LMFDB - Embedded newform 220.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [220,6,Mod(1,220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(220, 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("220.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	220.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:	$35.2844403589$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1761}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 440 $ x^2 - x - 440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-20.4821$ of defining polynomial
Character	$\chi$	$=$	220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.4821 q^{3} +25.0000 q^{5} +39.4107 q^{7} -111.161 q^{9} -121.000 q^{11} -1182.68 q^{13} +287.053 q^{15} -2195.77 q^{17} +1866.37 q^{19} +452.519 q^{21} +337.644 q^{23} +625.000 q^{25} -4066.52 q^{27} -5432.02 q^{29} +139.481 q^{31} -1389.34 q^{33} +985.267 q^{35} +5943.41 q^{37} -13579.7 q^{39} +4630.43 q^{41} -1155.36 q^{43} -2779.01 q^{45} +3413.28 q^{47} -15253.8 q^{49} -25212.1 q^{51} +5651.62 q^{53} -3025.00 q^{55} +21430.0 q^{57} +13627.0 q^{59} -34279.1 q^{61} -4380.91 q^{63} -29567.0 q^{65} -9963.03 q^{67} +3876.87 q^{69} -19869.0 q^{71} -51656.5 q^{73} +7176.33 q^{75} -4768.69 q^{77} +50960.7 q^{79} -19680.3 q^{81} -8628.02 q^{83} -54894.2 q^{85} -62371.2 q^{87} -29938.4 q^{89} -46610.2 q^{91} +1601.54 q^{93} +46659.4 q^{95} +48871.1 q^{97} +13450.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 19 q^{3} + 50 q^{5} - 131 q^{7} + 575 q^{9} - 242 q^{11} - 1610 q^{13} - 475 q^{15} - 321 q^{17} + 3439 q^{19} + 5647 q^{21} + 4536 q^{23} + 1250 q^{25} - 17575 q^{27} - 5031 q^{29} - 4463 q^{31}+ \cdots - 69575 q^{99}+O(q^{100}) $ 2 * q - 19 * q^3 + 50 * q^5 - 131 * q^7 + 575 * q^9 - 242 * q^11 - 1610 * q^13 - 475 * q^15 - 321 * q^17 + 3439 * q^19 + 5647 * q^21 + 4536 * q^23 + 1250 * q^25 - 17575 * q^27 - 5031 * q^29 - 4463 * q^31 + 2299 * q^33 - 3275 * q^35 - 2423 * q^37 - 554 * q^39 + 19668 * q^41 - 14900 * q^43 + 14375 * q^45 - 23052 * q^47 - 3021 * q^49 - 82359 * q^51 - 7203 * q^53 - 6050 * q^55 - 26507 * q^57 + 50838 * q^59 - 44177 * q^61 - 121310 * q^63 - 40250 * q^65 - 10610 * q^67 - 124098 * q^69 - 1089 * q^71 - 92654 * q^73 - 11875 * q^75 + 15851 * q^77 - 16334 * q^79 + 225350 * q^81 + 88410 * q^83 - 8025 * q^85 - 74595 * q^87 - 80817 * q^89 + 26210 * q^91 + 141895 * q^93 + 85975 * q^95 + 102694 * q^97 - 69575 * 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
$3$	11.4821	0.736579	0.368290	−	0.929711i	$-0.379943\pi$
$3$	11.4821	0.736579	0.368290	+	0.929711i	$0.379943\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	39.4107	0.303997	0.151998	−	0.988381i	$-0.451429\pi$
$7$	39.4107	0.303997	0.151998	+	0.988381i	$0.451429\pi$
$8$	0	0
$9$	−111.161	−0.457451
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−1182.68	−1.94092	−0.970461	−	0.241256i	$-0.922441\pi$
$13$	−1182.68	−1.94092	−0.970461	+	0.241256i	$0.922441\pi$
$14$	0	0
$15$	287.053	0.329408
$16$	0	0
$17$	−2195.77	−1.84274	−0.921370	−	0.388687i	$-0.872929\pi$
$17$	−2195.77	−1.84274	−0.921370	+	0.388687i	$0.872929\pi$
$18$	0	0
$19$	1866.37	1.18608	0.593041	−	0.805172i	$-0.297927\pi$
$19$	1866.37	1.18608	0.593041	+	0.805172i	$0.297927\pi$
$20$	0	0
$21$	452.519	0.223918
$22$	0	0
$23$	337.644	0.133088	0.0665440	−	0.997783i	$-0.478803\pi$
$23$	337.644	0.133088	0.0665440	+	0.997783i	$0.478803\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−4066.52	−1.07353
$28$	0	0
$29$	−5432.02	−1.19941	−0.599703	−	0.800223i	$-0.704715\pi$
$29$	−5432.02	−1.19941	−0.599703	+	0.800223i	$0.704715\pi$
$30$	0	0
$31$	139.481	0.0260682	0.0130341	−	0.999915i	$-0.495851\pi$
$31$	139.481	0.0260682	0.0130341	+	0.999915i	$0.495851\pi$
$32$	0	0
$33$	−1389.34	−0.222087
$34$	0	0
$35$	985.267	0.135951
$36$	0	0
$37$	5943.41	0.713726	0.356863	−	0.934157i	$-0.383846\pi$
$37$	5943.41	0.713726	0.356863	+	0.934157i	$0.383846\pi$
$38$	0	0
$39$	−13579.7	−1.42964
$40$	0	0
$41$	4630.43	0.430191	0.215096	−	0.976593i	$-0.430994\pi$
$41$	4630.43	0.430191	0.215096	+	0.976593i	$0.430994\pi$
$42$	0	0
$43$	−1155.36	−0.0952897	−0.0476449	−	0.998864i	$-0.515172\pi$
$43$	−1155.36	−0.0952897	−0.0476449	+	0.998864i	$0.515172\pi$
$44$	0	0
$45$	−2779.01	−0.204578
$46$	0	0
$47$	3413.28	0.225386	0.112693	−	0.993630i	$-0.464052\pi$
$47$	3413.28	0.225386	0.112693	+	0.993630i	$0.464052\pi$
$48$	0	0
$49$	−15253.8	−0.907586
$50$	0	0
$51$	−25212.1	−1.35732
$52$	0	0
$53$	5651.62	0.276365	0.138183	−	0.990407i	$-0.455874\pi$
$53$	5651.62	0.276365	0.138183	+	0.990407i	$0.455874\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	0	0
$57$	21430.0	0.873644
$58$	0	0
$59$	13627.0	0.509649	0.254825	−	0.966987i	$-0.417982\pi$
$59$	13627.0	0.509649	0.254825	+	0.966987i	$0.417982\pi$
$60$	0	0
$61$	−34279.1	−1.17952	−0.589760	−	0.807579i	$-0.700778\pi$
$61$	−34279.1	−1.17952	−0.589760	+	0.807579i	$0.700778\pi$
$62$	0	0
$63$	−4380.91	−0.139064
$64$	0	0
$65$	−29567.0	−0.868007
$66$	0	0
$67$	−9963.03	−0.271147	−0.135573	−	0.990767i	$-0.543288\pi$
$67$	−9963.03	−0.271147	−0.135573	+	0.990767i	$0.543288\pi$
$68$	0	0
$69$	3876.87	0.0980299
$70$	0	0
$71$	−19869.0	−0.467769	−0.233884	−	0.972264i	$-0.575144\pi$
$71$	−19869.0	−0.467769	−0.233884	+	0.972264i	$0.575144\pi$
$72$	0	0
$73$	−51656.5	−1.13453	−0.567267	−	0.823534i	$-0.691999\pi$
$73$	−51656.5	−1.13453	−0.567267	+	0.823534i	$0.691999\pi$
$74$	0	0
$75$	7176.33	0.147316
$76$	0	0
$77$	−4768.69	−0.0916584
$78$	0	0
$79$	50960.7	0.918686	0.459343	−	0.888259i	$-0.348085\pi$
$79$	50960.7	0.918686	0.459343	+	0.888259i	$0.348085\pi$
$80$	0	0
$81$	−19680.3	−0.333288
$82$	0	0
$83$	−8628.02	−0.137472	−0.0687362	−	0.997635i	$-0.521897\pi$
$83$	−8628.02	−0.137472	−0.0687362	+	0.997635i	$0.521897\pi$
$84$	0	0
$85$	−54894.2	−0.824098
$86$	0	0
$87$	−62371.2	−0.883457
$88$	0	0
$89$	−29938.4	−0.400640	−0.200320	−	0.979731i	$-0.564198\pi$
$89$	−29938.4	−0.400640	−0.200320	+	0.979731i	$0.564198\pi$
$90$	0	0
$91$	−46610.2	−0.590034
$92$	0	0
$93$	1601.54	0.0192013
$94$	0	0
$95$	46659.4	0.530432
$96$	0	0
$97$	48871.1	0.527379	0.263690	−	0.964608i	$-0.415061\pi$
$97$	48871.1	0.527379	0.263690	+	0.964608i	$0.415061\pi$
$98$	0	0
$99$	13450.4	0.137927
$100$	0	0
$101$	−46237.3	−0.451013	−0.225506	−	0.974242i	$-0.572404\pi$
$101$	−46237.3	−0.451013	−0.225506	+	0.974242i	$0.572404\pi$
$102$	0	0
$103$	−156417.	−1.45275	−0.726376	−	0.687297i	$-0.758797\pi$
$103$	−156417.	−1.45275	−0.726376	+	0.687297i	$0.758797\pi$
$104$	0	0
$105$	11313.0	0.100139
$106$	0	0
$107$	−144650.	−1.22140	−0.610702	−	0.791861i	$-0.709113\pi$
$107$	−144650.	−1.22140	−0.610702	+	0.791861i	$0.709113\pi$
$108$	0	0
$109$	−15491.8	−0.124892	−0.0624462	−	0.998048i	$-0.519890\pi$
$109$	−15491.8	−0.124892	−0.0624462	+	0.998048i	$0.519890\pi$
$110$	0	0
$111$	68243.0	0.525715
$112$	0	0
$113$	7502.99	0.0552762	0.0276381	−	0.999618i	$-0.491201\pi$
$113$	7502.99	0.0552762	0.0276381	+	0.999618i	$0.491201\pi$
$114$	0	0
$115$	8441.09	0.0595188
$116$	0	0
$117$	131467.	0.887877
$118$	0	0
$119$	−86536.7	−0.560187
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	53167.2	0.316870
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	282664.	1.55511	0.777556	−	0.628813i	$-0.216459\pi$
$127$	282664.	1.55511	0.777556	+	0.628813i	$0.216459\pi$
$128$	0	0
$129$	−13266.0	−0.0701884
$130$	0	0
$131$	148305.	0.755052	0.377526	−	0.925999i	$-0.376775\pi$
$131$	148305.	0.755052	0.377526	+	0.925999i	$0.376775\pi$
$132$	0	0
$133$	73555.1	0.360565
$134$	0	0
$135$	−101663.	−0.480096
$136$	0	0
$137$	94749.1	0.431294	0.215647	−	0.976471i	$-0.430814\pi$
$137$	94749.1	0.431294	0.215647	+	0.976471i	$0.430814\pi$
$138$	0	0
$139$	379277.	1.66502	0.832511	−	0.554009i	$-0.186903\pi$
$139$	379277.	1.66502	0.832511	+	0.554009i	$0.186903\pi$
$140$	0	0
$141$	39191.7	0.166015
$142$	0	0
$143$	143104.	0.585210
$144$	0	0
$145$	−135800.	−0.536391
$146$	0	0
$147$	−175146.	−0.668509
$148$	0	0
$149$	−346333.	−1.27799	−0.638996	−	0.769210i	$-0.720650\pi$
$149$	−346333.	−1.27799	−0.638996	+	0.769210i	$0.720650\pi$
$150$	0	0
$151$	−65850.9	−0.235028	−0.117514	−	0.993071i	$-0.537492\pi$
$151$	−65850.9	−0.235028	−0.117514	+	0.993071i	$0.537492\pi$
$152$	0	0
$153$	244083.	0.842963
$154$	0	0
$155$	3487.03	0.0116581
$156$	0	0
$157$	367291.	1.18922	0.594609	−	0.804015i	$-0.297307\pi$
$157$	367291.	1.18922	0.594609	+	0.804015i	$0.297307\pi$
$158$	0	0
$159$	64892.7	0.203565
$160$	0	0
$161$	13306.8	0.0404583
$162$	0	0
$163$	−481404.	−1.41919	−0.709596	−	0.704609i	$-0.751122\pi$
$163$	−481404.	−1.41919	−0.709596	+	0.704609i	$0.751122\pi$
$164$	0	0
$165$	−34733.5	−0.0993203
$166$	0	0
$167$	600777.	1.66695	0.833475	−	0.552558i	$-0.186348\pi$
$167$	600777.	1.66695	0.833475	+	0.552558i	$0.186348\pi$
$168$	0	0
$169$	1.02744e6	2.76718
$170$	0	0
$171$	−207467.	−0.542575
$172$	0	0
$173$	401433.	1.01976	0.509880	−	0.860245i	$-0.329690\pi$
$173$	401433.	1.01976	0.509880	+	0.860245i	$0.329690\pi$
$174$	0	0
$175$	24631.7	0.0607993
$176$	0	0
$177$	156468.	0.375397
$178$	0	0
$179$	547366.	1.27687	0.638433	−	0.769677i	$-0.279583\pi$
$179$	547366.	1.27687	0.638433	+	0.769677i	$0.279583\pi$
$180$	0	0
$181$	605472.	1.37372	0.686859	−	0.726791i	$-0.258989\pi$
$181$	605472.	1.37372	0.686859	+	0.726791i	$0.258989\pi$
$182$	0	0
$183$	−393597.	−0.868810
$184$	0	0
$185$	148585.	0.319188
$186$	0	0
$187$	265688.	0.555607
$188$	0	0
$189$	−160264.	−0.326349
$190$	0	0
$191$	49593.8	0.0983657	0.0491829	−	0.998790i	$-0.484338\pi$
$191$	49593.8	0.0983657	0.0491829	+	0.998790i	$0.484338\pi$
$192$	0	0
$193$	−899272.	−1.73779	−0.868896	−	0.494994i	$-0.835170\pi$
$193$	−899272.	−1.73779	−0.868896	+	0.494994i	$0.835170\pi$
$194$	0	0
$195$	−339492.	−0.639356
$196$	0	0
$197$	−279635.	−0.513365	−0.256683	−	0.966496i	$-0.582629\pi$
$197$	−279635.	−0.513365	−0.256683	+	0.966496i	$0.582629\pi$
$198$	0	0
$199$	−822226.	−1.47183	−0.735917	−	0.677072i	$-0.763248\pi$
$199$	−822226.	−1.47183	−0.735917	+	0.677072i	$0.763248\pi$
$200$	0	0
$201$	−114397.	−0.199721
$202$	0	0
$203$	−214079.	−0.364615
$204$	0	0
$205$	115761.	0.192387
$206$	0	0
$207$	−37532.7	−0.0608812
$208$	0	0
$209$	−225831.	−0.357617
$210$	0	0
$211$	−534284.	−0.826164	−0.413082	−	0.910694i	$-0.635548\pi$
$211$	−534284.	−0.826164	−0.413082	+	0.910694i	$0.635548\pi$
$212$	0	0
$213$	−228139.	−0.344549
$214$	0	0
$215$	−28884.0	−0.0426149
$216$	0	0
$217$	5497.05	0.00792466
$218$	0	0
$219$	−593126.	−0.835674
$220$	0	0
$221$	2.59689e6	3.57662
$222$	0	0
$223$	−25689.6	−0.0345936	−0.0172968	−	0.999850i	$-0.505506\pi$
$223$	−25689.6	−0.0345936	−0.0172968	+	0.999850i	$0.505506\pi$
$224$	0	0
$225$	−69475.4	−0.0914902
$226$	0	0
$227$	−519615.	−0.669295	−0.334647	−	0.942343i	$-0.608617\pi$
$227$	−519615.	−0.669295	−0.334647	+	0.942343i	$0.608617\pi$
$228$	0	0
$229$	566856.	0.714305	0.357153	−	0.934046i	$-0.383748\pi$
$229$	566856.	0.714305	0.357153	+	0.934046i	$0.383748\pi$
$230$	0	0
$231$	−54754.8	−0.0675137
$232$	0	0
$233$	−250125.	−0.301834	−0.150917	−	0.988546i	$-0.548223\pi$
$233$	−250125.	−0.301834	−0.150917	+	0.988546i	$0.548223\pi$
$234$	0	0
$235$	85332.0	0.100796
$236$	0	0
$237$	585137.	0.676685
$238$	0	0
$239$	−1.29750e6	−1.46931	−0.734654	−	0.678442i	$-0.762656\pi$
$239$	−1.29750e6	−1.46931	−0.734654	+	0.678442i	$0.762656\pi$
$240$	0	0
$241$	582411.	0.645932	0.322966	−	0.946411i	$-0.395320\pi$
$241$	582411.	0.645932	0.322966	+	0.946411i	$0.395320\pi$
$242$	0	0
$243$	762192.	0.828035
$244$	0	0
$245$	−381345.	−0.405885
$246$	0	0
$247$	−2.20732e6	−2.30209
$248$	0	0
$249$	−99068.1	−0.101259
$250$	0	0
$251$	1.75979e6	1.76310	0.881550	−	0.472091i	$-0.156501\pi$
$251$	1.75979e6	1.76310	0.881550	+	0.472091i	$0.156501\pi$
$252$	0	0
$253$	−40854.9	−0.0401275
$254$	0	0
$255$	−630302.	−0.607014
$256$	0	0
$257$	−539156.	−0.509192	−0.254596	−	0.967048i	$-0.581942\pi$
$257$	−539156.	−0.509192	−0.254596	+	0.967048i	$0.581942\pi$
$258$	0	0
$259$	234234.	0.216970
$260$	0	0
$261$	603826.	0.548669
$262$	0	0
$263$	−2.20351e6	−1.96438	−0.982192	−	0.187881i	$-0.939838\pi$
$263$	−2.20351e6	−1.96438	−0.982192	+	0.187881i	$0.939838\pi$
$264$	0	0
$265$	141291.	0.123594
$266$	0	0
$267$	−343757.	−0.295103
$268$	0	0
$269$	−731509.	−0.616366	−0.308183	−	0.951327i	$-0.599721\pi$
$269$	−731509.	−0.616366	−0.308183	+	0.951327i	$0.599721\pi$
$270$	0	0
$271$	446543.	0.369352	0.184676	−	0.982799i	$-0.440876\pi$
$271$	446543.	0.369352	0.184676	+	0.982799i	$0.440876\pi$
$272$	0	0
$273$	−535184.	−0.434607
$274$	0	0
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	1.60933e6	1.26022	0.630109	−	0.776507i	$-0.283010\pi$
$277$	1.60933e6	1.26022	0.630109	+	0.776507i	$0.283010\pi$
$278$	0	0
$279$	−15504.8	−0.0119249
$280$	0	0
$281$	−836866.	−0.632252	−0.316126	−	0.948717i	$-0.602382\pi$
$281$	−836866.	−0.632252	−0.316126	+	0.948717i	$0.602382\pi$
$282$	0	0
$283$	−1.43720e6	−1.06672	−0.533362	−	0.845887i	$-0.679072\pi$
$283$	−1.43720e6	−1.06672	−0.533362	+	0.845887i	$0.679072\pi$
$284$	0	0
$285$	535749.	0.390705
$286$	0	0
$287$	182488.	0.130777
$288$	0	0
$289$	3.40154e6	2.39569
$290$	0	0
$291$	561145.	0.388456
$292$	0	0
$293$	−977430.	−0.665146	−0.332573	−	0.943078i	$-0.607917\pi$
$293$	−977430.	−0.665146	−0.332573	+	0.943078i	$0.607917\pi$
$294$	0	0
$295$	340676.	0.227922
$296$	0	0
$297$	492049.	0.323681
$298$	0	0
$299$	−399324.	−0.258314
$300$	0	0
$301$	−45533.5	−0.0289677
$302$	0	0
$303$	−530903.	−0.332207
$304$	0	0
$305$	−856978.	−0.527497
$306$	0	0
$307$	242927.	0.147106	0.0735528	−	0.997291i	$-0.476566\pi$
$307$	242927.	0.147106	0.0735528	+	0.997291i	$0.476566\pi$
$308$	0	0
$309$	−1.79600e6	−1.07007
$310$	0	0
$311$	−1.93118e6	−1.13220	−0.566098	−	0.824338i	$-0.691548\pi$
$311$	−1.93118e6	−1.13220	−0.566098	+	0.824338i	$0.691548\pi$
$312$	0	0
$313$	−1.03171e6	−0.595249	−0.297625	−	0.954683i	$-0.596194\pi$
$313$	−1.03171e6	−0.595249	−0.297625	+	0.954683i	$0.596194\pi$
$314$	0	0
$315$	−109523.	−0.0621911
$316$	0	0
$317$	−2.05142e6	−1.14659	−0.573293	−	0.819351i	$-0.694334\pi$
$317$	−2.05142e6	−1.14659	−0.573293	+	0.819351i	$0.694334\pi$
$318$	0	0
$319$	657274.	0.361634
$320$	0	0
$321$	−1.66089e6	−0.899661
$322$	0	0
$323$	−4.09812e6	−2.18564
$324$	0	0
$325$	−739174.	−0.388185
$326$	0	0
$327$	−177879.	−0.0919931
$328$	0	0
$329$	134520.	0.0685166
$330$	0	0
$331$	1.58685e6	0.796095	0.398047	−	0.917365i	$-0.369688\pi$
$331$	1.58685e6	0.796095	0.398047	+	0.917365i	$0.369688\pi$
$332$	0	0
$333$	−660673.	−0.326494
$334$	0	0
$335$	−249076.	−0.121261
$336$	0	0
$337$	2.33349e6	1.11926	0.559630	−	0.828742i	$-0.310943\pi$
$337$	2.33349e6	1.11926	0.559630	+	0.828742i	$0.310943\pi$
$338$	0	0
$339$	86150.3	0.0407153
$340$	0	0
$341$	−16877.2	−0.00785987
$342$	0	0
$343$	−1.26354e6	−0.579900
$344$	0	0
$345$	96921.7	0.0438403
$346$	0	0
$347$	−194423.	−0.0866811	−0.0433406	−	0.999060i	$-0.513800\pi$
$347$	−194423.	−0.0866811	−0.0433406	+	0.999060i	$0.513800\pi$
$348$	0	0
$349$	−744025.	−0.326982	−0.163491	−	0.986545i	$-0.552275\pi$
$349$	−744025.	−0.326982	−0.163491	+	0.986545i	$0.552275\pi$
$350$	0	0
$351$	4.80938e6	2.08364
$352$	0	0
$353$	−257236.	−0.109874	−0.0549370	−	0.998490i	$-0.517496\pi$
$353$	−257236.	−0.109874	−0.0549370	+	0.998490i	$0.517496\pi$
$354$	0	0
$355$	−496726.	−0.209193
$356$	0	0
$357$	−993626.	−0.412622
$358$	0	0
$359$	−1.04798e6	−0.429157	−0.214579	−	0.976707i	$-0.568838\pi$
$359$	−1.04798e6	−0.429157	−0.214579	+	0.976707i	$0.568838\pi$
$360$	0	0
$361$	1.00726e6	0.406792
$362$	0	0
$363$	168110.	0.0669618
$364$	0	0
$365$	−1.29141e6	−0.507379
$366$	0	0
$367$	2.52808e6	0.979772	0.489886	−	0.871787i	$-0.337038\pi$
$367$	2.52808e6	0.979772	0.489886	+	0.871787i	$0.337038\pi$
$368$	0	0
$369$	−514721.	−0.196791
$370$	0	0
$371$	222734.	0.0840141
$372$	0	0
$373$	−398673.	−0.148370	−0.0741848	−	0.997245i	$-0.523635\pi$
$373$	−398673.	−0.148370	−0.0741848	+	0.997245i	$0.523635\pi$
$374$	0	0
$375$	179408.	0.0658817
$376$	0	0
$377$	6.42433e6	2.32795
$378$	0	0
$379$	−41188.8	−0.0147293	−0.00736463	−	0.999973i	$-0.502344\pi$
$379$	−41188.8	−0.0147293	−0.00736463	+	0.999973i	$0.502344\pi$
$380$	0	0
$381$	3.24559e6	1.14546
$382$	0	0
$383$	−2.53931e6	−0.884543	−0.442271	−	0.896881i	$-0.645827\pi$
$383$	−2.53931e6	−0.884543	−0.442271	+	0.896881i	$0.645827\pi$
$384$	0	0
$385$	−119217.	−0.0409909
$386$	0	0
$387$	128430.	0.0435904
$388$	0	0
$389$	−3.99822e6	−1.33965	−0.669827	−	0.742518i	$-0.733632\pi$
$389$	−3.99822e6	−1.33965	−0.669827	+	0.742518i	$0.733632\pi$
$390$	0	0
$391$	−741387.	−0.245247
$392$	0	0
$393$	1.70286e6	0.556156
$394$	0	0
$395$	1.27402e6	0.410849
$396$	0	0
$397$	−5.04861e6	−1.60766	−0.803832	−	0.594856i	$-0.797209\pi$
$397$	−5.04861e6	−1.60766	−0.803832	+	0.594856i	$0.797209\pi$
$398$	0	0
$399$	844570.	0.265585
$400$	0	0
$401$	4.71689e6	1.46485	0.732427	−	0.680845i	$-0.238387\pi$
$401$	4.71689e6	1.46485	0.732427	+	0.680845i	$0.238387\pi$
$402$	0	0
$403$	−164962.	−0.0505964
$404$	0	0
$405$	−492008.	−0.149051
$406$	0	0
$407$	−719152.	−0.215196
$408$	0	0
$409$	−1.19007e6	−0.351775	−0.175887	−	0.984410i	$-0.556279\pi$
$409$	−1.19007e6	−0.351775	−0.175887	+	0.984410i	$0.556279\pi$
$410$	0	0
$411$	1.08792e6	0.317682
$412$	0	0
$413$	537051.	0.154932
$414$	0	0
$415$	−215700.	−0.0614796
$416$	0	0
$417$	4.35491e6	1.22642
$418$	0	0
$419$	6.85301e6	1.90698	0.953491	−	0.301422i	$-0.0974614\pi$
$419$	6.85301e6	1.90698	0.953491	+	0.301422i	$0.0974614\pi$
$420$	0	0
$421$	6.62822e6	1.82260	0.911300	−	0.411743i	$-0.135080\pi$
$421$	6.62822e6	1.82260	0.911300	+	0.411743i	$0.135080\pi$
$422$	0	0
$423$	−379422.	−0.103103
$424$	0	0
$425$	−1.37235e6	−0.368548
$426$	0	0
$427$	−1.35096e6	−0.358570
$428$	0	0
$429$	1.64314e6	0.431054
$430$	0	0
$431$	−1.04575e6	−0.271165	−0.135582	−	0.990766i	$-0.543291\pi$
$431$	−1.04575e6	−0.271165	−0.135582	+	0.990766i	$0.543291\pi$
$432$	0	0
$433$	934093.	0.239425	0.119713	−	0.992809i	$-0.461803\pi$
$433$	934093.	0.239425	0.119713	+	0.992809i	$0.461803\pi$
$434$	0	0
$435$	−1.55928e6	−0.395094
$436$	0	0
$437$	630169.	0.157853
$438$	0	0
$439$	3.24978e6	0.804809	0.402404	−	0.915462i	$-0.368175\pi$
$439$	3.24978e6	0.804809	0.402404	+	0.915462i	$0.368175\pi$
$440$	0	0
$441$	1.69562e6	0.415176
$442$	0	0
$443$	3.11830e6	0.754933	0.377466	−	0.926023i	$-0.376795\pi$
$443$	3.11830e6	0.754933	0.377466	+	0.926023i	$0.376795\pi$
$444$	0	0
$445$	−748460.	−0.179171
$446$	0	0
$447$	−3.97664e6	−0.941343
$448$	0	0
$449$	3.36184e6	0.786976	0.393488	−	0.919330i	$-0.371268\pi$
$449$	3.36184e6	0.786976	0.393488	+	0.919330i	$0.371268\pi$
$450$	0	0
$451$	−560282.	−0.129708
$452$	0	0
$453$	−756109.	−0.173117
$454$	0	0
$455$	−1.16525e6	−0.263871
$456$	0	0
$457$	3.98953e6	0.893575	0.446787	−	0.894640i	$-0.352568\pi$
$457$	3.98953e6	0.893575	0.446787	+	0.894640i	$0.352568\pi$
$458$	0	0
$459$	8.92913e6	1.97823
$460$	0	0
$461$	−4.93053e6	−1.08054	−0.540271	−	0.841491i	$-0.681678\pi$
$461$	−4.93053e6	−1.08054	−0.540271	+	0.841491i	$0.681678\pi$
$462$	0	0
$463$	−6.61725e6	−1.43458	−0.717290	−	0.696774i	$-0.754618\pi$
$463$	−6.61725e6	−1.43458	−0.717290	+	0.696774i	$0.754618\pi$
$464$	0	0
$465$	40038.6	0.00858709
$466$	0	0
$467$	3.80218e6	0.806752	0.403376	−	0.915034i	$-0.367837\pi$
$467$	3.80218e6	0.806752	0.403376	+	0.915034i	$0.367837\pi$
$468$	0	0
$469$	−392650.	−0.0824277
$470$	0	0
$471$	4.21729e6	0.875954
$472$	0	0
$473$	139798.	0.0287309
$474$	0	0
$475$	1.16648e6	0.237216
$476$	0	0
$477$	−628237.	−0.126424
$478$	0	0
$479$	2.57373e6	0.512536	0.256268	−	0.966606i	$-0.417507\pi$
$479$	2.57373e6	0.512536	0.256268	+	0.966606i	$0.417507\pi$
$480$	0	0
$481$	−7.02914e6	−1.38529
$482$	0	0
$483$	152790.	0.0298007
$484$	0	0
$485$	1.22178e6	0.235851
$486$	0	0
$487$	9.87913e6	1.88754	0.943770	−	0.330603i	$-0.107252\pi$
$487$	9.87913e6	1.88754	0.943770	+	0.330603i	$0.107252\pi$
$488$	0	0
$489$	−5.52755e6	−1.04535
$490$	0	0
$491$	−9.45632e6	−1.77018	−0.885092	−	0.465415i	$-0.845905\pi$
$491$	−9.45632e6	−1.77018	−0.885092	+	0.465415i	$0.845905\pi$
$492$	0	0
$493$	1.19274e7	2.21019
$494$	0	0
$495$	336261.	0.0616827
$496$	0	0
$497$	−783053.	−0.142200
$498$	0	0
$499$	−1.64467e6	−0.295684	−0.147842	−	0.989011i	$-0.547233\pi$
$499$	−1.64467e6	−0.295684	−0.147842	+	0.989011i	$0.547233\pi$
$500$	0	0
$501$	6.89821e6	1.22784
$502$	0	0
$503$	−1.04146e7	−1.83536	−0.917682	−	0.397315i	$-0.869942\pi$
$503$	−1.04146e7	−1.83536	−0.917682	+	0.397315i	$0.869942\pi$
$504$	0	0
$505$	−1.15593e6	−0.201699
$506$	0	0
$507$	1.17972e7	2.03825
$508$	0	0
$509$	−8.06523e6	−1.37982	−0.689910	−	0.723896i	$-0.742350\pi$
$509$	−8.06523e6	−1.37982	−0.689910	+	0.723896i	$0.742350\pi$
$510$	0	0
$511$	−2.03582e6	−0.344895
$512$	0	0
$513$	−7.58965e6	−1.27329
$514$	0	0
$515$	−3.91043e6	−0.649691
$516$	0	0
$517$	−413007.	−0.0679565
$518$	0	0
$519$	4.60931e6	0.751134
$520$	0	0
$521$	−7.28702e6	−1.17613	−0.588065	−	0.808814i	$-0.700110\pi$
$521$	−7.28702e6	−1.17613	−0.588065	+	0.808814i	$0.700110\pi$
$522$	0	0
$523$	5.73656e6	0.917059	0.458530	−	0.888679i	$-0.348376\pi$
$523$	5.73656e6	0.917059	0.458530	+	0.888679i	$0.348376\pi$
$524$	0	0
$525$	282824.	0.0447835
$526$	0	0
$527$	−306268.	−0.0480370
$528$	0	0
$529$	−6.32234e6	−0.982288
$530$	0	0
$531$	−1.51479e6	−0.233140
$532$	0	0
$533$	−5.47631e6	−0.834968
$534$	0	0
$535$	−3.61625e6	−0.546228
$536$	0	0
$537$	6.28493e6	0.940513
$538$	0	0
$539$	1.84571e6	0.273647
$540$	0	0
$541$	5.90660e6	0.867650	0.433825	−	0.900997i	$-0.357164\pi$
$541$	5.90660e6	0.867650	0.433825	+	0.900997i	$0.357164\pi$
$542$	0	0
$543$	6.95211e6	1.01185
$544$	0	0
$545$	−387295.	−0.0558536
$546$	0	0
$547$	−5.14720e6	−0.735534	−0.367767	−	0.929918i	$-0.619878\pi$
$547$	−5.14720e6	−0.735534	−0.367767	+	0.929918i	$0.619878\pi$
$548$	0	0
$549$	3.81049e6	0.539572
$550$	0	0
$551$	−1.01382e7	−1.42259
$552$	0	0
$553$	2.00839e6	0.279278
$554$	0	0
$555$	1.70608e6	0.235107
$556$	0	0
$557$	−9.30363e6	−1.27062	−0.635308	−	0.772259i	$-0.719127\pi$
$557$	−9.30363e6	−1.27062	−0.635308	+	0.772259i	$0.719127\pi$
$558$	0	0
$559$	1.36642e6	0.184950
$560$	0	0
$561$	3.05066e6	0.409249
$562$	0	0
$563$	−1.00562e7	−1.33709	−0.668547	−	0.743670i	$-0.733083\pi$
$563$	−1.00562e7	−1.33709	−0.668547	+	0.743670i	$0.733083\pi$
$564$	0	0
$565$	187575.	0.0247203
$566$	0	0
$567$	−775614.	−0.101318
$568$	0	0
$569$	933127.	0.120826	0.0604130	−	0.998173i	$-0.480758\pi$
$569$	933127.	0.120826	0.0604130	+	0.998173i	$0.480758\pi$
$570$	0	0
$571$	1.07508e7	1.37990	0.689952	−	0.723855i	$-0.257632\pi$
$571$	1.07508e7	1.37990	0.689952	+	0.723855i	$0.257632\pi$
$572$	0	0
$573$	569442.	0.0724541
$574$	0	0
$575$	211027.	0.0266176
$576$	0	0
$577$	−387934.	−0.0485086	−0.0242543	−	0.999706i	$-0.507721\pi$
$577$	−387934.	−0.0485086	−0.0242543	+	0.999706i	$0.507721\pi$
$578$	0	0
$579$	−1.03256e7	−1.28002
$580$	0	0
$581$	−340036.	−0.0417912
$582$	0	0
$583$	−683846.	−0.0833273
$584$	0	0
$585$	3.28668e6	0.397071
$586$	0	0
$587$	−1.14459e7	−1.37105	−0.685525	−	0.728049i	$-0.740427\pi$
$587$	−1.14459e7	−1.37105	−0.685525	+	0.728049i	$0.740427\pi$
$588$	0	0
$589$	260324.	0.0309191
$590$	0	0
$591$	−3.21081e6	−0.378134
$592$	0	0
$593$	8.28060e6	0.966997	0.483499	−	0.875345i	$-0.339366\pi$
$593$	8.28060e6	0.966997	0.483499	+	0.875345i	$0.339366\pi$
$594$	0	0
$595$	−2.16342e6	−0.250523
$596$	0	0
$597$	−9.44091e6	−1.08412
$598$	0	0
$599$	−133370.	−0.0151876	−0.00759382	−	0.999971i	$-0.502417\pi$
$599$	−133370.	−0.0151876	−0.00759382	+	0.999971i	$0.502417\pi$
$600$	0	0
$601$	−8.78786e6	−0.992424	−0.496212	−	0.868201i	$-0.665276\pi$
$601$	−8.78786e6	−0.992424	−0.496212	+	0.868201i	$0.665276\pi$
$602$	0	0
$603$	1.10750e6	0.124036
$604$	0	0
$605$	366025.	0.0406558
$606$	0	0
$607$	−1.50026e7	−1.65271	−0.826354	−	0.563151i	$-0.809589\pi$
$607$	−1.50026e7	−1.65271	−0.826354	+	0.563151i	$0.809589\pi$
$608$	0	0
$609$	−2.45809e6	−0.268568
$610$	0	0
$611$	−4.03681e6	−0.437457
$612$	0	0
$613$	−1.21629e7	−1.30733	−0.653664	−	0.756785i	$-0.726769\pi$
$613$	−1.21629e7	−1.30733	−0.653664	+	0.756785i	$0.726769\pi$
$614$	0	0
$615$	1.32918e6	0.141709
$616$	0	0
$617$	2.77990e6	0.293979	0.146989	−	0.989138i	$-0.453042\pi$
$617$	2.77990e6	0.293979	0.146989	+	0.989138i	$0.453042\pi$
$618$	0	0
$619$	−2.84183e6	−0.298107	−0.149053	−	0.988829i	$-0.547623\pi$
$619$	−2.84183e6	−0.298107	−0.149053	+	0.988829i	$0.547623\pi$
$620$	0	0
$621$	−1.37303e6	−0.142874
$622$	0	0
$623$	−1.17989e6	−0.121793
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.59303e6	−0.263414
$628$	0	0
$629$	−1.30503e7	−1.31521
$630$	0	0
$631$	1.36584e7	1.36561	0.682806	−	0.730599i	$-0.260759\pi$
$631$	1.36584e7	1.36561	0.682806	+	0.730599i	$0.260759\pi$
$632$	0	0
$633$	−6.13473e6	−0.608535
$634$	0	0
$635$	7.06661e6	0.695468
$636$	0	0
$637$	1.80403e7	1.76155
$638$	0	0
$639$	2.20865e6	0.213981
$640$	0	0
$641$	1.47678e7	1.41961	0.709807	−	0.704396i	$-0.248782\pi$
$641$	1.47678e7	1.41961	0.709807	+	0.704396i	$0.248782\pi$
$642$	0	0
$643$	1.42372e7	1.35799	0.678995	−	0.734143i	$-0.262416\pi$
$643$	1.42372e7	1.35799	0.678995	+	0.734143i	$0.262416\pi$
$644$	0	0
$645$	−331650.	−0.0313892
$646$	0	0
$647$	−8.49593e6	−0.797903	−0.398951	−	0.916972i	$-0.630626\pi$
$647$	−8.49593e6	−0.797903	−0.398951	+	0.916972i	$0.630626\pi$
$648$	0	0
$649$	−1.64887e6	−0.153665
$650$	0	0
$651$	63117.9	0.00583714
$652$	0	0
$653$	−1.41649e7	−1.29996	−0.649979	−	0.759952i	$-0.725222\pi$
$653$	−1.41649e7	−1.29996	−0.649979	+	0.759952i	$0.725222\pi$
$654$	0	0
$655$	3.70762e6	0.337669
$656$	0	0
$657$	5.74216e6	0.518994
$658$	0	0
$659$	1.39876e7	1.25467	0.627335	−	0.778749i	$-0.284145\pi$
$659$	1.39876e7	1.25467	0.627335	+	0.778749i	$0.284145\pi$
$660$	0	0
$661$	−1.75134e7	−1.55908	−0.779538	−	0.626355i	$-0.784546\pi$
$661$	−1.75134e7	−1.55908	−0.779538	+	0.626355i	$0.784546\pi$
$662$	0	0
$663$	2.98178e7	2.63446
$664$	0	0
$665$	1.83888e6	0.161250
$666$	0	0
$667$	−1.83409e6	−0.159626
$668$	0	0
$669$	−294972.	−0.0254809
$670$	0	0
$671$	4.14777e6	0.355638
$672$	0	0
$673$	−1.13007e7	−0.961758	−0.480879	−	0.876787i	$-0.659682\pi$
$673$	−1.13007e7	−0.961758	−0.480879	+	0.876787i	$0.659682\pi$
$674$	0	0
$675$	−2.54157e6	−0.214706
$676$	0	0
$677$	−9.15138e6	−0.767388	−0.383694	−	0.923460i	$-0.625348\pi$
$677$	−9.15138e6	−0.767388	−0.383694	+	0.923460i	$0.625348\pi$
$678$	0	0
$679$	1.92604e6	0.160321
$680$	0	0
$681$	−5.96629e6	−0.492989
$682$	0	0
$683$	−3.66856e6	−0.300915	−0.150458	−	0.988616i	$-0.548075\pi$
$683$	−3.66856e6	−0.300915	−0.150458	+	0.988616i	$0.548075\pi$
$684$	0	0
$685$	2.36873e6	0.192881
$686$	0	0
$687$	6.50871e6	0.526142
$688$	0	0
$689$	−6.68405e6	−0.536404
$690$	0	0
$691$	−7.24875e6	−0.577521	−0.288761	−	0.957401i	$-0.593243\pi$
$691$	−7.24875e6	−0.577521	−0.288761	+	0.957401i	$0.593243\pi$
$692$	0	0
$693$	530091.	0.0419292
$694$	0	0
$695$	9.48193e6	0.744620
$696$	0	0
$697$	−1.01673e7	−0.792731
$698$	0	0
$699$	−2.87197e6	−0.222325
$700$	0	0
$701$	−1.67191e7	−1.28505	−0.642523	−	0.766267i	$-0.722112\pi$
$701$	−1.67191e7	−1.28505	−0.642523	+	0.766267i	$0.722112\pi$
$702$	0	0
$703$	1.10926e7	0.846537
$704$	0	0
$705$	979794.	0.0742441
$706$	0	0
$707$	−1.82224e6	−0.137106
$708$	0	0
$709$	9.11227e6	0.680786	0.340393	−	0.940283i	$-0.389440\pi$
$709$	9.11227e6	0.680786	0.340393	+	0.940283i	$0.389440\pi$
$710$	0	0
$711$	−5.66482e6	−0.420254
$712$	0	0
$713$	47095.0	0.00346937
$714$	0	0
$715$	3.57760e6	0.261714
$716$	0	0
$717$	−1.48981e7	−1.08226
$718$	0	0
$719$	−1.02074e7	−0.736368	−0.368184	−	0.929753i	$-0.620020\pi$
$719$	−1.02074e7	−0.736368	−0.368184	+	0.929753i	$0.620020\pi$
$720$	0	0
$721$	−6.16451e6	−0.441632
$722$	0	0
$723$	6.68732e6	0.475780
$724$	0	0
$725$	−3.39501e6	−0.239881
$726$	0	0
$727$	−3.96570e6	−0.278281	−0.139140	−	0.990273i	$-0.544434\pi$
$727$	−3.96570e6	−0.278281	−0.139140	+	0.990273i	$0.544434\pi$
$728$	0	0
$729$	1.35339e7	0.943201
$730$	0	0
$731$	2.53690e6	0.175594
$732$	0	0
$733$	−3.49709e6	−0.240407	−0.120203	−	0.992749i	$-0.538355\pi$
$733$	−3.49709e6	−0.240407	−0.120203	+	0.992749i	$0.538355\pi$
$734$	0	0
$735$	−4.37865e6	−0.298966
$736$	0	0
$737$	1.20553e6	0.0817539
$738$	0	0
$739$	−1.53549e7	−1.03428	−0.517138	−	0.855902i	$-0.673003\pi$
$739$	−1.53549e7	−1.03428	−0.517138	+	0.855902i	$0.673003\pi$
$740$	0	0
$741$	−2.53448e7	−1.69568
$742$	0	0
$743$	1.99982e7	1.32898	0.664491	−	0.747296i	$-0.268648\pi$
$743$	1.99982e7	1.32898	0.664491	+	0.747296i	$0.268648\pi$
$744$	0	0
$745$	−8.65832e6	−0.571535
$746$	0	0
$747$	959095.	0.0628869
$748$	0	0
$749$	−5.70076e6	−0.371303
$750$	0	0
$751$	1.89895e7	1.22861	0.614305	−	0.789069i	$-0.289436\pi$
$751$	1.89895e7	1.22861	0.614305	+	0.789069i	$0.289436\pi$
$752$	0	0
$753$	2.02062e7	1.29866
$754$	0	0
$755$	−1.64627e6	−0.105108
$756$	0	0
$757$	2.47833e7	1.57188	0.785939	−	0.618304i	$-0.212180\pi$
$757$	2.47833e7	1.57188	0.785939	+	0.618304i	$0.212180\pi$
$758$	0	0
$759$	−469101.	−0.0295571
$760$	0	0
$761$	2.62584e7	1.64364	0.821820	−	0.569747i	$-0.192959\pi$
$761$	2.62584e7	1.64364	0.821820	+	0.569747i	$0.192959\pi$
$762$	0	0
$763$	−610543.	−0.0379669
$764$	0	0
$765$	6.10207e6	0.376984
$766$	0	0
$767$	−1.61164e7	−0.989190
$768$	0	0
$769$	1.92359e7	1.17300	0.586500	−	0.809950i	$-0.300506\pi$
$769$	1.92359e7	1.17300	0.586500	+	0.809950i	$0.300506\pi$
$770$	0	0
$771$	−6.19066e6	−0.375060
$772$	0	0
$773$	−7.49718e6	−0.451283	−0.225642	−	0.974210i	$-0.572448\pi$
$773$	−7.49718e6	−0.451283	−0.225642	+	0.974210i	$0.572448\pi$
$774$	0	0
$775$	87175.8	0.00521365
$776$	0	0
$777$	2.68950e6	0.159816
$778$	0	0
$779$	8.64212e6	0.510242
$780$	0	0
$781$	2.40415e6	0.141038
$782$	0	0
$783$	2.20894e7	1.28760
$784$	0	0
$785$	9.18228e6	0.531835
$786$	0	0
$787$	1.48570e7	0.855053	0.427526	−	0.904003i	$-0.359385\pi$
$787$	1.48570e7	0.855053	0.427526	+	0.904003i	$0.359385\pi$
$788$	0	0
$789$	−2.53010e7	−1.44692
$790$	0	0
$791$	295698.	0.0168038
$792$	0	0
$793$	4.05412e7	2.28936
$794$	0	0
$795$	1.62232e6	0.0910370
$796$	0	0
$797$	−2.89236e7	−1.61290	−0.806450	−	0.591303i	$-0.798614\pi$
$797$	−2.89236e7	−1.61290	−0.806450	+	0.591303i	$0.798614\pi$
$798$	0	0
$799$	−7.49477e6	−0.415328
$800$	0	0
$801$	3.32797e6	0.183273
$802$	0	0
$803$	6.25043e6	0.342075
$804$	0	0
$805$	332669.	0.0180935
$806$	0	0
$807$	−8.39928e6	−0.454003
$808$	0	0
$809$	−1.97152e7	−1.05908	−0.529541	−	0.848285i	$-0.677636\pi$
$809$	−1.97152e7	−1.05908	−0.529541	+	0.848285i	$0.677636\pi$
$810$	0	0
$811$	−6.00642e6	−0.320674	−0.160337	−	0.987062i	$-0.551258\pi$
$811$	−6.00642e6	−0.320674	−0.160337	+	0.987062i	$0.551258\pi$
$812$	0	0
$813$	5.12727e6	0.272057
$814$	0	0
$815$	−1.20351e7	−0.634682
$816$	0	0
$817$	−2.15633e6	−0.113021
$818$	0	0
$819$	5.18121e6	0.269912
$820$	0	0
$821$	−3.12629e7	−1.61872	−0.809361	−	0.587312i	$-0.800186\pi$
$821$	−3.12629e7	−1.61872	−0.809361	+	0.587312i	$0.800186\pi$
$822$	0	0
$823$	1.43689e7	0.739474	0.369737	−	0.929137i	$-0.379448\pi$
$823$	1.43689e7	0.739474	0.369737	+	0.929137i	$0.379448\pi$
$824$	0	0
$825$	−868336.	−0.0444174
$826$	0	0
$827$	−3.13803e7	−1.59549	−0.797744	−	0.602997i	$-0.793973\pi$
$827$	−3.13803e7	−1.59549	−0.797744	+	0.602997i	$0.793973\pi$
$828$	0	0
$829$	1.74548e6	0.0882121	0.0441060	−	0.999027i	$-0.485956\pi$
$829$	1.74548e6	0.0882121	0.0441060	+	0.999027i	$0.485956\pi$
$830$	0	0
$831$	1.84785e7	0.928251
$832$	0	0
$833$	3.34938e7	1.67244
$834$	0	0
$835$	1.50194e7	0.745482
$836$	0	0
$837$	−567203.	−0.0279850
$838$	0	0
$839$	3.04376e7	1.49281	0.746406	−	0.665491i	$-0.231778\pi$
$839$	3.04376e7	1.49281	0.746406	+	0.665491i	$0.231778\pi$
$840$	0	0
$841$	8.99566e6	0.438574
$842$	0	0
$843$	−9.60901e6	−0.465704
$844$	0	0
$845$	2.56859e7	1.23752
$846$	0	0
$847$	577012.	0.0276361
$848$	0	0
$849$	−1.65022e7	−0.785726
$850$	0	0
$851$	2.00675e6	0.0949883
$852$	0	0
$853$	−2.05924e6	−0.0969022	−0.0484511	−	0.998826i	$-0.515429\pi$
$853$	−2.05924e6	−0.0969022	−0.0484511	+	0.998826i	$0.515429\pi$
$854$	0	0
$855$	−5.18668e6	−0.242647
$856$	0	0
$857$	2.90177e7	1.34962	0.674809	−	0.737993i	$-0.264226\pi$
$857$	2.90177e7	1.34962	0.674809	+	0.737993i	$0.264226\pi$
$858$	0	0
$859$	4.14336e7	1.91589	0.957943	−	0.286958i	$-0.0926440\pi$
$859$	4.14336e7	1.91589	0.957943	+	0.286958i	$0.0926440\pi$
$860$	0	0
$861$	2.09536e6	0.0963274
$862$	0	0
$863$	−3.45945e7	−1.58117	−0.790587	−	0.612349i	$-0.790225\pi$
$863$	−3.45945e7	−1.58117	−0.790587	+	0.612349i	$0.790225\pi$
$864$	0	0
$865$	1.00358e7	0.456051
$866$	0	0
$867$	3.90569e7	1.76462
$868$	0	0
$869$	−6.16624e6	−0.276994
$870$	0	0
$871$	1.17831e7	0.526275
$872$	0	0
$873$	−5.43254e6	−0.241250
$874$	0	0
$875$	615792.	0.0271903
$876$	0	0
$877$	7.03277e6	0.308764	0.154382	−	0.988011i	$-0.450661\pi$
$877$	7.03277e6	0.308764	0.154382	+	0.988011i	$0.450661\pi$
$878$	0	0
$879$	−1.12230e7	−0.489933
$880$	0	0
$881$	1.30800e7	0.567762	0.283881	−	0.958860i	$-0.408378\pi$
$881$	1.30800e7	0.567762	0.283881	+	0.958860i	$0.408378\pi$
$882$	0	0
$883$	1.87695e6	0.0810125	0.0405062	−	0.999179i	$-0.487103\pi$
$883$	1.87695e6	0.0810125	0.0405062	+	0.999179i	$0.487103\pi$
$884$	0	0
$885$	3.91169e6	0.167883
$886$	0	0
$887$	−5.77221e6	−0.246339	−0.123170	−	0.992386i	$-0.539306\pi$
$887$	−5.77221e6	−0.246339	−0.123170	+	0.992386i	$0.539306\pi$
$888$	0	0
$889$	1.11400e7	0.472749
$890$	0	0
$891$	2.38132e6	0.100490
$892$	0	0
$893$	6.37046e6	0.267327
$894$	0	0
$895$	1.36842e7	0.571032
$896$	0	0
$897$	−4.58509e6	−0.190268
$898$	0	0
$899$	−757665.	−0.0312664
$900$	0	0
$901$	−1.24096e7	−0.509269
$902$	0	0
$903$	−522822.	−0.0213370
$904$	0	0
$905$	1.51368e7	0.614345
$906$	0	0
$907$	−3.37177e7	−1.36094	−0.680472	−	0.732774i	$-0.738225\pi$
$907$	−3.37177e7	−1.36094	−0.680472	+	0.732774i	$0.738225\pi$
$908$	0	0
$909$	5.13976e6	0.206316
$910$	0	0
$911$	−1.06739e6	−0.0426116	−0.0213058	−	0.999773i	$-0.506782\pi$
$911$	−1.06739e6	−0.0426116	−0.0213058	+	0.999773i	$0.506782\pi$
$912$	0	0
$913$	1.04399e6	0.0414495
$914$	0	0
$915$	−9.83994e6	−0.388543
$916$	0	0
$917$	5.84479e6	0.229533
$918$	0	0
$919$	3.23736e7	1.26445	0.632225	−	0.774785i	$-0.282142\pi$
$919$	3.23736e7	1.26445	0.632225	+	0.774785i	$0.282142\pi$
$920$	0	0
$921$	2.78932e6	0.108355
$922$	0	0
$923$	2.34987e7	0.907903
$924$	0	0
$925$	3.71463e6	0.142745
$926$	0	0
$927$	1.73874e7	0.664563
$928$	0	0
$929$	−4.75549e7	−1.80782	−0.903911	−	0.427721i	$-0.859317\pi$
$929$	−4.75549e7	−1.80782	−0.903911	+	0.427721i	$0.859317\pi$
$930$	0	0
$931$	−2.84693e7	−1.07647
$932$	0	0
$933$	−2.21740e7	−0.833952
$934$	0	0
$935$	6.64220e6	0.248475
$936$	0	0
$937$	−5.71529e6	−0.212662	−0.106331	−	0.994331i	$-0.533910\pi$
$937$	−5.71529e6	−0.212662	−0.106331	+	0.994331i	$0.533910\pi$
$938$	0	0
$939$	−1.18463e7	−0.438448
$940$	0	0
$941$	−4.66044e7	−1.71575	−0.857873	−	0.513862i	$-0.828214\pi$
$941$	−4.66044e7	−1.71575	−0.857873	+	0.513862i	$0.828214\pi$
$942$	0	0
$943$	1.56344e6	0.0572533
$944$	0	0
$945$	−4.00661e6	−0.145948
$946$	0	0
$947$	1.59416e7	0.577641	0.288820	−	0.957383i	$-0.406737\pi$
$947$	1.59416e7	0.577641	0.288820	+	0.957383i	$0.406737\pi$
$948$	0	0
$949$	6.10930e7	2.20204
$950$	0	0
$951$	−2.35547e7	−0.844551
$952$	0	0
$953$	−2.31743e6	−0.0826560	−0.0413280	−	0.999146i	$-0.513159\pi$
$953$	−2.31743e6	−0.0826560	−0.0413280	+	0.999146i	$0.513159\pi$
$954$	0	0
$955$	1.23984e6	0.0439905
$956$	0	0
$957$	7.54691e6	0.266372
$958$	0	0
$959$	3.73412e6	0.131112
$960$	0	0
$961$	−2.86097e7	−0.999320
$962$	0	0
$963$	1.60794e7	0.558732
$964$	0	0
$965$	−2.24818e7	−0.777165
$966$	0	0
$967$	−3.54405e7	−1.21880	−0.609401	−	0.792862i	$-0.708590\pi$
$967$	−3.54405e7	−1.21880	−0.609401	+	0.792862i	$0.708590\pi$
$968$	0	0
$969$	−4.70552e7	−1.60990
$970$	0	0
$971$	−4.41218e7	−1.50178	−0.750888	−	0.660429i	$-0.770374\pi$
$971$	−4.41218e7	−1.50178	−0.750888	+	0.660429i	$0.770374\pi$
$972$	0	0
$973$	1.49476e7	0.506161
$974$	0	0
$975$	−8.48730e6	−0.285929
$976$	0	0
$977$	−1.53807e7	−0.515513	−0.257757	−	0.966210i	$-0.582983\pi$
$977$	−1.53807e7	−0.515513	−0.257757	+	0.966210i	$0.582983\pi$
$978$	0	0
$979$	3.62255e6	0.120797
$980$	0	0
$981$	1.72208e6	0.0571321
$982$	0	0
$983$	3.82728e7	1.26330	0.631650	−	0.775253i	$-0.282378\pi$
$983$	3.82728e7	1.26330	0.631650	+	0.775253i	$0.282378\pi$
$984$	0	0
$985$	−6.99088e6	−0.229584
$986$	0	0
$987$	1.54457e6	0.0504679
$988$	0	0
$989$	−390100.	−0.0126819
$990$	0	0
$991$	−3.12166e6	−0.100972	−0.0504860	−	0.998725i	$-0.516077\pi$
$991$	−3.12166e6	−0.100972	−0.0504860	+	0.998725i	$0.516077\pi$
$992$	0	0
$993$	1.82204e7	0.586387
$994$	0	0
$995$	−2.05557e7	−0.658224
$996$	0	0
$997$	1.26050e7	0.401610	0.200805	−	0.979631i	$-0.435644\pi$
$997$	1.26050e7	0.401610	0.200805	+	0.979631i	$0.435644\pi$
$998$	0	0
$999$	−2.41690e7	−0.766205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.6.a.a.1.2	✓	2
4.3	odd	2		880.6.a.h.1.1		2
5.2	odd	4		1100.6.b.b.749.2		4
5.3	odd	4		1100.6.b.b.749.3		4
5.4	even	2		1100.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.a.a.1.2	✓	2	1.1	even	1	trivial
880.6.a.h.1.1		2	4.3	odd	2
1100.6.a.c.1.1		2	5.4	even	2
1100.6.b.b.749.2		4	5.2	odd	4
1100.6.b.b.749.3		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)

\(f(q)\)	\(=\)	\(q+11.4821 q^{3} +25.0000 q^{5} +39.4107 q^{7} -111.161 q^{9} -121.000 q^{11} -1182.68 q^{13} +287.053 q^{15} -2195.77 q^{17} +1866.37 q^{19} +452.519 q^{21} +337.644 q^{23} +625.000 q^{25} -4066.52 q^{27} -5432.02 q^{29} +139.481 q^{31} -1389.34 q^{33} +985.267 q^{35} +5943.41 q^{37} -13579.7 q^{39} +4630.43 q^{41} -1155.36 q^{43} -2779.01 q^{45} +3413.28 q^{47} -15253.8 q^{49} -25212.1 q^{51} +5651.62 q^{53} -3025.00 q^{55} +21430.0 q^{57} +13627.0 q^{59} -34279.1 q^{61} -4380.91 q^{63} -29567.0 q^{65} -9963.03 q^{67} +3876.87 q^{69} -19869.0 q^{71} -51656.5 q^{73} +7176.33 q^{75} -4768.69 q^{77} +50960.7 q^{79} -19680.3 q^{81} -8628.02 q^{83} -54894.2 q^{85} -62371.2 q^{87} -29938.4 q^{89} -46610.2 q^{91} +1601.54 q^{93} +46659.4 q^{95} +48871.1 q^{97} +13450.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 19 q^{3} + 50 q^{5} - 131 q^{7} + 575 q^{9} - 242 q^{11} - 1610 q^{13} - 475 q^{15} - 321 q^{17} + 3439 q^{19} + 5647 q^{21} + 4536 q^{23} + 1250 q^{25} - 17575 q^{27} - 5031 q^{29} - 4463 q^{31}+ \cdots - 69575 q^{99}+O(q^{100}) \) 2 * q - 19 * q^3 + 50 * q^5 - 131 * q^7 + 575 * q^9 - 242 * q^11 - 1610 * q^13 - 475 * q^15 - 321 * q^17 + 3439 * q^19 + 5647 * q^21 + 4536 * q^23 + 1250 * q^25 - 17575 * q^27 - 5031 * q^29 - 4463 * q^31 + 2299 * q^33 - 3275 * q^35 - 2423 * q^37 - 554 * q^39 + 19668 * q^41 - 14900 * q^43 + 14375 * q^45 - 23052 * q^47 - 3021 * q^49 - 82359 * q^51 - 7203 * q^53 - 6050 * q^55 - 26507 * q^57 + 50838 * q^59 - 44177 * q^61 - 121310 * q^63 - 40250 * q^65 - 10610 * q^67 - 124098 * q^69 - 1089 * q^71 - 92654 * q^73 - 11875 * q^75 + 15851 * q^77 - 16334 * q^79 + 225350 * q^81 + 88410 * q^83 - 8025 * q^85 - 74595 * q^87 - 80817 * q^89 + 26210 * q^91 + 141895 * q^93 + 85975 * q^95 + 102694 * q^97 - 69575 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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