LMFDB - Embedded newform 6672.2.a.bs.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6672,2,Mod(1,6672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6672 = 2^{4} \cdot 3 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6672.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:	$53.2761882286$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 31 x^{10} + 142 x^{9} + 397 x^{8} - 1508 x^{7} - 2549 x^{6} + 7294 x^{5} + \cdots - 4912 $ x^12 - 5x^11 - 31x^10 + 142x^9 + 397x^8 - 1508x^7 - 2549x^6 + 7294x^5 + 7482x^4 - 15919x^3 - 5172x^2 + 14748*x - 4912
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3336)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.771577$ of defining polynomial
Character	$\chi$	$=$	6672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.228423 q^{5} +4.79785 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.228423 q^{5} +4.79785 q^{7} +1.00000 q^{9} +0.000464723 q^{11} -4.42847 q^{13} +0.228423 q^{15} -0.749938 q^{17} +3.92743 q^{19} +4.79785 q^{21} -3.00705 q^{23} -4.94782 q^{25} +1.00000 q^{27} +5.32754 q^{29} -0.301267 q^{31} +0.000464723 q^{33} +1.09594 q^{35} -8.13191 q^{37} -4.42847 q^{39} +2.90412 q^{41} +2.31306 q^{43} +0.228423 q^{45} +6.67634 q^{47} +16.0194 q^{49} -0.749938 q^{51} +10.9222 q^{53} +0.000106153 q^{55} +3.92743 q^{57} +9.57162 q^{59} +8.82072 q^{61} +4.79785 q^{63} -1.01156 q^{65} +8.01568 q^{67} -3.00705 q^{69} -13.5834 q^{71} +8.55020 q^{73} -4.94782 q^{75} +0.00222967 q^{77} +14.5372 q^{79} +1.00000 q^{81} -6.90438 q^{83} -0.171303 q^{85} +5.32754 q^{87} +6.30476 q^{89} -21.2471 q^{91} -0.301267 q^{93} +0.897116 q^{95} +12.5079 q^{97} +0.000464723 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9} + 10 q^{11} + 12 q^{13} + 7 q^{15} + 6 q^{17} + 5 q^{19} + 2 q^{21} + 8 q^{23} + 29 q^{25} + 12 q^{27} + 9 q^{29} + 10 q^{33} - 5 q^{35} + 23 q^{37} + 12 q^{39} + 19 q^{41} + 8 q^{43} + 7 q^{45} + 11 q^{47} + 22 q^{49} + 6 q^{51} + 19 q^{53} - 26 q^{55} + 5 q^{57} + 3 q^{59} + 27 q^{61} + 2 q^{63} + 12 q^{65} + 22 q^{67} + 8 q^{69} - q^{71} + 50 q^{73} + 29 q^{75} + q^{77} - 9 q^{79} + 12 q^{81} + 21 q^{83} + 35 q^{85} + 9 q^{87} + 26 q^{89} - 10 q^{91} - 30 q^{95} + 41 q^{97} + 10 q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9 + 10 * q^11 + 12 * q^13 + 7 * q^15 + 6 * q^17 + 5 * q^19 + 2 * q^21 + 8 * q^23 + 29 * q^25 + 12 * q^27 + 9 * q^29 + 10 * q^33 - 5 * q^35 + 23 * q^37 + 12 * q^39 + 19 * q^41 + 8 * q^43 + 7 * q^45 + 11 * q^47 + 22 * q^49 + 6 * q^51 + 19 * q^53 - 26 * q^55 + 5 * q^57 + 3 * q^59 + 27 * q^61 + 2 * q^63 + 12 * q^65 + 22 * q^67 + 8 * q^69 - q^71 + 50 * q^73 + 29 * q^75 + q^77 - 9 * q^79 + 12 * q^81 + 21 * q^83 + 35 * q^85 + 9 * q^87 + 26 * q^89 - 10 * q^91 - 30 * q^95 + 41 * q^97 + 10 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.228423	0.102154	0.0510769	−	0.998695i	$-0.483735\pi$
$5$	0.228423	0.102154	0.0510769	+	0.998695i	$0.483735\pi$
$6$	0	0
$7$	4.79785	1.81342	0.906708	−	0.421758i	$-0.138587\pi$
$7$	4.79785	1.81342	0.906708	+	0.421758i	$0.138587\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.000464723 0	0.000140119 0	7.00596e−5	−	1.00000i	$-0.499978\pi$
$11$	0.000464723 0	0.000140119 0	7.00596e−5		1.00000i	$0.499978\pi$
$12$	0	0
$13$	−4.42847	−1.22824	−0.614118	−	0.789214i	$-0.710488\pi$
$13$	−4.42847	−1.22824	−0.614118	+	0.789214i	$0.710488\pi$
$14$	0	0
$15$	0.228423	0.0589785
$16$	0	0
$17$	−0.749938	−0.181887	−0.0909433	−	0.995856i	$-0.528988\pi$
$17$	−0.749938	−0.181887	−0.0909433	+	0.995856i	$0.528988\pi$
$18$	0	0
$19$	3.92743	0.901015	0.450508	−	0.892773i	$-0.351243\pi$
$19$	3.92743	0.901015	0.450508	+	0.892773i	$0.351243\pi$
$20$	0	0
$21$	4.79785	1.04698
$22$	0	0
$23$	−3.00705	−0.627013	−0.313506	−	0.949586i	$-0.601504\pi$
$23$	−3.00705	−0.627013	−0.313506	+	0.949586i	$0.601504\pi$
$24$	0	0
$25$	−4.94782	−0.989565
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.32754	0.989299	0.494650	−	0.869092i	$-0.335296\pi$
$29$	5.32754	0.989299	0.494650	+	0.869092i	$0.335296\pi$
$30$	0	0
$31$	−0.301267	−0.0541092	−0.0270546	−	0.999634i	$-0.508613\pi$
$31$	−0.301267	−0.0541092	−0.0270546	+	0.999634i	$0.508613\pi$
$32$	0	0
$33$	0.000464723 0	8.08979e−5 0
$34$	0	0
$35$	1.09594	0.185247
$36$	0	0
$37$	−8.13191	−1.33688	−0.668439	−	0.743767i	$-0.733037\pi$
$37$	−8.13191	−1.33688	−0.668439	+	0.743767i	$0.733037\pi$
$38$	0	0
$39$	−4.42847	−0.709122
$40$	0	0
$41$	2.90412	0.453547	0.226774	−	0.973948i	$-0.427182\pi$
$41$	2.90412	0.453547	0.226774	+	0.973948i	$0.427182\pi$
$42$	0	0
$43$	2.31306	0.352738	0.176369	−	0.984324i	$-0.443565\pi$
$43$	2.31306	0.352738	0.176369	+	0.984324i	$0.443565\pi$
$44$	0	0
$45$	0.228423	0.0340513
$46$	0	0
$47$	6.67634	0.973844	0.486922	−	0.873445i	$-0.338120\pi$
$47$	6.67634	0.973844	0.486922	+	0.873445i	$0.338120\pi$
$48$	0	0
$49$	16.0194	2.28848
$50$	0	0
$51$	−0.749938	−0.105012
$52$	0	0
$53$	10.9222	1.50028	0.750141	−	0.661278i	$-0.229986\pi$
$53$	10.9222	1.50028	0.750141	+	0.661278i	$0.229986\pi$
$54$	0	0
$55$	0.000106153 0	1.43137e−5 0
$56$	0	0
$57$	3.92743	0.520201
$58$	0	0
$59$	9.57162	1.24612	0.623059	−	0.782175i	$-0.285890\pi$
$59$	9.57162	1.24612	0.623059	+	0.782175i	$0.285890\pi$
$60$	0	0
$61$	8.82072	1.12938	0.564689	−	0.825304i	$-0.308996\pi$
$61$	8.82072	1.12938	0.564689	+	0.825304i	$0.308996\pi$
$62$	0	0
$63$	4.79785	0.604472
$64$	0	0
$65$	−1.01156	−0.125469
$66$	0	0
$67$	8.01568	0.979271	0.489636	−	0.871927i	$-0.337130\pi$
$67$	8.01568	0.979271	0.489636	+	0.871927i	$0.337130\pi$
$68$	0	0
$69$	−3.00705	−0.362006
$70$	0	0
$71$	−13.5834	−1.61205	−0.806027	−	0.591878i	$-0.798387\pi$
$71$	−13.5834	−1.61205	−0.806027	+	0.591878i	$0.798387\pi$
$72$	0	0
$73$	8.55020	1.00073	0.500363	−	0.865816i	$-0.333200\pi$
$73$	8.55020	1.00073	0.500363	+	0.865816i	$0.333200\pi$
$74$	0	0
$75$	−4.94782	−0.571325
$76$	0	0
$77$	0.00222967	0.000254095 0
$78$	0	0
$79$	14.5372	1.63557	0.817783	−	0.575526i	$-0.195203\pi$
$79$	14.5372	1.63557	0.817783	+	0.575526i	$0.195203\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.90438	−0.757854	−0.378927	−	0.925426i	$-0.623707\pi$
$83$	−6.90438	−0.757854	−0.378927	+	0.925426i	$0.623707\pi$
$84$	0	0
$85$	−0.171303	−0.0185804
$86$	0	0
$87$	5.32754	0.571172
$88$	0	0
$89$	6.30476	0.668303	0.334152	−	0.942519i	$-0.391550\pi$
$89$	6.30476	0.668303	0.334152	+	0.942519i	$0.391550\pi$
$90$	0	0
$91$	−21.2471	−2.22730
$92$	0	0
$93$	−0.301267	−0.0312400
$94$	0	0
$95$	0.897116	0.0920422
$96$	0	0
$97$	12.5079	1.26999	0.634994	−	0.772517i	$-0.281002\pi$
$97$	12.5079	1.26999	0.634994	+	0.772517i	$0.281002\pi$
$98$	0	0
$99$	0.000464723 0	4.67064e−5 0
$100$	0	0
$101$	1.59996	0.159202	0.0796012	−	0.996827i	$-0.474635\pi$
$101$	1.59996	0.159202	0.0796012	+	0.996827i	$0.474635\pi$
$102$	0	0
$103$	−5.41659	−0.533712	−0.266856	−	0.963736i	$-0.585985\pi$
$103$	−5.41659	−0.533712	−0.266856	+	0.963736i	$0.585985\pi$
$104$	0	0
$105$	1.09594	0.106953
$106$	0	0
$107$	4.31872	0.417506	0.208753	−	0.977968i	$-0.433059\pi$
$107$	4.31872	0.417506	0.208753	+	0.977968i	$0.433059\pi$
$108$	0	0
$109$	17.6404	1.68964	0.844821	−	0.535049i	$-0.179707\pi$
$109$	17.6404	1.68964	0.844821	+	0.535049i	$0.179707\pi$
$110$	0	0
$111$	−8.13191	−0.771847
$112$	0	0
$113$	−18.4096	−1.73183	−0.865913	−	0.500195i	$-0.833262\pi$
$113$	−18.4096	−1.73183	−0.865913	+	0.500195i	$0.833262\pi$
$114$	0	0
$115$	−0.686878	−0.0640517
$116$	0	0
$117$	−4.42847	−0.409412
$118$	0	0
$119$	−3.59809	−0.329836
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	2.90412	0.261856
$124$	0	0
$125$	−2.27231	−0.203242
$126$	0	0
$127$	3.09963	0.275048	0.137524	−	0.990498i	$-0.456086\pi$
$127$	3.09963	0.275048	0.137524	+	0.990498i	$0.456086\pi$
$128$	0	0
$129$	2.31306	0.203654
$130$	0	0
$131$	−5.76193	−0.503423	−0.251711	−	0.967802i	$-0.580993\pi$
$131$	−5.76193	−0.503423	−0.251711	+	0.967802i	$0.580993\pi$
$132$	0	0
$133$	18.8432	1.63392
$134$	0	0
$135$	0.228423	0.0196595
$136$	0	0
$137$	−8.29558	−0.708739	−0.354370	−	0.935105i	$-0.615304\pi$
$137$	−8.29558	−0.708739	−0.354370	+	0.935105i	$0.615304\pi$
$138$	0	0
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0	0
$141$	6.67634	0.562249
$142$	0	0
$143$	−0.00205801	−0.000172099 0
$144$	0	0
$145$	1.21693	0.101061
$146$	0	0
$147$	16.0194	1.32125
$148$	0	0
$149$	11.0671	0.906656	0.453328	−	0.891344i	$-0.350237\pi$
$149$	11.0671	0.906656	0.453328	+	0.891344i	$0.350237\pi$
$150$	0	0
$151$	−1.06144	−0.0863786	−0.0431893	−	0.999067i	$-0.513752\pi$
$151$	−1.06144	−0.0863786	−0.0431893	+	0.999067i	$0.513752\pi$
$152$	0	0
$153$	−0.749938	−0.0606289
$154$	0	0
$155$	−0.0688163	−0.00552746
$156$	0	0
$157$	7.47219	0.596346	0.298173	−	0.954512i	$-0.403623\pi$
$157$	7.47219	0.596346	0.298173	+	0.954512i	$0.403623\pi$
$158$	0	0
$159$	10.9222	0.866188
$160$	0	0
$161$	−14.4274	−1.13704
$162$	0	0
$163$	16.6175	1.30159	0.650793	−	0.759255i	$-0.274437\pi$
$163$	16.6175	1.30159	0.650793	+	0.759255i	$0.274437\pi$
$164$	0	0
$165$	0.000106153 0	8.26403e−6 0
$166$	0	0
$167$	−9.13178	−0.706638	−0.353319	−	0.935503i	$-0.614947\pi$
$167$	−9.13178	−0.706638	−0.353319	+	0.935503i	$0.614947\pi$
$168$	0	0
$169$	6.61132	0.508563
$170$	0	0
$171$	3.92743	0.300338
$172$	0	0
$173$	12.4990	0.950284	0.475142	−	0.879909i	$-0.342397\pi$
$173$	12.4990	0.950284	0.475142	+	0.879909i	$0.342397\pi$
$174$	0	0
$175$	−23.7389	−1.79449
$176$	0	0
$177$	9.57162	0.719447
$178$	0	0
$179$	−21.7609	−1.62649	−0.813244	−	0.581922i	$-0.802301\pi$
$179$	−21.7609	−1.62649	−0.813244	+	0.581922i	$0.802301\pi$
$180$	0	0
$181$	2.09846	0.155977	0.0779885	−	0.996954i	$-0.475150\pi$
$181$	2.09846	0.155977	0.0779885	+	0.996954i	$0.475150\pi$
$182$	0	0
$183$	8.82072	0.652046
$184$	0	0
$185$	−1.85752	−0.136567
$186$	0	0
$187$	−0.000348513 0	−2.54858e−5 0
$188$	0	0
$189$	4.79785	0.348992
$190$	0	0
$191$	−24.2841	−1.75713	−0.878566	−	0.477620i	$-0.841500\pi$
$191$	−24.2841	−1.75713	−0.878566	+	0.477620i	$0.841500\pi$
$192$	0	0
$193$	22.2705	1.60307	0.801533	−	0.597951i	$-0.204018\pi$
$193$	22.2705	1.60307	0.801533	+	0.597951i	$0.204018\pi$
$194$	0	0
$195$	−1.01156	−0.0724395
$196$	0	0
$197$	5.21443	0.371513	0.185756	−	0.982596i	$-0.440527\pi$
$197$	5.21443	0.371513	0.185756	+	0.982596i	$0.440527\pi$
$198$	0	0
$199$	−3.38528	−0.239976	−0.119988	−	0.992775i	$-0.538286\pi$
$199$	−3.38528	−0.239976	−0.119988	+	0.992775i	$0.538286\pi$
$200$	0	0
$201$	8.01568	0.565382
$202$	0	0
$203$	25.5607	1.79401
$204$	0	0
$205$	0.663367	0.0463316
$206$	0	0
$207$	−3.00705	−0.209004
$208$	0	0
$209$	0.00182517	0.000126250 0
$210$	0	0
$211$	26.4595	1.82155	0.910773	−	0.412908i	$-0.135487\pi$
$211$	26.4595	1.82155	0.910773	+	0.412908i	$0.135487\pi$
$212$	0	0
$213$	−13.5834	−0.930720
$214$	0	0
$215$	0.528356	0.0360336
$216$	0	0
$217$	−1.44544	−0.0981225
$218$	0	0
$219$	8.55020	0.577769
$220$	0	0
$221$	3.32107	0.223400
$222$	0	0
$223$	−18.5175	−1.24002	−0.620011	−	0.784593i	$-0.712872\pi$
$223$	−18.5175	−1.24002	−0.620011	+	0.784593i	$0.712872\pi$
$224$	0	0
$225$	−4.94782	−0.329855
$226$	0	0
$227$	−6.33817	−0.420679	−0.210340	−	0.977628i	$-0.567457\pi$
$227$	−6.33817	−0.420679	−0.210340	+	0.977628i	$0.567457\pi$
$228$	0	0
$229$	−18.1085	−1.19665	−0.598323	−	0.801255i	$-0.704166\pi$
$229$	−18.1085	−1.19665	−0.598323	+	0.801255i	$0.704166\pi$
$230$	0	0
$231$	0.00222967	0.000146702 0
$232$	0	0
$233$	3.30673	0.216631	0.108315	−	0.994117i	$-0.465454\pi$
$233$	3.30673	0.216631	0.108315	+	0.994117i	$0.465454\pi$
$234$	0	0
$235$	1.52503	0.0994819
$236$	0	0
$237$	14.5372	0.944295
$238$	0	0
$239$	−5.74884	−0.371861	−0.185931	−	0.982563i	$-0.559530\pi$
$239$	−5.74884	−0.371861	−0.185931	+	0.982563i	$0.559530\pi$
$240$	0	0
$241$	10.8074	0.696169	0.348084	−	0.937463i	$-0.386832\pi$
$241$	10.8074	0.696169	0.348084	+	0.937463i	$0.386832\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.65919	0.233777
$246$	0	0
$247$	−17.3925	−1.10666
$248$	0	0
$249$	−6.90438	−0.437547
$250$	0	0
$251$	−1.70771	−0.107790	−0.0538949	−	0.998547i	$-0.517164\pi$
$251$	−1.70771	−0.107790	−0.0538949	+	0.998547i	$0.517164\pi$
$252$	0	0
$253$	−0.00139744	−8.78565e−5 0
$254$	0	0
$255$	−0.171303	−0.0107274
$256$	0	0
$257$	−29.4783	−1.83881	−0.919403	−	0.393318i	$-0.871327\pi$
$257$	−29.4783	−1.83881	−0.919403	+	0.393318i	$0.871327\pi$
$258$	0	0
$259$	−39.0157	−2.42432
$260$	0	0
$261$	5.32754	0.329766
$262$	0	0
$263$	23.3724	1.44120	0.720601	−	0.693350i	$-0.243866\pi$
$263$	23.3724	1.44120	0.720601	+	0.693350i	$0.243866\pi$
$264$	0	0
$265$	2.49488	0.153259
$266$	0	0
$267$	6.30476	0.385845
$268$	0	0
$269$	−16.0299	−0.977358	−0.488679	−	0.872464i	$-0.662521\pi$
$269$	−16.0299	−0.977358	−0.488679	+	0.872464i	$0.662521\pi$
$270$	0	0
$271$	−19.2935	−1.17200	−0.585999	−	0.810312i	$-0.699298\pi$
$271$	−19.2935	−1.17200	−0.585999	+	0.810312i	$0.699298\pi$
$272$	0	0
$273$	−21.2471	−1.28593
$274$	0	0
$275$	−0.00229937	−0.000138657 0
$276$	0	0
$277$	7.43257	0.446580	0.223290	−	0.974752i	$-0.428320\pi$
$277$	7.43257	0.446580	0.223290	+	0.974752i	$0.428320\pi$
$278$	0	0
$279$	−0.301267	−0.0180364
$280$	0	0
$281$	−9.44918	−0.563691	−0.281845	−	0.959460i	$-0.590947\pi$
$281$	−9.44918	−0.563691	−0.281845	+	0.959460i	$0.590947\pi$
$282$	0	0
$283$	−4.94927	−0.294203	−0.147102	−	0.989121i	$-0.546994\pi$
$283$	−4.94927	−0.294203	−0.147102	+	0.989121i	$0.546994\pi$
$284$	0	0
$285$	0.897116	0.0531406
$286$	0	0
$287$	13.9335	0.822470
$288$	0	0
$289$	−16.4376	−0.966917
$290$	0	0
$291$	12.5079	0.733228
$292$	0	0
$293$	−21.7068	−1.26812	−0.634061	−	0.773283i	$-0.718613\pi$
$293$	−21.7068	−1.26812	−0.634061	+	0.773283i	$0.718613\pi$
$294$	0	0
$295$	2.18638	0.127296
$296$	0	0
$297$	0.000464723 0	2.69660e−5 0
$298$	0	0
$299$	13.3166	0.770119
$300$	0	0
$301$	11.0977	0.639662
$302$	0	0
$303$	1.59996	0.0919156
$304$	0	0
$305$	2.01485	0.115370
$306$	0	0
$307$	−30.0709	−1.71624	−0.858119	−	0.513452i	$-0.828367\pi$
$307$	−30.0709	−1.71624	−0.858119	+	0.513452i	$0.828367\pi$
$308$	0	0
$309$	−5.41659	−0.308139
$310$	0	0
$311$	−23.1033	−1.31007	−0.655033	−	0.755600i	$-0.727345\pi$
$311$	−23.1033	−1.31007	−0.655033	+	0.755600i	$0.727345\pi$
$312$	0	0
$313$	14.6283	0.826838	0.413419	−	0.910541i	$-0.364335\pi$
$313$	14.6283	0.826838	0.413419	+	0.910541i	$0.364335\pi$
$314$	0	0
$315$	1.09594	0.0617492
$316$	0	0
$317$	−11.8156	−0.663632	−0.331816	−	0.943344i	$-0.607661\pi$
$317$	−11.8156	−0.663632	−0.331816	+	0.943344i	$0.607661\pi$
$318$	0	0
$319$	0.00247583	0.000138620 0
$320$	0	0
$321$	4.31872	0.241047
$322$	0	0
$323$	−2.94533	−0.163883
$324$	0	0
$325$	21.9113	1.21542
$326$	0	0
$327$	17.6404	0.975515
$328$	0	0
$329$	32.0321	1.76599
$330$	0	0
$331$	9.42852	0.518239	0.259119	−	0.965845i	$-0.416568\pi$
$331$	9.42852	0.518239	0.259119	+	0.965845i	$0.416568\pi$
$332$	0	0
$333$	−8.13191	−0.445626
$334$	0	0
$335$	1.83096	0.100036
$336$	0	0
$337$	11.9948	0.653399	0.326700	−	0.945128i	$-0.394063\pi$
$337$	11.9948	0.653399	0.326700	+	0.945128i	$0.394063\pi$
$338$	0	0
$339$	−18.4096	−0.999870
$340$	0	0
$341$	−0.000140006 0	−7.58174e−6 0
$342$	0	0
$343$	43.2735	2.33655
$344$	0	0
$345$	−0.686878	−0.0369803
$346$	0	0
$347$	35.0427	1.88119	0.940594	−	0.339533i	$-0.110269\pi$
$347$	35.0427	1.88119	0.940594	+	0.339533i	$0.110269\pi$
$348$	0	0
$349$	15.2844	0.818153	0.409077	−	0.912500i	$-0.365851\pi$
$349$	15.2844	0.818153	0.409077	+	0.912500i	$0.365851\pi$
$350$	0	0
$351$	−4.42847	−0.236374
$352$	0	0
$353$	32.8877	1.75044	0.875218	−	0.483729i	$-0.160718\pi$
$353$	32.8877	1.75044	0.875218	+	0.483729i	$0.160718\pi$
$354$	0	0
$355$	−3.10276	−0.164678
$356$	0	0
$357$	−3.59809	−0.190431
$358$	0	0
$359$	−27.6904	−1.46144	−0.730721	−	0.682676i	$-0.760816\pi$
$359$	−27.6904	−1.46144	−0.730721	+	0.682676i	$0.760816\pi$
$360$	0	0
$361$	−3.57526	−0.188171
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	1.95306	0.102228
$366$	0	0
$367$	−14.4753	−0.755607	−0.377804	−	0.925886i	$-0.623321\pi$
$367$	−14.4753	−0.755607	−0.377804	+	0.925886i	$0.623321\pi$
$368$	0	0
$369$	2.90412	0.151182
$370$	0	0
$371$	52.4031	2.72064
$372$	0	0
$373$	5.57684	0.288758	0.144379	−	0.989522i	$-0.453882\pi$
$373$	5.57684	0.288758	0.144379	+	0.989522i	$0.453882\pi$
$374$	0	0
$375$	−2.27231	−0.117342
$376$	0	0
$377$	−23.5928	−1.21509
$378$	0	0
$379$	21.4893	1.10383	0.551914	−	0.833901i	$-0.313897\pi$
$379$	21.4893	1.10383	0.551914	+	0.833901i	$0.313897\pi$
$380$	0	0
$381$	3.09963	0.158799
$382$	0	0
$383$	−4.20218	−0.214721	−0.107361	−	0.994220i	$-0.534240\pi$
$383$	−4.20218	−0.214721	−0.107361	+	0.994220i	$0.534240\pi$
$384$	0	0
$385$	0.000509308 0	2.59567e−5 0
$386$	0	0
$387$	2.31306	0.117579
$388$	0	0
$389$	−34.2476	−1.73642	−0.868210	−	0.496196i	$-0.834730\pi$
$389$	−34.2476	−1.73642	−0.868210	+	0.496196i	$0.834730\pi$
$390$	0	0
$391$	2.25510	0.114045
$392$	0	0
$393$	−5.76193	−0.290651
$394$	0	0
$395$	3.32064	0.167079
$396$	0	0
$397$	11.6234	0.583363	0.291681	−	0.956516i	$-0.405785\pi$
$397$	11.6234	0.583363	0.291681	+	0.956516i	$0.405785\pi$
$398$	0	0
$399$	18.8432	0.943342
$400$	0	0
$401$	3.16028	0.157817	0.0789084	−	0.996882i	$-0.474857\pi$
$401$	3.16028	0.157817	0.0789084	+	0.996882i	$0.474857\pi$
$402$	0	0
$403$	1.33415	0.0664588
$404$	0	0
$405$	0.228423	0.0113504
$406$	0	0
$407$	−0.00377909	−0.000187322 0
$408$	0	0
$409$	−11.3995	−0.563670	−0.281835	−	0.959463i	$-0.590943\pi$
$409$	−11.3995	−0.563670	−0.281835	+	0.959463i	$0.590943\pi$
$410$	0	0
$411$	−8.29558	−0.409191
$412$	0	0
$413$	45.9232	2.25973
$414$	0	0
$415$	−1.57712	−0.0774177
$416$	0	0
$417$	−1.00000	−0.0489702
$418$	0	0
$419$	−17.4646	−0.853204	−0.426602	−	0.904440i	$-0.640289\pi$
$419$	−17.4646	−0.853204	−0.426602	+	0.904440i	$0.640289\pi$
$420$	0	0
$421$	−2.01239	−0.0980779	−0.0490389	−	0.998797i	$-0.515616\pi$
$421$	−2.01239	−0.0980779	−0.0490389	+	0.998797i	$0.515616\pi$
$422$	0	0
$423$	6.67634	0.324615
$424$	0	0
$425$	3.71056	0.179989
$426$	0	0
$427$	42.3205	2.04803
$428$	0	0
$429$	−0.00205801	−9.93616e−5 0
$430$	0	0
$431$	−23.5898	−1.13628	−0.568139	−	0.822932i	$-0.692337\pi$
$431$	−23.5898	−1.13628	−0.568139	+	0.822932i	$0.692337\pi$
$432$	0	0
$433$	1.32696	0.0637695	0.0318848	−	0.999492i	$-0.489849\pi$
$433$	1.32696	0.0637695	0.0318848	+	0.999492i	$0.489849\pi$
$434$	0	0
$435$	1.21693	0.0583474
$436$	0	0
$437$	−11.8100	−0.564948
$438$	0	0
$439$	−6.76855	−0.323045	−0.161523	−	0.986869i	$-0.551640\pi$
$439$	−6.76855	−0.323045	−0.161523	+	0.986869i	$0.551640\pi$
$440$	0	0
$441$	16.0194	0.762827
$442$	0	0
$443$	31.9988	1.52031	0.760154	−	0.649743i	$-0.225124\pi$
$443$	31.9988	1.52031	0.760154	+	0.649743i	$0.225124\pi$
$444$	0	0
$445$	1.44015	0.0682698
$446$	0	0
$447$	11.0671	0.523458
$448$	0	0
$449$	−5.05038	−0.238342	−0.119171	−	0.992874i	$-0.538024\pi$
$449$	−5.05038	−0.238342	−0.119171	+	0.992874i	$0.538024\pi$
$450$	0	0
$451$	0.00134961	6.35507e−5 0
$452$	0	0
$453$	−1.06144	−0.0498707
$454$	0	0
$455$	−4.85333	−0.227528
$456$	0	0
$457$	26.1887	1.22506	0.612528	−	0.790449i	$-0.290153\pi$
$457$	26.1887	1.22506	0.612528	+	0.790449i	$0.290153\pi$
$458$	0	0
$459$	−0.749938	−0.0350041
$460$	0	0
$461$	−15.4665	−0.720345	−0.360173	−	0.932886i	$-0.617282\pi$
$461$	−15.4665	−0.720345	−0.360173	+	0.932886i	$0.617282\pi$
$462$	0	0
$463$	14.6457	0.680645	0.340322	−	0.940309i	$-0.389464\pi$
$463$	14.6457	0.680645	0.340322	+	0.940309i	$0.389464\pi$
$464$	0	0
$465$	−0.0688163	−0.00319128
$466$	0	0
$467$	−18.6479	−0.862924	−0.431462	−	0.902131i	$-0.642002\pi$
$467$	−18.6479	−0.862924	−0.431462	+	0.902131i	$0.642002\pi$
$468$	0	0
$469$	38.4580	1.77583
$470$	0	0
$471$	7.47219	0.344300
$472$	0	0
$473$	0.00107493	4.94254e−5 0
$474$	0	0
$475$	−19.4323	−0.891613
$476$	0	0
$477$	10.9222	0.500094
$478$	0	0
$479$	−14.1822	−0.648001	−0.324001	−	0.946057i	$-0.605028\pi$
$479$	−14.1822	−0.648001	−0.324001	+	0.946057i	$0.605028\pi$
$480$	0	0
$481$	36.0119	1.64200
$482$	0	0
$483$	−14.4274	−0.656468
$484$	0	0
$485$	2.85710	0.129734
$486$	0	0
$487$	3.56359	0.161482	0.0807408	−	0.996735i	$-0.474271\pi$
$487$	3.56359	0.161482	0.0807408	+	0.996735i	$0.474271\pi$
$488$	0	0
$489$	16.6175	0.751471
$490$	0	0
$491$	14.6644	0.661797	0.330898	−	0.943666i	$-0.392648\pi$
$491$	14.6644	0.661797	0.330898	+	0.943666i	$0.392648\pi$
$492$	0	0
$493$	−3.99532	−0.179940
$494$	0	0
$495$	0.000106153 0	4.77124e−6 0
$496$	0	0
$497$	−65.1712	−2.92333
$498$	0	0
$499$	−30.7442	−1.37630	−0.688149	−	0.725569i	$-0.741577\pi$
$499$	−30.7442	−1.37630	−0.688149	+	0.725569i	$0.741577\pi$
$500$	0	0
$501$	−9.13178	−0.407978
$502$	0	0
$503$	15.4574	0.689210	0.344605	−	0.938748i	$-0.388013\pi$
$503$	15.4574	0.689210	0.344605	+	0.938748i	$0.388013\pi$
$504$	0	0
$505$	0.365469	0.0162631
$506$	0	0
$507$	6.61132	0.293619
$508$	0	0
$509$	30.1985	1.33852	0.669262	−	0.743027i	$-0.266611\pi$
$509$	30.1985	1.33852	0.669262	+	0.743027i	$0.266611\pi$
$510$	0	0
$511$	41.0226	1.81473
$512$	0	0
$513$	3.92743	0.173400
$514$	0	0
$515$	−1.23727	−0.0545208
$516$	0	0
$517$	0.00310265	0.000136454 0
$518$	0	0
$519$	12.4990	0.548647
$520$	0	0
$521$	36.5558	1.60154	0.800768	−	0.598974i	$-0.204425\pi$
$521$	36.5558	1.60154	0.800768	+	0.598974i	$0.204425\pi$
$522$	0	0
$523$	−35.7750	−1.56433	−0.782166	−	0.623070i	$-0.785885\pi$
$523$	−35.7750	−1.56433	−0.782166	+	0.623070i	$0.785885\pi$
$524$	0	0
$525$	−23.7389	−1.03605
$526$	0	0
$527$	0.225932	0.00984174
$528$	0	0
$529$	−13.9577	−0.606855
$530$	0	0
$531$	9.57162	0.415373
$532$	0	0
$533$	−12.8608	−0.557063
$534$	0	0
$535$	0.986494	0.0426499
$536$	0	0
$537$	−21.7609	−0.939054
$538$	0	0
$539$	0.00744456	0.000320660 0
$540$	0	0
$541$	−32.0614	−1.37843	−0.689213	−	0.724559i	$-0.742044\pi$
$541$	−32.0614	−1.37843	−0.689213	+	0.724559i	$0.742044\pi$
$542$	0	0
$543$	2.09846	0.0900534
$544$	0	0
$545$	4.02947	0.172603
$546$	0	0
$547$	9.24671	0.395361	0.197680	−	0.980267i	$-0.436659\pi$
$547$	9.24671	0.395361	0.197680	+	0.980267i	$0.436659\pi$
$548$	0	0
$549$	8.82072	0.376459
$550$	0	0
$551$	20.9236	0.891374
$552$	0	0
$553$	69.7475	2.96596
$554$	0	0
$555$	−1.85752	−0.0788471
$556$	0	0
$557$	18.6654	0.790876	0.395438	−	0.918493i	$-0.370593\pi$
$557$	18.6654	0.790876	0.395438	+	0.918493i	$0.370593\pi$
$558$	0	0
$559$	−10.2433	−0.433246
$560$	0	0
$561$	−0.000348513 0	−1.47142e−5 0
$562$	0	0
$563$	−3.24798	−0.136886	−0.0684430	−	0.997655i	$-0.521803\pi$
$563$	−3.24798	−0.136886	−0.0684430	+	0.997655i	$0.521803\pi$
$564$	0	0
$565$	−4.20516	−0.176913
$566$	0	0
$567$	4.79785	0.201491
$568$	0	0
$569$	−28.8000	−1.20736	−0.603678	−	0.797228i	$-0.706299\pi$
$569$	−28.8000	−1.20736	−0.603678	+	0.797228i	$0.706299\pi$
$570$	0	0
$571$	12.9327	0.541218	0.270609	−	0.962689i	$-0.412775\pi$
$571$	12.9327	0.541218	0.270609	+	0.962689i	$0.412775\pi$
$572$	0	0
$573$	−24.2841	−1.01448
$574$	0	0
$575$	14.8783	0.620469
$576$	0	0
$577$	−26.6450	−1.10925	−0.554623	−	0.832102i	$-0.687138\pi$
$577$	−26.6450	−1.10925	−0.554623	+	0.832102i	$0.687138\pi$
$578$	0	0
$579$	22.2705	0.925531
$580$	0	0
$581$	−33.1262	−1.37431
$582$	0	0
$583$	0.00507580	0.000210218 0
$584$	0	0
$585$	−1.01156	−0.0418230
$586$	0	0
$587$	21.0478	0.868734	0.434367	−	0.900736i	$-0.356972\pi$
$587$	21.0478	0.868734	0.434367	+	0.900736i	$0.356972\pi$
$588$	0	0
$589$	−1.18321	−0.0487532
$590$	0	0
$591$	5.21443	0.214493
$592$	0	0
$593$	1.99109	0.0817641	0.0408821	−	0.999164i	$-0.486983\pi$
$593$	1.99109	0.0817641	0.0408821	+	0.999164i	$0.486983\pi$
$594$	0	0
$595$	−0.821886	−0.0336940
$596$	0	0
$597$	−3.38528	−0.138550
$598$	0	0
$599$	−9.30211	−0.380074	−0.190037	−	0.981777i	$-0.560861\pi$
$599$	−9.30211	−0.380074	−0.190037	+	0.981777i	$0.560861\pi$
$600$	0	0
$601$	−20.2444	−0.825787	−0.412894	−	0.910779i	$-0.635482\pi$
$601$	−20.2444	−0.825787	−0.412894	+	0.910779i	$0.635482\pi$
$602$	0	0
$603$	8.01568	0.326424
$604$	0	0
$605$	−2.51265	−0.102154
$606$	0	0
$607$	36.9384	1.49928	0.749641	−	0.661845i	$-0.230226\pi$
$607$	36.9384	1.49928	0.749641	+	0.661845i	$0.230226\pi$
$608$	0	0
$609$	25.5607	1.03577
$610$	0	0
$611$	−29.5659	−1.19611
$612$	0	0
$613$	−20.6863	−0.835513	−0.417756	−	0.908559i	$-0.637183\pi$
$613$	−20.6863	−0.835513	−0.417756	+	0.908559i	$0.637183\pi$
$614$	0	0
$615$	0.663367	0.0267495
$616$	0	0
$617$	40.6309	1.63574	0.817869	−	0.575405i	$-0.195155\pi$
$617$	40.6309	1.63574	0.817869	+	0.575405i	$0.195155\pi$
$618$	0	0
$619$	28.1924	1.13315	0.566573	−	0.824011i	$-0.308269\pi$
$619$	28.1924	1.13315	0.566573	+	0.824011i	$0.308269\pi$
$620$	0	0
$621$	−3.00705	−0.120669
$622$	0	0
$623$	30.2493	1.21191
$624$	0	0
$625$	24.2201	0.968803
$626$	0	0
$627$	0.00182517	7.28902e−5 0
$628$	0	0
$629$	6.09843	0.243160
$630$	0	0
$631$	4.68148	0.186367	0.0931834	−	0.995649i	$-0.470296\pi$
$631$	4.68148	0.186367	0.0931834	+	0.995649i	$0.470296\pi$
$632$	0	0
$633$	26.4595	1.05167
$634$	0	0
$635$	0.708027	0.0280972
$636$	0	0
$637$	−70.9412	−2.81079
$638$	0	0
$639$	−13.5834	−0.537352
$640$	0	0
$641$	10.1241	0.399877	0.199939	−	0.979808i	$-0.435926\pi$
$641$	10.1241	0.399877	0.199939	+	0.979808i	$0.435926\pi$
$642$	0	0
$643$	−12.5365	−0.494390	−0.247195	−	0.968966i	$-0.579509\pi$
$643$	−12.5365	−0.494390	−0.247195	+	0.968966i	$0.579509\pi$
$644$	0	0
$645$	0.528356	0.0208040
$646$	0	0
$647$	−4.07815	−0.160328	−0.0801642	−	0.996782i	$-0.525544\pi$
$647$	−4.07815	−0.160328	−0.0801642	+	0.996782i	$0.525544\pi$
$648$	0	0
$649$	0.00444815	0.000174605 0
$650$	0	0
$651$	−1.44544	−0.0566511
$652$	0	0
$653$	−4.02569	−0.157537	−0.0787687	−	0.996893i	$-0.525099\pi$
$653$	−4.02569	−0.157537	−0.0787687	+	0.996893i	$0.525099\pi$
$654$	0	0
$655$	−1.31616	−0.0514265
$656$	0	0
$657$	8.55020	0.333575
$658$	0	0
$659$	44.3561	1.72787	0.863934	−	0.503605i	$-0.167993\pi$
$659$	44.3561	1.72787	0.863934	+	0.503605i	$0.167993\pi$
$660$	0	0
$661$	−8.67464	−0.337404	−0.168702	−	0.985667i	$-0.553958\pi$
$661$	−8.67464	−0.337404	−0.168702	+	0.985667i	$0.553958\pi$
$662$	0	0
$663$	3.32107	0.128980
$664$	0	0
$665$	4.30423	0.166911
$666$	0	0
$667$	−16.0202	−0.620303
$668$	0	0
$669$	−18.5175	−0.715928
$670$	0	0
$671$	0.00409919	0.000158248 0
$672$	0	0
$673$	4.76488	0.183672	0.0918362	−	0.995774i	$-0.470726\pi$
$673$	4.76488	0.183672	0.0918362	+	0.995774i	$0.470726\pi$
$674$	0	0
$675$	−4.94782	−0.190442
$676$	0	0
$677$	−17.2902	−0.664515	−0.332258	−	0.943189i	$-0.607810\pi$
$677$	−17.2902	−0.664515	−0.332258	+	0.943189i	$0.607810\pi$
$678$	0	0
$679$	60.0112	2.30302
$680$	0	0
$681$	−6.33817	−0.242879
$682$	0	0
$683$	−7.61760	−0.291479	−0.145740	−	0.989323i	$-0.546556\pi$
$683$	−7.61760	−0.291479	−0.145740	+	0.989323i	$0.546556\pi$
$684$	0	0
$685$	−1.89490	−0.0724004
$686$	0	0
$687$	−18.1085	−0.690884
$688$	0	0
$689$	−48.3687	−1.84270
$690$	0	0
$691$	−14.7892	−0.562607	−0.281304	−	0.959619i	$-0.590767\pi$
$691$	−14.7892	−0.562607	−0.281304	+	0.959619i	$0.590767\pi$
$692$	0	0
$693$	0.00222967	8.46982e−5 0
$694$	0	0
$695$	−0.228423	−0.00866457
$696$	0	0
$697$	−2.17791	−0.0824941
$698$	0	0
$699$	3.30673	0.125072
$700$	0	0
$701$	−22.8989	−0.864878	−0.432439	−	0.901663i	$-0.642347\pi$
$701$	−22.8989	−0.864878	−0.432439	+	0.901663i	$0.642347\pi$
$702$	0	0
$703$	−31.9376	−1.20455
$704$	0	0
$705$	1.52503	0.0574359
$706$	0	0
$707$	7.67639	0.288700
$708$	0	0
$709$	−8.43270	−0.316697	−0.158348	−	0.987383i	$-0.550617\pi$
$709$	−8.43270	−0.316697	−0.158348	+	0.987383i	$0.550617\pi$
$710$	0	0
$711$	14.5372	0.545189
$712$	0	0
$713$	0.905925	0.0339271
$714$	0	0
$715$	−0.000470097 0	−1.75806e−5 0
$716$	0	0
$717$	−5.74884	−0.214694
$718$	0	0
$719$	30.8564	1.15075	0.575375	−	0.817890i	$-0.304856\pi$
$719$	30.8564	1.15075	0.575375	+	0.817890i	$0.304856\pi$
$720$	0	0
$721$	−25.9880	−0.967843
$722$	0	0
$723$	10.8074	0.401933
$724$	0	0
$725$	−26.3597	−0.978976
$726$	0	0
$727$	19.9824	0.741105	0.370552	−	0.928812i	$-0.379168\pi$
$727$	19.9824	0.741105	0.370552	+	0.928812i	$0.379168\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.73465	−0.0641584
$732$	0	0
$733$	−23.3509	−0.862486	−0.431243	−	0.902236i	$-0.641925\pi$
$733$	−23.3509	−0.862486	−0.431243	+	0.902236i	$0.641925\pi$
$734$	0	0
$735$	3.65919	0.134971
$736$	0	0
$737$	0.00372507	0.000137215 0
$738$	0	0
$739$	−32.2090	−1.18483	−0.592413	−	0.805634i	$-0.701825\pi$
$739$	−32.2090	−1.18483	−0.592413	+	0.805634i	$0.701825\pi$
$740$	0	0
$741$	−17.3925	−0.638930
$742$	0	0
$743$	25.4724	0.934491	0.467245	−	0.884128i	$-0.345246\pi$
$743$	25.4724	0.934491	0.467245	+	0.884128i	$0.345246\pi$
$744$	0	0
$745$	2.52799	0.0926183
$746$	0	0
$747$	−6.90438	−0.252618
$748$	0	0
$749$	20.7206	0.757113
$750$	0	0
$751$	−2.79254	−0.101901	−0.0509506	−	0.998701i	$-0.516225\pi$
$751$	−2.79254	−0.101901	−0.0509506	+	0.998701i	$0.516225\pi$
$752$	0	0
$753$	−1.70771	−0.0622325
$754$	0	0
$755$	−0.242457	−0.00882391
$756$	0	0
$757$	−35.7725	−1.30017	−0.650086	−	0.759860i	$-0.725267\pi$
$757$	−35.7725	−1.30017	−0.650086	+	0.759860i	$0.725267\pi$
$758$	0	0
$759$	−0.00139744	−5.07240e−5 0
$760$	0	0
$761$	−22.4566	−0.814049	−0.407025	−	0.913417i	$-0.633434\pi$
$761$	−22.4566	−0.814049	−0.407025	+	0.913417i	$0.633434\pi$
$762$	0	0
$763$	84.6359	3.06403
$764$	0	0
$765$	−0.171303	−0.00619347
$766$	0	0
$767$	−42.3876	−1.53053
$768$	0	0
$769$	25.3001	0.912346	0.456173	−	0.889891i	$-0.349220\pi$
$769$	25.3001	0.912346	0.456173	+	0.889891i	$0.349220\pi$
$770$	0	0
$771$	−29.4783	−1.06163
$772$	0	0
$773$	−33.4543	−1.20327	−0.601634	−	0.798772i	$-0.705483\pi$
$773$	−33.4543	−1.20327	−0.601634	+	0.798772i	$0.705483\pi$
$774$	0	0
$775$	1.49062	0.0535445
$776$	0	0
$777$	−39.0157	−1.39968
$778$	0	0
$779$	11.4057	0.408653
$780$	0	0
$781$	−0.00631252	−0.000225880 0
$782$	0	0
$783$	5.32754	0.190391
$784$	0	0
$785$	1.70682	0.0609190
$786$	0	0
$787$	19.4637	0.693807	0.346903	−	0.937901i	$-0.387233\pi$
$787$	19.4637	0.693807	0.346903	+	0.937901i	$0.387233\pi$
$788$	0	0
$789$	23.3724	0.832078
$790$	0	0
$791$	−88.3263	−3.14052
$792$	0	0
$793$	−39.0623	−1.38714
$794$	0	0
$795$	2.49488	0.0884844
$796$	0	0
$797$	−37.0576	−1.31265	−0.656325	−	0.754479i	$-0.727890\pi$
$797$	−37.0576	−1.31265	−0.656325	+	0.754479i	$0.727890\pi$
$798$	0	0
$799$	−5.00684	−0.177129
$800$	0	0
$801$	6.30476	0.222768
$802$	0	0
$803$	0.00397347	0.000140221 0
$804$	0	0
$805$	−3.29554	−0.116152
$806$	0	0
$807$	−16.0299	−0.564278
$808$	0	0
$809$	−17.5402	−0.616682	−0.308341	−	0.951276i	$-0.599774\pi$
$809$	−17.5402	−0.616682	−0.308341	+	0.951276i	$0.599774\pi$
$810$	0	0
$811$	−48.2958	−1.69590	−0.847948	−	0.530079i	$-0.822162\pi$
$811$	−48.2958	−1.69590	−0.847948	+	0.530079i	$0.822162\pi$
$812$	0	0
$813$	−19.2935	−0.676654
$814$	0	0
$815$	3.79583	0.132962
$816$	0	0
$817$	9.08439	0.317823
$818$	0	0
$819$	−21.2471	−0.742434
$820$	0	0
$821$	32.0021	1.11688	0.558440	−	0.829545i	$-0.311400\pi$
$821$	32.0021	1.11688	0.558440	+	0.829545i	$0.311400\pi$
$822$	0	0
$823$	−21.7340	−0.757601	−0.378801	−	0.925478i	$-0.623663\pi$
$823$	−21.7340	−0.757601	−0.378801	+	0.925478i	$0.623663\pi$
$824$	0	0
$825$	−0.00229937	−8.00537e−5 0
$826$	0	0
$827$	9.18630	0.319439	0.159719	−	0.987162i	$-0.448941\pi$
$827$	9.18630	0.319439	0.159719	+	0.987162i	$0.448941\pi$
$828$	0	0
$829$	−2.13144	−0.0740278	−0.0370139	−	0.999315i	$-0.511785\pi$
$829$	−2.13144	−0.0740278	−0.0370139	+	0.999315i	$0.511785\pi$
$830$	0	0
$831$	7.43257	0.257833
$832$	0	0
$833$	−12.0135	−0.416244
$834$	0	0
$835$	−2.08591	−0.0721858
$836$	0	0
$837$	−0.301267	−0.0104133
$838$	0	0
$839$	−23.6135	−0.815228	−0.407614	−	0.913154i	$-0.633639\pi$
$839$	−23.6135	−0.815228	−0.407614	+	0.913154i	$0.633639\pi$
$840$	0	0
$841$	−0.617319	−0.0212868
$842$	0	0
$843$	−9.44918	−0.325447
$844$	0	0
$845$	1.51018	0.0519516
$846$	0	0
$847$	−52.7763	−1.81342
$848$	0	0
$849$	−4.94927	−0.169858
$850$	0	0
$851$	24.4530	0.838239
$852$	0	0
$853$	−9.03236	−0.309262	−0.154631	−	0.987972i	$-0.549419\pi$
$853$	−9.03236	−0.309262	−0.154631	+	0.987972i	$0.549419\pi$
$854$	0	0
$855$	0.897116	0.0306807
$856$	0	0
$857$	−35.0418	−1.19701	−0.598503	−	0.801120i	$-0.704238\pi$
$857$	−35.0418	−1.19701	−0.598503	+	0.801120i	$0.704238\pi$
$858$	0	0
$859$	51.6274	1.76150	0.880752	−	0.473578i	$-0.157038\pi$
$859$	51.6274	1.76150	0.880752	+	0.473578i	$0.157038\pi$
$860$	0	0
$861$	13.9335	0.474853
$862$	0	0
$863$	−26.6883	−0.908482	−0.454241	−	0.890879i	$-0.650089\pi$
$863$	−26.6883	−0.908482	−0.454241	+	0.890879i	$0.650089\pi$
$864$	0	0
$865$	2.85507	0.0970752
$866$	0	0
$867$	−16.4376	−0.558250
$868$	0	0
$869$	0.00675578	0.000229174 0
$870$	0	0
$871$	−35.4972	−1.20278
$872$	0	0
$873$	12.5079	0.423330
$874$	0	0
$875$	−10.9022	−0.368562
$876$	0	0
$877$	−4.45459	−0.150421	−0.0752104	−	0.997168i	$-0.523963\pi$
$877$	−4.45459	−0.150421	−0.0752104	+	0.997168i	$0.523963\pi$
$878$	0	0
$879$	−21.7068	−0.732151
$880$	0	0
$881$	14.5224	0.489271	0.244635	−	0.969615i	$-0.421332\pi$
$881$	14.5224	0.489271	0.244635	+	0.969615i	$0.421332\pi$
$882$	0	0
$883$	46.3931	1.56125	0.780626	−	0.624999i	$-0.214900\pi$
$883$	46.3931	1.56125	0.780626	+	0.624999i	$0.214900\pi$
$884$	0	0
$885$	2.18638	0.0734943
$886$	0	0
$887$	22.2732	0.747860	0.373930	−	0.927457i	$-0.378010\pi$
$887$	22.2732	0.747860	0.373930	+	0.927457i	$0.378010\pi$
$888$	0	0
$889$	14.8716	0.498777
$890$	0	0
$891$	0.000464723 0	1.55688e−5 0
$892$	0	0
$893$	26.2209	0.877449
$894$	0	0
$895$	−4.97070	−0.166152
$896$	0	0
$897$	13.3166	0.444629
$898$	0	0
$899$	−1.60501	−0.0535302
$900$	0	0
$901$	−8.19098	−0.272881
$902$	0	0
$903$	11.0977	0.369309
$904$	0	0
$905$	0.479335	0.0159336
$906$	0	0
$907$	1.92022	0.0637599	0.0318799	−	0.999492i	$-0.489851\pi$
$907$	1.92022	0.0637599	0.0318799	+	0.999492i	$0.489851\pi$
$908$	0	0
$909$	1.59996	0.0530675
$910$	0	0
$911$	−0.640312	−0.0212145	−0.0106072	−	0.999944i	$-0.503376\pi$
$911$	−0.640312	−0.0212145	−0.0106072	+	0.999944i	$0.503376\pi$
$912$	0	0
$913$	−0.00320862	−0.000106190 0
$914$	0	0
$915$	2.01485	0.0666090
$916$	0	0
$917$	−27.6449	−0.912915
$918$	0	0
$919$	−14.1102	−0.465454	−0.232727	−	0.972542i	$-0.574765\pi$
$919$	−14.1102	−0.465454	−0.232727	+	0.972542i	$0.574765\pi$
$920$	0	0
$921$	−30.0709	−0.990870
$922$	0	0
$923$	60.1537	1.97998
$924$	0	0
$925$	40.2353	1.32293
$926$	0	0
$927$	−5.41659	−0.177904
$928$	0	0
$929$	−40.9706	−1.34420	−0.672101	−	0.740460i	$-0.734608\pi$
$929$	−40.9706	−1.34420	−0.672101	+	0.740460i	$0.734608\pi$
$930$	0	0
$931$	62.9150	2.06196
$932$	0	0
$933$	−23.1033	−0.756367
$934$	0	0
$935$	−7.96084e−5 0	−2.60347e−6 0
$936$	0	0
$937$	−18.7553	−0.612709	−0.306355	−	0.951917i	$-0.599109\pi$
$937$	−18.7553	−0.612709	−0.306355	+	0.951917i	$0.599109\pi$
$938$	0	0
$939$	14.6283	0.477375
$940$	0	0
$941$	22.3619	0.728978	0.364489	−	0.931208i	$-0.381244\pi$
$941$	22.3619	0.728978	0.364489	+	0.931208i	$0.381244\pi$
$942$	0	0
$943$	−8.73282	−0.284380
$944$	0	0
$945$	1.09594	0.0356509
$946$	0	0
$947$	−13.9203	−0.452350	−0.226175	−	0.974087i	$-0.572622\pi$
$947$	−13.9203	−0.452350	−0.226175	+	0.974087i	$0.572622\pi$
$948$	0	0
$949$	−37.8643	−1.22913
$950$	0	0
$951$	−11.8156	−0.383148
$952$	0	0
$953$	26.5113	0.858784	0.429392	−	0.903118i	$-0.358728\pi$
$953$	26.5113	0.858784	0.429392	+	0.903118i	$0.358728\pi$
$954$	0	0
$955$	−5.54704	−0.179498
$956$	0	0
$957$	0.00247583	8.00322e−5 0
$958$	0	0
$959$	−39.8009	−1.28524
$960$	0	0
$961$	−30.9092	−0.997072
$962$	0	0
$963$	4.31872	0.139169
$964$	0	0
$965$	5.08709	0.163759
$966$	0	0
$967$	−40.5273	−1.30327	−0.651635	−	0.758533i	$-0.725917\pi$
$967$	−40.5273	−1.30327	−0.651635	+	0.758533i	$0.725917\pi$
$968$	0	0
$969$	−2.94533	−0.0946177
$970$	0	0
$971$	−32.9563	−1.05762	−0.528809	−	0.848741i	$-0.677361\pi$
$971$	−32.9563	−1.05762	−0.528809	+	0.848741i	$0.677361\pi$
$972$	0	0
$973$	−4.79785	−0.153812
$974$	0	0
$975$	21.9113	0.701722
$976$	0	0
$977$	18.5137	0.592305	0.296152	−	0.955141i	$-0.404296\pi$
$977$	18.5137	0.592305	0.296152	+	0.955141i	$0.404296\pi$
$978$	0	0
$979$	0.00292997	9.36422e−5 0
$980$	0	0
$981$	17.6404	0.563214
$982$	0	0
$983$	−32.3057	−1.03039	−0.515196	−	0.857072i	$-0.672281\pi$
$983$	−32.3057	−1.03039	−0.515196	+	0.857072i	$0.672281\pi$
$984$	0	0
$985$	1.19109	0.0379514
$986$	0	0
$987$	32.0321	1.01959
$988$	0	0
$989$	−6.95548	−0.221171
$990$	0	0
$991$	−60.7734	−1.93053	−0.965266	−	0.261269i	$-0.915859\pi$
$991$	−60.7734	−1.93053	−0.965266	+	0.261269i	$0.915859\pi$
$992$	0	0
$993$	9.42852	0.299205
$994$	0	0
$995$	−0.773275	−0.0245145
$996$	0	0
$997$	−14.7345	−0.466646	−0.233323	−	0.972399i	$-0.574960\pi$
$997$	−14.7345	−0.466646	−0.233323	+	0.972399i	$0.574960\pi$
$998$	0	0
$999$	−8.13191	−0.257282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6672.2.a.bs.1.6		12
4.3	odd	2		3336.2.a.r.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3336.2.a.r.1.6	✓	12	4.3	odd	2
6672.2.a.bs.1.6		12	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:	\( N \)	\(=\)	\( 6672 = 2^{4} \cdot 3 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6672.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.228423 q^{5} +4.79785 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.228423 q^{5} +4.79785 q^{7} +1.00000 q^{9} +0.000464723 q^{11} -4.42847 q^{13} +0.228423 q^{15} -0.749938 q^{17} +3.92743 q^{19} +4.79785 q^{21} -3.00705 q^{23} -4.94782 q^{25} +1.00000 q^{27} +5.32754 q^{29} -0.301267 q^{31} +0.000464723 q^{33} +1.09594 q^{35} -8.13191 q^{37} -4.42847 q^{39} +2.90412 q^{41} +2.31306 q^{43} +0.228423 q^{45} +6.67634 q^{47} +16.0194 q^{49} -0.749938 q^{51} +10.9222 q^{53} +0.000106153 q^{55} +3.92743 q^{57} +9.57162 q^{59} +8.82072 q^{61} +4.79785 q^{63} -1.01156 q^{65} +8.01568 q^{67} -3.00705 q^{69} -13.5834 q^{71} +8.55020 q^{73} -4.94782 q^{75} +0.00222967 q^{77} +14.5372 q^{79} +1.00000 q^{81} -6.90438 q^{83} -0.171303 q^{85} +5.32754 q^{87} +6.30476 q^{89} -21.2471 q^{91} -0.301267 q^{93} +0.897116 q^{95} +12.5079 q^{97} +0.000464723 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 7 q^{5} + 2 q^{7} + 12 q^{9} + 10 q^{11} + 12 q^{13} + 7 q^{15} + 6 q^{17} + 5 q^{19} + 2 q^{21} + 8 q^{23} + 29 q^{25} + 12 q^{27} + 9 q^{29} + 10 q^{33} - 5 q^{35} + 23 q^{37} + 12 q^{39} + 19 q^{41} + 8 q^{43} + 7 q^{45} + 11 q^{47} + 22 q^{49} + 6 q^{51} + 19 q^{53} - 26 q^{55} + 5 q^{57} + 3 q^{59} + 27 q^{61} + 2 q^{63} + 12 q^{65} + 22 q^{67} + 8 q^{69} - q^{71} + 50 q^{73} + 29 q^{75} + q^{77} - 9 q^{79} + 12 q^{81} + 21 q^{83} + 35 q^{85} + 9 q^{87} + 26 q^{89} - 10 q^{91} - 30 q^{95} + 41 q^{97} + 10 q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 7 * q^5 + 2 * q^7 + 12 * q^9 + 10 * q^11 + 12 * q^13 + 7 * q^15 + 6 * q^17 + 5 * q^19 + 2 * q^21 + 8 * q^23 + 29 * q^25 + 12 * q^27 + 9 * q^29 + 10 * q^33 - 5 * q^35 + 23 * q^37 + 12 * q^39 + 19 * q^41 + 8 * q^43 + 7 * q^45 + 11 * q^47 + 22 * q^49 + 6 * q^51 + 19 * q^53 - 26 * q^55 + 5 * q^57 + 3 * q^59 + 27 * q^61 + 2 * q^63 + 12 * q^65 + 22 * q^67 + 8 * q^69 - q^71 + 50 * q^73 + 29 * q^75 + q^77 - 9 * q^79 + 12 * q^81 + 21 * q^83 + 35 * q^85 + 9 * q^87 + 26 * q^89 - 10 * q^91 - 30 * q^95 + 41 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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