LMFDB - Embedded newform 6975.2.a.bf.1.1

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)

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")

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);

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 = [3,1,0,5,0,0,-8,3,0,0,-2,0,-4,8,0,5,4] 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:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 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			1.1
Root			$-2.08613$ 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)

$f(q)$	$=$	$q-2.08613 q^{2} +2.35194 q^{4} -5.08613 q^{7} -0.734191 q^{8} +4.17226 q^{11} +1.08613 q^{13} +10.6103 q^{14} -3.17226 q^{16} +0.648061 q^{17} -2.70388 q^{19} -8.70388 q^{22} +7.52420 q^{23} -2.26581 q^{26} -11.9623 q^{28} -5.90645 q^{29} +1.00000 q^{31} +8.08613 q^{32} -1.35194 q^{34} -0.913870 q^{37} +5.64064 q^{38} -2.17226 q^{41} -8.17226 q^{43} +9.81290 q^{44} -15.6965 q^{46} -10.2281 q^{47} +18.8687 q^{49} +2.55451 q^{52} +5.52420 q^{53} +3.73419 q^{56} +12.3216 q^{58} -0.438069 q^{59} -2.00000 q^{61} -2.08613 q^{62} -10.5242 q^{64} -9.79001 q^{67} +1.52420 q^{68} -2.96969 q^{71} +1.25839 q^{73} +1.90645 q^{74} -6.35936 q^{76} -21.2207 q^{77} +10.8203 q^{79} +4.53162 q^{82} +14.2281 q^{83} +17.0484 q^{86} -3.06324 q^{88} -18.1903 q^{89} -5.52420 q^{91} +17.6965 q^{92} +21.3371 q^{94} +13.2207 q^{97} -39.3626 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 8 * q^7 + 3 * q^8 - 2 * q^11 - 4 * q^13 + 8 * q^14 + 5 * q^16 + 4 * q^17 - 4 * q^19 - 22 * q^22 + 6 * q^23 - 12 * q^26 - 10 * q^28 + 2 * q^29 + 3 * q^31 + 17 * q^32 - 2 * q^34 - 10 * q^37 - 8 * q^38 + 8 * q^41 - 10 * q^43 - 10 * q^44 - 16 * q^46 - 10 * q^47 + 11 * q^49 - 10 * q^52 + 6 * q^56 + 36 * q^58 + 8 * q^59 - 6 * q^61 + q^62 - 15 * q^64 - 18 * q^67 - 12 * q^68 - 10 * q^71 - 18 * q^73 - 14 * q^74 - 44 * q^76 - 16 * q^77 + 20 * q^79 + 24 * q^82 + 22 * q^83 + 18 * q^86 - 30 * q^88 - 8 * q^89 + 22 * q^92 + 8 * q^94 - 8 * q^97 - 57 * 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 $

$2$	−2.08613	−1.47512	−0.737558	−	0.675283i	$-0.764021\pi$
$2$	−2.08613	−1.47512	−0.737558	+	0.675283i	$0.764021\pi$
$3$	0	0
$3$	0	0
$4$	2.35194	1.17597
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−5.08613	−1.92238	−0.961188	−	0.275893i	$-0.911026\pi$
$7$	−5.08613	−1.92238	−0.961188	+	0.275893i	$0.911026\pi$
$8$	−0.734191	−0.259576
$9$	0	0
$10$	0	0
$11$	4.17226	1.25798	0.628992	−	0.777412i	$-0.283468\pi$
$11$	4.17226	1.25798	0.628992	+	0.777412i	$0.283468\pi$
$12$	0	0
$13$	1.08613	0.301238	0.150619	−	0.988592i	$-0.451873\pi$
$13$	1.08613	0.301238	0.150619	+	0.988592i	$0.451873\pi$
$14$	10.6103	2.83573
$15$	0	0
$16$	−3.17226	−0.793065
$17$	0.648061	0.157178	0.0785889	−	0.996907i	$-0.474959\pi$
$17$	0.648061	0.157178	0.0785889	+	0.996907i	$0.474959\pi$
$18$	0	0
$19$	−2.70388	−0.620312	−0.310156	−	0.950686i	$-0.600381\pi$
$19$	−2.70388	−0.620312	−0.310156	+	0.950686i	$0.600381\pi$
$20$	0	0
$21$	0	0
$22$	−8.70388	−1.85567
$23$	7.52420	1.56890	0.784452	−	0.620189i	$-0.212944\pi$
$23$	7.52420	1.56890	0.784452	+	0.620189i	$0.212944\pi$
$24$	0	0
$25$	0	0
$26$	−2.26581	−0.444362
$27$	0	0
$28$	−11.9623	−2.26066
$29$	−5.90645	−1.09680	−0.548400	−	0.836216i	$-0.684763\pi$
$29$	−5.90645	−1.09680	−0.548400	+	0.836216i	$0.684763\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	8.08613	1.42944
$33$	0	0
$34$	−1.35194	−0.231856
$35$	0	0
$36$	0	0
$37$	−0.913870	−0.150239	−0.0751196	−	0.997175i	$-0.523934\pi$
$37$	−0.913870	−0.150239	−0.0751196	+	0.997175i	$0.523934\pi$
$38$	5.64064	0.915033
$39$	0	0
$40$	0	0
$41$	−2.17226	−0.339250	−0.169625	−	0.985509i	$-0.554256\pi$
$41$	−2.17226	−0.339250	−0.169625	+	0.985509i	$0.554256\pi$
$42$	0	0
$43$	−8.17226	−1.24626	−0.623129	−	0.782119i	$-0.714139\pi$
$43$	−8.17226	−1.24626	−0.623129	+	0.782119i	$0.714139\pi$
$44$	9.81290	1.47935
$45$	0	0
$46$	−15.6965	−2.31432
$47$	−10.2281	−1.49192	−0.745959	−	0.665992i	$-0.768009\pi$
$47$	−10.2281	−1.49192	−0.745959	+	0.665992i	$0.768009\pi$
$48$	0	0
$49$	18.8687	2.69553
$50$	0	0
$51$	0	0
$52$	2.55451	0.354247
$53$	5.52420	0.758807	0.379404	−	0.925231i	$-0.376129\pi$
$53$	5.52420	0.758807	0.379404	+	0.925231i	$0.376129\pi$
$54$	0	0
$55$	0	0
$56$	3.73419	0.499002
$57$	0	0
$58$	12.3216	1.61791
$59$	−0.438069	−0.0570318	−0.0285159	−	0.999593i	$-0.509078\pi$
$59$	−0.438069	−0.0570318	−0.0285159	+	0.999593i	$0.509078\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−2.08613	−0.264939
$63$	0	0
$64$	−10.5242	−1.31552
$65$	0	0
$66$	0	0
$67$	−9.79001	−1.19604	−0.598020	−	0.801481i	$-0.704046\pi$
$67$	−9.79001	−1.19604	−0.598020	+	0.801481i	$0.704046\pi$
$68$	1.52420	0.184836
$69$	0	0
$70$	0	0
$71$	−2.96969	−0.352437	−0.176219	−	0.984351i	$-0.556387\pi$
$71$	−2.96969	−0.352437	−0.176219	+	0.984351i	$0.556387\pi$
$72$	0	0
$73$	1.25839	0.147283	0.0736417	−	0.997285i	$-0.476538\pi$
$73$	1.25839	0.147283	0.0736417	+	0.997285i	$0.476538\pi$
$74$	1.90645	0.221620
$75$	0	0
$76$	−6.35936	−0.729468
$77$	−21.2207	−2.41832
$78$	0	0
$79$	10.8203	1.21738	0.608691	−	0.793408i	$-0.291695\pi$
$79$	10.8203	1.21738	0.608691	+	0.793408i	$0.291695\pi$
$80$	0	0
$81$	0	0
$82$	4.53162	0.500433
$83$	14.2281	1.56173	0.780867	−	0.624697i	$-0.214778\pi$
$83$	14.2281	1.56173	0.780867	+	0.624697i	$0.214778\pi$
$84$	0	0
$85$	0	0
$86$	17.0484	1.83838
$87$	0	0
$88$	−3.06324	−0.326542
$89$	−18.1903	−1.92817	−0.964086	−	0.265589i	$-0.914434\pi$
$89$	−18.1903	−1.92817	−0.964086	+	0.265589i	$0.914434\pi$
$90$	0	0
$91$	−5.52420	−0.579093
$92$	17.6965	1.84498
$93$	0	0
$94$	21.3371	2.20075
$95$	0	0
$96$	0	0
$97$	13.2207	1.34235	0.671177	−	0.741297i	$-0.265789\pi$
$97$	13.2207	1.34235	0.671177	+	0.741297i	$0.265789\pi$
$98$	−39.3626	−3.97622
$99$	0	0
$100$	0	0
$101$	16.8761	1.67924	0.839619	−	0.543175i	$-0.182778\pi$
$101$	16.8761	1.67924	0.839619	+	0.543175i	$0.182778\pi$
$102$	0	0
$103$	0.493887	0.0486641	0.0243321	−	0.999704i	$-0.492254\pi$
$103$	0.493887	0.0486641	0.0243321	+	0.999704i	$0.492254\pi$
$104$	−0.797427	−0.0781942
$105$	0	0
$106$	−11.5242	−1.11933
$107$	9.69646	0.937392	0.468696	−	0.883359i	$-0.344724\pi$
$107$	9.69646	0.937392	0.468696	+	0.883359i	$0.344724\pi$
$108$	0	0
$109$	−16.4003	−1.57087	−0.785434	−	0.618946i	$-0.787560\pi$
$109$	−16.4003	−1.57087	−0.785434	+	0.618946i	$0.787560\pi$
$110$	0	0
$111$	0	0
$112$	16.1345	1.52457
$113$	12.5168	1.17748	0.588740	−	0.808323i	$-0.299624\pi$
$113$	12.5168	1.17748	0.588740	+	0.808323i	$0.299624\pi$
$114$	0	0
$115$	0	0
$116$	−13.8916	−1.28980
$117$	0	0
$118$	0.913870	0.0841285
$119$	−3.29612	−0.302155
$120$	0	0
$121$	6.40776	0.582523
$122$	4.17226	0.377739
$123$	0	0
$124$	2.35194	0.211210
$125$	0	0
$126$	0	0
$127$	7.58002	0.672618	0.336309	−	0.941752i	$-0.390821\pi$
$127$	7.58002	0.672618	0.336309	+	0.941752i	$0.390821\pi$
$128$	5.78259	0.511114
$129$	0	0
$130$	0	0
$131$	7.67357	0.670443	0.335221	−	0.942139i	$-0.391189\pi$
$131$	7.67357	0.670443	0.335221	+	0.942139i	$0.391189\pi$
$132$	0	0
$133$	13.7523	1.19247
$134$	20.4232	1.76430
$135$	0	0
$136$	−0.475800	−0.0407995
$137$	−4.75970	−0.406648	−0.203324	−	0.979111i	$-0.565175\pi$
$137$	−4.75970	−0.406648	−0.203324	+	0.979111i	$0.565175\pi$
$138$	0	0
$139$	10.2839	0.872269	0.436134	−	0.899882i	$-0.356347\pi$
$139$	10.2839	0.872269	0.436134	+	0.899882i	$0.356347\pi$
$140$	0	0
$141$	0	0
$142$	6.19515	0.519886
$143$	4.53162	0.378953
$144$	0	0
$145$	0	0
$146$	−2.62517	−0.217260
$147$	0	0
$148$	−2.14937	−0.176677
$149$	0.764504	0.0626306	0.0313153	−	0.999510i	$-0.490030\pi$
$149$	0.764504	0.0626306	0.0313153	+	0.999510i	$0.490030\pi$
$150$	0	0
$151$	−0.992582	−0.0807751	−0.0403876	−	0.999184i	$-0.512859\pi$
$151$	−0.992582	−0.0807751	−0.0403876	+	0.999184i	$0.512859\pi$
$152$	1.98516	0.161018
$153$	0	0
$154$	44.2691	3.56730
$155$	0	0
$156$	0	0
$157$	−15.9245	−1.27092	−0.635458	−	0.772135i	$-0.719189\pi$
$157$	−15.9245	−1.27092	−0.635458	+	0.772135i	$0.719189\pi$
$158$	−22.5726	−1.79578
$159$	0	0
$160$	0	0
$161$	−38.2691	−3.01602
$162$	0	0
$163$	1.25839	0.0985648	0.0492824	−	0.998785i	$-0.484307\pi$
$163$	1.25839	0.0985648	0.0492824	+	0.998785i	$0.484307\pi$
$164$	−5.10902	−0.398948
$165$	0	0
$166$	−29.6816	−2.30374
$167$	23.5800	1.82468	0.912338	−	0.409437i	$-0.134275\pi$
$167$	23.5800	1.82468	0.912338	+	0.409437i	$0.134275\pi$
$168$	0	0
$169$	−11.8203	−0.909255
$170$	0	0
$171$	0	0
$172$	−19.2207	−1.46556
$173$	−20.1723	−1.53367	−0.766834	−	0.641845i	$-0.778169\pi$
$173$	−20.1723	−1.53367	−0.766834	+	0.641845i	$0.778169\pi$
$174$	0	0
$175$	0	0
$176$	−13.2355	−0.997663
$177$	0	0
$178$	37.9474	2.84428
$179$	4.11164	0.307318	0.153659	−	0.988124i	$-0.450894\pi$
$179$	4.11164	0.307318	0.153659	+	0.988124i	$0.450894\pi$
$180$	0	0
$181$	10.3445	0.768902	0.384451	−	0.923145i	$-0.374391\pi$
$181$	10.3445	0.768902	0.384451	+	0.923145i	$0.374391\pi$
$182$	11.5242	0.854231
$183$	0	0
$184$	−5.52420	−0.407249
$185$	0	0
$186$	0	0
$187$	2.70388	0.197727
$188$	−24.0558	−1.75445
$189$	0	0
$190$	0	0
$191$	−3.56193	−0.257732	−0.128866	−	0.991662i	$-0.541134\pi$
$191$	−3.56193	−0.257732	−0.128866	+	0.991662i	$0.541134\pi$
$192$	0	0
$193$	3.58002	0.257695	0.128848	−	0.991664i	$-0.458872\pi$
$193$	3.58002	0.257695	0.128848	+	0.991664i	$0.458872\pi$
$194$	−27.5800	−1.98013
$195$	0	0
$196$	44.3781	3.16986
$197$	7.35194	0.523804	0.261902	−	0.965094i	$-0.415650\pi$
$197$	7.35194	0.523804	0.261902	+	0.965094i	$0.415650\pi$
$198$	0	0
$199$	25.1042	1.77959	0.889795	−	0.456360i	$-0.150847\pi$
$199$	25.1042	1.77959	0.889795	+	0.456360i	$0.150847\pi$
$200$	0	0
$201$	0	0
$202$	−35.2058	−2.47707
$203$	30.0410	2.10846
$204$	0	0
$205$	0	0
$206$	−1.03031	−0.0717853
$207$	0	0
$208$	−3.44549	−0.238902
$209$	−11.2813	−0.780343
$210$	0	0
$211$	−14.4051	−0.991691	−0.495846	−	0.868411i	$-0.665142\pi$
$211$	−14.4051	−0.991691	−0.495846	+	0.868411i	$0.665142\pi$
$212$	12.9926	0.892334
$213$	0	0
$214$	−20.2281	−1.38276
$215$	0	0
$216$	0	0
$217$	−5.08613	−0.345269
$218$	34.2132	2.31721
$219$	0	0
$220$	0	0
$221$	0.703878	0.0473480
$222$	0	0
$223$	−11.4078	−0.763920	−0.381960	−	0.924179i	$-0.624751\pi$
$223$	−11.4078	−0.763920	−0.381960	+	0.924179i	$0.624751\pi$
$224$	−41.1271	−2.74792
$225$	0	0
$226$	−26.1116	−1.73692
$227$	−15.3371	−1.01796	−0.508980	−	0.860779i	$-0.669977\pi$
$227$	−15.3371	−1.01796	−0.508980	+	0.860779i	$0.669977\pi$
$228$	0	0
$229$	−15.8277	−1.04593	−0.522963	−	0.852355i	$-0.675174\pi$
$229$	−15.8277	−1.04593	−0.522963	+	0.852355i	$0.675174\pi$
$230$	0	0
$231$	0	0
$232$	4.33646	0.284703
$233$	−27.9293	−1.82971	−0.914856	−	0.403780i	$-0.867696\pi$
$233$	−27.9293	−1.82971	−0.914856	+	0.403780i	$0.867696\pi$
$234$	0	0
$235$	0	0
$236$	−1.03031	−0.0670676
$237$	0	0
$238$	6.87614	0.445714
$239$	10.4562	0.676353	0.338176	−	0.941083i	$-0.390190\pi$
$239$	10.4562	0.676353	0.338176	+	0.941083i	$0.390190\pi$
$240$	0	0
$241$	−6.87614	−0.442931	−0.221466	−	0.975168i	$-0.571084\pi$
$241$	−6.87614	−0.442931	−0.221466	+	0.975168i	$0.571084\pi$
$242$	−13.3674	−0.859290
$243$	0	0
$244$	−4.70388	−0.301135
$245$	0	0
$246$	0	0
$247$	−2.93676	−0.186862
$248$	−0.734191	−0.0466212
$249$	0	0
$250$	0	0
$251$	−7.39292	−0.466637	−0.233318	−	0.972400i	$-0.574958\pi$
$251$	−7.39292	−0.466637	−0.233318	+	0.972400i	$0.574958\pi$
$252$	0	0
$253$	31.3929	1.97366
$254$	−15.8129	−0.992190
$255$	0	0
$256$	8.98516	0.561573
$257$	−6.28870	−0.392279	−0.196139	−	0.980576i	$-0.562840\pi$
$257$	−6.28870	−0.392279	−0.196139	+	0.980576i	$0.562840\pi$
$258$	0	0
$259$	4.64806	0.288816
$260$	0	0
$261$	0	0
$262$	−16.0081	−0.988981
$263$	−14.6284	−0.902027	−0.451013	−	0.892517i	$-0.648937\pi$
$263$	−14.6284	−0.902027	−0.451013	+	0.892517i	$0.648937\pi$
$264$	0	0
$265$	0	0
$266$	−28.6890	−1.75904
$267$	0	0
$268$	−23.0255	−1.40651
$269$	10.4381	0.636420	0.318210	−	0.948020i	$-0.396918\pi$
$269$	10.4381	0.636420	0.318210	+	0.948020i	$0.396918\pi$
$270$	0	0
$271$	−32.1574	−1.95342	−0.976712	−	0.214554i	$-0.931170\pi$
$271$	−32.1574	−1.95342	−0.976712	+	0.214554i	$0.931170\pi$
$272$	−2.05582	−0.124652
$273$	0	0
$274$	9.92935	0.599854
$275$	0	0
$276$	0	0
$277$	32.5397	1.95512	0.977560	−	0.210658i	$-0.0675607\pi$
$277$	32.5397	1.95512	0.977560	+	0.210658i	$0.0675607\pi$
$278$	−21.4535	−1.28670
$279$	0	0
$280$	0	0
$281$	3.82774	0.228344	0.114172	−	0.993461i	$-0.463579\pi$
$281$	3.82774	0.228344	0.114172	+	0.993461i	$0.463579\pi$
$282$	0	0
$283$	−6.03773	−0.358906	−0.179453	−	0.983767i	$-0.557433\pi$
$283$	−6.03773	−0.358906	−0.179453	+	0.983767i	$0.557433\pi$
$284$	−6.98452	−0.414455
$285$	0	0
$286$	−9.45355	−0.559000
$287$	11.0484	0.652166
$288$	0	0
$289$	−16.5800	−0.975295
$290$	0	0
$291$	0	0
$292$	2.95966	0.173201
$293$	9.33710	0.545479	0.272740	−	0.962088i	$-0.412070\pi$
$293$	9.33710	0.545479	0.272740	+	0.962088i	$0.412070\pi$
$294$	0	0
$295$	0	0
$296$	0.670955	0.0389985
$297$	0	0
$298$	−1.59485	−0.0923874
$299$	8.17226	0.472614
$300$	0	0
$301$	41.5652	2.39578
$302$	2.07065	0.119153
$303$	0	0
$304$	8.57741	0.491948
$305$	0	0
$306$	0	0
$307$	−26.4184	−1.50778	−0.753890	−	0.657001i	$-0.771825\pi$
$307$	−26.4184	−1.50778	−0.753890	+	0.657001i	$0.771825\pi$
$308$	−49.9097	−2.84387
$309$	0	0
$310$	0	0
$311$	6.36261	0.360790	0.180395	−	0.983594i	$-0.442262\pi$
$311$	6.36261	0.360790	0.180395	+	0.983594i	$0.442262\pi$
$312$	0	0
$313$	−8.02289	−0.453481	−0.226740	−	0.973955i	$-0.572807\pi$
$313$	−8.02289	−0.453481	−0.226740	+	0.973955i	$0.572807\pi$
$314$	33.2207	1.87475
$315$	0	0
$316$	25.4487	1.43160
$317$	4.75970	0.267331	0.133666	−	0.991026i	$-0.457325\pi$
$317$	4.75970	0.267331	0.133666	+	0.991026i	$0.457325\pi$
$318$	0	0
$319$	−24.6433	−1.37976
$320$	0	0
$321$	0	0
$322$	79.8342	4.44899
$323$	−1.75228	−0.0974993
$324$	0	0
$325$	0	0
$326$	−2.62517	−0.145395
$327$	0	0
$328$	1.59485	0.0880611
$329$	52.0213	2.86803
$330$	0	0
$331$	−21.9804	−1.20815	−0.604075	−	0.796928i	$-0.706457\pi$
$331$	−21.9804	−1.20815	−0.604075	+	0.796928i	$0.706457\pi$
$332$	33.4636	1.83655
$333$	0	0
$334$	−49.1910	−2.69161
$335$	0	0
$336$	0	0
$337$	−28.1952	−1.53589	−0.767944	−	0.640517i	$-0.778720\pi$
$337$	−28.1952	−1.53589	−0.767944	+	0.640517i	$0.778720\pi$
$338$	24.6587	1.34126
$339$	0	0
$340$	0	0
$341$	4.17226	0.225941
$342$	0	0
$343$	−60.3659	−3.25945
$344$	6.00000	0.323498
$345$	0	0
$346$	42.0820	2.26234
$347$	17.1090	0.918461	0.459230	−	0.888317i	$-0.348125\pi$
$347$	17.1090	0.918461	0.459230	+	0.888317i	$0.348125\pi$
$348$	0	0
$349$	−2.53643	−0.135772	−0.0678859	−	0.997693i	$-0.521625\pi$
$349$	−2.53643	−0.135772	−0.0678859	+	0.997693i	$0.521625\pi$
$350$	0	0
$351$	0	0
$352$	33.7374	1.79821
$353$	−25.2255	−1.34262	−0.671308	−	0.741178i	$-0.734267\pi$
$353$	−25.2255	−1.34262	−0.671308	+	0.741178i	$0.734267\pi$
$354$	0	0
$355$	0	0
$356$	−42.7826	−2.26747
$357$	0	0
$358$	−8.57741	−0.453330
$359$	−29.7704	−1.57122	−0.785610	−	0.618722i	$-0.787651\pi$
$359$	−29.7704	−1.57122	−0.785610	+	0.618722i	$0.787651\pi$
$360$	0	0
$361$	−11.6890	−0.615213
$362$	−21.5800	−1.13422
$363$	0	0
$364$	−12.9926	−0.680996
$365$	0	0
$366$	0	0
$367$	12.3445	0.644379	0.322189	−	0.946675i	$-0.395581\pi$
$367$	12.3445	0.644379	0.322189	+	0.946675i	$0.395581\pi$
$368$	−23.8687	−1.24424
$369$	0	0
$370$	0	0
$371$	−28.0968	−1.45871
$372$	0	0
$373$	−34.3807	−1.78016	−0.890082	−	0.455800i	$-0.849353\pi$
$373$	−34.3807	−1.78016	−0.890082	+	0.455800i	$0.849353\pi$
$374$	−5.64064	−0.291671
$375$	0	0
$376$	7.50936	0.387266
$377$	−6.41518	−0.330398
$378$	0	0
$379$	22.3297	1.14700	0.573499	−	0.819206i	$-0.305585\pi$
$379$	22.3297	1.14700	0.573499	+	0.819206i	$0.305585\pi$
$380$	0	0
$381$	0	0
$382$	7.43065	0.380185
$383$	15.4126	0.787545	0.393773	−	0.919208i	$-0.371170\pi$
$383$	15.4126	0.787545	0.393773	+	0.919208i	$0.371170\pi$
$384$	0	0
$385$	0	0
$386$	−7.46838	−0.380131
$387$	0	0
$388$	31.0942	1.57857
$389$	14.9549	0.758241	0.379121	−	0.925347i	$-0.376227\pi$
$389$	14.9549	0.758241	0.379121	+	0.925347i	$0.376227\pi$
$390$	0	0
$391$	4.87614	0.246597
$392$	−13.8532	−0.699695
$393$	0	0
$394$	−15.3371	−0.772672
$395$	0	0
$396$	0	0
$397$	33.0484	1.65865	0.829326	−	0.558765i	$-0.188725\pi$
$397$	33.0484	1.65865	0.829326	+	0.558765i	$0.188725\pi$
$398$	−52.3707	−2.62510
$399$	0	0
$400$	0	0
$401$	−3.56193	−0.177874	−0.0889372	−	0.996037i	$-0.528347\pi$
$401$	−3.56193	−0.177874	−0.0889372	+	0.996037i	$0.528347\pi$
$402$	0	0
$403$	1.08613	0.0541040
$404$	39.6917	1.97473
$405$	0	0
$406$	−62.6694	−3.11023
$407$	−3.81290	−0.188999
$408$	0	0
$409$	−13.6555	−0.675220	−0.337610	−	0.941286i	$-0.609618\pi$
$409$	−13.6555	−0.675220	−0.337610	+	0.941286i	$0.609618\pi$
$410$	0	0
$411$	0	0
$412$	1.16159	0.0572275
$413$	2.22808	0.109637
$414$	0	0
$415$	0	0
$416$	8.78259	0.430602
$417$	0	0
$418$	23.5342	1.15110
$419$	17.7948	0.869334	0.434667	−	0.900591i	$-0.356866\pi$
$419$	17.7948	0.869334	0.434667	+	0.900591i	$0.356866\pi$
$420$	0	0
$421$	8.70869	0.424435	0.212218	−	0.977222i	$-0.431931\pi$
$421$	8.70869	0.424435	0.212218	+	0.977222i	$0.431931\pi$
$422$	30.0510	1.46286
$423$	0	0
$424$	−4.05582	−0.196968
$425$	0	0
$426$	0	0
$427$	10.1723	0.492270
$428$	22.8055	1.10234
$429$	0	0
$430$	0	0
$431$	16.2658	0.783496	0.391748	−	0.920072i	$-0.371870\pi$
$431$	16.2658	0.783496	0.391748	+	0.920072i	$0.371870\pi$
$432$	0	0
$433$	7.85063	0.377277	0.188639	−	0.982047i	$-0.439592\pi$
$433$	7.85063	0.377277	0.188639	+	0.982047i	$0.439592\pi$
$434$	10.6103	0.509312
$435$	0	0
$436$	−38.5726	−1.84729
$437$	−20.3445	−0.973210
$438$	0	0
$439$	16.2233	0.774294	0.387147	−	0.922018i	$-0.373461\pi$
$439$	16.2233	0.774294	0.387147	+	0.922018i	$0.373461\pi$
$440$	0	0
$441$	0	0
$442$	−1.46838	−0.0698438
$443$	20.7449	0.985618	0.492809	−	0.870138i	$-0.335970\pi$
$443$	20.7449	0.985618	0.492809	+	0.870138i	$0.335970\pi$
$444$	0	0
$445$	0	0
$446$	23.7981	1.12687
$447$	0	0
$448$	53.5274	2.52893
$449$	24.7826	1.16956	0.584781	−	0.811191i	$-0.301180\pi$
$449$	24.7826	1.16956	0.584781	+	0.811191i	$0.301180\pi$
$450$	0	0
$451$	−9.06324	−0.426771
$452$	29.4387	1.38468
$453$	0	0
$454$	31.9952	1.50161
$455$	0	0
$456$	0	0
$457$	−27.8868	−1.30449	−0.652245	−	0.758008i	$-0.726173\pi$
$457$	−27.8868	−1.30449	−0.652245	+	0.758008i	$0.726173\pi$
$458$	33.0187	1.54286
$459$	0	0
$460$	0	0
$461$	29.4110	1.36981	0.684904	−	0.728634i	$-0.259844\pi$
$461$	29.4110	1.36981	0.684904	+	0.728634i	$0.259844\pi$
$462$	0	0
$463$	25.1239	1.16760	0.583802	−	0.811896i	$-0.301564\pi$
$463$	25.1239	1.16760	0.583802	+	0.811896i	$0.301564\pi$
$464$	18.7368	0.869834
$465$	0	0
$466$	58.2643	2.69904
$467$	−34.8055	−1.61061	−0.805303	−	0.592864i	$-0.797997\pi$
$467$	−34.8055	−1.61061	−0.805303	+	0.592864i	$0.797997\pi$
$468$	0	0
$469$	49.7933	2.29924
$470$	0	0
$471$	0	0
$472$	0.321627	0.0148041
$473$	−34.0968	−1.56777
$474$	0	0
$475$	0	0
$476$	−7.75228	−0.355325
$477$	0	0
$478$	−21.8129	−0.997699
$479$	−35.7704	−1.63439	−0.817195	−	0.576362i	$-0.804472\pi$
$479$	−35.7704	−1.63439	−0.817195	+	0.576362i	$0.804472\pi$
$480$	0	0
$481$	−0.992582	−0.0452578
$482$	14.3445	0.653375
$483$	0	0
$484$	15.0707	0.685030
$485$	0	0
$486$	0	0
$487$	21.5652	0.977212	0.488606	−	0.872505i	$-0.337506\pi$
$487$	21.5652	0.977212	0.488606	+	0.872505i	$0.337506\pi$
$488$	1.46838	0.0664705
$489$	0	0
$490$	0	0
$491$	−26.7497	−1.20720	−0.603598	−	0.797289i	$-0.706267\pi$
$491$	−26.7497	−1.20720	−0.603598	+	0.797289i	$0.706267\pi$
$492$	0	0
$493$	−3.82774	−0.172393
$494$	6.12647	0.275643
$495$	0	0
$496$	−3.17226	−0.142439
$497$	15.1042	0.677517
$498$	0	0
$499$	−10.9368	−0.489597	−0.244798	−	0.969574i	$-0.578722\pi$
$499$	−10.9368	−0.489597	−0.244798	+	0.969574i	$0.578722\pi$
$500$	0	0
$501$	0	0
$502$	15.4226	0.688344
$503$	−18.2281	−0.812750	−0.406375	−	0.913706i	$-0.633207\pi$
$503$	−18.2281	−0.812750	−0.406375	+	0.913706i	$0.633207\pi$
$504$	0	0
$505$	0	0
$506$	−65.4897	−2.91137
$507$	0	0
$508$	17.8277	0.790978
$509$	−10.6103	−0.470295	−0.235147	−	0.971960i	$-0.575557\pi$
$509$	−10.6103	−0.470295	−0.235147	+	0.971960i	$0.575557\pi$
$510$	0	0
$511$	−6.40034	−0.283134
$512$	−30.3094	−1.33950
$513$	0	0
$514$	13.1191	0.578657
$515$	0	0
$516$	0	0
$517$	−42.6742	−1.87681
$518$	−9.69646	−0.426038
$519$	0	0
$520$	0	0
$521$	25.6768	1.12492	0.562461	−	0.826824i	$-0.309855\pi$
$521$	25.6768	1.12492	0.562461	+	0.826824i	$0.309855\pi$
$522$	0	0
$523$	−20.1574	−0.881423	−0.440711	−	0.897649i	$-0.645274\pi$
$523$	−20.1574	−0.881423	−0.440711	+	0.897649i	$0.645274\pi$
$524$	18.0478	0.788420
$525$	0	0
$526$	30.5168	1.33059
$527$	0.648061	0.0282300
$528$	0	0
$529$	33.6136	1.46146
$530$	0	0
$531$	0	0
$532$	32.3445	1.40231
$533$	−2.35936	−0.102195
$534$	0	0
$535$	0	0
$536$	7.18774	0.310463
$537$	0	0
$538$	−21.7752	−0.938794
$539$	78.7252	3.39094
$540$	0	0
$541$	30.4610	1.30962	0.654810	−	0.755794i	$-0.272749\pi$
$541$	30.4610	1.30962	0.654810	+	0.755794i	$0.272749\pi$
$542$	67.0846	2.88153
$543$	0	0
$544$	5.24030	0.224676
$545$	0	0
$546$	0	0
$547$	11.8506	0.506697	0.253348	−	0.967375i	$-0.418468\pi$
$547$	11.8506	0.506697	0.253348	+	0.967375i	$0.418468\pi$
$548$	−11.1945	−0.478206
$549$	0	0
$550$	0	0
$551$	15.9703	0.680359
$552$	0	0
$553$	−55.0336	−2.34027
$554$	−67.8820	−2.88403
$555$	0	0
$556$	24.1871	1.02576
$557$	−3.95902	−0.167749	−0.0838745	−	0.996476i	$-0.526729\pi$
$557$	−3.95902	−0.167749	−0.0838745	+	0.996476i	$0.526729\pi$
$558$	0	0
$559$	−8.87614	−0.375421
$560$	0	0
$561$	0	0
$562$	−7.98516	−0.336834
$563$	−13.6965	−0.577237	−0.288618	−	0.957444i	$-0.593196\pi$
$563$	−13.6965	−0.577237	−0.288618	+	0.957444i	$0.593196\pi$
$564$	0	0
$565$	0	0
$566$	12.5955	0.529428
$567$	0	0
$568$	2.18032	0.0914841
$569$	−21.4865	−0.900760	−0.450380	−	0.892837i	$-0.648711\pi$
$569$	−21.4865	−0.900760	−0.450380	+	0.892837i	$0.648711\pi$
$570$	0	0
$571$	−3.46838	−0.145147	−0.0725736	−	0.997363i	$-0.523121\pi$
$571$	−3.46838	−0.145147	−0.0725736	+	0.997363i	$0.523121\pi$
$572$	10.6581	0.445637
$573$	0	0
$574$	−23.0484	−0.962022
$575$	0	0
$576$	0	0
$577$	−16.9516	−0.705704	−0.352852	−	0.935679i	$-0.614788\pi$
$577$	−16.9516	−0.705704	−0.352852	+	0.935679i	$0.614788\pi$
$578$	34.5881	1.43867
$579$	0	0
$580$	0	0
$581$	−72.3659	−3.00224
$582$	0	0
$583$	23.0484	0.954567
$584$	−0.923899	−0.0382312
$585$	0	0
$586$	−19.4784	−0.804646
$587$	6.97294	0.287804	0.143902	−	0.989592i	$-0.454035\pi$
$587$	6.97294	0.287804	0.143902	+	0.989592i	$0.454035\pi$
$588$	0	0
$589$	−2.70388	−0.111411
$590$	0	0
$591$	0	0
$592$	2.89903	0.119150
$593$	−27.9097	−1.14611	−0.573057	−	0.819515i	$-0.694243\pi$
$593$	−27.9097	−1.14611	−0.573057	+	0.819515i	$0.694243\pi$
$594$	0	0
$595$	0	0
$596$	1.79807	0.0736517
$597$	0	0
$598$	−17.0484	−0.697161
$599$	−30.4381	−1.24367	−0.621833	−	0.783150i	$-0.713612\pi$
$599$	−30.4381	−1.24367	−0.621833	+	0.783150i	$0.713612\pi$
$600$	0	0
$601$	−25.7013	−1.04838	−0.524188	−	0.851602i	$-0.675631\pi$
$601$	−25.7013	−1.04838	−0.524188	+	0.851602i	$0.675631\pi$
$602$	−86.7104	−3.53405
$603$	0	0
$604$	−2.33449	−0.0949891
$605$	0	0
$606$	0	0
$607$	22.7119	0.921849	0.460925	−	0.887439i	$-0.347518\pi$
$607$	22.7119	0.921849	0.460925	+	0.887439i	$0.347518\pi$
$608$	−21.8639	−0.886699
$609$	0	0
$610$	0	0
$611$	−11.1090	−0.449423
$612$	0	0
$613$	−30.4939	−1.23164	−0.615818	−	0.787888i	$-0.711174\pi$
$613$	−30.4939	−1.23164	−0.615818	+	0.787888i	$0.711174\pi$
$614$	55.1123	2.22415
$615$	0	0
$616$	15.5800	0.627737
$617$	10.6890	0.430325	0.215162	−	0.976578i	$-0.430972\pi$
$617$	10.6890	0.430325	0.215162	+	0.976578i	$0.430972\pi$
$618$	0	0
$619$	−12.3397	−0.495975	−0.247987	−	0.968763i	$-0.579769\pi$
$619$	−12.3397	−0.495975	−0.247987	+	0.968763i	$0.579769\pi$
$620$	0	0
$621$	0	0
$622$	−13.2732	−0.532208
$623$	92.5185	3.70667
$624$	0	0
$625$	0	0
$626$	16.7368	0.668937
$627$	0	0
$628$	−37.4535	−1.49456
$629$	−0.592243	−0.0236143
$630$	0	0
$631$	−23.1600	−0.921986	−0.460993	−	0.887404i	$-0.652507\pi$
$631$	−23.1600	−0.921986	−0.460993	+	0.887404i	$0.652507\pi$
$632$	−7.94418	−0.316003
$633$	0	0
$634$	−9.92935	−0.394345
$635$	0	0
$636$	0	0
$637$	20.4939	0.811997
$638$	51.4090	2.03530
$639$	0	0
$640$	0	0
$641$	−1.63739	−0.0646731	−0.0323366	−	0.999477i	$-0.510295\pi$
$641$	−1.63739	−0.0646731	−0.0323366	+	0.999477i	$0.510295\pi$
$642$	0	0
$643$	0.531618	0.0209650	0.0104825	−	0.999945i	$-0.496663\pi$
$643$	0.531618	0.0209650	0.0104825	+	0.999945i	$0.496663\pi$
$644$	−90.0065	−3.54675
$645$	0	0
$646$	3.65548	0.143823
$647$	−9.86391	−0.387790	−0.193895	−	0.981022i	$-0.562112\pi$
$647$	−9.86391	−0.387790	−0.193895	+	0.981022i	$0.562112\pi$
$648$	0	0
$649$	−1.82774	−0.0717451
$650$	0	0
$651$	0	0
$652$	2.95966	0.115909
$653$	−43.8081	−1.71434	−0.857172	−	0.515031i	$-0.827780\pi$
$653$	−43.8081	−1.71434	−0.857172	+	0.515031i	$0.827780\pi$
$654$	0	0
$655$	0	0
$656$	6.89098	0.269047
$657$	0	0
$658$	−108.523	−4.23068
$659$	−34.0639	−1.32694	−0.663470	−	0.748203i	$-0.730917\pi$
$659$	−34.0639	−1.32694	−0.663470	+	0.748203i	$0.730917\pi$
$660$	0	0
$661$	21.3567	0.830681	0.415341	−	0.909666i	$-0.363662\pi$
$661$	21.3567	0.830681	0.415341	+	0.909666i	$0.363662\pi$
$662$	45.8539	1.78216
$663$	0	0
$664$	−10.4461	−0.405388
$665$	0	0
$666$	0	0
$667$	−44.4413	−1.72077
$668$	55.4588	2.14576
$669$	0	0
$670$	0	0
$671$	−8.34452	−0.322137
$672$	0	0
$673$	−28.8990	−1.11398	−0.556988	−	0.830521i	$-0.688043\pi$
$673$	−28.8990	−1.11398	−0.556988	+	0.830521i	$0.688043\pi$
$674$	58.8188	2.26561
$675$	0	0
$676$	−27.8007	−1.06926
$677$	−27.8539	−1.07051	−0.535256	−	0.844690i	$-0.679785\pi$
$677$	−27.8539	−1.07051	−0.535256	+	0.844690i	$0.679785\pi$
$678$	0	0
$679$	−67.2420	−2.58051
$680$	0	0
$681$	0	0
$682$	−8.70388	−0.333289
$683$	15.7113	0.601176	0.300588	−	0.953754i	$-0.402817\pi$
$683$	15.7113	0.601176	0.300588	+	0.953754i	$0.402817\pi$
$684$	0	0
$685$	0	0
$686$	125.931	4.80807
$687$	0	0
$688$	25.9245	0.988364
$689$	6.00000	0.228582
$690$	0	0
$691$	7.92454	0.301464	0.150732	−	0.988575i	$-0.451837\pi$
$691$	7.92454	0.301464	0.150732	+	0.988575i	$0.451837\pi$
$692$	−47.4439	−1.80355
$693$	0	0
$694$	−35.6917	−1.35484
$695$	0	0
$696$	0	0
$697$	−1.40776	−0.0533226
$698$	5.29131	0.200279
$699$	0	0
$700$	0	0
$701$	−34.0968	−1.28782	−0.643909	−	0.765102i	$-0.722689\pi$
$701$	−34.0968	−1.28782	−0.643909	+	0.765102i	$0.722689\pi$
$702$	0	0
$703$	2.47099	0.0931953
$704$	−43.9097	−1.65491
$705$	0	0
$706$	52.6236	1.98052
$707$	−85.8342	−3.22813
$708$	0	0
$709$	−46.8826	−1.76071	−0.880357	−	0.474311i	$-0.842697\pi$
$709$	−46.8826	−1.76071	−0.880357	+	0.474311i	$0.842697\pi$
$710$	0	0
$711$	0	0
$712$	13.3552	0.500507
$713$	7.52420	0.281783
$714$	0	0
$715$	0	0
$716$	9.67032	0.361397
$717$	0	0
$718$	62.1049	2.31773
$719$	4.51678	0.168448	0.0842238	−	0.996447i	$-0.473159\pi$
$719$	4.51678	0.168448	0.0842238	+	0.996447i	$0.473159\pi$
$720$	0	0
$721$	−2.51197	−0.0935508
$722$	24.3849	0.907511
$723$	0	0
$724$	24.3297	0.904206
$725$	0	0
$726$	0	0
$727$	−4.72677	−0.175306	−0.0876531	−	0.996151i	$-0.527937\pi$
$727$	−4.72677	−0.175306	−0.0876531	+	0.996151i	$0.527937\pi$
$728$	4.05582	0.150319
$729$	0	0
$730$	0	0
$731$	−5.29612	−0.195884
$732$	0	0
$733$	1.87875	0.0693932	0.0346966	−	0.999398i	$-0.488954\pi$
$733$	1.87875	0.0693932	0.0346966	+	0.999398i	$0.488954\pi$
$734$	−25.7523	−0.950534
$735$	0	0
$736$	60.8417	2.24265
$737$	−40.8465	−1.50460
$738$	0	0
$739$	−16.1526	−0.594184	−0.297092	−	0.954849i	$-0.596017\pi$
$739$	−16.1526	−0.594184	−0.297092	+	0.954849i	$0.596017\pi$
$740$	0	0
$741$	0	0
$742$	58.6136	2.15177
$743$	−23.3175	−0.855435	−0.427717	−	0.903913i	$-0.640682\pi$
$743$	−23.3175	−0.855435	−0.427717	+	0.903913i	$0.640682\pi$
$744$	0	0
$745$	0	0
$746$	71.7226	2.62595
$747$	0	0
$748$	6.35936	0.232521
$749$	−49.3175	−1.80202
$750$	0	0
$751$	−3.46838	−0.126563	−0.0632815	−	0.997996i	$-0.520157\pi$
$751$	−3.46838	−0.126563	−0.0632815	+	0.997996i	$0.520157\pi$
$752$	32.4461	1.18319
$753$	0	0
$754$	13.3829	0.487376
$755$	0	0
$756$	0	0
$757$	−41.9326	−1.52407	−0.762033	−	0.647538i	$-0.775799\pi$
$757$	−41.9326	−1.52407	−0.762033	+	0.647538i	$0.775799\pi$
$758$	−46.5826	−1.69196
$759$	0	0
$760$	0	0
$761$	13.1877	0.478055	0.239028	−	0.971013i	$-0.423171\pi$
$761$	13.1877	0.478055	0.239028	+	0.971013i	$0.423171\pi$
$762$	0	0
$763$	83.4143	3.01980
$764$	−8.37744	−0.303085
$765$	0	0
$766$	−32.1526	−1.16172
$767$	−0.475800	−0.0171802
$768$	0	0
$769$	−47.6816	−1.71944	−0.859722	−	0.510763i	$-0.829363\pi$
$769$	−47.6816	−1.71944	−0.859722	+	0.510763i	$0.829363\pi$
$770$	0	0
$771$	0	0
$772$	8.41998	0.303042
$773$	30.6236	1.10145	0.550727	−	0.834685i	$-0.314350\pi$
$773$	30.6236	1.10145	0.550727	+	0.834685i	$0.314350\pi$
$774$	0	0
$775$	0	0
$776$	−9.70649	−0.348443
$777$	0	0
$778$	−31.1978	−1.11849
$779$	5.87353	0.210441
$780$	0	0
$781$	−12.3903	−0.443360
$782$	−10.1723	−0.363759
$783$	0	0
$784$	−59.8565	−2.13773
$785$	0	0
$786$	0	0
$787$	−26.9219	−0.959663	−0.479832	−	0.877361i	$-0.659302\pi$
$787$	−26.9219	−0.959663	−0.479832	+	0.877361i	$0.659302\pi$
$788$	17.2913	0.615978
$789$	0	0
$790$	0	0
$791$	−63.6620	−2.26356
$792$	0	0
$793$	−2.17226	−0.0771392
$794$	−68.9433	−2.44670
$795$	0	0
$796$	59.0436	2.09274
$797$	−23.9197	−0.847280	−0.423640	−	0.905831i	$-0.639248\pi$
$797$	−23.9197	−0.847280	−0.423640	+	0.905831i	$0.639248\pi$
$798$	0	0
$799$	−6.62842	−0.234497
$800$	0	0
$801$	0	0
$802$	7.43065	0.262385
$803$	5.25033	0.185280
$804$	0	0
$805$	0	0
$806$	−2.26581	−0.0798097
$807$	0	0
$808$	−12.3903	−0.435890
$809$	−34.8336	−1.22468	−0.612342	−	0.790593i	$-0.709772\pi$
$809$	−34.8336	−1.22468	−0.612342	+	0.790593i	$0.709772\pi$
$810$	0	0
$811$	−37.8277	−1.32831	−0.664156	−	0.747594i	$-0.731209\pi$
$811$	−37.8277	−1.32831	−0.664156	+	0.747594i	$0.731209\pi$
$812$	70.6546	2.47949
$813$	0	0
$814$	7.95421	0.278795
$815$	0	0
$816$	0	0
$817$	22.0968	0.773069
$818$	28.4871	0.996028
$819$	0	0
$820$	0	0
$821$	31.6439	1.10438	0.552190	−	0.833718i	$-0.313792\pi$
$821$	31.6439	1.10438	0.552190	+	0.833718i	$0.313792\pi$
$822$	0	0
$823$	−16.9729	−0.591639	−0.295820	−	0.955244i	$-0.595593\pi$
$823$	−16.9729	−0.591639	−0.295820	+	0.955244i	$0.595593\pi$
$824$	−0.362607	−0.0126320
$825$	0	0
$826$	−4.64806	−0.161727
$827$	−25.0532	−0.871185	−0.435593	−	0.900144i	$-0.643461\pi$
$827$	−25.0532	−0.871185	−0.435593	+	0.900144i	$0.643461\pi$
$828$	0	0
$829$	28.7497	0.998517	0.499259	−	0.866453i	$-0.333606\pi$
$829$	28.7497	0.998517	0.499259	+	0.866453i	$0.333606\pi$
$830$	0	0
$831$	0	0
$832$	−11.4307	−0.396287
$833$	12.2281	0.423678
$834$	0	0
$835$	0	0
$836$	−26.5329	−0.917659
$837$	0	0
$838$	−37.1223	−1.28237
$839$	7.83099	0.270356	0.135178	−	0.990821i	$-0.456839\pi$
$839$	7.83099	0.270356	0.135178	+	0.990821i	$0.456839\pi$
$840$	0	0
$841$	5.88617	0.202971
$842$	−18.1675	−0.626092
$843$	0	0
$844$	−33.8800	−1.16620
$845$	0	0
$846$	0	0
$847$	−32.5907	−1.11983
$848$	−17.5242	−0.601783
$849$	0	0
$850$	0	0
$851$	−6.87614	−0.235711
$852$	0	0
$853$	−35.1600	−1.20386	−0.601928	−	0.798550i	$-0.705601\pi$
$853$	−35.1600	−1.20386	−0.601928	+	0.798550i	$0.705601\pi$
$854$	−21.2207	−0.726156
$855$	0	0
$856$	−7.11905	−0.243324
$857$	10.6136	0.362553	0.181276	−	0.983432i	$-0.441977\pi$
$857$	10.6136	0.362553	0.181276	+	0.983432i	$0.441977\pi$
$858$	0	0
$859$	19.8491	0.677242	0.338621	−	0.940923i	$-0.390040\pi$
$859$	19.8491	0.677242	0.338621	+	0.940923i	$0.390040\pi$
$860$	0	0
$861$	0	0
$862$	−33.9326	−1.15575
$863$	−9.03356	−0.307506	−0.153753	−	0.988109i	$-0.549136\pi$
$863$	−9.03356	−0.307506	−0.153753	+	0.988109i	$0.549136\pi$
$864$	0	0
$865$	0	0
$866$	−16.3774	−0.556528
$867$	0	0
$868$	−11.9623	−0.406026
$869$	45.1452	1.53145
$870$	0	0
$871$	−10.6332	−0.360293
$872$	12.0410	0.407759
$873$	0	0
$874$	42.4413	1.43560
$875$	0	0
$876$	0	0
$877$	−33.6406	−1.13596	−0.567982	−	0.823041i	$-0.692276\pi$
$877$	−33.6406	−1.13596	−0.567982	+	0.823041i	$0.692276\pi$
$878$	−33.8439	−1.14217
$879$	0	0
$880$	0	0
$881$	−7.48647	−0.252226	−0.126113	−	0.992016i	$-0.540250\pi$
$881$	−7.48647	−0.252226	−0.126113	+	0.992016i	$0.540250\pi$
$882$	0	0
$883$	−17.7374	−0.596912	−0.298456	−	0.954423i	$-0.596472\pi$
$883$	−17.7374	−0.596912	−0.298456	+	0.954423i	$0.596472\pi$
$884$	1.65548	0.0556798
$885$	0	0
$886$	−43.2765	−1.45390
$887$	15.8687	0.532819	0.266410	−	0.963860i	$-0.414163\pi$
$887$	15.8687	0.532819	0.266410	+	0.963860i	$0.414163\pi$
$888$	0	0
$889$	−38.5530	−1.29302
$890$	0	0
$891$	0	0
$892$	−26.8304	−0.898347
$893$	27.6555	0.925455
$894$	0	0
$895$	0	0
$896$	−29.4110	−0.982553
$897$	0	0
$898$	−51.6997	−1.72524
$899$	−5.90645	−0.196991
$900$	0	0
$901$	3.58002	0.119268
$902$	18.9071	0.629537
$903$	0	0
$904$	−9.18971	−0.305645
$905$	0	0
$906$	0	0
$907$	−8.55451	−0.284048	−0.142024	−	0.989863i	$-0.545361\pi$
$907$	−8.55451	−0.284048	−0.142024	+	0.989863i	$0.545361\pi$
$908$	−36.0719	−1.19709
$909$	0	0
$910$	0	0
$911$	13.5800	0.449926	0.224963	−	0.974367i	$-0.427774\pi$
$911$	13.5800	0.449926	0.224963	+	0.974367i	$0.427774\pi$
$912$	0	0
$913$	59.3632	1.96464
$914$	58.1755	1.92427
$915$	0	0
$916$	−37.2259	−1.22998
$917$	−39.0288	−1.28884
$918$	0	0
$919$	52.3148	1.72571	0.862854	−	0.505454i	$-0.168675\pi$
$919$	52.3148	1.72571	0.862854	+	0.505454i	$0.168675\pi$
$920$	0	0
$921$	0	0
$922$	−61.3552	−2.02063
$923$	−3.22547	−0.106168
$924$	0	0
$925$	0	0
$926$	−52.4116	−1.72235
$927$	0	0
$928$	−47.7603	−1.56781
$929$	5.37483	0.176343	0.0881713	−	0.996105i	$-0.471898\pi$
$929$	5.37483	0.176343	0.0881713	+	0.996105i	$0.471898\pi$
$930$	0	0
$931$	−51.0187	−1.67207
$932$	−65.6881	−2.15169
$933$	0	0
$934$	72.6088	2.37583
$935$	0	0
$936$	0	0
$937$	−58.9433	−1.92559	−0.962796	−	0.270228i	$-0.912901\pi$
$937$	−58.9433	−1.92559	−0.962796	+	0.270228i	$0.912901\pi$
$938$	−103.875	−3.39165
$939$	0	0
$940$	0	0
$941$	4.62517	0.150776	0.0753881	−	0.997154i	$-0.475980\pi$
$941$	4.62517	0.150776	0.0753881	+	0.997154i	$0.475980\pi$
$942$	0	0
$943$	−16.3445	−0.532251
$944$	1.38967	0.0452299
$945$	0	0
$946$	71.1304	2.31265
$947$	−23.8591	−0.775317	−0.387658	−	0.921803i	$-0.626716\pi$
$947$	−23.8591	−0.775317	−0.387658	+	0.921803i	$0.626716\pi$
$948$	0	0
$949$	1.36678	0.0443674
$950$	0	0
$951$	0	0
$952$	2.41998	0.0784321
$953$	−15.8081	−0.512074	−0.256037	−	0.966667i	$-0.582417\pi$
$953$	−15.8081	−0.512074	−0.256037	+	0.966667i	$0.582417\pi$
$954$	0	0
$955$	0	0
$956$	24.5922	0.795370
$957$	0	0
$958$	74.6216	2.41092
$959$	24.2084	0.781731
$960$	0	0
$961$	1.00000	0.0322581
$962$	2.07065	0.0667606
$963$	0	0
$964$	−16.1723	−0.520873
$965$	0	0
$966$	0	0
$967$	−7.90320	−0.254150	−0.127075	−	0.991893i	$-0.540559\pi$
$967$	−7.90320	−0.254150	−0.127075	+	0.991893i	$0.540559\pi$
$968$	−4.70452	−0.151209
$969$	0	0
$970$	0	0
$971$	44.3478	1.42319	0.711594	−	0.702591i	$-0.247974\pi$
$971$	44.3478	1.42319	0.711594	+	0.702591i	$0.247974\pi$
$972$	0	0
$973$	−52.3052	−1.67683
$974$	−44.9878	−1.44150
$975$	0	0
$976$	6.34452	0.203083
$977$	23.3781	0.747931	0.373966	−	0.927443i	$-0.377998\pi$
$977$	23.3781	0.747931	0.373966	+	0.927443i	$0.377998\pi$
$978$	0	0
$979$	−75.8949	−2.42561
$980$	0	0
$981$	0	0
$982$	55.8033	1.78075
$983$	−35.7374	−1.13985	−0.569924	−	0.821698i	$-0.693027\pi$
$983$	−35.7374	−1.13985	−0.569924	+	0.821698i	$0.693027\pi$
$984$	0	0
$985$	0	0
$986$	7.98516	0.254299
$987$	0	0
$988$	−6.90709	−0.219744
$989$	−61.4897	−1.95526
$990$	0	0
$991$	61.4143	1.95089	0.975444	−	0.220247i	$-0.0706864\pi$
$991$	61.4143	1.95089	0.975444	+	0.220247i	$0.0706864\pi$
$992$	8.08613	0.256735
$993$	0	0
$994$	−31.5094	−0.999416
$995$	0	0
$996$	0	0
$997$	−52.8220	−1.67289	−0.836445	−	0.548051i	$-0.815370\pi$
$997$	−52.8220	−1.67289	−0.836445	+	0.548051i	$0.815370\pi$
$998$	22.8155	0.722212
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bf.1.1		3
3.2	odd	2		2325.2.a.r.1.3		3
5.4	even	2		1395.2.a.j.1.3		3
15.2	even	4		2325.2.c.k.1024.5		6
15.8	even	4		2325.2.c.k.1024.2		6
15.14	odd	2		465.2.a.e.1.1	✓	3
60.59	even	2		7440.2.a.bs.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.1	✓	3	15.14	odd	2
1395.2.a.j.1.3		3	5.4	even	2
2325.2.a.r.1.3		3	3.2	odd	2
2325.2.c.k.1024.2		6	15.8	even	4
2325.2.c.k.1024.5		6	15.2	even	4
6975.2.a.bf.1.1		3	1.1	even	1	trivial
7440.2.a.bs.1.1		3	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)

\(f(q)\)	\(=\)	\(q-2.08613 q^{2} +2.35194 q^{4} -5.08613 q^{7} -0.734191 q^{8} +4.17226 q^{11} +1.08613 q^{13} +10.6103 q^{14} -3.17226 q^{16} +0.648061 q^{17} -2.70388 q^{19} -8.70388 q^{22} +7.52420 q^{23} -2.26581 q^{26} -11.9623 q^{28} -5.90645 q^{29} +1.00000 q^{31} +8.08613 q^{32} -1.35194 q^{34} -0.913870 q^{37} +5.64064 q^{38} -2.17226 q^{41} -8.17226 q^{43} +9.81290 q^{44} -15.6965 q^{46} -10.2281 q^{47} +18.8687 q^{49} +2.55451 q^{52} +5.52420 q^{53} +3.73419 q^{56} +12.3216 q^{58} -0.438069 q^{59} -2.00000 q^{61} -2.08613 q^{62} -10.5242 q^{64} -9.79001 q^{67} +1.52420 q^{68} -2.96969 q^{71} +1.25839 q^{73} +1.90645 q^{74} -6.35936 q^{76} -21.2207 q^{77} +10.8203 q^{79} +4.53162 q^{82} +14.2281 q^{83} +17.0484 q^{86} -3.06324 q^{88} -18.1903 q^{89} -5.52420 q^{91} +17.6965 q^{92} +21.3371 q^{94} +13.2207 q^{97} -39.3626 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 8 * q^7 + 3 * q^8 - 2 * q^11 - 4 * q^13 + 8 * q^14 + 5 * q^16 + 4 * q^17 - 4 * q^19 - 22 * q^22 + 6 * q^23 - 12 * q^26 - 10 * q^28 + 2 * q^29 + 3 * q^31 + 17 * q^32 - 2 * q^34 - 10 * q^37 - 8 * q^38 + 8 * q^41 - 10 * q^43 - 10 * q^44 - 16 * q^46 - 10 * q^47 + 11 * q^49 - 10 * q^52 + 6 * q^56 + 36 * q^58 + 8 * q^59 - 6 * q^61 + q^62 - 15 * q^64 - 18 * q^67 - 12 * q^68 - 10 * q^71 - 18 * q^73 - 14 * q^74 - 44 * q^76 - 16 * q^77 + 20 * q^79 + 24 * q^82 + 22 * q^83 + 18 * q^86 - 30 * q^88 - 8 * q^89 + 22 * q^92 + 8 * q^94 - 8 * q^97 - 57 * q^98

Introduction

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