LMFDB - Embedded newform 3675.2.a.bb.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} -2.30278 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} -2.30278 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -3.30278 q^{12} +2.60555 q^{13} +0.302776 q^{16} -4.60555 q^{17} +2.30278 q^{18} -6.60555 q^{19} -6.90833 q^{22} -6.21110 q^{23} -3.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -7.60555 q^{29} +7.21110 q^{31} -5.30278 q^{32} +3.00000 q^{33} -10.6056 q^{34} +3.30278 q^{36} -4.21110 q^{37} -15.2111 q^{38} -2.60555 q^{39} +9.60555 q^{43} -9.90833 q^{44} -14.3028 q^{46} -10.6056 q^{47} -0.302776 q^{48} +4.60555 q^{51} +8.60555 q^{52} +3.21110 q^{53} -2.30278 q^{54} +6.60555 q^{57} -17.5139 q^{58} +10.6056 q^{59} +1.21110 q^{61} +16.6056 q^{62} -12.8167 q^{64} +6.90833 q^{66} +15.6056 q^{67} -15.2111 q^{68} +6.21110 q^{69} -3.00000 q^{71} +3.00000 q^{72} -0.605551 q^{73} -9.69722 q^{74} -21.8167 q^{76} -6.00000 q^{78} -14.8167 q^{79} +1.00000 q^{81} +3.21110 q^{83} +22.1194 q^{86} +7.60555 q^{87} -9.00000 q^{88} -7.81665 q^{89} -20.5139 q^{92} -7.21110 q^{93} -24.4222 q^{94} +5.30278 q^{96} +0.788897 q^{97} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} + 3 q^{4} - q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 - 2 * q^3 + 3 * q^4 - q^6 + 6 * q^8 + 2 * q^9	$ 2 q + q^{2} - 2 q^{3} + 3 q^{4} - q^{6} + 6 q^{8} + 2 q^{9} - 6 q^{11} - 3 q^{12} - 2 q^{13} - 3 q^{16} - 2 q^{17} + q^{18} - 6 q^{19} - 3 q^{22} + 2 q^{23} - 6 q^{24} + 12 q^{26} - 2 q^{27} - 8 q^{29} - 7 q^{32} + 6 q^{33} - 14 q^{34} + 3 q^{36} + 6 q^{37} - 16 q^{38} + 2 q^{39} + 12 q^{43} - 9 q^{44} - 25 q^{46} - 14 q^{47} + 3 q^{48} + 2 q^{51} + 10 q^{52} - 8 q^{53} - q^{54} + 6 q^{57} - 17 q^{58} + 14 q^{59} - 12 q^{61} + 26 q^{62} - 4 q^{64} + 3 q^{66} + 24 q^{67} - 16 q^{68} - 2 q^{69} - 6 q^{71} + 6 q^{72} + 6 q^{73} - 23 q^{74} - 22 q^{76} - 12 q^{78} - 8 q^{79} + 2 q^{81} - 8 q^{83} + 19 q^{86} + 8 q^{87} - 18 q^{88} + 6 q^{89} - 23 q^{92} - 20 q^{94} + 7 q^{96} + 16 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 + 3 * q^4 - q^6 + 6 * q^8 + 2 * q^9 - 6 * q^11 - 3 * q^12 - 2 * q^13 - 3 * q^16 - 2 * q^17 + q^18 - 6 * q^19 - 3 * q^22 + 2 * q^23 - 6 * q^24 + 12 * q^26 - 2 * q^27 - 8 * q^29 - 7 * q^32 + 6 * q^33 - 14 * q^34 + 3 * q^36 + 6 * q^37 - 16 * q^38 + 2 * q^39 + 12 * q^43 - 9 * q^44 - 25 * q^46 - 14 * q^47 + 3 * q^48 + 2 * q^51 + 10 * q^52 - 8 * q^53 - q^54 + 6 * q^57 - 17 * q^58 + 14 * q^59 - 12 * q^61 + 26 * q^62 - 4 * q^64 + 3 * q^66 + 24 * q^67 - 16 * q^68 - 2 * q^69 - 6 * q^71 + 6 * q^72 + 6 * q^73 - 23 * q^74 - 22 * q^76 - 12 * q^78 - 8 * q^79 + 2 * q^81 - 8 * q^83 + 19 * q^86 + 8 * q^87 - 18 * q^88 + 6 * q^89 - 23 * q^92 - 20 * q^94 + 7 * q^96 + 16 * q^97 - 6 * q^99

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 $

$2$	2.30278	1.62831	0.814154	−	0.580649i	$-0.197201\pi$
$2$	2.30278	1.62831	0.814154	+	0.580649i	$0.197201\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	−2.30278	−0.940104
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−3.30278	−0.953429
$13$	2.60555	0.722650	0.361325	−	0.932440i	$-0.382325\pi$
$13$	2.60555	0.722650	0.361325	+	0.932440i	$0.382325\pi$
$14$	0	0
$15$	0	0
$16$	0.302776	0.0756939
$17$	−4.60555	−1.11701	−0.558505	−	0.829501i	$-0.688625\pi$
$17$	−4.60555	−1.11701	−0.558505	+	0.829501i	$0.688625\pi$
$18$	2.30278	0.542769
$19$	−6.60555	−1.51542	−0.757709	−	0.652593i	$-0.773681\pi$
$19$	−6.60555	−1.51542	−0.757709	+	0.652593i	$0.773681\pi$
$20$	0	0
$21$	0	0
$22$	−6.90833	−1.47286
$23$	−6.21110	−1.29510	−0.647552	−	0.762021i	$-0.724207\pi$
$23$	−6.21110	−1.29510	−0.647552	+	0.762021i	$0.724207\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.60555	−1.41232	−0.706158	−	0.708055i	$-0.749573\pi$
$29$	−7.60555	−1.41232	−0.706158	+	0.708055i	$0.749573\pi$
$30$	0	0
$31$	7.21110	1.29515	0.647576	−	0.762001i	$-0.275783\pi$
$31$	7.21110	1.29515	0.647576	+	0.762001i	$0.275783\pi$
$32$	−5.30278	−0.937407
$33$	3.00000	0.522233
$34$	−10.6056	−1.81884
$35$	0	0
$36$	3.30278	0.550463
$37$	−4.21110	−0.692301	−0.346150	−	0.938179i	$-0.612511\pi$
$37$	−4.21110	−0.692301	−0.346150	+	0.938179i	$0.612511\pi$
$38$	−15.2111	−2.46757
$39$	−2.60555	−0.417222
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	9.60555	1.46483	0.732416	−	0.680857i	$-0.238392\pi$
$43$	9.60555	1.46483	0.732416	+	0.680857i	$0.238392\pi$
$44$	−9.90833	−1.49374
$45$	0	0
$46$	−14.3028	−2.10883
$47$	−10.6056	−1.54698	−0.773489	−	0.633809i	$-0.781490\pi$
$47$	−10.6056	−1.54698	−0.773489	+	0.633809i	$0.781490\pi$
$48$	−0.302776	−0.0437019
$49$	0	0
$50$	0	0
$51$	4.60555	0.644906
$52$	8.60555	1.19338
$53$	3.21110	0.441079	0.220539	−	0.975378i	$-0.429218\pi$
$53$	3.21110	0.441079	0.220539	+	0.975378i	$0.429218\pi$
$54$	−2.30278	−0.313368
$55$	0	0
$56$	0	0
$57$	6.60555	0.874927
$58$	−17.5139	−2.29968
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	1.21110	0.155066	0.0775329	−	0.996990i	$-0.475296\pi$
$61$	1.21110	0.155066	0.0775329	+	0.996990i	$0.475296\pi$
$62$	16.6056	2.10891
$63$	0	0
$64$	−12.8167	−1.60208
$65$	0	0
$66$	6.90833	0.850356
$67$	15.6056	1.90652	0.953261	−	0.302149i	$-0.0977040\pi$
$67$	15.6056	1.90652	0.953261	+	0.302149i	$0.0977040\pi$
$68$	−15.2111	−1.84462
$69$	6.21110	0.747729
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	3.00000	0.353553
$73$	−0.605551	−0.0708744	−0.0354372	−	0.999372i	$-0.511282\pi$
$73$	−0.605551	−0.0708744	−0.0354372	+	0.999372i	$0.511282\pi$
$74$	−9.69722	−1.12728
$75$	0	0
$76$	−21.8167	−2.50254
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−14.8167	−1.66700	−0.833502	−	0.552517i	$-0.813668\pi$
$79$	−14.8167	−1.66700	−0.833502	+	0.552517i	$0.813668\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.21110	0.352464	0.176232	−	0.984349i	$-0.443609\pi$
$83$	3.21110	0.352464	0.176232	+	0.984349i	$0.443609\pi$
$84$	0	0
$85$	0	0
$86$	22.1194	2.38520
$87$	7.60555	0.815401
$88$	−9.00000	−0.959403
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	0	0
$92$	−20.5139	−2.13872
$93$	−7.21110	−0.747757
$94$	−24.4222	−2.51896
$95$	0	0
$96$	5.30278	0.541212
$97$	0.788897	0.0801004	0.0400502	−	0.999198i	$-0.487248\pi$
$97$	0.788897	0.0801004	0.0400502	+	0.999198i	$0.487248\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−16.6056	−1.65231	−0.826157	−	0.563440i	$-0.809478\pi$
$101$	−16.6056	−1.65231	−0.826157	+	0.563440i	$0.809478\pi$
$102$	10.6056	1.05011
$103$	−3.81665	−0.376066	−0.188033	−	0.982163i	$-0.560211\pi$
$103$	−3.81665	−0.376066	−0.188033	+	0.982163i	$0.560211\pi$
$104$	7.81665	0.766486
$105$	0	0
$106$	7.39445	0.718212
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−3.30278	−0.317810
$109$	2.21110	0.211785	0.105893	−	0.994378i	$-0.466230\pi$
$109$	2.21110	0.211785	0.105893	+	0.994378i	$0.466230\pi$
$110$	0	0
$111$	4.21110	0.399700
$112$	0	0
$113$	−10.8167	−1.01755	−0.508773	−	0.860901i	$-0.669901\pi$
$113$	−10.8167	−1.01755	−0.508773	+	0.860901i	$0.669901\pi$
$114$	15.2111	1.42465
$115$	0	0
$116$	−25.1194	−2.33228
$117$	2.60555	0.240883
$118$	24.4222	2.24825
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.78890	0.252495
$123$	0	0
$124$	23.8167	2.13880
$125$	0	0
$126$	0	0
$127$	12.8167	1.13729	0.568647	−	0.822582i	$-0.307467\pi$
$127$	12.8167	1.13729	0.568647	+	0.822582i	$0.307467\pi$
$128$	−18.9083	−1.67128
$129$	−9.60555	−0.845722
$130$	0	0
$131$	7.39445	0.646056	0.323028	−	0.946389i	$-0.395299\pi$
$131$	7.39445	0.646056	0.323028	+	0.946389i	$0.395299\pi$
$132$	9.90833	0.862409
$133$	0	0
$134$	35.9361	3.10440
$135$	0	0
$136$	−13.8167	−1.18477
$137$	−3.21110	−0.274343	−0.137172	−	0.990547i	$-0.543801\pi$
$137$	−3.21110	−0.274343	−0.137172	+	0.990547i	$0.543801\pi$
$138$	14.3028	1.21753
$139$	−19.0278	−1.61391	−0.806957	−	0.590611i	$-0.798887\pi$
$139$	−19.0278	−1.61391	−0.806957	+	0.590611i	$0.798887\pi$
$140$	0	0
$141$	10.6056	0.893149
$142$	−6.90833	−0.579734
$143$	−7.81665	−0.653661
$144$	0.302776	0.0252313
$145$	0	0
$146$	−1.39445	−0.115405
$147$	0	0
$148$	−13.9083	−1.14326
$149$	−16.3944	−1.34309	−0.671543	−	0.740966i	$-0.734368\pi$
$149$	−16.3944	−1.34309	−0.671543	+	0.740966i	$0.734368\pi$
$150$	0	0
$151$	6.81665	0.554731	0.277366	−	0.960764i	$-0.410539\pi$
$151$	6.81665	0.554731	0.277366	+	0.960764i	$0.410539\pi$
$152$	−19.8167	−1.60734
$153$	−4.60555	−0.372337
$154$	0	0
$155$	0	0
$156$	−8.60555	−0.688996
$157$	−14.4222	−1.15102	−0.575509	−	0.817796i	$-0.695196\pi$
$157$	−14.4222	−1.15102	−0.575509	+	0.817796i	$0.695196\pi$
$158$	−34.1194	−2.71440
$159$	−3.21110	−0.254657
$160$	0	0
$161$	0	0
$162$	2.30278	0.180923
$163$	17.2111	1.34808	0.674039	−	0.738696i	$-0.264558\pi$
$163$	17.2111	1.34808	0.674039	+	0.738696i	$0.264558\pi$
$164$	0	0
$165$	0	0
$166$	7.39445	0.573921
$167$	17.0278	1.31765	0.658824	−	0.752297i	$-0.271054\pi$
$167$	17.0278	1.31765	0.658824	+	0.752297i	$0.271054\pi$
$168$	0	0
$169$	−6.21110	−0.477777
$170$	0	0
$171$	−6.60555	−0.505139
$172$	31.7250	2.41901
$173$	7.81665	0.594289	0.297145	−	0.954832i	$-0.403966\pi$
$173$	7.81665	0.594289	0.297145	+	0.954832i	$0.403966\pi$
$174$	17.5139	1.32772
$175$	0	0
$176$	−0.908327	−0.0684677
$177$	−10.6056	−0.797162
$178$	−18.0000	−1.34916
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−12.6056	−0.936963	−0.468482	−	0.883473i	$-0.655199\pi$
$181$	−12.6056	−0.936963	−0.468482	+	0.883473i	$0.655199\pi$
$182$	0	0
$183$	−1.21110	−0.0895273
$184$	−18.6333	−1.37367
$185$	0	0
$186$	−16.6056	−1.21758
$187$	13.8167	1.01037
$188$	−35.0278	−2.55466
$189$	0	0
$190$	0	0
$191$	−2.78890	−0.201798	−0.100899	−	0.994897i	$-0.532172\pi$
$191$	−2.78890	−0.201798	−0.100899	+	0.994897i	$0.532172\pi$
$192$	12.8167	0.924962
$193$	8.21110	0.591048	0.295524	−	0.955335i	$-0.404506\pi$
$193$	8.21110	0.591048	0.295524	+	0.955335i	$0.404506\pi$
$194$	1.81665	0.130428
$195$	0	0
$196$	0	0
$197$	16.8167	1.19814	0.599068	−	0.800698i	$-0.295538\pi$
$197$	16.8167	1.19814	0.599068	+	0.800698i	$0.295538\pi$
$198$	−6.90833	−0.490953
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−15.6056	−1.10073
$202$	−38.2389	−2.69048
$203$	0	0
$204$	15.2111	1.06499
$205$	0	0
$206$	−8.78890	−0.612352
$207$	−6.21110	−0.431701
$208$	0.788897	0.0547002
$209$	19.8167	1.37075
$210$	0	0
$211$	26.4222	1.81898	0.909490	−	0.415726i	$-0.136473\pi$
$211$	26.4222	1.81898	0.909490	+	0.415726i	$0.136473\pi$
$212$	10.6056	0.728392
$213$	3.00000	0.205557
$214$	0	0
$215$	0	0
$216$	−3.00000	−0.204124
$217$	0	0
$218$	5.09167	0.344852
$219$	0.605551	0.0409194
$220$	0	0
$221$	−12.0000	−0.807207
$222$	9.69722	0.650835
$223$	−15.3944	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$223$	−15.3944	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$224$	0	0
$225$	0	0
$226$	−24.9083	−1.65688
$227$	−22.6056	−1.50038	−0.750192	−	0.661221i	$-0.770039\pi$
$227$	−22.6056	−1.50038	−0.750192	+	0.661221i	$0.770039\pi$
$228$	21.8167	1.44484
$229$	7.21110	0.476523	0.238262	−	0.971201i	$-0.423422\pi$
$229$	7.21110	0.476523	0.238262	+	0.971201i	$0.423422\pi$
$230$	0	0
$231$	0	0
$232$	−22.8167	−1.49799
$233$	−1.18335	−0.0775236	−0.0387618	−	0.999248i	$-0.512341\pi$
$233$	−1.18335	−0.0775236	−0.0387618	+	0.999248i	$0.512341\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	35.0278	2.28011
$237$	14.8167	0.962445
$238$	0	0
$239$	14.7889	0.956614	0.478307	−	0.878193i	$-0.341251\pi$
$239$	14.7889	0.956614	0.478307	+	0.878193i	$0.341251\pi$
$240$	0	0
$241$	−9.39445	−0.605150	−0.302575	−	0.953126i	$-0.597846\pi$
$241$	−9.39445	−0.605150	−0.302575	+	0.953126i	$0.597846\pi$
$242$	−4.60555	−0.296056
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	0	0
$247$	−17.2111	−1.09512
$248$	21.6333	1.37372
$249$	−3.21110	−0.203495
$250$	0	0
$251$	13.8167	0.872099	0.436050	−	0.899923i	$-0.356377\pi$
$251$	13.8167	0.872099	0.436050	+	0.899923i	$0.356377\pi$
$252$	0	0
$253$	18.6333	1.17147
$254$	29.5139	1.85187
$255$	0	0
$256$	−17.9083	−1.11927
$257$	21.6333	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$258$	−22.1194	−1.37710
$259$	0	0
$260$	0	0
$261$	−7.60555	−0.470772
$262$	17.0278	1.05198
$263$	−5.78890	−0.356959	−0.178479	−	0.983944i	$-0.557118\pi$
$263$	−5.78890	−0.356959	−0.178479	+	0.983944i	$0.557118\pi$
$264$	9.00000	0.553912
$265$	0	0
$266$	0	0
$267$	7.81665	0.478371
$268$	51.5416	3.14841
$269$	3.21110	0.195784	0.0978922	−	0.995197i	$-0.468790\pi$
$269$	3.21110	0.195784	0.0978922	+	0.995197i	$0.468790\pi$
$270$	0	0
$271$	26.6056	1.61617	0.808086	−	0.589064i	$-0.200504\pi$
$271$	26.6056	1.61617	0.808086	+	0.589064i	$0.200504\pi$
$272$	−1.39445	−0.0845509
$273$	0	0
$274$	−7.39445	−0.446715
$275$	0	0
$276$	20.5139	1.23479
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−43.8167	−2.62795
$279$	7.21110	0.431717
$280$	0	0
$281$	−1.18335	−0.0705925	−0.0352963	−	0.999377i	$-0.511237\pi$
$281$	−1.18335	−0.0705925	−0.0352963	+	0.999377i	$0.511237\pi$
$282$	24.4222	1.45432
$283$	13.6333	0.810416	0.405208	−	0.914225i	$-0.367199\pi$
$283$	13.6333	0.810416	0.405208	+	0.914225i	$0.367199\pi$
$284$	−9.90833	−0.587951
$285$	0	0
$286$	−18.0000	−1.06436
$287$	0	0
$288$	−5.30278	−0.312469
$289$	4.21110	0.247712
$290$	0	0
$291$	−0.788897	−0.0462460
$292$	−2.00000	−0.117041
$293$	−10.6056	−0.619583	−0.309791	−	0.950805i	$-0.600259\pi$
$293$	−10.6056	−0.619583	−0.309791	+	0.950805i	$0.600259\pi$
$294$	0	0
$295$	0	0
$296$	−12.6333	−0.734296
$297$	3.00000	0.174078
$298$	−37.7527	−2.18696
$299$	−16.1833	−0.935907
$300$	0	0
$301$	0	0
$302$	15.6972	0.903274
$303$	16.6056	0.953964
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−10.6056	−0.606279
$307$	8.60555	0.491145	0.245572	−	0.969378i	$-0.421024\pi$
$307$	8.60555	0.491145	0.245572	+	0.969378i	$0.421024\pi$
$308$	0	0
$309$	3.81665	0.217122
$310$	0	0
$311$	−13.8167	−0.783471	−0.391735	−	0.920078i	$-0.628125\pi$
$311$	−13.8167	−0.783471	−0.391735	+	0.920078i	$0.628125\pi$
$312$	−7.81665	−0.442531
$313$	23.8167	1.34620	0.673098	−	0.739553i	$-0.264963\pi$
$313$	23.8167	1.34620	0.673098	+	0.739553i	$0.264963\pi$
$314$	−33.2111	−1.87421
$315$	0	0
$316$	−48.9361	−2.75287
$317$	−16.3944	−0.920804	−0.460402	−	0.887711i	$-0.652295\pi$
$317$	−16.3944	−0.920804	−0.460402	+	0.887711i	$0.652295\pi$
$318$	−7.39445	−0.414660
$319$	22.8167	1.27749
$320$	0	0
$321$	0	0
$322$	0	0
$323$	30.4222	1.69274
$324$	3.30278	0.183488
$325$	0	0
$326$	39.6333	2.19509
$327$	−2.21110	−0.122274
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−21.2389	−1.16739	−0.583697	−	0.811972i	$-0.698394\pi$
$331$	−21.2389	−1.16739	−0.583697	+	0.811972i	$0.698394\pi$
$332$	10.6056	0.582055
$333$	−4.21110	−0.230767
$334$	39.2111	2.14554
$335$	0	0
$336$	0	0
$337$	−7.21110	−0.392814	−0.196407	−	0.980522i	$-0.562927\pi$
$337$	−7.21110	−0.392814	−0.196407	+	0.980522i	$0.562927\pi$
$338$	−14.3028	−0.777968
$339$	10.8167	0.587480
$340$	0	0
$341$	−21.6333	−1.17151
$342$	−15.2111	−0.822522
$343$	0	0
$344$	28.8167	1.55369
$345$	0	0
$346$	18.0000	0.967686
$347$	30.2111	1.62182	0.810908	−	0.585173i	$-0.198973\pi$
$347$	30.2111	1.62182	0.810908	+	0.585173i	$0.198973\pi$
$348$	25.1194	1.34654
$349$	−31.4500	−1.68348	−0.841739	−	0.539885i	$-0.818468\pi$
$349$	−31.4500	−1.68348	−0.841739	+	0.539885i	$0.818468\pi$
$350$	0	0
$351$	−2.60555	−0.139074
$352$	15.9083	0.847917
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	−24.4222	−1.29803
$355$	0	0
$356$	−25.8167	−1.36828
$357$	0	0
$358$	0	0
$359$	24.6333	1.30010	0.650048	−	0.759893i	$-0.274749\pi$
$359$	24.6333	1.30010	0.650048	+	0.759893i	$0.274749\pi$
$360$	0	0
$361$	24.6333	1.29649
$362$	−29.0278	−1.52567
$363$	2.00000	0.104973
$364$	0	0
$365$	0	0
$366$	−2.78890	−0.145778
$367$	−14.4222	−0.752833	−0.376416	−	0.926451i	$-0.622844\pi$
$367$	−14.4222	−0.752833	−0.376416	+	0.926451i	$0.622844\pi$
$368$	−1.88057	−0.0980315
$369$	0	0
$370$	0	0
$371$	0	0
$372$	−23.8167	−1.23484
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	31.8167	1.64520
$375$	0	0
$376$	−31.8167	−1.64082
$377$	−19.8167	−1.02061
$378$	0	0
$379$	−14.8167	−0.761080	−0.380540	−	0.924764i	$-0.624262\pi$
$379$	−14.8167	−0.761080	−0.380540	+	0.924764i	$0.624262\pi$
$380$	0	0
$381$	−12.8167	−0.656617
$382$	−6.42221	−0.328589
$383$	26.2389	1.34074	0.670372	−	0.742026i	$-0.266135\pi$
$383$	26.2389	1.34074	0.670372	+	0.742026i	$0.266135\pi$
$384$	18.9083	0.964912
$385$	0	0
$386$	18.9083	0.962408
$387$	9.60555	0.488278
$388$	2.60555	0.132277
$389$	−28.8167	−1.46106	−0.730531	−	0.682879i	$-0.760727\pi$
$389$	−28.8167	−1.46106	−0.730531	+	0.682879i	$0.760727\pi$
$390$	0	0
$391$	28.6056	1.44664
$392$	0	0
$393$	−7.39445	−0.373001
$394$	38.7250	1.95094
$395$	0	0
$396$	−9.90833	−0.497912
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−18.4222	−0.923422
$399$	0	0
$400$	0	0
$401$	−16.8167	−0.839784	−0.419892	−	0.907574i	$-0.637932\pi$
$401$	−16.8167	−0.839784	−0.419892	+	0.907574i	$0.637932\pi$
$402$	−35.9361	−1.79233
$403$	18.7889	0.935942
$404$	−54.8444	−2.72861
$405$	0	0
$406$	0	0
$407$	12.6333	0.626210
$408$	13.8167	0.684026
$409$	11.8167	0.584296	0.292148	−	0.956373i	$-0.405630\pi$
$409$	11.8167	0.584296	0.292148	+	0.956373i	$0.405630\pi$
$410$	0	0
$411$	3.21110	0.158392
$412$	−12.6056	−0.621031
$413$	0	0
$414$	−14.3028	−0.702943
$415$	0	0
$416$	−13.8167	−0.677417
$417$	19.0278	0.931793
$418$	45.6333	2.23200
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−10.2111	−0.497659	−0.248829	−	0.968547i	$-0.580046\pi$
$421$	−10.2111	−0.497659	−0.248829	+	0.968547i	$0.580046\pi$
$422$	60.8444	2.96186
$423$	−10.6056	−0.515660
$424$	9.63331	0.467835
$425$	0	0
$426$	6.90833	0.334710
$427$	0	0
$428$	0	0
$429$	7.81665	0.377392
$430$	0	0
$431$	5.57779	0.268673	0.134336	−	0.990936i	$-0.457110\pi$
$431$	5.57779	0.268673	0.134336	+	0.990936i	$0.457110\pi$
$432$	−0.302776	−0.0145673
$433$	37.2111	1.78825	0.894126	−	0.447816i	$-0.147798\pi$
$433$	37.2111	1.78825	0.894126	+	0.447816i	$0.147798\pi$
$434$	0	0
$435$	0	0
$436$	7.30278	0.349740
$437$	41.0278	1.96262
$438$	1.39445	0.0666293
$439$	−6.60555	−0.315266	−0.157633	−	0.987498i	$-0.550386\pi$
$439$	−6.60555	−0.315266	−0.157633	+	0.987498i	$0.550386\pi$
$440$	0	0
$441$	0	0
$442$	−27.6333	−1.31438
$443$	15.6333	0.742761	0.371380	−	0.928481i	$-0.378885\pi$
$443$	15.6333	0.742761	0.371380	+	0.928481i	$0.378885\pi$
$444$	13.9083	0.660060
$445$	0	0
$446$	−35.4500	−1.67860
$447$	16.3944	0.775431
$448$	0	0
$449$	−34.8167	−1.64310	−0.821550	−	0.570137i	$-0.806890\pi$
$449$	−34.8167	−1.64310	−0.821550	+	0.570137i	$0.806890\pi$
$450$	0	0
$451$	0	0
$452$	−35.7250	−1.68036
$453$	−6.81665	−0.320274
$454$	−52.0555	−2.44309
$455$	0	0
$456$	19.8167	0.928000
$457$	−3.78890	−0.177237	−0.0886186	−	0.996066i	$-0.528245\pi$
$457$	−3.78890	−0.177237	−0.0886186	+	0.996066i	$0.528245\pi$
$458$	16.6056	0.775926
$459$	4.60555	0.214969
$460$	0	0
$461$	26.2389	1.22207	0.611033	−	0.791605i	$-0.290754\pi$
$461$	26.2389	1.22207	0.611033	+	0.791605i	$0.290754\pi$
$462$	0	0
$463$	17.2111	0.799868	0.399934	−	0.916544i	$-0.369033\pi$
$463$	17.2111	0.799868	0.399934	+	0.916544i	$0.369033\pi$
$464$	−2.30278	−0.106904
$465$	0	0
$466$	−2.72498	−0.126232
$467$	0.422205	0.0195373	0.00976866	−	0.999952i	$-0.496890\pi$
$467$	0.422205	0.0195373	0.00976866	+	0.999952i	$0.496890\pi$
$468$	8.60555	0.397792
$469$	0	0
$470$	0	0
$471$	14.4222	0.664540
$472$	31.8167	1.46448
$473$	−28.8167	−1.32499
$474$	34.1194	1.56716
$475$	0	0
$476$	0	0
$477$	3.21110	0.147026
$478$	34.0555	1.55766
$479$	−34.6056	−1.58117	−0.790584	−	0.612354i	$-0.790223\pi$
$479$	−34.6056	−1.58117	−0.790584	+	0.612354i	$0.790223\pi$
$480$	0	0
$481$	−10.9722	−0.500291
$482$	−21.6333	−0.985370
$483$	0	0
$484$	−6.60555	−0.300252
$485$	0	0
$486$	−2.30278	−0.104456
$487$	−20.8167	−0.943293	−0.471646	−	0.881788i	$-0.656340\pi$
$487$	−20.8167	−0.943293	−0.471646	+	0.881788i	$0.656340\pi$
$488$	3.63331	0.164472
$489$	−17.2111	−0.778313
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	35.0278	1.57757
$494$	−39.6333	−1.78319
$495$	0	0
$496$	2.18335	0.0980351
$497$	0	0
$498$	−7.39445	−0.331353
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−17.0278	−0.760744
$502$	31.8167	1.42005
$503$	8.78890	0.391878	0.195939	−	0.980616i	$-0.437225\pi$
$503$	8.78890	0.391878	0.195939	+	0.980616i	$0.437225\pi$
$504$	0	0
$505$	0	0
$506$	42.9083	1.90751
$507$	6.21110	0.275845
$508$	42.3305	1.87811
$509$	−1.39445	−0.0618079	−0.0309039	−	0.999522i	$-0.509839\pi$
$509$	−1.39445	−0.0618079	−0.0309039	+	0.999522i	$0.509839\pi$
$510$	0	0
$511$	0	0
$512$	−3.42221	−0.151242
$513$	6.60555	0.291642
$514$	49.8167	2.19732
$515$	0	0
$516$	−31.7250	−1.39661
$517$	31.8167	1.39929
$518$	0	0
$519$	−7.81665	−0.343113
$520$	0	0
$521$	36.4222	1.59569	0.797843	−	0.602865i	$-0.205974\pi$
$521$	36.4222	1.59569	0.797843	+	0.602865i	$0.205974\pi$
$522$	−17.5139	−0.766562
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	24.4222	1.06689
$525$	0	0
$526$	−13.3305	−0.581239
$527$	−33.2111	−1.44670
$528$	0.908327	0.0395299
$529$	15.5778	0.677295
$530$	0	0
$531$	10.6056	0.460242
$532$	0	0
$533$	0	0
$534$	18.0000	0.778936
$535$	0	0
$536$	46.8167	2.02217
$537$	0	0
$538$	7.39445	0.318797
$539$	0	0
$540$	0	0
$541$	−31.4222	−1.35095	−0.675473	−	0.737385i	$-0.736061\pi$
$541$	−31.4222	−1.35095	−0.675473	+	0.737385i	$0.736061\pi$
$542$	61.2666	2.63163
$543$	12.6056	0.540956
$544$	24.4222	1.04709
$545$	0	0
$546$	0	0
$547$	−29.6056	−1.26584	−0.632921	−	0.774216i	$-0.718144\pi$
$547$	−29.6056	−1.26584	−0.632921	+	0.774216i	$0.718144\pi$
$548$	−10.6056	−0.453047
$549$	1.21110	0.0516886
$550$	0	0
$551$	50.2389	2.14025
$552$	18.6333	0.793086
$553$	0	0
$554$	−23.0278	−0.978356
$555$	0	0
$556$	−62.8444	−2.66520
$557$	11.2389	0.476206	0.238103	−	0.971240i	$-0.423474\pi$
$557$	11.2389	0.476206	0.238103	+	0.971240i	$0.423474\pi$
$558$	16.6056	0.702969
$559$	25.0278	1.05856
$560$	0	0
$561$	−13.8167	−0.583340
$562$	−2.72498	−0.114946
$563$	−15.2111	−0.641072	−0.320536	−	0.947236i	$-0.603863\pi$
$563$	−15.2111	−0.641072	−0.320536	+	0.947236i	$0.603863\pi$
$564$	35.0278	1.47493
$565$	0	0
$566$	31.3944	1.31961
$567$	0	0
$568$	−9.00000	−0.377632
$569$	−1.18335	−0.0496085	−0.0248042	−	0.999692i	$-0.507896\pi$
$569$	−1.18335	−0.0496085	−0.0248042	+	0.999692i	$0.507896\pi$
$570$	0	0
$571$	−5.60555	−0.234585	−0.117293	−	0.993097i	$-0.537422\pi$
$571$	−5.60555	−0.234585	−0.117293	+	0.993097i	$0.537422\pi$
$572$	−25.8167	−1.07945
$573$	2.78890	0.116508
$574$	0	0
$575$	0	0
$576$	−12.8167	−0.534027
$577$	−30.6056	−1.27413	−0.637063	−	0.770812i	$-0.719851\pi$
$577$	−30.6056	−1.27413	−0.637063	+	0.770812i	$0.719851\pi$
$578$	9.69722	0.403351
$579$	−8.21110	−0.341242
$580$	0	0
$581$	0	0
$582$	−1.81665	−0.0753027
$583$	−9.63331	−0.398971
$584$	−1.81665	−0.0751737
$585$	0	0
$586$	−24.4222	−1.00887
$587$	−37.8167	−1.56086	−0.780430	−	0.625243i	$-0.785000\pi$
$587$	−37.8167	−1.56086	−0.780430	+	0.625243i	$0.785000\pi$
$588$	0	0
$589$	−47.6333	−1.96270
$590$	0	0
$591$	−16.8167	−0.691745
$592$	−1.27502	−0.0524030
$593$	−27.6333	−1.13476	−0.567382	−	0.823455i	$-0.692044\pi$
$593$	−27.6333	−1.13476	−0.567382	+	0.823455i	$0.692044\pi$
$594$	6.90833	0.283452
$595$	0	0
$596$	−54.1472	−2.21796
$597$	8.00000	0.327418
$598$	−37.2666	−1.52395
$599$	5.78890	0.236528	0.118264	−	0.992982i	$-0.462267\pi$
$599$	5.78890	0.236528	0.118264	+	0.992982i	$0.462267\pi$
$600$	0	0
$601$	−17.2111	−0.702056	−0.351028	−	0.936365i	$-0.614168\pi$
$601$	−17.2111	−0.702056	−0.351028	+	0.936365i	$0.614168\pi$
$602$	0	0
$603$	15.6056	0.635507
$604$	22.5139	0.916077
$605$	0	0
$606$	38.2389	1.55335
$607$	23.8167	0.966688	0.483344	−	0.875430i	$-0.339422\pi$
$607$	23.8167	0.966688	0.483344	+	0.875430i	$0.339422\pi$
$608$	35.0278	1.42056
$609$	0	0
$610$	0	0
$611$	−27.6333	−1.11792
$612$	−15.2111	−0.614872
$613$	−49.4222	−1.99614	−0.998072	−	0.0620663i	$-0.980231\pi$
$613$	−49.4222	−1.99614	−0.998072	+	0.0620663i	$0.980231\pi$
$614$	19.8167	0.799735
$615$	0	0
$616$	0	0
$617$	22.8167	0.918564	0.459282	−	0.888291i	$-0.348107\pi$
$617$	22.8167	0.918564	0.459282	+	0.888291i	$0.348107\pi$
$618$	8.78890	0.353541
$619$	2.60555	0.104726	0.0523630	−	0.998628i	$-0.483325\pi$
$619$	2.60555	0.104726	0.0523630	+	0.998628i	$0.483325\pi$
$620$	0	0
$621$	6.21110	0.249243
$622$	−31.8167	−1.27573
$623$	0	0
$624$	−0.788897	−0.0315812
$625$	0	0
$626$	54.8444	2.19202
$627$	−19.8167	−0.791401
$628$	−47.6333	−1.90078
$629$	19.3944	0.773307
$630$	0	0
$631$	22.0278	0.876911	0.438456	−	0.898753i	$-0.355526\pi$
$631$	22.0278	0.876911	0.438456	+	0.898753i	$0.355526\pi$
$632$	−44.4500	−1.76812
$633$	−26.4222	−1.05019
$634$	−37.7527	−1.49935
$635$	0	0
$636$	−10.6056	−0.420537
$637$	0	0
$638$	52.5416	2.08014
$639$	−3.00000	−0.118678
$640$	0	0
$641$	22.8167	0.901204	0.450602	−	0.892725i	$-0.351209\pi$
$641$	22.8167	0.901204	0.450602	+	0.892725i	$0.351209\pi$
$642$	0	0
$643$	−38.4222	−1.51522	−0.757612	−	0.652705i	$-0.773634\pi$
$643$	−38.4222	−1.51522	−0.757612	+	0.652705i	$0.773634\pi$
$644$	0	0
$645$	0	0
$646$	70.0555	2.75630
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	3.00000	0.117851
$649$	−31.8167	−1.24891
$650$	0	0
$651$	0	0
$652$	56.8444	2.22620
$653$	−3.21110	−0.125660	−0.0628301	−	0.998024i	$-0.520013\pi$
$653$	−3.21110	−0.125660	−0.0628301	+	0.998024i	$0.520013\pi$
$654$	−5.09167	−0.199100
$655$	0	0
$656$	0	0
$657$	−0.605551	−0.0236248
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	20.1833	0.785041	0.392521	−	0.919743i	$-0.371603\pi$
$661$	20.1833	0.785041	0.392521	+	0.919743i	$0.371603\pi$
$662$	−48.9083	−1.90088
$663$	12.0000	0.466041
$664$	9.63331	0.373845
$665$	0	0
$666$	−9.69722	−0.375760
$667$	47.2389	1.82910
$668$	56.2389	2.17595
$669$	15.3944	0.595184
$670$	0	0
$671$	−3.63331	−0.140262
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−16.6056	−0.639622
$675$	0	0
$676$	−20.5139	−0.788995
$677$	8.78890	0.337785	0.168892	−	0.985634i	$-0.445981\pi$
$677$	8.78890	0.337785	0.168892	+	0.985634i	$0.445981\pi$
$678$	24.9083	0.956599
$679$	0	0
$680$	0	0
$681$	22.6056	0.866247
$682$	−49.8167	−1.90758
$683$	−18.6333	−0.712984	−0.356492	−	0.934298i	$-0.616027\pi$
$683$	−18.6333	−0.712984	−0.356492	+	0.934298i	$0.616027\pi$
$684$	−21.8167	−0.834181
$685$	0	0
$686$	0	0
$687$	−7.21110	−0.275121
$688$	2.90833	0.110879
$689$	8.36669	0.318746
$690$	0	0
$691$	−41.2111	−1.56774	−0.783872	−	0.620922i	$-0.786758\pi$
$691$	−41.2111	−1.56774	−0.783872	+	0.620922i	$0.786758\pi$
$692$	25.8167	0.981402
$693$	0	0
$694$	69.5694	2.64082
$695$	0	0
$696$	22.8167	0.864863
$697$	0	0
$698$	−72.4222	−2.74122
$699$	1.18335	0.0447583
$700$	0	0
$701$	−20.7889	−0.785186	−0.392593	−	0.919712i	$-0.628422\pi$
$701$	−20.7889	−0.785186	−0.392593	+	0.919712i	$0.628422\pi$
$702$	−6.00000	−0.226455
$703$	27.8167	1.04912
$704$	38.4500	1.44914
$705$	0	0
$706$	69.0833	2.59998
$707$	0	0
$708$	−35.0278	−1.31642
$709$	−34.8444	−1.30861	−0.654305	−	0.756231i	$-0.727039\pi$
$709$	−34.8444	−1.30861	−0.654305	+	0.756231i	$0.727039\pi$
$710$	0	0
$711$	−14.8167	−0.555668
$712$	−23.4500	−0.878824
$713$	−44.7889	−1.67736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.7889	−0.552301
$718$	56.7250	2.11696
$719$	−26.7889	−0.999057	−0.499529	−	0.866297i	$-0.666493\pi$
$719$	−26.7889	−0.999057	−0.499529	+	0.866297i	$0.666493\pi$
$720$	0	0
$721$	0	0
$722$	56.7250	2.11109
$723$	9.39445	0.349383
$724$	−41.6333	−1.54729
$725$	0	0
$726$	4.60555	0.170928
$727$	23.3944	0.867652	0.433826	−	0.900997i	$-0.357163\pi$
$727$	23.3944	0.867652	0.433826	+	0.900997i	$0.357163\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−44.2389	−1.63623
$732$	−4.00000	−0.147844
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	−33.2111	−1.22584
$735$	0	0
$736$	32.9361	1.21404
$737$	−46.8167	−1.72451
$738$	0	0
$739$	3.18335	0.117101	0.0585506	−	0.998284i	$-0.481352\pi$
$739$	3.18335	0.117101	0.0585506	+	0.998284i	$0.481352\pi$
$740$	0	0
$741$	17.2111	0.632266
$742$	0	0
$743$	−27.6333	−1.01377	−0.506884	−	0.862014i	$-0.669203\pi$
$743$	−27.6333	−1.01377	−0.506884	+	0.862014i	$0.669203\pi$
$744$	−21.6333	−0.793116
$745$	0	0
$746$	−2.30278	−0.0843106
$747$	3.21110	0.117488
$748$	45.6333	1.66852
$749$	0	0
$750$	0	0
$751$	−0.366692	−0.0133808	−0.00669040	−	0.999978i	$-0.502130\pi$
$751$	−0.366692	−0.0133808	−0.00669040	+	0.999978i	$0.502130\pi$
$752$	−3.21110	−0.117097
$753$	−13.8167	−0.503507
$754$	−45.6333	−1.66187
$755$	0	0
$756$	0	0
$757$	41.0000	1.49017	0.745085	−	0.666969i	$-0.232409\pi$
$757$	41.0000	1.49017	0.745085	+	0.666969i	$0.232409\pi$
$758$	−34.1194	−1.23927
$759$	−18.6333	−0.676346
$760$	0	0
$761$	27.6333	1.00171	0.500853	−	0.865532i	$-0.333020\pi$
$761$	27.6333	1.00171	0.500853	+	0.865532i	$0.333020\pi$
$762$	−29.5139	−1.06917
$763$	0	0
$764$	−9.21110	−0.333246
$765$	0	0
$766$	60.4222	2.18314
$767$	27.6333	0.997781
$768$	17.9083	0.646211
$769$	31.6333	1.14073	0.570363	−	0.821393i	$-0.306802\pi$
$769$	31.6333	1.14073	0.570363	+	0.821393i	$0.306802\pi$
$770$	0	0
$771$	−21.6333	−0.779105
$772$	27.1194	0.976050
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	22.1194	0.795066
$775$	0	0
$776$	2.36669	0.0849593
$777$	0	0
$778$	−66.3583	−2.37906
$779$	0	0
$780$	0	0
$781$	9.00000	0.322045
$782$	65.8722	2.35558
$783$	7.60555	0.271800
$784$	0	0
$785$	0	0
$786$	−17.0278	−0.607360
$787$	−28.2389	−1.00661	−0.503303	−	0.864110i	$-0.667882\pi$
$787$	−28.2389	−1.00661	−0.503303	+	0.864110i	$0.667882\pi$
$788$	55.5416	1.97859
$789$	5.78890	0.206090
$790$	0	0
$791$	0	0
$792$	−9.00000	−0.319801
$793$	3.15559	0.112058
$794$	−4.60555	−0.163445
$795$	0	0
$796$	−26.4222	−0.936510
$797$	29.4500	1.04317	0.521586	−	0.853199i	$-0.325341\pi$
$797$	29.4500	1.04317	0.521586	+	0.853199i	$0.325341\pi$
$798$	0	0
$799$	48.8444	1.72799
$800$	0	0
$801$	−7.81665	−0.276188
$802$	−38.7250	−1.36743
$803$	1.81665	0.0641083
$804$	−51.5416	−1.81773
$805$	0	0
$806$	43.2666	1.52400
$807$	−3.21110	−0.113036
$808$	−49.8167	−1.75254
$809$	32.4500	1.14088	0.570440	−	0.821339i	$-0.306773\pi$
$809$	32.4500	1.14088	0.570440	+	0.821339i	$0.306773\pi$
$810$	0	0
$811$	−14.8444	−0.521258	−0.260629	−	0.965439i	$-0.583930\pi$
$811$	−14.8444	−0.521258	−0.260629	+	0.965439i	$0.583930\pi$
$812$	0	0
$813$	−26.6056	−0.933097
$814$	29.0917	1.01966
$815$	0	0
$816$	1.39445	0.0488155
$817$	−63.4500	−2.21983
$818$	27.2111	0.951414
$819$	0	0
$820$	0	0
$821$	49.2666	1.71942	0.859708	−	0.510785i	$-0.170645\pi$
$821$	49.2666	1.71942	0.859708	+	0.510785i	$0.170645\pi$
$822$	7.39445	0.257911
$823$	24.8167	0.865054	0.432527	−	0.901621i	$-0.357622\pi$
$823$	24.8167	0.865054	0.432527	+	0.901621i	$0.357622\pi$
$824$	−11.4500	−0.398878
$825$	0	0
$826$	0	0
$827$	−30.6333	−1.06522	−0.532612	−	0.846359i	$-0.678790\pi$
$827$	−30.6333	−1.06522	−0.532612	+	0.846359i	$0.678790\pi$
$828$	−20.5139	−0.712907
$829$	0.238859	0.00829591	0.00414796	−	0.999991i	$-0.498680\pi$
$829$	0.238859	0.00829591	0.00414796	+	0.999991i	$0.498680\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−33.3944	−1.15774
$833$	0	0
$834$	43.8167	1.51725
$835$	0	0
$836$	65.4500	2.26363
$837$	−7.21110	−0.249252
$838$	−13.8167	−0.477288
$839$	22.1833	0.765854	0.382927	−	0.923779i	$-0.374916\pi$
$839$	22.1833	0.765854	0.382927	+	0.923779i	$0.374916\pi$
$840$	0	0
$841$	28.8444	0.994635
$842$	−23.5139	−0.810342
$843$	1.18335	0.0407566
$844$	87.2666	3.00384
$845$	0	0
$846$	−24.4222	−0.839653
$847$	0	0
$848$	0.972244	0.0333870
$849$	−13.6333	−0.467894
$850$	0	0
$851$	26.1556	0.896602
$852$	9.90833	0.339454
$853$	6.78890	0.232447	0.116224	−	0.993223i	$-0.462921\pi$
$853$	6.78890	0.232447	0.116224	+	0.993223i	$0.462921\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.6333	−0.943936	−0.471968	−	0.881616i	$-0.656456\pi$
$857$	−27.6333	−0.943936	−0.471968	+	0.881616i	$0.656456\pi$
$858$	18.0000	0.614510
$859$	49.6333	1.69347	0.846733	−	0.532018i	$-0.178566\pi$
$859$	49.6333	1.69347	0.846733	+	0.532018i	$0.178566\pi$
$860$	0	0
$861$	0	0
$862$	12.8444	0.437482
$863$	5.36669	0.182684	0.0913422	−	0.995820i	$-0.470884\pi$
$863$	5.36669	0.182684	0.0913422	+	0.995820i	$0.470884\pi$
$864$	5.30278	0.180404
$865$	0	0
$866$	85.6888	2.91182
$867$	−4.21110	−0.143017
$868$	0	0
$869$	44.4500	1.50786
$870$	0	0
$871$	40.6611	1.37775
$872$	6.63331	0.224632
$873$	0.788897	0.0267001
$874$	94.4777	3.19576
$875$	0	0
$876$	2.00000	0.0675737
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−15.2111	−0.513350
$879$	10.6056	0.357716
$880$	0	0
$881$	−33.6333	−1.13313	−0.566567	−	0.824015i	$-0.691729\pi$
$881$	−33.6333	−1.13313	−0.566567	+	0.824015i	$0.691729\pi$
$882$	0	0
$883$	15.6056	0.525169	0.262584	−	0.964909i	$-0.415425\pi$
$883$	15.6056	0.525169	0.262584	+	0.964909i	$0.415425\pi$
$884$	−39.6333	−1.33301
$885$	0	0
$886$	36.0000	1.20944
$887$	14.7889	0.496563	0.248281	−	0.968688i	$-0.420134\pi$
$887$	14.7889	0.496563	0.248281	+	0.968688i	$0.420134\pi$
$888$	12.6333	0.423946
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−50.8444	−1.70240
$893$	70.0555	2.34432
$894$	37.7527	1.26264
$895$	0	0
$896$	0	0
$897$	16.1833	0.540346
$898$	−80.1749	−2.67547
$899$	−54.8444	−1.82916
$900$	0	0
$901$	−14.7889	−0.492690
$902$	0	0
$903$	0	0
$904$	−32.4500	−1.07927
$905$	0	0
$906$	−15.6972	−0.521505
$907$	−13.2111	−0.438667	−0.219334	−	0.975650i	$-0.570388\pi$
$907$	−13.2111	−0.438667	−0.219334	+	0.975650i	$0.570388\pi$
$908$	−74.6611	−2.47771
$909$	−16.6056	−0.550771
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	2.00000	0.0662266
$913$	−9.63331	−0.318816
$914$	−8.72498	−0.288597
$915$	0	0
$916$	23.8167	0.786924
$917$	0	0
$918$	10.6056	0.350035
$919$	−17.6056	−0.580754	−0.290377	−	0.956912i	$-0.593781\pi$
$919$	−17.6056	−0.580754	−0.290377	+	0.956912i	$0.593781\pi$
$920$	0	0
$921$	−8.60555	−0.283563
$922$	60.4222	1.98990
$923$	−7.81665	−0.257288
$924$	0	0
$925$	0	0
$926$	39.6333	1.30243
$927$	−3.81665	−0.125355
$928$	40.3305	1.32391
$929$	20.2389	0.664015	0.332008	−	0.943277i	$-0.392274\pi$
$929$	20.2389	0.664015	0.332008	+	0.943277i	$0.392274\pi$
$930$	0	0
$931$	0	0
$932$	−3.90833	−0.128022
$933$	13.8167	0.452337
$934$	0.972244	0.0318128
$935$	0	0
$936$	7.81665	0.255495
$937$	56.4777	1.84505	0.922523	−	0.385941i	$-0.126123\pi$
$937$	56.4777	1.84505	0.922523	+	0.385941i	$0.126123\pi$
$938$	0	0
$939$	−23.8167	−0.777227
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	33.2111	1.08208
$943$	0	0
$944$	3.21110	0.104512
$945$	0	0
$946$	−66.3583	−2.15749
$947$	−0.844410	−0.0274396	−0.0137198	−	0.999906i	$-0.504367\pi$
$947$	−0.844410	−0.0274396	−0.0137198	+	0.999906i	$0.504367\pi$
$948$	48.9361	1.58937
$949$	−1.57779	−0.0512174
$950$	0	0
$951$	16.3944	0.531626
$952$	0	0
$953$	−13.6056	−0.440727	−0.220364	−	0.975418i	$-0.570724\pi$
$953$	−13.6056	−0.440727	−0.220364	+	0.975418i	$0.570724\pi$
$954$	7.39445	0.239404
$955$	0	0
$956$	48.8444	1.57974
$957$	−22.8167	−0.737558
$958$	−79.6888	−2.57463
$959$	0	0
$960$	0	0
$961$	21.0000	0.677419
$962$	−25.2666	−0.814628
$963$	0	0
$964$	−31.0278	−0.999337
$965$	0	0
$966$	0	0
$967$	−40.8444	−1.31347	−0.656734	−	0.754122i	$-0.728063\pi$
$967$	−40.8444	−1.31347	−0.656734	+	0.754122i	$0.728063\pi$
$968$	−6.00000	−0.192847
$969$	−30.4222	−0.977302
$970$	0	0
$971$	−53.8722	−1.72884	−0.864420	−	0.502770i	$-0.832314\pi$
$971$	−53.8722	−1.72884	−0.864420	+	0.502770i	$0.832314\pi$
$972$	−3.30278	−0.105937
$973$	0	0
$974$	−47.9361	−1.53597
$975$	0	0
$976$	0.366692	0.0117375
$977$	−2.02776	−0.0648737	−0.0324368	−	0.999474i	$-0.510327\pi$
$977$	−2.02776	−0.0648737	−0.0324368	+	0.999474i	$0.510327\pi$
$978$	−39.6333	−1.26733
$979$	23.4500	0.749464
$980$	0	0
$981$	2.21110	0.0705951
$982$	−6.90833	−0.220454
$983$	−40.0555	−1.27757	−0.638786	−	0.769384i	$-0.720563\pi$
$983$	−40.0555	−1.27757	−0.638786	+	0.769384i	$0.720563\pi$
$984$	0	0
$985$	0	0
$986$	80.6611	2.56877
$987$	0	0
$988$	−56.8444	−1.80846
$989$	−59.6611	−1.89711
$990$	0	0
$991$	46.0278	1.46212	0.731060	−	0.682313i	$-0.239026\pi$
$991$	46.0278	1.46212	0.731060	+	0.682313i	$0.239026\pi$
$992$	−38.2389	−1.21408
$993$	21.2389	0.673995
$994$	0	0
$995$	0	0
$996$	−10.6056	−0.336050
$997$	15.4500	0.489305	0.244653	−	0.969611i	$-0.421326\pi$
$997$	15.4500	0.489305	0.244653	+	0.969611i	$0.421326\pi$
$998$	−64.4777	−2.04101
$999$	4.21110	0.133233

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bb.1.2		2
5.4	even	2		3675.2.a.w.1.1		2
7.6	odd	2		525.2.a.h.1.2	yes	2
21.20	even	2		1575.2.a.o.1.1		2
28.27	even	2		8400.2.a.cw.1.1		2
35.13	even	4		525.2.d.d.274.1		4
35.27	even	4		525.2.d.d.274.4		4
35.34	odd	2		525.2.a.f.1.1	✓	2
105.62	odd	4		1575.2.d.g.1324.1		4
105.83	odd	4		1575.2.d.g.1324.4		4
105.104	even	2		1575.2.a.t.1.2		2
140.139	even	2		8400.2.a.df.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.f.1.1	✓	2	35.34	odd	2
525.2.a.h.1.2	yes	2	7.6	odd	2
525.2.d.d.274.1		4	35.13	even	4
525.2.d.d.274.4		4	35.27	even	4
1575.2.a.o.1.1		2	21.20	even	2
1575.2.a.t.1.2		2	105.104	even	2
1575.2.d.g.1324.1		4	105.62	odd	4
1575.2.d.g.1324.4		4	105.83	odd	4
3675.2.a.w.1.1		2	5.4	even	2
3675.2.a.bb.1.2		2	1.1	even	1	trivial
8400.2.a.cw.1.1		2	28.27	even	2
8400.2.a.df.1.2		2	140.139	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:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} -2.30278 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} -2.30278 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -3.30278 q^{12} +2.60555 q^{13} +0.302776 q^{16} -4.60555 q^{17} +2.30278 q^{18} -6.60555 q^{19} -6.90833 q^{22} -6.21110 q^{23} -3.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -7.60555 q^{29} +7.21110 q^{31} -5.30278 q^{32} +3.00000 q^{33} -10.6056 q^{34} +3.30278 q^{36} -4.21110 q^{37} -15.2111 q^{38} -2.60555 q^{39} +9.60555 q^{43} -9.90833 q^{44} -14.3028 q^{46} -10.6056 q^{47} -0.302776 q^{48} +4.60555 q^{51} +8.60555 q^{52} +3.21110 q^{53} -2.30278 q^{54} +6.60555 q^{57} -17.5139 q^{58} +10.6056 q^{59} +1.21110 q^{61} +16.6056 q^{62} -12.8167 q^{64} +6.90833 q^{66} +15.6056 q^{67} -15.2111 q^{68} +6.21110 q^{69} -3.00000 q^{71} +3.00000 q^{72} -0.605551 q^{73} -9.69722 q^{74} -21.8167 q^{76} -6.00000 q^{78} -14.8167 q^{79} +1.00000 q^{81} +3.21110 q^{83} +22.1194 q^{86} +7.60555 q^{87} -9.00000 q^{88} -7.81665 q^{89} -20.5139 q^{92} -7.21110 q^{93} -24.4222 q^{94} +5.30278 q^{96} +0.788897 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} + 3 q^{4} - q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 + 3 * q^4 - q^6 + 6 * q^8 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} + 3 q^{4} - q^{6} + 6 q^{8} + 2 q^{9} - 6 q^{11} - 3 q^{12} - 2 q^{13} - 3 q^{16} - 2 q^{17} + q^{18} - 6 q^{19} - 3 q^{22} + 2 q^{23} - 6 q^{24} + 12 q^{26} - 2 q^{27} - 8 q^{29} - 7 q^{32} + 6 q^{33} - 14 q^{34} + 3 q^{36} + 6 q^{37} - 16 q^{38} + 2 q^{39} + 12 q^{43} - 9 q^{44} - 25 q^{46} - 14 q^{47} + 3 q^{48} + 2 q^{51} + 10 q^{52} - 8 q^{53} - q^{54} + 6 q^{57} - 17 q^{58} + 14 q^{59} - 12 q^{61} + 26 q^{62} - 4 q^{64} + 3 q^{66} + 24 q^{67} - 16 q^{68} - 2 q^{69} - 6 q^{71} + 6 q^{72} + 6 q^{73} - 23 q^{74} - 22 q^{76} - 12 q^{78} - 8 q^{79} + 2 q^{81} - 8 q^{83} + 19 q^{86} + 8 q^{87} - 18 q^{88} + 6 q^{89} - 23 q^{92} - 20 q^{94} + 7 q^{96} + 16 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 + 3 * q^4 - q^6 + 6 * q^8 + 2 * q^9 - 6 * q^11 - 3 * q^12 - 2 * q^13 - 3 * q^16 - 2 * q^17 + q^18 - 6 * q^19 - 3 * q^22 + 2 * q^23 - 6 * q^24 + 12 * q^26 - 2 * q^27 - 8 * q^29 - 7 * q^32 + 6 * q^33 - 14 * q^34 + 3 * q^36 + 6 * q^37 - 16 * q^38 + 2 * q^39 + 12 * q^43 - 9 * q^44 - 25 * q^46 - 14 * q^47 + 3 * q^48 + 2 * q^51 + 10 * q^52 - 8 * q^53 - q^54 + 6 * q^57 - 17 * q^58 + 14 * q^59 - 12 * q^61 + 26 * q^62 - 4 * q^64 + 3 * q^66 + 24 * q^67 - 16 * q^68 - 2 * q^69 - 6 * q^71 + 6 * q^72 + 6 * q^73 - 23 * q^74 - 22 * q^76 - 12 * q^78 - 8 * q^79 + 2 * q^81 - 8 * q^83 + 19 * q^86 + 8 * q^87 - 18 * q^88 + 6 * q^89 - 23 * q^92 - 20 * q^94 + 7 * q^96 + 16 * q^97 - 6 * q^99

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