LMFDB - Embedded newform 3675.2.a.bm.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			$-0.360409$ 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)

$f(q)$	$=$	$q-1.36041 q^{2} -1.00000 q^{3} -0.149286 q^{4} +1.36041 q^{6} +2.92391 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.36041 q^{2} -1.00000 q^{3} -0.149286 q^{4} +1.36041 q^{6} +2.92391 q^{8} +1.00000 q^{9} +0.625336 q^{11} +0.149286 q^{12} -0.211123 q^{13} -3.67914 q^{16} +0.904518 q^{17} -1.36041 q^{18} -0.808269 q^{19} -0.850714 q^{22} -5.75234 q^{23} -2.92391 q^{24} +0.287214 q^{26} -1.00000 q^{27} +3.22248 q^{29} +9.68428 q^{31} -0.842681 q^{32} -0.625336 q^{33} -1.23051 q^{34} -0.149286 q^{36} -2.89239 q^{37} +1.09958 q^{38} +0.211123 q^{39} +11.8478 q^{41} +9.94631 q^{43} -0.0933543 q^{44} +7.82553 q^{46} -1.36663 q^{47} +3.67914 q^{48} -0.904518 q^{51} +0.0315178 q^{52} -2.56864 q^{53} +1.36041 q^{54} +0.808269 q^{57} -4.38389 q^{58} -7.07833 q^{59} +5.17157 q^{61} -13.1746 q^{62} +8.50467 q^{64} +0.850714 q^{66} -14.6956 q^{67} -0.135032 q^{68} +5.75234 q^{69} -3.41345 q^{71} +2.92391 q^{72} -10.9148 q^{73} +3.93484 q^{74} +0.120664 q^{76} -0.287214 q^{78} -13.8387 q^{79} +1.00000 q^{81} -16.1179 q^{82} -10.0620 q^{83} -13.5311 q^{86} -3.22248 q^{87} +1.82843 q^{88} +7.02349 q^{89} +0.858746 q^{92} -9.68428 q^{93} +1.85918 q^{94} +0.842681 q^{96} -10.0114 q^{97} +0.625336 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.36041	−0.961955	−0.480977	−	0.876733i	$-0.659718\pi$
$2$	−1.36041	−0.961955	−0.480977	+	0.876733i	$0.659718\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.149286	−0.0746432
$5$	0	0
$5$	0	0
$6$	1.36041	0.555385
$7$	0	0
$7$	0	0
$8$	2.92391	1.03376
$9$	1.00000	0.333333
$10$	0	0
$11$	0.625336	0.188546	0.0942730	−	0.995546i	$-0.469947\pi$
$11$	0.625336	0.188546	0.0942730	+	0.995546i	$0.469947\pi$
$12$	0.149286	0.0430953
$13$	−0.211123	−0.0585550	−0.0292775	−	0.999571i	$-0.509321\pi$
$13$	−0.211123	−0.0585550	−0.0292775	+	0.999571i	$0.509321\pi$
$14$	0	0
$15$	0	0
$16$	−3.67914	−0.919785
$17$	0.904518	0.219378	0.109689	−	0.993966i	$-0.465015\pi$
$17$	0.904518	0.219378	0.109689	+	0.993966i	$0.465015\pi$
$18$	−1.36041	−0.320652
$19$	−0.808269	−0.185430	−0.0927148	−	0.995693i	$-0.529554\pi$
$19$	−0.808269	−0.185430	−0.0927148	+	0.995693i	$0.529554\pi$
$20$	0	0
$21$	0	0
$22$	−0.850714	−0.181373
$23$	−5.75234	−1.19945	−0.599723	−	0.800208i	$-0.704722\pi$
$23$	−5.75234	−1.19945	−0.599723	+	0.800208i	$0.704722\pi$
$24$	−2.92391	−0.596840
$25$	0	0
$26$	0.287214	0.0563272
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.22248	0.598400	0.299200	−	0.954190i	$-0.403280\pi$
$29$	3.22248	0.598400	0.299200	+	0.954190i	$0.403280\pi$
$30$	0	0
$31$	9.68428	1.73935	0.869674	−	0.493627i	$-0.164329\pi$
$31$	9.68428	1.73935	0.869674	+	0.493627i	$0.164329\pi$
$32$	−0.842681	−0.148966
$33$	−0.625336	−0.108857
$34$	−1.23051	−0.211031
$35$	0	0
$36$	−0.149286	−0.0248811
$37$	−2.89239	−0.475506	−0.237753	−	0.971326i	$-0.576411\pi$
$37$	−2.89239	−0.475506	−0.237753	+	0.971326i	$0.576411\pi$
$38$	1.09958	0.178375
$39$	0.211123	0.0338067
$40$	0	0
$41$	11.8478	1.85032	0.925159	−	0.379579i	$-0.123931\pi$
$41$	11.8478	1.85032	0.925159	+	0.379579i	$0.123931\pi$
$42$	0	0
$43$	9.94631	1.51680	0.758399	−	0.651791i	$-0.225982\pi$
$43$	9.94631	1.51680	0.758399	+	0.651791i	$0.225982\pi$
$44$	−0.0933543	−0.0140737
$45$	0	0
$46$	7.82553	1.15381
$47$	−1.36663	−0.199344	−0.0996718	−	0.995020i	$-0.531779\pi$
$47$	−1.36663	−0.199344	−0.0996718	+	0.995020i	$0.531779\pi$
$48$	3.67914	0.531038
$49$	0	0
$50$	0	0
$51$	−0.904518	−0.126658
$52$	0.0315178	0.00437073
$53$	−2.56864	−0.352829	−0.176415	−	0.984316i	$-0.556450\pi$
$53$	−2.56864	−0.352829	−0.176415	+	0.984316i	$0.556450\pi$
$54$	1.36041	0.185128
$55$	0	0
$56$	0	0
$57$	0.808269	0.107058
$58$	−4.38389	−0.575634
$59$	−7.07833	−0.921520	−0.460760	−	0.887525i	$-0.652423\pi$
$59$	−7.07833	−0.921520	−0.460760	+	0.887525i	$0.652423\pi$
$60$	0	0
$61$	5.17157	0.662152	0.331076	−	0.943604i	$-0.392588\pi$
$61$	5.17157	0.662152	0.331076	+	0.943604i	$0.392588\pi$
$62$	−13.1746	−1.67317
$63$	0	0
$64$	8.50467	1.06308
$65$	0	0
$66$	0.850714	0.104716
$67$	−14.6956	−1.79536	−0.897679	−	0.440650i	$-0.854748\pi$
$67$	−14.6956	−1.79536	−0.897679	+	0.440650i	$0.854748\pi$
$68$	−0.135032	−0.0163751
$69$	5.75234	0.692500
$70$	0	0
$71$	−3.41345	−0.405102	−0.202551	−	0.979272i	$-0.564923\pi$
$71$	−3.41345	−0.405102	−0.202551	+	0.979272i	$0.564923\pi$
$72$	2.92391	0.344586
$73$	−10.9148	−1.27748	−0.638740	−	0.769423i	$-0.720544\pi$
$73$	−10.9148	−1.27748	−0.638740	+	0.769423i	$0.720544\pi$
$74$	3.93484	0.457415
$75$	0	0
$76$	0.120664	0.0138411
$77$	0	0
$78$	−0.287214	−0.0325205
$79$	−13.8387	−1.55698	−0.778488	−	0.627660i	$-0.784013\pi$
$79$	−13.8387	−1.55698	−0.778488	+	0.627660i	$0.784013\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−16.1179	−1.77992
$83$	−10.0620	−1.10444	−0.552221	−	0.833698i	$-0.686220\pi$
$83$	−10.0620	−1.10444	−0.552221	+	0.833698i	$0.686220\pi$
$84$	0	0
$85$	0	0
$86$	−13.5311	−1.45909
$87$	−3.22248	−0.345486
$88$	1.82843	0.194911
$89$	7.02349	0.744488	0.372244	−	0.928135i	$-0.378588\pi$
$89$	7.02349	0.744488	0.372244	+	0.928135i	$0.378588\pi$
$90$	0	0
$91$	0	0
$92$	0.858746	0.0895304
$93$	−9.68428	−1.00421
$94$	1.85918	0.191760
$95$	0	0
$96$	0.842681	0.0860058
$97$	−10.0114	−1.01650	−0.508250	−	0.861210i	$-0.669707\pi$
$97$	−10.0114	−1.01650	−0.508250	+	0.861210i	$0.669707\pi$
$98$	0	0
$99$	0.625336	0.0628487
$100$	0	0
$101$	4.26706	0.424588	0.212294	−	0.977206i	$-0.431907\pi$
$101$	4.26706	0.424588	0.212294	+	0.977206i	$0.431907\pi$
$102$	1.23051	0.121839
$103$	7.63670	0.752466	0.376233	−	0.926525i	$-0.377219\pi$
$103$	7.63670	0.752466	0.376233	+	0.926525i	$0.377219\pi$
$104$	−0.617304	−0.0605317
$105$	0	0
$106$	3.49440	0.339406
$107$	16.8790	1.63176	0.815878	−	0.578224i	$-0.196254\pi$
$107$	16.8790	1.63176	0.815878	+	0.578224i	$0.196254\pi$
$108$	0.149286	0.0143651
$109$	−6.33403	−0.606690	−0.303345	−	0.952881i	$-0.598103\pi$
$109$	−6.33403	−0.606690	−0.303345	+	0.952881i	$0.598103\pi$
$110$	0	0
$111$	2.89239	0.274534
$112$	0	0
$113$	17.2570	1.62340	0.811701	−	0.584073i	$-0.198542\pi$
$113$	17.2570	1.62340	0.811701	+	0.584073i	$0.198542\pi$
$114$	−1.09958	−0.102985
$115$	0	0
$116$	−0.481073	−0.0446665
$117$	−0.211123	−0.0195183
$118$	9.62943	0.886461
$119$	0	0
$120$	0	0
$121$	−10.6090	−0.964450
$122$	−7.03546	−0.636960
$123$	−11.8478	−1.06828
$124$	−1.44573	−0.129831
$125$	0	0
$126$	0	0
$127$	−10.0597	−0.892655	−0.446327	−	0.894870i	$-0.647268\pi$
$127$	−10.0597	−0.892655	−0.446327	+	0.894870i	$0.647268\pi$
$128$	−9.88447	−0.873672
$129$	−9.94631	−0.875724
$130$	0	0
$131$	9.55651	0.834956	0.417478	−	0.908687i	$-0.362914\pi$
$131$	9.55651	0.834956	0.417478	+	0.908687i	$0.362914\pi$
$132$	0.0933543	0.00812544
$133$	0	0
$134$	19.9921	1.72705
$135$	0	0
$136$	2.64473	0.226784
$137$	0.180690	0.0154374	0.00771868	−	0.999970i	$-0.497543\pi$
$137$	0.180690	0.0154374	0.00771868	+	0.999970i	$0.497543\pi$
$138$	−7.82553	−0.666154
$139$	0.933026	0.0791382	0.0395691	−	0.999217i	$-0.487401\pi$
$139$	0.933026	0.0791382	0.0395691	+	0.999217i	$0.487401\pi$
$140$	0	0
$141$	1.36663	0.115091
$142$	4.64368	0.389689
$143$	−0.132023	−0.0110403
$144$	−3.67914	−0.306595
$145$	0	0
$146$	14.8486	1.22888
$147$	0	0
$148$	0.431795	0.0354933
$149$	−2.31070	−0.189300	−0.0946499	−	0.995511i	$-0.530173\pi$
$149$	−2.31070	−0.189300	−0.0946499	+	0.995511i	$0.530173\pi$
$150$	0	0
$151$	8.46589	0.688944	0.344472	−	0.938797i	$-0.388058\pi$
$151$	8.46589	0.688944	0.344472	+	0.938797i	$0.388058\pi$
$152$	−2.36330	−0.191689
$153$	0.904518	0.0731259
$154$	0	0
$155$	0	0
$156$	−0.0315178	−0.00252344
$157$	10.3088	0.822736	0.411368	−	0.911469i	$-0.365051\pi$
$157$	10.3088	0.822736	0.411368	+	0.911469i	$0.365051\pi$
$158$	18.8263	1.49774
$159$	2.56864	0.203706
$160$	0	0
$161$	0	0
$162$	−1.36041	−0.106884
$163$	−4.97635	−0.389778	−0.194889	−	0.980825i	$-0.562435\pi$
$163$	−4.97635	−0.389778	−0.194889	+	0.980825i	$0.562435\pi$
$164$	−1.76872	−0.138114
$165$	0	0
$166$	13.6884	1.06242
$167$	−1.61398	−0.124893	−0.0624467	−	0.998048i	$-0.519890\pi$
$167$	−1.61398	−0.124893	−0.0624467	+	0.998048i	$0.519890\pi$
$168$	0	0
$169$	−12.9554	−0.996571
$170$	0	0
$171$	−0.808269	−0.0618098
$172$	−1.48485	−0.113219
$173$	21.1978	1.61164	0.805818	−	0.592164i	$-0.201726\pi$
$173$	21.1978	1.61164	0.805818	+	0.592164i	$0.201726\pi$
$174$	4.38389	0.332342
$175$	0	0
$176$	−2.30070	−0.173422
$177$	7.07833	0.532040
$178$	−9.55482	−0.716164
$179$	21.2288	1.58672	0.793358	−	0.608755i	$-0.208331\pi$
$179$	21.2288	1.58672	0.793358	+	0.608755i	$0.208331\pi$
$180$	0	0
$181$	15.3941	1.14423	0.572116	−	0.820173i	$-0.306123\pi$
$181$	15.3941	1.14423	0.572116	+	0.820173i	$0.306123\pi$
$182$	0	0
$183$	−5.17157	−0.382294
$184$	−16.8193	−1.23994
$185$	0	0
$186$	13.1746	0.966007
$187$	0.565628	0.0413628
$188$	0.204020	0.0148797
$189$	0	0
$190$	0	0
$191$	−4.71656	−0.341279	−0.170639	−	0.985334i	$-0.554583\pi$
$191$	−4.71656	−0.341279	−0.170639	+	0.985334i	$0.554583\pi$
$192$	−8.50467	−0.613772
$193$	−4.56169	−0.328358	−0.164179	−	0.986431i	$-0.552497\pi$
$193$	−4.56169	−0.328358	−0.164179	+	0.986431i	$0.552497\pi$
$194$	13.6195	0.977826
$195$	0	0
$196$	0	0
$197$	4.41012	0.314208	0.157104	−	0.987582i	$-0.449784\pi$
$197$	4.41012	0.314208	0.157104	+	0.987582i	$0.449784\pi$
$198$	−0.850714	−0.0604576
$199$	22.9690	1.62823	0.814116	−	0.580703i	$-0.197222\pi$
$199$	22.9690	1.62823	0.814116	+	0.580703i	$0.197222\pi$
$200$	0	0
$201$	14.6956	1.03655
$202$	−5.80494	−0.408434
$203$	0	0
$204$	0.135032	0.00945415
$205$	0	0
$206$	−10.3890	−0.723838
$207$	−5.75234	−0.399815
$208$	0.776751	0.0538580
$209$	−0.505440	−0.0349620
$210$	0	0
$211$	10.5401	0.725612	0.362806	−	0.931865i	$-0.381819\pi$
$211$	10.5401	0.725612	0.362806	+	0.931865i	$0.381819\pi$
$212$	0.383463	0.0263363
$213$	3.41345	0.233886
$214$	−22.9624	−1.56968
$215$	0	0
$216$	−2.92391	−0.198947
$217$	0	0
$218$	8.61687	0.583608
$219$	10.9148	0.737553
$220$	0	0
$221$	−0.190964	−0.0128457
$222$	−3.93484	−0.264089
$223$	22.6507	1.51681	0.758403	−	0.651786i	$-0.225980\pi$
$223$	22.6507	1.51681	0.758403	+	0.651786i	$0.225980\pi$
$224$	0	0
$225$	0	0
$226$	−23.4766	−1.56164
$227$	−4.35326	−0.288936	−0.144468	−	0.989509i	$-0.546147\pi$
$227$	−4.35326	−0.288936	−0.144468	+	0.989509i	$0.546147\pi$
$228$	−0.120664	−0.00799114
$229$	−19.8813	−1.31379	−0.656895	−	0.753982i	$-0.728131\pi$
$229$	−19.8813	−1.31379	−0.656895	+	0.753982i	$0.728131\pi$
$230$	0	0
$231$	0	0
$232$	9.42225	0.618601
$233$	−5.87601	−0.384950	−0.192475	−	0.981302i	$-0.561651\pi$
$233$	−5.87601	−0.384950	−0.192475	+	0.981302i	$0.561651\pi$
$234$	0.287214	0.0187757
$235$	0	0
$236$	1.05670	0.0687853
$237$	13.8387	0.898920
$238$	0	0
$239$	−15.7587	−1.01934	−0.509672	−	0.860369i	$-0.670233\pi$
$239$	−15.7587	−1.01934	−0.509672	+	0.860369i	$0.670233\pi$
$240$	0	0
$241$	18.9101	1.21811	0.609053	−	0.793130i	$-0.291550\pi$
$241$	18.9101	1.21811	0.609053	+	0.793130i	$0.291550\pi$
$242$	14.4325	0.927758
$243$	−1.00000	−0.0641500
$244$	−0.772046	−0.0494252
$245$	0	0
$246$	16.1179	1.02764
$247$	0.170644	0.0108578
$248$	28.3160	1.79806
$249$	10.0620	0.637650
$250$	0	0
$251$	27.4916	1.73526	0.867628	−	0.497214i	$-0.165644\pi$
$251$	27.4916	1.73526	0.867628	+	0.497214i	$0.165644\pi$
$252$	0	0
$253$	−3.59715	−0.226151
$254$	13.6853	0.858693
$255$	0	0
$256$	−3.56242	−0.222651
$257$	20.1411	1.25636	0.628182	−	0.778066i	$-0.283799\pi$
$257$	20.1411	1.25636	0.628182	+	0.778066i	$0.283799\pi$
$258$	13.5311	0.842407
$259$	0	0
$260$	0	0
$261$	3.22248	0.199467
$262$	−13.0008	−0.803190
$263$	22.6641	1.39753	0.698765	−	0.715352i	$-0.253733\pi$
$263$	22.6641	1.39753	0.698765	+	0.715352i	$0.253733\pi$
$264$	−1.82843	−0.112532
$265$	0	0
$266$	0	0
$267$	−7.02349	−0.429830
$268$	2.19386	0.134011
$269$	17.5218	1.06832	0.534162	−	0.845382i	$-0.320627\pi$
$269$	17.5218	1.06832	0.534162	+	0.845382i	$0.320627\pi$
$270$	0	0
$271$	−14.5830	−0.885855	−0.442927	−	0.896558i	$-0.646060\pi$
$271$	−14.5830	−0.885855	−0.442927	+	0.896558i	$0.646060\pi$
$272$	−3.32785	−0.201780
$273$	0	0
$274$	−0.245812	−0.0148500
$275$	0	0
$276$	−0.858746	−0.0516904
$277$	0.176434	0.0106009	0.00530044	−	0.999986i	$-0.498313\pi$
$277$	0.176434	0.0106009	0.00530044	+	0.999986i	$0.498313\pi$
$278$	−1.26930	−0.0761274
$279$	9.68428	0.579783
$280$	0	0
$281$	18.8896	1.12686	0.563430	−	0.826164i	$-0.309482\pi$
$281$	18.8896	1.12686	0.563430	+	0.826164i	$0.309482\pi$
$282$	−1.85918	−0.110712
$283$	−13.6925	−0.813933	−0.406966	−	0.913443i	$-0.633413\pi$
$283$	−13.6925	−0.813933	−0.406966	+	0.913443i	$0.633413\pi$
$284$	0.509581	0.0302381
$285$	0	0
$286$	0.179605	0.0106203
$287$	0	0
$288$	−0.842681	−0.0496555
$289$	−16.1818	−0.951873
$290$	0	0
$291$	10.0114	0.586876
$292$	1.62943	0.0953552
$293$	−19.8191	−1.15785	−0.578924	−	0.815382i	$-0.696527\pi$
$293$	−19.8191	−1.15785	−0.578924	+	0.815382i	$0.696527\pi$
$294$	0	0
$295$	0	0
$296$	−8.45709	−0.491558
$297$	−0.625336	−0.0362857
$298$	3.14350	0.182098
$299$	1.21445	0.0702335
$300$	0	0
$301$	0	0
$302$	−11.5171	−0.662733
$303$	−4.26706	−0.245136
$304$	2.97373	0.170555
$305$	0	0
$306$	−1.23051	−0.0703438
$307$	2.98941	0.170615	0.0853073	−	0.996355i	$-0.472813\pi$
$307$	2.98941	0.170615	0.0853073	+	0.996355i	$0.472813\pi$
$308$	0	0
$309$	−7.63670	−0.434436
$310$	0	0
$311$	−24.3361	−1.37997	−0.689987	−	0.723822i	$-0.742384\pi$
$311$	−24.3361	−1.37997	−0.689987	+	0.723822i	$0.742384\pi$
$312$	0.617304	0.0349480
$313$	3.15141	0.178128	0.0890642	−	0.996026i	$-0.471612\pi$
$313$	3.15141	0.178128	0.0890642	+	0.996026i	$0.471612\pi$
$314$	−14.0243	−0.791434
$315$	0	0
$316$	2.06593	0.116218
$317$	26.7944	1.50493	0.752463	−	0.658634i	$-0.228866\pi$
$317$	26.7944	1.50493	0.752463	+	0.658634i	$0.228866\pi$
$318$	−3.49440	−0.195956
$319$	2.01514	0.112826
$320$	0	0
$321$	−16.8790	−0.942095
$322$	0	0
$323$	−0.731093	−0.0406791
$324$	−0.149286	−0.00829369
$325$	0	0
$326$	6.76988	0.374949
$327$	6.33403	0.350273
$328$	34.6419	1.91278
$329$	0	0
$330$	0	0
$331$	2.89239	0.158980	0.0794901	−	0.996836i	$-0.474671\pi$
$331$	2.89239	0.158980	0.0794901	+	0.996836i	$0.474671\pi$
$332$	1.50211	0.0824391
$333$	−2.89239	−0.158502
$334$	2.19567	0.120142
$335$	0	0
$336$	0	0
$337$	−22.8387	−1.24410	−0.622052	−	0.782976i	$-0.713701\pi$
$337$	−22.8387	−1.24410	−0.622052	+	0.782976i	$0.713701\pi$
$338$	17.6247	0.958656
$339$	−17.2570	−0.937272
$340$	0	0
$341$	6.05593	0.327947
$342$	1.09958	0.0594583
$343$	0	0
$344$	29.0821	1.56800
$345$	0	0
$346$	−28.8376	−1.55032
$347$	4.56230	0.244917	0.122459	−	0.992474i	$-0.460922\pi$
$347$	4.56230	0.244917	0.122459	+	0.992474i	$0.460922\pi$
$348$	0.481073	0.0257882
$349$	21.8680	1.17057	0.585283	−	0.810829i	$-0.300983\pi$
$349$	21.8680	1.17057	0.585283	+	0.810829i	$0.300983\pi$
$350$	0	0
$351$	0.211123	0.0112689
$352$	−0.526959	−0.0280870
$353$	29.5424	1.57238	0.786191	−	0.617984i	$-0.212050\pi$
$353$	29.5424	1.57238	0.786191	+	0.617984i	$0.212050\pi$
$354$	−9.62943	−0.511798
$355$	0	0
$356$	−1.04851	−0.0555710
$357$	0	0
$358$	−28.8799	−1.52635
$359$	6.34948	0.335113	0.167556	−	0.985862i	$-0.446412\pi$
$359$	6.34948	0.335113	0.167556	+	0.985862i	$0.446412\pi$
$360$	0	0
$361$	−18.3467	−0.965616
$362$	−20.9422	−1.10070
$363$	10.6090	0.556826
$364$	0	0
$365$	0	0
$366$	7.03546	0.367749
$367$	33.5595	1.75179	0.875896	−	0.482499i	$-0.160271\pi$
$367$	33.5595	1.75179	0.875896	+	0.482499i	$0.160271\pi$
$368$	21.1637	1.10323
$369$	11.8478	0.616773
$370$	0	0
$371$	0	0
$372$	1.44573	0.0749577
$373$	19.1845	0.993338	0.496669	−	0.867940i	$-0.334556\pi$
$373$	19.1845	0.993338	0.496669	+	0.867940i	$0.334556\pi$
$374$	−0.769486	−0.0397892
$375$	0	0
$376$	−3.99591	−0.206073
$377$	−0.680340	−0.0350393
$378$	0	0
$379$	−23.5586	−1.21012	−0.605062	−	0.796179i	$-0.706852\pi$
$379$	−23.5586	−1.21012	−0.605062	+	0.796179i	$0.706852\pi$
$380$	0	0
$381$	10.0597	0.515374
$382$	6.41646	0.328294
$383$	0.465123	0.0237667	0.0118833	−	0.999929i	$-0.496217\pi$
$383$	0.465123	0.0237667	0.0118833	+	0.999929i	$0.496217\pi$
$384$	9.88447	0.504415
$385$	0	0
$386$	6.20577	0.315865
$387$	9.94631	0.505599
$388$	1.49456	0.0758748
$389$	32.1506	1.63010	0.815049	−	0.579392i	$-0.196710\pi$
$389$	32.1506	1.63010	0.815049	+	0.579392i	$0.196710\pi$
$390$	0	0
$391$	−5.20309	−0.263132
$392$	0	0
$393$	−9.55651	−0.482062
$394$	−5.99957	−0.302254
$395$	0	0
$396$	−0.0933543	−0.00469123
$397$	6.31355	0.316868	0.158434	−	0.987370i	$-0.449355\pi$
$397$	6.31355	0.316868	0.158434	+	0.987370i	$0.449355\pi$
$398$	−31.2473	−1.56628
$399$	0	0
$400$	0	0
$401$	12.0912	0.603807	0.301904	−	0.953338i	$-0.402378\pi$
$401$	12.0912	0.603807	0.301904	+	0.953338i	$0.402378\pi$
$402$	−19.9921	−0.997114
$403$	−2.04457	−0.101847
$404$	−0.637013	−0.0316926
$405$	0	0
$406$	0	0
$407$	−1.80872	−0.0896548
$408$	−2.64473	−0.130934
$409$	24.3839	1.20570	0.602852	−	0.797853i	$-0.294031\pi$
$409$	24.3839	1.20570	0.602852	+	0.797853i	$0.294031\pi$
$410$	0	0
$411$	−0.180690	−0.00891277
$412$	−1.14006	−0.0561665
$413$	0	0
$414$	7.82553	0.384604
$415$	0	0
$416$	0.177909	0.00872272
$417$	−0.933026	−0.0456905
$418$	0.687605	0.0336319
$419$	−12.5084	−0.611078	−0.305539	−	0.952180i	$-0.598837\pi$
$419$	−12.5084	−0.611078	−0.305539	+	0.952180i	$0.598837\pi$
$420$	0	0
$421$	22.4804	1.09563	0.547814	−	0.836600i	$-0.315460\pi$
$421$	22.4804	1.09563	0.547814	+	0.836600i	$0.315460\pi$
$422$	−14.3389	−0.698006
$423$	−1.36663	−0.0664479
$424$	−7.51046	−0.364740
$425$	0	0
$426$	−4.64368	−0.224987
$427$	0	0
$428$	−2.51981	−0.121800
$429$	0.132023	0.00637412
$430$	0	0
$431$	−27.2433	−1.31227	−0.656133	−	0.754645i	$-0.727809\pi$
$431$	−27.2433	−1.31227	−0.656133	+	0.754645i	$0.727809\pi$
$432$	3.67914	0.177013
$433$	20.2358	0.972469	0.486234	−	0.873828i	$-0.338370\pi$
$433$	20.2358	0.972469	0.486234	+	0.873828i	$0.338370\pi$
$434$	0	0
$435$	0	0
$436$	0.945585	0.0452853
$437$	4.64943	0.222413
$438$	−14.8486	−0.709493
$439$	−8.82280	−0.421089	−0.210545	−	0.977584i	$-0.567524\pi$
$439$	−8.82280	−0.421089	−0.210545	+	0.977584i	$0.567524\pi$
$440$	0	0
$441$	0	0
$442$	0.259790	0.0123569
$443$	1.34647	0.0639729	0.0319864	−	0.999488i	$-0.489817\pi$
$443$	1.34647	0.0639729	0.0319864	+	0.999488i	$0.489817\pi$
$444$	−0.431795	−0.0204921
$445$	0	0
$446$	−30.8143	−1.45910
$447$	2.31070	0.109292
$448$	0	0
$449$	−22.3270	−1.05368	−0.526838	−	0.849966i	$-0.676623\pi$
$449$	−22.3270	−1.05368	−0.526838	+	0.849966i	$0.676623\pi$
$450$	0	0
$451$	7.40887	0.348870
$452$	−2.57624	−0.121176
$453$	−8.46589	−0.397762
$454$	5.92221	0.277943
$455$	0	0
$456$	2.36330	0.110672
$457$	−1.74933	−0.0818301	−0.0409150	−	0.999163i	$-0.513027\pi$
$457$	−1.74933	−0.0818301	−0.0409150	+	0.999163i	$0.513027\pi$
$458$	27.0467	1.26381
$459$	−0.904518	−0.0422193
$460$	0	0
$461$	−18.9086	−0.880659	−0.440330	−	0.897836i	$-0.645138\pi$
$461$	−18.9086	−0.880659	−0.440330	+	0.897836i	$0.645138\pi$
$462$	0	0
$463$	−16.9345	−0.787013	−0.393507	−	0.919322i	$-0.628738\pi$
$463$	−16.9345	−0.787013	−0.393507	+	0.919322i	$0.628738\pi$
$464$	−11.8560	−0.550399
$465$	0	0
$466$	7.99378	0.370305
$467$	25.3293	1.17210	0.586051	−	0.810274i	$-0.300682\pi$
$467$	25.3293	1.17210	0.586051	+	0.810274i	$0.300682\pi$
$468$	0.0315178	0.00145691
$469$	0	0
$470$	0	0
$471$	−10.3088	−0.475007
$472$	−20.6964	−0.952629
$473$	6.21979	0.285986
$474$	−18.8263	−0.864721
$475$	0	0
$476$	0	0
$477$	−2.56864	−0.117610
$478$	21.4382	0.980563
$479$	32.6636	1.49244	0.746220	−	0.665700i	$-0.231867\pi$
$479$	32.6636	1.49244	0.746220	+	0.665700i	$0.231867\pi$
$480$	0	0
$481$	0.610650	0.0278432
$482$	−25.7255	−1.17176
$483$	0	0
$484$	1.58377	0.0719897
$485$	0	0
$486$	1.36041	0.0617094
$487$	−26.8032	−1.21457	−0.607286	−	0.794484i	$-0.707742\pi$
$487$	−26.8032	−1.21457	−0.607286	+	0.794484i	$0.707742\pi$
$488$	15.1212	0.684505
$489$	4.97635	0.225038
$490$	0	0
$491$	−37.3134	−1.68393	−0.841965	−	0.539532i	$-0.818601\pi$
$491$	−37.3134	−1.68393	−0.841965	+	0.539532i	$0.818601\pi$
$492$	1.76872	0.0797400
$493$	2.91479	0.131276
$494$	−0.232146	−0.0104447
$495$	0	0
$496$	−35.6298	−1.59983
$497$	0	0
$498$	−13.6884	−0.613391
$499$	23.3137	1.04366	0.521832	−	0.853048i	$-0.325249\pi$
$499$	23.3137	1.04366	0.521832	+	0.853048i	$0.325249\pi$
$500$	0	0
$501$	1.61398	0.0721072
$502$	−37.3999	−1.66924
$503$	−9.11717	−0.406515	−0.203257	−	0.979125i	$-0.565153\pi$
$503$	−9.11717	−0.406515	−0.203257	+	0.979125i	$0.565153\pi$
$504$	0	0
$505$	0	0
$506$	4.89359	0.217547
$507$	12.9554	0.575371
$508$	1.50178	0.0666306
$509$	31.7539	1.40747	0.703734	−	0.710463i	$-0.251515\pi$
$509$	31.7539	1.40747	0.703734	+	0.710463i	$0.251515\pi$
$510$	0	0
$511$	0	0
$512$	24.6153	1.08785
$513$	0.808269	0.0356859
$514$	−27.4001	−1.20857
$515$	0	0
$516$	1.48485	0.0653668
$517$	−0.854604	−0.0375855
$518$	0	0
$519$	−21.1978	−0.930478
$520$	0	0
$521$	−16.5457	−0.724880	−0.362440	−	0.932007i	$-0.618056\pi$
$521$	−16.5457	−0.724880	−0.362440	+	0.932007i	$0.618056\pi$
$522$	−4.38389	−0.191878
$523$	−17.5844	−0.768912	−0.384456	−	0.923143i	$-0.625611\pi$
$523$	−17.5844	−0.768912	−0.384456	+	0.923143i	$0.625611\pi$
$524$	−1.42666	−0.0623238
$525$	0	0
$526$	−30.8325	−1.34436
$527$	8.75960	0.381574
$528$	2.30070	0.100125
$529$	10.0894	0.438668
$530$	0	0
$531$	−7.07833	−0.307173
$532$	0	0
$533$	−2.50135	−0.108345
$534$	9.55482	0.413477
$535$	0	0
$536$	−42.9687	−1.85597
$537$	−21.2288	−0.916091
$538$	−23.8369	−1.02768
$539$	0	0
$540$	0	0
$541$	26.1733	1.12528	0.562640	−	0.826702i	$-0.309786\pi$
$541$	26.1733	1.12528	0.562640	+	0.826702i	$0.309786\pi$
$542$	19.8389	0.852152
$543$	−15.3941	−0.660622
$544$	−0.762220	−0.0326799
$545$	0	0
$546$	0	0
$547$	3.76114	0.160815	0.0804073	−	0.996762i	$-0.474378\pi$
$547$	3.76114	0.160815	0.0804073	+	0.996762i	$0.474378\pi$
$548$	−0.0269745	−0.00115229
$549$	5.17157	0.220717
$550$	0	0
$551$	−2.60463	−0.110961
$552$	16.8193	0.715877
$553$	0	0
$554$	−0.240022	−0.0101976
$555$	0	0
$556$	−0.139288	−0.00590713
$557$	37.9105	1.60632	0.803160	−	0.595763i	$-0.203150\pi$
$557$	37.9105	1.60632	0.803160	+	0.595763i	$0.203150\pi$
$558$	−13.1746	−0.557725
$559$	−2.09989	−0.0888160
$560$	0	0
$561$	−0.565628	−0.0238808
$562$	−25.6976	−1.08399
$563$	−9.36362	−0.394630	−0.197315	−	0.980340i	$-0.563222\pi$
$563$	−9.36362	−0.394630	−0.197315	+	0.980340i	$0.563222\pi$
$564$	−0.204020	−0.00859077
$565$	0	0
$566$	18.6274	0.782966
$567$	0	0
$568$	−9.98061	−0.418777
$569$	22.5319	0.944588	0.472294	−	0.881441i	$-0.343426\pi$
$569$	22.5319	0.944588	0.472294	+	0.881441i	$0.343426\pi$
$570$	0	0
$571$	14.6898	0.614751	0.307375	−	0.951588i	$-0.400549\pi$
$571$	14.6898	0.614751	0.307375	+	0.951588i	$0.400549\pi$
$572$	0.0197092	0.000824084 0
$573$	4.71656	0.197037
$574$	0	0
$575$	0	0
$576$	8.50467	0.354361
$577$	27.7400	1.15483	0.577417	−	0.816450i	$-0.304061\pi$
$577$	27.7400	1.15483	0.577417	+	0.816450i	$0.304061\pi$
$578$	22.0139	0.915659
$579$	4.56169	0.189577
$580$	0	0
$581$	0	0
$582$	−13.6195	−0.564548
$583$	−1.60626	−0.0665246
$584$	−31.9139	−1.32061
$585$	0	0
$586$	26.9622	1.11380
$587$	36.8359	1.52038	0.760191	−	0.649700i	$-0.225106\pi$
$587$	36.8359	1.52038	0.760191	+	0.649700i	$0.225106\pi$
$588$	0	0
$589$	−7.82750	−0.322526
$590$	0	0
$591$	−4.41012	−0.181408
$592$	10.6415	0.437364
$593$	24.2036	0.993924	0.496962	−	0.867772i	$-0.334449\pi$
$593$	24.2036	0.993924	0.496962	+	0.867772i	$0.334449\pi$
$594$	0.850714	0.0349052
$595$	0	0
$596$	0.344956	0.0141299
$597$	−22.9690	−0.940060
$598$	−1.65215	−0.0675614
$599$	22.0912	0.902623	0.451312	−	0.892366i	$-0.350956\pi$
$599$	22.0912	0.902623	0.451312	+	0.892366i	$0.350956\pi$
$600$	0	0
$601$	−29.9672	−1.22239	−0.611195	−	0.791480i	$-0.709311\pi$
$601$	−29.9672	−1.22239	−0.611195	+	0.791480i	$0.709311\pi$
$602$	0	0
$603$	−14.6956	−0.598453
$604$	−1.26384	−0.0514250
$605$	0	0
$606$	5.80494	0.235810
$607$	20.9665	0.851003	0.425501	−	0.904958i	$-0.360098\pi$
$607$	20.9665	0.851003	0.425501	+	0.904958i	$0.360098\pi$
$608$	0.681113	0.0276228
$609$	0	0
$610$	0	0
$611$	0.288527	0.0116726
$612$	−0.135032	−0.00545835
$613$	−31.3288	−1.26536	−0.632680	−	0.774413i	$-0.718045\pi$
$613$	−31.3288	−1.26536	−0.632680	+	0.774413i	$0.718045\pi$
$614$	−4.06682	−0.164123
$615$	0	0
$616$	0	0
$617$	−0.143987	−0.00579671	−0.00289835	−	0.999996i	$-0.500923\pi$
$617$	−0.143987	−0.00579671	−0.00289835	+	0.999996i	$0.500923\pi$
$618$	10.3890	0.417908
$619$	−4.01011	−0.161180	−0.0805900	−	0.996747i	$-0.525680\pi$
$619$	−4.01011	−0.161180	−0.0805900	+	0.996747i	$0.525680\pi$
$620$	0	0
$621$	5.75234	0.230833
$622$	33.1071	1.32747
$623$	0	0
$624$	−0.776751	−0.0310949
$625$	0	0
$626$	−4.28721	−0.171352
$627$	0.505440	0.0201853
$628$	−1.53897	−0.0614116
$629$	−2.61622	−0.104316
$630$	0	0
$631$	5.80843	0.231230	0.115615	−	0.993294i	$-0.463116\pi$
$631$	5.80843	0.231230	0.115615	+	0.993294i	$0.463116\pi$
$632$	−40.4631	−1.60954
$633$	−10.5401	−0.418933
$634$	−36.4514	−1.44767
$635$	0	0
$636$	−0.383463	−0.0152053
$637$	0	0
$638$	−2.74141	−0.108533
$639$	−3.41345	−0.135034
$640$	0	0
$641$	−4.67384	−0.184606	−0.0923028	−	0.995731i	$-0.529423\pi$
$641$	−4.67384	−0.184606	−0.0923028	+	0.995731i	$0.529423\pi$
$642$	22.9624	0.906253
$643$	−27.0355	−1.06617	−0.533087	−	0.846060i	$-0.678968\pi$
$643$	−27.0355	−1.06617	−0.533087	+	0.846060i	$0.678968\pi$
$644$	0	0
$645$	0	0
$646$	0.994586	0.0391315
$647$	22.9840	0.903595	0.451797	−	0.892121i	$-0.350783\pi$
$647$	22.9840	0.903595	0.451797	+	0.892121i	$0.350783\pi$
$648$	2.92391	0.114862
$649$	−4.42634	−0.173749
$650$	0	0
$651$	0	0
$652$	0.742902	0.0290943
$653$	24.6559	0.964861	0.482430	−	0.875934i	$-0.339754\pi$
$653$	24.6559	0.964861	0.482430	+	0.875934i	$0.339754\pi$
$654$	−8.61687	−0.336946
$655$	0	0
$656$	−43.5898	−1.70190
$657$	−10.9148	−0.425827
$658$	0	0
$659$	14.7353	0.574008	0.287004	−	0.957929i	$-0.407341\pi$
$659$	14.7353	0.574008	0.287004	+	0.957929i	$0.407341\pi$
$660$	0	0
$661$	29.5233	1.14832	0.574162	−	0.818742i	$-0.305328\pi$
$661$	29.5233	1.14832	0.574162	+	0.818742i	$0.305328\pi$
$662$	−3.93484	−0.152932
$663$	0.190964	0.00741644
$664$	−29.4202	−1.14173
$665$	0	0
$666$	3.93484	0.152472
$667$	−18.5368	−0.717748
$668$	0.240945	0.00932244
$669$	−22.6507	−0.875728
$670$	0	0
$671$	3.23397	0.124846
$672$	0	0
$673$	33.6065	1.29544	0.647718	−	0.761881i	$-0.275724\pi$
$673$	33.6065	1.29544	0.647718	+	0.761881i	$0.275724\pi$
$674$	31.0700	1.19677
$675$	0	0
$676$	1.93407	0.0743873
$677$	1.19691	0.0460010	0.0230005	−	0.999735i	$-0.492678\pi$
$677$	1.19691	0.0460010	0.0230005	+	0.999735i	$0.492678\pi$
$678$	23.4766	0.901613
$679$	0	0
$680$	0	0
$681$	4.35326	0.166817
$682$	−8.23855	−0.315470
$683$	−2.10576	−0.0805745	−0.0402873	−	0.999188i	$-0.512827\pi$
$683$	−2.10576	−0.0805745	−0.0402873	+	0.999188i	$0.512827\pi$
$684$	0.120664	0.00461369
$685$	0	0
$686$	0	0
$687$	19.8813	0.758517
$688$	−36.5939	−1.39513
$689$	0.542298	0.0206599
$690$	0	0
$691$	−36.3961	−1.38457	−0.692287	−	0.721622i	$-0.743397\pi$
$691$	−36.3961	−1.38457	−0.692287	+	0.721622i	$0.743397\pi$
$692$	−3.16454	−0.120298
$693$	0	0
$694$	−6.20660	−0.235599
$695$	0	0
$696$	−9.42225	−0.357149
$697$	10.7166	0.405919
$698$	−29.7494	−1.12603
$699$	5.87601	0.222251
$700$	0	0
$701$	−44.0551	−1.66394	−0.831969	−	0.554823i	$-0.812786\pi$
$701$	−44.0551	−1.66394	−0.831969	+	0.554823i	$0.812786\pi$
$702$	−0.287214	−0.0108402
$703$	2.33783	0.0881729
$704$	5.31828	0.200440
$705$	0	0
$706$	−40.1897	−1.51256
$707$	0	0
$708$	−1.05670	−0.0397132
$709$	32.1400	1.20704	0.603521	−	0.797347i	$-0.293764\pi$
$709$	32.1400	1.20704	0.603521	+	0.797347i	$0.293764\pi$
$710$	0	0
$711$	−13.8387	−0.518992
$712$	20.5360	0.769620
$713$	−55.7072	−2.08625
$714$	0	0
$715$	0	0
$716$	−3.16917	−0.118438
$717$	15.7587	0.588518
$718$	−8.63789	−0.322363
$719$	−27.9109	−1.04090	−0.520450	−	0.853892i	$-0.674236\pi$
$719$	−27.9109	−1.04090	−0.520450	+	0.853892i	$0.674236\pi$
$720$	0	0
$721$	0	0
$722$	24.9590	0.928879
$723$	−18.9101	−0.703274
$724$	−2.29812	−0.0854091
$725$	0	0
$726$	−14.4325	−0.535641
$727$	44.6347	1.65541	0.827704	−	0.561164i	$-0.189646\pi$
$727$	44.6347	1.65541	0.827704	+	0.561164i	$0.189646\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.99661	0.332752
$732$	0.772046	0.0285356
$733$	−26.7598	−0.988397	−0.494198	−	0.869349i	$-0.664538\pi$
$733$	−26.7598	−0.988397	−0.494198	+	0.869349i	$0.664538\pi$
$734$	−45.6547	−1.68515
$735$	0	0
$736$	4.84739	0.178677
$737$	−9.18972	−0.338508
$738$	−16.1179	−0.593307
$739$	31.8841	1.17288	0.586439	−	0.809994i	$-0.300530\pi$
$739$	31.8841	1.17288	0.586439	+	0.809994i	$0.300530\pi$
$740$	0	0
$741$	−0.170644	−0.00626876
$742$	0	0
$743$	34.8572	1.27879	0.639393	−	0.768880i	$-0.279186\pi$
$743$	34.8572	1.27879	0.639393	+	0.768880i	$0.279186\pi$
$744$	−28.3160	−1.03811
$745$	0	0
$746$	−26.0988	−0.955546
$747$	−10.0620	−0.368148
$748$	−0.0844406	−0.00308745
$749$	0	0
$750$	0	0
$751$	10.4385	0.380908	0.190454	−	0.981696i	$-0.439004\pi$
$751$	10.4385	0.380908	0.190454	+	0.981696i	$0.439004\pi$
$752$	5.02803	0.183353
$753$	−27.4916	−1.00185
$754$	0.925541	0.0337062
$755$	0	0
$756$	0	0
$757$	7.19096	0.261360	0.130680	−	0.991425i	$-0.458284\pi$
$757$	7.19096	0.261360	0.130680	+	0.991425i	$0.458284\pi$
$758$	32.0493	1.16408
$759$	3.59715	0.130568
$760$	0	0
$761$	28.5401	1.03458	0.517289	−	0.855811i	$-0.326941\pi$
$761$	28.5401	1.03458	0.517289	+	0.855811i	$0.326941\pi$
$762$	−13.6853	−0.495767
$763$	0	0
$764$	0.704119	0.0254741
$765$	0	0
$766$	−0.632758	−0.0228625
$767$	1.49440	0.0539596
$768$	3.56242	0.128548
$769$	18.8426	0.679483	0.339741	−	0.940519i	$-0.389660\pi$
$769$	18.8426	0.679483	0.339741	+	0.940519i	$0.389660\pi$
$770$	0	0
$771$	−20.1411	−0.725362
$772$	0.680998	0.0245097
$773$	−39.6039	−1.42445	−0.712226	−	0.701950i	$-0.752313\pi$
$773$	−39.6039	−1.42445	−0.712226	+	0.701950i	$0.752313\pi$
$774$	−13.5311	−0.486364
$775$	0	0
$776$	−29.2723	−1.05081
$777$	0	0
$778$	−43.7379	−1.56808
$779$	−9.57622	−0.343104
$780$	0	0
$781$	−2.13455	−0.0763803
$782$	7.07833	0.253121
$783$	−3.22248	−0.115162
$784$	0	0
$785$	0	0
$786$	13.0008	0.463722
$787$	−37.4046	−1.33333	−0.666664	−	0.745358i	$-0.732279\pi$
$787$	−37.4046	−1.33333	−0.666664	+	0.745358i	$0.732279\pi$
$788$	−0.658371	−0.0234535
$789$	−22.6641	−0.806864
$790$	0	0
$791$	0	0
$792$	1.82843	0.0649703
$793$	−1.09184	−0.0387723
$794$	−8.58902	−0.304813
$795$	0	0
$796$	−3.42896	−0.121536
$797$	−21.9848	−0.778741	−0.389370	−	0.921081i	$-0.627307\pi$
$797$	−21.9848	−0.778741	−0.389370	+	0.921081i	$0.627307\pi$
$798$	0	0
$799$	−1.23614	−0.0437316
$800$	0	0
$801$	7.02349	0.248163
$802$	−16.4490	−0.580835
$803$	−6.82542	−0.240864
$804$	−2.19386	−0.0773715
$805$	0	0
$806$	2.78146	0.0979726
$807$	−17.5218	−0.616797
$808$	12.4765	0.438921
$809$	−50.0650	−1.76019	−0.880096	−	0.474795i	$-0.842522\pi$
$809$	−50.0650	−1.76019	−0.880096	+	0.474795i	$0.842522\pi$
$810$	0	0
$811$	−4.52428	−0.158869	−0.0794346	−	0.996840i	$-0.525311\pi$
$811$	−4.52428	−0.158869	−0.0794346	+	0.996840i	$0.525311\pi$
$812$	0	0
$813$	14.5830	0.511448
$814$	2.46060	0.0862439
$815$	0	0
$816$	3.32785	0.116498
$817$	−8.03929	−0.281259
$818$	−33.1720	−1.15983
$819$	0	0
$820$	0	0
$821$	41.2809	1.44071	0.720357	−	0.693603i	$-0.243978\pi$
$821$	41.2809	1.44071	0.720357	+	0.693603i	$0.243978\pi$
$822$	0.245812	0.00857368
$823$	−20.1761	−0.703293	−0.351647	−	0.936133i	$-0.614378\pi$
$823$	−20.1761	−0.703293	−0.351647	+	0.936133i	$0.614378\pi$
$824$	22.3290	0.777868
$825$	0	0
$826$	0	0
$827$	14.3883	0.500329	0.250165	−	0.968203i	$-0.419515\pi$
$827$	14.3883	0.500329	0.250165	+	0.968203i	$0.419515\pi$
$828$	0.858746	0.0298435
$829$	−31.6680	−1.09987	−0.549937	−	0.835206i	$-0.685348\pi$
$829$	−31.6680	−1.09987	−0.549937	+	0.835206i	$0.685348\pi$
$830$	0	0
$831$	−0.176434	−0.00612042
$832$	−1.79553	−0.0622488
$833$	0	0
$834$	1.26930	0.0439522
$835$	0	0
$836$	0.0754553	0.00260968
$837$	−9.68428	−0.334738
$838$	17.0166	0.587829
$839$	−30.8602	−1.06541	−0.532706	−	0.846301i	$-0.678825\pi$
$839$	−30.8602	−1.06541	−0.532706	+	0.846301i	$0.678825\pi$
$840$	0	0
$841$	−18.6156	−0.641918
$842$	−30.5826	−1.05395
$843$	−18.8896	−0.650593
$844$	−1.57350	−0.0541620
$845$	0	0
$846$	1.85918	0.0639199
$847$	0	0
$848$	9.45038	0.324527
$849$	13.6925	0.469924
$850$	0	0
$851$	16.6380	0.570344
$852$	−0.509581	−0.0174580
$853$	−0.143993	−0.00493024	−0.00246512	−	0.999997i	$-0.500785\pi$
$853$	−0.143993	−0.00493024	−0.00246512	+	0.999997i	$0.500785\pi$
$854$	0	0
$855$	0	0
$856$	49.3527	1.68684
$857$	−10.4122	−0.355674	−0.177837	−	0.984060i	$-0.556910\pi$
$857$	−10.4122	−0.355674	−0.177837	+	0.984060i	$0.556910\pi$
$858$	−0.179605	−0.00613162
$859$	8.16201	0.278484	0.139242	−	0.990258i	$-0.455533\pi$
$859$	8.16201	0.278484	0.139242	+	0.990258i	$0.455533\pi$
$860$	0	0
$861$	0	0
$862$	37.0621	1.26234
$863$	15.8130	0.538280	0.269140	−	0.963101i	$-0.413261\pi$
$863$	15.8130	0.538280	0.269140	+	0.963101i	$0.413261\pi$
$864$	0.842681	0.0286686
$865$	0	0
$866$	−27.5289	−0.935471
$867$	16.1818	0.549564
$868$	0	0
$869$	−8.65384	−0.293562
$870$	0	0
$871$	3.10259	0.105127
$872$	−18.5201	−0.627171
$873$	−10.0114	−0.338833
$874$	−6.32513	−0.213951
$875$	0	0
$876$	−1.62943	−0.0550534
$877$	−31.3204	−1.05761	−0.528807	−	0.848742i	$-0.677360\pi$
$877$	−31.3204	−1.05761	−0.528807	+	0.848742i	$0.677360\pi$
$878$	12.0026	0.405069
$879$	19.8191	0.668483
$880$	0	0
$881$	37.0691	1.24889	0.624445	−	0.781069i	$-0.285325\pi$
$881$	37.0691	1.24889	0.624445	+	0.781069i	$0.285325\pi$
$882$	0	0
$883$	24.2030	0.814497	0.407248	−	0.913317i	$-0.366488\pi$
$883$	24.2030	0.814497	0.407248	+	0.913317i	$0.366488\pi$
$884$	0.0285084	0.000958841 0
$885$	0	0
$886$	−1.83175	−0.0615390
$887$	15.4029	0.517177	0.258589	−	0.965988i	$-0.416743\pi$
$887$	15.4029	0.517177	0.258589	+	0.965988i	$0.416743\pi$
$888$	8.45709	0.283801
$889$	0	0
$890$	0	0
$891$	0.625336	0.0209496
$892$	−3.38145	−0.113219
$893$	1.10461	0.0369642
$894$	−3.14350	−0.105134
$895$	0	0
$896$	0	0
$897$	−1.21445	−0.0405493
$898$	30.3738	1.01359
$899$	31.2074	1.04083
$900$	0	0
$901$	−2.32338	−0.0774029
$902$	−10.0791	−0.335597
$903$	0	0
$904$	50.4579	1.67821
$905$	0	0
$906$	11.5171	0.382629
$907$	−3.00848	−0.0998950	−0.0499475	−	0.998752i	$-0.515905\pi$
$907$	−3.00848	−0.0998950	−0.0499475	+	0.998752i	$0.515905\pi$
$908$	0.649882	0.0215671
$909$	4.26706	0.141529
$910$	0	0
$911$	−28.9381	−0.958763	−0.479381	−	0.877607i	$-0.659139\pi$
$911$	−28.9381	−0.958763	−0.479381	+	0.877607i	$0.659139\pi$
$912$	−2.97373	−0.0984702
$913$	−6.29210	−0.208238
$914$	2.37980	0.0787168
$915$	0	0
$916$	2.96800	0.0980656
$917$	0	0
$918$	1.23051	0.0406130
$919$	28.9517	0.955029	0.477515	−	0.878624i	$-0.341538\pi$
$919$	28.9517	0.955029	0.477515	+	0.878624i	$0.341538\pi$
$920$	0	0
$921$	−2.98941	−0.0985043
$922$	25.7234	0.847154
$923$	0.720657	0.0237207
$924$	0	0
$925$	0	0
$926$	23.0379	0.757071
$927$	7.63670	0.250822
$928$	−2.71553	−0.0891415
$929$	−39.5868	−1.29880	−0.649400	−	0.760447i	$-0.724980\pi$
$929$	−39.5868	−1.29880	−0.649400	+	0.760447i	$0.724980\pi$
$930$	0	0
$931$	0	0
$932$	0.877208	0.0287339
$933$	24.3361	0.796728
$934$	−34.4582	−1.12751
$935$	0	0
$936$	−0.617304	−0.0201772
$937$	−12.9075	−0.421671	−0.210835	−	0.977522i	$-0.567618\pi$
$937$	−12.9075	−0.421671	−0.210835	+	0.977522i	$0.567618\pi$
$938$	0	0
$939$	−3.15141	−0.102843
$940$	0	0
$941$	7.14777	0.233011	0.116505	−	0.993190i	$-0.462831\pi$
$941$	7.14777	0.233011	0.116505	+	0.993190i	$0.462831\pi$
$942$	14.0243	0.456935
$943$	−68.1526	−2.21936
$944$	26.0422	0.847601
$945$	0	0
$946$	−8.46146	−0.275106
$947$	8.89123	0.288926	0.144463	−	0.989510i	$-0.453854\pi$
$947$	8.89123	0.288926	0.144463	+	0.989510i	$0.453854\pi$
$948$	−2.06593	−0.0670983
$949$	2.30436	0.0748028
$950$	0	0
$951$	−26.7944	−0.868869
$952$	0	0
$953$	5.47648	0.177401	0.0887003	−	0.996058i	$-0.471729\pi$
$953$	5.47648	0.177401	0.0887003	+	0.996058i	$0.471729\pi$
$954$	3.49440	0.113135
$955$	0	0
$956$	2.35256	0.0760871
$957$	−2.01514	−0.0651401
$958$	−44.4359	−1.43566
$959$	0	0
$960$	0	0
$961$	62.7852	2.02533
$962$	−0.830734	−0.0267839
$963$	16.8790	0.543919
$964$	−2.82302	−0.0909233
$965$	0	0
$966$	0	0
$967$	−52.7941	−1.69774	−0.848872	−	0.528598i	$-0.822718\pi$
$967$	−52.7941	−1.69774	−0.848872	+	0.528598i	$0.822718\pi$
$968$	−31.0196	−0.997008
$969$	0.731093	0.0234861
$970$	0	0
$971$	−47.6330	−1.52862	−0.764308	−	0.644852i	$-0.776919\pi$
$971$	−47.6330	−1.52862	−0.764308	+	0.644852i	$0.776919\pi$
$972$	0.149286	0.00478836
$973$	0	0
$974$	36.4634	1.16836
$975$	0	0
$976$	−19.0269	−0.609038
$977$	44.2561	1.41588	0.707939	−	0.706274i	$-0.249625\pi$
$977$	44.2561	1.41588	0.707939	+	0.706274i	$0.249625\pi$
$978$	−6.76988	−0.216477
$979$	4.39204	0.140370
$980$	0	0
$981$	−6.33403	−0.202230
$982$	50.7615	1.61986
$983$	−13.6315	−0.434778	−0.217389	−	0.976085i	$-0.569754\pi$
$983$	−13.6315	−0.434778	−0.217389	+	0.976085i	$0.569754\pi$
$984$	−34.6419	−1.10434
$985$	0	0
$986$	−3.96531	−0.126281
$987$	0	0
$988$	−0.0254748	−0.000810463 0
$989$	−57.2145	−1.81932
$990$	0	0
$991$	31.4182	0.998033	0.499016	−	0.866593i	$-0.333695\pi$
$991$	31.4182	0.998033	0.499016	+	0.866593i	$0.333695\pi$
$992$	−8.16076	−0.259104
$993$	−2.89239	−0.0917873
$994$	0	0
$995$	0	0
$996$	−1.50211	−0.0475963
$997$	−45.2107	−1.43184	−0.715919	−	0.698183i	$-0.753992\pi$
$997$	−45.2107	−1.43184	−0.715919	+	0.698183i	$0.753992\pi$
$998$	−31.7162	−1.00396
$999$	2.89239	0.0915112

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.36041 q^{2} -1.00000 q^{3} -0.149286 q^{4} +1.36041 q^{6} +2.92391 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.36041 q^{2} -1.00000 q^{3} -0.149286 q^{4} +1.36041 q^{6} +2.92391 q^{8} +1.00000 q^{9} +0.625336 q^{11} +0.149286 q^{12} -0.211123 q^{13} -3.67914 q^{16} +0.904518 q^{17} -1.36041 q^{18} -0.808269 q^{19} -0.850714 q^{22} -5.75234 q^{23} -2.92391 q^{24} +0.287214 q^{26} -1.00000 q^{27} +3.22248 q^{29} +9.68428 q^{31} -0.842681 q^{32} -0.625336 q^{33} -1.23051 q^{34} -0.149286 q^{36} -2.89239 q^{37} +1.09958 q^{38} +0.211123 q^{39} +11.8478 q^{41} +9.94631 q^{43} -0.0933543 q^{44} +7.82553 q^{46} -1.36663 q^{47} +3.67914 q^{48} -0.904518 q^{51} +0.0315178 q^{52} -2.56864 q^{53} +1.36041 q^{54} +0.808269 q^{57} -4.38389 q^{58} -7.07833 q^{59} +5.17157 q^{61} -13.1746 q^{62} +8.50467 q^{64} +0.850714 q^{66} -14.6956 q^{67} -0.135032 q^{68} +5.75234 q^{69} -3.41345 q^{71} +2.92391 q^{72} -10.9148 q^{73} +3.93484 q^{74} +0.120664 q^{76} -0.287214 q^{78} -13.8387 q^{79} +1.00000 q^{81} -16.1179 q^{82} -10.0620 q^{83} -13.5311 q^{86} -3.22248 q^{87} +1.82843 q^{88} +7.02349 q^{89} +0.858746 q^{92} -9.68428 q^{93} +1.85918 q^{94} +0.842681 q^{96} -10.0114 q^{97} +0.625336 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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