LMFDB - Embedded newform 1120.3.c.g.209.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,3,Mod(209,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1120.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1120.c (of order $2$, degree $1$, not 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:	$30.5177896084$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.19
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.26463i q^{3} +(4.99643 - 0.188883i) q^{5} +(6.12587 + 3.38729i) q^{7} -18.7163 q^{9} +O(q^{10})$	$q-5.26463i q^{3} +(4.99643 - 0.188883i) q^{5} +(6.12587 + 3.38729i) q^{7} -18.7163 q^{9} +9.50793i q^{11} +16.8405i q^{13} +(-0.994401 - 26.3044i) q^{15} +21.7673 q^{17} -7.78431 q^{19} +(17.8328 - 32.2504i) q^{21} -17.3018i q^{23} +(24.9286 - 1.88748i) q^{25} +51.1530i q^{27} -10.7974i q^{29} -52.3486i q^{31} +50.0558 q^{33} +(31.2473 + 15.7673i) q^{35} +16.3938 q^{37} +88.6589 q^{39} -53.8696i q^{41} +6.63054 q^{43} +(-93.5149 + 3.53520i) q^{45} +69.6204 q^{47} +(26.0525 + 41.5002i) q^{49} -114.597i q^{51} +71.7961 q^{53} +(1.79589 + 47.5057i) q^{55} +40.9815i q^{57} -3.57383 q^{59} +27.8212 q^{61} +(-114.654 - 63.3977i) q^{63} +(3.18088 + 84.1423i) q^{65} -62.1210 q^{67} -91.0874 q^{69} -78.8867 q^{71} -26.1926 q^{73} +(-9.93691 - 131.240i) q^{75} +(-32.2061 + 58.2443i) q^{77} +55.3047 q^{79} +100.855 q^{81} +62.8573i q^{83} +(108.759 - 4.11147i) q^{85} -56.8445 q^{87} +53.9148i q^{89} +(-57.0436 + 103.163i) q^{91} -275.596 q^{93} +(-38.8938 + 1.47033i) q^{95} -17.1560 q^{97} -177.954i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 224 q^{9}+O(q^{10}) $ 80 * q - 224 * q^9	$ 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) $ 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	5.26463i		−	1.75488i	−0.479689	−	0.877439i	$-0.659250\pi$
$3$		−	5.26463i		−	1.75488i	0.479689	−	0.877439i	$-0.340750\pi$
$4$		0			0
$5$	4.99643	−	0.188883i	0.999286	−	0.0377767i
$5$	4.99643	−	0.188883i	0.999286	−	0.0377767i
$6$		0			0
$7$	6.12587	+	3.38729i	0.875124	+	0.483899i
$7$	6.12587	+	3.38729i	0.875124	+	0.483899i
$8$		0			0
$9$	−18.7163			−2.07959
$10$		0			0
$11$			9.50793i			0.864357i	0.901788	+	0.432179i	$0.142255\pi$
$11$			9.50793i			0.864357i	−0.901788	+	0.432179i	$0.857745\pi$
$12$		0			0
$13$			16.8405i			1.29542i	0.761886	+	0.647711i	$0.224273\pi$
$13$			16.8405i			1.29542i	−0.761886	+	0.647711i	$0.775727\pi$
$14$		0			0
$15$	−0.994401	−	26.3044i	−0.0662934	−	1.75362i
$16$		0			0
$17$	21.7673			1.28043			0.640213	−	0.768197i	$-0.278846\pi$
$17$	21.7673			1.28043			0.640213	+	0.768197i	$0.278846\pi$
$18$		0			0
$19$	−7.78431			−0.409701			−0.204850	−	0.978793i	$-0.565671\pi$
$19$	−7.78431			−0.409701			−0.204850	+	0.978793i	$0.565671\pi$
$20$		0			0
$21$	17.8328	−	32.2504i	0.849182	−	1.53574i
$22$		0			0
$23$		−	17.3018i		−	0.752250i	−0.926569	−	0.376125i	$-0.877256\pi$
$23$		−	17.3018i		−	0.752250i	0.926569	−	0.376125i	$-0.122744\pi$
$24$		0			0
$25$	24.9286	−	1.88748i	0.997146	−	0.0754994i
$26$		0			0
$27$			51.1530i			1.89456i
$28$		0			0
$29$		−	10.7974i		−	0.372325i	−0.982519	−	0.186162i	$-0.940395\pi$
$29$		−	10.7974i		−	0.372325i	0.982519	−	0.186162i	$-0.0596051\pi$
$30$		0			0
$31$		−	52.3486i		−	1.68866i	−0.535820	−	0.844332i	$-0.679998\pi$
$31$		−	52.3486i		−	1.68866i	0.535820	−	0.844332i	$-0.320002\pi$
$32$		0			0
$33$	50.0558			1.51684
$34$		0			0
$35$	31.2473	+	15.7673i	0.892780	+	0.450494i
$36$		0			0
$37$	16.3938			0.443075			0.221537	−	0.975152i	$-0.428892\pi$
$37$	16.3938			0.443075			0.221537	+	0.975152i	$0.428892\pi$
$38$		0			0
$39$	88.6589			2.27331
$40$		0			0
$41$		−	53.8696i		−	1.31389i	−0.753938	−	0.656946i	$-0.771848\pi$
$41$		−	53.8696i		−	1.31389i	0.753938	−	0.656946i	$-0.228152\pi$
$42$		0			0
$43$	6.63054			0.154199			0.0770993	−	0.997023i	$-0.475434\pi$
$43$	6.63054			0.154199			0.0770993	+	0.997023i	$0.475434\pi$
$44$		0			0
$45$	−93.5149	+	3.53520i	−2.07811	+	0.0785601i
$46$		0			0
$47$	69.6204			1.48128			0.740642	−	0.671899i	$-0.234521\pi$
$47$	69.6204			1.48128			0.740642	+	0.671899i	$0.234521\pi$
$48$		0			0
$49$	26.0525	+	41.5002i	0.531684	+	0.846943i
$50$		0			0
$51$		−	114.597i		−	2.24699i
$52$		0			0
$53$	71.7961			1.35464			0.677322	−	0.735687i	$-0.263140\pi$
$53$	71.7961			1.35464			0.677322	+	0.735687i	$0.263140\pi$
$54$		0			0
$55$	1.79589	+	47.5057i	0.0326525	+	0.863740i
$56$		0			0
$57$			40.9815i			0.718974i
$58$		0			0
$59$	−3.57383			−0.0605734			−0.0302867	−	0.999541i	$-0.509642\pi$
$59$	−3.57383			−0.0605734			−0.0302867	+	0.999541i	$0.509642\pi$
$60$		0			0
$61$	27.8212			0.456086			0.228043	−	0.973651i	$-0.426767\pi$
$61$	27.8212			0.456086			0.228043	+	0.973651i	$0.426767\pi$
$62$		0			0
$63$	−114.654	−	63.3977i	−1.81990	−	1.00631i
$64$		0			0
$65$	3.18088	+	84.1423i	0.0489367	+	1.29450i
$66$		0			0
$67$	−62.1210			−0.927180			−0.463590	−	0.886050i	$-0.653439\pi$
$67$	−62.1210			−0.927180			−0.463590	+	0.886050i	$0.653439\pi$
$68$		0			0
$69$	−91.0874			−1.32011
$70$		0			0
$71$	−78.8867			−1.11108			−0.555540	−	0.831490i	$-0.687489\pi$
$71$	−78.8867			−1.11108			−0.555540	+	0.831490i	$0.687489\pi$
$72$		0			0
$73$	−26.1926			−0.358803			−0.179401	−	0.983776i	$-0.557416\pi$
$73$	−26.1926			−0.358803			−0.179401	+	0.983776i	$0.557416\pi$
$74$		0			0
$75$	−9.93691	−	131.240i	−0.132492	−	1.74987i
$76$		0			0
$77$	−32.2061	+	58.2443i	−0.418261	+	0.756420i
$78$		0			0
$79$	55.3047			0.700060			0.350030	−	0.936739i	$-0.386171\pi$
$79$	55.3047			0.700060			0.350030	+	0.936739i	$0.386171\pi$
$80$		0			0
$81$	100.855			1.24512
$82$		0			0
$83$			62.8573i			0.757316i	0.925537	+	0.378658i	$0.123614\pi$
$83$			62.8573i			0.757316i	−0.925537	+	0.378658i	$0.876386\pi$
$84$		0			0
$85$	108.759	−	4.11147i	1.27951	−	0.0483702i
$86$		0			0
$87$	−56.8445			−0.653385
$88$		0			0
$89$			53.9148i			0.605784i	0.953025	+	0.302892i	$0.0979522\pi$
$89$			53.9148i			0.605784i	−0.953025	+	0.302892i	$0.902048\pi$
$90$		0			0
$91$	−57.0436	+	103.163i	−0.626852	+	1.13365i
$92$		0			0
$93$	−275.596			−2.96340
$94$		0			0
$95$	−38.8938	+	1.47033i	−0.409408	+	0.0154771i
$96$		0			0
$97$	−17.1560			−0.176866			−0.0884331	−	0.996082i	$-0.528186\pi$
$97$	−17.1560			−0.176866			−0.0884331	+	0.996082i	$0.528186\pi$
$98$		0			0
$99$		−	177.954i		−	1.79751i
$100$		0			0
$101$	49.3961			0.489071			0.244535	−	0.969640i	$-0.421365\pi$
$101$	49.3961			0.489071			0.244535	+	0.969640i	$0.421365\pi$
$102$		0			0
$103$	35.5249			0.344902			0.172451	−	0.985018i	$-0.444831\pi$
$103$	35.5249			0.344902			0.172451	+	0.985018i	$0.444831\pi$
$104$		0			0
$105$	83.0089	−	164.505i	0.790561	−	1.56672i
$106$		0			0
$107$	150.957			1.41081			0.705405	−	0.708804i	$-0.250765\pi$
$107$	150.957			1.41081			0.705405	+	0.708804i	$0.250765\pi$
$108$		0			0
$109$			1.49558i			0.0137209i	0.999976	+	0.00686044i	$0.00218376\pi$
$109$			1.49558i			0.0137209i	−0.999976	+	0.00686044i	$0.997816\pi$
$110$		0			0
$111$		−	86.3072i		−	0.777542i
$112$		0			0
$113$		−	184.712i		−	1.63462i	−0.576197	−	0.817311i	$-0.695464\pi$
$113$		−	184.712i		−	1.63462i	0.576197	−	0.817311i	$-0.304536\pi$
$114$		0			0
$115$	−3.26801	−	86.4470i	−0.0284175	−	0.751713i
$116$		0			0
$117$		−	315.192i		−	2.69395i
$118$		0			0
$119$	133.343	+	73.7320i	1.12053	+	0.619597i
$120$		0			0
$121$	30.5992			0.252886
$122$		0			0
$123$	−283.604			−2.30572
$124$		0			0
$125$	124.198	−	14.1393i	0.993582	−	0.113114i
$126$		0			0
$127$			48.9383i			0.385341i	0.981264	+	0.192670i	$0.0617148\pi$
$127$			48.9383i			0.385341i	−0.981264	+	0.192670i	$0.938285\pi$
$128$		0			0
$129$		−	34.9073i		−	0.270599i
$130$		0			0
$131$	−121.072			−0.924214			−0.462107	−	0.886824i	$-0.652906\pi$
$131$	−121.072			−0.924214			−0.462107	+	0.886824i	$0.652906\pi$
$132$		0			0
$133$	−47.6857	−	26.3677i	−0.358539	−	0.198254i
$134$		0			0
$135$	9.66194	+	255.582i	0.0715699	+	1.89320i
$136$		0			0
$137$			202.587i			1.47874i	0.673300	+	0.739369i	$0.264876\pi$
$137$			202.587i			1.47874i	−0.673300	+	0.739369i	$0.735124\pi$
$138$		0			0
$139$	−176.207			−1.26768			−0.633840	−	0.773465i	$-0.718522\pi$
$139$	−176.207			−1.26768			−0.633840	+	0.773465i	$0.718522\pi$
$140$		0			0
$141$		−	366.526i		−	2.59947i
$142$		0			0
$143$	−160.118			−1.11971
$144$		0			0
$145$	−2.03945	−	53.9486i	−0.0140652	−	0.372059i
$146$		0			0
$147$	218.483	−	137.157i	1.48628	−	0.933041i
$148$		0			0
$149$		−	128.331i		−	0.861281i	−0.902524	−	0.430640i	$-0.858288\pi$
$149$		−	128.331i		−	0.861281i	0.902524	−	0.430640i	$-0.141712\pi$
$150$		0			0
$151$	189.463			1.25472			0.627361	−	0.778729i	$-0.284135\pi$
$151$	189.463			1.25472			0.627361	+	0.778729i	$0.284135\pi$
$152$		0			0
$153$	−407.403			−2.66277
$154$		0			0
$155$	−9.88777	−	261.556i	−0.0637921	−	1.68746i
$156$		0			0
$157$			20.0847i			0.127928i	0.997952	+	0.0639641i	$0.0203743\pi$
$157$			20.0847i			0.127928i	−0.997952	+	0.0639641i	$0.979626\pi$
$158$		0			0
$159$		−	377.980i		−	2.37723i
$160$		0			0
$161$	58.6061	−	105.988i	0.364013	−	0.658312i
$162$		0			0
$163$	−186.675			−1.14524			−0.572622	−	0.819819i	$-0.694074\pi$
$163$	−186.675			−1.14524			−0.572622	+	0.819819i	$0.694074\pi$
$164$		0			0
$165$	250.100	−	9.45469i	1.51576	−	0.0573012i
$166$		0			0
$167$	36.8456			0.220632			0.110316	−	0.993897i	$-0.464814\pi$
$167$	36.8456			0.220632			0.110316	+	0.993897i	$0.464814\pi$
$168$		0			0
$169$	−114.602			−0.678117
$170$		0			0
$171$	145.694			0.852011
$172$		0			0
$173$			193.813i			1.12031i	0.828389	+	0.560153i	$0.189258\pi$
$173$			193.813i			1.12031i	−0.828389	+	0.560153i	$0.810742\pi$
$174$		0			0
$175$	159.103	+	72.8781i	0.909160	+	0.416446i
$176$		0			0
$177$			18.8149i			0.106299i
$178$		0			0
$179$			185.148i			1.03435i	0.855880	+	0.517175i	$0.173016\pi$
$179$			185.148i			1.03435i	−0.855880	+	0.517175i	$0.826984\pi$
$180$		0			0
$181$	−96.1838			−0.531402			−0.265701	−	0.964055i	$-0.585603\pi$
$181$	−96.1838			−0.531402			−0.265701	+	0.964055i	$0.585603\pi$
$182$		0			0
$183$		−	146.469i		−	0.800375i
$184$		0			0
$185$	81.9103	−	3.09651i	0.442759	−	0.0167379i
$186$		0			0
$187$			206.962i			1.10675i
$188$		0			0
$189$	−173.270	+	313.357i	−0.916772	+	1.65797i
$190$		0			0
$191$	−216.967			−1.13595			−0.567977	−	0.823044i	$-0.692274\pi$
$191$	−216.967			−1.13595			−0.567977	+	0.823044i	$0.692274\pi$
$192$		0			0
$193$		−	364.330i		−	1.88772i	−0.330346	−	0.943860i	$-0.607165\pi$
$193$		−	364.330i		−	1.88772i	0.330346	−	0.943860i	$-0.392835\pi$
$194$		0			0
$195$	442.978	−	16.7462i	2.27168	−	0.0858779i
$196$		0			0
$197$	−53.5832			−0.271996			−0.135998	−	0.990709i	$-0.543424\pi$
$197$	−53.5832			−0.271996			−0.135998	+	0.990709i	$0.543424\pi$
$198$		0			0
$199$			306.342i			1.53941i	0.638401	+	0.769704i	$0.279596\pi$
$199$			306.342i			1.53941i	−0.638401	+	0.769704i	$0.720404\pi$
$200$		0			0
$201$			327.044i			1.62709i
$202$		0			0
$203$	36.5740	−	66.1436i	0.180168	−	0.325831i
$204$		0			0
$205$	−10.1751	−	269.156i	−0.0496344	−	1.31295i
$206$		0			0
$207$			323.826i			1.56438i
$208$		0			0
$209$		−	74.0127i		−	0.354128i
$210$		0			0
$211$			7.57377i			0.0358946i	0.999839	+	0.0179473i	$0.00571311\pi$
$211$			7.57377i			0.0358946i	−0.999839	+	0.0179473i	$0.994287\pi$
$212$		0			0
$213$			415.310i			1.94981i
$214$		0			0
$215$	33.1290	−	1.25240i	0.154088	−	0.00582510i
$216$		0			0
$217$	177.320	−	320.681i	0.817142	−	1.47779i
$218$		0			0
$219$			137.894i			0.629655i
$220$		0			0
$221$			366.571i			1.65869i
$222$		0			0
$223$	365.812			1.64041			0.820207	−	0.572067i	$-0.193858\pi$
$223$	365.812			1.64041			0.820207	+	0.572067i	$0.193858\pi$
$224$		0			0
$225$	−466.573	+	35.3268i	−2.07366	+	0.157008i
$226$		0			0
$227$			320.799i			1.41321i	0.707608	+	0.706605i	$0.249774\pi$
$227$			320.799i			1.41321i	−0.707608	+	0.706605i	$0.750226\pi$
$228$		0			0
$229$	−202.193			−0.882937			−0.441469	−	0.897277i	$-0.645542\pi$
$229$	−202.193			−0.882937			−0.441469	+	0.897277i	$0.645542\pi$
$230$		0			0
$231$	306.635	+	169.553i	1.32742	+	0.733997i
$232$		0			0
$233$		−	16.2649i		−	0.0698065i	−0.999391	−	0.0349032i	$-0.988888\pi$
$233$		−	16.2649i		−	0.0698065i	0.999391	−	0.0349032i	$-0.0111123\pi$
$234$		0			0
$235$	347.853	−	13.1501i	1.48023	−	0.0559580i
$236$		0			0
$237$		−	291.159i		−	1.22852i
$238$		0			0
$239$	−332.222			−1.39005			−0.695025	−	0.718986i	$-0.744607\pi$
$239$	−332.222			−1.39005			−0.695025	+	0.718986i	$0.744607\pi$
$240$		0			0
$241$			324.579i			1.34680i	0.739278	+	0.673400i	$0.235167\pi$
$241$			324.579i			1.34680i	−0.739278	+	0.673400i	$0.764833\pi$
$242$		0			0
$243$		−	70.5850i		−	0.290473i
$244$		0			0
$245$	138.008	+	202.432i	0.563300	+	0.826253i
$246$		0			0
$247$		−	131.092i		−	0.530735i
$248$		0			0
$249$	330.920			1.32900
$250$		0			0
$251$	−302.261			−1.20423			−0.602113	−	0.798411i	$-0.705675\pi$
$251$	−302.261			−1.20423			−0.602113	+	0.798411i	$0.705675\pi$
$252$		0			0
$253$	164.504			0.650213
$254$		0			0
$255$	−21.6454	−	572.574i	−0.0848838	−	2.24539i
$256$		0			0
$257$	26.7241			0.103985			0.0519924	−	0.998647i	$-0.483443\pi$
$257$	26.7241			0.103985			0.0519924	+	0.998647i	$0.483443\pi$
$258$		0			0
$259$	100.426	+	55.5304i	0.387746	+	0.214403i
$260$		0			0
$261$			202.088i			0.774285i
$262$		0			0
$263$		−	159.326i		−	0.605800i	−0.953022	−	0.302900i	$-0.902045\pi$
$263$		−	159.326i		−	0.605800i	0.953022	−	0.302900i	$-0.0979549\pi$
$264$		0			0
$265$	358.724	−	13.5611i	1.35368	−	0.0511739i
$266$		0			0
$267$	283.842			1.06308
$268$		0			0
$269$	120.174			0.446744			0.223372	−	0.974733i	$-0.428293\pi$
$269$	120.174			0.446744			0.223372	+	0.974733i	$0.428293\pi$
$270$		0			0
$271$		−	110.800i		−	0.408854i	−0.978882	−	0.204427i	$-0.934467\pi$
$271$		−	110.800i		−	0.408854i	0.978882	−	0.204427i	$-0.0655331\pi$
$272$		0			0
$273$	543.113	+	300.313i	1.98942	+	1.10005i
$274$		0			0
$275$	17.9461	+	237.020i	0.0652584	+	0.861890i
$276$		0			0
$277$	−406.197			−1.46641			−0.733207	−	0.680006i	$-0.761977\pi$
$277$	−406.197			−1.46641			−0.733207	+	0.680006i	$0.761977\pi$
$278$		0			0
$279$			979.774i			3.51174i
$280$		0			0
$281$	−283.218			−1.00789			−0.503946	−	0.863735i	$-0.668119\pi$
$281$	−283.218			−1.00789			−0.503946	+	0.863735i	$0.668119\pi$
$282$		0			0
$283$			244.520i			0.864027i	0.901867	+	0.432014i	$0.142197\pi$
$283$			244.520i			0.864027i	−0.901867	+	0.432014i	$0.857803\pi$
$284$		0			0
$285$	7.74073	+	204.761i	0.0271604	+	0.718461i
$286$		0			0
$287$	182.472	−	329.998i	0.635790	−	1.14982i
$288$		0			0
$289$	184.813			0.639492
$290$		0			0
$291$			90.3202i			0.310379i
$292$		0			0
$293$		−	27.5005i		−	0.0938584i	−0.998898	−	0.0469292i	$-0.985056\pi$
$293$		−	27.5005i		−	0.0938584i	0.998898	−	0.0469292i	$-0.0149435\pi$
$294$		0			0
$295$	−17.8564	+	0.675037i	−0.0605302	+	0.00228826i
$296$		0			0
$297$	−486.359			−1.63757
$298$		0			0
$299$	291.370			0.974481
$300$		0			0
$301$	40.6178	+	22.4595i	0.134943	+	0.0746164i
$302$		0			0
$303$		−	260.052i		−	0.858259i
$304$		0			0
$305$	139.007	−	5.25497i	0.455760	−	0.0172294i
$306$		0			0
$307$			135.158i			0.440254i	0.975471	+	0.220127i	$0.0706472\pi$
$307$			135.158i			0.440254i	−0.975471	+	0.220127i	$0.929353\pi$
$308$		0			0
$309$		−	187.026i		−	0.605261i
$310$		0			0
$311$			7.13693i			0.0229483i	0.999934	+	0.0114742i	$0.00365242\pi$
$311$			7.13693i			0.0229483i	−0.999934	+	0.0114742i	$0.996348\pi$
$312$		0			0
$313$	−367.602			−1.17445			−0.587224	−	0.809424i	$-0.699779\pi$
$313$	−367.602			−1.17445			−0.587224	+	0.809424i	$0.699779\pi$
$314$		0			0
$315$	−584.835	−	295.106i	−1.85662	−	0.936844i
$316$		0			0
$317$	438.224			1.38241			0.691206	−	0.722658i	$-0.257080\pi$
$317$	438.224			1.38241			0.691206	+	0.722658i	$0.257080\pi$
$318$		0			0
$319$	102.661			0.321822
$320$		0			0
$321$		−	794.731i		−	2.47580i
$322$		0			0
$323$	−169.443			−0.524592
$324$		0			0
$325$	31.7861	+	419.810i	0.0978035	+	1.29172i
$326$		0			0
$327$	7.87366			0.0240785
$328$		0			0
$329$	426.485	+	235.824i	1.29631	+	0.716791i
$330$		0			0
$331$		−	239.131i		−	0.722450i	−0.932479	−	0.361225i	$-0.882359\pi$
$331$		−	239.131i		−	0.722450i	0.932479	−	0.361225i	$-0.117641\pi$
$332$		0			0
$333$	−306.832			−0.921416
$334$		0			0
$335$	−310.383	+	11.7336i	−0.926518	+	0.0350257i
$336$		0			0
$337$		−	247.164i		−	0.733425i	−0.930334	−	0.366713i	$-0.880483\pi$
$337$		−	247.164i		−	0.733425i	0.930334	−	0.366713i	$-0.119517\pi$
$338$		0			0
$339$	−972.442			−2.86856
$340$		0			0
$341$	497.727			1.45961
$342$		0			0
$343$	19.0213	+	342.472i	0.0554557	+	0.998461i
$344$		0			0
$345$	−455.112	+	17.2049i	−1.31916	+	0.0498692i
$346$		0			0
$347$	−217.690			−0.627348			−0.313674	−	0.949531i	$-0.601560\pi$
$347$	−217.690			−0.627348			−0.313674	+	0.949531i	$0.601560\pi$
$348$		0			0
$349$	117.972			0.338029			0.169015	−	0.985614i	$-0.445941\pi$
$349$	117.972			0.338029			0.169015	+	0.985614i	$0.445941\pi$
$350$		0			0
$351$	−861.441			−2.45425
$352$		0			0
$353$	361.591			1.02434			0.512168	−	0.858885i	$-0.328842\pi$
$353$	361.591			1.02434			0.512168	+	0.858885i	$0.328842\pi$
$354$		0			0
$355$	−394.152	+	14.9004i	−1.11029	+	0.0419729i
$356$		0			0
$357$	388.172	−	702.004i	1.08732	−	1.96640i
$358$		0			0
$359$	−317.055			−0.883163			−0.441581	−	0.897221i	$-0.645582\pi$
$359$	−317.055			−0.883163			−0.441581	+	0.897221i	$0.645582\pi$
$360$		0			0
$361$	−300.404			−0.832145
$362$		0			0
$363$		−	161.094i		−	0.443784i
$364$		0			0
$365$	−130.870	+	4.94734i	−0.358547	+	0.0135544i
$366$		0			0
$367$	−148.629			−0.404983			−0.202492	−	0.979284i	$-0.564904\pi$
$367$	−148.629			−0.404983			−0.202492	+	0.979284i	$0.564904\pi$
$368$		0			0
$369$			1008.24i			2.73236i
$370$		0			0
$371$	439.814	+	243.194i	1.18548	+	0.655510i
$372$		0			0
$373$	172.124			0.461458			0.230729	−	0.973018i	$-0.425889\pi$
$373$	172.124			0.461458			0.230729	+	0.973018i	$0.425889\pi$
$374$		0			0
$375$	−74.4381	−	653.855i	−0.198502	−	1.74361i
$376$		0			0
$377$	181.834			0.482318
$378$		0			0
$379$			323.133i			0.852592i	0.904584	+	0.426296i	$0.140182\pi$
$379$			323.133i			0.852592i	−0.904584	+	0.426296i	$0.859818\pi$
$380$		0			0
$381$	257.642			0.676226
$382$		0			0
$383$	156.518			0.408662			0.204331	−	0.978902i	$-0.434498\pi$
$383$	156.518			0.408662			0.204331	+	0.978902i	$0.434498\pi$
$384$		0			0
$385$	−149.914	+	297.097i	−0.389388	+	0.771681i
$386$		0			0
$387$	−124.099			−0.320670
$388$		0			0
$389$			424.478i			1.09120i	0.838045	+	0.545602i	$0.183699\pi$
$389$			424.478i			1.09120i	−0.838045	+	0.545602i	$0.816301\pi$
$390$		0			0
$391$		−	376.612i		−	0.963201i
$392$		0			0
$393$			637.399i			1.62188i
$394$		0			0
$395$	276.326	−	10.4461i	0.699560	−	0.0264459i
$396$		0			0
$397$			125.289i			0.315590i	0.987472	+	0.157795i	$0.0504385\pi$
$397$			125.289i			0.315590i	−0.987472	+	0.157795i	$0.949562\pi$
$398$		0			0
$399$	−138.816	+	251.047i	−0.347911	+	0.629192i
$400$		0			0
$401$	−498.995			−1.24438			−0.622189	−	0.782867i	$-0.713756\pi$
$401$	−498.995			−1.24438			−0.622189	+	0.782867i	$0.713756\pi$
$402$		0			0
$403$	881.575			2.18753
$404$		0			0
$405$	503.913	−	19.0497i	1.24423	−	0.0470364i
$406$		0			0
$407$			155.871i			0.382975i
$408$		0			0
$409$		−	371.026i		−	0.907155i	−0.891217	−	0.453577i	$-0.850148\pi$
$409$		−	371.026i		−	0.907155i	0.891217	−	0.453577i	$-0.149852\pi$
$410$		0			0
$411$	1066.55			2.59500
$412$		0			0
$413$	−21.8928	−	12.1056i	−0.0530093	−	0.0293114i
$414$		0			0
$415$	11.8727	+	314.062i	0.0286089	+	0.756776i
$416$		0			0
$417$			927.667i			2.22462i
$418$		0			0
$419$	298.761			0.713033			0.356516	−	0.934289i	$-0.383964\pi$
$419$	298.761			0.713033			0.356516	+	0.934289i	$0.383964\pi$
$420$		0			0
$421$			39.1981i			0.0931070i	0.998916	+	0.0465535i	$0.0148238\pi$
$421$			39.1981i			0.0931070i	−0.998916	+	0.0465535i	$0.985176\pi$
$422$		0			0
$423$	−1303.04			−3.08047
$424$		0			0
$425$	542.628	−	41.0853i	1.27677	−	0.0966714i
$426$		0			0
$427$	170.429	+	94.2386i	0.399132	+	0.220699i
$428$		0			0
$429$			842.963i			1.96495i
$430$		0			0
$431$	336.264			0.780196			0.390098	−	0.920773i	$-0.372441\pi$
$431$	336.264			0.780196			0.390098	+	0.920773i	$0.372441\pi$
$432$		0			0
$433$	−295.132			−0.681597			−0.340799	−	0.940136i	$-0.610697\pi$
$433$	−295.132			−0.681597			−0.340799	+	0.940136i	$0.610697\pi$
$434$		0			0
$435$	−284.019	+	10.7370i	−0.652918	+	0.0246827i
$436$		0			0
$437$			134.682i			0.308197i
$438$		0			0
$439$		−	104.219i		−	0.237400i	−0.992930	−	0.118700i	$-0.962127\pi$
$439$		−	104.219i		−	0.237400i	0.992930	−	0.118700i	$-0.0378727\pi$
$440$		0			0
$441$	−487.608	−	776.732i	−1.10569	−	1.76130i
$442$		0			0
$443$	−219.958			−0.496518			−0.248259	−	0.968694i	$-0.579858\pi$
$443$	−219.958			−0.496518			−0.248259	+	0.968694i	$0.579858\pi$
$444$		0			0
$445$	10.1836	+	269.382i	0.0228845	+	0.605352i
$446$		0			0
$447$	−675.615			−1.51144
$448$		0			0
$449$	−274.759			−0.611936			−0.305968	−	0.952042i	$-0.598980\pi$
$449$	−274.759			−0.611936			−0.305968	+	0.952042i	$0.598980\pi$
$450$		0			0
$451$	512.188			1.13567
$452$		0			0
$453$		−	997.453i		−	2.20188i
$454$		0			0
$455$	−265.529	+	526.219i	−0.583579	+	1.15653i
$456$		0			0
$457$		−	172.854i		−	0.378237i	−0.981954	−	0.189119i	$-0.939437\pi$
$457$		−	172.854i		−	0.378237i	0.981954	−	0.189119i	$-0.0605630\pi$
$458$		0			0
$459$			1113.46i			2.42584i
$460$		0			0
$461$	−741.040			−1.60746			−0.803731	−	0.594993i	$-0.797155\pi$
$461$	−741.040			−1.60746			−0.803731	+	0.594993i	$0.797155\pi$
$462$		0			0
$463$			153.903i			0.332404i	0.986092	+	0.166202i	$0.0531504\pi$
$463$			153.903i			0.332404i	−0.986092	+	0.166202i	$0.946850\pi$
$464$		0			0
$465$	−1377.00	+	52.0555i	−2.96128	+	0.111947i
$466$		0			0
$467$		−	695.292i		−	1.48885i	−0.667708	−	0.744423i	$-0.732724\pi$
$467$		−	695.292i		−	1.48885i	0.667708	−	0.744423i	$-0.267276\pi$
$468$		0			0
$469$	−380.545	−	210.422i	−0.811397	−	0.448661i
$470$		0			0
$471$	105.739			0.224498
$472$		0			0
$473$			63.0427i			0.133283i
$474$		0			0
$475$	−194.052	+	14.6928i	−0.408531	+	0.0309321i
$476$		0			0
$477$	−1343.76			−2.81711
$478$		0			0
$479$			205.463i			0.428942i	0.976730	+	0.214471i	$0.0688028\pi$
$479$			205.463i			0.428942i	−0.976730	+	0.214471i	$0.931197\pi$
$480$		0			0
$481$			276.079i			0.573969i
$482$		0			0
$483$	−557.989	−	308.539i	−1.15526	−	0.638798i
$484$		0			0
$485$	−85.7189	+	3.24049i	−0.176740	+	0.00668141i
$486$		0			0
$487$			117.450i			0.241171i	0.992703	+	0.120586i	$0.0384772\pi$
$487$			117.450i			0.241171i	−0.992703	+	0.120586i	$0.961523\pi$
$488$		0			0
$489$			982.775i			2.00976i
$490$		0			0
$491$		−	324.679i		−	0.661261i	−0.943760	−	0.330631i	$-0.892739\pi$
$491$		−	324.679i		−	0.661261i	0.943760	−	0.330631i	$-0.107261\pi$
$492$		0			0
$493$		−	235.030i		−	0.476735i
$494$		0			0
$495$	−33.6125	−	889.134i	−0.0679040	−	1.79623i
$496$		0			0
$497$	−483.250	−	267.212i	−0.972334	−	0.537650i
$498$		0			0
$499$		−	684.255i		−	1.37125i	−0.727954	−	0.685626i	$-0.759529\pi$
$499$		−	684.255i		−	1.37125i	0.727954	−	0.685626i	$-0.240471\pi$
$500$		0			0
$501$		−	193.979i		−	0.387183i
$502$		0			0
$503$	628.664			1.24983			0.624915	−	0.780693i	$-0.285134\pi$
$503$	628.664			1.24983			0.624915	+	0.780693i	$0.285134\pi$
$504$		0			0
$505$	246.804	−	9.33010i	0.488722	−	0.0184755i
$506$		0			0
$507$			603.336i			1.19001i
$508$		0			0
$509$	−554.759			−1.08990			−0.544950	−	0.838469i	$-0.683451\pi$
$509$	−554.759			−1.08990			−0.544950	+	0.838469i	$0.683451\pi$
$510$		0			0
$511$	−160.452	−	88.7219i	−0.313997	−	0.173624i
$512$		0			0
$513$		−	398.191i		−	0.776200i
$514$		0			0
$515$	177.498	−	6.71007i	0.344656	−	0.0130293i
$516$		0			0
$517$			661.946i			1.28036i
$518$		0			0
$519$	1020.35			1.96600
$520$		0			0
$521$			255.396i			0.490204i	0.969497	+	0.245102i	$0.0788215\pi$
$521$			255.396i			0.490204i	−0.969497	+	0.245102i	$0.921178\pi$
$522$		0			0
$523$		−	849.106i		−	1.62353i	−0.583985	−	0.811765i	$-0.698507\pi$
$523$		−	849.106i		−	1.62353i	0.583985	−	0.811765i	$-0.301493\pi$
$524$		0			0
$525$	383.676	−	837.619i	0.730812	−	1.59546i
$526$		0			0
$527$		−	1139.48i		−	2.16221i
$528$		0			0
$529$	229.649			0.434120
$530$		0			0
$531$	66.8891			0.125968
$532$		0			0
$533$	907.189			1.70204
$534$		0			0
$535$	754.245	−	28.5132i	1.40980	−	0.0532957i
$536$		0			0
$537$	974.739			1.81516
$538$		0			0
$539$	−394.581	+	247.706i	−0.732061	+	0.459565i
$540$		0			0
$541$		−	654.500i		−	1.20980i	−0.796303	−	0.604899i	$-0.793214\pi$
$541$		−	654.500i		−	1.20980i	0.796303	−	0.604899i	$-0.206786\pi$
$542$		0			0
$543$			506.372i			0.932546i
$544$		0			0
$545$	0.282489	+	7.47254i	0.000518329	+	0.0137111i
$546$		0			0
$547$	−388.810			−0.710805			−0.355402	−	0.934713i	$-0.615656\pi$
$547$	−388.810			−0.710805			−0.355402	+	0.934713i	$0.615656\pi$
$548$		0			0
$549$	−520.712			−0.948474
$550$		0			0
$551$			84.0505i			0.152542i
$552$		0			0
$553$	338.790	+	187.333i	0.612639	+	0.338758i
$554$		0			0
$555$	−16.3020	−	431.228i	−0.0293729	−	0.776987i
$556$		0			0
$557$	−942.596			−1.69227			−0.846137	−	0.532966i	$-0.821077\pi$
$557$	−942.596			−1.69227			−0.846137	+	0.532966i	$0.821077\pi$
$558$		0			0
$559$			111.661i			0.199752i
$560$		0			0
$561$	1089.58			1.94220
$562$		0			0
$563$			364.114i			0.646739i	0.946273	+	0.323370i	$0.104816\pi$
$563$			364.114i			0.646739i	−0.946273	+	0.323370i	$0.895184\pi$
$564$		0			0
$565$	−34.8891	−	922.902i	−0.0617506	−	1.63346i
$566$		0			0
$567$	617.822	+	341.623i	1.08963	+	0.602511i
$568$		0			0
$569$	−350.135			−0.615352			−0.307676	−	0.951491i	$-0.599551\pi$
$569$	−350.135			−0.615352			−0.307676	+	0.951491i	$0.599551\pi$
$570$		0			0
$571$		−	800.701i		−	1.40228i	−0.713025	−	0.701139i	$-0.752675\pi$
$571$		−	800.701i		−	1.40228i	0.713025	−	0.701139i	$-0.247325\pi$
$572$		0			0
$573$			1142.25i			1.99346i
$574$		0			0
$575$	−32.6568	−	431.309i	−0.0567944	−	0.750103i
$576$		0			0
$577$	−546.338			−0.946859			−0.473429	−	0.880832i	$-0.656984\pi$
$577$	−546.338			−0.946859			−0.473429	+	0.880832i	$0.656984\pi$
$578$		0			0
$579$	−1918.06			−3.31272
$580$		0			0
$581$	−212.916	+	385.055i	−0.366464	+	0.662746i
$582$		0			0
$583$			682.633i			1.17090i
$584$		0			0
$585$	−59.5345	−	1574.84i	−0.101768	−	2.69203i
$586$		0			0
$587$			547.924i			0.933430i	0.884408	+	0.466715i	$0.154563\pi$
$587$			547.924i			0.933430i	−0.884408	+	0.466715i	$0.845437\pi$
$588$		0			0
$589$			407.498i			0.691847i
$590$		0			0
$591$			282.096i			0.477320i
$592$		0			0
$593$	826.816			1.39429			0.697147	−	0.716928i	$-0.254452\pi$
$593$	826.816			1.39429			0.697147	+	0.716928i	$0.254452\pi$
$594$		0			0
$595$	680.167	+	343.210i	1.14314	+	0.576824i
$596$		0			0
$597$	1612.78			2.70147
$598$		0			0
$599$	−66.8880			−0.111666			−0.0558331	−	0.998440i	$-0.517781\pi$
$599$	−66.8880			−0.111666			−0.0558331	+	0.998440i	$0.517781\pi$
$600$		0			0
$601$		−	796.784i		−	1.32576i	−0.748724	−	0.662882i	$-0.769333\pi$
$601$		−	796.784i		−	1.32576i	0.748724	−	0.662882i	$-0.230667\pi$
$602$		0			0
$603$	1162.68			1.92816
$604$		0			0
$605$	152.887	−	5.77968i	0.252706	−	0.00955319i
$606$		0			0
$607$	697.222			1.14864			0.574318	−	0.818632i	$-0.305267\pi$
$607$	697.222			1.14864			0.574318	+	0.818632i	$0.305267\pi$
$608$		0			0
$609$	−348.222	−	192.549i	−0.571793	−	0.316172i
$610$		0			0
$611$			1172.44i			1.91889i
$612$		0			0
$613$	772.158			1.25964			0.629819	−	0.776742i	$-0.283129\pi$
$613$	772.158			1.25964			0.629819	+	0.776742i	$0.283129\pi$
$614$		0			0
$615$	−1417.01	+	53.5680i	−2.30407	+	0.0871024i
$616$		0			0
$617$			672.411i			1.08981i	0.838499	+	0.544904i	$0.183434\pi$
$617$			672.411i			1.08981i	−0.838499	+	0.544904i	$0.816566\pi$
$618$		0			0
$619$	511.593			0.826483			0.413242	−	0.910621i	$-0.364396\pi$
$619$	511.593			0.826483			0.413242	+	0.910621i	$0.364396\pi$
$620$		0			0
$621$	885.037			1.42518
$622$		0			0
$623$	−182.625	+	330.275i	−0.293138	+	0.530136i
$624$		0			0
$625$	617.875	−	94.1049i	0.988600	−	0.150568i
$626$		0			0
$627$	−389.650			−0.621451
$628$		0			0
$629$	356.847			0.567325
$630$		0			0
$631$	286.544			0.454110			0.227055	−	0.973882i	$-0.427090\pi$
$631$	286.544			0.454110			0.227055	+	0.973882i	$0.427090\pi$
$632$		0			0
$633$	39.8731			0.0629907
$634$		0			0
$635$	9.24362	+	244.517i	0.0145569	+	0.385066i
$636$		0			0
$637$	−698.883	+	438.737i	−1.09715	+	0.688755i
$638$		0			0
$639$	1476.47			2.31060
$640$		0			0
$641$	−963.812			−1.50361			−0.751804	−	0.659387i	$-0.770816\pi$
$641$	−963.812			−1.50361			−0.751804	+	0.659387i	$0.770816\pi$
$642$		0			0
$643$			207.379i			0.322517i	0.986912	+	0.161259i	$0.0515553\pi$
$643$			207.379i			0.322517i	−0.986912	+	0.161259i	$0.948445\pi$
$644$		0			0
$645$	−6.59341	−	174.412i	−0.0102223	−	0.270406i
$646$		0			0
$647$	309.166			0.477846			0.238923	−	0.971038i	$-0.423206\pi$
$647$	309.166			0.477846			0.238923	+	0.971038i	$0.423206\pi$
$648$		0			0
$649$		−	33.9797i		−	0.0523571i
$650$		0			0
$651$	−1688.26	−	933.523i	−2.59334	−	1.43398i
$652$		0			0
$653$	−155.495			−0.238124			−0.119062	−	0.992887i	$-0.537989\pi$
$653$	−155.495			−0.238124			−0.119062	+	0.992887i	$0.537989\pi$
$654$		0			0
$655$	−604.928	+	22.8685i	−0.923554	+	0.0349137i
$656$		0			0
$657$	490.230			0.746164
$658$		0			0
$659$			245.819i			0.373018i	0.982453	+	0.186509i	$0.0597173\pi$
$659$			245.819i			0.373018i	−0.982453	+	0.186509i	$0.940283\pi$
$660$		0			0
$661$	−81.2555			−0.122928			−0.0614640	−	0.998109i	$-0.519577\pi$
$661$	−81.2555			−0.122928			−0.0614640	+	0.998109i	$0.519577\pi$
$662$		0			0
$663$	1929.86			2.91080
$664$		0			0
$665$	−243.239	−	122.737i	−0.365772	−	0.184568i
$666$		0			0
$667$	−186.814			−0.280082
$668$		0			0
$669$		−	1925.87i		−	2.87872i
$670$		0			0
$671$			264.522i			0.394221i
$672$		0			0
$673$		−	883.945i		−	1.31344i	−0.754134	−	0.656720i	$-0.771943\pi$
$673$		−	883.945i		−	1.31344i	0.754134	−	0.656720i	$-0.228057\pi$
$674$		0			0
$675$	96.5505	+	1275.17i	0.143038	+	1.88915i
$676$		0			0
$677$			16.0368i			0.0236880i	0.999930	+	0.0118440i	$0.00377016\pi$
$677$			16.0368i			0.0236880i	−0.999930	+	0.0118440i	$0.996230\pi$
$678$		0			0
$679$	−105.096	−	58.1124i	−0.154780	−	0.0855853i
$680$		0			0
$681$	1688.89			2.48001
$682$		0			0
$683$	242.497			0.355046			0.177523	−	0.984117i	$-0.443192\pi$
$683$	242.497			0.355046			0.177523	+	0.984117i	$0.443192\pi$
$684$		0			0
$685$	38.2653	+	1012.21i	0.0558618	+	1.47768i
$686$		0			0
$687$			1064.47i			1.54945i
$688$		0			0
$689$			1209.08i			1.75483i
$690$		0			0
$691$	283.360			0.410072			0.205036	−	0.978754i	$-0.434269\pi$
$691$	283.360			0.410072			0.205036	+	0.978754i	$0.434269\pi$
$692$		0			0
$693$	602.781	−	1090.12i	0.869814	−	1.57305i
$694$		0			0
$695$	−880.408	+	33.2826i	−1.26677	+	0.0478887i
$696$		0			0
$697$		−	1172.59i		−	1.68234i
$698$		0			0
$699$	−85.6288			−0.122502
$700$		0			0
$701$			1231.40i			1.75664i	0.478074	+	0.878320i	$0.341335\pi$
$701$			1231.40i			1.75664i	−0.478074	+	0.878320i	$0.658665\pi$
$702$		0			0
$703$	−127.614			−0.181528
$704$		0			0
$705$	−69.2306	−	1831.32i	−0.0981994	−	2.59762i
$706$		0			0
$707$	302.594	+	167.319i	0.427998	+	0.236661i
$708$		0			0
$709$			473.151i			0.667349i	0.942688	+	0.333675i	$0.108289\pi$
$709$			473.151i			0.667349i	−0.942688	+	0.333675i	$0.891711\pi$
$710$		0			0
$711$	−1035.10			−1.45584
$712$		0			0
$713$	−905.722			−1.27030
$714$		0			0
$715$	−800.019	+	30.2436i	−1.11891	+	0.0422988i
$716$		0			0
$717$			1749.03i			2.43937i
$718$		0			0
$719$			345.700i			0.480807i	0.970673	+	0.240403i	$0.0772797\pi$
$719$			345.700i			0.480807i	−0.970673	+	0.240403i	$0.922720\pi$
$720$		0			0
$721$	217.621	+	120.333i	0.301832	+	0.166898i
$722$		0			0
$723$	1708.79			2.36347
$724$		0			0
$725$	−20.3800	−	269.165i	−0.0281103	−	0.371262i
$726$		0			0
$727$	−691.369			−0.950989			−0.475495	−	0.879719i	$-0.657731\pi$
$727$	−691.369			−0.950989			−0.475495	+	0.879719i	$0.657731\pi$
$728$		0			0
$729$	536.087			0.735372
$730$		0			0
$731$	144.329			0.197440
$732$		0			0
$733$		−	388.815i		−	0.530443i	−0.964188	−	0.265221i	$-0.914555\pi$
$733$		−	388.815i		−	0.530443i	0.964188	−	0.265221i	$-0.0854451\pi$
$734$		0			0
$735$	1065.73	−	726.563i	1.44997	−	0.988522i
$736$		0			0
$737$		−	590.643i		−	0.801415i
$738$		0			0
$739$		−	1153.58i		−	1.56100i	−0.625154	−	0.780501i	$-0.714964\pi$
$739$		−	1153.58i		−	1.56100i	0.625154	−	0.780501i	$-0.285036\pi$
$740$		0			0
$741$	−690.149			−0.931375
$742$		0			0
$743$			100.750i			0.135599i	0.997699	+	0.0677993i	$0.0215977\pi$
$743$			100.750i			0.135599i	−0.997699	+	0.0677993i	$0.978402\pi$
$744$		0			0
$745$	−24.2396	−	641.196i	−0.0325363	−	0.860666i
$746$		0			0
$747$		−	1176.46i		−	1.57491i
$748$		0			0
$749$	924.741	+	511.334i	1.23463	+	0.682689i
$750$		0			0
$751$	239.295			0.318635			0.159317	−	0.987227i	$-0.449071\pi$
$751$	239.295			0.318635			0.159317	+	0.987227i	$0.449071\pi$
$752$		0			0
$753$			1591.29i			2.11327i
$754$		0			0
$755$	946.639	−	35.7864i	1.25383	−	0.0473992i
$756$		0			0
$757$	549.057			0.725307			0.362653	−	0.931924i	$-0.381871\pi$
$757$	549.057			0.725307			0.362653	+	0.931924i	$0.381871\pi$
$758$		0			0
$759$		−	866.053i		−	1.14104i
$760$		0			0
$761$			1103.71i			1.45035i	0.688567	+	0.725173i	$0.258240\pi$
$761$			1103.71i			1.45035i	−0.688567	+	0.725173i	$0.741760\pi$
$762$		0			0
$763$	−5.06595	+	9.16170i	−0.00663951	+	0.0120075i
$764$		0			0
$765$	−2035.56	+	76.9517i	−2.66087	+	0.100590i
$766$		0			0
$767$		−	60.1850i		−	0.0784681i
$768$		0			0
$769$		−	994.764i		−	1.29358i	−0.762668	−	0.646790i	$-0.776111\pi$
$769$		−	994.764i		−	1.29358i	0.762668	−	0.646790i	$-0.223889\pi$
$770$		0			0
$771$		−	140.693i		−	0.182481i
$772$		0			0
$773$			184.516i			0.238701i	0.992852	+	0.119350i	$0.0380812\pi$
$773$			184.516i			0.238701i	−0.992852	+	0.119350i	$0.961919\pi$
$774$		0			0
$775$	−98.8071	−	1304.98i	−0.127493	−	1.68384i
$776$		0			0
$777$	292.347	−	528.706i	0.376251	−	0.680446i
$778$		0			0
$779$			419.338i			0.538302i
$780$		0			0
$781$		−	750.050i		−	0.960371i
$782$		0			0
$783$	552.321			0.705390
$784$		0			0
$785$	3.79367	+	100.352i	0.00483270	+	0.127837i
$786$		0			0
$787$		−	303.686i		−	0.385878i	−0.981211	−	0.192939i	$-0.938198\pi$
$787$		−	303.686i		−	0.385878i	0.981211	−	0.192939i	$-0.0618019\pi$
$788$		0			0
$789$	−838.790			−1.06311
$790$		0			0
$791$	625.674	−	1131.52i	0.790991	−	1.43050i
$792$		0			0
$793$			468.523i			0.590823i
$794$		0			0
$795$	−71.3941	−	1888.55i	−0.0898039	−	2.37554i
$796$		0			0
$797$		−	407.980i		−	0.511895i	−0.966691	−	0.255947i	$-0.917613\pi$
$797$		−	407.980i		−	0.511895i	0.966691	−	0.255947i	$-0.0823874\pi$
$798$		0			0
$799$	1515.44			1.89668
$800$		0			0
$801$		−	1009.09i		−	1.25979i
$802$		0			0
$803$		−	249.037i		−	0.310134i
$804$		0			0
$805$	272.802	−	540.633i	0.338884	−	0.671594i
$806$		0			0
$807$		−	632.673i		−	0.783982i
$808$		0			0
$809$	−570.551			−0.705255			−0.352627	−	0.935764i	$-0.614712\pi$
$809$	−570.551			−0.705255			−0.352627	+	0.935764i	$0.614712\pi$
$810$		0			0
$811$	−1192.22			−1.47006			−0.735030	−	0.678034i	$-0.762832\pi$
$811$	−1192.22			−1.47006			−0.735030	+	0.678034i	$0.762832\pi$
$812$		0			0
$813$	−583.319			−0.717489
$814$		0			0
$815$	−932.708	+	35.2598i	−1.14443	+	0.0432635i
$816$		0			0
$817$	−51.6142			−0.0631752
$818$		0			0
$819$	1067.65	−	1930.83i	1.30360	−	2.35754i
$820$		0			0
$821$		−	1244.34i		−	1.51563i	−0.652467	−	0.757817i	$-0.726266\pi$
$821$		−	1244.34i		−	1.51563i	0.652467	−	0.757817i	$-0.273734\pi$
$822$		0			0
$823$		−	630.742i		−	0.766393i	−0.923667	−	0.383197i	$-0.874823\pi$
$823$		−	630.742i		−	0.766393i	0.923667	−	0.383197i	$-0.125177\pi$
$824$		0			0
$825$	1247.82	−	94.4795i	1.51251	−	0.114521i
$826$		0			0
$827$	−627.976			−0.759343			−0.379671	−	0.925121i	$-0.623963\pi$
$827$	−627.976			−0.759343			−0.379671	+	0.925121i	$0.623963\pi$
$828$		0			0
$829$	−1417.09			−1.70940			−0.854698	−	0.519126i	$-0.826257\pi$
$829$	−1417.09			−1.70940			−0.854698	+	0.519126i	$0.826257\pi$
$830$		0			0
$831$			2138.48i			2.57338i
$832$		0			0
$833$	567.092	+	903.345i	0.680783	+	1.08445i
$834$		0			0
$835$	184.097	−	6.95952i	0.220475	−	0.00833475i
$836$		0			0
$837$	2677.79			3.19927
$838$		0			0
$839$		−	507.188i		−	0.604515i	−0.953226	−	0.302257i	$-0.902260\pi$
$839$		−	507.188i		−	0.604515i	0.953226	−	0.302257i	$-0.0977402\pi$
$840$		0			0
$841$	724.416			0.861374
$842$		0			0
$843$			1491.04i			1.76873i
$844$		0			0
$845$	−572.599	+	21.6463i	−0.677632	+	0.0256170i
$846$		0			0
$847$	187.447	+	103.648i	0.221307	+	0.122371i
$848$		0			0
$849$	1287.31			1.51626
$850$		0			0
$851$		−	283.641i		−	0.333303i
$852$		0			0
$853$		−	1385.60i		−	1.62438i	−0.583392	−	0.812190i	$-0.698275\pi$
$853$		−	1385.60i		−	1.62438i	0.583392	−	0.812190i	$-0.301725\pi$
$854$		0			0
$855$	727.949	−	27.5191i	0.851403	−	0.0321861i
$856$		0			0
$857$	−1618.95			−1.88909			−0.944547	−	0.328377i	$-0.893498\pi$
$857$	−1618.95			−1.88909			−0.944547	+	0.328377i	$0.893498\pi$
$858$		0			0
$859$	254.737			0.296551			0.148276	−	0.988946i	$-0.452628\pi$
$859$	254.737			0.296551			0.148276	+	0.988946i	$0.452628\pi$
$860$		0			0
$861$	−1737.32	−	960.647i	−2.01779	−	1.11573i
$862$		0			0
$863$			1022.78i			1.18515i	0.805517	+	0.592573i	$0.201888\pi$
$863$			1022.78i			1.18515i	−0.805517	+	0.592573i	$0.798112\pi$
$864$		0			0
$865$	36.6080	+	968.373i	0.0423214	+	1.11951i
$866$		0			0
$867$		−	972.974i		−	1.12223i
$868$		0			0
$869$			525.834i			0.605102i
$870$		0			0
$871$		−	1046.15i		−	1.20109i
$872$		0			0
$873$	321.098			0.367810
$874$		0			0
$875$	808.713	+	334.078i	0.924243	+	0.381804i
$876$		0			0
$877$	−1060.41			−1.20913			−0.604566	−	0.796555i	$-0.706654\pi$
$877$	−1060.41			−1.20913			−0.604566	+	0.796555i	$0.706654\pi$
$878$		0			0
$879$	−144.780			−0.164710
$880$		0			0
$881$			852.759i			0.967944i	0.875083	+	0.483972i	$0.160806\pi$
$881$			852.759i			0.967944i	−0.875083	+	0.483972i	$0.839194\pi$
$882$		0			0
$883$	9.04631			0.0102450			0.00512249	−	0.999987i	$-0.498369\pi$
$883$	9.04631			0.0102450			0.00512249	+	0.999987i	$0.498369\pi$
$884$		0			0
$885$	3.55382	+	94.0074i	0.00401562	+	0.106223i
$886$		0			0
$887$	−1630.46			−1.83818			−0.919089	−	0.394051i	$-0.871073\pi$
$887$	−1630.46			−1.83818			−0.919089	+	0.394051i	$0.871073\pi$
$888$		0			0
$889$	−165.768	+	299.789i	−0.186466	+	0.337221i
$890$		0			0
$891$			958.918i			1.07623i
$892$		0			0
$893$	−541.947			−0.606883
$894$		0			0
$895$	34.9714	+	925.082i	0.0390742	+	1.03361i
$896$		0			0
$897$		−	1533.95i		−	1.71009i
$898$		0			0
$899$	−565.230			−0.628732
$900$		0			0
$901$	1562.80			1.73452
$902$		0			0
$903$	118.241	−	213.838i	0.130943	−	0.236808i
$904$		0			0
$905$	−480.576	+	18.1675i	−0.531023	+	0.0200746i
$906$		0			0
$907$	−1121.12			−1.23608			−0.618040	−	0.786147i	$-0.712073\pi$
$907$	−1121.12			−1.23608			−0.618040	+	0.786147i	$0.712073\pi$
$908$		0			0
$909$	−924.515			−1.01707
$910$		0			0
$911$	1192.51			1.30901			0.654507	−	0.756056i	$-0.272876\pi$
$911$	1192.51			1.30901			0.654507	+	0.756056i	$0.272876\pi$
$912$		0			0
$913$	−597.642			−0.654592
$914$		0			0
$915$	−27.6655	−	731.820i	−0.0302355	−	0.799803i
$916$		0			0
$917$	−741.671	−	410.106i	−0.808802	−	0.447226i
$918$		0			0
$919$	−1193.80			−1.29902			−0.649510	−	0.760353i	$-0.725026\pi$
$919$	−1193.80			−1.29902			−0.649510	+	0.760353i	$0.725026\pi$
$920$		0			0
$921$	711.557			0.772591
$922$		0			0
$923$		−	1328.49i		−	1.43932i
$924$		0			0
$925$	408.675	−	30.9430i	0.441810	−	0.0334519i
$926$		0			0
$927$	−664.897			−0.717257
$928$		0			0
$929$		−	1193.64i		−	1.28487i	−0.766341	−	0.642434i	$-0.777925\pi$
$929$		−	1193.64i		−	1.28487i	0.766341	−	0.642434i	$-0.222075\pi$
$930$		0			0
$931$	−202.801	−	323.050i	−0.217831	−	0.346993i
$932$		0			0
$933$	37.5733			0.0402715
$934$		0			0
$935$	39.0916	+	1034.07i	0.0418092	+	1.10596i
$936$		0			0
$937$	−233.497			−0.249196			−0.124598	−	0.992207i	$-0.539764\pi$
$937$	−233.497			−0.249196			−0.124598	+	0.992207i	$0.539764\pi$
$938$		0			0
$939$			1935.29i			2.06101i
$940$		0			0
$941$	582.111			0.618609			0.309304	−	0.950963i	$-0.399904\pi$
$941$	582.111			0.618609			0.309304	+	0.950963i	$0.399904\pi$
$942$		0			0
$943$	−932.038			−0.988376
$944$		0			0
$945$	−806.544	+	1598.39i	−0.853485	+	1.69142i
$946$		0			0
$947$	−1490.42			−1.57384			−0.786918	−	0.617057i	$-0.788325\pi$
$947$	−1490.42			−1.57384			−0.786918	+	0.617057i	$0.788325\pi$
$948$		0			0
$949$		−	441.096i		−	0.464801i
$950$		0			0
$951$		−	2307.09i		−	2.42596i
$952$		0			0
$953$			674.661i			0.707934i	0.935258	+	0.353967i	$0.115167\pi$
$953$			674.661i			0.707934i	−0.935258	+	0.353967i	$0.884833\pi$
$954$		0			0
$955$	−1084.06	+	40.9815i	−1.13514	+	0.0429125i
$956$		0			0
$957$		−	540.473i		−	0.564758i
$958$		0			0
$959$	−686.221	+	1241.02i	−0.715559	+	1.29408i
$960$		0			0
$961$	−1779.37			−1.85159
$962$		0			0
$963$	−2825.36			−2.93391
$964$		0			0
$965$	−68.8158	−	1820.35i	−0.0713117	−	1.88637i
$966$		0			0
$967$		−	316.851i		−	0.327664i	−0.986488	−	0.163832i	$-0.947615\pi$
$967$		−	316.851i		−	0.327664i	0.986488	−	0.163832i	$-0.0523855\pi$
$968$		0			0
$969$			892.055i			0.920594i
$970$		0			0
$971$	1735.31			1.78714			0.893569	−	0.448925i	$-0.148193\pi$
$971$	1735.31			1.78714			0.893569	+	0.448925i	$0.148193\pi$
$972$		0			0
$973$	−1079.42	−	596.865i	−1.10938	−	0.613428i
$974$		0			0
$975$	2210.15	−	167.342i	2.26682	−	0.171633i
$976$		0			0
$977$			216.917i			0.222024i	0.993819	+	0.111012i	$0.0354092\pi$
$977$			216.917i			0.222024i	−0.993819	+	0.111012i	$0.964591\pi$
$978$		0			0
$979$	−512.618			−0.523614
$980$		0			0
$981$		−	27.9917i		−	0.0285339i
$982$		0			0
$983$	−971.105			−0.987899			−0.493950	−	0.869491i	$-0.664447\pi$
$983$	−971.105			−0.987899			−0.493950	+	0.869491i	$0.664447\pi$
$984$		0			0
$985$	−267.725	+	10.1210i	−0.271802	+	0.0102751i
$986$		0			0
$987$	1241.53	−	2245.29i	1.25788	−	2.27486i
$988$		0			0
$989$		−	114.720i		−	0.115996i
$990$		0			0
$991$	869.891			0.877791			0.438896	−	0.898538i	$-0.355370\pi$
$991$	869.891			0.877791			0.438896	+	0.898538i	$0.355370\pi$
$992$		0			0
$993$	−1258.94			−1.26781
$994$		0			0
$995$	57.8629	+	1530.62i	0.0581537	+	1.53831i
$996$		0			0
$997$		−	666.914i		−	0.668921i	−0.942410	−	0.334461i	$-0.891446\pi$
$997$		−	666.914i		−	0.668921i	0.942410	−	0.334461i	$-0.108554\pi$
$998$		0			0
$999$			838.590i			0.839430i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.c.g.209.19		80
4.3	odd	2		280.3.c.g.69.60	yes	80
5.4	even	2	inner	1120.3.c.g.209.78		80
7.6	odd	2	inner	1120.3.c.g.209.66		80
8.3	odd	2		280.3.c.g.69.23	yes	80
8.5	even	2	inner	1120.3.c.g.209.60		80
20.19	odd	2		280.3.c.g.69.21	✓	80
28.27	even	2		280.3.c.g.69.59	yes	80
35.34	odd	2	inner	1120.3.c.g.209.59		80
40.19	odd	2		280.3.c.g.69.58	yes	80
40.29	even	2	inner	1120.3.c.g.209.65		80
56.13	odd	2	inner	1120.3.c.g.209.77		80
56.27	even	2		280.3.c.g.69.24	yes	80
140.139	even	2		280.3.c.g.69.22	yes	80
280.69	odd	2	inner	1120.3.c.g.209.20		80
280.139	even	2		280.3.c.g.69.57	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.21	✓	80	20.19	odd	2
280.3.c.g.69.22	yes	80	140.139	even	2
280.3.c.g.69.23	yes	80	8.3	odd	2
280.3.c.g.69.24	yes	80	56.27	even	2
280.3.c.g.69.57	yes	80	280.139	even	2
280.3.c.g.69.58	yes	80	40.19	odd	2
280.3.c.g.69.59	yes	80	28.27	even	2
280.3.c.g.69.60	yes	80	4.3	odd	2
1120.3.c.g.209.19		80	1.1	even	1	trivial
1120.3.c.g.209.20		80	280.69	odd	2	inner
1120.3.c.g.209.59		80	35.34	odd	2	inner
1120.3.c.g.209.60		80	8.5	even	2	inner
1120.3.c.g.209.65		80	40.29	even	2	inner
1120.3.c.g.209.66		80	7.6	odd	2	inner
1120.3.c.g.209.77		80	56.13	odd	2	inner
1120.3.c.g.209.78		80	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1120.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.26463i q^{3} +(4.99643 - 0.188883i) q^{5} +(6.12587 + 3.38729i) q^{7} -18.7163 q^{9} +O(q^{10})\)	\(q-5.26463i q^{3} +(4.99643 - 0.188883i) q^{5} +(6.12587 + 3.38729i) q^{7} -18.7163 q^{9} +9.50793i q^{11} +16.8405i q^{13} +(-0.994401 - 26.3044i) q^{15} +21.7673 q^{17} -7.78431 q^{19} +(17.8328 - 32.2504i) q^{21} -17.3018i q^{23} +(24.9286 - 1.88748i) q^{25} +51.1530i q^{27} -10.7974i q^{29} -52.3486i q^{31} +50.0558 q^{33} +(31.2473 + 15.7673i) q^{35} +16.3938 q^{37} +88.6589 q^{39} -53.8696i q^{41} +6.63054 q^{43} +(-93.5149 + 3.53520i) q^{45} +69.6204 q^{47} +(26.0525 + 41.5002i) q^{49} -114.597i q^{51} +71.7961 q^{53} +(1.79589 + 47.5057i) q^{55} +40.9815i q^{57} -3.57383 q^{59} +27.8212 q^{61} +(-114.654 - 63.3977i) q^{63} +(3.18088 + 84.1423i) q^{65} -62.1210 q^{67} -91.0874 q^{69} -78.8867 q^{71} -26.1926 q^{73} +(-9.93691 - 131.240i) q^{75} +(-32.2061 + 58.2443i) q^{77} +55.3047 q^{79} +100.855 q^{81} +62.8573i q^{83} +(108.759 - 4.11147i) q^{85} -56.8445 q^{87} +53.9148i q^{89} +(-57.0436 + 103.163i) q^{91} -275.596 q^{93} +(-38.8938 + 1.47033i) q^{95} -17.1560 q^{97} -177.954i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 80 * q - 224 * q^9	\( 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) \) 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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