LMFDB - Embedded newform 7623.2.a.ct.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
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} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.669744$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.669744 q^{2} -1.55144 q^{4} -2.14378 q^{5} +1.00000 q^{7} +2.37856 q^{8} +O(q^{10})$	$q-0.669744 q^{2} -1.55144 q^{4} -2.14378 q^{5} +1.00000 q^{7} +2.37856 q^{8} +1.43578 q^{10} +2.52826 q^{13} -0.669744 q^{14} +1.50986 q^{16} +1.79092 q^{17} -6.72740 q^{19} +3.32595 q^{20} +3.16429 q^{23} -0.404214 q^{25} -1.69329 q^{26} -1.55144 q^{28} +0.924170 q^{29} +3.00178 q^{31} -5.76834 q^{32} -1.19946 q^{34} -2.14378 q^{35} -1.50718 q^{37} +4.50564 q^{38} -5.09910 q^{40} -5.56104 q^{41} -8.42985 q^{43} -2.11926 q^{46} -4.39771 q^{47} +1.00000 q^{49} +0.270720 q^{50} -3.92246 q^{52} -0.667421 q^{53} +2.37856 q^{56} -0.618957 q^{58} +0.368360 q^{59} +5.01149 q^{61} -2.01043 q^{62} +0.843584 q^{64} -5.42003 q^{65} -0.902129 q^{67} -2.77851 q^{68} +1.43578 q^{70} +14.8694 q^{71} +8.03816 q^{73} +1.00942 q^{74} +10.4372 q^{76} +4.05668 q^{79} -3.23681 q^{80} +3.72447 q^{82} +4.05134 q^{83} -3.83934 q^{85} +5.64584 q^{86} +8.30727 q^{89} +2.52826 q^{91} -4.90921 q^{92} +2.94534 q^{94} +14.4221 q^{95} +8.51583 q^{97} -0.669744 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - 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 $

$2$	−0.669744	−0.473580	−0.236790	−	0.971561i	$-0.576095\pi$
$2$	−0.669744	−0.473580	−0.236790	+	0.971561i	$0.576095\pi$
$3$	0	0
$3$	0	0
$4$	−1.55144	−0.775722
$5$	−2.14378	−0.958727	−0.479363	−	0.877616i	$-0.659132\pi$
$5$	−2.14378	−0.958727	−0.479363	+	0.877616i	$0.659132\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.37856	0.840947
$9$	0	0
$10$	1.43578	0.454034
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.52826	0.701214	0.350607	−	0.936523i	$-0.385975\pi$
$13$	2.52826	0.701214	0.350607	+	0.936523i	$0.385975\pi$
$14$	−0.669744	−0.178997
$15$	0	0
$16$	1.50986	0.377465
$17$	1.79092	0.434362	0.217181	−	0.976131i	$-0.430314\pi$
$17$	1.79092	0.434362	0.217181	+	0.976131i	$0.430314\pi$
$18$	0	0
$19$	−6.72740	−1.54337	−0.771686	−	0.636004i	$-0.780586\pi$
$19$	−6.72740	−1.54337	−0.771686	+	0.636004i	$0.780586\pi$
$20$	3.32595	0.743705
$21$	0	0
$22$	0	0
$23$	3.16429	0.659799	0.329900	−	0.944016i	$-0.392985\pi$
$23$	3.16429	0.659799	0.329900	+	0.944016i	$0.392985\pi$
$24$	0	0
$25$	−0.404214	−0.0808427
$26$	−1.69329	−0.332081
$27$	0	0
$28$	−1.55144	−0.293195
$29$	0.924170	0.171614	0.0858070	−	0.996312i	$-0.472653\pi$
$29$	0.924170	0.171614	0.0858070	+	0.996312i	$0.472653\pi$
$30$	0	0
$31$	3.00178	0.539136	0.269568	−	0.962981i	$-0.413119\pi$
$31$	3.00178	0.539136	0.269568	+	0.962981i	$0.413119\pi$
$32$	−5.76834	−1.01971
$33$	0	0
$34$	−1.19946	−0.205705
$35$	−2.14378	−0.362365
$36$	0	0
$37$	−1.50718	−0.247779	−0.123889	−	0.992296i	$-0.539537\pi$
$37$	−1.50718	−0.247779	−0.123889	+	0.992296i	$0.539537\pi$
$38$	4.50564	0.730911
$39$	0	0
$40$	−5.09910	−0.806239
$41$	−5.56104	−0.868488	−0.434244	−	0.900795i	$-0.642984\pi$
$41$	−5.56104	−0.868488	−0.434244	+	0.900795i	$0.642984\pi$
$42$	0	0
$43$	−8.42985	−1.28554	−0.642770	−	0.766059i	$-0.722215\pi$
$43$	−8.42985	−1.28554	−0.642770	+	0.766059i	$0.722215\pi$
$44$	0	0
$45$	0	0
$46$	−2.11926	−0.312468
$47$	−4.39771	−0.641471	−0.320736	−	0.947169i	$-0.603930\pi$
$47$	−4.39771	−0.641471	−0.320736	+	0.947169i	$0.603930\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.270720	0.0382855
$51$	0	0
$52$	−3.92246	−0.543947
$53$	−0.667421	−0.0916773	−0.0458387	−	0.998949i	$-0.514596\pi$
$53$	−0.667421	−0.0916773	−0.0458387	+	0.998949i	$0.514596\pi$
$54$	0	0
$55$	0	0
$56$	2.37856	0.317848
$57$	0	0
$58$	−0.618957	−0.0812731
$59$	0.368360	0.0479564	0.0239782	−	0.999712i	$-0.492367\pi$
$59$	0.368360	0.0479564	0.0239782	+	0.999712i	$0.492367\pi$
$60$	0	0
$61$	5.01149	0.641655	0.320828	−	0.947138i	$-0.396039\pi$
$61$	5.01149	0.641655	0.320828	+	0.947138i	$0.396039\pi$
$62$	−2.01043	−0.255324
$63$	0	0
$64$	0.843584	0.105448
$65$	−5.42003	−0.672273
$66$	0	0
$67$	−0.902129	−0.110213	−0.0551063	−	0.998480i	$-0.517550\pi$
$67$	−0.902129	−0.110213	−0.0551063	+	0.998480i	$0.517550\pi$
$68$	−2.77851	−0.336944
$69$	0	0
$70$	1.43578	0.171609
$71$	14.8694	1.76467	0.882335	−	0.470623i	$-0.155971\pi$
$71$	14.8694	1.76467	0.882335	+	0.470623i	$0.155971\pi$
$72$	0	0
$73$	8.03816	0.940795	0.470398	−	0.882455i	$-0.344110\pi$
$73$	8.03816	0.940795	0.470398	+	0.882455i	$0.344110\pi$
$74$	1.00942	0.117343
$75$	0	0
$76$	10.4372	1.19723
$77$	0	0
$78$	0	0
$79$	4.05668	0.456412	0.228206	−	0.973613i	$-0.426714\pi$
$79$	4.05668	0.456412	0.228206	+	0.973613i	$0.426714\pi$
$80$	−3.23681	−0.361886
$81$	0	0
$82$	3.72447	0.411299
$83$	4.05134	0.444692	0.222346	−	0.974968i	$-0.428628\pi$
$83$	4.05134	0.444692	0.222346	+	0.974968i	$0.428628\pi$
$84$	0	0
$85$	−3.83934	−0.416434
$86$	5.64584	0.608807
$87$	0	0
$88$	0	0
$89$	8.30727	0.880569	0.440284	−	0.897858i	$-0.354878\pi$
$89$	8.30727	0.880569	0.440284	+	0.897858i	$0.354878\pi$
$90$	0	0
$91$	2.52826	0.265034
$92$	−4.90921	−0.511820
$93$	0	0
$94$	2.94534	0.303788
$95$	14.4221	1.47967
$96$	0	0
$97$	8.51583	0.864652	0.432326	−	0.901717i	$-0.357693\pi$
$97$	8.51583	0.864652	0.432326	+	0.901717i	$0.357693\pi$
$98$	−0.669744	−0.0676544
$99$	0	0
$100$	0.627115	0.0627115
$101$	4.02707	0.400708	0.200354	−	0.979724i	$-0.435791\pi$
$101$	4.02707	0.400708	0.200354	+	0.979724i	$0.435791\pi$
$102$	0	0
$103$	−17.6258	−1.73672	−0.868362	−	0.495930i	$-0.834827\pi$
$103$	−17.6258	−1.73672	−0.868362	+	0.495930i	$0.834827\pi$
$104$	6.01362	0.589684
$105$	0	0
$106$	0.447001	0.0434166
$107$	15.4037	1.48913	0.744567	−	0.667548i	$-0.232656\pi$
$107$	15.4037	1.48913	0.744567	+	0.667548i	$0.232656\pi$
$108$	0	0
$109$	−18.9265	−1.81283	−0.906416	−	0.422386i	$-0.861193\pi$
$109$	−18.9265	−1.81283	−0.906416	+	0.422386i	$0.861193\pi$
$110$	0	0
$111$	0	0
$112$	1.50986	0.142669
$113$	−1.68062	−0.158100	−0.0790498	−	0.996871i	$-0.525189\pi$
$113$	−1.68062	−0.158100	−0.0790498	+	0.996871i	$0.525189\pi$
$114$	0	0
$115$	−6.78353	−0.632567
$116$	−1.43380	−0.133125
$117$	0	0
$118$	−0.246707	−0.0227112
$119$	1.79092	0.164173
$120$	0	0
$121$	0	0
$122$	−3.35641	−0.303875
$123$	0	0
$124$	−4.65710	−0.418220
$125$	11.5854	1.03623
$126$	0	0
$127$	17.5669	1.55881	0.779406	−	0.626519i	$-0.215521\pi$
$127$	17.5669	1.55881	0.779406	+	0.626519i	$0.215521\pi$
$128$	10.9717	0.969769
$129$	0	0
$130$	3.63003	0.318375
$131$	−6.72557	−0.587616	−0.293808	−	0.955865i	$-0.594923\pi$
$131$	−6.72557	−0.587616	−0.293808	+	0.955865i	$0.594923\pi$
$132$	0	0
$133$	−6.72740	−0.583340
$134$	0.604195	0.0521945
$135$	0	0
$136$	4.25981	0.365275
$137$	−13.8676	−1.18479	−0.592396	−	0.805647i	$-0.701818\pi$
$137$	−13.8676	−1.18479	−0.592396	+	0.805647i	$0.701818\pi$
$138$	0	0
$139$	14.2707	1.21043	0.605214	−	0.796063i	$-0.293088\pi$
$139$	14.2707	1.21043	0.605214	+	0.796063i	$0.293088\pi$
$140$	3.32595	0.281094
$141$	0	0
$142$	−9.95867	−0.835713
$143$	0	0
$144$	0	0
$145$	−1.98122	−0.164531
$146$	−5.38351	−0.445542
$147$	0	0
$148$	2.33830	0.192207
$149$	−2.62292	−0.214878	−0.107439	−	0.994212i	$-0.534265\pi$
$149$	−2.62292	−0.214878	−0.107439	+	0.994212i	$0.534265\pi$
$150$	0	0
$151$	−2.98960	−0.243290	−0.121645	−	0.992574i	$-0.538817\pi$
$151$	−2.98960	−0.243290	−0.121645	+	0.992574i	$0.538817\pi$
$152$	−16.0015	−1.29789
$153$	0	0
$154$	0	0
$155$	−6.43516	−0.516884
$156$	0	0
$157$	11.2547	0.898222	0.449111	−	0.893476i	$-0.351741\pi$
$157$	11.2547	0.898222	0.449111	+	0.893476i	$0.351741\pi$
$158$	−2.71694	−0.216148
$159$	0	0
$160$	12.3660	0.977621
$161$	3.16429	0.249381
$162$	0	0
$163$	−18.2030	−1.42577	−0.712883	−	0.701283i	$-0.752611\pi$
$163$	−18.2030	−1.42577	−0.712883	+	0.701283i	$0.752611\pi$
$164$	8.62763	0.673705
$165$	0	0
$166$	−2.71336	−0.210598
$167$	−19.9312	−1.54233	−0.771163	−	0.636638i	$-0.780325\pi$
$167$	−19.9312	−1.54233	−0.771163	+	0.636638i	$0.780325\pi$
$168$	0	0
$169$	−6.60789	−0.508299
$170$	2.57137	0.197215
$171$	0	0
$172$	13.0784	0.997221
$173$	−5.99008	−0.455417	−0.227709	−	0.973729i	$-0.573123\pi$
$173$	−5.99008	−0.455417	−0.227709	+	0.973729i	$0.573123\pi$
$174$	0	0
$175$	−0.404214	−0.0305557
$176$	0	0
$177$	0	0
$178$	−5.56374	−0.417020
$179$	1.61894	0.121005	0.0605026	−	0.998168i	$-0.480730\pi$
$179$	1.61894	0.121005	0.0605026	+	0.998168i	$0.480730\pi$
$180$	0	0
$181$	2.42682	0.180384	0.0901921	−	0.995924i	$-0.471252\pi$
$181$	2.42682	0.180384	0.0901921	+	0.995924i	$0.471252\pi$
$182$	−1.69329	−0.125515
$183$	0	0
$184$	7.52643	0.554856
$185$	3.23106	0.237552
$186$	0	0
$187$	0	0
$188$	6.82279	0.497603
$189$	0	0
$190$	−9.65909	−0.700744
$191$	−12.7009	−0.919006	−0.459503	−	0.888176i	$-0.651973\pi$
$191$	−12.7009	−0.919006	−0.459503	+	0.888176i	$0.651973\pi$
$192$	0	0
$193$	−1.75931	−0.126638	−0.0633190	−	0.997993i	$-0.520169\pi$
$193$	−1.75931	−0.126638	−0.0633190	+	0.997993i	$0.520169\pi$
$194$	−5.70343	−0.409482
$195$	0	0
$196$	−1.55144	−0.110817
$197$	0.903053	0.0643399	0.0321699	−	0.999482i	$-0.489758\pi$
$197$	0.903053	0.0643399	0.0321699	+	0.999482i	$0.489758\pi$
$198$	0	0
$199$	−15.6296	−1.10795	−0.553976	−	0.832533i	$-0.686890\pi$
$199$	−15.6296	−1.10795	−0.553976	+	0.832533i	$0.686890\pi$
$200$	−0.961445	−0.0679845
$201$	0	0
$202$	−2.69710	−0.189768
$203$	0.924170	0.0648640
$204$	0	0
$205$	11.9216	0.832643
$206$	11.8048	0.822479
$207$	0	0
$208$	3.81733	0.264684
$209$	0	0
$210$	0	0
$211$	14.7687	1.01672	0.508360	−	0.861145i	$-0.330252\pi$
$211$	14.7687	1.01672	0.508360	+	0.861145i	$0.330252\pi$
$212$	1.03547	0.0711161
$213$	0	0
$214$	−10.3165	−0.705225
$215$	18.0717	1.23248
$216$	0	0
$217$	3.00178	0.203774
$218$	12.6759	0.858522
$219$	0	0
$220$	0	0
$221$	4.52791	0.304581
$222$	0	0
$223$	−1.84537	−0.123575	−0.0617875	−	0.998089i	$-0.519680\pi$
$223$	−1.84537	−0.123575	−0.0617875	+	0.998089i	$0.519680\pi$
$224$	−5.76834	−0.385413
$225$	0	0
$226$	1.12559	0.0748728
$227$	21.9399	1.45620	0.728102	−	0.685468i	$-0.240403\pi$
$227$	21.9399	1.45620	0.728102	+	0.685468i	$0.240403\pi$
$228$	0	0
$229$	−20.2518	−1.33828	−0.669138	−	0.743138i	$-0.733337\pi$
$229$	−20.2518	−1.33828	−0.669138	+	0.743138i	$0.733337\pi$
$230$	4.54323	0.299571
$231$	0	0
$232$	2.19819	0.144318
$233$	−21.0626	−1.37986	−0.689928	−	0.723878i	$-0.742358\pi$
$233$	−21.0626	−1.37986	−0.689928	+	0.723878i	$0.742358\pi$
$234$	0	0
$235$	9.42771	0.614996
$236$	−0.571490	−0.0372008
$237$	0	0
$238$	−1.19946	−0.0777493
$239$	−15.4965	−1.00239	−0.501194	−	0.865335i	$-0.667106\pi$
$239$	−15.4965	−1.00239	−0.501194	+	0.865335i	$0.667106\pi$
$240$	0	0
$241$	14.0848	0.907283	0.453641	−	0.891184i	$-0.350125\pi$
$241$	14.0848	0.907283	0.453641	+	0.891184i	$0.350125\pi$
$242$	0	0
$243$	0	0
$244$	−7.77504	−0.497746
$245$	−2.14378	−0.136961
$246$	0	0
$247$	−17.0086	−1.08223
$248$	7.13991	0.453385
$249$	0	0
$250$	−7.75928	−0.490740
$251$	1.07727	0.0679966	0.0339983	−	0.999422i	$-0.489176\pi$
$251$	1.07727	0.0679966	0.0339983	+	0.999422i	$0.489176\pi$
$252$	0	0
$253$	0	0
$254$	−11.7653	−0.738223
$255$	0	0
$256$	−9.03539	−0.564712
$257$	−13.0810	−0.815971	−0.407986	−	0.912988i	$-0.633769\pi$
$257$	−13.0810	−0.815971	−0.407986	+	0.912988i	$0.633769\pi$
$258$	0	0
$259$	−1.50718	−0.0936516
$260$	8.40887	0.521496
$261$	0	0
$262$	4.50441	0.278283
$263$	9.57216	0.590245	0.295122	−	0.955459i	$-0.404640\pi$
$263$	9.57216	0.590245	0.295122	+	0.955459i	$0.404640\pi$
$264$	0	0
$265$	1.43080	0.0878935
$266$	4.50564	0.276258
$267$	0	0
$268$	1.39960	0.0854943
$269$	4.76853	0.290743	0.145371	−	0.989377i	$-0.453562\pi$
$269$	4.76853	0.290743	0.145371	+	0.989377i	$0.453562\pi$
$270$	0	0
$271$	−20.0523	−1.21809	−0.609045	−	0.793136i	$-0.708447\pi$
$271$	−20.0523	−1.21809	−0.609045	+	0.793136i	$0.708447\pi$
$272$	2.70404	0.163957
$273$	0	0
$274$	9.28776	0.561094
$275$	0	0
$276$	0	0
$277$	−11.6024	−0.697124	−0.348562	−	0.937286i	$-0.613330\pi$
$277$	−11.6024	−0.697124	−0.348562	+	0.937286i	$0.613330\pi$
$278$	−9.55773	−0.573235
$279$	0	0
$280$	−5.09910	−0.304730
$281$	−12.0788	−0.720562	−0.360281	−	0.932844i	$-0.617319\pi$
$281$	−12.0788	−0.720562	−0.360281	+	0.932844i	$0.617319\pi$
$282$	0	0
$283$	21.9932	1.30736	0.653681	−	0.756771i	$-0.273224\pi$
$283$	21.9932	1.30736	0.653681	+	0.756771i	$0.273224\pi$
$284$	−23.0690	−1.36889
$285$	0	0
$286$	0	0
$287$	−5.56104	−0.328258
$288$	0	0
$289$	−13.7926	−0.811330
$290$	1.32691	0.0779187
$291$	0	0
$292$	−12.4707	−0.729795
$293$	−1.41721	−0.0827941	−0.0413971	−	0.999143i	$-0.513181\pi$
$293$	−1.41721	−0.0827941	−0.0413971	+	0.999143i	$0.513181\pi$
$294$	0	0
$295$	−0.789683	−0.0459771
$296$	−3.58491	−0.208369
$297$	0	0
$298$	1.75669	0.101762
$299$	8.00014	0.462660
$300$	0	0
$301$	−8.42985	−0.485888
$302$	2.00227	0.115217
$303$	0	0
$304$	−10.1574	−0.582570
$305$	−10.7435	−0.615172
$306$	0	0
$307$	29.4646	1.68163	0.840817	−	0.541319i	$-0.182075\pi$
$307$	29.4646	1.68163	0.840817	+	0.541319i	$0.182075\pi$
$308$	0	0
$309$	0	0
$310$	4.30991	0.244786
$311$	−26.8787	−1.52415	−0.762075	−	0.647489i	$-0.775819\pi$
$311$	−26.8787	−1.52415	−0.762075	+	0.647489i	$0.775819\pi$
$312$	0	0
$313$	−4.49270	−0.253942	−0.126971	−	0.991906i	$-0.540526\pi$
$313$	−4.49270	−0.253942	−0.126971	+	0.991906i	$0.540526\pi$
$314$	−7.53776	−0.425381
$315$	0	0
$316$	−6.29371	−0.354049
$317$	10.9501	0.615017	0.307508	−	0.951545i	$-0.400505\pi$
$317$	10.9501	0.615017	0.307508	+	0.951545i	$0.400505\pi$
$318$	0	0
$319$	0	0
$320$	−1.80846	−0.101096
$321$	0	0
$322$	−2.11926	−0.118102
$323$	−12.0482	−0.670382
$324$	0	0
$325$	−1.02196	−0.0566880
$326$	12.1913	0.675215
$327$	0	0
$328$	−13.2272	−0.730353
$329$	−4.39771	−0.242453
$330$	0	0
$331$	16.5226	0.908166	0.454083	−	0.890959i	$-0.349967\pi$
$331$	16.5226	0.908166	0.454083	+	0.890959i	$0.349967\pi$
$332$	−6.28543	−0.344958
$333$	0	0
$334$	13.3488	0.730415
$335$	1.93397	0.105664
$336$	0	0
$337$	−12.1207	−0.660258	−0.330129	−	0.943936i	$-0.607092\pi$
$337$	−12.1207	−0.660258	−0.330129	+	0.943936i	$0.607092\pi$
$338$	4.42559	0.240721
$339$	0	0
$340$	5.95651	0.323037
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−20.0509	−1.08107
$345$	0	0
$346$	4.01182	0.215677
$347$	24.2851	1.30369	0.651847	−	0.758350i	$-0.273994\pi$
$347$	24.2851	1.30369	0.651847	+	0.758350i	$0.273994\pi$
$348$	0	0
$349$	−3.31880	−0.177651	−0.0888257	−	0.996047i	$-0.528311\pi$
$349$	−3.31880	−0.177651	−0.0888257	+	0.996047i	$0.528311\pi$
$350$	0.270720	0.0144706
$351$	0	0
$352$	0	0
$353$	−20.3272	−1.08191	−0.540955	−	0.841051i	$-0.681937\pi$
$353$	−20.3272	−1.08191	−0.540955	+	0.841051i	$0.681937\pi$
$354$	0	0
$355$	−31.8766	−1.69184
$356$	−12.8883	−0.683076
$357$	0	0
$358$	−1.08427	−0.0573057
$359$	−29.0412	−1.53274	−0.766369	−	0.642401i	$-0.777938\pi$
$359$	−29.0412	−1.53274	−0.766369	+	0.642401i	$0.777938\pi$
$360$	0	0
$361$	26.2579	1.38200
$362$	−1.62535	−0.0854264
$363$	0	0
$364$	−3.92246	−0.205593
$365$	−17.2320	−0.901966
$366$	0	0
$367$	−22.7588	−1.18800	−0.593999	−	0.804465i	$-0.702452\pi$
$367$	−22.7588	−1.18800	−0.593999	+	0.804465i	$0.702452\pi$
$368$	4.77763	0.249051
$369$	0	0
$370$	−2.16398	−0.112500
$371$	−0.667421	−0.0346508
$372$	0	0
$373$	−22.2412	−1.15160	−0.575802	−	0.817589i	$-0.695310\pi$
$373$	−22.2412	−1.15160	−0.575802	+	0.817589i	$0.695310\pi$
$374$	0	0
$375$	0	0
$376$	−10.4602	−0.539444
$377$	2.33654	0.120338
$378$	0	0
$379$	33.4707	1.71927	0.859637	−	0.510906i	$-0.170690\pi$
$379$	33.4707	1.71927	0.859637	+	0.510906i	$0.170690\pi$
$380$	−22.3750	−1.14781
$381$	0	0
$382$	8.50637	0.435224
$383$	6.14592	0.314042	0.157021	−	0.987595i	$-0.449811\pi$
$383$	6.14592	0.314042	0.157021	+	0.987595i	$0.449811\pi$
$384$	0	0
$385$	0	0
$386$	1.17829	0.0599733
$387$	0	0
$388$	−13.2118	−0.670729
$389$	−7.40364	−0.375380	−0.187690	−	0.982228i	$-0.560100\pi$
$389$	−7.40364	−0.375380	−0.187690	+	0.982228i	$0.560100\pi$
$390$	0	0
$391$	5.66698	0.286592
$392$	2.37856	0.120135
$393$	0	0
$394$	−0.604814	−0.0304701
$395$	−8.69662	−0.437575
$396$	0	0
$397$	17.8079	0.893752	0.446876	−	0.894596i	$-0.352537\pi$
$397$	17.8079	0.893752	0.446876	+	0.894596i	$0.352537\pi$
$398$	10.4678	0.524704
$399$	0	0
$400$	−0.610307	−0.0305153
$401$	−25.6789	−1.28234	−0.641170	−	0.767398i	$-0.721551\pi$
$401$	−25.6789	−1.28234	−0.641170	+	0.767398i	$0.721551\pi$
$402$	0	0
$403$	7.58929	0.378050
$404$	−6.24777	−0.310838
$405$	0	0
$406$	−0.618957	−0.0307183
$407$	0	0
$408$	0	0
$409$	1.33754	0.0661373	0.0330687	−	0.999453i	$-0.489472\pi$
$409$	1.33754	0.0661373	0.0330687	+	0.999453i	$0.489472\pi$
$410$	−7.98444	−0.394323
$411$	0	0
$412$	27.3455	1.34722
$413$	0.368360	0.0181258
$414$	0	0
$415$	−8.68518	−0.426339
$416$	−14.5839	−0.715033
$417$	0	0
$418$	0	0
$419$	−37.4618	−1.83013	−0.915064	−	0.403310i	$-0.867860\pi$
$419$	−37.4618	−1.83013	−0.915064	+	0.403310i	$0.867860\pi$
$420$	0	0
$421$	8.26156	0.402644	0.201322	−	0.979525i	$-0.435476\pi$
$421$	8.26156	0.402644	0.201322	+	0.979525i	$0.435476\pi$
$422$	−9.89125	−0.481499
$423$	0	0
$424$	−1.58750	−0.0770958
$425$	−0.723914	−0.0351150
$426$	0	0
$427$	5.01149	0.242523
$428$	−23.8980	−1.15515
$429$	0	0
$430$	−12.1034	−0.583679
$431$	−32.6564	−1.57301	−0.786503	−	0.617587i	$-0.788110\pi$
$431$	−32.6564	−1.57301	−0.786503	+	0.617587i	$0.788110\pi$
$432$	0	0
$433$	15.7953	0.759072	0.379536	−	0.925177i	$-0.376084\pi$
$433$	15.7953	0.759072	0.379536	+	0.925177i	$0.376084\pi$
$434$	−2.01043	−0.0965035
$435$	0	0
$436$	29.3634	1.40625
$437$	−21.2874	−1.01832
$438$	0	0
$439$	−20.6942	−0.987678	−0.493839	−	0.869553i	$-0.664407\pi$
$439$	−20.6942	−0.987678	−0.493839	+	0.869553i	$0.664407\pi$
$440$	0	0
$441$	0	0
$442$	−3.03254	−0.144243
$443$	−30.0955	−1.42988	−0.714939	−	0.699186i	$-0.753546\pi$
$443$	−30.0955	−1.42988	−0.714939	+	0.699186i	$0.753546\pi$
$444$	0	0
$445$	−17.8089	−0.844225
$446$	1.23592	0.0585227
$447$	0	0
$448$	0.843584	0.0398556
$449$	−36.3944	−1.71756	−0.858779	−	0.512345i	$-0.828777\pi$
$449$	−36.3944	−1.71756	−0.858779	+	0.512345i	$0.828777\pi$
$450$	0	0
$451$	0	0
$452$	2.60739	0.122641
$453$	0	0
$454$	−14.6941	−0.689630
$455$	−5.42003	−0.254095
$456$	0	0
$457$	−9.79401	−0.458145	−0.229072	−	0.973409i	$-0.573569\pi$
$457$	−9.79401	−0.458145	−0.229072	+	0.973409i	$0.573569\pi$
$458$	13.5635	0.633781
$459$	0	0
$460$	10.5243	0.490696
$461$	21.8596	1.01810	0.509052	−	0.860736i	$-0.329996\pi$
$461$	21.8596	1.01810	0.509052	+	0.860736i	$0.329996\pi$
$462$	0	0
$463$	6.75889	0.314112	0.157056	−	0.987590i	$-0.449800\pi$
$463$	6.75889	0.314112	0.157056	+	0.987590i	$0.449800\pi$
$464$	1.39537	0.0647784
$465$	0	0
$466$	14.1065	0.653473
$467$	30.0232	1.38931	0.694654	−	0.719344i	$-0.255558\pi$
$467$	30.0232	1.38931	0.694654	+	0.719344i	$0.255558\pi$
$468$	0	0
$469$	−0.902129	−0.0416565
$470$	−6.31415	−0.291250
$471$	0	0
$472$	0.876166	0.0403288
$473$	0	0
$474$	0	0
$475$	2.71931	0.124770
$476$	−2.77851	−0.127353
$477$	0	0
$478$	10.3787	0.474711
$479$	6.00353	0.274308	0.137154	−	0.990550i	$-0.456204\pi$
$479$	6.00353	0.274308	0.137154	+	0.990550i	$0.456204\pi$
$480$	0	0
$481$	−3.81054	−0.173746
$482$	−9.43322	−0.429671
$483$	0	0
$484$	0	0
$485$	−18.2561	−0.828965
$486$	0	0
$487$	6.36723	0.288527	0.144263	−	0.989539i	$-0.453919\pi$
$487$	6.36723	0.288527	0.144263	+	0.989539i	$0.453919\pi$
$488$	11.9201	0.539598
$489$	0	0
$490$	1.43578	0.0648620
$491$	−12.3769	−0.558561	−0.279281	−	0.960210i	$-0.590096\pi$
$491$	−12.3769	−0.558561	−0.279281	+	0.960210i	$0.590096\pi$
$492$	0	0
$493$	1.65511	0.0745426
$494$	11.3914	0.512525
$495$	0	0
$496$	4.53228	0.203505
$497$	14.8694	0.666982
$498$	0	0
$499$	−14.2797	−0.639245	−0.319623	−	0.947545i	$-0.603556\pi$
$499$	−14.2797	−0.639245	−0.319623	+	0.947545i	$0.603556\pi$
$500$	−17.9741	−0.803828
$501$	0	0
$502$	−0.721494	−0.0322019
$503$	−5.92573	−0.264215	−0.132108	−	0.991235i	$-0.542174\pi$
$503$	−5.92573	−0.264215	−0.132108	+	0.991235i	$0.542174\pi$
$504$	0	0
$505$	−8.63314	−0.384170
$506$	0	0
$507$	0	0
$508$	−27.2541	−1.20920
$509$	−30.8735	−1.36844	−0.684222	−	0.729274i	$-0.739858\pi$
$509$	−30.8735	−1.36844	−0.684222	+	0.729274i	$0.739858\pi$
$510$	0	0
$511$	8.03816	0.355587
$512$	−15.8920	−0.702333
$513$	0	0
$514$	8.76093	0.386428
$515$	37.7859	1.66504
$516$	0	0
$517$	0	0
$518$	1.00942	0.0443516
$519$	0	0
$520$	−12.8919	−0.565346
$521$	−18.8870	−0.827453	−0.413726	−	0.910401i	$-0.635773\pi$
$521$	−18.8870	−0.827453	−0.413726	+	0.910401i	$0.635773\pi$
$522$	0	0
$523$	−7.94209	−0.347283	−0.173642	−	0.984809i	$-0.555553\pi$
$523$	−7.94209	−0.347283	−0.173642	+	0.984809i	$0.555553\pi$
$524$	10.4343	0.455826
$525$	0	0
$526$	−6.41090	−0.279528
$527$	5.37595	0.234180
$528$	0	0
$529$	−12.9873	−0.564665
$530$	−0.958271	−0.0416246
$531$	0	0
$532$	10.4372	0.452509
$533$	−14.0598	−0.608996
$534$	0	0
$535$	−33.0222	−1.42767
$536$	−2.14577	−0.0926830
$537$	0	0
$538$	−3.19370	−0.137690
$539$	0	0
$540$	0	0
$541$	7.60165	0.326820	0.163410	−	0.986558i	$-0.447751\pi$
$541$	7.60165	0.326820	0.163410	+	0.986558i	$0.447751\pi$
$542$	13.4299	0.576863
$543$	0	0
$544$	−10.3306	−0.442922
$545$	40.5743	1.73801
$546$	0	0
$547$	21.6989	0.927780	0.463890	−	0.885893i	$-0.346453\pi$
$547$	21.6989	0.927780	0.463890	+	0.885893i	$0.346453\pi$
$548$	21.5148	0.919068
$549$	0	0
$550$	0	0
$551$	−6.21726	−0.264864
$552$	0	0
$553$	4.05668	0.172508
$554$	7.77067	0.330144
$555$	0	0
$556$	−22.1402	−0.938955
$557$	40.6134	1.72084	0.860422	−	0.509582i	$-0.170200\pi$
$557$	40.6134	1.72084	0.860422	+	0.509582i	$0.170200\pi$
$558$	0	0
$559$	−21.3129	−0.901438
$560$	−3.23681	−0.136780
$561$	0	0
$562$	8.08972	0.341244
$563$	23.4293	0.987425	0.493713	−	0.869625i	$-0.335639\pi$
$563$	23.4293	0.987425	0.493713	+	0.869625i	$0.335639\pi$
$564$	0	0
$565$	3.60288	0.151574
$566$	−14.7298	−0.619141
$567$	0	0
$568$	35.3676	1.48399
$569$	11.1453	0.467237	0.233619	−	0.972328i	$-0.424943\pi$
$569$	11.1453	0.467237	0.233619	+	0.972328i	$0.424943\pi$
$570$	0	0
$571$	−6.15846	−0.257724	−0.128862	−	0.991663i	$-0.541132\pi$
$571$	−6.15846	−0.257724	−0.128862	+	0.991663i	$0.541132\pi$
$572$	0	0
$573$	0	0
$574$	3.72447	0.155456
$575$	−1.27905	−0.0533400
$576$	0	0
$577$	−14.4842	−0.602983	−0.301492	−	0.953469i	$-0.597485\pi$
$577$	−14.4842	−0.602983	−0.301492	+	0.953469i	$0.597485\pi$
$578$	9.23751	0.384230
$579$	0	0
$580$	3.07374	0.127630
$581$	4.05134	0.168078
$582$	0	0
$583$	0	0
$584$	19.1192	0.791159
$585$	0	0
$586$	0.949166	0.0392097
$587$	−15.8570	−0.654487	−0.327243	−	0.944940i	$-0.606120\pi$
$587$	−15.8570	−0.654487	−0.327243	+	0.944940i	$0.606120\pi$
$588$	0	0
$589$	−20.1942	−0.832087
$590$	0.528885	0.0217739
$591$	0	0
$592$	−2.27563	−0.0935279
$593$	−22.9285	−0.941560	−0.470780	−	0.882251i	$-0.656027\pi$
$593$	−22.9285	−0.941560	−0.470780	+	0.882251i	$0.656027\pi$
$594$	0	0
$595$	−3.83934	−0.157397
$596$	4.06931	0.166686
$597$	0	0
$598$	−5.35805	−0.219107
$599$	−13.1512	−0.537344	−0.268672	−	0.963232i	$-0.586585\pi$
$599$	−13.1512	−0.537344	−0.268672	+	0.963232i	$0.586585\pi$
$600$	0	0
$601$	27.3525	1.11573	0.557865	−	0.829932i	$-0.311621\pi$
$601$	27.3525	1.11573	0.557865	+	0.829932i	$0.311621\pi$
$602$	5.64584	0.230107
$603$	0	0
$604$	4.63819	0.188725
$605$	0	0
$606$	0	0
$607$	−46.8743	−1.90257	−0.951285	−	0.308313i	$-0.900236\pi$
$607$	−46.8743	−1.90257	−0.951285	+	0.308313i	$0.900236\pi$
$608$	38.8059	1.57379
$609$	0	0
$610$	7.19540	0.291333
$611$	−11.1186	−0.449809
$612$	0	0
$613$	−20.4694	−0.826752	−0.413376	−	0.910560i	$-0.635651\pi$
$613$	−20.4694	−0.826752	−0.413376	+	0.910560i	$0.635651\pi$
$614$	−19.7337	−0.796389
$615$	0	0
$616$	0	0
$617$	−44.1691	−1.77818	−0.889090	−	0.457733i	$-0.848662\pi$
$617$	−44.1691	−1.77818	−0.889090	+	0.457733i	$0.848662\pi$
$618$	0	0
$619$	0.681584	0.0273952	0.0136976	−	0.999906i	$-0.495640\pi$
$619$	0.681584	0.0273952	0.0136976	+	0.999906i	$0.495640\pi$
$620$	9.98378	0.400958
$621$	0	0
$622$	18.0018	0.721808
$623$	8.30727	0.332824
$624$	0	0
$625$	−22.8155	−0.912622
$626$	3.00896	0.120262
$627$	0	0
$628$	−17.4610	−0.696770
$629$	−2.69924	−0.107626
$630$	0	0
$631$	−36.8718	−1.46784	−0.733921	−	0.679234i	$-0.762312\pi$
$631$	−36.8718	−1.46784	−0.733921	+	0.679234i	$0.762312\pi$
$632$	9.64904	0.383818
$633$	0	0
$634$	−7.33374	−0.291260
$635$	−37.6596	−1.49447
$636$	0	0
$637$	2.52826	0.100173
$638$	0	0
$639$	0	0
$640$	−23.5209	−0.929744
$641$	20.5126	0.810201	0.405100	−	0.914272i	$-0.367237\pi$
$641$	20.5126	0.810201	0.405100	+	0.914272i	$0.367237\pi$
$642$	0	0
$643$	−7.63660	−0.301158	−0.150579	−	0.988598i	$-0.548114\pi$
$643$	−7.63660	−0.301158	−0.150579	+	0.988598i	$0.548114\pi$
$644$	−4.90921	−0.193450
$645$	0	0
$646$	8.06923	0.317480
$647$	−14.6828	−0.577241	−0.288621	−	0.957444i	$-0.593197\pi$
$647$	−14.6828	−0.577241	−0.288621	+	0.957444i	$0.593197\pi$
$648$	0	0
$649$	0	0
$650$	0.684450	0.0268463
$651$	0	0
$652$	28.2408	1.10600
$653$	−1.12703	−0.0441043	−0.0220521	−	0.999757i	$-0.507020\pi$
$653$	−1.12703	−0.0441043	−0.0220521	+	0.999757i	$0.507020\pi$
$654$	0	0
$655$	14.4181	0.563363
$656$	−8.39640	−0.327824
$657$	0	0
$658$	2.94534	0.114821
$659$	−10.0215	−0.390384	−0.195192	−	0.980765i	$-0.562533\pi$
$659$	−10.0215	−0.390384	−0.195192	+	0.980765i	$0.562533\pi$
$660$	0	0
$661$	15.7371	0.612101	0.306050	−	0.952015i	$-0.400992\pi$
$661$	15.7371	0.612101	0.306050	+	0.952015i	$0.400992\pi$
$662$	−11.0659	−0.430090
$663$	0	0
$664$	9.63635	0.373963
$665$	14.4221	0.559263
$666$	0	0
$667$	2.92434	0.113231
$668$	30.9222	1.19642
$669$	0	0
$670$	−1.29526	−0.0500403
$671$	0	0
$672$	0	0
$673$	32.0120	1.23397	0.616986	−	0.786974i	$-0.288353\pi$
$673$	32.0120	1.23397	0.616986	+	0.786974i	$0.288353\pi$
$674$	8.11779	0.312686
$675$	0	0
$676$	10.2518	0.394299
$677$	15.3400	0.589566	0.294783	−	0.955564i	$-0.404753\pi$
$677$	15.3400	0.589566	0.294783	+	0.955564i	$0.404753\pi$
$678$	0	0
$679$	8.51583	0.326808
$680$	−9.13208	−0.350199
$681$	0	0
$682$	0	0
$683$	−1.04764	−0.0400868	−0.0200434	−	0.999799i	$-0.506380\pi$
$683$	−1.04764	−0.0400868	−0.0200434	+	0.999799i	$0.506380\pi$
$684$	0	0
$685$	29.7291	1.13589
$686$	−0.669744	−0.0255709
$687$	0	0
$688$	−12.7279	−0.485247
$689$	−1.68742	−0.0642854
$690$	0	0
$691$	29.1883	1.11038	0.555188	−	0.831725i	$-0.312646\pi$
$691$	29.1883	1.11038	0.555188	+	0.831725i	$0.312646\pi$
$692$	9.29326	0.353277
$693$	0	0
$694$	−16.2648	−0.617404
$695$	−30.5933	−1.16047
$696$	0	0
$697$	−9.95937	−0.377238
$698$	2.22275	0.0841322
$699$	0	0
$700$	0.627115	0.0237027
$701$	−12.7785	−0.482637	−0.241318	−	0.970446i	$-0.577580\pi$
$701$	−12.7785	−0.482637	−0.241318	+	0.970446i	$0.577580\pi$
$702$	0	0
$703$	10.1394	0.382415
$704$	0	0
$705$	0	0
$706$	13.6141	0.512372
$707$	4.02707	0.151453
$708$	0	0
$709$	38.0710	1.42978	0.714892	−	0.699234i	$-0.246476\pi$
$709$	38.0710	1.42978	0.714892	+	0.699234i	$0.246476\pi$
$710$	21.3492	0.801220
$711$	0	0
$712$	19.7593	0.740512
$713$	9.49850	0.355722
$714$	0	0
$715$	0	0
$716$	−2.51169	−0.0938663
$717$	0	0
$718$	19.4502	0.725875
$719$	39.2694	1.46450	0.732250	−	0.681036i	$-0.238470\pi$
$719$	39.2694	1.46450	0.732250	+	0.681036i	$0.238470\pi$
$720$	0	0
$721$	−17.6258	−0.656420
$722$	−17.5861	−0.654486
$723$	0	0
$724$	−3.76507	−0.139928
$725$	−0.373562	−0.0138737
$726$	0	0
$727$	−28.4699	−1.05589	−0.527946	−	0.849278i	$-0.677037\pi$
$727$	−28.4699	−1.05589	−0.527946	+	0.849278i	$0.677037\pi$
$728$	6.01362	0.222879
$729$	0	0
$730$	11.5410	0.427153
$731$	−15.0972	−0.558389
$732$	0	0
$733$	−2.99337	−0.110563	−0.0552813	−	0.998471i	$-0.517606\pi$
$733$	−2.99337	−0.110563	−0.0552813	+	0.998471i	$0.517606\pi$
$734$	15.2426	0.562613
$735$	0	0
$736$	−18.2527	−0.672802
$737$	0	0
$738$	0	0
$739$	−50.7329	−1.86624	−0.933120	−	0.359566i	$-0.882925\pi$
$739$	−50.7329	−1.86624	−0.933120	+	0.359566i	$0.882925\pi$
$740$	−5.01280	−0.184274
$741$	0	0
$742$	0.447001	0.0164099
$743$	0.383877	0.0140831	0.00704154	−	0.999975i	$-0.497759\pi$
$743$	0.383877	0.0140831	0.00704154	+	0.999975i	$0.497759\pi$
$744$	0	0
$745$	5.62296	0.206009
$746$	14.8959	0.545378
$747$	0	0
$748$	0	0
$749$	15.4037	0.562840
$750$	0	0
$751$	−39.3570	−1.43616	−0.718078	−	0.695963i	$-0.754978\pi$
$751$	−39.3570	−1.43616	−0.718078	+	0.695963i	$0.754978\pi$
$752$	−6.63993	−0.242133
$753$	0	0
$754$	−1.56489	−0.0569898
$755$	6.40904	0.233249
$756$	0	0
$757$	11.3867	0.413856	0.206928	−	0.978356i	$-0.433653\pi$
$757$	11.3867	0.413856	0.206928	+	0.978356i	$0.433653\pi$
$758$	−22.4168	−0.814214
$759$	0	0
$760$	34.3037	1.24433
$761$	7.48165	0.271209	0.135605	−	0.990763i	$-0.456702\pi$
$761$	7.48165	0.271209	0.135605	+	0.990763i	$0.456702\pi$
$762$	0	0
$763$	−18.9265	−0.685186
$764$	19.7048	0.712893
$765$	0	0
$766$	−4.11619	−0.148724
$767$	0.931311	0.0336277
$768$	0	0
$769$	−26.8378	−0.967798	−0.483899	−	0.875124i	$-0.660780\pi$
$769$	−26.8378	−0.967798	−0.483899	+	0.875124i	$0.660780\pi$
$770$	0	0
$771$	0	0
$772$	2.72947	0.0982358
$773$	−4.46781	−0.160696	−0.0803479	−	0.996767i	$-0.525603\pi$
$773$	−4.46781	−0.160696	−0.0803479	+	0.996767i	$0.525603\pi$
$774$	0	0
$775$	−1.21336	−0.0435852
$776$	20.2554	0.727126
$777$	0	0
$778$	4.95855	0.177772
$779$	37.4113	1.34040
$780$	0	0
$781$	0	0
$782$	−3.79543	−0.135724
$783$	0	0
$784$	1.50986	0.0539236
$785$	−24.1276	−0.861150
$786$	0	0
$787$	−13.4859	−0.480721	−0.240361	−	0.970684i	$-0.577266\pi$
$787$	−13.4859	−0.480721	−0.240361	+	0.970684i	$0.577266\pi$
$788$	−1.40104	−0.0499098
$789$	0	0
$790$	5.82451	0.207227
$791$	−1.68062	−0.0597560
$792$	0	0
$793$	12.6704	0.449937
$794$	−11.9267	−0.423263
$795$	0	0
$796$	24.2484	0.859462
$797$	−52.5052	−1.85983	−0.929915	−	0.367775i	$-0.880120\pi$
$797$	−52.5052	−1.85983	−0.929915	+	0.367775i	$0.880120\pi$
$798$	0	0
$799$	−7.87594	−0.278631
$800$	2.33164	0.0824359
$801$	0	0
$802$	17.1983	0.607292
$803$	0	0
$804$	0	0
$805$	−6.78353	−0.239088
$806$	−5.08288	−0.179037
$807$	0	0
$808$	9.57861	0.336974
$809$	1.28984	0.0453484	0.0226742	−	0.999743i	$-0.492782\pi$
$809$	1.28984	0.0453484	0.0226742	+	0.999743i	$0.492782\pi$
$810$	0	0
$811$	−34.1878	−1.20049	−0.600247	−	0.799815i	$-0.704931\pi$
$811$	−34.1878	−1.20049	−0.600247	+	0.799815i	$0.704931\pi$
$812$	−1.43380	−0.0503164
$813$	0	0
$814$	0	0
$815$	39.0231	1.36692
$816$	0	0
$817$	56.7110	1.98407
$818$	−0.895813	−0.0313214
$819$	0	0
$820$	−18.4957	−0.645899
$821$	−39.6599	−1.38414	−0.692071	−	0.721830i	$-0.743301\pi$
$821$	−39.6599	−1.38414	−0.692071	+	0.721830i	$0.743301\pi$
$822$	0	0
$823$	−9.22714	−0.321638	−0.160819	−	0.986984i	$-0.551414\pi$
$823$	−9.22714	−0.321638	−0.160819	+	0.986984i	$0.551414\pi$
$824$	−41.9241	−1.46049
$825$	0	0
$826$	−0.246707	−0.00858403
$827$	25.6958	0.893530	0.446765	−	0.894651i	$-0.352576\pi$
$827$	25.6958	0.893530	0.446765	+	0.894651i	$0.352576\pi$
$828$	0	0
$829$	10.9385	0.379911	0.189955	−	0.981793i	$-0.439166\pi$
$829$	10.9385	0.379911	0.189955	+	0.981793i	$0.439166\pi$
$830$	5.81685	0.201906
$831$	0	0
$832$	2.13280	0.0739416
$833$	1.79092	0.0620517
$834$	0	0
$835$	42.7282	1.47867
$836$	0	0
$837$	0	0
$838$	25.0898	0.866712
$839$	34.7374	1.19927	0.599634	−	0.800275i	$-0.295313\pi$
$839$	34.7374	1.19927	0.599634	+	0.800275i	$0.295313\pi$
$840$	0	0
$841$	−28.1459	−0.970549
$842$	−5.53313	−0.190684
$843$	0	0
$844$	−22.9128	−0.788691
$845$	14.1659	0.487320
$846$	0	0
$847$	0	0
$848$	−1.00771	−0.0346050
$849$	0	0
$850$	0.484837	0.0166298
$851$	−4.76915	−0.163484
$852$	0	0
$853$	−20.6422	−0.706777	−0.353389	−	0.935477i	$-0.614971\pi$
$853$	−20.6422	−0.706777	−0.353389	+	0.935477i	$0.614971\pi$
$854$	−3.35641	−0.114854
$855$	0	0
$856$	36.6386	1.25228
$857$	34.1512	1.16658	0.583291	−	0.812263i	$-0.301765\pi$
$857$	34.1512	1.16658	0.583291	+	0.812263i	$0.301765\pi$
$858$	0	0
$859$	−33.4493	−1.14127	−0.570637	−	0.821202i	$-0.693304\pi$
$859$	−33.4493	−1.14127	−0.570637	+	0.821202i	$0.693304\pi$
$860$	−28.0373	−0.956063
$861$	0	0
$862$	21.8715	0.744945
$863$	34.0340	1.15853	0.579265	−	0.815139i	$-0.303340\pi$
$863$	34.0340	1.15853	0.579265	+	0.815139i	$0.303340\pi$
$864$	0	0
$865$	12.8414	0.436621
$866$	−10.5788	−0.359482
$867$	0	0
$868$	−4.65710	−0.158072
$869$	0	0
$870$	0	0
$871$	−2.28082	−0.0772826
$872$	−45.0178	−1.52450
$873$	0	0
$874$	14.2571	0.482254
$875$	11.5854	0.391659
$876$	0	0
$877$	7.86706	0.265652	0.132826	−	0.991139i	$-0.457595\pi$
$877$	7.86706	0.265652	0.132826	+	0.991139i	$0.457595\pi$
$878$	13.8598	0.467745
$879$	0	0
$880$	0	0
$881$	−13.3289	−0.449063	−0.224531	−	0.974467i	$-0.572085\pi$
$881$	−13.3289	−0.449063	−0.224531	+	0.974467i	$0.572085\pi$
$882$	0	0
$883$	−17.0998	−0.575454	−0.287727	−	0.957712i	$-0.592900\pi$
$883$	−17.0998	−0.575454	−0.287727	+	0.957712i	$0.592900\pi$
$884$	−7.02480	−0.236270
$885$	0	0
$886$	20.1563	0.677163
$887$	26.5185	0.890403	0.445201	−	0.895430i	$-0.353132\pi$
$887$	26.5185	0.890403	0.445201	+	0.895430i	$0.353132\pi$
$888$	0	0
$889$	17.5669	0.589176
$890$	11.9274	0.399808
$891$	0	0
$892$	2.86298	0.0958598
$893$	29.5851	0.990029
$894$	0	0
$895$	−3.47065	−0.116011
$896$	10.9717	0.366538
$897$	0	0
$898$	24.3749	0.813402
$899$	2.77416	0.0925233
$900$	0	0
$901$	−1.19530	−0.0398211
$902$	0	0
$903$	0	0
$904$	−3.99745	−0.132953
$905$	−5.20257	−0.172939
$906$	0	0
$907$	8.85256	0.293944	0.146972	−	0.989141i	$-0.453047\pi$
$907$	8.85256	0.293944	0.146972	+	0.989141i	$0.453047\pi$
$908$	−34.0386	−1.12961
$909$	0	0
$910$	3.63003	0.120334
$911$	55.8998	1.85204	0.926022	−	0.377469i	$-0.123206\pi$
$911$	55.8998	1.85204	0.926022	+	0.377469i	$0.123206\pi$
$912$	0	0
$913$	0	0
$914$	6.55948	0.216968
$915$	0	0
$916$	31.4195	1.03813
$917$	−6.72557	−0.222098
$918$	0	0
$919$	48.3776	1.59583	0.797915	−	0.602770i	$-0.205936\pi$
$919$	48.3776	1.59583	0.797915	+	0.602770i	$0.205936\pi$
$920$	−16.1350	−0.531955
$921$	0	0
$922$	−14.6404	−0.482155
$923$	37.5937	1.23741
$924$	0	0
$925$	0.609223	0.0200311
$926$	−4.52673	−0.148757
$927$	0	0
$928$	−5.33092	−0.174996
$929$	−0.864295	−0.0283566	−0.0141783	−	0.999899i	$-0.504513\pi$
$929$	−0.864295	−0.0283566	−0.0141783	+	0.999899i	$0.504513\pi$
$930$	0	0
$931$	−6.72740	−0.220482
$932$	32.6774	1.07038
$933$	0	0
$934$	−20.1079	−0.657949
$935$	0	0
$936$	0	0
$937$	53.4580	1.74640	0.873199	−	0.487364i	$-0.162042\pi$
$937$	53.4580	1.74640	0.873199	+	0.487364i	$0.162042\pi$
$938$	0.604195	0.0197277
$939$	0	0
$940$	−14.6266	−0.477066
$941$	−13.7000	−0.446607	−0.223304	−	0.974749i	$-0.571684\pi$
$941$	−13.7000	−0.446607	−0.223304	+	0.974749i	$0.571684\pi$
$942$	0	0
$943$	−17.5967	−0.573028
$944$	0.556173	0.0181019
$945$	0	0
$946$	0	0
$947$	−31.6444	−1.02830	−0.514152	−	0.857699i	$-0.671893\pi$
$947$	−31.6444	−1.02830	−0.514152	+	0.857699i	$0.671893\pi$
$948$	0	0
$949$	20.3226	0.659699
$950$	−1.82124	−0.0590888
$951$	0	0
$952$	4.25981	0.138061
$953$	−32.7379	−1.06049	−0.530243	−	0.847846i	$-0.677899\pi$
$953$	−32.7379	−1.06049	−0.530243	+	0.847846i	$0.677899\pi$
$954$	0	0
$955$	27.2280	0.881076
$956$	24.0420	0.777573
$957$	0	0
$958$	−4.02083	−0.129907
$959$	−13.8676	−0.447809
$960$	0	0
$961$	−21.9893	−0.709332
$962$	2.55209	0.0822827
$963$	0	0
$964$	−21.8518	−0.703799
$965$	3.77157	0.121411
$966$	0	0
$967$	−32.3487	−1.04026	−0.520132	−	0.854086i	$-0.674117\pi$
$967$	−32.3487	−1.04026	−0.520132	+	0.854086i	$0.674117\pi$
$968$	0	0
$969$	0	0
$970$	12.2269	0.392582
$971$	29.4270	0.944358	0.472179	−	0.881503i	$-0.343468\pi$
$971$	29.4270	0.944358	0.472179	+	0.881503i	$0.343468\pi$
$972$	0	0
$973$	14.2707	0.457498
$974$	−4.26441	−0.136641
$975$	0	0
$976$	7.56665	0.242203
$977$	−13.6944	−0.438124	−0.219062	−	0.975711i	$-0.570300\pi$
$977$	−13.6944	−0.438124	−0.219062	+	0.975711i	$0.570300\pi$
$978$	0	0
$979$	0	0
$980$	3.32595	0.106244
$981$	0	0
$982$	8.28935	0.264524
$983$	−17.0319	−0.543234	−0.271617	−	0.962405i	$-0.587558\pi$
$983$	−17.0319	−0.543234	−0.271617	+	0.962405i	$0.587558\pi$
$984$	0	0
$985$	−1.93595	−0.0616844
$986$	−1.10850	−0.0353019
$987$	0	0
$988$	26.3879	0.839512
$989$	−26.6744	−0.848198
$990$	0	0
$991$	23.2202	0.737614	0.368807	−	0.929506i	$-0.379766\pi$
$991$	23.2202	0.737614	0.368807	+	0.929506i	$0.379766\pi$
$992$	−17.3153	−0.549761
$993$	0	0
$994$	−9.95867	−0.315870
$995$	33.5064	1.06222
$996$	0	0
$997$	−18.3776	−0.582025	−0.291012	−	0.956719i	$-0.593992\pi$
$997$	−18.3776	−0.582025	−0.291012	+	0.956719i	$0.593992\pi$
$998$	9.56371	0.302734
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ct.1.4		8
3.2	odd	2		847.2.a.p.1.5		8
11.3	even	5		693.2.m.i.64.2		16
11.4	even	5		693.2.m.i.379.2		16
11.10	odd	2		7623.2.a.cw.1.5		8
21.20	even	2		5929.2.a.bt.1.5		8
33.2	even	10		847.2.f.v.323.3		16
33.5	odd	10		847.2.f.w.729.2		16
33.8	even	10		847.2.f.x.372.2		16
33.14	odd	10		77.2.f.b.64.3	✓	16
33.17	even	10		847.2.f.v.729.3		16
33.20	odd	10		847.2.f.w.323.2		16
33.26	odd	10		77.2.f.b.71.3	yes	16
33.29	even	10		847.2.f.x.148.2		16
33.32	even	2		847.2.a.o.1.4		8
231.26	even	30		539.2.q.f.214.3		32
231.47	even	30		539.2.q.f.361.2		32
231.59	even	30		539.2.q.f.324.2		32
231.80	even	30		539.2.q.f.471.3		32
231.125	even	10		539.2.f.e.148.3		16
231.146	even	10		539.2.f.e.295.3		16
231.158	odd	30		539.2.q.g.324.2		32
231.179	odd	30		539.2.q.g.471.3		32
231.191	odd	30		539.2.q.g.214.3		32
231.212	odd	30		539.2.q.g.361.2		32
231.230	odd	2		5929.2.a.bs.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.3	✓	16	33.14	odd	10
77.2.f.b.71.3	yes	16	33.26	odd	10
539.2.f.e.148.3		16	231.125	even	10
539.2.f.e.295.3		16	231.146	even	10
539.2.q.f.214.3		32	231.26	even	30
539.2.q.f.324.2		32	231.59	even	30
539.2.q.f.361.2		32	231.47	even	30
539.2.q.f.471.3		32	231.80	even	30
539.2.q.g.214.3		32	231.191	odd	30
539.2.q.g.324.2		32	231.158	odd	30
539.2.q.g.361.2		32	231.212	odd	30
539.2.q.g.471.3		32	231.179	odd	30
693.2.m.i.64.2		16	11.3	even	5
693.2.m.i.379.2		16	11.4	even	5
847.2.a.o.1.4		8	33.32	even	2
847.2.a.p.1.5		8	3.2	odd	2
847.2.f.v.323.3		16	33.2	even	10
847.2.f.v.729.3		16	33.17	even	10
847.2.f.w.323.2		16	33.20	odd	10
847.2.f.w.729.2		16	33.5	odd	10
847.2.f.x.148.2		16	33.29	even	10
847.2.f.x.372.2		16	33.8	even	10
5929.2.a.bs.1.4		8	231.230	odd	2
5929.2.a.bt.1.5		8	21.20	even	2
7623.2.a.ct.1.4		8	1.1	even	1	trivial
7623.2.a.cw.1.5		8	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-0.669744 q^{2} -1.55144 q^{4} -2.14378 q^{5} +1.00000 q^{7} +2.37856 q^{8} +O(q^{10})\)	\(q-0.669744 q^{2} -1.55144 q^{4} -2.14378 q^{5} +1.00000 q^{7} +2.37856 q^{8} +1.43578 q^{10} +2.52826 q^{13} -0.669744 q^{14} +1.50986 q^{16} +1.79092 q^{17} -6.72740 q^{19} +3.32595 q^{20} +3.16429 q^{23} -0.404214 q^{25} -1.69329 q^{26} -1.55144 q^{28} +0.924170 q^{29} +3.00178 q^{31} -5.76834 q^{32} -1.19946 q^{34} -2.14378 q^{35} -1.50718 q^{37} +4.50564 q^{38} -5.09910 q^{40} -5.56104 q^{41} -8.42985 q^{43} -2.11926 q^{46} -4.39771 q^{47} +1.00000 q^{49} +0.270720 q^{50} -3.92246 q^{52} -0.667421 q^{53} +2.37856 q^{56} -0.618957 q^{58} +0.368360 q^{59} +5.01149 q^{61} -2.01043 q^{62} +0.843584 q^{64} -5.42003 q^{65} -0.902129 q^{67} -2.77851 q^{68} +1.43578 q^{70} +14.8694 q^{71} +8.03816 q^{73} +1.00942 q^{74} +10.4372 q^{76} +4.05668 q^{79} -3.23681 q^{80} +3.72447 q^{82} +4.05134 q^{83} -3.83934 q^{85} +5.64584 q^{86} +8.30727 q^{89} +2.52826 q^{91} -4.90921 q^{92} +2.94534 q^{94} +14.4221 q^{95} +8.51583 q^{97} -0.669744 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - q^98

Introduction

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