LMFDB - Embedded newform 3675.2.a.ca.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.814115$ 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)

$f(q)$	$=$	$q+0.185885 q^{2} +1.00000 q^{3} -1.96545 q^{4} +0.185885 q^{6} -0.737118 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.185885 q^{2} +1.00000 q^{3} -1.96545 q^{4} +0.185885 q^{6} -0.737118 q^{8} +1.00000 q^{9} -5.19377 q^{11} -1.96545 q^{12} -2.77956 q^{13} +3.79387 q^{16} +2.56555 q^{17} +0.185885 q^{18} -5.08223 q^{19} -0.965447 q^{22} -2.09131 q^{23} -0.737118 q^{24} -0.516680 q^{26} +1.00000 q^{27} +4.66801 q^{29} -3.49910 q^{31} +2.17946 q^{32} -5.19377 q^{33} +0.476897 q^{34} -1.96545 q^{36} +6.20020 q^{37} -0.944711 q^{38} -2.77956 q^{39} +7.47424 q^{41} +10.4391 q^{43} +10.2081 q^{44} -0.388744 q^{46} +10.3387 q^{47} +3.79387 q^{48} +2.56555 q^{51} +5.46308 q^{52} -4.15399 q^{53} +0.185885 q^{54} -5.08223 q^{57} +0.867715 q^{58} -0.997340 q^{59} +10.8284 q^{61} -0.650431 q^{62} -7.18262 q^{64} -0.965447 q^{66} +5.94847 q^{67} -5.04244 q^{68} -2.09131 q^{69} -11.7991 q^{71} -0.737118 q^{72} -3.97600 q^{73} +1.15253 q^{74} +9.98884 q^{76} -0.516680 q^{78} +3.23888 q^{79} +1.00000 q^{81} +1.38935 q^{82} -16.7842 q^{83} +1.94047 q^{86} +4.66801 q^{87} +3.82843 q^{88} +4.68183 q^{89} +4.11036 q^{92} -3.49910 q^{93} +1.92181 q^{94} +2.17946 q^{96} +14.4476 q^{97} -5.19377 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9	$ 4 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{16} - 4 q^{17} + 2 q^{18} + 8 q^{19} + 6 q^{22} + 12 q^{26} + 4 q^{27} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{33} + 8 q^{34} + 2 q^{36} + 16 q^{37} + 4 q^{38} + 24 q^{41} + 20 q^{43} + 14 q^{44} - 6 q^{46} + 8 q^{47} - 6 q^{48} - 4 q^{51} + 16 q^{52} - 20 q^{53} + 2 q^{54} + 8 q^{57} - 6 q^{58} + 8 q^{59} + 32 q^{61} + 28 q^{62} - 12 q^{64} + 6 q^{66} + 12 q^{67} - 12 q^{68} + 4 q^{71} + 34 q^{74} + 40 q^{76} + 12 q^{78} + 4 q^{81} + 16 q^{82} - 20 q^{83} - 14 q^{86} - 4 q^{87} + 4 q^{88} + 8 q^{89} - 10 q^{92} + 8 q^{93} - 32 q^{94} + 2 q^{96} + 24 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9 - 4 * q^11 + 2 * q^12 - 6 * q^16 - 4 * q^17 + 2 * q^18 + 8 * q^19 + 6 * q^22 + 12 * q^26 + 4 * q^27 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^33 + 8 * q^34 + 2 * q^36 + 16 * q^37 + 4 * q^38 + 24 * q^41 + 20 * q^43 + 14 * q^44 - 6 * q^46 + 8 * q^47 - 6 * q^48 - 4 * q^51 + 16 * q^52 - 20 * q^53 + 2 * q^54 + 8 * q^57 - 6 * q^58 + 8 * q^59 + 32 * q^61 + 28 * q^62 - 12 * q^64 + 6 * q^66 + 12 * q^67 - 12 * q^68 + 4 * q^71 + 34 * q^74 + 40 * q^76 + 12 * q^78 + 4 * q^81 + 16 * q^82 - 20 * q^83 - 14 * q^86 - 4 * q^87 + 4 * q^88 + 8 * q^89 - 10 * q^92 + 8 * q^93 - 32 * q^94 + 2 * q^96 + 24 * q^97 - 4 * 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 $

$2$	0.185885	0.131441	0.0657204	−	0.997838i	$-0.479065\pi$
$2$	0.185885	0.131441	0.0657204	+	0.997838i	$0.479065\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.96545	−0.982723
$5$	0	0
$5$	0	0
$6$	0.185885	0.0758874
$7$	0	0
$7$	0	0
$8$	−0.737118	−0.260611
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.19377	−1.56598	−0.782991	−	0.622033i	$-0.786307\pi$
$11$	−5.19377	−1.56598	−0.782991	+	0.622033i	$0.786307\pi$
$12$	−1.96545	−0.567376
$13$	−2.77956	−0.770912	−0.385456	−	0.922726i	$-0.625956\pi$
$13$	−2.77956	−0.770912	−0.385456	+	0.922726i	$0.625956\pi$
$14$	0	0
$15$	0	0
$16$	3.79387	0.948468
$17$	2.56555	0.622236	0.311118	−	0.950371i	$-0.399297\pi$
$17$	2.56555	0.622236	0.311118	+	0.950371i	$0.399297\pi$
$18$	0.185885	0.0438136
$19$	−5.08223	−1.16594	−0.582971	−	0.812493i	$-0.698110\pi$
$19$	−5.08223	−1.16594	−0.582971	+	0.812493i	$0.698110\pi$
$20$	0	0
$21$	0	0
$22$	−0.965447	−0.205834
$23$	−2.09131	−0.436068	−0.218034	−	0.975941i	$-0.569964\pi$
$23$	−2.09131	−0.436068	−0.218034	+	0.975941i	$0.569964\pi$
$24$	−0.737118	−0.150464
$25$	0	0
$26$	−0.516680	−0.101329
$27$	1.00000	0.192450
$28$	0	0
$29$	4.66801	0.866828	0.433414	−	0.901195i	$-0.357309\pi$
$29$	4.66801	0.866828	0.433414	+	0.901195i	$0.357309\pi$
$30$	0	0
$31$	−3.49910	−0.628457	−0.314228	−	0.949347i	$-0.601746\pi$
$31$	−3.49910	−0.628457	−0.314228	+	0.949347i	$0.601746\pi$
$32$	2.17946	0.385278
$33$	−5.19377	−0.904120
$34$	0.476897	0.0817872
$35$	0	0
$36$	−1.96545	−0.327574
$37$	6.20020	1.01931	0.509653	−	0.860380i	$-0.329774\pi$
$37$	6.20020	1.01931	0.509653	+	0.860380i	$0.329774\pi$
$38$	−0.944711	−0.153252
$39$	−2.77956	−0.445086
$40$	0	0
$41$	7.47424	1.16728	0.583640	−	0.812013i	$-0.301628\pi$
$41$	7.47424	1.16728	0.583640	+	0.812013i	$0.301628\pi$
$42$	0	0
$43$	10.4391	1.59194	0.795972	−	0.605333i	$-0.206960\pi$
$43$	10.4391	1.59194	0.795972	+	0.605333i	$0.206960\pi$
$44$	10.2081	1.53893
$45$	0	0
$46$	−0.388744	−0.0573171
$47$	10.3387	1.50805	0.754026	−	0.656845i	$-0.228109\pi$
$47$	10.3387	1.50805	0.754026	+	0.656845i	$0.228109\pi$
$48$	3.79387	0.547599
$49$	0	0
$50$	0	0
$51$	2.56555	0.359248
$52$	5.46308	0.757593
$53$	−4.15399	−0.570595	−0.285297	−	0.958439i	$-0.592092\pi$
$53$	−4.15399	−0.570595	−0.285297	+	0.958439i	$0.592092\pi$
$54$	0.185885	0.0252958
$55$	0	0
$56$	0	0
$57$	−5.08223	−0.673157
$58$	0.867715	0.113937
$59$	−0.997340	−0.129843	−0.0649213	−	0.997890i	$-0.520680\pi$
$59$	−0.997340	−0.129843	−0.0649213	+	0.997890i	$0.520680\pi$
$60$	0	0
$61$	10.8284	1.38644	0.693219	−	0.720727i	$-0.256192\pi$
$61$	10.8284	1.38644	0.693219	+	0.720727i	$0.256192\pi$
$62$	−0.650431	−0.0826049
$63$	0	0
$64$	−7.18262	−0.897827
$65$	0	0
$66$	−0.965447	−0.118838
$67$	5.94847	0.726722	0.363361	−	0.931648i	$-0.381629\pi$
$67$	5.94847	0.726722	0.363361	+	0.931648i	$0.381629\pi$
$68$	−5.04244	−0.611486
$69$	−2.09131	−0.251764
$70$	0	0
$71$	−11.7991	−1.40030	−0.700148	−	0.713998i	$-0.746883\pi$
$71$	−11.7991	−1.40030	−0.700148	+	0.713998i	$0.746883\pi$
$72$	−0.737118	−0.0868702
$73$	−3.97600	−0.465355	−0.232678	−	0.972554i	$-0.574749\pi$
$73$	−3.97600	−0.465355	−0.232678	+	0.972554i	$0.574749\pi$
$74$	1.15253	0.133978
$75$	0	0
$76$	9.98884	1.14580
$77$	0	0
$78$	−0.516680	−0.0585025
$79$	3.23888	0.364402	0.182201	−	0.983261i	$-0.441678\pi$
$79$	3.23888	0.364402	0.182201	+	0.983261i	$0.441678\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.38935	0.153428
$83$	−16.7842	−1.84230	−0.921152	−	0.389204i	$-0.872750\pi$
$83$	−16.7842	−1.84230	−0.921152	+	0.389204i	$0.872750\pi$
$84$	0	0
$85$	0	0
$86$	1.94047	0.209246
$87$	4.66801	0.500463
$88$	3.82843	0.408112
$89$	4.68183	0.496273	0.248136	−	0.968725i	$-0.420182\pi$
$89$	4.68183	0.496273	0.248136	+	0.968725i	$0.420182\pi$
$90$	0	0
$91$	0	0
$92$	4.11036	0.428534
$93$	−3.49910	−0.362840
$94$	1.92181	0.198220
$95$	0	0
$96$	2.17946	0.222440
$97$	14.4476	1.46693	0.733464	−	0.679728i	$-0.237902\pi$
$97$	14.4476	1.46693	0.733464	+	0.679728i	$0.237902\pi$
$98$	0	0
$99$	−5.19377	−0.521994
$100$	0	0
$101$	13.3940	1.33275	0.666375	−	0.745617i	$-0.267845\pi$
$101$	13.3940	1.33275	0.666375	+	0.745617i	$0.267845\pi$
$102$	0.476897	0.0472199
$103$	−6.25380	−0.616205	−0.308103	−	0.951353i	$-0.599694\pi$
$103$	−6.25380	−0.616205	−0.308103	+	0.951353i	$0.599694\pi$
$104$	2.04887	0.200908
$105$	0	0
$106$	−0.772166	−0.0749994
$107$	20.0602	1.93929	0.969646	−	0.244515i	$-0.0786287\pi$
$107$	20.0602	1.93929	0.969646	+	0.244515i	$0.0786287\pi$
$108$	−1.96545	−0.189125
$109$	−4.94374	−0.473524	−0.236762	−	0.971568i	$-0.576086\pi$
$109$	−4.94374	−0.473524	−0.236762	+	0.971568i	$0.576086\pi$
$110$	0	0
$111$	6.20020	0.588497
$112$	0	0
$113$	6.27393	0.590201	0.295101	−	0.955466i	$-0.404647\pi$
$113$	6.27393	0.590201	0.295101	+	0.955466i	$0.404647\pi$
$114$	−0.944711	−0.0884803
$115$	0	0
$116$	−9.17473	−0.851852
$117$	−2.77956	−0.256971
$118$	−0.185391	−0.0170666
$119$	0	0
$120$	0	0
$121$	15.9753	1.45230
$122$	2.01285	0.182235
$123$	7.47424	0.673929
$124$	6.87729	0.617599
$125$	0	0
$126$	0	0
$127$	−8.51864	−0.755907	−0.377954	−	0.925825i	$-0.623372\pi$
$127$	−8.51864	−0.755907	−0.377954	+	0.925825i	$0.623372\pi$
$128$	−5.69407	−0.503289
$129$	10.4391	0.919109
$130$	0	0
$131$	9.61175	0.839783	0.419891	−	0.907574i	$-0.362068\pi$
$131$	9.61175	0.839783	0.419891	+	0.907574i	$0.362068\pi$
$132$	10.2081	0.888500
$133$	0	0
$134$	1.10573	0.0955209
$135$	0	0
$136$	−1.89111	−0.162161
$137$	−18.5415	−1.58411	−0.792055	−	0.610449i	$-0.790989\pi$
$137$	−18.5415	−1.58411	−0.792055	+	0.610449i	$0.790989\pi$
$138$	−0.388744	−0.0330921
$139$	11.4502	0.971196	0.485598	−	0.874182i	$-0.338602\pi$
$139$	11.4502	0.971196	0.485598	+	0.874182i	$0.338602\pi$
$140$	0	0
$141$	10.3387	0.870674
$142$	−2.19328	−0.184056
$143$	14.4364	1.20723
$144$	3.79387	0.316156
$145$	0	0
$146$	−0.739080	−0.0611667
$147$	0	0
$148$	−12.1862	−1.00170
$149$	0.834850	0.0683936	0.0341968	−	0.999415i	$-0.489113\pi$
$149$	0.834850	0.0683936	0.0341968	+	0.999415i	$0.489113\pi$
$150$	0	0
$151$	−9.78795	−0.796532	−0.398266	−	0.917270i	$-0.630388\pi$
$151$	−9.78795	−0.796532	−0.398266	+	0.917270i	$0.630388\pi$
$152$	3.74620	0.303857
$153$	2.56555	0.207412
$154$	0	0
$155$	0	0
$156$	5.46308	0.437396
$157$	−2.52044	−0.201153	−0.100577	−	0.994929i	$-0.532069\pi$
$157$	−2.52044	−0.201153	−0.100577	+	0.994929i	$0.532069\pi$
$158$	0.602060	0.0478973
$159$	−4.15399	−0.329433
$160$	0	0
$161$	0	0
$162$	0.185885	0.0146045
$163$	−23.8499	−1.86807	−0.934035	−	0.357181i	$-0.883738\pi$
$163$	−23.8499	−1.86807	−0.934035	+	0.357181i	$0.883738\pi$
$164$	−14.6902	−1.14711
$165$	0	0
$166$	−3.11993	−0.242154
$167$	−8.64135	−0.668688	−0.334344	−	0.942451i	$-0.608515\pi$
$167$	−8.64135	−0.668688	−0.334344	+	0.942451i	$0.608515\pi$
$168$	0	0
$169$	−5.27404	−0.405695
$170$	0	0
$171$	−5.08223	−0.388647
$172$	−20.5174	−1.56444
$173$	22.0399	1.67567	0.837833	−	0.545927i	$-0.183822\pi$
$173$	22.0399	1.67567	0.837833	+	0.545927i	$0.183822\pi$
$174$	0.867715	0.0657813
$175$	0	0
$176$	−19.7045	−1.48528
$177$	−0.997340	−0.0749646
$178$	0.870284	0.0652305
$179$	10.7816	0.805857	0.402929	−	0.915231i	$-0.367992\pi$
$179$	10.7816	0.805857	0.402929	+	0.915231i	$0.367992\pi$
$180$	0	0
$181$	22.4964	1.67215	0.836074	−	0.548617i	$-0.184846\pi$
$181$	22.4964	1.67215	0.836074	+	0.548617i	$0.184846\pi$
$182$	0	0
$183$	10.8284	0.800460
$184$	1.54154	0.113644
$185$	0	0
$186$	−0.650431	−0.0476919
$187$	−13.3249	−0.974411
$188$	−20.3201	−1.48200
$189$	0	0
$190$	0	0
$191$	25.1755	1.82163	0.910817	−	0.412809i	$-0.135452\pi$
$191$	25.1755	1.82163	0.910817	+	0.412809i	$0.135452\pi$
$192$	−7.18262	−0.518361
$193$	16.8489	1.21281	0.606407	−	0.795155i	$-0.292610\pi$
$193$	16.8489	1.21281	0.606407	+	0.795155i	$0.292610\pi$
$194$	2.68559	0.192814
$195$	0	0
$196$	0	0
$197$	−5.20662	−0.370956	−0.185478	−	0.982648i	$-0.559383\pi$
$197$	−5.20662	−0.370956	−0.185478	+	0.982648i	$0.559383\pi$
$198$	−0.965447	−0.0686113
$199$	11.4561	0.812099	0.406050	−	0.913851i	$-0.366906\pi$
$199$	11.4561	0.812099	0.406050	+	0.913851i	$0.366906\pi$
$200$	0	0
$201$	5.94847	0.419573
$202$	2.48974	0.175178
$203$	0	0
$204$	−5.04244	−0.353042
$205$	0	0
$206$	−1.16249	−0.0809945
$207$	−2.09131	−0.145356
$208$	−10.5453	−0.731185
$209$	26.3959	1.82584
$210$	0	0
$211$	−10.1698	−0.700116	−0.350058	−	0.936728i	$-0.613838\pi$
$211$	−10.1698	−0.700116	−0.350058	+	0.936728i	$0.613838\pi$
$212$	8.16445	0.560737
$213$	−11.7991	−0.808461
$214$	3.72889	0.254902
$215$	0	0
$216$	−0.737118	−0.0501546
$217$	0	0
$218$	−0.918969	−0.0622404
$219$	−3.97600	−0.268673
$220$	0	0
$221$	−7.13109	−0.479689
$222$	1.15253	0.0773525
$223$	20.8488	1.39614	0.698070	−	0.716029i	$-0.254042\pi$
$223$	20.8488	1.39614	0.698070	+	0.716029i	$0.254042\pi$
$224$	0	0
$225$	0	0
$226$	1.16623	0.0775765
$227$	−26.9217	−1.78686	−0.893428	−	0.449207i	$-0.851707\pi$
$227$	−26.9217	−1.78686	−0.893428	+	0.449207i	$0.851707\pi$
$228$	9.98884	0.661527
$229$	25.3257	1.67357	0.836786	−	0.547531i	$-0.184432\pi$
$229$	25.3257	1.67357	0.836786	+	0.547531i	$0.184432\pi$
$230$	0	0
$231$	0	0
$232$	−3.44088	−0.225905
$233$	−11.5813	−0.758718	−0.379359	−	0.925250i	$-0.623855\pi$
$233$	−11.5813	−0.758718	−0.379359	+	0.925250i	$0.623855\pi$
$234$	−0.516680	−0.0337764
$235$	0	0
$236$	1.96022	0.127599
$237$	3.23888	0.210388
$238$	0	0
$239$	3.97769	0.257295	0.128648	−	0.991690i	$-0.458936\pi$
$239$	3.97769	0.257295	0.128648	+	0.991690i	$0.458936\pi$
$240$	0	0
$241$	14.7614	0.950865	0.475433	−	0.879752i	$-0.342292\pi$
$241$	14.7614	0.950865	0.475433	+	0.879752i	$0.342292\pi$
$242$	2.96957	0.190891
$243$	1.00000	0.0641500
$244$	−21.2827	−1.36248
$245$	0	0
$246$	1.38935	0.0885818
$247$	14.1264	0.898839
$248$	2.57925	0.163783
$249$	−16.7842	−1.06365
$250$	0	0
$251$	−1.37166	−0.0865783	−0.0432891	−	0.999063i	$-0.513784\pi$
$251$	−1.37166	−0.0865783	−0.0432891	+	0.999063i	$0.513784\pi$
$252$	0	0
$253$	10.8618	0.682875
$254$	−1.58349	−0.0993570
$255$	0	0
$256$	13.3068	0.831674
$257$	24.0002	1.49709	0.748544	−	0.663085i	$-0.230753\pi$
$257$	24.0002	1.49709	0.748544	+	0.663085i	$0.230753\pi$
$258$	1.94047	0.120808
$259$	0	0
$260$	0	0
$261$	4.66801	0.288943
$262$	1.78668	0.110382
$263$	−19.4116	−1.19697	−0.598484	−	0.801135i	$-0.704230\pi$
$263$	−19.4116	−1.19697	−0.598484	+	0.801135i	$0.704230\pi$
$264$	3.82843	0.235623
$265$	0	0
$266$	0	0
$267$	4.68183	0.286523
$268$	−11.6914	−0.714166
$269$	11.3856	0.694192	0.347096	−	0.937830i	$-0.387168\pi$
$269$	11.3856	0.694192	0.347096	+	0.937830i	$0.387168\pi$
$270$	0	0
$271$	7.18528	0.436475	0.218237	−	0.975896i	$-0.429969\pi$
$271$	7.18528	0.436475	0.218237	+	0.975896i	$0.429969\pi$
$272$	9.73336	0.590171
$273$	0	0
$274$	−3.44660	−0.208217
$275$	0	0
$276$	4.11036	0.247414
$277$	9.00572	0.541101	0.270551	−	0.962706i	$-0.412794\pi$
$277$	9.00572	0.541101	0.270551	+	0.962706i	$0.412794\pi$
$278$	2.12843	0.127655
$279$	−3.49910	−0.209486
$280$	0	0
$281$	−2.39929	−0.143130	−0.0715649	−	0.997436i	$-0.522799\pi$
$281$	−2.39929	−0.143130	−0.0715649	+	0.997436i	$0.522799\pi$
$282$	1.92181	0.114442
$283$	−6.74077	−0.400697	−0.200349	−	0.979725i	$-0.564207\pi$
$283$	−6.74077	−0.400697	−0.200349	+	0.979725i	$0.564207\pi$
$284$	23.1905	1.37610
$285$	0	0
$286$	2.68352	0.158680
$287$	0	0
$288$	2.17946	0.128426
$289$	−10.4180	−0.612822
$290$	0	0
$291$	14.4476	0.846932
$292$	7.81461	0.457315
$293$	−29.7096	−1.73566	−0.867828	−	0.496865i	$-0.834484\pi$
$293$	−29.7096	−1.73566	−0.867828	+	0.496865i	$0.834484\pi$
$294$	0	0
$295$	0	0
$296$	−4.57028	−0.265642
$297$	−5.19377	−0.301373
$298$	0.155186	0.00898971
$299$	5.81292	0.336170
$300$	0	0
$301$	0	0
$302$	−1.81944	−0.104697
$303$	13.3940	0.769464
$304$	−19.2813	−1.10586
$305$	0	0
$306$	0.476897	0.0272624
$307$	12.6609	0.722595	0.361298	−	0.932451i	$-0.382334\pi$
$307$	12.6609	0.722595	0.361298	+	0.932451i	$0.382334\pi$
$308$	0	0
$309$	−6.25380	−0.355766
$310$	0	0
$311$	16.4899	0.935057	0.467528	−	0.883978i	$-0.345145\pi$
$311$	16.4899	0.935057	0.467528	+	0.883978i	$0.345145\pi$
$312$	2.04887	0.115994
$313$	−18.7391	−1.05920	−0.529598	−	0.848249i	$-0.677657\pi$
$313$	−18.7391	−1.05920	−0.529598	+	0.848249i	$0.677657\pi$
$314$	−0.468513	−0.0264397
$315$	0	0
$316$	−6.36584	−0.358107
$317$	−29.1065	−1.63478	−0.817392	−	0.576082i	$-0.804581\pi$
$317$	−29.1065	−1.63478	−0.817392	+	0.576082i	$0.804581\pi$
$318$	−0.772166	−0.0433009
$319$	−24.2446	−1.35744
$320$	0	0
$321$	20.0602	1.11965
$322$	0	0
$323$	−13.0387	−0.725492
$324$	−1.96545	−0.109191
$325$	0	0
$326$	−4.43335	−0.245541
$327$	−4.94374	−0.273389
$328$	−5.50940	−0.304206
$329$	0	0
$330$	0	0
$331$	6.20020	0.340794	0.170397	−	0.985376i	$-0.445495\pi$
$331$	6.20020	0.340794	0.170397	+	0.985376i	$0.445495\pi$
$332$	32.9884	1.81047
$333$	6.20020	0.339769
$334$	−1.60630	−0.0878928
$335$	0	0
$336$	0	0
$337$	5.76112	0.313828	0.156914	−	0.987612i	$-0.449845\pi$
$337$	5.76112	0.313828	0.156914	+	0.987612i	$0.449845\pi$
$338$	−0.980367	−0.0533249
$339$	6.27393	0.340753
$340$	0	0
$341$	18.1735	0.984152
$342$	−0.944711	−0.0510841
$343$	0	0
$344$	−7.69484	−0.414878
$345$	0	0
$346$	4.09690	0.220251
$347$	−9.73238	−0.522462	−0.261231	−	0.965276i	$-0.584128\pi$
$347$	−9.73238	−0.522462	−0.261231	+	0.965276i	$0.584128\pi$
$348$	−9.17473	−0.491817
$349$	7.56358	0.404869	0.202435	−	0.979296i	$-0.435115\pi$
$349$	7.56358	0.404869	0.202435	+	0.979296i	$0.435115\pi$
$350$	0	0
$351$	−2.77956	−0.148362
$352$	−11.3196	−0.603339
$353$	−0.564686	−0.0300552	−0.0150276	−	0.999887i	$-0.504784\pi$
$353$	−0.564686	−0.0300552	−0.0150276	+	0.999887i	$0.504784\pi$
$354$	−0.185391	−0.00985341
$355$	0	0
$356$	−9.20189	−0.487699
$357$	0	0
$358$	2.00415	0.105923
$359$	5.77048	0.304554	0.152277	−	0.988338i	$-0.451339\pi$
$359$	5.77048	0.304554	0.152277	+	0.988338i	$0.451339\pi$
$360$	0	0
$361$	6.82901	0.359422
$362$	4.18176	0.219788
$363$	15.9753	0.838486
$364$	0	0
$365$	0	0
$366$	2.01285	0.105213
$367$	−14.1329	−0.737731	−0.368865	−	0.929483i	$-0.620254\pi$
$367$	−14.1329	−0.737731	−0.368865	+	0.929483i	$0.620254\pi$
$368$	−7.93416	−0.413597
$369$	7.47424	0.389093
$370$	0	0
$371$	0	0
$372$	6.87729	0.356571
$373$	33.1322	1.71552	0.857761	−	0.514049i	$-0.171855\pi$
$373$	33.1322	1.71552	0.857761	+	0.514049i	$0.171855\pi$
$374$	−2.47690	−0.128077
$375$	0	0
$376$	−7.62083	−0.393015
$377$	−12.9750	−0.668248
$378$	0	0
$379$	15.8219	0.812716	0.406358	−	0.913714i	$-0.366799\pi$
$379$	15.8219	0.812716	0.406358	+	0.913714i	$0.366799\pi$
$380$	0	0
$381$	−8.51864	−0.436423
$382$	4.67976	0.239437
$383$	6.57463	0.335948	0.167974	−	0.985791i	$-0.446278\pi$
$383$	6.57463	0.335948	0.167974	+	0.985791i	$0.446278\pi$
$384$	−5.69407	−0.290574
$385$	0	0
$386$	3.13197	0.159413
$387$	10.4391	0.530648
$388$	−28.3959	−1.44159
$389$	−12.1973	−0.618427	−0.309214	−	0.950993i	$-0.600066\pi$
$389$	−12.1973	−0.618427	−0.309214	+	0.950993i	$0.600066\pi$
$390$	0	0
$391$	−5.36535	−0.271337
$392$	0	0
$393$	9.61175	0.484849
$394$	−0.967835	−0.0487588
$395$	0	0
$396$	10.2081	0.512976
$397$	12.2169	0.613151	0.306576	−	0.951846i	$-0.400817\pi$
$397$	12.2169	0.613151	0.306576	+	0.951846i	$0.400817\pi$
$398$	2.12952	0.106743
$399$	0	0
$400$	0	0
$401$	−11.9817	−0.598339	−0.299169	−	0.954200i	$-0.596710\pi$
$401$	−11.9817	−0.598339	−0.299169	+	0.954200i	$0.596710\pi$
$402$	1.10573	0.0551490
$403$	9.72596	0.484485
$404$	−26.3251	−1.30972
$405$	0	0
$406$	0	0
$407$	−32.2024	−1.59622
$408$	−1.89111	−0.0936239
$409$	2.92530	0.144647	0.0723234	−	0.997381i	$-0.476959\pi$
$409$	2.92530	0.144647	0.0723234	+	0.997381i	$0.476959\pi$
$410$	0	0
$411$	−18.5415	−0.914587
$412$	12.2915	0.605559
$413$	0	0
$414$	−0.388744	−0.0191057
$415$	0	0
$416$	−6.05795	−0.297015
$417$	11.4502	0.560720
$418$	4.90662	0.239990
$419$	33.8748	1.65489	0.827446	−	0.561545i	$-0.189793\pi$
$419$	33.8748	1.65489	0.827446	+	0.561545i	$0.189793\pi$
$420$	0	0
$421$	20.3489	0.991743	0.495872	−	0.868396i	$-0.334849\pi$
$421$	20.3489	0.991743	0.495872	+	0.868396i	$0.334849\pi$
$422$	−1.89041	−0.0920238
$423$	10.3387	0.502684
$424$	3.06198	0.148703
$425$	0	0
$426$	−2.19328	−0.106265
$427$	0	0
$428$	−39.4272	−1.90579
$429$	14.4364	0.696997
$430$	0	0
$431$	−32.9184	−1.58563	−0.792813	−	0.609465i	$-0.791384\pi$
$431$	−32.9184	−1.58563	−0.792813	+	0.609465i	$0.791384\pi$
$432$	3.79387	0.182533
$433$	9.22130	0.443147	0.221574	−	0.975144i	$-0.428881\pi$
$433$	9.22130	0.443147	0.221574	+	0.975144i	$0.428881\pi$
$434$	0	0
$435$	0	0
$436$	9.71666	0.465343
$437$	10.6285	0.508430
$438$	−0.739080	−0.0353146
$439$	−29.2190	−1.39455	−0.697274	−	0.716804i	$-0.745604\pi$
$439$	−29.2190	−1.39455	−0.697274	+	0.716804i	$0.745604\pi$
$440$	0	0
$441$	0	0
$442$	−1.32557	−0.0630507
$443$	−20.2493	−0.962075	−0.481037	−	0.876700i	$-0.659740\pi$
$443$	−20.2493	−0.962075	−0.481037	+	0.876700i	$0.659740\pi$
$444$	−12.1862	−0.578329
$445$	0	0
$446$	3.87549	0.183510
$447$	0.834850	0.0394871
$448$	0	0
$449$	31.2030	1.47256	0.736281	−	0.676676i	$-0.236580\pi$
$449$	31.2030	1.47256	0.736281	+	0.676676i	$0.236580\pi$
$450$	0	0
$451$	−38.8195	−1.82794
$452$	−12.3311	−0.580005
$453$	−9.78795	−0.459878
$454$	−5.00435	−0.234866
$455$	0	0
$456$	3.74620	0.175432
$457$	13.3875	0.626243	0.313122	−	0.949713i	$-0.398625\pi$
$457$	13.3875	0.626243	0.313122	+	0.949713i	$0.398625\pi$
$458$	4.70768	0.219975
$459$	2.56555	0.119749
$460$	0	0
$461$	−37.1880	−1.73202	−0.866010	−	0.500027i	$-0.833323\pi$
$461$	−37.1880	−1.73202	−0.866010	+	0.500027i	$0.833323\pi$
$462$	0	0
$463$	30.3980	1.41272	0.706358	−	0.707855i	$-0.250337\pi$
$463$	30.3980	1.41272	0.706358	+	0.707855i	$0.250337\pi$
$464$	17.7098	0.822159
$465$	0	0
$466$	−2.15280	−0.0997265
$467$	−34.6811	−1.60485	−0.802426	−	0.596752i	$-0.796458\pi$
$467$	−34.6811	−1.60485	−0.802426	+	0.596752i	$0.796458\pi$
$468$	5.46308	0.252531
$469$	0	0
$470$	0	0
$471$	−2.52044	−0.116136
$472$	0.735157	0.0338384
$473$	−54.2182	−2.49296
$474$	0.602060	0.0276535
$475$	0	0
$476$	0	0
$477$	−4.15399	−0.190198
$478$	0.739393	0.0338191
$479$	−28.8279	−1.31718	−0.658590	−	0.752502i	$-0.728847\pi$
$479$	−28.8279	−1.31718	−0.658590	+	0.752502i	$0.728847\pi$
$480$	0	0
$481$	−17.2338	−0.785795
$482$	2.74393	0.124982
$483$	0	0
$484$	−31.3986	−1.42721
$485$	0	0
$486$	0.185885	0.00843193
$487$	14.7483	0.668308	0.334154	−	0.942518i	$-0.391549\pi$
$487$	14.7483	0.668308	0.334154	+	0.942518i	$0.391549\pi$
$488$	−7.98183	−0.361321
$489$	−23.8499	−1.07853
$490$	0	0
$491$	12.3850	0.558929	0.279465	−	0.960156i	$-0.409843\pi$
$491$	12.3850	0.558929	0.279465	+	0.960156i	$0.409843\pi$
$492$	−14.6902	−0.662286
$493$	11.9760	0.539372
$494$	2.62588	0.118144
$495$	0	0
$496$	−13.2751	−0.596071
$497$	0	0
$498$	−3.11993	−0.139808
$499$	0.686292	0.0307226	0.0153613	−	0.999882i	$-0.495110\pi$
$499$	0.686292	0.0307226	0.0153613	+	0.999882i	$0.495110\pi$
$500$	0	0
$501$	−8.64135	−0.386067
$502$	−0.254971	−0.0113799
$503$	−19.7717	−0.881578	−0.440789	−	0.897611i	$-0.645301\pi$
$503$	−19.7717	−0.881578	−0.440789	+	0.897611i	$0.645301\pi$
$504$	0	0
$505$	0	0
$506$	2.01905	0.0897576
$507$	−5.27404	−0.234228
$508$	16.7429	0.742848
$509$	−38.4995	−1.70646	−0.853231	−	0.521533i	$-0.825360\pi$
$509$	−38.4995	−1.70646	−0.853231	+	0.521533i	$0.825360\pi$
$510$	0	0
$511$	0	0
$512$	13.8617	0.612605
$513$	−5.08223	−0.224386
$514$	4.46128	0.196779
$515$	0	0
$516$	−20.5174	−0.903230
$517$	−53.6968	−2.36158
$518$	0	0
$519$	22.0399	0.967446
$520$	0	0
$521$	4.84283	0.212168	0.106084	−	0.994357i	$-0.466169\pi$
$521$	4.84283	0.212168	0.106084	+	0.994357i	$0.466169\pi$
$522$	0.867715	0.0379789
$523$	38.7441	1.69416	0.847080	−	0.531465i	$-0.178358\pi$
$523$	38.7441	1.69416	0.847080	+	0.531465i	$0.178358\pi$
$524$	−18.8914	−0.825274
$525$	0	0
$526$	−3.60832	−0.157330
$527$	−8.97710	−0.391049
$528$	−19.7045	−0.857529
$529$	−18.6264	−0.809845
$530$	0	0
$531$	−0.997340	−0.0432808
$532$	0	0
$533$	−20.7751	−0.899869
$534$	0.870284	0.0376609
$535$	0	0
$536$	−4.38473	−0.189391
$537$	10.7816	0.465262
$538$	2.11641	0.0912451
$539$	0	0
$540$	0	0
$541$	−34.6766	−1.49086	−0.745431	−	0.666583i	$-0.767756\pi$
$541$	−34.6766	−1.49086	−0.745431	+	0.666583i	$0.767756\pi$
$542$	1.33564	0.0573706
$543$	22.4964	0.965415
$544$	5.59151	0.239734
$545$	0	0
$546$	0	0
$547$	18.4495	0.788845	0.394423	−	0.918929i	$-0.370945\pi$
$547$	18.4495	0.788845	0.394423	+	0.918929i	$0.370945\pi$
$548$	36.4424	1.55674
$549$	10.8284	0.462146
$550$	0	0
$551$	−23.7239	−1.01067
$552$	1.54154	0.0656124
$553$	0	0
$554$	1.67403	0.0711228
$555$	0	0
$556$	−22.5048	−0.954417
$557$	4.52326	0.191657	0.0958284	−	0.995398i	$-0.469450\pi$
$557$	4.52326	0.191657	0.0958284	+	0.995398i	$0.469450\pi$
$558$	−0.650431	−0.0275350
$559$	−29.0161	−1.22725
$560$	0	0
$561$	−13.3249	−0.562576
$562$	−0.445994	−0.0188131
$563$	37.8175	1.59382	0.796910	−	0.604098i	$-0.206467\pi$
$563$	37.8175	1.59382	0.796910	+	0.604098i	$0.206467\pi$
$564$	−20.3201	−0.855632
$565$	0	0
$566$	−1.25301	−0.0526679
$567$	0	0
$568$	8.69734	0.364932
$569$	−26.1930	−1.09807	−0.549033	−	0.835801i	$-0.685004\pi$
$569$	−26.1930	−1.09807	−0.549033	+	0.835801i	$0.685004\pi$
$570$	0	0
$571$	0.827841	0.0346441	0.0173220	−	0.999850i	$-0.494486\pi$
$571$	0.827841	0.0346441	0.0173220	+	0.999850i	$0.494486\pi$
$572$	−28.3740	−1.18638
$573$	25.1755	1.05172
$574$	0	0
$575$	0	0
$576$	−7.18262	−0.299276
$577$	4.49367	0.187074	0.0935369	−	0.995616i	$-0.470183\pi$
$577$	4.49367	0.187074	0.0935369	+	0.995616i	$0.470183\pi$
$578$	−1.93655	−0.0805498
$579$	16.8489	0.700218
$580$	0	0
$581$	0	0
$582$	2.68559	0.111321
$583$	21.5749	0.893541
$584$	2.93078	0.121277
$585$	0	0
$586$	−5.52259	−0.228136
$587$	4.83837	0.199701	0.0998504	−	0.995002i	$-0.468164\pi$
$587$	4.83837	0.199701	0.0998504	+	0.995002i	$0.468164\pi$
$588$	0	0
$589$	17.7832	0.732744
$590$	0	0
$591$	−5.20662	−0.214172
$592$	23.5228	0.966780
$593$	−7.35834	−0.302171	−0.151085	−	0.988521i	$-0.548277\pi$
$593$	−7.35834	−0.302171	−0.151085	+	0.988521i	$0.548277\pi$
$594$	−0.965447	−0.0396128
$595$	0	0
$596$	−1.64085	−0.0672120
$597$	11.4561	0.468866
$598$	1.08054	0.0441864
$599$	−1.98172	−0.0809709	−0.0404854	−	0.999180i	$-0.512890\pi$
$599$	−1.98172	−0.0809709	−0.0404854	+	0.999180i	$0.512890\pi$
$600$	0	0
$601$	11.5630	0.471666	0.235833	−	0.971794i	$-0.424218\pi$
$601$	11.5630	0.471666	0.235833	+	0.971794i	$0.424218\pi$
$602$	0	0
$603$	5.94847	0.242241
$604$	19.2377	0.782770
$605$	0	0
$606$	2.48974	0.101139
$607$	9.34973	0.379494	0.189747	−	0.981833i	$-0.439233\pi$
$607$	9.34973	0.379494	0.189747	+	0.981833i	$0.439233\pi$
$608$	−11.0765	−0.449212
$609$	0	0
$610$	0	0
$611$	−28.7370	−1.16257
$612$	−5.04244	−0.203829
$613$	−17.5583	−0.709173	−0.354587	−	0.935023i	$-0.615378\pi$
$613$	−17.5583	−0.709173	−0.354587	+	0.935023i	$0.615378\pi$
$614$	2.35347	0.0949785
$615$	0	0
$616$	0	0
$617$	−23.4974	−0.945971	−0.472985	−	0.881070i	$-0.656824\pi$
$617$	−23.4974	−0.945971	−0.472985	+	0.881070i	$0.656824\pi$
$618$	−1.16249	−0.0467622
$619$	38.5786	1.55060	0.775301	−	0.631591i	$-0.217598\pi$
$619$	38.5786	1.55060	0.775301	+	0.631591i	$0.217598\pi$
$620$	0	0
$621$	−2.09131	−0.0839213
$622$	3.06523	0.122905
$623$	0	0
$624$	−10.5453	−0.422150
$625$	0	0
$626$	−3.48332	−0.139221
$627$	26.3959	1.05415
$628$	4.95379	0.197678
$629$	15.9069	0.634249
$630$	0	0
$631$	41.2503	1.64215	0.821075	−	0.570821i	$-0.193375\pi$
$631$	41.2503	1.64215	0.821075	+	0.570821i	$0.193375\pi$
$632$	−2.38744	−0.0949671
$633$	−10.1698	−0.404212
$634$	−5.41047	−0.214877
$635$	0	0
$636$	8.16445	0.323742
$637$	0	0
$638$	−4.50672	−0.178423
$639$	−11.7991	−0.466765
$640$	0	0
$641$	28.2567	1.11607	0.558037	−	0.829816i	$-0.311555\pi$
$641$	28.2567	1.11607	0.558037	+	0.829816i	$0.311555\pi$
$642$	3.72889	0.147168
$643$	22.0128	0.868102	0.434051	−	0.900888i	$-0.357084\pi$
$643$	22.0128	0.868102	0.434051	+	0.900888i	$0.357084\pi$
$644$	0	0
$645$	0	0
$646$	−2.42370	−0.0953592
$647$	10.6918	0.420337	0.210168	−	0.977665i	$-0.432599\pi$
$647$	10.6918	0.420337	0.210168	+	0.977665i	$0.432599\pi$
$648$	−0.737118	−0.0289567
$649$	5.17996	0.203331
$650$	0	0
$651$	0	0
$652$	46.8758	1.83580
$653$	6.61164	0.258733	0.129367	−	0.991597i	$-0.458706\pi$
$653$	6.61164	0.258733	0.129367	+	0.991597i	$0.458706\pi$
$654$	−0.918969	−0.0359345
$655$	0	0
$656$	28.3563	1.10713
$657$	−3.97600	−0.155118
$658$	0	0
$659$	28.5086	1.11054	0.555268	−	0.831671i	$-0.312616\pi$
$659$	28.5086	1.11054	0.555268	+	0.831671i	$0.312616\pi$
$660$	0	0
$661$	26.3334	1.02425	0.512125	−	0.858911i	$-0.328859\pi$
$661$	26.3334	1.02425	0.512125	+	0.858911i	$0.328859\pi$
$662$	1.15253	0.0447942
$663$	−7.13109	−0.276949
$664$	12.3719	0.480124
$665$	0	0
$666$	1.15253	0.0446595
$667$	−9.76225	−0.377996
$668$	16.9841	0.657135
$669$	20.8488	0.806062
$670$	0	0
$671$	−56.2404	−2.17114
$672$	0	0
$673$	−9.49655	−0.366065	−0.183033	−	0.983107i	$-0.558591\pi$
$673$	−9.49655	−0.366065	−0.183033	+	0.983107i	$0.558591\pi$
$674$	1.07091	0.0412498
$675$	0	0
$676$	10.3658	0.398686
$677$	−9.15496	−0.351854	−0.175927	−	0.984403i	$-0.556292\pi$
$677$	−9.15496	−0.351854	−0.175927	+	0.984403i	$0.556292\pi$
$678$	1.16623	0.0447888
$679$	0	0
$680$	0	0
$681$	−26.9217	−1.03164
$682$	3.37819	0.129358
$683$	−5.84491	−0.223649	−0.111825	−	0.993728i	$-0.535669\pi$
$683$	−5.84491	−0.223649	−0.111825	+	0.993728i	$0.535669\pi$
$684$	9.98884	0.381933
$685$	0	0
$686$	0	0
$687$	25.3257	0.966237
$688$	39.6045	1.50991
$689$	11.5463	0.439878
$690$	0	0
$691$	−4.06280	−0.154556	−0.0772780	−	0.997010i	$-0.524623\pi$
$691$	−4.06280	−0.154556	−0.0772780	+	0.997010i	$0.524623\pi$
$692$	−43.3183	−1.64672
$693$	0	0
$694$	−1.80911	−0.0686728
$695$	0	0
$696$	−3.44088	−0.130426
$697$	19.1755	0.726324
$698$	1.40596	0.0532164
$699$	−11.5813	−0.438046
$700$	0	0
$701$	3.76284	0.142121	0.0710603	−	0.997472i	$-0.477362\pi$
$701$	3.76284	0.142121	0.0710603	+	0.997472i	$0.477362\pi$
$702$	−0.516680	−0.0195008
$703$	−31.5108	−1.18845
$704$	37.3049	1.40598
$705$	0	0
$706$	−0.104967	−0.00395048
$707$	0	0
$708$	1.96022	0.0736695
$709$	−27.8582	−1.04624	−0.523118	−	0.852260i	$-0.675231\pi$
$709$	−27.8582	−1.04624	−0.523118	+	0.852260i	$0.675231\pi$
$710$	0	0
$711$	3.23888	0.121467
$712$	−3.45106	−0.129334
$713$	7.31770	0.274050
$714$	0	0
$715$	0	0
$716$	−21.1907	−0.791935
$717$	3.97769	0.148549
$718$	1.07265	0.0400308
$719$	−12.5481	−0.467964	−0.233982	−	0.972241i	$-0.575176\pi$
$719$	−12.5481	−0.467964	−0.233982	+	0.972241i	$0.575176\pi$
$720$	0	0
$721$	0	0
$722$	1.26941	0.0472427
$723$	14.7614	0.548982
$724$	−44.2156	−1.64326
$725$	0	0
$726$	2.96957	0.110211
$727$	−7.44099	−0.275971	−0.137986	−	0.990434i	$-0.544063\pi$
$727$	−7.44099	−0.275971	−0.137986	+	0.990434i	$0.544063\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	26.7819	0.990565
$732$	−21.2827	−0.786631
$733$	−17.1013	−0.631649	−0.315825	−	0.948818i	$-0.602281\pi$
$733$	−17.1013	−0.631649	−0.315825	+	0.948818i	$0.602281\pi$
$734$	−2.62710	−0.0969679
$735$	0	0
$736$	−4.55793	−0.168007
$737$	−30.8950	−1.13803
$738$	1.38935	0.0511427
$739$	32.5514	1.19742	0.598712	−	0.800965i	$-0.295680\pi$
$739$	32.5514	1.19742	0.598712	+	0.800965i	$0.295680\pi$
$740$	0	0
$741$	14.1264	0.518945
$742$	0	0
$743$	0.890998	0.0326875	0.0163438	−	0.999866i	$-0.494797\pi$
$743$	0.890998	0.0326875	0.0163438	+	0.999866i	$0.494797\pi$
$744$	2.57925	0.0945599
$745$	0	0
$746$	6.15879	0.225489
$747$	−16.7842	−0.614101
$748$	26.1893	0.957576
$749$	0	0
$750$	0	0
$751$	−45.9273	−1.67591	−0.837955	−	0.545739i	$-0.816249\pi$
$751$	−45.9273	−1.67591	−0.837955	+	0.545739i	$0.816249\pi$
$752$	39.2237	1.43034
$753$	−1.37166	−0.0499860
$754$	−2.41187	−0.0878350
$755$	0	0
$756$	0	0
$757$	−14.1311	−0.513603	−0.256802	−	0.966464i	$-0.582669\pi$
$757$	−14.1311	−0.513603	−0.256802	+	0.966464i	$0.582669\pi$
$758$	2.94106	0.106824
$759$	10.8618	0.394258
$760$	0	0
$761$	7.83023	0.283846	0.141923	−	0.989878i	$-0.454672\pi$
$761$	7.83023	0.283846	0.141923	+	0.989878i	$0.454672\pi$
$762$	−1.58349	−0.0573638
$763$	0	0
$764$	−49.4811	−1.79016
$765$	0	0
$766$	1.22213	0.0441573
$767$	2.77217	0.100097
$768$	13.3068	0.480167
$769$	−33.0278	−1.19101	−0.595507	−	0.803350i	$-0.703049\pi$
$769$	−33.0278	−1.19101	−0.595507	+	0.803350i	$0.703049\pi$
$770$	0	0
$771$	24.0002	0.864345
$772$	−33.1157	−1.19186
$773$	22.0652	0.793629	0.396814	−	0.917899i	$-0.370116\pi$
$773$	22.0652	0.793629	0.396814	+	0.917899i	$0.370116\pi$
$774$	1.94047	0.0697488
$775$	0	0
$776$	−10.6496	−0.382297
$777$	0	0
$778$	−2.26730	−0.0812866
$779$	−37.9858	−1.36098
$780$	0	0
$781$	61.2819	2.19284
$782$	−0.997340	−0.0356648
$783$	4.66801	0.166821
$784$	0	0
$785$	0	0
$786$	1.78668	0.0637289
$787$	−10.9924	−0.391835	−0.195918	−	0.980620i	$-0.562769\pi$
$787$	−10.9924	−0.391835	−0.195918	+	0.980620i	$0.562769\pi$
$788$	10.2333	0.364547
$789$	−19.4116	−0.691070
$790$	0	0
$791$	0	0
$792$	3.82843	0.136037
$793$	−30.0983	−1.06882
$794$	2.27095	0.0805931
$795$	0	0
$796$	−22.5163	−0.798069
$797$	−22.9051	−0.811340	−0.405670	−	0.914020i	$-0.632962\pi$
$797$	−22.9051	−0.811340	−0.405670	+	0.914020i	$0.632962\pi$
$798$	0	0
$799$	26.5244	0.938365
$800$	0	0
$801$	4.68183	0.165424
$802$	−2.22723	−0.0786461
$803$	20.6504	0.728738
$804$	−11.6914	−0.412324
$805$	0	0
$806$	1.80791	0.0636810
$807$	11.3856	0.400792
$808$	−9.87294	−0.347329
$809$	36.2446	1.27429	0.637147	−	0.770743i	$-0.280115\pi$
$809$	36.2446	1.27429	0.637147	+	0.770743i	$0.280115\pi$
$810$	0	0
$811$	4.08626	0.143488	0.0717440	−	0.997423i	$-0.477144\pi$
$811$	4.08626	0.143488	0.0717440	+	0.997423i	$0.477144\pi$
$812$	0	0
$813$	7.18528	0.251999
$814$	−5.98596	−0.209808
$815$	0	0
$816$	9.73336	0.340736
$817$	−53.0537	−1.85612
$818$	0.543770	0.0190125
$819$	0	0
$820$	0	0
$821$	−22.8768	−0.798404	−0.399202	−	0.916863i	$-0.630713\pi$
$821$	−22.8768	−0.798404	−0.399202	+	0.916863i	$0.630713\pi$
$822$	−3.44660	−0.120214
$823$	9.29734	0.324085	0.162043	−	0.986784i	$-0.448192\pi$
$823$	9.29734	0.324085	0.162043	+	0.986784i	$0.448192\pi$
$824$	4.60979	0.160590
$825$	0	0
$826$	0	0
$827$	−16.3758	−0.569443	−0.284721	−	0.958610i	$-0.591901\pi$
$827$	−16.3758	−0.569443	−0.284721	+	0.958610i	$0.591901\pi$
$828$	4.11036	0.142845
$829$	42.2622	1.46783	0.733914	−	0.679243i	$-0.237692\pi$
$829$	42.2622	1.46783	0.733914	+	0.679243i	$0.237692\pi$
$830$	0	0
$831$	9.00572	0.312405
$832$	19.9645	0.692145
$833$	0	0
$834$	2.12843	0.0737015
$835$	0	0
$836$	−51.8798	−1.79430
$837$	−3.49910	−0.120947
$838$	6.29683	0.217520
$839$	24.3699	0.841341	0.420670	−	0.907214i	$-0.361795\pi$
$839$	24.3699	0.841341	0.420670	+	0.907214i	$0.361795\pi$
$840$	0	0
$841$	−7.20967	−0.248609
$842$	3.78256	0.130356
$843$	−2.39929	−0.0826361
$844$	19.9881	0.688020
$845$	0	0
$846$	1.92181	0.0660732
$847$	0	0
$848$	−15.7597	−0.541191
$849$	−6.74077	−0.231343
$850$	0	0
$851$	−12.9665	−0.444487
$852$	23.1905	0.794494
$853$	11.7674	0.402909	0.201455	−	0.979498i	$-0.435433\pi$
$853$	11.7674	0.402909	0.201455	+	0.979498i	$0.435433\pi$
$854$	0	0
$855$	0	0
$856$	−14.7867	−0.505400
$857$	−28.2270	−0.964217	−0.482108	−	0.876112i	$-0.660129\pi$
$857$	−28.2270	−0.964217	−0.482108	+	0.876112i	$0.660129\pi$
$858$	2.68352	0.0916138
$859$	39.4000	1.34431	0.672155	−	0.740411i	$-0.265369\pi$
$859$	39.4000	1.34431	0.672155	+	0.740411i	$0.265369\pi$
$860$	0	0
$861$	0	0
$862$	−6.11906	−0.208416
$863$	−9.34483	−0.318102	−0.159051	−	0.987270i	$-0.550843\pi$
$863$	−9.34483	−0.318102	−0.159051	+	0.987270i	$0.550843\pi$
$864$	2.17946	0.0741468
$865$	0	0
$866$	1.71410	0.0582476
$867$	−10.4180	−0.353813
$868$	0	0
$869$	−16.8220	−0.570647
$870$	0	0
$871$	−16.5341	−0.560238
$872$	3.64412	0.123406
$873$	14.4476	0.488976
$874$	1.97568	0.0668285
$875$	0	0
$876$	7.81461	0.264031
$877$	23.8713	0.806075	0.403038	−	0.915183i	$-0.367954\pi$
$877$	23.8713	0.806075	0.403038	+	0.915183i	$0.367954\pi$
$878$	−5.43139	−0.183301
$879$	−29.7096	−1.00208
$880$	0	0
$881$	−52.8655	−1.78108	−0.890541	−	0.454902i	$-0.849674\pi$
$881$	−52.8655	−1.78108	−0.890541	+	0.454902i	$0.849674\pi$
$882$	0	0
$883$	−49.7398	−1.67388	−0.836938	−	0.547298i	$-0.815657\pi$
$883$	−49.7398	−1.67388	−0.836938	+	0.547298i	$0.815657\pi$
$884$	14.0158	0.471402
$885$	0	0
$886$	−3.76405	−0.126456
$887$	−8.13821	−0.273254	−0.136627	−	0.990623i	$-0.543626\pi$
$887$	−8.13821	−0.273254	−0.136627	+	0.990623i	$0.543626\pi$
$888$	−4.57028	−0.153369
$889$	0	0
$890$	0	0
$891$	−5.19377	−0.173998
$892$	−40.9773	−1.37202
$893$	−52.5435	−1.75830
$894$	0.155186	0.00519021
$895$	0	0
$896$	0	0
$897$	5.81292	0.194088
$898$	5.80018	0.193555
$899$	−16.3338	−0.544764
$900$	0	0
$901$	−10.6573	−0.355045
$902$	−7.21598	−0.240266
$903$	0	0
$904$	−4.62463	−0.153813
$905$	0	0
$906$	−1.81944	−0.0604467
$907$	−38.4296	−1.27603	−0.638016	−	0.770023i	$-0.720245\pi$
$907$	−38.4296	−1.27603	−0.638016	+	0.770023i	$0.720245\pi$
$908$	52.9132	1.75598
$909$	13.3940	0.444250
$910$	0	0
$911$	19.4623	0.644814	0.322407	−	0.946601i	$-0.395508\pi$
$911$	19.4623	0.644814	0.322407	+	0.946601i	$0.395508\pi$
$912$	−19.2813	−0.638468
$913$	87.1733	2.88501
$914$	2.48855	0.0823139
$915$	0	0
$916$	−49.7764	−1.64466
$917$	0	0
$918$	0.476897	0.0157400
$919$	11.9846	0.395336	0.197668	−	0.980269i	$-0.436663\pi$
$919$	11.9846	0.395336	0.197668	+	0.980269i	$0.436663\pi$
$920$	0	0
$921$	12.6609	0.417190
$922$	−6.91271	−0.227658
$923$	32.7963	1.07950
$924$	0	0
$925$	0	0
$926$	5.65054	0.185688
$927$	−6.25380	−0.205402
$928$	10.1738	0.333970
$929$	12.8775	0.422496	0.211248	−	0.977433i	$-0.432247\pi$
$929$	12.8775	0.422496	0.211248	+	0.977433i	$0.432247\pi$
$930$	0	0
$931$	0	0
$932$	22.7625	0.745610
$933$	16.4899	0.539855
$934$	−6.44671	−0.210943
$935$	0	0
$936$	2.04887	0.0669693
$937$	−10.0444	−0.328136	−0.164068	−	0.986449i	$-0.552462\pi$
$937$	−10.0444	−0.328136	−0.164068	+	0.986449i	$0.552462\pi$
$938$	0	0
$939$	−18.7391	−0.611527
$940$	0	0
$941$	−11.9247	−0.388736	−0.194368	−	0.980929i	$-0.562266\pi$
$941$	−11.9247	−0.388736	−0.194368	+	0.980929i	$0.562266\pi$
$942$	−0.468513	−0.0152650
$943$	−15.6309	−0.509013
$944$	−3.78378	−0.123152
$945$	0	0
$946$	−10.0784	−0.327676
$947$	−36.3238	−1.18036	−0.590182	−	0.807270i	$-0.700944\pi$
$947$	−36.3238	−1.18036	−0.590182	+	0.807270i	$0.700944\pi$
$948$	−6.36584	−0.206753
$949$	11.0515	0.358748
$950$	0	0
$951$	−29.1065	−0.943843
$952$	0	0
$953$	−2.87294	−0.0930638	−0.0465319	−	0.998917i	$-0.514817\pi$
$953$	−2.87294	−0.0930638	−0.0465319	+	0.998917i	$0.514817\pi$
$954$	−0.772166	−0.0249998
$955$	0	0
$956$	−7.81793	−0.252850
$957$	−24.2446	−0.783717
$958$	−5.35868	−0.173131
$959$	0	0
$960$	0	0
$961$	−18.7563	−0.605042
$962$	−3.20352	−0.103286
$963$	20.0602	0.646430
$964$	−29.0127	−0.934437
$965$	0	0
$966$	0	0
$967$	28.0352	0.901550	0.450775	−	0.892638i	$-0.351148\pi$
$967$	28.0352	0.901550	0.450775	+	0.892638i	$0.351148\pi$
$968$	−11.7757	−0.378485
$969$	−13.0387	−0.418863
$970$	0	0
$971$	−36.9644	−1.18624	−0.593122	−	0.805113i	$-0.702105\pi$
$971$	−36.9644	−1.18624	−0.593122	+	0.805113i	$0.702105\pi$
$972$	−1.96545	−0.0630417
$973$	0	0
$974$	2.74149	0.0878430
$975$	0	0
$976$	41.0817	1.31499
$977$	−0.771290	−0.0246758	−0.0123379	−	0.999924i	$-0.503927\pi$
$977$	−0.771290	−0.0246758	−0.0123379	+	0.999924i	$0.503927\pi$
$978$	−4.43335	−0.141763
$979$	−24.3164	−0.777155
$980$	0	0
$981$	−4.94374	−0.157841
$982$	2.30220	0.0734661
$983$	0.0166084	0.000529726 0	0.000264863	−	1.00000i	$-0.499916\pi$
$983$	0.0166084	0.000529726 0	0.000264863		1.00000i	$0.499916\pi$
$984$	−5.50940	−0.175633
$985$	0	0
$986$	2.22616	0.0708954
$987$	0	0
$988$	−27.7646	−0.883310
$989$	−21.8313	−0.694196
$990$	0	0
$991$	−46.1847	−1.46711	−0.733553	−	0.679632i	$-0.762140\pi$
$991$	−46.1847	−1.46711	−0.733553	+	0.679632i	$0.762140\pi$
$992$	−7.62616	−0.242131
$993$	6.20020	0.196757
$994$	0	0
$995$	0	0
$996$	32.9884	1.04528
$997$	40.5235	1.28339	0.641696	−	0.766959i	$-0.278231\pi$
$997$	40.5235	1.28339	0.641696	+	0.766959i	$0.278231\pi$
$998$	0.127572	0.00403821
$999$	6.20020	0.196166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.ca.1.2	yes	4
5.4	even	2		3675.2.a.bm.1.3	✓	4
7.6	odd	2		3675.2.a.by.1.2	yes	4
35.34	odd	2		3675.2.a.bo.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3675.2.a.bm.1.3	✓	4	5.4	even	2
3675.2.a.bo.1.3	yes	4	35.34	odd	2
3675.2.a.by.1.2	yes	4	7.6	odd	2
3675.2.a.ca.1.2	yes	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)

\(f(q)\)	\(=\)	\(q+0.185885 q^{2} +1.00000 q^{3} -1.96545 q^{4} +0.185885 q^{6} -0.737118 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.185885 q^{2} +1.00000 q^{3} -1.96545 q^{4} +0.185885 q^{6} -0.737118 q^{8} +1.00000 q^{9} -5.19377 q^{11} -1.96545 q^{12} -2.77956 q^{13} +3.79387 q^{16} +2.56555 q^{17} +0.185885 q^{18} -5.08223 q^{19} -0.965447 q^{22} -2.09131 q^{23} -0.737118 q^{24} -0.516680 q^{26} +1.00000 q^{27} +4.66801 q^{29} -3.49910 q^{31} +2.17946 q^{32} -5.19377 q^{33} +0.476897 q^{34} -1.96545 q^{36} +6.20020 q^{37} -0.944711 q^{38} -2.77956 q^{39} +7.47424 q^{41} +10.4391 q^{43} +10.2081 q^{44} -0.388744 q^{46} +10.3387 q^{47} +3.79387 q^{48} +2.56555 q^{51} +5.46308 q^{52} -4.15399 q^{53} +0.185885 q^{54} -5.08223 q^{57} +0.867715 q^{58} -0.997340 q^{59} +10.8284 q^{61} -0.650431 q^{62} -7.18262 q^{64} -0.965447 q^{66} +5.94847 q^{67} -5.04244 q^{68} -2.09131 q^{69} -11.7991 q^{71} -0.737118 q^{72} -3.97600 q^{73} +1.15253 q^{74} +9.98884 q^{76} -0.516680 q^{78} +3.23888 q^{79} +1.00000 q^{81} +1.38935 q^{82} -16.7842 q^{83} +1.94047 q^{86} +4.66801 q^{87} +3.82843 q^{88} +4.68183 q^{89} +4.11036 q^{92} -3.49910 q^{93} +1.92181 q^{94} +2.17946 q^{96} +14.4476 q^{97} -5.19377 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9	\( 4 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9} - 4 q^{11} + 2 q^{12} - 6 q^{16} - 4 q^{17} + 2 q^{18} + 8 q^{19} + 6 q^{22} + 12 q^{26} + 4 q^{27} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{33} + 8 q^{34} + 2 q^{36} + 16 q^{37} + 4 q^{38} + 24 q^{41} + 20 q^{43} + 14 q^{44} - 6 q^{46} + 8 q^{47} - 6 q^{48} - 4 q^{51} + 16 q^{52} - 20 q^{53} + 2 q^{54} + 8 q^{57} - 6 q^{58} + 8 q^{59} + 32 q^{61} + 28 q^{62} - 12 q^{64} + 6 q^{66} + 12 q^{67} - 12 q^{68} + 4 q^{71} + 34 q^{74} + 40 q^{76} + 12 q^{78} + 4 q^{81} + 16 q^{82} - 20 q^{83} - 14 q^{86} - 4 q^{87} + 4 q^{88} + 8 q^{89} - 10 q^{92} + 8 q^{93} - 32 q^{94} + 2 q^{96} + 24 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 + 2 * q^6 + 4 * q^9 - 4 * q^11 + 2 * q^12 - 6 * q^16 - 4 * q^17 + 2 * q^18 + 8 * q^19 + 6 * q^22 + 12 * q^26 + 4 * q^27 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^33 + 8 * q^34 + 2 * q^36 + 16 * q^37 + 4 * q^38 + 24 * q^41 + 20 * q^43 + 14 * q^44 - 6 * q^46 + 8 * q^47 - 6 * q^48 - 4 * q^51 + 16 * q^52 - 20 * q^53 + 2 * q^54 + 8 * q^57 - 6 * q^58 + 8 * q^59 + 32 * q^61 + 28 * q^62 - 12 * q^64 + 6 * q^66 + 12 * q^67 - 12 * q^68 + 4 * q^71 + 34 * q^74 + 40 * q^76 + 12 * q^78 + 4 * q^81 + 16 * q^82 - 20 * q^83 - 14 * q^86 - 4 * q^87 + 4 * q^88 + 8 * q^89 - 10 * q^92 + 8 * q^93 - 32 * q^94 + 2 * q^96 + 24 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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