LMFDB - Embedded newform 6223.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,8,0,24,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86718$ of defining polynomial
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.86718 q^{2} +1.14215 q^{3} +1.48637 q^{4} -2.52504 q^{5} -2.13259 q^{6} +0.959047 q^{8} -1.69550 q^{9} +4.71472 q^{10} +2.14677 q^{11} +1.69765 q^{12} -3.10489 q^{13} -2.88397 q^{15} -4.76345 q^{16} -5.91789 q^{17} +3.16581 q^{18} -5.60200 q^{19} -3.75314 q^{20} -4.00841 q^{22} -7.11857 q^{23} +1.09537 q^{24} +1.37585 q^{25} +5.79739 q^{26} -5.36295 q^{27} -7.43778 q^{29} +5.38489 q^{30} -5.17447 q^{31} +6.97613 q^{32} +2.45192 q^{33} +11.0498 q^{34} -2.52014 q^{36} -4.49744 q^{37} +10.4600 q^{38} -3.54624 q^{39} -2.42164 q^{40} +6.07522 q^{41} -5.88817 q^{43} +3.19089 q^{44} +4.28122 q^{45} +13.2917 q^{46} -3.24407 q^{47} -5.44055 q^{48} -2.56896 q^{50} -6.75909 q^{51} -4.61500 q^{52} +10.8652 q^{53} +10.0136 q^{54} -5.42069 q^{55} -6.39830 q^{57} +13.8877 q^{58} -11.7201 q^{59} -4.28663 q^{60} +4.57418 q^{61} +9.66167 q^{62} -3.49880 q^{64} +7.83998 q^{65} -4.57818 q^{66} +3.13289 q^{67} -8.79615 q^{68} -8.13044 q^{69} +15.6126 q^{71} -1.62607 q^{72} +13.6309 q^{73} +8.39753 q^{74} +1.57142 q^{75} -8.32663 q^{76} +6.62146 q^{78} +11.7382 q^{79} +12.0279 q^{80} -1.03876 q^{81} -11.3435 q^{82} -11.0096 q^{83} +14.9429 q^{85} +10.9943 q^{86} -8.49503 q^{87} +2.05885 q^{88} -5.48084 q^{89} -7.99382 q^{90} -10.5808 q^{92} -5.90999 q^{93} +6.05728 q^{94} +14.1453 q^{95} +7.96775 q^{96} +1.03456 q^{97} -3.63985 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 8 q^{2} + 24 q^{4} - 3 q^{5} - 6 q^{6} + 24 q^{8} + 30 q^{9} + 8 q^{10} + 26 q^{11} + 4 q^{12} + 4 q^{13} + 10 q^{15} + 24 q^{16} - 4 q^{17} + 5 q^{18} - q^{19} + 2 q^{20} + q^{22} + 31 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) $ 20 * q + 8 * q^2 + 24 * q^4 - 3 * q^5 - 6 * q^6 + 24 * q^8 + 30 * q^9 + 8 * q^10 + 26 * q^11 + 4 * q^12 + 4 * q^13 + 10 * q^15 + 24 * q^16 - 4 * q^17 + 5 * q^18 - q^19 + 2 * q^20 + q^22 + 31 * q^23 + 6 * q^24 + 27 * q^25 - 4 * q^26 + 18 * q^27 + 16 * q^29 - 5 * q^30 - 6 * q^31 + 41 * q^32 + 18 * q^33 + 10 * q^34 + 18 * q^36 + 2 * q^37 - 3 * q^38 + 43 * q^39 + 38 * q^40 - 25 * q^41 + 13 * q^43 + 66 * q^44 + 2 * q^45 + 20 * q^46 - 19 * q^47 + 16 * q^48 - 4 * q^50 + 4 * q^51 - 20 * q^52 + 24 * q^53 - 5 * q^54 + 3 * q^55 - 4 * q^57 + 12 * q^58 - 23 * q^59 + 24 * q^60 + 27 * q^61 - 7 * q^62 + 2 * q^64 + 26 * q^65 - 26 * q^66 + 9 * q^67 + 25 * q^68 + 3 * q^69 + 63 * q^71 + 27 * q^72 + 21 * q^73 + 21 * q^74 + 52 * q^75 + 10 * q^76 - 70 * q^78 + 18 * q^79 + 23 * q^80 + 40 * q^81 + 42 * q^82 + q^83 - 41 * q^85 - 12 * q^86 + 9 * q^87 + 57 * q^88 + 16 * q^89 - q^90 + 17 * q^92 - 41 * q^93 - 7 * q^94 + 75 * q^95 + 81 * q^96 + 32 * q^97 + 15 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.86718	−1.32030	−0.660148	−	0.751135i	$-0.729507\pi$
$2$	−1.86718	−1.32030	−0.660148	+	0.751135i	$0.729507\pi$
$3$	1.14215	0.659418	0.329709	−	0.944083i	$-0.393049\pi$
$3$	1.14215	0.659418	0.329709	+	0.944083i	$0.393049\pi$
$4$	1.48637	0.743183
$5$	−2.52504	−1.12923	−0.564617	−	0.825353i	$-0.690976\pi$
$5$	−2.52504	−1.12923	−0.564617	+	0.825353i	$0.690976\pi$
$6$	−2.13259	−0.870627
$7$	0	0
$7$	0	0
$8$	0.959047	0.339074
$9$	−1.69550	−0.565168
$10$	4.71472	1.49092
$11$	2.14677	0.647275	0.323638	−	0.946181i	$-0.395094\pi$
$11$	2.14677	0.647275	0.323638	+	0.946181i	$0.395094\pi$
$12$	1.69765	0.490068
$13$	−3.10489	−0.861141	−0.430571	−	0.902557i	$-0.641688\pi$
$13$	−3.10489	−0.861141	−0.430571	+	0.902557i	$0.641688\pi$
$14$	0	0
$15$	−2.88397	−0.744637
$16$	−4.76345	−1.19086
$17$	−5.91789	−1.43530	−0.717649	−	0.696405i	$-0.754782\pi$
$17$	−5.91789	−1.43530	−0.717649	+	0.696405i	$0.754782\pi$
$18$	3.16581	0.746189
$19$	−5.60200	−1.28519	−0.642594	−	0.766207i	$-0.722142\pi$
$19$	−5.60200	−1.28519	−0.642594	+	0.766207i	$0.722142\pi$
$20$	−3.75314	−0.839228
$21$	0	0
$22$	−4.00841	−0.854595
$23$	−7.11857	−1.48432	−0.742162	−	0.670220i	$-0.766200\pi$
$23$	−7.11857	−1.48432	−0.742162	+	0.670220i	$0.766200\pi$
$24$	1.09537	0.223592
$25$	1.37585	0.275170
$26$	5.79739	1.13696
$27$	−5.36295	−1.03210
$28$	0	0
$29$	−7.43778	−1.38116	−0.690581	−	0.723255i	$-0.742645\pi$
$29$	−7.43778	−1.38116	−0.690581	+	0.723255i	$0.742645\pi$
$30$	5.38489	0.983142
$31$	−5.17447	−0.929361	−0.464681	−	0.885478i	$-0.653831\pi$
$31$	−5.17447	−0.929361	−0.464681	+	0.885478i	$0.653831\pi$
$32$	6.97613	1.23322
$33$	2.45192	0.426825
$34$	11.0498	1.89502
$35$	0	0
$36$	−2.52014	−0.420023
$37$	−4.49744	−0.739374	−0.369687	−	0.929156i	$-0.620535\pi$
$37$	−4.49744	−0.739374	−0.369687	+	0.929156i	$0.620535\pi$
$38$	10.4600	1.69683
$39$	−3.54624	−0.567852
$40$	−2.42164	−0.382894
$41$	6.07522	0.948791	0.474395	−	0.880312i	$-0.342667\pi$
$41$	6.07522	0.948791	0.474395	+	0.880312i	$0.342667\pi$
$42$	0	0
$43$	−5.88817	−0.897938	−0.448969	−	0.893547i	$-0.648209\pi$
$43$	−5.88817	−0.897938	−0.448969	+	0.893547i	$0.648209\pi$
$44$	3.19089	0.481044
$45$	4.28122	0.638207
$46$	13.2917	1.95975
$47$	−3.24407	−0.473197	−0.236598	−	0.971608i	$-0.576033\pi$
$47$	−3.24407	−0.473197	−0.236598	+	0.971608i	$0.576033\pi$
$48$	−5.44055	−0.785276
$49$	0	0
$50$	−2.56896	−0.363306
$51$	−6.75909	−0.946462
$52$	−4.61500	−0.639986
$53$	10.8652	1.49245	0.746223	−	0.665697i	$-0.231865\pi$
$53$	10.8652	1.49245	0.746223	+	0.665697i	$0.231865\pi$
$54$	10.0136	1.36268
$55$	−5.42069	−0.730925
$56$	0	0
$57$	−6.39830	−0.847476
$58$	13.8877	1.82354
$59$	−11.7201	−1.52583	−0.762915	−	0.646499i	$-0.776233\pi$
$59$	−11.7201	−1.52583	−0.762915	+	0.646499i	$0.776233\pi$
$60$	−4.28663	−0.553402
$61$	4.57418	0.585663	0.292832	−	0.956164i	$-0.405402\pi$
$61$	4.57418	0.585663	0.292832	+	0.956164i	$0.405402\pi$
$62$	9.66167	1.22703
$63$	0	0
$64$	−3.49880	−0.437350
$65$	7.83998	0.972430
$66$	−4.57818	−0.563535
$67$	3.13289	0.382743	0.191372	−	0.981518i	$-0.438706\pi$
$67$	3.13289	0.382743	0.191372	+	0.981518i	$0.438706\pi$
$68$	−8.79615	−1.06669
$69$	−8.13044	−0.978791
$70$	0	0
$71$	15.6126	1.85287	0.926437	−	0.376449i	$-0.122855\pi$
$71$	15.6126	1.85287	0.926437	+	0.376449i	$0.122855\pi$
$72$	−1.62607	−0.191634
$73$	13.6309	1.59538	0.797690	−	0.603068i	$-0.206055\pi$
$73$	13.6309	1.59538	0.797690	+	0.603068i	$0.206055\pi$
$74$	8.39753	0.976193
$75$	1.57142	0.181452
$76$	−8.32663	−0.955130
$77$	0	0
$78$	6.62146	0.749733
$79$	11.7382	1.32065	0.660326	−	0.750979i	$-0.270418\pi$
$79$	11.7382	1.32065	0.660326	+	0.750979i	$0.270418\pi$
$80$	12.0279	1.34476
$81$	−1.03876	−0.115417
$82$	−11.3435	−1.25268
$83$	−11.0096	−1.20846	−0.604229	−	0.796811i	$-0.706519\pi$
$83$	−11.0096	−1.20846	−0.604229	+	0.796811i	$0.706519\pi$
$84$	0	0
$85$	14.9429	1.62079
$86$	10.9943	1.18554
$87$	−8.49503	−0.910762
$88$	2.05885	0.219474
$89$	−5.48084	−0.580968	−0.290484	−	0.956880i	$-0.593816\pi$
$89$	−5.48084	−0.580968	−0.290484	+	0.956880i	$0.593816\pi$
$90$	−7.99382	−0.842623
$91$	0	0
$92$	−10.5808	−1.10313
$93$	−5.90999	−0.612838
$94$	6.05728	0.624760
$95$	14.1453	1.45128
$96$	7.96775	0.813205
$97$	1.03456	0.105043	0.0525217	−	0.998620i	$-0.483274\pi$
$97$	1.03456	0.105043	0.0525217	+	0.998620i	$0.483274\pi$
$98$	0	0
$99$	−3.63985	−0.365819
$100$	2.04502	0.204502
$101$	−13.0504	−1.29856	−0.649280	−	0.760549i	$-0.724930\pi$
$101$	−13.0504	−1.29856	−0.649280	+	0.760549i	$0.724930\pi$
$102$	12.6204	1.24961
$103$	9.26938	0.913339	0.456669	−	0.889636i	$-0.349042\pi$
$103$	9.26938	0.913339	0.456669	+	0.889636i	$0.349042\pi$
$104$	−2.97773	−0.291991
$105$	0	0
$106$	−20.2872	−1.97047
$107$	−8.65350	−0.836565	−0.418283	−	0.908317i	$-0.637368\pi$
$107$	−8.65350	−0.836565	−0.418283	+	0.908317i	$0.637368\pi$
$108$	−7.97131	−0.767039
$109$	−1.20206	−0.115136	−0.0575682	−	0.998342i	$-0.518335\pi$
$109$	−1.20206	−0.115136	−0.0575682	+	0.998342i	$0.518335\pi$
$110$	10.1214	0.965038
$111$	−5.13673	−0.487556
$112$	0	0
$113$	−1.24812	−0.117413	−0.0587067	−	0.998275i	$-0.518698\pi$
$113$	−1.24812	−0.117413	−0.0587067	+	0.998275i	$0.518698\pi$
$114$	11.9468	1.11892
$115$	17.9747	1.67615
$116$	−11.0553	−1.02646
$117$	5.26435	0.486689
$118$	21.8836	2.01455
$119$	0	0
$120$	−2.76586	−0.252487
$121$	−6.39138	−0.581035
$122$	−8.54082	−0.773249
$123$	6.93879	0.625650
$124$	−7.69115	−0.690686
$125$	9.15114	0.818503
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−7.41936	−0.655785
$129$	−6.72515	−0.592116
$130$	−14.6387	−1.28390
$131$	−10.6910	−0.934076	−0.467038	−	0.884237i	$-0.654679\pi$
$131$	−10.6910	−0.934076	−0.467038	+	0.884237i	$0.654679\pi$
$132$	3.64446	0.317209
$133$	0	0
$134$	−5.84967	−0.505335
$135$	13.5417	1.16548
$136$	−5.67553	−0.486673
$137$	8.97622	0.766891	0.383445	−	0.923564i	$-0.374737\pi$
$137$	8.97622	0.766891	0.383445	+	0.923564i	$0.374737\pi$
$138$	15.1810	1.29229
$139$	−16.3441	−1.38629	−0.693144	−	0.720799i	$-0.743775\pi$
$139$	−16.3441	−1.38629	−0.693144	+	0.720799i	$0.743775\pi$
$140$	0	0
$141$	−3.70520	−0.312035
$142$	−29.1516	−2.44634
$143$	−6.66548	−0.557395
$144$	8.07644	0.673037
$145$	18.7807	1.55965
$146$	−25.4514	−2.10637
$147$	0	0
$148$	−6.68484	−0.549490
$149$	21.0349	1.72325	0.861623	−	0.507549i	$-0.169448\pi$
$149$	21.0349	1.72325	0.861623	+	0.507549i	$0.169448\pi$
$150$	−2.93413	−0.239571
$151$	−19.6782	−1.60139	−0.800693	−	0.599075i	$-0.795535\pi$
$151$	−19.6782	−1.60139	−0.800693	+	0.599075i	$0.795535\pi$
$152$	−5.37258	−0.435774
$153$	10.0338	0.811185
$154$	0	0
$155$	13.0658	1.04947
$156$	−5.27101	−0.422018
$157$	−8.58647	−0.685275	−0.342637	−	0.939468i	$-0.611320\pi$
$157$	−8.58647	−0.685275	−0.342637	+	0.939468i	$0.611320\pi$
$158$	−21.9174	−1.74365
$159$	12.4096	0.984145
$160$	−17.6150	−1.39259
$161$	0	0
$162$	1.93954	0.152385
$163$	−7.72950	−0.605421	−0.302711	−	0.953082i	$-0.597892\pi$
$163$	−7.72950	−0.605421	−0.302711	+	0.953082i	$0.597892\pi$
$164$	9.03001	0.705125
$165$	−6.19121	−0.481985
$166$	20.5569	1.59552
$167$	1.90242	0.147214	0.0736070	−	0.997287i	$-0.476549\pi$
$167$	1.90242	0.147214	0.0736070	+	0.997287i	$0.476549\pi$
$168$	0	0
$169$	−3.35966	−0.258436
$170$	−27.9012	−2.13992
$171$	9.49822	0.726347
$172$	−8.75198	−0.667333
$173$	9.51257	0.723227	0.361614	−	0.932328i	$-0.382226\pi$
$173$	9.51257	0.723227	0.361614	+	0.932328i	$0.382226\pi$
$174$	15.8618	1.20248
$175$	0	0
$176$	−10.2260	−0.770815
$177$	−13.3861	−1.00616
$178$	10.2337	0.767050
$179$	9.54411	0.713360	0.356680	−	0.934227i	$-0.383909\pi$
$179$	9.54411	0.713360	0.356680	+	0.934227i	$0.383909\pi$
$180$	6.36347	0.474305
$181$	−21.9419	−1.63093	−0.815465	−	0.578807i	$-0.803519\pi$
$181$	−21.9419	−1.63093	−0.815465	+	0.578807i	$0.803519\pi$
$182$	0	0
$183$	5.22438	0.386197
$184$	−6.82704	−0.503296
$185$	11.3562	0.834926
$186$	11.0350	0.809127
$187$	−12.7043	−0.929033
$188$	−4.82188	−0.351672
$189$	0	0
$190$	−26.4119	−1.91612
$191$	−20.8863	−1.51128	−0.755640	−	0.654987i	$-0.772674\pi$
$191$	−20.8863	−1.51128	−0.755640	+	0.654987i	$0.772674\pi$
$192$	−3.99614	−0.288396
$193$	−14.8527	−1.06912	−0.534559	−	0.845131i	$-0.679522\pi$
$193$	−14.8527	−1.06912	−0.534559	+	0.845131i	$0.679522\pi$
$194$	−1.93171	−0.138689
$195$	8.95440	0.641238
$196$	0	0
$197$	13.6942	0.975669	0.487834	−	0.872936i	$-0.337787\pi$
$197$	13.6942	0.975669	0.487834	+	0.872936i	$0.337787\pi$
$198$	6.79627	0.482990
$199$	−17.7755	−1.26007	−0.630035	−	0.776567i	$-0.716959\pi$
$199$	−17.7755	−1.26007	−0.630035	+	0.776567i	$0.716959\pi$
$200$	1.31951	0.0933032
$201$	3.57822	0.252388
$202$	24.3674	1.71449
$203$	0	0
$204$	−10.0465	−0.703395
$205$	−15.3402	−1.07141
$206$	−17.3076	−1.20588
$207$	12.0696	0.838893
$208$	14.7900	1.02550
$209$	−12.0262	−0.831870
$210$	0	0
$211$	−7.22564	−0.497434	−0.248717	−	0.968576i	$-0.580009\pi$
$211$	−7.22564	−0.497434	−0.248717	+	0.968576i	$0.580009\pi$
$212$	16.1496	1.10916
$213$	17.8319	1.22182
$214$	16.1577	1.10451
$215$	14.8679	1.01398
$216$	−5.14332	−0.349959
$217$	0	0
$218$	2.24446	0.152014
$219$	15.5685	1.05202
$220$	−8.05713	−0.543212
$221$	18.3744	1.23600
$222$	9.59120	0.643719
$223$	1.99060	0.133300	0.0666501	−	0.997776i	$-0.478769\pi$
$223$	1.99060	0.133300	0.0666501	+	0.997776i	$0.478769\pi$
$224$	0	0
$225$	−2.33276	−0.155517
$226$	2.33047	0.155021
$227$	−1.85235	−0.122945	−0.0614723	−	0.998109i	$-0.519580\pi$
$227$	−1.85235	−0.122945	−0.0614723	+	0.998109i	$0.519580\pi$
$228$	−9.51022	−0.629830
$229$	4.03988	0.266963	0.133481	−	0.991051i	$-0.457384\pi$
$229$	4.03988	0.266963	0.133481	+	0.991051i	$0.457384\pi$
$230$	−33.5621	−2.21302
$231$	0	0
$232$	−7.13318	−0.468316
$233$	13.3853	0.876902	0.438451	−	0.898755i	$-0.355527\pi$
$233$	13.3853	0.876902	0.438451	+	0.898755i	$0.355527\pi$
$234$	−9.82950	−0.642575
$235$	8.19143	0.534350
$236$	−17.4204	−1.13397
$237$	13.4067	0.870862
$238$	0	0
$239$	−5.43938	−0.351844	−0.175922	−	0.984404i	$-0.556291\pi$
$239$	−5.43938	−0.351844	−0.175922	+	0.984404i	$0.556291\pi$
$240$	13.7376	0.886760
$241$	26.4016	1.70067	0.850337	−	0.526238i	$-0.176398\pi$
$241$	26.4016	1.70067	0.850337	+	0.526238i	$0.176398\pi$
$242$	11.9339	0.767138
$243$	14.9024	0.955992
$244$	6.79890	0.435255
$245$	0	0
$246$	−12.9560	−0.826043
$247$	17.3936	1.10673
$248$	−4.96256	−0.315123
$249$	−12.5745	−0.796878
$250$	−17.0868	−1.08067
$251$	20.9117	1.31993	0.659967	−	0.751294i	$-0.270570\pi$
$251$	20.9117	1.31993	0.659967	+	0.751294i	$0.270570\pi$
$252$	0	0
$253$	−15.2819	−0.960767
$254$	1.86718	0.117157
$255$	17.0670	1.06878
$256$	20.8509	1.30318
$257$	24.3901	1.52141	0.760705	−	0.649097i	$-0.224853\pi$
$257$	24.3901	1.52141	0.760705	+	0.649097i	$0.224853\pi$
$258$	12.5571	0.781769
$259$	0	0
$260$	11.6531	0.722694
$261$	12.6108	0.780588
$262$	19.9620	1.23326
$263$	8.75029	0.539566	0.269783	−	0.962921i	$-0.413048\pi$
$263$	8.75029	0.539566	0.269783	+	0.962921i	$0.413048\pi$
$264$	2.35151	0.144725
$265$	−27.4350	−1.68532
$266$	0	0
$267$	−6.25992	−0.383101
$268$	4.65662	0.284448
$269$	−0.734176	−0.0447635	−0.0223818	−	0.999749i	$-0.507125\pi$
$269$	−0.734176	−0.0447635	−0.0223818	+	0.999749i	$0.507125\pi$
$270$	−25.2848	−1.53878
$271$	−6.99503	−0.424918	−0.212459	−	0.977170i	$-0.568147\pi$
$271$	−6.99503	−0.424918	−0.212459	+	0.977170i	$0.568147\pi$
$272$	28.1895	1.70924
$273$	0	0
$274$	−16.7602	−1.01252
$275$	2.95363	0.178111
$276$	−12.0848	−0.727421
$277$	3.60962	0.216881	0.108441	−	0.994103i	$-0.465414\pi$
$277$	3.60962	0.216881	0.108441	+	0.994103i	$0.465414\pi$
$278$	30.5174	1.83031
$279$	8.77333	0.525245
$280$	0	0
$281$	24.7064	1.47386	0.736931	−	0.675968i	$-0.236274\pi$
$281$	24.7064	1.47386	0.736931	+	0.675968i	$0.236274\pi$
$282$	6.91829	0.411978
$283$	−23.5104	−1.39755	−0.698773	−	0.715343i	$-0.746270\pi$
$283$	−23.5104	−1.39755	−0.698773	+	0.715343i	$0.746270\pi$
$284$	23.2060	1.37703
$285$	16.1560	0.956999
$286$	12.4457	0.735927
$287$	0	0
$288$	−11.8281	−0.696975
$289$	18.0214	1.06008
$290$	−35.0670	−2.05921
$291$	1.18162	0.0692676
$292$	20.2606	1.18566
$293$	14.2321	0.831445	0.415723	−	0.909491i	$-0.363529\pi$
$293$	14.2321	0.831445	0.415723	+	0.909491i	$0.363529\pi$
$294$	0	0
$295$	29.5938	1.72302
$296$	−4.31325	−0.250703
$297$	−11.5130	−0.668053
$298$	−39.2760	−2.27520
$299$	22.1024	1.27821
$300$	2.33571	0.134852
$301$	0	0
$302$	36.7427	2.11430
$303$	−14.9054	−0.856294
$304$	26.6848	1.53048
$305$	−11.5500	−0.661351
$306$	−18.7349	−1.07100
$307$	2.43954	0.139232	0.0696159	−	0.997574i	$-0.477823\pi$
$307$	2.43954	0.139232	0.0696159	+	0.997574i	$0.477823\pi$
$308$	0	0
$309$	10.5870	0.602272
$310$	−24.3961	−1.38561
$311$	14.2486	0.807961	0.403981	−	0.914768i	$-0.367626\pi$
$311$	14.2486	0.807961	0.403981	+	0.914768i	$0.367626\pi$
$312$	−3.40101	−0.192544
$313$	−1.01920	−0.0576087	−0.0288043	−	0.999585i	$-0.509170\pi$
$313$	−1.01920	−0.0576087	−0.0288043	+	0.999585i	$0.509170\pi$
$314$	16.0325	0.904766
$315$	0	0
$316$	17.4473	0.981486
$317$	−26.7112	−1.50025	−0.750124	−	0.661297i	$-0.770006\pi$
$317$	−26.7112	−1.50025	−0.750124	+	0.661297i	$0.770006\pi$
$318$	−23.1710	−1.29936
$319$	−15.9672	−0.893991
$320$	8.83463	0.493871
$321$	−9.88356	−0.551646
$322$	0	0
$323$	33.1520	1.84463
$324$	−1.54397	−0.0857762
$325$	−4.27187	−0.236960
$326$	14.4324	0.799336
$327$	−1.37293	−0.0759231
$328$	5.82642	0.321710
$329$	0	0
$330$	11.5601	0.636364
$331$	3.21043	0.176461	0.0882305	−	0.996100i	$-0.471879\pi$
$331$	3.21043	0.176461	0.0882305	+	0.996100i	$0.471879\pi$
$332$	−16.3643	−0.898105
$333$	7.62542	0.417870
$334$	−3.55217	−0.194366
$335$	−7.91069	−0.432207
$336$	0	0
$337$	−15.9804	−0.870505	−0.435253	−	0.900308i	$-0.643341\pi$
$337$	−15.9804	−0.870505	−0.435253	+	0.900308i	$0.643341\pi$
$338$	6.27310	0.341212
$339$	−1.42554	−0.0774245
$340$	22.2107	1.20454
$341$	−11.1084	−0.601553
$342$	−17.7349	−0.958993
$343$	0	0
$344$	−5.64703	−0.304468
$345$	20.5297	1.10528
$346$	−17.7617	−0.954874
$347$	−4.64687	−0.249457	−0.124729	−	0.992191i	$-0.539806\pi$
$347$	−4.64687	−0.249457	−0.124729	+	0.992191i	$0.539806\pi$
$348$	−12.6267	−0.676863
$349$	12.8959	0.690300	0.345150	−	0.938547i	$-0.387828\pi$
$349$	12.8959	0.690300	0.345150	+	0.938547i	$0.387828\pi$
$350$	0	0
$351$	16.6514	0.888784
$352$	14.9761	0.798231
$353$	−6.04228	−0.321598	−0.160799	−	0.986987i	$-0.551407\pi$
$353$	−6.04228	−0.321598	−0.160799	+	0.986987i	$0.551407\pi$
$354$	24.9943	1.32843
$355$	−39.4225	−2.09233
$356$	−8.14654	−0.431766
$357$	0	0
$358$	−17.8206	−0.941847
$359$	27.1794	1.43447	0.717237	−	0.696830i	$-0.245407\pi$
$359$	27.1794	1.43447	0.717237	+	0.696830i	$0.245407\pi$
$360$	4.10589	0.216400
$361$	12.3824	0.651707
$362$	40.9695	2.15331
$363$	−7.29989	−0.383145
$364$	0	0
$365$	−34.4187	−1.80156
$366$	−9.75486	−0.509895
$367$	−32.2253	−1.68215	−0.841073	−	0.540922i	$-0.818076\pi$
$367$	−32.2253	−1.68215	−0.841073	+	0.540922i	$0.818076\pi$
$368$	33.9089	1.76763
$369$	−10.3006	−0.536226
$370$	−21.2041	−1.10235
$371$	0	0
$372$	−8.78441	−0.455451
$373$	−29.7787	−1.54188	−0.770941	−	0.636907i	$-0.780214\pi$
$373$	−29.7787	−1.54188	−0.770941	+	0.636907i	$0.780214\pi$
$374$	23.7213	1.22660
$375$	10.4519	0.539735
$376$	−3.11122	−0.160449
$377$	23.0935	1.18937
$378$	0	0
$379$	12.3300	0.633349	0.316675	−	0.948534i	$-0.397434\pi$
$379$	12.3300	0.633349	0.316675	+	0.948534i	$0.397434\pi$
$380$	21.0251	1.07857
$381$	−1.14215	−0.0585139
$382$	38.9985	1.99534
$383$	36.4315	1.86156	0.930782	−	0.365574i	$-0.119127\pi$
$383$	36.4315	1.86156	0.930782	+	0.365574i	$0.119127\pi$
$384$	−8.47399	−0.432436
$385$	0	0
$386$	27.7326	1.41155
$387$	9.98342	0.507486
$388$	1.53773	0.0780666
$389$	−35.0881	−1.77904	−0.889519	−	0.456898i	$-0.848960\pi$
$389$	−35.0881	−1.77904	−0.889519	+	0.456898i	$0.848960\pi$
$390$	−16.7195	−0.846625
$391$	42.1269	2.13045
$392$	0	0
$393$	−12.2107	−0.615946
$394$	−25.5695	−1.28817
$395$	−29.6395	−1.49133
$396$	−5.41016	−0.271871
$397$	−21.7798	−1.09310	−0.546548	−	0.837428i	$-0.684058\pi$
$397$	−21.7798	−1.09310	−0.546548	+	0.837428i	$0.684058\pi$
$398$	33.1900	1.66367
$399$	0	0
$400$	−6.55380	−0.327690
$401$	18.4781	0.922751	0.461375	−	0.887205i	$-0.347356\pi$
$401$	18.4781	0.922751	0.461375	+	0.887205i	$0.347356\pi$
$402$	−6.68118	−0.333227
$403$	16.0661	0.800312
$404$	−19.3976	−0.965068
$405$	2.62290	0.130333
$406$	0	0
$407$	−9.65496	−0.478578
$408$	−6.48228	−0.320921
$409$	26.3563	1.30324	0.651618	−	0.758547i	$-0.274091\pi$
$409$	26.3563	1.30324	0.651618	+	0.758547i	$0.274091\pi$
$410$	28.6430	1.41457
$411$	10.2522	0.505701
$412$	13.7777	0.678778
$413$	0	0
$414$	−22.5361	−1.10759
$415$	27.7997	1.36463
$416$	−21.6601	−1.06197
$417$	−18.6673	−0.914144
$418$	22.4551	1.09832
$419$	−7.75685	−0.378947	−0.189473	−	0.981886i	$-0.560678\pi$
$419$	−7.75685	−0.378947	−0.189473	+	0.981886i	$0.560678\pi$
$420$	0	0
$421$	−32.9882	−1.60775	−0.803873	−	0.594801i	$-0.797231\pi$
$421$	−32.9882	−1.60775	−0.803873	+	0.594801i	$0.797231\pi$
$422$	13.4916	0.656760
$423$	5.50034	0.267436
$424$	10.4202	0.506050
$425$	−8.14213	−0.394952
$426$	−33.2953	−1.61316
$427$	0	0
$428$	−12.8623	−0.621721
$429$	−7.61295	−0.367557
$430$	−27.7611	−1.33876
$431$	−0.521602	−0.0251247	−0.0125623	−	0.999921i	$-0.503999\pi$
$431$	−0.521602	−0.0251247	−0.0125623	+	0.999921i	$0.503999\pi$
$432$	25.5461	1.22909
$433$	−14.6517	−0.704117	−0.352059	−	0.935978i	$-0.614518\pi$
$433$	−14.6517	−0.704117	−0.352059	+	0.935978i	$0.614518\pi$
$434$	0	0
$435$	21.4503	1.02846
$436$	−1.78670	−0.0855675
$437$	39.8783	1.90764
$438$	−29.0692	−1.38898
$439$	−11.7950	−0.562946	−0.281473	−	0.959569i	$-0.590823\pi$
$439$	−11.7950	−0.562946	−0.281473	+	0.959569i	$0.590823\pi$
$440$	−5.19869	−0.247838
$441$	0	0
$442$	−34.3083	−1.63188
$443$	4.46547	0.212161	0.106080	−	0.994358i	$-0.466170\pi$
$443$	4.46547	0.212161	0.106080	+	0.994358i	$0.466170\pi$
$444$	−7.63506	−0.362344
$445$	13.8394	0.656049
$446$	−3.71680	−0.175996
$447$	24.0249	1.13634
$448$	0	0
$449$	2.18869	0.103291	0.0516454	−	0.998665i	$-0.483553\pi$
$449$	2.18869	0.103291	0.0516454	+	0.998665i	$0.483553\pi$
$450$	4.35569	0.205329
$451$	13.0421	0.614129
$452$	−1.85517	−0.0872597
$453$	−22.4753	−1.05598
$454$	3.45867	0.162323
$455$	0	0
$456$	−6.13627	−0.287357
$457$	−39.0672	−1.82749	−0.913743	−	0.406293i	$-0.866821\pi$
$457$	−39.0672	−1.82749	−0.913743	+	0.406293i	$0.866821\pi$
$458$	−7.54318	−0.352470
$459$	31.7373	1.48137
$460$	26.7170	1.24569
$461$	−23.5633	−1.09745	−0.548727	−	0.836002i	$-0.684887\pi$
$461$	−23.5633	−1.09745	−0.548727	+	0.836002i	$0.684887\pi$
$462$	0	0
$463$	11.8195	0.549298	0.274649	−	0.961544i	$-0.411438\pi$
$463$	11.8195	0.549298	0.274649	+	0.961544i	$0.411438\pi$
$464$	35.4295	1.64477
$465$	14.9230	0.692037
$466$	−24.9928	−1.15777
$467$	−16.5778	−0.767128	−0.383564	−	0.923514i	$-0.625303\pi$
$467$	−16.5778	−0.767128	−0.383564	+	0.923514i	$0.625303\pi$
$468$	7.82476	0.361699
$469$	0	0
$470$	−15.2949	−0.705501
$471$	−9.80700	−0.451882
$472$	−11.2401	−0.517370
$473$	−12.6405	−0.581213
$474$	−25.0328	−1.14980
$475$	−7.70752	−0.353645
$476$	0	0
$477$	−18.4219	−0.843482
$478$	10.1563	0.464538
$479$	−22.6906	−1.03676	−0.518380	−	0.855150i	$-0.673465\pi$
$479$	−22.6906	−1.03676	−0.518380	+	0.855150i	$0.673465\pi$
$480$	−20.1189	−0.918299
$481$	13.9640	0.636705
$482$	−49.2965	−2.24540
$483$	0	0
$484$	−9.49994	−0.431815
$485$	−2.61231	−0.118619
$486$	−27.8256	−1.26219
$487$	−21.8682	−0.990941	−0.495470	−	0.868625i	$-0.665004\pi$
$487$	−21.8682	−0.990941	−0.495470	+	0.868625i	$0.665004\pi$
$488$	4.38685	0.198583
$489$	−8.82822	−0.399226
$490$	0	0
$491$	40.6492	1.83447	0.917235	−	0.398346i	$-0.130416\pi$
$491$	40.6492	1.83447	0.917235	+	0.398346i	$0.130416\pi$
$492$	10.3136	0.464972
$493$	44.0159	1.98238
$494$	−32.4770	−1.46121
$495$	9.19080	0.413096
$496$	24.6483	1.10674
$497$	0	0
$498$	23.4789	1.05212
$499$	15.4222	0.690393	0.345197	−	0.938530i	$-0.387812\pi$
$499$	15.4222	0.690393	0.345197	+	0.938530i	$0.387812\pi$
$500$	13.6019	0.608297
$501$	2.17284	0.0970755
$502$	−39.0459	−1.74270
$503$	−14.5992	−0.650945	−0.325473	−	0.945551i	$-0.605523\pi$
$503$	−14.5992	−0.650945	−0.325473	+	0.945551i	$0.605523\pi$
$504$	0	0
$505$	32.9528	1.46638
$506$	28.5341	1.26850
$507$	−3.83722	−0.170417
$508$	−1.48637	−0.0659469
$509$	−16.9749	−0.752397	−0.376199	−	0.926539i	$-0.622769\pi$
$509$	−16.9749	−0.752397	−0.376199	+	0.926539i	$0.622769\pi$
$510$	−31.8672	−1.41110
$511$	0	0
$512$	−24.0937	−1.06480
$513$	30.0432	1.32644
$514$	−45.5407	−2.00871
$515$	−23.4056	−1.03137
$516$	−9.99604	−0.440051
$517$	−6.96428	−0.306289
$518$	0	0
$519$	10.8647	0.476909
$520$	7.51891	0.329726
$521$	−16.7832	−0.735283	−0.367642	−	0.929968i	$-0.619835\pi$
$521$	−16.7832	−0.735283	−0.367642	+	0.929968i	$0.619835\pi$
$522$	−23.5466	−1.03061
$523$	20.9119	0.914414	0.457207	−	0.889360i	$-0.348850\pi$
$523$	20.9119	0.914414	0.457207	+	0.889360i	$0.348850\pi$
$524$	−15.8907	−0.694189
$525$	0	0
$526$	−16.3384	−0.712387
$527$	30.6219	1.33391
$528$	−11.6796	−0.508289
$529$	27.6741	1.20322
$530$	51.2262	2.22512
$531$	19.8715	0.862350
$532$	0	0
$533$	−18.8629	−0.817043
$534$	11.6884	0.505807
$535$	21.8505	0.944678
$536$	3.00459	0.129778
$537$	10.9008	0.470403
$538$	1.37084	0.0591011
$539$	0	0
$540$	20.1279	0.866167
$541$	4.34159	0.186660	0.0933298	−	0.995635i	$-0.470249\pi$
$541$	4.34159	0.186660	0.0933298	+	0.995635i	$0.470249\pi$
$542$	13.0610	0.561018
$543$	−25.0609	−1.07546
$544$	−41.2839	−1.77003
$545$	3.03526	0.130016
$546$	0	0
$547$	−2.97082	−0.127023	−0.0635116	−	0.997981i	$-0.520230\pi$
$547$	−2.97082	−0.127023	−0.0635116	+	0.997981i	$0.520230\pi$
$548$	13.3420	0.569940
$549$	−7.75554	−0.330998
$550$	−5.51497	−0.235159
$551$	41.6665	1.77505
$552$	−7.79748	−0.331883
$553$	0	0
$554$	−6.73982	−0.286347
$555$	12.9705	0.550565
$556$	−24.2933	−1.03027
$557$	−37.9755	−1.60907	−0.804536	−	0.593904i	$-0.797586\pi$
$557$	−37.9755	−1.60907	−0.804536	+	0.593904i	$0.797586\pi$
$558$	−16.3814	−0.693480
$559$	18.2821	0.773252
$560$	0	0
$561$	−14.5102	−0.612621
$562$	−46.1314	−1.94594
$563$	25.8374	1.08892	0.544459	−	0.838787i	$-0.316735\pi$
$563$	25.8374	1.08892	0.544459	+	0.838787i	$0.316735\pi$
$564$	−5.50729	−0.231899
$565$	3.15156	0.132587
$566$	43.8981	1.84518
$567$	0	0
$568$	14.9732	0.628262
$569$	7.36191	0.308628	0.154314	−	0.988022i	$-0.450683\pi$
$569$	7.36191	0.308628	0.154314	+	0.988022i	$0.450683\pi$
$570$	−30.1662	−1.26352
$571$	−37.2421	−1.55854	−0.779268	−	0.626691i	$-0.784409\pi$
$571$	−37.2421	−1.55854	−0.779268	+	0.626691i	$0.784409\pi$
$572$	−9.90735	−0.414247
$573$	−23.8552	−0.996565
$574$	0	0
$575$	−9.79410	−0.408442
$576$	5.93223	0.247176
$577$	8.48323	0.353161	0.176581	−	0.984286i	$-0.443496\pi$
$577$	8.48323	0.353161	0.176581	+	0.984286i	$0.443496\pi$
$578$	−33.6492	−1.39962
$579$	−16.9639	−0.704995
$580$	27.9150	1.15911
$581$	0	0
$582$	−2.20629	−0.0914537
$583$	23.3250	0.966023
$584$	13.0727	0.540952
$585$	−13.2927	−0.549587
$586$	−26.5738	−1.09775
$587$	7.57932	0.312832	0.156416	−	0.987691i	$-0.450006\pi$
$587$	7.57932	0.312832	0.156416	+	0.987691i	$0.450006\pi$
$588$	0	0
$589$	28.9874	1.19440
$590$	−55.2571	−2.27490
$591$	15.6407	0.643374
$592$	21.4233	0.880492
$593$	18.4304	0.756848	0.378424	−	0.925632i	$-0.376466\pi$
$593$	18.4304	0.756848	0.378424	+	0.925632i	$0.376466\pi$
$594$	21.4969	0.882028
$595$	0	0
$596$	31.2656	1.28069
$597$	−20.3022	−0.830913
$598$	−41.2692	−1.68762
$599$	−6.29754	−0.257311	−0.128655	−	0.991689i	$-0.541066\pi$
$599$	−6.29754	−0.257311	−0.128655	+	0.991689i	$0.541066\pi$
$600$	1.50707	0.0615258
$601$	0.979702	0.0399629	0.0199814	−	0.999800i	$-0.493639\pi$
$601$	0.979702	0.0399629	0.0199814	+	0.999800i	$0.493639\pi$
$602$	0	0
$603$	−5.31183	−0.216314
$604$	−29.2490	−1.19012
$605$	16.1385	0.656125
$606$	27.8311	1.13056
$607$	−30.7475	−1.24800	−0.624001	−	0.781424i	$-0.714494\pi$
$607$	−30.7475	−1.24800	−0.624001	+	0.781424i	$0.714494\pi$
$608$	−39.0803	−1.58491
$609$	0	0
$610$	21.5660	0.873180
$611$	10.0725	0.407489
$612$	14.9139	0.602859
$613$	−5.30594	−0.214305	−0.107152	−	0.994243i	$-0.534173\pi$
$613$	−5.30594	−0.214305	−0.107152	+	0.994243i	$0.534173\pi$
$614$	−4.55506	−0.183827
$615$	−17.5208	−0.706505
$616$	0	0
$617$	31.1189	1.25280	0.626401	−	0.779501i	$-0.284527\pi$
$617$	31.1189	1.25280	0.626401	+	0.779501i	$0.284527\pi$
$618$	−19.7678	−0.795178
$619$	−2.21302	−0.0889489	−0.0444744	−	0.999011i	$-0.514161\pi$
$619$	−2.21302	−0.0889489	−0.0444744	+	0.999011i	$0.514161\pi$
$620$	19.4205	0.779946
$621$	38.1765	1.53197
$622$	−26.6046	−1.06675
$623$	0	0
$624$	16.8923	0.676233
$625$	−29.9863	−1.19945
$626$	1.90303	0.0760605
$627$	−13.7357	−0.548550
$628$	−12.7626	−0.509285
$629$	26.6153	1.06122
$630$	0	0
$631$	−30.3853	−1.20962	−0.604810	−	0.796370i	$-0.706751\pi$
$631$	−30.3853	−1.20962	−0.604810	+	0.796370i	$0.706751\pi$
$632$	11.2575	0.447799
$633$	−8.25274	−0.328017
$634$	49.8746	1.98077
$635$	2.52504	0.100203
$636$	18.4452	0.731400
$637$	0	0
$638$	29.8137	1.18033
$639$	−26.4712	−1.04719
$640$	18.7342	0.740535
$641$	−29.0454	−1.14722	−0.573612	−	0.819127i	$-0.694458\pi$
$641$	−29.0454	−1.14722	−0.573612	+	0.819127i	$0.694458\pi$
$642$	18.4544	0.728337
$643$	46.5068	1.83405	0.917024	−	0.398833i	$-0.130585\pi$
$643$	46.5068	1.83405	0.917024	+	0.398833i	$0.130585\pi$
$644$	0	0
$645$	16.9813	0.668638
$646$	−61.9008	−2.43546
$647$	40.6542	1.59828	0.799141	−	0.601144i	$-0.205288\pi$
$647$	40.6542	1.59828	0.799141	+	0.601144i	$0.205288\pi$
$648$	−0.996215	−0.0391350
$649$	−25.1604	−0.987632
$650$	7.97635	0.312858
$651$	0	0
$652$	−11.4889	−0.449939
$653$	−39.0488	−1.52810	−0.764049	−	0.645158i	$-0.776792\pi$
$653$	−39.0488	−1.52810	−0.764049	+	0.645158i	$0.776792\pi$
$654$	2.56350	0.100241
$655$	26.9952	1.05479
$656$	−28.9390	−1.12988
$657$	−23.1113	−0.901657
$658$	0	0
$659$	−43.6453	−1.70018	−0.850089	−	0.526639i	$-0.823452\pi$
$659$	−43.6453	−1.70018	−0.850089	+	0.526639i	$0.823452\pi$
$660$	−9.20241	−0.358203
$661$	8.95476	0.348300	0.174150	−	0.984719i	$-0.444282\pi$
$661$	8.95476	0.348300	0.174150	+	0.984719i	$0.444282\pi$
$662$	−5.99445	−0.232981
$663$	20.9862	0.815037
$664$	−10.5587	−0.409757
$665$	0	0
$666$	−14.2380	−0.551713
$667$	52.9464	2.05009
$668$	2.82770	0.109407
$669$	2.27355	0.0879005
$670$	14.7707	0.570641
$671$	9.81970	0.379085
$672$	0	0
$673$	24.6108	0.948678	0.474339	−	0.880342i	$-0.342687\pi$
$673$	24.6108	0.948678	0.474339	+	0.880342i	$0.342687\pi$
$674$	29.8382	1.14933
$675$	−7.37862	−0.284003
$676$	−4.99369	−0.192065
$677$	−28.2165	−1.08445	−0.542224	−	0.840234i	$-0.682418\pi$
$677$	−28.2165	−1.08445	−0.542224	+	0.840234i	$0.682418\pi$
$678$	2.66174	0.102223
$679$	0	0
$680$	14.3310	0.549568
$681$	−2.11565	−0.0810718
$682$	20.7414	0.794228
$683$	3.13372	0.119909	0.0599543	−	0.998201i	$-0.480904\pi$
$683$	3.13372	0.119909	0.0599543	+	0.998201i	$0.480904\pi$
$684$	14.1178	0.539809
$685$	−22.6654	−0.865999
$686$	0	0
$687$	4.61413	0.176040
$688$	28.0480	1.06932
$689$	−33.7351	−1.28521
$690$	−38.3327	−1.45930
$691$	15.3776	0.584992	0.292496	−	0.956267i	$-0.405514\pi$
$691$	15.3776	0.584992	0.292496	+	0.956267i	$0.405514\pi$
$692$	14.1392	0.537490
$693$	0	0
$694$	8.67656	0.329358
$695$	41.2696	1.56544
$696$	−8.14713	−0.308816
$697$	−35.9525	−1.36180
$698$	−24.0789	−0.911401
$699$	15.2880	0.578245
$700$	0	0
$701$	0.753947	0.0284762	0.0142381	−	0.999899i	$-0.495468\pi$
$701$	0.753947	0.0284762	0.0142381	+	0.999899i	$0.495468\pi$
$702$	−31.0911	−1.17346
$703$	25.1946	0.950234
$704$	−7.51112	−0.283086
$705$	9.35581	0.352360
$706$	11.2820	0.424605
$707$	0	0
$708$	−19.8966	−0.747761
$709$	−52.5784	−1.97462	−0.987312	−	0.158791i	$-0.949241\pi$
$709$	−52.5784	−1.97462	−0.987312	+	0.158791i	$0.949241\pi$
$710$	73.6090	2.76250
$711$	−19.9022	−0.746390
$712$	−5.25639	−0.196991
$713$	36.8348	1.37947
$714$	0	0
$715$	16.8306	0.629430
$716$	14.1860	0.530158
$717$	−6.21256	−0.232012
$718$	−50.7489	−1.89393
$719$	−42.2691	−1.57637	−0.788185	−	0.615439i	$-0.788979\pi$
$719$	−42.2691	−1.57637	−0.788185	+	0.615439i	$0.788979\pi$
$720$	−20.3934	−0.760016
$721$	0	0
$722$	−23.1202	−0.860446
$723$	30.1544	1.12146
$724$	−32.6137	−1.21208
$725$	−10.2333	−0.380054
$726$	13.6302	0.505865
$727$	−6.60122	−0.244826	−0.122413	−	0.992479i	$-0.539063\pi$
$727$	−6.60122	−0.244826	−0.122413	+	0.992479i	$0.539063\pi$
$728$	0	0
$729$	20.1370	0.745815
$730$	64.2660	2.37859
$731$	34.8456	1.28881
$732$	7.76534	0.287015
$733$	−11.4503	−0.422927	−0.211464	−	0.977386i	$-0.567823\pi$
$733$	−11.4503	−0.422927	−0.211464	+	0.977386i	$0.567823\pi$
$734$	60.1704	2.22093
$735$	0	0
$736$	−49.6601	−1.83049
$737$	6.72559	0.247740
$738$	19.2330	0.707977
$739$	−36.9250	−1.35831	−0.679154	−	0.733996i	$-0.737653\pi$
$739$	−36.9250	−1.35831	−0.679154	+	0.733996i	$0.737653\pi$
$740$	16.8795	0.620503
$741$	19.8660	0.729796
$742$	0	0
$743$	−19.9823	−0.733081	−0.366541	−	0.930402i	$-0.619458\pi$
$743$	−19.9823	−0.733081	−0.366541	+	0.930402i	$0.619458\pi$
$744$	−5.66796	−0.207797
$745$	−53.1141	−1.94595
$746$	55.6022	2.03574
$747$	18.6668	0.682981
$748$	−18.8833	−0.690442
$749$	0	0
$750$	−19.5157	−0.712611
$751$	8.90745	0.325037	0.162519	−	0.986705i	$-0.448038\pi$
$751$	8.90745	0.325037	0.162519	+	0.986705i	$0.448038\pi$
$752$	15.4530	0.563512
$753$	23.8842	0.870388
$754$	−43.1197	−1.57033
$755$	49.6882	1.80834
$756$	0	0
$757$	13.8060	0.501787	0.250893	−	0.968015i	$-0.419276\pi$
$757$	13.8060	0.501787	0.250893	+	0.968015i	$0.419276\pi$
$758$	−23.0223	−0.836209
$759$	−17.4542	−0.633547
$760$	13.5660	0.492091
$761$	−35.6626	−1.29277	−0.646383	−	0.763013i	$-0.723719\pi$
$761$	−35.6626	−1.29277	−0.646383	+	0.763013i	$0.723719\pi$
$762$	2.13259	0.0772557
$763$	0	0
$764$	−31.0447	−1.12316
$765$	−25.3358	−0.916018
$766$	−68.0243	−2.45782
$767$	36.3897	1.31396
$768$	23.8147	0.859341
$769$	3.21210	0.115831	0.0579156	−	0.998321i	$-0.481555\pi$
$769$	3.21210	0.115831	0.0579156	+	0.998321i	$0.481555\pi$
$770$	0	0
$771$	27.8570	1.00325
$772$	−22.0765	−0.794550
$773$	35.7577	1.28611	0.643057	−	0.765818i	$-0.277666\pi$
$773$	35.7577	1.28611	0.643057	+	0.765818i	$0.277666\pi$
$774$	−18.6409	−0.670032
$775$	−7.11930	−0.255733
$776$	0.992190	0.0356175
$777$	0	0
$778$	65.5159	2.34886
$779$	−34.0334	−1.21937
$780$	13.3095	0.476557
$781$	33.5166	1.19932
$782$	−78.6586	−2.81283
$783$	39.8884	1.42550
$784$	0	0
$785$	21.6812	0.773836
$786$	22.7995	0.813232
$787$	33.2160	1.18402	0.592012	−	0.805929i	$-0.298334\pi$
$787$	33.2160	1.18402	0.592012	+	0.805929i	$0.298334\pi$
$788$	20.3545	0.725101
$789$	9.99410	0.355799
$790$	55.3423	1.96899
$791$	0	0
$792$	−3.49079	−0.124040
$793$	−14.2023	−0.504339
$794$	40.6668	1.44321
$795$	−31.3348	−1.11133
$796$	−26.4209	−0.936463
$797$	8.21020	0.290820	0.145410	−	0.989371i	$-0.453550\pi$
$797$	8.21020	0.290820	0.145410	+	0.989371i	$0.453550\pi$
$798$	0	0
$799$	19.1981	0.679179
$800$	9.59811	0.339345
$801$	9.29279	0.328345
$802$	−34.5019	−1.21830
$803$	29.2625	1.03265
$804$	5.31854	0.187570
$805$	0	0
$806$	−29.9984	−1.05665
$807$	−0.838536	−0.0295179
$808$	−12.5159	−0.440308
$809$	49.0138	1.72323	0.861617	−	0.507559i	$-0.169452\pi$
$809$	49.0138	1.72323	0.861617	+	0.507559i	$0.169452\pi$
$810$	−4.89744	−0.172078
$811$	3.98700	0.140002	0.0700012	−	0.997547i	$-0.477700\pi$
$811$	3.98700	0.140002	0.0700012	+	0.997547i	$0.477700\pi$
$812$	0	0
$813$	−7.98934	−0.280199
$814$	18.0276	0.631865
$815$	19.5173	0.683663
$816$	32.1966	1.12711
$817$	32.9856	1.15402
$818$	−49.2120	−1.72066
$819$	0	0
$820$	−22.8012	−0.796252
$821$	24.3409	0.849502	0.424751	−	0.905310i	$-0.360362\pi$
$821$	24.3409	0.849502	0.424751	+	0.905310i	$0.360362\pi$
$822$	−19.1426	−0.667676
$823$	−3.33724	−0.116329	−0.0581645	−	0.998307i	$-0.518525\pi$
$823$	−3.33724	−0.116329	−0.0581645	+	0.998307i	$0.518525\pi$
$824$	8.88977	0.309690
$825$	3.37348	0.117450
$826$	0	0
$827$	8.40144	0.292147	0.146073	−	0.989274i	$-0.453336\pi$
$827$	8.40144	0.292147	0.146073	+	0.989274i	$0.453336\pi$
$828$	17.9398	0.623451
$829$	47.5492	1.65145	0.825726	−	0.564072i	$-0.190766\pi$
$829$	47.5492	1.65145	0.825726	+	0.564072i	$0.190766\pi$
$830$	−51.9070	−1.80172
$831$	4.12271	0.143015
$832$	10.8634	0.376620
$833$	0	0
$834$	34.8553	1.20694
$835$	−4.80371	−0.166239
$836$	−17.8753	−0.618232
$837$	27.7504	0.959194
$838$	14.4834	0.500322
$839$	−32.1450	−1.10977	−0.554884	−	0.831927i	$-0.687238\pi$
$839$	−32.1450	−1.10977	−0.554884	+	0.831927i	$0.687238\pi$
$840$	0	0
$841$	26.3206	0.907606
$842$	61.5949	2.12270
$843$	28.2183	0.971891
$844$	−10.7400	−0.369684
$845$	8.48330	0.291834
$846$	−10.2701	−0.353094
$847$	0	0
$848$	−51.7556	−1.77730
$849$	−26.8523	−0.921567
$850$	15.2028	0.521453
$851$	32.0153	1.09747
$852$	26.5047	0.908035
$853$	29.3210	1.00393	0.501966	−	0.864887i	$-0.332610\pi$
$853$	29.3210	1.00393	0.501966	+	0.864887i	$0.332610\pi$
$854$	0	0
$855$	−23.9834	−0.820216
$856$	−8.29911	−0.283658
$857$	4.67171	0.159583	0.0797913	−	0.996812i	$-0.474575\pi$
$857$	4.67171	0.159583	0.0797913	+	0.996812i	$0.474575\pi$
$858$	14.2148	0.485284
$859$	−19.8657	−0.677810	−0.338905	−	0.940821i	$-0.610056\pi$
$859$	−19.8657	−0.677810	−0.338905	+	0.940821i	$0.610056\pi$
$860$	22.0992	0.753575
$861$	0	0
$862$	0.973925	0.0331720
$863$	−51.5715	−1.75552	−0.877758	−	0.479105i	$-0.840961\pi$
$863$	−51.5715	−1.75552	−0.877758	+	0.479105i	$0.840961\pi$
$864$	−37.4126	−1.27280
$865$	−24.0197	−0.816693
$866$	27.3574	0.929644
$867$	20.5831	0.699037
$868$	0	0
$869$	25.1992	0.854825
$870$	−40.0516	−1.35788
$871$	−9.72727	−0.329596
$872$	−1.15283	−0.0390398
$873$	−1.75410	−0.0593672
$874$	−74.4599	−2.51865
$875$	0	0
$876$	23.1405	0.781845
$877$	−1.55320	−0.0524479	−0.0262240	−	0.999656i	$-0.508348\pi$
$877$	−1.55320	−0.0524479	−0.0262240	+	0.999656i	$0.508348\pi$
$878$	22.0235	0.743256
$879$	16.2551	0.548270
$880$	25.8212	0.870431
$881$	20.5004	0.690676	0.345338	−	0.938478i	$-0.387764\pi$
$881$	20.5004	0.690676	0.345338	+	0.938478i	$0.387764\pi$
$882$	0	0
$883$	0.206988	0.00696572	0.00348286	−	0.999994i	$-0.498891\pi$
$883$	0.206988	0.00696572	0.00348286	+	0.999994i	$0.498891\pi$
$884$	27.3111	0.918571
$885$	33.8005	1.13619
$886$	−8.33784	−0.280115
$887$	−7.01790	−0.235638	−0.117819	−	0.993035i	$-0.537590\pi$
$887$	−7.01790	−0.235638	−0.117819	+	0.993035i	$0.537590\pi$
$888$	−4.92636	−0.165318
$889$	0	0
$890$	−25.8406	−0.866180
$891$	−2.22997	−0.0747067
$892$	2.95876	0.0990665
$893$	18.1733	0.608147
$894$	−44.8589	−1.50031
$895$	−24.0993	−0.805551
$896$	0	0
$897$	25.2441	0.842877
$898$	−4.08668	−0.136374
$899$	38.4865	1.28360
$900$	−3.46734	−0.115578
$901$	−64.2988	−2.14210
$902$	−24.3520	−0.810832
$903$	0	0
$904$	−1.19701	−0.0398119
$905$	55.4043	1.84170
$906$	41.9655	1.39421
$907$	48.5183	1.61102	0.805512	−	0.592579i	$-0.201890\pi$
$907$	48.5183	1.61102	0.805512	+	0.592579i	$0.201890\pi$
$908$	−2.75326	−0.0913703
$909$	22.1270	0.733905
$910$	0	0
$911$	−37.7847	−1.25186	−0.625931	−	0.779878i	$-0.715281\pi$
$911$	−37.7847	−1.25186	−0.625931	+	0.779878i	$0.715281\pi$
$912$	30.4780	1.00923
$913$	−23.6350	−0.782204
$914$	72.9455	2.41282
$915$	−13.1918	−0.436107
$916$	6.00474	0.198402
$917$	0	0
$918$	−59.2593	−1.95585
$919$	−30.8794	−1.01862	−0.509309	−	0.860584i	$-0.670099\pi$
$919$	−30.8794	−1.01862	−0.509309	+	0.860584i	$0.670099\pi$
$920$	17.2386	0.568340
$921$	2.78631	0.0918120
$922$	43.9970	1.44896
$923$	−48.4754	−1.59559
$924$	0	0
$925$	−6.18780	−0.203454
$926$	−22.0691	−0.725237
$927$	−15.7163	−0.516190
$928$	−51.8869	−1.70327
$929$	38.4148	1.26035	0.630174	−	0.776454i	$-0.282984\pi$
$929$	38.4148	1.26035	0.630174	+	0.776454i	$0.282984\pi$
$930$	−27.8639	−0.913695
$931$	0	0
$932$	19.8955	0.651699
$933$	16.2739	0.532784
$934$	30.9537	1.01284
$935$	32.0790	1.04910
$936$	5.04876	0.165024
$937$	−9.54026	−0.311667	−0.155833	−	0.987783i	$-0.549806\pi$
$937$	−9.54026	−0.311667	−0.155833	+	0.987783i	$0.549806\pi$
$938$	0	0
$939$	−1.16408	−0.0379882
$940$	12.1755	0.397120
$941$	45.4254	1.48082	0.740412	−	0.672153i	$-0.234630\pi$
$941$	45.4254	1.48082	0.740412	+	0.672153i	$0.234630\pi$
$942$	18.3114	0.596619
$943$	−43.2469	−1.40831
$944$	55.8282	1.81705
$945$	0	0
$946$	23.6022	0.767374
$947$	−4.47693	−0.145481	−0.0727403	−	0.997351i	$-0.523174\pi$
$947$	−4.47693	−0.145481	−0.0727403	+	0.997351i	$0.523174\pi$
$948$	19.9273	0.647210
$949$	−42.3225	−1.37385
$950$	14.3913	0.466917
$951$	−30.5080	−0.989291
$952$	0	0
$953$	−29.6135	−0.959274	−0.479637	−	0.877467i	$-0.659232\pi$
$953$	−29.6135	−0.959274	−0.479637	+	0.877467i	$0.659232\pi$
$954$	34.3971	1.11365
$955$	52.7388	1.70659
$956$	−8.08491	−0.261485
$957$	−18.2369	−0.589514
$958$	42.3675	1.36883
$959$	0	0
$960$	10.0904	0.325667
$961$	−4.22491	−0.136287
$962$	−26.0734	−0.840640
$963$	14.6720	0.472800
$964$	39.2424	1.26391
$965$	37.5036	1.20728
$966$	0	0
$967$	53.1670	1.70974	0.854868	−	0.518846i	$-0.173638\pi$
$967$	53.1670	1.70974	0.854868	+	0.518846i	$0.173638\pi$
$968$	−6.12964	−0.197014
$969$	37.8644	1.21638
$970$	4.87765	0.156612
$971$	−4.77380	−0.153199	−0.0765993	−	0.997062i	$-0.524406\pi$
$971$	−4.77380	−0.153199	−0.0765993	+	0.997062i	$0.524406\pi$
$972$	22.1505	0.710477
$973$	0	0
$974$	40.8318	1.30834
$975$	−4.87909	−0.156256
$976$	−21.7889	−0.697444
$977$	20.3191	0.650065	0.325033	−	0.945703i	$-0.394625\pi$
$977$	20.3191	0.650065	0.325033	+	0.945703i	$0.394625\pi$
$978$	16.4839	0.527096
$979$	−11.7661	−0.376046
$980$	0	0
$981$	2.03810	0.0650714
$982$	−75.8993	−2.42205
$983$	2.20607	0.0703627	0.0351813	−	0.999381i	$-0.488799\pi$
$983$	2.20607	0.0703627	0.0351813	+	0.999381i	$0.488799\pi$
$984$	6.65462	0.212142
$985$	−34.5784	−1.10176
$986$	−82.1858	−2.61733
$987$	0	0
$988$	25.8533	0.822502
$989$	41.9154	1.33283
$990$	−17.1609	−0.545409
$991$	10.5849	0.336242	0.168121	−	0.985766i	$-0.446230\pi$
$991$	10.5849	0.336242	0.168121	+	0.985766i	$0.446230\pi$
$992$	−36.0977	−1.14610
$993$	3.66677	0.116362
$994$	0	0
$995$	44.8839	1.42291
$996$	−18.6904	−0.592227
$997$	−29.5991	−0.937413	−0.468706	−	0.883354i	$-0.655280\pi$
$997$	−29.5991	−0.937413	−0.468706	+	0.883354i	$0.655280\pi$
$998$	−28.7961	−0.911524
$999$	24.1195	0.763108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.l.1.3		20
7.6	odd	2		889.2.a.d.1.3	✓	20
21.20	even	2		8001.2.a.w.1.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.3	✓	20	7.6	odd	2
6223.2.a.l.1.3		20	1.1	even	1	trivial
8001.2.a.w.1.18		20	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86718 q^{2} +1.14215 q^{3} +1.48637 q^{4} -2.52504 q^{5} -2.13259 q^{6} +0.959047 q^{8} -1.69550 q^{9} +4.71472 q^{10} +2.14677 q^{11} +1.69765 q^{12} -3.10489 q^{13} -2.88397 q^{15} -4.76345 q^{16} -5.91789 q^{17} +3.16581 q^{18} -5.60200 q^{19} -3.75314 q^{20} -4.00841 q^{22} -7.11857 q^{23} +1.09537 q^{24} +1.37585 q^{25} +5.79739 q^{26} -5.36295 q^{27} -7.43778 q^{29} +5.38489 q^{30} -5.17447 q^{31} +6.97613 q^{32} +2.45192 q^{33} +11.0498 q^{34} -2.52014 q^{36} -4.49744 q^{37} +10.4600 q^{38} -3.54624 q^{39} -2.42164 q^{40} +6.07522 q^{41} -5.88817 q^{43} +3.19089 q^{44} +4.28122 q^{45} +13.2917 q^{46} -3.24407 q^{47} -5.44055 q^{48} -2.56896 q^{50} -6.75909 q^{51} -4.61500 q^{52} +10.8652 q^{53} +10.0136 q^{54} -5.42069 q^{55} -6.39830 q^{57} +13.8877 q^{58} -11.7201 q^{59} -4.28663 q^{60} +4.57418 q^{61} +9.66167 q^{62} -3.49880 q^{64} +7.83998 q^{65} -4.57818 q^{66} +3.13289 q^{67} -8.79615 q^{68} -8.13044 q^{69} +15.6126 q^{71} -1.62607 q^{72} +13.6309 q^{73} +8.39753 q^{74} +1.57142 q^{75} -8.32663 q^{76} +6.62146 q^{78} +11.7382 q^{79} +12.0279 q^{80} -1.03876 q^{81} -11.3435 q^{82} -11.0096 q^{83} +14.9429 q^{85} +10.9943 q^{86} -8.49503 q^{87} +2.05885 q^{88} -5.48084 q^{89} -7.99382 q^{90} -10.5808 q^{92} -5.90999 q^{93} +6.05728 q^{94} +14.1453 q^{95} +7.96775 q^{96} +1.03456 q^{97} -3.63985 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{2} + 24 q^{4} - 3 q^{5} - 6 q^{6} + 24 q^{8} + 30 q^{9} + 8 q^{10} + 26 q^{11} + 4 q^{12} + 4 q^{13} + 10 q^{15} + 24 q^{16} - 4 q^{17} + 5 q^{18} - q^{19} + 2 q^{20} + q^{22} + 31 q^{23}+ \cdots + 15 q^{99}+O(q^{100}) \) 20 * q + 8 * q^2 + 24 * q^4 - 3 * q^5 - 6 * q^6 + 24 * q^8 + 30 * q^9 + 8 * q^10 + 26 * q^11 + 4 * q^12 + 4 * q^13 + 10 * q^15 + 24 * q^16 - 4 * q^17 + 5 * q^18 - q^19 + 2 * q^20 + q^22 + 31 * q^23 + 6 * q^24 + 27 * q^25 - 4 * q^26 + 18 * q^27 + 16 * q^29 - 5 * q^30 - 6 * q^31 + 41 * q^32 + 18 * q^33 + 10 * q^34 + 18 * q^36 + 2 * q^37 - 3 * q^38 + 43 * q^39 + 38 * q^40 - 25 * q^41 + 13 * q^43 + 66 * q^44 + 2 * q^45 + 20 * q^46 - 19 * q^47 + 16 * q^48 - 4 * q^50 + 4 * q^51 - 20 * q^52 + 24 * q^53 - 5 * q^54 + 3 * q^55 - 4 * q^57 + 12 * q^58 - 23 * q^59 + 24 * q^60 + 27 * q^61 - 7 * q^62 + 2 * q^64 + 26 * q^65 - 26 * q^66 + 9 * q^67 + 25 * q^68 + 3 * q^69 + 63 * q^71 + 27 * q^72 + 21 * q^73 + 21 * q^74 + 52 * q^75 + 10 * q^76 - 70 * q^78 + 18 * q^79 + 23 * q^80 + 40 * q^81 + 42 * q^82 + q^83 - 41 * q^85 - 12 * q^86 + 9 * q^87 + 57 * q^88 + 16 * q^89 - q^90 + 17 * q^92 - 41 * q^93 - 7 * q^94 + 75 * q^95 + 81 * q^96 + 32 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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