LMFDB - Embedded newform 3675.2.a.bk.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:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
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			$-1.74912$ 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)

$f(q)$	$=$	$q-0.665096 q^{2} -1.00000 q^{3} -1.55765 q^{4} +0.665096 q^{6} +2.36618 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.665096 q^{2} -1.00000 q^{3} -1.55765 q^{4} +0.665096 q^{6} +2.36618 q^{8} +1.00000 q^{9} +5.61706 q^{11} +1.55765 q^{12} -6.44549 q^{13} +1.54156 q^{16} +0.947252 q^{17} -0.665096 q^{18} -6.91245 q^{19} -3.73588 q^{22} -1.53304 q^{23} -2.36618 q^{24} +4.28687 q^{26} -1.00000 q^{27} +8.99647 q^{29} +2.91245 q^{31} -5.75764 q^{32} -5.61706 q^{33} -0.630013 q^{34} -1.55765 q^{36} -6.16804 q^{37} +4.59744 q^{38} +6.44549 q^{39} -0.118824 q^{41} -5.11529 q^{43} -8.74940 q^{44} +1.01962 q^{46} +6.48528 q^{47} -1.54156 q^{48} -0.947252 q^{51} +10.0398 q^{52} +3.03127 q^{53} +0.665096 q^{54} +6.91245 q^{57} -5.98352 q^{58} +5.48881 q^{59} -9.41421 q^{61} -1.93706 q^{62} +0.746264 q^{64} +3.73588 q^{66} +8.72293 q^{67} -1.47548 q^{68} +1.53304 q^{69} -1.44902 q^{71} +2.36618 q^{72} +7.78510 q^{73} +4.10234 q^{74} +10.7672 q^{76} -4.28687 q^{78} -11.4325 q^{79} +1.00000 q^{81} +0.0790296 q^{82} -7.17157 q^{83} +3.40216 q^{86} -8.99647 q^{87} +13.2910 q^{88} +0.828427 q^{89} +2.38793 q^{92} -2.91245 q^{93} -4.31333 q^{94} +5.75764 q^{96} +7.93430 q^{97} +5.61706 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{6} - 12 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^6 - 12 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{6} - 12 q^{8} + 4 q^{9} + 8 q^{11} - 8 q^{12} + 12 q^{16} - 8 q^{17} - 4 q^{18} - 8 q^{19} + 12 q^{24} - 4 q^{27} + 8 q^{29} - 8 q^{31} - 28 q^{32} - 8 q^{33} - 8 q^{34} + 8 q^{36} - 8 q^{37} - 4 q^{38} + 8 q^{43} - 16 q^{44} + 12 q^{46} - 8 q^{47} - 12 q^{48} + 8 q^{51} + 32 q^{52} - 8 q^{53} + 4 q^{54} + 8 q^{57} + 24 q^{58} + 16 q^{59} - 32 q^{61} + 20 q^{62} + 24 q^{64} - 24 q^{68} - 8 q^{71} - 12 q^{72} - 32 q^{74} + 8 q^{76} + 4 q^{81} + 8 q^{82} - 40 q^{83} - 32 q^{86} - 8 q^{87} + 40 q^{88} - 8 q^{89} + 8 q^{92} + 8 q^{93} - 16 q^{94} + 28 q^{96} + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^6 - 12 * q^8 + 4 * q^9 + 8 * q^11 - 8 * q^12 + 12 * q^16 - 8 * q^17 - 4 * q^18 - 8 * q^19 + 12 * q^24 - 4 * q^27 + 8 * q^29 - 8 * q^31 - 28 * q^32 - 8 * q^33 - 8 * q^34 + 8 * q^36 - 8 * q^37 - 4 * q^38 + 8 * q^43 - 16 * q^44 + 12 * q^46 - 8 * q^47 - 12 * q^48 + 8 * q^51 + 32 * q^52 - 8 * q^53 + 4 * q^54 + 8 * q^57 + 24 * q^58 + 16 * q^59 - 32 * q^61 + 20 * q^62 + 24 * q^64 - 24 * q^68 - 8 * q^71 - 12 * q^72 - 32 * q^74 + 8 * q^76 + 4 * q^81 + 8 * q^82 - 40 * q^83 - 32 * q^86 - 8 * q^87 + 40 * q^88 - 8 * q^89 + 8 * q^92 + 8 * q^93 - 16 * q^94 + 28 * q^96 + 8 * 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 $

$2$	−0.665096	−0.470294	−0.235147	−	0.971960i	$-0.575557\pi$
$2$	−0.665096	−0.470294	−0.235147	+	0.971960i	$0.575557\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.55765	−0.778824
$5$	0	0
$5$	0	0
$6$	0.665096	0.271524
$7$	0	0
$7$	0	0
$8$	2.36618	0.836570
$9$	1.00000	0.333333
$10$	0	0
$11$	5.61706	1.69361	0.846804	−	0.531906i	$-0.178524\pi$
$11$	5.61706	1.69361	0.846804	+	0.531906i	$0.178524\pi$
$12$	1.55765	0.449654
$13$	−6.44549	−1.78766	−0.893828	−	0.448410i	$-0.851991\pi$
$13$	−6.44549	−1.78766	−0.893828	+	0.448410i	$0.851991\pi$
$14$	0	0
$15$	0	0
$16$	1.54156	0.385390
$17$	0.947252	0.229742	0.114871	−	0.993380i	$-0.463354\pi$
$17$	0.947252	0.229742	0.114871	+	0.993380i	$0.463354\pi$
$18$	−0.665096	−0.156765
$19$	−6.91245	−1.58582	−0.792912	−	0.609336i	$-0.791436\pi$
$19$	−6.91245	−1.58582	−0.792912	+	0.609336i	$0.791436\pi$
$20$	0	0
$21$	0	0
$22$	−3.73588	−0.796493
$23$	−1.53304	−0.319661	−0.159830	−	0.987145i	$-0.551095\pi$
$23$	−1.53304	−0.319661	−0.159830	+	0.987145i	$0.551095\pi$
$24$	−2.36618	−0.482994
$25$	0	0
$26$	4.28687	0.840724
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.99647	1.67060	0.835301	−	0.549792i	$-0.185293\pi$
$29$	8.99647	1.67060	0.835301	+	0.549792i	$0.185293\pi$
$30$	0	0
$31$	2.91245	0.523091	0.261546	−	0.965191i	$-0.415768\pi$
$31$	2.91245	0.523091	0.261546	+	0.965191i	$0.415768\pi$
$32$	−5.75764	−1.01782
$33$	−5.61706	−0.977805
$34$	−0.630013	−0.108046
$35$	0	0
$36$	−1.55765	−0.259608
$37$	−6.16804	−1.01402	−0.507010	−	0.861940i	$-0.669249\pi$
$37$	−6.16804	−1.01402	−0.507010	+	0.861940i	$0.669249\pi$
$38$	4.59744	0.745804
$39$	6.44549	1.03210
$40$	0	0
$41$	−0.118824	−0.0185573	−0.00927863	−	0.999957i	$-0.502954\pi$
$41$	−0.118824	−0.0185573	−0.00927863	+	0.999957i	$0.502954\pi$
$42$	0	0
$43$	−5.11529	−0.780075	−0.390038	−	0.920799i	$-0.627538\pi$
$43$	−5.11529	−0.780075	−0.390038	+	0.920799i	$0.627538\pi$
$44$	−8.74940	−1.31902
$45$	0	0
$46$	1.01962	0.150334
$47$	6.48528	0.945976	0.472988	−	0.881069i	$-0.343175\pi$
$47$	6.48528	0.945976	0.472988	+	0.881069i	$0.343175\pi$
$48$	−1.54156	−0.222505
$49$	0	0
$50$	0	0
$51$	−0.947252	−0.132642
$52$	10.0398	1.39227
$53$	3.03127	0.416377	0.208189	−	0.978089i	$-0.433243\pi$
$53$	3.03127	0.416377	0.208189	+	0.978089i	$0.433243\pi$
$54$	0.665096	0.0905081
$55$	0	0
$56$	0	0
$57$	6.91245	0.915576
$58$	−5.98352	−0.785674
$59$	5.48881	0.714582	0.357291	−	0.933993i	$-0.383700\pi$
$59$	5.48881	0.714582	0.357291	+	0.933993i	$0.383700\pi$
$60$	0	0
$61$	−9.41421	−1.20537	−0.602683	−	0.797981i	$-0.705902\pi$
$61$	−9.41421	−1.20537	−0.602683	+	0.797981i	$0.705902\pi$
$62$	−1.93706	−0.246007
$63$	0	0
$64$	0.746264	0.0932829
$65$	0	0
$66$	3.73588	0.459856
$67$	8.72293	1.06568	0.532838	−	0.846217i	$-0.321126\pi$
$67$	8.72293	1.06568	0.532838	+	0.846217i	$0.321126\pi$
$68$	−1.47548	−0.178929
$69$	1.53304	0.184556
$70$	0	0
$71$	−1.44902	−0.171967	−0.0859833	−	0.996297i	$-0.527403\pi$
$71$	−1.44902	−0.171967	−0.0859833	+	0.996297i	$0.527403\pi$
$72$	2.36618	0.278857
$73$	7.78510	0.911177	0.455589	−	0.890190i	$-0.349429\pi$
$73$	7.78510	0.911177	0.455589	+	0.890190i	$0.349429\pi$
$74$	4.10234	0.476887
$75$	0	0
$76$	10.7672	1.23508
$77$	0	0
$78$	−4.28687	−0.485392
$79$	−11.4325	−1.28626	−0.643130	−	0.765757i	$-0.722365\pi$
$79$	−11.4325	−1.28626	−0.643130	+	0.765757i	$0.722365\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.0790296	0.00872736
$83$	−7.17157	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$83$	−7.17157	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$84$	0	0
$85$	0	0
$86$	3.40216	0.366865
$87$	−8.99647	−0.964523
$88$	13.2910	1.41682
$89$	0.828427	0.0878131	0.0439065	−	0.999036i	$-0.486020\pi$
$89$	0.828427	0.0878131	0.0439065	+	0.999036i	$0.486020\pi$
$90$	0	0
$91$	0	0
$92$	2.38793	0.248959
$93$	−2.91245	−0.302007
$94$	−4.31333	−0.444887
$95$	0	0
$96$	5.75764	0.587637
$97$	7.93430	0.805606	0.402803	−	0.915287i	$-0.368036\pi$
$97$	7.93430	0.805606	0.402803	+	0.915287i	$0.368036\pi$
$98$	0	0
$99$	5.61706	0.564536
$100$	0	0
$101$	1.77568	0.176687	0.0883433	−	0.996090i	$-0.471843\pi$
$101$	1.77568	0.176687	0.0883433	+	0.996090i	$0.471843\pi$
$102$	0.630013	0.0623806
$103$	0.168043	0.0165578	0.00827889	−	0.999966i	$-0.497365\pi$
$103$	0.168043	0.0165578	0.00827889	+	0.999966i	$0.497365\pi$
$104$	−15.2512	−1.49550
$105$	0	0
$106$	−2.01609	−0.195820
$107$	−5.70108	−0.551144	−0.275572	−	0.961280i	$-0.588867\pi$
$107$	−5.70108	−0.551144	−0.275572	+	0.961280i	$0.588867\pi$
$108$	1.55765	0.149885
$109$	−3.65685	−0.350263	−0.175132	−	0.984545i	$-0.556035\pi$
$109$	−3.65685	−0.350263	−0.175132	+	0.984545i	$0.556035\pi$
$110$	0	0
$111$	6.16804	0.585445
$112$	0	0
$113$	17.5791	1.65370	0.826851	−	0.562421i	$-0.190130\pi$
$113$	17.5791	1.65370	0.826851	+	0.562421i	$0.190130\pi$
$114$	−4.59744	−0.430590
$115$	0	0
$116$	−14.0133	−1.30110
$117$	−6.44549	−0.595885
$118$	−3.65059	−0.336064
$119$	0	0
$120$	0	0
$121$	20.5514	1.86831
$122$	6.26136	0.566877
$123$	0.118824	0.0107140
$124$	−4.53657	−0.407396
$125$	0	0
$126$	0	0
$127$	14.2869	1.26775	0.633877	−	0.773434i	$-0.281462\pi$
$127$	14.2869	1.26775	0.633877	+	0.773434i	$0.281462\pi$
$128$	11.0189	0.973946
$129$	5.11529	0.450377
$130$	0	0
$131$	14.6926	1.28369	0.641847	−	0.766832i	$-0.278168\pi$
$131$	14.6926	1.28369	0.641847	+	0.766832i	$0.278168\pi$
$132$	8.74940	0.761537
$133$	0	0
$134$	−5.80159	−0.501181
$135$	0	0
$136$	2.24136	0.192195
$137$	−11.4370	−0.977126	−0.488563	−	0.872529i	$-0.662479\pi$
$137$	−11.4370	−0.977126	−0.488563	+	0.872529i	$0.662479\pi$
$138$	−1.01962	−0.0867956
$139$	−18.1466	−1.53917	−0.769586	−	0.638543i	$-0.779537\pi$
$139$	−18.1466	−1.53917	−0.769586	+	0.638543i	$0.779537\pi$
$140$	0	0
$141$	−6.48528	−0.546159
$142$	0.963735	0.0808748
$143$	−36.2047	−3.02759
$144$	1.54156	0.128463
$145$	0	0
$146$	−5.17784	−0.428521
$147$	0	0
$148$	9.60764	0.789743
$149$	−8.23059	−0.674276	−0.337138	−	0.941455i	$-0.609459\pi$
$149$	−8.23059	−0.674276	−0.337138	+	0.941455i	$0.609459\pi$
$150$	0	0
$151$	−9.79453	−0.797067	−0.398534	−	0.917154i	$-0.630481\pi$
$151$	−9.79453	−0.797067	−0.398534	+	0.917154i	$0.630481\pi$
$152$	−16.3561	−1.32665
$153$	0.947252	0.0765807
$154$	0	0
$155$	0	0
$156$	−10.0398	−0.803827
$157$	−8.10234	−0.646637	−0.323319	−	0.946290i	$-0.604799\pi$
$157$	−8.10234	−0.646637	−0.323319	+	0.946290i	$0.604799\pi$
$158$	7.60373	0.604920
$159$	−3.03127	−0.240396
$160$	0	0
$161$	0	0
$162$	−0.665096	−0.0522549
$163$	−17.8382	−1.39720	−0.698599	−	0.715514i	$-0.746193\pi$
$163$	−17.8382	−1.39720	−0.698599	+	0.715514i	$0.746193\pi$
$164$	0.185086	0.0144528
$165$	0	0
$166$	4.76978	0.370207
$167$	−15.5514	−1.20340	−0.601700	−	0.798722i	$-0.705510\pi$
$167$	−15.5514	−1.20340	−0.601700	+	0.798722i	$0.705510\pi$
$168$	0	0
$169$	28.5443	2.19572
$170$	0	0
$171$	−6.91245	−0.528608
$172$	7.96782	0.607541
$173$	−13.2645	−1.00848	−0.504240	−	0.863563i	$-0.668227\pi$
$173$	−13.2645	−1.00848	−0.504240	+	0.863563i	$0.668227\pi$
$174$	5.98352	0.453609
$175$	0	0
$176$	8.65903	0.652699
$177$	−5.48881	−0.412564
$178$	−0.550984	−0.0412980
$179$	−7.89060	−0.589771	−0.294886	−	0.955533i	$-0.595282\pi$
$179$	−7.89060	−0.589771	−0.294886	+	0.955533i	$0.595282\pi$
$180$	0	0
$181$	−24.8960	−1.85050	−0.925251	−	0.379355i	$-0.876146\pi$
$181$	−24.8960	−1.85050	−0.925251	+	0.379355i	$0.876146\pi$
$182$	0	0
$183$	9.41421	0.695919
$184$	−3.62744	−0.267418
$185$	0	0
$186$	1.93706	0.142032
$187$	5.32077	0.389093
$188$	−10.1018	−0.736748
$189$	0	0
$190$	0	0
$191$	7.03626	0.509126	0.254563	−	0.967056i	$-0.418068\pi$
$191$	7.03626	0.509126	0.254563	+	0.967056i	$0.418068\pi$
$192$	−0.746264	−0.0538569
$193$	−8.81510	−0.634525	−0.317262	−	0.948338i	$-0.602764\pi$
$193$	−8.81510	−0.634525	−0.317262	+	0.948338i	$0.602764\pi$
$194$	−5.27707	−0.378872
$195$	0	0
$196$	0	0
$197$	−22.5130	−1.60399	−0.801993	−	0.597333i	$-0.796227\pi$
$197$	−22.5130	−1.60399	−0.801993	+	0.597333i	$0.796227\pi$
$198$	−3.73588	−0.265498
$199$	−8.32167	−0.589908	−0.294954	−	0.955512i	$-0.595304\pi$
$199$	−8.32167	−0.589908	−0.294954	+	0.955512i	$0.595304\pi$
$200$	0	0
$201$	−8.72293	−0.615268
$202$	−1.18100	−0.0830946
$203$	0	0
$204$	1.47548	0.103305
$205$	0	0
$206$	−0.111765	−0.00778702
$207$	−1.53304	−0.106554
$208$	−9.93610	−0.688945
$209$	−38.8276	−2.68576
$210$	0	0
$211$	−8.65332	−0.595719	−0.297860	−	0.954610i	$-0.596273\pi$
$211$	−8.65332	−0.595719	−0.297860	+	0.954610i	$0.596273\pi$
$212$	−4.72165	−0.324285
$213$	1.44902	0.0992850
$214$	3.79177	0.259200
$215$	0	0
$216$	−2.36618	−0.160998
$217$	0	0
$218$	2.43216	0.164727
$219$	−7.78510	−0.526068
$220$	0	0
$221$	−6.10550	−0.410700
$222$	−4.10234	−0.275331
$223$	0.597838	0.0400342	0.0200171	−	0.999800i	$-0.493628\pi$
$223$	0.597838	0.0400342	0.0200171	+	0.999800i	$0.493628\pi$
$224$	0	0
$225$	0	0
$226$	−11.6918	−0.777726
$227$	−0.765881	−0.0508333	−0.0254167	−	0.999677i	$-0.508091\pi$
$227$	−0.765881	−0.0508333	−0.0254167	+	0.999677i	$0.508091\pi$
$228$	−10.7672	−0.713072
$229$	−4.92893	−0.325713	−0.162857	−	0.986650i	$-0.552071\pi$
$229$	−4.92893	−0.325713	−0.162857	+	0.986650i	$0.552071\pi$
$230$	0	0
$231$	0	0
$232$	21.2872	1.39758
$233$	−29.0938	−1.90600	−0.953000	−	0.302971i	$-0.902021\pi$
$233$	−29.0938	−1.90600	−0.953000	+	0.302971i	$0.902021\pi$
$234$	4.28687	0.280241
$235$	0	0
$236$	−8.54963	−0.556534
$237$	11.4325	0.742623
$238$	0	0
$239$	−21.1873	−1.37049	−0.685245	−	0.728313i	$-0.740305\pi$
$239$	−21.1873	−1.37049	−0.685245	+	0.728313i	$0.740305\pi$
$240$	0	0
$241$	0.261489	0.0168440	0.00842198	−	0.999965i	$-0.497319\pi$
$241$	0.261489	0.0168440	0.00842198	+	0.999965i	$0.497319\pi$
$242$	−13.6686	−0.878653
$243$	−1.00000	−0.0641500
$244$	14.6640	0.938768
$245$	0	0
$246$	−0.0790296	−0.00503875
$247$	44.5541	2.83491
$248$	6.89137	0.437602
$249$	7.17157	0.454480
$250$	0	0
$251$	19.7194	1.24468	0.622339	−	0.782748i	$-0.286183\pi$
$251$	19.7194	1.24468	0.622339	+	0.782748i	$0.286183\pi$
$252$	0	0
$253$	−8.61117	−0.541379
$254$	−9.50214	−0.596217
$255$	0	0
$256$	−8.82118	−0.551324
$257$	9.23412	0.576009	0.288004	−	0.957629i	$-0.407008\pi$
$257$	9.23412	0.576009	0.288004	+	0.957629i	$0.407008\pi$
$258$	−3.40216	−0.211809
$259$	0	0
$260$	0	0
$261$	8.99647	0.556868
$262$	−9.77196	−0.603714
$263$	17.2524	1.06383	0.531915	−	0.846797i	$-0.321472\pi$
$263$	17.2524	1.06383	0.531915	+	0.846797i	$0.321472\pi$
$264$	−13.2910	−0.818002
$265$	0	0
$266$	0	0
$267$	−0.828427	−0.0506989
$268$	−13.5872	−0.829973
$269$	11.5210	0.702447	0.351223	−	0.936292i	$-0.385766\pi$
$269$	11.5210	0.702447	0.351223	+	0.936292i	$0.385766\pi$
$270$	0	0
$271$	−18.2958	−1.11139	−0.555694	−	0.831387i	$-0.687547\pi$
$271$	−18.2958	−1.11139	−0.555694	+	0.831387i	$0.687547\pi$
$272$	1.46024	0.0885403
$273$	0	0
$274$	7.60668	0.459536
$275$	0	0
$276$	−2.38793	−0.143737
$277$	14.8776	0.893911	0.446956	−	0.894556i	$-0.352508\pi$
$277$	14.8776	0.893911	0.446956	+	0.894556i	$0.352508\pi$
$278$	12.0692	0.723863
$279$	2.91245	0.174364
$280$	0	0
$281$	−15.5443	−0.927295	−0.463648	−	0.886020i	$-0.653460\pi$
$281$	−15.5443	−0.927295	−0.463648	+	0.886020i	$0.653460\pi$
$282$	4.31333	0.256855
$283$	−6.30019	−0.374508	−0.187254	−	0.982312i	$-0.559959\pi$
$283$	−6.30019	−0.374508	−0.187254	+	0.982312i	$0.559959\pi$
$284$	2.25706	0.133932
$285$	0	0
$286$	24.0796	1.42386
$287$	0	0
$288$	−5.75764	−0.339272
$289$	−16.1027	−0.947219
$290$	0	0
$291$	−7.93430	−0.465117
$292$	−12.1264	−0.709646
$293$	6.65332	0.388691	0.194346	−	0.980933i	$-0.437742\pi$
$293$	6.65332	0.388691	0.194346	+	0.980933i	$0.437742\pi$
$294$	0	0
$295$	0	0
$296$	−14.5947	−0.848299
$297$	−5.61706	−0.325935
$298$	5.47413	0.317108
$299$	9.88118	0.571443
$300$	0	0
$301$	0	0
$302$	6.51430	0.374856
$303$	−1.77568	−0.102010
$304$	−10.6560	−0.611161
$305$	0	0
$306$	−0.630013	−0.0360155
$307$	−7.31371	−0.417415	−0.208708	−	0.977978i	$-0.566926\pi$
$307$	−7.31371	−0.417415	−0.208708	+	0.977978i	$0.566926\pi$
$308$	0	0
$309$	−0.168043	−0.00955964
$310$	0	0
$311$	−33.7686	−1.91484	−0.957421	−	0.288694i	$-0.906779\pi$
$311$	−33.7686	−1.91484	−0.957421	+	0.288694i	$0.906779\pi$
$312$	15.2512	0.863427
$313$	−17.9702	−1.01574	−0.507868	−	0.861435i	$-0.669566\pi$
$313$	−17.9702	−1.01574	−0.507868	+	0.861435i	$0.669566\pi$
$314$	5.38883	0.304110
$315$	0	0
$316$	17.8079	1.00177
$317$	−10.1769	−0.571594	−0.285797	−	0.958290i	$-0.592258\pi$
$317$	−10.1769	−0.571594	−0.285797	+	0.958290i	$0.592258\pi$
$318$	2.01609	0.113057
$319$	50.5337	2.82934
$320$	0	0
$321$	5.70108	0.318203
$322$	0	0
$323$	−6.54783	−0.364331
$324$	−1.55765	−0.0865360
$325$	0	0
$326$	11.8641	0.657094
$327$	3.65685	0.202225
$328$	−0.281160	−0.0155244
$329$	0	0
$330$	0	0
$331$	8.31724	0.457157	0.228578	−	0.973526i	$-0.426592\pi$
$331$	8.31724	0.457157	0.228578	+	0.973526i	$0.426592\pi$
$332$	11.1708	0.613076
$333$	−6.16804	−0.338007
$334$	10.3431	0.565952
$335$	0	0
$336$	0	0
$337$	−26.0233	−1.41758	−0.708790	−	0.705420i	$-0.750759\pi$
$337$	−26.0233	−1.41758	−0.708790	+	0.705420i	$0.750759\pi$
$338$	−18.9847	−1.03263
$339$	−17.5791	−0.954766
$340$	0	0
$341$	16.3594	0.885911
$342$	4.59744	0.248601
$343$	0	0
$344$	−12.1037	−0.652587
$345$	0	0
$346$	8.82216	0.474282
$347$	−9.77069	−0.524518	−0.262259	−	0.964998i	$-0.584467\pi$
$347$	−9.77069	−0.524518	−0.262259	+	0.964998i	$0.584467\pi$
$348$	14.0133	0.751193
$349$	19.9809	1.06955	0.534776	−	0.844994i	$-0.320396\pi$
$349$	19.9809	1.06955	0.534776	+	0.844994i	$0.320396\pi$
$350$	0	0
$351$	6.44549	0.344035
$352$	−32.3410	−1.72378
$353$	24.5667	1.30755	0.653776	−	0.756688i	$-0.273184\pi$
$353$	24.5667	1.30755	0.653776	+	0.756688i	$0.273184\pi$
$354$	3.65059	0.194026
$355$	0	0
$356$	−1.29040	−0.0683909
$357$	0	0
$358$	5.24801	0.277366
$359$	−4.03979	−0.213212	−0.106606	−	0.994301i	$-0.533998\pi$
$359$	−4.03979	−0.213212	−0.106606	+	0.994301i	$0.533998\pi$
$360$	0	0
$361$	28.7819	1.51484
$362$	16.5582	0.870280
$363$	−20.5514	−1.07867
$364$	0	0
$365$	0	0
$366$	−6.26136	−0.327286
$367$	5.27001	0.275092	0.137546	−	0.990495i	$-0.456078\pi$
$367$	5.27001	0.275092	0.137546	+	0.990495i	$0.456078\pi$
$368$	−2.36327	−0.123194
$369$	−0.118824	−0.00618575
$370$	0	0
$371$	0	0
$372$	4.53657	0.235210
$373$	27.3066	1.41388	0.706942	−	0.707271i	$-0.250074\pi$
$373$	27.3066	1.41388	0.706942	+	0.707271i	$0.250074\pi$
$374$	−3.53882	−0.182988
$375$	0	0
$376$	15.3453	0.791375
$377$	−57.9866	−2.98646
$378$	0	0
$379$	−12.1984	−0.626590	−0.313295	−	0.949656i	$-0.601433\pi$
$379$	−12.1984	−0.626590	−0.313295	+	0.949656i	$0.601433\pi$
$380$	0	0
$381$	−14.2869	−0.731938
$382$	−4.67979	−0.239439
$383$	13.4818	0.688885	0.344443	−	0.938807i	$-0.388068\pi$
$383$	13.4818	0.688885	0.344443	+	0.938807i	$0.388068\pi$
$384$	−11.0189	−0.562308
$385$	0	0
$386$	5.86289	0.298413
$387$	−5.11529	−0.260025
$388$	−12.3588	−0.627425
$389$	−33.0331	−1.67485	−0.837423	−	0.546556i	$-0.815939\pi$
$389$	−33.0331	−1.67485	−0.837423	+	0.546556i	$0.815939\pi$
$390$	0	0
$391$	−1.45217	−0.0734395
$392$	0	0
$393$	−14.6926	−0.741142
$394$	14.9733	0.754345
$395$	0	0
$396$	−8.74940	−0.439674
$397$	−21.0363	−1.05578	−0.527890	−	0.849313i	$-0.677017\pi$
$397$	−21.0363	−1.05578	−0.527890	+	0.849313i	$0.677017\pi$
$398$	5.53471	0.277430
$399$	0	0
$400$	0	0
$401$	6.82137	0.340643	0.170321	−	0.985389i	$-0.445519\pi$
$401$	6.82137	0.340643	0.170321	+	0.985389i	$0.445519\pi$
$402$	5.80159	0.289357
$403$	−18.7721	−0.935107
$404$	−2.76588	−0.137608
$405$	0	0
$406$	0	0
$407$	−34.6463	−1.71735
$408$	−2.24136	−0.110964
$409$	−13.7869	−0.681720	−0.340860	−	0.940114i	$-0.610718\pi$
$409$	−13.7869	−0.681720	−0.340860	+	0.940114i	$0.610718\pi$
$410$	0	0
$411$	11.4370	0.564144
$412$	−0.261752	−0.0128956
$413$	0	0
$414$	1.01962	0.0501115
$415$	0	0
$416$	37.1108	1.81951
$417$	18.1466	0.888641
$418$	25.8241	1.26310
$419$	−1.89450	−0.0925525	−0.0462763	−	0.998929i	$-0.514735\pi$
$419$	−1.89450	−0.0925525	−0.0462763	+	0.998929i	$0.514735\pi$
$420$	0	0
$421$	−4.81138	−0.234492	−0.117246	−	0.993103i	$-0.537407\pi$
$421$	−4.81138	−0.234492	−0.117246	+	0.993103i	$0.537407\pi$
$422$	5.75529	0.280163
$423$	6.48528	0.315325
$424$	7.17253	0.348329
$425$	0	0
$426$	−0.963735	−0.0466931
$427$	0	0
$428$	8.88027	0.429244
$429$	36.2047	1.74798
$430$	0	0
$431$	−18.4196	−0.887240	−0.443620	−	0.896215i	$-0.646306\pi$
$431$	−18.4196	−0.887240	−0.443620	+	0.896215i	$0.646306\pi$
$432$	−1.54156	−0.0741683
$433$	−14.7627	−0.709451	−0.354726	−	0.934970i	$-0.615426\pi$
$433$	−14.7627	−0.709451	−0.354726	+	0.934970i	$0.615426\pi$
$434$	0	0
$435$	0	0
$436$	5.69609	0.272793
$437$	10.5970	0.506925
$438$	5.17784	0.247407
$439$	−18.5434	−0.885028	−0.442514	−	0.896762i	$-0.645913\pi$
$439$	−18.5434	−0.885028	−0.442514	+	0.896762i	$0.645913\pi$
$440$	0	0
$441$	0	0
$442$	4.06074	0.193150
$443$	26.1068	1.24037	0.620185	−	0.784456i	$-0.287057\pi$
$443$	26.1068	1.24037	0.620185	+	0.784456i	$0.287057\pi$
$444$	−9.60764	−0.455958
$445$	0	0
$446$	−0.397620	−0.0188278
$447$	8.23059	0.389294
$448$	0	0
$449$	39.7631	1.87654	0.938268	−	0.345908i	$-0.112429\pi$
$449$	39.7631	1.87654	0.938268	+	0.345908i	$0.112429\pi$
$450$	0	0
$451$	−0.667444	−0.0314287
$452$	−27.3820	−1.28794
$453$	9.79453	0.460187
$454$	0.509384	0.0239066
$455$	0	0
$456$	16.3561	0.765943
$457$	8.22173	0.384596	0.192298	−	0.981337i	$-0.438406\pi$
$457$	8.22173	0.384596	0.192298	+	0.981337i	$0.438406\pi$
$458$	3.27821	0.153181
$459$	−0.947252	−0.0442139
$460$	0	0
$461$	−9.91155	−0.461627	−0.230813	−	0.972998i	$-0.574139\pi$
$461$	−9.91155	−0.461627	−0.230813	+	0.972998i	$0.574139\pi$
$462$	0	0
$463$	16.7285	0.777437	0.388719	−	0.921357i	$-0.372918\pi$
$463$	16.7285	0.777437	0.388719	+	0.921357i	$0.372918\pi$
$464$	13.8686	0.643833
$465$	0	0
$466$	19.3502	0.896380
$467$	28.7784	1.33171	0.665853	−	0.746083i	$-0.268068\pi$
$467$	28.7784	1.33171	0.665853	+	0.746083i	$0.268068\pi$
$468$	10.0398	0.464090
$469$	0	0
$470$	0	0
$471$	8.10234	0.373336
$472$	12.9875	0.597798
$473$	−28.7329	−1.32114
$474$	−7.60373	−0.349251
$475$	0	0
$476$	0	0
$477$	3.03127	0.138792
$478$	14.0916	0.644533
$479$	14.2173	0.649603	0.324802	−	0.945782i	$-0.394702\pi$
$479$	14.2173	0.649603	0.324802	+	0.945782i	$0.394702\pi$
$480$	0	0
$481$	39.7560	1.81272
$482$	−0.173915	−0.00792161
$483$	0	0
$484$	−32.0118	−1.45508
$485$	0	0
$486$	0.665096	0.0301694
$487$	9.56394	0.433383	0.216692	−	0.976240i	$-0.430473\pi$
$487$	9.56394	0.433383	0.216692	+	0.976240i	$0.430473\pi$
$488$	−22.2757	−1.00837
$489$	17.8382	0.806672
$490$	0	0
$491$	−25.0800	−1.13184	−0.565921	−	0.824459i	$-0.691479\pi$
$491$	−25.0800	−1.13184	−0.565921	+	0.824459i	$0.691479\pi$
$492$	−0.185086	−0.00834434
$493$	8.52192	0.383808
$494$	−29.6328	−1.33324
$495$	0	0
$496$	4.48971	0.201594
$497$	0	0
$498$	−4.76978	−0.213739
$499$	3.68903	0.165144	0.0825718	−	0.996585i	$-0.473687\pi$
$499$	3.68903	0.165144	0.0825718	+	0.996585i	$0.473687\pi$
$500$	0	0
$501$	15.5514	0.694783
$502$	−13.1153	−0.585364
$503$	41.8445	1.86575	0.932877	−	0.360195i	$-0.117290\pi$
$503$	41.8445	1.86575	0.932877	+	0.360195i	$0.117290\pi$
$504$	0	0
$505$	0	0
$506$	5.72725	0.254607
$507$	−28.5443	−1.26770
$508$	−22.2539	−0.987357
$509$	5.44958	0.241548	0.120774	−	0.992680i	$-0.461462\pi$
$509$	5.44958	0.241548	0.120774	+	0.992680i	$0.461462\pi$
$510$	0	0
$511$	0	0
$512$	−16.1710	−0.714662
$513$	6.91245	0.305192
$514$	−6.14158	−0.270893
$515$	0	0
$516$	−7.96782	−0.350764
$517$	36.4282	1.60211
$518$	0	0
$519$	13.2645	0.582246
$520$	0	0
$521$	−15.5817	−0.682648	−0.341324	−	0.939946i	$-0.610875\pi$
$521$	−15.5817	−0.682648	−0.341324	+	0.939946i	$0.610875\pi$
$522$	−5.98352	−0.261891
$523$	4.08845	0.178776	0.0893878	−	0.995997i	$-0.471509\pi$
$523$	4.08845	0.178776	0.0893878	+	0.995997i	$0.471509\pi$
$524$	−22.8858	−0.999772
$525$	0	0
$526$	−11.4745	−0.500313
$527$	2.75882	0.120176
$528$	−8.65903	−0.376836
$529$	−20.6498	−0.897817
$530$	0	0
$531$	5.48881	0.238194
$532$	0	0
$533$	0.765881	0.0331740
$534$	0.550984	0.0238434
$535$	0	0
$536$	20.6400	0.891512
$537$	7.89060	0.340505
$538$	−7.66256	−0.330357
$539$	0	0
$540$	0	0
$541$	−27.6764	−1.18990	−0.594952	−	0.803761i	$-0.702829\pi$
$541$	−27.6764	−1.18990	−0.594952	+	0.803761i	$0.702829\pi$
$542$	12.1684	0.522679
$543$	24.8960	1.06839
$544$	−5.45393	−0.233835
$545$	0	0
$546$	0	0
$547$	−10.1421	−0.433646	−0.216823	−	0.976211i	$-0.569570\pi$
$547$	−10.1421	−0.433646	−0.216823	+	0.976211i	$0.569570\pi$
$548$	17.8148	0.761009
$549$	−9.41421	−0.401789
$550$	0	0
$551$	−62.1876	−2.64928
$552$	3.62744	0.154394
$553$	0	0
$554$	−9.89506	−0.420401
$555$	0	0
$556$	28.2660	1.19874
$557$	28.0714	1.18943	0.594713	−	0.803938i	$-0.297266\pi$
$557$	28.0714	1.18943	0.594713	+	0.803938i	$0.297266\pi$
$558$	−1.93706	−0.0820022
$559$	32.9706	1.39451
$560$	0	0
$561$	−5.32077	−0.224643
$562$	10.3385	0.436101
$563$	−4.67037	−0.196833	−0.0984163	−	0.995145i	$-0.531378\pi$
$563$	−4.67037	−0.196833	−0.0984163	+	0.995145i	$0.531378\pi$
$564$	10.1018	0.425362
$565$	0	0
$566$	4.19023	0.176129
$567$	0	0
$568$	−3.42863	−0.143862
$569$	19.9311	0.835557	0.417778	−	0.908549i	$-0.362809\pi$
$569$	19.9311	0.835557	0.417778	+	0.908549i	$0.362809\pi$
$570$	0	0
$571$	41.9070	1.75375	0.876877	−	0.480714i	$-0.159622\pi$
$571$	41.9070	1.75375	0.876877	+	0.480714i	$0.159622\pi$
$572$	56.3941	2.35796
$573$	−7.03626	−0.293944
$574$	0	0
$575$	0	0
$576$	0.746264	0.0310943
$577$	−44.5965	−1.85658	−0.928288	−	0.371862i	$-0.878719\pi$
$577$	−44.5965	−1.85658	−0.928288	+	0.371862i	$0.878719\pi$
$578$	10.7099	0.445471
$579$	8.81510	0.366343
$580$	0	0
$581$	0	0
$582$	5.27707	0.218742
$583$	17.0268	0.705180
$584$	18.4209	0.762264
$585$	0	0
$586$	−4.42510	−0.182799
$587$	−23.5872	−0.973550	−0.486775	−	0.873527i	$-0.661827\pi$
$587$	−23.5872	−0.973550	−0.486775	+	0.873527i	$0.661827\pi$
$588$	0	0
$589$	−20.1322	−0.829531
$590$	0	0
$591$	22.5130	0.926062
$592$	−9.50841	−0.390793
$593$	−30.3627	−1.24685	−0.623424	−	0.781884i	$-0.714259\pi$
$593$	−30.3627	−1.24685	−0.623424	+	0.781884i	$0.714259\pi$
$594$	3.73588	0.153285
$595$	0	0
$596$	12.8204	0.525142
$597$	8.32167	0.340583
$598$	−6.57193	−0.268746
$599$	−12.5339	−0.512123	−0.256061	−	0.966661i	$-0.582425\pi$
$599$	−12.5339	−0.512123	−0.256061	+	0.966661i	$0.582425\pi$
$600$	0	0
$601$	−20.4107	−0.832569	−0.416285	−	0.909234i	$-0.636668\pi$
$601$	−20.4107	−0.832569	−0.416285	+	0.909234i	$0.636668\pi$
$602$	0	0
$603$	8.72293	0.355225
$604$	15.2564	0.620775
$605$	0	0
$606$	1.18100	0.0479747
$607$	−30.4253	−1.23492	−0.617462	−	0.786601i	$-0.711839\pi$
$607$	−30.4253	−1.23492	−0.617462	+	0.786601i	$0.711839\pi$
$608$	39.7994	1.61408
$609$	0	0
$610$	0	0
$611$	−41.8008	−1.69108
$612$	−1.47548	−0.0596429
$613$	−2.42900	−0.0981065	−0.0490533	−	0.998796i	$-0.515620\pi$
$613$	−2.42900	−0.0981065	−0.0490533	+	0.998796i	$0.515620\pi$
$614$	4.86432	0.196308
$615$	0	0
$616$	0	0
$617$	26.2324	1.05608	0.528039	−	0.849220i	$-0.322928\pi$
$617$	26.2324	1.05608	0.528039	+	0.849220i	$0.322928\pi$
$618$	0.111765	0.00449584
$619$	28.3772	1.14057	0.570287	−	0.821445i	$-0.306832\pi$
$619$	28.3772	1.14057	0.570287	+	0.821445i	$0.306832\pi$
$620$	0	0
$621$	1.53304	0.0615187
$622$	22.4594	0.900539
$623$	0	0
$624$	9.93610	0.397762
$625$	0	0
$626$	11.9519	0.477694
$627$	38.8276	1.55063
$628$	12.6206	0.503616
$629$	−5.84269	−0.232963
$630$	0	0
$631$	−8.96429	−0.356863	−0.178431	−	0.983952i	$-0.557102\pi$
$631$	−8.96429	−0.356863	−0.178431	+	0.983952i	$0.557102\pi$
$632$	−27.0514	−1.07605
$633$	8.65332	0.343939
$634$	6.76864	0.268817
$635$	0	0
$636$	4.72165	0.187226
$637$	0	0
$638$	−33.6098	−1.33062
$639$	−1.44902	−0.0573222
$640$	0	0
$641$	−18.2565	−0.721088	−0.360544	−	0.932742i	$-0.617409\pi$
$641$	−18.2565	−0.721088	−0.360544	+	0.932742i	$0.617409\pi$
$642$	−3.79177	−0.149649
$643$	5.10197	0.201202	0.100601	−	0.994927i	$-0.467923\pi$
$643$	5.10197	0.201202	0.100601	+	0.994927i	$0.467923\pi$
$644$	0	0
$645$	0	0
$646$	4.35493	0.171343
$647$	−13.8338	−0.543861	−0.271931	−	0.962317i	$-0.587662\pi$
$647$	−13.8338	−0.543861	−0.271931	+	0.962317i	$0.587662\pi$
$648$	2.36618	0.0929522
$649$	30.8310	1.21022
$650$	0	0
$651$	0	0
$652$	27.7857	1.08817
$653$	−4.37089	−0.171046	−0.0855231	−	0.996336i	$-0.527256\pi$
$653$	−4.37089	−0.171046	−0.0855231	+	0.996336i	$0.527256\pi$
$654$	−2.43216	−0.0951050
$655$	0	0
$656$	−0.183175	−0.00715178
$657$	7.78510	0.303726
$658$	0	0
$659$	−49.3365	−1.92188	−0.960938	−	0.276764i	$-0.910738\pi$
$659$	−49.3365	−1.92188	−0.960938	+	0.276764i	$0.910738\pi$
$660$	0	0
$661$	−6.77268	−0.263427	−0.131713	−	0.991288i	$-0.542048\pi$
$661$	−6.77268	−0.263427	−0.131713	+	0.991288i	$0.542048\pi$
$662$	−5.53176	−0.214998
$663$	6.10550	0.237118
$664$	−16.9692	−0.658533
$665$	0	0
$666$	4.10234	0.158962
$667$	−13.7919	−0.534026
$668$	24.2235	0.937236
$669$	−0.597838	−0.0231138
$670$	0	0
$671$	−52.8802	−2.04142
$672$	0	0
$673$	−26.5649	−1.02400	−0.512000	−	0.858985i	$-0.671095\pi$
$673$	−26.5649	−1.02400	−0.512000	+	0.858985i	$0.671095\pi$
$674$	17.3080	0.666679
$675$	0	0
$676$	−44.4619	−1.71007
$677$	−35.3522	−1.35869	−0.679347	−	0.733817i	$-0.737737\pi$
$677$	−35.3522	−1.35869	−0.679347	+	0.733817i	$0.737737\pi$
$678$	11.6918	0.449020
$679$	0	0
$680$	0	0
$681$	0.765881	0.0293486
$682$	−10.8806	−0.416639
$683$	39.2265	1.50096	0.750481	−	0.660892i	$-0.229822\pi$
$683$	39.2265	1.50096	0.750481	+	0.660892i	$0.229822\pi$
$684$	10.7672	0.411693
$685$	0	0
$686$	0	0
$687$	4.92893	0.188050
$688$	−7.88553	−0.300633
$689$	−19.5380	−0.744340
$690$	0	0
$691$	−42.3324	−1.61040	−0.805200	−	0.593003i	$-0.797942\pi$
$691$	−42.3324	−1.61040	−0.805200	+	0.593003i	$0.797942\pi$
$692$	20.6614	0.785428
$693$	0	0
$694$	6.49844	0.246678
$695$	0	0
$696$	−21.2872	−0.806891
$697$	−0.112557	−0.00426338
$698$	−13.2892	−0.503004
$699$	29.0938	1.10043
$700$	0	0
$701$	5.49062	0.207378	0.103689	−	0.994610i	$-0.466935\pi$
$701$	5.49062	0.207378	0.103689	+	0.994610i	$0.466935\pi$
$702$	−4.28687	−0.161797
$703$	42.6363	1.60806
$704$	4.19181	0.157985
$705$	0	0
$706$	−16.3392	−0.614934
$707$	0	0
$708$	8.54963	0.321315
$709$	5.78195	0.217146	0.108573	−	0.994089i	$-0.465372\pi$
$709$	5.78195	0.217146	0.108573	+	0.994089i	$0.465372\pi$
$710$	0	0
$711$	−11.4325	−0.428753
$712$	1.96021	0.0734618
$713$	−4.46489	−0.167212
$714$	0	0
$715$	0	0
$716$	12.2908	0.459328
$717$	21.1873	0.791253
$718$	2.68685	0.100272
$719$	46.7614	1.74390	0.871952	−	0.489591i	$-0.162854\pi$
$719$	46.7614	1.74390	0.871952	+	0.489591i	$0.162854\pi$
$720$	0	0
$721$	0	0
$722$	−19.1428	−0.712420
$723$	−0.261489	−0.00972486
$724$	38.7791	1.44122
$725$	0	0
$726$	13.6686	0.507290
$727$	3.32783	0.123422	0.0617111	−	0.998094i	$-0.480344\pi$
$727$	3.32783	0.123422	0.0617111	+	0.998094i	$0.480344\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.84547	−0.179216
$732$	−14.6640	−0.541998
$733$	3.94316	0.145644	0.0728220	−	0.997345i	$-0.476800\pi$
$733$	3.94316	0.145644	0.0728220	+	0.997345i	$0.476800\pi$
$734$	−3.50506	−0.129374
$735$	0	0
$736$	8.82668	0.325356
$737$	48.9972	1.80484
$738$	0.0790296	0.00290912
$739$	−31.9859	−1.17662	−0.588310	−	0.808636i	$-0.700206\pi$
$739$	−31.9859	−1.17662	−0.588310	+	0.808636i	$0.700206\pi$
$740$	0	0
$741$	−44.5541	−1.63674
$742$	0	0
$743$	−16.4170	−0.602280	−0.301140	−	0.953580i	$-0.597367\pi$
$743$	−16.4170	−0.602280	−0.301140	+	0.953580i	$0.597367\pi$
$744$	−6.89137	−0.252650
$745$	0	0
$746$	−18.1615	−0.664941
$747$	−7.17157	−0.262394
$748$	−8.28788	−0.303035
$749$	0	0
$750$	0	0
$751$	11.5702	0.422203	0.211101	−	0.977464i	$-0.432295\pi$
$751$	11.5702	0.422203	0.211101	+	0.977464i	$0.432295\pi$
$752$	9.99745	0.364569
$753$	−19.7194	−0.718615
$754$	38.5667	1.40452
$755$	0	0
$756$	0	0
$757$	−11.9178	−0.433160	−0.216580	−	0.976265i	$-0.569490\pi$
$757$	−11.9178	−0.433160	−0.216580	+	0.976265i	$0.569490\pi$
$758$	8.11312	0.294682
$759$	8.61117	0.312566
$760$	0	0
$761$	14.1492	0.512908	0.256454	−	0.966556i	$-0.417446\pi$
$761$	14.1492	0.512908	0.256454	+	0.966556i	$0.417446\pi$
$762$	9.50214	0.344226
$763$	0	0
$764$	−10.9600	−0.396520
$765$	0	0
$766$	−8.96666	−0.323979
$767$	−35.3781	−1.27743
$768$	8.82118	0.318307
$769$	40.2097	1.45000	0.724999	−	0.688750i	$-0.241840\pi$
$769$	40.2097	1.45000	0.724999	+	0.688750i	$0.241840\pi$
$770$	0	0
$771$	−9.23412	−0.332559
$772$	13.7308	0.494183
$773$	−54.0618	−1.94447	−0.972233	−	0.234015i	$-0.924814\pi$
$773$	−54.0618	−1.94447	−0.972233	+	0.234015i	$0.924814\pi$
$774$	3.40216	0.122288
$775$	0	0
$776$	18.7740	0.673946
$777$	0	0
$778$	21.9702	0.787669
$779$	0.821368	0.0294285
$780$	0	0
$781$	−8.13921	−0.291244
$782$	0.965834	0.0345382
$783$	−8.99647	−0.321508
$784$	0	0
$785$	0	0
$786$	9.77196	0.348554
$787$	23.8775	0.851140	0.425570	−	0.904926i	$-0.360074\pi$
$787$	23.8775	0.851140	0.425570	+	0.904926i	$0.360074\pi$
$788$	35.0674	1.24922
$789$	−17.2524	−0.614203
$790$	0	0
$791$	0	0
$792$	13.2910	0.472274
$793$	60.6792	2.15478
$794$	13.9911	0.496527
$795$	0	0
$796$	12.9622	0.459434
$797$	−4.63001	−0.164003	−0.0820017	−	0.996632i	$-0.526131\pi$
$797$	−4.63001	−0.164003	−0.0820017	+	0.996632i	$0.526131\pi$
$798$	0	0
$799$	6.14319	0.217331
$800$	0	0
$801$	0.828427	0.0292710
$802$	−4.53686	−0.160202
$803$	43.7294	1.54318
$804$	13.5872	0.479185
$805$	0	0
$806$	12.4853	0.439775
$807$	−11.5210	−0.405558
$808$	4.20157	0.147811
$809$	−0.908017	−0.0319242	−0.0159621	−	0.999873i	$-0.505081\pi$
$809$	−0.908017	−0.0319242	−0.0159621	+	0.999873i	$0.505081\pi$
$810$	0	0
$811$	3.39773	0.119310	0.0596552	−	0.998219i	$-0.481000\pi$
$811$	3.39773	0.119310	0.0596552	+	0.998219i	$0.481000\pi$
$812$	0	0
$813$	18.2958	0.641660
$814$	23.0431	0.807660
$815$	0	0
$816$	−1.46024	−0.0511188
$817$	35.3592	1.23706
$818$	9.16964	0.320609
$819$	0	0
$820$	0	0
$821$	0.193951	0.00676892	0.00338446	−	0.999994i	$-0.498923\pi$
$821$	0.193951	0.00676892	0.00338446	+	0.999994i	$0.498923\pi$
$822$	−7.60668	−0.265313
$823$	30.2869	1.05573	0.527867	−	0.849327i	$-0.322992\pi$
$823$	30.2869	1.05573	0.527867	+	0.849327i	$0.322992\pi$
$824$	0.397620	0.0138517
$825$	0	0
$826$	0	0
$827$	−8.23006	−0.286187	−0.143094	−	0.989709i	$-0.545705\pi$
$827$	−8.23006	−0.286187	−0.143094	+	0.989709i	$0.545705\pi$
$828$	2.38793	0.0829864
$829$	−7.80286	−0.271005	−0.135502	−	0.990777i	$-0.543265\pi$
$829$	−7.80286	−0.271005	−0.135502	+	0.990777i	$0.543265\pi$
$830$	0	0
$831$	−14.8776	−0.516100
$832$	−4.81003	−0.166758
$833$	0	0
$834$	−12.0692	−0.417923
$835$	0	0
$836$	60.4798	2.09174
$837$	−2.91245	−0.100669
$838$	1.26003	0.0435269
$839$	40.9409	1.41344	0.706719	−	0.707494i	$-0.250174\pi$
$839$	40.9409	1.41344	0.706719	+	0.707494i	$0.250174\pi$
$840$	0	0
$841$	51.9365	1.79091
$842$	3.20003	0.110280
$843$	15.5443	0.535374
$844$	13.4788	0.463960
$845$	0	0
$846$	−4.31333	−0.148296
$847$	0	0
$848$	4.67289	0.160468
$849$	6.30019	0.216222
$850$	0	0
$851$	9.45584	0.324142
$852$	−2.25706	−0.0773255
$853$	45.0488	1.54244	0.771221	−	0.636568i	$-0.219646\pi$
$853$	45.0488	1.54244	0.771221	+	0.636568i	$0.219646\pi$
$854$	0	0
$855$	0	0
$856$	−13.4898	−0.461071
$857$	34.4727	1.17757	0.588783	−	0.808291i	$-0.299607\pi$
$857$	34.4727	1.17757	0.588783	+	0.808291i	$0.299607\pi$
$858$	−24.0796	−0.822064
$859$	6.43497	0.219558	0.109779	−	0.993956i	$-0.464986\pi$
$859$	6.43497	0.219558	0.109779	+	0.993956i	$0.464986\pi$
$860$	0	0
$861$	0	0
$862$	12.2508	0.417264
$863$	34.8663	1.18686	0.593432	−	0.804884i	$-0.297773\pi$
$863$	34.8663	1.18686	0.593432	+	0.804884i	$0.297773\pi$
$864$	5.75764	0.195879
$865$	0	0
$866$	9.81863	0.333651
$867$	16.1027	0.546877
$868$	0	0
$869$	−64.2172	−2.17842
$870$	0	0
$871$	−56.2235	−1.90506
$872$	−8.65276	−0.293020
$873$	7.93430	0.268535
$874$	−7.04805	−0.238404
$875$	0	0
$876$	12.1264	0.409715
$877$	−21.6961	−0.732625	−0.366312	−	0.930492i	$-0.619380\pi$
$877$	−21.6961	−0.732625	−0.366312	+	0.930492i	$0.619380\pi$
$878$	12.3331	0.416223
$879$	−6.65332	−0.224411
$880$	0	0
$881$	3.32077	0.111880	0.0559398	−	0.998434i	$-0.482185\pi$
$881$	3.32077	0.111880	0.0559398	+	0.998434i	$0.482185\pi$
$882$	0	0
$883$	9.89450	0.332977	0.166488	−	0.986043i	$-0.446757\pi$
$883$	9.89450	0.332977	0.166488	+	0.986043i	$0.446757\pi$
$884$	9.51021	0.319863
$885$	0	0
$886$	−17.3635	−0.583339
$887$	−33.6864	−1.13108	−0.565540	−	0.824721i	$-0.691332\pi$
$887$	−33.6864	−1.13108	−0.565540	+	0.824721i	$0.691332\pi$
$888$	14.5947	0.489765
$889$	0	0
$890$	0	0
$891$	5.61706	0.188179
$892$	−0.931221	−0.0311796
$893$	−44.8292	−1.50015
$894$	−5.47413	−0.183082
$895$	0	0
$896$	0	0
$897$	−9.88118	−0.329923
$898$	−26.4463	−0.882524
$899$	26.2018	0.873878
$900$	0	0
$901$	2.87138	0.0956595
$902$	0.443914	0.0147807
$903$	0	0
$904$	41.5953	1.38344
$905$	0	0
$906$	−6.51430	−0.216423
$907$	17.2082	0.571389	0.285695	−	0.958321i	$-0.407776\pi$
$907$	17.2082	0.571389	0.285695	+	0.958321i	$0.407776\pi$
$908$	1.19297	0.0395902
$909$	1.77568	0.0588955
$910$	0	0
$911$	−23.0551	−0.763850	−0.381925	−	0.924193i	$-0.624739\pi$
$911$	−23.0551	−0.763850	−0.381925	+	0.924193i	$0.624739\pi$
$912$	10.6560	0.352854
$913$	−40.2832	−1.33318
$914$	−5.46824	−0.180873
$915$	0	0
$916$	7.67754	0.253673
$917$	0	0
$918$	0.630013	0.0207935
$919$	7.13568	0.235384	0.117692	−	0.993050i	$-0.462450\pi$
$919$	7.13568	0.235384	0.117692	+	0.993050i	$0.462450\pi$
$920$	0	0
$921$	7.31371	0.240995
$922$	6.59213	0.217100
$923$	9.33962	0.307417
$924$	0	0
$925$	0	0
$926$	−11.1260	−0.365624
$927$	0.168043	0.00551926
$928$	−51.7984	−1.70037
$929$	2.58372	0.0847691	0.0423845	−	0.999101i	$-0.486505\pi$
$929$	2.58372	0.0847691	0.0423845	+	0.999101i	$0.486505\pi$
$930$	0	0
$931$	0	0
$932$	45.3179	1.48444
$933$	33.7686	1.10553
$934$	−19.1404	−0.626293
$935$	0	0
$936$	−15.2512	−0.498500
$937$	−6.03979	−0.197311	−0.0986557	−	0.995122i	$-0.531454\pi$
$937$	−6.03979	−0.197311	−0.0986557	+	0.995122i	$0.531454\pi$
$938$	0	0
$939$	17.9702	0.586435
$940$	0	0
$941$	−37.1027	−1.20951	−0.604757	−	0.796410i	$-0.706730\pi$
$941$	−37.1027	−1.20951	−0.604757	+	0.796410i	$0.706730\pi$
$942$	−5.38883	−0.175578
$943$	0.182162	0.00593202
$944$	8.46133	0.275393
$945$	0	0
$946$	19.1101	0.621324
$947$	−17.6315	−0.572946	−0.286473	−	0.958088i	$-0.592483\pi$
$947$	−17.6315	−0.572946	−0.286473	+	0.958088i	$0.592483\pi$
$948$	−17.8079	−0.578372
$949$	−50.1788	−1.62887
$950$	0	0
$951$	10.1769	0.330010
$952$	0	0
$953$	−22.0366	−0.713836	−0.356918	−	0.934136i	$-0.616172\pi$
$953$	−22.0366	−0.713836	−0.356918	+	0.934136i	$0.616172\pi$
$954$	−2.01609	−0.0652732
$955$	0	0
$956$	33.0023	1.06737
$957$	−50.5337	−1.63352
$958$	−9.45584	−0.305504
$959$	0	0
$960$	0	0
$961$	−22.5176	−0.726376
$962$	−26.4416	−0.852511
$963$	−5.70108	−0.183715
$964$	−0.407307	−0.0131185
$965$	0	0
$966$	0	0
$967$	32.7088	1.05184	0.525922	−	0.850533i	$-0.323720\pi$
$967$	32.7088	1.05184	0.525922	+	0.850533i	$0.323720\pi$
$968$	48.6281	1.56297
$969$	6.54783	0.210347
$970$	0	0
$971$	−11.4684	−0.368039	−0.184020	−	0.982923i	$-0.558911\pi$
$971$	−11.4684	−0.368039	−0.184020	+	0.982923i	$0.558911\pi$
$972$	1.55765	0.0499616
$973$	0	0
$974$	−6.36094	−0.203818
$975$	0	0
$976$	−14.5126	−0.464536
$977$	−61.1404	−1.95606	−0.978028	−	0.208474i	$-0.933150\pi$
$977$	−61.1404	−1.95606	−0.978028	+	0.208474i	$0.933150\pi$
$978$	−11.8641	−0.379373
$979$	4.65332	0.148721
$980$	0	0
$981$	−3.65685	−0.116754
$982$	16.6806	0.532299
$983$	14.0259	0.447357	0.223678	−	0.974663i	$-0.428193\pi$
$983$	14.0259	0.447357	0.223678	+	0.974663i	$0.428193\pi$
$984$	0.281160	0.00896304
$985$	0	0
$986$	−5.66790	−0.180503
$987$	0	0
$988$	−69.3996	−2.20789
$989$	7.84194	0.249359
$990$	0	0
$991$	10.5353	0.334665	0.167332	−	0.985901i	$-0.446485\pi$
$991$	10.5353	0.334665	0.167332	+	0.985901i	$0.446485\pi$
$992$	−16.7688	−0.532411
$993$	−8.31724	−0.263940
$994$	0	0
$995$	0	0
$996$	−11.1708	−0.353960
$997$	−36.8971	−1.16854	−0.584271	−	0.811559i	$-0.698619\pi$
$997$	−36.8971	−1.16854	−0.584271	+	0.811559i	$0.698619\pi$
$998$	−2.45356	−0.0776660
$999$	6.16804	0.195148

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bk.1.3		4
5.4	even	2		735.2.a.o.1.2	yes	4
7.6	odd	2		3675.2.a.bl.1.3		4
15.14	odd	2		2205.2.a.bg.1.3		4
35.4	even	6		735.2.i.m.226.3		8
35.9	even	6		735.2.i.m.361.3		8
35.19	odd	6		735.2.i.n.361.3		8
35.24	odd	6		735.2.i.n.226.3		8
35.34	odd	2		735.2.a.n.1.2	✓	4
105.104	even	2		2205.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.a.n.1.2	✓	4	35.34	odd	2
735.2.a.o.1.2	yes	4	5.4	even	2
735.2.i.m.226.3		8	35.4	even	6
735.2.i.m.361.3		8	35.9	even	6
735.2.i.n.226.3		8	35.24	odd	6
735.2.i.n.361.3		8	35.19	odd	6
2205.2.a.bf.1.3		4	105.104	even	2
2205.2.a.bg.1.3		4	15.14	odd	2
3675.2.a.bk.1.3		4	1.1	even	1	trivial
3675.2.a.bl.1.3		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.665096 q^{2} -1.00000 q^{3} -1.55765 q^{4} +0.665096 q^{6} +2.36618 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.665096 q^{2} -1.00000 q^{3} -1.55765 q^{4} +0.665096 q^{6} +2.36618 q^{8} +1.00000 q^{9} +5.61706 q^{11} +1.55765 q^{12} -6.44549 q^{13} +1.54156 q^{16} +0.947252 q^{17} -0.665096 q^{18} -6.91245 q^{19} -3.73588 q^{22} -1.53304 q^{23} -2.36618 q^{24} +4.28687 q^{26} -1.00000 q^{27} +8.99647 q^{29} +2.91245 q^{31} -5.75764 q^{32} -5.61706 q^{33} -0.630013 q^{34} -1.55765 q^{36} -6.16804 q^{37} +4.59744 q^{38} +6.44549 q^{39} -0.118824 q^{41} -5.11529 q^{43} -8.74940 q^{44} +1.01962 q^{46} +6.48528 q^{47} -1.54156 q^{48} -0.947252 q^{51} +10.0398 q^{52} +3.03127 q^{53} +0.665096 q^{54} +6.91245 q^{57} -5.98352 q^{58} +5.48881 q^{59} -9.41421 q^{61} -1.93706 q^{62} +0.746264 q^{64} +3.73588 q^{66} +8.72293 q^{67} -1.47548 q^{68} +1.53304 q^{69} -1.44902 q^{71} +2.36618 q^{72} +7.78510 q^{73} +4.10234 q^{74} +10.7672 q^{76} -4.28687 q^{78} -11.4325 q^{79} +1.00000 q^{81} +0.0790296 q^{82} -7.17157 q^{83} +3.40216 q^{86} -8.99647 q^{87} +13.2910 q^{88} +0.828427 q^{89} +2.38793 q^{92} -2.91245 q^{93} -4.31333 q^{94} +5.75764 q^{96} +7.93430 q^{97} +5.61706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{6} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^6 - 12 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 8 q^{4} + 4 q^{6} - 12 q^{8} + 4 q^{9} + 8 q^{11} - 8 q^{12} + 12 q^{16} - 8 q^{17} - 4 q^{18} - 8 q^{19} + 12 q^{24} - 4 q^{27} + 8 q^{29} - 8 q^{31} - 28 q^{32} - 8 q^{33} - 8 q^{34} + 8 q^{36} - 8 q^{37} - 4 q^{38} + 8 q^{43} - 16 q^{44} + 12 q^{46} - 8 q^{47} - 12 q^{48} + 8 q^{51} + 32 q^{52} - 8 q^{53} + 4 q^{54} + 8 q^{57} + 24 q^{58} + 16 q^{59} - 32 q^{61} + 20 q^{62} + 24 q^{64} - 24 q^{68} - 8 q^{71} - 12 q^{72} - 32 q^{74} + 8 q^{76} + 4 q^{81} + 8 q^{82} - 40 q^{83} - 32 q^{86} - 8 q^{87} + 40 q^{88} - 8 q^{89} + 8 q^{92} + 8 q^{93} - 16 q^{94} + 28 q^{96} + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 8 * q^4 + 4 * q^6 - 12 * q^8 + 4 * q^9 + 8 * q^11 - 8 * q^12 + 12 * q^16 - 8 * q^17 - 4 * q^18 - 8 * q^19 + 12 * q^24 - 4 * q^27 + 8 * q^29 - 8 * q^31 - 28 * q^32 - 8 * q^33 - 8 * q^34 + 8 * q^36 - 8 * q^37 - 4 * q^38 + 8 * q^43 - 16 * q^44 + 12 * q^46 - 8 * q^47 - 12 * q^48 + 8 * q^51 + 32 * q^52 - 8 * q^53 + 4 * q^54 + 8 * q^57 + 24 * q^58 + 16 * q^59 - 32 * q^61 + 20 * q^62 + 24 * q^64 - 24 * q^68 - 8 * q^71 - 12 * q^72 - 32 * q^74 + 8 * q^76 + 4 * q^81 + 8 * q^82 - 40 * q^83 - 32 * q^86 - 8 * q^87 + 40 * q^88 - 8 * q^89 + 8 * q^92 + 8 * q^93 - 16 * q^94 + 28 * q^96 + 8 * q^99

Introduction

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