LMFDB - Embedded newform 27.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,8,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	27.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:	$8.43439568807$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{42}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 42 $ x^2 - 42
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.48074$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-19.4422 q^{2} +250.000 q^{4} -388.844 q^{5} -1261.00 q^{7} -2371.95 q^{8} +7560.00 q^{10} -1477.61 q^{11} +9581.00 q^{13} +24516.6 q^{14} +14116.0 q^{16} +21230.9 q^{17} -21931.0 q^{19} -97211.1 q^{20} +28728.0 q^{22} +85934.6 q^{23} +73075.0 q^{25} -186276. q^{26} -315250. q^{28} +32351.9 q^{29} -50908.0 q^{31} +29163.3 q^{32} -412776. q^{34} +490333. q^{35} +246467. q^{37} +426387. q^{38} +922320. q^{40} -610952. q^{41} +315512. q^{43} -369402. q^{44} -1.67076e6 q^{46} -425163. q^{47} +766578. q^{49} -1.42074e6 q^{50} +2.39525e6 q^{52} -127385. q^{53} +574560. q^{55} +2.99103e6 q^{56} -628992. q^{58} +964101. q^{59} -497953. q^{61} +989765. q^{62} -2.37385e6 q^{64} -3.72552e6 q^{65} +1.33636e6 q^{67} +5.30773e6 q^{68} -9.53316e6 q^{70} -901964. q^{71} +3.25079e6 q^{73} -4.79187e6 q^{74} -5.48275e6 q^{76} +1.86326e6 q^{77} +6.07548e6 q^{79} -5.48893e6 q^{80} +1.18783e7 q^{82} -8.19171e6 q^{83} -8.25552e6 q^{85} -6.13425e6 q^{86} +3.50482e6 q^{88} +1.30407e7 q^{89} -1.20816e7 q^{91} +2.14837e7 q^{92} +8.26610e6 q^{94} +8.52775e6 q^{95} +6.57063e6 q^{97} -1.49040e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 500 q^{4} - 2522 q^{7} + 15120 q^{10} + 19162 q^{13} + 28232 q^{16} - 43862 q^{19} + 57456 q^{22} + 146150 q^{25} - 630500 q^{28} - 101816 q^{31} - 825552 q^{34} + 492934 q^{37} + 1844640 q^{40} + 631024 q^{43}+ \cdots + 13141258 q^{97}+O(q^{100}) $ 2 * q + 500 * q^4 - 2522 * q^7 + 15120 * q^10 + 19162 * q^13 + 28232 * q^16 - 43862 * q^19 + 57456 * q^22 + 146150 * q^25 - 630500 * q^28 - 101816 * q^31 - 825552 * q^34 + 492934 * q^37 + 1844640 * q^40 + 631024 * q^43 - 3341520 * q^46 + 1533156 * q^49 + 4790500 * q^52 + 1149120 * q^55 - 1257984 * q^58 - 995906 * q^61 - 4747696 * q^64 + 2672722 * q^67 - 19066320 * q^70 + 6501586 * q^73 - 10965500 * q^76 + 12150970 * q^79 + 23756544 * q^82 - 16511040 * q^85 + 7009632 * q^88 - 24163282 * q^91 + 16532208 * q^94 + 13141258 * q^97

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 $

$2$	−19.4422	−1.71847	−0.859233	−	0.511585i	$-0.829059\pi$
$2$	−19.4422	−1.71847	−0.859233	+	0.511585i	$0.829059\pi$
$3$	0	0
$3$	0	0
$4$	250.000	1.95312
$5$	−388.844	−1.39117	−0.695586	−	0.718443i	$-0.744855\pi$
$5$	−388.844	−1.39117	−0.695586	+	0.718443i	$0.744855\pi$
$6$	0	0
$7$	−1261.00	−1.38954	−0.694771	−	0.719231i	$-0.744494\pi$
$7$	−1261.00	−1.38954	−0.694771	+	0.719231i	$0.744494\pi$
$8$	−2371.95	−1.63791
$9$	0	0
$10$	7560.00	2.39068
$11$	−1477.61	−0.334723	−0.167361	−	0.985896i	$-0.553525\pi$
$11$	−1477.61	−0.334723	−0.167361	+	0.985896i	$0.553525\pi$
$12$	0	0
$13$	9581.00	1.20951	0.604755	−	0.796412i	$-0.293271\pi$
$13$	9581.00	1.20951	0.604755	+	0.796412i	$0.293271\pi$
$14$	24516.6	2.38788
$15$	0	0
$16$	14116.0	0.861572
$17$	21230.9	1.04809	0.524043	−	0.851692i	$-0.324423\pi$
$17$	21230.9	1.04809	0.524043	+	0.851692i	$0.324423\pi$
$18$	0	0
$19$	−21931.0	−0.733535	−0.366767	−	0.930313i	$-0.619536\pi$
$19$	−21931.0	−0.733535	−0.366767	+	0.930313i	$0.619536\pi$
$20$	−97211.1	−2.71713
$21$	0	0
$22$	28728.0	0.575209
$23$	85934.6	1.47272	0.736361	−	0.676589i	$-0.236543\pi$
$23$	85934.6	1.47272	0.736361	+	0.676589i	$0.236543\pi$
$24$	0	0
$25$	73075.0	0.935360
$26$	−186276.	−2.07850
$27$	0	0
$28$	−315250.	−2.71395
$29$	32351.9	0.246324	0.123162	−	0.992387i	$-0.460697\pi$
$29$	32351.9	0.246324	0.123162	+	0.992387i	$0.460697\pi$
$30$	0	0
$31$	−50908.0	−0.306916	−0.153458	−	0.988155i	$-0.549041\pi$
$31$	−50908.0	−0.306916	−0.153458	+	0.988155i	$0.549041\pi$
$32$	29163.3	0.157330
$33$	0	0
$34$	−412776.	−1.80110
$35$	490333.	1.93309
$36$	0	0
$37$	246467.	0.799931	0.399966	−	0.916530i	$-0.369022\pi$
$37$	246467.	0.799931	0.399966	+	0.916530i	$0.369022\pi$
$38$	426387.	1.26055
$39$	0	0
$40$	922320.	2.27862
$41$	−610952.	−1.38441	−0.692204	−	0.721702i	$-0.743360\pi$
$41$	−610952.	−1.38441	−0.692204	+	0.721702i	$0.743360\pi$
$42$	0	0
$43$	315512.	0.605168	0.302584	−	0.953123i	$-0.402151\pi$
$43$	315512.	0.605168	0.302584	+	0.953123i	$0.402151\pi$
$44$	−369402.	−0.653755
$45$	0	0
$46$	−1.67076e6	−2.53082
$47$	−425163.	−0.597327	−0.298664	−	0.954358i	$-0.596541\pi$
$47$	−425163.	−0.597327	−0.298664	+	0.954358i	$0.596541\pi$
$48$	0	0
$49$	766578.	0.930829
$50$	−1.42074e6	−1.60738
$51$	0	0
$52$	2.39525e6	2.36232
$53$	−127385.	−0.117531	−0.0587657	−	0.998272i	$-0.518717\pi$
$53$	−127385.	−0.117531	−0.0587657	+	0.998272i	$0.518717\pi$
$54$	0	0
$55$	574560.	0.465657
$56$	2.99103e6	2.27595
$57$	0	0
$58$	−628992.	−0.423299
$59$	964101.	0.611140	0.305570	−	0.952170i	$-0.401153\pi$
$59$	964101.	0.611140	0.305570	+	0.952170i	$0.401153\pi$
$60$	0	0
$61$	−497953.	−0.280888	−0.140444	−	0.990089i	$-0.544853\pi$
$61$	−497953.	−0.280888	−0.140444	+	0.990089i	$0.544853\pi$
$62$	989765.	0.527425
$63$	0	0
$64$	−2.37385e6	−1.13194
$65$	−3.72552e6	−1.68264
$66$	0	0
$67$	1.33636e6	0.542828	0.271414	−	0.962463i	$-0.412509\pi$
$67$	1.33636e6	0.542828	0.271414	+	0.962463i	$0.412509\pi$
$68$	5.30773e6	2.04704
$69$	0	0
$70$	−9.53316e6	−3.32195
$71$	−901964.	−0.299078	−0.149539	−	0.988756i	$-0.547779\pi$
$71$	−901964.	−0.299078	−0.149539	+	0.988756i	$0.547779\pi$
$72$	0	0
$73$	3.25079e6	0.978046	0.489023	−	0.872271i	$-0.337353\pi$
$73$	3.25079e6	0.978046	0.489023	+	0.872271i	$0.337353\pi$
$74$	−4.79187e6	−1.37465
$75$	0	0
$76$	−5.48275e6	−1.43269
$77$	1.86326e6	0.465111
$78$	0	0
$79$	6.07548e6	1.38639	0.693196	−	0.720749i	$-0.256202\pi$
$79$	6.07548e6	1.38639	0.693196	+	0.720749i	$0.256202\pi$
$80$	−5.48893e6	−1.19860
$81$	0	0
$82$	1.18783e7	2.37906
$83$	−8.19171e6	−1.57254	−0.786269	−	0.617884i	$-0.787990\pi$
$83$	−8.19171e6	−1.57254	−0.786269	+	0.617884i	$0.787990\pi$
$84$	0	0
$85$	−8.25552e6	−1.45807
$86$	−6.13425e6	−1.03996
$87$	0	0
$88$	3.50482e6	0.548247
$89$	1.30407e7	1.96081	0.980404	−	0.196997i	$-0.0631190\pi$
$89$	1.30407e7	1.96081	0.980404	+	0.196997i	$0.0631190\pi$
$90$	0	0
$91$	−1.20816e7	−1.68066
$92$	2.14837e7	2.87641
$93$	0	0
$94$	8.26610e6	1.02649
$95$	8.52775e6	1.02047
$96$	0	0
$97$	6.57063e6	0.730981	0.365490	−	0.930815i	$-0.380901\pi$
$97$	6.57063e6	0.730981	0.365490	+	0.930815i	$0.380901\pi$
$98$	−1.49040e7	−1.59960
$99$	0	0
$100$	1.82688e7	1.82688
$101$	1.37657e7	1.32946	0.664728	−	0.747085i	$-0.268547\pi$
$101$	1.37657e7	1.32946	0.664728	+	0.747085i	$0.268547\pi$
$102$	0	0
$103$	412511.	0.0371968	0.0185984	−	0.999827i	$-0.494080\pi$
$103$	412511.	0.0371968	0.0185984	+	0.999827i	$0.494080\pi$
$104$	−2.27257e7	−1.98107
$105$	0	0
$106$	2.47666e6	0.201974
$107$	1.01453e7	0.800614	0.400307	−	0.916381i	$-0.368903\pi$
$107$	1.01453e7	0.800614	0.400307	+	0.916381i	$0.368903\pi$
$108$	0	0
$109$	1.84661e7	1.36578	0.682891	−	0.730520i	$-0.260722\pi$
$109$	1.84661e7	1.36578	0.682891	+	0.730520i	$0.260722\pi$
$110$	−1.11707e7	−0.800215
$111$	0	0
$112$	−1.78003e7	−1.19719
$113$	−2.05262e7	−1.33824	−0.669121	−	0.743153i	$-0.733329\pi$
$113$	−2.05262e7	−1.33824	−0.669121	+	0.743153i	$0.733329\pi$
$114$	0	0
$115$	−3.34152e7	−2.04881
$116$	8.08796e6	0.481101
$117$	0	0
$118$	−1.87443e7	−1.05022
$119$	−2.67722e7	−1.45636
$120$	0	0
$121$	−1.73038e7	−0.887961
$122$	9.68131e6	0.482697
$123$	0	0
$124$	−1.27270e7	−0.599446
$125$	1.96366e6	0.0899254
$126$	0	0
$127$	−1.70206e7	−0.737328	−0.368664	−	0.929563i	$-0.620185\pi$
$127$	−1.70206e7	−0.737328	−0.368664	+	0.929563i	$0.620185\pi$
$128$	4.24200e7	1.78787
$129$	0	0
$130$	7.24324e7	2.89155
$131$	4.00759e6	0.155752	0.0778760	−	0.996963i	$-0.475186\pi$
$131$	4.00759e6	0.155752	0.0778760	+	0.996963i	$0.475186\pi$
$132$	0	0
$133$	2.76550e7	1.01928
$134$	−2.59818e7	−0.932831
$135$	0	0
$136$	−5.03587e7	−1.71668
$137$	−9.85402e6	−0.327410	−0.163705	−	0.986509i	$-0.552344\pi$
$137$	−9.85402e6	−0.327410	−0.163705	+	0.986509i	$0.552344\pi$
$138$	0	0
$139$	−4.29592e7	−1.35676	−0.678382	−	0.734710i	$-0.737318\pi$
$139$	−4.29592e7	−1.35676	−0.678382	+	0.734710i	$0.737318\pi$
$140$	1.22583e8	3.77557
$141$	0	0
$142$	1.75362e7	0.513956
$143$	−1.41570e7	−0.404850
$144$	0	0
$145$	−1.25798e7	−0.342679
$146$	−6.32026e7	−1.68074
$147$	0	0
$148$	6.16167e7	1.56237
$149$	−1.90622e7	−0.472087	−0.236044	−	0.971742i	$-0.575851\pi$
$149$	−1.90622e7	−0.472087	−0.236044	+	0.971742i	$0.575851\pi$
$150$	0	0
$151$	6.80448e7	1.60833	0.804165	−	0.594405i	$-0.202613\pi$
$151$	6.80448e7	1.60833	0.804165	+	0.594405i	$0.202613\pi$
$152$	5.20193e7	1.20147
$153$	0	0
$154$	−3.62260e7	−0.799278
$155$	1.97953e7	0.426974
$156$	0	0
$157$	−1.23676e7	−0.255058	−0.127529	−	0.991835i	$-0.540705\pi$
$157$	−1.23676e7	−0.255058	−0.127529	+	0.991835i	$0.540705\pi$
$158$	−1.18121e8	−2.38247
$159$	0	0
$160$	−1.13400e7	−0.218873
$161$	−1.08364e8	−2.04641
$162$	0	0
$163$	2.03190e7	0.367490	0.183745	−	0.982974i	$-0.441178\pi$
$163$	2.03190e7	0.367490	0.183745	+	0.982974i	$0.441178\pi$
$164$	−1.52738e8	−2.70392
$165$	0	0
$166$	1.59265e8	2.70235
$167$	8.89076e7	1.47717	0.738587	−	0.674159i	$-0.235494\pi$
$167$	8.89076e7	1.47717	0.738587	+	0.674159i	$0.235494\pi$
$168$	0	0
$169$	2.90470e7	0.462912
$170$	1.60506e8	2.50564
$171$	0	0
$172$	7.88780e7	1.18197
$173$	1.30135e8	1.91088	0.955441	−	0.295182i	$-0.0953802\pi$
$173$	1.30135e8	1.91088	0.955441	+	0.295182i	$0.0953802\pi$
$174$	0	0
$175$	−9.21476e7	−1.29972
$176$	−2.08579e7	−0.288388
$177$	0	0
$178$	−2.53540e8	−3.36958
$179$	−4.09514e7	−0.533683	−0.266841	−	0.963740i	$-0.585980\pi$
$179$	−4.09514e7	−0.533683	−0.266841	+	0.963740i	$0.585980\pi$
$180$	0	0
$181$	6.43779e7	0.806979	0.403489	−	0.914984i	$-0.367797\pi$
$181$	6.43779e7	0.806979	0.403489	+	0.914984i	$0.367797\pi$
$182$	2.34894e8	2.88816
$183$	0	0
$184$	−2.03833e8	−2.41219
$185$	−9.58373e7	−1.11284
$186$	0	0
$187$	−3.13710e7	−0.350818
$188$	−1.06291e8	−1.16666
$189$	0	0
$190$	−1.65798e8	−1.75365
$191$	9.97479e7	1.03583	0.517913	−	0.855433i	$-0.326709\pi$
$191$	9.97479e7	1.03583	0.517913	+	0.855433i	$0.326709\pi$
$192$	0	0
$193$	1.97025e8	1.97274	0.986371	−	0.164539i	$-0.0526136\pi$
$193$	1.97025e8	1.97274	0.986371	+	0.164539i	$0.0526136\pi$
$194$	−1.27748e8	−1.25617
$195$	0	0
$196$	1.91644e8	1.81803
$197$	1.46469e8	1.36494	0.682472	−	0.730912i	$-0.260905\pi$
$197$	1.46469e8	1.36494	0.682472	+	0.730912i	$0.260905\pi$
$198$	0	0
$199$	−1.26743e7	−0.114009	−0.0570045	−	0.998374i	$-0.518155\pi$
$199$	−1.26743e7	−0.114009	−0.0570045	+	0.998374i	$0.518155\pi$
$200$	−1.73330e8	−1.53204
$201$	0	0
$202$	−2.67636e8	−2.28462
$203$	−4.07957e7	−0.342277
$204$	0	0
$205$	2.37565e8	1.92595
$206$	−8.02013e6	−0.0639214
$207$	0	0
$208$	1.35245e8	1.04208
$209$	3.24054e7	0.245531
$210$	0	0
$211$	−2.21343e8	−1.62210	−0.811051	−	0.584976i	$-0.801104\pi$
$211$	−2.21343e8	−1.62210	−0.811051	+	0.584976i	$0.801104\pi$
$212$	−3.18464e7	−0.229554
$213$	0	0
$214$	−1.97248e8	−1.37583
$215$	−1.22685e8	−0.841894
$216$	0	0
$217$	6.41950e7	0.426474
$218$	−3.59021e8	−2.34705
$219$	0	0
$220$	1.43640e8	0.909486
$221$	2.03413e8	1.26767
$222$	0	0
$223$	−1.35957e8	−0.820981	−0.410491	−	0.911865i	$-0.634643\pi$
$223$	−1.35957e8	−0.820981	−0.410491	+	0.911865i	$0.634643\pi$
$224$	−3.67750e7	−0.218617
$225$	0	0
$226$	3.99076e8	2.29972
$227$	−5.82817e7	−0.330706	−0.165353	−	0.986234i	$-0.552876\pi$
$227$	−5.82817e7	−0.330706	−0.165353	+	0.986234i	$0.552876\pi$
$228$	0	0
$229$	5.13523e6	0.0282577	0.0141288	−	0.999900i	$-0.495503\pi$
$229$	5.13523e6	0.0282577	0.0141288	+	0.999900i	$0.495503\pi$
$230$	6.49666e8	3.52081
$231$	0	0
$232$	−7.67370e7	−0.403457
$233$	5.10755e6	0.0264525	0.0132263	−	0.999913i	$-0.495790\pi$
$233$	5.10755e6	0.0264525	0.0132263	+	0.999913i	$0.495790\pi$
$234$	0	0
$235$	1.65322e8	0.830985
$236$	2.41025e8	1.19363
$237$	0	0
$238$	5.20511e8	2.50271
$239$	−2.65911e8	−1.25992	−0.629961	−	0.776627i	$-0.716929\pi$
$239$	−2.65911e8	−1.25992	−0.629961	+	0.776627i	$0.716929\pi$
$240$	0	0
$241$	1.79884e8	0.827813	0.413906	−	0.910319i	$-0.364164\pi$
$241$	1.79884e8	0.827813	0.413906	+	0.910319i	$0.364164\pi$
$242$	3.36425e8	1.52593
$243$	0	0
$244$	−1.24488e8	−0.548610
$245$	−2.98080e8	−1.29494
$246$	0	0
$247$	−2.10121e8	−0.887217
$248$	1.20751e8	0.502702
$249$	0	0
$250$	−3.81780e7	−0.154534
$251$	9.33796e7	0.372730	0.186365	−	0.982481i	$-0.440329\pi$
$251$	9.33796e7	0.372730	0.186365	+	0.982481i	$0.440329\pi$
$252$	0	0
$253$	−1.26978e8	−0.492953
$254$	3.30918e8	1.26707
$255$	0	0
$256$	−5.20886e8	−1.94045
$257$	−4.68071e7	−0.172007	−0.0860034	−	0.996295i	$-0.527410\pi$
$257$	−4.68071e7	−0.172007	−0.0860034	+	0.996295i	$0.527410\pi$
$258$	0	0
$259$	−3.10795e8	−1.11154
$260$	−9.31380e8	−3.28640
$261$	0	0
$262$	−7.79164e7	−0.267654
$263$	−2.90594e8	−0.985014	−0.492507	−	0.870309i	$-0.663919\pi$
$263$	−2.90594e8	−0.985014	−0.492507	+	0.870309i	$0.663919\pi$
$264$	0	0
$265$	4.95331e7	0.163507
$266$	−5.37674e8	−1.75159
$267$	0	0
$268$	3.34090e8	1.06021
$269$	−1.81315e8	−0.567938	−0.283969	−	0.958834i	$-0.591651\pi$
$269$	−1.81315e8	−0.567938	−0.283969	+	0.958834i	$0.591651\pi$
$270$	0	0
$271$	−5.68408e8	−1.73487	−0.867437	−	0.497547i	$-0.834234\pi$
$271$	−5.68408e8	−1.73487	−0.867437	+	0.497547i	$0.834234\pi$
$272$	2.99695e8	0.903003
$273$	0	0
$274$	1.91584e8	0.562642
$275$	−1.07976e8	−0.313086
$276$	0	0
$277$	2.49241e8	0.704595	0.352298	−	0.935888i	$-0.385401\pi$
$277$	2.49241e8	0.704595	0.352298	+	0.935888i	$0.385401\pi$
$278$	8.35222e8	2.33155
$279$	0	0
$280$	−1.16305e9	−3.16624
$281$	−1.31976e8	−0.354833	−0.177416	−	0.984136i	$-0.556774\pi$
$281$	−1.31976e8	−0.354833	−0.177416	+	0.984136i	$0.556774\pi$
$282$	0	0
$283$	4.79669e8	1.25802	0.629012	−	0.777395i	$-0.283460\pi$
$283$	4.79669e8	1.25802	0.629012	+	0.777395i	$0.283460\pi$
$284$	−2.25491e8	−0.584137
$285$	0	0
$286$	2.75243e8	0.695721
$287$	7.70411e8	1.92369
$288$	0	0
$289$	4.04127e7	0.0984863
$290$	2.44580e8	0.588882
$291$	0	0
$292$	8.12698e8	1.91025
$293$	4.76703e8	1.10716	0.553582	−	0.832795i	$-0.313261\pi$
$293$	4.76703e8	1.10716	0.553582	+	0.832795i	$0.313261\pi$
$294$	0	0
$295$	−3.74885e8	−0.850200
$296$	−5.84608e8	−1.31022
$297$	0	0
$298$	3.70612e8	0.811265
$299$	8.23340e8	1.78127
$300$	0	0
$301$	−3.97861e8	−0.840908
$302$	−1.32294e9	−2.76386
$303$	0	0
$304$	−3.09578e8	−0.631993
$305$	1.93626e8	0.390764
$306$	0	0
$307$	−1.08731e8	−0.214471	−0.107236	−	0.994234i	$-0.534200\pi$
$307$	−1.08731e8	−0.214471	−0.107236	+	0.994234i	$0.534200\pi$
$308$	4.65816e8	0.908421
$309$	0	0
$310$	−3.84864e8	−0.733740
$311$	2.78144e8	0.524335	0.262167	−	0.965022i	$-0.415563\pi$
$311$	2.78144e8	0.524335	0.262167	+	0.965022i	$0.415563\pi$
$312$	0	0
$313$	−8.31727e8	−1.53312	−0.766560	−	0.642173i	$-0.778033\pi$
$313$	−8.31727e8	−1.53312	−0.766560	+	0.642173i	$0.778033\pi$
$314$	2.40454e8	0.438308
$315$	0	0
$316$	1.51887e9	2.70780
$317$	−8.56343e8	−1.50987	−0.754936	−	0.655798i	$-0.772332\pi$
$317$	−8.56343e8	−1.50987	−0.754936	+	0.655798i	$0.772332\pi$
$318$	0	0
$319$	−4.78034e7	−0.0824501
$320$	9.23058e8	1.57472
$321$	0	0
$322$	2.10683e9	3.51669
$323$	−4.65615e8	−0.768808
$324$	0	0
$325$	7.00132e8	1.13133
$326$	−3.95046e8	−0.631519
$327$	0	0
$328$	1.44915e9	2.26754
$329$	5.36130e8	0.830012
$330$	0	0
$331$	−1.91069e7	−0.0289596	−0.0144798	−	0.999895i	$-0.504609\pi$
$331$	−1.91069e7	−0.0289596	−0.0144798	+	0.999895i	$0.504609\pi$
$332$	−2.04793e9	−3.07136
$333$	0	0
$334$	−1.72856e9	−2.53847
$335$	−5.19637e8	−0.755167
$336$	0	0
$337$	7.42680e8	1.05705	0.528527	−	0.848917i	$-0.322745\pi$
$337$	7.42680e8	1.05705	0.528527	+	0.848917i	$0.322745\pi$
$338$	−5.64739e8	−0.795499
$339$	0	0
$340$	−2.06388e9	−2.84779
$341$	7.52221e7	0.102732
$342$	0	0
$343$	7.18329e7	0.0961156
$344$	−7.48379e8	−0.991213
$345$	0	0
$346$	−2.53012e9	−3.28379
$347$	58637.7	7.53397e−5 0	3.76699e−5	−	1.00000i	$-0.499988\pi$
$347$	58637.7	7.53397e−5 0	3.76699e−5		1.00000i	$0.499988\pi$
$348$	0	0
$349$	8.21859e8	1.03492	0.517462	−	0.855706i	$-0.326877\pi$
$349$	8.21859e8	1.03492	0.517462	+	0.855706i	$0.326877\pi$
$350$	1.79155e9	2.23353
$351$	0	0
$352$	−4.30920e7	−0.0526620
$353$	8.01564e6	0.00969899	0.00484950	−	0.999988i	$-0.498456\pi$
$353$	8.01564e6	0.00969899	0.00484950	+	0.999988i	$0.498456\pi$
$354$	0	0
$355$	3.50724e8	0.416069
$356$	3.26017e9	3.82970
$357$	0	0
$358$	7.96186e8	0.917115
$359$	7.77296e8	0.886657	0.443329	−	0.896359i	$-0.353797\pi$
$359$	7.77296e8	0.886657	0.443329	+	0.896359i	$0.353797\pi$
$360$	0	0
$361$	−4.12903e8	−0.461926
$362$	−1.25165e9	−1.38677
$363$	0	0
$364$	−3.02041e9	−3.28255
$365$	−1.26405e9	−1.36063
$366$	0	0
$367$	−5.92955e8	−0.626167	−0.313084	−	0.949726i	$-0.601362\pi$
$367$	−5.92955e8	−0.626167	−0.313084	+	0.949726i	$0.601362\pi$
$368$	1.21305e9	1.26886
$369$	0	0
$370$	1.86329e9	1.91238
$371$	1.60633e8	0.163315
$372$	0	0
$373$	8.77986e8	0.876005	0.438002	−	0.898974i	$-0.355686\pi$
$373$	8.77986e8	0.876005	0.438002	+	0.898974i	$0.355686\pi$
$374$	6.09921e8	0.602870
$375$	0	0
$376$	1.00846e9	0.978370
$377$	3.09963e8	0.297931
$378$	0	0
$379$	−1.51623e8	−0.143063	−0.0715314	−	0.997438i	$-0.522789\pi$
$379$	−1.51623e8	−0.143063	−0.0715314	+	0.997438i	$0.522789\pi$
$380$	2.13194e9	1.99311
$381$	0	0
$382$	−1.93932e9	−1.78003
$383$	−1.54017e9	−1.40079	−0.700397	−	0.713754i	$-0.746993\pi$
$383$	−1.54017e9	−1.40079	−0.700397	+	0.713754i	$0.746993\pi$
$384$	0	0
$385$	−7.24520e8	−0.647050
$386$	−3.83060e9	−3.39009
$387$	0	0
$388$	1.64266e9	1.42770
$389$	1.21574e9	1.04717	0.523585	−	0.851973i	$-0.324594\pi$
$389$	1.21574e9	1.04717	0.523585	+	0.851973i	$0.324594\pi$
$390$	0	0
$391$	1.82447e9	1.54354
$392$	−1.81829e9	−1.52462
$393$	0	0
$394$	−2.84769e9	−2.34561
$395$	−2.36242e9	−1.92871
$396$	0	0
$397$	−8.04851e8	−0.645578	−0.322789	−	0.946471i	$-0.604620\pi$
$397$	−8.04851e8	−0.645578	−0.322789	+	0.946471i	$0.604620\pi$
$398$	2.46417e8	0.195921
$399$	0	0
$400$	1.03153e9	0.805880
$401$	9.50522e8	0.736134	0.368067	−	0.929799i	$-0.380020\pi$
$401$	9.50522e8	0.736134	0.368067	+	0.929799i	$0.380020\pi$
$402$	0	0
$403$	−4.87750e8	−0.371218
$404$	3.44143e9	2.59659
$405$	0	0
$406$	7.93159e8	0.588192
$407$	−3.64182e8	−0.267755
$408$	0	0
$409$	1.61635e8	0.116816	0.0584081	−	0.998293i	$-0.481398\pi$
$409$	1.61635e8	0.116816	0.0584081	+	0.998293i	$0.481398\pi$
$410$	−4.61880e9	−3.30968
$411$	0	0
$412$	1.03128e8	0.0726499
$413$	−1.21573e9	−0.849205
$414$	0	0
$415$	3.18530e9	2.18767
$416$	2.79414e8	0.190292
$417$	0	0
$418$	−6.30034e8	−0.421936
$419$	−6.65427e7	−0.0441928	−0.0220964	−	0.999756i	$-0.507034\pi$
$419$	−6.65427e7	−0.0441928	−0.0220964	+	0.999756i	$0.507034\pi$
$420$	0	0
$421$	2.04026e9	1.33259	0.666297	−	0.745686i	$-0.267878\pi$
$421$	2.04026e9	1.33259	0.666297	+	0.745686i	$0.267878\pi$
$422$	4.30341e9	2.78753
$423$	0	0
$424$	3.02152e8	0.192506
$425$	1.55145e9	0.980339
$426$	0	0
$427$	6.27919e8	0.390307
$428$	2.53634e9	1.56370
$429$	0	0
$430$	2.38527e9	1.44677
$431$	−3.39561e8	−0.204290	−0.102145	−	0.994770i	$-0.532571\pi$
$431$	−3.39561e8	−0.204290	−0.102145	+	0.994770i	$0.532571\pi$
$432$	0	0
$433$	−2.78104e9	−1.64626	−0.823132	−	0.567850i	$-0.807775\pi$
$433$	−2.78104e9	−1.64626	−0.823132	+	0.567850i	$0.807775\pi$
$434$	−1.24809e9	−0.732880
$435$	0	0
$436$	4.61651e9	2.66754
$437$	−1.88463e9	−1.08029
$438$	0	0
$439$	4.50331e8	0.254042	0.127021	−	0.991900i	$-0.459458\pi$
$439$	4.50331e8	0.254042	0.127021	+	0.991900i	$0.459458\pi$
$440$	−1.36283e9	−0.762705
$441$	0	0
$442$	−3.95481e9	−2.17845
$443$	2.30997e9	1.26239	0.631196	−	0.775624i	$-0.282565\pi$
$443$	2.30997e9	1.26239	0.631196	+	0.775624i	$0.282565\pi$
$444$	0	0
$445$	−5.07079e9	−2.72782
$446$	2.64330e9	1.41083
$447$	0	0
$448$	2.99342e9	1.57288
$449$	9.05716e8	0.472204	0.236102	−	0.971728i	$-0.424130\pi$
$449$	9.05716e8	0.472204	0.236102	+	0.971728i	$0.424130\pi$
$450$	0	0
$451$	9.02749e8	0.463392
$452$	−5.13156e9	−2.61376
$453$	0	0
$454$	1.13313e9	0.568307
$455$	4.69788e9	2.33809
$456$	0	0
$457$	−2.37798e9	−1.16547	−0.582735	−	0.812663i	$-0.698017\pi$
$457$	−2.37798e9	−1.16547	−0.582735	+	0.812663i	$0.698017\pi$
$458$	−9.98404e7	−0.0485598
$459$	0	0
$460$	−8.35380e9	−4.00158
$461$	2.40725e9	1.14437	0.572187	−	0.820123i	$-0.306095\pi$
$461$	2.40725e9	1.14437	0.572187	+	0.820123i	$0.306095\pi$
$462$	0	0
$463$	1.75705e9	0.822719	0.411359	−	0.911473i	$-0.365054\pi$
$463$	1.75705e9	0.822719	0.411359	+	0.911473i	$0.365054\pi$
$464$	4.56679e8	0.212226
$465$	0	0
$466$	−9.93021e7	−0.0454577
$467$	3.08270e8	0.140062	0.0700312	−	0.997545i	$-0.477690\pi$
$467$	3.08270e8	0.140062	0.0700312	+	0.997545i	$0.477690\pi$
$468$	0	0
$469$	−1.68515e9	−0.754282
$470$	−3.21423e9	−1.42802
$471$	0	0
$472$	−2.28680e9	−1.00099
$473$	−4.66203e8	−0.202564
$474$	0	0
$475$	−1.60261e9	−0.686119
$476$	−6.69304e9	−2.84446
$477$	0	0
$478$	5.16990e9	2.16513
$479$	−3.98618e9	−1.65723	−0.828614	−	0.559820i	$-0.810870\pi$
$479$	−3.98618e9	−1.65723	−0.828614	+	0.559820i	$0.810870\pi$
$480$	0	0
$481$	2.36140e9	0.967524
$482$	−3.49734e9	−1.42257
$483$	0	0
$484$	−4.32596e9	−1.73430
$485$	−2.55495e9	−1.01692
$486$	0	0
$487$	3.81385e9	1.49628	0.748139	−	0.663542i	$-0.230948\pi$
$487$	3.81385e9	1.49628	0.748139	+	0.663542i	$0.230948\pi$
$488$	1.18112e9	0.460071
$489$	0	0
$490$	5.79533e9	2.22532
$491$	2.55705e9	0.974885	0.487442	−	0.873155i	$-0.337930\pi$
$491$	2.55705e9	0.974885	0.487442	+	0.873155i	$0.337930\pi$
$492$	0	0
$493$	6.86859e8	0.258169
$494$	4.08522e9	1.52465
$495$	0	0
$496$	−7.18617e8	−0.264431
$497$	1.13738e9	0.415582
$498$	0	0
$499$	−2.04146e9	−0.735510	−0.367755	−	0.929923i	$-0.619873\pi$
$499$	−2.04146e9	−0.735510	−0.367755	+	0.929923i	$0.619873\pi$
$500$	4.90916e8	0.175635
$501$	0	0
$502$	−1.81551e9	−0.640523
$503$	1.57865e9	0.553094	0.276547	−	0.961000i	$-0.410810\pi$
$503$	1.57865e9	0.553094	0.276547	+	0.961000i	$0.410810\pi$
$504$	0	0
$505$	−5.35272e9	−1.84950
$506$	2.46873e9	0.847124
$507$	0	0
$508$	−4.25514e9	−1.44009
$509$	−9.25585e8	−0.311103	−0.155551	−	0.987828i	$-0.549715\pi$
$509$	−9.25585e8	−0.311103	−0.155551	+	0.987828i	$0.549715\pi$
$510$	0	0
$511$	−4.09925e9	−1.35904
$512$	4.69742e9	1.54673
$513$	0	0
$514$	9.10033e8	0.295588
$515$	−1.60403e8	−0.0517471
$516$	0	0
$517$	6.28224e8	0.199939
$518$	6.04254e9	1.91014
$519$	0	0
$520$	8.83675e9	2.75601
$521$	8.70376e7	0.0269634	0.0134817	−	0.999909i	$-0.495709\pi$
$521$	8.70376e7	0.0269634	0.0134817	+	0.999909i	$0.495709\pi$
$522$	0	0
$523$	9.76877e8	0.298596	0.149298	−	0.988792i	$-0.452299\pi$
$523$	9.76877e8	0.298596	0.149298	+	0.988792i	$0.452299\pi$
$524$	1.00190e9	0.304203
$525$	0	0
$526$	5.64980e9	1.69271
$527$	−1.08082e9	−0.321675
$528$	0	0
$529$	3.97993e9	1.16891
$530$	−9.63034e8	−0.280980
$531$	0	0
$532$	6.91375e9	1.99078
$533$	−5.85353e9	−1.67445
$534$	0	0
$535$	−3.94496e9	−1.11379
$536$	−3.16978e9	−0.889104
$537$	0	0
$538$	3.52517e9	0.975982
$539$	−1.13270e9	−0.311570
$540$	0	0
$541$	−1.62833e9	−0.442132	−0.221066	−	0.975259i	$-0.570954\pi$
$541$	−1.62833e9	−0.442132	−0.221066	+	0.975259i	$0.570954\pi$
$542$	1.10511e10	2.98132
$543$	0	0
$544$	6.19164e8	0.164896
$545$	−7.18042e9	−1.90004
$546$	0	0
$547$	4.71898e9	1.23280	0.616400	−	0.787433i	$-0.288590\pi$
$547$	4.71898e9	1.23280	0.616400	+	0.787433i	$0.288590\pi$
$548$	−2.46350e9	−0.639472
$549$	0	0
$550$	2.09930e9	0.538028
$551$	−7.09509e8	−0.180687
$552$	0	0
$553$	−7.66119e9	−1.92645
$554$	−4.84579e9	−1.21082
$555$	0	0
$556$	−1.07398e10	−2.64993
$557$	7.69391e9	1.88649	0.943244	−	0.332100i	$-0.107757\pi$
$557$	7.69391e9	1.88649	0.943244	+	0.332100i	$0.107757\pi$
$558$	0	0
$559$	3.02292e9	0.731957
$560$	6.92154e9	1.66550
$561$	0	0
$562$	2.56591e9	0.609768
$563$	−8.34538e9	−1.97091	−0.985455	−	0.169937i	$-0.945643\pi$
$563$	−8.34538e9	−1.97091	−0.985455	+	0.169937i	$0.945643\pi$
$564$	0	0
$565$	7.98152e9	1.86173
$566$	−9.32583e9	−2.16187
$567$	0	0
$568$	2.13941e9	0.489864
$569$	−7.13067e9	−1.62270	−0.811348	−	0.584563i	$-0.801266\pi$
$569$	−7.13067e9	−1.62270	−0.811348	+	0.584563i	$0.801266\pi$
$570$	0	0
$571$	−4.23679e8	−0.0952381	−0.0476191	−	0.998866i	$-0.515163\pi$
$571$	−4.23679e8	−0.0952381	−0.0476191	+	0.998866i	$0.515163\pi$
$572$	−3.53924e9	−0.790723
$573$	0	0
$574$	−1.49785e10	−3.30580
$575$	6.27967e9	1.37753
$576$	0	0
$577$	−5.22691e9	−1.13274	−0.566370	−	0.824151i	$-0.691653\pi$
$577$	−5.22691e9	−1.13274	−0.566370	+	0.824151i	$0.691653\pi$
$578$	−7.85713e8	−0.169245
$579$	0	0
$580$	−3.14496e9	−0.669294
$581$	1.03297e10	2.18511
$582$	0	0
$583$	1.88226e8	0.0393405
$584$	−7.71072e9	−1.60195
$585$	0	0
$586$	−9.26817e9	−1.90262
$587$	4.39679e9	0.897227	0.448614	−	0.893726i	$-0.351918\pi$
$587$	4.39679e9	0.897227	0.448614	+	0.893726i	$0.351918\pi$
$588$	0	0
$589$	1.11646e9	0.225134
$590$	7.28860e9	1.46104
$591$	0	0
$592$	3.47913e9	0.689199
$593$	−2.18339e9	−0.429971	−0.214986	−	0.976617i	$-0.568970\pi$
$593$	−2.18339e9	−0.429971	−0.214986	+	0.976617i	$0.568970\pi$
$594$	0	0
$595$	1.04102e10	2.02605
$596$	−4.76556e9	−0.922045
$597$	0	0
$598$	−1.60076e10	−3.06105
$599$	7.69910e8	0.146368	0.0731839	−	0.997318i	$-0.476684\pi$
$599$	7.69910e8	0.146368	0.0731839	+	0.997318i	$0.476684\pi$
$600$	0	0
$601$	−4.98027e9	−0.935820	−0.467910	−	0.883776i	$-0.654993\pi$
$601$	−4.98027e9	−0.935820	−0.467910	+	0.883776i	$0.654993\pi$
$602$	7.73529e9	1.44507
$603$	0	0
$604$	1.70112e10	3.14127
$605$	6.72850e9	1.23531
$606$	0	0
$607$	7.09267e9	1.28721	0.643605	−	0.765358i	$-0.277438\pi$
$607$	7.09267e9	1.28721	0.643605	+	0.765358i	$0.277438\pi$
$608$	−6.39581e8	−0.115407
$609$	0	0
$610$	−3.76452e9	−0.671515
$611$	−4.07348e9	−0.722473
$612$	0	0
$613$	6.27203e9	1.09976	0.549878	−	0.835245i	$-0.314674\pi$
$613$	6.27203e9	1.09976	0.549878	+	0.835245i	$0.314674\pi$
$614$	2.11397e9	0.368561
$615$	0	0
$616$	−4.41957e9	−0.761812
$617$	1.02766e10	1.76137	0.880683	−	0.473706i	$-0.157084\pi$
$617$	1.02766e10	1.76137	0.880683	+	0.473706i	$0.157084\pi$
$618$	0	0
$619$	−2.95459e9	−0.500703	−0.250351	−	0.968155i	$-0.580546\pi$
$619$	−2.95459e9	−0.500703	−0.250351	+	0.968155i	$0.580546\pi$
$620$	4.94882e9	0.833933
$621$	0	0
$622$	−5.40774e9	−0.901051
$623$	−1.64443e10	−2.72463
$624$	0	0
$625$	−6.47254e9	−1.06046
$626$	1.61706e10	2.63461
$627$	0	0
$628$	−3.09191e9	−0.498159
$629$	5.23272e9	0.838398
$630$	0	0
$631$	−5.81079e9	−0.920731	−0.460365	−	0.887730i	$-0.652282\pi$
$631$	−5.81079e9	−0.920731	−0.460365	+	0.887730i	$0.652282\pi$
$632$	−1.44108e10	−2.27079
$633$	0	0
$634$	1.66492e10	2.59466
$635$	6.61835e9	1.02575
$636$	0	0
$637$	7.34458e9	1.12585
$638$	9.29404e8	0.141688
$639$	0	0
$640$	−1.64948e10	−2.48723
$641$	1.54674e9	0.231961	0.115980	−	0.993252i	$-0.462999\pi$
$641$	1.54674e9	0.231961	0.115980	+	0.993252i	$0.462999\pi$
$642$	0	0
$643$	4.74870e9	0.704427	0.352213	−	0.935920i	$-0.385429\pi$
$643$	4.74870e9	0.704427	0.352213	+	0.935920i	$0.385429\pi$
$644$	−2.70909e10	−3.99689
$645$	0	0
$646$	9.05259e9	1.32117
$647$	−5.65810e8	−0.0821307	−0.0410653	−	0.999156i	$-0.513075\pi$
$647$	−5.65810e8	−0.0821307	−0.0410653	+	0.999156i	$0.513075\pi$
$648$	0	0
$649$	−1.42456e9	−0.204562
$650$	−1.36121e10	−1.94415
$651$	0	0
$652$	5.07975e9	0.717754
$653$	−5.35498e9	−0.752596	−0.376298	−	0.926499i	$-0.622803\pi$
$653$	−5.35498e9	−0.752596	−0.376298	+	0.926499i	$0.622803\pi$
$654$	0	0
$655$	−1.55833e9	−0.216678
$656$	−8.62420e9	−1.19277
$657$	0	0
$658$	−1.04236e10	−1.42635
$659$	9.80235e9	1.33423	0.667116	−	0.744954i	$-0.267529\pi$
$659$	9.80235e9	1.33423	0.667116	+	0.744954i	$0.267529\pi$
$660$	0	0
$661$	5.26059e9	0.708483	0.354242	−	0.935154i	$-0.384739\pi$
$661$	5.26059e9	0.708483	0.354242	+	0.935154i	$0.384739\pi$
$662$	3.71480e8	0.0497660
$663$	0	0
$664$	1.94303e10	2.57568
$665$	−1.07535e10	−1.41799
$666$	0	0
$667$	2.78014e9	0.362766
$668$	2.22269e10	2.88510
$669$	0	0
$670$	1.01029e10	1.29773
$671$	7.35780e8	0.0940197
$672$	0	0
$673$	1.29458e10	1.63710	0.818550	−	0.574435i	$-0.194778\pi$
$673$	1.29458e10	1.63710	0.818550	+	0.574435i	$0.194778\pi$
$674$	−1.44393e10	−1.81651
$675$	0	0
$676$	7.26176e9	0.904125
$677$	−5.78853e8	−0.0716981	−0.0358490	−	0.999357i	$-0.511414\pi$
$677$	−5.78853e8	−0.0716981	−0.0358490	+	0.999357i	$0.511414\pi$
$678$	0	0
$679$	−8.28556e9	−1.01573
$680$	1.95817e10	2.38819
$681$	0	0
$682$	−1.46249e9	−0.176541
$683$	4.24654e9	0.509992	0.254996	−	0.966942i	$-0.417926\pi$
$683$	4.24654e9	0.509992	0.254996	+	0.966942i	$0.417926\pi$
$684$	0	0
$685$	3.83168e9	0.455483
$686$	−1.39659e9	−0.165171
$687$	0	0
$688$	4.45377e9	0.521396
$689$	−1.22048e9	−0.142155
$690$	0	0
$691$	−1.36695e10	−1.57609	−0.788043	−	0.615620i	$-0.788906\pi$
$691$	−1.36695e10	−1.57609	−0.788043	+	0.615620i	$0.788906\pi$
$692$	3.25338e10	3.73219
$693$	0	0
$694$	−1.14005e6	−0.000129469 0
$695$	1.67044e10	1.88749
$696$	0	0
$697$	−1.29711e10	−1.45098
$698$	−1.59788e10	−1.77848
$699$	0	0
$700$	−2.30369e10	−2.53852
$701$	−1.78237e9	−0.195427	−0.0977134	−	0.995215i	$-0.531153\pi$
$701$	−1.78237e9	−0.195427	−0.0977134	+	0.995215i	$0.531153\pi$
$702$	0	0
$703$	−5.40527e9	−0.586778
$704$	3.50762e9	0.378886
$705$	0	0
$706$	−1.55842e8	−0.0166674
$707$	−1.73586e10	−1.84734
$708$	0	0
$709$	1.33988e10	1.41190	0.705951	−	0.708261i	$-0.250520\pi$
$709$	1.33988e10	1.41190	0.705951	+	0.708261i	$0.250520\pi$
$710$	−6.81884e9	−0.715001
$711$	0	0
$712$	−3.09318e10	−3.21163
$713$	−4.37476e9	−0.452003
$714$	0	0
$715$	5.50486e9	0.563216
$716$	−1.02378e10	−1.04235
$717$	0	0
$718$	−1.51124e10	−1.52369
$719$	−1.60918e9	−0.161455	−0.0807276	−	0.996736i	$-0.525724\pi$
$719$	−1.60918e9	−0.161455	−0.0807276	+	0.996736i	$0.525724\pi$
$720$	0	0
$721$	−5.20176e8	−0.0516865
$722$	8.02775e9	0.793805
$723$	0	0
$724$	1.60945e10	1.57613
$725$	2.36411e9	0.230401
$726$	0	0
$727$	−1.11466e10	−1.07590	−0.537949	−	0.842978i	$-0.680801\pi$
$727$	−1.11466e10	−1.07590	−0.537949	+	0.842978i	$0.680801\pi$
$728$	2.86571e10	2.75278
$729$	0	0
$730$	2.45760e10	2.33820
$731$	6.69861e9	0.634269
$732$	0	0
$733$	1.02155e10	0.958063	0.479031	−	0.877798i	$-0.340988\pi$
$733$	1.02155e10	0.958063	0.479031	+	0.877798i	$0.340988\pi$
$734$	1.15284e10	1.07605
$735$	0	0
$736$	2.50614e9	0.231704
$737$	−1.97462e9	−0.181697
$738$	0	0
$739$	−1.75748e9	−0.160189	−0.0800947	−	0.996787i	$-0.525522\pi$
$739$	−1.75748e9	−0.160189	−0.0800947	+	0.996787i	$0.525522\pi$
$740$	−2.39593e10	−2.17352
$741$	0	0
$742$	−3.12306e9	−0.280651
$743$	−7.57290e9	−0.677332	−0.338666	−	0.940907i	$-0.609976\pi$
$743$	−7.57290e9	−0.677332	−0.338666	+	0.940907i	$0.609976\pi$
$744$	0	0
$745$	7.41225e9	0.656754
$746$	−1.70700e10	−1.50538
$747$	0	0
$748$	−7.84274e9	−0.685192
$749$	−1.27933e10	−1.11249
$750$	0	0
$751$	−8.15251e9	−0.702347	−0.351173	−	0.936310i	$-0.614217\pi$
$751$	−8.15251e9	−0.702347	−0.351173	+	0.936310i	$0.614217\pi$
$752$	−6.00159e9	−0.514641
$753$	0	0
$754$	−6.02637e9	−0.511984
$755$	−2.64588e10	−2.23747
$756$	0	0
$757$	−1.65336e10	−1.38526	−0.692629	−	0.721294i	$-0.743548\pi$
$757$	−1.65336e10	−1.38526	−0.692629	+	0.721294i	$0.743548\pi$
$758$	2.94788e9	0.245848
$759$	0	0
$760$	−2.02274e10	−1.67145
$761$	−8.94719e9	−0.735936	−0.367968	−	0.929838i	$-0.619946\pi$
$761$	−8.94719e9	−0.735936	−0.367968	+	0.929838i	$0.619946\pi$
$762$	0	0
$763$	−2.32857e10	−1.89781
$764$	2.49370e10	2.02310
$765$	0	0
$766$	2.99444e10	2.40722
$767$	9.23705e9	0.739179
$768$	0	0
$769$	−3.24990e9	−0.257708	−0.128854	−	0.991664i	$-0.541130\pi$
$769$	−3.24990e9	−0.257708	−0.128854	+	0.991664i	$0.541130\pi$
$770$	1.40863e10	1.11193
$771$	0	0
$772$	4.92562e10	3.85301
$773$	−1.48477e10	−1.15619	−0.578096	−	0.815969i	$-0.696204\pi$
$773$	−1.48477e10	−1.15619	−0.578096	+	0.815969i	$0.696204\pi$
$774$	0	0
$775$	−3.72010e9	−0.287077
$776$	−1.55852e10	−1.19728
$777$	0	0
$778$	−2.36367e10	−1.79953
$779$	1.33988e10	1.01551
$780$	0	0
$781$	1.33275e9	0.100108
$782$	−3.54717e10	−2.65252
$783$	0	0
$784$	1.08210e10	0.801977
$785$	4.80909e9	0.354829
$786$	0	0
$787$	−3.58917e9	−0.262472	−0.131236	−	0.991351i	$-0.541895\pi$
$787$	−3.58917e9	−0.262472	−0.131236	+	0.991351i	$0.541895\pi$
$788$	3.66173e10	2.66590
$789$	0	0
$790$	4.59307e10	3.31442
$791$	2.58836e10	1.85955
$792$	0	0
$793$	−4.77089e9	−0.339737
$794$	1.56481e10	1.10940
$795$	0	0
$796$	−3.16858e9	−0.222674
$797$	4.27547e9	0.299144	0.149572	−	0.988751i	$-0.452210\pi$
$797$	4.27547e9	0.299144	0.149572	+	0.988751i	$0.452210\pi$
$798$	0	0
$799$	−9.02659e9	−0.626051
$800$	2.13111e9	0.147160
$801$	0	0
$802$	−1.84803e10	−1.26502
$803$	−4.80340e9	−0.327374
$804$	0	0
$805$	4.21366e10	2.84691
$806$	9.48294e9	0.637926
$807$	0	0
$808$	−3.26516e10	−2.17753
$809$	2.23500e10	1.48408	0.742040	−	0.670356i	$-0.233858\pi$
$809$	2.23500e10	1.48408	0.742040	+	0.670356i	$0.233858\pi$
$810$	0	0
$811$	1.46284e10	0.962998	0.481499	−	0.876447i	$-0.340093\pi$
$811$	1.46284e10	0.962998	0.481499	+	0.876447i	$0.340093\pi$
$812$	−1.01989e10	−0.668510
$813$	0	0
$814$	7.08050e9	0.460128
$815$	−7.90093e9	−0.511242
$816$	0	0
$817$	−6.91949e9	−0.443912
$818$	−3.14254e9	−0.200745
$819$	0	0
$820$	5.93914e10	3.76162
$821$	2.53690e10	1.59993	0.799967	−	0.600044i	$-0.204850\pi$
$821$	2.53690e10	1.59993	0.799967	+	0.600044i	$0.204850\pi$
$822$	0	0
$823$	6.42001e9	0.401455	0.200727	−	0.979647i	$-0.435669\pi$
$823$	6.42001e9	0.401455	0.200727	+	0.979647i	$0.435669\pi$
$824$	−9.78456e8	−0.0609251
$825$	0	0
$826$	2.36365e10	1.45933
$827$	1.32102e10	0.812157	0.406079	−	0.913838i	$-0.366896\pi$
$827$	1.32102e10	0.812157	0.406079	+	0.913838i	$0.366896\pi$
$828$	0	0
$829$	1.65694e10	1.01010	0.505051	−	0.863090i	$-0.331474\pi$
$829$	1.65694e10	1.01010	0.505051	+	0.863090i	$0.331474\pi$
$830$	−6.19293e10	−3.75944
$831$	0	0
$832$	−2.27438e10	−1.36909
$833$	1.62751e10	0.975590
$834$	0	0
$835$	−3.45712e10	−2.05500
$836$	8.10136e9	0.479552
$837$	0	0
$838$	1.29374e9	0.0759438
$839$	−1.93141e10	−1.12903	−0.564517	−	0.825422i	$-0.690937\pi$
$839$	−1.93141e10	−1.12903	−0.564517	+	0.825422i	$0.690937\pi$
$840$	0	0
$841$	−1.62032e10	−0.939325
$842$	−3.96672e10	−2.29002
$843$	0	0
$844$	−5.53359e10	−3.16817
$845$	−1.12948e10	−0.643990
$846$	0	0
$847$	2.18201e10	1.23386
$848$	−1.79817e9	−0.101262
$849$	0	0
$850$	−3.01636e10	−1.68468
$851$	2.11800e10	1.17808
$852$	0	0
$853$	6.27677e9	0.346270	0.173135	−	0.984898i	$-0.444610\pi$
$853$	6.27677e9	0.346270	0.173135	+	0.984898i	$0.444610\pi$
$854$	−1.22081e10	−0.670729
$855$	0	0
$856$	−2.40643e10	−1.31134
$857$	4.05896e9	0.220284	0.110142	−	0.993916i	$-0.464869\pi$
$857$	4.05896e9	0.220284	0.110142	+	0.993916i	$0.464869\pi$
$858$	0	0
$859$	−1.88283e10	−1.01352	−0.506762	−	0.862086i	$-0.669158\pi$
$859$	−1.88283e10	−1.01352	−0.506762	+	0.862086i	$0.669158\pi$
$860$	−3.06713e10	−1.64432
$861$	0	0
$862$	6.60183e9	0.351066
$863$	−3.10346e10	−1.64364	−0.821822	−	0.569744i	$-0.807042\pi$
$863$	−3.10346e10	−1.64364	−0.821822	+	0.569744i	$0.807042\pi$
$864$	0	0
$865$	−5.06024e10	−2.65837
$866$	5.40696e10	2.82905
$867$	0	0
$868$	1.60487e10	0.832956
$869$	−8.97719e9	−0.464057
$870$	0	0
$871$	1.28037e10	0.656555
$872$	−4.38006e10	−2.23703
$873$	0	0
$874$	3.66414e10	1.85645
$875$	−2.47618e9	−0.124955
$876$	0	0
$877$	2.85155e10	1.42752	0.713761	−	0.700389i	$-0.246990\pi$
$877$	2.85155e10	1.42752	0.713761	+	0.700389i	$0.246990\pi$
$878$	−8.75543e9	−0.436563
$879$	0	0
$880$	8.11049e9	0.401197
$881$	1.96153e10	0.966451	0.483225	−	0.875496i	$-0.339465\pi$
$881$	1.96153e10	0.966451	0.483225	+	0.875496i	$0.339465\pi$
$882$	0	0
$883$	1.47908e9	0.0722986	0.0361493	−	0.999346i	$-0.488491\pi$
$883$	1.47908e9	0.0722986	0.0361493	+	0.999346i	$0.488491\pi$
$884$	5.08533e10	2.47592
$885$	0	0
$886$	−4.49110e10	−2.16938
$887$	1.34099e10	0.645195	0.322598	−	0.946536i	$-0.395444\pi$
$887$	1.34099e10	0.645195	0.322598	+	0.946536i	$0.395444\pi$
$888$	0	0
$889$	2.14629e10	1.02455
$890$	9.85875e10	4.68767
$891$	0	0
$892$	−3.39891e10	−1.60348
$893$	9.32424e9	0.438161
$894$	0	0
$895$	1.59237e10	0.742444
$896$	−5.34916e10	−2.48432
$897$	0	0
$898$	−1.76091e10	−0.811467
$899$	−1.64697e9	−0.0756008
$900$	0	0
$901$	−2.70451e9	−0.123183
$902$	−1.75514e10	−0.796324
$903$	0	0
$904$	4.86872e10	2.19192
$905$	−2.50330e10	−1.12265
$906$	0	0
$907$	3.52251e10	1.56757	0.783785	−	0.621032i	$-0.213286\pi$
$907$	3.52251e10	1.56757	0.783785	+	0.621032i	$0.213286\pi$
$908$	−1.45704e10	−0.645910
$909$	0	0
$910$	−9.13372e10	−4.01793
$911$	−3.48713e10	−1.52811	−0.764054	−	0.645152i	$-0.776794\pi$
$911$	−3.48713e10	−1.52811	−0.764054	+	0.645152i	$0.776794\pi$
$912$	0	0
$913$	1.21041e10	0.526364
$914$	4.62331e10	2.00282
$915$	0	0
$916$	1.28381e9	0.0551907
$917$	−5.05357e9	−0.216424
$918$	0	0
$919$	2.34799e10	0.997910	0.498955	−	0.866628i	$-0.333717\pi$
$919$	2.34799e10	0.997910	0.498955	+	0.866628i	$0.333717\pi$
$920$	7.92592e10	3.35577
$921$	0	0
$922$	−4.68023e10	−1.96657
$923$	−8.64171e9	−0.361738
$924$	0	0
$925$	1.80106e10	0.748224
$926$	−3.41610e10	−1.41381
$927$	0	0
$928$	9.43488e8	0.0387542
$929$	1.39279e10	0.569943	0.284972	−	0.958536i	$-0.408016\pi$
$929$	1.39279e10	0.569943	0.284972	+	0.958536i	$0.408016\pi$
$930$	0	0
$931$	−1.68118e10	−0.682796
$932$	1.27689e9	0.0516651
$933$	0	0
$934$	−5.99345e9	−0.240693
$935$	1.21984e10	0.488049
$936$	0	0
$937$	1.06809e10	0.424148	0.212074	−	0.977254i	$-0.431978\pi$
$937$	1.06809e10	0.424148	0.212074	+	0.977254i	$0.431978\pi$
$938$	3.27631e10	1.29621
$939$	0	0
$940$	4.13305e10	1.62302
$941$	−1.37831e10	−0.539241	−0.269621	−	0.962967i	$-0.586898\pi$
$941$	−1.37831e10	−0.539241	−0.269621	+	0.962967i	$0.586898\pi$
$942$	0	0
$943$	−5.25020e10	−2.03885
$944$	1.36092e10	0.526541
$945$	0	0
$946$	9.06403e9	0.348099
$947$	−4.26474e10	−1.63180	−0.815901	−	0.578192i	$-0.803758\pi$
$947$	−4.26474e10	−1.63180	−0.815901	+	0.578192i	$0.803758\pi$
$948$	0	0
$949$	3.11458e10	1.18296
$950$	3.11583e10	1.17907
$951$	0	0
$952$	6.35023e10	2.38539
$953$	1.06780e10	0.399637	0.199819	−	0.979833i	$-0.435965\pi$
$953$	1.06780e10	0.399637	0.199819	+	0.979833i	$0.435965\pi$
$954$	0	0
$955$	−3.87864e10	−1.44101
$956$	−6.64777e10	−2.46078
$957$	0	0
$958$	7.75002e10	2.84789
$959$	1.24259e10	0.454950
$960$	0	0
$961$	−2.49210e10	−0.905802
$962$	−4.59109e10	−1.66266
$963$	0	0
$964$	4.49709e10	1.61682
$965$	−7.66120e10	−2.74442
$966$	0	0
$967$	3.96746e10	1.41098	0.705488	−	0.708721i	$-0.250728\pi$
$967$	3.96746e10	1.41098	0.705488	+	0.708721i	$0.250728\pi$
$968$	4.10439e10	1.45440
$969$	0	0
$970$	4.96740e10	1.74754
$971$	2.83156e10	0.992565	0.496282	−	0.868161i	$-0.334698\pi$
$971$	2.83156e10	0.992565	0.496282	+	0.868161i	$0.334698\pi$
$972$	0	0
$973$	5.41715e10	1.88528
$974$	−7.41497e10	−2.57130
$975$	0	0
$976$	−7.02910e9	−0.242006
$977$	5.38772e9	0.184831	0.0924153	−	0.995721i	$-0.470541\pi$
$977$	5.38772e9	0.184831	0.0924153	+	0.995721i	$0.470541\pi$
$978$	0	0
$979$	−1.92690e10	−0.656327
$980$	−7.45199e10	−2.52919
$981$	0	0
$982$	−4.97146e10	−1.67531
$983$	2.50997e10	0.842814	0.421407	−	0.906872i	$-0.361536\pi$
$983$	2.50997e10	0.842814	0.421407	+	0.906872i	$0.361536\pi$
$984$	0	0
$985$	−5.69537e10	−1.89887
$986$	−1.33541e10	−0.443654
$987$	0	0
$988$	−5.25302e10	−1.73285
$989$	2.71134e10	0.891245
$990$	0	0
$991$	−1.01878e10	−0.332524	−0.166262	−	0.986082i	$-0.553170\pi$
$991$	−1.01878e10	−0.332524	−0.166262	+	0.986082i	$0.553170\pi$
$992$	−1.48465e9	−0.0482872
$993$	0	0
$994$	−2.21131e10	−0.714163
$995$	4.92834e9	0.158606
$996$	0	0
$997$	1.65762e10	0.529727	0.264864	−	0.964286i	$-0.414673\pi$
$997$	1.65762e10	0.529727	0.264864	+	0.964286i	$0.414673\pi$
$998$	3.96905e10	1.26395
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.a.d.1.1	✓	2
3.2	odd	2	inner	27.8.a.d.1.2	yes	2
4.3	odd	2		432.8.a.o.1.1		2
9.2	odd	6		81.8.c.e.28.1		4
9.4	even	3		81.8.c.e.55.2		4
9.5	odd	6		81.8.c.e.55.1		4
9.7	even	3		81.8.c.e.28.2		4
12.11	even	2		432.8.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.a.d.1.1	✓	2	1.1	even	1	trivial
27.8.a.d.1.2	yes	2	3.2	odd	2	inner
81.8.c.e.28.1		4	9.2	odd	6
81.8.c.e.28.2		4	9.7	even	3
81.8.c.e.55.1		4	9.5	odd	6
81.8.c.e.55.2		4	9.4	even	3
432.8.a.o.1.1		2	4.3	odd	2
432.8.a.o.1.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-19.4422 q^{2} +250.000 q^{4} -388.844 q^{5} -1261.00 q^{7} -2371.95 q^{8} +7560.00 q^{10} -1477.61 q^{11} +9581.00 q^{13} +24516.6 q^{14} +14116.0 q^{16} +21230.9 q^{17} -21931.0 q^{19} -97211.1 q^{20} +28728.0 q^{22} +85934.6 q^{23} +73075.0 q^{25} -186276. q^{26} -315250. q^{28} +32351.9 q^{29} -50908.0 q^{31} +29163.3 q^{32} -412776. q^{34} +490333. q^{35} +246467. q^{37} +426387. q^{38} +922320. q^{40} -610952. q^{41} +315512. q^{43} -369402. q^{44} -1.67076e6 q^{46} -425163. q^{47} +766578. q^{49} -1.42074e6 q^{50} +2.39525e6 q^{52} -127385. q^{53} +574560. q^{55} +2.99103e6 q^{56} -628992. q^{58} +964101. q^{59} -497953. q^{61} +989765. q^{62} -2.37385e6 q^{64} -3.72552e6 q^{65} +1.33636e6 q^{67} +5.30773e6 q^{68} -9.53316e6 q^{70} -901964. q^{71} +3.25079e6 q^{73} -4.79187e6 q^{74} -5.48275e6 q^{76} +1.86326e6 q^{77} +6.07548e6 q^{79} -5.48893e6 q^{80} +1.18783e7 q^{82} -8.19171e6 q^{83} -8.25552e6 q^{85} -6.13425e6 q^{86} +3.50482e6 q^{88} +1.30407e7 q^{89} -1.20816e7 q^{91} +2.14837e7 q^{92} +8.26610e6 q^{94} +8.52775e6 q^{95} +6.57063e6 q^{97} -1.49040e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 500 q^{4} - 2522 q^{7} + 15120 q^{10} + 19162 q^{13} + 28232 q^{16} - 43862 q^{19} + 57456 q^{22} + 146150 q^{25} - 630500 q^{28} - 101816 q^{31} - 825552 q^{34} + 492934 q^{37} + 1844640 q^{40} + 631024 q^{43}+ \cdots + 13141258 q^{97}+O(q^{100}) \) 2 * q + 500 * q^4 - 2522 * q^7 + 15120 * q^10 + 19162 * q^13 + 28232 * q^16 - 43862 * q^19 + 57456 * q^22 + 146150 * q^25 - 630500 * q^28 - 101816 * q^31 - 825552 * q^34 + 492934 * q^37 + 1844640 * q^40 + 631024 * q^43 - 3341520 * q^46 + 1533156 * q^49 + 4790500 * q^52 + 1149120 * q^55 - 1257984 * q^58 - 995906 * q^61 - 4747696 * q^64 + 2672722 * q^67 - 19066320 * q^70 + 6501586 * q^73 - 10965500 * q^76 + 12150970 * q^79 + 23756544 * q^82 - 16511040 * q^85 + 7009632 * q^88 - 24163282 * q^91 + 16532208 * q^94 + 13141258 * q^97

Introduction

L-functions

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