LMFDB - Embedded newform 6975.2.a.cg.1.7

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 = [8,0,0,16,0,0,4,0,0,0,0,0,8,0,0,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.116450197504.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} + 58x^{4} - 62x^{2} + 1 $ x^8 - 14x^6 + 58x^4 - 62*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.127980$ 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)

$f(q)$	$=$	$q+2.10974 q^{2} +2.45100 q^{4} -2.98362 q^{7} +0.951486 q^{8} +5.06615 q^{11} +6.06661 q^{13} -6.29466 q^{14} -2.89461 q^{16} +2.18000 q^{17} +1.81298 q^{19} +10.6882 q^{22} -3.13149 q^{23} +12.7990 q^{26} -7.31285 q^{28} -4.11466 q^{29} -1.00000 q^{31} -8.00984 q^{32} +4.59923 q^{34} +3.08299 q^{37} +3.82492 q^{38} +1.15825 q^{41} +2.56538 q^{43} +12.4171 q^{44} -6.60662 q^{46} +3.02667 q^{47} +1.90199 q^{49} +14.8692 q^{52} +0.706146 q^{53} -2.83887 q^{56} -8.68086 q^{58} +7.07107 q^{59} -4.25965 q^{61} -2.10974 q^{62} -11.1094 q^{64} +1.64701 q^{67} +5.34317 q^{68} +5.93465 q^{71} +14.8692 q^{73} +6.50429 q^{74} +4.44361 q^{76} -15.1155 q^{77} -8.59923 q^{79} +2.44361 q^{82} +17.1331 q^{83} +5.41229 q^{86} +4.82037 q^{88} +9.06614 q^{89} -18.1005 q^{91} -7.67526 q^{92} +6.38548 q^{94} +8.54162 q^{97} +4.01271 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 4 q^{7} + 8 q^{13} + 12 q^{16} + 28 q^{22} + 36 q^{28} - 8 q^{31} - 28 q^{34} + 12 q^{37} + 52 q^{43} - 16 q^{46} + 8 q^{49} + 56 q^{52} + 16 q^{58} - 28 q^{61} + 48 q^{64} + 24 q^{67}+ \cdots + 44 q^{97}+O(q^{100}) $ 8 * q + 16 * q^4 + 4 * q^7 + 8 * q^13 + 12 * q^16 + 28 * q^22 + 36 * q^28 - 8 * q^31 - 28 * q^34 + 12 * q^37 + 52 * q^43 - 16 * q^46 + 8 * q^49 + 56 * q^52 + 16 * q^58 - 28 * q^61 + 48 * q^64 + 24 * q^67 + 56 * q^73 + 4 * q^76 - 4 * q^79 - 12 * q^82 + 52 * q^88 - 8 * q^91 - 64 * q^94 + 44 * q^97

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 $

$2$	2.10974	1.49181	0.745905	−	0.666052i	$-0.232017\pi$
$2$	2.10974	1.49181	0.745905	+	0.666052i	$0.232017\pi$
$3$	0	0
$3$	0	0
$4$	2.45100	1.22550
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.98362	−1.12770	−0.563851	−	0.825876i	$-0.690681\pi$
$7$	−2.98362	−1.12770	−0.563851	+	0.825876i	$0.690681\pi$
$8$	0.951486	0.336401
$9$	0	0
$10$	0	0
$11$	5.06615	1.52750	0.763750	−	0.645512i	$-0.223356\pi$
$11$	5.06615	1.52750	0.763750	+	0.645512i	$0.223356\pi$
$12$	0	0
$13$	6.06661	1.68257	0.841287	−	0.540589i	$-0.181798\pi$
$13$	6.06661	1.68257	0.841287	+	0.540589i	$0.181798\pi$
$14$	−6.29466	−1.68232
$15$	0	0
$16$	−2.89461	−0.723652
$17$	2.18000	0.528728	0.264364	−	0.964423i	$-0.414838\pi$
$17$	2.18000	0.528728	0.264364	+	0.964423i	$0.414838\pi$
$18$	0	0
$19$	1.81298	0.415926	0.207963	−	0.978137i	$-0.433317\pi$
$19$	1.81298	0.415926	0.207963	+	0.978137i	$0.433317\pi$
$20$	0	0
$21$	0	0
$22$	10.6882	2.27874
$23$	−3.13149	−0.652960	−0.326480	−	0.945204i	$-0.605863\pi$
$23$	−3.13149	−0.652960	−0.326480	+	0.945204i	$0.605863\pi$
$24$	0	0
$25$	0	0
$26$	12.7990	2.51008
$27$	0	0
$28$	−7.31285	−1.38200
$29$	−4.11466	−0.764073	−0.382037	−	0.924147i	$-0.624777\pi$
$29$	−4.11466	−0.764073	−0.382037	+	0.924147i	$0.624777\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−8.00984	−1.41595
$33$	0	0
$34$	4.59923	0.788762
$35$	0	0
$36$	0	0
$37$	3.08299	0.506840	0.253420	−	0.967356i	$-0.418445\pi$
$37$	3.08299	0.506840	0.253420	+	0.967356i	$0.418445\pi$
$38$	3.82492	0.620483
$39$	0	0
$40$	0	0
$41$	1.15825	0.180889	0.0904443	−	0.995902i	$-0.471171\pi$
$41$	1.15825	0.180889	0.0904443	+	0.995902i	$0.471171\pi$
$42$	0	0
$43$	2.56538	0.391217	0.195609	−	0.980682i	$-0.437332\pi$
$43$	2.56538	0.391217	0.195609	+	0.980682i	$0.437332\pi$
$44$	12.4171	1.87195
$45$	0	0
$46$	−6.60662	−0.974093
$47$	3.02667	0.441485	0.220743	−	0.975332i	$-0.429152\pi$
$47$	3.02667	0.441485	0.220743	+	0.975332i	$0.429152\pi$
$48$	0	0
$49$	1.90199	0.271713
$50$	0	0
$51$	0	0
$52$	14.8692	2.06199
$53$	0.706146	0.0969966	0.0484983	−	0.998823i	$-0.484556\pi$
$53$	0.706146	0.0969966	0.0484983	+	0.998823i	$0.484556\pi$
$54$	0	0
$55$	0	0
$56$	−2.83887	−0.379360
$57$	0	0
$58$	−8.68086	−1.13985
$59$	7.07107	0.920575	0.460287	−	0.887770i	$-0.347746\pi$
$59$	7.07107	0.920575	0.460287	+	0.887770i	$0.347746\pi$
$60$	0	0
$61$	−4.25965	−0.545393	−0.272696	−	0.962100i	$-0.587915\pi$
$61$	−4.25965	−0.545393	−0.272696	+	0.962100i	$0.587915\pi$
$62$	−2.10974	−0.267937
$63$	0	0
$64$	−11.1094	−1.38868
$65$	0	0
$66$	0	0
$67$	1.64701	0.201214	0.100607	−	0.994926i	$-0.467922\pi$
$67$	1.64701	0.201214	0.100607	+	0.994926i	$0.467922\pi$
$68$	5.34317	0.647955
$69$	0	0
$70$	0	0
$71$	5.93465	0.704314	0.352157	−	0.935941i	$-0.385448\pi$
$71$	5.93465	0.704314	0.352157	+	0.935941i	$0.385448\pi$
$72$	0	0
$73$	14.8692	1.74031	0.870156	−	0.492776i	$-0.164018\pi$
$73$	14.8692	1.74031	0.870156	+	0.492776i	$0.164018\pi$
$74$	6.50429	0.756109
$75$	0	0
$76$	4.44361	0.509717
$77$	−15.1155	−1.72257
$78$	0	0
$79$	−8.59923	−0.967489	−0.483745	−	0.875209i	$-0.660724\pi$
$79$	−8.59923	−0.967489	−0.483745	+	0.875209i	$0.660724\pi$
$80$	0	0
$81$	0	0
$82$	2.44361	0.269852
$83$	17.1331	1.88060	0.940301	−	0.340344i	$-0.110544\pi$
$83$	17.1331	1.88060	0.940301	+	0.340344i	$0.110544\pi$
$84$	0	0
$85$	0	0
$86$	5.41229	0.583622
$87$	0	0
$88$	4.82037	0.513853
$89$	9.06614	0.961009	0.480504	−	0.876992i	$-0.340454\pi$
$89$	9.06614	0.961009	0.480504	+	0.876992i	$0.340454\pi$
$90$	0	0
$91$	−18.1005	−1.89744
$92$	−7.67526	−0.800202
$93$	0	0
$94$	6.38548	0.658612
$95$	0	0
$96$	0	0
$97$	8.54162	0.867270	0.433635	−	0.901089i	$-0.357231\pi$
$97$	8.54162	0.867270	0.433635	+	0.901089i	$0.357231\pi$
$98$	4.01271	0.405345
$99$	0	0
$100$	0	0
$101$	−4.36000	−0.433836	−0.216918	−	0.976190i	$-0.569600\pi$
$101$	−4.36000	−0.433836	−0.216918	+	0.976190i	$0.569600\pi$
$102$	0	0
$103$	18.8971	1.86198	0.930991	−	0.365042i	$-0.118945\pi$
$103$	18.8971	1.86198	0.930991	+	0.365042i	$0.118945\pi$
$104$	5.77229	0.566020
$105$	0	0
$106$	1.48978	0.144700
$107$	12.7644	1.23398	0.616991	−	0.786971i	$-0.288352\pi$
$107$	12.7644	1.23398	0.616991	+	0.786971i	$0.288352\pi$
$108$	0	0
$109$	−6.61425	−0.633530	−0.316765	−	0.948504i	$-0.602597\pi$
$109$	−6.61425	−0.633530	−0.316765	+	0.948504i	$0.602597\pi$
$110$	0	0
$111$	0	0
$112$	8.63641	0.816064
$113$	−5.53097	−0.520310	−0.260155	−	0.965567i	$-0.583774\pi$
$113$	−5.53097	−0.520310	−0.260155	+	0.965567i	$0.583774\pi$
$114$	0	0
$115$	0	0
$116$	−10.0850	−0.936371
$117$	0	0
$118$	14.9181	1.37332
$119$	−6.50429	−0.596248
$120$	0	0
$121$	14.6658	1.33326
$122$	−8.98676	−0.813623
$123$	0	0
$124$	−2.45100	−0.220106
$125$	0	0
$126$	0	0
$127$	−7.68222	−0.681687	−0.340843	−	0.940120i	$-0.610713\pi$
$127$	−7.68222	−0.681687	−0.340843	+	0.940120i	$0.610713\pi$
$128$	−7.41836	−0.655696
$129$	0	0
$130$	0	0
$131$	9.02757	0.788742	0.394371	−	0.918951i	$-0.370962\pi$
$131$	9.02757	0.788742	0.394371	+	0.918951i	$0.370962\pi$
$132$	0	0
$133$	−5.40925	−0.469041
$134$	3.47476	0.300173
$135$	0	0
$136$	2.07424	0.177865
$137$	6.92185	0.591373	0.295687	−	0.955285i	$-0.404452\pi$
$137$	6.92185	0.591373	0.295687	+	0.955285i	$0.404452\pi$
$138$	0	0
$139$	−12.7324	−1.07995	−0.539976	−	0.841680i	$-0.681567\pi$
$139$	−12.7324	−1.07995	−0.539976	+	0.841680i	$0.681567\pi$
$140$	0	0
$141$	0	0
$142$	12.5206	1.05070
$143$	30.7343	2.57013
$144$	0	0
$145$	0	0
$146$	31.3702	2.59622
$147$	0	0
$148$	7.55639	0.621131
$149$	−10.8516	−0.888996	−0.444498	−	0.895780i	$-0.646618\pi$
$149$	−10.8516	−0.888996	−0.444498	+	0.895780i	$0.646618\pi$
$150$	0	0
$151$	−4.81298	−0.391675	−0.195837	−	0.980636i	$-0.562742\pi$
$151$	−4.81298	−0.391675	−0.195837	+	0.980636i	$0.562742\pi$
$152$	1.72503	0.139918
$153$	0	0
$154$	−31.8897	−2.56974
$155$	0	0
$156$	0	0
$157$	−22.0513	−1.75989	−0.879943	−	0.475078i	$-0.842420\pi$
$157$	−22.0513	−1.75989	−0.879943	+	0.475078i	$0.842420\pi$
$158$	−18.1421	−1.44331
$159$	0	0
$160$	0	0
$161$	9.34317	0.736345
$162$	0	0
$163$	11.3366	0.887952	0.443976	−	0.896039i	$-0.353568\pi$
$163$	11.3366	0.887952	0.443976	+	0.896039i	$0.353568\pi$
$164$	2.83887	0.221679
$165$	0	0
$166$	36.1464	2.80550
$167$	−15.4973	−1.19922	−0.599609	−	0.800293i	$-0.704677\pi$
$167$	−15.4973	−1.19922	−0.599609	+	0.800293i	$0.704677\pi$
$168$	0	0
$169$	23.8037	1.83106
$170$	0	0
$171$	0	0
$172$	6.28774	0.479436
$173$	20.4615	1.55566	0.777830	−	0.628475i	$-0.216321\pi$
$173$	20.4615	1.55566	0.777830	+	0.628475i	$0.216321\pi$
$174$	0	0
$175$	0	0
$176$	−14.6645	−1.10538
$177$	0	0
$178$	19.1272	1.43364
$179$	−10.2411	−0.765458	−0.382729	−	0.923861i	$-0.625016\pi$
$179$	−10.2411	−0.765458	−0.382729	+	0.923861i	$0.625016\pi$
$180$	0	0
$181$	15.1955	1.12947	0.564736	−	0.825271i	$-0.308978\pi$
$181$	15.1955	1.12947	0.564736	+	0.825271i	$0.308978\pi$
$182$	−38.1872	−2.83063
$183$	0	0
$184$	−2.97957	−0.219656
$185$	0	0
$186$	0	0
$187$	11.0442	0.807632
$188$	7.41836	0.541039
$189$	0	0
$190$	0	0
$191$	−10.6892	−0.773445	−0.386722	−	0.922196i	$-0.626393\pi$
$191$	−10.6892	−0.773445	−0.386722	+	0.922196i	$0.626393\pi$
$192$	0	0
$193$	8.44667	0.608005	0.304002	−	0.952671i	$-0.401677\pi$
$193$	8.44667	0.608005	0.304002	+	0.952671i	$0.401677\pi$
$194$	18.0206	1.29380
$195$	0	0
$196$	4.66178	0.332984
$197$	−24.6147	−1.75373	−0.876864	−	0.480739i	$-0.840368\pi$
$197$	−24.6147	−1.75373	−0.876864	+	0.480739i	$0.840368\pi$
$198$	0	0
$199$	2.82333	0.200141	0.100070	−	0.994980i	$-0.468093\pi$
$199$	2.82333	0.200141	0.100070	+	0.994980i	$0.468093\pi$
$200$	0	0
$201$	0	0
$202$	−9.19846	−0.647202
$203$	12.2766	0.861647
$204$	0	0
$205$	0	0
$206$	39.8679	2.77772
$207$	0	0
$208$	−17.5604	−1.21760
$209$	9.18482	0.635328
$210$	0	0
$211$	23.1379	1.59288	0.796439	−	0.604719i	$-0.206715\pi$
$211$	23.1379	1.59288	0.796439	+	0.604719i	$0.206715\pi$
$212$	1.73076	0.118869
$213$	0	0
$214$	26.9295	1.84087
$215$	0	0
$216$	0	0
$217$	2.98362	0.202541
$218$	−13.9543	−0.945107
$219$	0	0
$220$	0	0
$221$	13.2252	0.889624
$222$	0	0
$223$	5.25659	0.352007	0.176004	−	0.984390i	$-0.443683\pi$
$223$	5.25659	0.352007	0.176004	+	0.984390i	$0.443683\pi$
$224$	23.8983	1.59677
$225$	0	0
$226$	−11.6689	−0.776204
$227$	−18.7019	−1.24129	−0.620645	−	0.784092i	$-0.713129\pi$
$227$	−18.7019	−1.24129	−0.620645	+	0.784092i	$0.713129\pi$
$228$	0	0
$229$	11.6418	0.769314	0.384657	−	0.923060i	$-0.374320\pi$
$229$	11.6418	0.769314	0.384657	+	0.923060i	$0.374320\pi$
$230$	0	0
$231$	0	0
$232$	−3.91504	−0.257035
$233$	19.0936	1.25086	0.625432	−	0.780279i	$-0.284923\pi$
$233$	19.0936	1.25086	0.625432	+	0.780279i	$0.284923\pi$
$234$	0	0
$235$	0	0
$236$	17.3312	1.12816
$237$	0	0
$238$	−13.7224	−0.889489
$239$	−18.4446	−1.19308	−0.596541	−	0.802583i	$-0.703459\pi$
$239$	−18.4446	−1.19308	−0.596541	+	0.802583i	$0.703459\pi$
$240$	0	0
$241$	10.8515	0.699006	0.349503	−	0.936935i	$-0.386350\pi$
$241$	10.8515	0.699006	0.349503	+	0.936935i	$0.386350\pi$
$242$	30.9411	1.98897
$243$	0	0
$244$	−10.4404	−0.668378
$245$	0	0
$246$	0	0
$247$	10.9986	0.699827
$248$	−0.951486	−0.0604194
$249$	0	0
$250$	0	0
$251$	−12.8565	−0.811495	−0.405748	−	0.913985i	$-0.632989\pi$
$251$	−12.8565	−0.811495	−0.405748	+	0.913985i	$0.632989\pi$
$252$	0	0
$253$	−15.8646	−0.997397
$254$	−16.2075	−1.01695
$255$	0	0
$256$	6.56810	0.410506
$257$	12.0225	0.749946	0.374973	−	0.927036i	$-0.377652\pi$
$257$	12.0225	0.749946	0.374973	+	0.927036i	$0.377652\pi$
$258$	0	0
$259$	−9.19846	−0.571565
$260$	0	0
$261$	0	0
$262$	19.0458	1.17665
$263$	−3.38668	−0.208831	−0.104416	−	0.994534i	$-0.533297\pi$
$263$	−3.38668	−0.208831	−0.104416	+	0.994534i	$0.533297\pi$
$264$	0	0
$265$	0	0
$266$	−11.4121	−0.699721
$267$	0	0
$268$	4.03681	0.246588
$269$	30.9094	1.88458	0.942290	−	0.334799i	$-0.108668\pi$
$269$	30.9094	1.88458	0.942290	+	0.334799i	$0.108668\pi$
$270$	0	0
$271$	−22.2888	−1.35395	−0.676975	−	0.736006i	$-0.736710\pi$
$271$	−22.2888	−1.35395	−0.676975	+	0.736006i	$0.736710\pi$
$272$	−6.31024	−0.382615
$273$	0	0
$274$	14.6033	0.882217
$275$	0	0
$276$	0	0
$277$	28.9817	1.74134	0.870672	−	0.491864i	$-0.163684\pi$
$277$	28.9817	1.74134	0.870672	+	0.491864i	$0.163684\pi$
$278$	−26.8621	−1.61108
$279$	0	0
$280$	0	0
$281$	17.8910	1.06729	0.533643	−	0.845710i	$-0.320823\pi$
$281$	17.8910	1.06729	0.533643	+	0.845710i	$0.320823\pi$
$282$	0	0
$283$	−22.5144	−1.33834	−0.669170	−	0.743109i	$-0.733350\pi$
$283$	−22.5144	−1.33834	−0.669170	+	0.743109i	$0.733350\pi$
$284$	14.5458	0.863136
$285$	0	0
$286$	64.8414	3.83415
$287$	−3.45579	−0.203989
$288$	0	0
$289$	−12.2476	−0.720447
$290$	0	0
$291$	0	0
$292$	36.4445	2.13275
$293$	−5.51825	−0.322380	−0.161190	−	0.986923i	$-0.551533\pi$
$293$	−5.51825	−0.322380	−0.161190	+	0.986923i	$0.551533\pi$
$294$	0	0
$295$	0	0
$296$	2.93342	0.170501
$297$	0	0
$298$	−22.8940	−1.32621
$299$	−18.9975	−1.09865
$300$	0	0
$301$	−7.65413	−0.441177
$302$	−10.1541	−0.584305
$303$	0	0
$304$	−5.24787	−0.300986
$305$	0	0
$306$	0	0
$307$	−7.26398	−0.414577	−0.207289	−	0.978280i	$-0.566464\pi$
$307$	−7.26398	−0.414577	−0.207289	+	0.978280i	$0.566464\pi$
$308$	−37.0479	−2.11100
$309$	0	0
$310$	0	0
$311$	−12.4262	−0.704627	−0.352313	−	0.935882i	$-0.614605\pi$
$311$	−12.4262	−0.704627	−0.352313	+	0.935882i	$0.614605\pi$
$312$	0	0
$313$	−8.95655	−0.506254	−0.253127	−	0.967433i	$-0.581459\pi$
$313$	−8.95655	−0.506254	−0.253127	+	0.967433i	$0.581459\pi$
$314$	−46.5225	−2.62542
$315$	0	0
$316$	−21.0767	−1.18566
$317$	−24.3625	−1.36833	−0.684167	−	0.729325i	$-0.739834\pi$
$317$	−24.3625	−1.36833	−0.684167	+	0.729325i	$0.739834\pi$
$318$	0	0
$319$	−20.8455	−1.16712
$320$	0	0
$321$	0	0
$322$	19.7116	1.09849
$323$	3.95230	0.219912
$324$	0	0
$325$	0	0
$326$	23.9173	1.32466
$327$	0	0
$328$	1.10206	0.0608511
$329$	−9.03043	−0.497864
$330$	0	0
$331$	−10.3751	−0.570269	−0.285134	−	0.958488i	$-0.592038\pi$
$331$	−10.3751	−0.570269	−0.285134	+	0.958488i	$0.592038\pi$
$332$	41.9932	2.30467
$333$	0	0
$334$	−32.6953	−1.78901
$335$	0	0
$336$	0	0
$337$	20.1302	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$337$	20.1302	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$338$	50.2196	2.73159
$339$	0	0
$340$	0	0
$341$	−5.06615	−0.274347
$342$	0	0
$343$	15.2105	0.821291
$344$	2.44093	0.131606
$345$	0	0
$346$	43.1684	2.32075
$347$	31.7520	1.70454	0.852270	−	0.523103i	$-0.175226\pi$
$347$	31.7520	1.70454	0.852270	+	0.523103i	$0.175226\pi$
$348$	0	0
$349$	7.60958	0.407332	0.203666	−	0.979040i	$-0.434714\pi$
$349$	7.60958	0.407332	0.203666	+	0.979040i	$0.434714\pi$
$350$	0	0
$351$	0	0
$352$	−40.5790	−2.16287
$353$	−8.80298	−0.468535	−0.234268	−	0.972172i	$-0.575269\pi$
$353$	−8.80298	−0.468535	−0.234268	+	0.972172i	$0.575269\pi$
$354$	0	0
$355$	0	0
$356$	22.2211	1.17772
$357$	0	0
$358$	−21.6061	−1.14192
$359$	−1.08216	−0.0571142	−0.0285571	−	0.999592i	$-0.509091\pi$
$359$	−1.08216	−0.0571142	−0.0285571	+	0.999592i	$0.509091\pi$
$360$	0	0
$361$	−15.7131	−0.827005
$362$	32.0585	1.68496
$363$	0	0
$364$	−44.3642	−2.32531
$365$	0	0
$366$	0	0
$367$	−9.87823	−0.515639	−0.257820	−	0.966193i	$-0.583004\pi$
$367$	−9.87823	−0.515639	−0.257820	+	0.966193i	$0.583004\pi$
$368$	9.06442	0.472516
$369$	0	0
$370$	0	0
$371$	−2.10687	−0.109383
$372$	0	0
$373$	7.53729	0.390266	0.195133	−	0.980777i	$-0.437486\pi$
$373$	7.53729	0.390266	0.195133	+	0.980777i	$0.437486\pi$
$374$	23.3004	1.20483
$375$	0	0
$376$	2.87983	0.148516
$377$	−24.9620	−1.28561
$378$	0	0
$379$	27.3110	1.40287	0.701435	−	0.712733i	$-0.252543\pi$
$379$	27.3110	1.40287	0.701435	+	0.712733i	$0.252543\pi$
$380$	0	0
$381$	0	0
$382$	−22.5515	−1.15383
$383$	−34.0824	−1.74153	−0.870765	−	0.491699i	$-0.836376\pi$
$383$	−34.0824	−1.74153	−0.870765	+	0.491699i	$0.836376\pi$
$384$	0	0
$385$	0	0
$386$	17.8203	0.907028
$387$	0	0
$388$	20.9355	1.06284
$389$	−27.8827	−1.41371	−0.706855	−	0.707358i	$-0.749887\pi$
$389$	−27.8827	−1.41371	−0.706855	+	0.707358i	$0.749887\pi$
$390$	0	0
$391$	−6.82664	−0.345238
$392$	1.80972	0.0914047
$393$	0	0
$394$	−51.9307	−2.61623
$395$	0	0
$396$	0	0
$397$	−20.5171	−1.02972	−0.514862	−	0.857273i	$-0.672157\pi$
$397$	−20.5171	−1.02972	−0.514862	+	0.857273i	$0.672157\pi$
$398$	5.95649	0.298572
$399$	0	0
$400$	0	0
$401$	−22.2942	−1.11332	−0.556660	−	0.830740i	$-0.687917\pi$
$401$	−22.2942	−1.11332	−0.556660	+	0.830740i	$0.687917\pi$
$402$	0	0
$403$	−6.06661	−0.302199
$404$	−10.6863	−0.531666
$405$	0	0
$406$	25.9004	1.28541
$407$	15.6189	0.774198
$408$	0	0
$409$	−11.1168	−0.549692	−0.274846	−	0.961488i	$-0.588627\pi$
$409$	−11.1168	−0.549692	−0.274846	+	0.961488i	$0.588627\pi$
$410$	0	0
$411$	0	0
$412$	46.3166	2.28186
$413$	−21.0974	−1.03813
$414$	0	0
$415$	0	0
$416$	−48.5925	−2.38244
$417$	0	0
$418$	19.3776	0.947788
$419$	−8.25116	−0.403095	−0.201548	−	0.979479i	$-0.564597\pi$
$419$	−8.25116	−0.403095	−0.201548	+	0.979479i	$0.564597\pi$
$420$	0	0
$421$	20.6631	1.00706	0.503529	−	0.863978i	$-0.332035\pi$
$421$	20.6631	1.00706	0.503529	+	0.863978i	$0.332035\pi$
$422$	48.8149	2.37627
$423$	0	0
$424$	0.671888	0.0326298
$425$	0	0
$426$	0	0
$427$	12.7092	0.615041
$428$	31.2855	1.51224
$429$	0	0
$430$	0	0
$431$	0.100799	0.00485530	0.00242765	−	0.999997i	$-0.499227\pi$
$431$	0.100799	0.00485530	0.00242765	+	0.999997i	$0.499227\pi$
$432$	0	0
$433$	−14.0188	−0.673702	−0.336851	−	0.941558i	$-0.609362\pi$
$433$	−14.0188	−0.673702	−0.336851	+	0.941558i	$0.609362\pi$
$434$	6.29466	0.302153
$435$	0	0
$436$	−16.2115	−0.776390
$437$	−5.67732	−0.271583
$438$	0	0
$439$	8.96564	0.427906	0.213953	−	0.976844i	$-0.431366\pi$
$439$	8.96564	0.427906	0.213953	+	0.976844i	$0.431366\pi$
$440$	0	0
$441$	0	0
$442$	27.9017	1.32715
$443$	40.4058	1.91974	0.959869	−	0.280449i	$-0.0904834\pi$
$443$	40.4058	1.91974	0.959869	+	0.280449i	$0.0904834\pi$
$444$	0	0
$445$	0	0
$446$	11.0900	0.525128
$447$	0	0
$448$	33.1464	1.56602
$449$	8.12450	0.383419	0.191710	−	0.981452i	$-0.438597\pi$
$449$	8.12450	0.383419	0.191710	+	0.981452i	$0.438597\pi$
$450$	0	0
$451$	5.86788	0.276308
$452$	−13.5564	−0.637639
$453$	0	0
$454$	−39.4562	−1.85177
$455$	0	0
$456$	0	0
$457$	−0.134305	−0.00628252	−0.00314126	−	0.999995i	$-0.501000\pi$
$457$	−0.134305	−0.00628252	−0.00314126	+	0.999995i	$0.501000\pi$
$458$	24.5612	1.14767
$459$	0	0
$460$	0	0
$461$	−4.57547	−0.213101	−0.106550	−	0.994307i	$-0.533981\pi$
$461$	−4.57547	−0.213101	−0.106550	+	0.994307i	$0.533981\pi$
$462$	0	0
$463$	−22.4016	−1.04109	−0.520545	−	0.853834i	$-0.674271\pi$
$463$	−22.4016	−1.04109	−0.520545	+	0.853834i	$0.674271\pi$
$464$	11.9103	0.552923
$465$	0	0
$466$	40.2825	1.86605
$467$	−14.1640	−0.655431	−0.327715	−	0.944777i	$-0.606279\pi$
$467$	−14.1640	−0.655431	−0.327715	+	0.944777i	$0.606279\pi$
$468$	0	0
$469$	−4.91405	−0.226910
$470$	0	0
$471$	0	0
$472$	6.72802	0.309682
$473$	12.9966	0.597584
$474$	0	0
$475$	0	0
$476$	−15.9420	−0.730701
$477$	0	0
$478$	−38.9133	−1.77985
$479$	−9.49124	−0.433666	−0.216833	−	0.976209i	$-0.569573\pi$
$479$	−9.49124	−0.433666	−0.216833	+	0.976209i	$0.569573\pi$
$480$	0	0
$481$	18.7033	0.852795
$482$	22.8938	1.04279
$483$	0	0
$484$	35.9459	1.63391
$485$	0	0
$486$	0	0
$487$	−5.80228	−0.262927	−0.131463	−	0.991321i	$-0.541968\pi$
$487$	−5.80228	−0.262927	−0.131463	+	0.991321i	$0.541968\pi$
$488$	−4.05300	−0.183471
$489$	0	0
$490$	0	0
$491$	20.8128	0.939269	0.469634	−	0.882861i	$-0.344386\pi$
$491$	20.8128	0.939269	0.469634	+	0.882861i	$0.344386\pi$
$492$	0	0
$493$	−8.96996	−0.403987
$494$	23.2043	1.04401
$495$	0	0
$496$	2.89461	0.129972
$497$	−17.7068	−0.794257
$498$	0	0
$499$	−42.5553	−1.90504	−0.952518	−	0.304484i	$-0.901516\pi$
$499$	−42.5553	−1.90504	−0.952518	+	0.304484i	$0.901516\pi$
$500$	0	0
$501$	0	0
$502$	−27.1239	−1.21060
$503$	−22.0569	−0.983469	−0.491734	−	0.870745i	$-0.663637\pi$
$503$	−22.0569	−0.983469	−0.491734	+	0.870745i	$0.663637\pi$
$504$	0	0
$505$	0	0
$506$	−33.4701	−1.48793
$507$	0	0
$508$	−18.8291	−0.835406
$509$	8.36401	0.370728	0.185364	−	0.982670i	$-0.440654\pi$
$509$	8.36401	0.370728	0.185364	+	0.982670i	$0.440654\pi$
$510$	0	0
$511$	−44.3642	−1.96256
$512$	28.6937	1.26809
$513$	0	0
$514$	25.3644	1.11878
$515$	0	0
$516$	0	0
$517$	15.3335	0.674369
$518$	−19.4064	−0.852666
$519$	0	0
$520$	0	0
$521$	−30.0657	−1.31720	−0.658601	−	0.752492i	$-0.728851\pi$
$521$	−30.0657	−1.31720	−0.658601	+	0.752492i	$0.728851\pi$
$522$	0	0
$523$	−18.5635	−0.811726	−0.405863	−	0.913934i	$-0.633029\pi$
$523$	−18.5635	−0.811726	−0.405863	+	0.913934i	$0.633029\pi$
$524$	22.1265	0.966602
$525$	0	0
$526$	−7.14500	−0.311537
$527$	−2.18000	−0.0949623
$528$	0	0
$529$	−13.1938	−0.573643
$530$	0	0
$531$	0	0
$532$	−13.2580	−0.574809
$533$	7.02666	0.304359
$534$	0	0
$535$	0	0
$536$	1.56711	0.0676887
$537$	0	0
$538$	65.2107	2.81144
$539$	9.63578	0.415042
$540$	0	0
$541$	40.6074	1.74585	0.872925	−	0.487854i	$-0.162220\pi$
$541$	40.6074	1.74585	0.872925	+	0.487854i	$0.162220\pi$
$542$	−47.0236	−2.01984
$543$	0	0
$544$	−17.4614	−0.748653
$545$	0	0
$546$	0	0
$547$	−5.02809	−0.214986	−0.107493	−	0.994206i	$-0.534282\pi$
$547$	−5.02809	−0.214986	−0.107493	+	0.994206i	$0.534282\pi$
$548$	16.9654	0.724727
$549$	0	0
$550$	0	0
$551$	−7.45980	−0.317798
$552$	0	0
$553$	25.6568	1.09104
$554$	61.1439	2.59776
$555$	0	0
$556$	−31.2072	−1.32348
$557$	−34.8243	−1.47555	−0.737776	−	0.675046i	$-0.764124\pi$
$557$	−34.8243	−1.47555	−0.737776	+	0.675046i	$0.764124\pi$
$558$	0	0
$559$	15.5632	0.658252
$560$	0	0
$561$	0	0
$562$	37.7452	1.59219
$563$	−25.5236	−1.07569	−0.537846	−	0.843043i	$-0.680762\pi$
$563$	−25.5236	−1.07569	−0.537846	+	0.843043i	$0.680762\pi$
$564$	0	0
$565$	0	0
$566$	−47.4994	−1.99655
$567$	0	0
$568$	5.64674	0.236932
$569$	−37.9892	−1.59259	−0.796294	−	0.604909i	$-0.793209\pi$
$569$	−37.9892	−1.59259	−0.796294	+	0.604909i	$0.793209\pi$
$570$	0	0
$571$	−37.5218	−1.57024	−0.785118	−	0.619346i	$-0.787398\pi$
$571$	−37.5218	−1.57024	−0.785118	+	0.619346i	$0.787398\pi$
$572$	75.3297	3.14969
$573$	0	0
$574$	−7.29081	−0.304312
$575$	0	0
$576$	0	0
$577$	16.5662	0.689661	0.344831	−	0.938665i	$-0.387936\pi$
$577$	16.5662	0.689661	0.344831	+	0.938665i	$0.387936\pi$
$578$	−25.8392	−1.07477
$579$	0	0
$580$	0	0
$581$	−51.1187	−2.12076
$582$	0	0
$583$	3.57744	0.148162
$584$	14.1479	0.585443
$585$	0	0
$586$	−11.6421	−0.480930
$587$	−33.7126	−1.39147	−0.695733	−	0.718300i	$-0.744920\pi$
$587$	−33.7126	−1.39147	−0.695733	+	0.718300i	$0.744920\pi$
$588$	0	0
$589$	−1.81298	−0.0747026
$590$	0	0
$591$	0	0
$592$	−8.92403	−0.366776
$593$	−32.0775	−1.31726	−0.658632	−	0.752465i	$-0.728865\pi$
$593$	−32.0775	−1.31726	−0.658632	+	0.752465i	$0.728865\pi$
$594$	0	0
$595$	0	0
$596$	−26.5972	−1.08946
$597$	0	0
$598$	−40.0798	−1.63898
$599$	−40.7042	−1.66313	−0.831565	−	0.555427i	$-0.812555\pi$
$599$	−40.7042	−1.66313	−0.831565	+	0.555427i	$0.812555\pi$
$600$	0	0
$601$	−20.5550	−0.838457	−0.419229	−	0.907881i	$-0.637699\pi$
$601$	−20.5550	−0.838457	−0.419229	+	0.907881i	$0.637699\pi$
$602$	−16.1482	−0.658152
$603$	0	0
$604$	−11.7966	−0.479997
$605$	0	0
$606$	0	0
$607$	−6.80866	−0.276355	−0.138177	−	0.990407i	$-0.544124\pi$
$607$	−6.80866	−0.276355	−0.138177	+	0.990407i	$0.544124\pi$
$608$	−14.5217	−0.588932
$609$	0	0
$610$	0	0
$611$	18.3616	0.742831
$612$	0	0
$613$	43.9292	1.77428	0.887142	−	0.461497i	$-0.152687\pi$
$613$	43.9292	1.77428	0.887142	+	0.461497i	$0.152687\pi$
$614$	−15.3251	−0.618470
$615$	0	0
$616$	−14.3821	−0.579473
$617$	−31.5671	−1.27084	−0.635422	−	0.772165i	$-0.719174\pi$
$617$	−31.5671	−1.27084	−0.635422	+	0.772165i	$0.719174\pi$
$618$	0	0
$619$	−4.32456	−0.173819	−0.0869093	−	0.996216i	$-0.527699\pi$
$619$	−4.32456	−0.173819	−0.0869093	+	0.996216i	$0.527699\pi$
$620$	0	0
$621$	0	0
$622$	−26.2161	−1.05117
$623$	−27.0499	−1.08373
$624$	0	0
$625$	0	0
$626$	−18.8960	−0.755235
$627$	0	0
$628$	−54.0477	−2.15674
$629$	6.72091	0.267980
$630$	0	0
$631$	−27.3928	−1.09049	−0.545244	−	0.838277i	$-0.683563\pi$
$631$	−27.3928	−1.09049	−0.545244	+	0.838277i	$0.683563\pi$
$632$	−8.18205	−0.325464
$633$	0	0
$634$	−51.3985	−2.04130
$635$	0	0
$636$	0	0
$637$	11.5387	0.457178
$638$	−43.9785	−1.74113
$639$	0	0
$640$	0	0
$641$	12.0491	0.475912	0.237956	−	0.971276i	$-0.423523\pi$
$641$	12.0491	0.475912	0.237956	+	0.971276i	$0.423523\pi$
$642$	0	0
$643$	39.0155	1.53862	0.769311	−	0.638875i	$-0.220600\pi$
$643$	39.0155	1.53862	0.769311	+	0.638875i	$0.220600\pi$
$644$	22.9001	0.902390
$645$	0	0
$646$	8.33832	0.328067
$647$	11.3925	0.447885	0.223943	−	0.974602i	$-0.428107\pi$
$647$	11.3925	0.447885	0.223943	+	0.974602i	$0.428107\pi$
$648$	0	0
$649$	35.8231	1.40618
$650$	0	0
$651$	0	0
$652$	27.7860	1.08818
$653$	2.92070	0.114296	0.0571479	−	0.998366i	$-0.481799\pi$
$653$	2.92070	0.114296	0.0571479	+	0.998366i	$0.481799\pi$
$654$	0	0
$655$	0	0
$656$	−3.35269	−0.130900
$657$	0	0
$658$	−19.0519	−0.742719
$659$	−42.4187	−1.65240	−0.826199	−	0.563378i	$-0.809501\pi$
$659$	−42.4187	−1.65240	−0.826199	+	0.563378i	$0.809501\pi$
$660$	0	0
$661$	31.1005	1.20967	0.604834	−	0.796352i	$-0.293240\pi$
$661$	31.1005	1.20967	0.604834	+	0.796352i	$0.293240\pi$
$662$	−21.8888	−0.850733
$663$	0	0
$664$	16.3019	0.632637
$665$	0	0
$666$	0	0
$667$	12.8850	0.498909
$668$	−37.9839	−1.46964
$669$	0	0
$670$	0	0
$671$	−21.5800	−0.833088
$672$	0	0
$673$	6.87220	0.264904	0.132452	−	0.991189i	$-0.457715\pi$
$673$	6.87220	0.264904	0.132452	+	0.991189i	$0.457715\pi$
$674$	42.4696	1.63587
$675$	0	0
$676$	58.3428	2.24396
$677$	29.9106	1.14956	0.574780	−	0.818308i	$-0.305088\pi$
$677$	29.9106	1.14956	0.574780	+	0.818308i	$0.305088\pi$
$678$	0	0
$679$	−25.4849	−0.978022
$680$	0	0
$681$	0	0
$682$	−10.6882	−0.409274
$683$	−43.0397	−1.64687	−0.823435	−	0.567411i	$-0.807945\pi$
$683$	−43.0397	−1.64687	−0.823435	+	0.567411i	$0.807945\pi$
$684$	0	0
$685$	0	0
$686$	32.0902	1.22521
$687$	0	0
$688$	−7.42577	−0.283105
$689$	4.28391	0.163204
$690$	0	0
$691$	−43.2176	−1.64408	−0.822038	−	0.569432i	$-0.807163\pi$
$691$	−43.2176	−1.64408	−0.822038	+	0.569432i	$0.807163\pi$
$692$	50.1511	1.90646
$693$	0	0
$694$	66.9885	2.54285
$695$	0	0
$696$	0	0
$697$	2.52499	0.0956409
$698$	16.0542	0.607662
$699$	0	0
$700$	0	0
$701$	−36.7796	−1.38915	−0.694574	−	0.719422i	$-0.744407\pi$
$701$	−36.7796	−1.38915	−0.694574	+	0.719422i	$0.744407\pi$
$702$	0	0
$703$	5.58939	0.210808
$704$	−56.2821	−2.12121
$705$	0	0
$706$	−18.5720	−0.698966
$707$	13.0086	0.489238
$708$	0	0
$709$	26.7766	1.00562	0.502809	−	0.864398i	$-0.332300\pi$
$709$	26.7766	1.00562	0.502809	+	0.864398i	$0.332300\pi$
$710$	0	0
$711$	0	0
$712$	8.62631	0.323284
$713$	3.13149	0.117275
$714$	0	0
$715$	0	0
$716$	−25.1010	−0.938067
$717$	0	0
$718$	−2.28308	−0.0852036
$719$	−8.08880	−0.301661	−0.150831	−	0.988560i	$-0.548195\pi$
$719$	−8.08880	−0.301661	−0.150831	+	0.988560i	$0.548195\pi$
$720$	0	0
$721$	−56.3817	−2.09976
$722$	−33.1505	−1.23374
$723$	0	0
$724$	37.2441	1.38417
$725$	0	0
$726$	0	0
$727$	−2.33194	−0.0864870	−0.0432435	−	0.999065i	$-0.513769\pi$
$727$	−2.33194	−0.0864870	−0.0432435	+	0.999065i	$0.513769\pi$
$728$	−17.2223	−0.638302
$729$	0	0
$730$	0	0
$731$	5.59253	0.206847
$732$	0	0
$733$	4.98202	0.184015	0.0920075	−	0.995758i	$-0.470672\pi$
$733$	4.98202	0.184015	0.0920075	+	0.995758i	$0.470672\pi$
$734$	−20.8405	−0.769236
$735$	0	0
$736$	25.0827	0.924560
$737$	8.34399	0.307355
$738$	0	0
$739$	32.1408	1.18232	0.591160	−	0.806554i	$-0.298670\pi$
$739$	32.1408	1.18232	0.591160	+	0.806554i	$0.298670\pi$
$740$	0	0
$741$	0	0
$742$	−4.44495	−0.163179
$743$	−5.29548	−0.194272	−0.0971361	−	0.995271i	$-0.530968\pi$
$743$	−5.29548	−0.194272	−0.0971361	+	0.995271i	$0.530968\pi$
$744$	0	0
$745$	0	0
$746$	15.9017	0.582203
$747$	0	0
$748$	27.0693	0.989752
$749$	−38.0841	−1.39156
$750$	0	0
$751$	−6.51896	−0.237880	−0.118940	−	0.992901i	$-0.537950\pi$
$751$	−6.51896	−0.237880	−0.118940	+	0.992901i	$0.537950\pi$
$752$	−8.76102	−0.319481
$753$	0	0
$754$	−52.6634	−1.91789
$755$	0	0
$756$	0	0
$757$	31.2014	1.13404	0.567018	−	0.823706i	$-0.308097\pi$
$757$	31.2014	1.13404	0.567018	+	0.823706i	$0.308097\pi$
$758$	57.6190	2.09282
$759$	0	0
$760$	0	0
$761$	4.64286	0.168304	0.0841518	−	0.996453i	$-0.473182\pi$
$761$	4.64286	0.168304	0.0841518	+	0.996453i	$0.473182\pi$
$762$	0	0
$763$	19.7344	0.714434
$764$	−26.1992	−0.947855
$765$	0	0
$766$	−71.9050	−2.59803
$767$	42.8974	1.54893
$768$	0	0
$769$	−25.9088	−0.934294	−0.467147	−	0.884180i	$-0.654718\pi$
$769$	−25.9088	−0.934294	−0.467147	+	0.884180i	$0.654718\pi$
$770$	0	0
$771$	0	0
$772$	20.7028	0.745109
$773$	−10.2568	−0.368912	−0.184456	−	0.982841i	$-0.559052\pi$
$773$	−10.2568	−0.368912	−0.184456	+	0.982841i	$0.559052\pi$
$774$	0	0
$775$	0	0
$776$	8.12723	0.291750
$777$	0	0
$778$	−58.8253	−2.10899
$779$	2.09989	0.0752363
$780$	0	0
$781$	30.0658	1.07584
$782$	−14.4024	−0.515030
$783$	0	0
$784$	−5.50553	−0.196626
$785$	0	0
$786$	0	0
$787$	29.6757	1.05782	0.528912	−	0.848677i	$-0.322600\pi$
$787$	29.6757	1.05782	0.528912	+	0.848677i	$0.322600\pi$
$788$	−60.3306	−2.14919
$789$	0	0
$790$	0	0
$791$	16.5023	0.586755
$792$	0	0
$793$	−25.8416	−0.917664
$794$	−43.2857	−1.53615
$795$	0	0
$796$	6.91998	0.245272
$797$	41.4651	1.46877	0.734384	−	0.678734i	$-0.237471\pi$
$797$	41.4651	1.46877	0.734384	+	0.678734i	$0.237471\pi$
$798$	0	0
$799$	6.59814	0.233425
$800$	0	0
$801$	0	0
$802$	−47.0350	−1.66086
$803$	75.3297	2.65833
$804$	0	0
$805$	0	0
$806$	−12.7990	−0.450824
$807$	0	0
$808$	−4.14848	−0.145943
$809$	−14.8067	−0.520577	−0.260289	−	0.965531i	$-0.583818\pi$
$809$	−14.8067	−0.520577	−0.260289	+	0.965531i	$0.583818\pi$
$810$	0	0
$811$	54.3857	1.90974	0.954870	−	0.297023i	$-0.0959938\pi$
$811$	54.3857	1.90974	0.954870	+	0.297023i	$0.0959938\pi$
$812$	30.0899	1.05595
$813$	0	0
$814$	32.9517	1.15496
$815$	0	0
$816$	0	0
$817$	4.65099	0.162717
$818$	−23.4536	−0.820036
$819$	0	0
$820$	0	0
$821$	−34.1279	−1.19107	−0.595536	−	0.803329i	$-0.703060\pi$
$821$	−34.1279	−1.19107	−0.595536	+	0.803329i	$0.703060\pi$
$822$	0	0
$823$	11.8537	0.413194	0.206597	−	0.978426i	$-0.433761\pi$
$823$	11.8537	0.413194	0.206597	+	0.978426i	$0.433761\pi$
$824$	17.9803	0.626373
$825$	0	0
$826$	−44.5100	−1.54870
$827$	5.47968	0.190547	0.0952736	−	0.995451i	$-0.469627\pi$
$827$	5.47968	0.190547	0.0952736	+	0.995451i	$0.469627\pi$
$828$	0	0
$829$	32.5859	1.13176	0.565878	−	0.824489i	$-0.308537\pi$
$829$	32.5859	1.13176	0.565878	+	0.824489i	$0.308537\pi$
$830$	0	0
$831$	0	0
$832$	−67.3967	−2.33656
$833$	4.14635	0.143662
$834$	0	0
$835$	0	0
$836$	22.5120	0.778593
$837$	0	0
$838$	−17.4078	−0.601342
$839$	−27.2734	−0.941581	−0.470791	−	0.882245i	$-0.656031\pi$
$839$	−27.2734	−0.941581	−0.470791	+	0.882245i	$0.656031\pi$
$840$	0	0
$841$	−12.0696	−0.416192
$842$	43.5938	1.50234
$843$	0	0
$844$	56.7109	1.95207
$845$	0	0
$846$	0	0
$847$	−43.7573	−1.50352
$848$	−2.04401	−0.0701917
$849$	0	0
$850$	0	0
$851$	−9.65433	−0.330946
$852$	0	0
$853$	6.28774	0.215288	0.107644	−	0.994189i	$-0.465669\pi$
$853$	6.28774	0.215288	0.107644	+	0.994189i	$0.465669\pi$
$854$	26.8131	0.917525
$855$	0	0
$856$	12.1451	0.415113
$857$	27.9588	0.955055	0.477528	−	0.878617i	$-0.341533\pi$
$857$	27.9588	0.955055	0.477528	+	0.878617i	$0.341533\pi$
$858$	0	0
$859$	−1.40373	−0.0478948	−0.0239474	−	0.999713i	$-0.507623\pi$
$859$	−1.40373	−0.0478948	−0.0239474	+	0.999713i	$0.507623\pi$
$860$	0	0
$861$	0	0
$862$	0.212659	0.00724319
$863$	−25.4003	−0.864635	−0.432318	−	0.901721i	$-0.642304\pi$
$863$	−25.4003	−0.864635	−0.432318	+	0.901721i	$0.642304\pi$
$864$	0	0
$865$	0	0
$866$	−29.5761	−1.00504
$867$	0	0
$868$	7.31285	0.248214
$869$	−43.5650	−1.47784
$870$	0	0
$871$	9.99175	0.338558
$872$	−6.29337	−0.213120
$873$	0	0
$874$	−11.9777	−0.405151
$875$	0	0
$876$	0	0
$877$	18.9790	0.640874	0.320437	−	0.947270i	$-0.396170\pi$
$877$	18.9790	0.640874	0.320437	+	0.947270i	$0.396170\pi$
$878$	18.9152	0.638355
$879$	0	0
$880$	0	0
$881$	2.57571	0.0867780	0.0433890	−	0.999058i	$-0.486185\pi$
$881$	2.57571	0.0867780	0.0433890	+	0.999058i	$0.486185\pi$
$882$	0	0
$883$	43.2858	1.45668	0.728341	−	0.685215i	$-0.240292\pi$
$883$	43.2858	1.45668	0.728341	+	0.685215i	$0.240292\pi$
$884$	32.4149	1.09023
$885$	0	0
$886$	85.2457	2.86388
$887$	−15.9534	−0.535664	−0.267832	−	0.963466i	$-0.586307\pi$
$887$	−15.9534	−0.535664	−0.267832	+	0.963466i	$0.586307\pi$
$888$	0	0
$889$	22.9208	0.768740
$890$	0	0
$891$	0	0
$892$	12.8839	0.431385
$893$	5.48729	0.183625
$894$	0	0
$895$	0	0
$896$	22.1336	0.739431
$897$	0	0
$898$	17.1406	0.571989
$899$	4.11466	0.137232
$900$	0	0
$901$	1.53940	0.0512848
$902$	12.3797	0.412198
$903$	0	0
$904$	−5.26264	−0.175033
$905$	0	0
$906$	0	0
$907$	33.5496	1.11400	0.556998	−	0.830514i	$-0.311953\pi$
$907$	33.5496	1.11400	0.556998	+	0.830514i	$0.311953\pi$
$908$	−45.8384	−1.52120
$909$	0	0
$910$	0	0
$911$	34.8082	1.15325	0.576623	−	0.817010i	$-0.304370\pi$
$911$	34.8082	1.15325	0.576623	+	0.817010i	$0.304370\pi$
$912$	0	0
$913$	86.7988	2.87262
$914$	−0.283348	−0.00937233
$915$	0	0
$916$	28.5341	0.942793
$917$	−26.9348	−0.889467
$918$	0	0
$919$	−26.5211	−0.874852	−0.437426	−	0.899254i	$-0.644110\pi$
$919$	−26.5211	−0.874852	−0.437426	+	0.899254i	$0.644110\pi$
$920$	0	0
$921$	0	0
$922$	−9.65304	−0.317906
$923$	36.0032	1.18506
$924$	0	0
$925$	0	0
$926$	−47.2615	−1.55311
$927$	0	0
$928$	32.9578	1.08189
$929$	20.5318	0.673625	0.336813	−	0.941572i	$-0.390651\pi$
$929$	20.5318	0.673625	0.336813	+	0.941572i	$0.390651\pi$
$930$	0	0
$931$	3.44828	0.113013
$932$	46.7984	1.53293
$933$	0	0
$934$	−29.8823	−0.977778
$935$	0	0
$936$	0	0
$937$	−6.20585	−0.202736	−0.101368	−	0.994849i	$-0.532322\pi$
$937$	−6.20585	−0.202736	−0.101368	+	0.994849i	$0.532322\pi$
$938$	−10.3674	−0.338506
$939$	0	0
$940$	0	0
$941$	45.6569	1.48837	0.744186	−	0.667973i	$-0.232838\pi$
$941$	45.6569	1.48837	0.744186	+	0.667973i	$0.232838\pi$
$942$	0	0
$943$	−3.62705	−0.118113
$944$	−20.4680	−0.666176
$945$	0	0
$946$	27.4194	0.891483
$947$	−3.69128	−0.119950	−0.0599752	−	0.998200i	$-0.519102\pi$
$947$	−3.69128	−0.119950	−0.0599752	+	0.998200i	$0.519102\pi$
$948$	0	0
$949$	90.2058	2.92820
$950$	0	0
$951$	0	0
$952$	−6.18875	−0.200578
$953$	31.2003	1.01068	0.505338	−	0.862921i	$-0.331368\pi$
$953$	31.2003	1.01068	0.505338	+	0.862921i	$0.331368\pi$
$954$	0	0
$955$	0	0
$956$	−45.2076	−1.46212
$957$	0	0
$958$	−20.0240	−0.646947
$959$	−20.6522	−0.666893
$960$	0	0
$961$	1.00000	0.0322581
$962$	39.4590	1.27221
$963$	0	0
$964$	26.5970	0.856631
$965$	0	0
$966$	0	0
$967$	−3.39338	−0.109124	−0.0545619	−	0.998510i	$-0.517376\pi$
$967$	−3.39338	−0.109124	−0.0545619	+	0.998510i	$0.517376\pi$
$968$	13.9543	0.448509
$969$	0	0
$970$	0	0
$971$	−39.4364	−1.26558	−0.632788	−	0.774325i	$-0.718089\pi$
$971$	−39.4364	−1.26558	−0.632788	+	0.774325i	$0.718089\pi$
$972$	0	0
$973$	37.9888	1.21786
$974$	−12.2413	−0.392237
$975$	0	0
$976$	12.3300	0.394675
$977$	2.36895	0.0757893	0.0378947	−	0.999282i	$-0.487935\pi$
$977$	2.36895	0.0757893	0.0378947	+	0.999282i	$0.487935\pi$
$978$	0	0
$979$	45.9304	1.46794
$980$	0	0
$981$	0	0
$982$	43.9096	1.40121
$983$	56.5350	1.80319	0.901593	−	0.432586i	$-0.142399\pi$
$983$	56.5350	1.80319	0.901593	+	0.432586i	$0.142399\pi$
$984$	0	0
$985$	0	0
$986$	−18.9243	−0.602672
$987$	0	0
$988$	26.9576	0.857637
$989$	−8.03346	−0.255449
$990$	0	0
$991$	6.17089	0.196025	0.0980123	−	0.995185i	$-0.468752\pi$
$991$	6.17089	0.196025	0.0980123	+	0.995185i	$0.468752\pi$
$992$	8.00984	0.254313
$993$	0	0
$994$	−37.3566	−1.18488
$995$	0	0
$996$	0	0
$997$	−1.05813	−0.0335113	−0.0167556	−	0.999860i	$-0.505334\pi$
$997$	−1.05813	−0.0335113	−0.0167556	+	0.999860i	$0.505334\pi$
$998$	−89.7805	−2.84195
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cg.1.7	yes	8
3.2	odd	2	inner	6975.2.a.cg.1.2	yes	8
5.4	even	2		6975.2.a.cf.1.2	✓	8
15.14	odd	2		6975.2.a.cf.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6975.2.a.cf.1.2	✓	8	5.4	even	2
6975.2.a.cf.1.7	yes	8	15.14	odd	2
6975.2.a.cg.1.2	yes	8	3.2	odd	2	inner
6975.2.a.cg.1.7	yes	8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.10974 q^{2} +2.45100 q^{4} -2.98362 q^{7} +0.951486 q^{8} +5.06615 q^{11} +6.06661 q^{13} -6.29466 q^{14} -2.89461 q^{16} +2.18000 q^{17} +1.81298 q^{19} +10.6882 q^{22} -3.13149 q^{23} +12.7990 q^{26} -7.31285 q^{28} -4.11466 q^{29} -1.00000 q^{31} -8.00984 q^{32} +4.59923 q^{34} +3.08299 q^{37} +3.82492 q^{38} +1.15825 q^{41} +2.56538 q^{43} +12.4171 q^{44} -6.60662 q^{46} +3.02667 q^{47} +1.90199 q^{49} +14.8692 q^{52} +0.706146 q^{53} -2.83887 q^{56} -8.68086 q^{58} +7.07107 q^{59} -4.25965 q^{61} -2.10974 q^{62} -11.1094 q^{64} +1.64701 q^{67} +5.34317 q^{68} +5.93465 q^{71} +14.8692 q^{73} +6.50429 q^{74} +4.44361 q^{76} -15.1155 q^{77} -8.59923 q^{79} +2.44361 q^{82} +17.1331 q^{83} +5.41229 q^{86} +4.82037 q^{88} +9.06614 q^{89} -18.1005 q^{91} -7.67526 q^{92} +6.38548 q^{94} +8.54162 q^{97} +4.01271 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 4 q^{7} + 8 q^{13} + 12 q^{16} + 28 q^{22} + 36 q^{28} - 8 q^{31} - 28 q^{34} + 12 q^{37} + 52 q^{43} - 16 q^{46} + 8 q^{49} + 56 q^{52} + 16 q^{58} - 28 q^{61} + 48 q^{64} + 24 q^{67}+ \cdots + 44 q^{97}+O(q^{100}) \) 8 * q + 16 * q^4 + 4 * q^7 + 8 * q^13 + 12 * q^16 + 28 * q^22 + 36 * q^28 - 8 * q^31 - 28 * q^34 + 12 * q^37 + 52 * q^43 - 16 * q^46 + 8 * q^49 + 56 * q^52 + 16 * q^58 - 28 * q^61 + 48 * q^64 + 24 * q^67 + 56 * q^73 + 4 * q^76 - 4 * q^79 - 12 * q^82 + 52 * q^88 - 8 * q^91 - 64 * q^94 + 44 * q^97

Introduction

L-functions

Modular forms

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