LMFDB - Embedded newform 3100.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,2,Mod(1,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3100.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:	$24.7536246266$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.3418929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 9x^{2} + 18x - 3 $ x^5 - 2x^4 - 8x^3 + 9x^2 + 18x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.18729$ of defining polynomial
Character	$\chi$	$=$	3100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18729 q^{3} -2.76870 q^{7} -1.59033 q^{9} +O(q^{10})$	$q+1.18729 q^{3} -2.76870 q^{7} -1.59033 q^{9} -1.40304 q^{11} +3.57478 q^{13} +1.66582 q^{17} -2.85311 q^{19} -3.28726 q^{21} +3.04041 q^{23} -5.45007 q^{27} +8.24060 q^{29} -1.00000 q^{31} -1.66582 q^{33} -8.45900 q^{37} +4.24431 q^{39} +6.20031 q^{41} -6.55631 q^{43} -4.20285 q^{47} +0.665703 q^{49} +1.97782 q^{51} -9.42789 q^{53} -3.38748 q^{57} -12.9558 q^{59} -13.0338 q^{61} +4.40316 q^{63} -9.44345 q^{67} +3.60986 q^{69} -3.82863 q^{71} +7.81909 q^{73} +3.88460 q^{77} +0.515903 q^{79} -1.69984 q^{81} -5.13133 q^{83} +9.78401 q^{87} +5.03148 q^{89} -9.89749 q^{91} -1.18729 q^{93} -5.28089 q^{97} +2.23130 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} - 4 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 - 4 * q^7 + 6 * q^9	$ 5 q - 3 q^{3} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{17} + 6 q^{19} - 3 q^{21} - 14 q^{23} - 12 q^{27} + 10 q^{29} - 5 q^{31} + 3 q^{33} - 14 q^{37} + 3 q^{39} - 6 q^{41} - q^{43} - 18 q^{47} + 27 q^{49} - 15 q^{51} - 7 q^{53} - 6 q^{57} + 20 q^{59} - 17 q^{61} - 18 q^{63} - 13 q^{67} + 30 q^{69} - 6 q^{71} + q^{73} - 17 q^{77} - 15 q^{81} - 40 q^{83} - 21 q^{87} + 9 q^{89} - 20 q^{91} + 3 q^{93} - q^{97} + 21 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 - 4 * q^7 + 6 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^17 + 6 * q^19 - 3 * q^21 - 14 * q^23 - 12 * q^27 + 10 * q^29 - 5 * q^31 + 3 * q^33 - 14 * q^37 + 3 * q^39 - 6 * q^41 - q^43 - 18 * q^47 + 27 * q^49 - 15 * q^51 - 7 * q^53 - 6 * q^57 + 20 * q^59 - 17 * q^61 - 18 * q^63 - 13 * q^67 + 30 * q^69 - 6 * q^71 + q^73 - 17 * q^77 - 15 * q^81 - 40 * q^83 - 21 * q^87 + 9 * q^89 - 20 * q^91 + 3 * q^93 - q^97 + 21 * 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
$3$	1.18729	0.685484	0.342742	−	0.939429i	$-0.388644\pi$
$3$	1.18729	0.685484	0.342742	+	0.939429i	$0.388644\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.76870	−1.04647	−0.523235	−	0.852188i	$-0.675275\pi$
$7$	−2.76870	−1.04647	−0.523235	+	0.852188i	$0.675275\pi$
$8$	0	0
$9$	−1.59033	−0.530111
$10$	0	0
$11$	−1.40304	−0.423032	−0.211516	−	0.977375i	$-0.567840\pi$
$11$	−1.40304	−0.423032	−0.211516	+	0.977375i	$0.567840\pi$
$12$	0	0
$13$	3.57478	0.991465	0.495733	−	0.868475i	$-0.334900\pi$
$13$	3.57478	0.991465	0.495733	+	0.868475i	$0.334900\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.66582	0.404021	0.202010	−	0.979383i	$-0.435253\pi$
$17$	1.66582	0.404021	0.202010	+	0.979383i	$0.435253\pi$
$18$	0	0
$19$	−2.85311	−0.654549	−0.327275	−	0.944929i	$-0.606130\pi$
$19$	−2.85311	−0.654549	−0.327275	+	0.944929i	$0.606130\pi$
$20$	0	0
$21$	−3.28726	−0.717339
$22$	0	0
$23$	3.04041	0.633969	0.316984	−	0.948431i	$-0.397330\pi$
$23$	3.04041	0.633969	0.316984	+	0.948431i	$0.397330\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.45007	−1.04887
$28$	0	0
$29$	8.24060	1.53024	0.765120	−	0.643887i	$-0.222679\pi$
$29$	8.24060	1.53024	0.765120	+	0.643887i	$0.222679\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.66582	−0.289982
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.45900	−1.39065	−0.695326	−	0.718695i	$-0.744740\pi$
$37$	−8.45900	−1.39065	−0.695326	+	0.718695i	$0.744740\pi$
$38$	0	0
$39$	4.24431	0.679634
$40$	0	0
$41$	6.20031	0.968325	0.484163	−	0.874978i	$-0.339124\pi$
$41$	6.20031	0.968325	0.484163	+	0.874978i	$0.339124\pi$
$42$	0	0
$43$	−6.55631	−0.999828	−0.499914	−	0.866075i	$-0.666635\pi$
$43$	−6.55631	−0.999828	−0.499914	+	0.866075i	$0.666635\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.20285	−0.613048	−0.306524	−	0.951863i	$-0.599166\pi$
$47$	−4.20285	−0.613048	−0.306524	+	0.951863i	$0.599166\pi$
$48$	0	0
$49$	0.665703	0.0951004
$50$	0	0
$51$	1.97782	0.276950
$52$	0	0
$53$	−9.42789	−1.29502	−0.647510	−	0.762057i	$-0.724190\pi$
$53$	−9.42789	−1.29502	−0.647510	+	0.762057i	$0.724190\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.38748	−0.448683
$58$	0	0
$59$	−12.9558	−1.68670	−0.843348	−	0.537368i	$-0.819419\pi$
$59$	−12.9558	−1.68670	−0.843348	+	0.537368i	$0.819419\pi$
$60$	0	0
$61$	−13.0338	−1.66880	−0.834402	−	0.551156i	$-0.814187\pi$
$61$	−13.0338	−1.66880	−0.834402	+	0.551156i	$0.814187\pi$
$62$	0	0
$63$	4.40316	0.554746
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.44345	−1.15370	−0.576850	−	0.816850i	$-0.695718\pi$
$67$	−9.44345	−1.15370	−0.576850	+	0.816850i	$0.695718\pi$
$68$	0	0
$69$	3.60986	0.434576
$70$	0	0
$71$	−3.82863	−0.454375	−0.227188	−	0.973851i	$-0.572953\pi$
$71$	−3.82863	−0.454375	−0.227188	+	0.973851i	$0.572953\pi$
$72$	0	0
$73$	7.81909	0.915155	0.457578	−	0.889170i	$-0.348717\pi$
$73$	7.81909	0.915155	0.457578	+	0.889170i	$0.348717\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.88460	0.442691
$78$	0	0
$79$	0.515903	0.0580437	0.0290218	−	0.999579i	$-0.490761\pi$
$79$	0.515903	0.0580437	0.0290218	+	0.999579i	$0.490761\pi$
$80$	0	0
$81$	−1.69984	−0.188871
$82$	0	0
$83$	−5.13133	−0.563237	−0.281618	−	0.959526i	$-0.590871\pi$
$83$	−5.13133	−0.563237	−0.281618	+	0.959526i	$0.590871\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.78401	1.04896
$88$	0	0
$89$	5.03148	0.533336	0.266668	−	0.963788i	$-0.414077\pi$
$89$	5.03148	0.533336	0.266668	+	0.963788i	$0.414077\pi$
$90$	0	0
$91$	−9.89749	−1.03754
$92$	0	0
$93$	−1.18729	−0.123117
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.28089	−0.536193	−0.268097	−	0.963392i	$-0.586395\pi$
$97$	−5.28089	−0.536193	−0.268097	+	0.963392i	$0.586395\pi$
$98$	0	0
$99$	2.23130	0.224254
$100$	0	0
$101$	−14.5749	−1.45026	−0.725128	−	0.688614i	$-0.758220\pi$
$101$	−14.5749	−1.45026	−0.725128	+	0.688614i	$0.758220\pi$
$102$	0	0
$103$	−9.86613	−0.972138	−0.486069	−	0.873920i	$-0.661570\pi$
$103$	−9.86613	−0.972138	−0.486069	+	0.873920i	$0.661570\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8776	−1.05158	−0.525788	−	0.850615i	$-0.676230\pi$
$107$	−10.8776	−1.05158	−0.525788	+	0.850615i	$0.676230\pi$
$108$	0	0
$109$	7.89315	0.756027	0.378013	−	0.925800i	$-0.376607\pi$
$109$	7.89315	0.756027	0.378013	+	0.925800i	$0.376607\pi$
$110$	0	0
$111$	−10.0433	−0.953270
$112$	0	0
$113$	−16.0586	−1.51067	−0.755335	−	0.655339i	$-0.772526\pi$
$113$	−16.0586	−1.51067	−0.755335	+	0.655339i	$0.772526\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.68509	−0.525587
$118$	0	0
$119$	−4.61216	−0.422796
$120$	0	0
$121$	−9.03148	−0.821044
$122$	0	0
$123$	7.36159	0.663772
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.6903	−1.03735	−0.518673	−	0.854973i	$-0.673574\pi$
$127$	−11.6903	−1.03735	−0.518673	+	0.854973i	$0.673574\pi$
$128$	0	0
$129$	−7.78427	−0.685367
$130$	0	0
$131$	18.5779	1.62316	0.811580	−	0.584241i	$-0.198608\pi$
$131$	18.5779	1.62316	0.811580	+	0.584241i	$0.198608\pi$
$132$	0	0
$133$	7.89942	0.684966
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.70994	−0.402398	−0.201199	−	0.979550i	$-0.564484\pi$
$137$	−4.70994	−0.402398	−0.201199	+	0.979550i	$0.564484\pi$
$138$	0	0
$139$	10.8114	0.917012	0.458506	−	0.888691i	$-0.348385\pi$
$139$	10.8114	0.917012	0.458506	+	0.888691i	$0.348385\pi$
$140$	0	0
$141$	−4.99002	−0.420235
$142$	0	0
$143$	−5.01555	−0.419422
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.790385	0.0651899
$148$	0	0
$149$	11.8898	0.974051	0.487025	−	0.873388i	$-0.338082\pi$
$149$	11.8898	0.974051	0.487025	+	0.873388i	$0.338082\pi$
$150$	0	0
$151$	12.9431	1.05330	0.526648	−	0.850084i	$-0.323449\pi$
$151$	12.9431	1.05330	0.526648	+	0.850084i	$0.323449\pi$
$152$	0	0
$153$	−2.64921	−0.214176
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.6615	0.850880	0.425440	−	0.904987i	$-0.360119\pi$
$157$	10.6615	0.850880	0.425440	+	0.904987i	$0.360119\pi$
$158$	0	0
$159$	−11.1937	−0.887717
$160$	0	0
$161$	−8.41798	−0.663430
$162$	0	0
$163$	2.57887	0.201992	0.100996	−	0.994887i	$-0.467797\pi$
$163$	2.57887	0.201992	0.100996	+	0.994887i	$0.467797\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.80874	−0.449493	−0.224747	−	0.974417i	$-0.572155\pi$
$167$	−5.80874	−0.449493	−0.224747	+	0.974417i	$0.572155\pi$
$168$	0	0
$169$	−0.220957	−0.0169967
$170$	0	0
$171$	4.53740	0.346984
$172$	0	0
$173$	−3.39411	−0.258050	−0.129025	−	0.991641i	$-0.541185\pi$
$173$	−3.39411	−0.258050	−0.129025	+	0.991641i	$0.541185\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15.3823	−1.15620
$178$	0	0
$179$	23.2108	1.73486	0.867430	−	0.497559i	$-0.165770\pi$
$179$	23.2108	1.73486	0.867430	+	0.497559i	$0.165770\pi$
$180$	0	0
$181$	0.541116	0.0402208	0.0201104	−	0.999798i	$-0.493598\pi$
$181$	0.541116	0.0402208	0.0201104	+	0.999798i	$0.493598\pi$
$182$	0	0
$183$	−15.4749	−1.14394
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.33721	−0.170914
$188$	0	0
$189$	15.0896	1.09761
$190$	0	0
$191$	−3.40341	−0.246262	−0.123131	−	0.992390i	$-0.539294\pi$
$191$	−3.40341	−0.246262	−0.123131	+	0.992390i	$0.539294\pi$
$192$	0	0
$193$	26.6223	1.91632	0.958158	−	0.286241i	$-0.0924058\pi$
$193$	26.6223	1.91632	0.958158	+	0.286241i	$0.0924058\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.26148	0.446112	0.223056	−	0.974806i	$-0.428397\pi$
$197$	6.26148	0.446112	0.223056	+	0.974806i	$0.428397\pi$
$198$	0	0
$199$	3.44599	0.244280	0.122140	−	0.992513i	$-0.461024\pi$
$199$	3.44599	0.244280	0.122140	+	0.992513i	$0.461024\pi$
$200$	0	0
$201$	−11.2121	−0.790844
$202$	0	0
$203$	−22.8158	−1.60135
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.83526	−0.336074
$208$	0	0
$209$	4.00303	0.276895
$210$	0	0
$211$	15.8928	1.09411	0.547054	−	0.837097i	$-0.315749\pi$
$211$	15.8928	1.09411	0.547054	+	0.837097i	$0.315749\pi$
$212$	0	0
$213$	−4.54571	−0.311467
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.76870	0.187952
$218$	0	0
$219$	9.28356	0.627325
$220$	0	0
$221$	5.95494	0.400572
$222$	0	0
$223$	9.52718	0.637987	0.318993	−	0.947757i	$-0.396655\pi$
$223$	9.52718	0.637987	0.318993	+	0.947757i	$0.396655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.1351	−0.739060	−0.369530	−	0.929219i	$-0.620481\pi$
$227$	−11.1351	−0.739060	−0.369530	+	0.929219i	$0.620481\pi$
$228$	0	0
$229$	12.6866	0.838353	0.419177	−	0.907905i	$-0.362319\pi$
$229$	12.6866	0.838353	0.419177	+	0.907905i	$0.362319\pi$
$230$	0	0
$231$	4.61216	0.303458
$232$	0	0
$233$	4.10252	0.268765	0.134383	−	0.990930i	$-0.457095\pi$
$233$	4.10252	0.268765	0.134383	+	0.990930i	$0.457095\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.612529	0.0397880
$238$	0	0
$239$	0.261724	0.0169295	0.00846474	−	0.999964i	$-0.497306\pi$
$239$	0.261724	0.0169295	0.00846474	+	0.999964i	$0.497306\pi$
$240$	0	0
$241$	2.16220	0.139279	0.0696397	−	0.997572i	$-0.477815\pi$
$241$	2.16220	0.139279	0.0696397	+	0.997572i	$0.477815\pi$
$242$	0	0
$243$	14.3320	0.919399
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.1993	−0.648963
$248$	0	0
$249$	−6.09240	−0.386090
$250$	0	0
$251$	−3.96295	−0.250139	−0.125070	−	0.992148i	$-0.539915\pi$
$251$	−3.96295	−0.250139	−0.125070	+	0.992148i	$0.539915\pi$
$252$	0	0
$253$	−4.26581	−0.268189
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.4965	0.966645	0.483322	−	0.875442i	$-0.339430\pi$
$257$	15.4965	0.966645	0.483322	+	0.875442i	$0.339430\pi$
$258$	0	0
$259$	23.4204	1.45528
$260$	0	0
$261$	−13.1053	−0.811197
$262$	0	0
$263$	6.74590	0.415970	0.207985	−	0.978132i	$-0.433309\pi$
$263$	6.74590	0.415970	0.207985	+	0.978132i	$0.433309\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.97385	0.365594
$268$	0	0
$269$	11.1614	0.680522	0.340261	−	0.940331i	$-0.389485\pi$
$269$	11.1614	0.680522	0.340261	+	0.940331i	$0.389485\pi$
$270$	0	0
$271$	−5.01917	−0.304893	−0.152446	−	0.988312i	$-0.548715\pi$
$271$	−5.01917	−0.304893	−0.152446	+	0.988312i	$0.548715\pi$
$272$	0	0
$273$	−11.7512	−0.711217
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.6117	1.41869	0.709345	−	0.704862i	$-0.248991\pi$
$277$	23.6117	1.41869	0.709345	+	0.704862i	$0.248991\pi$
$278$	0	0
$279$	1.59033	0.0952107
$280$	0	0
$281$	4.49662	0.268246	0.134123	−	0.990965i	$-0.457178\pi$
$281$	4.49662	0.268246	0.134123	+	0.990965i	$0.457178\pi$
$282$	0	0
$283$	−7.83986	−0.466031	−0.233016	−	0.972473i	$-0.574859\pi$
$283$	−7.83986	−0.466031	−0.233016	+	0.972473i	$0.574859\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−17.1668	−1.01332
$288$	0	0
$289$	−14.2250	−0.836767
$290$	0	0
$291$	−6.26997	−0.367552
$292$	0	0
$293$	−12.4989	−0.730195	−0.365098	−	0.930969i	$-0.618964\pi$
$293$	−12.4989	−0.730195	−0.365098	+	0.930969i	$0.618964\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.64667	0.443705
$298$	0	0
$299$	10.8688	0.628558
$300$	0	0
$301$	18.1525	1.04629
$302$	0	0
$303$	−17.3047	−0.994128
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−19.4794	−1.11175	−0.555874	−	0.831266i	$-0.687616\pi$
$307$	−19.4794	−1.11175	−0.555874	+	0.831266i	$0.687616\pi$
$308$	0	0
$309$	−11.7140	−0.666386
$310$	0	0
$311$	14.9501	0.847740	0.423870	−	0.905723i	$-0.360671\pi$
$311$	14.9501	0.847740	0.423870	+	0.905723i	$0.360671\pi$
$312$	0	0
$313$	−19.5462	−1.10482	−0.552408	−	0.833574i	$-0.686291\pi$
$313$	−19.5462	−1.10482	−0.552408	+	0.833574i	$0.686291\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.7211	−1.72547	−0.862735	−	0.505657i	$-0.831250\pi$
$317$	−30.7211	−1.72547	−0.862735	+	0.505657i	$0.831250\pi$
$318$	0	0
$319$	−11.5619	−0.647341
$320$	0	0
$321$	−12.9149	−0.720840
$322$	0	0
$323$	−4.75277	−0.264451
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.37149	0.518244
$328$	0	0
$329$	11.6364	0.641537
$330$	0	0
$331$	12.2843	0.675205	0.337603	−	0.941289i	$-0.390384\pi$
$331$	12.2843	0.675205	0.337603	+	0.941289i	$0.390384\pi$
$332$	0	0
$333$	13.4526	0.737200
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.3971	0.729787	0.364894	−	0.931049i	$-0.381105\pi$
$337$	13.3971	0.729787	0.364894	+	0.931049i	$0.381105\pi$
$338$	0	0
$339$	−19.0663	−1.03554
$340$	0	0
$341$	1.40304	0.0759788
$342$	0	0
$343$	17.5378	0.946951
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.5649	−1.21135	−0.605674	−	0.795713i	$-0.707096\pi$
$347$	−22.5649	−1.21135	−0.605674	+	0.795713i	$0.707096\pi$
$348$	0	0
$349$	−24.7586	−1.32530	−0.662649	−	0.748930i	$-0.730568\pi$
$349$	−24.7586	−1.32530	−0.662649	+	0.748930i	$0.730568\pi$
$350$	0	0
$351$	−19.4828	−1.03992
$352$	0	0
$353$	10.1173	0.538491	0.269245	−	0.963072i	$-0.413226\pi$
$353$	10.1173	0.538491	0.269245	+	0.963072i	$0.413226\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.47599	−0.289820
$358$	0	0
$359$	−7.78623	−0.410941	−0.205471	−	0.978663i	$-0.565873\pi$
$359$	−7.78623	−0.410941	−0.205471	+	0.978663i	$0.565873\pi$
$360$	0	0
$361$	−10.8597	−0.571565
$362$	0	0
$363$	−10.7230	−0.562813
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.5784	0.552187	0.276093	−	0.961131i	$-0.410960\pi$
$367$	10.5784	0.552187	0.276093	+	0.961131i	$0.410960\pi$
$368$	0	0
$369$	−9.86055	−0.513320
$370$	0	0
$371$	26.1030	1.35520
$372$	0	0
$373$	0.602415	0.0311918	0.0155959	−	0.999878i	$-0.495035\pi$
$373$	0.602415	0.0311918	0.0155959	+	0.999878i	$0.495035\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	29.4583	1.51718
$378$	0	0
$379$	4.30380	0.221072	0.110536	−	0.993872i	$-0.464743\pi$
$379$	4.30380	0.221072	0.110536	+	0.993872i	$0.464743\pi$
$380$	0	0
$381$	−13.8798	−0.711085
$382$	0	0
$383$	−22.8069	−1.16538	−0.582690	−	0.812695i	$-0.698000\pi$
$383$	−22.8069	−1.16538	−0.582690	+	0.812695i	$0.698000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.4267	0.530020
$388$	0	0
$389$	−22.3464	−1.13301	−0.566503	−	0.824060i	$-0.691704\pi$
$389$	−22.3464	−1.13301	−0.566503	+	0.824060i	$0.691704\pi$
$390$	0	0
$391$	5.06477	0.256137
$392$	0	0
$393$	22.0575	1.11265
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−26.8027	−1.34519	−0.672595	−	0.740011i	$-0.734820\pi$
$397$	−26.8027	−1.34519	−0.672595	+	0.740011i	$0.734820\pi$
$398$	0	0
$399$	9.37893	0.469534
$400$	0	0
$401$	10.5205	0.525369	0.262684	−	0.964882i	$-0.415392\pi$
$401$	10.5205	0.525369	0.262684	+	0.964882i	$0.415392\pi$
$402$	0	0
$403$	−3.57478	−0.178072
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.8683	0.588290
$408$	0	0
$409$	24.3043	1.20177	0.600884	−	0.799336i	$-0.294815\pi$
$409$	24.3043	1.20177	0.600884	+	0.799336i	$0.294815\pi$
$410$	0	0
$411$	−5.59209	−0.275837
$412$	0	0
$413$	35.8706	1.76508
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.8363	0.628597
$418$	0	0
$419$	−21.3479	−1.04291	−0.521456	−	0.853278i	$-0.674611\pi$
$419$	−21.3479	−1.04291	−0.521456	+	0.853278i	$0.674611\pi$
$420$	0	0
$421$	−26.5186	−1.29244	−0.646219	−	0.763152i	$-0.723651\pi$
$421$	−26.5186	−1.29244	−0.646219	+	0.763152i	$0.723651\pi$
$422$	0	0
$423$	6.68393	0.324984
$424$	0	0
$425$	0	0
$426$	0	0
$427$	36.0866	1.74635
$428$	0	0
$429$	−5.95494	−0.287507
$430$	0	0
$431$	−13.4498	−0.647852	−0.323926	−	0.946082i	$-0.605003\pi$
$431$	−13.4498	−0.647852	−0.323926	+	0.946082i	$0.605003\pi$
$432$	0	0
$433$	22.2731	1.07038	0.535188	−	0.844733i	$-0.320241\pi$
$433$	22.2731	1.07038	0.535188	+	0.844733i	$0.320241\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.67463	−0.414964
$438$	0	0
$439$	−29.6126	−1.41333	−0.706667	−	0.707546i	$-0.749802\pi$
$439$	−29.6126	−1.41333	−0.706667	+	0.707546i	$0.749802\pi$
$440$	0	0
$441$	−1.05869	−0.0504138
$442$	0	0
$443$	−11.0037	−0.522802	−0.261401	−	0.965230i	$-0.584185\pi$
$443$	−11.0037	−0.522802	−0.261401	+	0.965230i	$0.584185\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.1167	0.667697
$448$	0	0
$449$	−6.68465	−0.315468	−0.157734	−	0.987482i	$-0.550419\pi$
$449$	−6.68465	−0.315468	−0.157734	+	0.987482i	$0.550419\pi$
$450$	0	0
$451$	−8.69927	−0.409633
$452$	0	0
$453$	15.3673	0.722018
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.2057	−1.17907	−0.589535	−	0.807743i	$-0.700689\pi$
$457$	−25.2057	−1.17907	−0.589535	+	0.807743i	$0.700689\pi$
$458$	0	0
$459$	−9.07884	−0.423764
$460$	0	0
$461$	34.0021	1.58364	0.791818	−	0.610757i	$-0.209135\pi$
$461$	34.0021	1.58364	0.791818	+	0.610757i	$0.209135\pi$
$462$	0	0
$463$	−28.5486	−1.32677	−0.663384	−	0.748279i	$-0.730880\pi$
$463$	−28.5486	−1.32677	−0.663384	+	0.748279i	$0.730880\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.5511	1.32119	0.660594	−	0.750743i	$-0.270305\pi$
$467$	28.5511	1.32119	0.660594	+	0.750743i	$0.270305\pi$
$468$	0	0
$469$	26.1461	1.20731
$470$	0	0
$471$	12.6583	0.583265
$472$	0	0
$473$	9.19876	0.422959
$474$	0	0
$475$	0	0
$476$	0	0
$477$	14.9935	0.686505
$478$	0	0
$479$	−35.9971	−1.64475	−0.822374	−	0.568947i	$-0.807351\pi$
$479$	−35.9971	−1.64475	−0.822374	+	0.568947i	$0.807351\pi$
$480$	0	0
$481$	−30.2391	−1.37878
$482$	0	0
$483$	−9.99462	−0.454771
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.5036	1.42757	0.713783	−	0.700367i	$-0.246980\pi$
$487$	31.5036	1.42757	0.713783	+	0.700367i	$0.246980\pi$
$488$	0	0
$489$	3.06187	0.138463
$490$	0	0
$491$	29.5694	1.33445	0.667224	−	0.744857i	$-0.267482\pi$
$491$	29.5694	1.33445	0.667224	+	0.744857i	$0.267482\pi$
$492$	0	0
$493$	13.7274	0.618249
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.6003	0.475490
$498$	0	0
$499$	12.8839	0.576761	0.288380	−	0.957516i	$-0.406883\pi$
$499$	12.8839	0.576761	0.288380	+	0.957516i	$0.406883\pi$
$500$	0	0
$501$	−6.89668	−0.308121
$502$	0	0
$503$	−25.8536	−1.15275	−0.576377	−	0.817184i	$-0.695534\pi$
$503$	−25.8536	−1.15275	−0.576377	+	0.817184i	$0.695534\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−0.262341	−0.0116510
$508$	0	0
$509$	−2.02461	−0.0897391	−0.0448695	−	0.998993i	$-0.514287\pi$
$509$	−2.02461	−0.0897391	−0.0448695	+	0.998993i	$0.514287\pi$
$510$	0	0
$511$	−21.6487	−0.957683
$512$	0	0
$513$	15.5497	0.686535
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.89676	0.259339
$518$	0	0
$519$	−4.02981	−0.176889
$520$	0	0
$521$	26.4387	1.15830	0.579150	−	0.815221i	$-0.303385\pi$
$521$	26.4387	1.15830	0.579150	+	0.815221i	$0.303385\pi$
$522$	0	0
$523$	12.0293	0.526005	0.263002	−	0.964795i	$-0.415287\pi$
$523$	12.0293	0.526005	0.263002	+	0.964795i	$0.415287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.66582	−0.0725643
$528$	0	0
$529$	−13.7559	−0.598084
$530$	0	0
$531$	20.6040	0.894136
$532$	0	0
$533$	22.1647	0.960061
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.5581	1.18922
$538$	0	0
$539$	−0.934007	−0.0402305
$540$	0	0
$541$	−29.2871	−1.25915	−0.629575	−	0.776940i	$-0.716771\pi$
$541$	−29.2871	−1.25915	−0.629575	+	0.776940i	$0.716771\pi$
$542$	0	0
$543$	0.642464	0.0275707
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−19.4869	−0.833199	−0.416600	−	0.909090i	$-0.636778\pi$
$547$	−19.4869	−0.833199	−0.416600	+	0.909090i	$0.636778\pi$
$548$	0	0
$549$	20.7281	0.884652
$550$	0	0
$551$	−23.5114	−1.00162
$552$	0	0
$553$	−1.42838	−0.0607410
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.9493	−0.548679	−0.274340	−	0.961633i	$-0.588459\pi$
$557$	−12.9493	−0.548679	−0.274340	+	0.961633i	$0.588459\pi$
$558$	0	0
$559$	−23.4374	−0.991295
$560$	0	0
$561$	−2.77496	−0.117159
$562$	0	0
$563$	−14.4115	−0.607372	−0.303686	−	0.952772i	$-0.598217\pi$
$563$	−14.4115	−0.607372	−0.303686	+	0.952772i	$0.598217\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.70635	0.197648
$568$	0	0
$569$	0.231111	0.00968867	0.00484434	−	0.999988i	$-0.498458\pi$
$569$	0.231111	0.00968867	0.00484434	+	0.999988i	$0.498458\pi$
$570$	0	0
$571$	46.0866	1.92867	0.964333	−	0.264693i	$-0.0852705\pi$
$571$	46.0866	1.92867	0.964333	+	0.264693i	$0.0852705\pi$
$572$	0	0
$573$	−4.04085	−0.168809
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.36752	0.348344	0.174172	−	0.984715i	$-0.444275\pi$
$577$	8.36752	0.348344	0.174172	+	0.984715i	$0.444275\pi$
$578$	0	0
$579$	31.6085	1.31360
$580$	0	0
$581$	14.2071	0.589411
$582$	0	0
$583$	13.2277	0.547835
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.8902	−0.738410	−0.369205	−	0.929348i	$-0.620370\pi$
$587$	−17.8902	−0.738410	−0.369205	+	0.929348i	$0.620370\pi$
$588$	0	0
$589$	2.85311	0.117561
$590$	0	0
$591$	7.43422	0.305803
$592$	0	0
$593$	20.0274	0.822428	0.411214	−	0.911539i	$-0.365105\pi$
$593$	20.0274	0.822428	0.411214	+	0.911539i	$0.365105\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.09140	0.167450
$598$	0	0
$599$	−11.2706	−0.460503	−0.230252	−	0.973131i	$-0.573955\pi$
$599$	−11.2706	−0.460503	−0.230252	+	0.973131i	$0.573955\pi$
$600$	0	0
$601$	41.3857	1.68816	0.844080	−	0.536218i	$-0.180147\pi$
$601$	41.3857	1.68816	0.844080	+	0.536218i	$0.180147\pi$
$602$	0	0
$603$	15.0182	0.611589
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.3612	−1.06997	−0.534984	−	0.844862i	$-0.679682\pi$
$607$	−26.3612	−1.06997	−0.534984	+	0.844862i	$0.679682\pi$
$608$	0	0
$609$	−27.0890	−1.09770
$610$	0	0
$611$	−15.0243	−0.607816
$612$	0	0
$613$	−15.2023	−0.614014	−0.307007	−	0.951707i	$-0.599328\pi$
$613$	−15.2023	−0.614014	−0.307007	+	0.951707i	$0.599328\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.33885	0.174675	0.0873377	−	0.996179i	$-0.472164\pi$
$617$	4.33885	0.174675	0.0873377	+	0.996179i	$0.472164\pi$
$618$	0	0
$619$	−26.3637	−1.05965	−0.529824	−	0.848107i	$-0.677742\pi$
$619$	−26.3637	−1.05965	−0.529824	+	0.848107i	$0.677742\pi$
$620$	0	0
$621$	−16.5704	−0.664949
$622$	0	0
$623$	−13.9307	−0.558120
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.75277	0.189807
$628$	0	0
$629$	−14.0912	−0.561852
$630$	0	0
$631$	1.08262	0.0430986	0.0215493	−	0.999768i	$-0.493140\pi$
$631$	1.08262	0.0430986	0.0215493	+	0.999768i	$0.493140\pi$
$632$	0	0
$633$	18.8695	0.749994
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.37974	0.0942888
$638$	0	0
$639$	6.08880	0.240869
$640$	0	0
$641$	41.5471	1.64101	0.820505	−	0.571639i	$-0.193692\pi$
$641$	41.5471	1.64101	0.820505	+	0.571639i	$0.193692\pi$
$642$	0	0
$643$	9.63971	0.380153	0.190077	−	0.981769i	$-0.439126\pi$
$643$	9.63971	0.380153	0.190077	+	0.981769i	$0.439126\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.8932	−1.33248	−0.666240	−	0.745737i	$-0.732097\pi$
$647$	−33.8932	−1.33248	−0.666240	+	0.745737i	$0.732097\pi$
$648$	0	0
$649$	18.1774	0.713526
$650$	0	0
$651$	3.28726	0.128838
$652$	0	0
$653$	10.6859	0.418172	0.209086	−	0.977897i	$-0.432951\pi$
$653$	10.6859	0.418172	0.209086	+	0.977897i	$0.432951\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−12.4350	−0.485134
$658$	0	0
$659$	−34.2886	−1.33569	−0.667847	−	0.744298i	$-0.732784\pi$
$659$	−34.2886	−1.33569	−0.667847	+	0.744298i	$0.732784\pi$
$660$	0	0
$661$	3.40224	0.132332	0.0661659	−	0.997809i	$-0.478923\pi$
$661$	3.40224	0.132332	0.0661659	+	0.997809i	$0.478923\pi$
$662$	0	0
$663$	7.07026	0.274586
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.0548	0.970125
$668$	0	0
$669$	11.3116	0.437330
$670$	0	0
$671$	18.2869	0.705958
$672$	0	0
$673$	41.3344	1.59333	0.796663	−	0.604424i	$-0.206597\pi$
$673$	41.3344	1.59333	0.796663	+	0.604424i	$0.206597\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.0729	0.387134	0.193567	−	0.981087i	$-0.437994\pi$
$677$	10.0729	0.387134	0.193567	+	0.981087i	$0.437994\pi$
$678$	0	0
$679$	14.6212	0.561110
$680$	0	0
$681$	−13.2206	−0.506614
$682$	0	0
$683$	5.91429	0.226304	0.113152	−	0.993578i	$-0.463905\pi$
$683$	5.91429	0.226304	0.113152	+	0.993578i	$0.463905\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15.0627	0.574678
$688$	0	0
$689$	−33.7026	−1.28397
$690$	0	0
$691$	19.6565	0.747770	0.373885	−	0.927475i	$-0.378025\pi$
$691$	19.6565	0.747770	0.373885	+	0.927475i	$0.378025\pi$
$692$	0	0
$693$	−6.17780	−0.234675
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.3286	0.391223
$698$	0	0
$699$	4.87090	0.184234
$700$	0	0
$701$	−10.3363	−0.390397	−0.195198	−	0.980764i	$-0.562535\pi$
$701$	−10.3363	−0.390397	−0.195198	+	0.980764i	$0.562535\pi$
$702$	0	0
$703$	24.1345	0.910250
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.3535	1.51765
$708$	0	0
$709$	−45.7547	−1.71835	−0.859177	−	0.511678i	$-0.829024\pi$
$709$	−45.7547	−1.71835	−0.859177	+	0.511678i	$0.829024\pi$
$710$	0	0
$711$	−0.820458	−0.0307696
$712$	0	0
$713$	−3.04041	−0.113864
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.310743	0.0116049
$718$	0	0
$719$	−48.1601	−1.79607	−0.898035	−	0.439924i	$-0.855005\pi$
$719$	−48.1601	−1.79607	−0.898035	+	0.439924i	$0.855005\pi$
$720$	0	0
$721$	27.3164	1.01731
$722$	0	0
$723$	2.56717	0.0954739
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12.8283	0.475774	0.237887	−	0.971293i	$-0.423545\pi$
$727$	12.8283	0.475774	0.237887	+	0.971293i	$0.423545\pi$
$728$	0	0
$729$	22.1158	0.819105
$730$	0	0
$731$	−10.9216	−0.403951
$732$	0	0
$733$	23.1228	0.854058	0.427029	−	0.904238i	$-0.359560\pi$
$733$	23.1228	0.854058	0.427029	+	0.904238i	$0.359560\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.2495	0.488053
$738$	0	0
$739$	−30.9000	−1.13668	−0.568338	−	0.822795i	$-0.692414\pi$
$739$	−30.9000	−1.13668	−0.568338	+	0.822795i	$0.692414\pi$
$740$	0	0
$741$	−12.1095	−0.444854
$742$	0	0
$743$	19.3986	0.711667	0.355833	−	0.934549i	$-0.384197\pi$
$743$	19.3986	0.711667	0.355833	+	0.934549i	$0.384197\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.16053	0.298578
$748$	0	0
$749$	30.1168	1.10044
$750$	0	0
$751$	19.3054	0.704465	0.352232	−	0.935913i	$-0.385423\pi$
$751$	19.3054	0.704465	0.352232	+	0.935913i	$0.385423\pi$
$752$	0	0
$753$	−4.70518	−0.171466
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.3214	1.57454	0.787271	−	0.616607i	$-0.211493\pi$
$757$	43.3214	1.57454	0.787271	+	0.616607i	$0.211493\pi$
$758$	0	0
$759$	−5.06477	−0.183840
$760$	0	0
$761$	−31.0269	−1.12472	−0.562361	−	0.826892i	$-0.690107\pi$
$761$	−31.0269	−1.12472	−0.562361	+	0.826892i	$0.690107\pi$
$762$	0	0
$763$	−21.8538	−0.791159
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−46.3139	−1.67230
$768$	0	0
$769$	34.3285	1.23792	0.618958	−	0.785424i	$-0.287555\pi$
$769$	34.3285	1.23792	0.618958	+	0.785424i	$0.287555\pi$
$770$	0	0
$771$	18.3989	0.662620
$772$	0	0
$773$	−14.8797	−0.535185	−0.267593	−	0.963532i	$-0.586228\pi$
$773$	−14.8797	−0.535185	−0.267593	+	0.963532i	$0.586228\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.8069	0.997569
$778$	0	0
$779$	−17.6902	−0.633817
$780$	0	0
$781$	5.37172	0.192215
$782$	0	0
$783$	−44.9119	−1.60502
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.3320	−0.439589	−0.219795	−	0.975546i	$-0.570539\pi$
$787$	−12.3320	−0.439589	−0.219795	+	0.975546i	$0.570539\pi$
$788$	0	0
$789$	8.00937	0.285141
$790$	0	0
$791$	44.4615	1.58087
$792$	0	0
$793$	−46.5929	−1.65456
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.1782	−1.13981	−0.569906	−	0.821710i	$-0.693020\pi$
$797$	−32.1782	−1.13981	−0.569906	+	0.821710i	$0.693020\pi$
$798$	0	0
$799$	−7.00119	−0.247684
$800$	0	0
$801$	−8.00173	−0.282727
$802$	0	0
$803$	−10.9705	−0.387140
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13.2519	0.466488
$808$	0	0
$809$	−2.64407	−0.0929604	−0.0464802	−	0.998919i	$-0.514800\pi$
$809$	−2.64407	−0.0929604	−0.0464802	+	0.998919i	$0.514800\pi$
$810$	0	0
$811$	−30.2848	−1.06344	−0.531721	−	0.846919i	$-0.678455\pi$
$811$	−30.2848	−1.06344	−0.531721	+	0.846919i	$0.678455\pi$
$812$	0	0
$813$	−5.95923	−0.208999
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.7059	0.654437
$818$	0	0
$819$	15.7403	0.550011
$820$	0	0
$821$	57.1572	1.99480	0.997401	−	0.0720522i	$-0.0229548\pi$
$821$	57.1572	1.99480	0.997401	+	0.0720522i	$0.0229548\pi$
$822$	0	0
$823$	40.2662	1.40359	0.701796	−	0.712378i	$-0.252382\pi$
$823$	40.2662	1.40359	0.701796	+	0.712378i	$0.252382\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.09558	−0.0728704	−0.0364352	−	0.999336i	$-0.511600\pi$
$827$	−2.09558	−0.0728704	−0.0364352	+	0.999336i	$0.511600\pi$
$828$	0	0
$829$	34.3922	1.19449	0.597245	−	0.802059i	$-0.296262\pi$
$829$	34.3922	1.19449	0.597245	+	0.802059i	$0.296262\pi$
$830$	0	0
$831$	28.0340	0.972490
$832$	0	0
$833$	1.10894	0.0384225
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.45007	0.188382
$838$	0	0
$839$	−48.2326	−1.66517	−0.832587	−	0.553894i	$-0.813141\pi$
$839$	−48.2326	−1.66517	−0.832587	+	0.553894i	$0.813141\pi$
$840$	0	0
$841$	38.9075	1.34164
$842$	0	0
$843$	5.33881	0.183878
$844$	0	0
$845$	0	0
$846$	0	0
$847$	25.0055	0.859198
$848$	0	0
$849$	−9.30822	−0.319457
$850$	0	0
$851$	−25.7188	−0.881630
$852$	0	0
$853$	8.14503	0.278881	0.139440	−	0.990230i	$-0.455470\pi$
$853$	8.14503	0.278881	0.139440	+	0.990230i	$0.455470\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.32650	0.250268	0.125134	−	0.992140i	$-0.460064\pi$
$857$	7.32650	0.250268	0.125134	+	0.992140i	$0.460064\pi$
$858$	0	0
$859$	36.7117	1.25259	0.626294	−	0.779587i	$-0.284571\pi$
$859$	36.7117	1.25259	0.626294	+	0.779587i	$0.284571\pi$
$860$	0	0
$861$	−20.3820	−0.694618
$862$	0	0
$863$	−31.9983	−1.08924	−0.544618	−	0.838684i	$-0.683325\pi$
$863$	−31.9983	−1.08924	−0.544618	+	0.838684i	$0.683325\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.8893	−0.573591
$868$	0	0
$869$	−0.723833	−0.0245543
$870$	0	0
$871$	−33.7582	−1.14385
$872$	0	0
$873$	8.39837	0.284242
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.7897	0.364344	0.182172	−	0.983267i	$-0.441687\pi$
$877$	10.7897	0.364344	0.182172	+	0.983267i	$0.441687\pi$
$878$	0	0
$879$	−14.8399	−0.500538
$880$	0	0
$881$	32.4394	1.09291	0.546455	−	0.837489i	$-0.315977\pi$
$881$	32.4394	1.09291	0.546455	+	0.837489i	$0.315977\pi$
$882$	0	0
$883$	−13.5500	−0.455993	−0.227997	−	0.973662i	$-0.573218\pi$
$883$	−13.5500	−0.455993	−0.227997	+	0.973662i	$0.573218\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.3939	0.886221	0.443111	−	0.896467i	$-0.353875\pi$
$887$	26.3939	0.886221	0.443111	+	0.896467i	$0.353875\pi$
$888$	0	0
$889$	32.3669	1.08555
$890$	0	0
$891$	2.38494	0.0798986
$892$	0	0
$893$	11.9912	0.401270
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.9044	0.430867
$898$	0	0
$899$	−8.24060	−0.274839
$900$	0	0
$901$	−15.7052	−0.523215
$902$	0	0
$903$	21.5523	0.717216
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53.7213	1.78379	0.891893	−	0.452246i	$-0.149377\pi$
$907$	53.7213	1.78379	0.891893	+	0.452246i	$0.149377\pi$
$908$	0	0
$909$	23.1789	0.768797
$910$	0	0
$911$	30.8467	1.02200	0.510999	−	0.859581i	$-0.329276\pi$
$911$	30.8467	1.02200	0.510999	+	0.859581i	$0.329276\pi$
$912$	0	0
$913$	7.19946	0.238267
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−51.4367	−1.69859
$918$	0	0
$919$	5.79252	0.191078	0.0955388	−	0.995426i	$-0.469543\pi$
$919$	5.79252	0.191078	0.0955388	+	0.995426i	$0.469543\pi$
$920$	0	0
$921$	−23.1278	−0.762086
$922$	0	0
$923$	−13.6865	−0.450497
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.6904	0.515341
$928$	0	0
$929$	−5.29659	−0.173776	−0.0868878	−	0.996218i	$-0.527692\pi$
$929$	−5.29659	−0.173776	−0.0868878	+	0.996218i	$0.527692\pi$
$930$	0	0
$931$	−1.89933	−0.0622479
$932$	0	0
$933$	17.7501	0.581113
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.5112	0.637403	0.318701	−	0.947855i	$-0.396753\pi$
$937$	19.5112	0.637403	0.318701	+	0.947855i	$0.396753\pi$
$938$	0	0
$939$	−23.2071	−0.757335
$940$	0	0
$941$	−32.9560	−1.07434	−0.537168	−	0.843476i	$-0.680506\pi$
$941$	−32.9560	−1.07434	−0.537168	+	0.843476i	$0.680506\pi$
$942$	0	0
$943$	18.8515	0.613888
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.0379	1.52853	0.764263	−	0.644905i	$-0.223103\pi$
$947$	47.0379	1.52853	0.764263	+	0.644905i	$0.223103\pi$
$948$	0	0
$949$	27.9515	0.907345
$950$	0	0
$951$	−36.4750	−1.18278
$952$	0	0
$953$	−9.00688	−0.291762	−0.145881	−	0.989302i	$-0.546602\pi$
$953$	−9.00688	−0.291762	−0.145881	+	0.989302i	$0.546602\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.7274	−0.443742
$958$	0	0
$959$	13.0404	0.421097
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	17.2990	0.557452
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−32.7556	−1.05335	−0.526675	−	0.850067i	$-0.676561\pi$
$967$	−32.7556	−1.05335	−0.526675	+	0.850067i	$0.676561\pi$
$968$	0	0
$969$	−5.64294	−0.181277
$970$	0	0
$971$	−46.9171	−1.50564	−0.752821	−	0.658225i	$-0.771308\pi$
$971$	−46.9171	−1.50564	−0.752821	+	0.658225i	$0.771308\pi$
$972$	0	0
$973$	−29.9335	−0.959626
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.9605	1.02251	0.511253	−	0.859430i	$-0.329182\pi$
$977$	31.9605	1.02251	0.511253	+	0.859430i	$0.329182\pi$
$978$	0	0
$979$	−7.05936	−0.225618
$980$	0	0
$981$	−12.5527	−0.400778
$982$	0	0
$983$	−36.8703	−1.17598	−0.587990	−	0.808868i	$-0.700081\pi$
$983$	−36.8703	−1.17598	−0.587990	+	0.808868i	$0.700081\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.8159	0.439764
$988$	0	0
$989$	−19.9339	−0.633860
$990$	0	0
$991$	26.8105	0.851665	0.425832	−	0.904802i	$-0.359981\pi$
$991$	26.8105	0.851665	0.425832	+	0.904802i	$0.359981\pi$
$992$	0	0
$993$	14.5850	0.462843
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.5656	1.03136	0.515681	−	0.856781i	$-0.327539\pi$
$997$	32.5656	1.03136	0.515681	+	0.856781i	$0.327539\pi$
$998$	0	0
$999$	46.1022	1.45861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.2.a.j.1.4	✓	5
5.2	odd	4		3100.2.c.h.249.4		10
5.3	odd	4		3100.2.c.h.249.7		10
5.4	even	2		3100.2.a.m.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.2.a.j.1.4	✓	5	1.1	even	1	trivial
3100.2.a.m.1.2	yes	5	5.4	even	2
3100.2.c.h.249.4		10	5.2	odd	4
3100.2.c.h.249.7		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3100.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18729 q^{3} -2.76870 q^{7} -1.59033 q^{9} +O(q^{10})\)	\(q+1.18729 q^{3} -2.76870 q^{7} -1.59033 q^{9} -1.40304 q^{11} +3.57478 q^{13} +1.66582 q^{17} -2.85311 q^{19} -3.28726 q^{21} +3.04041 q^{23} -5.45007 q^{27} +8.24060 q^{29} -1.00000 q^{31} -1.66582 q^{33} -8.45900 q^{37} +4.24431 q^{39} +6.20031 q^{41} -6.55631 q^{43} -4.20285 q^{47} +0.665703 q^{49} +1.97782 q^{51} -9.42789 q^{53} -3.38748 q^{57} -12.9558 q^{59} -13.0338 q^{61} +4.40316 q^{63} -9.44345 q^{67} +3.60986 q^{69} -3.82863 q^{71} +7.81909 q^{73} +3.88460 q^{77} +0.515903 q^{79} -1.69984 q^{81} -5.13133 q^{83} +9.78401 q^{87} +5.03148 q^{89} -9.89749 q^{91} -1.18729 q^{93} -5.28089 q^{97} +2.23130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 - 4 * q^7 + 6 * q^9	\( 5 q - 3 q^{3} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{13} - 3 q^{17} + 6 q^{19} - 3 q^{21} - 14 q^{23} - 12 q^{27} + 10 q^{29} - 5 q^{31} + 3 q^{33} - 14 q^{37} + 3 q^{39} - 6 q^{41} - q^{43} - 18 q^{47} + 27 q^{49} - 15 q^{51} - 7 q^{53} - 6 q^{57} + 20 q^{59} - 17 q^{61} - 18 q^{63} - 13 q^{67} + 30 q^{69} - 6 q^{71} + q^{73} - 17 q^{77} - 15 q^{81} - 40 q^{83} - 21 q^{87} + 9 q^{89} - 20 q^{91} + 3 q^{93} - q^{97} + 21 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 - 4 * q^7 + 6 * q^9 - 2 * q^11 - 2 * q^13 - 3 * q^17 + 6 * q^19 - 3 * q^21 - 14 * q^23 - 12 * q^27 + 10 * q^29 - 5 * q^31 + 3 * q^33 - 14 * q^37 + 3 * q^39 - 6 * q^41 - q^43 - 18 * q^47 + 27 * q^49 - 15 * q^51 - 7 * q^53 - 6 * q^57 + 20 * q^59 - 17 * q^61 - 18 * q^63 - 13 * q^67 + 30 * q^69 - 6 * q^71 + q^73 - 17 * q^77 - 15 * q^81 - 40 * q^83 - 21 * q^87 + 9 * q^89 - 20 * q^91 + 3 * q^93 - q^97 + 21 * q^99

Introduction

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