LMFDB - Embedded newform 855.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,2,Mod(1,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,10,-2,0,-2,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.64575 q^{2} +5.00000 q^{4} -1.00000 q^{5} -3.64575 q^{7} -7.93725 q^{8} +2.64575 q^{10} -5.64575 q^{11} -5.64575 q^{13} +9.64575 q^{14} +11.0000 q^{16} +4.00000 q^{17} -1.00000 q^{19} -5.00000 q^{20} +14.9373 q^{22} +1.29150 q^{23} +1.00000 q^{25} +14.9373 q^{26} -18.2288 q^{28} +6.93725 q^{29} +6.00000 q^{31} -13.2288 q^{32} -10.5830 q^{34} +3.64575 q^{35} -1.64575 q^{37} +2.64575 q^{38} +7.93725 q^{40} +4.35425 q^{41} +0.354249 q^{43} -28.2288 q^{44} -3.41699 q^{46} -9.29150 q^{47} +6.29150 q^{49} -2.64575 q^{50} -28.2288 q^{52} -0.708497 q^{53} +5.64575 q^{55} +28.9373 q^{56} -18.3542 q^{58} -0.708497 q^{59} -0.708497 q^{61} -15.8745 q^{62} +13.0000 q^{64} +5.64575 q^{65} +14.5830 q^{67} +20.0000 q^{68} -9.64575 q^{70} +3.29150 q^{71} +10.0000 q^{73} +4.35425 q^{74} -5.00000 q^{76} +20.5830 q^{77} -14.5830 q^{79} -11.0000 q^{80} -11.5203 q^{82} -6.00000 q^{83} -4.00000 q^{85} -0.937254 q^{86} +44.8118 q^{88} +1.06275 q^{89} +20.5830 q^{91} +6.45751 q^{92} +24.5830 q^{94} +1.00000 q^{95} +12.9373 q^{97} -16.6458 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 6 q^{13} + 14 q^{14} + 22 q^{16} + 8 q^{17} - 2 q^{19} - 10 q^{20} + 14 q^{22} - 8 q^{23} + 2 q^{25} + 14 q^{26} - 10 q^{28} - 2 q^{29} + 12 q^{31} + 2 q^{35}+ \cdots - 28 q^{98}+O(q^{100}) $ 2 * q + 10 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 6 * q^13 + 14 * q^14 + 22 * q^16 + 8 * q^17 - 2 * q^19 - 10 * q^20 + 14 * q^22 - 8 * q^23 + 2 * q^25 + 14 * q^26 - 10 * q^28 - 2 * q^29 + 12 * q^31 + 2 * q^35 + 2 * q^37 + 14 * q^41 + 6 * q^43 - 30 * q^44 - 28 * q^46 - 8 * q^47 + 2 * q^49 - 30 * q^52 - 12 * q^53 + 6 * q^55 + 42 * q^56 - 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 + 6 * q^65 + 8 * q^67 + 40 * q^68 - 14 * q^70 - 4 * q^71 + 20 * q^73 + 14 * q^74 - 10 * q^76 + 20 * q^77 - 8 * q^79 - 22 * q^80 + 14 * q^82 - 12 * q^83 - 8 * q^85 + 14 * q^86 + 42 * q^88 + 18 * q^89 + 20 * q^91 - 40 * q^92 + 28 * q^94 + 2 * q^95 + 10 * q^97 - 28 * q^98

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 $

$2$	−2.64575	−1.87083	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	−2.64575	−1.87083	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	2.50000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.64575	−1.37796	−0.688982	−	0.724778i	$-0.741942\pi$
$7$	−3.64575	−1.37796	−0.688982	+	0.724778i	$0.741942\pi$
$8$	−7.93725	−2.80624
$9$	0	0
$10$	2.64575	0.836660
$11$	−5.64575	−1.70226	−0.851129	−	0.524957i	$-0.824082\pi$
$11$	−5.64575	−1.70226	−0.851129	+	0.524957i	$0.824082\pi$
$12$	0	0
$13$	−5.64575	−1.56585	−0.782925	−	0.622116i	$-0.786273\pi$
$13$	−5.64575	−1.56585	−0.782925	+	0.622116i	$0.786273\pi$
$14$	9.64575	2.57794
$15$	0	0
$16$	11.0000	2.75000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−5.00000	−1.11803
$21$	0	0
$22$	14.9373	3.18463
$23$	1.29150	0.269297	0.134648	−	0.990893i	$-0.457009\pi$
$23$	1.29150	0.269297	0.134648	+	0.990893i	$0.457009\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	14.9373	2.92944
$27$	0	0
$28$	−18.2288	−3.44491
$29$	6.93725	1.28822	0.644108	−	0.764935i	$-0.277229\pi$
$29$	6.93725	1.28822	0.644108	+	0.764935i	$0.277229\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−13.2288	−2.33854
$33$	0	0
$34$	−10.5830	−1.81497
$35$	3.64575	0.616244
$36$	0	0
$37$	−1.64575	−0.270560	−0.135280	−	0.990807i	$-0.543193\pi$
$37$	−1.64575	−0.270560	−0.135280	+	0.990807i	$0.543193\pi$
$38$	2.64575	0.429198
$39$	0	0
$40$	7.93725	1.25499
$41$	4.35425	0.680019	0.340010	−	0.940422i	$-0.389570\pi$
$41$	4.35425	0.680019	0.340010	+	0.940422i	$0.389570\pi$
$42$	0	0
$43$	0.354249	0.0540224	0.0270112	−	0.999635i	$-0.491401\pi$
$43$	0.354249	0.0540224	0.0270112	+	0.999635i	$0.491401\pi$
$44$	−28.2288	−4.25565
$45$	0	0
$46$	−3.41699	−0.503808
$47$	−9.29150	−1.35530	−0.677652	−	0.735382i	$-0.737003\pi$
$47$	−9.29150	−1.35530	−0.677652	+	0.735382i	$0.737003\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	−2.64575	−0.374166
$51$	0	0
$52$	−28.2288	−3.91462
$53$	−0.708497	−0.0973196	−0.0486598	−	0.998815i	$-0.515495\pi$
$53$	−0.708497	−0.0973196	−0.0486598	+	0.998815i	$0.515495\pi$
$54$	0	0
$55$	5.64575	0.761273
$56$	28.9373	3.86690
$57$	0	0
$58$	−18.3542	−2.41003
$59$	−0.708497	−0.0922385	−0.0461193	−	0.998936i	$-0.514685\pi$
$59$	−0.708497	−0.0922385	−0.0461193	+	0.998936i	$0.514685\pi$
$60$	0	0
$61$	−0.708497	−0.0907138	−0.0453569	−	0.998971i	$-0.514443\pi$
$61$	−0.708497	−0.0907138	−0.0453569	+	0.998971i	$0.514443\pi$
$62$	−15.8745	−2.01606
$63$	0	0
$64$	13.0000	1.62500
$65$	5.64575	0.700269
$66$	0	0
$67$	14.5830	1.78160	0.890799	−	0.454398i	$-0.150146\pi$
$67$	14.5830	1.78160	0.890799	+	0.454398i	$0.150146\pi$
$68$	20.0000	2.42536
$69$	0	0
$70$	−9.64575	−1.15289
$71$	3.29150	0.390629	0.195315	−	0.980741i	$-0.437427\pi$
$71$	3.29150	0.390629	0.195315	+	0.980741i	$0.437427\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	4.35425	0.506171
$75$	0	0
$76$	−5.00000	−0.573539
$77$	20.5830	2.34565
$78$	0	0
$79$	−14.5830	−1.64072	−0.820358	−	0.571850i	$-0.806226\pi$
$79$	−14.5830	−1.64072	−0.820358	+	0.571850i	$0.806226\pi$
$80$	−11.0000	−1.22984
$81$	0	0
$82$	−11.5203	−1.27220
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−0.937254	−0.101067
$87$	0	0
$88$	44.8118	4.77695
$89$	1.06275	0.112651	0.0563254	−	0.998412i	$-0.482062\pi$
$89$	1.06275	0.112651	0.0563254	+	0.998412i	$0.482062\pi$
$90$	0	0
$91$	20.5830	2.15769
$92$	6.45751	0.673242
$93$	0	0
$94$	24.5830	2.53554
$95$	1.00000	0.102598
$96$	0	0
$97$	12.9373	1.31358	0.656790	−	0.754074i	$-0.271914\pi$
$97$	12.9373	1.31358	0.656790	+	0.754074i	$0.271914\pi$
$98$	−16.6458	−1.68147
$99$	0	0
$100$	5.00000	0.500000
$101$	1.29150	0.128509	0.0642547	−	0.997934i	$-0.479533\pi$
$101$	1.29150	0.128509	0.0642547	+	0.997934i	$0.479533\pi$
$102$	0	0
$103$	10.5830	1.04277	0.521387	−	0.853320i	$-0.325415\pi$
$103$	10.5830	1.04277	0.521387	+	0.853320i	$0.325415\pi$
$104$	44.8118	4.39415
$105$	0	0
$106$	1.87451	0.182068
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−5.29150	−0.506834	−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150	−0.506834	−0.253417	+	0.967357i	$0.581554\pi$
$110$	−14.9373	−1.42421
$111$	0	0
$112$	−40.1033	−3.78940
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−1.29150	−0.120433
$116$	34.6863	3.22054
$117$	0	0
$118$	1.87451	0.172562
$119$	−14.5830	−1.33682
$120$	0	0
$121$	20.8745	1.89768
$122$	1.87451	0.169710
$123$	0	0
$124$	30.0000	2.69408
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−7.93725	−0.701561
$129$	0	0
$130$	−14.9373	−1.31008
$131$	−12.2288	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$131$	−12.2288	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$132$	0	0
$133$	3.64575	0.316127
$134$	−38.5830	−3.33306
$135$	0	0
$136$	−31.7490	−2.72246
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−13.8745	−1.17682	−0.588410	−	0.808563i	$-0.700246\pi$
$139$	−13.8745	−1.17682	−0.588410	+	0.808563i	$0.700246\pi$
$140$	18.2288	1.54061
$141$	0	0
$142$	−8.70850	−0.730801
$143$	31.8745	2.66548
$144$	0	0
$145$	−6.93725	−0.576108
$146$	−26.4575	−2.18964
$147$	0	0
$148$	−8.22876	−0.676400
$149$	−0.583005	−0.0477617	−0.0238808	−	0.999715i	$-0.507602\pi$
$149$	−0.583005	−0.0477617	−0.0238808	+	0.999715i	$0.507602\pi$
$150$	0	0
$151$	12.5830	1.02399	0.511995	−	0.858988i	$-0.328907\pi$
$151$	12.5830	1.02399	0.511995	+	0.858988i	$0.328907\pi$
$152$	7.93725	0.643796
$153$	0	0
$154$	−54.4575	−4.38831
$155$	−6.00000	−0.481932
$156$	0	0
$157$	−2.70850	−0.216162	−0.108081	−	0.994142i	$-0.534471\pi$
$157$	−2.70850	−0.216162	−0.108081	+	0.994142i	$0.534471\pi$
$158$	38.5830	3.06950
$159$	0	0
$160$	13.2288	1.04583
$161$	−4.70850	−0.371082
$162$	0	0
$163$	−7.64575	−0.598861	−0.299431	−	0.954118i	$-0.596797\pi$
$163$	−7.64575	−0.598861	−0.299431	+	0.954118i	$0.596797\pi$
$164$	21.7712	1.70005
$165$	0	0
$166$	15.8745	1.23210
$167$	10.7085	0.828648	0.414324	−	0.910129i	$-0.364018\pi$
$167$	10.7085	0.828648	0.414324	+	0.910129i	$0.364018\pi$
$168$	0	0
$169$	18.8745	1.45189
$170$	10.5830	0.811679
$171$	0	0
$172$	1.77124	0.135056
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−3.64575	−0.275593
$176$	−62.1033	−4.68121
$177$	0	0
$178$	−2.81176	−0.210750
$179$	−3.29150	−0.246018	−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150	−0.246018	−0.123009	+	0.992406i	$0.539254\pi$
$180$	0	0
$181$	6.70850	0.498639	0.249319	−	0.968421i	$-0.419793\pi$
$181$	6.70850	0.498639	0.249319	+	0.968421i	$0.419793\pi$
$182$	−54.4575	−4.03666
$183$	0	0
$184$	−10.2510	−0.755713
$185$	1.64575	0.120998
$186$	0	0
$187$	−22.5830	−1.65143
$188$	−46.4575	−3.38826
$189$	0	0
$190$	−2.64575	−0.191943
$191$	20.2288	1.46370	0.731851	−	0.681465i	$-0.238657\pi$
$191$	20.2288	1.46370	0.731851	+	0.681465i	$0.238657\pi$
$192$	0	0
$193$	−6.35425	−0.457389	−0.228694	−	0.973498i	$-0.573446\pi$
$193$	−6.35425	−0.457389	−0.228694	+	0.973498i	$0.573446\pi$
$194$	−34.2288	−2.45748
$195$	0	0
$196$	31.4575	2.24697
$197$	−17.1660	−1.22303	−0.611514	−	0.791234i	$-0.709439\pi$
$197$	−17.1660	−1.22303	−0.611514	+	0.791234i	$0.709439\pi$
$198$	0	0
$199$	−3.29150	−0.233328	−0.116664	−	0.993171i	$-0.537220\pi$
$199$	−3.29150	−0.233328	−0.116664	+	0.993171i	$0.537220\pi$
$200$	−7.93725	−0.561249
$201$	0	0
$202$	−3.41699	−0.240419
$203$	−25.2915	−1.77512
$204$	0	0
$205$	−4.35425	−0.304114
$206$	−28.0000	−1.95085
$207$	0	0
$208$	−62.1033	−4.30609
$209$	5.64575	0.390525
$210$	0	0
$211$	−18.5830	−1.27931	−0.639653	−	0.768663i	$-0.720922\pi$
$211$	−18.5830	−1.27931	−0.639653	+	0.768663i	$0.720922\pi$
$212$	−3.54249	−0.243299
$213$	0	0
$214$	0	0
$215$	−0.354249	−0.0241596
$216$	0	0
$217$	−21.8745	−1.48494
$218$	14.0000	0.948200
$219$	0	0
$220$	28.2288	1.90318
$221$	−22.5830	−1.51910
$222$	0	0
$223$	−18.5830	−1.24441	−0.622205	−	0.782854i	$-0.713763\pi$
$223$	−18.5830	−1.24441	−0.622205	+	0.782854i	$0.713763\pi$
$224$	48.2288	3.22242
$225$	0	0
$226$	−10.5830	−0.703971
$227$	21.2915	1.41317	0.706583	−	0.707630i	$-0.250236\pi$
$227$	21.2915	1.41317	0.706583	+	0.707630i	$0.250236\pi$
$228$	0	0
$229$	19.2915	1.27482	0.637409	−	0.770525i	$-0.280006\pi$
$229$	19.2915	1.27482	0.637409	+	0.770525i	$0.280006\pi$
$230$	3.41699	0.225310
$231$	0	0
$232$	−55.0627	−3.61505
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	9.29150	0.606111
$236$	−3.54249	−0.230596
$237$	0	0
$238$	38.5830	2.50096
$239$	−10.3542	−0.669761	−0.334880	−	0.942261i	$-0.608696\pi$
$239$	−10.3542	−0.669761	−0.334880	+	0.942261i	$0.608696\pi$
$240$	0	0
$241$	−4.58301	−0.295217	−0.147609	−	0.989046i	$-0.547158\pi$
$241$	−4.58301	−0.295217	−0.147609	+	0.989046i	$0.547158\pi$
$242$	−55.2288	−3.55024
$243$	0	0
$244$	−3.54249	−0.226784
$245$	−6.29150	−0.401949
$246$	0	0
$247$	5.64575	0.359231
$248$	−47.6235	−3.02410
$249$	0	0
$250$	2.64575	0.167332
$251$	21.6458	1.36627	0.683134	−	0.730293i	$-0.260617\pi$
$251$	21.6458	1.36627	0.683134	+	0.730293i	$0.260617\pi$
$252$	0	0
$253$	−7.29150	−0.458413
$254$	−10.5830	−0.664037
$255$	0	0
$256$	−5.00000	−0.312500
$257$	26.5830	1.65820	0.829101	−	0.559099i	$-0.188853\pi$
$257$	26.5830	1.65820	0.829101	+	0.559099i	$0.188853\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	28.2288	1.75067
$261$	0	0
$262$	32.3542	1.99885
$263$	16.5830	1.02255	0.511276	−	0.859417i	$-0.329173\pi$
$263$	16.5830	1.02255	0.511276	+	0.859417i	$0.329173\pi$
$264$	0	0
$265$	0.708497	0.0435226
$266$	−9.64575	−0.591419
$267$	0	0
$268$	72.9150	4.45399
$269$	22.2288	1.35531	0.677656	−	0.735379i	$-0.262996\pi$
$269$	22.2288	1.35531	0.677656	+	0.735379i	$0.262996\pi$
$270$	0	0
$271$	−15.2915	−0.928893	−0.464446	−	0.885601i	$-0.653747\pi$
$271$	−15.2915	−0.928893	−0.464446	+	0.885601i	$0.653747\pi$
$272$	44.0000	2.66789
$273$	0	0
$274$	15.8745	0.959014
$275$	−5.64575	−0.340452
$276$	0	0
$277$	−20.5830	−1.23671	−0.618356	−	0.785898i	$-0.712201\pi$
$277$	−20.5830	−1.23671	−0.618356	+	0.785898i	$0.712201\pi$
$278$	36.7085	2.20163
$279$	0	0
$280$	−28.9373	−1.72933
$281$	5.77124	0.344284	0.172142	−	0.985072i	$-0.444931\pi$
$281$	5.77124	0.344284	0.172142	+	0.985072i	$0.444931\pi$
$282$	0	0
$283$	25.5203	1.51702	0.758511	−	0.651660i	$-0.225927\pi$
$283$	25.5203	1.51702	0.758511	+	0.651660i	$0.225927\pi$
$284$	16.4575	0.976574
$285$	0	0
$286$	−84.3320	−4.98666
$287$	−15.8745	−0.937043
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	18.3542	1.07780
$291$	0	0
$292$	50.0000	2.92603
$293$	−26.5830	−1.55300	−0.776498	−	0.630120i	$-0.783006\pi$
$293$	−26.5830	−1.55300	−0.776498	+	0.630120i	$0.783006\pi$
$294$	0	0
$295$	0.708497	0.0412503
$296$	13.0627	0.759257
$297$	0	0
$298$	1.54249	0.0893539
$299$	−7.29150	−0.421678
$300$	0	0
$301$	−1.29150	−0.0744410
$302$	−33.2915	−1.91571
$303$	0	0
$304$	−11.0000	−0.630893
$305$	0.708497	0.0405684
$306$	0	0
$307$	28.4575	1.62416	0.812078	−	0.583549i	$-0.198336\pi$
$307$	28.4575	1.62416	0.812078	+	0.583549i	$0.198336\pi$
$308$	102.915	5.86413
$309$	0	0
$310$	15.8745	0.901611
$311$	−7.06275	−0.400492	−0.200246	−	0.979746i	$-0.564174\pi$
$311$	−7.06275	−0.400492	−0.200246	+	0.979746i	$0.564174\pi$
$312$	0	0
$313$	6.70850	0.379187	0.189593	−	0.981863i	$-0.439283\pi$
$313$	6.70850	0.379187	0.189593	+	0.981863i	$0.439283\pi$
$314$	7.16601	0.404401
$315$	0	0
$316$	−72.9150	−4.10179
$317$	32.4575	1.82300	0.911498	−	0.411305i	$-0.134927\pi$
$317$	32.4575	1.82300	0.911498	+	0.411305i	$0.134927\pi$
$318$	0	0
$319$	−39.1660	−2.19288
$320$	−13.0000	−0.726722
$321$	0	0
$322$	12.4575	0.694230
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−5.64575	−0.313170
$326$	20.2288	1.12037
$327$	0	0
$328$	−34.5608	−1.90830
$329$	33.8745	1.86756
$330$	0	0
$331$	32.5830	1.79092	0.895462	−	0.445138i	$-0.146845\pi$
$331$	32.5830	1.79092	0.895462	+	0.445138i	$0.146845\pi$
$332$	−30.0000	−1.64646
$333$	0	0
$334$	−28.3320	−1.55026
$335$	−14.5830	−0.796755
$336$	0	0
$337$	−0.937254	−0.0510555	−0.0255277	−	0.999674i	$-0.508127\pi$
$337$	−0.937254	−0.0510555	−0.0255277	+	0.999674i	$0.508127\pi$
$338$	−49.9373	−2.71623
$339$	0	0
$340$	−20.0000	−1.08465
$341$	−33.8745	−1.83441
$342$	0	0
$343$	2.58301	0.139469
$344$	−2.81176	−0.151600
$345$	0	0
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	−19.1660	−1.02593	−0.512967	−	0.858409i	$-0.671454\pi$
$349$	−19.1660	−1.02593	−0.512967	+	0.858409i	$0.671454\pi$
$350$	9.64575	0.515587
$351$	0	0
$352$	74.6863	3.98079
$353$	20.5830	1.09552	0.547761	−	0.836635i	$-0.315480\pi$
$353$	20.5830	1.09552	0.547761	+	0.836635i	$0.315480\pi$
$354$	0	0
$355$	−3.29150	−0.174695
$356$	5.31373	0.281627
$357$	0	0
$358$	8.70850	0.460258
$359$	−22.1033	−1.16657	−0.583283	−	0.812269i	$-0.698232\pi$
$359$	−22.1033	−1.16657	−0.583283	+	0.812269i	$0.698232\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.7490	−0.932868
$363$	0	0
$364$	102.915	5.39421
$365$	−10.0000	−0.523424
$366$	0	0
$367$	−3.64575	−0.190307	−0.0951533	−	0.995463i	$-0.530334\pi$
$367$	−3.64575	−0.190307	−0.0951533	+	0.995463i	$0.530334\pi$
$368$	14.2065	0.740567
$369$	0	0
$370$	−4.35425	−0.226367
$371$	2.58301	0.134103
$372$	0	0
$373$	20.9373	1.08409	0.542045	−	0.840349i	$-0.317650\pi$
$373$	20.9373	1.08409	0.542045	+	0.840349i	$0.317650\pi$
$374$	59.7490	3.08955
$375$	0	0
$376$	73.7490	3.80332
$377$	−39.1660	−2.01715
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	5.00000	0.256495
$381$	0	0
$382$	−53.5203	−2.73833
$383$	−2.58301	−0.131985	−0.0659927	−	0.997820i	$-0.521021\pi$
$383$	−2.58301	−0.131985	−0.0659927	+	0.997820i	$0.521021\pi$
$384$	0	0
$385$	−20.5830	−1.04901
$386$	16.8118	0.855696
$387$	0	0
$388$	64.6863	3.28395
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	5.16601	0.261256
$392$	−49.9373	−2.52221
$393$	0	0
$394$	45.4170	2.28808
$395$	14.5830	0.733751
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	8.70850	0.436518
$399$	0	0
$400$	11.0000	0.550000
$401$	−10.9373	−0.546180	−0.273090	−	0.961988i	$-0.588046\pi$
$401$	−10.9373	−0.546180	−0.273090	+	0.961988i	$0.588046\pi$
$402$	0	0
$403$	−33.8745	−1.68741
$404$	6.45751	0.321273
$405$	0	0
$406$	66.9150	3.32094
$407$	9.29150	0.460563
$408$	0	0
$409$	−17.2915	−0.855010	−0.427505	−	0.904013i	$-0.640607\pi$
$409$	−17.2915	−0.855010	−0.427505	+	0.904013i	$0.640607\pi$
$410$	11.5203	0.568945
$411$	0	0
$412$	52.9150	2.60694
$413$	2.58301	0.127101
$414$	0	0
$415$	6.00000	0.294528
$416$	74.6863	3.66180
$417$	0	0
$418$	−14.9373	−0.730605
$419$	−23.0627	−1.12669	−0.563344	−	0.826222i	$-0.690486\pi$
$419$	−23.0627	−1.12669	−0.563344	+	0.826222i	$0.690486\pi$
$420$	0	0
$421$	7.41699	0.361482	0.180741	−	0.983531i	$-0.442150\pi$
$421$	7.41699	0.361482	0.180741	+	0.983531i	$0.442150\pi$
$422$	49.1660	2.39336
$423$	0	0
$424$	5.62352	0.273102
$425$	4.00000	0.194029
$426$	0	0
$427$	2.58301	0.125000
$428$	0	0
$429$	0	0
$430$	0.937254	0.0451984
$431$	7.29150	0.351219	0.175610	−	0.984460i	$-0.443810\pi$
$431$	7.29150	0.351219	0.175610	+	0.984460i	$0.443810\pi$
$432$	0	0
$433$	−22.3542	−1.07428	−0.537138	−	0.843494i	$-0.680495\pi$
$433$	−22.3542	−1.07428	−0.537138	+	0.843494i	$0.680495\pi$
$434$	57.8745	2.77807
$435$	0	0
$436$	−26.4575	−1.26709
$437$	−1.29150	−0.0617809
$438$	0	0
$439$	−22.5830	−1.07783	−0.538914	−	0.842361i	$-0.681165\pi$
$439$	−22.5830	−1.07783	−0.538914	+	0.842361i	$0.681165\pi$
$440$	−44.8118	−2.13632
$441$	0	0
$442$	59.7490	2.84197
$443$	−37.2915	−1.77177	−0.885886	−	0.463902i	$-0.846449\pi$
$443$	−37.2915	−1.77177	−0.885886	+	0.463902i	$0.846449\pi$
$444$	0	0
$445$	−1.06275	−0.0503790
$446$	49.1660	2.32808
$447$	0	0
$448$	−47.3948	−2.23919
$449$	9.77124	0.461133	0.230567	−	0.973057i	$-0.425942\pi$
$449$	9.77124	0.461133	0.230567	+	0.973057i	$0.425942\pi$
$450$	0	0
$451$	−24.5830	−1.15757
$452$	20.0000	0.940721
$453$	0	0
$454$	−56.3320	−2.64379
$455$	−20.5830	−0.964946
$456$	0	0
$457$	31.8745	1.49103	0.745513	−	0.666491i	$-0.232204\pi$
$457$	31.8745	1.49103	0.745513	+	0.666491i	$0.232204\pi$
$458$	−51.0405	−2.38497
$459$	0	0
$460$	−6.45751	−0.301083
$461$	−25.7490	−1.19925	−0.599626	−	0.800281i	$-0.704684\pi$
$461$	−25.7490	−1.19925	−0.599626	+	0.800281i	$0.704684\pi$
$462$	0	0
$463$	1.52026	0.0706524	0.0353262	−	0.999376i	$-0.488753\pi$
$463$	1.52026	0.0706524	0.0353262	+	0.999376i	$0.488753\pi$
$464$	76.3098	3.54259
$465$	0	0
$466$	−47.6235	−2.20612
$467$	11.8745	0.549487	0.274743	−	0.961518i	$-0.411407\pi$
$467$	11.8745	0.549487	0.274743	+	0.961518i	$0.411407\pi$
$468$	0	0
$469$	−53.1660	−2.45498
$470$	−24.5830	−1.13393
$471$	0	0
$472$	5.62352	0.258844
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−72.9150	−3.34205
$477$	0	0
$478$	27.3948	1.25301
$479$	−9.64575	−0.440726	−0.220363	−	0.975418i	$-0.570724\pi$
$479$	−9.64575	−0.440726	−0.220363	+	0.975418i	$0.570724\pi$
$480$	0	0
$481$	9.29150	0.423656
$482$	12.1255	0.552301
$483$	0	0
$484$	104.373	4.74421
$485$	−12.9373	−0.587450
$486$	0	0
$487$	−13.8745	−0.628714	−0.314357	−	0.949305i	$-0.601789\pi$
$487$	−13.8745	−0.628714	−0.314357	+	0.949305i	$0.601789\pi$
$488$	5.62352	0.254565
$489$	0	0
$490$	16.6458	0.751978
$491$	32.2288	1.45446	0.727232	−	0.686392i	$-0.240807\pi$
$491$	32.2288	1.45446	0.727232	+	0.686392i	$0.240807\pi$
$492$	0	0
$493$	27.7490	1.24975
$494$	−14.9373	−0.672059
$495$	0	0
$496$	66.0000	2.96349
$497$	−12.0000	−0.538274
$498$	0	0
$499$	43.0405	1.92676	0.963379	−	0.268143i	$-0.0864100\pi$
$499$	43.0405	1.92676	0.963379	+	0.268143i	$0.0864100\pi$
$500$	−5.00000	−0.223607
$501$	0	0
$502$	−57.2693	−2.55605
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−1.29150	−0.0574711
$506$	19.2915	0.857612
$507$	0	0
$508$	20.0000	0.887357
$509$	−24.1033	−1.06836	−0.534179	−	0.845371i	$-0.679379\pi$
$509$	−24.1033	−1.06836	−0.534179	+	0.845371i	$0.679379\pi$
$510$	0	0
$511$	−36.4575	−1.61279
$512$	29.1033	1.28619
$513$	0	0
$514$	−70.3320	−3.10221
$515$	−10.5830	−0.466343
$516$	0	0
$517$	52.4575	2.30708
$518$	−15.8745	−0.697486
$519$	0	0
$520$	−44.8118	−1.96513
$521$	36.1033	1.58171	0.790856	−	0.612002i	$-0.209635\pi$
$521$	36.1033	1.58171	0.790856	+	0.612002i	$0.209635\pi$
$522$	0	0
$523$	−12.7085	−0.555704	−0.277852	−	0.960624i	$-0.589622\pi$
$523$	−12.7085	−0.555704	−0.277852	+	0.960624i	$0.589622\pi$
$524$	−61.1438	−2.67108
$525$	0	0
$526$	−43.8745	−1.91302
$527$	24.0000	1.04546
$528$	0	0
$529$	−21.3320	−0.927479
$530$	−1.87451	−0.0814234
$531$	0	0
$532$	18.2288	0.790317
$533$	−24.5830	−1.06481
$534$	0	0
$535$	0	0
$536$	−115.749	−4.99960
$537$	0	0
$538$	−58.8118	−2.53556
$539$	−35.5203	−1.52997
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	40.4575	1.73780
$543$	0	0
$544$	−52.9150	−2.26871
$545$	5.29150	0.226663
$546$	0	0
$547$	−29.8745	−1.27734	−0.638671	−	0.769480i	$-0.720515\pi$
$547$	−29.8745	−1.27734	−0.638671	+	0.769480i	$0.720515\pi$
$548$	−30.0000	−1.28154
$549$	0	0
$550$	14.9373	0.636927
$551$	−6.93725	−0.295537
$552$	0	0
$553$	53.1660	2.26085
$554$	54.4575	2.31368
$555$	0	0
$556$	−69.3725	−2.94205
$557$	32.5830	1.38059	0.690293	−	0.723530i	$-0.257482\pi$
$557$	32.5830	1.38059	0.690293	+	0.723530i	$0.257482\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	40.1033	1.69467
$561$	0	0
$562$	−15.2693	−0.644095
$563$	−39.8745	−1.68051	−0.840255	−	0.542191i	$-0.817595\pi$
$563$	−39.8745	−1.68051	−0.840255	+	0.542191i	$0.817595\pi$
$564$	0	0
$565$	−4.00000	−0.168281
$566$	−67.5203	−2.83809
$567$	0	0
$568$	−26.1255	−1.09620
$569$	22.9373	0.961580	0.480790	−	0.876836i	$-0.340350\pi$
$569$	22.9373	0.961580	0.480790	+	0.876836i	$0.340350\pi$
$570$	0	0
$571$	27.2915	1.14211	0.571057	−	0.820910i	$-0.306534\pi$
$571$	27.2915	1.14211	0.571057	+	0.820910i	$0.306534\pi$
$572$	159.373	6.66370
$573$	0	0
$574$	42.0000	1.75305
$575$	1.29150	0.0538594
$576$	0	0
$577$	−2.70850	−0.112756	−0.0563781	−	0.998409i	$-0.517955\pi$
$577$	−2.70850	−0.112756	−0.0563781	+	0.998409i	$0.517955\pi$
$578$	2.64575	0.110049
$579$	0	0
$580$	−34.6863	−1.44027
$581$	21.8745	0.907508
$582$	0	0
$583$	4.00000	0.165663
$584$	−79.3725	−3.28446
$585$	0	0
$586$	70.3320	2.90539
$587$	22.7085	0.937280	0.468640	−	0.883389i	$-0.344744\pi$
$587$	22.7085	0.937280	0.468640	+	0.883389i	$0.344744\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	−1.87451	−0.0771723
$591$	0	0
$592$	−18.1033	−0.744040
$593$	−24.5830	−1.00950	−0.504752	−	0.863265i	$-0.668416\pi$
$593$	−24.5830	−1.00950	−0.504752	+	0.863265i	$0.668416\pi$
$594$	0	0
$595$	14.5830	0.597845
$596$	−2.91503	−0.119404
$597$	0	0
$598$	19.2915	0.788888
$599$	−30.5830	−1.24959	−0.624794	−	0.780790i	$-0.714817\pi$
$599$	−30.5830	−1.24959	−0.624794	+	0.780790i	$0.714817\pi$
$600$	0	0
$601$	33.2915	1.35799	0.678994	−	0.734143i	$-0.262416\pi$
$601$	33.2915	1.35799	0.678994	+	0.734143i	$0.262416\pi$
$602$	3.41699	0.139266
$603$	0	0
$604$	62.9150	2.55998
$605$	−20.8745	−0.848669
$606$	0	0
$607$	31.0405	1.25990	0.629948	−	0.776637i	$-0.283076\pi$
$607$	31.0405	1.25990	0.629948	+	0.776637i	$0.283076\pi$
$608$	13.2288	0.536497
$609$	0	0
$610$	−1.87451	−0.0758966
$611$	52.4575	2.12220
$612$	0	0
$613$	22.4575	0.907050	0.453525	−	0.891243i	$-0.350166\pi$
$613$	22.4575	0.907050	0.453525	+	0.891243i	$0.350166\pi$
$614$	−75.2915	−3.03852
$615$	0	0
$616$	−163.373	−6.58247
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	12.7085	0.510798	0.255399	−	0.966836i	$-0.417793\pi$
$619$	12.7085	0.510798	0.255399	+	0.966836i	$0.417793\pi$
$620$	−30.0000	−1.20483
$621$	0	0
$622$	18.6863	0.749251
$623$	−3.87451	−0.155229
$624$	0	0
$625$	1.00000	0.0400000
$626$	−17.7490	−0.709393
$627$	0	0
$628$	−13.5425	−0.540404
$629$	−6.58301	−0.262482
$630$	0	0
$631$	6.58301	0.262065	0.131033	−	0.991378i	$-0.458171\pi$
$631$	6.58301	0.262065	0.131033	+	0.991378i	$0.458171\pi$
$632$	115.749	4.60425
$633$	0	0
$634$	−85.8745	−3.41051
$635$	−4.00000	−0.158735
$636$	0	0
$637$	−35.5203	−1.40736
$638$	103.624	4.10249
$639$	0	0
$640$	7.93725	0.313748
$641$	1.06275	0.0419759	0.0209880	−	0.999780i	$-0.493319\pi$
$641$	1.06275	0.0419759	0.0209880	+	0.999780i	$0.493319\pi$
$642$	0	0
$643$	−8.35425	−0.329459	−0.164730	−	0.986339i	$-0.552675\pi$
$643$	−8.35425	−0.329459	−0.164730	+	0.986339i	$0.552675\pi$
$644$	−23.5425	−0.927704
$645$	0	0
$646$	10.5830	0.416383
$647$	24.5830	0.966458	0.483229	−	0.875494i	$-0.339464\pi$
$647$	24.5830	0.966458	0.483229	+	0.875494i	$0.339464\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	14.9373	0.585887
$651$	0	0
$652$	−38.2288	−1.49715
$653$	−30.5830	−1.19681	−0.598403	−	0.801195i	$-0.704198\pi$
$653$	−30.5830	−1.19681	−0.598403	+	0.801195i	$0.704198\pi$
$654$	0	0
$655$	12.2288	0.477817
$656$	47.8967	1.87005
$657$	0	0
$658$	−89.6235	−3.49389
$659$	−46.5830	−1.81462	−0.907308	−	0.420466i	$-0.861866\pi$
$659$	−46.5830	−1.81462	−0.907308	+	0.420466i	$0.861866\pi$
$660$	0	0
$661$	−9.29150	−0.361398	−0.180699	−	0.983538i	$-0.557836\pi$
$661$	−9.29150	−0.361398	−0.180699	+	0.983538i	$0.557836\pi$
$662$	−86.2065	−3.35051
$663$	0	0
$664$	47.6235	1.84815
$665$	−3.64575	−0.141376
$666$	0	0
$667$	8.95948	0.346913
$668$	53.5425	2.07162
$669$	0	0
$670$	38.5830	1.49059
$671$	4.00000	0.154418
$672$	0	0
$673$	−28.9373	−1.11545	−0.557725	−	0.830026i	$-0.688325\pi$
$673$	−28.9373	−1.11545	−0.557725	+	0.830026i	$0.688325\pi$
$674$	2.47974	0.0955160
$675$	0	0
$676$	94.3725	3.62971
$677$	12.4575	0.478781	0.239391	−	0.970923i	$-0.423052\pi$
$677$	12.4575	0.478781	0.239391	+	0.970923i	$0.423052\pi$
$678$	0	0
$679$	−47.1660	−1.81007
$680$	31.7490	1.21752
$681$	0	0
$682$	89.6235	3.43186
$683$	43.7490	1.67401	0.837005	−	0.547196i	$-0.184305\pi$
$683$	43.7490	1.67401	0.837005	+	0.547196i	$0.184305\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−6.83399	−0.260923
$687$	0	0
$688$	3.89674	0.148562
$689$	4.00000	0.152388
$690$	0	0
$691$	31.0405	1.18084	0.590418	−	0.807097i	$-0.298963\pi$
$691$	31.0405	1.18084	0.590418	+	0.807097i	$0.298963\pi$
$692$	0	0
$693$	0	0
$694$	68.7895	2.61122
$695$	13.8745	0.526290
$696$	0	0
$697$	17.4170	0.659716
$698$	50.7085	1.91934
$699$	0	0
$700$	−18.2288	−0.688982
$701$	4.58301	0.173098	0.0865489	−	0.996248i	$-0.472416\pi$
$701$	4.58301	0.173098	0.0865489	+	0.996248i	$0.472416\pi$
$702$	0	0
$703$	1.64575	0.0620707
$704$	−73.3948	−2.76617
$705$	0	0
$706$	−54.4575	−2.04954
$707$	−4.70850	−0.177081
$708$	0	0
$709$	−15.2915	−0.574284	−0.287142	−	0.957888i	$-0.592705\pi$
$709$	−15.2915	−0.574284	−0.287142	+	0.957888i	$0.592705\pi$
$710$	8.70850	0.326824
$711$	0	0
$712$	−8.43529	−0.316126
$713$	7.74902	0.290203
$714$	0	0
$715$	−31.8745	−1.19204
$716$	−16.4575	−0.615046
$717$	0	0
$718$	58.4797	2.18244
$719$	14.8118	0.552386	0.276193	−	0.961102i	$-0.410927\pi$
$719$	14.8118	0.552386	0.276193	+	0.961102i	$0.410927\pi$
$720$	0	0
$721$	−38.5830	−1.43691
$722$	−2.64575	−0.0984647
$723$	0	0
$724$	33.5425	1.24660
$725$	6.93725	0.257643
$726$	0	0
$727$	−44.1033	−1.63570	−0.817850	−	0.575432i	$-0.804834\pi$
$727$	−44.1033	−1.63570	−0.817850	+	0.575432i	$0.804834\pi$
$728$	−163.373	−6.05499
$729$	0	0
$730$	26.4575	0.979236
$731$	1.41699	0.0524094
$732$	0	0
$733$	28.5830	1.05574	0.527869	−	0.849326i	$-0.322991\pi$
$733$	28.5830	1.05574	0.527869	+	0.849326i	$0.322991\pi$
$734$	9.64575	0.356031
$735$	0	0
$736$	−17.0850	−0.629760
$737$	−82.3320	−3.03274
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	8.22876	0.302495
$741$	0	0
$742$	−6.83399	−0.250884
$743$	−10.5830	−0.388253	−0.194126	−	0.980977i	$-0.562187\pi$
$743$	−10.5830	−0.388253	−0.194126	+	0.980977i	$0.562187\pi$
$744$	0	0
$745$	0.583005	0.0213597
$746$	−55.3948	−2.02815
$747$	0	0
$748$	−112.915	−4.12858
$749$	0	0
$750$	0	0
$751$	−23.4170	−0.854498	−0.427249	−	0.904134i	$-0.640517\pi$
$751$	−23.4170	−0.854498	−0.427249	+	0.904134i	$0.640517\pi$
$752$	−102.207	−3.72709
$753$	0	0
$754$	103.624	3.77375
$755$	−12.5830	−0.457942
$756$	0	0
$757$	7.87451	0.286204	0.143102	−	0.989708i	$-0.454292\pi$
$757$	7.87451	0.286204	0.143102	+	0.989708i	$0.454292\pi$
$758$	−26.4575	−0.960980
$759$	0	0
$760$	−7.93725	−0.287914
$761$	37.7490	1.36840	0.684200	−	0.729294i	$-0.260151\pi$
$761$	37.7490	1.36840	0.684200	+	0.729294i	$0.260151\pi$
$762$	0	0
$763$	19.2915	0.698399
$764$	101.144	3.65925
$765$	0	0
$766$	6.83399	0.246922
$767$	4.00000	0.144432
$768$	0	0
$769$	45.7490	1.64975	0.824876	−	0.565314i	$-0.191245\pi$
$769$	45.7490	1.64975	0.824876	+	0.565314i	$0.191245\pi$
$770$	54.4575	1.96251
$771$	0	0
$772$	−31.7712	−1.14347
$773$	19.2915	0.693867	0.346934	−	0.937890i	$-0.387223\pi$
$773$	19.2915	0.693867	0.346934	+	0.937890i	$0.387223\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−102.686	−3.68622
$777$	0	0
$778$	−15.8745	−0.569129
$779$	−4.35425	−0.156007
$780$	0	0
$781$	−18.5830	−0.664952
$782$	−13.6680	−0.488766
$783$	0	0
$784$	69.2065	2.47166
$785$	2.70850	0.0966704
$786$	0	0
$787$	6.12549	0.218350	0.109175	−	0.994023i	$-0.465179\pi$
$787$	6.12549	0.218350	0.109175	+	0.994023i	$0.465179\pi$
$788$	−85.8301	−3.05757
$789$	0	0
$790$	−38.5830	−1.37272
$791$	−14.5830	−0.518512
$792$	0	0
$793$	4.00000	0.142044
$794$	5.29150	0.187788
$795$	0	0
$796$	−16.4575	−0.583321
$797$	−40.0000	−1.41687	−0.708436	−	0.705775i	$-0.750599\pi$
$797$	−40.0000	−1.41687	−0.708436	+	0.705775i	$0.750599\pi$
$798$	0	0
$799$	−37.1660	−1.31484
$800$	−13.2288	−0.467707
$801$	0	0
$802$	28.9373	1.02181
$803$	−56.4575	−1.99234
$804$	0	0
$805$	4.70850	0.165953
$806$	89.6235	3.15685
$807$	0	0
$808$	−10.2510	−0.360628
$809$	40.5830	1.42682	0.713411	−	0.700746i	$-0.247149\pi$
$809$	40.5830	1.42682	0.713411	+	0.700746i	$0.247149\pi$
$810$	0	0
$811$	−46.3320	−1.62694	−0.813469	−	0.581609i	$-0.802423\pi$
$811$	−46.3320	−1.62694	−0.813469	+	0.581609i	$0.802423\pi$
$812$	−126.458	−4.43779
$813$	0	0
$814$	−24.5830	−0.861634
$815$	7.64575	0.267819
$816$	0	0
$817$	−0.354249	−0.0123936
$818$	45.7490	1.59958
$819$	0	0
$820$	−21.7712	−0.760285
$821$	47.6235	1.66207	0.831036	−	0.556218i	$-0.187748\pi$
$821$	47.6235	1.66207	0.831036	+	0.556218i	$0.187748\pi$
$822$	0	0
$823$	−22.2288	−0.774846	−0.387423	−	0.921902i	$-0.626635\pi$
$823$	−22.2288	−0.774846	−0.387423	+	0.921902i	$0.626635\pi$
$824$	−84.0000	−2.92628
$825$	0	0
$826$	−6.83399	−0.237785
$827$	22.4575	0.780924	0.390462	−	0.920619i	$-0.372315\pi$
$827$	22.4575	0.780924	0.390462	+	0.920619i	$0.372315\pi$
$828$	0	0
$829$	13.2915	0.461633	0.230816	−	0.972997i	$-0.425860\pi$
$829$	13.2915	0.461633	0.230816	+	0.972997i	$0.425860\pi$
$830$	−15.8745	−0.551012
$831$	0	0
$832$	−73.3948	−2.54451
$833$	25.1660	0.871951
$834$	0	0
$835$	−10.7085	−0.370583
$836$	28.2288	0.976312
$837$	0	0
$838$	61.0183	2.10784
$839$	−40.4575	−1.39675	−0.698374	−	0.715733i	$-0.746093\pi$
$839$	−40.4575	−1.39675	−0.698374	+	0.715733i	$0.746093\pi$
$840$	0	0
$841$	19.1255	0.659500
$842$	−19.6235	−0.676271
$843$	0	0
$844$	−92.9150	−3.19827
$845$	−18.8745	−0.649303
$846$	0	0
$847$	−76.1033	−2.61494
$848$	−7.79347	−0.267629
$849$	0	0
$850$	−10.5830	−0.362994
$851$	−2.12549	−0.0728609
$852$	0	0
$853$	25.2915	0.865965	0.432982	−	0.901402i	$-0.357461\pi$
$853$	25.2915	0.865965	0.432982	+	0.901402i	$0.357461\pi$
$854$	−6.83399	−0.233854
$855$	0	0
$856$	0	0
$857$	43.0405	1.47024	0.735118	−	0.677939i	$-0.237127\pi$
$857$	43.0405	1.47024	0.735118	+	0.677939i	$0.237127\pi$
$858$	0	0
$859$	50.3320	1.71731	0.858653	−	0.512557i	$-0.171302\pi$
$859$	50.3320	1.71731	0.858653	+	0.512557i	$0.171302\pi$
$860$	−1.77124	−0.0603989
$861$	0	0
$862$	−19.2915	−0.657071
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	0	0
$866$	59.1438	2.00979
$867$	0	0
$868$	−109.373	−3.71235
$869$	82.3320	2.79292
$870$	0	0
$871$	−82.3320	−2.78971
$872$	42.0000	1.42230
$873$	0	0
$874$	3.41699	0.115582
$875$	3.64575	0.123249
$876$	0	0
$877$	−40.9373	−1.38235	−0.691176	−	0.722686i	$-0.742907\pi$
$877$	−40.9373	−1.38235	−0.691176	+	0.722686i	$0.742907\pi$
$878$	59.7490	2.01643
$879$	0	0
$880$	62.1033	2.09350
$881$	31.8745	1.07388	0.536940	−	0.843621i	$-0.319580\pi$
$881$	31.8745	1.07388	0.536940	+	0.843621i	$0.319580\pi$
$882$	0	0
$883$	−36.8118	−1.23881	−0.619407	−	0.785070i	$-0.712627\pi$
$883$	−36.8118	−1.23881	−0.619407	+	0.785070i	$0.712627\pi$
$884$	−112.915	−3.79774
$885$	0	0
$886$	98.6640	3.31468
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−14.5830	−0.489098
$890$	2.81176	0.0942505
$891$	0	0
$892$	−92.9150	−3.11103
$893$	9.29150	0.310928
$894$	0	0
$895$	3.29150	0.110023
$896$	28.9373	0.966726
$897$	0	0
$898$	−25.8523	−0.862702
$899$	41.6235	1.38822
$900$	0	0
$901$	−2.83399	−0.0944139
$902$	65.0405	2.16561
$903$	0	0
$904$	−31.7490	−1.05596
$905$	−6.70850	−0.222998
$906$	0	0
$907$	27.0405	0.897866	0.448933	−	0.893566i	$-0.351804\pi$
$907$	27.0405	0.897866	0.448933	+	0.893566i	$0.351804\pi$
$908$	106.458	3.53292
$909$	0	0
$910$	54.4575	1.80525
$911$	−5.41699	−0.179473	−0.0897365	−	0.995966i	$-0.528602\pi$
$911$	−5.41699	−0.179473	−0.0897365	+	0.995966i	$0.528602\pi$
$912$	0	0
$913$	33.8745	1.12108
$914$	−84.3320	−2.78946
$915$	0	0
$916$	96.4575	3.18705
$917$	44.5830	1.47226
$918$	0	0
$919$	57.1660	1.88573	0.942866	−	0.333171i	$-0.108119\pi$
$919$	57.1660	1.88573	0.942866	+	0.333171i	$0.108119\pi$
$920$	10.2510	0.337965
$921$	0	0
$922$	68.1255	2.24359
$923$	−18.5830	−0.611667
$924$	0	0
$925$	−1.64575	−0.0541120
$926$	−4.02223	−0.132179
$927$	0	0
$928$	−91.7712	−3.01254
$929$	−19.8745	−0.652061	−0.326031	−	0.945359i	$-0.605711\pi$
$929$	−19.8745	−0.652061	−0.326031	+	0.945359i	$0.605711\pi$
$930$	0	0
$931$	−6.29150	−0.206196
$932$	90.0000	2.94805
$933$	0	0
$934$	−31.4170	−1.02800
$935$	22.5830	0.738543
$936$	0	0
$937$	51.8745	1.69467	0.847333	−	0.531062i	$-0.178207\pi$
$937$	51.8745	1.69467	0.847333	+	0.531062i	$0.178207\pi$
$938$	140.664	4.59284
$939$	0	0
$940$	46.4575	1.51528
$941$	38.2288	1.24622	0.623111	−	0.782133i	$-0.285869\pi$
$941$	38.2288	1.24622	0.623111	+	0.782133i	$0.285869\pi$
$942$	0	0
$943$	5.62352	0.183127
$944$	−7.79347	−0.253656
$945$	0	0
$946$	5.29150	0.172042
$947$	−32.5830	−1.05881	−0.529403	−	0.848371i	$-0.677584\pi$
$947$	−32.5830	−1.05881	−0.529403	+	0.848371i	$0.677584\pi$
$948$	0	0
$949$	−56.4575	−1.83269
$950$	2.64575	0.0858395
$951$	0	0
$952$	115.749	3.75145
$953$	−10.5830	−0.342817	−0.171409	−	0.985200i	$-0.554832\pi$
$953$	−10.5830	−0.342817	−0.171409	+	0.985200i	$0.554832\pi$
$954$	0	0
$955$	−20.2288	−0.654587
$956$	−51.7712	−1.67440
$957$	0	0
$958$	25.5203	0.824522
$959$	21.8745	0.706365
$960$	0	0
$961$	5.00000	0.161290
$962$	−24.5830	−0.792588
$963$	0	0
$964$	−22.9150	−0.738043
$965$	6.35425	0.204551
$966$	0	0
$967$	−36.3542	−1.16907	−0.584537	−	0.811367i	$-0.698724\pi$
$967$	−36.3542	−1.16907	−0.584537	+	0.811367i	$0.698724\pi$
$968$	−165.686	−5.32536
$969$	0	0
$970$	34.2288	1.09902
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	50.5830	1.62162
$974$	36.7085	1.17622
$975$	0	0
$976$	−7.79347	−0.249463
$977$	−23.0405	−0.737131	−0.368566	−	0.929602i	$-0.620151\pi$
$977$	−23.0405	−0.737131	−0.368566	+	0.929602i	$0.620151\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	−31.4575	−1.00487
$981$	0	0
$982$	−85.2693	−2.72105
$983$	18.4575	0.588703	0.294352	−	0.955697i	$-0.404896\pi$
$983$	18.4575	0.588703	0.294352	+	0.955697i	$0.404896\pi$
$984$	0	0
$985$	17.1660	0.546955
$986$	−73.4170	−2.33807
$987$	0	0
$988$	28.2288	0.898076
$989$	0.457513	0.0145481
$990$	0	0
$991$	27.7490	0.881477	0.440738	−	0.897636i	$-0.354717\pi$
$991$	27.7490	0.881477	0.440738	+	0.897636i	$0.354717\pi$
$992$	−79.3725	−2.52008
$993$	0	0
$994$	31.7490	1.00702
$995$	3.29150	0.104348
$996$	0	0
$997$	37.2915	1.18103	0.590517	−	0.807025i	$-0.298924\pi$
$997$	37.2915	1.18103	0.590517	+	0.807025i	$0.298924\pi$
$998$	−113.875	−3.60463
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.g.1.1		2
3.2	odd	2		285.2.a.d.1.2	✓	2
5.4	even	2		4275.2.a.u.1.2		2
12.11	even	2		4560.2.a.bo.1.2		2
15.2	even	4		1425.2.c.i.799.4		4
15.8	even	4		1425.2.c.i.799.1		4
15.14	odd	2		1425.2.a.p.1.1		2
57.56	even	2		5415.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.d.1.2	✓	2	3.2	odd	2
855.2.a.g.1.1		2	1.1	even	1	trivial
1425.2.a.p.1.1		2	15.14	odd	2
1425.2.c.i.799.1		4	15.8	even	4
1425.2.c.i.799.4		4	15.2	even	4
4275.2.a.u.1.2		2	5.4	even	2
4560.2.a.bo.1.2		2	12.11	even	2
5415.2.a.s.1.1		2	57.56	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64575 q^{2} +5.00000 q^{4} -1.00000 q^{5} -3.64575 q^{7} -7.93725 q^{8} +2.64575 q^{10} -5.64575 q^{11} -5.64575 q^{13} +9.64575 q^{14} +11.0000 q^{16} +4.00000 q^{17} -1.00000 q^{19} -5.00000 q^{20} +14.9373 q^{22} +1.29150 q^{23} +1.00000 q^{25} +14.9373 q^{26} -18.2288 q^{28} +6.93725 q^{29} +6.00000 q^{31} -13.2288 q^{32} -10.5830 q^{34} +3.64575 q^{35} -1.64575 q^{37} +2.64575 q^{38} +7.93725 q^{40} +4.35425 q^{41} +0.354249 q^{43} -28.2288 q^{44} -3.41699 q^{46} -9.29150 q^{47} +6.29150 q^{49} -2.64575 q^{50} -28.2288 q^{52} -0.708497 q^{53} +5.64575 q^{55} +28.9373 q^{56} -18.3542 q^{58} -0.708497 q^{59} -0.708497 q^{61} -15.8745 q^{62} +13.0000 q^{64} +5.64575 q^{65} +14.5830 q^{67} +20.0000 q^{68} -9.64575 q^{70} +3.29150 q^{71} +10.0000 q^{73} +4.35425 q^{74} -5.00000 q^{76} +20.5830 q^{77} -14.5830 q^{79} -11.0000 q^{80} -11.5203 q^{82} -6.00000 q^{83} -4.00000 q^{85} -0.937254 q^{86} +44.8118 q^{88} +1.06275 q^{89} +20.5830 q^{91} +6.45751 q^{92} +24.5830 q^{94} +1.00000 q^{95} +12.9373 q^{97} -16.6458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{11} - 6 q^{13} + 14 q^{14} + 22 q^{16} + 8 q^{17} - 2 q^{19} - 10 q^{20} + 14 q^{22} - 8 q^{23} + 2 q^{25} + 14 q^{26} - 10 q^{28} - 2 q^{29} + 12 q^{31} + 2 q^{35}+ \cdots - 28 q^{98}+O(q^{100}) \) 2 * q + 10 * q^4 - 2 * q^5 - 2 * q^7 - 6 * q^11 - 6 * q^13 + 14 * q^14 + 22 * q^16 + 8 * q^17 - 2 * q^19 - 10 * q^20 + 14 * q^22 - 8 * q^23 + 2 * q^25 + 14 * q^26 - 10 * q^28 - 2 * q^29 + 12 * q^31 + 2 * q^35 + 2 * q^37 + 14 * q^41 + 6 * q^43 - 30 * q^44 - 28 * q^46 - 8 * q^47 + 2 * q^49 - 30 * q^52 - 12 * q^53 + 6 * q^55 + 42 * q^56 - 42 * q^58 - 12 * q^59 - 12 * q^61 + 26 * q^64 + 6 * q^65 + 8 * q^67 + 40 * q^68 - 14 * q^70 - 4 * q^71 + 20 * q^73 + 14 * q^74 - 10 * q^76 + 20 * q^77 - 8 * q^79 - 22 * q^80 + 14 * q^82 - 12 * q^83 - 8 * q^85 + 14 * q^86 + 42 * q^88 + 18 * q^89 + 20 * q^91 - 40 * q^92 + 28 * q^94 + 2 * q^95 + 10 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

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