LMFDB - Embedded newform 882.5.b.i.197.5

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:	$8$
Coefficient field:	8.0.4494128644096.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 967x^{4} + 279841 $ x^8 + 967*x^4 + 279841
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.5
Root			$-3.01913 + 3.72624i$ of defining polynomial
Character	$\chi$	$=$	882.197
Dual form			882.5.b.i.197.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} -29.6182i q^{5} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} -29.6182i q^{5} -22.6274i q^{8} +83.7729 q^{10} -12.1879i q^{11} +89.7640 q^{13} +64.0000 q^{16} -54.0909i q^{17} +208.352 q^{19} +236.945i q^{20} +34.4727 q^{22} +702.170i q^{23} -252.236 q^{25} +253.891i q^{26} -746.345i q^{29} +948.732 q^{31} +181.019i q^{32} +152.992 q^{34} +1025.96 q^{37} +589.309i q^{38} -670.183 q^{40} +2862.53i q^{41} -2827.38 q^{43} +97.5035i q^{44} -1986.04 q^{46} -684.691i q^{47} -713.432i q^{50} -718.112 q^{52} -2854.68i q^{53} -360.985 q^{55} +2110.98 q^{58} -913.636i q^{59} +973.081 q^{61} +2683.42i q^{62} -512.000 q^{64} -2658.64i q^{65} +3436.14 q^{67} +432.727i q^{68} -2055.75i q^{71} +8107.87 q^{73} +2901.86i q^{74} -1666.82 q^{76} -6146.51 q^{79} -1895.56i q^{80} -8096.45 q^{82} +44.6917i q^{83} -1602.07 q^{85} -7997.04i q^{86} -275.782 q^{88} -12429.1i q^{89} -5617.36i q^{92} +1936.60 q^{94} -6171.01i q^{95} -1257.36 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} + 512 q^{16} - 640 q^{22} - 1560 q^{25} - 1408 q^{37} - 10256 q^{43} - 320 q^{46} - 512 q^{58} - 4096 q^{64} - 9600 q^{67} + 17680 q^{79} - 32048 q^{85} + 5120 q^{88}+O(q^{100}) $ 8 * q - 64 * q^4 + 512 * q^16 - 640 * q^22 - 1560 * q^25 - 1408 * q^37 - 10256 * q^43 - 320 * q^46 - 512 * q^58 - 4096 * q^64 - 9600 * q^67 + 17680 * q^79 - 32048 * q^85 + 5120 * q^88

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
$5$	− 29.6182i	− 1.18473i	−0.805671	−	0.592364i	$-0.798195\pi$
$5$	− 29.6182i	− 1.18473i	0.805671	−	0.592364i	$-0.201805\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 22.6274i	− 0.353553i
$9$	0	0
$10$	83.7729	0.837729
$11$	− 12.1879i	− 0.100727i	−0.998731	−	0.0503634i	$-0.983962\pi$
$11$	− 12.1879i	− 0.100727i	0.998731	−	0.0503634i	$-0.0160380\pi$
$12$	0	0
$13$	89.7640	0.531148	0.265574	−	0.964091i	$-0.414439\pi$
$13$	89.7640	0.531148	0.265574	+	0.964091i	$0.414439\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	0.250000
$17$	− 54.0909i	− 0.187166i	−0.995611	−	0.0935828i	$-0.970168\pi$
$17$	− 54.0909i	− 0.187166i	0.995611	−	0.0935828i	$-0.0298320\pi$
$18$	0	0
$19$	208.352	0.577153	0.288576	−	0.957457i	$-0.406818\pi$
$19$	208.352	0.577153	0.288576	+	0.957457i	$0.406818\pi$
$20$	236.945i	0.592364i
$21$	0	0
$22$	34.4727	0.0712246
$23$	702.170i	1.32735i	0.748020	+	0.663677i	$0.231005\pi$
$23$	702.170i	1.32735i	−0.748020	+	0.663677i	$0.768995\pi$
$24$	0	0
$25$	−252.236	−0.403578
$26$	253.891i	0.375578i
$27$	0	0
$28$	0	0
$29$	− 746.345i	− 0.887449i	−0.896163	−	0.443725i	$-0.853657\pi$
$29$	− 746.345i	− 0.887449i	0.896163	−	0.443725i	$-0.146343\pi$
$30$	0	0
$31$	948.732	0.987234	0.493617	−	0.869679i	$-0.335675\pi$
$31$	948.732	0.987234	0.493617	+	0.869679i	$0.335675\pi$
$32$	181.019i	0.176777i
$33$	0	0
$34$	152.992	0.132346
$35$	0	0
$36$	0	0
$37$	1025.96	0.749425	0.374713	−	0.927141i	$-0.377741\pi$
$37$	1025.96	0.749425	0.374713	+	0.927141i	$0.377741\pi$
$38$	589.309i	0.408109i
$39$	0	0
$40$	−670.183	−0.418864
$41$	2862.53i	1.70287i	0.524459	+	0.851436i	$0.324268\pi$
$41$	2862.53i	1.70287i	−0.524459	+	0.851436i	$0.675732\pi$
$42$	0	0
$43$	−2827.38	−1.52914	−0.764570	−	0.644540i	$-0.777049\pi$
$43$	−2827.38	−1.52914	−0.764570	+	0.644540i	$0.777049\pi$
$44$	97.5035i	0.0503634i
$45$	0	0
$46$	−1986.04	−0.938580
$47$	− 684.691i	− 0.309955i	−0.987918	−	0.154978i	$-0.950469\pi$
$47$	− 684.691i	− 0.309955i	0.987918	−	0.154978i	$-0.0495305\pi$
$48$	0	0
$49$	0	0
$50$	− 713.432i	− 0.285373i
$51$	0	0
$52$	−718.112	−0.265574
$53$	− 2854.68i	− 1.01626i	−0.861280	−	0.508131i	$-0.830337\pi$
$53$	− 2854.68i	− 1.01626i	0.861280	−	0.508131i	$-0.169663\pi$
$54$	0	0
$55$	−360.985	−0.119334
$56$	0	0
$57$	0	0
$58$	2110.98	0.627521
$59$	− 913.636i	− 0.262464i	−0.991352	−	0.131232i	$-0.958107\pi$
$59$	− 913.636i	− 0.262464i	0.991352	−	0.131232i	$-0.0418932\pi$
$60$	0	0
$61$	973.081	0.261511	0.130755	−	0.991415i	$-0.458260\pi$
$61$	973.081	0.261511	0.130755	+	0.991415i	$0.458260\pi$
$62$	2683.42i	0.698080i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 2658.64i	− 0.629265i
$66$	0	0
$67$	3436.14	0.765459	0.382729	−	0.923860i	$-0.374984\pi$
$67$	3436.14	0.765459	0.382729	+	0.923860i	$0.374984\pi$
$68$	432.727i	0.0935828i
$69$	0	0
$70$	0	0
$71$	− 2055.75i	− 0.407807i	−0.978991	−	0.203903i	$-0.934637\pi$
$71$	− 2055.75i	− 0.407807i	0.978991	−	0.203903i	$-0.0653628\pi$
$72$	0	0
$73$	8107.87	1.52146	0.760731	−	0.649068i	$-0.224841\pi$
$73$	8107.87	1.52146	0.760731	+	0.649068i	$0.224841\pi$
$74$	2901.86i	0.529924i
$75$	0	0
$76$	−1666.82	−0.288576
$77$	0	0
$78$	0	0
$79$	−6146.51	−0.984859	−0.492430	−	0.870352i	$-0.663891\pi$
$79$	−6146.51	−0.984859	−0.492430	+	0.870352i	$0.663891\pi$
$80$	− 1895.56i	− 0.296182i
$81$	0	0
$82$	−8096.45	−1.20411
$83$	44.6917i	0.00648740i	0.999995	+	0.00324370i	$0.00103250\pi$
$83$	44.6917i	0.00648740i	−0.999995	+	0.00324370i	$0.998967\pi$
$84$	0	0
$85$	−1602.07	−0.221740
$86$	− 7997.04i	− 1.08127i
$87$	0	0
$88$	−275.782	−0.0356123
$89$	− 12429.1i	− 1.56914i	−0.620041	−	0.784569i	$-0.712884\pi$
$89$	− 12429.1i	− 1.56914i	0.620041	−	0.784569i	$-0.287116\pi$
$90$	0	0
$91$	0	0
$92$	− 5617.36i	− 0.663677i
$93$	0	0
$94$	1936.60	0.219171
$95$	− 6171.01i	− 0.683769i
$96$	0	0
$97$	−1257.36	−0.133634	−0.0668171	−	0.997765i	$-0.521284\pi$
$97$	−1257.36	−0.133634	−0.0668171	+	0.997765i	$0.521284\pi$
$98$	0	0
$99$	0	0
$100$	2017.89	0.201789
$101$	− 6684.93i	− 0.655321i	−0.944796	−	0.327660i	$-0.893740\pi$
$101$	− 6684.93i	− 0.655321i	0.944796	−	0.327660i	$-0.106260\pi$
$102$	0	0
$103$	−16606.1	−1.56529	−0.782644	−	0.622470i	$-0.786129\pi$
$103$	−16606.1	−1.56529	−0.782644	+	0.622470i	$0.786129\pi$
$104$	− 2031.13i	− 0.187789i
$105$	0	0
$106$	8074.25	0.718606
$107$	− 21604.8i	− 1.88704i	−0.331311	−	0.943522i	$-0.607491\pi$
$107$	− 21604.8i	− 1.88704i	0.331311	−	0.943522i	$-0.392509\pi$
$108$	0	0
$109$	10467.2	0.881006	0.440503	−	0.897751i	$-0.354800\pi$
$109$	10467.2	0.881006	0.440503	+	0.897751i	$0.354800\pi$
$110$	− 1021.02i	− 0.0843817i
$111$	0	0
$112$	0	0
$113$	− 6651.97i	− 0.520947i	−0.965481	−	0.260474i	$-0.916121\pi$
$113$	− 6651.97i	− 0.520947i	0.965481	−	0.260474i	$-0.0838787\pi$
$114$	0	0
$115$	20797.0	1.57255
$116$	5970.76i	0.443725i
$117$	0	0
$118$	2584.15	0.185590
$119$	0	0
$120$	0	0
$121$	14492.5	0.989854
$122$	2752.29i	0.184916i
$123$	0	0
$124$	−7589.85	−0.493617
$125$	− 11040.6i	− 0.706597i
$126$	0	0
$127$	4245.60	0.263228	0.131614	−	0.991301i	$-0.457984\pi$
$127$	4245.60	0.263228	0.131614	+	0.991301i	$0.457984\pi$
$128$	− 1448.15i	− 0.0883883i
$129$	0	0
$130$	7519.78	0.444958
$131$	5664.58i	0.330085i	0.986286	+	0.165042i	$0.0527761\pi$
$131$	5664.58i	0.330085i	−0.986286	+	0.165042i	$0.947224\pi$
$132$	0	0
$133$	0	0
$134$	9718.88i	0.541261i
$135$	0	0
$136$	−1223.94	−0.0661731
$137$	− 18646.0i	− 0.993448i	−0.867909	−	0.496724i	$-0.834536\pi$
$137$	− 18646.0i	− 0.993448i	0.867909	−	0.496724i	$-0.165464\pi$
$138$	0	0
$139$	−19179.6	−0.992680	−0.496340	−	0.868128i	$-0.665323\pi$
$139$	−19179.6	−0.992680	−0.496340	+	0.868128i	$0.665323\pi$
$140$	0	0
$141$	0	0
$142$	5814.55	0.288363
$143$	− 1094.04i	− 0.0535008i
$144$	0	0
$145$	−22105.4	−1.05138
$146$	22932.5i	1.07584i
$147$	0	0
$148$	−8207.71	−0.374713
$149$	− 24865.0i	− 1.11999i	−0.828495	−	0.559997i	$-0.810802\pi$
$149$	− 24865.0i	− 1.11999i	0.828495	−	0.559997i	$-0.189198\pi$
$150$	0	0
$151$	30568.0	1.34064	0.670321	−	0.742072i	$-0.266157\pi$
$151$	30568.0	1.34064	0.670321	+	0.742072i	$0.266157\pi$
$152$	− 4714.47i	− 0.204054i
$153$	0	0
$154$	0	0
$155$	− 28099.7i	− 1.16960i
$156$	0	0
$157$	21546.1	0.874118	0.437059	−	0.899433i	$-0.356020\pi$
$157$	21546.1	0.874118	0.437059	+	0.899433i	$0.356020\pi$
$158$	− 17384.9i	− 0.696401i
$159$	0	0
$160$	5361.46	0.209432
$161$	0	0
$162$	0	0
$163$	38834.8	1.46166	0.730828	−	0.682561i	$-0.239134\pi$
$163$	38834.8	1.46166	0.730828	+	0.682561i	$0.239134\pi$
$164$	− 22900.2i	− 0.851436i
$165$	0	0
$166$	−126.407	−0.00458729
$167$	32444.5i	1.16334i	0.813424	+	0.581671i	$0.197601\pi$
$167$	32444.5i	1.16334i	−0.813424	+	0.581671i	$0.802399\pi$
$168$	0	0
$169$	−20503.4	−0.717882
$170$	− 4531.35i	− 0.156794i
$171$	0	0
$172$	22619.1	0.764570
$173$	47953.3i	1.60224i	0.598506	+	0.801118i	$0.295761\pi$
$173$	47953.3i	1.60224i	−0.598506	+	0.801118i	$0.704239\pi$
$174$	0	0
$175$	0	0
$176$	− 780.028i	− 0.0251817i
$177$	0	0
$178$	35154.9	1.10955
$179$	− 32643.0i	− 1.01879i	−0.860533	−	0.509395i	$-0.829869\pi$
$179$	− 32643.0i	− 1.01879i	0.860533	−	0.509395i	$-0.170131\pi$
$180$	0	0
$181$	1100.28	0.0335850	0.0167925	−	0.999859i	$-0.494655\pi$
$181$	1100.28	0.0335850	0.0167925	+	0.999859i	$0.494655\pi$
$182$	0	0
$183$	0	0
$184$	15888.3	0.469290
$185$	− 30387.2i	− 0.887865i
$186$	0	0
$187$	−659.256	−0.0188526
$188$	5477.53i	0.154978i
$189$	0	0
$190$	17454.3	0.483497
$191$	25712.8i	0.704827i	0.935844	+	0.352414i	$0.114639\pi$
$191$	25712.8i	0.704827i	−0.935844	+	0.352414i	$0.885361\pi$
$192$	0	0
$193$	−57332.9	−1.53918	−0.769589	−	0.638539i	$-0.779539\pi$
$193$	−57332.9	−1.53918	−0.769589	+	0.638539i	$0.779539\pi$
$194$	− 3556.36i	− 0.0944937i
$195$	0	0
$196$	0	0
$197$	− 6098.06i	− 0.157130i	−0.996909	−	0.0785651i	$-0.974966\pi$
$197$	− 6098.06i	− 0.157130i	0.996909	−	0.0785651i	$-0.0250338\pi$
$198$	0	0
$199$	43535.0	1.09934	0.549671	−	0.835381i	$-0.314753\pi$
$199$	43535.0	1.09934	0.549671	+	0.835381i	$0.314753\pi$
$200$	5707.46i	0.142686i
$201$	0	0
$202$	18907.8	0.463382
$203$	0	0
$204$	0	0
$205$	84782.8	2.01744
$206$	− 46969.2i	− 1.10683i
$207$	0	0
$208$	5744.89	0.132787
$209$	− 2539.38i	− 0.0581348i
$210$	0	0
$211$	53968.1	1.21219	0.606097	−	0.795391i	$-0.292734\pi$
$211$	53968.1	1.21219	0.606097	+	0.795391i	$0.292734\pi$
$212$	22837.4i	0.508131i
$213$	0	0
$214$	61107.5	1.33434
$215$	83741.9i	1.81161i
$216$	0	0
$217$	0	0
$218$	29605.8i	0.622966i
$219$	0	0
$220$	2887.88	0.0596669
$221$	− 4855.41i	− 0.0994126i
$222$	0	0
$223$	−89644.3	−1.80266	−0.901328	−	0.433136i	$-0.857407\pi$
$223$	−89644.3	−1.80266	−0.901328	+	0.433136i	$0.857407\pi$
$224$	0	0
$225$	0	0
$226$	18814.6	0.368365
$227$	− 30223.9i	− 0.586541i	−0.956029	−	0.293271i	$-0.905256\pi$
$227$	− 30223.9i	− 0.586541i	0.956029	−	0.293271i	$-0.0947437\pi$
$228$	0	0
$229$	−4926.57	−0.0939451	−0.0469725	−	0.998896i	$-0.514957\pi$
$229$	−4926.57	−0.0939451	−0.0469725	+	0.998896i	$0.514957\pi$
$230$	58822.8i	1.11196i
$231$	0	0
$232$	−16887.9	−0.313761
$233$	− 77307.2i	− 1.42399i	−0.702182	−	0.711997i	$-0.747791\pi$
$233$	− 77307.2i	− 1.42399i	0.702182	−	0.711997i	$-0.252209\pi$
$234$	0	0
$235$	−20279.3	−0.367212
$236$	7309.09i	0.131232i
$237$	0	0
$238$	0	0
$239$	− 19508.6i	− 0.341532i	−0.985312	−	0.170766i	$-0.945376\pi$
$239$	− 19508.6i	− 0.341532i	0.985312	−	0.170766i	$-0.0546242\pi$
$240$	0	0
$241$	101947.	1.75525	0.877625	−	0.479348i	$-0.159127\pi$
$241$	101947.	1.75525	0.877625	+	0.479348i	$0.159127\pi$
$242$	40990.9i	0.699933i
$243$	0	0
$244$	−7784.65	−0.130755
$245$	0	0
$246$	0	0
$247$	18702.5	0.306553
$248$	− 21467.3i	− 0.349040i
$249$	0	0
$250$	31227.5	0.499640
$251$	− 34172.7i	− 0.542416i	−0.962521	−	0.271208i	$-0.912577\pi$
$251$	− 34172.7i	− 0.542416i	0.962521	−	0.271208i	$-0.0874231\pi$
$252$	0	0
$253$	8558.00	0.133700
$254$	12008.4i	0.186130i
$255$	0	0
$256$	4096.00	0.0625000
$257$	29325.2i	0.443991i	0.975048	+	0.221996i	$0.0712571\pi$
$257$	29325.2i	0.443991i	−0.975048	+	0.221996i	$0.928743\pi$
$258$	0	0
$259$	0	0
$260$	21269.2i	0.314633i
$261$	0	0
$262$	−16021.9	−0.233405
$263$	− 40944.8i	− 0.591953i	−0.955195	−	0.295977i	$-0.904355\pi$
$263$	− 40944.8i	− 0.591953i	0.955195	−	0.295977i	$-0.0956451\pi$
$264$	0	0
$265$	−84550.4	−1.20399
$266$	0	0
$267$	0	0
$268$	−27489.2	−0.382729
$269$	− 15448.7i	− 0.213495i	−0.994286	−	0.106748i	$-0.965956\pi$
$269$	− 15448.7i	− 0.213495i	0.994286	−	0.106748i	$-0.0340437\pi$
$270$	0	0
$271$	−64041.0	−0.872005	−0.436003	−	0.899945i	$-0.643606\pi$
$271$	−64041.0	−0.872005	−0.436003	+	0.899945i	$0.643606\pi$
$272$	− 3461.82i	− 0.0467914i
$273$	0	0
$274$	52738.9	0.702474
$275$	3074.24i	0.0406511i
$276$	0	0
$277$	80158.9	1.04470	0.522350	−	0.852731i	$-0.325055\pi$
$277$	80158.9	1.04470	0.522350	+	0.852731i	$0.325055\pi$
$278$	− 54248.0i	− 0.701930i
$279$	0	0
$280$	0	0
$281$	112942.i	1.43035i	0.698943	+	0.715177i	$0.253654\pi$
$281$	112942.i	1.43035i	−0.698943	+	0.715177i	$0.746346\pi$
$282$	0	0
$283$	−10235.6	−0.127803	−0.0639013	−	0.997956i	$-0.520354\pi$
$283$	−10235.6	−0.127803	−0.0639013	+	0.997956i	$0.520354\pi$
$284$	16446.0i	0.203903i
$285$	0	0
$286$	3094.41	0.0378308
$287$	0	0
$288$	0	0
$289$	80595.2	0.964969
$290$	− 62523.4i	− 0.743441i
$291$	0	0
$292$	−64862.9	−0.760731
$293$	− 158722.i	− 1.84885i	−0.381360	−	0.924427i	$-0.624544\pi$
$293$	− 158722.i	− 1.84885i	0.381360	−	0.924427i	$-0.375456\pi$
$294$	0	0
$295$	−27060.2	−0.310948
$296$	− 23214.9i	− 0.264962i
$297$	0	0
$298$	70328.8	0.791955
$299$	63029.5i	0.705020i
$300$	0	0
$301$	0	0
$302$	86459.2i	0.947976i
$303$	0	0
$304$	13334.5	0.144288
$305$	− 28820.9i	− 0.309819i
$306$	0	0
$307$	−151461.	−1.60703	−0.803517	−	0.595282i	$-0.797040\pi$
$307$	−151461.	−1.60703	−0.803517	+	0.595282i	$0.797040\pi$
$308$	0	0
$309$	0	0
$310$	79478.0	0.827034
$311$	3971.61i	0.0410626i	0.999789	+	0.0205313i	$0.00653577\pi$
$311$	3971.61i	0.0410626i	−0.999789	+	0.0205313i	$0.993464\pi$
$312$	0	0
$313$	177954.	1.81643	0.908215	−	0.418503i	$-0.137445\pi$
$313$	177954.	1.81643	0.908215	+	0.418503i	$0.137445\pi$
$314$	60941.7i	0.618095i
$315$	0	0
$316$	49172.1	0.492430
$317$	− 98283.0i	− 0.978047i	−0.872270	−	0.489024i	$-0.837353\pi$
$317$	− 98283.0i	− 0.978047i	0.872270	−	0.489024i	$-0.162647\pi$
$318$	0	0
$319$	−9096.40	−0.0893899
$320$	15164.5i	0.148091i
$321$	0	0
$322$	0	0
$323$	− 11270.0i	− 0.108023i
$324$	0	0
$325$	−22641.7	−0.214360
$326$	109841.i	1.03355i
$327$	0	0
$328$	64771.6	0.602056
$329$	0	0
$330$	0	0
$331$	−199886.	−1.82442	−0.912212	−	0.409718i	$-0.865627\pi$
$331$	−199886.	−1.82442	−0.912212	+	0.409718i	$0.865627\pi$
$332$	− 357.534i	− 0.00324370i
$333$	0	0
$334$	−91766.8	−0.822608
$335$	− 101772.i	− 0.906860i
$336$	0	0
$337$	−23590.0	−0.207715	−0.103857	−	0.994592i	$-0.533119\pi$
$337$	−23590.0	−0.207715	−0.103857	+	0.994592i	$0.533119\pi$
$338$	− 57992.5i	− 0.507619i
$339$	0	0
$340$	12816.6	0.110870
$341$	− 11563.1i	− 0.0994409i
$342$	0	0
$343$	0	0
$344$	63976.3i	0.540633i
$345$	0	0
$346$	−135632.	−1.13295
$347$	− 91756.5i	− 0.762040i	−0.924567	−	0.381020i	$-0.875573\pi$
$347$	− 91756.5i	− 0.762040i	0.924567	−	0.381020i	$-0.124427\pi$
$348$	0	0
$349$	118272.	0.971030	0.485515	−	0.874228i	$-0.338632\pi$
$349$	118272.	0.971030	0.485515	+	0.874228i	$0.338632\pi$
$350$	0	0
$351$	0	0
$352$	2206.25	0.0178061
$353$	14663.6i	0.117677i	0.998268	+	0.0588385i	$0.0187397\pi$
$353$	14663.6i	0.117677i	−0.998268	+	0.0588385i	$0.981260\pi$
$354$	0	0
$355$	−60887.7	−0.483139
$356$	99433.1i	0.784569i
$357$	0	0
$358$	92328.4	0.720393
$359$	99010.8i	0.768234i	0.923284	+	0.384117i	$0.125494\pi$
$359$	99010.8i	0.768234i	−0.923284	+	0.384117i	$0.874506\pi$
$360$	0	0
$361$	−86910.4	−0.666895
$362$	3112.06i	0.0237482i
$363$	0	0
$364$	0	0
$365$	− 240140.i	− 1.80252i
$366$	0	0
$367$	99418.1	0.738131	0.369065	−	0.929403i	$-0.379678\pi$
$367$	99418.1	0.738131	0.369065	+	0.929403i	$0.379678\pi$
$368$	44938.9i	0.331838i
$369$	0	0
$370$	85947.9	0.627815
$371$	0	0
$372$	0	0
$373$	−27662.0	−0.198822	−0.0994112	−	0.995046i	$-0.531696\pi$
$373$	−27662.0	−0.198822	−0.0994112	+	0.995046i	$0.531696\pi$
$374$	− 1864.66i	− 0.0133308i
$375$	0	0
$376$	−15492.8	−0.109586
$377$	− 66994.8i	− 0.471366i
$378$	0	0
$379$	51727.6	0.360117	0.180059	−	0.983656i	$-0.442371\pi$
$379$	51727.6	0.360117	0.180059	+	0.983656i	$0.442371\pi$
$380$	49368.1i	0.341884i
$381$	0	0
$382$	−72726.8	−0.498388
$383$	− 72865.5i	− 0.496734i	−0.968666	−	0.248367i	$-0.920106\pi$
$383$	− 72865.5i	− 0.496734i	0.968666	−	0.248367i	$-0.0798939\pi$
$384$	0	0
$385$	0	0
$386$	− 162162.i	− 1.08836i
$387$	0	0
$388$	10058.9	0.0668171
$389$	− 133336.i	− 0.881144i	−0.897717	−	0.440572i	$-0.854776\pi$
$389$	− 133336.i	− 0.881144i	0.897717	−	0.440572i	$-0.145224\pi$
$390$	0	0
$391$	37981.0	0.248435
$392$	0	0
$393$	0	0
$394$	17247.9	0.111108
$395$	182048.i	1.16679i
$396$	0	0
$397$	117614.	0.746239	0.373120	−	0.927783i	$-0.378288\pi$
$397$	117614.	0.746239	0.373120	+	0.927783i	$0.378288\pi$
$398$	123136.i	0.777352i
$399$	0	0
$400$	−16143.1	−0.100895
$401$	61273.2i	0.381050i	0.981682	+	0.190525i	$0.0610190\pi$
$401$	61273.2i	0.381050i	−0.981682	+	0.190525i	$0.938981\pi$
$402$	0	0
$403$	85161.9	0.524367
$404$	53479.4i	0.327660i
$405$	0	0
$406$	0	0
$407$	− 12504.4i	− 0.0754872i
$408$	0	0
$409$	−140593.	−0.840459	−0.420229	−	0.907418i	$-0.638050\pi$
$409$	−140593.	−0.840459	−0.420229	+	0.907418i	$0.638050\pi$
$410$	239802.i	1.42654i
$411$	0	0
$412$	132849.	0.782644
$413$	0	0
$414$	0	0
$415$	1323.69	0.00768580
$416$	16249.0i	0.0938945i
$417$	0	0
$418$	7182.46	0.0411075
$419$	− 22935.9i	− 0.130644i	−0.997864	−	0.0653218i	$-0.979193\pi$
$419$	− 22935.9i	− 0.130644i	0.997864	−	0.0653218i	$-0.0208074\pi$
$420$	0	0
$421$	−281334.	−1.58730	−0.793648	−	0.608378i	$-0.791821\pi$
$421$	−281334.	−1.58730	−0.793648	+	0.608378i	$0.791821\pi$
$422$	152645.i	0.857150i
$423$	0	0
$424$	−64594.0	−0.359303
$425$	13643.7i	0.0755360i
$426$	0	0
$427$	0	0
$428$	172838.i	0.943522i
$429$	0	0
$430$	−236858.	−1.28100
$431$	− 289139.i	− 1.55651i	−0.627948	−	0.778255i	$-0.716105\pi$
$431$	− 289139.i	− 1.55651i	0.627948	−	0.778255i	$-0.283895\pi$
$432$	0	0
$433$	184347.	0.983241	0.491620	−	0.870810i	$-0.336405\pi$
$433$	184347.	0.983241	0.491620	+	0.870810i	$0.336405\pi$
$434$	0	0
$435$	0	0
$436$	−83737.9	−0.440503
$437$	146299.i	0.766086i
$438$	0	0
$439$	198176.	1.02830	0.514152	−	0.857699i	$-0.328107\pi$
$439$	198176.	1.02830	0.514152	+	0.857699i	$0.328107\pi$
$440$	8168.15i	0.0421909i
$441$	0	0
$442$	13733.2	0.0702953
$443$	215377.i	1.09747i	0.835997	+	0.548734i	$0.184890\pi$
$443$	215377.i	1.09747i	−0.835997	+	0.548734i	$0.815110\pi$
$444$	0	0
$445$	−368129.	−1.85900
$446$	− 253552.i	− 1.27467i
$447$	0	0
$448$	0	0
$449$	− 105196.i	− 0.521803i	−0.965365	−	0.260901i	$-0.915980\pi$
$449$	− 105196.i	− 0.521803i	0.965365	−	0.260901i	$-0.0840198\pi$
$450$	0	0
$451$	34888.3	0.171525
$452$	53215.8i	0.260474i
$453$	0	0
$454$	85486.1	0.414747
$455$	0	0
$456$	0	0
$457$	−194321.	−0.930436	−0.465218	−	0.885196i	$-0.654024\pi$
$457$	−194321.	−0.930436	−0.465218	+	0.885196i	$0.654024\pi$
$458$	− 13934.5i	− 0.0664292i
$459$	0	0
$460$	−166376.	−0.786276
$461$	− 3888.36i	− 0.0182963i	−0.999958	−	0.00914817i	$-0.997088\pi$
$461$	− 3888.36i	− 0.0182963i	0.999958	−	0.00914817i	$-0.00291199\pi$
$462$	0	0
$463$	−242175.	−1.12971	−0.564856	−	0.825190i	$-0.691068\pi$
$463$	−242175.	−1.12971	−0.564856	+	0.825190i	$0.691068\pi$
$464$	− 47766.1i	− 0.221862i
$465$	0	0
$466$	218658.	1.00692
$467$	− 128127.i	− 0.587499i	−0.955882	−	0.293750i	$-0.905097\pi$
$467$	− 128127.i	− 0.587499i	0.955882	−	0.293750i	$-0.0949032\pi$
$468$	0	0
$469$	0	0
$470$	− 57358.5i	− 0.259658i
$471$	0	0
$472$	−20673.2	−0.0927950
$473$	34460.0i	0.154025i
$474$	0	0
$475$	−52554.0	−0.232926
$476$	0	0
$477$	0	0
$478$	55178.8	0.241499
$479$	439156.i	1.91403i	0.290042	+	0.957014i	$0.406331\pi$
$479$	439156.i	1.91403i	−0.290042	+	0.957014i	$0.593669\pi$
$480$	0	0
$481$	92094.5	0.398056
$482$	288349.i	1.24115i
$483$	0	0
$484$	−115940.	−0.494927
$485$	37240.8i	0.158320i
$486$	0	0
$487$	−163736.	−0.690378	−0.345189	−	0.938533i	$-0.612185\pi$
$487$	−163736.	−0.690378	−0.345189	+	0.938533i	$0.612185\pi$
$488$	− 22018.3i	− 0.0924580i
$489$	0	0
$490$	0	0
$491$	− 251791.i	− 1.04442i	−0.852816	−	0.522212i	$-0.825107\pi$
$491$	− 251791.i	− 1.04442i	0.852816	−	0.522212i	$-0.174893\pi$
$492$	0	0
$493$	−40370.4	−0.166100
$494$	52898.7i	0.216766i
$495$	0	0
$496$	60718.8	0.246808
$497$	0	0
$498$	0	0
$499$	−357783.	−1.43687	−0.718437	−	0.695592i	$-0.755142\pi$
$499$	−357783.	−1.43687	−0.718437	+	0.695592i	$0.755142\pi$
$500$	88324.6i	0.353299i
$501$	0	0
$502$	96655.1	0.383546
$503$	26255.3i	0.103772i	0.998653	+	0.0518862i	$0.0165233\pi$
$503$	26255.3i	0.103772i	−0.998653	+	0.0518862i	$0.983477\pi$
$504$	0	0
$505$	−197995.	−0.776376
$506$	24205.7i	0.0945402i
$507$	0	0
$508$	−33964.8	−0.131614
$509$	− 260684.i	− 1.00619i	−0.864231	−	0.503094i	$-0.832195\pi$
$509$	− 260684.i	− 1.00619i	0.864231	−	0.503094i	$-0.167805\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	0.0441942i
$513$	0	0
$514$	−82944.1	−0.313949
$515$	491843.i	1.85444i
$516$	0	0
$517$	−8344.97	−0.0312208
$518$	0	0
$519$	0	0
$520$	−60158.3	−0.222479
$521$	369684.i	1.36193i	0.732316	+	0.680965i	$0.238439\pi$
$521$	369684.i	1.36193i	−0.732316	+	0.680965i	$0.761561\pi$
$522$	0	0
$523$	−43115.9	−0.157628	−0.0788142	−	0.996889i	$-0.525113\pi$
$523$	−43115.9	−0.157628	−0.0788142	+	0.996889i	$0.525113\pi$
$524$	− 45316.6i	− 0.165042i
$525$	0	0
$526$	115809.	0.418574
$527$	− 51317.7i	− 0.184776i
$528$	0	0
$529$	−213201.	−0.761866
$530$	− 239145.i	− 0.851352i
$531$	0	0
$532$	0	0
$533$	256952.i	0.904476i
$534$	0	0
$535$	−639894.	−2.23563
$536$	− 77751.1i	− 0.270631i
$537$	0	0
$538$	43695.7	0.150964
$539$	0	0
$540$	0	0
$541$	−361793.	−1.23613	−0.618066	−	0.786126i	$-0.712084\pi$
$541$	−361793.	−1.23613	−0.618066	+	0.786126i	$0.712084\pi$
$542$	− 181135.i	− 0.616601i
$543$	0	0
$544$	9791.50	0.0330865
$545$	− 310020.i	− 1.04375i
$546$	0	0
$547$	139100.	0.464893	0.232446	−	0.972609i	$-0.425327\pi$
$547$	139100.	0.464893	0.232446	+	0.972609i	$0.425327\pi$
$548$	149168.i	0.496724i
$549$	0	0
$550$	−8695.27	−0.0287447
$551$	− 155503.i	− 0.512194i
$552$	0	0
$553$	0	0
$554$	226723.i	0.738715i
$555$	0	0
$556$	153436.	0.496340
$557$	− 110498.i	− 0.356158i	−0.984016	−	0.178079i	$-0.943012\pi$
$557$	− 110498.i	− 0.356158i	0.984016	−	0.178079i	$-0.0569882\pi$
$558$	0	0
$559$	−253797.	−0.812200
$560$	0	0
$561$	0	0
$562$	−319449.	−1.01141
$563$	− 44113.7i	− 0.139173i	−0.997576	−	0.0695867i	$-0.977832\pi$
$563$	− 44113.7i	− 0.139173i	0.997576	−	0.0695867i	$-0.0221681\pi$
$564$	0	0
$565$	−197019.	−0.617180
$566$	− 28950.6i	− 0.0903701i
$567$	0	0
$568$	−46516.4	−0.144181
$569$	− 193693.i	− 0.598259i	−0.954212	−	0.299130i	$-0.903304\pi$
$569$	− 193693.i	− 0.598259i	0.954212	−	0.299130i	$-0.0966963\pi$
$570$	0	0
$571$	−546392.	−1.67584	−0.837919	−	0.545794i	$-0.816228\pi$
$571$	−546392.	−1.67584	−0.837919	+	0.545794i	$0.816228\pi$
$572$	8752.30i	0.0267504i
$573$	0	0
$574$	0	0
$575$	− 177113.i	− 0.535691i
$576$	0	0
$577$	−88240.7	−0.265043	−0.132522	−	0.991180i	$-0.542307\pi$
$577$	−88240.7	−0.265043	−0.132522	+	0.991180i	$0.542307\pi$
$578$	227958.i	0.682336i
$579$	0	0
$580$	176843.	0.525692
$581$	0	0
$582$	0	0
$583$	−34792.7	−0.102365
$584$	− 183460.i	− 0.537918i
$585$	0	0
$586$	448934.	1.30734
$587$	− 453587.i	− 1.31639i	−0.752848	−	0.658194i	$-0.771321\pi$
$587$	− 453587.i	− 1.31639i	0.752848	−	0.658194i	$-0.228679\pi$
$588$	0	0
$589$	197670.	0.569785
$590$	− 76537.9i	− 0.219873i
$591$	0	0
$592$	65661.7	0.187356
$593$	120463.i	0.342565i	0.985222	+	0.171283i	$0.0547912\pi$
$593$	120463.i	0.342565i	−0.985222	+	0.171283i	$0.945209\pi$
$594$	0	0
$595$	0	0
$596$	198920.i	0.559997i
$597$	0	0
$598$	−178274.	−0.498525
$599$	381099.i	1.06215i	0.847326	+	0.531073i	$0.178211\pi$
$599$	381099.i	1.06215i	−0.847326	+	0.531073i	$0.821789\pi$
$600$	0	0
$601$	−52665.9	−0.145808	−0.0729039	−	0.997339i	$-0.523227\pi$
$601$	−52665.9	−0.145808	−0.0729039	+	0.997339i	$0.523227\pi$
$602$	0	0
$603$	0	0
$604$	−244544.	−0.670321
$605$	− 429240.i	− 1.17271i
$606$	0	0
$607$	241219.	0.654689	0.327344	−	0.944905i	$-0.393846\pi$
$607$	241219.	0.654689	0.327344	+	0.944905i	$0.393846\pi$
$608$	37715.8i	0.102027i
$609$	0	0
$610$	81517.8	0.219075
$611$	− 61460.6i	− 0.164632i
$612$	0	0
$613$	−141201.	−0.375766	−0.187883	−	0.982191i	$-0.560163\pi$
$613$	−141201.	−0.375766	−0.187883	+	0.982191i	$0.560163\pi$
$614$	− 428397.i	− 1.13634i
$615$	0	0
$616$	0	0
$617$	− 157714.i	− 0.414286i	−0.978311	−	0.207143i	$-0.933583\pi$
$617$	− 157714.i	− 0.414286i	0.978311	−	0.207143i	$-0.0664166\pi$
$618$	0	0
$619$	237066.	0.618710	0.309355	−	0.950947i	$-0.399887\pi$
$619$	237066.	0.618710	0.309355	+	0.950947i	$0.399887\pi$
$620$	224798.i	0.584801i
$621$	0	0
$622$	−11233.4	−0.0290356
$623$	0	0
$624$	0	0
$625$	−484650.	−1.24070
$626$	503330.i	1.28441i
$627$	0	0
$628$	−172369.	−0.437059
$629$	− 55495.3i	− 0.140267i
$630$	0	0
$631$	−162254.	−0.407508	−0.203754	−	0.979022i	$-0.565314\pi$
$631$	−162254.	−0.407508	−0.203754	+	0.979022i	$0.565314\pi$
$632$	139080.i	0.348200i
$633$	0	0
$634$	277986.	0.691584
$635$	− 125747.i	− 0.311853i
$636$	0	0
$637$	0	0
$638$	− 25728.5i	− 0.0632082i
$639$	0	0
$640$	−42891.7	−0.104716
$641$	− 63766.9i	− 0.155196i	−0.996985	−	0.0775978i	$-0.975275\pi$
$641$	− 63766.9i	− 0.155196i	0.996985	−	0.0775978i	$-0.0247250\pi$
$642$	0	0
$643$	478884.	1.15827	0.579133	−	0.815233i	$-0.303391\pi$
$643$	478884.	1.15827	0.579133	+	0.815233i	$0.303391\pi$
$644$	0	0
$645$	0	0
$646$	31876.2	0.0763839
$647$	750554.i	1.79297i	0.443074	+	0.896485i	$0.353888\pi$
$647$	750554.i	1.79297i	−0.443074	+	0.896485i	$0.646112\pi$
$648$	0	0
$649$	−11135.3	−0.0264371
$650$	− 64040.5i	− 0.151575i
$651$	0	0
$652$	−310678.	−0.730828
$653$	157611.i	0.369624i	0.982774	+	0.184812i	$0.0591677\pi$
$653$	157611.i	0.369624i	−0.982774	+	0.184812i	$0.940832\pi$
$654$	0	0
$655$	167775.	0.391060
$656$	183202.i	0.425718i
$657$	0	0
$658$	0	0
$659$	401806.i	0.925221i	0.886562	+	0.462610i	$0.153087\pi$
$659$	401806.i	0.925221i	−0.886562	+	0.462610i	$0.846913\pi$
$660$	0	0
$661$	−753475.	−1.72451	−0.862256	−	0.506473i	$-0.830949\pi$
$661$	−753475.	−1.72451	−0.862256	+	0.506473i	$0.830949\pi$
$662$	− 565362.i	− 1.29006i
$663$	0	0
$664$	1011.26	0.00229364
$665$	0	0
$666$	0	0
$667$	524061.	1.17796
$668$	− 259556.i	− 0.581671i
$669$	0	0
$670$	287856.	0.641247
$671$	− 11859.9i	− 0.0263411i
$672$	0	0
$673$	325865.	0.719461	0.359731	−	0.933056i	$-0.382869\pi$
$673$	325865.	0.719461	0.359731	+	0.933056i	$0.382869\pi$
$674$	− 66722.5i	− 0.146877i
$675$	0	0
$676$	164027.	0.358941
$677$	437096.i	0.953672i	0.878992	+	0.476836i	$0.158216\pi$
$677$	437096.i	0.953672i	−0.878992	+	0.476836i	$0.841784\pi$
$678$	0	0
$679$	0	0
$680$	36250.8i	0.0783970i
$681$	0	0
$682$	32705.3	0.0703153
$683$	− 181278.i	− 0.388600i	−0.980942	−	0.194300i	$-0.937756\pi$
$683$	− 181278.i	− 0.388600i	0.980942	−	0.194300i	$-0.0622436\pi$
$684$	0	0
$685$	−552261.	−1.17696
$686$	0	0
$687$	0	0
$688$	−180952.	−0.382285
$689$	− 256247.i	− 0.539785i
$690$	0	0
$691$	−40690.2	−0.0852185	−0.0426093	−	0.999092i	$-0.513567\pi$
$691$	−40690.2	−0.0852185	−0.0426093	+	0.999092i	$0.513567\pi$
$692$	− 383627.i	− 0.801118i
$693$	0	0
$694$	259527.	0.538844
$695$	568064.i	1.17605i
$696$	0	0
$697$	154837.	0.318719
$698$	334525.i	0.686622i
$699$	0	0
$700$	0	0
$701$	204400.i	0.415953i	0.978134	+	0.207976i	$0.0666877\pi$
$701$	204400.i	0.415953i	−0.978134	+	0.207976i	$0.933312\pi$
$702$	0	0
$703$	213762.	0.432533
$704$	6240.23i	0.0125908i
$705$	0	0
$706$	−41475.0	−0.0832102
$707$	0	0
$708$	0	0
$709$	318463.	0.633529	0.316764	−	0.948504i	$-0.397404\pi$
$709$	318463.	0.633529	0.316764	+	0.948504i	$0.397404\pi$
$710$	− 172216.i	− 0.341631i
$711$	0	0
$712$	−281239.	−0.554774
$713$	666171.i	1.31041i
$714$	0	0
$715$	−32403.4	−0.0633838
$716$	261144.i	0.509395i
$717$	0	0
$718$	−280045.	−0.543224
$719$	472806.i	0.914587i	0.889316	+	0.457294i	$0.151181\pi$
$719$	472806.i	0.914587i	−0.889316	+	0.457294i	$0.848819\pi$
$720$	0	0
$721$	0	0
$722$	− 245820.i	− 0.471566i
$723$	0	0
$724$	−8802.23	−0.0167925
$725$	188255.i	0.358155i
$726$	0	0
$727$	293710.	0.555713	0.277857	−	0.960623i	$-0.410376\pi$
$727$	293710.	0.555713	0.277857	+	0.960623i	$0.410376\pi$
$728$	0	0
$729$	0	0
$730$	679219.	1.27457
$731$	152936.i	0.286203i
$732$	0	0
$733$	764139.	1.42221	0.711106	−	0.703085i	$-0.248195\pi$
$733$	764139.	1.42221	0.711106	+	0.703085i	$0.248195\pi$
$734$	281197.i	0.521937i
$735$	0	0
$736$	−127106.	−0.234645
$737$	− 41879.5i	− 0.0771022i
$738$	0	0
$739$	397149.	0.727217	0.363609	−	0.931552i	$-0.381545\pi$
$739$	397149.	0.727217	0.363609	+	0.931552i	$0.381545\pi$
$740$	243097.i	0.443932i
$741$	0	0
$742$	0	0
$743$	507011.i	0.918417i	0.888328	+	0.459209i	$0.151867\pi$
$743$	507011.i	0.918417i	−0.888328	+	0.459209i	$0.848133\pi$
$744$	0	0
$745$	−736456.	−1.32689
$746$	− 78239.8i	− 0.140589i
$747$	0	0
$748$	5274.05	0.00942630
$749$	0	0
$750$	0	0
$751$	577783.	1.02444	0.512218	−	0.858855i	$-0.328824\pi$
$751$	577783.	1.02444	0.512218	+	0.858855i	$0.328824\pi$
$752$	− 43820.2i	− 0.0774888i
$753$	0	0
$754$	189490.	0.333306
$755$	− 905367.i	− 1.58829i
$756$	0	0
$757$	340607.	0.594376	0.297188	−	0.954819i	$-0.403951\pi$
$757$	340607.	0.594376	0.297188	+	0.954819i	$0.403951\pi$
$758$	146308.i	0.254641i
$759$	0	0
$760$	−139634.	−0.241749
$761$	366397.i	0.632677i	0.948646	+	0.316339i	$0.102454\pi$
$761$	366397.i	0.632677i	−0.948646	+	0.316339i	$0.897546\pi$
$762$	0	0
$763$	0	0
$764$	− 205702.i	− 0.352414i
$765$	0	0
$766$	206095.	0.351244
$767$	− 82011.6i	− 0.139407i
$768$	0	0
$769$	535120.	0.904896	0.452448	−	0.891791i	$-0.350551\pi$
$769$	535120.	0.904896	0.452448	+	0.891791i	$0.350551\pi$
$770$	0	0
$771$	0	0
$772$	458663.	0.769589
$773$	− 14415.7i	− 0.0241256i	−0.999927	−	0.0120628i	$-0.996160\pi$
$773$	− 14415.7i	− 0.0241256i	0.999927	−	0.0120628i	$-0.00383980\pi$
$774$	0	0
$775$	−239305.	−0.398426
$776$	28450.9i	0.0472468i
$777$	0	0
$778$	377130.	0.623063
$779$	596414.i	0.982817i
$780$	0	0
$781$	−25055.4	−0.0410770
$782$	107426.i	0.175670i
$783$	0	0
$784$	0	0
$785$	− 638157.i	− 1.03559i
$786$	0	0
$787$	1.07509e6	1.73579	0.867894	−	0.496750i	$-0.165473\pi$
$787$	1.07509e6	1.73579	0.867894	+	0.496750i	$0.165473\pi$
$788$	48784.5i	0.0785651i
$789$	0	0
$790$	−514910.	−0.825045
$791$	0	0
$792$	0	0
$793$	87347.6	0.138901
$794$	332663.i	0.527671i
$795$	0	0
$796$	−348280.	−0.549671
$797$	920775.i	1.44956i	0.688979	+	0.724782i	$0.258059\pi$
$797$	920775.i	1.44956i	−0.688979	+	0.724782i	$0.741941\pi$
$798$	0	0
$799$	−37035.5	−0.0580130
$800$	− 45659.7i	− 0.0713432i
$801$	0	0
$802$	−173307.	−0.269443
$803$	− 98818.2i	− 0.153252i
$804$	0	0
$805$	0	0
$806$	240874.i	0.370783i
$807$	0	0
$808$	−151263.	−0.231691
$809$	− 893640.i	− 1.36542i	−0.730691	−	0.682709i	$-0.760802\pi$
$809$	− 893640.i	− 1.36542i	0.730691	−	0.682709i	$-0.239198\pi$
$810$	0	0
$811$	476483.	0.724445	0.362222	−	0.932092i	$-0.382018\pi$
$811$	476483.	0.724445	0.362222	+	0.932092i	$0.382018\pi$
$812$	0	0
$813$	0	0
$814$	35367.7	0.0533775
$815$	− 1.15021e6i	− 1.73166i
$816$	0	0
$817$	−589091.	−0.882548
$818$	− 397656.i	− 0.594294i
$819$	0	0
$820$	−678263.	−1.00872
$821$	62659.3i	0.0929606i	0.998919	+	0.0464803i	$0.0148005\pi$
$821$	62659.3i	0.0929606i	−0.998919	+	0.0464803i	$0.985200\pi$
$822$	0	0
$823$	784222.	1.15782	0.578908	−	0.815393i	$-0.303479\pi$
$823$	784222.	1.15782	0.578908	+	0.815393i	$0.303479\pi$
$824$	375754.i	0.553413i
$825$	0	0
$826$	0	0
$827$	207577.i	0.303507i	0.988418	+	0.151754i	$0.0484920\pi$
$827$	207577.i	0.303507i	−0.988418	+	0.151754i	$0.951508\pi$
$828$	0	0
$829$	1.12614e6	1.63864	0.819318	−	0.573339i	$-0.194352\pi$
$829$	1.12614e6	1.63864	0.819318	+	0.573339i	$0.194352\pi$
$830$	3743.95i	0.00543468i
$831$	0	0
$832$	−45959.1	−0.0663935
$833$	0	0
$834$	0	0
$835$	960946.	1.37824
$836$	20315.1i	0.0290674i
$837$	0	0
$838$	64872.6	0.0923790
$839$	930536.i	1.32193i	0.750415	+	0.660967i	$0.229853\pi$
$839$	930536.i	1.32193i	−0.750415	+	0.660967i	$0.770147\pi$
$840$	0	0
$841$	150251.	0.212434
$842$	− 795732.i	− 1.12239i
$843$	0	0
$844$	−431744.	−0.606097
$845$	607274.i	0.850494i
$846$	0	0
$847$	0	0
$848$	− 182699.i	− 0.254065i
$849$	0	0
$850$	−38590.2	−0.0534120
$851$	720400.i	0.994752i
$852$	0	0
$853$	−532024.	−0.731195	−0.365598	−	0.930773i	$-0.619135\pi$
$853$	−532024.	−0.731195	−0.365598	+	0.930773i	$0.619135\pi$
$854$	0	0
$855$	0	0
$856$	−488860.	−0.667171
$857$	− 1.37499e6i	− 1.87213i	−0.351825	−	0.936066i	$-0.614439\pi$
$857$	− 1.37499e6i	− 1.87213i	0.351825	−	0.936066i	$-0.385561\pi$
$858$	0	0
$859$	767617.	1.04030	0.520149	−	0.854075i	$-0.325876\pi$
$859$	767617.	1.04030	0.520149	+	0.854075i	$0.325876\pi$
$860$	− 669935.i	− 0.905807i
$861$	0	0
$862$	817808.	1.10062
$863$	− 908202.i	− 1.21944i	−0.792616	−	0.609721i	$-0.791282\pi$
$863$	− 908202.i	− 1.21944i	0.792616	−	0.609721i	$-0.208718\pi$
$864$	0	0
$865$	1.42029e6	1.89821
$866$	521412.i	0.695256i
$867$	0	0
$868$	0	0
$869$	74913.3i	0.0992017i
$870$	0	0
$871$	308442.	0.406572
$872$	− 236847.i	− 0.311483i
$873$	0	0
$874$	−413795.	−0.541704
$875$	0	0
$876$	0	0
$877$	−1.13533e6	−1.47613	−0.738063	−	0.674732i	$-0.764259\pi$
$877$	−1.13533e6	−1.47613	−0.738063	+	0.674732i	$0.764259\pi$
$878$	560526.i	0.727121i
$879$	0	0
$880$	−23103.0	−0.0298334
$881$	− 263319.i	− 0.339258i	−0.985508	−	0.169629i	$-0.945743\pi$
$881$	− 263319.i	− 0.339258i	0.985508	−	0.169629i	$-0.0542569\pi$
$882$	0	0
$883$	852498.	1.09338	0.546691	−	0.837334i	$-0.315887\pi$
$883$	852498.	1.09338	0.546691	+	0.837334i	$0.315887\pi$
$884$	38843.3i	0.0497063i
$885$	0	0
$886$	−609178.	−0.776027
$887$	− 835509.i	− 1.06195i	−0.847388	−	0.530974i	$-0.821826\pi$
$887$	− 835509.i	− 1.06195i	0.847388	−	0.530974i	$-0.178174\pi$
$888$	0	0
$889$	0	0
$890$	− 1.04122e6i	− 1.31451i
$891$	0	0
$892$	717155.	0.901328
$893$	− 142657.i	− 0.178892i
$894$	0	0
$895$	−966827.	−1.20699
$896$	0	0
$897$	0	0
$898$	297539.	0.368970
$899$	− 708081.i	− 0.876120i
$900$	0	0
$901$	−154412.	−0.190209
$902$	98679.0i	0.121286i
$903$	0	0
$904$	−150517.	−0.184183
$905$	− 32588.2i	− 0.0397891i
$906$	0	0
$907$	−331060.	−0.402432	−0.201216	−	0.979547i	$-0.564489\pi$
$907$	−331060.	−0.402432	−0.201216	+	0.979547i	$0.564489\pi$
$908$	241791.i	0.293271i
$909$	0	0
$910$	0	0
$911$	1.42629e6i	1.71859i	0.511484	+	0.859293i	$0.329096\pi$
$911$	1.42629e6i	1.71859i	−0.511484	+	0.859293i	$0.670904\pi$
$912$	0	0
$913$	544.700	0.000653455 0
$914$	− 549622.i	− 0.657918i
$915$	0	0
$916$	39412.6	0.0469725
$917$	0	0
$918$	0	0
$919$	1.37601e6	1.62926	0.814632	−	0.579979i	$-0.196939\pi$
$919$	1.37601e6	1.62926	0.814632	+	0.579979i	$0.196939\pi$
$920$	− 470582.i	− 0.555981i
$921$	0	0
$922$	10997.9	0.0129375
$923$	− 184533.i	− 0.216606i
$924$	0	0
$925$	−258785.	−0.302452
$926$	− 684974.i	− 0.798826i
$927$	0	0
$928$	135103.	0.156880
$929$	− 460552.i	− 0.533639i	−0.963747	−	0.266819i	$-0.914027\pi$
$929$	− 460552.i	− 0.533639i	0.963747	−	0.266819i	$-0.0859727\pi$
$930$	0	0
$931$	0	0
$932$	618458.i	0.711997i
$933$	0	0
$934$	362398.	0.415425
$935$	19526.0i	0.0223352i
$936$	0	0
$937$	−605549.	−0.689715	−0.344858	−	0.938655i	$-0.612073\pi$
$937$	−605549.	−0.689715	−0.344858	+	0.938655i	$0.612073\pi$
$938$	0	0
$939$	0	0
$940$	162234.	0.183606
$941$	− 1.09177e6i	− 1.23297i	−0.787366	−	0.616486i	$-0.788556\pi$
$941$	− 1.09177e6i	− 1.23297i	0.787366	−	0.616486i	$-0.211444\pi$
$942$	0	0
$943$	−2.00998e6	−2.26031
$944$	− 58472.7i	− 0.0656159i
$945$	0	0
$946$	−97467.5	−0.108912
$947$	− 936998.i	− 1.04481i	−0.852697	−	0.522407i	$-0.825034\pi$
$947$	− 936998.i	− 1.04481i	0.852697	−	0.522407i	$-0.174966\pi$
$948$	0	0
$949$	727794.	0.808120
$950$	− 148645.i	− 0.164704i
$951$	0	0
$952$	0	0
$953$	431397.i	0.474997i	0.971388	+	0.237499i	$0.0763275\pi$
$953$	431397.i	0.474997i	−0.971388	+	0.237499i	$0.923673\pi$
$954$	0	0
$955$	761567.	0.835028
$956$	156069.i	0.170766i
$957$	0	0
$958$	−1.24212e6	−1.35342
$959$	0	0
$960$	0	0
$961$	−23429.4	−0.0253696
$962$	260483.i	0.281468i
$963$	0	0
$964$	−815573.	−0.877625
$965$	1.69809e6i	1.82351i
$966$	0	0
$967$	−1.17576e6	−1.25738	−0.628689	−	0.777657i	$-0.716408\pi$
$967$	−1.17576e6	−1.25738	−0.628689	+	0.777657i	$0.716408\pi$
$968$	− 327927.i	− 0.349966i
$969$	0	0
$970$	−105333.	−0.111949
$971$	− 1.27276e6i	− 1.34992i	−0.737852	−	0.674962i	$-0.764160\pi$
$971$	− 1.27276e6i	− 1.34992i	0.737852	−	0.674962i	$-0.235840\pi$
$972$	0	0
$973$	0	0
$974$	− 463116.i	− 0.488171i
$975$	0	0
$976$	62277.2	0.0653777
$977$	295959.i	0.310058i	0.987910	+	0.155029i	$0.0495470\pi$
$977$	295959.i	0.310058i	−0.987910	+	0.155029i	$0.950453\pi$
$978$	0	0
$979$	−151486.	−0.158054
$980$	0	0
$981$	0	0
$982$	712171.	0.738519
$983$	1.01949e6i	1.05505i	0.849539	+	0.527526i	$0.176880\pi$
$983$	1.01949e6i	1.05505i	−0.849539	+	0.527526i	$0.823120\pi$
$984$	0	0
$985$	−180614.	−0.186156
$986$	− 114185.i	− 0.117450i
$987$	0	0
$988$	−149620.	−0.153277
$989$	− 1.98530e6i	− 2.02971i
$990$	0	0
$991$	1.29572e6	1.31937	0.659683	−	0.751544i	$-0.270691\pi$
$991$	1.29572e6	1.31937	0.659683	+	0.751544i	$0.270691\pi$
$992$	171739.i	0.174520i
$993$	0	0
$994$	0	0
$995$	− 1.28943e6i	− 1.30242i
$996$	0	0
$997$	559812.	0.563186	0.281593	−	0.959534i	$-0.409137\pi$
$997$	559812.	0.563186	0.281593	+	0.959534i	$0.409137\pi$
$998$	− 1.01196e6i	− 1.01602i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.5.b.i.197.5	yes	8
3.2	odd	2	inner	882.5.b.i.197.4	yes	8
7.6	odd	2	inner	882.5.b.i.197.8	yes	8
21.20	even	2	inner	882.5.b.i.197.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.5.b.i.197.1	✓	8	21.20	even	2	inner
882.5.b.i.197.4	yes	8	3.2	odd	2	inner
882.5.b.i.197.5	yes	8	1.1	even	1	trivial
882.5.b.i.197.8	yes	8	7.6	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)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} -29.6182i q^{5} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} -29.6182i q^{5} -22.6274i q^{8} +83.7729 q^{10} -12.1879i q^{11} +89.7640 q^{13} +64.0000 q^{16} -54.0909i q^{17} +208.352 q^{19} +236.945i q^{20} +34.4727 q^{22} +702.170i q^{23} -252.236 q^{25} +253.891i q^{26} -746.345i q^{29} +948.732 q^{31} +181.019i q^{32} +152.992 q^{34} +1025.96 q^{37} +589.309i q^{38} -670.183 q^{40} +2862.53i q^{41} -2827.38 q^{43} +97.5035i q^{44} -1986.04 q^{46} -684.691i q^{47} -713.432i q^{50} -718.112 q^{52} -2854.68i q^{53} -360.985 q^{55} +2110.98 q^{58} -913.636i q^{59} +973.081 q^{61} +2683.42i q^{62} -512.000 q^{64} -2658.64i q^{65} +3436.14 q^{67} +432.727i q^{68} -2055.75i q^{71} +8107.87 q^{73} +2901.86i q^{74} -1666.82 q^{76} -6146.51 q^{79} -1895.56i q^{80} -8096.45 q^{82} +44.6917i q^{83} -1602.07 q^{85} -7997.04i q^{86} -275.782 q^{88} -12429.1i q^{89} -5617.36i q^{92} +1936.60 q^{94} -6171.01i q^{95} -1257.36 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} + 512 q^{16} - 640 q^{22} - 1560 q^{25} - 1408 q^{37} - 10256 q^{43} - 320 q^{46} - 512 q^{58} - 4096 q^{64} - 9600 q^{67} + 17680 q^{79} - 32048 q^{85} + 5120 q^{88}+O(q^{100}) \) 8 * q - 64 * q^4 + 512 * q^16 - 640 * q^22 - 1560 * q^25 - 1408 * q^37 - 10256 * q^43 - 320 * q^46 - 512 * q^58 - 4096 * q^64 - 9600 * q^67 + 17680 * q^79 - 32048 * q^85 + 5120 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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