LMFDB - Embedded newform 1104.6.a.v.1.5

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			1.5
Root			$-37.9449$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} +41.9449 q^{5} -134.859 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +41.9449 q^{5} -134.859 q^{7} +81.0000 q^{9} -423.021 q^{11} +797.218 q^{13} +377.504 q^{15} -2104.38 q^{17} -1560.66 q^{19} -1213.73 q^{21} +529.000 q^{23} -1365.63 q^{25} +729.000 q^{27} +5241.09 q^{29} +1856.77 q^{31} -3807.19 q^{33} -5656.64 q^{35} -64.5054 q^{37} +7174.97 q^{39} +7202.32 q^{41} +4271.12 q^{43} +3397.53 q^{45} +16072.3 q^{47} +1379.94 q^{49} -18939.4 q^{51} +10449.2 q^{53} -17743.6 q^{55} -14046.0 q^{57} -12980.0 q^{59} +54474.7 q^{61} -10923.6 q^{63} +33439.2 q^{65} -53517.7 q^{67} +4761.00 q^{69} -3723.77 q^{71} -1013.01 q^{73} -12290.7 q^{75} +57048.2 q^{77} +21690.2 q^{79} +6561.00 q^{81} +17331.9 q^{83} -88268.0 q^{85} +47169.8 q^{87} +25715.1 q^{89} -107512. q^{91} +16711.0 q^{93} -65461.8 q^{95} +128285. q^{97} -34264.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	41.9449	0.750333	0.375166	−	0.926958i	$-0.377586\pi$
$5$	41.9449	0.750333	0.375166	+	0.926958i	$0.377586\pi$
$6$	0	0
$7$	−134.859	−1.04024	−0.520121	−	0.854092i	$-0.674113\pi$
$7$	−134.859	−1.04024	−0.520121	+	0.854092i	$0.674113\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−423.021	−1.05410	−0.527048	−	0.849836i	$-0.676701\pi$
$11$	−423.021	−1.05410	−0.527048	+	0.849836i	$0.676701\pi$
$12$	0	0
$13$	797.218	1.30834	0.654168	−	0.756350i	$-0.273019\pi$
$13$	797.218	1.30834	0.654168	+	0.756350i	$0.273019\pi$
$14$	0	0
$15$	377.504	0.433205
$16$	0	0
$17$	−2104.38	−1.76605	−0.883024	−	0.469329i	$-0.844496\pi$
$17$	−2104.38	−1.76605	−0.883024	+	0.469329i	$0.844496\pi$
$18$	0	0
$19$	−1560.66	−0.991802	−0.495901	−	0.868379i	$-0.665162\pi$
$19$	−1560.66	−0.991802	−0.495901	+	0.868379i	$0.665162\pi$
$20$	0	0
$21$	−1213.73	−0.600585
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1365.63	−0.437001
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	5241.09	1.15725	0.578625	−	0.815594i	$-0.303590\pi$
$29$	5241.09	1.15725	0.578625	+	0.815594i	$0.303590\pi$
$30$	0	0
$31$	1856.77	0.347020	0.173510	−	0.984832i	$-0.444489\pi$
$31$	1856.77	0.347020	0.173510	+	0.984832i	$0.444489\pi$
$32$	0	0
$33$	−3807.19	−0.608583
$34$	0	0
$35$	−5656.64	−0.780528
$36$	0	0
$37$	−64.5054	−0.00774625	−0.00387312	−	0.999992i	$-0.501233\pi$
$37$	−64.5054	−0.00774625	−0.00387312	+	0.999992i	$0.501233\pi$
$38$	0	0
$39$	7174.97	0.755368
$40$	0	0
$41$	7202.32	0.669134	0.334567	−	0.942372i	$-0.391410\pi$
$41$	7202.32	0.669134	0.334567	+	0.942372i	$0.391410\pi$
$42$	0	0
$43$	4271.12	0.352266	0.176133	−	0.984366i	$-0.443641\pi$
$43$	4271.12	0.352266	0.176133	+	0.984366i	$0.443641\pi$
$44$	0	0
$45$	3397.53	0.250111
$46$	0	0
$47$	16072.3	1.06129	0.530645	−	0.847594i	$-0.321950\pi$
$47$	16072.3	1.06129	0.530645	+	0.847594i	$0.321950\pi$
$48$	0	0
$49$	1379.94	0.0821053
$50$	0	0
$51$	−18939.4	−1.01963
$52$	0	0
$53$	10449.2	0.510969	0.255484	−	0.966813i	$-0.417765\pi$
$53$	10449.2	0.510969	0.255484	+	0.966813i	$0.417765\pi$
$54$	0	0
$55$	−17743.6	−0.790923
$56$	0	0
$57$	−14046.0	−0.572617
$58$	0	0
$59$	−12980.0	−0.485451	−0.242725	−	0.970095i	$-0.578041\pi$
$59$	−12980.0	−0.485451	−0.242725	+	0.970095i	$0.578041\pi$
$60$	0	0
$61$	54474.7	1.87444	0.937218	−	0.348745i	$-0.113392\pi$
$61$	54474.7	1.87444	0.937218	+	0.348745i	$0.113392\pi$
$62$	0	0
$63$	−10923.6	−0.346748
$64$	0	0
$65$	33439.2	0.981687
$66$	0	0
$67$	−53517.7	−1.45650	−0.728251	−	0.685311i	$-0.759666\pi$
$67$	−53517.7	−1.45650	−0.728251	+	0.685311i	$0.759666\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−3723.77	−0.0876671	−0.0438335	−	0.999039i	$-0.513957\pi$
$71$	−3723.77	−0.0876671	−0.0438335	+	0.999039i	$0.513957\pi$
$72$	0	0
$73$	−1013.01	−0.0222487	−0.0111244	−	0.999938i	$-0.503541\pi$
$73$	−1013.01	−0.0222487	−0.0111244	+	0.999938i	$0.503541\pi$
$74$	0	0
$75$	−12290.7	−0.252303
$76$	0	0
$77$	57048.2	1.09652
$78$	0	0
$79$	21690.2	0.391017	0.195508	−	0.980702i	$-0.437364\pi$
$79$	21690.2	0.391017	0.195508	+	0.980702i	$0.437364\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	17331.9	0.276153	0.138077	−	0.990422i	$-0.455908\pi$
$83$	17331.9	0.276153	0.138077	+	0.990422i	$0.455908\pi$
$84$	0	0
$85$	−88268.0	−1.32512
$86$	0	0
$87$	47169.8	0.668138
$88$	0	0
$89$	25715.1	0.344123	0.172061	−	0.985086i	$-0.444957\pi$
$89$	25715.1	0.344123	0.172061	+	0.985086i	$0.444957\pi$
$90$	0	0
$91$	−107512.	−1.36099
$92$	0	0
$93$	16711.0	0.200352
$94$	0	0
$95$	−65461.8	−0.744181
$96$	0	0
$97$	128285.	1.38435	0.692174	−	0.721731i	$-0.256653\pi$
$97$	128285.	1.38435	0.692174	+	0.721731i	$0.256653\pi$
$98$	0	0
$99$	−34264.7	−0.351365
$100$	0	0
$101$	164186.	1.60152	0.800760	−	0.598985i	$-0.204429\pi$
$101$	164186.	1.60152	0.800760	+	0.598985i	$0.204429\pi$
$102$	0	0
$103$	176900.	1.64299	0.821496	−	0.570214i	$-0.193140\pi$
$103$	176900.	1.64299	0.821496	+	0.570214i	$0.193140\pi$
$104$	0	0
$105$	−50909.8	−0.450638
$106$	0	0
$107$	44353.3	0.374512	0.187256	−	0.982311i	$-0.440041\pi$
$107$	44353.3	0.374512	0.187256	+	0.982311i	$0.440041\pi$
$108$	0	0
$109$	179407.	1.44635	0.723174	−	0.690666i	$-0.242683\pi$
$109$	179407.	1.44635	0.723174	+	0.690666i	$0.242683\pi$
$110$	0	0
$111$	−580.548	−0.00447230
$112$	0	0
$113$	101391.	0.746970	0.373485	−	0.927636i	$-0.378163\pi$
$113$	101391.	0.746970	0.373485	+	0.927636i	$0.378163\pi$
$114$	0	0
$115$	22188.8	0.156455
$116$	0	0
$117$	64574.7	0.436112
$118$	0	0
$119$	283795.	1.83712
$120$	0	0
$121$	17895.7	0.111118
$122$	0	0
$123$	64820.9	0.386324
$124$	0	0
$125$	−188359.	−1.07823
$126$	0	0
$127$	−43993.6	−0.242036	−0.121018	−	0.992650i	$-0.538616\pi$
$127$	−43993.6	−0.242036	−0.121018	+	0.992650i	$0.538616\pi$
$128$	0	0
$129$	38440.1	0.203381
$130$	0	0
$131$	138301.	0.704121	0.352060	−	0.935977i	$-0.385481\pi$
$131$	138301.	0.704121	0.352060	+	0.935977i	$0.385481\pi$
$132$	0	0
$133$	210469.	1.03171
$134$	0	0
$135$	30577.8	0.144402
$136$	0	0
$137$	−163483.	−0.744167	−0.372084	−	0.928199i	$-0.621357\pi$
$137$	−163483.	−0.744167	−0.372084	+	0.928199i	$0.621357\pi$
$138$	0	0
$139$	67488.8	0.296275	0.148137	−	0.988967i	$-0.452672\pi$
$139$	67488.8	0.296275	0.148137	+	0.988967i	$0.452672\pi$
$140$	0	0
$141$	144651.	0.612736
$142$	0	0
$143$	−337240.	−1.37911
$144$	0	0
$145$	219837.	0.868322
$146$	0	0
$147$	12419.5	0.0474035
$148$	0	0
$149$	−467498.	−1.72510	−0.862550	−	0.505971i	$-0.831134\pi$
$149$	−467498.	−1.72510	−0.862550	+	0.505971i	$0.831134\pi$
$150$	0	0
$151$	−249887.	−0.891870	−0.445935	−	0.895065i	$-0.647129\pi$
$151$	−249887.	−0.891870	−0.445935	+	0.895065i	$0.647129\pi$
$152$	0	0
$153$	−170455.	−0.588682
$154$	0	0
$155$	77882.1	0.260381
$156$	0	0
$157$	−130946.	−0.423978	−0.211989	−	0.977272i	$-0.567994\pi$
$157$	−130946.	−0.423978	−0.211989	+	0.977272i	$0.567994\pi$
$158$	0	0
$159$	94043.0	0.295008
$160$	0	0
$161$	−71340.4	−0.216906
$162$	0	0
$163$	−352469.	−1.03909	−0.519544	−	0.854444i	$-0.673898\pi$
$163$	−352469.	−1.03909	−0.519544	+	0.854444i	$0.673898\pi$
$164$	0	0
$165$	−159692.	−0.456639
$166$	0	0
$167$	18627.2	0.0516840	0.0258420	−	0.999666i	$-0.491773\pi$
$167$	18627.2	0.0516840	0.0258420	+	0.999666i	$0.491773\pi$
$168$	0	0
$169$	264264.	0.711741
$170$	0	0
$171$	−126414.	−0.330601
$172$	0	0
$173$	−461123.	−1.17139	−0.585695	−	0.810532i	$-0.699178\pi$
$173$	−461123.	−1.17139	−0.585695	+	0.810532i	$0.699178\pi$
$174$	0	0
$175$	184167.	0.454587
$176$	0	0
$177$	−116820.	−0.280275
$178$	0	0
$179$	−310824.	−0.725074	−0.362537	−	0.931969i	$-0.618089\pi$
$179$	−310824.	−0.725074	−0.362537	+	0.931969i	$0.618089\pi$
$180$	0	0
$181$	644785.	1.46291	0.731457	−	0.681888i	$-0.238841\pi$
$181$	644785.	1.46291	0.731457	+	0.681888i	$0.238841\pi$
$182$	0	0
$183$	490273.	1.08221
$184$	0	0
$185$	−2705.67	−0.00581226
$186$	0	0
$187$	890198.	1.86158
$188$	0	0
$189$	−98312.2	−0.200195
$190$	0	0
$191$	22415.7	0.0444600	0.0222300	−	0.999753i	$-0.492923\pi$
$191$	22415.7	0.0444600	0.0222300	+	0.999753i	$0.492923\pi$
$192$	0	0
$193$	634901.	1.22691	0.613455	−	0.789729i	$-0.289779\pi$
$193$	634901.	1.22691	0.613455	+	0.789729i	$0.289779\pi$
$194$	0	0
$195$	300953.	0.566777
$196$	0	0
$197$	−558484.	−1.02529	−0.512643	−	0.858602i	$-0.671334\pi$
$197$	−558484.	−1.02529	−0.512643	+	0.858602i	$0.671334\pi$
$198$	0	0
$199$	−44451.2	−0.0795704	−0.0397852	−	0.999208i	$-0.512667\pi$
$199$	−44451.2	−0.0795704	−0.0397852	+	0.999208i	$0.512667\pi$
$200$	0	0
$201$	−481660.	−0.840911
$202$	0	0
$203$	−706809.	−1.20382
$204$	0	0
$205$	302100.	0.502073
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	660193.	1.04545
$210$	0	0
$211$	467308.	0.722599	0.361300	−	0.932450i	$-0.382333\pi$
$211$	467308.	0.722599	0.361300	+	0.932450i	$0.382333\pi$
$212$	0	0
$213$	−33513.9	−0.0506146
$214$	0	0
$215$	179152.	0.264317
$216$	0	0
$217$	−250403.	−0.360985
$218$	0	0
$219$	−9117.06	−0.0128453
$220$	0	0
$221$	−1.67765e6	−2.31058
$222$	0	0
$223$	726815.	0.978727	0.489364	−	0.872080i	$-0.337229\pi$
$223$	726815.	0.978727	0.489364	+	0.872080i	$0.337229\pi$
$224$	0	0
$225$	−110616.	−0.145667
$226$	0	0
$227$	564950.	0.727688	0.363844	−	0.931460i	$-0.381464\pi$
$227$	564950.	0.727688	0.363844	+	0.931460i	$0.381464\pi$
$228$	0	0
$229$	−955868.	−1.20451	−0.602254	−	0.798305i	$-0.705730\pi$
$229$	−955868.	−1.20451	−0.602254	+	0.798305i	$0.705730\pi$
$230$	0	0
$231$	513434.	0.633074
$232$	0	0
$233$	657297.	0.793181	0.396590	−	0.917996i	$-0.370193\pi$
$233$	657297.	0.793181	0.396590	+	0.917996i	$0.370193\pi$
$234$	0	0
$235$	674151.	0.796320
$236$	0	0
$237$	195212.	0.225754
$238$	0	0
$239$	1.30368e6	1.47631	0.738153	−	0.674634i	$-0.235698\pi$
$239$	1.30368e6	1.47631	0.738153	+	0.674634i	$0.235698\pi$
$240$	0	0
$241$	646612.	0.717135	0.358568	−	0.933504i	$-0.383265\pi$
$241$	646612.	0.717135	0.358568	+	0.933504i	$0.383265\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	57881.6	0.0616063
$246$	0	0
$247$	−1.24419e6	−1.29761
$248$	0	0
$249$	155987.	0.159437
$250$	0	0
$251$	1.51598e6	1.51883	0.759413	−	0.650608i	$-0.225486\pi$
$251$	1.51598e6	1.51883	0.759413	+	0.650608i	$0.225486\pi$
$252$	0	0
$253$	−223778.	−0.219794
$254$	0	0
$255$	−794412.	−0.765060
$256$	0	0
$257$	995626.	0.940294	0.470147	−	0.882588i	$-0.344201\pi$
$257$	995626.	0.940294	0.470147	+	0.882588i	$0.344201\pi$
$258$	0	0
$259$	8699.13	0.00805798
$260$	0	0
$261$	424529.	0.385750
$262$	0	0
$263$	−503455.	−0.448819	−0.224410	−	0.974495i	$-0.572045\pi$
$263$	−503455.	−0.448819	−0.224410	+	0.974495i	$0.572045\pi$
$264$	0	0
$265$	438291.	0.383396
$266$	0	0
$267$	231436.	0.198679
$268$	0	0
$269$	1.76828e6	1.48994	0.744971	−	0.667096i	$-0.232463\pi$
$269$	1.76828e6	1.48994	0.744971	+	0.667096i	$0.232463\pi$
$270$	0	0
$271$	448289.	0.370796	0.185398	−	0.982664i	$-0.440643\pi$
$271$	448289.	0.370796	0.185398	+	0.982664i	$0.440643\pi$
$272$	0	0
$273$	−967609.	−0.785766
$274$	0	0
$275$	577689.	0.460641
$276$	0	0
$277$	−202497.	−0.158569	−0.0792845	−	0.996852i	$-0.525264\pi$
$277$	−202497.	−0.158569	−0.0792845	+	0.996852i	$0.525264\pi$
$278$	0	0
$279$	150399.	0.115673
$280$	0	0
$281$	−2.47348e6	−1.86871	−0.934356	−	0.356340i	$-0.884024\pi$
$281$	−2.47348e6	−1.86871	−0.934356	+	0.356340i	$0.884024\pi$
$282$	0	0
$283$	−1.42836e6	−1.06016	−0.530081	−	0.847947i	$-0.677838\pi$
$283$	−1.42836e6	−1.06016	−0.530081	+	0.847947i	$0.677838\pi$
$284$	0	0
$285$	−589156.	−0.429653
$286$	0	0
$287$	−971298.	−0.696062
$288$	0	0
$289$	3.00857e6	2.11892
$290$	0	0
$291$	1.15456e6	0.799253
$292$	0	0
$293$	−1.23071e6	−0.837503	−0.418752	−	0.908101i	$-0.637532\pi$
$293$	−1.23071e6	−0.837503	−0.418752	+	0.908101i	$0.637532\pi$
$294$	0	0
$295$	−544445.	−0.364250
$296$	0	0
$297$	−308382.	−0.202861
$298$	0	0
$299$	421729.	0.272807
$300$	0	0
$301$	−575999.	−0.366442
$302$	0	0
$303$	1.47767e6	0.924638
$304$	0	0
$305$	2.28494e6	1.40645
$306$	0	0
$307$	−2.89972e6	−1.75594	−0.877972	−	0.478712i	$-0.841104\pi$
$307$	−2.89972e6	−1.75594	−0.877972	+	0.478712i	$0.841104\pi$
$308$	0	0
$309$	1.59210e6	0.948582
$310$	0	0
$311$	1.82826e6	1.07186	0.535928	−	0.844264i	$-0.319962\pi$
$311$	1.82826e6	1.07186	0.535928	+	0.844264i	$0.319962\pi$
$312$	0	0
$313$	1.99396e6	1.15042	0.575208	−	0.818007i	$-0.304921\pi$
$313$	1.99396e6	1.15042	0.575208	+	0.818007i	$0.304921\pi$
$314$	0	0
$315$	−458188.	−0.260176
$316$	0	0
$317$	1.54409e6	0.863028	0.431514	−	0.902106i	$-0.357979\pi$
$317$	1.54409e6	0.863028	0.431514	+	0.902106i	$0.357979\pi$
$318$	0	0
$319$	−2.21709e6	−1.21985
$320$	0	0
$321$	399180.	0.216225
$322$	0	0
$323$	3.28423e6	1.75157
$324$	0	0
$325$	−1.08870e6	−0.571744
$326$	0	0
$327$	1.61466e6	0.835049
$328$	0	0
$329$	−2.16750e6	−1.10400
$330$	0	0
$331$	−3.59523e6	−1.80367	−0.901833	−	0.432084i	$-0.857778\pi$
$331$	−3.59523e6	−1.80367	−0.901833	+	0.432084i	$0.857778\pi$
$332$	0	0
$333$	−5224.93	−0.00258208
$334$	0	0
$335$	−2.24480e6	−1.09286
$336$	0	0
$337$	46799.3	0.0224473	0.0112237	−	0.999937i	$-0.496427\pi$
$337$	46799.3	0.0224473	0.0112237	+	0.999937i	$0.496427\pi$
$338$	0	0
$339$	912519.	0.431264
$340$	0	0
$341$	−785454.	−0.365793
$342$	0	0
$343$	2.08048e6	0.954833
$344$	0	0
$345$	199700.	0.0903294
$346$	0	0
$347$	564028.	0.251464	0.125732	−	0.992064i	$-0.459872\pi$
$347$	564028.	0.251464	0.125732	+	0.992064i	$0.459872\pi$
$348$	0	0
$349$	3.80931e6	1.67410	0.837052	−	0.547124i	$-0.184277\pi$
$349$	3.80931e6	1.67410	0.837052	+	0.547124i	$0.184277\pi$
$350$	0	0
$351$	581172.	0.251789
$352$	0	0
$353$	2.94219e6	1.25671	0.628354	−	0.777928i	$-0.283729\pi$
$353$	2.94219e6	1.25671	0.628354	+	0.777928i	$0.283729\pi$
$354$	0	0
$355$	−156193.	−0.0657795
$356$	0	0
$357$	2.55415e6	1.06066
$358$	0	0
$359$	2.32868e6	0.953618	0.476809	−	0.879007i	$-0.341793\pi$
$359$	2.32868e6	0.953618	0.476809	+	0.879007i	$0.341793\pi$
$360$	0	0
$361$	−40432.2	−0.0163290
$362$	0	0
$363$	161061.	0.0641541
$364$	0	0
$365$	−42490.5	−0.0166940
$366$	0	0
$367$	3.75436e6	1.45502	0.727512	−	0.686095i	$-0.240677\pi$
$367$	3.75436e6	1.45502	0.727512	+	0.686095i	$0.240677\pi$
$368$	0	0
$369$	583388.	0.223045
$370$	0	0
$371$	−1.40917e6	−0.531531
$372$	0	0
$373$	982222.	0.365542	0.182771	−	0.983155i	$-0.441493\pi$
$373$	982222.	0.365542	0.182771	+	0.983155i	$0.441493\pi$
$374$	0	0
$375$	−1.69523e6	−0.622516
$376$	0	0
$377$	4.17830e6	1.51407
$378$	0	0
$379$	−2.27127e6	−0.812216	−0.406108	−	0.913825i	$-0.633114\pi$
$379$	−2.27127e6	−0.812216	−0.406108	+	0.913825i	$0.633114\pi$
$380$	0	0
$381$	−395942.	−0.139740
$382$	0	0
$383$	1.46506e6	0.510338	0.255169	−	0.966896i	$-0.417869\pi$
$383$	1.46506e6	0.510338	0.255169	+	0.966896i	$0.417869\pi$
$384$	0	0
$385$	2.39288e6	0.822752
$386$	0	0
$387$	345961.	0.117422
$388$	0	0
$389$	5.19411e6	1.74035	0.870176	−	0.492740i	$-0.164005\pi$
$389$	5.19411e6	1.74035	0.870176	+	0.492740i	$0.164005\pi$
$390$	0	0
$391$	−1.11322e6	−0.368246
$392$	0	0
$393$	1.24471e6	0.406524
$394$	0	0
$395$	909792.	0.293393
$396$	0	0
$397$	2.00207e6	0.637535	0.318767	−	0.947833i	$-0.396731\pi$
$397$	2.00207e6	0.637535	0.318767	+	0.947833i	$0.396731\pi$
$398$	0	0
$399$	1.89422e6	0.595661
$400$	0	0
$401$	1.95868e6	0.608279	0.304139	−	0.952628i	$-0.401631\pi$
$401$	1.95868e6	0.608279	0.304139	+	0.952628i	$0.401631\pi$
$402$	0	0
$403$	1.48025e6	0.454019
$404$	0	0
$405$	275200.	0.0833703
$406$	0	0
$407$	27287.1	0.00816529
$408$	0	0
$409$	−968304.	−0.286222	−0.143111	−	0.989707i	$-0.545711\pi$
$409$	−968304.	−0.286222	−0.143111	+	0.989707i	$0.545711\pi$
$410$	0	0
$411$	−1.47134e6	−0.429645
$412$	0	0
$413$	1.75047e6	0.504987
$414$	0	0
$415$	726983.	0.207207
$416$	0	0
$417$	607400.	0.171054
$418$	0	0
$419$	−4.50856e6	−1.25459	−0.627296	−	0.778781i	$-0.715838\pi$
$419$	−4.50856e6	−1.25459	−0.627296	+	0.778781i	$0.715838\pi$
$420$	0	0
$421$	−6.26378e6	−1.72239	−0.861195	−	0.508275i	$-0.830283\pi$
$421$	−6.26378e6	−1.72239	−0.861195	+	0.508275i	$0.830283\pi$
$422$	0	0
$423$	1.30186e6	0.353763
$424$	0	0
$425$	2.87380e6	0.771764
$426$	0	0
$427$	−7.34641e6	−1.94987
$428$	0	0
$429$	−3.03516e6	−0.796230
$430$	0	0
$431$	4.47109e6	1.15937	0.579683	−	0.814842i	$-0.303176\pi$
$431$	4.47109e6	1.15937	0.579683	+	0.814842i	$0.303176\pi$
$432$	0	0
$433$	−6.65931e6	−1.70691	−0.853453	−	0.521170i	$-0.825496\pi$
$433$	−6.65931e6	−1.70691	−0.853453	+	0.521170i	$0.825496\pi$
$434$	0	0
$435$	1.97853e6	0.501326
$436$	0	0
$437$	−825590.	−0.206805
$438$	0	0
$439$	5.29654e6	1.31169	0.655844	−	0.754896i	$-0.272313\pi$
$439$	5.29654e6	1.31169	0.655844	+	0.754896i	$0.272313\pi$
$440$	0	0
$441$	111775.	0.0273684
$442$	0	0
$443$	799674.	0.193599	0.0967996	−	0.995304i	$-0.469139\pi$
$443$	799674.	0.193599	0.0967996	+	0.995304i	$0.469139\pi$
$444$	0	0
$445$	1.07862e6	0.258206
$446$	0	0
$447$	−4.20749e6	−0.995987
$448$	0	0
$449$	−4.14650e6	−0.970657	−0.485328	−	0.874332i	$-0.661300\pi$
$449$	−4.14650e6	−0.970657	−0.485328	+	0.874332i	$0.661300\pi$
$450$	0	0
$451$	−3.04673e6	−0.705331
$452$	0	0
$453$	−2.24899e6	−0.514922
$454$	0	0
$455$	−4.50958e6	−1.02119
$456$	0	0
$457$	−5.47249e6	−1.22573	−0.612865	−	0.790188i	$-0.709983\pi$
$457$	−5.47249e6	−1.22573	−0.612865	+	0.790188i	$0.709983\pi$
$458$	0	0
$459$	−1.53409e6	−0.339876
$460$	0	0
$461$	−6.18737e6	−1.35598	−0.677990	−	0.735071i	$-0.737149\pi$
$461$	−6.18737e6	−1.35598	−0.677990	+	0.735071i	$0.737149\pi$
$462$	0	0
$463$	−3.17047e6	−0.687339	−0.343670	−	0.939091i	$-0.611670\pi$
$463$	−3.17047e6	−0.687339	−0.343670	+	0.939091i	$0.611670\pi$
$464$	0	0
$465$	700939.	0.150331
$466$	0	0
$467$	5.89569e6	1.25096	0.625478	−	0.780242i	$-0.284904\pi$
$467$	5.89569e6	1.25096	0.625478	+	0.780242i	$0.284904\pi$
$468$	0	0
$469$	7.21735e6	1.51511
$470$	0	0
$471$	−1.17851e6	−0.244784
$472$	0	0
$473$	−1.80677e6	−0.371322
$474$	0	0
$475$	2.13128e6	0.433418
$476$	0	0
$477$	846387.	0.170323
$478$	0	0
$479$	−3.29939e6	−0.657044	−0.328522	−	0.944496i	$-0.606551\pi$
$479$	−3.29939e6	−0.657044	−0.328522	+	0.944496i	$0.606551\pi$
$480$	0	0
$481$	−51424.9	−0.0101347
$482$	0	0
$483$	−642064.	−0.125231
$484$	0	0
$485$	5.38088e6	1.03872
$486$	0	0
$487$	6.21633e6	1.18771	0.593857	−	0.804571i	$-0.297605\pi$
$487$	6.21633e6	1.18771	0.593857	+	0.804571i	$0.297605\pi$
$488$	0	0
$489$	−3.17222e6	−0.599917
$490$	0	0
$491$	7.60160e6	1.42299	0.711494	−	0.702693i	$-0.248019\pi$
$491$	7.60160e6	1.42299	0.711494	+	0.702693i	$0.248019\pi$
$492$	0	0
$493$	−1.10293e7	−2.04376
$494$	0	0
$495$	−1.43723e6	−0.263641
$496$	0	0
$497$	502183.	0.0911951
$498$	0	0
$499$	1.04939e7	1.88662	0.943309	−	0.331917i	$-0.107695\pi$
$499$	1.04939e7	1.88662	0.943309	+	0.331917i	$0.107695\pi$
$500$	0	0
$501$	167645.	0.0298398
$502$	0	0
$503$	−1.88002e6	−0.331316	−0.165658	−	0.986183i	$-0.552975\pi$
$503$	−1.88002e6	−0.331316	−0.165658	+	0.986183i	$0.552975\pi$
$504$	0	0
$505$	6.88676e6	1.20167
$506$	0	0
$507$	2.37838e6	0.410924
$508$	0	0
$509$	505872.	0.0865459	0.0432730	−	0.999063i	$-0.486221\pi$
$509$	505872.	0.0865459	0.0432730	+	0.999063i	$0.486221\pi$
$510$	0	0
$511$	136613.	0.0231441
$512$	0	0
$513$	−1.13772e6	−0.190872
$514$	0	0
$515$	7.42006e6	1.23279
$516$	0	0
$517$	−6.79893e6	−1.11870
$518$	0	0
$519$	−4.15010e6	−0.676302
$520$	0	0
$521$	−4.95995e6	−0.800540	−0.400270	−	0.916397i	$-0.631084\pi$
$521$	−4.95995e6	−0.800540	−0.400270	+	0.916397i	$0.631084\pi$
$522$	0	0
$523$	1.15127e6	0.184044	0.0920222	−	0.995757i	$-0.470667\pi$
$523$	1.15127e6	0.184044	0.0920222	+	0.995757i	$0.470667\pi$
$524$	0	0
$525$	1.65750e6	0.262456
$526$	0	0
$527$	−3.90736e6	−0.612854
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−1.05138e6	−0.161817
$532$	0	0
$533$	5.74182e6	0.875451
$534$	0	0
$535$	1.86039e6	0.281009
$536$	0	0
$537$	−2.79742e6	−0.418622
$538$	0	0
$539$	−583745.	−0.0865469
$540$	0	0
$541$	1.04033e6	0.152819	0.0764094	−	0.997077i	$-0.475654\pi$
$541$	1.04033e6	0.152819	0.0764094	+	0.997077i	$0.475654\pi$
$542$	0	0
$543$	5.80307e6	0.844614
$544$	0	0
$545$	7.52520e6	1.08524
$546$	0	0
$547$	7.80794e6	1.11575	0.557877	−	0.829924i	$-0.311616\pi$
$547$	7.80794e6	1.11575	0.557877	+	0.829924i	$0.311616\pi$
$548$	0	0
$549$	4.41245e6	0.624812
$550$	0	0
$551$	−8.17958e6	−1.14776
$552$	0	0
$553$	−2.92512e6	−0.406753
$554$	0	0
$555$	−24351.0	−0.00335571
$556$	0	0
$557$	6.93037e6	0.946495	0.473248	−	0.880929i	$-0.343082\pi$
$557$	6.93037e6	0.946495	0.473248	+	0.880929i	$0.343082\pi$
$558$	0	0
$559$	3.40502e6	0.460882
$560$	0	0
$561$	8.01178e6	1.07479
$562$	0	0
$563$	362187.	0.0481572	0.0240786	−	0.999710i	$-0.492335\pi$
$563$	362187.	0.0481572	0.0240786	+	0.999710i	$0.492335\pi$
$564$	0	0
$565$	4.25283e6	0.560476
$566$	0	0
$567$	−884810.	−0.115583
$568$	0	0
$569$	8.89769e6	1.15212	0.576058	−	0.817408i	$-0.304590\pi$
$569$	8.89769e6	1.15212	0.576058	+	0.817408i	$0.304590\pi$
$570$	0	0
$571$	−1.27496e7	−1.63646	−0.818232	−	0.574888i	$-0.805046\pi$
$571$	−1.27496e7	−1.63646	−0.818232	+	0.574888i	$0.805046\pi$
$572$	0	0
$573$	201741.	0.0256690
$574$	0	0
$575$	−722417.	−0.0911210
$576$	0	0
$577$	5.91353e6	0.739447	0.369724	−	0.929142i	$-0.379452\pi$
$577$	5.91353e6	0.739447	0.369724	+	0.929142i	$0.379452\pi$
$578$	0	0
$579$	5.71411e6	0.708357
$580$	0	0
$581$	−2.33736e6	−0.287266
$582$	0	0
$583$	−4.42024e6	−0.538610
$584$	0	0
$585$	2.70858e6	0.327229
$586$	0	0
$587$	8.60773e6	1.03108	0.515541	−	0.856865i	$-0.327591\pi$
$587$	8.60773e6	1.03108	0.515541	+	0.856865i	$0.327591\pi$
$588$	0	0
$589$	−2.89780e6	−0.344175
$590$	0	0
$591$	−5.02636e6	−0.591950
$592$	0	0
$593$	−6.60718e6	−0.771577	−0.385788	−	0.922587i	$-0.626071\pi$
$593$	−6.60718e6	−0.771577	−0.385788	+	0.922587i	$0.626071\pi$
$594$	0	0
$595$	1.19037e7	1.37845
$596$	0	0
$597$	−400061.	−0.0459400
$598$	0	0
$599$	1.06699e7	1.21505	0.607525	−	0.794301i	$-0.292162\pi$
$599$	1.06699e7	1.21505	0.607525	+	0.794301i	$0.292162\pi$
$600$	0	0
$601$	−1.55199e7	−1.75267	−0.876337	−	0.481698i	$-0.840020\pi$
$601$	−1.55199e7	−1.75267	−0.876337	+	0.481698i	$0.840020\pi$
$602$	0	0
$603$	−4.33494e6	−0.485500
$604$	0	0
$605$	750633.	0.0833756
$606$	0	0
$607$	1.23871e7	1.36458	0.682290	−	0.731081i	$-0.260984\pi$
$607$	1.23871e7	1.36458	0.682290	+	0.731081i	$0.260984\pi$
$608$	0	0
$609$	−6.36128e6	−0.695026
$610$	0	0
$611$	1.28131e7	1.38852
$612$	0	0
$613$	−4.15666e6	−0.446780	−0.223390	−	0.974729i	$-0.571712\pi$
$613$	−4.15666e6	−0.446780	−0.223390	+	0.974729i	$0.571712\pi$
$614$	0	0
$615$	2.71890e6	0.289872
$616$	0	0
$617$	−4.54846e6	−0.481007	−0.240504	−	0.970648i	$-0.577313\pi$
$617$	−4.54846e6	−0.481007	−0.240504	+	0.970648i	$0.577313\pi$
$618$	0	0
$619$	1.30894e7	1.37307	0.686535	−	0.727097i	$-0.259131\pi$
$619$	1.30894e7	1.37307	0.686535	+	0.727097i	$0.259131\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−3.46791e6	−0.357971
$624$	0	0
$625$	−3.63310e6	−0.372029
$626$	0	0
$627$	5.94174e6	0.603593
$628$	0	0
$629$	135744.	0.0136802
$630$	0	0
$631$	−1.81114e7	−1.81083	−0.905415	−	0.424528i	$-0.860440\pi$
$631$	−1.81114e7	−1.81083	−0.905415	+	0.424528i	$0.860440\pi$
$632$	0	0
$633$	4.20578e6	0.417193
$634$	0	0
$635$	−1.84530e6	−0.181608
$636$	0	0
$637$	1.10012e6	0.107421
$638$	0	0
$639$	−301625.	−0.0292224
$640$	0	0
$641$	−1.11746e7	−1.07421	−0.537103	−	0.843517i	$-0.680481\pi$
$641$	−1.11746e7	−1.07421	−0.537103	+	0.843517i	$0.680481\pi$
$642$	0	0
$643$	1.05124e7	1.00271	0.501353	−	0.865243i	$-0.332836\pi$
$643$	1.05124e7	1.00271	0.501353	+	0.865243i	$0.332836\pi$
$644$	0	0
$645$	1.61236e6	0.152603
$646$	0	0
$647$	6.15646e6	0.578190	0.289095	−	0.957300i	$-0.406646\pi$
$647$	6.15646e6	0.578190	0.289095	+	0.957300i	$0.406646\pi$
$648$	0	0
$649$	5.49082e6	0.511712
$650$	0	0
$651$	−2.25362e6	−0.208415
$652$	0	0
$653$	−3.41560e6	−0.313461	−0.156731	−	0.987641i	$-0.550095\pi$
$653$	−3.41560e6	−0.313461	−0.156731	+	0.987641i	$0.550095\pi$
$654$	0	0
$655$	5.80102e6	0.528325
$656$	0	0
$657$	−82053.6	−0.00741625
$658$	0	0
$659$	6.63725e6	0.595353	0.297677	−	0.954667i	$-0.403788\pi$
$659$	6.63725e6	0.595353	0.297677	+	0.954667i	$0.403788\pi$
$660$	0	0
$661$	4.89807e6	0.436035	0.218018	−	0.975945i	$-0.430041\pi$
$661$	4.89807e6	0.436035	0.218018	+	0.975945i	$0.430041\pi$
$662$	0	0
$663$	−1.50989e7	−1.33401
$664$	0	0
$665$	8.82811e6	0.774129
$666$	0	0
$667$	2.77254e6	0.241303
$668$	0	0
$669$	6.54134e6	0.565069
$670$	0	0
$671$	−2.30440e7	−1.97583
$672$	0	0
$673$	−8.31280e6	−0.707473	−0.353736	−	0.935345i	$-0.615089\pi$
$673$	−8.31280e6	−0.707473	−0.353736	+	0.935345i	$0.615089\pi$
$674$	0	0
$675$	−995543.	−0.0841009
$676$	0	0
$677$	−9.87077e6	−0.827712	−0.413856	−	0.910342i	$-0.635818\pi$
$677$	−9.87077e6	−0.827712	−0.413856	+	0.910342i	$0.635818\pi$
$678$	0	0
$679$	−1.73003e7	−1.44006
$680$	0	0
$681$	5.08455e6	0.420131
$682$	0	0
$683$	−2.18298e6	−0.179059	−0.0895297	−	0.995984i	$-0.528536\pi$
$683$	−2.18298e6	−0.179059	−0.0895297	+	0.995984i	$0.528536\pi$
$684$	0	0
$685$	−6.85726e6	−0.558373
$686$	0	0
$687$	−8.60281e6	−0.695422
$688$	0	0
$689$	8.33031e6	0.668518
$690$	0	0
$691$	−1.90186e7	−1.51524	−0.757622	−	0.652693i	$-0.773639\pi$
$691$	−1.90186e7	−1.51524	−0.757622	+	0.652693i	$0.773639\pi$
$692$	0	0
$693$	4.62090e6	0.365505
$694$	0	0
$695$	2.83081e6	0.222305
$696$	0	0
$697$	−1.51564e7	−1.18172
$698$	0	0
$699$	5.91568e6	0.457943
$700$	0	0
$701$	−1.42517e7	−1.09539	−0.547697	−	0.836677i	$-0.684495\pi$
$701$	−1.42517e7	−1.09539	−0.547697	+	0.836677i	$0.684495\pi$
$702$	0	0
$703$	100671.	0.00768274
$704$	0	0
$705$	6.06736e6	0.459756
$706$	0	0
$707$	−2.21419e7	−1.66597
$708$	0	0
$709$	5.72320e6	0.427586	0.213793	−	0.976879i	$-0.431418\pi$
$709$	5.72320e6	0.427586	0.213793	+	0.976879i	$0.431418\pi$
$710$	0	0
$711$	1.75690e6	0.130339
$712$	0	0
$713$	982233.	0.0723587
$714$	0	0
$715$	−1.41455e7	−1.03479
$716$	0	0
$717$	1.17331e7	0.852345
$718$	0	0
$719$	−1.63170e7	−1.17712	−0.588558	−	0.808455i	$-0.700304\pi$
$719$	−1.63170e7	−1.17712	−0.588558	+	0.808455i	$0.700304\pi$
$720$	0	0
$721$	−2.38566e7	−1.70911
$722$	0	0
$723$	5.81951e6	0.414038
$724$	0	0
$725$	−7.15738e6	−0.505719
$726$	0	0
$727$	4.51481e6	0.316813	0.158407	−	0.987374i	$-0.449364\pi$
$727$	4.51481e6	0.316813	0.158407	+	0.987374i	$0.449364\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−8.98807e6	−0.622118
$732$	0	0
$733$	−2.34501e7	−1.61208	−0.806038	−	0.591864i	$-0.798392\pi$
$733$	−2.34501e7	−1.61208	−0.806038	+	0.591864i	$0.798392\pi$
$734$	0	0
$735$	520934.	0.0355684
$736$	0	0
$737$	2.26391e7	1.53529
$738$	0	0
$739$	−1.08453e7	−0.730518	−0.365259	−	0.930906i	$-0.619020\pi$
$739$	−1.08453e7	−0.730518	−0.365259	+	0.930906i	$0.619020\pi$
$740$	0	0
$741$	−1.11977e7	−0.749175
$742$	0	0
$743$	−1.83039e7	−1.21638	−0.608192	−	0.793790i	$-0.708105\pi$
$743$	−1.83039e7	−1.21638	−0.608192	+	0.793790i	$0.708105\pi$
$744$	0	0
$745$	−1.96092e7	−1.29440
$746$	0	0
$747$	1.40388e6	0.0920511
$748$	0	0
$749$	−5.98144e6	−0.389584
$750$	0	0
$751$	2.06140e6	0.133371	0.0666857	−	0.997774i	$-0.478758\pi$
$751$	2.06140e6	0.133371	0.0666857	+	0.997774i	$0.478758\pi$
$752$	0	0
$753$	1.36438e7	0.876895
$754$	0	0
$755$	−1.04815e7	−0.669200
$756$	0	0
$757$	−3.03416e7	−1.92441	−0.962206	−	0.272322i	$-0.912208\pi$
$757$	−3.03416e7	−1.92441	−0.962206	+	0.272322i	$0.912208\pi$
$758$	0	0
$759$	−2.01400e6	−0.126898
$760$	0	0
$761$	2.34279e7	1.46647	0.733233	−	0.679978i	$-0.238011\pi$
$761$	2.34279e7	1.46647	0.733233	+	0.679978i	$0.238011\pi$
$762$	0	0
$763$	−2.41946e7	−1.50455
$764$	0	0
$765$	−7.14971e6	−0.441708
$766$	0	0
$767$	−1.03479e7	−0.635132
$768$	0	0
$769$	−1.49807e7	−0.913517	−0.456759	−	0.889591i	$-0.650990\pi$
$769$	−1.49807e7	−0.913517	−0.456759	+	0.889591i	$0.650990\pi$
$770$	0	0
$771$	8.96064e6	0.542879
$772$	0	0
$773$	2.09634e7	1.26187	0.630933	−	0.775837i	$-0.282672\pi$
$773$	2.09634e7	1.26187	0.630933	+	0.775837i	$0.282672\pi$
$774$	0	0
$775$	−2.53566e6	−0.151648
$776$	0	0
$777$	78292.1	0.00465228
$778$	0	0
$779$	−1.12404e7	−0.663648
$780$	0	0
$781$	1.57523e6	0.0924095
$782$	0	0
$783$	3.82076e6	0.222713
$784$	0	0
$785$	−5.49251e6	−0.318124
$786$	0	0
$787$	−221080.	−0.0127237	−0.00636184	−	0.999980i	$-0.502025\pi$
$787$	−221080.	−0.0127237	−0.00636184	+	0.999980i	$0.502025\pi$
$788$	0	0
$789$	−4.53110e6	−0.259126
$790$	0	0
$791$	−1.36735e7	−0.777031
$792$	0	0
$793$	4.34283e7	2.45239
$794$	0	0
$795$	3.94462e6	0.221354
$796$	0	0
$797$	−1.86121e7	−1.03789	−0.518943	−	0.854809i	$-0.673674\pi$
$797$	−1.86121e7	−1.03789	−0.518943	+	0.854809i	$0.673674\pi$
$798$	0	0
$799$	−3.38223e7	−1.87429
$800$	0	0
$801$	2.08292e6	0.114708
$802$	0	0
$803$	428523.	0.0234523
$804$	0	0
$805$	−2.99236e6	−0.162751
$806$	0	0
$807$	1.59145e7	0.860219
$808$	0	0
$809$	1.85674e7	0.997426	0.498713	−	0.866767i	$-0.333806\pi$
$809$	1.85674e7	0.997426	0.498713	+	0.866767i	$0.333806\pi$
$810$	0	0
$811$	151943.	0.00811202	0.00405601	−	0.999992i	$-0.498709\pi$
$811$	151943.	0.00811202	0.00405601	+	0.999992i	$0.498709\pi$
$812$	0	0
$813$	4.03460e6	0.214079
$814$	0	0
$815$	−1.47843e7	−0.779661
$816$	0	0
$817$	−6.66578e6	−0.349378
$818$	0	0
$819$	−8.70848e6	−0.453662
$820$	0	0
$821$	−2.10720e7	−1.09106	−0.545529	−	0.838092i	$-0.683671\pi$
$821$	−2.10720e7	−1.09106	−0.545529	+	0.838092i	$0.683671\pi$
$822$	0	0
$823$	−6.62153e6	−0.340768	−0.170384	−	0.985378i	$-0.554501\pi$
$823$	−6.62153e6	−0.340768	−0.170384	+	0.985378i	$0.554501\pi$
$824$	0	0
$825$	5.19920e6	0.265951
$826$	0	0
$827$	6.70260e6	0.340784	0.170392	−	0.985376i	$-0.445497\pi$
$827$	6.70260e6	0.340784	0.170392	+	0.985376i	$0.445497\pi$
$828$	0	0
$829$	2.45517e7	1.24078	0.620390	−	0.784293i	$-0.286974\pi$
$829$	2.45517e7	1.24078	0.620390	+	0.784293i	$0.286974\pi$
$830$	0	0
$831$	−1.82247e6	−0.0915499
$832$	0	0
$833$	−2.90393e6	−0.145002
$834$	0	0
$835$	781315.	0.0387802
$836$	0	0
$837$	1.35359e6	0.0667841
$838$	0	0
$839$	3.00991e6	0.147621	0.0738107	−	0.997272i	$-0.476484\pi$
$839$	3.00991e6	0.147621	0.0738107	+	0.997272i	$0.476484\pi$
$840$	0	0
$841$	6.95792e6	0.339226
$842$	0	0
$843$	−2.22613e7	−1.07890
$844$	0	0
$845$	1.10845e7	0.534042
$846$	0	0
$847$	−2.41340e6	−0.115590
$848$	0	0
$849$	−1.28553e7	−0.612085
$850$	0	0
$851$	−34123.3	−0.00161520
$852$	0	0
$853$	−1.04283e7	−0.490729	−0.245364	−	0.969431i	$-0.578908\pi$
$853$	−1.04283e7	−0.490729	−0.245364	+	0.969431i	$0.578908\pi$
$854$	0	0
$855$	−5.30240e6	−0.248060
$856$	0	0
$857$	−1.58259e7	−0.736064	−0.368032	−	0.929813i	$-0.619968\pi$
$857$	−1.58259e7	−0.736064	−0.368032	+	0.929813i	$0.619968\pi$
$858$	0	0
$859$	−8.75498e6	−0.404830	−0.202415	−	0.979300i	$-0.564879\pi$
$859$	−8.75498e6	−0.404830	−0.202415	+	0.979300i	$0.564879\pi$
$860$	0	0
$861$	−8.74168e6	−0.401871
$862$	0	0
$863$	3.06343e7	1.40017	0.700084	−	0.714060i	$-0.253146\pi$
$863$	3.06343e7	1.40017	0.700084	+	0.714060i	$0.253146\pi$
$864$	0	0
$865$	−1.93417e7	−0.878932
$866$	0	0
$867$	2.70771e7	1.22336
$868$	0	0
$869$	−9.17540e6	−0.412169
$870$	0	0
$871$	−4.26653e7	−1.90559
$872$	0	0
$873$	1.03910e7	0.461449
$874$	0	0
$875$	2.54019e7	1.12162
$876$	0	0
$877$	3.72186e7	1.63403	0.817017	−	0.576613i	$-0.195626\pi$
$877$	3.72186e7	1.63403	0.817017	+	0.576613i	$0.195626\pi$
$878$	0	0
$879$	−1.10764e7	−0.483533
$880$	0	0
$881$	−1.35137e7	−0.586589	−0.293295	−	0.956022i	$-0.594752\pi$
$881$	−1.35137e7	−0.586589	−0.293295	+	0.956022i	$0.594752\pi$
$882$	0	0
$883$	4.01534e7	1.73309	0.866543	−	0.499103i	$-0.166337\pi$
$883$	4.01534e7	1.73309	0.866543	+	0.499103i	$0.166337\pi$
$884$	0	0
$885$	−4.90001e6	−0.210300
$886$	0	0
$887$	−137718.	−0.00587736	−0.00293868	−	0.999996i	$-0.500935\pi$
$887$	−137718.	−0.00587736	−0.00293868	+	0.999996i	$0.500935\pi$
$888$	0	0
$889$	5.93293e6	0.251776
$890$	0	0
$891$	−2.77544e6	−0.117122
$892$	0	0
$893$	−2.50835e7	−1.05259
$894$	0	0
$895$	−1.30375e7	−0.544047
$896$	0	0
$897$	3.79556e6	0.157505
$898$	0	0
$899$	9.73153e6	0.401589
$900$	0	0
$901$	−2.19891e7	−0.902395
$902$	0	0
$903$	−5.18399e6	−0.211566
$904$	0	0
$905$	2.70454e7	1.09767
$906$	0	0
$907$	−2.07740e7	−0.838497	−0.419249	−	0.907871i	$-0.637706\pi$
$907$	−2.07740e7	−0.838497	−0.419249	+	0.907871i	$0.637706\pi$
$908$	0	0
$909$	1.32991e7	0.533840
$910$	0	0
$911$	3.56241e6	0.142216	0.0711079	−	0.997469i	$-0.477347\pi$
$911$	3.56241e6	0.142216	0.0711079	+	0.997469i	$0.477347\pi$
$912$	0	0
$913$	−7.33174e6	−0.291092
$914$	0	0
$915$	2.05644e7	0.812014
$916$	0	0
$917$	−1.86511e7	−0.732457
$918$	0	0
$919$	−1.16502e7	−0.455036	−0.227518	−	0.973774i	$-0.573061\pi$
$919$	−1.16502e7	−0.455036	−0.227518	+	0.973774i	$0.573061\pi$
$920$	0	0
$921$	−2.60975e7	−1.01379
$922$	0	0
$923$	−2.96866e6	−0.114698
$924$	0	0
$925$	88090.3	0.00338512
$926$	0	0
$927$	1.43289e7	0.547664
$928$	0	0
$929$	1.53955e7	0.585267	0.292633	−	0.956225i	$-0.405468\pi$
$929$	1.53955e7	0.585267	0.292633	+	0.956225i	$0.405468\pi$
$930$	0	0
$931$	−2.15363e6	−0.0814322
$932$	0	0
$933$	1.64543e7	0.618836
$934$	0	0
$935$	3.73392e7	1.39681
$936$	0	0
$937$	−1.31302e7	−0.488564	−0.244282	−	0.969704i	$-0.578552\pi$
$937$	−1.31302e7	−0.488564	−0.244282	+	0.969704i	$0.578552\pi$
$938$	0	0
$939$	1.79456e7	0.664194
$940$	0	0
$941$	1.94995e7	0.717876	0.358938	−	0.933361i	$-0.383139\pi$
$941$	1.94995e7	0.717876	0.358938	+	0.933361i	$0.383139\pi$
$942$	0	0
$943$	3.81003e6	0.139524
$944$	0	0
$945$	−4.12369e6	−0.150213
$946$	0	0
$947$	−5.80449e6	−0.210324	−0.105162	−	0.994455i	$-0.533536\pi$
$947$	−5.80449e6	−0.210324	−0.105162	+	0.994455i	$0.533536\pi$
$948$	0	0
$949$	−807588.	−0.0291088
$950$	0	0
$951$	1.38968e7	0.498270
$952$	0	0
$953$	−2.33745e7	−0.833701	−0.416851	−	0.908975i	$-0.636866\pi$
$953$	−2.33745e7	−0.833701	−0.416851	+	0.908975i	$0.636866\pi$
$954$	0	0
$955$	940224.	0.0333598
$956$	0	0
$957$	−1.99538e7	−0.704282
$958$	0	0
$959$	2.20471e7	0.774115
$960$	0	0
$961$	−2.51815e7	−0.879577
$962$	0	0
$963$	3.59262e6	0.124837
$964$	0	0
$965$	2.66309e7	0.920591
$966$	0	0
$967$	−1.89613e7	−0.652081	−0.326041	−	0.945356i	$-0.605715\pi$
$967$	−1.89613e7	−0.652081	−0.326041	+	0.945356i	$0.605715\pi$
$968$	0	0
$969$	2.95581e7	1.01127
$970$	0	0
$971$	4.71884e7	1.60615	0.803077	−	0.595875i	$-0.203195\pi$
$971$	4.71884e7	1.60615	0.803077	+	0.595875i	$0.203195\pi$
$972$	0	0
$973$	−9.10148e6	−0.308198
$974$	0	0
$975$	−9.79833e6	−0.330096
$976$	0	0
$977$	−2.72456e7	−0.913186	−0.456593	−	0.889676i	$-0.650930\pi$
$977$	−2.72456e7	−0.913186	−0.456593	+	0.889676i	$0.650930\pi$
$978$	0	0
$979$	−1.08780e7	−0.362738
$980$	0	0
$981$	1.45320e7	0.482116
$982$	0	0
$983$	−4.43758e6	−0.146475	−0.0732373	−	0.997315i	$-0.523333\pi$
$983$	−4.43758e6	−0.146475	−0.0732373	+	0.997315i	$0.523333\pi$
$984$	0	0
$985$	−2.34255e7	−0.769306
$986$	0	0
$987$	−1.95075e7	−0.637394
$988$	0	0
$989$	2.25942e6	0.0734525
$990$	0	0
$991$	−3.61430e7	−1.16907	−0.584534	−	0.811369i	$-0.698723\pi$
$991$	−3.61430e7	−1.16907	−0.584534	+	0.811369i	$0.698723\pi$
$992$	0	0
$993$	−3.23570e7	−1.04135
$994$	0	0
$995$	−1.86450e6	−0.0597042
$996$	0	0
$997$	1.52960e6	0.0487349	0.0243675	−	0.999703i	$-0.492243\pi$
$997$	1.52960e6	0.0487349	0.0243675	+	0.999703i	$0.492243\pi$
$998$	0	0
$999$	−47024.4	−0.00149077

Twists

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

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

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +41.9449 q^{5} -134.859 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +41.9449 q^{5} -134.859 q^{7} +81.0000 q^{9} -423.021 q^{11} +797.218 q^{13} +377.504 q^{15} -2104.38 q^{17} -1560.66 q^{19} -1213.73 q^{21} +529.000 q^{23} -1365.63 q^{25} +729.000 q^{27} +5241.09 q^{29} +1856.77 q^{31} -3807.19 q^{33} -5656.64 q^{35} -64.5054 q^{37} +7174.97 q^{39} +7202.32 q^{41} +4271.12 q^{43} +3397.53 q^{45} +16072.3 q^{47} +1379.94 q^{49} -18939.4 q^{51} +10449.2 q^{53} -17743.6 q^{55} -14046.0 q^{57} -12980.0 q^{59} +54474.7 q^{61} -10923.6 q^{63} +33439.2 q^{65} -53517.7 q^{67} +4761.00 q^{69} -3723.77 q^{71} -1013.01 q^{73} -12290.7 q^{75} +57048.2 q^{77} +21690.2 q^{79} +6561.00 q^{81} +17331.9 q^{83} -88268.0 q^{85} +47169.8 q^{87} +25715.1 q^{89} -107512. q^{91} +16711.0 q^{93} -65461.8 q^{95} +128285. q^{97} -34264.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{3} + 22 q^{5} - 134 q^{7} + 486 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more