LMFDB - Embedded newform 1100.6.a.m.1.14

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2272 x^{12} + 1983198 x^{10} - 827062096 x^{8} + 165415157329 x^{6} - 13843733383152 x^{4} + \cdots - 27\!\cdots\!24 $ x^14 - 2272x^12 + 1983198x^10 - 827062096x^8 + 165415157329x^6 - 13843733383152x^4 + 402818574022128x^2 - 2766293773242624
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{2}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$26.2077$ 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)

$f(q)$	$=$	$q+26.2077 q^{3} +64.0890 q^{7} +443.842 q^{9} +121.000 q^{11} -528.657 q^{13} +1541.39 q^{17} +3107.81 q^{19} +1679.62 q^{21} -1789.78 q^{23} +5263.61 q^{27} +4892.43 q^{29} +1172.26 q^{31} +3171.13 q^{33} -12632.9 q^{37} -13854.9 q^{39} +1520.48 q^{41} -11538.1 q^{43} +19780.6 q^{47} -12699.6 q^{49} +40396.2 q^{51} +16273.8 q^{53} +81448.4 q^{57} +39459.2 q^{59} +18830.5 q^{61} +28445.4 q^{63} -40415.1 q^{67} -46906.1 q^{69} -65200.4 q^{71} -46369.3 q^{73} +7754.77 q^{77} -20145.5 q^{79} +30093.4 q^{81} +85967.1 q^{83} +128219. q^{87} +12067.3 q^{89} -33881.1 q^{91} +30722.2 q^{93} +137100. q^{97} +53704.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 1142 q^{9} + 1694 q^{11} + 4540 q^{19} + 3824 q^{21} + 9972 q^{29} + 19076 q^{31} + 13616 q^{39} + 15052 q^{41} + 55346 q^{49} - 13380 q^{51} + 2108 q^{59} + 11660 q^{61} - 1620 q^{69} - 89284 q^{71}+ \cdots + 138182 q^{99}+O(q^{100}) $ 14 * q + 1142 * q^9 + 1694 * q^11 + 4540 * q^19 + 3824 * q^21 + 9972 * q^29 + 19076 * q^31 + 13616 * q^39 + 15052 * q^41 + 55346 * q^49 - 13380 * q^51 + 2108 * q^59 + 11660 * q^61 - 1620 * q^69 - 89284 * q^71 - 19432 * q^79 - 94714 * q^81 + 38836 * q^89 + 39632 * q^91 + 138182 * 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
$3$	26.2077	1.68122	0.840612	−	0.541638i	$-0.182196\pi$
$3$	26.2077	1.68122	0.840612	+	0.541638i	$0.182196\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	64.0890	0.494354	0.247177	−	0.968970i	$-0.420497\pi$
$7$	64.0890	0.494354	0.247177	+	0.968970i	$0.420497\pi$
$8$	0	0
$9$	443.842	1.82651
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−528.657	−0.867592	−0.433796	−	0.901011i	$-0.642826\pi$
$13$	−528.657	−0.867592	−0.433796	+	0.901011i	$0.642826\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1541.39	1.29357	0.646785	−	0.762672i	$-0.276113\pi$
$17$	1541.39	1.29357	0.646785	+	0.762672i	$0.276113\pi$
$18$	0	0
$19$	3107.81	1.97501	0.987507	−	0.157576i	$-0.0503678\pi$
$19$	3107.81	1.97501	0.987507	+	0.157576i	$0.0503678\pi$
$20$	0	0
$21$	1679.62	0.831120
$22$	0	0
$23$	−1789.78	−0.705474	−0.352737	−	0.935722i	$-0.614749\pi$
$23$	−1789.78	−0.705474	−0.352737	+	0.935722i	$0.614749\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5263.61	1.38955
$28$	0	0
$29$	4892.43	1.08026	0.540131	−	0.841581i	$-0.318375\pi$
$29$	4892.43	1.08026	0.540131	+	0.841581i	$0.318375\pi$
$30$	0	0
$31$	1172.26	0.219089	0.109544	−	0.993982i	$-0.465061\pi$
$31$	1172.26	0.219089	0.109544	+	0.993982i	$0.465061\pi$
$32$	0	0
$33$	3171.13	0.506908
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12632.9	−1.51705	−0.758525	−	0.651644i	$-0.774080\pi$
$37$	−12632.9	−1.51705	−0.758525	+	0.651644i	$0.774080\pi$
$38$	0	0
$39$	−13854.9	−1.45862
$40$	0	0
$41$	1520.48	0.141261	0.0706305	−	0.997503i	$-0.477499\pi$
$41$	1520.48	0.141261	0.0706305	+	0.997503i	$0.477499\pi$
$42$	0	0
$43$	−11538.1	−0.951620	−0.475810	−	0.879548i	$-0.657845\pi$
$43$	−11538.1	−0.951620	−0.475810	+	0.879548i	$0.657845\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	19780.6	1.30615	0.653076	−	0.757292i	$-0.273478\pi$
$47$	19780.6	1.30615	0.653076	+	0.757292i	$0.273478\pi$
$48$	0	0
$49$	−12699.6	−0.755614
$50$	0	0
$51$	40396.2	2.17478
$52$	0	0
$53$	16273.8	0.795790	0.397895	−	0.917431i	$-0.369741\pi$
$53$	16273.8	0.795790	0.397895	+	0.917431i	$0.369741\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	81448.4	3.32044
$58$	0	0
$59$	39459.2	1.47577	0.737884	−	0.674927i	$-0.235825\pi$
$59$	39459.2	1.47577	0.737884	+	0.674927i	$0.235825\pi$
$60$	0	0
$61$	18830.5	0.647943	0.323972	−	0.946067i	$-0.394982\pi$
$61$	18830.5	0.647943	0.323972	+	0.946067i	$0.394982\pi$
$62$	0	0
$63$	28445.4	0.902944
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−40415.1	−1.09991	−0.549955	−	0.835194i	$-0.685355\pi$
$67$	−40415.1	−1.09991	−0.549955	+	0.835194i	$0.685355\pi$
$68$	0	0
$69$	−46906.1	−1.18606
$70$	0	0
$71$	−65200.4	−1.53499	−0.767493	−	0.641057i	$-0.778496\pi$
$71$	−65200.4	−1.53499	−0.767493	+	0.641057i	$0.778496\pi$
$72$	0	0
$73$	−46369.3	−1.01841	−0.509205	−	0.860645i	$-0.670061\pi$
$73$	−46369.3	−1.01841	−0.509205	+	0.860645i	$0.670061\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7754.77	0.149053
$78$	0	0
$79$	−20145.5	−0.363170	−0.181585	−	0.983375i	$-0.558123\pi$
$79$	−20145.5	−0.363170	−0.181585	+	0.983375i	$0.558123\pi$
$80$	0	0
$81$	30093.4	0.509634
$82$	0	0
$83$	85967.1	1.36974	0.684869	−	0.728667i	$-0.259860\pi$
$83$	85967.1	1.36974	0.684869	+	0.728667i	$0.259860\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	128219.	1.81616
$88$	0	0
$89$	12067.3	0.161486	0.0807430	−	0.996735i	$-0.474271\pi$
$89$	12067.3	0.161486	0.0807430	+	0.996735i	$0.474271\pi$
$90$	0	0
$91$	−33881.1	−0.428898
$92$	0	0
$93$	30722.2	0.368337
$94$	0	0
$95$	0	0
$96$	0	0
$97$	137100.	1.47947	0.739737	−	0.672896i	$-0.234950\pi$
$97$	137100.	1.47947	0.739737	+	0.672896i	$0.234950\pi$
$98$	0	0
$99$	53704.9	0.550714
$100$	0	0
$101$	−47877.0	−0.467008	−0.233504	−	0.972356i	$-0.575019\pi$
$101$	−47877.0	−0.467008	−0.233504	+	0.972356i	$0.575019\pi$
$102$	0	0
$103$	83825.2	0.778541	0.389271	−	0.921123i	$-0.372727\pi$
$103$	83825.2	0.778541	0.389271	+	0.921123i	$0.372727\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	25557.2	0.215801	0.107901	−	0.994162i	$-0.465587\pi$
$107$	25557.2	0.215801	0.107901	+	0.994162i	$0.465587\pi$
$108$	0	0
$109$	54744.9	0.441344	0.220672	−	0.975348i	$-0.429175\pi$
$109$	54744.9	0.441344	0.220672	+	0.975348i	$0.429175\pi$
$110$	0	0
$111$	−331080.	−2.55050
$112$	0	0
$113$	72063.0	0.530904	0.265452	−	0.964124i	$-0.414479\pi$
$113$	72063.0	0.530904	0.265452	+	0.964124i	$0.414479\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−234640.	−1.58467
$118$	0	0
$119$	98786.0	0.639482
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	39848.4	0.237491
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−113193.	−0.622745	−0.311372	−	0.950288i	$-0.600789\pi$
$127$	−113193.	−0.622745	−0.311372	+	0.950288i	$0.600789\pi$
$128$	0	0
$129$	−302387.	−1.59989
$130$	0	0
$131$	142156.	0.723746	0.361873	−	0.932227i	$-0.382137\pi$
$131$	142156.	0.723746	0.361873	+	0.932227i	$0.382137\pi$
$132$	0	0
$133$	199176.	0.976356
$134$	0	0
$135$	0	0
$136$	0	0
$137$	206954.	0.942045	0.471022	−	0.882121i	$-0.343885\pi$
$137$	206954.	0.942045	0.471022	+	0.882121i	$0.343885\pi$
$138$	0	0
$139$	432970.	1.90073	0.950367	−	0.311132i	$-0.100708\pi$
$139$	432970.	1.90073	0.950367	+	0.311132i	$0.100708\pi$
$140$	0	0
$141$	518402.	2.19593
$142$	0	0
$143$	−63967.5	−0.261589
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−332827.	−1.27036
$148$	0	0
$149$	−171827.	−0.634052	−0.317026	−	0.948417i	$-0.602684\pi$
$149$	−171827.	−0.634052	−0.317026	+	0.948417i	$0.602684\pi$
$150$	0	0
$151$	295056.	1.05308	0.526542	−	0.850149i	$-0.323488\pi$
$151$	295056.	1.05308	0.526542	+	0.850149i	$0.323488\pi$
$152$	0	0
$153$	684134.	2.36272
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−133587.	−0.432529	−0.216264	−	0.976335i	$-0.569387\pi$
$157$	−133587.	−0.432529	−0.216264	+	0.976335i	$0.569387\pi$
$158$	0	0
$159$	426498.	1.33790
$160$	0	0
$161$	−114705.	−0.348754
$162$	0	0
$163$	326639.	0.962939	0.481470	−	0.876463i	$-0.340103\pi$
$163$	326639.	0.962939	0.481470	+	0.876463i	$0.340103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−614485.	−1.70498	−0.852491	−	0.522741i	$-0.824909\pi$
$167$	−614485.	−1.70498	−0.852491	+	0.522741i	$0.824909\pi$
$168$	0	0
$169$	−91814.9	−0.247284
$170$	0	0
$171$	1.37938e6	3.60739
$172$	0	0
$173$	500953.	1.27257	0.636285	−	0.771454i	$-0.280470\pi$
$173$	500953.	1.27257	0.636285	+	0.771454i	$0.280470\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.03413e6	2.48110
$178$	0	0
$179$	305612.	0.712915	0.356457	−	0.934312i	$-0.383984\pi$
$179$	305612.	0.712915	0.356457	+	0.934312i	$0.383984\pi$
$180$	0	0
$181$	−52085.1	−0.118173	−0.0590864	−	0.998253i	$-0.518819\pi$
$181$	−52085.1	−0.118173	−0.0590864	+	0.998253i	$0.518819\pi$
$182$	0	0
$183$	493503.	1.08934
$184$	0	0
$185$	0	0
$186$	0	0
$187$	186508.	0.390026
$188$	0	0
$189$	337340.	0.686930
$190$	0	0
$191$	−894767.	−1.77471	−0.887354	−	0.461089i	$-0.847459\pi$
$191$	−894767.	−1.77471	−0.887354	+	0.461089i	$0.847459\pi$
$192$	0	0
$193$	−456312.	−0.881797	−0.440899	−	0.897557i	$-0.645340\pi$
$193$	−456312.	−0.881797	−0.440899	+	0.897557i	$0.645340\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	423709.	0.777861	0.388931	−	0.921267i	$-0.372845\pi$
$197$	423709.	0.777861	0.388931	+	0.921267i	$0.372845\pi$
$198$	0	0
$199$	1.09199e6	1.95472	0.977360	−	0.211581i	$-0.0678613\pi$
$199$	1.09199e6	1.95472	0.977360	+	0.211581i	$0.0678613\pi$
$200$	0	0
$201$	−1.05919e6	−1.84919
$202$	0	0
$203$	313551.	0.534032
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−794382.	−1.28856
$208$	0	0
$209$	376045.	0.595489
$210$	0	0
$211$	−741511.	−1.14660	−0.573299	−	0.819346i	$-0.694337\pi$
$211$	−741511.	−1.14660	−0.573299	+	0.819346i	$0.694337\pi$
$212$	0	0
$213$	−1.70875e6	−2.58065
$214$	0	0
$215$	0	0
$216$	0	0
$217$	75129.0	0.108307
$218$	0	0
$219$	−1.21523e6	−1.71218
$220$	0	0
$221$	−814866.	−1.12229
$222$	0	0
$223$	−1.01755e6	−1.37023	−0.685113	−	0.728436i	$-0.740247\pi$
$223$	−1.01755e6	−1.37023	−0.685113	+	0.728436i	$0.740247\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−51060.6	−0.0657690	−0.0328845	−	0.999459i	$-0.510469\pi$
$227$	−51060.6	−0.0657690	−0.0328845	+	0.999459i	$0.510469\pi$
$228$	0	0
$229$	405426.	0.510885	0.255443	−	0.966824i	$-0.417779\pi$
$229$	405426.	0.510885	0.255443	+	0.966824i	$0.417779\pi$
$230$	0	0
$231$	203234.	0.250592
$232$	0	0
$233$	−175866.	−0.212223	−0.106111	−	0.994354i	$-0.533840\pi$
$233$	−175866.	−0.212223	−0.106111	+	0.994354i	$0.533840\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−527967.	−0.610570
$238$	0	0
$239$	310598.	0.351726	0.175863	−	0.984415i	$-0.443728\pi$
$239$	310598.	0.351726	0.175863	+	0.984415i	$0.443728\pi$
$240$	0	0
$241$	−62501.7	−0.0693185	−0.0346592	−	0.999399i	$-0.511035\pi$
$241$	−62501.7	−0.0693185	−0.0346592	+	0.999399i	$0.511035\pi$
$242$	0	0
$243$	−490381.	−0.532743
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.64296e6	−1.71351
$248$	0	0
$249$	2.25300e6	2.30283
$250$	0	0
$251$	1.65573e6	1.65885	0.829423	−	0.558621i	$-0.188670\pi$
$251$	1.65573e6	1.65885	0.829423	+	0.558621i	$0.188670\pi$
$252$	0	0
$253$	−216564.	−0.212708
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11379.3	−0.0107469	−0.00537344	−	0.999986i	$-0.501710\pi$
$257$	−11379.3	−0.0107469	−0.00537344	+	0.999986i	$0.501710\pi$
$258$	0	0
$259$	−809632.	−0.749960
$260$	0	0
$261$	2.17147e6	1.97311
$262$	0	0
$263$	−1.42857e6	−1.27354	−0.636771	−	0.771053i	$-0.719730\pi$
$263$	−1.42857e6	−1.27354	−0.636771	+	0.771053i	$0.719730\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	316256.	0.271494
$268$	0	0
$269$	−1.52305e6	−1.28331	−0.641657	−	0.766991i	$-0.721753\pi$
$269$	−1.52305e6	−1.28331	−0.641657	+	0.766991i	$0.721753\pi$
$270$	0	0
$271$	−629994.	−0.521090	−0.260545	−	0.965462i	$-0.583902\pi$
$271$	−629994.	−0.521090	−0.260545	+	0.965462i	$0.583902\pi$
$272$	0	0
$273$	−887944.	−0.721073
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.27661e6	1.78274	0.891371	−	0.453275i	$-0.149744\pi$
$277$	2.27661e6	1.78274	0.891371	+	0.453275i	$0.149744\pi$
$278$	0	0
$279$	520299.	0.400168
$280$	0	0
$281$	496611.	0.375189	0.187595	−	0.982247i	$-0.439931\pi$
$281$	496611.	0.375189	0.187595	+	0.982247i	$0.439931\pi$
$282$	0	0
$283$	−1.50166e6	−1.11456	−0.557282	−	0.830323i	$-0.688156\pi$
$283$	−1.50166e6	−1.11456	−0.557282	+	0.830323i	$0.688156\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	97446.3	0.0698330
$288$	0	0
$289$	956023.	0.673323
$290$	0	0
$291$	3.59307e6	2.48733
$292$	0	0
$293$	−1.42593e6	−0.970355	−0.485177	−	0.874416i	$-0.661245\pi$
$293$	−1.42593e6	−0.970355	−0.485177	+	0.874416i	$0.661245\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	636897.	0.418965
$298$	0	0
$299$	946182.	0.612064
$300$	0	0
$301$	−739466.	−0.470438
$302$	0	0
$303$	−1.25475e6	−0.785144
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.86573e6	−1.12981	−0.564903	−	0.825157i	$-0.691086\pi$
$307$	−1.86573e6	−1.12981	−0.564903	+	0.825157i	$0.691086\pi$
$308$	0	0
$309$	2.19686e6	1.30890
$310$	0	0
$311$	61240.1	0.0359034	0.0179517	−	0.999839i	$-0.494285\pi$
$311$	61240.1	0.0359034	0.0179517	+	0.999839i	$0.494285\pi$
$312$	0	0
$313$	−152008.	−0.0877011	−0.0438506	−	0.999038i	$-0.513963\pi$
$313$	−152008.	−0.0877011	−0.0438506	+	0.999038i	$0.513963\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.21712e6	0.680277	0.340138	−	0.940375i	$-0.389526\pi$
$317$	1.21712e6	0.680277	0.340138	+	0.940375i	$0.389526\pi$
$318$	0	0
$319$	591984.	0.325711
$320$	0	0
$321$	669795.	0.362810
$322$	0	0
$323$	4.79034e6	2.55482
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.43474e6	0.741999
$328$	0	0
$329$	1.26772e6	0.645702
$330$	0	0
$331$	−2.90401e6	−1.45690	−0.728448	−	0.685101i	$-0.759758\pi$
$331$	−2.90401e6	−1.45690	−0.728448	+	0.685101i	$0.759758\pi$
$332$	0	0
$333$	−5.60703e6	−2.77091
$334$	0	0
$335$	0	0
$336$	0	0
$337$	577357.	0.276930	0.138465	−	0.990367i	$-0.455783\pi$
$337$	577357.	0.276930	0.138465	+	0.990367i	$0.455783\pi$
$338$	0	0
$339$	1.88860e6	0.892568
$340$	0	0
$341$	141844.	0.0660577
$342$	0	0
$343$	−1.89105e6	−0.867895
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.99345e6	−1.33459	−0.667296	−	0.744793i	$-0.732548\pi$
$347$	−2.99345e6	−1.33459	−0.667296	+	0.744793i	$0.732548\pi$
$348$	0	0
$349$	2.05019e6	0.901012	0.450506	−	0.892773i	$-0.351244\pi$
$349$	2.05019e6	0.901012	0.450506	+	0.892773i	$0.351244\pi$
$350$	0	0
$351$	−2.78264e6	−1.20556
$352$	0	0
$353$	566987.	0.242179	0.121089	−	0.992642i	$-0.461361\pi$
$353$	566987.	0.242179	0.121089	+	0.992642i	$0.461361\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.58895e6	1.07511
$358$	0	0
$359$	1.82570e6	0.747643	0.373821	−	0.927501i	$-0.378047\pi$
$359$	1.82570e6	0.747643	0.373821	+	0.927501i	$0.378047\pi$
$360$	0	0
$361$	7.18237e6	2.90068
$362$	0	0
$363$	383707.	0.152838
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.33009e6	−0.515485	−0.257742	−	0.966214i	$-0.582979\pi$
$367$	−1.33009e6	−0.515485	−0.257742	+	0.966214i	$0.582979\pi$
$368$	0	0
$369$	674856.	0.258015
$370$	0	0
$371$	1.04297e6	0.393402
$372$	0	0
$373$	475043.	0.176791	0.0883956	−	0.996085i	$-0.471826\pi$
$373$	475043.	0.176791	0.0883956	+	0.996085i	$0.471826\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.58642e6	−0.937227
$378$	0	0
$379$	1.05251e6	0.376382	0.188191	−	0.982132i	$-0.439738\pi$
$379$	1.05251e6	0.376382	0.188191	+	0.982132i	$0.439738\pi$
$380$	0	0
$381$	−2.96652e6	−1.04697
$382$	0	0
$383$	−1.05438e6	−0.367283	−0.183641	−	0.982993i	$-0.558789\pi$
$383$	−1.05438e6	−0.367283	−0.183641	+	0.982993i	$0.558789\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.12110e6	−1.73815
$388$	0	0
$389$	4.41847e6	1.48046	0.740231	−	0.672352i	$-0.234716\pi$
$389$	4.41847e6	1.48046	0.740231	+	0.672352i	$0.234716\pi$
$390$	0	0
$391$	−2.75875e6	−0.912580
$392$	0	0
$393$	3.72557e6	1.21678
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.02930e6	−0.327766	−0.163883	−	0.986480i	$-0.552402\pi$
$397$	−1.02930e6	−0.327766	−0.163883	+	0.986480i	$0.552402\pi$
$398$	0	0
$399$	5.21995e6	1.64147
$400$	0	0
$401$	4.82524e6	1.49850	0.749252	−	0.662285i	$-0.230413\pi$
$401$	4.82524e6	1.49850	0.749252	+	0.662285i	$0.230413\pi$
$402$	0	0
$403$	−619724.	−0.190080
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.52859e6	−0.457408
$408$	0	0
$409$	−5.24511e6	−1.55041	−0.775205	−	0.631710i	$-0.782353\pi$
$409$	−5.24511e6	−1.55041	−0.775205	+	0.631710i	$0.782353\pi$
$410$	0	0
$411$	5.42377e6	1.58379
$412$	0	0
$413$	2.52890e6	0.729552
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.13472e7	3.19556
$418$	0	0
$419$	−588049.	−0.163636	−0.0818180	−	0.996647i	$-0.526073\pi$
$419$	−588049.	−0.163636	−0.0818180	+	0.996647i	$0.526073\pi$
$420$	0	0
$421$	4.83665e6	1.32996	0.664982	−	0.746860i	$-0.268439\pi$
$421$	4.83665e6	1.32996	0.664982	+	0.746860i	$0.268439\pi$
$422$	0	0
$423$	8.77945e6	2.38570
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.20683e6	0.320314
$428$	0	0
$429$	−1.67644e6	−0.439789
$430$	0	0
$431$	−3.40144e6	−0.882003	−0.441002	−	0.897506i	$-0.645377\pi$
$431$	−3.40144e6	−0.882003	−0.441002	+	0.897506i	$0.645377\pi$
$432$	0	0
$433$	5.81707e6	1.49102	0.745512	−	0.666493i	$-0.232205\pi$
$433$	5.81707e6	1.49102	0.745512	+	0.666493i	$0.232205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.56230e6	−1.39332
$438$	0	0
$439$	−7.33653e6	−1.81689	−0.908446	−	0.418001i	$-0.862731\pi$
$439$	−7.33653e6	−1.81689	−0.908446	+	0.418001i	$0.862731\pi$
$440$	0	0
$441$	−5.63662e6	−1.38014
$442$	0	0
$443$	−1.29709e6	−0.314022	−0.157011	−	0.987597i	$-0.550186\pi$
$443$	−1.29709e6	−0.314022	−0.157011	+	0.987597i	$0.550186\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4.50317e6	−1.06598
$448$	0	0
$449$	4.47446e6	1.04743	0.523715	−	0.851894i	$-0.324546\pi$
$449$	4.47446e6	1.04743	0.523715	+	0.851894i	$0.324546\pi$
$450$	0	0
$451$	183979.	0.0425918
$452$	0	0
$453$	7.73275e6	1.77047
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.49383e6	1.23051	0.615254	−	0.788329i	$-0.289054\pi$
$457$	5.49383e6	1.23051	0.615254	+	0.788329i	$0.289054\pi$
$458$	0	0
$459$	8.11327e6	1.79748
$460$	0	0
$461$	−4.40705e6	−0.965819	−0.482909	−	0.875670i	$-0.660420\pi$
$461$	−4.40705e6	−0.965819	−0.482909	+	0.875670i	$0.660420\pi$
$462$	0	0
$463$	5.34156e6	1.15802	0.579009	−	0.815321i	$-0.303440\pi$
$463$	5.34156e6	1.15802	0.579009	+	0.815321i	$0.303440\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.24423e6	−0.900548	−0.450274	−	0.892890i	$-0.648674\pi$
$467$	−4.24423e6	−0.900548	−0.450274	+	0.892890i	$0.648674\pi$
$468$	0	0
$469$	−2.59017e6	−0.543745
$470$	0	0
$471$	−3.50100e6	−0.727177
$472$	0	0
$473$	−1.39611e6	−0.286924
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.22299e6	1.45352
$478$	0	0
$479$	127031.	0.0252970	0.0126485	−	0.999920i	$-0.495974\pi$
$479$	127031.	0.0252970	0.0126485	+	0.999920i	$0.495974\pi$
$480$	0	0
$481$	6.67849e6	1.31618
$482$	0	0
$483$	−3.00616e6	−0.586333
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.88474e6	1.12436	0.562179	−	0.827015i	$-0.309963\pi$
$487$	5.88474e6	1.12436	0.562179	+	0.827015i	$0.309963\pi$
$488$	0	0
$489$	8.56044e6	1.61892
$490$	0	0
$491$	670926.	0.125595	0.0627973	−	0.998026i	$-0.479998\pi$
$491$	670926.	0.125595	0.0627973	+	0.998026i	$0.479998\pi$
$492$	0	0
$493$	7.54113e6	1.39740
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.17863e6	−0.758827
$498$	0	0
$499$	−4.23099e6	−0.760661	−0.380330	−	0.924851i	$-0.624190\pi$
$499$	−4.23099e6	−0.760661	−0.380330	+	0.924851i	$0.624190\pi$
$500$	0	0
$501$	−1.61042e7	−2.86646
$502$	0	0
$503$	−4.42693e6	−0.780157	−0.390079	−	0.920782i	$-0.627552\pi$
$503$	−4.42693e6	−0.780157	−0.390079	+	0.920782i	$0.627552\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.40626e6	−0.415740
$508$	0	0
$509$	−7.57091e6	−1.29525	−0.647625	−	0.761959i	$-0.724238\pi$
$509$	−7.57091e6	−1.29525	−0.647625	+	0.761959i	$0.724238\pi$
$510$	0	0
$511$	−2.97176e6	−0.503456
$512$	0	0
$513$	1.63583e7	2.74438
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.39345e6	0.393820
$518$	0	0
$519$	1.31288e7	2.13948
$520$	0	0
$521$	−4.67098e6	−0.753900	−0.376950	−	0.926234i	$-0.623027\pi$
$521$	−4.67098e6	−0.753900	−0.376950	+	0.926234i	$0.623027\pi$
$522$	0	0
$523$	−3.45852e6	−0.552887	−0.276444	−	0.961030i	$-0.589156\pi$
$523$	−3.45852e6	−0.552887	−0.276444	+	0.961030i	$0.589156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.80691e6	0.283407
$528$	0	0
$529$	−3.23302e6	−0.502306
$530$	0	0
$531$	1.75137e7	2.69551
$532$	0	0
$533$	−803815.	−0.122557
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.00938e6	1.19857
$538$	0	0
$539$	−1.53665e6	−0.227826
$540$	0	0
$541$	−9.09362e6	−1.33581	−0.667904	−	0.744248i	$-0.732808\pi$
$541$	−9.09362e6	−1.33581	−0.667904	+	0.744248i	$0.732808\pi$
$542$	0	0
$543$	−1.36503e6	−0.198675
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−5.08979e6	−0.727331	−0.363665	−	0.931530i	$-0.618475\pi$
$547$	−5.08979e6	−0.727331	−0.363665	+	0.931530i	$0.618475\pi$
$548$	0	0
$549$	8.35777e6	1.18348
$550$	0	0
$551$	1.52047e7	2.13353
$552$	0	0
$553$	−1.29110e6	−0.179535
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.80352e6	−1.33889	−0.669443	−	0.742863i	$-0.733467\pi$
$557$	−9.80352e6	−1.33889	−0.669443	+	0.742863i	$0.733467\pi$
$558$	0	0
$559$	6.09970e6	0.825618
$560$	0	0
$561$	4.88794e6	0.655721
$562$	0	0
$563$	1.28530e7	1.70897	0.854484	−	0.519478i	$-0.173874\pi$
$563$	1.28530e7	1.70897	0.854484	+	0.519478i	$0.173874\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.92865e6	0.251940
$568$	0	0
$569$	1.43558e6	0.185886	0.0929429	−	0.995671i	$-0.470373\pi$
$569$	1.43558e6	0.185886	0.0929429	+	0.995671i	$0.470373\pi$
$570$	0	0
$571$	−4.63588e6	−0.595034	−0.297517	−	0.954716i	$-0.596159\pi$
$571$	−4.63588e6	−0.595034	−0.297517	+	0.954716i	$0.596159\pi$
$572$	0	0
$573$	−2.34498e7	−2.98368
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−591099.	−0.0739130	−0.0369565	−	0.999317i	$-0.511766\pi$
$577$	−591099.	−0.0739130	−0.0369565	+	0.999317i	$0.511766\pi$
$578$	0	0
$579$	−1.19589e7	−1.48250
$580$	0	0
$581$	5.50955e6	0.677135
$582$	0	0
$583$	1.96913e6	0.239940
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.87938e6	−0.344908	−0.172454	−	0.985018i	$-0.555170\pi$
$587$	−2.87938e6	−0.344908	−0.172454	+	0.985018i	$0.555170\pi$
$588$	0	0
$589$	3.64316e6	0.432703
$590$	0	0
$591$	1.11044e7	1.30776
$592$	0	0
$593$	1.87004e6	0.218381	0.109190	−	0.994021i	$-0.465174\pi$
$593$	1.87004e6	0.218381	0.109190	+	0.994021i	$0.465174\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.86184e7	3.28632
$598$	0	0
$599$	−415288.	−0.0472915	−0.0236457	−	0.999720i	$-0.507527\pi$
$599$	−415288.	−0.0472915	−0.0236457	+	0.999720i	$0.507527\pi$
$600$	0	0
$601$	1.31366e7	1.48353	0.741764	−	0.670661i	$-0.233990\pi$
$601$	1.31366e7	1.48353	0.741764	+	0.670661i	$0.233990\pi$
$602$	0	0
$603$	−1.79380e7	−2.00900
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18756.3	0.00206622	0.00103311	−	0.999999i	$-0.499671\pi$
$607$	18756.3	0.00206622	0.00103311	+	0.999999i	$0.499671\pi$
$608$	0	0
$609$	8.21744e6	0.897828
$610$	0	0
$611$	−1.04571e7	−1.13321
$612$	0	0
$613$	−5.55708e6	−0.597304	−0.298652	−	0.954362i	$-0.596537\pi$
$613$	−5.55708e6	−0.597304	−0.298652	+	0.954362i	$0.596537\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.68269e7	−1.77947	−0.889735	−	0.456478i	$-0.849111\pi$
$617$	−1.68269e7	−1.77947	−0.889735	+	0.456478i	$0.849111\pi$
$618$	0	0
$619$	−6.25620e6	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$619$	−6.25620e6	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$620$	0	0
$621$	−9.42073e6	−0.980292
$622$	0	0
$623$	773381.	0.0798313
$624$	0	0
$625$	0	0
$626$	0	0
$627$	9.85526e6	1.00115
$628$	0	0
$629$	−1.94723e7	−1.96241
$630$	0	0
$631$	1.28950e7	1.28928	0.644641	−	0.764486i	$-0.277007\pi$
$631$	1.28950e7	1.28928	0.644641	+	0.764486i	$0.277007\pi$
$632$	0	0
$633$	−1.94333e7	−1.92769
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.71373e6	0.655565
$638$	0	0
$639$	−2.89387e7	−2.80367
$640$	0	0
$641$	−1.14483e7	−1.10051	−0.550257	−	0.834995i	$-0.685470\pi$
$641$	−1.14483e7	−1.10051	−0.550257	+	0.834995i	$0.685470\pi$
$642$	0	0
$643$	−9.95257e6	−0.949309	−0.474655	−	0.880172i	$-0.657427\pi$
$643$	−9.95257e6	−0.949309	−0.474655	+	0.880172i	$0.657427\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.71410e6	0.442729	0.221365	−	0.975191i	$-0.428949\pi$
$647$	4.71410e6	0.442729	0.221365	+	0.975191i	$0.428949\pi$
$648$	0	0
$649$	4.77456e6	0.444961
$650$	0	0
$651$	1.96896e6	0.182089
$652$	0	0
$653$	1.66636e7	1.52928	0.764639	−	0.644459i	$-0.222917\pi$
$653$	1.66636e7	1.52928	0.764639	+	0.644459i	$0.222917\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−2.05806e7	−1.86014
$658$	0	0
$659$	7.73348e6	0.693684	0.346842	−	0.937924i	$-0.387254\pi$
$659$	7.73348e6	0.693684	0.346842	+	0.937924i	$0.387254\pi$
$660$	0	0
$661$	8.58080e6	0.763878	0.381939	−	0.924188i	$-0.375256\pi$
$661$	8.58080e6	0.763878	0.381939	+	0.924188i	$0.375256\pi$
$662$	0	0
$663$	−2.13557e7	−1.88682
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.75639e6	−0.762097
$668$	0	0
$669$	−2.66675e7	−2.30366
$670$	0	0
$671$	2.27849e6	0.195362
$672$	0	0
$673$	1.06321e7	0.904859	0.452429	−	0.891800i	$-0.350557\pi$
$673$	1.06321e7	0.904859	0.452429	+	0.891800i	$0.350557\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.42422e7	−1.19427	−0.597137	−	0.802139i	$-0.703695\pi$
$677$	−1.42422e7	−1.19427	−0.597137	+	0.802139i	$0.703695\pi$
$678$	0	0
$679$	8.78658e6	0.731384
$680$	0	0
$681$	−1.33818e6	−0.110572
$682$	0	0
$683$	1.10334e7	0.905020	0.452510	−	0.891759i	$-0.350529\pi$
$683$	1.10334e7	0.905020	0.452510	+	0.891759i	$0.350529\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.06253e7	0.858912
$688$	0	0
$689$	−8.60324e6	−0.690421
$690$	0	0
$691$	−3.50968e6	−0.279623	−0.139811	−	0.990178i	$-0.544650\pi$
$691$	−3.50968e6	−0.279623	−0.139811	+	0.990178i	$0.544650\pi$
$692$	0	0
$693$	3.44189e6	0.272248
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.34366e6	0.182731
$698$	0	0
$699$	−4.60904e6	−0.356794
$700$	0	0
$701$	−2.06987e7	−1.59092	−0.795458	−	0.606009i	$-0.792770\pi$
$701$	−2.06987e7	−1.59092	−0.795458	+	0.606009i	$0.792770\pi$
$702$	0	0
$703$	−3.92607e7	−2.99620
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.06839e6	−0.230867
$708$	0	0
$709$	1.39218e7	1.04011	0.520057	−	0.854132i	$-0.325911\pi$
$709$	1.39218e7	1.04011	0.520057	+	0.854132i	$0.325911\pi$
$710$	0	0
$711$	−8.94142e6	−0.663335
$712$	0	0
$713$	−2.09809e6	−0.154561
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.14006e6	0.591330
$718$	0	0
$719$	−3.96939e6	−0.286353	−0.143176	−	0.989697i	$-0.545732\pi$
$719$	−3.96939e6	−0.286353	−0.143176	+	0.989697i	$0.545732\pi$
$720$	0	0
$721$	5.37227e6	0.384875
$722$	0	0
$723$	−1.63802e6	−0.116540
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.28442e7	0.901300	0.450650	−	0.892701i	$-0.351192\pi$
$727$	1.28442e7	0.901300	0.450650	+	0.892701i	$0.351192\pi$
$728$	0	0
$729$	−2.01644e7	−1.40529
$730$	0	0
$731$	−1.77847e7	−1.23099
$732$	0	0
$733$	−1.51085e7	−1.03863	−0.519316	−	0.854582i	$-0.673813\pi$
$733$	−1.51085e7	−1.03863	−0.519316	+	0.854582i	$0.673813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.89023e6	−0.331635
$738$	0	0
$739$	2.27094e7	1.52966	0.764829	−	0.644234i	$-0.222824\pi$
$739$	2.27094e7	1.52966	0.764829	+	0.644234i	$0.222824\pi$
$740$	0	0
$741$	−4.30583e7	−2.88079
$742$	0	0
$743$	−1.54143e6	−0.102436	−0.0512179	−	0.998688i	$-0.516310\pi$
$743$	−1.54143e6	−0.102436	−0.0512179	+	0.998688i	$0.516310\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.81559e7	2.50184
$748$	0	0
$749$	1.63793e6	0.106682
$750$	0	0
$751$	−7.76182e6	−0.502185	−0.251093	−	0.967963i	$-0.580790\pi$
$751$	−7.76182e6	−0.502185	−0.251093	+	0.967963i	$0.580790\pi$
$752$	0	0
$753$	4.33929e7	2.78889
$754$	0	0
$755$	0	0
$756$	0	0
$757$	69075.5	0.00438111	0.00219055	−	0.999998i	$-0.499303\pi$
$757$	69075.5	0.00438111	0.00219055	+	0.999998i	$0.499303\pi$
$758$	0	0
$759$	−5.67564e6	−0.357610
$760$	0	0
$761$	−9.54346e6	−0.597371	−0.298686	−	0.954352i	$-0.596548\pi$
$761$	−9.54346e6	−0.597371	−0.298686	+	0.954352i	$0.596548\pi$
$762$	0	0
$763$	3.50855e6	0.218180
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.08604e7	−1.28036
$768$	0	0
$769$	−1.65278e7	−1.00786	−0.503929	−	0.863745i	$-0.668113\pi$
$769$	−1.65278e7	−1.00786	−0.503929	+	0.863745i	$0.668113\pi$
$770$	0	0
$771$	−298225.	−0.0180679
$772$	0	0
$773$	−2.71610e7	−1.63492	−0.817460	−	0.575985i	$-0.804619\pi$
$773$	−2.71610e7	−1.63492	−0.817460	+	0.575985i	$0.804619\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.12186e7	−1.26085
$778$	0	0
$779$	4.72537e6	0.278992
$780$	0	0
$781$	−7.88925e6	−0.462816
$782$	0	0
$783$	2.57518e7	1.50108
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.98471e7	−1.71777	−0.858886	−	0.512167i	$-0.828843\pi$
$787$	−2.98471e7	−1.71777	−0.858886	+	0.512167i	$0.828843\pi$
$788$	0	0
$789$	−3.74396e7	−2.14111
$790$	0	0
$791$	4.61844e6	0.262455
$792$	0	0
$793$	−9.95487e6	−0.562150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.31780e6	−0.519598	−0.259799	−	0.965663i	$-0.583656\pi$
$797$	−9.31780e6	−0.519598	−0.259799	+	0.965663i	$0.583656\pi$
$798$	0	0
$799$	3.04895e7	1.68960
$800$	0	0
$801$	5.35598e6	0.294956
$802$	0	0
$803$	−5.61068e6	−0.307062
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.99156e7	−2.15754
$808$	0	0
$809$	3.50340e7	1.88200	0.940998	−	0.338411i	$-0.109889\pi$
$809$	3.50340e7	1.88200	0.940998	+	0.338411i	$0.109889\pi$
$810$	0	0
$811$	−3.42814e7	−1.83023	−0.915117	−	0.403190i	$-0.867902\pi$
$811$	−3.42814e7	−1.83023	−0.915117	+	0.403190i	$0.867902\pi$
$812$	0	0
$813$	−1.65107e7	−0.876069
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.58582e7	−1.87946
$818$	0	0
$819$	−1.50379e7	−0.783387
$820$	0	0
$821$	2.38494e7	1.23487	0.617433	−	0.786623i	$-0.288173\pi$
$821$	2.38494e7	1.23487	0.617433	+	0.786623i	$0.288173\pi$
$822$	0	0
$823$	−5.89850e6	−0.303558	−0.151779	−	0.988414i	$-0.548500\pi$
$823$	−5.89850e6	−0.303558	−0.151779	+	0.988414i	$0.548500\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.39931e7	0.711460	0.355730	−	0.934589i	$-0.384232\pi$
$827$	1.39931e7	0.711460	0.355730	+	0.934589i	$0.384232\pi$
$828$	0	0
$829$	−1.94439e7	−0.982644	−0.491322	−	0.870978i	$-0.663486\pi$
$829$	−1.94439e7	−0.982644	−0.491322	+	0.870978i	$0.663486\pi$
$830$	0	0
$831$	5.96646e7	2.99719
$832$	0	0
$833$	−1.95750e7	−0.977440
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.17033e6	0.304435
$838$	0	0
$839$	−1.46211e7	−0.717092	−0.358546	−	0.933512i	$-0.616727\pi$
$839$	−1.46211e7	−0.717092	−0.358546	+	0.933512i	$0.616727\pi$
$840$	0	0
$841$	3.42470e6	0.166968
$842$	0	0
$843$	1.30150e7	0.630777
$844$	0	0
$845$	0	0
$846$	0	0
$847$	938327.	0.0449413
$848$	0	0
$849$	−3.93550e7	−1.87383
$850$	0	0
$851$	2.26102e7	1.07024
$852$	0	0
$853$	−1.92349e7	−0.905144	−0.452572	−	0.891728i	$-0.649493\pi$
$853$	−1.92349e7	−0.905144	−0.452572	+	0.891728i	$0.649493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.91181e6	−0.181939	−0.0909694	−	0.995854i	$-0.528997\pi$
$857$	−3.91181e6	−0.181939	−0.0909694	+	0.995854i	$0.528997\pi$
$858$	0	0
$859$	−1.32707e7	−0.613638	−0.306819	−	0.951768i	$-0.599265\pi$
$859$	−1.32707e7	−0.613638	−0.306819	+	0.951768i	$0.599265\pi$
$860$	0	0
$861$	2.55384e6	0.117405
$862$	0	0
$863$	2.05251e7	0.938121	0.469061	−	0.883166i	$-0.344593\pi$
$863$	2.05251e7	0.938121	0.469061	+	0.883166i	$0.344593\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.50551e7	1.13201
$868$	0	0
$869$	−2.43760e6	−0.109500
$870$	0	0
$871$	2.13657e7	0.954273
$872$	0	0
$873$	6.08507e7	2.70228
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.33070e7	1.02326	0.511632	−	0.859205i	$-0.329041\pi$
$877$	2.33070e7	1.02326	0.511632	+	0.859205i	$0.329041\pi$
$878$	0	0
$879$	−3.73704e7	−1.63138
$880$	0	0
$881$	−9.31106e6	−0.404165	−0.202083	−	0.979368i	$-0.564771\pi$
$881$	−9.31106e6	−0.404165	−0.202083	+	0.979368i	$0.564771\pi$
$882$	0	0
$883$	−3.85172e7	−1.66247	−0.831233	−	0.555925i	$-0.812364\pi$
$883$	−3.85172e7	−1.66247	−0.831233	+	0.555925i	$0.812364\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.82081e7	−1.20383	−0.601915	−	0.798560i	$-0.705595\pi$
$887$	−2.82081e7	−1.20383	−0.601915	+	0.798560i	$0.705595\pi$
$888$	0	0
$889$	−7.25442e6	−0.307856
$890$	0	0
$891$	3.64130e6	0.153660
$892$	0	0
$893$	6.14742e7	2.57967
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.47972e7	1.02902
$898$	0	0
$899$	5.73520e6	0.236673
$900$	0	0
$901$	2.50842e7	1.02941
$902$	0	0
$903$	−1.93797e7	−0.790911
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.41463e7	0.570985	0.285492	−	0.958381i	$-0.407843\pi$
$907$	1.41463e7	0.570985	0.285492	+	0.958381i	$0.407843\pi$
$908$	0	0
$909$	−2.12499e7	−0.852995
$910$	0	0
$911$	−3.41700e7	−1.36411	−0.682055	−	0.731301i	$-0.738913\pi$
$911$	−3.41700e7	−1.36411	−0.682055	+	0.731301i	$0.738913\pi$
$912$	0	0
$913$	1.04020e7	0.412991
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.11062e6	0.357787
$918$	0	0
$919$	−9.77047e6	−0.381616	−0.190808	−	0.981627i	$-0.561111\pi$
$919$	−9.77047e6	−0.381616	−0.190808	+	0.981627i	$0.561111\pi$
$920$	0	0
$921$	−4.88966e7	−1.89946
$922$	0	0
$923$	3.44687e7	1.33174
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.72052e7	1.42202
$928$	0	0
$929$	−3.67368e7	−1.39657	−0.698285	−	0.715820i	$-0.746053\pi$
$929$	−3.67368e7	−1.39657	−0.698285	+	0.715820i	$0.746053\pi$
$930$	0	0
$931$	−3.94679e7	−1.49235
$932$	0	0
$933$	1.60496e6	0.0603616
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.55132e7	−1.32142	−0.660711	−	0.750641i	$-0.729745\pi$
$937$	−3.55132e7	−1.32142	−0.660711	+	0.750641i	$0.729745\pi$
$938$	0	0
$939$	−3.98377e6	−0.147445
$940$	0	0
$941$	8.61941e6	0.317325	0.158662	−	0.987333i	$-0.449282\pi$
$941$	8.61941e6	0.317325	0.158662	+	0.987333i	$0.449282\pi$
$942$	0	0
$943$	−2.72134e6	−0.0996560
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.88715e6	−0.213319	−0.106660	−	0.994296i	$-0.534016\pi$
$947$	−5.88715e6	−0.213319	−0.106660	+	0.994296i	$0.534016\pi$
$948$	0	0
$949$	2.45134e7	0.883565
$950$	0	0
$951$	3.18979e7	1.14370
$952$	0	0
$953$	−8.65335e6	−0.308640	−0.154320	−	0.988021i	$-0.549319\pi$
$953$	−8.65335e6	−0.308640	−0.154320	+	0.988021i	$0.549319\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.55145e7	0.547594
$958$	0	0
$959$	1.32634e7	0.465704
$960$	0	0
$961$	−2.72550e7	−0.952000
$962$	0	0
$963$	1.13434e7	0.394163
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.86242e7	−1.32829	−0.664146	−	0.747603i	$-0.731205\pi$
$967$	−3.86242e7	−1.32829	−0.664146	+	0.747603i	$0.731205\pi$
$968$	0	0
$969$	1.25544e8	4.29522
$970$	0	0
$971$	−1.98124e7	−0.674357	−0.337179	−	0.941441i	$-0.609473\pi$
$971$	−1.98124e7	−0.674357	−0.337179	+	0.941441i	$0.609473\pi$
$972$	0	0
$973$	2.77486e7	0.939636
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.08355e7	−1.36868	−0.684340	−	0.729163i	$-0.739910\pi$
$977$	−4.08355e7	−1.36868	−0.684340	+	0.729163i	$0.739910\pi$
$978$	0	0
$979$	1.46014e6	0.0486899
$980$	0	0
$981$	2.42981e7	0.806121
$982$	0	0
$983$	−1.60569e7	−0.530003	−0.265002	−	0.964248i	$-0.585372\pi$
$983$	−1.60569e7	−0.530003	−0.265002	+	0.964248i	$0.585372\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.32239e7	1.08557
$988$	0	0
$989$	2.06507e7	0.671343
$990$	0	0
$991$	3.67047e6	0.118724	0.0593619	−	0.998237i	$-0.481093\pi$
$991$	3.67047e6	0.118724	0.0593619	+	0.998237i	$0.481093\pi$
$992$	0	0
$993$	−7.61075e7	−2.44937
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.14427e7	−0.364579	−0.182289	−	0.983245i	$-0.558351\pi$
$997$	−1.14427e7	−0.364579	−0.182289	+	0.983245i	$0.558351\pi$
$998$	0	0
$999$	−6.64949e7	−2.10802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.m.1.14		14
5.2	odd	4		220.6.b.b.89.1	✓	14
5.3	odd	4		220.6.b.b.89.14	yes	14
5.4	even	2	inner	1100.6.a.m.1.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.b.b.89.1	✓	14	5.2	odd	4
220.6.b.b.89.14	yes	14	5.3	odd	4
1100.6.a.m.1.1		14	5.4	even	2	inner
1100.6.a.m.1.14		14	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+26.2077 q^{3} +64.0890 q^{7} +443.842 q^{9} +121.000 q^{11} -528.657 q^{13} +1541.39 q^{17} +3107.81 q^{19} +1679.62 q^{21} -1789.78 q^{23} +5263.61 q^{27} +4892.43 q^{29} +1172.26 q^{31} +3171.13 q^{33} -12632.9 q^{37} -13854.9 q^{39} +1520.48 q^{41} -11538.1 q^{43} +19780.6 q^{47} -12699.6 q^{49} +40396.2 q^{51} +16273.8 q^{53} +81448.4 q^{57} +39459.2 q^{59} +18830.5 q^{61} +28445.4 q^{63} -40415.1 q^{67} -46906.1 q^{69} -65200.4 q^{71} -46369.3 q^{73} +7754.77 q^{77} -20145.5 q^{79} +30093.4 q^{81} +85967.1 q^{83} +128219. q^{87} +12067.3 q^{89} -33881.1 q^{91} +30722.2 q^{93} +137100. q^{97} +53704.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 1142 q^{9} + 1694 q^{11} + 4540 q^{19} + 3824 q^{21} + 9972 q^{29} + 19076 q^{31} + 13616 q^{39} + 15052 q^{41} + 55346 q^{49} - 13380 q^{51} + 2108 q^{59} + 11660 q^{61} - 1620 q^{69} - 89284 q^{71}+ \cdots + 138182 q^{99}+O(q^{100}) \) 14 * q + 1142 * q^9 + 1694 * q^11 + 4540 * q^19 + 3824 * q^21 + 9972 * q^29 + 19076 * q^31 + 13616 * q^39 + 15052 * q^41 + 55346 * q^49 - 13380 * q^51 + 2108 * q^59 + 11660 * q^61 - 1620 * q^69 - 89284 * q^71 - 19432 * q^79 - 94714 * q^81 + 38836 * q^89 + 39632 * q^91 + 138182 * q^99

Introduction

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