LMFDB - Embedded newform 1104.6.a.v.1.2

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8826x^{4} + 29484x^{3} + 14987840x^{2} + 92913200x - 1288596000 $ x^6 - 2x^5 - 8826x^4 + 29484x^3 + 14987840x^2 + 92913200*x - 1288596000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$54.9743$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} -50.9743 q^{5} +180.186 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -50.9743 q^{5} +180.186 q^{7} +81.0000 q^{9} -138.332 q^{11} -830.295 q^{13} -458.768 q^{15} -1281.42 q^{17} +2646.25 q^{19} +1621.67 q^{21} +529.000 q^{23} -526.625 q^{25} +729.000 q^{27} +2853.36 q^{29} -3881.75 q^{31} -1244.99 q^{33} -9184.83 q^{35} -3089.64 q^{37} -7472.65 q^{39} +2277.91 q^{41} -9590.03 q^{43} -4128.92 q^{45} +25946.6 q^{47} +15659.9 q^{49} -11532.8 q^{51} +18223.8 q^{53} +7051.38 q^{55} +23816.2 q^{57} +33059.8 q^{59} +9546.02 q^{61} +14595.0 q^{63} +42323.7 q^{65} -14312.9 q^{67} +4761.00 q^{69} -21458.4 q^{71} +39143.1 q^{73} -4739.62 q^{75} -24925.5 q^{77} -85813.8 q^{79} +6561.00 q^{81} -8698.94 q^{83} +65319.3 q^{85} +25680.3 q^{87} +87801.1 q^{89} -149607. q^{91} -34935.8 q^{93} -134890. q^{95} -65841.4 q^{97} -11204.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 54 q^{3} + 22 q^{5} - 134 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q + 54 * q^3 + 22 * q^5 - 134 * q^7 + 486 * q^9	$ 6 q + 54 q^{3} + 22 q^{5} - 134 q^{7} + 486 q^{9} + 68 q^{11} - 340 q^{13} + 198 q^{15} + 222 q^{17} + 2314 q^{19} - 1206 q^{21} + 3174 q^{23} - 1014 q^{25} + 4374 q^{27} - 4432 q^{29} - 756 q^{31} + 612 q^{33} + 3596 q^{35} + 13908 q^{37} - 3060 q^{39} + 16884 q^{41} - 20838 q^{43} + 1782 q^{45} - 31900 q^{47} + 57070 q^{49} + 1998 q^{51} + 60914 q^{53} - 47728 q^{55} + 20826 q^{57} - 57332 q^{59} + 90252 q^{61} - 10854 q^{63} + 113220 q^{65} - 69138 q^{67} + 28566 q^{69} - 109800 q^{71} + 208204 q^{73} - 9126 q^{75} + 175368 q^{77} - 178066 q^{79} + 39366 q^{81} - 92692 q^{83} + 274648 q^{85} - 39888 q^{87} + 240354 q^{89} - 77796 q^{91} - 6804 q^{93} - 30740 q^{95} + 290104 q^{97} + 5508 q^{99}+O(q^{100}) $ 6 * q + 54 * q^3 + 22 * q^5 - 134 * q^7 + 486 * q^9 + 68 * q^11 - 340 * q^13 + 198 * q^15 + 222 * q^17 + 2314 * q^19 - 1206 * q^21 + 3174 * q^23 - 1014 * q^25 + 4374 * q^27 - 4432 * q^29 - 756 * q^31 + 612 * q^33 + 3596 * q^35 + 13908 * q^37 - 3060 * q^39 + 16884 * q^41 - 20838 * q^43 + 1782 * q^45 - 31900 * q^47 + 57070 * q^49 + 1998 * q^51 + 60914 * q^53 - 47728 * q^55 + 20826 * q^57 - 57332 * q^59 + 90252 * q^61 - 10854 * q^63 + 113220 * q^65 - 69138 * q^67 + 28566 * q^69 - 109800 * q^71 + 208204 * q^73 - 9126 * q^75 + 175368 * q^77 - 178066 * q^79 + 39366 * q^81 - 92692 * q^83 + 274648 * q^85 - 39888 * q^87 + 240354 * q^89 - 77796 * q^91 - 6804 * q^93 - 30740 * q^95 + 290104 * q^97 + 5508 * 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
$5$	−50.9743	−0.911855	−0.455928	−	0.890017i	$-0.650692\pi$
$5$	−50.9743	−0.911855	−0.455928	+	0.890017i	$0.650692\pi$
$6$	0	0
$7$	180.186	1.38987	0.694936	−	0.719071i	$-0.255433\pi$
$7$	180.186	1.38987	0.694936	+	0.719071i	$0.255433\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−138.332	−0.344700	−0.172350	−	0.985036i	$-0.555136\pi$
$11$	−138.332	−0.344700	−0.172350	+	0.985036i	$0.555136\pi$
$12$	0	0
$13$	−830.295	−1.36262	−0.681309	−	0.731996i	$-0.738589\pi$
$13$	−830.295	−1.36262	−0.681309	+	0.731996i	$0.738589\pi$
$14$	0	0
$15$	−458.768	−0.526460
$16$	0	0
$17$	−1281.42	−1.07540	−0.537698	−	0.843138i	$-0.680706\pi$
$17$	−1281.42	−1.07540	−0.537698	+	0.843138i	$0.680706\pi$
$18$	0	0
$19$	2646.25	1.68169	0.840846	−	0.541275i	$-0.182058\pi$
$19$	2646.25	1.68169	0.840846	+	0.541275i	$0.182058\pi$
$20$	0	0
$21$	1621.67	0.802443
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−526.625	−0.168520
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	2853.36	0.630032	0.315016	−	0.949086i	$-0.397990\pi$
$29$	2853.36	0.630032	0.315016	+	0.949086i	$0.397990\pi$
$30$	0	0
$31$	−3881.75	−0.725477	−0.362738	−	0.931891i	$-0.618158\pi$
$31$	−3881.75	−0.725477	−0.362738	+	0.931891i	$0.618158\pi$
$32$	0	0
$33$	−1244.99	−0.199013
$34$	0	0
$35$	−9184.83	−1.26736
$36$	0	0
$37$	−3089.64	−0.371025	−0.185512	−	0.982642i	$-0.559394\pi$
$37$	−3089.64	−0.371025	−0.185512	+	0.982642i	$0.559394\pi$
$38$	0	0
$39$	−7472.65	−0.786708
$40$	0	0
$41$	2277.91	0.211630	0.105815	−	0.994386i	$-0.466255\pi$
$41$	2277.91	0.211630	0.105815	+	0.994386i	$0.466255\pi$
$42$	0	0
$43$	−9590.03	−0.790950	−0.395475	−	0.918477i	$-0.629420\pi$
$43$	−9590.03	−0.790950	−0.395475	+	0.918477i	$0.629420\pi$
$44$	0	0
$45$	−4128.92	−0.303952
$46$	0	0
$47$	25946.6	1.71331	0.856655	−	0.515890i	$-0.172539\pi$
$47$	25946.6	1.71331	0.856655	+	0.515890i	$0.172539\pi$
$48$	0	0
$49$	15659.9	0.931746
$50$	0	0
$51$	−11532.8	−0.620880
$52$	0	0
$53$	18223.8	0.891148	0.445574	−	0.895245i	$-0.353000\pi$
$53$	18223.8	0.891148	0.445574	+	0.895245i	$0.353000\pi$
$54$	0	0
$55$	7051.38	0.314317
$56$	0	0
$57$	23816.2	0.970925
$58$	0	0
$59$	33059.8	1.23643	0.618216	−	0.786008i	$-0.287856\pi$
$59$	33059.8	1.23643	0.618216	+	0.786008i	$0.287856\pi$
$60$	0	0
$61$	9546.02	0.328472	0.164236	−	0.986421i	$-0.447484\pi$
$61$	9546.02	0.328472	0.164236	+	0.986421i	$0.447484\pi$
$62$	0	0
$63$	14595.0	0.463291
$64$	0	0
$65$	42323.7	1.24251
$66$	0	0
$67$	−14312.9	−0.389529	−0.194765	−	0.980850i	$-0.562394\pi$
$67$	−14312.9	−0.389529	−0.194765	+	0.980850i	$0.562394\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−21458.4	−0.505187	−0.252593	−	0.967573i	$-0.581283\pi$
$71$	−21458.4	−0.505187	−0.252593	+	0.967573i	$0.581283\pi$
$72$	0	0
$73$	39143.1	0.859703	0.429851	−	0.902900i	$-0.358566\pi$
$73$	39143.1	0.859703	0.429851	+	0.902900i	$0.358566\pi$
$74$	0	0
$75$	−4739.62	−0.0972950
$76$	0	0
$77$	−24925.5	−0.479089
$78$	0	0
$79$	−85813.8	−1.54700	−0.773499	−	0.633798i	$-0.781495\pi$
$79$	−85813.8	−1.54700	−0.773499	+	0.633798i	$0.781495\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−8698.94	−0.138602	−0.0693012	−	0.997596i	$-0.522077\pi$
$83$	−8698.94	−0.138602	−0.0693012	+	0.997596i	$0.522077\pi$
$84$	0	0
$85$	65319.3	0.980605
$86$	0	0
$87$	25680.3	0.363749
$88$	0	0
$89$	87801.1	1.17497	0.587483	−	0.809237i	$-0.300119\pi$
$89$	87801.1	1.17497	0.587483	+	0.809237i	$0.300119\pi$
$90$	0	0
$91$	−149607.	−1.89386
$92$	0	0
$93$	−34935.8	−0.418854
$94$	0	0
$95$	−134890.	−1.53346
$96$	0	0
$97$	−65841.4	−0.710510	−0.355255	−	0.934770i	$-0.615606\pi$
$97$	−65841.4	−0.710510	−0.355255	+	0.934770i	$0.615606\pi$
$98$	0	0
$99$	−11204.9	−0.114900
$100$	0	0
$101$	−68831.9	−0.671408	−0.335704	−	0.941968i	$-0.608974\pi$
$101$	−68831.9	−0.671408	−0.335704	+	0.941968i	$0.608974\pi$
$102$	0	0
$103$	81029.9	0.752579	0.376290	−	0.926502i	$-0.377200\pi$
$103$	81029.9	0.752579	0.376290	+	0.926502i	$0.377200\pi$
$104$	0	0
$105$	−82663.5	−0.731712
$106$	0	0
$107$	158186.	1.33570	0.667851	−	0.744295i	$-0.267214\pi$
$107$	158186.	1.33570	0.667851	+	0.744295i	$0.267214\pi$
$108$	0	0
$109$	30129.2	0.242896	0.121448	−	0.992598i	$-0.461246\pi$
$109$	30129.2	0.242896	0.121448	+	0.992598i	$0.461246\pi$
$110$	0	0
$111$	−27806.7	−0.214211
$112$	0	0
$113$	48769.6	0.359296	0.179648	−	0.983731i	$-0.442504\pi$
$113$	48769.6	0.359296	0.179648	+	0.983731i	$0.442504\pi$
$114$	0	0
$115$	−26965.4	−0.190135
$116$	0	0
$117$	−67253.9	−0.454206
$118$	0	0
$119$	−230893.	−1.49466
$120$	0	0
$121$	−141915.	−0.881182
$122$	0	0
$123$	20501.2	0.122185
$124$	0	0
$125$	186139.	1.06552
$126$	0	0
$127$	−211778.	−1.16512	−0.582561	−	0.812787i	$-0.697949\pi$
$127$	−211778.	−1.16512	−0.582561	+	0.812787i	$0.697949\pi$
$128$	0	0
$129$	−86310.3	−0.456655
$130$	0	0
$131$	286898.	1.46066	0.730329	−	0.683095i	$-0.239367\pi$
$131$	286898.	1.46066	0.730329	+	0.683095i	$0.239367\pi$
$132$	0	0
$133$	476816.	2.33734
$134$	0	0
$135$	−37160.2	−0.175487
$136$	0	0
$137$	−35605.1	−0.162073	−0.0810366	−	0.996711i	$-0.525823\pi$
$137$	−35605.1	−0.162073	−0.0810366	+	0.996711i	$0.525823\pi$
$138$	0	0
$139$	55243.4	0.242518	0.121259	−	0.992621i	$-0.461307\pi$
$139$	55243.4	0.242518	0.121259	+	0.992621i	$0.461307\pi$
$140$	0	0
$141$	233520.	0.989180
$142$	0	0
$143$	114857.	0.469695
$144$	0	0
$145$	−145448.	−0.574498
$146$	0	0
$147$	140939.	0.537944
$148$	0	0
$149$	82819.5	0.305610	0.152805	−	0.988256i	$-0.451169\pi$
$149$	82819.5	0.305610	0.152805	+	0.988256i	$0.451169\pi$
$150$	0	0
$151$	188868.	0.674085	0.337043	−	0.941489i	$-0.390573\pi$
$151$	188868.	0.674085	0.337043	+	0.941489i	$0.390573\pi$
$152$	0	0
$153$	−103795.	−0.358465
$154$	0	0
$155$	197869.	0.661530
$156$	0	0
$157$	−330235.	−1.06924	−0.534619	−	0.845093i	$-0.679545\pi$
$157$	−330235.	−1.06924	−0.534619	+	0.845093i	$0.679545\pi$
$158$	0	0
$159$	164014.	0.514505
$160$	0	0
$161$	95318.2	0.289808
$162$	0	0
$163$	291976.	0.860751	0.430375	−	0.902650i	$-0.358381\pi$
$163$	291976.	0.860751	0.430375	+	0.902650i	$0.358381\pi$
$164$	0	0
$165$	63462.4	0.181471
$166$	0	0
$167$	190164.	0.527639	0.263820	−	0.964572i	$-0.415018\pi$
$167$	190164.	0.527639	0.263820	+	0.964572i	$0.415018\pi$
$168$	0	0
$169$	318097.	0.856726
$170$	0	0
$171$	214346.	0.560564
$172$	0	0
$173$	362708.	0.921388	0.460694	−	0.887559i	$-0.347601\pi$
$173$	362708.	0.921388	0.460694	+	0.887559i	$0.347601\pi$
$174$	0	0
$175$	−94890.2	−0.234221
$176$	0	0
$177$	297538.	0.713855
$178$	0	0
$179$	634047.	1.47907	0.739535	−	0.673118i	$-0.235045\pi$
$179$	634047.	1.47907	0.739535	+	0.673118i	$0.235045\pi$
$180$	0	0
$181$	506088.	1.14823	0.574116	−	0.818774i	$-0.305346\pi$
$181$	506088.	1.14823	0.574116	+	0.818774i	$0.305346\pi$
$182$	0	0
$183$	85914.2	0.189643
$184$	0	0
$185$	157492.	0.338321
$186$	0	0
$187$	177261.	0.370689
$188$	0	0
$189$	131355.	0.267481
$190$	0	0
$191$	125454.	0.248829	0.124415	−	0.992230i	$-0.460295\pi$
$191$	125454.	0.248829	0.124415	+	0.992230i	$0.460295\pi$
$192$	0	0
$193$	−573441.	−1.10814	−0.554071	−	0.832470i	$-0.686926\pi$
$193$	−573441.	−1.10814	−0.554071	+	0.832470i	$0.686926\pi$
$194$	0	0
$195$	380913.	0.717363
$196$	0	0
$197$	−456088.	−0.837304	−0.418652	−	0.908147i	$-0.637497\pi$
$197$	−456088.	−0.837304	−0.418652	+	0.908147i	$0.637497\pi$
$198$	0	0
$199$	−591900.	−1.05954	−0.529768	−	0.848143i	$-0.677721\pi$
$199$	−591900.	−1.05954	−0.529768	+	0.848143i	$0.677721\pi$
$200$	0	0
$201$	−128816.	−0.224895
$202$	0	0
$203$	514135.	0.875664
$204$	0	0
$205$	−116115.	−0.192976
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−366061.	−0.579680
$210$	0	0
$211$	−1.14607e6	−1.77217	−0.886083	−	0.463526i	$-0.846584\pi$
$211$	−1.14607e6	−1.77217	−0.886083	+	0.463526i	$0.846584\pi$
$212$	0	0
$213$	−193126.	−0.291670
$214$	0	0
$215$	488845.	0.721232
$216$	0	0
$217$	−699436.	−1.00832
$218$	0	0
$219$	352288.	0.496350
$220$	0	0
$221$	1.06395e6	1.46535
$222$	0	0
$223$	762232.	1.02642	0.513210	−	0.858263i	$-0.328456\pi$
$223$	762232.	1.02642	0.513210	+	0.858263i	$0.328456\pi$
$224$	0	0
$225$	−42656.6	−0.0561733
$226$	0	0
$227$	1.04279e6	1.34317	0.671585	−	0.740927i	$-0.265614\pi$
$227$	1.04279e6	1.34317	0.671585	+	0.740927i	$0.265614\pi$
$228$	0	0
$229$	1.13577e6	1.43121	0.715603	−	0.698507i	$-0.246152\pi$
$229$	1.13577e6	1.43121	0.715603	+	0.698507i	$0.246152\pi$
$230$	0	0
$231$	−224329.	−0.276602
$232$	0	0
$233$	−689400.	−0.831920	−0.415960	−	0.909383i	$-0.636554\pi$
$233$	−689400.	−0.831920	−0.415960	+	0.909383i	$0.636554\pi$
$234$	0	0
$235$	−1.32261e6	−1.56229
$236$	0	0
$237$	−772325.	−0.893159
$238$	0	0
$239$	1.47487e6	1.67016	0.835079	−	0.550130i	$-0.185422\pi$
$239$	1.47487e6	1.67016	0.835079	+	0.550130i	$0.185422\pi$
$240$	0	0
$241$	1.09992e6	1.21988	0.609940	−	0.792448i	$-0.291194\pi$
$241$	1.09992e6	1.21988	0.609940	+	0.792448i	$0.291194\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−798250.	−0.849618
$246$	0	0
$247$	−2.19717e6	−2.29150
$248$	0	0
$249$	−78290.4	−0.0800222
$250$	0	0
$251$	939717.	0.941483	0.470742	−	0.882271i	$-0.343986\pi$
$251$	939717.	0.941483	0.470742	+	0.882271i	$0.343986\pi$
$252$	0	0
$253$	−73177.7	−0.0718750
$254$	0	0
$255$	587874.	0.566153
$256$	0	0
$257$	553442.	0.522684	0.261342	−	0.965246i	$-0.415835\pi$
$257$	553442.	0.522684	0.261342	+	0.965246i	$0.415835\pi$
$258$	0	0
$259$	−556708.	−0.515677
$260$	0	0
$261$	231123.	0.210011
$262$	0	0
$263$	648275.	0.577923	0.288961	−	0.957341i	$-0.406690\pi$
$263$	648275.	0.577923	0.288961	+	0.957341i	$0.406690\pi$
$264$	0	0
$265$	−928946.	−0.812598
$266$	0	0
$267$	790210.	0.678366
$268$	0	0
$269$	−1.68338e6	−1.41841	−0.709203	−	0.705005i	$-0.750945\pi$
$269$	−1.68338e6	−1.41841	−0.709203	+	0.705005i	$0.750945\pi$
$270$	0	0
$271$	2.15964e6	1.78632	0.893159	−	0.449742i	$-0.148484\pi$
$271$	2.15964e6	1.78632	0.893159	+	0.449742i	$0.148484\pi$
$272$	0	0
$273$	−1.34646e6	−1.09342
$274$	0	0
$275$	72849.2	0.0580889
$276$	0	0
$277$	1.86667e6	1.46174	0.730868	−	0.682519i	$-0.239115\pi$
$277$	1.86667e6	1.46174	0.730868	+	0.682519i	$0.239115\pi$
$278$	0	0
$279$	−314422.	−0.241826
$280$	0	0
$281$	1.05454e6	0.796702	0.398351	−	0.917233i	$-0.369583\pi$
$281$	1.05454e6	0.796702	0.398351	+	0.917233i	$0.369583\pi$
$282$	0	0
$283$	1.71840e6	1.27544	0.637718	−	0.770270i	$-0.279878\pi$
$283$	1.71840e6	1.27544	0.637718	+	0.770270i	$0.279878\pi$
$284$	0	0
$285$	−1.21401e6	−0.885343
$286$	0	0
$287$	410447.	0.294139
$288$	0	0
$289$	222174.	0.156476
$290$	0	0
$291$	−592573.	−0.410213
$292$	0	0
$293$	831590.	0.565901	0.282950	−	0.959135i	$-0.408687\pi$
$293$	831590.	0.565901	0.282950	+	0.959135i	$0.408687\pi$
$294$	0	0
$295$	−1.68520e6	−1.12745
$296$	0	0
$297$	−100844.	−0.0663376
$298$	0	0
$299$	−439226.	−0.284125
$300$	0	0
$301$	−1.72799e6	−1.09932
$302$	0	0
$303$	−619487.	−0.387638
$304$	0	0
$305$	−486602.	−0.299519
$306$	0	0
$307$	1.37090e6	0.830156	0.415078	−	0.909786i	$-0.363754\pi$
$307$	1.37090e6	0.830156	0.415078	+	0.909786i	$0.363754\pi$
$308$	0	0
$309$	729269.	0.434502
$310$	0	0
$311$	−2.91469e6	−1.70880	−0.854401	−	0.519614i	$-0.826076\pi$
$311$	−2.91469e6	−1.70880	−0.854401	+	0.519614i	$0.826076\pi$
$312$	0	0
$313$	788520.	0.454938	0.227469	−	0.973785i	$-0.426955\pi$
$313$	788520.	0.454938	0.227469	+	0.973785i	$0.426955\pi$
$314$	0	0
$315$	−743971.	−0.422454
$316$	0	0
$317$	−786686.	−0.439697	−0.219848	−	0.975534i	$-0.570556\pi$
$317$	−786686.	−0.439697	−0.219848	+	0.975534i	$0.570556\pi$
$318$	0	0
$319$	−394712.	−0.217172
$320$	0	0
$321$	1.42368e6	0.771168
$322$	0	0
$323$	−3.39095e6	−1.80848
$324$	0	0
$325$	437254.	0.229628
$326$	0	0
$327$	271163.	0.140236
$328$	0	0
$329$	4.67521e6	2.38128
$330$	0	0
$331$	−1.18589e6	−0.594941	−0.297471	−	0.954731i	$-0.596143\pi$
$331$	−1.18589e6	−0.594941	−0.297471	+	0.954731i	$0.596143\pi$
$332$	0	0
$333$	−250260.	−0.123675
$334$	0	0
$335$	729589.	0.355194
$336$	0	0
$337$	2.21972e6	1.06469	0.532345	−	0.846527i	$-0.321311\pi$
$337$	2.21972e6	1.06469	0.532345	+	0.846527i	$0.321311\pi$
$338$	0	0
$339$	438926.	0.207440
$340$	0	0
$341$	536971.	0.250072
$342$	0	0
$343$	−206699.	−0.0948641
$344$	0	0
$345$	−242688.	−0.109774
$346$	0	0
$347$	−4.36667e6	−1.94682	−0.973412	−	0.229061i	$-0.926434\pi$
$347$	−4.36667e6	−1.94682	−0.973412	+	0.229061i	$0.926434\pi$
$348$	0	0
$349$	3.15051e6	1.38458	0.692288	−	0.721621i	$-0.256603\pi$
$349$	3.15051e6	1.38458	0.692288	+	0.721621i	$0.256603\pi$
$350$	0	0
$351$	−605285.	−0.262236
$352$	0	0
$353$	−628885.	−0.268618	−0.134309	−	0.990940i	$-0.542881\pi$
$353$	−628885.	−0.268618	−0.134309	+	0.990940i	$0.542881\pi$
$354$	0	0
$355$	1.09383e6	0.460657
$356$	0	0
$357$	−2.07804e6	−0.862944
$358$	0	0
$359$	−1.61256e6	−0.660360	−0.330180	−	0.943918i	$-0.607109\pi$
$359$	−1.61256e6	−0.660360	−0.330180	+	0.943918i	$0.607109\pi$
$360$	0	0
$361$	4.52652e6	1.82809
$362$	0	0
$363$	−1.27724e6	−0.508751
$364$	0	0
$365$	−1.99529e6	−0.783924
$366$	0	0
$367$	−1.25277e6	−0.485518	−0.242759	−	0.970087i	$-0.578052\pi$
$367$	−1.25277e6	−0.485518	−0.242759	+	0.970087i	$0.578052\pi$
$368$	0	0
$369$	184511.	0.0705434
$370$	0	0
$371$	3.28367e6	1.23858
$372$	0	0
$373$	4.50965e6	1.67831	0.839153	−	0.543895i	$-0.183051\pi$
$373$	4.50965e6	1.67831	0.839153	+	0.543895i	$0.183051\pi$
$374$	0	0
$375$	1.67525e6	0.615179
$376$	0	0
$377$	−2.36913e6	−0.858492
$378$	0	0
$379$	4.60502e6	1.64677	0.823386	−	0.567481i	$-0.192082\pi$
$379$	4.60502e6	1.64677	0.823386	+	0.567481i	$0.192082\pi$
$380$	0	0
$381$	−1.90600e6	−0.672684
$382$	0	0
$383$	1.15190e6	0.401252	0.200626	−	0.979668i	$-0.435702\pi$
$383$	1.15190e6	0.401252	0.200626	+	0.979668i	$0.435702\pi$
$384$	0	0
$385$	1.27056e6	0.436860
$386$	0	0
$387$	−776793.	−0.263650
$388$	0	0
$389$	−4.14288e6	−1.38812	−0.694062	−	0.719915i	$-0.744181\pi$
$389$	−4.14288e6	−1.38812	−0.694062	+	0.719915i	$0.744181\pi$
$390$	0	0
$391$	−677870.	−0.224236
$392$	0	0
$393$	2.58208e6	0.843311
$394$	0	0
$395$	4.37430e6	1.41064
$396$	0	0
$397$	−1.56709e6	−0.499020	−0.249510	−	0.968372i	$-0.580270\pi$
$397$	−1.56709e6	−0.499020	−0.249510	+	0.968372i	$0.580270\pi$
$398$	0	0
$399$	4.29134e6	1.34946
$400$	0	0
$401$	5.83201e6	1.81116	0.905581	−	0.424173i	$-0.139435\pi$
$401$	5.83201e6	1.81116	0.905581	+	0.424173i	$0.139435\pi$
$402$	0	0
$403$	3.22300e6	0.988547
$404$	0	0
$405$	−334442.	−0.101317
$406$	0	0
$407$	427396.	0.127892
$408$	0	0
$409$	3.26726e6	0.965775	0.482887	−	0.875682i	$-0.339588\pi$
$409$	3.26726e6	0.965775	0.482887	+	0.875682i	$0.339588\pi$
$410$	0	0
$411$	−320446.	−0.0935730
$412$	0	0
$413$	5.95690e6	1.71848
$414$	0	0
$415$	443422.	0.126385
$416$	0	0
$417$	497191.	0.140018
$418$	0	0
$419$	1.47204e6	0.409622	0.204811	−	0.978802i	$-0.434342\pi$
$419$	1.47204e6	0.409622	0.204811	+	0.978802i	$0.434342\pi$
$420$	0	0
$421$	4.44395e6	1.22198	0.610990	−	0.791638i	$-0.290772\pi$
$421$	4.44395e6	1.22198	0.610990	+	0.791638i	$0.290772\pi$
$422$	0	0
$423$	2.10168e6	0.571103
$424$	0	0
$425$	674826.	0.181226
$426$	0	0
$427$	1.72006e6	0.456534
$428$	0	0
$429$	1.03371e6	0.271178
$430$	0	0
$431$	−2.59093e6	−0.671835	−0.335917	−	0.941891i	$-0.609046\pi$
$431$	−2.59093e6	−0.671835	−0.335917	+	0.941891i	$0.609046\pi$
$432$	0	0
$433$	−2.65601e6	−0.680785	−0.340393	−	0.940283i	$-0.610560\pi$
$433$	−2.65601e6	−0.680785	−0.340393	+	0.940283i	$0.610560\pi$
$434$	0	0
$435$	−1.30903e6	−0.331686
$436$	0	0
$437$	1.39986e6	0.350657
$438$	0	0
$439$	−1.52100e6	−0.376676	−0.188338	−	0.982104i	$-0.560310\pi$
$439$	−1.52100e6	−0.376676	−0.188338	+	0.982104i	$0.560310\pi$
$440$	0	0
$441$	1.26845e6	0.310582
$442$	0	0
$443$	2.59058e6	0.627173	0.313586	−	0.949560i	$-0.398469\pi$
$443$	2.59058e6	0.627173	0.313586	+	0.949560i	$0.398469\pi$
$444$	0	0
$445$	−4.47560e6	−1.07140
$446$	0	0
$447$	745376.	0.176444
$448$	0	0
$449$	108145.	0.0253158	0.0126579	−	0.999920i	$-0.495971\pi$
$449$	108145.	0.0253158	0.0126579	+	0.999920i	$0.495971\pi$
$450$	0	0
$451$	−315109.	−0.0729490
$452$	0	0
$453$	1.69981e6	0.389183
$454$	0	0
$455$	7.62612e6	1.72693
$456$	0	0
$457$	−2.49861e6	−0.559639	−0.279820	−	0.960053i	$-0.590275\pi$
$457$	−2.49861e6	−0.559639	−0.279820	+	0.960053i	$0.590275\pi$
$458$	0	0
$459$	−934153.	−0.206960
$460$	0	0
$461$	−2.66308e6	−0.583622	−0.291811	−	0.956476i	$-0.594258\pi$
$461$	−2.66308e6	−0.583622	−0.291811	+	0.956476i	$0.594258\pi$
$462$	0	0
$463$	−5.40125e6	−1.17096	−0.585479	−	0.810687i	$-0.699094\pi$
$463$	−5.40125e6	−1.17096	−0.585479	+	0.810687i	$0.699094\pi$
$464$	0	0
$465$	1.78082e6	0.381934
$466$	0	0
$467$	3.72337e6	0.790030	0.395015	−	0.918675i	$-0.370739\pi$
$467$	3.72337e6	0.790030	0.395015	+	0.918675i	$0.370739\pi$
$468$	0	0
$469$	−2.57898e6	−0.541396
$470$	0	0
$471$	−2.97212e6	−0.617324
$472$	0	0
$473$	1.32661e6	0.272641
$474$	0	0
$475$	−1.39358e6	−0.283399
$476$	0	0
$477$	1.47613e6	0.297049
$478$	0	0
$479$	204721.	0.0407684	0.0203842	−	0.999792i	$-0.493511\pi$
$479$	204721.	0.0407684	0.0203842	+	0.999792i	$0.493511\pi$
$480$	0	0
$481$	2.56531e6	0.505565
$482$	0	0
$483$	857864.	0.167321
$484$	0	0
$485$	3.35622e6	0.647882
$486$	0	0
$487$	−3.63241e6	−0.694021	−0.347010	−	0.937861i	$-0.612803\pi$
$487$	−3.63241e6	−0.694021	−0.347010	+	0.937861i	$0.612803\pi$
$488$	0	0
$489$	2.62778e6	0.496955
$490$	0	0
$491$	9.48030e6	1.77467	0.887336	−	0.461123i	$-0.152553\pi$
$491$	9.48030e6	1.77467	0.887336	+	0.461123i	$0.152553\pi$
$492$	0	0
$493$	−3.65635e6	−0.677533
$494$	0	0
$495$	571162.	0.104772
$496$	0	0
$497$	−3.86650e6	−0.702145
$498$	0	0
$499$	1.00339e7	1.80393	0.901964	−	0.431811i	$-0.142125\pi$
$499$	1.00339e7	1.80393	0.901964	+	0.431811i	$0.142125\pi$
$500$	0	0
$501$	1.71148e6	0.304633
$502$	0	0
$503$	−9.57742e6	−1.68783	−0.843914	−	0.536478i	$-0.819755\pi$
$503$	−9.57742e6	−1.68783	−0.843914	+	0.536478i	$0.819755\pi$
$504$	0	0
$505$	3.50866e6	0.612227
$506$	0	0
$507$	2.86287e6	0.494631
$508$	0	0
$509$	1.01936e7	1.74394	0.871971	−	0.489558i	$-0.162842\pi$
$509$	1.01936e7	1.74394	0.871971	+	0.489558i	$0.162842\pi$
$510$	0	0
$511$	7.05303e6	1.19488
$512$	0	0
$513$	1.92911e6	0.323642
$514$	0	0
$515$	−4.13044e6	−0.686243
$516$	0	0
$517$	−3.58925e6	−0.590578
$518$	0	0
$519$	3.26438e6	0.531963
$520$	0	0
$521$	−5.32791e6	−0.859930	−0.429965	−	0.902846i	$-0.641474\pi$
$521$	−5.32791e6	−0.859930	−0.429965	+	0.902846i	$0.641474\pi$
$522$	0	0
$523$	−4.45611e6	−0.712364	−0.356182	−	0.934417i	$-0.615922\pi$
$523$	−4.45611e6	−0.712364	−0.356182	+	0.934417i	$0.615922\pi$
$524$	0	0
$525$	−854012.	−0.135228
$526$	0	0
$527$	4.97414e6	0.780175
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.67785e6	0.412144
$532$	0	0
$533$	−1.89134e6	−0.288371
$534$	0	0
$535$	−8.06343e6	−1.21797
$536$	0	0
$537$	5.70642e6	0.853942
$538$	0	0
$539$	−2.16626e6	−0.321173
$540$	0	0
$541$	−1.32342e7	−1.94404	−0.972020	−	0.234899i	$-0.924524\pi$
$541$	−1.32342e7	−1.94404	−0.972020	+	0.234899i	$0.924524\pi$
$542$	0	0
$543$	4.55480e6	0.662933
$544$	0	0
$545$	−1.53581e6	−0.221486
$546$	0	0
$547$	1.78019e6	0.254389	0.127195	−	0.991878i	$-0.459403\pi$
$547$	1.78019e6	0.254389	0.127195	+	0.991878i	$0.459403\pi$
$548$	0	0
$549$	773228.	0.109491
$550$	0	0
$551$	7.55071e6	1.05952
$552$	0	0
$553$	−1.54624e7	−2.15013
$554$	0	0
$555$	1.41743e6	0.195330
$556$	0	0
$557$	−9.96008e6	−1.36027	−0.680135	−	0.733087i	$-0.738079\pi$
$557$	−9.96008e6	−1.36027	−0.680135	+	0.733087i	$0.738079\pi$
$558$	0	0
$559$	7.96256e6	1.07776
$560$	0	0
$561$	1.59535e6	0.214018
$562$	0	0
$563$	−608704.	−0.0809347	−0.0404674	−	0.999181i	$-0.512885\pi$
$563$	−608704.	−0.0809347	−0.0404674	+	0.999181i	$0.512885\pi$
$564$	0	0
$565$	−2.48599e6	−0.327626
$566$	0	0
$567$	1.18220e6	0.154430
$568$	0	0
$569$	9.68623e6	1.25422	0.627110	−	0.778930i	$-0.284238\pi$
$569$	9.68623e6	1.25422	0.627110	+	0.778930i	$0.284238\pi$
$570$	0	0
$571$	1.17662e7	1.51024	0.755122	−	0.655584i	$-0.227577\pi$
$571$	1.17662e7	1.51024	0.755122	+	0.655584i	$0.227577\pi$
$572$	0	0
$573$	1.12909e6	0.143662
$574$	0	0
$575$	−278585.	−0.0351388
$576$	0	0
$577$	−1.60162e6	−0.200271	−0.100136	−	0.994974i	$-0.531928\pi$
$577$	−1.60162e6	−0.200271	−0.100136	+	0.994974i	$0.531928\pi$
$578$	0	0
$579$	−5.16096e6	−0.639786
$580$	0	0
$581$	−1.56742e6	−0.192640
$582$	0	0
$583$	−2.52094e6	−0.307179
$584$	0	0
$585$	3.42822e6	0.414170
$586$	0	0
$587$	−1.17511e7	−1.40762	−0.703808	−	0.710391i	$-0.748518\pi$
$587$	−1.17511e7	−1.40762	−0.703808	+	0.710391i	$0.748518\pi$
$588$	0	0
$589$	−1.02721e7	−1.22003
$590$	0	0
$591$	−4.10479e6	−0.483418
$592$	0	0
$593$	1.51446e6	0.176856	0.0884280	−	0.996083i	$-0.471816\pi$
$593$	1.51446e6	0.176856	0.0884280	+	0.996083i	$0.471816\pi$
$594$	0	0
$595$	1.17696e7	1.36292
$596$	0	0
$597$	−5.32710e6	−0.611723
$598$	0	0
$599$	−6.87169e6	−0.782521	−0.391261	−	0.920280i	$-0.627961\pi$
$599$	−6.87169e6	−0.782521	−0.391261	+	0.920280i	$0.627961\pi$
$600$	0	0
$601$	−3.19438e6	−0.360745	−0.180373	−	0.983598i	$-0.557730\pi$
$601$	−3.19438e6	−0.360745	−0.180373	+	0.983598i	$0.557730\pi$
$602$	0	0
$603$	−1.15934e6	−0.129843
$604$	0	0
$605$	7.23402e6	0.803510
$606$	0	0
$607$	−1.09869e7	−1.21033	−0.605164	−	0.796100i	$-0.706893\pi$
$607$	−1.09869e7	−1.21033	−0.605164	+	0.796100i	$0.706893\pi$
$608$	0	0
$609$	4.62722e6	0.505565
$610$	0	0
$611$	−2.15433e7	−2.33459
$612$	0	0
$613$	−7.97638e6	−0.857343	−0.428671	−	0.903460i	$-0.641018\pi$
$613$	−7.97638e6	−0.857343	−0.428671	+	0.903460i	$0.641018\pi$
$614$	0	0
$615$	−1.04503e6	−0.111415
$616$	0	0
$617$	2.06078e6	0.217931	0.108965	−	0.994046i	$-0.465246\pi$
$617$	2.06078e6	0.217931	0.108965	+	0.994046i	$0.465246\pi$
$618$	0	0
$619$	−6.29233e6	−0.660063	−0.330031	−	0.943970i	$-0.607059\pi$
$619$	−6.29233e6	−0.660063	−0.330031	+	0.943970i	$0.607059\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	1.58205e7	1.63305
$624$	0	0
$625$	−7.84259e6	−0.803081
$626$	0	0
$627$	−3.29455e6	−0.334678
$628$	0	0
$629$	3.95911e6	0.398999
$630$	0	0
$631$	5.48136e6	0.548043	0.274021	−	0.961724i	$-0.411646\pi$
$631$	5.48136e6	0.548043	0.274021	+	0.961724i	$0.411646\pi$
$632$	0	0
$633$	−1.03146e7	−1.02316
$634$	0	0
$635$	1.07952e7	1.06242
$636$	0	0
$637$	−1.30023e7	−1.26961
$638$	0	0
$639$	−1.73813e6	−0.168396
$640$	0	0
$641$	1.19465e6	0.114841	0.0574205	−	0.998350i	$-0.481712\pi$
$641$	1.19465e6	0.114841	0.0574205	+	0.998350i	$0.481712\pi$
$642$	0	0
$643$	−581312.	−0.0554475	−0.0277237	−	0.999616i	$-0.508826\pi$
$643$	−581312.	−0.0554475	−0.0277237	+	0.999616i	$0.508826\pi$
$644$	0	0
$645$	4.39960e6	0.416403
$646$	0	0
$647$	−5.05090e6	−0.474360	−0.237180	−	0.971466i	$-0.576223\pi$
$647$	−5.05090e6	−0.474360	−0.237180	+	0.971466i	$0.576223\pi$
$648$	0	0
$649$	−4.57324e6	−0.426199
$650$	0	0
$651$	−6.29492e6	−0.582154
$652$	0	0
$653$	8.87589e6	0.814571	0.407286	−	0.913301i	$-0.366475\pi$
$653$	8.87589e6	0.814571	0.407286	+	0.913301i	$0.366475\pi$
$654$	0	0
$655$	−1.46244e7	−1.33191
$656$	0	0
$657$	3.17059e6	0.286568
$658$	0	0
$659$	−2.13244e7	−1.91277	−0.956387	−	0.292102i	$-0.905645\pi$
$659$	−2.13244e7	−1.91277	−0.956387	+	0.292102i	$0.905645\pi$
$660$	0	0
$661$	−8.42892e6	−0.750358	−0.375179	−	0.926952i	$-0.622419\pi$
$661$	−8.42892e6	−0.750358	−0.375179	+	0.926952i	$0.622419\pi$
$662$	0	0
$663$	9.57559e6	0.846022
$664$	0	0
$665$	−2.43053e7	−2.13131
$666$	0	0
$667$	1.50943e6	0.131371
$668$	0	0
$669$	6.86009e6	0.592604
$670$	0	0
$671$	−1.32052e6	−0.113224
$672$	0	0
$673$	1.05852e7	0.900868	0.450434	−	0.892810i	$-0.351269\pi$
$673$	1.05852e7	0.900868	0.450434	+	0.892810i	$0.351269\pi$
$674$	0	0
$675$	−383910.	−0.0324317
$676$	0	0
$677$	5.84365e6	0.490018	0.245009	−	0.969521i	$-0.421209\pi$
$677$	5.84365e6	0.490018	0.245009	+	0.969521i	$0.421209\pi$
$678$	0	0
$679$	−1.18637e7	−0.987518
$680$	0	0
$681$	9.38509e6	0.775480
$682$	0	0
$683$	7.50330e6	0.615461	0.307730	−	0.951474i	$-0.400431\pi$
$683$	7.50330e6	0.615461	0.307730	+	0.951474i	$0.400431\pi$
$684$	0	0
$685$	1.81495e6	0.147787
$686$	0	0
$687$	1.02219e7	0.826308
$688$	0	0
$689$	−1.51311e7	−1.21429
$690$	0	0
$691$	535122.	0.0426342	0.0213171	−	0.999773i	$-0.493214\pi$
$691$	535122.	0.0426342	0.0213171	+	0.999773i	$0.493214\pi$
$692$	0	0
$693$	−2.01896e6	−0.159696
$694$	0	0
$695$	−2.81599e6	−0.221141
$696$	0	0
$697$	−2.91896e6	−0.227586
$698$	0	0
$699$	−6.20460e6	−0.480309
$700$	0	0
$701$	−7.38121e6	−0.567326	−0.283663	−	0.958924i	$-0.591550\pi$
$701$	−7.38121e6	−0.567326	−0.283663	+	0.958924i	$0.591550\pi$
$702$	0	0
$703$	−8.17594e6	−0.623949
$704$	0	0
$705$	−1.19035e7	−0.901989
$706$	0	0
$707$	−1.24025e7	−0.933171
$708$	0	0
$709$	−1.96801e7	−1.47032	−0.735160	−	0.677894i	$-0.762893\pi$
$709$	−1.96801e7	−1.47032	−0.735160	+	0.677894i	$0.762893\pi$
$710$	0	0
$711$	−6.95092e6	−0.515666
$712$	0	0
$713$	−2.05345e6	−0.151272
$714$	0	0
$715$	−5.85473e6	−0.428293
$716$	0	0
$717$	1.32738e7	0.964266
$718$	0	0
$719$	1.09901e6	0.0792826	0.0396413	−	0.999214i	$-0.487378\pi$
$719$	1.09901e6	0.0792826	0.0396413	+	0.999214i	$0.487378\pi$
$720$	0	0
$721$	1.46004e7	1.04599
$722$	0	0
$723$	9.89925e6	0.704298
$724$	0	0
$725$	−1.50265e6	−0.106173
$726$	0	0
$727$	1.37020e6	0.0961495	0.0480748	−	0.998844i	$-0.484691\pi$
$727$	1.37020e6	0.0961495	0.0480748	+	0.998844i	$0.484691\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.22888e7	0.850584
$732$	0	0
$733$	9.23015e6	0.634525	0.317263	−	0.948338i	$-0.397236\pi$
$733$	9.23015e6	0.634525	0.317263	+	0.948338i	$0.397236\pi$
$734$	0	0
$735$	−7.18425e6	−0.490527
$736$	0	0
$737$	1.97993e6	0.134271
$738$	0	0
$739$	−2.25859e7	−1.52134	−0.760668	−	0.649141i	$-0.775129\pi$
$739$	−2.25859e7	−1.52134	−0.760668	+	0.649141i	$0.775129\pi$
$740$	0	0
$741$	−1.97745e7	−1.32300
$742$	0	0
$743$	2.00349e7	1.33142	0.665710	−	0.746211i	$-0.268129\pi$
$743$	2.00349e7	1.33142	0.665710	+	0.746211i	$0.268129\pi$
$744$	0	0
$745$	−4.22166e6	−0.278672
$746$	0	0
$747$	−704614.	−0.0462008
$748$	0	0
$749$	2.85029e7	1.85646
$750$	0	0
$751$	2.86134e7	1.85127	0.925636	−	0.378416i	$-0.123531\pi$
$751$	2.86134e7	1.85127	0.925636	+	0.378416i	$0.123531\pi$
$752$	0	0
$753$	8.45745e6	0.543566
$754$	0	0
$755$	−9.62738e6	−0.614668
$756$	0	0
$757$	−4.40043e6	−0.279097	−0.139549	−	0.990215i	$-0.544565\pi$
$757$	−4.40043e6	−0.279097	−0.139549	+	0.990215i	$0.544565\pi$
$758$	0	0
$759$	−658600.	−0.0414970
$760$	0	0
$761$	2.63202e7	1.64751	0.823754	−	0.566948i	$-0.191876\pi$
$761$	2.63202e7	1.64751	0.823754	+	0.566948i	$0.191876\pi$
$762$	0	0
$763$	5.42885e6	0.337595
$764$	0	0
$765$	5.29086e6	0.326868
$766$	0	0
$767$	−2.74494e7	−1.68478
$768$	0	0
$769$	2.72165e7	1.65965	0.829825	−	0.558023i	$-0.188440\pi$
$769$	2.72165e7	1.65965	0.829825	+	0.558023i	$0.188440\pi$
$770$	0	0
$771$	4.98098e6	0.301772
$772$	0	0
$773$	−1.80212e7	−1.08476	−0.542381	−	0.840133i	$-0.682477\pi$
$773$	−1.80212e7	−1.08476	−0.542381	+	0.840133i	$0.682477\pi$
$774$	0	0
$775$	2.04423e6	0.122257
$776$	0	0
$777$	−5.01037e6	−0.297726
$778$	0	0
$779$	6.02792e6	0.355897
$780$	0	0
$781$	2.96839e6	0.174138
$782$	0	0
$783$	2.08010e6	0.121250
$784$	0	0
$785$	1.68335e7	0.974990
$786$	0	0
$787$	−1.51040e7	−0.869269	−0.434634	−	0.900607i	$-0.643122\pi$
$787$	−1.51040e7	−0.869269	−0.434634	+	0.900607i	$0.643122\pi$
$788$	0	0
$789$	5.83447e6	0.333664
$790$	0	0
$791$	8.78757e6	0.499376
$792$	0	0
$793$	−7.92601e6	−0.447581
$794$	0	0
$795$	−8.36051e6	−0.469154
$796$	0	0
$797$	2.00633e7	1.11881	0.559404	−	0.828895i	$-0.311030\pi$
$797$	2.00633e7	1.11881	0.559404	+	0.828895i	$0.311030\pi$
$798$	0	0
$799$	−3.32484e7	−1.84249
$800$	0	0
$801$	7.11189e6	0.391655
$802$	0	0
$803$	−5.41475e6	−0.296340
$804$	0	0
$805$	−4.85877e6	−0.264263
$806$	0	0
$807$	−1.51504e7	−0.818917
$808$	0	0
$809$	4.74708e6	0.255009	0.127504	−	0.991838i	$-0.459303\pi$
$809$	4.74708e6	0.255009	0.127504	+	0.991838i	$0.459303\pi$
$810$	0	0
$811$	1.36204e6	0.0727173	0.0363586	−	0.999339i	$-0.488424\pi$
$811$	1.36204e6	0.0727173	0.0363586	+	0.999339i	$0.488424\pi$
$812$	0	0
$813$	1.94368e7	1.03133
$814$	0	0
$815$	−1.48832e7	−0.784880
$816$	0	0
$817$	−2.53776e7	−1.33013
$818$	0	0
$819$	−1.21182e7	−0.631288
$820$	0	0
$821$	806927.	0.0417808	0.0208904	−	0.999782i	$-0.493350\pi$
$821$	806927.	0.0417808	0.0208904	+	0.999782i	$0.493350\pi$
$822$	0	0
$823$	−1.71185e7	−0.880982	−0.440491	−	0.897757i	$-0.645196\pi$
$823$	−1.71185e7	−0.880982	−0.440491	+	0.897757i	$0.645196\pi$
$824$	0	0
$825$	655643.	0.0335376
$826$	0	0
$827$	2.49549e7	1.26880	0.634398	−	0.773006i	$-0.281248\pi$
$827$	2.49549e7	1.26880	0.634398	+	0.773006i	$0.281248\pi$
$828$	0	0
$829$	−9.98894e6	−0.504816	−0.252408	−	0.967621i	$-0.581222\pi$
$829$	−9.98894e6	−0.504816	−0.252408	+	0.967621i	$0.581222\pi$
$830$	0	0
$831$	1.68001e7	0.843934
$832$	0	0
$833$	−2.00668e7	−1.00200
$834$	0	0
$835$	−9.69347e6	−0.481131
$836$	0	0
$837$	−2.82980e6	−0.139618
$838$	0	0
$839$	−4.04293e7	−1.98286	−0.991428	−	0.130651i	$-0.958293\pi$
$839$	−4.04293e7	−1.98286	−0.991428	+	0.130651i	$0.958293\pi$
$840$	0	0
$841$	−1.23695e7	−0.603060
$842$	0	0
$843$	9.49083e6	0.459976
$844$	0	0
$845$	−1.62147e7	−0.781211
$846$	0	0
$847$	−2.55711e7	−1.22473
$848$	0	0
$849$	1.54656e7	0.736374
$850$	0	0
$851$	−1.63442e6	−0.0773640
$852$	0	0
$853$	3.32363e7	1.56401	0.782005	−	0.623272i	$-0.214197\pi$
$853$	3.32363e7	1.56401	0.782005	+	0.623272i	$0.214197\pi$
$854$	0	0
$855$	−1.09261e7	−0.511153
$856$	0	0
$857$	−1.77481e7	−0.825467	−0.412733	−	0.910852i	$-0.635426\pi$
$857$	−1.77481e7	−0.825467	−0.412733	+	0.910852i	$0.635426\pi$
$858$	0	0
$859$	2.18771e7	1.01160	0.505798	−	0.862652i	$-0.331198\pi$
$859$	2.18771e7	1.01160	0.505798	+	0.862652i	$0.331198\pi$
$860$	0	0
$861$	3.69403e6	0.169821
$862$	0	0
$863$	3.37458e7	1.54238	0.771192	−	0.636602i	$-0.219661\pi$
$863$	3.37458e7	1.54238	0.771192	+	0.636602i	$0.219661\pi$
$864$	0	0
$865$	−1.84888e7	−0.840172
$866$	0	0
$867$	1.99957e6	0.0903416
$868$	0	0
$869$	1.18708e7	0.533250
$870$	0	0
$871$	1.18839e7	0.530779
$872$	0	0
$873$	−5.33316e6	−0.236837
$874$	0	0
$875$	3.35395e7	1.48094
$876$	0	0
$877$	−1.01912e7	−0.447432	−0.223716	−	0.974654i	$-0.571819\pi$
$877$	−1.01912e7	−0.447432	−0.223716	+	0.974654i	$0.571819\pi$
$878$	0	0
$879$	7.48431e6	0.326723
$880$	0	0
$881$	−1.45608e7	−0.632042	−0.316021	−	0.948752i	$-0.602347\pi$
$881$	−1.45608e7	−0.632042	−0.316021	+	0.948752i	$0.602347\pi$
$882$	0	0
$883$	−2.32169e7	−1.00208	−0.501039	−	0.865424i	$-0.667049\pi$
$883$	−2.32169e7	−1.00208	−0.501039	+	0.865424i	$0.667049\pi$
$884$	0	0
$885$	−1.51668e7	−0.650932
$886$	0	0
$887$	−7.11076e6	−0.303464	−0.151732	−	0.988422i	$-0.548485\pi$
$887$	−7.11076e6	−0.303464	−0.151732	+	0.988422i	$0.548485\pi$
$888$	0	0
$889$	−3.81594e7	−1.61937
$890$	0	0
$891$	−907598.	−0.0383000
$892$	0	0
$893$	6.86611e7	2.88126
$894$	0	0
$895$	−3.23201e7	−1.34870
$896$	0	0
$897$	−3.95303e6	−0.164040
$898$	0	0
$899$	−1.10761e7	−0.457073
$900$	0	0
$901$	−2.33523e7	−0.958337
$902$	0	0
$903$	−1.55519e7	−0.634693
$904$	0	0
$905$	−2.57975e7	−1.04702
$906$	0	0
$907$	−1.60759e7	−0.648867	−0.324434	−	0.945908i	$-0.605174\pi$
$907$	−1.60759e7	−0.648867	−0.324434	+	0.945908i	$0.605174\pi$
$908$	0	0
$909$	−5.57539e6	−0.223803
$910$	0	0
$911$	−650087.	−0.0259523	−0.0129761	−	0.999916i	$-0.504131\pi$
$911$	−650087.	−0.0259523	−0.0129761	+	0.999916i	$0.504131\pi$
$912$	0	0
$913$	1.20334e6	0.0477763
$914$	0	0
$915$	−4.37941e6	−0.172927
$916$	0	0
$917$	5.16948e7	2.03013
$918$	0	0
$919$	−3.20732e7	−1.25272	−0.626360	−	0.779534i	$-0.715456\pi$
$919$	−3.20732e7	−1.25272	−0.626360	+	0.779534i	$0.715456\pi$
$920$	0	0
$921$	1.23381e7	0.479291
$922$	0	0
$923$	1.78168e7	0.688376
$924$	0	0
$925$	1.62708e6	0.0625251
$926$	0	0
$927$	6.56342e6	0.250860
$928$	0	0
$929$	−1.26393e7	−0.480489	−0.240245	−	0.970712i	$-0.577228\pi$
$929$	−1.26393e7	−0.480489	−0.240245	+	0.970712i	$0.577228\pi$
$930$	0	0
$931$	4.14399e7	1.56691
$932$	0	0
$933$	−2.62322e7	−0.986577
$934$	0	0
$935$	−9.03576e6	−0.338015
$936$	0	0
$937$	−4.22662e7	−1.57270	−0.786348	−	0.617784i	$-0.788031\pi$
$937$	−4.22662e7	−1.57270	−0.786348	+	0.617784i	$0.788031\pi$
$938$	0	0
$939$	7.09668e6	0.262658
$940$	0	0
$941$	−4.57761e7	−1.68525	−0.842625	−	0.538500i	$-0.818991\pi$
$941$	−4.57761e7	−1.68525	−0.842625	+	0.538500i	$0.818991\pi$
$942$	0	0
$943$	1.20502e6	0.0441279
$944$	0	0
$945$	−6.69574e6	−0.243904
$946$	0	0
$947$	5.06216e7	1.83426	0.917131	−	0.398587i	$-0.130499\pi$
$947$	5.06216e7	1.83426	0.917131	+	0.398587i	$0.130499\pi$
$948$	0	0
$949$	−3.25003e7	−1.17145
$950$	0	0
$951$	−7.08017e6	−0.253859
$952$	0	0
$953$	−3.80332e7	−1.35653	−0.678266	−	0.734816i	$-0.737268\pi$
$953$	−3.80332e7	−1.35653	−0.678266	+	0.734816i	$0.737268\pi$
$954$	0	0
$955$	−6.39493e6	−0.226896
$956$	0	0
$957$	−3.55241e6	−0.125384
$958$	0	0
$959$	−6.41554e6	−0.225261
$960$	0	0
$961$	−1.35612e7	−0.473684
$962$	0	0
$963$	1.28131e7	0.445234
$964$	0	0
$965$	2.92307e7	1.01046
$966$	0	0
$967$	−1.21551e7	−0.418017	−0.209008	−	0.977914i	$-0.567024\pi$
$967$	−1.21551e7	−0.418017	−0.209008	+	0.977914i	$0.567024\pi$
$968$	0	0
$969$	−3.05185e7	−1.04413
$970$	0	0
$971$	−4.68313e7	−1.59400	−0.797000	−	0.603979i	$-0.793581\pi$
$971$	−4.68313e7	−1.59400	−0.797000	+	0.603979i	$0.793581\pi$
$972$	0	0
$973$	9.95407e6	0.337069
$974$	0	0
$975$	3.93529e6	0.132576
$976$	0	0
$977$	−5.87159e6	−0.196798	−0.0983988	−	0.995147i	$-0.531372\pi$
$977$	−5.87159e6	−0.196798	−0.0983988	+	0.995147i	$0.531372\pi$
$978$	0	0
$979$	−1.21457e7	−0.405011
$980$	0	0
$981$	2.44046e6	0.0809655
$982$	0	0
$983$	−1.25982e7	−0.415839	−0.207919	−	0.978146i	$-0.566669\pi$
$983$	−1.25982e7	−0.415839	−0.207919	+	0.978146i	$0.566669\pi$
$984$	0	0
$985$	2.32487e7	0.763500
$986$	0	0
$987$	4.20769e7	1.37483
$988$	0	0
$989$	−5.07313e6	−0.164924
$990$	0	0
$991$	3.36841e6	0.108953	0.0544767	−	0.998515i	$-0.482651\pi$
$991$	3.36841e6	0.108953	0.0544767	+	0.998515i	$0.482651\pi$
$992$	0	0
$993$	−1.06730e7	−0.343489
$994$	0	0
$995$	3.01717e7	0.966143
$996$	0	0
$997$	−6.75104e6	−0.215096	−0.107548	−	0.994200i	$-0.534300\pi$
$997$	−6.75104e6	−0.215096	−0.107548	+	0.994200i	$0.534300\pi$
$998$	0	0
$999$	−2.25234e6	−0.0714038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.v.1.2		6
4.3	odd	2		276.6.a.c.1.2	✓	6
12.11	even	2		828.6.a.f.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.6.a.c.1.2	✓	6	4.3	odd	2
828.6.a.f.1.5		6	12.11	even	2
1104.6.a.v.1.2		6	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)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -50.9743 q^{5} +180.186 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -50.9743 q^{5} +180.186 q^{7} +81.0000 q^{9} -138.332 q^{11} -830.295 q^{13} -458.768 q^{15} -1281.42 q^{17} +2646.25 q^{19} +1621.67 q^{21} +529.000 q^{23} -526.625 q^{25} +729.000 q^{27} +2853.36 q^{29} -3881.75 q^{31} -1244.99 q^{33} -9184.83 q^{35} -3089.64 q^{37} -7472.65 q^{39} +2277.91 q^{41} -9590.03 q^{43} -4128.92 q^{45} +25946.6 q^{47} +15659.9 q^{49} -11532.8 q^{51} +18223.8 q^{53} +7051.38 q^{55} +23816.2 q^{57} +33059.8 q^{59} +9546.02 q^{61} +14595.0 q^{63} +42323.7 q^{65} -14312.9 q^{67} +4761.00 q^{69} -21458.4 q^{71} +39143.1 q^{73} -4739.62 q^{75} -24925.5 q^{77} -85813.8 q^{79} +6561.00 q^{81} -8698.94 q^{83} +65319.3 q^{85} +25680.3 q^{87} +87801.1 q^{89} -149607. q^{91} -34935.8 q^{93} -134890. q^{95} -65841.4 q^{97} -11204.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{3} + 22 q^{5} - 134 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q + 54 * q^3 + 22 * q^5 - 134 * q^7 + 486 * q^9	\( 6 q + 54 q^{3} + 22 q^{5} - 134 q^{7} + 486 q^{9} + 68 q^{11} - 340 q^{13} + 198 q^{15} + 222 q^{17} + 2314 q^{19} - 1206 q^{21} + 3174 q^{23} - 1014 q^{25} + 4374 q^{27} - 4432 q^{29} - 756 q^{31} + 612 q^{33} + 3596 q^{35} + 13908 q^{37} - 3060 q^{39} + 16884 q^{41} - 20838 q^{43} + 1782 q^{45} - 31900 q^{47} + 57070 q^{49} + 1998 q^{51} + 60914 q^{53} - 47728 q^{55} + 20826 q^{57} - 57332 q^{59} + 90252 q^{61} - 10854 q^{63} + 113220 q^{65} - 69138 q^{67} + 28566 q^{69} - 109800 q^{71} + 208204 q^{73} - 9126 q^{75} + 175368 q^{77} - 178066 q^{79} + 39366 q^{81} - 92692 q^{83} + 274648 q^{85} - 39888 q^{87} + 240354 q^{89} - 77796 q^{91} - 6804 q^{93} - 30740 q^{95} + 290104 q^{97} + 5508 q^{99}+O(q^{100}) \) 6 * q + 54 * q^3 + 22 * q^5 - 134 * q^7 + 486 * q^9 + 68 * q^11 - 340 * q^13 + 198 * q^15 + 222 * q^17 + 2314 * q^19 - 1206 * q^21 + 3174 * q^23 - 1014 * q^25 + 4374 * q^27 - 4432 * q^29 - 756 * q^31 + 612 * q^33 + 3596 * q^35 + 13908 * q^37 - 3060 * q^39 + 16884 * q^41 - 20838 * q^43 + 1782 * q^45 - 31900 * q^47 + 57070 * q^49 + 1998 * q^51 + 60914 * q^53 - 47728 * q^55 + 20826 * q^57 - 57332 * q^59 + 90252 * q^61 - 10854 * q^63 + 113220 * q^65 - 69138 * q^67 + 28566 * q^69 - 109800 * q^71 + 208204 * q^73 - 9126 * q^75 + 175368 * q^77 - 178066 * q^79 + 39366 * q^81 - 92692 * q^83 + 274648 * q^85 - 39888 * q^87 + 240354 * q^89 - 77796 * q^91 - 6804 * q^93 - 30740 * q^95 + 290104 * q^97 + 5508 * q^99

Introduction

L-functions

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Database

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