LMFDB - Embedded newform 27.7.b.b.26.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,7,Mod(26,27)]

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([1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	27.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.21146025774$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.7.b.b.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.00000i q^{2} +28.0000 q^{4} -240.000i q^{5} +299.000 q^{7} +552.000i q^{8} +1440.00 q^{10} -624.000i q^{11} +2495.00 q^{13} +1794.00i q^{14} -1520.00 q^{16} +1872.00i q^{17} -2509.00 q^{19} -6720.00i q^{20} +3744.00 q^{22} -14352.0i q^{23} -41975.0 q^{25} +14970.0i q^{26} +8372.00 q^{28} +23712.0i q^{29} +5330.00 q^{31} +26208.0i q^{32} -11232.0 q^{34} -71760.0i q^{35} +32591.0 q^{37} -15054.0i q^{38} +132480. q^{40} +66144.0i q^{41} -70630.0 q^{43} -17472.0i q^{44} +86112.0 q^{46} +3984.00i q^{47} -28248.0 q^{49} -251850. i q^{50} +69860.0 q^{52} +190944. i q^{53} -149760. q^{55} +165048. i q^{56} -142272. q^{58} +237360. i q^{59} -61801.0 q^{61} +31980.0i q^{62} -254528. q^{64} -598800. i q^{65} -430261. q^{67} +52416.0i q^{68} +430560. q^{70} -251712. i q^{71} +251615. q^{73} +195546. i q^{74} -70252.0 q^{76} -186576. i q^{77} +660827. q^{79} +364800. i q^{80} -396864. q^{82} +797856. i q^{83} +449280. q^{85} -423780. i q^{86} +344448. q^{88} -270576. i q^{89} +746005. q^{91} -401856. i q^{92} -23904.0 q^{94} +602160. i q^{95} +220727. q^{97} -169488. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 56 q^{4} + 598 q^{7} + 2880 q^{10} + 4990 q^{13} - 3040 q^{16} - 5018 q^{19} + 7488 q^{22} - 83950 q^{25} + 16744 q^{28} + 10660 q^{31} - 22464 q^{34} + 65182 q^{37} + 264960 q^{40} - 141260 q^{43}+ \cdots + 441454 q^{97}+O(q^{100}) $ 2 * q + 56 * q^4 + 598 * q^7 + 2880 * q^10 + 4990 * q^13 - 3040 * q^16 - 5018 * q^19 + 7488 * q^22 - 83950 * q^25 + 16744 * q^28 + 10660 * q^31 - 22464 * q^34 + 65182 * q^37 + 264960 * q^40 - 141260 * q^43 + 172224 * q^46 - 56496 * q^49 + 139720 * q^52 - 299520 * q^55 - 284544 * q^58 - 123602 * q^61 - 509056 * q^64 - 860522 * q^67 + 861120 * q^70 + 503230 * q^73 - 140504 * q^76 + 1321654 * q^79 - 793728 * q^82 + 898560 * q^85 + 688896 * q^88 + 1492010 * q^91 - 47808 * q^94 + 441454 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	6.00000i	0.750000i	0.927025	+	0.375000i	$0.122357\pi$
$2$	6.00000i	0.750000i	−0.927025	+	0.375000i	$0.877643\pi$
$3$	0	0
$3$	0	0
$4$	28.0000	0.437500
$5$	− 240.000i	− 1.92000i	−0.280000	−	0.960000i	$-0.590334\pi$
$5$	− 240.000i	− 1.92000i	0.280000	−	0.960000i	$-0.409666\pi$
$6$	0	0
$7$	299.000	0.871720	0.435860	−	0.900014i	$-0.356444\pi$
$7$	299.000	0.871720	0.435860	+	0.900014i	$0.356444\pi$
$8$	552.000i	1.07812i
$9$	0	0
$10$	1440.00	1.44000
$11$	− 624.000i	− 0.468820i	−0.972138	−	0.234410i	$-0.924684\pi$
$11$	− 624.000i	− 0.468820i	0.972138	−	0.234410i	$-0.0753159\pi$
$12$	0	0
$13$	2495.00	1.13564	0.567820	−	0.823153i	$-0.307787\pi$
$13$	2495.00	1.13564	0.567820	+	0.823153i	$0.307787\pi$
$14$	1794.00i	0.653790i
$15$	0	0
$16$	−1520.00	−0.371094
$17$	1872.00i	0.381030i	0.981684	+	0.190515i	$0.0610158\pi$
$17$	1872.00i	0.381030i	−0.981684	+	0.190515i	$0.938984\pi$
$18$	0	0
$19$	−2509.00	−0.365797	−0.182898	−	0.983132i	$-0.558548\pi$
$19$	−2509.00	−0.365797	−0.182898	+	0.983132i	$0.558548\pi$
$20$	− 6720.00i	− 0.840000i
$21$	0	0
$22$	3744.00	0.351615
$23$	− 14352.0i	− 1.17958i	−0.807555	−	0.589792i	$-0.799210\pi$
$23$	− 14352.0i	− 1.17958i	0.807555	−	0.589792i	$-0.200790\pi$
$24$	0	0
$25$	−41975.0	−2.68640
$26$	14970.0i	0.851730i
$27$	0	0
$28$	8372.00	0.381378
$29$	23712.0i	0.972242i	0.873892	+	0.486121i	$0.161588\pi$
$29$	23712.0i	0.972242i	−0.873892	+	0.486121i	$0.838412\pi$
$30$	0	0
$31$	5330.00	0.178913	0.0894565	−	0.995991i	$-0.471487\pi$
$31$	5330.00	0.178913	0.0894565	+	0.995991i	$0.471487\pi$
$32$	26208.0i	0.799805i
$33$	0	0
$34$	−11232.0	−0.285772
$35$	− 71760.0i	− 1.67370i
$36$	0	0
$37$	32591.0	0.643417	0.321708	−	0.946839i	$-0.395743\pi$
$37$	32591.0	0.643417	0.321708	+	0.946839i	$0.395743\pi$
$38$	− 15054.0i	− 0.274348i
$39$	0	0
$40$	132480.	2.07000
$41$	66144.0i	0.959707i	0.877348	+	0.479854i	$0.159310\pi$
$41$	66144.0i	0.959707i	−0.877348	+	0.479854i	$0.840690\pi$
$42$	0	0
$43$	−70630.0	−0.888349	−0.444175	−	0.895940i	$-0.646503\pi$
$43$	−70630.0	−0.888349	−0.444175	+	0.895940i	$0.646503\pi$
$44$	− 17472.0i	− 0.205109i
$45$	0	0
$46$	86112.0	0.884688
$47$	3984.00i	0.0383730i	0.999816	+	0.0191865i	$0.00610763\pi$
$47$	3984.00i	0.0383730i	−0.999816	+	0.0191865i	$0.993892\pi$
$48$	0	0
$49$	−28248.0	−0.240104
$50$	− 251850.i	− 2.01480i
$51$	0	0
$52$	69860.0	0.496842
$53$	190944.i	1.28256i	0.767306	+	0.641281i	$0.221597\pi$
$53$	190944.i	1.28256i	−0.767306	+	0.641281i	$0.778403\pi$
$54$	0	0
$55$	−149760.	−0.900135
$56$	165048.i	0.939823i
$57$	0	0
$58$	−142272.	−0.729181
$59$	237360.i	1.15572i	0.816137	+	0.577858i	$0.196111\pi$
$59$	237360.i	1.15572i	−0.816137	+	0.577858i	$0.803889\pi$
$60$	0	0
$61$	−61801.0	−0.272274	−0.136137	−	0.990690i	$-0.543469\pi$
$61$	−61801.0	−0.272274	−0.136137	+	0.990690i	$0.543469\pi$
$62$	31980.0i	0.134185i
$63$	0	0
$64$	−254528.	−0.970947
$65$	− 598800.i	− 2.18043i
$66$	0	0
$67$	−430261.	−1.43056	−0.715282	−	0.698835i	$-0.753702\pi$
$67$	−430261.	−1.43056	−0.715282	+	0.698835i	$0.753702\pi$
$68$	52416.0i	0.166701i
$69$	0	0
$70$	430560.	1.25528
$71$	− 251712.i	− 0.703281i	−0.936135	−	0.351640i	$-0.885624\pi$
$71$	− 251712.i	− 0.703281i	0.936135	−	0.351640i	$-0.114376\pi$
$72$	0	0
$73$	251615.	0.646797	0.323398	−	0.946263i	$-0.395175\pi$
$73$	251615.	0.646797	0.323398	+	0.946263i	$0.395175\pi$
$74$	195546.i	0.482563i
$75$	0	0
$76$	−70252.0	−0.160036
$77$	− 186576.i	− 0.408680i
$78$	0	0
$79$	660827.	1.34031	0.670157	−	0.742219i	$-0.266227\pi$
$79$	660827.	1.34031	0.670157	+	0.742219i	$0.266227\pi$
$80$	364800.i	0.712500i
$81$	0	0
$82$	−396864.	−0.719781
$83$	797856.i	1.39537i	0.716403	+	0.697686i	$0.245787\pi$
$83$	797856.i	1.39537i	−0.716403	+	0.697686i	$0.754213\pi$
$84$	0	0
$85$	449280.	0.731577
$86$	− 423780.i	− 0.666262i
$87$	0	0
$88$	344448.	0.505447
$89$	− 270576.i	− 0.383813i	−0.981413	−	0.191906i	$-0.938533\pi$
$89$	− 270576.i	− 0.383813i	0.981413	−	0.191906i	$-0.0614670\pi$
$90$	0	0
$91$	746005.	0.989960
$92$	− 401856.i	− 0.516068i
$93$	0	0
$94$	−23904.0	−0.0287798
$95$	602160.i	0.702330i
$96$	0	0
$97$	220727.	0.241847	0.120923	−	0.992662i	$-0.461414\pi$
$97$	220727.	0.241847	0.120923	+	0.992662i	$0.461414\pi$
$98$	− 169488.i	− 0.180078i
$99$	0	0
$100$	−1.17530e6	−1.17530
$101$	289536.i	0.281021i	0.990079	+	0.140510i	$0.0448743\pi$
$101$	289536.i	0.281021i	−0.990079	+	0.140510i	$0.955126\pi$
$102$	0	0
$103$	−594205.	−0.543782	−0.271891	−	0.962328i	$-0.587649\pi$
$103$	−594205.	−0.543782	−0.271891	+	0.962328i	$0.587649\pi$
$104$	1.37724e6i	1.22436i
$105$	0	0
$106$	−1.14566e6	−0.961922
$107$	− 1.99368e6i	− 1.62744i	−0.581259	−	0.813718i	$-0.697440\pi$
$107$	− 1.99368e6i	− 1.62744i	0.581259	−	0.813718i	$-0.302560\pi$
$108$	0	0
$109$	1.51135e6	1.16704	0.583521	−	0.812098i	$-0.301674\pi$
$109$	1.51135e6	1.16704	0.583521	+	0.812098i	$0.301674\pi$
$110$	− 898560.i	− 0.675101i
$111$	0	0
$112$	−454480.	−0.323490
$113$	− 1.65048e6i	− 1.14387i	−0.820301	−	0.571933i	$-0.806194\pi$
$113$	− 1.65048e6i	− 1.14387i	0.820301	−	0.571933i	$-0.193806\pi$
$114$	0	0
$115$	−3.44448e6	−2.26480
$116$	663936.i	0.425356i
$117$	0	0
$118$	−1.42416e6	−0.866788
$119$	559728.i	0.332151i
$120$	0	0
$121$	1.38218e6	0.780207
$122$	− 370806.i	− 0.204205i
$123$	0	0
$124$	149240.	0.0782745
$125$	6.32400e6i	3.23789i
$126$	0	0
$127$	1.41135e6	0.689005	0.344502	−	0.938785i	$-0.388048\pi$
$127$	1.41135e6	0.689005	0.344502	+	0.938785i	$0.388048\pi$
$128$	150144.i	0.0715942i
$129$	0	0
$130$	3.59280e6	1.63532
$131$	− 323232.i	− 0.143781i	−0.997413	−	0.0718903i	$-0.977097\pi$
$131$	− 323232.i	− 0.143781i	0.997413	−	0.0718903i	$-0.0229032\pi$
$132$	0	0
$133$	−750191.	−0.318872
$134$	− 2.58157e6i	− 1.07292i
$135$	0	0
$136$	−1.03334e6	−0.410798
$137$	− 2.13845e6i	− 0.831643i	−0.909446	−	0.415822i	$-0.863494\pi$
$137$	− 2.13845e6i	− 0.831643i	0.909446	−	0.415822i	$-0.136506\pi$
$138$	0	0
$139$	−111229.	−0.0414165	−0.0207083	−	0.999786i	$-0.506592\pi$
$139$	−111229.	−0.0414165	−0.0207083	+	0.999786i	$0.506592\pi$
$140$	− 2.00928e6i	− 0.732245i
$141$	0	0
$142$	1.51027e6	0.527461
$143$	− 1.55688e6i	− 0.532411i
$144$	0	0
$145$	5.69088e6	1.86670
$146$	1.50969e6i	0.485098i
$147$	0	0
$148$	912548.	0.281495
$149$	3.43661e6i	1.03889i	0.854503	+	0.519447i	$0.173862\pi$
$149$	3.43661e6i	1.03889i	−0.854503	+	0.519447i	$0.826138\pi$
$150$	0	0
$151$	−3.93932e6	−1.14417	−0.572086	−	0.820194i	$-0.693865\pi$
$151$	−3.93932e6	−1.14417	−0.572086	+	0.820194i	$0.693865\pi$
$152$	− 1.38497e6i	− 0.394375i
$153$	0	0
$154$	1.11946e6	0.306510
$155$	− 1.27920e6i	− 0.343513i
$156$	0	0
$157$	431066.	0.111390	0.0556948	−	0.998448i	$-0.482263\pi$
$157$	431066.	0.111390	0.0556948	+	0.998448i	$0.482263\pi$
$158$	3.96496e6i	1.00524i
$159$	0	0
$160$	6.28992e6	1.53563
$161$	− 4.29125e6i	− 1.02827i
$162$	0	0
$163$	−7.94353e6	−1.83422	−0.917109	−	0.398637i	$-0.869483\pi$
$163$	−7.94353e6	−1.83422	−0.917109	+	0.398637i	$0.869483\pi$
$164$	1.85203e6i	0.419872i
$165$	0	0
$166$	−4.78714e6	−1.04653
$167$	− 3.53832e6i	− 0.759710i	−0.925046	−	0.379855i	$-0.875974\pi$
$167$	− 3.53832e6i	− 0.759710i	0.925046	−	0.379855i	$-0.124026\pi$
$168$	0	0
$169$	1.39822e6	0.289677
$170$	2.69568e6i	0.548683i
$171$	0	0
$172$	−1.97764e6	−0.388653
$173$	1.76842e6i	0.341544i	0.985311	+	0.170772i	$0.0546261\pi$
$173$	1.76842e6i	0.341544i	−0.985311	+	0.170772i	$0.945374\pi$
$174$	0	0
$175$	−1.25505e7	−2.34179
$176$	948480.i	0.173976i
$177$	0	0
$178$	1.62346e6	0.287859
$179$	3.82637e6i	0.667156i	0.942722	+	0.333578i	$0.108256\pi$
$179$	3.82637e6i	0.667156i	−0.942722	+	0.333578i	$0.891744\pi$
$180$	0	0
$181$	−2.32721e6	−0.392464	−0.196232	−	0.980558i	$-0.562871\pi$
$181$	−2.32721e6	−0.392464	−0.196232	+	0.980558i	$0.562871\pi$
$182$	4.47603e6i	0.742470i
$183$	0	0
$184$	7.92230e6	1.27174
$185$	− 7.82184e6i	− 1.23536i
$186$	0	0
$187$	1.16813e6	0.178635
$188$	111552.i	0.0167882i
$189$	0	0
$190$	−3.61296e6	−0.526747
$191$	5.16235e6i	0.740879i	0.928857	+	0.370440i	$0.120793\pi$
$191$	5.16235e6i	0.740879i	−0.928857	+	0.370440i	$0.879207\pi$
$192$	0	0
$193$	−3.47554e6	−0.483448	−0.241724	−	0.970345i	$-0.577713\pi$
$193$	−3.47554e6	−0.483448	−0.241724	+	0.970345i	$0.577713\pi$
$194$	1.32436e6i	0.181385i
$195$	0	0
$196$	−790944.	−0.105046
$197$	− 7.36920e6i	− 0.963877i	−0.876205	−	0.481939i	$-0.839933\pi$
$197$	− 7.36920e6i	− 0.963877i	0.876205	−	0.481939i	$-0.160067\pi$
$198$	0	0
$199$	8.43100e6	1.06984	0.534921	−	0.844902i	$-0.320341\pi$
$199$	8.43100e6	1.06984	0.534921	+	0.844902i	$0.320341\pi$
$200$	− 2.31702e7i	− 2.89628i
$201$	0	0
$202$	−1.73722e6	−0.210766
$203$	7.08989e6i	0.847523i
$204$	0	0
$205$	1.58746e7	1.84264
$206$	− 3.56523e6i	− 0.407836i
$207$	0	0
$208$	−3.79240e6	−0.421429
$209$	1.56562e6i	0.171493i
$210$	0	0
$211$	4.10012e6	0.436465	0.218233	−	0.975897i	$-0.429971\pi$
$211$	4.10012e6	0.436465	0.218233	+	0.975897i	$0.429971\pi$
$212$	5.34643e6i	0.561121i
$213$	0	0
$214$	1.19621e7	1.22058
$215$	1.69512e7i	1.70563i
$216$	0	0
$217$	1.59367e6	0.155962
$218$	9.06812e6i	0.875282i
$219$	0	0
$220$	−4.19328e6	−0.393809
$221$	4.67064e6i	0.432713i
$222$	0	0
$223$	−1.45364e7	−1.31081	−0.655407	−	0.755276i	$-0.727503\pi$
$223$	−1.45364e7	−1.31081	−0.655407	+	0.755276i	$0.727503\pi$
$224$	7.83619e6i	0.697206i
$225$	0	0
$226$	9.90288e6	0.857899
$227$	− 5.91514e6i	− 0.505693i	−0.967506	−	0.252847i	$-0.918633\pi$
$227$	− 5.91514e6i	− 0.505693i	0.967506	−	0.252847i	$-0.0813668\pi$
$228$	0	0
$229$	1.91699e7	1.59630	0.798149	−	0.602460i	$-0.205813\pi$
$229$	1.91699e7	1.59630	0.798149	+	0.602460i	$0.205813\pi$
$230$	− 2.06669e7i	− 1.69860i
$231$	0	0
$232$	−1.30890e7	−1.04820
$233$	− 7.40563e6i	− 0.585456i	−0.956196	−	0.292728i	$-0.905437\pi$
$233$	− 7.40563e6i	− 0.585456i	0.956196	−	0.292728i	$-0.0945631\pi$
$234$	0	0
$235$	956160.	0.0736762
$236$	6.64608e6i	0.505626i
$237$	0	0
$238$	−3.35837e6	−0.249114
$239$	− 1.48687e7i	− 1.08913i	−0.838720	−	0.544563i	$-0.816695\pi$
$239$	− 1.48687e7i	− 1.08913i	0.838720	−	0.544563i	$-0.183305\pi$
$240$	0	0
$241$	−4.22995e6	−0.302193	−0.151097	−	0.988519i	$-0.548280\pi$
$241$	−4.22995e6	−0.302193	−0.151097	+	0.988519i	$0.548280\pi$
$242$	8.29311e6i	0.585156i
$243$	0	0
$244$	−1.73043e6	−0.119120
$245$	6.77952e6i	0.461000i
$246$	0	0
$247$	−6.25996e6	−0.415413
$248$	2.94216e6i	0.192891i
$249$	0	0
$250$	−3.79440e7	−2.42842
$251$	− 1.32500e7i	− 0.837906i	−0.908008	−	0.418953i	$-0.862397\pi$
$251$	− 1.32500e7i	− 0.837906i	0.908008	−	0.418953i	$-0.137603\pi$
$252$	0	0
$253$	−8.95565e6	−0.553013
$254$	8.46808e6i	0.516754i
$255$	0	0
$256$	−1.71907e7	−1.02464
$257$	3.21672e7i	1.89502i	0.319726	+	0.947510i	$0.396409\pi$
$257$	3.21672e7i	1.89502i	−0.319726	+	0.947510i	$0.603591\pi$
$258$	0	0
$259$	9.74471e6	0.560880
$260$	− 1.67664e7i	− 0.953937i
$261$	0	0
$262$	1.93939e6	0.107835
$263$	− 2.91608e7i	− 1.60299i	−0.597999	−	0.801497i	$-0.704037\pi$
$263$	− 2.91608e7i	− 1.60299i	0.597999	−	0.801497i	$-0.295963\pi$
$264$	0	0
$265$	4.58266e7	2.46252
$266$	− 4.50115e6i	− 0.239154i
$267$	0	0
$268$	−1.20473e7	−0.625872
$269$	− 1.84710e7i	− 0.948930i	−0.880274	−	0.474465i	$-0.842642\pi$
$269$	− 1.84710e7i	− 0.948930i	0.880274	−	0.474465i	$-0.157358\pi$
$270$	0	0
$271$	−2.27473e7	−1.14294	−0.571468	−	0.820624i	$-0.693626\pi$
$271$	−2.27473e7	−1.14294	−0.571468	+	0.820624i	$0.693626\pi$
$272$	− 2.84544e6i	− 0.141398i
$273$	0	0
$274$	1.28307e7	0.623732
$275$	2.61924e7i	1.25944i
$276$	0	0
$277$	−2.17765e7	−1.02459	−0.512293	−	0.858811i	$-0.671204\pi$
$277$	−2.17765e7	−1.02459	−0.512293	+	0.858811i	$0.671204\pi$
$278$	− 667374.i	− 0.0310624i
$279$	0	0
$280$	3.96115e7	1.80446
$281$	3.01957e7i	1.36090i	0.732794	+	0.680451i	$0.238216\pi$
$281$	3.01957e7i	1.36090i	−0.732794	+	0.680451i	$0.761784\pi$
$282$	0	0
$283$	−1.97788e7	−0.872649	−0.436325	−	0.899789i	$-0.643720\pi$
$283$	−1.97788e7	−0.872649	−0.436325	+	0.899789i	$0.643720\pi$
$284$	− 7.04794e6i	− 0.307685i
$285$	0	0
$286$	9.34128e6	0.399308
$287$	1.97771e7i	0.836596i
$288$	0	0
$289$	2.06332e7	0.854816
$290$	3.41453e7i	1.40003i
$291$	0	0
$292$	7.04522e6	0.282974
$293$	− 1.99512e7i	− 0.793168i	−0.917998	−	0.396584i	$-0.870196\pi$
$293$	− 1.99512e7i	− 0.793168i	0.917998	−	0.396584i	$-0.129804\pi$
$294$	0	0
$295$	5.69664e7	2.21898
$296$	1.79902e7i	0.693684i
$297$	0	0
$298$	−2.06196e7	−0.779170
$299$	− 3.58082e7i	− 1.33958i
$300$	0	0
$301$	−2.11184e7	−0.774392
$302$	− 2.36360e7i	− 0.858128i
$303$	0	0
$304$	3.81368e6	0.135745
$305$	1.48322e7i	0.522766i
$306$	0	0
$307$	2.43404e7	0.841226	0.420613	−	0.907240i	$-0.361815\pi$
$307$	2.43404e7	0.841226	0.420613	+	0.907240i	$0.361815\pi$
$308$	− 5.22413e6i	− 0.178798i
$309$	0	0
$310$	7.67520e6	0.257635
$311$	− 5.31455e7i	− 1.76679i	−0.468629	−	0.883395i	$-0.655252\pi$
$311$	− 5.31455e7i	− 1.76679i	0.468629	−	0.883395i	$-0.344748\pi$
$312$	0	0
$313$	−3.17868e6	−0.103661	−0.0518303	−	0.998656i	$-0.516506\pi$
$313$	−3.17868e6	−0.103661	−0.0518303	+	0.998656i	$0.516506\pi$
$314$	2.58640e6i	0.0835422i
$315$	0	0
$316$	1.85032e7	0.586387
$317$	− 3.70569e7i	− 1.16330i	−0.813440	−	0.581649i	$-0.802408\pi$
$317$	− 3.70569e7i	− 1.16330i	0.813440	−	0.581649i	$-0.197592\pi$
$318$	0	0
$319$	1.47963e7	0.455807
$320$	6.10867e7i	1.86422i
$321$	0	0
$322$	2.57475e7	0.771200
$323$	− 4.69685e6i	− 0.139380i
$324$	0	0
$325$	−1.04728e8	−3.05078
$326$	− 4.76612e7i	− 1.37566i
$327$	0	0
$328$	−3.65115e7	−1.03468
$329$	1.19122e6i	0.0334505i
$330$	0	0
$331$	5.46748e6	0.150766	0.0753829	−	0.997155i	$-0.475982\pi$
$331$	5.46748e6	0.150766	0.0753829	+	0.997155i	$0.475982\pi$
$332$	2.23400e7i	0.610476i
$333$	0	0
$334$	2.12299e7	0.569782
$335$	1.03263e8i	2.74668i
$336$	0	0
$337$	−4.09219e7	−1.06922	−0.534609	−	0.845099i	$-0.679541\pi$
$337$	−4.09219e7	−1.06922	−0.534609	+	0.845099i	$0.679541\pi$
$338$	8.38930e6i	0.217258i
$339$	0	0
$340$	1.25798e7	0.320065
$341$	− 3.32592e6i	− 0.0838781i
$342$	0	0
$343$	−4.36232e7	−1.08102
$344$	− 3.89878e7i	− 0.957752i
$345$	0	0
$346$	−1.06105e7	−0.256158
$347$	5.36465e7i	1.28397i	0.766719	+	0.641983i	$0.221888\pi$
$347$	5.36465e7i	1.28397i	−0.766719	+	0.641983i	$0.778112\pi$
$348$	0	0
$349$	1.60887e7	0.378482	0.189241	−	0.981931i	$-0.439397\pi$
$349$	1.60887e7	0.378482	0.189241	+	0.981931i	$0.439397\pi$
$350$	− 7.53032e7i	− 1.75634i
$351$	0	0
$352$	1.63538e7	0.374965
$353$	4.11130e6i	0.0934662i	0.998907	+	0.0467331i	$0.0148810\pi$
$353$	4.11130e6i	0.0934662i	−0.998907	+	0.0467331i	$0.985119\pi$
$354$	0	0
$355$	−6.04109e7	−1.35030
$356$	− 7.57613e6i	− 0.167918i
$357$	0	0
$358$	−2.29582e7	−0.500367
$359$	1.41496e7i	0.305816i	0.988240	+	0.152908i	$0.0488638\pi$
$359$	1.41496e7i	0.305816i	−0.988240	+	0.152908i	$0.951136\pi$
$360$	0	0
$361$	−4.07508e7	−0.866193
$362$	− 1.39633e7i	− 0.294348i
$363$	0	0
$364$	2.08881e7	0.433107
$365$	− 6.03876e7i	− 1.24185i
$366$	0	0
$367$	4.13390e7	0.836300	0.418150	−	0.908378i	$-0.362679\pi$
$367$	4.13390e7	0.836300	0.418150	+	0.908378i	$0.362679\pi$
$368$	2.18150e7i	0.437736i
$369$	0	0
$370$	4.69310e7	0.926520
$371$	5.70923e7i	1.11804i
$372$	0	0
$373$	5.83380e7	1.12415	0.562076	−	0.827085i	$-0.310003\pi$
$373$	5.83380e7	1.12415	0.562076	+	0.827085i	$0.310003\pi$
$374$	7.00877e6i	0.133976i
$375$	0	0
$376$	−2.19917e6	−0.0413709
$377$	5.91614e7i	1.10412i
$378$	0	0
$379$	7.49065e7	1.37595	0.687974	−	0.725735i	$-0.258500\pi$
$379$	7.49065e7	1.37595	0.687974	+	0.725735i	$0.258500\pi$
$380$	1.68605e7i	0.307269i
$381$	0	0
$382$	−3.09741e7	−0.555660
$383$	− 2.84444e7i	− 0.506292i	−0.967428	−	0.253146i	$-0.918535\pi$
$383$	− 2.84444e7i	− 0.506292i	0.967428	−	0.253146i	$-0.0814652\pi$
$384$	0	0
$385$	−4.47782e7	−0.784666
$386$	− 2.08532e7i	− 0.362586i
$387$	0	0
$388$	6.18036e6	0.105808
$389$	− 6.11364e7i	− 1.03861i	−0.854590	−	0.519303i	$-0.826192\pi$
$389$	− 6.11364e7i	− 1.03861i	0.854590	−	0.519303i	$-0.173808\pi$
$390$	0	0
$391$	2.68669e7	0.449457
$392$	− 1.55929e7i	− 0.258862i
$393$	0	0
$394$	4.42152e7	0.722908
$395$	− 1.58598e8i	− 2.57340i
$396$	0	0
$397$	−3.70144e7	−0.591561	−0.295780	−	0.955256i	$-0.595580\pi$
$397$	−3.70144e7	−0.591561	−0.295780	+	0.955256i	$0.595580\pi$
$398$	5.05860e7i	0.802381i
$399$	0	0
$400$	6.38020e7	0.996906
$401$	− 2.08769e7i	− 0.323768i	−0.986810	−	0.161884i	$-0.948243\pi$
$401$	− 2.08769e7i	− 0.323768i	0.986810	−	0.161884i	$-0.0517570\pi$
$402$	0	0
$403$	1.32984e7	0.203181
$404$	8.10701e6i	0.122947i
$405$	0	0
$406$	−4.25393e7	−0.635642
$407$	− 2.03368e7i	− 0.301647i
$408$	0	0
$409$	9.10530e7	1.33084	0.665418	−	0.746471i	$-0.268254\pi$
$409$	9.10530e7	1.33084	0.665418	+	0.746471i	$0.268254\pi$
$410$	9.52474e7i	1.38198i
$411$	0	0
$412$	−1.66377e7	−0.237905
$413$	7.09706e7i	1.00746i
$414$	0	0
$415$	1.91485e8	2.67912
$416$	6.53890e7i	0.908290i
$417$	0	0
$418$	−9.39370e6	−0.128620
$419$	1.15211e8i	1.56622i	0.621885	+	0.783108i	$0.286367\pi$
$419$	1.15211e8i	1.56622i	−0.621885	+	0.783108i	$0.713633\pi$
$420$	0	0
$421$	−1.31590e8	−1.76350	−0.881750	−	0.471718i	$-0.843634\pi$
$421$	−1.31590e8	−1.76350	−0.881750	+	0.471718i	$0.843634\pi$
$422$	2.46007e7i	0.327349i
$423$	0	0
$424$	−1.05401e8	−1.38276
$425$	− 7.85772e7i	− 1.02360i
$426$	0	0
$427$	−1.84785e7	−0.237347
$428$	− 5.58230e7i	− 0.712004i
$429$	0	0
$430$	−1.01707e8	−1.27922
$431$	7.15543e7i	0.893725i	0.894603	+	0.446863i	$0.147459\pi$
$431$	7.15543e7i	0.893725i	−0.894603	+	0.446863i	$0.852541\pi$
$432$	0	0
$433$	1.83937e7	0.226572	0.113286	−	0.993562i	$-0.463862\pi$
$433$	1.83937e7	0.226572	0.113286	+	0.993562i	$0.463862\pi$
$434$	9.56202e6i	0.116972i
$435$	0	0
$436$	4.23179e7	0.510581
$437$	3.60092e7i	0.431488i
$438$	0	0
$439$	−9.90611e7	−1.17087	−0.585436	−	0.810719i	$-0.699077\pi$
$439$	−9.90611e7	−1.17087	−0.585436	+	0.810719i	$0.699077\pi$
$440$	− 8.26675e7i	− 0.970458i
$441$	0	0
$442$	−2.80238e7	−0.324534
$443$	− 1.35397e8i	− 1.55739i	−0.627403	−	0.778695i	$-0.715882\pi$
$443$	− 1.35397e8i	− 1.55739i	0.627403	−	0.778695i	$-0.284118\pi$
$444$	0	0
$445$	−6.49382e7	−0.736920
$446$	− 8.72182e7i	− 0.983111i
$447$	0	0
$448$	−7.61039e7	−0.846394
$449$	6.85948e7i	0.757796i	0.925438	+	0.378898i	$0.123697\pi$
$449$	6.85948e7i	0.757796i	−0.925438	+	0.378898i	$0.876303\pi$
$450$	0	0
$451$	4.12739e7	0.449930
$452$	− 4.62134e7i	− 0.500441i
$453$	0	0
$454$	3.54908e7	0.379270
$455$	− 1.79041e8i	− 1.90072i
$456$	0	0
$457$	7.12649e7	0.746667	0.373334	−	0.927697i	$-0.378215\pi$
$457$	7.12649e7	0.746667	0.373334	+	0.927697i	$0.378215\pi$
$458$	1.15020e8i	1.19722i
$459$	0	0
$460$	−9.64454e7	−0.990851
$461$	1.56559e8i	1.59800i	0.601332	+	0.798999i	$0.294637\pi$
$461$	1.56559e8i	1.59800i	−0.601332	+	0.798999i	$0.705363\pi$
$462$	0	0
$463$	−553813.	−0.00557982	−0.00278991	−	0.999996i	$-0.500888\pi$
$463$	−553813.	−0.00557982	−0.00278991	+	0.999996i	$0.500888\pi$
$464$	− 3.60422e7i	− 0.360793i
$465$	0	0
$466$	4.44338e7	0.439092
$467$	− 3.14739e7i	− 0.309030i	−0.987990	−	0.154515i	$-0.950619\pi$
$467$	− 3.14739e7i	− 0.309030i	0.987990	−	0.154515i	$-0.0493815\pi$
$468$	0	0
$469$	−1.28648e8	−1.24705
$470$	5.73696e6i	0.0552571i
$471$	0	0
$472$	−1.31023e8	−1.24601
$473$	4.40731e7i	0.416476i
$474$	0	0
$475$	1.05315e8	0.982676
$476$	1.56724e7i	0.145316i
$477$	0	0
$478$	8.92120e7	0.816845
$479$	6.67838e7i	0.607666i	0.952725	+	0.303833i	$0.0982665\pi$
$479$	6.67838e7i	0.607666i	−0.952725	+	0.303833i	$0.901734\pi$
$480$	0	0
$481$	8.13145e7	0.730690
$482$	− 2.53797e7i	− 0.226645i
$483$	0	0
$484$	3.87012e7	0.341341
$485$	− 5.29745e7i	− 0.464346i
$486$	0	0
$487$	2.00508e8	1.73598	0.867992	−	0.496579i	$-0.165411\pi$
$487$	2.00508e8	1.73598	0.867992	+	0.496579i	$0.165411\pi$
$488$	− 3.41142e7i	− 0.293545i
$489$	0	0
$490$	−4.06771e7	−0.345750
$491$	8.61838e7i	0.728083i	0.931383	+	0.364042i	$0.118603\pi$
$491$	8.61838e7i	0.728083i	−0.931383	+	0.364042i	$0.881397\pi$
$492$	0	0
$493$	−4.43889e7	−0.370453
$494$	− 3.75597e7i	− 0.311560i
$495$	0	0
$496$	−8.10160e6	−0.0663935
$497$	− 7.52619e7i	− 0.613064i
$498$	0	0
$499$	1.06673e7	0.0858525	0.0429263	−	0.999078i	$-0.486332\pi$
$499$	1.06673e7	0.0858525	0.0429263	+	0.999078i	$0.486332\pi$
$500$	1.77072e8i	1.41658i
$501$	0	0
$502$	7.95001e7	0.628429
$503$	− 1.07893e8i	− 0.847790i	−0.905711	−	0.423895i	$-0.860663\pi$
$503$	− 1.07893e8i	− 0.847790i	0.905711	−	0.423895i	$-0.139337\pi$
$504$	0	0
$505$	6.94886e7	0.539560
$506$	− 5.37339e7i	− 0.414760i
$507$	0	0
$508$	3.95177e7	0.301440
$509$	1.54033e8i	1.16805i	0.811736	+	0.584025i	$0.198523\pi$
$509$	1.54033e8i	1.16805i	−0.811736	+	0.584025i	$0.801477\pi$
$510$	0	0
$511$	7.52329e7	0.563826
$512$	− 9.35347e7i	− 0.696888i
$513$	0	0
$514$	−1.93003e8	−1.42127
$515$	1.42609e8i	1.04406i
$516$	0	0
$517$	2.48602e6	0.0179900
$518$	5.84683e7i	0.420660i
$519$	0	0
$520$	3.30538e8	2.35077
$521$	− 1.85442e8i	− 1.31128i	−0.755074	−	0.655640i	$-0.772399\pi$
$521$	− 1.85442e8i	− 1.31128i	0.755074	−	0.655640i	$-0.227601\pi$
$522$	0	0
$523$	−1.46950e7	−0.102723	−0.0513613	−	0.998680i	$-0.516356\pi$
$523$	−1.46950e7	−0.102723	−0.0513613	+	0.998680i	$0.516356\pi$
$524$	− 9.05050e6i	− 0.0629040i
$525$	0	0
$526$	1.74965e8	1.20224
$527$	9.97776e6i	0.0681712i
$528$	0	0
$529$	−5.79440e7	−0.391419
$530$	2.74959e8i	1.84689i
$531$	0	0
$532$	−2.10053e7	−0.139507
$533$	1.65029e8i	1.08988i
$534$	0	0
$535$	−4.78483e8	−3.12468
$536$	− 2.37504e8i	− 1.54233i
$537$	0	0
$538$	1.10826e8	0.711697
$539$	1.76268e7i	0.112566i
$540$	0	0
$541$	−1.03570e7	−0.0654098	−0.0327049	−	0.999465i	$-0.510412\pi$
$541$	−1.03570e7	−0.0654098	−0.0327049	+	0.999465i	$0.510412\pi$
$542$	− 1.36484e8i	− 0.857202i
$543$	0	0
$544$	−4.90614e7	−0.304750
$545$	− 3.62725e8i	− 2.24072i
$546$	0	0
$547$	8.03197e7	0.490750	0.245375	−	0.969428i	$-0.421089\pi$
$547$	8.03197e7	0.490750	0.245375	+	0.969428i	$0.421089\pi$
$548$	− 5.98765e7i	− 0.363844i
$549$	0	0
$550$	−1.57154e8	−0.944579
$551$	− 5.94934e7i	− 0.355643i
$552$	0	0
$553$	1.97587e8	1.16838
$554$	− 1.30659e8i	− 0.768439i
$555$	0	0
$556$	−3.11441e6	−0.0181197
$557$	− 3.87935e7i	− 0.224488i	−0.993681	−	0.112244i	$-0.964196\pi$
$557$	− 3.87935e7i	− 0.224488i	0.993681	−	0.112244i	$-0.0358038\pi$
$558$	0	0
$559$	−1.76222e8	−1.00884
$560$	1.09075e8i	0.621101i
$561$	0	0
$562$	−1.81174e8	−1.02068
$563$	2.45635e8i	1.37647i	0.725490	+	0.688233i	$0.241613\pi$
$563$	2.45635e8i	1.37647i	−0.725490	+	0.688233i	$0.758387\pi$
$564$	0	0
$565$	−3.96115e8	−2.19622
$566$	− 1.18673e8i	− 0.654487i
$567$	0	0
$568$	1.38945e8	0.758225
$569$	− 3.37004e7i	− 0.182935i	−0.995808	−	0.0914677i	$-0.970844\pi$
$569$	− 3.37004e7i	− 0.182935i	0.995808	−	0.0914677i	$-0.0291558\pi$
$570$	0	0
$571$	−2.36969e8	−1.27287	−0.636435	−	0.771330i	$-0.719592\pi$
$571$	−2.36969e8	−1.27287	−0.636435	+	0.771330i	$0.719592\pi$
$572$	− 4.35926e7i	− 0.232930i
$573$	0	0
$574$	−1.18662e8	−0.627447
$575$	6.02425e8i	3.16883i
$576$	0	0
$577$	4.09015e6	0.0212918	0.0106459	−	0.999943i	$-0.496611\pi$
$577$	4.09015e6	0.0212918	0.0106459	+	0.999943i	$0.496611\pi$
$578$	1.23799e8i	0.641112i
$579$	0	0
$580$	1.59345e8	0.816683
$581$	2.38559e8i	1.21637i
$582$	0	0
$583$	1.19149e8	0.601291
$584$	1.38891e8i	0.697328i
$585$	0	0
$586$	1.19707e8	0.594876
$587$	− 3.64756e8i	− 1.80338i	−0.432379	−	0.901692i	$-0.642326\pi$
$587$	− 3.64756e8i	− 1.80338i	0.432379	−	0.901692i	$-0.357674\pi$
$588$	0	0
$589$	−1.33730e7	−0.0654458
$590$	3.41798e8i	1.66423i
$591$	0	0
$592$	−4.95383e7	−0.238768
$593$	− 3.38297e8i	− 1.62231i	−0.584832	−	0.811154i	$-0.698840\pi$
$593$	− 3.38297e8i	− 1.62231i	0.584832	−	0.811154i	$-0.301160\pi$
$594$	0	0
$595$	1.34335e8	0.637731
$596$	9.62250e7i	0.454516i
$597$	0	0
$598$	2.14849e8	1.00469
$599$	− 2.52534e8i	− 1.17500i	−0.809223	−	0.587502i	$-0.800111\pi$
$599$	− 2.52534e8i	− 1.17500i	0.809223	−	0.587502i	$-0.199889\pi$
$600$	0	0
$601$	2.67857e8	1.23390	0.616949	−	0.787003i	$-0.288368\pi$
$601$	2.67857e8	1.23390	0.616949	+	0.787003i	$0.288368\pi$
$602$	− 1.26710e8i	− 0.580794i
$603$	0	0
$604$	−1.10301e8	−0.500575
$605$	− 3.31724e8i	− 1.49800i
$606$	0	0
$607$	−2.72153e8	−1.21688	−0.608438	−	0.793601i	$-0.708204\pi$
$607$	−2.72153e8	−1.21688	−0.608438	+	0.793601i	$0.708204\pi$
$608$	− 6.57559e7i	− 0.292566i
$609$	0	0
$610$	−8.89934e7	−0.392074
$611$	9.94008e6i	0.0435779i
$612$	0	0
$613$	−4.91383e7	−0.213323	−0.106662	−	0.994295i	$-0.534016\pi$
$613$	−4.91383e7	−0.213323	−0.106662	+	0.994295i	$0.534016\pi$
$614$	1.46042e8i	0.630919i
$615$	0	0
$616$	1.02990e8	0.440608
$617$	1.24682e8i	0.530823i	0.964135	+	0.265412i	$0.0855078\pi$
$617$	1.24682e8i	0.530823i	−0.964135	+	0.265412i	$0.914492\pi$
$618$	0	0
$619$	−4.53573e8	−1.91239	−0.956193	−	0.292738i	$-0.905434\pi$
$619$	−4.53573e8	−1.91239	−0.956193	+	0.292738i	$0.905434\pi$
$620$	− 3.58176e7i	− 0.150287i
$621$	0	0
$622$	3.18873e8	1.32509
$623$	− 8.09022e7i	− 0.334577i
$624$	0	0
$625$	8.61901e8	3.53034
$626$	− 1.90721e7i	− 0.0777455i
$627$	0	0
$628$	1.20698e7	0.0487330
$629$	6.10104e7i	0.245161i
$630$	0	0
$631$	−5.31523e7	−0.211560	−0.105780	−	0.994390i	$-0.533734\pi$
$631$	−5.31523e7	−0.211560	−0.105780	+	0.994390i	$0.533734\pi$
$632$	3.64777e8i	1.44503i
$633$	0	0
$634$	2.22341e8	0.872473
$635$	− 3.38723e8i	− 1.32289i
$636$	0	0
$637$	−7.04788e7	−0.272672
$638$	8.87777e7i	0.341855i
$639$	0	0
$640$	3.60346e7	0.137461
$641$	3.65442e8i	1.38754i	0.720199	+	0.693768i	$0.244051\pi$
$641$	3.65442e8i	1.38754i	−0.720199	+	0.693768i	$0.755949\pi$
$642$	0	0
$643$	1.52561e8	0.573867	0.286933	−	0.957951i	$-0.407364\pi$
$643$	1.52561e8	0.573867	0.286933	+	0.957951i	$0.407364\pi$
$644$	− 1.20155e8i	− 0.449867i
$645$	0	0
$646$	2.81811e7	0.104535
$647$	− 1.22990e7i	− 0.0454107i	−0.999742	−	0.0227054i	$-0.992772\pi$
$647$	− 1.22990e7i	− 0.0454107i	0.999742	−	0.0227054i	$-0.00722796\pi$
$648$	0	0
$649$	1.48113e8	0.541824
$650$	− 6.28366e8i	− 2.28809i
$651$	0	0
$652$	−2.22419e8	−0.802470
$653$	− 5.72096e7i	− 0.205461i	−0.994709	−	0.102730i	$-0.967242\pi$
$653$	− 5.72096e7i	− 0.205461i	0.994709	−	0.102730i	$-0.0327579\pi$
$654$	0	0
$655$	−7.75757e7	−0.276059
$656$	− 1.00539e8i	− 0.356141i
$657$	0	0
$658$	−7.14730e6	−0.0250879
$659$	2.30093e8i	0.803982i	0.915644	+	0.401991i	$0.131682\pi$
$659$	2.30093e8i	0.803982i	−0.915644	+	0.401991i	$0.868318\pi$
$660$	0	0
$661$	4.81046e7	0.166564	0.0832822	−	0.996526i	$-0.473460\pi$
$661$	4.81046e7	0.166564	0.0832822	+	0.996526i	$0.473460\pi$
$662$	3.28048e7i	0.113074i
$663$	0	0
$664$	−4.40417e8	−1.50439
$665$	1.80046e8i	0.612235i
$666$	0	0
$667$	3.40315e8	1.14684
$668$	− 9.90730e7i	− 0.332373i
$669$	0	0
$670$	−6.19576e8	−2.06001
$671$	3.85638e7i	0.127648i
$672$	0	0
$673$	2.30653e8	0.756681	0.378341	−	0.925666i	$-0.376495\pi$
$673$	2.30653e8	0.756681	0.378341	+	0.925666i	$0.376495\pi$
$674$	− 2.45532e8i	− 0.801914i
$675$	0	0
$676$	3.91500e7	0.126734
$677$	5.64246e8i	1.81846i	0.416299	+	0.909228i	$0.363327\pi$
$677$	5.64246e8i	1.81846i	−0.416299	+	0.909228i	$0.636673\pi$
$678$	0	0
$679$	6.59974e7	0.210823
$680$	2.48003e8i	0.788732i
$681$	0	0
$682$	1.99555e7	0.0629086
$683$	− 2.05363e8i	− 0.644555i	−0.946645	−	0.322278i	$-0.895552\pi$
$683$	− 2.05363e8i	− 0.644555i	0.946645	−	0.322278i	$-0.104448\pi$
$684$	0	0
$685$	−5.13228e8	−1.59675
$686$	− 2.61739e8i	− 0.810768i
$687$	0	0
$688$	1.07358e8	0.329661
$689$	4.76405e8i	1.45653i
$690$	0	0
$691$	−1.08669e8	−0.329360	−0.164680	−	0.986347i	$-0.552659\pi$
$691$	−1.08669e8	−0.329360	−0.164680	+	0.986347i	$0.552659\pi$
$692$	4.95156e7i	0.149425i
$693$	0	0
$694$	−3.21879e8	−0.962974
$695$	2.66950e7i	0.0795197i
$696$	0	0
$697$	−1.23822e8	−0.365677
$698$	9.65324e7i	0.283862i
$699$	0	0
$700$	−3.51415e8	−1.02453
$701$	1.08271e8i	0.314310i	0.987574	+	0.157155i	$0.0502321\pi$
$701$	1.08271e8i	0.314310i	−0.987574	+	0.157155i	$0.949768\pi$
$702$	0	0
$703$	−8.17708e7	−0.235360
$704$	1.58825e8i	0.455200i
$705$	0	0
$706$	−2.46678e7	−0.0700997
$707$	8.65713e7i	0.244971i
$708$	0	0
$709$	1.68535e8	0.472881	0.236440	−	0.971646i	$-0.424019\pi$
$709$	1.68535e8	0.472881	0.236440	+	0.971646i	$0.424019\pi$
$710$	− 3.62465e8i	− 1.01272i
$711$	0	0
$712$	1.49358e8	0.413798
$713$	− 7.64962e7i	− 0.211043i
$714$	0	0
$715$	−3.73651e8	−1.02223
$716$	1.07138e8i	0.291881i
$717$	0	0
$718$	−8.48975e7	−0.229362
$719$	− 6.10328e8i	− 1.64201i	−0.570918	−	0.821007i	$-0.693413\pi$
$719$	− 6.10328e8i	− 1.64201i	0.570918	−	0.821007i	$-0.306587\pi$
$720$	0	0
$721$	−1.77667e8	−0.474025
$722$	− 2.44505e8i	− 0.649645i
$723$	0	0
$724$	−6.51619e7	−0.171703
$725$	− 9.95311e8i	− 2.61183i
$726$	0	0
$727$	6.15925e8	1.60297	0.801483	−	0.598017i	$-0.204045\pi$
$727$	6.15925e8	1.60297	0.801483	+	0.598017i	$0.204045\pi$
$728$	4.11795e8i	1.06730i
$729$	0	0
$730$	3.62326e8	0.931388
$731$	− 1.32219e8i	− 0.338488i
$732$	0	0
$733$	3.65043e8	0.926897	0.463449	−	0.886124i	$-0.346612\pi$
$733$	3.65043e8	0.926897	0.463449	+	0.886124i	$0.346612\pi$
$734$	2.48034e8i	0.627225i
$735$	0	0
$736$	3.76137e8	0.943437
$737$	2.68483e8i	0.670678i
$738$	0	0
$739$	−7.10760e8	−1.76112	−0.880562	−	0.473931i	$-0.842835\pi$
$739$	−7.10760e8	−1.76112	−0.880562	+	0.473931i	$0.842835\pi$
$740$	− 2.19012e8i	− 0.540470i
$741$	0	0
$742$	−3.42554e8	−0.838526
$743$	− 2.11394e8i	− 0.515378i	−0.966228	−	0.257689i	$-0.917039\pi$
$743$	− 2.11394e8i	− 0.515378i	0.966228	−	0.257689i	$-0.0829611\pi$
$744$	0	0
$745$	8.24786e8	1.99468
$746$	3.50028e8i	0.843115i
$747$	0	0
$748$	3.27076e7	0.0781526
$749$	− 5.96110e8i	− 1.41867i
$750$	0	0
$751$	−1.90625e8	−0.450050	−0.225025	−	0.974353i	$-0.572246\pi$
$751$	−1.90625e8	−0.450050	−0.225025	+	0.974353i	$0.572246\pi$
$752$	− 6.05568e6i	− 0.0142400i
$753$	0	0
$754$	−3.54969e8	−0.828087
$755$	9.45438e8i	2.19681i
$756$	0	0
$757$	6.27405e8	1.44631	0.723153	−	0.690688i	$-0.242692\pi$
$757$	6.27405e8	1.44631	0.723153	+	0.690688i	$0.242692\pi$
$758$	4.49439e8i	1.03196i
$759$	0	0
$760$	−3.32392e8	−0.757199
$761$	7.99042e7i	0.181307i	0.995882	+	0.0906537i	$0.0288956\pi$
$761$	7.99042e7i	0.181307i	−0.995882	+	0.0906537i	$0.971104\pi$
$762$	0	0
$763$	4.51895e8	1.01733
$764$	1.44546e8i	0.324135i
$765$	0	0
$766$	1.70666e8	0.379719
$767$	5.92213e8i	1.31248i
$768$	0	0
$769$	6.64475e8	1.46117	0.730583	−	0.682824i	$-0.239249\pi$
$769$	6.64475e8	1.46117	0.730583	+	0.682824i	$0.239249\pi$
$770$	− 2.68669e8i	− 0.588499i
$771$	0	0
$772$	−9.73150e7	−0.211509
$773$	6.85987e6i	0.0148517i	0.999972	+	0.00742587i	$0.00236375\pi$
$773$	6.85987e6i	0.0148517i	−0.999972	+	0.00742587i	$0.997636\pi$
$774$	0	0
$775$	−2.23727e8	−0.480632
$776$	1.21841e8i	0.260741i
$777$	0	0
$778$	3.66818e8	0.778955
$779$	− 1.65955e8i	− 0.351058i
$780$	0	0
$781$	−1.57068e8	−0.329712
$782$	1.61202e8i	0.337093i
$783$	0	0
$784$	4.29370e7	0.0891011
$785$	− 1.03456e8i	− 0.213868i
$786$	0	0
$787$	−3.05590e8	−0.626923	−0.313462	−	0.949601i	$-0.601489\pi$
$787$	−3.05590e8	−0.626923	−0.313462	+	0.949601i	$0.601489\pi$
$788$	− 2.06338e8i	− 0.421696i
$789$	0	0
$790$	9.51591e8	1.93005
$791$	− 4.93494e8i	− 0.997131i
$792$	0	0
$793$	−1.54193e8	−0.309205
$794$	− 2.22086e8i	− 0.443670i
$795$	0	0
$796$	2.36068e8	0.468056
$797$	2.01445e8i	0.397906i	0.980009	+	0.198953i	$0.0637542\pi$
$797$	2.01445e8i	0.397906i	−0.980009	+	0.198953i	$0.936246\pi$
$798$	0	0
$799$	−7.45805e6	−0.0146213
$800$	− 1.10008e9i	− 2.14860i
$801$	0	0
$802$	1.25262e8	0.242826
$803$	− 1.57008e8i	− 0.303232i
$804$	0	0
$805$	−1.02990e9	−1.97427
$806$	7.97901e7i	0.152386i
$807$	0	0
$808$	−1.59824e8	−0.302976
$809$	1.11534e8i	0.210650i	0.994438	+	0.105325i	$0.0335882\pi$
$809$	1.11534e8i	0.210650i	−0.994438	+	0.105325i	$0.966412\pi$
$810$	0	0
$811$	−1.72124e8	−0.322685	−0.161343	−	0.986898i	$-0.551582\pi$
$811$	−1.72124e8	−0.322685	−0.161343	+	0.986898i	$0.551582\pi$
$812$	1.98517e8i	0.370791i
$813$	0	0
$814$	1.22021e8	0.226235
$815$	1.90645e9i	3.52170i
$816$	0	0
$817$	1.77211e8	0.324955
$818$	5.46318e8i	0.998126i
$819$	0	0
$820$	4.44488e8	0.806154
$821$	− 8.62483e8i	− 1.55855i	−0.626681	−	0.779276i	$-0.715587\pi$
$821$	− 8.62483e8i	− 1.55855i	0.626681	−	0.779276i	$-0.284413\pi$
$822$	0	0
$823$	−5.89009e6	−0.0105663	−0.00528315	−	0.999986i	$-0.501682\pi$
$823$	−5.89009e6	−0.0105663	−0.00528315	+	0.999986i	$0.501682\pi$
$824$	− 3.28001e8i	− 0.586265i
$825$	0	0
$826$	−4.25824e8	−0.755596
$827$	− 7.60618e8i	− 1.34478i	−0.740199	−	0.672388i	$-0.765269\pi$
$827$	− 7.60618e8i	− 1.34478i	0.740199	−	0.672388i	$-0.234731\pi$
$828$	0	0
$829$	−1.06302e9	−1.86585	−0.932926	−	0.360067i	$-0.882754\pi$
$829$	−1.06302e9	−1.86585	−0.932926	+	0.360067i	$0.882754\pi$
$830$	1.14891e9i	2.00934i
$831$	0	0
$832$	−6.35047e8	−1.10265
$833$	− 5.28803e7i	− 0.0914868i
$834$	0	0
$835$	−8.49197e8	−1.45864
$836$	4.38372e7i	0.0750282i
$837$	0	0
$838$	−6.91266e8	−1.17466
$839$	− 4.66992e8i	− 0.790722i	−0.918526	−	0.395361i	$-0.870619\pi$
$839$	− 4.66992e8i	− 0.790722i	0.918526	−	0.395361i	$-0.129381\pi$
$840$	0	0
$841$	3.25644e7	0.0547463
$842$	− 7.89538e8i	− 1.32262i
$843$	0	0
$844$	1.14803e8	0.190953
$845$	− 3.35572e8i	− 0.556180i
$846$	0	0
$847$	4.13273e8	0.680122
$848$	− 2.90235e8i	− 0.475951i
$849$	0	0
$850$	4.71463e8	0.767699
$851$	− 4.67746e8i	− 0.758964i
$852$	0	0
$853$	−1.02054e9	−1.64430	−0.822150	−	0.569272i	$-0.807225\pi$
$853$	−1.02054e9	−1.64430	−0.822150	+	0.569272i	$0.807225\pi$
$854$	− 1.10871e8i	− 0.178010i
$855$	0	0
$856$	1.10051e9	1.75458
$857$	− 6.31488e7i	− 0.100328i	−0.998741	−	0.0501641i	$-0.984026\pi$
$857$	− 6.31488e7i	− 0.100328i	0.998741	−	0.0501641i	$-0.0159744\pi$
$858$	0	0
$859$	1.03556e9	1.63378	0.816892	−	0.576791i	$-0.195695\pi$
$859$	1.03556e9	1.63378	0.816892	+	0.576791i	$0.195695\pi$
$860$	4.74634e8i	0.746214i
$861$	0	0
$862$	−4.29326e8	−0.670294
$863$	− 4.60575e8i	− 0.716585i	−0.933609	−	0.358292i	$-0.883359\pi$
$863$	− 4.60575e8i	− 0.716585i	0.933609	−	0.358292i	$-0.116641\pi$
$864$	0	0
$865$	4.24420e8	0.655764
$866$	1.10362e8i	0.169929i
$867$	0	0
$868$	4.46228e7	0.0682334
$869$	− 4.12356e8i	− 0.628367i
$870$	0	0
$871$	−1.07350e9	−1.62461
$872$	8.34267e8i	1.25822i
$873$	0	0
$874$	−2.16055e8	−0.323616
$875$	1.89088e9i	2.82253i
$876$	0	0
$877$	−9.04580e8	−1.34106	−0.670530	−	0.741883i	$-0.733933\pi$
$877$	−9.04580e8	−1.34106	−0.670530	+	0.741883i	$0.733933\pi$
$878$	− 5.94366e8i	− 0.878154i
$879$	0	0
$880$	2.27635e8	0.334035
$881$	1.06708e9i	1.56051i	0.625460	+	0.780257i	$0.284912\pi$
$881$	1.06708e9i	1.56051i	−0.625460	+	0.780257i	$0.715088\pi$
$882$	0	0
$883$	−8.52780e7	−0.123867	−0.0619334	−	0.998080i	$-0.519727\pi$
$883$	−8.52780e7	−0.123867	−0.0619334	+	0.998080i	$0.519727\pi$
$884$	1.30778e8i	0.189312i
$885$	0	0
$886$	8.12381e8	1.16804
$887$	2.43209e8i	0.348505i	0.984701	+	0.174252i	$0.0557509\pi$
$887$	2.43209e8i	0.348505i	−0.984701	+	0.174252i	$0.944249\pi$
$888$	0	0
$889$	4.21992e8	0.600619
$890$	− 3.89629e8i	− 0.552690i
$891$	0	0
$892$	−4.07018e8	−0.573481
$893$	− 9.99586e6i	− 0.0140367i
$894$	0	0
$895$	9.18328e8	1.28094
$896$	4.48931e7i	0.0624101i
$897$	0	0
$898$	−4.11569e8	−0.568347
$899$	1.26385e8i	0.173947i
$900$	0	0
$901$	−3.57447e8	−0.488695
$902$	2.47643e8i	0.337448i
$903$	0	0
$904$	9.11065e8	1.23323
$905$	5.58530e8i	0.753531i
$906$	0	0
$907$	8.62425e8	1.15584	0.577922	−	0.816092i	$-0.303864\pi$
$907$	8.62425e8	1.15584	0.577922	+	0.816092i	$0.303864\pi$
$908$	− 1.65624e8i	− 0.221241i
$909$	0	0
$910$	1.07425e9	1.42554
$911$	− 1.14635e9i	− 1.51622i	−0.652126	−	0.758111i	$-0.726123\pi$
$911$	− 1.14635e9i	− 1.51622i	0.652126	−	0.758111i	$-0.273877\pi$
$912$	0	0
$913$	4.97862e8	0.654179
$914$	4.27589e8i	0.560000i
$915$	0	0
$916$	5.36758e8	0.698381
$917$	− 9.66464e7i	− 0.125336i
$918$	0	0
$919$	8.23240e8	1.06067	0.530335	−	0.847788i	$-0.322066\pi$
$919$	8.23240e8	1.06067	0.530335	+	0.847788i	$0.322066\pi$
$920$	− 1.90135e9i	− 2.44174i
$921$	0	0
$922$	−9.39356e8	−1.19850
$923$	− 6.28021e8i	− 0.798674i
$924$	0	0
$925$	−1.36801e9	−1.72848
$926$	− 3.32288e6i	− 0.00418486i
$927$	0	0
$928$	−6.21444e8	−0.777603
$929$	7.28737e8i	0.908916i	0.890768	+	0.454458i	$0.150167\pi$
$929$	7.28737e8i	0.908916i	−0.890768	+	0.454458i	$0.849833\pi$
$930$	0	0
$931$	7.08742e7	0.0878293
$932$	− 2.07358e8i	− 0.256137i
$933$	0	0
$934$	1.88844e8	0.231772
$935$	− 2.80351e8i	− 0.342978i
$936$	0	0
$937$	5.88557e8	0.715434	0.357717	−	0.933830i	$-0.383555\pi$
$937$	5.88557e8	0.715434	0.357717	+	0.933830i	$0.383555\pi$
$938$	− 7.71888e8i	− 0.935289i
$939$	0	0
$940$	2.67725e7	0.0322333
$941$	9.73643e8i	1.16851i	0.811572	+	0.584253i	$0.198612\pi$
$941$	9.73643e8i	1.16851i	−0.811572	+	0.584253i	$0.801388\pi$
$942$	0	0
$943$	9.49299e8	1.13206
$944$	− 3.60787e8i	− 0.428879i
$945$	0	0
$946$	−2.64439e8	−0.312357
$947$	− 3.90596e8i	− 0.459916i	−0.973201	−	0.229958i	$-0.926141\pi$
$947$	− 3.90596e8i	− 0.459916i	0.973201	−	0.229958i	$-0.0738588\pi$
$948$	0	0
$949$	6.27779e8	0.734528
$950$	6.31892e8i	0.737007i
$951$	0	0
$952$	−3.08970e8	−0.358101
$953$	4.41062e7i	0.0509590i	0.999675	+	0.0254795i	$0.00811125\pi$
$953$	4.41062e7i	0.0509590i	−0.999675	+	0.0254795i	$0.991889\pi$
$954$	0	0
$955$	1.23896e9	1.42249
$956$	− 4.16323e8i	− 0.476493i
$957$	0	0
$958$	−4.00703e8	−0.455749
$959$	− 6.39396e8i	− 0.724960i
$960$	0	0
$961$	−8.59095e8	−0.967990
$962$	4.87887e8i	0.548017i
$963$	0	0
$964$	−1.18439e8	−0.132209
$965$	8.34129e8i	0.928221i
$966$	0	0
$967$	−9.63378e8	−1.06541	−0.532705	−	0.846301i	$-0.678825\pi$
$967$	−9.63378e8	−1.06541	−0.532705	+	0.846301i	$0.678825\pi$
$968$	7.62966e8i	0.841161i
$969$	0	0
$970$	3.17847e8	0.348259
$971$	7.36207e8i	0.804160i	0.915605	+	0.402080i	$0.131713\pi$
$971$	7.36207e8i	0.804160i	−0.915605	+	0.402080i	$0.868287\pi$
$972$	0	0
$973$	−3.32575e7	−0.0361036
$974$	1.20305e9i	1.30199i
$975$	0	0
$976$	9.39375e7	0.101039
$977$	− 1.60897e9i	− 1.72530i	−0.505805	−	0.862648i	$-0.668804\pi$
$977$	− 1.60897e9i	− 1.72530i	0.505805	−	0.862648i	$-0.331196\pi$
$978$	0	0
$979$	−1.68839e8	−0.179939
$980$	1.89827e8i	0.201687i
$981$	0	0
$982$	−5.17103e8	−0.546062
$983$	4.46977e8i	0.470570i	0.971926	+	0.235285i	$0.0756024\pi$
$983$	4.46977e8i	0.470570i	−0.971926	+	0.235285i	$0.924398\pi$
$984$	0	0
$985$	−1.76861e9	−1.85064
$986$	− 2.66333e8i	− 0.277840i
$987$	0	0
$988$	−1.75279e8	−0.181743
$989$	1.01368e9i	1.04788i
$990$	0	0
$991$	−1.40384e9	−1.44244	−0.721220	−	0.692706i	$-0.756418\pi$
$991$	−1.40384e9	−1.44244	−0.721220	+	0.692706i	$0.756418\pi$
$992$	1.39689e8i	0.143096i
$993$	0	0
$994$	4.51571e8	0.459798
$995$	− 2.02344e9i	− 2.05410i
$996$	0	0
$997$	1.33142e9	1.34348	0.671738	−	0.740789i	$-0.265548\pi$
$997$	1.33142e9	1.34348	0.671738	+	0.740789i	$0.265548\pi$
$998$	6.40038e7i	0.0643894i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.7.b.b.26.2	yes	2
3.2	odd	2	inner	27.7.b.b.26.1	✓	2
4.3	odd	2		432.7.e.d.161.1		2
9.2	odd	6		81.7.d.b.53.1		4
9.4	even	3		81.7.d.b.26.1		4
9.5	odd	6		81.7.d.b.26.2		4
9.7	even	3		81.7.d.b.53.2		4
12.11	even	2		432.7.e.d.161.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.b.b.26.1	✓	2	3.2	odd	2	inner
27.7.b.b.26.2	yes	2	1.1	even	1	trivial
81.7.d.b.26.1		4	9.4	even	3
81.7.d.b.26.2		4	9.5	odd	6
81.7.d.b.53.1		4	9.2	odd	6
81.7.d.b.53.2		4	9.7	even	3
432.7.e.d.161.1		2	4.3	odd	2
432.7.e.d.161.2		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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{2} +28.0000 q^{4} -240.000i q^{5} +299.000 q^{7} +552.000i q^{8} +1440.00 q^{10} -624.000i q^{11} +2495.00 q^{13} +1794.00i q^{14} -1520.00 q^{16} +1872.00i q^{17} -2509.00 q^{19} -6720.00i q^{20} +3744.00 q^{22} -14352.0i q^{23} -41975.0 q^{25} +14970.0i q^{26} +8372.00 q^{28} +23712.0i q^{29} +5330.00 q^{31} +26208.0i q^{32} -11232.0 q^{34} -71760.0i q^{35} +32591.0 q^{37} -15054.0i q^{38} +132480. q^{40} +66144.0i q^{41} -70630.0 q^{43} -17472.0i q^{44} +86112.0 q^{46} +3984.00i q^{47} -28248.0 q^{49} -251850. i q^{50} +69860.0 q^{52} +190944. i q^{53} -149760. q^{55} +165048. i q^{56} -142272. q^{58} +237360. i q^{59} -61801.0 q^{61} +31980.0i q^{62} -254528. q^{64} -598800. i q^{65} -430261. q^{67} +52416.0i q^{68} +430560. q^{70} -251712. i q^{71} +251615. q^{73} +195546. i q^{74} -70252.0 q^{76} -186576. i q^{77} +660827. q^{79} +364800. i q^{80} -396864. q^{82} +797856. i q^{83} +449280. q^{85} -423780. i q^{86} +344448. q^{88} -270576. i q^{89} +746005. q^{91} -401856. i q^{92} -23904.0 q^{94} +602160. i q^{95} +220727. q^{97} -169488. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 56 q^{4} + 598 q^{7} + 2880 q^{10} + 4990 q^{13} - 3040 q^{16} - 5018 q^{19} + 7488 q^{22} - 83950 q^{25} + 16744 q^{28} + 10660 q^{31} - 22464 q^{34} + 65182 q^{37} + 264960 q^{40} - 141260 q^{43}+ \cdots + 441454 q^{97}+O(q^{100}) \) 2 * q + 56 * q^4 + 598 * q^7 + 2880 * q^10 + 4990 * q^13 - 3040 * q^16 - 5018 * q^19 + 7488 * q^22 - 83950 * q^25 + 16744 * q^28 + 10660 * q^31 - 22464 * q^34 + 65182 * q^37 + 264960 * q^40 - 141260 * q^43 + 172224 * q^46 - 56496 * q^49 + 139720 * q^52 - 299520 * q^55 - 284544 * q^58 - 123602 * q^61 - 509056 * q^64 - 860522 * q^67 + 861120 * q^70 + 503230 * q^73 - 140504 * q^76 + 1321654 * q^79 - 793728 * q^82 + 898560 * q^85 + 688896 * q^88 + 1492010 * q^91 - 47808 * q^94 + 441454 * q^97

Introduction

L-functions

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Database

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