LMFDB - Embedded newform 8004.2.a.j.1.2

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} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.12066$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.12066 q^{5} -0.760812 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.12066 q^{5} -0.760812 q^{7} +1.00000 q^{9} +3.50921 q^{11} -3.64935 q^{13} -3.12066 q^{15} +6.89390 q^{17} -2.74332 q^{19} -0.760812 q^{21} +1.00000 q^{23} +4.73851 q^{25} +1.00000 q^{27} -1.00000 q^{29} -5.32027 q^{31} +3.50921 q^{33} +2.37424 q^{35} +5.93131 q^{37} -3.64935 q^{39} +6.52429 q^{41} -1.04224 q^{43} -3.12066 q^{45} -8.77868 q^{47} -6.42116 q^{49} +6.89390 q^{51} +6.18193 q^{53} -10.9510 q^{55} -2.74332 q^{57} -11.2513 q^{59} +0.576123 q^{61} -0.760812 q^{63} +11.3884 q^{65} +3.98419 q^{67} +1.00000 q^{69} -6.42449 q^{71} +7.28163 q^{73} +4.73851 q^{75} -2.66985 q^{77} +6.45544 q^{79} +1.00000 q^{81} +2.58965 q^{83} -21.5135 q^{85} -1.00000 q^{87} -10.7495 q^{89} +2.77647 q^{91} -5.32027 q^{93} +8.56096 q^{95} +17.1691 q^{97} +3.50921 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * 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
$5$	−3.12066	−1.39560	−0.697800	−	0.716292i	$-0.745838\pi$
$5$	−3.12066	−1.39560	−0.697800	+	0.716292i	$0.745838\pi$
$6$	0	0
$7$	−0.760812	−0.287560	−0.143780	−	0.989610i	$-0.545926\pi$
$7$	−0.760812	−0.287560	−0.143780	+	0.989610i	$0.545926\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.50921	1.05807	0.529033	−	0.848601i	$-0.322555\pi$
$11$	3.50921	1.05807	0.529033	+	0.848601i	$0.322555\pi$
$12$	0	0
$13$	−3.64935	−1.01215	−0.506074	−	0.862490i	$-0.668904\pi$
$13$	−3.64935	−1.01215	−0.506074	+	0.862490i	$0.668904\pi$
$14$	0	0
$15$	−3.12066	−0.805751
$16$	0	0
$17$	6.89390	1.67202	0.836008	−	0.548717i	$-0.184884\pi$
$17$	6.89390	1.67202	0.836008	+	0.548717i	$0.184884\pi$
$18$	0	0
$19$	−2.74332	−0.629360	−0.314680	−	0.949198i	$-0.601897\pi$
$19$	−2.74332	−0.629360	−0.314680	+	0.949198i	$0.601897\pi$
$20$	0	0
$21$	−0.760812	−0.166023
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.73851	0.947702
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−5.32027	−0.955549	−0.477774	−	0.878483i	$-0.658556\pi$
$31$	−5.32027	−0.955549	−0.477774	+	0.878483i	$0.658556\pi$
$32$	0	0
$33$	3.50921	0.610875
$34$	0	0
$35$	2.37424	0.401319
$36$	0	0
$37$	5.93131	0.975102	0.487551	−	0.873095i	$-0.337890\pi$
$37$	5.93131	0.975102	0.487551	+	0.873095i	$0.337890\pi$
$38$	0	0
$39$	−3.64935	−0.584364
$40$	0	0
$41$	6.52429	1.01892	0.509462	−	0.860493i	$-0.329845\pi$
$41$	6.52429	1.01892	0.509462	+	0.860493i	$0.329845\pi$
$42$	0	0
$43$	−1.04224	−0.158940	−0.0794702	−	0.996837i	$-0.525323\pi$
$43$	−1.04224	−0.158940	−0.0794702	+	0.996837i	$0.525323\pi$
$44$	0	0
$45$	−3.12066	−0.465200
$46$	0	0
$47$	−8.77868	−1.28050	−0.640251	−	0.768166i	$-0.721170\pi$
$47$	−8.77868	−1.28050	−0.640251	+	0.768166i	$0.721170\pi$
$48$	0	0
$49$	−6.42116	−0.917309
$50$	0	0
$51$	6.89390	0.965339
$52$	0	0
$53$	6.18193	0.849153	0.424576	−	0.905392i	$-0.360423\pi$
$53$	6.18193	0.849153	0.424576	+	0.905392i	$0.360423\pi$
$54$	0	0
$55$	−10.9510	−1.47664
$56$	0	0
$57$	−2.74332	−0.363361
$58$	0	0
$59$	−11.2513	−1.46480	−0.732398	−	0.680876i	$-0.761599\pi$
$59$	−11.2513	−1.46480	−0.732398	+	0.680876i	$0.761599\pi$
$60$	0	0
$61$	0.576123	0.0737650	0.0368825	−	0.999320i	$-0.488257\pi$
$61$	0.576123	0.0737650	0.0368825	+	0.999320i	$0.488257\pi$
$62$	0	0
$63$	−0.760812	−0.0958534
$64$	0	0
$65$	11.3884	1.41255
$66$	0	0
$67$	3.98419	0.486746	0.243373	−	0.969933i	$-0.421746\pi$
$67$	3.98419	0.486746	0.243373	+	0.969933i	$0.421746\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−6.42449	−0.762447	−0.381224	−	0.924483i	$-0.624497\pi$
$71$	−6.42449	−0.762447	−0.381224	+	0.924483i	$0.624497\pi$
$72$	0	0
$73$	7.28163	0.852250	0.426125	−	0.904664i	$-0.359878\pi$
$73$	7.28163	0.852250	0.426125	+	0.904664i	$0.359878\pi$
$74$	0	0
$75$	4.73851	0.547156
$76$	0	0
$77$	−2.66985	−0.304258
$78$	0	0
$79$	6.45544	0.726294	0.363147	−	0.931732i	$-0.381702\pi$
$79$	6.45544	0.726294	0.363147	+	0.931732i	$0.381702\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.58965	0.284252	0.142126	−	0.989849i	$-0.454606\pi$
$83$	2.58965	0.284252	0.142126	+	0.989849i	$0.454606\pi$
$84$	0	0
$85$	−21.5135	−2.33347
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−10.7495	−1.13944	−0.569722	−	0.821837i	$-0.692949\pi$
$89$	−10.7495	−1.13944	−0.569722	+	0.821837i	$0.692949\pi$
$90$	0	0
$91$	2.77647	0.291053
$92$	0	0
$93$	−5.32027	−0.551686
$94$	0	0
$95$	8.56096	0.878336
$96$	0	0
$97$	17.1691	1.74326	0.871631	−	0.490163i	$-0.163063\pi$
$97$	17.1691	1.74326	0.871631	+	0.490163i	$0.163063\pi$
$98$	0	0
$99$	3.50921	0.352689
$100$	0	0
$101$	6.05636	0.602630	0.301315	−	0.953525i	$-0.402574\pi$
$101$	6.05636	0.602630	0.301315	+	0.953525i	$0.402574\pi$
$102$	0	0
$103$	−10.3203	−1.01689	−0.508445	−	0.861094i	$-0.669780\pi$
$103$	−10.3203	−1.01689	−0.508445	+	0.861094i	$0.669780\pi$
$104$	0	0
$105$	2.37424	0.231702
$106$	0	0
$107$	12.8482	1.24208	0.621041	−	0.783778i	$-0.286710\pi$
$107$	12.8482	1.24208	0.621041	+	0.783778i	$0.286710\pi$
$108$	0	0
$109$	−9.57760	−0.917368	−0.458684	−	0.888600i	$-0.651679\pi$
$109$	−9.57760	−0.917368	−0.458684	+	0.888600i	$0.651679\pi$
$110$	0	0
$111$	5.93131	0.562975
$112$	0	0
$113$	−17.2072	−1.61872	−0.809359	−	0.587314i	$-0.800185\pi$
$113$	−17.2072	−1.61872	−0.809359	+	0.587314i	$0.800185\pi$
$114$	0	0
$115$	−3.12066	−0.291003
$116$	0	0
$117$	−3.64935	−0.337382
$118$	0	0
$119$	−5.24497	−0.480805
$120$	0	0
$121$	1.31456	0.119506
$122$	0	0
$123$	6.52429	0.588276
$124$	0	0
$125$	0.816024	0.0729874
$126$	0	0
$127$	12.8327	1.13872	0.569360	−	0.822088i	$-0.307191\pi$
$127$	12.8327	1.13872	0.569360	+	0.822088i	$0.307191\pi$
$128$	0	0
$129$	−1.04224	−0.0917643
$130$	0	0
$131$	19.1889	1.67654	0.838272	−	0.545252i	$-0.183566\pi$
$131$	19.1889	1.67654	0.838272	+	0.545252i	$0.183566\pi$
$132$	0	0
$133$	2.08715	0.180979
$134$	0	0
$135$	−3.12066	−0.268584
$136$	0	0
$137$	17.2757	1.47596	0.737982	−	0.674821i	$-0.235779\pi$
$137$	17.2757	1.47596	0.737982	+	0.674821i	$0.235779\pi$
$138$	0	0
$139$	8.12399	0.689068	0.344534	−	0.938774i	$-0.388037\pi$
$139$	8.12399	0.689068	0.344534	+	0.938774i	$0.388037\pi$
$140$	0	0
$141$	−8.77868	−0.739298
$142$	0	0
$143$	−12.8063	−1.07092
$144$	0	0
$145$	3.12066	0.259157
$146$	0	0
$147$	−6.42116	−0.529609
$148$	0	0
$149$	−2.15116	−0.176230	−0.0881148	−	0.996110i	$-0.528084\pi$
$149$	−2.15116	−0.176230	−0.0881148	+	0.996110i	$0.528084\pi$
$150$	0	0
$151$	11.9382	0.971521	0.485760	−	0.874092i	$-0.338543\pi$
$151$	11.9382	0.971521	0.485760	+	0.874092i	$0.338543\pi$
$152$	0	0
$153$	6.89390	0.557339
$154$	0	0
$155$	16.6027	1.33356
$156$	0	0
$157$	4.17674	0.333340	0.166670	−	0.986013i	$-0.446699\pi$
$157$	4.17674	0.333340	0.166670	+	0.986013i	$0.446699\pi$
$158$	0	0
$159$	6.18193	0.490259
$160$	0	0
$161$	−0.760812	−0.0599604
$162$	0	0
$163$	6.80419	0.532945	0.266473	−	0.963842i	$-0.414142\pi$
$163$	6.80419	0.532945	0.266473	+	0.963842i	$0.414142\pi$
$164$	0	0
$165$	−10.9510	−0.852538
$166$	0	0
$167$	−2.97280	−0.230042	−0.115021	−	0.993363i	$-0.536694\pi$
$167$	−2.97280	−0.230042	−0.115021	+	0.993363i	$0.536694\pi$
$168$	0	0
$169$	0.317749	0.0244423
$170$	0	0
$171$	−2.74332	−0.209787
$172$	0	0
$173$	1.57666	0.119871	0.0599355	−	0.998202i	$-0.480911\pi$
$173$	1.57666	0.119871	0.0599355	+	0.998202i	$0.480911\pi$
$174$	0	0
$175$	−3.60512	−0.272521
$176$	0	0
$177$	−11.2513	−0.845701
$178$	0	0
$179$	1.09555	0.0818853	0.0409427	−	0.999161i	$-0.486964\pi$
$179$	1.09555	0.0818853	0.0409427	+	0.999161i	$0.486964\pi$
$180$	0	0
$181$	15.7252	1.16885	0.584423	−	0.811449i	$-0.301321\pi$
$181$	15.7252	1.16885	0.584423	+	0.811449i	$0.301321\pi$
$182$	0	0
$183$	0.576123	0.0425882
$184$	0	0
$185$	−18.5096	−1.36085
$186$	0	0
$187$	24.1922	1.76911
$188$	0	0
$189$	−0.760812	−0.0553410
$190$	0	0
$191$	1.34219	0.0971173	0.0485586	−	0.998820i	$-0.484537\pi$
$191$	1.34219	0.0971173	0.0485586	+	0.998820i	$0.484537\pi$
$192$	0	0
$193$	15.9686	1.14944	0.574722	−	0.818349i	$-0.305110\pi$
$193$	15.9686	1.14944	0.574722	+	0.818349i	$0.305110\pi$
$194$	0	0
$195$	11.3884	0.815538
$196$	0	0
$197$	10.8821	0.775317	0.387659	−	0.921803i	$-0.373284\pi$
$197$	10.8821	0.775317	0.387659	+	0.921803i	$0.373284\pi$
$198$	0	0
$199$	20.5109	1.45398	0.726990	−	0.686648i	$-0.240919\pi$
$199$	20.5109	1.45398	0.726990	+	0.686648i	$0.240919\pi$
$200$	0	0
$201$	3.98419	0.281023
$202$	0	0
$203$	0.760812	0.0533986
$204$	0	0
$205$	−20.3601	−1.42201
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−9.62688	−0.665906
$210$	0	0
$211$	−8.39339	−0.577825	−0.288912	−	0.957356i	$-0.593294\pi$
$211$	−8.39339	−0.577825	−0.288912	+	0.957356i	$0.593294\pi$
$212$	0	0
$213$	−6.42449	−0.440199
$214$	0	0
$215$	3.25248	0.221817
$216$	0	0
$217$	4.04773	0.274778
$218$	0	0
$219$	7.28163	0.492047
$220$	0	0
$221$	−25.1582	−1.69233
$222$	0	0
$223$	−2.46867	−0.165314	−0.0826572	−	0.996578i	$-0.526341\pi$
$223$	−2.46867	−0.165314	−0.0826572	+	0.996578i	$0.526341\pi$
$224$	0	0
$225$	4.73851	0.315901
$226$	0	0
$227$	−18.3394	−1.21723	−0.608615	−	0.793466i	$-0.708275\pi$
$227$	−18.3394	−1.21723	−0.608615	+	0.793466i	$0.708275\pi$
$228$	0	0
$229$	2.30393	0.152248	0.0761240	−	0.997098i	$-0.475746\pi$
$229$	2.30393	0.152248	0.0761240	+	0.997098i	$0.475746\pi$
$230$	0	0
$231$	−2.66985	−0.175663
$232$	0	0
$233$	−3.12350	−0.204627	−0.102314	−	0.994752i	$-0.532625\pi$
$233$	−3.12350	−0.204627	−0.102314	+	0.994752i	$0.532625\pi$
$234$	0	0
$235$	27.3953	1.78707
$236$	0	0
$237$	6.45544	0.419326
$238$	0	0
$239$	−2.43709	−0.157642	−0.0788211	−	0.996889i	$-0.525116\pi$
$239$	−2.43709	−0.157642	−0.0788211	+	0.996889i	$0.525116\pi$
$240$	0	0
$241$	−4.96582	−0.319877	−0.159938	−	0.987127i	$-0.551130\pi$
$241$	−4.96582	−0.319877	−0.159938	+	0.987127i	$0.551130\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	20.0383	1.28020
$246$	0	0
$247$	10.0113	0.637006
$248$	0	0
$249$	2.58965	0.164113
$250$	0	0
$251$	9.69649	0.612037	0.306018	−	0.952026i	$-0.401003\pi$
$251$	9.69649	0.612037	0.306018	+	0.952026i	$0.401003\pi$
$252$	0	0
$253$	3.50921	0.220622
$254$	0	0
$255$	−21.5135	−1.34723
$256$	0	0
$257$	19.9656	1.24542	0.622710	−	0.782453i	$-0.286032\pi$
$257$	19.9656	1.24542	0.622710	+	0.782453i	$0.286032\pi$
$258$	0	0
$259$	−4.51262	−0.280400
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−12.3305	−0.760332	−0.380166	−	0.924918i	$-0.624133\pi$
$263$	−12.3305	−0.760332	−0.380166	+	0.924918i	$0.624133\pi$
$264$	0	0
$265$	−19.2917	−1.18508
$266$	0	0
$267$	−10.7495	−0.657859
$268$	0	0
$269$	−16.4593	−1.00354	−0.501772	−	0.865000i	$-0.667318\pi$
$269$	−16.4593	−1.00354	−0.501772	+	0.865000i	$0.667318\pi$
$270$	0	0
$271$	0.668966	0.0406368	0.0203184	−	0.999794i	$-0.493532\pi$
$271$	0.668966	0.0406368	0.0203184	+	0.999794i	$0.493532\pi$
$272$	0	0
$273$	2.77647	0.168040
$274$	0	0
$275$	16.6284	1.00273
$276$	0	0
$277$	4.27134	0.256640	0.128320	−	0.991733i	$-0.459042\pi$
$277$	4.27134	0.256640	0.128320	+	0.991733i	$0.459042\pi$
$278$	0	0
$279$	−5.32027	−0.318516
$280$	0	0
$281$	−0.0728644	−0.00434673	−0.00217336	−	0.999998i	$-0.500692\pi$
$281$	−0.0728644	−0.00434673	−0.00217336	+	0.999998i	$0.500692\pi$
$282$	0	0
$283$	2.10058	0.124867	0.0624334	−	0.998049i	$-0.480114\pi$
$283$	2.10058	0.124867	0.0624334	+	0.998049i	$0.480114\pi$
$284$	0	0
$285$	8.56096	0.507108
$286$	0	0
$287$	−4.96376	−0.293002
$288$	0	0
$289$	30.5259	1.79564
$290$	0	0
$291$	17.1691	1.00647
$292$	0	0
$293$	19.0717	1.11418	0.557090	−	0.830452i	$-0.311918\pi$
$293$	19.0717	1.11418	0.557090	+	0.830452i	$0.311918\pi$
$294$	0	0
$295$	35.1115	2.04427
$296$	0	0
$297$	3.50921	0.203625
$298$	0	0
$299$	−3.64935	−0.211047
$300$	0	0
$301$	0.792951	0.0457049
$302$	0	0
$303$	6.05636	0.347929
$304$	0	0
$305$	−1.79788	−0.102946
$306$	0	0
$307$	26.6122	1.51884	0.759418	−	0.650603i	$-0.225484\pi$
$307$	26.6122	1.51884	0.759418	+	0.650603i	$0.225484\pi$
$308$	0	0
$309$	−10.3203	−0.587102
$310$	0	0
$311$	−3.92862	−0.222772	−0.111386	−	0.993777i	$-0.535529\pi$
$311$	−3.92862	−0.222772	−0.111386	+	0.993777i	$0.535529\pi$
$312$	0	0
$313$	−10.1563	−0.574069	−0.287034	−	0.957920i	$-0.592669\pi$
$313$	−10.1563	−0.574069	−0.287034	+	0.957920i	$0.592669\pi$
$314$	0	0
$315$	2.37424	0.133773
$316$	0	0
$317$	21.2960	1.19610	0.598051	−	0.801458i	$-0.295942\pi$
$317$	21.2960	1.19610	0.598051	+	0.801458i	$0.295942\pi$
$318$	0	0
$319$	−3.50921	−0.196478
$320$	0	0
$321$	12.8482	0.717116
$322$	0	0
$323$	−18.9122	−1.05230
$324$	0	0
$325$	−17.2925	−0.959214
$326$	0	0
$327$	−9.57760	−0.529642
$328$	0	0
$329$	6.67893	0.368221
$330$	0	0
$331$	21.2770	1.16949	0.584745	−	0.811217i	$-0.301195\pi$
$331$	21.2770	1.16949	0.584745	+	0.811217i	$0.301195\pi$
$332$	0	0
$333$	5.93131	0.325034
$334$	0	0
$335$	−12.4333	−0.679303
$336$	0	0
$337$	16.0286	0.873135	0.436567	−	0.899672i	$-0.356194\pi$
$337$	16.0286	0.873135	0.436567	+	0.899672i	$0.356194\pi$
$338$	0	0
$339$	−17.2072	−0.934567
$340$	0	0
$341$	−18.6699	−1.01103
$342$	0	0
$343$	10.2110	0.551342
$344$	0	0
$345$	−3.12066	−0.168011
$346$	0	0
$347$	11.0678	0.594150	0.297075	−	0.954854i	$-0.403989\pi$
$347$	11.0678	0.594150	0.297075	+	0.954854i	$0.403989\pi$
$348$	0	0
$349$	34.6536	1.85496	0.927482	−	0.373869i	$-0.121969\pi$
$349$	34.6536	1.85496	0.927482	+	0.373869i	$0.121969\pi$
$350$	0	0
$351$	−3.64935	−0.194788
$352$	0	0
$353$	22.7684	1.21184	0.605919	−	0.795527i	$-0.292806\pi$
$353$	22.7684	1.21184	0.605919	+	0.795527i	$0.292806\pi$
$354$	0	0
$355$	20.0487	1.06407
$356$	0	0
$357$	−5.24497	−0.277593
$358$	0	0
$359$	17.5444	0.925960	0.462980	−	0.886369i	$-0.346780\pi$
$359$	17.5444	0.925960	0.462980	+	0.886369i	$0.346780\pi$
$360$	0	0
$361$	−11.4742	−0.603905
$362$	0	0
$363$	1.31456	0.0689967
$364$	0	0
$365$	−22.7235	−1.18940
$366$	0	0
$367$	−9.26701	−0.483734	−0.241867	−	0.970309i	$-0.577760\pi$
$367$	−9.26701	−0.483734	−0.241867	+	0.970309i	$0.577760\pi$
$368$	0	0
$369$	6.52429	0.339641
$370$	0	0
$371$	−4.70329	−0.244182
$372$	0	0
$373$	27.4289	1.42022	0.710108	−	0.704093i	$-0.248646\pi$
$373$	27.4289	1.42022	0.710108	+	0.704093i	$0.248646\pi$
$374$	0	0
$375$	0.816024	0.0421393
$376$	0	0
$377$	3.64935	0.187951
$378$	0	0
$379$	21.1334	1.08555	0.542776	−	0.839878i	$-0.317373\pi$
$379$	21.1334	1.08555	0.542776	+	0.839878i	$0.317373\pi$
$380$	0	0
$381$	12.8327	0.657441
$382$	0	0
$383$	−15.9421	−0.814602	−0.407301	−	0.913294i	$-0.633530\pi$
$383$	−15.9421	−0.814602	−0.407301	+	0.913294i	$0.633530\pi$
$384$	0	0
$385$	8.33170	0.424623
$386$	0	0
$387$	−1.04224	−0.0529801
$388$	0	0
$389$	−4.34366	−0.220233	−0.110116	−	0.993919i	$-0.535122\pi$
$389$	−4.34366	−0.220233	−0.110116	+	0.993919i	$0.535122\pi$
$390$	0	0
$391$	6.89390	0.348640
$392$	0	0
$393$	19.1889	0.967953
$394$	0	0
$395$	−20.1452	−1.01362
$396$	0	0
$397$	−30.9210	−1.55188	−0.775941	−	0.630806i	$-0.782724\pi$
$397$	−30.9210	−1.55188	−0.775941	+	0.630806i	$0.782724\pi$
$398$	0	0
$399$	2.08715	0.104488
$400$	0	0
$401$	−12.6909	−0.633754	−0.316877	−	0.948467i	$-0.602634\pi$
$401$	−12.6909	−0.633754	−0.316877	+	0.948467i	$0.602634\pi$
$402$	0	0
$403$	19.4155	0.967156
$404$	0	0
$405$	−3.12066	−0.155067
$406$	0	0
$407$	20.8142	1.03172
$408$	0	0
$409$	7.13402	0.352754	0.176377	−	0.984323i	$-0.443562\pi$
$409$	7.13402	0.352754	0.176377	+	0.984323i	$0.443562\pi$
$410$	0	0
$411$	17.2757	0.852148
$412$	0	0
$413$	8.56014	0.421217
$414$	0	0
$415$	−8.08143	−0.396702
$416$	0	0
$417$	8.12399	0.397834
$418$	0	0
$419$	−38.3203	−1.87207	−0.936034	−	0.351909i	$-0.885533\pi$
$419$	−38.3203	−1.87207	−0.936034	+	0.351909i	$0.885533\pi$
$420$	0	0
$421$	15.7125	0.765779	0.382890	−	0.923794i	$-0.374929\pi$
$421$	15.7125	0.765779	0.382890	+	0.923794i	$0.374929\pi$
$422$	0	0
$423$	−8.77868	−0.426834
$424$	0	0
$425$	32.6668	1.58457
$426$	0	0
$427$	−0.438321	−0.0212119
$428$	0	0
$429$	−12.8063	−0.618296
$430$	0	0
$431$	−9.14367	−0.440435	−0.220217	−	0.975451i	$-0.570677\pi$
$431$	−9.14367	−0.440435	−0.220217	+	0.975451i	$0.570677\pi$
$432$	0	0
$433$	−30.3032	−1.45628	−0.728139	−	0.685430i	$-0.759614\pi$
$433$	−30.3032	−1.45628	−0.728139	+	0.685430i	$0.759614\pi$
$434$	0	0
$435$	3.12066	0.149624
$436$	0	0
$437$	−2.74332	−0.131231
$438$	0	0
$439$	20.9626	1.00049	0.500245	−	0.865884i	$-0.333243\pi$
$439$	20.9626	1.00049	0.500245	+	0.865884i	$0.333243\pi$
$440$	0	0
$441$	−6.42116	−0.305770
$442$	0	0
$443$	−5.54152	−0.263286	−0.131643	−	0.991297i	$-0.542025\pi$
$443$	−5.54152	−0.263286	−0.131643	+	0.991297i	$0.542025\pi$
$444$	0	0
$445$	33.5455	1.59021
$446$	0	0
$447$	−2.15116	−0.101746
$448$	0	0
$449$	−39.2609	−1.85284	−0.926418	−	0.376497i	$-0.877128\pi$
$449$	−39.2609	−1.85284	−0.926418	+	0.376497i	$0.877128\pi$
$450$	0	0
$451$	22.8951	1.07809
$452$	0	0
$453$	11.9382	0.560908
$454$	0	0
$455$	−8.66442	−0.406194
$456$	0	0
$457$	−0.674951	−0.0315729	−0.0157864	−	0.999875i	$-0.505025\pi$
$457$	−0.674951	−0.0315729	−0.0157864	+	0.999875i	$0.505025\pi$
$458$	0	0
$459$	6.89390	0.321780
$460$	0	0
$461$	−10.6701	−0.496956	−0.248478	−	0.968638i	$-0.579930\pi$
$461$	−10.6701	−0.496956	−0.248478	+	0.968638i	$0.579930\pi$
$462$	0	0
$463$	11.2011	0.520557	0.260279	−	0.965534i	$-0.416186\pi$
$463$	11.2011	0.520557	0.260279	+	0.965534i	$0.416186\pi$
$464$	0	0
$465$	16.6027	0.769934
$466$	0	0
$467$	35.2217	1.62987	0.814934	−	0.579554i	$-0.196773\pi$
$467$	35.2217	1.62987	0.814934	+	0.579554i	$0.196773\pi$
$468$	0	0
$469$	−3.03122	−0.139969
$470$	0	0
$471$	4.17674	0.192454
$472$	0	0
$473$	−3.65745	−0.168170
$474$	0	0
$475$	−12.9992	−0.596446
$476$	0	0
$477$	6.18193	0.283051
$478$	0	0
$479$	−31.8883	−1.45701	−0.728507	−	0.685038i	$-0.759785\pi$
$479$	−31.8883	−1.45701	−0.728507	+	0.685038i	$0.759785\pi$
$480$	0	0
$481$	−21.6454	−0.986947
$482$	0	0
$483$	−0.760812	−0.0346182
$484$	0	0
$485$	−53.5790	−2.43290
$486$	0	0
$487$	−4.29418	−0.194588	−0.0972939	−	0.995256i	$-0.531019\pi$
$487$	−4.29418	−0.194588	−0.0972939	+	0.995256i	$0.531019\pi$
$488$	0	0
$489$	6.80419	0.307696
$490$	0	0
$491$	21.3458	0.963320	0.481660	−	0.876358i	$-0.340034\pi$
$491$	21.3458	0.963320	0.481660	+	0.876358i	$0.340034\pi$
$492$	0	0
$493$	−6.89390	−0.310486
$494$	0	0
$495$	−10.9510	−0.492213
$496$	0	0
$497$	4.88784	0.219249
$498$	0	0
$499$	38.3671	1.71755	0.858774	−	0.512354i	$-0.171226\pi$
$499$	38.3671	1.71755	0.858774	+	0.512354i	$0.171226\pi$
$500$	0	0
$501$	−2.97280	−0.132815
$502$	0	0
$503$	2.11484	0.0942959	0.0471480	−	0.998888i	$-0.484987\pi$
$503$	2.11484	0.0942959	0.0471480	+	0.998888i	$0.484987\pi$
$504$	0	0
$505$	−18.8998	−0.841031
$506$	0	0
$507$	0.317749	0.0141117
$508$	0	0
$509$	37.6267	1.66777	0.833887	−	0.551935i	$-0.186110\pi$
$509$	37.6267	1.66777	0.833887	+	0.551935i	$0.186110\pi$
$510$	0	0
$511$	−5.53995	−0.245073
$512$	0	0
$513$	−2.74332	−0.121120
$514$	0	0
$515$	32.2062	1.41917
$516$	0	0
$517$	−30.8062	−1.35486
$518$	0	0
$519$	1.57666	0.0692075
$520$	0	0
$521$	−12.1293	−0.531392	−0.265696	−	0.964057i	$-0.585602\pi$
$521$	−12.1293	−0.531392	−0.265696	+	0.964057i	$0.585602\pi$
$522$	0	0
$523$	31.5939	1.38151	0.690753	−	0.723091i	$-0.257279\pi$
$523$	31.5939	1.38151	0.690753	+	0.723091i	$0.257279\pi$
$524$	0	0
$525$	−3.60512	−0.157340
$526$	0	0
$527$	−36.6774	−1.59769
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.2513	−0.488266
$532$	0	0
$533$	−23.8094	−1.03130
$534$	0	0
$535$	−40.0948	−1.73345
$536$	0	0
$537$	1.09555	0.0472765
$538$	0	0
$539$	−22.5332	−0.970575
$540$	0	0
$541$	26.3451	1.13266	0.566331	−	0.824178i	$-0.308362\pi$
$541$	26.3451	1.13266	0.566331	+	0.824178i	$0.308362\pi$
$542$	0	0
$543$	15.7252	0.674834
$544$	0	0
$545$	29.8884	1.28028
$546$	0	0
$547$	13.4948	0.576994	0.288497	−	0.957481i	$-0.406844\pi$
$547$	13.4948	0.576994	0.288497	+	0.957481i	$0.406844\pi$
$548$	0	0
$549$	0.576123	0.0245883
$550$	0	0
$551$	2.74332	0.116869
$552$	0	0
$553$	−4.91138	−0.208853
$554$	0	0
$555$	−18.5096	−0.785689
$556$	0	0
$557$	−35.1330	−1.48863	−0.744317	−	0.667827i	$-0.767225\pi$
$557$	−35.1330	−1.48863	−0.744317	+	0.667827i	$0.767225\pi$
$558$	0	0
$559$	3.80351	0.160871
$560$	0	0
$561$	24.1922	1.02139
$562$	0	0
$563$	−1.86137	−0.0784475	−0.0392237	−	0.999230i	$-0.512489\pi$
$563$	−1.86137	−0.0784475	−0.0392237	+	0.999230i	$0.512489\pi$
$564$	0	0
$565$	53.6978	2.25908
$566$	0	0
$567$	−0.760812	−0.0319511
$568$	0	0
$569$	33.6725	1.41163	0.705813	−	0.708399i	$-0.250582\pi$
$569$	33.6725	1.41163	0.705813	+	0.708399i	$0.250582\pi$
$570$	0	0
$571$	5.50754	0.230483	0.115242	−	0.993337i	$-0.463236\pi$
$571$	5.50754	0.230483	0.115242	+	0.993337i	$0.463236\pi$
$572$	0	0
$573$	1.34219	0.0560707
$574$	0	0
$575$	4.73851	0.197609
$576$	0	0
$577$	−22.0105	−0.916309	−0.458154	−	0.888873i	$-0.651489\pi$
$577$	−22.0105	−0.916309	−0.458154	+	0.888873i	$0.651489\pi$
$578$	0	0
$579$	15.9686	0.663632
$580$	0	0
$581$	−1.97024	−0.0817394
$582$	0	0
$583$	21.6937	0.898461
$584$	0	0
$585$	11.3884	0.470851
$586$	0	0
$587$	−1.14883	−0.0474171	−0.0237086	−	0.999719i	$-0.507547\pi$
$587$	−1.14883	−0.0474171	−0.0237086	+	0.999719i	$0.507547\pi$
$588$	0	0
$589$	14.5952	0.601384
$590$	0	0
$591$	10.8821	0.447630
$592$	0	0
$593$	4.59279	0.188603	0.0943017	−	0.995544i	$-0.469938\pi$
$593$	4.59279	0.188603	0.0943017	+	0.995544i	$0.469938\pi$
$594$	0	0
$595$	16.3677	0.671012
$596$	0	0
$597$	20.5109	0.839456
$598$	0	0
$599$	−17.0701	−0.697465	−0.348733	−	0.937222i	$-0.613388\pi$
$599$	−17.0701	−0.697465	−0.348733	+	0.937222i	$0.613388\pi$
$600$	0	0
$601$	46.7860	1.90844	0.954220	−	0.299106i	$-0.0966883\pi$
$601$	46.7860	1.90844	0.954220	+	0.299106i	$0.0966883\pi$
$602$	0	0
$603$	3.98419	0.162249
$604$	0	0
$605$	−4.10230	−0.166782
$606$	0	0
$607$	17.9793	0.729756	0.364878	−	0.931055i	$-0.381111\pi$
$607$	17.9793	0.729756	0.364878	+	0.931055i	$0.381111\pi$
$608$	0	0
$609$	0.760812	0.0308297
$610$	0	0
$611$	32.0365	1.29606
$612$	0	0
$613$	6.23189	0.251704	0.125852	−	0.992049i	$-0.459834\pi$
$613$	6.23189	0.251704	0.125852	+	0.992049i	$0.459834\pi$
$614$	0	0
$615$	−20.3601	−0.820998
$616$	0	0
$617$	−9.37601	−0.377464	−0.188732	−	0.982029i	$-0.560438\pi$
$617$	−9.37601	−0.377464	−0.188732	+	0.982029i	$0.560438\pi$
$618$	0	0
$619$	−25.5678	−1.02766	−0.513828	−	0.857893i	$-0.671773\pi$
$619$	−25.5678	−1.02766	−0.513828	+	0.857893i	$0.671773\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	8.17835	0.327659
$624$	0	0
$625$	−26.2391	−1.04956
$626$	0	0
$627$	−9.62688	−0.384461
$628$	0	0
$629$	40.8899	1.63039
$630$	0	0
$631$	−19.5977	−0.780173	−0.390087	−	0.920778i	$-0.627555\pi$
$631$	−19.5977	−0.780173	−0.390087	+	0.920778i	$0.627555\pi$
$632$	0	0
$633$	−8.39339	−0.333607
$634$	0	0
$635$	−40.0466	−1.58920
$636$	0	0
$637$	23.4331	0.928452
$638$	0	0
$639$	−6.42449	−0.254149
$640$	0	0
$641$	17.5793	0.694339	0.347169	−	0.937802i	$-0.387143\pi$
$641$	17.5793	0.694339	0.347169	+	0.937802i	$0.387143\pi$
$642$	0	0
$643$	6.21692	0.245171	0.122586	−	0.992458i	$-0.460881\pi$
$643$	6.21692	0.245171	0.122586	+	0.992458i	$0.460881\pi$
$644$	0	0
$645$	3.25248	0.128066
$646$	0	0
$647$	−6.28683	−0.247161	−0.123580	−	0.992335i	$-0.539438\pi$
$647$	−6.28683	−0.247161	−0.123580	+	0.992335i	$0.539438\pi$
$648$	0	0
$649$	−39.4832	−1.54985
$650$	0	0
$651$	4.04773	0.158643
$652$	0	0
$653$	−0.335306	−0.0131215	−0.00656077	−	0.999978i	$-0.502088\pi$
$653$	−0.335306	−0.0131215	−0.00656077	+	0.999978i	$0.502088\pi$
$654$	0	0
$655$	−59.8821	−2.33979
$656$	0	0
$657$	7.28163	0.284083
$658$	0	0
$659$	38.7356	1.50892	0.754462	−	0.656343i	$-0.227898\pi$
$659$	38.7356	1.50892	0.754462	+	0.656343i	$0.227898\pi$
$660$	0	0
$661$	−36.2085	−1.40835	−0.704174	−	0.710027i	$-0.748683\pi$
$661$	−36.2085	−1.40835	−0.704174	+	0.710027i	$0.748683\pi$
$662$	0	0
$663$	−25.1582	−0.977065
$664$	0	0
$665$	−6.51329	−0.252574
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−2.46867	−0.0954443
$670$	0	0
$671$	2.02174	0.0780483
$672$	0	0
$673$	−25.6364	−0.988211	−0.494105	−	0.869402i	$-0.664504\pi$
$673$	−25.6364	−0.988211	−0.494105	+	0.869402i	$0.664504\pi$
$674$	0	0
$675$	4.73851	0.182385
$676$	0	0
$677$	−36.2195	−1.39203	−0.696015	−	0.718027i	$-0.745045\pi$
$677$	−36.2195	−1.39203	−0.696015	+	0.718027i	$0.745045\pi$
$678$	0	0
$679$	−13.0625	−0.501292
$680$	0	0
$681$	−18.3394	−0.702768
$682$	0	0
$683$	0.374532	0.0143311	0.00716554	−	0.999974i	$-0.497719\pi$
$683$	0.374532	0.0143311	0.00716554	+	0.999974i	$0.497719\pi$
$684$	0	0
$685$	−53.9116	−2.05986
$686$	0	0
$687$	2.30393	0.0879004
$688$	0	0
$689$	−22.5600	−0.859468
$690$	0	0
$691$	−50.4676	−1.91988	−0.959938	−	0.280212i	$-0.909595\pi$
$691$	−50.4676	−1.91988	−0.959938	+	0.280212i	$0.909595\pi$
$692$	0	0
$693$	−2.66985	−0.101419
$694$	0	0
$695$	−25.3522	−0.961664
$696$	0	0
$697$	44.9778	1.70366
$698$	0	0
$699$	−3.12350	−0.118142
$700$	0	0
$701$	28.2080	1.06540	0.532701	−	0.846304i	$-0.321177\pi$
$701$	28.2080	1.06540	0.532701	+	0.846304i	$0.321177\pi$
$702$	0	0
$703$	−16.2715	−0.613690
$704$	0	0
$705$	27.3953	1.03177
$706$	0	0
$707$	−4.60775	−0.173292
$708$	0	0
$709$	17.4552	0.655544	0.327772	−	0.944757i	$-0.393702\pi$
$709$	17.4552	0.655544	0.327772	+	0.944757i	$0.393702\pi$
$710$	0	0
$711$	6.45544	0.242098
$712$	0	0
$713$	−5.32027	−0.199246
$714$	0	0
$715$	39.9642	1.49458
$716$	0	0
$717$	−2.43709	−0.0910147
$718$	0	0
$719$	−22.2018	−0.827986	−0.413993	−	0.910280i	$-0.635866\pi$
$719$	−22.2018	−0.827986	−0.413993	+	0.910280i	$0.635866\pi$
$720$	0	0
$721$	7.85182	0.292417
$722$	0	0
$723$	−4.96582	−0.184681
$724$	0	0
$725$	−4.73851	−0.175984
$726$	0	0
$727$	−20.8883	−0.774706	−0.387353	−	0.921932i	$-0.626610\pi$
$727$	−20.8883	−0.774706	−0.387353	+	0.921932i	$0.626610\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.18511	−0.265751
$732$	0	0
$733$	−21.7263	−0.802480	−0.401240	−	0.915973i	$-0.631421\pi$
$733$	−21.7263	−0.802480	−0.401240	+	0.915973i	$0.631421\pi$
$734$	0	0
$735$	20.0383	0.739122
$736$	0	0
$737$	13.9814	0.515010
$738$	0	0
$739$	−10.6222	−0.390742	−0.195371	−	0.980729i	$-0.562591\pi$
$739$	−10.6222	−0.390742	−0.195371	+	0.980729i	$0.562591\pi$
$740$	0	0
$741$	10.0113	0.367775
$742$	0	0
$743$	−37.0613	−1.35965	−0.679825	−	0.733375i	$-0.737944\pi$
$743$	−37.0613	−1.35965	−0.679825	+	0.733375i	$0.737944\pi$
$744$	0	0
$745$	6.71302	0.245946
$746$	0	0
$747$	2.58965	0.0947505
$748$	0	0
$749$	−9.77507	−0.357173
$750$	0	0
$751$	0.703673	0.0256774	0.0128387	−	0.999918i	$-0.495913\pi$
$751$	0.703673	0.0256774	0.0128387	+	0.999918i	$0.495913\pi$
$752$	0	0
$753$	9.69649	0.353360
$754$	0	0
$755$	−37.2552	−1.35586
$756$	0	0
$757$	10.9505	0.398001	0.199001	−	0.979999i	$-0.436230\pi$
$757$	10.9505	0.398001	0.199001	+	0.979999i	$0.436230\pi$
$758$	0	0
$759$	3.50921	0.127376
$760$	0	0
$761$	22.3228	0.809201	0.404600	−	0.914494i	$-0.367411\pi$
$761$	22.3228	0.809201	0.404600	+	0.914494i	$0.367411\pi$
$762$	0	0
$763$	7.28676	0.263798
$764$	0	0
$765$	−21.5135	−0.777823
$766$	0	0
$767$	41.0600	1.48259
$768$	0	0
$769$	−22.0265	−0.794297	−0.397149	−	0.917754i	$-0.630000\pi$
$769$	−22.0265	−0.794297	−0.397149	+	0.917754i	$0.630000\pi$
$770$	0	0
$771$	19.9656	0.719043
$772$	0	0
$773$	23.8648	0.858356	0.429178	−	0.903220i	$-0.358803\pi$
$773$	23.8648	0.858356	0.429178	+	0.903220i	$0.358803\pi$
$774$	0	0
$775$	−25.2101	−0.905575
$776$	0	0
$777$	−4.51262	−0.161889
$778$	0	0
$779$	−17.8982	−0.641270
$780$	0	0
$781$	−22.5449	−0.806720
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−13.0342	−0.465209
$786$	0	0
$787$	−37.9888	−1.35415	−0.677077	−	0.735912i	$-0.736754\pi$
$787$	−37.9888	−1.35415	−0.677077	+	0.735912i	$0.736754\pi$
$788$	0	0
$789$	−12.3305	−0.438978
$790$	0	0
$791$	13.0915	0.465479
$792$	0	0
$793$	−2.10247	−0.0746610
$794$	0	0
$795$	−19.2917	−0.684205
$796$	0	0
$797$	−39.8803	−1.41263	−0.706317	−	0.707896i	$-0.749645\pi$
$797$	−39.8803	−1.41263	−0.706317	+	0.707896i	$0.749645\pi$
$798$	0	0
$799$	−60.5194	−2.14102
$800$	0	0
$801$	−10.7495	−0.379815
$802$	0	0
$803$	25.5528	0.901737
$804$	0	0
$805$	2.37424	0.0836808
$806$	0	0
$807$	−16.4593	−0.579396
$808$	0	0
$809$	40.9930	1.44124	0.720618	−	0.693332i	$-0.243858\pi$
$809$	40.9930	1.44124	0.720618	+	0.693332i	$0.243858\pi$
$810$	0	0
$811$	−3.05091	−0.107132	−0.0535659	−	0.998564i	$-0.517059\pi$
$811$	−3.05091	−0.107132	−0.0535659	+	0.998564i	$0.517059\pi$
$812$	0	0
$813$	0.668966	0.0234617
$814$	0	0
$815$	−21.2336	−0.743779
$816$	0	0
$817$	2.85920	0.100031
$818$	0	0
$819$	2.77647	0.0970177
$820$	0	0
$821$	35.0548	1.22342	0.611710	−	0.791082i	$-0.290482\pi$
$821$	35.0548	1.22342	0.611710	+	0.791082i	$0.290482\pi$
$822$	0	0
$823$	−10.3018	−0.359097	−0.179549	−	0.983749i	$-0.557464\pi$
$823$	−10.3018	−0.359097	−0.179549	+	0.983749i	$0.557464\pi$
$824$	0	0
$825$	16.6284	0.578928
$826$	0	0
$827$	−20.8974	−0.726674	−0.363337	−	0.931658i	$-0.618363\pi$
$827$	−20.8974	−0.726674	−0.363337	+	0.931658i	$0.618363\pi$
$828$	0	0
$829$	4.34240	0.150818	0.0754089	−	0.997153i	$-0.475974\pi$
$829$	4.34240	0.150818	0.0754089	+	0.997153i	$0.475974\pi$
$830$	0	0
$831$	4.27134	0.148171
$832$	0	0
$833$	−44.2669	−1.53376
$834$	0	0
$835$	9.27710	0.321047
$836$	0	0
$837$	−5.32027	−0.183895
$838$	0	0
$839$	−37.9123	−1.30888	−0.654439	−	0.756114i	$-0.727095\pi$
$839$	−37.9123	−1.30888	−0.654439	+	0.756114i	$0.727095\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−0.0728644	−0.00250958
$844$	0	0
$845$	−0.991587	−0.0341116
$846$	0	0
$847$	−1.00014	−0.0343651
$848$	0	0
$849$	2.10058	0.0720919
$850$	0	0
$851$	5.93131	0.203323
$852$	0	0
$853$	−17.6823	−0.605431	−0.302715	−	0.953081i	$-0.597893\pi$
$853$	−17.6823	−0.605431	−0.302715	+	0.953081i	$0.597893\pi$
$854$	0	0
$855$	8.56096	0.292779
$856$	0	0
$857$	0.407348	0.0139148	0.00695738	−	0.999976i	$-0.497785\pi$
$857$	0.407348	0.0139148	0.00695738	+	0.999976i	$0.497785\pi$
$858$	0	0
$859$	−22.5200	−0.768373	−0.384186	−	0.923256i	$-0.625518\pi$
$859$	−22.5200	−0.768373	−0.384186	+	0.923256i	$0.625518\pi$
$860$	0	0
$861$	−4.96376	−0.169165
$862$	0	0
$863$	22.5349	0.767098	0.383549	−	0.923521i	$-0.374702\pi$
$863$	22.5349	0.767098	0.383549	+	0.923521i	$0.374702\pi$
$864$	0	0
$865$	−4.92020	−0.167292
$866$	0	0
$867$	30.5259	1.03671
$868$	0	0
$869$	22.6535	0.768467
$870$	0	0
$871$	−14.5397	−0.492659
$872$	0	0
$873$	17.1691	0.581087
$874$	0	0
$875$	−0.620841	−0.0209883
$876$	0	0
$877$	15.0439	0.507997	0.253999	−	0.967205i	$-0.418254\pi$
$877$	15.0439	0.507997	0.253999	+	0.967205i	$0.418254\pi$
$878$	0	0
$879$	19.0717	0.643272
$880$	0	0
$881$	4.02843	0.135721	0.0678607	−	0.997695i	$-0.478383\pi$
$881$	4.02843	0.135721	0.0678607	+	0.997695i	$0.478383\pi$
$882$	0	0
$883$	26.3602	0.887092	0.443546	−	0.896252i	$-0.353720\pi$
$883$	26.3602	0.887092	0.443546	+	0.896252i	$0.353720\pi$
$884$	0	0
$885$	35.1115	1.18026
$886$	0	0
$887$	−17.4075	−0.584486	−0.292243	−	0.956344i	$-0.594402\pi$
$887$	−17.4075	−0.584486	−0.292243	+	0.956344i	$0.594402\pi$
$888$	0	0
$889$	−9.76330	−0.327451
$890$	0	0
$891$	3.50921	0.117563
$892$	0	0
$893$	24.0827	0.805898
$894$	0	0
$895$	−3.41884	−0.114279
$896$	0	0
$897$	−3.64935	−0.121848
$898$	0	0
$899$	5.32027	0.177441
$900$	0	0
$901$	42.6176	1.41980
$902$	0	0
$903$	0.792951	0.0263878
$904$	0	0
$905$	−49.0730	−1.63124
$906$	0	0
$907$	−36.5466	−1.21351	−0.606756	−	0.794889i	$-0.707529\pi$
$907$	−36.5466	−1.21351	−0.606756	+	0.794889i	$0.707529\pi$
$908$	0	0
$909$	6.05636	0.200877
$910$	0	0
$911$	32.4069	1.07369	0.536845	−	0.843681i	$-0.319616\pi$
$911$	32.4069	1.07369	0.536845	+	0.843681i	$0.319616\pi$
$912$	0	0
$913$	9.08765	0.300757
$914$	0	0
$915$	−1.79788	−0.0594362
$916$	0	0
$917$	−14.5992	−0.482107
$918$	0	0
$919$	−4.49915	−0.148413	−0.0742066	−	0.997243i	$-0.523642\pi$
$919$	−4.49915	−0.148413	−0.0742066	+	0.997243i	$0.523642\pi$
$920$	0	0
$921$	26.6122	0.876900
$922$	0	0
$923$	23.4452	0.771709
$924$	0	0
$925$	28.1056	0.924106
$926$	0	0
$927$	−10.3203	−0.338963
$928$	0	0
$929$	−28.8104	−0.945240	−0.472620	−	0.881266i	$-0.656692\pi$
$929$	−28.8104	−0.945240	−0.472620	+	0.881266i	$0.656692\pi$
$930$	0	0
$931$	17.6153	0.577318
$932$	0	0
$933$	−3.92862	−0.128617
$934$	0	0
$935$	−75.4954	−2.46897
$936$	0	0
$937$	−6.81247	−0.222554	−0.111277	−	0.993789i	$-0.535494\pi$
$937$	−6.81247	−0.222554	−0.111277	+	0.993789i	$0.535494\pi$
$938$	0	0
$939$	−10.1563	−0.331439
$940$	0	0
$941$	3.27509	0.106765	0.0533824	−	0.998574i	$-0.483000\pi$
$941$	3.27509	0.106765	0.0533824	+	0.998574i	$0.483000\pi$
$942$	0	0
$943$	6.52429	0.212460
$944$	0	0
$945$	2.37424	0.0772339
$946$	0	0
$947$	−26.3180	−0.855219	−0.427610	−	0.903964i	$-0.640644\pi$
$947$	−26.3180	−0.855219	−0.427610	+	0.903964i	$0.640644\pi$
$948$	0	0
$949$	−26.5732	−0.862602
$950$	0	0
$951$	21.2960	0.690570
$952$	0	0
$953$	49.3912	1.59994	0.799968	−	0.600042i	$-0.204850\pi$
$953$	49.3912	1.59994	0.799968	+	0.600042i	$0.204850\pi$
$954$	0	0
$955$	−4.18851	−0.135537
$956$	0	0
$957$	−3.50921	−0.113437
$958$	0	0
$959$	−13.1436	−0.424428
$960$	0	0
$961$	−2.69473	−0.0869269
$962$	0	0
$963$	12.8482	0.414027
$964$	0	0
$965$	−49.8325	−1.60417
$966$	0	0
$967$	−32.0459	−1.03053	−0.515264	−	0.857031i	$-0.672306\pi$
$967$	−32.0459	−1.03053	−0.515264	+	0.857031i	$0.672306\pi$
$968$	0	0
$969$	−18.9122	−0.607546
$970$	0	0
$971$	8.64105	0.277304	0.138652	−	0.990341i	$-0.455723\pi$
$971$	8.64105	0.277304	0.138652	+	0.990341i	$0.455723\pi$
$972$	0	0
$973$	−6.18084	−0.198149
$974$	0	0
$975$	−17.2925	−0.553802
$976$	0	0
$977$	6.10450	0.195300	0.0976502	−	0.995221i	$-0.468867\pi$
$977$	6.10450	0.195300	0.0976502	+	0.995221i	$0.468867\pi$
$978$	0	0
$979$	−37.7223	−1.20561
$980$	0	0
$981$	−9.57760	−0.305789
$982$	0	0
$983$	−37.5267	−1.19692	−0.598458	−	0.801154i	$-0.704220\pi$
$983$	−37.5267	−1.19692	−0.598458	+	0.801154i	$0.704220\pi$
$984$	0	0
$985$	−33.9593	−1.08203
$986$	0	0
$987$	6.67893	0.212593
$988$	0	0
$989$	−1.04224	−0.0331414
$990$	0	0
$991$	−21.9560	−0.697457	−0.348728	−	0.937224i	$-0.613386\pi$
$991$	−21.9560	−0.697457	−0.348728	+	0.937224i	$0.613386\pi$
$992$	0	0
$993$	21.2770	0.675205
$994$	0	0
$995$	−64.0076	−2.02918
$996$	0	0
$997$	−4.70970	−0.149158	−0.0745788	−	0.997215i	$-0.523761\pi$
$997$	−4.70970	−0.149158	−0.0745788	+	0.997215i	$0.523761\pi$
$998$	0	0
$999$	5.93131	0.187658

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.2	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.12066 q^{5} -0.760812 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.12066 q^{5} -0.760812 q^{7} +1.00000 q^{9} +3.50921 q^{11} -3.64935 q^{13} -3.12066 q^{15} +6.89390 q^{17} -2.74332 q^{19} -0.760812 q^{21} +1.00000 q^{23} +4.73851 q^{25} +1.00000 q^{27} -1.00000 q^{29} -5.32027 q^{31} +3.50921 q^{33} +2.37424 q^{35} +5.93131 q^{37} -3.64935 q^{39} +6.52429 q^{41} -1.04224 q^{43} -3.12066 q^{45} -8.77868 q^{47} -6.42116 q^{49} +6.89390 q^{51} +6.18193 q^{53} -10.9510 q^{55} -2.74332 q^{57} -11.2513 q^{59} +0.576123 q^{61} -0.760812 q^{63} +11.3884 q^{65} +3.98419 q^{67} +1.00000 q^{69} -6.42449 q^{71} +7.28163 q^{73} +4.73851 q^{75} -2.66985 q^{77} +6.45544 q^{79} +1.00000 q^{81} +2.58965 q^{83} -21.5135 q^{85} -1.00000 q^{87} -10.7495 q^{89} +2.77647 q^{91} -5.32027 q^{93} +8.56096 q^{95} +17.1691 q^{97} +3.50921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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