LMFDB - Embedded newform 3025.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +1.00000 q^{12} +3.46410 q^{13} -3.00000 q^{14} -5.00000 q^{16} -6.92820 q^{17} -3.46410 q^{18} +3.46410 q^{19} -1.73205 q^{21} -1.73205 q^{24} +6.00000 q^{26} -5.00000 q^{27} -1.73205 q^{28} -8.00000 q^{31} -5.19615 q^{32} -12.0000 q^{34} -2.00000 q^{36} +8.00000 q^{37} +6.00000 q^{38} +3.46410 q^{39} -12.1244 q^{41} -3.00000 q^{42} +8.66025 q^{43} -9.00000 q^{47} -5.00000 q^{48} -4.00000 q^{49} -6.92820 q^{51} +3.46410 q^{52} -6.00000 q^{53} -8.66025 q^{54} +3.00000 q^{56} +3.46410 q^{57} -12.0000 q^{59} -8.66025 q^{61} -13.8564 q^{62} +3.46410 q^{63} +1.00000 q^{64} +5.00000 q^{67} -6.92820 q^{68} -12.0000 q^{71} +3.46410 q^{72} +13.8564 q^{74} +3.46410 q^{76} +6.00000 q^{78} +10.3923 q^{79} +1.00000 q^{81} -21.0000 q^{82} +3.46410 q^{83} -1.73205 q^{84} +15.0000 q^{86} +3.00000 q^{89} -6.00000 q^{91} -8.00000 q^{93} -15.5885 q^{94} -5.19615 q^{96} +10.0000 q^{97} -6.92820 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9	$ 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9} + 2 q^{12} - 6 q^{14} - 10 q^{16} + 12 q^{26} - 10 q^{27} - 16 q^{31} - 24 q^{34} - 4 q^{36} + 16 q^{37} + 12 q^{38} - 6 q^{42} - 18 q^{47} - 10 q^{48} - 8 q^{49} - 12 q^{53} + 6 q^{56} - 24 q^{59} + 2 q^{64} + 10 q^{67} - 24 q^{71} + 12 q^{78} + 2 q^{81} - 42 q^{82} + 30 q^{86} + 6 q^{89} - 12 q^{91} - 16 q^{93} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9 + 2 * q^12 - 6 * q^14 - 10 * q^16 + 12 * q^26 - 10 * q^27 - 16 * q^31 - 24 * q^34 - 4 * q^36 + 16 * q^37 + 12 * q^38 - 6 * q^42 - 18 * q^47 - 10 * q^48 - 8 * q^49 - 12 * q^53 + 6 * q^56 - 24 * q^59 + 2 * q^64 + 10 * q^67 - 24 * q^71 + 12 * q^78 + 2 * q^81 - 42 * q^82 + 30 * q^86 + 6 * q^89 - 12 * q^91 - 16 * q^93 + 20 * q^97

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.73205	0.707107
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	−1.73205	−0.612372
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	−3.46410	−0.816497
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	−1.73205	−0.377964
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	6.00000	1.17670
$27$	−5.00000	−0.962250
$28$	−1.73205	−0.327327
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	−12.0000	−2.05798
$35$	0	0
$36$	−2.00000	−0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	6.00000	0.973329
$39$	3.46410	0.554700
$40$	0	0
$41$	−12.1244	−1.89351	−0.946753	−	0.321960i	$-0.895658\pi$
$41$	−12.1244	−1.89351	−0.946753	+	0.321960i	$0.895658\pi$
$42$	−3.00000	−0.462910
$43$	8.66025	1.32068	0.660338	−	0.750968i	$-0.270413\pi$
$43$	8.66025	1.32068	0.660338	+	0.750968i	$0.270413\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−5.00000	−0.721688
$49$	−4.00000	−0.571429
$50$	0	0
$51$	−6.92820	−0.970143
$52$	3.46410	0.480384
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−8.66025	−1.17851
$55$	0	0
$56$	3.00000	0.400892
$57$	3.46410	0.458831
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−8.66025	−1.10883	−0.554416	−	0.832240i	$-0.687058\pi$
$61$	−8.66025	−1.10883	−0.554416	+	0.832240i	$0.687058\pi$
$62$	−13.8564	−1.75977
$63$	3.46410	0.436436
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−6.92820	−0.840168
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	3.46410	0.408248
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	13.8564	1.61077
$75$	0	0
$76$	3.46410	0.397360
$77$	0	0
$78$	6.00000	0.679366
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−21.0000	−2.31906
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	−1.73205	−0.188982
$85$	0	0
$86$	15.0000	1.61749
$87$	0	0
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−8.00000	−0.829561
$94$	−15.5885	−1.60783
$95$	0	0
$96$	−5.19615	−0.530330
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−6.92820	−0.699854
$99$	0	0
$100$	0	0
$101$	1.73205	0.172345	0.0861727	−	0.996280i	$-0.472536\pi$
$101$	1.73205	0.172345	0.0861727	+	0.996280i	$0.472536\pi$
$102$	−12.0000	−1.18818
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−10.3923	−1.00939
$107$	1.73205	0.167444	0.0837218	−	0.996489i	$-0.473319\pi$
$107$	1.73205	0.167444	0.0837218	+	0.996489i	$0.473319\pi$
$108$	−5.00000	−0.481125
$109$	−1.73205	−0.165900	−0.0829502	−	0.996554i	$-0.526434\pi$
$109$	−1.73205	−0.165900	−0.0829502	+	0.996554i	$0.526434\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	8.66025	0.818317
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	6.00000	0.561951
$115$	0	0
$116$	0	0
$117$	−6.92820	−0.640513
$118$	−20.7846	−1.91338
$119$	12.0000	1.10004
$120$	0	0
$121$	0	0
$122$	−15.0000	−1.35804
$123$	−12.1244	−1.09322
$124$	−8.00000	−0.718421
$125$	0	0
$126$	6.00000	0.534522
$127$	1.73205	0.153695	0.0768473	−	0.997043i	$-0.475515\pi$
$127$	1.73205	0.153695	0.0768473	+	0.997043i	$0.475515\pi$
$128$	12.1244	1.07165
$129$	8.66025	0.762493
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	8.66025	0.748132
$135$	0	0
$136$	12.0000	1.02899
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−13.8564	−1.17529	−0.587643	−	0.809121i	$-0.699944\pi$
$139$	−13.8564	−1.17529	−0.587643	+	0.809121i	$0.699944\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	−20.7846	−1.74421
$143$	0	0
$144$	10.0000	0.833333
$145$	0	0
$146$	0	0
$147$	−4.00000	−0.329914
$148$	8.00000	0.657596
$149$	19.0526	1.56085	0.780423	−	0.625252i	$-0.215004\pi$
$149$	19.0526	1.56085	0.780423	+	0.625252i	$0.215004\pi$
$150$	0	0
$151$	20.7846	1.69143	0.845714	−	0.533637i	$-0.179175\pi$
$151$	20.7846	1.69143	0.845714	+	0.533637i	$0.179175\pi$
$152$	−6.00000	−0.486664
$153$	13.8564	1.12022
$154$	0	0
$155$	0	0
$156$	3.46410	0.277350
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	18.0000	1.43200
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	1.73205	0.136083
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−12.1244	−0.946753
$165$	0	0
$166$	6.00000	0.465690
$167$	−5.19615	−0.402090	−0.201045	−	0.979582i	$-0.564434\pi$
$167$	−5.19615	−0.402090	−0.201045	+	0.979582i	$0.564434\pi$
$168$	3.00000	0.231455
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−6.92820	−0.529813
$172$	8.66025	0.660338
$173$	10.3923	0.790112	0.395056	−	0.918657i	$-0.370725\pi$
$173$	10.3923	0.790112	0.395056	+	0.918657i	$0.370725\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	5.19615	0.389468
$179$	−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
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	−10.3923	−0.770329
$183$	−8.66025	−0.640184
$184$	0	0
$185$	0	0
$186$	−13.8564	−1.01600
$187$	0	0
$188$	−9.00000	−0.656392
$189$	8.66025	0.629941
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	1.00000	0.0721688
$193$	−3.46410	−0.249351	−0.124676	−	0.992198i	$-0.539789\pi$
$193$	−3.46410	−0.249351	−0.124676	+	0.992198i	$0.539789\pi$
$194$	17.3205	1.24354
$195$	0	0
$196$	−4.00000	−0.285714
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	3.00000	0.211079
$203$	0	0
$204$	−6.92820	−0.485071
$205$	0	0
$206$	6.92820	0.482711
$207$	0	0
$208$	−17.3205	−1.20096
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−6.00000	−0.412082
$213$	−12.0000	−0.822226
$214$	3.00000	0.205076
$215$	0	0
$216$	8.66025	0.589256
$217$	13.8564	0.940634
$218$	−3.00000	−0.203186
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	13.8564	0.929981
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	9.00000	0.601338
$225$	0	0
$226$	10.3923	0.691286
$227$	19.0526	1.26456	0.632281	−	0.774739i	$-0.282119\pi$
$227$	19.0526	1.26456	0.632281	+	0.774739i	$0.282119\pi$
$228$	3.46410	0.229416
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.46410	−0.226941	−0.113470	−	0.993541i	$-0.536197\pi$
$233$	−3.46410	−0.226941	−0.113470	+	0.993541i	$0.536197\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	−12.0000	−0.781133
$237$	10.3923	0.675053
$238$	20.7846	1.34727
$239$	−3.46410	−0.224074	−0.112037	−	0.993704i	$-0.535738\pi$
$239$	−3.46410	−0.224074	−0.112037	+	0.993704i	$0.535738\pi$
$240$	0	0
$241$	19.0526	1.22728	0.613642	−	0.789585i	$-0.289704\pi$
$241$	19.0526	1.22728	0.613642	+	0.789585i	$0.289704\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	−8.66025	−0.554416
$245$	0	0
$246$	−21.0000	−1.33891
$247$	12.0000	0.763542
$248$	13.8564	0.879883
$249$	3.46410	0.219529
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	3.46410	0.218218
$253$	0	0
$254$	3.00000	0.188237
$255$	0	0
$256$	19.0000	1.18750
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	15.0000	0.933859
$259$	−13.8564	−0.860995
$260$	0	0
$261$	0	0
$262$	6.00000	0.370681
$263$	24.2487	1.49524	0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487	1.49524	0.747620	+	0.664127i	$0.231197\pi$
$264$	0	0
$265$	0	0
$266$	−10.3923	−0.637193
$267$	3.00000	0.183597
$268$	5.00000	0.305424
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	13.8564	0.841717	0.420858	−	0.907126i	$-0.361729\pi$
$271$	13.8564	0.841717	0.420858	+	0.907126i	$0.361729\pi$
$272$	34.6410	2.10042
$273$	−6.00000	−0.363137
$274$	−31.1769	−1.88347
$275$	0	0
$276$	0	0
$277$	10.3923	0.624413	0.312207	−	0.950014i	$-0.398932\pi$
$277$	10.3923	0.624413	0.312207	+	0.950014i	$0.398932\pi$
$278$	−24.0000	−1.43942
$279$	16.0000	0.957895
$280$	0	0
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	−15.5885	−0.928279
$283$	−5.19615	−0.308879	−0.154440	−	0.988002i	$-0.549357\pi$
$283$	−5.19615	−0.308879	−0.154440	+	0.988002i	$0.549357\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	21.0000	1.23959
$288$	10.3923	0.612372
$289$	31.0000	1.82353
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	−6.92820	−0.404750	−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820	−0.404750	−0.202375	+	0.979308i	$0.564866\pi$
$294$	−6.92820	−0.404061
$295$	0	0
$296$	−13.8564	−0.805387
$297$	0	0
$298$	33.0000	1.91164
$299$	0	0
$300$	0	0
$301$	−15.0000	−0.864586
$302$	36.0000	2.07157
$303$	1.73205	0.0995037
$304$	−17.3205	−0.993399
$305$	0	0
$306$	24.0000	1.37199
$307$	−10.3923	−0.593120	−0.296560	−	0.955014i	$-0.595840\pi$
$307$	−10.3923	−0.593120	−0.296560	+	0.955014i	$0.595840\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	−6.00000	−0.339683
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−6.92820	−0.390981
$315$	0	0
$316$	10.3923	0.584613
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−10.3923	−0.582772
$319$	0	0
$320$	0	0
$321$	1.73205	0.0966736
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	32.9090	1.82266
$327$	−1.73205	−0.0957826
$328$	21.0000	1.15953
$329$	15.5885	0.859419
$330$	0	0
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	3.46410	0.190117
$333$	−16.0000	−0.876795
$334$	−9.00000	−0.492458
$335$	0	0
$336$	8.66025	0.472456
$337$	10.3923	0.566105	0.283052	−	0.959104i	$-0.408653\pi$
$337$	10.3923	0.566105	0.283052	+	0.959104i	$0.408653\pi$
$338$	−1.73205	−0.0942111
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	−12.0000	−0.648886
$343$	19.0526	1.02874
$344$	−15.0000	−0.808746
$345$	0	0
$346$	18.0000	0.967686
$347$	−12.1244	−0.650870	−0.325435	−	0.945564i	$-0.605511\pi$
$347$	−12.1244	−0.650870	−0.325435	+	0.945564i	$0.605511\pi$
$348$	0	0
$349$	−34.6410	−1.85429	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	−34.6410	−1.85429	−0.927146	+	0.374701i	$0.877745\pi$
$350$	0	0
$351$	−17.3205	−0.924500
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	−20.7846	−1.10469
$355$	0	0
$356$	3.00000	0.159000
$357$	12.0000	0.635107
$358$	−31.1769	−1.64775
$359$	3.46410	0.182828	0.0914141	−	0.995813i	$-0.470861\pi$
$359$	3.46410	0.182828	0.0914141	+	0.995813i	$0.470861\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	19.0526	1.00138
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−15.0000	−0.784063
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	0	0
$369$	24.2487	1.26234
$370$	0	0
$371$	10.3923	0.539542
$372$	−8.00000	−0.414781
$373$	−13.8564	−0.717458	−0.358729	−	0.933442i	$-0.616790\pi$
$373$	−13.8564	−0.717458	−0.358729	+	0.933442i	$0.616790\pi$
$374$	0	0
$375$	0	0
$376$	15.5885	0.803913
$377$	0	0
$378$	15.0000	0.771517
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	1.73205	0.0887357
$382$	−10.3923	−0.531717
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	12.1244	0.618718
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−17.3205	−0.880451
$388$	10.0000	0.507673
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	0	0
$392$	6.92820	0.349927
$393$	3.46410	0.174741
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−24.2487	−1.21548
$399$	−6.00000	−0.300376
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	8.66025	0.431934
$403$	−27.7128	−1.38047
$404$	1.73205	0.0861727
$405$	0	0
$406$	0	0
$407$	0	0
$408$	12.0000	0.594089
$409$	−5.19615	−0.256933	−0.128467	−	0.991714i	$-0.541006\pi$
$409$	−5.19615	−0.256933	−0.128467	+	0.991714i	$0.541006\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	4.00000	0.197066
$413$	20.7846	1.02274
$414$	0	0
$415$	0	0
$416$	−18.0000	−0.882523
$417$	−13.8564	−0.678551
$418$	0	0
$419$	18.0000	0.879358	0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000	0.879358	0.439679	+	0.898155i	$0.355092\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	0	0
$423$	18.0000	0.875190
$424$	10.3923	0.504695
$425$	0	0
$426$	−20.7846	−1.00702
$427$	15.0000	0.725901
$428$	1.73205	0.0837218
$429$	0	0
$430$	0	0
$431$	−38.1051	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$431$	−38.1051	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$432$	25.0000	1.20281
$433$	−32.0000	−1.53782	−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000	−1.53782	−0.768911	+	0.639356i	$0.779201\pi$
$434$	24.0000	1.15204
$435$	0	0
$436$	−1.73205	−0.0829502
$437$	0	0
$438$	0	0
$439$	27.7128	1.32266	0.661330	−	0.750095i	$-0.269992\pi$
$439$	27.7128	1.32266	0.661330	+	0.750095i	$0.269992\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	−41.5692	−1.97725
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	32.9090	1.55828
$447$	19.0526	0.901155
$448$	−1.73205	−0.0818317
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	20.7846	0.976546
$454$	33.0000	1.54877
$455$	0	0
$456$	−6.00000	−0.280976
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	12.1244	0.566534
$459$	34.6410	1.61690
$460$	0	0
$461$	12.1244	0.564688	0.282344	−	0.959313i	$-0.408888\pi$
$461$	12.1244	0.564688	0.282344	+	0.959313i	$0.408888\pi$
$462$	0	0
$463$	−41.0000	−1.90543	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000	−1.90543	−0.952716	+	0.303863i	$0.901724\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	−6.92820	−0.320256
$469$	−8.66025	−0.399893
$470$	0	0
$471$	−4.00000	−0.184310
$472$	20.7846	0.956689
$473$	0	0
$474$	18.0000	0.826767
$475$	0	0
$476$	12.0000	0.550019
$477$	12.0000	0.549442
$478$	−6.00000	−0.274434
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	27.7128	1.26360
$482$	33.0000	1.50311
$483$	0	0
$484$	0	0
$485$	0	0
$486$	27.7128	1.25708
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	15.0000	0.679018
$489$	19.0000	0.859210
$490$	0	0
$491$	−24.2487	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$491$	−24.2487	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$492$	−12.1244	−0.546608
$493$	0	0
$494$	20.7846	0.935144
$495$	0	0
$496$	40.0000	1.79605
$497$	20.7846	0.932317
$498$	6.00000	0.268866
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	0	0
$501$	−5.19615	−0.232147
$502$	0	0
$503$	8.66025	0.386142	0.193071	−	0.981185i	$-0.438155\pi$
$503$	8.66025	0.386142	0.193071	+	0.981185i	$0.438155\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	1.73205	0.0768473
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	−17.3205	−0.764719
$514$	−20.7846	−0.916770
$515$	0	0
$516$	8.66025	0.381246
$517$	0	0
$518$	−24.0000	−1.05450
$519$	10.3923	0.456172
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	−17.3205	−0.757373	−0.378686	−	0.925525i	$-0.623624\pi$
$523$	−17.3205	−0.757373	−0.378686	+	0.925525i	$0.623624\pi$
$524$	3.46410	0.151330
$525$	0	0
$526$	42.0000	1.83129
$527$	55.4256	2.41438
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	24.0000	1.04151
$532$	−6.00000	−0.260133
$533$	−42.0000	−1.81922
$534$	5.19615	0.224860
$535$	0	0
$536$	−8.66025	−0.374066
$537$	−18.0000	−0.776757
$538$	−25.9808	−1.12011
$539$	0	0
$540$	0	0
$541$	5.19615	0.223400	0.111700	−	0.993742i	$-0.464370\pi$
$541$	5.19615	0.223400	0.111700	+	0.993742i	$0.464370\pi$
$542$	24.0000	1.03089
$543$	11.0000	0.472055
$544$	36.0000	1.54349
$545$	0	0
$546$	−10.3923	−0.444750
$547$	−45.0333	−1.92549	−0.962743	−	0.270418i	$-0.912838\pi$
$547$	−45.0333	−1.92549	−0.962743	+	0.270418i	$0.912838\pi$
$548$	−18.0000	−0.768922
$549$	17.3205	0.739221
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−18.0000	−0.765438
$554$	18.0000	0.764747
$555$	0	0
$556$	−13.8564	−0.587643
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	27.7128	1.17318
$559$	30.0000	1.26886
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	−15.5885	−0.656975	−0.328488	−	0.944508i	$-0.606539\pi$
$563$	−15.5885	−0.656975	−0.328488	+	0.944508i	$0.606539\pi$
$564$	−9.00000	−0.378968
$565$	0	0
$566$	−9.00000	−0.378298
$567$	−1.73205	−0.0727393
$568$	20.7846	0.872103
$569$	1.73205	0.0726113	0.0363057	−	0.999341i	$-0.488441\pi$
$569$	1.73205	0.0726113	0.0363057	+	0.999341i	$0.488441\pi$
$570$	0	0
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	36.3731	1.51818
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	53.6936	2.23336
$579$	−3.46410	−0.143963
$580$	0	0
$581$	−6.00000	−0.248922
$582$	17.3205	0.717958
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−12.0000	−0.495715
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	−4.00000	−0.164957
$589$	−27.7128	−1.14189
$590$	0	0
$591$	−10.3923	−0.427482
$592$	−40.0000	−1.64399
$593$	24.2487	0.995775	0.497888	−	0.867242i	$-0.334109\pi$
$593$	24.2487	0.995775	0.497888	+	0.867242i	$0.334109\pi$
$594$	0	0
$595$	0	0
$596$	19.0526	0.780423
$597$	−14.0000	−0.572982
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−20.7846	−0.847822	−0.423911	−	0.905704i	$-0.639343\pi$
$601$	−20.7846	−0.847822	−0.423911	+	0.905704i	$0.639343\pi$
$602$	−25.9808	−1.05890
$603$	−10.0000	−0.407231
$604$	20.7846	0.845714
$605$	0	0
$606$	3.00000	0.121867
$607$	−3.46410	−0.140604	−0.0703018	−	0.997526i	$-0.522396\pi$
$607$	−3.46410	−0.140604	−0.0703018	+	0.997526i	$0.522396\pi$
$608$	−18.0000	−0.729996
$609$	0	0
$610$	0	0
$611$	−31.1769	−1.26128
$612$	13.8564	0.560112
$613$	−10.3923	−0.419741	−0.209871	−	0.977729i	$-0.567304\pi$
$613$	−10.3923	−0.419741	−0.209871	+	0.977729i	$0.567304\pi$
$614$	−18.0000	−0.726421
$615$	0	0
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	6.92820	0.278693
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	−10.3923	−0.416693
$623$	−5.19615	−0.208179
$624$	−17.3205	−0.693375
$625$	0	0
$626$	−17.3205	−0.692267
$627$	0	0
$628$	−4.00000	−0.159617
$629$	−55.4256	−2.20996
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−18.0000	−0.716002
$633$	0	0
$634$	10.3923	0.412731
$635$	0	0
$636$	−6.00000	−0.237915
$637$	−13.8564	−0.549011
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	3.00000	0.118401
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	0	0
$645$	0	0
$646$	−41.5692	−1.63552
$647$	−33.0000	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$648$	−1.73205	−0.0680414
$649$	0	0
$650$	0	0
$651$	13.8564	0.543075
$652$	19.0000	0.744097
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−3.00000	−0.117309
$655$	0	0
$656$	60.6218	2.36688
$657$	0	0
$658$	27.0000	1.05257
$659$	−3.46410	−0.134942	−0.0674711	−	0.997721i	$-0.521493\pi$
$659$	−3.46410	−0.134942	−0.0674711	+	0.997721i	$0.521493\pi$
$660$	0	0
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	−58.8897	−2.28881
$663$	−24.0000	−0.932083
$664$	−6.00000	−0.232845
$665$	0	0
$666$	−27.7128	−1.07385
$667$	0	0
$668$	−5.19615	−0.201045
$669$	19.0000	0.734582
$670$	0	0
$671$	0	0
$672$	9.00000	0.347183
$673$	−20.7846	−0.801188	−0.400594	−	0.916256i	$-0.631196\pi$
$673$	−20.7846	−0.801188	−0.400594	+	0.916256i	$0.631196\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	27.7128	1.06509	0.532545	−	0.846402i	$-0.321236\pi$
$677$	27.7128	1.06509	0.532545	+	0.846402i	$0.321236\pi$
$678$	10.3923	0.399114
$679$	−17.3205	−0.664700
$680$	0	0
$681$	19.0526	0.730096
$682$	0	0
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	−6.92820	−0.264906
$685$	0	0
$686$	33.0000	1.25995
$687$	7.00000	0.267067
$688$	−43.3013	−1.65085
$689$	−20.7846	−0.791831
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	10.3923	0.395056
$693$	0	0
$694$	−21.0000	−0.797149
$695$	0	0
$696$	0	0
$697$	84.0000	3.18173
$698$	−60.0000	−2.27103
$699$	−3.46410	−0.131024
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−30.0000	−1.13228
$703$	27.7128	1.04521
$704$	0	0
$705$	0	0
$706$	−41.5692	−1.56448
$707$	−3.00000	−0.112827
$708$	−12.0000	−0.450988
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	−20.7846	−0.779484
$712$	−5.19615	−0.194734
$713$	0	0
$714$	20.7846	0.777844
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−3.46410	−0.129369
$718$	6.00000	0.223918
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−6.92820	−0.258020
$722$	−12.1244	−0.451222
$723$	19.0526	0.708572
$724$	11.0000	0.408812
$725$	0	0
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	10.3923	0.385164
$729$	13.0000	0.481481
$730$	0	0
$731$	−60.0000	−2.21918
$732$	−8.66025	−0.320092
$733$	−20.7846	−0.767697	−0.383849	−	0.923396i	$-0.625402\pi$
$733$	−20.7846	−0.767697	−0.383849	+	0.923396i	$0.625402\pi$
$734$	−29.4449	−1.08683
$735$	0	0
$736$	0	0
$737$	0	0
$738$	42.0000	1.54604
$739$	−3.46410	−0.127429	−0.0637145	−	0.997968i	$-0.520295\pi$
$739$	−3.46410	−0.127429	−0.0637145	+	0.997968i	$0.520295\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	18.0000	0.660801
$743$	5.19615	0.190628	0.0953142	−	0.995447i	$-0.469614\pi$
$743$	5.19615	0.190628	0.0953142	+	0.995447i	$0.469614\pi$
$744$	13.8564	0.508001
$745$	0	0
$746$	−24.0000	−0.878702
$747$	−6.92820	−0.253490
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	−14.0000	−0.510867	−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000	−0.510867	−0.255434	+	0.966827i	$0.582218\pi$
$752$	45.0000	1.64098
$753$	0	0
$754$	0	0
$755$	0	0
$756$	8.66025	0.314970
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−58.8897	−2.13897
$759$	0	0
$760$	0	0
$761$	−34.6410	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$761$	−34.6410	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$762$	3.00000	0.108679
$763$	3.00000	0.108607
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−41.5692	−1.50196
$767$	−41.5692	−1.50098
$768$	19.0000	0.685603
$769$	−27.7128	−0.999350	−0.499675	−	0.866213i	$-0.666547\pi$
$769$	−27.7128	−0.999350	−0.499675	+	0.866213i	$0.666547\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−3.46410	−0.124676
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	−30.0000	−1.07833
$775$	0	0
$776$	−17.3205	−0.621770
$777$	−13.8564	−0.497096
$778$	5.19615	0.186291
$779$	−42.0000	−1.50481
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	20.0000	0.714286
$785$	0	0
$786$	6.00000	0.214013
$787$	−25.9808	−0.926114	−0.463057	−	0.886328i	$-0.653248\pi$
$787$	−25.9808	−0.926114	−0.463057	+	0.886328i	$0.653248\pi$
$788$	−10.3923	−0.370211
$789$	24.2487	0.863277
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	−30.0000	−1.06533
$794$	−34.6410	−1.22936
$795$	0	0
$796$	−14.0000	−0.496217
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	−10.3923	−0.367884
$799$	62.3538	2.20592
$800$	0	0
$801$	−6.00000	−0.212000
$802$	46.7654	1.65134
$803$	0	0
$804$	5.00000	0.176336
$805$	0	0
$806$	−48.0000	−1.69073
$807$	−15.0000	−0.528025
$808$	−3.00000	−0.105540
$809$	48.4974	1.70508	0.852539	−	0.522663i	$-0.175061\pi$
$809$	48.4974	1.70508	0.852539	+	0.522663i	$0.175061\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	0	0
$813$	13.8564	0.485965
$814$	0	0
$815$	0	0
$816$	34.6410	1.21268
$817$	30.0000	1.04957
$818$	−9.00000	−0.314678
$819$	12.0000	0.419314
$820$	0	0
$821$	−29.4449	−1.02763	−0.513816	−	0.857900i	$-0.671769\pi$
$821$	−29.4449	−1.02763	−0.513816	+	0.857900i	$0.671769\pi$
$822$	−31.1769	−1.08742
$823$	−17.0000	−0.592583	−0.296291	−	0.955098i	$-0.595750\pi$
$823$	−17.0000	−0.592583	−0.296291	+	0.955098i	$0.595750\pi$
$824$	−6.92820	−0.241355
$825$	0	0
$826$	36.0000	1.25260
$827$	29.4449	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$827$	29.4449	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$828$	0	0
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	0	0
$831$	10.3923	0.360505
$832$	3.46410	0.120096
$833$	27.7128	0.960192
$834$	−24.0000	−0.831052
$835$	0	0
$836$	0	0
$837$	40.0000	1.38260
$838$	31.1769	1.07699
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−29.4449	−1.01474
$843$	6.92820	0.238620
$844$	0	0
$845$	0	0
$846$	31.1769	1.07188
$847$	0	0
$848$	30.0000	1.03020
$849$	−5.19615	−0.178331
$850$	0	0
$851$	0	0
$852$	−12.0000	−0.411113
$853$	−34.6410	−1.18609	−0.593043	−	0.805171i	$-0.702074\pi$
$853$	−34.6410	−1.18609	−0.593043	+	0.805171i	$0.702074\pi$
$854$	25.9808	0.889043
$855$	0	0
$856$	−3.00000	−0.102538
$857$	−10.3923	−0.354994	−0.177497	−	0.984121i	$-0.556800\pi$
$857$	−10.3923	−0.354994	−0.177497	+	0.984121i	$0.556800\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	21.0000	0.715678
$862$	−66.0000	−2.24797
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	25.9808	0.883883
$865$	0	0
$866$	−55.4256	−1.88344
$867$	31.0000	1.05282
$868$	13.8564	0.470317
$869$	0	0
$870$	0	0
$871$	17.3205	0.586883
$872$	3.00000	0.101593
$873$	−20.0000	−0.676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.7128	−0.935795	−0.467898	−	0.883783i	$-0.654988\pi$
$877$	−27.7128	−0.935795	−0.467898	+	0.883783i	$0.654988\pi$
$878$	48.0000	1.61992
$879$	−6.92820	−0.233682
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	13.8564	0.466569
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	25.9808	0.872841
$887$	−32.9090	−1.10497	−0.552487	−	0.833521i	$-0.686321\pi$
$887$	−32.9090	−1.10497	−0.552487	+	0.833521i	$0.686321\pi$
$888$	−13.8564	−0.464991
$889$	−3.00000	−0.100617
$890$	0	0
$891$	0	0
$892$	19.0000	0.636167
$893$	−31.1769	−1.04330
$894$	33.0000	1.10369
$895$	0	0
$896$	−21.0000	−0.701561
$897$	0	0
$898$	−25.9808	−0.866989
$899$	0	0
$900$	0	0
$901$	41.5692	1.38487
$902$	0	0
$903$	−15.0000	−0.499169
$904$	−10.3923	−0.345643
$905$	0	0
$906$	36.0000	1.19602
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	19.0526	0.632281
$909$	−3.46410	−0.114897
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	−17.3205	−0.573539
$913$	0	0
$914$	36.0000	1.19077
$915$	0	0
$916$	7.00000	0.231287
$917$	−6.00000	−0.198137
$918$	60.0000	1.98030
$919$	−38.1051	−1.25697	−0.628486	−	0.777821i	$-0.716325\pi$
$919$	−38.1051	−1.25697	−0.628486	+	0.777821i	$0.716325\pi$
$920$	0	0
$921$	−10.3923	−0.342438
$922$	21.0000	0.691598
$923$	−41.5692	−1.36827
$924$	0	0
$925$	0	0
$926$	−71.0141	−2.33367
$927$	−8.00000	−0.262754
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	−3.46410	−0.113470
$933$	−6.00000	−0.196431
$934$	−5.19615	−0.170023
$935$	0	0
$936$	12.0000	0.392232
$937$	−24.2487	−0.792171	−0.396085	−	0.918214i	$-0.629632\pi$
$937$	−24.2487	−0.792171	−0.396085	+	0.918214i	$0.629632\pi$
$938$	−15.0000	−0.489767
$939$	−10.0000	−0.326338
$940$	0	0
$941$	50.2295	1.63743	0.818717	−	0.574197i	$-0.194686\pi$
$941$	50.2295	1.63743	0.818717	+	0.574197i	$0.194686\pi$
$942$	−6.92820	−0.225733
$943$	0	0
$944$	60.0000	1.95283
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	10.3923	0.337526
$949$	0	0
$950$	0	0
$951$	6.00000	0.194563
$952$	−20.7846	−0.673633
$953$	55.4256	1.79541	0.897706	−	0.440595i	$-0.145232\pi$
$953$	55.4256	1.79541	0.897706	+	0.440595i	$0.145232\pi$
$954$	20.7846	0.672927
$955$	0	0
$956$	−3.46410	−0.112037
$957$	0	0
$958$	−18.0000	−0.581554
$959$	31.1769	1.00676
$960$	0	0
$961$	33.0000	1.06452
$962$	48.0000	1.54758
$963$	−3.46410	−0.111629
$964$	19.0526	0.613642
$965$	0	0
$966$	0	0
$967$	38.1051	1.22538	0.612689	−	0.790324i	$-0.290088\pi$
$967$	38.1051	1.22538	0.612689	+	0.790324i	$0.290088\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	16.0000	0.513200
$973$	24.0000	0.769405
$974$	34.6410	1.10997
$975$	0	0
$976$	43.3013	1.38604
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	32.9090	1.05231
$979$	0	0
$980$	0	0
$981$	3.46410	0.110600
$982$	−42.0000	−1.34027
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	21.0000	0.669456
$985$	0	0
$986$	0	0
$987$	15.5885	0.496186
$988$	12.0000	0.381771
$989$	0	0
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	41.5692	1.31982
$993$	−34.0000	−1.07896
$994$	36.0000	1.14185
$995$	0	0
$996$	3.46410	0.109764
$997$	−45.0333	−1.42622	−0.713110	−	0.701052i	$-0.752714\pi$
$997$	−45.0333	−1.42622	−0.713110	+	0.701052i	$0.752714\pi$
$998$	−65.8179	−2.08343
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.l.1.2		2
5.4	even	2		605.2.a.e.1.1	✓	2
11.10	odd	2	inner	3025.2.a.l.1.1		2
15.14	odd	2		5445.2.a.u.1.2		2
20.19	odd	2		9680.2.a.bu.1.1		2
55.4	even	10		605.2.g.i.511.1		8
55.9	even	10		605.2.g.i.81.2		8
55.14	even	10		605.2.g.i.251.1		8
55.19	odd	10		605.2.g.i.251.2		8
55.24	odd	10		605.2.g.i.81.1		8
55.29	odd	10		605.2.g.i.511.2		8
55.39	odd	10		605.2.g.i.366.1		8
55.49	even	10		605.2.g.i.366.2		8
55.54	odd	2		605.2.a.e.1.2	yes	2
165.164	even	2		5445.2.a.u.1.1		2
220.219	even	2		9680.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.e.1.1	✓	2	5.4	even	2
605.2.a.e.1.2	yes	2	55.54	odd	2
605.2.g.i.81.1		8	55.24	odd	10
605.2.g.i.81.2		8	55.9	even	10
605.2.g.i.251.1		8	55.14	even	10
605.2.g.i.251.2		8	55.19	odd	10
605.2.g.i.366.1		8	55.39	odd	10
605.2.g.i.366.2		8	55.49	even	10
605.2.g.i.511.1		8	55.4	even	10
605.2.g.i.511.2		8	55.29	odd	10
3025.2.a.l.1.1		2	11.10	odd	2	inner
3025.2.a.l.1.2		2	1.1	even	1	trivial
5445.2.a.u.1.1		2	165.164	even	2
5445.2.a.u.1.2		2	15.14	odd	2
9680.2.a.bu.1.1		2	20.19	odd	2
9680.2.a.bu.1.2		2	220.219	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -1.73205 q^{7} -1.73205 q^{8} -2.00000 q^{9} +1.00000 q^{12} +3.46410 q^{13} -3.00000 q^{14} -5.00000 q^{16} -6.92820 q^{17} -3.46410 q^{18} +3.46410 q^{19} -1.73205 q^{21} -1.73205 q^{24} +6.00000 q^{26} -5.00000 q^{27} -1.73205 q^{28} -8.00000 q^{31} -5.19615 q^{32} -12.0000 q^{34} -2.00000 q^{36} +8.00000 q^{37} +6.00000 q^{38} +3.46410 q^{39} -12.1244 q^{41} -3.00000 q^{42} +8.66025 q^{43} -9.00000 q^{47} -5.00000 q^{48} -4.00000 q^{49} -6.92820 q^{51} +3.46410 q^{52} -6.00000 q^{53} -8.66025 q^{54} +3.00000 q^{56} +3.46410 q^{57} -12.0000 q^{59} -8.66025 q^{61} -13.8564 q^{62} +3.46410 q^{63} +1.00000 q^{64} +5.00000 q^{67} -6.92820 q^{68} -12.0000 q^{71} +3.46410 q^{72} +13.8564 q^{74} +3.46410 q^{76} +6.00000 q^{78} +10.3923 q^{79} +1.00000 q^{81} -21.0000 q^{82} +3.46410 q^{83} -1.73205 q^{84} +15.0000 q^{86} +3.00000 q^{89} -6.00000 q^{91} -8.00000 q^{93} -15.5885 q^{94} -5.19615 q^{96} +10.0000 q^{97} -6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9	\( 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9} + 2 q^{12} - 6 q^{14} - 10 q^{16} + 12 q^{26} - 10 q^{27} - 16 q^{31} - 24 q^{34} - 4 q^{36} + 16 q^{37} + 12 q^{38} - 6 q^{42} - 18 q^{47} - 10 q^{48} - 8 q^{49} - 12 q^{53} + 6 q^{56} - 24 q^{59} + 2 q^{64} + 10 q^{67} - 24 q^{71} + 12 q^{78} + 2 q^{81} - 42 q^{82} + 30 q^{86} + 6 q^{89} - 12 q^{91} - 16 q^{93} + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9 + 2 * q^12 - 6 * q^14 - 10 * q^16 + 12 * q^26 - 10 * q^27 - 16 * q^31 - 24 * q^34 - 4 * q^36 + 16 * q^37 + 12 * q^38 - 6 * q^42 - 18 * q^47 - 10 * q^48 - 8 * q^49 - 12 * q^53 + 6 * q^56 - 24 * q^59 + 2 * q^64 + 10 * q^67 - 24 * q^71 + 12 * q^78 + 2 * q^81 - 42 * q^82 + 30 * q^86 + 6 * q^89 - 12 * q^91 - 16 * q^93 + 20 * q^97

Introduction

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