LMFDB - Embedded newform 88.6.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,6,Mod(1,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("88.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	88.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:	$14.1137761435$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1784453.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 368x - 2705 $ x^3 - 368*x - 2705
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.4642$ of defining polynomial
Character	$\chi$	$=$	88.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.3952 q^{3} -2.46146 q^{5} +36.9338 q^{7} +401.918 q^{9} +121.000 q^{11} +816.111 q^{13} -62.5093 q^{15} +382.011 q^{17} -1397.89 q^{19} +937.941 q^{21} +377.087 q^{23} -3118.94 q^{25} +4035.75 q^{27} +4821.96 q^{29} +6199.66 q^{31} +3072.82 q^{33} -90.9110 q^{35} -3441.50 q^{37} +20725.3 q^{39} -5808.45 q^{41} +8703.19 q^{43} -989.304 q^{45} +2281.27 q^{47} -15442.9 q^{49} +9701.25 q^{51} -28337.3 q^{53} -297.837 q^{55} -35499.7 q^{57} -35539.9 q^{59} -48370.5 q^{61} +14844.3 q^{63} -2008.82 q^{65} -48310.7 q^{67} +9576.20 q^{69} +66892.9 q^{71} +80865.0 q^{73} -79206.2 q^{75} +4468.99 q^{77} -58924.1 q^{79} +4822.76 q^{81} -15396.9 q^{83} -940.304 q^{85} +122455. q^{87} -46488.8 q^{89} +30142.1 q^{91} +157442. q^{93} +3440.84 q^{95} -158298. q^{97} +48632.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 14 q^{3} + 56 q^{5} + 112 q^{7} + 145 q^{9} + 363 q^{11} + 450 q^{13} + 994 q^{15} + 1274 q^{17} + 2416 q^{19} + 2064 q^{21} + 4042 q^{23} + 4103 q^{25} + 6398 q^{27} + 2086 q^{29} + 10034 q^{31}+ \cdots + 17545 q^{99}+O(q^{100}) $ 3 * q + 14 * q^3 + 56 * q^5 + 112 * q^7 + 145 * q^9 + 363 * q^11 + 450 * q^13 + 994 * q^15 + 1274 * q^17 + 2416 * q^19 + 2064 * q^21 + 4042 * q^23 + 4103 * q^25 + 6398 * q^27 + 2086 * q^29 + 10034 * q^31 + 1694 * q^33 + 15256 * q^35 - 2916 * q^37 + 17188 * q^39 - 9450 * q^41 + 17296 * q^43 - 24334 * q^45 + 22312 * q^47 - 31533 * q^49 + 8952 * q^51 - 50990 * q^53 + 6776 * q^55 - 51324 * q^57 - 16654 * q^59 - 50674 * q^61 - 12504 * q^63 - 60312 * q^65 - 59966 * q^67 - 34558 * q^69 - 16954 * q^71 - 50850 * q^73 - 39116 * q^75 + 13552 * q^77 + 7580 * q^79 + 3475 * q^81 + 1664 * q^83 + 61812 * q^85 + 83536 * q^87 - 41920 * q^89 - 36824 * q^91 + 99042 * q^93 + 164916 * q^95 - 93356 * q^97 + 17545 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	25.3952	1.62910	0.814552	−	0.580090i	$-0.196983\pi$
$3$	25.3952	1.62910	0.814552	+	0.580090i	$0.196983\pi$
$4$	0	0
$5$	−2.46146	−0.0440319	−0.0220160	−	0.999758i	$-0.507008\pi$
$5$	−2.46146	−0.0440319	−0.0220160	+	0.999758i	$0.507008\pi$
$6$	0	0
$7$	36.9338	0.284891	0.142445	−	0.989803i	$-0.454503\pi$
$7$	36.9338	0.284891	0.142445	+	0.989803i	$0.454503\pi$
$8$	0	0
$9$	401.918	1.65398
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	816.111	1.33934	0.669670	−	0.742659i	$-0.266436\pi$
$13$	816.111	1.33934	0.669670	+	0.742659i	$0.266436\pi$
$14$	0	0
$15$	−62.5093	−0.0717326
$16$	0	0
$17$	382.011	0.320593	0.160296	−	0.987069i	$-0.448755\pi$
$17$	382.011	0.320593	0.160296	+	0.987069i	$0.448755\pi$
$18$	0	0
$19$	−1397.89	−0.888358	−0.444179	−	0.895938i	$-0.646505\pi$
$19$	−1397.89	−0.888358	−0.444179	+	0.895938i	$0.646505\pi$
$20$	0	0
$21$	937.941	0.464117
$22$	0	0
$23$	377.087	0.148635	0.0743176	−	0.997235i	$-0.476322\pi$
$23$	377.087	0.148635	0.0743176	+	0.997235i	$0.476322\pi$
$24$	0	0
$25$	−3118.94	−0.998061
$26$	0	0
$27$	4035.75	1.06540
$28$	0	0
$29$	4821.96	1.06470	0.532352	−	0.846523i	$-0.321308\pi$
$29$	4821.96	1.06470	0.532352	+	0.846523i	$0.321308\pi$
$30$	0	0
$31$	6199.66	1.15868	0.579340	−	0.815086i	$-0.303310\pi$
$31$	6199.66	1.15868	0.579340	+	0.815086i	$0.303310\pi$
$32$	0	0
$33$	3072.82	0.491194
$34$	0	0
$35$	−90.9110	−0.0125443
$36$	0	0
$37$	−3441.50	−0.413279	−0.206639	−	0.978417i	$-0.566253\pi$
$37$	−3441.50	−0.413279	−0.206639	+	0.978417i	$0.566253\pi$
$38$	0	0
$39$	20725.3	2.18192
$40$	0	0
$41$	−5808.45	−0.539636	−0.269818	−	0.962911i	$-0.586963\pi$
$41$	−5808.45	−0.539636	−0.269818	+	0.962911i	$0.586963\pi$
$42$	0	0
$43$	8703.19	0.717807	0.358903	−	0.933375i	$-0.383151\pi$
$43$	8703.19	0.717807	0.358903	+	0.933375i	$0.383151\pi$
$44$	0	0
$45$	−989.304	−0.0728280
$46$	0	0
$47$	2281.27	0.150637	0.0753186	−	0.997160i	$-0.476003\pi$
$47$	2281.27	0.150637	0.0753186	+	0.997160i	$0.476003\pi$
$48$	0	0
$49$	−15442.9	−0.918837
$50$	0	0
$51$	9701.25	0.522279
$52$	0	0
$53$	−28337.3	−1.38570	−0.692850	−	0.721081i	$-0.743645\pi$
$53$	−28337.3	−1.38570	−0.692850	+	0.721081i	$0.743645\pi$
$54$	0	0
$55$	−297.837	−0.0132761
$56$	0	0
$57$	−35499.7	−1.44723
$58$	0	0
$59$	−35539.9	−1.32919	−0.664593	−	0.747206i	$-0.731395\pi$
$59$	−35539.9	−1.32919	−0.664593	+	0.747206i	$0.731395\pi$
$60$	0	0
$61$	−48370.5	−1.66439	−0.832196	−	0.554481i	$-0.812917\pi$
$61$	−48370.5	−1.66439	−0.832196	+	0.554481i	$0.812917\pi$
$62$	0	0
$63$	14844.3	0.471204
$64$	0	0
$65$	−2008.82	−0.0589737
$66$	0	0
$67$	−48310.7	−1.31479	−0.657395	−	0.753546i	$-0.728342\pi$
$67$	−48310.7	−1.31479	−0.657395	+	0.753546i	$0.728342\pi$
$68$	0	0
$69$	9576.20	0.242142
$70$	0	0
$71$	66892.9	1.57483	0.787416	−	0.616423i	$-0.211419\pi$
$71$	66892.9	1.57483	0.787416	+	0.616423i	$0.211419\pi$
$72$	0	0
$73$	80865.0	1.77604	0.888022	−	0.459802i	$-0.152080\pi$
$73$	80865.0	1.77604	0.888022	+	0.459802i	$0.152080\pi$
$74$	0	0
$75$	−79206.2	−1.62595
$76$	0	0
$77$	4468.99	0.0858978
$78$	0	0
$79$	−58924.1	−1.06225	−0.531123	−	0.847295i	$-0.678230\pi$
$79$	−58924.1	−1.06225	−0.531123	+	0.847295i	$0.678230\pi$
$80$	0	0
$81$	4822.76	0.0816738
$82$	0	0
$83$	−15396.9	−0.245323	−0.122662	−	0.992449i	$-0.539143\pi$
$83$	−15396.9	−0.245323	−0.122662	+	0.992449i	$0.539143\pi$
$84$	0	0
$85$	−940.304	−0.0141163
$86$	0	0
$87$	122455.	1.73451
$88$	0	0
$89$	−46488.8	−0.622118	−0.311059	−	0.950391i	$-0.600684\pi$
$89$	−46488.8	−0.622118	−0.311059	+	0.950391i	$0.600684\pi$
$90$	0	0
$91$	30142.1	0.381566
$92$	0	0
$93$	157442.	1.88761
$94$	0	0
$95$	3440.84	0.0391161
$96$	0	0
$97$	−158298.	−1.70823	−0.854113	−	0.520088i	$-0.825899\pi$
$97$	−158298.	−1.70823	−0.854113	+	0.520088i	$0.825899\pi$
$98$	0	0
$99$	48632.0	0.498694
$100$	0	0
$101$	42922.8	0.418683	0.209341	−	0.977843i	$-0.432868\pi$
$101$	42922.8	0.418683	0.209341	+	0.977843i	$0.432868\pi$
$102$	0	0
$103$	26872.7	0.249585	0.124792	−	0.992183i	$-0.460174\pi$
$103$	26872.7	0.249585	0.124792	+	0.992183i	$0.460174\pi$
$104$	0	0
$105$	−2308.70	−0.0204360
$106$	0	0
$107$	−52819.1	−0.445996	−0.222998	−	0.974819i	$-0.571584\pi$
$107$	−52819.1	−0.445996	−0.222998	+	0.974819i	$0.571584\pi$
$108$	0	0
$109$	−22677.6	−0.182823	−0.0914115	−	0.995813i	$-0.529138\pi$
$109$	−22677.6	−0.182823	−0.0914115	+	0.995813i	$0.529138\pi$
$110$	0	0
$111$	−87397.6	−0.673274
$112$	0	0
$113$	−111227.	−0.819437	−0.409719	−	0.912212i	$-0.634373\pi$
$113$	−111227.	−0.819437	−0.409719	+	0.912212i	$0.634373\pi$
$114$	0	0
$115$	−928.183	−0.00654469
$116$	0	0
$117$	328009.	2.21524
$118$	0	0
$119$	14109.1	0.0913339
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−147507.	−0.879123
$124$	0	0
$125$	15369.2	0.0879785
$126$	0	0
$127$	58642.3	0.322628	0.161314	−	0.986903i	$-0.448427\pi$
$127$	58642.3	0.322628	0.161314	+	0.986903i	$0.448427\pi$
$128$	0	0
$129$	221020.	1.16938
$130$	0	0
$131$	330826.	1.68431	0.842154	−	0.539237i	$-0.181287\pi$
$131$	330826.	1.68431	0.842154	+	0.539237i	$0.181287\pi$
$132$	0	0
$133$	−51629.3	−0.253085
$134$	0	0
$135$	−9933.83	−0.0469118
$136$	0	0
$137$	7944.43	0.0361627	0.0180814	−	0.999837i	$-0.494244\pi$
$137$	7944.43	0.0361627	0.0180814	+	0.999837i	$0.494244\pi$
$138$	0	0
$139$	322874.	1.41741	0.708707	−	0.705503i	$-0.249279\pi$
$139$	322874.	1.41741	0.708707	+	0.705503i	$0.249279\pi$
$140$	0	0
$141$	57933.4	0.245404
$142$	0	0
$143$	98749.4	0.403826
$144$	0	0
$145$	−11869.1	−0.0468810
$146$	0	0
$147$	−392176.	−1.49688
$148$	0	0
$149$	392580.	1.44865	0.724323	−	0.689461i	$-0.242153\pi$
$149$	392580.	1.44865	0.724323	+	0.689461i	$0.242153\pi$
$150$	0	0
$151$	−367134.	−1.31033	−0.655167	−	0.755484i	$-0.727402\pi$
$151$	−367134.	−1.31033	−0.655167	+	0.755484i	$0.727402\pi$
$152$	0	0
$153$	153537.	0.530254
$154$	0	0
$155$	−15260.2	−0.0510189
$156$	0	0
$157$	−442244.	−1.43190	−0.715950	−	0.698151i	$-0.754006\pi$
$157$	−442244.	−1.43190	−0.715950	+	0.698151i	$0.754006\pi$
$158$	0	0
$159$	−719633.	−2.25745
$160$	0	0
$161$	13927.2	0.0423448
$162$	0	0
$163$	5132.92	0.0151320	0.00756598	−	0.999971i	$-0.497592\pi$
$163$	5132.92	0.0151320	0.00756598	+	0.999971i	$0.497592\pi$
$164$	0	0
$165$	−7563.63	−0.0216282
$166$	0	0
$167$	−295614.	−0.820226	−0.410113	−	0.912035i	$-0.634511\pi$
$167$	−295614.	−0.820226	−0.410113	+	0.912035i	$0.634511\pi$
$168$	0	0
$169$	294744.	0.793831
$170$	0	0
$171$	−561836.	−1.46933
$172$	0	0
$173$	492410.	1.25087	0.625435	−	0.780276i	$-0.284922\pi$
$173$	492410.	1.25087	0.625435	+	0.780276i	$0.284922\pi$
$174$	0	0
$175$	−115194.	−0.284339
$176$	0	0
$177$	−902543.	−2.16538
$178$	0	0
$179$	629780.	1.46912	0.734559	−	0.678545i	$-0.237389\pi$
$179$	629780.	1.46912	0.734559	+	0.678545i	$0.237389\pi$
$180$	0	0
$181$	−267374.	−0.606629	−0.303314	−	0.952891i	$-0.598093\pi$
$181$	−267374.	−0.606629	−0.303314	+	0.952891i	$0.598093\pi$
$182$	0	0
$183$	−1.22838e6	−2.71147
$184$	0	0
$185$	8471.10	0.0181974
$186$	0	0
$187$	46223.3	0.0966623
$188$	0	0
$189$	149055.	0.303524
$190$	0	0
$191$	801769.	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$191$	801769.	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$192$	0	0
$193$	−903981.	−1.74689	−0.873446	−	0.486921i	$-0.838120\pi$
$193$	−903981.	−1.74689	−0.873446	+	0.486921i	$0.838120\pi$
$194$	0	0
$195$	−51014.5	−0.0960743
$196$	0	0
$197$	27824.7	0.0510816	0.0255408	−	0.999674i	$-0.491869\pi$
$197$	27824.7	0.0510816	0.0255408	+	0.999674i	$0.491869\pi$
$198$	0	0
$199$	875292.	1.56682	0.783412	−	0.621502i	$-0.213477\pi$
$199$	875292.	1.56682	0.783412	+	0.621502i	$0.213477\pi$
$200$	0	0
$201$	−1.22686e6	−2.14193
$202$	0	0
$203$	178093.	0.303324
$204$	0	0
$205$	14297.3	0.0237612
$206$	0	0
$207$	151558.	0.245840
$208$	0	0
$209$	−169144.	−0.267850
$210$	0	0
$211$	556951.	0.861213	0.430606	−	0.902540i	$-0.358300\pi$
$211$	556951.	0.861213	0.430606	+	0.902540i	$0.358300\pi$
$212$	0	0
$213$	1.69876e6	2.56556
$214$	0	0
$215$	−21422.6	−0.0316064
$216$	0	0
$217$	228977.	0.330098
$218$	0	0
$219$	2.05359e6	2.89336
$220$	0	0
$221$	311763.	0.429382
$222$	0	0
$223$	880406.	1.18555	0.592777	−	0.805367i	$-0.298032\pi$
$223$	880406.	1.18555	0.592777	+	0.805367i	$0.298032\pi$
$224$	0	0
$225$	−1.25356e6	−1.65077
$226$	0	0
$227$	−536468.	−0.691001	−0.345501	−	0.938419i	$-0.612291\pi$
$227$	−536468.	−0.691001	−0.345501	+	0.938419i	$0.612291\pi$
$228$	0	0
$229$	456255.	0.574935	0.287468	−	0.957790i	$-0.407187\pi$
$229$	456255.	0.574935	0.287468	+	0.957790i	$0.407187\pi$
$230$	0	0
$231$	113491.	0.139937
$232$	0	0
$233$	731816.	0.883104	0.441552	−	0.897236i	$-0.354428\pi$
$233$	731816.	0.883104	0.441552	+	0.897236i	$0.354428\pi$
$234$	0	0
$235$	−5615.26	−0.00663285
$236$	0	0
$237$	−1.49639e6	−1.73051
$238$	0	0
$239$	685553.	0.776330	0.388165	−	0.921590i	$-0.373109\pi$
$239$	685553.	0.776330	0.388165	+	0.921590i	$0.373109\pi$
$240$	0	0
$241$	−1.16895e6	−1.29644	−0.648219	−	0.761454i	$-0.724486\pi$
$241$	−1.16895e6	−1.29644	−0.648219	+	0.761454i	$0.724486\pi$
$242$	0	0
$243$	−858212.	−0.932349
$244$	0	0
$245$	38012.1	0.0404582
$246$	0	0
$247$	−1.14083e6	−1.18981
$248$	0	0
$249$	−391009.	−0.399657
$250$	0	0
$251$	1.62769e6	1.63075	0.815375	−	0.578933i	$-0.196531\pi$
$251$	1.62769e6	1.63075	0.815375	+	0.578933i	$0.196531\pi$
$252$	0	0
$253$	45627.5	0.0448152
$254$	0	0
$255$	−23879.2	−0.0229969
$256$	0	0
$257$	866219.	0.818078	0.409039	−	0.912517i	$-0.365864\pi$
$257$	866219.	0.818078	0.409039	+	0.912517i	$0.365864\pi$
$258$	0	0
$259$	−127107.	−0.117739
$260$	0	0
$261$	1.93803e6	1.76100
$262$	0	0
$263$	1.36434e6	1.21628	0.608139	−	0.793831i	$-0.291916\pi$
$263$	1.36434e6	1.21628	0.608139	+	0.793831i	$0.291916\pi$
$264$	0	0
$265$	69751.2	0.0610151
$266$	0	0
$267$	−1.18059e6	−1.01350
$268$	0	0
$269$	−1.62684e6	−1.37077	−0.685386	−	0.728180i	$-0.740366\pi$
$269$	−1.62684e6	−1.37077	−0.685386	+	0.728180i	$0.740366\pi$
$270$	0	0
$271$	−1.32521e6	−1.09613	−0.548066	−	0.836435i	$-0.684636\pi$
$271$	−1.32521e6	−1.09613	−0.548066	+	0.836435i	$0.684636\pi$
$272$	0	0
$273$	765464.	0.621610
$274$	0	0
$275$	−377392.	−0.300927
$276$	0	0
$277$	49591.7	0.0388338	0.0194169	−	0.999811i	$-0.493819\pi$
$277$	49591.7	0.0388338	0.0194169	+	0.999811i	$0.493819\pi$
$278$	0	0
$279$	2.49175e6	1.91644
$280$	0	0
$281$	1.18685e6	0.896667	0.448334	−	0.893866i	$-0.352018\pi$
$281$	1.18685e6	0.896667	0.448334	+	0.893866i	$0.352018\pi$
$282$	0	0
$283$	−307811.	−0.228464	−0.114232	−	0.993454i	$-0.536441\pi$
$283$	−307811.	−0.228464	−0.114232	+	0.993454i	$0.536441\pi$
$284$	0	0
$285$	87381.0	0.0637243
$286$	0	0
$287$	−214528.	−0.153737
$288$	0	0
$289$	−1.27392e6	−0.897220
$290$	0	0
$291$	−4.02001e6	−2.78288
$292$	0	0
$293$	1.91593e6	1.30380	0.651898	−	0.758307i	$-0.273973\pi$
$293$	1.91593e6	1.30380	0.651898	+	0.758307i	$0.273973\pi$
$294$	0	0
$295$	87479.9	0.0585266
$296$	0	0
$297$	488325.	0.321232
$298$	0	0
$299$	307745.	0.199073
$300$	0	0
$301$	321442.	0.204497
$302$	0	0
$303$	1.09004e6	0.682078
$304$	0	0
$305$	119062.	0.0732864
$306$	0	0
$307$	1.59722e6	0.967206	0.483603	−	0.875288i	$-0.339328\pi$
$307$	1.59722e6	0.967206	0.483603	+	0.875288i	$0.339328\pi$
$308$	0	0
$309$	682438.	0.406600
$310$	0	0
$311$	−2.41343e6	−1.41492	−0.707462	−	0.706751i	$-0.750160\pi$
$311$	−2.41343e6	−1.41492	−0.707462	+	0.706751i	$0.750160\pi$
$312$	0	0
$313$	−314389.	−0.181387	−0.0906935	−	0.995879i	$-0.528908\pi$
$313$	−314389.	−0.181387	−0.0906935	+	0.995879i	$0.528908\pi$
$314$	0	0
$315$	−36538.7	−0.0207480
$316$	0	0
$317$	1.75459e6	0.980681	0.490341	−	0.871531i	$-0.336872\pi$
$317$	1.75459e6	0.980681	0.490341	+	0.871531i	$0.336872\pi$
$318$	0	0
$319$	583458.	0.321020
$320$	0	0
$321$	−1.34135e6	−0.726575
$322$	0	0
$323$	−534008.	−0.284801
$324$	0	0
$325$	−2.54540e6	−1.33674
$326$	0	0
$327$	−575903.	−0.297838
$328$	0	0
$329$	84256.0	0.0429152
$330$	0	0
$331$	−2.42629e6	−1.21723	−0.608614	−	0.793466i	$-0.708274\pi$
$331$	−2.42629e6	−1.21723	−0.608614	+	0.793466i	$0.708274\pi$
$332$	0	0
$333$	−1.38320e6	−0.683555
$334$	0	0
$335$	118915.	0.0578928
$336$	0	0
$337$	−781584.	−0.374888	−0.187444	−	0.982275i	$-0.560020\pi$
$337$	−781584.	−0.374888	−0.187444	+	0.982275i	$0.560020\pi$
$338$	0	0
$339$	−2.82464e6	−1.33495
$340$	0	0
$341$	750159.	0.349355
$342$	0	0
$343$	−1.19111e6	−0.546659
$344$	0	0
$345$	−23571.4	−0.0106620
$346$	0	0
$347$	−2.27802e6	−1.01562	−0.507812	−	0.861468i	$-0.669546\pi$
$347$	−2.27802e6	−1.01562	−0.507812	+	0.861468i	$0.669546\pi$
$348$	0	0
$349$	−1.81344e6	−0.796964	−0.398482	−	0.917176i	$-0.630463\pi$
$349$	−1.81344e6	−0.796964	−0.398482	+	0.917176i	$0.630463\pi$
$350$	0	0
$351$	3.29362e6	1.42694
$352$	0	0
$353$	−4.19809e6	−1.79314	−0.896572	−	0.442897i	$-0.853951\pi$
$353$	−4.19809e6	−1.79314	−0.896572	+	0.442897i	$0.853951\pi$
$354$	0	0
$355$	−164654.	−0.0693428
$356$	0	0
$357$	358304.	0.148792
$358$	0	0
$359$	947065.	0.387832	0.193916	−	0.981018i	$-0.437881\pi$
$359$	947065.	0.387832	0.193916	+	0.981018i	$0.437881\pi$
$360$	0	0
$361$	−522009.	−0.210819
$362$	0	0
$363$	371812.	0.148100
$364$	0	0
$365$	−199046.	−0.0782026
$366$	0	0
$367$	815817.	0.316175	0.158088	−	0.987425i	$-0.449467\pi$
$367$	815817.	0.316175	0.158088	+	0.987425i	$0.449467\pi$
$368$	0	0
$369$	−2.33452e6	−0.892547
$370$	0	0
$371$	−1.04660e6	−0.394773
$372$	0	0
$373$	3.21610e6	1.19690	0.598450	−	0.801160i	$-0.295783\pi$
$373$	3.21610e6	1.19690	0.598450	+	0.801160i	$0.295783\pi$
$374$	0	0
$375$	390304.	0.143326
$376$	0	0
$377$	3.93526e6	1.42600
$378$	0	0
$379$	867175.	0.310105	0.155052	−	0.987906i	$-0.450445\pi$
$379$	867175.	0.310105	0.155052	+	0.987906i	$0.450445\pi$
$380$	0	0
$381$	1.48924e6	0.525595
$382$	0	0
$383$	−5.49995e6	−1.91585	−0.957925	−	0.287019i	$-0.907336\pi$
$383$	−5.49995e6	−1.91585	−0.957925	+	0.287019i	$0.907336\pi$
$384$	0	0
$385$	−11000.2	−0.00378225
$386$	0	0
$387$	3.49797e6	1.18724
$388$	0	0
$389$	2.66553e6	0.893118	0.446559	−	0.894754i	$-0.352649\pi$
$389$	2.66553e6	0.893118	0.446559	+	0.894754i	$0.352649\pi$
$390$	0	0
$391$	144051.	0.0476513
$392$	0	0
$393$	8.40140e6	2.74391
$394$	0	0
$395$	145039.	0.0467727
$396$	0	0
$397$	−2.66718e6	−0.849329	−0.424665	−	0.905351i	$-0.639608\pi$
$397$	−2.66718e6	−0.849329	−0.424665	+	0.905351i	$0.639608\pi$
$398$	0	0
$399$	−1.31114e6	−0.412302
$400$	0	0
$401$	4.43714e6	1.37798	0.688988	−	0.724772i	$-0.258055\pi$
$401$	4.43714e6	1.37798	0.688988	+	0.724772i	$0.258055\pi$
$402$	0	0
$403$	5.05961e6	1.55187
$404$	0	0
$405$	−11871.0	−0.00359625
$406$	0	0
$407$	−416421.	−0.124608
$408$	0	0
$409$	−4.42419e6	−1.30775	−0.653877	−	0.756601i	$-0.726858\pi$
$409$	−4.42419e6	−1.30775	−0.653877	+	0.756601i	$0.726858\pi$
$410$	0	0
$411$	201751.	0.0589129
$412$	0	0
$413$	−1.31262e6	−0.378673
$414$	0	0
$415$	37898.9	0.0108021
$416$	0	0
$417$	8.19947e6	2.30911
$418$	0	0
$419$	4.31614e6	1.20105	0.600524	−	0.799607i	$-0.294959\pi$
$419$	4.31614e6	1.20105	0.600524	+	0.799607i	$0.294959\pi$
$420$	0	0
$421$	3.16156e6	0.869353	0.434677	−	0.900587i	$-0.356863\pi$
$421$	3.16156e6	0.869353	0.434677	+	0.900587i	$0.356863\pi$
$422$	0	0
$423$	916883.	0.249151
$424$	0	0
$425$	−1.19147e6	−0.319971
$426$	0	0
$427$	−1.78650e6	−0.474170
$428$	0	0
$429$	2.50776e6	0.657875
$430$	0	0
$431$	−3.95469e6	−1.02546	−0.512731	−	0.858549i	$-0.671366\pi$
$431$	−3.95469e6	−1.02546	−0.512731	+	0.858549i	$0.671366\pi$
$432$	0	0
$433$	−934155.	−0.239441	−0.119721	−	0.992808i	$-0.538200\pi$
$433$	−934155.	−0.239441	−0.119721	+	0.992808i	$0.538200\pi$
$434$	0	0
$435$	−301418.	−0.0763740
$436$	0	0
$437$	−527125.	−0.132041
$438$	0	0
$439$	133074.	0.0329557	0.0164779	−	0.999864i	$-0.494755\pi$
$439$	133074.	0.0329557	0.0164779	+	0.999864i	$0.494755\pi$
$440$	0	0
$441$	−6.20677e6	−1.51974
$442$	0	0
$443$	−918705.	−0.222416	−0.111208	−	0.993797i	$-0.535472\pi$
$443$	−918705.	−0.222416	−0.111208	+	0.993797i	$0.535472\pi$
$444$	0	0
$445$	114430.	0.0273931
$446$	0	0
$447$	9.96965e6	2.36000
$448$	0	0
$449$	2.33313e6	0.546163	0.273081	−	0.961991i	$-0.411957\pi$
$449$	2.33313e6	0.546163	0.273081	+	0.961991i	$0.411957\pi$
$450$	0	0
$451$	−702822.	−0.162706
$452$	0	0
$453$	−9.32345e6	−2.13467
$454$	0	0
$455$	−74193.4	−0.0168011
$456$	0	0
$457$	6.72738e6	1.50680	0.753400	−	0.657562i	$-0.228412\pi$
$457$	6.72738e6	1.50680	0.753400	+	0.657562i	$0.228412\pi$
$458$	0	0
$459$	1.54170e6	0.341561
$460$	0	0
$461$	−3.00613e6	−0.658802	−0.329401	−	0.944190i	$-0.606847\pi$
$461$	−3.00613e6	−0.658802	−0.329401	+	0.944190i	$0.606847\pi$
$462$	0	0
$463$	−2.96527e6	−0.642852	−0.321426	−	0.946935i	$-0.604162\pi$
$463$	−2.96527e6	−0.642852	−0.321426	+	0.946935i	$0.604162\pi$
$464$	0	0
$465$	−387537.	−0.0831152
$466$	0	0
$467$	−2.58822e6	−0.549173	−0.274586	−	0.961562i	$-0.588541\pi$
$467$	−2.58822e6	−0.549173	−0.274586	+	0.961562i	$0.588541\pi$
$468$	0	0
$469$	−1.78430e6	−0.374572
$470$	0	0
$471$	−1.12309e7	−2.33272
$472$	0	0
$473$	1.05309e6	0.216427
$474$	0	0
$475$	4.35993e6	0.886636
$476$	0	0
$477$	−1.13893e7	−2.29192
$478$	0	0
$479$	859744.	0.171210	0.0856052	−	0.996329i	$-0.472718\pi$
$479$	859744.	0.171210	0.0856052	+	0.996329i	$0.472718\pi$
$480$	0	0
$481$	−2.80864e6	−0.553520
$482$	0	0
$483$	353685.	0.0689841
$484$	0	0
$485$	389643.	0.0752165
$486$	0	0
$487$	−1.82420e6	−0.348539	−0.174269	−	0.984698i	$-0.555756\pi$
$487$	−1.82420e6	−0.348539	−0.174269	+	0.984698i	$0.555756\pi$
$488$	0	0
$489$	130352.	0.0246516
$490$	0	0
$491$	5.09714e6	0.954164	0.477082	−	0.878859i	$-0.341694\pi$
$491$	5.09714e6	0.954164	0.477082	+	0.878859i	$0.341694\pi$
$492$	0	0
$493$	1.84204e6	0.341336
$494$	0	0
$495$	−119706.	−0.0219585
$496$	0	0
$497$	2.47061e6	0.448655
$498$	0	0
$499$	1.07238e7	1.92796	0.963980	−	0.265975i	$-0.0856938\pi$
$499$	1.07238e7	1.92796	0.963980	+	0.265975i	$0.0856938\pi$
$500$	0	0
$501$	−7.50718e6	−1.33623
$502$	0	0
$503$	−6.61070e6	−1.16500	−0.582502	−	0.812829i	$-0.697926\pi$
$503$	−6.61070e6	−1.16500	−0.582502	+	0.812829i	$0.697926\pi$
$504$	0	0
$505$	−105653.	−0.0184354
$506$	0	0
$507$	7.48509e6	1.29323
$508$	0	0
$509$	241509.	0.0413179	0.0206589	−	0.999787i	$-0.493424\pi$
$509$	241509.	0.0413179	0.0206589	+	0.999787i	$0.493424\pi$
$510$	0	0
$511$	2.98665e6	0.505978
$512$	0	0
$513$	−5.64152e6	−0.946461
$514$	0	0
$515$	−66146.0	−0.0109897
$516$	0	0
$517$	276034.	0.0454188
$518$	0	0
$519$	1.25049e7	2.03780
$520$	0	0
$521$	−6.03899e6	−0.974698	−0.487349	−	0.873207i	$-0.662036\pi$
$521$	−6.03899e6	−0.974698	−0.487349	+	0.873207i	$0.662036\pi$
$522$	0	0
$523$	1.02194e7	1.63369	0.816847	−	0.576854i	$-0.195720\pi$
$523$	1.02194e7	1.63369	0.816847	+	0.576854i	$0.195720\pi$
$524$	0	0
$525$	−2.92538e6	−0.463217
$526$	0	0
$527$	2.36834e6	0.371464
$528$	0	0
$529$	−6.29415e6	−0.977908
$530$	0	0
$531$	−1.42841e7	−2.19845
$532$	0	0
$533$	−4.74034e6	−0.722756
$534$	0	0
$535$	130012.	0.0196381
$536$	0	0
$537$	1.59934e7	2.39335
$538$	0	0
$539$	−1.86859e6	−0.277040
$540$	0	0
$541$	−2.77400e6	−0.407487	−0.203744	−	0.979024i	$-0.565311\pi$
$541$	−2.77400e6	−0.407487	−0.203744	+	0.979024i	$0.565311\pi$
$542$	0	0
$543$	−6.79003e6	−0.988261
$544$	0	0
$545$	55820.0	0.00805005
$546$	0	0
$547$	597638.	0.0854024	0.0427012	−	0.999088i	$-0.486404\pi$
$547$	597638.	0.0854024	0.0427012	+	0.999088i	$0.486404\pi$
$548$	0	0
$549$	−1.94409e7	−2.75287
$550$	0	0
$551$	−6.74056e6	−0.945839
$552$	0	0
$553$	−2.17629e6	−0.302624
$554$	0	0
$555$	215126.	0.0296455
$556$	0	0
$557$	2.44709e6	0.334204	0.167102	−	0.985940i	$-0.446559\pi$
$557$	2.44709e6	0.334204	0.167102	+	0.985940i	$0.446559\pi$
$558$	0	0
$559$	7.10277e6	0.961387
$560$	0	0
$561$	1.17385e6	0.157473
$562$	0	0
$563$	−6.74672e6	−0.897061	−0.448530	−	0.893768i	$-0.648052\pi$
$563$	−6.74672e6	−0.897061	−0.448530	+	0.893768i	$0.648052\pi$
$564$	0	0
$565$	273782.	0.0360814
$566$	0	0
$567$	178123.	0.0232681
$568$	0	0
$569$	−1.93041e6	−0.249959	−0.124980	−	0.992159i	$-0.539887\pi$
$569$	−1.93041e6	−0.249959	−0.124980	+	0.992159i	$0.539887\pi$
$570$	0	0
$571$	−1.05760e6	−0.135748	−0.0678739	−	0.997694i	$-0.521622\pi$
$571$	−1.05760e6	−0.135748	−0.0678739	+	0.997694i	$0.521622\pi$
$572$	0	0
$573$	2.03611e7	2.59068
$574$	0	0
$575$	−1.17611e6	−0.148347
$576$	0	0
$577$	1.29491e7	1.61920	0.809602	−	0.586979i	$-0.199683\pi$
$577$	1.29491e7	1.61920	0.809602	+	0.586979i	$0.199683\pi$
$578$	0	0
$579$	−2.29568e7	−2.84587
$580$	0	0
$581$	−568667.	−0.0698904
$582$	0	0
$583$	−3.42882e6	−0.417804
$584$	0	0
$585$	−807381.	−0.0975414
$586$	0	0
$587$	7.35799e6	0.881382	0.440691	−	0.897659i	$-0.354733\pi$
$587$	7.35799e6	0.881382	0.440691	+	0.897659i	$0.354733\pi$
$588$	0	0
$589$	−8.66643e6	−1.02932
$590$	0	0
$591$	706614.	0.0832173
$592$	0	0
$593$	−230604.	−0.0269296	−0.0134648	−	0.999909i	$-0.504286\pi$
$593$	−230604.	−0.0269296	−0.0134648	+	0.999909i	$0.504286\pi$
$594$	0	0
$595$	−34729.0	−0.00402161
$596$	0	0
$597$	2.22282e7	2.55252
$598$	0	0
$599$	−1.22697e7	−1.39723	−0.698614	−	0.715499i	$-0.746200\pi$
$599$	−1.22697e7	−1.39723	−0.698614	+	0.715499i	$0.746200\pi$
$600$	0	0
$601$	−5.66439e6	−0.639686	−0.319843	−	0.947471i	$-0.603630\pi$
$601$	−5.66439e6	−0.639686	−0.319843	+	0.947471i	$0.603630\pi$
$602$	0	0
$603$	−1.94169e7	−2.17464
$604$	0	0
$605$	−36038.2	−0.00400290
$606$	0	0
$607$	1.43205e7	1.57756	0.788780	−	0.614676i	$-0.210713\pi$
$607$	1.43205e7	1.57756	0.788780	+	0.614676i	$0.210713\pi$
$608$	0	0
$609$	4.52272e6	0.494147
$610$	0	0
$611$	1.86177e6	0.201754
$612$	0	0
$613$	−723067.	−0.0777191	−0.0388595	−	0.999245i	$-0.512372\pi$
$613$	−723067.	−0.0777191	−0.0388595	+	0.999245i	$0.512372\pi$
$614$	0	0
$615$	363082.	0.0387095
$616$	0	0
$617$	−900971.	−0.0952791	−0.0476396	−	0.998865i	$-0.515170\pi$
$617$	−900971.	−0.0952791	−0.0476396	+	0.998865i	$0.515170\pi$
$618$	0	0
$619$	1.04107e7	1.09207	0.546037	−	0.837761i	$-0.316136\pi$
$619$	1.04107e7	1.09207	0.546037	+	0.837761i	$0.316136\pi$
$620$	0	0
$621$	1.52183e6	0.158357
$622$	0	0
$623$	−1.71700e6	−0.177236
$624$	0	0
$625$	9.70886e6	0.994187
$626$	0	0
$627$	−4.29546e6	−0.436356
$628$	0	0
$629$	−1.31469e6	−0.132494
$630$	0	0
$631$	−2.63321e6	−0.263277	−0.131638	−	0.991298i	$-0.542024\pi$
$631$	−2.63321e6	−0.263277	−0.131638	+	0.991298i	$0.542024\pi$
$632$	0	0
$633$	1.41439e7	1.40301
$634$	0	0
$635$	−144346.	−0.0142059
$636$	0	0
$637$	−1.26031e7	−1.23064
$638$	0	0
$639$	2.68854e7	2.60474
$640$	0	0
$641$	1.15325e7	1.10861	0.554304	−	0.832314i	$-0.312985\pi$
$641$	1.15325e7	1.10861	0.554304	+	0.832314i	$0.312985\pi$
$642$	0	0
$643$	3.88676e6	0.370732	0.185366	−	0.982670i	$-0.440653\pi$
$643$	3.88676e6	0.370732	0.185366	+	0.982670i	$0.440653\pi$
$644$	0	0
$645$	−544031.	−0.0514901
$646$	0	0
$647$	9.17393e6	0.861579	0.430789	−	0.902453i	$-0.358235\pi$
$647$	9.17393e6	0.861579	0.430789	+	0.902453i	$0.358235\pi$
$648$	0	0
$649$	−4.30032e6	−0.400765
$650$	0	0
$651$	5.81492e6	0.537763
$652$	0	0
$653$	9.62692e6	0.883496	0.441748	−	0.897139i	$-0.354359\pi$
$653$	9.62692e6	0.883496	0.441748	+	0.897139i	$0.354359\pi$
$654$	0	0
$655$	−814315.	−0.0741633
$656$	0	0
$657$	3.25011e7	2.93754
$658$	0	0
$659$	−1.20253e7	−1.07865	−0.539327	−	0.842096i	$-0.681321\pi$
$659$	−1.20253e7	−1.07865	−0.539327	+	0.842096i	$0.681321\pi$
$660$	0	0
$661$	5.52924e6	0.492223	0.246111	−	0.969242i	$-0.420847\pi$
$661$	5.52924e6	0.492223	0.246111	+	0.969242i	$0.420847\pi$
$662$	0	0
$663$	7.91730e6	0.699509
$664$	0	0
$665$	127083.	0.0111438
$666$	0	0
$667$	1.81830e6	0.158252
$668$	0	0
$669$	2.23581e7	1.93139
$670$	0	0
$671$	−5.85283e6	−0.501833
$672$	0	0
$673$	4.66768e6	0.397249	0.198625	−	0.980076i	$-0.436353\pi$
$673$	4.66768e6	0.397249	0.198625	+	0.980076i	$0.436353\pi$
$674$	0	0
$675$	−1.25873e7	−1.06334
$676$	0	0
$677$	−1.48229e7	−1.24297	−0.621487	−	0.783425i	$-0.713471\pi$
$677$	−1.48229e7	−1.24297	−0.621487	+	0.783425i	$0.713471\pi$
$678$	0	0
$679$	−5.84653e6	−0.486658
$680$	0	0
$681$	−1.36237e7	−1.12571
$682$	0	0
$683$	−1.69744e6	−0.139234	−0.0696168	−	0.997574i	$-0.522178\pi$
$683$	−1.69744e6	−0.139234	−0.0696168	+	0.997574i	$0.522178\pi$
$684$	0	0
$685$	−19554.9	−0.00159231
$686$	0	0
$687$	1.15867e7	0.936629
$688$	0	0
$689$	−2.31264e7	−1.85592
$690$	0	0
$691$	−8.43993e6	−0.672425	−0.336213	−	0.941786i	$-0.609146\pi$
$691$	−8.43993e6	−0.672425	−0.336213	+	0.941786i	$0.609146\pi$
$692$	0	0
$693$	1.79616e6	0.142073
$694$	0	0
$695$	−794742.	−0.0624114
$696$	0	0
$697$	−2.21889e6	−0.173003
$698$	0	0
$699$	1.85846e7	1.43867
$700$	0	0
$701$	−1.84917e7	−1.42128	−0.710642	−	0.703554i	$-0.751595\pi$
$701$	−1.84917e7	−1.42128	−0.710642	+	0.703554i	$0.751595\pi$
$702$	0	0
$703$	4.81082e6	0.367140
$704$	0	0
$705$	−142601.	−0.0108056
$706$	0	0
$707$	1.58530e6	0.119279
$708$	0	0
$709$	−2.05168e7	−1.53283	−0.766415	−	0.642346i	$-0.777961\pi$
$709$	−2.05168e7	−1.53283	−0.766415	+	0.642346i	$0.777961\pi$
$710$	0	0
$711$	−2.36826e7	−1.75693
$712$	0	0
$713$	2.33781e6	0.172221
$714$	0	0
$715$	−243068.	−0.0177812
$716$	0	0
$717$	1.74098e7	1.26472
$718$	0	0
$719$	−4.89582e6	−0.353186	−0.176593	−	0.984284i	$-0.556508\pi$
$719$	−4.89582e6	−0.353186	−0.176593	+	0.984284i	$0.556508\pi$
$720$	0	0
$721$	992510.	0.0711044
$722$	0	0
$723$	−2.96856e7	−2.11203
$724$	0	0
$725$	−1.50394e7	−1.06264
$726$	0	0
$727$	8.08996e6	0.567689	0.283844	−	0.958870i	$-0.408390\pi$
$727$	8.08996e6	0.567689	0.283844	+	0.958870i	$0.408390\pi$
$728$	0	0
$729$	−2.29664e7	−1.60057
$730$	0	0
$731$	3.32471e6	0.230123
$732$	0	0
$733$	2.08651e7	1.43437	0.717183	−	0.696885i	$-0.245431\pi$
$733$	2.08651e7	1.43437	0.717183	+	0.696885i	$0.245431\pi$
$734$	0	0
$735$	965325.	0.0659106
$736$	0	0
$737$	−5.84560e6	−0.396424
$738$	0	0
$739$	−8.45704e6	−0.569649	−0.284825	−	0.958580i	$-0.591935\pi$
$739$	−8.45704e6	−0.569649	−0.284825	+	0.958580i	$0.591935\pi$
$740$	0	0
$741$	−2.89717e7	−1.93833
$742$	0	0
$743$	4.50054e6	0.299083	0.149542	−	0.988755i	$-0.452220\pi$
$743$	4.50054e6	0.299083	0.149542	+	0.988755i	$0.452220\pi$
$744$	0	0
$745$	−966319.	−0.0637867
$746$	0	0
$747$	−6.18830e6	−0.405760
$748$	0	0
$749$	−1.95081e6	−0.127060
$750$	0	0
$751$	1.42022e7	0.918871	0.459435	−	0.888211i	$-0.348052\pi$
$751$	1.42022e7	0.918871	0.459435	+	0.888211i	$0.348052\pi$
$752$	0	0
$753$	4.13356e7	2.65666
$754$	0	0
$755$	903685.	0.0576965
$756$	0	0
$757$	−4.47342e6	−0.283727	−0.141863	−	0.989886i	$-0.545309\pi$
$757$	−4.47342e6	−0.283727	−0.141863	+	0.989886i	$0.545309\pi$
$758$	0	0
$759$	1.15872e6	0.0730086
$760$	0	0
$761$	−1.78407e7	−1.11673	−0.558366	−	0.829594i	$-0.688572\pi$
$761$	−1.78407e7	−1.11673	−0.558366	+	0.829594i	$0.688572\pi$
$762$	0	0
$763$	−837569.	−0.0520846
$764$	0	0
$765$	−377925.	−0.0233481
$766$	0	0
$767$	−2.90045e7	−1.78023
$768$	0	0
$769$	−1.22169e7	−0.744984	−0.372492	−	0.928036i	$-0.621497\pi$
$769$	−1.22169e7	−0.744984	−0.372492	+	0.928036i	$0.621497\pi$
$770$	0	0
$771$	2.19978e7	1.33273
$772$	0	0
$773$	−1.31589e7	−0.792083	−0.396041	−	0.918233i	$-0.629616\pi$
$773$	−1.31589e7	−0.792083	−0.396041	+	0.918233i	$0.629616\pi$
$774$	0	0
$775$	−1.93364e7	−1.15643
$776$	0	0
$777$	−3.22792e6	−0.191810
$778$	0	0
$779$	8.11956e6	0.479390
$780$	0	0
$781$	8.09404e6	0.474829
$782$	0	0
$783$	1.94602e7	1.13434
$784$	0	0
$785$	1.08857e6	0.0630493
$786$	0	0
$787$	1.18114e7	0.679773	0.339887	−	0.940466i	$-0.389611\pi$
$787$	1.18114e7	0.679773	0.339887	+	0.940466i	$0.389611\pi$
$788$	0	0
$789$	3.46477e7	1.98144
$790$	0	0
$791$	−4.10805e6	−0.233450
$792$	0	0
$793$	−3.94757e7	−2.22919
$794$	0	0
$795$	1.77135e6	0.0993999
$796$	0	0
$797$	3.15545e7	1.75961	0.879804	−	0.475336i	$-0.157673\pi$
$797$	3.15545e7	1.75961	0.879804	+	0.475336i	$0.157673\pi$
$798$	0	0
$799$	871470.	0.0482932
$800$	0	0
$801$	−1.86846e7	−1.02897
$802$	0	0
$803$	9.78467e6	0.535497
$804$	0	0
$805$	−34281.3	−0.00186452
$806$	0	0
$807$	−4.13141e7	−2.23313
$808$	0	0
$809$	3.43127e7	1.84324	0.921622	−	0.388088i	$-0.126864\pi$
$809$	3.43127e7	1.84324	0.921622	+	0.388088i	$0.126864\pi$
$810$	0	0
$811$	−1.57834e7	−0.842651	−0.421325	−	0.906910i	$-0.638435\pi$
$811$	−1.57834e7	−0.842651	−0.421325	+	0.906910i	$0.638435\pi$
$812$	0	0
$813$	−3.36541e7	−1.78571
$814$	0	0
$815$	−12634.5	−0.000666289 0
$816$	0	0
$817$	−1.21661e7	−0.637670
$818$	0	0
$819$	1.21146e7	0.631103
$820$	0	0
$821$	3.27662e7	1.69655	0.848277	−	0.529553i	$-0.177640\pi$
$821$	3.27662e7	1.69655	0.848277	+	0.529553i	$0.177640\pi$
$822$	0	0
$823$	316695.	0.0162983	0.00814913	−	0.999967i	$-0.497406\pi$
$823$	316695.	0.0162983	0.00814913	+	0.999967i	$0.497406\pi$
$824$	0	0
$825$	−9.58395e6	−0.490241
$826$	0	0
$827$	268265.	0.0136395	0.00681977	−	0.999977i	$-0.497829\pi$
$827$	268265.	0.0136395	0.00681977	+	0.999977i	$0.497829\pi$
$828$	0	0
$829$	3.25083e7	1.64289	0.821443	−	0.570291i	$-0.193170\pi$
$829$	3.25083e7	1.64289	0.821443	+	0.570291i	$0.193170\pi$
$830$	0	0
$831$	1.25939e6	0.0632642
$832$	0	0
$833$	−5.89935e6	−0.294572
$834$	0	0
$835$	727641.	0.0361161
$836$	0	0
$837$	2.50203e7	1.23446
$838$	0	0
$839$	1.67263e7	0.820343	0.410172	−	0.912008i	$-0.365469\pi$
$839$	1.67263e7	0.820343	0.410172	+	0.912008i	$0.365469\pi$
$840$	0	0
$841$	2.74019e6	0.133595
$842$	0	0
$843$	3.01404e7	1.46076
$844$	0	0
$845$	−725500.	−0.0349539
$846$	0	0
$847$	540747.	0.0258992
$848$	0	0
$849$	−7.81694e6	−0.372192
$850$	0	0
$851$	−1.29774e6	−0.0614277
$852$	0	0
$853$	−4.62988e6	−0.217870	−0.108935	−	0.994049i	$-0.534744\pi$
$853$	−4.62988e6	−0.217870	−0.108935	+	0.994049i	$0.534744\pi$
$854$	0	0
$855$	1.38294e6	0.0646974
$856$	0	0
$857$	−5.74853e6	−0.267365	−0.133682	−	0.991024i	$-0.542680\pi$
$857$	−5.74853e6	−0.267365	−0.133682	+	0.991024i	$0.542680\pi$
$858$	0	0
$859$	1.56347e7	0.722948	0.361474	−	0.932382i	$-0.382274\pi$
$859$	1.56347e7	0.722948	0.361474	+	0.932382i	$0.382274\pi$
$860$	0	0
$861$	−5.44799e6	−0.250454
$862$	0	0
$863$	6.87214e6	0.314098	0.157049	−	0.987591i	$-0.449802\pi$
$863$	6.87214e6	0.314098	0.157049	+	0.987591i	$0.449802\pi$
$864$	0	0
$865$	−1.21205e6	−0.0550782
$866$	0	0
$867$	−3.23516e7	−1.46167
$868$	0	0
$869$	−7.12981e6	−0.320279
$870$	0	0
$871$	−3.94269e7	−1.76095
$872$	0	0
$873$	−6.36226e7	−2.82537
$874$	0	0
$875$	567643.	0.0250643
$876$	0	0
$877$	−2.66074e7	−1.16816	−0.584082	−	0.811695i	$-0.698545\pi$
$877$	−2.66074e7	−1.16816	−0.584082	+	0.811695i	$0.698545\pi$
$878$	0	0
$879$	4.86554e7	2.12402
$880$	0	0
$881$	−3.06349e7	−1.32977	−0.664886	−	0.746945i	$-0.731520\pi$
$881$	−3.06349e7	−1.32977	−0.664886	+	0.746945i	$0.731520\pi$
$882$	0	0
$883$	3.43836e7	1.48405	0.742027	−	0.670369i	$-0.233864\pi$
$883$	3.43836e7	1.48405	0.742027	+	0.670369i	$0.233864\pi$
$884$	0	0
$885$	2.22157e6	0.0953460
$886$	0	0
$887$	4.68639e6	0.200000	0.0999998	−	0.994987i	$-0.468116\pi$
$887$	4.68639e6	0.200000	0.0999998	+	0.994987i	$0.468116\pi$
$888$	0	0
$889$	2.16588e6	0.0919138
$890$	0	0
$891$	583553.	0.0246256
$892$	0	0
$893$	−3.18896e6	−0.133820
$894$	0	0
$895$	−1.55018e6	−0.0646881
$896$	0	0
$897$	7.81524e6	0.324311
$898$	0	0
$899$	2.98946e7	1.23365
$900$	0	0
$901$	−1.08252e7	−0.444245
$902$	0	0
$903$	8.16309e6	0.333146
$904$	0	0
$905$	658130.	0.0267110
$906$	0	0
$907$	1.65093e7	0.666364	0.333182	−	0.942862i	$-0.391878\pi$
$907$	1.65093e7	0.666364	0.333182	+	0.942862i	$0.391878\pi$
$908$	0	0
$909$	1.72514e7	0.692493
$910$	0	0
$911$	−1.04813e7	−0.418425	−0.209213	−	0.977870i	$-0.567090\pi$
$911$	−1.04813e7	−0.418425	−0.209213	+	0.977870i	$0.567090\pi$
$912$	0	0
$913$	−1.86303e6	−0.0739678
$914$	0	0
$915$	3.02360e6	0.119391
$916$	0	0
$917$	1.22187e7	0.479844
$918$	0	0
$919$	−2.17428e6	−0.0849234	−0.0424617	−	0.999098i	$-0.513520\pi$
$919$	−2.17428e6	−0.0849234	−0.0424617	+	0.999098i	$0.513520\pi$
$920$	0	0
$921$	4.05618e7	1.57568
$922$	0	0
$923$	5.45920e7	2.10923
$924$	0	0
$925$	1.07338e7	0.412477
$926$	0	0
$927$	1.08006e7	0.412809
$928$	0	0
$929$	4.11976e7	1.56615	0.783073	−	0.621929i	$-0.213651\pi$
$929$	4.11976e7	1.56615	0.783073	+	0.621929i	$0.213651\pi$
$930$	0	0
$931$	2.15874e7	0.816257
$932$	0	0
$933$	−6.12895e7	−2.30506
$934$	0	0
$935$	−113777.	−0.00425623
$936$	0	0
$937$	−932491.	−0.0346973	−0.0173486	−	0.999850i	$-0.505523\pi$
$937$	−932491.	−0.0346973	−0.0173486	+	0.999850i	$0.505523\pi$
$938$	0	0
$939$	−7.98398e6	−0.295498
$940$	0	0
$941$	−2.61944e7	−0.964350	−0.482175	−	0.876075i	$-0.660153\pi$
$941$	−2.61944e7	−0.964350	−0.482175	+	0.876075i	$0.660153\pi$
$942$	0	0
$943$	−2.19029e6	−0.0802088
$944$	0	0
$945$	−366894.	−0.0133647
$946$	0	0
$947$	−3.25549e7	−1.17962	−0.589809	−	0.807542i	$-0.700797\pi$
$947$	−3.25549e7	−1.17962	−0.589809	+	0.807542i	$0.700797\pi$
$948$	0	0
$949$	6.59948e7	2.37873
$950$	0	0
$951$	4.45583e7	1.59763
$952$	0	0
$953$	3.34686e7	1.19373	0.596865	−	0.802342i	$-0.296413\pi$
$953$	3.34686e7	1.19373	0.596865	+	0.802342i	$0.296413\pi$
$954$	0	0
$955$	−1.97352e6	−0.0700218
$956$	0	0
$957$	1.48170e7	0.522976
$958$	0	0
$959$	293418.	0.0103024
$960$	0	0
$961$	9.80667e6	0.342541
$962$	0	0
$963$	−2.12289e7	−0.737670
$964$	0	0
$965$	2.22511e6	0.0769190
$966$	0	0
$967$	−2.92544e7	−1.00606	−0.503031	−	0.864268i	$-0.667782\pi$
$967$	−2.92544e7	−1.00606	−0.503031	+	0.864268i	$0.667782\pi$
$968$	0	0
$969$	−1.35613e7	−0.463971
$970$	0	0
$971$	−79377.9	−0.00270179	−0.00135090	−	0.999999i	$-0.500430\pi$
$971$	−79377.9	−0.00270179	−0.00135090	+	0.999999i	$0.500430\pi$
$972$	0	0
$973$	1.19250e7	0.403808
$974$	0	0
$975$	−6.46411e7	−2.17769
$976$	0	0
$977$	1.24149e7	0.416110	0.208055	−	0.978117i	$-0.433287\pi$
$977$	1.24149e7	0.416110	0.208055	+	0.978117i	$0.433287\pi$
$978$	0	0
$979$	−5.62514e6	−0.187576
$980$	0	0
$981$	−9.11453e6	−0.302386
$982$	0	0
$983$	3.03404e7	1.00147	0.500734	−	0.865601i	$-0.333063\pi$
$983$	3.03404e7	1.00147	0.500734	+	0.865601i	$0.333063\pi$
$984$	0	0
$985$	−68489.3	−0.00224922
$986$	0	0
$987$	2.13970e6	0.0699133
$988$	0	0
$989$	3.28186e6	0.106691
$990$	0	0
$991$	−3.86025e7	−1.24862	−0.624311	−	0.781176i	$-0.714620\pi$
$991$	−3.86025e7	−1.24862	−0.624311	+	0.781176i	$0.714620\pi$
$992$	0	0
$993$	−6.16161e7	−1.98299
$994$	0	0
$995$	−2.15450e6	−0.0689903
$996$	0	0
$997$	5.17340e7	1.64831	0.824154	−	0.566366i	$-0.191651\pi$
$997$	5.17340e7	1.64831	0.824154	+	0.566366i	$0.191651\pi$
$998$	0	0
$999$	−1.38890e7	−0.440309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.6.a.b.1.3	✓	3
3.2	odd	2		792.6.a.f.1.2		3
4.3	odd	2		176.6.a.j.1.1		3
8.3	odd	2		704.6.a.s.1.3		3
8.5	even	2		704.6.a.r.1.1		3
11.10	odd	2		968.6.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.a.b.1.3	✓	3	1.1	even	1	trivial
176.6.a.j.1.1		3	4.3	odd	2
704.6.a.r.1.1		3	8.5	even	2
704.6.a.s.1.3		3	8.3	odd	2
792.6.a.f.1.2		3	3.2	odd	2
968.6.a.c.1.3		3	11.10	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:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	88.a (trivial)

\(f(q)\)	\(=\)	\(q+25.3952 q^{3} -2.46146 q^{5} +36.9338 q^{7} +401.918 q^{9} +121.000 q^{11} +816.111 q^{13} -62.5093 q^{15} +382.011 q^{17} -1397.89 q^{19} +937.941 q^{21} +377.087 q^{23} -3118.94 q^{25} +4035.75 q^{27} +4821.96 q^{29} +6199.66 q^{31} +3072.82 q^{33} -90.9110 q^{35} -3441.50 q^{37} +20725.3 q^{39} -5808.45 q^{41} +8703.19 q^{43} -989.304 q^{45} +2281.27 q^{47} -15442.9 q^{49} +9701.25 q^{51} -28337.3 q^{53} -297.837 q^{55} -35499.7 q^{57} -35539.9 q^{59} -48370.5 q^{61} +14844.3 q^{63} -2008.82 q^{65} -48310.7 q^{67} +9576.20 q^{69} +66892.9 q^{71} +80865.0 q^{73} -79206.2 q^{75} +4468.99 q^{77} -58924.1 q^{79} +4822.76 q^{81} -15396.9 q^{83} -940.304 q^{85} +122455. q^{87} -46488.8 q^{89} +30142.1 q^{91} +157442. q^{93} +3440.84 q^{95} -158298. q^{97} +48632.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 14 q^{3} + 56 q^{5} + 112 q^{7} + 145 q^{9} + 363 q^{11} + 450 q^{13} + 994 q^{15} + 1274 q^{17} + 2416 q^{19} + 2064 q^{21} + 4042 q^{23} + 4103 q^{25} + 6398 q^{27} + 2086 q^{29} + 10034 q^{31}+ \cdots + 17545 q^{99}+O(q^{100}) \) 3 * q + 14 * q^3 + 56 * q^5 + 112 * q^7 + 145 * q^9 + 363 * q^11 + 450 * q^13 + 994 * q^15 + 1274 * q^17 + 2416 * q^19 + 2064 * q^21 + 4042 * q^23 + 4103 * q^25 + 6398 * q^27 + 2086 * q^29 + 10034 * q^31 + 1694 * q^33 + 15256 * q^35 - 2916 * q^37 + 17188 * q^39 - 9450 * q^41 + 17296 * q^43 - 24334 * q^45 + 22312 * q^47 - 31533 * q^49 + 8952 * q^51 - 50990 * q^53 + 6776 * q^55 - 51324 * q^57 - 16654 * q^59 - 50674 * q^61 - 12504 * q^63 - 60312 * q^65 - 59966 * q^67 - 34558 * q^69 - 16954 * q^71 - 50850 * q^73 - 39116 * q^75 + 13552 * q^77 + 7580 * q^79 + 3475 * q^81 + 1664 * q^83 + 61812 * q^85 + 83536 * q^87 - 41920 * q^89 - 36824 * q^91 + 99042 * q^93 + 164916 * q^95 - 93356 * q^97 + 17545 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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