LMFDB - Embedded newform 6975.2.a.ch.1.5

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 = [10,0,0,12,0,0,0,0,0,0,-16,0,0,6,0,8,0] 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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 88x^{6} - 183x^{4} + 92x^{2} - 4 $ x^10 - 16x^8 + 88x^6 - 183x^4 + 92x^2 - 4
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			1.5
Root			$-0.218967$ 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)

$f(q)$	$=$	$q-0.218967 q^{2} -1.95205 q^{4} -4.66797 q^{7} +0.865369 q^{8} +2.85310 q^{11} -1.26481 q^{13} +1.02213 q^{14} +3.71462 q^{16} +4.33394 q^{17} -4.96269 q^{19} -0.624733 q^{22} +0.202144 q^{23} +0.276952 q^{26} +9.11213 q^{28} -4.57614 q^{29} -1.00000 q^{31} -2.54412 q^{32} -0.948989 q^{34} +2.16527 q^{37} +1.08667 q^{38} -10.0137 q^{41} +9.47017 q^{43} -5.56940 q^{44} -0.0442629 q^{46} +7.86899 q^{47} +14.7900 q^{49} +2.46898 q^{52} +9.93609 q^{53} -4.03952 q^{56} +1.00202 q^{58} +5.58310 q^{59} -9.27559 q^{61} +0.218967 q^{62} -6.87216 q^{64} +2.37082 q^{67} -8.46008 q^{68} +7.06471 q^{71} -6.73840 q^{73} -0.474122 q^{74} +9.68744 q^{76} -13.3182 q^{77} +0.948989 q^{79} +2.19267 q^{82} -6.68941 q^{83} -2.07365 q^{86} +2.46898 q^{88} -3.42386 q^{89} +5.90411 q^{91} -0.394597 q^{92} -1.72305 q^{94} +14.0656 q^{97} -3.23851 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 12 q^{4} - 16 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} - 28 q^{26} - 32 q^{29} - 10 q^{31} - 28 q^{34} - 36 q^{41} - 52 q^{44} + 8 q^{46} + 22 q^{49} - 12 q^{56} - 12 q^{59} - 70 q^{64} - 12 q^{71}+ \cdots - 48 q^{94}+O(q^{100}) $ 10 * q + 12 * q^4 - 16 * q^11 + 6 * q^14 + 8 * q^16 - 4 * q^19 - 28 * q^26 - 32 * q^29 - 10 * q^31 - 28 * q^34 - 36 * q^41 - 52 * q^44 + 8 * q^46 + 22 * q^49 - 12 * q^56 - 12 * q^59 - 70 * q^64 - 12 * q^71 - 28 * q^74 - 30 * q^76 + 28 * q^79 - 36 * q^86 - 48 * q^89 - 4 * q^91 - 48 * q^94

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.218967	−0.154833	−0.0774165	−	0.996999i	$-0.524667\pi$
$2$	−0.218967	−0.154833	−0.0774165	+	0.996999i	$0.524667\pi$
$3$	0	0
$3$	0	0
$4$	−1.95205	−0.976027
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.66797	−1.76433	−0.882164	−	0.470942i	$-0.843914\pi$
$7$	−4.66797	−1.76433	−0.882164	+	0.470942i	$0.843914\pi$
$8$	0.865369	0.305954
$9$	0	0
$10$	0	0
$11$	2.85310	0.860241	0.430120	−	0.902772i	$-0.358471\pi$
$11$	2.85310	0.860241	0.430120	+	0.902772i	$0.358471\pi$
$12$	0	0
$13$	−1.26481	−0.350796	−0.175398	−	0.984498i	$-0.556121\pi$
$13$	−1.26481	−0.350796	−0.175398	+	0.984498i	$0.556121\pi$
$14$	1.02213	0.273176
$15$	0	0
$16$	3.71462	0.928655
$17$	4.33394	1.05113	0.525567	−	0.850752i	$-0.323853\pi$
$17$	4.33394	1.05113	0.525567	+	0.850752i	$0.323853\pi$
$18$	0	0
$19$	−4.96269	−1.13852	−0.569260	−	0.822158i	$-0.692770\pi$
$19$	−4.96269	−1.13852	−0.569260	+	0.822158i	$0.692770\pi$
$20$	0	0
$21$	0	0
$22$	−0.624733	−0.133194
$23$	0.202144	0.0421500	0.0210750	−	0.999778i	$-0.493291\pi$
$23$	0.202144	0.0421500	0.0210750	+	0.999778i	$0.493291\pi$
$24$	0	0
$25$	0	0
$26$	0.276952	0.0543147
$27$	0	0
$28$	9.11213	1.72203
$29$	−4.57614	−0.849769	−0.424884	−	0.905248i	$-0.639685\pi$
$29$	−4.57614	−0.849769	−0.424884	+	0.905248i	$0.639685\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−2.54412	−0.449740
$33$	0	0
$34$	−0.948989	−0.162750
$35$	0	0
$36$	0	0
$37$	2.16527	0.355968	0.177984	−	0.984033i	$-0.443043\pi$
$37$	2.16527	0.355968	0.177984	+	0.984033i	$0.443043\pi$
$38$	1.08667	0.176280
$39$	0	0
$40$	0	0
$41$	−10.0137	−1.56388	−0.781939	−	0.623355i	$-0.785769\pi$
$41$	−10.0137	−1.56388	−0.781939	+	0.623355i	$0.785769\pi$
$42$	0	0
$43$	9.47017	1.44419	0.722093	−	0.691796i	$-0.243180\pi$
$43$	9.47017	1.44419	0.722093	+	0.691796i	$0.243180\pi$
$44$	−5.56940	−0.839618
$45$	0	0
$46$	−0.0442629	−0.00652622
$47$	7.86899	1.14781	0.573905	−	0.818922i	$-0.305428\pi$
$47$	7.86899	1.14781	0.573905	+	0.818922i	$0.305428\pi$
$48$	0	0
$49$	14.7900	2.11285
$50$	0	0
$51$	0	0
$52$	2.46898	0.342386
$53$	9.93609	1.36483	0.682414	−	0.730966i	$-0.260930\pi$
$53$	9.93609	1.36483	0.682414	+	0.730966i	$0.260930\pi$
$54$	0	0
$55$	0	0
$56$	−4.03952	−0.539803
$57$	0	0
$58$	1.00202	0.131572
$59$	5.58310	0.726858	0.363429	−	0.931622i	$-0.381606\pi$
$59$	5.58310	0.726858	0.363429	+	0.931622i	$0.381606\pi$
$60$	0	0
$61$	−9.27559	−1.18762	−0.593809	−	0.804606i	$-0.702376\pi$
$61$	−9.27559	−1.18762	−0.593809	+	0.804606i	$0.702376\pi$
$62$	0.218967	0.0278088
$63$	0	0
$64$	−6.87216	−0.859020
$65$	0	0
$66$	0	0
$67$	2.37082	0.289641	0.144821	−	0.989458i	$-0.453739\pi$
$67$	2.37082	0.289641	0.144821	+	0.989458i	$0.453739\pi$
$68$	−8.46008	−1.02594
$69$	0	0
$70$	0	0
$71$	7.06471	0.838427	0.419214	−	0.907888i	$-0.362306\pi$
$71$	7.06471	0.838427	0.419214	+	0.907888i	$0.362306\pi$
$72$	0	0
$73$	−6.73840	−0.788670	−0.394335	−	0.918967i	$-0.629025\pi$
$73$	−6.73840	−0.788670	−0.394335	+	0.918967i	$0.629025\pi$
$74$	−0.474122	−0.0551155
$75$	0	0
$76$	9.68744	1.11123
$77$	−13.3182	−1.51775
$78$	0	0
$79$	0.948989	0.106770	0.0533848	−	0.998574i	$-0.482999\pi$
$79$	0.948989	0.106770	0.0533848	+	0.998574i	$0.482999\pi$
$80$	0	0
$81$	0	0
$82$	2.19267	0.242140
$83$	−6.68941	−0.734258	−0.367129	−	0.930170i	$-0.619659\pi$
$83$	−6.68941	−0.734258	−0.367129	+	0.930170i	$0.619659\pi$
$84$	0	0
$85$	0	0
$86$	−2.07365	−0.223608
$87$	0	0
$88$	2.46898	0.263194
$89$	−3.42386	−0.362928	−0.181464	−	0.983398i	$-0.558084\pi$
$89$	−3.42386	−0.362928	−0.181464	+	0.983398i	$0.558084\pi$
$90$	0	0
$91$	5.90411	0.618919
$92$	−0.394597	−0.0411396
$93$	0	0
$94$	−1.72305	−0.177719
$95$	0	0
$96$	0	0
$97$	14.0656	1.42814	0.714070	−	0.700074i	$-0.246850\pi$
$97$	14.0656	1.42814	0.714070	+	0.700074i	$0.246850\pi$
$98$	−3.23851	−0.327139
$99$	0	0
$100$	0	0
$101$	14.0191	1.39495	0.697476	−	0.716608i	$-0.254307\pi$
$101$	14.0191	1.39495	0.697476	+	0.716608i	$0.254307\pi$
$102$	0	0
$103$	5.57183	0.549009	0.274504	−	0.961586i	$-0.411486\pi$
$103$	5.57183	0.549009	0.274504	+	0.961586i	$0.411486\pi$
$104$	−1.09453	−0.107327
$105$	0	0
$106$	−2.17568	−0.211320
$107$	0.830201	0.0802586	0.0401293	−	0.999194i	$-0.487223\pi$
$107$	0.830201	0.0802586	0.0401293	+	0.999194i	$0.487223\pi$
$108$	0	0
$109$	10.6341	1.01856	0.509282	−	0.860600i	$-0.329911\pi$
$109$	10.6341	1.01856	0.509282	+	0.860600i	$0.329911\pi$
$110$	0	0
$111$	0	0
$112$	−17.3397	−1.63845
$113$	−3.96626	−0.373114	−0.186557	−	0.982444i	$-0.559733\pi$
$113$	−3.96626	−0.373114	−0.186557	+	0.982444i	$0.559733\pi$
$114$	0	0
$115$	0	0
$116$	8.93288	0.829397
$117$	0	0
$118$	−1.22251	−0.112542
$119$	−20.2307	−1.85455
$120$	0	0
$121$	−2.85984	−0.259986
$122$	2.03105	0.183882
$123$	0	0
$124$	1.95205	0.175300
$125$	0	0
$126$	0	0
$127$	6.29706	0.558774	0.279387	−	0.960179i	$-0.409869\pi$
$127$	6.29706	0.558774	0.279387	+	0.960179i	$0.409869\pi$
$128$	6.59301	0.582745
$129$	0	0
$130$	0	0
$131$	1.77089	0.154723	0.0773615	−	0.997003i	$-0.475350\pi$
$131$	1.77089	0.154723	0.0773615	+	0.997003i	$0.475350\pi$
$132$	0	0
$133$	23.1657	2.00872
$134$	−0.519130	−0.0448460
$135$	0	0
$136$	3.75045	0.321599
$137$	3.17311	0.271097	0.135548	−	0.990771i	$-0.456720\pi$
$137$	3.17311	0.271097	0.135548	+	0.990771i	$0.456720\pi$
$138$	0	0
$139$	−14.8015	−1.25544	−0.627722	−	0.778438i	$-0.716013\pi$
$139$	−14.8015	−1.25544	−0.627722	+	0.778438i	$0.716013\pi$
$140$	0	0
$141$	0	0
$142$	−1.54694	−0.129816
$143$	−3.60863	−0.301769
$144$	0	0
$145$	0	0
$146$	1.47549	0.122112
$147$	0	0
$148$	−4.22672	−0.347434
$149$	6.55571	0.537065	0.268532	−	0.963271i	$-0.413461\pi$
$149$	6.55571	0.537065	0.268532	+	0.963271i	$0.413461\pi$
$150$	0	0
$151$	−6.35695	−0.517321	−0.258661	−	0.965968i	$-0.583281\pi$
$151$	−6.35695	−0.517321	−0.258661	+	0.965968i	$0.583281\pi$
$152$	−4.29456	−0.348335
$153$	0	0
$154$	2.91624	0.234997
$155$	0	0
$156$	0	0
$157$	−3.56197	−0.284276	−0.142138	−	0.989847i	$-0.545398\pi$
$157$	−3.56197	−0.284276	−0.142138	+	0.989847i	$0.545398\pi$
$158$	−0.207797	−0.0165315
$159$	0	0
$160$	0	0
$161$	−0.943605	−0.0743665
$162$	0	0
$163$	−5.51585	−0.432035	−0.216017	−	0.976390i	$-0.569307\pi$
$163$	−5.51585	−0.432035	−0.216017	+	0.976390i	$0.569307\pi$
$164$	19.5473	1.52639
$165$	0	0
$166$	1.46476	0.113687
$167$	4.82105	0.373064	0.186532	−	0.982449i	$-0.440275\pi$
$167$	4.82105	0.373064	0.186532	+	0.982449i	$0.440275\pi$
$168$	0	0
$169$	−11.4003	−0.876942
$170$	0	0
$171$	0	0
$172$	−18.4863	−1.40956
$173$	−6.37404	−0.484609	−0.242305	−	0.970200i	$-0.577903\pi$
$173$	−6.37404	−0.484609	−0.242305	+	0.970200i	$0.577903\pi$
$174$	0	0
$175$	0	0
$176$	10.5982	0.798867
$177$	0	0
$178$	0.749711	0.0561932
$179$	−7.52588	−0.562511	−0.281255	−	0.959633i	$-0.590751\pi$
$179$	−7.52588	−0.562511	−0.281255	+	0.959633i	$0.590751\pi$
$180$	0	0
$181$	5.53126	0.411135	0.205568	−	0.978643i	$-0.434096\pi$
$181$	5.53126	0.411135	0.205568	+	0.978643i	$0.434096\pi$
$182$	−1.29280	−0.0958290
$183$	0	0
$184$	0.174930	0.0128960
$185$	0	0
$186$	0	0
$187$	12.3651	0.904229
$188$	−15.3607	−1.12029
$189$	0	0
$190$	0	0
$191$	−16.2206	−1.17368	−0.586840	−	0.809703i	$-0.699628\pi$
$191$	−16.2206	−1.17368	−0.586840	+	0.809703i	$0.699628\pi$
$192$	0	0
$193$	−5.39768	−0.388534	−0.194267	−	0.980949i	$-0.562233\pi$
$193$	−5.39768	−0.388534	−0.194267	+	0.980949i	$0.562233\pi$
$194$	−3.07989	−0.221123
$195$	0	0
$196$	−28.8708	−2.06220
$197$	−24.5744	−1.75085	−0.875425	−	0.483354i	$-0.839419\pi$
$197$	−24.5744	−1.75085	−0.875425	+	0.483354i	$0.839419\pi$
$198$	0	0
$199$	−6.15904	−0.436602	−0.218301	−	0.975881i	$-0.570052\pi$
$199$	−6.15904	−0.436602	−0.218301	+	0.975881i	$0.570052\pi$
$200$	0	0
$201$	0	0
$202$	−3.06972	−0.215984
$203$	21.3613	1.49927
$204$	0	0
$205$	0	0
$206$	−1.22005	−0.0850047
$207$	0	0
$208$	−4.69829	−0.325768
$209$	−14.1590	−0.979401
$210$	0	0
$211$	−24.1523	−1.66271	−0.831357	−	0.555739i	$-0.812435\pi$
$211$	−24.1523	−1.66271	−0.831357	+	0.555739i	$0.812435\pi$
$212$	−19.3958	−1.33211
$213$	0	0
$214$	−0.181787	−0.0124267
$215$	0	0
$216$	0	0
$217$	4.66797	0.316883
$218$	−2.32852	−0.157707
$219$	0	0
$220$	0	0
$221$	−5.48162	−0.368733
$222$	0	0
$223$	−0.726303	−0.0486368	−0.0243184	−	0.999704i	$-0.507742\pi$
$223$	−0.726303	−0.0486368	−0.0243184	+	0.999704i	$0.507742\pi$
$224$	11.8759	0.793490
$225$	0	0
$226$	0.868479	0.0577704
$227$	−11.0387	−0.732664	−0.366332	−	0.930484i	$-0.619386\pi$
$227$	−11.0387	−0.732664	−0.366332	+	0.930484i	$0.619386\pi$
$228$	0	0
$229$	−24.1348	−1.59487	−0.797437	−	0.603403i	$-0.793811\pi$
$229$	−24.1348	−1.59487	−0.797437	+	0.603403i	$0.793811\pi$
$230$	0	0
$231$	0	0
$232$	−3.96005	−0.259990
$233$	−15.2264	−0.997514	−0.498757	−	0.866742i	$-0.666210\pi$
$233$	−15.2264	−0.997514	−0.498757	+	0.866742i	$0.666210\pi$
$234$	0	0
$235$	0	0
$236$	−10.8985	−0.709432
$237$	0	0
$238$	4.42985	0.287145
$239$	−15.0199	−0.971557	−0.485779	−	0.874082i	$-0.661464\pi$
$239$	−15.0199	−0.971557	−0.485779	+	0.874082i	$0.661464\pi$
$240$	0	0
$241$	21.1878	1.36483	0.682413	−	0.730967i	$-0.260930\pi$
$241$	21.1878	1.36483	0.682413	+	0.730967i	$0.260930\pi$
$242$	0.626211	0.0402544
$243$	0	0
$244$	18.1064	1.15915
$245$	0	0
$246$	0	0
$247$	6.27687	0.399388
$248$	−0.865369	−0.0549510
$249$	0	0
$250$	0	0
$251$	−20.9301	−1.32110	−0.660550	−	0.750782i	$-0.729677\pi$
$251$	−20.9301	−1.32110	−0.660550	+	0.750782i	$0.729677\pi$
$252$	0	0
$253$	0.576738	0.0362592
$254$	−1.37885	−0.0865166
$255$	0	0
$256$	12.3007	0.768792
$257$	−15.7405	−0.981865	−0.490933	−	0.871198i	$-0.663344\pi$
$257$	−15.7405	−0.981865	−0.490933	+	0.871198i	$0.663344\pi$
$258$	0	0
$259$	−10.1074	−0.628044
$260$	0	0
$261$	0	0
$262$	−0.387765	−0.0239562
$263$	−18.2152	−1.12320	−0.561599	−	0.827410i	$-0.689814\pi$
$263$	−18.2152	−1.12320	−0.561599	+	0.827410i	$0.689814\pi$
$264$	0	0
$265$	0	0
$266$	−5.07252	−0.311016
$267$	0	0
$268$	−4.62796	−0.282698
$269$	−21.6110	−1.31765	−0.658824	−	0.752297i	$-0.728946\pi$
$269$	−21.6110	−1.31765	−0.658824	+	0.752297i	$0.728946\pi$
$270$	0	0
$271$	26.6143	1.61670	0.808352	−	0.588699i	$-0.200360\pi$
$271$	26.6143	1.61670	0.808352	+	0.588699i	$0.200360\pi$
$272$	16.0989	0.976141
$273$	0	0
$274$	−0.694805	−0.0419747
$275$	0	0
$276$	0	0
$277$	14.8867	0.894455	0.447228	−	0.894420i	$-0.352411\pi$
$277$	14.8867	0.894455	0.447228	+	0.894420i	$0.352411\pi$
$278$	3.24103	0.194384
$279$	0	0
$280$	0	0
$281$	−23.3906	−1.39536	−0.697682	−	0.716407i	$-0.745785\pi$
$281$	−23.3906	−1.39536	−0.697682	+	0.716407i	$0.745785\pi$
$282$	0	0
$283$	−19.0733	−1.13379	−0.566895	−	0.823790i	$-0.691855\pi$
$283$	−19.0733	−1.13379	−0.566895	+	0.823790i	$0.691855\pi$
$284$	−13.7907	−0.818328
$285$	0	0
$286$	0.790170	0.0467237
$287$	46.7437	2.75919
$288$	0	0
$289$	1.78302	0.104883
$290$	0	0
$291$	0	0
$292$	13.1537	0.769763
$293$	7.15193	0.417820	0.208910	−	0.977935i	$-0.433008\pi$
$293$	7.15193	0.417820	0.208910	+	0.977935i	$0.433008\pi$
$294$	0	0
$295$	0	0
$296$	1.87375	0.108910
$297$	0	0
$298$	−1.43548	−0.0831554
$299$	−0.255675	−0.0147861
$300$	0	0
$301$	−44.2065	−2.54802
$302$	1.39196	0.0800984
$303$	0	0
$304$	−18.4345	−1.05729
$305$	0	0
$306$	0	0
$307$	16.4729	0.940158	0.470079	−	0.882624i	$-0.344225\pi$
$307$	16.4729	0.940158	0.470079	+	0.882624i	$0.344225\pi$
$308$	25.9978	1.48136
$309$	0	0
$310$	0	0
$311$	20.8279	1.18104	0.590521	−	0.807022i	$-0.298922\pi$
$311$	20.8279	1.18104	0.590521	+	0.807022i	$0.298922\pi$
$312$	0	0
$313$	0.692658	0.0391513	0.0195757	−	0.999808i	$-0.493768\pi$
$313$	0.692658	0.0391513	0.0195757	+	0.999808i	$0.493768\pi$
$314$	0.779953	0.0440153
$315$	0	0
$316$	−1.85248	−0.104210
$317$	8.85235	0.497197	0.248599	−	0.968607i	$-0.420030\pi$
$317$	8.85235	0.497197	0.248599	+	0.968607i	$0.420030\pi$
$318$	0	0
$319$	−13.0562	−0.731006
$320$	0	0
$321$	0	0
$322$	0.206618	0.0115144
$323$	−21.5080	−1.19674
$324$	0	0
$325$	0	0
$326$	1.20779	0.0668932
$327$	0	0
$328$	−8.66555	−0.478475
$329$	−36.7322	−2.02511
$330$	0	0
$331$	7.27095	0.399648	0.199824	−	0.979832i	$-0.435963\pi$
$331$	7.27095	0.399648	0.199824	+	0.979832i	$0.435963\pi$
$332$	13.0581	0.716655
$333$	0	0
$334$	−1.05565	−0.0577626
$335$	0	0
$336$	0	0
$337$	−30.1366	−1.64164	−0.820822	−	0.571184i	$-0.806484\pi$
$337$	−30.1366	−1.64164	−0.820822	+	0.571184i	$0.806484\pi$
$338$	2.49628	0.135780
$339$	0	0
$340$	0	0
$341$	−2.85310	−0.154504
$342$	0	0
$343$	−36.3634	−1.96344
$344$	8.19519	0.441855
$345$	0	0
$346$	1.39570	0.0750335
$347$	−6.66482	−0.357786	−0.178893	−	0.983869i	$-0.557252\pi$
$347$	−6.66482	−0.357786	−0.178893	+	0.983869i	$0.557252\pi$
$348$	0	0
$349$	−13.5952	−0.727735	−0.363867	−	0.931451i	$-0.618544\pi$
$349$	−13.5952	−0.727735	−0.363867	+	0.931451i	$0.618544\pi$
$350$	0	0
$351$	0	0
$352$	−7.25861	−0.386885
$353$	31.7589	1.69036	0.845179	−	0.534484i	$-0.179494\pi$
$353$	31.7589	1.69036	0.845179	+	0.534484i	$0.179494\pi$
$354$	0	0
$355$	0	0
$356$	6.68355	0.354227
$357$	0	0
$358$	1.64792	0.0870952
$359$	8.65181	0.456625	0.228313	−	0.973588i	$-0.426679\pi$
$359$	8.65181	0.456625	0.228313	+	0.973588i	$0.426679\pi$
$360$	0	0
$361$	5.62831	0.296227
$362$	−1.21116	−0.0636573
$363$	0	0
$364$	−11.5251	−0.604081
$365$	0	0
$366$	0	0
$367$	24.0252	1.25410	0.627052	−	0.778978i	$-0.284262\pi$
$367$	24.0252	1.25410	0.627052	+	0.778978i	$0.284262\pi$
$368$	0.750890	0.0391428
$369$	0	0
$370$	0	0
$371$	−46.3814	−2.40800
$372$	0	0
$373$	−1.90355	−0.0985623	−0.0492811	−	0.998785i	$-0.515693\pi$
$373$	−1.90355	−0.0985623	−0.0492811	+	0.998785i	$0.515693\pi$
$374$	−2.70756	−0.140004
$375$	0	0
$376$	6.80958	0.351177
$377$	5.78796	0.298095
$378$	0	0
$379$	18.2619	0.938051	0.469026	−	0.883185i	$-0.344605\pi$
$379$	18.2619	0.938051	0.469026	+	0.883185i	$0.344605\pi$
$380$	0	0
$381$	0	0
$382$	3.55177	0.181724
$383$	−14.4420	−0.737950	−0.368975	−	0.929439i	$-0.620291\pi$
$383$	−14.4420	−0.737950	−0.368975	+	0.929439i	$0.620291\pi$
$384$	0	0
$385$	0	0
$386$	1.18191	0.0601578
$387$	0	0
$388$	−27.4567	−1.39390
$389$	−18.9771	−0.962179	−0.481090	−	0.876671i	$-0.659759\pi$
$389$	−18.9771	−0.962179	−0.481090	+	0.876671i	$0.659759\pi$
$390$	0	0
$391$	0.876082	0.0443054
$392$	12.7988	0.646436
$393$	0	0
$394$	5.38097	0.271089
$395$	0	0
$396$	0	0
$397$	−32.0694	−1.60952	−0.804758	−	0.593603i	$-0.797705\pi$
$397$	−32.0694	−1.60952	−0.804758	+	0.593603i	$0.797705\pi$
$398$	1.34862	0.0676005
$399$	0	0
$400$	0	0
$401$	−26.5380	−1.32524	−0.662622	−	0.748954i	$-0.730557\pi$
$401$	−26.5380	−1.32524	−0.662622	+	0.748954i	$0.730557\pi$
$402$	0	0
$403$	1.26481	0.0630048
$404$	−27.3660	−1.36151
$405$	0	0
$406$	−4.67742	−0.232136
$407$	6.17771	0.306218
$408$	0	0
$409$	10.3506	0.511804	0.255902	−	0.966703i	$-0.417627\pi$
$409$	10.3506	0.511804	0.255902	+	0.966703i	$0.417627\pi$
$410$	0	0
$411$	0	0
$412$	−10.8765	−0.535847
$413$	−26.0618	−1.28242
$414$	0	0
$415$	0	0
$416$	3.21783	0.157767
$417$	0	0
$418$	3.10036	0.151644
$419$	5.17650	0.252889	0.126444	−	0.991974i	$-0.459643\pi$
$419$	5.17650	0.252889	0.126444	+	0.991974i	$0.459643\pi$
$420$	0	0
$421$	−2.26938	−0.110603	−0.0553014	−	0.998470i	$-0.517612\pi$
$421$	−2.26938	−0.110603	−0.0553014	+	0.998470i	$0.517612\pi$
$422$	5.28856	0.257443
$423$	0	0
$424$	8.59838	0.417574
$425$	0	0
$426$	0	0
$427$	43.2982	2.09535
$428$	−1.62060	−0.0783345
$429$	0	0
$430$	0	0
$431$	−20.4376	−0.984444	−0.492222	−	0.870470i	$-0.663815\pi$
$431$	−20.4376	−0.984444	−0.492222	+	0.870470i	$0.663815\pi$
$432$	0	0
$433$	−22.3750	−1.07527	−0.537637	−	0.843177i	$-0.680683\pi$
$433$	−22.3750	−1.07527	−0.537637	+	0.843177i	$0.680683\pi$
$434$	−1.02213	−0.0490639
$435$	0	0
$436$	−20.7584	−0.994145
$437$	−1.00318	−0.0479886
$438$	0	0
$439$	12.9681	0.618932	0.309466	−	0.950910i	$-0.399850\pi$
$439$	12.9681	0.618932	0.309466	+	0.950910i	$0.399850\pi$
$440$	0	0
$441$	0	0
$442$	1.20029	0.0570921
$443$	13.2540	0.629718	0.314859	−	0.949139i	$-0.398043\pi$
$443$	13.2540	0.629718	0.314859	+	0.949139i	$0.398043\pi$
$444$	0	0
$445$	0	0
$446$	0.159036	0.00753058
$447$	0	0
$448$	32.0791	1.51559
$449$	−16.7430	−0.790149	−0.395075	−	0.918649i	$-0.629281\pi$
$449$	−16.7430	−0.790149	−0.395075	+	0.918649i	$0.629281\pi$
$450$	0	0
$451$	−28.5701	−1.34531
$452$	7.74235	0.364169
$453$	0	0
$454$	2.41711	0.113440
$455$	0	0
$456$	0	0
$457$	−16.0229	−0.749522	−0.374761	−	0.927122i	$-0.622275\pi$
$457$	−16.0229	−0.749522	−0.374761	+	0.927122i	$0.622275\pi$
$458$	5.28472	0.246939
$459$	0	0
$460$	0	0
$461$	17.5746	0.818529	0.409264	−	0.912416i	$-0.365785\pi$
$461$	17.5746	0.818529	0.409264	+	0.912416i	$0.365785\pi$
$462$	0	0
$463$	31.0652	1.44372	0.721861	−	0.692038i	$-0.243287\pi$
$463$	31.0652	1.44372	0.721861	+	0.692038i	$0.243287\pi$
$464$	−16.9986	−0.789142
$465$	0	0
$466$	3.33407	0.154448
$467$	34.6213	1.60208	0.801042	−	0.598608i	$-0.204279\pi$
$467$	34.6213	1.60208	0.801042	+	0.598608i	$0.204279\pi$
$468$	0	0
$469$	−11.0669	−0.511022
$470$	0	0
$471$	0	0
$472$	4.83144	0.222385
$473$	27.0193	1.24235
$474$	0	0
$475$	0	0
$476$	39.4914	1.81009
$477$	0	0
$478$	3.28886	0.150429
$479$	−15.4384	−0.705399	−0.352699	−	0.935737i	$-0.614736\pi$
$479$	−15.4384	−0.705399	−0.352699	+	0.935737i	$0.614736\pi$
$480$	0	0
$481$	−2.73865	−0.124872
$482$	−4.63943	−0.211320
$483$	0	0
$484$	5.58257	0.253753
$485$	0	0
$486$	0	0
$487$	33.3765	1.51243	0.756216	−	0.654322i	$-0.227046\pi$
$487$	33.3765	1.51243	0.756216	+	0.654322i	$0.227046\pi$
$488$	−8.02680	−0.363356
$489$	0	0
$490$	0	0
$491$	4.33512	0.195641	0.0978207	−	0.995204i	$-0.468813\pi$
$491$	4.33512	0.195641	0.0978207	+	0.995204i	$0.468813\pi$
$492$	0	0
$493$	−19.8327	−0.893221
$494$	−1.37443	−0.0618384
$495$	0	0
$496$	−3.71462	−0.166791
$497$	−32.9779	−1.47926
$498$	0	0
$499$	−0.804195	−0.0360007	−0.0180003	−	0.999838i	$-0.505730\pi$
$499$	−0.804195	−0.0360007	−0.0180003	+	0.999838i	$0.505730\pi$
$500$	0	0
$501$	0	0
$502$	4.58301	0.204550
$503$	17.2621	0.769679	0.384839	−	0.922984i	$-0.374257\pi$
$503$	17.2621	0.769679	0.384839	+	0.922984i	$0.374257\pi$
$504$	0	0
$505$	0	0
$506$	−0.126286	−0.00561412
$507$	0	0
$508$	−12.2922	−0.545378
$509$	34.6134	1.53421	0.767105	−	0.641522i	$-0.221697\pi$
$509$	34.6134	1.53421	0.767105	+	0.641522i	$0.221697\pi$
$510$	0	0
$511$	31.4547	1.39147
$512$	−15.8795	−0.701780
$513$	0	0
$514$	3.44665	0.152025
$515$	0	0
$516$	0	0
$517$	22.4510	0.987393
$518$	2.21319	0.0972419
$519$	0	0
$520$	0	0
$521$	−24.5912	−1.07736	−0.538681	−	0.842510i	$-0.681077\pi$
$521$	−24.5912	−1.07736	−0.538681	+	0.842510i	$0.681077\pi$
$522$	0	0
$523$	5.40341	0.236275	0.118137	−	0.992997i	$-0.462308\pi$
$523$	5.40341	0.236275	0.118137	+	0.992997i	$0.462308\pi$
$524$	−3.45686	−0.151014
$525$	0	0
$526$	3.98853	0.173908
$527$	−4.33394	−0.188789
$528$	0	0
$529$	−22.9591	−0.998223
$530$	0	0
$531$	0	0
$532$	−45.2207	−1.96057
$533$	12.6654	0.548601
$534$	0	0
$535$	0	0
$536$	2.05163	0.0886169
$537$	0	0
$538$	4.73210	0.204015
$539$	42.1972	1.81756
$540$	0	0
$541$	−7.96512	−0.342447	−0.171224	−	0.985232i	$-0.554772\pi$
$541$	−7.96512	−0.342447	−0.171224	+	0.985232i	$0.554772\pi$
$542$	−5.82765	−0.250319
$543$	0	0
$544$	−11.0260	−0.472738
$545$	0	0
$546$	0	0
$547$	−34.8058	−1.48819	−0.744094	−	0.668075i	$-0.767119\pi$
$547$	−34.8058	−1.48819	−0.744094	+	0.668075i	$0.767119\pi$
$548$	−6.19407	−0.264598
$549$	0	0
$550$	0	0
$551$	22.7100	0.967478
$552$	0	0
$553$	−4.42985	−0.188377
$554$	−3.25969	−0.138491
$555$	0	0
$556$	28.8933	1.22535
$557$	2.70011	0.114407	0.0572037	−	0.998363i	$-0.481782\pi$
$557$	2.70011	0.114407	0.0572037	+	0.998363i	$0.481782\pi$
$558$	0	0
$559$	−11.9780	−0.506614
$560$	0	0
$561$	0	0
$562$	5.12176	0.216048
$563$	26.5742	1.11997	0.559984	−	0.828504i	$-0.310807\pi$
$563$	26.5742	1.11997	0.559984	+	0.828504i	$0.310807\pi$
$564$	0	0
$565$	0	0
$566$	4.17642	0.175548
$567$	0	0
$568$	6.11358	0.256520
$569$	21.5907	0.905130	0.452565	−	0.891731i	$-0.350509\pi$
$569$	21.5907	0.905130	0.452565	+	0.891731i	$0.350509\pi$
$570$	0	0
$571$	−9.83045	−0.411392	−0.205696	−	0.978616i	$-0.565946\pi$
$571$	−9.83045	−0.411392	−0.205696	+	0.978616i	$0.565946\pi$
$572$	7.04424	0.294534
$573$	0	0
$574$	−10.2353	−0.427214
$575$	0	0
$576$	0	0
$577$	1.13695	0.0473319	0.0236660	−	0.999720i	$-0.492466\pi$
$577$	1.13695	0.0473319	0.0236660	+	0.999720i	$0.492466\pi$
$578$	−0.390422	−0.0162394
$579$	0	0
$580$	0	0
$581$	31.2260	1.29547
$582$	0	0
$583$	28.3486	1.17408
$584$	−5.83120	−0.241297
$585$	0	0
$586$	−1.56604	−0.0646923
$587$	10.9542	0.452127	0.226063	−	0.974113i	$-0.427414\pi$
$587$	10.9542	0.452127	0.226063	+	0.974113i	$0.427414\pi$
$588$	0	0
$589$	4.96269	0.204484
$590$	0	0
$591$	0	0
$592$	8.04314	0.330571
$593$	−15.6655	−0.643304	−0.321652	−	0.946858i	$-0.604238\pi$
$593$	−15.6655	−0.643304	−0.321652	+	0.946858i	$0.604238\pi$
$594$	0	0
$595$	0	0
$596$	−12.7971	−0.524190
$597$	0	0
$598$	0.0559843	0.00228937
$599$	−7.67744	−0.313692	−0.156846	−	0.987623i	$-0.550133\pi$
$599$	−7.67744	−0.313692	−0.156846	+	0.987623i	$0.550133\pi$
$600$	0	0
$601$	16.1466	0.658634	0.329317	−	0.944219i	$-0.393182\pi$
$601$	16.1466	0.658634	0.329317	+	0.944219i	$0.393182\pi$
$602$	9.67976	0.394517
$603$	0	0
$604$	12.4091	0.504920
$605$	0	0
$606$	0	0
$607$	−0.457937	−0.0185871	−0.00929355	−	0.999957i	$-0.502958\pi$
$607$	−0.457937	−0.0185871	−0.00929355	+	0.999957i	$0.502958\pi$
$608$	12.6257	0.512038
$609$	0	0
$610$	0	0
$611$	−9.95279	−0.402647
$612$	0	0
$613$	−0.964974	−0.0389749	−0.0194875	−	0.999810i	$-0.506203\pi$
$613$	−0.964974	−0.0389749	−0.0194875	+	0.999810i	$0.506203\pi$
$614$	−3.60702	−0.145567
$615$	0	0
$616$	−11.5251	−0.464361
$617$	−32.2373	−1.29783	−0.648913	−	0.760863i	$-0.724776\pi$
$617$	−32.2373	−1.29783	−0.648913	+	0.760863i	$0.724776\pi$
$618$	0	0
$619$	48.8949	1.96525	0.982627	−	0.185592i	$-0.0594203\pi$
$619$	48.8949	1.96525	0.982627	+	0.185592i	$0.0594203\pi$
$620$	0	0
$621$	0	0
$622$	−4.56062	−0.182864
$623$	15.9825	0.640324
$624$	0	0
$625$	0	0
$626$	−0.151669	−0.00606192
$627$	0	0
$628$	6.95315	0.277461
$629$	9.38413	0.374170
$630$	0	0
$631$	−26.0230	−1.03596	−0.517979	−	0.855393i	$-0.673316\pi$
$631$	−26.0230	−1.03596	−0.517979	+	0.855393i	$0.673316\pi$
$632$	0.821225	0.0326666
$633$	0	0
$634$	−1.93837	−0.0769825
$635$	0	0
$636$	0	0
$637$	−18.7065	−0.741180
$638$	2.85887	0.113184
$639$	0	0
$640$	0	0
$641$	−35.9254	−1.41897	−0.709484	−	0.704722i	$-0.751072\pi$
$641$	−35.9254	−1.41897	−0.709484	+	0.704722i	$0.751072\pi$
$642$	0	0
$643$	−28.0941	−1.10792	−0.553961	−	0.832543i	$-0.686884\pi$
$643$	−28.0941	−1.10792	−0.553961	+	0.832543i	$0.686884\pi$
$644$	1.84197	0.0725837
$645$	0	0
$646$	4.70954	0.185294
$647$	32.7912	1.28915	0.644577	−	0.764539i	$-0.277033\pi$
$647$	32.7912	1.28915	0.644577	+	0.764539i	$0.277033\pi$
$648$	0	0
$649$	15.9291	0.625273
$650$	0	0
$651$	0	0
$652$	10.7672	0.421677
$653$	16.3946	0.641570	0.320785	−	0.947152i	$-0.396053\pi$
$653$	16.3946	0.641570	0.320785	+	0.947152i	$0.396053\pi$
$654$	0	0
$655$	0	0
$656$	−37.1971	−1.45230
$657$	0	0
$658$	8.04314	0.313554
$659$	−18.4501	−0.718714	−0.359357	−	0.933200i	$-0.617004\pi$
$659$	−18.4501	−0.718714	−0.359357	+	0.933200i	$0.617004\pi$
$660$	0	0
$661$	16.7104	0.649960	0.324980	−	0.945721i	$-0.394642\pi$
$661$	16.7104	0.649960	0.324980	+	0.945721i	$0.394642\pi$
$662$	−1.59210	−0.0618786
$663$	0	0
$664$	−5.78881	−0.224649
$665$	0	0
$666$	0	0
$667$	−0.925042	−0.0358178
$668$	−9.41094	−0.364120
$669$	0	0
$670$	0	0
$671$	−26.4641	−1.02164
$672$	0	0
$673$	34.2660	1.32086	0.660428	−	0.750890i	$-0.270375\pi$
$673$	34.2660	1.32086	0.660428	+	0.750890i	$0.270375\pi$
$674$	6.59891	0.254181
$675$	0	0
$676$	22.2539	0.855919
$677$	−0.964974	−0.0370870	−0.0185435	−	0.999828i	$-0.505903\pi$
$677$	−0.964974	−0.0370870	−0.0185435	+	0.999828i	$0.505903\pi$
$678$	0	0
$679$	−65.6576	−2.51971
$680$	0	0
$681$	0	0
$682$	0.624733	0.0239223
$683$	−5.95960	−0.228038	−0.114019	−	0.993479i	$-0.536372\pi$
$683$	−5.95960	−0.228038	−0.114019	+	0.993479i	$0.536372\pi$
$684$	0	0
$685$	0	0
$686$	7.96238	0.304005
$687$	0	0
$688$	35.1781	1.34115
$689$	−12.5673	−0.478775
$690$	0	0
$691$	0.0278309	0.00105874	0.000529369	−	1.00000i	$-0.499831\pi$
$691$	0.0278309	0.00105874	0.000529369		1.00000i	$0.499831\pi$
$692$	12.4425	0.472992
$693$	0	0
$694$	1.45938	0.0553971
$695$	0	0
$696$	0	0
$697$	−43.3988	−1.64385
$698$	2.97690	0.112677
$699$	0	0
$700$	0	0
$701$	24.9176	0.941126	0.470563	−	0.882366i	$-0.344051\pi$
$701$	24.9176	0.941126	0.470563	+	0.882366i	$0.344051\pi$
$702$	0	0
$703$	−10.7456	−0.405276
$704$	−19.6069	−0.738964
$705$	0	0
$706$	−6.95416	−0.261723
$707$	−65.4407	−2.46115
$708$	0	0
$709$	−10.2192	−0.383790	−0.191895	−	0.981415i	$-0.561463\pi$
$709$	−10.2192	−0.383790	−0.191895	+	0.981415i	$0.561463\pi$
$710$	0	0
$711$	0	0
$712$	−2.96290	−0.111039
$713$	−0.202144	−0.00757037
$714$	0	0
$715$	0	0
$716$	14.6909	0.549025
$717$	0	0
$718$	−1.89446	−0.0707006
$719$	−20.4032	−0.760912	−0.380456	−	0.924799i	$-0.624233\pi$
$719$	−20.4032	−0.760912	−0.380456	+	0.924799i	$0.624233\pi$
$720$	0	0
$721$	−26.0092	−0.968632
$722$	−1.23241	−0.0458657
$723$	0	0
$724$	−10.7973	−0.401279
$725$	0	0
$726$	0	0
$727$	−14.4475	−0.535828	−0.267914	−	0.963443i	$-0.586334\pi$
$727$	−14.4475	−0.535828	−0.267914	+	0.963443i	$0.586334\pi$
$728$	5.10923	0.189361
$729$	0	0
$730$	0	0
$731$	41.0431	1.51803
$732$	0	0
$733$	−26.1389	−0.965463	−0.482732	−	0.875768i	$-0.660355\pi$
$733$	−26.1389	−0.965463	−0.482732	+	0.875768i	$0.660355\pi$
$734$	−5.26071	−0.194177
$735$	0	0
$736$	−0.514279	−0.0189566
$737$	6.76416	0.249161
$738$	0	0
$739$	9.63328	0.354366	0.177183	−	0.984178i	$-0.443302\pi$
$739$	9.63328	0.354366	0.177183	+	0.984178i	$0.443302\pi$
$740$	0	0
$741$	0	0
$742$	10.1560	0.372838
$743$	−37.7418	−1.38461	−0.692306	−	0.721604i	$-0.743405\pi$
$743$	−37.7418	−1.38461	−0.692306	+	0.721604i	$0.743405\pi$
$744$	0	0
$745$	0	0
$746$	0.416815	0.0152607
$747$	0	0
$748$	−24.1374	−0.882551
$749$	−3.87536	−0.141602
$750$	0	0
$751$	−40.4235	−1.47508	−0.737538	−	0.675306i	$-0.764012\pi$
$751$	−40.4235	−1.47508	−0.737538	+	0.675306i	$0.764012\pi$
$752$	29.2303	1.06592
$753$	0	0
$754$	−1.26737	−0.0461550
$755$	0	0
$756$	0	0
$757$	−2.89675	−0.105284	−0.0526421	−	0.998613i	$-0.516764\pi$
$757$	−2.89675	−0.105284	−0.0526421	+	0.998613i	$0.516764\pi$
$758$	−3.99875	−0.145241
$759$	0	0
$760$	0	0
$761$	−40.7822	−1.47835	−0.739177	−	0.673511i	$-0.764785\pi$
$761$	−40.7822	−1.47835	−0.739177	+	0.673511i	$0.764785\pi$
$762$	0	0
$763$	−49.6397	−1.79708
$764$	31.6634	1.14554
$765$	0	0
$766$	3.16231	0.114259
$767$	−7.06157	−0.254978
$768$	0	0
$769$	−4.84478	−0.174707	−0.0873536	−	0.996177i	$-0.527841\pi$
$769$	−4.84478	−0.174707	−0.0873536	+	0.996177i	$0.527841\pi$
$770$	0	0
$771$	0	0
$772$	10.5366	0.379219
$773$	−50.6773	−1.82274	−0.911368	−	0.411593i	$-0.864973\pi$
$773$	−50.6773	−1.82274	−0.911368	+	0.411593i	$0.864973\pi$
$774$	0	0
$775$	0	0
$776$	12.1719	0.436946
$777$	0	0
$778$	4.15537	0.148977
$779$	49.6949	1.78051
$780$	0	0
$781$	20.1563	0.721249
$782$	−0.191833	−0.00685993
$783$	0	0
$784$	54.9391	1.96211
$785$	0	0
$786$	0	0
$787$	−35.5198	−1.26615	−0.633073	−	0.774092i	$-0.718207\pi$
$787$	−35.5198	−1.26615	−0.633073	+	0.774092i	$0.718207\pi$
$788$	47.9705	1.70888
$789$	0	0
$790$	0	0
$791$	18.5144	0.658296
$792$	0	0
$793$	11.7319	0.416611
$794$	7.02213	0.249206
$795$	0	0
$796$	12.0228	0.426136
$797$	34.1721	1.21044	0.605219	−	0.796059i	$-0.293086\pi$
$797$	34.1721	1.21044	0.605219	+	0.796059i	$0.293086\pi$
$798$	0	0
$799$	34.1037	1.20650
$800$	0	0
$801$	0	0
$802$	5.81095	0.205192
$803$	−19.2253	−0.678446
$804$	0	0
$805$	0	0
$806$	−0.276952	−0.00975521
$807$	0	0
$808$	12.1317	0.426791
$809$	−42.9659	−1.51060	−0.755300	−	0.655379i	$-0.772509\pi$
$809$	−42.9659	−1.51060	−0.755300	+	0.655379i	$0.772509\pi$
$810$	0	0
$811$	14.5692	0.511594	0.255797	−	0.966731i	$-0.417662\pi$
$811$	14.5692	0.511594	0.255797	+	0.966731i	$0.417662\pi$
$812$	−41.6984	−1.46333
$813$	0	0
$814$	−1.35271	−0.0474126
$815$	0	0
$816$	0	0
$817$	−46.9975	−1.64424
$818$	−2.26644	−0.0792442
$819$	0	0
$820$	0	0
$821$	−3.81820	−0.133256	−0.0666280	−	0.997778i	$-0.521224\pi$
$821$	−3.81820	−0.133256	−0.0666280	+	0.997778i	$0.521224\pi$
$822$	0	0
$823$	24.4163	0.851097	0.425549	−	0.904936i	$-0.360081\pi$
$823$	24.4163	0.851097	0.425549	+	0.904936i	$0.360081\pi$
$824$	4.82169	0.167972
$825$	0	0
$826$	5.70666	0.198560
$827$	0.571160	0.0198612	0.00993059	−	0.999951i	$-0.496839\pi$
$827$	0.571160	0.0198612	0.00993059	+	0.999951i	$0.496839\pi$
$828$	0	0
$829$	21.5894	0.749832	0.374916	−	0.927059i	$-0.377672\pi$
$829$	21.5894	0.749832	0.374916	+	0.927059i	$0.377672\pi$
$830$	0	0
$831$	0	0
$832$	8.69199	0.301341
$833$	64.0988	2.22089
$834$	0	0
$835$	0	0
$836$	27.6392	0.955922
$837$	0	0
$838$	−1.13348	−0.0391555
$839$	46.1708	1.59399	0.796996	−	0.603985i	$-0.206421\pi$
$839$	46.1708	1.59399	0.796996	+	0.603985i	$0.206421\pi$
$840$	0	0
$841$	−8.05891	−0.277893
$842$	0.496919	0.0171249
$843$	0	0
$844$	47.1466	1.62285
$845$	0	0
$846$	0	0
$847$	13.3497	0.458700
$848$	36.9088	1.26745
$849$	0	0
$850$	0	0
$851$	0.437697	0.0150040
$852$	0	0
$853$	26.1761	0.896252	0.448126	−	0.893970i	$-0.352092\pi$
$853$	26.1761	0.896252	0.448126	+	0.893970i	$0.352092\pi$
$854$	−9.48087	−0.324429
$855$	0	0
$856$	0.718430	0.0245554
$857$	−10.2378	−0.349717	−0.174859	−	0.984594i	$-0.555947\pi$
$857$	−10.2378	−0.349717	−0.174859	+	0.984594i	$0.555947\pi$
$858$	0	0
$859$	−45.4941	−1.55224	−0.776119	−	0.630587i	$-0.782814\pi$
$859$	−45.4941	−1.55224	−0.776119	+	0.630587i	$0.782814\pi$
$860$	0	0
$861$	0	0
$862$	4.47515	0.152424
$863$	−20.6205	−0.701929	−0.350965	−	0.936389i	$-0.614146\pi$
$863$	−20.6205	−0.701929	−0.350965	+	0.936389i	$0.614146\pi$
$864$	0	0
$865$	0	0
$866$	4.89938	0.166488
$867$	0	0
$868$	−9.11213	−0.309286
$869$	2.70756	0.0918475
$870$	0	0
$871$	−2.99864	−0.101605
$872$	9.20243	0.311634
$873$	0	0
$874$	0.219663	0.00743022
$875$	0	0
$876$	0	0
$877$	−28.6425	−0.967190	−0.483595	−	0.875292i	$-0.660669\pi$
$877$	−28.6425	−0.967190	−0.483595	+	0.875292i	$0.660669\pi$
$878$	−2.83958	−0.0958312
$879$	0	0
$880$	0	0
$881$	45.9038	1.54654	0.773269	−	0.634078i	$-0.218620\pi$
$881$	45.9038	1.54654	0.773269	+	0.634078i	$0.218620\pi$
$882$	0	0
$883$	12.5872	0.423593	0.211797	−	0.977314i	$-0.432069\pi$
$883$	12.5872	0.423593	0.211797	+	0.977314i	$0.432069\pi$
$884$	10.7004	0.359894
$885$	0	0
$886$	−2.90219	−0.0975010
$887$	−30.2580	−1.01596	−0.507982	−	0.861368i	$-0.669608\pi$
$887$	−30.2580	−1.01596	−0.507982	+	0.861368i	$0.669608\pi$
$888$	0	0
$889$	−29.3945	−0.985860
$890$	0	0
$891$	0	0
$892$	1.41778	0.0474708
$893$	−39.0514	−1.30680
$894$	0	0
$895$	0	0
$896$	−30.7760	−1.02815
$897$	0	0
$898$	3.66615	0.122341
$899$	4.57614	0.152623
$900$	0	0
$901$	43.0624	1.43462
$902$	6.25590	0.208299
$903$	0	0
$904$	−3.43228	−0.114156
$905$	0	0
$906$	0	0
$907$	57.4764	1.90847	0.954237	−	0.299051i	$-0.0966701\pi$
$907$	57.4764	1.90847	0.954237	+	0.299051i	$0.0966701\pi$
$908$	21.5481	0.715099
$909$	0	0
$910$	0	0
$911$	43.5394	1.44253	0.721263	−	0.692662i	$-0.243562\pi$
$911$	43.5394	1.44253	0.721263	+	0.692662i	$0.243562\pi$
$912$	0	0
$913$	−19.0855	−0.631639
$914$	3.50849	0.116051
$915$	0	0
$916$	47.1124	1.55664
$917$	−8.26645	−0.272982
$918$	0	0
$919$	−27.2870	−0.900115	−0.450057	−	0.893000i	$-0.648596\pi$
$919$	−27.2870	−0.900115	−0.450057	+	0.893000i	$0.648596\pi$
$920$	0	0
$921$	0	0
$922$	−3.84825	−0.126735
$923$	−8.93553	−0.294117
$924$	0	0
$925$	0	0
$926$	−6.80225	−0.223536
$927$	0	0
$928$	11.6422	0.382175
$929$	3.52955	0.115801	0.0579004	−	0.998322i	$-0.481559\pi$
$929$	3.52955	0.115801	0.0579004	+	0.998322i	$0.481559\pi$
$930$	0	0
$931$	−73.3981	−2.40552
$932$	29.7227	0.973600
$933$	0	0
$934$	−7.58093	−0.248056
$935$	0	0
$936$	0	0
$937$	−10.2378	−0.334455	−0.167227	−	0.985918i	$-0.553481\pi$
$937$	−10.2378	−0.334455	−0.167227	+	0.985918i	$0.553481\pi$
$938$	2.42329	0.0791231
$939$	0	0
$940$	0	0
$941$	−16.0807	−0.524215	−0.262108	−	0.965039i	$-0.584418\pi$
$941$	−16.0807	−0.524215	−0.262108	+	0.965039i	$0.584418\pi$
$942$	0	0
$943$	−2.02421	−0.0659175
$944$	20.7391	0.675000
$945$	0	0
$946$	−5.91633	−0.192356
$947$	−22.8462	−0.742402	−0.371201	−	0.928552i	$-0.621054\pi$
$947$	−22.8462	−0.742402	−0.371201	+	0.928552i	$0.621054\pi$
$948$	0	0
$949$	8.52281	0.276662
$950$	0	0
$951$	0	0
$952$	−17.5070	−0.567406
$953$	−26.0704	−0.844502	−0.422251	−	0.906479i	$-0.638760\pi$
$953$	−26.0704	−0.844502	−0.422251	+	0.906479i	$0.638760\pi$
$954$	0	0
$955$	0	0
$956$	29.3197	0.948266
$957$	0	0
$958$	3.38050	0.109219
$959$	−14.8120	−0.478304
$960$	0	0
$961$	1.00000	0.0322581
$962$	0.599675	0.0193343
$963$	0	0
$964$	−41.3597	−1.33211
$965$	0	0
$966$	0	0
$967$	−24.8659	−0.799635	−0.399817	−	0.916595i	$-0.630926\pi$
$967$	−24.8659	−0.799635	−0.399817	+	0.916595i	$0.630926\pi$
$968$	−2.47482	−0.0795437
$969$	0	0
$970$	0	0
$971$	42.6042	1.36723	0.683617	−	0.729841i	$-0.260406\pi$
$971$	42.6042	1.36723	0.683617	+	0.729841i	$0.260406\pi$
$972$	0	0
$973$	69.0928	2.21501
$974$	−7.30834	−0.234174
$975$	0	0
$976$	−34.4553	−1.10289
$977$	−39.2452	−1.25556	−0.627782	−	0.778389i	$-0.716037\pi$
$977$	−39.2452	−1.25556	−0.627782	+	0.778389i	$0.716037\pi$
$978$	0	0
$979$	−9.76859	−0.312205
$980$	0	0
$981$	0	0
$982$	−0.949249	−0.0302917
$983$	−1.59529	−0.0508817	−0.0254408	−	0.999676i	$-0.508099\pi$
$983$	−1.59529	−0.0508817	−0.0254408	+	0.999676i	$0.508099\pi$
$984$	0	0
$985$	0	0
$986$	4.34271	0.138300
$987$	0	0
$988$	−12.2528	−0.389813
$989$	1.91434	0.0608725
$990$	0	0
$991$	6.18939	0.196612	0.0983062	−	0.995156i	$-0.468658\pi$
$991$	6.18939	0.196612	0.0983062	+	0.995156i	$0.468658\pi$
$992$	2.54412	0.0807758
$993$	0	0
$994$	7.22107	0.229038
$995$	0	0
$996$	0	0
$997$	−6.61719	−0.209569	−0.104784	−	0.994495i	$-0.533415\pi$
$997$	−6.61719	−0.209569	−0.104784	+	0.994495i	$0.533415\pi$
$998$	0.176092	0.00557409
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ch.1.5		10
3.2	odd	2		775.2.a.l.1.6		10
5.2	odd	4		1395.2.c.e.559.5		10
5.3	odd	4		1395.2.c.e.559.6		10
5.4	even	2	inner	6975.2.a.ch.1.6		10
15.2	even	4		155.2.b.b.94.6	yes	10
15.8	even	4		155.2.b.b.94.5	✓	10
15.14	odd	2		775.2.a.l.1.5		10
60.23	odd	4		2480.2.d.g.1489.9		10
60.47	odd	4		2480.2.d.g.1489.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.b.b.94.5	✓	10	15.8	even	4
155.2.b.b.94.6	yes	10	15.2	even	4
775.2.a.l.1.5		10	15.14	odd	2
775.2.a.l.1.6		10	3.2	odd	2
1395.2.c.e.559.5		10	5.2	odd	4
1395.2.c.e.559.6		10	5.3	odd	4
2480.2.d.g.1489.2		10	60.47	odd	4
2480.2.d.g.1489.9		10	60.23	odd	4
6975.2.a.ch.1.5		10	1.1	even	1	trivial
6975.2.a.ch.1.6		10	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.218967 q^{2} -1.95205 q^{4} -4.66797 q^{7} +0.865369 q^{8} +2.85310 q^{11} -1.26481 q^{13} +1.02213 q^{14} +3.71462 q^{16} +4.33394 q^{17} -4.96269 q^{19} -0.624733 q^{22} +0.202144 q^{23} +0.276952 q^{26} +9.11213 q^{28} -4.57614 q^{29} -1.00000 q^{31} -2.54412 q^{32} -0.948989 q^{34} +2.16527 q^{37} +1.08667 q^{38} -10.0137 q^{41} +9.47017 q^{43} -5.56940 q^{44} -0.0442629 q^{46} +7.86899 q^{47} +14.7900 q^{49} +2.46898 q^{52} +9.93609 q^{53} -4.03952 q^{56} +1.00202 q^{58} +5.58310 q^{59} -9.27559 q^{61} +0.218967 q^{62} -6.87216 q^{64} +2.37082 q^{67} -8.46008 q^{68} +7.06471 q^{71} -6.73840 q^{73} -0.474122 q^{74} +9.68744 q^{76} -13.3182 q^{77} +0.948989 q^{79} +2.19267 q^{82} -6.68941 q^{83} -2.07365 q^{86} +2.46898 q^{88} -3.42386 q^{89} +5.90411 q^{91} -0.394597 q^{92} -1.72305 q^{94} +14.0656 q^{97} -3.23851 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 12 q^{4} - 16 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} - 28 q^{26} - 32 q^{29} - 10 q^{31} - 28 q^{34} - 36 q^{41} - 52 q^{44} + 8 q^{46} + 22 q^{49} - 12 q^{56} - 12 q^{59} - 70 q^{64} - 12 q^{71}+ \cdots - 48 q^{94}+O(q^{100}) \) 10 * q + 12 * q^4 - 16 * q^11 + 6 * q^14 + 8 * q^16 - 4 * q^19 - 28 * q^26 - 32 * q^29 - 10 * q^31 - 28 * q^34 - 36 * q^41 - 52 * q^44 + 8 * q^46 + 22 * q^49 - 12 * q^56 - 12 * q^59 - 70 * q^64 - 12 * q^71 - 28 * q^74 - 30 * q^76 + 28 * q^79 - 36 * q^86 - 48 * q^89 - 4 * q^91 - 48 * q^94

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