LMFDB - Embedded newform 1100.6.a.g.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 809x^{3} + 562x^{2} + 88864x + 320704 $ x^5 - 2x^4 - 809x^3 + 562x^2 + 88864x + 320704
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 220)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.57706$ 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)

$f(q)$	$=$	$q-11.5771 q^{3} +76.8335 q^{7} -108.972 q^{9} +121.000 q^{11} +121.483 q^{13} -2220.42 q^{17} +1824.09 q^{19} -889.507 q^{21} -513.500 q^{23} +4074.80 q^{27} +2859.46 q^{29} +7688.01 q^{31} -1400.82 q^{33} -5056.28 q^{37} -1406.42 q^{39} -10557.6 q^{41} -16899.1 q^{43} +17656.1 q^{47} -10903.6 q^{49} +25705.9 q^{51} -30205.1 q^{53} -21117.6 q^{57} -11262.7 q^{59} +21940.3 q^{61} -8372.68 q^{63} -13810.2 q^{67} +5944.82 q^{69} +63315.4 q^{71} +35545.8 q^{73} +9296.86 q^{77} +29780.8 q^{79} -20694.1 q^{81} +94482.7 q^{83} -33104.1 q^{87} +112063. q^{89} +9333.97 q^{91} -89004.6 q^{93} -36803.2 q^{97} -13185.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 18 q^{3} - 130 q^{7} + 471 q^{9} + 605 q^{11} - 600 q^{13} + 176 q^{17} + 102 q^{19} + 1468 q^{21} - 4434 q^{23} - 7764 q^{27} - 742 q^{29} + 3132 q^{31} - 2178 q^{33} - 11296 q^{37} + 12032 q^{39}+ \cdots + 56991 q^{99}+O(q^{100}) $ 5 * q - 18 * q^3 - 130 * q^7 + 471 * q^9 + 605 * q^11 - 600 * q^13 + 176 * q^17 + 102 * q^19 + 1468 * q^21 - 4434 * q^23 - 7764 * q^27 - 742 * q^29 + 3132 * q^31 - 2178 * q^33 - 11296 * q^37 + 12032 * q^39 + 7154 * q^41 - 18956 * q^43 - 32370 * q^47 + 34263 * q^49 + 29014 * q^51 - 13328 * q^53 - 34168 * q^57 + 72728 * q^59 + 73402 * q^61 - 73700 * q^63 - 61018 * q^67 + 47124 * q^69 + 86120 * q^71 - 62784 * q^73 - 15730 * q^77 + 93320 * q^79 + 133785 * q^81 - 107548 * q^83 + 18114 * q^87 + 22164 * q^89 + 28404 * q^91 - 135290 * q^93 - 26934 * q^97 + 56991 * 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
$3$	−11.5771	−0.742669	−0.371334	−	0.928499i	$-0.621100\pi$
$3$	−11.5771	−0.742669	−0.371334	+	0.928499i	$0.621100\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	76.8335	0.592660	0.296330	−	0.955086i	$-0.404237\pi$
$7$	76.8335	0.592660	0.296330	+	0.955086i	$0.404237\pi$
$8$	0	0
$9$	−108.972	−0.448443
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	121.483	0.199369	0.0996844	−	0.995019i	$-0.468217\pi$
$13$	121.483	0.199369	0.0996844	+	0.995019i	$0.468217\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2220.42	−1.86342	−0.931712	−	0.363197i	$-0.881685\pi$
$17$	−2220.42	−1.86342	−0.931712	+	0.363197i	$0.881685\pi$
$18$	0	0
$19$	1824.09	1.15921	0.579605	−	0.814898i	$-0.303207\pi$
$19$	1824.09	1.15921	0.579605	+	0.814898i	$0.303207\pi$
$20$	0	0
$21$	−889.507	−0.440150
$22$	0	0
$23$	−513.500	−0.202405	−0.101202	−	0.994866i	$-0.532269\pi$
$23$	−513.500	−0.202405	−0.101202	+	0.994866i	$0.532269\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4074.80	1.07571
$28$	0	0
$29$	2859.46	0.631377	0.315688	−	0.948863i	$-0.397765\pi$
$29$	2859.46	0.631377	0.315688	+	0.948863i	$0.397765\pi$
$30$	0	0
$31$	7688.01	1.43684	0.718422	−	0.695607i	$-0.244865\pi$
$31$	7688.01	1.43684	0.718422	+	0.695607i	$0.244865\pi$
$32$	0	0
$33$	−1400.82	−0.223923
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5056.28	−0.607193	−0.303596	−	0.952801i	$-0.598187\pi$
$37$	−5056.28	−0.607193	−0.303596	+	0.952801i	$0.598187\pi$
$38$	0	0
$39$	−1406.42	−0.148065
$40$	0	0
$41$	−10557.6	−0.980858	−0.490429	−	0.871481i	$-0.663160\pi$
$41$	−10557.6	−0.980858	−0.490429	+	0.871481i	$0.663160\pi$
$42$	0	0
$43$	−16899.1	−1.39377	−0.696886	−	0.717182i	$-0.745432\pi$
$43$	−16899.1	−1.39377	−0.696886	+	0.717182i	$0.745432\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	17656.1	1.16587	0.582936	−	0.812518i	$-0.301904\pi$
$47$	17656.1	1.16587	0.582936	+	0.812518i	$0.301904\pi$
$48$	0	0
$49$	−10903.6	−0.648754
$50$	0	0
$51$	25705.9	1.38391
$52$	0	0
$53$	−30205.1	−1.47703	−0.738517	−	0.674235i	$-0.764474\pi$
$53$	−30205.1	−1.47703	−0.738517	+	0.674235i	$0.764474\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−21117.6	−0.860909
$58$	0	0
$59$	−11262.7	−0.421223	−0.210611	−	0.977570i	$-0.567545\pi$
$59$	−11262.7	−0.421223	−0.210611	+	0.977570i	$0.567545\pi$
$60$	0	0
$61$	21940.3	0.754948	0.377474	−	0.926020i	$-0.376793\pi$
$61$	21940.3	0.754948	0.377474	+	0.926020i	$0.376793\pi$
$62$	0	0
$63$	−8372.68	−0.265774
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13810.2	−0.375850	−0.187925	−	0.982183i	$-0.560176\pi$
$67$	−13810.2	−0.375850	−0.187925	+	0.982183i	$0.560176\pi$
$68$	0	0
$69$	5944.82	0.150320
$70$	0	0
$71$	63315.4	1.49061	0.745304	−	0.666725i	$-0.232305\pi$
$71$	63315.4	1.49061	0.745304	+	0.666725i	$0.232305\pi$
$72$	0	0
$73$	35545.8	0.780695	0.390347	−	0.920668i	$-0.372355\pi$
$73$	35545.8	0.780695	0.390347	+	0.920668i	$0.372355\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9296.86	0.178694
$78$	0	0
$79$	29780.8	0.536870	0.268435	−	0.963298i	$-0.413494\pi$
$79$	29780.8	0.536870	0.268435	+	0.963298i	$0.413494\pi$
$80$	0	0
$81$	−20694.1	−0.350456
$82$	0	0
$83$	94482.7	1.50542	0.752709	−	0.658354i	$-0.228747\pi$
$83$	94482.7	1.50542	0.752709	+	0.658354i	$0.228747\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−33104.1	−0.468904
$88$	0	0
$89$	112063.	1.49964	0.749819	−	0.661643i	$-0.230141\pi$
$89$	112063.	1.49964	0.749819	+	0.661643i	$0.230141\pi$
$90$	0	0
$91$	9333.97	0.118158
$92$	0	0
$93$	−89004.6	−1.06710
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−36803.2	−0.397152	−0.198576	−	0.980086i	$-0.563632\pi$
$97$	−36803.2	−0.397152	−0.198576	+	0.980086i	$0.563632\pi$
$98$	0	0
$99$	−13185.6	−0.135211
$100$	0	0
$101$	−20799.0	−0.202880	−0.101440	−	0.994842i	$-0.532345\pi$
$101$	−20799.0	−0.202880	−0.101440	+	0.994842i	$0.532345\pi$
$102$	0	0
$103$	−101114.	−0.939110	−0.469555	−	0.882903i	$-0.655586\pi$
$103$	−101114.	−0.939110	−0.469555	+	0.882903i	$0.655586\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	177051.	1.49499	0.747497	−	0.664265i	$-0.231255\pi$
$107$	177051.	1.49499	0.747497	+	0.664265i	$0.231255\pi$
$108$	0	0
$109$	−101393.	−0.817410	−0.408705	−	0.912667i	$-0.634020\pi$
$109$	−101393.	−0.817410	−0.408705	+	0.912667i	$0.634020\pi$
$110$	0	0
$111$	58536.9	0.450943
$112$	0	0
$113$	−178588.	−1.31570	−0.657849	−	0.753150i	$-0.728533\pi$
$113$	−178588.	−1.31570	−0.657849	+	0.753150i	$0.728533\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13238.2	−0.0894056
$118$	0	0
$119$	−170602.	−1.10438
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	122226.	0.728452
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19588.0	0.107766	0.0538829	−	0.998547i	$-0.482840\pi$
$127$	19588.0	0.107766	0.0538829	+	0.998547i	$0.482840\pi$
$128$	0	0
$129$	195642.	1.03511
$130$	0	0
$131$	258238.	1.31474	0.657372	−	0.753566i	$-0.271668\pi$
$131$	258238.	1.31474	0.657372	+	0.753566i	$0.271668\pi$
$132$	0	0
$133$	140151.	0.687017
$134$	0	0
$135$	0	0
$136$	0	0
$137$	276147.	1.25701	0.628506	−	0.777805i	$-0.283667\pi$
$137$	276147.	1.25701	0.628506	+	0.777805i	$0.283667\pi$
$138$	0	0
$139$	−51378.7	−0.225552	−0.112776	−	0.993620i	$-0.535974\pi$
$139$	−51378.7	−0.225552	−0.112776	+	0.993620i	$0.535974\pi$
$140$	0	0
$141$	−204406.	−0.865856
$142$	0	0
$143$	14699.4	0.0601120
$144$	0	0
$145$	0	0
$146$	0	0
$147$	126232.	0.481809
$148$	0	0
$149$	−450876.	−1.66376	−0.831881	−	0.554954i	$-0.812736\pi$
$149$	−450876.	−1.66376	−0.831881	+	0.554954i	$0.812736\pi$
$150$	0	0
$151$	−484744.	−1.73009	−0.865047	−	0.501690i	$-0.832712\pi$
$151$	−484744.	−1.73009	−0.865047	+	0.501690i	$0.832712\pi$
$152$	0	0
$153$	241962.	0.835640
$154$	0	0
$155$	0	0
$156$	0	0
$157$	297124.	0.962031	0.481015	−	0.876712i	$-0.340268\pi$
$157$	297124.	0.962031	0.481015	+	0.876712i	$0.340268\pi$
$158$	0	0
$159$	349686.	1.09695
$160$	0	0
$161$	−39454.0	−0.119957
$162$	0	0
$163$	−545387.	−1.60781	−0.803907	−	0.594755i	$-0.797249\pi$
$163$	−545387.	−1.60781	−0.803907	+	0.594755i	$0.797249\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	167813.	0.465623	0.232812	−	0.972522i	$-0.425207\pi$
$167$	167813.	0.465623	0.232812	+	0.972522i	$0.425207\pi$
$168$	0	0
$169$	−356535.	−0.960252
$170$	0	0
$171$	−198774.	−0.519840
$172$	0	0
$173$	435664.	1.10672	0.553358	−	0.832943i	$-0.313346\pi$
$173$	435664.	1.10672	0.553358	+	0.832943i	$0.313346\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	130389.	0.312829
$178$	0	0
$179$	−536924.	−1.25251	−0.626253	−	0.779620i	$-0.715412\pi$
$179$	−536924.	−1.25251	−0.626253	+	0.779620i	$0.715412\pi$
$180$	0	0
$181$	−503910.	−1.14329	−0.571645	−	0.820501i	$-0.693695\pi$
$181$	−503910.	−1.14329	−0.571645	+	0.820501i	$0.693695\pi$
$182$	0	0
$183$	−254004.	−0.560677
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−268670.	−0.561844
$188$	0	0
$189$	313081.	0.637533
$190$	0	0
$191$	471596.	0.935377	0.467688	−	0.883893i	$-0.345087\pi$
$191$	471596.	0.935377	0.467688	+	0.883893i	$0.345087\pi$
$192$	0	0
$193$	−171663.	−0.331728	−0.165864	−	0.986149i	$-0.553041\pi$
$193$	−171663.	−0.331728	−0.165864	+	0.986149i	$0.553041\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	44206.4	0.0811558	0.0405779	−	0.999176i	$-0.487080\pi$
$197$	44206.4	0.0811558	0.0405779	+	0.999176i	$0.487080\pi$
$198$	0	0
$199$	558881.	1.00043	0.500215	−	0.865901i	$-0.333254\pi$
$199$	558881.	1.00043	0.500215	+	0.865901i	$0.333254\pi$
$200$	0	0
$201$	159882.	0.279132
$202$	0	0
$203$	219702.	0.374192
$204$	0	0
$205$	0	0
$206$	0	0
$207$	55957.0	0.0907671
$208$	0	0
$209$	220715.	0.349515
$210$	0	0
$211$	−510035.	−0.788667	−0.394333	−	0.918967i	$-0.629024\pi$
$211$	−510035.	−0.788667	−0.394333	+	0.918967i	$0.629024\pi$
$212$	0	0
$213$	−733006.	−1.10703
$214$	0	0
$215$	0	0
$216$	0	0
$217$	590697.	0.851561
$218$	0	0
$219$	−411516.	−0.579798
$220$	0	0
$221$	−269743.	−0.371509
$222$	0	0
$223$	−1.45648e6	−1.96130	−0.980648	−	0.195777i	$-0.937277\pi$
$223$	−1.45648e6	−1.96130	−0.980648	+	0.195777i	$0.937277\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−710880.	−0.915654	−0.457827	−	0.889041i	$-0.651372\pi$
$227$	−710880.	−0.915654	−0.457827	+	0.889041i	$0.651372\pi$
$228$	0	0
$229$	−968984.	−1.22103	−0.610517	−	0.792003i	$-0.709038\pi$
$229$	−968984.	−1.22103	−0.610517	+	0.792003i	$0.709038\pi$
$230$	0	0
$231$	−107630.	−0.132710
$232$	0	0
$233$	−115072.	−0.138861	−0.0694306	−	0.997587i	$-0.522118\pi$
$233$	−115072.	−0.138861	−0.0694306	+	0.997587i	$0.522118\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−344774.	−0.398716
$238$	0	0
$239$	1.19934e6	1.35815	0.679076	−	0.734068i	$-0.262381\pi$
$239$	1.19934e6	1.35815	0.679076	+	0.734068i	$0.262381\pi$
$240$	0	0
$241$	−1.56629e6	−1.73712	−0.868561	−	0.495583i	$-0.834955\pi$
$241$	−1.56629e6	−1.73712	−0.868561	+	0.495583i	$0.834955\pi$
$242$	0	0
$243$	−750599.	−0.815441
$244$	0	0
$245$	0	0
$246$	0	0
$247$	221596.	0.231110
$248$	0	0
$249$	−1.09383e6	−1.11803
$250$	0	0
$251$	311142.	0.311727	0.155864	−	0.987779i	$-0.450184\pi$
$251$	311142.	0.311727	0.155864	+	0.987779i	$0.450184\pi$
$252$	0	0
$253$	−62133.5	−0.0610274
$254$	0	0
$255$	0	0
$256$	0	0
$257$	385662.	0.364228	0.182114	−	0.983277i	$-0.441706\pi$
$257$	385662.	0.364228	0.182114	+	0.983277i	$0.441706\pi$
$258$	0	0
$259$	−388492.	−0.359859
$260$	0	0
$261$	−311600.	−0.283136
$262$	0	0
$263$	−1.37022e6	−1.22152	−0.610758	−	0.791817i	$-0.709135\pi$
$263$	−1.37022e6	−1.22152	−0.610758	+	0.791817i	$0.709135\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.29736e6	−1.11373
$268$	0	0
$269$	84063.6	0.0708316	0.0354158	−	0.999373i	$-0.488724\pi$
$269$	84063.6	0.0708316	0.0354158	+	0.999373i	$0.488724\pi$
$270$	0	0
$271$	−253210.	−0.209439	−0.104719	−	0.994502i	$-0.533394\pi$
$271$	−253210.	−0.209439	−0.104719	+	0.994502i	$0.533394\pi$
$272$	0	0
$273$	−108060.	−0.0877523
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−750352.	−0.587578	−0.293789	−	0.955870i	$-0.594916\pi$
$277$	−750352.	−0.587578	−0.293789	+	0.955870i	$0.594916\pi$
$278$	0	0
$279$	−837775.	−0.644343
$280$	0	0
$281$	−2.44406e6	−1.84648	−0.923242	−	0.384218i	$-0.874471\pi$
$281$	−2.44406e6	−1.84648	−0.923242	+	0.384218i	$0.874471\pi$
$282$	0	0
$283$	446166.	0.331155	0.165577	−	0.986197i	$-0.447051\pi$
$283$	446166.	0.331155	0.165577	+	0.986197i	$0.447051\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−811178.	−0.581315
$288$	0	0
$289$	3.51039e6	2.47235
$290$	0	0
$291$	426073.	0.294952
$292$	0	0
$293$	−2.37475e6	−1.61603	−0.808014	−	0.589164i	$-0.799457\pi$
$293$	−2.37475e6	−1.61603	−0.808014	+	0.589164i	$0.799457\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	493050.	0.324340
$298$	0	0
$299$	−62381.6	−0.0403532
$300$	0	0
$301$	−1.29842e6	−0.826033
$302$	0	0
$303$	240791.	0.150672
$304$	0	0
$305$	0	0
$306$	0	0
$307$	154947.	0.0938293	0.0469147	−	0.998899i	$-0.485061\pi$
$307$	154947.	0.0938293	0.0469147	+	0.998899i	$0.485061\pi$
$308$	0	0
$309$	1.17060e6	0.697448
$310$	0	0
$311$	281180.	0.164848	0.0824239	−	0.996597i	$-0.473734\pi$
$311$	281180.	0.164848	0.0824239	+	0.996597i	$0.473734\pi$
$312$	0	0
$313$	576972.	0.332885	0.166443	−	0.986051i	$-0.446772\pi$
$313$	576972.	0.332885	0.166443	+	0.986051i	$0.446772\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	331855.	0.185481	0.0927406	−	0.995690i	$-0.470437\pi$
$317$	331855.	0.185481	0.0927406	+	0.995690i	$0.470437\pi$
$318$	0	0
$319$	345994.	0.190367
$320$	0	0
$321$	−2.04973e6	−1.11029
$322$	0	0
$323$	−4.05024e6	−2.16010
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.17383e6	0.607065
$328$	0	0
$329$	1.35658e6	0.690965
$330$	0	0
$331$	−1.63063e6	−0.818059	−0.409029	−	0.912521i	$-0.634133\pi$
$331$	−1.63063e6	−0.818059	−0.409029	+	0.912521i	$0.634133\pi$
$332$	0	0
$333$	550991.	0.272291
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.07071e6	0.513567	0.256783	−	0.966469i	$-0.417337\pi$
$337$	1.07071e6	0.513567	0.256783	+	0.966469i	$0.417337\pi$
$338$	0	0
$339$	2.06752e6	0.977128
$340$	0	0
$341$	930249.	0.433225
$342$	0	0
$343$	−2.12910e6	−0.977151
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−726632.	−0.323960	−0.161980	−	0.986794i	$-0.551788\pi$
$347$	−726632.	−0.323960	−0.161980	+	0.986794i	$0.551788\pi$
$348$	0	0
$349$	−276008.	−0.121299	−0.0606496	−	0.998159i	$-0.519317\pi$
$349$	−276008.	−0.121299	−0.0606496	+	0.998159i	$0.519317\pi$
$350$	0	0
$351$	495019.	0.214464
$352$	0	0
$353$	224676.	0.0959666	0.0479833	−	0.998848i	$-0.484721\pi$
$353$	224676.	0.0959666	0.0479833	+	0.998848i	$0.484721\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.97507e6	0.820187
$358$	0	0
$359$	−208507.	−0.0853856	−0.0426928	−	0.999088i	$-0.513594\pi$
$359$	−208507.	−0.0853856	−0.0426928	+	0.999088i	$0.513594\pi$
$360$	0	0
$361$	851202.	0.343768
$362$	0	0
$363$	−169500.	−0.0675154
$364$	0	0
$365$	0	0
$366$	0	0
$367$	261402.	0.101308	0.0506540	−	0.998716i	$-0.483869\pi$
$367$	261402.	0.101308	0.0506540	+	0.998716i	$0.483869\pi$
$368$	0	0
$369$	1.15048e6	0.439859
$370$	0	0
$371$	−2.32076e6	−0.875379
$372$	0	0
$373$	−3.16739e6	−1.17877	−0.589386	−	0.807851i	$-0.700630\pi$
$373$	−3.16739e6	−1.17877	−0.589386	+	0.807851i	$0.700630\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	347376.	0.125877
$378$	0	0
$379$	−3.01152e6	−1.07693	−0.538465	−	0.842648i	$-0.680996\pi$
$379$	−3.01152e6	−1.07693	−0.538465	+	0.842648i	$0.680996\pi$
$380$	0	0
$381$	−226772.	−0.0800343
$382$	0	0
$383$	−612371.	−0.213313	−0.106657	−	0.994296i	$-0.534015\pi$
$383$	−612371.	−0.213313	−0.106657	+	0.994296i	$0.534015\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.84152e6	0.625027
$388$	0	0
$389$	−5.20939e6	−1.74547	−0.872735	−	0.488194i	$-0.837656\pi$
$389$	−5.20939e6	−1.74547	−0.872735	+	0.488194i	$0.837656\pi$
$390$	0	0
$391$	1.14018e6	0.377166
$392$	0	0
$393$	−2.98963e6	−0.976420
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.25602e6	−0.399964	−0.199982	−	0.979800i	$-0.564088\pi$
$397$	−1.25602e6	−0.399964	−0.199982	+	0.979800i	$0.564088\pi$
$398$	0	0
$399$	−1.62254e6	−0.510226
$400$	0	0
$401$	−3.67519e6	−1.14135	−0.570675	−	0.821176i	$-0.693318\pi$
$401$	−3.67519e6	−1.14135	−0.570675	+	0.821176i	$0.693318\pi$
$402$	0	0
$403$	933963.	0.286462
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−611810.	−0.183076
$408$	0	0
$409$	−4.45766e6	−1.31765	−0.658823	−	0.752298i	$-0.728945\pi$
$409$	−4.45766e6	−1.31765	−0.658823	+	0.752298i	$0.728945\pi$
$410$	0	0
$411$	−3.19697e6	−0.933544
$412$	0	0
$413$	−865351.	−0.249642
$414$	0	0
$415$	0	0
$416$	0	0
$417$	594815.	0.167510
$418$	0	0
$419$	−140989.	−0.0392328	−0.0196164	−	0.999808i	$-0.506245\pi$
$419$	−140989.	−0.0392328	−0.0196164	+	0.999808i	$0.506245\pi$
$420$	0	0
$421$	−3.13175e6	−0.861155	−0.430577	−	0.902554i	$-0.641690\pi$
$421$	−3.13175e6	−0.861155	−0.430577	+	0.902554i	$0.641690\pi$
$422$	0	0
$423$	−1.92402e6	−0.522827
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.68575e6	0.447428
$428$	0	0
$429$	−170176.	−0.0446433
$430$	0	0
$431$	443009.	0.114873	0.0574367	−	0.998349i	$-0.481707\pi$
$431$	443009.	0.114873	0.0574367	+	0.998349i	$0.481707\pi$
$432$	0	0
$433$	6.69167e6	1.71520	0.857599	−	0.514318i	$-0.171955\pi$
$433$	6.69167e6	1.71520	0.857599	+	0.514318i	$0.171955\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−936670.	−0.234630
$438$	0	0
$439$	−5.17022e6	−1.28041	−0.640203	−	0.768205i	$-0.721150\pi$
$439$	−5.17022e6	−1.28041	−0.640203	+	0.768205i	$0.721150\pi$
$440$	0	0
$441$	1.18818e6	0.290929
$442$	0	0
$443$	−4.04670e6	−0.979696	−0.489848	−	0.871808i	$-0.662948\pi$
$443$	−4.04670e6	−0.979696	−0.489848	+	0.871808i	$0.662948\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.21981e6	1.23562
$448$	0	0
$449$	6.26809e6	1.46730	0.733651	−	0.679527i	$-0.237815\pi$
$449$	6.26809e6	1.46730	0.733651	+	0.679527i	$0.237815\pi$
$450$	0	0
$451$	−1.27747e6	−0.295740
$452$	0	0
$453$	5.61191e6	1.28489
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.15794e6	1.15528	0.577638	−	0.816293i	$-0.303974\pi$
$457$	5.15794e6	1.15528	0.577638	+	0.816293i	$0.303974\pi$
$458$	0	0
$459$	−9.04774e6	−2.00451
$460$	0	0
$461$	5.86780e6	1.28595	0.642973	−	0.765889i	$-0.277701\pi$
$461$	5.86780e6	1.28595	0.642973	+	0.765889i	$0.277701\pi$
$462$	0	0
$463$	−523571.	−0.113507	−0.0567536	−	0.998388i	$-0.518075\pi$
$463$	−523571.	−0.113507	−0.0567536	+	0.998388i	$0.518075\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.07992e6	0.865684	0.432842	−	0.901470i	$-0.357511\pi$
$467$	4.07992e6	0.865684	0.432842	+	0.901470i	$0.357511\pi$
$468$	0	0
$469$	−1.06109e6	−0.222751
$470$	0	0
$471$	−3.43983e6	−0.714470
$472$	0	0
$473$	−2.04479e6	−0.420238
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.29150e6	0.662366
$478$	0	0
$479$	1.06505e6	0.212096	0.106048	−	0.994361i	$-0.466180\pi$
$479$	1.06505e6	0.212096	0.106048	+	0.994361i	$0.466180\pi$
$480$	0	0
$481$	−614252.	−0.121055
$482$	0	0
$483$	456762.	0.0890886
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−5.98861e6	−1.14420	−0.572102	−	0.820182i	$-0.693872\pi$
$487$	−5.98861e6	−1.14420	−0.572102	+	0.820182i	$0.693872\pi$
$488$	0	0
$489$	6.31398e6	1.19407
$490$	0	0
$491$	−7.86776e6	−1.47281	−0.736407	−	0.676539i	$-0.763479\pi$
$491$	−7.86776e6	−1.47281	−0.736407	+	0.676539i	$0.763479\pi$
$492$	0	0
$493$	−6.34918e6	−1.17652
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.86474e6	0.883423
$498$	0	0
$499$	5.79961e6	1.04267	0.521335	−	0.853352i	$-0.325434\pi$
$499$	5.79961e6	1.04267	0.521335	+	0.853352i	$0.325434\pi$
$500$	0	0
$501$	−1.94278e6	−0.345804
$502$	0	0
$503$	−6.93841e6	−1.22276	−0.611378	−	0.791338i	$-0.709385\pi$
$503$	−6.93841e6	−1.22276	−0.611378	+	0.791338i	$0.709385\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.12763e6	0.713149
$508$	0	0
$509$	6.50634e6	1.11312	0.556560	−	0.830807i	$-0.312121\pi$
$509$	6.50634e6	1.11312	0.556560	+	0.830807i	$0.312121\pi$
$510$	0	0
$511$	2.73111e6	0.462687
$512$	0	0
$513$	7.43279e6	1.24698
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.13639e6	0.351523
$518$	0	0
$519$	−5.04371e6	−0.821924
$520$	0	0
$521$	4.60095e6	0.742597	0.371299	−	0.928514i	$-0.378913\pi$
$521$	4.60095e6	0.742597	0.371299	+	0.928514i	$0.378913\pi$
$522$	0	0
$523$	−3.47107e6	−0.554893	−0.277446	−	0.960741i	$-0.589488\pi$
$523$	−3.47107e6	−0.554893	−0.277446	+	0.960741i	$0.589488\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.70706e7	−2.67745
$528$	0	0
$529$	−6.17266e6	−0.959032
$530$	0	0
$531$	1.22731e6	0.188894
$532$	0	0
$533$	−1.28257e6	−0.195552
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.21600e6	0.930198
$538$	0	0
$539$	−1.31934e6	−0.195607
$540$	0	0
$541$	6.06748e6	0.891283	0.445641	−	0.895212i	$-0.352976\pi$
$541$	6.06748e6	0.891283	0.445641	+	0.895212i	$0.352976\pi$
$542$	0	0
$543$	5.83379e6	0.849086
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.18230e6	0.311851	0.155925	−	0.987769i	$-0.450164\pi$
$547$	2.18230e6	0.311851	0.155925	+	0.987769i	$0.450164\pi$
$548$	0	0
$549$	−2.39087e6	−0.338551
$550$	0	0
$551$	5.21590e6	0.731898
$552$	0	0
$553$	2.28817e6	0.318181
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.47241e6	0.747379	0.373689	−	0.927554i	$-0.378093\pi$
$557$	5.47241e6	0.747379	0.373689	+	0.927554i	$0.378093\pi$
$558$	0	0
$559$	−2.05295e6	−0.277875
$560$	0	0
$561$	3.11041e6	0.417264
$562$	0	0
$563$	2.72211e6	0.361939	0.180969	−	0.983489i	$-0.442077\pi$
$563$	2.72211e6	0.361939	0.180969	+	0.983489i	$0.442077\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.59000e6	−0.207701
$568$	0	0
$569$	−5.31250e6	−0.687889	−0.343945	−	0.938990i	$-0.611763\pi$
$569$	−5.31250e6	−0.687889	−0.343945	+	0.938990i	$0.611763\pi$
$570$	0	0
$571$	7.65194e6	0.982158	0.491079	−	0.871115i	$-0.336603\pi$
$571$	7.65194e6	0.982158	0.491079	+	0.871115i	$0.336603\pi$
$572$	0	0
$573$	−5.45969e6	−0.694675
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.08703e6	−1.01123	−0.505614	−	0.862760i	$-0.668734\pi$
$577$	−8.08703e6	−1.01123	−0.505614	+	0.862760i	$0.668734\pi$
$578$	0	0
$579$	1.98735e6	0.246364
$580$	0	0
$581$	7.25944e6	0.892201
$582$	0	0
$583$	−3.65482e6	−0.445343
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.49611e6	−0.658355	−0.329178	−	0.944268i	$-0.606771\pi$
$587$	−5.49611e6	−0.658355	−0.329178	+	0.944268i	$0.606771\pi$
$588$	0	0
$589$	1.40236e7	1.66560
$590$	0	0
$591$	−511780.	−0.0602719
$592$	0	0
$593$	−1.45110e7	−1.69457	−0.847284	−	0.531140i	$-0.821764\pi$
$593$	−1.45110e7	−1.69457	−0.847284	+	0.531140i	$0.821764\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.47020e6	−0.742989
$598$	0	0
$599$	5.75533e6	0.655395	0.327698	−	0.944783i	$-0.393727\pi$
$599$	5.75533e6	0.655395	0.327698	+	0.944783i	$0.393727\pi$
$600$	0	0
$601$	−1.51057e7	−1.70590	−0.852950	−	0.521992i	$-0.825189\pi$
$601$	−1.51057e7	−1.70590	−0.852950	+	0.521992i	$0.825189\pi$
$602$	0	0
$603$	1.50492e6	0.168547
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.49543e6	0.495222	0.247611	−	0.968860i	$-0.420355\pi$
$607$	4.49543e6	0.495222	0.247611	+	0.968860i	$0.420355\pi$
$608$	0	0
$609$	−2.54351e6	−0.277901
$610$	0	0
$611$	2.14492e6	0.232438
$612$	0	0
$613$	1.08995e6	0.117154	0.0585769	−	0.998283i	$-0.481344\pi$
$613$	1.08995e6	0.117154	0.0585769	+	0.998283i	$0.481344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.51849e6	−0.160583	−0.0802916	−	0.996771i	$-0.525585\pi$
$617$	−1.51849e6	−0.160583	−0.0802916	+	0.996771i	$0.525585\pi$
$618$	0	0
$619$	8.26864e6	0.867376	0.433688	−	0.901063i	$-0.357212\pi$
$619$	8.26864e6	0.867376	0.433688	+	0.901063i	$0.357212\pi$
$620$	0	0
$621$	−2.09241e6	−0.217730
$622$	0	0
$623$	8.61018e6	0.888775
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.55523e6	−0.259574
$628$	0	0
$629$	1.12270e7	1.13146
$630$	0	0
$631$	−7.69787e6	−0.769657	−0.384829	−	0.922988i	$-0.625739\pi$
$631$	−7.69787e6	−0.769657	−0.384829	+	0.922988i	$0.625739\pi$
$632$	0	0
$633$	5.90470e6	0.585718
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.32460e6	−0.129341
$638$	0	0
$639$	−6.89958e6	−0.668452
$640$	0	0
$641$	1.84592e7	1.77447	0.887233	−	0.461322i	$-0.152625\pi$
$641$	1.84592e7	1.77447	0.887233	+	0.461322i	$0.152625\pi$
$642$	0	0
$643$	1.47148e7	1.40355	0.701773	−	0.712401i	$-0.252392\pi$
$643$	1.47148e7	1.40355	0.701773	+	0.712401i	$0.252392\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.04257e6	−0.285746	−0.142873	−	0.989741i	$-0.545634\pi$
$647$	−3.04257e6	−0.285746	−0.142873	+	0.989741i	$0.545634\pi$
$648$	0	0
$649$	−1.36278e6	−0.127003
$650$	0	0
$651$	−6.83854e6	−0.632428
$652$	0	0
$653$	1.83611e6	0.168506	0.0842530	−	0.996444i	$-0.473150\pi$
$653$	1.83611e6	0.168506	0.0842530	+	0.996444i	$0.473150\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.87348e6	−0.350097
$658$	0	0
$659$	8.93871e6	0.801791	0.400896	−	0.916124i	$-0.368699\pi$
$659$	8.93871e6	0.801791	0.400896	+	0.916124i	$0.368699\pi$
$660$	0	0
$661$	−1.18561e7	−1.05545	−0.527727	−	0.849414i	$-0.676956\pi$
$661$	−1.18561e7	−1.05545	−0.527727	+	0.849414i	$0.676956\pi$
$662$	0	0
$663$	3.12283e6	0.275908
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.46833e6	−0.127794
$668$	0	0
$669$	1.68618e7	1.45659
$670$	0	0
$671$	2.65477e6	0.227626
$672$	0	0
$673$	−3.59378e6	−0.305854	−0.152927	−	0.988237i	$-0.548870\pi$
$673$	−3.59378e6	−0.305854	−0.152927	+	0.988237i	$0.548870\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.92010e7	−1.61009	−0.805047	−	0.593211i	$-0.797860\pi$
$677$	−1.92010e7	−1.61009	−0.805047	+	0.593211i	$0.797860\pi$
$678$	0	0
$679$	−2.82772e6	−0.235376
$680$	0	0
$681$	8.22990e6	0.680028
$682$	0	0
$683$	1.23373e7	1.01197	0.505986	−	0.862542i	$-0.331129\pi$
$683$	1.23373e7	1.01197	0.505986	+	0.862542i	$0.331129\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.12180e7	0.906824
$688$	0	0
$689$	−3.66941e6	−0.294475
$690$	0	0
$691$	1.21370e7	0.966977	0.483489	−	0.875351i	$-0.339369\pi$
$691$	1.21370e7	0.966977	0.483489	+	0.875351i	$0.339369\pi$
$692$	0	0
$693$	−1.01309e6	−0.0801340
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.34423e7	1.82775
$698$	0	0
$699$	1.33220e6	0.103128
$700$	0	0
$701$	−8.01721e6	−0.616209	−0.308104	−	0.951353i	$-0.599695\pi$
$701$	−8.01721e6	−0.616209	−0.308104	+	0.951353i	$0.599695\pi$
$702$	0	0
$703$	−9.22310e6	−0.703864
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.59806e6	−0.120239
$708$	0	0
$709$	−6.18927e6	−0.462406	−0.231203	−	0.972906i	$-0.574266\pi$
$709$	−6.18927e6	−0.462406	−0.231203	+	0.972906i	$0.574266\pi$
$710$	0	0
$711$	−3.24526e6	−0.240755
$712$	0	0
$713$	−3.94780e6	−0.290824
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.38849e7	−1.00866
$718$	0	0
$719$	−2.13491e7	−1.54013	−0.770066	−	0.637964i	$-0.779777\pi$
$719$	−2.13491e7	−1.54013	−0.770066	+	0.637964i	$0.779777\pi$
$720$	0	0
$721$	−7.76892e6	−0.556573
$722$	0	0
$723$	1.81331e7	1.29011
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.54941e7	−1.08725	−0.543626	−	0.839327i	$-0.682949\pi$
$727$	−1.54941e7	−1.08725	−0.543626	+	0.839327i	$0.682949\pi$
$728$	0	0
$729$	1.37184e7	0.956058
$730$	0	0
$731$	3.75229e7	2.59719
$732$	0	0
$733$	1.77958e7	1.22337	0.611684	−	0.791102i	$-0.290492\pi$
$733$	1.77958e7	1.22337	0.611684	+	0.791102i	$0.290492\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.67104e6	−0.113323
$738$	0	0
$739$	−2.52496e7	−1.70076	−0.850382	−	0.526165i	$-0.823629\pi$
$739$	−2.52496e7	−1.70076	−0.850382	+	0.526165i	$0.823629\pi$
$740$	0	0
$741$	−2.56543e6	−0.171638
$742$	0	0
$743$	−1.95019e6	−0.129600	−0.0648000	−	0.997898i	$-0.520641\pi$
$743$	−1.95019e6	−0.129600	−0.0648000	+	0.997898i	$0.520641\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.02959e7	−0.675094
$748$	0	0
$749$	1.36035e7	0.886024
$750$	0	0
$751$	−8.89009e6	−0.575183	−0.287592	−	0.957753i	$-0.592855\pi$
$751$	−8.89009e6	−0.575183	−0.287592	+	0.957753i	$0.592855\pi$
$752$	0	0
$753$	−3.60211e6	−0.231510
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−2.56944e7	−1.62967	−0.814833	−	0.579696i	$-0.803171\pi$
$757$	−2.56944e7	−1.62967	−0.814833	+	0.579696i	$0.803171\pi$
$758$	0	0
$759$	719324.	0.0453231
$760$	0	0
$761$	1.54059e7	0.964328	0.482164	−	0.876081i	$-0.339851\pi$
$761$	1.54059e7	0.964328	0.482164	+	0.876081i	$0.339851\pi$
$762$	0	0
$763$	−7.79035e6	−0.484446
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.36822e6	−0.0839787
$768$	0	0
$769$	8.55760e6	0.521839	0.260919	−	0.965361i	$-0.415974\pi$
$769$	8.55760e6	0.521839	0.260919	+	0.965361i	$0.415974\pi$
$770$	0	0
$771$	−4.46483e6	−0.270501
$772$	0	0
$773$	−2.10537e7	−1.26730	−0.633651	−	0.773619i	$-0.718444\pi$
$773$	−2.10537e7	−1.26730	−0.633651	+	0.773619i	$0.718444\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.49759e6	0.267256
$778$	0	0
$779$	−1.92580e7	−1.13702
$780$	0	0
$781$	7.66116e6	0.449435
$782$	0	0
$783$	1.16517e7	0.679180
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.82890e6	0.162810	0.0814048	−	0.996681i	$-0.474059\pi$
$787$	2.82890e6	0.162810	0.0814048	+	0.996681i	$0.474059\pi$
$788$	0	0
$789$	1.58631e7	0.907183
$790$	0	0
$791$	−1.37215e7	−0.779762
$792$	0	0
$793$	2.66537e6	0.150513
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.97926e6	0.221899	0.110950	−	0.993826i	$-0.464611\pi$
$797$	3.97926e6	0.221899	0.110950	+	0.993826i	$0.464611\pi$
$798$	0	0
$799$	−3.92039e7	−2.17251
$800$	0	0
$801$	−1.22117e7	−0.672502
$802$	0	0
$803$	4.30104e6	0.235388
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−973209.	−0.0526044
$808$	0	0
$809$	−1.32714e7	−0.712927	−0.356463	−	0.934309i	$-0.616018\pi$
$809$	−1.32714e7	−0.712927	−0.356463	+	0.934309i	$0.616018\pi$
$810$	0	0
$811$	−1.57062e7	−0.838533	−0.419267	−	0.907863i	$-0.637713\pi$
$811$	−1.57062e7	−0.838533	−0.419267	+	0.907863i	$0.637713\pi$
$812$	0	0
$813$	2.93143e6	0.155544
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.08254e7	−1.61567
$818$	0	0
$819$	−1.01714e6	−0.0529871
$820$	0	0
$821$	1.78852e7	0.926054	0.463027	−	0.886344i	$-0.346763\pi$
$821$	1.78852e7	0.926054	0.463027	+	0.886344i	$0.346763\pi$
$822$	0	0
$823$	3.05955e7	1.57455	0.787277	−	0.616599i	$-0.211490\pi$
$823$	3.05955e7	1.57455	0.787277	+	0.616599i	$0.211490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.00955e7	1.53016	0.765082	−	0.643932i	$-0.222698\pi$
$827$	3.00955e7	1.53016	0.765082	+	0.643932i	$0.222698\pi$
$828$	0	0
$829$	−1.52740e6	−0.0771910	−0.0385955	−	0.999255i	$-0.512288\pi$
$829$	−1.52740e6	−0.0771910	−0.0385955	+	0.999255i	$0.512288\pi$
$830$	0	0
$831$	8.68688e6	0.436376
$832$	0	0
$833$	2.42105e7	1.20890
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.13271e7	1.54563
$838$	0	0
$839$	2.05366e7	1.00722	0.503610	−	0.863931i	$-0.332005\pi$
$839$	2.05366e7	1.00722	0.503610	+	0.863931i	$0.332005\pi$
$840$	0	0
$841$	−1.23347e7	−0.601363
$842$	0	0
$843$	2.82950e7	1.37133
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.12492e6	0.0538782
$848$	0	0
$849$	−5.16530e6	−0.245938
$850$	0	0
$851$	2.59640e6	0.122899
$852$	0	0
$853$	2.21750e7	1.04350	0.521748	−	0.853099i	$-0.325280\pi$
$853$	2.21750e7	1.04350	0.521748	+	0.853099i	$0.325280\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.29311e7	1.06653	0.533265	−	0.845948i	$-0.320965\pi$
$857$	2.29311e7	1.06653	0.533265	+	0.845948i	$0.320965\pi$
$858$	0	0
$859$	1.38827e7	0.641934	0.320967	−	0.947090i	$-0.395992\pi$
$859$	1.38827e7	0.641934	0.320967	+	0.947090i	$0.395992\pi$
$860$	0	0
$861$	9.39106e6	0.431725
$862$	0	0
$863$	2.67227e7	1.22139	0.610693	−	0.791867i	$-0.290891\pi$
$863$	2.67227e7	1.22139	0.610693	+	0.791867i	$0.290891\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.06400e7	−1.83614
$868$	0	0
$869$	3.60348e6	0.161872
$870$	0	0
$871$	−1.67771e6	−0.0749327
$872$	0	0
$873$	4.01051e6	0.178100
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.32981e6	0.0583835	0.0291917	−	0.999574i	$-0.490707\pi$
$877$	1.32981e6	0.0583835	0.0291917	+	0.999574i	$0.490707\pi$
$878$	0	0
$879$	2.74926e7	1.20017
$880$	0	0
$881$	−1.01801e7	−0.441889	−0.220944	−	0.975286i	$-0.570914\pi$
$881$	−1.01801e7	−0.441889	−0.220944	+	0.975286i	$0.570914\pi$
$882$	0	0
$883$	−1.84048e7	−0.794381	−0.397190	−	0.917736i	$-0.630015\pi$
$883$	−1.84048e7	−0.794381	−0.397190	+	0.917736i	$0.630015\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.43853e7	−0.613919	−0.306960	−	0.951723i	$-0.599312\pi$
$887$	−1.43853e7	−0.613919	−0.306960	+	0.951723i	$0.599312\pi$
$888$	0	0
$889$	1.50502e6	0.0638685
$890$	0	0
$891$	−2.50398e6	−0.105666
$892$	0	0
$893$	3.22063e7	1.35149
$894$	0	0
$895$	0	0
$896$	0	0
$897$	722195.	0.0299691
$898$	0	0
$899$	2.19835e7	0.907190
$900$	0	0
$901$	6.70679e7	2.75234
$902$	0	0
$903$	1.50318e7	0.613469
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.93449e7	0.780816	0.390408	−	0.920642i	$-0.372334\pi$
$907$	1.93449e7	0.780816	0.390408	+	0.920642i	$0.372334\pi$
$908$	0	0
$909$	2.26650e6	0.0909799
$910$	0	0
$911$	2.01366e7	0.803876	0.401938	−	0.915667i	$-0.368337\pi$
$911$	2.01366e7	0.803876	0.401938	+	0.915667i	$0.368337\pi$
$912$	0	0
$913$	1.14324e7	0.453900
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.98413e7	0.779197
$918$	0	0
$919$	−2.67069e7	−1.04312	−0.521561	−	0.853214i	$-0.674650\pi$
$919$	−2.67069e7	−1.04312	−0.521561	+	0.853214i	$0.674650\pi$
$920$	0	0
$921$	−1.79384e6	−0.0696841
$922$	0	0
$923$	7.69174e6	0.297181
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.10185e7	0.421137
$928$	0	0
$929$	−1.25962e7	−0.478850	−0.239425	−	0.970915i	$-0.576959\pi$
$929$	−1.25962e7	−0.478850	−0.239425	+	0.970915i	$0.576959\pi$
$930$	0	0
$931$	−1.98892e7	−0.752042
$932$	0	0
$933$	−3.25524e6	−0.122427
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.72205e7	1.38495	0.692474	−	0.721443i	$-0.256521\pi$
$937$	3.72205e7	1.38495	0.692474	+	0.721443i	$0.256521\pi$
$938$	0	0
$939$	−6.67965e6	−0.247223
$940$	0	0
$941$	−2.81993e7	−1.03816	−0.519080	−	0.854726i	$-0.673725\pi$
$941$	−2.81993e7	−1.03816	−0.519080	+	0.854726i	$0.673725\pi$
$942$	0	0
$943$	5.42134e6	0.198530
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.19943e7	−1.15930	−0.579652	−	0.814864i	$-0.696811\pi$
$947$	−3.19943e7	−1.15930	−0.579652	+	0.814864i	$0.696811\pi$
$948$	0	0
$949$	4.31821e6	0.155646
$950$	0	0
$951$	−3.84190e6	−0.137751
$952$	0	0
$953$	−4.59883e7	−1.64027	−0.820134	−	0.572171i	$-0.806101\pi$
$953$	−4.59883e7	−1.64027	−0.820134	+	0.572171i	$0.806101\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.00560e6	−0.141380
$958$	0	0
$959$	2.12174e7	0.744981
$960$	0	0
$961$	3.04764e7	1.06452
$962$	0	0
$963$	−1.92936e7	−0.670420
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.26125e7	−1.12155	−0.560773	−	0.827969i	$-0.689496\pi$
$967$	−3.26125e7	−1.12155	−0.560773	+	0.827969i	$0.689496\pi$
$968$	0	0
$969$	4.68898e7	1.60424
$970$	0	0
$971$	3.45824e7	1.17708	0.588541	−	0.808467i	$-0.299703\pi$
$971$	3.45824e7	1.17708	0.588541	+	0.808467i	$0.299703\pi$
$972$	0	0
$973$	−3.94761e6	−0.133676
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.39377e7	0.802317	0.401158	−	0.916009i	$-0.368608\pi$
$977$	2.39377e7	0.802317	0.401158	+	0.916009i	$0.368608\pi$
$978$	0	0
$979$	1.35596e7	0.452158
$980$	0	0
$981$	1.10489e7	0.366562
$982$	0	0
$983$	−748485.	−0.0247058	−0.0123529	−	0.999924i	$-0.503932\pi$
$983$	−748485.	−0.0247058	−0.0123529	+	0.999924i	$0.503932\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.57052e7	−0.513158
$988$	0	0
$989$	8.67768e6	0.282106
$990$	0	0
$991$	5.06401e7	1.63799	0.818994	−	0.573803i	$-0.194532\pi$
$991$	5.06401e7	1.63799	0.818994	+	0.573803i	$0.194532\pi$
$992$	0	0
$993$	1.88779e7	0.607547
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.22859e6	0.198450	0.0992252	−	0.995065i	$-0.468364\pi$
$997$	6.22859e6	0.198450	0.0992252	+	0.995065i	$0.468364\pi$
$998$	0	0
$999$	−2.06033e7	−0.653166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.a.g.1.2		5
5.2	odd	4		1100.6.b.g.749.8		10
5.3	odd	4		1100.6.b.g.749.3		10
5.4	even	2		220.6.a.e.1.4	✓	5
20.19	odd	2		880.6.a.p.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.6.a.e.1.4	✓	5	5.4	even	2
880.6.a.p.1.2		5	20.19	odd	2
1100.6.a.g.1.2		5	1.1	even	1	trivial
1100.6.b.g.749.3		10	5.3	odd	4
1100.6.b.g.749.8		10	5.2	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.a (trivial)

\(f(q)\)	\(=\)	\(q-11.5771 q^{3} +76.8335 q^{7} -108.972 q^{9} +121.000 q^{11} +121.483 q^{13} -2220.42 q^{17} +1824.09 q^{19} -889.507 q^{21} -513.500 q^{23} +4074.80 q^{27} +2859.46 q^{29} +7688.01 q^{31} -1400.82 q^{33} -5056.28 q^{37} -1406.42 q^{39} -10557.6 q^{41} -16899.1 q^{43} +17656.1 q^{47} -10903.6 q^{49} +25705.9 q^{51} -30205.1 q^{53} -21117.6 q^{57} -11262.7 q^{59} +21940.3 q^{61} -8372.68 q^{63} -13810.2 q^{67} +5944.82 q^{69} +63315.4 q^{71} +35545.8 q^{73} +9296.86 q^{77} +29780.8 q^{79} -20694.1 q^{81} +94482.7 q^{83} -33104.1 q^{87} +112063. q^{89} +9333.97 q^{91} -89004.6 q^{93} -36803.2 q^{97} -13185.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 18 q^{3} - 130 q^{7} + 471 q^{9} + 605 q^{11} - 600 q^{13} + 176 q^{17} + 102 q^{19} + 1468 q^{21} - 4434 q^{23} - 7764 q^{27} - 742 q^{29} + 3132 q^{31} - 2178 q^{33} - 11296 q^{37} + 12032 q^{39}+ \cdots + 56991 q^{99}+O(q^{100}) \) 5 * q - 18 * q^3 - 130 * q^7 + 471 * q^9 + 605 * q^11 - 600 * q^13 + 176 * q^17 + 102 * q^19 + 1468 * q^21 - 4434 * q^23 - 7764 * q^27 - 742 * q^29 + 3132 * q^31 - 2178 * q^33 - 11296 * q^37 + 12032 * q^39 + 7154 * q^41 - 18956 * q^43 - 32370 * q^47 + 34263 * q^49 + 29014 * q^51 - 13328 * q^53 - 34168 * q^57 + 72728 * q^59 + 73402 * q^61 - 73700 * q^63 - 61018 * q^67 + 47124 * q^69 + 86120 * q^71 - 62784 * q^73 - 15730 * q^77 + 93320 * q^79 + 133785 * q^81 - 107548 * q^83 + 18114 * q^87 + 22164 * q^89 + 28404 * q^91 - 135290 * q^93 - 26934 * q^97 + 56991 * q^99

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