LMFDB - Embedded newform 8004.2.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.309856$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.309856 q^{5} -1.15608 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.309856 q^{5} -1.15608 q^{7} +1.00000 q^{9} +5.92157 q^{11} -4.60261 q^{13} +0.309856 q^{15} -7.33718 q^{17} -3.96569 q^{19} +1.15608 q^{21} +1.00000 q^{23} -4.90399 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.27559 q^{31} -5.92157 q^{33} +0.358219 q^{35} +2.80212 q^{37} +4.60261 q^{39} -4.27213 q^{41} +5.11611 q^{43} -0.309856 q^{45} +3.04361 q^{47} -5.66347 q^{49} +7.33718 q^{51} -13.6612 q^{53} -1.83483 q^{55} +3.96569 q^{57} -0.903422 q^{59} +11.0272 q^{61} -1.15608 q^{63} +1.42615 q^{65} +7.41964 q^{67} -1.00000 q^{69} +3.44912 q^{71} -4.89506 q^{73} +4.90399 q^{75} -6.84582 q^{77} +14.5506 q^{79} +1.00000 q^{81} -15.6780 q^{83} +2.27347 q^{85} -1.00000 q^{87} +5.84517 q^{89} +5.32099 q^{91} -7.27559 q^{93} +1.22879 q^{95} +5.92869 q^{97} +5.92157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.309856	−0.138572	−0.0692859	−	0.997597i	$-0.522072\pi$
$5$	−0.309856	−0.138572	−0.0692859	+	0.997597i	$0.522072\pi$
$6$	0	0
$7$	−1.15608	−0.436958	−0.218479	−	0.975842i	$-0.570110\pi$
$7$	−1.15608	−0.436958	−0.218479	+	0.975842i	$0.570110\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.92157	1.78542	0.892710	−	0.450632i	$-0.148802\pi$
$11$	5.92157	1.78542	0.892710	+	0.450632i	$0.148802\pi$
$12$	0	0
$13$	−4.60261	−1.27653	−0.638267	−	0.769815i	$-0.720348\pi$
$13$	−4.60261	−1.27653	−0.638267	+	0.769815i	$0.720348\pi$
$14$	0	0
$15$	0.309856	0.0800045
$16$	0	0
$17$	−7.33718	−1.77953	−0.889764	−	0.456421i	$-0.849131\pi$
$17$	−7.33718	−1.77953	−0.889764	+	0.456421i	$0.849131\pi$
$18$	0	0
$19$	−3.96569	−0.909791	−0.454896	−	0.890545i	$-0.650323\pi$
$19$	−3.96569	−0.909791	−0.454896	+	0.890545i	$0.650323\pi$
$20$	0	0
$21$	1.15608	0.252278
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.90399	−0.980798
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.27559	1.30673	0.653367	−	0.757041i	$-0.273356\pi$
$31$	7.27559	1.30673	0.653367	+	0.757041i	$0.273356\pi$
$32$	0	0
$33$	−5.92157	−1.03081
$34$	0	0
$35$	0.358219	0.0605501
$36$	0	0
$37$	2.80212	0.460666	0.230333	−	0.973112i	$-0.426018\pi$
$37$	2.80212	0.460666	0.230333	+	0.973112i	$0.426018\pi$
$38$	0	0
$39$	4.60261	0.737007
$40$	0	0
$41$	−4.27213	−0.667194	−0.333597	−	0.942716i	$-0.608263\pi$
$41$	−4.27213	−0.667194	−0.333597	+	0.942716i	$0.608263\pi$
$42$	0	0
$43$	5.11611	0.780199	0.390100	−	0.920773i	$-0.372441\pi$
$43$	5.11611	0.780199	0.390100	+	0.920773i	$0.372441\pi$
$44$	0	0
$45$	−0.309856	−0.0461906
$46$	0	0
$47$	3.04361	0.443956	0.221978	−	0.975052i	$-0.428749\pi$
$47$	3.04361	0.443956	0.221978	+	0.975052i	$0.428749\pi$
$48$	0	0
$49$	−5.66347	−0.809068
$50$	0	0
$51$	7.33718	1.02741
$52$	0	0
$53$	−13.6612	−1.87652	−0.938258	−	0.345936i	$-0.887561\pi$
$53$	−13.6612	−1.87652	−0.938258	+	0.345936i	$0.887561\pi$
$54$	0	0
$55$	−1.83483	−0.247409
$56$	0	0
$57$	3.96569	0.525268
$58$	0	0
$59$	−0.903422	−0.117615	−0.0588077	−	0.998269i	$-0.518730\pi$
$59$	−0.903422	−0.117615	−0.0588077	+	0.998269i	$0.518730\pi$
$60$	0	0
$61$	11.0272	1.41189	0.705946	−	0.708266i	$-0.250522\pi$
$61$	11.0272	1.41189	0.705946	+	0.708266i	$0.250522\pi$
$62$	0	0
$63$	−1.15608	−0.145653
$64$	0	0
$65$	1.42615	0.176892
$66$	0	0
$67$	7.41964	0.906454	0.453227	−	0.891395i	$-0.350273\pi$
$67$	7.41964	0.906454	0.453227	+	0.891395i	$0.350273\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	3.44912	0.409335	0.204667	−	0.978832i	$-0.434389\pi$
$71$	3.44912	0.409335	0.204667	+	0.978832i	$0.434389\pi$
$72$	0	0
$73$	−4.89506	−0.572923	−0.286461	−	0.958092i	$-0.592479\pi$
$73$	−4.89506	−0.572923	−0.286461	+	0.958092i	$0.592479\pi$
$74$	0	0
$75$	4.90399	0.566264
$76$	0	0
$77$	−6.84582	−0.780153
$78$	0	0
$79$	14.5506	1.63707	0.818535	−	0.574456i	$-0.194786\pi$
$79$	14.5506	1.63707	0.818535	+	0.574456i	$0.194786\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.6780	−1.72088	−0.860441	−	0.509550i	$-0.829812\pi$
$83$	−15.6780	−1.72088	−0.860441	+	0.509550i	$0.829812\pi$
$84$	0	0
$85$	2.27347	0.246593
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	5.84517	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$89$	5.84517	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$90$	0	0
$91$	5.32099	0.557792
$92$	0	0
$93$	−7.27559	−0.754443
$94$	0	0
$95$	1.22879	0.126071
$96$	0	0
$97$	5.92869	0.601967	0.300984	−	0.953629i	$-0.402685\pi$
$97$	5.92869	0.601967	0.300984	+	0.953629i	$0.402685\pi$
$98$	0	0
$99$	5.92157	0.595140
$100$	0	0
$101$	10.5052	1.04531	0.522655	−	0.852545i	$-0.324942\pi$
$101$	10.5052	1.04531	0.522655	+	0.852545i	$0.324942\pi$
$102$	0	0
$103$	−4.57362	−0.450652	−0.225326	−	0.974283i	$-0.572345\pi$
$103$	−4.57362	−0.450652	−0.225326	+	0.974283i	$0.572345\pi$
$104$	0	0
$105$	−0.358219	−0.0349586
$106$	0	0
$107$	−6.80910	−0.658261	−0.329130	−	0.944284i	$-0.606756\pi$
$107$	−6.80910	−0.658261	−0.329130	+	0.944284i	$0.606756\pi$
$108$	0	0
$109$	2.49451	0.238931	0.119465	−	0.992838i	$-0.461882\pi$
$109$	2.49451	0.238931	0.119465	+	0.992838i	$0.461882\pi$
$110$	0	0
$111$	−2.80212	−0.265966
$112$	0	0
$113$	9.51872	0.895446	0.447723	−	0.894172i	$-0.352235\pi$
$113$	9.51872	0.895446	0.447723	+	0.894172i	$0.352235\pi$
$114$	0	0
$115$	−0.309856	−0.0288942
$116$	0	0
$117$	−4.60261	−0.425511
$118$	0	0
$119$	8.48238	0.777579
$120$	0	0
$121$	24.0649	2.18772
$122$	0	0
$123$	4.27213	0.385205
$124$	0	0
$125$	3.06881	0.274483
$126$	0	0
$127$	−17.2784	−1.53321	−0.766604	−	0.642120i	$-0.778055\pi$
$127$	−17.2784	−1.53321	−0.766604	+	0.642120i	$0.778055\pi$
$128$	0	0
$129$	−5.11611	−0.450448
$130$	0	0
$131$	8.59137	0.750632	0.375316	−	0.926897i	$-0.377534\pi$
$131$	8.59137	0.750632	0.375316	+	0.926897i	$0.377534\pi$
$132$	0	0
$133$	4.58466	0.397540
$134$	0	0
$135$	0.309856	0.0266682
$136$	0	0
$137$	4.88945	0.417734	0.208867	−	0.977944i	$-0.433022\pi$
$137$	4.88945	0.417734	0.208867	+	0.977944i	$0.433022\pi$
$138$	0	0
$139$	−20.2173	−1.71481	−0.857404	−	0.514644i	$-0.827924\pi$
$139$	−20.2173	−1.71481	−0.857404	+	0.514644i	$0.827924\pi$
$140$	0	0
$141$	−3.04361	−0.256318
$142$	0	0
$143$	−27.2547	−2.27915
$144$	0	0
$145$	−0.309856	−0.0257322
$146$	0	0
$147$	5.66347	0.467115
$148$	0	0
$149$	−12.2316	−1.00205	−0.501025	−	0.865433i	$-0.667043\pi$
$149$	−12.2316	−1.00205	−0.501025	+	0.865433i	$0.667043\pi$
$150$	0	0
$151$	13.3244	1.08433	0.542164	−	0.840273i	$-0.317605\pi$
$151$	13.3244	1.08433	0.542164	+	0.840273i	$0.317605\pi$
$152$	0	0
$153$	−7.33718	−0.593176
$154$	0	0
$155$	−2.25439	−0.181077
$156$	0	0
$157$	9.60039	0.766194	0.383097	−	0.923708i	$-0.374857\pi$
$157$	9.60039	0.766194	0.383097	+	0.923708i	$0.374857\pi$
$158$	0	0
$159$	13.6612	1.08341
$160$	0	0
$161$	−1.15608	−0.0911120
$162$	0	0
$163$	10.2315	0.801396	0.400698	−	0.916210i	$-0.368768\pi$
$163$	10.2315	0.801396	0.400698	+	0.916210i	$0.368768\pi$
$164$	0	0
$165$	1.83483	0.142842
$166$	0	0
$167$	5.87711	0.454784	0.227392	−	0.973803i	$-0.426980\pi$
$167$	5.87711	0.454784	0.227392	+	0.973803i	$0.426980\pi$
$168$	0	0
$169$	8.18401	0.629539
$170$	0	0
$171$	−3.96569	−0.303264
$172$	0	0
$173$	12.3777	0.941057	0.470529	−	0.882385i	$-0.344063\pi$
$173$	12.3777	0.941057	0.470529	+	0.882385i	$0.344063\pi$
$174$	0	0
$175$	5.66941	0.428567
$176$	0	0
$177$	0.903422	0.0679053
$178$	0	0
$179$	−16.0178	−1.19722	−0.598612	−	0.801039i	$-0.704281\pi$
$179$	−16.0178	−1.19722	−0.598612	+	0.801039i	$0.704281\pi$
$180$	0	0
$181$	−2.94379	−0.218810	−0.109405	−	0.993997i	$-0.534895\pi$
$181$	−2.94379	−0.218810	−0.109405	+	0.993997i	$0.534895\pi$
$182$	0	0
$183$	−11.0272	−0.815156
$184$	0	0
$185$	−0.868255	−0.0638354
$186$	0	0
$187$	−43.4476	−3.17720
$188$	0	0
$189$	1.15608	0.0840926
$190$	0	0
$191$	22.1472	1.60252	0.801259	−	0.598317i	$-0.204164\pi$
$191$	22.1472	1.60252	0.801259	+	0.598317i	$0.204164\pi$
$192$	0	0
$193$	9.91772	0.713893	0.356947	−	0.934125i	$-0.383818\pi$
$193$	9.91772	0.713893	0.356947	+	0.934125i	$0.383818\pi$
$194$	0	0
$195$	−1.42615	−0.102128
$196$	0	0
$197$	2.07980	0.148179	0.0740897	−	0.997252i	$-0.476395\pi$
$197$	2.07980	0.148179	0.0740897	+	0.997252i	$0.476395\pi$
$198$	0	0
$199$	−11.4598	−0.812361	−0.406181	−	0.913793i	$-0.633140\pi$
$199$	−11.4598	−0.812361	−0.406181	+	0.913793i	$0.633140\pi$
$200$	0	0
$201$	−7.41964	−0.523341
$202$	0	0
$203$	−1.15608	−0.0811411
$204$	0	0
$205$	1.32375	0.0924544
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−23.4831	−1.62436
$210$	0	0
$211$	−10.7296	−0.738659	−0.369329	−	0.929299i	$-0.620413\pi$
$211$	−10.7296	−0.738659	−0.369329	+	0.929299i	$0.620413\pi$
$212$	0	0
$213$	−3.44912	−0.236330
$214$	0	0
$215$	−1.58526	−0.108114
$216$	0	0
$217$	−8.41118	−0.570988
$218$	0	0
$219$	4.89506	0.330777
$220$	0	0
$221$	33.7702	2.27163
$222$	0	0
$223$	−6.24882	−0.418452	−0.209226	−	0.977867i	$-0.567094\pi$
$223$	−6.24882	−0.418452	−0.209226	+	0.977867i	$0.567094\pi$
$224$	0	0
$225$	−4.90399	−0.326933
$226$	0	0
$227$	23.1179	1.53439	0.767196	−	0.641413i	$-0.221651\pi$
$227$	23.1179	1.53439	0.767196	+	0.641413i	$0.221651\pi$
$228$	0	0
$229$	−6.58061	−0.434859	−0.217429	−	0.976076i	$-0.569767\pi$
$229$	−6.58061	−0.434859	−0.217429	+	0.976076i	$0.569767\pi$
$230$	0	0
$231$	6.84582	0.450422
$232$	0	0
$233$	−16.8724	−1.10535	−0.552673	−	0.833398i	$-0.686392\pi$
$233$	−16.8724	−1.10535	−0.552673	+	0.833398i	$0.686392\pi$
$234$	0	0
$235$	−0.943081	−0.0615198
$236$	0	0
$237$	−14.5506	−0.945163
$238$	0	0
$239$	22.1356	1.43183	0.715915	−	0.698187i	$-0.246010\pi$
$239$	22.1356	1.43183	0.715915	+	0.698187i	$0.246010\pi$
$240$	0	0
$241$	−4.15266	−0.267496	−0.133748	−	0.991015i	$-0.542701\pi$
$241$	−4.15266	−0.267496	−0.133748	+	0.991015i	$0.542701\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.75486	0.112114
$246$	0	0
$247$	18.2525	1.16138
$248$	0	0
$249$	15.6780	0.993552
$250$	0	0
$251$	10.8013	0.681769	0.340885	−	0.940105i	$-0.389273\pi$
$251$	10.8013	0.681769	0.340885	+	0.940105i	$0.389273\pi$
$252$	0	0
$253$	5.92157	0.372286
$254$	0	0
$255$	−2.27347	−0.142370
$256$	0	0
$257$	27.1696	1.69479	0.847397	−	0.530959i	$-0.178168\pi$
$257$	27.1696	1.69479	0.847397	+	0.530959i	$0.178168\pi$
$258$	0	0
$259$	−3.23949	−0.201292
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−17.0608	−1.05202	−0.526008	−	0.850480i	$-0.676312\pi$
$263$	−17.0608	−1.05202	−0.526008	+	0.850480i	$0.676312\pi$
$264$	0	0
$265$	4.23302	0.260032
$266$	0	0
$267$	−5.84517	−0.357718
$268$	0	0
$269$	−1.06857	−0.0651521	−0.0325761	−	0.999469i	$-0.510371\pi$
$269$	−1.06857	−0.0651521	−0.0325761	+	0.999469i	$0.510371\pi$
$270$	0	0
$271$	−29.0640	−1.76551	−0.882756	−	0.469832i	$-0.844314\pi$
$271$	−29.0640	−1.76551	−0.882756	+	0.469832i	$0.844314\pi$
$272$	0	0
$273$	−5.32099	−0.322041
$274$	0	0
$275$	−29.0393	−1.75114
$276$	0	0
$277$	−19.3666	−1.16363	−0.581813	−	0.813323i	$-0.697656\pi$
$277$	−19.3666	−1.16363	−0.581813	+	0.813323i	$0.697656\pi$
$278$	0	0
$279$	7.27559	0.435578
$280$	0	0
$281$	−10.4904	−0.625808	−0.312904	−	0.949785i	$-0.601302\pi$
$281$	−10.4904	−0.625808	−0.312904	+	0.949785i	$0.601302\pi$
$282$	0	0
$283$	5.42301	0.322364	0.161182	−	0.986925i	$-0.448469\pi$
$283$	5.42301	0.322364	0.161182	+	0.986925i	$0.448469\pi$
$284$	0	0
$285$	−1.22879	−0.0727874
$286$	0	0
$287$	4.93893	0.291536
$288$	0	0
$289$	36.8342	2.16672
$290$	0	0
$291$	−5.92869	−0.347546
$292$	0	0
$293$	9.55426	0.558166	0.279083	−	0.960267i	$-0.409970\pi$
$293$	9.55426	0.558166	0.279083	+	0.960267i	$0.409970\pi$
$294$	0	0
$295$	0.279931	0.0162982
$296$	0	0
$297$	−5.92157	−0.343604
$298$	0	0
$299$	−4.60261	−0.266176
$300$	0	0
$301$	−5.91464	−0.340914
$302$	0	0
$303$	−10.5052	−0.603510
$304$	0	0
$305$	−3.41685	−0.195648
$306$	0	0
$307$	−16.7330	−0.955004	−0.477502	−	0.878631i	$-0.658458\pi$
$307$	−16.7330	−0.955004	−0.477502	+	0.878631i	$0.658458\pi$
$308$	0	0
$309$	4.57362	0.260184
$310$	0	0
$311$	22.8155	1.29375	0.646873	−	0.762597i	$-0.276076\pi$
$311$	22.8155	1.29375	0.646873	+	0.762597i	$0.276076\pi$
$312$	0	0
$313$	2.54313	0.143746	0.0718730	−	0.997414i	$-0.477102\pi$
$313$	2.54313	0.143746	0.0718730	+	0.997414i	$0.477102\pi$
$314$	0	0
$315$	0.358219	0.0201834
$316$	0	0
$317$	−16.4581	−0.924378	−0.462189	−	0.886781i	$-0.652936\pi$
$317$	−16.4581	−0.924378	−0.462189	+	0.886781i	$0.652936\pi$
$318$	0	0
$319$	5.92157	0.331544
$320$	0	0
$321$	6.80910	0.380047
$322$	0	0
$323$	29.0970	1.61900
$324$	0	0
$325$	22.5711	1.25202
$326$	0	0
$327$	−2.49451	−0.137947
$328$	0	0
$329$	−3.51866	−0.193990
$330$	0	0
$331$	28.3233	1.55679	0.778394	−	0.627777i	$-0.216035\pi$
$331$	28.3233	1.55679	0.778394	+	0.627777i	$0.216035\pi$
$332$	0	0
$333$	2.80212	0.153555
$334$	0	0
$335$	−2.29902	−0.125609
$336$	0	0
$337$	31.1542	1.69708	0.848539	−	0.529133i	$-0.177483\pi$
$337$	31.1542	1.69708	0.848539	+	0.529133i	$0.177483\pi$
$338$	0	0
$339$	−9.51872	−0.516986
$340$	0	0
$341$	43.0829	2.33307
$342$	0	0
$343$	14.6400	0.790487
$344$	0	0
$345$	0.309856	0.0166821
$346$	0	0
$347$	25.8412	1.38723	0.693615	−	0.720346i	$-0.256017\pi$
$347$	25.8412	1.38723	0.693615	+	0.720346i	$0.256017\pi$
$348$	0	0
$349$	26.8116	1.43519	0.717596	−	0.696459i	$-0.245242\pi$
$349$	26.8116	1.43519	0.717596	+	0.696459i	$0.245242\pi$
$350$	0	0
$351$	4.60261	0.245669
$352$	0	0
$353$	32.1950	1.71357	0.856783	−	0.515678i	$-0.172460\pi$
$353$	32.1950	1.71357	0.856783	+	0.515678i	$0.172460\pi$
$354$	0	0
$355$	−1.06873	−0.0567223
$356$	0	0
$357$	−8.48238	−0.448935
$358$	0	0
$359$	17.4132	0.919035	0.459517	−	0.888169i	$-0.348022\pi$
$359$	17.4132	0.919035	0.459517	+	0.888169i	$0.348022\pi$
$360$	0	0
$361$	−3.27332	−0.172280
$362$	0	0
$363$	−24.0649	−1.26308
$364$	0	0
$365$	1.51676	0.0793910
$366$	0	0
$367$	−2.75102	−0.143602	−0.0718010	−	0.997419i	$-0.522875\pi$
$367$	−2.75102	−0.143602	−0.0718010	+	0.997419i	$0.522875\pi$
$368$	0	0
$369$	−4.27213	−0.222398
$370$	0	0
$371$	15.7935	0.819959
$372$	0	0
$373$	−18.8136	−0.974131	−0.487066	−	0.873365i	$-0.661933\pi$
$373$	−18.8136	−0.974131	−0.487066	+	0.873365i	$0.661933\pi$
$374$	0	0
$375$	−3.06881	−0.158473
$376$	0	0
$377$	−4.60261	−0.237046
$378$	0	0
$379$	17.7700	0.912781	0.456391	−	0.889780i	$-0.349142\pi$
$379$	17.7700	0.912781	0.456391	+	0.889780i	$0.349142\pi$
$380$	0	0
$381$	17.2784	0.885198
$382$	0	0
$383$	14.2681	0.729068	0.364534	−	0.931190i	$-0.381228\pi$
$383$	14.2681	0.729068	0.364534	+	0.931190i	$0.381228\pi$
$384$	0	0
$385$	2.12122	0.108107
$386$	0	0
$387$	5.11611	0.260066
$388$	0	0
$389$	20.0206	1.01509	0.507543	−	0.861626i	$-0.330554\pi$
$389$	20.0206	1.01509	0.507543	+	0.861626i	$0.330554\pi$
$390$	0	0
$391$	−7.33718	−0.371057
$392$	0	0
$393$	−8.59137	−0.433378
$394$	0	0
$395$	−4.50859	−0.226852
$396$	0	0
$397$	29.3789	1.47448	0.737242	−	0.675628i	$-0.236128\pi$
$397$	29.3789	1.47448	0.737242	+	0.675628i	$0.236128\pi$
$398$	0	0
$399$	−4.58466	−0.229520
$400$	0	0
$401$	−16.0068	−0.799342	−0.399671	−	0.916659i	$-0.630876\pi$
$401$	−16.0068	−0.799342	−0.399671	+	0.916659i	$0.630876\pi$
$402$	0	0
$403$	−33.4867	−1.66809
$404$	0	0
$405$	−0.309856	−0.0153969
$406$	0	0
$407$	16.5930	0.822482
$408$	0	0
$409$	−13.7904	−0.681890	−0.340945	−	0.940083i	$-0.610747\pi$
$409$	−13.7904	−0.681890	−0.340945	+	0.940083i	$0.610747\pi$
$410$	0	0
$411$	−4.88945	−0.241179
$412$	0	0
$413$	1.04443	0.0513930
$414$	0	0
$415$	4.85792	0.238466
$416$	0	0
$417$	20.2173	0.990045
$418$	0	0
$419$	−16.2513	−0.793926	−0.396963	−	0.917835i	$-0.629936\pi$
$419$	−16.2513	−0.793926	−0.396963	+	0.917835i	$0.629936\pi$
$420$	0	0
$421$	−1.53538	−0.0748296	−0.0374148	−	0.999300i	$-0.511912\pi$
$421$	−1.53538	−0.0748296	−0.0374148	+	0.999300i	$0.511912\pi$
$422$	0	0
$423$	3.04361	0.147985
$424$	0	0
$425$	35.9815	1.74536
$426$	0	0
$427$	−12.7484	−0.616937
$428$	0	0
$429$	27.2547	1.31587
$430$	0	0
$431$	10.1108	0.487022	0.243511	−	0.969898i	$-0.421701\pi$
$431$	10.1108	0.487022	0.243511	+	0.969898i	$0.421701\pi$
$432$	0	0
$433$	0.641277	0.0308178	0.0154089	−	0.999881i	$-0.495095\pi$
$433$	0.641277	0.0308178	0.0154089	+	0.999881i	$0.495095\pi$
$434$	0	0
$435$	0.309856	0.0148565
$436$	0	0
$437$	−3.96569	−0.189705
$438$	0	0
$439$	−33.3571	−1.59205	−0.796025	−	0.605264i	$-0.793068\pi$
$439$	−33.3571	−1.59205	−0.796025	+	0.605264i	$0.793068\pi$
$440$	0	0
$441$	−5.66347	−0.269689
$442$	0	0
$443$	30.6287	1.45522	0.727608	−	0.685993i	$-0.240632\pi$
$443$	30.6287	1.45522	0.727608	+	0.685993i	$0.240632\pi$
$444$	0	0
$445$	−1.81116	−0.0858572
$446$	0	0
$447$	12.2316	0.578533
$448$	0	0
$449$	16.9021	0.797658	0.398829	−	0.917025i	$-0.369417\pi$
$449$	16.9021	0.797658	0.398829	+	0.917025i	$0.369417\pi$
$450$	0	0
$451$	−25.2977	−1.19122
$452$	0	0
$453$	−13.3244	−0.626037
$454$	0	0
$455$	−1.64874	−0.0772942
$456$	0	0
$457$	28.7154	1.34325	0.671624	−	0.740892i	$-0.265597\pi$
$457$	28.7154	1.34325	0.671624	+	0.740892i	$0.265597\pi$
$458$	0	0
$459$	7.33718	0.342470
$460$	0	0
$461$	7.39735	0.344529	0.172264	−	0.985051i	$-0.444892\pi$
$461$	7.39735	0.344529	0.172264	+	0.985051i	$0.444892\pi$
$462$	0	0
$463$	−32.3696	−1.50434	−0.752172	−	0.658966i	$-0.770994\pi$
$463$	−32.3696	−1.50434	−0.752172	+	0.658966i	$0.770994\pi$
$464$	0	0
$465$	2.25439	0.104545
$466$	0	0
$467$	−11.6083	−0.537168	−0.268584	−	0.963256i	$-0.586556\pi$
$467$	−11.6083	−0.537168	−0.268584	+	0.963256i	$0.586556\pi$
$468$	0	0
$469$	−8.57772	−0.396082
$470$	0	0
$471$	−9.60039	−0.442362
$472$	0	0
$473$	30.2954	1.39298
$474$	0	0
$475$	19.4477	0.892321
$476$	0	0
$477$	−13.6612	−0.625505
$478$	0	0
$479$	28.3318	1.29451	0.647256	−	0.762272i	$-0.275916\pi$
$479$	28.3318	1.29451	0.647256	+	0.762272i	$0.275916\pi$
$480$	0	0
$481$	−12.8971	−0.588056
$482$	0	0
$483$	1.15608	0.0526036
$484$	0	0
$485$	−1.83704	−0.0834157
$486$	0	0
$487$	26.6567	1.20793	0.603964	−	0.797011i	$-0.293587\pi$
$487$	26.6567	1.20793	0.603964	+	0.797011i	$0.293587\pi$
$488$	0	0
$489$	−10.2315	−0.462686
$490$	0	0
$491$	37.9235	1.71146	0.855732	−	0.517420i	$-0.173108\pi$
$491$	37.9235	1.71146	0.855732	+	0.517420i	$0.173108\pi$
$492$	0	0
$493$	−7.33718	−0.330450
$494$	0	0
$495$	−1.83483	−0.0824696
$496$	0	0
$497$	−3.98746	−0.178862
$498$	0	0
$499$	8.59056	0.384566	0.192283	−	0.981339i	$-0.438411\pi$
$499$	8.59056	0.384566	0.192283	+	0.981339i	$0.438411\pi$
$500$	0	0
$501$	−5.87711	−0.262570
$502$	0	0
$503$	1.87519	0.0836106	0.0418053	−	0.999126i	$-0.486689\pi$
$503$	1.87519	0.0836106	0.0418053	+	0.999126i	$0.486689\pi$
$504$	0	0
$505$	−3.25511	−0.144850
$506$	0	0
$507$	−8.18401	−0.363465
$508$	0	0
$509$	42.0374	1.86328	0.931638	−	0.363387i	$-0.118380\pi$
$509$	42.0374	1.86328	0.931638	+	0.363387i	$0.118380\pi$
$510$	0	0
$511$	5.65909	0.250343
$512$	0	0
$513$	3.96569	0.175089
$514$	0	0
$515$	1.41716	0.0624477
$516$	0	0
$517$	18.0229	0.792648
$518$	0	0
$519$	−12.3777	−0.543320
$520$	0	0
$521$	7.62850	0.334211	0.167105	−	0.985939i	$-0.446558\pi$
$521$	7.62850	0.334211	0.167105	+	0.985939i	$0.446558\pi$
$522$	0	0
$523$	28.1008	1.22876	0.614381	−	0.789009i	$-0.289406\pi$
$523$	28.1008	1.22876	0.614381	+	0.789009i	$0.289406\pi$
$524$	0	0
$525$	−5.66941	−0.247434
$526$	0	0
$527$	−53.3823	−2.32537
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.903422	−0.0392052
$532$	0	0
$533$	19.6629	0.851696
$534$	0	0
$535$	2.10984	0.0912164
$536$	0	0
$537$	16.0178	0.691218
$538$	0	0
$539$	−33.5366	−1.44453
$540$	0	0
$541$	9.06933	0.389921	0.194960	−	0.980811i	$-0.437542\pi$
$541$	9.06933	0.389921	0.194960	+	0.980811i	$0.437542\pi$
$542$	0	0
$543$	2.94379	0.126330
$544$	0	0
$545$	−0.772939	−0.0331091
$546$	0	0
$547$	14.1909	0.606759	0.303379	−	0.952870i	$-0.401885\pi$
$547$	14.1909	0.606759	0.303379	+	0.952870i	$0.401885\pi$
$548$	0	0
$549$	11.0272	0.470631
$550$	0	0
$551$	−3.96569	−0.168944
$552$	0	0
$553$	−16.8217	−0.715331
$554$	0	0
$555$	0.868255	0.0368554
$556$	0	0
$557$	−7.80664	−0.330778	−0.165389	−	0.986228i	$-0.552888\pi$
$557$	−7.80664	−0.330778	−0.165389	+	0.986228i	$0.552888\pi$
$558$	0	0
$559$	−23.5474	−0.995951
$560$	0	0
$561$	43.4476	1.83436
$562$	0	0
$563$	−1.73244	−0.0730135	−0.0365068	−	0.999333i	$-0.511623\pi$
$563$	−1.73244	−0.0730135	−0.0365068	+	0.999333i	$0.511623\pi$
$564$	0	0
$565$	−2.94944	−0.124084
$566$	0	0
$567$	−1.15608	−0.0485509
$568$	0	0
$569$	39.3501	1.64964	0.824821	−	0.565394i	$-0.191276\pi$
$569$	39.3501	1.64964	0.824821	+	0.565394i	$0.191276\pi$
$570$	0	0
$571$	−16.2007	−0.677977	−0.338988	−	0.940791i	$-0.610085\pi$
$571$	−16.2007	−0.677977	−0.338988	+	0.940791i	$0.610085\pi$
$572$	0	0
$573$	−22.1472	−0.925214
$574$	0	0
$575$	−4.90399	−0.204510
$576$	0	0
$577$	31.6855	1.31908	0.659542	−	0.751668i	$-0.270750\pi$
$577$	31.6855	1.31908	0.659542	+	0.751668i	$0.270750\pi$
$578$	0	0
$579$	−9.91772	−0.412166
$580$	0	0
$581$	18.1250	0.751953
$582$	0	0
$583$	−80.8960	−3.35037
$584$	0	0
$585$	1.42615	0.0589639
$586$	0	0
$587$	34.4254	1.42089	0.710445	−	0.703753i	$-0.248494\pi$
$587$	34.4254	1.42089	0.710445	+	0.703753i	$0.248494\pi$
$588$	0	0
$589$	−28.8527	−1.18886
$590$	0	0
$591$	−2.07980	−0.0855514
$592$	0	0
$593$	−38.7103	−1.58964	−0.794821	−	0.606844i	$-0.792435\pi$
$593$	−38.7103	−1.58964	−0.794821	+	0.606844i	$0.792435\pi$
$594$	0	0
$595$	−2.62832	−0.107751
$596$	0	0
$597$	11.4598	0.469017
$598$	0	0
$599$	−26.7623	−1.09348	−0.546739	−	0.837303i	$-0.684131\pi$
$599$	−26.7623	−1.09348	−0.546739	+	0.837303i	$0.684131\pi$
$600$	0	0
$601$	−5.97759	−0.243831	−0.121915	−	0.992540i	$-0.538904\pi$
$601$	−5.97759	−0.243831	−0.121915	+	0.992540i	$0.538904\pi$
$602$	0	0
$603$	7.41964	0.302151
$604$	0	0
$605$	−7.45667	−0.303157
$606$	0	0
$607$	27.0115	1.09636	0.548182	−	0.836359i	$-0.315320\pi$
$607$	27.0115	1.09636	0.548182	+	0.836359i	$0.315320\pi$
$608$	0	0
$609$	1.15608	0.0468468
$610$	0	0
$611$	−14.0085	−0.566725
$612$	0	0
$613$	19.8569	0.802014	0.401007	−	0.916075i	$-0.368660\pi$
$613$	19.8569	0.802014	0.401007	+	0.916075i	$0.368660\pi$
$614$	0	0
$615$	−1.32375	−0.0533786
$616$	0	0
$617$	−1.56892	−0.0631622	−0.0315811	−	0.999501i	$-0.510054\pi$
$617$	−1.56892	−0.0631622	−0.0315811	+	0.999501i	$0.510054\pi$
$618$	0	0
$619$	40.2654	1.61840	0.809201	−	0.587532i	$-0.199900\pi$
$619$	40.2654	1.61840	0.809201	+	0.587532i	$0.199900\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−6.75749	−0.270733
$624$	0	0
$625$	23.5691	0.942762
$626$	0	0
$627$	23.4831	0.937824
$628$	0	0
$629$	−20.5597	−0.819769
$630$	0	0
$631$	7.30495	0.290805	0.145403	−	0.989373i	$-0.453552\pi$
$631$	7.30495	0.290805	0.145403	+	0.989373i	$0.453552\pi$
$632$	0	0
$633$	10.7296	0.426465
$634$	0	0
$635$	5.35381	0.212460
$636$	0	0
$637$	26.0668	1.03280
$638$	0	0
$639$	3.44912	0.136445
$640$	0	0
$641$	−2.70083	−0.106676	−0.0533381	−	0.998577i	$-0.516986\pi$
$641$	−2.70083	−0.106676	−0.0533381	+	0.998577i	$0.516986\pi$
$642$	0	0
$643$	−42.7930	−1.68759	−0.843796	−	0.536664i	$-0.819684\pi$
$643$	−42.7930	−1.68759	−0.843796	+	0.536664i	$0.819684\pi$
$644$	0	0
$645$	1.58526	0.0624194
$646$	0	0
$647$	−44.4022	−1.74563	−0.872815	−	0.488051i	$-0.837708\pi$
$647$	−44.4022	−1.74563	−0.872815	+	0.488051i	$0.837708\pi$
$648$	0	0
$649$	−5.34967	−0.209993
$650$	0	0
$651$	8.41118	0.329660
$652$	0	0
$653$	−8.71982	−0.341233	−0.170617	−	0.985338i	$-0.554576\pi$
$653$	−8.71982	−0.341233	−0.170617	+	0.985338i	$0.554576\pi$
$654$	0	0
$655$	−2.66209	−0.104016
$656$	0	0
$657$	−4.89506	−0.190974
$658$	0	0
$659$	42.7102	1.66375	0.831876	−	0.554961i	$-0.187267\pi$
$659$	42.7102	1.66375	0.831876	+	0.554961i	$0.187267\pi$
$660$	0	0
$661$	−28.8654	−1.12273	−0.561366	−	0.827568i	$-0.689724\pi$
$661$	−28.8654	−1.12273	−0.561366	+	0.827568i	$0.689724\pi$
$662$	0	0
$663$	−33.7702	−1.31153
$664$	0	0
$665$	−1.42059	−0.0550879
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	6.24882	0.241593
$670$	0	0
$671$	65.2984	2.52082
$672$	0	0
$673$	−35.2189	−1.35759	−0.678794	−	0.734329i	$-0.737497\pi$
$673$	−35.2189	−1.35759	−0.678794	+	0.734329i	$0.737497\pi$
$674$	0	0
$675$	4.90399	0.188755
$676$	0	0
$677$	−22.7503	−0.874366	−0.437183	−	0.899373i	$-0.644024\pi$
$677$	−22.7503	−0.874366	−0.437183	+	0.899373i	$0.644024\pi$
$678$	0	0
$679$	−6.85405	−0.263034
$680$	0	0
$681$	−23.1179	−0.885882
$682$	0	0
$683$	24.2731	0.928785	0.464392	−	0.885630i	$-0.346273\pi$
$683$	24.2731	0.928785	0.464392	+	0.885630i	$0.346273\pi$
$684$	0	0
$685$	−1.51503	−0.0578862
$686$	0	0
$687$	6.58061	0.251066
$688$	0	0
$689$	62.8774	2.39544
$690$	0	0
$691$	1.94632	0.0740417	0.0370208	−	0.999314i	$-0.488213\pi$
$691$	1.94632	0.0740417	0.0370208	+	0.999314i	$0.488213\pi$
$692$	0	0
$693$	−6.84582	−0.260051
$694$	0	0
$695$	6.26445	0.237624
$696$	0	0
$697$	31.3454	1.18729
$698$	0	0
$699$	16.8724	0.638171
$700$	0	0
$701$	−40.3281	−1.52317	−0.761586	−	0.648064i	$-0.775579\pi$
$701$	−40.3281	−1.52317	−0.761586	+	0.648064i	$0.775579\pi$
$702$	0	0
$703$	−11.1123	−0.419110
$704$	0	0
$705$	0.943081	0.0355185
$706$	0	0
$707$	−12.1449	−0.456756
$708$	0	0
$709$	−44.4185	−1.66817	−0.834085	−	0.551636i	$-0.814004\pi$
$709$	−44.4185	−1.66817	−0.834085	+	0.551636i	$0.814004\pi$
$710$	0	0
$711$	14.5506	0.545690
$712$	0	0
$713$	7.27559	0.272473
$714$	0	0
$715$	8.44502	0.315826
$716$	0	0
$717$	−22.1356	−0.826668
$718$	0	0
$719$	−41.9951	−1.56615	−0.783076	−	0.621926i	$-0.786350\pi$
$719$	−41.9951	−1.56615	−0.783076	+	0.621926i	$0.786350\pi$
$720$	0	0
$721$	5.28748	0.196916
$722$	0	0
$723$	4.15266	0.154439
$724$	0	0
$725$	−4.90399	−0.182130
$726$	0	0
$727$	−46.4995	−1.72457	−0.862285	−	0.506424i	$-0.830967\pi$
$727$	−46.4995	−1.72457	−0.862285	+	0.506424i	$0.830967\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−37.5378	−1.38839
$732$	0	0
$733$	27.7531	1.02508	0.512542	−	0.858662i	$-0.328704\pi$
$733$	27.7531	1.02508	0.512542	+	0.858662i	$0.328704\pi$
$734$	0	0
$735$	−1.75486	−0.0647291
$736$	0	0
$737$	43.9359	1.61840
$738$	0	0
$739$	−21.0418	−0.774034	−0.387017	−	0.922073i	$-0.626495\pi$
$739$	−21.0418	−0.774034	−0.387017	+	0.922073i	$0.626495\pi$
$740$	0	0
$741$	−18.2525	−0.670523
$742$	0	0
$743$	−6.20977	−0.227814	−0.113907	−	0.993491i	$-0.536337\pi$
$743$	−6.20977	−0.227814	−0.113907	+	0.993491i	$0.536337\pi$
$744$	0	0
$745$	3.79003	0.138856
$746$	0	0
$747$	−15.6780	−0.573627
$748$	0	0
$749$	7.87188	0.287632
$750$	0	0
$751$	−35.9054	−1.31021	−0.655103	−	0.755540i	$-0.727375\pi$
$751$	−35.9054	−1.31021	−0.655103	+	0.755540i	$0.727375\pi$
$752$	0	0
$753$	−10.8013	−0.393620
$754$	0	0
$755$	−4.12866	−0.150257
$756$	0	0
$757$	42.7030	1.55207	0.776033	−	0.630692i	$-0.217229\pi$
$757$	42.7030	1.55207	0.776033	+	0.630692i	$0.217229\pi$
$758$	0	0
$759$	−5.92157	−0.214939
$760$	0	0
$761$	−11.0786	−0.401599	−0.200799	−	0.979632i	$-0.564354\pi$
$761$	−11.0786	−0.401599	−0.200799	+	0.979632i	$0.564354\pi$
$762$	0	0
$763$	−2.88386	−0.104403
$764$	0	0
$765$	2.27347	0.0821975
$766$	0	0
$767$	4.15810	0.150140
$768$	0	0
$769$	22.8258	0.823121	0.411561	−	0.911382i	$-0.364984\pi$
$769$	22.8258	0.823121	0.411561	+	0.911382i	$0.364984\pi$
$770$	0	0
$771$	−27.1696	−0.978490
$772$	0	0
$773$	−27.8064	−1.00013	−0.500063	−	0.865989i	$-0.666690\pi$
$773$	−27.8064	−1.00013	−0.500063	+	0.865989i	$0.666690\pi$
$774$	0	0
$775$	−35.6794	−1.28164
$776$	0	0
$777$	3.23949	0.116216
$778$	0	0
$779$	16.9419	0.607007
$780$	0	0
$781$	20.4242	0.730834
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−2.97474	−0.106173
$786$	0	0
$787$	−27.9091	−0.994853	−0.497426	−	0.867506i	$-0.665722\pi$
$787$	−27.9091	−0.994853	−0.497426	+	0.867506i	$0.665722\pi$
$788$	0	0
$789$	17.0608	0.607382
$790$	0	0
$791$	−11.0044	−0.391272
$792$	0	0
$793$	−50.7540	−1.80233
$794$	0	0
$795$	−4.23302	−0.150130
$796$	0	0
$797$	−26.2558	−0.930028	−0.465014	−	0.885303i	$-0.653951\pi$
$797$	−26.2558	−0.930028	−0.465014	+	0.885303i	$0.653951\pi$
$798$	0	0
$799$	−22.3315	−0.790032
$800$	0	0
$801$	5.84517	0.206529
$802$	0	0
$803$	−28.9864	−1.02291
$804$	0	0
$805$	0.358219	0.0126256
$806$	0	0
$807$	1.06857	0.0376156
$808$	0	0
$809$	−35.1355	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$809$	−35.1355	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$810$	0	0
$811$	−31.0505	−1.09033	−0.545165	−	0.838329i	$-0.683533\pi$
$811$	−31.0505	−1.09033	−0.545165	+	0.838329i	$0.683533\pi$
$812$	0	0
$813$	29.0640	1.01932
$814$	0	0
$815$	−3.17031	−0.111051
$816$	0	0
$817$	−20.2889	−0.709818
$818$	0	0
$819$	5.32099	0.185931
$820$	0	0
$821$	34.0706	1.18907	0.594536	−	0.804069i	$-0.297336\pi$
$821$	34.0706	1.18907	0.594536	+	0.804069i	$0.297336\pi$
$822$	0	0
$823$	−31.8488	−1.11018	−0.555090	−	0.831790i	$-0.687316\pi$
$823$	−31.8488	−1.11018	−0.555090	+	0.831790i	$0.687316\pi$
$824$	0	0
$825$	29.0393	1.01102
$826$	0	0
$827$	8.76305	0.304721	0.152361	−	0.988325i	$-0.451313\pi$
$827$	8.76305	0.304721	0.152361	+	0.988325i	$0.451313\pi$
$828$	0	0
$829$	−17.2199	−0.598073	−0.299037	−	0.954242i	$-0.596665\pi$
$829$	−17.2199	−0.598073	−0.299037	+	0.954242i	$0.596665\pi$
$830$	0	0
$831$	19.3666	0.671820
$832$	0	0
$833$	41.5539	1.43976
$834$	0	0
$835$	−1.82106	−0.0630203
$836$	0	0
$837$	−7.27559	−0.251481
$838$	0	0
$839$	−11.2209	−0.387389	−0.193694	−	0.981062i	$-0.562047\pi$
$839$	−11.2209	−0.387389	−0.193694	+	0.981062i	$0.562047\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	10.4904	0.361310
$844$	0	0
$845$	−2.53587	−0.0872364
$846$	0	0
$847$	−27.8210	−0.955942
$848$	0	0
$849$	−5.42301	−0.186117
$850$	0	0
$851$	2.80212	0.0960556
$852$	0	0
$853$	−18.2145	−0.623654	−0.311827	−	0.950139i	$-0.600941\pi$
$853$	−18.2145	−0.623654	−0.311827	+	0.950139i	$0.600941\pi$
$854$	0	0
$855$	1.22879	0.0420238
$856$	0	0
$857$	23.0162	0.786218	0.393109	−	0.919492i	$-0.371400\pi$
$857$	23.0162	0.786218	0.393109	+	0.919492i	$0.371400\pi$
$858$	0	0
$859$	50.4794	1.72233	0.861167	−	0.508322i	$-0.169734\pi$
$859$	50.4794	1.72233	0.861167	+	0.508322i	$0.169734\pi$
$860$	0	0
$861$	−4.93893	−0.168318
$862$	0	0
$863$	27.9165	0.950288	0.475144	−	0.879908i	$-0.342396\pi$
$863$	27.9165	0.950288	0.475144	+	0.879908i	$0.342396\pi$
$864$	0	0
$865$	−3.83530	−0.130404
$866$	0	0
$867$	−36.8342	−1.25096
$868$	0	0
$869$	86.1624	2.92286
$870$	0	0
$871$	−34.1497	−1.15712
$872$	0	0
$873$	5.92869	0.200656
$874$	0	0
$875$	−3.54780	−0.119937
$876$	0	0
$877$	16.9914	0.573757	0.286879	−	0.957967i	$-0.407382\pi$
$877$	16.9914	0.573757	0.286879	+	0.957967i	$0.407382\pi$
$878$	0	0
$879$	−9.55426	−0.322257
$880$	0	0
$881$	17.4171	0.586796	0.293398	−	0.955990i	$-0.405214\pi$
$881$	17.4171	0.586796	0.293398	+	0.955990i	$0.405214\pi$
$882$	0	0
$883$	50.7274	1.70711	0.853556	−	0.521001i	$-0.174441\pi$
$883$	50.7274	1.70711	0.853556	+	0.521001i	$0.174441\pi$
$884$	0	0
$885$	−0.279931	−0.00940977
$886$	0	0
$887$	−18.3574	−0.616382	−0.308191	−	0.951325i	$-0.599724\pi$
$887$	−18.3574	−0.616382	−0.308191	+	0.951325i	$0.599724\pi$
$888$	0	0
$889$	19.9752	0.669948
$890$	0	0
$891$	5.92157	0.198380
$892$	0	0
$893$	−12.0700	−0.403907
$894$	0	0
$895$	4.96320	0.165902
$896$	0	0
$897$	4.60261	0.153677
$898$	0	0
$899$	7.27559	0.242654
$900$	0	0
$901$	100.235	3.33931
$902$	0	0
$903$	5.91464	0.196827
$904$	0	0
$905$	0.912152	0.0303210
$906$	0	0
$907$	40.0700	1.33050	0.665250	−	0.746620i	$-0.268325\pi$
$907$	40.0700	1.33050	0.665250	+	0.746620i	$0.268325\pi$
$908$	0	0
$909$	10.5052	0.348436
$910$	0	0
$911$	28.0643	0.929813	0.464906	−	0.885360i	$-0.346088\pi$
$911$	28.0643	0.929813	0.464906	+	0.885360i	$0.346088\pi$
$912$	0	0
$913$	−92.8382	−3.07250
$914$	0	0
$915$	3.41685	0.112958
$916$	0	0
$917$	−9.93233	−0.327995
$918$	0	0
$919$	16.9774	0.560033	0.280016	−	0.959995i	$-0.409660\pi$
$919$	16.9774	0.560033	0.280016	+	0.959995i	$0.409660\pi$
$920$	0	0
$921$	16.7330	0.551372
$922$	0	0
$923$	−15.8749	−0.522530
$924$	0	0
$925$	−13.7416	−0.451821
$926$	0	0
$927$	−4.57362	−0.150217
$928$	0	0
$929$	−21.0340	−0.690104	−0.345052	−	0.938584i	$-0.612139\pi$
$929$	−21.0340	−0.690104	−0.345052	+	0.938584i	$0.612139\pi$
$930$	0	0
$931$	22.4596	0.736083
$932$	0	0
$933$	−22.8155	−0.746945
$934$	0	0
$935$	13.4625	0.440271
$936$	0	0
$937$	26.9518	0.880476	0.440238	−	0.897881i	$-0.354894\pi$
$937$	26.9518	0.880476	0.440238	+	0.897881i	$0.354894\pi$
$938$	0	0
$939$	−2.54313	−0.0829918
$940$	0	0
$941$	−54.3406	−1.77145	−0.885726	−	0.464209i	$-0.846339\pi$
$941$	−54.3406	−1.77145	−0.885726	+	0.464209i	$0.846339\pi$
$942$	0	0
$943$	−4.27213	−0.139120
$944$	0	0
$945$	−0.358219	−0.0116529
$946$	0	0
$947$	17.3397	0.563465	0.281733	−	0.959493i	$-0.409091\pi$
$947$	17.3397	0.563465	0.281733	+	0.959493i	$0.409091\pi$
$948$	0	0
$949$	22.5300	0.731356
$950$	0	0
$951$	16.4581	0.533690
$952$	0	0
$953$	−27.6489	−0.895635	−0.447818	−	0.894125i	$-0.647799\pi$
$953$	−27.6489	−0.895635	−0.447818	+	0.894125i	$0.647799\pi$
$954$	0	0
$955$	−6.86246	−0.222064
$956$	0	0
$957$	−5.92157	−0.191417
$958$	0	0
$959$	−5.65261	−0.182532
$960$	0	0
$961$	21.9342	0.707555
$962$	0	0
$963$	−6.80910	−0.219420
$964$	0	0
$965$	−3.07307	−0.0989255
$966$	0	0
$967$	−19.4879	−0.626688	−0.313344	−	0.949640i	$-0.601449\pi$
$967$	−19.4879	−0.626688	−0.313344	+	0.949640i	$0.601449\pi$
$968$	0	0
$969$	−29.0970	−0.934729
$970$	0	0
$971$	−18.9686	−0.608730	−0.304365	−	0.952555i	$-0.598444\pi$
$971$	−18.9686	−0.608730	−0.304365	+	0.952555i	$0.598444\pi$
$972$	0	0
$973$	23.3728	0.749299
$974$	0	0
$975$	−22.5711	−0.722855
$976$	0	0
$977$	12.5571	0.401738	0.200869	−	0.979618i	$-0.435623\pi$
$977$	12.5571	0.401738	0.200869	+	0.979618i	$0.435623\pi$
$978$	0	0
$979$	34.6125	1.10622
$980$	0	0
$981$	2.49451	0.0796436
$982$	0	0
$983$	19.9515	0.636353	0.318176	−	0.948032i	$-0.396930\pi$
$983$	19.9515	0.636353	0.318176	+	0.948032i	$0.396930\pi$
$984$	0	0
$985$	−0.644438	−0.0205335
$986$	0	0
$987$	3.51866	0.112000
$988$	0	0
$989$	5.11611	0.162683
$990$	0	0
$991$	−44.7719	−1.42223	−0.711113	−	0.703078i	$-0.751809\pi$
$991$	−44.7719	−1.42223	−0.711113	+	0.703078i	$0.751809\pi$
$992$	0	0
$993$	−28.3233	−0.898812
$994$	0	0
$995$	3.55088	0.112570
$996$	0	0
$997$	−29.2775	−0.927228	−0.463614	−	0.886037i	$-0.653448\pi$
$997$	−29.2775	−0.927228	−0.463614	+	0.886037i	$0.653448\pi$
$998$	0	0
$999$	−2.80212	−0.0886553

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.6	✓	16	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.309856 q^{5} -1.15608 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.309856 q^{5} -1.15608 q^{7} +1.00000 q^{9} +5.92157 q^{11} -4.60261 q^{13} +0.309856 q^{15} -7.33718 q^{17} -3.96569 q^{19} +1.15608 q^{21} +1.00000 q^{23} -4.90399 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.27559 q^{31} -5.92157 q^{33} +0.358219 q^{35} +2.80212 q^{37} +4.60261 q^{39} -4.27213 q^{41} +5.11611 q^{43} -0.309856 q^{45} +3.04361 q^{47} -5.66347 q^{49} +7.33718 q^{51} -13.6612 q^{53} -1.83483 q^{55} +3.96569 q^{57} -0.903422 q^{59} +11.0272 q^{61} -1.15608 q^{63} +1.42615 q^{65} +7.41964 q^{67} -1.00000 q^{69} +3.44912 q^{71} -4.89506 q^{73} +4.90399 q^{75} -6.84582 q^{77} +14.5506 q^{79} +1.00000 q^{81} -15.6780 q^{83} +2.27347 q^{85} -1.00000 q^{87} +5.84517 q^{89} +5.32099 q^{91} -7.27559 q^{93} +1.22879 q^{95} +5.92869 q^{97} +5.92157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

L-functions

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Database

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