LMFDB - Embedded newform 2320.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.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:	$18.5252932689$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.462598 q^{3} -1.00000 q^{5} +2.86081 q^{7} -2.78600 q^{9} -5.72161 q^{11} +0.462598 q^{13} -0.462598 q^{15} +1.93561 q^{17} +8.36842 q^{19} +1.32340 q^{21} -8.18421 q^{23} +1.00000 q^{25} -2.67660 q^{27} -1.00000 q^{29} -3.78600 q^{31} -2.64681 q^{33} -2.86081 q^{35} +3.07480 q^{37} +0.213997 q^{39} +3.72161 q^{41} -9.78600 q^{43} +2.78600 q^{45} -8.64681 q^{47} +1.18421 q^{49} +0.895410 q^{51} -2.58242 q^{53} +5.72161 q^{55} +3.87122 q^{57} -11.7562 q^{59} -9.62743 q^{61} -7.97021 q^{63} -0.462598 q^{65} -13.2936 q^{67} -3.78600 q^{69} +5.29362 q^{71} +9.63640 q^{73} +0.462598 q^{75} -16.3684 q^{77} +5.90582 q^{79} +7.11982 q^{81} -4.92520 q^{83} -1.93561 q^{85} -0.462598 q^{87} -5.07480 q^{89} +1.32340 q^{91} -1.75140 q^{93} -8.36842 q^{95} -1.66618 q^{97} +15.9404 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 2 q^{9} - 6 q^{11} - q^{13} + q^{15} + 5 q^{17} - 2 q^{19} - 4 q^{21} - 11 q^{23} + 3 q^{25} - 16 q^{27} - 3 q^{29} - q^{31} + 8 q^{33} - 3 q^{35} + 14 q^{37} + 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 2 * q^9 - 6 * q^11 - q^13 + q^15 + 5 * q^17 - 2 * q^19 - 4 * q^21 - 11 * q^23 + 3 * q^25 - 16 * q^27 - 3 * q^29 - q^31 + 8 * q^33 - 3 * q^35 + 14 * q^37 + 11 * q^39 - 19 * q^43 - 2 * q^45 - 10 * q^47 - 10 * q^49 - 26 * q^51 + 9 * q^53 + 6 * q^55 + 10 * q^57 - q^59 + 7 * q^61 + q^65 - 8 * q^67 - q^69 - 16 * q^71 + 9 * q^73 - q^75 - 22 * q^77 - 7 * q^79 + 7 * q^81 - 10 * q^83 - 5 * q^85 + q^87 - 20 * q^89 - 4 * q^91 - 18 * q^93 + 2 * q^95 - 9 * 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 $

$2$	0	0
$2$	0	0
$3$	0.462598	0.267081	0.133541	−	0.991043i	$-0.457365\pi$
$3$	0.462598	0.267081	0.133541	+	0.991043i	$0.457365\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.86081	1.08128	0.540641	−	0.841253i	$-0.318182\pi$
$7$	2.86081	1.08128	0.540641	+	0.841253i	$0.318182\pi$
$8$	0	0
$9$	−2.78600	−0.928668
$10$	0	0
$11$	−5.72161	−1.72513	−0.862565	−	0.505946i	$-0.831144\pi$
$11$	−5.72161	−1.72513	−0.862565	+	0.505946i	$0.831144\pi$
$12$	0	0
$13$	0.462598	0.128302	0.0641509	−	0.997940i	$-0.479566\pi$
$13$	0.462598	0.128302	0.0641509	+	0.997940i	$0.479566\pi$
$14$	0	0
$15$	−0.462598	−0.119442
$16$	0	0
$17$	1.93561	0.469454	0.234727	−	0.972061i	$-0.424580\pi$
$17$	1.93561	0.469454	0.234727	+	0.972061i	$0.424580\pi$
$18$	0	0
$19$	8.36842	1.91985	0.959924	−	0.280262i	$-0.0904212\pi$
$19$	8.36842	1.91985	0.959924	+	0.280262i	$0.0904212\pi$
$20$	0	0
$21$	1.32340	0.288790
$22$	0	0
$23$	−8.18421	−1.70653	−0.853263	−	0.521481i	$-0.825380\pi$
$23$	−8.18421	−1.70653	−0.853263	+	0.521481i	$0.825380\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.67660	−0.515111
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.78600	−0.679986	−0.339993	−	0.940428i	$-0.610425\pi$
$31$	−3.78600	−0.679986	−0.339993	+	0.940428i	$0.610425\pi$
$32$	0	0
$33$	−2.64681	−0.460750
$34$	0	0
$35$	−2.86081	−0.483564
$36$	0	0
$37$	3.07480	0.505495	0.252747	−	0.967532i	$-0.418666\pi$
$37$	3.07480	0.505495	0.252747	+	0.967532i	$0.418666\pi$
$38$	0	0
$39$	0.213997	0.0342670
$40$	0	0
$41$	3.72161	0.581218	0.290609	−	0.956842i	$-0.406142\pi$
$41$	3.72161	0.581218	0.290609	+	0.956842i	$0.406142\pi$
$42$	0	0
$43$	−9.78600	−1.49235	−0.746176	−	0.665749i	$-0.768112\pi$
$43$	−9.78600	−1.49235	−0.746176	+	0.665749i	$0.768112\pi$
$44$	0	0
$45$	2.78600	0.415313
$46$	0	0
$47$	−8.64681	−1.26127	−0.630633	−	0.776081i	$-0.717205\pi$
$47$	−8.64681	−1.26127	−0.630633	+	0.776081i	$0.717205\pi$
$48$	0	0
$49$	1.18421	0.169173
$50$	0	0
$51$	0.895410	0.125382
$52$	0	0
$53$	−2.58242	−0.354722	−0.177361	−	0.984146i	$-0.556756\pi$
$53$	−2.58242	−0.354722	−0.177361	+	0.984146i	$0.556756\pi$
$54$	0	0
$55$	5.72161	0.771502
$56$	0	0
$57$	3.87122	0.512755
$58$	0	0
$59$	−11.7562	−1.53053	−0.765264	−	0.643716i	$-0.777392\pi$
$59$	−11.7562	−1.53053	−0.765264	+	0.643716i	$0.777392\pi$
$60$	0	0
$61$	−9.62743	−1.23267	−0.616333	−	0.787485i	$-0.711383\pi$
$61$	−9.62743	−1.23267	−0.616333	+	0.787485i	$0.711383\pi$
$62$	0	0
$63$	−7.97021	−1.00415
$64$	0	0
$65$	−0.462598	−0.0573783
$66$	0	0
$67$	−13.2936	−1.62407	−0.812037	−	0.583606i	$-0.801641\pi$
$67$	−13.2936	−1.62407	−0.812037	+	0.583606i	$0.801641\pi$
$68$	0	0
$69$	−3.78600	−0.455781
$70$	0	0
$71$	5.29362	0.628237	0.314118	−	0.949384i	$-0.398291\pi$
$71$	5.29362	0.628237	0.314118	+	0.949384i	$0.398291\pi$
$72$	0	0
$73$	9.63640	1.12785	0.563927	−	0.825824i	$-0.309290\pi$
$73$	9.63640	1.12785	0.563927	+	0.825824i	$0.309290\pi$
$74$	0	0
$75$	0.462598	0.0534163
$76$	0	0
$77$	−16.3684	−1.86535
$78$	0	0
$79$	5.90582	0.664457	0.332228	−	0.943199i	$-0.392200\pi$
$79$	5.90582	0.664457	0.332228	+	0.943199i	$0.392200\pi$
$80$	0	0
$81$	7.11982	0.791091
$82$	0	0
$83$	−4.92520	−0.540611	−0.270305	−	0.962775i	$-0.587125\pi$
$83$	−4.92520	−0.540611	−0.270305	+	0.962775i	$0.587125\pi$
$84$	0	0
$85$	−1.93561	−0.209946
$86$	0	0
$87$	−0.462598	−0.0495958
$88$	0	0
$89$	−5.07480	−0.537928	−0.268964	−	0.963150i	$-0.586681\pi$
$89$	−5.07480	−0.537928	−0.268964	+	0.963150i	$0.586681\pi$
$90$	0	0
$91$	1.32340	0.138730
$92$	0	0
$93$	−1.75140	−0.181612
$94$	0	0
$95$	−8.36842	−0.858582
$96$	0	0
$97$	−1.66618	−0.169175	−0.0845877	−	0.996416i	$-0.526957\pi$
$97$	−1.66618	−0.169175	−0.0845877	+	0.996416i	$0.526957\pi$
$98$	0	0
$99$	15.9404	1.60207
$100$	0	0
$101$	−13.2590	−1.31932	−0.659661	−	0.751564i	$-0.729300\pi$
$101$	−13.2590	−1.31932	−0.659661	+	0.751564i	$0.729300\pi$
$102$	0	0
$103$	14.8864	1.46681	0.733403	−	0.679795i	$-0.237931\pi$
$103$	14.8864	1.46681	0.733403	+	0.679795i	$0.237931\pi$
$104$	0	0
$105$	−1.32340	−0.129151
$106$	0	0
$107$	−1.48197	−0.143268	−0.0716339	−	0.997431i	$-0.522821\pi$
$107$	−1.48197	−0.143268	−0.0716339	+	0.997431i	$0.522821\pi$
$108$	0	0
$109$	17.0152	1.62976	0.814882	−	0.579627i	$-0.196802\pi$
$109$	17.0152	1.62976	0.814882	+	0.579627i	$0.196802\pi$
$110$	0	0
$111$	1.42240	0.135008
$112$	0	0
$113$	−5.40862	−0.508800	−0.254400	−	0.967099i	$-0.581878\pi$
$113$	−5.40862	−0.508800	−0.254400	+	0.967099i	$0.581878\pi$
$114$	0	0
$115$	8.18421	0.763182
$116$	0	0
$117$	−1.28880	−0.119150
$118$	0	0
$119$	5.53740	0.507613
$120$	0	0
$121$	21.7368	1.97608
$122$	0	0
$123$	1.72161	0.155232
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.0900	−1.07282	−0.536408	−	0.843959i	$-0.680219\pi$
$127$	−12.0900	−1.07282	−0.536408	+	0.843959i	$0.680219\pi$
$128$	0	0
$129$	−4.52699	−0.398579
$130$	0	0
$131$	−8.49720	−0.742404	−0.371202	−	0.928552i	$-0.621054\pi$
$131$	−8.49720	−0.742404	−0.371202	+	0.928552i	$0.621054\pi$
$132$	0	0
$133$	23.9404	2.07590
$134$	0	0
$135$	2.67660	0.230365
$136$	0	0
$137$	−3.25901	−0.278436	−0.139218	−	0.990262i	$-0.544459\pi$
$137$	−3.25901	−0.278436	−0.139218	+	0.990262i	$0.544459\pi$
$138$	0	0
$139$	−22.6724	−1.92305	−0.961526	−	0.274714i	$-0.911417\pi$
$139$	−22.6724	−1.92305	−0.961526	+	0.274714i	$0.911417\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−2.64681	−0.221337
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0.547814	0.0451829
$148$	0	0
$149$	11.6620	0.955391	0.477696	−	0.878525i	$-0.341472\pi$
$149$	11.6620	0.955391	0.477696	+	0.878525i	$0.341472\pi$
$150$	0	0
$151$	−4.66763	−0.379847	−0.189923	−	0.981799i	$-0.560824\pi$
$151$	−4.66763	−0.379847	−0.189923	+	0.981799i	$0.560824\pi$
$152$	0	0
$153$	−5.39261	−0.435967
$154$	0	0
$155$	3.78600	0.304099
$156$	0	0
$157$	−15.3836	−1.22775	−0.613874	−	0.789404i	$-0.710390\pi$
$157$	−15.3836	−1.22775	−0.613874	+	0.789404i	$0.710390\pi$
$158$	0	0
$159$	−1.19462	−0.0947397
$160$	0	0
$161$	−23.4134	−1.84524
$162$	0	0
$163$	0.646809	0.0506620	0.0253310	−	0.999679i	$-0.491936\pi$
$163$	0.646809	0.0506620	0.0253310	+	0.999679i	$0.491936\pi$
$164$	0	0
$165$	2.64681	0.206054
$166$	0	0
$167$	14.5616	1.12681	0.563405	−	0.826181i	$-0.309491\pi$
$167$	14.5616	1.12681	0.563405	+	0.826181i	$0.309491\pi$
$168$	0	0
$169$	−12.7860	−0.983539
$170$	0	0
$171$	−23.3144	−1.78290
$172$	0	0
$173$	6.98062	0.530727	0.265364	−	0.964148i	$-0.414508\pi$
$173$	6.98062	0.530727	0.265364	+	0.964148i	$0.414508\pi$
$174$	0	0
$175$	2.86081	0.216257
$176$	0	0
$177$	−5.43841	−0.408776
$178$	0	0
$179$	11.7562	0.878701	0.439350	−	0.898316i	$-0.355209\pi$
$179$	11.7562	0.878701	0.439350	+	0.898316i	$0.355209\pi$
$180$	0	0
$181$	−5.57201	−0.414164	−0.207082	−	0.978324i	$-0.566397\pi$
$181$	−5.57201	−0.414164	−0.207082	+	0.978324i	$0.566397\pi$
$182$	0	0
$183$	−4.45364	−0.329222
$184$	0	0
$185$	−3.07480	−0.226064
$186$	0	0
$187$	−11.0748	−0.809870
$188$	0	0
$189$	−7.65722	−0.556981
$190$	0	0
$191$	13.0796	0.946408	0.473204	−	0.880953i	$-0.343097\pi$
$191$	13.0796	0.946408	0.473204	+	0.880953i	$0.343097\pi$
$192$	0	0
$193$	−4.18421	−0.301186	−0.150593	−	0.988596i	$-0.548118\pi$
$193$	−4.18421	−0.301186	−0.150593	+	0.988596i	$0.548118\pi$
$194$	0	0
$195$	−0.213997	−0.0153247
$196$	0	0
$197$	−1.28880	−0.0918232	−0.0459116	−	0.998946i	$-0.514619\pi$
$197$	−1.28880	−0.0918232	−0.0459116	+	0.998946i	$0.514619\pi$
$198$	0	0
$199$	−3.87122	−0.274423	−0.137212	−	0.990542i	$-0.543814\pi$
$199$	−3.87122	−0.274423	−0.137212	+	0.990542i	$0.543814\pi$
$200$	0	0
$201$	−6.14961	−0.433760
$202$	0	0
$203$	−2.86081	−0.200789
$204$	0	0
$205$	−3.72161	−0.259929
$206$	0	0
$207$	22.8012	1.58480
$208$	0	0
$209$	−47.8809	−3.31199
$210$	0	0
$211$	−25.1648	−1.73242	−0.866209	−	0.499681i	$-0.833450\pi$
$211$	−25.1648	−1.73242	−0.866209	+	0.499681i	$0.833450\pi$
$212$	0	0
$213$	2.44882	0.167790
$214$	0	0
$215$	9.78600	0.667400
$216$	0	0
$217$	−10.8310	−0.735257
$218$	0	0
$219$	4.45778	0.301229
$220$	0	0
$221$	0.895410	0.0602318
$222$	0	0
$223$	5.96540	0.399472	0.199736	−	0.979850i	$-0.435991\pi$
$223$	5.96540	0.399472	0.199736	+	0.979850i	$0.435991\pi$
$224$	0	0
$225$	−2.78600	−0.185734
$226$	0	0
$227$	−12.7964	−0.849328	−0.424664	−	0.905351i	$-0.639608\pi$
$227$	−12.7964	−0.849328	−0.424664	+	0.905351i	$0.639608\pi$
$228$	0	0
$229$	−11.3788	−0.751934	−0.375967	−	0.926633i	$-0.622689\pi$
$229$	−11.3788	−0.751934	−0.375967	+	0.926633i	$0.622689\pi$
$230$	0	0
$231$	−7.57201	−0.498201
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	8.64681	0.564056
$236$	0	0
$237$	2.73202	0.177464
$238$	0	0
$239$	8.79641	0.568993	0.284496	−	0.958677i	$-0.408174\pi$
$239$	8.79641	0.568993	0.284496	+	0.958677i	$0.408174\pi$
$240$	0	0
$241$	−5.62743	−0.362495	−0.181247	−	0.983438i	$-0.558013\pi$
$241$	−5.62743	−0.362495	−0.181247	+	0.983438i	$0.558013\pi$
$242$	0	0
$243$	11.3234	0.726397
$244$	0	0
$245$	−1.18421	−0.0756564
$246$	0	0
$247$	3.87122	0.246320
$248$	0	0
$249$	−2.27839	−0.144387
$250$	0	0
$251$	11.0748	0.699035	0.349518	−	0.936930i	$-0.386345\pi$
$251$	11.0748	0.699035	0.349518	+	0.936930i	$0.386345\pi$
$252$	0	0
$253$	46.8269	2.94398
$254$	0	0
$255$	−0.895410	−0.0560727
$256$	0	0
$257$	−13.8712	−0.865263	−0.432631	−	0.901571i	$-0.642415\pi$
$257$	−13.8712	−0.865263	−0.432631	+	0.901571i	$0.642415\pi$
$258$	0	0
$259$	8.79641	0.546583
$260$	0	0
$261$	2.78600	0.172449
$262$	0	0
$263$	8.09003	0.498853	0.249426	−	0.968394i	$-0.419758\pi$
$263$	8.09003	0.498853	0.249426	+	0.968394i	$0.419758\pi$
$264$	0	0
$265$	2.58242	0.158637
$266$	0	0
$267$	−2.34760	−0.143671
$268$	0	0
$269$	4.80123	0.292736	0.146368	−	0.989230i	$-0.453242\pi$
$269$	4.80123	0.292736	0.146368	+	0.989230i	$0.453242\pi$
$270$	0	0
$271$	25.0361	1.52083	0.760416	−	0.649436i	$-0.224995\pi$
$271$	25.0361	1.52083	0.760416	+	0.649436i	$0.224995\pi$
$272$	0	0
$273$	0.612205	0.0370523
$274$	0	0
$275$	−5.72161	−0.345026
$276$	0	0
$277$	22.1801	1.33267	0.666335	−	0.745652i	$-0.267862\pi$
$277$	22.1801	1.33267	0.666335	+	0.745652i	$0.267862\pi$
$278$	0	0
$279$	10.5478	0.631481
$280$	0	0
$281$	−3.80683	−0.227096	−0.113548	−	0.993532i	$-0.536222\pi$
$281$	−3.80683	−0.227096	−0.113548	+	0.993532i	$0.536222\pi$
$282$	0	0
$283$	4.92520	0.292773	0.146386	−	0.989228i	$-0.453236\pi$
$283$	4.92520	0.292773	0.146386	+	0.989228i	$0.453236\pi$
$284$	0	0
$285$	−3.87122	−0.229311
$286$	0	0
$287$	10.6468	0.628461
$288$	0	0
$289$	−13.2534	−0.779613
$290$	0	0
$291$	−0.770774	−0.0451836
$292$	0	0
$293$	22.2188	1.29804	0.649018	−	0.760773i	$-0.275180\pi$
$293$	22.2188	1.29804	0.649018	+	0.760773i	$0.275180\pi$
$294$	0	0
$295$	11.7562	0.684473
$296$	0	0
$297$	15.3144	0.888634
$298$	0	0
$299$	−3.78600	−0.218950
$300$	0	0
$301$	−27.9959	−1.61365
$302$	0	0
$303$	−6.13360	−0.352366
$304$	0	0
$305$	9.62743	0.551265
$306$	0	0
$307$	21.9404	1.25221	0.626103	−	0.779740i	$-0.284649\pi$
$307$	21.9404	1.25221	0.626103	+	0.779740i	$0.284649\pi$
$308$	0	0
$309$	6.88645	0.391756
$310$	0	0
$311$	−19.1994	−1.08870	−0.544350	−	0.838858i	$-0.683224\pi$
$311$	−19.1994	−1.08870	−0.544350	+	0.838858i	$0.683224\pi$
$312$	0	0
$313$	−8.88645	−0.502292	−0.251146	−	0.967949i	$-0.580807\pi$
$313$	−8.88645	−0.502292	−0.251146	+	0.967949i	$0.580807\pi$
$314$	0	0
$315$	7.97021	0.449071
$316$	0	0
$317$	3.87122	0.217429	0.108715	−	0.994073i	$-0.465327\pi$
$317$	3.87122	0.217429	0.108715	+	0.994073i	$0.465327\pi$
$318$	0	0
$319$	5.72161	0.320349
$320$	0	0
$321$	−0.685559	−0.0382641
$322$	0	0
$323$	16.1980	0.901280
$324$	0	0
$325$	0.462598	0.0256603
$326$	0	0
$327$	7.87122	0.435279
$328$	0	0
$329$	−24.7368	−1.36379
$330$	0	0
$331$	−21.4224	−1.17748	−0.588741	−	0.808322i	$-0.700376\pi$
$331$	−21.4224	−1.17748	−0.588741	+	0.808322i	$0.700376\pi$
$332$	0	0
$333$	−8.56641	−0.469436
$334$	0	0
$335$	13.2936	0.726308
$336$	0	0
$337$	27.1004	1.47626	0.738128	−	0.674661i	$-0.235710\pi$
$337$	27.1004	1.47626	0.738128	+	0.674661i	$0.235710\pi$
$338$	0	0
$339$	−2.50202	−0.135891
$340$	0	0
$341$	21.6620	1.17307
$342$	0	0
$343$	−16.6378	−0.898359
$344$	0	0
$345$	3.78600	0.203832
$346$	0	0
$347$	−26.5180	−1.42356	−0.711781	−	0.702401i	$-0.752111\pi$
$347$	−26.5180	−1.42356	−0.711781	+	0.702401i	$0.752111\pi$
$348$	0	0
$349$	−27.2340	−1.45780	−0.728902	−	0.684618i	$-0.759969\pi$
$349$	−27.2340	−1.45780	−0.728902	+	0.684618i	$0.759969\pi$
$350$	0	0
$351$	−1.23819	−0.0660896
$352$	0	0
$353$	34.6773	1.84568	0.922842	−	0.385178i	$-0.125860\pi$
$353$	34.6773	1.84568	0.922842	+	0.385178i	$0.125860\pi$
$354$	0	0
$355$	−5.29362	−0.280956
$356$	0	0
$357$	2.56159	0.135574
$358$	0	0
$359$	20.9806	1.10732	0.553658	−	0.832744i	$-0.313232\pi$
$359$	20.9806	1.10732	0.553658	+	0.832744i	$0.313232\pi$
$360$	0	0
$361$	51.0305	2.68581
$362$	0	0
$363$	10.0554	0.527773
$364$	0	0
$365$	−9.63640	−0.504392
$366$	0	0
$367$	3.97918	0.207711	0.103856	−	0.994592i	$-0.466882\pi$
$367$	3.97918	0.207711	0.103856	+	0.994592i	$0.466882\pi$
$368$	0	0
$369$	−10.3684	−0.539758
$370$	0	0
$371$	−7.38780	−0.383555
$372$	0	0
$373$	−17.1697	−0.889011	−0.444505	−	0.895776i	$-0.646621\pi$
$373$	−17.1697	−0.889011	−0.444505	+	0.895776i	$0.646621\pi$
$374$	0	0
$375$	−0.462598	−0.0238885
$376$	0	0
$377$	−0.462598	−0.0238250
$378$	0	0
$379$	−7.27279	−0.373578	−0.186789	−	0.982400i	$-0.559808\pi$
$379$	−7.27279	−0.373578	−0.186789	+	0.982400i	$0.559808\pi$
$380$	0	0
$381$	−5.59283	−0.286529
$382$	0	0
$383$	18.5616	0.948453	0.474227	−	0.880403i	$-0.342728\pi$
$383$	18.5616	0.948453	0.474227	+	0.880403i	$0.342728\pi$
$384$	0	0
$385$	16.3684	0.834212
$386$	0	0
$387$	27.2638	1.38590
$388$	0	0
$389$	4.40717	0.223452	0.111726	−	0.993739i	$-0.464362\pi$
$389$	4.40717	0.223452	0.111726	+	0.993739i	$0.464362\pi$
$390$	0	0
$391$	−15.8414	−0.801136
$392$	0	0
$393$	−3.93079	−0.198282
$394$	0	0
$395$	−5.90582	−0.297154
$396$	0	0
$397$	29.1697	1.46398	0.731991	−	0.681314i	$-0.238591\pi$
$397$	29.1697	1.46398	0.731991	+	0.681314i	$0.238591\pi$
$398$	0	0
$399$	11.0748	0.554434
$400$	0	0
$401$	−24.3338	−1.21517	−0.607586	−	0.794254i	$-0.707862\pi$
$401$	−24.3338	−1.21517	−0.607586	+	0.794254i	$0.707862\pi$
$402$	0	0
$403$	−1.75140	−0.0872434
$404$	0	0
$405$	−7.11982	−0.353787
$406$	0	0
$407$	−17.5928	−0.872044
$408$	0	0
$409$	12.9044	0.638080	0.319040	−	0.947741i	$-0.396640\pi$
$409$	12.9044	0.638080	0.319040	+	0.947741i	$0.396640\pi$
$410$	0	0
$411$	−1.50761	−0.0743651
$412$	0	0
$413$	−33.6323	−1.65493
$414$	0	0
$415$	4.92520	0.241768
$416$	0	0
$417$	−10.4882	−0.513611
$418$	0	0
$419$	10.3732	0.506766	0.253383	−	0.967366i	$-0.418457\pi$
$419$	10.3732	0.506766	0.253383	+	0.967366i	$0.418457\pi$
$420$	0	0
$421$	−5.74244	−0.279869	−0.139935	−	0.990161i	$-0.544689\pi$
$421$	−5.74244	−0.279869	−0.139935	+	0.990161i	$0.544689\pi$
$422$	0	0
$423$	24.0900	1.17130
$424$	0	0
$425$	1.93561	0.0938908
$426$	0	0
$427$	−27.5422	−1.33286
$428$	0	0
$429$	−1.22441	−0.0591150
$430$	0	0
$431$	2.81724	0.135702	0.0678508	−	0.997695i	$-0.478386\pi$
$431$	2.81724	0.135702	0.0678508	+	0.997695i	$0.478386\pi$
$432$	0	0
$433$	18.6468	0.896108	0.448054	−	0.894006i	$-0.352117\pi$
$433$	18.6468	0.896108	0.448054	+	0.894006i	$0.352117\pi$
$434$	0	0
$435$	0.462598	0.0221799
$436$	0	0
$437$	−68.4889	−3.27627
$438$	0	0
$439$	−40.4197	−1.92913	−0.964564	−	0.263851i	$-0.915007\pi$
$439$	−40.4197	−1.92913	−0.964564	+	0.263851i	$0.915007\pi$
$440$	0	0
$441$	−3.29921	−0.157105
$442$	0	0
$443$	39.1607	1.86058	0.930290	−	0.366824i	$-0.119555\pi$
$443$	39.1607	1.86058	0.930290	+	0.366824i	$0.119555\pi$
$444$	0	0
$445$	5.07480	0.240569
$446$	0	0
$447$	5.39484	0.255167
$448$	0	0
$449$	5.61365	0.264925	0.132462	−	0.991188i	$-0.457712\pi$
$449$	5.61365	0.264925	0.132462	+	0.991188i	$0.457712\pi$
$450$	0	0
$451$	−21.2936	−1.00268
$452$	0	0
$453$	−2.15924	−0.101450
$454$	0	0
$455$	−1.32340	−0.0620421
$456$	0	0
$457$	−8.32967	−0.389646	−0.194823	−	0.980838i	$-0.562413\pi$
$457$	−8.32967	−0.389646	−0.194823	+	0.980838i	$0.562413\pi$
$458$	0	0
$459$	−5.18084	−0.241821
$460$	0	0
$461$	30.1365	1.40360	0.701798	−	0.712376i	$-0.252381\pi$
$461$	30.1365	1.40360	0.701798	+	0.712376i	$0.252381\pi$
$462$	0	0
$463$	15.1857	0.705737	0.352869	−	0.935673i	$-0.385206\pi$
$463$	15.1857	0.705737	0.352869	+	0.935673i	$0.385206\pi$
$464$	0	0
$465$	1.75140	0.0812192
$466$	0	0
$467$	−13.7860	−0.637940	−0.318970	−	0.947765i	$-0.603337\pi$
$467$	−13.7860	−0.637940	−0.318970	+	0.947765i	$0.603337\pi$
$468$	0	0
$469$	−38.0305	−1.75608
$470$	0	0
$471$	−7.11645	−0.327909
$472$	0	0
$473$	55.9917	2.57450
$474$	0	0
$475$	8.36842	0.383969
$476$	0	0
$477$	7.19462	0.329419
$478$	0	0
$479$	−17.7652	−0.811712	−0.405856	−	0.913937i	$-0.633027\pi$
$479$	−17.7652	−0.811712	−0.405856	+	0.913937i	$0.633027\pi$
$480$	0	0
$481$	1.42240	0.0648558
$482$	0	0
$483$	−10.8310	−0.492828
$484$	0	0
$485$	1.66618	0.0756575
$486$	0	0
$487$	7.62743	0.345632	0.172816	−	0.984954i	$-0.444713\pi$
$487$	7.62743	0.345632	0.172816	+	0.984954i	$0.444713\pi$
$488$	0	0
$489$	0.299213	0.0135309
$490$	0	0
$491$	16.1801	0.730196	0.365098	−	0.930969i	$-0.381035\pi$
$491$	16.1801	0.730196	0.365098	+	0.930969i	$0.381035\pi$
$492$	0	0
$493$	−1.93561	−0.0871754
$494$	0	0
$495$	−15.9404	−0.716469
$496$	0	0
$497$	15.1440	0.679302
$498$	0	0
$499$	33.3074	1.49104	0.745522	−	0.666481i	$-0.232200\pi$
$499$	33.3074	1.49104	0.745522	+	0.666481i	$0.232200\pi$
$500$	0	0
$501$	6.73617	0.300950
$502$	0	0
$503$	−28.2701	−1.26050	−0.630251	−	0.776392i	$-0.717048\pi$
$503$	−28.2701	−1.26050	−0.630251	+	0.776392i	$0.717048\pi$
$504$	0	0
$505$	13.2590	0.590018
$506$	0	0
$507$	−5.91478	−0.262685
$508$	0	0
$509$	12.7160	0.563628	0.281814	−	0.959469i	$-0.409064\pi$
$509$	12.7160	0.563628	0.281814	+	0.959469i	$0.409064\pi$
$510$	0	0
$511$	27.5679	1.21953
$512$	0	0
$513$	−22.3989	−0.988935
$514$	0	0
$515$	−14.8864	−0.655975
$516$	0	0
$517$	49.4737	2.17585
$518$	0	0
$519$	3.22923	0.141747
$520$	0	0
$521$	−11.2084	−0.491049	−0.245524	−	0.969390i	$-0.578960\pi$
$521$	−11.2084	−0.491049	−0.245524	+	0.969390i	$0.578960\pi$
$522$	0	0
$523$	−39.5541	−1.72958	−0.864790	−	0.502134i	$-0.832548\pi$
$523$	−39.5541	−1.72958	−0.864790	+	0.502134i	$0.832548\pi$
$524$	0	0
$525$	1.32340	0.0577581
$526$	0	0
$527$	−7.32822	−0.319222
$528$	0	0
$529$	43.9813	1.91223
$530$	0	0
$531$	32.7528	1.42135
$532$	0	0
$533$	1.72161	0.0745713
$534$	0	0
$535$	1.48197	0.0640713
$536$	0	0
$537$	5.43841	0.234685
$538$	0	0
$539$	−6.77559	−0.291845
$540$	0	0
$541$	30.1365	1.29567	0.647835	−	0.761781i	$-0.275675\pi$
$541$	30.1365	1.29567	0.647835	+	0.761781i	$0.275675\pi$
$542$	0	0
$543$	−2.57760	−0.110615
$544$	0	0
$545$	−17.0152	−0.728852
$546$	0	0
$547$	−44.5485	−1.90476	−0.952378	−	0.304920i	$-0.901370\pi$
$547$	−44.5485	−1.90476	−0.952378	+	0.304920i	$0.901370\pi$
$548$	0	0
$549$	26.8221	1.14474
$550$	0	0
$551$	−8.36842	−0.356507
$552$	0	0
$553$	16.8954	0.718466
$554$	0	0
$555$	−1.42240	−0.0603775
$556$	0	0
$557$	−23.3490	−0.989331	−0.494665	−	0.869084i	$-0.664709\pi$
$557$	−23.3490	−0.989331	−0.494665	+	0.869084i	$0.664709\pi$
$558$	0	0
$559$	−4.52699	−0.191471
$560$	0	0
$561$	−5.12319	−0.216301
$562$	0	0
$563$	−39.1607	−1.65043	−0.825213	−	0.564821i	$-0.808945\pi$
$563$	−39.1607	−1.65043	−0.825213	+	0.564821i	$0.808945\pi$
$564$	0	0
$565$	5.40862	0.227542
$566$	0	0
$567$	20.3684	0.855393
$568$	0	0
$569$	47.2340	1.98015	0.990077	−	0.140526i	$-0.0448792\pi$
$569$	47.2340	1.98015	0.990077	+	0.140526i	$0.0448792\pi$
$570$	0	0
$571$	27.3795	1.14580	0.572898	−	0.819626i	$-0.305819\pi$
$571$	27.3795	1.14580	0.572898	+	0.819626i	$0.305819\pi$
$572$	0	0
$573$	6.05061	0.252768
$574$	0	0
$575$	−8.18421	−0.341305
$576$	0	0
$577$	−19.6572	−0.818341	−0.409170	−	0.912458i	$-0.634182\pi$
$577$	−19.6572	−0.818341	−0.409170	+	0.912458i	$0.634182\pi$
$578$	0	0
$579$	−1.93561	−0.0804412
$580$	0	0
$581$	−14.0900	−0.584553
$582$	0	0
$583$	14.7756	0.611942
$584$	0	0
$585$	1.28880	0.0532853
$586$	0	0
$587$	−26.9460	−1.11218	−0.556091	−	0.831122i	$-0.687699\pi$
$587$	−26.9460	−1.11218	−0.556091	+	0.831122i	$0.687699\pi$
$588$	0	0
$589$	−31.6829	−1.30547
$590$	0	0
$591$	−0.596197	−0.0245243
$592$	0	0
$593$	3.73954	0.153564	0.0767822	−	0.997048i	$-0.475535\pi$
$593$	3.73954	0.153564	0.0767822	+	0.997048i	$0.475535\pi$
$594$	0	0
$595$	−5.53740	−0.227011
$596$	0	0
$597$	−1.79082	−0.0732934
$598$	0	0
$599$	21.2084	0.866552	0.433276	−	0.901261i	$-0.357358\pi$
$599$	21.2084	0.866552	0.433276	+	0.901261i	$0.357358\pi$
$600$	0	0
$601$	−32.1413	−1.31107	−0.655536	−	0.755164i	$-0.727557\pi$
$601$	−32.1413	−1.31107	−0.655536	+	0.755164i	$0.727557\pi$
$602$	0	0
$603$	37.0361	1.50822
$604$	0	0
$605$	−21.7368	−0.883728
$606$	0	0
$607$	5.76036	0.233806	0.116903	−	0.993143i	$-0.462703\pi$
$607$	5.76036	0.233806	0.116903	+	0.993143i	$0.462703\pi$
$608$	0	0
$609$	−1.32340	−0.0536270
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−26.4633	−1.06884	−0.534421	−	0.845219i	$-0.679470\pi$
$613$	−26.4633	−1.06884	−0.534421	+	0.845219i	$0.679470\pi$
$614$	0	0
$615$	−1.72161	−0.0694221
$616$	0	0
$617$	−16.8221	−0.677230	−0.338615	−	0.940925i	$-0.609958\pi$
$617$	−16.8221	−0.677230	−0.338615	+	0.940925i	$0.609958\pi$
$618$	0	0
$619$	25.8325	1.03829	0.519147	−	0.854685i	$-0.326250\pi$
$619$	25.8325	1.03829	0.519147	+	0.854685i	$0.326250\pi$
$620$	0	0
$621$	21.9058	0.879050
$622$	0	0
$623$	−14.5180	−0.581652
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−22.1496	−0.884570
$628$	0	0
$629$	5.95162	0.237306
$630$	0	0
$631$	−38.0900	−1.51634	−0.758170	−	0.652057i	$-0.773906\pi$
$631$	−38.0900	−1.51634	−0.758170	+	0.652057i	$0.773906\pi$
$632$	0	0
$633$	−11.6412	−0.462697
$634$	0	0
$635$	12.0900	0.479778
$636$	0	0
$637$	0.547814	0.0217052
$638$	0	0
$639$	−14.7480	−0.583423
$640$	0	0
$641$	14.5664	0.575339	0.287669	−	0.957730i	$-0.407120\pi$
$641$	14.5664	0.575339	0.287669	+	0.957730i	$0.407120\pi$
$642$	0	0
$643$	40.5664	1.59978	0.799892	−	0.600145i	$-0.204890\pi$
$643$	40.5664	1.59978	0.799892	+	0.600145i	$0.204890\pi$
$644$	0	0
$645$	4.52699	0.178250
$646$	0	0
$647$	40.7368	1.60153	0.800765	−	0.598978i	$-0.204426\pi$
$647$	40.7368	1.60153	0.800765	+	0.598978i	$0.204426\pi$
$648$	0	0
$649$	67.2645	2.64036
$650$	0	0
$651$	−5.01041	−0.196374
$652$	0	0
$653$	33.1648	1.29784	0.648920	−	0.760857i	$-0.275221\pi$
$653$	33.1648	1.29784	0.648920	+	0.760857i	$0.275221\pi$
$654$	0	0
$655$	8.49720	0.332013
$656$	0	0
$657$	−26.8470	−1.04740
$658$	0	0
$659$	−4.12878	−0.160835	−0.0804173	−	0.996761i	$-0.525625\pi$
$659$	−4.12878	−0.160835	−0.0804173	+	0.996761i	$0.525625\pi$
$660$	0	0
$661$	6.66763	0.259341	0.129670	−	0.991557i	$-0.458608\pi$
$661$	6.66763	0.259341	0.129670	+	0.991557i	$0.458608\pi$
$662$	0	0
$663$	0.414215	0.0160868
$664$	0	0
$665$	−23.9404	−0.928370
$666$	0	0
$667$	8.18421	0.316894
$668$	0	0
$669$	2.75958	0.106692
$670$	0	0
$671$	55.0844	2.12651
$672$	0	0
$673$	−10.4280	−0.401970	−0.200985	−	0.979594i	$-0.564414\pi$
$673$	−10.4280	−0.401970	−0.200985	+	0.979594i	$0.564414\pi$
$674$	0	0
$675$	−2.67660	−0.103022
$676$	0	0
$677$	43.3657	1.66668	0.833340	−	0.552761i	$-0.186426\pi$
$677$	43.3657	1.66668	0.833340	+	0.552761i	$0.186426\pi$
$678$	0	0
$679$	−4.76663	−0.182926
$680$	0	0
$681$	−5.91960	−0.226840
$682$	0	0
$683$	6.81724	0.260854	0.130427	−	0.991458i	$-0.458365\pi$
$683$	6.81724	0.260854	0.130427	+	0.991458i	$0.458365\pi$
$684$	0	0
$685$	3.25901	0.124520
$686$	0	0
$687$	−5.26383	−0.200828
$688$	0	0
$689$	−1.19462	−0.0455115
$690$	0	0
$691$	0.950838	0.0361716	0.0180858	−	0.999836i	$-0.494243\pi$
$691$	0.950838	0.0361716	0.0180858	+	0.999836i	$0.494243\pi$
$692$	0	0
$693$	45.6025	1.73229
$694$	0	0
$695$	22.6724	0.860015
$696$	0	0
$697$	7.20359	0.272855
$698$	0	0
$699$	8.32677	0.314947
$700$	0	0
$701$	−49.2132	−1.85876	−0.929379	−	0.369127i	$-0.879657\pi$
$701$	−49.2132	−1.85876	−0.929379	+	0.369127i	$0.879657\pi$
$702$	0	0
$703$	25.7312	0.970472
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	−37.9315	−1.42656
$708$	0	0
$709$	39.6204	1.48797	0.743987	−	0.668194i	$-0.232932\pi$
$709$	39.6204	1.48797	0.743987	+	0.668194i	$0.232932\pi$
$710$	0	0
$711$	−16.4536	−0.617060
$712$	0	0
$713$	30.9854	1.16041
$714$	0	0
$715$	2.64681	0.0989850
$716$	0	0
$717$	4.06921	0.151967
$718$	0	0
$719$	−3.33237	−0.124276	−0.0621382	−	0.998068i	$-0.519792\pi$
$719$	−3.33237	−0.124276	−0.0621382	+	0.998068i	$0.519792\pi$
$720$	0	0
$721$	42.5872	1.58603
$722$	0	0
$723$	−2.60324	−0.0968156
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−2.36842	−0.0878398	−0.0439199	−	0.999035i	$-0.513985\pi$
$727$	−2.36842	−0.0878398	−0.0439199	+	0.999035i	$0.513985\pi$
$728$	0	0
$729$	−16.1213	−0.597084
$730$	0	0
$731$	−18.9419	−0.700591
$732$	0	0
$733$	30.3297	1.12025	0.560126	−	0.828407i	$-0.310753\pi$
$733$	30.3297	1.12025	0.560126	+	0.828407i	$0.310753\pi$
$734$	0	0
$735$	−0.547814	−0.0202064
$736$	0	0
$737$	76.0609	2.80174
$738$	0	0
$739$	−33.5928	−1.23573	−0.617866	−	0.786283i	$-0.712003\pi$
$739$	−33.5928	−1.23573	−0.617866	+	0.786283i	$0.712003\pi$
$740$	0	0
$741$	1.79082	0.0657874
$742$	0	0
$743$	21.9404	0.804916	0.402458	−	0.915438i	$-0.368156\pi$
$743$	21.9404	0.804916	0.402458	+	0.915438i	$0.368156\pi$
$744$	0	0
$745$	−11.6620	−0.427264
$746$	0	0
$747$	13.7216	0.502048
$748$	0	0
$749$	−4.23964	−0.154913
$750$	0	0
$751$	−40.9944	−1.49591	−0.747954	−	0.663751i	$-0.768963\pi$
$751$	−40.9944	−1.49591	−0.747954	+	0.663751i	$0.768963\pi$
$752$	0	0
$753$	5.12319	0.186699
$754$	0	0
$755$	4.66763	0.169873
$756$	0	0
$757$	14.0388	0.510247	0.255124	−	0.966908i	$-0.417884\pi$
$757$	14.0388	0.510247	0.255124	+	0.966908i	$0.417884\pi$
$758$	0	0
$759$	21.6620	0.786282
$760$	0	0
$761$	−28.9300	−1.04871	−0.524356	−	0.851499i	$-0.675694\pi$
$761$	−28.9300	−1.04871	−0.524356	+	0.851499i	$0.675694\pi$
$762$	0	0
$763$	48.6773	1.76224
$764$	0	0
$765$	5.39261	0.194970
$766$	0	0
$767$	−5.43841	−0.196369
$768$	0	0
$769$	−16.7756	−0.604943	−0.302472	−	0.953158i	$-0.597812\pi$
$769$	−16.7756	−0.604943	−0.302472	+	0.953158i	$0.597812\pi$
$770$	0	0
$771$	−6.41680	−0.231096
$772$	0	0
$773$	−8.05957	−0.289883	−0.144941	−	0.989440i	$-0.546299\pi$
$773$	−8.05957	−0.289883	−0.144941	+	0.989440i	$0.546299\pi$
$774$	0	0
$775$	−3.78600	−0.135997
$776$	0	0
$777$	4.06921	0.145982
$778$	0	0
$779$	31.1440	1.11585
$780$	0	0
$781$	−30.2880	−1.08379
$782$	0	0
$783$	2.67660	0.0956537
$784$	0	0
$785$	15.3836	0.549066
$786$	0	0
$787$	33.0636	1.17859	0.589295	−	0.807918i	$-0.299405\pi$
$787$	33.0636	1.17859	0.589295	+	0.807918i	$0.299405\pi$
$788$	0	0
$789$	3.74244	0.133234
$790$	0	0
$791$	−15.4730	−0.550157
$792$	0	0
$793$	−4.45364	−0.158153
$794$	0	0
$795$	1.19462	0.0423689
$796$	0	0
$797$	21.2936	0.754259	0.377129	−	0.926161i	$-0.376911\pi$
$797$	21.2936	0.754259	0.377129	+	0.926161i	$0.376911\pi$
$798$	0	0
$799$	−16.7368	−0.592107
$800$	0	0
$801$	14.1384	0.499556
$802$	0	0
$803$	−55.1357	−1.94570
$804$	0	0
$805$	23.4134	0.825215
$806$	0	0
$807$	2.22104	0.0781844
$808$	0	0
$809$	11.3532	0.399157	0.199578	−	0.979882i	$-0.436043\pi$
$809$	11.3532	0.399157	0.199578	+	0.979882i	$0.436043\pi$
$810$	0	0
$811$	−14.2445	−0.500190	−0.250095	−	0.968221i	$-0.580462\pi$
$811$	−14.2445	−0.500190	−0.250095	+	0.968221i	$0.580462\pi$
$812$	0	0
$813$	11.5816	0.406186
$814$	0	0
$815$	−0.646809	−0.0226567
$816$	0	0
$817$	−81.8934	−2.86509
$818$	0	0
$819$	−3.68701	−0.128834
$820$	0	0
$821$	−20.2188	−0.705641	−0.352821	−	0.935691i	$-0.614777\pi$
$821$	−20.2188	−0.705641	−0.352821	+	0.935691i	$0.614777\pi$
$822$	0	0
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	−2.64681	−0.0921500
$826$	0	0
$827$	15.5589	0.541036	0.270518	−	0.962715i	$-0.412805\pi$
$827$	15.5589	0.541036	0.270518	+	0.962715i	$0.412805\pi$
$828$	0	0
$829$	−27.5589	−0.957160	−0.478580	−	0.878044i	$-0.658848\pi$
$829$	−27.5589	−0.957160	−0.478580	+	0.878044i	$0.658848\pi$
$830$	0	0
$831$	10.2605	0.355932
$832$	0	0
$833$	2.29217	0.0794189
$834$	0	0
$835$	−14.5616	−0.503925
$836$	0	0
$837$	10.1336	0.350268
$838$	0	0
$839$	−3.95835	−0.136657	−0.0683287	−	0.997663i	$-0.521767\pi$
$839$	−3.95835	−0.136657	−0.0683287	+	0.997663i	$0.521767\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−1.76103	−0.0606532
$844$	0	0
$845$	12.7860	0.439852
$846$	0	0
$847$	62.1849	2.13670
$848$	0	0
$849$	2.27839	0.0781941
$850$	0	0
$851$	−25.1648	−0.862639
$852$	0	0
$853$	3.63158	0.124343	0.0621715	−	0.998065i	$-0.480197\pi$
$853$	3.63158	0.124343	0.0621715	+	0.998065i	$0.480197\pi$
$854$	0	0
$855$	23.3144	0.797337
$856$	0	0
$857$	−39.9196	−1.36363	−0.681814	−	0.731526i	$-0.738809\pi$
$857$	−39.9196	−1.36363	−0.681814	+	0.731526i	$0.738809\pi$
$858$	0	0
$859$	−26.1772	−0.893153	−0.446577	−	0.894745i	$-0.647357\pi$
$859$	−26.1772	−0.893153	−0.446577	+	0.894745i	$0.647357\pi$
$860$	0	0
$861$	4.92520	0.167850
$862$	0	0
$863$	24.9211	0.848322	0.424161	−	0.905587i	$-0.360569\pi$
$863$	24.9211	0.848322	0.424161	+	0.905587i	$0.360569\pi$
$864$	0	0
$865$	−6.98062	−0.237348
$866$	0	0
$867$	−6.13101	−0.208220
$868$	0	0
$869$	−33.7908	−1.14628
$870$	0	0
$871$	−6.14961	−0.208371
$872$	0	0
$873$	4.64199	0.157108
$874$	0	0
$875$	−2.86081	−0.0967129
$876$	0	0
$877$	−45.6371	−1.54105	−0.770527	−	0.637407i	$-0.780007\pi$
$877$	−45.6371	−1.54105	−0.770527	+	0.637407i	$0.780007\pi$
$878$	0	0
$879$	10.2784	0.346681
$880$	0	0
$881$	−39.7216	−1.33826	−0.669128	−	0.743148i	$-0.733332\pi$
$881$	−39.7216	−1.33826	−0.669128	+	0.743148i	$0.733332\pi$
$882$	0	0
$883$	−55.2132	−1.85807	−0.929036	−	0.369988i	$-0.879362\pi$
$883$	−55.2132	−1.85807	−0.929036	+	0.369988i	$0.879362\pi$
$884$	0	0
$885$	5.43841	0.182810
$886$	0	0
$887$	−8.13168	−0.273035	−0.136518	−	0.990638i	$-0.543591\pi$
$887$	−8.13168	−0.273035	−0.136518	+	0.990638i	$0.543591\pi$
$888$	0	0
$889$	−34.5872	−1.16002
$890$	0	0
$891$	−40.7368	−1.36474
$892$	0	0
$893$	−72.3601	−2.42144
$894$	0	0
$895$	−11.7562	−0.392967
$896$	0	0
$897$	−1.75140	−0.0584775
$898$	0	0
$899$	3.78600	0.126270
$900$	0	0
$901$	−4.99855	−0.166526
$902$	0	0
$903$	−12.9508	−0.430977
$904$	0	0
$905$	5.57201	0.185220
$906$	0	0
$907$	9.78600	0.324939	0.162469	−	0.986714i	$-0.448054\pi$
$907$	9.78600	0.324939	0.162469	+	0.986714i	$0.448054\pi$
$908$	0	0
$909$	36.9396	1.22521
$910$	0	0
$911$	−39.3795	−1.30470	−0.652351	−	0.757917i	$-0.726217\pi$
$911$	−39.3795	−1.30470	−0.652351	+	0.757917i	$0.726217\pi$
$912$	0	0
$913$	28.1801	0.932624
$914$	0	0
$915$	4.45364	0.147233
$916$	0	0
$917$	−24.3088	−0.802749
$918$	0	0
$919$	0.994404	0.0328024	0.0164012	−	0.999865i	$-0.494779\pi$
$919$	0.994404	0.0328024	0.0164012	+	0.999865i	$0.494779\pi$
$920$	0	0
$921$	10.1496	0.334441
$922$	0	0
$923$	2.44882	0.0806039
$924$	0	0
$925$	3.07480	0.101099
$926$	0	0
$927$	−41.4737	−1.36217
$928$	0	0
$929$	22.6933	0.744542	0.372271	−	0.928124i	$-0.378579\pi$
$929$	22.6933	0.744542	0.372271	+	0.928124i	$0.378579\pi$
$930$	0	0
$931$	9.90997	0.324786
$932$	0	0
$933$	−8.88163	−0.290772
$934$	0	0
$935$	11.0748	0.362185
$936$	0	0
$937$	−34.0096	−1.11105	−0.555523	−	0.831501i	$-0.687482\pi$
$937$	−34.0096	−1.11105	−0.555523	+	0.831501i	$0.687482\pi$
$938$	0	0
$939$	−4.11086	−0.134153
$940$	0	0
$941$	−31.3532	−1.02208	−0.511042	−	0.859555i	$-0.670740\pi$
$941$	−31.3532	−1.02208	−0.511042	+	0.859555i	$0.670740\pi$
$942$	0	0
$943$	−30.4585	−0.991864
$944$	0	0
$945$	7.65722	0.249089
$946$	0	0
$947$	−2.24378	−0.0729132	−0.0364566	−	0.999335i	$-0.511607\pi$
$947$	−2.24378	−0.0729132	−0.0364566	+	0.999335i	$0.511607\pi$
$948$	0	0
$949$	4.45778	0.144706
$950$	0	0
$951$	1.79082	0.0580713
$952$	0	0
$953$	−1.56237	−0.0506102	−0.0253051	−	0.999680i	$-0.508056\pi$
$953$	−1.56237	−0.0506102	−0.0253051	+	0.999680i	$0.508056\pi$
$954$	0	0
$955$	−13.0796	−0.423247
$956$	0	0
$957$	2.64681	0.0855592
$958$	0	0
$959$	−9.32340	−0.301068
$960$	0	0
$961$	−16.6662	−0.537619
$962$	0	0
$963$	4.12878	0.133048
$964$	0	0
$965$	4.18421	0.134694
$966$	0	0
$967$	58.8060	1.89108	0.945538	−	0.325513i	$-0.105537\pi$
$967$	58.8060	1.89108	0.945538	+	0.325513i	$0.105537\pi$
$968$	0	0
$969$	7.49316	0.240715
$970$	0	0
$971$	15.5541	0.499154	0.249577	−	0.968355i	$-0.419708\pi$
$971$	15.5541	0.499154	0.249577	+	0.968355i	$0.419708\pi$
$972$	0	0
$973$	−64.8615	−2.07936
$974$	0	0
$975$	0.213997	0.00685340
$976$	0	0
$977$	−35.9612	−1.15050	−0.575251	−	0.817977i	$-0.695096\pi$
$977$	−35.9612	−1.15050	−0.575251	+	0.817977i	$0.695096\pi$
$978$	0	0
$979$	29.0361	0.927996
$980$	0	0
$981$	−47.4045	−1.51351
$982$	0	0
$983$	50.0609	1.59670	0.798348	−	0.602197i	$-0.205708\pi$
$983$	50.0609	1.59670	0.798348	+	0.602197i	$0.205708\pi$
$984$	0	0
$985$	1.28880	0.0410646
$986$	0	0
$987$	−11.4432	−0.364242
$988$	0	0
$989$	80.0907	2.54674
$990$	0	0
$991$	−20.0096	−0.635627	−0.317813	−	0.948153i	$-0.602949\pi$
$991$	−20.0096	−0.635627	−0.317813	+	0.948153i	$0.602949\pi$
$992$	0	0
$993$	−9.90997	−0.314483
$994$	0	0
$995$	3.87122	0.122726
$996$	0	0
$997$	19.7521	0.625554	0.312777	−	0.949827i	$-0.398741\pi$
$997$	19.7521	0.625554	0.312777	+	0.949827i	$0.398741\pi$
$998$	0	0
$999$	−8.23000	−0.260386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.a.p.1.2		3
4.3	odd	2		1160.2.a.g.1.2	✓	3
8.3	odd	2		9280.2.a.bo.1.2		3
8.5	even	2		9280.2.a.bq.1.2		3
20.19	odd	2		5800.2.a.q.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.g.1.2	✓	3	4.3	odd	2
2320.2.a.p.1.2		3	1.1	even	1	trivial
5800.2.a.q.1.2		3	20.19	odd	2
9280.2.a.bo.1.2		3	8.3	odd	2
9280.2.a.bq.1.2		3	8.5	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q+0.462598 q^{3} -1.00000 q^{5} +2.86081 q^{7} -2.78600 q^{9} -5.72161 q^{11} +0.462598 q^{13} -0.462598 q^{15} +1.93561 q^{17} +8.36842 q^{19} +1.32340 q^{21} -8.18421 q^{23} +1.00000 q^{25} -2.67660 q^{27} -1.00000 q^{29} -3.78600 q^{31} -2.64681 q^{33} -2.86081 q^{35} +3.07480 q^{37} +0.213997 q^{39} +3.72161 q^{41} -9.78600 q^{43} +2.78600 q^{45} -8.64681 q^{47} +1.18421 q^{49} +0.895410 q^{51} -2.58242 q^{53} +5.72161 q^{55} +3.87122 q^{57} -11.7562 q^{59} -9.62743 q^{61} -7.97021 q^{63} -0.462598 q^{65} -13.2936 q^{67} -3.78600 q^{69} +5.29362 q^{71} +9.63640 q^{73} +0.462598 q^{75} -16.3684 q^{77} +5.90582 q^{79} +7.11982 q^{81} -4.92520 q^{83} -1.93561 q^{85} -0.462598 q^{87} -5.07480 q^{89} +1.32340 q^{91} -1.75140 q^{93} -8.36842 q^{95} -1.66618 q^{97} +15.9404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 2 q^{9} - 6 q^{11} - q^{13} + q^{15} + 5 q^{17} - 2 q^{19} - 4 q^{21} - 11 q^{23} + 3 q^{25} - 16 q^{27} - 3 q^{29} - q^{31} + 8 q^{33} - 3 q^{35} + 14 q^{37} + 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 2 * q^9 - 6 * q^11 - q^13 + q^15 + 5 * q^17 - 2 * q^19 - 4 * q^21 - 11 * q^23 + 3 * q^25 - 16 * q^27 - 3 * q^29 - q^31 + 8 * q^33 - 3 * q^35 + 14 * q^37 + 11 * q^39 - 19 * q^43 - 2 * q^45 - 10 * q^47 - 10 * q^49 - 26 * q^51 + 9 * q^53 + 6 * q^55 + 10 * q^57 - q^59 + 7 * q^61 + q^65 - 8 * q^67 - q^69 - 16 * q^71 + 9 * q^73 - q^75 - 22 * q^77 - 7 * q^79 + 7 * q^81 - 10 * q^83 - 5 * q^85 + q^87 - 20 * q^89 - 4 * q^91 - 18 * q^93 + 2 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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