LMFDB - Embedded newform 8008.2.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 11x^{7} + 46x^{6} + 37x^{5} - 169x^{4} - 18x^{3} + 195x^{2} - 72x + 7 $ x^9 - 4x^8 - 11x^7 + 46x^6 + 37x^5 - 169x^4 - 18x^3 + 195x^2 - 72x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.17419$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.174190 q^{3} -1.75196 q^{5} -1.00000 q^{7} -2.96966 q^{9} +O(q^{10})$	$q+0.174190 q^{3} -1.75196 q^{5} -1.00000 q^{7} -2.96966 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.305174 q^{15} +2.54810 q^{17} +5.20989 q^{19} -0.174190 q^{21} +1.41725 q^{23} -1.93063 q^{25} -1.03986 q^{27} +5.76230 q^{29} -3.14844 q^{31} -0.174190 q^{33} +1.75196 q^{35} -5.05529 q^{37} +0.174190 q^{39} +1.64287 q^{41} -3.81044 q^{43} +5.20273 q^{45} +5.32923 q^{47} +1.00000 q^{49} +0.443854 q^{51} +6.31040 q^{53} +1.75196 q^{55} +0.907511 q^{57} -5.51800 q^{59} -9.18973 q^{61} +2.96966 q^{63} -1.75196 q^{65} +9.05824 q^{67} +0.246871 q^{69} -2.65527 q^{71} -6.24560 q^{73} -0.336297 q^{75} +1.00000 q^{77} +8.13699 q^{79} +8.72784 q^{81} -3.74833 q^{83} -4.46417 q^{85} +1.00374 q^{87} +15.4453 q^{89} -1.00000 q^{91} -0.548427 q^{93} -9.12753 q^{95} -10.3878 q^{97} +2.96966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9} - 9 q^{11} + 9 q^{13} - q^{15} + 13 q^{17} - 5 q^{19} + 5 q^{21} - 7 q^{23} + 14 q^{25} - 23 q^{27} - 19 q^{29} + 2 q^{31} + 5 q^{33} + 5 q^{35} + 4 q^{37} - 5 q^{39} + 10 q^{41} - 17 q^{43} + 11 q^{45} - 5 q^{47} + 9 q^{49} - 9 q^{51} - 24 q^{53} + 5 q^{55} - 4 q^{57} - 19 q^{59} - 12 q^{63} - 5 q^{65} + 2 q^{67} - 2 q^{69} - 2 q^{71} + 34 q^{73} - 46 q^{75} + 9 q^{77} + 5 q^{79} + 37 q^{81} - 24 q^{83} + 33 q^{85} + 41 q^{87} - 11 q^{89} - 9 q^{91} + 53 q^{93} - 23 q^{95} - 12 q^{99}+O(q^{100}) $ 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9 - 9 * q^11 + 9 * q^13 - q^15 + 13 * q^17 - 5 * q^19 + 5 * q^21 - 7 * q^23 + 14 * q^25 - 23 * q^27 - 19 * q^29 + 2 * q^31 + 5 * q^33 + 5 * q^35 + 4 * q^37 - 5 * q^39 + 10 * q^41 - 17 * q^43 + 11 * q^45 - 5 * q^47 + 9 * q^49 - 9 * q^51 - 24 * q^53 + 5 * q^55 - 4 * q^57 - 19 * q^59 - 12 * q^63 - 5 * q^65 + 2 * q^67 - 2 * q^69 - 2 * q^71 + 34 * q^73 - 46 * q^75 + 9 * q^77 + 5 * q^79 + 37 * q^81 - 24 * q^83 + 33 * q^85 + 41 * q^87 - 11 * q^89 - 9 * q^91 + 53 * q^93 - 23 * q^95 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.174190	0.100569	0.0502844	−	0.998735i	$-0.483987\pi$
$3$	0.174190	0.100569	0.0502844	+	0.998735i	$0.483987\pi$
$4$	0	0
$5$	−1.75196	−0.783501	−0.391751	−	0.920071i	$-0.628130\pi$
$5$	−1.75196	−0.783501	−0.391751	+	0.920071i	$0.628130\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.96966	−0.989886
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.305174	−0.0787957
$16$	0	0
$17$	2.54810	0.618005	0.309002	−	0.951061i	$-0.400005\pi$
$17$	2.54810	0.618005	0.309002	+	0.951061i	$0.400005\pi$
$18$	0	0
$19$	5.20989	1.19523	0.597615	−	0.801783i	$-0.296115\pi$
$19$	5.20989	1.19523	0.597615	+	0.801783i	$0.296115\pi$
$20$	0	0
$21$	−0.174190	−0.0380114
$22$	0	0
$23$	1.41725	0.295516	0.147758	−	0.989024i	$-0.452794\pi$
$23$	1.41725	0.295516	0.147758	+	0.989024i	$0.452794\pi$
$24$	0	0
$25$	−1.93063	−0.386126
$26$	0	0
$27$	−1.03986	−0.200120
$28$	0	0
$29$	5.76230	1.07003	0.535016	−	0.844842i	$-0.320306\pi$
$29$	5.76230	1.07003	0.535016	+	0.844842i	$0.320306\pi$
$30$	0	0
$31$	−3.14844	−0.565477	−0.282738	−	0.959197i	$-0.591243\pi$
$31$	−3.14844	−0.565477	−0.282738	+	0.959197i	$0.591243\pi$
$32$	0	0
$33$	−0.174190	−0.0303226
$34$	0	0
$35$	1.75196	0.296136
$36$	0	0
$37$	−5.05529	−0.831084	−0.415542	−	0.909574i	$-0.636408\pi$
$37$	−5.05529	−0.831084	−0.415542	+	0.909574i	$0.636408\pi$
$38$	0	0
$39$	0.174190	0.0278927
$40$	0	0
$41$	1.64287	0.256574	0.128287	−	0.991737i	$-0.459052\pi$
$41$	1.64287	0.256574	0.128287	+	0.991737i	$0.459052\pi$
$42$	0	0
$43$	−3.81044	−0.581087	−0.290544	−	0.956862i	$-0.593836\pi$
$43$	−3.81044	−0.581087	−0.290544	+	0.956862i	$0.593836\pi$
$44$	0	0
$45$	5.20273	0.775577
$46$	0	0
$47$	5.32923	0.777348	0.388674	−	0.921375i	$-0.372933\pi$
$47$	5.32923	0.777348	0.388674	+	0.921375i	$0.372933\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.443854	0.0621519
$52$	0	0
$53$	6.31040	0.866800	0.433400	−	0.901202i	$-0.357314\pi$
$53$	6.31040	0.866800	0.433400	+	0.901202i	$0.357314\pi$
$54$	0	0
$55$	1.75196	0.236234
$56$	0	0
$57$	0.907511	0.120203
$58$	0	0
$59$	−5.51800	−0.718382	−0.359191	−	0.933264i	$-0.616947\pi$
$59$	−5.51800	−0.718382	−0.359191	+	0.933264i	$0.616947\pi$
$60$	0	0
$61$	−9.18973	−1.17662	−0.588312	−	0.808634i	$-0.700207\pi$
$61$	−9.18973	−1.17662	−0.588312	+	0.808634i	$0.700207\pi$
$62$	0	0
$63$	2.96966	0.374142
$64$	0	0
$65$	−1.75196	−0.217304
$66$	0	0
$67$	9.05824	1.10664	0.553320	−	0.832969i	$-0.313361\pi$
$67$	9.05824	1.10664	0.553320	+	0.832969i	$0.313361\pi$
$68$	0	0
$69$	0.246871	0.0297197
$70$	0	0
$71$	−2.65527	−0.315123	−0.157561	−	0.987509i	$-0.550363\pi$
$71$	−2.65527	−0.315123	−0.157561	+	0.987509i	$0.550363\pi$
$72$	0	0
$73$	−6.24560	−0.730992	−0.365496	−	0.930813i	$-0.619101\pi$
$73$	−6.24560	−0.730992	−0.365496	+	0.930813i	$0.619101\pi$
$74$	0	0
$75$	−0.336297	−0.0388322
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.13699	0.915483	0.457741	−	0.889085i	$-0.348659\pi$
$79$	8.13699	0.915483	0.457741	+	0.889085i	$0.348659\pi$
$80$	0	0
$81$	8.72784	0.969760
$82$	0	0
$83$	−3.74833	−0.411433	−0.205716	−	0.978612i	$-0.565952\pi$
$83$	−3.74833	−0.411433	−0.205716	+	0.978612i	$0.565952\pi$
$84$	0	0
$85$	−4.46417	−0.484207
$86$	0	0
$87$	1.00374	0.107612
$88$	0	0
$89$	15.4453	1.63720	0.818601	−	0.574362i	$-0.194750\pi$
$89$	15.4453	1.63720	0.818601	+	0.574362i	$0.194750\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.548427	−0.0568693
$94$	0	0
$95$	−9.12753	−0.936464
$96$	0	0
$97$	−10.3878	−1.05472	−0.527359	−	0.849642i	$-0.676818\pi$
$97$	−10.3878	−1.05472	−0.527359	+	0.849642i	$0.676818\pi$
$98$	0	0
$99$	2.96966	0.298462
$100$	0	0
$101$	−11.4970	−1.14400	−0.571998	−	0.820255i	$-0.693832\pi$
$101$	−11.4970	−1.14400	−0.571998	+	0.820255i	$0.693832\pi$
$102$	0	0
$103$	7.87897	0.776338	0.388169	−	0.921588i	$-0.373108\pi$
$103$	7.87897	0.776338	0.388169	+	0.921588i	$0.373108\pi$
$104$	0	0
$105$	0.305174	0.0297820
$106$	0	0
$107$	16.2852	1.57435	0.787175	−	0.616730i	$-0.211543\pi$
$107$	16.2852	1.57435	0.787175	+	0.616730i	$0.211543\pi$
$108$	0	0
$109$	10.8782	1.04194	0.520969	−	0.853576i	$-0.325571\pi$
$109$	10.8782	1.04194	0.520969	+	0.853576i	$0.325571\pi$
$110$	0	0
$111$	−0.880581	−0.0835811
$112$	0	0
$113$	−13.2083	−1.24253	−0.621264	−	0.783601i	$-0.713381\pi$
$113$	−13.2083	−1.24253	−0.621264	+	0.783601i	$0.713381\pi$
$114$	0	0
$115$	−2.48296	−0.231537
$116$	0	0
$117$	−2.96966	−0.274545
$118$	0	0
$119$	−2.54810	−0.233584
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.286172	0.0258033
$124$	0	0
$125$	12.1422	1.08603
$126$	0	0
$127$	−13.4459	−1.19313	−0.596565	−	0.802565i	$-0.703468\pi$
$127$	−13.4459	−1.19313	−0.596565	+	0.802565i	$0.703468\pi$
$128$	0	0
$129$	−0.663741	−0.0584392
$130$	0	0
$131$	−16.5644	−1.44724	−0.723618	−	0.690201i	$-0.757522\pi$
$131$	−16.5644	−1.44724	−0.723618	+	0.690201i	$0.757522\pi$
$132$	0	0
$133$	−5.20989	−0.451755
$134$	0	0
$135$	1.82179	0.156794
$136$	0	0
$137$	0.312561	0.0267039	0.0133519	−	0.999911i	$-0.495750\pi$
$137$	0.312561	0.0267039	0.0133519	+	0.999911i	$0.495750\pi$
$138$	0	0
$139$	−2.57651	−0.218536	−0.109268	−	0.994012i	$-0.534851\pi$
$139$	−2.57651	−0.218536	−0.109268	+	0.994012i	$0.534851\pi$
$140$	0	0
$141$	0.928299	0.0781769
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−10.0953	−0.838371
$146$	0	0
$147$	0.174190	0.0143670
$148$	0	0
$149$	4.98586	0.408458	0.204229	−	0.978923i	$-0.434531\pi$
$149$	4.98586	0.408458	0.204229	+	0.978923i	$0.434531\pi$
$150$	0	0
$151$	−3.18391	−0.259103	−0.129551	−	0.991573i	$-0.541354\pi$
$151$	−3.18391	−0.259103	−0.129551	+	0.991573i	$0.541354\pi$
$152$	0	0
$153$	−7.56698	−0.611754
$154$	0	0
$155$	5.51595	0.443052
$156$	0	0
$157$	−3.53287	−0.281954	−0.140977	−	0.990013i	$-0.545024\pi$
$157$	−3.53287	−0.281954	−0.140977	+	0.990013i	$0.545024\pi$
$158$	0	0
$159$	1.09921	0.0871730
$160$	0	0
$161$	−1.41725	−0.111695
$162$	0	0
$163$	15.0669	1.18013	0.590067	−	0.807354i	$-0.299101\pi$
$163$	15.0669	1.18013	0.590067	+	0.807354i	$0.299101\pi$
$164$	0	0
$165$	0.305174	0.0237578
$166$	0	0
$167$	−5.64148	−0.436551	−0.218276	−	0.975887i	$-0.570043\pi$
$167$	−5.64148	−0.436551	−0.218276	+	0.975887i	$0.570043\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−15.4716	−1.18314
$172$	0	0
$173$	−10.9895	−0.835519	−0.417759	−	0.908558i	$-0.637185\pi$
$173$	−10.9895	−0.835519	−0.417759	+	0.908558i	$0.637185\pi$
$174$	0	0
$175$	1.93063	0.145942
$176$	0	0
$177$	−0.961181	−0.0722468
$178$	0	0
$179$	−14.1618	−1.05850	−0.529250	−	0.848466i	$-0.677527\pi$
$179$	−14.1618	−1.05850	−0.529250	+	0.848466i	$0.677527\pi$
$180$	0	0
$181$	8.04595	0.598050	0.299025	−	0.954245i	$-0.403339\pi$
$181$	8.04595	0.598050	0.299025	+	0.954245i	$0.403339\pi$
$182$	0	0
$183$	−1.60076	−0.118332
$184$	0	0
$185$	8.85667	0.651155
$186$	0	0
$187$	−2.54810	−0.186335
$188$	0	0
$189$	1.03986	0.0756384
$190$	0	0
$191$	−2.40360	−0.173918	−0.0869592	−	0.996212i	$-0.527715\pi$
$191$	−2.40360	−0.173918	−0.0869592	+	0.996212i	$0.527715\pi$
$192$	0	0
$193$	−21.5161	−1.54876	−0.774381	−	0.632719i	$-0.781939\pi$
$193$	−21.5161	−1.54876	−0.774381	+	0.632719i	$0.781939\pi$
$194$	0	0
$195$	−0.305174	−0.0218540
$196$	0	0
$197$	−7.73482	−0.551083	−0.275542	−	0.961289i	$-0.588857\pi$
$197$	−7.73482	−0.551083	−0.275542	+	0.961289i	$0.588857\pi$
$198$	0	0
$199$	−15.5840	−1.10472	−0.552360	−	0.833606i	$-0.686273\pi$
$199$	−15.5840	−1.10472	−0.552360	+	0.833606i	$0.686273\pi$
$200$	0	0
$201$	1.57786	0.111293
$202$	0	0
$203$	−5.76230	−0.404434
$204$	0	0
$205$	−2.87825	−0.201026
$206$	0	0
$207$	−4.20874	−0.292528
$208$	0	0
$209$	−5.20989	−0.360376
$210$	0	0
$211$	−14.2671	−0.982188	−0.491094	−	0.871107i	$-0.663403\pi$
$211$	−14.2671	−0.982188	−0.491094	+	0.871107i	$0.663403\pi$
$212$	0	0
$213$	−0.462522	−0.0316915
$214$	0	0
$215$	6.67575	0.455282
$216$	0	0
$217$	3.14844	0.213730
$218$	0	0
$219$	−1.08792	−0.0735149
$220$	0	0
$221$	2.54810	0.171404
$222$	0	0
$223$	−10.2223	−0.684534	−0.342267	−	0.939603i	$-0.611195\pi$
$223$	−10.2223	−0.684534	−0.342267	+	0.939603i	$0.611195\pi$
$224$	0	0
$225$	5.73331	0.382221
$226$	0	0
$227$	−3.89710	−0.258660	−0.129330	−	0.991602i	$-0.541283\pi$
$227$	−3.89710	−0.258660	−0.129330	+	0.991602i	$0.541283\pi$
$228$	0	0
$229$	−0.687206	−0.0454119	−0.0227059	−	0.999742i	$-0.507228\pi$
$229$	−0.687206	−0.0454119	−0.0227059	+	0.999742i	$0.507228\pi$
$230$	0	0
$231$	0.174190	0.0114609
$232$	0	0
$233$	−7.95321	−0.521032	−0.260516	−	0.965470i	$-0.583893\pi$
$233$	−7.95321	−0.521032	−0.260516	+	0.965470i	$0.583893\pi$
$234$	0	0
$235$	−9.33661	−0.609053
$236$	0	0
$237$	1.41738	0.0920689
$238$	0	0
$239$	−25.1785	−1.62866	−0.814332	−	0.580400i	$-0.802896\pi$
$239$	−25.1785	−1.62866	−0.814332	+	0.580400i	$0.802896\pi$
$240$	0	0
$241$	8.01716	0.516431	0.258215	−	0.966087i	$-0.416866\pi$
$241$	8.01716	0.516431	0.258215	+	0.966087i	$0.416866\pi$
$242$	0	0
$243$	4.63987	0.297648
$244$	0	0
$245$	−1.75196	−0.111929
$246$	0	0
$247$	5.20989	0.331497
$248$	0	0
$249$	−0.652922	−0.0413773
$250$	0	0
$251$	−10.7069	−0.675815	−0.337908	−	0.941179i	$-0.609719\pi$
$251$	−10.7069	−0.675815	−0.337908	+	0.941179i	$0.609719\pi$
$252$	0	0
$253$	−1.41725	−0.0891016
$254$	0	0
$255$	−0.777614	−0.0486961
$256$	0	0
$257$	−27.8092	−1.73469	−0.867346	−	0.497705i	$-0.834176\pi$
$257$	−27.8092	−1.73469	−0.867346	+	0.497705i	$0.834176\pi$
$258$	0	0
$259$	5.05529	0.314120
$260$	0	0
$261$	−17.1121	−1.05921
$262$	0	0
$263$	−21.8959	−1.35016	−0.675078	−	0.737746i	$-0.735890\pi$
$263$	−21.8959	−1.35016	−0.675078	+	0.737746i	$0.735890\pi$
$264$	0	0
$265$	−11.0556	−0.679139
$266$	0	0
$267$	2.69043	0.164651
$268$	0	0
$269$	−18.8632	−1.15011	−0.575054	−	0.818115i	$-0.695019\pi$
$269$	−18.8632	−1.15011	−0.575054	+	0.818115i	$0.695019\pi$
$270$	0	0
$271$	−19.0168	−1.15519	−0.577593	−	0.816325i	$-0.696008\pi$
$271$	−19.0168	−1.15519	−0.577593	+	0.816325i	$0.696008\pi$
$272$	0	0
$273$	−0.174190	−0.0105425
$274$	0	0
$275$	1.93063	0.116421
$276$	0	0
$277$	27.4061	1.64667	0.823336	−	0.567555i	$-0.192110\pi$
$277$	27.4061	1.64667	0.823336	+	0.567555i	$0.192110\pi$
$278$	0	0
$279$	9.34979	0.559758
$280$	0	0
$281$	−17.7386	−1.05820	−0.529099	−	0.848560i	$-0.677470\pi$
$281$	−17.7386	−1.05820	−0.529099	+	0.848560i	$0.677470\pi$
$282$	0	0
$283$	−7.74248	−0.460243	−0.230121	−	0.973162i	$-0.573912\pi$
$283$	−7.74248	−0.460243	−0.230121	+	0.973162i	$0.573912\pi$
$284$	0	0
$285$	−1.58993	−0.0941790
$286$	0	0
$287$	−1.64287	−0.0969758
$288$	0	0
$289$	−10.5072	−0.618070
$290$	0	0
$291$	−1.80945	−0.106072
$292$	0	0
$293$	8.72037	0.509449	0.254725	−	0.967014i	$-0.418015\pi$
$293$	8.72037	0.509449	0.254725	+	0.967014i	$0.418015\pi$
$294$	0	0
$295$	9.66732	0.562853
$296$	0	0
$297$	1.03986	0.0603385
$298$	0	0
$299$	1.41725	0.0819615
$300$	0	0
$301$	3.81044	0.219630
$302$	0	0
$303$	−2.00267	−0.115050
$304$	0	0
$305$	16.1001	0.921886
$306$	0	0
$307$	−14.6278	−0.834852	−0.417426	−	0.908711i	$-0.637068\pi$
$307$	−14.6278	−0.834852	−0.417426	+	0.908711i	$0.637068\pi$
$308$	0	0
$309$	1.37244	0.0780753
$310$	0	0
$311$	26.9520	1.52831	0.764153	−	0.645035i	$-0.223157\pi$
$311$	26.9520	1.52831	0.764153	+	0.645035i	$0.223157\pi$
$312$	0	0
$313$	23.5551	1.33141	0.665706	−	0.746214i	$-0.268131\pi$
$313$	23.5551	1.33141	0.665706	+	0.746214i	$0.268131\pi$
$314$	0	0
$315$	−5.20273	−0.293140
$316$	0	0
$317$	−15.6534	−0.879180	−0.439590	−	0.898198i	$-0.644876\pi$
$317$	−15.6534	−0.879180	−0.439590	+	0.898198i	$0.644876\pi$
$318$	0	0
$319$	−5.76230	−0.322627
$320$	0	0
$321$	2.83672	0.158330
$322$	0	0
$323$	13.2753	0.738658
$324$	0	0
$325$	−1.93063	−0.107092
$326$	0	0
$327$	1.89487	0.104786
$328$	0	0
$329$	−5.32923	−0.293810
$330$	0	0
$331$	−2.86839	−0.157661	−0.0788306	−	0.996888i	$-0.525119\pi$
$331$	−2.86839	−0.157661	−0.0788306	+	0.996888i	$0.525119\pi$
$332$	0	0
$333$	15.0125	0.822678
$334$	0	0
$335$	−15.8697	−0.867054
$336$	0	0
$337$	−18.6229	−1.01445	−0.507226	−	0.861813i	$-0.669329\pi$
$337$	−18.6229	−1.01445	−0.507226	+	0.861813i	$0.669329\pi$
$338$	0	0
$339$	−2.30075	−0.124959
$340$	0	0
$341$	3.14844	0.170498
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.432508	−0.0232854
$346$	0	0
$347$	−28.2962	−1.51902	−0.759510	−	0.650496i	$-0.774561\pi$
$347$	−28.2962	−1.51902	−0.759510	+	0.650496i	$0.774561\pi$
$348$	0	0
$349$	5.25080	0.281069	0.140534	−	0.990076i	$-0.455118\pi$
$349$	5.25080	0.281069	0.140534	+	0.990076i	$0.455118\pi$
$350$	0	0
$351$	−1.03986	−0.0555034
$352$	0	0
$353$	−7.23498	−0.385079	−0.192540	−	0.981289i	$-0.561672\pi$
$353$	−7.23498	−0.385079	−0.192540	+	0.981289i	$0.561672\pi$
$354$	0	0
$355$	4.65193	0.246899
$356$	0	0
$357$	−0.443854	−0.0234912
$358$	0	0
$359$	3.78142	0.199575	0.0997877	−	0.995009i	$-0.468184\pi$
$359$	3.78142	0.199575	0.0997877	+	0.995009i	$0.468184\pi$
$360$	0	0
$361$	8.14295	0.428576
$362$	0	0
$363$	0.174190	0.00914261
$364$	0	0
$365$	10.9420	0.572733
$366$	0	0
$367$	21.5470	1.12474	0.562371	−	0.826885i	$-0.309889\pi$
$367$	21.5470	1.12474	0.562371	+	0.826885i	$0.309889\pi$
$368$	0	0
$369$	−4.87877	−0.253979
$370$	0	0
$371$	−6.31040	−0.327620
$372$	0	0
$373$	3.21391	0.166410	0.0832049	−	0.996532i	$-0.473484\pi$
$373$	3.21391	0.166410	0.0832049	+	0.996532i	$0.473484\pi$
$374$	0	0
$375$	2.11505	0.109221
$376$	0	0
$377$	5.76230	0.296774
$378$	0	0
$379$	−19.3747	−0.995210	−0.497605	−	0.867404i	$-0.665787\pi$
$379$	−19.3747	−0.995210	−0.497605	+	0.867404i	$0.665787\pi$
$380$	0	0
$381$	−2.34214	−0.119992
$382$	0	0
$383$	−12.3220	−0.629626	−0.314813	−	0.949154i	$-0.601942\pi$
$383$	−12.3220	−0.629626	−0.314813	+	0.949154i	$0.601942\pi$
$384$	0	0
$385$	−1.75196	−0.0892882
$386$	0	0
$387$	11.3157	0.575210
$388$	0	0
$389$	−12.7535	−0.646627	−0.323313	−	0.946292i	$-0.604797\pi$
$389$	−12.7535	−0.646627	−0.323313	+	0.946292i	$0.604797\pi$
$390$	0	0
$391$	3.61129	0.182631
$392$	0	0
$393$	−2.88535	−0.145547
$394$	0	0
$395$	−14.2557	−0.717282
$396$	0	0
$397$	29.2750	1.46927	0.734636	−	0.678461i	$-0.237353\pi$
$397$	29.2750	1.46927	0.734636	+	0.678461i	$0.237353\pi$
$398$	0	0
$399$	−0.907511	−0.0454324
$400$	0	0
$401$	−0.516861	−0.0258108	−0.0129054	−	0.999917i	$-0.504108\pi$
$401$	−0.516861	−0.0258108	−0.0129054	+	0.999917i	$0.504108\pi$
$402$	0	0
$403$	−3.14844	−0.156835
$404$	0	0
$405$	−15.2908	−0.759808
$406$	0	0
$407$	5.05529	0.250581
$408$	0	0
$409$	18.4344	0.911522	0.455761	−	0.890102i	$-0.349367\pi$
$409$	18.4344	0.911522	0.455761	+	0.890102i	$0.349367\pi$
$410$	0	0
$411$	0.0544450	0.00268557
$412$	0	0
$413$	5.51800	0.271523
$414$	0	0
$415$	6.56693	0.322358
$416$	0	0
$417$	−0.448802	−0.0219779
$418$	0	0
$419$	16.2659	0.794644	0.397322	−	0.917679i	$-0.369940\pi$
$419$	16.2659	0.794644	0.397322	+	0.917679i	$0.369940\pi$
$420$	0	0
$421$	26.0236	1.26831	0.634156	−	0.773205i	$-0.281348\pi$
$421$	26.0236	1.26831	0.634156	+	0.773205i	$0.281348\pi$
$422$	0	0
$423$	−15.8260	−0.769486
$424$	0	0
$425$	−4.91944	−0.238628
$426$	0	0
$427$	9.18973	0.444722
$428$	0	0
$429$	−0.174190	−0.00840998
$430$	0	0
$431$	28.3483	1.36549	0.682745	−	0.730657i	$-0.260786\pi$
$431$	28.3483	1.36549	0.682745	+	0.730657i	$0.260786\pi$
$432$	0	0
$433$	11.0872	0.532815	0.266407	−	0.963861i	$-0.414163\pi$
$433$	11.0872	0.532815	0.266407	+	0.963861i	$0.414163\pi$
$434$	0	0
$435$	−1.75851	−0.0843139
$436$	0	0
$437$	7.38370	0.353210
$438$	0	0
$439$	27.7071	1.32239	0.661193	−	0.750216i	$-0.270051\pi$
$439$	27.7071	1.32239	0.661193	+	0.750216i	$0.270051\pi$
$440$	0	0
$441$	−2.96966	−0.141412
$442$	0	0
$443$	−24.2902	−1.15406	−0.577031	−	0.816722i	$-0.695789\pi$
$443$	−24.2902	−1.15406	−0.577031	+	0.816722i	$0.695789\pi$
$444$	0	0
$445$	−27.0596	−1.28275
$446$	0	0
$447$	0.868488	0.0410781
$448$	0	0
$449$	−29.7068	−1.40195	−0.700974	−	0.713186i	$-0.747251\pi$
$449$	−29.7068	−1.40195	−0.700974	+	0.713186i	$0.747251\pi$
$450$	0	0
$451$	−1.64287	−0.0773599
$452$	0	0
$453$	−0.554605	−0.0260576
$454$	0	0
$455$	1.75196	0.0821332
$456$	0	0
$457$	35.0062	1.63752	0.818760	−	0.574135i	$-0.194662\pi$
$457$	35.0062	1.63752	0.818760	+	0.574135i	$0.194662\pi$
$458$	0	0
$459$	−2.64965	−0.123675
$460$	0	0
$461$	−22.0832	−1.02852	−0.514258	−	0.857636i	$-0.671933\pi$
$461$	−22.0832	−1.02852	−0.514258	+	0.857636i	$0.671933\pi$
$462$	0	0
$463$	−10.9277	−0.507855	−0.253928	−	0.967223i	$-0.581723\pi$
$463$	−10.9277	−0.507855	−0.253928	+	0.967223i	$0.581723\pi$
$464$	0	0
$465$	0.960824	0.0445571
$466$	0	0
$467$	−5.58412	−0.258402	−0.129201	−	0.991618i	$-0.541241\pi$
$467$	−5.58412	−0.258402	−0.129201	+	0.991618i	$0.541241\pi$
$468$	0	0
$469$	−9.05824	−0.418271
$470$	0	0
$471$	−0.615392	−0.0283557
$472$	0	0
$473$	3.81044	0.175204
$474$	0	0
$475$	−10.0584	−0.461510
$476$	0	0
$477$	−18.7397	−0.858033
$478$	0	0
$479$	−14.9713	−0.684054	−0.342027	−	0.939690i	$-0.611114\pi$
$479$	−14.9713	−0.684054	−0.342027	+	0.939690i	$0.611114\pi$
$480$	0	0
$481$	−5.05529	−0.230501
$482$	0	0
$483$	−0.246871	−0.0112330
$484$	0	0
$485$	18.1990	0.826373
$486$	0	0
$487$	23.4158	1.06107	0.530535	−	0.847663i	$-0.321991\pi$
$487$	23.4158	1.06107	0.530535	+	0.847663i	$0.321991\pi$
$488$	0	0
$489$	2.62451	0.118685
$490$	0	0
$491$	−37.8052	−1.70612	−0.853062	−	0.521810i	$-0.825257\pi$
$491$	−37.8052	−1.70612	−0.853062	+	0.521810i	$0.825257\pi$
$492$	0	0
$493$	14.6829	0.661285
$494$	0	0
$495$	−5.20273	−0.233845
$496$	0	0
$497$	2.65527	0.119105
$498$	0	0
$499$	−14.8644	−0.665422	−0.332711	−	0.943029i	$-0.607963\pi$
$499$	−14.8644	−0.665422	−0.332711	+	0.943029i	$0.607963\pi$
$500$	0	0
$501$	−0.982691	−0.0439034
$502$	0	0
$503$	−20.4821	−0.913251	−0.456625	−	0.889659i	$-0.650942\pi$
$503$	−20.4821	−0.913251	−0.456625	+	0.889659i	$0.650942\pi$
$504$	0	0
$505$	20.1423	0.896322
$506$	0	0
$507$	0.174190	0.00773606
$508$	0	0
$509$	−24.2240	−1.07371	−0.536854	−	0.843675i	$-0.680387\pi$
$509$	−24.2240	−1.07371	−0.536854	+	0.843675i	$0.680387\pi$
$510$	0	0
$511$	6.24560	0.276289
$512$	0	0
$513$	−5.41753	−0.239190
$514$	0	0
$515$	−13.8036	−0.608261
$516$	0	0
$517$	−5.32923	−0.234379
$518$	0	0
$519$	−1.91427	−0.0840271
$520$	0	0
$521$	28.3934	1.24394	0.621968	−	0.783042i	$-0.286333\pi$
$521$	28.3934	1.24394	0.621968	+	0.783042i	$0.286333\pi$
$522$	0	0
$523$	21.7585	0.951434	0.475717	−	0.879599i	$-0.342189\pi$
$523$	21.7585	0.951434	0.475717	+	0.879599i	$0.342189\pi$
$524$	0	0
$525$	0.336297	0.0146772
$526$	0	0
$527$	−8.02254	−0.349467
$528$	0	0
$529$	−20.9914	−0.912670
$530$	0	0
$531$	16.3866	0.711116
$532$	0	0
$533$	1.64287	0.0711607
$534$	0	0
$535$	−28.5310	−1.23350
$536$	0	0
$537$	−2.46684	−0.106452
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	10.6098	0.456151	0.228075	−	0.973643i	$-0.426757\pi$
$541$	10.6098	0.456151	0.228075	+	0.973643i	$0.426757\pi$
$542$	0	0
$543$	1.40152	0.0601452
$544$	0	0
$545$	−19.0581	−0.816360
$546$	0	0
$547$	−33.3564	−1.42622	−0.713108	−	0.701054i	$-0.752713\pi$
$547$	−33.3564	−1.42622	−0.713108	+	0.701054i	$0.752713\pi$
$548$	0	0
$549$	27.2904	1.16472
$550$	0	0
$551$	30.0209	1.27894
$552$	0	0
$553$	−8.13699	−0.346020
$554$	0	0
$555$	1.54274	0.0654858
$556$	0	0
$557$	−26.0470	−1.10365	−0.551823	−	0.833961i	$-0.686068\pi$
$557$	−26.0470	−1.10365	−0.551823	+	0.833961i	$0.686068\pi$
$558$	0	0
$559$	−3.81044	−0.161165
$560$	0	0
$561$	−0.443854	−0.0187395
$562$	0	0
$563$	13.3523	0.562734	0.281367	−	0.959600i	$-0.409212\pi$
$563$	13.3523	0.562734	0.281367	+	0.959600i	$0.409212\pi$
$564$	0	0
$565$	23.1404	0.973522
$566$	0	0
$567$	−8.72784	−0.366535
$568$	0	0
$569$	40.3159	1.69013	0.845066	−	0.534662i	$-0.179561\pi$
$569$	40.3159	1.69013	0.845066	+	0.534662i	$0.179561\pi$
$570$	0	0
$571$	−12.7349	−0.532937	−0.266469	−	0.963844i	$-0.585857\pi$
$571$	−12.7349	−0.532937	−0.266469	+	0.963844i	$0.585857\pi$
$572$	0	0
$573$	−0.418683	−0.0174908
$574$	0	0
$575$	−2.73618	−0.114107
$576$	0	0
$577$	39.9469	1.66301	0.831506	−	0.555515i	$-0.187479\pi$
$577$	39.9469	1.66301	0.831506	+	0.555515i	$0.187479\pi$
$578$	0	0
$579$	−3.74789	−0.155757
$580$	0	0
$581$	3.74833	0.155507
$582$	0	0
$583$	−6.31040	−0.261350
$584$	0	0
$585$	5.20273	0.215106
$586$	0	0
$587$	−42.4713	−1.75298	−0.876489	−	0.481423i	$-0.840120\pi$
$587$	−42.4713	−1.75298	−0.876489	+	0.481423i	$0.840120\pi$
$588$	0	0
$589$	−16.4030	−0.675875
$590$	0	0
$591$	−1.34733	−0.0554218
$592$	0	0
$593$	45.2229	1.85708	0.928540	−	0.371232i	$-0.121065\pi$
$593$	45.2229	1.85708	0.928540	+	0.371232i	$0.121065\pi$
$594$	0	0
$595$	4.46417	0.183013
$596$	0	0
$597$	−2.71458	−0.111100
$598$	0	0
$599$	2.85107	0.116492	0.0582458	−	0.998302i	$-0.481449\pi$
$599$	2.85107	0.116492	0.0582458	+	0.998302i	$0.481449\pi$
$600$	0	0
$601$	13.3355	0.543966	0.271983	−	0.962302i	$-0.412321\pi$
$601$	13.3355	0.543966	0.271983	+	0.962302i	$0.412321\pi$
$602$	0	0
$603$	−26.8999	−1.09545
$604$	0	0
$605$	−1.75196	−0.0712274
$606$	0	0
$607$	−8.30385	−0.337043	−0.168521	−	0.985698i	$-0.553899\pi$
$607$	−8.30385	−0.337043	−0.168521	+	0.985698i	$0.553899\pi$
$608$	0	0
$609$	−1.00374	−0.0406734
$610$	0	0
$611$	5.32923	0.215598
$612$	0	0
$613$	23.3799	0.944305	0.472152	−	0.881517i	$-0.343477\pi$
$613$	23.3799	0.944305	0.472152	+	0.881517i	$0.343477\pi$
$614$	0	0
$615$	−0.501363	−0.0202169
$616$	0	0
$617$	−11.1541	−0.449047	−0.224523	−	0.974469i	$-0.572083\pi$
$617$	−11.1541	−0.449047	−0.224523	+	0.974469i	$0.572083\pi$
$618$	0	0
$619$	13.8569	0.556954	0.278477	−	0.960443i	$-0.410170\pi$
$619$	13.8569	0.556954	0.278477	+	0.960443i	$0.410170\pi$
$620$	0	0
$621$	−1.47373	−0.0591388
$622$	0	0
$623$	−15.4453	−0.618804
$624$	0	0
$625$	−11.6195	−0.464780
$626$	0	0
$627$	−0.907511	−0.0362425
$628$	0	0
$629$	−12.8814	−0.513614
$630$	0	0
$631$	−9.93840	−0.395641	−0.197821	−	0.980238i	$-0.563386\pi$
$631$	−9.93840	−0.395641	−0.197821	+	0.980238i	$0.563386\pi$
$632$	0	0
$633$	−2.48519	−0.0987774
$634$	0	0
$635$	23.5567	0.934818
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	7.88524	0.311935
$640$	0	0
$641$	−21.7984	−0.860986	−0.430493	−	0.902594i	$-0.641660\pi$
$641$	−21.7984	−0.860986	−0.430493	+	0.902594i	$0.641660\pi$
$642$	0	0
$643$	−28.8614	−1.13818	−0.569091	−	0.822274i	$-0.692705\pi$
$643$	−28.8614	−1.13818	−0.569091	+	0.822274i	$0.692705\pi$
$644$	0	0
$645$	1.16285	0.0457872
$646$	0	0
$647$	3.68074	0.144705	0.0723524	−	0.997379i	$-0.476949\pi$
$647$	3.68074	0.144705	0.0723524	+	0.997379i	$0.476949\pi$
$648$	0	0
$649$	5.51800	0.216600
$650$	0	0
$651$	0.548427	0.0214946
$652$	0	0
$653$	12.5175	0.489847	0.244924	−	0.969542i	$-0.421237\pi$
$653$	12.5175	0.489847	0.244924	+	0.969542i	$0.421237\pi$
$654$	0	0
$655$	29.0201	1.13391
$656$	0	0
$657$	18.5473	0.723599
$658$	0	0
$659$	20.8669	0.812858	0.406429	−	0.913682i	$-0.366774\pi$
$659$	20.8669	0.812858	0.406429	+	0.913682i	$0.366774\pi$
$660$	0	0
$661$	31.5362	1.22661	0.613307	−	0.789844i	$-0.289839\pi$
$661$	31.5362	1.22661	0.613307	+	0.789844i	$0.289839\pi$
$662$	0	0
$663$	0.443854	0.0172378
$664$	0	0
$665$	9.12753	0.353950
$666$	0	0
$667$	8.16660	0.316212
$668$	0	0
$669$	−1.78062	−0.0688427
$670$	0	0
$671$	9.18973	0.354766
$672$	0	0
$673$	0.0585840	0.00225825	0.00112912	−	0.999999i	$-0.499641\pi$
$673$	0.0585840	0.00225825	0.00112912	+	0.999999i	$0.499641\pi$
$674$	0	0
$675$	2.00758	0.0772717
$676$	0	0
$677$	−8.75392	−0.336440	−0.168220	−	0.985749i	$-0.553802\pi$
$677$	−8.75392	−0.336440	−0.168220	+	0.985749i	$0.553802\pi$
$678$	0	0
$679$	10.3878	0.398646
$680$	0	0
$681$	−0.678836	−0.0260131
$682$	0	0
$683$	7.63351	0.292088	0.146044	−	0.989278i	$-0.453346\pi$
$683$	7.63351	0.292088	0.146044	+	0.989278i	$0.453346\pi$
$684$	0	0
$685$	−0.547594	−0.0209225
$686$	0	0
$687$	−0.119705	−0.00456702
$688$	0	0
$689$	6.31040	0.240407
$690$	0	0
$691$	−35.3766	−1.34579	−0.672895	−	0.739738i	$-0.734949\pi$
$691$	−35.3766	−1.34579	−0.672895	+	0.739738i	$0.734949\pi$
$692$	0	0
$693$	−2.96966	−0.112808
$694$	0	0
$695$	4.51394	0.171223
$696$	0	0
$697$	4.18620	0.158564
$698$	0	0
$699$	−1.38537	−0.0523995
$700$	0	0
$701$	9.57698	0.361717	0.180859	−	0.983509i	$-0.442112\pi$
$701$	9.57698	0.361717	0.180859	+	0.983509i	$0.442112\pi$
$702$	0	0
$703$	−26.3375	−0.993337
$704$	0	0
$705$	−1.62634	−0.0612517
$706$	0	0
$707$	11.4970	0.432390
$708$	0	0
$709$	−34.3297	−1.28928	−0.644640	−	0.764486i	$-0.722993\pi$
$709$	−34.3297	−1.28928	−0.644640	+	0.764486i	$0.722993\pi$
$710$	0	0
$711$	−24.1641	−0.906223
$712$	0	0
$713$	−4.46212	−0.167108
$714$	0	0
$715$	1.75196	0.0655196
$716$	0	0
$717$	−4.38585	−0.163793
$718$	0	0
$719$	−34.5630	−1.28898	−0.644492	−	0.764611i	$-0.722931\pi$
$719$	−34.5630	−1.28898	−0.644492	+	0.764611i	$0.722931\pi$
$720$	0	0
$721$	−7.87897	−0.293428
$722$	0	0
$723$	1.39651	0.0519368
$724$	0	0
$725$	−11.1249	−0.413167
$726$	0	0
$727$	−7.53555	−0.279478	−0.139739	−	0.990188i	$-0.544626\pi$
$727$	−7.53555	−0.279478	−0.139739	+	0.990188i	$0.544626\pi$
$728$	0	0
$729$	−25.3753	−0.939826
$730$	0	0
$731$	−9.70938	−0.359114
$732$	0	0
$733$	27.5830	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$733$	27.5830	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$734$	0	0
$735$	−0.305174	−0.0112565
$736$	0	0
$737$	−9.05824	−0.333665
$738$	0	0
$739$	−25.5939	−0.941485	−0.470743	−	0.882271i	$-0.656014\pi$
$739$	−25.5939	−0.941485	−0.470743	+	0.882271i	$0.656014\pi$
$740$	0	0
$741$	0.907511	0.0333383
$742$	0	0
$743$	−6.68765	−0.245346	−0.122673	−	0.992447i	$-0.539147\pi$
$743$	−6.68765	−0.245346	−0.122673	+	0.992447i	$0.539147\pi$
$744$	0	0
$745$	−8.73504	−0.320027
$746$	0	0
$747$	11.1313	0.407271
$748$	0	0
$749$	−16.2852	−0.595048
$750$	0	0
$751$	20.7689	0.757867	0.378934	−	0.925424i	$-0.376291\pi$
$751$	20.7689	0.757867	0.378934	+	0.925424i	$0.376291\pi$
$752$	0	0
$753$	−1.86504	−0.0679659
$754$	0	0
$755$	5.57809	0.203007
$756$	0	0
$757$	8.72570	0.317141	0.158571	−	0.987348i	$-0.449311\pi$
$757$	8.72570	0.317141	0.158571	+	0.987348i	$0.449311\pi$
$758$	0	0
$759$	−0.246871	−0.00896083
$760$	0	0
$761$	−19.3523	−0.701522	−0.350761	−	0.936465i	$-0.614077\pi$
$761$	−19.3523	−0.701522	−0.350761	+	0.936465i	$0.614077\pi$
$762$	0	0
$763$	−10.8782	−0.393816
$764$	0	0
$765$	13.2571	0.479310
$766$	0	0
$767$	−5.51800	−0.199243
$768$	0	0
$769$	33.7713	1.21782	0.608912	−	0.793238i	$-0.291606\pi$
$769$	33.7713	1.21782	0.608912	+	0.793238i	$0.291606\pi$
$770$	0	0
$771$	−4.84409	−0.174456
$772$	0	0
$773$	−21.5296	−0.774365	−0.387182	−	0.922003i	$-0.626552\pi$
$773$	−21.5296	−0.774365	−0.387182	+	0.922003i	$0.626552\pi$
$774$	0	0
$775$	6.07848	0.218345
$776$	0	0
$777$	0.880581	0.0315907
$778$	0	0
$779$	8.55919	0.306665
$780$	0	0
$781$	2.65527	0.0950130
$782$	0	0
$783$	−5.99196	−0.214135
$784$	0	0
$785$	6.18946	0.220911
$786$	0	0
$787$	1.25044	0.0445735	0.0222867	−	0.999752i	$-0.492905\pi$
$787$	1.25044	0.0445735	0.0222867	+	0.999752i	$0.492905\pi$
$788$	0	0
$789$	−3.81404	−0.135783
$790$	0	0
$791$	13.2083	0.469632
$792$	0	0
$793$	−9.18973	−0.326337
$794$	0	0
$795$	−1.92577	−0.0683001
$796$	0	0
$797$	−11.1932	−0.396483	−0.198241	−	0.980153i	$-0.563523\pi$
$797$	−11.1932	−0.396483	−0.198241	+	0.980153i	$0.563523\pi$
$798$	0	0
$799$	13.5794	0.480405
$800$	0	0
$801$	−45.8674	−1.62064
$802$	0	0
$803$	6.24560	0.220402
$804$	0	0
$805$	2.48296	0.0875129
$806$	0	0
$807$	−3.28578	−0.115665
$808$	0	0
$809$	36.3918	1.27947	0.639734	−	0.768596i	$-0.279044\pi$
$809$	36.3918	1.27947	0.639734	+	0.768596i	$0.279044\pi$
$810$	0	0
$811$	33.6300	1.18091	0.590454	−	0.807071i	$-0.298949\pi$
$811$	33.6300	1.18091	0.590454	+	0.807071i	$0.298949\pi$
$812$	0	0
$813$	−3.31253	−0.116176
$814$	0	0
$815$	−26.3967	−0.924636
$816$	0	0
$817$	−19.8520	−0.694533
$818$	0	0
$819$	2.96966	0.103768
$820$	0	0
$821$	11.8608	0.413946	0.206973	−	0.978347i	$-0.433639\pi$
$821$	11.8608	0.413946	0.206973	+	0.978347i	$0.433639\pi$
$822$	0	0
$823$	−21.5322	−0.750565	−0.375282	−	0.926911i	$-0.622454\pi$
$823$	−21.5322	−0.750565	−0.375282	+	0.926911i	$0.622454\pi$
$824$	0	0
$825$	0.336297	0.0117084
$826$	0	0
$827$	−32.0736	−1.11531	−0.557653	−	0.830074i	$-0.688298\pi$
$827$	−32.0736	−1.11531	−0.557653	+	0.830074i	$0.688298\pi$
$828$	0	0
$829$	−16.6089	−0.576850	−0.288425	−	0.957502i	$-0.593132\pi$
$829$	−16.6089	−0.576850	−0.288425	+	0.957502i	$0.593132\pi$
$830$	0	0
$831$	4.77387	0.165604
$832$	0	0
$833$	2.54810	0.0882864
$834$	0	0
$835$	9.88366	0.342038
$836$	0	0
$837$	3.27392	0.113163
$838$	0	0
$839$	25.3657	0.875723	0.437861	−	0.899042i	$-0.355736\pi$
$839$	25.3657	0.875723	0.437861	+	0.899042i	$0.355736\pi$
$840$	0	0
$841$	4.20411	0.144969
$842$	0	0
$843$	−3.08990	−0.106422
$844$	0	0
$845$	−1.75196	−0.0602693
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−1.34866	−0.0462860
$850$	0	0
$851$	−7.16459	−0.245599
$852$	0	0
$853$	15.8980	0.544336	0.272168	−	0.962250i	$-0.412259\pi$
$853$	15.8980	0.544336	0.272168	+	0.962250i	$0.412259\pi$
$854$	0	0
$855$	27.1056	0.926993
$856$	0	0
$857$	28.5862	0.976486	0.488243	−	0.872708i	$-0.337638\pi$
$857$	28.5862	0.976486	0.488243	+	0.872708i	$0.337638\pi$
$858$	0	0
$859$	−30.2897	−1.03347	−0.516735	−	0.856145i	$-0.672853\pi$
$859$	−30.2897	−1.03347	−0.516735	+	0.856145i	$0.672853\pi$
$860$	0	0
$861$	−0.286172	−0.00975273
$862$	0	0
$863$	−11.9666	−0.407348	−0.203674	−	0.979039i	$-0.565288\pi$
$863$	−11.9666	−0.407348	−0.203674	+	0.979039i	$0.565288\pi$
$864$	0	0
$865$	19.2532	0.654630
$866$	0	0
$867$	−1.83025	−0.0621585
$868$	0	0
$869$	−8.13699	−0.276028
$870$	0	0
$871$	9.05824	0.306927
$872$	0	0
$873$	30.8481	1.04405
$874$	0	0
$875$	−12.1422	−0.410481
$876$	0	0
$877$	29.8487	1.00792	0.503960	−	0.863727i	$-0.331876\pi$
$877$	29.8487	1.00792	0.503960	+	0.863727i	$0.331876\pi$
$878$	0	0
$879$	1.51900	0.0512347
$880$	0	0
$881$	−7.84926	−0.264448	−0.132224	−	0.991220i	$-0.542212\pi$
$881$	−7.84926	−0.264448	−0.132224	+	0.991220i	$0.542212\pi$
$882$	0	0
$883$	−47.1529	−1.58682	−0.793411	−	0.608686i	$-0.791697\pi$
$883$	−47.1529	−1.58682	−0.793411	+	0.608686i	$0.791697\pi$
$884$	0	0
$885$	1.68395	0.0566054
$886$	0	0
$887$	−2.25714	−0.0757873	−0.0378937	−	0.999282i	$-0.512065\pi$
$887$	−2.25714	−0.0757873	−0.0378937	+	0.999282i	$0.512065\pi$
$888$	0	0
$889$	13.4459	0.450961
$890$	0	0
$891$	−8.72784	−0.292394
$892$	0	0
$893$	27.7647	0.929110
$894$	0	0
$895$	24.8109	0.829335
$896$	0	0
$897$	0.246871	0.00824277
$898$	0	0
$899$	−18.1423	−0.605078
$900$	0	0
$901$	16.0795	0.535686
$902$	0	0
$903$	0.663741	0.0220879
$904$	0	0
$905$	−14.0962	−0.468573
$906$	0	0
$907$	−0.743618	−0.0246914	−0.0123457	−	0.999924i	$-0.503930\pi$
$907$	−0.743618	−0.0246914	−0.0123457	+	0.999924i	$0.503930\pi$
$908$	0	0
$909$	34.1422	1.13243
$910$	0	0
$911$	−9.28805	−0.307727	−0.153863	−	0.988092i	$-0.549172\pi$
$911$	−9.28805	−0.307727	−0.153863	+	0.988092i	$0.549172\pi$
$912$	0	0
$913$	3.74833	0.124052
$914$	0	0
$915$	2.80447	0.0927129
$916$	0	0
$917$	16.5644	0.547004
$918$	0	0
$919$	−43.7574	−1.44342	−0.721711	−	0.692194i	$-0.756644\pi$
$919$	−43.7574	−1.44342	−0.721711	+	0.692194i	$0.756644\pi$
$920$	0	0
$921$	−2.54802	−0.0839600
$922$	0	0
$923$	−2.65527	−0.0873993
$924$	0	0
$925$	9.75989	0.320903
$926$	0	0
$927$	−23.3978	−0.768486
$928$	0	0
$929$	49.8797	1.63650	0.818250	−	0.574862i	$-0.194944\pi$
$929$	49.8797	1.63650	0.818250	+	0.574862i	$0.194944\pi$
$930$	0	0
$931$	5.20989	0.170747
$932$	0	0
$933$	4.69477	0.153700
$934$	0	0
$935$	4.46417	0.145994
$936$	0	0
$937$	26.6698	0.871263	0.435632	−	0.900125i	$-0.356525\pi$
$937$	26.6698	0.871263	0.435632	+	0.900125i	$0.356525\pi$
$938$	0	0
$939$	4.10306	0.133898
$940$	0	0
$941$	24.0660	0.784528	0.392264	−	0.919853i	$-0.371692\pi$
$941$	24.0660	0.784528	0.392264	+	0.919853i	$0.371692\pi$
$942$	0	0
$943$	2.32836	0.0758218
$944$	0	0
$945$	−1.82179	−0.0592627
$946$	0	0
$947$	29.3656	0.954255	0.477127	−	0.878834i	$-0.341678\pi$
$947$	29.3656	0.954255	0.477127	+	0.878834i	$0.341678\pi$
$948$	0	0
$949$	−6.24560	−0.202741
$950$	0	0
$951$	−2.72666	−0.0884181
$952$	0	0
$953$	5.51335	0.178595	0.0892974	−	0.996005i	$-0.471538\pi$
$953$	5.51335	0.178595	0.0892974	+	0.996005i	$0.471538\pi$
$954$	0	0
$955$	4.21102	0.136265
$956$	0	0
$957$	−1.00374	−0.0324462
$958$	0	0
$959$	−0.312561	−0.0100931
$960$	0	0
$961$	−21.0873	−0.680236
$962$	0	0
$963$	−48.3615	−1.55843
$964$	0	0
$965$	37.6954	1.21346
$966$	0	0
$967$	−4.98035	−0.160157	−0.0800786	−	0.996789i	$-0.525517\pi$
$967$	−4.98035	−0.160157	−0.0800786	+	0.996789i	$0.525517\pi$
$968$	0	0
$969$	2.31243	0.0742859
$970$	0	0
$971$	21.6266	0.694032	0.347016	−	0.937859i	$-0.387195\pi$
$971$	21.6266	0.694032	0.347016	+	0.937859i	$0.387195\pi$
$972$	0	0
$973$	2.57651	0.0825990
$974$	0	0
$975$	−0.336297	−0.0107701
$976$	0	0
$977$	−34.7536	−1.11187	−0.555934	−	0.831227i	$-0.687639\pi$
$977$	−34.7536	−1.11187	−0.555934	+	0.831227i	$0.687639\pi$
$978$	0	0
$979$	−15.4453	−0.493635
$980$	0	0
$981$	−32.3044	−1.03140
$982$	0	0
$983$	−16.5349	−0.527382	−0.263691	−	0.964607i	$-0.584940\pi$
$983$	−16.5349	−0.527382	−0.263691	+	0.964607i	$0.584940\pi$
$984$	0	0
$985$	13.5511	0.431774
$986$	0	0
$987$	−0.928299	−0.0295481
$988$	0	0
$989$	−5.40034	−0.171721
$990$	0	0
$991$	38.7841	1.23202	0.616008	−	0.787740i	$-0.288749\pi$
$991$	38.7841	1.23202	0.616008	+	0.787740i	$0.288749\pi$
$992$	0	0
$993$	−0.499646	−0.0158558
$994$	0	0
$995$	27.3026	0.865549
$996$	0	0
$997$	−30.7451	−0.973708	−0.486854	−	0.873483i	$-0.661856\pi$
$997$	−30.7451	−0.973708	−0.486854	+	0.873483i	$0.661856\pi$
$998$	0	0
$999$	5.25677	0.166317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.m.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.m.1.6	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.174190 q^{3} -1.75196 q^{5} -1.00000 q^{7} -2.96966 q^{9} +O(q^{10})\)	\(q+0.174190 q^{3} -1.75196 q^{5} -1.00000 q^{7} -2.96966 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.305174 q^{15} +2.54810 q^{17} +5.20989 q^{19} -0.174190 q^{21} +1.41725 q^{23} -1.93063 q^{25} -1.03986 q^{27} +5.76230 q^{29} -3.14844 q^{31} -0.174190 q^{33} +1.75196 q^{35} -5.05529 q^{37} +0.174190 q^{39} +1.64287 q^{41} -3.81044 q^{43} +5.20273 q^{45} +5.32923 q^{47} +1.00000 q^{49} +0.443854 q^{51} +6.31040 q^{53} +1.75196 q^{55} +0.907511 q^{57} -5.51800 q^{59} -9.18973 q^{61} +2.96966 q^{63} -1.75196 q^{65} +9.05824 q^{67} +0.246871 q^{69} -2.65527 q^{71} -6.24560 q^{73} -0.336297 q^{75} +1.00000 q^{77} +8.13699 q^{79} +8.72784 q^{81} -3.74833 q^{83} -4.46417 q^{85} +1.00374 q^{87} +15.4453 q^{89} -1.00000 q^{91} -0.548427 q^{93} -9.12753 q^{95} -10.3878 q^{97} +2.96966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9} - 9 q^{11} + 9 q^{13} - q^{15} + 13 q^{17} - 5 q^{19} + 5 q^{21} - 7 q^{23} + 14 q^{25} - 23 q^{27} - 19 q^{29} + 2 q^{31} + 5 q^{33} + 5 q^{35} + 4 q^{37} - 5 q^{39} + 10 q^{41} - 17 q^{43} + 11 q^{45} - 5 q^{47} + 9 q^{49} - 9 q^{51} - 24 q^{53} + 5 q^{55} - 4 q^{57} - 19 q^{59} - 12 q^{63} - 5 q^{65} + 2 q^{67} - 2 q^{69} - 2 q^{71} + 34 q^{73} - 46 q^{75} + 9 q^{77} + 5 q^{79} + 37 q^{81} - 24 q^{83} + 33 q^{85} + 41 q^{87} - 11 q^{89} - 9 q^{91} + 53 q^{93} - 23 q^{95} - 12 q^{99}+O(q^{100}) \) 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9 - 9 * q^11 + 9 * q^13 - q^15 + 13 * q^17 - 5 * q^19 + 5 * q^21 - 7 * q^23 + 14 * q^25 - 23 * q^27 - 19 * q^29 + 2 * q^31 + 5 * q^33 + 5 * q^35 + 4 * q^37 - 5 * q^39 + 10 * q^41 - 17 * q^43 + 11 * q^45 - 5 * q^47 + 9 * q^49 - 9 * q^51 - 24 * q^53 + 5 * q^55 - 4 * q^57 - 19 * q^59 - 12 * q^63 - 5 * q^65 + 2 * q^67 - 2 * q^69 - 2 * q^71 + 34 * q^73 - 46 * q^75 + 9 * q^77 + 5 * q^79 + 37 * q^81 - 24 * q^83 + 33 * q^85 + 41 * q^87 - 11 * q^89 - 9 * q^91 + 53 * q^93 - 23 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more