LMFDB - Embedded newform 2178.3.c.m.485.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,3,Mod(485,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2178.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2178.c (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:	$59.3462015777$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 102x^{6} + 3599x^{4} + 51708x^{2} + 249001 $ x^8 + 102x^6 + 3599x^4 + 51708*x^2 + 249001
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 198)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.3
Root			$5.21424i$ of defining polynomial
Character	$\chi$	$=$	2178.485
Dual form			2178.3.c.m.485.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +2.34854i q^{5} +9.61011 q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +2.34854i q^{5} +9.61011 q^{7} +2.82843i q^{8} +3.32134 q^{10} -5.69539 q^{13} -13.5908i q^{14} +4.00000 q^{16} -14.8398i q^{17} -30.9235 q^{19} -4.69709i q^{20} +5.53385i q^{23} +19.4843 q^{25} +8.05449i q^{26} -19.2202 q^{28} -21.4717i q^{29} -36.8254 q^{31} -5.65685i q^{32} -20.9866 q^{34} +22.5698i q^{35} +4.13407 q^{37} +43.7325i q^{38} -6.64268 q^{40} -64.0163i q^{41} -28.5124 q^{43} +7.82605 q^{46} +9.03037i q^{47} +43.3543 q^{49} -27.5550i q^{50} +11.3908 q^{52} +46.3337i q^{53} +27.1815i q^{56} -30.3655 q^{58} -59.4034i q^{59} -1.97531 q^{61} +52.0790i q^{62} -8.00000 q^{64} -13.3759i q^{65} -100.516 q^{67} +29.6796i q^{68} +31.9185 q^{70} -90.9809i q^{71} +108.839 q^{73} -5.84645i q^{74} +61.8471 q^{76} +130.407 q^{79} +9.39417i q^{80} -90.5327 q^{82} -85.8693i q^{83} +34.8519 q^{85} +40.3226i q^{86} -131.551i q^{89} -54.7333 q^{91} -11.0677i q^{92} +12.7709 q^{94} -72.6253i q^{95} -19.5114 q^{97} -61.3122i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 - 8 * q^7	$ 8 q - 16 q^{4} - 8 q^{7} + 12 q^{10} + 36 q^{13} + 32 q^{16} - 72 q^{19} - 64 q^{25} + 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} - 24 q^{40} + 220 q^{43} + 68 q^{46} + 56 q^{49} - 72 q^{52} + 112 q^{58} + 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} + 488 q^{73} + 144 q^{76} + 368 q^{79} - 388 q^{82} - 248 q^{85} - 356 q^{91} + 120 q^{94} - 832 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 - 8 * q^7 + 12 * q^10 + 36 * q^13 + 32 * q^16 - 72 * q^19 - 64 * q^25 + 16 * q^28 - 88 * q^31 - 56 * q^34 + 108 * q^37 - 24 * q^40 + 220 * q^43 + 68 * q^46 + 56 * q^49 - 72 * q^52 + 112 * q^58 + 160 * q^61 - 64 * q^64 - 276 * q^67 - 92 * q^70 + 488 * q^73 + 144 * q^76 + 368 * q^79 - 388 * q^82 - 248 * q^85 - 356 * q^91 + 120 * q^94 - 832 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	2.34854i	0.469709i	0.972031	+	0.234854i	$0.0754613\pi$
$5$	2.34854i	0.469709i	−0.972031	+	0.234854i	$0.924539\pi$
$6$	0	0
$7$	9.61011	1.37287	0.686437	−	0.727190i	$-0.259174\pi$
$7$	9.61011	1.37287	0.686437	+	0.727190i	$0.259174\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	3.32134	0.332134
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.69539	−0.438107	−0.219053	−	0.975713i	$-0.570297\pi$
$13$	−5.69539	−0.438107	−0.219053	+	0.975713i	$0.570297\pi$
$14$	− 13.5908i	− 0.970768i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 14.8398i	− 0.872929i	−0.899721	−	0.436464i	$-0.856230\pi$
$17$	− 14.8398i	− 0.872929i	0.899721	−	0.436464i	$-0.143770\pi$
$18$	0	0
$19$	−30.9235	−1.62755	−0.813777	−	0.581177i	$-0.802592\pi$
$19$	−30.9235	−1.62755	−0.813777	+	0.581177i	$0.802592\pi$
$20$	− 4.69709i	− 0.234854i
$21$	0	0
$22$	0	0
$23$	5.53385i	0.240602i	0.992737	+	0.120301i	$0.0383860\pi$
$23$	5.53385i	0.240602i	−0.992737	+	0.120301i	$0.961614\pi$
$24$	0	0
$25$	19.4843	0.779374
$26$	8.05449i	0.309788i
$27$	0	0
$28$	−19.2202	−0.686437
$29$	− 21.4717i	− 0.740402i	−0.928952	−	0.370201i	$-0.879289\pi$
$29$	− 21.4717i	− 0.740402i	0.928952	−	0.370201i	$-0.120711\pi$
$30$	0	0
$31$	−36.8254	−1.18792	−0.593959	−	0.804496i	$-0.702436\pi$
$31$	−36.8254	−1.18792	−0.593959	+	0.804496i	$0.702436\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−20.9866	−0.617254
$35$	22.5698i	0.644850i
$36$	0	0
$37$	4.13407	0.111732	0.0558658	−	0.998438i	$-0.482208\pi$
$37$	4.13407	0.111732	0.0558658	+	0.998438i	$0.482208\pi$
$38$	43.7325i	1.15085i
$39$	0	0
$40$	−6.64268	−0.166067
$41$	− 64.0163i	− 1.56137i	−0.624923	−	0.780686i	$-0.714870\pi$
$41$	− 64.0163i	− 1.56137i	0.624923	−	0.780686i	$-0.285130\pi$
$42$	0	0
$43$	−28.5124	−0.663078	−0.331539	−	0.943442i	$-0.607568\pi$
$43$	−28.5124	−0.663078	−0.331539	+	0.943442i	$0.607568\pi$
$44$	0	0
$45$	0	0
$46$	7.82605	0.170132
$47$	9.03037i	0.192135i	0.995375	+	0.0960677i	$0.0306265\pi$
$47$	9.03037i	0.192135i	−0.995375	+	0.0960677i	$0.969373\pi$
$48$	0	0
$49$	43.3543	0.884781
$50$	− 27.5550i	− 0.551100i
$51$	0	0
$52$	11.3908	0.219053
$53$	46.3337i	0.874220i	0.899408	+	0.437110i	$0.143998\pi$
$53$	46.3337i	0.874220i	−0.899408	+	0.437110i	$0.856002\pi$
$54$	0	0
$55$	0	0
$56$	27.1815i	0.485384i
$57$	0	0
$58$	−30.3655	−0.523543
$59$	− 59.4034i	− 1.00684i	−0.864043	−	0.503419i	$-0.832075\pi$
$59$	− 59.4034i	− 1.00684i	0.864043	−	0.503419i	$-0.167925\pi$
$60$	0	0
$61$	−1.97531	−0.0323821	−0.0161911	−	0.999869i	$-0.505154\pi$
$61$	−1.97531	−0.0323821	−0.0161911	+	0.999869i	$0.505154\pi$
$62$	52.0790i	0.839984i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 13.3759i	− 0.205782i
$66$	0	0
$67$	−100.516	−1.50024	−0.750118	−	0.661304i	$-0.770003\pi$
$67$	−100.516	−1.50024	−0.750118	+	0.661304i	$0.770003\pi$
$68$	29.6796i	0.436464i
$69$	0	0
$70$	31.9185	0.455978
$71$	− 90.9809i	− 1.28142i	−0.767782	−	0.640711i	$-0.778640\pi$
$71$	− 90.9809i	− 1.28142i	0.767782	−	0.640711i	$-0.221360\pi$
$72$	0	0
$73$	108.839	1.49094	0.745470	−	0.666539i	$-0.232225\pi$
$73$	108.839	1.49094	0.745470	+	0.666539i	$0.232225\pi$
$74$	− 5.84645i	− 0.0790061i
$75$	0	0
$76$	61.8471	0.813777
$77$	0	0
$78$	0	0
$79$	130.407	1.65073	0.825363	−	0.564603i	$-0.190971\pi$
$79$	130.407	1.65073	0.825363	+	0.564603i	$0.190971\pi$
$80$	9.39417i	0.117427i
$81$	0	0
$82$	−90.5327	−1.10406
$83$	− 85.8693i	− 1.03457i	−0.855813	−	0.517285i	$-0.826943\pi$
$83$	− 85.8693i	− 1.03457i	0.855813	−	0.517285i	$-0.173057\pi$
$84$	0	0
$85$	34.8519	0.410022
$86$	40.3226i	0.468867i
$87$	0	0
$88$	0	0
$89$	− 131.551i	− 1.47810i	−0.673648	−	0.739052i	$-0.735274\pi$
$89$	− 131.551i	− 1.47810i	0.673648	−	0.739052i	$-0.264726\pi$
$90$	0	0
$91$	−54.7333	−0.601465
$92$	− 11.0677i	− 0.120301i
$93$	0	0
$94$	12.7709	0.135860
$95$	− 72.6253i	− 0.764476i
$96$	0	0
$97$	−19.5114	−0.201149	−0.100574	−	0.994930i	$-0.532068\pi$
$97$	−19.5114	−0.201149	−0.100574	+	0.994930i	$0.532068\pi$
$98$	− 61.3122i	− 0.625634i
$99$	0	0
$100$	−38.9687	−0.389687
$101$	65.7089i	0.650583i	0.945614	+	0.325291i	$0.105462\pi$
$101$	65.7089i	0.650583i	−0.945614	+	0.325291i	$0.894538\pi$
$102$	0	0
$103$	157.546	1.52957	0.764787	−	0.644283i	$-0.222844\pi$
$103$	157.546	1.52957	0.764787	+	0.644283i	$0.222844\pi$
$104$	− 16.1090i	− 0.154894i
$105$	0	0
$106$	65.5257	0.618167
$107$	69.7102i	0.651497i	0.945456	+	0.325749i	$0.105616\pi$
$107$	69.7102i	0.651497i	−0.945456	+	0.325749i	$0.894384\pi$
$108$	0	0
$109$	16.2205	0.148812	0.0744058	−	0.997228i	$-0.476294\pi$
$109$	16.2205	0.148812	0.0744058	+	0.997228i	$0.476294\pi$
$110$	0	0
$111$	0	0
$112$	38.4404	0.343218
$113$	− 144.913i	− 1.28242i	−0.767366	−	0.641210i	$-0.778433\pi$
$113$	− 144.913i	− 1.28242i	0.767366	−	0.641210i	$-0.221567\pi$
$114$	0	0
$115$	−12.9965	−0.113013
$116$	42.9433i	0.370201i
$117$	0	0
$118$	−84.0091	−0.711942
$119$	− 142.612i	− 1.19842i
$120$	0	0
$121$	0	0
$122$	2.79351i	0.0228976i
$123$	0	0
$124$	73.6509	0.593959
$125$	104.473i	0.835787i
$126$	0	0
$127$	153.892	1.21175	0.605876	−	0.795559i	$-0.292823\pi$
$127$	153.892	1.21175	0.605876	+	0.795559i	$0.292823\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−18.9163	−0.145510
$131$	− 157.050i	− 1.19886i	−0.800428	−	0.599428i	$-0.795395\pi$
$131$	− 157.050i	− 1.19886i	0.800428	−	0.599428i	$-0.204605\pi$
$132$	0	0
$133$	−297.179	−2.23443
$134$	142.151i	1.06083i
$135$	0	0
$136$	41.9733	0.308627
$137$	70.3072i	0.513191i	0.966519	+	0.256596i	$0.0826009\pi$
$137$	70.3072i	0.513191i	−0.966519	+	0.256596i	$0.917399\pi$
$138$	0	0
$139$	2.40432	0.0172973	0.00864863	−	0.999963i	$-0.497247\pi$
$139$	2.40432	0.0172973	0.00864863	+	0.999963i	$0.497247\pi$
$140$	− 45.1395i	− 0.322425i
$141$	0	0
$142$	−128.666	−0.906102
$143$	0	0
$144$	0	0
$145$	50.4271	0.347773
$146$	− 153.921i	− 1.05425i
$147$	0	0
$148$	−8.26813	−0.0558658
$149$	− 278.285i	− 1.86769i	−0.357682	−	0.933843i	$-0.616433\pi$
$149$	− 278.285i	− 1.86769i	0.357682	−	0.933843i	$-0.383567\pi$
$150$	0	0
$151$	−277.186	−1.83567	−0.917833	−	0.396966i	$-0.870063\pi$
$151$	−277.186	−1.83567	−0.917833	+	0.396966i	$0.870063\pi$
$152$	− 87.4650i	− 0.575427i
$153$	0	0
$154$	0	0
$155$	− 86.4861i	− 0.557975i
$156$	0	0
$157$	−259.461	−1.65262	−0.826310	−	0.563216i	$-0.809564\pi$
$157$	−259.461	−1.65262	−0.826310	+	0.563216i	$0.809564\pi$
$158$	− 184.424i	− 1.16724i
$159$	0	0
$160$	13.2854	0.0830335
$161$	53.1809i	0.330316i
$162$	0	0
$163$	−140.181	−0.860004	−0.430002	−	0.902828i	$-0.641487\pi$
$163$	−140.181	−0.860004	−0.430002	+	0.902828i	$0.641487\pi$
$164$	128.033i	0.780686i
$165$	0	0
$166$	−121.438	−0.731552
$167$	14.0031i	0.0838507i	0.999121	+	0.0419253i	$0.0133492\pi$
$167$	14.0031i	0.0838507i	−0.999121	+	0.0419253i	$0.986651\pi$
$168$	0	0
$169$	−136.563	−0.808063
$170$	− 49.2880i	− 0.289930i
$171$	0	0
$172$	57.0247	0.331539
$173$	− 172.287i	− 0.995877i	−0.867212	−	0.497939i	$-0.834090\pi$
$173$	− 172.287i	− 0.995877i	0.867212	−	0.497939i	$-0.165910\pi$
$174$	0	0
$175$	187.247	1.06998
$176$	0	0
$177$	0	0
$178$	−186.042	−1.04518
$179$	− 34.5647i	− 0.193099i	−0.995328	−	0.0965495i	$-0.969219\pi$
$179$	− 34.5647i	− 0.193099i	0.995328	−	0.0965495i	$-0.0307806\pi$
$180$	0	0
$181$	−17.3070	−0.0956190	−0.0478095	−	0.998856i	$-0.515224\pi$
$181$	−17.3070	−0.0956190	−0.0478095	+	0.998856i	$0.515224\pi$
$182$	77.4046i	0.425300i
$183$	0	0
$184$	−15.6521	−0.0850658
$185$	9.70903i	0.0524813i
$186$	0	0
$187$	0	0
$188$	− 18.0607i	− 0.0960677i
$189$	0	0
$190$	−102.708	−0.540566
$191$	197.127i	1.03208i	0.856564	+	0.516040i	$0.172594\pi$
$191$	197.127i	1.03208i	−0.856564	+	0.516040i	$0.827406\pi$
$192$	0	0
$193$	−259.207	−1.34304	−0.671521	−	0.740985i	$-0.734359\pi$
$193$	−259.207	−1.34304	−0.671521	+	0.740985i	$0.734359\pi$
$194$	27.5933i	0.142234i
$195$	0	0
$196$	−86.7085	−0.442390
$197$	160.457i	0.814504i	0.913316	+	0.407252i	$0.133513\pi$
$197$	160.457i	0.814504i	−0.913316	+	0.407252i	$0.866487\pi$
$198$	0	0
$199$	27.0388	0.135873	0.0679366	−	0.997690i	$-0.478358\pi$
$199$	27.0388	0.135873	0.0679366	+	0.997690i	$0.478358\pi$
$200$	55.1100i	0.275550i
$201$	0	0
$202$	92.9264	0.460032
$203$	− 206.345i	− 1.01648i
$204$	0	0
$205$	150.345	0.733390
$206$	− 222.804i	− 1.08157i
$207$	0	0
$208$	−22.7815	−0.109527
$209$	0	0
$210$	0	0
$211$	292.716	1.38728	0.693640	−	0.720322i	$-0.256006\pi$
$211$	292.716	1.38728	0.693640	+	0.720322i	$0.256006\pi$
$212$	− 92.6673i	− 0.437110i
$213$	0	0
$214$	98.5852	0.460678
$215$	− 66.9625i	− 0.311454i
$216$	0	0
$217$	−353.897	−1.63086
$218$	− 22.9392i	− 0.105226i
$219$	0	0
$220$	0	0
$221$	84.5183i	0.382436i
$222$	0	0
$223$	−324.546	−1.45536	−0.727682	−	0.685915i	$-0.759402\pi$
$223$	−324.546	−1.45536	−0.727682	+	0.685915i	$0.759402\pi$
$224$	− 54.3630i	− 0.242692i
$225$	0	0
$226$	−204.938	−0.906807
$227$	371.930i	1.63846i	0.573466	+	0.819229i	$0.305598\pi$
$227$	371.930i	1.63846i	−0.573466	+	0.819229i	$0.694402\pi$
$228$	0	0
$229$	−325.102	−1.41966	−0.709829	−	0.704374i	$-0.751228\pi$
$229$	−325.102	−1.41966	−0.709829	+	0.704374i	$0.751228\pi$
$230$	18.3798i	0.0799122i
$231$	0	0
$232$	60.7310	0.261772
$233$	− 407.958i	− 1.75089i	−0.483313	−	0.875447i	$-0.660567\pi$
$233$	− 407.958i	− 1.75089i	0.483313	−	0.875447i	$-0.339433\pi$
$234$	0	0
$235$	−21.2082	−0.0902477
$236$	118.807i	0.503419i
$237$	0	0
$238$	−201.684	−0.847411
$239$	− 143.450i	− 0.600211i	−0.953906	−	0.300106i	$-0.902978\pi$
$239$	− 143.450i	− 0.600211i	0.953906	−	0.300106i	$-0.0970219\pi$
$240$	0	0
$241$	−88.3220	−0.366481	−0.183241	−	0.983068i	$-0.558659\pi$
$241$	−88.3220	−0.366481	−0.183241	+	0.983068i	$0.558659\pi$
$242$	0	0
$243$	0	0
$244$	3.95062	0.0161911
$245$	101.819i	0.415589i
$246$	0	0
$247$	176.121	0.713042
$248$	− 104.158i	− 0.419992i
$249$	0	0
$250$	147.748	0.590991
$251$	− 433.831i	− 1.72841i	−0.503141	−	0.864204i	$-0.667822\pi$
$251$	− 433.831i	− 1.72841i	0.503141	−	0.864204i	$-0.332178\pi$
$252$	0	0
$253$	0	0
$254$	− 217.637i	− 0.856838i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 19.2563i	− 0.0749273i	−0.999298	−	0.0374636i	$-0.988072\pi$
$257$	− 19.2563i	− 0.0749273i	0.999298	−	0.0374636i	$-0.0119278\pi$
$258$	0	0
$259$	39.7288	0.153393
$260$	26.7517i	0.102891i
$261$	0	0
$262$	−222.103	−0.847720
$263$	− 136.538i	− 0.519155i	−0.965722	−	0.259577i	$-0.916417\pi$
$263$	− 136.538i	− 0.519155i	0.965722	−	0.259577i	$-0.0835833\pi$
$264$	0	0
$265$	−108.817	−0.410629
$266$	420.274i	1.57998i
$267$	0	0
$268$	201.032	0.750118
$269$	75.4425i	0.280455i	0.990119	+	0.140228i	$0.0447835\pi$
$269$	75.4425i	0.280455i	−0.990119	+	0.140228i	$0.955217\pi$
$270$	0	0
$271$	147.879	0.545680	0.272840	−	0.962059i	$-0.412037\pi$
$271$	147.879	0.545680	0.272840	+	0.962059i	$0.412037\pi$
$272$	− 59.3592i	− 0.218232i
$273$	0	0
$274$	99.4294	0.362881
$275$	0	0
$276$	0	0
$277$	434.723	1.56940	0.784699	−	0.619877i	$-0.212818\pi$
$277$	434.723	1.56940	0.784699	+	0.619877i	$0.212818\pi$
$278$	− 3.40022i	− 0.0122310i
$279$	0	0
$280$	−63.8369	−0.227989
$281$	457.590i	1.62843i	0.580561	+	0.814217i	$0.302833\pi$
$281$	457.590i	1.62843i	−0.580561	+	0.814217i	$0.697167\pi$
$282$	0	0
$283$	−117.401	−0.414846	−0.207423	−	0.978251i	$-0.566508\pi$
$283$	−117.401	−0.414846	−0.207423	+	0.978251i	$0.566508\pi$
$284$	181.962i	0.640711i
$285$	0	0
$286$	0	0
$287$	− 615.203i	− 2.14357i
$288$	0	0
$289$	68.7806	0.237995
$290$	− 71.3147i	− 0.245913i
$291$	0	0
$292$	−217.677	−0.745470
$293$	− 402.745i	− 1.37456i	−0.726394	−	0.687278i	$-0.758805\pi$
$293$	− 402.745i	− 1.37456i	0.726394	−	0.687278i	$-0.241195\pi$
$294$	0	0
$295$	139.511	0.472920
$296$	11.6929i	0.0395031i
$297$	0	0
$298$	−393.555	−1.32065
$299$	− 31.5174i	− 0.105409i
$300$	0	0
$301$	−274.007	−0.910322
$302$	392.000i	1.29801i
$303$	0	0
$304$	−123.694	−0.406889
$305$	− 4.63910i	− 0.0152102i
$306$	0	0
$307$	−226.573	−0.738022	−0.369011	−	0.929425i	$-0.620304\pi$
$307$	−226.573	−0.738022	−0.369011	+	0.929425i	$0.620304\pi$
$308$	0	0
$309$	0	0
$310$	−122.310	−0.394548
$311$	17.0565i	0.0548439i	0.999624	+	0.0274220i	$0.00872977\pi$
$311$	17.0565i	0.0548439i	−0.999624	+	0.0274220i	$0.991270\pi$
$312$	0	0
$313$	−375.490	−1.19965	−0.599823	−	0.800132i	$-0.704763\pi$
$313$	−375.490	−1.19965	−0.599823	+	0.800132i	$0.704763\pi$
$314$	366.934i	1.16858i
$315$	0	0
$316$	−260.815	−0.825363
$317$	205.707i	0.648918i	0.945900	+	0.324459i	$0.105182\pi$
$317$	205.707i	0.648918i	−0.945900	+	0.324459i	$0.894818\pi$
$318$	0	0
$319$	0	0
$320$	− 18.7883i	− 0.0587136i
$321$	0	0
$322$	75.2092	0.233569
$323$	458.899i	1.42074i
$324$	0	0
$325$	−110.971	−0.341449
$326$	198.245i	0.608115i
$327$	0	0
$328$	181.065	0.552028
$329$	86.7828i	0.263778i
$330$	0	0
$331$	−112.857	−0.340959	−0.170480	−	0.985361i	$-0.554532\pi$
$331$	−112.857	−0.340959	−0.170480	+	0.985361i	$0.554532\pi$
$332$	171.739i	0.517285i
$333$	0	0
$334$	19.8033	0.0592914
$335$	− 236.066i	− 0.704674i
$336$	0	0
$337$	381.826	1.13302	0.566508	−	0.824057i	$-0.308294\pi$
$337$	381.826	1.13302	0.566508	+	0.824057i	$0.308294\pi$
$338$	193.129i	0.571387i
$339$	0	0
$340$	−69.7038	−0.205011
$341$	0	0
$342$	0	0
$343$	−54.2563	−0.158182
$344$	− 80.6451i	− 0.234434i
$345$	0	0
$346$	−243.650	−0.704192
$347$	− 168.869i	− 0.486654i	−0.969944	−	0.243327i	$-0.921761\pi$
$347$	− 168.869i	− 0.486654i	0.969944	−	0.243327i	$-0.0782388\pi$
$348$	0	0
$349$	334.728	0.959107	0.479553	−	0.877513i	$-0.340799\pi$
$349$	334.728	0.959107	0.479553	+	0.877513i	$0.340799\pi$
$350$	− 264.807i	− 0.756591i
$351$	0	0
$352$	0	0
$353$	− 192.539i	− 0.545435i	−0.962094	−	0.272718i	$-0.912078\pi$
$353$	− 192.539i	− 0.545435i	0.962094	−	0.272718i	$-0.0879224\pi$
$354$	0	0
$355$	213.673	0.601895
$356$	263.103i	0.739052i
$357$	0	0
$358$	−48.8819	−0.136542
$359$	253.211i	0.705323i	0.935751	+	0.352661i	$0.114723\pi$
$359$	253.211i	0.705323i	−0.935751	+	0.352661i	$0.885277\pi$
$360$	0	0
$361$	595.265	1.64893
$362$	24.4759i	0.0676129i
$363$	0	0
$364$	109.467	0.300732
$365$	255.612i	0.700308i
$366$	0	0
$367$	−164.919	−0.449369	−0.224685	−	0.974432i	$-0.572135\pi$
$367$	−164.919	−0.449369	−0.224685	+	0.974432i	$0.572135\pi$
$368$	22.1354i	0.0601506i
$369$	0	0
$370$	13.7306	0.0371099
$371$	445.272i	1.20019i
$372$	0	0
$373$	−131.076	−0.351410	−0.175705	−	0.984443i	$-0.556221\pi$
$373$	−131.076	−0.351410	−0.175705	+	0.984443i	$0.556221\pi$
$374$	0	0
$375$	0	0
$376$	−25.5417	−0.0679301
$377$	122.289i	0.324375i
$378$	0	0
$379$	314.813	0.830640	0.415320	−	0.909675i	$-0.363670\pi$
$379$	314.813	0.830640	0.415320	+	0.909675i	$0.363670\pi$
$380$	145.251i	0.382238i
$381$	0	0
$382$	278.780	0.729791
$383$	297.350i	0.776371i	0.921581	+	0.388186i	$0.126898\pi$
$383$	297.350i	0.776371i	−0.921581	+	0.388186i	$0.873102\pi$
$384$	0	0
$385$	0	0
$386$	366.574i	0.949675i
$387$	0	0
$388$	39.0228	0.100574
$389$	681.481i	1.75188i	0.482420	+	0.875940i	$0.339758\pi$
$389$	681.481i	1.75188i	−0.482420	+	0.875940i	$0.660242\pi$
$390$	0	0
$391$	82.1212	0.210029
$392$	122.624i	0.312817i
$393$	0	0
$394$	226.921	0.575941
$395$	306.267i	0.775360i
$396$	0	0
$397$	437.138	1.10110	0.550551	−	0.834801i	$-0.314417\pi$
$397$	437.138	1.10110	0.550551	+	0.834801i	$0.314417\pi$
$398$	− 38.2386i	− 0.0960768i
$399$	0	0
$400$	77.9374	0.194843
$401$	− 400.524i	− 0.998813i	−0.866368	−	0.499406i	$-0.833551\pi$
$401$	− 400.524i	− 0.998813i	0.866368	−	0.499406i	$-0.166449\pi$
$402$	0	0
$403$	209.735	0.520434
$404$	− 131.418i	− 0.325291i
$405$	0	0
$406$	−291.816	−0.718759
$407$	0	0
$408$	0	0
$409$	720.455	1.76150	0.880751	−	0.473579i	$-0.157038\pi$
$409$	720.455	1.76150	0.880751	+	0.473579i	$0.157038\pi$
$410$	− 212.620i	− 0.518585i
$411$	0	0
$412$	−315.092	−0.764787
$413$	− 570.873i	− 1.38226i
$414$	0	0
$415$	201.668	0.485947
$416$	32.2180i	0.0774470i
$417$	0	0
$418$	0	0
$419$	− 318.471i	− 0.760075i	−0.924971	−	0.380037i	$-0.875911\pi$
$419$	− 318.471i	− 0.760075i	0.924971	−	0.380037i	$-0.124089\pi$
$420$	0	0
$421$	−681.633	−1.61908	−0.809540	−	0.587065i	$-0.800283\pi$
$421$	−681.633	−1.61908	−0.809540	+	0.587065i	$0.800283\pi$
$422$	− 413.963i	− 0.980955i
$423$	0	0
$424$	−131.051	−0.309083
$425$	− 289.144i	− 0.680338i
$426$	0	0
$427$	−18.9830	−0.0444566
$428$	− 139.420i	− 0.325749i
$429$	0	0
$430$	−94.6993	−0.220231
$431$	− 49.3177i	− 0.114426i	−0.998362	−	0.0572131i	$-0.981779\pi$
$431$	− 49.3177i	− 0.114426i	0.998362	−	0.0572131i	$-0.0182214\pi$
$432$	0	0
$433$	202.124	0.466799	0.233400	−	0.972381i	$-0.425015\pi$
$433$	202.124	0.466799	0.233400	+	0.972381i	$0.425015\pi$
$434$	500.485i	1.15319i
$435$	0	0
$436$	−32.4409	−0.0744058
$437$	− 171.126i	− 0.391593i
$438$	0	0
$439$	−5.40666	−0.0123158	−0.00615792	−	0.999981i	$-0.501960\pi$
$439$	−5.40666	−0.0123158	−0.00615792	+	0.999981i	$0.501960\pi$
$440$	0	0
$441$	0	0
$442$	119.527	0.270423
$443$	530.107i	1.19663i	0.801261	+	0.598315i	$0.204163\pi$
$443$	530.107i	1.19663i	−0.801261	+	0.598315i	$0.795837\pi$
$444$	0	0
$445$	308.954	0.694278
$446$	458.978i	1.02910i
$447$	0	0
$448$	−76.8809	−0.171609
$449$	− 392.847i	− 0.874938i	−0.899233	−	0.437469i	$-0.855875\pi$
$449$	− 392.847i	− 0.874938i	0.899233	−	0.437469i	$-0.144125\pi$
$450$	0	0
$451$	0	0
$452$	289.827i	0.641210i
$453$	0	0
$454$	525.988	1.15856
$455$	− 128.544i	− 0.282513i
$456$	0	0
$457$	66.5693	0.145666	0.0728329	−	0.997344i	$-0.476796\pi$
$457$	66.5693	0.145666	0.0728329	+	0.997344i	$0.476796\pi$
$458$	459.763i	1.00385i
$459$	0	0
$460$	25.9930	0.0565065
$461$	− 203.321i	− 0.441044i	−0.975382	−	0.220522i	$-0.929224\pi$
$461$	− 203.321i	− 0.441044i	0.975382	−	0.220522i	$-0.0707761\pi$
$462$	0	0
$463$	226.138	0.488418	0.244209	−	0.969723i	$-0.421472\pi$
$463$	226.138	0.488418	0.244209	+	0.969723i	$0.421472\pi$
$464$	− 85.8866i	− 0.185101i
$465$	0	0
$466$	−576.940	−1.23807
$467$	− 448.547i	− 0.960487i	−0.877135	−	0.480244i	$-0.840548\pi$
$467$	− 448.547i	− 0.960487i	0.877135	−	0.480244i	$-0.159452\pi$
$468$	0	0
$469$	−965.968	−2.05963
$470$	29.9929i	0.0638148i
$471$	0	0
$472$	168.018	0.355971
$473$	0	0
$474$	0	0
$475$	−602.525	−1.26847
$476$	285.224i	0.599210i
$477$	0	0
$478$	−202.870	−0.424413
$479$	245.104i	0.511699i	0.966717	+	0.255849i	$0.0823551\pi$
$479$	245.104i	0.511699i	−0.966717	+	0.255849i	$0.917645\pi$
$480$	0	0
$481$	−23.5451	−0.0489503
$482$	124.906i	0.259142i
$483$	0	0
$484$	0	0
$485$	− 45.8234i	− 0.0944812i
$486$	0	0
$487$	−33.9288	−0.0696690	−0.0348345	−	0.999393i	$-0.511090\pi$
$487$	−33.9288	−0.0696690	−0.0348345	+	0.999393i	$0.511090\pi$
$488$	− 5.58702i	− 0.0114488i
$489$	0	0
$490$	143.994	0.293866
$491$	− 242.389i	− 0.493665i	−0.969058	−	0.246832i	$-0.920610\pi$
$491$	− 242.389i	− 0.493665i	0.969058	−	0.246832i	$-0.0793897\pi$
$492$	0	0
$493$	−318.635	−0.646318
$494$	− 249.073i	− 0.504197i
$495$	0	0
$496$	−147.302	−0.296979
$497$	− 874.337i	− 1.75923i
$498$	0	0
$499$	−208.879	−0.418594	−0.209297	−	0.977852i	$-0.567118\pi$
$499$	−208.879	−0.418594	−0.209297	+	0.977852i	$0.567118\pi$
$500$	− 208.947i	− 0.417894i
$501$	0	0
$502$	−613.529	−1.22217
$503$	517.116i	1.02806i	0.857771	+	0.514031i	$0.171849\pi$
$503$	517.116i	1.02806i	−0.857771	+	0.514031i	$0.828151\pi$
$504$	0	0
$505$	−154.320	−0.305584
$506$	0	0
$507$	0	0
$508$	−307.785	−0.605876
$509$	236.337i	0.464317i	0.972678	+	0.232158i	$0.0745787\pi$
$509$	236.337i	0.464317i	−0.972678	+	0.232158i	$0.925421\pi$
$510$	0	0
$511$	1045.95	2.04687
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−27.2325	−0.0529816
$515$	370.004i	0.718454i
$516$	0	0
$517$	0	0
$518$	− 56.1851i	− 0.108465i
$519$	0	0
$520$	37.8326	0.0727551
$521$	− 1010.01i	− 1.93860i	−0.245885	−	0.969299i	$-0.579079\pi$
$521$	− 1010.01i	− 1.93860i	0.245885	−	0.969299i	$-0.420921\pi$
$522$	0	0
$523$	−424.292	−0.811266	−0.405633	−	0.914036i	$-0.632949\pi$
$523$	−424.292	−0.811266	−0.405633	+	0.914036i	$0.632949\pi$
$524$	314.100i	0.599428i
$525$	0	0
$526$	−193.094	−0.367098
$527$	546.482i	1.03697i
$528$	0	0
$529$	498.376	0.942111
$530$	153.890i	0.290358i
$531$	0	0
$532$	594.357	1.11721
$533$	364.597i	0.684047i
$534$	0	0
$535$	−163.717	−0.306014
$536$	− 284.302i	− 0.530413i
$537$	0	0
$538$	106.692	0.198312
$539$	0	0
$540$	0	0
$541$	−362.339	−0.669757	−0.334879	−	0.942261i	$-0.608695\pi$
$541$	−362.339	−0.669757	−0.334879	+	0.942261i	$0.608695\pi$
$542$	− 209.133i	− 0.385854i
$543$	0	0
$544$	−83.9465	−0.154313
$545$	38.0945i	0.0698981i
$546$	0	0
$547$	−357.399	−0.653380	−0.326690	−	0.945132i	$-0.605933\pi$
$547$	−357.399	−0.653380	−0.326690	+	0.945132i	$0.605933\pi$
$548$	− 140.614i	− 0.256596i
$549$	0	0
$550$	0	0
$551$	663.980i	1.20504i
$552$	0	0
$553$	1253.23	2.26624
$554$	− 614.791i	− 1.10973i
$555$	0	0
$556$	−4.80864	−0.00864863
$557$	186.255i	0.334390i	0.985924	+	0.167195i	$0.0534709\pi$
$557$	186.255i	0.334390i	−0.985924	+	0.167195i	$0.946529\pi$
$558$	0	0
$559$	162.389	0.290499
$560$	90.2791i	0.161213i
$561$	0	0
$562$	647.130	1.15148
$563$	− 975.364i	− 1.73244i	−0.499662	−	0.866220i	$-0.666542\pi$
$563$	− 975.364i	− 1.73244i	0.499662	−	0.866220i	$-0.333458\pi$
$564$	0	0
$565$	340.335	0.602363
$566$	166.031i	0.293340i
$567$	0	0
$568$	257.333	0.453051
$569$	− 438.959i	− 0.771458i	−0.922612	−	0.385729i	$-0.873950\pi$
$569$	− 438.959i	− 0.771458i	0.922612	−	0.385729i	$-0.126050\pi$
$570$	0	0
$571$	971.709	1.70177	0.850884	−	0.525354i	$-0.176067\pi$
$571$	971.709	1.70177	0.850884	+	0.525354i	$0.176067\pi$
$572$	0	0
$573$	0	0
$574$	−870.029	−1.51573
$575$	107.823i	0.187519i
$576$	0	0
$577$	151.825	0.263128	0.131564	−	0.991308i	$-0.458000\pi$
$577$	151.825	0.263128	0.131564	+	0.991308i	$0.458000\pi$
$578$	− 97.2704i	− 0.168288i
$579$	0	0
$580$	−100.854	−0.173887
$581$	− 825.214i	− 1.42033i
$582$	0	0
$583$	0	0
$584$	307.842i	0.527127i
$585$	0	0
$586$	−569.567	−0.971958
$587$	186.925i	0.318441i	0.987243	+	0.159220i	$0.0508980\pi$
$587$	186.925i	0.318441i	−0.987243	+	0.159220i	$0.949102\pi$
$588$	0	0
$589$	1138.77	1.93340
$590$	− 197.299i	− 0.334405i
$591$	0	0
$592$	16.5363	0.0279329
$593$	− 391.300i	− 0.659865i	−0.944005	−	0.329933i	$-0.892974\pi$
$593$	− 391.300i	− 0.659865i	0.944005	−	0.329933i	$-0.107026\pi$
$594$	0	0
$595$	334.931	0.562909
$596$	556.571i	0.933843i
$597$	0	0
$598$	−44.5724	−0.0745357
$599$	251.068i	0.419146i	0.977793	+	0.209573i	$0.0672073\pi$
$599$	251.068i	0.419146i	−0.977793	+	0.209573i	$0.932793\pi$
$600$	0	0
$601$	217.862	0.362499	0.181249	−	0.983437i	$-0.441986\pi$
$601$	217.862	0.362499	0.181249	+	0.983437i	$0.441986\pi$
$602$	387.504i	0.643695i
$603$	0	0
$604$	554.371	0.917833
$605$	0	0
$606$	0	0
$607$	−185.261	−0.305207	−0.152603	−	0.988288i	$-0.548766\pi$
$607$	−185.261	−0.305207	−0.152603	+	0.988288i	$0.548766\pi$
$608$	174.930i	0.287714i
$609$	0	0
$610$	−6.56068	−0.0107552
$611$	− 51.4314i	− 0.0841758i
$612$	0	0
$613$	768.009	1.25287	0.626435	−	0.779474i	$-0.284513\pi$
$613$	768.009	1.25287	0.626435	+	0.779474i	$0.284513\pi$
$614$	320.422i	0.521860i
$615$	0	0
$616$	0	0
$617$	995.596i	1.61361i	0.590819	+	0.806804i	$0.298805\pi$
$617$	995.596i	1.61361i	−0.590819	+	0.806804i	$0.701195\pi$
$618$	0	0
$619$	−249.197	−0.402580	−0.201290	−	0.979532i	$-0.564513\pi$
$619$	−249.197	−0.402580	−0.201290	+	0.979532i	$0.564513\pi$
$620$	172.972i	0.278988i
$621$	0	0
$622$	24.1215	0.0387805
$623$	− 1264.22i	− 2.02925i
$624$	0	0
$625$	241.748	0.386797
$626$	531.022i	0.848279i
$627$	0	0
$628$	518.923	0.826310
$629$	− 61.3487i	− 0.0975337i
$630$	0	0
$631$	−936.069	−1.48347	−0.741735	−	0.670693i	$-0.765997\pi$
$631$	−936.069	−1.48347	−0.741735	+	0.670693i	$0.765997\pi$
$632$	368.848i	0.583619i
$633$	0	0
$634$	290.913	0.458854
$635$	361.423i	0.569170i
$636$	0	0
$637$	−246.919	−0.387628
$638$	0	0
$639$	0	0
$640$	−26.5707	−0.0415168
$641$	− 699.799i	− 1.09173i	−0.837873	−	0.545865i	$-0.816201\pi$
$641$	− 699.799i	− 1.09173i	0.837873	−	0.545865i	$-0.183799\pi$
$642$	0	0
$643$	−865.457	−1.34597	−0.672983	−	0.739658i	$-0.734987\pi$
$643$	−865.457	−1.34597	−0.672983	+	0.739658i	$0.734987\pi$
$644$	− 106.362i	− 0.165158i
$645$	0	0
$646$	648.981	1.00461
$647$	842.371i	1.30196i	0.759093	+	0.650982i	$0.225643\pi$
$647$	842.371i	1.30196i	−0.759093	+	0.650982i	$0.774357\pi$
$648$	0	0
$649$	0	0
$650$	156.936i	0.241441i
$651$	0	0
$652$	280.361	0.430002
$653$	474.974i	0.727373i	0.931521	+	0.363686i	$0.118482\pi$
$653$	474.974i	0.727373i	−0.931521	+	0.363686i	$0.881518\pi$
$654$	0	0
$655$	368.839	0.563113
$656$	− 256.065i	− 0.390343i
$657$	0	0
$658$	122.729	0.186519
$659$	− 295.557i	− 0.448494i	−0.974532	−	0.224247i	$-0.928008\pi$
$659$	− 295.557i	− 0.448494i	0.974532	−	0.224247i	$-0.0719922\pi$
$660$	0	0
$661$	561.030	0.848760	0.424380	−	0.905484i	$-0.360492\pi$
$661$	561.030	0.848760	0.424380	+	0.905484i	$0.360492\pi$
$662$	159.605i	0.241095i
$663$	0	0
$664$	242.875	0.365776
$665$	− 697.937i	− 1.04953i
$666$	0	0
$667$	118.821	0.178142
$668$	− 28.0061i	− 0.0419253i
$669$	0	0
$670$	−333.847	−0.498280
$671$	0	0
$672$	0	0
$673$	−492.287	−0.731481	−0.365740	−	0.930717i	$-0.619184\pi$
$673$	−492.287	−0.731481	−0.365740	+	0.930717i	$0.619184\pi$
$674$	− 539.984i	− 0.801163i
$675$	0	0
$676$	273.125	0.404031
$677$	− 776.442i	− 1.14689i	−0.819245	−	0.573443i	$-0.805607\pi$
$677$	− 776.442i	− 1.14689i	0.819245	−	0.573443i	$-0.194393\pi$
$678$	0	0
$679$	−187.507	−0.276151
$680$	98.5760i	0.144965i
$681$	0	0
$682$	0	0
$683$	416.608i	0.609967i	0.952358	+	0.304984i	$0.0986510\pi$
$683$	416.608i	0.609967i	−0.952358	+	0.304984i	$0.901349\pi$
$684$	0	0
$685$	−165.120	−0.241050
$686$	76.7300i	0.111851i
$687$	0	0
$688$	−114.049	−0.165770
$689$	− 263.888i	− 0.383001i
$690$	0	0
$691$	−98.0216	−0.141855	−0.0709274	−	0.997481i	$-0.522596\pi$
$691$	−98.0216	−0.141855	−0.0709274	+	0.997481i	$0.522596\pi$
$692$	344.574i	0.497939i
$693$	0	0
$694$	−238.817	−0.344116
$695$	5.64664i	0.00812467i
$696$	0	0
$697$	−949.988	−1.36297
$698$	− 473.377i	− 0.678191i
$699$	0	0
$700$	−374.493	−0.534991
$701$	994.327i	1.41844i	0.704987	+	0.709220i	$0.250953\pi$
$701$	994.327i	1.41844i	−0.704987	+	0.709220i	$0.749047\pi$
$702$	0	0
$703$	−127.840	−0.181849
$704$	0	0
$705$	0	0
$706$	−272.291	−0.385681
$707$	631.470i	0.893168i
$708$	0	0
$709$	−549.455	−0.774972	−0.387486	−	0.921876i	$-0.626656\pi$
$709$	−549.455	−0.774972	−0.387486	+	0.921876i	$0.626656\pi$
$710$	− 302.179i	− 0.425604i
$711$	0	0
$712$	372.083	0.522589
$713$	− 203.787i	− 0.285816i
$714$	0	0
$715$	0	0
$716$	69.1295i	0.0965495i
$717$	0	0
$718$	358.094	0.498738
$719$	− 26.2828i	− 0.0365546i	−0.999833	−	0.0182773i	$-0.994182\pi$
$719$	− 26.2828i	− 0.0365546i	0.999833	−	0.0182773i	$-0.00581817\pi$
$720$	0	0
$721$	1514.04	2.09991
$722$	− 841.832i	− 1.16597i
$723$	0	0
$724$	34.6141	0.0478095
$725$	− 418.361i	− 0.577050i
$726$	0	0
$727$	58.7452	0.0808049	0.0404024	−	0.999183i	$-0.487136\pi$
$727$	58.7452	0.0808049	0.0404024	+	0.999183i	$0.487136\pi$
$728$	− 154.809i	− 0.212650i
$729$	0	0
$730$	361.490	0.495192
$731$	423.117i	0.578820i
$732$	0	0
$733$	−802.094	−1.09426	−0.547131	−	0.837047i	$-0.684280\pi$
$733$	−802.094	−1.09426	−0.547131	+	0.837047i	$0.684280\pi$
$734$	233.230i	0.317752i
$735$	0	0
$736$	31.3042	0.0425329
$737$	0	0
$738$	0	0
$739$	−684.875	−0.926759	−0.463379	−	0.886160i	$-0.653363\pi$
$739$	−684.875	−0.926759	−0.463379	+	0.886160i	$0.653363\pi$
$740$	− 19.4181i	− 0.0262406i
$741$	0	0
$742$	629.709	0.848665
$743$	1090.75i	1.46803i	0.679131	+	0.734017i	$0.262357\pi$
$743$	1090.75i	1.46803i	−0.679131	+	0.734017i	$0.737643\pi$
$744$	0	0
$745$	653.565	0.877269
$746$	185.370i	0.248485i
$747$	0	0
$748$	0	0
$749$	669.923i	0.894423i
$750$	0	0
$751$	950.324	1.26541	0.632706	−	0.774392i	$-0.281944\pi$
$751$	950.324	1.26541	0.632706	+	0.774392i	$0.281944\pi$
$752$	36.1215i	0.0480339i
$753$	0	0
$754$	172.943	0.229368
$755$	− 650.982i	− 0.862228i
$756$	0	0
$757$	1190.52	1.57268	0.786339	−	0.617795i	$-0.211974\pi$
$757$	1190.52	1.57268	0.786339	+	0.617795i	$0.211974\pi$
$758$	− 445.212i	− 0.587351i
$759$	0	0
$760$	205.415	0.270283
$761$	− 621.960i	− 0.817293i	−0.912693	−	0.408646i	$-0.866001\pi$
$761$	− 621.960i	− 0.817293i	0.912693	−	0.408646i	$-0.133999\pi$
$762$	0	0
$763$	155.881	0.204300
$764$	− 394.255i	− 0.516040i
$765$	0	0
$766$	420.517	0.548977
$767$	338.325i	0.441102i
$768$	0	0
$769$	130.220	0.169336	0.0846682	−	0.996409i	$-0.473017\pi$
$769$	130.220	0.169336	0.0846682	+	0.996409i	$0.473017\pi$
$770$	0	0
$771$	0	0
$772$	518.415	0.671521
$773$	1370.67i	1.77319i	0.462550	+	0.886593i	$0.346935\pi$
$773$	1370.67i	1.77319i	−0.462550	+	0.886593i	$0.653065\pi$
$774$	0	0
$775$	−717.520	−0.925832
$776$	− 55.1866i	− 0.0711168i
$777$	0	0
$778$	963.760	1.23877
$779$	1979.61i	2.54122i
$780$	0	0
$781$	0	0
$782$	− 116.137i	− 0.148513i
$783$	0	0
$784$	173.417	0.221195
$785$	− 609.356i	− 0.776250i
$786$	0	0
$787$	34.3817	0.0436870	0.0218435	−	0.999761i	$-0.493046\pi$
$787$	34.3817	0.0436870	0.0218435	+	0.999761i	$0.493046\pi$
$788$	− 320.915i	− 0.407252i
$789$	0	0
$790$	433.127	0.548262
$791$	− 1392.63i	− 1.76060i
$792$	0	0
$793$	11.2502	0.0141868
$794$	− 618.206i	− 0.778597i
$795$	0	0
$796$	−54.0775	−0.0679366
$797$	1163.45i	1.45978i	0.683564	+	0.729891i	$0.260429\pi$
$797$	1163.45i	1.45978i	−0.683564	+	0.729891i	$0.739571\pi$
$798$	0	0
$799$	134.009	0.167721
$800$	− 110.220i	− 0.137775i
$801$	0	0
$802$	−566.426	−0.706267
$803$	0	0
$804$	0	0
$805$	−124.898	−0.155152
$806$	− 296.610i	− 0.368003i
$807$	0	0
$808$	−185.853	−0.230016
$809$	− 719.682i	− 0.889595i	−0.895631	−	0.444797i	$-0.853276\pi$
$809$	− 719.682i	− 0.889595i	0.895631	−	0.444797i	$-0.146724\pi$
$810$	0	0
$811$	543.992	0.670766	0.335383	−	0.942082i	$-0.391134\pi$
$811$	543.992	0.670766	0.335383	+	0.942082i	$0.391134\pi$
$812$	412.690i	0.508239i
$813$	0	0
$814$	0	0
$815$	− 329.220i	− 0.403951i
$816$	0	0
$817$	881.703	1.07920
$818$	− 1018.88i	− 1.24557i
$819$	0	0
$820$	−300.690	−0.366695
$821$	594.222i	0.723778i	0.932221	+	0.361889i	$0.117868\pi$
$821$	594.222i	0.723778i	−0.932221	+	0.361889i	$0.882132\pi$
$822$	0	0
$823$	−1045.31	−1.27012	−0.635059	−	0.772463i	$-0.719024\pi$
$823$	−1045.31	−1.27012	−0.635059	+	0.772463i	$0.719024\pi$
$824$	445.608i	0.540786i
$825$	0	0
$826$	−807.337	−0.977406
$827$	1059.79i	1.28149i	0.767754	+	0.640745i	$0.221374\pi$
$827$	1059.79i	1.28149i	−0.767754	+	0.640745i	$0.778626\pi$
$828$	0	0
$829$	1574.88	1.89974	0.949868	−	0.312650i	$-0.101217\pi$
$829$	1574.88	1.89974	0.949868	+	0.312650i	$0.101217\pi$
$830$	− 285.201i	− 0.343616i
$831$	0	0
$832$	45.5631	0.0547633
$833$	− 643.368i	− 0.772351i
$834$	0	0
$835$	−32.8868	−0.0393854
$836$	0	0
$837$	0	0
$838$	−450.386	−0.537454
$839$	− 587.713i	− 0.700493i	−0.936658	−	0.350246i	$-0.886098\pi$
$839$	− 587.713i	− 0.700493i	0.936658	−	0.350246i	$-0.113902\pi$
$840$	0	0
$841$	379.968	0.451805
$842$	963.974i	1.14486i
$843$	0	0
$844$	−585.432	−0.693640
$845$	− 320.723i	− 0.379554i
$846$	0	0
$847$	0	0
$848$	185.335i	0.218555i
$849$	0	0
$850$	−408.911	−0.481072
$851$	22.8773i	0.0268829i
$852$	0	0
$853$	−372.721	−0.436953	−0.218477	−	0.975842i	$-0.570109\pi$
$853$	−372.721	−0.436953	−0.218477	+	0.975842i	$0.570109\pi$
$854$	26.8460i	0.0314355i
$855$	0	0
$856$	−197.170	−0.230339
$857$	1144.22i	1.33515i	0.744542	+	0.667575i	$0.232668\pi$
$857$	1144.22i	1.33515i	−0.744542	+	0.667575i	$0.767332\pi$
$858$	0	0
$859$	516.697	0.601510	0.300755	−	0.953701i	$-0.402761\pi$
$859$	516.697	0.601510	0.300755	+	0.953701i	$0.402761\pi$
$860$	133.925i	0.155727i
$861$	0	0
$862$	−69.7457	−0.0809115
$863$	− 1115.22i	− 1.29226i	−0.763227	−	0.646130i	$-0.776386\pi$
$863$	− 1115.22i	− 1.29226i	0.763227	−	0.646130i	$-0.223614\pi$
$864$	0	0
$865$	404.623	0.467772
$866$	− 285.847i	− 0.330077i
$867$	0	0
$868$	707.793	0.815430
$869$	0	0
$870$	0	0
$871$	572.476	0.657263
$872$	45.8784i	0.0526129i
$873$	0	0
$874$	−242.009	−0.276898
$875$	1004.00i	1.14743i
$876$	0	0
$877$	1282.57	1.46245	0.731227	−	0.682134i	$-0.238948\pi$
$877$	1282.57	1.46245	0.731227	+	0.682134i	$0.238948\pi$
$878$	7.64617i	0.00870862i
$879$	0	0
$880$	0	0
$881$	1334.54i	1.51480i	0.652953	+	0.757398i	$0.273530\pi$
$881$	1334.54i	1.51480i	−0.652953	+	0.757398i	$0.726470\pi$
$882$	0	0
$883$	276.465	0.313097	0.156549	−	0.987670i	$-0.449963\pi$
$883$	276.465	0.313097	0.156549	+	0.987670i	$0.449963\pi$
$884$	− 169.037i	− 0.191218i
$885$	0	0
$886$	749.684	0.846145
$887$	927.906i	1.04612i	0.852297	+	0.523059i	$0.175209\pi$
$887$	927.906i	1.04612i	−0.852297	+	0.523059i	$0.824791\pi$
$888$	0	0
$889$	1478.92	1.66358
$890$	− 436.927i	− 0.490929i
$891$	0	0
$892$	649.092	0.727682
$893$	− 279.251i	− 0.312711i
$894$	0	0
$895$	81.1768	0.0907003
$896$	108.726i	0.121346i
$897$	0	0
$898$	−555.570	−0.618675
$899$	790.703i	0.879536i
$900$	0	0
$901$	687.582	0.763132
$902$	0	0
$903$	0	0
$904$	409.877	0.453404
$905$	− 40.6463i	− 0.0449131i
$906$	0	0
$907$	343.887	0.379147	0.189574	−	0.981867i	$-0.439289\pi$
$907$	343.887	0.379147	0.189574	+	0.981867i	$0.439289\pi$
$908$	− 743.860i	− 0.819229i
$909$	0	0
$910$	−181.788	−0.199767
$911$	1603.13i	1.75975i	0.475204	+	0.879876i	$0.342374\pi$
$911$	1603.13i	1.75975i	−0.475204	+	0.879876i	$0.657626\pi$
$912$	0	0
$913$	0	0
$914$	− 94.1431i	− 0.103001i
$915$	0	0
$916$	650.203	0.709829
$917$	− 1509.27i	− 1.64588i
$918$	0	0
$919$	−520.149	−0.565994	−0.282997	−	0.959121i	$-0.591329\pi$
$919$	−520.149	−0.565994	−0.282997	+	0.959121i	$0.591329\pi$
$920$	− 36.7596i	− 0.0399561i
$921$	0	0
$922$	−287.540	−0.311865
$923$	518.172i	0.561399i
$924$	0	0
$925$	80.5496	0.0870806
$926$	− 319.807i	− 0.345364i
$927$	0	0
$928$	−121.462	−0.130886
$929$	494.615i	0.532417i	0.963916	+	0.266208i	$0.0857709\pi$
$929$	494.615i	0.532417i	−0.963916	+	0.266208i	$0.914229\pi$
$930$	0	0
$931$	−1340.67	−1.44003
$932$	815.917i	0.875447i
$933$	0	0
$934$	−634.342	−0.679167
$935$	0	0
$936$	0	0
$937$	−30.4477	−0.0324949	−0.0162474	−	0.999868i	$-0.505172\pi$
$937$	−30.4477	−0.0324949	−0.0162474	+	0.999868i	$0.505172\pi$
$938$	1366.08i	1.45638i
$939$	0	0
$940$	42.4164	0.0451238
$941$	− 248.976i	− 0.264587i	−0.991211	−	0.132293i	$-0.957766\pi$
$941$	− 248.976i	− 0.264587i	0.991211	−	0.132293i	$-0.0422341\pi$
$942$	0	0
$943$	354.257	0.375670
$944$	− 237.614i	− 0.251709i
$945$	0	0
$946$	0	0
$947$	250.989i	0.265036i	0.991181	+	0.132518i	$0.0423062\pi$
$947$	250.989i	0.265036i	−0.991181	+	0.132518i	$0.957694\pi$
$948$	0	0
$949$	−619.878	−0.653191
$950$	852.099i	0.896946i
$951$	0	0
$952$	403.368	0.423706
$953$	406.326i	0.426365i	0.977012	+	0.213183i	$0.0683829\pi$
$953$	406.326i	0.426365i	−0.977012	+	0.213183i	$0.931617\pi$
$954$	0	0
$955$	−462.962	−0.484777
$956$	286.901i	0.300106i
$957$	0	0
$958$	346.629	0.361826
$959$	675.660i	0.704547i
$960$	0	0
$961$	395.113	0.411148
$962$	33.2978i	0.0346131i
$963$	0	0
$964$	176.644	0.183241
$965$	− 608.759i	− 0.630839i
$966$	0	0
$967$	381.721	0.394748	0.197374	−	0.980328i	$-0.436759\pi$
$967$	381.721	0.394748	0.197374	+	0.980328i	$0.436759\pi$
$968$	0	0
$969$	0	0
$970$	−64.8041	−0.0668083
$971$	− 1034.36i	− 1.06525i	−0.846351	−	0.532625i	$-0.821206\pi$
$971$	− 1034.36i	− 1.06525i	0.846351	−	0.532625i	$-0.178794\pi$
$972$	0	0
$973$	23.1058	0.0237469
$974$	47.9826i	0.0492634i
$975$	0	0
$976$	−7.90124	−0.00809554
$977$	− 1193.01i	− 1.22109i	−0.791981	−	0.610546i	$-0.790950\pi$
$977$	− 1193.01i	− 1.22109i	0.791981	−	0.610546i	$-0.209050\pi$
$978$	0	0
$979$	0	0
$980$	− 203.639i	− 0.207795i
$981$	0	0
$982$	−342.790	−0.349074
$983$	652.142i	0.663420i	0.943381	+	0.331710i	$0.107626\pi$
$983$	652.142i	0.663420i	−0.943381	+	0.331710i	$0.892374\pi$
$984$	0	0
$985$	−376.841	−0.382579
$986$	450.618i	0.457016i
$987$	0	0
$988$	−352.243	−0.356521
$989$	− 157.783i	− 0.159538i
$990$	0	0
$991$	1089.72	1.09961	0.549806	−	0.835292i	$-0.314702\pi$
$991$	1089.72	1.09961	0.549806	+	0.835292i	$0.314702\pi$
$992$	208.316i	0.209996i
$993$	0	0
$994$	−1236.50	−1.24396
$995$	63.5017i	0.0638208i
$996$	0	0
$997$	839.869	0.842396	0.421198	−	0.906969i	$-0.361610\pi$
$997$	839.869	0.842396	0.421198	+	0.906969i	$0.361610\pi$
$998$	295.399i	0.295991i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.3.c.m.485.3		8
3.2	odd	2	inner	2178.3.c.m.485.6		8
11.2	odd	10		198.3.k.a.125.3	yes	16
11.6	odd	10		198.3.k.a.179.2	yes	16
11.10	odd	2		2178.3.c.p.485.7		8
33.2	even	10		198.3.k.a.125.2	✓	16
33.17	even	10		198.3.k.a.179.3	yes	16
33.32	even	2		2178.3.c.p.485.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.3.k.a.125.2	✓	16	33.2	even	10
198.3.k.a.125.3	yes	16	11.2	odd	10
198.3.k.a.179.2	yes	16	11.6	odd	10
198.3.k.a.179.3	yes	16	33.17	even	10
2178.3.c.m.485.3		8	1.1	even	1	trivial
2178.3.c.m.485.6		8	3.2	odd	2	inner
2178.3.c.p.485.2		8	33.32	even	2
2178.3.c.p.485.7		8	11.10	odd	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2178.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.34854i q^{5} +9.61011 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.34854i q^{5} +9.61011 q^{7} +2.82843i q^{8} +3.32134 q^{10} -5.69539 q^{13} -13.5908i q^{14} +4.00000 q^{16} -14.8398i q^{17} -30.9235 q^{19} -4.69709i q^{20} +5.53385i q^{23} +19.4843 q^{25} +8.05449i q^{26} -19.2202 q^{28} -21.4717i q^{29} -36.8254 q^{31} -5.65685i q^{32} -20.9866 q^{34} +22.5698i q^{35} +4.13407 q^{37} +43.7325i q^{38} -6.64268 q^{40} -64.0163i q^{41} -28.5124 q^{43} +7.82605 q^{46} +9.03037i q^{47} +43.3543 q^{49} -27.5550i q^{50} +11.3908 q^{52} +46.3337i q^{53} +27.1815i q^{56} -30.3655 q^{58} -59.4034i q^{59} -1.97531 q^{61} +52.0790i q^{62} -8.00000 q^{64} -13.3759i q^{65} -100.516 q^{67} +29.6796i q^{68} +31.9185 q^{70} -90.9809i q^{71} +108.839 q^{73} -5.84645i q^{74} +61.8471 q^{76} +130.407 q^{79} +9.39417i q^{80} -90.5327 q^{82} -85.8693i q^{83} +34.8519 q^{85} +40.3226i q^{86} -131.551i q^{89} -54.7333 q^{91} -11.0677i q^{92} +12.7709 q^{94} -72.6253i q^{95} -19.5114 q^{97} -61.3122i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 - 8 * q^7	\( 8 q - 16 q^{4} - 8 q^{7} + 12 q^{10} + 36 q^{13} + 32 q^{16} - 72 q^{19} - 64 q^{25} + 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} - 24 q^{40} + 220 q^{43} + 68 q^{46} + 56 q^{49} - 72 q^{52} + 112 q^{58} + 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} + 488 q^{73} + 144 q^{76} + 368 q^{79} - 388 q^{82} - 248 q^{85} - 356 q^{91} + 120 q^{94} - 832 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 - 8 * q^7 + 12 * q^10 + 36 * q^13 + 32 * q^16 - 72 * q^19 - 64 * q^25 + 16 * q^28 - 88 * q^31 - 56 * q^34 + 108 * q^37 - 24 * q^40 + 220 * q^43 + 68 * q^46 + 56 * q^49 - 72 * q^52 + 112 * q^58 + 160 * q^61 - 64 * q^64 - 276 * q^67 - 92 * q^70 + 488 * q^73 + 144 * q^76 + 368 * q^79 - 388 * q^82 - 248 * q^85 - 356 * q^91 + 120 * q^94 - 832 * q^97

Introduction

L-functions

Modular forms

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Database

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