LMFDB - Embedded newform 882.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

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([0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.44622$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -3.44622 q^{5} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} -3.44622 q^{5} -8.00000 q^{8} +6.89244 q^{10} +36.1236 q^{11} +10.2311 q^{13} +16.0000 q^{16} +118.462 q^{17} -38.6613 q^{19} -13.7849 q^{20} -72.2471 q^{22} -36.2471 q^{23} -113.124 q^{25} -20.4622 q^{26} -12.1236 q^{29} -145.494 q^{31} -32.0000 q^{32} -236.924 q^{34} -1.37066 q^{37} +77.3227 q^{38} +27.5698 q^{40} +168.000 q^{41} +299.371 q^{43} +144.494 q^{44} +72.4942 q^{46} +502.709 q^{47} +226.247 q^{50} +40.9244 q^{52} -625.112 q^{53} -124.490 q^{55} +24.2471 q^{58} -42.2195 q^{59} -439.172 q^{61} +290.988 q^{62} +64.0000 q^{64} -35.2587 q^{65} -763.618 q^{67} +473.849 q^{68} +1020.49 q^{71} +579.284 q^{73} +2.74132 q^{74} -154.645 q^{76} +942.730 q^{79} -55.1396 q^{80} -336.000 q^{82} -474.714 q^{83} -408.247 q^{85} -598.741 q^{86} -288.988 q^{88} +821.904 q^{89} -144.988 q^{92} -1005.42 q^{94} +133.236 q^{95} -1108.16 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 + 7 * q^5 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 16 q^{8} - 14 q^{10} - 25 q^{11} - 49 q^{13} + 32 q^{16} + 98 q^{17} - 119 q^{19} + 28 q^{20} + 50 q^{22} + 122 q^{23} - 129 q^{25} + 98 q^{26} + 73 q^{29} + 98 q^{31} - 64 q^{32} - 196 q^{34} + 289 q^{37} + 238 q^{38} - 56 q^{40} + 336 q^{41} + 307 q^{43} - 100 q^{44} - 244 q^{46} + 672 q^{47} + 258 q^{50} - 196 q^{52} - 375 q^{53} - 763 q^{55} - 146 q^{58} + 763 q^{59} - 406 q^{61} - 196 q^{62} + 128 q^{64} - 654 q^{65} - 1041 q^{67} + 392 q^{68} + 1652 q^{71} - 189 q^{73} - 578 q^{74} - 476 q^{76} + 524 q^{79} + 112 q^{80} - 672 q^{82} + 287 q^{83} - 622 q^{85} - 614 q^{86} + 200 q^{88} + 2394 q^{89} + 488 q^{92} - 1344 q^{94} - 706 q^{95} - 63 q^{97}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 + 7 * q^5 - 16 * q^8 - 14 * q^10 - 25 * q^11 - 49 * q^13 + 32 * q^16 + 98 * q^17 - 119 * q^19 + 28 * q^20 + 50 * q^22 + 122 * q^23 - 129 * q^25 + 98 * q^26 + 73 * q^29 + 98 * q^31 - 64 * q^32 - 196 * q^34 + 289 * q^37 + 238 * q^38 - 56 * q^40 + 336 * q^41 + 307 * q^43 - 100 * q^44 - 244 * q^46 + 672 * q^47 + 258 * q^50 - 196 * q^52 - 375 * q^53 - 763 * q^55 - 146 * q^58 + 763 * q^59 - 406 * q^61 - 196 * q^62 + 128 * q^64 - 654 * q^65 - 1041 * q^67 + 392 * q^68 + 1652 * q^71 - 189 * q^73 - 578 * q^74 - 476 * q^76 + 524 * q^79 + 112 * q^80 - 672 * q^82 + 287 * q^83 - 622 * q^85 - 614 * q^86 + 200 * q^88 + 2394 * q^89 + 488 * q^92 - 1344 * q^94 - 706 * q^95 - 63 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−3.44622	−0.308239	−0.154120	−	0.988052i	$-0.549254\pi$
$5$	−3.44622	−0.308239	−0.154120	+	0.988052i	$0.549254\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	6.89244	0.217958
$11$	36.1236	0.990151	0.495076	−	0.868850i	$-0.335140\pi$
$11$	36.1236	0.990151	0.495076	+	0.868850i	$0.335140\pi$
$12$	0	0
$13$	10.2311	0.218277	0.109138	−	0.994027i	$-0.465191\pi$
$13$	10.2311	0.218277	0.109138	+	0.994027i	$0.465191\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	118.462	1.69008	0.845038	−	0.534705i	$-0.179577\pi$
$17$	118.462	1.69008	0.845038	+	0.534705i	$0.179577\pi$
$18$	0	0
$19$	−38.6613	−0.466817	−0.233408	−	0.972379i	$-0.574988\pi$
$19$	−38.6613	−0.466817	−0.233408	+	0.972379i	$0.574988\pi$
$20$	−13.7849	−0.154120
$21$	0	0
$22$	−72.2471	−0.700143
$23$	−36.2471	−0.328611	−0.164305	−	0.986410i	$-0.552538\pi$
$23$	−36.2471	−0.328611	−0.164305	+	0.986410i	$0.552538\pi$
$24$	0	0
$25$	−113.124	−0.904988
$26$	−20.4622	−0.154345
$27$	0	0
$28$	0	0
$29$	−12.1236	−0.0776306	−0.0388153	−	0.999246i	$-0.512358\pi$
$29$	−12.1236	−0.0776306	−0.0388153	+	0.999246i	$0.512358\pi$
$30$	0	0
$31$	−145.494	−0.842953	−0.421476	−	0.906839i	$-0.638488\pi$
$31$	−145.494	−0.842953	−0.421476	+	0.906839i	$0.638488\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−236.924	−1.19506
$35$	0	0
$36$	0	0
$37$	−1.37066	−0.00609015	−0.00304507	−	0.999995i	$-0.500969\pi$
$37$	−1.37066	−0.00609015	−0.00304507	+	0.999995i	$0.500969\pi$
$38$	77.3227	0.330089
$39$	0	0
$40$	27.5698	0.108979
$41$	168.000	0.639932	0.319966	−	0.947429i	$-0.396329\pi$
$41$	168.000	0.639932	0.319966	+	0.947429i	$0.396329\pi$
$42$	0	0
$43$	299.371	1.06171	0.530856	−	0.847462i	$-0.321871\pi$
$43$	299.371	1.06171	0.530856	+	0.847462i	$0.321871\pi$
$44$	144.494	0.495076
$45$	0	0
$46$	72.4942	0.232363
$47$	502.709	1.56016	0.780082	−	0.625678i	$-0.215177\pi$
$47$	502.709	1.56016	0.780082	+	0.625678i	$0.215177\pi$
$48$	0	0
$49$	0	0
$50$	226.247	0.639923
$51$	0	0
$52$	40.9244	0.109138
$53$	−625.112	−1.62011	−0.810054	−	0.586355i	$-0.800562\pi$
$53$	−625.112	−1.62011	−0.810054	+	0.586355i	$0.800562\pi$
$54$	0	0
$55$	−124.490	−0.305204
$56$	0	0
$57$	0	0
$58$	24.2471	0.0548931
$59$	−42.2195	−0.0931613	−0.0465806	−	0.998915i	$-0.514832\pi$
$59$	−42.2195	−0.0931613	−0.0465806	+	0.998915i	$0.514832\pi$
$60$	0	0
$61$	−439.172	−0.921806	−0.460903	−	0.887451i	$-0.652474\pi$
$61$	−439.172	−0.921806	−0.460903	+	0.887451i	$0.652474\pi$
$62$	290.988	0.596058
$63$	0	0
$64$	64.0000	0.125000
$65$	−35.2587	−0.0672815
$66$	0	0
$67$	−763.618	−1.39240	−0.696200	−	0.717848i	$-0.745127\pi$
$67$	−763.618	−1.39240	−0.696200	+	0.717848i	$0.745127\pi$
$68$	473.849	0.845038
$69$	0	0
$70$	0	0
$71$	1020.49	1.70578	0.852890	−	0.522091i	$-0.174848\pi$
$71$	1020.49	1.70578	0.852890	+	0.522091i	$0.174848\pi$
$72$	0	0
$73$	579.284	0.928767	0.464384	−	0.885634i	$-0.346276\pi$
$73$	579.284	0.928767	0.464384	+	0.885634i	$0.346276\pi$
$74$	2.74132	0.00430638
$75$	0	0
$76$	−154.645	−0.233408
$77$	0	0
$78$	0	0
$79$	942.730	1.34260	0.671300	−	0.741186i	$-0.265736\pi$
$79$	942.730	1.34260	0.671300	+	0.741186i	$0.265736\pi$
$80$	−55.1396	−0.0770599
$81$	0	0
$82$	−336.000	−0.452500
$83$	−474.714	−0.627790	−0.313895	−	0.949458i	$-0.601634\pi$
$83$	−474.714	−0.627790	−0.313895	+	0.949458i	$0.601634\pi$
$84$	0	0
$85$	−408.247	−0.520948
$86$	−598.741	−0.750743
$87$	0	0
$88$	−288.988	−0.350071
$89$	821.904	0.978895	0.489447	−	0.872033i	$-0.337198\pi$
$89$	821.904	0.978895	0.489447	+	0.872033i	$0.337198\pi$
$90$	0	0
$91$	0	0
$92$	−144.988	−0.164305
$93$	0	0
$94$	−1005.42	−1.10320
$95$	133.236	0.143891
$96$	0	0
$97$	−1108.16	−1.15997	−0.579985	−	0.814627i	$-0.696942\pi$
$97$	−1108.16	−1.15997	−0.579985	+	0.814627i	$0.696942\pi$
$98$	0	0
$99$	0	0
$100$	−452.494	−0.452494
$101$	1885.92	1.85798	0.928991	−	0.370102i	$-0.120677\pi$
$101$	1885.92	1.85798	0.928991	+	0.370102i	$0.120677\pi$
$102$	0	0
$103$	482.144	0.461234	0.230617	−	0.973045i	$-0.425926\pi$
$103$	482.144	0.461234	0.230617	+	0.973045i	$0.425926\pi$
$104$	−81.8489	−0.0771725
$105$	0	0
$106$	1250.22	1.14559
$107$	−1730.59	−1.56358	−0.781789	−	0.623543i	$-0.785693\pi$
$107$	−1730.59	−1.56358	−0.781789	+	0.623543i	$0.785693\pi$
$108$	0	0
$109$	1148.83	1.00952	0.504761	−	0.863259i	$-0.331580\pi$
$109$	1148.83	1.00952	0.504761	+	0.863259i	$0.331580\pi$
$110$	248.980	0.215812
$111$	0	0
$112$	0	0
$113$	1331.51	1.10847	0.554237	−	0.832359i	$-0.313010\pi$
$113$	1331.51	1.10847	0.554237	+	0.832359i	$0.313010\pi$
$114$	0	0
$115$	124.916	0.101291
$116$	−48.4942	−0.0388153
$117$	0	0
$118$	84.4391	0.0658750
$119$	0	0
$120$	0	0
$121$	−26.0888	−0.0196009
$122$	878.343	0.651815
$123$	0	0
$124$	−581.977	−0.421476
$125$	820.627	0.587193
$126$	0	0
$127$	−830.236	−0.580090	−0.290045	−	0.957013i	$-0.593670\pi$
$127$	−830.236	−0.580090	−0.290045	+	0.957013i	$0.593670\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	70.5174	0.0475752
$131$	1972.49	1.31555	0.657776	−	0.753214i	$-0.271497\pi$
$131$	1972.49	1.31555	0.657776	+	0.753214i	$0.271497\pi$
$132$	0	0
$133$	0	0
$134$	1527.24	0.984575
$135$	0	0
$136$	−947.698	−0.597532
$137$	−68.7876	−0.0428972	−0.0214486	−	0.999770i	$-0.506828\pi$
$137$	−68.7876	−0.0428972	−0.0214486	+	0.999770i	$0.506828\pi$
$138$	0	0
$139$	1864.48	1.13772	0.568860	−	0.822435i	$-0.307385\pi$
$139$	1864.48	1.13772	0.568860	+	0.822435i	$0.307385\pi$
$140$	0	0
$141$	0	0
$142$	−2040.99	−1.20617
$143$	369.584	0.216127
$144$	0	0
$145$	41.7805	0.0239288
$146$	−1158.57	−0.656738
$147$	0	0
$148$	−5.48265	−0.00304507
$149$	−20.2934	−0.0111577	−0.00557885	−	0.999984i	$-0.501776\pi$
$149$	−20.2934	−0.0111577	−0.00557885	+	0.999984i	$0.501776\pi$
$150$	0	0
$151$	−1580.32	−0.851689	−0.425844	−	0.904796i	$-0.640023\pi$
$151$	−1580.32	−0.851689	−0.425844	+	0.904796i	$0.640023\pi$
$152$	309.291	0.165045
$153$	0	0
$154$	0	0
$155$	501.405	0.259831
$156$	0	0
$157$	3384.35	1.72039	0.860193	−	0.509968i	$-0.170343\pi$
$157$	3384.35	1.72039	0.860193	+	0.509968i	$0.170343\pi$
$158$	−1885.46	−0.949361
$159$	0	0
$160$	110.279	0.0544896
$161$	0	0
$162$	0	0
$163$	1571.71	0.755249	0.377624	−	0.925959i	$-0.376741\pi$
$163$	1571.71	0.755249	0.377624	+	0.925959i	$0.376741\pi$
$164$	672.000	0.319966
$165$	0	0
$166$	949.428	0.443915
$167$	2473.92	1.14633	0.573167	−	0.819439i	$-0.305715\pi$
$167$	2473.92	1.14633	0.573167	+	0.819439i	$0.305715\pi$
$168$	0	0
$169$	−2092.32	−0.952355
$170$	816.494	0.368366
$171$	0	0
$172$	1197.48	0.530856
$173$	4151.34	1.82440	0.912198	−	0.409749i	$-0.134384\pi$
$173$	4151.34	1.82440	0.912198	+	0.409749i	$0.134384\pi$
$174$	0	0
$175$	0	0
$176$	577.977	0.247538
$177$	0	0
$178$	−1643.81	−0.692183
$179$	−4217.93	−1.76125	−0.880623	−	0.473818i	$-0.842875\pi$
$179$	−4217.93	−1.76125	−0.880623	+	0.473818i	$0.842875\pi$
$180$	0	0
$181$	3504.65	1.43922	0.719609	−	0.694380i	$-0.244321\pi$
$181$	3504.65	1.43922	0.719609	+	0.694380i	$0.244321\pi$
$182$	0	0
$183$	0	0
$184$	289.977	0.116181
$185$	4.72361	0.00187722
$186$	0	0
$187$	4279.28	1.67343
$188$	2010.84	0.780082
$189$	0	0
$190$	−266.471	−0.101747
$191$	1982.97	0.751216	0.375608	−	0.926779i	$-0.377434\pi$
$191$	1982.97	0.751216	0.375608	+	0.926779i	$0.377434\pi$
$192$	0	0
$193$	2111.74	0.787598	0.393799	−	0.919197i	$-0.371161\pi$
$193$	2111.74	0.787598	0.393799	+	0.919197i	$0.371161\pi$
$194$	2216.33	0.820222
$195$	0	0
$196$	0	0
$197$	3932.65	1.42228	0.711141	−	0.703049i	$-0.248179\pi$
$197$	3932.65	1.42228	0.711141	+	0.703049i	$0.248179\pi$
$198$	0	0
$199$	−552.704	−0.196885	−0.0984426	−	0.995143i	$-0.531386\pi$
$199$	−552.704	−0.196885	−0.0984426	+	0.995143i	$0.531386\pi$
$200$	904.988	0.319962
$201$	0	0
$202$	−3771.84	−1.31379
$203$	0	0
$204$	0	0
$205$	−578.965	−0.197252
$206$	−964.288	−0.326141
$207$	0	0
$208$	163.698	0.0545692
$209$	−1396.58	−0.462219
$210$	0	0
$211$	1720.90	0.561476	0.280738	−	0.959784i	$-0.409421\pi$
$211$	1720.90	0.561476	0.280738	+	0.959784i	$0.409421\pi$
$212$	−2500.45	−0.810054
$213$	0	0
$214$	3461.19	1.10562
$215$	−1031.70	−0.327261
$216$	0	0
$217$	0	0
$218$	−2297.66	−0.713840
$219$	0	0
$220$	−497.959	−0.152602
$221$	1212.00	0.368905
$222$	0	0
$223$	4788.43	1.43793	0.718963	−	0.695049i	$-0.244617\pi$
$223$	4788.43	1.43793	0.718963	+	0.695049i	$0.244617\pi$
$224$	0	0
$225$	0	0
$226$	−2663.01	−0.783809
$227$	4087.59	1.19517	0.597584	−	0.801807i	$-0.296128\pi$
$227$	4087.59	1.19517	0.597584	+	0.801807i	$0.296128\pi$
$228$	0	0
$229$	−4066.61	−1.17349	−0.586745	−	0.809772i	$-0.699591\pi$
$229$	−4066.61	−1.17349	−0.586745	+	0.809772i	$0.699591\pi$
$230$	−249.831	−0.0716234
$231$	0	0
$232$	96.9884	0.0274466
$233$	178.270	0.0501239	0.0250620	−	0.999686i	$-0.492022\pi$
$233$	178.270	0.0501239	0.0250620	+	0.999686i	$0.492022\pi$
$234$	0	0
$235$	−1732.45	−0.480904
$236$	−168.878	−0.0465806
$237$	0	0
$238$	0	0
$239$	−2118.67	−0.573412	−0.286706	−	0.958019i	$-0.592560\pi$
$239$	−2118.67	−0.573412	−0.286706	+	0.958019i	$0.592560\pi$
$240$	0	0
$241$	2398.31	0.641031	0.320516	−	0.947243i	$-0.396144\pi$
$241$	2398.31	0.641031	0.320516	+	0.947243i	$0.396144\pi$
$242$	52.1777	0.0138600
$243$	0	0
$244$	−1756.69	−0.460903
$245$	0	0
$246$	0	0
$247$	−395.548	−0.101895
$248$	1163.95	0.298029
$249$	0	0
$250$	−1641.25	−0.415208
$251$	−1550.76	−0.389973	−0.194986	−	0.980806i	$-0.562466\pi$
$251$	−1550.76	−0.389973	−0.194986	+	0.980806i	$0.562466\pi$
$252$	0	0
$253$	−1309.37	−0.325374
$254$	1660.47	0.410186
$255$	0	0
$256$	256.000	0.0625000
$257$	−5395.44	−1.30957	−0.654783	−	0.755817i	$-0.727240\pi$
$257$	−5395.44	−1.30957	−0.654783	+	0.755817i	$0.727240\pi$
$258$	0	0
$259$	0	0
$260$	−141.035	−0.0336408
$261$	0	0
$262$	−3944.98	−0.930235
$263$	3581.93	0.839815	0.419907	−	0.907567i	$-0.362063\pi$
$263$	3581.93	0.839815	0.419907	+	0.907567i	$0.362063\pi$
$264$	0	0
$265$	2154.27	0.499381
$266$	0	0
$267$	0	0
$268$	−3054.47	−0.696200
$269$	−1175.55	−0.266448	−0.133224	−	0.991086i	$-0.542533\pi$
$269$	−1175.55	−0.266448	−0.133224	+	0.991086i	$0.542533\pi$
$270$	0	0
$271$	259.327	0.0581291	0.0290646	−	0.999578i	$-0.490747\pi$
$271$	259.327	0.0581291	0.0290646	+	0.999578i	$0.490747\pi$
$272$	1895.40	0.422519
$273$	0	0
$274$	137.575	0.0303329
$275$	−4086.42	−0.896075
$276$	0	0
$277$	−1298.92	−0.281750	−0.140875	−	0.990027i	$-0.544992\pi$
$277$	−1298.92	−0.281750	−0.140875	+	0.990027i	$0.544992\pi$
$278$	−3728.96	−0.804489
$279$	0	0
$280$	0	0
$281$	−4524.25	−0.960477	−0.480238	−	0.877138i	$-0.659450\pi$
$281$	−4524.25	−0.960477	−0.480238	+	0.877138i	$0.659450\pi$
$282$	0	0
$283$	6478.05	1.36071	0.680354	−	0.732884i	$-0.261826\pi$
$283$	6478.05	1.36071	0.680354	+	0.732884i	$0.261826\pi$
$284$	4081.98	0.852890
$285$	0	0
$286$	−739.168	−0.152825
$287$	0	0
$288$	0	0
$289$	9120.30	1.85636
$290$	−83.5609	−0.0169202
$291$	0	0
$292$	2317.13	0.464384
$293$	−2890.91	−0.576413	−0.288207	−	0.957568i	$-0.593059\pi$
$293$	−2890.91	−0.576413	−0.288207	+	0.957568i	$0.593059\pi$
$294$	0	0
$295$	145.498	0.0287160
$296$	10.9653	0.00215319
$297$	0	0
$298$	40.5868	0.00788969
$299$	−370.848	−0.0717281
$300$	0	0
$301$	0	0
$302$	3160.65	0.602235
$303$	0	0
$304$	−618.581	−0.116704
$305$	1513.48	0.284137
$306$	0	0
$307$	−3137.42	−0.583264	−0.291632	−	0.956531i	$-0.594198\pi$
$307$	−3137.42	−0.583264	−0.291632	+	0.956531i	$0.594198\pi$
$308$	0	0
$309$	0	0
$310$	−1002.81	−0.183728
$311$	−7914.95	−1.44314	−0.721568	−	0.692343i	$-0.756578\pi$
$311$	−7914.95	−1.44314	−0.721568	+	0.692343i	$0.756578\pi$
$312$	0	0
$313$	7616.61	1.37545	0.687726	−	0.725971i	$-0.258609\pi$
$313$	7616.61	1.37545	0.687726	+	0.725971i	$0.258609\pi$
$314$	−6768.70	−1.21650
$315$	0	0
$316$	3770.92	0.671300
$317$	−1991.49	−0.352849	−0.176425	−	0.984314i	$-0.556453\pi$
$317$	−1991.49	−0.352849	−0.176425	+	0.984314i	$0.556453\pi$
$318$	0	0
$319$	−437.946	−0.0768660
$320$	−220.558	−0.0385299
$321$	0	0
$322$	0	0
$323$	−4579.91	−0.788956
$324$	0	0
$325$	−1157.38	−0.197538
$326$	−3143.41	−0.534042
$327$	0	0
$328$	−1344.00	−0.226250
$329$	0	0
$330$	0	0
$331$	2848.63	0.473036	0.236518	−	0.971627i	$-0.423994\pi$
$331$	2848.63	0.473036	0.236518	+	0.971627i	$0.423994\pi$
$332$	−1898.86	−0.313895
$333$	0	0
$334$	−4947.84	−0.810581
$335$	2631.60	0.429192
$336$	0	0
$337$	−5813.87	−0.939768	−0.469884	−	0.882728i	$-0.655704\pi$
$337$	−5813.87	−0.939768	−0.469884	+	0.882728i	$0.655704\pi$
$338$	4184.65	0.673417
$339$	0	0
$340$	−1632.99	−0.260474
$341$	−5255.77	−0.834650
$342$	0	0
$343$	0	0
$344$	−2394.97	−0.375372
$345$	0	0
$346$	−8302.68	−1.29004
$347$	3435.10	0.531430	0.265715	−	0.964052i	$-0.414392\pi$
$347$	3435.10	0.531430	0.265715	+	0.964052i	$0.414392\pi$
$348$	0	0
$349$	−3034.99	−0.465499	−0.232750	−	0.972537i	$-0.574772\pi$
$349$	−3034.99	−0.465499	−0.232750	+	0.972537i	$0.574772\pi$
$350$	0	0
$351$	0	0
$352$	−1155.95	−0.175036
$353$	−8220.44	−1.23946	−0.619730	−	0.784815i	$-0.712758\pi$
$353$	−8220.44	−1.23946	−0.619730	+	0.784815i	$0.712758\pi$
$354$	0	0
$355$	−3516.85	−0.525789
$356$	3287.62	0.489447
$357$	0	0
$358$	8435.86	1.24539
$359$	9901.59	1.45567	0.727836	−	0.685752i	$-0.240526\pi$
$359$	9901.59	1.45567	0.727836	+	0.685752i	$0.240526\pi$
$360$	0	0
$361$	−5364.30	−0.782082
$362$	−7009.29	−1.01768
$363$	0	0
$364$	0	0
$365$	−1996.34	−0.286283
$366$	0	0
$367$	11028.0	1.56855	0.784276	−	0.620412i	$-0.213035\pi$
$367$	11028.0	1.56855	0.784276	+	0.620412i	$0.213035\pi$
$368$	−579.954	−0.0821527
$369$	0	0
$370$	−9.44721	−0.00132740
$371$	0	0
$372$	0	0
$373$	−2746.37	−0.381238	−0.190619	−	0.981664i	$-0.561050\pi$
$373$	−2746.37	−0.381238	−0.190619	+	0.981664i	$0.561050\pi$
$374$	−8558.55	−1.18329
$375$	0	0
$376$	−4021.67	−0.551601
$377$	−124.037	−0.0169450
$378$	0	0
$379$	−4184.22	−0.567095	−0.283547	−	0.958958i	$-0.591511\pi$
$379$	−4184.22	−0.567095	−0.283547	+	0.958958i	$0.591511\pi$
$380$	532.942	0.0719457
$381$	0	0
$382$	−3965.93	−0.531190
$383$	−3060.58	−0.408324	−0.204162	−	0.978937i	$-0.565447\pi$
$383$	−3060.58	−0.408324	−0.204162	+	0.978937i	$0.565447\pi$
$384$	0	0
$385$	0	0
$386$	−4223.48	−0.556916
$387$	0	0
$388$	−4432.66	−0.579985
$389$	−4919.61	−0.641219	−0.320610	−	0.947211i	$-0.603888\pi$
$389$	−4919.61	−0.641219	−0.320610	+	0.947211i	$0.603888\pi$
$390$	0	0
$391$	−4293.91	−0.555377
$392$	0	0
$393$	0	0
$394$	−7865.30	−1.00571
$395$	−3248.86	−0.413842
$396$	0	0
$397$	−13173.3	−1.66536	−0.832678	−	0.553757i	$-0.813194\pi$
$397$	−13173.3	−1.66536	−0.832678	+	0.553757i	$0.813194\pi$
$398$	1105.41	0.139219
$399$	0	0
$400$	−1809.98	−0.226247
$401$	2662.38	0.331553	0.165777	−	0.986163i	$-0.446987\pi$
$401$	2662.38	0.331553	0.165777	+	0.986163i	$0.446987\pi$
$402$	0	0
$403$	−1488.57	−0.183997
$404$	7543.69	0.928991
$405$	0	0
$406$	0	0
$407$	−49.5132	−0.00603016
$408$	0	0
$409$	−12992.2	−1.57071	−0.785357	−	0.619043i	$-0.787521\pi$
$409$	−12992.2	−1.57071	−0.785357	+	0.619043i	$0.787521\pi$
$410$	1157.93	0.139478
$411$	0	0
$412$	1928.58	0.230617
$413$	0	0
$414$	0	0
$415$	1635.97	0.193510
$416$	−327.396	−0.0385863
$417$	0	0
$418$	2793.17	0.326838
$419$	−7236.96	−0.843791	−0.421896	−	0.906644i	$-0.638635\pi$
$419$	−7236.96	−0.843791	−0.421896	+	0.906644i	$0.638635\pi$
$420$	0	0
$421$	3706.07	0.429032	0.214516	−	0.976720i	$-0.431183\pi$
$421$	3706.07	0.429032	0.214516	+	0.976720i	$0.431183\pi$
$422$	−3441.79	−0.397023
$423$	0	0
$424$	5000.90	0.572795
$425$	−13400.9	−1.52950
$426$	0	0
$427$	0	0
$428$	−6922.38	−0.781789
$429$	0	0
$430$	2063.40	0.231409
$431$	6382.38	0.713291	0.356645	−	0.934240i	$-0.383920\pi$
$431$	6382.38	0.713291	0.356645	+	0.934240i	$0.383920\pi$
$432$	0	0
$433$	−7275.62	−0.807492	−0.403746	−	0.914871i	$-0.632292\pi$
$433$	−7275.62	−0.807492	−0.403746	+	0.914871i	$0.632292\pi$
$434$	0	0
$435$	0	0
$436$	4595.32	0.504761
$437$	1401.36	0.153401
$438$	0	0
$439$	9760.36	1.06113	0.530566	−	0.847644i	$-0.321980\pi$
$439$	9760.36	1.06113	0.530566	+	0.847644i	$0.321980\pi$
$440$	995.918	0.107906
$441$	0	0
$442$	−2424.00	−0.260855
$443$	5733.16	0.614877	0.307439	−	0.951568i	$-0.400528\pi$
$443$	5733.16	0.614877	0.307439	+	0.951568i	$0.400528\pi$
$444$	0	0
$445$	−2832.46	−0.301734
$446$	−9576.87	−1.01677
$447$	0	0
$448$	0	0
$449$	−16628.4	−1.74775	−0.873877	−	0.486147i	$-0.838402\pi$
$449$	−16628.4	−1.74775	−0.873877	+	0.486147i	$0.838402\pi$
$450$	0	0
$451$	6068.76	0.633629
$452$	5326.02	0.554237
$453$	0	0
$454$	−8175.18	−0.845111
$455$	0	0
$456$	0	0
$457$	17119.3	1.75231	0.876157	−	0.482027i	$-0.160099\pi$
$457$	17119.3	1.75231	0.876157	+	0.482027i	$0.160099\pi$
$458$	8133.22	0.829782
$459$	0	0
$460$	499.662	0.0506454
$461$	6956.95	0.702858	0.351429	−	0.936215i	$-0.385696\pi$
$461$	6956.95	0.702858	0.351429	+	0.936215i	$0.385696\pi$
$462$	0	0
$463$	6594.47	0.661924	0.330962	−	0.943644i	$-0.392627\pi$
$463$	6594.47	0.661924	0.330962	+	0.943644i	$0.392627\pi$
$464$	−193.977	−0.0194077
$465$	0	0
$466$	−356.540	−0.0354430
$467$	−14465.8	−1.43340	−0.716700	−	0.697381i	$-0.754348\pi$
$467$	−14465.8	−1.43340	−0.716700	+	0.697381i	$0.754348\pi$
$468$	0	0
$469$	0	0
$470$	3464.90	0.340050
$471$	0	0
$472$	337.756	0.0329375
$473$	10814.3	1.05125
$474$	0	0
$475$	4373.51	0.422464
$476$	0	0
$477$	0	0
$478$	4237.34	0.405464
$479$	4075.47	0.388754	0.194377	−	0.980927i	$-0.437732\pi$
$479$	4075.47	0.388754	0.194377	+	0.980927i	$0.437732\pi$
$480$	0	0
$481$	−14.0234	−0.00132934
$482$	−4796.61	−0.453277
$483$	0	0
$484$	−104.355	−0.00980047
$485$	3818.98	0.357548
$486$	0	0
$487$	4378.79	0.407437	0.203719	−	0.979029i	$-0.434697\pi$
$487$	4378.79	0.407437	0.203719	+	0.979029i	$0.434697\pi$
$488$	3513.37	0.325908
$489$	0	0
$490$	0	0
$491$	−6612.37	−0.607764	−0.303882	−	0.952710i	$-0.598283\pi$
$491$	−6612.37	−0.607764	−0.303882	+	0.952710i	$0.598283\pi$
$492$	0	0
$493$	−1436.18	−0.131202
$494$	791.097	0.0720509
$495$	0	0
$496$	−2327.91	−0.210738
$497$	0	0
$498$	0	0
$499$	−11456.4	−1.02778	−0.513888	−	0.857857i	$-0.671795\pi$
$499$	−11456.4	−1.02778	−0.513888	+	0.857857i	$0.671795\pi$
$500$	3282.51	0.293596
$501$	0	0
$502$	3101.52	0.275752
$503$	−7697.10	−0.682300	−0.341150	−	0.940009i	$-0.610816\pi$
$503$	−7697.10	−0.682300	−0.341150	+	0.940009i	$0.610816\pi$
$504$	0	0
$505$	−6499.30	−0.572704
$506$	2618.75	0.230074
$507$	0	0
$508$	−3320.94	−0.290045
$509$	4623.97	0.402659	0.201330	−	0.979524i	$-0.435474\pi$
$509$	4623.97	0.402659	0.201330	+	0.979524i	$0.435474\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	10790.9	0.926003
$515$	−1661.58	−0.142170
$516$	0	0
$517$	18159.6	1.54480
$518$	0	0
$519$	0	0
$520$	282.069	0.0237876
$521$	3486.70	0.293196	0.146598	−	0.989196i	$-0.453168\pi$
$521$	3486.70	0.293196	0.146598	+	0.989196i	$0.453168\pi$
$522$	0	0
$523$	−4415.13	−0.369140	−0.184570	−	0.982819i	$-0.559089\pi$
$523$	−4415.13	−0.369140	−0.184570	+	0.982819i	$0.559089\pi$
$524$	7889.96	0.657776
$525$	0	0
$526$	−7163.86	−0.593839
$527$	−17235.6	−1.42465
$528$	0	0
$529$	−10853.1	−0.892015
$530$	−4308.55	−0.353116
$531$	0	0
$532$	0	0
$533$	1718.83	0.139682
$534$	0	0
$535$	5964.01	0.481957
$536$	6108.94	0.492288
$537$	0	0
$538$	2351.10	0.188407
$539$	0	0
$540$	0	0
$541$	−10804.5	−0.858634	−0.429317	−	0.903154i	$-0.641246\pi$
$541$	−10804.5	−0.858634	−0.429317	+	0.903154i	$0.641246\pi$
$542$	−518.654	−0.0411035
$543$	0	0
$544$	−3790.79	−0.298766
$545$	−3959.12	−0.311175
$546$	0	0
$547$	18783.4	1.46823	0.734113	−	0.679027i	$-0.237598\pi$
$547$	18783.4	1.46823	0.734113	+	0.679027i	$0.237598\pi$
$548$	−275.150	−0.0214486
$549$	0	0
$550$	8172.85	0.633621
$551$	468.713	0.0362393
$552$	0	0
$553$	0	0
$554$	2597.85	0.199227
$555$	0	0
$556$	7457.91	0.568860
$557$	1451.14	0.110389	0.0551944	−	0.998476i	$-0.482422\pi$
$557$	1451.14	0.110389	0.0551944	+	0.998476i	$0.482422\pi$
$558$	0	0
$559$	3062.89	0.231747
$560$	0	0
$561$	0	0
$562$	9048.49	0.679160
$563$	−16781.1	−1.25620	−0.628098	−	0.778134i	$-0.716166\pi$
$563$	−16781.1	−1.25620	−0.628098	+	0.778134i	$0.716166\pi$
$564$	0	0
$565$	−4588.66	−0.341675
$566$	−12956.1	−0.962165
$567$	0	0
$568$	−8163.95	−0.603084
$569$	14722.8	1.08473	0.542367	−	0.840142i	$-0.317528\pi$
$569$	14722.8	1.08473	0.542367	+	0.840142i	$0.317528\pi$
$570$	0	0
$571$	9717.12	0.712169	0.356085	−	0.934454i	$-0.384111\pi$
$571$	9717.12	0.712169	0.356085	+	0.934454i	$0.384111\pi$
$572$	1478.34	0.108064
$573$	0	0
$574$	0	0
$575$	4100.40	0.297389
$576$	0	0
$577$	−20188.9	−1.45663	−0.728313	−	0.685244i	$-0.759695\pi$
$577$	−20188.9	−1.45663	−0.728313	+	0.685244i	$0.759695\pi$
$578$	−18240.6	−1.31264
$579$	0	0
$580$	167.122	0.0119644
$581$	0	0
$582$	0	0
$583$	−22581.3	−1.60415
$584$	−4634.27	−0.328369
$585$	0	0
$586$	5781.83	0.407586
$587$	21720.8	1.52728	0.763641	−	0.645641i	$-0.223410\pi$
$587$	21720.8	1.52728	0.763641	+	0.645641i	$0.223410\pi$
$588$	0	0
$589$	5625.00	0.393504
$590$	−290.996	−0.0203053
$591$	0	0
$592$	−21.9306	−0.00152254
$593$	19524.3	1.35205	0.676025	−	0.736879i	$-0.263701\pi$
$593$	19524.3	1.35205	0.676025	+	0.736879i	$0.263701\pi$
$594$	0	0
$595$	0	0
$596$	−81.1735	−0.00557885
$597$	0	0
$598$	741.696	0.0507194
$599$	−11598.7	−0.791167	−0.395583	−	0.918430i	$-0.629458\pi$
$599$	−11598.7	−0.791167	−0.395583	+	0.918430i	$0.629458\pi$
$600$	0	0
$601$	−9335.68	−0.633628	−0.316814	−	0.948488i	$-0.602613\pi$
$601$	−9335.68	−0.633628	−0.316814	+	0.948488i	$0.602613\pi$
$602$	0	0
$603$	0	0
$604$	−6321.30	−0.425844
$605$	89.9080	0.00604178
$606$	0	0
$607$	15150.9	1.01311	0.506553	−	0.862209i	$-0.330919\pi$
$607$	15150.9	1.01311	0.506553	+	0.862209i	$0.330919\pi$
$608$	1237.16	0.0825223
$609$	0	0
$610$	−3026.97	−0.200915
$611$	5143.27	0.340548
$612$	0	0
$613$	24793.4	1.63360	0.816798	−	0.576923i	$-0.195747\pi$
$613$	24793.4	1.63360	0.816798	+	0.576923i	$0.195747\pi$
$614$	6274.84	0.412430
$615$	0	0
$616$	0	0
$617$	25194.6	1.64391	0.821957	−	0.569550i	$-0.192882\pi$
$617$	25194.6	1.64391	0.821957	+	0.569550i	$0.192882\pi$
$618$	0	0
$619$	−6046.47	−0.392614	−0.196307	−	0.980542i	$-0.562895\pi$
$619$	−6046.47	−0.392614	−0.196307	+	0.980542i	$0.562895\pi$
$620$	2005.62	0.129916
$621$	0	0
$622$	15829.9	1.02045
$623$	0	0
$624$	0	0
$625$	11312.4	0.723992
$626$	−15233.2	−0.972591
$627$	0	0
$628$	13537.4	0.860193
$629$	−162.372	−0.0102928
$630$	0	0
$631$	−6537.14	−0.412424	−0.206212	−	0.978507i	$-0.566114\pi$
$631$	−6537.14	−0.412424	−0.206212	+	0.978507i	$0.566114\pi$
$632$	−7541.84	−0.474681
$633$	0	0
$634$	3982.98	0.249502
$635$	2861.18	0.178807
$636$	0	0
$637$	0	0
$638$	875.892	0.0543525
$639$	0	0
$640$	441.116	0.0272448
$641$	−9246.78	−0.569775	−0.284888	−	0.958561i	$-0.591956\pi$
$641$	−9246.78	−0.569775	−0.284888	+	0.958561i	$0.591956\pi$
$642$	0	0
$643$	−157.563	−0.00966355	−0.00483178	−	0.999988i	$-0.501538\pi$
$643$	−157.563	−0.00966355	−0.00483178	+	0.999988i	$0.501538\pi$
$644$	0	0
$645$	0	0
$646$	9159.81	0.557876
$647$	12706.0	0.772062	0.386031	−	0.922486i	$-0.373846\pi$
$647$	12706.0	0.772062	0.386031	+	0.922486i	$0.373846\pi$
$648$	0	0
$649$	−1525.12	−0.0922437
$650$	2314.76	0.139680
$651$	0	0
$652$	6286.83	0.377624
$653$	−17104.8	−1.02506	−0.512529	−	0.858670i	$-0.671291\pi$
$653$	−17104.8	−1.02506	−0.512529	+	0.858670i	$0.671291\pi$
$654$	0	0
$655$	−6797.64	−0.405505
$656$	2688.00	0.159983
$657$	0	0
$658$	0	0
$659$	12518.0	0.739956	0.369978	−	0.929040i	$-0.379365\pi$
$659$	12518.0	0.739956	0.369978	+	0.929040i	$0.379365\pi$
$660$	0	0
$661$	−4780.48	−0.281300	−0.140650	−	0.990059i	$-0.544919\pi$
$661$	−4780.48	−0.281300	−0.140650	+	0.990059i	$0.544919\pi$
$662$	−5697.26	−0.334487
$663$	0	0
$664$	3797.71	0.221957
$665$	0	0
$666$	0	0
$667$	439.444	0.0255102
$668$	9895.69	0.573167
$669$	0	0
$670$	−5263.19	−0.303485
$671$	−15864.4	−0.912727
$672$	0	0
$673$	−28447.0	−1.62935	−0.814673	−	0.579920i	$-0.803084\pi$
$673$	−28447.0	−1.62935	−0.814673	+	0.579920i	$0.803084\pi$
$674$	11627.7	0.664516
$675$	0	0
$676$	−8369.30	−0.476178
$677$	15878.5	0.901421	0.450710	−	0.892670i	$-0.351171\pi$
$677$	15878.5	0.901421	0.450710	+	0.892670i	$0.351171\pi$
$678$	0	0
$679$	0	0
$680$	3265.98	0.184183
$681$	0	0
$682$	10511.5	0.590187
$683$	−2438.46	−0.136610	−0.0683052	−	0.997664i	$-0.521759\pi$
$683$	−2438.46	−0.136610	−0.0683052	+	0.997664i	$0.521759\pi$
$684$	0	0
$685$	237.057	0.0132226
$686$	0	0
$687$	0	0
$688$	4789.93	0.265428
$689$	−6395.59	−0.353632
$690$	0	0
$691$	22486.9	1.23798	0.618990	−	0.785399i	$-0.287542\pi$
$691$	22486.9	1.23798	0.618990	+	0.785399i	$0.287542\pi$
$692$	16605.4	0.912198
$693$	0	0
$694$	−6870.21	−0.375777
$695$	−6425.41	−0.350690
$696$	0	0
$697$	19901.7	1.08153
$698$	6069.98	0.329158
$699$	0	0
$700$	0	0
$701$	−3916.29	−0.211008	−0.105504	−	0.994419i	$-0.533646\pi$
$701$	−3916.29	−0.211008	−0.105504	+	0.994419i	$0.533646\pi$
$702$	0	0
$703$	52.9916	0.00284298
$704$	2311.91	0.123769
$705$	0	0
$706$	16440.9	0.876431
$707$	0	0
$708$	0	0
$709$	−22910.3	−1.21356	−0.606780	−	0.794869i	$-0.707539\pi$
$709$	−22910.3	−1.21356	−0.606780	+	0.794869i	$0.707539\pi$
$710$	7033.70	0.371789
$711$	0	0
$712$	−6575.23	−0.346092
$713$	5273.74	0.277003
$714$	0	0
$715$	−1273.67	−0.0666189
$716$	−16871.7	−0.880623
$717$	0	0
$718$	−19803.2	−1.02932
$719$	20472.7	1.06190	0.530949	−	0.847404i	$-0.321836\pi$
$719$	20472.7	1.06190	0.530949	+	0.847404i	$0.321836\pi$
$720$	0	0
$721$	0	0
$722$	10728.6	0.553016
$723$	0	0
$724$	14018.6	0.719609
$725$	1371.46	0.0702548
$726$	0	0
$727$	11208.5	0.571802	0.285901	−	0.958259i	$-0.407707\pi$
$727$	11208.5	0.571802	0.285901	+	0.958259i	$0.407707\pi$
$728$	0	0
$729$	0	0
$730$	3992.68	0.202432
$731$	35464.1	1.79437
$732$	0	0
$733$	−35438.0	−1.78572	−0.892860	−	0.450334i	$-0.851305\pi$
$733$	−35438.0	−1.78572	−0.892860	+	0.450334i	$0.851305\pi$
$734$	−22056.1	−1.10913
$735$	0	0
$736$	1159.91	0.0580907
$737$	−27584.6	−1.37869
$738$	0	0
$739$	−17935.4	−0.892779	−0.446389	−	0.894839i	$-0.647290\pi$
$739$	−17935.4	−0.892779	−0.446389	+	0.894839i	$0.647290\pi$
$740$	18.8944	0.000938612 0
$741$	0	0
$742$	0	0
$743$	19031.0	0.939676	0.469838	−	0.882753i	$-0.344312\pi$
$743$	19031.0	0.939676	0.469838	+	0.882753i	$0.344312\pi$
$744$	0	0
$745$	69.9355	0.00343925
$746$	5492.75	0.269576
$747$	0	0
$748$	17117.1	0.836716
$749$	0	0
$750$	0	0
$751$	−2728.13	−0.132558	−0.0662790	−	0.997801i	$-0.521113\pi$
$751$	−2728.13	−0.132558	−0.0662790	+	0.997801i	$0.521113\pi$
$752$	8043.35	0.390041
$753$	0	0
$754$	248.075	0.0119819
$755$	5446.15	0.262524
$756$	0	0
$757$	−6617.58	−0.317728	−0.158864	−	0.987300i	$-0.550783\pi$
$757$	−6617.58	−0.317728	−0.158864	+	0.987300i	$0.550783\pi$
$758$	8368.44	0.400997
$759$	0	0
$760$	−1065.88	−0.0508733
$761$	−600.108	−0.0285860	−0.0142930	−	0.999898i	$-0.504550\pi$
$761$	−600.108	−0.0285860	−0.0142930	+	0.999898i	$0.504550\pi$
$762$	0	0
$763$	0	0
$764$	7931.86	0.375608
$765$	0	0
$766$	6121.16	0.288729
$767$	−431.953	−0.0203350
$768$	0	0
$769$	−28329.2	−1.32845	−0.664223	−	0.747534i	$-0.731238\pi$
$769$	−28329.2	−1.32845	−0.664223	+	0.747534i	$0.731238\pi$
$770$	0	0
$771$	0	0
$772$	8446.97	0.393799
$773$	23837.0	1.10913	0.554565	−	0.832140i	$-0.312884\pi$
$773$	23837.0	1.10913	0.554565	+	0.832140i	$0.312884\pi$
$774$	0	0
$775$	16458.8	0.762862
$776$	8865.32	0.410111
$777$	0	0
$778$	9839.23	0.453411
$779$	−6495.10	−0.298731
$780$	0	0
$781$	36863.9	1.68898
$782$	8587.83	0.392711
$783$	0	0
$784$	0	0
$785$	−11663.2	−0.530291
$786$	0	0
$787$	−536.910	−0.0243186	−0.0121593	−	0.999926i	$-0.503871\pi$
$787$	−536.910	−0.0243186	−0.0121593	+	0.999926i	$0.503871\pi$
$788$	15730.6	0.711141
$789$	0	0
$790$	6497.71	0.292631
$791$	0	0
$792$	0	0
$793$	−4493.21	−0.201209
$794$	26346.5	1.17758
$795$	0	0
$796$	−2210.82	−0.0984426
$797$	−8857.47	−0.393661	−0.196830	−	0.980438i	$-0.563065\pi$
$797$	−8857.47	−0.393661	−0.196830	+	0.980438i	$0.563065\pi$
$798$	0	0
$799$	59552.1	2.63680
$800$	3619.95	0.159981
$801$	0	0
$802$	−5324.76	−0.234444
$803$	20925.8	0.919620
$804$	0	0
$805$	0	0
$806$	2977.13	0.130106
$807$	0	0
$808$	−15087.4	−0.656896
$809$	−21832.9	−0.948832	−0.474416	−	0.880301i	$-0.657341\pi$
$809$	−21832.9	−0.948832	−0.474416	+	0.880301i	$0.657341\pi$
$810$	0	0
$811$	−37942.3	−1.64283	−0.821415	−	0.570330i	$-0.806815\pi$
$811$	−37942.3	−1.64283	−0.821415	+	0.570330i	$0.806815\pi$
$812$	0	0
$813$	0	0
$814$	99.0264	0.00426397
$815$	−5416.45	−0.232798
$816$	0	0
$817$	−11574.1	−0.495625
$818$	25984.4	1.11066
$819$	0	0
$820$	−2315.86	−0.0986261
$821$	18077.3	0.768456	0.384228	−	0.923238i	$-0.374468\pi$
$821$	18077.3	0.768456	0.384228	+	0.923238i	$0.374468\pi$
$822$	0	0
$823$	−4902.75	−0.207654	−0.103827	−	0.994595i	$-0.533109\pi$
$823$	−4902.75	−0.207654	−0.103827	+	0.994595i	$0.533109\pi$
$824$	−3857.15	−0.163071
$825$	0	0
$826$	0	0
$827$	19578.9	0.823247	0.411623	−	0.911354i	$-0.364962\pi$
$827$	19578.9	0.823247	0.411623	+	0.911354i	$0.364962\pi$
$828$	0	0
$829$	−37930.2	−1.58911	−0.794554	−	0.607194i	$-0.792295\pi$
$829$	−37930.2	−1.58911	−0.794554	+	0.607194i	$0.792295\pi$
$830$	−3271.94	−0.136832
$831$	0	0
$832$	654.791	0.0272846
$833$	0	0
$834$	0	0
$835$	−8525.68	−0.353345
$836$	−5586.34	−0.231110
$837$	0	0
$838$	14473.9	0.596651
$839$	−5954.22	−0.245009	−0.122504	−	0.992468i	$-0.539093\pi$
$839$	−5954.22	−0.245009	−0.122504	+	0.992468i	$0.539093\pi$
$840$	0	0
$841$	−24242.0	−0.993973
$842$	−7412.13	−0.303372
$843$	0	0
$844$	6883.58	0.280738
$845$	7210.61	0.293553
$846$	0	0
$847$	0	0
$848$	−10001.8	−0.405027
$849$	0	0
$850$	26801.7	1.08152
$851$	49.6825	0.00200129
$852$	0	0
$853$	−26196.2	−1.05151	−0.525757	−	0.850635i	$-0.676218\pi$
$853$	−26196.2	−1.05151	−0.525757	+	0.850635i	$0.676218\pi$
$854$	0	0
$855$	0	0
$856$	13844.8	0.552808
$857$	36877.2	1.46990	0.734949	−	0.678122i	$-0.237206\pi$
$857$	36877.2	1.46990	0.734949	+	0.678122i	$0.237206\pi$
$858$	0	0
$859$	34478.2	1.36948	0.684740	−	0.728788i	$-0.259916\pi$
$859$	34478.2	1.36948	0.684740	+	0.728788i	$0.259916\pi$
$860$	−4126.79	−0.163631
$861$	0	0
$862$	−12764.8	−0.504373
$863$	1679.69	0.0662542	0.0331271	−	0.999451i	$-0.489453\pi$
$863$	1679.69	0.0662542	0.0331271	+	0.999451i	$0.489453\pi$
$864$	0	0
$865$	−14306.4	−0.562351
$866$	14551.2	0.570983
$867$	0	0
$868$	0	0
$869$	34054.7	1.32938
$870$	0	0
$871$	−7812.66	−0.303929
$872$	−9190.64	−0.356920
$873$	0	0
$874$	−2802.72	−0.108471
$875$	0	0
$876$	0	0
$877$	2503.82	0.0964060	0.0482030	−	0.998838i	$-0.484651\pi$
$877$	2503.82	0.0964060	0.0482030	+	0.998838i	$0.484651\pi$
$878$	−19520.7	−0.750333
$879$	0	0
$880$	−1991.84	−0.0763009
$881$	34886.0	1.33410	0.667048	−	0.745014i	$-0.267557\pi$
$881$	34886.0	1.33410	0.667048	+	0.745014i	$0.267557\pi$
$882$	0	0
$883$	35334.0	1.34664	0.673320	−	0.739351i	$-0.264868\pi$
$883$	35334.0	1.34664	0.673320	+	0.739351i	$0.264868\pi$
$884$	4848.00	0.184452
$885$	0	0
$886$	−11466.3	−0.434784
$887$	26652.8	1.00892	0.504460	−	0.863435i	$-0.331692\pi$
$887$	26652.8	1.00892	0.504460	+	0.863435i	$0.331692\pi$
$888$	0	0
$889$	0	0
$890$	5664.93	0.213358
$891$	0	0
$892$	19153.7	0.718963
$893$	−19435.4	−0.728311
$894$	0	0
$895$	14535.9	0.542885
$896$	0	0
$897$	0	0
$898$	33256.7	1.23585
$899$	1763.91	0.0654389
$900$	0	0
$901$	−74052.2	−2.73811
$902$	−12137.5	−0.448043
$903$	0	0
$904$	−10652.0	−0.391905
$905$	−12077.8	−0.443624
$906$	0	0
$907$	−46055.0	−1.68603	−0.843016	−	0.537888i	$-0.819222\pi$
$907$	−46055.0	−1.68603	−0.843016	+	0.537888i	$0.819222\pi$
$908$	16350.4	0.597584
$909$	0	0
$910$	0	0
$911$	−14227.1	−0.517416	−0.258708	−	0.965956i	$-0.583297\pi$
$911$	−14227.1	−0.517416	−0.258708	+	0.965956i	$0.583297\pi$
$912$	0	0
$913$	−17148.3	−0.621607
$914$	−34238.6	−1.23907
$915$	0	0
$916$	−16266.4	−0.586745
$917$	0	0
$918$	0	0
$919$	6686.95	0.240024	0.120012	−	0.992772i	$-0.461707\pi$
$919$	6686.95	0.240024	0.120012	+	0.992772i	$0.461707\pi$
$920$	−999.325	−0.0358117
$921$	0	0
$922$	−13913.9	−0.496996
$923$	10440.8	0.372332
$924$	0	0
$925$	155.054	0.00551151
$926$	−13188.9	−0.468051
$927$	0	0
$928$	387.954	0.0137233
$929$	13539.6	0.478172	0.239086	−	0.970998i	$-0.423152\pi$
$929$	13539.6	0.478172	0.239086	+	0.970998i	$0.423152\pi$
$930$	0	0
$931$	0	0
$932$	713.081	0.0250620
$933$	0	0
$934$	28931.6	1.01357
$935$	−14747.3	−0.515818
$936$	0	0
$937$	−12017.0	−0.418974	−0.209487	−	0.977811i	$-0.567179\pi$
$937$	−12017.0	−0.418974	−0.209487	+	0.977811i	$0.567179\pi$
$938$	0	0
$939$	0	0
$940$	−6929.79	−0.240452
$941$	−55239.5	−1.91366	−0.956831	−	0.290646i	$-0.906130\pi$
$941$	−55239.5	−1.91366	−0.956831	+	0.290646i	$0.906130\pi$
$942$	0	0
$943$	−6089.51	−0.210288
$944$	−675.513	−0.0232903
$945$	0	0
$946$	−21628.7	−0.743349
$947$	47989.8	1.64674	0.823368	−	0.567508i	$-0.192093\pi$
$947$	47989.8	1.64674	0.823368	+	0.567508i	$0.192093\pi$
$948$	0	0
$949$	5926.71	0.202728
$950$	−8747.02	−0.298727
$951$	0	0
$952$	0	0
$953$	19900.0	0.676415	0.338207	−	0.941072i	$-0.390179\pi$
$953$	19900.0	0.676415	0.338207	+	0.941072i	$0.390179\pi$
$954$	0	0
$955$	−6833.74	−0.231555
$956$	−8474.69	−0.286706
$957$	0	0
$958$	−8150.94	−0.274890
$959$	0	0
$960$	0	0
$961$	−8622.43	−0.289431
$962$	28.0468	0.000939984 0
$963$	0	0
$964$	9593.23	0.320516
$965$	−7277.53	−0.242769
$966$	0	0
$967$	−423.293	−0.0140767	−0.00703836	−	0.999975i	$-0.502240\pi$
$967$	−423.293	−0.0140767	−0.00703836	+	0.999975i	$0.502240\pi$
$968$	208.711	0.00692998
$969$	0	0
$970$	−7637.96	−0.252825
$971$	−1744.60	−0.0576590	−0.0288295	−	0.999584i	$-0.509178\pi$
$971$	−1744.60	−0.0576590	−0.0288295	+	0.999584i	$0.509178\pi$
$972$	0	0
$973$	0	0
$974$	−8757.58	−0.288102
$975$	0	0
$976$	−7026.74	−0.230451
$977$	3661.34	0.119894	0.0599472	−	0.998202i	$-0.480907\pi$
$977$	3661.34	0.119894	0.0599472	+	0.998202i	$0.480907\pi$
$978$	0	0
$979$	29690.1	0.969254
$980$	0	0
$981$	0	0
$982$	13224.7	0.429754
$983$	27194.1	0.882358	0.441179	−	0.897419i	$-0.354560\pi$
$983$	27194.1	0.882358	0.441179	+	0.897419i	$0.354560\pi$
$984$	0	0
$985$	−13552.8	−0.438404
$986$	2872.37	0.0927736
$987$	0	0
$988$	−1582.19	−0.0509477
$989$	−10851.3	−0.348890
$990$	0	0
$991$	−13539.7	−0.434008	−0.217004	−	0.976171i	$-0.569629\pi$
$991$	−13539.7	−0.434008	−0.217004	+	0.976171i	$0.569629\pi$
$992$	4655.81	0.149014
$993$	0	0
$994$	0	0
$995$	1904.74	0.0606878
$996$	0	0
$997$	22863.8	0.726283	0.363142	−	0.931734i	$-0.381704\pi$
$997$	22863.8	0.726283	0.363142	+	0.931734i	$0.381704\pi$
$998$	22912.9	0.726747
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.ba.1.1		2
3.2	odd	2		882.4.a.bd.1.2		2
7.2	even	3		126.4.g.f.109.2	yes	4
7.3	odd	6		882.4.g.bj.667.1		4
7.4	even	3		126.4.g.f.37.2	yes	4
7.5	odd	6		882.4.g.bj.361.1		4
7.6	odd	2		882.4.a.u.1.2		2
21.2	odd	6		126.4.g.e.109.1	yes	4
21.5	even	6		882.4.g.z.361.2		4
21.11	odd	6		126.4.g.e.37.1	✓	4
21.17	even	6		882.4.g.z.667.2		4
21.20	even	2		882.4.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.g.e.37.1	✓	4	21.11	odd	6
126.4.g.e.109.1	yes	4	21.2	odd	6
126.4.g.f.37.2	yes	4	7.4	even	3
126.4.g.f.109.2	yes	4	7.2	even	3
882.4.a.u.1.2		2	7.6	odd	2
882.4.a.ba.1.1		2	1.1	even	1	trivial
882.4.a.bd.1.2		2	3.2	odd	2
882.4.a.bh.1.1		2	21.20	even	2
882.4.g.z.361.2		4	21.5	even	6
882.4.g.z.667.2		4	21.17	even	6
882.4.g.bj.361.1		4	7.5	odd	6
882.4.g.bj.667.1		4	7.3	odd	6

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -3.44622 q^{5} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -3.44622 q^{5} -8.00000 q^{8} +6.89244 q^{10} +36.1236 q^{11} +10.2311 q^{13} +16.0000 q^{16} +118.462 q^{17} -38.6613 q^{19} -13.7849 q^{20} -72.2471 q^{22} -36.2471 q^{23} -113.124 q^{25} -20.4622 q^{26} -12.1236 q^{29} -145.494 q^{31} -32.0000 q^{32} -236.924 q^{34} -1.37066 q^{37} +77.3227 q^{38} +27.5698 q^{40} +168.000 q^{41} +299.371 q^{43} +144.494 q^{44} +72.4942 q^{46} +502.709 q^{47} +226.247 q^{50} +40.9244 q^{52} -625.112 q^{53} -124.490 q^{55} +24.2471 q^{58} -42.2195 q^{59} -439.172 q^{61} +290.988 q^{62} +64.0000 q^{64} -35.2587 q^{65} -763.618 q^{67} +473.849 q^{68} +1020.49 q^{71} +579.284 q^{73} +2.74132 q^{74} -154.645 q^{76} +942.730 q^{79} -55.1396 q^{80} -336.000 q^{82} -474.714 q^{83} -408.247 q^{85} -598.741 q^{86} -288.988 q^{88} +821.904 q^{89} -144.988 q^{92} -1005.42 q^{94} +133.236 q^{95} -1108.16 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 + 7 * q^5 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} + 7 q^{5} - 16 q^{8} - 14 q^{10} - 25 q^{11} - 49 q^{13} + 32 q^{16} + 98 q^{17} - 119 q^{19} + 28 q^{20} + 50 q^{22} + 122 q^{23} - 129 q^{25} + 98 q^{26} + 73 q^{29} + 98 q^{31} - 64 q^{32} - 196 q^{34} + 289 q^{37} + 238 q^{38} - 56 q^{40} + 336 q^{41} + 307 q^{43} - 100 q^{44} - 244 q^{46} + 672 q^{47} + 258 q^{50} - 196 q^{52} - 375 q^{53} - 763 q^{55} - 146 q^{58} + 763 q^{59} - 406 q^{61} - 196 q^{62} + 128 q^{64} - 654 q^{65} - 1041 q^{67} + 392 q^{68} + 1652 q^{71} - 189 q^{73} - 578 q^{74} - 476 q^{76} + 524 q^{79} + 112 q^{80} - 672 q^{82} + 287 q^{83} - 622 q^{85} - 614 q^{86} + 200 q^{88} + 2394 q^{89} + 488 q^{92} - 1344 q^{94} - 706 q^{95} - 63 q^{97}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 + 7 * q^5 - 16 * q^8 - 14 * q^10 - 25 * q^11 - 49 * q^13 + 32 * q^16 + 98 * q^17 - 119 * q^19 + 28 * q^20 + 50 * q^22 + 122 * q^23 - 129 * q^25 + 98 * q^26 + 73 * q^29 + 98 * q^31 - 64 * q^32 - 196 * q^34 + 289 * q^37 + 238 * q^38 - 56 * q^40 + 336 * q^41 + 307 * q^43 - 100 * q^44 - 244 * q^46 + 672 * q^47 + 258 * q^50 - 196 * q^52 - 375 * q^53 - 763 * q^55 - 146 * q^58 + 763 * q^59 - 406 * q^61 - 196 * q^62 + 128 * q^64 - 654 * q^65 - 1041 * q^67 + 392 * q^68 + 1652 * q^71 - 189 * q^73 - 578 * q^74 - 476 * q^76 + 524 * q^79 + 112 * q^80 - 672 * q^82 + 287 * q^83 - 622 * q^85 - 614 * q^86 + 200 * q^88 + 2394 * q^89 + 488 * q^92 - 1344 * q^94 - 706 * q^95 - 63 * q^97

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