LMFDB - Embedded newform 882.5.b.h.197.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,5,Mod(197,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("882.197");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	882.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:	$91.1723074400$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 13x^{4} - 660x^{3} + 702x^{2} - 5832x + 157464 $ x^6 - 2x^5 + 13x^4 - 660x^3 + 702x^2 - 5832*x + 157464
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.3
Root			$7.22197 + 1.35763i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.h.197.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +42.3810i q^{5} +22.6274i q^{8} +O(q^{10})$	$q-2.82843i q^{2} -8.00000 q^{4} +42.3810i q^{5} +22.6274i q^{8} +119.872 q^{10} +183.208i q^{11} +98.5795 q^{13} +64.0000 q^{16} +532.698i q^{17} -296.907 q^{19} -339.048i q^{20} +518.190 q^{22} +878.048i q^{23} -1171.15 q^{25} -278.825i q^{26} +849.688i q^{29} +411.986 q^{31} -181.019i q^{32} +1506.70 q^{34} +1453.60 q^{37} +839.780i q^{38} -958.973 q^{40} -2521.02i q^{41} +2087.63 q^{43} -1465.66i q^{44} +2483.49 q^{46} -944.584i q^{47} +3312.52i q^{50} -788.636 q^{52} -206.078i q^{53} -7764.53 q^{55} +2403.28 q^{58} -2534.52i q^{59} -179.340 q^{61} -1165.27i q^{62} -512.000 q^{64} +4177.90i q^{65} +607.526 q^{67} -4261.58i q^{68} +3054.58i q^{71} +4429.43 q^{73} -4111.40i q^{74} +2375.26 q^{76} -1336.50 q^{79} +2712.39i q^{80} -7130.52 q^{82} +4342.55i q^{83} -22576.3 q^{85} -5904.70i q^{86} -4145.52 q^{88} +5076.01i q^{89} -7024.38i q^{92} -2671.69 q^{94} -12583.2i q^{95} -1487.59 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 48 q^{4}+O(q^{10}) $ 6 * q - 48 * q^4	$ 6 q - 48 q^{4} + 104 q^{10} + 86 q^{13} + 384 q^{16} + 174 q^{19} + 472 q^{22} - 2170 q^{25} + 1682 q^{31} + 1744 q^{34} + 6018 q^{37} - 832 q^{40} - 2130 q^{43} - 656 q^{46} - 688 q^{52} - 12680 q^{55} + 7040 q^{58} - 20632 q^{61} - 3072 q^{64} - 74 q^{67} - 2942 q^{73} - 1392 q^{76} + 10526 q^{79} - 25464 q^{82} - 33056 q^{85} - 3776 q^{88} - 11640 q^{94} - 46436 q^{97}+O(q^{100}) $ 6 * q - 48 * q^4 + 104 * q^10 + 86 * q^13 + 384 * q^16 + 174 * q^19 + 472 * q^22 - 2170 * q^25 + 1682 * q^31 + 1744 * q^34 + 6018 * q^37 - 832 * q^40 - 2130 * q^43 - 656 * q^46 - 688 * q^52 - 12680 * q^55 + 7040 * q^58 - 20632 * q^61 - 3072 * q^64 - 74 * q^67 - 2942 * q^73 - 1392 * q^76 + 10526 * q^79 - 25464 * q^82 - 33056 * q^85 - 3776 * q^88 - 11640 * q^94 - 46436 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	42.3810i	1.69524i	0.530603	+	0.847620i	$0.321966\pi$
$5$	42.3810i	1.69524i	−0.530603	+	0.847620i	$0.678034\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	22.6274i	0.353553i
$9$	0	0
$10$	119.872	1.19872
$11$	183.208i	1.51411i	0.653349	+	0.757056i	$0.273363\pi$
$11$	183.208i	1.51411i	−0.653349	+	0.757056i	$0.726637\pi$
$12$	0	0
$13$	98.5795	0.583311	0.291655	−	0.956524i	$-0.405794\pi$
$13$	98.5795	0.583311	0.291655	+	0.956524i	$0.405794\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	532.698i	1.84325i	0.388087	+	0.921623i	$0.373136\pi$
$17$	532.698i	1.84325i	−0.388087	+	0.921623i	$0.626864\pi$
$18$	0	0
$19$	−296.907	−0.822457	−0.411229	−	0.911532i	$-0.634900\pi$
$19$	−296.907	−0.822457	−0.411229	+	0.911532i	$0.634900\pi$
$20$	− 339.048i	− 0.847620i
$21$	0	0
$22$	518.190	1.07064
$23$	878.048i	1.65983i	0.557893	+	0.829913i	$0.311610\pi$
$23$	878.048i	1.65983i	−0.557893	+	0.829913i	$0.688390\pi$
$24$	0	0
$25$	−1171.15	−1.87384
$26$	− 278.825i	− 0.412463i
$27$	0	0
$28$	0	0
$29$	849.688i	1.01033i	0.863022	+	0.505166i	$0.168569\pi$
$29$	849.688i	1.01033i	−0.863022	+	0.505166i	$0.831431\pi$
$30$	0	0
$31$	411.986	0.428705	0.214353	−	0.976756i	$-0.431236\pi$
$31$	411.986	0.428705	0.214353	+	0.976756i	$0.431236\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	1506.70	1.30337
$35$	0	0
$36$	0	0
$37$	1453.60	1.06180	0.530899	−	0.847435i	$-0.321854\pi$
$37$	1453.60	1.06180	0.530899	+	0.847435i	$0.321854\pi$
$38$	839.780i	0.581565i
$39$	0	0
$40$	−958.973	−0.599358
$41$	− 2521.02i	− 1.49972i	−0.661599	−	0.749858i	$-0.730122\pi$
$41$	− 2521.02i	− 1.49972i	0.661599	−	0.749858i	$-0.269878\pi$
$42$	0	0
$43$	2087.63	1.12906	0.564529	−	0.825413i	$-0.309058\pi$
$43$	2087.63	1.12906	0.564529	+	0.825413i	$0.309058\pi$
$44$	− 1465.66i	− 0.757056i
$45$	0	0
$46$	2483.49	1.17367
$47$	− 944.584i	− 0.427607i	−0.976877	−	0.213803i	$-0.931415\pi$
$47$	− 944.584i	− 0.427607i	0.976877	−	0.213803i	$-0.0685852\pi$
$48$	0	0
$49$	0	0
$50$	3312.52i	1.32501i
$51$	0	0
$52$	−788.636	−0.291655
$53$	− 206.078i	− 0.0733634i	−0.999327	−	0.0366817i	$-0.988321\pi$
$53$	− 206.078i	− 0.0733634i	0.999327	−	0.0366817i	$-0.0116788\pi$
$54$	0	0
$55$	−7764.53	−2.56679
$56$	0	0
$57$	0	0
$58$	2403.28	0.714412
$59$	− 2534.52i	− 0.728102i	−0.931379	−	0.364051i	$-0.881393\pi$
$59$	− 2534.52i	− 0.728102i	0.931379	−	0.364051i	$-0.118607\pi$
$60$	0	0
$61$	−179.340	−0.0481969	−0.0240984	−	0.999710i	$-0.507672\pi$
$61$	−179.340	−0.0481969	−0.0240984	+	0.999710i	$0.507672\pi$
$62$	− 1165.27i	− 0.303140i
$63$	0	0
$64$	−512.000	−0.125000
$65$	4177.90i	0.988852i
$66$	0	0
$67$	607.526	0.135337	0.0676683	−	0.997708i	$-0.478444\pi$
$67$	607.526	0.135337	0.0676683	+	0.997708i	$0.478444\pi$
$68$	− 4261.58i	− 0.921623i
$69$	0	0
$70$	0	0
$71$	3054.58i	0.605947i	0.952999	+	0.302973i	$0.0979793\pi$
$71$	3054.58i	0.605947i	−0.952999	+	0.302973i	$0.902021\pi$
$72$	0	0
$73$	4429.43	0.831194	0.415597	−	0.909549i	$-0.363573\pi$
$73$	4429.43	0.831194	0.415597	+	0.909549i	$0.363573\pi$
$74$	− 4111.40i	− 0.750804i
$75$	0	0
$76$	2375.26	0.411229
$77$	0	0
$78$	0	0
$79$	−1336.50	−0.214148	−0.107074	−	0.994251i	$-0.534148\pi$
$79$	−1336.50	−0.214148	−0.107074	+	0.994251i	$0.534148\pi$
$80$	2712.39i	0.423810i
$81$	0	0
$82$	−7130.52	−1.06046
$83$	4342.55i	0.630360i	0.949032	+	0.315180i	$0.102065\pi$
$83$	4342.55i	0.630360i	−0.949032	+	0.315180i	$0.897935\pi$
$84$	0	0
$85$	−22576.3	−3.12474
$86$	− 5904.70i	− 0.798364i
$87$	0	0
$88$	−4145.52	−0.535320
$89$	5076.01i	0.640830i	0.947277	+	0.320415i	$0.103822\pi$
$89$	5076.01i	0.640830i	−0.947277	+	0.320415i	$0.896178\pi$
$90$	0	0
$91$	0	0
$92$	− 7024.38i	− 0.829913i
$93$	0	0
$94$	−2671.69	−0.302364
$95$	− 12583.2i	− 1.39426i
$96$	0	0
$97$	−1487.59	−0.158103	−0.0790516	−	0.996871i	$-0.525189\pi$
$97$	−1487.59	−0.158103	−0.0790516	+	0.996871i	$0.525189\pi$
$98$	0	0
$99$	0	0
$100$	9369.21	0.936921
$101$	− 8218.33i	− 0.805640i	−0.915279	−	0.402820i	$-0.868030\pi$
$101$	− 8218.33i	− 0.805640i	0.915279	−	0.402820i	$-0.131970\pi$
$102$	0	0
$103$	−8166.70	−0.769790	−0.384895	−	0.922960i	$-0.625762\pi$
$103$	−8166.70	−0.769790	−0.384895	+	0.922960i	$0.625762\pi$
$104$	2230.60i	0.206231i
$105$	0	0
$106$	−582.876	−0.0518758
$107$	377.281i	0.0329532i	0.999864	+	0.0164766i	$0.00524490\pi$
$107$	377.281i	0.0329532i	−0.999864	+	0.0164766i	$0.994755\pi$
$108$	0	0
$109$	−8738.45	−0.735498	−0.367749	−	0.929925i	$-0.619871\pi$
$109$	−8738.45	−0.735498	−0.367749	+	0.929925i	$0.619871\pi$
$110$	21961.4i	1.81499i
$111$	0	0
$112$	0	0
$113$	4273.72i	0.334695i	0.985898	+	0.167347i	$0.0535202\pi$
$113$	4273.72i	0.334695i	−0.985898	+	0.167347i	$0.946480\pi$
$114$	0	0
$115$	−37212.6	−2.81380
$116$	− 6797.51i	− 0.505166i
$117$	0	0
$118$	−7168.71	−0.514846
$119$	0	0
$120$	0	0
$121$	−18924.1	−1.29254
$122$	507.251i	0.0340803i
$123$	0	0
$124$	−3295.89	−0.214353
$125$	− 23146.4i	− 1.48137i
$126$	0	0
$127$	20922.8	1.29722	0.648608	−	0.761123i	$-0.275352\pi$
$127$	20922.8	1.29722	0.648608	+	0.761123i	$0.275352\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	11816.9	0.699224
$131$	− 1201.70i	− 0.0700253i	−0.999387	−	0.0350126i	$-0.988853\pi$
$131$	− 1201.70i	− 0.0700253i	0.999387	−	0.0350126i	$-0.0111471\pi$
$132$	0	0
$133$	0	0
$134$	− 1718.34i	− 0.0956974i
$135$	0	0
$136$	−12053.6	−0.651686
$137$	6159.07i	0.328151i	0.986448	+	0.164076i	$0.0524641\pi$
$137$	6159.07i	0.328151i	−0.986448	+	0.164076i	$0.947536\pi$
$138$	0	0
$139$	−5739.19	−0.297044	−0.148522	−	0.988909i	$-0.547452\pi$
$139$	−5739.19	−0.297044	−0.148522	+	0.988909i	$0.547452\pi$
$140$	0	0
$141$	0	0
$142$	8639.65	0.428469
$143$	18060.5i	0.883198i
$144$	0	0
$145$	−36010.7	−1.71275
$146$	− 12528.3i	− 0.587743i
$147$	0	0
$148$	−11628.8	−0.530899
$149$	− 36220.0i	− 1.63146i	−0.578435	−	0.815729i	$-0.696336\pi$
$149$	− 36220.0i	− 1.63146i	0.578435	−	0.815729i	$-0.303664\pi$
$150$	0	0
$151$	22424.5	0.983487	0.491744	−	0.870740i	$-0.336360\pi$
$151$	22424.5	0.983487	0.491744	+	0.870740i	$0.336360\pi$
$152$	− 6718.24i	− 0.290783i
$153$	0	0
$154$	0	0
$155$	17460.4i	0.726759i
$156$	0	0
$157$	−19835.9	−0.804734	−0.402367	−	0.915478i	$-0.631812\pi$
$157$	−19835.9	−0.804734	−0.402367	+	0.915478i	$0.631812\pi$
$158$	3780.19i	0.151426i
$159$	0	0
$160$	7671.78	0.299679
$161$	0	0
$162$	0	0
$163$	20936.9	0.788018	0.394009	−	0.919106i	$-0.371088\pi$
$163$	20936.9	0.788018	0.394009	+	0.919106i	$0.371088\pi$
$164$	20168.2i	0.749858i
$165$	0	0
$166$	12282.6	0.445732
$167$	44262.5i	1.58709i	0.608509	+	0.793547i	$0.291768\pi$
$167$	44262.5i	1.58709i	−0.608509	+	0.793547i	$0.708232\pi$
$168$	0	0
$169$	−18843.1	−0.659749
$170$	63855.4i	2.20953i
$171$	0	0
$172$	−16701.0	−0.564529
$173$	6544.86i	0.218679i	0.994004	+	0.109340i	$0.0348736\pi$
$173$	6544.86i	0.218679i	−0.994004	+	0.109340i	$0.965126\pi$
$174$	0	0
$175$	0	0
$176$	11725.3i	0.378528i
$177$	0	0
$178$	14357.1	0.453135
$179$	− 31602.8i	− 0.986324i	−0.869937	−	0.493162i	$-0.835841\pi$
$179$	− 31602.8i	− 0.986324i	0.869937	−	0.493162i	$-0.164159\pi$
$180$	0	0
$181$	−27665.7	−0.844471	−0.422236	−	0.906486i	$-0.638755\pi$
$181$	−27665.7	−0.844471	−0.422236	+	0.906486i	$0.638755\pi$
$182$	0	0
$183$	0	0
$184$	−19868.0	−0.586837
$185$	61605.1i	1.80000i
$186$	0	0
$187$	−97594.3	−2.79088
$188$	7556.67i	0.213803i
$189$	0	0
$190$	−35590.7	−0.985893
$191$	− 18287.6i	− 0.501292i	−0.968079	−	0.250646i	$-0.919357\pi$
$191$	− 18287.6i	− 0.501292i	0.968079	−	0.250646i	$-0.0806431\pi$
$192$	0	0
$193$	49479.7	1.32835	0.664175	−	0.747577i	$-0.268783\pi$
$193$	49479.7	1.32835	0.664175	+	0.747577i	$0.268783\pi$
$194$	4207.55i	0.111796i
$195$	0	0
$196$	0	0
$197$	76174.6i	1.96281i	0.191957	+	0.981403i	$0.438517\pi$
$197$	76174.6i	1.96281i	−0.191957	+	0.981403i	$0.561483\pi$
$198$	0	0
$199$	25984.9	0.656169	0.328084	−	0.944648i	$-0.393597\pi$
$199$	25984.9	0.656169	0.328084	+	0.944648i	$0.393597\pi$
$200$	− 26500.1i	− 0.662503i
$201$	0	0
$202$	−23245.0	−0.569673
$203$	0	0
$204$	0	0
$205$	106843.	2.54238
$206$	23098.9i	0.544324i
$207$	0	0
$208$	6309.09	0.145828
$209$	− 54395.7i	− 1.24529i
$210$	0	0
$211$	−14981.1	−0.336496	−0.168248	−	0.985745i	$-0.553811\pi$
$211$	−14981.1	−0.336496	−0.168248	+	0.985745i	$0.553811\pi$
$212$	1648.62i	0.0366817i
$213$	0	0
$214$	1067.11	0.0233014
$215$	88475.8i	1.91402i
$216$	0	0
$217$	0	0
$218$	24716.1i	0.520076i
$219$	0	0
$220$	62116.2	1.28339
$221$	52513.1i	1.07518i
$222$	0	0
$223$	41965.9	0.843891	0.421946	−	0.906621i	$-0.361347\pi$
$223$	41965.9	0.843891	0.421946	+	0.906621i	$0.361347\pi$
$224$	0	0
$225$	0	0
$226$	12087.9	0.236665
$227$	− 93712.4i	− 1.81863i	−0.416104	−	0.909317i	$-0.636605\pi$
$227$	− 93712.4i	− 1.81863i	0.416104	−	0.909317i	$-0.363395\pi$
$228$	0	0
$229$	23109.1	0.440668	0.220334	−	0.975425i	$-0.429285\pi$
$229$	23109.1	0.440668	0.220334	+	0.975425i	$0.429285\pi$
$230$	105253.i	1.98966i
$231$	0	0
$232$	−19226.3	−0.357206
$233$	− 1170.24i	− 0.0215558i	−0.999942	−	0.0107779i	$-0.996569\pi$
$233$	− 1170.24i	− 0.0215558i	0.999942	−	0.0107779i	$-0.00343077\pi$
$234$	0	0
$235$	40032.4	0.724897
$236$	20276.2i	0.364051i
$237$	0	0
$238$	0	0
$239$	68537.8i	1.19987i	0.800049	+	0.599935i	$0.204807\pi$
$239$	68537.8i	1.19987i	−0.800049	+	0.599935i	$0.795193\pi$
$240$	0	0
$241$	−104154.	−1.79326	−0.896630	−	0.442780i	$-0.853992\pi$
$241$	−104154.	−1.79326	−0.896630	+	0.442780i	$0.853992\pi$
$242$	53525.3i	0.913962i
$243$	0	0
$244$	1434.72	0.0240984
$245$	0	0
$246$	0	0
$247$	−29268.9	−0.479748
$248$	9322.17i	0.151570i
$249$	0	0
$250$	−65468.0	−1.04749
$251$	− 4966.17i	− 0.0788268i	−0.999223	−	0.0394134i	$-0.987451\pi$
$251$	− 4966.17i	− 0.0788268i	0.999223	−	0.0394134i	$-0.0125489\pi$
$252$	0	0
$253$	−160865.	−2.51316
$254$	− 59178.6i	− 0.917270i
$255$	0	0
$256$	4096.00	0.0625000
$257$	10161.5i	0.153847i	0.997037	+	0.0769237i	$0.0245098\pi$
$257$	10161.5i	0.153847i	−0.997037	+	0.0769237i	$0.975490\pi$
$258$	0	0
$259$	0	0
$260$	− 33423.2i	− 0.494426i
$261$	0	0
$262$	−3398.93	−0.0495153
$263$	− 27879.7i	− 0.403066i	−0.979482	−	0.201533i	$-0.935408\pi$
$263$	− 27879.7i	− 0.403066i	0.979482	−	0.201533i	$-0.0645923\pi$
$264$	0	0
$265$	8733.79	0.124369
$266$	0	0
$267$	0	0
$268$	−4860.21	−0.0676683
$269$	47018.9i	0.649782i	0.945751	+	0.324891i	$0.105328\pi$
$269$	47018.9i	0.649782i	−0.945751	+	0.324891i	$0.894672\pi$
$270$	0	0
$271$	21354.5	0.290771	0.145386	−	0.989375i	$-0.453558\pi$
$271$	21354.5	0.290771	0.145386	+	0.989375i	$0.453558\pi$
$272$	34092.7i	0.460811i
$273$	0	0
$274$	17420.5	0.232038
$275$	− 214564.i	− 2.83721i
$276$	0	0
$277$	126041.	1.64268	0.821339	−	0.570441i	$-0.193228\pi$
$277$	126041.	1.64268	0.821339	+	0.570441i	$0.193228\pi$
$278$	16232.9i	0.210042i
$279$	0	0
$280$	0	0
$281$	− 135130.i	− 1.71135i	−0.517511	−	0.855677i	$-0.673141\pi$
$281$	− 135130.i	− 1.71135i	0.517511	−	0.855677i	$-0.326859\pi$
$282$	0	0
$283$	−74843.7	−0.934506	−0.467253	−	0.884124i	$-0.654756\pi$
$283$	−74843.7	−0.934506	−0.467253	+	0.884124i	$0.654756\pi$
$284$	− 24436.6i	− 0.302973i
$285$	0	0
$286$	51082.9	0.624515
$287$	0	0
$288$	0	0
$289$	−200246.	−2.39755
$290$	101854.i	1.21110i
$291$	0	0
$292$	−35435.4	−0.415597
$293$	− 15629.6i	− 0.182059i	−0.995848	−	0.0910294i	$-0.970984\pi$
$293$	− 15629.6i	− 0.182059i	0.995848	−	0.0910294i	$-0.0290157\pi$
$294$	0	0
$295$	107416.	1.23431
$296$	32891.2i	0.375402i
$297$	0	0
$298$	−102446.	−1.15361
$299$	86557.5i	0.968194i
$300$	0	0
$301$	0	0
$302$	− 63426.0i	− 0.695431i
$303$	0	0
$304$	−19002.1	−0.205614
$305$	− 7600.63i	− 0.0817053i
$306$	0	0
$307$	18949.2	0.201055	0.100527	−	0.994934i	$-0.467947\pi$
$307$	18949.2	0.201055	0.100527	+	0.994934i	$0.467947\pi$
$308$	0	0
$309$	0	0
$310$	49385.4	0.513896
$311$	− 175703.i	− 1.81660i	−0.418324	−	0.908298i	$-0.637382\pi$
$311$	− 175703.i	− 1.81660i	0.418324	−	0.908298i	$-0.362618\pi$
$312$	0	0
$313$	76733.3	0.783241	0.391620	−	0.920127i	$-0.371915\pi$
$313$	76733.3	0.783241	0.391620	+	0.920127i	$0.371915\pi$
$314$	56104.4i	0.569033i
$315$	0	0
$316$	10692.0	0.107074
$317$	108410.i	1.07882i	0.842043	+	0.539410i	$0.181353\pi$
$317$	108410.i	1.07882i	−0.842043	+	0.539410i	$0.818647\pi$
$318$	0	0
$319$	−155669.	−1.52976
$320$	− 21699.1i	− 0.211905i
$321$	0	0
$322$	0	0
$323$	− 158162.i	− 1.51599i
$324$	0	0
$325$	−115451.	−1.09303
$326$	− 59218.4i	− 0.557213i
$327$	0	0
$328$	57044.2	0.530229
$329$	0	0
$330$	0	0
$331$	148875.	1.35884	0.679418	−	0.733752i	$-0.262232\pi$
$331$	148875.	1.35884	0.679418	+	0.733752i	$0.262232\pi$
$332$	− 34740.4i	− 0.315180i
$333$	0	0
$334$	125193.	1.12224
$335$	25747.6i	0.229428i
$336$	0	0
$337$	−91425.4	−0.805020	−0.402510	−	0.915416i	$-0.631862\pi$
$337$	−91425.4	−0.805020	−0.402510	+	0.915416i	$0.631862\pi$
$338$	53296.3i	0.466513i
$339$	0	0
$340$	180610.	1.56237
$341$	75479.0i	0.649108i
$342$	0	0
$343$	0	0
$344$	47237.6i	0.399182i
$345$	0	0
$346$	18511.6	0.154630
$347$	151339.i	1.25687i	0.777861	+	0.628436i	$0.216305\pi$
$347$	151339.i	1.25687i	−0.777861	+	0.628436i	$0.783695\pi$
$348$	0	0
$349$	34479.8	0.283083	0.141542	−	0.989932i	$-0.454794\pi$
$349$	34479.8	0.283083	0.141542	+	0.989932i	$0.454794\pi$
$350$	0	0
$351$	0	0
$352$	33164.1	0.267660
$353$	− 687.270i	− 0.00551541i	−0.999996	−	0.00275771i	$-0.999122\pi$
$353$	− 687.270i	− 0.00551541i	0.999996	−	0.00275771i	$-0.000877806\pi$
$354$	0	0
$355$	−129456.	−1.02723
$356$	− 40608.1i	− 0.320415i
$357$	0	0
$358$	−89386.2	−0.697436
$359$	− 43991.1i	− 0.341331i	−0.985329	−	0.170665i	$-0.945408\pi$
$359$	− 43991.1i	− 0.341331i	0.985329	−	0.170665i	$-0.0545917\pi$
$360$	0	0
$361$	−42167.2	−0.323564
$362$	78250.5i	0.597131i
$363$	0	0
$364$	0	0
$365$	187724.i	1.40907i
$366$	0	0
$367$	129464.	0.961210	0.480605	−	0.876937i	$-0.340417\pi$
$367$	129464.	0.961210	0.480605	+	0.876937i	$0.340417\pi$
$368$	56195.1i	0.414956i
$369$	0	0
$370$	174245.	1.27279
$371$	0	0
$372$	0	0
$373$	−246017.	−1.76827	−0.884134	−	0.467234i	$-0.845251\pi$
$373$	−246017.	−1.76827	−0.884134	+	0.467234i	$0.845251\pi$
$374$	276038.i	1.97345i
$375$	0	0
$376$	21373.5	0.151182
$377$	83761.8i	0.589337i
$378$	0	0
$379$	200968.	1.39910	0.699550	−	0.714583i	$-0.253384\pi$
$379$	200968.	1.39910	0.699550	+	0.714583i	$0.253384\pi$
$380$	100666.i	0.697132i
$381$	0	0
$382$	−51725.3	−0.354467
$383$	− 184923.i	− 1.26065i	−0.776333	−	0.630323i	$-0.782922\pi$
$383$	− 184923.i	− 1.26065i	0.776333	−	0.630323i	$-0.217078\pi$
$384$	0	0
$385$	0	0
$386$	− 139950.i	− 0.939285i
$387$	0	0
$388$	11900.7	0.0790516
$389$	38207.6i	0.252494i	0.991999	+	0.126247i	$0.0402931\pi$
$389$	38207.6i	0.252494i	−0.991999	+	0.126247i	$0.959707\pi$
$390$	0	0
$391$	−467734.	−3.05947
$392$	0	0
$393$	0	0
$394$	215454.	1.38791
$395$	− 56642.3i	− 0.363033i
$396$	0	0
$397$	−93376.7	−0.592458	−0.296229	−	0.955117i	$-0.595729\pi$
$397$	−93376.7	−0.592458	−0.296229	+	0.955117i	$0.595729\pi$
$398$	− 73496.5i	− 0.463981i
$399$	0	0
$400$	−74953.7	−0.468460
$401$	90647.6i	0.563725i	0.959455	+	0.281863i	$0.0909522\pi$
$401$	90647.6i	0.563725i	−0.959455	+	0.281863i	$0.909048\pi$
$402$	0	0
$403$	40613.3	0.250068
$404$	65746.7i	0.402820i
$405$	0	0
$406$	0	0
$407$	266311.i	1.60768i
$408$	0	0
$409$	−35944.4	−0.214874	−0.107437	−	0.994212i	$-0.534264\pi$
$409$	−35944.4	−0.214874	−0.107437	+	0.994212i	$0.534264\pi$
$410$	− 302199.i	− 1.79773i
$411$	0	0
$412$	65333.6	0.384895
$413$	0	0
$414$	0	0
$415$	−184042.	−1.06861
$416$	− 17844.8i	− 0.103116i
$417$	0	0
$418$	−153854.	−0.880555
$419$	− 106411.i	− 0.606119i	−0.952972	−	0.303059i	$-0.901992\pi$
$419$	− 106411.i	− 0.606119i	0.952972	−	0.303059i	$-0.0980081\pi$
$420$	0	0
$421$	147852.	0.834188	0.417094	−	0.908863i	$-0.363049\pi$
$421$	147852.	0.834188	0.417094	+	0.908863i	$0.363049\pi$
$422$	42373.0i	0.237938i
$423$	0	0
$424$	4663.01	0.0259379
$425$	− 623870.i	− 3.45395i
$426$	0	0
$427$	0	0
$428$	− 3018.25i	− 0.0164766i
$429$	0	0
$430$	250247.	1.35342
$431$	− 51592.2i	− 0.277734i	−0.990311	−	0.138867i	$-0.955654\pi$
$431$	− 51592.2i	− 0.277734i	0.990311	−	0.138867i	$-0.0443461\pi$
$432$	0	0
$433$	−252999.	−1.34941	−0.674703	−	0.738090i	$-0.735728\pi$
$433$	−252999.	−1.34941	−0.674703	+	0.738090i	$0.735728\pi$
$434$	0	0
$435$	0	0
$436$	69907.6	0.367749
$437$	− 260699.i	− 1.36514i
$438$	0	0
$439$	277801.	1.44147	0.720734	−	0.693211i	$-0.243805\pi$
$439$	277801.	1.44147	0.720734	+	0.693211i	$0.243805\pi$
$440$	− 175691.i	− 0.907496i
$441$	0	0
$442$	148529.	0.760270
$443$	− 35615.5i	− 0.181481i	−0.995875	−	0.0907405i	$-0.971077\pi$
$443$	− 35615.5i	− 0.181481i	0.995875	−	0.0907405i	$-0.0289234\pi$
$444$	0	0
$445$	−215127.	−1.08636
$446$	− 118697.i	− 0.596721i
$447$	0	0
$448$	0	0
$449$	199858.i	0.991352i	0.868507	+	0.495676i	$0.165080\pi$
$449$	199858.i	0.991352i	−0.868507	+	0.495676i	$0.834920\pi$
$450$	0	0
$451$	461870.	2.27074
$452$	− 34189.8i	− 0.167347i
$453$	0	0
$454$	−265059.	−1.28597
$455$	0	0
$456$	0	0
$457$	−93011.6	−0.445354	−0.222677	−	0.974892i	$-0.571479\pi$
$457$	−93011.6	−0.445354	−0.222677	+	0.974892i	$0.571479\pi$
$458$	− 65362.3i	− 0.311599i
$459$	0	0
$460$	297700.	1.40690
$461$	− 292074.i	− 1.37433i	−0.726501	−	0.687165i	$-0.758855\pi$
$461$	− 292074.i	− 1.37433i	0.726501	−	0.687165i	$-0.241145\pi$
$462$	0	0
$463$	276636.	1.29047	0.645233	−	0.763986i	$-0.276760\pi$
$463$	276636.	1.29047	0.645233	+	0.763986i	$0.276760\pi$
$464$	54380.1i	0.252583i
$465$	0	0
$466$	−3309.94	−0.0152422
$467$	− 110153.i	− 0.505082i	−0.967586	−	0.252541i	$-0.918734\pi$
$467$	− 110153.i	− 0.505082i	0.967586	−	0.252541i	$-0.0812663\pi$
$468$	0	0
$469$	0	0
$470$	− 113229.i	− 0.512579i
$471$	0	0
$472$	57349.7	0.257423
$473$	382469.i	1.70952i
$474$	0	0
$475$	347723.	1.54115
$476$	0	0
$477$	0	0
$478$	193854.	0.848436
$479$	33570.7i	0.146315i	0.997320	+	0.0731576i	$0.0233076\pi$
$479$	33570.7i	0.146315i	−0.997320	+	0.0731576i	$0.976692\pi$
$480$	0	0
$481$	143295.	0.619357
$482$	294593.i	1.26803i
$483$	0	0
$484$	151392.	0.646269
$485$	− 63045.7i	− 0.268023i
$486$	0	0
$487$	320321.	1.35060	0.675301	−	0.737542i	$-0.264014\pi$
$487$	320321.	1.35060	0.675301	+	0.737542i	$0.264014\pi$
$488$	− 4058.01i	− 0.0170402i
$489$	0	0
$490$	0	0
$491$	− 79130.6i	− 0.328233i	−0.986441	−	0.164116i	$-0.947523\pi$
$491$	− 79130.6i	− 0.328233i	0.986441	−	0.164116i	$-0.0524772\pi$
$492$	0	0
$493$	−452627.	−1.86229
$494$	82785.1i	0.339233i
$495$	0	0
$496$	26367.1	0.107176
$497$	0	0
$498$	0	0
$499$	108772.	0.436832	0.218416	−	0.975856i	$-0.429911\pi$
$499$	108772.	0.436832	0.218416	+	0.975856i	$0.429911\pi$
$500$	185172.i	0.740686i
$501$	0	0
$502$	−14046.4	−0.0557390
$503$	− 120085.i	− 0.474627i	−0.971433	−	0.237314i	$-0.923733\pi$
$503$	− 120085.i	− 0.474627i	0.971433	−	0.237314i	$-0.0762669\pi$
$504$	0	0
$505$	348301.	1.36575
$506$	454995.i	1.77707i
$507$	0	0
$508$	−167382.	−0.648608
$509$	293527.i	1.13295i	0.824078	+	0.566477i	$0.191694\pi$
$509$	293527.i	1.13295i	−0.824078	+	0.566477i	$0.808306\pi$
$510$	0	0
$511$	0	0
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	28741.0	0.108786
$515$	− 346113.i	− 1.30498i
$516$	0	0
$517$	173055.	0.647445
$518$	0	0
$519$	0	0
$520$	−94535.1	−0.349612
$521$	− 189633.i	− 0.698616i	−0.937008	−	0.349308i	$-0.886417\pi$
$521$	− 189633.i	− 0.698616i	0.937008	−	0.349308i	$-0.113583\pi$
$522$	0	0
$523$	326757.	1.19460	0.597298	−	0.802019i	$-0.296241\pi$
$523$	326757.	1.19460	0.597298	+	0.802019i	$0.296241\pi$
$524$	9613.63i	0.0350126i
$525$	0	0
$526$	−78855.6	−0.285011
$527$	219464.i	0.790209i
$528$	0	0
$529$	−491127.	−1.75502
$530$	− 24702.9i	− 0.0879419i
$531$	0	0
$532$	0	0
$533$	− 248521.i	− 0.874800i
$534$	0	0
$535$	−15989.5	−0.0558636
$536$	13746.7i	0.0478487i
$537$	0	0
$538$	132990.	0.459466
$539$	0	0
$540$	0	0
$541$	9625.48	0.0328873	0.0164436	−	0.999865i	$-0.494766\pi$
$541$	9625.48	0.0328873	0.0164436	+	0.999865i	$0.494766\pi$
$542$	− 60399.8i	− 0.205606i
$543$	0	0
$544$	96428.6	0.325843
$545$	− 370345.i	− 1.24685i
$546$	0	0
$547$	−50092.8	−0.167418	−0.0837088	−	0.996490i	$-0.526677\pi$
$547$	−50092.8	−0.167418	−0.0837088	+	0.996490i	$0.526677\pi$
$548$	− 49272.6i	− 0.164076i
$549$	0	0
$550$	−606878.	−2.00621
$551$	− 252279.i	− 0.830954i
$552$	0	0
$553$	0	0
$554$	− 356498.i	− 1.16155i
$555$	0	0
$556$	45913.5	0.148522
$557$	− 485203.i	− 1.56391i	−0.623332	−	0.781957i	$-0.714221\pi$
$557$	− 485203.i	− 1.56391i	0.623332	−	0.781957i	$-0.285779\pi$
$558$	0	0
$559$	205797.	0.658591
$560$	0	0
$561$	0	0
$562$	−382206.	−1.21011
$563$	− 268806.i	− 0.848051i	−0.905650	−	0.424026i	$-0.860617\pi$
$563$	− 268806.i	− 0.848051i	0.905650	−	0.424026i	$-0.139383\pi$
$564$	0	0
$565$	−181125.	−0.567389
$566$	211690.i	0.660796i
$567$	0	0
$568$	−69117.2	−0.214234
$569$	− 557477.i	− 1.72188i	−0.508707	−	0.860940i	$-0.669876\pi$
$569$	− 557477.i	− 1.72188i	0.508707	−	0.860940i	$-0.330124\pi$
$570$	0	0
$571$	−250764.	−0.769118	−0.384559	−	0.923100i	$-0.625647\pi$
$571$	−250764.	−0.769118	−0.384559	+	0.923100i	$0.625647\pi$
$572$	− 144484.i	− 0.441599i
$573$	0	0
$574$	0	0
$575$	− 1.02833e6i	− 3.11025i
$576$	0	0
$577$	540693.	1.62405	0.812025	−	0.583623i	$-0.198365\pi$
$577$	540693.	1.62405	0.812025	+	0.583623i	$0.198365\pi$
$578$	566381.i	1.69533i
$579$	0	0
$580$	288085.	0.856377
$581$	0	0
$582$	0	0
$583$	37755.0	0.111080
$584$	100227.i	0.293871i
$585$	0	0
$586$	−44207.1	−0.128735
$587$	− 620569.i	− 1.80100i	−0.434855	−	0.900500i	$-0.643201\pi$
$587$	− 620569.i	− 1.80100i	0.434855	−	0.900500i	$-0.356799\pi$
$588$	0	0
$589$	−122321.	−0.352592
$590$	− 303817.i	− 0.872788i
$591$	0	0
$592$	93030.4	0.265449
$593$	− 564272.i	− 1.60464i	−0.596891	−	0.802322i	$-0.703597\pi$
$593$	− 564272.i	− 1.60464i	0.596891	−	0.802322i	$-0.296403\pi$
$594$	0	0
$595$	0	0
$596$	289760.i	0.815729i
$597$	0	0
$598$	244822.	0.684616
$599$	573482.i	1.59833i	0.601113	+	0.799164i	$0.294724\pi$
$599$	573482.i	1.59833i	−0.601113	+	0.799164i	$0.705276\pi$
$600$	0	0
$601$	320557.	0.887475	0.443738	−	0.896157i	$-0.353652\pi$
$601$	320557.	0.887475	0.443738	+	0.896157i	$0.353652\pi$
$602$	0	0
$603$	0	0
$604$	−179396.	−0.491744
$605$	− 802021.i	− 2.19116i
$606$	0	0
$607$	168262.	0.456678	0.228339	−	0.973582i	$-0.426671\pi$
$607$	168262.	0.456678	0.228339	+	0.973582i	$0.426671\pi$
$608$	53745.9i	0.145391i
$609$	0	0
$610$	−21497.8	−0.0577744
$611$	− 93116.6i	− 0.249428i
$612$	0	0
$613$	147877.	0.393533	0.196766	−	0.980450i	$-0.436956\pi$
$613$	147877.	0.393533	0.196766	+	0.980450i	$0.436956\pi$
$614$	− 53596.5i	− 0.142167i
$615$	0	0
$616$	0	0
$617$	− 271848.i	− 0.714096i	−0.934086	−	0.357048i	$-0.883783\pi$
$617$	− 271848.i	− 0.714096i	0.934086	−	0.357048i	$-0.116217\pi$
$618$	0	0
$619$	−394399.	−1.02933	−0.514664	−	0.857392i	$-0.672083\pi$
$619$	−394399.	−1.02933	−0.514664	+	0.857392i	$0.672083\pi$
$620$	− 139683.i	− 0.363379i
$621$	0	0
$622$	−496963.	−1.28453
$623$	0	0
$624$	0	0
$625$	249000.	0.637441
$626$	− 217035.i	− 0.553835i
$627$	0	0
$628$	158687.	0.402367
$629$	774330.i	1.95715i
$630$	0	0
$631$	−31733.3	−0.0796996	−0.0398498	−	0.999206i	$-0.512688\pi$
$631$	−31733.3	−0.0796996	−0.0398498	+	0.999206i	$0.512688\pi$
$632$	− 30241.6i	− 0.0757129i
$633$	0	0
$634$	306628.	0.762841
$635$	886729.i	2.19909i
$636$	0	0
$637$	0	0
$638$	440300.i	1.08170i
$639$	0	0
$640$	−61374.3	−0.149840
$641$	225770.i	0.549477i	0.961519	+	0.274738i	$0.0885912\pi$
$641$	225770.i	0.549477i	−0.961519	+	0.274738i	$0.911409\pi$
$642$	0	0
$643$	−212365.	−0.513642	−0.256821	−	0.966459i	$-0.582675\pi$
$643$	−212365.	−0.513642	−0.256821	+	0.966459i	$0.582675\pi$
$644$	0	0
$645$	0	0
$646$	−447349.	−1.07197
$647$	300467.i	0.717775i	0.933381	+	0.358887i	$0.116844\pi$
$647$	300467.i	0.717775i	−0.933381	+	0.358887i	$0.883156\pi$
$648$	0	0
$649$	464344.	1.10243
$650$	326546.i	0.772890i
$651$	0	0
$652$	−167495.	−0.394009
$653$	343010.i	0.804416i	0.915548	+	0.402208i	$0.131757\pi$
$653$	343010.i	0.804416i	−0.915548	+	0.402208i	$0.868243\pi$
$654$	0	0
$655$	50929.4	0.118710
$656$	− 161345.i	− 0.374929i
$657$	0	0
$658$	0	0
$659$	694015.i	1.59808i	0.601279	+	0.799039i	$0.294658\pi$
$659$	694015.i	1.59808i	−0.601279	+	0.799039i	$0.705342\pi$
$660$	0	0
$661$	−255860.	−0.585598	−0.292799	−	0.956174i	$-0.594587\pi$
$661$	−255860.	−0.585598	−0.292799	+	0.956174i	$0.594587\pi$
$662$	− 421083.i	− 0.960842i
$663$	0	0
$664$	−98260.7	−0.222866
$665$	0	0
$666$	0	0
$667$	−746067.	−1.67697
$668$	− 354100.i	− 0.793547i
$669$	0	0
$670$	72825.1	0.162230
$671$	− 32856.6i	− 0.0729755i
$672$	0	0
$673$	357750.	0.789858	0.394929	−	0.918712i	$-0.370769\pi$
$673$	357750.	0.789858	0.394929	+	0.918712i	$0.370769\pi$
$674$	258590.i	0.569235i
$675$	0	0
$676$	150745.	0.329874
$677$	260845.i	0.569122i	0.958658	+	0.284561i	$0.0918479\pi$
$677$	260845.i	0.569122i	−0.958658	+	0.284561i	$0.908152\pi$
$678$	0	0
$679$	0	0
$680$	− 510843.i	− 1.10476i
$681$	0	0
$682$	213487.	0.458989
$683$	431562.i	0.925129i	0.886586	+	0.462564i	$0.153071\pi$
$683$	431562.i	0.925129i	−0.886586	+	0.462564i	$0.846929\pi$
$684$	0	0
$685$	−261028.	−0.556296
$686$	0	0
$687$	0	0
$688$	133608.	0.282264
$689$	− 20315.0i	− 0.0427936i
$690$	0	0
$691$	412365.	0.863626	0.431813	−	0.901963i	$-0.357874\pi$
$691$	412365.	0.863626	0.431813	+	0.901963i	$0.357874\pi$
$692$	− 52358.8i	− 0.109340i
$693$	0	0
$694$	428050.	0.888743
$695$	− 243233.i	− 0.503562i
$696$	0	0
$697$	1.34294e6	2.76434
$698$	− 97523.7i	− 0.200170i
$699$	0	0
$700$	0	0
$701$	264478.i	0.538212i	0.963111	+	0.269106i	$0.0867281\pi$
$701$	264478.i	0.538212i	−0.963111	+	0.269106i	$0.913272\pi$
$702$	0	0
$703$	−431584.	−0.873283
$704$	− 93802.3i	− 0.189264i
$705$	0	0
$706$	−1943.89	−0.00389999
$707$	0	0
$708$	0	0
$709$	781250.	1.55417	0.777083	−	0.629398i	$-0.216699\pi$
$709$	781250.	1.55417	0.777083	+	0.629398i	$0.216699\pi$
$710$	366157.i	0.726358i
$711$	0	0
$712$	−114857.	−0.226567
$713$	361743.i	0.711576i
$714$	0	0
$715$	−765423.	−1.49723
$716$	252822.i	0.493162i
$717$	0	0
$718$	−124426.	−0.241357
$719$	− 326204.i	− 0.631003i	−0.948925	−	0.315501i	$-0.897827\pi$
$719$	− 326204.i	− 0.631003i	0.948925	−	0.315501i	$-0.102173\pi$
$720$	0	0
$721$	0	0
$722$	119267.i	0.228794i
$723$	0	0
$724$	221326.	0.422236
$725$	− 995114.i	− 1.89320i
$726$	0	0
$727$	−939021.	−1.77667	−0.888335	−	0.459197i	$-0.848137\pi$
$727$	−939021.	−1.77667	−0.888335	+	0.459197i	$0.848137\pi$
$728$	0	0
$729$	0	0
$730$	530963.	0.996365
$731$	1.11207e6i	2.08113i
$732$	0	0
$733$	−463705.	−0.863045	−0.431523	−	0.902102i	$-0.642023\pi$
$733$	−463705.	−0.863045	−0.431523	+	0.902102i	$0.642023\pi$
$734$	− 366181.i	− 0.679678i
$735$	0	0
$736$	158944.	0.293418
$737$	111303.i	0.204915i
$738$	0	0
$739$	−746132.	−1.36624	−0.683120	−	0.730307i	$-0.739377\pi$
$739$	−746132.	−1.36624	−0.683120	+	0.730307i	$0.739377\pi$
$740$	− 492841.i	− 0.900001i
$741$	0	0
$742$	0	0
$743$	529606.i	0.959347i	0.877447	+	0.479673i	$0.159245\pi$
$743$	529606.i	0.959347i	−0.877447	+	0.479673i	$0.840755\pi$
$744$	0	0
$745$	1.53504e6	2.76571
$746$	695842.i	1.25035i
$747$	0	0
$748$	780755.	1.39544
$749$	0	0
$750$	0	0
$751$	273042.	0.484117	0.242058	−	0.970262i	$-0.422177\pi$
$751$	273042.	0.484117	0.242058	+	0.970262i	$0.422177\pi$
$752$	− 60453.4i	− 0.106902i
$753$	0	0
$754$	236914.	0.416724
$755$	950373.i	1.66725i
$756$	0	0
$757$	452899.	0.790332	0.395166	−	0.918610i	$-0.370687\pi$
$757$	452899.	0.790332	0.395166	+	0.918610i	$0.370687\pi$
$758$	− 568424.i	− 0.989314i
$759$	0	0
$760$	284726.	0.492946
$761$	− 160056.i	− 0.276377i	−0.990406	−	0.138188i	$-0.955872\pi$
$761$	− 160056.i	− 0.276377i	0.990406	−	0.138188i	$-0.0441279\pi$
$762$	0	0
$763$	0	0
$764$	146301.i	0.250646i
$765$	0	0
$766$	−523041.	−0.891411
$767$	− 249852.i	− 0.424710i
$768$	0	0
$769$	874888.	1.47945	0.739724	−	0.672911i	$-0.234956\pi$
$769$	874888.	1.47945	0.739724	+	0.672911i	$0.234956\pi$
$770$	0	0
$771$	0	0
$772$	−395838.	−0.664175
$773$	1.01098e6i	1.69193i	0.533235	+	0.845967i	$0.320976\pi$
$773$	1.01098e6i	1.69193i	−0.533235	+	0.845967i	$0.679024\pi$
$774$	0	0
$775$	−482498.	−0.803326
$776$	− 33660.4i	− 0.0558979i
$777$	0	0
$778$	108067.	0.178540
$779$	748509.i	1.23345i
$780$	0	0
$781$	−559622.	−0.917472
$782$	1.32295e6i	2.16337i
$783$	0	0
$784$	0	0
$785$	− 840665.i	− 1.36422i
$786$	0	0
$787$	29307.3	0.0473180	0.0236590	−	0.999720i	$-0.492468\pi$
$787$	29307.3	0.0473180	0.0236590	+	0.999720i	$0.492468\pi$
$788$	− 609397.i	− 0.981403i
$789$	0	0
$790$	−160209.	−0.256703
$791$	0	0
$792$	0	0
$793$	−17679.3	−0.0281137
$794$	264109.i	0.418931i
$795$	0	0
$796$	−207879.	−0.328084
$797$	− 279387.i	− 0.439835i	−0.975519	−	0.219917i	$-0.929421\pi$
$797$	− 279387.i	− 0.439835i	0.975519	−	0.219917i	$-0.0705788\pi$
$798$	0	0
$799$	503178.	0.788184
$800$	212001.i	0.331252i
$801$	0	0
$802$	256390.	0.398614
$803$	811506.i	1.25852i
$804$	0	0
$805$	0	0
$806$	− 114872.i	− 0.176825i
$807$	0	0
$808$	185960.	0.284837
$809$	1.19514e6i	1.82609i	0.407858	+	0.913045i	$0.366276\pi$
$809$	1.19514e6i	1.82609i	−0.407858	+	0.913045i	$0.633724\pi$
$810$	0	0
$811$	980429.	1.49065	0.745323	−	0.666703i	$-0.232295\pi$
$811$	980429.	1.49065	0.745323	+	0.666703i	$0.232295\pi$
$812$	0	0
$813$	0	0
$814$	753241.	1.13680
$815$	887326.i	1.33588i
$816$	0	0
$817$	−619831.	−0.928601
$818$	101666.i	0.151939i
$819$	0	0
$820$	−854748.	−1.27119
$821$	− 117968.i	− 0.175017i	−0.996164	−	0.0875083i	$-0.972110\pi$
$821$	− 117968.i	− 0.175017i	0.996164	−	0.0875083i	$-0.0278904\pi$
$822$	0	0
$823$	986833.	1.45695	0.728474	−	0.685074i	$-0.240230\pi$
$823$	986833.	1.45695	0.728474	+	0.685074i	$0.240230\pi$
$824$	− 184791.i	− 0.272162i
$825$	0	0
$826$	0	0
$827$	1.06396e6i	1.55565i	0.628479	+	0.777826i	$0.283678\pi$
$827$	1.06396e6i	1.55565i	−0.628479	+	0.777826i	$0.716322\pi$
$828$	0	0
$829$	−584733.	−0.850842	−0.425421	−	0.904996i	$-0.639874\pi$
$829$	−584733.	−0.850842	−0.425421	+	0.904996i	$0.639874\pi$
$830$	520549.i	0.755623i
$831$	0	0
$832$	−50472.7	−0.0729138
$833$	0	0
$834$	0	0
$835$	−1.87589e6	−2.69051
$836$	435165.i	0.622647i
$837$	0	0
$838$	−300975.	−0.428591
$839$	433601.i	0.615979i	0.951390	+	0.307990i	$0.0996562\pi$
$839$	433601.i	0.615979i	−0.951390	+	0.307990i	$0.900344\pi$
$840$	0	0
$841$	−14689.5	−0.0207689
$842$	− 418190.i	− 0.589860i
$843$	0	0
$844$	119849.	0.168248
$845$	− 798589.i	− 1.11843i
$846$	0	0
$847$	0	0
$848$	− 13189.0i	− 0.0183409i
$849$	0	0
$850$	−1.76457e6	−2.44231
$851$	1.27633e6i	1.76240i
$852$	0	0
$853$	−673527.	−0.925671	−0.462836	−	0.886444i	$-0.653168\pi$
$853$	−673527.	−0.925671	−0.462836	+	0.886444i	$0.653168\pi$
$854$	0	0
$855$	0	0
$856$	−8536.89	−0.0116507
$857$	− 1.38444e6i	− 1.88501i	−0.334196	−	0.942504i	$-0.608465\pi$
$857$	− 1.38444e6i	− 1.88501i	0.334196	−	0.942504i	$-0.391535\pi$
$858$	0	0
$859$	−844486.	−1.14447	−0.572237	−	0.820088i	$-0.693924\pi$
$859$	−844486.	−1.14447	−0.572237	+	0.820088i	$0.693924\pi$
$860$	− 707806.i	− 0.957012i
$861$	0	0
$862$	−145925.	−0.196388
$863$	450856.i	0.605363i	0.953092	+	0.302682i	$0.0978819\pi$
$863$	450856.i	0.605363i	−0.953092	+	0.302682i	$0.902118\pi$
$864$	0	0
$865$	−277378.	−0.370714
$866$	715588.i	0.954173i
$867$	0	0
$868$	0	0
$869$	− 244857.i	− 0.324245i
$870$	0	0
$871$	59889.6	0.0789433
$872$	− 197729.i	− 0.260038i
$873$	0	0
$874$	−737367.	−0.965297
$875$	0	0
$876$	0	0
$877$	620239.	0.806417	0.403209	−	0.915108i	$-0.367895\pi$
$877$	620239.	0.806417	0.403209	+	0.915108i	$0.367895\pi$
$878$	− 785741.i	− 1.01927i
$879$	0	0
$880$	−496930.	−0.641697
$881$	− 294871.i	− 0.379909i	−0.981793	−	0.189955i	$-0.939166\pi$
$881$	− 294871.i	− 0.379909i	0.981793	−	0.189955i	$-0.0608341\pi$
$882$	0	0
$883$	−982084.	−1.25958	−0.629792	−	0.776764i	$-0.716860\pi$
$883$	−982084.	−1.25958	−0.629792	+	0.776764i	$0.716860\pi$
$884$	− 420105.i	− 0.537592i
$885$	0	0
$886$	−100736.	−0.128326
$887$	901828.i	1.14624i	0.819470	+	0.573121i	$0.194268\pi$
$887$	901828.i	1.14624i	−0.819470	+	0.573121i	$0.805732\pi$
$888$	0	0
$889$	0	0
$890$	608470.i	0.768173i
$891$	0	0
$892$	−335727.	−0.421946
$893$	280454.i	0.351688i
$894$	0	0
$895$	1.33936e6	1.67206
$896$	0	0
$897$	0	0
$898$	565283.	0.700992
$899$	350060.i	0.433134i
$900$	0	0
$901$	109777.	0.135227
$902$	− 1.30637e6i	− 1.60565i
$903$	0	0
$904$	−96703.2	−0.118333
$905$	− 1.17250e6i	− 1.43158i
$906$	0	0
$907$	−1.06695e6	−1.29696	−0.648481	−	0.761230i	$-0.724596\pi$
$907$	−1.06695e6	−1.29696	−0.648481	+	0.761230i	$0.724596\pi$
$908$	749699.i	0.909317i
$909$	0	0
$910$	0	0
$911$	524695.i	0.632223i	0.948722	+	0.316111i	$0.102377\pi$
$911$	524695.i	0.632223i	−0.948722	+	0.316111i	$0.897623\pi$
$912$	0	0
$913$	−795589.	−0.954437
$914$	263077.i	0.314912i
$915$	0	0
$916$	−184873.	−0.220334
$917$	0	0
$918$	0	0
$919$	939251.	1.11212	0.556058	−	0.831143i	$-0.312313\pi$
$919$	939251.	1.11212	0.556058	+	0.831143i	$0.312313\pi$
$920$	− 842024.i	− 0.994830i
$921$	0	0
$922$	−826110.	−0.971798
$923$	301119.i	0.353455i
$924$	0	0
$925$	−1.70239e6	−1.98964
$926$	− 782444.i	− 0.912497i
$927$	0	0
$928$	153810.	0.178603
$929$	233874.i	0.270989i	0.990778	+	0.135494i	$0.0432622\pi$
$929$	233874.i	0.270989i	−0.990778	+	0.135494i	$0.956738\pi$
$930$	0	0
$931$	0	0
$932$	9361.93i	0.0107779i
$933$	0	0
$934$	−311559.	−0.357147
$935$	− 4.13615e6i	− 4.73122i
$936$	0	0
$937$	−805148.	−0.917057	−0.458529	−	0.888680i	$-0.651623\pi$
$937$	−805148.	−0.917057	−0.458529	+	0.888680i	$0.651623\pi$
$938$	0	0
$939$	0	0
$940$	−320259.	−0.362448
$941$	− 1.54723e6i	− 1.74733i	−0.486528	−	0.873665i	$-0.661737\pi$
$941$	− 1.54723e6i	− 1.74733i	0.486528	−	0.873665i	$-0.338263\pi$
$942$	0	0
$943$	2.21358e6	2.48927
$944$	− 162209.i	− 0.182025i
$945$	0	0
$946$	1.08179e6	1.20881
$947$	− 458912.i	− 0.511717i	−0.966714	−	0.255858i	$-0.917642\pi$
$947$	− 458912.i	− 0.511717i	0.966714	−	0.255858i	$-0.0823581\pi$
$948$	0	0
$949$	436651.	0.484844
$950$	− 983509.i	− 1.08976i
$951$	0	0
$952$	0	0
$953$	− 1.69252e6i	− 1.86358i	−0.363001	−	0.931789i	$-0.618248\pi$
$953$	− 1.69252e6i	− 1.86358i	0.363001	−	0.931789i	$-0.381752\pi$
$954$	0	0
$955$	775049.	0.849811
$956$	− 548302.i	− 0.599935i
$957$	0	0
$958$	94952.2	0.103460
$959$	0	0
$960$	0	0
$961$	−753789.	−0.816212
$962$	− 405300.i	− 0.437952i
$963$	0	0
$964$	833235.	0.896630
$965$	2.09700e6i	2.25187i
$966$	0	0
$967$	−1.52926e6	−1.63542	−0.817709	−	0.575632i	$-0.804756\pi$
$967$	−1.52926e6	−1.63542	−0.817709	+	0.575632i	$0.804756\pi$
$968$	− 428202.i	− 0.456981i
$969$	0	0
$970$	−178320.	−0.189521
$971$	230724.i	0.244712i	0.992486	+	0.122356i	$0.0390449\pi$
$971$	230724.i	0.244712i	−0.992486	+	0.122356i	$0.960955\pi$
$972$	0	0
$973$	0	0
$974$	− 906005.i	− 0.955020i
$975$	0	0
$976$	−11477.8	−0.0120492
$977$	284089.i	0.297622i	0.988866	+	0.148811i	$0.0475446\pi$
$977$	284089.i	0.297622i	−0.988866	+	0.148811i	$0.952455\pi$
$978$	0	0
$979$	−929964.	−0.970288
$980$	0	0
$981$	0	0
$982$	−223815.	−0.232095
$983$	1.36297e6i	1.41052i	0.708950	+	0.705259i	$0.249169\pi$
$983$	1.36297e6i	1.41052i	−0.708950	+	0.705259i	$0.750831\pi$
$984$	0	0
$985$	−3.22836e6	−3.32743
$986$	1.28022e6i	1.31684i
$987$	0	0
$988$	234152.	0.239874
$989$	1.83304e6i	1.87404i
$990$	0	0
$991$	1.19040e6	1.21212	0.606061	−	0.795418i	$-0.292749\pi$
$991$	1.19040e6	1.21212	0.606061	+	0.795418i	$0.292749\pi$
$992$	− 74577.4i	− 0.0757851i
$993$	0	0
$994$	0	0
$995$	1.10127e6i	1.11236i
$996$	0	0
$997$	−371292.	−0.373530	−0.186765	−	0.982405i	$-0.559800\pi$
$997$	−371292.	−0.373530	−0.186765	+	0.982405i	$0.559800\pi$
$998$	− 307652.i	− 0.308887i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.h.197.3		6
3.2	odd	2	inner	882.5.b.h.197.4		6
7.2	even	3		126.5.s.b.53.3	✓	12
7.4	even	3		126.5.s.b.107.4	yes	12
7.6	odd	2		882.5.b.e.197.1		6
21.2	odd	6		126.5.s.b.53.4	yes	12
21.11	odd	6		126.5.s.b.107.3	yes	12
21.20	even	2		882.5.b.e.197.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.5.s.b.53.3	✓	12	7.2	even	3
126.5.s.b.53.4	yes	12	21.2	odd	6
126.5.s.b.107.3	yes	12	21.11	odd	6
126.5.s.b.107.4	yes	12	7.4	even	3
882.5.b.e.197.1		6	7.6	odd	2
882.5.b.e.197.6		6	21.20	even	2
882.5.b.h.197.3		6	1.1	even	1	trivial
882.5.b.h.197.4		6	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	882.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +42.3810i q^{5} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +42.3810i q^{5} +22.6274i q^{8} +119.872 q^{10} +183.208i q^{11} +98.5795 q^{13} +64.0000 q^{16} +532.698i q^{17} -296.907 q^{19} -339.048i q^{20} +518.190 q^{22} +878.048i q^{23} -1171.15 q^{25} -278.825i q^{26} +849.688i q^{29} +411.986 q^{31} -181.019i q^{32} +1506.70 q^{34} +1453.60 q^{37} +839.780i q^{38} -958.973 q^{40} -2521.02i q^{41} +2087.63 q^{43} -1465.66i q^{44} +2483.49 q^{46} -944.584i q^{47} +3312.52i q^{50} -788.636 q^{52} -206.078i q^{53} -7764.53 q^{55} +2403.28 q^{58} -2534.52i q^{59} -179.340 q^{61} -1165.27i q^{62} -512.000 q^{64} +4177.90i q^{65} +607.526 q^{67} -4261.58i q^{68} +3054.58i q^{71} +4429.43 q^{73} -4111.40i q^{74} +2375.26 q^{76} -1336.50 q^{79} +2712.39i q^{80} -7130.52 q^{82} +4342.55i q^{83} -22576.3 q^{85} -5904.70i q^{86} -4145.52 q^{88} +5076.01i q^{89} -7024.38i q^{92} -2671.69 q^{94} -12583.2i q^{95} -1487.59 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 48 q^{4}+O(q^{10}) \) 6 * q - 48 * q^4	\( 6 q - 48 q^{4} + 104 q^{10} + 86 q^{13} + 384 q^{16} + 174 q^{19} + 472 q^{22} - 2170 q^{25} + 1682 q^{31} + 1744 q^{34} + 6018 q^{37} - 832 q^{40} - 2130 q^{43} - 656 q^{46} - 688 q^{52} - 12680 q^{55} + 7040 q^{58} - 20632 q^{61} - 3072 q^{64} - 74 q^{67} - 2942 q^{73} - 1392 q^{76} + 10526 q^{79} - 25464 q^{82} - 33056 q^{85} - 3776 q^{88} - 11640 q^{94} - 46436 q^{97}+O(q^{100}) \) 6 * q - 48 * q^4 + 104 * q^10 + 86 * q^13 + 384 * q^16 + 174 * q^19 + 472 * q^22 - 2170 * q^25 + 1682 * q^31 + 1744 * q^34 + 6018 * q^37 - 832 * q^40 - 2130 * q^43 - 656 * q^46 - 688 * q^52 - 12680 * q^55 + 7040 * q^58 - 20632 * q^61 - 3072 * q^64 - 74 * q^67 - 2942 * q^73 - 1392 * q^76 + 10526 * q^79 - 25464 * q^82 - 33056 * q^85 - 3776 * q^88 - 11640 * q^94 - 46436 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more