LMFDB - Embedded newform 1104.6.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 223x^{2} - 168x + 7047 $ x^4 - 2x^3 - 223x^2 - 168*x + 7047
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$15.3152$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -83.2716 q^{5} -164.449 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -83.2716 q^{5} -164.449 q^{7} +81.0000 q^{9} +32.5897 q^{11} -436.774 q^{13} +749.445 q^{15} -338.357 q^{17} -1036.70 q^{19} +1480.04 q^{21} +529.000 q^{23} +3809.17 q^{25} -729.000 q^{27} -312.291 q^{29} +4642.38 q^{31} -293.307 q^{33} +13693.9 q^{35} +2029.03 q^{37} +3930.97 q^{39} -12690.5 q^{41} +3123.20 q^{43} -6745.00 q^{45} +25980.1 q^{47} +10236.5 q^{49} +3045.22 q^{51} -15914.7 q^{53} -2713.80 q^{55} +9330.31 q^{57} +29796.9 q^{59} +21253.9 q^{61} -13320.4 q^{63} +36370.9 q^{65} +33411.8 q^{67} -4761.00 q^{69} +28793.2 q^{71} +32969.7 q^{73} -34282.5 q^{75} -5359.34 q^{77} +81177.2 q^{79} +6561.00 q^{81} -77651.1 q^{83} +28175.6 q^{85} +2810.62 q^{87} -450.374 q^{89} +71827.1 q^{91} -41781.4 q^{93} +86327.8 q^{95} -56189.4 q^{97} +2639.76 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} - 50 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) $ 4 * q - 36 * q^3 - 50 * q^5 + 62 * q^7 + 324 * q^9	$ 4 q - 36 q^{3} - 50 q^{5} + 62 q^{7} + 324 q^{9} + 392 q^{11} - 396 q^{13} + 450 q^{15} - 794 q^{17} + 374 q^{19} - 558 q^{21} + 2116 q^{23} - 3736 q^{25} - 2916 q^{27} - 10028 q^{29} + 7092 q^{31} - 3528 q^{33} + 11496 q^{35} - 16436 q^{37} + 3564 q^{39} - 20384 q^{41} + 18206 q^{43} - 4050 q^{45} + 30872 q^{47} - 952 q^{49} + 7146 q^{51} - 11150 q^{53} + 17864 q^{55} - 3366 q^{57} + 16080 q^{59} + 156 q^{61} + 5022 q^{63} - 2196 q^{65} + 12122 q^{67} - 19044 q^{69} + 69840 q^{71} + 3008 q^{73} + 33624 q^{75} - 86856 q^{77} + 106522 q^{79} + 26244 q^{81} + 137928 q^{83} + 15300 q^{85} + 90252 q^{87} - 121878 q^{89} + 120148 q^{91} - 63828 q^{93} + 186624 q^{95} - 100956 q^{97} + 31752 q^{99}+O(q^{100}) $ 4 * q - 36 * q^3 - 50 * q^5 + 62 * q^7 + 324 * q^9 + 392 * q^11 - 396 * q^13 + 450 * q^15 - 794 * q^17 + 374 * q^19 - 558 * q^21 + 2116 * q^23 - 3736 * q^25 - 2916 * q^27 - 10028 * q^29 + 7092 * q^31 - 3528 * q^33 + 11496 * q^35 - 16436 * q^37 + 3564 * q^39 - 20384 * q^41 + 18206 * q^43 - 4050 * q^45 + 30872 * q^47 - 952 * q^49 + 7146 * q^51 - 11150 * q^53 + 17864 * q^55 - 3366 * q^57 + 16080 * q^59 + 156 * q^61 + 5022 * q^63 - 2196 * q^65 + 12122 * q^67 - 19044 * q^69 + 69840 * q^71 + 3008 * q^73 + 33624 * q^75 - 86856 * q^77 + 106522 * q^79 + 26244 * q^81 + 137928 * q^83 + 15300 * q^85 + 90252 * q^87 - 121878 * q^89 + 120148 * q^91 - 63828 * q^93 + 186624 * q^95 - 100956 * q^97 + 31752 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−83.2716	−1.48961	−0.744804	−	0.667283i	$-0.767457\pi$
$5$	−83.2716	−1.48961	−0.744804	+	0.667283i	$0.767457\pi$
$6$	0	0
$7$	−164.449	−1.26849	−0.634244	−	0.773133i	$-0.718688\pi$
$7$	−164.449	−1.26849	−0.634244	+	0.773133i	$0.718688\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	32.5897	0.0812079	0.0406040	−	0.999175i	$-0.487072\pi$
$11$	32.5897	0.0812079	0.0406040	+	0.999175i	$0.487072\pi$
$12$	0	0
$13$	−436.774	−0.716801	−0.358401	−	0.933568i	$-0.616678\pi$
$13$	−436.774	−0.716801	−0.358401	+	0.933568i	$0.616678\pi$
$14$	0	0
$15$	749.445	0.860026
$16$	0	0
$17$	−338.357	−0.283958	−0.141979	−	0.989870i	$-0.545346\pi$
$17$	−338.357	−0.283958	−0.141979	+	0.989870i	$0.545346\pi$
$18$	0	0
$19$	−1036.70	−0.658824	−0.329412	−	0.944186i	$-0.606851\pi$
$19$	−1036.70	−0.658824	−0.329412	+	0.944186i	$0.606851\pi$
$20$	0	0
$21$	1480.04	0.732362
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	3809.17	1.21893
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−312.291	−0.0689548	−0.0344774	−	0.999405i	$-0.510977\pi$
$29$	−312.291	−0.0689548	−0.0344774	+	0.999405i	$0.510977\pi$
$30$	0	0
$31$	4642.38	0.867634	0.433817	−	0.901001i	$-0.357166\pi$
$31$	4642.38	0.867634	0.433817	+	0.901001i	$0.357166\pi$
$32$	0	0
$33$	−293.307	−0.0468854
$34$	0	0
$35$	13693.9	1.88955
$36$	0	0
$37$	2029.03	0.243660	0.121830	−	0.992551i	$-0.461124\pi$
$37$	2029.03	0.243660	0.121830	+	0.992551i	$0.461124\pi$
$38$	0	0
$39$	3930.97	0.413845
$40$	0	0
$41$	−12690.5	−1.17901	−0.589505	−	0.807765i	$-0.700677\pi$
$41$	−12690.5	−1.17901	−0.589505	+	0.807765i	$0.700677\pi$
$42$	0	0
$43$	3123.20	0.257590	0.128795	−	0.991671i	$-0.458889\pi$
$43$	3123.20	0.257590	0.128795	+	0.991671i	$0.458889\pi$
$44$	0	0
$45$	−6745.00	−0.496536
$46$	0	0
$47$	25980.1	1.71552	0.857762	−	0.514047i	$-0.171854\pi$
$47$	25980.1	1.71552	0.857762	+	0.514047i	$0.171854\pi$
$48$	0	0
$49$	10236.5	0.609061
$50$	0	0
$51$	3045.22	0.163943
$52$	0	0
$53$	−15914.7	−0.778229	−0.389115	−	0.921189i	$-0.627219\pi$
$53$	−15914.7	−0.778229	−0.389115	+	0.921189i	$0.627219\pi$
$54$	0	0
$55$	−2713.80	−0.120968
$56$	0	0
$57$	9330.31	0.380372
$58$	0	0
$59$	29796.9	1.11440	0.557200	−	0.830378i	$-0.311876\pi$
$59$	29796.9	1.11440	0.557200	+	0.830378i	$0.311876\pi$
$60$	0	0
$61$	21253.9	0.731331	0.365665	−	0.930746i	$-0.380841\pi$
$61$	21253.9	0.731331	0.365665	+	0.930746i	$0.380841\pi$
$62$	0	0
$63$	−13320.4	−0.422829
$64$	0	0
$65$	36370.9	1.06775
$66$	0	0
$67$	33411.8	0.909312	0.454656	−	0.890667i	$-0.349762\pi$
$67$	33411.8	0.909312	0.454656	+	0.890667i	$0.349762\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	28793.2	0.677867	0.338933	−	0.940810i	$-0.389934\pi$
$71$	28793.2	0.677867	0.338933	+	0.940810i	$0.389934\pi$
$72$	0	0
$73$	32969.7	0.724115	0.362057	−	0.932156i	$-0.382074\pi$
$73$	32969.7	0.724115	0.362057	+	0.932156i	$0.382074\pi$
$74$	0	0
$75$	−34282.5	−0.703752
$76$	0	0
$77$	−5359.34	−0.103011
$78$	0	0
$79$	81177.2	1.46341	0.731706	−	0.681621i	$-0.238725\pi$
$79$	81177.2	1.46341	0.731706	+	0.681621i	$0.238725\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−77651.1	−1.23723	−0.618617	−	0.785692i	$-0.712307\pi$
$83$	−77651.1	−1.23723	−0.618617	+	0.785692i	$0.712307\pi$
$84$	0	0
$85$	28175.6	0.422986
$86$	0	0
$87$	2810.62	0.0398110
$88$	0	0
$89$	−450.374	−0.00602697	−0.00301348	−	0.999995i	$-0.500959\pi$
$89$	−450.374	−0.00602697	−0.00301348	+	0.999995i	$0.500959\pi$
$90$	0	0
$91$	71827.1	0.909253
$92$	0	0
$93$	−41781.4	−0.500929
$94$	0	0
$95$	86327.8	0.981390
$96$	0	0
$97$	−56189.4	−0.606352	−0.303176	−	0.952935i	$-0.598047\pi$
$97$	−56189.4	−0.606352	−0.303176	+	0.952935i	$0.598047\pi$
$98$	0	0
$99$	2639.76	0.0270693
$100$	0	0
$101$	−22418.9	−0.218681	−0.109340	−	0.994004i	$-0.534874\pi$
$101$	−22418.9	−0.218681	−0.109340	+	0.994004i	$0.534874\pi$
$102$	0	0
$103$	134342.	1.24773	0.623864	−	0.781533i	$-0.285562\pi$
$103$	134342.	1.24773	0.623864	+	0.781533i	$0.285562\pi$
$104$	0	0
$105$	−123245.	−1.09093
$106$	0	0
$107$	131103.	1.10701	0.553506	−	0.832845i	$-0.313290\pi$
$107$	131103.	1.10701	0.553506	+	0.832845i	$0.313290\pi$
$108$	0	0
$109$	−86629.9	−0.698396	−0.349198	−	0.937049i	$-0.613546\pi$
$109$	−86629.9	−0.698396	−0.349198	+	0.937049i	$0.613546\pi$
$110$	0	0
$111$	−18261.3	−0.140677
$112$	0	0
$113$	−95560.9	−0.704019	−0.352009	−	0.935996i	$-0.614502\pi$
$113$	−95560.9	−0.704019	−0.352009	+	0.935996i	$0.614502\pi$
$114$	0	0
$115$	−44050.7	−0.310605
$116$	0	0
$117$	−35378.7	−0.238934
$118$	0	0
$119$	55642.6	0.360197
$120$	0	0
$121$	−159989.	−0.993405
$122$	0	0
$123$	114214.	0.680702
$124$	0	0
$125$	−56971.7	−0.326125
$126$	0	0
$127$	−60517.1	−0.332942	−0.166471	−	0.986046i	$-0.553237\pi$
$127$	−60517.1	−0.332942	−0.166471	+	0.986046i	$0.553237\pi$
$128$	0	0
$129$	−28108.8	−0.148720
$130$	0	0
$131$	−255035.	−1.29844	−0.649220	−	0.760600i	$-0.724905\pi$
$131$	−255035.	−1.29844	−0.649220	+	0.760600i	$0.724905\pi$
$132$	0	0
$133$	170485.	0.835710
$134$	0	0
$135$	60705.0	0.286675
$136$	0	0
$137$	163898.	0.746058	0.373029	−	0.927820i	$-0.378319\pi$
$137$	163898.	0.746058	0.373029	+	0.927820i	$0.378319\pi$
$138$	0	0
$139$	−77221.9	−0.339003	−0.169502	−	0.985530i	$-0.554216\pi$
$139$	−77221.9	−0.339003	−0.169502	+	0.985530i	$0.554216\pi$
$140$	0	0
$141$	−233821.	−0.990458
$142$	0	0
$143$	−14234.3	−0.0582099
$144$	0	0
$145$	26005.0	0.102716
$146$	0	0
$147$	−92128.4	−0.351642
$148$	0	0
$149$	−202219.	−0.746202	−0.373101	−	0.927791i	$-0.621706\pi$
$149$	−202219.	−0.746202	−0.373101	+	0.927791i	$0.621706\pi$
$150$	0	0
$151$	−498507.	−1.77922	−0.889609	−	0.456723i	$-0.849023\pi$
$151$	−498507.	−1.77922	−0.889609	+	0.456723i	$0.849023\pi$
$152$	0	0
$153$	−27407.0	−0.0946525
$154$	0	0
$155$	−386579.	−1.29243
$156$	0	0
$157$	489182.	1.58388	0.791938	−	0.610601i	$-0.209072\pi$
$157$	489182.	1.58388	0.791938	+	0.610601i	$0.209072\pi$
$158$	0	0
$159$	143232.	0.449311
$160$	0	0
$161$	−86993.6	−0.264498
$162$	0	0
$163$	−463516.	−1.36646	−0.683228	−	0.730205i	$-0.739424\pi$
$163$	−463516.	−1.36646	−0.683228	+	0.730205i	$0.739424\pi$
$164$	0	0
$165$	24424.2	0.0698409
$166$	0	0
$167$	268393.	0.744697	0.372349	−	0.928093i	$-0.378553\pi$
$167$	268393.	0.744697	0.372349	+	0.928093i	$0.378553\pi$
$168$	0	0
$169$	−180521.	−0.486196
$170$	0	0
$171$	−83972.8	−0.219608
$172$	0	0
$173$	513106.	1.30344	0.651722	−	0.758458i	$-0.274047\pi$
$173$	513106.	1.30344	0.651722	+	0.758458i	$0.274047\pi$
$174$	0	0
$175$	−626414.	−1.54620
$176$	0	0
$177$	−268172.	−0.643399
$178$	0	0
$179$	−20653.6	−0.0481796	−0.0240898	−	0.999710i	$-0.507669\pi$
$179$	−20653.6	−0.0481796	−0.0240898	+	0.999710i	$0.507669\pi$
$180$	0	0
$181$	25034.6	0.0567994	0.0283997	−	0.999597i	$-0.490959\pi$
$181$	25034.6	0.0567994	0.0283997	+	0.999597i	$0.490959\pi$
$182$	0	0
$183$	−191285.	−0.422234
$184$	0	0
$185$	−168961.	−0.362958
$186$	0	0
$187$	−11027.0	−0.0230596
$188$	0	0
$189$	119883.	0.244121
$190$	0	0
$191$	830177.	1.64660	0.823299	−	0.567608i	$-0.192131\pi$
$191$	830177.	1.64660	0.823299	+	0.567608i	$0.192131\pi$
$192$	0	0
$193$	−232480.	−0.449254	−0.224627	−	0.974445i	$-0.572116\pi$
$193$	−232480.	−0.449254	−0.224627	+	0.974445i	$0.572116\pi$
$194$	0	0
$195$	−327338.	−0.616467
$196$	0	0
$197$	−24305.8	−0.0446216	−0.0223108	−	0.999751i	$-0.507102\pi$
$197$	−24305.8	−0.0446216	−0.0223108	+	0.999751i	$0.507102\pi$
$198$	0	0
$199$	344497.	0.616669	0.308335	−	0.951278i	$-0.400228\pi$
$199$	344497.	0.616669	0.308335	+	0.951278i	$0.400228\pi$
$200$	0	0
$201$	−300706.	−0.524991
$202$	0	0
$203$	51355.9	0.0874683
$204$	0	0
$205$	1.05675e6	1.75626
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−33785.8	−0.0535017
$210$	0	0
$211$	−1.11790e6	−1.72861	−0.864303	−	0.502972i	$-0.832240\pi$
$211$	−1.11790e6	−1.72861	−0.864303	+	0.502972i	$0.832240\pi$
$212$	0	0
$213$	−259139.	−0.391367
$214$	0	0
$215$	−260074.	−0.383708
$216$	0	0
$217$	−763435.	−1.10058
$218$	0	0
$219$	−296727.	−0.418068
$220$	0	0
$221$	147786.	0.203541
$222$	0	0
$223$	−321178.	−0.432498	−0.216249	−	0.976338i	$-0.569382\pi$
$223$	−321178.	−0.432498	−0.216249	+	0.976338i	$0.569382\pi$
$224$	0	0
$225$	308543.	0.406311
$226$	0	0
$227$	184272.	0.237353	0.118676	−	0.992933i	$-0.462135\pi$
$227$	184272.	0.237353	0.118676	+	0.992933i	$0.462135\pi$
$228$	0	0
$229$	253160.	0.319012	0.159506	−	0.987197i	$-0.449010\pi$
$229$	253160.	0.319012	0.159506	+	0.987197i	$0.449010\pi$
$230$	0	0
$231$	48234.1	0.0594736
$232$	0	0
$233$	−593979.	−0.716773	−0.358386	−	0.933573i	$-0.616673\pi$
$233$	−593979.	−0.716773	−0.358386	+	0.933573i	$0.616673\pi$
$234$	0	0
$235$	−2.16341e6	−2.55546
$236$	0	0
$237$	−730595.	−0.844901
$238$	0	0
$239$	397785.	0.450457	0.225229	−	0.974306i	$-0.427687\pi$
$239$	397785.	0.450457	0.225229	+	0.974306i	$0.427687\pi$
$240$	0	0
$241$	1.33402e6	1.47952	0.739758	−	0.672873i	$-0.234940\pi$
$241$	1.33402e6	1.47952	0.739758	+	0.672873i	$0.234940\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−852410.	−0.907263
$246$	0	0
$247$	452804.	0.472246
$248$	0	0
$249$	698860.	0.714318
$250$	0	0
$251$	−118840.	−0.119064	−0.0595318	−	0.998226i	$-0.518961\pi$
$251$	−118840.	−0.119064	−0.0595318	+	0.998226i	$0.518961\pi$
$252$	0	0
$253$	17239.9	0.0169330
$254$	0	0
$255$	−253580.	−0.244211
$256$	0	0
$257$	−369890.	−0.349333	−0.174667	−	0.984628i	$-0.555885\pi$
$257$	−369890.	−0.349333	−0.174667	+	0.984628i	$0.555885\pi$
$258$	0	0
$259$	−333672.	−0.309080
$260$	0	0
$261$	−25295.6	−0.0229849
$262$	0	0
$263$	−787257.	−0.701822	−0.350911	−	0.936409i	$-0.614128\pi$
$263$	−787257.	−0.701822	−0.350911	+	0.936409i	$0.614128\pi$
$264$	0	0
$265$	1.32524e6	1.15926
$266$	0	0
$267$	4053.37	0.00347967
$268$	0	0
$269$	1.69297e6	1.42649	0.713246	−	0.700914i	$-0.247224\pi$
$269$	1.69297e6	1.42649	0.713246	+	0.700914i	$0.247224\pi$
$270$	0	0
$271$	417238.	0.345113	0.172556	−	0.985000i	$-0.444797\pi$
$271$	417238.	0.345113	0.172556	+	0.985000i	$0.444797\pi$
$272$	0	0
$273$	−646444.	−0.524958
$274$	0	0
$275$	124140.	0.0989871
$276$	0	0
$277$	−1.29975e6	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$277$	−1.29975e6	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$278$	0	0
$279$	376033.	0.289211
$280$	0	0
$281$	2.00546e6	1.51513	0.757563	−	0.652763i	$-0.226390\pi$
$281$	2.00546e6	1.51513	0.757563	+	0.652763i	$0.226390\pi$
$282$	0	0
$283$	1.85482e6	1.37669	0.688344	−	0.725384i	$-0.258338\pi$
$283$	1.85482e6	1.37669	0.688344	+	0.725384i	$0.258338\pi$
$284$	0	0
$285$	−776950.	−0.566606
$286$	0	0
$287$	2.08693e6	1.49556
$288$	0	0
$289$	−1.30537e6	−0.919368
$290$	0	0
$291$	505704.	0.350078
$292$	0	0
$293$	−1.29646e6	−0.882249	−0.441124	−	0.897446i	$-0.645420\pi$
$293$	−1.29646e6	−0.882249	−0.441124	+	0.897446i	$0.645420\pi$
$294$	0	0
$295$	−2.48124e6	−1.66002
$296$	0	0
$297$	−23757.9	−0.0156285
$298$	0	0
$299$	−231054.	−0.149463
$300$	0	0
$301$	−513607.	−0.326749
$302$	0	0
$303$	201770.	0.126255
$304$	0	0
$305$	−1.76985e6	−1.08940
$306$	0	0
$307$	−2.78102e6	−1.68407	−0.842033	−	0.539427i	$-0.818641\pi$
$307$	−2.78102e6	−1.68407	−0.842033	+	0.539427i	$0.818641\pi$
$308$	0	0
$309$	−1.20908e6	−0.720376
$310$	0	0
$311$	−402399.	−0.235915	−0.117957	−	0.993019i	$-0.537635\pi$
$311$	−402399.	−0.235915	−0.117957	+	0.993019i	$0.537635\pi$
$312$	0	0
$313$	−184602.	−0.106506	−0.0532531	−	0.998581i	$-0.516959\pi$
$313$	−184602.	−0.106506	−0.0532531	+	0.998581i	$0.516959\pi$
$314$	0	0
$315$	1.10921e6	0.629850
$316$	0	0
$317$	−464126.	−0.259411	−0.129705	−	0.991553i	$-0.541403\pi$
$317$	−464126.	−0.259411	−0.129705	+	0.991553i	$0.541403\pi$
$318$	0	0
$319$	−10177.5	−0.00559967
$320$	0	0
$321$	−1.17993e6	−0.639134
$322$	0	0
$323$	350776.	0.187078
$324$	0	0
$325$	−1.66375e6	−0.873733
$326$	0	0
$327$	779670.	0.403219
$328$	0	0
$329$	−4.27241e6	−2.17612
$330$	0	0
$331$	−3.89668e6	−1.95490	−0.977449	−	0.211170i	$-0.932273\pi$
$331$	−3.89668e6	−1.95490	−0.977449	+	0.211170i	$0.932273\pi$
$332$	0	0
$333$	164352.	0.0812200
$334$	0	0
$335$	−2.78226e6	−1.35452
$336$	0	0
$337$	−2.50584e6	−1.20193	−0.600964	−	0.799276i	$-0.705217\pi$
$337$	−2.50584e6	−1.20193	−0.600964	+	0.799276i	$0.705217\pi$
$338$	0	0
$339$	860048.	0.406465
$340$	0	0
$341$	151294.	0.0704588
$342$	0	0
$343$	1.08051e6	0.495901
$344$	0	0
$345$	396456.	0.179328
$346$	0	0
$347$	1.39592e6	0.622352	0.311176	−	0.950352i	$-0.399277\pi$
$347$	1.39592e6	0.622352	0.311176	+	0.950352i	$0.399277\pi$
$348$	0	0
$349$	429529.	0.188768	0.0943842	−	0.995536i	$-0.469912\pi$
$349$	429529.	0.188768	0.0943842	+	0.995536i	$0.469912\pi$
$350$	0	0
$351$	318408.	0.137948
$352$	0	0
$353$	2.94172e6	1.25651	0.628253	−	0.778009i	$-0.283770\pi$
$353$	2.94172e6	1.25651	0.628253	+	0.778009i	$0.283770\pi$
$354$	0	0
$355$	−2.39766e6	−1.00976
$356$	0	0
$357$	−500783.	−0.207960
$358$	0	0
$359$	−2.77650e6	−1.13700	−0.568501	−	0.822682i	$-0.692477\pi$
$359$	−2.77650e6	−1.13700	−0.568501	+	0.822682i	$0.692477\pi$
$360$	0	0
$361$	−1.40135e6	−0.565951
$362$	0	0
$363$	1.43990e6	0.573543
$364$	0	0
$365$	−2.74544e6	−1.07865
$366$	0	0
$367$	−1.58193e6	−0.613085	−0.306543	−	0.951857i	$-0.599172\pi$
$367$	−1.58193e6	−0.613085	−0.306543	+	0.951857i	$0.599172\pi$
$368$	0	0
$369$	−1.02793e6	−0.393003
$370$	0	0
$371$	2.61715e6	0.987174
$372$	0	0
$373$	−225951.	−0.0840896	−0.0420448	−	0.999116i	$-0.513387\pi$
$373$	−225951.	−0.0840896	−0.0420448	+	0.999116i	$0.513387\pi$
$374$	0	0
$375$	512745.	0.188288
$376$	0	0
$377$	136401.	0.0494268
$378$	0	0
$379$	1.15512e6	0.413075	0.206537	−	0.978439i	$-0.433780\pi$
$379$	1.15512e6	0.413075	0.206537	+	0.978439i	$0.433780\pi$
$380$	0	0
$381$	544654.	0.192224
$382$	0	0
$383$	4.84750e6	1.68858	0.844288	−	0.535889i	$-0.180023\pi$
$383$	4.84750e6	1.68858	0.844288	+	0.535889i	$0.180023\pi$
$384$	0	0
$385$	446281.	0.153446
$386$	0	0
$387$	252979.	0.0858632
$388$	0	0
$389$	−571852.	−0.191606	−0.0958030	−	0.995400i	$-0.530542\pi$
$389$	−571852.	−0.191606	−0.0958030	+	0.995400i	$0.530542\pi$
$390$	0	0
$391$	−178991.	−0.0592093
$392$	0	0
$393$	2.29532e6	0.749655
$394$	0	0
$395$	−6.75976e6	−2.17991
$396$	0	0
$397$	−3.83250e6	−1.22041	−0.610205	−	0.792244i	$-0.708913\pi$
$397$	−3.83250e6	−1.22041	−0.610205	+	0.792244i	$0.708913\pi$
$398$	0	0
$399$	−1.53436e6	−0.482498
$400$	0	0
$401$	1.63298e6	0.507130	0.253565	−	0.967318i	$-0.418397\pi$
$401$	1.63298e6	0.507130	0.253565	+	0.967318i	$0.418397\pi$
$402$	0	0
$403$	−2.02767e6	−0.621921
$404$	0	0
$405$	−546345.	−0.165512
$406$	0	0
$407$	66125.5	0.0197871
$408$	0	0
$409$	1.68068e6	0.496795	0.248397	−	0.968658i	$-0.420096\pi$
$409$	1.68068e6	0.496795	0.248397	+	0.968658i	$0.420096\pi$
$410$	0	0
$411$	−1.47508e6	−0.430737
$412$	0	0
$413$	−4.90007e6	−1.41360
$414$	0	0
$415$	6.46613e6	1.84300
$416$	0	0
$417$	694997.	0.195724
$418$	0	0
$419$	567667.	0.157964	0.0789821	−	0.996876i	$-0.474833\pi$
$419$	567667.	0.157964	0.0789821	+	0.996876i	$0.474833\pi$
$420$	0	0
$421$	−6.34694e6	−1.74526	−0.872628	−	0.488386i	$-0.837586\pi$
$421$	−6.34694e6	−1.74526	−0.872628	+	0.488386i	$0.837586\pi$
$422$	0	0
$423$	2.10439e6	0.571841
$424$	0	0
$425$	−1.28886e6	−0.346125
$426$	0	0
$427$	−3.49518e6	−0.927684
$428$	0	0
$429$	128109.	0.0336075
$430$	0	0
$431$	4.06109e6	1.05305	0.526526	−	0.850159i	$-0.323494\pi$
$431$	4.06109e6	1.05305	0.526526	+	0.850159i	$0.323494\pi$
$432$	0	0
$433$	1.61805e6	0.414738	0.207369	−	0.978263i	$-0.433510\pi$
$433$	1.61805e6	0.414738	0.207369	+	0.978263i	$0.433510\pi$
$434$	0	0
$435$	−234045.	−0.0593029
$436$	0	0
$437$	−548415.	−0.137374
$438$	0	0
$439$	−419021.	−0.103771	−0.0518854	−	0.998653i	$-0.516523\pi$
$439$	−419021.	−0.103771	−0.0518854	+	0.998653i	$0.516523\pi$
$440$	0	0
$441$	829156.	0.203020
$442$	0	0
$443$	3.43341e6	0.831221	0.415611	−	0.909543i	$-0.363568\pi$
$443$	3.43341e6	0.831221	0.415611	+	0.909543i	$0.363568\pi$
$444$	0	0
$445$	37503.4	0.00897782
$446$	0	0
$447$	1.81997e6	0.430820
$448$	0	0
$449$	477283.	0.111727	0.0558637	−	0.998438i	$-0.482209\pi$
$449$	477283.	0.111727	0.0558637	+	0.998438i	$0.482209\pi$
$450$	0	0
$451$	−413578.	−0.0957450
$452$	0	0
$453$	4.48656e6	1.02723
$454$	0	0
$455$	−5.98116e6	−1.35443
$456$	0	0
$457$	5.99396e6	1.34253	0.671264	−	0.741218i	$-0.265752\pi$
$457$	5.99396e6	1.34253	0.671264	+	0.741218i	$0.265752\pi$
$458$	0	0
$459$	246663.	0.0546477
$460$	0	0
$461$	−4.60012e6	−1.00813	−0.504066	−	0.863665i	$-0.668163\pi$
$461$	−4.60012e6	−1.00813	−0.504066	+	0.863665i	$0.668163\pi$
$462$	0	0
$463$	5.85666e6	1.26969	0.634845	−	0.772639i	$-0.281064\pi$
$463$	5.85666e6	1.26969	0.634845	+	0.772639i	$0.281064\pi$
$464$	0	0
$465$	3.47921e6	0.746188
$466$	0	0
$467$	−2.39360e6	−0.507877	−0.253938	−	0.967220i	$-0.581726\pi$
$467$	−2.39360e6	−0.507877	−0.253938	+	0.967220i	$0.581726\pi$
$468$	0	0
$469$	−5.49454e6	−1.15345
$470$	0	0
$471$	−4.40264e6	−0.914452
$472$	0	0
$473$	101784.	0.0209183
$474$	0	0
$475$	−3.94897e6	−0.803063
$476$	0	0
$477$	−1.28909e6	−0.259410
$478$	0	0
$479$	−2.19342e6	−0.436800	−0.218400	−	0.975859i	$-0.570084\pi$
$479$	−2.19342e6	−0.436800	−0.218400	+	0.975859i	$0.570084\pi$
$480$	0	0
$481$	−886229.	−0.174656
$482$	0	0
$483$	782942.	0.152708
$484$	0	0
$485$	4.67898e6	0.903227
$486$	0	0
$487$	6.02160e6	1.15051	0.575254	−	0.817975i	$-0.304903\pi$
$487$	6.02160e6	1.15051	0.575254	+	0.817975i	$0.304903\pi$
$488$	0	0
$489$	4.17164e6	0.788924
$490$	0	0
$491$	−1.15918e6	−0.216993	−0.108496	−	0.994097i	$-0.534604\pi$
$491$	−1.15918e6	−0.216993	−0.108496	+	0.994097i	$0.534604\pi$
$492$	0	0
$493$	105666.	0.0195802
$494$	0	0
$495$	−219818.	−0.0403227
$496$	0	0
$497$	−4.73502e6	−0.859866
$498$	0	0
$499$	1.13220e6	0.203550	0.101775	−	0.994807i	$-0.467548\pi$
$499$	1.13220e6	0.203550	0.101775	+	0.994807i	$0.467548\pi$
$500$	0	0
$501$	−2.41554e6	−0.429951
$502$	0	0
$503$	171289.	0.0301862	0.0150931	−	0.999886i	$-0.495196\pi$
$503$	171289.	0.0301862	0.0150931	+	0.999886i	$0.495196\pi$
$504$	0	0
$505$	1.86686e6	0.325748
$506$	0	0
$507$	1.62469e6	0.280706
$508$	0	0
$509$	−4.37189e6	−0.747954	−0.373977	−	0.927438i	$-0.622006\pi$
$509$	−4.37189e6	−0.747954	−0.373977	+	0.927438i	$0.622006\pi$
$510$	0	0
$511$	−5.42183e6	−0.918530
$512$	0	0
$513$	755755.	0.126791
$514$	0	0
$515$	−1.11869e7	−1.85863
$516$	0	0
$517$	846684.	0.139314
$518$	0	0
$519$	−4.61796e6	−0.752543
$520$	0	0
$521$	−8.06001e6	−1.30089	−0.650446	−	0.759553i	$-0.725418\pi$
$521$	−8.06001e6	−1.30089	−0.650446	+	0.759553i	$0.725418\pi$
$522$	0	0
$523$	3.42273e6	0.547164	0.273582	−	0.961849i	$-0.411791\pi$
$523$	3.42273e6	0.547164	0.273582	+	0.961849i	$0.411791\pi$
$524$	0	0
$525$	5.63773e6	0.892700
$526$	0	0
$527$	−1.57078e6	−0.246371
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.41355e6	0.371467
$532$	0	0
$533$	5.54286e6	0.845116
$534$	0	0
$535$	−1.09171e7	−1.64902
$536$	0	0
$537$	185883.	0.0278165
$538$	0	0
$539$	333604.	0.0494606
$540$	0	0
$541$	1.57720e6	0.231682	0.115841	−	0.993268i	$-0.463044\pi$
$541$	1.57720e6	0.231682	0.115841	+	0.993268i	$0.463044\pi$
$542$	0	0
$543$	−225311.	−0.0327931
$544$	0	0
$545$	7.21382e6	1.04034
$546$	0	0
$547$	−6.50079e6	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$547$	−6.50079e6	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$548$	0	0
$549$	1.72157e6	0.243777
$550$	0	0
$551$	323752.	0.0454291
$552$	0	0
$553$	−1.33495e7	−1.85632
$554$	0	0
$555$	1.52065e6	0.209554
$556$	0	0
$557$	2.50762e6	0.342471	0.171235	−	0.985230i	$-0.445224\pi$
$557$	2.50762e6	0.342471	0.171235	+	0.985230i	$0.445224\pi$
$558$	0	0
$559$	−1.36413e6	−0.184641
$560$	0	0
$561$	99242.7	0.0133135
$562$	0	0
$563$	−5.30561e6	−0.705447	−0.352723	−	0.935728i	$-0.614744\pi$
$563$	−5.30561e6	−0.705447	−0.352723	+	0.935728i	$0.614744\pi$
$564$	0	0
$565$	7.95752e6	1.04871
$566$	0	0
$567$	−1.07895e6	−0.140943
$568$	0	0
$569$	8.13573e6	1.05345	0.526727	−	0.850034i	$-0.323419\pi$
$569$	8.13573e6	1.05345	0.526727	+	0.850034i	$0.323419\pi$
$570$	0	0
$571$	6.45827e6	0.828946	0.414473	−	0.910062i	$-0.363966\pi$
$571$	6.45827e6	0.828946	0.414473	+	0.910062i	$0.363966\pi$
$572$	0	0
$573$	−7.47160e6	−0.950663
$574$	0	0
$575$	2.01505e6	0.254165
$576$	0	0
$577$	1.06129e7	1.32707	0.663537	−	0.748143i	$-0.269054\pi$
$577$	1.06129e7	1.32707	0.663537	+	0.748143i	$0.269054\pi$
$578$	0	0
$579$	2.09232e6	0.259377
$580$	0	0
$581$	1.27696e7	1.56942
$582$	0	0
$583$	−518654.	−0.0631984
$584$	0	0
$585$	2.94604e6	0.355918
$586$	0	0
$587$	7.04476e6	0.843861	0.421931	−	0.906628i	$-0.361353\pi$
$587$	7.04476e6	0.843861	0.421931	+	0.906628i	$0.361353\pi$
$588$	0	0
$589$	−4.81276e6	−0.571618
$590$	0	0
$591$	218753.	0.0257623
$592$	0	0
$593$	9.87775e6	1.15351	0.576755	−	0.816917i	$-0.304319\pi$
$593$	9.87775e6	1.15351	0.576755	+	0.816917i	$0.304319\pi$
$594$	0	0
$595$	−4.63345e6	−0.536552
$596$	0	0
$597$	−3.10047e6	−0.356034
$598$	0	0
$599$	−1.13399e7	−1.29135	−0.645675	−	0.763613i	$-0.723424\pi$
$599$	−1.13399e7	−1.29135	−0.645675	+	0.763613i	$0.723424\pi$
$600$	0	0
$601$	1.50395e7	1.69842	0.849212	−	0.528053i	$-0.177078\pi$
$601$	1.50395e7	1.69842	0.849212	+	0.528053i	$0.177078\pi$
$602$	0	0
$603$	2.70636e6	0.303104
$604$	0	0
$605$	1.33225e7	1.47978
$606$	0	0
$607$	−1.93367e6	−0.213016	−0.106508	−	0.994312i	$-0.533967\pi$
$607$	−1.93367e6	−0.213016	−0.106508	+	0.994312i	$0.533967\pi$
$608$	0	0
$609$	−462203.	−0.0504998
$610$	0	0
$611$	−1.13475e7	−1.22969
$612$	0	0
$613$	−1.34894e7	−1.44991	−0.724957	−	0.688795i	$-0.758140\pi$
$613$	−1.34894e7	−1.44991	−0.724957	+	0.688795i	$0.758140\pi$
$614$	0	0
$615$	−9.51079e6	−1.01398
$616$	0	0
$617$	−2.47429e6	−0.261660	−0.130830	−	0.991405i	$-0.541764\pi$
$617$	−2.47429e6	−0.261660	−0.130830	+	0.991405i	$0.541764\pi$
$618$	0	0
$619$	4.20087e6	0.440669	0.220335	−	0.975424i	$-0.429285\pi$
$619$	4.20087e6	0.440669	0.220335	+	0.975424i	$0.429285\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	74063.6	0.00764513
$624$	0	0
$625$	−7.15952e6	−0.733135
$626$	0	0
$627$	304072.	0.0308892
$628$	0	0
$629$	−686538.	−0.0691891
$630$	0	0
$631$	1.39472e7	1.39449	0.697244	−	0.716834i	$-0.254410\pi$
$631$	1.39472e7	1.39449	0.697244	+	0.716834i	$0.254410\pi$
$632$	0	0
$633$	1.00611e7	0.998011
$634$	0	0
$635$	5.03936e6	0.495954
$636$	0	0
$637$	−4.47104e6	−0.436576
$638$	0	0
$639$	2.33225e6	0.225956
$640$	0	0
$641$	−1.67533e7	−1.61048	−0.805241	−	0.592948i	$-0.797964\pi$
$641$	−1.67533e7	−1.61048	−0.805241	+	0.592948i	$0.797964\pi$
$642$	0	0
$643$	1.27897e7	1.21992	0.609960	−	0.792432i	$-0.291185\pi$
$643$	1.27897e7	1.21992	0.609960	+	0.792432i	$0.291185\pi$
$644$	0	0
$645$	2.34067e6	0.221534
$646$	0	0
$647$	1.60993e6	0.151198	0.0755989	−	0.997138i	$-0.475913\pi$
$647$	1.60993e6	0.151198	0.0755989	+	0.997138i	$0.475913\pi$
$648$	0	0
$649$	971072.	0.0904981
$650$	0	0
$651$	6.87092e6	0.635422
$652$	0	0
$653$	1.31696e7	1.20862	0.604310	−	0.796749i	$-0.293449\pi$
$653$	1.31696e7	1.20862	0.604310	+	0.796749i	$0.293449\pi$
$654$	0	0
$655$	2.12372e7	1.93417
$656$	0	0
$657$	2.67054e6	0.241372
$658$	0	0
$659$	6.21132e6	0.557147	0.278574	−	0.960415i	$-0.410138\pi$
$659$	6.21132e6	0.557147	0.278574	+	0.960415i	$0.410138\pi$
$660$	0	0
$661$	−9.82802e6	−0.874908	−0.437454	−	0.899241i	$-0.644120\pi$
$661$	−9.82802e6	−0.874908	−0.437454	+	0.899241i	$0.644120\pi$
$662$	0	0
$663$	−1.33007e6	−0.117515
$664$	0	0
$665$	−1.41965e7	−1.24488
$666$	0	0
$667$	−165202.	−0.0143781
$668$	0	0
$669$	2.89061e6	0.249703
$670$	0	0
$671$	692658.	0.0593899
$672$	0	0
$673$	−1.23121e7	−1.04783	−0.523917	−	0.851769i	$-0.675530\pi$
$673$	−1.23121e7	−1.04783	−0.523917	+	0.851769i	$0.675530\pi$
$674$	0	0
$675$	−2.77688e6	−0.234584
$676$	0	0
$677$	1.83306e7	1.53711	0.768556	−	0.639783i	$-0.220976\pi$
$677$	1.83306e7	1.53711	0.768556	+	0.639783i	$0.220976\pi$
$678$	0	0
$679$	9.24029e6	0.769150
$680$	0	0
$681$	−1.65845e6	−0.137036
$682$	0	0
$683$	1.05744e7	0.867366	0.433683	−	0.901066i	$-0.357214\pi$
$683$	1.05744e7	0.867366	0.433683	+	0.901066i	$0.357214\pi$
$684$	0	0
$685$	−1.36481e7	−1.11133
$686$	0	0
$687$	−2.27844e6	−0.184182
$688$	0	0
$689$	6.95111e6	0.557836
$690$	0	0
$691$	1.35547e7	1.07993	0.539966	−	0.841687i	$-0.318437\pi$
$691$	1.35547e7	1.07993	0.539966	+	0.841687i	$0.318437\pi$
$692$	0	0
$693$	−434107.	−0.0343371
$694$	0	0
$695$	6.43040e6	0.504982
$696$	0	0
$697$	4.29391e6	0.334789
$698$	0	0
$699$	5.34581e6	0.413829
$700$	0	0
$701$	−1.10553e7	−0.849721	−0.424860	−	0.905259i	$-0.639677\pi$
$701$	−1.10553e7	−0.849721	−0.424860	+	0.905259i	$0.639677\pi$
$702$	0	0
$703$	−2.10350e6	−0.160529
$704$	0	0
$705$	1.94707e7	1.47539
$706$	0	0
$707$	3.68676e6	0.277394
$708$	0	0
$709$	945639.	0.0706496	0.0353248	−	0.999376i	$-0.488753\pi$
$709$	945639.	0.0706496	0.0353248	+	0.999376i	$0.488753\pi$
$710$	0	0
$711$	6.57535e6	0.487804
$712$	0	0
$713$	2.45582e6	0.180914
$714$	0	0
$715$	1.18532e6	0.0867100
$716$	0	0
$717$	−3.58006e6	−0.260072
$718$	0	0
$719$	−2.42489e7	−1.74932	−0.874659	−	0.484738i	$-0.838915\pi$
$719$	−2.42489e7	−1.74932	−0.874659	+	0.484738i	$0.838915\pi$
$720$	0	0
$721$	−2.20925e7	−1.58273
$722$	0	0
$723$	−1.20062e7	−0.854199
$724$	0	0
$725$	−1.18957e6	−0.0840513
$726$	0	0
$727$	1.89322e7	1.32851	0.664257	−	0.747505i	$-0.268748\pi$
$727$	1.89322e7	1.32851	0.664257	+	0.747505i	$0.268748\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.05676e6	−0.0731446
$732$	0	0
$733$	−4.94007e6	−0.339604	−0.169802	−	0.985478i	$-0.554313\pi$
$733$	−4.94007e6	−0.339604	−0.169802	+	0.985478i	$0.554313\pi$
$734$	0	0
$735$	7.67169e6	0.523808
$736$	0	0
$737$	1.08888e6	0.0738433
$738$	0	0
$739$	−2.09183e7	−1.40901	−0.704506	−	0.709698i	$-0.748832\pi$
$739$	−2.09183e7	−1.40901	−0.704506	+	0.709698i	$0.748832\pi$
$740$	0	0
$741$	−4.07524e6	−0.272651
$742$	0	0
$743$	−1.15866e7	−0.769986	−0.384993	−	0.922919i	$-0.625796\pi$
$743$	−1.15866e7	−0.769986	−0.384993	+	0.922919i	$0.625796\pi$
$744$	0	0
$745$	1.68391e7	1.11155
$746$	0	0
$747$	−6.28974e6	−0.412412
$748$	0	0
$749$	−2.15597e7	−1.40423
$750$	0	0
$751$	−2.39410e7	−1.54897	−0.774485	−	0.632592i	$-0.781991\pi$
$751$	−2.39410e7	−1.54897	−0.774485	+	0.632592i	$0.781991\pi$
$752$	0	0
$753$	1.06956e6	0.0687414
$754$	0	0
$755$	4.15115e7	2.65034
$756$	0	0
$757$	8.44645e6	0.535716	0.267858	−	0.963458i	$-0.413684\pi$
$757$	8.44645e6	0.535716	0.267858	+	0.963458i	$0.413684\pi$
$758$	0	0
$759$	−155159.	−0.00977629
$760$	0	0
$761$	1.33497e7	0.835622	0.417811	−	0.908534i	$-0.362797\pi$
$761$	1.33497e7	0.835622	0.417811	+	0.908534i	$0.362797\pi$
$762$	0	0
$763$	1.42462e7	0.885907
$764$	0	0
$765$	2.28222e6	0.140995
$766$	0	0
$767$	−1.30145e7	−0.798803
$768$	0	0
$769$	6.70510e6	0.408874	0.204437	−	0.978880i	$-0.434464\pi$
$769$	6.70510e6	0.408874	0.204437	+	0.978880i	$0.434464\pi$
$770$	0	0
$771$	3.32901e6	0.201688
$772$	0	0
$773$	1.06782e7	0.642759	0.321379	−	0.946950i	$-0.395853\pi$
$773$	1.06782e7	0.642759	0.321379	+	0.946950i	$0.395853\pi$
$774$	0	0
$775$	1.76836e7	1.05759
$776$	0	0
$777$	3.00305e6	0.178447
$778$	0	0
$779$	1.31562e7	0.776760
$780$	0	0
$781$	938362.	0.0550482
$782$	0	0
$783$	227660.	0.0132703
$784$	0	0
$785$	−4.07350e7	−2.35936
$786$	0	0
$787$	−2.94509e7	−1.69497	−0.847485	−	0.530820i	$-0.821884\pi$
$787$	−2.94509e7	−1.69497	−0.847485	+	0.530820i	$0.821884\pi$
$788$	0	0
$789$	7.08531e6	0.405197
$790$	0	0
$791$	1.57149e7	0.893039
$792$	0	0
$793$	−9.28315e6	−0.524219
$794$	0	0
$795$	−1.19272e7	−0.669297
$796$	0	0
$797$	1.06760e7	0.595336	0.297668	−	0.954670i	$-0.403791\pi$
$797$	1.06760e7	0.595336	0.297668	+	0.954670i	$0.403791\pi$
$798$	0	0
$799$	−8.79057e6	−0.487136
$800$	0	0
$801$	−36480.3	−0.00200899
$802$	0	0
$803$	1.07447e6	0.0588038
$804$	0	0
$805$	7.24410e6	0.393998
$806$	0	0
$807$	−1.52368e7	−0.823586
$808$	0	0
$809$	1.57444e7	0.845772	0.422886	−	0.906183i	$-0.361017\pi$
$809$	1.57444e7	0.845772	0.422886	+	0.906183i	$0.361017\pi$
$810$	0	0
$811$	−2.46104e7	−1.31391	−0.656956	−	0.753929i	$-0.728156\pi$
$811$	−2.46104e7	−1.31391	−0.656956	+	0.753929i	$0.728156\pi$
$812$	0	0
$813$	−3.75514e6	−0.199251
$814$	0	0
$815$	3.85977e7	2.03548
$816$	0	0
$817$	−3.23782e6	−0.169706
$818$	0	0
$819$	5.81800e6	0.303084
$820$	0	0
$821$	3.36048e7	1.73998	0.869988	−	0.493072i	$-0.164126\pi$
$821$	3.36048e7	1.73998	0.869988	+	0.493072i	$0.164126\pi$
$822$	0	0
$823$	−1.79190e6	−0.0922176	−0.0461088	−	0.998936i	$-0.514682\pi$
$823$	−1.79190e6	−0.0922176	−0.0461088	+	0.998936i	$0.514682\pi$
$824$	0	0
$825$	−1.11726e6	−0.0571502
$826$	0	0
$827$	2.09124e7	1.06326	0.531631	−	0.846976i	$-0.321579\pi$
$827$	2.09124e7	1.06326	0.531631	+	0.846976i	$0.321579\pi$
$828$	0	0
$829$	−4.51789e6	−0.228323	−0.114161	−	0.993462i	$-0.536418\pi$
$829$	−4.51789e6	−0.228323	−0.114161	+	0.993462i	$0.536418\pi$
$830$	0	0
$831$	1.16978e7	0.587626
$832$	0	0
$833$	−3.46359e6	−0.172948
$834$	0	0
$835$	−2.23495e7	−1.10931
$836$	0	0
$837$	−3.38430e6	−0.166976
$838$	0	0
$839$	1.86006e6	0.0912266	0.0456133	−	0.998959i	$-0.485476\pi$
$839$	1.86006e6	0.0912266	0.0456133	+	0.998959i	$0.485476\pi$
$840$	0	0
$841$	−2.04136e7	−0.995245
$842$	0	0
$843$	−1.80492e7	−0.874758
$844$	0	0
$845$	1.50323e7	0.724242
$846$	0	0
$847$	2.63100e7	1.26012
$848$	0	0
$849$	−1.66934e7	−0.794831
$850$	0	0
$851$	1.07336e6	0.0508066
$852$	0	0
$853$	−3.24375e7	−1.52642	−0.763211	−	0.646149i	$-0.776378\pi$
$853$	−3.24375e7	−1.52642	−0.763211	+	0.646149i	$0.776378\pi$
$854$	0	0
$855$	6.99255e6	0.327130
$856$	0	0
$857$	2.05228e7	0.954520	0.477260	−	0.878762i	$-0.341630\pi$
$857$	2.05228e7	0.954520	0.477260	+	0.878762i	$0.341630\pi$
$858$	0	0
$859$	2.04496e7	0.945589	0.472795	−	0.881173i	$-0.343245\pi$
$859$	2.04496e7	0.945589	0.472795	+	0.881173i	$0.343245\pi$
$860$	0	0
$861$	−1.87824e7	−0.863462
$862$	0	0
$863$	4.05332e7	1.85261	0.926306	−	0.376772i	$-0.122966\pi$
$863$	4.05332e7	1.85261	0.926306	+	0.376772i	$0.122966\pi$
$864$	0	0
$865$	−4.27272e7	−1.94162
$866$	0	0
$867$	1.17483e7	0.530797
$868$	0	0
$869$	2.64554e6	0.118841
$870$	0	0
$871$	−1.45934e7	−0.651796
$872$	0	0
$873$	−4.55134e6	−0.202117
$874$	0	0
$875$	9.36894e6	0.413686
$876$	0	0
$877$	2.21082e7	0.970632	0.485316	−	0.874339i	$-0.338705\pi$
$877$	2.21082e7	0.970632	0.485316	+	0.874339i	$0.338705\pi$
$878$	0	0
$879$	1.16682e7	0.509367
$880$	0	0
$881$	−1.41950e7	−0.616161	−0.308080	−	0.951360i	$-0.599687\pi$
$881$	−1.41950e7	−0.616161	−0.308080	+	0.951360i	$0.599687\pi$
$882$	0	0
$883$	−5.14719e6	−0.222161	−0.111081	−	0.993811i	$-0.535431\pi$
$883$	−5.14719e6	−0.222161	−0.111081	+	0.993811i	$0.535431\pi$
$884$	0	0
$885$	2.23311e7	0.958413
$886$	0	0
$887$	1.58855e7	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$887$	1.58855e7	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$888$	0	0
$889$	9.95198e6	0.422333
$890$	0	0
$891$	213821.	0.00902310
$892$	0	0
$893$	−2.69336e7	−1.13023
$894$	0	0
$895$	1.71986e6	0.0717688
$896$	0	0
$897$	2.07948e6	0.0862927
$898$	0	0
$899$	−1.44977e6	−0.0598275
$900$	0	0
$901$	5.38484e6	0.220984
$902$	0	0
$903$	4.62246e6	0.188649
$904$	0	0
$905$	−2.08467e6	−0.0846088
$906$	0	0
$907$	−2.98738e7	−1.20579	−0.602896	−	0.797819i	$-0.705987\pi$
$907$	−2.98738e7	−1.20579	−0.602896	+	0.797819i	$0.705987\pi$
$908$	0	0
$909$	−1.81593e6	−0.0728935
$910$	0	0
$911$	4.60066e7	1.83664	0.918321	−	0.395837i	$-0.129545\pi$
$911$	4.60066e7	1.83664	0.918321	+	0.395837i	$0.129545\pi$
$912$	0	0
$913$	−2.53062e6	−0.100473
$914$	0	0
$915$	1.59286e7	0.628963
$916$	0	0
$917$	4.19403e7	1.64706
$918$	0	0
$919$	−3.41213e7	−1.33271	−0.666357	−	0.745633i	$-0.732147\pi$
$919$	−3.41213e7	−1.33271	−0.666357	+	0.745633i	$0.732147\pi$
$920$	0	0
$921$	2.50292e7	0.972295
$922$	0	0
$923$	−1.25761e7	−0.485896
$924$	0	0
$925$	7.72892e6	0.297005
$926$	0	0
$927$	1.08817e7	0.415909
$928$	0	0
$929$	−3.39707e7	−1.29141	−0.645707	−	0.763585i	$-0.723437\pi$
$929$	−3.39707e7	−1.29141	−0.645707	+	0.763585i	$0.723437\pi$
$930$	0	0
$931$	−1.06122e7	−0.401264
$932$	0	0
$933$	3.62159e6	0.136206
$934$	0	0
$935$	918233.	0.0343498
$936$	0	0
$937$	5.97794e6	0.222435	0.111217	−	0.993796i	$-0.464525\pi$
$937$	5.97794e6	0.222435	0.111217	+	0.993796i	$0.464525\pi$
$938$	0	0
$939$	1.66141e6	0.0614913
$940$	0	0
$941$	4.66689e7	1.71812	0.859061	−	0.511873i	$-0.171048\pi$
$941$	4.66689e7	1.71812	0.859061	+	0.511873i	$0.171048\pi$
$942$	0	0
$943$	−6.71325e6	−0.245841
$944$	0	0
$945$	−9.98289e6	−0.363644
$946$	0	0
$947$	−4.05480e7	−1.46925	−0.734624	−	0.678475i	$-0.762641\pi$
$947$	−4.05480e7	−1.46925	−0.734624	+	0.678475i	$0.762641\pi$
$948$	0	0
$949$	−1.44003e7	−0.519046
$950$	0	0
$951$	4.17713e6	0.149771
$952$	0	0
$953$	3.14464e7	1.12160	0.560801	−	0.827951i	$-0.310493\pi$
$953$	3.14464e7	1.12160	0.560801	+	0.827951i	$0.310493\pi$
$954$	0	0
$955$	−6.91302e7	−2.45279
$956$	0	0
$957$	91597.1	0.00323297
$958$	0	0
$959$	−2.69529e7	−0.946366
$960$	0	0
$961$	−7.07745e6	−0.247211
$962$	0	0
$963$	1.06193e7	0.369004
$964$	0	0
$965$	1.93590e7	0.669212
$966$	0	0
$967$	5.37683e6	0.184910	0.0924549	−	0.995717i	$-0.470529\pi$
$967$	5.37683e6	0.184910	0.0924549	+	0.995717i	$0.470529\pi$
$968$	0	0
$969$	−3.15698e6	−0.108010
$970$	0	0
$971$	−737807.	−0.0251128	−0.0125564	−	0.999921i	$-0.503997\pi$
$971$	−737807.	−0.0251128	−0.0125564	+	0.999921i	$0.503997\pi$
$972$	0	0
$973$	1.26991e7	0.430021
$974$	0	0
$975$	1.49737e7	0.504450
$976$	0	0
$977$	2.91860e7	0.978224	0.489112	−	0.872221i	$-0.337321\pi$
$977$	2.91860e7	0.978224	0.489112	+	0.872221i	$0.337321\pi$
$978$	0	0
$979$	−14677.6	−0.000489437 0
$980$	0	0
$981$	−7.01703e6	−0.232799
$982$	0	0
$983$	−3.29664e7	−1.08815	−0.544074	−	0.839037i	$-0.683119\pi$
$983$	−3.29664e7	−1.08815	−0.544074	+	0.839037i	$0.683119\pi$
$984$	0	0
$985$	2.02399e6	0.0664687
$986$	0	0
$987$	3.84517e7	1.25638
$988$	0	0
$989$	1.65217e6	0.0537112
$990$	0	0
$991$	−1.23139e7	−0.398301	−0.199150	−	0.979969i	$-0.563818\pi$
$991$	−1.23139e7	−0.398301	−0.199150	+	0.979969i	$0.563818\pi$
$992$	0	0
$993$	3.50701e7	1.12866
$994$	0	0
$995$	−2.86868e7	−0.918596
$996$	0	0
$997$	−3.55209e7	−1.13174	−0.565870	−	0.824495i	$-0.691460\pi$
$997$	−3.55209e7	−1.13174	−0.565870	+	0.824495i	$0.691460\pi$
$998$	0	0
$999$	−1.47916e6	−0.0468924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.l.1.1		4
4.3	odd	2		276.6.a.b.1.1	✓	4
12.11	even	2		828.6.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.6.a.b.1.1	✓	4	4.3	odd	2
828.6.a.d.1.4		4	12.11	even	2
1104.6.a.l.1.1		4	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -83.2716 q^{5} -164.449 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -83.2716 q^{5} -164.449 q^{7} +81.0000 q^{9} +32.5897 q^{11} -436.774 q^{13} +749.445 q^{15} -338.357 q^{17} -1036.70 q^{19} +1480.04 q^{21} +529.000 q^{23} +3809.17 q^{25} -729.000 q^{27} -312.291 q^{29} +4642.38 q^{31} -293.307 q^{33} +13693.9 q^{35} +2029.03 q^{37} +3930.97 q^{39} -12690.5 q^{41} +3123.20 q^{43} -6745.00 q^{45} +25980.1 q^{47} +10236.5 q^{49} +3045.22 q^{51} -15914.7 q^{53} -2713.80 q^{55} +9330.31 q^{57} +29796.9 q^{59} +21253.9 q^{61} -13320.4 q^{63} +36370.9 q^{65} +33411.8 q^{67} -4761.00 q^{69} +28793.2 q^{71} +32969.7 q^{73} -34282.5 q^{75} -5359.34 q^{77} +81177.2 q^{79} +6561.00 q^{81} -77651.1 q^{83} +28175.6 q^{85} +2810.62 q^{87} -450.374 q^{89} +71827.1 q^{91} -41781.4 q^{93} +86327.8 q^{95} -56189.4 q^{97} +2639.76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} - 50 q^{5} + 62 q^{7} + 324 q^{9}+O(q^{10}) \) 4 * q - 36 * q^3 - 50 * q^5 + 62 * q^7 + 324 * q^9	\( 4 q - 36 q^{3} - 50 q^{5} + 62 q^{7} + 324 q^{9} + 392 q^{11} - 396 q^{13} + 450 q^{15} - 794 q^{17} + 374 q^{19} - 558 q^{21} + 2116 q^{23} - 3736 q^{25} - 2916 q^{27} - 10028 q^{29} + 7092 q^{31} - 3528 q^{33} + 11496 q^{35} - 16436 q^{37} + 3564 q^{39} - 20384 q^{41} + 18206 q^{43} - 4050 q^{45} + 30872 q^{47} - 952 q^{49} + 7146 q^{51} - 11150 q^{53} + 17864 q^{55} - 3366 q^{57} + 16080 q^{59} + 156 q^{61} + 5022 q^{63} - 2196 q^{65} + 12122 q^{67} - 19044 q^{69} + 69840 q^{71} + 3008 q^{73} + 33624 q^{75} - 86856 q^{77} + 106522 q^{79} + 26244 q^{81} + 137928 q^{83} + 15300 q^{85} + 90252 q^{87} - 121878 q^{89} + 120148 q^{91} - 63828 q^{93} + 186624 q^{95} - 100956 q^{97} + 31752 q^{99}+O(q^{100}) \) 4 * q - 36 * q^3 - 50 * q^5 + 62 * q^7 + 324 * q^9 + 392 * q^11 - 396 * q^13 + 450 * q^15 - 794 * q^17 + 374 * q^19 - 558 * q^21 + 2116 * q^23 - 3736 * q^25 - 2916 * q^27 - 10028 * q^29 + 7092 * q^31 - 3528 * q^33 + 11496 * q^35 - 16436 * q^37 + 3564 * q^39 - 20384 * q^41 + 18206 * q^43 - 4050 * q^45 + 30872 * q^47 - 952 * q^49 + 7146 * q^51 - 11150 * q^53 + 17864 * q^55 - 3366 * q^57 + 16080 * q^59 + 156 * q^61 + 5022 * q^63 - 2196 * q^65 + 12122 * q^67 - 19044 * q^69 + 69840 * q^71 + 3008 * q^73 + 33624 * q^75 - 86856 * q^77 + 106522 * q^79 + 26244 * q^81 + 137928 * q^83 + 15300 * q^85 + 90252 * q^87 - 121878 * q^89 + 120148 * q^91 - 63828 * q^93 + 186624 * q^95 - 100956 * q^97 + 31752 * q^99

Introduction

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