LMFDB - Embedded newform 1100.6.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(1,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, 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("1100.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.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:	$176.422201794$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2261 x^{10} + 1852999 x^{8} - 695664339 x^{6} + 119577150000 x^{4} - 7602696450000 x^{2} + 69155856000000 $ x^12 - 2261x^10 + 1852999x^8 - 695664339x^6 + 119577150000x^4 - 7602696450000*x^2 + 69155856000000
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-22.2039$ of defining polynomial
Character	$\chi$	$=$	1100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-22.2039 q^{3} +145.246 q^{7} +250.011 q^{9} -121.000 q^{11} -141.152 q^{13} +406.646 q^{17} +1523.82 q^{19} -3225.01 q^{21} -2335.80 q^{23} -155.675 q^{27} -3360.56 q^{29} +2203.15 q^{31} +2686.67 q^{33} -370.951 q^{37} +3134.11 q^{39} -16177.7 q^{41} -11748.8 q^{43} -15065.4 q^{47} +4289.30 q^{49} -9029.11 q^{51} +36849.8 q^{53} -33834.7 q^{57} +32061.1 q^{59} -38914.6 q^{61} +36313.0 q^{63} +30507.3 q^{67} +51863.8 q^{69} +20260.6 q^{71} +32793.2 q^{73} -17574.7 q^{77} -42482.3 q^{79} -57296.1 q^{81} -5067.48 q^{83} +74617.4 q^{87} +58149.8 q^{89} -20501.7 q^{91} -48918.5 q^{93} +120536. q^{97} -30251.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 1606 q^{9} - 1452 q^{11} - 2100 q^{19} - 916 q^{21} + 9932 q^{29} - 23382 q^{31} - 16400 q^{39} - 16700 q^{41} + 4888 q^{49} - 59376 q^{51} - 90722 q^{59} + 27700 q^{61} - 31118 q^{69} - 26758 q^{71}+ \cdots - 194326 q^{99}+O(q^{100}) $ 12 * q + 1606 * q^9 - 1452 * q^11 - 2100 * q^19 - 916 * q^21 + 9932 * q^29 - 23382 * q^31 - 16400 * q^39 - 16700 * q^41 + 4888 * q^49 - 59376 * q^51 - 90722 * q^59 + 27700 * q^61 - 31118 * q^69 - 26758 * q^71 - 84852 * q^79 + 224296 * q^81 + 12598 * q^89 + 64368 * q^91 - 194326 * 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
$3$	−22.2039	−1.42438	−0.712189	−	0.701988i	$-0.752296\pi$
$3$	−22.2039	−1.42438	−0.712189	+	0.701988i	$0.752296\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	145.246	1.12036	0.560181	−	0.828370i	$-0.310732\pi$
$7$	145.246	1.12036	0.560181	+	0.828370i	$0.310732\pi$
$8$	0	0
$9$	250.011	1.02885
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−141.152	−0.231647	−0.115824	−	0.993270i	$-0.536951\pi$
$13$	−141.152	−0.231647	−0.115824	+	0.993270i	$0.536951\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	406.646	0.341267	0.170634	−	0.985335i	$-0.445419\pi$
$17$	406.646	0.341267	0.170634	+	0.985335i	$0.445419\pi$
$18$	0	0
$19$	1523.82	0.968389	0.484195	−	0.874960i	$-0.339113\pi$
$19$	1523.82	0.968389	0.484195	+	0.874960i	$0.339113\pi$
$20$	0	0
$21$	−3225.01	−1.59582
$22$	0	0
$23$	−2335.80	−0.920696	−0.460348	−	0.887739i	$-0.652275\pi$
$23$	−2335.80	−0.920696	−0.460348	+	0.887739i	$0.652275\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−155.675	−0.0410968
$28$	0	0
$29$	−3360.56	−0.742022	−0.371011	−	0.928628i	$-0.620989\pi$
$29$	−3360.56	−0.742022	−0.371011	+	0.928628i	$0.620989\pi$
$30$	0	0
$31$	2203.15	0.411756	0.205878	−	0.978578i	$-0.433995\pi$
$31$	2203.15	0.411756	0.205878	+	0.978578i	$0.433995\pi$
$32$	0	0
$33$	2686.67	0.429466
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−370.951	−0.0445464	−0.0222732	−	0.999752i	$-0.507090\pi$
$37$	−370.951	−0.0445464	−0.0222732	+	0.999752i	$0.507090\pi$
$38$	0	0
$39$	3134.11	0.329953
$40$	0	0
$41$	−16177.7	−1.50299	−0.751496	−	0.659738i	$-0.770667\pi$
$41$	−16177.7	−1.50299	−0.751496	+	0.659738i	$0.770667\pi$
$42$	0	0
$43$	−11748.8	−0.968999	−0.484500	−	0.874792i	$-0.660998\pi$
$43$	−11748.8	−0.968999	−0.484500	+	0.874792i	$0.660998\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−15065.4	−0.994802	−0.497401	−	0.867521i	$-0.665712\pi$
$47$	−15065.4	−0.994802	−0.497401	+	0.867521i	$0.665712\pi$
$48$	0	0
$49$	4289.30	0.255209
$50$	0	0
$51$	−9029.11	−0.486093
$52$	0	0
$53$	36849.8	1.80196	0.900981	−	0.433859i	$-0.142848\pi$
$53$	36849.8	1.80196	0.900981	+	0.433859i	$0.142848\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−33834.7	−1.37935
$58$	0	0
$59$	32061.1	1.19908	0.599539	−	0.800345i	$-0.295350\pi$
$59$	32061.1	1.19908	0.599539	+	0.800345i	$0.295350\pi$
$60$	0	0
$61$	−38914.6	−1.33902	−0.669511	−	0.742802i	$-0.733496\pi$
$61$	−38914.6	−1.33902	−0.669511	+	0.742802i	$0.733496\pi$
$62$	0	0
$63$	36313.0	1.15269
$64$	0	0
$65$	0	0
$66$	0	0
$67$	30507.3	0.830266	0.415133	−	0.909761i	$-0.363735\pi$
$67$	30507.3	0.830266	0.415133	+	0.909761i	$0.363735\pi$
$68$	0	0
$69$	51863.8	1.31142
$70$	0	0
$71$	20260.6	0.476986	0.238493	−	0.971144i	$-0.423347\pi$
$71$	20260.6	0.476986	0.238493	+	0.971144i	$0.423347\pi$
$72$	0	0
$73$	32793.2	0.720238	0.360119	−	0.932906i	$-0.382736\pi$
$73$	32793.2	0.720238	0.360119	+	0.932906i	$0.382736\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−17574.7	−0.337802
$78$	0	0
$79$	−42482.3	−0.765843	−0.382922	−	0.923781i	$-0.625082\pi$
$79$	−42482.3	−0.765843	−0.382922	+	0.923781i	$0.625082\pi$
$80$	0	0
$81$	−57296.1	−0.970315
$82$	0	0
$83$	−5067.48	−0.0807415	−0.0403707	−	0.999185i	$-0.512854\pi$
$83$	−5067.48	−0.0807415	−0.0403707	+	0.999185i	$0.512854\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	74617.4	1.05692
$88$	0	0
$89$	58149.8	0.778168	0.389084	−	0.921202i	$-0.372792\pi$
$89$	58149.8	0.778168	0.389084	+	0.921202i	$0.372792\pi$
$90$	0	0
$91$	−20501.7	−0.259529
$92$	0	0
$93$	−48918.5	−0.586497
$94$	0	0
$95$	0	0
$96$	0	0
$97$	120536.	1.30073	0.650367	−	0.759620i	$-0.274615\pi$
$97$	120536.	1.30073	0.650367	+	0.759620i	$0.274615\pi$
$98$	0	0
$99$	−30251.3	−0.310211
$100$	0	0
$101$	109585.	1.06893	0.534463	−	0.845192i	$-0.320514\pi$
$101$	109585.	1.06893	0.534463	+	0.845192i	$0.320514\pi$
$102$	0	0
$103$	23480.5	0.218079	0.109040	−	0.994037i	$-0.465222\pi$
$103$	23480.5	0.218079	0.109040	+	0.994037i	$0.465222\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	136967.	1.15653	0.578266	−	0.815848i	$-0.303729\pi$
$107$	136967.	1.15653	0.578266	+	0.815848i	$0.303729\pi$
$108$	0	0
$109$	−172412.	−1.38996	−0.694980	−	0.719029i	$-0.744587\pi$
$109$	−172412.	−1.38996	−0.694980	+	0.719029i	$0.744587\pi$
$110$	0	0
$111$	8236.55	0.0634509
$112$	0	0
$113$	231311.	1.70412	0.852060	−	0.523444i	$-0.175353\pi$
$113$	231311.	1.70412	0.852060	+	0.523444i	$0.175353\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−35289.5	−0.238331
$118$	0	0
$119$	59063.6	0.382342
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	359207.	2.14083
$124$	0	0
$125$	0	0
$126$	0	0
$127$	66234.4	0.364397	0.182198	−	0.983262i	$-0.441679\pi$
$127$	66234.4	0.364397	0.182198	+	0.983262i	$0.441679\pi$
$128$	0	0
$129$	260869.	1.38022
$130$	0	0
$131$	−173181.	−0.881701	−0.440851	−	0.897581i	$-0.645323\pi$
$131$	−173181.	−0.881701	−0.440851	+	0.897581i	$0.645323\pi$
$132$	0	0
$133$	221328.	1.08495
$134$	0	0
$135$	0	0
$136$	0	0
$137$	206649.	0.940658	0.470329	−	0.882491i	$-0.344135\pi$
$137$	206649.	0.940658	0.470329	+	0.882491i	$0.344135\pi$
$138$	0	0
$139$	−292347.	−1.28340	−0.641699	−	0.766956i	$-0.721770\pi$
$139$	−292347.	−1.28340	−0.641699	+	0.766956i	$0.721770\pi$
$140$	0	0
$141$	334510.	1.41697
$142$	0	0
$143$	17079.3	0.0698443
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−95239.1	−0.363514
$148$	0	0
$149$	−24029.1	−0.0886691	−0.0443345	−	0.999017i	$-0.514117\pi$
$149$	−24029.1	−0.0886691	−0.0443345	+	0.999017i	$0.514117\pi$
$150$	0	0
$151$	−158677.	−0.566332	−0.283166	−	0.959071i	$-0.591385\pi$
$151$	−158677.	−0.566332	−0.283166	+	0.959071i	$0.591385\pi$
$152$	0	0
$153$	101666.	0.351113
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−231069.	−0.748155	−0.374078	−	0.927397i	$-0.622041\pi$
$157$	−231069.	−0.748155	−0.374078	+	0.927397i	$0.622041\pi$
$158$	0	0
$159$	−818208.	−2.56667
$160$	0	0
$161$	−339265.	−1.03151
$162$	0	0
$163$	−148771.	−0.438579	−0.219290	−	0.975660i	$-0.570374\pi$
$163$	−148771.	−0.438579	−0.219290	+	0.975660i	$0.570374\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	250798.	0.695879	0.347939	−	0.937517i	$-0.386882\pi$
$167$	250798.	0.695879	0.347939	+	0.937517i	$0.386882\pi$
$168$	0	0
$169$	−351369.	−0.946340
$170$	0	0
$171$	380972.	0.996330
$172$	0	0
$173$	−261843.	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$173$	−261843.	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−711879.	−1.70794
$178$	0	0
$179$	−510446.	−1.19074	−0.595370	−	0.803451i	$-0.702995\pi$
$179$	−510446.	−1.19074	−0.595370	+	0.803451i	$0.702995\pi$
$180$	0	0
$181$	272631.	0.618557	0.309278	−	0.950972i	$-0.399913\pi$
$181$	272631.	0.618557	0.309278	+	0.950972i	$0.399913\pi$
$182$	0	0
$183$	864053.	1.90727
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−49204.2	−0.102896
$188$	0	0
$189$	−22611.1	−0.0460433
$190$	0	0
$191$	177582.	0.352220	0.176110	−	0.984370i	$-0.443648\pi$
$191$	177582.	0.352220	0.176110	+	0.984370i	$0.443648\pi$
$192$	0	0
$193$	−977580.	−1.88912	−0.944559	−	0.328343i	$-0.893510\pi$
$193$	−977580.	−1.88912	−0.944559	+	0.328343i	$0.893510\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	845804.	1.55276	0.776380	−	0.630265i	$-0.217054\pi$
$197$	845804.	1.55276	0.776380	+	0.630265i	$0.217054\pi$
$198$	0	0
$199$	−308607.	−0.552425	−0.276212	−	0.961097i	$-0.589079\pi$
$199$	−308607.	−0.552425	−0.276212	+	0.961097i	$0.589079\pi$
$200$	0	0
$201$	−677380.	−1.18261
$202$	0	0
$203$	−488107.	−0.831333
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−583976.	−0.947260
$208$	0	0
$209$	−184382.	−0.291980
$210$	0	0
$211$	−188506.	−0.291486	−0.145743	−	0.989322i	$-0.546557\pi$
$211$	−188506.	−0.291486	−0.145743	+	0.989322i	$0.546557\pi$
$212$	0	0
$213$	−449862.	−0.679408
$214$	0	0
$215$	0	0
$216$	0	0
$217$	319998.	0.461316
$218$	0	0
$219$	−728135.	−1.02589
$220$	0	0
$221$	−57398.7	−0.0790536
$222$	0	0
$223$	−476529.	−0.641693	−0.320846	−	0.947131i	$-0.603967\pi$
$223$	−476529.	−0.641693	−0.320846	+	0.947131i	$0.603967\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.31691e6	−1.69626	−0.848129	−	0.529790i	$-0.822271\pi$
$227$	−1.31691e6	−1.69626	−0.848129	+	0.529790i	$0.822271\pi$
$228$	0	0
$229$	1.26942e6	1.59961	0.799807	−	0.600258i	$-0.204935\pi$
$229$	1.26942e6	1.59961	0.799807	+	0.600258i	$0.204935\pi$
$230$	0	0
$231$	390227.	0.481157
$232$	0	0
$233$	−1.30982e6	−1.58060	−0.790298	−	0.612723i	$-0.790074\pi$
$233$	−1.30982e6	−1.58060	−0.790298	+	0.612723i	$0.790074\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	943270.	1.09085
$238$	0	0
$239$	−1.07615e6	−1.21865	−0.609325	−	0.792920i	$-0.708560\pi$
$239$	−1.07615e6	−1.21865	−0.609325	+	0.792920i	$0.708560\pi$
$240$	0	0
$241$	−623008.	−0.690957	−0.345478	−	0.938427i	$-0.612283\pi$
$241$	−623008.	−0.690957	−0.345478	+	0.938427i	$0.612283\pi$
$242$	0	0
$243$	1.31002e6	1.42319
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−215090.	−0.224325
$248$	0	0
$249$	112518.	0.115006
$250$	0	0
$251$	1.38338e6	1.38598	0.692988	−	0.720949i	$-0.256294\pi$
$251$	1.38338e6	1.38598	0.692988	+	0.720949i	$0.256294\pi$
$252$	0	0
$253$	282632.	0.277600
$254$	0	0
$255$	0	0
$256$	0	0
$257$	77981.9	0.0736480	0.0368240	−	0.999322i	$-0.488276\pi$
$257$	77981.9	0.0736480	0.0368240	+	0.999322i	$0.488276\pi$
$258$	0	0
$259$	−53879.1	−0.0499081
$260$	0	0
$261$	−840178.	−0.763432
$262$	0	0
$263$	−737014.	−0.657032	−0.328516	−	0.944498i	$-0.606549\pi$
$263$	−737014.	−0.657032	−0.328516	+	0.944498i	$0.606549\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.29115e6	−1.10841
$268$	0	0
$269$	−465456.	−0.392192	−0.196096	−	0.980585i	$-0.562826\pi$
$269$	−465456.	−0.392192	−0.196096	+	0.980585i	$0.562826\pi$
$270$	0	0
$271$	−14432.1	−0.0119373	−0.00596865	−	0.999982i	$-0.501900\pi$
$271$	−14432.1	−0.0119373	−0.00596865	+	0.999982i	$0.501900\pi$
$272$	0	0
$273$	455216.	0.369667
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−332445.	−0.260328	−0.130164	−	0.991493i	$-0.541550\pi$
$277$	−332445.	−0.260328	−0.130164	+	0.991493i	$0.541550\pi$
$278$	0	0
$279$	550813.	0.423637
$280$	0	0
$281$	−1.39699e6	−1.05542	−0.527712	−	0.849423i	$-0.676950\pi$
$281$	−1.39699e6	−1.05542	−0.527712	+	0.849423i	$0.676950\pi$
$282$	0	0
$283$	−1.00599e6	−0.746670	−0.373335	−	0.927697i	$-0.621786\pi$
$283$	−1.00599e6	−0.746670	−0.373335	+	0.927697i	$0.621786\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.34974e6	−1.68389
$288$	0	0
$289$	−1.25450e6	−0.883537
$290$	0	0
$291$	−2.67637e6	−1.85274
$292$	0	0
$293$	562267.	0.382625	0.191313	−	0.981529i	$-0.438726\pi$
$293$	562267.	0.382625	0.191313	+	0.981529i	$0.438726\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18836.6	0.0123912
$298$	0	0
$299$	329702.	0.213277
$300$	0	0
$301$	−1.70647e6	−1.08563
$302$	0	0
$303$	−2.43321e6	−1.52255
$304$	0	0
$305$	0	0
$306$	0	0
$307$	703593.	0.426065	0.213032	−	0.977045i	$-0.431666\pi$
$307$	703593.	0.426065	0.213032	+	0.977045i	$0.431666\pi$
$308$	0	0
$309$	−521358.	−0.310628
$310$	0	0
$311$	1.87852e6	1.10133	0.550663	−	0.834728i	$-0.314375\pi$
$311$	1.87852e6	1.10133	0.550663	+	0.834728i	$0.314375\pi$
$312$	0	0
$313$	−1.76874e6	−1.02048	−0.510239	−	0.860033i	$-0.670443\pi$
$313$	−1.76874e6	−1.02048	−0.510239	+	0.860033i	$0.670443\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.21492e6	1.23797	0.618986	−	0.785402i	$-0.287544\pi$
$317$	2.21492e6	1.23797	0.618986	+	0.785402i	$0.287544\pi$
$318$	0	0
$319$	406628.	0.223728
$320$	0	0
$321$	−3.04120e6	−1.64734
$322$	0	0
$323$	619656.	0.330479
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.82822e6	1.97983
$328$	0	0
$329$	−2.18819e6	−1.11454
$330$	0	0
$331$	−2.75290e6	−1.38108	−0.690542	−	0.723293i	$-0.742628\pi$
$331$	−2.75290e6	−1.38108	−0.690542	+	0.723293i	$0.742628\pi$
$332$	0	0
$333$	−92742.0	−0.0458317
$334$	0	0
$335$	0	0
$336$	0	0
$337$	70335.7	0.0337366	0.0168683	−	0.999858i	$-0.494630\pi$
$337$	70335.7	0.0337366	0.0168683	+	0.999858i	$0.494630\pi$
$338$	0	0
$339$	−5.13599e6	−2.42731
$340$	0	0
$341$	−266581.	−0.124149
$342$	0	0
$343$	−1.81814e6	−0.834435
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−171567.	−0.0764910	−0.0382455	−	0.999268i	$-0.512177\pi$
$347$	−171567.	−0.0764910	−0.0382455	+	0.999268i	$0.512177\pi$
$348$	0	0
$349$	1.31108e6	0.576189	0.288094	−	0.957602i	$-0.406978\pi$
$349$	1.31108e6	0.576189	0.288094	+	0.957602i	$0.406978\pi$
$350$	0	0
$351$	21973.7	0.00951997
$352$	0	0
$353$	2.22601e6	0.950803	0.475401	−	0.879769i	$-0.342303\pi$
$353$	2.22601e6	0.950803	0.475401	+	0.879769i	$0.342303\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.31144e6	−0.544600
$358$	0	0
$359$	−667547.	−0.273367	−0.136683	−	0.990615i	$-0.543644\pi$
$359$	−667547.	−0.273367	−0.136683	+	0.990615i	$0.543644\pi$
$360$	0	0
$361$	−154068.	−0.0622222
$362$	0	0
$363$	−325087.	−0.129489
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.03416e6	0.400797	0.200398	−	0.979714i	$-0.435776\pi$
$367$	1.03416e6	0.400797	0.200398	+	0.979714i	$0.435776\pi$
$368$	0	0
$369$	−4.04460e6	−1.54636
$370$	0	0
$371$	5.35228e6	2.01885
$372$	0	0
$373$	−1.08319e6	−0.403118	−0.201559	−	0.979476i	$-0.564601\pi$
$373$	−1.08319e6	−0.403118	−0.201559	+	0.979476i	$0.564601\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	474349.	0.171888
$378$	0	0
$379$	4.30085e6	1.53800	0.768999	−	0.639250i	$-0.220755\pi$
$379$	4.30085e6	1.53800	0.768999	+	0.639250i	$0.220755\pi$
$380$	0	0
$381$	−1.47066e6	−0.519039
$382$	0	0
$383$	−1.63889e6	−0.570890	−0.285445	−	0.958395i	$-0.592141\pi$
$383$	−1.63889e6	−0.570890	−0.285445	+	0.958395i	$0.592141\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.93734e6	−0.996957
$388$	0	0
$389$	245593.	0.0822891	0.0411446	−	0.999153i	$-0.486900\pi$
$389$	245593.	0.0822891	0.0411446	+	0.999153i	$0.486900\pi$
$390$	0	0
$391$	−949844.	−0.314203
$392$	0	0
$393$	3.84528e6	1.25588
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.16926e6	−0.372336	−0.186168	−	0.982518i	$-0.559607\pi$
$397$	−1.16926e6	−0.372336	−0.186168	+	0.982518i	$0.559607\pi$
$398$	0	0
$399$	−4.91434e6	−1.54537
$400$	0	0
$401$	3.64067e6	1.13063	0.565314	−	0.824875i	$-0.308755\pi$
$401$	3.64067e6	1.13063	0.565314	+	0.824875i	$0.308755\pi$
$402$	0	0
$403$	−310978.	−0.0953823
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44885.1	0.0134312
$408$	0	0
$409$	−4.72150e6	−1.39563	−0.697817	−	0.716276i	$-0.745845\pi$
$409$	−4.72150e6	−1.39563	−0.697817	+	0.716276i	$0.745845\pi$
$410$	0	0
$411$	−4.58840e6	−1.33985
$412$	0	0
$413$	4.65673e6	1.34340
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.49123e6	1.82804
$418$	0	0
$419$	−6.42193e6	−1.78702	−0.893512	−	0.449039i	$-0.851767\pi$
$419$	−6.42193e6	−1.78702	−0.893512	+	0.449039i	$0.851767\pi$
$420$	0	0
$421$	4.98652e6	1.37117	0.685586	−	0.727991i	$-0.259546\pi$
$421$	4.98652e6	1.37117	0.685586	+	0.727991i	$0.259546\pi$
$422$	0	0
$423$	−3.76652e6	−1.02350
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.65217e6	−1.50019
$428$	0	0
$429$	−379227.	−0.0994847
$430$	0	0
$431$	−4.59132e6	−1.19054	−0.595270	−	0.803526i	$-0.702955\pi$
$431$	−4.59132e6	−1.19054	−0.595270	+	0.803526i	$0.702955\pi$
$432$	0	0
$433$	5.83501e6	1.49562	0.747811	−	0.663911i	$-0.231105\pi$
$433$	5.83501e6	1.49562	0.747811	+	0.663911i	$0.231105\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.55934e6	−0.891592
$438$	0	0
$439$	−2.79892e6	−0.693154	−0.346577	−	0.938022i	$-0.612656\pi$
$439$	−2.79892e6	−0.693154	−0.346577	+	0.938022i	$0.612656\pi$
$440$	0	0
$441$	1.07237e6	0.262573
$442$	0	0
$443$	4.80858e6	1.16415	0.582073	−	0.813137i	$-0.302242\pi$
$443$	4.80858e6	1.16415	0.582073	+	0.813137i	$0.302242\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	533539.	0.126298
$448$	0	0
$449$	−3.39735e6	−0.795289	−0.397644	−	0.917540i	$-0.630172\pi$
$449$	−3.39735e6	−0.795289	−0.397644	+	0.917540i	$0.630172\pi$
$450$	0	0
$451$	1.95750e6	0.453169
$452$	0	0
$453$	3.52324e6	0.806671
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.03755e6	−0.680350	−0.340175	−	0.940362i	$-0.610486\pi$
$457$	−3.03755e6	−0.680350	−0.340175	+	0.940362i	$0.610486\pi$
$458$	0	0
$459$	−63304.5	−0.0140250
$460$	0	0
$461$	4.30824e6	0.944163	0.472082	−	0.881555i	$-0.343503\pi$
$461$	4.30824e6	0.944163	0.472082	+	0.881555i	$0.343503\pi$
$462$	0	0
$463$	−6.59969e6	−1.43077	−0.715387	−	0.698728i	$-0.753750\pi$
$463$	−6.59969e6	−1.43077	−0.715387	+	0.698728i	$0.753750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.18419e6	−1.52435	−0.762176	−	0.647370i	$-0.775869\pi$
$467$	−7.18419e6	−1.52435	−0.762176	+	0.647370i	$0.775869\pi$
$468$	0	0
$469$	4.43106e6	0.930198
$470$	0	0
$471$	5.13061e6	1.06566
$472$	0	0
$473$	1.42161e6	0.292164
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.21286e6	1.85395
$478$	0	0
$479$	−2.88740e6	−0.575000	−0.287500	−	0.957781i	$-0.592824\pi$
$479$	−2.88740e6	−0.575000	−0.287500	+	0.957781i	$0.592824\pi$
$480$	0	0
$481$	52360.4	0.0103191
$482$	0	0
$483$	7.53299e6	1.46926
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.20372e6	−0.994241	−0.497120	−	0.867682i	$-0.665609\pi$
$487$	−5.20372e6	−0.994241	−0.497120	+	0.867682i	$0.665609\pi$
$488$	0	0
$489$	3.30328e6	0.624703
$490$	0	0
$491$	−4.75750e6	−0.890584	−0.445292	−	0.895386i	$-0.646900\pi$
$491$	−4.75750e6	−0.890584	−0.445292	+	0.895386i	$0.646900\pi$
$492$	0	0
$493$	−1.36656e6	−0.253228
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.94276e6	0.534396
$498$	0	0
$499$	−3.74618e6	−0.673499	−0.336750	−	0.941594i	$-0.609328\pi$
$499$	−3.74618e6	−0.673499	−0.336750	+	0.941594i	$0.609328\pi$
$500$	0	0
$501$	−5.56869e6	−0.991194
$502$	0	0
$503$	2.56728e6	0.452433	0.226216	−	0.974077i	$-0.427364\pi$
$503$	2.56728e6	0.452433	0.226216	+	0.974077i	$0.427364\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.80175e6	1.34795
$508$	0	0
$509$	−3.52410e6	−0.602912	−0.301456	−	0.953480i	$-0.597473\pi$
$509$	−3.52410e6	−0.602912	−0.301456	+	0.953480i	$0.597473\pi$
$510$	0	0
$511$	4.76307e6	0.806927
$512$	0	0
$513$	−237220.	−0.0397977
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.82292e6	0.299944
$518$	0	0
$519$	5.81393e6	0.947439
$520$	0	0
$521$	6.31287e6	1.01890	0.509451	−	0.860499i	$-0.329848\pi$
$521$	6.31287e6	1.01890	0.509451	+	0.860499i	$0.329848\pi$
$522$	0	0
$523$	9.14634e6	1.46215	0.731077	−	0.682295i	$-0.239018\pi$
$523$	9.14634e6	1.46215	0.731077	+	0.682295i	$0.239018\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	895904.	0.140519
$528$	0	0
$529$	−980382.	−0.152320
$530$	0	0
$531$	8.01562e6	1.23368
$532$	0	0
$533$	2.28350e6	0.348164
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.13339e7	1.69607
$538$	0	0
$539$	−519006.	−0.0769485
$540$	0	0
$541$	−4.31851e6	−0.634367	−0.317183	−	0.948364i	$-0.602737\pi$
$541$	−4.31851e6	−0.634367	−0.317183	+	0.948364i	$0.602737\pi$
$542$	0	0
$543$	−6.05347e6	−0.881058
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.59069e6	0.227310	0.113655	−	0.993520i	$-0.463744\pi$
$547$	1.59069e6	0.227310	0.113655	+	0.993520i	$0.463744\pi$
$548$	0	0
$549$	−9.72907e6	−1.37766
$550$	0	0
$551$	−5.12090e6	−0.718566
$552$	0	0
$553$	−6.17037e6	−0.858021
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.37230e7	1.87418	0.937091	−	0.349085i	$-0.113508\pi$
$557$	1.37230e7	1.87418	0.937091	+	0.349085i	$0.113508\pi$
$558$	0	0
$559$	1.65837e6	0.224466
$560$	0	0
$561$	1.09252e6	0.146563
$562$	0	0
$563$	−6.60134e6	−0.877731	−0.438865	−	0.898553i	$-0.644619\pi$
$563$	−6.60134e6	−0.877731	−0.438865	+	0.898553i	$0.644619\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−8.32201e6	−1.08710
$568$	0	0
$569$	8.38245e6	1.08540	0.542701	−	0.839926i	$-0.317402\pi$
$569$	8.38245e6	1.08540	0.542701	+	0.839926i	$0.317402\pi$
$570$	0	0
$571$	−1.38366e7	−1.77598	−0.887992	−	0.459858i	$-0.847900\pi$
$571$	−1.38366e7	−1.77598	−0.887992	+	0.459858i	$0.847900\pi$
$572$	0	0
$573$	−3.94300e6	−0.501695
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.10028e6	−0.512713	−0.256357	−	0.966582i	$-0.582522\pi$
$577$	−4.10028e6	−0.512713	−0.256357	+	0.966582i	$0.582522\pi$
$578$	0	0
$579$	2.17060e7	2.69082
$580$	0	0
$581$	−736029.	−0.0904596
$582$	0	0
$583$	−4.45883e6	−0.543312
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.62565e6	0.314516	0.157258	−	0.987558i	$-0.449735\pi$
$587$	2.62565e6	0.314516	0.157258	+	0.987558i	$0.449735\pi$
$588$	0	0
$589$	3.35721e6	0.398741
$590$	0	0
$591$	−1.87801e7	−2.21172
$592$	0	0
$593$	7.12981e6	0.832610	0.416305	−	0.909225i	$-0.363325\pi$
$593$	7.12981e6	0.832610	0.416305	+	0.909225i	$0.363325\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.85226e6	0.786861
$598$	0	0
$599$	−1.15567e7	−1.31603	−0.658014	−	0.753006i	$-0.728603\pi$
$599$	−1.15567e7	−1.31603	−0.658014	+	0.753006i	$0.728603\pi$
$600$	0	0
$601$	−1.28012e7	−1.44566	−0.722830	−	0.691026i	$-0.757159\pi$
$601$	−1.28012e7	−1.44566	−0.722830	+	0.691026i	$0.757159\pi$
$602$	0	0
$603$	7.62717e6	0.854221
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.57677e7	−1.73699	−0.868495	−	0.495698i	$-0.834912\pi$
$607$	−1.57677e7	−1.73699	−0.868495	+	0.495698i	$0.834912\pi$
$608$	0	0
$609$	1.08379e7	1.18413
$610$	0	0
$611$	2.12651e6	0.230443
$612$	0	0
$613$	1.67536e7	1.80076	0.900380	−	0.435104i	$-0.143288\pi$
$613$	1.67536e7	1.80076	0.900380	+	0.435104i	$0.143288\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.66348e6	−0.598922	−0.299461	−	0.954108i	$-0.596807\pi$
$617$	−5.66348e6	−0.598922	−0.299461	+	0.954108i	$0.596807\pi$
$618$	0	0
$619$	−4.69997e6	−0.493024	−0.246512	−	0.969140i	$-0.579285\pi$
$619$	−4.69997e6	−0.493024	−0.246512	+	0.969140i	$0.579285\pi$
$620$	0	0
$621$	363625.	0.0378377
$622$	0	0
$623$	8.44601e6	0.871830
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.09400e6	0.415890
$628$	0	0
$629$	−150846.	−0.0152022
$630$	0	0
$631$	−1.86062e7	−1.86031	−0.930154	−	0.367170i	$-0.880327\pi$
$631$	−1.86062e7	−1.86031	−0.930154	+	0.367170i	$0.880327\pi$
$632$	0	0
$633$	4.18555e6	0.415187
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−605442.	−0.0591186
$638$	0	0
$639$	5.06536e6	0.490748
$640$	0	0
$641$	1.28204e7	1.23241	0.616205	−	0.787586i	$-0.288669\pi$
$641$	1.28204e7	1.23241	0.616205	+	0.787586i	$0.288669\pi$
$642$	0	0
$643$	−6.61605e6	−0.631061	−0.315531	−	0.948915i	$-0.602182\pi$
$643$	−6.61605e6	−0.631061	−0.315531	+	0.948915i	$0.602182\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.21325e6	−0.583523	−0.291761	−	0.956491i	$-0.594241\pi$
$647$	−6.21325e6	−0.583523	−0.291761	+	0.956491i	$0.594241\pi$
$648$	0	0
$649$	−3.87939e6	−0.361536
$650$	0	0
$651$	−7.10520e6	−0.657088
$652$	0	0
$653$	−6.12578e6	−0.562184	−0.281092	−	0.959681i	$-0.590697\pi$
$653$	−6.12578e6	−0.562184	−0.281092	+	0.959681i	$0.590697\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	8.19866e6	0.741019
$658$	0	0
$659$	−8.25333e6	−0.740313	−0.370157	−	0.928969i	$-0.620696\pi$
$659$	−8.25333e6	−0.740313	−0.370157	+	0.928969i	$0.620696\pi$
$660$	0	0
$661$	1.14858e7	1.02249	0.511244	−	0.859436i	$-0.329185\pi$
$661$	1.14858e7	1.02249	0.511244	+	0.859436i	$0.329185\pi$
$662$	0	0
$663$	1.27447e6	0.112602
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.84960e6	0.683177
$668$	0	0
$669$	1.05808e7	0.914013
$670$	0	0
$671$	4.70866e6	0.403730
$672$	0	0
$673$	−1.96340e7	−1.67098	−0.835489	−	0.549507i	$-0.814816\pi$
$673$	−1.96340e7	−1.67098	−0.835489	+	0.549507i	$0.814816\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.13339e7	−1.78896	−0.894478	−	0.447113i	$-0.852452\pi$
$677$	−2.13339e7	−1.78896	−0.894478	+	0.447113i	$0.852452\pi$
$678$	0	0
$679$	1.75074e7	1.45729
$680$	0	0
$681$	2.92405e7	2.41611
$682$	0	0
$683$	−2.10491e7	−1.72656	−0.863281	−	0.504723i	$-0.831595\pi$
$683$	−2.10491e7	−1.72656	−0.863281	+	0.504723i	$0.831595\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.81859e7	−2.27845
$688$	0	0
$689$	−5.20141e6	−0.417420
$690$	0	0
$691$	2.00592e7	1.59816	0.799078	−	0.601228i	$-0.205322\pi$
$691$	2.00592e7	1.59816	0.799078	+	0.601228i	$0.205322\pi$
$692$	0	0
$693$	−4.39388e6	−0.347548
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.57859e6	−0.512921
$698$	0	0
$699$	2.90830e7	2.25137
$700$	0	0
$701$	6.95349e6	0.534451	0.267225	−	0.963634i	$-0.413893\pi$
$701$	6.95349e6	0.534451	0.267225	+	0.963634i	$0.413893\pi$
$702$	0	0
$703$	−565264.	−0.0431383
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.59167e7	1.19758
$708$	0	0
$709$	7.23498e6	0.540533	0.270266	−	0.962786i	$-0.412888\pi$
$709$	7.23498e6	0.540533	0.270266	+	0.962786i	$0.412888\pi$
$710$	0	0
$711$	−1.06210e7	−0.787940
$712$	0	0
$713$	−5.14612e6	−0.379102
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.38947e7	1.73582
$718$	0	0
$719$	1.95178e7	1.40802	0.704010	−	0.710190i	$-0.251391\pi$
$719$	1.95178e7	1.40802	0.704010	+	0.710190i	$0.251391\pi$
$720$	0	0
$721$	3.41044e6	0.244328
$722$	0	0
$723$	1.38332e7	0.984183
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.00979e6	−0.211203	−0.105601	−	0.994409i	$-0.533677\pi$
$727$	−3.00979e6	−0.211203	−0.105601	+	0.994409i	$0.533677\pi$
$728$	0	0
$729$	−1.51646e7	−1.05685
$730$	0	0
$731$	−4.77762e6	−0.330687
$732$	0	0
$733$	−2.27046e7	−1.56082	−0.780411	−	0.625266i	$-0.784990\pi$
$733$	−2.27046e7	−1.56082	−0.780411	+	0.625266i	$0.784990\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.69139e6	−0.250335
$738$	0	0
$739$	1.77673e7	1.19677	0.598384	−	0.801210i	$-0.295810\pi$
$739$	1.77673e7	1.19677	0.598384	+	0.801210i	$0.295810\pi$
$740$	0	0
$741$	4.77582e6	0.319523
$742$	0	0
$743$	−1.36226e7	−0.905293	−0.452646	−	0.891690i	$-0.649520\pi$
$743$	−1.36226e7	−0.905293	−0.452646	+	0.891690i	$0.649520\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.26693e6	−0.0830711
$748$	0	0
$749$	1.98939e7	1.29573
$750$	0	0
$751$	−7.65196e6	−0.495077	−0.247539	−	0.968878i	$-0.579622\pi$
$751$	−7.65196e6	−0.495077	−0.247539	+	0.968878i	$0.579622\pi$
$752$	0	0
$753$	−3.07163e7	−1.97415
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.69844e7	−1.71148	−0.855742	−	0.517403i	$-0.826899\pi$
$757$	−2.69844e7	−1.71148	−0.855742	+	0.517403i	$0.826899\pi$
$758$	0	0
$759$	−6.27551e6	−0.395408
$760$	0	0
$761$	2.02615e7	1.26826	0.634131	−	0.773226i	$-0.281358\pi$
$761$	2.02615e7	1.26826	0.634131	+	0.773226i	$0.281358\pi$
$762$	0	0
$763$	−2.50422e7	−1.55726
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.52547e6	−0.277763
$768$	0	0
$769$	−6.94784e6	−0.423676	−0.211838	−	0.977305i	$-0.567945\pi$
$769$	−6.94784e6	−0.423676	−0.211838	+	0.977305i	$0.567945\pi$
$770$	0	0
$771$	−1.73150e6	−0.104903
$772$	0	0
$773$	5.68170e6	0.342003	0.171001	−	0.985271i	$-0.445300\pi$
$773$	5.68170e6	0.342003	0.171001	+	0.985271i	$0.445300\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.19632e6	0.0710880
$778$	0	0
$779$	−2.46519e7	−1.45548
$780$	0	0
$781$	−2.45153e6	−0.143817
$782$	0	0
$783$	523154.	0.0304948
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.17645e7	1.82812	0.914062	−	0.405574i	$-0.132928\pi$
$787$	3.17645e7	1.82812	0.914062	+	0.405574i	$0.132928\pi$
$788$	0	0
$789$	1.63646e7	0.935862
$790$	0	0
$791$	3.35969e7	1.90923
$792$	0	0
$793$	5.49285e6	0.310181
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.83331e7	−1.57997	−0.789983	−	0.613128i	$-0.789911\pi$
$797$	−2.83331e7	−1.57997	−0.789983	+	0.613128i	$0.789911\pi$
$798$	0	0
$799$	−6.12629e6	−0.339493
$800$	0	0
$801$	1.45381e7	0.800620
$802$	0	0
$803$	−3.96797e6	−0.217160
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.03349e7	0.558629
$808$	0	0
$809$	−2.88144e7	−1.54788	−0.773942	−	0.633256i	$-0.781718\pi$
$809$	−2.88144e7	−1.54788	−0.773942	+	0.633256i	$0.781718\pi$
$810$	0	0
$811$	−2.05812e7	−1.09880	−0.549400	−	0.835560i	$-0.685144\pi$
$811$	−2.05812e7	−1.09880	−0.549400	+	0.835560i	$0.685144\pi$
$812$	0	0
$813$	320448.	0.0170032
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.79031e7	−0.938368
$818$	0	0
$819$	−5.12564e6	−0.267017
$820$	0	0
$821$	−1.68229e7	−0.871050	−0.435525	−	0.900177i	$-0.643437\pi$
$821$	−1.68229e7	−0.871050	−0.435525	+	0.900177i	$0.643437\pi$
$822$	0	0
$823$	3.10268e6	0.159675	0.0798377	−	0.996808i	$-0.474560\pi$
$823$	3.10268e6	0.159675	0.0798377	+	0.996808i	$0.474560\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.89411e7	−0.963032	−0.481516	−	0.876437i	$-0.659914\pi$
$827$	−1.89411e7	−0.963032	−0.481516	+	0.876437i	$0.659914\pi$
$828$	0	0
$829$	2.70629e7	1.36769	0.683845	−	0.729627i	$-0.260306\pi$
$829$	2.70629e7	1.36769	0.683845	+	0.729627i	$0.260306\pi$
$830$	0	0
$831$	7.38156e6	0.370805
$832$	0	0
$833$	1.74423e6	0.0870945
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−342975.	−0.0169219
$838$	0	0
$839$	−1.75449e7	−0.860492	−0.430246	−	0.902712i	$-0.641573\pi$
$839$	−1.75449e7	−0.860492	−0.430246	+	0.902712i	$0.641573\pi$
$840$	0	0
$841$	−9.21777e6	−0.449403
$842$	0	0
$843$	3.10185e7	1.50332
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.12654e6	0.101851
$848$	0	0
$849$	2.23369e7	1.06354
$850$	0	0
$851$	866468.	0.0410137
$852$	0	0
$853$	3.18236e7	1.49753	0.748767	−	0.662833i	$-0.230646\pi$
$853$	3.18236e7	1.49753	0.748767	+	0.662833i	$0.230646\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.08624e7	0.505213	0.252607	−	0.967569i	$-0.418712\pi$
$857$	1.08624e7	0.505213	0.252607	+	0.967569i	$0.418712\pi$
$858$	0	0
$859$	1.60192e6	0.0740726	0.0370363	−	0.999314i	$-0.488208\pi$
$859$	1.60192e6	0.0740726	0.0370363	+	0.999314i	$0.488208\pi$
$860$	0	0
$861$	5.21732e7	2.39850
$862$	0	0
$863$	−3.43393e7	−1.56951	−0.784755	−	0.619806i	$-0.787211\pi$
$863$	−3.43393e7	−1.56951	−0.784755	+	0.619806i	$0.787211\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.78546e7	1.25849
$868$	0	0
$869$	5.14035e6	0.230910
$870$	0	0
$871$	−4.30616e6	−0.192329
$872$	0	0
$873$	3.01354e7	1.33826
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.72149e7	0.755798	0.377899	−	0.925847i	$-0.376647\pi$
$877$	1.72149e7	0.755798	0.377899	+	0.925847i	$0.376647\pi$
$878$	0	0
$879$	−1.24845e7	−0.545003
$880$	0	0
$881$	1.06181e7	0.460901	0.230451	−	0.973084i	$-0.425980\pi$
$881$	1.06181e7	0.460901	0.230451	+	0.973084i	$0.425980\pi$
$882$	0	0
$883$	1.72907e7	0.746296	0.373148	−	0.927772i	$-0.378278\pi$
$883$	1.72907e7	0.746296	0.373148	+	0.927772i	$0.378278\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.43856e7	−1.04070	−0.520348	−	0.853954i	$-0.674198\pi$
$887$	−2.43856e7	−1.04070	−0.520348	+	0.853954i	$0.674198\pi$
$888$	0	0
$889$	9.62027e6	0.408256
$890$	0	0
$891$	6.93283e6	0.292561
$892$	0	0
$893$	−2.29570e7	−0.963355
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−7.32065e6	−0.303787
$898$	0	0
$899$	−7.40383e6	−0.305532
$900$	0	0
$901$	1.49848e7	0.614950
$902$	0	0
$903$	3.78901e7	1.54635
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.46330e7	1.39789	0.698944	−	0.715177i	$-0.253654\pi$
$907$	3.46330e7	1.39789	0.698944	+	0.715177i	$0.253654\pi$
$908$	0	0
$909$	2.73975e7	1.09977
$910$	0	0
$911$	−1.56080e7	−0.623089	−0.311544	−	0.950232i	$-0.600846\pi$
$911$	−1.56080e7	−0.623089	−0.311544	+	0.950232i	$0.600846\pi$
$912$	0	0
$913$	613165.	0.0243445
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.51538e7	−0.987824
$918$	0	0
$919$	7.50778e6	0.293240	0.146620	−	0.989193i	$-0.453161\pi$
$919$	7.50778e6	0.293240	0.146620	+	0.989193i	$0.453161\pi$
$920$	0	0
$921$	−1.56225e7	−0.606877
$922$	0	0
$923$	−2.85981e6	−0.110493
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.87039e6	0.224372
$928$	0	0
$929$	1.14687e7	0.435987	0.217993	−	0.975950i	$-0.430049\pi$
$929$	1.14687e7	0.435987	0.217993	+	0.975950i	$0.430049\pi$
$930$	0	0
$931$	6.53613e6	0.247142
$932$	0	0
$933$	−4.17105e7	−1.56870
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.23242e7	−0.830669	−0.415334	−	0.909669i	$-0.636335\pi$
$937$	−2.23242e7	−0.830669	−0.415334	+	0.909669i	$0.636335\pi$
$938$	0	0
$939$	3.92729e7	1.45355
$940$	0	0
$941$	4.70356e7	1.73162	0.865810	−	0.500373i	$-0.166804\pi$
$941$	4.70356e7	1.73162	0.865810	+	0.500373i	$0.166804\pi$
$942$	0	0
$943$	3.77878e7	1.38380
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.48250e7	0.537180	0.268590	−	0.963255i	$-0.413442\pi$
$947$	1.48250e7	0.537180	0.268590	+	0.963255i	$0.413442\pi$
$948$	0	0
$949$	−4.62881e6	−0.166841
$950$	0	0
$951$	−4.91798e7	−1.76334
$952$	0	0
$953$	4.10518e7	1.46420	0.732100	−	0.681198i	$-0.238541\pi$
$953$	4.10518e7	1.46420	0.732100	+	0.681198i	$0.238541\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.02871e6	−0.318673
$958$	0	0
$959$	3.00148e7	1.05388
$960$	0	0
$961$	−2.37753e7	−0.830457
$962$	0	0
$963$	3.42434e7	1.18990
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.77022e6	0.267219	0.133610	−	0.991034i	$-0.457343\pi$
$967$	7.77022e6	0.267219	0.133610	+	0.991034i	$0.457343\pi$
$968$	0	0
$969$	−1.37588e7	−0.470728
$970$	0	0
$971$	−2.09597e7	−0.713408	−0.356704	−	0.934217i	$-0.616100\pi$
$971$	−2.09597e7	−0.713408	−0.356704	+	0.934217i	$0.616100\pi$
$972$	0	0
$973$	−4.24621e7	−1.43787
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.04224e7	1.01967	0.509833	−	0.860274i	$-0.329707\pi$
$977$	3.04224e7	1.01967	0.509833	+	0.860274i	$0.329707\pi$
$978$	0	0
$979$	−7.03613e6	−0.234627
$980$	0	0
$981$	−4.31050e7	−1.43006
$982$	0	0
$983$	−5.02785e7	−1.65958	−0.829791	−	0.558074i	$-0.811540\pi$
$983$	−5.02785e7	−1.65958	−0.829791	+	0.558074i	$0.811540\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.85862e7	1.58752
$988$	0	0
$989$	2.74429e7	0.892153
$990$	0	0
$991$	5.47442e7	1.77074	0.885369	−	0.464890i	$-0.153906\pi$
$991$	5.47442e7	1.77074	0.885369	+	0.464890i	$0.153906\pi$
$992$	0	0
$993$	6.11249e7	1.96718
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.39172e6	0.235509	0.117754	−	0.993043i	$-0.462430\pi$
$997$	7.39172e6	0.235509	0.117754	+	0.993043i	$0.462430\pi$
$998$	0	0
$999$	57747.7	0.00183072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.l.1.2		12
5.2	odd	4		220.6.b.a.89.11	yes	12
5.3	odd	4		220.6.b.a.89.2	✓	12
5.4	even	2	inner	1100.6.a.l.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.b.a.89.2	✓	12	5.3	odd	4
220.6.b.a.89.11	yes	12	5.2	odd	4
1100.6.a.l.1.2		12	1.1	even	1	trivial
1100.6.a.l.1.11		12	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q-22.2039 q^{3} +145.246 q^{7} +250.011 q^{9} -121.000 q^{11} -141.152 q^{13} +406.646 q^{17} +1523.82 q^{19} -3225.01 q^{21} -2335.80 q^{23} -155.675 q^{27} -3360.56 q^{29} +2203.15 q^{31} +2686.67 q^{33} -370.951 q^{37} +3134.11 q^{39} -16177.7 q^{41} -11748.8 q^{43} -15065.4 q^{47} +4289.30 q^{49} -9029.11 q^{51} +36849.8 q^{53} -33834.7 q^{57} +32061.1 q^{59} -38914.6 q^{61} +36313.0 q^{63} +30507.3 q^{67} +51863.8 q^{69} +20260.6 q^{71} +32793.2 q^{73} -17574.7 q^{77} -42482.3 q^{79} -57296.1 q^{81} -5067.48 q^{83} +74617.4 q^{87} +58149.8 q^{89} -20501.7 q^{91} -48918.5 q^{93} +120536. q^{97} -30251.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 1606 q^{9} - 1452 q^{11} - 2100 q^{19} - 916 q^{21} + 9932 q^{29} - 23382 q^{31} - 16400 q^{39} - 16700 q^{41} + 4888 q^{49} - 59376 q^{51} - 90722 q^{59} + 27700 q^{61} - 31118 q^{69} - 26758 q^{71}+ \cdots - 194326 q^{99}+O(q^{100}) \) 12 * q + 1606 * q^9 - 1452 * q^11 - 2100 * q^19 - 916 * q^21 + 9932 * q^29 - 23382 * q^31 - 16400 * q^39 - 16700 * q^41 + 4888 * q^49 - 59376 * q^51 - 90722 * q^59 + 27700 * q^61 - 31118 * q^69 - 26758 * q^71 - 84852 * q^79 + 224296 * q^81 + 12598 * q^89 + 64368 * q^91 - 194326 * q^99

Introduction

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