LMFDB - Embedded newform 8004.2.a.f.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:	$1$
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} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 $ x^9 - x^8 - 17x^7 + 4x^6 + 75x^5 + x^4 - 118x^3 - 26x^2 + 60x + 24
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			$-0.808455$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.808455 q^{5} -2.11918 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.808455 q^{5} -2.11918 q^{7} +1.00000 q^{9} -0.503028 q^{11} -3.72396 q^{13} +0.808455 q^{15} +2.72926 q^{17} +6.39117 q^{19} -2.11918 q^{21} -1.00000 q^{23} -4.34640 q^{25} +1.00000 q^{27} -1.00000 q^{29} +0.365919 q^{31} -0.503028 q^{33} -1.71326 q^{35} +2.18071 q^{37} -3.72396 q^{39} -8.11013 q^{41} -9.75936 q^{43} +0.808455 q^{45} -3.69437 q^{47} -2.50907 q^{49} +2.72926 q^{51} +7.96941 q^{53} -0.406676 q^{55} +6.39117 q^{57} -8.52659 q^{59} -4.14009 q^{61} -2.11918 q^{63} -3.01065 q^{65} +11.4755 q^{67} -1.00000 q^{69} -15.3581 q^{71} -6.69271 q^{73} -4.34640 q^{75} +1.06601 q^{77} +14.4850 q^{79} +1.00000 q^{81} -0.758561 q^{83} +2.20648 q^{85} -1.00000 q^{87} +8.63314 q^{89} +7.89175 q^{91} +0.365919 q^{93} +5.16697 q^{95} +4.82346 q^{97} -0.503028 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * 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
$5$	0.808455	0.361552	0.180776	−	0.983524i	$-0.442139\pi$
$5$	0.808455	0.361552	0.180776	+	0.983524i	$0.442139\pi$
$6$	0	0
$7$	−2.11918	−0.800975	−0.400488	−	0.916302i	$-0.631159\pi$
$7$	−2.11918	−0.800975	−0.400488	+	0.916302i	$0.631159\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.503028	−0.151669	−0.0758343	−	0.997120i	$-0.524162\pi$
$11$	−0.503028	−0.151669	−0.0758343	+	0.997120i	$0.524162\pi$
$12$	0	0
$13$	−3.72396	−1.03284	−0.516421	−	0.856335i	$-0.672736\pi$
$13$	−3.72396	−1.03284	−0.516421	+	0.856335i	$0.672736\pi$
$14$	0	0
$15$	0.808455	0.208742
$16$	0	0
$17$	2.72926	0.661943	0.330971	−	0.943641i	$-0.392624\pi$
$17$	2.72926	0.661943	0.330971	+	0.943641i	$0.392624\pi$
$18$	0	0
$19$	6.39117	1.46623	0.733117	−	0.680102i	$-0.238065\pi$
$19$	6.39117	1.46623	0.733117	+	0.680102i	$0.238065\pi$
$20$	0	0
$21$	−2.11918	−0.462443
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.34640	−0.869280
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.365919	0.0657210	0.0328605	−	0.999460i	$-0.489538\pi$
$31$	0.365919	0.0657210	0.0328605	+	0.999460i	$0.489538\pi$
$32$	0	0
$33$	−0.503028	−0.0875659
$34$	0	0
$35$	−1.71326	−0.289594
$36$	0	0
$37$	2.18071	0.358506	0.179253	−	0.983803i	$-0.442632\pi$
$37$	2.18071	0.358506	0.179253	+	0.983803i	$0.442632\pi$
$38$	0	0
$39$	−3.72396	−0.596311
$40$	0	0
$41$	−8.11013	−1.26659	−0.633295	−	0.773911i	$-0.718298\pi$
$41$	−8.11013	−1.26659	−0.633295	+	0.773911i	$0.718298\pi$
$42$	0	0
$43$	−9.75936	−1.48829	−0.744144	−	0.668019i	$-0.767143\pi$
$43$	−9.75936	−1.48829	−0.744144	+	0.668019i	$0.767143\pi$
$44$	0	0
$45$	0.808455	0.120517
$46$	0	0
$47$	−3.69437	−0.538880	−0.269440	−	0.963017i	$-0.586839\pi$
$47$	−3.69437	−0.538880	−0.269440	+	0.963017i	$0.586839\pi$
$48$	0	0
$49$	−2.50907	−0.358439
$50$	0	0
$51$	2.72926	0.382173
$52$	0	0
$53$	7.96941	1.09468	0.547341	−	0.836909i	$-0.315640\pi$
$53$	7.96941	1.09468	0.547341	+	0.836909i	$0.315640\pi$
$54$	0	0
$55$	−0.406676	−0.0548361
$56$	0	0
$57$	6.39117	0.846531
$58$	0	0
$59$	−8.52659	−1.11007	−0.555034	−	0.831828i	$-0.687295\pi$
$59$	−8.52659	−1.11007	−0.555034	+	0.831828i	$0.687295\pi$
$60$	0	0
$61$	−4.14009	−0.530084	−0.265042	−	0.964237i	$-0.585386\pi$
$61$	−4.14009	−0.530084	−0.265042	+	0.964237i	$0.585386\pi$
$62$	0	0
$63$	−2.11918	−0.266992
$64$	0	0
$65$	−3.01065	−0.373426
$66$	0	0
$67$	11.4755	1.40195	0.700976	−	0.713185i	$-0.252748\pi$
$67$	11.4755	1.40195	0.700976	+	0.713185i	$0.252748\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−15.3581	−1.82267	−0.911336	−	0.411664i	$-0.864948\pi$
$71$	−15.3581	−1.82267	−0.911336	+	0.411664i	$0.864948\pi$
$72$	0	0
$73$	−6.69271	−0.783323	−0.391661	−	0.920109i	$-0.628099\pi$
$73$	−6.69271	−0.783323	−0.391661	+	0.920109i	$0.628099\pi$
$74$	0	0
$75$	−4.34640	−0.501879
$76$	0	0
$77$	1.06601	0.121483
$78$	0	0
$79$	14.4850	1.62969	0.814844	−	0.579681i	$-0.196823\pi$
$79$	14.4850	1.62969	0.814844	+	0.579681i	$0.196823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.758561	−0.0832629	−0.0416314	−	0.999133i	$-0.513256\pi$
$83$	−0.758561	−0.0832629	−0.0416314	+	0.999133i	$0.513256\pi$
$84$	0	0
$85$	2.20648	0.239327
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	8.63314	0.915111	0.457555	−	0.889181i	$-0.348725\pi$
$89$	8.63314	0.915111	0.457555	+	0.889181i	$0.348725\pi$
$90$	0	0
$91$	7.89175	0.827280
$92$	0	0
$93$	0.365919	0.0379440
$94$	0	0
$95$	5.16697	0.530120
$96$	0	0
$97$	4.82346	0.489748	0.244874	−	0.969555i	$-0.421253\pi$
$97$	4.82346	0.489748	0.244874	+	0.969555i	$0.421253\pi$
$98$	0	0
$99$	−0.503028	−0.0505562
$100$	0	0
$101$	3.00902	0.299409	0.149704	−	0.988731i	$-0.452168\pi$
$101$	3.00902	0.299409	0.149704	+	0.988731i	$0.452168\pi$
$102$	0	0
$103$	−13.1223	−1.29298	−0.646490	−	0.762922i	$-0.723764\pi$
$103$	−13.1223	−1.29298	−0.646490	+	0.762922i	$0.723764\pi$
$104$	0	0
$105$	−1.71326	−0.167197
$106$	0	0
$107$	−0.583801	−0.0564382	−0.0282191	−	0.999602i	$-0.508984\pi$
$107$	−0.583801	−0.0564382	−0.0282191	+	0.999602i	$0.508984\pi$
$108$	0	0
$109$	9.60016	0.919529	0.459764	−	0.888041i	$-0.347934\pi$
$109$	9.60016	0.919529	0.459764	+	0.888041i	$0.347934\pi$
$110$	0	0
$111$	2.18071	0.206984
$112$	0	0
$113$	−1.43969	−0.135435	−0.0677174	−	0.997705i	$-0.521572\pi$
$113$	−1.43969	−0.135435	−0.0677174	+	0.997705i	$0.521572\pi$
$114$	0	0
$115$	−0.808455	−0.0753888
$116$	0	0
$117$	−3.72396	−0.344280
$118$	0	0
$119$	−5.78380	−0.530200
$120$	0	0
$121$	−10.7470	−0.976997
$122$	0	0
$123$	−8.11013	−0.731266
$124$	0	0
$125$	−7.55614	−0.675842
$126$	0	0
$127$	15.1799	1.34700	0.673499	−	0.739188i	$-0.264791\pi$
$127$	15.1799	1.34700	0.673499	+	0.739188i	$0.264791\pi$
$128$	0	0
$129$	−9.75936	−0.859264
$130$	0	0
$131$	−7.73687	−0.675973	−0.337987	−	0.941151i	$-0.609746\pi$
$131$	−7.73687	−0.675973	−0.337987	+	0.941151i	$0.609746\pi$
$132$	0	0
$133$	−13.5440	−1.17442
$134$	0	0
$135$	0.808455	0.0695807
$136$	0	0
$137$	−13.4394	−1.14821	−0.574104	−	0.818782i	$-0.694650\pi$
$137$	−13.4394	−1.14821	−0.574104	+	0.818782i	$0.694650\pi$
$138$	0	0
$139$	−10.4763	−0.888585	−0.444293	−	0.895882i	$-0.646545\pi$
$139$	−10.4763	−0.888585	−0.444293	+	0.895882i	$0.646545\pi$
$140$	0	0
$141$	−3.69437	−0.311122
$142$	0	0
$143$	1.87326	0.156650
$144$	0	0
$145$	−0.808455	−0.0671385
$146$	0	0
$147$	−2.50907	−0.206945
$148$	0	0
$149$	12.7406	1.04375	0.521876	−	0.853021i	$-0.325232\pi$
$149$	12.7406	1.04375	0.521876	+	0.853021i	$0.325232\pi$
$150$	0	0
$151$	−22.6411	−1.84251	−0.921253	−	0.388963i	$-0.872833\pi$
$151$	−22.6411	−1.84251	−0.921253	+	0.388963i	$0.872833\pi$
$152$	0	0
$153$	2.72926	0.220648
$154$	0	0
$155$	0.295829	0.0237615
$156$	0	0
$157$	11.9624	0.954707	0.477354	−	0.878711i	$-0.341596\pi$
$157$	11.9624	0.954707	0.477354	+	0.878711i	$0.341596\pi$
$158$	0	0
$159$	7.96941	0.632015
$160$	0	0
$161$	2.11918	0.167015
$162$	0	0
$163$	−20.5898	−1.61272	−0.806358	−	0.591428i	$-0.798564\pi$
$163$	−20.5898	−1.61272	−0.806358	+	0.591428i	$0.798564\pi$
$164$	0	0
$165$	−0.406676	−0.0316596
$166$	0	0
$167$	3.57254	0.276452	0.138226	−	0.990401i	$-0.455860\pi$
$167$	3.57254	0.276452	0.138226	+	0.990401i	$0.455860\pi$
$168$	0	0
$169$	0.867887	0.0667605
$170$	0	0
$171$	6.39117	0.488745
$172$	0	0
$173$	−2.23953	−0.170269	−0.0851343	−	0.996369i	$-0.527132\pi$
$173$	−2.23953	−0.170269	−0.0851343	+	0.996369i	$0.527132\pi$
$174$	0	0
$175$	9.21081	0.696272
$176$	0	0
$177$	−8.52659	−0.640898
$178$	0	0
$179$	1.96377	0.146779	0.0733894	−	0.997303i	$-0.476618\pi$
$179$	1.96377	0.146779	0.0733894	+	0.997303i	$0.476618\pi$
$180$	0	0
$181$	1.57604	0.117146	0.0585731	−	0.998283i	$-0.481345\pi$
$181$	1.57604	0.117146	0.0585731	+	0.998283i	$0.481345\pi$
$182$	0	0
$183$	−4.14009	−0.306044
$184$	0	0
$185$	1.76300	0.129619
$186$	0	0
$187$	−1.37289	−0.100396
$188$	0	0
$189$	−2.11918	−0.154148
$190$	0	0
$191$	−13.6885	−0.990464	−0.495232	−	0.868761i	$-0.664917\pi$
$191$	−13.6885	−0.990464	−0.495232	+	0.868761i	$0.664917\pi$
$192$	0	0
$193$	−15.9632	−1.14905	−0.574527	−	0.818486i	$-0.694814\pi$
$193$	−15.9632	−1.14905	−0.574527	+	0.818486i	$0.694814\pi$
$194$	0	0
$195$	−3.01065	−0.215597
$196$	0	0
$197$	−3.50397	−0.249647	−0.124824	−	0.992179i	$-0.539836\pi$
$197$	−3.50397	−0.249647	−0.124824	+	0.992179i	$0.539836\pi$
$198$	0	0
$199$	−5.42153	−0.384322	−0.192161	−	0.981363i	$-0.561550\pi$
$199$	−5.42153	−0.384322	−0.192161	+	0.981363i	$0.561550\pi$
$200$	0	0
$201$	11.4755	0.809418
$202$	0	0
$203$	2.11918	0.148737
$204$	0	0
$205$	−6.55668	−0.457938
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−3.21494	−0.222382
$210$	0	0
$211$	20.1459	1.38690	0.693450	−	0.720505i	$-0.256090\pi$
$211$	20.1459	1.38690	0.693450	+	0.720505i	$0.256090\pi$
$212$	0	0
$213$	−15.3581	−1.05232
$214$	0	0
$215$	−7.89000	−0.538094
$216$	0	0
$217$	−0.775448	−0.0526409
$218$	0	0
$219$	−6.69271	−0.452251
$220$	0	0
$221$	−10.1637	−0.683682
$222$	0	0
$223$	13.7246	0.919069	0.459535	−	0.888160i	$-0.348016\pi$
$223$	13.7246	0.919069	0.459535	+	0.888160i	$0.348016\pi$
$224$	0	0
$225$	−4.34640	−0.289760
$226$	0	0
$227$	−7.82157	−0.519136	−0.259568	−	0.965725i	$-0.583580\pi$
$227$	−7.82157	−0.519136	−0.259568	+	0.965725i	$0.583580\pi$
$228$	0	0
$229$	−15.7428	−1.04031	−0.520157	−	0.854070i	$-0.674127\pi$
$229$	−15.7428	−1.04031	−0.520157	+	0.854070i	$0.674127\pi$
$230$	0	0
$231$	1.06601	0.0701382
$232$	0	0
$233$	−17.0283	−1.11556	−0.557781	−	0.829988i	$-0.688347\pi$
$233$	−17.0283	−1.11556	−0.557781	+	0.829988i	$0.688347\pi$
$234$	0	0
$235$	−2.98673	−0.194833
$236$	0	0
$237$	14.4850	0.940901
$238$	0	0
$239$	−3.28621	−0.212567	−0.106284	−	0.994336i	$-0.533895\pi$
$239$	−3.28621	−0.212567	−0.106284	+	0.994336i	$0.533895\pi$
$240$	0	0
$241$	−24.3514	−1.56861	−0.784304	−	0.620376i	$-0.786980\pi$
$241$	−24.3514	−1.56861	−0.784304	+	0.620376i	$0.786980\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.02847	−0.129594
$246$	0	0
$247$	−23.8005	−1.51439
$248$	0	0
$249$	−0.758561	−0.0480718
$250$	0	0
$251$	−7.15939	−0.451897	−0.225948	−	0.974139i	$-0.572548\pi$
$251$	−7.15939	−0.451897	−0.225948	+	0.974139i	$0.572548\pi$
$252$	0	0
$253$	0.503028	0.0316251
$254$	0	0
$255$	2.20648	0.138175
$256$	0	0
$257$	−10.6342	−0.663346	−0.331673	−	0.943394i	$-0.607613\pi$
$257$	−10.6342	−0.663346	−0.331673	+	0.943394i	$0.607613\pi$
$258$	0	0
$259$	−4.62131	−0.287154
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−3.54207	−0.218413	−0.109207	−	0.994019i	$-0.534831\pi$
$263$	−3.54207	−0.218413	−0.109207	+	0.994019i	$0.534831\pi$
$264$	0	0
$265$	6.44291	0.395785
$266$	0	0
$267$	8.63314	0.528339
$268$	0	0
$269$	−7.49318	−0.456867	−0.228434	−	0.973559i	$-0.573360\pi$
$269$	−7.49318	−0.456867	−0.228434	+	0.973559i	$0.573360\pi$
$270$	0	0
$271$	18.7656	1.13993	0.569966	−	0.821668i	$-0.306956\pi$
$271$	18.7656	1.13993	0.569966	+	0.821668i	$0.306956\pi$
$272$	0	0
$273$	7.89175	0.477630
$274$	0	0
$275$	2.18636	0.131843
$276$	0	0
$277$	−7.94073	−0.477112	−0.238556	−	0.971129i	$-0.576674\pi$
$277$	−7.94073	−0.477112	−0.238556	+	0.971129i	$0.576674\pi$
$278$	0	0
$279$	0.365919	0.0219070
$280$	0	0
$281$	−21.8351	−1.30257	−0.651287	−	0.758832i	$-0.725770\pi$
$281$	−21.8351	−1.30257	−0.651287	+	0.758832i	$0.725770\pi$
$282$	0	0
$283$	−0.821050	−0.0488064	−0.0244032	−	0.999702i	$-0.507769\pi$
$283$	−0.821050	−0.0488064	−0.0244032	+	0.999702i	$0.507769\pi$
$284$	0	0
$285$	5.16697	0.306065
$286$	0	0
$287$	17.1868	1.01451
$288$	0	0
$289$	−9.55114	−0.561832
$290$	0	0
$291$	4.82346	0.282756
$292$	0	0
$293$	7.86387	0.459412	0.229706	−	0.973260i	$-0.426224\pi$
$293$	7.86387	0.459412	0.229706	+	0.973260i	$0.426224\pi$
$294$	0	0
$295$	−6.89336	−0.401347
$296$	0	0
$297$	−0.503028	−0.0291886
$298$	0	0
$299$	3.72396	0.215362
$300$	0	0
$301$	20.6819	1.19208
$302$	0	0
$303$	3.00902	0.172864
$304$	0	0
$305$	−3.34707	−0.191653
$306$	0	0
$307$	3.56292	0.203347	0.101673	−	0.994818i	$-0.467580\pi$
$307$	3.56292	0.203347	0.101673	+	0.994818i	$0.467580\pi$
$308$	0	0
$309$	−13.1223	−0.746503
$310$	0	0
$311$	31.9740	1.81308	0.906540	−	0.422119i	$-0.138714\pi$
$311$	31.9740	1.81308	0.906540	+	0.422119i	$0.138714\pi$
$312$	0	0
$313$	−26.4123	−1.49291	−0.746456	−	0.665434i	$-0.768246\pi$
$313$	−26.4123	−1.49291	−0.746456	+	0.665434i	$0.768246\pi$
$314$	0	0
$315$	−1.71326	−0.0965314
$316$	0	0
$317$	−10.9760	−0.616472	−0.308236	−	0.951310i	$-0.599739\pi$
$317$	−10.9760	−0.616472	−0.308236	+	0.951310i	$0.599739\pi$
$318$	0	0
$319$	0.503028	0.0281642
$320$	0	0
$321$	−0.583801	−0.0325846
$322$	0	0
$323$	17.4432	0.970564
$324$	0	0
$325$	16.1858	0.897828
$326$	0	0
$327$	9.60016	0.530890
$328$	0	0
$329$	7.82905	0.431629
$330$	0	0
$331$	−3.88297	−0.213427	−0.106714	−	0.994290i	$-0.534033\pi$
$331$	−3.88297	−0.213427	−0.106714	+	0.994290i	$0.534033\pi$
$332$	0	0
$333$	2.18071	0.119502
$334$	0	0
$335$	9.27740	0.506879
$336$	0	0
$337$	−26.4335	−1.43992	−0.719962	−	0.694013i	$-0.755841\pi$
$337$	−26.4335	−1.43992	−0.719962	+	0.694013i	$0.755841\pi$
$338$	0	0
$339$	−1.43969	−0.0781933
$340$	0	0
$341$	−0.184067	−0.00996781
$342$	0	0
$343$	20.1514	1.08808
$344$	0	0
$345$	−0.808455	−0.0435257
$346$	0	0
$347$	−0.587539	−0.0315407	−0.0157704	−	0.999876i	$-0.505020\pi$
$347$	−0.587539	−0.0315407	−0.0157704	+	0.999876i	$0.505020\pi$
$348$	0	0
$349$	12.1333	0.649481	0.324741	−	0.945803i	$-0.394723\pi$
$349$	12.1333	0.649481	0.324741	+	0.945803i	$0.394723\pi$
$350$	0	0
$351$	−3.72396	−0.198770
$352$	0	0
$353$	19.8410	1.05603	0.528014	−	0.849235i	$-0.322937\pi$
$353$	19.8410	1.05603	0.528014	+	0.849235i	$0.322937\pi$
$354$	0	0
$355$	−12.4163	−0.658990
$356$	0	0
$357$	−5.78380	−0.306111
$358$	0	0
$359$	1.57434	0.0830903	0.0415451	−	0.999137i	$-0.486772\pi$
$359$	1.57434	0.0830903	0.0415451	+	0.999137i	$0.486772\pi$
$360$	0	0
$361$	21.8470	1.14984
$362$	0	0
$363$	−10.7470	−0.564069
$364$	0	0
$365$	−5.41075	−0.283212
$366$	0	0
$367$	4.86125	0.253755	0.126877	−	0.991918i	$-0.459505\pi$
$367$	4.86125	0.253755	0.126877	+	0.991918i	$0.459505\pi$
$368$	0	0
$369$	−8.11013	−0.422197
$370$	0	0
$371$	−16.8886	−0.876814
$372$	0	0
$373$	−15.4873	−0.801899	−0.400950	−	0.916100i	$-0.631320\pi$
$373$	−15.4873	−0.801899	−0.400950	+	0.916100i	$0.631320\pi$
$374$	0	0
$375$	−7.55614	−0.390198
$376$	0	0
$377$	3.72396	0.191794
$378$	0	0
$379$	7.95479	0.408610	0.204305	−	0.978907i	$-0.434507\pi$
$379$	7.95479	0.408610	0.204305	+	0.978907i	$0.434507\pi$
$380$	0	0
$381$	15.1799	0.777689
$382$	0	0
$383$	18.2245	0.931231	0.465615	−	0.884987i	$-0.345833\pi$
$383$	18.2245	0.931231	0.465615	+	0.884987i	$0.345833\pi$
$384$	0	0
$385$	0.861819	0.0439224
$386$	0	0
$387$	−9.75936	−0.496096
$388$	0	0
$389$	−11.4854	−0.582333	−0.291167	−	0.956672i	$-0.594043\pi$
$389$	−11.4854	−0.582333	−0.291167	+	0.956672i	$0.594043\pi$
$390$	0	0
$391$	−2.72926	−0.138025
$392$	0	0
$393$	−7.73687	−0.390273
$394$	0	0
$395$	11.7105	0.589217
$396$	0	0
$397$	−12.3460	−0.619628	−0.309814	−	0.950797i	$-0.600267\pi$
$397$	−12.3460	−0.619628	−0.309814	+	0.950797i	$0.600267\pi$
$398$	0	0
$399$	−13.5440	−0.678050
$400$	0	0
$401$	−32.8871	−1.64230	−0.821152	−	0.570710i	$-0.806668\pi$
$401$	−32.8871	−1.64230	−0.821152	+	0.570710i	$0.806668\pi$
$402$	0	0
$403$	−1.36267	−0.0678793
$404$	0	0
$405$	0.808455	0.0401724
$406$	0	0
$407$	−1.09696	−0.0543741
$408$	0	0
$409$	3.03882	0.150260	0.0751301	−	0.997174i	$-0.476063\pi$
$409$	3.03882	0.150260	0.0751301	+	0.997174i	$0.476063\pi$
$410$	0	0
$411$	−13.4394	−0.662918
$412$	0	0
$413$	18.0694	0.889136
$414$	0	0
$415$	−0.613262	−0.0301039
$416$	0	0
$417$	−10.4763	−0.513025
$418$	0	0
$419$	12.7499	0.622872	0.311436	−	0.950267i	$-0.399190\pi$
$419$	12.7499	0.622872	0.311436	+	0.950267i	$0.399190\pi$
$420$	0	0
$421$	−16.8970	−0.823509	−0.411755	−	0.911295i	$-0.635084\pi$
$421$	−16.8970	−0.823509	−0.411755	+	0.911295i	$0.635084\pi$
$422$	0	0
$423$	−3.69437	−0.179627
$424$	0	0
$425$	−11.8625	−0.575414
$426$	0	0
$427$	8.77360	0.424584
$428$	0	0
$429$	1.87326	0.0904417
$430$	0	0
$431$	2.23267	0.107544	0.0537719	−	0.998553i	$-0.482876\pi$
$431$	2.23267	0.107544	0.0537719	+	0.998553i	$0.482876\pi$
$432$	0	0
$433$	16.2794	0.782338	0.391169	−	0.920319i	$-0.372071\pi$
$433$	16.2794	0.782338	0.391169	+	0.920319i	$0.372071\pi$
$434$	0	0
$435$	−0.808455	−0.0387624
$436$	0	0
$437$	−6.39117	−0.305731
$438$	0	0
$439$	−20.2293	−0.965493	−0.482747	−	0.875760i	$-0.660361\pi$
$439$	−20.2293	−0.965493	−0.482747	+	0.875760i	$0.660361\pi$
$440$	0	0
$441$	−2.50907	−0.119480
$442$	0	0
$443$	30.6934	1.45829	0.729144	−	0.684360i	$-0.239918\pi$
$443$	30.6934	1.45829	0.729144	+	0.684360i	$0.239918\pi$
$444$	0	0
$445$	6.97950	0.330860
$446$	0	0
$447$	12.7406	0.602611
$448$	0	0
$449$	28.2800	1.33462	0.667308	−	0.744782i	$-0.267447\pi$
$449$	28.2800	1.33462	0.667308	+	0.744782i	$0.267447\pi$
$450$	0	0
$451$	4.07962	0.192102
$452$	0	0
$453$	−22.6411	−1.06377
$454$	0	0
$455$	6.38012	0.299105
$456$	0	0
$457$	−8.45765	−0.395632	−0.197816	−	0.980239i	$-0.563385\pi$
$457$	−8.45765	−0.395632	−0.197816	+	0.980239i	$0.563385\pi$
$458$	0	0
$459$	2.72926	0.127391
$460$	0	0
$461$	−5.14575	−0.239661	−0.119831	−	0.992794i	$-0.538235\pi$
$461$	−5.14575	−0.239661	−0.119831	+	0.992794i	$0.538235\pi$
$462$	0	0
$463$	8.80611	0.409254	0.204627	−	0.978840i	$-0.434402\pi$
$463$	8.80611	0.409254	0.204627	+	0.978840i	$0.434402\pi$
$464$	0	0
$465$	0.295829	0.0137187
$466$	0	0
$467$	−8.13419	−0.376405	−0.188203	−	0.982130i	$-0.560266\pi$
$467$	−8.13419	−0.376405	−0.188203	+	0.982130i	$0.560266\pi$
$468$	0	0
$469$	−24.3186	−1.12293
$470$	0	0
$471$	11.9624	0.551201
$472$	0	0
$473$	4.90923	0.225727
$474$	0	0
$475$	−27.7786	−1.27457
$476$	0	0
$477$	7.96941	0.364894
$478$	0	0
$479$	41.2610	1.88526	0.942632	−	0.333833i	$-0.108342\pi$
$479$	41.2610	1.88526	0.942632	+	0.333833i	$0.108342\pi$
$480$	0	0
$481$	−8.12087	−0.370280
$482$	0	0
$483$	2.11918	0.0964261
$484$	0	0
$485$	3.89955	0.177070
$486$	0	0
$487$	5.04942	0.228811	0.114406	−	0.993434i	$-0.463504\pi$
$487$	5.04942	0.228811	0.114406	+	0.993434i	$0.463504\pi$
$488$	0	0
$489$	−20.5898	−0.931102
$490$	0	0
$491$	−16.5566	−0.747191	−0.373595	−	0.927592i	$-0.621875\pi$
$491$	−16.5566	−0.747191	−0.373595	+	0.927592i	$0.621875\pi$
$492$	0	0
$493$	−2.72926	−0.122920
$494$	0	0
$495$	−0.406676	−0.0182787
$496$	0	0
$497$	32.5466	1.45991
$498$	0	0
$499$	−26.2552	−1.17534	−0.587672	−	0.809099i	$-0.699956\pi$
$499$	−26.2552	−1.17534	−0.587672	+	0.809099i	$0.699956\pi$
$500$	0	0
$501$	3.57254	0.159610
$502$	0	0
$503$	−17.7303	−0.790555	−0.395278	−	0.918562i	$-0.629352\pi$
$503$	−17.7303	−0.790555	−0.395278	+	0.918562i	$0.629352\pi$
$504$	0	0
$505$	2.43266	0.108252
$506$	0	0
$507$	0.867887	0.0385442
$508$	0	0
$509$	−3.39744	−0.150589	−0.0752944	−	0.997161i	$-0.523990\pi$
$509$	−3.39744	−0.150589	−0.0752944	+	0.997161i	$0.523990\pi$
$510$	0	0
$511$	14.1831	0.627422
$512$	0	0
$513$	6.39117	0.282177
$514$	0	0
$515$	−10.6088	−0.467480
$516$	0	0
$517$	1.85837	0.0817312
$518$	0	0
$519$	−2.23953	−0.0983046
$520$	0	0
$521$	−1.30472	−0.0571609	−0.0285805	−	0.999591i	$-0.509099\pi$
$521$	−1.30472	−0.0571609	−0.0285805	+	0.999591i	$0.509099\pi$
$522$	0	0
$523$	−17.7933	−0.778048	−0.389024	−	0.921228i	$-0.627188\pi$
$523$	−17.7933	−0.778048	−0.389024	+	0.921228i	$0.627188\pi$
$524$	0	0
$525$	9.21081	0.401993
$526$	0	0
$527$	0.998688	0.0435035
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.52659	−0.370022
$532$	0	0
$533$	30.2018	1.30819
$534$	0	0
$535$	−0.471977	−0.0204053
$536$	0	0
$537$	1.96377	0.0847428
$538$	0	0
$539$	1.26213	0.0543639
$540$	0	0
$541$	13.6321	0.586091	0.293046	−	0.956098i	$-0.405331\pi$
$541$	13.6321	0.586091	0.293046	+	0.956098i	$0.405331\pi$
$542$	0	0
$543$	1.57604	0.0676344
$544$	0	0
$545$	7.76130	0.332457
$546$	0	0
$547$	17.1234	0.732143	0.366071	−	0.930587i	$-0.380703\pi$
$547$	17.1234	0.732143	0.366071	+	0.930587i	$0.380703\pi$
$548$	0	0
$549$	−4.14009	−0.176695
$550$	0	0
$551$	−6.39117	−0.272273
$552$	0	0
$553$	−30.6963	−1.30534
$554$	0	0
$555$	1.76300	0.0748353
$556$	0	0
$557$	31.7737	1.34629	0.673146	−	0.739509i	$-0.264942\pi$
$557$	31.7737	1.34629	0.673146	+	0.739509i	$0.264942\pi$
$558$	0	0
$559$	36.3435	1.53717
$560$	0	0
$561$	−1.37289	−0.0579637
$562$	0	0
$563$	13.6670	0.575996	0.287998	−	0.957631i	$-0.407010\pi$
$563$	13.6670	0.575996	0.287998	+	0.957631i	$0.407010\pi$
$564$	0	0
$565$	−1.16393	−0.0489667
$566$	0	0
$567$	−2.11918	−0.0889972
$568$	0	0
$569$	1.93666	0.0811892	0.0405946	−	0.999176i	$-0.487075\pi$
$569$	1.93666	0.0811892	0.0405946	+	0.999176i	$0.487075\pi$
$570$	0	0
$571$	6.84672	0.286526	0.143263	−	0.989685i	$-0.454241\pi$
$571$	6.84672	0.286526	0.143263	+	0.989685i	$0.454241\pi$
$572$	0	0
$573$	−13.6885	−0.571845
$574$	0	0
$575$	4.34640	0.181257
$576$	0	0
$577$	−16.7255	−0.696292	−0.348146	−	0.937440i	$-0.613189\pi$
$577$	−16.7255	−0.696292	−0.348146	+	0.937440i	$0.613189\pi$
$578$	0	0
$579$	−15.9632	−0.663407
$580$	0	0
$581$	1.60753	0.0666915
$582$	0	0
$583$	−4.00884	−0.166029
$584$	0	0
$585$	−3.01065	−0.124475
$586$	0	0
$587$	32.0601	1.32326	0.661632	−	0.749829i	$-0.269864\pi$
$587$	32.0601	1.32326	0.661632	+	0.749829i	$0.269864\pi$
$588$	0	0
$589$	2.33865	0.0963623
$590$	0	0
$591$	−3.50397	−0.144134
$592$	0	0
$593$	19.2792	0.791700	0.395850	−	0.918315i	$-0.370450\pi$
$593$	19.2792	0.791700	0.395850	+	0.918315i	$0.370450\pi$
$594$	0	0
$595$	−4.67594	−0.191695
$596$	0	0
$597$	−5.42153	−0.221888
$598$	0	0
$599$	14.5525	0.594598	0.297299	−	0.954784i	$-0.403914\pi$
$599$	14.5525	0.594598	0.297299	+	0.954784i	$0.403914\pi$
$600$	0	0
$601$	−20.7650	−0.847021	−0.423511	−	0.905891i	$-0.639202\pi$
$601$	−20.7650	−0.847021	−0.423511	+	0.905891i	$0.639202\pi$
$602$	0	0
$603$	11.4755	0.467317
$604$	0	0
$605$	−8.68843	−0.353235
$606$	0	0
$607$	14.8584	0.603082	0.301541	−	0.953453i	$-0.402499\pi$
$607$	14.8584	0.603082	0.301541	+	0.953453i	$0.402499\pi$
$608$	0	0
$609$	2.11918	0.0858736
$610$	0	0
$611$	13.7577	0.556577
$612$	0	0
$613$	−20.6032	−0.832156	−0.416078	−	0.909329i	$-0.636596\pi$
$613$	−20.6032	−0.832156	−0.416078	+	0.909329i	$0.636596\pi$
$614$	0	0
$615$	−6.55668	−0.264391
$616$	0	0
$617$	20.8814	0.840656	0.420328	−	0.907372i	$-0.361915\pi$
$617$	20.8814	0.840656	0.420328	+	0.907372i	$0.361915\pi$
$618$	0	0
$619$	−13.8227	−0.555581	−0.277791	−	0.960642i	$-0.589602\pi$
$619$	−13.8227	−0.555581	−0.277791	+	0.960642i	$0.589602\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−18.2952	−0.732981
$624$	0	0
$625$	15.6232	0.624928
$626$	0	0
$627$	−3.21494	−0.128392
$628$	0	0
$629$	5.95172	0.237310
$630$	0	0
$631$	0.701913	0.0279427	0.0139714	−	0.999902i	$-0.495553\pi$
$631$	0.701913	0.0279427	0.0139714	+	0.999902i	$0.495553\pi$
$632$	0	0
$633$	20.1459	0.800727
$634$	0	0
$635$	12.2723	0.487010
$636$	0	0
$637$	9.34368	0.370210
$638$	0	0
$639$	−15.3581	−0.607557
$640$	0	0
$641$	−5.38656	−0.212756	−0.106378	−	0.994326i	$-0.533925\pi$
$641$	−5.38656	−0.212756	−0.106378	+	0.994326i	$0.533925\pi$
$642$	0	0
$643$	−14.6971	−0.579596	−0.289798	−	0.957088i	$-0.593588\pi$
$643$	−14.6971	−0.579596	−0.289798	+	0.957088i	$0.593588\pi$
$644$	0	0
$645$	−7.89000	−0.310669
$646$	0	0
$647$	−33.1737	−1.30419	−0.652096	−	0.758137i	$-0.726110\pi$
$647$	−33.1737	−1.30419	−0.652096	+	0.758137i	$0.726110\pi$
$648$	0	0
$649$	4.28911	0.168362
$650$	0	0
$651$	−0.775448	−0.0303922
$652$	0	0
$653$	2.31563	0.0906174	0.0453087	−	0.998973i	$-0.485573\pi$
$653$	2.31563	0.0906174	0.0453087	+	0.998973i	$0.485573\pi$
$654$	0	0
$655$	−6.25491	−0.244399
$656$	0	0
$657$	−6.69271	−0.261108
$658$	0	0
$659$	−42.5754	−1.65850	−0.829250	−	0.558877i	$-0.811232\pi$
$659$	−42.5754	−1.65850	−0.829250	+	0.558877i	$0.811232\pi$
$660$	0	0
$661$	−13.0540	−0.507742	−0.253871	−	0.967238i	$-0.581704\pi$
$661$	−13.0540	−0.507742	−0.253871	+	0.967238i	$0.581704\pi$
$662$	0	0
$663$	−10.1637	−0.394724
$664$	0	0
$665$	−10.9497	−0.424613
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	13.7246	0.530625
$670$	0	0
$671$	2.08258	0.0803971
$672$	0	0
$673$	−34.8478	−1.34328	−0.671642	−	0.740876i	$-0.734411\pi$
$673$	−34.8478	−1.34328	−0.671642	+	0.740876i	$0.734411\pi$
$674$	0	0
$675$	−4.34640	−0.167293
$676$	0	0
$677$	−1.20798	−0.0464264	−0.0232132	−	0.999731i	$-0.507390\pi$
$677$	−1.20798	−0.0464264	−0.0232132	+	0.999731i	$0.507390\pi$
$678$	0	0
$679$	−10.2218	−0.392276
$680$	0	0
$681$	−7.82157	−0.299723
$682$	0	0
$683$	35.5345	1.35969	0.679845	−	0.733355i	$-0.262047\pi$
$683$	35.5345	1.35969	0.679845	+	0.733355i	$0.262047\pi$
$684$	0	0
$685$	−10.8652	−0.415137
$686$	0	0
$687$	−15.7428	−0.600626
$688$	0	0
$689$	−29.6778	−1.13063
$690$	0	0
$691$	−19.4855	−0.741265	−0.370632	−	0.928780i	$-0.620859\pi$
$691$	−19.4855	−0.741265	−0.370632	+	0.928780i	$0.620859\pi$
$692$	0	0
$693$	1.06601	0.0404943
$694$	0	0
$695$	−8.46959	−0.321270
$696$	0	0
$697$	−22.1347	−0.838410
$698$	0	0
$699$	−17.0283	−0.644070
$700$	0	0
$701$	3.77525	0.142589	0.0712947	−	0.997455i	$-0.477287\pi$
$701$	3.77525	0.142589	0.0712947	+	0.997455i	$0.477287\pi$
$702$	0	0
$703$	13.9373	0.525654
$704$	0	0
$705$	−2.98673	−0.112487
$706$	0	0
$707$	−6.37666	−0.239819
$708$	0	0
$709$	6.48519	0.243556	0.121778	−	0.992557i	$-0.461140\pi$
$709$	6.48519	0.243556	0.121778	+	0.992557i	$0.461140\pi$
$710$	0	0
$711$	14.4850	0.543229
$712$	0	0
$713$	−0.365919	−0.0137038
$714$	0	0
$715$	1.51444	0.0566370
$716$	0	0
$717$	−3.28621	−0.122726
$718$	0	0
$719$	−30.7460	−1.14663	−0.573316	−	0.819334i	$-0.694343\pi$
$719$	−30.7460	−1.14663	−0.573316	+	0.819334i	$0.694343\pi$
$720$	0	0
$721$	27.8086	1.03565
$722$	0	0
$723$	−24.3514	−0.905637
$724$	0	0
$725$	4.34640	0.161421
$726$	0	0
$727$	24.5659	0.911100	0.455550	−	0.890210i	$-0.349443\pi$
$727$	24.5659	0.911100	0.455550	+	0.890210i	$0.349443\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.6358	−0.985162
$732$	0	0
$733$	9.20163	0.339870	0.169935	−	0.985455i	$-0.445644\pi$
$733$	9.20163	0.339870	0.169935	+	0.985455i	$0.445644\pi$
$734$	0	0
$735$	−2.02847	−0.0748213
$736$	0	0
$737$	−5.77249	−0.212632
$738$	0	0
$739$	−15.0557	−0.553832	−0.276916	−	0.960894i	$-0.589312\pi$
$739$	−15.0557	−0.553832	−0.276916	+	0.960894i	$0.589312\pi$
$740$	0	0
$741$	−23.8005	−0.874332
$742$	0	0
$743$	−0.563417	−0.0206698	−0.0103349	−	0.999947i	$-0.503290\pi$
$743$	−0.563417	−0.0206698	−0.0103349	+	0.999947i	$0.503290\pi$
$744$	0	0
$745$	10.3002	0.377371
$746$	0	0
$747$	−0.758561	−0.0277543
$748$	0	0
$749$	1.23718	0.0452056
$750$	0	0
$751$	−25.6888	−0.937397	−0.468698	−	0.883358i	$-0.655277\pi$
$751$	−25.6888	−0.937397	−0.468698	+	0.883358i	$0.655277\pi$
$752$	0	0
$753$	−7.15939	−0.260903
$754$	0	0
$755$	−18.3043	−0.666162
$756$	0	0
$757$	36.7558	1.33591	0.667957	−	0.744200i	$-0.267169\pi$
$757$	36.7558	1.33591	0.667957	+	0.744200i	$0.267169\pi$
$758$	0	0
$759$	0.503028	0.0182588
$760$	0	0
$761$	6.02427	0.218379	0.109190	−	0.994021i	$-0.465174\pi$
$761$	6.02427	0.218379	0.109190	+	0.994021i	$0.465174\pi$
$762$	0	0
$763$	−20.3445	−0.736520
$764$	0	0
$765$	2.20648	0.0797756
$766$	0	0
$767$	31.7527	1.14652
$768$	0	0
$769$	48.0015	1.73098	0.865490	−	0.500927i	$-0.167007\pi$
$769$	48.0015	1.73098	0.865490	+	0.500927i	$0.167007\pi$
$770$	0	0
$771$	−10.6342	−0.382983
$772$	0	0
$773$	5.38577	0.193713	0.0968564	−	0.995298i	$-0.469121\pi$
$773$	5.38577	0.193713	0.0968564	+	0.995298i	$0.469121\pi$
$774$	0	0
$775$	−1.59043	−0.0571299
$776$	0	0
$777$	−4.62131	−0.165789
$778$	0	0
$779$	−51.8332	−1.85712
$780$	0	0
$781$	7.72556	0.276442
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	9.67110	0.345176
$786$	0	0
$787$	−35.8416	−1.27762	−0.638808	−	0.769367i	$-0.720572\pi$
$787$	−35.8416	−1.27762	−0.638808	+	0.769367i	$0.720572\pi$
$788$	0	0
$789$	−3.54207	−0.126101
$790$	0	0
$791$	3.05097	0.108480
$792$	0	0
$793$	15.4175	0.547492
$794$	0	0
$795$	6.44291	0.228506
$796$	0	0
$797$	32.9828	1.16831	0.584156	−	0.811641i	$-0.301426\pi$
$797$	32.9828	1.16831	0.584156	+	0.811641i	$0.301426\pi$
$798$	0	0
$799$	−10.0829	−0.356708
$800$	0	0
$801$	8.63314	0.305037
$802$	0	0
$803$	3.36662	0.118805
$804$	0	0
$805$	1.71326	0.0603846
$806$	0	0
$807$	−7.49318	−0.263772
$808$	0	0
$809$	−20.1484	−0.708379	−0.354189	−	0.935174i	$-0.615243\pi$
$809$	−20.1484	−0.708379	−0.354189	+	0.935174i	$0.615243\pi$
$810$	0	0
$811$	19.0553	0.669121	0.334560	−	0.942374i	$-0.391412\pi$
$811$	19.0553	0.669121	0.334560	+	0.942374i	$0.391412\pi$
$812$	0	0
$813$	18.7656	0.658140
$814$	0	0
$815$	−16.6459	−0.583081
$816$	0	0
$817$	−62.3737	−2.18218
$818$	0	0
$819$	7.89175	0.275760
$820$	0	0
$821$	46.7070	1.63008	0.815042	−	0.579402i	$-0.196714\pi$
$821$	46.7070	1.63008	0.815042	+	0.579402i	$0.196714\pi$
$822$	0	0
$823$	43.2132	1.50632	0.753158	−	0.657839i	$-0.228529\pi$
$823$	43.2132	1.50632	0.753158	+	0.657839i	$0.228529\pi$
$824$	0	0
$825$	2.18636	0.0761193
$826$	0	0
$827$	35.9639	1.25059	0.625293	−	0.780390i	$-0.284979\pi$
$827$	35.9639	1.25059	0.625293	+	0.780390i	$0.284979\pi$
$828$	0	0
$829$	−11.4790	−0.398681	−0.199341	−	0.979930i	$-0.563880\pi$
$829$	−11.4790	−0.398681	−0.199341	+	0.979930i	$0.563880\pi$
$830$	0	0
$831$	−7.94073	−0.275461
$832$	0	0
$833$	−6.84791	−0.237266
$834$	0	0
$835$	2.88824	0.0999517
$836$	0	0
$837$	0.365919	0.0126480
$838$	0	0
$839$	11.0098	0.380102	0.190051	−	0.981774i	$-0.439135\pi$
$839$	11.0098	0.380102	0.190051	+	0.981774i	$0.439135\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−21.8351	−0.752041
$844$	0	0
$845$	0.701648	0.0241374
$846$	0	0
$847$	22.7748	0.782550
$848$	0	0
$849$	−0.821050	−0.0281784
$850$	0	0
$851$	−2.18071	−0.0747537
$852$	0	0
$853$	−34.8739	−1.19406	−0.597030	−	0.802219i	$-0.703653\pi$
$853$	−34.8739	−1.19406	−0.597030	+	0.802219i	$0.703653\pi$
$854$	0	0
$855$	5.16697	0.176707
$856$	0	0
$857$	18.5247	0.632793	0.316396	−	0.948627i	$-0.397527\pi$
$857$	18.5247	0.632793	0.316396	+	0.948627i	$0.397527\pi$
$858$	0	0
$859$	29.6409	1.01133	0.505666	−	0.862729i	$-0.331247\pi$
$859$	29.6409	1.01133	0.505666	+	0.862729i	$0.331247\pi$
$860$	0	0
$861$	17.1868	0.585726
$862$	0	0
$863$	45.8424	1.56049	0.780247	−	0.625471i	$-0.215093\pi$
$863$	45.8424	1.56049	0.780247	+	0.625471i	$0.215093\pi$
$864$	0	0
$865$	−1.81056	−0.0615610
$866$	0	0
$867$	−9.55114	−0.324374
$868$	0	0
$869$	−7.28635	−0.247173
$870$	0	0
$871$	−42.7342	−1.44799
$872$	0	0
$873$	4.82346	0.163249
$874$	0	0
$875$	16.0128	0.541333
$876$	0	0
$877$	29.3173	0.989974	0.494987	−	0.868900i	$-0.335173\pi$
$877$	29.3173	0.989974	0.494987	+	0.868900i	$0.335173\pi$
$878$	0	0
$879$	7.86387	0.265242
$880$	0	0
$881$	6.28234	0.211658	0.105829	−	0.994384i	$-0.466250\pi$
$881$	6.28234	0.211658	0.105829	+	0.994384i	$0.466250\pi$
$882$	0	0
$883$	0.277737	0.00934658	0.00467329	−	0.999989i	$-0.498512\pi$
$883$	0.277737	0.00934658	0.00467329	+	0.999989i	$0.498512\pi$
$884$	0	0
$885$	−6.89336	−0.231718
$886$	0	0
$887$	49.9494	1.67714	0.838569	−	0.544795i	$-0.183393\pi$
$887$	49.9494	1.67714	0.838569	+	0.544795i	$0.183393\pi$
$888$	0	0
$889$	−32.1689	−1.07891
$890$	0	0
$891$	−0.503028	−0.0168521
$892$	0	0
$893$	−23.6114	−0.790124
$894$	0	0
$895$	1.58762	0.0530682
$896$	0	0
$897$	3.72396	0.124339
$898$	0	0
$899$	−0.365919	−0.0122041
$900$	0	0
$901$	21.7506	0.724618
$902$	0	0
$903$	20.6819	0.688249
$904$	0	0
$905$	1.27416	0.0423544
$906$	0	0
$907$	29.2204	0.970248	0.485124	−	0.874445i	$-0.338774\pi$
$907$	29.2204	0.970248	0.485124	+	0.874445i	$0.338774\pi$
$908$	0	0
$909$	3.00902	0.0998030
$910$	0	0
$911$	53.2930	1.76568	0.882838	−	0.469678i	$-0.155630\pi$
$911$	53.2930	1.76568	0.882838	+	0.469678i	$0.155630\pi$
$912$	0	0
$913$	0.381577	0.0126284
$914$	0	0
$915$	−3.34707	−0.110651
$916$	0	0
$917$	16.3958	0.541438
$918$	0	0
$919$	2.23365	0.0736815	0.0368407	−	0.999321i	$-0.488271\pi$
$919$	2.23365	0.0736815	0.0368407	+	0.999321i	$0.488271\pi$
$920$	0	0
$921$	3.56292	0.117402
$922$	0	0
$923$	57.1930	1.88253
$924$	0	0
$925$	−9.47823	−0.311642
$926$	0	0
$927$	−13.1223	−0.430993
$928$	0	0
$929$	−35.2579	−1.15678	−0.578388	−	0.815762i	$-0.696318\pi$
$929$	−35.2579	−1.15678	−0.578388	+	0.815762i	$0.696318\pi$
$930$	0	0
$931$	−16.0359	−0.525555
$932$	0	0
$933$	31.9740	1.04678
$934$	0	0
$935$	−1.10992	−0.0362984
$936$	0	0
$937$	−24.6478	−0.805208	−0.402604	−	0.915374i	$-0.631895\pi$
$937$	−24.6478	−0.805208	−0.402604	+	0.915374i	$0.631895\pi$
$938$	0	0
$939$	−26.4123	−0.861934
$940$	0	0
$941$	−40.9182	−1.33389	−0.666947	−	0.745105i	$-0.732399\pi$
$941$	−40.9182	−1.33389	−0.666947	+	0.745105i	$0.732399\pi$
$942$	0	0
$943$	8.11013	0.264102
$944$	0	0
$945$	−1.71326	−0.0557324
$946$	0	0
$947$	56.7703	1.84479	0.922393	−	0.386252i	$-0.126230\pi$
$947$	56.7703	1.84479	0.922393	+	0.386252i	$0.126230\pi$
$948$	0	0
$949$	24.9234	0.809048
$950$	0	0
$951$	−10.9760	−0.355921
$952$	0	0
$953$	48.5922	1.57405	0.787027	−	0.616918i	$-0.211619\pi$
$953$	48.5922	1.57405	0.787027	+	0.616918i	$0.211619\pi$
$954$	0	0
$955$	−11.0665	−0.358104
$956$	0	0
$957$	0.503028	0.0162606
$958$	0	0
$959$	28.4806	0.919686
$960$	0	0
$961$	−30.8661	−0.995681
$962$	0	0
$963$	−0.583801	−0.0188127
$964$	0	0
$965$	−12.9055	−0.415443
$966$	0	0
$967$	−16.7731	−0.539386	−0.269693	−	0.962946i	$-0.586922\pi$
$967$	−16.7731	−0.539386	−0.269693	+	0.962946i	$0.586922\pi$
$968$	0	0
$969$	17.4432	0.560355
$970$	0	0
$971$	4.03878	0.129611	0.0648054	−	0.997898i	$-0.479357\pi$
$971$	4.03878	0.129611	0.0648054	+	0.997898i	$0.479357\pi$
$972$	0	0
$973$	22.2011	0.711735
$974$	0	0
$975$	16.1858	0.518361
$976$	0	0
$977$	−41.8623	−1.33929	−0.669647	−	0.742679i	$-0.733555\pi$
$977$	−41.8623	−1.33929	−0.669647	+	0.742679i	$0.733555\pi$
$978$	0	0
$979$	−4.34271	−0.138794
$980$	0	0
$981$	9.60016	0.306510
$982$	0	0
$983$	−13.0167	−0.415169	−0.207585	−	0.978217i	$-0.566560\pi$
$983$	−13.0167	−0.415169	−0.207585	+	0.978217i	$0.566560\pi$
$984$	0	0
$985$	−2.83280	−0.0902605
$986$	0	0
$987$	7.82905	0.249201
$988$	0	0
$989$	9.75936	0.310330
$990$	0	0
$991$	4.60542	0.146296	0.0731480	−	0.997321i	$-0.476695\pi$
$991$	4.60542	0.146296	0.0731480	+	0.997321i	$0.476695\pi$
$992$	0	0
$993$	−3.88297	−0.123222
$994$	0	0
$995$	−4.38306	−0.138952
$996$	0	0
$997$	−12.2900	−0.389227	−0.194614	−	0.980880i	$-0.562345\pi$
$997$	−12.2900	−0.389227	−0.194614	+	0.980880i	$0.562345\pi$
$998$	0	0
$999$	2.18071	0.0689945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.f.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.f.1.6	✓	9	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.808455 q^{5} -2.11918 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.808455 q^{5} -2.11918 q^{7} +1.00000 q^{9} -0.503028 q^{11} -3.72396 q^{13} +0.808455 q^{15} +2.72926 q^{17} +6.39117 q^{19} -2.11918 q^{21} -1.00000 q^{23} -4.34640 q^{25} +1.00000 q^{27} -1.00000 q^{29} +0.365919 q^{31} -0.503028 q^{33} -1.71326 q^{35} +2.18071 q^{37} -3.72396 q^{39} -8.11013 q^{41} -9.75936 q^{43} +0.808455 q^{45} -3.69437 q^{47} -2.50907 q^{49} +2.72926 q^{51} +7.96941 q^{53} -0.406676 q^{55} +6.39117 q^{57} -8.52659 q^{59} -4.14009 q^{61} -2.11918 q^{63} -3.01065 q^{65} +11.4755 q^{67} -1.00000 q^{69} -15.3581 q^{71} -6.69271 q^{73} -4.34640 q^{75} +1.06601 q^{77} +14.4850 q^{79} +1.00000 q^{81} -0.758561 q^{83} +2.20648 q^{85} -1.00000 q^{87} +8.63314 q^{89} +7.89175 q^{91} +0.365919 q^{93} +5.16697 q^{95} +4.82346 q^{97} -0.503028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - q^5 - 5 * q^7 + 9 * q^9 - 8 * q^11 - q^13 - q^15 - 2 * q^17 - 11 * q^19 - 5 * q^21 - 9 * q^23 - 10 * q^25 + 9 * q^27 - 9 * q^29 - 8 * q^33 + q^35 - 2 * q^37 - q^39 - 3 * q^41 - 19 * q^43 - q^45 - 3 * q^47 - 6 * q^49 - 2 * q^51 - 9 * q^53 - 7 * q^55 - 11 * q^57 - 2 * q^59 - 25 * q^61 - 5 * q^63 - 12 * q^65 - 20 * q^67 - 9 * q^69 + 9 * q^71 - 11 * q^73 - 10 * q^75 - 19 * q^77 + 4 * q^79 + 9 * q^81 - 9 * q^83 - 50 * q^85 - 9 * q^87 - 29 * q^89 - 38 * q^91 + 23 * q^95 - 43 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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