LMFDB - Embedded newform 3675.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.517638 q^{2} +1.00000 q^{3} -1.73205 q^{4} +0.517638 q^{6} -1.93185 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.517638 q^{2} +1.00000 q^{3} -1.73205 q^{4} +0.517638 q^{6} -1.93185 q^{8} +1.00000 q^{9} +3.46410 q^{11} -1.73205 q^{12} -4.00000 q^{13} +2.46410 q^{16} -4.00000 q^{17} +0.517638 q^{18} +5.27792 q^{19} +1.79315 q^{22} -3.48477 q^{23} -1.93185 q^{24} -2.07055 q^{26} +1.00000 q^{27} -4.92820 q^{29} -7.34847 q^{31} +5.13922 q^{32} +3.46410 q^{33} -2.07055 q^{34} -1.73205 q^{36} -10.5558 q^{37} +2.73205 q^{38} -4.00000 q^{39} +8.48528 q^{41} -6.00000 q^{44} -1.80385 q^{46} -6.00000 q^{47} +2.46410 q^{48} -4.00000 q^{51} +6.92820 q^{52} +7.34847 q^{53} +0.517638 q^{54} +5.27792 q^{57} -2.55103 q^{58} -0.757875 q^{59} +0.656339 q^{61} -3.80385 q^{62} -2.26795 q^{64} +1.79315 q^{66} +12.6264 q^{67} +6.92820 q^{68} -3.48477 q^{69} -6.39230 q^{71} -1.93185 q^{72} +2.92820 q^{73} -5.46410 q^{74} -9.14162 q^{76} -2.07055 q^{78} -4.53590 q^{79} +1.00000 q^{81} +4.39230 q^{82} -6.00000 q^{83} -4.92820 q^{87} -6.69213 q^{88} -15.4548 q^{89} +6.03579 q^{92} -7.34847 q^{93} -3.10583 q^{94} +5.13922 q^{96} -18.9282 q^{97} +3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} - 16 q^{13} - 4 q^{16} - 16 q^{17} + 4 q^{27} + 8 q^{29} + 4 q^{38} - 16 q^{39} - 24 q^{44} - 28 q^{46} - 24 q^{47} - 4 q^{48} - 16 q^{51} - 36 q^{62} - 16 q^{64} + 16 q^{71} - 16 q^{73} - 8 q^{74} - 32 q^{79} + 4 q^{81} - 24 q^{82} - 24 q^{83} + 8 q^{87} - 48 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 - 16 * q^13 - 4 * q^16 - 16 * q^17 + 4 * q^27 + 8 * q^29 + 4 * q^38 - 16 * q^39 - 24 * q^44 - 28 * q^46 - 24 * q^47 - 4 * q^48 - 16 * q^51 - 36 * q^62 - 16 * q^64 + 16 * q^71 - 16 * q^73 - 8 * q^74 - 32 * q^79 + 4 * q^81 - 24 * q^82 - 24 * q^83 + 8 * q^87 - 48 * 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 $

$2$	0.517638	0.366025	0.183013	−	0.983111i	$-0.441415\pi$
$2$	0.517638	0.366025	0.183013	+	0.983111i	$0.441415\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.73205	−0.866025
$5$	0	0
$5$	0	0
$6$	0.517638	0.211325
$7$	0	0
$7$	0	0
$8$	−1.93185	−0.683013
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	−1.73205	−0.500000
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	2.46410	0.616025
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0.517638	0.122008
$19$	5.27792	1.21084	0.605419	−	0.795907i	$-0.293006\pi$
$19$	5.27792	1.21084	0.605419	+	0.795907i	$0.293006\pi$
$20$	0	0
$21$	0	0
$22$	1.79315	0.382301
$23$	−3.48477	−0.726624	−0.363312	−	0.931668i	$-0.618354\pi$
$23$	−3.48477	−0.726624	−0.363312	+	0.931668i	$0.618354\pi$
$24$	−1.93185	−0.394338
$25$	0	0
$26$	−2.07055	−0.406069
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.92820	−0.915144	−0.457572	−	0.889172i	$-0.651281\pi$
$29$	−4.92820	−0.915144	−0.457572	+	0.889172i	$0.651281\pi$
$30$	0	0
$31$	−7.34847	−1.31982	−0.659912	−	0.751343i	$-0.729406\pi$
$31$	−7.34847	−1.31982	−0.659912	+	0.751343i	$0.729406\pi$
$32$	5.13922	0.908494
$33$	3.46410	0.603023
$34$	−2.07055	−0.355097
$35$	0	0
$36$	−1.73205	−0.288675
$37$	−10.5558	−1.73537	−0.867684	−	0.497116i	$-0.834392\pi$
$37$	−10.5558	−1.73537	−0.867684	+	0.497116i	$0.834392\pi$
$38$	2.73205	0.443197
$39$	−4.00000	−0.640513
$40$	0	0
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	−1.80385	−0.265963
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	2.46410	0.355662
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	6.92820	0.960769
$53$	7.34847	1.00939	0.504695	−	0.863298i	$-0.331605\pi$
$53$	7.34847	1.00939	0.504695	+	0.863298i	$0.331605\pi$
$54$	0.517638	0.0704416
$55$	0	0
$56$	0	0
$57$	5.27792	0.699077
$58$	−2.55103	−0.334966
$59$	−0.757875	−0.0986669	−0.0493334	−	0.998782i	$-0.515710\pi$
$59$	−0.757875	−0.0986669	−0.0493334	+	0.998782i	$0.515710\pi$
$60$	0	0
$61$	0.656339	0.0840356	0.0420178	−	0.999117i	$-0.486621\pi$
$61$	0.656339	0.0840356	0.0420178	+	0.999117i	$0.486621\pi$
$62$	−3.80385	−0.483089
$63$	0	0
$64$	−2.26795	−0.283494
$65$	0	0
$66$	1.79315	0.220722
$67$	12.6264	1.54256	0.771279	−	0.636497i	$-0.219617\pi$
$67$	12.6264	1.54256	0.771279	+	0.636497i	$0.219617\pi$
$68$	6.92820	0.840168
$69$	−3.48477	−0.419517
$70$	0	0
$71$	−6.39230	−0.758627	−0.379314	−	0.925268i	$-0.623840\pi$
$71$	−6.39230	−0.758627	−0.379314	+	0.925268i	$0.623840\pi$
$72$	−1.93185	−0.227671
$73$	2.92820	0.342720	0.171360	−	0.985208i	$-0.445184\pi$
$73$	2.92820	0.342720	0.171360	+	0.985208i	$0.445184\pi$
$74$	−5.46410	−0.635189
$75$	0	0
$76$	−9.14162	−1.04862
$77$	0	0
$78$	−2.07055	−0.234444
$79$	−4.53590	−0.510328	−0.255164	−	0.966898i	$-0.582130\pi$
$79$	−4.53590	−0.510328	−0.255164	+	0.966898i	$0.582130\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.39230	0.485049
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.92820	−0.528359
$88$	−6.69213	−0.713384
$89$	−15.4548	−1.63821	−0.819103	−	0.573646i	$-0.805529\pi$
$89$	−15.4548	−1.63821	−0.819103	+	0.573646i	$0.805529\pi$
$90$	0	0
$91$	0	0
$92$	6.03579	0.629275
$93$	−7.34847	−0.762001
$94$	−3.10583	−0.320342
$95$	0	0
$96$	5.13922	0.524519
$97$	−18.9282	−1.92187	−0.960934	−	0.276778i	$-0.910733\pi$
$97$	−18.9282	−1.92187	−0.960934	+	0.276778i	$0.910733\pi$
$98$	0	0
$99$	3.46410	0.348155
$100$	0	0
$101$	−6.96953	−0.693494	−0.346747	−	0.937959i	$-0.612714\pi$
$101$	−6.96953	−0.693494	−0.346747	+	0.937959i	$0.612714\pi$
$102$	−2.07055	−0.205015
$103$	−13.8564	−1.36531	−0.682656	−	0.730740i	$-0.739175\pi$
$103$	−13.8564	−1.36531	−0.682656	+	0.730740i	$0.739175\pi$
$104$	7.72741	0.757735
$105$	0	0
$106$	3.80385	0.369462
$107$	−15.5563	−1.50389	−0.751945	−	0.659226i	$-0.770884\pi$
$107$	−15.5563	−1.50389	−0.751945	+	0.659226i	$0.770884\pi$
$108$	−1.73205	−0.166667
$109$	15.8564	1.51877	0.759384	−	0.650643i	$-0.225500\pi$
$109$	15.8564	1.51877	0.759384	+	0.650643i	$0.225500\pi$
$110$	0	0
$111$	−10.5558	−1.00192
$112$	0	0
$113$	5.27792	0.496505	0.248252	−	0.968695i	$-0.420144\pi$
$113$	5.27792	0.496505	0.248252	+	0.968695i	$0.420144\pi$
$114$	2.73205	0.255880
$115$	0	0
$116$	8.53590	0.792538
$117$	−4.00000	−0.369800
$118$	−0.392305	−0.0361146
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.339746	0.0307592
$123$	8.48528	0.765092
$124$	12.7279	1.14300
$125$	0	0
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	−11.4524	−1.01226
$129$	0	0
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	6.53590	0.564616
$135$	0	0
$136$	7.72741	0.662620
$137$	−10.1769	−0.869471	−0.434735	−	0.900558i	$-0.643158\pi$
$137$	−10.1769	−0.869471	−0.434735	+	0.900558i	$0.643158\pi$
$138$	−1.80385	−0.153554
$139$	−0.378937	−0.0321410	−0.0160705	−	0.999871i	$-0.505116\pi$
$139$	−0.378937	−0.0321410	−0.0160705	+	0.999871i	$0.505116\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−3.30890	−0.277677
$143$	−13.8564	−1.15873
$144$	2.46410	0.205342
$145$	0	0
$146$	1.51575	0.125444
$147$	0	0
$148$	18.2832	1.50287
$149$	7.07180	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$149$	7.07180	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$150$	0	0
$151$	−19.4641	−1.58397	−0.791983	−	0.610543i	$-0.790951\pi$
$151$	−19.4641	−1.58397	−0.791983	+	0.610543i	$0.790951\pi$
$152$	−10.1962	−0.827017
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	6.92820	0.554700
$157$	−5.07180	−0.404773	−0.202387	−	0.979306i	$-0.564870\pi$
$157$	−5.07180	−0.404773	−0.202387	+	0.979306i	$0.564870\pi$
$158$	−2.34795	−0.186793
$159$	7.34847	0.582772
$160$	0	0
$161$	0	0
$162$	0.517638	0.0406695
$163$	22.4243	1.75641	0.878205	−	0.478284i	$-0.158741\pi$
$163$	22.4243	1.75641	0.878205	+	0.478284i	$0.158741\pi$
$164$	−14.6969	−1.14764
$165$	0	0
$166$	−3.10583	−0.241059
$167$	−18.9282	−1.46471	−0.732354	−	0.680924i	$-0.761578\pi$
$167$	−18.9282	−1.46471	−0.732354	+	0.680924i	$0.761578\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	5.27792	0.403612
$172$	0	0
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	−2.55103	−0.193393
$175$	0	0
$176$	8.53590	0.643418
$177$	−0.757875	−0.0569654
$178$	−8.00000	−0.599625
$179$	10.3923	0.776757	0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923	0.776757	0.388379	+	0.921500i	$0.373035\pi$
$180$	0	0
$181$	−6.31319	−0.469256	−0.234628	−	0.972085i	$-0.575387\pi$
$181$	−6.31319	−0.469256	−0.234628	+	0.972085i	$0.575387\pi$
$182$	0	0
$183$	0.656339	0.0485180
$184$	6.73205	0.496293
$185$	0	0
$186$	−3.80385	−0.278912
$187$	−13.8564	−1.01328
$188$	10.3923	0.757937
$189$	0	0
$190$	0	0
$191$	−7.46410	−0.540083	−0.270042	−	0.962849i	$-0.587037\pi$
$191$	−7.46410	−0.540083	−0.270042	+	0.962849i	$0.587037\pi$
$192$	−2.26795	−0.163675
$193$	6.41473	0.461742	0.230871	−	0.972984i	$-0.425842\pi$
$193$	6.41473	0.461742	0.230871	+	0.972984i	$0.425842\pi$
$194$	−9.79796	−0.703452
$195$	0	0
$196$	0	0
$197$	22.8033	1.62467	0.812333	−	0.583193i	$-0.198197\pi$
$197$	22.8033	1.62467	0.812333	+	0.583193i	$0.198197\pi$
$198$	1.79315	0.127434
$199$	8.10634	0.574643	0.287322	−	0.957834i	$-0.407235\pi$
$199$	8.10634	0.574643	0.287322	+	0.957834i	$0.407235\pi$
$200$	0	0
$201$	12.6264	0.890597
$202$	−3.60770	−0.253837
$203$	0	0
$204$	6.92820	0.485071
$205$	0	0
$206$	−7.17260	−0.499739
$207$	−3.48477	−0.242208
$208$	−9.85641	−0.683419
$209$	18.2832	1.26468
$210$	0	0
$211$	−26.9282	−1.85381	−0.926907	−	0.375291i	$-0.877543\pi$
$211$	−26.9282	−1.85381	−0.926907	+	0.375291i	$0.877543\pi$
$212$	−12.7279	−0.874157
$213$	−6.39230	−0.437994
$214$	−8.05256	−0.550462
$215$	0	0
$216$	−1.93185	−0.131446
$217$	0	0
$218$	8.20788	0.555908
$219$	2.92820	0.197870
$220$	0	0
$221$	16.0000	1.07628
$222$	−5.46410	−0.366726
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	0	0
$226$	2.73205	0.181733
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−9.14162	−0.605419
$229$	−13.4858	−0.891167	−0.445583	−	0.895240i	$-0.647004\pi$
$229$	−13.4858	−0.891167	−0.445583	+	0.895240i	$0.647004\pi$
$230$	0	0
$231$	0	0
$232$	9.52056	0.625055
$233$	5.27792	0.345768	0.172884	−	0.984942i	$-0.444691\pi$
$233$	5.27792	0.345768	0.172884	+	0.984942i	$0.444691\pi$
$234$	−2.07055	−0.135356
$235$	0	0
$236$	1.31268	0.0854480
$237$	−4.53590	−0.294638
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	0	0
$241$	14.0406	0.904435	0.452217	−	0.891908i	$-0.350633\pi$
$241$	14.0406	0.904435	0.452217	+	0.891908i	$0.350633\pi$
$242$	0.517638	0.0332750
$243$	1.00000	0.0641500
$244$	−1.13681	−0.0727769
$245$	0	0
$246$	4.39230	0.280043
$247$	−21.1117	−1.34330
$248$	14.1962	0.901457
$249$	−6.00000	−0.380235
$250$	0	0
$251$	20.3538	1.28472	0.642360	−	0.766403i	$-0.277955\pi$
$251$	20.3538	1.28472	0.642360	+	0.766403i	$0.277955\pi$
$252$	0	0
$253$	−12.0716	−0.758934
$254$	−1.46410	−0.0918659
$255$	0	0
$256$	−1.39230	−0.0870191
$257$	3.46410	0.216085	0.108042	−	0.994146i	$-0.465542\pi$
$257$	3.46410	0.216085	0.108042	+	0.994146i	$0.465542\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−4.92820	−0.305048
$262$	−2.92820	−0.180905
$263$	−7.62587	−0.470231	−0.235116	−	0.971967i	$-0.575547\pi$
$263$	−7.62587	−0.470231	−0.235116	+	0.971967i	$0.575547\pi$
$264$	−6.69213	−0.411872
$265$	0	0
$266$	0	0
$267$	−15.4548	−0.945819
$268$	−21.8695	−1.33589
$269$	30.9096	1.88459	0.942297	−	0.334779i	$-0.108662\pi$
$269$	30.9096	1.88459	0.942297	+	0.334779i	$0.108662\pi$
$270$	0	0
$271$	−1.69161	−0.102758	−0.0513791	−	0.998679i	$-0.516362\pi$
$271$	−1.69161	−0.102758	−0.0513791	+	0.998679i	$0.516362\pi$
$272$	−9.85641	−0.597632
$273$	0	0
$274$	−5.26795	−0.318248
$275$	0	0
$276$	6.03579	0.363312
$277$	11.3137	0.679775	0.339887	−	0.940466i	$-0.389611\pi$
$277$	11.3137	0.679775	0.339887	+	0.940466i	$0.389611\pi$
$278$	−0.196152	−0.0117644
$279$	−7.34847	−0.439941
$280$	0	0
$281$	27.8564	1.66177	0.830887	−	0.556441i	$-0.187834\pi$
$281$	27.8564	1.66177	0.830887	+	0.556441i	$0.187834\pi$
$282$	−3.10583	−0.184949
$283$	−2.14359	−0.127423	−0.0637117	−	0.997968i	$-0.520294\pi$
$283$	−2.14359	−0.127423	−0.0637117	+	0.997968i	$0.520294\pi$
$284$	11.0718	0.656990
$285$	0	0
$286$	−7.17260	−0.424125
$287$	0	0
$288$	5.13922	0.302831
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−18.9282	−1.10959
$292$	−5.07180	−0.296804
$293$	−11.4641	−0.669740	−0.334870	−	0.942264i	$-0.608692\pi$
$293$	−11.4641	−0.669740	−0.334870	+	0.942264i	$0.608692\pi$
$294$	0	0
$295$	0	0
$296$	20.3923	1.18528
$297$	3.46410	0.201008
$298$	3.66063	0.212055
$299$	13.9391	0.806117
$300$	0	0
$301$	0	0
$302$	−10.0754	−0.579772
$303$	−6.96953	−0.400389
$304$	13.0053	0.745906
$305$	0	0
$306$	−2.07055	−0.118366
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−13.8564	−0.788263
$310$	0	0
$311$	4.14110	0.234821	0.117410	−	0.993083i	$-0.462541\pi$
$311$	4.14110	0.234821	0.117410	+	0.993083i	$0.462541\pi$
$312$	7.72741	0.437478
$313$	2.92820	0.165512	0.0827559	−	0.996570i	$-0.473628\pi$
$313$	2.92820	0.165512	0.0827559	+	0.996570i	$0.473628\pi$
$314$	−2.62536	−0.148157
$315$	0	0
$316$	7.85641	0.441957
$317$	4.72311	0.265277	0.132638	−	0.991165i	$-0.457655\pi$
$317$	4.72311	0.265277	0.132638	+	0.991165i	$0.457655\pi$
$318$	3.80385	0.213309
$319$	−17.0718	−0.955837
$320$	0	0
$321$	−15.5563	−0.868271
$322$	0	0
$323$	−21.1117	−1.17468
$324$	−1.73205	−0.0962250
$325$	0	0
$326$	11.6077	0.642891
$327$	15.8564	0.876861
$328$	−16.3923	−0.905114
$329$	0	0
$330$	0	0
$331$	−8.78461	−0.482846	−0.241423	−	0.970420i	$-0.577614\pi$
$331$	−8.78461	−0.482846	−0.241423	+	0.970420i	$0.577614\pi$
$332$	10.3923	0.570352
$333$	−10.5558	−0.578456
$334$	−9.79796	−0.536120
$335$	0	0
$336$	0	0
$337$	17.7284	0.965730	0.482865	−	0.875695i	$-0.339596\pi$
$337$	17.7284	0.965730	0.482865	+	0.875695i	$0.339596\pi$
$338$	1.55291	0.0844674
$339$	5.27792	0.286657
$340$	0	0
$341$	−25.4558	−1.37851
$342$	2.73205	0.147732
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−8.28221	−0.445254
$347$	31.0112	1.66477	0.832383	−	0.554200i	$-0.186976\pi$
$347$	31.0112	1.66477	0.832383	+	0.554200i	$0.186976\pi$
$348$	8.53590	0.457572
$349$	−17.6269	−0.943546	−0.471773	−	0.881720i	$-0.656386\pi$
$349$	−17.6269	−0.943546	−0.471773	+	0.881720i	$0.656386\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	17.8028	0.948891
$353$	−11.4641	−0.610173	−0.305086	−	0.952325i	$-0.598685\pi$
$353$	−11.4641	−0.610173	−0.305086	+	0.952325i	$0.598685\pi$
$354$	−0.392305	−0.0208508
$355$	0	0
$356$	26.7685	1.41873
$357$	0	0
$358$	5.37945	0.284313
$359$	15.4641	0.816164	0.408082	−	0.912945i	$-0.366198\pi$
$359$	15.4641	0.816164	0.408082	+	0.912945i	$0.366198\pi$
$360$	0	0
$361$	8.85641	0.466127
$362$	−3.26795	−0.171760
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	0.339746	0.0177588
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−8.58682	−0.447619
$369$	8.48528	0.441726
$370$	0	0
$371$	0	0
$372$	12.7279	0.659912
$373$	−7.17260	−0.371383	−0.185692	−	0.982608i	$-0.559453\pi$
$373$	−7.17260	−0.371383	−0.185692	+	0.982608i	$0.559453\pi$
$374$	−7.17260	−0.370887
$375$	0	0
$376$	11.5911	0.597766
$377$	19.7128	1.01526
$378$	0	0
$379$	11.4641	0.588871	0.294436	−	0.955671i	$-0.404868\pi$
$379$	11.4641	0.588871	0.294436	+	0.955671i	$0.404868\pi$
$380$	0	0
$381$	−2.82843	−0.144905
$382$	−3.86370	−0.197684
$383$	23.8564	1.21901	0.609503	−	0.792784i	$-0.291369\pi$
$383$	23.8564	1.21901	0.609503	+	0.792784i	$0.291369\pi$
$384$	−11.4524	−0.584428
$385$	0	0
$386$	3.32051	0.169009
$387$	0	0
$388$	32.7846	1.66439
$389$	11.0718	0.561362	0.280681	−	0.959801i	$-0.409440\pi$
$389$	11.0718	0.561362	0.280681	+	0.959801i	$0.409440\pi$
$390$	0	0
$391$	13.9391	0.704929
$392$	0	0
$393$	−5.65685	−0.285351
$394$	11.8038	0.594669
$395$	0	0
$396$	−6.00000	−0.301511
$397$	−23.7128	−1.19011	−0.595056	−	0.803684i	$-0.702870\pi$
$397$	−23.7128	−1.19011	−0.595056	+	0.803684i	$0.702870\pi$
$398$	4.19615	0.210334
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	6.53590	0.325981
$403$	29.3939	1.46421
$404$	12.0716	0.600584
$405$	0	0
$406$	0	0
$407$	−36.5665	−1.81253
$408$	7.72741	0.382564
$409$	−11.4152	−0.564448	−0.282224	−	0.959349i	$-0.591072\pi$
$409$	−11.4152	−0.564448	−0.282224	+	0.959349i	$0.591072\pi$
$410$	0	0
$411$	−10.1769	−0.501989
$412$	24.0000	1.18240
$413$	0	0
$414$	−1.80385	−0.0886543
$415$	0	0
$416$	−20.5569	−1.00788
$417$	−0.378937	−0.0185566
$418$	9.46410	0.462904
$419$	−21.1117	−1.03137	−0.515686	−	0.856778i	$-0.672463\pi$
$419$	−21.1117	−1.03137	−0.515686	+	0.856778i	$0.672463\pi$
$420$	0	0
$421$	12.7846	0.623084	0.311542	−	0.950232i	$-0.399155\pi$
$421$	12.7846	0.623084	0.311542	+	0.950232i	$0.399155\pi$
$422$	−13.9391	−0.678543
$423$	−6.00000	−0.291730
$424$	−14.1962	−0.689426
$425$	0	0
$426$	−3.30890	−0.160317
$427$	0	0
$428$	26.9444	1.30241
$429$	−13.8564	−0.668994
$430$	0	0
$431$	37.3205	1.79767	0.898833	−	0.438292i	$-0.144416\pi$
$431$	37.3205	1.79767	0.898833	+	0.438292i	$0.144416\pi$
$432$	2.46410	0.118554
$433$	17.8564	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$433$	17.8564	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$434$	0	0
$435$	0	0
$436$	−27.4641	−1.31529
$437$	−18.3923	−0.879823
$438$	1.51575	0.0724253
$439$	13.0053	0.620710	0.310355	−	0.950621i	$-0.399552\pi$
$439$	13.0053	0.620710	0.310355	+	0.950621i	$0.399552\pi$
$440$	0	0
$441$	0	0
$442$	8.28221	0.393945
$443$	−12.5249	−0.595074	−0.297537	−	0.954710i	$-0.596165\pi$
$443$	−12.5249	−0.595074	−0.297537	+	0.954710i	$0.596165\pi$
$444$	18.2832	0.867684
$445$	0	0
$446$	−4.14110	−0.196087
$447$	7.07180	0.334485
$448$	0	0
$449$	35.8564	1.69217	0.846084	−	0.533049i	$-0.178954\pi$
$449$	35.8564	1.69217	0.846084	+	0.533049i	$0.178954\pi$
$450$	0	0
$451$	29.3939	1.38410
$452$	−9.14162	−0.429986
$453$	−19.4641	−0.914503
$454$	−2.07055	−0.0971758
$455$	0	0
$456$	−10.1962	−0.477479
$457$	21.8695	1.02301	0.511507	−	0.859279i	$-0.329087\pi$
$457$	21.8695	1.02301	0.511507	+	0.859279i	$0.329087\pi$
$458$	−6.98076	−0.326190
$459$	−4.00000	−0.186704
$460$	0	0
$461$	32.4254	1.51020	0.755100	−	0.655609i	$-0.227588\pi$
$461$	32.4254	1.51020	0.755100	+	0.655609i	$0.227588\pi$
$462$	0	0
$463$	−22.4243	−1.04215	−0.521074	−	0.853512i	$-0.674468\pi$
$463$	−22.4243	−1.04215	−0.521074	+	0.853512i	$0.674468\pi$
$464$	−12.1436	−0.563752
$465$	0	0
$466$	2.73205	0.126560
$467$	9.85641	0.456100	0.228050	−	0.973649i	$-0.426765\pi$
$467$	9.85641	0.456100	0.228050	+	0.973649i	$0.426765\pi$
$468$	6.92820	0.320256
$469$	0	0
$470$	0	0
$471$	−5.07180	−0.233696
$472$	1.46410	0.0673907
$473$	0	0
$474$	−2.34795	−0.107845
$475$	0	0
$476$	0	0
$477$	7.34847	0.336463
$478$	−4.41851	−0.202098
$479$	−25.2528	−1.15383	−0.576914	−	0.816805i	$-0.695743\pi$
$479$	−25.2528	−1.15383	−0.576914	+	0.816805i	$0.695743\pi$
$480$	0	0
$481$	42.2233	1.92522
$482$	7.26795	0.331046
$483$	0	0
$484$	−1.73205	−0.0787296
$485$	0	0
$486$	0.517638	0.0234805
$487$	−15.4548	−0.700324	−0.350162	−	0.936689i	$-0.613874\pi$
$487$	−15.4548	−0.700324	−0.350162	+	0.936689i	$0.613874\pi$
$488$	−1.26795	−0.0573974
$489$	22.4243	1.01406
$490$	0	0
$491$	41.3205	1.86477	0.932384	−	0.361469i	$-0.117725\pi$
$491$	41.3205	1.86477	0.932384	+	0.361469i	$0.117725\pi$
$492$	−14.6969	−0.662589
$493$	19.7128	0.887820
$494$	−10.9282	−0.491683
$495$	0	0
$496$	−18.1074	−0.813045
$497$	0	0
$498$	−3.10583	−0.139176
$499$	−32.2487	−1.44365	−0.721825	−	0.692075i	$-0.756697\pi$
$499$	−32.2487	−1.44365	−0.721825	+	0.692075i	$0.756697\pi$
$500$	0	0
$501$	−18.9282	−0.845650
$502$	10.5359	0.470240
$503$	−23.8564	−1.06370	−0.531852	−	0.846837i	$-0.678504\pi$
$503$	−23.8564	−1.06370	−0.531852	+	0.846837i	$0.678504\pi$
$504$	0	0
$505$	0	0
$506$	−6.24871	−0.277789
$507$	3.00000	0.133235
$508$	4.89898	0.217357
$509$	−3.03150	−0.134369	−0.0671844	−	0.997741i	$-0.521402\pi$
$509$	−3.03150	−0.134369	−0.0671844	+	0.997741i	$0.521402\pi$
$510$	0	0
$511$	0	0
$512$	22.1841	0.980408
$513$	5.27792	0.233026
$514$	1.79315	0.0790925
$515$	0	0
$516$	0	0
$517$	−20.7846	−0.914106
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−28.2843	−1.23916	−0.619578	−	0.784935i	$-0.712696\pi$
$521$	−28.2843	−1.23916	−0.619578	+	0.784935i	$0.712696\pi$
$522$	−2.55103	−0.111655
$523$	43.7128	1.91143	0.955714	−	0.294297i	$-0.0950856\pi$
$523$	43.7128	1.91143	0.955714	+	0.294297i	$0.0950856\pi$
$524$	9.79796	0.428026
$525$	0	0
$526$	−3.94744	−0.172117
$527$	29.3939	1.28042
$528$	8.53590	0.371477
$529$	−10.8564	−0.472018
$530$	0	0
$531$	−0.757875	−0.0328890
$532$	0	0
$533$	−33.9411	−1.47015
$534$	−8.00000	−0.346194
$535$	0	0
$536$	−24.3923	−1.05359
$537$	10.3923	0.448461
$538$	16.0000	0.689809
$539$	0	0
$540$	0	0
$541$	26.6410	1.14539	0.572693	−	0.819770i	$-0.305899\pi$
$541$	26.6410	1.14539	0.572693	+	0.819770i	$0.305899\pi$
$542$	−0.875644	−0.0376121
$543$	−6.31319	−0.270925
$544$	−20.5569	−0.881368
$545$	0	0
$546$	0	0
$547$	−10.0010	−0.427613	−0.213807	−	0.976876i	$-0.568586\pi$
$547$	−10.0010	−0.427613	−0.213807	+	0.976876i	$0.568586\pi$
$548$	17.6269	0.752984
$549$	0.656339	0.0280119
$550$	0	0
$551$	−26.0106	−1.10809
$552$	6.73205	0.286535
$553$	0	0
$554$	5.85641	0.248815
$555$	0	0
$556$	0.656339	0.0278350
$557$	−8.86422	−0.375589	−0.187795	−	0.982208i	$-0.560134\pi$
$557$	−8.86422	−0.375589	−0.187795	+	0.982208i	$0.560134\pi$
$558$	−3.80385	−0.161030
$559$	0	0
$560$	0	0
$561$	−13.8564	−0.585018
$562$	14.4195	0.608251
$563$	−4.14359	−0.174632	−0.0873158	−	0.996181i	$-0.527829\pi$
$563$	−4.14359	−0.174632	−0.0873158	+	0.996181i	$0.527829\pi$
$564$	10.3923	0.437595
$565$	0	0
$566$	−1.10961	−0.0466402
$567$	0	0
$568$	12.3490	0.518152
$569$	−15.8564	−0.664735	−0.332368	−	0.943150i	$-0.607847\pi$
$569$	−15.8564	−0.664735	−0.332368	+	0.943150i	$0.607847\pi$
$570$	0	0
$571$	−15.7128	−0.657561	−0.328780	−	0.944406i	$-0.606638\pi$
$571$	−15.7128	−0.657561	−0.328780	+	0.944406i	$0.606638\pi$
$572$	24.0000	1.00349
$573$	−7.46410	−0.311817
$574$	0	0
$575$	0	0
$576$	−2.26795	−0.0944979
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−0.517638	−0.0215309
$579$	6.41473	0.266587
$580$	0	0
$581$	0	0
$582$	−9.79796	−0.406138
$583$	25.4558	1.05427
$584$	−5.65685	−0.234082
$585$	0	0
$586$	−5.93426	−0.245142
$587$	4.14359	0.171024	0.0855122	−	0.996337i	$-0.472747\pi$
$587$	4.14359	0.171024	0.0855122	+	0.996337i	$0.472747\pi$
$588$	0	0
$589$	−38.7846	−1.59809
$590$	0	0
$591$	22.8033	0.938002
$592$	−26.0106	−1.06903
$593$	−25.3205	−1.03979	−0.519894	−	0.854231i	$-0.674029\pi$
$593$	−25.3205	−1.03979	−0.519894	+	0.854231i	$0.674029\pi$
$594$	1.79315	0.0735739
$595$	0	0
$596$	−12.2487	−0.501727
$597$	8.10634	0.331771
$598$	7.21539	0.295059
$599$	29.3205	1.19800	0.599002	−	0.800748i	$-0.295564\pi$
$599$	29.3205	1.19800	0.599002	+	0.800748i	$0.295564\pi$
$600$	0	0
$601$	−25.1512	−1.02594	−0.512970	−	0.858406i	$-0.671455\pi$
$601$	−25.1512	−1.02594	−0.512970	+	0.858406i	$0.671455\pi$
$602$	0	0
$603$	12.6264	0.514186
$604$	33.7128	1.37175
$605$	0	0
$606$	−3.60770	−0.146553
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	27.1244	1.10004
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	6.92820	0.280056
$613$	−9.79796	−0.395736	−0.197868	−	0.980229i	$-0.563402\pi$
$613$	−9.79796	−0.395736	−0.197868	+	0.980229i	$0.563402\pi$
$614$	2.07055	0.0835607
$615$	0	0
$616$	0	0
$617$	−27.1475	−1.09292	−0.546458	−	0.837487i	$-0.684024\pi$
$617$	−27.1475	−1.09292	−0.546458	+	0.837487i	$0.684024\pi$
$618$	−7.17260	−0.288524
$619$	−20.7327	−0.833319	−0.416659	−	0.909063i	$-0.636799\pi$
$619$	−20.7327	−0.833319	−0.416659	+	0.909063i	$0.636799\pi$
$620$	0	0
$621$	−3.48477	−0.139839
$622$	2.14359	0.0859503
$623$	0	0
$624$	−9.85641	−0.394572
$625$	0	0
$626$	1.51575	0.0605815
$627$	18.2832	0.730162
$628$	8.78461	0.350544
$629$	42.2233	1.68355
$630$	0	0
$631$	−25.3205	−1.00799	−0.503997	−	0.863706i	$-0.668138\pi$
$631$	−25.3205	−1.00799	−0.503997	+	0.863706i	$0.668138\pi$
$632$	8.76268	0.348561
$633$	−26.9282	−1.07030
$634$	2.44486	0.0970979
$635$	0	0
$636$	−12.7279	−0.504695
$637$	0	0
$638$	−8.83701	−0.349861
$639$	−6.39230	−0.252876
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−8.05256	−0.317809
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	−10.9282	−0.429964
$647$	42.9282	1.68768	0.843841	−	0.536593i	$-0.180289\pi$
$647$	42.9282	1.68768	0.843841	+	0.536593i	$0.180289\pi$
$648$	−1.93185	−0.0758903
$649$	−2.62536	−0.103054
$650$	0	0
$651$	0	0
$652$	−38.8401	−1.52110
$653$	3.96524	0.155172	0.0775859	−	0.996986i	$-0.475279\pi$
$653$	3.96524	0.155172	0.0775859	+	0.996986i	$0.475279\pi$
$654$	8.20788	0.320954
$655$	0	0
$656$	20.9086	0.816344
$657$	2.92820	0.114240
$658$	0	0
$659$	10.3923	0.404827	0.202413	−	0.979300i	$-0.435122\pi$
$659$	10.3923	0.404827	0.202413	+	0.979300i	$0.435122\pi$
$660$	0	0
$661$	13.4858	0.524537	0.262268	−	0.964995i	$-0.415529\pi$
$661$	13.4858	0.524537	0.262268	+	0.964995i	$0.415529\pi$
$662$	−4.54725	−0.176734
$663$	16.0000	0.621389
$664$	11.5911	0.449822
$665$	0	0
$666$	−5.46410	−0.211730
$667$	17.1736	0.664966
$668$	32.7846	1.26847
$669$	−8.00000	−0.309298
$670$	0	0
$671$	2.27362	0.0877723
$672$	0	0
$673$	−21.8695	−0.843009	−0.421504	−	0.906826i	$-0.638498\pi$
$673$	−21.8695	−0.843009	−0.421504	+	0.906826i	$0.638498\pi$
$674$	9.17691	0.353482
$675$	0	0
$676$	−5.19615	−0.199852
$677$	23.1769	0.890761	0.445381	−	0.895341i	$-0.353068\pi$
$677$	23.1769	0.890761	0.445381	+	0.895341i	$0.353068\pi$
$678$	2.73205	0.104924
$679$	0	0
$680$	0	0
$681$	−4.00000	−0.153280
$682$	−13.1769	−0.504570
$683$	42.3249	1.61952	0.809758	−	0.586764i	$-0.199598\pi$
$683$	42.3249	1.61952	0.809758	+	0.586764i	$0.199598\pi$
$684$	−9.14162	−0.349539
$685$	0	0
$686$	0	0
$687$	−13.4858	−0.514515
$688$	0	0
$689$	−29.3939	−1.11982
$690$	0	0
$691$	10.9348	0.415978	0.207989	−	0.978131i	$-0.433308\pi$
$691$	10.9348	0.415978	0.207989	+	0.978131i	$0.433308\pi$
$692$	27.7128	1.05348
$693$	0	0
$694$	16.0526	0.609347
$695$	0	0
$696$	9.52056	0.360876
$697$	−33.9411	−1.28561
$698$	−9.12436	−0.345362
$699$	5.27792	0.199629
$700$	0	0
$701$	−11.0718	−0.418176	−0.209088	−	0.977897i	$-0.567050\pi$
$701$	−11.0718	−0.418176	−0.209088	+	0.977897i	$0.567050\pi$
$702$	−2.07055	−0.0781480
$703$	−55.7128	−2.10125
$704$	−7.85641	−0.296099
$705$	0	0
$706$	−5.93426	−0.223339
$707$	0	0
$708$	1.31268	0.0493334
$709$	−10.9282	−0.410417	−0.205209	−	0.978718i	$-0.565787\pi$
$709$	−10.9282	−0.410417	−0.205209	+	0.978718i	$0.565787\pi$
$710$	0	0
$711$	−4.53590	−0.170109
$712$	29.8564	1.11892
$713$	25.6077	0.959016
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	−8.53590	−0.318779
$718$	8.00481	0.298737
$719$	14.6969	0.548103	0.274052	−	0.961715i	$-0.411636\pi$
$719$	14.6969	0.548103	0.274052	+	0.961715i	$0.411636\pi$
$720$	0	0
$721$	0	0
$722$	4.58441	0.170614
$723$	14.0406	0.522176
$724$	10.9348	0.406388
$725$	0	0
$726$	0.517638	0.0192114
$727$	23.7128	0.879460	0.439730	−	0.898130i	$-0.355074\pi$
$727$	23.7128	0.879460	0.439730	+	0.898130i	$0.355074\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−1.13681	−0.0420178
$733$	−17.8564	−0.659541	−0.329771	−	0.944061i	$-0.606971\pi$
$733$	−17.8564	−0.659541	−0.329771	+	0.944061i	$0.606971\pi$
$734$	−2.07055	−0.0764255
$735$	0	0
$736$	−17.9090	−0.660133
$737$	43.7391	1.61115
$738$	4.39230	0.161683
$739$	−25.8564	−0.951143	−0.475572	−	0.879677i	$-0.657759\pi$
$739$	−25.8564	−0.951143	−0.475572	+	0.879677i	$0.657759\pi$
$740$	0	0
$741$	−21.1117	−0.775556
$742$	0	0
$743$	51.1619	1.87695	0.938474	−	0.345351i	$-0.112240\pi$
$743$	51.1619	1.87695	0.938474	+	0.345351i	$0.112240\pi$
$744$	14.1962	0.520456
$745$	0	0
$746$	−3.71281	−0.135936
$747$	−6.00000	−0.219529
$748$	24.0000	0.877527
$749$	0	0
$750$	0	0
$751$	−36.7846	−1.34229	−0.671145	−	0.741326i	$-0.734197\pi$
$751$	−36.7846	−1.34229	−0.671145	+	0.741326i	$0.734197\pi$
$752$	−14.7846	−0.539139
$753$	20.3538	0.741733
$754$	10.2041	0.371612
$755$	0	0
$756$	0	0
$757$	−34.2929	−1.24640	−0.623198	−	0.782064i	$-0.714167\pi$
$757$	−34.2929	−1.24640	−0.623198	+	0.782064i	$0.714167\pi$
$758$	5.93426	0.215542
$759$	−12.0716	−0.438171
$760$	0	0
$761$	18.0802	0.655406	0.327703	−	0.944781i	$-0.393726\pi$
$761$	18.0802	0.655406	0.327703	+	0.944781i	$0.393726\pi$
$762$	−1.46410	−0.0530388
$763$	0	0
$764$	12.9282	0.467726
$765$	0	0
$766$	12.3490	0.446187
$767$	3.03150	0.109461
$768$	−1.39230	−0.0502405
$769$	39.2934	1.41696	0.708478	−	0.705733i	$-0.249382\pi$
$769$	39.2934	1.41696	0.708478	+	0.705733i	$0.249382\pi$
$770$	0	0
$771$	3.46410	0.124757
$772$	−11.1106	−0.399881
$773$	−37.8564	−1.36160	−0.680800	−	0.732469i	$-0.738368\pi$
$773$	−37.8564	−1.36160	−0.680800	+	0.732469i	$0.738368\pi$
$774$	0	0
$775$	0	0
$776$	36.5665	1.31266
$777$	0	0
$778$	5.73118	0.205473
$779$	44.7846	1.60458
$780$	0	0
$781$	−22.1436	−0.792360
$782$	7.21539	0.258022
$783$	−4.92820	−0.176120
$784$	0	0
$785$	0	0
$786$	−2.92820	−0.104446
$787$	45.8564	1.63460	0.817302	−	0.576209i	$-0.195469\pi$
$787$	45.8564	1.63460	0.817302	+	0.576209i	$0.195469\pi$
$788$	−39.4964	−1.40700
$789$	−7.62587	−0.271488
$790$	0	0
$791$	0	0
$792$	−6.69213	−0.237795
$793$	−2.62536	−0.0932291
$794$	−12.2747	−0.435611
$795$	0	0
$796$	−14.0406	−0.497656
$797$	10.1436	0.359305	0.179652	−	0.983730i	$-0.442503\pi$
$797$	10.1436	0.359305	0.179652	+	0.983730i	$0.442503\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−15.4548	−0.546069
$802$	1.03528	0.0365569
$803$	10.1436	0.357960
$804$	−21.8695	−0.771279
$805$	0	0
$806$	15.2154	0.535939
$807$	30.9096	1.08807
$808$	13.4641	0.473665
$809$	21.7128	0.763382	0.381691	−	0.924290i	$-0.375342\pi$
$809$	21.7128	0.763382	0.381691	+	0.924290i	$0.375342\pi$
$810$	0	0
$811$	−25.6317	−0.900051	−0.450026	−	0.893016i	$-0.648585\pi$
$811$	−25.6317	−0.900051	−0.450026	+	0.893016i	$0.648585\pi$
$812$	0	0
$813$	−1.69161	−0.0593275
$814$	−18.9282	−0.663433
$815$	0	0
$816$	−9.85641	−0.345043
$817$	0	0
$818$	−5.90897	−0.206602
$819$	0	0
$820$	0	0
$821$	18.7846	0.655587	0.327794	−	0.944749i	$-0.393695\pi$
$821$	18.7846	0.655587	0.327794	+	0.944749i	$0.393695\pi$
$822$	−5.26795	−0.183741
$823$	−30.7066	−1.07036	−0.535182	−	0.844737i	$-0.679757\pi$
$823$	−30.7066	−1.07036	−0.535182	+	0.844737i	$0.679757\pi$
$824$	26.7685	0.932526
$825$	0	0
$826$	0	0
$827$	8.18067	0.284470	0.142235	−	0.989833i	$-0.454571\pi$
$827$	8.18067	0.284470	0.142235	+	0.989833i	$0.454571\pi$
$828$	6.03579	0.209758
$829$	−6.51626	−0.226319	−0.113160	−	0.993577i	$-0.536097\pi$
$829$	−6.51626	−0.226319	−0.113160	+	0.993577i	$0.536097\pi$
$830$	0	0
$831$	11.3137	0.392468
$832$	9.07180	0.314508
$833$	0	0
$834$	−0.196152	−0.00679220
$835$	0	0
$836$	−31.6675	−1.09524
$837$	−7.34847	−0.254000
$838$	−10.9282	−0.377509
$839$	−20.3538	−0.702691	−0.351345	−	0.936246i	$-0.614276\pi$
$839$	−20.3538	−0.702691	−0.351345	+	0.936246i	$0.614276\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	6.61780	0.228064
$843$	27.8564	0.959426
$844$	46.6410	1.60545
$845$	0	0
$846$	−3.10583	−0.106781
$847$	0	0
$848$	18.1074	0.621810
$849$	−2.14359	−0.0735679
$850$	0	0
$851$	36.7846	1.26096
$852$	11.0718	0.379314
$853$	−13.0718	−0.447570	−0.223785	−	0.974639i	$-0.571841\pi$
$853$	−13.0718	−0.447570	−0.223785	+	0.974639i	$0.571841\pi$
$854$	0	0
$855$	0	0
$856$	30.0526	1.02718
$857$	9.85641	0.336688	0.168344	−	0.985728i	$-0.446158\pi$
$857$	9.85641	0.336688	0.168344	+	0.985728i	$0.446158\pi$
$858$	−7.17260	−0.244869
$859$	−3.41044	−0.116363	−0.0581813	−	0.998306i	$-0.518530\pi$
$859$	−3.41044	−0.116363	−0.0581813	+	0.998306i	$0.518530\pi$
$860$	0	0
$861$	0	0
$862$	19.3185	0.657991
$863$	15.9081	0.541517	0.270759	−	0.962647i	$-0.412725\pi$
$863$	15.9081	0.541517	0.270759	+	0.962647i	$0.412725\pi$
$864$	5.13922	0.174840
$865$	0	0
$866$	9.24316	0.314095
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−15.7128	−0.533021
$870$	0	0
$871$	−50.5055	−1.71132
$872$	−30.6322	−1.03734
$873$	−18.9282	−0.640623
$874$	−9.52056	−0.322038
$875$	0	0
$876$	−5.07180	−0.171360
$877$	33.9411	1.14611	0.573055	−	0.819517i	$-0.305758\pi$
$877$	33.9411	1.14611	0.573055	+	0.819517i	$0.305758\pi$
$878$	6.73205	0.227196
$879$	−11.4641	−0.386675
$880$	0	0
$881$	−29.5969	−0.997147	−0.498573	−	0.866848i	$-0.666143\pi$
$881$	−29.5969	−0.997147	−0.498573	+	0.866848i	$0.666143\pi$
$882$	0	0
$883$	−29.3939	−0.989183	−0.494591	−	0.869126i	$-0.664682\pi$
$883$	−29.3939	−0.989183	−0.494591	+	0.869126i	$0.664682\pi$
$884$	−27.7128	−0.932083
$885$	0	0
$886$	−6.48334	−0.217812
$887$	−13.7128	−0.460431	−0.230216	−	0.973140i	$-0.573943\pi$
$887$	−13.7128	−0.460431	−0.230216	+	0.973140i	$0.573943\pi$
$888$	20.3923	0.684321
$889$	0	0
$890$	0	0
$891$	3.46410	0.116052
$892$	13.8564	0.463947
$893$	−31.6675	−1.05971
$894$	3.66063	0.122430
$895$	0	0
$896$	0	0
$897$	13.9391	0.465412
$898$	18.5606	0.619377
$899$	36.2147	1.20783
$900$	0	0
$901$	−29.3939	−0.979252
$902$	15.2154	0.506617
$903$	0	0
$904$	−10.1962	−0.339119
$905$	0	0
$906$	−10.0754	−0.334731
$907$	−6.76646	−0.224677	−0.112338	−	0.993670i	$-0.535834\pi$
$907$	−6.76646	−0.224677	−0.112338	+	0.993670i	$0.535834\pi$
$908$	6.92820	0.229920
$909$	−6.96953	−0.231165
$910$	0	0
$911$	9.60770	0.318317	0.159159	−	0.987253i	$-0.449122\pi$
$911$	9.60770	0.318317	0.159159	+	0.987253i	$0.449122\pi$
$912$	13.0053	0.430649
$913$	−20.7846	−0.687870
$914$	11.3205	0.374449
$915$	0	0
$916$	23.3581	0.771773
$917$	0	0
$918$	−2.07055	−0.0683384
$919$	−9.07180	−0.299251	−0.149625	−	0.988743i	$-0.547807\pi$
$919$	−9.07180	−0.299251	−0.149625	+	0.988743i	$0.547807\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	16.7846	0.552772
$923$	25.5692	0.841621
$924$	0	0
$925$	0	0
$926$	−11.6077	−0.381453
$927$	−13.8564	−0.455104
$928$	−25.3271	−0.831403
$929$	−18.0802	−0.593191	−0.296596	−	0.955003i	$-0.595851\pi$
$929$	−18.0802	−0.593191	−0.296596	+	0.955003i	$0.595851\pi$
$930$	0	0
$931$	0	0
$932$	−9.14162	−0.299444
$933$	4.14110	0.135574
$934$	5.10205	0.166944
$935$	0	0
$936$	7.72741	0.252578
$937$	16.7846	0.548329	0.274165	−	0.961683i	$-0.411599\pi$
$937$	16.7846	0.548329	0.274165	+	0.961683i	$0.411599\pi$
$938$	0	0
$939$	2.92820	0.0955583
$940$	0	0
$941$	0.203072	0.00661996	0.00330998	−	0.999995i	$-0.498946\pi$
$941$	0.203072	0.00661996	0.00330998	+	0.999995i	$0.498946\pi$
$942$	−2.62536	−0.0855387
$943$	−29.5692	−0.962906
$944$	−1.86748	−0.0607813
$945$	0	0
$946$	0	0
$947$	−18.3848	−0.597425	−0.298712	−	0.954343i	$-0.596557\pi$
$947$	−18.3848	−0.597425	−0.298712	+	0.954343i	$0.596557\pi$
$948$	7.85641	0.255164
$949$	−11.7128	−0.380214
$950$	0	0
$951$	4.72311	0.153157
$952$	0	0
$953$	−53.9160	−1.74651	−0.873255	−	0.487264i	$-0.837995\pi$
$953$	−53.9160	−1.74651	−0.873255	+	0.487264i	$0.837995\pi$
$954$	3.80385	0.123154
$955$	0	0
$956$	14.7846	0.478168
$957$	−17.0718	−0.551853
$958$	−13.0718	−0.422331
$959$	0	0
$960$	0	0
$961$	23.0000	0.741935
$962$	21.8564	0.704679
$963$	−15.5563	−0.501296
$964$	−24.3190	−0.783263
$965$	0	0
$966$	0	0
$967$	−10.0010	−0.321611	−0.160806	−	0.986986i	$-0.551409\pi$
$967$	−10.0010	−0.321611	−0.160806	+	0.986986i	$0.551409\pi$
$968$	−1.93185	−0.0620921
$969$	−21.1117	−0.678204
$970$	0	0
$971$	−11.6654	−0.374362	−0.187181	−	0.982325i	$-0.559935\pi$
$971$	−11.6654	−0.374362	−0.187181	+	0.982325i	$0.559935\pi$
$972$	−1.73205	−0.0555556
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	1.61729	0.0517680
$977$	−45.6338	−1.45995	−0.729977	−	0.683472i	$-0.760469\pi$
$977$	−45.6338	−1.45995	−0.729977	+	0.683472i	$0.760469\pi$
$978$	11.6077	0.371173
$979$	−53.5370	−1.71105
$980$	0	0
$981$	15.8564	0.506256
$982$	21.3891	0.682553
$983$	24.7846	0.790506	0.395253	−	0.918572i	$-0.370657\pi$
$983$	24.7846	0.790506	0.395253	+	0.918572i	$0.370657\pi$
$984$	−16.3923	−0.522568
$985$	0	0
$986$	10.2041	0.324965
$987$	0	0
$988$	36.5665	1.16333
$989$	0	0
$990$	0	0
$991$	12.7846	0.406117	0.203058	−	0.979167i	$-0.434912\pi$
$991$	12.7846	0.406117	0.203058	+	0.979167i	$0.434912\pi$
$992$	−37.7654	−1.19905
$993$	−8.78461	−0.278771
$994$	0	0
$995$	0	0
$996$	10.3923	0.329293
$997$	−41.8564	−1.32561	−0.662803	−	0.748794i	$-0.730633\pi$
$997$	−41.8564	−1.32561	−0.662803	+	0.748794i	$0.730633\pi$
$998$	−16.6932	−0.528413
$999$	−10.5558	−0.333972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bu.1.3		4
5.2	odd	4		735.2.d.f.589.5	yes	8
5.3	odd	4		735.2.d.f.589.4	yes	8
5.4	even	2		3675.2.a.bs.1.2		4
7.6	odd	2		3675.2.a.bs.1.3		4
15.2	even	4		2205.2.d.t.1324.3		8
15.8	even	4		2205.2.d.t.1324.5		8
35.2	odd	12		735.2.q.d.214.3		8
35.3	even	12		735.2.q.c.79.3		8
35.12	even	12		735.2.q.c.214.3		8
35.13	even	4		735.2.d.f.589.3	✓	8
35.17	even	12		735.2.q.d.79.2		8
35.18	odd	12		735.2.q.d.79.3		8
35.23	odd	12		735.2.q.c.214.2		8
35.27	even	4		735.2.d.f.589.6	yes	8
35.32	odd	12		735.2.q.c.79.2		8
35.33	even	12		735.2.q.d.214.2		8
35.34	odd	2	inner	3675.2.a.bu.1.2		4
105.62	odd	4		2205.2.d.t.1324.4		8
105.83	odd	4		2205.2.d.t.1324.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.d.f.589.3	✓	8	35.13	even	4
735.2.d.f.589.4	yes	8	5.3	odd	4
735.2.d.f.589.5	yes	8	5.2	odd	4
735.2.d.f.589.6	yes	8	35.27	even	4
735.2.q.c.79.2		8	35.32	odd	12
735.2.q.c.79.3		8	35.3	even	12
735.2.q.c.214.2		8	35.23	odd	12
735.2.q.c.214.3		8	35.12	even	12
735.2.q.d.79.2		8	35.17	even	12
735.2.q.d.79.3		8	35.18	odd	12
735.2.q.d.214.2		8	35.33	even	12
735.2.q.d.214.3		8	35.2	odd	12
2205.2.d.t.1324.3		8	15.2	even	4
2205.2.d.t.1324.4		8	105.62	odd	4
2205.2.d.t.1324.5		8	15.8	even	4
2205.2.d.t.1324.6		8	105.83	odd	4
3675.2.a.bs.1.2		4	5.4	even	2
3675.2.a.bs.1.3		4	7.6	odd	2
3675.2.a.bu.1.2		4	35.34	odd	2	inner
3675.2.a.bu.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+0.517638 q^{2} +1.00000 q^{3} -1.73205 q^{4} +0.517638 q^{6} -1.93185 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.517638 q^{2} +1.00000 q^{3} -1.73205 q^{4} +0.517638 q^{6} -1.93185 q^{8} +1.00000 q^{9} +3.46410 q^{11} -1.73205 q^{12} -4.00000 q^{13} +2.46410 q^{16} -4.00000 q^{17} +0.517638 q^{18} +5.27792 q^{19} +1.79315 q^{22} -3.48477 q^{23} -1.93185 q^{24} -2.07055 q^{26} +1.00000 q^{27} -4.92820 q^{29} -7.34847 q^{31} +5.13922 q^{32} +3.46410 q^{33} -2.07055 q^{34} -1.73205 q^{36} -10.5558 q^{37} +2.73205 q^{38} -4.00000 q^{39} +8.48528 q^{41} -6.00000 q^{44} -1.80385 q^{46} -6.00000 q^{47} +2.46410 q^{48} -4.00000 q^{51} +6.92820 q^{52} +7.34847 q^{53} +0.517638 q^{54} +5.27792 q^{57} -2.55103 q^{58} -0.757875 q^{59} +0.656339 q^{61} -3.80385 q^{62} -2.26795 q^{64} +1.79315 q^{66} +12.6264 q^{67} +6.92820 q^{68} -3.48477 q^{69} -6.39230 q^{71} -1.93185 q^{72} +2.92820 q^{73} -5.46410 q^{74} -9.14162 q^{76} -2.07055 q^{78} -4.53590 q^{79} +1.00000 q^{81} +4.39230 q^{82} -6.00000 q^{83} -4.92820 q^{87} -6.69213 q^{88} -15.4548 q^{89} +6.03579 q^{92} -7.34847 q^{93} -3.10583 q^{94} +5.13922 q^{96} -18.9282 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} - 16 q^{13} - 4 q^{16} - 16 q^{17} + 4 q^{27} + 8 q^{29} + 4 q^{38} - 16 q^{39} - 24 q^{44} - 28 q^{46} - 24 q^{47} - 4 q^{48} - 16 q^{51} - 36 q^{62} - 16 q^{64} + 16 q^{71} - 16 q^{73} - 8 q^{74} - 32 q^{79} + 4 q^{81} - 24 q^{82} - 24 q^{83} + 8 q^{87} - 48 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 - 16 * q^13 - 4 * q^16 - 16 * q^17 + 4 * q^27 + 8 * q^29 + 4 * q^38 - 16 * q^39 - 24 * q^44 - 28 * q^46 - 24 * q^47 - 4 * q^48 - 16 * q^51 - 36 * q^62 - 16 * q^64 + 16 * q^71 - 16 * q^73 - 8 * q^74 - 32 * q^79 + 4 * q^81 - 24 * q^82 - 24 * q^83 + 8 * q^87 - 48 * q^97

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