LMFDB - Embedded newform 6525.2.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 $ x^8 - 2x^7 - 12x^6 + 23x^5 + 36x^4 - 62x^3 - 15x^2 + 14*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2175)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.72810$ 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)

$f(q)$	$=$	$q-2.72810 q^{2} +5.44251 q^{4} +4.61695 q^{7} -9.39150 q^{8} +1.87758 q^{11} +5.48400 q^{13} -12.5955 q^{14} +14.7359 q^{16} +5.74056 q^{17} -5.40455 q^{19} -5.12221 q^{22} +0.137016 q^{23} -14.9609 q^{26} +25.1278 q^{28} -1.00000 q^{29} -8.18167 q^{31} -21.4180 q^{32} -15.6608 q^{34} -4.45785 q^{37} +14.7441 q^{38} +0.215243 q^{41} +7.17115 q^{43} +10.2187 q^{44} -0.373793 q^{46} +9.91293 q^{47} +14.3162 q^{49} +29.8467 q^{52} +1.14754 q^{53} -43.3601 q^{56} +2.72810 q^{58} +0.244913 q^{59} +8.75284 q^{61} +22.3204 q^{62} +28.9585 q^{64} -6.53295 q^{67} +31.2431 q^{68} +0.277412 q^{71} -11.4933 q^{73} +12.1614 q^{74} -29.4143 q^{76} +8.66866 q^{77} +1.66223 q^{79} -0.587203 q^{82} -9.58514 q^{83} -19.5636 q^{86} -17.6333 q^{88} -16.3332 q^{89} +25.3193 q^{91} +0.745712 q^{92} -27.0434 q^{94} +8.04128 q^{97} -39.0559 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 12 q^{4} + 2 q^{7} + 3 q^{8} - 6 q^{11} + 6 q^{13} - 9 q^{14} + 32 q^{16} + 12 q^{17} + 3 q^{22} + 14 q^{23} - 18 q^{26} + 14 q^{28} - 8 q^{29} + 8 q^{31} - 2 q^{32} - 13 q^{34} + 4 q^{37}+ \cdots + 29 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 + 12 * q^4 + 2 * q^7 + 3 * q^8 - 6 * q^11 + 6 * q^13 - 9 * q^14 + 32 * q^16 + 12 * q^17 + 3 * q^22 + 14 * q^23 - 18 * q^26 + 14 * q^28 - 8 * q^29 + 8 * q^31 - 2 * q^32 - 13 * q^34 + 4 * q^37 + 26 * q^38 - 2 * q^41 + 2 * q^43 + 15 * q^44 + 24 * q^46 + 12 * q^47 + 38 * q^49 + 49 * q^52 + 4 * q^53 - 58 * q^56 - 2 * q^58 - 18 * q^59 + 12 * q^61 - 4 * q^62 + 21 * q^64 + 26 * q^67 + 81 * q^68 - 24 * q^71 - 14 * q^73 + 22 * q^74 - 26 * q^77 + 10 * q^79 + 48 * q^82 + 40 * q^83 - 8 * q^86 - 10 * q^88 - 34 * q^89 + 26 * q^91 - 18 * q^92 - 43 * q^94 + 30 * q^97 + 29 * q^98

Coefficient data

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

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

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

$2$	−2.72810	−1.92906	−0.964528	−	0.263981i	$-0.914964\pi$
$2$	−2.72810	−1.92906	−0.964528	+	0.263981i	$0.914964\pi$
$3$	0	0
$3$	0	0
$4$	5.44251	2.72126
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.61695	1.74504	0.872521	−	0.488577i	$-0.162484\pi$
$7$	4.61695	1.74504	0.872521	+	0.488577i	$0.162484\pi$
$8$	−9.39150	−3.32040
$9$	0	0
$10$	0	0
$11$	1.87758	0.566110	0.283055	−	0.959104i	$-0.408652\pi$
$11$	1.87758	0.566110	0.283055	+	0.959104i	$0.408652\pi$
$12$	0	0
$13$	5.48400	1.52099	0.760494	−	0.649345i	$-0.224957\pi$
$13$	5.48400	1.52099	0.760494	+	0.649345i	$0.224957\pi$
$14$	−12.5955	−3.36628
$15$	0	0
$16$	14.7359	3.68398
$17$	5.74056	1.39229	0.696145	−	0.717901i	$-0.254897\pi$
$17$	5.74056	1.39229	0.696145	+	0.717901i	$0.254897\pi$
$18$	0	0
$19$	−5.40455	−1.23989	−0.619944	−	0.784646i	$-0.712845\pi$
$19$	−5.40455	−1.23989	−0.619944	+	0.784646i	$0.712845\pi$
$20$	0	0
$21$	0	0
$22$	−5.12221	−1.09206
$23$	0.137016	0.0285698	0.0142849	−	0.999898i	$-0.495453\pi$
$23$	0.137016	0.0285698	0.0142849	+	0.999898i	$0.495453\pi$
$24$	0	0
$25$	0	0
$26$	−14.9609	−2.93407
$27$	0	0
$28$	25.1278	4.74870
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.18167	−1.46947	−0.734736	−	0.678353i	$-0.762694\pi$
$31$	−8.18167	−1.46947	−0.734736	+	0.678353i	$0.762694\pi$
$32$	−21.4180	−3.78620
$33$	0	0
$34$	−15.6608	−2.68581
$35$	0	0
$36$	0	0
$37$	−4.45785	−0.732866	−0.366433	−	0.930444i	$-0.619421\pi$
$37$	−4.45785	−0.732866	−0.366433	+	0.930444i	$0.619421\pi$
$38$	14.7441	2.39181
$39$	0	0
$40$	0	0
$41$	0.215243	0.0336153	0.0168076	−	0.999859i	$-0.494650\pi$
$41$	0.215243	0.0336153	0.0168076	+	0.999859i	$0.494650\pi$
$42$	0	0
$43$	7.17115	1.09359	0.546795	−	0.837266i	$-0.315848\pi$
$43$	7.17115	1.09359	0.546795	+	0.837266i	$0.315848\pi$
$44$	10.2187	1.54053
$45$	0	0
$46$	−0.373793	−0.0551128
$47$	9.91293	1.44595	0.722975	−	0.690874i	$-0.242774\pi$
$47$	9.91293	1.44595	0.722975	+	0.690874i	$0.242774\pi$
$48$	0	0
$49$	14.3162	2.04517
$50$	0	0
$51$	0	0
$52$	29.8467	4.13899
$53$	1.14754	0.157627	0.0788134	−	0.996889i	$-0.474887\pi$
$53$	1.14754	0.157627	0.0788134	+	0.996889i	$0.474887\pi$
$54$	0	0
$55$	0	0
$56$	−43.3601	−5.79423
$57$	0	0
$58$	2.72810	0.358217
$59$	0.244913	0.0318849	0.0159425	−	0.999873i	$-0.494925\pi$
$59$	0.244913	0.0318849	0.0159425	+	0.999873i	$0.494925\pi$
$60$	0	0
$61$	8.75284	1.12069	0.560343	−	0.828260i	$-0.310669\pi$
$61$	8.75284	1.12069	0.560343	+	0.828260i	$0.310669\pi$
$62$	22.3204	2.83469
$63$	0	0
$64$	28.9585	3.61981
$65$	0	0
$66$	0	0
$67$	−6.53295	−0.798127	−0.399064	−	0.916923i	$-0.630665\pi$
$67$	−6.53295	−0.798127	−0.399064	+	0.916923i	$0.630665\pi$
$68$	31.2431	3.78878
$69$	0	0
$70$	0	0
$71$	0.277412	0.0329227	0.0164614	−	0.999865i	$-0.494760\pi$
$71$	0.277412	0.0329227	0.0164614	+	0.999865i	$0.494760\pi$
$72$	0	0
$73$	−11.4933	−1.34519	−0.672594	−	0.740011i	$-0.734820\pi$
$73$	−11.4933	−1.34519	−0.672594	+	0.740011i	$0.734820\pi$
$74$	12.1614	1.41374
$75$	0	0
$76$	−29.4143	−3.37405
$77$	8.66866	0.987886
$78$	0	0
$79$	1.66223	0.187015	0.0935076	−	0.995619i	$-0.470192\pi$
$79$	1.66223	0.187015	0.0935076	+	0.995619i	$0.470192\pi$
$80$	0	0
$81$	0	0
$82$	−0.587203	−0.0648457
$83$	−9.58514	−1.05211	−0.526053	−	0.850452i	$-0.676329\pi$
$83$	−9.58514	−1.05211	−0.526053	+	0.850452i	$0.676329\pi$
$84$	0	0
$85$	0	0
$86$	−19.5636	−2.10960
$87$	0	0
$88$	−17.6333	−1.87971
$89$	−16.3332	−1.73132	−0.865659	−	0.500634i	$-0.833100\pi$
$89$	−16.3332	−1.73132	−0.865659	+	0.500634i	$0.833100\pi$
$90$	0	0
$91$	25.3193	2.65419
$92$	0.745712	0.0777458
$93$	0	0
$94$	−27.0434	−2.78932
$95$	0	0
$96$	0	0
$97$	8.04128	0.816468	0.408234	−	0.912877i	$-0.366145\pi$
$97$	8.04128	0.816468	0.408234	+	0.912877i	$0.366145\pi$
$98$	−39.0559	−3.94525
$99$	0	0
$100$	0	0
$101$	12.3239	1.22627	0.613135	−	0.789978i	$-0.289908\pi$
$101$	12.3239	1.22627	0.613135	+	0.789978i	$0.289908\pi$
$102$	0	0
$103$	−0.660912	−0.0651216	−0.0325608	−	0.999470i	$-0.510366\pi$
$103$	−0.660912	−0.0651216	−0.0325608	+	0.999470i	$0.510366\pi$
$104$	−51.5030	−5.05028
$105$	0	0
$106$	−3.13060	−0.304071
$107$	14.5868	1.41016	0.705079	−	0.709128i	$-0.250911\pi$
$107$	14.5868	1.41016	0.705079	+	0.709128i	$0.250911\pi$
$108$	0	0
$109$	15.5050	1.48511	0.742557	−	0.669783i	$-0.233613\pi$
$109$	15.5050	1.48511	0.742557	+	0.669783i	$0.233613\pi$
$110$	0	0
$111$	0	0
$112$	68.0349	6.42869
$113$	7.80660	0.734383	0.367191	−	0.930145i	$-0.380319\pi$
$113$	7.80660	0.734383	0.367191	+	0.930145i	$0.380319\pi$
$114$	0	0
$115$	0	0
$116$	−5.44251	−0.505325
$117$	0	0
$118$	−0.668145	−0.0615078
$119$	26.5038	2.42960
$120$	0	0
$121$	−7.47471	−0.679519
$122$	−23.8786	−2.16187
$123$	0	0
$124$	−44.5289	−3.99881
$125$	0	0
$126$	0	0
$127$	19.7106	1.74903	0.874515	−	0.484999i	$-0.161180\pi$
$127$	19.7106	1.74903	0.874515	+	0.484999i	$0.161180\pi$
$128$	−36.1656	−3.19662
$129$	0	0
$130$	0	0
$131$	1.05738	0.0923834	0.0461917	−	0.998933i	$-0.485291\pi$
$131$	1.05738	0.0923834	0.0461917	+	0.998933i	$0.485291\pi$
$132$	0	0
$133$	−24.9525	−2.16366
$134$	17.8225	1.53963
$135$	0	0
$136$	−53.9125	−4.62296
$137$	−3.21467	−0.274648	−0.137324	−	0.990526i	$-0.543850\pi$
$137$	−3.21467	−0.274648	−0.137324	+	0.990526i	$0.543850\pi$
$138$	0	0
$139$	8.48410	0.719612	0.359806	−	0.933027i	$-0.382843\pi$
$139$	8.48410	0.719612	0.359806	+	0.933027i	$0.382843\pi$
$140$	0	0
$141$	0	0
$142$	−0.756807	−0.0635098
$143$	10.2966	0.861046
$144$	0	0
$145$	0	0
$146$	31.3548	2.59494
$147$	0	0
$148$	−24.2619	−1.99432
$149$	5.93205	0.485972	0.242986	−	0.970030i	$-0.421873\pi$
$149$	5.93205	0.485972	0.242986	+	0.970030i	$0.421873\pi$
$150$	0	0
$151$	8.06281	0.656142	0.328071	−	0.944653i	$-0.393601\pi$
$151$	8.06281	0.656142	0.328071	+	0.944653i	$0.393601\pi$
$152$	50.7568	4.11692
$153$	0	0
$154$	−23.6490	−1.90569
$155$	0	0
$156$	0	0
$157$	−9.87127	−0.787813	−0.393906	−	0.919151i	$-0.628877\pi$
$157$	−9.87127	−0.787813	−0.393906	+	0.919151i	$0.628877\pi$
$158$	−4.53472	−0.360763
$159$	0	0
$160$	0	0
$161$	0.632596	0.0498555
$162$	0	0
$163$	1.25711	0.0984643	0.0492321	−	0.998787i	$-0.484323\pi$
$163$	1.25711	0.0984643	0.0492321	+	0.998787i	$0.484323\pi$
$164$	1.17146	0.0914757
$165$	0	0
$166$	26.1492	2.02957
$167$	15.1462	1.17205	0.586024	−	0.810294i	$-0.300692\pi$
$167$	15.1462	1.17205	0.586024	+	0.810294i	$0.300692\pi$
$168$	0	0
$169$	17.0742	1.31340
$170$	0	0
$171$	0	0
$172$	39.0291	2.97594
$173$	−2.34929	−0.178613	−0.0893065	−	0.996004i	$-0.528465\pi$
$173$	−2.34929	−0.178613	−0.0893065	+	0.996004i	$0.528465\pi$
$174$	0	0
$175$	0	0
$176$	27.6678	2.08554
$177$	0	0
$178$	44.5586	3.33981
$179$	17.7814	1.32904	0.664520	−	0.747270i	$-0.268636\pi$
$179$	17.7814	1.32904	0.664520	+	0.747270i	$0.268636\pi$
$180$	0	0
$181$	−2.69871	−0.200594	−0.100297	−	0.994958i	$-0.531979\pi$
$181$	−2.69871	−0.200594	−0.100297	+	0.994958i	$0.531979\pi$
$182$	−69.0735	−5.12007
$183$	0	0
$184$	−1.28679	−0.0948632
$185$	0	0
$186$	0	0
$187$	10.7783	0.788190
$188$	53.9512	3.93480
$189$	0	0
$190$	0	0
$191$	−10.7997	−0.781442	−0.390721	−	0.920509i	$-0.627774\pi$
$191$	−10.7997	−0.781442	−0.390721	+	0.920509i	$0.627774\pi$
$192$	0	0
$193$	−4.18447	−0.301205	−0.150602	−	0.988594i	$-0.548121\pi$
$193$	−4.18447	−0.301205	−0.150602	+	0.988594i	$0.548121\pi$
$194$	−21.9374	−1.57501
$195$	0	0
$196$	77.9160	5.56543
$197$	−5.76856	−0.410993	−0.205496	−	0.978658i	$-0.565881\pi$
$197$	−5.76856	−0.410993	−0.205496	+	0.978658i	$0.565881\pi$
$198$	0	0
$199$	−5.45094	−0.386407	−0.193203	−	0.981159i	$-0.561888\pi$
$199$	−5.45094	−0.386407	−0.193203	+	0.981159i	$0.561888\pi$
$200$	0	0
$201$	0	0
$202$	−33.6207	−2.36554
$203$	−4.61695	−0.324046
$204$	0	0
$205$	0	0
$206$	1.80303	0.125623
$207$	0	0
$208$	80.8117	5.60328
$209$	−10.1474	−0.701914
$210$	0	0
$211$	0.863332	0.0594342	0.0297171	−	0.999558i	$-0.490539\pi$
$211$	0.863332	0.0594342	0.0297171	+	0.999558i	$0.490539\pi$
$212$	6.24550	0.428943
$213$	0	0
$214$	−39.7942	−2.72027
$215$	0	0
$216$	0	0
$217$	−37.7743	−2.56429
$218$	−42.2993	−2.86487
$219$	0	0
$220$	0	0
$221$	31.4812	2.11766
$222$	0	0
$223$	−0.615197	−0.0411967	−0.0205983	−	0.999788i	$-0.506557\pi$
$223$	−0.615197	−0.0411967	−0.0205983	+	0.999788i	$0.506557\pi$
$224$	−98.8856	−6.60707
$225$	0	0
$226$	−21.2972	−1.41667
$227$	23.7230	1.57455	0.787275	−	0.616602i	$-0.211491\pi$
$227$	23.7230	1.57455	0.787275	+	0.616602i	$0.211491\pi$
$228$	0	0
$229$	−20.9808	−1.38645	−0.693226	−	0.720721i	$-0.743811\pi$
$229$	−20.9808	−1.38645	−0.693226	+	0.720721i	$0.743811\pi$
$230$	0	0
$231$	0	0
$232$	9.39150	0.616582
$233$	7.95482	0.521138	0.260569	−	0.965455i	$-0.416090\pi$
$233$	7.95482	0.521138	0.260569	+	0.965455i	$0.416090\pi$
$234$	0	0
$235$	0	0
$236$	1.33294	0.0867670
$237$	0	0
$238$	−72.3051	−4.68684
$239$	−17.4888	−1.13125	−0.565627	−	0.824661i	$-0.691366\pi$
$239$	−17.4888	−1.13125	−0.565627	+	0.824661i	$0.691366\pi$
$240$	0	0
$241$	−14.7743	−0.951699	−0.475849	−	0.879527i	$-0.657859\pi$
$241$	−14.7743	−0.951699	−0.475849	+	0.879527i	$0.657859\pi$
$242$	20.3917	1.31083
$243$	0	0
$244$	47.6375	3.04968
$245$	0	0
$246$	0	0
$247$	−29.6385	−1.88585
$248$	76.8382	4.87923
$249$	0	0
$250$	0	0
$251$	3.30996	0.208923	0.104461	−	0.994529i	$-0.466688\pi$
$251$	3.30996	0.208923	0.104461	+	0.994529i	$0.466688\pi$
$252$	0	0
$253$	0.257258	0.0161737
$254$	−53.7723	−3.37398
$255$	0	0
$256$	40.7463	2.54664
$257$	30.8668	1.92542	0.962709	−	0.270539i	$-0.0872020\pi$
$257$	30.8668	1.92542	0.962709	+	0.270539i	$0.0872020\pi$
$258$	0	0
$259$	−20.5816	−1.27888
$260$	0	0
$261$	0	0
$262$	−2.88463	−0.178213
$263$	−19.4833	−1.20139	−0.600697	−	0.799477i	$-0.705110\pi$
$263$	−19.4833	−1.20139	−0.600697	+	0.799477i	$0.705110\pi$
$264$	0	0
$265$	0	0
$266$	68.0728	4.17381
$267$	0	0
$268$	−35.5557	−2.17191
$269$	−20.8385	−1.27055	−0.635273	−	0.772287i	$-0.719113\pi$
$269$	−20.8385	−1.27055	−0.635273	+	0.772287i	$0.719113\pi$
$270$	0	0
$271$	24.5849	1.49343	0.746713	−	0.665146i	$-0.231631\pi$
$271$	24.5849	1.49343	0.746713	+	0.665146i	$0.231631\pi$
$272$	84.5924	5.12916
$273$	0	0
$274$	8.76994	0.529811
$275$	0	0
$276$	0	0
$277$	1.89904	0.114102	0.0570510	−	0.998371i	$-0.481830\pi$
$277$	1.89904	0.114102	0.0570510	+	0.998371i	$0.481830\pi$
$278$	−23.1455	−1.38817
$279$	0	0
$280$	0	0
$281$	−32.4916	−1.93828	−0.969142	−	0.246503i	$-0.920719\pi$
$281$	−32.4916	−1.93828	−0.969142	+	0.246503i	$0.920719\pi$
$282$	0	0
$283$	−11.9151	−0.708279	−0.354140	−	0.935193i	$-0.615226\pi$
$283$	−11.9151	−0.708279	−0.354140	+	0.935193i	$0.615226\pi$
$284$	1.50982	0.0895912
$285$	0	0
$286$	−28.0902	−1.66101
$287$	0.993764	0.0586600
$288$	0	0
$289$	15.9540	0.938472
$290$	0	0
$291$	0	0
$292$	−62.5524	−3.66060
$293$	−20.7359	−1.21140	−0.605702	−	0.795692i	$-0.707108\pi$
$293$	−20.7359	−1.21140	−0.605702	+	0.795692i	$0.707108\pi$
$294$	0	0
$295$	0	0
$296$	41.8659	2.43341
$297$	0	0
$298$	−16.1832	−0.937467
$299$	0.751396	0.0434544
$300$	0	0
$301$	33.1088	1.90836
$302$	−21.9961	−1.26574
$303$	0	0
$304$	−79.6409	−4.56772
$305$	0	0
$306$	0	0
$307$	−16.9755	−0.968842	−0.484421	−	0.874835i	$-0.660970\pi$
$307$	−16.9755	−0.968842	−0.484421	+	0.874835i	$0.660970\pi$
$308$	47.1793	2.68829
$309$	0	0
$310$	0	0
$311$	20.9019	1.18524	0.592619	−	0.805483i	$-0.298094\pi$
$311$	20.9019	1.18524	0.592619	+	0.805483i	$0.298094\pi$
$312$	0	0
$313$	12.7754	0.722110	0.361055	−	0.932545i	$-0.382417\pi$
$313$	12.7754	0.722110	0.361055	+	0.932545i	$0.382417\pi$
$314$	26.9298	1.51973
$315$	0	0
$316$	9.04669	0.508916
$317$	−15.0552	−0.845582	−0.422791	−	0.906227i	$-0.638950\pi$
$317$	−15.0552	−0.845582	−0.422791	+	0.906227i	$0.638950\pi$
$318$	0	0
$319$	−1.87758	−0.105124
$320$	0	0
$321$	0	0
$322$	−1.72578	−0.0961741
$323$	−31.0251	−1.72628
$324$	0	0
$325$	0	0
$326$	−3.42951	−0.189943
$327$	0	0
$328$	−2.02145	−0.111616
$329$	45.7675	2.52324
$330$	0	0
$331$	−17.7325	−0.974668	−0.487334	−	0.873216i	$-0.662031\pi$
$331$	−17.7325	−0.974668	−0.487334	+	0.873216i	$0.662031\pi$
$332$	−52.1672	−2.86305
$333$	0	0
$334$	−41.3203	−2.26095
$335$	0	0
$336$	0	0
$337$	−17.5905	−0.958217	−0.479108	−	0.877756i	$-0.659040\pi$
$337$	−17.5905	−0.958217	−0.479108	+	0.877756i	$0.659040\pi$
$338$	−46.5801	−2.53363
$339$	0	0
$340$	0	0
$341$	−15.3617	−0.831883
$342$	0	0
$343$	33.7784	1.82386
$344$	−67.3479	−3.63115
$345$	0	0
$346$	6.40909	0.344555
$347$	21.0789	1.13157	0.565787	−	0.824552i	$-0.308573\pi$
$347$	21.0789	1.13157	0.565787	+	0.824552i	$0.308573\pi$
$348$	0	0
$349$	−0.723154	−0.0387096	−0.0193548	−	0.999813i	$-0.506161\pi$
$349$	−0.723154	−0.0387096	−0.0193548	+	0.999813i	$0.506161\pi$
$350$	0	0
$351$	0	0
$352$	−40.2139	−2.14341
$353$	−5.93891	−0.316096	−0.158048	−	0.987431i	$-0.550520\pi$
$353$	−5.93891	−0.316096	−0.158048	+	0.987431i	$0.550520\pi$
$354$	0	0
$355$	0	0
$356$	−88.8938	−4.71136
$357$	0	0
$358$	−48.5092	−2.56379
$359$	2.30973	0.121903	0.0609514	−	0.998141i	$-0.480587\pi$
$359$	2.30973	0.121903	0.0609514	+	0.998141i	$0.480587\pi$
$360$	0	0
$361$	10.2091	0.537323
$362$	7.36234	0.386956
$363$	0	0
$364$	137.801	7.22272
$365$	0	0
$366$	0	0
$367$	22.9235	1.19660	0.598298	−	0.801274i	$-0.295844\pi$
$367$	22.9235	1.19660	0.598298	+	0.801274i	$0.295844\pi$
$368$	2.01906	0.105251
$369$	0	0
$370$	0	0
$371$	5.29813	0.275065
$372$	0	0
$373$	−2.83182	−0.146626	−0.0733129	−	0.997309i	$-0.523357\pi$
$373$	−2.83182	−0.146626	−0.0733129	+	0.997309i	$0.523357\pi$
$374$	−29.4043	−1.52046
$375$	0	0
$376$	−93.0973	−4.80113
$377$	−5.48400	−0.282440
$378$	0	0
$379$	−29.2539	−1.50267	−0.751335	−	0.659921i	$-0.770590\pi$
$379$	−29.2539	−1.50267	−0.751335	+	0.659921i	$0.770590\pi$
$380$	0	0
$381$	0	0
$382$	29.4627	1.50744
$383$	2.99063	0.152814	0.0764070	−	0.997077i	$-0.475655\pi$
$383$	2.99063	0.152814	0.0764070	+	0.997077i	$0.475655\pi$
$384$	0	0
$385$	0	0
$386$	11.4156	0.581040
$387$	0	0
$388$	43.7648	2.22182
$389$	8.57339	0.434688	0.217344	−	0.976095i	$-0.430261\pi$
$389$	8.57339	0.434688	0.217344	+	0.976095i	$0.430261\pi$
$390$	0	0
$391$	0.786549	0.0397775
$392$	−134.451	−6.79078
$393$	0	0
$394$	15.7372	0.792828
$395$	0	0
$396$	0	0
$397$	9.29605	0.466556	0.233278	−	0.972410i	$-0.425055\pi$
$397$	9.29605	0.466556	0.233278	+	0.972410i	$0.425055\pi$
$398$	14.8707	0.745400
$399$	0	0
$400$	0	0
$401$	−19.5392	−0.975739	−0.487870	−	0.872916i	$-0.662226\pi$
$401$	−19.5392	−0.975739	−0.487870	+	0.872916i	$0.662226\pi$
$402$	0	0
$403$	−44.8683	−2.23505
$404$	67.0728	3.33700
$405$	0	0
$406$	12.5955	0.625103
$407$	−8.36995	−0.414883
$408$	0	0
$409$	37.9582	1.87691	0.938456	−	0.345398i	$-0.112256\pi$
$409$	37.9582	1.87691	0.938456	+	0.345398i	$0.112256\pi$
$410$	0	0
$411$	0	0
$412$	−3.59702	−0.177213
$413$	1.13075	0.0556405
$414$	0	0
$415$	0	0
$416$	−117.456	−5.75876
$417$	0	0
$418$	27.6832	1.35403
$419$	10.0952	0.493184	0.246592	−	0.969119i	$-0.420689\pi$
$419$	10.0952	0.493184	0.246592	+	0.969119i	$0.420689\pi$
$420$	0	0
$421$	13.8521	0.675111	0.337556	−	0.941306i	$-0.390400\pi$
$421$	13.8521	0.675111	0.337556	+	0.941306i	$0.390400\pi$
$422$	−2.35525	−0.114652
$423$	0	0
$424$	−10.7771	−0.523384
$425$	0	0
$426$	0	0
$427$	40.4114	1.95565
$428$	79.3888	3.83740
$429$	0	0
$430$	0	0
$431$	−18.2149	−0.877380	−0.438690	−	0.898638i	$-0.644557\pi$
$431$	−18.2149	−0.877380	−0.438690	+	0.898638i	$0.644557\pi$
$432$	0	0
$433$	6.73476	0.323652	0.161826	−	0.986819i	$-0.448262\pi$
$433$	6.73476	0.323652	0.161826	+	0.986819i	$0.448262\pi$
$434$	103.052	4.94666
$435$	0	0
$436$	84.3864	4.04137
$437$	−0.740510	−0.0354234
$438$	0	0
$439$	30.9590	1.47759	0.738795	−	0.673930i	$-0.235395\pi$
$439$	30.9590	1.47759	0.738795	+	0.673930i	$0.235395\pi$
$440$	0	0
$441$	0	0
$442$	−85.8838	−4.08508
$443$	−21.4432	−1.01880	−0.509398	−	0.860531i	$-0.670132\pi$
$443$	−21.4432	−1.01880	−0.509398	+	0.860531i	$0.670132\pi$
$444$	0	0
$445$	0	0
$446$	1.67832	0.0794707
$447$	0	0
$448$	133.700	6.31672
$449$	−25.6895	−1.21236	−0.606180	−	0.795327i	$-0.707299\pi$
$449$	−25.6895	−1.21236	−0.606180	+	0.795327i	$0.707299\pi$
$450$	0	0
$451$	0.404134	0.0190299
$452$	42.4875	1.99844
$453$	0	0
$454$	−64.7186	−3.03739
$455$	0	0
$456$	0	0
$457$	30.3278	1.41868	0.709338	−	0.704868i	$-0.248994\pi$
$457$	30.3278	1.41868	0.709338	+	0.704868i	$0.248994\pi$
$458$	57.2377	2.67454
$459$	0	0
$460$	0	0
$461$	31.0026	1.44393	0.721967	−	0.691928i	$-0.243238\pi$
$461$	31.0026	1.44393	0.721967	+	0.691928i	$0.243238\pi$
$462$	0	0
$463$	−27.6990	−1.28728	−0.643640	−	0.765328i	$-0.722577\pi$
$463$	−27.6990	−1.28728	−0.643640	+	0.765328i	$0.722577\pi$
$464$	−14.7359	−0.684097
$465$	0	0
$466$	−21.7015	−1.00530
$467$	4.03243	0.186598	0.0932992	−	0.995638i	$-0.470259\pi$
$467$	4.03243	0.186598	0.0932992	+	0.995638i	$0.470259\pi$
$468$	0	0
$469$	−30.1623	−1.39277
$470$	0	0
$471$	0	0
$472$	−2.30010	−0.105871
$473$	13.4644	0.619093
$474$	0	0
$475$	0	0
$476$	144.248	6.61157
$477$	0	0
$478$	47.7110	2.18225
$479$	−1.32186	−0.0603973	−0.0301986	−	0.999544i	$-0.509614\pi$
$479$	−1.32186	−0.0603973	−0.0301986	+	0.999544i	$0.509614\pi$
$480$	0	0
$481$	−24.4468	−1.11468
$482$	40.3058	1.83588
$483$	0	0
$484$	−40.6812	−1.84915
$485$	0	0
$486$	0	0
$487$	−9.89390	−0.448335	−0.224168	−	0.974551i	$-0.571966\pi$
$487$	−9.89390	−0.448335	−0.224168	+	0.974551i	$0.571966\pi$
$488$	−82.2024	−3.72113
$489$	0	0
$490$	0	0
$491$	−17.1305	−0.773089	−0.386545	−	0.922271i	$-0.626331\pi$
$491$	−17.1305	−0.773089	−0.386545	+	0.922271i	$0.626331\pi$
$492$	0	0
$493$	−5.74056	−0.258542
$494$	80.8568	3.63792
$495$	0	0
$496$	−120.564	−5.41350
$497$	1.28080	0.0574516
$498$	0	0
$499$	−7.95477	−0.356104	−0.178052	−	0.984021i	$-0.556980\pi$
$499$	−7.95477	−0.356104	−0.178052	+	0.984021i	$0.556980\pi$
$500$	0	0
$501$	0	0
$502$	−9.02989	−0.403024
$503$	−13.3471	−0.595119	−0.297559	−	0.954703i	$-0.596173\pi$
$503$	−13.3471	−0.595119	−0.297559	+	0.954703i	$0.596173\pi$
$504$	0	0
$505$	0	0
$506$	−0.701825	−0.0311999
$507$	0	0
$508$	107.275	4.75956
$509$	5.55324	0.246143	0.123071	−	0.992398i	$-0.460726\pi$
$509$	5.55324	0.246143	0.123071	+	0.992398i	$0.460726\pi$
$510$	0	0
$511$	−53.0639	−2.34741
$512$	−38.8286	−1.71600
$513$	0	0
$514$	−84.2076	−3.71424
$515$	0	0
$516$	0	0
$517$	18.6123	0.818567
$518$	56.1487	2.46703
$519$	0	0
$520$	0	0
$521$	−41.5095	−1.81857	−0.909283	−	0.416179i	$-0.863369\pi$
$521$	−41.5095	−1.81857	−0.909283	+	0.416179i	$0.863369\pi$
$522$	0	0
$523$	−11.6261	−0.508375	−0.254188	−	0.967155i	$-0.581808\pi$
$523$	−11.6261	−0.508375	−0.254188	+	0.967155i	$0.581808\pi$
$524$	5.75479	0.251399
$525$	0	0
$526$	53.1525	2.31756
$527$	−46.9674	−2.04593
$528$	0	0
$529$	−22.9812	−0.999184
$530$	0	0
$531$	0	0
$532$	−135.804	−5.88786
$533$	1.18039	0.0511284
$534$	0	0
$535$	0	0
$536$	61.3543	2.65010
$537$	0	0
$538$	56.8495	2.45096
$539$	26.8797	1.15779
$540$	0	0
$541$	−5.29830	−0.227792	−0.113896	−	0.993493i	$-0.536333\pi$
$541$	−5.29830	−0.227792	−0.113896	+	0.993493i	$0.536333\pi$
$542$	−67.0700	−2.88090
$543$	0	0
$544$	−122.951	−5.27149
$545$	0	0
$546$	0	0
$547$	−6.78093	−0.289932	−0.144966	−	0.989437i	$-0.546307\pi$
$547$	−6.78093	−0.289932	−0.144966	+	0.989437i	$0.546307\pi$
$548$	−17.4959	−0.747387
$549$	0	0
$550$	0	0
$551$	5.40455	0.230241
$552$	0	0
$553$	7.67441	0.326349
$554$	−5.18076	−0.220109
$555$	0	0
$556$	46.1748	1.95825
$557$	−0.186307	−0.00789410	−0.00394705	−	0.999992i	$-0.501256\pi$
$557$	−0.186307	−0.00789410	−0.00394705	+	0.999992i	$0.501256\pi$
$558$	0	0
$559$	39.3266	1.66334
$560$	0	0
$561$	0	0
$562$	88.6401	3.73906
$563$	27.1382	1.14374	0.571869	−	0.820345i	$-0.306219\pi$
$563$	27.1382	1.14374	0.571869	+	0.820345i	$0.306219\pi$
$564$	0	0
$565$	0	0
$566$	32.5056	1.36631
$567$	0	0
$568$	−2.60532	−0.109317
$569$	−22.8795	−0.959157	−0.479578	−	0.877499i	$-0.659210\pi$
$569$	−22.8795	−0.959157	−0.479578	+	0.877499i	$0.659210\pi$
$570$	0	0
$571$	38.3621	1.60541	0.802703	−	0.596379i	$-0.203394\pi$
$571$	38.3621	1.60541	0.802703	+	0.596379i	$0.203394\pi$
$572$	56.0395	2.34313
$573$	0	0
$574$	−2.71108	−0.113158
$575$	0	0
$576$	0	0
$577$	−12.0497	−0.501637	−0.250818	−	0.968034i	$-0.580700\pi$
$577$	−12.0497	−0.501637	−0.250818	+	0.968034i	$0.580700\pi$
$578$	−43.5241	−1.81036
$579$	0	0
$580$	0	0
$581$	−44.2541	−1.83597
$582$	0	0
$583$	2.15459	0.0892341
$584$	107.939	4.46656
$585$	0	0
$586$	56.5695	2.33687
$587$	23.6704	0.976982	0.488491	−	0.872569i	$-0.337547\pi$
$587$	23.6704	0.976982	0.488491	+	0.872569i	$0.337547\pi$
$588$	0	0
$589$	44.2183	1.82198
$590$	0	0
$591$	0	0
$592$	−65.6905	−2.69986
$593$	−25.9414	−1.06529	−0.532643	−	0.846340i	$-0.678801\pi$
$593$	−25.9414	−1.06529	−0.532643	+	0.846340i	$0.678801\pi$
$594$	0	0
$595$	0	0
$596$	32.2852	1.32245
$597$	0	0
$598$	−2.04988	−0.0838259
$599$	17.2886	0.706393	0.353196	−	0.935549i	$-0.385095\pi$
$599$	17.2886	0.706393	0.353196	+	0.935549i	$0.385095\pi$
$600$	0	0
$601$	−30.8997	−1.26042	−0.630212	−	0.776423i	$-0.717032\pi$
$601$	−30.8997	−1.26042	−0.630212	+	0.776423i	$0.717032\pi$
$602$	−90.3240	−3.68133
$603$	0	0
$604$	43.8820	1.78553
$605$	0	0
$606$	0	0
$607$	4.07152	0.165258	0.0826290	−	0.996580i	$-0.473668\pi$
$607$	4.07152	0.165258	0.0826290	+	0.996580i	$0.473668\pi$
$608$	115.754	4.69446
$609$	0	0
$610$	0	0
$611$	54.3625	2.19927
$612$	0	0
$613$	32.3595	1.30699	0.653494	−	0.756931i	$-0.273302\pi$
$613$	32.3595	1.30699	0.653494	+	0.756931i	$0.273302\pi$
$614$	46.3108	1.86895
$615$	0	0
$616$	−81.4118	−3.28017
$617$	33.4957	1.34849	0.674243	−	0.738509i	$-0.264470\pi$
$617$	33.4957	1.34849	0.674243	+	0.738509i	$0.264470\pi$
$618$	0	0
$619$	−16.6857	−0.670656	−0.335328	−	0.942101i	$-0.608847\pi$
$619$	−16.6857	−0.670656	−0.335328	+	0.942101i	$0.608847\pi$
$620$	0	0
$621$	0	0
$622$	−57.0224	−2.28639
$623$	−75.4096	−3.02122
$624$	0	0
$625$	0	0
$626$	−34.8526	−1.39299
$627$	0	0
$628$	−53.7245	−2.14384
$629$	−25.5905	−1.02036
$630$	0	0
$631$	9.91590	0.394746	0.197373	−	0.980328i	$-0.436759\pi$
$631$	9.91590	0.394746	0.197373	+	0.980328i	$0.436759\pi$
$632$	−15.6108	−0.620965
$633$	0	0
$634$	41.0719	1.63118
$635$	0	0
$636$	0	0
$637$	78.5099	3.11068
$638$	5.12221	0.202790
$639$	0	0
$640$	0	0
$641$	22.9862	0.907902	0.453951	−	0.891027i	$-0.350014\pi$
$641$	22.9862	0.907902	0.453951	+	0.891027i	$0.350014\pi$
$642$	0	0
$643$	−12.9117	−0.509190	−0.254595	−	0.967048i	$-0.581942\pi$
$643$	−12.9117	−0.509190	−0.254595	+	0.967048i	$0.581942\pi$
$644$	3.44291	0.135670
$645$	0	0
$646$	84.6395	3.33010
$647$	−16.6096	−0.652989	−0.326494	−	0.945199i	$-0.605867\pi$
$647$	−16.6096	−0.652989	−0.326494	+	0.945199i	$0.605867\pi$
$648$	0	0
$649$	0.459842	0.0180504
$650$	0	0
$651$	0	0
$652$	6.84182	0.267947
$653$	21.5638	0.843858	0.421929	−	0.906629i	$-0.361353\pi$
$653$	21.5638	0.843858	0.421929	+	0.906629i	$0.361353\pi$
$654$	0	0
$655$	0	0
$656$	3.17180	0.123838
$657$	0	0
$658$	−124.858	−4.86747
$659$	−31.7863	−1.23822	−0.619110	−	0.785305i	$-0.712506\pi$
$659$	−31.7863	−1.23822	−0.619110	+	0.785305i	$0.712506\pi$
$660$	0	0
$661$	30.0046	1.16704	0.583521	−	0.812098i	$-0.301674\pi$
$661$	30.0046	1.16704	0.583521	+	0.812098i	$0.301674\pi$
$662$	48.3760	1.88019
$663$	0	0
$664$	90.0189	3.49341
$665$	0	0
$666$	0	0
$667$	−0.137016	−0.00530529
$668$	82.4334	3.18944
$669$	0	0
$670$	0	0
$671$	16.4341	0.634432
$672$	0	0
$673$	−35.0842	−1.35240	−0.676199	−	0.736719i	$-0.736374\pi$
$673$	−35.0842	−1.35240	−0.676199	+	0.736719i	$0.736374\pi$
$674$	47.9886	1.84845
$675$	0	0
$676$	92.9267	3.57410
$677$	−42.1433	−1.61970	−0.809849	−	0.586638i	$-0.800451\pi$
$677$	−42.1433	−1.61970	−0.809849	+	0.586638i	$0.800451\pi$
$678$	0	0
$679$	37.1261	1.42477
$680$	0	0
$681$	0	0
$682$	41.9082	1.60475
$683$	31.8895	1.22022	0.610109	−	0.792318i	$-0.291126\pi$
$683$	31.8895	1.22022	0.610109	+	0.792318i	$0.291126\pi$
$684$	0	0
$685$	0	0
$686$	−92.1508	−3.51833
$687$	0	0
$688$	105.673	4.02876
$689$	6.29311	0.239748
$690$	0	0
$691$	11.2963	0.429732	0.214866	−	0.976644i	$-0.431069\pi$
$691$	11.2963	0.429732	0.214866	+	0.976644i	$0.431069\pi$
$692$	−12.7860	−0.486052
$693$	0	0
$694$	−57.5052	−2.18287
$695$	0	0
$696$	0	0
$697$	1.23561	0.0468022
$698$	1.97283	0.0746729
$699$	0	0
$700$	0	0
$701$	1.39354	0.0526332	0.0263166	−	0.999654i	$-0.491622\pi$
$701$	1.39354	0.0526332	0.0263166	+	0.999654i	$0.491622\pi$
$702$	0	0
$703$	24.0927	0.908672
$704$	54.3717	2.04921
$705$	0	0
$706$	16.2019	0.609767
$707$	56.8986	2.13989
$708$	0	0
$709$	5.24967	0.197156	0.0985778	−	0.995129i	$-0.468571\pi$
$709$	5.24967	0.197156	0.0985778	+	0.995129i	$0.468571\pi$
$710$	0	0
$711$	0	0
$712$	153.394	5.74867
$713$	−1.12102	−0.0419826
$714$	0	0
$715$	0	0
$716$	96.7752	3.61666
$717$	0	0
$718$	−6.30116	−0.235157
$719$	−46.4801	−1.73341	−0.866707	−	0.498817i	$-0.833768\pi$
$719$	−46.4801	−1.73341	−0.866707	+	0.498817i	$0.833768\pi$
$720$	0	0
$721$	−3.05140	−0.113640
$722$	−27.8515	−1.03653
$723$	0	0
$724$	−14.6878	−0.545866
$725$	0	0
$726$	0	0
$727$	43.3101	1.60628	0.803141	−	0.595790i	$-0.203161\pi$
$727$	43.3101	1.60628	0.803141	+	0.595790i	$0.203161\pi$
$728$	−237.786	−8.81295
$729$	0	0
$730$	0	0
$731$	41.1664	1.52259
$732$	0	0
$733$	18.6793	0.689936	0.344968	−	0.938614i	$-0.387890\pi$
$733$	18.6793	0.689936	0.344968	+	0.938614i	$0.387890\pi$
$734$	−62.5375	−2.30830
$735$	0	0
$736$	−2.93461	−0.108171
$737$	−12.2661	−0.451828
$738$	0	0
$739$	41.3256	1.52019	0.760093	−	0.649815i	$-0.225153\pi$
$739$	41.3256	1.52019	0.760093	+	0.649815i	$0.225153\pi$
$740$	0	0
$741$	0	0
$742$	−14.4538	−0.530616
$743$	−32.3390	−1.18640	−0.593202	−	0.805054i	$-0.702136\pi$
$743$	−32.3390	−1.18640	−0.593202	+	0.805054i	$0.702136\pi$
$744$	0	0
$745$	0	0
$746$	7.72547	0.282849
$747$	0	0
$748$	58.6612	2.14487
$749$	67.3464	2.46078
$750$	0	0
$751$	−9.34454	−0.340987	−0.170494	−	0.985359i	$-0.554536\pi$
$751$	−9.34454	−0.340987	−0.170494	+	0.985359i	$0.554536\pi$
$752$	146.076	5.32685
$753$	0	0
$754$	14.9609	0.544843
$755$	0	0
$756$	0	0
$757$	5.01932	0.182430	0.0912151	−	0.995831i	$-0.470925\pi$
$757$	5.01932	0.182430	0.0912151	+	0.995831i	$0.470925\pi$
$758$	79.8074	2.89873
$759$	0	0
$760$	0	0
$761$	34.9103	1.26550	0.632748	−	0.774358i	$-0.281927\pi$
$761$	34.9103	1.26550	0.632748	+	0.774358i	$0.281927\pi$
$762$	0	0
$763$	71.5859	2.59159
$764$	−58.7777	−2.12650
$765$	0	0
$766$	−8.15872	−0.294787
$767$	1.34310	0.0484965
$768$	0	0
$769$	37.4889	1.35189	0.675943	−	0.736954i	$-0.263737\pi$
$769$	37.4889	1.35189	0.675943	+	0.736954i	$0.263737\pi$
$770$	0	0
$771$	0	0
$772$	−22.7740	−0.819655
$773$	−33.7332	−1.21330	−0.606649	−	0.794970i	$-0.707487\pi$
$773$	−33.7332	−1.21330	−0.606649	+	0.794970i	$0.707487\pi$
$774$	0	0
$775$	0	0
$776$	−75.5197	−2.71100
$777$	0	0
$778$	−23.3890	−0.838538
$779$	−1.16329	−0.0416792
$780$	0	0
$781$	0.520862	0.0186379
$782$	−2.14578	−0.0767330
$783$	0	0
$784$	210.962	7.53436
$785$	0	0
$786$	0	0
$787$	−11.5508	−0.411740	−0.205870	−	0.978579i	$-0.566002\pi$
$787$	−11.5508	−0.411740	−0.205870	+	0.978579i	$0.566002\pi$
$788$	−31.3955	−1.11842
$789$	0	0
$790$	0	0
$791$	36.0426	1.28153
$792$	0	0
$793$	48.0006	1.70455
$794$	−25.3605	−0.900012
$795$	0	0
$796$	−29.6668	−1.05151
$797$	9.11795	0.322974	0.161487	−	0.986875i	$-0.448371\pi$
$797$	9.11795	0.322974	0.161487	+	0.986875i	$0.448371\pi$
$798$	0	0
$799$	56.9058	2.01318
$800$	0	0
$801$	0	0
$802$	53.3047	1.88226
$803$	−21.5795	−0.761525
$804$	0	0
$805$	0	0
$806$	122.405	4.31153
$807$	0	0
$808$	−115.740	−4.07171
$809$	11.9809	0.421227	0.210613	−	0.977569i	$-0.432454\pi$
$809$	11.9809	0.421227	0.210613	+	0.977569i	$0.432454\pi$
$810$	0	0
$811$	−19.2172	−0.674807	−0.337403	−	0.941360i	$-0.609549\pi$
$811$	−19.2172	−0.674807	−0.337403	+	0.941360i	$0.609549\pi$
$812$	−25.1278	−0.881812
$813$	0	0
$814$	22.8340	0.800332
$815$	0	0
$816$	0	0
$817$	−38.7568	−1.35593
$818$	−103.554	−3.62067
$819$	0	0
$820$	0	0
$821$	10.5611	0.368585	0.184292	−	0.982871i	$-0.441001\pi$
$821$	10.5611	0.368585	0.184292	+	0.982871i	$0.441001\pi$
$822$	0	0
$823$	29.1655	1.01665	0.508323	−	0.861166i	$-0.330266\pi$
$823$	29.1655	1.01665	0.508323	+	0.861166i	$0.330266\pi$
$824$	6.20696	0.216230
$825$	0	0
$826$	−3.08479	−0.107334
$827$	40.5140	1.40881	0.704405	−	0.709798i	$-0.251214\pi$
$827$	40.5140	1.40881	0.704405	+	0.709798i	$0.251214\pi$
$828$	0	0
$829$	51.3925	1.78494	0.892468	−	0.451111i	$-0.148972\pi$
$829$	51.3925	1.78494	0.892468	+	0.451111i	$0.148972\pi$
$830$	0	0
$831$	0	0
$832$	158.808	5.50569
$833$	82.1829	2.84747
$834$	0	0
$835$	0	0
$836$	−55.2276	−1.91009
$837$	0	0
$838$	−27.5407	−0.951379
$839$	30.2352	1.04383	0.521917	−	0.852996i	$-0.325217\pi$
$839$	30.2352	1.04383	0.521917	+	0.852996i	$0.325217\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−37.7899	−1.30233
$843$	0	0
$844$	4.69869	0.161736
$845$	0	0
$846$	0	0
$847$	−34.5103	−1.18579
$848$	16.9100	0.580693
$849$	0	0
$850$	0	0
$851$	−0.610797	−0.0209379
$852$	0	0
$853$	−0.163336	−0.00559252	−0.00279626	−	0.999996i	$-0.500890\pi$
$853$	−0.163336	−0.00559252	−0.00279626	+	0.999996i	$0.500890\pi$
$854$	−110.246	−3.77255
$855$	0	0
$856$	−136.992	−4.68229
$857$	−0.869730	−0.0297094	−0.0148547	−	0.999890i	$-0.504729\pi$
$857$	−0.869730	−0.0297094	−0.0148547	+	0.999890i	$0.504729\pi$
$858$	0	0
$859$	−50.5362	−1.72427	−0.862136	−	0.506677i	$-0.830874\pi$
$859$	−50.5362	−1.72427	−0.862136	+	0.506677i	$0.830874\pi$
$860$	0	0
$861$	0	0
$862$	49.6920	1.69251
$863$	43.1139	1.46761	0.733807	−	0.679358i	$-0.237742\pi$
$863$	43.1139	1.46761	0.733807	+	0.679358i	$0.237742\pi$
$864$	0	0
$865$	0	0
$866$	−18.3731	−0.624342
$867$	0	0
$868$	−205.587	−6.97809
$869$	3.12096	0.105871
$870$	0	0
$871$	−35.8267	−1.21394
$872$	−145.616	−4.93117
$873$	0	0
$874$	2.02018	0.0683337
$875$	0	0
$876$	0	0
$877$	−3.23901	−0.109374	−0.0546868	−	0.998504i	$-0.517416\pi$
$877$	−3.23901	−0.109374	−0.0546868	+	0.998504i	$0.517416\pi$
$878$	−84.4590	−2.85035
$879$	0	0
$880$	0	0
$881$	−9.49218	−0.319800	−0.159900	−	0.987133i	$-0.551117\pi$
$881$	−9.49218	−0.319800	−0.159900	+	0.987133i	$0.551117\pi$
$882$	0	0
$883$	−40.4268	−1.36047	−0.680235	−	0.732994i	$-0.738122\pi$
$883$	−40.4268	−1.36047	−0.680235	+	0.732994i	$0.738122\pi$
$884$	171.337	5.76268
$885$	0	0
$886$	58.4991	1.96532
$887$	−17.5249	−0.588430	−0.294215	−	0.955739i	$-0.595058\pi$
$887$	−17.5249	−0.588430	−0.294215	+	0.955739i	$0.595058\pi$
$888$	0	0
$889$	91.0026	3.05213
$890$	0	0
$891$	0	0
$892$	−3.34822	−0.112107
$893$	−53.5749	−1.79282
$894$	0	0
$895$	0	0
$896$	−166.975	−5.57823
$897$	0	0
$898$	70.0833	2.33871
$899$	8.18167	0.272874
$900$	0	0
$901$	6.58752	0.219462
$902$	−1.10252	−0.0367098
$903$	0	0
$904$	−73.3157	−2.43844
$905$	0	0
$906$	0	0
$907$	34.8878	1.15843	0.579216	−	0.815174i	$-0.303359\pi$
$907$	34.8878	1.15843	0.579216	+	0.815174i	$0.303359\pi$
$908$	129.113	4.28475
$909$	0	0
$910$	0	0
$911$	−43.5212	−1.44192	−0.720960	−	0.692976i	$-0.756299\pi$
$911$	−43.5212	−1.44192	−0.720960	+	0.692976i	$0.756299\pi$
$912$	0	0
$913$	−17.9968	−0.595608
$914$	−82.7373	−2.73671
$915$	0	0
$916$	−114.188	−3.77289
$917$	4.88185	0.161213
$918$	0	0
$919$	−38.2543	−1.26189	−0.630946	−	0.775827i	$-0.717333\pi$
$919$	−38.2543	−1.26189	−0.630946	+	0.775827i	$0.717333\pi$
$920$	0	0
$921$	0	0
$922$	−84.5780	−2.78543
$923$	1.52133	0.0500751
$924$	0	0
$925$	0	0
$926$	75.5655	2.48324
$927$	0	0
$928$	21.4180	0.703079
$929$	−40.3492	−1.32381	−0.661906	−	0.749586i	$-0.730252\pi$
$929$	−40.3492	−1.32381	−0.661906	+	0.749586i	$0.730252\pi$
$930$	0	0
$931$	−77.3725	−2.53578
$932$	43.2942	1.41815
$933$	0	0
$934$	−11.0009	−0.359959
$935$	0	0
$936$	0	0
$937$	−4.95384	−0.161835	−0.0809174	−	0.996721i	$-0.525785\pi$
$937$	−4.95384	−0.161835	−0.0809174	+	0.996721i	$0.525785\pi$
$938$	82.2857	2.68672
$939$	0	0
$940$	0	0
$941$	47.5220	1.54917	0.774587	−	0.632467i	$-0.217958\pi$
$941$	47.5220	1.54917	0.774587	+	0.632467i	$0.217958\pi$
$942$	0	0
$943$	0.0294917	0.000960383 0
$944$	3.60901	0.117463
$945$	0	0
$946$	−36.7321	−1.19426
$947$	37.1547	1.20736	0.603682	−	0.797225i	$-0.293700\pi$
$947$	37.1547	1.20736	0.603682	+	0.797225i	$0.293700\pi$
$948$	0	0
$949$	−63.0292	−2.04601
$950$	0	0
$951$	0	0
$952$	−248.911	−8.06725
$953$	−44.5351	−1.44263	−0.721316	−	0.692606i	$-0.756463\pi$
$953$	−44.5351	−1.44263	−0.721316	+	0.692606i	$0.756463\pi$
$954$	0	0
$955$	0	0
$956$	−95.1828	−3.07843
$957$	0	0
$958$	3.60616	0.116510
$959$	−14.8420	−0.479272
$960$	0	0
$961$	35.9398	1.15935
$962$	66.6933	2.15028
$963$	0	0
$964$	−80.4095	−2.58982
$965$	0	0
$966$	0	0
$967$	48.7291	1.56702	0.783511	−	0.621379i	$-0.213427\pi$
$967$	48.7291	1.56702	0.783511	+	0.621379i	$0.213427\pi$
$968$	70.1988	2.25627
$969$	0	0
$970$	0	0
$971$	51.7448	1.66057	0.830285	−	0.557339i	$-0.188178\pi$
$971$	51.7448	1.66057	0.830285	+	0.557339i	$0.188178\pi$
$972$	0	0
$973$	39.1706	1.25575
$974$	26.9915	0.864864
$975$	0	0
$976$	128.981	4.12858
$977$	−37.7726	−1.20845	−0.604227	−	0.796813i	$-0.706518\pi$
$977$	−37.7726	−1.20845	−0.604227	+	0.796813i	$0.706518\pi$
$978$	0	0
$979$	−30.6669	−0.980117
$980$	0	0
$981$	0	0
$982$	46.7337	1.49133
$983$	21.1318	0.673999	0.336999	−	0.941505i	$-0.390588\pi$
$983$	21.1318	0.673999	0.336999	+	0.941505i	$0.390588\pi$
$984$	0	0
$985$	0	0
$986$	15.6608	0.498742
$987$	0	0
$988$	−161.308	−5.13189
$989$	0.982563	0.0312437
$990$	0	0
$991$	11.0241	0.350191	0.175096	−	0.984551i	$-0.443977\pi$
$991$	11.0241	0.350191	0.175096	+	0.984551i	$0.443977\pi$
$992$	175.235	5.56371
$993$	0	0
$994$	−3.49414	−0.110827
$995$	0	0
$996$	0	0
$997$	28.2785	0.895589	0.447795	−	0.894136i	$-0.352210\pi$
$997$	28.2785	0.895589	0.447795	+	0.894136i	$0.352210\pi$
$998$	21.7014	0.686945
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bz.1.1		8
3.2	odd	2		2175.2.a.bc.1.8	✓	8
5.4	even	2		6525.2.a.by.1.8		8
15.2	even	4		2175.2.c.p.349.16		16
15.8	even	4		2175.2.c.p.349.1		16
15.14	odd	2		2175.2.a.bd.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.8	✓	8	3.2	odd	2
2175.2.a.bd.1.1	yes	8	15.14	odd	2
2175.2.c.p.349.1		16	15.8	even	4
2175.2.c.p.349.16		16	15.2	even	4
6525.2.a.by.1.8		8	5.4	even	2
6525.2.a.bz.1.1		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.72810 q^{2} +5.44251 q^{4} +4.61695 q^{7} -9.39150 q^{8} +1.87758 q^{11} +5.48400 q^{13} -12.5955 q^{14} +14.7359 q^{16} +5.74056 q^{17} -5.40455 q^{19} -5.12221 q^{22} +0.137016 q^{23} -14.9609 q^{26} +25.1278 q^{28} -1.00000 q^{29} -8.18167 q^{31} -21.4180 q^{32} -15.6608 q^{34} -4.45785 q^{37} +14.7441 q^{38} +0.215243 q^{41} +7.17115 q^{43} +10.2187 q^{44} -0.373793 q^{46} +9.91293 q^{47} +14.3162 q^{49} +29.8467 q^{52} +1.14754 q^{53} -43.3601 q^{56} +2.72810 q^{58} +0.244913 q^{59} +8.75284 q^{61} +22.3204 q^{62} +28.9585 q^{64} -6.53295 q^{67} +31.2431 q^{68} +0.277412 q^{71} -11.4933 q^{73} +12.1614 q^{74} -29.4143 q^{76} +8.66866 q^{77} +1.66223 q^{79} -0.587203 q^{82} -9.58514 q^{83} -19.5636 q^{86} -17.6333 q^{88} -16.3332 q^{89} +25.3193 q^{91} +0.745712 q^{92} -27.0434 q^{94} +8.04128 q^{97} -39.0559 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 12 q^{4} + 2 q^{7} + 3 q^{8} - 6 q^{11} + 6 q^{13} - 9 q^{14} + 32 q^{16} + 12 q^{17} + 3 q^{22} + 14 q^{23} - 18 q^{26} + 14 q^{28} - 8 q^{29} + 8 q^{31} - 2 q^{32} - 13 q^{34} + 4 q^{37}+ \cdots + 29 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 + 12 * q^4 + 2 * q^7 + 3 * q^8 - 6 * q^11 + 6 * q^13 - 9 * q^14 + 32 * q^16 + 12 * q^17 + 3 * q^22 + 14 * q^23 - 18 * q^26 + 14 * q^28 - 8 * q^29 + 8 * q^31 - 2 * q^32 - 13 * q^34 + 4 * q^37 + 26 * q^38 - 2 * q^41 + 2 * q^43 + 15 * q^44 + 24 * q^46 + 12 * q^47 + 38 * q^49 + 49 * q^52 + 4 * q^53 - 58 * q^56 - 2 * q^58 - 18 * q^59 + 12 * q^61 - 4 * q^62 + 21 * q^64 + 26 * q^67 + 81 * q^68 - 24 * q^71 - 14 * q^73 + 22 * q^74 - 26 * q^77 + 10 * q^79 + 48 * q^82 + 40 * q^83 - 8 * q^86 - 10 * q^88 - 34 * q^89 + 26 * q^91 - 18 * q^92 - 43 * q^94 + 30 * q^97 + 29 * q^98

Introduction

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