LMFDB - Embedded newform 1089.3.c.k.604.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.523388583936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 $ x^8 + 28x^6 + 262x^4 + 948*x^2 + 1089
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 11 $
Twist minimal:	no (minimal twist has level 121)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.7
Root			$2.89067i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.k.604.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.89067i q^{2} -4.35597 q^{4} +1.96778 q^{5} +9.76707i q^{7} -1.02900i q^{8} +O(q^{10})$	$q+2.89067i q^{2} -4.35597 q^{4} +1.96778 q^{5} +9.76707i q^{7} -1.02900i q^{8} +5.68820i q^{10} +3.56414i q^{13} -28.2334 q^{14} -14.4494 q^{16} -4.57982i q^{17} +21.8968i q^{19} -8.57160 q^{20} +35.9581 q^{23} -21.1278 q^{25} -10.3028 q^{26} -42.5451i q^{28} -6.35955i q^{29} +4.74568 q^{31} -45.8844i q^{32} +13.2388 q^{34} +19.2194i q^{35} -48.8126 q^{37} -63.2963 q^{38} -2.02485i q^{40} +55.1891i q^{41} +35.4777i q^{43} +103.943i q^{46} +50.1176 q^{47} -46.3956 q^{49} -61.0736i q^{50} -15.5253i q^{52} -21.9347 q^{53} +10.0504 q^{56} +18.3834 q^{58} -32.5364 q^{59} -79.0369i q^{61} +13.7182i q^{62} +74.8392 q^{64} +7.01346i q^{65} +33.2026 q^{67} +19.9496i q^{68} -55.5571 q^{70} -94.4045 q^{71} -14.3988i q^{73} -141.101i q^{74} -95.3817i q^{76} -19.7648i q^{79} -28.4332 q^{80} -159.533 q^{82} -63.1249i q^{83} -9.01208i q^{85} -102.554 q^{86} -42.3567 q^{89} -34.8112 q^{91} -156.632 q^{92} +144.874i q^{94} +43.0880i q^{95} -176.984 q^{97} -134.114i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} + 4 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 + 4 * q^5	$ 8 q - 24 q^{4} + 4 q^{5} + 4 q^{14} - 24 q^{16} - 52 q^{20} + 12 q^{23} - 16 q^{25} - 168 q^{26} - 116 q^{31} - 180 q^{34} - 4 q^{37} + 132 q^{38} + 244 q^{47} + 88 q^{49} - 268 q^{53} + 12 q^{56} + 88 q^{58} + 56 q^{59} - 40 q^{64} + 284 q^{67} - 188 q^{70} - 272 q^{71} + 356 q^{80} - 180 q^{82} - 336 q^{86} + 24 q^{89} + 140 q^{91} + 156 q^{92} + 152 q^{97}+O(q^{100}) $ 8 * q - 24 * q^4 + 4 * q^5 + 4 * q^14 - 24 * q^16 - 52 * q^20 + 12 * q^23 - 16 * q^25 - 168 * q^26 - 116 * q^31 - 180 * q^34 - 4 * q^37 + 132 * q^38 + 244 * q^47 + 88 * q^49 - 268 * q^53 + 12 * q^56 + 88 * q^58 + 56 * q^59 - 40 * q^64 + 284 * q^67 - 188 * q^70 - 272 * q^71 + 356 * q^80 - 180 * q^82 - 336 * q^86 + 24 * q^89 + 140 * q^91 + 156 * q^92 + 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.89067i	1.44534i	0.691196	+	0.722668i	$0.257084\pi$
$2$	2.89067i	1.44534i	−0.691196	+	0.722668i	$0.742916\pi$
$3$	0	0
$3$	0	0
$4$	−4.35597	−1.08899
$5$	1.96778	0.393556	0.196778	−	0.980448i	$-0.436952\pi$
$5$	1.96778	0.393556	0.196778	+	0.980448i	$0.436952\pi$
$6$	0	0
$7$	9.76707i	1.39530i	0.716441	+	0.697648i	$0.245770\pi$
$7$	9.76707i	1.39530i	−0.716441	+	0.697648i	$0.754230\pi$
$8$	− 1.02900i	− 0.128625i
$9$	0	0
$10$	5.68820i	0.568820i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.56414i	0.274165i	0.990560	+	0.137082i	$0.0437725\pi$
$13$	3.56414i	0.274165i	−0.990560	+	0.137082i	$0.956227\pi$
$14$	−28.2334	−2.01667
$15$	0	0
$16$	−14.4494	−0.903087
$17$	− 4.57982i	− 0.269401i	−0.990886	−	0.134701i	$-0.956993\pi$
$17$	− 4.57982i	− 0.269401i	0.990886	−	0.134701i	$-0.0430073\pi$
$18$	0	0
$19$	21.8968i	1.15246i	0.817287	+	0.576231i	$0.195477\pi$
$19$	21.8968i	1.15246i	−0.817287	+	0.576231i	$0.804523\pi$
$20$	−8.57160	−0.428580
$21$	0	0
$22$	0	0
$23$	35.9581	1.56339	0.781697	−	0.623658i	$-0.214354\pi$
$23$	35.9581	1.56339	0.781697	+	0.623658i	$0.214354\pi$
$24$	0	0
$25$	−21.1278	−0.845114
$26$	−10.3028	−0.396260
$27$	0	0
$28$	− 42.5451i	− 1.51947i
$29$	− 6.35955i	− 0.219295i	−0.993971	−	0.109647i	$-0.965028\pi$
$29$	− 6.35955i	− 0.219295i	0.993971	−	0.109647i	$-0.0349722\pi$
$30$	0	0
$31$	4.74568	0.153086	0.0765432	−	0.997066i	$-0.475612\pi$
$31$	4.74568	0.153086	0.0765432	+	0.997066i	$0.475612\pi$
$32$	− 45.8844i	− 1.43389i
$33$	0	0
$34$	13.2388	0.389375
$35$	19.2194i	0.549127i
$36$	0	0
$37$	−48.8126	−1.31926	−0.659629	−	0.751591i	$-0.729287\pi$
$37$	−48.8126	−1.31926	−0.659629	+	0.751591i	$0.729287\pi$
$38$	−63.2963	−1.66569
$39$	0	0
$40$	− 2.02485i	− 0.0506214i
$41$	55.1891i	1.34607i	0.739608	+	0.673037i	$0.235011\pi$
$41$	55.1891i	1.34607i	−0.739608	+	0.673037i	$0.764989\pi$
$42$	0	0
$43$	35.4777i	0.825064i	0.910943	+	0.412532i	$0.135355\pi$
$43$	35.4777i	0.825064i	−0.910943	+	0.412532i	$0.864645\pi$
$44$	0	0
$45$	0	0
$46$	103.943i	2.25963i
$47$	50.1176	1.06633	0.533166	−	0.846011i	$-0.321002\pi$
$47$	50.1176	1.06633	0.533166	+	0.846011i	$0.321002\pi$
$48$	0	0
$49$	−46.3956	−0.946849
$50$	− 61.0736i	− 1.22147i
$51$	0	0
$52$	− 15.5253i	− 0.298564i
$53$	−21.9347	−0.413862	−0.206931	−	0.978356i	$-0.566348\pi$
$53$	−21.9347	−0.413862	−0.206931	+	0.978356i	$0.566348\pi$
$54$	0	0
$55$	0	0
$56$	10.0504	0.179471
$57$	0	0
$58$	18.3834	0.316955
$59$	−32.5364	−0.551464	−0.275732	−	0.961235i	$-0.588920\pi$
$59$	−32.5364	−0.551464	−0.275732	+	0.961235i	$0.588920\pi$
$60$	0	0
$61$	− 79.0369i	− 1.29569i	−0.761773	−	0.647844i	$-0.775671\pi$
$61$	− 79.0369i	− 1.29569i	0.761773	−	0.647844i	$-0.224329\pi$
$62$	13.7182i	0.221261i
$63$	0	0
$64$	74.8392	1.16936
$65$	7.01346i	0.107899i
$66$	0	0
$67$	33.2026	0.495561	0.247780	−	0.968816i	$-0.420299\pi$
$67$	33.2026	0.495561	0.247780	+	0.968816i	$0.420299\pi$
$68$	19.9496i	0.293376i
$69$	0	0
$70$	−55.5571	−0.793672
$71$	−94.4045	−1.32964	−0.664821	−	0.747003i	$-0.731492\pi$
$71$	−94.4045	−1.32964	−0.664821	+	0.747003i	$0.731492\pi$
$72$	0	0
$73$	− 14.3988i	− 0.197243i	−0.995125	−	0.0986216i	$-0.968557\pi$
$73$	− 14.3988i	− 0.197243i	0.995125	−	0.0986216i	$-0.0314433\pi$
$74$	− 141.101i	− 1.90677i
$75$	0	0
$76$	− 95.3817i	− 1.25502i
$77$	0	0
$78$	0	0
$79$	− 19.7648i	− 0.250187i	−0.992145	−	0.125093i	$-0.960077\pi$
$79$	− 19.7648i	− 0.250187i	0.992145	−	0.125093i	$-0.0399231\pi$
$80$	−28.4332	−0.355415
$81$	0	0
$82$	−159.533	−1.94553
$83$	− 63.1249i	− 0.760540i	−0.924875	−	0.380270i	$-0.875831\pi$
$83$	− 63.1249i	− 0.760540i	0.924875	−	0.380270i	$-0.124169\pi$
$84$	0	0
$85$	− 9.01208i	− 0.106025i
$86$	−102.554	−1.19249
$87$	0	0
$88$	0	0
$89$	−42.3567	−0.475918	−0.237959	−	0.971275i	$-0.576478\pi$
$89$	−42.3567	−0.475918	−0.237959	+	0.971275i	$0.576478\pi$
$90$	0	0
$91$	−34.8112	−0.382541
$92$	−156.632	−1.70253
$93$	0	0
$94$	144.874i	1.54121i
$95$	43.0880i	0.453558i
$96$	0	0
$97$	−176.984	−1.82458	−0.912289	−	0.409546i	$-0.865687\pi$
$97$	−176.984	−1.82458	−0.912289	+	0.409546i	$0.865687\pi$
$98$	− 134.114i	− 1.36851i
$99$	0	0
$100$	92.0323	0.920323
$101$	− 19.1338i	− 0.189443i	−0.995504	−	0.0947216i	$-0.969804\pi$
$101$	− 19.1338i	− 0.189443i	0.995504	−	0.0947216i	$-0.0301961\pi$
$102$	0	0
$103$	30.6916	0.297976	0.148988	−	0.988839i	$-0.452398\pi$
$103$	30.6916	0.297976	0.148988	+	0.988839i	$0.452398\pi$
$104$	3.66752	0.0352646
$105$	0	0
$106$	− 63.4059i	− 0.598169i
$107$	− 133.133i	− 1.24424i	−0.782923	−	0.622119i	$-0.786272\pi$
$107$	− 133.133i	− 1.24424i	0.782923	−	0.622119i	$-0.213728\pi$
$108$	0	0
$109$	88.6261i	0.813084i	0.913632	+	0.406542i	$0.133265\pi$
$109$	88.6261i	0.813084i	−0.913632	+	0.406542i	$0.866735\pi$
$110$	0	0
$111$	0	0
$112$	− 141.128i	− 1.26007i
$113$	96.1145	0.850571	0.425285	−	0.905059i	$-0.360174\pi$
$113$	96.1145	0.850571	0.425285	+	0.905059i	$0.360174\pi$
$114$	0	0
$115$	70.7576	0.615283
$116$	27.7020i	0.238811i
$117$	0	0
$118$	− 94.0519i	− 0.797050i
$119$	44.7314	0.375894
$120$	0	0
$121$	0	0
$122$	228.470	1.87270
$123$	0	0
$124$	−20.6721	−0.166710
$125$	−90.7695	−0.726156
$126$	0	0
$127$	− 100.254i	− 0.789399i	−0.918810	−	0.394699i	$-0.870849\pi$
$127$	− 100.254i	− 0.789399i	0.918810	−	0.394699i	$-0.129151\pi$
$128$	32.7977i	0.256232i
$129$	0	0
$130$	−20.2736	−0.155951
$131$	16.5475i	0.126317i	0.998004	+	0.0631585i	$0.0201174\pi$
$131$	16.5475i	0.126317i	−0.998004	+	0.0631585i	$0.979883\pi$
$132$	0	0
$133$	−213.867	−1.60802
$134$	95.9776i	0.716251i
$135$	0	0
$136$	−4.71265	−0.0346519
$137$	144.419	1.05415	0.527076	−	0.849818i	$-0.323288\pi$
$137$	144.419	1.05415	0.527076	+	0.849818i	$0.323288\pi$
$138$	0	0
$139$	203.558i	1.46445i	0.681064	+	0.732224i	$0.261518\pi$
$139$	203.558i	1.46445i	−0.681064	+	0.732224i	$0.738482\pi$
$140$	− 83.7194i	− 0.597996i
$141$	0	0
$142$	− 272.892i	− 1.92178i
$143$	0	0
$144$	0	0
$145$	− 12.5142i	− 0.0863048i
$146$	41.6220	0.285083
$147$	0	0
$148$	212.626	1.43666
$149$	23.4611i	0.157457i	0.996896	+	0.0787284i	$0.0250860\pi$
$149$	23.4611i	0.157457i	−0.996896	+	0.0787284i	$0.974914\pi$
$150$	0	0
$151$	4.95067i	0.0327859i	0.999866	+	0.0163929i	$0.00521827\pi$
$151$	4.95067i	0.0327859i	−0.999866	+	0.0163929i	$0.994782\pi$
$152$	22.5319	0.148236
$153$	0	0
$154$	0	0
$155$	9.33846	0.0602481
$156$	0	0
$157$	248.087	1.58017	0.790087	−	0.612995i	$-0.210035\pi$
$157$	248.087	1.58017	0.790087	+	0.612995i	$0.210035\pi$
$158$	57.1334	0.361604
$159$	0	0
$160$	− 90.2905i	− 0.564316i
$161$	351.205i	2.18140i
$162$	0	0
$163$	−225.335	−1.38242	−0.691212	−	0.722652i	$-0.742923\pi$
$163$	−225.335	−1.38242	−0.691212	+	0.722652i	$0.742923\pi$
$164$	− 240.402i	− 1.46587i
$165$	0	0
$166$	182.473	1.09924
$167$	− 162.351i	− 0.972163i	−0.873913	−	0.486082i	$-0.838426\pi$
$167$	− 162.351i	− 0.972163i	0.873913	−	0.486082i	$-0.161574\pi$
$168$	0	0
$169$	156.297	0.924834
$170$	26.0510	0.153241
$171$	0	0
$172$	− 154.540i	− 0.898489i
$173$	297.580i	1.72011i	0.510198	+	0.860057i	$0.329572\pi$
$173$	297.580i	1.72011i	−0.510198	+	0.860057i	$0.670428\pi$
$174$	0	0
$175$	− 206.357i	− 1.17918i
$176$	0	0
$177$	0	0
$178$	− 122.439i	− 0.687861i
$179$	−8.28975	−0.0463115	−0.0231557	−	0.999732i	$-0.507371\pi$
$179$	−8.28975	−0.0463115	−0.0231557	+	0.999732i	$0.507371\pi$
$180$	0	0
$181$	48.5182	0.268056	0.134028	−	0.990978i	$-0.457209\pi$
$181$	48.5182	0.268056	0.134028	+	0.990978i	$0.457209\pi$
$182$	− 100.628i	− 0.552900i
$183$	0	0
$184$	− 37.0010i	− 0.201092i
$185$	−96.0524	−0.519202
$186$	0	0
$187$	0	0
$188$	−218.311	−1.16123
$189$	0	0
$190$	−124.553	−0.655543
$191$	186.644	0.977195	0.488597	−	0.872509i	$-0.337509\pi$
$191$	186.644	0.977195	0.488597	+	0.872509i	$0.337509\pi$
$192$	0	0
$193$	351.672i	1.82213i	0.412260	+	0.911066i	$0.364739\pi$
$193$	351.672i	1.82213i	−0.412260	+	0.911066i	$0.635261\pi$
$194$	− 511.603i	− 2.63713i
$195$	0	0
$196$	202.098	1.03111
$197$	− 230.585i	− 1.17048i	−0.810860	−	0.585240i	$-0.801000\pi$
$197$	− 230.585i	− 1.17048i	0.810860	−	0.585240i	$-0.199000\pi$
$198$	0	0
$199$	−24.0096	−0.120651	−0.0603256	−	0.998179i	$-0.519214\pi$
$199$	−24.0096	−0.120651	−0.0603256	+	0.998179i	$0.519214\pi$
$200$	21.7406i	0.108703i
$201$	0	0
$202$	55.3094	0.273809
$203$	62.1142	0.305981
$204$	0	0
$205$	108.600i	0.529756i
$206$	88.7192i	0.430676i
$207$	0	0
$208$	− 51.4997i	− 0.247595i
$209$	0	0
$210$	0	0
$211$	− 79.1038i	− 0.374900i	−0.982274	−	0.187450i	$-0.939978\pi$
$211$	− 79.1038i	− 0.374900i	0.982274	−	0.187450i	$-0.0600222\pi$
$212$	95.5469	0.450693
$213$	0	0
$214$	384.845	1.79834
$215$	69.8124i	0.324709i
$216$	0	0
$217$	46.3514i	0.213601i
$218$	−256.189	−1.17518
$219$	0	0
$220$	0	0
$221$	16.3231	0.0738604
$222$	0	0
$223$	61.2860	0.274825	0.137413	−	0.990514i	$-0.456121\pi$
$223$	61.2860	0.274825	0.137413	+	0.990514i	$0.456121\pi$
$224$	448.156	2.00070
$225$	0	0
$226$	277.835i	1.22936i
$227$	279.478i	1.23118i	0.788067	+	0.615589i	$0.211082\pi$
$227$	279.478i	1.23118i	−0.788067	+	0.615589i	$0.788918\pi$
$228$	0	0
$229$	201.902	0.881670	0.440835	−	0.897588i	$-0.354682\pi$
$229$	201.902	0.881670	0.440835	+	0.897588i	$0.354682\pi$
$230$	204.537i	0.889290i
$231$	0	0
$232$	−6.54400	−0.0282069
$233$	207.589i	0.890940i	0.895297	+	0.445470i	$0.146963\pi$
$233$	207.589i	0.890940i	−0.895297	+	0.445470i	$0.853037\pi$
$234$	0	0
$235$	98.6205	0.419662
$236$	141.728	0.600541
$237$	0	0
$238$	129.304i	0.543293i
$239$	− 177.609i	− 0.743136i	−0.928406	−	0.371568i	$-0.878820\pi$
$239$	− 177.609i	− 0.743136i	0.928406	−	0.371568i	$-0.121180\pi$
$240$	0	0
$241$	170.909i	0.709165i	0.935025	+	0.354583i	$0.115377\pi$
$241$	170.909i	0.709165i	−0.935025	+	0.354583i	$0.884623\pi$
$242$	0	0
$243$	0	0
$244$	344.283i	1.41100i
$245$	−91.2963	−0.372638
$246$	0	0
$247$	−78.0432	−0.315964
$248$	− 4.88332i	− 0.0196908i
$249$	0	0
$250$	− 262.385i	− 1.04954i
$251$	−60.1778	−0.239752	−0.119876	−	0.992789i	$-0.538250\pi$
$251$	−60.1778	−0.239752	−0.119876	+	0.992789i	$0.538250\pi$
$252$	0	0
$253$	0	0
$254$	289.800	1.14095
$255$	0	0
$256$	204.549	0.799021
$257$	−399.412	−1.55413	−0.777067	−	0.629418i	$-0.783293\pi$
$257$	−399.412	−1.55413	−0.777067	+	0.629418i	$0.783293\pi$
$258$	0	0
$259$	− 476.756i	− 1.84076i
$260$	− 30.5504i	− 0.117502i
$261$	0	0
$262$	−47.8335	−0.182571
$263$	522.890i	1.98817i	0.108585	+	0.994087i	$0.465368\pi$
$263$	522.890i	1.98817i	−0.108585	+	0.994087i	$0.534632\pi$
$264$	0	0
$265$	−43.1626	−0.162878
$266$	− 618.219i	− 2.32413i
$267$	0	0
$268$	−144.629	−0.539662
$269$	−205.035	−0.762212	−0.381106	−	0.924531i	$-0.624457\pi$
$269$	−205.035	−0.762212	−0.381106	+	0.924531i	$0.624457\pi$
$270$	0	0
$271$	176.980i	0.653063i	0.945186	+	0.326532i	$0.105880\pi$
$271$	176.980i	0.653063i	−0.945186	+	0.326532i	$0.894120\pi$
$272$	66.1756i	0.243293i
$273$	0	0
$274$	417.468i	1.52360i
$275$	0	0
$276$	0	0
$277$	− 105.558i	− 0.381075i	−0.981680	−	0.190537i	$-0.938977\pi$
$277$	− 105.558i	− 0.381075i	0.981680	−	0.190537i	$-0.0610231\pi$
$278$	−588.420	−2.11662
$279$	0	0
$280$	19.7769	0.0706317
$281$	309.347i	1.10088i	0.834875	+	0.550440i	$0.185540\pi$
$281$	309.347i	1.10088i	−0.834875	+	0.550440i	$0.814460\pi$
$282$	0	0
$283$	191.644i	0.677186i	0.940933	+	0.338593i	$0.109951\pi$
$283$	191.644i	0.677186i	−0.940933	+	0.338593i	$0.890049\pi$
$284$	411.224	1.44797
$285$	0	0
$286$	0	0
$287$	−539.035	−1.87817
$288$	0	0
$289$	268.025	0.927423
$290$	36.1744	0.124739
$291$	0	0
$292$	62.7206i	0.214797i
$293$	− 274.800i	− 0.937883i	−0.883229	−	0.468942i	$-0.844635\pi$
$293$	− 274.800i	− 0.937883i	0.883229	−	0.468942i	$-0.155365\pi$
$294$	0	0
$295$	−64.0244	−0.217032
$296$	50.2283i	0.169690i
$297$	0	0
$298$	−67.8182	−0.227578
$299$	128.160i	0.428628i
$300$	0	0
$301$	−346.513	−1.15121
$302$	−14.3107	−0.0473866
$303$	0	0
$304$	− 316.395i	− 1.04077i
$305$	− 155.527i	− 0.509926i
$306$	0	0
$307$	347.487i	1.13188i	0.824446	+	0.565940i	$0.191487\pi$
$307$	347.487i	1.13188i	−0.824446	+	0.565940i	$0.808513\pi$
$308$	0	0
$309$	0	0
$310$	26.9944i	0.0870787i
$311$	356.252	1.14550	0.572752	−	0.819729i	$-0.305876\pi$
$311$	356.252	1.14550	0.572752	+	0.819729i	$0.305876\pi$
$312$	0	0
$313$	−100.468	−0.320984	−0.160492	−	0.987037i	$-0.551308\pi$
$313$	−100.468	−0.320984	−0.160492	+	0.987037i	$0.551308\pi$
$314$	717.139i	2.28388i
$315$	0	0
$316$	86.0948i	0.272452i
$317$	398.921	1.25843	0.629214	−	0.777232i	$-0.283377\pi$
$317$	398.921	1.25843	0.629214	+	0.777232i	$0.283377\pi$
$318$	0	0
$319$	0	0
$320$	147.267	0.460210
$321$	0	0
$322$	−1015.22	−3.15285
$323$	100.283	0.310474
$324$	0	0
$325$	− 75.3027i	− 0.231701i
$326$	− 651.370i	− 1.99807i
$327$	0	0
$328$	56.7898	0.173140
$329$	489.502i	1.48785i
$330$	0	0
$331$	−459.126	−1.38709	−0.693544	−	0.720415i	$-0.743952\pi$
$331$	−459.126	−1.38709	−0.693544	+	0.720415i	$0.743952\pi$
$332$	274.970i	0.828224i
$333$	0	0
$334$	469.304	1.40510
$335$	65.3353	0.195031
$336$	0	0
$337$	− 176.542i	− 0.523865i	−0.965086	−	0.261932i	$-0.915640\pi$
$337$	− 176.542i	− 0.523865i	0.965086	−	0.261932i	$-0.0843597\pi$
$338$	451.803i	1.33669i
$339$	0	0
$340$	39.2564i	0.115460i
$341$	0	0
$342$	0	0
$343$	25.4375i	0.0741617i
$344$	36.5067	0.106124
$345$	0	0
$346$	−860.205	−2.48614
$347$	− 582.098i	− 1.67751i	−0.544505	−	0.838757i	$-0.683283\pi$
$347$	− 582.098i	− 1.67751i	0.544505	−	0.838757i	$-0.316717\pi$
$348$	0	0
$349$	177.164i	0.507634i	0.967252	+	0.253817i	$0.0816862\pi$
$349$	177.164i	0.507634i	−0.967252	+	0.253817i	$0.918314\pi$
$350$	596.510	1.70431
$351$	0	0
$352$	0	0
$353$	−85.7497	−0.242917	−0.121458	−	0.992597i	$-0.538757\pi$
$353$	−85.7497	−0.242917	−0.121458	+	0.992597i	$0.538757\pi$
$354$	0	0
$355$	−185.767	−0.523288
$356$	184.505	0.518271
$357$	0	0
$358$	− 23.9629i	− 0.0669356i
$359$	184.626i	0.514278i	0.966374	+	0.257139i	$0.0827799\pi$
$359$	184.626i	0.514278i	−0.966374	+	0.257139i	$0.917220\pi$
$360$	0	0
$361$	−118.468	−0.328166
$362$	140.250i	0.387431i
$363$	0	0
$364$	151.637	0.416585
$365$	− 28.3336i	− 0.0776263i
$366$	0	0
$367$	−423.712	−1.15453	−0.577264	−	0.816558i	$-0.695880\pi$
$367$	−423.712	−1.15453	−0.577264	+	0.816558i	$0.695880\pi$
$368$	−519.572	−1.41188
$369$	0	0
$370$	− 277.656i	− 0.750421i
$371$	− 214.237i	− 0.577459i
$372$	0	0
$373$	252.191i	0.676114i	0.941126	+	0.338057i	$0.109770\pi$
$373$	252.191i	0.676114i	−0.941126	+	0.338057i	$0.890230\pi$
$374$	0	0
$375$	0	0
$376$	− 51.5712i	− 0.137158i
$377$	22.6664	0.0601230
$378$	0	0
$379$	−112.440	−0.296675	−0.148338	−	0.988937i	$-0.547392\pi$
$379$	−112.440	−0.296675	−0.148338	+	0.988937i	$0.547392\pi$
$380$	− 187.690i	− 0.493922i
$381$	0	0
$382$	539.527i	1.41237i
$383$	−281.037	−0.733777	−0.366888	−	0.930265i	$-0.619577\pi$
$383$	−281.037	−0.733777	−0.366888	+	0.930265i	$0.619577\pi$
$384$	0	0
$385$	0	0
$386$	−1016.57	−2.63359
$387$	0	0
$388$	770.938	1.98695
$389$	−78.8469	−0.202691	−0.101346	−	0.994851i	$-0.532315\pi$
$389$	−78.8469	−0.202691	−0.101346	+	0.994851i	$0.532315\pi$
$390$	0	0
$391$	− 164.682i	− 0.421180i
$392$	47.7412i	0.121789i
$393$	0	0
$394$	666.544	1.69174
$395$	− 38.8927i	− 0.0984625i
$396$	0	0
$397$	180.370	0.454332	0.227166	−	0.973856i	$-0.427054\pi$
$397$	180.370	0.454332	0.227166	+	0.973856i	$0.427054\pi$
$398$	− 69.4038i	− 0.174381i
$399$	0	0
$400$	305.284	0.763211
$401$	730.307	1.82122	0.910608	−	0.413272i	$-0.135614\pi$
$401$	730.307	1.82122	0.910608	+	0.413272i	$0.135614\pi$
$402$	0	0
$403$	16.9143i	0.0419709i
$404$	83.3462i	0.206302i
$405$	0	0
$406$	179.552i	0.442245i
$407$	0	0
$408$	0	0
$409$	431.310i	1.05455i	0.849696	+	0.527273i	$0.176786\pi$
$409$	431.310i	1.05455i	−0.849696	+	0.527273i	$0.823214\pi$
$410$	−313.927	−0.765675
$411$	0	0
$412$	−133.692	−0.324494
$413$	− 317.785i	− 0.769455i
$414$	0	0
$415$	− 124.216i	− 0.299315i
$416$	163.539	0.393122
$417$	0	0
$418$	0	0
$419$	724.604	1.72937	0.864683	−	0.502318i	$-0.167519\pi$
$419$	724.604	1.72937	0.864683	+	0.502318i	$0.167519\pi$
$420$	0	0
$421$	−274.593	−0.652239	−0.326120	−	0.945329i	$-0.605741\pi$
$421$	−274.593	−0.652239	−0.326120	+	0.945329i	$0.605741\pi$
$422$	228.663	0.541856
$423$	0	0
$424$	22.5709i	0.0532332i
$425$	96.7617i	0.227675i
$426$	0	0
$427$	771.959	1.80787
$428$	579.926i	1.35497i
$429$	0	0
$430$	−201.805	−0.469313
$431$	696.003i	1.61486i	0.589966	+	0.807428i	$0.299141\pi$
$431$	696.003i	1.61486i	−0.589966	+	0.807428i	$0.700859\pi$
$432$	0	0
$433$	129.798	0.299764	0.149882	−	0.988704i	$-0.452111\pi$
$433$	129.798	0.299764	0.149882	+	0.988704i	$0.452111\pi$
$434$	−133.987	−0.308725
$435$	0	0
$436$	− 386.053i	− 0.885443i
$437$	787.365i	1.80175i
$438$	0	0
$439$	− 124.092i	− 0.282669i	−0.989962	−	0.141334i	$-0.954861\pi$
$439$	− 124.092i	− 0.282669i	0.989962	−	0.141334i	$-0.0451393\pi$
$440$	0	0
$441$	0	0
$442$	47.1848i	0.106753i
$443$	263.341	0.594448	0.297224	−	0.954808i	$-0.403939\pi$
$443$	263.341	0.594448	0.297224	+	0.954808i	$0.403939\pi$
$444$	0	0
$445$	−83.3486	−0.187300
$446$	177.158i	0.397214i
$447$	0	0
$448$	730.959i	1.63161i
$449$	284.975	0.634687	0.317344	−	0.948311i	$-0.397209\pi$
$449$	284.975	0.634687	0.317344	+	0.948311i	$0.397209\pi$
$450$	0	0
$451$	0	0
$452$	−418.672	−0.926266
$453$	0	0
$454$	−807.878	−1.77947
$455$	−68.5009	−0.150551
$456$	0	0
$457$	162.520i	0.355624i	0.984064	+	0.177812i	$0.0569019\pi$
$457$	162.520i	0.355624i	−0.984064	+	0.177812i	$0.943098\pi$
$458$	583.633i	1.27431i
$459$	0	0
$460$	−308.218	−0.670039
$461$	385.620i	0.836487i	0.908335	+	0.418243i	$0.137354\pi$
$461$	385.620i	0.836487i	−0.908335	+	0.418243i	$0.862646\pi$
$462$	0	0
$463$	457.098	0.987253	0.493626	−	0.869674i	$-0.335671\pi$
$463$	457.098	0.987253	0.493626	+	0.869674i	$0.335671\pi$
$464$	91.8916i	0.198042i
$465$	0	0
$466$	−600.071	−1.28771
$467$	202.999	0.434688	0.217344	−	0.976095i	$-0.430261\pi$
$467$	202.999	0.434688	0.217344	+	0.976095i	$0.430261\pi$
$468$	0	0
$469$	324.292i	0.691453i
$470$	285.079i	0.606552i
$471$	0	0
$472$	33.4801i	0.0709323i
$473$	0	0
$474$	0	0
$475$	− 462.631i	− 0.973960i
$476$	−194.849	−0.409346
$477$	0	0
$478$	513.410	1.07408
$479$	− 395.984i	− 0.826688i	−0.910575	−	0.413344i	$-0.864361\pi$
$479$	− 395.984i	− 0.826688i	0.910575	−	0.413344i	$-0.135639\pi$
$480$	0	0
$481$	− 173.975i	− 0.361695i
$482$	−494.041	−1.02498
$483$	0	0
$484$	0	0
$485$	−348.266	−0.718074
$486$	0	0
$487$	−202.304	−0.415408	−0.207704	−	0.978192i	$-0.566599\pi$
$487$	−202.304	−0.415408	−0.207704	+	0.978192i	$0.566599\pi$
$488$	−81.3293	−0.166658
$489$	0	0
$490$	− 263.908i	− 0.538587i
$491$	− 233.814i	− 0.476199i	−0.971241	−	0.238100i	$-0.923475\pi$
$491$	− 233.814i	− 0.476199i	0.971241	−	0.238100i	$-0.0765245\pi$
$492$	0	0
$493$	−29.1256	−0.0590783
$494$	− 225.597i	− 0.456675i
$495$	0	0
$496$	−68.5722	−0.138250
$497$	− 922.055i	− 1.85524i
$498$	0	0
$499$	183.744	0.368225	0.184112	−	0.982905i	$-0.441059\pi$
$499$	183.744	0.368225	0.184112	+	0.982905i	$0.441059\pi$
$500$	395.389	0.790779
$501$	0	0
$502$	− 173.954i	− 0.346522i
$503$	214.737i	0.426913i	0.976953	+	0.213457i	$0.0684722\pi$
$503$	214.737i	0.426913i	−0.976953	+	0.213457i	$0.931528\pi$
$504$	0	0
$505$	− 37.6510i	− 0.0745565i
$506$	0	0
$507$	0	0
$508$	436.702i	0.859650i
$509$	207.887	0.408423	0.204211	−	0.978927i	$-0.434537\pi$
$509$	207.887	0.408423	0.204211	+	0.978927i	$0.434537\pi$
$510$	0	0
$511$	140.634	0.275212
$512$	722.476i	1.41109i
$513$	0	0
$514$	− 1154.57i	− 2.24624i
$515$	60.3943	0.117270
$516$	0	0
$517$	0	0
$518$	1378.14	2.66051
$519$	0	0
$520$	7.21687	0.0138786
$521$	153.554	0.294729	0.147365	−	0.989082i	$-0.452921\pi$
$521$	153.554	0.294729	0.147365	+	0.989082i	$0.452921\pi$
$522$	0	0
$523$	916.304i	1.75201i	0.482298	+	0.876007i	$0.339802\pi$
$523$	916.304i	1.75201i	−0.482298	+	0.876007i	$0.660198\pi$
$524$	− 72.0806i	− 0.137558i
$525$	0	0
$526$	−1511.50	−2.87358
$527$	− 21.7344i	− 0.0412417i
$528$	0	0
$529$	763.982	1.44420
$530$	− 124.769i	− 0.235413i
$531$	0	0
$532$	931.600	1.75113
$533$	−196.702	−0.369047
$534$	0	0
$535$	− 261.977i	− 0.489677i
$536$	− 34.1656i	− 0.0637417i
$537$	0	0
$538$	− 592.689i	− 1.10165i
$539$	0	0
$540$	0	0
$541$	647.885i	1.19757i	0.800910	+	0.598785i	$0.204350\pi$
$541$	647.885i	1.19757i	−0.800910	+	0.598785i	$0.795650\pi$
$542$	−511.591	−0.943896
$543$	0	0
$544$	−210.142	−0.386291
$545$	174.397i	0.319994i
$546$	0	0
$547$	105.774i	0.193371i	0.995315	+	0.0966853i	$0.0308240\pi$
$547$	105.774i	0.193371i	−0.995315	+	0.0966853i	$0.969176\pi$
$548$	−629.085	−1.14797
$549$	0	0
$550$	0	0
$551$	139.254	0.252729
$552$	0	0
$553$	193.044	0.349084
$554$	305.133	0.550781
$555$	0	0
$556$	− 886.694i	− 1.59477i
$557$	823.074i	1.47769i	0.673875	+	0.738845i	$0.264629\pi$
$557$	823.074i	1.47769i	−0.673875	+	0.738845i	$0.735371\pi$
$558$	0	0
$559$	−126.448	−0.226204
$560$	− 277.709i	− 0.495909i
$561$	0	0
$562$	−894.221	−1.59114
$563$	715.741i	1.27130i	0.771978	+	0.635649i	$0.219267\pi$
$563$	715.741i	1.27130i	−0.771978	+	0.635649i	$0.780733\pi$
$564$	0	0
$565$	189.132	0.334747
$566$	−553.979	−0.978761
$567$	0	0
$568$	97.1426i	0.171026i
$569$	− 936.082i	− 1.64514i	−0.568667	−	0.822568i	$-0.692541\pi$
$569$	− 936.082i	− 1.64514i	0.568667	−	0.822568i	$-0.307459\pi$
$570$	0	0
$571$	28.0926i	0.0491990i	0.999697	+	0.0245995i	$0.00783105\pi$
$571$	28.0926i	0.0491990i	−0.999697	+	0.0245995i	$0.992169\pi$
$572$	0	0
$573$	0	0
$574$	− 1558.17i	− 2.71459i
$575$	−759.716	−1.32125
$576$	0	0
$577$	718.462	1.24517	0.622584	−	0.782553i	$-0.286083\pi$
$577$	718.462	1.24517	0.622584	+	0.782553i	$0.286083\pi$
$578$	774.773i	1.34044i
$579$	0	0
$580$	54.5115i	0.0939854i
$581$	616.545	1.06118
$582$	0	0
$583$	0	0
$584$	−14.8164	−0.0253705
$585$	0	0
$586$	794.356	1.35556
$587$	−231.920	−0.395094	−0.197547	−	0.980293i	$-0.563297\pi$
$587$	−231.920	−0.395094	−0.197547	+	0.980293i	$0.563297\pi$
$588$	0	0
$589$	103.915i	0.176426i
$590$	− 185.074i	− 0.313684i
$591$	0	0
$592$	705.312	1.19140
$593$	− 1036.70i	− 1.74822i	−0.485725	−	0.874112i	$-0.661444\pi$
$593$	− 1036.70i	− 1.74822i	0.485725	−	0.874112i	$-0.338556\pi$
$594$	0	0
$595$	88.0216	0.147936
$596$	− 102.196i	− 0.171469i
$597$	0	0
$598$	−370.468	−0.619511
$599$	804.253	1.34266	0.671330	−	0.741159i	$-0.265723\pi$
$599$	804.253	1.34266	0.671330	+	0.741159i	$0.265723\pi$
$600$	0	0
$601$	912.870i	1.51892i	0.650555	+	0.759459i	$0.274536\pi$
$601$	912.870i	1.51892i	−0.650555	+	0.759459i	$0.725464\pi$
$602$	− 1001.66i	− 1.66388i
$603$	0	0
$604$	− 21.5650i	− 0.0357036i
$605$	0	0
$606$	0	0
$607$	141.759i	0.233540i	0.993159	+	0.116770i	$0.0372541\pi$
$607$	141.759i	0.233540i	−0.993159	+	0.116770i	$0.962746\pi$
$608$	1004.72	1.65250
$609$	0	0
$610$	449.578	0.737014
$611$	178.626i	0.292351i
$612$	0	0
$613$	83.5679i	0.136326i	0.997674	+	0.0681631i	$0.0217138\pi$
$613$	83.5679i	0.136326i	−0.997674	+	0.0681631i	$0.978286\pi$
$614$	−1004.47	−1.63595
$615$	0	0
$616$	0	0
$617$	−312.117	−0.505862	−0.252931	−	0.967484i	$-0.581395\pi$
$617$	−312.117	−0.505862	−0.252931	+	0.967484i	$0.581395\pi$
$618$	0	0
$619$	769.067	1.24243	0.621217	−	0.783638i	$-0.286638\pi$
$619$	769.067	1.24243	0.621217	+	0.783638i	$0.286638\pi$
$620$	−40.6781	−0.0656098
$621$	0	0
$622$	1029.81i	1.65564i
$623$	− 413.700i	− 0.664046i
$624$	0	0
$625$	349.582	0.559331
$626$	− 290.420i	− 0.463930i
$627$	0	0
$628$	−1080.66	−1.72080
$629$	223.553i	0.355410i
$630$	0	0
$631$	−489.886	−0.776364	−0.388182	−	0.921583i	$-0.626897\pi$
$631$	−489.886	−0.776364	−0.388182	+	0.921583i	$0.626897\pi$
$632$	−20.3380	−0.0321804
$633$	0	0
$634$	1153.15i	1.81885i
$635$	− 197.277i	− 0.310673i
$636$	0	0
$637$	− 165.361i	− 0.259593i
$638$	0	0
$639$	0	0
$640$	64.5387i	0.100842i
$641$	580.637	0.905830	0.452915	−	0.891554i	$-0.350384\pi$
$641$	580.637	0.905830	0.452915	+	0.891554i	$0.350384\pi$
$642$	0	0
$643$	1072.29	1.66764	0.833820	−	0.552037i	$-0.186149\pi$
$643$	1072.29	1.66764	0.833820	+	0.552037i	$0.186149\pi$
$644$	− 1529.84i	− 2.37553i
$645$	0	0
$646$	289.886i	0.448740i
$647$	178.944	0.276576	0.138288	−	0.990392i	$-0.455840\pi$
$647$	178.944	0.276576	0.138288	+	0.990392i	$0.455840\pi$
$648$	0	0
$649$	0	0
$650$	217.675	0.334885
$651$	0	0
$652$	981.554	1.50545
$653$	−764.640	−1.17096	−0.585482	−	0.810685i	$-0.699095\pi$
$653$	−764.640	−1.17096	−0.585482	+	0.810685i	$0.699095\pi$
$654$	0	0
$655$	32.5619i	0.0497129i
$656$	− 797.448i	− 1.21562i
$657$	0	0
$658$	−1414.99	−2.15044
$659$	− 709.815i	− 1.07711i	−0.842591	−	0.538555i	$-0.818971\pi$
$659$	− 709.815i	− 1.07711i	0.842591	−	0.538555i	$-0.181029\pi$
$660$	0	0
$661$	22.3956	0.0338814	0.0169407	−	0.999856i	$-0.494607\pi$
$661$	22.3956	0.0338814	0.0169407	+	0.999856i	$0.494607\pi$
$662$	− 1327.18i	− 2.00481i
$663$	0	0
$664$	−64.9557	−0.0978249
$665$	−420.844	−0.632847
$666$	0	0
$667$	− 228.677i	− 0.342844i
$668$	707.198i	1.05868i
$669$	0	0
$670$	188.863i	0.281885i
$671$	0	0
$672$	0	0
$673$	− 507.729i	− 0.754427i	−0.926126	−	0.377214i	$-0.876882\pi$
$673$	− 507.729i	− 0.754427i	0.926126	−	0.377214i	$-0.123118\pi$
$674$	510.326	0.757160
$675$	0	0
$676$	−680.825	−1.00714
$677$	− 412.152i	− 0.608791i	−0.952546	−	0.304396i	$-0.901545\pi$
$677$	− 412.152i	− 0.608791i	0.952546	−	0.304396i	$-0.0984545\pi$
$678$	0	0
$679$	− 1728.62i	− 2.54583i
$680$	−9.27347	−0.0136375
$681$	0	0
$682$	0	0
$683$	−933.884	−1.36733	−0.683663	−	0.729798i	$-0.739614\pi$
$683$	−933.884	−1.36733	−0.683663	+	0.729798i	$0.739614\pi$
$684$	0	0
$685$	284.185	0.414868
$686$	−73.5314	−0.107189
$687$	0	0
$688$	− 512.631i	− 0.745104i
$689$	− 78.1784i	− 0.113466i
$690$	0	0
$691$	462.272	0.668991	0.334495	−	0.942397i	$-0.391434\pi$
$691$	462.272	0.668991	0.334495	+	0.942397i	$0.391434\pi$
$692$	− 1296.25i	− 1.87319i
$693$	0	0
$694$	1682.65	2.42457
$695$	400.558i	0.576342i
$696$	0	0
$697$	252.756	0.362634
$698$	−512.124	−0.733702
$699$	0	0
$700$	898.886i	1.28412i
$701$	− 772.395i	− 1.10185i	−0.834556	−	0.550923i	$-0.814276\pi$
$701$	− 772.395i	− 1.10185i	0.834556	−	0.550923i	$-0.185724\pi$
$702$	0	0
$703$	− 1068.84i	− 1.52039i
$704$	0	0
$705$	0	0
$706$	− 247.874i	− 0.351096i
$707$	186.881	0.264329
$708$	0	0
$709$	−591.112	−0.833726	−0.416863	−	0.908969i	$-0.636871\pi$
$709$	−591.112	−0.833726	−0.416863	+	0.908969i	$0.636871\pi$
$710$	− 536.992i	− 0.756327i
$711$	0	0
$712$	43.5852i	0.0612152i
$713$	170.645	0.239334
$714$	0	0
$715$	0	0
$716$	36.1099	0.0504329
$717$	0	0
$718$	−533.693	−0.743304
$719$	884.700	1.23046	0.615229	−	0.788348i	$-0.289063\pi$
$719$	884.700	1.23046	0.615229	+	0.788348i	$0.289063\pi$
$720$	0	0
$721$	299.767i	0.415765i
$722$	− 342.452i	− 0.474310i
$723$	0	0
$724$	−211.344	−0.291912
$725$	134.364i	0.185329i
$726$	0	0
$727$	−848.076	−1.16654	−0.583271	−	0.812277i	$-0.698228\pi$
$727$	−848.076	−1.16654	−0.583271	+	0.812277i	$0.698228\pi$
$728$	35.8209i	0.0492045i
$729$	0	0
$730$	81.9031	0.112196
$731$	162.482	0.222273
$732$	0	0
$733$	− 1207.24i	− 1.64698i	−0.567330	−	0.823490i	$-0.692024\pi$
$733$	− 1207.24i	− 1.64698i	0.567330	−	0.823490i	$-0.307976\pi$
$734$	− 1224.81i	− 1.66868i
$735$	0	0
$736$	− 1649.91i	− 2.24173i
$737$	0	0
$738$	0	0
$739$	449.700i	0.608525i	0.952588	+	0.304262i	$0.0984100\pi$
$739$	449.700i	0.608525i	−0.952588	+	0.304262i	$0.901590\pi$
$740$	418.402	0.565408
$741$	0	0
$742$	619.290	0.834622
$743$	669.684i	0.901325i	0.892694	+	0.450662i	$0.148812\pi$
$743$	669.684i	0.901325i	−0.892694	+	0.450662i	$0.851188\pi$
$744$	0	0
$745$	46.1662i	0.0619681i
$746$	−729.000	−0.977211
$747$	0	0
$748$	0	0
$749$	1300.32	1.73608
$750$	0	0
$751$	−354.590	−0.472157	−0.236078	−	0.971734i	$-0.575862\pi$
$751$	−354.590	−0.472157	−0.236078	+	0.971734i	$0.575862\pi$
$752$	−724.169	−0.962991
$753$	0	0
$754$	65.5210i	0.0868979i
$755$	9.74182i	0.0129031i
$756$	0	0
$757$	559.010	0.738455	0.369227	−	0.929339i	$-0.379622\pi$
$757$	559.010	0.738455	0.369227	+	0.929339i	$0.379622\pi$
$758$	− 325.026i	− 0.428795i
$759$	0	0
$760$	44.3377	0.0583391
$761$	− 987.303i	− 1.29738i	−0.761055	−	0.648688i	$-0.775318\pi$
$761$	− 987.303i	− 1.29738i	0.761055	−	0.648688i	$-0.224682\pi$
$762$	0	0
$763$	−865.617	−1.13449
$764$	−813.017	−1.06416
$765$	0	0
$766$	− 812.384i	− 1.06055i
$767$	− 115.964i	− 0.151192i
$768$	0	0
$769$	− 567.708i	− 0.738241i	−0.929381	−	0.369121i	$-0.879659\pi$
$769$	− 567.708i	− 0.738241i	0.929381	−	0.369121i	$-0.120341\pi$
$770$	0	0
$771$	0	0
$772$	− 1531.87i	− 1.98429i
$773$	−307.804	−0.398194	−0.199097	−	0.979980i	$-0.563801\pi$
$773$	−307.804	−0.398194	−0.199097	+	0.979980i	$0.563801\pi$
$774$	0	0
$775$	−100.266	−0.129375
$776$	182.117i	0.234687i
$777$	0	0
$778$	− 227.920i	− 0.292957i
$779$	−1208.46	−1.55130
$780$	0	0
$781$	0	0
$782$	476.040	0.608747
$783$	0	0
$784$	670.388	0.855086
$785$	488.181	0.621887
$786$	0	0
$787$	− 1218.24i	− 1.54796i	−0.633210	−	0.773980i	$-0.718263\pi$
$787$	− 1218.24i	− 1.54796i	0.633210	−	0.773980i	$-0.281737\pi$
$788$	1004.42i	1.27465i
$789$	0	0
$790$	112.426	0.142311
$791$	938.756i	1.18680i
$792$	0	0
$793$	281.699	0.355232
$794$	521.390i	0.656662i
$795$	0	0
$796$	104.585	0.131388
$797$	−914.292	−1.14717	−0.573583	−	0.819147i	$-0.694447\pi$
$797$	−914.292	−1.14717	−0.573583	+	0.819147i	$0.694447\pi$
$798$	0	0
$799$	− 229.530i	− 0.287271i
$800$	969.439i	1.21180i
$801$	0	0
$802$	2111.08i	2.63227i
$803$	0	0
$804$	0	0
$805$	691.094i	0.858502i
$806$	−48.8936	−0.0606621
$807$	0	0
$808$	−19.6887	−0.0243672
$809$	856.779i	1.05906i	0.848291	+	0.529530i	$0.177632\pi$
$809$	856.779i	1.05906i	−0.848291	+	0.529530i	$0.822368\pi$
$810$	0	0
$811$	− 174.129i	− 0.214709i	−0.994221	−	0.107355i	$-0.965762\pi$
$811$	− 174.129i	− 0.214709i	0.994221	−	0.107355i	$-0.0342380\pi$
$812$	−270.568	−0.333211
$813$	0	0
$814$	0	0
$815$	−443.410	−0.544062
$816$	0	0
$817$	−776.847	−0.950854
$818$	−1246.77	−1.52417
$819$	0	0
$820$	− 473.059i	− 0.576901i
$821$	1174.76i	1.43089i	0.698670	+	0.715444i	$0.253775\pi$
$821$	1174.76i	1.43089i	−0.698670	+	0.715444i	$0.746225\pi$
$822$	0	0
$823$	−845.210	−1.02699	−0.513493	−	0.858094i	$-0.671649\pi$
$823$	−845.210	−1.02699	−0.513493	+	0.858094i	$0.671649\pi$
$824$	− 31.5818i	− 0.0383274i
$825$	0	0
$826$	918.611	1.11212
$827$	197.867i	0.239259i	0.992819	+	0.119630i	$0.0381707\pi$
$827$	197.867i	0.239259i	−0.992819	+	0.119630i	$0.961829\pi$
$828$	0	0
$829$	1339.10	1.61532	0.807662	−	0.589646i	$-0.200733\pi$
$829$	1339.10	1.61532	0.807662	+	0.589646i	$0.200733\pi$
$830$	359.067	0.432611
$831$	0	0
$832$	266.738i	0.320598i
$833$	212.484i	0.255082i
$834$	0	0
$835$	− 319.472i	− 0.382601i
$836$	0	0
$837$	0	0
$838$	2094.59i	2.49951i
$839$	−492.672	−0.587213	−0.293606	−	0.955926i	$-0.594856\pi$
$839$	−492.672	−0.587213	−0.293606	+	0.955926i	$0.594856\pi$
$840$	0	0
$841$	800.556	0.951910
$842$	− 793.757i	− 0.942704i
$843$	0	0
$844$	344.574i	0.408263i
$845$	307.558	0.363974
$846$	0	0
$847$	0	0
$848$	316.943	0.373753
$849$	0	0
$850$	−279.706	−0.329066
$851$	−1755.21	−2.06252
$852$	0	0
$853$	745.303i	0.873743i	0.899524	+	0.436872i	$0.143914\pi$
$853$	745.303i	0.873743i	−0.899524	+	0.436872i	$0.856086\pi$
$854$	2231.48i	2.61297i
$855$	0	0
$856$	−136.995	−0.160041
$857$	− 892.710i	− 1.04167i	−0.853658	−	0.520834i	$-0.825621\pi$
$857$	− 892.710i	− 1.04167i	0.853658	−	0.520834i	$-0.174379\pi$
$858$	0	0
$859$	−900.724	−1.04857	−0.524287	−	0.851542i	$-0.675668\pi$
$859$	−900.724	−1.04857	−0.524287	+	0.851542i	$0.675668\pi$
$860$	− 304.101i	− 0.353606i
$861$	0	0
$862$	−2011.91	−2.33401
$863$	1215.12	1.40802	0.704011	−	0.710189i	$-0.251391\pi$
$863$	1215.12	1.40802	0.704011	+	0.710189i	$0.251391\pi$
$864$	0	0
$865$	585.572i	0.676961i
$866$	375.203i	0.433260i
$867$	0	0
$868$	− 201.905i	− 0.232610i
$869$	0	0
$870$	0	0
$871$	118.339i	0.135865i
$872$	91.1966	0.104583
$873$	0	0
$874$	−2276.01	−2.60413
$875$	− 886.551i	− 1.01320i
$876$	0	0
$877$	− 331.351i	− 0.377823i	−0.981994	−	0.188912i	$-0.939504\pi$
$877$	− 331.351i	− 0.377823i	0.981994	−	0.188912i	$-0.0604960\pi$
$878$	358.708	0.408551
$879$	0	0
$880$	0	0
$881$	883.396	1.00272	0.501360	−	0.865239i	$-0.332833\pi$
$881$	883.396	1.00272	0.501360	+	0.865239i	$0.332833\pi$
$882$	0	0
$883$	−1390.87	−1.57517	−0.787585	−	0.616207i	$-0.788669\pi$
$883$	−1390.87	−1.57517	−0.787585	+	0.616207i	$0.788669\pi$
$884$	−71.1032	−0.0804335
$885$	0	0
$886$	761.231i	0.859177i
$887$	− 1165.12i	− 1.31355i	−0.754086	−	0.656776i	$-0.771920\pi$
$887$	− 1165.12i	− 1.31355i	0.754086	−	0.656776i	$-0.228080\pi$
$888$	0	0
$889$	979.184	1.10144
$890$	− 240.933i	− 0.270712i
$891$	0	0
$892$	−266.960	−0.299283
$893$	1097.41i	1.22891i
$894$	0	0
$895$	−16.3124	−0.0182262
$896$	−320.338	−0.357520
$897$	0	0
$898$	823.768i	0.917336i
$899$	− 30.1804i	− 0.0335711i
$900$	0	0
$901$	100.457i	0.111495i
$902$	0	0
$903$	0	0
$904$	− 98.9022i	− 0.109405i
$905$	95.4732	0.105495
$906$	0	0
$907$	86.1958	0.0950340	0.0475170	−	0.998870i	$-0.484869\pi$
$907$	86.1958	0.0950340	0.0475170	+	0.998870i	$0.484869\pi$
$908$	− 1217.40i	− 1.34075i
$909$	0	0
$910$	− 198.013i	− 0.217597i
$911$	−453.721	−0.498047	−0.249023	−	0.968497i	$-0.580110\pi$
$911$	−453.721	−0.498047	−0.249023	+	0.968497i	$0.580110\pi$
$912$	0	0
$913$	0	0
$914$	−469.792	−0.513996
$915$	0	0
$916$	−879.482	−0.960133
$917$	−161.621	−0.176250
$918$	0	0
$919$	− 1495.30i	− 1.62710i	−0.581495	−	0.813550i	$-0.697532\pi$
$919$	− 1495.30i	− 1.62710i	0.581495	−	0.813550i	$-0.302468\pi$
$920$	− 72.8098i	− 0.0791411i
$921$	0	0
$922$	−1114.70	−1.20900
$923$	− 336.471i	− 0.364541i
$924$	0	0
$925$	1031.30	1.11492
$926$	1321.32i	1.42691i
$927$	0	0
$928$	−291.804	−0.314444
$929$	−1130.36	−1.21675	−0.608377	−	0.793648i	$-0.708179\pi$
$929$	−1130.36	−1.21675	−0.608377	+	0.793648i	$0.708179\pi$
$930$	0	0
$931$	− 1015.91i	− 1.09121i
$932$	− 904.253i	− 0.970228i
$933$	0	0
$934$	586.804i	0.628270i
$935$	0	0
$936$	0	0
$937$	536.813i	0.572906i	0.958094	+	0.286453i	$0.0924762\pi$
$937$	536.813i	0.572906i	−0.958094	+	0.286453i	$0.907524\pi$
$938$	−937.420	−0.999382
$939$	0	0
$940$	−429.588	−0.457009
$941$	945.191i	1.00445i	0.864736	+	0.502227i	$0.167486\pi$
$941$	945.191i	1.00445i	−0.864736	+	0.502227i	$0.832514\pi$
$942$	0	0
$943$	1984.49i	2.10445i
$944$	470.131	0.498020
$945$	0	0
$946$	0	0
$947$	488.006	0.515318	0.257659	−	0.966236i	$-0.417049\pi$
$947$	488.006	0.515318	0.257659	+	0.966236i	$0.417049\pi$
$948$	0	0
$949$	51.3192	0.0540772
$950$	1337.31	1.40770
$951$	0	0
$952$	− 46.0288i	− 0.0483496i
$953$	− 493.289i	− 0.517617i	−0.965929	−	0.258809i	$-0.916670\pi$
$953$	− 493.289i	− 0.517617i	0.965929	−	0.258809i	$-0.0833299\pi$
$954$	0	0
$955$	367.275	0.384581
$956$	773.662i	0.809270i
$957$	0	0
$958$	1144.66	1.19484
$959$	1410.55i	1.47085i
$960$	0	0
$961$	−938.479	−0.976565
$962$	502.905	0.522770
$963$	0	0
$964$	− 744.475i	− 0.772277i
$965$	692.012i	0.717111i
$966$	0	0
$967$	657.808i	0.680257i	0.940379	+	0.340128i	$0.110471\pi$
$967$	657.808i	0.680257i	−0.940379	+	0.340128i	$0.889529\pi$
$968$	0	0
$969$	0	0
$970$	− 1006.72i	− 1.03786i
$971$	−492.510	−0.507219	−0.253610	−	0.967307i	$-0.581618\pi$
$971$	−492.510	−0.507219	−0.253610	+	0.967307i	$0.581618\pi$
$972$	0	0
$973$	−1988.17	−2.04334
$974$	− 584.794i	− 0.600404i
$975$	0	0
$976$	1142.04i	1.17012i
$977$	786.861	0.805385	0.402693	−	0.915335i	$-0.368074\pi$
$977$	786.861	0.805385	0.402693	+	0.915335i	$0.368074\pi$
$978$	0	0
$979$	0	0
$980$	397.684	0.405800
$981$	0	0
$982$	675.879	0.688268
$983$	765.442	0.778680	0.389340	−	0.921094i	$-0.372703\pi$
$983$	765.442	0.778680	0.389340	+	0.921094i	$0.372703\pi$
$984$	0	0
$985$	− 453.740i	− 0.460650i
$986$	− 84.1925i	− 0.0853880i
$987$	0	0
$988$	339.954	0.344083
$989$	1275.71i	1.28990i
$990$	0	0
$991$	757.254	0.764131	0.382065	−	0.924135i	$-0.375213\pi$
$991$	757.254	0.764131	0.382065	+	0.924135i	$0.375213\pi$
$992$	− 217.753i	− 0.219509i
$993$	0	0
$994$	2665.36	2.68145
$995$	−47.2456	−0.0474830
$996$	0	0
$997$	− 365.876i	− 0.366977i	−0.983022	−	0.183489i	$-0.941261\pi$
$997$	− 365.876i	− 0.366977i	0.983022	−	0.183489i	$-0.0587390\pi$
$998$	531.144i	0.532208i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.k.604.7		8
3.2	odd	2		121.3.b.c.120.2	✓	8
11.10	odd	2	inner	1089.3.c.k.604.2		8
33.2	even	10		121.3.d.f.40.7		32
33.5	odd	10		121.3.d.f.118.7		32
33.8	even	10		121.3.d.f.112.7		32
33.14	odd	10		121.3.d.f.112.2		32
33.17	even	10		121.3.d.f.118.2		32
33.20	odd	10		121.3.d.f.40.2		32
33.26	odd	10		121.3.d.f.94.7		32
33.29	even	10		121.3.d.f.94.2		32
33.32	even	2		121.3.b.c.120.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.3.b.c.120.2	✓	8	3.2	odd	2
121.3.b.c.120.7	yes	8	33.32	even	2
121.3.d.f.40.2		32	33.20	odd	10
121.3.d.f.40.7		32	33.2	even	10
121.3.d.f.94.2		32	33.29	even	10
121.3.d.f.94.7		32	33.26	odd	10
121.3.d.f.112.2		32	33.14	odd	10
121.3.d.f.112.7		32	33.8	even	10
121.3.d.f.118.2		32	33.17	even	10
121.3.d.f.118.7		32	33.5	odd	10
1089.3.c.k.604.2		8	11.10	odd	2	inner
1089.3.c.k.604.7		8	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.89067i q^{2} -4.35597 q^{4} +1.96778 q^{5} +9.76707i q^{7} -1.02900i q^{8} +O(q^{10})\)	\(q+2.89067i q^{2} -4.35597 q^{4} +1.96778 q^{5} +9.76707i q^{7} -1.02900i q^{8} +5.68820i q^{10} +3.56414i q^{13} -28.2334 q^{14} -14.4494 q^{16} -4.57982i q^{17} +21.8968i q^{19} -8.57160 q^{20} +35.9581 q^{23} -21.1278 q^{25} -10.3028 q^{26} -42.5451i q^{28} -6.35955i q^{29} +4.74568 q^{31} -45.8844i q^{32} +13.2388 q^{34} +19.2194i q^{35} -48.8126 q^{37} -63.2963 q^{38} -2.02485i q^{40} +55.1891i q^{41} +35.4777i q^{43} +103.943i q^{46} +50.1176 q^{47} -46.3956 q^{49} -61.0736i q^{50} -15.5253i q^{52} -21.9347 q^{53} +10.0504 q^{56} +18.3834 q^{58} -32.5364 q^{59} -79.0369i q^{61} +13.7182i q^{62} +74.8392 q^{64} +7.01346i q^{65} +33.2026 q^{67} +19.9496i q^{68} -55.5571 q^{70} -94.4045 q^{71} -14.3988i q^{73} -141.101i q^{74} -95.3817i q^{76} -19.7648i q^{79} -28.4332 q^{80} -159.533 q^{82} -63.1249i q^{83} -9.01208i q^{85} -102.554 q^{86} -42.3567 q^{89} -34.8112 q^{91} -156.632 q^{92} +144.874i q^{94} +43.0880i q^{95} -176.984 q^{97} -134.114i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} + 4 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 + 4 * q^5	\( 8 q - 24 q^{4} + 4 q^{5} + 4 q^{14} - 24 q^{16} - 52 q^{20} + 12 q^{23} - 16 q^{25} - 168 q^{26} - 116 q^{31} - 180 q^{34} - 4 q^{37} + 132 q^{38} + 244 q^{47} + 88 q^{49} - 268 q^{53} + 12 q^{56} + 88 q^{58} + 56 q^{59} - 40 q^{64} + 284 q^{67} - 188 q^{70} - 272 q^{71} + 356 q^{80} - 180 q^{82} - 336 q^{86} + 24 q^{89} + 140 q^{91} + 156 q^{92} + 152 q^{97}+O(q^{100}) \) 8 * q - 24 * q^4 + 4 * q^5 + 4 * q^14 - 24 * q^16 - 52 * q^20 + 12 * q^23 - 16 * q^25 - 168 * q^26 - 116 * q^31 - 180 * q^34 - 4 * q^37 + 132 * q^38 + 244 * q^47 + 88 * q^49 - 268 * q^53 + 12 * q^56 + 88 * q^58 + 56 * q^59 - 40 * q^64 + 284 * q^67 - 188 * q^70 - 272 * q^71 + 356 * q^80 - 180 * q^82 - 336 * q^86 + 24 * q^89 + 140 * q^91 + 156 * q^92 + 152 * q^97

Introduction

L-functions

Modular forms

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Database

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