LMFDB - Embedded newform 6975.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6975,2,Mod(1,6975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6975, 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("6975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.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:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.20308.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 12 $ x^4 - x^3 - 8x^2 + 4x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.62946$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.62946 q^{2} +0.655151 q^{4} +2.97431 q^{7} +2.19138 q^{8} +O(q^{10})$	$q-1.62946 q^{2} +0.655151 q^{4} +2.97431 q^{7} +2.19138 q^{8} +1.71539 q^{11} -6.97431 q^{13} -4.84653 q^{14} -4.88108 q^{16} -2.76812 q^{17} +5.47600 q^{19} -2.79516 q^{22} +0.127779 q^{23} +11.3644 q^{26} +1.94862 q^{28} +7.07977 q^{29} +1.00000 q^{31} +3.57078 q^{32} +4.51054 q^{34} -8.02704 q^{37} -8.92294 q^{38} -4.50168 q^{41} -4.76812 q^{43} +1.12384 q^{44} -0.208211 q^{46} +2.10546 q^{47} +1.84653 q^{49} -4.56923 q^{52} +5.20957 q^{53} +6.51785 q^{56} -11.5362 q^{58} -13.2968 q^{59} -0.105460 q^{61} -1.62946 q^{62} +3.94371 q^{64} +2.28461 q^{67} -1.81353 q^{68} -5.31916 q^{71} +0.640336 q^{73} +13.0798 q^{74} +3.58761 q^{76} +5.10209 q^{77} +4.92294 q^{79} +7.33533 q^{82} +1.87021 q^{83} +7.76947 q^{86} +3.75906 q^{88} +11.4922 q^{89} -20.7438 q^{91} +0.0837146 q^{92} -3.43077 q^{94} -6.84653 q^{97} -3.00886 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 9 q^{4} - 9 q^{8}+O(q^{10}) $ 4 * q - q^2 + 9 * q^4 - 9 * q^8	$ 4 q - q^{2} + 9 q^{4} - 9 q^{8} + 6 q^{11} - 16 q^{13} - 8 q^{14} + 11 q^{16} + q^{17} + 5 q^{19} + 24 q^{22} + 12 q^{26} - 16 q^{28} - 6 q^{29} + 4 q^{31} - 29 q^{32} - 18 q^{34} - 9 q^{37} - 13 q^{41} - 7 q^{43} - 20 q^{44} - 26 q^{46} - 14 q^{47} - 4 q^{49} - 20 q^{52} + 11 q^{53} + 4 q^{56} - 22 q^{58} - 13 q^{59} + 22 q^{61} - q^{62} + 47 q^{64} + 10 q^{67} + 30 q^{68} - 3 q^{71} - 9 q^{73} + 18 q^{74} + 14 q^{76} + 8 q^{77} - 16 q^{79} - 6 q^{82} - 17 q^{83} - 16 q^{86} + 44 q^{88} + 12 q^{89} - 24 q^{91} - 10 q^{92} - 12 q^{94} - 16 q^{97} + 19 q^{98}+O(q^{100}) $ 4 * q - q^2 + 9 * q^4 - 9 * q^8 + 6 * q^11 - 16 * q^13 - 8 * q^14 + 11 * q^16 + q^17 + 5 * q^19 + 24 * q^22 + 12 * q^26 - 16 * q^28 - 6 * q^29 + 4 * q^31 - 29 * q^32 - 18 * q^34 - 9 * q^37 - 13 * q^41 - 7 * q^43 - 20 * q^44 - 26 * q^46 - 14 * q^47 - 4 * q^49 - 20 * q^52 + 11 * q^53 + 4 * q^56 - 22 * q^58 - 13 * q^59 + 22 * q^61 - q^62 + 47 * q^64 + 10 * q^67 + 30 * q^68 - 3 * q^71 - 9 * q^73 + 18 * q^74 + 14 * q^76 + 8 * q^77 - 16 * q^79 - 6 * q^82 - 17 * q^83 - 16 * q^86 + 44 * q^88 + 12 * q^89 - 24 * q^91 - 10 * q^92 - 12 * q^94 - 16 * q^97 + 19 * q^98

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 $

$2$	−1.62946	−1.15220	−0.576102	−	0.817378i	$-0.695427\pi$
$2$	−1.62946	−1.15220	−0.576102	+	0.817378i	$0.695427\pi$
$3$	0	0
$3$	0	0
$4$	0.655151	0.327576
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.97431	1.12418	0.562092	−	0.827075i	$-0.309997\pi$
$7$	2.97431	1.12418	0.562092	+	0.827075i	$0.309997\pi$
$8$	2.19138	0.774771
$9$	0	0
$10$	0	0
$11$	1.71539	0.517208	0.258604	−	0.965983i	$-0.416738\pi$
$11$	1.71539	0.517208	0.258604	+	0.965983i	$0.416738\pi$
$12$	0	0
$13$	−6.97431	−1.93433	−0.967163	−	0.254157i	$-0.918202\pi$
$13$	−6.97431	−1.93433	−0.967163	+	0.254157i	$0.918202\pi$
$14$	−4.84653	−1.29529
$15$	0	0
$16$	−4.88108	−1.22027
$17$	−2.76812	−0.671367	−0.335683	−	0.941975i	$-0.608967\pi$
$17$	−2.76812	−0.671367	−0.335683	+	0.941975i	$0.608967\pi$
$18$	0	0
$19$	5.47600	1.25628	0.628140	−	0.778100i	$-0.283817\pi$
$19$	5.47600	1.25628	0.628140	+	0.778100i	$0.283817\pi$
$20$	0	0
$21$	0	0
$22$	−2.79516	−0.595930
$23$	0.127779	0.0266438	0.0133219	−	0.999911i	$-0.495759\pi$
$23$	0.127779	0.0266438	0.0133219	+	0.999911i	$0.495759\pi$
$24$	0	0
$25$	0	0
$26$	11.3644	2.22874
$27$	0	0
$28$	1.94862	0.368255
$29$	7.07977	1.31468	0.657340	−	0.753594i	$-0.271681\pi$
$29$	7.07977	1.31468	0.657340	+	0.753594i	$0.271681\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.57078	0.631230
$33$	0	0
$34$	4.51054	0.773552
$35$	0	0
$36$	0	0
$37$	−8.02704	−1.31964	−0.659819	−	0.751425i	$-0.729367\pi$
$37$	−8.02704	−1.31964	−0.659819	+	0.751425i	$0.729367\pi$
$38$	−8.92294	−1.44749
$39$	0	0
$40$	0	0
$41$	−4.50168	−0.703045	−0.351522	−	0.936179i	$-0.614336\pi$
$41$	−4.50168	−0.703045	−0.351522	+	0.936179i	$0.614336\pi$
$42$	0	0
$43$	−4.76812	−0.727131	−0.363565	−	0.931569i	$-0.618441\pi$
$43$	−4.76812	−0.727131	−0.363565	+	0.931569i	$0.618441\pi$
$44$	1.12384	0.169425
$45$	0	0
$46$	−0.208211	−0.0306991
$47$	2.10546	0.307113	0.153556	−	0.988140i	$-0.450927\pi$
$47$	2.10546	0.307113	0.153556	+	0.988140i	$0.450927\pi$
$48$	0	0
$49$	1.84653	0.263790
$50$	0	0
$51$	0	0
$52$	−4.56923	−0.633638
$53$	5.20957	0.715589	0.357794	−	0.933800i	$-0.383529\pi$
$53$	5.20957	0.715589	0.357794	+	0.933800i	$0.383529\pi$
$54$	0	0
$55$	0	0
$56$	6.51785	0.870985
$57$	0	0
$58$	−11.5362	−1.51478
$59$	−13.2968	−1.73110	−0.865551	−	0.500821i	$-0.833031\pi$
$59$	−13.2968	−1.73110	−0.865551	+	0.500821i	$0.833031\pi$
$60$	0	0
$61$	−0.105460	−0.0135028	−0.00675140	−	0.999977i	$-0.502149\pi$
$61$	−0.105460	−0.0135028	−0.00675140	+	0.999977i	$0.502149\pi$
$62$	−1.62946	−0.206942
$63$	0	0
$64$	3.94371	0.492964
$65$	0	0
$66$	0	0
$67$	2.28461	0.279110	0.139555	−	0.990214i	$-0.455433\pi$
$67$	2.28461	0.279110	0.139555	+	0.990214i	$0.455433\pi$
$68$	−1.81353	−0.219923
$69$	0	0
$70$	0	0
$71$	−5.31916	−0.631268	−0.315634	−	0.948881i	$-0.602217\pi$
$71$	−5.31916	−0.631268	−0.315634	+	0.948881i	$0.602217\pi$
$72$	0	0
$73$	0.640336	0.0749457	0.0374728	−	0.999298i	$-0.488069\pi$
$73$	0.640336	0.0749457	0.0374728	+	0.999298i	$0.488069\pi$
$74$	13.0798	1.52049
$75$	0	0
$76$	3.58761	0.411527
$77$	5.10209	0.581437
$78$	0	0
$79$	4.92294	0.553874	0.276937	−	0.960888i	$-0.410681\pi$
$79$	4.92294	0.553874	0.276937	+	0.960888i	$0.410681\pi$
$80$	0	0
$81$	0	0
$82$	7.33533	0.810052
$83$	1.87021	0.205282	0.102641	−	0.994718i	$-0.467271\pi$
$83$	1.87021	0.205282	0.102641	+	0.994718i	$0.467271\pi$
$84$	0	0
$85$	0	0
$86$	7.76947	0.837803
$87$	0	0
$88$	3.75906	0.400718
$89$	11.4922	1.21817	0.609084	−	0.793106i	$-0.291537\pi$
$89$	11.4922	1.21817	0.609084	+	0.793106i	$0.291537\pi$
$90$	0	0
$91$	−20.7438	−2.17454
$92$	0.0837146	0.00872785
$93$	0	0
$94$	−3.43077	−0.353857
$95$	0	0
$96$	0	0
$97$	−6.84653	−0.695160	−0.347580	−	0.937650i	$-0.612997\pi$
$97$	−6.84653	−0.695160	−0.347580	+	0.937650i	$0.612997\pi$
$98$	−3.00886	−0.303941
$99$	0	0
$100$	0	0
$101$	−18.2712	−1.81805	−0.909024	−	0.416744i	$-0.863171\pi$
$101$	−18.2712	−1.81805	−0.909024	+	0.416744i	$0.863171\pi$
$102$	0	0
$103$	1.20484	0.118717	0.0593583	−	0.998237i	$-0.481095\pi$
$103$	1.20484	0.118717	0.0593583	+	0.998237i	$0.481095\pi$
$104$	−15.2834	−1.49866
$105$	0	0
$106$	−8.48880	−0.824505
$107$	−5.79179	−0.559913	−0.279957	−	0.960013i	$-0.590320\pi$
$107$	−5.79179	−0.559913	−0.279957	+	0.960013i	$0.590320\pi$
$108$	0	0
$109$	−5.42462	−0.519584	−0.259792	−	0.965665i	$-0.583654\pi$
$109$	−5.42462	−0.519584	−0.259792	+	0.965665i	$0.583654\pi$
$110$	0	0
$111$	0	0
$112$	−14.5179	−1.37181
$113$	−2.61329	−0.245838	−0.122919	−	0.992417i	$-0.539226\pi$
$113$	−2.61329	−0.245838	−0.122919	+	0.992417i	$0.539226\pi$
$114$	0	0
$115$	0	0
$116$	4.63832	0.430657
$117$	0	0
$118$	21.6667	1.99458
$119$	−8.23324	−0.754740
$120$	0	0
$121$	−8.05745	−0.732496
$122$	0.171843	0.0155580
$123$	0	0
$124$	0.655151	0.0588343
$125$	0	0
$126$	0	0
$127$	6.66738	0.591634	0.295817	−	0.955245i	$-0.404408\pi$
$127$	6.66738	0.591634	0.295817	+	0.955245i	$0.404408\pi$
$128$	−13.5677	−1.19923
$129$	0	0
$130$	0	0
$131$	−14.8336	−1.29602	−0.648011	−	0.761631i	$-0.724399\pi$
$131$	−14.8336	−1.29602	−0.648011	+	0.761631i	$0.724399\pi$
$132$	0	0
$133$	16.2873	1.41229
$134$	−3.72270	−0.321592
$135$	0	0
$136$	−6.06600	−0.520155
$137$	−6.10074	−0.521221	−0.260611	−	0.965444i	$-0.583924\pi$
$137$	−6.10074	−0.521221	−0.260611	+	0.965444i	$0.583924\pi$
$138$	0	0
$139$	0.112771	0.00956514	0.00478257	−	0.999989i	$-0.498478\pi$
$139$	0.112771	0.00956514	0.00478257	+	0.999989i	$0.498478\pi$
$140$	0	0
$141$	0	0
$142$	8.66738	0.727350
$143$	−11.9636	−1.00045
$144$	0	0
$145$	0	0
$146$	−1.04340	−0.0863528
$147$	0	0
$148$	−5.25893	−0.432281
$149$	7.98654	0.654283	0.327141	−	0.944975i	$-0.393915\pi$
$149$	7.98654	0.654283	0.327141	+	0.944975i	$0.393915\pi$
$150$	0	0
$151$	−20.8566	−1.69728	−0.848641	−	0.528969i	$-0.822579\pi$
$151$	−20.8566	−1.69728	−0.848641	+	0.528969i	$0.822579\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	−8.31367	−0.669935
$155$	0	0
$156$	0	0
$157$	−15.0798	−1.20350	−0.601748	−	0.798686i	$-0.705529\pi$
$157$	−15.0798	−1.20350	−0.601748	+	0.798686i	$0.705529\pi$
$158$	−8.02175	−0.638176
$159$	0	0
$160$	0	0
$161$	0.380055	0.0299525
$162$	0	0
$163$	−4.27730	−0.335024	−0.167512	−	0.985870i	$-0.553573\pi$
$163$	−4.27730	−0.335024	−0.167512	+	0.985870i	$0.553573\pi$
$164$	−2.94928	−0.230300
$165$	0	0
$166$	−3.04743	−0.236527
$167$	−14.5599	−1.12668	−0.563340	−	0.826225i	$-0.690484\pi$
$167$	−14.5599	−1.12668	−0.563340	+	0.826225i	$0.690484\pi$
$168$	0	0
$169$	35.6410	2.74162
$170$	0	0
$171$	0	0
$172$	−3.12384	−0.238190
$173$	23.5463	1.79019	0.895094	−	0.445877i	$-0.147108\pi$
$173$	23.5463	1.79019	0.895094	+	0.445877i	$0.147108\pi$
$174$	0	0
$175$	0	0
$176$	−8.37293	−0.631133
$177$	0	0
$178$	−18.7261	−1.40358
$179$	8.86885	0.662889	0.331445	−	0.943475i	$-0.392464\pi$
$179$	8.86885	0.662889	0.331445	+	0.943475i	$0.392464\pi$
$180$	0	0
$181$	13.1088	0.974372	0.487186	−	0.873298i	$-0.338023\pi$
$181$	13.1088	0.974372	0.487186	+	0.873298i	$0.338023\pi$
$182$	33.8012	2.50551
$183$	0	0
$184$	0.280013	0.0206428
$185$	0	0
$186$	0	0
$187$	−4.74838	−0.347236
$188$	1.37939	0.100603
$189$	0	0
$190$	0	0
$191$	−8.53286	−0.617416	−0.308708	−	0.951157i	$-0.599897\pi$
$191$	−8.53286	−0.617416	−0.308708	+	0.951157i	$0.599897\pi$
$192$	0	0
$193$	−16.1595	−1.16319	−0.581595	−	0.813479i	$-0.697571\pi$
$193$	−16.1595	−1.16319	−0.581595	+	0.813479i	$0.697571\pi$
$194$	11.1562	0.800967
$195$	0	0
$196$	1.20976	0.0864113
$197$	−1.02569	−0.0730772	−0.0365386	−	0.999332i	$-0.511633\pi$
$197$	−1.02569	−0.0730772	−0.0365386	+	0.999332i	$0.511633\pi$
$198$	0	0
$199$	26.6701	1.89059	0.945296	−	0.326213i	$-0.105773\pi$
$199$	26.6701	1.89059	0.945296	+	0.326213i	$0.105773\pi$
$200$	0	0
$201$	0	0
$202$	29.7722	2.09476
$203$	21.0575	1.47794
$204$	0	0
$205$	0	0
$206$	−1.96325	−0.136786
$207$	0	0
$208$	34.0422	2.36040
$209$	9.39344	0.649758
$210$	0	0
$211$	15.0061	1.03306	0.516531	−	0.856269i	$-0.327223\pi$
$211$	15.0061	1.03306	0.516531	+	0.856269i	$0.327223\pi$
$212$	3.41305	0.234409
$213$	0	0
$214$	9.43751	0.645135
$215$	0	0
$216$	0	0
$217$	2.97431	0.201909
$218$	8.83922	0.598668
$219$	0	0
$220$	0	0
$221$	19.3057	1.29864
$222$	0	0
$223$	8.36574	0.560211	0.280106	−	0.959969i	$-0.409630\pi$
$223$	8.36574	0.560211	0.280106	+	0.959969i	$0.409630\pi$
$224$	10.6206	0.709619
$225$	0	0
$226$	4.25827	0.283256
$227$	14.7147	0.976651	0.488325	−	0.872662i	$-0.337608\pi$
$227$	14.7147	0.976651	0.488325	+	0.872662i	$0.337608\pi$
$228$	0	0
$229$	−18.4659	−1.22026	−0.610131	−	0.792301i	$-0.708883\pi$
$229$	−18.4659	−1.22026	−0.610131	+	0.792301i	$0.708883\pi$
$230$	0	0
$231$	0	0
$232$	15.5145	1.01858
$233$	−0.0736944	−0.00482788	−0.00241394	−	0.999997i	$-0.500768\pi$
$233$	−0.0736944	−0.00482788	−0.00241394	+	0.999997i	$0.500768\pi$
$234$	0	0
$235$	0	0
$236$	−8.71144	−0.567067
$237$	0	0
$238$	13.4158	0.869615
$239$	−20.0758	−1.29860	−0.649299	−	0.760533i	$-0.724938\pi$
$239$	−20.0758	−1.29860	−0.649299	+	0.760533i	$0.724938\pi$
$240$	0	0
$241$	−3.49217	−0.224950	−0.112475	−	0.993655i	$-0.535878\pi$
$241$	−3.49217	−0.224950	−0.112475	+	0.993655i	$0.535878\pi$
$242$	13.1293	0.843985
$243$	0	0
$244$	−0.0690924	−0.00442319
$245$	0	0
$246$	0	0
$247$	−38.1913	−2.43005
$248$	2.19138	0.139153
$249$	0	0
$250$	0	0
$251$	−18.9520	−1.19624	−0.598120	−	0.801407i	$-0.704085\pi$
$251$	−18.9520	−1.19624	−0.598120	+	0.801407i	$0.704085\pi$
$252$	0	0
$253$	0.219190	0.0137804
$254$	−10.8642	−0.681684
$255$	0	0
$256$	14.2206	0.888789
$257$	6.73376	0.420041	0.210020	−	0.977697i	$-0.432647\pi$
$257$	6.73376	0.420041	0.210020	+	0.977697i	$0.432647\pi$
$258$	0	0
$259$	−23.8749	−1.48352
$260$	0	0
$261$	0	0
$262$	24.1709	1.49328
$263$	−13.5342	−0.834556	−0.417278	−	0.908779i	$-0.637016\pi$
$263$	−13.5342	−0.834556	−0.417278	+	0.908779i	$0.637016\pi$
$264$	0	0
$265$	0	0
$266$	−26.5396	−1.62725
$267$	0	0
$268$	1.49677	0.0914297
$269$	7.04340	0.429444	0.214722	−	0.976675i	$-0.431115\pi$
$269$	7.04340	0.429444	0.214722	+	0.976675i	$0.431115\pi$
$270$	0	0
$271$	11.8208	0.718065	0.359033	−	0.933325i	$-0.383107\pi$
$271$	11.8208	0.718065	0.359033	+	0.933325i	$0.383107\pi$
$272$	13.5114	0.819248
$273$	0	0
$274$	9.94093	0.600553
$275$	0	0
$276$	0	0
$277$	22.9350	1.37803	0.689014	−	0.724748i	$-0.258044\pi$
$277$	22.9350	1.37803	0.689014	+	0.724748i	$0.258044\pi$
$278$	−0.183757	−0.0110210
$279$	0	0
$280$	0	0
$281$	14.4603	0.862631	0.431315	−	0.902201i	$-0.358050\pi$
$281$	14.4603	0.862631	0.431315	+	0.902201i	$0.358050\pi$
$282$	0	0
$283$	27.7985	1.65245	0.826225	−	0.563340i	$-0.190484\pi$
$283$	27.7985	1.65245	0.826225	+	0.563340i	$0.190484\pi$
$284$	−3.48486	−0.206788
$285$	0	0
$286$	19.4943	1.15272
$287$	−13.3894	−0.790352
$288$	0	0
$289$	−9.33754	−0.549267
$290$	0	0
$291$	0	0
$292$	0.419517	0.0245504
$293$	−2.95930	−0.172884	−0.0864422	−	0.996257i	$-0.527550\pi$
$293$	−2.95930	−0.172884	−0.0864422	+	0.996257i	$0.527550\pi$
$294$	0	0
$295$	0	0
$296$	−17.5903	−1.02242
$297$	0	0
$298$	−13.0138	−0.753868
$299$	−0.891171	−0.0515377
$300$	0	0
$301$	−14.1819	−0.817429
$302$	33.9850	1.95562
$303$	0	0
$304$	−26.7288	−1.53300
$305$	0	0
$306$	0	0
$307$	10.6160	0.605887	0.302944	−	0.953008i	$-0.402031\pi$
$307$	10.6160	0.605887	0.302944	+	0.953008i	$0.402031\pi$
$308$	3.34264	0.190465
$309$	0	0
$310$	0	0
$311$	−15.6579	−0.887876	−0.443938	−	0.896058i	$-0.646419\pi$
$311$	−15.6579	−0.887876	−0.443938	+	0.896058i	$0.646419\pi$
$312$	0	0
$313$	22.2994	1.26043	0.630217	−	0.776419i	$-0.282966\pi$
$313$	22.2994	1.26043	0.630217	+	0.776419i	$0.282966\pi$
$314$	24.5719	1.38667
$315$	0	0
$316$	3.22527	0.181436
$317$	7.84990	0.440894	0.220447	−	0.975399i	$-0.429248\pi$
$317$	7.84990	0.440894	0.220447	+	0.975399i	$0.429248\pi$
$318$	0	0
$319$	12.1445	0.679964
$320$	0	0
$321$	0	0
$322$	−0.619285	−0.0345114
$323$	−15.1582	−0.843424
$324$	0	0
$325$	0	0
$326$	6.96971	0.386017
$327$	0	0
$328$	−9.86491	−0.544699
$329$	6.26230	0.345252
$330$	0	0
$331$	−15.6199	−0.858550	−0.429275	−	0.903174i	$-0.641231\pi$
$331$	−15.6199	−0.858550	−0.429275	+	0.903174i	$0.641231\pi$
$332$	1.22527	0.0672453
$333$	0	0
$334$	23.7248	1.29817
$335$	0	0
$336$	0	0
$337$	13.6660	0.744436	0.372218	−	0.928145i	$-0.378597\pi$
$337$	13.6660	0.744436	0.372218	+	0.928145i	$0.378597\pi$
$338$	−58.0758	−3.15890
$339$	0	0
$340$	0	0
$341$	1.71539	0.0928933
$342$	0	0
$343$	−15.3280	−0.827635
$344$	−10.4488	−0.563359
$345$	0	0
$346$	−38.3678	−2.06266
$347$	17.7181	0.951157	0.475579	−	0.879673i	$-0.342239\pi$
$347$	17.7181	0.951157	0.475579	+	0.879673i	$0.342239\pi$
$348$	0	0
$349$	20.0529	1.07341	0.536704	−	0.843770i	$-0.319669\pi$
$349$	20.0529	1.07341	0.536704	+	0.843770i	$0.319669\pi$
$350$	0	0
$351$	0	0
$352$	6.12526	0.326477
$353$	2.40508	0.128010	0.0640048	−	0.997950i	$-0.479613\pi$
$353$	2.40508	0.128010	0.0640048	+	0.997950i	$0.479613\pi$
$354$	0	0
$355$	0	0
$356$	7.52911	0.399042
$357$	0	0
$358$	−14.4515	−0.763784
$359$	−10.9843	−0.579731	−0.289865	−	0.957067i	$-0.593611\pi$
$359$	−10.9843	−0.579731	−0.289865	+	0.957067i	$0.593611\pi$
$360$	0	0
$361$	10.9865	0.578239
$362$	−21.3604	−1.12268
$363$	0	0
$364$	−13.5903	−0.712326
$365$	0	0
$366$	0	0
$367$	−19.2196	−1.00325	−0.501627	−	0.865084i	$-0.667265\pi$
$367$	−19.2196	−1.00325	−0.501627	+	0.865084i	$0.667265\pi$
$368$	−0.623700	−0.0325126
$369$	0	0
$370$	0	0
$371$	15.4949	0.804454
$372$	0	0
$373$	−20.9079	−1.08257	−0.541286	−	0.840839i	$-0.682062\pi$
$373$	−20.9079	−1.08257	−0.541286	+	0.840839i	$0.682062\pi$
$374$	7.73732	0.400087
$375$	0	0
$376$	4.61387	0.237942
$377$	−49.3765	−2.54302
$378$	0	0
$379$	−27.2996	−1.40228	−0.701142	−	0.713022i	$-0.747326\pi$
$379$	−27.2996	−1.40228	−0.701142	+	0.713022i	$0.747326\pi$
$380$	0	0
$381$	0	0
$382$	13.9040	0.711390
$383$	−25.5883	−1.30750	−0.653751	−	0.756710i	$-0.726805\pi$
$383$	−25.5883	−1.30750	−0.653751	+	0.756710i	$0.726805\pi$
$384$	0	0
$385$	0	0
$386$	26.3314	1.34023
$387$	0	0
$388$	−4.48551	−0.227718
$389$	6.23267	0.316009	0.158004	−	0.987438i	$-0.449494\pi$
$389$	6.23267	0.316009	0.158004	+	0.987438i	$0.449494\pi$
$390$	0	0
$391$	−0.353707	−0.0178877
$392$	4.04646	0.204377
$393$	0	0
$394$	1.67132	0.0841999
$395$	0	0
$396$	0	0
$397$	8.29231	0.416179	0.208090	−	0.978110i	$-0.433275\pi$
$397$	8.29231	0.416179	0.208090	+	0.978110i	$0.433275\pi$
$398$	−43.4579	−2.17835
$399$	0	0
$400$	0	0
$401$	−29.8263	−1.48945	−0.744726	−	0.667370i	$-0.767420\pi$
$401$	−29.8263	−1.48945	−0.744726	+	0.667370i	$0.767420\pi$
$402$	0	0
$403$	−6.97431	−0.347415
$404$	−11.9704	−0.595548
$405$	0	0
$406$	−34.3124	−1.70289
$407$	−13.7695	−0.682527
$408$	0	0
$409$	7.18252	0.355153	0.177576	−	0.984107i	$-0.443174\pi$
$409$	7.18252	0.355153	0.177576	+	0.984107i	$0.443174\pi$
$410$	0	0
$411$	0	0
$412$	0.789354	0.0388887
$413$	−39.5490	−1.94608
$414$	0	0
$415$	0	0
$416$	−24.9037	−1.22100
$417$	0	0
$418$	−15.3063	−0.748654
$419$	17.7997	0.869572	0.434786	−	0.900534i	$-0.356824\pi$
$419$	17.7997	0.869572	0.434786	+	0.900534i	$0.356824\pi$
$420$	0	0
$421$	−31.1941	−1.52031	−0.760153	−	0.649744i	$-0.774876\pi$
$421$	−31.1941	−1.52031	−0.760153	+	0.649744i	$0.774876\pi$
$422$	−24.4519	−1.19030
$423$	0	0
$424$	11.4161	0.554417
$425$	0	0
$426$	0	0
$427$	−0.313671	−0.0151796
$428$	−3.79450	−0.183414
$429$	0	0
$430$	0	0
$431$	−30.7526	−1.48130	−0.740651	−	0.671890i	$-0.765483\pi$
$431$	−30.7526	−1.48130	−0.740651	+	0.671890i	$0.765483\pi$
$432$	0	0
$433$	−15.7991	−0.759256	−0.379628	−	0.925139i	$-0.623948\pi$
$433$	−15.7991	−0.759256	−0.379628	+	0.925139i	$0.623948\pi$
$434$	−4.84653	−0.232641
$435$	0	0
$436$	−3.55395	−0.170203
$437$	0.699718	0.0334720
$438$	0	0
$439$	−24.4503	−1.16695	−0.583475	−	0.812131i	$-0.698307\pi$
$439$	−24.4503	−1.16695	−0.583475	+	0.812131i	$0.698307\pi$
$440$	0	0
$441$	0	0
$442$	−31.4579	−1.49630
$443$	5.28067	0.250892	0.125446	−	0.992100i	$-0.459964\pi$
$443$	5.28067	0.250892	0.125446	+	0.992100i	$0.459964\pi$
$444$	0	0
$445$	0	0
$446$	−13.6317	−0.645478
$447$	0	0
$448$	11.7298	0.554182
$449$	−21.4308	−1.01138	−0.505690	−	0.862715i	$-0.668762\pi$
$449$	−21.4308	−1.01138	−0.505690	+	0.862715i	$0.668762\pi$
$450$	0	0
$451$	−7.72212	−0.363621
$452$	−1.71210	−0.0805305
$453$	0	0
$454$	−23.9771	−1.12530
$455$	0	0
$456$	0	0
$457$	25.1920	1.17843	0.589215	−	0.807976i	$-0.299437\pi$
$457$	25.1920	1.17843	0.589215	+	0.807976i	$0.299437\pi$
$458$	30.0895	1.40599
$459$	0	0
$460$	0	0
$461$	−19.0975	−0.889459	−0.444729	−	0.895665i	$-0.646700\pi$
$461$	−19.0975	−0.889459	−0.444729	+	0.895665i	$0.646700\pi$
$462$	0	0
$463$	−28.8933	−1.34279	−0.671393	−	0.741102i	$-0.734304\pi$
$463$	−28.8933	−1.34279	−0.671393	+	0.741102i	$0.734304\pi$
$464$	−34.5569	−1.60427
$465$	0	0
$466$	0.120082	0.00556271
$467$	−9.34938	−0.432638	−0.216319	−	0.976323i	$-0.569405\pi$
$467$	−9.34938	−0.432638	−0.216319	+	0.976323i	$0.569405\pi$
$468$	0	0
$469$	6.79516	0.313771
$470$	0	0
$471$	0	0
$472$	−29.1385	−1.34121
$473$	−8.17915	−0.376078
$474$	0	0
$475$	0	0
$476$	−5.39402	−0.247234
$477$	0	0
$478$	32.7128	1.49625
$479$	−27.8309	−1.27162	−0.635812	−	0.771844i	$-0.719335\pi$
$479$	−27.8309	−1.27162	−0.635812	+	0.771844i	$0.719335\pi$
$480$	0	0
$481$	55.9831	2.55261
$482$	5.69036	0.259189
$483$	0	0
$484$	−5.27885	−0.239948
$485$	0	0
$486$	0	0
$487$	−3.08910	−0.139980	−0.0699902	−	0.997548i	$-0.522297\pi$
$487$	−3.08910	−0.139980	−0.0699902	+	0.997548i	$0.522297\pi$
$488$	−0.231103	−0.0104616
$489$	0	0
$490$	0	0
$491$	−2.65062	−0.119621	−0.0598104	−	0.998210i	$-0.519050\pi$
$491$	−2.65062	−0.119621	−0.0598104	+	0.998210i	$0.519050\pi$
$492$	0	0
$493$	−19.5976	−0.882633
$494$	62.2313	2.79992
$495$	0	0
$496$	−4.88108	−0.219167
$497$	−15.8208	−0.709662
$498$	0	0
$499$	−2.46320	−0.110268	−0.0551339	−	0.998479i	$-0.517559\pi$
$499$	−2.46320	−0.110268	−0.0551339	+	0.998479i	$0.517559\pi$
$500$	0	0
$501$	0	0
$502$	30.8816	1.37831
$503$	10.2846	0.458568	0.229284	−	0.973360i	$-0.426361\pi$
$503$	10.2846	0.458568	0.229284	+	0.973360i	$0.426361\pi$
$504$	0	0
$505$	0	0
$506$	−0.357163	−0.0158778
$507$	0	0
$508$	4.36814	0.193805
$509$	42.7097	1.89308	0.946538	−	0.322593i	$-0.104554\pi$
$509$	42.7097	1.89308	0.946538	+	0.322593i	$0.104554\pi$
$510$	0	0
$511$	1.90456	0.0842528
$512$	3.96337	0.175158
$513$	0	0
$514$	−10.9724	−0.483973
$515$	0	0
$516$	0	0
$517$	3.61168	0.158841
$518$	38.9033	1.70931
$519$	0	0
$520$	0	0
$521$	31.6142	1.38504	0.692521	−	0.721398i	$-0.256500\pi$
$521$	31.6142	1.38504	0.692521	+	0.721398i	$0.256500\pi$
$522$	0	0
$523$	−33.7769	−1.47696	−0.738480	−	0.674275i	$-0.764456\pi$
$523$	−33.7769	−1.47696	−0.738480	+	0.674275i	$0.764456\pi$
$524$	−9.71828	−0.424545
$525$	0	0
$526$	22.0535	0.961579
$527$	−2.76812	−0.120581
$528$	0	0
$529$	−22.9837	−0.999290
$530$	0	0
$531$	0	0
$532$	10.6707	0.462632
$533$	31.3962	1.35992
$534$	0	0
$535$	0	0
$536$	5.00646	0.216246
$537$	0	0
$538$	−11.4770	−0.494807
$539$	3.16752	0.136435
$540$	0	0
$541$	15.6350	0.672199	0.336100	−	0.941826i	$-0.390892\pi$
$541$	15.6350	0.672199	0.336100	+	0.941826i	$0.390892\pi$
$542$	−19.2616	−0.827358
$543$	0	0
$544$	−9.88432	−0.423787
$545$	0	0
$546$	0	0
$547$	−23.5289	−1.00602	−0.503012	−	0.864279i	$-0.667775\pi$
$547$	−23.5289	−1.00602	−0.503012	+	0.864279i	$0.667775\pi$
$548$	−3.99691	−0.170739
$549$	0	0
$550$	0	0
$551$	38.7688	1.65161
$552$	0	0
$553$	14.6424	0.622656
$554$	−37.3717	−1.58777
$555$	0	0
$556$	0.0738822	0.00313331
$557$	−6.90793	−0.292698	−0.146349	−	0.989233i	$-0.546752\pi$
$557$	−6.90793	−0.292698	−0.146349	+	0.989233i	$0.546752\pi$
$558$	0	0
$559$	33.2543	1.40651
$560$	0	0
$561$	0	0
$562$	−23.5626	−0.993927
$563$	−17.0351	−0.717945	−0.358973	−	0.933348i	$-0.616873\pi$
$563$	−17.0351	−0.717945	−0.358973	+	0.933348i	$0.616873\pi$
$564$	0	0
$565$	0	0
$566$	−45.2967	−1.90396
$567$	0	0
$568$	−11.6563	−0.489088
$569$	−36.6801	−1.53771	−0.768855	−	0.639423i	$-0.779173\pi$
$569$	−36.6801	−1.53771	−0.768855	+	0.639423i	$0.779173\pi$
$570$	0	0
$571$	7.24335	0.303125	0.151562	−	0.988448i	$-0.451570\pi$
$571$	7.24335	0.303125	0.151562	+	0.988448i	$0.451570\pi$
$572$	−7.83799	−0.327723
$573$	0	0
$574$	21.8176	0.910647
$575$	0	0
$576$	0	0
$577$	10.0981	0.420391	0.210196	−	0.977659i	$-0.432590\pi$
$577$	10.0981	0.420391	0.210196	+	0.977659i	$0.432590\pi$
$578$	15.2152	0.632868
$579$	0	0
$580$	0	0
$581$	5.56258	0.230775
$582$	0	0
$583$	8.93641	0.370108
$584$	1.40322	0.0580657
$585$	0	0
$586$	4.82208	0.199198
$587$	−21.9290	−0.905107	−0.452554	−	0.891737i	$-0.649487\pi$
$587$	−21.9290	−0.905107	−0.452554	+	0.891737i	$0.649487\pi$
$588$	0	0
$589$	5.47600	0.225635
$590$	0	0
$591$	0	0
$592$	39.1806	1.61031
$593$	−21.5212	−0.883771	−0.441885	−	0.897072i	$-0.645690\pi$
$593$	−21.5212	−0.883771	−0.441885	+	0.897072i	$0.645690\pi$
$594$	0	0
$595$	0	0
$596$	5.23239	0.214327
$597$	0	0
$598$	1.45213	0.0593820
$599$	3.72270	0.152105	0.0760526	−	0.997104i	$-0.475768\pi$
$599$	3.72270	0.152105	0.0760526	+	0.997104i	$0.475768\pi$
$600$	0	0
$601$	−32.2142	−1.31404	−0.657022	−	0.753871i	$-0.728184\pi$
$601$	−32.2142	−1.31404	−0.657022	+	0.753871i	$0.728184\pi$
$602$	23.1088	0.941846
$603$	0	0
$604$	−13.6642	−0.555988
$605$	0	0
$606$	0	0
$607$	−32.9324	−1.33668	−0.668342	−	0.743854i	$-0.732996\pi$
$607$	−32.9324	−1.33668	−0.668342	+	0.743854i	$0.732996\pi$
$608$	19.5536	0.793002
$609$	0	0
$610$	0	0
$611$	−14.6841	−0.594057
$612$	0	0
$613$	−19.8619	−0.802216	−0.401108	−	0.916031i	$-0.631375\pi$
$613$	−19.8619	−0.802216	−0.401108	+	0.916031i	$0.631375\pi$
$614$	−17.2984	−0.698106
$615$	0	0
$616$	11.1806	0.450480
$617$	−14.1251	−0.568654	−0.284327	−	0.958727i	$-0.591770\pi$
$617$	−14.1251	−0.568654	−0.284327	+	0.958727i	$0.591770\pi$
$618$	0	0
$619$	−30.6478	−1.23184	−0.615919	−	0.787810i	$-0.711215\pi$
$619$	−30.6478	−1.23184	−0.615919	+	0.787810i	$0.711215\pi$
$620$	0	0
$621$	0	0
$622$	25.5139	1.02301
$623$	34.1813	1.36944
$624$	0	0
$625$	0	0
$626$	−36.3360	−1.45228
$627$	0	0
$628$	−9.87953	−0.394236
$629$	22.2198	0.885961
$630$	0	0
$631$	−4.67198	−0.185989	−0.0929943	−	0.995667i	$-0.529644\pi$
$631$	−4.67198	−0.185989	−0.0929943	+	0.995667i	$0.529644\pi$
$632$	10.7880	0.429125
$633$	0	0
$634$	−12.7911	−0.508001
$635$	0	0
$636$	0	0
$637$	−12.8783	−0.510257
$638$	−19.7891	−0.783457
$639$	0	0
$640$	0	0
$641$	0.427402	0.0168813	0.00844067	−	0.999964i	$-0.497313\pi$
$641$	0.427402	0.0168813	0.00844067	+	0.999964i	$0.497313\pi$
$642$	0	0
$643$	37.4755	1.47789	0.738945	−	0.673765i	$-0.235324\pi$
$643$	37.4755	1.47789	0.738945	+	0.673765i	$0.235324\pi$
$644$	0.248993	0.00981171
$645$	0	0
$646$	24.6997	0.971797
$647$	−7.60791	−0.299098	−0.149549	−	0.988754i	$-0.547782\pi$
$647$	−7.60791	−0.299098	−0.149549	+	0.988754i	$0.547782\pi$
$648$	0	0
$649$	−22.8092	−0.895340
$650$	0	0
$651$	0	0
$652$	−2.80228	−0.109746
$653$	−9.97308	−0.390277	−0.195138	−	0.980776i	$-0.562516\pi$
$653$	−9.97308	−0.390277	−0.195138	+	0.980776i	$0.562516\pi$
$654$	0	0
$655$	0	0
$656$	21.9731	0.857905
$657$	0	0
$658$	−10.2042	−0.397800
$659$	26.4130	1.02890	0.514452	−	0.857519i	$-0.327995\pi$
$659$	26.4130	1.02890	0.514452	+	0.857519i	$0.327995\pi$
$660$	0	0
$661$	25.4910	0.991485	0.495743	−	0.868470i	$-0.334896\pi$
$661$	25.4910	0.991485	0.495743	+	0.868470i	$0.334896\pi$
$662$	25.4521	0.989225
$663$	0	0
$664$	4.09834	0.159046
$665$	0	0
$666$	0	0
$667$	0.904647	0.0350281
$668$	−9.53894	−0.369073
$669$	0	0
$670$	0	0
$671$	−0.180905	−0.00698375
$672$	0	0
$673$	−30.2988	−1.16793	−0.583966	−	0.811778i	$-0.698500\pi$
$673$	−30.2988	−1.16793	−0.583966	+	0.811778i	$0.698500\pi$
$674$	−22.2683	−0.857742
$675$	0	0
$676$	23.3503	0.898087
$677$	16.1111	0.619202	0.309601	−	0.950867i	$-0.399805\pi$
$677$	16.1111	0.619202	0.309601	+	0.950867i	$0.399805\pi$
$678$	0	0
$679$	−20.3637	−0.781488
$680$	0	0
$681$	0	0
$682$	−2.79516	−0.107032
$683$	−4.68912	−0.179424	−0.0897122	−	0.995968i	$-0.528595\pi$
$683$	−4.68912	−0.179424	−0.0897122	+	0.995968i	$0.528595\pi$
$684$	0	0
$685$	0	0
$686$	24.9764	0.953605
$687$	0	0
$688$	23.2735	0.887296
$689$	−36.3331	−1.38418
$690$	0	0
$691$	−19.1696	−0.729247	−0.364624	−	0.931155i	$-0.618802\pi$
$691$	−19.1696	−0.729247	−0.364624	+	0.931155i	$0.618802\pi$
$692$	15.4264	0.586422
$693$	0	0
$694$	−28.8710	−1.09593
$695$	0	0
$696$	0	0
$697$	12.4612	0.472001
$698$	−32.6755	−1.23679
$699$	0	0
$700$	0	0
$701$	−30.9764	−1.16996	−0.584982	−	0.811046i	$-0.698898\pi$
$701$	−30.9764	−1.16996	−0.584982	+	0.811046i	$0.698898\pi$
$702$	0	0
$703$	−43.9561	−1.65783
$704$	6.76498	0.254965
$705$	0	0
$706$	−3.91899	−0.147493
$707$	−54.3441	−2.04382
$708$	0	0
$709$	−17.6781	−0.663913	−0.331957	−	0.943295i	$-0.607709\pi$
$709$	−17.6781	−0.663913	−0.331957	+	0.943295i	$0.607709\pi$
$710$	0	0
$711$	0	0
$712$	25.1837	0.943800
$713$	0.127779	0.00478536
$714$	0	0
$715$	0	0
$716$	5.81044	0.217146
$717$	0	0
$718$	17.8986	0.667969
$719$	−0.0363678	−0.00135629	−0.000678145	−	1.00000i	$-0.500216\pi$
$719$	−0.0363678	−0.00135629	−0.000678145		1.00000i	$0.500216\pi$
$720$	0	0
$721$	3.58358	0.133459
$722$	−17.9022	−0.666250
$723$	0	0
$724$	8.58827	0.319180
$725$	0	0
$726$	0	0
$727$	−49.9904	−1.85404	−0.927021	−	0.375010i	$-0.877639\pi$
$727$	−49.9904	−1.85404	−0.927021	+	0.375010i	$0.877639\pi$
$728$	−45.4575	−1.68477
$729$	0	0
$730$	0	0
$731$	13.1987	0.488171
$732$	0	0
$733$	17.0602	0.630131	0.315066	−	0.949070i	$-0.397973\pi$
$733$	17.0602	0.630131	0.315066	+	0.949070i	$0.397973\pi$
$734$	31.3176	1.15595
$735$	0	0
$736$	0.456270	0.0168184
$737$	3.91899	0.144358
$738$	0	0
$739$	−11.6240	−0.427595	−0.213797	−	0.976878i	$-0.568583\pi$
$739$	−11.6240	−0.427595	−0.213797	+	0.976878i	$0.568583\pi$
$740$	0	0
$741$	0	0
$742$	−25.2483	−0.926895
$743$	40.8980	1.50040	0.750202	−	0.661209i	$-0.229956\pi$
$743$	40.8980	1.50040	0.750202	+	0.661209i	$0.229956\pi$
$744$	0	0
$745$	0	0
$746$	34.0687	1.24734
$747$	0	0
$748$	−3.11091	−0.113746
$749$	−17.2266	−0.629446
$750$	0	0
$751$	−9.73770	−0.355334	−0.177667	−	0.984091i	$-0.556855\pi$
$751$	−9.73770	−0.355334	−0.177667	+	0.984091i	$0.556855\pi$
$752$	−10.2769	−0.374761
$753$	0	0
$754$	80.4573	2.93008
$755$	0	0
$756$	0	0
$757$	−15.7898	−0.573889	−0.286945	−	0.957947i	$-0.592640\pi$
$757$	−15.7898	−0.573889	−0.286945	+	0.957947i	$0.592640\pi$
$758$	44.4836	1.61572
$759$	0	0
$760$	0	0
$761$	50.7315	1.83901	0.919507	−	0.393073i	$-0.128588\pi$
$761$	50.7315	1.83901	0.919507	+	0.393073i	$0.128588\pi$
$762$	0	0
$763$	−16.1345	−0.584109
$764$	−5.59032	−0.202251
$765$	0	0
$766$	41.6952	1.50651
$767$	92.7363	3.34851
$768$	0	0
$769$	−3.91071	−0.141024	−0.0705119	−	0.997511i	$-0.522463\pi$
$769$	−3.91071	−0.141024	−0.0705119	+	0.997511i	$0.522463\pi$
$770$	0	0
$771$	0	0
$772$	−10.5869	−0.381033
$773$	−26.2213	−0.943116	−0.471558	−	0.881835i	$-0.656308\pi$
$773$	−26.2213	−0.943116	−0.471558	+	0.881835i	$0.656308\pi$
$774$	0	0
$775$	0	0
$776$	−15.0034	−0.538590
$777$	0	0
$778$	−10.1559	−0.364107
$779$	−24.6512	−0.883221
$780$	0	0
$781$	−9.12441	−0.326497
$782$	0.576353	0.0206103
$783$	0	0
$784$	−9.01307	−0.321896
$785$	0	0
$786$	0	0
$787$	12.6377	0.450487	0.225244	−	0.974302i	$-0.427682\pi$
$787$	12.6377	0.450487	0.225244	+	0.974302i	$0.427682\pi$
$788$	−0.671981	−0.0239383
$789$	0	0
$790$	0	0
$791$	−7.77275	−0.276367
$792$	0	0
$793$	0.735512	0.0261188
$794$	−13.5120	−0.479524
$795$	0	0
$796$	17.4729	0.619312
$797$	23.0461	0.816335	0.408168	−	0.912907i	$-0.366168\pi$
$797$	23.0461	0.816335	0.408168	+	0.912907i	$0.366168\pi$
$798$	0	0
$799$	−5.82816	−0.206185
$800$	0	0
$801$	0	0
$802$	48.6008	1.71615
$803$	1.09842	0.0387625
$804$	0	0
$805$	0	0
$806$	11.3644	0.400293
$807$	0	0
$808$	−40.0391	−1.40857
$809$	−19.1161	−0.672088	−0.336044	−	0.941846i	$-0.609089\pi$
$809$	−19.1161	−0.672088	−0.336044	+	0.941846i	$0.609089\pi$
$810$	0	0
$811$	−18.5288	−0.650636	−0.325318	−	0.945605i	$-0.605471\pi$
$811$	−18.5288	−0.650636	−0.325318	+	0.945605i	$0.605471\pi$
$812$	13.7958	0.484138
$813$	0	0
$814$	22.4368	0.786411
$815$	0	0
$816$	0	0
$817$	−26.1102	−0.913480
$818$	−11.7037	−0.409209
$819$	0	0
$820$	0	0
$821$	36.0522	1.25823	0.629115	−	0.777312i	$-0.283417\pi$
$821$	36.0522	1.25823	0.629115	+	0.777312i	$0.283417\pi$
$822$	0	0
$823$	−17.3204	−0.603753	−0.301876	−	0.953347i	$-0.597613\pi$
$823$	−17.3204	−0.603753	−0.301876	+	0.953347i	$0.597613\pi$
$824$	2.64027	0.0919782
$825$	0	0
$826$	64.4436	2.24228
$827$	−23.1732	−0.805811	−0.402906	−	0.915242i	$-0.632000\pi$
$827$	−23.1732	−0.805811	−0.402906	+	0.915242i	$0.632000\pi$
$828$	0	0
$829$	23.0938	0.802082	0.401041	−	0.916060i	$-0.368649\pi$
$829$	23.0938	0.802082	0.401041	+	0.916060i	$0.368649\pi$
$830$	0	0
$831$	0	0
$832$	−27.5047	−0.953552
$833$	−5.11142	−0.177100
$834$	0	0
$835$	0	0
$836$	6.15413	0.212845
$837$	0	0
$838$	−29.0039	−1.00192
$839$	43.7883	1.51174	0.755871	−	0.654721i	$-0.227214\pi$
$839$	43.7883	1.51174	0.755871	+	0.654721i	$0.227214\pi$
$840$	0	0
$841$	21.1232	0.728385
$842$	50.8296	1.75170
$843$	0	0
$844$	9.83125	0.338406
$845$	0	0
$846$	0	0
$847$	−23.9654	−0.823460
$848$	−25.4283	−0.873212
$849$	0	0
$850$	0	0
$851$	−1.02569	−0.0351601
$852$	0	0
$853$	−30.5569	−1.04625	−0.523125	−	0.852256i	$-0.675234\pi$
$853$	−30.5569	−1.04625	−0.523125	+	0.852256i	$0.675234\pi$
$854$	0.511116	0.0174900
$855$	0	0
$856$	−12.6920	−0.433804
$857$	54.5160	1.86223	0.931115	−	0.364726i	$-0.118837\pi$
$857$	54.5160	1.86223	0.931115	+	0.364726i	$0.118837\pi$
$858$	0	0
$859$	−49.3771	−1.68473	−0.842363	−	0.538911i	$-0.818836\pi$
$859$	−49.3771	−1.68473	−0.842363	+	0.538911i	$0.818836\pi$
$860$	0	0
$861$	0	0
$862$	50.1103	1.70676
$863$	−29.6647	−1.00980	−0.504899	−	0.863178i	$-0.668470\pi$
$863$	−29.6647	−1.00980	−0.504899	+	0.863178i	$0.668470\pi$
$864$	0	0
$865$	0	0
$866$	25.7441	0.874819
$867$	0	0
$868$	1.94862	0.0661406
$869$	8.44473	0.286468
$870$	0	0
$871$	−15.9336	−0.539890
$872$	−11.8874	−0.402559
$873$	0	0
$874$	−1.14016	−0.0385666
$875$	0	0
$876$	0	0
$877$	28.6701	0.968120	0.484060	−	0.875035i	$-0.339162\pi$
$877$	28.6701	0.968120	0.484060	+	0.875035i	$0.339162\pi$
$878$	39.8409	1.34456
$879$	0	0
$880$	0	0
$881$	−26.8689	−0.905235	−0.452617	−	0.891705i	$-0.649510\pi$
$881$	−26.8689	−0.905235	−0.452617	+	0.891705i	$0.649510\pi$
$882$	0	0
$883$	−21.4596	−0.722172	−0.361086	−	0.932533i	$-0.617594\pi$
$883$	−21.4596	−0.722172	−0.361086	+	0.932533i	$0.617594\pi$
$884$	12.6482	0.425403
$885$	0	0
$886$	−8.60466	−0.289079
$887$	−7.94131	−0.266643	−0.133322	−	0.991073i	$-0.542564\pi$
$887$	−7.94131	−0.266643	−0.133322	+	0.991073i	$0.542564\pi$
$888$	0	0
$889$	19.8309	0.665106
$890$	0	0
$891$	0	0
$892$	5.48083	0.183512
$893$	11.5295	0.385820
$894$	0	0
$895$	0	0
$896$	−40.3545	−1.34815
$897$	0	0
$898$	34.9207	1.16532
$899$	7.07977	0.236124
$900$	0	0
$901$	−14.4207	−0.480422
$902$	12.5829	0.418965
$903$	0	0
$904$	−5.72673	−0.190468
$905$	0	0
$906$	0	0
$907$	45.4658	1.50967	0.754834	−	0.655916i	$-0.227717\pi$
$907$	45.4658	1.50967	0.754834	+	0.655916i	$0.227717\pi$
$908$	9.64037	0.319927
$909$	0	0
$910$	0	0
$911$	33.6098	1.11354	0.556771	−	0.830666i	$-0.312040\pi$
$911$	33.6098	1.11354	0.556771	+	0.830666i	$0.312040\pi$
$912$	0	0
$913$	3.20812	0.106173
$914$	−41.0494	−1.35779
$915$	0	0
$916$	−12.0980	−0.399728
$917$	−44.1199	−1.45697
$918$	0	0
$919$	−24.3699	−0.803888	−0.401944	−	0.915664i	$-0.631665\pi$
$919$	−24.3699	−0.803888	−0.401944	+	0.915664i	$0.631665\pi$
$920$	0	0
$921$	0	0
$922$	31.1187	1.02484
$923$	37.0975	1.22108
$924$	0	0
$925$	0	0
$926$	47.0806	1.54716
$927$	0	0
$928$	25.2803	0.829866
$929$	9.10480	0.298719	0.149359	−	0.988783i	$-0.452279\pi$
$929$	9.10480	0.298719	0.149359	+	0.988783i	$0.452279\pi$
$930$	0	0
$931$	10.1116	0.331395
$932$	−0.0482810	−0.00158150
$933$	0	0
$934$	15.2345	0.498487
$935$	0	0
$936$	0	0
$937$	1.74387	0.0569697	0.0284849	−	0.999594i	$-0.490932\pi$
$937$	1.74387	0.0569697	0.0284849	+	0.999594i	$0.490932\pi$
$938$	−11.0725	−0.361529
$939$	0	0
$940$	0	0
$941$	−44.3041	−1.44427	−0.722136	−	0.691751i	$-0.756839\pi$
$941$	−44.3041	−1.44427	−0.722136	+	0.691751i	$0.756839\pi$
$942$	0	0
$943$	−0.575221	−0.0187318
$944$	64.9029	2.11241
$945$	0	0
$946$	13.3276	0.433319
$947$	17.9601	0.583624	0.291812	−	0.956476i	$-0.405742\pi$
$947$	17.9601	0.583624	0.291812	+	0.956476i	$0.405742\pi$
$948$	0	0
$949$	−4.46591	−0.144969
$950$	0	0
$951$	0	0
$952$	−18.0422	−0.584750
$953$	−21.7642	−0.705011	−0.352505	−	0.935810i	$-0.614670\pi$
$953$	−21.7642	−0.705011	−0.352505	+	0.935810i	$0.614670\pi$
$954$	0	0
$955$	0	0
$956$	−13.1527	−0.425389
$957$	0	0
$958$	45.3494	1.46517
$959$	−18.1455	−0.585949
$960$	0	0
$961$	1.00000	0.0322581
$962$	−91.2224	−2.94113
$963$	0	0
$964$	−2.28790	−0.0736882
$965$	0	0
$966$	0	0
$967$	29.2152	0.939499	0.469750	−	0.882800i	$-0.344344\pi$
$967$	29.2152	0.939499	0.469750	+	0.882800i	$0.344344\pi$
$968$	−17.6570	−0.567516
$969$	0	0
$970$	0	0
$971$	4.11489	0.132053	0.0660266	−	0.997818i	$-0.478968\pi$
$971$	4.11489	0.132053	0.0660266	+	0.997818i	$0.478968\pi$
$972$	0	0
$973$	0.335417	0.0107530
$974$	5.03357	0.161286
$975$	0	0
$976$	0.514759	0.0164770
$977$	−42.0412	−1.34502	−0.672509	−	0.740089i	$-0.734783\pi$
$977$	−42.0412	−1.34502	−0.672509	+	0.740089i	$0.734783\pi$
$978$	0	0
$979$	19.7135	0.630046
$980$	0	0
$981$	0	0
$982$	4.31909	0.137828
$983$	44.2806	1.41233	0.706166	−	0.708047i	$-0.250423\pi$
$983$	44.2806	1.41233	0.706166	+	0.708047i	$0.250423\pi$
$984$	0	0
$985$	0	0
$986$	31.9336	1.01697
$987$	0	0
$988$	−25.0211	−0.796027
$989$	−0.609265	−0.0193735
$990$	0	0
$991$	8.37142	0.265927	0.132964	−	0.991121i	$-0.457551\pi$
$991$	8.37142	0.265927	0.132964	+	0.991121i	$0.457551\pi$
$992$	3.57078	0.113372
$993$	0	0
$994$	25.7795	0.817676
$995$	0	0
$996$	0	0
$997$	−53.8984	−1.70698	−0.853490	−	0.521109i	$-0.825518\pi$
$997$	−53.8984	−1.70698	−0.853490	+	0.521109i	$0.825518\pi$
$998$	4.01369	0.127051
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bj.1.2		4
3.2	odd	2		775.2.a.g.1.3		4
5.4	even	2		1395.2.a.m.1.3		4
15.2	even	4		775.2.b.e.249.6		8
15.8	even	4		775.2.b.e.249.3		8
15.14	odd	2		155.2.a.d.1.2	✓	4
60.59	even	2		2480.2.a.z.1.4		4
105.104	even	2		7595.2.a.q.1.2		4
120.29	odd	2		9920.2.a.ch.1.4		4
120.59	even	2		9920.2.a.cd.1.1		4
465.464	even	2		4805.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.d.1.2	✓	4	15.14	odd	2
775.2.a.g.1.3		4	3.2	odd	2
775.2.b.e.249.3		8	15.8	even	4
775.2.b.e.249.6		8	15.2	even	4
1395.2.a.m.1.3		4	5.4	even	2
2480.2.a.z.1.4		4	60.59	even	2
4805.2.a.j.1.2		4	465.464	even	2
6975.2.a.bj.1.2		4	1.1	even	1	trivial
7595.2.a.q.1.2		4	105.104	even	2
9920.2.a.cd.1.1		4	120.59	even	2
9920.2.a.ch.1.4		4	120.29	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q-1.62946 q^{2} +0.655151 q^{4} +2.97431 q^{7} +2.19138 q^{8} +O(q^{10})\)	\(q-1.62946 q^{2} +0.655151 q^{4} +2.97431 q^{7} +2.19138 q^{8} +1.71539 q^{11} -6.97431 q^{13} -4.84653 q^{14} -4.88108 q^{16} -2.76812 q^{17} +5.47600 q^{19} -2.79516 q^{22} +0.127779 q^{23} +11.3644 q^{26} +1.94862 q^{28} +7.07977 q^{29} +1.00000 q^{31} +3.57078 q^{32} +4.51054 q^{34} -8.02704 q^{37} -8.92294 q^{38} -4.50168 q^{41} -4.76812 q^{43} +1.12384 q^{44} -0.208211 q^{46} +2.10546 q^{47} +1.84653 q^{49} -4.56923 q^{52} +5.20957 q^{53} +6.51785 q^{56} -11.5362 q^{58} -13.2968 q^{59} -0.105460 q^{61} -1.62946 q^{62} +3.94371 q^{64} +2.28461 q^{67} -1.81353 q^{68} -5.31916 q^{71} +0.640336 q^{73} +13.0798 q^{74} +3.58761 q^{76} +5.10209 q^{77} +4.92294 q^{79} +7.33533 q^{82} +1.87021 q^{83} +7.76947 q^{86} +3.75906 q^{88} +11.4922 q^{89} -20.7438 q^{91} +0.0837146 q^{92} -3.43077 q^{94} -6.84653 q^{97} -3.00886 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 9 q^{4} - 9 q^{8}+O(q^{10}) \) 4 * q - q^2 + 9 * q^4 - 9 * q^8	\( 4 q - q^{2} + 9 q^{4} - 9 q^{8} + 6 q^{11} - 16 q^{13} - 8 q^{14} + 11 q^{16} + q^{17} + 5 q^{19} + 24 q^{22} + 12 q^{26} - 16 q^{28} - 6 q^{29} + 4 q^{31} - 29 q^{32} - 18 q^{34} - 9 q^{37} - 13 q^{41} - 7 q^{43} - 20 q^{44} - 26 q^{46} - 14 q^{47} - 4 q^{49} - 20 q^{52} + 11 q^{53} + 4 q^{56} - 22 q^{58} - 13 q^{59} + 22 q^{61} - q^{62} + 47 q^{64} + 10 q^{67} + 30 q^{68} - 3 q^{71} - 9 q^{73} + 18 q^{74} + 14 q^{76} + 8 q^{77} - 16 q^{79} - 6 q^{82} - 17 q^{83} - 16 q^{86} + 44 q^{88} + 12 q^{89} - 24 q^{91} - 10 q^{92} - 12 q^{94} - 16 q^{97} + 19 q^{98}+O(q^{100}) \) 4 * q - q^2 + 9 * q^4 - 9 * q^8 + 6 * q^11 - 16 * q^13 - 8 * q^14 + 11 * q^16 + q^17 + 5 * q^19 + 24 * q^22 + 12 * q^26 - 16 * q^28 - 6 * q^29 + 4 * q^31 - 29 * q^32 - 18 * q^34 - 9 * q^37 - 13 * q^41 - 7 * q^43 - 20 * q^44 - 26 * q^46 - 14 * q^47 - 4 * q^49 - 20 * q^52 + 11 * q^53 + 4 * q^56 - 22 * q^58 - 13 * q^59 + 22 * q^61 - q^62 + 47 * q^64 + 10 * q^67 + 30 * q^68 - 3 * q^71 - 9 * q^73 + 18 * q^74 + 14 * q^76 + 8 * q^77 - 16 * q^79 - 6 * q^82 - 17 * q^83 - 16 * q^86 + 44 * q^88 + 12 * q^89 - 24 * q^91 - 10 * q^92 - 12 * q^94 - 16 * q^97 + 19 * q^98

Introduction

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