LMFDB - Embedded newform 8024.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.01806 q^{3} +3.78696 q^{5} -1.07498 q^{7} +6.10866 q^{9} +O(q^{10})$	$q-3.01806 q^{3} +3.78696 q^{5} -1.07498 q^{7} +6.10866 q^{9} +2.37858 q^{11} -2.27700 q^{13} -11.4293 q^{15} +1.00000 q^{17} +7.34775 q^{19} +3.24436 q^{21} -5.69920 q^{23} +9.34105 q^{25} -9.38211 q^{27} +4.79232 q^{29} -3.39730 q^{31} -7.17870 q^{33} -4.07092 q^{35} +3.12222 q^{37} +6.87212 q^{39} -0.961505 q^{41} -0.566857 q^{43} +23.1332 q^{45} +4.78110 q^{47} -5.84441 q^{49} -3.01806 q^{51} +7.75767 q^{53} +9.00759 q^{55} -22.1759 q^{57} +1.00000 q^{59} +9.49944 q^{61} -6.56671 q^{63} -8.62291 q^{65} +2.32111 q^{67} +17.2005 q^{69} -6.98671 q^{71} +5.74766 q^{73} -28.1918 q^{75} -2.55694 q^{77} +0.540861 q^{79} +9.98974 q^{81} -11.8325 q^{83} +3.78696 q^{85} -14.4635 q^{87} +2.87621 q^{89} +2.44774 q^{91} +10.2532 q^{93} +27.8256 q^{95} +8.14144 q^{97} +14.5300 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.01806	−1.74248	−0.871238	−	0.490862i	$-0.836682\pi$
$3$	−3.01806	−1.74248	−0.871238	+	0.490862i	$0.836682\pi$
$4$	0	0
$5$	3.78696	1.69358	0.846790	−	0.531928i	$-0.178532\pi$
$5$	3.78696	1.69358	0.846790	+	0.531928i	$0.178532\pi$
$6$	0	0
$7$	−1.07498	−0.406306	−0.203153	−	0.979147i	$-0.565119\pi$
$7$	−1.07498	−0.406306	−0.203153	+	0.979147i	$0.565119\pi$
$8$	0	0
$9$	6.10866	2.03622
$10$	0	0
$11$	2.37858	0.717170	0.358585	−	0.933497i	$-0.383259\pi$
$11$	2.37858	0.717170	0.358585	+	0.933497i	$0.383259\pi$
$12$	0	0
$13$	−2.27700	−0.631527	−0.315763	−	0.948838i	$-0.602261\pi$
$13$	−2.27700	−0.631527	−0.315763	+	0.948838i	$0.602261\pi$
$14$	0	0
$15$	−11.4293	−2.95102
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	7.34775	1.68569	0.842844	−	0.538158i	$-0.180879\pi$
$19$	7.34775	1.68569	0.842844	+	0.538158i	$0.180879\pi$
$20$	0	0
$21$	3.24436	0.707978
$22$	0	0
$23$	−5.69920	−1.18837	−0.594183	−	0.804330i	$-0.702524\pi$
$23$	−5.69920	−1.18837	−0.594183	+	0.804330i	$0.702524\pi$
$24$	0	0
$25$	9.34105	1.86821
$26$	0	0
$27$	−9.38211	−1.80559
$28$	0	0
$29$	4.79232	0.889911	0.444956	−	0.895553i	$-0.353219\pi$
$29$	4.79232	0.889911	0.444956	+	0.895553i	$0.353219\pi$
$30$	0	0
$31$	−3.39730	−0.610173	−0.305086	−	0.952325i	$-0.598685\pi$
$31$	−3.39730	−0.610173	−0.305086	+	0.952325i	$0.598685\pi$
$32$	0	0
$33$	−7.17870	−1.24965
$34$	0	0
$35$	−4.07092	−0.688111
$36$	0	0
$37$	3.12222	0.513290	0.256645	−	0.966506i	$-0.417383\pi$
$37$	3.12222	0.513290	0.256645	+	0.966506i	$0.417383\pi$
$38$	0	0
$39$	6.87212	1.10042
$40$	0	0
$41$	−0.961505	−0.150162	−0.0750809	−	0.997177i	$-0.523922\pi$
$41$	−0.961505	−0.150162	−0.0750809	+	0.997177i	$0.523922\pi$
$42$	0	0
$43$	−0.566857	−0.0864449	−0.0432224	−	0.999065i	$-0.513762\pi$
$43$	−0.566857	−0.0864449	−0.0432224	+	0.999065i	$0.513762\pi$
$44$	0	0
$45$	23.1332	3.44850
$46$	0	0
$47$	4.78110	0.697395	0.348698	−	0.937235i	$-0.386624\pi$
$47$	4.78110	0.697395	0.348698	+	0.937235i	$0.386624\pi$
$48$	0	0
$49$	−5.84441	−0.834916
$50$	0	0
$51$	−3.01806	−0.422612
$52$	0	0
$53$	7.75767	1.06560	0.532799	−	0.846242i	$-0.321140\pi$
$53$	7.75767	1.06560	0.532799	+	0.846242i	$0.321140\pi$
$54$	0	0
$55$	9.00759	1.21458
$56$	0	0
$57$	−22.1759	−2.93727
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	9.49944	1.21628	0.608139	−	0.793830i	$-0.291916\pi$
$61$	9.49944	1.21628	0.608139	+	0.793830i	$0.291916\pi$
$62$	0	0
$63$	−6.56671	−0.827328
$64$	0	0
$65$	−8.62291	−1.06954
$66$	0	0
$67$	2.32111	0.283569	0.141784	−	0.989898i	$-0.454716\pi$
$67$	2.32111	0.283569	0.141784	+	0.989898i	$0.454716\pi$
$68$	0	0
$69$	17.2005	2.07070
$70$	0	0
$71$	−6.98671	−0.829170	−0.414585	−	0.910011i	$-0.636073\pi$
$71$	−6.98671	−0.829170	−0.414585	+	0.910011i	$0.636073\pi$
$72$	0	0
$73$	5.74766	0.672713	0.336356	−	0.941735i	$-0.390805\pi$
$73$	5.74766	0.672713	0.336356	+	0.941735i	$0.390805\pi$
$74$	0	0
$75$	−28.1918	−3.25531
$76$	0	0
$77$	−2.55694	−0.291390
$78$	0	0
$79$	0.540861	0.0608516	0.0304258	−	0.999537i	$-0.490314\pi$
$79$	0.540861	0.0608516	0.0304258	+	0.999537i	$0.490314\pi$
$80$	0	0
$81$	9.98974	1.10997
$82$	0	0
$83$	−11.8325	−1.29879	−0.649394	−	0.760452i	$-0.724977\pi$
$83$	−11.8325	−1.29879	−0.649394	+	0.760452i	$0.724977\pi$
$84$	0	0
$85$	3.78696	0.410753
$86$	0	0
$87$	−14.4635	−1.55065
$88$	0	0
$89$	2.87621	0.304878	0.152439	−	0.988313i	$-0.451287\pi$
$89$	2.87621	0.304878	0.152439	+	0.988313i	$0.451287\pi$
$90$	0	0
$91$	2.44774	0.256593
$92$	0	0
$93$	10.2532	1.06321
$94$	0	0
$95$	27.8256	2.85485
$96$	0	0
$97$	8.14144	0.826638	0.413319	−	0.910586i	$-0.364369\pi$
$97$	8.14144	0.826638	0.413319	+	0.910586i	$0.364369\pi$
$98$	0	0
$99$	14.5300	1.46032
$100$	0	0
$101$	5.30417	0.527784	0.263892	−	0.964552i	$-0.414994\pi$
$101$	5.30417	0.527784	0.263892	+	0.964552i	$0.414994\pi$
$102$	0	0
$103$	−7.10693	−0.700266	−0.350133	−	0.936700i	$-0.613864\pi$
$103$	−7.10693	−0.700266	−0.350133	+	0.936700i	$0.613864\pi$
$104$	0	0
$105$	12.2863	1.19902
$106$	0	0
$107$	2.23541	0.216105	0.108052	−	0.994145i	$-0.465539\pi$
$107$	2.23541	0.216105	0.108052	+	0.994145i	$0.465539\pi$
$108$	0	0
$109$	14.4272	1.38187	0.690936	−	0.722916i	$-0.257199\pi$
$109$	14.4272	1.38187	0.690936	+	0.722916i	$0.257199\pi$
$110$	0	0
$111$	−9.42304	−0.894395
$112$	0	0
$113$	19.8746	1.86964	0.934821	−	0.355118i	$-0.115559\pi$
$113$	19.8746	1.86964	0.934821	+	0.355118i	$0.115559\pi$
$114$	0	0
$115$	−21.5826	−2.01259
$116$	0	0
$117$	−13.9094	−1.28593
$118$	0	0
$119$	−1.07498	−0.0985436
$120$	0	0
$121$	−5.34234	−0.485668
$122$	0	0
$123$	2.90188	0.261653
$124$	0	0
$125$	16.4394	1.47038
$126$	0	0
$127$	13.9354	1.23657	0.618283	−	0.785955i	$-0.287828\pi$
$127$	13.9354	1.23657	0.618283	+	0.785955i	$0.287828\pi$
$128$	0	0
$129$	1.71081	0.150628
$130$	0	0
$131$	17.3982	1.52009	0.760043	−	0.649873i	$-0.225178\pi$
$131$	17.3982	1.52009	0.760043	+	0.649873i	$0.225178\pi$
$132$	0	0
$133$	−7.89871	−0.684905
$134$	0	0
$135$	−35.5296	−3.05791
$136$	0	0
$137$	−20.9553	−1.79033	−0.895167	−	0.445732i	$-0.852944\pi$
$137$	−20.9553	−1.79033	−0.895167	+	0.445732i	$0.852944\pi$
$138$	0	0
$139$	−1.33431	−0.113175	−0.0565874	−	0.998398i	$-0.518022\pi$
$139$	−1.33431	−0.113175	−0.0565874	+	0.998398i	$0.518022\pi$
$140$	0	0
$141$	−14.4296	−1.21519
$142$	0	0
$143$	−5.41604	−0.452912
$144$	0	0
$145$	18.1483	1.50714
$146$	0	0
$147$	17.6388	1.45482
$148$	0	0
$149$	−22.8396	−1.87109	−0.935547	−	0.353203i	$-0.885092\pi$
$149$	−22.8396	−1.87109	−0.935547	+	0.353203i	$0.885092\pi$
$150$	0	0
$151$	9.84475	0.801154	0.400577	−	0.916263i	$-0.368810\pi$
$151$	9.84475	0.801154	0.400577	+	0.916263i	$0.368810\pi$
$152$	0	0
$153$	6.10866	0.493856
$154$	0	0
$155$	−12.8654	−1.03338
$156$	0	0
$157$	−7.81650	−0.623825	−0.311912	−	0.950111i	$-0.600970\pi$
$157$	−7.81650	−0.623825	−0.311912	+	0.950111i	$0.600970\pi$
$158$	0	0
$159$	−23.4131	−1.85678
$160$	0	0
$161$	6.12655	0.482840
$162$	0	0
$163$	2.34778	0.183892	0.0919460	−	0.995764i	$-0.470691\pi$
$163$	2.34778	0.183892	0.0919460	+	0.995764i	$0.470691\pi$
$164$	0	0
$165$	−27.1854	−2.11638
$166$	0	0
$167$	−3.47493	−0.268898	−0.134449	−	0.990920i	$-0.542927\pi$
$167$	−3.47493	−0.268898	−0.134449	+	0.990920i	$0.542927\pi$
$168$	0	0
$169$	−7.81526	−0.601174
$170$	0	0
$171$	44.8849	3.43243
$172$	0	0
$173$	−11.9873	−0.911380	−0.455690	−	0.890139i	$-0.650607\pi$
$173$	−11.9873	−0.911380	−0.455690	+	0.890139i	$0.650607\pi$
$174$	0	0
$175$	−10.0415	−0.759065
$176$	0	0
$177$	−3.01806	−0.226851
$178$	0	0
$179$	9.11231	0.681086	0.340543	−	0.940229i	$-0.389389\pi$
$179$	9.11231	0.681086	0.340543	+	0.940229i	$0.389389\pi$
$180$	0	0
$181$	−12.7948	−0.951027	−0.475514	−	0.879708i	$-0.657738\pi$
$181$	−12.7948	−0.951027	−0.475514	+	0.879708i	$0.657738\pi$
$182$	0	0
$183$	−28.6698	−2.11933
$184$	0	0
$185$	11.8237	0.869297
$186$	0	0
$187$	2.37858	0.173939
$188$	0	0
$189$	10.0856	0.733620
$190$	0	0
$191$	−16.5324	−1.19625	−0.598123	−	0.801404i	$-0.704087\pi$
$191$	−16.5324	−1.19625	−0.598123	+	0.801404i	$0.704087\pi$
$192$	0	0
$193$	0.445398	0.0320605	0.0160302	−	0.999872i	$-0.494897\pi$
$193$	0.445398	0.0320605	0.0160302	+	0.999872i	$0.494897\pi$
$194$	0	0
$195$	26.0244	1.86365
$196$	0	0
$197$	−17.3829	−1.23848	−0.619241	−	0.785201i	$-0.712560\pi$
$197$	−17.3829	−1.23848	−0.619241	+	0.785201i	$0.712560\pi$
$198$	0	0
$199$	−12.5738	−0.891334	−0.445667	−	0.895199i	$-0.647034\pi$
$199$	−12.5738	−0.891334	−0.445667	+	0.895199i	$0.647034\pi$
$200$	0	0
$201$	−7.00524	−0.494112
$202$	0	0
$203$	−5.15167	−0.361576
$204$	0	0
$205$	−3.64118	−0.254311
$206$	0	0
$207$	−34.8145	−2.41977
$208$	0	0
$209$	17.4772	1.20892
$210$	0	0
$211$	−15.4054	−1.06055	−0.530274	−	0.847826i	$-0.677911\pi$
$211$	−15.4054	−1.06055	−0.530274	+	0.847826i	$0.677911\pi$
$212$	0	0
$213$	21.0863	1.44481
$214$	0	0
$215$	−2.14666	−0.146401
$216$	0	0
$217$	3.65204	0.247917
$218$	0	0
$219$	−17.3468	−1.17219
$220$	0	0
$221$	−2.27700	−0.153168
$222$	0	0
$223$	11.8174	0.791351	0.395676	−	0.918390i	$-0.370510\pi$
$223$	11.8174	0.791351	0.395676	+	0.918390i	$0.370510\pi$
$224$	0	0
$225$	57.0613	3.80409
$226$	0	0
$227$	−19.5854	−1.29993	−0.649966	−	0.759964i	$-0.725217\pi$
$227$	−19.5854	−1.29993	−0.649966	+	0.759964i	$0.725217\pi$
$228$	0	0
$229$	22.9227	1.51478	0.757388	−	0.652965i	$-0.226475\pi$
$229$	22.9227	1.51478	0.757388	+	0.652965i	$0.226475\pi$
$230$	0	0
$231$	7.71698	0.507740
$232$	0	0
$233$	17.6809	1.15832	0.579158	−	0.815215i	$-0.303381\pi$
$233$	17.6809	1.15832	0.579158	+	0.815215i	$0.303381\pi$
$234$	0	0
$235$	18.1058	1.18109
$236$	0	0
$237$	−1.63235	−0.106032
$238$	0	0
$239$	11.1960	0.724209	0.362104	−	0.932138i	$-0.382058\pi$
$239$	11.1960	0.724209	0.362104	+	0.932138i	$0.382058\pi$
$240$	0	0
$241$	−1.92646	−0.124094	−0.0620470	−	0.998073i	$-0.519763\pi$
$241$	−1.92646	−0.124094	−0.0620470	+	0.998073i	$0.519763\pi$
$242$	0	0
$243$	−2.00328	−0.128510
$244$	0	0
$245$	−22.1325	−1.41400
$246$	0	0
$247$	−16.7308	−1.06456
$248$	0	0
$249$	35.7112	2.26310
$250$	0	0
$251$	27.3313	1.72513	0.862567	−	0.505943i	$-0.168855\pi$
$251$	27.3313	1.72513	0.862567	+	0.505943i	$0.168855\pi$
$252$	0	0
$253$	−13.5560	−0.852259
$254$	0	0
$255$	−11.4293	−0.715727
$256$	0	0
$257$	−16.6535	−1.03882	−0.519409	−	0.854526i	$-0.673848\pi$
$257$	−16.6535	−1.03882	−0.519409	+	0.854526i	$0.673848\pi$
$258$	0	0
$259$	−3.35634	−0.208553
$260$	0	0
$261$	29.2746	1.81206
$262$	0	0
$263$	−0.160996	−0.00992744	−0.00496372	−	0.999988i	$-0.501580\pi$
$263$	−0.160996	−0.00992744	−0.00496372	+	0.999988i	$0.501580\pi$
$264$	0	0
$265$	29.3780	1.80468
$266$	0	0
$267$	−8.68057	−0.531243
$268$	0	0
$269$	7.20075	0.439037	0.219519	−	0.975608i	$-0.429551\pi$
$269$	7.20075	0.439037	0.219519	+	0.975608i	$0.429551\pi$
$270$	0	0
$271$	−12.8749	−0.782092	−0.391046	−	0.920371i	$-0.627887\pi$
$271$	−12.8749	−0.782092	−0.391046	+	0.920371i	$0.627887\pi$
$272$	0	0
$273$	−7.38742	−0.447107
$274$	0	0
$275$	22.2185	1.33982
$276$	0	0
$277$	−4.27539	−0.256883	−0.128442	−	0.991717i	$-0.540998\pi$
$277$	−4.27539	−0.256883	−0.128442	+	0.991717i	$0.540998\pi$
$278$	0	0
$279$	−20.7529	−1.24245
$280$	0	0
$281$	21.4493	1.27956	0.639779	−	0.768559i	$-0.279026\pi$
$281$	21.4493	1.27956	0.639779	+	0.768559i	$0.279026\pi$
$282$	0	0
$283$	2.08014	0.123652	0.0618259	−	0.998087i	$-0.480308\pi$
$283$	2.08014	0.123652	0.0618259	+	0.998087i	$0.480308\pi$
$284$	0	0
$285$	−83.9792	−4.97450
$286$	0	0
$287$	1.03360	0.0610116
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−24.5713	−1.44040
$292$	0	0
$293$	3.38532	0.197772	0.0988862	−	0.995099i	$-0.468472\pi$
$293$	3.38532	0.197772	0.0988862	+	0.995099i	$0.468472\pi$
$294$	0	0
$295$	3.78696	0.220485
$296$	0	0
$297$	−22.3161	−1.29491
$298$	0	0
$299$	12.9771	0.750484
$300$	0	0
$301$	0.609362	0.0351231
$302$	0	0
$303$	−16.0083	−0.919651
$304$	0	0
$305$	35.9740	2.05986
$306$	0	0
$307$	−15.3636	−0.876847	−0.438424	−	0.898768i	$-0.644463\pi$
$307$	−15.3636	−0.876847	−0.438424	+	0.898768i	$0.644463\pi$
$308$	0	0
$309$	21.4491	1.22020
$310$	0	0
$311$	30.2124	1.71319	0.856594	−	0.515992i	$-0.172576\pi$
$311$	30.2124	1.71319	0.856594	+	0.515992i	$0.172576\pi$
$312$	0	0
$313$	−16.3946	−0.926676	−0.463338	−	0.886182i	$-0.653348\pi$
$313$	−16.3946	−0.926676	−0.463338	+	0.886182i	$0.653348\pi$
$314$	0	0
$315$	−24.8679	−1.40115
$316$	0	0
$317$	−7.66739	−0.430643	−0.215322	−	0.976543i	$-0.569080\pi$
$317$	−7.66739	−0.430643	−0.215322	+	0.976543i	$0.569080\pi$
$318$	0	0
$319$	11.3989	0.638217
$320$	0	0
$321$	−6.74658	−0.376558
$322$	0	0
$323$	7.34775	0.408840
$324$	0	0
$325$	−21.2696	−1.17982
$326$	0	0
$327$	−43.5420	−2.40788
$328$	0	0
$329$	−5.13960	−0.283356
$330$	0	0
$331$	−30.0500	−1.65170	−0.825848	−	0.563893i	$-0.809303\pi$
$331$	−30.0500	−1.65170	−0.825848	+	0.563893i	$0.809303\pi$
$332$	0	0
$333$	19.0726	1.04517
$334$	0	0
$335$	8.78995	0.480246
$336$	0	0
$337$	6.66554	0.363095	0.181548	−	0.983382i	$-0.441889\pi$
$337$	6.66554	0.363095	0.181548	+	0.983382i	$0.441889\pi$
$338$	0	0
$339$	−59.9826	−3.25781
$340$	0	0
$341$	−8.08076	−0.437598
$342$	0	0
$343$	13.8075	0.745537
$344$	0	0
$345$	65.1376	3.50689
$346$	0	0
$347$	−9.45683	−0.507669	−0.253835	−	0.967248i	$-0.581692\pi$
$347$	−9.45683	−0.507669	−0.253835	+	0.967248i	$0.581692\pi$
$348$	0	0
$349$	14.1558	0.757742	0.378871	−	0.925450i	$-0.376312\pi$
$349$	14.1558	0.757742	0.378871	+	0.925450i	$0.376312\pi$
$350$	0	0
$351$	21.3631	1.14028
$352$	0	0
$353$	3.96392	0.210978	0.105489	−	0.994420i	$-0.466359\pi$
$353$	3.96392	0.210978	0.105489	+	0.994420i	$0.466359\pi$
$354$	0	0
$355$	−26.4584	−1.40427
$356$	0	0
$357$	3.24436	0.171710
$358$	0	0
$359$	23.1409	1.22133	0.610665	−	0.791889i	$-0.290902\pi$
$359$	23.1409	1.22133	0.610665	+	0.791889i	$0.290902\pi$
$360$	0	0
$361$	34.9894	1.84155
$362$	0	0
$363$	16.1235	0.846264
$364$	0	0
$365$	21.7662	1.13929
$366$	0	0
$367$	7.18521	0.375065	0.187533	−	0.982258i	$-0.439951\pi$
$367$	7.18521	0.375065	0.187533	+	0.982258i	$0.439951\pi$
$368$	0	0
$369$	−5.87351	−0.305763
$370$	0	0
$371$	−8.33937	−0.432959
$372$	0	0
$373$	−6.82134	−0.353195	−0.176598	−	0.984283i	$-0.556509\pi$
$373$	−6.82134	−0.353195	−0.176598	+	0.984283i	$0.556509\pi$
$374$	0	0
$375$	−49.6150	−2.56211
$376$	0	0
$377$	−10.9121	−0.562003
$378$	0	0
$379$	30.9161	1.58805	0.794026	−	0.607883i	$-0.207981\pi$
$379$	30.9161	1.58805	0.794026	+	0.607883i	$0.207981\pi$
$380$	0	0
$381$	−42.0578	−2.15469
$382$	0	0
$383$	−13.1523	−0.672049	−0.336024	−	0.941853i	$-0.609082\pi$
$383$	−13.1523	−0.672049	−0.336024	+	0.941853i	$0.609082\pi$
$384$	0	0
$385$	−9.68302	−0.493492
$386$	0	0
$387$	−3.46274	−0.176021
$388$	0	0
$389$	20.5972	1.04432	0.522160	−	0.852847i	$-0.325126\pi$
$389$	20.5972	1.04432	0.522160	+	0.852847i	$0.325126\pi$
$390$	0	0
$391$	−5.69920	−0.288221
$392$	0	0
$393$	−52.5087	−2.64871
$394$	0	0
$395$	2.04822	0.103057
$396$	0	0
$397$	2.13439	0.107122	0.0535610	−	0.998565i	$-0.482943\pi$
$397$	2.13439	0.107122	0.0535610	+	0.998565i	$0.482943\pi$
$398$	0	0
$399$	23.8387	1.19343
$400$	0	0
$401$	31.1377	1.55494	0.777472	−	0.628917i	$-0.216502\pi$
$401$	31.1377	1.55494	0.777472	+	0.628917i	$0.216502\pi$
$402$	0	0
$403$	7.73566	0.385341
$404$	0	0
$405$	37.8307	1.87982
$406$	0	0
$407$	7.42646	0.368116
$408$	0	0
$409$	−34.0133	−1.68185	−0.840926	−	0.541151i	$-0.817989\pi$
$409$	−34.0133	−1.68185	−0.840926	+	0.541151i	$0.817989\pi$
$410$	0	0
$411$	63.2443	3.11961
$412$	0	0
$413$	−1.07498	−0.0528965
$414$	0	0
$415$	−44.8092	−2.19960
$416$	0	0
$417$	4.02703	0.197204
$418$	0	0
$419$	26.5608	1.29758	0.648791	−	0.760967i	$-0.275275\pi$
$419$	26.5608	1.29758	0.648791	+	0.760967i	$0.275275\pi$
$420$	0	0
$421$	31.5851	1.53937	0.769683	−	0.638427i	$-0.220414\pi$
$421$	31.5851	1.53937	0.769683	+	0.638427i	$0.220414\pi$
$422$	0	0
$423$	29.2061	1.42005
$424$	0	0
$425$	9.34105	0.453108
$426$	0	0
$427$	−10.2117	−0.494181
$428$	0	0
$429$	16.3459	0.789188
$430$	0	0
$431$	15.9560	0.768572	0.384286	−	0.923214i	$-0.374448\pi$
$431$	15.9560	0.768572	0.384286	+	0.923214i	$0.374448\pi$
$432$	0	0
$433$	26.6350	1.28000	0.639999	−	0.768376i	$-0.278935\pi$
$433$	26.6350	1.28000	0.639999	+	0.768376i	$0.278935\pi$
$434$	0	0
$435$	−54.7726	−2.62615
$436$	0	0
$437$	−41.8763	−2.00321
$438$	0	0
$439$	19.2058	0.916642	0.458321	−	0.888787i	$-0.348451\pi$
$439$	19.2058	0.916642	0.458321	+	0.888787i	$0.348451\pi$
$440$	0	0
$441$	−35.7015	−1.70007
$442$	0	0
$443$	−3.39809	−0.161448	−0.0807242	−	0.996736i	$-0.525723\pi$
$443$	−3.39809	−0.161448	−0.0807242	+	0.996736i	$0.525723\pi$
$444$	0	0
$445$	10.8921	0.516335
$446$	0	0
$447$	68.9312	3.26033
$448$	0	0
$449$	21.1131	0.996390	0.498195	−	0.867065i	$-0.333996\pi$
$449$	21.1131	0.996390	0.498195	+	0.867065i	$0.333996\pi$
$450$	0	0
$451$	−2.28702	−0.107692
$452$	0	0
$453$	−29.7120	−1.39599
$454$	0	0
$455$	9.26949	0.434560
$456$	0	0
$457$	−12.4017	−0.580126	−0.290063	−	0.957008i	$-0.593676\pi$
$457$	−12.4017	−0.580126	−0.290063	+	0.957008i	$0.593676\pi$
$458$	0	0
$459$	−9.38211	−0.437919
$460$	0	0
$461$	−33.0745	−1.54043	−0.770217	−	0.637782i	$-0.779852\pi$
$461$	−33.0745	−1.54043	−0.770217	+	0.637782i	$0.779852\pi$
$462$	0	0
$463$	−8.54751	−0.397236	−0.198618	−	0.980077i	$-0.563645\pi$
$463$	−8.54751	−0.397236	−0.198618	+	0.980077i	$0.563645\pi$
$464$	0	0
$465$	38.8286	1.80063
$466$	0	0
$467$	19.7442	0.913654	0.456827	−	0.889556i	$-0.348986\pi$
$467$	19.7442	0.913654	0.456827	+	0.889556i	$0.348986\pi$
$468$	0	0
$469$	−2.49516	−0.115216
$470$	0	0
$471$	23.5906	1.08700
$472$	0	0
$473$	−1.34832	−0.0619957
$474$	0	0
$475$	68.6357	3.14922
$476$	0	0
$477$	47.3890	2.16979
$478$	0	0
$479$	6.19763	0.283177	0.141589	−	0.989926i	$-0.454779\pi$
$479$	6.19763	0.283177	0.141589	+	0.989926i	$0.454779\pi$
$480$	0	0
$481$	−7.10931	−0.324156
$482$	0	0
$483$	−18.4903	−0.841336
$484$	0	0
$485$	30.8313	1.39998
$486$	0	0
$487$	−38.8027	−1.75832	−0.879160	−	0.476527i	$-0.841895\pi$
$487$	−38.8027	−1.75832	−0.879160	+	0.476527i	$0.841895\pi$
$488$	0	0
$489$	−7.08572	−0.320427
$490$	0	0
$491$	32.4015	1.46226	0.731130	−	0.682238i	$-0.238993\pi$
$491$	32.4015	1.46226	0.731130	+	0.682238i	$0.238993\pi$
$492$	0	0
$493$	4.79232	0.215835
$494$	0	0
$495$	55.0243	2.47316
$496$	0	0
$497$	7.51060	0.336897
$498$	0	0
$499$	−7.76196	−0.347473	−0.173736	−	0.984792i	$-0.555584\pi$
$499$	−7.76196	−0.347473	−0.173736	+	0.984792i	$0.555584\pi$
$500$	0	0
$501$	10.4875	0.468549
$502$	0	0
$503$	−0.928787	−0.0414126	−0.0207063	−	0.999786i	$-0.506591\pi$
$503$	−0.928787	−0.0414126	−0.0207063	+	0.999786i	$0.506591\pi$
$504$	0	0
$505$	20.0867	0.893845
$506$	0	0
$507$	23.5869	1.04753
$508$	0	0
$509$	−10.2373	−0.453759	−0.226880	−	0.973923i	$-0.572852\pi$
$509$	−10.2373	−0.453759	−0.226880	+	0.973923i	$0.572852\pi$
$510$	0	0
$511$	−6.17864	−0.273327
$512$	0	0
$513$	−68.9373	−3.04366
$514$	0	0
$515$	−26.9136	−1.18596
$516$	0	0
$517$	11.3722	0.500151
$518$	0	0
$519$	36.1784	1.58806
$520$	0	0
$521$	32.6626	1.43097	0.715486	−	0.698627i	$-0.246205\pi$
$521$	32.6626	1.43097	0.715486	+	0.698627i	$0.246205\pi$
$522$	0	0
$523$	25.1946	1.10169	0.550843	−	0.834609i	$-0.314306\pi$
$523$	25.1946	1.10169	0.550843	+	0.834609i	$0.314306\pi$
$524$	0	0
$525$	30.3057	1.32265
$526$	0	0
$527$	−3.39730	−0.147989
$528$	0	0
$529$	9.48087	0.412212
$530$	0	0
$531$	6.10866	0.265093
$532$	0	0
$533$	2.18935	0.0948312
$534$	0	0
$535$	8.46539	0.365991
$536$	0	0
$537$	−27.5014	−1.18677
$538$	0	0
$539$	−13.9014	−0.598776
$540$	0	0
$541$	−13.1456	−0.565172	−0.282586	−	0.959242i	$-0.591192\pi$
$541$	−13.1456	−0.565172	−0.282586	+	0.959242i	$0.591192\pi$
$542$	0	0
$543$	38.6153	1.65714
$544$	0	0
$545$	54.6350	2.34031
$546$	0	0
$547$	−8.62300	−0.368693	−0.184346	−	0.982861i	$-0.559017\pi$
$547$	−8.62300	−0.368693	−0.184346	+	0.982861i	$0.559017\pi$
$548$	0	0
$549$	58.0288	2.47661
$550$	0	0
$551$	35.2127	1.50011
$552$	0	0
$553$	−0.581417	−0.0247244
$554$	0	0
$555$	−35.6847	−1.51473
$556$	0	0
$557$	−18.8922	−0.800489	−0.400244	−	0.916408i	$-0.631075\pi$
$557$	−18.8922	−0.800489	−0.400244	+	0.916408i	$0.631075\pi$
$558$	0	0
$559$	1.29073	0.0545923
$560$	0	0
$561$	−7.17870	−0.303085
$562$	0	0
$563$	16.9728	0.715317	0.357658	−	0.933852i	$-0.383575\pi$
$563$	16.9728	0.715317	0.357658	+	0.933852i	$0.383575\pi$
$564$	0	0
$565$	75.2642	3.16639
$566$	0	0
$567$	−10.7388	−0.450988
$568$	0	0
$569$	−28.9201	−1.21239	−0.606197	−	0.795314i	$-0.707306\pi$
$569$	−28.9201	−1.21239	−0.606197	+	0.795314i	$0.707306\pi$
$570$	0	0
$571$	32.5125	1.36060	0.680302	−	0.732932i	$-0.261849\pi$
$571$	32.5125	1.36060	0.680302	+	0.732932i	$0.261849\pi$
$572$	0	0
$573$	49.8958	2.08443
$574$	0	0
$575$	−53.2365	−2.22012
$576$	0	0
$577$	−1.34973	−0.0561902	−0.0280951	−	0.999605i	$-0.508944\pi$
$577$	−1.34973	−0.0561902	−0.0280951	+	0.999605i	$0.508944\pi$
$578$	0	0
$579$	−1.34424	−0.0558645
$580$	0	0
$581$	12.7198	0.527705
$582$	0	0
$583$	18.4523	0.764215
$584$	0	0
$585$	−52.6744	−2.17782
$586$	0	0
$587$	40.0684	1.65380	0.826901	−	0.562348i	$-0.190102\pi$
$587$	40.0684	1.65380	0.826901	+	0.562348i	$0.190102\pi$
$588$	0	0
$589$	−24.9625	−1.02856
$590$	0	0
$591$	52.4627	2.15803
$592$	0	0
$593$	11.4533	0.470332	0.235166	−	0.971955i	$-0.424437\pi$
$593$	11.4533	0.470332	0.235166	+	0.971955i	$0.424437\pi$
$594$	0	0
$595$	−4.07092	−0.166891
$596$	0	0
$597$	37.9485	1.55313
$598$	0	0
$599$	43.7962	1.78947	0.894733	−	0.446602i	$-0.147366\pi$
$599$	43.7962	1.78947	0.894733	+	0.446602i	$0.147366\pi$
$600$	0	0
$601$	−20.0971	−0.819776	−0.409888	−	0.912136i	$-0.634432\pi$
$601$	−20.0971	−0.819776	−0.409888	+	0.912136i	$0.634432\pi$
$602$	0	0
$603$	14.1789	0.577408
$604$	0	0
$605$	−20.2312	−0.822517
$606$	0	0
$607$	42.0317	1.70601	0.853007	−	0.521899i	$-0.174776\pi$
$607$	42.0317	1.70601	0.853007	+	0.521899i	$0.174776\pi$
$608$	0	0
$609$	15.5480	0.630037
$610$	0	0
$611$	−10.8866	−0.440424
$612$	0	0
$613$	−5.52211	−0.223036	−0.111518	−	0.993762i	$-0.535571\pi$
$613$	−5.52211	−0.223036	−0.111518	+	0.993762i	$0.535571\pi$
$614$	0	0
$615$	10.9893	0.443131
$616$	0	0
$617$	12.2310	0.492403	0.246201	−	0.969219i	$-0.420818\pi$
$617$	12.2310	0.492403	0.246201	+	0.969219i	$0.420818\pi$
$618$	0	0
$619$	10.5795	0.425227	0.212613	−	0.977136i	$-0.431802\pi$
$619$	10.5795	0.425227	0.212613	+	0.977136i	$0.431802\pi$
$620$	0	0
$621$	53.4705	2.14570
$622$	0	0
$623$	−3.09188	−0.123874
$624$	0	0
$625$	15.5500	0.622000
$626$	0	0
$627$	−52.7472	−2.10652
$628$	0	0
$629$	3.12222	0.124491
$630$	0	0
$631$	20.4705	0.814919	0.407460	−	0.913223i	$-0.366415\pi$
$631$	20.4705	0.814919	0.407460	+	0.913223i	$0.366415\pi$
$632$	0	0
$633$	46.4942	1.84798
$634$	0	0
$635$	52.7728	2.09422
$636$	0	0
$637$	13.3077	0.527272
$638$	0	0
$639$	−42.6794	−1.68837
$640$	0	0
$641$	−24.4657	−0.966337	−0.483168	−	0.875527i	$-0.660514\pi$
$641$	−24.4657	−0.966337	−0.483168	+	0.875527i	$0.660514\pi$
$642$	0	0
$643$	−12.1977	−0.481030	−0.240515	−	0.970645i	$-0.577316\pi$
$643$	−12.1977	−0.481030	−0.240515	+	0.970645i	$0.577316\pi$
$644$	0	0
$645$	6.47875	0.255101
$646$	0	0
$647$	−38.5575	−1.51585	−0.757926	−	0.652341i	$-0.773787\pi$
$647$	−38.5575	−1.51585	−0.757926	+	0.652341i	$0.773787\pi$
$648$	0	0
$649$	2.37858	0.0933675
$650$	0	0
$651$	−11.0221	−0.431989
$652$	0	0
$653$	33.7779	1.32183	0.660915	−	0.750460i	$-0.270168\pi$
$653$	33.7779	1.32183	0.660915	+	0.750460i	$0.270168\pi$
$654$	0	0
$655$	65.8862	2.57439
$656$	0	0
$657$	35.1105	1.36979
$658$	0	0
$659$	−4.00595	−0.156050	−0.0780248	−	0.996951i	$-0.524861\pi$
$659$	−4.00595	−0.156050	−0.0780248	+	0.996951i	$0.524861\pi$
$660$	0	0
$661$	33.8862	1.31802	0.659010	−	0.752134i	$-0.270976\pi$
$661$	33.8862	1.31802	0.659010	+	0.752134i	$0.270976\pi$
$662$	0	0
$663$	6.87212	0.266891
$664$	0	0
$665$	−29.9121	−1.15994
$666$	0	0
$667$	−27.3124	−1.05754
$668$	0	0
$669$	−35.6656	−1.37891
$670$	0	0
$671$	22.5952	0.872278
$672$	0	0
$673$	−23.7373	−0.915006	−0.457503	−	0.889208i	$-0.651256\pi$
$673$	−23.7373	−0.915006	−0.457503	+	0.889208i	$0.651256\pi$
$674$	0	0
$675$	−87.6388	−3.37322
$676$	0	0
$677$	−16.9812	−0.652640	−0.326320	−	0.945259i	$-0.605809\pi$
$677$	−16.9812	−0.652640	−0.326320	+	0.945259i	$0.605809\pi$
$678$	0	0
$679$	−8.75192	−0.335868
$680$	0	0
$681$	59.1100	2.26510
$682$	0	0
$683$	49.8018	1.90561	0.952806	−	0.303581i	$-0.0981824\pi$
$683$	49.8018	1.90561	0.952806	+	0.303581i	$0.0981824\pi$
$684$	0	0
$685$	−79.3569	−3.03207
$686$	0	0
$687$	−69.1820	−2.63946
$688$	0	0
$689$	−17.6642	−0.672954
$690$	0	0
$691$	−13.5700	−0.516229	−0.258114	−	0.966114i	$-0.583101\pi$
$691$	−13.5700	−0.516229	−0.258114	+	0.966114i	$0.583101\pi$
$692$	0	0
$693$	−15.6195	−0.593334
$694$	0	0
$695$	−5.05298	−0.191671
$696$	0	0
$697$	−0.961505	−0.0364196
$698$	0	0
$699$	−53.3620	−2.01834
$700$	0	0
$701$	−5.16774	−0.195183	−0.0975914	−	0.995227i	$-0.531114\pi$
$701$	−5.16774	−0.195183	−0.0975914	+	0.995227i	$0.531114\pi$
$702$	0	0
$703$	22.9413	0.865247
$704$	0	0
$705$	−54.6444	−2.05803
$706$	0	0
$707$	−5.70189	−0.214442
$708$	0	0
$709$	−40.7610	−1.53081	−0.765406	−	0.643547i	$-0.777462\pi$
$709$	−40.7610	−1.53081	−0.765406	+	0.643547i	$0.777462\pi$
$710$	0	0
$711$	3.30394	0.123907
$712$	0	0
$713$	19.3619	0.725108
$714$	0	0
$715$	−20.5103	−0.767042
$716$	0	0
$717$	−33.7901	−1.26192
$718$	0	0
$719$	6.48665	0.241911	0.120956	−	0.992658i	$-0.461404\pi$
$719$	6.48665	0.241911	0.120956	+	0.992658i	$0.461404\pi$
$720$	0	0
$721$	7.63983	0.284522
$722$	0	0
$723$	5.81415	0.216231
$724$	0	0
$725$	44.7653	1.66254
$726$	0	0
$727$	22.7054	0.842097	0.421048	−	0.907038i	$-0.361662\pi$
$727$	22.7054	0.842097	0.421048	+	0.907038i	$0.361662\pi$
$728$	0	0
$729$	−23.9232	−0.886045
$730$	0	0
$731$	−0.566857	−0.0209660
$732$	0	0
$733$	−22.5564	−0.833141	−0.416570	−	0.909103i	$-0.636768\pi$
$733$	−22.5564	−0.833141	−0.416570	+	0.909103i	$0.636768\pi$
$734$	0	0
$735$	66.7972	2.46385
$736$	0	0
$737$	5.52095	0.203367
$738$	0	0
$739$	−11.6030	−0.426824	−0.213412	−	0.976962i	$-0.568458\pi$
$739$	−11.6030	−0.426824	−0.213412	+	0.976962i	$0.568458\pi$
$740$	0	0
$741$	50.4946	1.85496
$742$	0	0
$743$	−11.9149	−0.437115	−0.218558	−	0.975824i	$-0.570135\pi$
$743$	−11.9149	−0.437115	−0.218558	+	0.975824i	$0.570135\pi$
$744$	0	0
$745$	−86.4926	−3.16884
$746$	0	0
$747$	−72.2808	−2.64462
$748$	0	0
$749$	−2.40303	−0.0878047
$750$	0	0
$751$	7.50072	0.273705	0.136853	−	0.990591i	$-0.456301\pi$
$751$	7.50072	0.273705	0.136853	+	0.990591i	$0.456301\pi$
$752$	0	0
$753$	−82.4873	−3.00600
$754$	0	0
$755$	37.2816	1.35682
$756$	0	0
$757$	11.3769	0.413499	0.206750	−	0.978394i	$-0.433711\pi$
$757$	11.3769	0.413499	0.206750	+	0.978394i	$0.433711\pi$
$758$	0	0
$759$	40.9128	1.48504
$760$	0	0
$761$	−9.96955	−0.361396	−0.180698	−	0.983539i	$-0.557836\pi$
$761$	−9.96955	−0.361396	−0.180698	+	0.983539i	$0.557836\pi$
$762$	0	0
$763$	−15.5090	−0.561462
$764$	0	0
$765$	23.1332	0.836384
$766$	0	0
$767$	−2.27700	−0.0822178
$768$	0	0
$769$	47.6159	1.71707	0.858537	−	0.512752i	$-0.171374\pi$
$769$	47.6159	1.71707	0.858537	+	0.512752i	$0.171374\pi$
$770$	0	0
$771$	50.2613	1.81012
$772$	0	0
$773$	−29.6174	−1.06526	−0.532632	−	0.846347i	$-0.678797\pi$
$773$	−29.6174	−1.06526	−0.532632	+	0.846347i	$0.678797\pi$
$774$	0	0
$775$	−31.7344	−1.13993
$776$	0	0
$777$	10.1296	0.363398
$778$	0	0
$779$	−7.06489	−0.253126
$780$	0	0
$781$	−16.6185	−0.594656
$782$	0	0
$783$	−44.9621	−1.60681
$784$	0	0
$785$	−29.6008	−1.05650
$786$	0	0
$787$	21.6150	0.770490	0.385245	−	0.922814i	$-0.374117\pi$
$787$	21.6150	0.770490	0.385245	+	0.922814i	$0.374117\pi$
$788$	0	0
$789$	0.485895	0.0172983
$790$	0	0
$791$	−21.3648	−0.759647
$792$	0	0
$793$	−21.6302	−0.768112
$794$	0	0
$795$	−88.6644	−3.14460
$796$	0	0
$797$	24.4702	0.866778	0.433389	−	0.901207i	$-0.357318\pi$
$797$	24.4702	0.866778	0.433389	+	0.901207i	$0.357318\pi$
$798$	0	0
$799$	4.78110	0.169143
$800$	0	0
$801$	17.5698	0.620799
$802$	0	0
$803$	13.6713	0.482449
$804$	0	0
$805$	23.2010	0.817727
$806$	0	0
$807$	−21.7323	−0.765012
$808$	0	0
$809$	−13.9685	−0.491105	−0.245553	−	0.969383i	$-0.578969\pi$
$809$	−13.9685	−0.491105	−0.245553	+	0.969383i	$0.578969\pi$
$810$	0	0
$811$	−52.8441	−1.85561	−0.927803	−	0.373069i	$-0.878305\pi$
$811$	−52.8441	−1.85561	−0.927803	+	0.373069i	$0.878305\pi$
$812$	0	0
$813$	38.8570	1.36278
$814$	0	0
$815$	8.89093	0.311436
$816$	0	0
$817$	−4.16512	−0.145719
$818$	0	0
$819$	14.9524	0.522480
$820$	0	0
$821$	−9.18853	−0.320682	−0.160341	−	0.987062i	$-0.551259\pi$
$821$	−9.18853	−0.320682	−0.160341	+	0.987062i	$0.551259\pi$
$822$	0	0
$823$	31.3959	1.09439	0.547196	−	0.837005i	$-0.315695\pi$
$823$	31.3959	1.09439	0.547196	+	0.837005i	$0.315695\pi$
$824$	0	0
$825$	−67.0566	−2.33461
$826$	0	0
$827$	11.3390	0.394294	0.197147	−	0.980374i	$-0.436832\pi$
$827$	11.3390	0.394294	0.197147	+	0.980374i	$0.436832\pi$
$828$	0	0
$829$	15.5149	0.538854	0.269427	−	0.963021i	$-0.413166\pi$
$829$	15.5149	0.538854	0.269427	+	0.963021i	$0.413166\pi$
$830$	0	0
$831$	12.9034	0.447613
$832$	0	0
$833$	−5.84441	−0.202497
$834$	0	0
$835$	−13.1594	−0.455401
$836$	0	0
$837$	31.8738	1.10172
$838$	0	0
$839$	3.08820	0.106617	0.0533083	−	0.998578i	$-0.483023\pi$
$839$	3.08820	0.106617	0.0533083	+	0.998578i	$0.483023\pi$
$840$	0	0
$841$	−6.03368	−0.208058
$842$	0	0
$843$	−64.7352	−2.22960
$844$	0	0
$845$	−29.5961	−1.01814
$846$	0	0
$847$	5.74293	0.197330
$848$	0	0
$849$	−6.27799	−0.215460
$850$	0	0
$851$	−17.7942	−0.609976
$852$	0	0
$853$	−15.7716	−0.540010	−0.270005	−	0.962859i	$-0.587025\pi$
$853$	−15.7716	−0.540010	−0.270005	+	0.962859i	$0.587025\pi$
$854$	0	0
$855$	169.977	5.81310
$856$	0	0
$857$	46.7583	1.59723	0.798617	−	0.601840i	$-0.205566\pi$
$857$	46.7583	1.59723	0.798617	+	0.601840i	$0.205566\pi$
$858$	0	0
$859$	38.2531	1.30518	0.652590	−	0.757711i	$-0.273682\pi$
$859$	38.2531	1.30518	0.652590	+	0.757711i	$0.273682\pi$
$860$	0	0
$861$	−3.11947	−0.106311
$862$	0	0
$863$	27.0736	0.921596	0.460798	−	0.887505i	$-0.347563\pi$
$863$	27.0736	0.921596	0.460798	+	0.887505i	$0.347563\pi$
$864$	0	0
$865$	−45.3955	−1.54349
$866$	0	0
$867$	−3.01806	−0.102499
$868$	0	0
$869$	1.28648	0.0436409
$870$	0	0
$871$	−5.28517	−0.179081
$872$	0	0
$873$	49.7333	1.68322
$874$	0	0
$875$	−17.6721	−0.597425
$876$	0	0
$877$	−22.6875	−0.766102	−0.383051	−	0.923727i	$-0.625127\pi$
$877$	−22.6875	−0.766102	−0.383051	+	0.923727i	$0.625127\pi$
$878$	0	0
$879$	−10.2171	−0.344614
$880$	0	0
$881$	34.2699	1.15458	0.577291	−	0.816539i	$-0.304110\pi$
$881$	34.2699	1.15458	0.577291	+	0.816539i	$0.304110\pi$
$882$	0	0
$883$	30.9552	1.04173	0.520863	−	0.853641i	$-0.325610\pi$
$883$	30.9552	1.04173	0.520863	+	0.853641i	$0.325610\pi$
$884$	0	0
$885$	−11.4293	−0.384190
$886$	0	0
$887$	53.7299	1.80407	0.902036	−	0.431660i	$-0.142072\pi$
$887$	53.7299	1.80407	0.902036	+	0.431660i	$0.142072\pi$
$888$	0	0
$889$	−14.9803	−0.502424
$890$	0	0
$891$	23.7614	0.796038
$892$	0	0
$893$	35.1303	1.17559
$894$	0	0
$895$	34.5079	1.15347
$896$	0	0
$897$	−39.1656	−1.30770
$898$	0	0
$899$	−16.2809	−0.543000
$900$	0	0
$901$	7.75767	0.258446
$902$	0	0
$903$	−1.83909	−0.0612010
$904$	0	0
$905$	−48.4532	−1.61064
$906$	0	0
$907$	−0.969266	−0.0321839	−0.0160920	−	0.999871i	$-0.505122\pi$
$907$	−0.969266	−0.0321839	−0.0160920	+	0.999871i	$0.505122\pi$
$908$	0	0
$909$	32.4014	1.07469
$910$	0	0
$911$	−3.60197	−0.119339	−0.0596693	−	0.998218i	$-0.519005\pi$
$911$	−3.60197	−0.119339	−0.0596693	+	0.998218i	$0.519005\pi$
$912$	0	0
$913$	−28.1446	−0.931451
$914$	0	0
$915$	−108.571	−3.58926
$916$	0	0
$917$	−18.7028	−0.617619
$918$	0	0
$919$	−12.6297	−0.416614	−0.208307	−	0.978063i	$-0.566795\pi$
$919$	−12.6297	−0.416614	−0.208307	+	0.978063i	$0.566795\pi$
$920$	0	0
$921$	46.3682	1.52788
$922$	0	0
$923$	15.9088	0.523643
$924$	0	0
$925$	29.1648	0.958934
$926$	0	0
$927$	−43.4138	−1.42590
$928$	0	0
$929$	24.4508	0.802205	0.401103	−	0.916033i	$-0.368627\pi$
$929$	24.4508	0.802205	0.401103	+	0.916033i	$0.368627\pi$
$930$	0	0
$931$	−42.9432	−1.40741
$932$	0	0
$933$	−91.1827	−2.98519
$934$	0	0
$935$	9.00759	0.294580
$936$	0	0
$937$	11.8119	0.385879	0.192940	−	0.981211i	$-0.438198\pi$
$937$	11.8119	0.385879	0.192940	+	0.981211i	$0.438198\pi$
$938$	0	0
$939$	49.4797	1.61471
$940$	0	0
$941$	0.680332	0.0221782	0.0110891	−	0.999939i	$-0.496470\pi$
$941$	0.680332	0.0221782	0.0110891	+	0.999939i	$0.496470\pi$
$942$	0	0
$943$	5.47981	0.178447
$944$	0	0
$945$	38.1938	1.24244
$946$	0	0
$947$	−19.0399	−0.618715	−0.309357	−	0.950946i	$-0.600114\pi$
$947$	−19.0399	−0.618715	−0.309357	+	0.950946i	$0.600114\pi$
$948$	0	0
$949$	−13.0874	−0.424836
$950$	0	0
$951$	23.1406	0.750385
$952$	0	0
$953$	0.217929	0.00705940	0.00352970	−	0.999994i	$-0.498876\pi$
$953$	0.217929	0.00705940	0.00352970	+	0.999994i	$0.498876\pi$
$954$	0	0
$955$	−62.6077	−2.02594
$956$	0	0
$957$	−34.4026	−1.11208
$958$	0	0
$959$	22.5266	0.727423
$960$	0	0
$961$	−19.4584	−0.627689
$962$	0	0
$963$	13.6553	0.440037
$964$	0	0
$965$	1.68670	0.0542969
$966$	0	0
$967$	11.4628	0.368620	0.184310	−	0.982868i	$-0.440995\pi$
$967$	11.4628	0.368620	0.184310	+	0.982868i	$0.440995\pi$
$968$	0	0
$969$	−22.1759	−0.712393
$970$	0	0
$971$	−28.1705	−0.904033	−0.452017	−	0.892010i	$-0.649295\pi$
$971$	−28.1705	−0.904033	−0.452017	+	0.892010i	$0.649295\pi$
$972$	0	0
$973$	1.43436	0.0459836
$974$	0	0
$975$	64.1928	2.05582
$976$	0	0
$977$	−23.7286	−0.759144	−0.379572	−	0.925162i	$-0.623929\pi$
$977$	−23.7286	−0.759144	−0.379572	+	0.925162i	$0.623929\pi$
$978$	0	0
$979$	6.84131	0.218649
$980$	0	0
$981$	88.1306	2.81379
$982$	0	0
$983$	−49.7609	−1.58713	−0.793563	−	0.608488i	$-0.791776\pi$
$983$	−49.7609	−1.58713	−0.793563	+	0.608488i	$0.791776\pi$
$984$	0	0
$985$	−65.8285	−2.09747
$986$	0	0
$987$	15.5116	0.493740
$988$	0	0
$989$	3.23063	0.102728
$990$	0	0
$991$	−8.35442	−0.265387	−0.132693	−	0.991157i	$-0.542363\pi$
$991$	−8.35442	−0.265387	−0.132693	+	0.991157i	$0.542363\pi$
$992$	0	0
$993$	90.6925	2.87804
$994$	0	0
$995$	−47.6165	−1.50955
$996$	0	0
$997$	−0.910155	−0.0288249	−0.0144125	−	0.999896i	$-0.504588\pi$
$997$	−0.910155	−0.0288249	−0.0144125	+	0.999896i	$0.504588\pi$
$998$	0	0
$999$	−29.2930	−0.926790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.2	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.2	✓	32	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.01806 q^{3} +3.78696 q^{5} -1.07498 q^{7} +6.10866 q^{9} +O(q^{10})\)	\(q-3.01806 q^{3} +3.78696 q^{5} -1.07498 q^{7} +6.10866 q^{9} +2.37858 q^{11} -2.27700 q^{13} -11.4293 q^{15} +1.00000 q^{17} +7.34775 q^{19} +3.24436 q^{21} -5.69920 q^{23} +9.34105 q^{25} -9.38211 q^{27} +4.79232 q^{29} -3.39730 q^{31} -7.17870 q^{33} -4.07092 q^{35} +3.12222 q^{37} +6.87212 q^{39} -0.961505 q^{41} -0.566857 q^{43} +23.1332 q^{45} +4.78110 q^{47} -5.84441 q^{49} -3.01806 q^{51} +7.75767 q^{53} +9.00759 q^{55} -22.1759 q^{57} +1.00000 q^{59} +9.49944 q^{61} -6.56671 q^{63} -8.62291 q^{65} +2.32111 q^{67} +17.2005 q^{69} -6.98671 q^{71} +5.74766 q^{73} -28.1918 q^{75} -2.55694 q^{77} +0.540861 q^{79} +9.98974 q^{81} -11.8325 q^{83} +3.78696 q^{85} -14.4635 q^{87} +2.87621 q^{89} +2.44774 q^{91} +10.2532 q^{93} +27.8256 q^{95} +8.14144 q^{97} +14.5300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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