LMFDB - Embedded newform 882.4.a.bi.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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -3.89949 q^{5} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} -3.89949 q^{5} +8.00000 q^{8} -7.79899 q^{10} +61.3970 q^{11} -53.6985 q^{13} +16.0000 q^{16} +32.1005 q^{17} +55.7990 q^{19} -15.5980 q^{20} +122.794 q^{22} +94.6030 q^{23} -109.794 q^{25} -107.397 q^{26} -138.191 q^{29} -132.603 q^{31} +32.0000 q^{32} +64.2010 q^{34} +149.206 q^{37} +111.598 q^{38} -31.1960 q^{40} +427.497 q^{41} +437.588 q^{43} +245.588 q^{44} +189.206 q^{46} +57.0051 q^{47} -219.588 q^{50} -214.794 q^{52} +263.588 q^{53} -239.417 q^{55} -276.382 q^{58} +451.799 q^{59} -579.307 q^{61} -265.206 q^{62} +64.0000 q^{64} +209.397 q^{65} +309.588 q^{67} +128.402 q^{68} +1058.98 q^{71} -1193.66 q^{73} +298.412 q^{74} +223.196 q^{76} +1319.56 q^{79} -62.3919 q^{80} +854.995 q^{82} +1190.33 q^{83} -125.176 q^{85} +875.176 q^{86} +491.176 q^{88} -233.085 q^{89} +378.412 q^{92} +114.010 q^{94} -217.588 q^{95} +1609.44 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8} + 24 q^{10} + 4 q^{11} - 48 q^{13} + 32 q^{16} + 84 q^{17} + 72 q^{19} + 48 q^{20} + 8 q^{22} + 308 q^{23} + 18 q^{25} - 96 q^{26} + 80 q^{29} - 384 q^{31} + 64 q^{32} + 168 q^{34} + 536 q^{37} + 144 q^{38} + 96 q^{40} + 756 q^{41} + 400 q^{43} + 16 q^{44} + 616 q^{46} + 312 q^{47} + 36 q^{50} - 192 q^{52} + 52 q^{53} - 1152 q^{55} + 160 q^{58} + 864 q^{59} - 1416 q^{61} - 768 q^{62} + 128 q^{64} + 300 q^{65} + 144 q^{67} + 336 q^{68} + 1524 q^{71} - 744 q^{73} + 1072 q^{74} + 288 q^{76} + 976 q^{79} + 192 q^{80} + 1512 q^{82} - 312 q^{83} + 700 q^{85} + 800 q^{86} + 32 q^{88} + 108 q^{89} + 1232 q^{92} + 624 q^{94} + 40 q^{95} + 744 q^{97}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8 + 24 * q^10 + 4 * q^11 - 48 * q^13 + 32 * q^16 + 84 * q^17 + 72 * q^19 + 48 * q^20 + 8 * q^22 + 308 * q^23 + 18 * q^25 - 96 * q^26 + 80 * q^29 - 384 * q^31 + 64 * q^32 + 168 * q^34 + 536 * q^37 + 144 * q^38 + 96 * q^40 + 756 * q^41 + 400 * q^43 + 16 * q^44 + 616 * q^46 + 312 * q^47 + 36 * q^50 - 192 * q^52 + 52 * q^53 - 1152 * q^55 + 160 * q^58 + 864 * q^59 - 1416 * q^61 - 768 * q^62 + 128 * q^64 + 300 * q^65 + 144 * q^67 + 336 * q^68 + 1524 * q^71 - 744 * q^73 + 1072 * q^74 + 288 * q^76 + 976 * q^79 + 192 * q^80 + 1512 * q^82 - 312 * q^83 + 700 * q^85 + 800 * q^86 + 32 * q^88 + 108 * q^89 + 1232 * q^92 + 624 * q^94 + 40 * q^95 + 744 * 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
$5$	−3.89949	−0.348781	−0.174391	−	0.984677i	$-0.555796\pi$
$5$	−3.89949	−0.348781	−0.174391	+	0.984677i	$0.555796\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	−7.79899	−0.246626
$11$	61.3970	1.68290	0.841449	−	0.540336i	$-0.181703\pi$
$11$	61.3970	1.68290	0.841449	+	0.540336i	$0.181703\pi$
$12$	0	0
$13$	−53.6985	−1.14564	−0.572818	−	0.819682i	$-0.694150\pi$
$13$	−53.6985	−1.14564	−0.572818	+	0.819682i	$0.694150\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	32.1005	0.457972	0.228986	−	0.973430i	$-0.426459\pi$
$17$	32.1005	0.457972	0.228986	+	0.973430i	$0.426459\pi$
$18$	0	0
$19$	55.7990	0.673746	0.336873	−	0.941550i	$-0.390631\pi$
$19$	55.7990	0.673746	0.336873	+	0.941550i	$0.390631\pi$
$20$	−15.5980	−0.174391
$21$	0	0
$22$	122.794	1.18999
$23$	94.6030	0.857656	0.428828	−	0.903386i	$-0.358927\pi$
$23$	94.6030	0.857656	0.428828	+	0.903386i	$0.358927\pi$
$24$	0	0
$25$	−109.794	−0.878352
$26$	−107.397	−0.810088
$27$	0	0
$28$	0	0
$29$	−138.191	−0.884876	−0.442438	−	0.896799i	$-0.645886\pi$
$29$	−138.191	−0.884876	−0.442438	+	0.896799i	$0.645886\pi$
$30$	0	0
$31$	−132.603	−0.768265	−0.384132	−	0.923278i	$-0.625499\pi$
$31$	−132.603	−0.768265	−0.384132	+	0.923278i	$0.625499\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	64.2010	0.323835
$35$	0	0
$36$	0	0
$37$	149.206	0.662955	0.331477	−	0.943463i	$-0.392453\pi$
$37$	149.206	0.662955	0.331477	+	0.943463i	$0.392453\pi$
$38$	111.598	0.476410
$39$	0	0
$40$	−31.1960	−0.123313
$41$	427.497	1.62839	0.814194	−	0.580593i	$-0.197179\pi$
$41$	427.497	1.62839	0.814194	+	0.580593i	$0.197179\pi$
$42$	0	0
$43$	437.588	1.55190	0.775948	−	0.630797i	$-0.217272\pi$
$43$	437.588	1.55190	0.775948	+	0.630797i	$0.217272\pi$
$44$	245.588	0.841449
$45$	0	0
$46$	189.206	0.606455
$47$	57.0051	0.176916	0.0884579	−	0.996080i	$-0.471806\pi$
$47$	57.0051	0.176916	0.0884579	+	0.996080i	$0.471806\pi$
$48$	0	0
$49$	0	0
$50$	−219.588	−0.621088
$51$	0	0
$52$	−214.794	−0.572818
$53$	263.588	0.683143	0.341572	−	0.939856i	$-0.389041\pi$
$53$	263.588	0.683143	0.341572	+	0.939856i	$0.389041\pi$
$54$	0	0
$55$	−239.417	−0.586964
$56$	0	0
$57$	0	0
$58$	−276.382	−0.625702
$59$	451.799	0.996936	0.498468	−	0.866908i	$-0.333896\pi$
$59$	451.799	0.996936	0.498468	+	0.866908i	$0.333896\pi$
$60$	0	0
$61$	−579.307	−1.21594	−0.607972	−	0.793958i	$-0.708017\pi$
$61$	−579.307	−1.21594	−0.607972	+	0.793958i	$0.708017\pi$
$62$	−265.206	−0.543245
$63$	0	0
$64$	64.0000	0.125000
$65$	209.397	0.399577
$66$	0	0
$67$	309.588	0.564510	0.282255	−	0.959339i	$-0.408918\pi$
$67$	309.588	0.564510	0.282255	+	0.959339i	$0.408918\pi$
$68$	128.402	0.228986
$69$	0	0
$70$	0	0
$71$	1058.98	1.77012	0.885059	−	0.465479i	$-0.154118\pi$
$71$	1058.98	1.77012	0.885059	+	0.465479i	$0.154118\pi$
$72$	0	0
$73$	−1193.66	−1.91380	−0.956898	−	0.290424i	$-0.906204\pi$
$73$	−1193.66	−1.91380	−0.956898	+	0.290424i	$0.906204\pi$
$74$	298.412	0.468780
$75$	0	0
$76$	223.196	0.336873
$77$	0	0
$78$	0	0
$79$	1319.56	1.87926	0.939632	−	0.342187i	$-0.111168\pi$
$79$	1319.56	1.87926	0.939632	+	0.342187i	$0.111168\pi$
$80$	−62.3919	−0.0871954
$81$	0	0
$82$	854.995	1.15144
$83$	1190.33	1.57417	0.787083	−	0.616847i	$-0.211590\pi$
$83$	1190.33	1.57417	0.787083	+	0.616847i	$0.211590\pi$
$84$	0	0
$85$	−125.176	−0.159732
$86$	875.176	1.09736
$87$	0	0
$88$	491.176	0.594994
$89$	−233.085	−0.277607	−0.138803	−	0.990320i	$-0.544326\pi$
$89$	−233.085	−0.277607	−0.138803	+	0.990320i	$0.544326\pi$
$90$	0	0
$91$	0	0
$92$	378.412	0.428828
$93$	0	0
$94$	114.010	0.125098
$95$	−217.588	−0.234990
$96$	0	0
$97$	1609.44	1.68468	0.842338	−	0.538950i	$-0.181179\pi$
$97$	1609.44	1.68468	0.842338	+	0.538950i	$0.181179\pi$
$98$	0	0
$99$	0	0
$100$	−439.176	−0.439176
$101$	−1479.26	−1.45734	−0.728671	−	0.684864i	$-0.759861\pi$
$101$	−1479.26	−1.45734	−0.728671	+	0.684864i	$0.759861\pi$
$102$	0	0
$103$	1145.35	1.09567	0.547837	−	0.836585i	$-0.315452\pi$
$103$	1145.35	1.09567	0.547837	+	0.836585i	$0.315452\pi$
$104$	−429.588	−0.405044
$105$	0	0
$106$	527.176	0.483055
$107$	−436.955	−0.394785	−0.197392	−	0.980325i	$-0.563247\pi$
$107$	−436.955	−0.394785	−0.197392	+	0.980325i	$0.563247\pi$
$108$	0	0
$109$	−166.352	−0.146180	−0.0730898	−	0.997325i	$-0.523286\pi$
$109$	−166.352	−0.146180	−0.0730898	+	0.997325i	$0.523286\pi$
$110$	−478.834	−0.415046
$111$	0	0
$112$	0	0
$113$	−490.824	−0.408609	−0.204305	−	0.978907i	$-0.565493\pi$
$113$	−490.824	−0.408609	−0.204305	+	0.978907i	$0.565493\pi$
$114$	0	0
$115$	−368.904	−0.299135
$116$	−552.764	−0.442438
$117$	0	0
$118$	903.598	0.704940
$119$	0	0
$120$	0	0
$121$	2438.59	1.83215
$122$	−1158.61	−0.859802
$123$	0	0
$124$	−530.412	−0.384132
$125$	915.578	0.655134
$126$	0	0
$127$	−2616.70	−1.82831	−0.914153	−	0.405369i	$-0.867143\pi$
$127$	−2616.70	−1.82831	−0.914153	+	0.405369i	$0.867143\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	418.794	0.282544
$131$	177.588	0.118442	0.0592211	−	0.998245i	$-0.481138\pi$
$131$	177.588	0.118442	0.0592211	+	0.998245i	$0.481138\pi$
$132$	0	0
$133$	0	0
$134$	619.176	0.399169
$135$	0	0
$136$	256.804	0.161917
$137$	−27.0152	−0.0168472	−0.00842358	−	0.999965i	$-0.502681\pi$
$137$	−27.0152	−0.0168472	−0.00842358	+	0.999965i	$0.502681\pi$
$138$	0	0
$139$	922.754	0.563071	0.281536	−	0.959551i	$-0.409156\pi$
$139$	922.754	0.563071	0.281536	+	0.959551i	$0.409156\pi$
$140$	0	0
$141$	0	0
$142$	2117.97	1.25166
$143$	−3296.92	−1.92799
$144$	0	0
$145$	538.875	0.308628
$146$	−2387.32	−1.35326
$147$	0	0
$148$	596.824	0.331477
$149$	−746.703	−0.410552	−0.205276	−	0.978704i	$-0.565809\pi$
$149$	−746.703	−0.410552	−0.205276	+	0.978704i	$0.565809\pi$
$150$	0	0
$151$	2073.15	1.11729	0.558643	−	0.829408i	$-0.311322\pi$
$151$	2073.15	1.11729	0.558643	+	0.829408i	$0.311322\pi$
$152$	446.392	0.238205
$153$	0	0
$154$	0	0
$155$	517.085	0.267956
$156$	0	0
$157$	−1566.22	−0.796166	−0.398083	−	0.917349i	$-0.630324\pi$
$157$	−1566.22	−0.796166	−0.398083	+	0.917349i	$0.630324\pi$
$158$	2639.12	1.32884
$159$	0	0
$160$	−124.784	−0.0616564
$161$	0	0
$162$	0	0
$163$	98.7333	0.0474441	0.0237221	−	0.999719i	$-0.492448\pi$
$163$	98.7333	0.0474441	0.0237221	+	0.999719i	$0.492448\pi$
$164$	1709.99	0.814194
$165$	0	0
$166$	2380.66	1.11310
$167$	−2231.36	−1.03394	−0.516969	−	0.856004i	$-0.672940\pi$
$167$	−2231.36	−1.03394	−0.516969	+	0.856004i	$0.672940\pi$
$168$	0	0
$169$	686.527	0.312484
$170$	−250.352	−0.112948
$171$	0	0
$172$	1750.35	0.775948
$173$	2100.84	0.923261	0.461631	−	0.887072i	$-0.347265\pi$
$173$	2100.84	0.923261	0.461631	+	0.887072i	$0.347265\pi$
$174$	0	0
$175$	0	0
$176$	982.352	0.420725
$177$	0	0
$178$	−466.171	−0.196298
$179$	−1722.54	−0.719267	−0.359634	−	0.933094i	$-0.617098\pi$
$179$	−1722.54	−0.719267	−0.359634	+	0.933094i	$0.617098\pi$
$180$	0	0
$181$	−1655.00	−0.679644	−0.339822	−	0.940490i	$-0.610367\pi$
$181$	−1655.00	−0.679644	−0.339822	+	0.940490i	$0.610367\pi$
$182$	0	0
$183$	0	0
$184$	756.824	0.303227
$185$	−581.828	−0.231226
$186$	0	0
$187$	1970.87	0.770720
$188$	228.020	0.0884579
$189$	0	0
$190$	−435.176	−0.166163
$191$	1007.69	0.381747	0.190874	−	0.981615i	$-0.438868\pi$
$191$	1007.69	0.381747	0.190874	+	0.981615i	$0.438868\pi$
$192$	0	0
$193$	−7.64849	−0.00285259	−0.00142630	−	0.999999i	$-0.500454\pi$
$193$	−7.64849	−0.00285259	−0.00142630	+	0.999999i	$0.500454\pi$
$194$	3218.87	1.19125
$195$	0	0
$196$	0	0
$197$	−2689.88	−0.972822	−0.486411	−	0.873730i	$-0.661694\pi$
$197$	−2689.88	−0.972822	−0.486411	+	0.873730i	$0.661694\pi$
$198$	0	0
$199$	−867.497	−0.309021	−0.154511	−	0.987991i	$-0.549380\pi$
$199$	−867.497	−0.309021	−0.154511	+	0.987991i	$0.549380\pi$
$200$	−878.352	−0.310544
$201$	0	0
$202$	−2958.51	−1.03050
$203$	0	0
$204$	0	0
$205$	−1667.02	−0.567951
$206$	2290.69	0.774758
$207$	0	0
$208$	−859.176	−0.286409
$209$	3425.89	1.13385
$210$	0	0
$211$	162.030	0.0528655	0.0264328	−	0.999651i	$-0.491585\pi$
$211$	162.030	0.0528655	0.0264328	+	0.999651i	$0.491585\pi$
$212$	1054.35	0.341572
$213$	0	0
$214$	−873.909	−0.279155
$215$	−1706.37	−0.541272
$216$	0	0
$217$	0	0
$218$	−332.703	−0.103365
$219$	0	0
$220$	−957.669	−0.293482
$221$	−1723.75	−0.524669
$222$	0	0
$223$	4577.85	1.37469	0.687344	−	0.726332i	$-0.258777\pi$
$223$	4577.85	1.37469	0.687344	+	0.726332i	$0.258777\pi$
$224$	0	0
$225$	0	0
$226$	−981.648	−0.288930
$227$	−2218.19	−0.648575	−0.324287	−	0.945959i	$-0.605125\pi$
$227$	−2218.19	−0.648575	−0.324287	+	0.945959i	$0.605125\pi$
$228$	0	0
$229$	−785.217	−0.226588	−0.113294	−	0.993562i	$-0.536140\pi$
$229$	−785.217	−0.226588	−0.113294	+	0.993562i	$0.536140\pi$
$230$	−737.808	−0.211520
$231$	0	0
$232$	−1105.53	−0.312851
$233$	5369.12	1.50963	0.754813	−	0.655940i	$-0.227727\pi$
$233$	5369.12	1.50963	0.754813	+	0.655940i	$0.227727\pi$
$234$	0	0
$235$	−222.291	−0.0617049
$236$	1807.20	0.498468
$237$	0	0
$238$	0	0
$239$	−3713.28	−1.00499	−0.502493	−	0.864581i	$-0.667584\pi$
$239$	−3713.28	−1.00499	−0.502493	+	0.864581i	$0.667584\pi$
$240$	0	0
$241$	−6998.62	−1.87063	−0.935313	−	0.353821i	$-0.884882\pi$
$241$	−6998.62	−1.87063	−0.935313	+	0.353821i	$0.884882\pi$
$242$	4877.18	1.29552
$243$	0	0
$244$	−2317.23	−0.607972
$245$	0	0
$246$	0	0
$247$	−2996.32	−0.771868
$248$	−1060.82	−0.271623
$249$	0	0
$250$	1831.16	0.463250
$251$	3722.75	0.936168	0.468084	−	0.883684i	$-0.344945\pi$
$251$	3722.75	0.936168	0.468084	+	0.883684i	$0.344945\pi$
$252$	0	0
$253$	5808.34	1.44335
$254$	−5233.41	−1.29281
$255$	0	0
$256$	256.000	0.0625000
$257$	1230.15	0.298578	0.149289	−	0.988794i	$-0.452301\pi$
$257$	1230.15	0.298578	0.149289	+	0.988794i	$0.452301\pi$
$258$	0	0
$259$	0	0
$260$	837.588	0.199788
$261$	0	0
$262$	355.176	0.0837513
$263$	−2388.63	−0.560036	−0.280018	−	0.959995i	$-0.590340\pi$
$263$	−2388.63	−0.560036	−0.280018	+	0.959995i	$0.590340\pi$
$264$	0	0
$265$	−1027.86	−0.238268
$266$	0	0
$267$	0	0
$268$	1238.35	0.282255
$269$	6702.31	1.51913	0.759567	−	0.650429i	$-0.225411\pi$
$269$	6702.31	1.51913	0.759567	+	0.650429i	$0.225411\pi$
$270$	0	0
$271$	−4950.37	−1.10964	−0.554822	−	0.831969i	$-0.687214\pi$
$271$	−4950.37	−1.10964	−0.554822	+	0.831969i	$0.687214\pi$
$272$	513.608	0.114493
$273$	0	0
$274$	−54.0303	−0.0119127
$275$	−6741.02	−1.47818
$276$	0	0
$277$	−3705.18	−0.803691	−0.401846	−	0.915707i	$-0.631631\pi$
$277$	−3705.18	−0.803691	−0.401846	+	0.915707i	$0.631631\pi$
$278$	1845.51	0.398152
$279$	0	0
$280$	0	0
$281$	−9324.74	−1.97960	−0.989800	−	0.142465i	$-0.954497\pi$
$281$	−9324.74	−1.97960	−0.989800	+	0.142465i	$0.954497\pi$
$282$	0	0
$283$	5569.49	1.16986	0.584932	−	0.811082i	$-0.301121\pi$
$283$	5569.49	1.16986	0.584932	+	0.811082i	$0.301121\pi$
$284$	4235.94	0.885059
$285$	0	0
$286$	−6593.85	−1.36330
$287$	0	0
$288$	0	0
$289$	−3882.56	−0.790262
$290$	1077.75	0.218233
$291$	0	0
$292$	−4774.63	−0.956898
$293$	−1665.31	−0.332042	−0.166021	−	0.986122i	$-0.553092\pi$
$293$	−1665.31	−0.332042	−0.166021	+	0.986122i	$0.553092\pi$
$294$	0	0
$295$	−1761.79	−0.347713
$296$	1193.65	0.234390
$297$	0	0
$298$	−1493.41	−0.290304
$299$	−5080.04	−0.982563
$300$	0	0
$301$	0	0
$302$	4146.29	0.790040
$303$	0	0
$304$	892.784	0.168436
$305$	2259.00	0.424099
$306$	0	0
$307$	5303.32	0.985916	0.492958	−	0.870053i	$-0.335916\pi$
$307$	5303.32	0.985916	0.492958	+	0.870053i	$0.335916\pi$
$308$	0	0
$309$	0	0
$310$	1034.17	0.189474
$311$	1125.99	0.205302	0.102651	−	0.994717i	$-0.467267\pi$
$311$	1125.99	0.205302	0.102651	+	0.994717i	$0.467267\pi$
$312$	0	0
$313$	8299.51	1.49877	0.749386	−	0.662133i	$-0.230349\pi$
$313$	8299.51	1.49877	0.749386	+	0.662133i	$0.230349\pi$
$314$	−3132.44	−0.562974
$315$	0	0
$316$	5278.23	0.939632
$317$	4278.76	0.758105	0.379053	−	0.925375i	$-0.376250\pi$
$317$	4278.76	0.758105	0.379053	+	0.925375i	$0.376250\pi$
$318$	0	0
$319$	−8484.50	−1.48916
$320$	−249.568	−0.0435977
$321$	0	0
$322$	0	0
$323$	1791.18	0.308556
$324$	0	0
$325$	5895.77	1.00627
$326$	197.467	0.0335481
$327$	0	0
$328$	3419.98	0.575722
$329$	0	0
$330$	0	0
$331$	−1707.12	−0.283479	−0.141739	−	0.989904i	$-0.545270\pi$
$331$	−1707.12	−0.283479	−0.141739	+	0.989904i	$0.545270\pi$
$332$	4761.33	0.787083
$333$	0	0
$334$	−4462.71	−0.731104
$335$	−1207.24	−0.196891
$336$	0	0
$337$	1710.67	0.276517	0.138259	−	0.990396i	$-0.455849\pi$
$337$	1710.67	0.276517	0.138259	+	0.990396i	$0.455849\pi$
$338$	1373.05	0.220960
$339$	0	0
$340$	−500.703	−0.0798660
$341$	−8141.42	−1.29291
$342$	0	0
$343$	0	0
$344$	3500.70	0.548678
$345$	0	0
$346$	4201.69	0.652844
$347$	−8910.30	−1.37847	−0.689236	−	0.724537i	$-0.742054\pi$
$347$	−8910.30	−1.37847	−0.689236	+	0.724537i	$0.742054\pi$
$348$	0	0
$349$	−5378.68	−0.824969	−0.412485	−	0.910965i	$-0.635339\pi$
$349$	−5378.68	−0.824969	−0.412485	+	0.910965i	$0.635339\pi$
$350$	0	0
$351$	0	0
$352$	1964.70	0.297497
$353$	−4252.56	−0.641193	−0.320596	−	0.947216i	$-0.603883\pi$
$353$	−4252.56	−0.641193	−0.320596	+	0.947216i	$0.603883\pi$
$354$	0	0
$355$	−4129.51	−0.617384
$356$	−932.341	−0.138803
$357$	0	0
$358$	−3445.08	−0.508599
$359$	−4903.89	−0.720940	−0.360470	−	0.932771i	$-0.617384\pi$
$359$	−4903.89	−0.720940	−0.360470	+	0.932771i	$0.617384\pi$
$360$	0	0
$361$	−3745.47	−0.546067
$362$	−3310.01	−0.480581
$363$	0	0
$364$	0	0
$365$	4654.66	0.667497
$366$	0	0
$367$	4041.57	0.574845	0.287423	−	0.957804i	$-0.407202\pi$
$367$	4041.57	0.574845	0.287423	+	0.957804i	$0.407202\pi$
$368$	1513.65	0.214414
$369$	0	0
$370$	−1163.66	−0.163502
$371$	0	0
$372$	0	0
$373$	−7451.35	−1.03436	−0.517180	−	0.855877i	$-0.673018\pi$
$373$	−7451.35	−1.03436	−0.517180	+	0.855877i	$0.673018\pi$
$374$	3941.75	0.544981
$375$	0	0
$376$	456.040	0.0625492
$377$	7420.64	1.01375
$378$	0	0
$379$	−12564.4	−1.70288	−0.851438	−	0.524456i	$-0.824269\pi$
$379$	−12564.4	−1.70288	−0.851438	+	0.524456i	$0.824269\pi$
$380$	−870.352	−0.117495
$381$	0	0
$382$	2015.38	0.269936
$383$	−4289.93	−0.572337	−0.286169	−	0.958179i	$-0.592382\pi$
$383$	−4289.93	−0.572337	−0.286169	+	0.958179i	$0.592382\pi$
$384$	0	0
$385$	0	0
$386$	−15.2970	−0.00201709
$387$	0	0
$388$	6437.75	0.842338
$389$	−5662.77	−0.738082	−0.369041	−	0.929413i	$-0.620314\pi$
$389$	−5662.77	−0.738082	−0.369041	+	0.929413i	$0.620314\pi$
$390$	0	0
$391$	3036.81	0.392782
$392$	0	0
$393$	0	0
$394$	−5379.76	−0.687889
$395$	−5145.61	−0.655452
$396$	0	0
$397$	−14561.4	−1.84084	−0.920421	−	0.390928i	$-0.872154\pi$
$397$	−14561.4	−1.84084	−0.920421	+	0.390928i	$0.872154\pi$
$398$	−1734.99	−0.218511
$399$	0	0
$400$	−1756.70	−0.219588
$401$	3742.19	0.466025	0.233013	−	0.972474i	$-0.425142\pi$
$401$	3742.19	0.466025	0.233013	+	0.972474i	$0.425142\pi$
$402$	0	0
$403$	7120.58	0.880152
$404$	−5917.02	−0.728671
$405$	0	0
$406$	0	0
$407$	9160.80	1.11569
$408$	0	0
$409$	3517.17	0.425215	0.212608	−	0.977138i	$-0.431804\pi$
$409$	3517.17	0.425215	0.212608	+	0.977138i	$0.431804\pi$
$410$	−3334.05	−0.401602
$411$	0	0
$412$	4581.39	0.547837
$413$	0	0
$414$	0	0
$415$	−4641.69	−0.549040
$416$	−1718.35	−0.202522
$417$	0	0
$418$	6851.78	0.801750
$419$	−7579.52	−0.883732	−0.441866	−	0.897081i	$-0.645683\pi$
$419$	−7579.52	−0.883732	−0.441866	+	0.897081i	$0.645683\pi$
$420$	0	0
$421$	−4980.87	−0.576610	−0.288305	−	0.957539i	$-0.593092\pi$
$421$	−4980.87	−0.576610	−0.288305	+	0.957539i	$0.593092\pi$
$422$	324.061	0.0373816
$423$	0	0
$424$	2108.70	0.241528
$425$	−3524.44	−0.402260
$426$	0	0
$427$	0	0
$428$	−1747.82	−0.197392
$429$	0	0
$430$	−3412.74	−0.382737
$431$	−14203.3	−1.58736	−0.793678	−	0.608339i	$-0.791836\pi$
$431$	−14203.3	−1.58736	−0.793678	+	0.608339i	$0.791836\pi$
$432$	0	0
$433$	−3874.82	−0.430051	−0.215026	−	0.976608i	$-0.568983\pi$
$433$	−3874.82	−0.430051	−0.215026	+	0.976608i	$0.568983\pi$
$434$	0	0
$435$	0	0
$436$	−665.406	−0.0730898
$437$	5278.75	0.577842
$438$	0	0
$439$	−7763.82	−0.844070	−0.422035	−	0.906579i	$-0.638684\pi$
$439$	−7763.82	−0.844070	−0.422035	+	0.906579i	$0.638684\pi$
$440$	−1915.34	−0.207523
$441$	0	0
$442$	−3447.50	−0.370997
$443$	−9662.24	−1.03627	−0.518134	−	0.855299i	$-0.673373\pi$
$443$	−9662.24	−1.03627	−0.518134	+	0.855299i	$0.673373\pi$
$444$	0	0
$445$	908.915	0.0968241
$446$	9155.70	0.972051
$447$	0	0
$448$	0	0
$449$	10942.6	1.15014	0.575069	−	0.818105i	$-0.304975\pi$
$449$	10942.6	1.15014	0.575069	+	0.818105i	$0.304975\pi$
$450$	0	0
$451$	26247.0	2.74041
$452$	−1963.30	−0.204305
$453$	0	0
$454$	−4436.38	−0.458612
$455$	0	0
$456$	0	0
$457$	13618.3	1.39396	0.696979	−	0.717091i	$-0.254527\pi$
$457$	13618.3	1.39396	0.696979	+	0.717091i	$0.254527\pi$
$458$	−1570.43	−0.160222
$459$	0	0
$460$	−1475.62	−0.149567
$461$	11955.8	1.20789	0.603947	−	0.797025i	$-0.293594\pi$
$461$	11955.8	1.20789	0.603947	+	0.797025i	$0.293594\pi$
$462$	0	0
$463$	648.503	0.0650939	0.0325470	−	0.999470i	$-0.489638\pi$
$463$	648.503	0.0650939	0.0325470	+	0.999470i	$0.489638\pi$
$464$	−2211.05	−0.221219
$465$	0	0
$466$	10738.2	1.06747
$467$	2784.74	0.275937	0.137969	−	0.990437i	$-0.455943\pi$
$467$	2784.74	0.275937	0.137969	+	0.990437i	$0.455943\pi$
$468$	0	0
$469$	0	0
$470$	−444.582	−0.0436320
$471$	0	0
$472$	3614.39	0.352470
$473$	26866.6	2.61168
$474$	0	0
$475$	−6126.39	−0.591785
$476$	0	0
$477$	0	0
$478$	−7426.55	−0.710633
$479$	11113.4	1.06009	0.530046	−	0.847969i	$-0.322175\pi$
$479$	11113.4	1.06009	0.530046	+	0.847969i	$0.322175\pi$
$480$	0	0
$481$	−8012.14	−0.759505
$482$	−13997.2	−1.32273
$483$	0	0
$484$	9754.35	0.916074
$485$	−6275.99	−0.587584
$486$	0	0
$487$	−3786.27	−0.352305	−0.176152	−	0.984363i	$-0.556365\pi$
$487$	−3786.27	−0.352305	−0.176152	+	0.984363i	$0.556365\pi$
$488$	−4634.45	−0.429901
$489$	0	0
$490$	0	0
$491$	9582.12	0.880723	0.440361	−	0.897821i	$-0.354850\pi$
$491$	9582.12	0.880723	0.440361	+	0.897821i	$0.354850\pi$
$492$	0	0
$493$	−4436.00	−0.405248
$494$	−5992.64	−0.545793
$495$	0	0
$496$	−2121.65	−0.192066
$497$	0	0
$498$	0	0
$499$	5581.55	0.500730	0.250365	−	0.968152i	$-0.419449\pi$
$499$	5581.55	0.500730	0.250365	+	0.968152i	$0.419449\pi$
$500$	3662.31	0.327567
$501$	0	0
$502$	7445.51	0.661970
$503$	−14116.3	−1.25132	−0.625661	−	0.780095i	$-0.715171\pi$
$503$	−14116.3	−1.25132	−0.625661	+	0.780095i	$0.715171\pi$
$504$	0	0
$505$	5768.35	0.508294
$506$	11616.7	1.02060
$507$	0	0
$508$	−10466.8	−0.914153
$509$	16787.6	1.46188	0.730941	−	0.682441i	$-0.239081\pi$
$509$	16787.6	1.46188	0.730941	+	0.682441i	$0.239081\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	2460.30	0.211127
$515$	−4466.27	−0.382150
$516$	0	0
$517$	3499.94	0.297731
$518$	0	0
$519$	0	0
$520$	1675.18	0.141272
$521$	−5598.61	−0.470786	−0.235393	−	0.971900i	$-0.575638\pi$
$521$	−5598.61	−0.470786	−0.235393	+	0.971900i	$0.575638\pi$
$522$	0	0
$523$	13270.7	1.10954	0.554769	−	0.832004i	$-0.312807\pi$
$523$	13270.7	1.10954	0.554769	+	0.832004i	$0.312807\pi$
$524$	710.352	0.0592211
$525$	0	0
$526$	−4777.27	−0.396005
$527$	−4256.62	−0.351843
$528$	0	0
$529$	−3217.27	−0.264426
$530$	−2055.72	−0.168481
$531$	0	0
$532$	0	0
$533$	−22956.0	−1.86554
$534$	0	0
$535$	1703.90	0.137694
$536$	2476.70	0.199584
$537$	0	0
$538$	13404.6	1.07419
$539$	0	0
$540$	0	0
$541$	−23017.5	−1.82921	−0.914603	−	0.404353i	$-0.867497\pi$
$541$	−23017.5	−1.82921	−0.914603	+	0.404353i	$0.867497\pi$
$542$	−9900.74	−0.784637
$543$	0	0
$544$	1027.22	0.0809587
$545$	648.687	0.0509848
$546$	0	0
$547$	4475.84	0.349859	0.174930	−	0.984581i	$-0.444030\pi$
$547$	4475.84	0.349859	0.174930	+	0.984581i	$0.444030\pi$
$548$	−108.061	−0.00842358
$549$	0	0
$550$	−13482.0	−1.04523
$551$	−7710.91	−0.596181
$552$	0	0
$553$	0	0
$554$	−7410.35	−0.568295
$555$	0	0
$556$	3691.01	0.281536
$557$	−1541.70	−0.117278	−0.0586390	−	0.998279i	$-0.518676\pi$
$557$	−1541.70	−0.117278	−0.0586390	+	0.998279i	$0.518676\pi$
$558$	0	0
$559$	−23497.8	−1.77791
$560$	0	0
$561$	0	0
$562$	−18649.5	−1.39979
$563$	13079.7	0.979115	0.489557	−	0.871971i	$-0.337158\pi$
$563$	13079.7	0.979115	0.489557	+	0.871971i	$0.337158\pi$
$564$	0	0
$565$	1913.97	0.142515
$566$	11139.0	0.827219
$567$	0	0
$568$	8471.88	0.625831
$569$	−11411.2	−0.840740	−0.420370	−	0.907353i	$-0.638100\pi$
$569$	−11411.2	−0.840740	−0.420370	+	0.907353i	$0.638100\pi$
$570$	0	0
$571$	2311.74	0.169428	0.0847139	−	0.996405i	$-0.473002\pi$
$571$	2311.74	0.169428	0.0847139	+	0.996405i	$0.473002\pi$
$572$	−13187.7	−0.963995
$573$	0	0
$574$	0	0
$575$	−10386.8	−0.753324
$576$	0	0
$577$	−25097.0	−1.81075	−0.905373	−	0.424617i	$-0.860409\pi$
$577$	−25097.0	−1.81075	−0.905373	+	0.424617i	$0.860409\pi$
$578$	−7765.12	−0.558800
$579$	0	0
$580$	2155.50	0.154314
$581$	0	0
$582$	0	0
$583$	16183.5	1.14966
$584$	−9549.26	−0.676629
$585$	0	0
$586$	−3330.61	−0.234789
$587$	19789.1	1.39145	0.695726	−	0.718307i	$-0.255083\pi$
$587$	19789.1	1.39145	0.695726	+	0.718307i	$0.255083\pi$
$588$	0	0
$589$	−7399.12	−0.517615
$590$	−3523.58	−0.245870
$591$	0	0
$592$	2387.30	0.165739
$593$	−2774.13	−0.192108	−0.0960540	−	0.995376i	$-0.530622\pi$
$593$	−2774.13	−0.192108	−0.0960540	+	0.995376i	$0.530622\pi$
$594$	0	0
$595$	0	0
$596$	−2986.81	−0.205276
$597$	0	0
$598$	−10160.1	−0.694777
$599$	26190.3	1.78649	0.893245	−	0.449570i	$-0.148423\pi$
$599$	26190.3	1.78649	0.893245	+	0.449570i	$0.148423\pi$
$600$	0	0
$601$	11038.6	0.749211	0.374605	−	0.927184i	$-0.377778\pi$
$601$	11038.6	0.749211	0.374605	+	0.927184i	$0.377778\pi$
$602$	0	0
$603$	0	0
$604$	8292.58	0.558643
$605$	−9509.26	−0.639019
$606$	0	0
$607$	1854.39	0.123999	0.0619996	−	0.998076i	$-0.480252\pi$
$607$	1854.39	0.123999	0.0619996	+	0.998076i	$0.480252\pi$
$608$	1785.57	0.119103
$609$	0	0
$610$	4518.01	0.299883
$611$	−3061.08	−0.202681
$612$	0	0
$613$	−14856.8	−0.978894	−0.489447	−	0.872033i	$-0.662801\pi$
$613$	−14856.8	−0.978894	−0.489447	+	0.872033i	$0.662801\pi$
$614$	10606.6	0.697148
$615$	0	0
$616$	0	0
$617$	−7600.00	−0.495890	−0.247945	−	0.968774i	$-0.579755\pi$
$617$	−7600.00	−0.495890	−0.247945	+	0.968774i	$0.579755\pi$
$618$	0	0
$619$	−22685.3	−1.47302	−0.736509	−	0.676427i	$-0.763527\pi$
$619$	−22685.3	−1.47302	−0.736509	+	0.676427i	$0.763527\pi$
$620$	2068.34	0.133978
$621$	0	0
$622$	2251.98	0.145171
$623$	0	0
$624$	0	0
$625$	10154.0	0.649853
$626$	16599.0	1.05979
$627$	0	0
$628$	−6264.88	−0.398083
$629$	4789.59	0.303614
$630$	0	0
$631$	−12024.1	−0.758595	−0.379297	−	0.925275i	$-0.623834\pi$
$631$	−12024.1	−0.758595	−0.379297	+	0.925275i	$0.623834\pi$
$632$	10556.5	0.664420
$633$	0	0
$634$	8557.53	0.536061
$635$	10203.8	0.637679
$636$	0	0
$637$	0	0
$638$	−16969.0	−1.05299
$639$	0	0
$640$	−499.135	−0.0308282
$641$	−11320.6	−0.697559	−0.348779	−	0.937205i	$-0.613404\pi$
$641$	−11320.6	−0.697559	−0.348779	+	0.937205i	$0.613404\pi$
$642$	0	0
$643$	−16843.6	−1.03304	−0.516521	−	0.856275i	$-0.672773\pi$
$643$	−16843.6	−1.03304	−0.516521	+	0.856275i	$0.672773\pi$
$644$	0	0
$645$	0	0
$646$	3582.35	0.218182
$647$	7719.32	0.469054	0.234527	−	0.972110i	$-0.424646\pi$
$647$	7719.32	0.469054	0.234527	+	0.972110i	$0.424646\pi$
$648$	0	0
$649$	27739.1	1.67774
$650$	11791.5	0.711542
$651$	0	0
$652$	394.933	0.0237221
$653$	30803.2	1.84597	0.922987	−	0.384832i	$-0.125741\pi$
$653$	30803.2	1.84597	0.922987	+	0.384832i	$0.125741\pi$
$654$	0	0
$655$	−692.503	−0.0413104
$656$	6839.96	0.407097
$657$	0	0
$658$	0	0
$659$	−9760.68	−0.576968	−0.288484	−	0.957485i	$-0.593151\pi$
$659$	−9760.68	−0.576968	−0.288484	+	0.957485i	$0.593151\pi$
$660$	0	0
$661$	−18071.1	−1.06336	−0.531682	−	0.846944i	$-0.678440\pi$
$661$	−18071.1	−1.06336	−0.531682	+	0.846944i	$0.678440\pi$
$662$	−3414.23	−0.200450
$663$	0	0
$664$	9522.65	0.556552
$665$	0	0
$666$	0	0
$667$	−13073.3	−0.758920
$668$	−8925.43	−0.516969
$669$	0	0
$670$	−2414.47	−0.139223
$671$	−35567.7	−2.04631
$672$	0	0
$673$	26591.1	1.52305	0.761524	−	0.648137i	$-0.224451\pi$
$673$	26591.1	1.52305	0.761524	+	0.648137i	$0.224451\pi$
$674$	3421.35	0.195527
$675$	0	0
$676$	2746.11	0.156242
$677$	33080.3	1.87796	0.938979	−	0.343975i	$-0.111773\pi$
$677$	33080.3	1.87796	0.938979	+	0.343975i	$0.111773\pi$
$678$	0	0
$679$	0	0
$680$	−1001.41	−0.0564738
$681$	0	0
$682$	−16282.8	−0.914227
$683$	10466.1	0.586344	0.293172	−	0.956060i	$-0.405289\pi$
$683$	10466.1	0.586344	0.293172	+	0.956060i	$0.405289\pi$
$684$	0	0
$685$	105.345	0.00587597
$686$	0	0
$687$	0	0
$688$	7001.41	0.387974
$689$	−14154.3	−0.782634
$690$	0	0
$691$	−13269.9	−0.730551	−0.365275	−	0.930900i	$-0.619025\pi$
$691$	−13269.9	−0.730551	−0.365275	+	0.930900i	$0.619025\pi$
$692$	8403.38	0.461631
$693$	0	0
$694$	−17820.6	−0.974727
$695$	−3598.27	−0.196389
$696$	0	0
$697$	13722.9	0.745755
$698$	−10757.4	−0.583341
$699$	0	0
$700$	0	0
$701$	15169.9	0.817344	0.408672	−	0.912681i	$-0.365992\pi$
$701$	15169.9	0.817344	0.408672	+	0.912681i	$0.365992\pi$
$702$	0	0
$703$	8325.55	0.446663
$704$	3929.41	0.210362
$705$	0	0
$706$	−8505.13	−0.453392
$707$	0	0
$708$	0	0
$709$	26317.7	1.39405	0.697026	−	0.717046i	$-0.254506\pi$
$709$	26317.7	1.39405	0.697026	+	0.717046i	$0.254506\pi$
$710$	−8259.01	−0.436557
$711$	0	0
$712$	−1864.68	−0.0981488
$713$	−12544.6	−0.658907
$714$	0	0
$715$	12856.3	0.672447
$716$	−6890.17	−0.359634
$717$	0	0
$718$	−9807.78	−0.509781
$719$	23013.1	1.19366	0.596831	−	0.802367i	$-0.296426\pi$
$719$	23013.1	1.19366	0.596831	+	0.802367i	$0.296426\pi$
$720$	0	0
$721$	0	0
$722$	−7490.95	−0.386128
$723$	0	0
$724$	−6620.02	−0.339822
$725$	15172.5	0.777232
$726$	0	0
$727$	16265.4	0.829780	0.414890	−	0.909872i	$-0.363820\pi$
$727$	16265.4	0.829780	0.414890	+	0.909872i	$0.363820\pi$
$728$	0	0
$729$	0	0
$730$	9309.33	0.471991
$731$	14046.8	0.710724
$732$	0	0
$733$	866.444	0.0436601	0.0218300	−	0.999762i	$-0.493051\pi$
$733$	866.444	0.0436601	0.0218300	+	0.999762i	$0.493051\pi$
$734$	8083.14	0.406477
$735$	0	0
$736$	3027.30	0.151614
$737$	19007.8	0.950013
$738$	0	0
$739$	10990.1	0.547058	0.273529	−	0.961864i	$-0.411809\pi$
$739$	10990.1	0.547058	0.273529	+	0.961864i	$0.411809\pi$
$740$	−2327.31	−0.115613
$741$	0	0
$742$	0	0
$743$	22416.4	1.10683	0.553417	−	0.832905i	$-0.313324\pi$
$743$	22416.4	1.10683	0.553417	+	0.832905i	$0.313324\pi$
$744$	0	0
$745$	2911.76	0.143193
$746$	−14902.7	−0.731403
$747$	0	0
$748$	7883.49	0.385360
$749$	0	0
$750$	0	0
$751$	−19717.9	−0.958080	−0.479040	−	0.877793i	$-0.659015\pi$
$751$	−19717.9	−0.958080	−0.479040	+	0.877793i	$0.659015\pi$
$752$	912.081	0.0442289
$753$	0	0
$754$	14841.3	0.716827
$755$	−8084.22	−0.389689
$756$	0	0
$757$	839.321	0.0402981	0.0201490	−	0.999797i	$-0.493586\pi$
$757$	839.321	0.0402981	0.0201490	+	0.999797i	$0.493586\pi$
$758$	−25128.8	−1.20411
$759$	0	0
$760$	−1740.70	−0.0830815
$761$	18364.5	0.874786	0.437393	−	0.899270i	$-0.355902\pi$
$761$	18364.5	0.874786	0.437393	+	0.899270i	$0.355902\pi$
$762$	0	0
$763$	0	0
$764$	4030.75	0.190874
$765$	0	0
$766$	−8579.86	−0.404704
$767$	−24260.9	−1.14213
$768$	0	0
$769$	14890.8	0.698277	0.349138	−	0.937071i	$-0.386474\pi$
$769$	14890.8	0.698277	0.349138	+	0.937071i	$0.386474\pi$
$770$	0	0
$771$	0	0
$772$	−30.5939	−0.00142630
$773$	−16606.2	−0.772683	−0.386341	−	0.922356i	$-0.626261\pi$
$773$	−16606.2	−0.772683	−0.386341	+	0.922356i	$0.626261\pi$
$774$	0	0
$775$	14559.0	0.674807
$776$	12875.5	0.595623
$777$	0	0
$778$	−11325.5	−0.521903
$779$	23853.9	1.09712
$780$	0	0
$781$	65018.5	2.97893
$782$	6073.61	0.277739
$783$	0	0
$784$	0	0
$785$	6107.47	0.277688
$786$	0	0
$787$	−3156.20	−0.142956	−0.0714781	−	0.997442i	$-0.522772\pi$
$787$	−3156.20	−0.142956	−0.0714781	+	0.997442i	$0.522772\pi$
$788$	−10759.5	−0.486411
$789$	0	0
$790$	−10291.2	−0.463475
$791$	0	0
$792$	0	0
$793$	31107.9	1.39303
$794$	−29122.8	−1.30167
$795$	0	0
$796$	−3469.99	−0.154511
$797$	−13514.8	−0.600652	−0.300326	−	0.953837i	$-0.597096\pi$
$797$	−13514.8	−0.600652	−0.300326	+	0.953837i	$0.597096\pi$
$798$	0	0
$799$	1829.89	0.0810224
$800$	−3513.41	−0.155272
$801$	0	0
$802$	7484.38	0.329530
$803$	−73287.0	−3.22072
$804$	0	0
$805$	0	0
$806$	14241.2	0.622362
$807$	0	0
$808$	−11834.0	−0.515248
$809$	26758.1	1.16287	0.581437	−	0.813591i	$-0.302491\pi$
$809$	26758.1	1.16287	0.581437	+	0.813591i	$0.302491\pi$
$810$	0	0
$811$	15920.7	0.689338	0.344669	−	0.938724i	$-0.387991\pi$
$811$	15920.7	0.689338	0.344669	+	0.938724i	$0.387991\pi$
$812$	0	0
$813$	0	0
$814$	18321.6	0.788909
$815$	−385.010	−0.0165476
$816$	0	0
$817$	24417.0	1.04558
$818$	7034.35	0.300673
$819$	0	0
$820$	−6668.10	−0.283976
$821$	19306.2	0.820696	0.410348	−	0.911929i	$-0.365407\pi$
$821$	19306.2	0.820696	0.410348	+	0.911929i	$0.365407\pi$
$822$	0	0
$823$	791.000	0.0335025	0.0167512	−	0.999860i	$-0.494668\pi$
$823$	791.000	0.0335025	0.0167512	+	0.999860i	$0.494668\pi$
$824$	9162.77	0.387379
$825$	0	0
$826$	0	0
$827$	−29537.6	−1.24199	−0.620993	−	0.783816i	$-0.713270\pi$
$827$	−29537.6	−1.24199	−0.620993	+	0.783816i	$0.713270\pi$
$828$	0	0
$829$	5766.52	0.241592	0.120796	−	0.992677i	$-0.461455\pi$
$829$	5766.52	0.241592	0.120796	+	0.992677i	$0.461455\pi$
$830$	−9283.38	−0.388230
$831$	0	0
$832$	−3436.70	−0.143205
$833$	0	0
$834$	0	0
$835$	8701.16	0.360618
$836$	13703.6	0.566923
$837$	0	0
$838$	−15159.0	−0.624893
$839$	29726.4	1.22321	0.611603	−	0.791165i	$-0.290525\pi$
$839$	29726.4	1.22321	0.611603	+	0.791165i	$0.290525\pi$
$840$	0	0
$841$	−5292.27	−0.216994
$842$	−9961.75	−0.407725
$843$	0	0
$844$	648.121	0.0264328
$845$	−2677.11	−0.108989
$846$	0	0
$847$	0	0
$848$	4217.41	0.170786
$849$	0	0
$850$	−7048.88	−0.284441
$851$	14115.3	0.568587
$852$	0	0
$853$	17829.6	0.715677	0.357838	−	0.933784i	$-0.383514\pi$
$853$	17829.6	0.715677	0.357838	+	0.933784i	$0.383514\pi$
$854$	0	0
$855$	0	0
$856$	−3495.64	−0.139578
$857$	−39682.4	−1.58171	−0.790856	−	0.612003i	$-0.790364\pi$
$857$	−39682.4	−1.58171	−0.790856	+	0.612003i	$0.790364\pi$
$858$	0	0
$859$	2195.13	0.0871909	0.0435955	−	0.999049i	$-0.486119\pi$
$859$	2195.13	0.0871909	0.0435955	+	0.999049i	$0.486119\pi$
$860$	−6825.49	−0.270636
$861$	0	0
$862$	−28406.6	−1.12243
$863$	−31917.1	−1.25894	−0.629472	−	0.777023i	$-0.716729\pi$
$863$	−31917.1	−1.25894	−0.629472	+	0.777023i	$0.716729\pi$
$864$	0	0
$865$	−8192.23	−0.322016
$866$	−7749.65	−0.304092
$867$	0	0
$868$	0	0
$869$	81016.8	3.16261
$870$	0	0
$871$	−16624.4	−0.646724
$872$	−1330.81	−0.0516823
$873$	0	0
$874$	10557.5	0.408596
$875$	0	0
$876$	0	0
$877$	28842.9	1.11055	0.555276	−	0.831666i	$-0.312613\pi$
$877$	28842.9	1.11055	0.555276	+	0.831666i	$0.312613\pi$
$878$	−15527.6	−0.596848
$879$	0	0
$880$	−3830.67	−0.146741
$881$	−15350.2	−0.587015	−0.293508	−	0.955957i	$-0.594823\pi$
$881$	−15350.2	−0.587015	−0.293508	+	0.955957i	$0.594823\pi$
$882$	0	0
$883$	6089.64	0.232087	0.116043	−	0.993244i	$-0.462979\pi$
$883$	6089.64	0.232087	0.116043	+	0.993244i	$0.462979\pi$
$884$	−6894.99	−0.262335
$885$	0	0
$886$	−19324.5	−0.732752
$887$	−384.441	−0.0145527	−0.00727637	−	0.999974i	$-0.502316\pi$
$887$	−384.441	−0.0145527	−0.00727637	+	0.999974i	$0.502316\pi$
$888$	0	0
$889$	0	0
$890$	1817.83	0.0684650
$891$	0	0
$892$	18311.4	0.687344
$893$	3180.82	0.119196
$894$	0	0
$895$	6717.05	0.250867
$896$	0	0
$897$	0	0
$898$	21885.2	0.813271
$899$	18324.5	0.679819
$900$	0	0
$901$	8461.30	0.312860
$902$	52494.1	1.93776
$903$	0	0
$904$	−3926.59	−0.144465
$905$	6453.68	0.237047
$906$	0	0
$907$	−7267.93	−0.266073	−0.133036	−	0.991111i	$-0.542473\pi$
$907$	−7267.93	−0.266073	−0.133036	+	0.991111i	$0.542473\pi$
$908$	−8872.76	−0.324287
$909$	0	0
$910$	0	0
$911$	8535.12	0.310408	0.155204	−	0.987882i	$-0.450397\pi$
$911$	8535.12	0.310408	0.155204	+	0.987882i	$0.450397\pi$
$912$	0	0
$913$	73082.7	2.64916
$914$	27236.7	0.985678
$915$	0	0
$916$	−3140.87	−0.113294
$917$	0	0
$918$	0	0
$919$	−10851.7	−0.389516	−0.194758	−	0.980851i	$-0.562392\pi$
$919$	−10851.7	−0.389516	−0.194758	+	0.980851i	$0.562392\pi$
$920$	−2951.23	−0.105760
$921$	0	0
$922$	23911.7	0.854110
$923$	−56865.9	−2.02791
$924$	0	0
$925$	−16381.9	−0.582307
$926$	1297.01	0.0460284
$927$	0	0
$928$	−4422.11	−0.156425
$929$	−1560.64	−0.0551161	−0.0275581	−	0.999620i	$-0.508773\pi$
$929$	−1560.64	−0.0551161	−0.0275581	+	0.999620i	$0.508773\pi$
$930$	0	0
$931$	0	0
$932$	21476.5	0.754813
$933$	0	0
$934$	5569.49	0.195117
$935$	−7685.41	−0.268813
$936$	0	0
$937$	−11978.4	−0.417627	−0.208813	−	0.977956i	$-0.566960\pi$
$937$	−11978.4	−0.417627	−0.208813	+	0.977956i	$0.566960\pi$
$938$	0	0
$939$	0	0
$940$	−889.164	−0.0308525
$941$	24597.7	0.852137	0.426068	−	0.904691i	$-0.359898\pi$
$941$	24597.7	0.852137	0.426068	+	0.904691i	$0.359898\pi$
$942$	0	0
$943$	40442.6	1.39660
$944$	7228.78	0.249234
$945$	0	0
$946$	53733.1	1.84674
$947$	10834.6	0.371783	0.185891	−	0.982570i	$-0.440483\pi$
$947$	10834.6	0.371783	0.185891	+	0.982570i	$0.440483\pi$
$948$	0	0
$949$	64097.6	2.19252
$950$	−12252.8	−0.418456
$951$	0	0
$952$	0	0
$953$	−701.418	−0.0238417	−0.0119209	−	0.999929i	$-0.503795\pi$
$953$	−701.418	−0.0238417	−0.0119209	+	0.999929i	$0.503795\pi$
$954$	0	0
$955$	−3929.47	−0.133146
$956$	−14853.1	−0.502493
$957$	0	0
$958$	22226.8	0.749599
$959$	0	0
$960$	0	0
$961$	−12207.4	−0.409769
$962$	−16024.3	−0.537051
$963$	0	0
$964$	−27994.5	−0.935313
$965$	29.8252	0.000994931 0
$966$	0	0
$967$	42402.5	1.41011	0.705053	−	0.709154i	$-0.250923\pi$
$967$	42402.5	1.41011	0.705053	+	0.709154i	$0.250923\pi$
$968$	19508.7	0.647762
$969$	0	0
$970$	−12552.0	−0.415484
$971$	15463.4	0.511064	0.255532	−	0.966801i	$-0.417749\pi$
$971$	15463.4	0.511064	0.255532	+	0.966801i	$0.417749\pi$
$972$	0	0
$973$	0	0
$974$	−7572.55	−0.249117
$975$	0	0
$976$	−9268.91	−0.303986
$977$	34484.1	1.12922	0.564609	−	0.825359i	$-0.309027\pi$
$977$	34484.1	1.12922	0.564609	+	0.825359i	$0.309027\pi$
$978$	0	0
$979$	−14310.7	−0.467184
$980$	0	0
$981$	0	0
$982$	19164.2	0.622765
$983$	7335.36	0.238008	0.119004	−	0.992894i	$-0.462030\pi$
$983$	7335.36	0.238008	0.119004	+	0.992894i	$0.462030\pi$
$984$	0	0
$985$	10489.2	0.339302
$986$	−8872.00	−0.286554
$987$	0	0
$988$	−11985.3	−0.385934
$989$	41397.1	1.33099
$990$	0	0
$991$	−10123.0	−0.324487	−0.162243	−	0.986751i	$-0.551873\pi$
$991$	−10123.0	−0.324487	−0.162243	+	0.986751i	$0.551873\pi$
$992$	−4243.30	−0.135811
$993$	0	0
$994$	0	0
$995$	3382.80	0.107781
$996$	0	0
$997$	−56669.2	−1.80013	−0.900066	−	0.435755i	$-0.856482\pi$
$997$	−56669.2	−1.80013	−0.900066	+	0.435755i	$0.856482\pi$
$998$	11163.1	0.354070
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.bi.1.1		2
3.2	odd	2		294.4.a.k.1.2	yes	2
7.2	even	3		882.4.g.y.361.2		4
7.3	odd	6		882.4.g.bd.667.1		4
7.4	even	3		882.4.g.y.667.2		4
7.5	odd	6		882.4.g.bd.361.1		4
7.6	odd	2		882.4.a.bc.1.2		2
12.11	even	2		2352.4.a.bn.1.2		2
21.2	odd	6		294.4.e.n.67.1		4
21.5	even	6		294.4.e.o.67.2		4
21.11	odd	6		294.4.e.n.79.1		4
21.17	even	6		294.4.e.o.79.2		4
21.20	even	2		294.4.a.j.1.1	✓	2
84.83	odd	2		2352.4.a.cd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.j.1.1	✓	2	21.20	even	2
294.4.a.k.1.2	yes	2	3.2	odd	2
294.4.e.n.67.1		4	21.2	odd	6
294.4.e.n.79.1		4	21.11	odd	6
294.4.e.o.67.2		4	21.5	even	6
294.4.e.o.79.2		4	21.17	even	6
882.4.a.bc.1.2		2	7.6	odd	2
882.4.a.bi.1.1		2	1.1	even	1	trivial
882.4.g.y.361.2		4	7.2	even	3
882.4.g.y.667.2		4	7.4	even	3
882.4.g.bd.361.1		4	7.5	odd	6
882.4.g.bd.667.1		4	7.3	odd	6
2352.4.a.bn.1.2		2	12.11	even	2
2352.4.a.cd.1.1		2	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -3.89949 q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -3.89949 q^{5} +8.00000 q^{8} -7.79899 q^{10} +61.3970 q^{11} -53.6985 q^{13} +16.0000 q^{16} +32.1005 q^{17} +55.7990 q^{19} -15.5980 q^{20} +122.794 q^{22} +94.6030 q^{23} -109.794 q^{25} -107.397 q^{26} -138.191 q^{29} -132.603 q^{31} +32.0000 q^{32} +64.2010 q^{34} +149.206 q^{37} +111.598 q^{38} -31.1960 q^{40} +427.497 q^{41} +437.588 q^{43} +245.588 q^{44} +189.206 q^{46} +57.0051 q^{47} -219.588 q^{50} -214.794 q^{52} +263.588 q^{53} -239.417 q^{55} -276.382 q^{58} +451.799 q^{59} -579.307 q^{61} -265.206 q^{62} +64.0000 q^{64} +209.397 q^{65} +309.588 q^{67} +128.402 q^{68} +1058.98 q^{71} -1193.66 q^{73} +298.412 q^{74} +223.196 q^{76} +1319.56 q^{79} -62.3919 q^{80} +854.995 q^{82} +1190.33 q^{83} -125.176 q^{85} +875.176 q^{86} +491.176 q^{88} -233.085 q^{89} +378.412 q^{92} +114.010 q^{94} -217.588 q^{95} +1609.44 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 16 q^{8} + 24 q^{10} + 4 q^{11} - 48 q^{13} + 32 q^{16} + 84 q^{17} + 72 q^{19} + 48 q^{20} + 8 q^{22} + 308 q^{23} + 18 q^{25} - 96 q^{26} + 80 q^{29} - 384 q^{31} + 64 q^{32} + 168 q^{34} + 536 q^{37} + 144 q^{38} + 96 q^{40} + 756 q^{41} + 400 q^{43} + 16 q^{44} + 616 q^{46} + 312 q^{47} + 36 q^{50} - 192 q^{52} + 52 q^{53} - 1152 q^{55} + 160 q^{58} + 864 q^{59} - 1416 q^{61} - 768 q^{62} + 128 q^{64} + 300 q^{65} + 144 q^{67} + 336 q^{68} + 1524 q^{71} - 744 q^{73} + 1072 q^{74} + 288 q^{76} + 976 q^{79} + 192 q^{80} + 1512 q^{82} - 312 q^{83} + 700 q^{85} + 800 q^{86} + 32 q^{88} + 108 q^{89} + 1232 q^{92} + 624 q^{94} + 40 q^{95} + 744 q^{97}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 16 * q^8 + 24 * q^10 + 4 * q^11 - 48 * q^13 + 32 * q^16 + 84 * q^17 + 72 * q^19 + 48 * q^20 + 8 * q^22 + 308 * q^23 + 18 * q^25 - 96 * q^26 + 80 * q^29 - 384 * q^31 + 64 * q^32 + 168 * q^34 + 536 * q^37 + 144 * q^38 + 96 * q^40 + 756 * q^41 + 400 * q^43 + 16 * q^44 + 616 * q^46 + 312 * q^47 + 36 * q^50 - 192 * q^52 + 52 * q^53 - 1152 * q^55 + 160 * q^58 + 864 * q^59 - 1416 * q^61 - 768 * q^62 + 128 * q^64 + 300 * q^65 + 144 * q^67 + 336 * q^68 + 1524 * q^71 - 744 * q^73 + 1072 * q^74 + 288 * q^76 + 976 * q^79 + 192 * q^80 + 1512 * q^82 - 312 * q^83 + 700 * q^85 + 800 * q^86 + 32 * q^88 + 108 * q^89 + 1232 * q^92 + 624 * q^94 + 40 * q^95 + 744 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more