LMFDB - Embedded newform 3675.2.a.br.1.4

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:	4.4.88404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 5x + 4 $ x^4 - x^3 - 7x^2 + 5x + 4
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			1.4
Root			$-2.43736$ 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)

$f(q)$	$=$	$q+2.43736 q^{2} +1.00000 q^{3} +3.94072 q^{4} +2.43736 q^{6} +4.73024 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.43736 q^{2} +1.00000 q^{3} +3.94072 q^{4} +2.43736 q^{6} +4.73024 q^{8} +1.00000 q^{9} +4.58184 q^{11} +3.94072 q^{12} +1.35888 q^{13} +3.64784 q^{16} +4.29288 q^{17} +2.43736 q^{18} -6.81544 q^{19} +11.1676 q^{22} +3.88144 q^{23} +4.73024 q^{24} +3.31208 q^{26} +1.00000 q^{27} +4.00000 q^{29} -8.10832 q^{31} -0.569365 q^{32} +4.58184 q^{33} +10.4633 q^{34} +3.94072 q^{36} -11.7562 q^{37} -16.6117 q^{38} +1.35888 q^{39} +8.17432 q^{41} +3.52256 q^{43} +18.0558 q^{44} +9.46047 q^{46} -3.29960 q^{47} +3.64784 q^{48} +4.29288 q^{51} +5.35497 q^{52} +9.45656 q^{53} +2.43736 q^{54} -6.81544 q^{57} +9.74944 q^{58} +0.700398 q^{59} -5.00000 q^{61} -19.7629 q^{62} -8.68344 q^{64} +11.1676 q^{66} +1.94072 q^{67} +16.9170 q^{68} +3.88144 q^{69} +11.8680 q^{71} +4.73024 q^{72} -6.16368 q^{73} -28.6540 q^{74} -26.8578 q^{76} +3.31208 q^{78} -6.16760 q^{79} +1.00000 q^{81} +19.9238 q^{82} -13.0490 q^{83} +8.58575 q^{86} +4.00000 q^{87} +21.6732 q^{88} +0.292877 q^{89} +15.2957 q^{92} -8.10832 q^{93} -8.04232 q^{94} -0.569365 q^{96} +2.00672 q^{97} +4.58184 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 4 q^{3} + 7 q^{4} - q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - q^2 + 4 * q^3 + 7 * q^4 - q^6 - 3 * q^8 + 4 * q^9	$ 4 q - q^{2} + 4 q^{3} + 7 q^{4} - q^{6} - 3 q^{8} + 4 q^{9} + 8 q^{11} + 7 q^{12} + 7 q^{13} + 17 q^{16} + 6 q^{17} - q^{18} + 3 q^{19} + 12 q^{22} - 2 q^{23} - 3 q^{24} - 19 q^{26} + 4 q^{27} + 16 q^{29} + 9 q^{31} - 17 q^{32} + 8 q^{33} + 14 q^{34} + 7 q^{36} - 8 q^{37} - 27 q^{38} + 7 q^{39} + 4 q^{41} - 5 q^{43} + 26 q^{44} - 6 q^{46} - 6 q^{47} + 17 q^{48} + 6 q^{51} + 35 q^{52} + 6 q^{53} - q^{54} + 3 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} - 44 q^{62} + 21 q^{64} + 12 q^{66} - q^{67} - 8 q^{68} - 2 q^{69} + 22 q^{71} - 3 q^{72} - 4 q^{73} - 21 q^{74} - 23 q^{76} - 19 q^{78} + 8 q^{79} + 4 q^{81} + 8 q^{82} - 2 q^{83} + 12 q^{86} + 16 q^{87} - 28 q^{88} - 10 q^{89} + 66 q^{92} + 9 q^{93} + 22 q^{94} - 17 q^{96} + 12 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q - q^2 + 4 * q^3 + 7 * q^4 - q^6 - 3 * q^8 + 4 * q^9 + 8 * q^11 + 7 * q^12 + 7 * q^13 + 17 * q^16 + 6 * q^17 - q^18 + 3 * q^19 + 12 * q^22 - 2 * q^23 - 3 * q^24 - 19 * q^26 + 4 * q^27 + 16 * q^29 + 9 * q^31 - 17 * q^32 + 8 * q^33 + 14 * q^34 + 7 * q^36 - 8 * q^37 - 27 * q^38 + 7 * q^39 + 4 * q^41 - 5 * q^43 + 26 * q^44 - 6 * q^46 - 6 * q^47 + 17 * q^48 + 6 * q^51 + 35 * q^52 + 6 * q^53 - q^54 + 3 * q^57 - 4 * q^58 + 10 * q^59 - 20 * q^61 - 44 * q^62 + 21 * q^64 + 12 * q^66 - q^67 - 8 * q^68 - 2 * q^69 + 22 * q^71 - 3 * q^72 - 4 * q^73 - 21 * q^74 - 23 * q^76 - 19 * q^78 + 8 * q^79 + 4 * q^81 + 8 * q^82 - 2 * q^83 + 12 * q^86 + 16 * q^87 - 28 * q^88 - 10 * q^89 + 66 * q^92 + 9 * q^93 + 22 * q^94 - 17 * q^96 + 12 * q^97 + 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 $

$2$	2.43736	1.72347	0.861737	−	0.507356i	$-0.169377\pi$
$2$	2.43736	1.72347	0.861737	+	0.507356i	$0.169377\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.94072	1.97036
$5$	0	0
$5$	0	0
$6$	2.43736	0.995048
$7$	0	0
$7$	0	0
$8$	4.73024	1.67239
$9$	1.00000	0.333333
$10$	0	0
$11$	4.58184	1.38148	0.690739	−	0.723104i	$-0.257286\pi$
$11$	4.58184	1.38148	0.690739	+	0.723104i	$0.257286\pi$
$12$	3.94072	1.13759
$13$	1.35888	0.376885	0.188443	−	0.982084i	$-0.439656\pi$
$13$	1.35888	0.376885	0.188443	+	0.982084i	$0.439656\pi$
$14$	0	0
$15$	0	0
$16$	3.64784	0.911961
$17$	4.29288	1.04118	0.520588	−	0.853808i	$-0.325713\pi$
$17$	4.29288	1.04118	0.520588	+	0.853808i	$0.325713\pi$
$18$	2.43736	0.574491
$19$	−6.81544	−1.56357	−0.781785	−	0.623548i	$-0.785690\pi$
$19$	−6.81544	−1.56357	−0.781785	+	0.623548i	$0.785690\pi$
$20$	0	0
$21$	0	0
$22$	11.1676	2.38094
$23$	3.88144	0.809337	0.404668	−	0.914463i	$-0.367387\pi$
$23$	3.88144	0.809337	0.404668	+	0.914463i	$0.367387\pi$
$24$	4.73024	0.965556
$25$	0	0
$26$	3.31208	0.649552
$27$	1.00000	0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−8.10832	−1.45630	−0.728148	−	0.685419i	$-0.759619\pi$
$31$	−8.10832	−1.45630	−0.728148	+	0.685419i	$0.759619\pi$
$32$	−0.569365	−0.100650
$33$	4.58184	0.797596
$34$	10.4633	1.79444
$35$	0	0
$36$	3.94072	0.656787
$37$	−11.7562	−1.93270	−0.966351	−	0.257228i	$-0.917191\pi$
$37$	−11.7562	−1.93270	−0.966351	+	0.257228i	$0.917191\pi$
$38$	−16.6117	−2.69477
$39$	1.35888	0.217595
$40$	0	0
$41$	8.17432	1.27661	0.638307	−	0.769782i	$-0.279635\pi$
$41$	8.17432	1.27661	0.638307	+	0.769782i	$0.279635\pi$
$42$	0	0
$43$	3.52256	0.537186	0.268593	−	0.963254i	$-0.413441\pi$
$43$	3.52256	0.537186	0.268593	+	0.963254i	$0.413441\pi$
$44$	18.0558	2.72201
$45$	0	0
$46$	9.46047	1.39487
$47$	−3.29960	−0.481296	−0.240648	−	0.970612i	$-0.577360\pi$
$47$	−3.29960	−0.481296	−0.240648	+	0.970612i	$0.577360\pi$
$48$	3.64784	0.526521
$49$	0	0
$50$	0	0
$51$	4.29288	0.601123
$52$	5.35497	0.742600
$53$	9.45656	1.29896	0.649479	−	0.760379i	$-0.274987\pi$
$53$	9.45656	1.29896	0.649479	+	0.760379i	$0.274987\pi$
$54$	2.43736	0.331683
$55$	0	0
$56$	0	0
$57$	−6.81544	−0.902727
$58$	9.74944	1.28016
$59$	0.700398	0.0911841	0.0455921	−	0.998960i	$-0.485483\pi$
$59$	0.700398	0.0911841	0.0455921	+	0.998960i	$0.485483\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	−19.7629	−2.50989
$63$	0	0
$64$	−8.68344	−1.08543
$65$	0	0
$66$	11.1676	1.37464
$67$	1.94072	0.237097	0.118548	−	0.992948i	$-0.462176\pi$
$67$	1.94072	0.237097	0.118548	+	0.992948i	$0.462176\pi$
$68$	16.9170	2.05149
$69$	3.88144	0.467271
$70$	0	0
$71$	11.8680	1.40847	0.704236	−	0.709966i	$-0.251290\pi$
$71$	11.8680	1.40847	0.704236	+	0.709966i	$0.251290\pi$
$72$	4.73024	0.557464
$73$	−6.16368	−0.721405	−0.360702	−	0.932681i	$-0.617463\pi$
$73$	−6.16368	−0.721405	−0.360702	+	0.932681i	$0.617463\pi$
$74$	−28.6540	−3.33096
$75$	0	0
$76$	−26.8578	−3.08080
$77$	0	0
$78$	3.31208	0.375019
$79$	−6.16760	−0.693909	−0.346954	−	0.937882i	$-0.612784\pi$
$79$	−6.16760	−0.693909	−0.346954	+	0.937882i	$0.612784\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	19.9238	2.20021
$83$	−13.0490	−1.43232	−0.716159	−	0.697937i	$-0.754102\pi$
$83$	−13.0490	−1.43232	−0.716159	+	0.697937i	$0.754102\pi$
$84$	0	0
$85$	0	0
$86$	8.58575	0.925826
$87$	4.00000	0.428845
$88$	21.6732	2.31037
$89$	0.292877	0.0310449	0.0155225	−	0.999880i	$-0.495059\pi$
$89$	0.292877	0.0310449	0.0155225	+	0.999880i	$0.495059\pi$
$90$	0	0
$91$	0	0
$92$	15.2957	1.59469
$93$	−8.10832	−0.840793
$94$	−8.04232	−0.829502
$95$	0	0
$96$	−0.569365	−0.0581105
$97$	2.00672	0.203752	0.101876	−	0.994797i	$-0.467516\pi$
$97$	2.00672	0.203752	0.101876	+	0.994797i	$0.467516\pi$
$98$	0	0
$99$	4.58184	0.460492
$100$	0	0
$101$	6.29288	0.626165	0.313082	−	0.949726i	$-0.398638\pi$
$101$	6.29288	0.626165	0.313082	+	0.949726i	$0.398638\pi$
$102$	10.4633	1.03602
$103$	3.41143	0.336139	0.168069	−	0.985775i	$-0.446247\pi$
$103$	3.41143	0.336139	0.168069	+	0.985775i	$0.446247\pi$
$104$	6.42782	0.630300
$105$	0	0
$106$	23.0490	2.23872
$107$	−3.16760	−0.306223	−0.153112	−	0.988209i	$-0.548929\pi$
$107$	−3.16760	−0.306223	−0.153112	+	0.988209i	$0.548929\pi$
$108$	3.94072	0.379196
$109$	10.2336	0.980201	0.490101	−	0.871666i	$-0.336960\pi$
$109$	10.2336	0.980201	0.490101	+	0.871666i	$0.336960\pi$
$110$	0	0
$111$	−11.7562	−1.11585
$112$	0	0
$113$	3.16368	0.297614	0.148807	−	0.988866i	$-0.452457\pi$
$113$	3.16368	0.297614	0.148807	+	0.988866i	$0.452457\pi$
$114$	−16.6117	−1.55583
$115$	0	0
$116$	15.7629	1.46355
$117$	1.35888	0.125628
$118$	1.70712	0.157153
$119$	0	0
$120$	0	0
$121$	9.99328	0.908480
$122$	−12.1868	−1.10334
$123$	8.17432	0.737054
$124$	−31.9526	−2.86943
$125$	0	0
$126$	0	0
$127$	−0.0106369	−0.000943870 0	−0.000471935	−	1.00000i	$-0.500150\pi$
$127$	−0.0106369	−0.000943870 0	−0.000471935		1.00000i	$0.500150\pi$
$128$	−20.0259	−1.77006
$129$	3.52256	0.310144
$130$	0	0
$131$	13.7629	1.20247	0.601235	−	0.799073i	$-0.294676\pi$
$131$	13.7629	1.20247	0.601235	+	0.799073i	$0.294676\pi$
$132$	18.0558	1.57155
$133$	0	0
$134$	4.73024	0.408630
$135$	0	0
$136$	20.3063	1.74125
$137$	−9.04513	−0.772777	−0.386389	−	0.922336i	$-0.626278\pi$
$137$	−9.04513	−0.772777	−0.386389	+	0.922336i	$0.626278\pi$
$138$	9.46047	0.805329
$139$	−21.3313	−1.80930	−0.904648	−	0.426160i	$-0.859866\pi$
$139$	−21.3313	−1.80930	−0.904648	+	0.426160i	$0.859866\pi$
$140$	0	0
$141$	−3.29960	−0.277877
$142$	28.9266	2.42746
$143$	6.22617	0.520659
$144$	3.64784	0.303987
$145$	0	0
$146$	−15.0231	−1.24332
$147$	0	0
$148$	−46.3278	−3.80812
$149$	−15.8052	−1.29481	−0.647406	−	0.762145i	$-0.724146\pi$
$149$	−15.8052	−1.29481	−0.647406	+	0.762145i	$0.724146\pi$
$150$	0	0
$151$	4.86408	0.395833	0.197917	−	0.980219i	$-0.436582\pi$
$151$	4.86408	0.395833	0.197917	+	0.980219i	$0.436582\pi$
$152$	−32.2386	−2.61490
$153$	4.29288	0.347059
$154$	0	0
$155$	0	0
$156$	5.35497	0.428741
$157$	6.57903	0.525064	0.262532	−	0.964923i	$-0.415443\pi$
$157$	6.57903	0.525064	0.262532	+	0.964923i	$0.415443\pi$
$158$	−15.0327	−1.19593
$159$	9.45656	0.749954
$160$	0	0
$161$	0	0
$162$	2.43736	0.191497
$163$	−11.4633	−0.897874	−0.448937	−	0.893563i	$-0.648197\pi$
$163$	−11.4633	−0.897874	−0.448937	+	0.893563i	$0.648197\pi$
$164$	32.2127	2.51539
$165$	0	0
$166$	−31.8052	−2.46856
$167$	−20.9170	−1.61861	−0.809304	−	0.587390i	$-0.800156\pi$
$167$	−20.9170	−1.61861	−0.809304	+	0.587390i	$0.800156\pi$
$168$	0	0
$169$	−11.1534	−0.857957
$170$	0	0
$171$	−6.81544	−0.521190
$172$	13.8814	1.05845
$173$	11.1637	0.848759	0.424380	−	0.905484i	$-0.360492\pi$
$173$	11.1637	0.848759	0.424380	+	0.905484i	$0.360492\pi$
$174$	9.74944	0.739103
$175$	0	0
$176$	16.7138	1.25985
$177$	0.700398	0.0526452
$178$	0.713847	0.0535051
$179$	−17.5123	−1.30893	−0.654466	−	0.756091i	$-0.727107\pi$
$179$	−17.5123	−1.30893	−0.654466	+	0.756091i	$0.727107\pi$
$180$	0	0
$181$	20.2653	1.50631	0.753153	−	0.657845i	$-0.228532\pi$
$181$	20.2653	1.50631	0.753153	+	0.657845i	$0.228532\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	18.3601	1.35353
$185$	0	0
$186$	−19.7629	−1.44909
$187$	19.6693	1.43836
$188$	−13.0028	−0.948328
$189$	0	0
$190$	0	0
$191$	−4.13592	−0.299265	−0.149632	−	0.988742i	$-0.547809\pi$
$191$	−4.13592	−0.299265	−0.149632	+	0.988742i	$0.547809\pi$
$192$	−8.68344	−0.626673
$193$	4.76640	0.343093	0.171547	−	0.985176i	$-0.445124\pi$
$193$	4.76640	0.343093	0.171547	+	0.985176i	$0.445124\pi$
$194$	4.89111	0.351161
$195$	0	0
$196$	0	0
$197$	5.30351	0.377860	0.188930	−	0.981991i	$-0.439498\pi$
$197$	5.30351	0.377860	0.188930	+	0.981991i	$0.439498\pi$
$198$	11.1676	0.793647
$199$	−2.27833	−0.161506	−0.0807532	−	0.996734i	$-0.525733\pi$
$199$	−2.27833	−0.161506	−0.0807532	+	0.996734i	$0.525733\pi$
$200$	0	0
$201$	1.94072	0.136888
$202$	15.3380	1.07918
$203$	0	0
$204$	16.9170	1.18443
$205$	0	0
$206$	8.31489	0.579326
$207$	3.88144	0.269779
$208$	4.95698	0.343705
$209$	−31.2273	−2.16004
$210$	0	0
$211$	−14.8052	−1.01923	−0.509616	−	0.860402i	$-0.670213\pi$
$211$	−14.8052	−1.01923	−0.509616	+	0.860402i	$0.670213\pi$
$212$	37.2657	2.55942
$213$	11.8680	0.813182
$214$	−7.72057	−0.527767
$215$	0	0
$216$	4.73024	0.321852
$217$	0	0
$218$	24.9430	1.68935
$219$	−6.16368	−0.416503
$220$	0	0
$221$	5.83350	0.392404
$222$	−28.6540	−1.92313
$223$	−6.63479	−0.444299	−0.222149	−	0.975013i	$-0.571307\pi$
$223$	−6.63479	−0.444299	−0.222149	+	0.975013i	$0.571307\pi$
$224$	0	0
$225$	0	0
$226$	7.71104	0.512930
$227$	9.03168	0.599454	0.299727	−	0.954025i	$-0.403104\pi$
$227$	9.03168	0.599454	0.299727	+	0.954025i	$0.403104\pi$
$228$	−26.8578	−1.77870
$229$	16.8328	1.11234	0.556171	−	0.831068i	$-0.312270\pi$
$229$	16.8328	1.11234	0.556171	+	0.831068i	$0.312270\pi$
$230$	0	0
$231$	0	0
$232$	18.9209	1.24222
$233$	−18.2929	−1.19841	−0.599203	−	0.800597i	$-0.704516\pi$
$233$	−18.2929	−1.19841	−0.599203	+	0.800597i	$0.704516\pi$
$234$	3.31208	0.216517
$235$	0	0
$236$	2.76008	0.179666
$237$	−6.16760	−0.400628
$238$	0	0
$239$	11.6309	0.752339	0.376170	−	0.926551i	$-0.377241\pi$
$239$	11.6309	0.752339	0.376170	+	0.926551i	$0.377241\pi$
$240$	0	0
$241$	−1.76961	−0.113991	−0.0569953	−	0.998374i	$-0.518152\pi$
$241$	−1.76961	−0.113991	−0.0569953	+	0.998374i	$0.518152\pi$
$242$	24.3572	1.56574
$243$	1.00000	0.0641500
$244$	−19.7036	−1.26139
$245$	0	0
$246$	19.9238	1.27029
$247$	−9.26137	−0.589287
$248$	−38.3543	−2.43550
$249$	−13.0490	−0.826949
$250$	0	0
$251$	−10.7004	−0.675403	−0.337702	−	0.941253i	$-0.609649\pi$
$251$	−10.7004	−0.675403	−0.337702	+	0.941253i	$0.609649\pi$
$252$	0	0
$253$	17.7842	1.11808
$254$	−0.0259259	−0.00162674
$255$	0	0
$256$	−31.4435	−1.96522
$257$	−26.1609	−1.63187	−0.815935	−	0.578143i	$-0.803777\pi$
$257$	−26.1609	−1.63187	−0.815935	+	0.578143i	$0.803777\pi$
$258$	8.58575	0.534526
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	33.5451	2.07242
$263$	−13.3803	−0.825066	−0.412533	−	0.910943i	$-0.635356\pi$
$263$	−13.3803	−0.825066	−0.412533	+	0.910943i	$0.635356\pi$
$264$	21.6732	1.33389
$265$	0	0
$266$	0	0
$267$	0.292877	0.0179238
$268$	7.64784	0.467166
$269$	−2.87081	−0.175036	−0.0875181	−	0.996163i	$-0.527894\pi$
$269$	−2.87081	−0.175036	−0.0875181	+	0.996163i	$0.527894\pi$
$270$	0	0
$271$	29.3419	1.78239	0.891197	−	0.453616i	$-0.149866\pi$
$271$	29.3419	1.78239	0.891197	+	0.453616i	$0.149866\pi$
$272$	15.6597	0.949512
$273$	0	0
$274$	−22.0462	−1.33186
$275$	0	0
$276$	15.2957	0.920692
$277$	−19.9830	−1.20066	−0.600332	−	0.799751i	$-0.704965\pi$
$277$	−19.9830	−1.20066	−0.600332	+	0.799751i	$0.704965\pi$
$278$	−51.9920	−3.11827
$279$	−8.10832	−0.485432
$280$	0	0
$281$	7.12919	0.425292	0.212646	−	0.977129i	$-0.431792\pi$
$281$	7.12919	0.425292	0.212646	+	0.977129i	$0.431792\pi$
$282$	−8.04232	−0.478913
$283$	−22.8751	−1.35978	−0.679892	−	0.733312i	$-0.737973\pi$
$283$	−22.8751	−1.35978	−0.679892	+	0.733312i	$0.737973\pi$
$284$	46.7685	2.77520
$285$	0	0
$286$	15.1754	0.897341
$287$	0	0
$288$	−0.569365	−0.0335501
$289$	1.42879	0.0840468
$290$	0	0
$291$	2.00672	0.117636
$292$	−24.2894	−1.42143
$293$	3.63088	0.212118	0.106059	−	0.994360i	$-0.466177\pi$
$293$	3.63088	0.212118	0.106059	+	0.994360i	$0.466177\pi$
$294$	0	0
$295$	0	0
$296$	−55.6094	−3.23223
$297$	4.58184	0.265865
$298$	−38.5230	−2.23158
$299$	5.27442	0.305027
$300$	0	0
$301$	0	0
$302$	11.8555	0.682208
$303$	6.29288	0.361516
$304$	−24.8617	−1.42591
$305$	0	0
$306$	10.4633	0.598146
$307$	6.22687	0.355387	0.177693	−	0.984086i	$-0.443137\pi$
$307$	6.22687	0.355387	0.177693	+	0.984086i	$0.443137\pi$
$308$	0	0
$309$	3.41143	0.194070
$310$	0	0
$311$	−21.6309	−1.22657	−0.613287	−	0.789860i	$-0.710153\pi$
$311$	−21.6309	−1.22657	−0.613287	+	0.789860i	$0.710153\pi$
$312$	6.42782	0.363904
$313$	−23.8962	−1.35069	−0.675345	−	0.737502i	$-0.736005\pi$
$313$	−23.8962	−1.35069	−0.675345	+	0.737502i	$0.736005\pi$
$314$	16.0355	0.904933
$315$	0	0
$316$	−24.3048	−1.36725
$317$	−8.57231	−0.481469	−0.240734	−	0.970591i	$-0.577388\pi$
$317$	−8.57231	−0.481469	−0.240734	+	0.970591i	$0.577388\pi$
$318$	23.0490	1.29253
$319$	18.3274	1.02614
$320$	0	0
$321$	−3.16760	−0.176798
$322$	0	0
$323$	−29.2579	−1.62795
$324$	3.94072	0.218929
$325$	0	0
$326$	−27.9401	−1.54746
$327$	10.2336	0.565919
$328$	38.6665	2.13500
$329$	0	0
$330$	0	0
$331$	13.5095	0.742550	0.371275	−	0.928523i	$-0.378921\pi$
$331$	13.5095	0.742550	0.371275	+	0.928523i	$0.378921\pi$
$332$	−51.4226	−2.82218
$333$	−11.7562	−0.644234
$334$	−50.9823	−2.78963
$335$	0	0
$336$	0	0
$337$	−11.1016	−0.604742	−0.302371	−	0.953190i	$-0.597778\pi$
$337$	−11.1016	−0.604742	−0.302371	+	0.953190i	$0.597778\pi$
$338$	−27.1850	−1.47867
$339$	3.16368	0.171828
$340$	0	0
$341$	−37.1510	−2.01184
$342$	−16.6117	−0.898257
$343$	0	0
$344$	16.6626	0.898385
$345$	0	0
$346$	27.2099	1.46281
$347$	−27.3842	−1.47006	−0.735031	−	0.678033i	$-0.762833\pi$
$347$	−27.3842	−1.47006	−0.735031	+	0.678033i	$0.762833\pi$
$348$	15.7629	0.844979
$349$	6.17823	0.330713	0.165357	−	0.986234i	$-0.447122\pi$
$349$	6.17823	0.330713	0.165357	+	0.986234i	$0.447122\pi$
$350$	0	0
$351$	1.35888	0.0725316
$352$	−2.60874	−0.139046
$353$	7.80520	0.415429	0.207715	−	0.978189i	$-0.433397\pi$
$353$	7.80520	0.415429	0.207715	+	0.978189i	$0.433397\pi$
$354$	1.70712	0.0907326
$355$	0	0
$356$	1.15415	0.0611697
$357$	0	0
$358$	−42.6838	−2.25591
$359$	4.11464	0.217163	0.108581	−	0.994088i	$-0.465369\pi$
$359$	4.11464	0.217163	0.108581	+	0.994088i	$0.465369\pi$
$360$	0	0
$361$	27.4502	1.44475
$362$	49.3938	2.59608
$363$	9.99328	0.524511
$364$	0	0
$365$	0	0
$366$	−12.1868	−0.637014
$367$	21.8115	1.13855	0.569276	−	0.822146i	$-0.307223\pi$
$367$	21.8115	1.13855	0.569276	+	0.822146i	$0.307223\pi$
$368$	14.1589	0.738084
$369$	8.17432	0.425538
$370$	0	0
$371$	0	0
$372$	−31.9526	−1.65667
$373$	−2.93791	−0.152119	−0.0760596	−	0.997103i	$-0.524234\pi$
$373$	−2.93791	−0.152119	−0.0760596	+	0.997103i	$0.524234\pi$
$374$	47.9411	2.47898
$375$	0	0
$376$	−15.6079	−0.804916
$377$	5.43552	0.279943
$378$	0	0
$379$	−8.85384	−0.454791	−0.227396	−	0.973802i	$-0.573021\pi$
$379$	−8.85384	−0.454791	−0.227396	+	0.973802i	$0.573021\pi$
$380$	0	0
$381$	−0.0106369	−0.000544944 0
$382$	−10.0807	−0.515774
$383$	17.6387	0.901296	0.450648	−	0.892702i	$-0.351193\pi$
$383$	17.6387	0.901296	0.450648	+	0.892702i	$0.351193\pi$
$384$	−20.0259	−1.02194
$385$	0	0
$386$	11.6174	0.591312
$387$	3.52256	0.179062
$388$	7.90794	0.401465
$389$	18.1878	0.922157	0.461078	−	0.887359i	$-0.347463\pi$
$389$	18.1878	0.922157	0.461078	+	0.887359i	$0.347463\pi$
$390$	0	0
$391$	16.6626	0.842662
$392$	0	0
$393$	13.7629	0.694246
$394$	12.9266	0.651231
$395$	0	0
$396$	18.0558	0.907336
$397$	10.5237	0.528168	0.264084	−	0.964500i	$-0.414930\pi$
$397$	10.5237	0.528168	0.264084	+	0.964500i	$0.414930\pi$
$398$	−5.55310	−0.278352
$399$	0	0
$400$	0	0
$401$	3.58857	0.179204	0.0896022	−	0.995978i	$-0.471440\pi$
$401$	3.58857	0.179204	0.0896022	+	0.995978i	$0.471440\pi$
$402$	4.73024	0.235923
$403$	−11.0182	−0.548857
$404$	24.7985	1.23377
$405$	0	0
$406$	0	0
$407$	−53.8649	−2.66998
$408$	20.3063	1.00531
$409$	−22.5304	−1.11406	−0.557028	−	0.830494i	$-0.688058\pi$
$409$	−22.5304	−1.11406	−0.557028	+	0.830494i	$0.688058\pi$
$410$	0	0
$411$	−9.04513	−0.446163
$412$	13.4435	0.662314
$413$	0	0
$414$	9.46047	0.464957
$415$	0	0
$416$	−0.773698	−0.0379337
$417$	−21.3313	−1.04460
$418$	−76.1121	−3.72276
$419$	39.4055	1.92508	0.962542	−	0.271131i	$-0.0873976\pi$
$419$	39.4055	1.92508	0.962542	+	0.271131i	$0.0873976\pi$
$420$	0	0
$421$	−8.57121	−0.417735	−0.208867	−	0.977944i	$-0.566978\pi$
$421$	−8.57121	−0.417735	−0.208867	+	0.977944i	$0.566978\pi$
$422$	−36.0856	−1.75662
$423$	−3.29960	−0.160432
$424$	44.7318	2.17237
$425$	0	0
$426$	28.9266	1.40150
$427$	0	0
$428$	−12.4826	−0.603370
$429$	6.22617	0.300602
$430$	0	0
$431$	1.28224	0.0617633	0.0308817	−	0.999523i	$-0.490168\pi$
$431$	1.28224	0.0617633	0.0308817	+	0.999523i	$0.490168\pi$
$432$	3.64784	0.175507
$433$	−5.21233	−0.250488	−0.125244	−	0.992126i	$-0.539971\pi$
$433$	−5.21233	−0.250488	−0.125244	+	0.992126i	$0.539971\pi$
$434$	0	0
$435$	0	0
$436$	40.3278	1.93135
$437$	−26.4537	−1.26545
$438$	−15.0231	−0.717832
$439$	19.6281	0.936797	0.468398	−	0.883517i	$-0.344831\pi$
$439$	19.6281	0.936797	0.468398	+	0.883517i	$0.344831\pi$
$440$	0	0
$441$	0	0
$442$	14.2183	0.676298
$443$	18.0347	0.856856	0.428428	−	0.903576i	$-0.359068\pi$
$443$	18.0347	0.856856	0.428428	+	0.903576i	$0.359068\pi$
$444$	−46.3278	−2.19862
$445$	0	0
$446$	−16.1714	−0.765737
$447$	−15.8052	−0.747560
$448$	0	0
$449$	−6.99719	−0.330218	−0.165109	−	0.986275i	$-0.552798\pi$
$449$	−6.99719	−0.330218	−0.165109	+	0.986275i	$0.552798\pi$
$450$	0	0
$451$	37.4534	1.76361
$452$	12.4672	0.586408
$453$	4.86408	0.228535
$454$	22.0134	1.03314
$455$	0	0
$456$	−32.2386	−1.50971
$457$	−37.8408	−1.77012	−0.885059	−	0.465479i	$-0.845882\pi$
$457$	−37.8408	−1.77012	−0.885059	+	0.465479i	$0.845882\pi$
$458$	41.0276	1.91709
$459$	4.29288	0.200374
$460$	0	0
$461$	24.0903	1.12200	0.560998	−	0.827817i	$-0.310418\pi$
$461$	24.0903	1.12200	0.560998	+	0.827817i	$0.310418\pi$
$462$	0	0
$463$	−23.1154	−1.07427	−0.537133	−	0.843498i	$-0.680493\pi$
$463$	−23.1154	−1.07427	−0.537133	+	0.843498i	$0.680493\pi$
$464$	14.5914	0.677388
$465$	0	0
$466$	−44.5863	−2.06542
$467$	−12.5645	−0.581415	−0.290707	−	0.956812i	$-0.593891\pi$
$467$	−12.5645	−0.581415	−0.290707	+	0.956812i	$0.593891\pi$
$468$	5.35497	0.247533
$469$	0	0
$470$	0	0
$471$	6.57903	0.303146
$472$	3.31305	0.152496
$473$	16.1398	0.742110
$474$	−15.0327	−0.690472
$475$	0	0
$476$	0	0
$477$	9.45656	0.432986
$478$	28.3486	1.29664
$479$	−2.03081	−0.0927901	−0.0463950	−	0.998923i	$-0.514773\pi$
$479$	−2.03081	−0.0927901	−0.0463950	+	0.998923i	$0.514773\pi$
$480$	0	0
$481$	−15.9752	−0.728407
$482$	−4.31318	−0.196460
$483$	0	0
$484$	39.3807	1.79003
$485$	0	0
$486$	2.43736	0.110561
$487$	−27.5226	−1.24717	−0.623583	−	0.781757i	$-0.714324\pi$
$487$	−27.5226	−1.24717	−0.623583	+	0.781757i	$0.714324\pi$
$488$	−23.6512	−1.07064
$489$	−11.4633	−0.518388
$490$	0	0
$491$	0.459373	0.0207312	0.0103656	−	0.999946i	$-0.496700\pi$
$491$	0.459373	0.0207312	0.0103656	+	0.999946i	$0.496700\pi$
$492$	32.2127	1.45226
$493$	17.1715	0.773366
$494$	−22.5733	−1.01562
$495$	0	0
$496$	−29.5779	−1.32809
$497$	0	0
$498$	−31.8052	−1.42523
$499$	−11.6383	−0.521002	−0.260501	−	0.965474i	$-0.583888\pi$
$499$	−11.6383	−0.521002	−0.260501	+	0.965474i	$0.583888\pi$
$500$	0	0
$501$	−20.9170	−0.934504
$502$	−26.0807	−1.16404
$503$	32.4254	1.44578	0.722890	−	0.690964i	$-0.242813\pi$
$503$	32.4254	1.44578	0.722890	+	0.690964i	$0.242813\pi$
$504$	0	0
$505$	0	0
$506$	43.3464	1.92698
$507$	−11.1534	−0.495342
$508$	−0.0419170	−0.00185977
$509$	17.0395	0.755263	0.377631	−	0.925956i	$-0.376739\pi$
$509$	17.0395	0.755263	0.377631	+	0.925956i	$0.376739\pi$
$510$	0	0
$511$	0	0
$512$	−36.5873	−1.61695
$513$	−6.81544	−0.300909
$514$	−63.7635	−2.81249
$515$	0	0
$516$	13.8814	0.611096
$517$	−15.1183	−0.664900
$518$	0	0
$519$	11.1637	0.490031
$520$	0	0
$521$	2.61265	0.114462	0.0572312	−	0.998361i	$-0.481773\pi$
$521$	2.61265	0.114462	0.0572312	+	0.998361i	$0.481773\pi$
$522$	9.74944	0.426721
$523$	28.0032	1.22449	0.612247	−	0.790666i	$-0.290266\pi$
$523$	28.0032	1.22449	0.612247	+	0.790666i	$0.290266\pi$
$524$	54.2357	2.36930
$525$	0	0
$526$	−32.6127	−1.42198
$527$	−34.8080	−1.51626
$528$	16.7138	0.727377
$529$	−7.93440	−0.344974
$530$	0	0
$531$	0.700398	0.0303947
$532$	0	0
$533$	11.1079	0.481137
$534$	0.713847	0.0308912
$535$	0	0
$536$	9.18007	0.396519
$537$	−17.5123	−0.755713
$538$	−6.99719	−0.301670
$539$	0	0
$540$	0	0
$541$	−7.45696	−0.320600	−0.160300	−	0.987068i	$-0.551246\pi$
$541$	−7.45696	−0.320600	−0.160300	+	0.987068i	$0.551246\pi$
$542$	71.5168	3.07191
$543$	20.2653	0.869666
$544$	−2.44421	−0.104795
$545$	0	0
$546$	0	0
$547$	16.7640	0.716776	0.358388	−	0.933573i	$-0.383326\pi$
$547$	16.7640	0.716776	0.358388	+	0.933573i	$0.383326\pi$
$548$	−35.6443	−1.52265
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−27.2618	−1.16139
$552$	18.3601	0.781460
$553$	0	0
$554$	−48.7058	−2.06931
$555$	0	0
$556$	−84.0606	−3.56497
$557$	15.0451	0.637483	0.318741	−	0.947842i	$-0.396740\pi$
$557$	15.0451	0.637483	0.318741	+	0.947842i	$0.396740\pi$
$558$	−19.7629	−0.836630
$559$	4.78674	0.202458
$560$	0	0
$561$	19.6693	0.830438
$562$	17.3764	0.732980
$563$	−21.1676	−0.892108	−0.446054	−	0.895006i	$-0.647171\pi$
$563$	−21.1676	−0.892108	−0.446054	+	0.895006i	$0.647171\pi$
$564$	−13.0028	−0.547517
$565$	0	0
$566$	−55.7549	−2.34355
$567$	0	0
$568$	56.1384	2.35552
$569$	44.9767	1.88552	0.942761	−	0.333469i	$-0.108219\pi$
$569$	44.9767	1.88552	0.942761	+	0.333469i	$0.108219\pi$
$570$	0	0
$571$	39.2238	1.64146	0.820732	−	0.571314i	$-0.193566\pi$
$571$	39.2238	1.64146	0.820732	+	0.571314i	$0.193566\pi$
$572$	24.5356	1.02589
$573$	−4.13592	−0.172780
$574$	0	0
$575$	0	0
$576$	−8.68344	−0.361810
$577$	−7.06319	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$577$	−7.06319	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$578$	3.48249	0.144852
$579$	4.76640	0.198085
$580$	0	0
$581$	0	0
$582$	4.89111	0.202743
$583$	43.3285	1.79448
$584$	−29.1557	−1.20647
$585$	0	0
$586$	8.84976	0.365580
$587$	13.7629	0.568055	0.284028	−	0.958816i	$-0.408329\pi$
$587$	13.7629	0.568055	0.284028	+	0.958816i	$0.408329\pi$
$588$	0	0
$589$	55.2618	2.27702
$590$	0	0
$591$	5.30351	0.218157
$592$	−42.8847	−1.76255
$593$	29.6309	1.21679	0.608397	−	0.793633i	$-0.291813\pi$
$593$	29.6309	1.21679	0.608397	+	0.793633i	$0.291813\pi$
$594$	11.1676	0.458212
$595$	0	0
$596$	−62.2839	−2.55125
$597$	−2.27833	−0.0932458
$598$	12.8556	0.525706
$599$	33.1810	1.35574	0.677870	−	0.735181i	$-0.262903\pi$
$599$	33.1810	1.35574	0.677870	+	0.735181i	$0.262903\pi$
$600$	0	0
$601$	27.1161	1.10609	0.553045	−	0.833151i	$-0.313466\pi$
$601$	27.1161	1.10609	0.553045	+	0.833151i	$0.313466\pi$
$602$	0	0
$603$	1.94072	0.0790323
$604$	19.1680	0.779935
$605$	0	0
$606$	15.3380	0.623064
$607$	23.5684	0.956612	0.478306	−	0.878193i	$-0.341251\pi$
$607$	23.5684	0.956612	0.478306	+	0.878193i	$0.341251\pi$
$608$	3.88047	0.157374
$609$	0	0
$610$	0	0
$611$	−4.48376	−0.181394
$612$	16.9170	0.683831
$613$	−18.5190	−0.747977	−0.373989	−	0.927433i	$-0.622010\pi$
$613$	−18.5190	−0.747977	−0.373989	+	0.927433i	$0.622010\pi$
$614$	15.1771	0.612499
$615$	0	0
$616$	0	0
$617$	24.5653	0.988961	0.494480	−	0.869189i	$-0.335358\pi$
$617$	24.5653	0.988961	0.494480	+	0.869189i	$0.335358\pi$
$618$	8.31489	0.334474
$619$	21.2720	0.854994	0.427497	−	0.904017i	$-0.359395\pi$
$619$	21.2720	0.854994	0.427497	+	0.904017i	$0.359395\pi$
$620$	0	0
$621$	3.88144	0.155757
$622$	−52.7222	−2.11397
$623$	0	0
$624$	4.95698	0.198438
$625$	0	0
$626$	−58.2435	−2.32788
$627$	−31.2273	−1.24710
$628$	25.9261	1.03457
$629$	−50.4678	−2.01228
$630$	0	0
$631$	14.3899	0.572851	0.286426	−	0.958102i	$-0.407533\pi$
$631$	14.3899	0.572851	0.286426	+	0.958102i	$0.407533\pi$
$632$	−29.1742	−1.16049
$633$	−14.8052	−0.588454
$634$	−20.8938	−0.829798
$635$	0	0
$636$	37.2657	1.47768
$637$	0	0
$638$	44.6704	1.76852
$639$	11.8680	0.469491
$640$	0	0
$641$	−27.7155	−1.09470	−0.547348	−	0.836905i	$-0.684363\pi$
$641$	−27.7155	−1.09470	−0.547348	+	0.836905i	$0.684363\pi$
$642$	−7.72057	−0.304707
$643$	−44.6065	−1.75911	−0.879554	−	0.475799i	$-0.842159\pi$
$643$	−44.6065	−1.75911	−0.879554	+	0.475799i	$0.842159\pi$
$644$	0	0
$645$	0	0
$646$	−71.3119	−2.80573
$647$	13.2023	0.519037	0.259518	−	0.965738i	$-0.416436\pi$
$647$	13.2023	0.519037	0.259518	+	0.965738i	$0.416436\pi$
$648$	4.73024	0.185821
$649$	3.20911	0.125969
$650$	0	0
$651$	0	0
$652$	−45.1736	−1.76914
$653$	−12.4114	−0.485697	−0.242848	−	0.970064i	$-0.578082\pi$
$653$	−12.4114	−0.485697	−0.242848	+	0.970064i	$0.578082\pi$
$654$	24.9430	0.975347
$655$	0	0
$656$	29.8187	1.16422
$657$	−6.16368	−0.240468
$658$	0	0
$659$	18.3391	0.714390	0.357195	−	0.934030i	$-0.383733\pi$
$659$	18.3391	0.714390	0.357195	+	0.934030i	$0.383733\pi$
$660$	0	0
$661$	−42.9277	−1.66969	−0.834846	−	0.550483i	$-0.814444\pi$
$661$	−42.9277	−1.66969	−0.834846	+	0.550483i	$0.814444\pi$
$662$	32.9275	1.27977
$663$	5.83350	0.226555
$664$	−61.7250	−2.39540
$665$	0	0
$666$	−28.6540	−1.11032
$667$	15.5258	0.601160
$668$	−82.4282	−3.18924
$669$	−6.63479	−0.256516
$670$	0	0
$671$	−22.9092	−0.884400
$672$	0	0
$673$	−17.9400	−0.691537	−0.345768	−	0.938320i	$-0.612382\pi$
$673$	−17.9400	−0.691537	−0.345768	+	0.938320i	$0.612382\pi$
$674$	−27.0586	−1.04226
$675$	0	0
$676$	−43.9526	−1.69049
$677$	25.7077	0.988027	0.494013	−	0.869454i	$-0.335529\pi$
$677$	25.7077	0.988027	0.494013	+	0.869454i	$0.335529\pi$
$678$	7.71104	0.296141
$679$	0	0
$680$	0	0
$681$	9.03168	0.346095
$682$	−90.5504	−3.46735
$683$	16.3100	0.624085	0.312043	−	0.950068i	$-0.398987\pi$
$683$	16.3100	0.624085	0.312043	+	0.950068i	$0.398987\pi$
$684$	−26.8578	−1.02693
$685$	0	0
$686$	0	0
$687$	16.8328	0.642211
$688$	12.8498	0.489893
$689$	12.8503	0.489559
$690$	0	0
$691$	16.8890	0.642489	0.321245	−	0.946996i	$-0.395899\pi$
$691$	16.8890	0.642489	0.321245	+	0.946996i	$0.395899\pi$
$692$	43.9930	1.67236
$693$	0	0
$694$	−66.7452	−2.53361
$695$	0	0
$696$	18.9209	0.717197
$697$	35.0914	1.32918
$698$	15.0586	0.569976
$699$	−18.2929	−0.691900
$700$	0	0
$701$	31.2329	1.17965	0.589825	−	0.807531i	$-0.299197\pi$
$701$	31.2329	1.17965	0.589825	+	0.807531i	$0.299197\pi$
$702$	3.31208	0.125006
$703$	80.1234	3.02191
$704$	−39.7861	−1.49950
$705$	0	0
$706$	19.0241	0.715981
$707$	0	0
$708$	2.76008	0.103730
$709$	16.9064	0.634933	0.317467	−	0.948269i	$-0.397168\pi$
$709$	16.9064	0.634933	0.317467	+	0.948269i	$0.397168\pi$
$710$	0	0
$711$	−6.16760	−0.231303
$712$	1.38538	0.0519193
$713$	−31.4720	−1.17863
$714$	0	0
$715$	0	0
$716$	−69.0112	−2.57907
$717$	11.6309	0.434363
$718$	10.0289	0.374274
$719$	−12.7638	−0.476008	−0.238004	−	0.971264i	$-0.576493\pi$
$719$	−12.7638	−0.476008	−0.238004	+	0.971264i	$0.576493\pi$
$720$	0	0
$721$	0	0
$722$	66.9061	2.48999
$723$	−1.76961	−0.0658126
$724$	79.8598	2.96797
$725$	0	0
$726$	24.3572	0.903981
$727$	−12.5649	−0.466006	−0.233003	−	0.972476i	$-0.574855\pi$
$727$	−12.5649	−0.466006	−0.233003	+	0.972476i	$0.574855\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.1219	0.559305
$732$	−19.7036	−0.728266
$733$	−38.5202	−1.42277	−0.711387	−	0.702800i	$-0.751933\pi$
$733$	−38.5202	−1.42277	−0.711387	+	0.702800i	$0.751933\pi$
$734$	53.1625	1.96227
$735$	0	0
$736$	−2.20996	−0.0814601
$737$	8.89208	0.327544
$738$	19.9238	0.733404
$739$	−4.56729	−0.168011	−0.0840053	−	0.996465i	$-0.526771\pi$
$739$	−4.56729	−0.168011	−0.0840053	+	0.996465i	$0.526771\pi$
$740$	0	0
$741$	−9.26137	−0.340225
$742$	0	0
$743$	36.7043	1.34655	0.673275	−	0.739392i	$-0.264887\pi$
$743$	36.7043	1.34655	0.673275	+	0.739392i	$0.264887\pi$
$744$	−38.3543	−1.40614
$745$	0	0
$746$	−7.16074	−0.262173
$747$	−13.0490	−0.477439
$748$	77.5112	2.83409
$749$	0	0
$750$	0	0
$751$	−54.4092	−1.98542	−0.992710	−	0.120530i	$-0.961541\pi$
$751$	−54.4092	−1.98542	−0.992710	+	0.120530i	$0.961541\pi$
$752$	−12.0364	−0.438924
$753$	−10.7004	−0.389944
$754$	13.2483	0.482475
$755$	0	0
$756$	0	0
$757$	−13.1704	−0.478687	−0.239343	−	0.970935i	$-0.576932\pi$
$757$	−13.1704	−0.478687	−0.239343	+	0.970935i	$0.576932\pi$
$758$	−21.5800	−0.783821
$759$	17.7842	0.645524
$760$	0	0
$761$	48.2724	1.74987	0.874937	−	0.484238i	$-0.160903\pi$
$761$	48.2724	1.74987	0.874937	+	0.484238i	$0.160903\pi$
$762$	−0.0259259	−0.000939196 0
$763$	0	0
$764$	−16.2985	−0.589659
$765$	0	0
$766$	42.9919	1.55336
$767$	0.951757	0.0343660
$768$	−31.4435	−1.13462
$769$	19.7846	0.713452	0.356726	−	0.934209i	$-0.383893\pi$
$769$	19.7846	0.713452	0.356726	+	0.934209i	$0.383893\pi$
$770$	0	0
$771$	−26.1609	−0.942161
$772$	18.7831	0.676017
$773$	−15.7707	−0.567233	−0.283617	−	0.958938i	$-0.591534\pi$
$773$	−15.7707	−0.567233	−0.283617	+	0.958938i	$0.591534\pi$
$774$	8.58575	0.308609
$775$	0	0
$776$	9.49228	0.340753
$777$	0	0
$778$	44.3301	1.58931
$779$	−55.7116	−1.99608
$780$	0	0
$781$	54.3773	1.94577
$782$	40.6127	1.45231
$783$	4.00000	0.142948
$784$	0	0
$785$	0	0
$786$	33.5451	1.19651
$787$	14.6937	0.523773	0.261886	−	0.965099i	$-0.415655\pi$
$787$	14.6937	0.523773	0.261886	+	0.965099i	$0.415655\pi$
$788$	20.8997	0.744520
$789$	−13.3803	−0.476352
$790$	0	0
$791$	0	0
$792$	21.6732	0.770124
$793$	−6.79440	−0.241276
$794$	25.6500	0.910283
$795$	0	0
$796$	−8.97826	−0.318226
$797$	25.3938	0.899493	0.449747	−	0.893156i	$-0.351514\pi$
$797$	25.3938	0.899493	0.449747	+	0.893156i	$0.351514\pi$
$798$	0	0
$799$	−14.1648	−0.501114
$800$	0	0
$801$	0.292877	0.0103483
$802$	8.74663	0.308854
$803$	−28.2410	−0.996604
$804$	7.64784	0.269719
$805$	0	0
$806$	−26.8554	−0.945941
$807$	−2.87081	−0.101057
$808$	29.7668	1.04719
$809$	33.3938	1.17406	0.587031	−	0.809564i	$-0.300297\pi$
$809$	33.3938	1.17406	0.587031	+	0.809564i	$0.300297\pi$
$810$	0	0
$811$	−33.3276	−1.17029	−0.585145	−	0.810929i	$-0.698962\pi$
$811$	−33.3276	−1.17029	−0.585145	+	0.810929i	$0.698962\pi$
$812$	0	0
$813$	29.3419	1.02907
$814$	−131.288	−4.60164
$815$	0	0
$816$	15.6597	0.548201
$817$	−24.0078	−0.839927
$818$	−54.9147	−1.92005
$819$	0	0
$820$	0	0
$821$	52.1538	1.82018	0.910091	−	0.414409i	$-0.136012\pi$
$821$	52.1538	1.82018	0.910091	+	0.414409i	$0.136012\pi$
$822$	−22.0462	−0.768951
$823$	17.0345	0.593785	0.296893	−	0.954911i	$-0.404050\pi$
$823$	17.0345	0.593785	0.296893	+	0.954911i	$0.404050\pi$
$824$	16.1369	0.562155
$825$	0	0
$826$	0	0
$827$	−11.2054	−0.389651	−0.194826	−	0.980838i	$-0.562414\pi$
$827$	−11.2054	−0.389651	−0.194826	+	0.980838i	$0.562414\pi$
$828$	15.2957	0.531562
$829$	20.6693	0.717874	0.358937	−	0.933362i	$-0.383139\pi$
$829$	20.6693	0.717874	0.358937	+	0.933362i	$0.383139\pi$
$830$	0	0
$831$	−19.9830	−0.693204
$832$	−11.7997	−0.409083
$833$	0	0
$834$	−51.9920	−1.80034
$835$	0	0
$836$	−123.058	−4.25605
$837$	−8.10832	−0.280264
$838$	96.0454	3.31783
$839$	−10.3486	−0.357275	−0.178637	−	0.983915i	$-0.557169\pi$
$839$	−10.3486	−0.357275	−0.178637	+	0.983915i	$0.557169\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−20.8911	−0.719955
$843$	7.12919	0.245542
$844$	−58.3432	−2.00826
$845$	0	0
$846$	−8.04232	−0.276501
$847$	0	0
$848$	34.4961	1.18460
$849$	−22.8751	−0.785072
$850$	0	0
$851$	−45.6309	−1.56421
$852$	46.7685	1.60226
$853$	−50.5890	−1.73213	−0.866067	−	0.499929i	$-0.833360\pi$
$853$	−50.5890	−1.73213	−0.866067	+	0.499929i	$0.833360\pi$
$854$	0	0
$855$	0	0
$856$	−14.9835	−0.512125
$857$	−35.8814	−1.22569	−0.612843	−	0.790204i	$-0.709974\pi$
$857$	−35.8814	−1.22569	−0.612843	+	0.790204i	$0.709974\pi$
$858$	15.1754	0.518080
$859$	43.1850	1.47345	0.736726	−	0.676192i	$-0.236371\pi$
$859$	43.1850	1.47345	0.736726	+	0.676192i	$0.236371\pi$
$860$	0	0
$861$	0	0
$862$	3.12528	0.106447
$863$	23.6135	0.803814	0.401907	−	0.915681i	$-0.368348\pi$
$863$	23.6135	0.803814	0.401907	+	0.915681i	$0.368348\pi$
$864$	−0.569365	−0.0193702
$865$	0	0
$866$	−12.7043	−0.431710
$867$	1.42879	0.0485244
$868$	0	0
$869$	−28.2590	−0.958619
$870$	0	0
$871$	2.63721	0.0893584
$872$	48.4073	1.63928
$873$	2.00672	0.0679173
$874$	−64.4773	−2.18098
$875$	0	0
$876$	−24.2894	−0.820662
$877$	32.8397	1.10892	0.554459	−	0.832211i	$-0.312925\pi$
$877$	32.8397	1.10892	0.554459	+	0.832211i	$0.312925\pi$
$878$	47.8407	1.61454
$879$	3.63088	0.122467
$880$	0	0
$881$	−9.72896	−0.327777	−0.163889	−	0.986479i	$-0.552404\pi$
$881$	−9.72896	−0.327777	−0.163889	+	0.986479i	$0.552404\pi$
$882$	0	0
$883$	54.5366	1.83530	0.917651	−	0.397387i	$-0.130083\pi$
$883$	54.5366	1.83530	0.917651	+	0.397387i	$0.130083\pi$
$884$	22.9882	0.773177
$885$	0	0
$886$	43.9571	1.47677
$887$	−49.0168	−1.64582	−0.822912	−	0.568169i	$-0.807652\pi$
$887$	−49.0168	−1.64582	−0.822912	+	0.568169i	$0.807652\pi$
$888$	−55.6094	−1.86613
$889$	0	0
$890$	0	0
$891$	4.58184	0.153497
$892$	−26.1459	−0.875429
$893$	22.4882	0.752540
$894$	−38.5230	−1.28840
$895$	0	0
$896$	0	0
$897$	5.27442	0.176108
$898$	−17.0547	−0.569121
$899$	−32.4333	−1.08171
$900$	0	0
$901$	40.5959	1.35244
$902$	91.2875	3.03954
$903$	0	0
$904$	14.9650	0.497728
$905$	0	0
$906$	11.8555	0.393873
$907$	3.23768	0.107505	0.0537527	−	0.998554i	$-0.482882\pi$
$907$	3.23768	0.107505	0.0537527	+	0.998554i	$0.482882\pi$
$908$	35.5913	1.18114
$909$	6.29288	0.208722
$910$	0	0
$911$	17.8758	0.592252	0.296126	−	0.955149i	$-0.404305\pi$
$911$	17.8758	0.592252	0.296126	+	0.955149i	$0.404305\pi$
$912$	−24.8617	−0.823252
$913$	−59.7886	−1.97872
$914$	−92.2316	−3.05075
$915$	0	0
$916$	66.3334	2.19172
$917$	0	0
$918$	10.4633	0.345340
$919$	−6.26160	−0.206551	−0.103276	−	0.994653i	$-0.532932\pi$
$919$	−6.26160	−0.206551	−0.103276	+	0.994653i	$0.532932\pi$
$920$	0	0
$921$	6.22687	0.205182
$922$	58.7166	1.93373
$923$	16.1272	0.530833
$924$	0	0
$925$	0	0
$926$	−56.3406	−1.85147
$927$	3.41143	0.112046
$928$	−2.27746	−0.0747613
$929$	−9.57512	−0.314149	−0.157075	−	0.987587i	$-0.550206\pi$
$929$	−9.57512	−0.314149	−0.157075	+	0.987587i	$0.550206\pi$
$930$	0	0
$931$	0	0
$932$	−72.0871	−2.36129
$933$	−21.6309	−0.708163
$934$	−30.6242	−1.00205
$935$	0	0
$936$	6.42782	0.210100
$937$	55.8679	1.82512	0.912562	−	0.408939i	$-0.134101\pi$
$937$	55.8679	1.82512	0.912562	+	0.408939i	$0.134101\pi$
$938$	0	0
$939$	−23.8962	−0.779822
$940$	0	0
$941$	27.2830	0.889402	0.444701	−	0.895679i	$-0.353310\pi$
$941$	27.2830	0.889402	0.444701	+	0.895679i	$0.353310\pi$
$942$	16.0355	0.522464
$943$	31.7282	1.03321
$944$	2.55494	0.0831564
$945$	0	0
$946$	39.3386	1.27901
$947$	52.1671	1.69520	0.847601	−	0.530634i	$-0.178046\pi$
$947$	52.1671	1.69520	0.847601	+	0.530634i	$0.178046\pi$
$948$	−24.3048	−0.789383
$949$	−8.37571	−0.271887
$950$	0	0
$951$	−8.57231	−0.277976
$952$	0	0
$953$	−27.4644	−0.889659	−0.444829	−	0.895615i	$-0.646736\pi$
$953$	−27.4644	−0.889659	−0.444829	+	0.895615i	$0.646736\pi$
$954$	23.0490	0.746240
$955$	0	0
$956$	45.8341	1.48238
$957$	18.3274	0.592440
$958$	−4.94981	−0.159921
$959$	0	0
$960$	0	0
$961$	34.7448	1.12080
$962$	−38.9373	−1.25539
$963$	−3.16760	−0.102074
$964$	−6.97355	−0.224603
$965$	0	0
$966$	0	0
$967$	−45.8901	−1.47573	−0.737864	−	0.674950i	$-0.764165\pi$
$967$	−45.8901	−1.47573	−0.737864	+	0.674950i	$0.764165\pi$
$968$	47.2706	1.51933
$969$	−29.2579	−0.939898
$970$	0	0
$971$	39.0569	1.25339	0.626697	−	0.779263i	$-0.284406\pi$
$971$	39.0569	1.25339	0.626697	+	0.779263i	$0.284406\pi$
$972$	3.94072	0.126399
$973$	0	0
$974$	−67.0824	−2.14946
$975$	0	0
$976$	−18.2392	−0.583823
$977$	9.12137	0.291818	0.145909	−	0.989298i	$-0.453389\pi$
$977$	9.12137	0.291818	0.145909	+	0.989298i	$0.453389\pi$
$978$	−27.9401	−0.893427
$979$	1.34192	0.0428879
$980$	0	0
$981$	10.2336	0.326734
$982$	1.11966	0.0357297
$983$	−45.7038	−1.45772	−0.728862	−	0.684661i	$-0.759950\pi$
$983$	−45.7038	−1.45772	−0.728862	+	0.684661i	$0.759950\pi$
$984$	38.6665	1.23264
$985$	0	0
$986$	41.8531	1.33288
$987$	0	0
$988$	−36.4965	−1.16111
$989$	13.6726	0.434764
$990$	0	0
$991$	−12.7798	−0.405965	−0.202983	−	0.979182i	$-0.565064\pi$
$991$	−12.7798	−0.405965	−0.202983	+	0.979182i	$0.565064\pi$
$992$	4.61659	0.146577
$993$	13.5095	0.428711
$994$	0	0
$995$	0	0
$996$	−51.4226	−1.62939
$997$	4.09136	0.129575	0.0647873	−	0.997899i	$-0.479363\pi$
$997$	4.09136	0.129575	0.0647873	+	0.997899i	$0.479363\pi$
$998$	−28.3667	−0.897934
$999$	−11.7562	−0.371949

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.br.1.4		4
5.4	even	2		3675.2.a.bw.1.1		4
7.2	even	3		525.2.i.j.151.1	yes	8
7.4	even	3		525.2.i.j.226.1	yes	8
7.6	odd	2		3675.2.a.bq.1.4		4
35.2	odd	12		525.2.r.h.424.7		16
35.4	even	6		525.2.i.i.226.4	yes	8
35.9	even	6		525.2.i.i.151.4	✓	8
35.18	odd	12		525.2.r.h.499.7		16
35.23	odd	12		525.2.r.h.424.2		16
35.32	odd	12		525.2.r.h.499.2		16
35.34	odd	2		3675.2.a.bx.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.i.i.151.4	✓	8	35.9	even	6
525.2.i.i.226.4	yes	8	35.4	even	6
525.2.i.j.151.1	yes	8	7.2	even	3
525.2.i.j.226.1	yes	8	7.4	even	3
525.2.r.h.424.2		16	35.23	odd	12
525.2.r.h.424.7		16	35.2	odd	12
525.2.r.h.499.2		16	35.32	odd	12
525.2.r.h.499.7		16	35.18	odd	12
3675.2.a.bq.1.4		4	7.6	odd	2
3675.2.a.br.1.4		4	1.1	even	1	trivial
3675.2.a.bw.1.1		4	5.4	even	2
3675.2.a.bx.1.1		4	35.34	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)

\(f(q)\)	\(=\)	\(q+2.43736 q^{2} +1.00000 q^{3} +3.94072 q^{4} +2.43736 q^{6} +4.73024 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.43736 q^{2} +1.00000 q^{3} +3.94072 q^{4} +2.43736 q^{6} +4.73024 q^{8} +1.00000 q^{9} +4.58184 q^{11} +3.94072 q^{12} +1.35888 q^{13} +3.64784 q^{16} +4.29288 q^{17} +2.43736 q^{18} -6.81544 q^{19} +11.1676 q^{22} +3.88144 q^{23} +4.73024 q^{24} +3.31208 q^{26} +1.00000 q^{27} +4.00000 q^{29} -8.10832 q^{31} -0.569365 q^{32} +4.58184 q^{33} +10.4633 q^{34} +3.94072 q^{36} -11.7562 q^{37} -16.6117 q^{38} +1.35888 q^{39} +8.17432 q^{41} +3.52256 q^{43} +18.0558 q^{44} +9.46047 q^{46} -3.29960 q^{47} +3.64784 q^{48} +4.29288 q^{51} +5.35497 q^{52} +9.45656 q^{53} +2.43736 q^{54} -6.81544 q^{57} +9.74944 q^{58} +0.700398 q^{59} -5.00000 q^{61} -19.7629 q^{62} -8.68344 q^{64} +11.1676 q^{66} +1.94072 q^{67} +16.9170 q^{68} +3.88144 q^{69} +11.8680 q^{71} +4.73024 q^{72} -6.16368 q^{73} -28.6540 q^{74} -26.8578 q^{76} +3.31208 q^{78} -6.16760 q^{79} +1.00000 q^{81} +19.9238 q^{82} -13.0490 q^{83} +8.58575 q^{86} +4.00000 q^{87} +21.6732 q^{88} +0.292877 q^{89} +15.2957 q^{92} -8.10832 q^{93} -8.04232 q^{94} -0.569365 q^{96} +2.00672 q^{97} +4.58184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 4 q^{3} + 7 q^{4} - q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - q^2 + 4 * q^3 + 7 * q^4 - q^6 - 3 * q^8 + 4 * q^9	\( 4 q - q^{2} + 4 q^{3} + 7 q^{4} - q^{6} - 3 q^{8} + 4 q^{9} + 8 q^{11} + 7 q^{12} + 7 q^{13} + 17 q^{16} + 6 q^{17} - q^{18} + 3 q^{19} + 12 q^{22} - 2 q^{23} - 3 q^{24} - 19 q^{26} + 4 q^{27} + 16 q^{29} + 9 q^{31} - 17 q^{32} + 8 q^{33} + 14 q^{34} + 7 q^{36} - 8 q^{37} - 27 q^{38} + 7 q^{39} + 4 q^{41} - 5 q^{43} + 26 q^{44} - 6 q^{46} - 6 q^{47} + 17 q^{48} + 6 q^{51} + 35 q^{52} + 6 q^{53} - q^{54} + 3 q^{57} - 4 q^{58} + 10 q^{59} - 20 q^{61} - 44 q^{62} + 21 q^{64} + 12 q^{66} - q^{67} - 8 q^{68} - 2 q^{69} + 22 q^{71} - 3 q^{72} - 4 q^{73} - 21 q^{74} - 23 q^{76} - 19 q^{78} + 8 q^{79} + 4 q^{81} + 8 q^{82} - 2 q^{83} + 12 q^{86} + 16 q^{87} - 28 q^{88} - 10 q^{89} + 66 q^{92} + 9 q^{93} + 22 q^{94} - 17 q^{96} + 12 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q - q^2 + 4 * q^3 + 7 * q^4 - q^6 - 3 * q^8 + 4 * q^9 + 8 * q^11 + 7 * q^12 + 7 * q^13 + 17 * q^16 + 6 * q^17 - q^18 + 3 * q^19 + 12 * q^22 - 2 * q^23 - 3 * q^24 - 19 * q^26 + 4 * q^27 + 16 * q^29 + 9 * q^31 - 17 * q^32 + 8 * q^33 + 14 * q^34 + 7 * q^36 - 8 * q^37 - 27 * q^38 + 7 * q^39 + 4 * q^41 - 5 * q^43 + 26 * q^44 - 6 * q^46 - 6 * q^47 + 17 * q^48 + 6 * q^51 + 35 * q^52 + 6 * q^53 - q^54 + 3 * q^57 - 4 * q^58 + 10 * q^59 - 20 * q^61 - 44 * q^62 + 21 * q^64 + 12 * q^66 - q^67 - 8 * q^68 - 2 * q^69 + 22 * q^71 - 3 * q^72 - 4 * q^73 - 21 * q^74 - 23 * q^76 - 19 * q^78 + 8 * q^79 + 4 * q^81 + 8 * q^82 - 2 * q^83 + 12 * q^86 + 16 * q^87 - 28 * q^88 - 10 * q^89 + 66 * q^92 + 9 * q^93 + 22 * q^94 - 17 * q^96 + 12 * q^97 + 8 * q^99

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