LMFDB - Embedded newform 7875.2.a.r.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7875,2,Mod(1,7875)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7875 = 3^{2} \cdot 5^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7875.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$62.8821915918$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3438125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 12x^{3} + 6x^{2} - 15x + 5 $ x^6 - 2x^5 - 6x^4 + 12x^3 + 6x^2 - 15*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 875)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.07183$ of defining polynomial
Character	$\chi$	$=$	7875.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.07183 q^{2} +2.29250 q^{4} +1.00000 q^{7} +0.606002 q^{8} +5.64479 q^{11} +2.66243 q^{13} +2.07183 q^{14} -3.32946 q^{16} -3.53065 q^{17} +7.65560 q^{19} +11.6951 q^{22} -1.96850 q^{23} +5.51611 q^{26} +2.29250 q^{28} -4.22052 q^{29} -1.41712 q^{31} -8.11008 q^{32} -7.31492 q^{34} +7.73719 q^{37} +15.8611 q^{38} -5.08612 q^{41} +9.51820 q^{43} +12.9407 q^{44} -4.07841 q^{46} +9.88720 q^{47} +1.00000 q^{49} +6.10361 q^{52} -1.55322 q^{53} +0.606002 q^{56} -8.74421 q^{58} +7.12144 q^{59} -7.41536 q^{61} -2.93603 q^{62} -10.1438 q^{64} +1.63863 q^{67} -8.09400 q^{68} -15.0816 q^{71} +10.2736 q^{73} +16.0302 q^{74} +17.5504 q^{76} +5.64479 q^{77} -0.312520 q^{79} -10.5376 q^{82} -5.65811 q^{83} +19.7201 q^{86} +3.42076 q^{88} +0.145384 q^{89} +2.66243 q^{91} -4.51278 q^{92} +20.4846 q^{94} -0.367025 q^{97} +2.07183 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 4 q^{4} + 6 q^{7} + 3 q^{11} + 16 q^{13} - 2 q^{14} - 8 q^{16} - 8 q^{17} + 9 q^{19} + 9 q^{22} - 9 q^{23} - 8 q^{26} + 4 q^{28} + 9 q^{29} + 15 q^{31} - 7 q^{32} - 4 q^{34} + 31 q^{37}+ \cdots - 2 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 + 3 * q^11 + 16 * q^13 - 2 * q^14 - 8 * q^16 - 8 * q^17 + 9 * q^19 + 9 * q^22 - 9 * q^23 - 8 * q^26 + 4 * q^28 + 9 * q^29 + 15 * q^31 - 7 * q^32 - 4 * q^34 + 31 * q^37 + 10 * q^38 - 9 * q^41 + 26 * q^43 + 22 * q^44 - 12 * q^46 + 6 * q^49 + 7 * q^52 - 5 * q^53 - 25 * q^58 - 5 * q^59 - q^61 - 9 * q^62 - 18 * q^64 + 34 * q^67 - 22 * q^68 + 7 * q^71 + 10 * q^73 - 31 * q^74 - 8 * q^76 + 3 * q^77 - 22 * q^79 + 29 * q^82 + 25 * q^83 + 37 * q^86 + 4 * q^89 + 16 * q^91 + 34 * q^92 + 26 * q^94 + 40 * q^97 - 2 * 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 $

$2$	2.07183	1.46501	0.732504	−	0.680763i	$-0.238352\pi$
$2$	2.07183	1.46501	0.732504	+	0.680763i	$0.238352\pi$
$3$	0	0
$3$	0	0
$4$	2.29250	1.14625
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.606002	0.214254
$9$	0	0
$10$	0	0
$11$	5.64479	1.70197	0.850985	−	0.525191i	$-0.176006\pi$
$11$	5.64479	1.70197	0.850985	+	0.525191i	$0.176006\pi$
$12$	0	0
$13$	2.66243	0.738425	0.369213	−	0.929345i	$-0.379627\pi$
$13$	2.66243	0.738425	0.369213	+	0.929345i	$0.379627\pi$
$14$	2.07183	0.553721
$15$	0	0
$16$	−3.32946	−0.832364
$17$	−3.53065	−0.856308	−0.428154	−	0.903706i	$-0.640836\pi$
$17$	−3.53065	−0.856308	−0.428154	+	0.903706i	$0.640836\pi$
$18$	0	0
$19$	7.65560	1.75632	0.878158	−	0.478371i	$-0.158772\pi$
$19$	7.65560	1.75632	0.878158	+	0.478371i	$0.158772\pi$
$20$	0	0
$21$	0	0
$22$	11.6951	2.49340
$23$	−1.96850	−0.410461	−0.205230	−	0.978714i	$-0.565794\pi$
$23$	−1.96850	−0.410461	−0.205230	+	0.978714i	$0.565794\pi$
$24$	0	0
$25$	0	0
$26$	5.51611	1.08180
$27$	0	0
$28$	2.29250	0.433241
$29$	−4.22052	−0.783730	−0.391865	−	0.920023i	$-0.628170\pi$
$29$	−4.22052	−0.783730	−0.391865	+	0.920023i	$0.628170\pi$
$30$	0	0
$31$	−1.41712	−0.254521	−0.127261	−	0.991869i	$-0.540618\pi$
$31$	−1.41712	−0.254521	−0.127261	+	0.991869i	$0.540618\pi$
$32$	−8.11008	−1.43367
$33$	0	0
$34$	−7.31492	−1.25450
$35$	0	0
$36$	0	0
$37$	7.73719	1.27199	0.635993	−	0.771695i	$-0.280591\pi$
$37$	7.73719	1.27199	0.635993	+	0.771695i	$0.280591\pi$
$38$	15.8611	2.57302
$39$	0	0
$40$	0	0
$41$	−5.08612	−0.794318	−0.397159	−	0.917750i	$-0.630004\pi$
$41$	−5.08612	−0.794318	−0.397159	+	0.917750i	$0.630004\pi$
$42$	0	0
$43$	9.51820	1.45151	0.725756	−	0.687952i	$-0.241490\pi$
$43$	9.51820	1.45151	0.725756	+	0.687952i	$0.241490\pi$
$44$	12.9407	1.95088
$45$	0	0
$46$	−4.07841	−0.601328
$47$	9.88720	1.44220	0.721098	−	0.692833i	$-0.243638\pi$
$47$	9.88720	1.44220	0.721098	+	0.692833i	$0.243638\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	6.10361	0.846418
$53$	−1.55322	−0.213351	−0.106675	−	0.994294i	$-0.534021\pi$
$53$	−1.55322	−0.213351	−0.106675	+	0.994294i	$0.534021\pi$
$54$	0	0
$55$	0	0
$56$	0.606002	0.0809804
$57$	0	0
$58$	−8.74421	−1.14817
$59$	7.12144	0.927132	0.463566	−	0.886062i	$-0.346570\pi$
$59$	7.12144	0.927132	0.463566	+	0.886062i	$0.346570\pi$
$60$	0	0
$61$	−7.41536	−0.949440	−0.474720	−	0.880137i	$-0.657451\pi$
$61$	−7.41536	−0.949440	−0.474720	+	0.880137i	$0.657451\pi$
$62$	−2.93603	−0.372876
$63$	0	0
$64$	−10.1438	−1.26798
$65$	0	0
$66$	0	0
$67$	1.63863	0.200191	0.100095	−	0.994978i	$-0.468085\pi$
$67$	1.63863	0.200191	0.100095	+	0.994978i	$0.468085\pi$
$68$	−8.09400	−0.981541
$69$	0	0
$70$	0	0
$71$	−15.0816	−1.78985	−0.894926	−	0.446214i	$-0.852772\pi$
$71$	−15.0816	−1.78985	−0.894926	+	0.446214i	$0.852772\pi$
$72$	0	0
$73$	10.2736	1.20243	0.601215	−	0.799088i	$-0.294684\pi$
$73$	10.2736	1.20243	0.601215	+	0.799088i	$0.294684\pi$
$74$	16.0302	1.86347
$75$	0	0
$76$	17.5504	2.01317
$77$	5.64479	0.643284
$78$	0	0
$79$	−0.312520	−0.0351612	−0.0175806	−	0.999845i	$-0.505596\pi$
$79$	−0.312520	−0.0351612	−0.0175806	+	0.999845i	$0.505596\pi$
$80$	0	0
$81$	0	0
$82$	−10.5376	−1.16368
$83$	−5.65811	−0.621059	−0.310529	−	0.950564i	$-0.600506\pi$
$83$	−5.65811	−0.621059	−0.310529	+	0.950564i	$0.600506\pi$
$84$	0	0
$85$	0	0
$86$	19.7201	2.12648
$87$	0	0
$88$	3.42076	0.364654
$89$	0.145384	0.0154107	0.00770536	−	0.999970i	$-0.497547\pi$
$89$	0.145384	0.0154107	0.00770536	+	0.999970i	$0.497547\pi$
$90$	0	0
$91$	2.66243	0.279098
$92$	−4.51278	−0.470490
$93$	0	0
$94$	20.4846	2.11283
$95$	0	0
$96$	0	0
$97$	−0.367025	−0.0372657	−0.0186329	−	0.999826i	$-0.505931\pi$
$97$	−0.367025	−0.0372657	−0.0186329	+	0.999826i	$0.505931\pi$
$98$	2.07183	0.209287
$99$	0	0
$100$	0	0
$101$	−1.66635	−0.165808	−0.0829038	−	0.996558i	$-0.526419\pi$
$101$	−1.66635	−0.165808	−0.0829038	+	0.996558i	$0.526419\pi$
$102$	0	0
$103$	3.31627	0.326761	0.163381	−	0.986563i	$-0.447760\pi$
$103$	3.31627	0.326761	0.163381	+	0.986563i	$0.447760\pi$
$104$	1.61344	0.158211
$105$	0	0
$106$	−3.21801	−0.312560
$107$	15.7112	1.51886	0.759431	−	0.650588i	$-0.225477\pi$
$107$	15.7112	1.51886	0.759431	+	0.650588i	$0.225477\pi$
$108$	0	0
$109$	6.17895	0.591836	0.295918	−	0.955213i	$-0.404374\pi$
$109$	6.17895	0.591836	0.295918	+	0.955213i	$0.404374\pi$
$110$	0	0
$111$	0	0
$112$	−3.32946	−0.314604
$113$	−9.50083	−0.893763	−0.446881	−	0.894593i	$-0.647465\pi$
$113$	−9.50083	−0.893763	−0.446881	+	0.894593i	$0.647465\pi$
$114$	0	0
$115$	0	0
$116$	−9.67552	−0.898349
$117$	0	0
$118$	14.7544	1.35826
$119$	−3.53065	−0.323654
$120$	0	0
$121$	20.8637	1.89670
$122$	−15.3634	−1.39094
$123$	0	0
$124$	−3.24873	−0.291745
$125$	0	0
$126$	0	0
$127$	18.1670	1.61206	0.806031	−	0.591874i	$-0.201612\pi$
$127$	18.1670	1.61206	0.806031	+	0.591874i	$0.201612\pi$
$128$	−4.79617	−0.423926
$129$	0	0
$130$	0	0
$131$	−7.31187	−0.638841	−0.319420	−	0.947613i	$-0.603488\pi$
$131$	−7.31187	−0.638841	−0.319420	+	0.947613i	$0.603488\pi$
$132$	0	0
$133$	7.65560	0.663825
$134$	3.39498	0.293281
$135$	0	0
$136$	−2.13958	−0.183468
$137$	−12.4190	−1.06103	−0.530514	−	0.847676i	$-0.678001\pi$
$137$	−12.4190	−1.06103	−0.530514	+	0.847676i	$0.678001\pi$
$138$	0	0
$139$	15.7921	1.33946	0.669732	−	0.742603i	$-0.266409\pi$
$139$	15.7921	1.33946	0.669732	+	0.742603i	$0.266409\pi$
$140$	0	0
$141$	0	0
$142$	−31.2465	−2.62215
$143$	15.0289	1.25678
$144$	0	0
$145$	0	0
$146$	21.2851	1.76157
$147$	0	0
$148$	17.7375	1.45801
$149$	14.5889	1.19517	0.597585	−	0.801806i	$-0.296127\pi$
$149$	14.5889	1.19517	0.597585	+	0.801806i	$0.296127\pi$
$150$	0	0
$151$	−21.1896	−1.72439	−0.862195	−	0.506577i	$-0.830910\pi$
$151$	−21.1896	−1.72439	−0.862195	+	0.506577i	$0.830910\pi$
$152$	4.63931	0.376298
$153$	0	0
$154$	11.6951	0.942416
$155$	0	0
$156$	0	0
$157$	4.27712	0.341352	0.170676	−	0.985327i	$-0.445405\pi$
$157$	4.27712	0.341352	0.170676	+	0.985327i	$0.445405\pi$
$158$	−0.647489	−0.0515114
$159$	0	0
$160$	0	0
$161$	−1.96850	−0.155140
$162$	0	0
$163$	−10.8420	−0.849208	−0.424604	−	0.905379i	$-0.639587\pi$
$163$	−10.8420	−0.849208	−0.424604	+	0.905379i	$0.639587\pi$
$164$	−11.6599	−0.910485
$165$	0	0
$166$	−11.7227	−0.909856
$167$	21.6347	1.67414	0.837071	−	0.547094i	$-0.184266\pi$
$167$	21.6347	1.67414	0.837071	+	0.547094i	$0.184266\pi$
$168$	0	0
$169$	−5.91147	−0.454728
$170$	0	0
$171$	0	0
$172$	21.8204	1.66379
$173$	−2.10250	−0.159850	−0.0799250	−	0.996801i	$-0.525468\pi$
$173$	−2.10250	−0.159850	−0.0799250	+	0.996801i	$0.525468\pi$
$174$	0	0
$175$	0	0
$176$	−18.7941	−1.41666
$177$	0	0
$178$	0.301212	0.0225768
$179$	−4.83168	−0.361137	−0.180568	−	0.983562i	$-0.557794\pi$
$179$	−4.83168	−0.361137	−0.180568	+	0.983562i	$0.557794\pi$
$180$	0	0
$181$	−18.4553	−1.37177	−0.685884	−	0.727711i	$-0.740584\pi$
$181$	−18.4553	−1.37177	−0.685884	+	0.727711i	$0.740584\pi$
$182$	5.51611	0.408881
$183$	0	0
$184$	−1.19292	−0.0879429
$185$	0	0
$186$	0	0
$187$	−19.9298	−1.45741
$188$	22.6664	1.65311
$189$	0	0
$190$	0	0
$191$	−0.201771	−0.0145996	−0.00729980	−	0.999973i	$-0.502324\pi$
$191$	−0.201771	−0.0145996	−0.00729980	+	0.999973i	$0.502324\pi$
$192$	0	0
$193$	−21.4027	−1.54060	−0.770298	−	0.637684i	$-0.779893\pi$
$193$	−21.4027	−1.54060	−0.770298	+	0.637684i	$0.779893\pi$
$194$	−0.760415	−0.0545946
$195$	0	0
$196$	2.29250	0.163750
$197$	−1.13408	−0.0807997	−0.0403999	−	0.999184i	$-0.512863\pi$
$197$	−1.13408	−0.0807997	−0.0403999	+	0.999184i	$0.512863\pi$
$198$	0	0
$199$	13.2595	0.939942	0.469971	−	0.882682i	$-0.344264\pi$
$199$	13.2595	0.939942	0.469971	+	0.882682i	$0.344264\pi$
$200$	0	0
$201$	0	0
$202$	−3.45239	−0.242910
$203$	−4.22052	−0.296222
$204$	0	0
$205$	0	0
$206$	6.87075	0.478708
$207$	0	0
$208$	−8.86444	−0.614638
$209$	43.2143	2.98920
$210$	0	0
$211$	−8.00814	−0.551303	−0.275652	−	0.961258i	$-0.588894\pi$
$211$	−8.00814	−0.551303	−0.275652	+	0.961258i	$0.588894\pi$
$212$	−3.56074	−0.244553
$213$	0	0
$214$	32.5511	2.22515
$215$	0	0
$216$	0	0
$217$	−1.41712	−0.0962000
$218$	12.8018	0.867045
$219$	0	0
$220$	0	0
$221$	−9.40011	−0.632319
$222$	0	0
$223$	20.7848	1.39185	0.695926	−	0.718114i	$-0.254994\pi$
$223$	20.7848	1.39185	0.695926	+	0.718114i	$0.254994\pi$
$224$	−8.11008	−0.541878
$225$	0	0
$226$	−19.6841	−1.30937
$227$	15.2668	1.01329	0.506647	−	0.862153i	$-0.330885\pi$
$227$	15.2668	1.01329	0.506647	+	0.862153i	$0.330885\pi$
$228$	0	0
$229$	−12.2904	−0.812170	−0.406085	−	0.913835i	$-0.633106\pi$
$229$	−12.2904	−0.812170	−0.406085	+	0.913835i	$0.633106\pi$
$230$	0	0
$231$	0	0
$232$	−2.55764	−0.167917
$233$	−19.3493	−1.26762	−0.633808	−	0.773490i	$-0.718509\pi$
$233$	−19.3493	−1.26762	−0.633808	+	0.773490i	$0.718509\pi$
$234$	0	0
$235$	0	0
$236$	16.3259	1.06272
$237$	0	0
$238$	−7.31492	−0.474156
$239$	22.9749	1.48612	0.743061	−	0.669224i	$-0.233373\pi$
$239$	22.9749	1.48612	0.743061	+	0.669224i	$0.233373\pi$
$240$	0	0
$241$	−21.8943	−1.41033	−0.705167	−	0.709042i	$-0.749128\pi$
$241$	−21.8943	−1.41033	−0.705167	+	0.709042i	$0.749128\pi$
$242$	43.2261	2.77868
$243$	0	0
$244$	−16.9997	−1.08829
$245$	0	0
$246$	0	0
$247$	20.3825	1.29691
$248$	−0.858775	−0.0545323
$249$	0	0
$250$	0	0
$251$	16.4481	1.03819	0.519096	−	0.854716i	$-0.326269\pi$
$251$	16.4481	1.03819	0.519096	+	0.854716i	$0.326269\pi$
$252$	0	0
$253$	−11.1118	−0.698591
$254$	37.6390	2.36168
$255$	0	0
$256$	10.3508	0.646925
$257$	12.8294	0.800276	0.400138	−	0.916455i	$-0.368962\pi$
$257$	12.8294	0.800276	0.400138	+	0.916455i	$0.368962\pi$
$258$	0	0
$259$	7.73719	0.480766
$260$	0	0
$261$	0	0
$262$	−15.1490	−0.935907
$263$	24.5042	1.51099	0.755496	−	0.655153i	$-0.227396\pi$
$263$	24.5042	1.51099	0.755496	+	0.655153i	$0.227396\pi$
$264$	0	0
$265$	0	0
$266$	15.8611	0.972509
$267$	0	0
$268$	3.75656	0.229468
$269$	−4.79818	−0.292550	−0.146275	−	0.989244i	$-0.546728\pi$
$269$	−4.79818	−0.292550	−0.146275	+	0.989244i	$0.546728\pi$
$270$	0	0
$271$	18.0445	1.09613	0.548064	−	0.836437i	$-0.315365\pi$
$271$	18.0445	1.09613	0.548064	+	0.836437i	$0.315365\pi$
$272$	11.7551	0.712760
$273$	0	0
$274$	−25.7301	−1.55441
$275$	0	0
$276$	0	0
$277$	−3.55774	−0.213764	−0.106882	−	0.994272i	$-0.534087\pi$
$277$	−3.55774	−0.213764	−0.106882	+	0.994272i	$0.534087\pi$
$278$	32.7185	1.96233
$279$	0	0
$280$	0	0
$281$	18.5017	1.10372	0.551860	−	0.833937i	$-0.313918\pi$
$281$	18.5017	1.10372	0.551860	+	0.833937i	$0.313918\pi$
$282$	0	0
$283$	20.7504	1.23348	0.616742	−	0.787165i	$-0.288452\pi$
$283$	20.7504	1.23348	0.616742	+	0.787165i	$0.288452\pi$
$284$	−34.5744	−2.05161
$285$	0	0
$286$	31.1373	1.84119
$287$	−5.08612	−0.300224
$288$	0	0
$289$	−4.53452	−0.266736
$290$	0	0
$291$	0	0
$292$	23.5521	1.37828
$293$	−17.5663	−1.02623	−0.513116	−	0.858319i	$-0.671509\pi$
$293$	−17.5663	−1.02623	−0.513116	+	0.858319i	$0.671509\pi$
$294$	0	0
$295$	0	0
$296$	4.68876	0.272528
$297$	0	0
$298$	30.2258	1.75093
$299$	−5.24099	−0.303095
$300$	0	0
$301$	9.51820	0.548620
$302$	−43.9014	−2.52624
$303$	0	0
$304$	−25.4890	−1.46189
$305$	0	0
$306$	0	0
$307$	1.11832	0.0638256	0.0319128	−	0.999491i	$-0.489840\pi$
$307$	1.11832	0.0638256	0.0319128	+	0.999491i	$0.489840\pi$
$308$	12.9407	0.737363
$309$	0	0
$310$	0	0
$311$	6.81298	0.386329	0.193164	−	0.981166i	$-0.438125\pi$
$311$	6.81298	0.386329	0.193164	+	0.981166i	$0.438125\pi$
$312$	0	0
$313$	25.0853	1.41791	0.708954	−	0.705255i	$-0.249168\pi$
$313$	25.0853	1.41791	0.708954	+	0.705255i	$0.249168\pi$
$314$	8.86149	0.500083
$315$	0	0
$316$	−0.716450	−0.0403035
$317$	−11.6886	−0.656499	−0.328249	−	0.944591i	$-0.606459\pi$
$317$	−11.6886	−0.656499	−0.328249	+	0.944591i	$0.606459\pi$
$318$	0	0
$319$	−23.8239	−1.33388
$320$	0	0
$321$	0	0
$322$	−4.07841	−0.227281
$323$	−27.0293	−1.50395
$324$	0	0
$325$	0	0
$326$	−22.4627	−1.24410
$327$	0	0
$328$	−3.08220	−0.170186
$329$	9.88720	0.545099
$330$	0	0
$331$	−0.294057	−0.0161628	−0.00808140	−	0.999967i	$-0.502572\pi$
$331$	−0.294057	−0.0161628	−0.00808140	+	0.999967i	$0.502572\pi$
$332$	−12.9712	−0.711887
$333$	0	0
$334$	44.8235	2.45263
$335$	0	0
$336$	0	0
$337$	−17.6896	−0.963613	−0.481806	−	0.876278i	$-0.660019\pi$
$337$	−17.6896	−0.963613	−0.481806	+	0.876278i	$0.660019\pi$
$338$	−12.2476	−0.666180
$339$	0	0
$340$	0	0
$341$	−7.99932	−0.433188
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.76805	0.310992
$345$	0	0
$346$	−4.35603	−0.234182
$347$	−20.7630	−1.11462	−0.557309	−	0.830305i	$-0.688166\pi$
$347$	−20.7630	−1.11462	−0.557309	+	0.830305i	$0.688166\pi$
$348$	0	0
$349$	2.08491	0.111603	0.0558013	−	0.998442i	$-0.482229\pi$
$349$	2.08491	0.111603	0.0558013	+	0.998442i	$0.482229\pi$
$350$	0	0
$351$	0	0
$352$	−45.7797	−2.44007
$353$	9.47731	0.504426	0.252213	−	0.967672i	$-0.418842\pi$
$353$	9.47731	0.504426	0.252213	+	0.967672i	$0.418842\pi$
$354$	0	0
$355$	0	0
$356$	0.333293	0.0176645
$357$	0	0
$358$	−10.0104	−0.529068
$359$	11.0842	0.585001	0.292501	−	0.956265i	$-0.405513\pi$
$359$	11.0842	0.585001	0.292501	+	0.956265i	$0.405513\pi$
$360$	0	0
$361$	39.6083	2.08465
$362$	−38.2362	−2.00965
$363$	0	0
$364$	6.10361	0.319916
$365$	0	0
$366$	0	0
$367$	−3.17535	−0.165752	−0.0828759	−	0.996560i	$-0.526411\pi$
$367$	−3.17535	−0.165752	−0.0828759	+	0.996560i	$0.526411\pi$
$368$	6.55403	0.341653
$369$	0	0
$370$	0	0
$371$	−1.55322	−0.0806390
$372$	0	0
$373$	8.77299	0.454248	0.227124	−	0.973866i	$-0.427068\pi$
$373$	8.77299	0.454248	0.227124	+	0.973866i	$0.427068\pi$
$374$	−41.2912	−2.13512
$375$	0	0
$376$	5.99166	0.308996
$377$	−11.2368	−0.578726
$378$	0	0
$379$	−34.8301	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$379$	−34.8301	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$380$	0	0
$381$	0	0
$382$	−0.418035	−0.0213885
$383$	36.4660	1.86333	0.931663	−	0.363323i	$-0.118358\pi$
$383$	36.4660	1.86333	0.931663	+	0.363323i	$0.118358\pi$
$384$	0	0
$385$	0	0
$386$	−44.3427	−2.25699
$387$	0	0
$388$	−0.841403	−0.0427158
$389$	−9.17665	−0.465275	−0.232637	−	0.972564i	$-0.574735\pi$
$389$	−9.17665	−0.465275	−0.232637	+	0.972564i	$0.574735\pi$
$390$	0	0
$391$	6.95008	0.351481
$392$	0.606002	0.0306077
$393$	0	0
$394$	−2.34962	−0.118372
$395$	0	0
$396$	0	0
$397$	−15.4092	−0.773368	−0.386684	−	0.922212i	$-0.626380\pi$
$397$	−15.4092	−0.773368	−0.386684	+	0.922212i	$0.626380\pi$
$398$	27.4715	1.37702
$399$	0	0
$400$	0	0
$401$	−9.05929	−0.452400	−0.226200	−	0.974081i	$-0.572630\pi$
$401$	−9.05929	−0.452400	−0.226200	+	0.974081i	$0.572630\pi$
$402$	0	0
$403$	−3.77297	−0.187945
$404$	−3.82009	−0.190057
$405$	0	0
$406$	−8.74421	−0.433968
$407$	43.6749	2.16488
$408$	0	0
$409$	−31.3284	−1.54909	−0.774544	−	0.632519i	$-0.782021\pi$
$409$	−31.3284	−1.54909	−0.774544	+	0.632519i	$0.782021\pi$
$410$	0	0
$411$	0	0
$412$	7.60253	0.374550
$413$	7.12144	0.350423
$414$	0	0
$415$	0	0
$416$	−21.5925	−1.05866
$417$	0	0
$418$	89.5328	4.37919
$419$	10.6205	0.518843	0.259421	−	0.965764i	$-0.416468\pi$
$419$	10.6205	0.518843	0.259421	+	0.965764i	$0.416468\pi$
$420$	0	0
$421$	13.6324	0.664404	0.332202	−	0.943208i	$-0.392208\pi$
$421$	13.6324	0.664404	0.332202	+	0.943208i	$0.392208\pi$
$422$	−16.5915	−0.807664
$423$	0	0
$424$	−0.941253	−0.0457113
$425$	0	0
$426$	0	0
$427$	−7.41536	−0.358855
$428$	36.0179	1.74099
$429$	0	0
$430$	0	0
$431$	−30.1719	−1.45333	−0.726664	−	0.686993i	$-0.758930\pi$
$431$	−30.1719	−1.45333	−0.726664	+	0.686993i	$0.758930\pi$
$432$	0	0
$433$	−34.7562	−1.67027	−0.835137	−	0.550042i	$-0.814612\pi$
$433$	−34.7562	−1.67027	−0.835137	+	0.550042i	$0.814612\pi$
$434$	−2.93603	−0.140934
$435$	0	0
$436$	14.1652	0.678391
$437$	−15.0701	−0.720899
$438$	0	0
$439$	33.1711	1.58317	0.791586	−	0.611058i	$-0.209256\pi$
$439$	33.1711	1.58317	0.791586	+	0.611058i	$0.209256\pi$
$440$	0	0
$441$	0	0
$442$	−19.4755	−0.926353
$443$	21.5802	1.02531	0.512654	−	0.858595i	$-0.328662\pi$
$443$	21.5802	1.02531	0.512654	+	0.858595i	$0.328662\pi$
$444$	0	0
$445$	0	0
$446$	43.0626	2.03907
$447$	0	0
$448$	−10.1438	−0.479251
$449$	−37.4583	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$449$	−37.4583	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$450$	0	0
$451$	−28.7101	−1.35190
$452$	−21.7806	−1.02447
$453$	0	0
$454$	31.6303	1.48448
$455$	0	0
$456$	0	0
$457$	10.6567	0.498499	0.249250	−	0.968439i	$-0.419816\pi$
$457$	10.6567	0.498499	0.249250	+	0.968439i	$0.419816\pi$
$458$	−25.4636	−1.18984
$459$	0	0
$460$	0	0
$461$	−28.6007	−1.33207	−0.666033	−	0.745922i	$-0.732009\pi$
$461$	−28.6007	−1.33207	−0.666033	+	0.745922i	$0.732009\pi$
$462$	0	0
$463$	−5.04199	−0.234321	−0.117161	−	0.993113i	$-0.537379\pi$
$463$	−5.04199	−0.234321	−0.117161	+	0.993113i	$0.537379\pi$
$464$	14.0520	0.652349
$465$	0	0
$466$	−40.0886	−1.85707
$467$	−18.2071	−0.842522	−0.421261	−	0.906939i	$-0.638412\pi$
$467$	−18.2071	−0.842522	−0.421261	+	0.906939i	$0.638412\pi$
$468$	0	0
$469$	1.63863	0.0756651
$470$	0	0
$471$	0	0
$472$	4.31561	0.198642
$473$	53.7283	2.47043
$474$	0	0
$475$	0	0
$476$	−8.09400	−0.370988
$477$	0	0
$478$	47.6001	2.17718
$479$	13.6088	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$479$	13.6088	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$480$	0	0
$481$	20.5997	0.939267
$482$	−45.3613	−2.06615
$483$	0	0
$484$	47.8299	2.17409
$485$	0	0
$486$	0	0
$487$	−18.4643	−0.836695	−0.418348	−	0.908287i	$-0.637391\pi$
$487$	−18.4643	−0.836695	−0.418348	+	0.908287i	$0.637391\pi$
$488$	−4.49373	−0.203421
$489$	0	0
$490$	0	0
$491$	16.6303	0.750513	0.375257	−	0.926921i	$-0.377555\pi$
$491$	16.6303	0.750513	0.375257	+	0.926921i	$0.377555\pi$
$492$	0	0
$493$	14.9012	0.671115
$494$	42.2292	1.89998
$495$	0	0
$496$	4.71822	0.211854
$497$	−15.0816	−0.676501
$498$	0	0
$499$	−21.2574	−0.951612	−0.475806	−	0.879550i	$-0.657844\pi$
$499$	−21.2574	−0.951612	−0.475806	+	0.879550i	$0.657844\pi$
$500$	0	0
$501$	0	0
$502$	34.0777	1.52096
$503$	−27.5789	−1.22968	−0.614841	−	0.788651i	$-0.710780\pi$
$503$	−27.5789	−1.22968	−0.614841	+	0.788651i	$0.710780\pi$
$504$	0	0
$505$	0	0
$506$	−23.0218	−1.02344
$507$	0	0
$508$	41.6478	1.84782
$509$	8.74846	0.387769	0.193884	−	0.981024i	$-0.437891\pi$
$509$	8.74846	0.387769	0.193884	+	0.981024i	$0.437891\pi$
$510$	0	0
$511$	10.2736	0.454476
$512$	31.0375	1.37168
$513$	0	0
$514$	26.5804	1.17241
$515$	0	0
$516$	0	0
$517$	55.8112	2.45457
$518$	16.0302	0.704326
$519$	0	0
$520$	0	0
$521$	12.2800	0.537997	0.268998	−	0.963141i	$-0.413307\pi$
$521$	12.2800	0.537997	0.268998	+	0.963141i	$0.413307\pi$
$522$	0	0
$523$	−43.3065	−1.89366	−0.946831	−	0.321730i	$-0.895736\pi$
$523$	−43.3065	−1.89366	−0.946831	+	0.321730i	$0.895736\pi$
$524$	−16.7624	−0.732270
$525$	0	0
$526$	50.7686	2.21362
$527$	5.00334	0.217949
$528$	0	0
$529$	−19.1250	−0.831522
$530$	0	0
$531$	0	0
$532$	17.5504	0.760908
$533$	−13.5414	−0.586544
$534$	0	0
$535$	0	0
$536$	0.993016	0.0428917
$537$	0	0
$538$	−9.94102	−0.428588
$539$	5.64479	0.243138
$540$	0	0
$541$	−23.7452	−1.02088	−0.510442	−	0.859912i	$-0.670518\pi$
$541$	−23.7452	−1.02088	−0.510442	+	0.859912i	$0.670518\pi$
$542$	37.3853	1.60584
$543$	0	0
$544$	28.6339	1.22767
$545$	0	0
$546$	0	0
$547$	−10.3120	−0.440909	−0.220455	−	0.975397i	$-0.570754\pi$
$547$	−10.3120	−0.440909	−0.220455	+	0.975397i	$0.570754\pi$
$548$	−28.4705	−1.21620
$549$	0	0
$550$	0	0
$551$	−32.3106	−1.37648
$552$	0	0
$553$	−0.312520	−0.0132897
$554$	−7.37104	−0.313165
$555$	0	0
$556$	36.2032	1.53536
$557$	−12.7410	−0.539852	−0.269926	−	0.962881i	$-0.586999\pi$
$557$	−12.7410	−0.539852	−0.269926	+	0.962881i	$0.586999\pi$
$558$	0	0
$559$	25.3415	1.07183
$560$	0	0
$561$	0	0
$562$	38.3325	1.61696
$563$	5.61906	0.236815	0.118408	−	0.992965i	$-0.462221\pi$
$563$	5.61906	0.236815	0.118408	+	0.992965i	$0.462221\pi$
$564$	0	0
$565$	0	0
$566$	42.9914	1.80706
$567$	0	0
$568$	−9.13946	−0.383483
$569$	−19.4497	−0.815375	−0.407687	−	0.913122i	$-0.633665\pi$
$569$	−19.4497	−0.815375	−0.407687	+	0.913122i	$0.633665\pi$
$570$	0	0
$571$	−21.8140	−0.912889	−0.456445	−	0.889752i	$-0.650877\pi$
$571$	−21.8140	−0.912889	−0.456445	+	0.889752i	$0.650877\pi$
$572$	34.4536	1.44058
$573$	0	0
$574$	−10.5376	−0.439830
$575$	0	0
$576$	0	0
$577$	−12.6085	−0.524897	−0.262448	−	0.964946i	$-0.584530\pi$
$577$	−12.6085	−0.524897	−0.262448	+	0.964946i	$0.584530\pi$
$578$	−9.39477	−0.390771
$579$	0	0
$580$	0	0
$581$	−5.65811	−0.234738
$582$	0	0
$583$	−8.76759	−0.363116
$584$	6.22580	0.257625
$585$	0	0
$586$	−36.3944	−1.50344
$587$	−44.3430	−1.83023	−0.915115	−	0.403193i	$-0.867900\pi$
$587$	−44.3430	−1.83023	−0.915115	+	0.403193i	$0.867900\pi$
$588$	0	0
$589$	−10.8489	−0.447020
$590$	0	0
$591$	0	0
$592$	−25.7606	−1.05876
$593$	−13.0923	−0.537636	−0.268818	−	0.963191i	$-0.586633\pi$
$593$	−13.0923	−0.537636	−0.268818	+	0.963191i	$0.586633\pi$
$594$	0	0
$595$	0	0
$596$	33.4450	1.36996
$597$	0	0
$598$	−10.8585	−0.444036
$599$	−1.43860	−0.0587794	−0.0293897	−	0.999568i	$-0.509356\pi$
$599$	−1.43860	−0.0587794	−0.0293897	+	0.999568i	$0.509356\pi$
$600$	0	0
$601$	6.23535	0.254345	0.127173	−	0.991881i	$-0.459410\pi$
$601$	6.23535	0.254345	0.127173	+	0.991881i	$0.459410\pi$
$602$	19.7201	0.803732
$603$	0	0
$604$	−48.5772	−1.97658
$605$	0	0
$606$	0	0
$607$	−28.1334	−1.14190	−0.570949	−	0.820985i	$-0.693425\pi$
$607$	−28.1334	−1.14190	−0.570949	+	0.820985i	$0.693425\pi$
$608$	−62.0876	−2.51798
$609$	0	0
$610$	0	0
$611$	26.3240	1.06495
$612$	0	0
$613$	36.2031	1.46223	0.731114	−	0.682255i	$-0.239001\pi$
$613$	36.2031	1.46223	0.731114	+	0.682255i	$0.239001\pi$
$614$	2.31696	0.0935050
$615$	0	0
$616$	3.42076	0.137826
$617$	−9.37570	−0.377451	−0.188726	−	0.982030i	$-0.560436\pi$
$617$	−9.37570	−0.377451	−0.188726	+	0.982030i	$0.560436\pi$
$618$	0	0
$619$	1.86560	0.0749846	0.0374923	−	0.999297i	$-0.488063\pi$
$619$	1.86560	0.0749846	0.0374923	+	0.999297i	$0.488063\pi$
$620$	0	0
$621$	0	0
$622$	14.1154	0.565974
$623$	0.145384	0.00582470
$624$	0	0
$625$	0	0
$626$	51.9727	2.07725
$627$	0	0
$628$	9.80528	0.391273
$629$	−27.3173	−1.08921
$630$	0	0
$631$	−28.5504	−1.13657	−0.568286	−	0.822831i	$-0.692393\pi$
$631$	−28.5504	−1.13657	−0.568286	+	0.822831i	$0.692393\pi$
$632$	−0.189388	−0.00753343
$633$	0	0
$634$	−24.2169	−0.961776
$635$	0	0
$636$	0	0
$637$	2.66243	0.105489
$638$	−49.3593	−1.95415
$639$	0	0
$640$	0	0
$641$	−15.5304	−0.613415	−0.306708	−	0.951804i	$-0.599227\pi$
$641$	−15.5304	−0.613415	−0.306708	+	0.951804i	$0.599227\pi$
$642$	0	0
$643$	−27.3884	−1.08009	−0.540047	−	0.841635i	$-0.681593\pi$
$643$	−27.3884	−1.08009	−0.540047	+	0.841635i	$0.681593\pi$
$644$	−4.51278	−0.177828
$645$	0	0
$646$	−56.0001	−2.20330
$647$	35.1556	1.38211	0.691055	−	0.722802i	$-0.257146\pi$
$647$	35.1556	1.38211	0.691055	+	0.722802i	$0.257146\pi$
$648$	0	0
$649$	40.1990	1.57795
$650$	0	0
$651$	0	0
$652$	−24.8551	−0.973402
$653$	−36.3322	−1.42179	−0.710895	−	0.703298i	$-0.751710\pi$
$653$	−36.3322	−1.42179	−0.710895	+	0.703298i	$0.751710\pi$
$654$	0	0
$655$	0	0
$656$	16.9340	0.661162
$657$	0	0
$658$	20.4846	0.798574
$659$	−27.1360	−1.05707	−0.528534	−	0.848912i	$-0.677258\pi$
$659$	−27.1360	−1.05707	−0.528534	+	0.848912i	$0.677258\pi$
$660$	0	0
$661$	24.7823	0.963918	0.481959	−	0.876194i	$-0.339925\pi$
$661$	24.7823	0.963918	0.481959	+	0.876194i	$0.339925\pi$
$662$	−0.609236	−0.0236786
$663$	0	0
$664$	−3.42883	−0.133064
$665$	0	0
$666$	0	0
$667$	8.30809	0.321690
$668$	49.5974	1.91898
$669$	0	0
$670$	0	0
$671$	−41.8582	−1.61592
$672$	0	0
$673$	3.94512	0.152073	0.0760366	−	0.997105i	$-0.475773\pi$
$673$	3.94512	0.152073	0.0760366	+	0.997105i	$0.475773\pi$
$674$	−36.6499	−1.41170
$675$	0	0
$676$	−13.5520	−0.521231
$677$	−48.4953	−1.86383	−0.931913	−	0.362683i	$-0.881861\pi$
$677$	−48.4953	−1.86383	−0.931913	+	0.362683i	$0.881861\pi$
$678$	0	0
$679$	−0.367025	−0.0140851
$680$	0	0
$681$	0	0
$682$	−16.5733	−0.634623
$683$	−8.26648	−0.316308	−0.158154	−	0.987414i	$-0.550554\pi$
$683$	−8.26648	−0.316308	−0.158154	+	0.987414i	$0.550554\pi$
$684$	0	0
$685$	0	0
$686$	2.07183	0.0791030
$687$	0	0
$688$	−31.6904	−1.20819
$689$	−4.13533	−0.157544
$690$	0	0
$691$	−51.4752	−1.95821	−0.979105	−	0.203357i	$-0.934815\pi$
$691$	−51.4752	−1.95821	−0.979105	+	0.203357i	$0.934815\pi$
$692$	−4.81997	−0.183228
$693$	0	0
$694$	−43.0175	−1.63292
$695$	0	0
$696$	0	0
$697$	17.9573	0.680181
$698$	4.31958	0.163499
$699$	0	0
$700$	0	0
$701$	1.42968	0.0539983	0.0269992	−	0.999635i	$-0.491405\pi$
$701$	1.42968	0.0539983	0.0269992	+	0.999635i	$0.491405\pi$
$702$	0	0
$703$	59.2329	2.23401
$704$	−57.2598	−2.15806
$705$	0	0
$706$	19.6354	0.738988
$707$	−1.66635	−0.0626694
$708$	0	0
$709$	−23.0609	−0.866069	−0.433034	−	0.901377i	$-0.642557\pi$
$709$	−23.0609	−0.866069	−0.433034	+	0.901377i	$0.642557\pi$
$710$	0	0
$711$	0	0
$712$	0.0881033	0.00330181
$713$	2.78959	0.104471
$714$	0	0
$715$	0	0
$716$	−11.0766	−0.413952
$717$	0	0
$718$	22.9646	0.857031
$719$	3.89005	0.145074	0.0725372	−	0.997366i	$-0.476890\pi$
$719$	3.89005	0.145074	0.0725372	+	0.997366i	$0.476890\pi$
$720$	0	0
$721$	3.31627	0.123504
$722$	82.0618	3.05402
$723$	0	0
$724$	−42.3086	−1.57239
$725$	0	0
$726$	0	0
$727$	−22.8374	−0.846994	−0.423497	−	0.905898i	$-0.639198\pi$
$727$	−22.8374	−0.846994	−0.423497	+	0.905898i	$0.639198\pi$
$728$	1.61344	0.0597980
$729$	0	0
$730$	0	0
$731$	−33.6054	−1.24294
$732$	0	0
$733$	−0.906984	−0.0335002	−0.0167501	−	0.999860i	$-0.505332\pi$
$733$	−0.906984	−0.0335002	−0.0167501	+	0.999860i	$0.505332\pi$
$734$	−6.57879	−0.242828
$735$	0	0
$736$	15.9647	0.588467
$737$	9.24975	0.340719
$738$	0	0
$739$	−28.6839	−1.05515	−0.527577	−	0.849507i	$-0.676899\pi$
$739$	−28.6839	−1.05515	−0.527577	+	0.849507i	$0.676899\pi$
$740$	0	0
$741$	0	0
$742$	−3.21801	−0.118137
$743$	32.4052	1.18883	0.594416	−	0.804158i	$-0.297383\pi$
$743$	32.4052	1.18883	0.594416	+	0.804158i	$0.297383\pi$
$744$	0	0
$745$	0	0
$746$	18.1762	0.665477
$747$	0	0
$748$	−45.6889	−1.67055
$749$	15.7112	0.574076
$750$	0	0
$751$	21.8542	0.797471	0.398736	−	0.917066i	$-0.369449\pi$
$751$	21.8542	0.797471	0.398736	+	0.917066i	$0.369449\pi$
$752$	−32.9190	−1.20043
$753$	0	0
$754$	−23.2808	−0.847838
$755$	0	0
$756$	0	0
$757$	−17.0985	−0.621456	−0.310728	−	0.950499i	$-0.600573\pi$
$757$	−17.0985	−0.621456	−0.310728	+	0.950499i	$0.600573\pi$
$758$	−72.1621	−2.62105
$759$	0	0
$760$	0	0
$761$	−11.3866	−0.412763	−0.206382	−	0.978472i	$-0.566169\pi$
$761$	−11.3866	−0.412763	−0.206382	+	0.978472i	$0.566169\pi$
$762$	0	0
$763$	6.17895	0.223693
$764$	−0.462558	−0.0167348
$765$	0	0
$766$	75.5515	2.72979
$767$	18.9603	0.684618
$768$	0	0
$769$	−12.7333	−0.459174	−0.229587	−	0.973288i	$-0.573738\pi$
$769$	−12.7333	−0.459174	−0.229587	+	0.973288i	$0.573738\pi$
$770$	0	0
$771$	0	0
$772$	−49.0655	−1.76591
$773$	−19.4489	−0.699527	−0.349764	−	0.936838i	$-0.613738\pi$
$773$	−19.4489	−0.699527	−0.349764	+	0.936838i	$0.613738\pi$
$774$	0	0
$775$	0	0
$776$	−0.222418	−0.00798434
$777$	0	0
$778$	−19.0125	−0.681631
$779$	−38.9373	−1.39507
$780$	0	0
$781$	−85.1323	−3.04627
$782$	14.3994	0.514922
$783$	0	0
$784$	−3.32946	−0.118909
$785$	0	0
$786$	0	0
$787$	10.7453	0.383028	0.191514	−	0.981490i	$-0.438660\pi$
$787$	10.7453	0.383028	0.191514	+	0.981490i	$0.438660\pi$
$788$	−2.59987	−0.0926165
$789$	0	0
$790$	0	0
$791$	−9.50083	−0.337811
$792$	0	0
$793$	−19.7429	−0.701090
$794$	−31.9254	−1.13299
$795$	0	0
$796$	30.3974	1.07741
$797$	27.0668	0.958756	0.479378	−	0.877609i	$-0.340862\pi$
$797$	27.0668	0.958756	0.479378	+	0.877609i	$0.340862\pi$
$798$	0	0
$799$	−34.9082	−1.23496
$800$	0	0
$801$	0	0
$802$	−18.7694	−0.662769
$803$	57.9921	2.04650
$804$	0	0
$805$	0	0
$806$	−7.81697	−0.275341
$807$	0	0
$808$	−1.00981	−0.0355250
$809$	−50.7467	−1.78416	−0.892080	−	0.451877i	$-0.850755\pi$
$809$	−50.7467	−1.78416	−0.892080	+	0.451877i	$0.850755\pi$
$810$	0	0
$811$	3.77490	0.132555	0.0662773	−	0.997801i	$-0.478888\pi$
$811$	3.77490	0.132555	0.0662773	+	0.997801i	$0.478888\pi$
$812$	−9.67552	−0.339544
$813$	0	0
$814$	90.4870	3.17157
$815$	0	0
$816$	0	0
$817$	72.8675	2.54931
$818$	−64.9072	−2.26943
$819$	0	0
$820$	0	0
$821$	−18.8686	−0.658520	−0.329260	−	0.944239i	$-0.606799\pi$
$821$	−18.8686	−0.658520	−0.329260	+	0.944239i	$0.606799\pi$
$822$	0	0
$823$	−12.1906	−0.424938	−0.212469	−	0.977168i	$-0.568150\pi$
$823$	−12.1906	−0.424938	−0.212469	+	0.977168i	$0.568150\pi$
$824$	2.00966	0.0700100
$825$	0	0
$826$	14.7544	0.513372
$827$	42.8315	1.48940	0.744698	−	0.667401i	$-0.232593\pi$
$827$	42.8315	1.48940	0.744698	+	0.667401i	$0.232593\pi$
$828$	0	0
$829$	−0.686961	−0.0238592	−0.0119296	−	0.999929i	$-0.503797\pi$
$829$	−0.686961	−0.0238592	−0.0119296	+	0.999929i	$0.503797\pi$
$830$	0	0
$831$	0	0
$832$	−27.0072	−0.936308
$833$	−3.53065	−0.122330
$834$	0	0
$835$	0	0
$836$	99.0686	3.42636
$837$	0	0
$838$	22.0038	0.760109
$839$	5.32102	0.183702	0.0918510	−	0.995773i	$-0.470722\pi$
$839$	5.32102	0.183702	0.0918510	+	0.995773i	$0.470722\pi$
$840$	0	0
$841$	−11.1872	−0.385767
$842$	28.2441	0.973357
$843$	0	0
$844$	−18.3586	−0.631930
$845$	0	0
$846$	0	0
$847$	20.8637	0.716885
$848$	5.17137	0.177585
$849$	0	0
$850$	0	0
$851$	−15.2307	−0.522101
$852$	0	0
$853$	1.59236	0.0545213	0.0272607	−	0.999628i	$-0.491322\pi$
$853$	1.59236	0.0545213	0.0272607	+	0.999628i	$0.491322\pi$
$854$	−15.3634	−0.525725
$855$	0	0
$856$	9.52104	0.325423
$857$	12.4779	0.426237	0.213118	−	0.977026i	$-0.431638\pi$
$857$	12.4779	0.426237	0.213118	+	0.977026i	$0.431638\pi$
$858$	0	0
$859$	9.79301	0.334133	0.167067	−	0.985946i	$-0.446570\pi$
$859$	9.79301	0.334133	0.167067	+	0.985946i	$0.446570\pi$
$860$	0	0
$861$	0	0
$862$	−62.5111	−2.12914
$863$	−28.8405	−0.981742	−0.490871	−	0.871232i	$-0.663321\pi$
$863$	−28.8405	−0.981742	−0.490871	+	0.871232i	$0.663321\pi$
$864$	0	0
$865$	0	0
$866$	−72.0090	−2.44696
$867$	0	0
$868$	−3.24873	−0.110269
$869$	−1.76411	−0.0598433
$870$	0	0
$871$	4.36275	0.147826
$872$	3.74446	0.126803
$873$	0	0
$874$	−31.2227	−1.05612
$875$	0	0
$876$	0	0
$877$	31.3594	1.05893	0.529466	−	0.848331i	$-0.322392\pi$
$877$	31.3594	1.05893	0.529466	+	0.848331i	$0.322392\pi$
$878$	68.7251	2.31936
$879$	0	0
$880$	0	0
$881$	−44.6106	−1.50297	−0.751484	−	0.659751i	$-0.770662\pi$
$881$	−44.6106	−1.50297	−0.751484	+	0.659751i	$0.770662\pi$
$882$	0	0
$883$	−5.27121	−0.177390	−0.0886952	−	0.996059i	$-0.528270\pi$
$883$	−5.27121	−0.177390	−0.0886952	+	0.996059i	$0.528270\pi$
$884$	−21.5497	−0.724795
$885$	0	0
$886$	44.7107	1.50208
$887$	33.7405	1.13290	0.566448	−	0.824097i	$-0.308317\pi$
$887$	33.7405	1.13290	0.566448	+	0.824097i	$0.308317\pi$
$888$	0	0
$889$	18.1670	0.609302
$890$	0	0
$891$	0	0
$892$	47.6490	1.59541
$893$	75.6925	2.53295
$894$	0	0
$895$	0	0
$896$	−4.79617	−0.160229
$897$	0	0
$898$	−77.6074	−2.58979
$899$	5.98096	0.199476
$900$	0	0
$901$	5.48386	0.182694
$902$	−59.4825	−1.98055
$903$	0	0
$904$	−5.75752	−0.191492
$905$	0	0
$906$	0	0
$907$	22.3236	0.741243	0.370621	−	0.928784i	$-0.379145\pi$
$907$	22.3236	0.741243	0.370621	+	0.928784i	$0.379145\pi$
$908$	34.9991	1.16149
$909$	0	0
$910$	0	0
$911$	−13.7738	−0.456347	−0.228173	−	0.973621i	$-0.573275\pi$
$911$	−13.7738	−0.456347	−0.228173	+	0.973621i	$0.573275\pi$
$912$	0	0
$913$	−31.9389	−1.05702
$914$	22.0789	0.730305
$915$	0	0
$916$	−28.1756	−0.930948
$917$	−7.31187	−0.241459
$918$	0	0
$919$	24.5320	0.809237	0.404619	−	0.914486i	$-0.367404\pi$
$919$	24.5320	0.809237	0.404619	+	0.914486i	$0.367404\pi$
$920$	0	0
$921$	0	0
$922$	−59.2559	−1.95149
$923$	−40.1536	−1.32167
$924$	0	0
$925$	0	0
$926$	−10.4462	−0.343282
$927$	0	0
$928$	34.2287	1.12361
$929$	−16.7756	−0.550391	−0.275195	−	0.961388i	$-0.588743\pi$
$929$	−16.7756	−0.550391	−0.275195	+	0.961388i	$0.588743\pi$
$930$	0	0
$931$	7.65560	0.250902
$932$	−44.3582	−1.45300
$933$	0	0
$934$	−37.7220	−1.23430
$935$	0	0
$936$	0	0
$937$	20.4645	0.668545	0.334272	−	0.942477i	$-0.391509\pi$
$937$	20.4645	0.668545	0.334272	+	0.942477i	$0.391509\pi$
$938$	3.39498	0.110850
$939$	0	0
$940$	0	0
$941$	9.23872	0.301174	0.150587	−	0.988597i	$-0.451884\pi$
$941$	9.23872	0.301174	0.150587	+	0.988597i	$0.451884\pi$
$942$	0	0
$943$	10.0120	0.326036
$944$	−23.7105	−0.771711
$945$	0	0
$946$	111.316	3.61920
$947$	−18.7089	−0.607957	−0.303978	−	0.952679i	$-0.598315\pi$
$947$	−18.7089	−0.607957	−0.303978	+	0.952679i	$0.598315\pi$
$948$	0	0
$949$	27.3526	0.887904
$950$	0	0
$951$	0	0
$952$	−2.13958	−0.0693442
$953$	19.1469	0.620229	0.310114	−	0.950699i	$-0.399633\pi$
$953$	19.1469	0.620229	0.310114	+	0.950699i	$0.399633\pi$
$954$	0	0
$955$	0	0
$956$	52.6698	1.70346
$957$	0	0
$958$	28.1951	0.910941
$959$	−12.4190	−0.401031
$960$	0	0
$961$	−28.9918	−0.935219
$962$	42.6792	1.37603
$963$	0	0
$964$	−50.1925	−1.61659
$965$	0	0
$966$	0	0
$967$	1.09729	0.0352865	0.0176433	−	0.999844i	$-0.494384\pi$
$967$	1.09729	0.0352865	0.0176433	+	0.999844i	$0.494384\pi$
$968$	12.6434	0.406376
$969$	0	0
$970$	0	0
$971$	5.47636	0.175745	0.0878724	−	0.996132i	$-0.471993\pi$
$971$	5.47636	0.175745	0.0878724	+	0.996132i	$0.471993\pi$
$972$	0	0
$973$	15.7921	0.506270
$974$	−38.2549	−1.22576
$975$	0	0
$976$	24.6891	0.790280
$977$	12.2420	0.391656	0.195828	−	0.980638i	$-0.437261\pi$
$977$	12.2420	0.391656	0.195828	+	0.980638i	$0.437261\pi$
$978$	0	0
$979$	0.820665	0.0262286
$980$	0	0
$981$	0	0
$982$	34.4551	1.09951
$983$	12.6229	0.402610	0.201305	−	0.979529i	$-0.435482\pi$
$983$	12.6229	0.402610	0.201305	+	0.979529i	$0.435482\pi$
$984$	0	0
$985$	0	0
$986$	30.8727	0.983188
$987$	0	0
$988$	46.7268	1.48658
$989$	−18.7366	−0.595788
$990$	0	0
$991$	−20.8040	−0.660861	−0.330431	−	0.943830i	$-0.607194\pi$
$991$	−20.8040	−0.660861	−0.330431	+	0.943830i	$0.607194\pi$
$992$	11.4929	0.364901
$993$	0	0
$994$	−31.2465	−0.991079
$995$	0	0
$996$	0	0
$997$	36.4050	1.15296	0.576478	−	0.817113i	$-0.304427\pi$
$997$	36.4050	1.15296	0.576478	+	0.817113i	$0.304427\pi$
$998$	−44.0418	−1.39412
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7875.2.a.r.1.6		6
3.2	odd	2		875.2.a.g.1.1	yes	6
5.4	even	2		7875.2.a.s.1.1		6
15.2	even	4		875.2.b.d.624.2		12
15.8	even	4		875.2.b.d.624.11		12
15.14	odd	2		875.2.a.f.1.6	✓	6
21.20	even	2		6125.2.a.u.1.1		6
105.104	even	2		6125.2.a.r.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.f.1.6	✓	6	15.14	odd	2
875.2.a.g.1.1	yes	6	3.2	odd	2
875.2.b.d.624.2		12	15.2	even	4
875.2.b.d.624.11		12	15.8	even	4
6125.2.a.r.1.6		6	105.104	even	2
6125.2.a.u.1.1		6	21.20	even	2
7875.2.a.r.1.6		6	1.1	even	1	trivial
7875.2.a.s.1.1		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7875 = 3^{2} \cdot 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7875.a (trivial)

\(f(q)\)	\(=\)	\(q+2.07183 q^{2} +2.29250 q^{4} +1.00000 q^{7} +0.606002 q^{8} +5.64479 q^{11} +2.66243 q^{13} +2.07183 q^{14} -3.32946 q^{16} -3.53065 q^{17} +7.65560 q^{19} +11.6951 q^{22} -1.96850 q^{23} +5.51611 q^{26} +2.29250 q^{28} -4.22052 q^{29} -1.41712 q^{31} -8.11008 q^{32} -7.31492 q^{34} +7.73719 q^{37} +15.8611 q^{38} -5.08612 q^{41} +9.51820 q^{43} +12.9407 q^{44} -4.07841 q^{46} +9.88720 q^{47} +1.00000 q^{49} +6.10361 q^{52} -1.55322 q^{53} +0.606002 q^{56} -8.74421 q^{58} +7.12144 q^{59} -7.41536 q^{61} -2.93603 q^{62} -10.1438 q^{64} +1.63863 q^{67} -8.09400 q^{68} -15.0816 q^{71} +10.2736 q^{73} +16.0302 q^{74} +17.5504 q^{76} +5.64479 q^{77} -0.312520 q^{79} -10.5376 q^{82} -5.65811 q^{83} +19.7201 q^{86} +3.42076 q^{88} +0.145384 q^{89} +2.66243 q^{91} -4.51278 q^{92} +20.4846 q^{94} -0.367025 q^{97} +2.07183 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 4 q^{4} + 6 q^{7} + 3 q^{11} + 16 q^{13} - 2 q^{14} - 8 q^{16} - 8 q^{17} + 9 q^{19} + 9 q^{22} - 9 q^{23} - 8 q^{26} + 4 q^{28} + 9 q^{29} + 15 q^{31} - 7 q^{32} - 4 q^{34} + 31 q^{37}+ \cdots - 2 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 + 3 * q^11 + 16 * q^13 - 2 * q^14 - 8 * q^16 - 8 * q^17 + 9 * q^19 + 9 * q^22 - 9 * q^23 - 8 * q^26 + 4 * q^28 + 9 * q^29 + 15 * q^31 - 7 * q^32 - 4 * q^34 + 31 * q^37 + 10 * q^38 - 9 * q^41 + 26 * q^43 + 22 * q^44 - 12 * q^46 + 6 * q^49 + 7 * q^52 - 5 * q^53 - 25 * q^58 - 5 * q^59 - q^61 - 9 * q^62 - 18 * q^64 + 34 * q^67 - 22 * q^68 + 7 * q^71 + 10 * q^73 - 31 * q^74 - 8 * q^76 + 3 * q^77 - 22 * q^79 + 29 * q^82 + 25 * q^83 + 37 * q^86 + 4 * q^89 + 16 * q^91 + 34 * q^92 + 26 * q^94 + 40 * q^97 - 2 * q^98

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