LMFDB - Embedded newform 6525.2.a.bm.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] 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("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,0,4,0,0,6,-3,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.240881.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 5x - 7 $ x^5 - 2x^4 - 5x^3 + 9x^2 + 5x - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 725)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.88481$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.88481 q^{2} +1.55251 q^{4} +1.70908 q^{7} -0.843434 q^{8} +2.85130 q^{11} +6.22128 q^{13} +3.22128 q^{14} -4.69473 q^{16} +1.40381 q^{17} +6.74028 q^{19} +5.37416 q^{22} -1.42153 q^{23} +11.7259 q^{26} +2.65336 q^{28} -1.00000 q^{29} -1.29510 q^{31} -7.16181 q^{32} +2.64591 q^{34} -0.989830 q^{37} +12.7042 q^{38} -6.32213 q^{41} +6.03720 q^{43} +4.42667 q^{44} -2.67932 q^{46} -8.97549 q^{47} -4.07906 q^{49} +9.65860 q^{52} -2.11288 q^{53} -1.44149 q^{56} -1.88481 q^{58} +10.0504 q^{59} +5.73236 q^{61} -2.44101 q^{62} -4.10919 q^{64} -2.45274 q^{67} +2.17943 q^{68} -5.04282 q^{71} +5.55668 q^{73} -1.86564 q^{74} +10.4644 q^{76} +4.87309 q^{77} +9.37892 q^{79} -11.9160 q^{82} -5.06744 q^{83} +11.3790 q^{86} -2.40488 q^{88} +1.75693 q^{89} +10.6326 q^{91} -2.20694 q^{92} -16.9171 q^{94} +14.6037 q^{97} -7.68825 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} + 6 q^{7} - 3 q^{8} + 2 q^{11} + 4 q^{13} - 11 q^{14} - 10 q^{16} - 9 q^{17} + 2 q^{19} + 4 q^{22} - q^{23} + 16 q^{26} + 10 q^{28} - 5 q^{29} - q^{31} + 2 q^{32} + 3 q^{34} + 14 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 - 3 * q^8 + 2 * q^11 + 4 * q^13 - 11 * q^14 - 10 * q^16 - 9 * q^17 + 2 * q^19 + 4 * q^22 - q^23 + 16 * q^26 + 10 * q^28 - 5 * q^29 - q^31 + 2 * q^32 + 3 * q^34 + 14 * q^37 + 3 * q^38 - 5 * q^41 + 28 * q^43 - 7 * q^44 - 20 * q^46 - 15 * q^47 - 3 * q^49 + 6 * q^52 + 8 * q^53 + 16 * q^56 + 2 * q^58 + 11 * q^59 - 5 * q^61 + 6 * q^62 - 5 * q^64 + 23 * q^67 - q^68 + 5 * q^71 + 16 * q^73 - 8 * q^74 + 16 * q^76 + 30 * q^77 - 10 * q^79 - 19 * q^82 - 9 * q^83 + 13 * q^86 + 35 * q^88 + 18 * q^89 + q^91 + 27 * q^92 - 13 * q^94 + 23 * q^97 - 3 * q^98

Coefficient data

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

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

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

$2$	1.88481	1.33276	0.666381	−	0.745611i	$-0.267842\pi$
$2$	1.88481	1.33276	0.666381	+	0.745611i	$0.267842\pi$
$3$	0	0
$3$	0	0
$4$	1.55251	0.776255
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.70908	0.645970	0.322985	−	0.946404i	$-0.395314\pi$
$7$	1.70908	0.645970	0.322985	+	0.946404i	$0.395314\pi$
$8$	−0.843434	−0.298199
$9$	0	0
$10$	0	0
$11$	2.85130	0.859699	0.429849	−	0.902901i	$-0.358567\pi$
$11$	2.85130	0.859699	0.429849	+	0.902901i	$0.358567\pi$
$12$	0	0
$13$	6.22128	1.72547	0.862737	−	0.505653i	$-0.168748\pi$
$13$	6.22128	1.72547	0.862737	+	0.505653i	$0.168748\pi$
$14$	3.22128	0.860924
$15$	0	0
$16$	−4.69473	−1.17368
$17$	1.40381	0.340474	0.170237	−	0.985403i	$-0.445547\pi$
$17$	1.40381	0.340474	0.170237	+	0.985403i	$0.445547\pi$
$18$	0	0
$19$	6.74028	1.54633	0.773163	−	0.634207i	$-0.218673\pi$
$19$	6.74028	1.54633	0.773163	+	0.634207i	$0.218673\pi$
$20$	0	0
$21$	0	0
$22$	5.37416	1.14577
$23$	−1.42153	−0.296410	−0.148205	−	0.988957i	$-0.547349\pi$
$23$	−1.42153	−0.296410	−0.148205	+	0.988957i	$0.547349\pi$
$24$	0	0
$25$	0	0
$26$	11.7259	2.29965
$27$	0	0
$28$	2.65336	0.501437
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.29510	−0.232606	−0.116303	−	0.993214i	$-0.537104\pi$
$31$	−1.29510	−0.232606	−0.116303	+	0.993214i	$0.537104\pi$
$32$	−7.16181	−1.26604
$33$	0	0
$34$	2.64591	0.453770
$35$	0	0
$36$	0	0
$37$	−0.989830	−0.162727	−0.0813635	−	0.996684i	$-0.525927\pi$
$37$	−0.989830	−0.162727	−0.0813635	+	0.996684i	$0.525927\pi$
$38$	12.7042	2.06089
$39$	0	0
$40$	0	0
$41$	−6.32213	−0.987351	−0.493675	−	0.869646i	$-0.664347\pi$
$41$	−6.32213	−0.987351	−0.493675	+	0.869646i	$0.664347\pi$
$42$	0	0
$43$	6.03720	0.920665	0.460332	−	0.887747i	$-0.347730\pi$
$43$	6.03720	0.920665	0.460332	+	0.887747i	$0.347730\pi$
$44$	4.42667	0.667346
$45$	0	0
$46$	−2.67932	−0.395044
$47$	−8.97549	−1.30921	−0.654605	−	0.755971i	$-0.727165\pi$
$47$	−8.97549	−1.30921	−0.654605	+	0.755971i	$0.727165\pi$
$48$	0	0
$49$	−4.07906	−0.582723
$50$	0	0
$51$	0	0
$52$	9.65860	1.33941
$53$	−2.11288	−0.290227	−0.145113	−	0.989415i	$-0.546355\pi$
$53$	−2.11288	−0.290227	−0.145113	+	0.989415i	$0.546355\pi$
$54$	0	0
$55$	0	0
$56$	−1.44149	−0.192628
$57$	0	0
$58$	−1.88481	−0.247488
$59$	10.0504	1.30845	0.654224	−	0.756301i	$-0.272995\pi$
$59$	10.0504	1.30845	0.654224	+	0.756301i	$0.272995\pi$
$60$	0	0
$61$	5.73236	0.733953	0.366977	−	0.930230i	$-0.380393\pi$
$61$	5.73236	0.733953	0.366977	+	0.930230i	$0.380393\pi$
$62$	−2.44101	−0.310009
$63$	0	0
$64$	−4.10919	−0.513649
$65$	0	0
$66$	0	0
$67$	−2.45274	−0.299650	−0.149825	−	0.988713i	$-0.547871\pi$
$67$	−2.45274	−0.299650	−0.149825	+	0.988713i	$0.547871\pi$
$68$	2.17943	0.264294
$69$	0	0
$70$	0	0
$71$	−5.04282	−0.598473	−0.299236	−	0.954179i	$-0.596732\pi$
$71$	−5.04282	−0.598473	−0.299236	+	0.954179i	$0.596732\pi$
$72$	0	0
$73$	5.55668	0.650360	0.325180	−	0.945652i	$-0.394575\pi$
$73$	5.55668	0.650360	0.325180	+	0.945652i	$0.394575\pi$
$74$	−1.86564	−0.216876
$75$	0	0
$76$	10.4644	1.20034
$77$	4.87309	0.555340
$78$	0	0
$79$	9.37892	1.05521	0.527606	−	0.849489i	$-0.323090\pi$
$79$	9.37892	1.05521	0.527606	+	0.849489i	$0.323090\pi$
$80$	0	0
$81$	0	0
$82$	−11.9160	−1.31590
$83$	−5.06744	−0.556224	−0.278112	−	0.960549i	$-0.589709\pi$
$83$	−5.06744	−0.556224	−0.278112	+	0.960549i	$0.589709\pi$
$84$	0	0
$85$	0	0
$86$	11.3790	1.22703
$87$	0	0
$88$	−2.40488	−0.256361
$89$	1.75693	0.186234	0.0931171	−	0.995655i	$-0.470317\pi$
$89$	1.75693	0.186234	0.0931171	+	0.995655i	$0.470317\pi$
$90$	0	0
$91$	10.6326	1.11460
$92$	−2.20694	−0.230089
$93$	0	0
$94$	−16.9171	−1.74486
$95$	0	0
$96$	0	0
$97$	14.6037	1.48278	0.741389	−	0.671075i	$-0.234167\pi$
$97$	14.6037	1.48278	0.741389	+	0.671075i	$0.234167\pi$
$98$	−7.68825	−0.776631
$99$	0	0
$100$	0	0
$101$	−15.8868	−1.58079	−0.790397	−	0.612595i	$-0.790126\pi$
$101$	−15.8868	−1.58079	−0.790397	+	0.612595i	$0.790126\pi$
$102$	0	0
$103$	11.6322	1.14616	0.573079	−	0.819500i	$-0.305749\pi$
$103$	11.6322	1.14616	0.573079	+	0.819500i	$0.305749\pi$
$104$	−5.24724	−0.514534
$105$	0	0
$106$	−3.98239	−0.386804
$107$	−13.0887	−1.26533	−0.632664	−	0.774426i	$-0.718039\pi$
$107$	−13.0887	−1.26533	−0.632664	+	0.774426i	$0.718039\pi$
$108$	0	0
$109$	10.5634	1.01179	0.505894	−	0.862596i	$-0.331163\pi$
$109$	10.5634	1.01179	0.505894	+	0.862596i	$0.331163\pi$
$110$	0	0
$111$	0	0
$112$	−8.02365	−0.758164
$113$	18.4952	1.73988	0.869940	−	0.493157i	$-0.164157\pi$
$113$	18.4952	1.73988	0.869940	+	0.493157i	$0.164157\pi$
$114$	0	0
$115$	0	0
$116$	−1.55251	−0.144147
$117$	0	0
$118$	18.9430	1.74385
$119$	2.39922	0.219936
$120$	0	0
$121$	−2.87010	−0.260918
$122$	10.8044	0.978185
$123$	0	0
$124$	−2.01065	−0.180562
$125$	0	0
$126$	0	0
$127$	−2.95697	−0.262389	−0.131194	−	0.991357i	$-0.541881\pi$
$127$	−2.95697	−0.262389	−0.131194	+	0.991357i	$0.541881\pi$
$128$	6.57858	0.581470
$129$	0	0
$130$	0	0
$131$	20.9785	1.83290	0.916450	−	0.400150i	$-0.131042\pi$
$131$	20.9785	1.83290	0.916450	+	0.400150i	$0.131042\pi$
$132$	0	0
$133$	11.5197	0.998880
$134$	−4.62294	−0.399362
$135$	0	0
$136$	−1.18402	−0.101529
$137$	19.5953	1.67414	0.837071	−	0.547094i	$-0.184266\pi$
$137$	19.5953	1.67414	0.837071	+	0.547094i	$0.184266\pi$
$138$	0	0
$139$	−21.2859	−1.80545	−0.902725	−	0.430218i	$-0.858437\pi$
$139$	−21.2859	−1.80545	−0.902725	+	0.430218i	$0.858437\pi$
$140$	0	0
$141$	0	0
$142$	−9.50476	−0.797622
$143$	17.7387	1.48339
$144$	0	0
$145$	0	0
$146$	10.4733	0.866776
$147$	0	0
$148$	−1.53672	−0.126318
$149$	16.6123	1.36093	0.680466	−	0.732779i	$-0.261777\pi$
$149$	16.6123	1.36093	0.680466	+	0.732779i	$0.261777\pi$
$150$	0	0
$151$	8.74718	0.711835	0.355918	−	0.934517i	$-0.384168\pi$
$151$	8.74718	0.711835	0.355918	+	0.934517i	$0.384168\pi$
$152$	−5.68498	−0.461113
$153$	0	0
$154$	9.18484	0.740136
$155$	0	0
$156$	0	0
$157$	−20.0971	−1.60392	−0.801960	−	0.597378i	$-0.796209\pi$
$157$	−20.0971	−1.60392	−0.801960	+	0.597378i	$0.796209\pi$
$158$	17.6775	1.40635
$159$	0	0
$160$	0	0
$161$	−2.42950	−0.191472
$162$	0	0
$163$	−3.21100	−0.251505	−0.125753	−	0.992062i	$-0.540135\pi$
$163$	−3.21100	−0.251505	−0.125753	+	0.992062i	$0.540135\pi$
$164$	−9.81517	−0.766436
$165$	0	0
$166$	−9.55117	−0.741315
$167$	−6.83823	−0.529158	−0.264579	−	0.964364i	$-0.585233\pi$
$167$	−6.83823	−0.529158	−0.264579	+	0.964364i	$0.585233\pi$
$168$	0	0
$169$	25.7044	1.97726
$170$	0	0
$171$	0	0
$172$	9.37282	0.714671
$173$	−3.47907	−0.264509	−0.132254	−	0.991216i	$-0.542222\pi$
$173$	−3.47907	−0.264509	−0.132254	+	0.991216i	$0.542222\pi$
$174$	0	0
$175$	0	0
$176$	−13.3861	−1.00901
$177$	0	0
$178$	3.31148	0.248206
$179$	23.3359	1.74421	0.872105	−	0.489319i	$-0.162755\pi$
$179$	23.3359	1.74421	0.872105	+	0.489319i	$0.162755\pi$
$180$	0	0
$181$	−3.34240	−0.248439	−0.124219	−	0.992255i	$-0.539643\pi$
$181$	−3.34240	−0.248439	−0.124219	+	0.992255i	$0.539643\pi$
$182$	20.0405	1.48550
$183$	0	0
$184$	1.19897	0.0883891
$185$	0	0
$186$	0	0
$187$	4.00268	0.292705
$188$	−13.9345	−1.01628
$189$	0	0
$190$	0	0
$191$	−4.61979	−0.334276	−0.167138	−	0.985934i	$-0.553453\pi$
$191$	−4.61979	−0.334276	−0.167138	+	0.985934i	$0.553453\pi$
$192$	0	0
$193$	24.9023	1.79251	0.896254	−	0.443542i	$-0.146278\pi$
$193$	24.9023	1.79251	0.896254	+	0.443542i	$0.146278\pi$
$194$	27.5252	1.97619
$195$	0	0
$196$	−6.33278	−0.452342
$197$	5.96322	0.424862	0.212431	−	0.977176i	$-0.431862\pi$
$197$	5.96322	0.424862	0.212431	+	0.977176i	$0.431862\pi$
$198$	0	0
$199$	−19.0033	−1.34711	−0.673555	−	0.739137i	$-0.735233\pi$
$199$	−19.0033	−1.34711	−0.673555	+	0.739137i	$0.735233\pi$
$200$	0	0
$201$	0	0
$202$	−29.9436	−2.10682
$203$	−1.70908	−0.119954
$204$	0	0
$205$	0	0
$206$	21.9245	1.52755
$207$	0	0
$208$	−29.2073	−2.02516
$209$	19.2186	1.32938
$210$	0	0
$211$	−8.60278	−0.592240	−0.296120	−	0.955151i	$-0.595693\pi$
$211$	−8.60278	−0.592240	−0.296120	+	0.955151i	$0.595693\pi$
$212$	−3.28027	−0.225290
$213$	0	0
$214$	−24.6696	−1.68638
$215$	0	0
$216$	0	0
$217$	−2.21342	−0.150257
$218$	19.9100	1.34847
$219$	0	0
$220$	0	0
$221$	8.73349	0.587478
$222$	0	0
$223$	−7.77513	−0.520661	−0.260331	−	0.965520i	$-0.583832\pi$
$223$	−7.77513	−0.520661	−0.260331	+	0.965520i	$0.583832\pi$
$224$	−12.2401	−0.817825
$225$	0	0
$226$	34.8599	2.31885
$227$	11.4310	0.758699	0.379349	−	0.925253i	$-0.376148\pi$
$227$	11.4310	0.758699	0.379349	+	0.925253i	$0.376148\pi$
$228$	0	0
$229$	−6.14881	−0.406325	−0.203162	−	0.979145i	$-0.565122\pi$
$229$	−6.14881	−0.406325	−0.203162	+	0.979145i	$0.565122\pi$
$230$	0	0
$231$	0	0
$232$	0.843434	0.0553742
$233$	0.829745	0.0543584	0.0271792	−	0.999631i	$-0.491348\pi$
$233$	0.829745	0.0543584	0.0271792	+	0.999631i	$0.491348\pi$
$234$	0	0
$235$	0	0
$236$	15.6033	1.01569
$237$	0	0
$238$	4.52207	0.293122
$239$	0.473622	0.0306361	0.0153180	−	0.999883i	$-0.495124\pi$
$239$	0.473622	0.0306361	0.0153180	+	0.999883i	$0.495124\pi$
$240$	0	0
$241$	−22.0803	−1.42232	−0.711160	−	0.703030i	$-0.751830\pi$
$241$	−22.0803	−1.42232	−0.711160	+	0.703030i	$0.751830\pi$
$242$	−5.40958	−0.347741
$243$	0	0
$244$	8.89954	0.569735
$245$	0	0
$246$	0	0
$247$	41.9332	2.66815
$248$	1.09233	0.0693630
$249$	0	0
$250$	0	0
$251$	−6.98079	−0.440624	−0.220312	−	0.975429i	$-0.570708\pi$
$251$	−6.98079	−0.440624	−0.220312	+	0.975429i	$0.570708\pi$
$252$	0	0
$253$	−4.05321	−0.254823
$254$	−5.57333	−0.349702
$255$	0	0
$256$	20.6178	1.28861
$257$	−5.35099	−0.333785	−0.166893	−	0.985975i	$-0.553373\pi$
$257$	−5.35099	−0.333785	−0.166893	+	0.985975i	$0.553373\pi$
$258$	0	0
$259$	−1.69169	−0.105117
$260$	0	0
$261$	0	0
$262$	39.5405	2.44282
$263$	−8.56128	−0.527911	−0.263955	−	0.964535i	$-0.585027\pi$
$263$	−8.56128	−0.527911	−0.263955	+	0.964535i	$0.585027\pi$
$264$	0	0
$265$	0	0
$266$	21.7124	1.33127
$267$	0	0
$268$	−3.80790	−0.232604
$269$	7.36998	0.449356	0.224678	−	0.974433i	$-0.427867\pi$
$269$	7.36998	0.449356	0.224678	+	0.974433i	$0.427867\pi$
$270$	0	0
$271$	−5.49008	−0.333499	−0.166749	−	0.985999i	$-0.553327\pi$
$271$	−5.49008	−0.333499	−0.166749	+	0.985999i	$0.553327\pi$
$272$	−6.59051	−0.399608
$273$	0	0
$274$	36.9335	2.23123
$275$	0	0
$276$	0	0
$277$	−24.0013	−1.44210	−0.721051	−	0.692882i	$-0.756341\pi$
$277$	−24.0013	−1.44210	−0.721051	+	0.692882i	$0.756341\pi$
$278$	−40.1200	−2.40624
$279$	0	0
$280$	0	0
$281$	−10.0364	−0.598722	−0.299361	−	0.954140i	$-0.596773\pi$
$281$	−10.0364	−0.598722	−0.299361	+	0.954140i	$0.596773\pi$
$282$	0	0
$283$	−8.41013	−0.499930	−0.249965	−	0.968255i	$-0.580419\pi$
$283$	−8.41013	−0.499930	−0.249965	+	0.968255i	$0.580419\pi$
$284$	−7.82903	−0.464568
$285$	0	0
$286$	33.4342	1.97700
$287$	−10.8050	−0.637799
$288$	0	0
$289$	−15.0293	−0.884078
$290$	0	0
$291$	0	0
$292$	8.62680	0.504846
$293$	6.38571	0.373057	0.186529	−	0.982450i	$-0.440276\pi$
$293$	6.38571	0.373057	0.186529	+	0.982450i	$0.440276\pi$
$294$	0	0
$295$	0	0
$296$	0.834856	0.0485250
$297$	0	0
$298$	31.3110	1.81380
$299$	−8.84375	−0.511447
$300$	0	0
$301$	10.3180	0.594722
$302$	16.4868	0.948707
$303$	0	0
$304$	−31.6438	−1.81490
$305$	0	0
$306$	0	0
$307$	28.7573	1.64126	0.820632	−	0.571458i	$-0.193622\pi$
$307$	28.7573	1.64126	0.820632	+	0.571458i	$0.193622\pi$
$308$	7.56551	0.431085
$309$	0	0
$310$	0	0
$311$	−4.20464	−0.238423	−0.119211	−	0.992869i	$-0.538037\pi$
$311$	−4.20464	−0.238423	−0.119211	+	0.992869i	$0.538037\pi$
$312$	0	0
$313$	−18.2272	−1.03026	−0.515132	−	0.857111i	$-0.672257\pi$
$313$	−18.2272	−1.03026	−0.515132	+	0.857111i	$0.672257\pi$
$314$	−37.8791	−2.13764
$315$	0	0
$316$	14.5609	0.819113
$317$	−13.4593	−0.755951	−0.377975	−	0.925816i	$-0.623380\pi$
$317$	−13.4593	−0.755951	−0.377975	+	0.925816i	$0.623380\pi$
$318$	0	0
$319$	−2.85130	−0.159642
$320$	0	0
$321$	0	0
$322$	−4.57915	−0.255186
$323$	9.46207	0.526483
$324$	0	0
$325$	0	0
$326$	−6.05214	−0.335197
$327$	0	0
$328$	5.33230	0.294427
$329$	−15.3398	−0.845710
$330$	0	0
$331$	15.2501	0.838223	0.419112	−	0.907935i	$-0.362342\pi$
$331$	15.2501	0.838223	0.419112	+	0.907935i	$0.362342\pi$
$332$	−7.86726	−0.431772
$333$	0	0
$334$	−12.8888	−0.705242
$335$	0	0
$336$	0	0
$337$	−5.07452	−0.276426	−0.138213	−	0.990402i	$-0.544136\pi$
$337$	−5.07452	−0.276426	−0.138213	+	0.990402i	$0.544136\pi$
$338$	48.4479	2.63522
$339$	0	0
$340$	0	0
$341$	−3.69271	−0.199971
$342$	0	0
$343$	−18.9350	−1.02239
$344$	−5.09198	−0.274541
$345$	0	0
$346$	−6.55738	−0.352527
$347$	−28.0386	−1.50519	−0.752596	−	0.658482i	$-0.771199\pi$
$347$	−28.0386	−1.50519	−0.752596	+	0.658482i	$0.771199\pi$
$348$	0	0
$349$	−27.7759	−1.48681	−0.743406	−	0.668840i	$-0.766791\pi$
$349$	−27.7759	−1.48681	−0.743406	+	0.668840i	$0.766791\pi$
$350$	0	0
$351$	0	0
$352$	−20.4205	−1.08841
$353$	31.4223	1.67244	0.836221	−	0.548392i	$-0.184760\pi$
$353$	31.4223	1.67244	0.836221	+	0.548392i	$0.184760\pi$
$354$	0	0
$355$	0	0
$356$	2.72765	0.144565
$357$	0	0
$358$	43.9838	2.32462
$359$	−27.0194	−1.42603	−0.713014	−	0.701150i	$-0.752671\pi$
$359$	−27.0194	−1.42603	−0.713014	+	0.701150i	$0.752671\pi$
$360$	0	0
$361$	26.4314	1.39113
$362$	−6.29980	−0.331110
$363$	0	0
$364$	16.5073	0.865217
$365$	0	0
$366$	0	0
$367$	−20.3243	−1.06092	−0.530460	−	0.847710i	$-0.677981\pi$
$367$	−20.3243	−1.06092	−0.530460	+	0.847710i	$0.677981\pi$
$368$	6.67371	0.347891
$369$	0	0
$370$	0	0
$371$	−3.61108	−0.187478
$372$	0	0
$373$	15.9724	0.827021	0.413511	−	0.910499i	$-0.364302\pi$
$373$	15.9724	0.827021	0.413511	+	0.910499i	$0.364302\pi$
$374$	7.54429	0.390106
$375$	0	0
$376$	7.57023	0.390405
$377$	−6.22128	−0.320412
$378$	0	0
$379$	29.4840	1.51449	0.757247	−	0.653129i	$-0.226544\pi$
$379$	29.4840	1.51449	0.757247	+	0.653129i	$0.226544\pi$
$380$	0	0
$381$	0	0
$382$	−8.70742	−0.445510
$383$	35.1376	1.79545	0.897723	−	0.440560i	$-0.145220\pi$
$383$	35.1376	1.79545	0.897723	+	0.440560i	$0.145220\pi$
$384$	0	0
$385$	0	0
$386$	46.9361	2.38899
$387$	0	0
$388$	22.6724	1.15101
$389$	−28.3776	−1.43880	−0.719401	−	0.694595i	$-0.755583\pi$
$389$	−28.3776	−1.43880	−0.719401	+	0.694595i	$0.755583\pi$
$390$	0	0
$391$	−1.99556	−0.100920
$392$	3.44042	0.173767
$393$	0	0
$394$	11.2395	0.566240
$395$	0	0
$396$	0	0
$397$	24.5435	1.23180	0.615901	−	0.787823i	$-0.288792\pi$
$397$	24.5435	1.23180	0.615901	+	0.787823i	$0.288792\pi$
$398$	−35.8176	−1.79538
$399$	0	0
$400$	0	0
$401$	−4.95811	−0.247596	−0.123798	−	0.992307i	$-0.539507\pi$
$401$	−4.95811	−0.247596	−0.123798	+	0.992307i	$0.539507\pi$
$402$	0	0
$403$	−8.05717	−0.401356
$404$	−24.6644	−1.22710
$405$	0	0
$406$	−3.22128	−0.159870
$407$	−2.82230	−0.139896
$408$	0	0
$409$	−29.6350	−1.46535	−0.732677	−	0.680576i	$-0.761729\pi$
$409$	−29.6350	−1.46535	−0.732677	+	0.680576i	$0.761729\pi$
$410$	0	0
$411$	0	0
$412$	18.0591	0.889710
$413$	17.1769	0.845218
$414$	0	0
$415$	0	0
$416$	−44.5557	−2.18452
$417$	0	0
$418$	36.2233	1.77174
$419$	−27.1403	−1.32589	−0.662945	−	0.748668i	$-0.730694\pi$
$419$	−27.1403	−1.32589	−0.662945	+	0.748668i	$0.730694\pi$
$420$	0	0
$421$	−27.0310	−1.31741	−0.658705	−	0.752401i	$-0.728896\pi$
$421$	−27.0310	−1.31741	−0.658705	+	0.752401i	$0.728896\pi$
$422$	−16.2146	−0.789314
$423$	0	0
$424$	1.78208	0.0865454
$425$	0	0
$426$	0	0
$427$	9.79703	0.474112
$428$	−20.3203	−0.982217
$429$	0	0
$430$	0	0
$431$	27.7496	1.33665	0.668327	−	0.743868i	$-0.267011\pi$
$431$	27.7496	1.33665	0.668327	+	0.743868i	$0.267011\pi$
$432$	0	0
$433$	−14.3507	−0.689649	−0.344825	−	0.938667i	$-0.612062\pi$
$433$	−14.3507	−0.689649	−0.344825	+	0.938667i	$0.612062\pi$
$434$	−4.17187	−0.200256
$435$	0	0
$436$	16.3997	0.785405
$437$	−9.58152	−0.458346
$438$	0	0
$439$	19.0540	0.909398	0.454699	−	0.890645i	$-0.349747\pi$
$439$	19.0540	0.909398	0.454699	+	0.890645i	$0.349747\pi$
$440$	0	0
$441$	0	0
$442$	16.4610	0.782969
$443$	−36.2772	−1.72358	−0.861791	−	0.507263i	$-0.830657\pi$
$443$	−36.2772	−1.72358	−0.861791	+	0.507263i	$0.830657\pi$
$444$	0	0
$445$	0	0
$446$	−14.6546	−0.693918
$447$	0	0
$448$	−7.02292	−0.331802
$449$	−3.04282	−0.143600	−0.0717998	−	0.997419i	$-0.522874\pi$
$449$	−3.04282	−0.143600	−0.0717998	+	0.997419i	$0.522874\pi$
$450$	0	0
$451$	−18.0263	−0.848825
$452$	28.7140	1.35059
$453$	0	0
$454$	21.5452	1.01117
$455$	0	0
$456$	0	0
$457$	−24.8217	−1.16111	−0.580556	−	0.814220i	$-0.697165\pi$
$457$	−24.8217	−1.16111	−0.580556	+	0.814220i	$0.697165\pi$
$458$	−11.5893	−0.541534
$459$	0	0
$460$	0	0
$461$	14.1662	0.659786	0.329893	−	0.944018i	$-0.392987\pi$
$461$	14.1662	0.659786	0.329893	+	0.944018i	$0.392987\pi$
$462$	0	0
$463$	9.03538	0.419910	0.209955	−	0.977711i	$-0.432668\pi$
$463$	9.03538	0.419910	0.209955	+	0.977711i	$0.432668\pi$
$464$	4.69473	0.217948
$465$	0	0
$466$	1.56391	0.0724468
$467$	5.28374	0.244503	0.122251	−	0.992499i	$-0.460989\pi$
$467$	5.28374	0.244503	0.122251	+	0.992499i	$0.460989\pi$
$468$	0	0
$469$	−4.19191	−0.193565
$470$	0	0
$471$	0	0
$472$	−8.47683	−0.390178
$473$	17.2139	0.791495
$474$	0	0
$475$	0	0
$476$	3.72481	0.170726
$477$	0	0
$478$	0.892688	0.0408306
$479$	16.3846	0.748634	0.374317	−	0.927301i	$-0.377877\pi$
$479$	16.3846	0.748634	0.374317	+	0.927301i	$0.377877\pi$
$480$	0	0
$481$	−6.15801	−0.280781
$482$	−41.6173	−1.89561
$483$	0	0
$484$	−4.45585	−0.202539
$485$	0	0
$486$	0	0
$487$	2.74404	0.124344	0.0621722	−	0.998065i	$-0.480197\pi$
$487$	2.74404	0.124344	0.0621722	+	0.998065i	$0.480197\pi$
$488$	−4.83487	−0.218864
$489$	0	0
$490$	0	0
$491$	−9.46362	−0.427087	−0.213544	−	0.976934i	$-0.568501\pi$
$491$	−9.46362	−0.427087	−0.213544	+	0.976934i	$0.568501\pi$
$492$	0	0
$493$	−1.40381	−0.0632244
$494$	79.0361	3.55600
$495$	0	0
$496$	6.08013	0.273006
$497$	−8.61857	−0.386595
$498$	0	0
$499$	−36.0076	−1.61192	−0.805960	−	0.591970i	$-0.798350\pi$
$499$	−36.0076	−1.61192	−0.805960	+	0.591970i	$0.798350\pi$
$500$	0	0
$501$	0	0
$502$	−13.1575	−0.587247
$503$	−21.6537	−0.965490	−0.482745	−	0.875761i	$-0.660360\pi$
$503$	−21.6537	−0.965490	−0.482745	+	0.875761i	$0.660360\pi$
$504$	0	0
$505$	0	0
$506$	−7.63953	−0.339619
$507$	0	0
$508$	−4.59073	−0.203681
$509$	−16.5219	−0.732321	−0.366160	−	0.930552i	$-0.619328\pi$
$509$	−16.5219	−0.732321	−0.366160	+	0.930552i	$0.619328\pi$
$510$	0	0
$511$	9.49679	0.420113
$512$	25.7034	1.13594
$513$	0	0
$514$	−10.0856	−0.444857
$515$	0	0
$516$	0	0
$517$	−25.5918	−1.12553
$518$	−3.18852	−0.140096
$519$	0	0
$520$	0	0
$521$	−4.77373	−0.209141	−0.104571	−	0.994517i	$-0.533347\pi$
$521$	−4.77373	−0.209141	−0.104571	+	0.994517i	$0.533347\pi$
$522$	0	0
$523$	18.4589	0.807150	0.403575	−	0.914947i	$-0.367767\pi$
$523$	18.4589	0.807150	0.403575	+	0.914947i	$0.367767\pi$
$524$	32.5693	1.42280
$525$	0	0
$526$	−16.1364	−0.703580
$527$	−1.81807	−0.0791963
$528$	0	0
$529$	−20.9792	−0.912141
$530$	0	0
$531$	0	0
$532$	17.8844	0.775386
$533$	−39.3318	−1.70365
$534$	0	0
$535$	0	0
$536$	2.06872	0.0893552
$537$	0	0
$538$	13.8910	0.598885
$539$	−11.6306	−0.500966
$540$	0	0
$541$	−43.5067	−1.87050	−0.935250	−	0.353989i	$-0.884825\pi$
$541$	−43.5067	−1.87050	−0.935250	+	0.353989i	$0.884825\pi$
$542$	−10.3478	−0.444474
$543$	0	0
$544$	−10.0538	−0.431054
$545$	0	0
$546$	0	0
$547$	5.97641	0.255533	0.127766	−	0.991804i	$-0.459219\pi$
$547$	5.97641	0.255533	0.127766	+	0.991804i	$0.459219\pi$
$548$	30.4219	1.29956
$549$	0	0
$550$	0	0
$551$	−6.74028	−0.287146
$552$	0	0
$553$	16.0293	0.681635
$554$	−45.2380	−1.92198
$555$	0	0
$556$	−33.0466	−1.40149
$557$	30.0628	1.27380	0.636900	−	0.770946i	$-0.280216\pi$
$557$	30.0628	1.27380	0.636900	+	0.770946i	$0.280216\pi$
$558$	0	0
$559$	37.5592	1.58858
$560$	0	0
$561$	0	0
$562$	−18.9167	−0.797954
$563$	−18.5651	−0.782425	−0.391212	−	0.920300i	$-0.627944\pi$
$563$	−18.5651	−0.782425	−0.391212	+	0.920300i	$0.627944\pi$
$564$	0	0
$565$	0	0
$566$	−15.8515	−0.666288
$567$	0	0
$568$	4.25329	0.178464
$569$	30.6606	1.28536	0.642679	−	0.766135i	$-0.277823\pi$
$569$	30.6606	1.28536	0.642679	+	0.766135i	$0.277823\pi$
$570$	0	0
$571$	−6.78472	−0.283932	−0.141966	−	0.989872i	$-0.545342\pi$
$571$	−6.78472	−0.283932	−0.141966	+	0.989872i	$0.545342\pi$
$572$	27.5396	1.15149
$573$	0	0
$574$	−20.3654	−0.850034
$575$	0	0
$576$	0	0
$577$	27.7326	1.15453	0.577263	−	0.816559i	$-0.304121\pi$
$577$	27.7326	1.15453	0.577263	+	0.816559i	$0.304121\pi$
$578$	−28.3274	−1.17827
$579$	0	0
$580$	0	0
$581$	−8.66064	−0.359304
$582$	0	0
$583$	−6.02446	−0.249508
$584$	−4.68670	−0.193937
$585$	0	0
$586$	12.0359	0.497197
$587$	16.6049	0.685359	0.342679	−	0.939452i	$-0.388666\pi$
$587$	16.6049	0.685359	0.342679	+	0.939452i	$0.388666\pi$
$588$	0	0
$589$	−8.72932	−0.359685
$590$	0	0
$591$	0	0
$592$	4.64699	0.190990
$593$	20.5406	0.843499	0.421750	−	0.906712i	$-0.361416\pi$
$593$	20.5406	0.843499	0.421750	+	0.906712i	$0.361416\pi$
$594$	0	0
$595$	0	0
$596$	25.7908	1.05643
$597$	0	0
$598$	−16.6688	−0.681637
$599$	−31.3560	−1.28117	−0.640586	−	0.767886i	$-0.721309\pi$
$599$	−31.3560	−1.28117	−0.640586	+	0.767886i	$0.721309\pi$
$600$	0	0
$601$	14.3674	0.586057	0.293029	−	0.956104i	$-0.405337\pi$
$601$	14.3674	0.586057	0.293029	+	0.956104i	$0.405337\pi$
$602$	19.4475	0.792623
$603$	0	0
$604$	13.5801	0.552566
$605$	0	0
$606$	0	0
$607$	25.1167	1.01945	0.509727	−	0.860336i	$-0.329746\pi$
$607$	25.1167	1.01945	0.509727	+	0.860336i	$0.329746\pi$
$608$	−48.2726	−1.95771
$609$	0	0
$610$	0	0
$611$	−55.8390	−2.25901
$612$	0	0
$613$	−36.4069	−1.47046	−0.735230	−	0.677818i	$-0.762926\pi$
$613$	−36.4069	−1.47046	−0.735230	+	0.677818i	$0.762926\pi$
$614$	54.2020	2.18741
$615$	0	0
$616$	−4.11013	−0.165602
$617$	28.9915	1.16715	0.583576	−	0.812058i	$-0.301653\pi$
$617$	28.9915	1.16715	0.583576	+	0.812058i	$0.301653\pi$
$618$	0	0
$619$	−38.6097	−1.55185	−0.775927	−	0.630823i	$-0.782717\pi$
$619$	−38.6097	−1.55185	−0.775927	+	0.630823i	$0.782717\pi$
$620$	0	0
$621$	0	0
$622$	−7.92494	−0.317761
$623$	3.00273	0.120302
$624$	0	0
$625$	0	0
$626$	−34.3549	−1.37310
$627$	0	0
$628$	−31.2009	−1.24505
$629$	−1.38953	−0.0554043
$630$	0	0
$631$	12.4946	0.497401	0.248700	−	0.968580i	$-0.419997\pi$
$631$	12.4946	0.497401	0.248700	+	0.968580i	$0.419997\pi$
$632$	−7.91050	−0.314663
$633$	0	0
$634$	−25.3683	−1.00750
$635$	0	0
$636$	0	0
$637$	−25.3770	−1.00547
$638$	−5.37416	−0.212765
$639$	0	0
$640$	0	0
$641$	−18.6523	−0.736723	−0.368361	−	0.929683i	$-0.620081\pi$
$641$	−18.6523	−0.736723	−0.368361	+	0.929683i	$0.620081\pi$
$642$	0	0
$643$	9.99161	0.394030	0.197015	−	0.980400i	$-0.436875\pi$
$643$	9.99161	0.394030	0.197015	+	0.980400i	$0.436875\pi$
$644$	−3.77183	−0.148631
$645$	0	0
$646$	17.8342	0.701677
$647$	6.59170	0.259146	0.129573	−	0.991570i	$-0.458639\pi$
$647$	6.59170	0.259146	0.129573	+	0.991570i	$0.458639\pi$
$648$	0	0
$649$	28.6566	1.12487
$650$	0	0
$651$	0	0
$652$	−4.98512	−0.195232
$653$	47.5008	1.85885	0.929425	−	0.369011i	$-0.120304\pi$
$653$	47.5008	1.85885	0.929425	+	0.369011i	$0.120304\pi$
$654$	0	0
$655$	0	0
$656$	29.6807	1.15884
$657$	0	0
$658$	−28.9126	−1.12713
$659$	−32.7065	−1.27407	−0.637033	−	0.770837i	$-0.719838\pi$
$659$	−32.7065	−1.27407	−0.637033	+	0.770837i	$0.719838\pi$
$660$	0	0
$661$	−37.6003	−1.46248	−0.731242	−	0.682119i	$-0.761059\pi$
$661$	−37.6003	−1.46248	−0.731242	+	0.682119i	$0.761059\pi$
$662$	28.7436	1.11715
$663$	0	0
$664$	4.27405	0.165866
$665$	0	0
$666$	0	0
$667$	1.42153	0.0550419
$668$	−10.6164	−0.410762
$669$	0	0
$670$	0	0
$671$	16.3447	0.630979
$672$	0	0
$673$	47.2395	1.82095	0.910475	−	0.413564i	$-0.135716\pi$
$673$	47.2395	1.82095	0.910475	+	0.413564i	$0.135716\pi$
$674$	−9.56450	−0.368411
$675$	0	0
$676$	39.9063	1.53486
$677$	−17.3323	−0.666134	−0.333067	−	0.942903i	$-0.608083\pi$
$677$	−17.3323	−0.666134	−0.333067	+	0.942903i	$0.608083\pi$
$678$	0	0
$679$	24.9588	0.957830
$680$	0	0
$681$	0	0
$682$	−6.96005	−0.266514
$683$	−4.70774	−0.180137	−0.0900683	−	0.995936i	$-0.528709\pi$
$683$	−4.70774	−0.180137	−0.0900683	+	0.995936i	$0.528709\pi$
$684$	0	0
$685$	0	0
$686$	−35.6888	−1.36260
$687$	0	0
$688$	−28.3431	−1.08057
$689$	−13.1449	−0.500779
$690$	0	0
$691$	−2.88811	−0.109869	−0.0549345	−	0.998490i	$-0.517495\pi$
$691$	−2.88811	−0.109869	−0.0549345	+	0.998490i	$0.517495\pi$
$692$	−5.40129	−0.205326
$693$	0	0
$694$	−52.8475	−2.00606
$695$	0	0
$696$	0	0
$697$	−8.87506	−0.336167
$698$	−52.3524	−1.98157
$699$	0	0
$700$	0	0
$701$	−39.1914	−1.48024	−0.740120	−	0.672475i	$-0.765231\pi$
$701$	−39.1914	−1.48024	−0.740120	+	0.672475i	$0.765231\pi$
$702$	0	0
$703$	−6.67173	−0.251629
$704$	−11.7165	−0.441584
$705$	0	0
$706$	59.2251	2.22897
$707$	−27.1517	−1.02115
$708$	0	0
$709$	35.0367	1.31583	0.657915	−	0.753092i	$-0.271439\pi$
$709$	35.0367	1.31583	0.657915	+	0.753092i	$0.271439\pi$
$710$	0	0
$711$	0	0
$712$	−1.48185	−0.0555349
$713$	1.84102	0.0689468
$714$	0	0
$715$	0	0
$716$	36.2293	1.35395
$717$	0	0
$718$	−50.9264	−1.90056
$719$	−9.93705	−0.370590	−0.185295	−	0.982683i	$-0.559324\pi$
$719$	−9.93705	−0.370590	−0.185295	+	0.982683i	$0.559324\pi$
$720$	0	0
$721$	19.8804	0.740383
$722$	49.8182	1.85404
$723$	0	0
$724$	−5.18912	−0.192852
$725$	0	0
$726$	0	0
$727$	2.37679	0.0881504	0.0440752	−	0.999028i	$-0.485966\pi$
$727$	2.37679	0.0881504	0.0440752	+	0.999028i	$0.485966\pi$
$728$	−8.96793	−0.332374
$729$	0	0
$730$	0	0
$731$	8.47508	0.313462
$732$	0	0
$733$	−32.6214	−1.20490	−0.602450	−	0.798157i	$-0.705809\pi$
$733$	−32.6214	−1.20490	−0.602450	+	0.798157i	$0.705809\pi$
$734$	−38.3075	−1.41396
$735$	0	0
$736$	10.1807	0.375267
$737$	−6.99349	−0.257608
$738$	0	0
$739$	16.6544	0.612642	0.306321	−	0.951928i	$-0.400902\pi$
$739$	16.6544	0.612642	0.306321	+	0.951928i	$0.400902\pi$
$740$	0	0
$741$	0	0
$742$	−6.80620	−0.249863
$743$	35.2240	1.29224	0.646121	−	0.763235i	$-0.276390\pi$
$743$	35.2240	1.29224	0.646121	+	0.763235i	$0.276390\pi$
$744$	0	0
$745$	0	0
$746$	30.1050	1.10222
$747$	0	0
$748$	6.21420	0.227214
$749$	−22.3695	−0.817364
$750$	0	0
$751$	−46.9948	−1.71487	−0.857433	−	0.514596i	$-0.827942\pi$
$751$	−46.9948	−1.71487	−0.857433	+	0.514596i	$0.827942\pi$
$752$	42.1375	1.53660
$753$	0	0
$754$	−11.7259	−0.427034
$755$	0	0
$756$	0	0
$757$	16.5681	0.602179	0.301089	−	0.953596i	$-0.402650\pi$
$757$	16.5681	0.602179	0.301089	+	0.953596i	$0.402650\pi$
$758$	55.5718	2.01846
$759$	0	0
$760$	0	0
$761$	−43.1593	−1.56453	−0.782263	−	0.622949i	$-0.785934\pi$
$761$	−43.1593	−1.56453	−0.782263	+	0.622949i	$0.785934\pi$
$762$	0	0
$763$	18.0536	0.653585
$764$	−7.17226	−0.259483
$765$	0	0
$766$	66.2277	2.39290
$767$	62.5262	2.25769
$768$	0	0
$769$	−16.7337	−0.603434	−0.301717	−	0.953398i	$-0.597560\pi$
$769$	−16.7337	−0.603434	−0.301717	+	0.953398i	$0.597560\pi$
$770$	0	0
$771$	0	0
$772$	38.6611	1.39144
$773$	−8.41758	−0.302759	−0.151380	−	0.988476i	$-0.548372\pi$
$773$	−8.41758	−0.302759	−0.151380	+	0.988476i	$0.548372\pi$
$774$	0	0
$775$	0	0
$776$	−12.3172	−0.442163
$777$	0	0
$778$	−53.4864	−1.91758
$779$	−42.6129	−1.52677
$780$	0	0
$781$	−14.3786	−0.514507
$782$	−3.76125	−0.134502
$783$	0	0
$784$	19.1501	0.683932
$785$	0	0
$786$	0	0
$787$	50.6781	1.80648	0.903240	−	0.429136i	$-0.141182\pi$
$787$	50.6781	1.80648	0.903240	+	0.429136i	$0.141182\pi$
$788$	9.25795	0.329801
$789$	0	0
$790$	0	0
$791$	31.6097	1.12391
$792$	0	0
$793$	35.6626	1.26642
$794$	46.2598	1.64170
$795$	0	0
$796$	−29.5028	−1.04570
$797$	21.8932	0.775498	0.387749	−	0.921765i	$-0.373253\pi$
$797$	21.8932	0.775498	0.387749	+	0.921765i	$0.373253\pi$
$798$	0	0
$799$	−12.5999	−0.445751
$800$	0	0
$801$	0	0
$802$	−9.34509	−0.329987
$803$	15.8438	0.559114
$804$	0	0
$805$	0	0
$806$	−15.1862	−0.534912
$807$	0	0
$808$	13.3995	0.471391
$809$	8.70213	0.305951	0.152975	−	0.988230i	$-0.451115\pi$
$809$	8.70213	0.305951	0.152975	+	0.988230i	$0.451115\pi$
$810$	0	0
$811$	39.0632	1.37170	0.685848	−	0.727745i	$-0.259431\pi$
$811$	39.0632	1.37170	0.685848	+	0.727745i	$0.259431\pi$
$812$	−2.65336	−0.0931146
$813$	0	0
$814$	−5.31950	−0.186448
$815$	0	0
$816$	0	0
$817$	40.6925	1.42365
$818$	−55.8563	−1.95297
$819$	0	0
$820$	0	0
$821$	−16.7509	−0.584612	−0.292306	−	0.956325i	$-0.594423\pi$
$821$	−16.7509	−0.584612	−0.292306	+	0.956325i	$0.594423\pi$
$822$	0	0
$823$	38.3755	1.33768	0.668842	−	0.743405i	$-0.266790\pi$
$823$	38.3755	1.33768	0.668842	+	0.743405i	$0.266790\pi$
$824$	−9.81101	−0.341783
$825$	0	0
$826$	32.3751	1.12647
$827$	−6.70137	−0.233029	−0.116515	−	0.993189i	$-0.537172\pi$
$827$	−6.70137	−0.233029	−0.116515	+	0.993189i	$0.537172\pi$
$828$	0	0
$829$	7.98795	0.277433	0.138716	−	0.990332i	$-0.455702\pi$
$829$	7.98795	0.277433	0.138716	+	0.990332i	$0.455702\pi$
$830$	0	0
$831$	0	0
$832$	−25.5644	−0.886288
$833$	−5.72622	−0.198402
$834$	0	0
$835$	0	0
$836$	29.8370	1.03193
$837$	0	0
$838$	−51.1543	−1.76710
$839$	13.9407	0.481286	0.240643	−	0.970614i	$-0.422642\pi$
$839$	13.9407	0.481286	0.240643	+	0.970614i	$0.422642\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−50.9483	−1.75580
$843$	0	0
$844$	−13.3559	−0.459729
$845$	0	0
$846$	0	0
$847$	−4.90521	−0.168545
$848$	9.91943	0.340635
$849$	0	0
$850$	0	0
$851$	1.40707	0.0482339
$852$	0	0
$853$	−5.78046	−0.197919	−0.0989597	−	0.995091i	$-0.531551\pi$
$853$	−5.78046	−0.197919	−0.0989597	+	0.995091i	$0.531551\pi$
$854$	18.4655	0.631878
$855$	0	0
$856$	11.0394	0.377319
$857$	33.2410	1.13549	0.567745	−	0.823205i	$-0.307816\pi$
$857$	33.2410	1.13549	0.567745	+	0.823205i	$0.307816\pi$
$858$	0	0
$859$	1.19425	0.0407473	0.0203736	−	0.999792i	$-0.493514\pi$
$859$	1.19425	0.0407473	0.0203736	+	0.999792i	$0.493514\pi$
$860$	0	0
$861$	0	0
$862$	52.3028	1.78144
$863$	42.0216	1.43043	0.715215	−	0.698904i	$-0.246329\pi$
$863$	42.0216	1.43043	0.715215	+	0.698904i	$0.246329\pi$
$864$	0	0
$865$	0	0
$866$	−27.0483	−0.919138
$867$	0	0
$868$	−3.43635	−0.116637
$869$	26.7421	0.907164
$870$	0	0
$871$	−15.2592	−0.517037
$872$	−8.90951	−0.301714
$873$	0	0
$874$	−18.0593	−0.610866
$875$	0	0
$876$	0	0
$877$	−22.8893	−0.772917	−0.386458	−	0.922307i	$-0.626302\pi$
$877$	−22.8893	−0.772917	−0.386458	+	0.922307i	$0.626302\pi$
$878$	35.9132	1.21201
$879$	0	0
$880$	0	0
$881$	−13.6751	−0.460727	−0.230363	−	0.973105i	$-0.573991\pi$
$881$	−13.6751	−0.460727	−0.230363	+	0.973105i	$0.573991\pi$
$882$	0	0
$883$	1.49072	0.0501668	0.0250834	−	0.999685i	$-0.492015\pi$
$883$	1.49072	0.0501668	0.0250834	+	0.999685i	$0.492015\pi$
$884$	13.5588	0.456033
$885$	0	0
$886$	−68.3757	−2.29713
$887$	2.58705	0.0868646	0.0434323	−	0.999056i	$-0.486171\pi$
$887$	2.58705	0.0868646	0.0434323	+	0.999056i	$0.486171\pi$
$888$	0	0
$889$	−5.05369	−0.169495
$890$	0	0
$891$	0	0
$892$	−12.0710	−0.404166
$893$	−60.4973	−2.02447
$894$	0	0
$895$	0	0
$896$	11.2433	0.375612
$897$	0	0
$898$	−5.73514	−0.191384
$899$	1.29510	0.0431939
$900$	0	0
$901$	−2.96609	−0.0988146
$902$	−33.9761	−1.13128
$903$	0	0
$904$	−15.5995	−0.518831
$905$	0	0
$906$	0	0
$907$	18.3747	0.610121	0.305060	−	0.952333i	$-0.401323\pi$
$907$	18.3747	0.610121	0.305060	+	0.952333i	$0.401323\pi$
$908$	17.7467	0.588944
$909$	0	0
$910$	0	0
$911$	−32.8875	−1.08961	−0.544806	−	0.838562i	$-0.683397\pi$
$911$	−32.8875	−1.08961	−0.544806	+	0.838562i	$0.683397\pi$
$912$	0	0
$913$	−14.4488	−0.478185
$914$	−46.7843	−1.54749
$915$	0	0
$916$	−9.54608	−0.315412
$917$	35.8538	1.18400
$918$	0	0
$919$	28.2361	0.931422	0.465711	−	0.884937i	$-0.345799\pi$
$919$	28.2361	0.931422	0.465711	+	0.884937i	$0.345799\pi$
$920$	0	0
$921$	0	0
$922$	26.7006	0.879337
$923$	−31.3728	−1.03265
$924$	0	0
$925$	0	0
$926$	17.0300	0.559640
$927$	0	0
$928$	7.16181	0.235098
$929$	44.5794	1.46260	0.731301	−	0.682055i	$-0.238913\pi$
$929$	44.5794	1.46260	0.731301	+	0.682055i	$0.238913\pi$
$930$	0	0
$931$	−27.4940	−0.901080
$932$	1.28819	0.0421960
$933$	0	0
$934$	9.95886	0.325864
$935$	0	0
$936$	0	0
$937$	−47.0602	−1.53739	−0.768695	−	0.639615i	$-0.779094\pi$
$937$	−47.0602	−1.53739	−0.768695	+	0.639615i	$0.779094\pi$
$938$	−7.90096	−0.257976
$939$	0	0
$940$	0	0
$941$	45.5100	1.48358	0.741792	−	0.670630i	$-0.233976\pi$
$941$	45.5100	1.48358	0.741792	+	0.670630i	$0.233976\pi$
$942$	0	0
$943$	8.98710	0.292660
$944$	−47.1838	−1.53570
$945$	0	0
$946$	32.4449	1.05487
$947$	−9.26597	−0.301104	−0.150552	−	0.988602i	$-0.548105\pi$
$947$	−9.26597	−0.301104	−0.150552	+	0.988602i	$0.548105\pi$
$948$	0	0
$949$	34.5697	1.12218
$950$	0	0
$951$	0	0
$952$	−2.02358	−0.0655846
$953$	17.7750	0.575788	0.287894	−	0.957662i	$-0.407045\pi$
$953$	17.7750	0.575788	0.287894	+	0.957662i	$0.407045\pi$
$954$	0	0
$955$	0	0
$956$	0.735303	0.0237814
$957$	0	0
$958$	30.8820	0.997751
$959$	33.4899	1.08145
$960$	0	0
$961$	−29.3227	−0.945894
$962$	−11.6067	−0.374215
$963$	0	0
$964$	−34.2800	−1.10408
$965$	0	0
$966$	0	0
$967$	−45.7913	−1.47255	−0.736275	−	0.676683i	$-0.763417\pi$
$967$	−45.7913	−1.47255	−0.736275	+	0.676683i	$0.763417\pi$
$968$	2.42074	0.0778054
$969$	0	0
$970$	0	0
$971$	14.0095	0.449587	0.224794	−	0.974406i	$-0.427829\pi$
$971$	14.0095	0.449587	0.224794	+	0.974406i	$0.427829\pi$
$972$	0	0
$973$	−36.3793	−1.16627
$974$	5.17199	0.165721
$975$	0	0
$976$	−26.9119	−0.861428
$977$	−51.0534	−1.63334	−0.816671	−	0.577103i	$-0.804183\pi$
$977$	−51.0534	−1.63334	−0.816671	+	0.577103i	$0.804183\pi$
$978$	0	0
$979$	5.00953	0.160105
$980$	0	0
$981$	0	0
$982$	−17.8371	−0.569206
$983$	−4.02331	−0.128324	−0.0641618	−	0.997940i	$-0.520437\pi$
$983$	−4.02331	−0.128324	−0.0641618	+	0.997940i	$0.520437\pi$
$984$	0	0
$985$	0	0
$986$	−2.64591	−0.0842630
$987$	0	0
$988$	65.1017	2.07116
$989$	−8.58207	−0.272894
$990$	0	0
$991$	−49.5829	−1.57505	−0.787526	−	0.616282i	$-0.788638\pi$
$991$	−49.5829	−1.57505	−0.787526	+	0.616282i	$0.788638\pi$
$992$	9.27524	0.294489
$993$	0	0
$994$	−16.2444	−0.515240
$995$	0	0
$996$	0	0
$997$	34.8740	1.10447	0.552235	−	0.833688i	$-0.313775\pi$
$997$	34.8740	1.10447	0.552235	+	0.833688i	$0.313775\pi$
$998$	−67.8674	−2.14831
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bm.1.5		5
3.2	odd	2		725.2.a.k.1.1	yes	5
5.4	even	2		6525.2.a.bq.1.1		5
15.2	even	4		725.2.b.f.349.2		10
15.8	even	4		725.2.b.f.349.9		10
15.14	odd	2		725.2.a.h.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.h.1.5	✓	5	15.14	odd	2
725.2.a.k.1.1	yes	5	3.2	odd	2
725.2.b.f.349.2		10	15.2	even	4
725.2.b.f.349.9		10	15.8	even	4
6525.2.a.bm.1.5		5	1.1	even	1	trivial
6525.2.a.bq.1.1		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.88481 q^{2} +1.55251 q^{4} +1.70908 q^{7} -0.843434 q^{8} +2.85130 q^{11} +6.22128 q^{13} +3.22128 q^{14} -4.69473 q^{16} +1.40381 q^{17} +6.74028 q^{19} +5.37416 q^{22} -1.42153 q^{23} +11.7259 q^{26} +2.65336 q^{28} -1.00000 q^{29} -1.29510 q^{31} -7.16181 q^{32} +2.64591 q^{34} -0.989830 q^{37} +12.7042 q^{38} -6.32213 q^{41} +6.03720 q^{43} +4.42667 q^{44} -2.67932 q^{46} -8.97549 q^{47} -4.07906 q^{49} +9.65860 q^{52} -2.11288 q^{53} -1.44149 q^{56} -1.88481 q^{58} +10.0504 q^{59} +5.73236 q^{61} -2.44101 q^{62} -4.10919 q^{64} -2.45274 q^{67} +2.17943 q^{68} -5.04282 q^{71} +5.55668 q^{73} -1.86564 q^{74} +10.4644 q^{76} +4.87309 q^{77} +9.37892 q^{79} -11.9160 q^{82} -5.06744 q^{83} +11.3790 q^{86} -2.40488 q^{88} +1.75693 q^{89} +10.6326 q^{91} -2.20694 q^{92} -16.9171 q^{94} +14.6037 q^{97} -7.68825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} + 6 q^{7} - 3 q^{8} + 2 q^{11} + 4 q^{13} - 11 q^{14} - 10 q^{16} - 9 q^{17} + 2 q^{19} + 4 q^{22} - q^{23} + 16 q^{26} + 10 q^{28} - 5 q^{29} - q^{31} + 2 q^{32} + 3 q^{34} + 14 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 + 6 * q^7 - 3 * q^8 + 2 * q^11 + 4 * q^13 - 11 * q^14 - 10 * q^16 - 9 * q^17 + 2 * q^19 + 4 * q^22 - q^23 + 16 * q^26 + 10 * q^28 - 5 * q^29 - q^31 + 2 * q^32 + 3 * q^34 + 14 * q^37 + 3 * q^38 - 5 * q^41 + 28 * q^43 - 7 * q^44 - 20 * q^46 - 15 * q^47 - 3 * q^49 + 6 * q^52 + 8 * q^53 + 16 * q^56 + 2 * q^58 + 11 * q^59 - 5 * q^61 + 6 * q^62 - 5 * q^64 + 23 * q^67 - q^68 + 5 * q^71 + 16 * q^73 - 8 * q^74 + 16 * q^76 + 30 * q^77 - 10 * q^79 - 19 * q^82 - 9 * q^83 + 13 * q^86 + 35 * q^88 + 18 * q^89 + q^91 + 27 * q^92 - 13 * q^94 + 23 * q^97 - 3 * q^98

Introduction

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