LMFDB - Embedded newform 8008.2.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 $ x^9 - 2x^8 - 12x^7 + 20x^6 + 47x^5 - 55x^4 - 68x^3 + 37x^2 + 21x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.390946$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.390946 q^{3} -3.55681 q^{5} +1.00000 q^{7} -2.84716 q^{9} +O(q^{10})$	$q-0.390946 q^{3} -3.55681 q^{5} +1.00000 q^{7} -2.84716 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.39052 q^{15} -6.24378 q^{17} +1.56572 q^{19} -0.390946 q^{21} +1.70359 q^{23} +7.65092 q^{25} +2.28592 q^{27} +8.46788 q^{29} +3.26930 q^{31} +0.390946 q^{33} -3.55681 q^{35} +0.0330987 q^{37} -0.390946 q^{39} +7.60036 q^{41} -3.65244 q^{43} +10.1268 q^{45} -3.76297 q^{47} +1.00000 q^{49} +2.44098 q^{51} +5.93611 q^{53} +3.55681 q^{55} -0.612110 q^{57} -7.89174 q^{59} -1.11331 q^{61} -2.84716 q^{63} -3.55681 q^{65} +0.0566581 q^{67} -0.666011 q^{69} +2.92064 q^{71} +6.41687 q^{73} -2.99110 q^{75} -1.00000 q^{77} +3.75635 q^{79} +7.64781 q^{81} -1.54220 q^{83} +22.2079 q^{85} -3.31048 q^{87} +2.79072 q^{89} +1.00000 q^{91} -1.27812 q^{93} -5.56896 q^{95} +1.64308 q^{97} +2.84716 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - 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
$3$	−0.390946	−0.225713	−0.112856	−	0.993611i	$-0.536000\pi$
$3$	−0.390946	−0.225713	−0.112856	+	0.993611i	$0.536000\pi$
$4$	0	0
$5$	−3.55681	−1.59066	−0.795328	−	0.606180i	$-0.792701\pi$
$5$	−3.55681	−1.59066	−0.795328	+	0.606180i	$0.792701\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.84716	−0.949054
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.39052	0.359031
$16$	0	0
$17$	−6.24378	−1.51434	−0.757169	−	0.653219i	$-0.773418\pi$
$17$	−6.24378	−1.51434	−0.757169	+	0.653219i	$0.773418\pi$
$18$	0	0
$19$	1.56572	0.359200	0.179600	−	0.983740i	$-0.442520\pi$
$19$	1.56572	0.359200	0.179600	+	0.983740i	$0.442520\pi$
$20$	0	0
$21$	−0.390946	−0.0853114
$22$	0	0
$23$	1.70359	0.355223	0.177611	−	0.984101i	$-0.443163\pi$
$23$	1.70359	0.355223	0.177611	+	0.984101i	$0.443163\pi$
$24$	0	0
$25$	7.65092	1.53018
$26$	0	0
$27$	2.28592	0.439926
$28$	0	0
$29$	8.46788	1.57245	0.786223	−	0.617943i	$-0.212034\pi$
$29$	8.46788	1.57245	0.786223	+	0.617943i	$0.212034\pi$
$30$	0	0
$31$	3.26930	0.587183	0.293592	−	0.955931i	$-0.405149\pi$
$31$	3.26930	0.587183	0.293592	+	0.955931i	$0.405149\pi$
$32$	0	0
$33$	0.390946	0.0680549
$34$	0	0
$35$	−3.55681	−0.601211
$36$	0	0
$37$	0.0330987	0.00544139	0.00272069	−	0.999996i	$-0.499134\pi$
$37$	0.0330987	0.00544139	0.00272069	+	0.999996i	$0.499134\pi$
$38$	0	0
$39$	−0.390946	−0.0626014
$40$	0	0
$41$	7.60036	1.18698	0.593488	−	0.804843i	$-0.297750\pi$
$41$	7.60036	1.18698	0.593488	+	0.804843i	$0.297750\pi$
$42$	0	0
$43$	−3.65244	−0.556992	−0.278496	−	0.960437i	$-0.589836\pi$
$43$	−3.65244	−0.556992	−0.278496	+	0.960437i	$0.589836\pi$
$44$	0	0
$45$	10.1268	1.50962
$46$	0	0
$47$	−3.76297	−0.548886	−0.274443	−	0.961603i	$-0.588493\pi$
$47$	−3.76297	−0.548886	−0.274443	+	0.961603i	$0.588493\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.44098	0.341805
$52$	0	0
$53$	5.93611	0.815387	0.407694	−	0.913119i	$-0.366333\pi$
$53$	5.93611	0.815387	0.407694	+	0.913119i	$0.366333\pi$
$54$	0	0
$55$	3.55681	0.479601
$56$	0	0
$57$	−0.612110	−0.0810760
$58$	0	0
$59$	−7.89174	−1.02742	−0.513709	−	0.857965i	$-0.671729\pi$
$59$	−7.89174	−1.02742	−0.513709	+	0.857965i	$0.671729\pi$
$60$	0	0
$61$	−1.11331	−0.142545	−0.0712725	−	0.997457i	$-0.522706\pi$
$61$	−1.11331	−0.142545	−0.0712725	+	0.997457i	$0.522706\pi$
$62$	0	0
$63$	−2.84716	−0.358709
$64$	0	0
$65$	−3.55681	−0.441168
$66$	0	0
$67$	0.0566581	0.00692189	0.00346095	−	0.999994i	$-0.498898\pi$
$67$	0.0566581	0.00692189	0.00346095	+	0.999994i	$0.498898\pi$
$68$	0	0
$69$	−0.666011	−0.0801783
$70$	0	0
$71$	2.92064	0.346616	0.173308	−	0.984868i	$-0.444554\pi$
$71$	2.92064	0.346616	0.173308	+	0.984868i	$0.444554\pi$
$72$	0	0
$73$	6.41687	0.751038	0.375519	−	0.926815i	$-0.377464\pi$
$73$	6.41687	0.751038	0.375519	+	0.926815i	$0.377464\pi$
$74$	0	0
$75$	−2.99110	−0.345382
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	3.75635	0.422622	0.211311	−	0.977419i	$-0.432227\pi$
$79$	3.75635	0.422622	0.211311	+	0.977419i	$0.432227\pi$
$80$	0	0
$81$	7.64781	0.849757
$82$	0	0
$83$	−1.54220	−0.169279	−0.0846394	−	0.996412i	$-0.526974\pi$
$83$	−1.54220	−0.169279	−0.0846394	+	0.996412i	$0.526974\pi$
$84$	0	0
$85$	22.2079	2.40879
$86$	0	0
$87$	−3.31048	−0.354921
$88$	0	0
$89$	2.79072	0.295816	0.147908	−	0.989001i	$-0.452746\pi$
$89$	2.79072	0.295816	0.147908	+	0.989001i	$0.452746\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.27812	−0.132535
$94$	0	0
$95$	−5.56896	−0.571363
$96$	0	0
$97$	1.64308	0.166830	0.0834149	−	0.996515i	$-0.473417\pi$
$97$	1.64308	0.166830	0.0834149	+	0.996515i	$0.473417\pi$
$98$	0	0
$99$	2.84716	0.286150
$100$	0	0
$101$	7.98664	0.794701	0.397350	−	0.917667i	$-0.369930\pi$
$101$	7.98664	0.794701	0.397350	+	0.917667i	$0.369930\pi$
$102$	0	0
$103$	−5.60451	−0.552229	−0.276115	−	0.961125i	$-0.589047\pi$
$103$	−5.60451	−0.552229	−0.276115	+	0.961125i	$0.589047\pi$
$104$	0	0
$105$	1.39052	0.135701
$106$	0	0
$107$	10.0463	0.971211	0.485606	−	0.874178i	$-0.338599\pi$
$107$	10.0463	0.971211	0.485606	+	0.874178i	$0.338599\pi$
$108$	0	0
$109$	−11.7659	−1.12697	−0.563484	−	0.826127i	$-0.690539\pi$
$109$	−11.7659	−1.12697	−0.563484	+	0.826127i	$0.690539\pi$
$110$	0	0
$111$	−0.0129398	−0.00122819
$112$	0	0
$113$	−17.1689	−1.61511	−0.807555	−	0.589792i	$-0.799210\pi$
$113$	−17.1689	−1.61511	−0.807555	+	0.589792i	$0.799210\pi$
$114$	0	0
$115$	−6.05935	−0.565037
$116$	0	0
$117$	−2.84716	−0.263220
$118$	0	0
$119$	−6.24378	−0.572366
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.97133	−0.267916
$124$	0	0
$125$	−9.42884	−0.843341
$126$	0	0
$127$	−8.51577	−0.755652	−0.377826	−	0.925877i	$-0.623328\pi$
$127$	−8.51577	−0.755652	−0.377826	+	0.925877i	$0.623328\pi$
$128$	0	0
$129$	1.42791	0.125720
$130$	0	0
$131$	−16.0651	−1.40362	−0.701808	−	0.712366i	$-0.747623\pi$
$131$	−16.0651	−1.40362	−0.701808	+	0.712366i	$0.747623\pi$
$132$	0	0
$133$	1.56572	0.135765
$134$	0	0
$135$	−8.13060	−0.699771
$136$	0	0
$137$	−17.2124	−1.47056	−0.735278	−	0.677765i	$-0.762949\pi$
$137$	−17.2124	−1.47056	−0.735278	+	0.677765i	$0.762949\pi$
$138$	0	0
$139$	0.436481	0.0370218	0.0185109	−	0.999829i	$-0.494107\pi$
$139$	0.436481	0.0370218	0.0185109	+	0.999829i	$0.494107\pi$
$140$	0	0
$141$	1.47112	0.123891
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−30.1187	−2.50122
$146$	0	0
$147$	−0.390946	−0.0322447
$148$	0	0
$149$	−4.59074	−0.376088	−0.188044	−	0.982161i	$-0.560215\pi$
$149$	−4.59074	−0.376088	−0.188044	+	0.982161i	$0.560215\pi$
$150$	0	0
$151$	22.3211	1.81647	0.908233	−	0.418465i	$-0.137432\pi$
$151$	22.3211	1.81647	0.908233	+	0.418465i	$0.137432\pi$
$152$	0	0
$153$	17.7770	1.43719
$154$	0	0
$155$	−11.6283	−0.934006
$156$	0	0
$157$	8.50865	0.679065	0.339532	−	0.940594i	$-0.389731\pi$
$157$	8.50865	0.679065	0.339532	+	0.940594i	$0.389731\pi$
$158$	0	0
$159$	−2.32070	−0.184043
$160$	0	0
$161$	1.70359	0.134262
$162$	0	0
$163$	−6.39648	−0.501011	−0.250506	−	0.968115i	$-0.580597\pi$
$163$	−6.39648	−0.501011	−0.250506	+	0.968115i	$0.580597\pi$
$164$	0	0
$165$	−1.39052	−0.108252
$166$	0	0
$167$	−14.4829	−1.12072	−0.560360	−	0.828249i	$-0.689337\pi$
$167$	−14.4829	−1.12072	−0.560360	+	0.828249i	$0.689337\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.45785	−0.340900
$172$	0	0
$173$	−21.6357	−1.64493	−0.822467	−	0.568813i	$-0.807403\pi$
$173$	−21.6357	−1.64493	−0.822467	+	0.568813i	$0.807403\pi$
$174$	0	0
$175$	7.65092	0.578355
$176$	0	0
$177$	3.08524	0.231901
$178$	0	0
$179$	16.8418	1.25882	0.629409	−	0.777074i	$-0.283297\pi$
$179$	16.8418	1.25882	0.629409	+	0.777074i	$0.283297\pi$
$180$	0	0
$181$	2.61505	0.194375	0.0971875	−	0.995266i	$-0.469015\pi$
$181$	2.61505	0.194375	0.0971875	+	0.995266i	$0.469015\pi$
$182$	0	0
$183$	0.435245	0.0321742
$184$	0	0
$185$	−0.117726	−0.00865537
$186$	0	0
$187$	6.24378	0.456590
$188$	0	0
$189$	2.28592	0.166276
$190$	0	0
$191$	5.96429	0.431561	0.215780	−	0.976442i	$-0.430770\pi$
$191$	5.96429	0.431561	0.215780	+	0.976442i	$0.430770\pi$
$192$	0	0
$193$	−8.19687	−0.590024	−0.295012	−	0.955494i	$-0.595324\pi$
$193$	−8.19687	−0.590024	−0.295012	+	0.955494i	$0.595324\pi$
$194$	0	0
$195$	1.39052	0.0995773
$196$	0	0
$197$	20.4318	1.45571	0.727853	−	0.685733i	$-0.240518\pi$
$197$	20.4318	1.45571	0.727853	+	0.685733i	$0.240518\pi$
$198$	0	0
$199$	−27.1357	−1.92360	−0.961800	−	0.273752i	$-0.911735\pi$
$199$	−27.1357	−1.92360	−0.961800	+	0.273752i	$0.911735\pi$
$200$	0	0
$201$	−0.0221503	−0.00156236
$202$	0	0
$203$	8.46788	0.594328
$204$	0	0
$205$	−27.0331	−1.88807
$206$	0	0
$207$	−4.85039	−0.337125
$208$	0	0
$209$	−1.56572	−0.108303
$210$	0	0
$211$	18.9846	1.30695	0.653476	−	0.756947i	$-0.273310\pi$
$211$	18.9846	1.30695	0.653476	+	0.756947i	$0.273310\pi$
$212$	0	0
$213$	−1.14181	−0.0782357
$214$	0	0
$215$	12.9911	0.885983
$216$	0	0
$217$	3.26930	0.221934
$218$	0	0
$219$	−2.50865	−0.169519
$220$	0	0
$221$	−6.24378	−0.420002
$222$	0	0
$223$	10.8953	0.729605	0.364803	−	0.931085i	$-0.381136\pi$
$223$	10.8953	0.729605	0.364803	+	0.931085i	$0.381136\pi$
$224$	0	0
$225$	−21.7834	−1.45223
$226$	0	0
$227$	−16.7119	−1.10921	−0.554605	−	0.832114i	$-0.687131\pi$
$227$	−16.7119	−1.10921	−0.554605	+	0.832114i	$0.687131\pi$
$228$	0	0
$229$	5.00730	0.330892	0.165446	−	0.986219i	$-0.447094\pi$
$229$	5.00730	0.330892	0.165446	+	0.986219i	$0.447094\pi$
$230$	0	0
$231$	0.390946	0.0257224
$232$	0	0
$233$	−8.47721	−0.555360	−0.277680	−	0.960674i	$-0.589566\pi$
$233$	−8.47721	−0.555360	−0.277680	+	0.960674i	$0.589566\pi$
$234$	0	0
$235$	13.3842	0.873088
$236$	0	0
$237$	−1.46853	−0.0953912
$238$	0	0
$239$	9.95054	0.643647	0.321824	−	0.946800i	$-0.395704\pi$
$239$	9.95054	0.643647	0.321824	+	0.946800i	$0.395704\pi$
$240$	0	0
$241$	6.90189	0.444590	0.222295	−	0.974980i	$-0.428645\pi$
$241$	6.90189	0.444590	0.222295	+	0.974980i	$0.428645\pi$
$242$	0	0
$243$	−9.84765	−0.631727
$244$	0	0
$245$	−3.55681	−0.227236
$246$	0	0
$247$	1.56572	0.0996242
$248$	0	0
$249$	0.602918	0.0382084
$250$	0	0
$251$	−7.87813	−0.497263	−0.248632	−	0.968598i	$-0.579981\pi$
$251$	−7.87813	−0.497263	−0.248632	+	0.968598i	$0.579981\pi$
$252$	0	0
$253$	−1.70359	−0.107104
$254$	0	0
$255$	−8.68211	−0.543695
$256$	0	0
$257$	0.992999	0.0619416	0.0309708	−	0.999520i	$-0.490140\pi$
$257$	0.992999	0.0619416	0.0309708	+	0.999520i	$0.490140\pi$
$258$	0	0
$259$	0.0330987	0.00205665
$260$	0	0
$261$	−24.1094	−1.49234
$262$	0	0
$263$	−13.0261	−0.803223	−0.401611	−	0.915810i	$-0.631550\pi$
$263$	−13.0261	−0.803223	−0.401611	+	0.915810i	$0.631550\pi$
$264$	0	0
$265$	−21.1136	−1.29700
$266$	0	0
$267$	−1.09102	−0.0667694
$268$	0	0
$269$	−19.6771	−1.19974	−0.599868	−	0.800099i	$-0.704780\pi$
$269$	−19.6771	−1.19974	−0.599868	+	0.800099i	$0.704780\pi$
$270$	0	0
$271$	−3.47243	−0.210935	−0.105468	−	0.994423i	$-0.533634\pi$
$271$	−3.47243	−0.210935	−0.105468	+	0.994423i	$0.533634\pi$
$272$	0	0
$273$	−0.390946	−0.0236611
$274$	0	0
$275$	−7.65092	−0.461368
$276$	0	0
$277$	19.1530	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$277$	19.1530	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$278$	0	0
$279$	−9.30822	−0.557268
$280$	0	0
$281$	−27.3543	−1.63182	−0.815911	−	0.578177i	$-0.803764\pi$
$281$	−27.3543	−1.63182	−0.815911	+	0.578177i	$0.803764\pi$
$282$	0	0
$283$	24.4704	1.45461	0.727306	−	0.686314i	$-0.240772\pi$
$283$	24.4704	1.45461	0.727306	+	0.686314i	$0.240772\pi$
$284$	0	0
$285$	2.17716	0.128964
$286$	0	0
$287$	7.60036	0.448635
$288$	0	0
$289$	21.9847	1.29322
$290$	0	0
$291$	−0.642356	−0.0376556
$292$	0	0
$293$	−18.9660	−1.10800	−0.554002	−	0.832515i	$-0.686900\pi$
$293$	−18.9660	−1.10800	−0.554002	+	0.832515i	$0.686900\pi$
$294$	0	0
$295$	28.0695	1.63427
$296$	0	0
$297$	−2.28592	−0.132643
$298$	0	0
$299$	1.70359	0.0985211
$300$	0	0
$301$	−3.65244	−0.210523
$302$	0	0
$303$	−3.12235	−0.179374
$304$	0	0
$305$	3.95984	0.226740
$306$	0	0
$307$	−10.0624	−0.574293	−0.287146	−	0.957887i	$-0.592707\pi$
$307$	−10.0624	−0.574293	−0.287146	+	0.957887i	$0.592707\pi$
$308$	0	0
$309$	2.19106	0.124645
$310$	0	0
$311$	9.64578	0.546962	0.273481	−	0.961877i	$-0.411825\pi$
$311$	9.64578	0.546962	0.273481	+	0.961877i	$0.411825\pi$
$312$	0	0
$313$	−23.6980	−1.33949	−0.669744	−	0.742592i	$-0.733596\pi$
$313$	−23.6980	−1.33949	−0.669744	+	0.742592i	$0.733596\pi$
$314$	0	0
$315$	10.1268	0.570582
$316$	0	0
$317$	−6.20765	−0.348656	−0.174328	−	0.984688i	$-0.555775\pi$
$317$	−6.20765	−0.348656	−0.174328	+	0.984688i	$0.555775\pi$
$318$	0	0
$319$	−8.46788	−0.474110
$320$	0	0
$321$	−3.92755	−0.219215
$322$	0	0
$323$	−9.77598	−0.543950
$324$	0	0
$325$	7.65092	0.424397
$326$	0	0
$327$	4.59983	0.254371
$328$	0	0
$329$	−3.76297	−0.207459
$330$	0	0
$331$	20.8934	1.14841	0.574204	−	0.818712i	$-0.305312\pi$
$331$	20.8934	1.14841	0.574204	+	0.818712i	$0.305312\pi$
$332$	0	0
$333$	−0.0942372	−0.00516417
$334$	0	0
$335$	−0.201522	−0.0110103
$336$	0	0
$337$	13.6548	0.743826	0.371913	−	0.928267i	$-0.378702\pi$
$337$	13.6548	0.743826	0.371913	+	0.928267i	$0.378702\pi$
$338$	0	0
$339$	6.71209	0.364551
$340$	0	0
$341$	−3.26930	−0.177042
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.36888	0.127536
$346$	0	0
$347$	−13.4825	−0.723778	−0.361889	−	0.932221i	$-0.617868\pi$
$347$	−13.4825	−0.723778	−0.361889	+	0.932221i	$0.617868\pi$
$348$	0	0
$349$	9.69048	0.518719	0.259360	−	0.965781i	$-0.416489\pi$
$349$	9.69048	0.518719	0.259360	+	0.965781i	$0.416489\pi$
$350$	0	0
$351$	2.28592	0.122014
$352$	0	0
$353$	25.8306	1.37482	0.687412	−	0.726268i	$-0.258747\pi$
$353$	25.8306	1.37482	0.687412	+	0.726268i	$0.258747\pi$
$354$	0	0
$355$	−10.3882	−0.551347
$356$	0	0
$357$	2.44098	0.129190
$358$	0	0
$359$	−25.1020	−1.32483	−0.662417	−	0.749136i	$-0.730469\pi$
$359$	−25.1020	−1.32483	−0.662417	+	0.749136i	$0.730469\pi$
$360$	0	0
$361$	−16.5485	−0.870975
$362$	0	0
$363$	−0.390946	−0.0205193
$364$	0	0
$365$	−22.8236	−1.19464
$366$	0	0
$367$	−33.1785	−1.73190	−0.865951	−	0.500129i	$-0.833286\pi$
$367$	−33.1785	−1.73190	−0.865951	+	0.500129i	$0.833286\pi$
$368$	0	0
$369$	−21.6394	−1.12650
$370$	0	0
$371$	5.93611	0.308187
$372$	0	0
$373$	−21.5477	−1.11570	−0.557850	−	0.829942i	$-0.688373\pi$
$373$	−21.5477	−1.11570	−0.557850	+	0.829942i	$0.688373\pi$
$374$	0	0
$375$	3.68617	0.190353
$376$	0	0
$377$	8.46788	0.436118
$378$	0	0
$379$	19.0925	0.980713	0.490357	−	0.871522i	$-0.336867\pi$
$379$	19.0925	0.980713	0.490357	+	0.871522i	$0.336867\pi$
$380$	0	0
$381$	3.32921	0.170560
$382$	0	0
$383$	−18.7892	−0.960084	−0.480042	−	0.877245i	$-0.659379\pi$
$383$	−18.7892	−0.960084	−0.480042	+	0.877245i	$0.659379\pi$
$384$	0	0
$385$	3.55681	0.181272
$386$	0	0
$387$	10.3991	0.528616
$388$	0	0
$389$	26.9408	1.36595	0.682977	−	0.730440i	$-0.260685\pi$
$389$	26.9408	1.36595	0.682977	+	0.730440i	$0.260685\pi$
$390$	0	0
$391$	−10.6368	−0.537927
$392$	0	0
$393$	6.28059	0.316814
$394$	0	0
$395$	−13.3606	−0.672246
$396$	0	0
$397$	7.68221	0.385559	0.192780	−	0.981242i	$-0.438250\pi$
$397$	7.68221	0.385559	0.192780	+	0.981242i	$0.438250\pi$
$398$	0	0
$399$	−0.612110	−0.0306438
$400$	0	0
$401$	−15.0770	−0.752909	−0.376455	−	0.926435i	$-0.622857\pi$
$401$	−15.0770	−0.752909	−0.376455	+	0.926435i	$0.622857\pi$
$402$	0	0
$403$	3.26930	0.162855
$404$	0	0
$405$	−27.2018	−1.35167
$406$	0	0
$407$	−0.0330987	−0.00164064
$408$	0	0
$409$	2.11128	0.104396	0.0521980	−	0.998637i	$-0.483377\pi$
$409$	2.11128	0.104396	0.0521980	+	0.998637i	$0.483377\pi$
$410$	0	0
$411$	6.72913	0.331923
$412$	0	0
$413$	−7.89174	−0.388327
$414$	0	0
$415$	5.48533	0.269264
$416$	0	0
$417$	−0.170640	−0.00835630
$418$	0	0
$419$	−19.1786	−0.936936	−0.468468	−	0.883480i	$-0.655194\pi$
$419$	−19.1786	−0.936936	−0.468468	+	0.883480i	$0.655194\pi$
$420$	0	0
$421$	19.3037	0.940803	0.470401	−	0.882453i	$-0.344109\pi$
$421$	19.3037	0.940803	0.470401	+	0.882453i	$0.344109\pi$
$422$	0	0
$423$	10.7138	0.520922
$424$	0	0
$425$	−47.7706	−2.31722
$426$	0	0
$427$	−1.11331	−0.0538769
$428$	0	0
$429$	0.390946	0.0188750
$430$	0	0
$431$	−24.0880	−1.16028	−0.580139	−	0.814517i	$-0.697002\pi$
$431$	−24.0880	−1.16028	−0.580139	+	0.814517i	$0.697002\pi$
$432$	0	0
$433$	22.8651	1.09883	0.549413	−	0.835551i	$-0.314851\pi$
$433$	22.8651	1.09883	0.549413	+	0.835551i	$0.314851\pi$
$434$	0	0
$435$	11.7748	0.564557
$436$	0	0
$437$	2.66734	0.127596
$438$	0	0
$439$	−15.2033	−0.725612	−0.362806	−	0.931865i	$-0.618181\pi$
$439$	−15.2033	−0.725612	−0.362806	+	0.931865i	$0.618181\pi$
$440$	0	0
$441$	−2.84716	−0.135579
$442$	0	0
$443$	−33.8949	−1.61040	−0.805198	−	0.593005i	$-0.797941\pi$
$443$	−33.8949	−1.61040	−0.805198	+	0.593005i	$0.797941\pi$
$444$	0	0
$445$	−9.92607	−0.470541
$446$	0	0
$447$	1.79473	0.0848878
$448$	0	0
$449$	−15.7398	−0.742809	−0.371404	−	0.928471i	$-0.621124\pi$
$449$	−15.7398	−0.742809	−0.371404	+	0.928471i	$0.621124\pi$
$450$	0	0
$451$	−7.60036	−0.357887
$452$	0	0
$453$	−8.72634	−0.409999
$454$	0	0
$455$	−3.55681	−0.166746
$456$	0	0
$457$	−29.1105	−1.36173	−0.680865	−	0.732409i	$-0.738396\pi$
$457$	−29.1105	−1.36173	−0.680865	+	0.732409i	$0.738396\pi$
$458$	0	0
$459$	−14.2728	−0.666197
$460$	0	0
$461$	−16.9470	−0.789302	−0.394651	−	0.918831i	$-0.629134\pi$
$461$	−16.9470	−0.789302	−0.394651	+	0.918831i	$0.629134\pi$
$462$	0	0
$463$	26.2656	1.22066	0.610332	−	0.792145i	$-0.291036\pi$
$463$	26.2656	1.22066	0.610332	+	0.792145i	$0.291036\pi$
$464$	0	0
$465$	4.54603	0.210817
$466$	0	0
$467$	5.69461	0.263515	0.131758	−	0.991282i	$-0.457938\pi$
$467$	5.69461	0.263515	0.131758	+	0.991282i	$0.457938\pi$
$468$	0	0
$469$	0.0566581	0.00261623
$470$	0	0
$471$	−3.32642	−0.153274
$472$	0	0
$473$	3.65244	0.167939
$474$	0	0
$475$	11.9792	0.549642
$476$	0	0
$477$	−16.9011	−0.773846
$478$	0	0
$479$	−14.9183	−0.681633	−0.340817	−	0.940130i	$-0.610704\pi$
$479$	−14.9183	−0.681633	−0.340817	+	0.940130i	$0.610704\pi$
$480$	0	0
$481$	0.0330987	0.00150917
$482$	0	0
$483$	−0.666011	−0.0303045
$484$	0	0
$485$	−5.84414	−0.265369
$486$	0	0
$487$	−2.30811	−0.104590	−0.0522952	−	0.998632i	$-0.516654\pi$
$487$	−2.30811	−0.104590	−0.0522952	+	0.998632i	$0.516654\pi$
$488$	0	0
$489$	2.50068	0.113085
$490$	0	0
$491$	0.366552	0.0165423	0.00827113	−	0.999966i	$-0.497367\pi$
$491$	0.366552	0.0165423	0.00827113	+	0.999966i	$0.497367\pi$
$492$	0	0
$493$	−52.8715	−2.38121
$494$	0	0
$495$	−10.1268	−0.455167
$496$	0	0
$497$	2.92064	0.131009
$498$	0	0
$499$	35.1551	1.57376	0.786878	−	0.617108i	$-0.211696\pi$
$499$	35.1551	1.57376	0.786878	+	0.617108i	$0.211696\pi$
$500$	0	0
$501$	5.66203	0.252961
$502$	0	0
$503$	3.05811	0.136354	0.0681772	−	0.997673i	$-0.478282\pi$
$503$	3.05811	0.136354	0.0681772	+	0.997673i	$0.478282\pi$
$504$	0	0
$505$	−28.4070	−1.26410
$506$	0	0
$507$	−0.390946	−0.0173625
$508$	0	0
$509$	−0.441285	−0.0195596	−0.00977981	−	0.999952i	$-0.503113\pi$
$509$	−0.441285	−0.0195596	−0.00977981	+	0.999952i	$0.503113\pi$
$510$	0	0
$511$	6.41687	0.283866
$512$	0	0
$513$	3.57911	0.158021
$514$	0	0
$515$	19.9342	0.878406
$516$	0	0
$517$	3.76297	0.165495
$518$	0	0
$519$	8.45840	0.371283
$520$	0	0
$521$	−24.1984	−1.06015	−0.530075	−	0.847951i	$-0.677836\pi$
$521$	−24.1984	−1.06015	−0.530075	+	0.847951i	$0.677836\pi$
$522$	0	0
$523$	−7.51834	−0.328754	−0.164377	−	0.986398i	$-0.552561\pi$
$523$	−7.51834	−0.328754	−0.164377	+	0.986398i	$0.552561\pi$
$524$	0	0
$525$	−2.99110	−0.130542
$526$	0	0
$527$	−20.4128	−0.889194
$528$	0	0
$529$	−20.0978	−0.873817
$530$	0	0
$531$	22.4691	0.975074
$532$	0	0
$533$	7.60036	0.329208
$534$	0	0
$535$	−35.7328	−1.54486
$536$	0	0
$537$	−6.58425	−0.284131
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−23.6879	−1.01842	−0.509212	−	0.860641i	$-0.670063\pi$
$541$	−23.6879	−1.01842	−0.509212	+	0.860641i	$0.670063\pi$
$542$	0	0
$543$	−1.02234	−0.0438729
$544$	0	0
$545$	41.8491	1.79262
$546$	0	0
$547$	0.805406	0.0344367	0.0172183	−	0.999852i	$-0.494519\pi$
$547$	0.805406	0.0344367	0.0172183	+	0.999852i	$0.494519\pi$
$548$	0	0
$549$	3.16978	0.135283
$550$	0	0
$551$	13.2583	0.564822
$552$	0	0
$553$	3.75635	0.159736
$554$	0	0
$555$	0.0460244	0.00195363
$556$	0	0
$557$	−23.3835	−0.990792	−0.495396	−	0.868667i	$-0.664977\pi$
$557$	−23.3835	−0.990792	−0.495396	+	0.868667i	$0.664977\pi$
$558$	0	0
$559$	−3.65244	−0.154482
$560$	0	0
$561$	−2.44098	−0.103058
$562$	0	0
$563$	−2.92834	−0.123415	−0.0617074	−	0.998094i	$-0.519655\pi$
$563$	−2.92834	−0.123415	−0.0617074	+	0.998094i	$0.519655\pi$
$564$	0	0
$565$	61.0664	2.56908
$566$	0	0
$567$	7.64781	0.321178
$568$	0	0
$569$	−27.3011	−1.14452	−0.572262	−	0.820071i	$-0.693934\pi$
$569$	−27.3011	−1.14452	−0.572262	+	0.820071i	$0.693934\pi$
$570$	0	0
$571$	33.1864	1.38881	0.694405	−	0.719585i	$-0.255668\pi$
$571$	33.1864	1.38881	0.694405	+	0.719585i	$0.255668\pi$
$572$	0	0
$573$	−2.33171	−0.0974088
$574$	0	0
$575$	13.0340	0.543556
$576$	0	0
$577$	−19.9464	−0.830380	−0.415190	−	0.909735i	$-0.636285\pi$
$577$	−19.9464	−0.830380	−0.415190	+	0.909735i	$0.636285\pi$
$578$	0	0
$579$	3.20453	0.133176
$580$	0	0
$581$	−1.54220	−0.0639814
$582$	0	0
$583$	−5.93611	−0.245848
$584$	0	0
$585$	10.1268	0.418693
$586$	0	0
$587$	21.8531	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$587$	21.8531	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$588$	0	0
$589$	5.11879	0.210916
$590$	0	0
$591$	−7.98774	−0.328572
$592$	0	0
$593$	−15.1816	−0.623432	−0.311716	−	0.950175i	$-0.600904\pi$
$593$	−15.1816	−0.623432	−0.311716	+	0.950175i	$0.600904\pi$
$594$	0	0
$595$	22.2079	0.910437
$596$	0	0
$597$	10.6086	0.434181
$598$	0	0
$599$	11.0829	0.452836	0.226418	−	0.974030i	$-0.427299\pi$
$599$	11.0829	0.452836	0.226418	+	0.974030i	$0.427299\pi$
$600$	0	0
$601$	−24.4568	−0.997612	−0.498806	−	0.866714i	$-0.666228\pi$
$601$	−24.4568	−0.997612	−0.498806	+	0.866714i	$0.666228\pi$
$602$	0	0
$603$	−0.161315	−0.00656925
$604$	0	0
$605$	−3.55681	−0.144605
$606$	0	0
$607$	27.1293	1.10114	0.550571	−	0.834788i	$-0.314410\pi$
$607$	27.1293	1.10114	0.550571	+	0.834788i	$0.314410\pi$
$608$	0	0
$609$	−3.31048	−0.134147
$610$	0	0
$611$	−3.76297	−0.152234
$612$	0	0
$613$	−14.0155	−0.566080	−0.283040	−	0.959108i	$-0.591343\pi$
$613$	−14.0155	−0.566080	−0.283040	+	0.959108i	$0.591343\pi$
$614$	0	0
$615$	10.5685	0.426162
$616$	0	0
$617$	46.4769	1.87109	0.935544	−	0.353211i	$-0.114910\pi$
$617$	46.4769	1.87109	0.935544	+	0.353211i	$0.114910\pi$
$618$	0	0
$619$	−4.37657	−0.175909	−0.0879546	−	0.996124i	$-0.528033\pi$
$619$	−4.37657	−0.175909	−0.0879546	+	0.996124i	$0.528033\pi$
$620$	0	0
$621$	3.89427	0.156272
$622$	0	0
$623$	2.79072	0.111808
$624$	0	0
$625$	−4.71798	−0.188719
$626$	0	0
$627$	0.612110	0.0244453
$628$	0	0
$629$	−0.206661	−0.00824010
$630$	0	0
$631$	−3.49502	−0.139134	−0.0695672	−	0.997577i	$-0.522162\pi$
$631$	−3.49502	−0.139134	−0.0695672	+	0.997577i	$0.522162\pi$
$632$	0	0
$633$	−7.42195	−0.294996
$634$	0	0
$635$	30.2890	1.20198
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−8.31553	−0.328957
$640$	0	0
$641$	−10.3188	−0.407566	−0.203783	−	0.979016i	$-0.565324\pi$
$641$	−10.3188	−0.407566	−0.203783	+	0.979016i	$0.565324\pi$
$642$	0	0
$643$	29.4330	1.16072	0.580362	−	0.814358i	$-0.302911\pi$
$643$	29.4330	1.16072	0.580362	+	0.814358i	$0.302911\pi$
$644$	0	0
$645$	−5.07880	−0.199978
$646$	0	0
$647$	−11.9358	−0.469245	−0.234623	−	0.972087i	$-0.575385\pi$
$647$	−11.9358	−0.469245	−0.234623	+	0.972087i	$0.575385\pi$
$648$	0	0
$649$	7.89174	0.309778
$650$	0	0
$651$	−1.27812	−0.0500934
$652$	0	0
$653$	12.2405	0.479007	0.239504	−	0.970895i	$-0.423015\pi$
$653$	12.2405	0.479007	0.239504	+	0.970895i	$0.423015\pi$
$654$	0	0
$655$	57.1406	2.23267
$656$	0	0
$657$	−18.2699	−0.712775
$658$	0	0
$659$	−7.76991	−0.302673	−0.151336	−	0.988482i	$-0.548358\pi$
$659$	−7.76991	−0.302673	−0.151336	+	0.988482i	$0.548358\pi$
$660$	0	0
$661$	−6.70746	−0.260890	−0.130445	−	0.991456i	$-0.541641\pi$
$661$	−6.70746	−0.260890	−0.130445	+	0.991456i	$0.541641\pi$
$662$	0	0
$663$	2.44098	0.0947997
$664$	0	0
$665$	−5.56896	−0.215955
$666$	0	0
$667$	14.4258	0.558568
$668$	0	0
$669$	−4.25949	−0.164681
$670$	0	0
$671$	1.11331	0.0429789
$672$	0	0
$673$	23.6939	0.913332	0.456666	−	0.889638i	$-0.349043\pi$
$673$	23.6939	0.913332	0.456666	+	0.889638i	$0.349043\pi$
$674$	0	0
$675$	17.4894	0.673168
$676$	0	0
$677$	−9.60994	−0.369340	−0.184670	−	0.982801i	$-0.559122\pi$
$677$	−9.60994	−0.369340	−0.184670	+	0.982801i	$0.559122\pi$
$678$	0	0
$679$	1.64308	0.0630557
$680$	0	0
$681$	6.53347	0.250363
$682$	0	0
$683$	−19.6527	−0.751991	−0.375996	−	0.926621i	$-0.622699\pi$
$683$	−19.6527	−0.751991	−0.375996	+	0.926621i	$0.622699\pi$
$684$	0	0
$685$	61.2214	2.33915
$686$	0	0
$687$	−1.95759	−0.0746865
$688$	0	0
$689$	5.93611	0.226148
$690$	0	0
$691$	39.6040	1.50661	0.753304	−	0.657672i	$-0.228459\pi$
$691$	39.6040	1.50661	0.753304	+	0.657672i	$0.228459\pi$
$692$	0	0
$693$	2.84716	0.108155
$694$	0	0
$695$	−1.55248	−0.0588890
$696$	0	0
$697$	−47.4549	−1.79748
$698$	0	0
$699$	3.31413	0.125352
$700$	0	0
$701$	32.4284	1.22480	0.612402	−	0.790547i	$-0.290204\pi$
$701$	32.4284	1.22480	0.612402	+	0.790547i	$0.290204\pi$
$702$	0	0
$703$	0.0518231	0.00195455
$704$	0	0
$705$	−5.23250	−0.197067
$706$	0	0
$707$	7.98664	0.300369
$708$	0	0
$709$	−17.2676	−0.648498	−0.324249	−	0.945972i	$-0.605112\pi$
$709$	−17.2676	−0.648498	−0.324249	+	0.945972i	$0.605112\pi$
$710$	0	0
$711$	−10.6949	−0.401091
$712$	0	0
$713$	5.56954	0.208581
$714$	0	0
$715$	3.55681	0.133017
$716$	0	0
$717$	−3.89012	−0.145279
$718$	0	0
$719$	−13.4389	−0.501187	−0.250593	−	0.968092i	$-0.580626\pi$
$719$	−13.4389	−0.501187	−0.250593	+	0.968092i	$0.580626\pi$
$720$	0	0
$721$	−5.60451	−0.208723
$722$	0	0
$723$	−2.69826	−0.100350
$724$	0	0
$725$	64.7871	2.40613
$726$	0	0
$727$	−3.56478	−0.132210	−0.0661052	−	0.997813i	$-0.521057\pi$
$727$	−3.56478	−0.132210	−0.0661052	+	0.997813i	$0.521057\pi$
$728$	0	0
$729$	−19.0935	−0.707168
$730$	0	0
$731$	22.8050	0.843474
$732$	0	0
$733$	−37.8334	−1.39741	−0.698704	−	0.715410i	$-0.746240\pi$
$733$	−37.8334	−1.39741	−0.698704	+	0.715410i	$0.746240\pi$
$734$	0	0
$735$	1.39052	0.0512902
$736$	0	0
$737$	−0.0566581	−0.00208703
$738$	0	0
$739$	7.17042	0.263768	0.131884	−	0.991265i	$-0.457897\pi$
$739$	7.17042	0.263768	0.131884	+	0.991265i	$0.457897\pi$
$740$	0	0
$741$	−0.612110	−0.0224864
$742$	0	0
$743$	−34.2394	−1.25612	−0.628061	−	0.778164i	$-0.716151\pi$
$743$	−34.2394	−1.25612	−0.628061	+	0.778164i	$0.716151\pi$
$744$	0	0
$745$	16.3284	0.598226
$746$	0	0
$747$	4.39090	0.160655
$748$	0	0
$749$	10.0463	0.367083
$750$	0	0
$751$	−33.9726	−1.23968	−0.619839	−	0.784729i	$-0.712802\pi$
$751$	−33.9726	−1.23968	−0.619839	+	0.784729i	$0.712802\pi$
$752$	0	0
$753$	3.07992	0.112239
$754$	0	0
$755$	−79.3920	−2.88937
$756$	0	0
$757$	9.16751	0.333199	0.166599	−	0.986025i	$-0.446721\pi$
$757$	9.16751	0.333199	0.166599	+	0.986025i	$0.446721\pi$
$758$	0	0
$759$	0.666011	0.0241747
$760$	0	0
$761$	5.00798	0.181539	0.0907695	−	0.995872i	$-0.471067\pi$
$761$	5.00798	0.181539	0.0907695	+	0.995872i	$0.471067\pi$
$762$	0	0
$763$	−11.7659	−0.425954
$764$	0	0
$765$	−63.2296	−2.28607
$766$	0	0
$767$	−7.89174	−0.284954
$768$	0	0
$769$	−23.6611	−0.853242	−0.426621	−	0.904430i	$-0.640296\pi$
$769$	−23.6611	−0.853242	−0.426621	+	0.904430i	$0.640296\pi$
$770$	0	0
$771$	−0.388209	−0.0139810
$772$	0	0
$773$	1.37267	0.0493714	0.0246857	−	0.999695i	$-0.492142\pi$
$773$	1.37267	0.0493714	0.0246857	+	0.999695i	$0.492142\pi$
$774$	0	0
$775$	25.0131	0.898499
$776$	0	0
$777$	−0.0129398	−0.000464212 0
$778$	0	0
$779$	11.9000	0.426362
$780$	0	0
$781$	−2.92064	−0.104509
$782$	0	0
$783$	19.3569	0.691760
$784$	0	0
$785$	−30.2637	−1.08016
$786$	0	0
$787$	2.23550	0.0796869	0.0398435	−	0.999206i	$-0.487314\pi$
$787$	2.23550	0.0796869	0.0398435	+	0.999206i	$0.487314\pi$
$788$	0	0
$789$	5.09250	0.181298
$790$	0	0
$791$	−17.1689	−0.610454
$792$	0	0
$793$	−1.11331	−0.0395349
$794$	0	0
$795$	8.25429	0.292749
$796$	0	0
$797$	17.2871	0.612342	0.306171	−	0.951977i	$-0.400952\pi$
$797$	17.2871	0.612342	0.306171	+	0.951977i	$0.400952\pi$
$798$	0	0
$799$	23.4952	0.831199
$800$	0	0
$801$	−7.94563	−0.280745
$802$	0	0
$803$	−6.41687	−0.226446
$804$	0	0
$805$	−6.05935	−0.213564
$806$	0	0
$807$	7.69270	0.270796
$808$	0	0
$809$	14.8925	0.523591	0.261796	−	0.965123i	$-0.415685\pi$
$809$	14.8925	0.523591	0.261796	+	0.965123i	$0.415685\pi$
$810$	0	0
$811$	54.5443	1.91531	0.957655	−	0.287919i	$-0.0929634\pi$
$811$	54.5443	1.91531	0.957655	+	0.287919i	$0.0929634\pi$
$812$	0	0
$813$	1.35753	0.0476107
$814$	0	0
$815$	22.7511	0.796936
$816$	0	0
$817$	−5.71869	−0.200072
$818$	0	0
$819$	−2.84716	−0.0994879
$820$	0	0
$821$	−33.5365	−1.17043	−0.585216	−	0.810877i	$-0.698990\pi$
$821$	−33.5365	−1.17043	−0.585216	+	0.810877i	$0.698990\pi$
$822$	0	0
$823$	−26.3349	−0.917977	−0.458989	−	0.888442i	$-0.651788\pi$
$823$	−26.3349	−0.917977	−0.458989	+	0.888442i	$0.651788\pi$
$824$	0	0
$825$	2.99110	0.104137
$826$	0	0
$827$	−7.62451	−0.265130	−0.132565	−	0.991174i	$-0.542321\pi$
$827$	−7.62451	−0.265130	−0.132565	+	0.991174i	$0.542321\pi$
$828$	0	0
$829$	−14.9670	−0.519827	−0.259914	−	0.965632i	$-0.583694\pi$
$829$	−14.9670	−0.519827	−0.259914	+	0.965632i	$0.583694\pi$
$830$	0	0
$831$	−7.48779	−0.259749
$832$	0	0
$833$	−6.24378	−0.216334
$834$	0	0
$835$	51.5129	1.78268
$836$	0	0
$837$	7.47336	0.258317
$838$	0	0
$839$	−32.8696	−1.13479	−0.567393	−	0.823447i	$-0.692048\pi$
$839$	−32.8696	−1.13479	−0.567393	+	0.823447i	$0.692048\pi$
$840$	0	0
$841$	42.7049	1.47258
$842$	0	0
$843$	10.6941	0.368323
$844$	0	0
$845$	−3.55681	−0.122358
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−9.56658	−0.328324
$850$	0	0
$851$	0.0563865	0.00193290
$852$	0	0
$853$	−10.4809	−0.358858	−0.179429	−	0.983771i	$-0.557425\pi$
$853$	−10.4809	−0.358858	−0.179429	+	0.983771i	$0.557425\pi$
$854$	0	0
$855$	15.8557	0.542255
$856$	0	0
$857$	−15.1300	−0.516830	−0.258415	−	0.966034i	$-0.583200\pi$
$857$	−15.1300	−0.516830	−0.258415	+	0.966034i	$0.583200\pi$
$858$	0	0
$859$	17.2888	0.589886	0.294943	−	0.955515i	$-0.404699\pi$
$859$	17.2888	0.589886	0.294943	+	0.955515i	$0.404699\pi$
$860$	0	0
$861$	−2.97133	−0.101263
$862$	0	0
$863$	28.2712	0.962362	0.481181	−	0.876621i	$-0.340208\pi$
$863$	28.2712	0.962362	0.481181	+	0.876621i	$0.340208\pi$
$864$	0	0
$865$	76.9543	2.61652
$866$	0	0
$867$	−8.59484	−0.291896
$868$	0	0
$869$	−3.75635	−0.127425
$870$	0	0
$871$	0.0566581	0.00191979
$872$	0	0
$873$	−4.67812	−0.158330
$874$	0	0
$875$	−9.42884	−0.318753
$876$	0	0
$877$	−49.2407	−1.66274	−0.831371	−	0.555718i	$-0.812443\pi$
$877$	−49.2407	−1.66274	−0.831371	+	0.555718i	$0.812443\pi$
$878$	0	0
$879$	7.41467	0.250091
$880$	0	0
$881$	−24.8847	−0.838387	−0.419193	−	0.907897i	$-0.637687\pi$
$881$	−24.8847	−0.838387	−0.419193	+	0.907897i	$0.637687\pi$
$882$	0	0
$883$	48.0225	1.61609	0.808043	−	0.589124i	$-0.200527\pi$
$883$	48.0225	1.61609	0.808043	+	0.589124i	$0.200527\pi$
$884$	0	0
$885$	−10.9736	−0.368875
$886$	0	0
$887$	−2.78187	−0.0934059	−0.0467030	−	0.998909i	$-0.514871\pi$
$887$	−2.78187	−0.0934059	−0.0467030	+	0.998909i	$0.514871\pi$
$888$	0	0
$889$	−8.51577	−0.285610
$890$	0	0
$891$	−7.64781	−0.256211
$892$	0	0
$893$	−5.89175	−0.197160
$894$	0	0
$895$	−59.9033	−2.00235
$896$	0	0
$897$	−0.666011	−0.0222375
$898$	0	0
$899$	27.6840	0.923313
$900$	0	0
$901$	−37.0637	−1.23477
$902$	0	0
$903$	1.42791	0.0475178
$904$	0	0
$905$	−9.30124	−0.309184
$906$	0	0
$907$	−11.0419	−0.366639	−0.183320	−	0.983053i	$-0.558684\pi$
$907$	−11.0419	−0.366639	−0.183320	+	0.983053i	$0.558684\pi$
$908$	0	0
$909$	−22.7393	−0.754214
$910$	0	0
$911$	−8.95807	−0.296794	−0.148397	−	0.988928i	$-0.547411\pi$
$911$	−8.95807	−0.296794	−0.148397	+	0.988928i	$0.547411\pi$
$912$	0	0
$913$	1.54220	0.0510395
$914$	0	0
$915$	−1.54808	−0.0511781
$916$	0	0
$917$	−16.0651	−0.530517
$918$	0	0
$919$	1.42000	0.0468414	0.0234207	−	0.999726i	$-0.492544\pi$
$919$	1.42000	0.0468414	0.0234207	+	0.999726i	$0.492544\pi$
$920$	0	0
$921$	3.93386	0.129625
$922$	0	0
$923$	2.92064	0.0961341
$924$	0	0
$925$	0.253235	0.00832633
$926$	0	0
$927$	15.9570	0.524095
$928$	0	0
$929$	18.4390	0.604964	0.302482	−	0.953155i	$-0.402185\pi$
$929$	18.4390	0.604964	0.302482	+	0.953155i	$0.402185\pi$
$930$	0	0
$931$	1.56572	0.0513143
$932$	0	0
$933$	−3.77098	−0.123456
$934$	0	0
$935$	−22.2079	−0.726277
$936$	0	0
$937$	−31.5917	−1.03206	−0.516029	−	0.856571i	$-0.672590\pi$
$937$	−31.5917	−1.03206	−0.516029	+	0.856571i	$0.672590\pi$
$938$	0	0
$939$	9.26462	0.302339
$940$	0	0
$941$	−25.5841	−0.834016	−0.417008	−	0.908903i	$-0.636921\pi$
$941$	−25.5841	−0.834016	−0.417008	+	0.908903i	$0.636921\pi$
$942$	0	0
$943$	12.9479	0.421641
$944$	0	0
$945$	−8.13060	−0.264489
$946$	0	0
$947$	25.3322	0.823187	0.411593	−	0.911368i	$-0.364972\pi$
$947$	25.3322	0.823187	0.411593	+	0.911368i	$0.364972\pi$
$948$	0	0
$949$	6.41687	0.208300
$950$	0	0
$951$	2.42686	0.0786962
$952$	0	0
$953$	28.3728	0.919086	0.459543	−	0.888155i	$-0.348013\pi$
$953$	28.3728	0.919086	0.459543	+	0.888155i	$0.348013\pi$
$954$	0	0
$955$	−21.2139	−0.686465
$956$	0	0
$957$	3.31048	0.107013
$958$	0	0
$959$	−17.2124	−0.555818
$960$	0	0
$961$	−20.3117	−0.655216
$962$	0	0
$963$	−28.6034	−0.921732
$964$	0	0
$965$	29.1547	0.938524
$966$	0	0
$967$	41.3701	1.33037	0.665187	−	0.746677i	$-0.268352\pi$
$967$	41.3701	1.33037	0.665187	+	0.746677i	$0.268352\pi$
$968$	0	0
$969$	3.82188	0.122776
$970$	0	0
$971$	17.4953	0.561451	0.280726	−	0.959788i	$-0.409425\pi$
$971$	17.4953	0.561451	0.280726	+	0.959788i	$0.409425\pi$
$972$	0	0
$973$	0.436481	0.0139929
$974$	0	0
$975$	−2.99110	−0.0957918
$976$	0	0
$977$	27.7964	0.889286	0.444643	−	0.895708i	$-0.353331\pi$
$977$	27.7964	0.889286	0.444643	+	0.895708i	$0.353331\pi$
$978$	0	0
$979$	−2.79072	−0.0891918
$980$	0	0
$981$	33.4994	1.06955
$982$	0	0
$983$	28.8755	0.920987	0.460493	−	0.887663i	$-0.347673\pi$
$983$	28.8755	0.920987	0.460493	+	0.887663i	$0.347673\pi$
$984$	0	0
$985$	−72.6722	−2.31553
$986$	0	0
$987$	1.47112	0.0468262
$988$	0	0
$989$	−6.22226	−0.197856
$990$	0	0
$991$	−33.8170	−1.07423	−0.537117	−	0.843508i	$-0.680486\pi$
$991$	−33.8170	−1.07423	−0.537117	+	0.843508i	$0.680486\pi$
$992$	0	0
$993$	−8.16821	−0.259210
$994$	0	0
$995$	96.5167	3.05979
$996$	0	0
$997$	−58.3957	−1.84941	−0.924705	−	0.380684i	$-0.875689\pi$
$997$	−58.3957	−1.84941	−0.924705	+	0.380684i	$0.875689\pi$
$998$	0	0
$999$	0.0756610	0.00239381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.o.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.o.1.5	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.390946 q^{3} -3.55681 q^{5} +1.00000 q^{7} -2.84716 q^{9} +O(q^{10})\)	\(q-0.390946 q^{3} -3.55681 q^{5} +1.00000 q^{7} -2.84716 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.39052 q^{15} -6.24378 q^{17} +1.56572 q^{19} -0.390946 q^{21} +1.70359 q^{23} +7.65092 q^{25} +2.28592 q^{27} +8.46788 q^{29} +3.26930 q^{31} +0.390946 q^{33} -3.55681 q^{35} +0.0330987 q^{37} -0.390946 q^{39} +7.60036 q^{41} -3.65244 q^{43} +10.1268 q^{45} -3.76297 q^{47} +1.00000 q^{49} +2.44098 q^{51} +5.93611 q^{53} +3.55681 q^{55} -0.612110 q^{57} -7.89174 q^{59} -1.11331 q^{61} -2.84716 q^{63} -3.55681 q^{65} +0.0566581 q^{67} -0.666011 q^{69} +2.92064 q^{71} +6.41687 q^{73} -2.99110 q^{75} -1.00000 q^{77} +3.75635 q^{79} +7.64781 q^{81} -1.54220 q^{83} +22.2079 q^{85} -3.31048 q^{87} +2.79072 q^{89} +1.00000 q^{91} -1.27812 q^{93} -5.56896 q^{95} +1.64308 q^{97} +2.84716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more