LMFDB - Embedded newform 6975.2.a.ba.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:	$0$
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	$-0.482503\pi$
$7$	0.290725	0.109884	0.0549418	+	0.998490i	$0.482503\pi$
$8$	−3.07838	−1.08837
$9$	0	0
$10$	0	0
$11$	−0.460811	−0.138940	−0.0694699	−	0.997584i	$-0.522131\pi$
$11$	−0.460811	−0.138940	−0.0694699	+	0.997584i	$0.522131\pi$
$12$	0	0
$13$	−1.80098	−0.499503	−0.249752	−	0.968310i	$-0.580349\pi$
$13$	−1.80098	−0.499503	−0.249752	+	0.968310i	$0.580349\pi$
$14$	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	$-0.849493\pi$
$29$	−9.58864	−1.78057	−0.890283	+	0.455408i	$0.849493\pi$
$30$	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	$-0.308691\pi$
$37$	6.87936	1.13096	0.565480	+	0.824762i	$0.308691\pi$
$38$	3.51026	0.569439
$39$	0	0
$40$	0	0
$41$	3.32684	0.519565	0.259783	−	0.965667i	$-0.416349\pi$
$41$	3.32684	0.519565	0.259783	+	0.965667i	$0.416349\pi$
$42$	0	0
$43$	10.8371	1.65264	0.826321	−	0.563199i	$-0.190430\pi$
$43$	10.8371	1.65264	0.826321	+	0.563199i	$0.190430\pi$
$44$	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	$-0.520458\pi$
$59$	−0.986669	−0.128453	−0.0642267	+	0.997935i	$0.520458\pi$
$60$	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	$-0.499437\pi$
$67$	0.0289294	0.00353429	0.00176715	+	0.999998i	$0.499437\pi$
$68$	1.95055	0.236539
$69$	0	0
$70$	0	0
$71$	13.9421	1.65463	0.827314	−	0.561740i	$-0.189868\pi$
$71$	13.9421	1.65463	0.827314	+	0.561740i	$0.189868\pi$
$72$	0	0
$73$	−9.91548	−1.16052	−0.580260	−	0.814432i	$-0.697049\pi$
$73$	−9.91548	−1.16052	−0.580260	+	0.814432i	$0.697049\pi$
$74$	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	$-0.405154\pi$
$89$	5.53919	0.587153	0.293576	+	0.955936i	$0.405154\pi$
$90$	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	$-0.458636\pi$
$97$	2.55252	0.259169	0.129585	+	0.991568i	$0.458636\pi$
$98$	−8.09171	−0.817386
$99$	0	0
$100$	0	0
$101$	−9.07838	−0.903332	−0.451666	−	0.892187i	$-0.649170\pi$
$101$	−9.07838	−0.903332	−0.451666	+	0.892187i	$0.649170\pi$
$102$	0	0
$103$	18.3340	1.80651	0.903253	−	0.429109i	$-0.141172\pi$
$103$	18.3340	1.80651	0.903253	+	0.429109i	$0.141172\pi$
$104$	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	$-0.319632\pi$
$127$	12.0989	1.07360	0.536802	+	0.843708i	$0.319632\pi$
$128$	3.31965	0.293419
$129$	0	0
$130$	0	0
$131$	12.1639	1.06277	0.531384	−	0.847131i	$-0.321672\pi$
$131$	12.1639	1.06277	0.531384	+	0.847131i	$0.321672\pi$
$132$	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	$-0.118337\pi$
$149$	22.7454	1.86338	0.931688	+	0.363261i	$0.118337\pi$
$150$	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	$-0.308593\pi$
$157$	14.1773	1.13147	0.565735	+	0.824587i	$0.308593\pi$
$158$	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	$-0.391937\pi$
$163$	8.50307	0.666012	0.333006	+	0.942925i	$0.391937\pi$
$164$	−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	$-0.633373\pi$
$179$	−10.8865	−0.813699	−0.406849	+	0.913495i	$0.633373\pi$
$180$	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	$-0.385942\pi$
$191$	9.69368	0.701410	0.350705	+	0.936486i	$0.385942\pi$
$192$	0	0
$193$	4.95055	0.356349	0.178174	−	0.983999i	$-0.442981\pi$
$193$	4.95055	0.356349	0.178174	+	0.983999i	$0.442981\pi$
$194$	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	$-0.405835\pi$
$223$	8.70701	0.583064	0.291532	+	0.956561i	$0.405835\pi$
$224$	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	$-0.507142\pi$
$239$	−0.693677	−0.0448702	−0.0224351	+	0.999748i	$0.507142\pi$
$240$	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	$-0.502558\pi$
$251$	−0.254607	−0.0160707	−0.00803534	+	0.999968i	$0.502558\pi$
$252$	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	$-0.361607\pi$
$269$	13.8166	0.842412	0.421206	+	0.906965i	$0.361607\pi$
$270$	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	$-0.392587\pi$
$277$	11.0205	0.662159	0.331079	+	0.943603i	$0.392587\pi$
$278$	−15.8348	−0.949710
$279$	0	0
$280$	0	0
$281$	−8.15676	−0.486591	−0.243296	−	0.969952i	$-0.578229\pi$
$281$	−8.15676	−0.486591	−0.243296	+	0.969952i	$0.578229\pi$
$282$	0	0
$283$	−18.9155	−1.12441	−0.562204	−	0.826998i	$-0.690047\pi$
$283$	−18.9155	−1.12441	−0.562204	+	0.826998i	$0.690047\pi$
$284$	−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	$-0.414142\pi$
$307$	9.33791	0.532942	0.266471	+	0.963843i	$0.414142\pi$
$308$	0.0845208	0.00481602
$309$	0	0
$310$	0	0
$311$	24.6153	1.39581	0.697903	−	0.716193i	$-0.254117\pi$
$311$	24.6153	1.39581	0.697903	+	0.716193i	$0.254117\pi$
$312$	0	0
$313$	9.61757	0.543617	0.271809	−	0.962351i	$-0.412378\pi$
$313$	9.61757	0.543617	0.271809	+	0.962351i	$0.412378\pi$
$314$	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	$-0.242321\pi$
$337$	26.5802	1.44792	0.723959	+	0.689843i	$0.242321\pi$
$338$	−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	$-0.460359\pi$
$359$	4.70701	0.248426	0.124213	+	0.992256i	$0.460359\pi$
$360$	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	$-0.336142\pi$
$367$	18.8638	0.984680	0.492340	+	0.870403i	$0.336142\pi$
$368$	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	$-0.346750\pi$
$373$	17.8865	0.926130	0.463065	+	0.886324i	$0.346750\pi$
$374$	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	$-0.720070\pi$
$389$	−25.1506	−1.27519	−0.637593	+	0.770373i	$0.720070\pi$
$390$	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	$-0.378082\pi$
$397$	14.8927	0.747443	0.373721	+	0.927541i	$0.378082\pi$
$398$	−9.28005	−0.465167
$399$	0	0
$400$	0	0
$401$	−20.9988	−1.04863	−0.524314	−	0.851525i	$-0.675678\pi$
$401$	−20.9988	−1.04863	−0.524314	+	0.851525i	$0.675678\pi$
$402$	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	$-0.592263\pi$
$419$	−11.7009	−0.571625	−0.285812	+	0.958286i	$0.592263\pi$
$420$	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	$-0.744666\pi$
$431$	−28.8638	−1.39032	−0.695159	+	0.718856i	$0.744666\pi$
$432$	0	0
$433$	26.8371	1.28971	0.644854	−	0.764305i	$-0.276918\pi$
$433$	26.8371	1.28971	0.644854	+	0.764305i	$0.276918\pi$
$434$	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	$-0.671076\pi$
$449$	−21.6959	−1.02389	−0.511947	+	0.859017i	$0.671076\pi$
$450$	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	$-0.565745\pi$
$457$	−8.76818	−0.410158	−0.205079	+	0.978745i	$0.565745\pi$
$458$	26.2739	1.22770
$459$	0	0
$460$	0	0
$461$	27.0423	1.25948	0.629742	−	0.776805i	$-0.283161\pi$
$461$	27.0423	1.25948	0.629742	+	0.776805i	$0.283161\pi$
$462$	0	0
$463$	22.0989	1.02702	0.513511	−	0.858083i	$-0.328344\pi$
$463$	22.0989	1.02702	0.513511	+	0.858083i	$0.328344\pi$
$464$	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	$-0.730853\pi$
$479$	−29.0349	−1.32664	−0.663319	+	0.748337i	$0.730853\pi$
$480$	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	$-0.407560\pi$
$487$	12.6381	0.572686	0.286343	+	0.958127i	$0.407560\pi$
$488$	20.5646	0.930917
$489$	0	0
$490$	0	0
$491$	−13.1145	−0.591849	−0.295924	−	0.955211i	$-0.595628\pi$
$491$	−13.1145	−0.591849	−0.295924	+	0.955211i	$0.595628\pi$
$492$	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	$-0.606212\pi$
$509$	−14.7782	−0.655032	−0.327516	+	0.944846i	$0.606212\pi$
$510$	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	$-0.588212\pi$
$521$	−12.4897	−0.547185	−0.273593	+	0.961846i	$0.588212\pi$
$522$	0	0
$523$	6.71154	0.293475	0.146738	−	0.989175i	$-0.453123\pi$
$523$	6.71154	0.293475	0.146738	+	0.989175i	$0.453123\pi$
$524$	−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	$-0.336760\pi$
$547$	22.9506	0.981295	0.490647	+	0.871358i	$0.336760\pi$
$548$	−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	$-0.654013\pi$
$569$	−22.1929	−0.930374	−0.465187	+	0.885213i	$0.654013\pi$
$570$	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	$-0.136062\pi$
$577$	43.7191	1.82005	0.910025	+	0.414553i	$0.136062\pi$
$578$	−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	$-0.542421\pi$
$599$	−6.50412	−0.265751	−0.132875	+	0.991133i	$0.542421\pi$
$600$	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	$-0.856326\pi$
$607$	−44.3400	−1.79970	−0.899852	+	0.436194i	$0.856326\pi$
$608$	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	$-0.501472\pi$
$613$	−0.228990	−0.00924883	−0.00462441	+	0.999989i	$0.501472\pi$
$614$	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	$-0.627871\pi$
$641$	−19.7987	−0.782002	−0.391001	+	0.920390i	$0.627871\pi$
$642$	0	0
$643$	−29.3763	−1.15849	−0.579244	−	0.815154i	$-0.696652\pi$
$643$	−29.3763	−1.15849	−0.579244	+	0.815154i	$0.696652\pi$
$644$	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	$-0.756234\pi$
$659$	−37.0082	−1.44164	−0.720818	+	0.693124i	$0.756234\pi$
$660$	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	$-0.486593\pi$
$673$	2.18464	0.0842117	0.0421058	+	0.999113i	$0.486593\pi$
$674$	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	$-0.266289\pi$
$701$	35.4791	1.34003	0.670013	+	0.742349i	$0.266289\pi$
$702$	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	$-0.284314\pi$
$719$	33.6209	1.25385	0.626924	+	0.779081i	$0.284314\pi$
$720$	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	$-0.663835\pi$
$727$	−26.5464	−0.984551	−0.492275	+	0.870440i	$0.663835\pi$
$728$	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	$-0.547481\pi$
$733$	−8.04718	−0.297229	−0.148615	+	0.988895i	$0.547481\pi$
$734$	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	$-0.801856\pi$
$757$	−44.7058	−1.62486	−0.812430	+	0.583059i	$0.801856\pi$
$758$	13.2351	0.480722
$759$	0	0
$760$	0	0
$761$	−27.0121	−0.979188	−0.489594	−	0.871950i	$-0.662855\pi$
$761$	−27.0121	−0.979188	−0.489594	+	0.871950i	$0.662855\pi$
$762$	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	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	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	$-0.243843\pi$
$809$	40.9949	1.44130	0.720652	+	0.693297i	$0.243843\pi$
$810$	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	$-0.632566\pi$
$821$	−23.1822	−0.809064	−0.404532	+	0.914524i	$0.632566\pi$
$822$	0	0
$823$	−24.3356	−0.848287	−0.424144	−	0.905595i	$-0.639425\pi$
$823$	−24.3356	−0.848287	−0.424144	+	0.905595i	$0.639425\pi$
$824$	−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	$-0.518158\pi$
$839$	−3.30283	−0.114026	−0.0570132	+	0.998373i	$0.518158\pi$
$840$	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	$-0.558231\pi$
$853$	−10.6264	−0.363840	−0.181920	+	0.983313i	$0.558231\pi$
$854$	−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	$-0.402486\pi$
$877$	17.8622	0.603162	0.301581	+	0.953441i	$0.402486\pi$
$878$	−8.80590	−0.297185
$879$	0	0
$880$	0	0
$881$	37.7998	1.27351	0.636753	−	0.771068i	$-0.280277\pi$
$881$	37.7998	1.27351	0.636753	+	0.771068i	$0.280277\pi$
$882$	0	0
$883$	55.7140	1.87493	0.937463	−	0.348085i	$-0.113168\pi$
$883$	55.7140	1.87493	0.937463	+	0.348085i	$0.113168\pi$
$884$	−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	$-0.528541\pi$
$907$	−5.39350	−0.179088	−0.0895441	+	0.995983i	$0.528541\pi$
$908$	−1.43991	−0.0477850
$909$	0	0
$910$	0	0
$911$	−51.8131	−1.71664	−0.858322	−	0.513111i	$-0.828493\pi$
$911$	−51.8131	−1.71664	−0.858322	+	0.513111i	$0.828493\pi$
$912$	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	$-0.546800\pi$
$929$	−8.93042	−0.292998	−0.146499	+	0.989211i	$0.546800\pi$
$930$	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	$-0.479295\pi$
$937$	3.97948	0.130004	0.0650020	+	0.997885i	$0.479295\pi$
$938$	0.00984100	0.000321320 0
$939$	0	0
$940$	0	0
$941$	−19.7682	−0.644424	−0.322212	−	0.946668i	$-0.604426\pi$
$941$	−19.7682	−0.644424	−0.322212	+	0.946668i	$0.604426\pi$
$942$	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	$-0.427050\pi$
$967$	14.1289	0.454354	0.227177	+	0.973854i	$0.427050\pi$
$968$	33.2085	1.06736
$969$	0	0
$970$	0	0
$971$	8.62968	0.276939	0.138470	−	0.990367i	$-0.455782\pi$
$971$	8.62968	0.276939	0.138470	+	0.990367i	$0.455782\pi$
$972$	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	$-0.536904\pi$
$997$	−7.30510	−0.231355	−0.115677	+	0.993287i	$0.536904\pi$
$998$	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.ba.1.3	yes	3
3.2	odd	2		6975.2.a.bh.1.1	yes	3
5.4	even	2		6975.2.a.bg.1.1	yes	3
15.14	odd	2		6975.2.a.z.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.z.1.3	✓	3	15.14	odd	2
6975.2.a.ba.1.3	yes	3	1.1	even	1	trivial
6975.2.a.bg.1.1	yes	3	5.4	even	2
6975.2.a.bh.1.1	yes	3	3.2	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 $\F_{q}$
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

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Database

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