LMFDB - Embedded newform 6975.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [3,-2,0,2,0,0,-8,-6,0,0,3,0,-4,10,0,4,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ 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)

$f(q)$	$=$	$q+1.17009 q^{2} -0.630898 q^{4} -0.290725 q^{7} -3.07838 q^{8} +0.460811 q^{11} +1.80098 q^{13} -0.340173 q^{14} -2.34017 q^{16} -3.09171 q^{17} +3.00000 q^{19} +0.539189 q^{22} -2.46081 q^{23} +2.10731 q^{26} +0.183417 q^{28} +9.58864 q^{29} +1.00000 q^{31} +3.41855 q^{32} -3.61757 q^{34} -6.87936 q^{37} +3.51026 q^{38} -3.32684 q^{41} -10.8371 q^{43} -0.290725 q^{44} -2.87936 q^{46} +10.3402 q^{47} -6.91548 q^{49} -1.13624 q^{52} -1.92881 q^{53} +0.894960 q^{56} +11.2195 q^{58} +0.986669 q^{59} -6.68035 q^{61} +1.17009 q^{62} +8.68035 q^{64} -0.0289294 q^{67} +1.95055 q^{68} -13.9421 q^{71} +9.91548 q^{73} -8.04945 q^{74} -1.89269 q^{76} -0.133969 q^{77} +16.6381 q^{79} -3.89269 q^{82} -10.0628 q^{83} -12.6803 q^{86} -1.41855 q^{88} -5.53919 q^{89} -0.523590 q^{91} +1.55252 q^{92} +12.0989 q^{94} -2.55252 q^{97} -8.09171 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 2 q^{4} - 8 q^{7} - 6 q^{8} + 3 q^{11} - 4 q^{13} + 10 q^{14} + 4 q^{16} - 7 q^{17} + 9 q^{19} - 9 q^{23} + 18 q^{26} - 4 q^{28} + 9 q^{29} + 3 q^{31} - 4 q^{32} - 6 q^{34} - 8 q^{37} - 6 q^{38}+ \cdots - 22 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 2 * q^4 - 8 * q^7 - 6 * q^8 + 3 * q^11 - 4 * q^13 + 10 * q^14 + 4 * q^16 - 7 * q^17 + 9 * q^19 - 9 * q^23 + 18 * q^26 - 4 * q^28 + 9 * q^29 + 3 * q^31 - 4 * q^32 - 6 * q^34 - 8 * q^37 - 6 * q^38 + 2 * q^41 - 4 * q^43 - 8 * q^44 + 4 * q^46 + 20 * q^47 + 11 * q^49 - 30 * q^52 + 25 * q^53 + 4 * q^56 + 10 * q^58 + 2 * q^59 + 2 * q^61 - 2 * q^62 + 4 * q^64 - 15 * q^67 + 24 * q^68 - 12 * q^71 - 2 * q^73 - 6 * q^74 + 6 * q^76 - 14 * q^77 + 12 * q^79 - 13 * q^83 - 16 * q^86 + 10 * q^88 - 15 * q^89 + 14 * q^91 + 4 * q^92 - 7 * q^97 - 22 * 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 $

$2$	1.17009	0.827376	0.413688	−	0.910419i	$-0.364240\pi$
$2$	1.17009	0.827376	0.413688	+	0.910419i	$0.364240\pi$
$3$	0	0
$3$	0	0
$4$	−0.630898	−0.315449
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.290725	−0.109884	−0.0549418	−	0.998490i	$-0.517497\pi$
$7$	−0.290725	−0.109884	−0.0549418	+	0.998490i	$0.517497\pi$
$8$	−3.07838	−1.08837
$9$	0	0
$10$	0	0
$11$	0.460811	0.138940	0.0694699	−	0.997584i	$-0.477869\pi$
$11$	0.460811	0.138940	0.0694699	+	0.997584i	$0.477869\pi$
$12$	0	0
$13$	1.80098	0.499503	0.249752	−	0.968310i	$-0.419651\pi$
$13$	1.80098	0.499503	0.249752	+	0.968310i	$0.419651\pi$
$14$	−0.340173	−0.0909151
$15$	0	0
$16$	−2.34017	−0.585043
$17$	−3.09171	−0.749850	−0.374925	−	0.927055i	$-0.622331\pi$
$17$	−3.09171	−0.749850	−0.374925	+	0.927055i	$0.622331\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	0.539189	0.114955
$23$	−2.46081	−0.513115	−0.256557	−	0.966529i	$-0.582588\pi$
$23$	−2.46081	−0.513115	−0.256557	+	0.966529i	$0.582588\pi$
$24$	0	0
$25$	0	0
$26$	2.10731	0.413277
$27$	0	0
$28$	0.183417	0.0346626
$29$	9.58864	1.78057	0.890283	−	0.455408i	$-0.150507\pi$
$29$	9.58864	1.78057	0.890283	+	0.455408i	$0.150507\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.41855	0.604320
$33$	0	0
$34$	−3.61757	−0.620408
$35$	0	0
$36$	0	0
$37$	−6.87936	−1.13096	−0.565480	−	0.824762i	$-0.691309\pi$
$37$	−6.87936	−1.13096	−0.565480	+	0.824762i	$0.691309\pi$
$38$	3.51026	0.569439
$39$	0	0
$40$	0	0
$41$	−3.32684	−0.519565	−0.259783	−	0.965667i	$-0.583651\pi$
$41$	−3.32684	−0.519565	−0.259783	+	0.965667i	$0.583651\pi$
$42$	0	0
$43$	−10.8371	−1.65264	−0.826321	−	0.563199i	$-0.809570\pi$
$43$	−10.8371	−1.65264	−0.826321	+	0.563199i	$0.809570\pi$
$44$	−0.290725	−0.0438284
$45$	0	0
$46$	−2.87936	−0.424539
$47$	10.3402	1.50827	0.754135	−	0.656720i	$-0.228057\pi$
$47$	10.3402	1.50827	0.754135	+	0.656720i	$0.228057\pi$
$48$	0	0
$49$	−6.91548	−0.987926
$50$	0	0
$51$	0	0
$52$	−1.13624	−0.157568
$53$	−1.92881	−0.264942	−0.132471	−	0.991187i	$-0.542291\pi$
$53$	−1.92881	−0.264942	−0.132471	+	0.991187i	$0.542291\pi$
$54$	0	0
$55$	0	0
$56$	0.894960	0.119594
$57$	0	0
$58$	11.2195	1.47320
$59$	0.986669	0.128453	0.0642267	−	0.997935i	$-0.479542\pi$
$59$	0.986669	0.128453	0.0642267	+	0.997935i	$0.479542\pi$
$60$	0	0
$61$	−6.68035	−0.855331	−0.427665	−	0.903937i	$-0.640664\pi$
$61$	−6.68035	−0.855331	−0.427665	+	0.903937i	$0.640664\pi$
$62$	1.17009	0.148601
$63$	0	0
$64$	8.68035	1.08504
$65$	0	0
$66$	0	0
$67$	−0.0289294	−0.00353429	−0.00176715	−	0.999998i	$-0.500563\pi$
$67$	−0.0289294	−0.00353429	−0.00176715	+	0.999998i	$0.500563\pi$
$68$	1.95055	0.236539
$69$	0	0
$70$	0	0
$71$	−13.9421	−1.65463	−0.827314	−	0.561740i	$-0.810132\pi$
$71$	−13.9421	−1.65463	−0.827314	+	0.561740i	$0.810132\pi$
$72$	0	0
$73$	9.91548	1.16052	0.580260	−	0.814432i	$-0.302951\pi$
$73$	9.91548	1.16052	0.580260	+	0.814432i	$0.302951\pi$
$74$	−8.04945	−0.935729
$75$	0	0
$76$	−1.89269	−0.217107
$77$	−0.133969	−0.0152672
$78$	0	0
$79$	16.6381	1.87193	0.935965	−	0.352092i	$-0.114530\pi$
$79$	16.6381	1.87193	0.935965	+	0.352092i	$0.114530\pi$
$80$	0	0
$81$	0	0
$82$	−3.89269	−0.429876
$83$	−10.0628	−1.10453	−0.552267	−	0.833667i	$-0.686237\pi$
$83$	−10.0628	−1.10453	−0.552267	+	0.833667i	$0.686237\pi$
$84$	0	0
$85$	0	0
$86$	−12.6803	−1.36736
$87$	0	0
$88$	−1.41855	−0.151218
$89$	−5.53919	−0.587153	−0.293576	−	0.955936i	$-0.594846\pi$
$89$	−5.53919	−0.587153	−0.293576	+	0.955936i	$0.594846\pi$
$90$	0	0
$91$	−0.523590	−0.0548872
$92$	1.55252	0.161861
$93$	0	0
$94$	12.0989	1.24791
$95$	0	0
$96$	0	0
$97$	−2.55252	−0.259169	−0.129585	−	0.991568i	$-0.541364\pi$
$97$	−2.55252	−0.259169	−0.129585	+	0.991568i	$0.541364\pi$
$98$	−8.09171	−0.817386
$99$	0	0
$100$	0	0
$101$	9.07838	0.903332	0.451666	−	0.892187i	$-0.350830\pi$
$101$	9.07838	0.903332	0.451666	+	0.892187i	$0.350830\pi$
$102$	0	0
$103$	−18.3340	−1.80651	−0.903253	−	0.429109i	$-0.858828\pi$
$103$	−18.3340	−1.80651	−0.903253	+	0.429109i	$0.858828\pi$
$104$	−5.54411	−0.543645
$105$	0	0
$106$	−2.25687	−0.219207
$107$	11.0133	1.06470	0.532349	−	0.846525i	$-0.321309\pi$
$107$	11.0133	1.06470	0.532349	+	0.846525i	$0.321309\pi$
$108$	0	0
$109$	13.3112	1.27499	0.637493	−	0.770456i	$-0.279972\pi$
$109$	13.3112	1.27499	0.637493	+	0.770456i	$0.279972\pi$
$110$	0	0
$111$	0	0
$112$	0.680346	0.0642866
$113$	−10.4391	−0.982025	−0.491013	−	0.871153i	$-0.663373\pi$
$113$	−10.4391	−0.982025	−0.491013	+	0.871153i	$0.663373\pi$
$114$	0	0
$115$	0	0
$116$	−6.04945	−0.561677
$117$	0	0
$118$	1.15449	0.106279
$119$	0.898836	0.0823962
$120$	0	0
$121$	−10.7877	−0.980696
$122$	−7.81658	−0.707680
$123$	0	0
$124$	−0.630898	−0.0566563
$125$	0	0
$126$	0	0
$127$	−12.0989	−1.07360	−0.536802	−	0.843708i	$-0.680368\pi$
$127$	−12.0989	−1.07360	−0.536802	+	0.843708i	$0.680368\pi$
$128$	3.31965	0.293419
$129$	0	0
$130$	0	0
$131$	−12.1639	−1.06277	−0.531384	−	0.847131i	$-0.678328\pi$
$131$	−12.1639	−1.06277	−0.531384	+	0.847131i	$0.678328\pi$
$132$	0	0
$133$	−0.872174	−0.0756271
$134$	−0.0338499	−0.00292419
$135$	0	0
$136$	9.51745	0.816114
$137$	2.85043	0.243529	0.121764	−	0.992559i	$-0.461145\pi$
$137$	2.85043	0.243529	0.121764	+	0.992559i	$0.461145\pi$
$138$	0	0
$139$	−13.5330	−1.14786	−0.573929	−	0.818905i	$-0.694581\pi$
$139$	−13.5330	−1.14786	−0.573929	+	0.818905i	$0.694581\pi$
$140$	0	0
$141$	0	0
$142$	−16.3135	−1.36900
$143$	0.829914	0.0694009
$144$	0	0
$145$	0	0
$146$	11.6020	0.960186
$147$	0	0
$148$	4.34017	0.356760
$149$	−22.7454	−1.86338	−0.931688	−	0.363261i	$-0.881663\pi$
$149$	−22.7454	−1.86338	−0.931688	+	0.363261i	$0.881663\pi$
$150$	0	0
$151$	−10.8638	−0.884081	−0.442040	−	0.896995i	$-0.645745\pi$
$151$	−10.8638	−0.884081	−0.442040	+	0.896995i	$0.645745\pi$
$152$	−9.23513	−0.749068
$153$	0	0
$154$	−0.156755	−0.0126317
$155$	0	0
$156$	0	0
$157$	−14.1773	−1.13147	−0.565735	−	0.824587i	$-0.691407\pi$
$157$	−14.1773	−1.13147	−0.565735	+	0.824587i	$0.691407\pi$
$158$	19.4680	1.54879
$159$	0	0
$160$	0	0
$161$	0.715418	0.0563829
$162$	0	0
$163$	−8.50307	−0.666012	−0.333006	−	0.942925i	$-0.608063\pi$
$163$	−8.50307	−0.666012	−0.333006	+	0.942925i	$0.608063\pi$
$164$	2.09890	0.163896
$165$	0	0
$166$	−11.7743	−0.913865
$167$	5.94214	0.459817	0.229908	−	0.973212i	$-0.426157\pi$
$167$	5.94214	0.459817	0.229908	+	0.973212i	$0.426157\pi$
$168$	0	0
$169$	−9.75646	−0.750497
$170$	0	0
$171$	0	0
$172$	6.83710	0.521324
$173$	−19.0856	−1.45105	−0.725524	−	0.688197i	$-0.758403\pi$
$173$	−19.0856	−1.45105	−0.725524	+	0.688197i	$0.758403\pi$
$174$	0	0
$175$	0	0
$176$	−1.07838	−0.0812858
$177$	0	0
$178$	−6.48133	−0.485796
$179$	10.8865	0.813699	0.406849	−	0.913495i	$-0.366627\pi$
$179$	10.8865	0.813699	0.406849	+	0.913495i	$0.366627\pi$
$180$	0	0
$181$	−10.3246	−0.767420	−0.383710	−	0.923454i	$-0.625354\pi$
$181$	−10.3246	−0.767420	−0.383710	+	0.923454i	$0.625354\pi$
$182$	−0.612646	−0.0454124
$183$	0	0
$184$	7.57531	0.558459
$185$	0	0
$186$	0	0
$187$	−1.42469	−0.104184
$188$	−6.52359	−0.475782
$189$	0	0
$190$	0	0
$191$	−9.69368	−0.701410	−0.350705	−	0.936486i	$-0.614058\pi$
$191$	−9.69368	−0.701410	−0.350705	+	0.936486i	$0.614058\pi$
$192$	0	0
$193$	−4.95055	−0.356349	−0.178174	−	0.983999i	$-0.557019\pi$
$193$	−4.95055	−0.356349	−0.178174	+	0.983999i	$0.557019\pi$
$194$	−2.98667	−0.214430
$195$	0	0
$196$	4.36296	0.311640
$197$	−4.65368	−0.331561	−0.165781	−	0.986163i	$-0.553014\pi$
$197$	−4.65368	−0.331561	−0.165781	+	0.986163i	$0.553014\pi$
$198$	0	0
$199$	−7.93108	−0.562219	−0.281110	−	0.959676i	$-0.590702\pi$
$199$	−7.93108	−0.562219	−0.281110	+	0.959676i	$0.590702\pi$
$200$	0	0
$201$	0	0
$202$	10.6225	0.747396
$203$	−2.78765	−0.195655
$204$	0	0
$205$	0	0
$206$	−21.4524	−1.49466
$207$	0	0
$208$	−4.21461	−0.292231
$209$	1.38243	0.0956249
$210$	0	0
$211$	8.26180	0.568765	0.284383	−	0.958711i	$-0.408211\pi$
$211$	8.26180	0.568765	0.284383	+	0.958711i	$0.408211\pi$
$212$	1.21688	0.0835758
$213$	0	0
$214$	12.8865	0.880906
$215$	0	0
$216$	0	0
$217$	−0.290725	−0.0197357
$218$	15.5753	1.05489
$219$	0	0
$220$	0	0
$221$	−5.56812	−0.374552
$222$	0	0
$223$	−8.70701	−0.583064	−0.291532	−	0.956561i	$-0.594165\pi$
$223$	−8.70701	−0.583064	−0.291532	+	0.956561i	$0.594165\pi$
$224$	−0.993857	−0.0664049
$225$	0	0
$226$	−12.2146	−0.812504
$227$	2.28231	0.151483	0.0757413	−	0.997128i	$-0.475868\pi$
$227$	2.28231	0.151483	0.0757413	+	0.997128i	$0.475868\pi$
$228$	0	0
$229$	22.4547	1.48385	0.741923	−	0.670485i	$-0.233914\pi$
$229$	22.4547	1.48385	0.741923	+	0.670485i	$0.233914\pi$
$230$	0	0
$231$	0	0
$232$	−29.5174	−1.93792
$233$	23.0928	1.51286	0.756428	−	0.654077i	$-0.226943\pi$
$233$	23.0928	1.51286	0.756428	+	0.654077i	$0.226943\pi$
$234$	0	0
$235$	0	0
$236$	−0.622487	−0.0405205
$237$	0	0
$238$	1.05172	0.0681726
$239$	0.693677	0.0448702	0.0224351	−	0.999748i	$-0.492858\pi$
$239$	0.693677	0.0448702	0.0224351	+	0.999748i	$0.492858\pi$
$240$	0	0
$241$	15.8154	1.01876	0.509378	−	0.860543i	$-0.329875\pi$
$241$	15.8154	1.01876	0.509378	+	0.860543i	$0.329875\pi$
$242$	−12.6225	−0.811404
$243$	0	0
$244$	4.21461	0.269813
$245$	0	0
$246$	0	0
$247$	5.40295	0.343782
$248$	−3.07838	−0.195477
$249$	0	0
$250$	0	0
$251$	0.254607	0.0160707	0.00803534	−	0.999968i	$-0.497442\pi$
$251$	0.254607	0.0160707	0.00803534	+	0.999968i	$0.497442\pi$
$252$	0	0
$253$	−1.13397	−0.0712920
$254$	−14.1568	−0.888274
$255$	0	0
$256$	−13.4764	−0.842276
$257$	31.0010	1.93379	0.966896	−	0.255171i	$-0.0821317\pi$
$257$	31.0010	1.93379	0.966896	+	0.255171i	$0.0821317\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	−14.2329	−0.879309
$263$	18.3402	1.13090	0.565452	−	0.824781i	$-0.308702\pi$
$263$	18.3402	1.13090	0.565452	+	0.824781i	$0.308702\pi$
$264$	0	0
$265$	0	0
$266$	−1.02052	−0.0625720
$267$	0	0
$268$	0.0182515	0.00111489
$269$	−13.8166	−0.842412	−0.421206	−	0.906965i	$-0.638393\pi$
$269$	−13.8166	−0.842412	−0.421206	+	0.906965i	$0.638393\pi$
$270$	0	0
$271$	−8.11450	−0.492920	−0.246460	−	0.969153i	$-0.579267\pi$
$271$	−8.11450	−0.492920	−0.246460	+	0.969153i	$0.579267\pi$
$272$	7.23513	0.438694
$273$	0	0
$274$	3.33525	0.201490
$275$	0	0
$276$	0	0
$277$	−11.0205	−0.662159	−0.331079	−	0.943603i	$-0.607413\pi$
$277$	−11.0205	−0.662159	−0.331079	+	0.943603i	$0.607413\pi$
$278$	−15.8348	−0.949710
$279$	0	0
$280$	0	0
$281$	8.15676	0.486591	0.243296	−	0.969952i	$-0.421771\pi$
$281$	8.15676	0.486591	0.243296	+	0.969952i	$0.421771\pi$
$282$	0	0
$283$	18.9155	1.12441	0.562204	−	0.826998i	$-0.309953\pi$
$283$	18.9155	1.12441	0.562204	+	0.826998i	$0.309953\pi$
$284$	8.79606	0.521950
$285$	0	0
$286$	0.971071	0.0574206
$287$	0.967195	0.0570917
$288$	0	0
$289$	−7.44134	−0.437726
$290$	0	0
$291$	0	0
$292$	−6.25565	−0.366084
$293$	0.190605	0.0111353	0.00556764	−	0.999985i	$-0.498228\pi$
$293$	0.190605	0.0111353	0.00556764	+	0.999985i	$0.498228\pi$
$294$	0	0
$295$	0	0
$296$	21.1773	1.23090
$297$	0	0
$298$	−26.6141	−1.54171
$299$	−4.43188	−0.256302
$300$	0	0
$301$	3.15061	0.181598
$302$	−12.7115	−0.731467
$303$	0	0
$304$	−7.02052	−0.402654
$305$	0	0
$306$	0	0
$307$	−9.33791	−0.532942	−0.266471	−	0.963843i	$-0.585858\pi$
$307$	−9.33791	−0.532942	−0.266471	+	0.963843i	$0.585858\pi$
$308$	0.0845208	0.00481602
$309$	0	0
$310$	0	0
$311$	−24.6153	−1.39581	−0.697903	−	0.716193i	$-0.745883\pi$
$311$	−24.6153	−1.39581	−0.697903	+	0.716193i	$0.745883\pi$
$312$	0	0
$313$	−9.61757	−0.543617	−0.271809	−	0.962351i	$-0.587622\pi$
$313$	−9.61757	−0.543617	−0.271809	+	0.962351i	$0.587622\pi$
$314$	−16.5886	−0.936151
$315$	0	0
$316$	−10.4969	−0.590498
$317$	−15.4836	−0.869645	−0.434823	−	0.900516i	$-0.643189\pi$
$317$	−15.4836	−0.869645	−0.434823	+	0.900516i	$0.643189\pi$
$318$	0	0
$319$	4.41855	0.247391
$320$	0	0
$321$	0	0
$322$	0.837101	0.0466498
$323$	−9.27513	−0.516082
$324$	0	0
$325$	0	0
$326$	−9.94933	−0.551042
$327$	0	0
$328$	10.2413	0.565480
$329$	−3.00614	−0.165734
$330$	0	0
$331$	10.4813	0.576106	0.288053	−	0.957614i	$-0.406992\pi$
$331$	10.4813	0.576106	0.288053	+	0.957614i	$0.406992\pi$
$332$	6.34858	0.348424
$333$	0	0
$334$	6.95282	0.380441
$335$	0	0
$336$	0	0
$337$	−26.5802	−1.44792	−0.723959	−	0.689843i	$-0.757679\pi$
$337$	−26.5802	−1.44792	−0.723959	+	0.689843i	$0.757679\pi$
$338$	−11.4159	−0.620943
$339$	0	0
$340$	0	0
$341$	0.460811	0.0249543
$342$	0	0
$343$	4.04557	0.218440
$344$	33.3607	1.79869
$345$	0	0
$346$	−22.3318	−1.20056
$347$	−16.0494	−0.861580	−0.430790	−	0.902452i	$-0.641765\pi$
$347$	−16.0494	−0.861580	−0.430790	+	0.902452i	$0.641765\pi$
$348$	0	0
$349$	−1.60811	−0.0860802	−0.0430401	−	0.999073i	$-0.513704\pi$
$349$	−1.60811	−0.0860802	−0.0430401	+	0.999073i	$0.513704\pi$
$350$	0	0
$351$	0	0
$352$	1.57531	0.0839641
$353$	−19.1061	−1.01691	−0.508457	−	0.861087i	$-0.669784\pi$
$353$	−19.1061	−1.01691	−0.508457	+	0.861087i	$0.669784\pi$
$354$	0	0
$355$	0	0
$356$	3.49466	0.185217
$357$	0	0
$358$	12.7382	0.673235
$359$	−4.70701	−0.248426	−0.124213	−	0.992256i	$-0.539641\pi$
$359$	−4.70701	−0.248426	−0.124213	+	0.992256i	$0.539641\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−12.0806	−0.634945
$363$	0	0
$364$	0.330332	0.0173141
$365$	0	0
$366$	0	0
$367$	−18.8638	−0.984680	−0.492340	−	0.870403i	$-0.663858\pi$
$367$	−18.8638	−0.984680	−0.492340	+	0.870403i	$0.663858\pi$
$368$	5.75872	0.300194
$369$	0	0
$370$	0	0
$371$	0.560753	0.0291128
$372$	0	0
$373$	−17.8865	−0.926130	−0.463065	−	0.886324i	$-0.653250\pi$
$373$	−17.8865	−0.926130	−0.463065	+	0.886324i	$0.653250\pi$
$374$	−1.66701	−0.0861993
$375$	0	0
$376$	−31.8310	−1.64156
$377$	17.2690	0.889398
$378$	0	0
$379$	11.3112	0.581020	0.290510	−	0.956872i	$-0.406175\pi$
$379$	11.3112	0.581020	0.290510	+	0.956872i	$0.406175\pi$
$380$	0	0
$381$	0	0
$382$	−11.3424	−0.580330
$383$	−34.1845	−1.74674	−0.873372	−	0.487053i	$-0.838072\pi$
$383$	−34.1845	−1.74674	−0.873372	+	0.487053i	$0.838072\pi$
$384$	0	0
$385$	0	0
$386$	−5.79257	−0.294834
$387$	0	0
$388$	1.61038	0.0817546
$389$	25.1506	1.27519	0.637593	−	0.770373i	$-0.279930\pi$
$389$	25.1506	1.27519	0.637593	+	0.770373i	$0.279930\pi$
$390$	0	0
$391$	7.60811	0.384759
$392$	21.2885	1.07523
$393$	0	0
$394$	−5.44521	−0.274326
$395$	0	0
$396$	0	0
$397$	−14.8927	−0.747443	−0.373721	−	0.927541i	$-0.621918\pi$
$397$	−14.8927	−0.747443	−0.373721	+	0.927541i	$0.621918\pi$
$398$	−9.28005	−0.465167
$399$	0	0
$400$	0	0
$401$	20.9988	1.04863	0.524314	−	0.851525i	$-0.324322\pi$
$401$	20.9988	1.04863	0.524314	+	0.851525i	$0.324322\pi$
$402$	0	0
$403$	1.80098	0.0897134
$404$	−5.72753	−0.284955
$405$	0	0
$406$	−3.26180	−0.161880
$407$	−3.17009	−0.157135
$408$	0	0
$409$	23.7009	1.17193	0.585966	−	0.810336i	$-0.300715\pi$
$409$	23.7009	1.17193	0.585966	+	0.810336i	$0.300715\pi$
$410$	0	0
$411$	0	0
$412$	11.5669	0.569860
$413$	−0.286849	−0.0141149
$414$	0	0
$415$	0	0
$416$	6.15676	0.301860
$417$	0	0
$418$	1.61757	0.0791178
$419$	11.7009	0.571625	0.285812	−	0.958286i	$-0.407737\pi$
$419$	11.7009	0.571625	0.285812	+	0.958286i	$0.407737\pi$
$420$	0	0
$421$	3.18568	0.155261	0.0776304	−	0.996982i	$-0.475265\pi$
$421$	3.18568	0.155261	0.0776304	+	0.996982i	$0.475265\pi$
$422$	9.66701	0.470583
$423$	0	0
$424$	5.93761	0.288356
$425$	0	0
$426$	0	0
$427$	1.94214	0.0939868
$428$	−6.94828	−0.335858
$429$	0	0
$430$	0	0
$431$	28.8638	1.39032	0.695159	−	0.718856i	$-0.255334\pi$
$431$	28.8638	1.39032	0.695159	+	0.718856i	$0.255334\pi$
$432$	0	0
$433$	−26.8371	−1.28971	−0.644854	−	0.764305i	$-0.723082\pi$
$433$	−26.8371	−1.28971	−0.644854	+	0.764305i	$0.723082\pi$
$434$	−0.340173	−0.0163288
$435$	0	0
$436$	−8.39803	−0.402193
$437$	−7.38243	−0.353150
$438$	0	0
$439$	−7.52586	−0.359190	−0.179595	−	0.983741i	$-0.557479\pi$
$439$	−7.52586	−0.359190	−0.179595	+	0.983741i	$0.557479\pi$
$440$	0	0
$441$	0	0
$442$	−6.51518	−0.309896
$443$	−7.72034	−0.366804	−0.183402	−	0.983038i	$-0.558711\pi$
$443$	−7.72034	−0.366804	−0.183402	+	0.983038i	$0.558711\pi$
$444$	0	0
$445$	0	0
$446$	−10.1880	−0.482414
$447$	0	0
$448$	−2.52359	−0.119228
$449$	21.6959	1.02389	0.511947	−	0.859017i	$-0.328924\pi$
$449$	21.6959	1.02389	0.511947	+	0.859017i	$0.328924\pi$
$450$	0	0
$451$	−1.53305	−0.0721883
$452$	6.58598	0.309779
$453$	0	0
$454$	2.67050	0.125333
$455$	0	0
$456$	0	0
$457$	8.76818	0.410158	0.205079	−	0.978745i	$-0.434255\pi$
$457$	8.76818	0.410158	0.205079	+	0.978745i	$0.434255\pi$
$458$	26.2739	1.22770
$459$	0	0
$460$	0	0
$461$	−27.0423	−1.25948	−0.629742	−	0.776805i	$-0.716839\pi$
$461$	−27.0423	−1.25948	−0.629742	+	0.776805i	$0.716839\pi$
$462$	0	0
$463$	−22.0989	−1.02702	−0.513511	−	0.858083i	$-0.671656\pi$
$463$	−22.0989	−1.02702	−0.513511	+	0.858083i	$0.671656\pi$
$464$	−22.4391	−1.04171
$465$	0	0
$466$	27.0205	1.25170
$467$	14.4969	0.670838	0.335419	−	0.942069i	$-0.391122\pi$
$467$	14.4969	0.670838	0.335419	+	0.942069i	$0.391122\pi$
$468$	0	0
$469$	0.00841049	0.000388360 0
$470$	0	0
$471$	0	0
$472$	−3.03734	−0.139805
$473$	−4.99386	−0.229618
$474$	0	0
$475$	0	0
$476$	−0.567073	−0.0259918
$477$	0	0
$478$	0.811662	0.0371246
$479$	29.0349	1.32664	0.663319	−	0.748337i	$-0.269147\pi$
$479$	29.0349	1.32664	0.663319	+	0.748337i	$0.269147\pi$
$480$	0	0
$481$	−12.3896	−0.564918
$482$	18.5053	0.842895
$483$	0	0
$484$	6.80590	0.309359
$485$	0	0
$486$	0	0
$487$	−12.6381	−0.572686	−0.286343	−	0.958127i	$-0.592440\pi$
$487$	−12.6381	−0.572686	−0.286343	+	0.958127i	$0.592440\pi$
$488$	20.5646	0.930917
$489$	0	0
$490$	0	0
$491$	13.1145	0.591849	0.295924	−	0.955211i	$-0.404372\pi$
$491$	13.1145	0.591849	0.295924	+	0.955211i	$0.404372\pi$
$492$	0	0
$493$	−29.6453	−1.33516
$494$	6.32192	0.284437
$495$	0	0
$496$	−2.34017	−0.105077
$497$	4.05332	0.181816
$498$	0	0
$499$	27.5741	1.23439	0.617193	−	0.786812i	$-0.288270\pi$
$499$	27.5741	1.23439	0.617193	+	0.786812i	$0.288270\pi$
$500$	0	0
$501$	0	0
$502$	0.297913	0.0132965
$503$	37.5897	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$503$	37.5897	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$504$	0	0
$505$	0	0
$506$	−1.32684	−0.0589853
$507$	0	0
$508$	7.63317	0.338667
$509$	14.7782	0.655032	0.327516	−	0.944846i	$-0.393788\pi$
$509$	14.7782	0.655032	0.327516	+	0.944846i	$0.393788\pi$
$510$	0	0
$511$	−2.88267	−0.127522
$512$	−22.4079	−0.990297
$513$	0	0
$514$	36.2739	1.59997
$515$	0	0
$516$	0	0
$517$	4.76487	0.209559
$518$	2.34017	0.102821
$519$	0	0
$520$	0	0
$521$	12.4897	0.547185	0.273593	−	0.961846i	$-0.411788\pi$
$521$	12.4897	0.547185	0.273593	+	0.961846i	$0.411788\pi$
$522$	0	0
$523$	−6.71154	−0.293475	−0.146738	−	0.989175i	$-0.546877\pi$
$523$	−6.71154	−0.293475	−0.146738	+	0.989175i	$0.546877\pi$
$524$	7.67420	0.335249
$525$	0	0
$526$	21.4596	0.935683
$527$	−3.09171	−0.134677
$528$	0	0
$529$	−16.9444	−0.736713
$530$	0	0
$531$	0	0
$532$	0.550252	0.0238565
$533$	−5.99159	−0.259525
$534$	0	0
$535$	0	0
$536$	0.0890557	0.00384662
$537$	0	0
$538$	−16.1666	−0.696991
$539$	−3.18673	−0.137262
$540$	0	0
$541$	13.3584	0.574324	0.287162	−	0.957882i	$-0.407288\pi$
$541$	13.3584	0.574324	0.287162	+	0.957882i	$0.407288\pi$
$542$	−9.49466	−0.407831
$543$	0	0
$544$	−10.5692	−0.453149
$545$	0	0
$546$	0	0
$547$	−22.9506	−0.981295	−0.490647	−	0.871358i	$-0.663240\pi$
$547$	−22.9506	−0.981295	−0.490647	+	0.871358i	$0.663240\pi$
$548$	−1.79833	−0.0768209
$549$	0	0
$550$	0	0
$551$	28.7659	1.22547
$552$	0	0
$553$	−4.83710	−0.205694
$554$	−12.8950	−0.547854
$555$	0	0
$556$	8.53797	0.362090
$557$	−30.4173	−1.28882	−0.644412	−	0.764679i	$-0.722898\pi$
$557$	−30.4173	−1.28882	−0.644412	+	0.764679i	$0.722898\pi$
$558$	0	0
$559$	−19.5174	−0.825500
$560$	0	0
$561$	0	0
$562$	9.54411	0.402594
$563$	2.30179	0.0970088	0.0485044	−	0.998823i	$-0.484555\pi$
$563$	2.30179	0.0970088	0.0485044	+	0.998823i	$0.484555\pi$
$564$	0	0
$565$	0	0
$566$	22.1327	0.930309
$567$	0	0
$568$	42.9192	1.80085
$569$	22.1929	0.930374	0.465187	−	0.885213i	$-0.345987\pi$
$569$	22.1929	0.930374	0.465187	+	0.885213i	$0.345987\pi$
$570$	0	0
$571$	−4.02666	−0.168511	−0.0842553	−	0.996444i	$-0.526851\pi$
$571$	−4.02666	−0.168511	−0.0842553	+	0.996444i	$0.526851\pi$
$572$	−0.523590	−0.0218924
$573$	0	0
$574$	1.13170	0.0472363
$575$	0	0
$576$	0	0
$577$	−43.7191	−1.82005	−0.910025	−	0.414553i	$-0.863938\pi$
$577$	−43.7191	−1.82005	−0.910025	+	0.414553i	$0.863938\pi$
$578$	−8.70701	−0.362164
$579$	0	0
$580$	0	0
$581$	2.92550	0.121370
$582$	0	0
$583$	−0.888817	−0.0368110
$584$	−30.5236	−1.26308
$585$	0	0
$586$	0.223025	0.00921307
$587$	29.5848	1.22109	0.610547	−	0.791980i	$-0.290950\pi$
$587$	29.5848	1.22109	0.610547	+	0.791980i	$0.290950\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	0	0
$591$	0	0
$592$	16.0989	0.661661
$593$	46.5886	1.91317	0.956583	−	0.291460i	$-0.0941410\pi$
$593$	46.5886	1.91317	0.956583	+	0.291460i	$0.0941410\pi$
$594$	0	0
$595$	0	0
$596$	14.3500	0.587799
$597$	0	0
$598$	−5.18568	−0.212058
$599$	6.50412	0.265751	0.132875	−	0.991133i	$-0.457579\pi$
$599$	6.50412	0.265751	0.132875	+	0.991133i	$0.457579\pi$
$600$	0	0
$601$	−19.3496	−0.789288	−0.394644	−	0.918834i	$-0.629132\pi$
$601$	−19.3496	−0.789288	−0.394644	+	0.918834i	$0.629132\pi$
$602$	3.68649	0.150250
$603$	0	0
$604$	6.85392	0.278882
$605$	0	0
$606$	0	0
$607$	44.3400	1.79970	0.899852	−	0.436194i	$-0.143674\pi$
$607$	44.3400	1.79970	0.899852	+	0.436194i	$0.143674\pi$
$608$	10.2557	0.415922
$609$	0	0
$610$	0	0
$611$	18.6225	0.753385
$612$	0	0
$613$	0.228990	0.00924883	0.00462441	−	0.999989i	$-0.498528\pi$
$613$	0.228990	0.00924883	0.00462441	+	0.999989i	$0.498528\pi$
$614$	−10.9262	−0.440944
$615$	0	0
$616$	0.412408	0.0166164
$617$	7.51026	0.302352	0.151176	−	0.988507i	$-0.451694\pi$
$617$	7.51026	0.302352	0.151176	+	0.988507i	$0.451694\pi$
$618$	0	0
$619$	−37.5441	−1.50903	−0.754513	−	0.656286i	$-0.772127\pi$
$619$	−37.5441	−1.50903	−0.754513	+	0.656286i	$0.772127\pi$
$620$	0	0
$621$	0	0
$622$	−28.8020	−1.15486
$623$	1.61038	0.0645185
$624$	0	0
$625$	0	0
$626$	−11.2534	−0.449776
$627$	0	0
$628$	8.94441	0.356921
$629$	21.2690	0.848050
$630$	0	0
$631$	26.7226	1.06381	0.531905	−	0.846804i	$-0.321476\pi$
$631$	26.7226	1.06381	0.531905	+	0.846804i	$0.321476\pi$
$632$	−51.2183	−2.03736
$633$	0	0
$634$	−18.1171	−0.719524
$635$	0	0
$636$	0	0
$637$	−12.4547	−0.493472
$638$	5.17009	0.204686
$639$	0	0
$640$	0	0
$641$	19.7987	0.782002	0.391001	−	0.920390i	$-0.372129\pi$
$641$	19.7987	0.782002	0.391001	+	0.920390i	$0.372129\pi$
$642$	0	0
$643$	29.3763	1.15849	0.579244	−	0.815154i	$-0.303348\pi$
$643$	29.3763	1.15849	0.579244	+	0.815154i	$0.303348\pi$
$644$	−0.451356	−0.0177859
$645$	0	0
$646$	−10.8527	−0.426994
$647$	−45.1867	−1.77647	−0.888237	−	0.459386i	$-0.848070\pi$
$647$	−45.1867	−1.77647	−0.888237	+	0.459386i	$0.848070\pi$
$648$	0	0
$649$	0.454668	0.0178473
$650$	0	0
$651$	0	0
$652$	5.36457	0.210093
$653$	−0.704355	−0.0275635	−0.0137818	−	0.999905i	$-0.504387\pi$
$653$	−0.704355	−0.0275635	−0.0137818	+	0.999905i	$0.504387\pi$
$654$	0	0
$655$	0	0
$656$	7.78539	0.303968
$657$	0	0
$658$	−3.51745	−0.137124
$659$	37.0082	1.44164	0.720818	−	0.693124i	$-0.243766\pi$
$659$	37.0082	1.44164	0.720818	+	0.693124i	$0.243766\pi$
$660$	0	0
$661$	−41.9009	−1.62976	−0.814879	−	0.579632i	$-0.803196\pi$
$661$	−41.9009	−1.62976	−0.814879	+	0.579632i	$0.803196\pi$
$662$	12.2641	0.476656
$663$	0	0
$664$	30.9770	1.20214
$665$	0	0
$666$	0	0
$667$	−23.5958	−0.913634
$668$	−3.74888	−0.145049
$669$	0	0
$670$	0	0
$671$	−3.07838	−0.118839
$672$	0	0
$673$	−2.18464	−0.0842117	−0.0421058	−	0.999113i	$-0.513407\pi$
$673$	−2.18464	−0.0842117	−0.0421058	+	0.999113i	$0.513407\pi$
$674$	−31.1012	−1.19797
$675$	0	0
$676$	6.15532	0.236743
$677$	7.18115	0.275994	0.137997	−	0.990433i	$-0.455934\pi$
$677$	7.18115	0.275994	0.137997	+	0.990433i	$0.455934\pi$
$678$	0	0
$679$	0.742080	0.0284784
$680$	0	0
$681$	0	0
$682$	0.539189	0.0206466
$683$	−35.6430	−1.36384	−0.681921	−	0.731426i	$-0.738855\pi$
$683$	−35.6430	−1.36384	−0.681921	+	0.731426i	$0.738855\pi$
$684$	0	0
$685$	0	0
$686$	4.73367	0.180732
$687$	0	0
$688$	25.3607	0.966867
$689$	−3.47376	−0.132340
$690$	0	0
$691$	20.9877	0.798410	0.399205	−	0.916862i	$-0.369286\pi$
$691$	20.9877	0.798410	0.399205	+	0.916862i	$0.369286\pi$
$692$	12.0410	0.457732
$693$	0	0
$694$	−18.7792	−0.712850
$695$	0	0
$696$	0	0
$697$	10.2856	0.389596
$698$	−1.88163	−0.0712207
$699$	0	0
$700$	0	0
$701$	−35.4791	−1.34003	−0.670013	−	0.742349i	$-0.733711\pi$
$701$	−35.4791	−1.34003	−0.670013	+	0.742349i	$0.733711\pi$
$702$	0	0
$703$	−20.6381	−0.778380
$704$	4.00000	0.150756
$705$	0	0
$706$	−22.3558	−0.841371
$707$	−2.63931	−0.0992614
$708$	0	0
$709$	3.71646	0.139575	0.0697874	−	0.997562i	$-0.477768\pi$
$709$	3.71646	0.139575	0.0697874	+	0.997562i	$0.477768\pi$
$710$	0	0
$711$	0	0
$712$	17.0517	0.639040
$713$	−2.46081	−0.0921581
$714$	0	0
$715$	0	0
$716$	−6.86830	−0.256680
$717$	0	0
$718$	−5.50761	−0.205542
$719$	−33.6209	−1.25385	−0.626924	−	0.779081i	$-0.715686\pi$
$719$	−33.6209	−1.25385	−0.626924	+	0.779081i	$0.715686\pi$
$720$	0	0
$721$	5.33015	0.198505
$722$	−11.7009	−0.435461
$723$	0	0
$724$	6.51375	0.242082
$725$	0	0
$726$	0	0
$727$	26.5464	0.984551	0.492275	−	0.870440i	$-0.336165\pi$
$727$	26.5464	0.984551	0.492275	+	0.870440i	$0.336165\pi$
$728$	1.61181	0.0597376
$729$	0	0
$730$	0	0
$731$	33.5052	1.23923
$732$	0	0
$733$	8.04718	0.297229	0.148615	−	0.988895i	$-0.452519\pi$
$733$	8.04718	0.297229	0.148615	+	0.988895i	$0.452519\pi$
$734$	−22.0722	−0.814701
$735$	0	0
$736$	−8.41241	−0.310085
$737$	−0.0133310	−0.000491054 0
$738$	0	0
$739$	34.2544	1.26007	0.630035	−	0.776567i	$-0.283041\pi$
$739$	34.2544	1.26007	0.630035	+	0.776567i	$0.283041\pi$
$740$	0	0
$741$	0	0
$742$	0.656129	0.0240873
$743$	24.5909	0.902153	0.451076	−	0.892485i	$-0.351040\pi$
$743$	24.5909	0.902153	0.451076	+	0.892485i	$0.351040\pi$
$744$	0	0
$745$	0	0
$746$	−20.9288	−0.766258
$747$	0	0
$748$	0.898836	0.0328647
$749$	−3.20185	−0.116993
$750$	0	0
$751$	0.824993	0.0301044	0.0150522	−	0.999887i	$-0.495209\pi$
$751$	0.824993	0.0301044	0.0150522	+	0.999887i	$0.495209\pi$
$752$	−24.1978	−0.882403
$753$	0	0
$754$	20.2062	0.735867
$755$	0	0
$756$	0	0
$757$	44.7058	1.62486	0.812430	−	0.583059i	$-0.198144\pi$
$757$	44.7058	1.62486	0.812430	+	0.583059i	$0.198144\pi$
$758$	13.2351	0.480722
$759$	0	0
$760$	0	0
$761$	27.0121	0.979188	0.489594	−	0.871950i	$-0.337145\pi$
$761$	27.0121	0.979188	0.489594	+	0.871950i	$0.337145\pi$
$762$	0	0
$763$	−3.86991	−0.140100
$764$	6.11572	0.221259
$765$	0	0
$766$	−39.9988	−1.44521
$767$	1.77698	0.0641629
$768$	0	0
$769$	−17.5464	−0.632739	−0.316369	−	0.948636i	$-0.602464\pi$
$769$	−17.5464	−0.632739	−0.316369	+	0.948636i	$0.602464\pi$
$770$	0	0
$771$	0	0
$772$	3.12329	0.112410
$773$	−6.88550	−0.247654	−0.123827	−	0.992304i	$-0.539517\pi$
$773$	−6.88550	−0.247654	−0.123827	+	0.992304i	$0.539517\pi$
$774$	0	0
$775$	0	0
$776$	7.85762	0.282072
$777$	0	0
$778$	29.4284	1.05506
$779$	−9.98053	−0.357589
$780$	0	0
$781$	−6.42469	−0.229894
$782$	8.90215	0.318340
$783$	0	0
$784$	16.1834	0.577979
$785$	0	0
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	2.93600	0.104591
$789$	0	0
$790$	0	0
$791$	3.03489	0.107908
$792$	0	0
$793$	−12.0312	−0.427240
$794$	−17.4257	−0.618416
$795$	0	0
$796$	5.00370	0.177351
$797$	−46.9420	−1.66277	−0.831385	−	0.555697i	$-0.812451\pi$
$797$	−46.9420	−1.66277	−0.831385	+	0.555697i	$0.812451\pi$
$798$	0	0
$799$	−31.9688	−1.13097
$800$	0	0
$801$	0	0
$802$	24.5704	0.867610
$803$	4.56916	0.161242
$804$	0	0
$805$	0	0
$806$	2.10731	0.0742267
$807$	0	0
$808$	−27.9467	−0.983161
$809$	−40.9949	−1.44130	−0.720652	−	0.693297i	$-0.756157\pi$
$809$	−40.9949	−1.44130	−0.720652	+	0.693297i	$0.756157\pi$
$810$	0	0
$811$	44.8164	1.57372	0.786858	−	0.617134i	$-0.211706\pi$
$811$	44.8164	1.57372	0.786858	+	0.617134i	$0.211706\pi$
$812$	1.75872	0.0617191
$813$	0	0
$814$	−3.70928	−0.130010
$815$	0	0
$816$	0	0
$817$	−32.5113	−1.13743
$818$	27.7321	0.969629
$819$	0	0
$820$	0	0
$821$	23.1822	0.809064	0.404532	−	0.914524i	$-0.367434\pi$
$821$	23.1822	0.809064	0.404532	+	0.914524i	$0.367434\pi$
$822$	0	0
$823$	24.3356	0.848287	0.424144	−	0.905595i	$-0.360575\pi$
$823$	24.3356	0.848287	0.424144	+	0.905595i	$0.360575\pi$
$824$	56.4391	1.96615
$825$	0	0
$826$	−0.335638	−0.0116783
$827$	−14.7126	−0.511607	−0.255803	−	0.966729i	$-0.582340\pi$
$827$	−14.7126	−0.511607	−0.255803	+	0.966729i	$0.582340\pi$
$828$	0	0
$829$	−24.2979	−0.843901	−0.421951	−	0.906619i	$-0.638654\pi$
$829$	−24.2979	−0.843901	−0.421951	+	0.906619i	$0.638654\pi$
$830$	0	0
$831$	0	0
$832$	15.6332	0.541982
$833$	21.3806	0.740796
$834$	0	0
$835$	0	0
$836$	−0.872174	−0.0301648
$837$	0	0
$838$	13.6910	0.472949
$839$	3.30283	0.114026	0.0570132	−	0.998373i	$-0.481842\pi$
$839$	3.30283	0.114026	0.0570132	+	0.998373i	$0.481842\pi$
$840$	0	0
$841$	62.9420	2.17041
$842$	3.72753	0.128459
$843$	0	0
$844$	−5.21235	−0.179416
$845$	0	0
$846$	0	0
$847$	3.13624	0.107762
$848$	4.51375	0.155003
$849$	0	0
$850$	0	0
$851$	16.9288	0.580312
$852$	0	0
$853$	10.6264	0.363840	0.181920	−	0.983313i	$-0.441769\pi$
$853$	10.6264	0.363840	0.181920	+	0.983313i	$0.441769\pi$
$854$	2.27247	0.0777624
$855$	0	0
$856$	−33.9032	−1.15879
$857$	27.6092	0.943111	0.471555	−	0.881836i	$-0.343693\pi$
$857$	27.6092	0.943111	0.471555	+	0.881836i	$0.343693\pi$
$858$	0	0
$859$	−6.79606	−0.231879	−0.115939	−	0.993256i	$-0.536988\pi$
$859$	−6.79606	−0.231879	−0.115939	+	0.993256i	$0.536988\pi$
$860$	0	0
$861$	0	0
$862$	33.7731	1.15032
$863$	−1.42309	−0.0484424	−0.0242212	−	0.999707i	$-0.507711\pi$
$863$	−1.42309	−0.0484424	−0.0242212	+	0.999707i	$0.507711\pi$
$864$	0	0
$865$	0	0
$866$	−31.4017	−1.06707
$867$	0	0
$868$	0.183417	0.00622559
$869$	7.66701	0.260086
$870$	0	0
$871$	−0.0521014	−0.00176539
$872$	−40.9770	−1.38766
$873$	0	0
$874$	−8.63809	−0.292188
$875$	0	0
$876$	0	0
$877$	−17.8622	−0.603162	−0.301581	−	0.953441i	$-0.597514\pi$
$877$	−17.8622	−0.603162	−0.301581	+	0.953441i	$0.597514\pi$
$878$	−8.80590	−0.297185
$879$	0	0
$880$	0	0
$881$	−37.7998	−1.27351	−0.636753	−	0.771068i	$-0.719723\pi$
$881$	−37.7998	−1.27351	−0.636753	+	0.771068i	$0.719723\pi$
$882$	0	0
$883$	−55.7140	−1.87493	−0.937463	−	0.348085i	$-0.886832\pi$
$883$	−55.7140	−1.87493	−0.937463	+	0.348085i	$0.886832\pi$
$884$	3.51291	0.118152
$885$	0	0
$886$	−9.03346	−0.303485
$887$	24.3908	0.818964	0.409482	−	0.912318i	$-0.365709\pi$
$887$	24.3908	0.818964	0.409482	+	0.912318i	$0.365709\pi$
$888$	0	0
$889$	3.51745	0.117971
$890$	0	0
$891$	0	0
$892$	5.49323	0.183927
$893$	31.0205	1.03806
$894$	0	0
$895$	0	0
$896$	−0.965105	−0.0322419
$897$	0	0
$898$	25.3861	0.847146
$899$	9.58864	0.319799
$900$	0	0
$901$	5.96332	0.198667
$902$	−1.79380	−0.0597269
$903$	0	0
$904$	32.1354	1.06881
$905$	0	0
$906$	0	0
$907$	5.39350	0.179088	0.0895441	−	0.995983i	$-0.471459\pi$
$907$	5.39350	0.179088	0.0895441	+	0.995983i	$0.471459\pi$
$908$	−1.43991	−0.0477850
$909$	0	0
$910$	0	0
$911$	51.8131	1.71664	0.858322	−	0.513111i	$-0.171507\pi$
$911$	51.8131	1.71664	0.858322	+	0.513111i	$0.171507\pi$
$912$	0	0
$913$	−4.63704	−0.153464
$914$	10.2595	0.339355
$915$	0	0
$916$	−14.1666	−0.468078
$917$	3.53636	0.116781
$918$	0	0
$919$	−39.6186	−1.30690	−0.653449	−	0.756971i	$-0.726678\pi$
$919$	−39.6186	−1.30690	−0.653449	+	0.756971i	$0.726678\pi$
$920$	0	0
$921$	0	0
$922$	−31.6418	−1.04207
$923$	−25.1096	−0.826492
$924$	0	0
$925$	0	0
$926$	−25.8576	−0.849734
$927$	0	0
$928$	32.7792	1.07603
$929$	8.93042	0.292998	0.146499	−	0.989211i	$-0.453200\pi$
$929$	8.93042	0.292998	0.146499	+	0.989211i	$0.453200\pi$
$930$	0	0
$931$	−20.7464	−0.679937
$932$	−14.5692	−0.477229
$933$	0	0
$934$	16.9627	0.555035
$935$	0	0
$936$	0	0
$937$	−3.97948	−0.130004	−0.0650020	−	0.997885i	$-0.520705\pi$
$937$	−3.97948	−0.130004	−0.0650020	+	0.997885i	$0.520705\pi$
$938$	0.00984100	0.000321320 0
$939$	0	0
$940$	0	0
$941$	19.7682	0.644424	0.322212	−	0.946668i	$-0.395574\pi$
$941$	19.7682	0.644424	0.322212	+	0.946668i	$0.395574\pi$
$942$	0	0
$943$	8.18673	0.266597
$944$	−2.30898	−0.0751508
$945$	0	0
$946$	−5.84324	−0.189980
$947$	−4.37629	−0.142210	−0.0711052	−	0.997469i	$-0.522653\pi$
$947$	−4.37629	−0.142210	−0.0711052	+	0.997469i	$0.522653\pi$
$948$	0	0
$949$	17.8576	0.579683
$950$	0	0
$951$	0	0
$952$	−2.76696	−0.0896776
$953$	−9.15836	−0.296669	−0.148334	−	0.988937i	$-0.547391\pi$
$953$	−9.15836	−0.296669	−0.148334	+	0.988937i	$0.547391\pi$
$954$	0	0
$955$	0	0
$956$	−0.437639	−0.0141543
$957$	0	0
$958$	33.9733	1.09763
$959$	−0.828691	−0.0267598
$960$	0	0
$961$	1.00000	0.0322581
$962$	−14.4969	−0.467400
$963$	0	0
$964$	−9.97787	−0.321366
$965$	0	0
$966$	0	0
$967$	−14.1289	−0.454354	−0.227177	−	0.973854i	$-0.572950\pi$
$967$	−14.1289	−0.454354	−0.227177	+	0.973854i	$0.572950\pi$
$968$	33.2085	1.06736
$969$	0	0
$970$	0	0
$971$	−8.62968	−0.276939	−0.138470	−	0.990367i	$-0.544218\pi$
$971$	−8.62968	−0.276939	−0.138470	+	0.990367i	$0.544218\pi$
$972$	0	0
$973$	3.93439	0.126131
$974$	−14.7877	−0.473827
$975$	0	0
$976$	15.6332	0.500405
$977$	2.79872	0.0895389	0.0447694	−	0.998997i	$-0.485745\pi$
$977$	2.79872	0.0895389	0.0447694	+	0.998997i	$0.485745\pi$
$978$	0	0
$979$	−2.55252	−0.0815789
$980$	0	0
$981$	0	0
$982$	15.3451	0.489682
$983$	22.0712	0.703962	0.351981	−	0.936007i	$-0.385508\pi$
$983$	22.0712	0.703962	0.351981	+	0.936007i	$0.385508\pi$
$984$	0	0
$985$	0	0
$986$	−34.6875	−1.10468
$987$	0	0
$988$	−3.40871	−0.108446
$989$	26.6681	0.847995
$990$	0	0
$991$	17.8855	0.568152	0.284076	−	0.958802i	$-0.408313\pi$
$991$	17.8855	0.568152	0.284076	+	0.958802i	$0.408313\pi$
$992$	3.41855	0.108539
$993$	0	0
$994$	4.74274	0.150431
$995$	0	0
$996$	0	0
$997$	7.30510	0.231355	0.115677	−	0.993287i	$-0.463096\pi$
$997$	7.30510	0.231355	0.115677	+	0.993287i	$0.463096\pi$
$998$	32.2641	1.02130
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.z.1.3	✓	3
3.2	odd	2		6975.2.a.bg.1.1	yes	3
5.4	even	2		6975.2.a.bh.1.1	yes	3
15.14	odd	2		6975.2.a.ba.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.z.1.3	✓	3	1.1	even	1	trivial
6975.2.a.ba.1.3	yes	3	15.14	odd	2
6975.2.a.bg.1.1	yes	3	3.2	odd	2
6975.2.a.bh.1.1	yes	3	5.4	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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17009 q^{2} -0.630898 q^{4} -0.290725 q^{7} -3.07838 q^{8} +0.460811 q^{11} +1.80098 q^{13} -0.340173 q^{14} -2.34017 q^{16} -3.09171 q^{17} +3.00000 q^{19} +0.539189 q^{22} -2.46081 q^{23} +2.10731 q^{26} +0.183417 q^{28} +9.58864 q^{29} +1.00000 q^{31} +3.41855 q^{32} -3.61757 q^{34} -6.87936 q^{37} +3.51026 q^{38} -3.32684 q^{41} -10.8371 q^{43} -0.290725 q^{44} -2.87936 q^{46} +10.3402 q^{47} -6.91548 q^{49} -1.13624 q^{52} -1.92881 q^{53} +0.894960 q^{56} +11.2195 q^{58} +0.986669 q^{59} -6.68035 q^{61} +1.17009 q^{62} +8.68035 q^{64} -0.0289294 q^{67} +1.95055 q^{68} -13.9421 q^{71} +9.91548 q^{73} -8.04945 q^{74} -1.89269 q^{76} -0.133969 q^{77} +16.6381 q^{79} -3.89269 q^{82} -10.0628 q^{83} -12.6803 q^{86} -1.41855 q^{88} -5.53919 q^{89} -0.523590 q^{91} +1.55252 q^{92} +12.0989 q^{94} -2.55252 q^{97} -8.09171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 2 q^{4} - 8 q^{7} - 6 q^{8} + 3 q^{11} - 4 q^{13} + 10 q^{14} + 4 q^{16} - 7 q^{17} + 9 q^{19} - 9 q^{23} + 18 q^{26} - 4 q^{28} + 9 q^{29} + 3 q^{31} - 4 q^{32} - 6 q^{34} - 8 q^{37} - 6 q^{38}+ \cdots - 22 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 2 * q^4 - 8 * q^7 - 6 * q^8 + 3 * q^11 - 4 * q^13 + 10 * q^14 + 4 * q^16 - 7 * q^17 + 9 * q^19 - 9 * q^23 + 18 * q^26 - 4 * q^28 + 9 * q^29 + 3 * q^31 - 4 * q^32 - 6 * q^34 - 8 * q^37 - 6 * q^38 + 2 * q^41 - 4 * q^43 - 8 * q^44 + 4 * q^46 + 20 * q^47 + 11 * q^49 - 30 * q^52 + 25 * q^53 + 4 * q^56 + 10 * q^58 + 2 * q^59 + 2 * q^61 - 2 * q^62 + 4 * q^64 - 15 * q^67 + 24 * q^68 - 12 * q^71 - 2 * q^73 - 6 * q^74 + 6 * q^76 - 14 * q^77 + 12 * q^79 - 13 * q^83 - 16 * q^86 + 10 * q^88 - 15 * q^89 + 14 * q^91 + 4 * q^92 - 7 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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