LMFDB - Embedded newform 6975.2.a.v.1.2

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 = [2,0,0,2,0,0,6,0,0,0,-4,0,6,6,0,-10,4] 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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +4.73205 q^{7} -1.73205 q^{8} +1.46410 q^{11} +1.26795 q^{13} +8.19615 q^{14} -5.00000 q^{16} -1.46410 q^{17} +5.46410 q^{19} +2.53590 q^{22} -0.535898 q^{23} +2.19615 q^{26} +4.73205 q^{28} -0.732051 q^{29} -1.00000 q^{31} -5.19615 q^{32} -2.53590 q^{34} +6.73205 q^{37} +9.46410 q^{38} -3.46410 q^{41} +4.00000 q^{43} +1.46410 q^{44} -0.928203 q^{46} -6.00000 q^{47} +15.3923 q^{49} +1.26795 q^{52} +12.3923 q^{53} -8.19615 q^{56} -1.26795 q^{58} +12.1962 q^{59} -10.3923 q^{61} -1.73205 q^{62} +1.00000 q^{64} +10.1962 q^{67} -1.46410 q^{68} +3.80385 q^{71} -5.66025 q^{73} +11.6603 q^{74} +5.46410 q^{76} +6.92820 q^{77} -10.0000 q^{79} -6.00000 q^{82} +15.4641 q^{83} +6.92820 q^{86} -2.53590 q^{88} -7.26795 q^{89} +6.00000 q^{91} -0.535898 q^{92} -10.3923 q^{94} +2.00000 q^{97} +26.6603 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{7} - 4 q^{11} + 6 q^{13} + 6 q^{14} - 10 q^{16} + 4 q^{17} + 4 q^{19} + 12 q^{22} - 8 q^{23} - 6 q^{26} + 6 q^{28} + 2 q^{29} - 2 q^{31} - 12 q^{34} + 10 q^{37} + 12 q^{38} + 8 q^{43}+ \cdots + 36 q^{98}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^7 - 4 * q^11 + 6 * q^13 + 6 * q^14 - 10 * q^16 + 4 * q^17 + 4 * q^19 + 12 * q^22 - 8 * q^23 - 6 * q^26 + 6 * q^28 + 2 * q^29 - 2 * q^31 - 12 * q^34 + 10 * q^37 + 12 * q^38 + 8 * q^43 - 4 * q^44 + 12 * q^46 - 12 * q^47 + 10 * q^49 + 6 * q^52 + 4 * q^53 - 6 * q^56 - 6 * q^58 + 14 * q^59 + 2 * q^64 + 10 * q^67 + 4 * q^68 + 18 * q^71 + 6 * q^73 + 6 * q^74 + 4 * q^76 - 20 * q^79 - 12 * q^82 + 24 * q^83 - 12 * q^88 - 18 * q^89 + 12 * q^91 - 8 * q^92 + 4 * q^97 + 36 * 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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	−1.73205	−0.612372
$9$	0	0
$10$	0	0
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	0	0
$13$	1.26795	0.351666	0.175833	−	0.984420i	$-0.443738\pi$
$13$	1.26795	0.351666	0.175833	+	0.984420i	$0.443738\pi$
$14$	8.19615	2.19051
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−1.46410	−0.355097	−0.177548	−	0.984112i	$-0.556817\pi$
$17$	−1.46410	−0.355097	−0.177548	+	0.984112i	$0.556817\pi$
$18$	0	0
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	0	0
$21$	0	0
$22$	2.53590	0.540655
$23$	−0.535898	−0.111743	−0.0558713	−	0.998438i	$-0.517794\pi$
$23$	−0.535898	−0.111743	−0.0558713	+	0.998438i	$0.517794\pi$
$24$	0	0
$25$	0	0
$26$	2.19615	0.430701
$27$	0	0
$28$	4.73205	0.894274
$29$	−0.732051	−0.135938	−0.0679692	−	0.997687i	$-0.521652\pi$
$29$	−0.732051	−0.135938	−0.0679692	+	0.997687i	$0.521652\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.19615	−0.918559
$33$	0	0
$34$	−2.53590	−0.434903
$35$	0	0
$36$	0	0
$37$	6.73205	1.10674	0.553371	−	0.832935i	$-0.313341\pi$
$37$	6.73205	1.10674	0.553371	+	0.832935i	$0.313341\pi$
$38$	9.46410	1.53528
$39$	0	0
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.46410	0.220722
$45$	0	0
$46$	−0.928203	−0.136856
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	0	0
$52$	1.26795	0.175833
$53$	12.3923	1.70221	0.851107	−	0.524992i	$-0.175932\pi$
$53$	12.3923	1.70221	0.851107	+	0.524992i	$0.175932\pi$
$54$	0	0
$55$	0	0
$56$	−8.19615	−1.09526
$57$	0	0
$58$	−1.26795	−0.166490
$59$	12.1962	1.58780	0.793902	−	0.608046i	$-0.208046\pi$
$59$	12.1962	1.58780	0.793902	+	0.608046i	$0.208046\pi$
$60$	0	0
$61$	−10.3923	−1.33060	−0.665299	−	0.746577i	$-0.731696\pi$
$61$	−10.3923	−1.33060	−0.665299	+	0.746577i	$0.731696\pi$
$62$	−1.73205	−0.219971
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.1962	1.24566	0.622829	−	0.782358i	$-0.285983\pi$
$67$	10.1962	1.24566	0.622829	+	0.782358i	$0.285983\pi$
$68$	−1.46410	−0.177548
$69$	0	0
$70$	0	0
$71$	3.80385	0.451434	0.225717	−	0.974193i	$-0.427528\pi$
$71$	3.80385	0.451434	0.225717	+	0.974193i	$0.427528\pi$
$72$	0	0
$73$	−5.66025	−0.662483	−0.331241	−	0.943546i	$-0.607467\pi$
$73$	−5.66025	−0.662483	−0.331241	+	0.943546i	$0.607467\pi$
$74$	11.6603	1.35548
$75$	0	0
$76$	5.46410	0.626775
$77$	6.92820	0.789542
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	0	0
$82$	−6.00000	−0.662589
$83$	15.4641	1.69741	0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641	1.69741	0.848703	+	0.528870i	$0.177384\pi$
$84$	0	0
$85$	0	0
$86$	6.92820	0.747087
$87$	0	0
$88$	−2.53590	−0.270328
$89$	−7.26795	−0.770401	−0.385201	−	0.922833i	$-0.625868\pi$
$89$	−7.26795	−0.770401	−0.385201	+	0.922833i	$0.625868\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	−0.535898	−0.0558713
$93$	0	0
$94$	−10.3923	−1.07188
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	26.6603	2.69309
$99$	0	0
$100$	0	0
$101$	−4.53590	−0.451339	−0.225669	−	0.974204i	$-0.572457\pi$
$101$	−4.53590	−0.451339	−0.225669	+	0.974204i	$0.572457\pi$
$102$	0	0
$103$	−3.26795	−0.322001	−0.161000	−	0.986954i	$-0.551472\pi$
$103$	−3.26795	−0.322001	−0.161000	+	0.986954i	$0.551472\pi$
$104$	−2.19615	−0.215350
$105$	0	0
$106$	21.4641	2.08478
$107$	−15.8564	−1.53290	−0.766448	−	0.642306i	$-0.777978\pi$
$107$	−15.8564	−1.53290	−0.766448	+	0.642306i	$0.777978\pi$
$108$	0	0
$109$	−2.53590	−0.242895	−0.121448	−	0.992598i	$-0.538754\pi$
$109$	−2.53590	−0.242895	−0.121448	+	0.992598i	$0.538754\pi$
$110$	0	0
$111$	0	0
$112$	−23.6603	−2.23568
$113$	−4.92820	−0.463606	−0.231803	−	0.972763i	$-0.574463\pi$
$113$	−4.92820	−0.463606	−0.231803	+	0.972763i	$0.574463\pi$
$114$	0	0
$115$	0	0
$116$	−0.732051	−0.0679692
$117$	0	0
$118$	21.1244	1.94465
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−8.85641	−0.805128
$122$	−18.0000	−1.62964
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	0	0
$127$	10.5359	0.934910	0.467455	−	0.884017i	$-0.345171\pi$
$127$	10.5359	0.934910	0.467455	+	0.884017i	$0.345171\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	−6.73205	−0.588182	−0.294091	−	0.955777i	$-0.595017\pi$
$131$	−6.73205	−0.588182	−0.294091	+	0.955777i	$0.595017\pi$
$132$	0	0
$133$	25.8564	2.24203
$134$	17.6603	1.52561
$135$	0	0
$136$	2.53590	0.217451
$137$	18.9282	1.61715	0.808573	−	0.588396i	$-0.200240\pi$
$137$	18.9282	1.61715	0.808573	+	0.588396i	$0.200240\pi$
$138$	0	0
$139$	9.85641	0.836009	0.418005	−	0.908445i	$-0.362730\pi$
$139$	9.85641	0.836009	0.418005	+	0.908445i	$0.362730\pi$
$140$	0	0
$141$	0	0
$142$	6.58846	0.552891
$143$	1.85641	0.155241
$144$	0	0
$145$	0	0
$146$	−9.80385	−0.811372
$147$	0	0
$148$	6.73205	0.553371
$149$	−21.3205	−1.74664	−0.873322	−	0.487143i	$-0.838039\pi$
$149$	−21.3205	−1.74664	−0.873322	+	0.487143i	$0.838039\pi$
$150$	0	0
$151$	−20.9282	−1.70311	−0.851557	−	0.524263i	$-0.824341\pi$
$151$	−20.9282	−1.70311	−0.851557	+	0.524263i	$0.824341\pi$
$152$	−9.46410	−0.767640
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	12.9282	1.03178	0.515891	−	0.856654i	$-0.327461\pi$
$157$	12.9282	1.03178	0.515891	+	0.856654i	$0.327461\pi$
$158$	−17.3205	−1.37795
$159$	0	0
$160$	0	0
$161$	−2.53590	−0.199857
$162$	0	0
$163$	14.1962	1.11193	0.555964	−	0.831206i	$-0.312349\pi$
$163$	14.1962	1.11193	0.555964	+	0.831206i	$0.312349\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	−11.3923	−0.876331
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	−22.7846	−1.73228	−0.866141	−	0.499800i	$-0.833407\pi$
$173$	−22.7846	−1.73228	−0.866141	+	0.499800i	$0.833407\pi$
$174$	0	0
$175$	0	0
$176$	−7.32051	−0.551804
$177$	0	0
$178$	−12.5885	−0.943545
$179$	9.07180	0.678058	0.339029	−	0.940776i	$-0.389902\pi$
$179$	9.07180	0.678058	0.339029	+	0.940776i	$0.389902\pi$
$180$	0	0
$181$	−10.3923	−0.772454	−0.386227	−	0.922404i	$-0.626222\pi$
$181$	−10.3923	−0.772454	−0.386227	+	0.922404i	$0.626222\pi$
$182$	10.3923	0.770329
$183$	0	0
$184$	0.928203	0.0684280
$185$	0	0
$186$	0	0
$187$	−2.14359	−0.156755
$188$	−6.00000	−0.437595
$189$	0	0
$190$	0	0
$191$	11.8038	0.854096	0.427048	−	0.904229i	$-0.359553\pi$
$191$	11.8038	0.854096	0.427048	+	0.904229i	$0.359553\pi$
$192$	0	0
$193$	23.8564	1.71722	0.858611	−	0.512628i	$-0.171328\pi$
$193$	23.8564	1.71722	0.858611	+	0.512628i	$0.171328\pi$
$194$	3.46410	0.248708
$195$	0	0
$196$	15.3923	1.09945
$197$	13.0718	0.931327	0.465663	−	0.884962i	$-0.345816\pi$
$197$	13.0718	0.931327	0.465663	+	0.884962i	$0.345816\pi$
$198$	0	0
$199$	16.9282	1.20001	0.600004	−	0.799997i	$-0.295166\pi$
$199$	16.9282	1.20001	0.600004	+	0.799997i	$0.295166\pi$
$200$	0	0
$201$	0	0
$202$	−7.85641	−0.552775
$203$	−3.46410	−0.243132
$204$	0	0
$205$	0	0
$206$	−5.66025	−0.394369
$207$	0	0
$208$	−6.33975	−0.439582
$209$	8.00000	0.553372
$210$	0	0
$211$	7.32051	0.503965	0.251982	−	0.967732i	$-0.418918\pi$
$211$	7.32051	0.503965	0.251982	+	0.967732i	$0.418918\pi$
$212$	12.3923	0.851107
$213$	0	0
$214$	−27.4641	−1.87741
$215$	0	0
$216$	0	0
$217$	−4.73205	−0.321233
$218$	−4.39230	−0.297484
$219$	0	0
$220$	0	0
$221$	−1.85641	−0.124875
$222$	0	0
$223$	0.392305	0.0262707	0.0131353	−	0.999914i	$-0.495819\pi$
$223$	0.392305	0.0262707	0.0131353	+	0.999914i	$0.495819\pi$
$224$	−24.5885	−1.64289
$225$	0	0
$226$	−8.53590	−0.567800
$227$	13.3205	0.884113	0.442057	−	0.896987i	$-0.354249\pi$
$227$	13.3205	0.884113	0.442057	+	0.896987i	$0.354249\pi$
$228$	0	0
$229$	18.7846	1.24132	0.620661	−	0.784079i	$-0.286864\pi$
$229$	18.7846	1.24132	0.620661	+	0.784079i	$0.286864\pi$
$230$	0	0
$231$	0	0
$232$	1.26795	0.0832449
$233$	10.9282	0.715930	0.357965	−	0.933735i	$-0.383471\pi$
$233$	10.9282	0.715930	0.357965	+	0.933735i	$0.383471\pi$
$234$	0	0
$235$	0	0
$236$	12.1962	0.793902
$237$	0	0
$238$	−12.0000	−0.777844
$239$	13.8564	0.896296	0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564	0.896296	0.448148	+	0.893959i	$0.352084\pi$
$240$	0	0
$241$	23.8564	1.53673	0.768363	−	0.640014i	$-0.221072\pi$
$241$	23.8564	1.53673	0.768363	+	0.640014i	$0.221072\pi$
$242$	−15.3397	−0.986076
$243$	0	0
$244$	−10.3923	−0.665299
$245$	0	0
$246$	0	0
$247$	6.92820	0.440831
$248$	1.73205	0.109985
$249$	0	0
$250$	0	0
$251$	−2.92820	−0.184827	−0.0924133	−	0.995721i	$-0.529458\pi$
$251$	−2.92820	−0.184827	−0.0924133	+	0.995721i	$0.529458\pi$
$252$	0	0
$253$	−0.784610	−0.0493280
$254$	18.2487	1.14503
$255$	0	0
$256$	19.0000	1.18750
$257$	−9.85641	−0.614826	−0.307413	−	0.951576i	$-0.599463\pi$
$257$	−9.85641	−0.614826	−0.307413	+	0.951576i	$0.599463\pi$
$258$	0	0
$259$	31.8564	1.97946
$260$	0	0
$261$	0	0
$262$	−11.6603	−0.720373
$263$	−17.0718	−1.05269	−0.526346	−	0.850270i	$-0.676438\pi$
$263$	−17.0718	−1.05269	−0.526346	+	0.850270i	$0.676438\pi$
$264$	0	0
$265$	0	0
$266$	44.7846	2.74592
$267$	0	0
$268$	10.1962	0.622829
$269$	−9.80385	−0.597751	−0.298876	−	0.954292i	$-0.596612\pi$
$269$	−9.80385	−0.597751	−0.298876	+	0.954292i	$0.596612\pi$
$270$	0	0
$271$	−16.7846	−1.01959	−0.509796	−	0.860295i	$-0.670279\pi$
$271$	−16.7846	−1.01959	−0.509796	+	0.860295i	$0.670279\pi$
$272$	7.32051	0.443871
$273$	0	0
$274$	32.7846	1.98059
$275$	0	0
$276$	0	0
$277$	15.1244	0.908734	0.454367	−	0.890814i	$-0.349865\pi$
$277$	15.1244	0.908734	0.454367	+	0.890814i	$0.349865\pi$
$278$	17.0718	1.02390
$279$	0	0
$280$	0	0
$281$	16.5359	0.986449	0.493224	−	0.869902i	$-0.335818\pi$
$281$	16.5359	0.986449	0.493224	+	0.869902i	$0.335818\pi$
$282$	0	0
$283$	−7.26795	−0.432035	−0.216017	−	0.976390i	$-0.569307\pi$
$283$	−7.26795	−0.432035	−0.216017	+	0.976390i	$0.569307\pi$
$284$	3.80385	0.225717
$285$	0	0
$286$	3.21539	0.190130
$287$	−16.3923	−0.967607
$288$	0	0
$289$	−14.8564	−0.873906
$290$	0	0
$291$	0	0
$292$	−5.66025	−0.331241
$293$	5.07180	0.296298	0.148149	−	0.988965i	$-0.452669\pi$
$293$	5.07180	0.296298	0.148149	+	0.988965i	$0.452669\pi$
$294$	0	0
$295$	0	0
$296$	−11.6603	−0.677738
$297$	0	0
$298$	−36.9282	−2.13919
$299$	−0.679492	−0.0392960
$300$	0	0
$301$	18.9282	1.09100
$302$	−36.2487	−2.08588
$303$	0	0
$304$	−27.3205	−1.56694
$305$	0	0
$306$	0	0
$307$	−26.5885	−1.51748	−0.758742	−	0.651392i	$-0.774186\pi$
$307$	−26.5885	−1.51748	−0.758742	+	0.651392i	$0.774186\pi$
$308$	6.92820	0.394771
$309$	0	0
$310$	0	0
$311$	−13.6603	−0.774602	−0.387301	−	0.921953i	$-0.626593\pi$
$311$	−13.6603	−0.774602	−0.387301	+	0.921953i	$0.626593\pi$
$312$	0	0
$313$	21.6603	1.22431	0.612155	−	0.790738i	$-0.290303\pi$
$313$	21.6603	1.22431	0.612155	+	0.790738i	$0.290303\pi$
$314$	22.3923	1.26367
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−22.9282	−1.28778	−0.643888	−	0.765120i	$-0.722680\pi$
$317$	−22.9282	−1.28778	−0.643888	+	0.765120i	$0.722680\pi$
$318$	0	0
$319$	−1.07180	−0.0600091
$320$	0	0
$321$	0	0
$322$	−4.39230	−0.244774
$323$	−8.00000	−0.445132
$324$	0	0
$325$	0	0
$326$	24.5885	1.36183
$327$	0	0
$328$	6.00000	0.331295
$329$	−28.3923	−1.56532
$330$	0	0
$331$	24.9282	1.37018	0.685089	−	0.728459i	$-0.259763\pi$
$331$	24.9282	1.37018	0.685089	+	0.728459i	$0.259763\pi$
$332$	15.4641	0.848703
$333$	0	0
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	−18.0526	−0.983386	−0.491693	−	0.870769i	$-0.663622\pi$
$337$	−18.0526	−0.983386	−0.491693	+	0.870769i	$0.663622\pi$
$338$	−19.7321	−1.07328
$339$	0	0
$340$	0	0
$341$	−1.46410	−0.0792855
$342$	0	0
$343$	39.7128	2.14429
$344$	−6.92820	−0.373544
$345$	0	0
$346$	−39.4641	−2.12160
$347$	−12.7846	−0.686314	−0.343157	−	0.939278i	$-0.611496\pi$
$347$	−12.7846	−0.686314	−0.343157	+	0.939278i	$0.611496\pi$
$348$	0	0
$349$	6.53590	0.349859	0.174929	−	0.984581i	$-0.444030\pi$
$349$	6.53590	0.349859	0.174929	+	0.984581i	$0.444030\pi$
$350$	0	0
$351$	0	0
$352$	−7.60770	−0.405492
$353$	−4.39230	−0.233779	−0.116889	−	0.993145i	$-0.537292\pi$
$353$	−4.39230	−0.233779	−0.116889	+	0.993145i	$0.537292\pi$
$354$	0	0
$355$	0	0
$356$	−7.26795	−0.385201
$357$	0	0
$358$	15.7128	0.830448
$359$	1.66025	0.0876249	0.0438124	−	0.999040i	$-0.486050\pi$
$359$	1.66025	0.0876249	0.0438124	+	0.999040i	$0.486050\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−18.0000	−0.946059
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	21.8564	1.14090	0.570448	−	0.821334i	$-0.306770\pi$
$367$	21.8564	1.14090	0.570448	+	0.821334i	$0.306770\pi$
$368$	2.67949	0.139678
$369$	0	0
$370$	0	0
$371$	58.6410	3.04449
$372$	0	0
$373$	−26.3923	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$373$	−26.3923	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$374$	−3.71281	−0.191985
$375$	0	0
$376$	10.3923	0.535942
$377$	−0.928203	−0.0478049
$378$	0	0
$379$	32.3923	1.66388	0.831940	−	0.554865i	$-0.187230\pi$
$379$	32.3923	1.66388	0.831940	+	0.554865i	$0.187230\pi$
$380$	0	0
$381$	0	0
$382$	20.4449	1.04605
$383$	−13.3205	−0.680646	−0.340323	−	0.940309i	$-0.610536\pi$
$383$	−13.3205	−0.680646	−0.340323	+	0.940309i	$0.610536\pi$
$384$	0	0
$385$	0	0
$386$	41.3205	2.10316
$387$	0	0
$388$	2.00000	0.101535
$389$	−12.3397	−0.625650	−0.312825	−	0.949811i	$-0.601275\pi$
$389$	−12.3397	−0.625650	−0.312825	+	0.949811i	$0.601275\pi$
$390$	0	0
$391$	0.784610	0.0396794
$392$	−26.6603	−1.34655
$393$	0	0
$394$	22.6410	1.14064
$395$	0	0
$396$	0	0
$397$	−23.8564	−1.19732	−0.598659	−	0.801004i	$-0.704300\pi$
$397$	−23.8564	−1.19732	−0.598659	+	0.801004i	$0.704300\pi$
$398$	29.3205	1.46970
$399$	0	0
$400$	0	0
$401$	−9.51666	−0.475239	−0.237620	−	0.971358i	$-0.576367\pi$
$401$	−9.51666	−0.475239	−0.237620	+	0.971358i	$0.576367\pi$
$402$	0	0
$403$	−1.26795	−0.0631610
$404$	−4.53590	−0.225669
$405$	0	0
$406$	−6.00000	−0.297775
$407$	9.85641	0.488564
$408$	0	0
$409$	−17.3205	−0.856444	−0.428222	−	0.903674i	$-0.640860\pi$
$409$	−17.3205	−0.856444	−0.428222	+	0.903674i	$0.640860\pi$
$410$	0	0
$411$	0	0
$412$	−3.26795	−0.161000
$413$	57.7128	2.83986
$414$	0	0
$415$	0	0
$416$	−6.58846	−0.323026
$417$	0	0
$418$	13.8564	0.677739
$419$	27.1244	1.32511	0.662556	−	0.749013i	$-0.269472\pi$
$419$	27.1244	1.32511	0.662556	+	0.749013i	$0.269472\pi$
$420$	0	0
$421$	−15.3205	−0.746676	−0.373338	−	0.927695i	$-0.621787\pi$
$421$	−15.3205	−0.746676	−0.373338	+	0.927695i	$0.621787\pi$
$422$	12.6795	0.617228
$423$	0	0
$424$	−21.4641	−1.04239
$425$	0	0
$426$	0	0
$427$	−49.1769	−2.37984
$428$	−15.8564	−0.766448
$429$	0	0
$430$	0	0
$431$	−8.87564	−0.427525	−0.213762	−	0.976886i	$-0.568572\pi$
$431$	−8.87564	−0.427525	−0.213762	+	0.976886i	$0.568572\pi$
$432$	0	0
$433$	−15.5167	−0.745683	−0.372842	−	0.927895i	$-0.621617\pi$
$433$	−15.5167	−0.745683	−0.372842	+	0.927895i	$0.621617\pi$
$434$	−8.19615	−0.393428
$435$	0	0
$436$	−2.53590	−0.121448
$437$	−2.92820	−0.140075
$438$	0	0
$439$	−35.7128	−1.70448	−0.852240	−	0.523151i	$-0.824756\pi$
$439$	−35.7128	−1.70448	−0.852240	+	0.523151i	$0.824756\pi$
$440$	0	0
$441$	0	0
$442$	−3.21539	−0.152941
$443$	30.3923	1.44398	0.721991	−	0.691902i	$-0.243227\pi$
$443$	30.3923	1.44398	0.721991	+	0.691902i	$0.243227\pi$
$444$	0	0
$445$	0	0
$446$	0.679492	0.0321749
$447$	0	0
$448$	4.73205	0.223568
$449$	−40.7321	−1.92226	−0.961132	−	0.276089i	$-0.910962\pi$
$449$	−40.7321	−1.92226	−0.961132	+	0.276089i	$0.910962\pi$
$450$	0	0
$451$	−5.07180	−0.238822
$452$	−4.92820	−0.231803
$453$	0	0
$454$	23.0718	1.08281
$455$	0	0
$456$	0	0
$457$	−32.5885	−1.52442	−0.762212	−	0.647328i	$-0.775887\pi$
$457$	−32.5885	−1.52442	−0.762212	+	0.647328i	$0.775887\pi$
$458$	32.5359	1.52030
$459$	0	0
$460$	0	0
$461$	0.732051	0.0340950	0.0170475	−	0.999855i	$-0.494573\pi$
$461$	0.732051	0.0340950	0.0170475	+	0.999855i	$0.494573\pi$
$462$	0	0
$463$	−39.7128	−1.84561	−0.922805	−	0.385266i	$-0.874110\pi$
$463$	−39.7128	−1.84561	−0.922805	+	0.385266i	$0.874110\pi$
$464$	3.66025	0.169923
$465$	0	0
$466$	18.9282	0.876832
$467$	2.00000	0.0925490	0.0462745	−	0.998929i	$-0.485265\pi$
$467$	2.00000	0.0925490	0.0462745	+	0.998929i	$0.485265\pi$
$468$	0	0
$469$	48.2487	2.22792
$470$	0	0
$471$	0	0
$472$	−21.1244	−0.972327
$473$	5.85641	0.269278
$474$	0	0
$475$	0	0
$476$	−6.92820	−0.317554
$477$	0	0
$478$	24.0000	1.09773
$479$	27.8038	1.27039	0.635195	−	0.772352i	$-0.280920\pi$
$479$	27.8038	1.27039	0.635195	+	0.772352i	$0.280920\pi$
$480$	0	0
$481$	8.53590	0.389203
$482$	41.3205	1.88210
$483$	0	0
$484$	−8.85641	−0.402564
$485$	0	0
$486$	0	0
$487$	−10.5359	−0.477427	−0.238714	−	0.971090i	$-0.576726\pi$
$487$	−10.5359	−0.477427	−0.238714	+	0.971090i	$0.576726\pi$
$488$	18.0000	0.814822
$489$	0	0
$490$	0	0
$491$	−24.3923	−1.10081	−0.550405	−	0.834898i	$-0.685527\pi$
$491$	−24.3923	−1.10081	−0.550405	+	0.834898i	$0.685527\pi$
$492$	0	0
$493$	1.07180	0.0482713
$494$	12.0000	0.539906
$495$	0	0
$496$	5.00000	0.224507
$497$	18.0000	0.807410
$498$	0	0
$499$	−14.9282	−0.668278	−0.334139	−	0.942524i	$-0.608446\pi$
$499$	−14.9282	−0.668278	−0.334139	+	0.942524i	$0.608446\pi$
$500$	0	0
$501$	0	0
$502$	−5.07180	−0.226365
$503$	−11.8564	−0.528651	−0.264326	−	0.964434i	$-0.585149\pi$
$503$	−11.8564	−0.528651	−0.264326	+	0.964434i	$0.585149\pi$
$504$	0	0
$505$	0	0
$506$	−1.35898	−0.0604142
$507$	0	0
$508$	10.5359	0.467455
$509$	−16.7321	−0.741635	−0.370818	−	0.928706i	$-0.620922\pi$
$509$	−16.7321	−0.741635	−0.370818	+	0.928706i	$0.620922\pi$
$510$	0	0
$511$	−26.7846	−1.18488
$512$	8.66025	0.382733
$513$	0	0
$514$	−17.0718	−0.753005
$515$	0	0
$516$	0	0
$517$	−8.78461	−0.386347
$518$	55.1769	2.42433
$519$	0	0
$520$	0	0
$521$	−21.3205	−0.934068	−0.467034	−	0.884239i	$-0.654678\pi$
$521$	−21.3205	−0.934068	−0.467034	+	0.884239i	$0.654678\pi$
$522$	0	0
$523$	11.6077	0.507569	0.253785	−	0.967261i	$-0.418325\pi$
$523$	11.6077	0.507569	0.253785	+	0.967261i	$0.418325\pi$
$524$	−6.73205	−0.294091
$525$	0	0
$526$	−29.5692	−1.28928
$527$	1.46410	0.0637773
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	0	0
$532$	25.8564	1.12102
$533$	−4.39230	−0.190252
$534$	0	0
$535$	0	0
$536$	−17.6603	−0.762807
$537$	0	0
$538$	−16.9808	−0.732093
$539$	22.5359	0.970690
$540$	0	0
$541$	−6.53590	−0.281000	−0.140500	−	0.990081i	$-0.544871\pi$
$541$	−6.53590	−0.281000	−0.140500	+	0.990081i	$0.544871\pi$
$542$	−29.0718	−1.24874
$543$	0	0
$544$	7.60770	0.326177
$545$	0	0
$546$	0	0
$547$	34.1962	1.46212	0.731061	−	0.682312i	$-0.239025\pi$
$547$	34.1962	1.46212	0.731061	+	0.682312i	$0.239025\pi$
$548$	18.9282	0.808573
$549$	0	0
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	−47.3205	−2.01227
$554$	26.1962	1.11297
$555$	0	0
$556$	9.85641	0.418005
$557$	−9.46410	−0.401007	−0.200503	−	0.979693i	$-0.564258\pi$
$557$	−9.46410	−0.401007	−0.200503	+	0.979693i	$0.564258\pi$
$558$	0	0
$559$	5.07180	0.214514
$560$	0	0
$561$	0	0
$562$	28.6410	1.20815
$563$	40.6410	1.71281	0.856407	−	0.516301i	$-0.172691\pi$
$563$	40.6410	1.71281	0.856407	+	0.516301i	$0.172691\pi$
$564$	0	0
$565$	0	0
$566$	−12.5885	−0.529132
$567$	0	0
$568$	−6.58846	−0.276446
$569$	28.7321	1.20451	0.602255	−	0.798304i	$-0.294269\pi$
$569$	28.7321	1.20451	0.602255	+	0.798304i	$0.294269\pi$
$570$	0	0
$571$	−23.7128	−0.992350	−0.496175	−	0.868222i	$-0.665263\pi$
$571$	−23.7128	−0.992350	−0.496175	+	0.868222i	$0.665263\pi$
$572$	1.85641	0.0776203
$573$	0	0
$574$	−28.3923	−1.18507
$575$	0	0
$576$	0	0
$577$	−5.32051	−0.221496	−0.110748	−	0.993849i	$-0.535325\pi$
$577$	−5.32051	−0.221496	−0.110748	+	0.993849i	$0.535325\pi$
$578$	−25.7321	−1.07031
$579$	0	0
$580$	0	0
$581$	73.1769	3.03589
$582$	0	0
$583$	18.1436	0.751431
$584$	9.80385	0.405686
$585$	0	0
$586$	8.78461	0.362889
$587$	10.9282	0.451055	0.225528	−	0.974237i	$-0.427589\pi$
$587$	10.9282	0.451055	0.225528	+	0.974237i	$0.427589\pi$
$588$	0	0
$589$	−5.46410	−0.225144
$590$	0	0
$591$	0	0
$592$	−33.6603	−1.38343
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	0	0
$596$	−21.3205	−0.873322
$597$	0	0
$598$	−1.17691	−0.0481276
$599$	−41.3731	−1.69046	−0.845229	−	0.534405i	$-0.820536\pi$
$599$	−41.3731	−1.69046	−0.845229	+	0.534405i	$0.820536\pi$
$600$	0	0
$601$	−37.3205	−1.52234	−0.761168	−	0.648555i	$-0.775374\pi$
$601$	−37.3205	−1.52234	−0.761168	+	0.648555i	$0.775374\pi$
$602$	32.7846	1.33620
$603$	0	0
$604$	−20.9282	−0.851557
$605$	0	0
$606$	0	0
$607$	14.8756	0.603784	0.301892	−	0.953342i	$-0.402382\pi$
$607$	14.8756	0.603784	0.301892	+	0.953342i	$0.402382\pi$
$608$	−28.3923	−1.15146
$609$	0	0
$610$	0	0
$611$	−7.60770	−0.307774
$612$	0	0
$613$	10.4449	0.421864	0.210932	−	0.977501i	$-0.432350\pi$
$613$	10.4449	0.421864	0.210932	+	0.977501i	$0.432350\pi$
$614$	−46.0526	−1.85853
$615$	0	0
$616$	−12.0000	−0.483494
$617$	−34.7846	−1.40038	−0.700188	−	0.713959i	$-0.746900\pi$
$617$	−34.7846	−1.40038	−0.700188	+	0.713959i	$0.746900\pi$
$618$	0	0
$619$	15.8564	0.637323	0.318661	−	0.947869i	$-0.396767\pi$
$619$	15.8564	0.637323	0.318661	+	0.947869i	$0.396767\pi$
$620$	0	0
$621$	0	0
$622$	−23.6603	−0.948690
$623$	−34.3923	−1.37790
$624$	0	0
$625$	0	0
$626$	37.5167	1.49947
$627$	0	0
$628$	12.9282	0.515891
$629$	−9.85641	−0.393001
$630$	0	0
$631$	2.92820	0.116570	0.0582850	−	0.998300i	$-0.481437\pi$
$631$	2.92820	0.116570	0.0582850	+	0.998300i	$0.481437\pi$
$632$	17.3205	0.688973
$633$	0	0
$634$	−39.7128	−1.57720
$635$	0	0
$636$	0	0
$637$	19.5167	0.773278
$638$	−1.85641	−0.0734958
$639$	0	0
$640$	0	0
$641$	49.5167	1.95579	0.977895	−	0.209095i	$-0.0670519\pi$
$641$	49.5167	1.95579	0.977895	+	0.209095i	$0.0670519\pi$
$642$	0	0
$643$	−46.2487	−1.82387	−0.911936	−	0.410333i	$-0.865412\pi$
$643$	−46.2487	−1.82387	−0.911936	+	0.410333i	$0.865412\pi$
$644$	−2.53590	−0.0999284
$645$	0	0
$646$	−13.8564	−0.545173
$647$	3.21539	0.126410	0.0632050	−	0.998001i	$-0.479868\pi$
$647$	3.21539	0.126410	0.0632050	+	0.998001i	$0.479868\pi$
$648$	0	0
$649$	17.8564	0.700925
$650$	0	0
$651$	0	0
$652$	14.1962	0.555964
$653$	−25.8564	−1.01184	−0.505920	−	0.862581i	$-0.668847\pi$
$653$	−25.8564	−1.01184	−0.505920	+	0.862581i	$0.668847\pi$
$654$	0	0
$655$	0	0
$656$	17.3205	0.676252
$657$	0	0
$658$	−49.1769	−1.91712
$659$	39.1244	1.52407	0.762034	−	0.647537i	$-0.224201\pi$
$659$	39.1244	1.52407	0.762034	+	0.647537i	$0.224201\pi$
$660$	0	0
$661$	−44.6410	−1.73633	−0.868167	−	0.496272i	$-0.834702\pi$
$661$	−44.6410	−1.73633	−0.868167	+	0.496272i	$0.834702\pi$
$662$	43.1769	1.67812
$663$	0	0
$664$	−26.7846	−1.03944
$665$	0	0
$666$	0	0
$667$	0.392305	0.0151901
$668$	13.8564	0.536120
$669$	0	0
$670$	0	0
$671$	−15.2154	−0.587384
$672$	0	0
$673$	9.26795	0.357253	0.178627	−	0.983917i	$-0.442835\pi$
$673$	9.26795	0.357253	0.178627	+	0.983917i	$0.442835\pi$
$674$	−31.2679	−1.20440
$675$	0	0
$676$	−11.3923	−0.438166
$677$	28.3923	1.09120	0.545602	−	0.838044i	$-0.316301\pi$
$677$	28.3923	1.09120	0.545602	+	0.838044i	$0.316301\pi$
$678$	0	0
$679$	9.46410	0.363199
$680$	0	0
$681$	0	0
$682$	−2.53590	−0.0971046
$683$	−19.1769	−0.733784	−0.366892	−	0.930263i	$-0.619578\pi$
$683$	−19.1769	−0.733784	−0.366892	+	0.930263i	$0.619578\pi$
$684$	0	0
$685$	0	0
$686$	68.7846	2.62621
$687$	0	0
$688$	−20.0000	−0.762493
$689$	15.7128	0.598610
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−22.7846	−0.866141
$693$	0	0
$694$	−22.1436	−0.840559
$695$	0	0
$696$	0	0
$697$	5.07180	0.192108
$698$	11.3205	0.428488
$699$	0	0
$700$	0	0
$701$	−6.78461	−0.256251	−0.128126	−	0.991758i	$-0.540896\pi$
$701$	−6.78461	−0.256251	−0.128126	+	0.991758i	$0.540896\pi$
$702$	0	0
$703$	36.7846	1.38736
$704$	1.46410	0.0551804
$705$	0	0
$706$	−7.60770	−0.286319
$707$	−21.4641	−0.807241
$708$	0	0
$709$	32.9282	1.23664	0.618322	−	0.785925i	$-0.287813\pi$
$709$	32.9282	1.23664	0.618322	+	0.785925i	$0.287813\pi$
$710$	0	0
$711$	0	0
$712$	12.5885	0.471772
$713$	0.535898	0.0200696
$714$	0	0
$715$	0	0
$716$	9.07180	0.339029
$717$	0	0
$718$	2.87564	0.107318
$719$	9.46410	0.352951	0.176476	−	0.984305i	$-0.443530\pi$
$719$	9.46410	0.352951	0.176476	+	0.984305i	$0.443530\pi$
$720$	0	0
$721$	−15.4641	−0.575913
$722$	18.8038	0.699807
$723$	0	0
$724$	−10.3923	−0.386227
$725$	0	0
$726$	0	0
$727$	−17.5167	−0.649657	−0.324828	−	0.945773i	$-0.605307\pi$
$727$	−17.5167	−0.649657	−0.324828	+	0.945773i	$0.605307\pi$
$728$	−10.3923	−0.385164
$729$	0	0
$730$	0	0
$731$	−5.85641	−0.216607
$732$	0	0
$733$	−8.92820	−0.329771	−0.164885	−	0.986313i	$-0.552725\pi$
$733$	−8.92820	−0.329771	−0.164885	+	0.986313i	$0.552725\pi$
$734$	37.8564	1.39731
$735$	0	0
$736$	2.78461	0.102642
$737$	14.9282	0.549887
$738$	0	0
$739$	−39.5692	−1.45558	−0.727789	−	0.685802i	$-0.759452\pi$
$739$	−39.5692	−1.45558	−0.727789	+	0.685802i	$0.759452\pi$
$740$	0	0
$741$	0	0
$742$	101.569	3.72872
$743$	48.4974	1.77920	0.889599	−	0.456743i	$-0.150984\pi$
$743$	48.4974	1.77920	0.889599	+	0.456743i	$0.150984\pi$
$744$	0	0
$745$	0	0
$746$	−45.7128	−1.67366
$747$	0	0
$748$	−2.14359	−0.0783775
$749$	−75.0333	−2.74166
$750$	0	0
$751$	−8.78461	−0.320555	−0.160277	−	0.987072i	$-0.551239\pi$
$751$	−8.78461	−0.320555	−0.160277	+	0.987072i	$0.551239\pi$
$752$	30.0000	1.09399
$753$	0	0
$754$	−1.60770	−0.0585488
$755$	0	0
$756$	0	0
$757$	−22.7321	−0.826210	−0.413105	−	0.910683i	$-0.635556\pi$
$757$	−22.7321	−0.826210	−0.413105	+	0.910683i	$0.635556\pi$
$758$	56.1051	2.03783
$759$	0	0
$760$	0	0
$761$	31.6603	1.14768	0.573842	−	0.818966i	$-0.305452\pi$
$761$	31.6603	1.14768	0.573842	+	0.818966i	$0.305452\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	11.8038	0.427048
$765$	0	0
$766$	−23.0718	−0.833618
$767$	15.4641	0.558376
$768$	0	0
$769$	−36.3923	−1.31234	−0.656170	−	0.754613i	$-0.727825\pi$
$769$	−36.3923	−1.31234	−0.656170	+	0.754613i	$0.727825\pi$
$770$	0	0
$771$	0	0
$772$	23.8564	0.858611
$773$	−17.1769	−0.617811	−0.308905	−	0.951093i	$-0.599963\pi$
$773$	−17.1769	−0.617811	−0.308905	+	0.951093i	$0.599963\pi$
$774$	0	0
$775$	0	0
$776$	−3.46410	−0.124354
$777$	0	0
$778$	−21.3731	−0.766262
$779$	−18.9282	−0.678173
$780$	0	0
$781$	5.56922	0.199282
$782$	1.35898	0.0485972
$783$	0	0
$784$	−76.9615	−2.74863
$785$	0	0
$786$	0	0
$787$	53.4641	1.90579	0.952895	−	0.303301i	$-0.0980889\pi$
$787$	53.4641	1.90579	0.952895	+	0.303301i	$0.0980889\pi$
$788$	13.0718	0.465663
$789$	0	0
$790$	0	0
$791$	−23.3205	−0.829182
$792$	0	0
$793$	−13.1769	−0.467926
$794$	−41.3205	−1.46641
$795$	0	0
$796$	16.9282	0.600004
$797$	−1.85641	−0.0657573	−0.0328786	−	0.999459i	$-0.510467\pi$
$797$	−1.85641	−0.0657573	−0.0328786	+	0.999459i	$0.510467\pi$
$798$	0	0
$799$	8.78461	0.310777
$800$	0	0
$801$	0	0
$802$	−16.4833	−0.582047
$803$	−8.28719	−0.292448
$804$	0	0
$805$	0	0
$806$	−2.19615	−0.0773562
$807$	0	0
$808$	7.85641	0.276387
$809$	28.8372	1.01386	0.506930	−	0.861987i	$-0.330780\pi$
$809$	28.8372	1.01386	0.506930	+	0.861987i	$0.330780\pi$
$810$	0	0
$811$	−30.2487	−1.06218	−0.531088	−	0.847317i	$-0.678217\pi$
$811$	−30.2487	−1.06218	−0.531088	+	0.847317i	$0.678217\pi$
$812$	−3.46410	−0.121566
$813$	0	0
$814$	17.0718	0.598366
$815$	0	0
$816$	0	0
$817$	21.8564	0.764659
$818$	−30.0000	−1.04893
$819$	0	0
$820$	0	0
$821$	−17.1244	−0.597644	−0.298822	−	0.954309i	$-0.596594\pi$
$821$	−17.1244	−0.597644	−0.298822	+	0.954309i	$0.596594\pi$
$822$	0	0
$823$	23.7128	0.826577	0.413288	−	0.910600i	$-0.364380\pi$
$823$	23.7128	0.826577	0.413288	+	0.910600i	$0.364380\pi$
$824$	5.66025	0.197184
$825$	0	0
$826$	99.9615	3.47811
$827$	−34.1051	−1.18595	−0.592976	−	0.805220i	$-0.702047\pi$
$827$	−34.1051	−1.18595	−0.592976	+	0.805220i	$0.702047\pi$
$828$	0	0
$829$	−54.1051	−1.87915	−0.939574	−	0.342345i	$-0.888779\pi$
$829$	−54.1051	−1.87915	−0.939574	+	0.342345i	$0.888779\pi$
$830$	0	0
$831$	0	0
$832$	1.26795	0.0439582
$833$	−22.5359	−0.780823
$834$	0	0
$835$	0	0
$836$	8.00000	0.276686
$837$	0	0
$838$	46.9808	1.62292
$839$	44.5885	1.53936	0.769682	−	0.638427i	$-0.220415\pi$
$839$	44.5885	1.53936	0.769682	+	0.638427i	$0.220415\pi$
$840$	0	0
$841$	−28.4641	−0.981521
$842$	−26.5359	−0.914487
$843$	0	0
$844$	7.32051	0.251982
$845$	0	0
$846$	0	0
$847$	−41.9090	−1.44001
$848$	−61.9615	−2.12777
$849$	0	0
$850$	0	0
$851$	−3.60770	−0.123670
$852$	0	0
$853$	−52.2487	−1.78896	−0.894481	−	0.447106i	$-0.852455\pi$
$853$	−52.2487	−1.78896	−0.894481	+	0.447106i	$0.852455\pi$
$854$	−85.1769	−2.91469
$855$	0	0
$856$	27.4641	0.938704
$857$	9.71281	0.331783	0.165892	−	0.986144i	$-0.446950\pi$
$857$	9.71281	0.331783	0.165892	+	0.986144i	$0.446950\pi$
$858$	0	0
$859$	39.7128	1.35498	0.677492	−	0.735530i	$-0.263067\pi$
$859$	39.7128	1.35498	0.677492	+	0.735530i	$0.263067\pi$
$860$	0	0
$861$	0	0
$862$	−15.3731	−0.523609
$863$	27.7128	0.943355	0.471678	−	0.881771i	$-0.343649\pi$
$863$	27.7128	0.943355	0.471678	+	0.881771i	$0.343649\pi$
$864$	0	0
$865$	0	0
$866$	−26.8756	−0.913272
$867$	0	0
$868$	−4.73205	−0.160616
$869$	−14.6410	−0.496662
$870$	0	0
$871$	12.9282	0.438055
$872$	4.39230	0.148742
$873$	0	0
$874$	−5.07180	−0.171556
$875$	0	0
$876$	0	0
$877$	34.1051	1.15165	0.575824	−	0.817574i	$-0.304681\pi$
$877$	34.1051	1.15165	0.575824	+	0.817574i	$0.304681\pi$
$878$	−61.8564	−2.08755
$879$	0	0
$880$	0	0
$881$	−48.8372	−1.64537	−0.822683	−	0.568500i	$-0.807524\pi$
$881$	−48.8372	−1.64537	−0.822683	+	0.568500i	$0.807524\pi$
$882$	0	0
$883$	19.3205	0.650187	0.325093	−	0.945682i	$-0.394604\pi$
$883$	19.3205	0.650187	0.325093	+	0.945682i	$0.394604\pi$
$884$	−1.85641	−0.0624377
$885$	0	0
$886$	52.6410	1.76851
$887$	−15.4641	−0.519234	−0.259617	−	0.965712i	$-0.583596\pi$
$887$	−15.4641	−0.519234	−0.259617	+	0.965712i	$0.583596\pi$
$888$	0	0
$889$	49.8564	1.67213
$890$	0	0
$891$	0	0
$892$	0.392305	0.0131353
$893$	−32.7846	−1.09710
$894$	0	0
$895$	0	0
$896$	57.3731	1.91670
$897$	0	0
$898$	−70.5500	−2.35428
$899$	0.732051	0.0244153
$900$	0	0
$901$	−18.1436	−0.604451
$902$	−8.78461	−0.292496
$903$	0	0
$904$	8.53590	0.283900
$905$	0	0
$906$	0	0
$907$	−20.7321	−0.688396	−0.344198	−	0.938897i	$-0.611849\pi$
$907$	−20.7321	−0.688396	−0.344198	+	0.938897i	$0.611849\pi$
$908$	13.3205	0.442057
$909$	0	0
$910$	0	0
$911$	9.46410	0.313560	0.156780	−	0.987634i	$-0.449889\pi$
$911$	9.46410	0.313560	0.156780	+	0.987634i	$0.449889\pi$
$912$	0	0
$913$	22.6410	0.749308
$914$	−56.4449	−1.86703
$915$	0	0
$916$	18.7846	0.620661
$917$	−31.8564	−1.05199
$918$	0	0
$919$	−10.5359	−0.347547	−0.173774	−	0.984786i	$-0.555596\pi$
$919$	−10.5359	−0.347547	−0.173774	+	0.984786i	$0.555596\pi$
$920$	0	0
$921$	0	0
$922$	1.26795	0.0417577
$923$	4.82309	0.158754
$924$	0	0
$925$	0	0
$926$	−68.7846	−2.26040
$927$	0	0
$928$	3.80385	0.124867
$929$	50.1962	1.64688	0.823441	−	0.567402i	$-0.192051\pi$
$929$	50.1962	1.64688	0.823441	+	0.567402i	$0.192051\pi$
$930$	0	0
$931$	84.1051	2.75643
$932$	10.9282	0.357965
$933$	0	0
$934$	3.46410	0.113349
$935$	0	0
$936$	0	0
$937$	0.143594	0.00469100	0.00234550	−	0.999997i	$-0.499253\pi$
$937$	0.143594	0.00469100	0.00234550	+	0.999997i	$0.499253\pi$
$938$	83.5692	2.72863
$939$	0	0
$940$	0	0
$941$	6.58846	0.214778	0.107389	−	0.994217i	$-0.465751\pi$
$941$	6.58846	0.214778	0.107389	+	0.994217i	$0.465751\pi$
$942$	0	0
$943$	1.85641	0.0604529
$944$	−60.9808	−1.98475
$945$	0	0
$946$	10.1436	0.329797
$947$	23.1769	0.753149	0.376574	−	0.926386i	$-0.377102\pi$
$947$	23.1769	0.753149	0.376574	+	0.926386i	$0.377102\pi$
$948$	0	0
$949$	−7.17691	−0.232973
$950$	0	0
$951$	0	0
$952$	12.0000	0.388922
$953$	−44.7846	−1.45072	−0.725358	−	0.688372i	$-0.758326\pi$
$953$	−44.7846	−1.45072	−0.725358	+	0.688372i	$0.758326\pi$
$954$	0	0
$955$	0	0
$956$	13.8564	0.448148
$957$	0	0
$958$	48.1577	1.55590
$959$	89.5692	2.89234
$960$	0	0
$961$	1.00000	0.0322581
$962$	14.7846	0.476675
$963$	0	0
$964$	23.8564	0.768363
$965$	0	0
$966$	0	0
$967$	20.6795	0.665008	0.332504	−	0.943102i	$-0.392107\pi$
$967$	20.6795	0.665008	0.332504	+	0.943102i	$0.392107\pi$
$968$	15.3397	0.493038
$969$	0	0
$970$	0	0
$971$	−27.8038	−0.892268	−0.446134	−	0.894966i	$-0.647200\pi$
$971$	−27.8038	−0.892268	−0.446134	+	0.894966i	$0.647200\pi$
$972$	0	0
$973$	46.6410	1.49524
$974$	−18.2487	−0.584726
$975$	0	0
$976$	51.9615	1.66325
$977$	−20.9282	−0.669553	−0.334776	−	0.942298i	$-0.608661\pi$
$977$	−20.9282	−0.669553	−0.334776	+	0.942298i	$0.608661\pi$
$978$	0	0
$979$	−10.6410	−0.340088
$980$	0	0
$981$	0	0
$982$	−42.2487	−1.34821
$983$	−45.5692	−1.45343	−0.726716	−	0.686938i	$-0.758954\pi$
$983$	−45.5692	−1.45343	−0.726716	+	0.686938i	$0.758954\pi$
$984$	0	0
$985$	0	0
$986$	1.85641	0.0591200
$987$	0	0
$988$	6.92820	0.220416
$989$	−2.14359	−0.0681623
$990$	0	0
$991$	18.1436	0.576350	0.288175	−	0.957578i	$-0.406951\pi$
$991$	18.1436	0.576350	0.288175	+	0.957578i	$0.406951\pi$
$992$	5.19615	0.164978
$993$	0	0
$994$	31.1769	0.988872
$995$	0	0
$996$	0	0
$997$	29.6077	0.937685	0.468843	−	0.883282i	$-0.344671\pi$
$997$	29.6077	0.937685	0.468843	+	0.883282i	$0.344671\pi$
$998$	−25.8564	−0.818470
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.v.1.2		2
3.2	odd	2		2325.2.a.m.1.1		2
5.4	even	2		1395.2.a.f.1.1		2
15.2	even	4		2325.2.c.j.1024.1		4
15.8	even	4		2325.2.c.j.1024.4		4
15.14	odd	2		465.2.a.d.1.2	✓	2
60.59	even	2		7440.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.d.1.2	✓	2	15.14	odd	2
1395.2.a.f.1.1		2	5.4	even	2
2325.2.a.m.1.1		2	3.2	odd	2
2325.2.c.j.1024.1		4	15.2	even	4
2325.2.c.j.1024.4		4	15.8	even	4
6975.2.a.v.1.2		2	1.1	even	1	trivial
7440.2.a.bk.1.2		2	60.59	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)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +4.73205 q^{7} -1.73205 q^{8} +1.46410 q^{11} +1.26795 q^{13} +8.19615 q^{14} -5.00000 q^{16} -1.46410 q^{17} +5.46410 q^{19} +2.53590 q^{22} -0.535898 q^{23} +2.19615 q^{26} +4.73205 q^{28} -0.732051 q^{29} -1.00000 q^{31} -5.19615 q^{32} -2.53590 q^{34} +6.73205 q^{37} +9.46410 q^{38} -3.46410 q^{41} +4.00000 q^{43} +1.46410 q^{44} -0.928203 q^{46} -6.00000 q^{47} +15.3923 q^{49} +1.26795 q^{52} +12.3923 q^{53} -8.19615 q^{56} -1.26795 q^{58} +12.1962 q^{59} -10.3923 q^{61} -1.73205 q^{62} +1.00000 q^{64} +10.1962 q^{67} -1.46410 q^{68} +3.80385 q^{71} -5.66025 q^{73} +11.6603 q^{74} +5.46410 q^{76} +6.92820 q^{77} -10.0000 q^{79} -6.00000 q^{82} +15.4641 q^{83} +6.92820 q^{86} -2.53590 q^{88} -7.26795 q^{89} +6.00000 q^{91} -0.535898 q^{92} -10.3923 q^{94} +2.00000 q^{97} +26.6603 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{7} - 4 q^{11} + 6 q^{13} + 6 q^{14} - 10 q^{16} + 4 q^{17} + 4 q^{19} + 12 q^{22} - 8 q^{23} - 6 q^{26} + 6 q^{28} + 2 q^{29} - 2 q^{31} - 12 q^{34} + 10 q^{37} + 12 q^{38} + 8 q^{43}+ \cdots + 36 q^{98}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^7 - 4 * q^11 + 6 * q^13 + 6 * q^14 - 10 * q^16 + 4 * q^17 + 4 * q^19 + 12 * q^22 - 8 * q^23 - 6 * q^26 + 6 * q^28 + 2 * q^29 - 2 * q^31 - 12 * q^34 + 10 * q^37 + 12 * q^38 + 8 * q^43 - 4 * q^44 + 12 * q^46 - 12 * q^47 + 10 * q^49 + 6 * q^52 + 4 * q^53 - 6 * q^56 - 6 * q^58 + 14 * q^59 + 2 * q^64 + 10 * q^67 + 4 * q^68 + 18 * q^71 + 6 * q^73 + 6 * q^74 + 4 * q^76 - 20 * q^79 - 12 * q^82 + 24 * q^83 - 12 * q^88 - 18 * q^89 + 12 * q^91 - 8 * q^92 + 4 * q^97 + 36 * q^98

Introduction

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