0
$525$	−1.54055e9	−0.464640
$526$	0	0
$527$	−3.35154e9	−0.997488
$528$	0	0
$529$	−3.08146e9	−0.905026
$530$	0	0
$531$	1.87168e9	0.542500
$532$	0	0
$533$	1.13128e9	0.323611
$534$	0	0
$535$	−6.57707e9	−1.85693
$536$	0	0
$537$	2.98643e9	0.832227
$538$	0	0
$539$	5.54033e6	0.00152397
$540$	0	0
$541$	5.32934e8	0.144705	0.0723525	−	0.997379i	$-0.476949\pi$
$541$	5.32934e8	0.144705	0.0723525	+	0.997379i	$0.476949\pi$
$542$	0	0
$543$	−3.55385e9	−0.952576
$544$	0	0
$545$	−1.04636e10	−2.76881
$546$	0	0
$547$	−6.64324e9	−1.73550	−0.867750	−	0.497001i	$-0.834434\pi$
$547$	−6.64324e9	−1.73550	−0.867750	+	0.497001i	$0.834434\pi$
$548$	0	0
$549$	−1.79413e9	−0.462753
$550$	0	0
$551$	−4.75148e8	−0.121004
$552$	0	0
$553$	4.03691e7	0.0101511
$554$	0	0
$555$	−5.85077e9	−1.45274
$556$	0	0
$557$	3.80213e9	0.932253	0.466127	−	0.884718i	$-0.345649\pi$
$557$	3.80213e9	0.932253	0.466127	+	0.884718i	$0.345649\pi$
$558$	0	0
$559$	−2.01701e9	−0.488390
$560$	0	0
$561$	1.86180e7	0.00445208
$562$	0	0
$563$	−5.42835e9	−1.28200	−0.641001	−	0.767540i	$-0.721481\pi$
$563$	−5.42835e9	−1.28200	−0.641001	+	0.767540i	$0.721481\pi$
$564$	0	0
$565$	1.10703e10	2.58221
$566$	0	0
$567$	1.82284e8	0.0419961
$568$	0	0
$569$	2.79698e9	0.636496	0.318248	−	0.948007i	$-0.396905\pi$
$569$	2.79698e9	0.636496	0.318248	+	0.948007i	$0.396905\pi$
$570$	0	0
$571$	7.26727e8	0.163360	0.0816799	−	0.996659i	$-0.473971\pi$
$571$	7.26727e8	0.163360	0.0816799	+	0.996659i	$0.473971\pi$
$572$	0	0
$573$	−1.21093e9	−0.268891
$574$	0	0
$575$	−2.99134e9	−0.656189
$576$	0	0
$577$	1.90887e9	0.413677	0.206838	−	0.978375i	$-0.433683\pi$
$577$	1.90887e9	0.413677	0.206838	+	0.978375i	$0.433683\pi$
$578$	0	0
$579$	−3.27246e8	−0.0700647
$580$	0	0
$581$	−1.74709e8	−0.0369572
$582$	0	0
$583$	9.64070e7	0.0201497
$584$	0	0
$585$	1.30648e9	0.269810
$586$	0	0
$587$	−3.18149e9	−0.649228	−0.324614	−	0.945847i	$-0.605234\pi$
$587$	−3.18149e9	−0.649228	−0.324614	+	0.945847i	$0.605234\pi$
$588$	0	0
$589$	−7.77293e8	−0.156740
$590$	0	0
$591$	1.51298e9	0.301494
$592$	0	0
$593$	−5.59654e9	−1.10212	−0.551059	−	0.834466i	$-0.685776\pi$
$593$	−5.59654e9	−1.10212	−0.551059	+	0.834466i	$0.685776\pi$
$594$	0	0
$595$	−2.48331e9	−0.483305
$596$	0	0
$597$	2.90316e9	0.558420
$598$	0	0
$599$	7.17615e9	1.36426	0.682130	−	0.731231i	$-0.261054\pi$
$599$	7.17615e9	1.36426	0.682130	+	0.731231i	$0.261054\pi$
$600$	0	0
$601$	7.05414e9	1.32551	0.662755	−	0.748836i	$-0.269387\pi$
$601$	7.05414e9	1.32551	0.662755	+	0.748836i	$0.269387\pi$
$602$	0	0
$603$	1.57085e9	0.291759
$604$	0	0
$605$	−9.63417e9	−1.76877
$606$	0	0
$607$	8.51078e9	1.54457	0.772287	−	0.635274i	$-0.219113\pi$
$607$	8.51078e9	1.54457	0.772287	+	0.635274i	$0.219113\pi$
$608$	0	0
$609$	−1.29576e9	−0.232469
$610$	0	0
$611$	−2.87878e9	−0.510580
$612$	0	0
$613$	−4.00820e8	−0.0702811	−0.0351405	−	0.999382i	$-0.511188\pi$
$613$	−4.00820e8	−0.0702811	−0.0351405	+	0.999382i	$0.511188\pi$
$614$	0	0
$615$	−4.16665e9	−0.722311
$616$	0	0
$617$	−1.69988e9	−0.291353	−0.145677	−	0.989332i	$-0.546536\pi$
$617$	−1.69988e9	−0.291353	−0.145677	+	0.989332i	$0.546536\pi$
$618$	0	0
$619$	−4.88473e9	−0.827796	−0.413898	−	0.910323i	$-0.635833\pi$
$619$	−4.88473e9	−0.827796	−0.413898	+	0.910323i	$0.635833\pi$
$620$	0	0
$621$	3.53949e8	0.0593090
$622$	0	0
$623$	−1.42699e9	−0.236435
$624$	0	0
$625$	8.57211e9	1.40445
$626$	0	0
$627$	4.31790e6	0.000699578 0
$628$	0	0
$629$	−6.41735e9	−1.02820
$630$	0	0
$631$	−1.03079e10	−1.63331	−0.816653	−	0.577129i	$-0.804173\pi$
$631$	−1.03079e10	−1.63331	−0.816653	+	0.577129i	$0.804173\pi$
$632$	0	0
$633$	6.63868e9	1.04032
$634$	0	0
$635$	−1.70883e10	−2.64844
$636$	0	0
$637$	4.26431e8	0.0653673
$638$	0	0
$639$	−1.74029e9	−0.263857
$640$	0	0
$641$	6.20053e9	0.929878	0.464939	−	0.885343i	$-0.346076\pi$
$641$	6.20053e9	0.929878	0.464939	+	0.885343i	$0.346076\pi$
$642$	0	0
$643$	−1.16268e10	−1.72474	−0.862368	−	0.506282i	$-0.831020\pi$
$643$	−1.16268e10	−1.72474	−0.862368	+	0.506282i	$0.831020\pi$
$644$	0	0
$645$	7.42893e9	1.09010
$646$	0	0
$647$	9.25127e9	1.34288	0.671439	−	0.741060i	$-0.265677\pi$
$647$	9.25127e9	1.34288	0.671439	+	0.741060i	$0.265677\pi$
$648$	0	0
$649$	1.20907e8	0.0173618
$650$	0	0
$651$	−2.11973e9	−0.301126
$652$	0	0
$653$	1.81381e9	0.254915	0.127457	−	0.991844i	$-0.459318\pi$
$653$	1.81381e9	0.254915	0.127457	+	0.991844i	$0.459318\pi$
$654$	0	0
$655$	2.00921e10	2.79370
$656$	0	0
$657$	−1.43800e9	−0.197824
$658$	0	0
$659$	−8.29325e9	−1.12882	−0.564412	−	0.825493i	$-0.690897\pi$
$659$	−8.29325e9	−1.12882	−0.564412	+	0.825493i	$0.690897\pi$
$660$	0	0
$661$	−1.00591e10	−1.35473	−0.677367	−	0.735645i	$-0.736879\pi$
$661$	−1.00591e10	−1.35473	−0.677367	+	0.735645i	$0.736879\pi$
$662$	0	0
$663$	1.43300e9	0.190963
$664$	0	0
$665$	−5.75931e8	−0.0759442
$666$	0	0
$667$	−2.51604e9	−0.328304
$668$	0	0
$669$	7.28125e9	0.940188
$670$	0	0
$671$	−1.15897e8	−0.0148096
$672$	0	0
$673$	−1.24551e10	−1.57505	−0.787524	−	0.616284i	$-0.788638\pi$
$673$	−1.24551e10	−1.57505	−0.787524	+	0.616284i	$0.788638\pi$
$674$	0	0
$675$	−3.27422e9	−0.409774
$676$	0	0
$677$	−1.07400e10	−1.33029	−0.665144	−	0.746715i	$-0.731630\pi$
$677$	−1.07400e10	−1.33029	−0.665144	+	0.746715i	$0.731630\pi$
$678$	0	0
$679$	2.88406e9	0.353558
$680$	0	0
$681$	1.87668e9	0.227707
$682$	0	0
$683$	7.36716e9	0.884764	0.442382	−	0.896827i	$-0.354133\pi$
$683$	7.36716e9	0.884764	0.442382	+	0.896827i	$0.354133\pi$
$684$	0	0
$685$	1.65775e10	1.97061
$686$	0	0
$687$	6.43016e9	0.756612
$688$	0	0
$689$	7.42030e9	0.864279
$690$	0	0
$691$	7.60126e9	0.876420	0.438210	−	0.898873i	$-0.355613\pi$
$691$	7.60126e9	0.876420	0.438210	+	0.898873i	$0.355613\pi$
$692$	0	0
$693$	1.17752e7	0.00134401
$694$	0	0
$695$	1.69123e10	1.91098
$696$	0	0
$697$	−4.57014e9	−0.511228
$698$	0	0
$699$	7.70337e9	0.853121
$700$	0	0
$701$	−7.13734e9	−0.782571	−0.391285	−	0.920269i	$-0.627969\pi$
$701$	−7.13734e9	−0.782571	−0.391285	+	0.920269i	$0.627969\pi$
$702$	0	0
$703$	−1.48832e9	−0.161567
$704$	0	0
$705$	1.06030e10	1.13963
$706$	0	0
$707$	2.89359e9	0.307942
$708$	0	0
$709$	2.65758e9	0.280042	0.140021	−	0.990149i	$-0.455283\pi$
$709$	2.65758e9	0.280042	0.140021	+	0.990149i	$0.455283\pi$
$710$	0	0
$711$	8.57990e7	0.00895239
$712$	0	0
$713$	−4.11597e9	−0.425265
$714$	0	0
$715$	8.43962e7	0.00863479
$716$	0	0
$717$	−7.41381e9	−0.751146
$718$	0	0
$719$	−7.05904e9	−0.708263	−0.354131	−	0.935196i	$-0.615223\pi$
$719$	−7.05904e9	−0.708263	−0.354131	+	0.935196i	$0.615223\pi$
$720$	0	0
$721$	−7.57327e7	−0.00752506
$722$	0	0
$723$	−7.12118e9	−0.700758
$724$	0	0
$725$	2.32747e10	2.26830
$726$	0	0
$727$	6.39475e9	0.617238	0.308619	−	0.951186i	$-0.400133\pi$
$727$	6.39475e9	0.617238	0.308619	+	0.951186i	$0.400133\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	8.14834e9	0.771540
$732$	0	0
$733$	−1.09324e10	−1.02530	−0.512651	−	0.858597i	$-0.671337\pi$
$733$	−1.09324e10	−1.02530	−0.512651	+	0.858597i	$0.671337\pi$
$734$	0	0
$735$	−1.57061e9	−0.145902
$736$	0	0
$737$	1.01474e8	0.00933724
$738$	0	0
$739$	5.28969e9	0.482142	0.241071	−	0.970507i	$-0.422501\pi$
$739$	5.28969e9	0.482142	0.241071	+	0.970507i	$0.422501\pi$
$740$	0	0
$741$	3.32342e8	0.0300069
$742$	0	0
$743$	−6.16099e9	−0.551048	−0.275524	−	0.961294i	$-0.588851\pi$
$743$	−6.16099e9	−0.551048	−0.275524	+	0.961294i	$0.588851\pi$
$744$	0	0
$745$	1.09642e10	0.971471
$746$	0	0
$747$	−3.71321e8	−0.0325932
$748$	0	0
$749$	−4.56259e9	−0.396758
$750$	0	0
$751$	4.20146e9	0.361960	0.180980	−	0.983487i	$-0.442073\pi$
$751$	4.20146e9	0.361960	0.180980	+	0.983487i	$0.442073\pi$
$752$	0	0
$753$	6.30019e9	0.537739
$754$	0	0
$755$	−3.05932e10	−2.58709
$756$	0	0
$757$	1.33579e10	1.11918	0.559592	−	0.828768i	$-0.310958\pi$
$757$	1.33579e10	1.11918	0.559592	+	0.828768i	$0.310958\pi$
$758$	0	0
$759$	2.28645e7	0.00189808
$760$	0	0
$761$	5.02956e9	0.413699	0.206849	−	0.978373i	$-0.433679\pi$
$761$	5.02956e9	0.413699	0.206849	+	0.978373i	$0.433679\pi$
$762$	0	0
$763$	−7.25871e9	−0.591594
$764$	0	0
$765$	−5.27794e9	−0.426235
$766$	0	0
$767$	9.30601e9	0.744697
$768$	0	0
$769$	−5.26581e9	−0.417564	−0.208782	−	0.977962i	$-0.566950\pi$
$769$	−5.26581e9	−0.417564	−0.208782	+	0.977962i	$0.566950\pi$
$770$	0	0
$771$	3.51748e9	0.276402
$772$	0	0
$773$	−1.89624e10	−1.47661	−0.738303	−	0.674469i	$-0.764373\pi$
$773$	−1.89624e10	−1.47661	−0.738303	+	0.674469i	$0.764373\pi$
$774$	0	0
$775$	3.80750e10	2.93822
$776$	0	0
$777$	−4.05875e9	−0.310398
$778$	0	0
$779$	−1.05991e9	−0.0803319
$780$	0	0
$781$	−1.12419e8	−0.00844428
$782$	0	0
$783$	−2.75397e9	−0.205018
$784$	0	0
$785$	−9.91743e9	−0.731737
$786$	0	0
$787$	−1.07037e10	−0.782747	−0.391373	−	0.920232i	$-0.628000\pi$
$787$	−1.07037e10	−0.782747	−0.391373	+	0.920232i	$0.628000\pi$
$788$	0	0
$789$	5.97061e9	0.432762
$790$	0	0
$791$	7.67962e9	0.551724
$792$	0	0
$793$	−8.92043e9	−0.635228
$794$	0	0
$795$	−2.73300e10	−1.92910
$796$	0	0
$797$	1.72938e10	1.21000	0.605002	−	0.796224i	$-0.293172\pi$
$797$	1.72938e10	1.21000	0.605002	+	0.796224i	$0.293172\pi$
$798$	0	0
$799$	1.16297e10	0.806596
$800$	0	0
$801$	−3.03287e9	−0.208516
$802$	0	0
$803$	−9.28920e7	−0.00633102
$804$	0	0
$805$	−3.04971e9	−0.206050
$806$	0	0
$807$	4.22126e9	0.282738
$808$	0	0
$809$	4.94516e9	0.328368	0.164184	−	0.986430i	$-0.447501\pi$
$809$	4.94516e9	0.328368	0.164184	+	0.986430i	$0.447501\pi$
$810$	0	0
$811$	−1.58525e10	−1.04358	−0.521789	−	0.853074i	$-0.674735\pi$
$811$	−1.58525e10	−1.04358	−0.521789	+	0.853074i	$0.674735\pi$
$812$	0	0
$813$	−1.29242e10	−0.843502
$814$	0	0
$815$	1.06526e10	0.689293
$816$	0	0
$817$	1.88977e9	0.121236
$818$	0	0
$819$	9.06321e8	0.0576485
$820$	0	0
$821$	3.27707e9	0.206674	0.103337	−	0.994646i	$-0.467048\pi$
$821$	3.27707e9	0.206674	0.103337	+	0.994646i	$0.467048\pi$
$822$	0	0
$823$	−1.90874e10	−1.19357	−0.596784	−	0.802402i	$-0.703555\pi$
$823$	−1.90874e10	−1.19357	−0.596784	+	0.802402i	$0.703555\pi$
$824$	0	0
$825$	−2.11508e8	−0.0131141
$826$	0	0
$827$	1.54063e9	0.0947173	0.0473587	−	0.998878i	$-0.484920\pi$
$827$	1.54063e9	0.0947173	0.0473587	+	0.998878i	$0.484920\pi$
$828$	0	0
$829$	−1.94769e10	−1.18735	−0.593675	−	0.804705i	$-0.702323\pi$
$829$	−1.94769e10	−1.18735	−0.593675	+	0.804705i	$0.702323\pi$
$830$	0	0
$831$	1.32897e10	0.803360
$832$	0	0
$833$	−1.72270e9	−0.103265
$834$	0	0
$835$	−1.72151e10	−1.02331
$836$	0	0
$837$	−4.50520e9	−0.265568
$838$	0	0
$839$	1.55492e10	0.908950	0.454475	−	0.890759i	$-0.349827\pi$
$839$	1.55492e10	0.908950	0.454475	+	0.890759i	$0.349827\pi$
$840$	0	0
$841$	2.32663e9	0.134878
$842$	0	0
$843$	3.79000e9	0.217893
$844$	0	0
$845$	−2.45296e10	−1.39860
$846$	0	0
$847$	−6.68334e9	−0.377921
$848$	0	0
$849$	−4.53349e9	−0.254247
$850$	0	0
$851$	−7.88105e9	−0.438360
$852$	0	0
$853$	1.01163e10	0.558083	0.279041	−	0.960279i	$-0.409983\pi$
$853$	1.01163e10	0.558083	0.279041	+	0.960279i	$0.409983\pi$
$854$	0	0
$855$	−1.22406e9	−0.0669765
$856$	0	0
$857$	−1.22085e10	−0.662566	−0.331283	−	0.943531i	$-0.607481\pi$
$857$	−1.22085e10	−0.662566	−0.331283	+	0.943531i	$0.607481\pi$
$858$	0	0
$859$	7.41595e9	0.399200	0.199600	−	0.979877i	$-0.436036\pi$
$859$	7.41595e9	0.399200	0.199600	+	0.979877i	$0.436036\pi$
$860$	0	0
$861$	−2.89045e9	−0.154332
$862$	0	0
$863$	−1.01350e10	−0.536767	−0.268384	−	0.963312i	$-0.586489\pi$
$863$	−1.01350e10	−0.536767	−0.268384	+	0.963312i	$0.586489\pi$
$864$	0	0
$865$	−4.37349e10	−2.29759
$866$	0	0
$867$	5.29010e9	0.275675
$868$	0	0
$869$	5.54246e6	0.000286506 0
$870$	0	0
$871$	7.81030e9	0.400501
$872$	0	0
$873$	6.12968e9	0.311808
$874$	0	0
$875$	1.49620e10	0.755024
$876$	0	0
$877$	−3.80796e10	−1.90631	−0.953155	−	0.302482i	$-0.902185\pi$
$877$	−3.80796e10	−1.90631	−0.953155	+	0.302482i	$0.902185\pi$
$878$	0	0
$879$	−1.15822e10	−0.575215
$880$	0	0
$881$	6.81088e9	0.335573	0.167787	−	0.985823i	$-0.446338\pi$
$881$	6.81088e9	0.335573	0.167787	+	0.985823i	$0.446338\pi$
$882$	0	0
$883$	1.00815e10	0.492792	0.246396	−	0.969169i	$-0.420754\pi$
$883$	1.00815e10	0.492792	0.246396	+	0.969169i	$0.420754\pi$
$884$	0	0
$885$	−3.42754e10	−1.66219
$886$	0	0
$887$	1.01793e10	0.489761	0.244881	−	0.969553i	$-0.421251\pi$
$887$	1.01793e10	0.489761	0.244881	+	0.969553i	$0.421251\pi$
$888$	0	0
$889$	−1.18543e10	−0.565875
$890$	0	0
$891$	2.50266e7	0.00118531
$892$	0	0
$893$	2.69718e9	0.126745
$894$	0	0
$895$	−5.46894e10	−2.54990
$896$	0	0
$897$	1.75984e9	0.0814142
$898$	0	0
$899$	3.20251e10	1.47005
$900$	0	0
$901$	−2.99766e10	−1.36536
$902$	0	0
$903$	5.15354e9	0.232916
$904$	0	0
$905$	6.50804e10	2.91864
$906$	0	0
$907$	2.44845e10	1.08960	0.544799	−	0.838567i	$-0.316606\pi$
$907$	2.44845e10	1.08960	0.544799	+	0.838567i	$0.316606\pi$
$908$	0	0
$909$	6.14993e9	0.271579
$910$	0	0
$911$	1.64923e10	0.722717	0.361358	−	0.932427i	$-0.382313\pi$
$911$	1.64923e10	0.722717	0.361358	+	0.932427i	$0.382313\pi$
$912$	0	0
$913$	−2.39866e7	−0.00104309
$914$	0	0
$915$	3.28552e10	1.41785
$916$	0	0
$917$	1.39381e10	0.596913
$918$	0	0
$919$	2.67297e9	0.113603	0.0568016	−	0.998385i	$-0.481910\pi$
$919$	2.67297e9	0.113603	0.0568016	+	0.998385i	$0.481910\pi$
$920$	0	0
$921$	−9.03666e9	−0.381153
$922$	0	0
$923$	−8.65275e9	−0.362200
$924$	0	0
$925$	7.29039e10	3.02869
$926$	0	0
$927$	−1.60959e8	−0.00663648
$928$	0	0
$929$	−3.34696e10	−1.36961	−0.684803	−	0.728728i	$-0.740112\pi$
$929$	−3.34696e10	−1.36961	−0.684803	+	0.728728i	$0.740112\pi$
$930$	0	0
$931$	−3.99530e8	−0.0162265
$932$	0	0
$933$	−2.06005e10	−0.830409
$934$	0	0
$935$	−3.40945e8	−0.0136409
$936$	0	0
$937$	2.18466e10	0.867552	0.433776	−	0.901021i	$-0.357181\pi$
$937$	2.18466e10	0.867552	0.433776	+	0.901021i	$0.357181\pi$
$938$	0	0
$939$	1.25624e8	0.00495156
$940$	0	0
$941$	2.23623e10	0.874890	0.437445	−	0.899245i	$-0.355883\pi$
$941$	2.23623e10	0.874890	0.437445	+	0.899245i	$0.355883\pi$
$942$	0	0
$943$	−5.61252e9	−0.217955
$944$	0	0
$945$	−3.33811e9	−0.128674
$946$	0	0
$947$	4.50399e10	1.72335	0.861673	−	0.507464i	$-0.169417\pi$
$947$	4.50399e10	1.72335	0.861673	+	0.507464i	$0.169417\pi$
$948$	0	0
$949$	−7.14975e9	−0.271556
$950$	0	0
$951$	1.34950e10	0.508794
$952$	0	0
$953$	−4.76726e10	−1.78420	−0.892100	−	0.451837i	$-0.850769\pi$
$953$	−4.76726e10	−1.78420	−0.892100	+	0.451837i	$0.850769\pi$
$954$	0	0
$955$	2.21753e10	0.823866
$956$	0	0
$957$	−1.77901e8	−0.00656126
$958$	0	0
$959$	1.15000e10	0.421049
$960$	0	0
$961$	2.48771e10	0.904208
$962$	0	0
$963$	−9.69717e9	−0.349907
$964$	0	0
$965$	5.99274e9	0.214674
$966$	0	0
$967$	−8.80378e9	−0.313095	−0.156548	−	0.987670i	$-0.550036\pi$
$967$	−8.80378e9	−0.313095	−0.156548	+	0.987670i	$0.550036\pi$
$968$	0	0
$969$	−1.34260e9	−0.0474038
$970$	0	0
$971$	2.02237e9	0.0708912	0.0354456	−	0.999372i	$-0.488715\pi$
$971$	2.02237e9	0.0708912	0.0354456	+	0.999372i	$0.488715\pi$
$972$	0	0
$973$	1.17322e10	0.408306
$974$	0	0
$975$	−1.62795e10	−0.562502
$976$	0	0
$977$	9.09480e9	0.312006	0.156003	−	0.987757i	$-0.450139\pi$
$977$	9.09480e9	0.312006	0.156003	+	0.987757i	$0.450139\pi$
$978$	0	0
$979$	−1.95918e8	−0.00667319
$980$	0	0
$981$	−1.54274e10	−0.521737
$982$	0	0
$983$	1.69966e10	0.570724	0.285362	−	0.958420i	$-0.407886\pi$
$983$	1.69966e10	0.570724	0.285362	+	0.958420i	$0.407886\pi$
$984$	0	0
$985$	−2.77068e10	−0.923760
$986$	0	0
$987$	7.35540e9	0.243498
$988$	0	0
$989$	1.00068e10	0.328935
$990$	0	0
$991$	2.75492e10	0.899190	0.449595	−	0.893233i	$-0.351568\pi$
$991$	2.75492e10	0.899190	0.449595	+	0.893233i	$0.351568\pi$
$992$	0	0
$993$	1.05168e10	0.340850
$994$	0	0
$995$	−5.31646e10	−1.71097
$996$	0	0
$997$	2.23696e10	0.714866	0.357433	−	0.933939i	$-0.383652\pi$
$997$	2.23696e10	0.714866	0.357433	+	0.933939i	$0.383652\pi$
$998$	0	0
$999$	−8.62632e9	−0.273745

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.8.a.c.1.2	✓	2
3.2	odd	2		252.8.a.c.1.1		2
4.3	odd	2		336.8.a.q.1.2		2
7.2	even	3		588.8.i.k.361.1		4
7.3	odd	6		588.8.i.j.373.2		4
7.4	even	3		588.8.i.k.373.1		4
7.5	odd	6		588.8.i.j.361.2		4
7.6	odd	2		588.8.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.8.a.c.1.2	✓	2	1.1	even	1	trivial
252.8.a.c.1.1		2	3.2	odd	2
336.8.a.q.1.2		2	4.3	odd	2
588.8.a.f.1.1		2	7.6	odd	2
588.8.i.j.361.2		4	7.5	odd	6
588.8.i.j.373.2		4	7.3	odd	6
588.8.i.k.361.1		4	7.2	even	3
588.8.i.k.373.1		4	7.4	even	3

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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +494.442 q^{5} +343.000 q^{7} +729.000 q^{9} +47.0920 q^{11} +3624.60 q^{13} -13349.9 q^{15} -14642.7 q^{17} -3395.95 q^{19} -9261.00 q^{21} -17982.5 q^{23} +166348. q^{25} -19683.0 q^{27} +139916. q^{29} +228888. q^{31} -1271.48 q^{33} +169594. q^{35} +438263. q^{37} -97864.2 q^{39} +312110. q^{41} -556478. q^{43} +360448. q^{45} -794234. q^{47} +117649. q^{49} +395353. q^{51} +2.04720e6 q^{53} +23284.3 q^{55} +91690.7 q^{57} +2.56746e6 q^{59} -2.46108e6 q^{61} +250047. q^{63} +1.79215e6 q^{65} +2.15480e6 q^{67} +485527. q^{69} -2.38723e6 q^{71} -1.97256e6 q^{73} -4.49139e6 q^{75} +16152.6 q^{77} +117694. q^{79} +531441. q^{81} -509356. q^{83} -7.23997e6 q^{85} -3.77773e6 q^{87} -4.16031e6 q^{89} +1.24324e6 q^{91} -6.17998e6 q^{93} -1.67910e6 q^{95} +8.40834e6 q^{97} +34330.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} + 264 q^{5} + 686 q^{7} + 1458 q^{9} - 4980 q^{11} - 10148 q^{13} - 7128 q^{15} + 17832 q^{17} + 6256 q^{19} - 18522 q^{21} + 14052 q^{23} + 141326 q^{25} - 39366 q^{27} + 243588 q^{29} + 470824 q^{31}+ \cdots - 3630420 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 + 264 * q^5 + 686 * q^7 + 1458 * q^9 - 4980 * q^11 - 10148 * q^13 - 7128 * q^15 + 17832 * q^17 + 6256 * q^19 - 18522 * q^21 + 14052 * q^23 + 141326 * q^25 - 39366 * q^27 + 243588 * q^29 + 470824 * q^31 + 134460 * q^33 + 90552 * q^35 + 311116 * q^37 + 273996 * q^39 + 919248 * q^41 - 112616 * q^43 + 192456 * q^45 - 102456 * q^47 + 235298 * q^49 - 481464 * q^51 + 2720028 * q^53 + 1181736 * q^55 - 168912 * q^57 - 19008 * q^59 - 925148 * q^61 + 500094 * q^63 + 4965936 * q^65 - 2053424 * q^67 - 379404 * q^69 - 869508 * q^71 + 3505228 * q^73 - 3815802 * q^75 - 1708140 * q^77 - 6640856 * q^79 + 1062882 * q^81 + 7856760 * q^83 - 14723496 * q^85 - 6576876 * q^87 - 9330384 * q^89 - 3480764 * q^91 - 12712248 * q^93 - 3903312 * q^95 - 2220212 * q^97 - 3630420 * q^99

Introduction

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