LMFDB - Embedded newform 9675.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9675,2,Mod(1,9675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9675, 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("9675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9675 = 3^{2} \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9675.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:	$77.2552639556$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +0.585786 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +0.585786 q^{7} -2.82843 q^{8} -1.82843 q^{11} -3.82843 q^{13} +0.828427 q^{14} -4.00000 q^{16} +7.82843 q^{17} -4.82843 q^{19} -2.58579 q^{22} -4.65685 q^{23} -5.41421 q^{26} -4.24264 q^{29} -3.00000 q^{31} +11.0711 q^{34} +8.48528 q^{37} -6.82843 q^{38} +3.82843 q^{41} -1.00000 q^{43} -6.58579 q^{46} +6.00000 q^{47} -6.65685 q^{49} +8.17157 q^{53} -1.65685 q^{56} -6.00000 q^{58} -0.828427 q^{59} +8.24264 q^{61} -4.24264 q^{62} +8.00000 q^{64} -9.48528 q^{67} +8.82843 q^{71} +7.75736 q^{73} +12.0000 q^{74} -1.07107 q^{77} -0.828427 q^{79} +5.41421 q^{82} +14.6569 q^{83} -1.41421 q^{86} +5.17157 q^{88} +10.2426 q^{89} -2.24264 q^{91} +8.48528 q^{94} +3.82843 q^{97} -9.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7}+O(q^{10}) $ 2 * q + 4 * q^7	$ 2 q + 4 q^{7} + 2 q^{11} - 2 q^{13} - 4 q^{14} - 8 q^{16} + 10 q^{17} - 4 q^{19} - 8 q^{22} + 2 q^{23} - 8 q^{26} - 6 q^{31} + 8 q^{34} - 8 q^{38} + 2 q^{41} - 2 q^{43} - 16 q^{46} + 12 q^{47} - 2 q^{49} + 22 q^{53} + 8 q^{56} - 12 q^{58} + 4 q^{59} + 8 q^{61} + 16 q^{64} - 2 q^{67} + 12 q^{71} + 24 q^{73} + 24 q^{74} + 12 q^{77} + 4 q^{79} + 8 q^{82} + 18 q^{83} + 16 q^{88} + 12 q^{89} + 4 q^{91} + 2 q^{97} - 16 q^{98}+O(q^{100}) $ 2 * q + 4 * q^7 + 2 * q^11 - 2 * q^13 - 4 * q^14 - 8 * q^16 + 10 * q^17 - 4 * q^19 - 8 * q^22 + 2 * q^23 - 8 * q^26 - 6 * q^31 + 8 * q^34 - 8 * q^38 + 2 * q^41 - 2 * q^43 - 16 * q^46 + 12 * q^47 - 2 * q^49 + 22 * q^53 + 8 * q^56 - 12 * q^58 + 4 * q^59 + 8 * q^61 + 16 * q^64 - 2 * q^67 + 12 * q^71 + 24 * q^73 + 24 * q^74 + 12 * q^77 + 4 * q^79 + 8 * q^82 + 18 * q^83 + 16 * q^88 + 12 * q^89 + 4 * q^91 + 2 * q^97 - 16 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	0	0
$11$	−1.82843	−0.551292	−0.275646	−	0.961259i	$-0.588892\pi$
$11$	−1.82843	−0.551292	−0.275646	+	0.961259i	$0.588892\pi$
$12$	0	0
$13$	−3.82843	−1.06181	−0.530907	−	0.847430i	$-0.678149\pi$
$13$	−3.82843	−1.06181	−0.530907	+	0.847430i	$0.678149\pi$
$14$	0.828427	0.221406
$15$	0	0
$16$	−4.00000	−1.00000
$17$	7.82843	1.89867	0.949336	−	0.314262i	$-0.101757\pi$
$17$	7.82843	1.89867	0.949336	+	0.314262i	$0.101757\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	0	0
$22$	−2.58579	−0.551292
$23$	−4.65685	−0.971021	−0.485511	−	0.874231i	$-0.661366\pi$
$23$	−4.65685	−0.971021	−0.485511	+	0.874231i	$0.661366\pi$
$24$	0	0
$25$	0	0
$26$	−5.41421	−1.06181
$27$	0	0
$28$	0	0
$29$	−4.24264	−0.787839	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	−4.24264	−0.787839	−0.393919	+	0.919145i	$0.628881\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	0	0
$34$	11.0711	1.89867
$35$	0	0
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	−6.82843	−1.10772
$39$	0	0
$40$	0	0
$41$	3.82843	0.597900	0.298950	−	0.954269i	$-0.403364\pi$
$41$	3.82843	0.597900	0.298950	+	0.954269i	$0.403364\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	0	0
$46$	−6.58579	−0.971021
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.17157	1.12245	0.561226	−	0.827663i	$-0.310330\pi$
$53$	8.17157	1.12245	0.561226	+	0.827663i	$0.310330\pi$
$54$	0	0
$55$	0	0
$56$	−1.65685	−0.221406
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−0.828427	−0.107852	−0.0539260	−	0.998545i	$-0.517174\pi$
$59$	−0.828427	−0.107852	−0.0539260	+	0.998545i	$0.517174\pi$
$60$	0	0
$61$	8.24264	1.05536	0.527681	−	0.849443i	$-0.323062\pi$
$61$	8.24264	1.05536	0.527681	+	0.849443i	$0.323062\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	−9.48528	−1.15881	−0.579406	−	0.815039i	$-0.696715\pi$
$67$	−9.48528	−1.15881	−0.579406	+	0.815039i	$0.696715\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.82843	1.04774	0.523871	−	0.851798i	$-0.324487\pi$
$71$	8.82843	1.04774	0.523871	+	0.851798i	$0.324487\pi$
$72$	0	0
$73$	7.75736	0.907930	0.453965	−	0.891019i	$-0.350009\pi$
$73$	7.75736	0.907930	0.453965	+	0.891019i	$0.350009\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	0	0
$77$	−1.07107	−0.122060
$78$	0	0
$79$	−0.828427	−0.0932053	−0.0466027	−	0.998914i	$-0.514839\pi$
$79$	−0.828427	−0.0932053	−0.0466027	+	0.998914i	$0.514839\pi$
$80$	0	0
$81$	0	0
$82$	5.41421	0.597900
$83$	14.6569	1.60880	0.804399	−	0.594089i	$-0.202487\pi$
$83$	14.6569	1.60880	0.804399	+	0.594089i	$0.202487\pi$
$84$	0	0
$85$	0	0
$86$	−1.41421	−0.152499
$87$	0	0
$88$	5.17157	0.551292
$89$	10.2426	1.08572	0.542859	−	0.839824i	$-0.317342\pi$
$89$	10.2426	1.08572	0.542859	+	0.839824i	$0.317342\pi$
$90$	0	0
$91$	−2.24264	−0.235093
$92$	0	0
$93$	0	0
$94$	8.48528	0.875190
$95$	0	0
$96$	0	0
$97$	3.82843	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$97$	3.82843	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$98$	−9.41421	−0.950979
$99$	0	0
$100$	0	0
$101$	−0.171573	−0.0170721	−0.00853607	−	0.999964i	$-0.502717\pi$
$101$	−0.171573	−0.0170721	−0.00853607	+	0.999964i	$0.502717\pi$
$102$	0	0
$103$	−17.4853	−1.72288	−0.861438	−	0.507863i	$-0.830436\pi$
$103$	−17.4853	−1.72288	−0.861438	+	0.507863i	$0.830436\pi$
$104$	10.8284	1.06181
$105$	0	0
$106$	11.5563	1.12245
$107$	−11.6569	−1.12691	−0.563455	−	0.826147i	$-0.690528\pi$
$107$	−11.6569	−1.12691	−0.563455	+	0.826147i	$0.690528\pi$
$108$	0	0
$109$	13.9706	1.33814	0.669069	−	0.743201i	$-0.266693\pi$
$109$	13.9706	1.33814	0.669069	+	0.743201i	$0.266693\pi$
$110$	0	0
$111$	0	0
$112$	−2.34315	−0.221406
$113$	−1.17157	−0.110212	−0.0551062	−	0.998481i	$-0.517550\pi$
$113$	−1.17157	−0.110212	−0.0551062	+	0.998481i	$0.517550\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−1.17157	−0.107852
$119$	4.58579	0.420378
$120$	0	0
$121$	−7.65685	−0.696078
$122$	11.6569	1.05536
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.82843	0.162247	0.0811233	−	0.996704i	$-0.474149\pi$
$127$	1.82843	0.162247	0.0811233	+	0.996704i	$0.474149\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	0	0
$131$	1.65685	0.144760	0.0723800	−	0.997377i	$-0.476941\pi$
$131$	1.65685	0.144760	0.0723800	+	0.997377i	$0.476941\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	−13.4142	−1.15881
$135$	0	0
$136$	−22.1421	−1.89867
$137$	2.48528	0.212332	0.106166	−	0.994348i	$-0.466143\pi$
$137$	2.48528	0.212332	0.106166	+	0.994348i	$0.466143\pi$
$138$	0	0
$139$	−11.4853	−0.974169	−0.487084	−	0.873355i	$-0.661940\pi$
$139$	−11.4853	−0.974169	−0.487084	+	0.873355i	$0.661940\pi$
$140$	0	0
$141$	0	0
$142$	12.4853	1.04774
$143$	7.00000	0.585369
$144$	0	0
$145$	0	0
$146$	10.9706	0.907930
$147$	0	0
$148$	0	0
$149$	8.48528	0.695141	0.347571	−	0.937654i	$-0.387007\pi$
$149$	8.48528	0.695141	0.347571	+	0.937654i	$0.387007\pi$
$150$	0	0
$151$	9.75736	0.794043	0.397021	−	0.917809i	$-0.370044\pi$
$151$	9.75736	0.794043	0.397021	+	0.917809i	$0.370044\pi$
$152$	13.6569	1.10772
$153$	0	0
$154$	−1.51472	−0.122060
$155$	0	0
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−1.17157	−0.0932053
$159$	0	0
$160$	0	0
$161$	−2.72792	−0.214990
$162$	0	0
$163$	20.2426	1.58553	0.792763	−	0.609530i	$-0.208642\pi$
$163$	20.2426	1.58553	0.792763	+	0.609530i	$0.208642\pi$
$164$	0	0
$165$	0	0
$166$	20.7279	1.60880
$167$	8.31371	0.643334	0.321667	−	0.946853i	$-0.395757\pi$
$167$	8.31371	0.643334	0.321667	+	0.946853i	$0.395757\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.3431	0.938432	0.469216	−	0.883083i	$-0.344537\pi$
$173$	12.3431	0.938432	0.469216	+	0.883083i	$0.344537\pi$
$174$	0	0
$175$	0	0
$176$	7.31371	0.551292
$177$	0	0
$178$	14.4853	1.08572
$179$	7.41421	0.554164	0.277082	−	0.960846i	$-0.410633\pi$
$179$	7.41421	0.554164	0.277082	+	0.960846i	$0.410633\pi$
$180$	0	0
$181$	−15.3137	−1.13826	−0.569129	−	0.822248i	$-0.692720\pi$
$181$	−15.3137	−1.13826	−0.569129	+	0.822248i	$0.692720\pi$
$182$	−3.17157	−0.235093
$183$	0	0
$184$	13.1716	0.971021
$185$	0	0
$186$	0	0
$187$	−14.3137	−1.04672
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.1421	−1.60215	−0.801074	−	0.598565i	$-0.795738\pi$
$191$	−22.1421	−1.60215	−0.801074	+	0.598565i	$0.795738\pi$
$192$	0	0
$193$	17.9706	1.29355	0.646775	−	0.762681i	$-0.276117\pi$
$193$	17.9706	1.29355	0.646775	+	0.762681i	$0.276117\pi$
$194$	5.41421	0.388718
$195$	0	0
$196$	0	0
$197$	14.1421	1.00759	0.503793	−	0.863825i	$-0.331938\pi$
$197$	14.1421	1.00759	0.503793	+	0.863825i	$0.331938\pi$
$198$	0	0
$199$	−3.65685	−0.259228	−0.129614	−	0.991565i	$-0.541374\pi$
$199$	−3.65685	−0.259228	−0.129614	+	0.991565i	$0.541374\pi$
$200$	0	0
$201$	0	0
$202$	−0.242641	−0.0170721
$203$	−2.48528	−0.174433
$204$	0	0
$205$	0	0
$206$	−24.7279	−1.72288
$207$	0	0
$208$	15.3137	1.06181
$209$	8.82843	0.610675
$210$	0	0
$211$	28.1421	1.93738	0.968692	−	0.248265i	$-0.0798602\pi$
$211$	28.1421	1.93738	0.968692	+	0.248265i	$0.0798602\pi$
$212$	0	0
$213$	0	0
$214$	−16.4853	−1.12691
$215$	0	0
$216$	0	0
$217$	−1.75736	−0.119297
$218$	19.7574	1.33814
$219$	0	0
$220$	0	0
$221$	−29.9706	−2.01604
$222$	0	0
$223$	−23.8995	−1.60043	−0.800214	−	0.599714i	$-0.795281\pi$
$223$	−23.8995	−1.60043	−0.800214	+	0.599714i	$0.795281\pi$
$224$	0	0
$225$	0	0
$226$	−1.65685	−0.110212
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	7.97056	0.526710	0.263355	−	0.964699i	$-0.415171\pi$
$229$	7.97056	0.526710	0.263355	+	0.964699i	$0.415171\pi$
$230$	0	0
$231$	0	0
$232$	12.0000	0.787839
$233$	−0.828427	−0.0542721	−0.0271360	−	0.999632i	$-0.508639\pi$
$233$	−0.828427	−0.0542721	−0.0271360	+	0.999632i	$0.508639\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	6.48528	0.420378
$239$	2.48528	0.160759	0.0803797	−	0.996764i	$-0.474387\pi$
$239$	2.48528	0.160759	0.0803797	+	0.996764i	$0.474387\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−10.8284	−0.696078
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.4853	1.17619
$248$	8.48528	0.538816
$249$	0	0
$250$	0	0
$251$	5.14214	0.324569	0.162284	−	0.986744i	$-0.448114\pi$
$251$	5.14214	0.324569	0.162284	+	0.986744i	$0.448114\pi$
$252$	0	0
$253$	8.51472	0.535316
$254$	2.58579	0.162247
$255$	0	0
$256$	0	0
$257$	−10.2426	−0.638918	−0.319459	−	0.947600i	$-0.603501\pi$
$257$	−10.2426	−0.638918	−0.319459	+	0.947600i	$0.603501\pi$
$258$	0	0
$259$	4.97056	0.308856
$260$	0	0
$261$	0	0
$262$	2.34315	0.144760
$263$	−28.1421	−1.73532	−0.867659	−	0.497159i	$-0.834376\pi$
$263$	−28.1421	−1.73532	−0.867659	+	0.497159i	$0.834376\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	7.97056	0.484177	0.242089	−	0.970254i	$-0.422168\pi$
$271$	7.97056	0.484177	0.242089	+	0.970254i	$0.422168\pi$
$272$	−31.3137	−1.89867
$273$	0	0
$274$	3.51472	0.212332
$275$	0	0
$276$	0	0
$277$	11.8995	0.714971	0.357486	−	0.933919i	$-0.383634\pi$
$277$	11.8995	0.714971	0.357486	+	0.933919i	$0.383634\pi$
$278$	−16.2426	−0.974169
$279$	0	0
$280$	0	0
$281$	−2.65685	−0.158495	−0.0792473	−	0.996855i	$-0.525252\pi$
$281$	−2.65685	−0.158495	−0.0792473	+	0.996855i	$0.525252\pi$
$282$	0	0
$283$	4.31371	0.256423	0.128212	−	0.991747i	$-0.459076\pi$
$283$	4.31371	0.256423	0.128212	+	0.991747i	$0.459076\pi$
$284$	0	0
$285$	0	0
$286$	9.89949	0.585369
$287$	2.24264	0.132379
$288$	0	0
$289$	44.2843	2.60496
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.6569	1.26521	0.632603	−	0.774476i	$-0.281986\pi$
$293$	21.6569	1.26521	0.632603	+	0.774476i	$0.281986\pi$
$294$	0	0
$295$	0	0
$296$	−24.0000	−1.39497
$297$	0	0
$298$	12.0000	0.695141
$299$	17.8284	1.03104
$300$	0	0
$301$	−0.585786	−0.0337642
$302$	13.7990	0.794043
$303$	0	0
$304$	19.3137	1.10772
$305$	0	0
$306$	0	0
$307$	12.7990	0.730477	0.365238	−	0.930914i	$-0.380987\pi$
$307$	12.7990	0.730477	0.365238	+	0.930914i	$0.380987\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.9706	−1.13243	−0.566213	−	0.824259i	$-0.691592\pi$
$311$	−19.9706	−1.13243	−0.566213	+	0.824259i	$0.691592\pi$
$312$	0	0
$313$	−17.2132	−0.972948	−0.486474	−	0.873695i	$-0.661717\pi$
$313$	−17.2132	−0.972948	−0.486474	+	0.873695i	$0.661717\pi$
$314$	14.1421	0.798087
$315$	0	0
$316$	0	0
$317$	20.1716	1.13295	0.566474	−	0.824079i	$-0.308307\pi$
$317$	20.1716	1.13295	0.566474	+	0.824079i	$0.308307\pi$
$318$	0	0
$319$	7.75736	0.434329
$320$	0	0
$321$	0	0
$322$	−3.85786	−0.214990
$323$	−37.7990	−2.10319
$324$	0	0
$325$	0	0
$326$	28.6274	1.58553
$327$	0	0
$328$	−10.8284	−0.597900
$329$	3.51472	0.193773
$330$	0	0
$331$	18.5858	1.02157	0.510784	−	0.859709i	$-0.329355\pi$
$331$	18.5858	1.02157	0.510784	+	0.859709i	$0.329355\pi$
$332$	0	0
$333$	0	0
$334$	11.7574	0.643334
$335$	0	0
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	2.34315	0.127450
$339$	0	0
$340$	0	0
$341$	5.48528	0.297045
$342$	0	0
$343$	−8.00000	−0.431959
$344$	2.82843	0.152499
$345$	0	0
$346$	17.4558	0.938432
$347$	−7.41421	−0.398016	−0.199008	−	0.979998i	$-0.563772\pi$
$347$	−7.41421	−0.398016	−0.199008	+	0.979998i	$0.563772\pi$
$348$	0	0
$349$	−11.7574	−0.629357	−0.314679	−	0.949198i	$-0.601897\pi$
$349$	−11.7574	−0.629357	−0.314679	+	0.949198i	$0.601897\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.171573	0.00913190	0.00456595	−	0.999990i	$-0.498547\pi$
$353$	0.171573	0.00913190	0.00456595	+	0.999990i	$0.498547\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	10.4853	0.554164
$359$	32.6569	1.72356	0.861781	−	0.507280i	$-0.169349\pi$
$359$	32.6569	1.72356	0.861781	+	0.507280i	$0.169349\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	−21.6569	−1.13826
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	29.7990	1.55549	0.777747	−	0.628577i	$-0.216362\pi$
$367$	29.7990	1.55549	0.777747	+	0.628577i	$0.216362\pi$
$368$	18.6274	0.971021
$369$	0	0
$370$	0	0
$371$	4.78680	0.248518
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	−20.2426	−1.04672
$375$	0	0
$376$	−16.9706	−0.875190
$377$	16.2426	0.836539
$378$	0	0
$379$	7.68629	0.394818	0.197409	−	0.980321i	$-0.436747\pi$
$379$	7.68629	0.394818	0.197409	+	0.980321i	$0.436747\pi$
$380$	0	0
$381$	0	0
$382$	−31.3137	−1.60215
$383$	−3.51472	−0.179594	−0.0897969	−	0.995960i	$-0.528622\pi$
$383$	−3.51472	−0.179594	−0.0897969	+	0.995960i	$0.528622\pi$
$384$	0	0
$385$	0	0
$386$	25.4142	1.29355
$387$	0	0
$388$	0	0
$389$	−28.6274	−1.45147	−0.725734	−	0.687976i	$-0.758500\pi$
$389$	−28.6274	−1.45147	−0.725734	+	0.687976i	$0.758500\pi$
$390$	0	0
$391$	−36.4558	−1.84365
$392$	18.8284	0.950979
$393$	0	0
$394$	20.0000	1.00759
$395$	0	0
$396$	0	0
$397$	21.4558	1.07684	0.538419	−	0.842677i	$-0.319022\pi$
$397$	21.4558	1.07684	0.538419	+	0.842677i	$0.319022\pi$
$398$	−5.17157	−0.259228
$399$	0	0
$400$	0	0
$401$	29.4853	1.47242	0.736212	−	0.676751i	$-0.236612\pi$
$401$	29.4853	1.47242	0.736212	+	0.676751i	$0.236612\pi$
$402$	0	0
$403$	11.4853	0.572123
$404$	0	0
$405$	0	0
$406$	−3.51472	−0.174433
$407$	−15.5147	−0.769036
$408$	0	0
$409$	36.8701	1.82311	0.911554	−	0.411181i	$-0.134884\pi$
$409$	36.8701	1.82311	0.911554	+	0.411181i	$0.134884\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.485281	−0.0238791
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	12.4853	0.610675
$419$	4.10051	0.200323	0.100161	−	0.994971i	$-0.468064\pi$
$419$	4.10051	0.200323	0.100161	+	0.994971i	$0.468064\pi$
$420$	0	0
$421$	−4.34315	−0.211672	−0.105836	−	0.994384i	$-0.533752\pi$
$421$	−4.34315	−0.211672	−0.105836	+	0.994384i	$0.533752\pi$
$422$	39.7990	1.93738
$423$	0	0
$424$	−23.1127	−1.12245
$425$	0	0
$426$	0	0
$427$	4.82843	0.233664
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.2843	1.60325	0.801623	−	0.597829i	$-0.203970\pi$
$431$	33.2843	1.60325	0.801623	+	0.597829i	$0.203970\pi$
$432$	0	0
$433$	−30.2426	−1.45337	−0.726684	−	0.686972i	$-0.758940\pi$
$433$	−30.2426	−1.45337	−0.726684	+	0.686972i	$0.758940\pi$
$434$	−2.48528	−0.119297
$435$	0	0
$436$	0	0
$437$	22.4853	1.07562
$438$	0	0
$439$	−5.48528	−0.261798	−0.130899	−	0.991396i	$-0.541786\pi$
$439$	−5.48528	−0.261798	−0.130899	+	0.991396i	$0.541786\pi$
$440$	0	0
$441$	0	0
$442$	−42.3848	−2.01604
$443$	36.1421	1.71716	0.858582	−	0.512676i	$-0.171346\pi$
$443$	36.1421	1.71716	0.858582	+	0.512676i	$0.171346\pi$
$444$	0	0
$445$	0	0
$446$	−33.7990	−1.60043
$447$	0	0
$448$	4.68629	0.221406
$449$	9.21320	0.434798	0.217399	−	0.976083i	$-0.430243\pi$
$449$	9.21320	0.434798	0.217399	+	0.976083i	$0.430243\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	0	0
$453$	0	0
$454$	−24.9706	−1.17193
$455$	0	0
$456$	0	0
$457$	24.7279	1.15672	0.578362	−	0.815780i	$-0.303692\pi$
$457$	24.7279	1.15672	0.578362	+	0.815780i	$0.303692\pi$
$458$	11.2721	0.526710
$459$	0	0
$460$	0	0
$461$	2.62742	0.122371	0.0611855	−	0.998126i	$-0.480512\pi$
$461$	2.62742	0.122371	0.0611855	+	0.998126i	$0.480512\pi$
$462$	0	0
$463$	5.27208	0.245014	0.122507	−	0.992468i	$-0.460907\pi$
$463$	5.27208	0.245014	0.122507	+	0.992468i	$0.460907\pi$
$464$	16.9706	0.787839
$465$	0	0
$466$	−1.17157	−0.0542721
$467$	12.3431	0.571173	0.285586	−	0.958353i	$-0.407812\pi$
$467$	12.3431	0.571173	0.285586	+	0.958353i	$0.407812\pi$
$468$	0	0
$469$	−5.55635	−0.256568
$470$	0	0
$471$	0	0
$472$	2.34315	0.107852
$473$	1.82843	0.0840712
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	3.51472	0.160759
$479$	−4.65685	−0.212777	−0.106389	−	0.994325i	$-0.533929\pi$
$479$	−4.65685	−0.212777	−0.106389	+	0.994325i	$0.533929\pi$
$480$	0	0
$481$	−32.4853	−1.48120
$482$	5.65685	0.257663
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−33.6569	−1.52514	−0.762569	−	0.646907i	$-0.776062\pi$
$487$	−33.6569	−1.52514	−0.762569	+	0.646907i	$0.776062\pi$
$488$	−23.3137	−1.05536
$489$	0	0
$490$	0	0
$491$	−10.5858	−0.477730	−0.238865	−	0.971053i	$-0.576775\pi$
$491$	−10.5858	−0.477730	−0.238865	+	0.971053i	$0.576775\pi$
$492$	0	0
$493$	−33.2132	−1.49585
$494$	26.1421	1.17619
$495$	0	0
$496$	12.0000	0.538816
$497$	5.17157	0.231977
$498$	0	0
$499$	−34.2426	−1.53291	−0.766456	−	0.642297i	$-0.777981\pi$
$499$	−34.2426	−1.53291	−0.766456	+	0.642297i	$0.777981\pi$
$500$	0	0
$501$	0	0
$502$	7.27208	0.324569
$503$	42.7696	1.90700	0.953500	−	0.301393i	$-0.0974516\pi$
$503$	42.7696	1.90700	0.953500	+	0.301393i	$0.0974516\pi$
$504$	0	0
$505$	0	0
$506$	12.0416	0.535316
$507$	0	0
$508$	0	0
$509$	−17.4853	−0.775021	−0.387511	−	0.921865i	$-0.626665\pi$
$509$	−17.4853	−0.775021	−0.387511	+	0.921865i	$0.626665\pi$
$510$	0	0
$511$	4.54416	0.201022
$512$	−22.6274	−1.00000
$513$	0	0
$514$	−14.4853	−0.638918
$515$	0	0
$516$	0	0
$517$	−10.9706	−0.482485
$518$	7.02944	0.308856
$519$	0	0
$520$	0	0
$521$	−31.0711	−1.36125	−0.680624	−	0.732633i	$-0.738291\pi$
$521$	−31.0711	−1.36125	−0.680624	+	0.732633i	$0.738291\pi$
$522$	0	0
$523$	−3.21320	−0.140504	−0.0702518	−	0.997529i	$-0.522380\pi$
$523$	−3.21320	−0.140504	−0.0702518	+	0.997529i	$0.522380\pi$
$524$	0	0
$525$	0	0
$526$	−39.7990	−1.73532
$527$	−23.4853	−1.02303
$528$	0	0
$529$	−1.31371	−0.0571178
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14.6569	−0.634859
$534$	0	0
$535$	0	0
$536$	26.8284	1.15881
$537$	0	0
$538$	4.24264	0.182913
$539$	12.1716	0.524267
$540$	0	0
$541$	−30.7990	−1.32415	−0.662076	−	0.749437i	$-0.730324\pi$
$541$	−30.7990	−1.32415	−0.662076	+	0.749437i	$0.730324\pi$
$542$	11.2721	0.484177
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.00000	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$547$	−9.00000	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.4853	0.872702
$552$	0	0
$553$	−0.485281	−0.0206363
$554$	16.8284	0.714971
$555$	0	0
$556$	0	0
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	0	0
$559$	3.82843	0.161925
$560$	0	0
$561$	0	0
$562$	−3.75736	−0.158495
$563$	−5.62742	−0.237167	−0.118584	−	0.992944i	$-0.537835\pi$
$563$	−5.62742	−0.237167	−0.118584	+	0.992944i	$0.537835\pi$
$564$	0	0
$565$	0	0
$566$	6.10051	0.256423
$567$	0	0
$568$	−24.9706	−1.04774
$569$	−24.6569	−1.03367	−0.516835	−	0.856085i	$-0.672890\pi$
$569$	−24.6569	−1.03367	−0.516835	+	0.856085i	$0.672890\pi$
$570$	0	0
$571$	11.0711	0.463310	0.231655	−	0.972798i	$-0.425586\pi$
$571$	11.0711	0.463310	0.231655	+	0.972798i	$0.425586\pi$
$572$	0	0
$573$	0	0
$574$	3.17157	0.132379
$575$	0	0
$576$	0	0
$577$	−18.9289	−0.788022	−0.394011	−	0.919106i	$-0.628913\pi$
$577$	−18.9289	−0.788022	−0.394011	+	0.919106i	$0.628913\pi$
$578$	62.6274	2.60496
$579$	0	0
$580$	0	0
$581$	8.58579	0.356198
$582$	0	0
$583$	−14.9411	−0.618798
$584$	−21.9411	−0.907930
$585$	0	0
$586$	30.6274	1.26521
$587$	−5.79899	−0.239350	−0.119675	−	0.992813i	$-0.538185\pi$
$587$	−5.79899	−0.239350	−0.119675	+	0.992813i	$0.538185\pi$
$588$	0	0
$589$	14.4853	0.596856
$590$	0	0
$591$	0	0
$592$	−33.9411	−1.39497
$593$	5.07107	0.208244	0.104122	−	0.994565i	$-0.466797\pi$
$593$	5.07107	0.208244	0.104122	+	0.994565i	$0.466797\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	25.2132	1.03104
$599$	−25.3431	−1.03549	−0.517746	−	0.855534i	$-0.673229\pi$
$599$	−25.3431	−1.03549	−0.517746	+	0.855534i	$0.673229\pi$
$600$	0	0
$601$	8.97056	0.365917	0.182958	−	0.983121i	$-0.441433\pi$
$601$	8.97056	0.365917	0.182958	+	0.983121i	$0.441433\pi$
$602$	−0.828427	−0.0337642
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−0.970563	−0.0393939	−0.0196970	−	0.999806i	$-0.506270\pi$
$607$	−0.970563	−0.0393939	−0.0196970	+	0.999806i	$0.506270\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.9706	−0.929289
$612$	0	0
$613$	−1.17157	−0.0473194	−0.0236597	−	0.999720i	$-0.507532\pi$
$613$	−1.17157	−0.0473194	−0.0236597	+	0.999720i	$0.507532\pi$
$614$	18.1005	0.730477
$615$	0	0
$616$	3.02944	0.122060
$617$	−13.9706	−0.562434	−0.281217	−	0.959644i	$-0.590738\pi$
$617$	−13.9706	−0.562434	−0.281217	+	0.959644i	$0.590738\pi$
$618$	0	0
$619$	−4.97056	−0.199784	−0.0998919	−	0.994998i	$-0.531850\pi$
$619$	−4.97056	−0.199784	−0.0998919	+	0.994998i	$0.531850\pi$
$620$	0	0
$621$	0	0
$622$	−28.2426	−1.13243
$623$	6.00000	0.240385
$624$	0	0
$625$	0	0
$626$	−24.3431	−0.972948
$627$	0	0
$628$	0	0
$629$	66.4264	2.64859
$630$	0	0
$631$	−26.7696	−1.06568	−0.532840	−	0.846216i	$-0.678875\pi$
$631$	−26.7696	−1.06568	−0.532840	+	0.846216i	$0.678875\pi$
$632$	2.34315	0.0932053
$633$	0	0
$634$	28.5269	1.13295
$635$	0	0
$636$	0	0
$637$	25.4853	1.00976
$638$	10.9706	0.434329
$639$	0	0
$640$	0	0
$641$	−1.55635	−0.0614721	−0.0307360	−	0.999528i	$-0.509785\pi$
$641$	−1.55635	−0.0614721	−0.0307360	+	0.999528i	$0.509785\pi$
$642$	0	0
$643$	9.51472	0.375224	0.187612	−	0.982243i	$-0.439925\pi$
$643$	9.51472	0.375224	0.187612	+	0.982243i	$0.439925\pi$
$644$	0	0
$645$	0	0
$646$	−53.4558	−2.10319
$647$	−6.82843	−0.268453	−0.134227	−	0.990951i	$-0.542855\pi$
$647$	−6.82843	−0.268453	−0.134227	+	0.990951i	$0.542855\pi$
$648$	0	0
$649$	1.51472	0.0594579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.8284	0.815079	0.407540	−	0.913188i	$-0.366387\pi$
$653$	20.8284	0.815079	0.407540	+	0.913188i	$0.366387\pi$
$654$	0	0
$655$	0	0
$656$	−15.3137	−0.597900
$657$	0	0
$658$	4.97056	0.193773
$659$	6.31371	0.245947	0.122974	−	0.992410i	$-0.460757\pi$
$659$	6.31371	0.245947	0.122974	+	0.992410i	$0.460757\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	26.2843	1.02157
$663$	0	0
$664$	−41.4558	−1.60880
$665$	0	0
$666$	0	0
$667$	19.7574	0.765008
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.0711	−0.581812
$672$	0	0
$673$	−8.72792	−0.336437	−0.168218	−	0.985750i	$-0.553801\pi$
$673$	−8.72792	−0.336437	−0.168218	+	0.985750i	$0.553801\pi$
$674$	7.07107	0.272367
$675$	0	0
$676$	0	0
$677$	16.8284	0.646769	0.323384	−	0.946268i	$-0.395179\pi$
$677$	16.8284	0.646769	0.323384	+	0.946268i	$0.395179\pi$
$678$	0	0
$679$	2.24264	0.0860647
$680$	0	0
$681$	0	0
$682$	7.75736	0.297045
$683$	4.45584	0.170498	0.0852491	−	0.996360i	$-0.472831\pi$
$683$	4.45584	0.170498	0.0852491	+	0.996360i	$0.472831\pi$
$684$	0	0
$685$	0	0
$686$	−11.3137	−0.431959
$687$	0	0
$688$	4.00000	0.152499
$689$	−31.2843	−1.19184
$690$	0	0
$691$	−1.27208	−0.0483921	−0.0241961	−	0.999707i	$-0.507703\pi$
$691$	−1.27208	−0.0483921	−0.0241961	+	0.999707i	$0.507703\pi$
$692$	0	0
$693$	0	0
$694$	−10.4853	−0.398016
$695$	0	0
$696$	0	0
$697$	29.9706	1.13522
$698$	−16.6274	−0.629357
$699$	0	0
$700$	0	0
$701$	18.3431	0.692811	0.346406	−	0.938085i	$-0.387402\pi$
$701$	18.3431	0.692811	0.346406	+	0.938085i	$0.387402\pi$
$702$	0	0
$703$	−40.9706	−1.54523
$704$	−14.6274	−0.551292
$705$	0	0
$706$	0.242641	0.00913190
$707$	−0.100505	−0.00377988
$708$	0	0
$709$	24.1127	0.905571	0.452786	−	0.891619i	$-0.350430\pi$
$709$	24.1127	0.905571	0.452786	+	0.891619i	$0.350430\pi$
$710$	0	0
$711$	0	0
$712$	−28.9706	−1.08572
$713$	13.9706	0.523202
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	46.1838	1.72356
$719$	−22.6274	−0.843860	−0.421930	−	0.906628i	$-0.638647\pi$
$719$	−22.6274	−0.843860	−0.421930	+	0.906628i	$0.638647\pi$
$720$	0	0
$721$	−10.2426	−0.381456
$722$	6.10051	0.227037
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.97056	−0.332700	−0.166350	−	0.986067i	$-0.553198\pi$
$727$	−8.97056	−0.332700	−0.166350	+	0.986067i	$0.553198\pi$
$728$	6.34315	0.235093
$729$	0	0
$730$	0	0
$731$	−7.82843	−0.289545
$732$	0	0
$733$	16.9706	0.626822	0.313411	−	0.949618i	$-0.398528\pi$
$733$	16.9706	0.626822	0.313411	+	0.949618i	$0.398528\pi$
$734$	42.1421	1.55549
$735$	0	0
$736$	0	0
$737$	17.3431	0.638843
$738$	0	0
$739$	11.4558	0.421410	0.210705	−	0.977550i	$-0.432424\pi$
$739$	11.4558	0.421410	0.210705	+	0.977550i	$0.432424\pi$
$740$	0	0
$741$	0	0
$742$	6.76955	0.248518
$743$	−47.1127	−1.72840	−0.864199	−	0.503151i	$-0.832174\pi$
$743$	−47.1127	−1.72840	−0.864199	+	0.503151i	$0.832174\pi$
$744$	0	0
$745$	0	0
$746$	12.0000	0.439351
$747$	0	0
$748$	0	0
$749$	−6.82843	−0.249505
$750$	0	0
$751$	−20.2426	−0.738664	−0.369332	−	0.929297i	$-0.620414\pi$
$751$	−20.2426	−0.738664	−0.369332	+	0.929297i	$0.620414\pi$
$752$	−24.0000	−0.875190
$753$	0	0
$754$	22.9706	0.836539
$755$	0	0
$756$	0	0
$757$	−20.4853	−0.744550	−0.372275	−	0.928122i	$-0.621422\pi$
$757$	−20.4853	−0.744550	−0.372275	+	0.928122i	$0.621422\pi$
$758$	10.8701	0.394818
$759$	0	0
$760$	0	0
$761$	43.1127	1.56283	0.781417	−	0.624009i	$-0.214497\pi$
$761$	43.1127	1.56283	0.781417	+	0.624009i	$0.214497\pi$
$762$	0	0
$763$	8.18377	0.296272
$764$	0	0
$765$	0	0
$766$	−4.97056	−0.179594
$767$	3.17157	0.114519
$768$	0	0
$769$	−28.7696	−1.03746	−0.518728	−	0.854939i	$-0.673594\pi$
$769$	−28.7696	−1.03746	−0.518728	+	0.854939i	$0.673594\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.8995	1.00348	0.501738	−	0.865020i	$-0.332694\pi$
$773$	27.8995	1.00348	0.501738	+	0.865020i	$0.332694\pi$
$774$	0	0
$775$	0	0
$776$	−10.8284	−0.388718
$777$	0	0
$778$	−40.4853	−1.45147
$779$	−18.4853	−0.662304
$780$	0	0
$781$	−16.1421	−0.577611
$782$	−51.5563	−1.84365
$783$	0	0
$784$	26.6274	0.950979
$785$	0	0
$786$	0	0
$787$	29.7990	1.06222	0.531110	−	0.847303i	$-0.321775\pi$
$787$	29.7990	1.06222	0.531110	+	0.847303i	$0.321775\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.686292	−0.0244017
$792$	0	0
$793$	−31.5563	−1.12060
$794$	30.3431	1.07684
$795$	0	0
$796$	0	0
$797$	12.6863	0.449372	0.224686	−	0.974431i	$-0.427864\pi$
$797$	12.6863	0.449372	0.224686	+	0.974431i	$0.427864\pi$
$798$	0	0
$799$	46.9706	1.66170
$800$	0	0
$801$	0	0
$802$	41.6985	1.47242
$803$	−14.1838	−0.500534
$804$	0	0
$805$	0	0
$806$	16.2426	0.572123
$807$	0	0
$808$	0.485281	0.0170721
$809$	−5.65685	−0.198884	−0.0994422	−	0.995043i	$-0.531706\pi$
$809$	−5.65685	−0.198884	−0.0994422	+	0.995043i	$0.531706\pi$
$810$	0	0
$811$	48.7279	1.71107	0.855534	−	0.517746i	$-0.173229\pi$
$811$	48.7279	1.71107	0.855534	+	0.517746i	$0.173229\pi$
$812$	0	0
$813$	0	0
$814$	−21.9411	−0.769036
$815$	0	0
$816$	0	0
$817$	4.82843	0.168925
$818$	52.1421	1.82311
$819$	0	0
$820$	0	0
$821$	−10.1127	−0.352936	−0.176468	−	0.984306i	$-0.556467\pi$
$821$	−10.1127	−0.352936	−0.176468	+	0.984306i	$0.556467\pi$
$822$	0	0
$823$	43.3431	1.51085	0.755424	−	0.655237i	$-0.227431\pi$
$823$	43.3431	1.51085	0.755424	+	0.655237i	$0.227431\pi$
$824$	49.4558	1.72288
$825$	0	0
$826$	−0.686292	−0.0238791
$827$	9.65685	0.335802	0.167901	−	0.985804i	$-0.446301\pi$
$827$	9.65685	0.335802	0.167901	+	0.985804i	$0.446301\pi$
$828$	0	0
$829$	−15.7990	−0.548722	−0.274361	−	0.961627i	$-0.588466\pi$
$829$	−15.7990	−0.548722	−0.274361	+	0.961627i	$0.588466\pi$
$830$	0	0
$831$	0	0
$832$	−30.6274	−1.06181
$833$	−52.1127	−1.80560
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	5.79899	0.200323
$839$	−4.87006	−0.168133	−0.0840665	−	0.996460i	$-0.526791\pi$
$839$	−4.87006	−0.168133	−0.0840665	+	0.996460i	$0.526791\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	−6.14214	−0.211672
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.48528	−0.154116
$848$	−32.6863	−1.12245
$849$	0	0
$850$	0	0
$851$	−39.5147	−1.35455
$852$	0	0
$853$	32.5980	1.11613	0.558067	−	0.829796i	$-0.311543\pi$
$853$	32.5980	1.11613	0.558067	+	0.829796i	$0.311543\pi$
$854$	6.82843	0.233664
$855$	0	0
$856$	32.9706	1.12691
$857$	−3.65685	−0.124916	−0.0624579	−	0.998048i	$-0.519894\pi$
$857$	−3.65685	−0.124916	−0.0624579	+	0.998048i	$0.519894\pi$
$858$	0	0
$859$	−16.9706	−0.579028	−0.289514	−	0.957174i	$-0.593494\pi$
$859$	−16.9706	−0.579028	−0.289514	+	0.957174i	$0.593494\pi$
$860$	0	0
$861$	0	0
$862$	47.0711	1.60325
$863$	43.2548	1.47241	0.736206	−	0.676758i	$-0.236616\pi$
$863$	43.2548	1.47241	0.736206	+	0.676758i	$0.236616\pi$
$864$	0	0
$865$	0	0
$866$	−42.7696	−1.45337
$867$	0	0
$868$	0	0
$869$	1.51472	0.0513833
$870$	0	0
$871$	36.3137	1.23044
$872$	−39.5147	−1.33814
$873$	0	0
$874$	31.7990	1.07562
$875$	0	0
$876$	0	0
$877$	2.79899	0.0945152	0.0472576	−	0.998883i	$-0.484952\pi$
$877$	2.79899	0.0945152	0.0472576	+	0.998883i	$0.484952\pi$
$878$	−7.75736	−0.261798
$879$	0	0
$880$	0	0
$881$	54.2548	1.82789	0.913946	−	0.405836i	$-0.133020\pi$
$881$	54.2548	1.82789	0.913946	+	0.405836i	$0.133020\pi$
$882$	0	0
$883$	41.9706	1.41242	0.706211	−	0.708001i	$-0.250403\pi$
$883$	41.9706	1.41242	0.706211	+	0.708001i	$0.250403\pi$
$884$	0	0
$885$	0	0
$886$	51.1127	1.71716
$887$	25.0294	0.840406	0.420203	−	0.907430i	$-0.361959\pi$
$887$	25.0294	0.840406	0.420203	+	0.907430i	$0.361959\pi$
$888$	0	0
$889$	1.07107	0.0359225
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−28.9706	−0.969463
$894$	0	0
$895$	0	0
$896$	6.62742	0.221406
$897$	0	0
$898$	13.0294	0.434798
$899$	12.7279	0.424500
$900$	0	0
$901$	63.9706	2.13117
$902$	−9.89949	−0.329617
$903$	0	0
$904$	3.31371	0.110212
$905$	0	0
$906$	0	0
$907$	−13.9706	−0.463885	−0.231942	−	0.972730i	$-0.574508\pi$
$907$	−13.9706	−0.463885	−0.231942	+	0.972730i	$0.574508\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	0	0
$913$	−26.7990	−0.886917
$914$	34.9706	1.15672
$915$	0	0
$916$	0	0
$917$	0.970563	0.0320508
$918$	0	0
$919$	−29.4853	−0.972630	−0.486315	−	0.873784i	$-0.661659\pi$
$919$	−29.4853	−0.972630	−0.486315	+	0.873784i	$0.661659\pi$
$920$	0	0
$921$	0	0
$922$	3.71573	0.122371
$923$	−33.7990	−1.11251
$924$	0	0
$925$	0	0
$926$	7.45584	0.245014
$927$	0	0
$928$	0	0
$929$	−44.8284	−1.47077	−0.735386	−	0.677648i	$-0.762999\pi$
$929$	−44.8284	−1.47077	−0.735386	+	0.677648i	$0.762999\pi$
$930$	0	0
$931$	32.1421	1.05342
$932$	0	0
$933$	0	0
$934$	17.4558	0.571173
$935$	0	0
$936$	0	0
$937$	−31.0122	−1.01312	−0.506562	−	0.862203i	$-0.669084\pi$
$937$	−31.0122	−1.01312	−0.506562	+	0.862203i	$0.669084\pi$
$938$	−7.85786	−0.256568
$939$	0	0
$940$	0	0
$941$	−13.6274	−0.444241	−0.222121	−	0.975019i	$-0.571298\pi$
$941$	−13.6274	−0.444241	−0.222121	+	0.975019i	$0.571298\pi$
$942$	0	0
$943$	−17.8284	−0.580573
$944$	3.31371	0.107852
$945$	0	0
$946$	2.58579	0.0840712
$947$	−0.171573	−0.00557537	−0.00278768	−	0.999996i	$-0.500887\pi$
$947$	−0.171573	−0.00557537	−0.00278768	+	0.999996i	$0.500887\pi$
$948$	0	0
$949$	−29.6985	−0.964054
$950$	0	0
$951$	0	0
$952$	−12.9706	−0.420378
$953$	24.0416	0.778785	0.389392	−	0.921072i	$-0.372685\pi$
$953$	24.0416	0.778785	0.389392	+	0.921072i	$0.372685\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−6.58579	−0.212777
$959$	1.45584	0.0470117
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−45.9411	−1.48120
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.9411	−0.866368	−0.433184	−	0.901305i	$-0.642610\pi$
$967$	−26.9411	−0.866368	−0.433184	+	0.901305i	$0.642610\pi$
$968$	21.6569	0.696078
$969$	0	0
$970$	0	0
$971$	−25.1421	−0.806850	−0.403425	−	0.915013i	$-0.632180\pi$
$971$	−25.1421	−0.806850	−0.403425	+	0.915013i	$0.632180\pi$
$972$	0	0
$973$	−6.72792	−0.215687
$974$	−47.5980	−1.52514
$975$	0	0
$976$	−32.9706	−1.05536
$977$	−0.686292	−0.0219564	−0.0109782	−	0.999940i	$-0.503495\pi$
$977$	−0.686292	−0.0219564	−0.0109782	+	0.999940i	$0.503495\pi$
$978$	0	0
$979$	−18.7279	−0.598547
$980$	0	0
$981$	0	0
$982$	−14.9706	−0.477730
$983$	2.52691	0.0805960	0.0402980	−	0.999188i	$-0.487169\pi$
$983$	2.52691	0.0805960	0.0402980	+	0.999188i	$0.487169\pi$
$984$	0	0
$985$	0	0
$986$	−46.9706	−1.49585
$987$	0	0
$988$	0	0
$989$	4.65685	0.148079
$990$	0	0
$991$	−19.5563	−0.621228	−0.310614	−	0.950536i	$-0.600535\pi$
$991$	−19.5563	−0.621228	−0.310614	+	0.950536i	$0.600535\pi$
$992$	0	0
$993$	0	0
$994$	7.31371	0.231977
$995$	0	0
$996$	0	0
$997$	−17.0711	−0.540646	−0.270323	−	0.962770i	$-0.587131\pi$
$997$	−17.0711	−0.540646	−0.270323	+	0.962770i	$0.587131\pi$
$998$	−48.4264	−1.53291
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9675.2.a.bf.1.2		2
3.2	odd	2		1075.2.a.i.1.1		2
5.4	even	2		387.2.a.h.1.1		2
15.2	even	4		1075.2.b.f.474.2		4
15.8	even	4		1075.2.b.f.474.3		4
15.14	odd	2		43.2.a.b.1.2	✓	2
20.19	odd	2		6192.2.a.bd.1.2		2
60.59	even	2		688.2.a.f.1.2		2
105.104	even	2		2107.2.a.b.1.2		2
120.29	odd	2		2752.2.a.l.1.2		2
120.59	even	2		2752.2.a.m.1.1		2
165.164	even	2		5203.2.a.f.1.1		2
195.194	odd	2		7267.2.a.b.1.1		2
645.644	even	2		1849.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.a.b.1.2	✓	2	15.14	odd	2
387.2.a.h.1.1		2	5.4	even	2
688.2.a.f.1.2		2	60.59	even	2
1075.2.a.i.1.1		2	3.2	odd	2
1075.2.b.f.474.2		4	15.2	even	4
1075.2.b.f.474.3		4	15.8	even	4
1849.2.a.f.1.1		2	645.644	even	2
2107.2.a.b.1.2		2	105.104	even	2
2752.2.a.l.1.2		2	120.29	odd	2
2752.2.a.m.1.1		2	120.59	even	2
5203.2.a.f.1.1		2	165.164	even	2
6192.2.a.bd.1.2		2	20.19	odd	2
7267.2.a.b.1.1		2	195.194	odd	2
9675.2.a.bf.1.2		2	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:	\( N \)	\(=\)	\( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9675.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +0.585786 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +0.585786 q^{7} -2.82843 q^{8} -1.82843 q^{11} -3.82843 q^{13} +0.828427 q^{14} -4.00000 q^{16} +7.82843 q^{17} -4.82843 q^{19} -2.58579 q^{22} -4.65685 q^{23} -5.41421 q^{26} -4.24264 q^{29} -3.00000 q^{31} +11.0711 q^{34} +8.48528 q^{37} -6.82843 q^{38} +3.82843 q^{41} -1.00000 q^{43} -6.58579 q^{46} +6.00000 q^{47} -6.65685 q^{49} +8.17157 q^{53} -1.65685 q^{56} -6.00000 q^{58} -0.828427 q^{59} +8.24264 q^{61} -4.24264 q^{62} +8.00000 q^{64} -9.48528 q^{67} +8.82843 q^{71} +7.75736 q^{73} +12.0000 q^{74} -1.07107 q^{77} -0.828427 q^{79} +5.41421 q^{82} +14.6569 q^{83} -1.41421 q^{86} +5.17157 q^{88} +10.2426 q^{89} -2.24264 q^{91} +8.48528 q^{94} +3.82843 q^{97} -9.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7}+O(q^{10}) \) 2 * q + 4 * q^7	\( 2 q + 4 q^{7} + 2 q^{11} - 2 q^{13} - 4 q^{14} - 8 q^{16} + 10 q^{17} - 4 q^{19} - 8 q^{22} + 2 q^{23} - 8 q^{26} - 6 q^{31} + 8 q^{34} - 8 q^{38} + 2 q^{41} - 2 q^{43} - 16 q^{46} + 12 q^{47} - 2 q^{49} + 22 q^{53} + 8 q^{56} - 12 q^{58} + 4 q^{59} + 8 q^{61} + 16 q^{64} - 2 q^{67} + 12 q^{71} + 24 q^{73} + 24 q^{74} + 12 q^{77} + 4 q^{79} + 8 q^{82} + 18 q^{83} + 16 q^{88} + 12 q^{89} + 4 q^{91} + 2 q^{97} - 16 q^{98}+O(q^{100}) \) 2 * q + 4 * q^7 + 2 * q^11 - 2 * q^13 - 4 * q^14 - 8 * q^16 + 10 * q^17 - 4 * q^19 - 8 * q^22 + 2 * q^23 - 8 * q^26 - 6 * q^31 + 8 * q^34 - 8 * q^38 + 2 * q^41 - 2 * q^43 - 16 * q^46 + 12 * q^47 - 2 * q^49 + 22 * q^53 + 8 * q^56 - 12 * q^58 + 4 * q^59 + 8 * q^61 + 16 * q^64 - 2 * q^67 + 12 * q^71 + 24 * q^73 + 24 * q^74 + 12 * q^77 + 4 * q^79 + 8 * q^82 + 18 * q^83 + 16 * q^88 + 12 * q^89 + 4 * q^91 + 2 * q^97 - 16 * q^98

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