LMFDB - Embedded newform 882.4.a.v.1.2

Error: no document with id 273130155 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [882,4,Mod(1,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,-4,0,8,-5,0,0,-16,0,10,-67,0,41] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.0396846251$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1345}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 336 $ x^2 - x - 336
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.8371$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +15.8371 q^{5} -8.00000 q^{8} -31.6742 q^{10} -51.8371 q^{11} +38.8371 q^{13} +16.0000 q^{16} -27.3485 q^{17} +76.5114 q^{19} +63.3485 q^{20} +103.674 q^{22} -147.348 q^{23} +125.814 q^{25} -77.6742 q^{26} -240.208 q^{29} -296.674 q^{31} -32.0000 q^{32} +54.6970 q^{34} -161.534 q^{37} -153.023 q^{38} -126.697 q^{40} +102.977 q^{41} -328.557 q^{43} -207.348 q^{44} +294.697 q^{46} -67.9546 q^{47} -251.629 q^{50} +155.348 q^{52} +66.4886 q^{53} -820.951 q^{55} +480.417 q^{58} +461.928 q^{59} +185.348 q^{61} +593.348 q^{62} +64.0000 q^{64} +615.068 q^{65} +545.208 q^{67} -109.394 q^{68} +130.742 q^{71} +181.299 q^{73} +323.068 q^{74} +306.045 q^{76} -409.697 q^{79} +253.394 q^{80} -205.955 q^{82} -347.928 q^{83} -433.121 q^{85} +657.114 q^{86} +414.697 q^{88} -1157.16 q^{89} -589.394 q^{92} +135.909 q^{94} +1211.72 q^{95} +1618.30 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 5 q^{5} - 16 q^{8} + 10 q^{10} - 67 q^{11} + 41 q^{13} + 32 q^{16} + 92 q^{17} + 43 q^{19} - 20 q^{20} + 134 q^{22} - 148 q^{23} + 435 q^{25} - 82 q^{26} - 77 q^{29} - 520 q^{31}+ \cdots + 1953 q^{97}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 5 * q^5 - 16 * q^8 + 10 * q^10 - 67 * q^11 + 41 * q^13 + 32 * q^16 + 92 * q^17 + 43 * q^19 - 20 * q^20 + 134 * q^22 - 148 * q^23 + 435 * q^25 - 82 * q^26 - 77 * q^29 - 520 * q^31 - 64 * q^32 - 184 * q^34 + 7 * q^37 - 86 * q^38 + 40 * q^40 + 426 * q^41 - 107 * q^43 - 268 * q^44 + 296 * q^46 - 576 * q^47 - 870 * q^50 + 164 * q^52 + 243 * q^53 - 505 * q^55 + 154 * q^58 + 7 * q^59 + 224 * q^61 + 1040 * q^62 + 128 * q^64 + 570 * q^65 + 687 * q^67 + 368 * q^68 - 472 * q^71 - 921 * q^73 - 14 * q^74 + 172 * q^76 - 526 * q^79 - 80 * q^80 - 852 * q^82 + 221 * q^83 - 2920 * q^85 + 214 * q^86 + 536 * q^88 - 774 * q^89 - 592 * q^92 + 1152 * q^94 + 1910 * q^95 + 1953 * q^97

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
$5$	15.8371	1.41652	0.708258	−	0.705954i	$-0.249482\pi$
$5$	15.8371	1.41652	0.708258	+	0.705954i	$0.249482\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−31.6742	−1.00163
$11$	−51.8371	−1.42086	−0.710431	−	0.703767i	$-0.751500\pi$
$11$	−51.8371	−1.42086	−0.710431	+	0.703767i	$0.751500\pi$
$12$	0	0
$13$	38.8371	0.828575	0.414288	−	0.910146i	$-0.364031\pi$
$13$	38.8371	0.828575	0.414288	+	0.910146i	$0.364031\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	−27.3485	−0.390175	−0.195088	−	0.980786i	$-0.562499\pi$
$17$	−27.3485	−0.390175	−0.195088	+	0.980786i	$0.562499\pi$
$18$	0	0
$19$	76.5114	0.923837	0.461919	−	0.886922i	$-0.347161\pi$
$19$	76.5114	0.923837	0.461919	+	0.886922i	$0.347161\pi$
$20$	63.3485	0.708258
$21$	0	0
$22$	103.674	1.00470
$23$	−147.348	−1.33584	−0.667919	−	0.744234i	$-0.732815\pi$
$23$	−147.348	−1.33584	−0.667919	+	0.744234i	$0.732815\pi$
$24$	0	0
$25$	125.814	1.00652
$26$	−77.6742	−0.585891
$27$	0	0
$28$	0	0
$29$	−240.208	−1.53812	−0.769061	−	0.639175i	$-0.779276\pi$
$29$	−240.208	−1.53812	−0.769061	+	0.639175i	$0.779276\pi$
$30$	0	0
$31$	−296.674	−1.71885	−0.859424	−	0.511264i	$-0.829177\pi$
$31$	−296.674	−1.71885	−0.859424	+	0.511264i	$0.829177\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	54.6970	0.275896
$35$	0	0
$36$	0	0
$37$	−161.534	−0.717731	−0.358865	−	0.933389i	$-0.616836\pi$
$37$	−161.534	−0.717731	−0.358865	+	0.933389i	$0.616836\pi$
$38$	−153.023	−0.653252
$39$	0	0
$40$	−126.697	−0.500814
$41$	102.977	0.392252	0.196126	−	0.980579i	$-0.437164\pi$
$41$	102.977	0.392252	0.196126	+	0.980579i	$0.437164\pi$
$42$	0	0
$43$	−328.557	−1.16522	−0.582610	−	0.812752i	$-0.697968\pi$
$43$	−328.557	−1.16522	−0.582610	+	0.812752i	$0.697968\pi$
$44$	−207.348	−0.710431
$45$	0	0
$46$	294.697	0.944580
$47$	−67.9546	−0.210898	−0.105449	−	0.994425i	$-0.533628\pi$
$47$	−67.9546	−0.210898	−0.105449	+	0.994425i	$0.533628\pi$
$48$	0	0
$49$	0	0
$50$	−251.629	−0.711714
$51$	0	0
$52$	155.348	0.414288
$53$	66.4886	0.172319	0.0861596	−	0.996281i	$-0.472541\pi$
$53$	66.4886	0.172319	0.0861596	+	0.996281i	$0.472541\pi$
$54$	0	0
$55$	−820.951	−2.01267
$56$	0	0
$57$	0	0
$58$	480.417	1.08762
$59$	461.928	1.01929	0.509643	−	0.860386i	$-0.329777\pi$
$59$	461.928	1.01929	0.509643	+	0.860386i	$0.329777\pi$
$60$	0	0
$61$	185.348	0.389040	0.194520	−	0.980899i	$-0.437685\pi$
$61$	185.348	0.389040	0.194520	+	0.980899i	$0.437685\pi$
$62$	593.348	1.21541
$63$	0	0
$64$	64.0000	0.125000
$65$	615.068	1.17369
$66$	0	0
$67$	545.208	0.994146	0.497073	−	0.867709i	$-0.334408\pi$
$67$	545.208	0.994146	0.497073	+	0.867709i	$0.334408\pi$
$68$	−109.394	−0.195088
$69$	0	0
$70$	0	0
$71$	130.742	0.218539	0.109270	−	0.994012i	$-0.465149\pi$
$71$	130.742	0.218539	0.109270	+	0.994012i	$0.465149\pi$
$72$	0	0
$73$	181.299	0.290678	0.145339	−	0.989382i	$-0.453573\pi$
$73$	181.299	0.290678	0.145339	+	0.989382i	$0.453573\pi$
$74$	323.068	0.507512
$75$	0	0
$76$	306.045	0.461919
$77$	0	0
$78$	0	0
$79$	−409.697	−0.583475	−0.291737	−	0.956498i	$-0.594233\pi$
$79$	−409.697	−0.583475	−0.291737	+	0.956498i	$0.594233\pi$
$80$	253.394	0.354129
$81$	0	0
$82$	−205.955	−0.277364
$83$	−347.928	−0.460121	−0.230061	−	0.973176i	$-0.573892\pi$
$83$	−347.928	−0.460121	−0.230061	+	0.973176i	$0.573892\pi$
$84$	0	0
$85$	−433.121	−0.552689
$86$	657.114	0.823935
$87$	0	0
$88$	414.697	0.502351
$89$	−1157.16	−1.37819	−0.689093	−	0.724673i	$-0.741991\pi$
$89$	−1157.16	−1.37819	−0.689093	+	0.724673i	$0.741991\pi$
$90$	0	0
$91$	0	0
$92$	−589.394	−0.667919
$93$	0	0
$94$	135.909	0.149127
$95$	1211.72	1.30863
$96$	0	0
$97$	1618.30	1.69395	0.846976	−	0.531631i	$-0.178421\pi$
$97$	1618.30	1.69395	0.846976	+	0.531631i	$0.178421\pi$
$98$	0	0
$99$	0	0
$100$	503.258	0.503258
$101$	−718.742	−0.708094	−0.354047	−	0.935228i	$-0.615195\pi$
$101$	−718.742	−0.708094	−0.354047	+	0.935228i	$0.615195\pi$
$102$	0	0
$103$	−1611.58	−1.54169	−0.770843	−	0.637025i	$-0.780165\pi$
$103$	−1611.58	−1.54169	−0.770843	+	0.637025i	$0.780165\pi$
$104$	−310.697	−0.292946
$105$	0	0
$106$	−132.977	−0.121848
$107$	−934.670	−0.844467	−0.422234	−	0.906487i	$-0.638754\pi$
$107$	−934.670	−0.844467	−0.422234	+	0.906487i	$0.638754\pi$
$108$	0	0
$109$	−1197.02	−1.05187	−0.525934	−	0.850525i	$-0.676284\pi$
$109$	−1197.02	−1.05187	−0.525934	+	0.850525i	$0.676284\pi$
$110$	1641.90	1.42317
$111$	0	0
$112$	0	0
$113$	2384.64	1.98521	0.992604	−	0.121400i	$-0.0387384\pi$
$113$	2384.64	1.98521	0.992604	+	0.121400i	$0.0387384\pi$
$114$	0	0
$115$	−2333.58	−1.89224
$116$	−960.833	−0.769061
$117$	0	0
$118$	−923.856	−0.720744
$119$	0	0
$120$	0	0
$121$	1356.09	1.01885
$122$	−370.697	−0.275093
$123$	0	0
$124$	−1186.70	−0.859424
$125$	12.8977	0.00922883
$126$	0	0
$127$	−2673.92	−1.86829	−0.934143	−	0.356898i	$-0.883834\pi$
$127$	−2673.92	−1.86829	−0.934143	+	0.356898i	$0.883834\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−1230.14	−0.829924
$131$	38.8598	0.0259176	0.0129588	−	0.999916i	$-0.495875\pi$
$131$	38.8598	0.0259176	0.0129588	+	0.999916i	$0.495875\pi$
$132$	0	0
$133$	0	0
$134$	−1090.42	−0.702968
$135$	0	0
$136$	218.788	0.137948
$137$	768.144	0.479029	0.239514	−	0.970893i	$-0.423012\pi$
$137$	768.144	0.479029	0.239514	+	0.970893i	$0.423012\pi$
$138$	0	0
$139$	1052.55	0.642274	0.321137	−	0.947033i	$-0.395935\pi$
$139$	1052.55	0.642274	0.321137	+	0.947033i	$0.395935\pi$
$140$	0	0
$141$	0	0
$142$	−261.485	−0.154530
$143$	−2013.20	−1.17729
$144$	0	0
$145$	−3804.21	−2.17877
$146$	−362.598	−0.205540
$147$	0	0
$148$	−646.136	−0.358865
$149$	−360.977	−0.198473	−0.0992363	−	0.995064i	$-0.531640\pi$
$149$	−360.977	−0.198473	−0.0992363	+	0.995064i	$0.531640\pi$
$150$	0	0
$151$	1548.39	0.834478	0.417239	−	0.908797i	$-0.362998\pi$
$151$	1548.39	0.834478	0.417239	+	0.908797i	$0.362998\pi$
$152$	−612.091	−0.326626
$153$	0	0
$154$	0	0
$155$	−4698.47	−2.43477
$156$	0	0
$157$	−967.068	−0.491595	−0.245798	−	0.969321i	$-0.579050\pi$
$157$	−967.068	−0.491595	−0.245798	+	0.969321i	$0.579050\pi$
$158$	819.394	0.412579
$159$	0	0
$160$	−506.788	−0.250407
$161$	0	0
$162$	0	0
$163$	−1326.50	−0.637420	−0.318710	−	0.947852i	$-0.603250\pi$
$163$	−1326.50	−0.637420	−0.318710	+	0.947852i	$0.603250\pi$
$164$	411.909	0.196126
$165$	0	0
$166$	695.856	0.325355
$167$	−1416.70	−0.656451	−0.328225	−	0.944599i	$-0.606451\pi$
$167$	−1416.70	−0.656451	−0.328225	+	0.944599i	$0.606451\pi$
$168$	0	0
$169$	−688.678	−0.313463
$170$	866.242	0.390810
$171$	0	0
$172$	−1314.23	−0.582610
$173$	1036.60	0.455556	0.227778	−	0.973713i	$-0.426854\pi$
$173$	1036.60	0.455556	0.227778	+	0.973713i	$0.426854\pi$
$174$	0	0
$175$	0	0
$176$	−829.394	−0.355215
$177$	0	0
$178$	2314.32	0.974525
$179$	767.432	0.320450	0.160225	−	0.987081i	$-0.448778\pi$
$179$	767.432	0.320450	0.160225	+	0.987081i	$0.448778\pi$
$180$	0	0
$181$	−3957.71	−1.62527	−0.812636	−	0.582772i	$-0.801968\pi$
$181$	−3957.71	−1.62527	−0.812636	+	0.582772i	$0.801968\pi$
$182$	0	0
$183$	0	0
$184$	1178.79	0.472290
$185$	−2558.23	−1.01668
$186$	0	0
$187$	1417.67	0.554385
$188$	−271.818	−0.105449
$189$	0	0
$190$	−2423.44	−0.925341
$191$	1805.30	0.683909	0.341954	−	0.939717i	$-0.388911\pi$
$191$	1805.30	0.683909	0.341954	+	0.939717i	$0.388911\pi$
$192$	0	0
$193$	3370.84	1.25719	0.628597	−	0.777731i	$-0.283630\pi$
$193$	3370.84	1.25719	0.628597	+	0.777731i	$0.283630\pi$
$194$	−3236.60	−1.19781
$195$	0	0
$196$	0	0
$197$	4612.31	1.66809	0.834044	−	0.551697i	$-0.186020\pi$
$197$	4612.31	1.66809	0.834044	+	0.551697i	$0.186020\pi$
$198$	0	0
$199$	−2229.86	−0.794326	−0.397163	−	0.917748i	$-0.630005\pi$
$199$	−2229.86	−0.794326	−0.397163	+	0.917748i	$0.630005\pi$
$200$	−1006.52	−0.355857
$201$	0	0
$202$	1437.48	0.500698
$203$	0	0
$204$	0	0
$205$	1630.86	0.555631
$206$	3223.16	1.09014
$207$	0	0
$208$	621.394	0.207144
$209$	−3966.13	−1.31265
$210$	0	0
$211$	912.614	0.297758	0.148879	−	0.988855i	$-0.452434\pi$
$211$	912.614	0.297758	0.148879	+	0.988855i	$0.452434\pi$
$212$	265.955	0.0861596
$213$	0	0
$214$	1869.34	0.597129
$215$	−5203.39	−1.65055
$216$	0	0
$217$	0	0
$218$	2394.04	0.743783
$219$	0	0
$220$	−3283.80	−1.00634
$221$	−1062.14	−0.323290
$222$	0	0
$223$	−4319.47	−1.29710	−0.648549	−	0.761173i	$-0.724624\pi$
$223$	−4319.47	−1.29710	−0.648549	+	0.761173i	$0.724624\pi$
$224$	0	0
$225$	0	0
$226$	−4769.29	−1.40375
$227$	−2061.28	−0.602697	−0.301349	−	0.953514i	$-0.597437\pi$
$227$	−2061.28	−0.602697	−0.301349	+	0.953514i	$0.597437\pi$
$228$	0	0
$229$	−3474.63	−1.00266	−0.501332	−	0.865255i	$-0.667157\pi$
$229$	−3474.63	−1.00266	−0.501332	+	0.865255i	$0.667157\pi$
$230$	4667.15	1.33801
$231$	0	0
$232$	1921.67	0.543809
$233$	−776.099	−0.218214	−0.109107	−	0.994030i	$-0.534799\pi$
$233$	−776.099	−0.218214	−0.109107	+	0.994030i	$0.534799\pi$
$234$	0	0
$235$	−1076.20	−0.298740
$236$	1847.71	0.509643
$237$	0	0
$238$	0	0
$239$	−2006.80	−0.543133	−0.271567	−	0.962420i	$-0.587542\pi$
$239$	−2006.80	−0.543133	−0.271567	+	0.962420i	$0.587542\pi$
$240$	0	0
$241$	805.648	0.215337	0.107669	−	0.994187i	$-0.465661\pi$
$241$	805.648	0.215337	0.107669	+	0.994187i	$0.465661\pi$
$242$	−2712.17	−0.720435
$243$	0	0
$244$	741.394	0.194520
$245$	0	0
$246$	0	0
$247$	2971.48	0.765469
$248$	2373.39	0.607704
$249$	0	0
$250$	−25.7954	−0.00652577
$251$	−1421.78	−0.357539	−0.178769	−	0.983891i	$-0.557212\pi$
$251$	−1421.78	−0.357539	−0.178769	+	0.983891i	$0.557212\pi$
$252$	0	0
$253$	7638.12	1.89804
$254$	5347.85	1.32108
$255$	0	0
$256$	256.000	0.0625000
$257$	1465.82	0.355779	0.177890	−	0.984050i	$-0.443073\pi$
$257$	1465.82	0.355779	0.177890	+	0.984050i	$0.443073\pi$
$258$	0	0
$259$	0	0
$260$	2460.27	0.586845
$261$	0	0
$262$	−77.7197	−0.0183265
$263$	−6991.39	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$263$	−6991.39	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$264$	0	0
$265$	1052.99	0.244093
$266$	0	0
$267$	0	0
$268$	2180.83	0.497073
$269$	−808.958	−0.183357	−0.0916786	−	0.995789i	$-0.529223\pi$
$269$	−808.958	−0.183357	−0.0916786	+	0.995789i	$0.529223\pi$
$270$	0	0
$271$	6661.77	1.49326	0.746630	−	0.665239i	$-0.231670\pi$
$271$	6661.77	1.49326	0.746630	+	0.665239i	$0.231670\pi$
$272$	−437.576	−0.0975438
$273$	0	0
$274$	−1536.29	−0.338725
$275$	−6521.86	−1.43012
$276$	0	0
$277$	−7531.47	−1.63365	−0.816827	−	0.576883i	$-0.804269\pi$
$277$	−7531.47	−1.63365	−0.816827	+	0.576883i	$0.804269\pi$
$278$	−2105.10	−0.454156
$279$	0	0
$280$	0	0
$281$	−1690.19	−0.358819	−0.179410	−	0.983774i	$-0.557419\pi$
$281$	−1690.19	−0.358819	−0.179410	+	0.983774i	$0.557419\pi$
$282$	0	0
$283$	3178.23	0.667584	0.333792	−	0.942647i	$-0.391672\pi$
$283$	3178.23	0.667584	0.333792	+	0.942647i	$0.391672\pi$
$284$	522.970	0.109270
$285$	0	0
$286$	4026.41	0.832470
$287$	0	0
$288$	0	0
$289$	−4165.06	−0.847763
$290$	7608.42	1.54063
$291$	0	0
$292$	725.197	0.145339
$293$	2176.53	0.433974	0.216987	−	0.976174i	$-0.430377\pi$
$293$	2176.53	0.433974	0.216987	+	0.976174i	$0.430377\pi$
$294$	0	0
$295$	7315.61	1.44383
$296$	1292.27	0.253756
$297$	0	0
$298$	721.955	0.140341
$299$	−5722.59	−1.10684
$300$	0	0
$301$	0	0
$302$	−3096.78	−0.590065
$303$	0	0
$304$	1224.18	0.230959
$305$	2935.39	0.551081
$306$	0	0
$307$	623.504	0.115913	0.0579564	−	0.998319i	$-0.481542\pi$
$307$	623.504	0.115913	0.0579564	+	0.998319i	$0.481542\pi$
$308$	0	0
$309$	0	0
$310$	9396.93	1.72164
$311$	−467.992	−0.0853293	−0.0426647	−	0.999089i	$-0.513585\pi$
$311$	−467.992	−0.0853293	−0.0426647	+	0.999089i	$0.513585\pi$
$312$	0	0
$313$	3612.81	0.652422	0.326211	−	0.945297i	$-0.394228\pi$
$313$	3612.81	0.652422	0.326211	+	0.945297i	$0.394228\pi$
$314$	1934.14	0.347610
$315$	0	0
$316$	−1638.79	−0.291737
$317$	−4531.74	−0.802927	−0.401463	−	0.915875i	$-0.631498\pi$
$317$	−4531.74	−0.802927	−0.401463	+	0.915875i	$0.631498\pi$
$318$	0	0
$319$	12451.7	2.18546
$320$	1013.58	0.177064
$321$	0	0
$322$	0	0
$323$	−2092.47	−0.360459
$324$	0	0
$325$	4886.27	0.833974
$326$	2653.00	0.450724
$327$	0	0
$328$	−823.818	−0.138682
$329$	0	0
$330$	0	0
$331$	−1237.06	−0.205422	−0.102711	−	0.994711i	$-0.532752\pi$
$331$	−1237.06	−0.205422	−0.102711	+	0.994711i	$0.532752\pi$
$332$	−1391.71	−0.230061
$333$	0	0
$334$	2833.39	0.464181
$335$	8634.53	1.40822
$336$	0	0
$337$	−1867.83	−0.301921	−0.150960	−	0.988540i	$-0.548237\pi$
$337$	−1867.83	−0.301921	−0.150960	+	0.988540i	$0.548237\pi$
$338$	1377.36	0.221652
$339$	0	0
$340$	−1732.48	−0.276345
$341$	15378.7	2.44224
$342$	0	0
$343$	0	0
$344$	2628.45	0.411967
$345$	0	0
$346$	−2073.20	−0.322126
$347$	63.3637	0.00980271	0.00490136	−	0.999988i	$-0.498440\pi$
$347$	63.3637	0.00980271	0.00490136	+	0.999988i	$0.498440\pi$
$348$	0	0
$349$	1223.79	0.187702	0.0938508	−	0.995586i	$-0.470082\pi$
$349$	1223.79	0.187702	0.0938508	+	0.995586i	$0.470082\pi$
$350$	0	0
$351$	0	0
$352$	1658.79	0.251175
$353$	4515.61	0.680855	0.340428	−	0.940271i	$-0.389428\pi$
$353$	4515.61	0.680855	0.340428	+	0.940271i	$0.389428\pi$
$354$	0	0
$355$	2070.58	0.309564
$356$	−4628.64	−0.689093
$357$	0	0
$358$	−1534.86	−0.226592
$359$	−2228.49	−0.327619	−0.163810	−	0.986492i	$-0.552378\pi$
$359$	−2228.49	−0.327619	−0.163810	+	0.986492i	$0.552378\pi$
$360$	0	0
$361$	−1005.01	−0.146524
$362$	7915.42	1.14924
$363$	0	0
$364$	0	0
$365$	2871.26	0.411749
$366$	0	0
$367$	−1437.34	−0.204438	−0.102219	−	0.994762i	$-0.532594\pi$
$367$	−1437.34	−0.204438	−0.102219	+	0.994762i	$0.532594\pi$
$368$	−2357.58	−0.333960
$369$	0	0
$370$	5116.47	0.718899
$371$	0	0
$372$	0	0
$373$	−12237.4	−1.69874	−0.849370	−	0.527798i	$-0.823018\pi$
$373$	−12237.4	−1.69874	−0.849370	+	0.527798i	$0.823018\pi$
$374$	−2835.33	−0.392010
$375$	0	0
$376$	543.636	0.0745636
$377$	−9329.00	−1.27445
$378$	0	0
$379$	−10647.0	−1.44301	−0.721503	−	0.692411i	$-0.756549\pi$
$379$	−10647.0	−1.44301	−0.721503	+	0.692411i	$0.756549\pi$
$380$	4846.88	0.654315
$381$	0	0
$382$	−3610.59	−0.483597
$383$	6714.81	0.895851	0.447925	−	0.894071i	$-0.352163\pi$
$383$	6714.81	0.895851	0.447925	+	0.894071i	$0.352163\pi$
$384$	0	0
$385$	0	0
$386$	−6741.68	−0.888970
$387$	0	0
$388$	6473.20	0.846976
$389$	10653.1	1.38852	0.694258	−	0.719727i	$-0.255733\pi$
$389$	10653.1	1.38852	0.694258	+	0.719727i	$0.255733\pi$
$390$	0	0
$391$	4029.76	0.521211
$392$	0	0
$393$	0	0
$394$	−9224.62	−1.17952
$395$	−6488.42	−0.826501
$396$	0	0
$397$	3221.04	0.407203	0.203601	−	0.979054i	$-0.434735\pi$
$397$	3221.04	0.407203	0.203601	+	0.979054i	$0.434735\pi$
$398$	4459.73	0.561673
$399$	0	0
$400$	2013.03	0.251629
$401$	−12485.0	−1.55479	−0.777395	−	0.629012i	$-0.783460\pi$
$401$	−12485.0	−1.55479	−0.777395	+	0.629012i	$0.783460\pi$
$402$	0	0
$403$	−11522.0	−1.42419
$404$	−2874.97	−0.354047
$405$	0	0
$406$	0	0
$407$	8373.46	1.01980
$408$	0	0
$409$	7037.39	0.850798	0.425399	−	0.905006i	$-0.360134\pi$
$409$	7037.39	0.850798	0.425399	+	0.905006i	$0.360134\pi$
$410$	−3261.73	−0.392891
$411$	0	0
$412$	−6446.32	−0.770843
$413$	0	0
$414$	0	0
$415$	−5510.18	−0.651769
$416$	−1242.79	−0.146473
$417$	0	0
$418$	7932.26	0.928180
$419$	1549.66	0.180682	0.0903410	−	0.995911i	$-0.471204\pi$
$419$	1549.66	0.180682	0.0903410	+	0.995911i	$0.471204\pi$
$420$	0	0
$421$	5531.63	0.640369	0.320184	−	0.947355i	$-0.396255\pi$
$421$	5531.63	0.640369	0.320184	+	0.947355i	$0.396255\pi$
$422$	−1825.23	−0.210547
$423$	0	0
$424$	−531.909	−0.0609240
$425$	−3440.83	−0.392717
$426$	0	0
$427$	0	0
$428$	−3738.68	−0.422234
$429$	0	0
$430$	10406.8	1.16712
$431$	2029.93	0.226864	0.113432	−	0.993546i	$-0.463816\pi$
$431$	2029.93	0.226864	0.113432	+	0.993546i	$0.463816\pi$
$432$	0	0
$433$	−327.739	−0.0363744	−0.0181872	−	0.999835i	$-0.505789\pi$
$433$	−327.739	−0.0363744	−0.0181872	+	0.999835i	$0.505789\pi$
$434$	0	0
$435$	0	0
$436$	−4788.08	−0.525934
$437$	−11273.8	−1.23410
$438$	0	0
$439$	7908.68	0.859819	0.429910	−	0.902872i	$-0.358545\pi$
$439$	7908.68	0.859819	0.429910	+	0.902872i	$0.358545\pi$
$440$	6567.61	0.711587
$441$	0	0
$442$	2124.27	0.228600
$443$	2920.82	0.313256	0.156628	−	0.987658i	$-0.449938\pi$
$443$	2920.82	0.313256	0.156628	+	0.987658i	$0.449938\pi$
$444$	0	0
$445$	−18326.1	−1.95222
$446$	8638.93	0.917187
$447$	0	0
$448$	0	0
$449$	10240.2	1.07631	0.538156	−	0.842845i	$-0.319121\pi$
$449$	10240.2	1.07631	0.538156	+	0.842845i	$0.319121\pi$
$450$	0	0
$451$	−5338.05	−0.557336
$452$	9538.58	0.992604
$453$	0	0
$454$	4122.57	0.426171
$455$	0	0
$456$	0	0
$457$	5892.63	0.603163	0.301582	−	0.953440i	$-0.402485\pi$
$457$	5892.63	0.603163	0.301582	+	0.953440i	$0.402485\pi$
$458$	6949.26	0.708991
$459$	0	0
$460$	−9334.30	−0.946118
$461$	12643.4	1.27735	0.638677	−	0.769475i	$-0.279482\pi$
$461$	12643.4	1.27735	0.638677	+	0.769475i	$0.279482\pi$
$462$	0	0
$463$	15093.2	1.51499	0.757494	−	0.652842i	$-0.226423\pi$
$463$	15093.2	1.51499	0.757494	+	0.652842i	$0.226423\pi$
$464$	−3843.33	−0.384531
$465$	0	0
$466$	1552.20	0.154301
$467$	2820.23	0.279454	0.139727	−	0.990190i	$-0.455378\pi$
$467$	2820.23	0.279454	0.139727	+	0.990190i	$0.455378\pi$
$468$	0	0
$469$	0	0
$470$	2152.41	0.211241
$471$	0	0
$472$	−3695.42	−0.360372
$473$	17031.4	1.65562
$474$	0	0
$475$	9626.23	0.929856
$476$	0	0
$477$	0	0
$478$	4013.59	0.384053
$479$	16448.5	1.56900	0.784500	−	0.620129i	$-0.212920\pi$
$479$	16448.5	1.56900	0.784500	+	0.620129i	$0.212920\pi$
$480$	0	0
$481$	−6273.52	−0.594694
$482$	−1611.30	−0.152267
$483$	0	0
$484$	5424.35	0.509424
$485$	25629.2	2.39951
$486$	0	0
$487$	6331.07	0.589093	0.294546	−	0.955637i	$-0.404831\pi$
$487$	6331.07	0.589093	0.294546	+	0.955637i	$0.404831\pi$
$488$	−1482.79	−0.137546
$489$	0	0
$490$	0	0
$491$	9286.90	0.853588	0.426794	−	0.904349i	$-0.359643\pi$
$491$	9286.90	0.853588	0.426794	+	0.904349i	$0.359643\pi$
$492$	0	0
$493$	6569.33	0.600138
$494$	−5942.96	−0.541268
$495$	0	0
$496$	−4746.79	−0.429712
$497$	0	0
$498$	0	0
$499$	−243.451	−0.0218404	−0.0109202	−	0.999940i	$-0.503476\pi$
$499$	−243.451	−0.0218404	−0.0109202	+	0.999940i	$0.503476\pi$
$500$	51.5907	0.00461442
$501$	0	0
$502$	2843.57	0.252818
$503$	−8499.30	−0.753409	−0.376705	−	0.926333i	$-0.622943\pi$
$503$	−8499.30	−0.753409	−0.376705	+	0.926333i	$0.622943\pi$
$504$	0	0
$505$	−11382.8	−1.00303
$506$	−15276.2	−1.34212
$507$	0	0
$508$	−10695.7	−0.934143
$509$	7683.10	0.669052	0.334526	−	0.942387i	$-0.391424\pi$
$509$	7683.10	0.669052	0.334526	+	0.942387i	$0.391424\pi$
$510$	0	0
$511$	0	0
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−2931.64	−0.251574
$515$	−25522.8	−2.18382
$516$	0	0
$517$	3522.57	0.299656
$518$	0	0
$519$	0	0
$520$	−4920.55	−0.414962
$521$	21530.6	1.81051	0.905253	−	0.424874i	$-0.139681\pi$
$521$	21530.6	1.81051	0.905253	+	0.424874i	$0.139681\pi$
$522$	0	0
$523$	−16847.1	−1.40855	−0.704274	−	0.709929i	$-0.748727\pi$
$523$	−16847.1	−1.40855	−0.704274	+	0.709929i	$0.748727\pi$
$524$	155.439	0.0129588
$525$	0	0
$526$	13982.8	1.15908
$527$	8113.59	0.670652
$528$	0	0
$529$	9544.58	0.784464
$530$	−2105.98	−0.172600
$531$	0	0
$532$	0	0
$533$	3999.34	0.325011
$534$	0	0
$535$	−14802.5	−1.19620
$536$	−4361.67	−0.351484
$537$	0	0
$538$	1617.92	0.129653
$539$	0	0
$540$	0	0
$541$	17440.0	1.38596	0.692981	−	0.720955i	$-0.256297\pi$
$541$	17440.0	1.38596	0.692981	+	0.720955i	$0.256297\pi$
$542$	−13323.5	−1.05589
$543$	0	0
$544$	875.151	0.0689739
$545$	−18957.3	−1.48999
$546$	0	0
$547$	11520.7	0.900530	0.450265	−	0.892895i	$-0.351329\pi$
$547$	11520.7	0.900530	0.450265	+	0.892895i	$0.351329\pi$
$548$	3072.58	0.239514
$549$	0	0
$550$	13043.7	1.01125
$551$	−18378.7	−1.42098
$552$	0	0
$553$	0	0
$554$	15062.9	1.15517
$555$	0	0
$556$	4210.20	0.321137
$557$	−11493.0	−0.874282	−0.437141	−	0.899393i	$-0.644009\pi$
$557$	−11493.0	−0.874282	−0.437141	+	0.899393i	$0.644009\pi$
$558$	0	0
$559$	−12760.2	−0.965472
$560$	0	0
$561$	0	0
$562$	3380.38	0.253724
$563$	−18111.3	−1.35577	−0.677886	−	0.735167i	$-0.737104\pi$
$563$	−18111.3	−1.35577	−0.677886	+	0.735167i	$0.737104\pi$
$564$	0	0
$565$	37765.9	2.81208
$566$	−6356.46	−0.472053
$567$	0	0
$568$	−1045.94	−0.0772652
$569$	−4417.61	−0.325476	−0.162738	−	0.986669i	$-0.552033\pi$
$569$	−4417.61	−0.325476	−0.162738	+	0.986669i	$0.552033\pi$
$570$	0	0
$571$	13219.7	0.968878	0.484439	−	0.874825i	$-0.339024\pi$
$571$	13219.7	0.968878	0.484439	+	0.874825i	$0.339024\pi$
$572$	−8052.82	−0.588646
$573$	0	0
$574$	0	0
$575$	−18538.6	−1.34454
$576$	0	0
$577$	−17496.4	−1.26236	−0.631182	−	0.775634i	$-0.717430\pi$
$577$	−17496.4	−1.26236	−0.631182	+	0.775634i	$0.717430\pi$
$578$	8330.12	0.599459
$579$	0	0
$580$	−15216.8	−1.08939
$581$	0	0
$582$	0	0
$583$	−3446.58	−0.244842
$584$	−1450.39	−0.102770
$585$	0	0
$586$	−4353.07	−0.306866
$587$	4280.53	0.300982	0.150491	−	0.988611i	$-0.451915\pi$
$587$	4280.53	0.300982	0.150491	+	0.988611i	$0.451915\pi$
$588$	0	0
$589$	−22699.0	−1.58794
$590$	−14631.2	−1.02095
$591$	0	0
$592$	−2584.55	−0.179433
$593$	1590.93	0.110172	0.0550858	−	0.998482i	$-0.482457\pi$
$593$	1590.93	0.110172	0.0550858	+	0.998482i	$0.482457\pi$
$594$	0	0
$595$	0	0
$596$	−1443.91	−0.0992363
$597$	0	0
$598$	11445.2	0.782656
$599$	13922.8	0.949703	0.474851	−	0.880066i	$-0.342502\pi$
$599$	13922.8	0.949703	0.474851	+	0.880066i	$0.342502\pi$
$600$	0	0
$601$	12559.7	0.852446	0.426223	−	0.904618i	$-0.359844\pi$
$601$	12559.7	0.852446	0.426223	+	0.904618i	$0.359844\pi$
$602$	0	0
$603$	0	0
$604$	6193.56	0.417239
$605$	21476.5	1.44321
$606$	0	0
$607$	7678.37	0.513436	0.256718	−	0.966486i	$-0.417359\pi$
$607$	7678.37	0.513436	0.256718	+	0.966486i	$0.417359\pi$
$608$	−2448.36	−0.163313
$609$	0	0
$610$	−5870.77	−0.389673
$611$	−2639.16	−0.174745
$612$	0	0
$613$	6158.37	0.405766	0.202883	−	0.979203i	$-0.434969\pi$
$613$	6158.37	0.405766	0.202883	+	0.979203i	$0.434969\pi$
$614$	−1247.01	−0.0819628
$615$	0	0
$616$	0	0
$617$	−8813.12	−0.575045	−0.287523	−	0.957774i	$-0.592832\pi$
$617$	−8813.12	−0.575045	−0.287523	+	0.957774i	$0.592832\pi$
$618$	0	0
$619$	23189.9	1.50579	0.752894	−	0.658142i	$-0.228658\pi$
$619$	23189.9	1.50579	0.752894	+	0.658142i	$0.228658\pi$
$620$	−18793.9	−1.21739
$621$	0	0
$622$	935.985	0.0603369
$623$	0	0
$624$	0	0
$625$	−15522.5	−0.993442
$626$	−7225.62	−0.461332
$627$	0	0
$628$	−3868.27	−0.245798
$629$	4417.71	0.280041
$630$	0	0
$631$	7936.94	0.500736	0.250368	−	0.968151i	$-0.419448\pi$
$631$	7936.94	0.500736	0.250368	+	0.968151i	$0.419448\pi$
$632$	3277.58	0.206289
$633$	0	0
$634$	9063.48	0.567755
$635$	−42347.3	−2.64646
$636$	0	0
$637$	0	0
$638$	−24903.4	−1.54535
$639$	0	0
$640$	−2027.15	−0.125203
$641$	−32114.6	−1.97886	−0.989432	−	0.144996i	$-0.953683\pi$
$641$	−32114.6	−1.97886	−0.989432	+	0.144996i	$0.953683\pi$
$642$	0	0
$643$	−24786.7	−1.52021	−0.760104	−	0.649802i	$-0.774852\pi$
$643$	−24786.7	−1.52021	−0.760104	+	0.649802i	$0.774852\pi$
$644$	0	0
$645$	0	0
$646$	4184.94	0.254883
$647$	−7545.59	−0.458497	−0.229249	−	0.973368i	$-0.573627\pi$
$647$	−7545.59	−0.458497	−0.229249	+	0.973368i	$0.573627\pi$
$648$	0	0
$649$	−23945.0	−1.44827
$650$	−9772.54	−0.589708
$651$	0	0
$652$	−5306.00	−0.318710
$653$	4888.99	0.292988	0.146494	−	0.989212i	$-0.453201\pi$
$653$	4888.99	0.292988	0.146494	+	0.989212i	$0.453201\pi$
$654$	0	0
$655$	615.428	0.0367126
$656$	1647.64	0.0980631
$657$	0	0
$658$	0	0
$659$	−25895.9	−1.53075	−0.765374	−	0.643586i	$-0.777446\pi$
$659$	−25895.9	−1.53075	−0.765374	+	0.643586i	$0.777446\pi$
$660$	0	0
$661$	8183.37	0.481537	0.240769	−	0.970583i	$-0.422601\pi$
$661$	8183.37	0.481537	0.240769	+	0.970583i	$0.422601\pi$
$662$	2474.11	0.145256
$663$	0	0
$664$	2783.42	0.162677
$665$	0	0
$666$	0	0
$667$	35394.3	2.05468
$668$	−5666.79	−0.328225
$669$	0	0
$670$	−17269.1	−0.995764
$671$	−9607.93	−0.552772
$672$	0	0
$673$	−4635.02	−0.265478	−0.132739	−	0.991151i	$-0.542377\pi$
$673$	−4635.02	−0.265478	−0.132739	+	0.991151i	$0.542377\pi$
$674$	3735.67	0.213490
$675$	0	0
$676$	−2754.71	−0.156731
$677$	24385.8	1.38437	0.692187	−	0.721718i	$-0.256647\pi$
$677$	24385.8	1.38437	0.692187	+	0.721718i	$0.256647\pi$
$678$	0	0
$679$	0	0
$680$	3464.97	0.195405
$681$	0	0
$682$	−30757.5	−1.72693
$683$	18393.4	1.03046	0.515230	−	0.857052i	$-0.327706\pi$
$683$	18393.4	1.03046	0.515230	+	0.857052i	$0.327706\pi$
$684$	0	0
$685$	12165.2	0.678552
$686$	0	0
$687$	0	0
$688$	−5256.91	−0.291305
$689$	2582.23	0.142779
$690$	0	0
$691$	−14898.9	−0.820231	−0.410116	−	0.912034i	$-0.634512\pi$
$691$	−14898.9	−0.820231	−0.410116	+	0.912034i	$0.634512\pi$
$692$	4146.39	0.227778
$693$	0	0
$694$	−126.727	−0.00693157
$695$	16669.3	0.909791
$696$	0	0
$697$	−2816.27	−0.153047
$698$	−2447.58	−0.132725
$699$	0	0
$700$	0	0
$701$	5725.70	0.308497	0.154249	−	0.988032i	$-0.450704\pi$
$701$	5725.70	0.308497	0.154249	+	0.988032i	$0.450704\pi$
$702$	0	0
$703$	−12359.2	−0.663067
$704$	−3317.58	−0.177608
$705$	0	0
$706$	−9031.23	−0.481437
$707$	0	0
$708$	0	0
$709$	−23456.8	−1.24251	−0.621255	−	0.783609i	$-0.713377\pi$
$709$	−23456.8	−1.24251	−0.621255	+	0.783609i	$0.713377\pi$
$710$	−4141.17	−0.218895
$711$	0	0
$712$	9257.27	0.487263
$713$	43714.5	2.29610
$714$	0	0
$715$	−31883.4	−1.66765
$716$	3069.73	0.160225
$717$	0	0
$718$	4456.98	0.231662
$719$	8459.00	0.438759	0.219379	−	0.975640i	$-0.429597\pi$
$719$	8459.00	0.438759	0.219379	+	0.975640i	$0.429597\pi$
$720$	0	0
$721$	0	0
$722$	2010.02	0.103608
$723$	0	0
$724$	−15830.8	−0.812636
$725$	−30221.7	−1.54814
$726$	0	0
$727$	−11822.2	−0.603111	−0.301555	−	0.953449i	$-0.597506\pi$
$727$	−11822.2	−0.603111	−0.301555	+	0.953449i	$0.597506\pi$
$728$	0	0
$729$	0	0
$730$	−5742.52	−0.291151
$731$	8985.53	0.454640
$732$	0	0
$733$	5028.95	0.253409	0.126704	−	0.991941i	$-0.459560\pi$
$733$	5028.95	0.253409	0.126704	+	0.991941i	$0.459560\pi$
$734$	2874.68	0.144559
$735$	0	0
$736$	4715.15	0.236145
$737$	−28262.0	−1.41254
$738$	0	0
$739$	−17743.9	−0.883247	−0.441624	−	0.897200i	$-0.645597\pi$
$739$	−17743.9	−0.883247	−0.441624	+	0.897200i	$0.645597\pi$
$740$	−10232.9	−0.508338
$741$	0	0
$742$	0	0
$743$	13202.3	0.651877	0.325938	−	0.945391i	$-0.394320\pi$
$743$	13202.3	0.651877	0.325938	+	0.945391i	$0.394320\pi$
$744$	0	0
$745$	−5716.84	−0.281139
$746$	24474.9	1.20119
$747$	0	0
$748$	5670.67	0.277193
$749$	0	0
$750$	0	0
$751$	15601.0	0.758040	0.379020	−	0.925388i	$-0.376261\pi$
$751$	15601.0	0.758040	0.379020	+	0.925388i	$0.376261\pi$
$752$	−1087.27	−0.0527244
$753$	0	0
$754$	18658.0	0.901173
$755$	24522.0	1.18205
$756$	0	0
$757$	2948.08	0.141545	0.0707725	−	0.997492i	$-0.477454\pi$
$757$	2948.08	0.141545	0.0707725	+	0.997492i	$0.477454\pi$
$758$	21294.0	1.02036
$759$	0	0
$760$	−9693.76	−0.462670
$761$	−1697.03	−0.0808375	−0.0404187	−	0.999183i	$-0.512869\pi$
$761$	−1697.03	−0.0808375	−0.0404187	+	0.999183i	$0.512869\pi$
$762$	0	0
$763$	0	0
$764$	7221.18	0.341954
$765$	0	0
$766$	−13429.6	−0.633462
$767$	17940.0	0.844556
$768$	0	0
$769$	−96.7799	−0.00453833	−0.00226916	−	0.999997i	$-0.500722\pi$
$769$	−96.7799	−0.00453833	−0.00226916	+	0.999997i	$0.500722\pi$
$770$	0	0
$771$	0	0
$772$	13483.4	0.628597
$773$	−36326.8	−1.69028	−0.845138	−	0.534548i	$-0.820482\pi$
$773$	−36326.8	−1.69028	−0.845138	+	0.534548i	$0.820482\pi$
$774$	0	0
$775$	−37325.9	−1.73005
$776$	−12946.4	−0.598903
$777$	0	0
$778$	−21306.2	−0.981828
$779$	7878.93	0.362377
$780$	0	0
$781$	−6777.31	−0.310514
$782$	−8059.52	−0.368552
$783$	0	0
$784$	0	0
$785$	−15315.6	−0.696352
$786$	0	0
$787$	7096.46	0.321425	0.160713	−	0.987001i	$-0.448621\pi$
$787$	7096.46	0.321425	0.160713	+	0.987001i	$0.448621\pi$
$788$	18449.2	0.834044
$789$	0	0
$790$	12976.8	0.584424
$791$	0	0
$792$	0	0
$793$	7198.40	0.322349
$794$	−6442.08	−0.287936
$795$	0	0
$796$	−8919.45	−0.397163
$797$	−40289.6	−1.79063	−0.895314	−	0.445436i	$-0.853049\pi$
$797$	−40289.6	−1.79063	−0.895314	+	0.445436i	$0.853049\pi$
$798$	0	0
$799$	1858.45	0.0822871
$800$	−4026.06	−0.177928
$801$	0	0
$802$	24970.0	1.09940
$803$	−9398.03	−0.413013
$804$	0	0
$805$	0	0
$806$	23043.9	1.00706
$807$	0	0
$808$	5749.94	0.250349
$809$	11566.0	0.502642	0.251321	−	0.967904i	$-0.419135\pi$
$809$	11566.0	0.502642	0.251321	+	0.967904i	$0.419135\pi$
$810$	0	0
$811$	18014.2	0.779981	0.389991	−	0.920819i	$-0.372478\pi$
$811$	18014.2	0.779981	0.389991	+	0.920819i	$0.372478\pi$
$812$	0	0
$813$	0	0
$814$	−16746.9	−0.721105
$815$	−21007.9	−0.902915
$816$	0	0
$817$	−25138.3	−1.07647
$818$	−14074.8	−0.601605
$819$	0	0
$820$	6523.45	0.277816
$821$	16792.4	0.713837	0.356918	−	0.934136i	$-0.383827\pi$
$821$	16792.4	0.713837	0.356918	+	0.934136i	$0.383827\pi$
$822$	0	0
$823$	−8819.03	−0.373526	−0.186763	−	0.982405i	$-0.559800\pi$
$823$	−8819.03	−0.373526	−0.186763	+	0.982405i	$0.559800\pi$
$824$	12892.6	0.545068
$825$	0	0
$826$	0	0
$827$	−8250.13	−0.346898	−0.173449	−	0.984843i	$-0.555491\pi$
$827$	−8250.13	−0.346898	−0.173449	+	0.984843i	$0.555491\pi$
$828$	0	0
$829$	−21550.5	−0.902871	−0.451435	−	0.892304i	$-0.649088\pi$
$829$	−21550.5	−0.902871	−0.451435	+	0.892304i	$0.649088\pi$
$830$	11020.4	0.460870
$831$	0	0
$832$	2485.58	0.103572
$833$	0	0
$834$	0	0
$835$	−22436.4	−0.929873
$836$	−15864.5	−0.656323
$837$	0	0
$838$	−3099.32	−0.127762
$839$	−30130.4	−1.23983	−0.619916	−	0.784669i	$-0.712833\pi$
$839$	−30130.4	−1.23983	−0.619916	+	0.784669i	$0.712833\pi$
$840$	0	0
$841$	33311.0	1.36582
$842$	−11063.3	−0.452809
$843$	0	0
$844$	3650.45	0.148879
$845$	−10906.7	−0.444025
$846$	0	0
$847$	0	0
$848$	1063.82	0.0430798
$849$	0	0
$850$	6881.67	0.277693
$851$	23801.8	0.958772
$852$	0	0
$853$	−40738.6	−1.63525	−0.817623	−	0.575754i	$-0.804709\pi$
$853$	−40738.6	−1.63525	−0.817623	+	0.575754i	$0.804709\pi$
$854$	0	0
$855$	0	0
$856$	7477.36	0.298564
$857$	36508.2	1.45519	0.727594	−	0.686008i	$-0.240638\pi$
$857$	36508.2	1.45519	0.727594	+	0.686008i	$0.240638\pi$
$858$	0	0
$859$	−21980.8	−0.873081	−0.436541	−	0.899685i	$-0.643796\pi$
$859$	−21980.8	−0.873081	−0.436541	+	0.899685i	$0.643796\pi$
$860$	−20813.6	−0.825276
$861$	0	0
$862$	−4059.86	−0.160417
$863$	23426.1	0.924024	0.462012	−	0.886874i	$-0.347128\pi$
$863$	23426.1	0.924024	0.462012	+	0.886874i	$0.347128\pi$
$864$	0	0
$865$	16416.7	0.645301
$866$	655.477	0.0257206
$867$	0	0
$868$	0	0
$869$	21237.5	0.829037
$870$	0	0
$871$	21174.3	0.823725
$872$	9576.15	0.371892
$873$	0	0
$874$	22547.7	0.872639
$875$	0	0
$876$	0	0
$877$	307.401	0.0118360	0.00591802	−	0.999982i	$-0.498116\pi$
$877$	307.401	0.0118360	0.00591802	+	0.999982i	$0.498116\pi$
$878$	−15817.4	−0.607984
$879$	0	0
$880$	−13135.2	−0.503168
$881$	19941.7	0.762605	0.381302	−	0.924450i	$-0.375476\pi$
$881$	19941.7	0.762605	0.381302	+	0.924450i	$0.375476\pi$
$882$	0	0
$883$	−37524.1	−1.43011	−0.715056	−	0.699068i	$-0.753599\pi$
$883$	−37524.1	−1.43011	−0.715056	+	0.699068i	$0.753599\pi$
$884$	−4248.55	−0.161645
$885$	0	0
$886$	−5841.64	−0.221505
$887$	2880.20	0.109028	0.0545140	−	0.998513i	$-0.482639\pi$
$887$	2880.20	0.109028	0.0545140	+	0.998513i	$0.482639\pi$
$888$	0	0
$889$	0	0
$890$	36652.1	1.38043
$891$	0	0
$892$	−17277.9	−0.648549
$893$	−5199.30	−0.194835
$894$	0	0
$895$	12153.9	0.453922
$896$	0	0
$897$	0	0
$898$	−20480.4	−0.761068
$899$	71263.6	2.64380
$900$	0	0
$901$	−1818.36	−0.0672347
$902$	10676.1	0.394096
$903$	0	0
$904$	−19077.2	−0.701877
$905$	−62678.7	−2.30222
$906$	0	0
$907$	18319.2	0.670651	0.335326	−	0.942102i	$-0.391154\pi$
$907$	18319.2	0.670651	0.335326	+	0.942102i	$0.391154\pi$
$908$	−8245.14	−0.301349
$909$	0	0
$910$	0	0
$911$	−46150.7	−1.67842	−0.839210	−	0.543807i	$-0.816982\pi$
$911$	−46150.7	−1.67842	−0.839210	+	0.543807i	$0.816982\pi$
$912$	0	0
$913$	18035.6	0.653769
$914$	−11785.3	−0.426501
$915$	0	0
$916$	−13898.5	−0.501332
$917$	0	0
$918$	0	0
$919$	47264.3	1.69652	0.848261	−	0.529578i	$-0.177650\pi$
$919$	47264.3	1.69652	0.848261	+	0.529578i	$0.177650\pi$
$920$	18668.6	0.669006
$921$	0	0
$922$	−25286.7	−0.903226
$923$	5077.66	0.181076
$924$	0	0
$925$	−20323.3	−0.722407
$926$	−30186.4	−1.07126
$927$	0	0
$928$	7686.67	0.271904
$929$	53271.5	1.88136	0.940678	−	0.339301i	$-0.110190\pi$
$929$	53271.5	1.88136	0.940678	+	0.339301i	$0.110190\pi$
$930$	0	0
$931$	0	0
$932$	−3104.39	−0.109107
$933$	0	0
$934$	−5640.47	−0.197604
$935$	22451.8	0.785295
$936$	0	0
$937$	17197.8	0.599602	0.299801	−	0.954002i	$-0.403080\pi$
$937$	17197.8	0.599602	0.299801	+	0.954002i	$0.403080\pi$
$938$	0	0
$939$	0	0
$940$	−4304.82	−0.149370
$941$	−28834.9	−0.998926	−0.499463	−	0.866335i	$-0.666469\pi$
$941$	−28834.9	−0.998926	−0.499463	+	0.866335i	$0.666469\pi$
$942$	0	0
$943$	−15173.5	−0.523986
$944$	7390.85	0.254822
$945$	0	0
$946$	−34062.9	−1.17070
$947$	51841.2	1.77890	0.889448	−	0.457037i	$-0.151089\pi$
$947$	51841.2	1.77890	0.889448	+	0.457037i	$0.151089\pi$
$948$	0	0
$949$	7041.14	0.240848
$950$	−19252.5	−0.657508
$951$	0	0
$952$	0	0
$953$	5887.31	0.200114	0.100057	−	0.994982i	$-0.468097\pi$
$953$	5887.31	0.200114	0.100057	+	0.994982i	$0.468097\pi$
$954$	0	0
$955$	28590.7	0.968767
$956$	−8027.18	−0.271567
$957$	0	0
$958$	−32897.0	−1.10945
$959$	0	0
$960$	0	0
$961$	58224.6	1.95444
$962$	12547.0	0.420512
$963$	0	0
$964$	3222.59	0.107669
$965$	53384.4	1.78083
$966$	0	0
$967$	−36620.0	−1.21781	−0.608904	−	0.793244i	$-0.708391\pi$
$967$	−36620.0	−1.21781	−0.608904	+	0.793244i	$0.708391\pi$
$968$	−10848.7	−0.360217
$969$	0	0
$970$	−51258.4	−1.69671
$971$	42364.9	1.40016	0.700079	−	0.714065i	$-0.253148\pi$
$971$	42364.9	1.40016	0.700079	+	0.714065i	$0.253148\pi$
$972$	0	0
$973$	0	0
$974$	−12662.1	−0.416551
$975$	0	0
$976$	2965.58	0.0972600
$977$	−5022.32	−0.164461	−0.0822304	−	0.996613i	$-0.526204\pi$
$977$	−5022.32	−0.164461	−0.0822304	+	0.996613i	$0.526204\pi$
$978$	0	0
$979$	59983.8	1.95821
$980$	0	0
$981$	0	0
$982$	−18573.8	−0.603578
$983$	−28292.8	−0.918005	−0.459003	−	0.888435i	$-0.651793\pi$
$983$	−28292.8	−0.918005	−0.459003	+	0.888435i	$0.651793\pi$
$984$	0	0
$985$	73045.7	2.36287
$986$	−13138.7	−0.424361
$987$	0	0
$988$	11885.9	0.382734
$989$	48412.3	1.55655
$990$	0	0
$991$	36401.1	1.16682	0.583410	−	0.812178i	$-0.301718\pi$
$991$	36401.1	1.16682	0.583410	+	0.812178i	$0.301718\pi$
$992$	9493.58	0.303852
$993$	0	0
$994$	0	0
$995$	−35314.6	−1.12517
$996$	0	0
$997$	−2357.19	−0.0748777	−0.0374389	−	0.999299i	$-0.511920\pi$
$997$	−2357.19	−0.0748777	−0.0374389	+	0.999299i	$0.511920\pi$
$998$	486.902	0.0154435
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.v.1.2		2
3.2	odd	2		294.4.a.n.1.1		2
7.2	even	3		126.4.g.g.109.1		4
7.3	odd	6		882.4.g.bf.667.2		4
7.4	even	3		126.4.g.g.37.1		4
7.5	odd	6		882.4.g.bf.361.2		4
7.6	odd	2		882.4.a.z.1.1		2
12.11	even	2		2352.4.a.bq.1.1		2
21.2	odd	6		42.4.e.c.25.2	✓	4
21.5	even	6		294.4.e.l.67.1		4
21.11	odd	6		42.4.e.c.37.2	yes	4
21.17	even	6		294.4.e.l.79.1		4
21.20	even	2		294.4.a.m.1.2		2
84.11	even	6		336.4.q.j.289.2		4
84.23	even	6		336.4.q.j.193.2		4
84.83	odd	2		2352.4.a.ca.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.c.25.2	✓	4	21.2	odd	6
42.4.e.c.37.2	yes	4	21.11	odd	6
126.4.g.g.37.1		4	7.4	even	3
126.4.g.g.109.1		4	7.2	even	3
294.4.a.m.1.2		2	21.20	even	2
294.4.a.n.1.1		2	3.2	odd	2
294.4.e.l.67.1		4	21.5	even	6
294.4.e.l.79.1		4	21.17	even	6
336.4.q.j.193.2		4	84.23	even	6
336.4.q.j.289.2		4	84.11	even	6
882.4.a.v.1.2		2	1.1	even	1	trivial
882.4.a.z.1.1		2	7.6	odd	2
882.4.g.bf.361.2		4	7.5	odd	6
882.4.g.bf.667.2		4	7.3	odd	6
2352.4.a.bq.1.1		2	12.11	even	2
2352.4.a.ca.1.2		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +15.8371 q^{5} -8.00000 q^{8} -31.6742 q^{10} -51.8371 q^{11} +38.8371 q^{13} +16.0000 q^{16} -27.3485 q^{17} +76.5114 q^{19} +63.3485 q^{20} +103.674 q^{22} -147.348 q^{23} +125.814 q^{25} -77.6742 q^{26} -240.208 q^{29} -296.674 q^{31} -32.0000 q^{32} +54.6970 q^{34} -161.534 q^{37} -153.023 q^{38} -126.697 q^{40} +102.977 q^{41} -328.557 q^{43} -207.348 q^{44} +294.697 q^{46} -67.9546 q^{47} -251.629 q^{50} +155.348 q^{52} +66.4886 q^{53} -820.951 q^{55} +480.417 q^{58} +461.928 q^{59} +185.348 q^{61} +593.348 q^{62} +64.0000 q^{64} +615.068 q^{65} +545.208 q^{67} -109.394 q^{68} +130.742 q^{71} +181.299 q^{73} +323.068 q^{74} +306.045 q^{76} -409.697 q^{79} +253.394 q^{80} -205.955 q^{82} -347.928 q^{83} -433.121 q^{85} +657.114 q^{86} +414.697 q^{88} -1157.16 q^{89} -589.394 q^{92} +135.909 q^{94} +1211.72 q^{95} +1618.30 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 5 q^{5} - 16 q^{8} + 10 q^{10} - 67 q^{11} + 41 q^{13} + 32 q^{16} + 92 q^{17} + 43 q^{19} - 20 q^{20} + 134 q^{22} - 148 q^{23} + 435 q^{25} - 82 q^{26} - 77 q^{29} - 520 q^{31}+ \cdots + 1953 q^{97}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 5 * q^5 - 16 * q^8 + 10 * q^10 - 67 * q^11 + 41 * q^13 + 32 * q^16 + 92 * q^17 + 43 * q^19 - 20 * q^20 + 134 * q^22 - 148 * q^23 + 435 * q^25 - 82 * q^26 - 77 * q^29 - 520 * q^31 - 64 * q^32 - 184 * q^34 + 7 * q^37 - 86 * q^38 + 40 * q^40 + 426 * q^41 - 107 * q^43 - 268 * q^44 + 296 * q^46 - 576 * q^47 - 870 * q^50 + 164 * q^52 + 243 * q^53 - 505 * q^55 + 154 * q^58 + 7 * q^59 + 224 * q^61 + 1040 * q^62 + 128 * q^64 + 570 * q^65 + 687 * q^67 + 368 * q^68 - 472 * q^71 - 921 * q^73 - 14 * q^74 + 172 * q^76 - 526 * q^79 - 80 * q^80 - 852 * q^82 + 221 * q^83 - 2920 * q^85 + 214 * q^86 + 536 * q^88 - 774 * q^89 - 592 * q^92 + 1152 * q^94 + 1910 * q^95 + 1953 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more