LMFDB - Embedded newform 207.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,2,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("207.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	207.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:	$1.65290332184$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} +3.41421 q^{5} -0.585786 q^{7} +1.58579 q^{8} +O(q^{10})$	$q-0.414214 q^{2} -1.82843 q^{4} +3.41421 q^{5} -0.585786 q^{7} +1.58579 q^{8} -1.41421 q^{10} +2.82843 q^{11} +0.242641 q^{14} +3.00000 q^{16} +4.58579 q^{17} +2.24264 q^{19} -6.24264 q^{20} -1.17157 q^{22} +1.00000 q^{23} +6.65685 q^{25} +1.07107 q^{28} -8.48528 q^{29} -8.48528 q^{31} -4.41421 q^{32} -1.89949 q^{34} -2.00000 q^{35} +0.828427 q^{37} -0.928932 q^{38} +5.41421 q^{40} +9.65685 q^{41} -10.2426 q^{43} -5.17157 q^{44} -0.414214 q^{46} -11.6569 q^{47} -6.65685 q^{49} -2.75736 q^{50} +9.07107 q^{53} +9.65685 q^{55} -0.928932 q^{56} +3.51472 q^{58} -3.65685 q^{59} +4.82843 q^{61} +3.51472 q^{62} -4.17157 q^{64} +11.4142 q^{67} -8.38478 q^{68} +0.828427 q^{70} -2.34315 q^{71} -9.31371 q^{73} -0.343146 q^{74} -4.10051 q^{76} -1.65685 q^{77} -11.8995 q^{79} +10.2426 q^{80} -4.00000 q^{82} +1.17157 q^{83} +15.6569 q^{85} +4.24264 q^{86} +4.48528 q^{88} -1.07107 q^{89} -1.82843 q^{92} +4.82843 q^{94} +7.65685 q^{95} -10.0000 q^{97} +2.75736 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} - 8 q^{14} + 6 q^{16} + 12 q^{17} - 4 q^{19} - 4 q^{20} - 8 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{28} - 6 q^{32} + 16 q^{34} - 4 q^{35} - 4 q^{37} - 16 q^{38} + 8 q^{40} + 8 q^{41} - 12 q^{43} - 16 q^{44} + 2 q^{46} - 12 q^{47} - 2 q^{49} - 14 q^{50} + 4 q^{53} + 8 q^{55} - 16 q^{56} + 24 q^{58} + 4 q^{59} + 4 q^{61} + 24 q^{62} - 14 q^{64} + 20 q^{67} + 20 q^{68} - 4 q^{70} - 16 q^{71} + 4 q^{73} - 12 q^{74} - 28 q^{76} + 8 q^{77} - 4 q^{79} + 12 q^{80} - 8 q^{82} + 8 q^{83} + 20 q^{85} - 8 q^{88} + 12 q^{89} + 2 q^{92} + 4 q^{94} + 4 q^{95} - 20 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 - 8 * q^14 + 6 * q^16 + 12 * q^17 - 4 * q^19 - 4 * q^20 - 8 * q^22 + 2 * q^23 + 2 * q^25 - 12 * q^28 - 6 * q^32 + 16 * q^34 - 4 * q^35 - 4 * q^37 - 16 * q^38 + 8 * q^40 + 8 * q^41 - 12 * q^43 - 16 * q^44 + 2 * q^46 - 12 * q^47 - 2 * q^49 - 14 * q^50 + 4 * q^53 + 8 * q^55 - 16 * q^56 + 24 * q^58 + 4 * q^59 + 4 * q^61 + 24 * q^62 - 14 * q^64 + 20 * q^67 + 20 * q^68 - 4 * q^70 - 16 * q^71 + 4 * q^73 - 12 * q^74 - 28 * q^76 + 8 * q^77 - 4 * q^79 + 12 * q^80 - 8 * q^82 + 8 * q^83 + 20 * q^85 - 8 * q^88 + 12 * q^89 + 2 * q^92 + 4 * q^94 + 4 * q^95 - 20 * q^97 + 14 * 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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	1.58579	0.560660
$9$	0	0
$10$	−1.41421	−0.447214
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0.242641	0.0648485
$15$	0	0
$16$	3.00000	0.750000
$17$	4.58579	1.11222	0.556108	−	0.831110i	$-0.312294\pi$
$17$	4.58579	1.11222	0.556108	+	0.831110i	$0.312294\pi$
$18$	0	0
$19$	2.24264	0.514497	0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264	0.514497	0.257249	+	0.966345i	$0.417184\pi$
$20$	−6.24264	−1.39590
$21$	0	0
$22$	−1.17157	−0.249780
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	0	0
$28$	1.07107	0.202413
$29$	−8.48528	−1.57568	−0.787839	−	0.615882i	$-0.788800\pi$
$29$	−8.48528	−1.57568	−0.787839	+	0.615882i	$0.788800\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	−1.89949	−0.325761
$35$	−2.00000	−0.338062
$36$	0	0
$37$	0.828427	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$37$	0.828427	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$38$	−0.928932	−0.150693
$39$	0	0
$40$	5.41421	0.856062
$41$	9.65685	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$41$	9.65685	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$42$	0	0
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	−5.17157	−0.779644
$45$	0	0
$46$	−0.414214	−0.0610725
$47$	−11.6569	−1.70033	−0.850163	−	0.526519i	$-0.823497\pi$
$47$	−11.6569	−1.70033	−0.850163	+	0.526519i	$0.823497\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	−2.75736	−0.389949
$51$	0	0
$52$	0	0
$53$	9.07107	1.24601	0.623003	−	0.782219i	$-0.285912\pi$
$53$	9.07107	1.24601	0.623003	+	0.782219i	$0.285912\pi$
$54$	0	0
$55$	9.65685	1.30213
$56$	−0.928932	−0.124134
$57$	0	0
$58$	3.51472	0.461505
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	4.82843	0.618217	0.309108	−	0.951027i	$-0.399969\pi$
$61$	4.82843	0.618217	0.309108	+	0.951027i	$0.399969\pi$
$62$	3.51472	0.446370
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	0	0
$67$	11.4142	1.39447	0.697234	−	0.716844i	$-0.254414\pi$
$67$	11.4142	1.39447	0.697234	+	0.716844i	$0.254414\pi$
$68$	−8.38478	−1.01680
$69$	0	0
$70$	0.828427	0.0990160
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	−9.31371	−1.09009	−0.545044	−	0.838408i	$-0.683487\pi$
$73$	−9.31371	−1.09009	−0.545044	+	0.838408i	$0.683487\pi$
$74$	−0.343146	−0.0398899
$75$	0	0
$76$	−4.10051	−0.470360
$77$	−1.65685	−0.188816
$78$	0	0
$79$	−11.8995	−1.33880	−0.669399	−	0.742903i	$-0.733448\pi$
$79$	−11.8995	−1.33880	−0.669399	+	0.742903i	$0.733448\pi$
$80$	10.2426	1.14516
$81$	0	0
$82$	−4.00000	−0.441726
$83$	1.17157	0.128597	0.0642984	−	0.997931i	$-0.479519\pi$
$83$	1.17157	0.128597	0.0642984	+	0.997931i	$0.479519\pi$
$84$	0	0
$85$	15.6569	1.69822
$86$	4.24264	0.457496
$87$	0	0
$88$	4.48528	0.478133
$89$	−1.07107	−0.113533	−0.0567665	−	0.998387i	$-0.518079\pi$
$89$	−1.07107	−0.113533	−0.0567665	+	0.998387i	$0.518079\pi$
$90$	0	0
$91$	0	0
$92$	−1.82843	−0.190627
$93$	0	0
$94$	4.82843	0.498014
$95$	7.65685	0.785577
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	2.75736	0.278535
$99$	0	0
$100$	−12.1716	−1.21716
$101$	11.3137	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$102$	0	0
$103$	−0.585786	−0.0577193	−0.0288596	−	0.999583i	$-0.509188\pi$
$103$	−0.585786	−0.0577193	−0.0288596	+	0.999583i	$0.509188\pi$
$104$	0	0
$105$	0	0
$106$	−3.75736	−0.364947
$107$	8.48528	0.820303	0.410152	−	0.912017i	$-0.365476\pi$
$107$	8.48528	0.820303	0.410152	+	0.912017i	$0.365476\pi$
$108$	0	0
$109$	14.4853	1.38744	0.693719	−	0.720246i	$-0.255971\pi$
$109$	14.4853	1.38744	0.693719	+	0.720246i	$0.255971\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−1.75736	−0.166055
$113$	12.5858	1.18397	0.591986	−	0.805949i	$-0.298344\pi$
$113$	12.5858	1.18397	0.591986	+	0.805949i	$0.298344\pi$
$114$	0	0
$115$	3.41421	0.318377
$116$	15.5147	1.44051
$117$	0	0
$118$	1.51472	0.139441
$119$	−2.68629	−0.246252
$120$	0	0
$121$	−3.00000	−0.272727
$122$	−2.00000	−0.181071
$123$	0	0
$124$	15.5147	1.39326
$125$	5.65685	0.505964
$126$	0	0
$127$	12.4853	1.10789	0.553945	−	0.832553i	$-0.313122\pi$
$127$	12.4853	1.10789	0.553945	+	0.832553i	$0.313122\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	−1.31371	−0.113913
$134$	−4.72792	−0.408430
$135$	0	0
$136$	7.27208	0.623576
$137$	3.41421	0.291696	0.145848	−	0.989307i	$-0.453409\pi$
$137$	3.41421	0.291696	0.145848	+	0.989307i	$0.453409\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	3.65685	0.309061
$141$	0	0
$142$	0.970563	0.0814478
$143$	0	0
$144$	0	0
$145$	−28.9706	−2.40587
$146$	3.85786	0.319279
$147$	0	0
$148$	−1.51472	−0.124509
$149$	−19.4142	−1.59047	−0.795237	−	0.606298i	$-0.792654\pi$
$149$	−19.4142	−1.59047	−0.795237	+	0.606298i	$0.792654\pi$
$150$	0	0
$151$	1.65685	0.134833	0.0674164	−	0.997725i	$-0.478524\pi$
$151$	1.65685	0.134833	0.0674164	+	0.997725i	$0.478524\pi$
$152$	3.55635	0.288458
$153$	0	0
$154$	0.686292	0.0553029
$155$	−28.9706	−2.32697
$156$	0	0
$157$	−18.4853	−1.47529	−0.737643	−	0.675191i	$-0.764061\pi$
$157$	−18.4853	−1.47529	−0.737643	+	0.675191i	$0.764061\pi$
$158$	4.92893	0.392125
$159$	0	0
$160$	−15.0711	−1.19147
$161$	−0.585786	−0.0461664
$162$	0	0
$163$	13.1716	1.03168	0.515839	−	0.856686i	$-0.327480\pi$
$163$	13.1716	1.03168	0.515839	+	0.856686i	$0.327480\pi$
$164$	−17.6569	−1.37877
$165$	0	0
$166$	−0.485281	−0.0376651
$167$	5.65685	0.437741	0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685	0.437741	0.218870	+	0.975754i	$0.429763\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−6.48528	−0.497398
$171$	0	0
$172$	18.7279	1.42799
$173$	5.17157	0.393187	0.196594	−	0.980485i	$-0.437012\pi$
$173$	5.17157	0.393187	0.196594	+	0.980485i	$0.437012\pi$
$174$	0	0
$175$	−3.89949	−0.294774
$176$	8.48528	0.639602
$177$	0	0
$178$	0.443651	0.0332530
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−16.8284	−1.25085	−0.625424	−	0.780285i	$-0.715074\pi$
$181$	−16.8284	−1.25085	−0.625424	+	0.780285i	$0.715074\pi$
$182$	0	0
$183$	0	0
$184$	1.58579	0.116906
$185$	2.82843	0.207950
$186$	0	0
$187$	12.9706	0.948501
$188$	21.3137	1.55446
$189$	0	0
$190$	−3.17157	−0.230090
$191$	−23.3137	−1.68692	−0.843460	−	0.537191i	$-0.819485\pi$
$191$	−23.3137	−1.68692	−0.843460	+	0.537191i	$0.819485\pi$
$192$	0	0
$193$	9.65685	0.695116	0.347558	−	0.937659i	$-0.387011\pi$
$193$	9.65685	0.695116	0.347558	+	0.937659i	$0.387011\pi$
$194$	4.14214	0.297388
$195$	0	0
$196$	12.1716	0.869398
$197$	−14.8284	−1.05648	−0.528241	−	0.849095i	$-0.677148\pi$
$197$	−14.8284	−1.05648	−0.528241	+	0.849095i	$0.677148\pi$
$198$	0	0
$199$	6.24264	0.442529	0.221265	−	0.975214i	$-0.428982\pi$
$199$	6.24264	0.442529	0.221265	+	0.975214i	$0.428982\pi$
$200$	10.5563	0.746447
$201$	0	0
$202$	−4.68629	−0.329726
$203$	4.97056	0.348865
$204$	0	0
$205$	32.9706	2.30276
$206$	0.242641	0.0169056
$207$	0	0
$208$	0	0
$209$	6.34315	0.438765
$210$	0	0
$211$	4.48528	0.308780	0.154390	−	0.988010i	$-0.450659\pi$
$211$	4.48528	0.308780	0.154390	+	0.988010i	$0.450659\pi$
$212$	−16.5858	−1.13912
$213$	0	0
$214$	−3.51472	−0.240261
$215$	−34.9706	−2.38497
$216$	0	0
$217$	4.97056	0.337424
$218$	−6.00000	−0.406371
$219$	0	0
$220$	−17.6569	−1.19042
$221$	0	0
$222$	0	0
$223$	20.9706	1.40429	0.702146	−	0.712033i	$-0.252225\pi$
$223$	20.9706	1.40429	0.702146	+	0.712033i	$0.252225\pi$
$224$	2.58579	0.172770
$225$	0	0
$226$	−5.21320	−0.346777
$227$	22.1421	1.46963	0.734813	−	0.678270i	$-0.237270\pi$
$227$	22.1421	1.46963	0.734813	+	0.678270i	$0.237270\pi$
$228$	0	0
$229$	−2.48528	−0.164232	−0.0821160	−	0.996623i	$-0.526168\pi$
$229$	−2.48528	−0.164232	−0.0821160	+	0.996623i	$0.526168\pi$
$230$	−1.41421	−0.0932505
$231$	0	0
$232$	−13.4558	−0.883419
$233$	10.3431	0.677602	0.338801	−	0.940858i	$-0.389979\pi$
$233$	10.3431	0.677602	0.338801	+	0.940858i	$0.389979\pi$
$234$	0	0
$235$	−39.7990	−2.59620
$236$	6.68629	0.435241
$237$	0	0
$238$	1.11270	0.0721255
$239$	3.31371	0.214346	0.107173	−	0.994240i	$-0.465820\pi$
$239$	3.31371	0.214346	0.107173	+	0.994240i	$0.465820\pi$
$240$	0	0
$241$	10.9706	0.706676	0.353338	−	0.935496i	$-0.385047\pi$
$241$	10.9706	0.706676	0.353338	+	0.935496i	$0.385047\pi$
$242$	1.24264	0.0798800
$243$	0	0
$244$	−8.82843	−0.565182
$245$	−22.7279	−1.45203
$246$	0	0
$247$	0	0
$248$	−13.4558	−0.854447
$249$	0	0
$250$	−2.34315	−0.148194
$251$	−3.51472	−0.221847	−0.110924	−	0.993829i	$-0.535381\pi$
$251$	−3.51472	−0.221847	−0.110924	+	0.993829i	$0.535381\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	−5.17157	−0.324493
$255$	0	0
$256$	3.97056	0.248160
$257$	17.6569	1.10140	0.550702	−	0.834702i	$-0.314360\pi$
$257$	17.6569	1.10140	0.550702	+	0.834702i	$0.314360\pi$
$258$	0	0
$259$	−0.485281	−0.0301539
$260$	0	0
$261$	0	0
$262$	7.02944	0.434280
$263$	0.686292	0.0423185	0.0211593	−	0.999776i	$-0.493264\pi$
$263$	0.686292	0.0423185	0.0211593	+	0.999776i	$0.493264\pi$
$264$	0	0
$265$	30.9706	1.90251
$266$	0.544156	0.0333643
$267$	0	0
$268$	−20.8701	−1.27484
$269$	−10.1421	−0.618377	−0.309188	−	0.951001i	$-0.600057\pi$
$269$	−10.1421	−0.618377	−0.309188	+	0.951001i	$0.600057\pi$
$270$	0	0
$271$	4.48528	0.272461	0.136231	−	0.990677i	$-0.456501\pi$
$271$	4.48528	0.272461	0.136231	+	0.990677i	$0.456501\pi$
$272$	13.7574	0.834162
$273$	0	0
$274$	−1.41421	−0.0854358
$275$	18.8284	1.13540
$276$	0	0
$277$	−3.65685	−0.219719	−0.109860	−	0.993947i	$-0.535040\pi$
$277$	−3.65685	−0.219719	−0.109860	+	0.993947i	$0.535040\pi$
$278$	1.65685	0.0993715
$279$	0	0
$280$	−3.17157	−0.189538
$281$	11.4142	0.680915	0.340457	−	0.940260i	$-0.389418\pi$
$281$	11.4142	0.680915	0.340457	+	0.940260i	$0.389418\pi$
$282$	0	0
$283$	−27.2132	−1.61766	−0.808829	−	0.588045i	$-0.799898\pi$
$283$	−27.2132	−1.61766	−0.808829	+	0.588045i	$0.799898\pi$
$284$	4.28427	0.254225
$285$	0	0
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	4.02944	0.237026
$290$	12.0000	0.704664
$291$	0	0
$292$	17.0294	0.996572
$293$	4.58579	0.267905	0.133952	−	0.990988i	$-0.457233\pi$
$293$	4.58579	0.267905	0.133952	+	0.990988i	$0.457233\pi$
$294$	0	0
$295$	−12.4853	−0.726921
$296$	1.31371	0.0763578
$297$	0	0
$298$	8.04163	0.465839
$299$	0	0
$300$	0	0
$301$	6.00000	0.345834
$302$	−0.686292	−0.0394916
$303$	0	0
$304$	6.72792	0.385873
$305$	16.4853	0.943944
$306$	0	0
$307$	5.17157	0.295157	0.147579	−	0.989050i	$-0.452852\pi$
$307$	5.17157	0.295157	0.147579	+	0.989050i	$0.452852\pi$
$308$	3.02944	0.172618
$309$	0	0
$310$	12.0000	0.681554
$311$	13.3137	0.754951	0.377476	−	0.926020i	$-0.376792\pi$
$311$	13.3137	0.754951	0.377476	+	0.926020i	$0.376792\pi$
$312$	0	0
$313$	−21.3137	−1.20472	−0.602361	−	0.798224i	$-0.705773\pi$
$313$	−21.3137	−1.20472	−0.602361	+	0.798224i	$0.705773\pi$
$314$	7.65685	0.432101
$315$	0	0
$316$	21.7574	1.22395
$317$	25.4558	1.42974	0.714871	−	0.699256i	$-0.246485\pi$
$317$	25.4558	1.42974	0.714871	+	0.699256i	$0.246485\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	−14.2426	−0.796188
$321$	0	0
$322$	0.242641	0.0135218
$323$	10.2843	0.572232
$324$	0	0
$325$	0	0
$326$	−5.45584	−0.302171
$327$	0	0
$328$	15.3137	0.845558
$329$	6.82843	0.376463
$330$	0	0
$331$	−1.17157	−0.0643955	−0.0321977	−	0.999482i	$-0.510251\pi$
$331$	−1.17157	−0.0643955	−0.0321977	+	0.999482i	$0.510251\pi$
$332$	−2.14214	−0.117565
$333$	0	0
$334$	−2.34315	−0.128211
$335$	38.9706	2.12919
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	5.38478	0.292893
$339$	0	0
$340$	−28.6274	−1.55254
$341$	−24.0000	−1.29967
$342$	0	0
$343$	8.00000	0.431959
$344$	−16.2426	−0.875744
$345$	0	0
$346$	−2.14214	−0.115162
$347$	2.68629	0.144208	0.0721038	−	0.997397i	$-0.477029\pi$
$347$	2.68629	0.144208	0.0721038	+	0.997397i	$0.477029\pi$
$348$	0	0
$349$	12.9706	0.694298	0.347149	−	0.937810i	$-0.387150\pi$
$349$	12.9706	0.694298	0.347149	+	0.937810i	$0.387150\pi$
$350$	1.61522	0.0863373
$351$	0	0
$352$	−12.4853	−0.665468
$353$	34.1421	1.81720	0.908601	−	0.417665i	$-0.137151\pi$
$353$	34.1421	1.81720	0.908601	+	0.417665i	$0.137151\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	1.95837	0.103793
$357$	0	0
$358$	7.45584	0.394054
$359$	−24.2843	−1.28167	−0.640837	−	0.767677i	$-0.721413\pi$
$359$	−24.2843	−1.28167	−0.640837	+	0.767677i	$0.721413\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	6.97056	0.366365
$363$	0	0
$364$	0	0
$365$	−31.7990	−1.66444
$366$	0	0
$367$	−17.5563	−0.916434	−0.458217	−	0.888840i	$-0.651512\pi$
$367$	−17.5563	−0.916434	−0.458217	+	0.888840i	$0.651512\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	−1.17157	−0.0609072
$371$	−5.31371	−0.275874
$372$	0	0
$373$	−10.4853	−0.542907	−0.271454	−	0.962452i	$-0.587504\pi$
$373$	−10.4853	−0.542907	−0.271454	+	0.962452i	$0.587504\pi$
$374$	−5.37258	−0.277810
$375$	0	0
$376$	−18.4853	−0.953306
$377$	0	0
$378$	0	0
$379$	3.41421	0.175376	0.0876882	−	0.996148i	$-0.472052\pi$
$379$	3.41421	0.175376	0.0876882	+	0.996148i	$0.472052\pi$
$380$	−14.0000	−0.718185
$381$	0	0
$382$	9.65685	0.494088
$383$	10.3431	0.528510	0.264255	−	0.964453i	$-0.414874\pi$
$383$	10.3431	0.528510	0.264255	+	0.964453i	$0.414874\pi$
$384$	0	0
$385$	−5.65685	−0.288300
$386$	−4.00000	−0.203595
$387$	0	0
$388$	18.2843	0.928243
$389$	15.8995	0.806136	0.403068	−	0.915170i	$-0.367944\pi$
$389$	15.8995	0.806136	0.403068	+	0.915170i	$0.367944\pi$
$390$	0	0
$391$	4.58579	0.231913
$392$	−10.5563	−0.533176
$393$	0	0
$394$	6.14214	0.309436
$395$	−40.6274	−2.04419
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−2.58579	−0.129614
$399$	0	0
$400$	19.9706	0.998528
$401$	−18.0416	−0.900956	−0.450478	−	0.892788i	$-0.648746\pi$
$401$	−18.0416	−0.900956	−0.450478	+	0.892788i	$0.648746\pi$
$402$	0	0
$403$	0	0
$404$	−20.6863	−1.02918
$405$	0	0
$406$	−2.05887	−0.102180
$407$	2.34315	0.116145
$408$	0	0
$409$	25.6569	1.26865	0.634325	−	0.773067i	$-0.281278\pi$
$409$	25.6569	1.26865	0.634325	+	0.773067i	$0.281278\pi$
$410$	−13.6569	−0.674464
$411$	0	0
$412$	1.07107	0.0527677
$413$	2.14214	0.105408
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	0	0
$418$	−2.62742	−0.128511
$419$	3.51472	0.171705	0.0858526	−	0.996308i	$-0.472639\pi$
$419$	3.51472	0.171705	0.0858526	+	0.996308i	$0.472639\pi$
$420$	0	0
$421$	11.4558	0.558324	0.279162	−	0.960244i	$-0.409943\pi$
$421$	11.4558	0.558324	0.279162	+	0.960244i	$0.409943\pi$
$422$	−1.85786	−0.0904394
$423$	0	0
$424$	14.3848	0.698586
$425$	30.5269	1.48077
$426$	0	0
$427$	−2.82843	−0.136877
$428$	−15.5147	−0.749932
$429$	0	0
$430$	14.4853	0.698542
$431$	31.3137	1.50833	0.754164	−	0.656686i	$-0.228042\pi$
$431$	31.3137	1.50833	0.754164	+	0.656686i	$0.228042\pi$
$432$	0	0
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	−2.05887	−0.0988291
$435$	0	0
$436$	−26.4853	−1.26841
$437$	2.24264	0.107280
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	15.3137	0.730052
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	−3.65685	−0.173352
$446$	−8.68629	−0.411308
$447$	0	0
$448$	2.44365	0.115452
$449$	−34.6274	−1.63417	−0.817084	−	0.576518i	$-0.804411\pi$
$449$	−34.6274	−1.63417	−0.817084	+	0.576518i	$0.804411\pi$
$450$	0	0
$451$	27.3137	1.28615
$452$	−23.0122	−1.08240
$453$	0	0
$454$	−9.17157	−0.430443
$455$	0	0
$456$	0	0
$457$	1.02944	0.0481550	0.0240775	−	0.999710i	$-0.492335\pi$
$457$	1.02944	0.0481550	0.0240775	+	0.999710i	$0.492335\pi$
$458$	1.02944	0.0481024
$459$	0	0
$460$	−6.24264	−0.291065
$461$	−14.8284	−0.690629	−0.345314	−	0.938487i	$-0.612228\pi$
$461$	−14.8284	−0.690629	−0.345314	+	0.938487i	$0.612228\pi$
$462$	0	0
$463$	−24.9706	−1.16048	−0.580240	−	0.814445i	$-0.697041\pi$
$463$	−24.9706	−1.16048	−0.580240	+	0.814445i	$0.697041\pi$
$464$	−25.4558	−1.18176
$465$	0	0
$466$	−4.28427	−0.198465
$467$	34.1421	1.57991	0.789955	−	0.613165i	$-0.210104\pi$
$467$	34.1421	1.57991	0.789955	+	0.613165i	$0.210104\pi$
$468$	0	0
$469$	−6.68629	−0.308744
$470$	16.4853	0.760409
$471$	0	0
$472$	−5.79899	−0.266920
$473$	−28.9706	−1.33207
$474$	0	0
$475$	14.9289	0.684986
$476$	4.91169	0.225127
$477$	0	0
$478$	−1.37258	−0.0627805
$479$	39.3137	1.79629	0.898145	−	0.439700i	$-0.144915\pi$
$479$	39.3137	1.79629	0.898145	+	0.439700i	$0.144915\pi$
$480$	0	0
$481$	0	0
$482$	−4.54416	−0.206981
$483$	0	0
$484$	5.48528	0.249331
$485$	−34.1421	−1.55031
$486$	0	0
$487$	−24.9706	−1.13152	−0.565762	−	0.824569i	$-0.691418\pi$
$487$	−24.9706	−1.13152	−0.565762	+	0.824569i	$0.691418\pi$
$488$	7.65685	0.346610
$489$	0	0
$490$	9.41421	0.425291
$491$	29.3137	1.32291	0.661455	−	0.749985i	$-0.269939\pi$
$491$	29.3137	1.32291	0.661455	+	0.749985i	$0.269939\pi$
$492$	0	0
$493$	−38.9117	−1.75249
$494$	0	0
$495$	0	0
$496$	−25.4558	−1.14300
$497$	1.37258	0.0615688
$498$	0	0
$499$	−7.51472	−0.336405	−0.168203	−	0.985752i	$-0.553796\pi$
$499$	−7.51472	−0.336405	−0.168203	+	0.985752i	$0.553796\pi$
$500$	−10.3431	−0.462560
$501$	0	0
$502$	1.45584	0.0649775
$503$	6.34315	0.282827	0.141413	−	0.989951i	$-0.454835\pi$
$503$	6.34315	0.282827	0.141413	+	0.989951i	$0.454835\pi$
$504$	0	0
$505$	38.6274	1.71890
$506$	−1.17157	−0.0520828
$507$	0	0
$508$	−22.8284	−1.01285
$509$	−9.85786	−0.436942	−0.218471	−	0.975843i	$-0.570107\pi$
$509$	−9.85786	−0.436942	−0.218471	+	0.975843i	$0.570107\pi$
$510$	0	0
$511$	5.45584	0.241352
$512$	−22.7574	−1.00574
$513$	0	0
$514$	−7.31371	−0.322594
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	−32.9706	−1.45004
$518$	0.201010	0.00883188
$519$	0	0
$520$	0	0
$521$	−6.72792	−0.294756	−0.147378	−	0.989080i	$-0.547083\pi$
$521$	−6.72792	−0.294756	−0.147378	+	0.989080i	$0.547083\pi$
$522$	0	0
$523$	23.6985	1.03626	0.518131	−	0.855301i	$-0.326628\pi$
$523$	23.6985	1.03626	0.518131	+	0.855301i	$0.326628\pi$
$524$	31.0294	1.35553
$525$	0	0
$526$	−0.284271	−0.0123948
$527$	−38.9117	−1.69502
$528$	0	0
$529$	1.00000	0.0434783
$530$	−12.8284	−0.557231
$531$	0	0
$532$	2.40202	0.104141
$533$	0	0
$534$	0	0
$535$	28.9706	1.25251
$536$	18.1005	0.781823
$537$	0	0
$538$	4.20101	0.181118
$539$	−18.8284	−0.810998
$540$	0	0
$541$	−28.9706	−1.24554	−0.622771	−	0.782404i	$-0.713993\pi$
$541$	−28.9706	−1.24554	−0.622771	+	0.782404i	$0.713993\pi$
$542$	−1.85786	−0.0798021
$543$	0	0
$544$	−20.2426	−0.867896
$545$	49.4558	2.11846
$546$	0	0
$547$	−4.48528	−0.191777	−0.0958884	−	0.995392i	$-0.530569\pi$
$547$	−4.48528	−0.191777	−0.0958884	+	0.995392i	$0.530569\pi$
$548$	−6.24264	−0.266672
$549$	0	0
$550$	−7.79899	−0.332550
$551$	−19.0294	−0.810681
$552$	0	0
$553$	6.97056	0.296418
$554$	1.51472	0.0643542
$555$	0	0
$556$	7.31371	0.310170
$557$	−13.7574	−0.582918	−0.291459	−	0.956583i	$-0.594141\pi$
$557$	−13.7574	−0.582918	−0.291459	+	0.956583i	$0.594141\pi$
$558$	0	0
$559$	0	0
$560$	−6.00000	−0.253546
$561$	0	0
$562$	−4.72792	−0.199435
$563$	26.8284	1.13068	0.565342	−	0.824857i	$-0.308744\pi$
$563$	26.8284	1.13068	0.565342	+	0.824857i	$0.308744\pi$
$564$	0	0
$565$	42.9706	1.80779
$566$	11.2721	0.473801
$567$	0	0
$568$	−3.71573	−0.155909
$569$	11.4142	0.478509	0.239254	−	0.970957i	$-0.423097\pi$
$569$	11.4142	0.478509	0.239254	+	0.970957i	$0.423097\pi$
$570$	0	0
$571$	10.2426	0.428641	0.214321	−	0.976763i	$-0.431246\pi$
$571$	10.2426	0.428641	0.214321	+	0.976763i	$0.431246\pi$
$572$	0	0
$573$	0	0
$574$	2.34315	0.0978010
$575$	6.65685	0.277610
$576$	0	0
$577$	24.2843	1.01097	0.505484	−	0.862836i	$-0.331314\pi$
$577$	24.2843	1.01097	0.505484	+	0.862836i	$0.331314\pi$
$578$	−1.66905	−0.0694232
$579$	0	0
$580$	52.9706	2.19948
$581$	−0.686292	−0.0284722
$582$	0	0
$583$	25.6569	1.06260
$584$	−14.7696	−0.611168
$585$	0	0
$586$	−1.89949	−0.0784674
$587$	40.9706	1.69104	0.845518	−	0.533947i	$-0.179292\pi$
$587$	40.9706	1.69104	0.845518	+	0.533947i	$0.179292\pi$
$588$	0	0
$589$	−19.0294	−0.784094
$590$	5.17157	0.212910
$591$	0	0
$592$	2.48528	0.102144
$593$	23.3137	0.957379	0.478690	−	0.877984i	$-0.341112\pi$
$593$	23.3137	0.957379	0.478690	+	0.877984i	$0.341112\pi$
$594$	0	0
$595$	−9.17157	−0.375998
$596$	35.4975	1.45403
$597$	0	0
$598$	0	0
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	0	0
$601$	−9.65685	−0.393911	−0.196956	−	0.980412i	$-0.563105\pi$
$601$	−9.65685	−0.393911	−0.196956	+	0.980412i	$0.563105\pi$
$602$	−2.48528	−0.101293
$603$	0	0
$604$	−3.02944	−0.123266
$605$	−10.2426	−0.416423
$606$	0	0
$607$	−25.6569	−1.04138	−0.520690	−	0.853746i	$-0.674325\pi$
$607$	−25.6569	−1.04138	−0.520690	+	0.853746i	$0.674325\pi$
$608$	−9.89949	−0.401478
$609$	0	0
$610$	−6.82843	−0.276475
$611$	0	0
$612$	0	0
$613$	−11.4558	−0.462697	−0.231349	−	0.972871i	$-0.574314\pi$
$613$	−11.4558	−0.462697	−0.231349	+	0.972871i	$0.574314\pi$
$614$	−2.14214	−0.0864496
$615$	0	0
$616$	−2.62742	−0.105862
$617$	−2.44365	−0.0983777	−0.0491888	−	0.998789i	$-0.515664\pi$
$617$	−2.44365	−0.0983777	−0.0491888	+	0.998789i	$0.515664\pi$
$618$	0	0
$619$	−13.7574	−0.552955	−0.276477	−	0.961020i	$-0.589167\pi$
$619$	−13.7574	−0.552955	−0.276477	+	0.961020i	$0.589167\pi$
$620$	52.9706	2.12735
$621$	0	0
$622$	−5.51472	−0.221120
$623$	0.627417	0.0251369
$624$	0	0
$625$	−13.9706	−0.558823
$626$	8.82843	0.352855
$627$	0	0
$628$	33.7990	1.34873
$629$	3.79899	0.151476
$630$	0	0
$631$	28.8701	1.14930	0.574649	−	0.818400i	$-0.305138\pi$
$631$	28.8701	1.14930	0.574649	+	0.818400i	$0.305138\pi$
$632$	−18.8701	−0.750611
$633$	0	0
$634$	−10.5442	−0.418762
$635$	42.6274	1.69162
$636$	0	0
$637$	0	0
$638$	9.94113	0.393573
$639$	0	0
$640$	36.0416	1.42467
$641$	−17.0711	−0.674267	−0.337133	−	0.941457i	$-0.609457\pi$
$641$	−17.0711	−0.674267	−0.337133	+	0.941457i	$0.609457\pi$
$642$	0	0
$643$	28.3848	1.11939	0.559693	−	0.828700i	$-0.310919\pi$
$643$	28.3848	1.11939	0.559693	+	0.828700i	$0.310919\pi$
$644$	1.07107	0.0422060
$645$	0	0
$646$	−4.25988	−0.167603
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−10.3431	−0.406004
$650$	0	0
$651$	0	0
$652$	−24.0833	−0.943173
$653$	−10.3431	−0.404759	−0.202379	−	0.979307i	$-0.564867\pi$
$653$	−10.3431	−0.404759	−0.202379	+	0.979307i	$0.564867\pi$
$654$	0	0
$655$	−57.9411	−2.26395
$656$	28.9706	1.13111
$657$	0	0
$658$	−2.82843	−0.110264
$659$	5.85786	0.228190	0.114095	−	0.993470i	$-0.463603\pi$
$659$	5.85786	0.228190	0.114095	+	0.993470i	$0.463603\pi$
$660$	0	0
$661$	−40.1421	−1.56135	−0.780674	−	0.624938i	$-0.785124\pi$
$661$	−40.1421	−1.56135	−0.780674	+	0.624938i	$0.785124\pi$
$662$	0.485281	0.0188610
$663$	0	0
$664$	1.85786	0.0720991
$665$	−4.48528	−0.173932
$666$	0	0
$667$	−8.48528	−0.328551
$668$	−10.3431	−0.400188
$669$	0	0
$670$	−16.1421	−0.623625
$671$	13.6569	0.527217
$672$	0	0
$673$	−7.02944	−0.270965	−0.135482	−	0.990780i	$-0.543258\pi$
$673$	−7.02944	−0.270965	−0.135482	+	0.990780i	$0.543258\pi$
$674$	−9.11270	−0.351008
$675$	0	0
$676$	23.7696	0.914214
$677$	−23.6985	−0.910807	−0.455403	−	0.890285i	$-0.650505\pi$
$677$	−23.6985	−0.910807	−0.455403	+	0.890285i	$0.650505\pi$
$678$	0	0
$679$	5.85786	0.224804
$680$	24.8284	0.952127
$681$	0	0
$682$	9.94113	0.380665
$683$	−21.6569	−0.828676	−0.414338	−	0.910123i	$-0.635987\pi$
$683$	−21.6569	−0.828676	−0.414338	+	0.910123i	$0.635987\pi$
$684$	0	0
$685$	11.6569	0.445386
$686$	−3.31371	−0.126518
$687$	0	0
$688$	−30.7279	−1.17149
$689$	0	0
$690$	0	0
$691$	−17.1716	−0.653237	−0.326619	−	0.945156i	$-0.605909\pi$
$691$	−17.1716	−0.653237	−0.326619	+	0.945156i	$0.605909\pi$
$692$	−9.45584	−0.359457
$693$	0	0
$694$	−1.11270	−0.0422375
$695$	−13.6569	−0.518034
$696$	0	0
$697$	44.2843	1.67739
$698$	−5.37258	−0.203355
$699$	0	0
$700$	7.12994	0.269486
$701$	−15.8995	−0.600516	−0.300258	−	0.953858i	$-0.597073\pi$
$701$	−15.8995	−0.600516	−0.300258	+	0.953858i	$0.597073\pi$
$702$	0	0
$703$	1.85786	0.0700707
$704$	−11.7990	−0.444691
$705$	0	0
$706$	−14.1421	−0.532246
$707$	−6.62742	−0.249250
$708$	0	0
$709$	9.79899	0.368009	0.184004	−	0.982925i	$-0.441094\pi$
$709$	9.79899	0.368009	0.184004	+	0.982925i	$0.441094\pi$
$710$	3.31371	0.124361
$711$	0	0
$712$	−1.69848	−0.0636534
$713$	−8.48528	−0.317776
$714$	0	0
$715$	0	0
$716$	32.9117	1.22997
$717$	0	0
$718$	10.0589	0.375394
$719$	18.6863	0.696881	0.348441	−	0.937331i	$-0.386711\pi$
$719$	18.6863	0.696881	0.348441	+	0.937331i	$0.386711\pi$
$720$	0	0
$721$	0.343146	0.0127794
$722$	5.78680	0.215362
$723$	0	0
$724$	30.7696	1.14354
$725$	−56.4853	−2.09781
$726$	0	0
$727$	46.2426	1.71504	0.857522	−	0.514447i	$-0.172003\pi$
$727$	46.2426	1.71504	0.857522	+	0.514447i	$0.172003\pi$
$728$	0	0
$729$	0	0
$730$	13.1716	0.487502
$731$	−46.9706	−1.73727
$732$	0	0
$733$	29.7990	1.10065	0.550325	−	0.834950i	$-0.314504\pi$
$733$	29.7990	1.10065	0.550325	+	0.834950i	$0.314504\pi$
$734$	7.27208	0.268417
$735$	0	0
$736$	−4.41421	−0.162710
$737$	32.2843	1.18921
$738$	0	0
$739$	20.6863	0.760958	0.380479	−	0.924790i	$-0.375759\pi$
$739$	20.6863	0.760958	0.380479	+	0.924790i	$0.375759\pi$
$740$	−5.17157	−0.190111
$741$	0	0
$742$	2.20101	0.0808016
$743$	−40.9706	−1.50306	−0.751532	−	0.659697i	$-0.770685\pi$
$743$	−40.9706	−1.50306	−0.751532	+	0.659697i	$0.770685\pi$
$744$	0	0
$745$	−66.2843	−2.42847
$746$	4.34315	0.159014
$747$	0	0
$748$	−23.7157	−0.867133
$749$	−4.97056	−0.181620
$750$	0	0
$751$	49.3553	1.80100	0.900501	−	0.434854i	$-0.143200\pi$
$751$	49.3553	1.80100	0.900501	+	0.434854i	$0.143200\pi$
$752$	−34.9706	−1.27525
$753$	0	0
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	−4.14214	−0.150548	−0.0752742	−	0.997163i	$-0.523983\pi$
$757$	−4.14214	−0.150548	−0.0752742	+	0.997163i	$0.523983\pi$
$758$	−1.41421	−0.0513665
$759$	0	0
$760$	12.1421	0.440442
$761$	7.79899	0.282713	0.141357	−	0.989959i	$-0.454854\pi$
$761$	7.79899	0.282713	0.141357	+	0.989959i	$0.454854\pi$
$762$	0	0
$763$	−8.48528	−0.307188
$764$	42.6274	1.54221
$765$	0	0
$766$	−4.28427	−0.154797
$767$	0	0
$768$	0	0
$769$	−1.02944	−0.0371225	−0.0185612	−	0.999828i	$-0.505909\pi$
$769$	−1.02944	−0.0371225	−0.0185612	+	0.999828i	$0.505909\pi$
$770$	2.34315	0.0844411
$771$	0	0
$772$	−17.6569	−0.635484
$773$	−28.3848	−1.02093	−0.510465	−	0.859899i	$-0.670527\pi$
$773$	−28.3848	−1.02093	−0.510465	+	0.859899i	$0.670527\pi$
$774$	0	0
$775$	−56.4853	−2.02901
$776$	−15.8579	−0.569264
$777$	0	0
$778$	−6.58579	−0.236112
$779$	21.6569	0.775937
$780$	0	0
$781$	−6.62742	−0.237148
$782$	−1.89949	−0.0679258
$783$	0	0
$784$	−19.9706	−0.713234
$785$	−63.1127	−2.25259
$786$	0	0
$787$	31.6985	1.12993	0.564964	−	0.825115i	$-0.308890\pi$
$787$	31.6985	1.12993	0.564964	+	0.825115i	$0.308890\pi$
$788$	27.1127	0.965850
$789$	0	0
$790$	16.8284	0.598729
$791$	−7.37258	−0.262139
$792$	0	0
$793$	0	0
$794$	10.7696	0.382197
$795$	0	0
$796$	−11.4142	−0.404566
$797$	−22.9289	−0.812184	−0.406092	−	0.913832i	$-0.633109\pi$
$797$	−22.9289	−0.812184	−0.406092	+	0.913832i	$0.633109\pi$
$798$	0	0
$799$	−53.4558	−1.89113
$800$	−29.3848	−1.03891
$801$	0	0
$802$	7.47309	0.263884
$803$	−26.3431	−0.929629
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	0	0
$808$	17.9411	0.631167
$809$	43.1127	1.51576	0.757881	−	0.652393i	$-0.226235\pi$
$809$	43.1127	1.51576	0.757881	+	0.652393i	$0.226235\pi$
$810$	0	0
$811$	−16.7696	−0.588859	−0.294429	−	0.955673i	$-0.595130\pi$
$811$	−16.7696	−0.588859	−0.294429	+	0.955673i	$0.595130\pi$
$812$	−9.08831	−0.318937
$813$	0	0
$814$	−0.970563	−0.0340182
$815$	44.9706	1.57525
$816$	0	0
$817$	−22.9706	−0.803638
$818$	−10.6274	−0.371579
$819$	0	0
$820$	−60.2843	−2.10522
$821$	11.3137	0.394851	0.197426	−	0.980318i	$-0.436742\pi$
$821$	11.3137	0.394851	0.197426	+	0.980318i	$0.436742\pi$
$822$	0	0
$823$	32.4853	1.13237	0.566183	−	0.824280i	$-0.308420\pi$
$823$	32.4853	1.13237	0.566183	+	0.824280i	$0.308420\pi$
$824$	−0.928932	−0.0323609
$825$	0	0
$826$	−0.887302	−0.0308732
$827$	−31.7990	−1.10576	−0.552880	−	0.833261i	$-0.686471\pi$
$827$	−31.7990	−1.10576	−0.552880	+	0.833261i	$0.686471\pi$
$828$	0	0
$829$	19.0294	0.660920	0.330460	−	0.943820i	$-0.392796\pi$
$829$	19.0294	0.660920	0.330460	+	0.943820i	$0.392796\pi$
$830$	−1.65685	−0.0575103
$831$	0	0
$832$	0	0
$833$	−30.5269	−1.05769
$834$	0	0
$835$	19.3137	0.668378
$836$	−11.5980	−0.401125
$837$	0	0
$838$	−1.45584	−0.0502913
$839$	−19.0294	−0.656969	−0.328485	−	0.944509i	$-0.606538\pi$
$839$	−19.0294	−0.656969	−0.328485	+	0.944509i	$0.606538\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	−4.74517	−0.163529
$843$	0	0
$844$	−8.20101	−0.282290
$845$	−44.3848	−1.52688
$846$	0	0
$847$	1.75736	0.0603836
$848$	27.2132	0.934505
$849$	0	0
$850$	−12.6447	−0.433708
$851$	0.828427	0.0283981
$852$	0	0
$853$	−15.9411	−0.545814	−0.272907	−	0.962040i	$-0.587985\pi$
$853$	−15.9411	−0.545814	−0.272907	+	0.962040i	$0.587985\pi$
$854$	1.17157	0.0400904
$855$	0	0
$856$	13.4558	0.459911
$857$	18.3431	0.626590	0.313295	−	0.949656i	$-0.398567\pi$
$857$	18.3431	0.626590	0.313295	+	0.949656i	$0.398567\pi$
$858$	0	0
$859$	48.9706	1.67085	0.835427	−	0.549601i	$-0.185220\pi$
$859$	48.9706	1.67085	0.835427	+	0.549601i	$0.185220\pi$
$860$	63.9411	2.18037
$861$	0	0
$862$	−12.9706	−0.441779
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	17.6569	0.600351
$866$	−3.17157	−0.107774
$867$	0	0
$868$	−9.08831	−0.308477
$869$	−33.6569	−1.14173
$870$	0	0
$871$	0	0
$872$	22.9706	0.777881
$873$	0	0
$874$	−0.928932	−0.0314216
$875$	−3.31371	−0.112024
$876$	0	0
$877$	−32.3431	−1.09215	−0.546075	−	0.837736i	$-0.683879\pi$
$877$	−32.3431	−1.09215	−0.546075	+	0.837736i	$0.683879\pi$
$878$	−9.94113	−0.335497
$879$	0	0
$880$	28.9706	0.976597
$881$	1.07107	0.0360852	0.0180426	−	0.999837i	$-0.494257\pi$
$881$	1.07107	0.0360852	0.0180426	+	0.999837i	$0.494257\pi$
$882$	0	0
$883$	−28.4853	−0.958606	−0.479303	−	0.877649i	$-0.659111\pi$
$883$	−28.4853	−0.958606	−0.479303	+	0.877649i	$0.659111\pi$
$884$	0	0
$885$	0	0
$886$	9.94113	0.333979
$887$	11.6569	0.391399	0.195699	−	0.980664i	$-0.437302\pi$
$887$	11.6569	0.391399	0.195699	+	0.980664i	$0.437302\pi$
$888$	0	0
$889$	−7.31371	−0.245294
$890$	1.51472	0.0507735
$891$	0	0
$892$	−38.3431	−1.28382
$893$	−26.1421	−0.874813
$894$	0	0
$895$	−61.4558	−2.05424
$896$	−6.18377	−0.206585
$897$	0	0
$898$	14.3431	0.478637
$899$	72.0000	2.40133
$900$	0	0
$901$	41.5980	1.38583
$902$	−11.3137	−0.376705
$903$	0	0
$904$	19.9584	0.663805
$905$	−57.4558	−1.90990
$906$	0	0
$907$	23.6985	0.786895	0.393448	−	0.919347i	$-0.371282\pi$
$907$	23.6985	0.786895	0.393448	+	0.919347i	$0.371282\pi$
$908$	−40.4853	−1.34355
$909$	0	0
$910$	0	0
$911$	−58.6274	−1.94241	−0.971206	−	0.238239i	$-0.923430\pi$
$911$	−58.6274	−1.94241	−0.971206	+	0.238239i	$0.923430\pi$
$912$	0	0
$913$	3.31371	0.109668
$914$	−0.426407	−0.0141043
$915$	0	0
$916$	4.54416	0.150143
$917$	9.94113	0.328285
$918$	0	0
$919$	39.2132	1.29352	0.646762	−	0.762692i	$-0.276123\pi$
$919$	39.2132	1.29352	0.646762	+	0.762692i	$0.276123\pi$
$920$	5.41421	0.178501
$921$	0	0
$922$	6.14214	0.202280
$923$	0	0
$924$	0	0
$925$	5.51472	0.181323
$926$	10.3431	0.339897
$927$	0	0
$928$	37.4558	1.22955
$929$	0.686292	0.0225165	0.0112582	−	0.999937i	$-0.496416\pi$
$929$	0.686292	0.0225165	0.0112582	+	0.999937i	$0.496416\pi$
$930$	0	0
$931$	−14.9289	−0.489276
$932$	−18.9117	−0.619473
$933$	0	0
$934$	−14.1421	−0.462745
$935$	44.2843	1.44825
$936$	0	0
$937$	20.3431	0.664582	0.332291	−	0.943177i	$-0.392178\pi$
$937$	20.3431	0.664582	0.332291	+	0.943177i	$0.392178\pi$
$938$	2.76955	0.0904291
$939$	0	0
$940$	72.7696	2.37348
$941$	12.3848	0.403732	0.201866	−	0.979413i	$-0.435299\pi$
$941$	12.3848	0.403732	0.201866	+	0.979413i	$0.435299\pi$
$942$	0	0
$943$	9.65685	0.314470
$944$	−10.9706	−0.357061
$945$	0	0
$946$	12.0000	0.390154
$947$	25.3137	0.822585	0.411292	−	0.911503i	$-0.365077\pi$
$947$	25.3137	0.822585	0.411292	+	0.911503i	$0.365077\pi$
$948$	0	0
$949$	0	0
$950$	−6.18377	−0.200628
$951$	0	0
$952$	−4.25988	−0.138064
$953$	51.0122	1.65245	0.826224	−	0.563342i	$-0.190485\pi$
$953$	51.0122	1.65245	0.826224	+	0.563342i	$0.190485\pi$
$954$	0	0
$955$	−79.5980	−2.57573
$956$	−6.05887	−0.195958
$957$	0	0
$958$	−16.2843	−0.526121
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	−20.0589	−0.646053
$965$	32.9706	1.06136
$966$	0	0
$967$	−32.4853	−1.04466	−0.522328	−	0.852745i	$-0.674936\pi$
$967$	−32.4853	−1.04466	−0.522328	+	0.852745i	$0.674936\pi$
$968$	−4.75736	−0.152907
$969$	0	0
$970$	14.1421	0.454077
$971$	−55.1127	−1.76865	−0.884325	−	0.466871i	$-0.845381\pi$
$971$	−55.1127	−1.76865	−0.884325	+	0.466871i	$0.845381\pi$
$972$	0	0
$973$	2.34315	0.0751178
$974$	10.3431	0.331416
$975$	0	0
$976$	14.4853	0.463663
$977$	−18.0416	−0.577203	−0.288601	−	0.957449i	$-0.593190\pi$
$977$	−18.0416	−0.577203	−0.288601	+	0.957449i	$0.593190\pi$
$978$	0	0
$979$	−3.02944	−0.0968212
$980$	41.5563	1.32747
$981$	0	0
$982$	−12.1421	−0.387471
$983$	0.686292	0.0218893	0.0109446	−	0.999940i	$-0.496516\pi$
$983$	0.686292	0.0218893	0.0109446	+	0.999940i	$0.496516\pi$
$984$	0	0
$985$	−50.6274	−1.61312
$986$	16.1177	0.513294
$987$	0	0
$988$	0	0
$989$	−10.2426	−0.325697
$990$	0	0
$991$	−25.4558	−0.808632	−0.404316	−	0.914619i	$-0.632490\pi$
$991$	−25.4558	−0.808632	−0.404316	+	0.914619i	$0.632490\pi$
$992$	37.4558	1.18922
$993$	0	0
$994$	−0.568542	−0.0180331
$995$	21.3137	0.675690
$996$	0	0
$997$	36.9706	1.17087	0.585435	−	0.810720i	$-0.300924\pi$
$997$	36.9706	1.17087	0.585435	+	0.810720i	$0.300924\pi$
$998$	3.11270	0.0985307
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.2.a.e.1.1	yes	2
3.2	odd	2		207.2.a.b.1.2	✓	2
4.3	odd	2		3312.2.a.be.1.2		2
5.4	even	2		5175.2.a.bc.1.2		2
12.11	even	2		3312.2.a.u.1.1		2
15.14	odd	2		5175.2.a.bo.1.1		2
23.22	odd	2		4761.2.a.z.1.1		2
69.68	even	2		4761.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.a.b.1.2	✓	2	3.2	odd	2
207.2.a.e.1.1	yes	2	1.1	even	1	trivial
3312.2.a.u.1.1		2	12.11	even	2
3312.2.a.be.1.2		2	4.3	odd	2
4761.2.a.k.1.2		2	69.68	even	2
4761.2.a.z.1.1		2	23.22	odd	2
5175.2.a.bc.1.2		2	5.4	even	2
5175.2.a.bo.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} +3.41421 q^{5} -0.585786 q^{7} +1.58579 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -1.82843 q^{4} +3.41421 q^{5} -0.585786 q^{7} +1.58579 q^{8} -1.41421 q^{10} +2.82843 q^{11} +0.242641 q^{14} +3.00000 q^{16} +4.58579 q^{17} +2.24264 q^{19} -6.24264 q^{20} -1.17157 q^{22} +1.00000 q^{23} +6.65685 q^{25} +1.07107 q^{28} -8.48528 q^{29} -8.48528 q^{31} -4.41421 q^{32} -1.89949 q^{34} -2.00000 q^{35} +0.828427 q^{37} -0.928932 q^{38} +5.41421 q^{40} +9.65685 q^{41} -10.2426 q^{43} -5.17157 q^{44} -0.414214 q^{46} -11.6569 q^{47} -6.65685 q^{49} -2.75736 q^{50} +9.07107 q^{53} +9.65685 q^{55} -0.928932 q^{56} +3.51472 q^{58} -3.65685 q^{59} +4.82843 q^{61} +3.51472 q^{62} -4.17157 q^{64} +11.4142 q^{67} -8.38478 q^{68} +0.828427 q^{70} -2.34315 q^{71} -9.31371 q^{73} -0.343146 q^{74} -4.10051 q^{76} -1.65685 q^{77} -11.8995 q^{79} +10.2426 q^{80} -4.00000 q^{82} +1.17157 q^{83} +15.6569 q^{85} +4.24264 q^{86} +4.48528 q^{88} -1.07107 q^{89} -1.82843 q^{92} +4.82843 q^{94} +7.65685 q^{95} -10.0000 q^{97} +2.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} - 8 q^{14} + 6 q^{16} + 12 q^{17} - 4 q^{19} - 4 q^{20} - 8 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{28} - 6 q^{32} + 16 q^{34} - 4 q^{35} - 4 q^{37} - 16 q^{38} + 8 q^{40} + 8 q^{41} - 12 q^{43} - 16 q^{44} + 2 q^{46} - 12 q^{47} - 2 q^{49} - 14 q^{50} + 4 q^{53} + 8 q^{55} - 16 q^{56} + 24 q^{58} + 4 q^{59} + 4 q^{61} + 24 q^{62} - 14 q^{64} + 20 q^{67} + 20 q^{68} - 4 q^{70} - 16 q^{71} + 4 q^{73} - 12 q^{74} - 28 q^{76} + 8 q^{77} - 4 q^{79} + 12 q^{80} - 8 q^{82} + 8 q^{83} + 20 q^{85} - 8 q^{88} + 12 q^{89} + 2 q^{92} + 4 q^{94} + 4 q^{95} - 20 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 - 8 * q^14 + 6 * q^16 + 12 * q^17 - 4 * q^19 - 4 * q^20 - 8 * q^22 + 2 * q^23 + 2 * q^25 - 12 * q^28 - 6 * q^32 + 16 * q^34 - 4 * q^35 - 4 * q^37 - 16 * q^38 + 8 * q^40 + 8 * q^41 - 12 * q^43 - 16 * q^44 + 2 * q^46 - 12 * q^47 - 2 * q^49 - 14 * q^50 + 4 * q^53 + 8 * q^55 - 16 * q^56 + 24 * q^58 + 4 * q^59 + 4 * q^61 + 24 * q^62 - 14 * q^64 + 20 * q^67 + 20 * q^68 - 4 * q^70 - 16 * q^71 + 4 * q^73 - 12 * q^74 - 28 * q^76 + 8 * q^77 - 4 * q^79 + 12 * q^80 - 8 * q^82 + 8 * q^83 + 20 * q^85 - 8 * q^88 + 12 * q^89 + 2 * q^92 + 4 * q^94 + 4 * q^95 - 20 * q^97 + 14 * q^98

Introduction

L-functions

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