LMFDB - Embedded newform 4004.2.a.g.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.246302029.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 $ x^6 - 2x^5 - 9x^4 + 14x^3 + 15x^2 - 13*x - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.55484$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.55484 q^{3} -1.87627 q^{5} +1.00000 q^{7} +3.52720 q^{9} +O(q^{10})$	$q+2.55484 q^{3} -1.87627 q^{5} +1.00000 q^{7} +3.52720 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.79356 q^{15} -1.96704 q^{17} -8.28099 q^{19} +2.55484 q^{21} -0.362456 q^{23} -1.47962 q^{25} +1.34691 q^{27} -7.79356 q^{29} +7.22999 q^{31} -2.55484 q^{33} -1.87627 q^{35} -1.16350 q^{37} -2.55484 q^{39} +6.10842 q^{41} -2.22534 q^{43} -6.61797 q^{45} -7.69090 q^{47} +1.00000 q^{49} -5.02547 q^{51} -3.04884 q^{53} +1.87627 q^{55} -21.1566 q^{57} +3.83433 q^{59} -13.6910 q^{61} +3.52720 q^{63} +1.87627 q^{65} -10.2644 q^{67} -0.926017 q^{69} +13.6978 q^{71} +5.85012 q^{73} -3.78018 q^{75} -1.00000 q^{77} -11.0204 q^{79} -7.14047 q^{81} +3.50750 q^{83} +3.69070 q^{85} -19.9113 q^{87} -3.89863 q^{89} -1.00000 q^{91} +18.4714 q^{93} +15.5374 q^{95} -2.05403 q^{97} -3.52720 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * 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
$3$	2.55484	1.47504	0.737518	−	0.675327i	$-0.235998\pi$
$3$	2.55484	1.47504	0.737518	+	0.675327i	$0.235998\pi$
$4$	0	0
$5$	−1.87627	−0.839093	−0.419546	−	0.907734i	$-0.637811\pi$
$5$	−1.87627	−0.839093	−0.419546	+	0.907734i	$0.637811\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.52720	1.17573
$10$	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
$15$	−4.79356	−1.23769
$16$	0	0
$17$	−1.96704	−0.477078	−0.238539	−	0.971133i	$-0.576668\pi$
$17$	−1.96704	−0.477078	−0.238539	+	0.971133i	$0.576668\pi$
$18$	0	0
$19$	−8.28099	−1.89979	−0.949895	−	0.312570i	$-0.898810\pi$
$19$	−8.28099	−1.89979	−0.949895	+	0.312570i	$0.898810\pi$
$20$	0	0
$21$	2.55484	0.557511
$22$	0	0
$23$	−0.362456	−0.0755773	−0.0377887	−	0.999286i	$-0.512031\pi$
$23$	−0.362456	−0.0755773	−0.0377887	+	0.999286i	$0.512031\pi$
$24$	0	0
$25$	−1.47962	−0.295923
$26$	0	0
$27$	1.34691	0.259212
$28$	0	0
$29$	−7.79356	−1.44723	−0.723614	−	0.690205i	$-0.757520\pi$
$29$	−7.79356	−1.44723	−0.723614	+	0.690205i	$0.757520\pi$
$30$	0	0
$31$	7.22999	1.29854	0.649272	−	0.760556i	$-0.275074\pi$
$31$	7.22999	1.29854	0.649272	+	0.760556i	$0.275074\pi$
$32$	0	0
$33$	−2.55484	−0.444740
$34$	0	0
$35$	−1.87627	−0.317147
$36$	0	0
$37$	−1.16350	−0.191279	−0.0956393	−	0.995416i	$-0.530490\pi$
$37$	−1.16350	−0.191279	−0.0956393	+	0.995416i	$0.530490\pi$
$38$	0	0
$39$	−2.55484	−0.409102
$40$	0	0
$41$	6.10842	0.953975	0.476988	−	0.878910i	$-0.341729\pi$
$41$	6.10842	0.953975	0.476988	+	0.878910i	$0.341729\pi$
$42$	0	0
$43$	−2.22534	−0.339361	−0.169681	−	0.985499i	$-0.554274\pi$
$43$	−2.22534	−0.339361	−0.169681	+	0.985499i	$0.554274\pi$
$44$	0	0
$45$	−6.61797	−0.986549
$46$	0	0
$47$	−7.69090	−1.12183	−0.560917	−	0.827872i	$-0.689551\pi$
$47$	−7.69090	−1.12183	−0.560917	+	0.827872i	$0.689551\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.02547	−0.703707
$52$	0	0
$53$	−3.04884	−0.418790	−0.209395	−	0.977831i	$-0.567149\pi$
$53$	−3.04884	−0.418790	−0.209395	+	0.977831i	$0.567149\pi$
$54$	0	0
$55$	1.87627	0.252996
$56$	0	0
$57$	−21.1566	−2.80226
$58$	0	0
$59$	3.83433	0.499188	0.249594	−	0.968351i	$-0.419703\pi$
$59$	3.83433	0.499188	0.249594	+	0.968351i	$0.419703\pi$
$60$	0	0
$61$	−13.6910	−1.75296	−0.876478	−	0.481441i	$-0.840113\pi$
$61$	−13.6910	−1.75296	−0.876478	+	0.481441i	$0.840113\pi$
$62$	0	0
$63$	3.52720	0.444385
$64$	0	0
$65$	1.87627	0.232723
$66$	0	0
$67$	−10.2644	−1.25400	−0.627001	−	0.779019i	$-0.715718\pi$
$67$	−10.2644	−1.25400	−0.627001	+	0.779019i	$0.715718\pi$
$68$	0	0
$69$	−0.926017	−0.111479
$70$	0	0
$71$	13.6978	1.62563	0.812817	−	0.582519i	$-0.197933\pi$
$71$	13.6978	1.62563	0.812817	+	0.582519i	$0.197933\pi$
$72$	0	0
$73$	5.85012	0.684705	0.342353	−	0.939572i	$-0.388776\pi$
$73$	5.85012	0.684705	0.342353	+	0.939572i	$0.388776\pi$
$74$	0	0
$75$	−3.78018	−0.436497
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−11.0204	−1.23989	−0.619946	−	0.784645i	$-0.712845\pi$
$79$	−11.0204	−1.23989	−0.619946	+	0.784645i	$0.712845\pi$
$80$	0	0
$81$	−7.14047	−0.793385
$82$	0	0
$83$	3.50750	0.384998	0.192499	−	0.981297i	$-0.438341\pi$
$83$	3.50750	0.384998	0.192499	+	0.981297i	$0.438341\pi$
$84$	0	0
$85$	3.69070	0.400313
$86$	0	0
$87$	−19.9113	−2.13471
$88$	0	0
$89$	−3.89863	−0.413254	−0.206627	−	0.978420i	$-0.566249\pi$
$89$	−3.89863	−0.413254	−0.206627	+	0.978420i	$0.566249\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	18.4714	1.91540
$94$	0	0
$95$	15.5374	1.59410
$96$	0	0
$97$	−2.05403	−0.208555	−0.104277	−	0.994548i	$-0.533253\pi$
$97$	−2.05403	−0.208555	−0.104277	+	0.994548i	$0.533253\pi$
$98$	0	0
$99$	−3.52720	−0.354497
$100$	0	0
$101$	2.57072	0.255796	0.127898	−	0.991787i	$-0.459177\pi$
$101$	2.57072	0.255796	0.127898	+	0.991787i	$0.459177\pi$
$102$	0	0
$103$	2.65991	0.262089	0.131044	−	0.991377i	$-0.458167\pi$
$103$	2.65991	0.262089	0.131044	+	0.991377i	$0.458167\pi$
$104$	0	0
$105$	−4.79356	−0.467804
$106$	0	0
$107$	−3.04339	−0.294216	−0.147108	−	0.989120i	$-0.546996\pi$
$107$	−3.04339	−0.294216	−0.147108	+	0.989120i	$0.546996\pi$
$108$	0	0
$109$	−4.44540	−0.425792	−0.212896	−	0.977075i	$-0.568290\pi$
$109$	−4.44540	−0.425792	−0.212896	+	0.977075i	$0.568290\pi$
$110$	0	0
$111$	−2.97256	−0.282143
$112$	0	0
$113$	12.4382	1.17009	0.585046	−	0.811000i	$-0.301076\pi$
$113$	12.4382	1.17009	0.585046	+	0.811000i	$0.301076\pi$
$114$	0	0
$115$	0.680065	0.0634164
$116$	0	0
$117$	−3.52720	−0.326090
$118$	0	0
$119$	−1.96704	−0.180318
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.6060	1.40715
$124$	0	0
$125$	12.1575	1.08740
$126$	0	0
$127$	17.4283	1.54651	0.773255	−	0.634096i	$-0.218628\pi$
$127$	17.4283	1.54651	0.773255	+	0.634096i	$0.218628\pi$
$128$	0	0
$129$	−5.68538	−0.500570
$130$	0	0
$131$	−19.7907	−1.72912	−0.864561	−	0.502527i	$-0.832404\pi$
$131$	−19.7907	−1.72912	−0.864561	+	0.502527i	$0.832404\pi$
$132$	0	0
$133$	−8.28099	−0.718053
$134$	0	0
$135$	−2.52716	−0.217503
$136$	0	0
$137$	8.55608	0.730995	0.365498	−	0.930812i	$-0.380899\pi$
$137$	8.55608	0.730995	0.365498	+	0.930812i	$0.380899\pi$
$138$	0	0
$139$	−13.6123	−1.15458	−0.577288	−	0.816540i	$-0.695889\pi$
$139$	−13.6123	−1.15458	−0.577288	+	0.816540i	$0.695889\pi$
$140$	0	0
$141$	−19.6490	−1.65474
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	14.6228	1.21436
$146$	0	0
$147$	2.55484	0.210719
$148$	0	0
$149$	9.27331	0.759699	0.379849	−	0.925048i	$-0.375976\pi$
$149$	9.27331	0.759699	0.379849	+	0.925048i	$0.375976\pi$
$150$	0	0
$151$	15.9916	1.30138	0.650689	−	0.759345i	$-0.274480\pi$
$151$	15.9916	1.30138	0.650689	+	0.759345i	$0.274480\pi$
$152$	0	0
$153$	−6.93815	−0.560916
$154$	0	0
$155$	−13.5654	−1.08960
$156$	0	0
$157$	−13.9882	−1.11638	−0.558192	−	0.829712i	$-0.688505\pi$
$157$	−13.9882	−1.11638	−0.558192	+	0.829712i	$0.688505\pi$
$158$	0	0
$159$	−7.78929	−0.617730
$160$	0	0
$161$	−0.362456	−0.0285655
$162$	0	0
$163$	−8.13861	−0.637465	−0.318733	−	0.947845i	$-0.603257\pi$
$163$	−8.13861	−0.637465	−0.318733	+	0.947845i	$0.603257\pi$
$164$	0	0
$165$	4.79356	0.373178
$166$	0	0
$167$	0.374658	0.0289919	0.0144960	−	0.999895i	$-0.495386\pi$
$167$	0.374658	0.0289919	0.0144960	+	0.999895i	$0.495386\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−29.2087	−2.23364
$172$	0	0
$173$	13.1508	0.999839	0.499920	−	0.866072i	$-0.333363\pi$
$173$	13.1508	0.999839	0.499920	+	0.866072i	$0.333363\pi$
$174$	0	0
$175$	−1.47962	−0.111848
$176$	0	0
$177$	9.79610	0.736320
$178$	0	0
$179$	−1.52137	−0.113713	−0.0568563	−	0.998382i	$-0.518108\pi$
$179$	−1.52137	−0.113713	−0.0568563	+	0.998382i	$0.518108\pi$
$180$	0	0
$181$	11.5003	0.854814	0.427407	−	0.904059i	$-0.359427\pi$
$181$	11.5003	0.854814	0.427407	+	0.904059i	$0.359427\pi$
$182$	0	0
$183$	−34.9784	−2.58567
$184$	0	0
$185$	2.18304	0.160501
$186$	0	0
$187$	1.96704	0.143844
$188$	0	0
$189$	1.34691	0.0979730
$190$	0	0
$191$	−4.68143	−0.338737	−0.169368	−	0.985553i	$-0.554173\pi$
$191$	−4.68143	−0.338737	−0.169368	+	0.985553i	$0.554173\pi$
$192$	0	0
$193$	−12.7884	−0.920526	−0.460263	−	0.887783i	$-0.652245\pi$
$193$	−12.7884	−0.920526	−0.460263	+	0.887783i	$0.652245\pi$
$194$	0	0
$195$	4.79356	0.343274
$196$	0	0
$197$	3.44010	0.245097	0.122549	−	0.992463i	$-0.460893\pi$
$197$	3.44010	0.245097	0.122549	+	0.992463i	$0.460893\pi$
$198$	0	0
$199$	−16.2599	−1.15263	−0.576315	−	0.817227i	$-0.695510\pi$
$199$	−16.2599	−1.15263	−0.576315	+	0.817227i	$0.695510\pi$
$200$	0	0
$201$	−26.2240	−1.84970
$202$	0	0
$203$	−7.79356	−0.547001
$204$	0	0
$205$	−11.4610	−0.800474
$206$	0	0
$207$	−1.27845	−0.0888587
$208$	0	0
$209$	8.28099	0.572808
$210$	0	0
$211$	1.98076	0.136361	0.0681804	−	0.997673i	$-0.478281\pi$
$211$	1.98076	0.136361	0.0681804	+	0.997673i	$0.478281\pi$
$212$	0	0
$213$	34.9958	2.39787
$214$	0	0
$215$	4.17534	0.284756
$216$	0	0
$217$	7.22999	0.490804
$218$	0	0
$219$	14.9461	1.00997
$220$	0	0
$221$	1.96704	0.132318
$222$	0	0
$223$	7.75791	0.519508	0.259754	−	0.965675i	$-0.416359\pi$
$223$	7.75791	0.519508	0.259754	+	0.965675i	$0.416359\pi$
$224$	0	0
$225$	−5.21890	−0.347926
$226$	0	0
$227$	−17.0303	−1.13034	−0.565171	−	0.824974i	$-0.691190\pi$
$227$	−17.0303	−1.13034	−0.565171	+	0.824974i	$0.691190\pi$
$228$	0	0
$229$	−1.11375	−0.0735987	−0.0367994	−	0.999323i	$-0.511716\pi$
$229$	−1.11375	−0.0735987	−0.0367994	+	0.999323i	$0.511716\pi$
$230$	0	0
$231$	−2.55484	−0.168096
$232$	0	0
$233$	−3.11475	−0.204054	−0.102027	−	0.994782i	$-0.532533\pi$
$233$	−3.11475	−0.204054	−0.102027	+	0.994782i	$0.532533\pi$
$234$	0	0
$235$	14.4302	0.941322
$236$	0	0
$237$	−28.1553	−1.82889
$238$	0	0
$239$	−8.93068	−0.577678	−0.288839	−	0.957378i	$-0.593269\pi$
$239$	−8.93068	−0.577678	−0.288839	+	0.957378i	$0.593269\pi$
$240$	0	0
$241$	1.98582	0.127918	0.0639588	−	0.997953i	$-0.479627\pi$
$241$	1.98582	0.127918	0.0639588	+	0.997953i	$0.479627\pi$
$242$	0	0
$243$	−22.2835	−1.42948
$244$	0	0
$245$	−1.87627	−0.119870
$246$	0	0
$247$	8.28099	0.526907
$248$	0	0
$249$	8.96109	0.567886
$250$	0	0
$251$	−1.40499	−0.0886825	−0.0443412	−	0.999016i	$-0.514119\pi$
$251$	−1.40499	−0.0886825	−0.0443412	+	0.999016i	$0.514119\pi$
$252$	0	0
$253$	0.362456	0.0227874
$254$	0	0
$255$	9.42914	0.590476
$256$	0	0
$257$	−5.35372	−0.333956	−0.166978	−	0.985961i	$-0.553401\pi$
$257$	−5.35372	−0.333956	−0.166978	+	0.985961i	$0.553401\pi$
$258$	0	0
$259$	−1.16350	−0.0722965
$260$	0	0
$261$	−27.4894	−1.70155
$262$	0	0
$263$	−10.9685	−0.676349	−0.338175	−	0.941083i	$-0.609809\pi$
$263$	−10.9685	−0.676349	−0.338175	+	0.941083i	$0.609809\pi$
$264$	0	0
$265$	5.72044	0.351404
$266$	0	0
$267$	−9.96038	−0.609565
$268$	0	0
$269$	13.1620	0.802500	0.401250	−	0.915969i	$-0.368576\pi$
$269$	13.1620	0.802500	0.401250	+	0.915969i	$0.368576\pi$
$270$	0	0
$271$	9.11532	0.553716	0.276858	−	0.960911i	$-0.410707\pi$
$271$	9.11532	0.553716	0.276858	+	0.960911i	$0.410707\pi$
$272$	0	0
$273$	−2.55484	−0.154626
$274$	0	0
$275$	1.47962	0.0892242
$276$	0	0
$277$	−8.64299	−0.519307	−0.259653	−	0.965702i	$-0.583608\pi$
$277$	−8.64299	−0.519307	−0.259653	+	0.965702i	$0.583608\pi$
$278$	0	0
$279$	25.5016	1.52674
$280$	0	0
$281$	−14.6735	−0.875347	−0.437674	−	0.899134i	$-0.644197\pi$
$281$	−14.6735	−0.875347	−0.437674	+	0.899134i	$0.644197\pi$
$282$	0	0
$283$	−21.5580	−1.28149	−0.640744	−	0.767755i	$-0.721374\pi$
$283$	−21.5580	−1.28149	−0.640744	+	0.767755i	$0.721374\pi$
$284$	0	0
$285$	39.6954	2.35136
$286$	0	0
$287$	6.10842	0.360569
$288$	0	0
$289$	−13.1307	−0.772397
$290$	0	0
$291$	−5.24770	−0.307626
$292$	0	0
$293$	−5.92201	−0.345968	−0.172984	−	0.984925i	$-0.555341\pi$
$293$	−5.92201	−0.345968	−0.172984	+	0.984925i	$0.555341\pi$
$294$	0	0
$295$	−7.19424	−0.418865
$296$	0	0
$297$	−1.34691	−0.0781554
$298$	0	0
$299$	0.362456	0.0209614
$300$	0	0
$301$	−2.22534	−0.128266
$302$	0	0
$303$	6.56777	0.377308
$304$	0	0
$305$	25.6881	1.47089
$306$	0	0
$307$	−2.74561	−0.156700	−0.0783502	−	0.996926i	$-0.524965\pi$
$307$	−2.74561	−0.156700	−0.0783502	+	0.996926i	$0.524965\pi$
$308$	0	0
$309$	6.79563	0.386590
$310$	0	0
$311$	−12.3246	−0.698866	−0.349433	−	0.936961i	$-0.613626\pi$
$311$	−12.3246	−0.698866	−0.349433	+	0.936961i	$0.613626\pi$
$312$	0	0
$313$	−0.496679	−0.0280739	−0.0140370	−	0.999901i	$-0.504468\pi$
$313$	−0.496679	−0.0280739	−0.0140370	+	0.999901i	$0.504468\pi$
$314$	0	0
$315$	−6.61797	−0.372880
$316$	0	0
$317$	6.34988	0.356645	0.178322	−	0.983972i	$-0.442933\pi$
$317$	6.34988	0.356645	0.178322	+	0.983972i	$0.442933\pi$
$318$	0	0
$319$	7.79356	0.436356
$320$	0	0
$321$	−7.77537	−0.433979
$322$	0	0
$323$	16.2891	0.906347
$324$	0	0
$325$	1.47962	0.0820743
$326$	0	0
$327$	−11.3573	−0.628059
$328$	0	0
$329$	−7.69090	−0.424013
$330$	0	0
$331$	−19.5405	−1.07404	−0.537022	−	0.843568i	$-0.680451\pi$
$331$	−19.5405	−1.07404	−0.537022	+	0.843568i	$0.680451\pi$
$332$	0	0
$333$	−4.10390	−0.224892
$334$	0	0
$335$	19.2589	1.05222
$336$	0	0
$337$	21.9730	1.19695	0.598474	−	0.801142i	$-0.295774\pi$
$337$	21.9730	1.19695	0.598474	+	0.801142i	$0.295774\pi$
$338$	0	0
$339$	31.7777	1.72593
$340$	0	0
$341$	−7.22999	−0.391526
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.73746	0.0935415
$346$	0	0
$347$	15.7646	0.846288	0.423144	−	0.906062i	$-0.360926\pi$
$347$	15.7646	0.846288	0.423144	+	0.906062i	$0.360926\pi$
$348$	0	0
$349$	21.5507	1.15358	0.576791	−	0.816892i	$-0.304305\pi$
$349$	21.5507	1.15358	0.576791	+	0.816892i	$0.304305\pi$
$350$	0	0
$351$	−1.34691	−0.0718925
$352$	0	0
$353$	14.5825	0.776149	0.388075	−	0.921628i	$-0.373140\pi$
$353$	14.5825	0.776149	0.388075	+	0.921628i	$0.373140\pi$
$354$	0	0
$355$	−25.7008	−1.36406
$356$	0	0
$357$	−5.02547	−0.265976
$358$	0	0
$359$	4.46616	0.235715	0.117858	−	0.993031i	$-0.462397\pi$
$359$	4.46616	0.235715	0.117858	+	0.993031i	$0.462397\pi$
$360$	0	0
$361$	49.5748	2.60920
$362$	0	0
$363$	2.55484	0.134094
$364$	0	0
$365$	−10.9764	−0.574531
$366$	0	0
$367$	−8.73981	−0.456214	−0.228107	−	0.973636i	$-0.573254\pi$
$367$	−8.73981	−0.456214	−0.228107	+	0.973636i	$0.573254\pi$
$368$	0	0
$369$	21.5456	1.12162
$370$	0	0
$371$	−3.04884	−0.158288
$372$	0	0
$373$	4.06727	0.210595	0.105298	−	0.994441i	$-0.466421\pi$
$373$	4.06727	0.210595	0.105298	+	0.994441i	$0.466421\pi$
$374$	0	0
$375$	31.0604	1.60395
$376$	0	0
$377$	7.79356	0.401389
$378$	0	0
$379$	4.92584	0.253023	0.126512	−	0.991965i	$-0.459622\pi$
$379$	4.92584	0.253023	0.126512	+	0.991965i	$0.459622\pi$
$380$	0	0
$381$	44.5264	2.28116
$382$	0	0
$383$	−4.88786	−0.249758	−0.124879	−	0.992172i	$-0.539854\pi$
$383$	−4.88786	−0.249758	−0.124879	+	0.992172i	$0.539854\pi$
$384$	0	0
$385$	1.87627	0.0956235
$386$	0	0
$387$	−7.84921	−0.398998
$388$	0	0
$389$	28.8846	1.46451	0.732253	−	0.681032i	$-0.238469\pi$
$389$	28.8846	1.46451	0.732253	+	0.681032i	$0.238469\pi$
$390$	0	0
$391$	0.712967	0.0360563
$392$	0	0
$393$	−50.5621	−2.55052
$394$	0	0
$395$	20.6772	1.04038
$396$	0	0
$397$	7.96000	0.399501	0.199751	−	0.979847i	$-0.435987\pi$
$397$	7.96000	0.399501	0.199751	+	0.979847i	$0.435987\pi$
$398$	0	0
$399$	−21.1566	−1.05915
$400$	0	0
$401$	24.2626	1.21162	0.605808	−	0.795611i	$-0.292850\pi$
$401$	24.2626	1.21162	0.605808	+	0.795611i	$0.292850\pi$
$402$	0	0
$403$	−7.22999	−0.360151
$404$	0	0
$405$	13.3974	0.665724
$406$	0	0
$407$	1.16350	0.0576727
$408$	0	0
$409$	−20.8104	−1.02901	−0.514505	−	0.857487i	$-0.672024\pi$
$409$	−20.8104	−1.02901	−0.514505	+	0.857487i	$0.672024\pi$
$410$	0	0
$411$	21.8594	1.07824
$412$	0	0
$413$	3.83433	0.188675
$414$	0	0
$415$	−6.58101	−0.323049
$416$	0	0
$417$	−34.7771	−1.70304
$418$	0	0
$419$	1.25316	0.0612211	0.0306106	−	0.999531i	$-0.490255\pi$
$419$	1.25316	0.0612211	0.0306106	+	0.999531i	$0.490255\pi$
$420$	0	0
$421$	−24.3045	−1.18453	−0.592265	−	0.805743i	$-0.701766\pi$
$421$	−24.3045	−1.18453	−0.592265	+	0.805743i	$0.701766\pi$
$422$	0	0
$423$	−27.1273	−1.31898
$424$	0	0
$425$	2.91047	0.141178
$426$	0	0
$427$	−13.6910	−0.662555
$428$	0	0
$429$	2.55484	0.123349
$430$	0	0
$431$	39.9407	1.92388	0.961938	−	0.273269i	$-0.0881050\pi$
$431$	39.9407	1.92388	0.961938	+	0.273269i	$0.0881050\pi$
$432$	0	0
$433$	−37.6468	−1.80919	−0.904596	−	0.426271i	$-0.859827\pi$
$433$	−37.6468	−1.80919	−0.904596	+	0.426271i	$0.859827\pi$
$434$	0	0
$435$	37.3589	1.79122
$436$	0	0
$437$	3.00150	0.143581
$438$	0	0
$439$	4.91735	0.234692	0.117346	−	0.993091i	$-0.462561\pi$
$439$	4.91735	0.234692	0.117346	+	0.993091i	$0.462561\pi$
$440$	0	0
$441$	3.52720	0.167962
$442$	0	0
$443$	23.7809	1.12986	0.564932	−	0.825137i	$-0.308902\pi$
$443$	23.7809	1.12986	0.564932	+	0.825137i	$0.308902\pi$
$444$	0	0
$445$	7.31488	0.346759
$446$	0	0
$447$	23.6918	1.12058
$448$	0	0
$449$	−9.58825	−0.452498	−0.226249	−	0.974070i	$-0.572646\pi$
$449$	−9.58825	−0.452498	−0.226249	+	0.974070i	$0.572646\pi$
$450$	0	0
$451$	−6.10842	−0.287634
$452$	0	0
$453$	40.8559	1.91958
$454$	0	0
$455$	1.87627	0.0879608
$456$	0	0
$457$	−25.1265	−1.17537	−0.587684	−	0.809090i	$-0.699960\pi$
$457$	−25.1265	−1.17537	−0.587684	+	0.809090i	$0.699960\pi$
$458$	0	0
$459$	−2.64942	−0.123664
$460$	0	0
$461$	−17.3804	−0.809488	−0.404744	−	0.914430i	$-0.632639\pi$
$461$	−17.3804	−0.809488	−0.404744	+	0.914430i	$0.632639\pi$
$462$	0	0
$463$	10.7750	0.500756	0.250378	−	0.968148i	$-0.419445\pi$
$463$	10.7750	0.500756	0.250378	+	0.968148i	$0.419445\pi$
$464$	0	0
$465$	−34.6574	−1.60720
$466$	0	0
$467$	14.4418	0.668287	0.334143	−	0.942522i	$-0.391553\pi$
$467$	14.4418	0.668287	0.334143	+	0.942522i	$0.391553\pi$
$468$	0	0
$469$	−10.2644	−0.473968
$470$	0	0
$471$	−35.7377	−1.64671
$472$	0	0
$473$	2.22534	0.102321
$474$	0	0
$475$	12.2527	0.562192
$476$	0	0
$477$	−10.7539	−0.492385
$478$	0	0
$479$	22.6231	1.03368	0.516838	−	0.856083i	$-0.327109\pi$
$479$	22.6231	1.03368	0.516838	+	0.856083i	$0.327109\pi$
$480$	0	0
$481$	1.16350	0.0530511
$482$	0	0
$483$	−0.926017	−0.0421352
$484$	0	0
$485$	3.85390	0.174997
$486$	0	0
$487$	−8.79699	−0.398630	−0.199315	−	0.979936i	$-0.563872\pi$
$487$	−8.79699	−0.398630	−0.199315	+	0.979936i	$0.563872\pi$
$488$	0	0
$489$	−20.7928	−0.940285
$490$	0	0
$491$	−10.4070	−0.469662	−0.234831	−	0.972036i	$-0.575454\pi$
$491$	−10.4070	−0.469662	−0.234831	+	0.972036i	$0.575454\pi$
$492$	0	0
$493$	15.3303	0.690441
$494$	0	0
$495$	6.61797	0.297456
$496$	0	0
$497$	13.6978	0.614432
$498$	0	0
$499$	33.5474	1.50179	0.750895	−	0.660422i	$-0.229623\pi$
$499$	33.5474	1.50179	0.750895	+	0.660422i	$0.229623\pi$
$500$	0	0
$501$	0.957191	0.0427642
$502$	0	0
$503$	19.0643	0.850034	0.425017	−	0.905185i	$-0.360268\pi$
$503$	19.0643	0.850034	0.425017	+	0.905185i	$0.360268\pi$
$504$	0	0
$505$	−4.82336	−0.214637
$506$	0	0
$507$	2.55484	0.113464
$508$	0	0
$509$	4.48788	0.198922	0.0994610	−	0.995041i	$-0.468288\pi$
$509$	4.48788	0.198922	0.0994610	+	0.995041i	$0.468288\pi$
$510$	0	0
$511$	5.85012	0.258794
$512$	0	0
$513$	−11.1537	−0.492448
$514$	0	0
$515$	−4.99070	−0.219917
$516$	0	0
$517$	7.69090	0.338245
$518$	0	0
$519$	33.5982	1.47480
$520$	0	0
$521$	18.1983	0.797281	0.398640	−	0.917107i	$-0.369482\pi$
$521$	18.1983	0.797281	0.398640	+	0.917107i	$0.369482\pi$
$522$	0	0
$523$	−33.2252	−1.45284	−0.726418	−	0.687253i	$-0.758817\pi$
$523$	−33.2252	−1.45284	−0.726418	+	0.687253i	$0.758817\pi$
$524$	0	0
$525$	−3.78018	−0.164980
$526$	0	0
$527$	−14.2217	−0.619507
$528$	0	0
$529$	−22.8686	−0.994288
$530$	0	0
$531$	13.5244	0.586911
$532$	0	0
$533$	−6.10842	−0.264585
$534$	0	0
$535$	5.71022	0.246874
$536$	0	0
$537$	−3.88686	−0.167730
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−18.2784	−0.785851	−0.392925	−	0.919570i	$-0.628537\pi$
$541$	−18.2784	−0.785851	−0.392925	+	0.919570i	$0.628537\pi$
$542$	0	0
$543$	29.3815	1.26088
$544$	0	0
$545$	8.34077	0.357279
$546$	0	0
$547$	−16.8820	−0.721823	−0.360911	−	0.932600i	$-0.617534\pi$
$547$	−16.8820	−0.721823	−0.360911	+	0.932600i	$0.617534\pi$
$548$	0	0
$549$	−48.2910	−2.06101
$550$	0	0
$551$	64.5384	2.74943
$552$	0	0
$553$	−11.0204	−0.468635
$554$	0	0
$555$	5.57732	0.236744
$556$	0	0
$557$	11.4642	0.485753	0.242876	−	0.970057i	$-0.421909\pi$
$557$	11.4642	0.485753	0.242876	+	0.970057i	$0.421909\pi$
$558$	0	0
$559$	2.22534	0.0941218
$560$	0	0
$561$	5.02547	0.212176
$562$	0	0
$563$	−6.73960	−0.284040	−0.142020	−	0.989864i	$-0.545360\pi$
$563$	−6.73960	−0.284040	−0.142020	+	0.989864i	$0.545360\pi$
$564$	0	0
$565$	−23.3375	−0.981816
$566$	0	0
$567$	−7.14047	−0.299871
$568$	0	0
$569$	−47.1803	−1.97790	−0.988950	−	0.148252i	$-0.952635\pi$
$569$	−47.1803	−1.97790	−0.988950	+	0.148252i	$0.952635\pi$
$570$	0	0
$571$	45.8800	1.92002	0.960008	−	0.279971i	$-0.0903250\pi$
$571$	45.8800	1.92002	0.960008	+	0.279971i	$0.0903250\pi$
$572$	0	0
$573$	−11.9603	−0.499649
$574$	0	0
$575$	0.536296	0.0223651
$576$	0	0
$577$	13.3765	0.556872	0.278436	−	0.960455i	$-0.410184\pi$
$577$	13.3765	0.556872	0.278436	+	0.960455i	$0.410184\pi$
$578$	0	0
$579$	−32.6722	−1.35781
$580$	0	0
$581$	3.50750	0.145515
$582$	0	0
$583$	3.04884	0.126270
$584$	0	0
$585$	6.61797	0.273619
$586$	0	0
$587$	−5.00657	−0.206643	−0.103322	−	0.994648i	$-0.532947\pi$
$587$	−5.00657	−0.206643	−0.103322	+	0.994648i	$0.532947\pi$
$588$	0	0
$589$	−59.8715	−2.46696
$590$	0	0
$591$	8.78890	0.361527
$592$	0	0
$593$	−44.1739	−1.81400	−0.907001	−	0.421128i	$-0.861634\pi$
$593$	−44.1739	−1.81400	−0.907001	+	0.421128i	$0.861634\pi$
$594$	0	0
$595$	3.69070	0.151304
$596$	0	0
$597$	−41.5413	−1.70017
$598$	0	0
$599$	11.4137	0.466352	0.233176	−	0.972435i	$-0.425088\pi$
$599$	11.4137	0.466352	0.233176	+	0.972435i	$0.425088\pi$
$600$	0	0
$601$	0.735736	0.0300113	0.0150056	−	0.999887i	$-0.495223\pi$
$601$	0.735736	0.0300113	0.0150056	+	0.999887i	$0.495223\pi$
$602$	0	0
$603$	−36.2047	−1.47437
$604$	0	0
$605$	−1.87627	−0.0762812
$606$	0	0
$607$	−22.5579	−0.915599	−0.457799	−	0.889056i	$-0.651362\pi$
$607$	−22.5579	−0.915599	−0.457799	+	0.889056i	$0.651362\pi$
$608$	0	0
$609$	−19.9113	−0.806846
$610$	0	0
$611$	7.69090	0.311141
$612$	0	0
$613$	37.9491	1.53275	0.766375	−	0.642394i	$-0.222059\pi$
$613$	37.9491	1.53275	0.766375	+	0.642394i	$0.222059\pi$
$614$	0	0
$615$	−29.2811	−1.18073
$616$	0	0
$617$	−42.3359	−1.70438	−0.852189	−	0.523234i	$-0.824725\pi$
$617$	−42.3359	−1.70438	−0.852189	+	0.523234i	$0.824725\pi$
$618$	0	0
$619$	−27.7690	−1.11613	−0.558064	−	0.829798i	$-0.688456\pi$
$619$	−27.7690	−1.11613	−0.558064	+	0.829798i	$0.688456\pi$
$620$	0	0
$621$	−0.488194	−0.0195906
$622$	0	0
$623$	−3.89863	−0.156195
$624$	0	0
$625$	−15.4127	−0.616506
$626$	0	0
$627$	21.1566	0.844913
$628$	0	0
$629$	2.28866	0.0912548
$630$	0	0
$631$	41.4634	1.65063	0.825316	−	0.564671i	$-0.190997\pi$
$631$	41.4634	1.65063	0.825316	+	0.564671i	$0.190997\pi$
$632$	0	0
$633$	5.06051	0.201137
$634$	0	0
$635$	−32.7001	−1.29766
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	48.3150	1.91131
$640$	0	0
$641$	32.6339	1.28896	0.644481	−	0.764620i	$-0.277073\pi$
$641$	32.6339	1.28896	0.644481	+	0.764620i	$0.277073\pi$
$642$	0	0
$643$	9.03643	0.356362	0.178181	−	0.983998i	$-0.442979\pi$
$643$	9.03643	0.356362	0.178181	+	0.983998i	$0.442979\pi$
$644$	0	0
$645$	10.6673	0.420025
$646$	0	0
$647$	47.0160	1.84839	0.924195	−	0.381920i	$-0.124737\pi$
$647$	47.0160	1.84839	0.924195	+	0.381920i	$0.124737\pi$
$648$	0	0
$649$	−3.83433	−0.150511
$650$	0	0
$651$	18.4714	0.723953
$652$	0	0
$653$	7.30976	0.286053	0.143026	−	0.989719i	$-0.454317\pi$
$653$	7.30976	0.286053	0.143026	+	0.989719i	$0.454317\pi$
$654$	0	0
$655$	37.1327	1.45089
$656$	0	0
$657$	20.6345	0.805030
$658$	0	0
$659$	−42.8899	−1.67075	−0.835376	−	0.549678i	$-0.814750\pi$
$659$	−42.8899	−1.67075	−0.835376	+	0.549678i	$0.814750\pi$
$660$	0	0
$661$	−10.4420	−0.406148	−0.203074	−	0.979163i	$-0.565093\pi$
$661$	−10.4420	−0.406148	−0.203074	+	0.979163i	$0.565093\pi$
$662$	0	0
$663$	5.02547	0.195173
$664$	0	0
$665$	15.5374	0.602513
$666$	0	0
$667$	2.82482	0.109378
$668$	0	0
$669$	19.8202	0.766293
$670$	0	0
$671$	13.6910	0.528536
$672$	0	0
$673$	24.1045	0.929161	0.464580	−	0.885531i	$-0.346205\pi$
$673$	24.1045	0.929161	0.464580	+	0.885531i	$0.346205\pi$
$674$	0	0
$675$	−1.99290	−0.0767068
$676$	0	0
$677$	50.0332	1.92293	0.961467	−	0.274922i	$-0.0886520\pi$
$677$	50.0332	1.92293	0.961467	+	0.274922i	$0.0886520\pi$
$678$	0	0
$679$	−2.05403	−0.0788263
$680$	0	0
$681$	−43.5097	−1.66730
$682$	0	0
$683$	−26.3993	−1.01014	−0.505070	−	0.863078i	$-0.668533\pi$
$683$	−26.3993	−1.01014	−0.505070	+	0.863078i	$0.668533\pi$
$684$	0	0
$685$	−16.0535	−0.613373
$686$	0	0
$687$	−2.84545	−0.108561
$688$	0	0
$689$	3.04884	0.116151
$690$	0	0
$691$	19.9904	0.760473	0.380236	−	0.924889i	$-0.375843\pi$
$691$	19.9904	0.760473	0.380236	+	0.924889i	$0.375843\pi$
$692$	0	0
$693$	−3.52720	−0.133987
$694$	0	0
$695$	25.5403	0.968797
$696$	0	0
$697$	−12.0155	−0.455120
$698$	0	0
$699$	−7.95769	−0.300987
$700$	0	0
$701$	−30.6947	−1.15932	−0.579661	−	0.814858i	$-0.696815\pi$
$701$	−30.6947	−1.15932	−0.579661	+	0.814858i	$0.696815\pi$
$702$	0	0
$703$	9.63495	0.363389
$704$	0	0
$705$	36.8668	1.38848
$706$	0	0
$707$	2.57072	0.0966818
$708$	0	0
$709$	−27.6986	−1.04024	−0.520122	−	0.854092i	$-0.674113\pi$
$709$	−27.6986	−1.04024	−0.520122	+	0.854092i	$0.674113\pi$
$710$	0	0
$711$	−38.8711	−1.45778
$712$	0	0
$713$	−2.62055	−0.0981405
$714$	0	0
$715$	−1.87627	−0.0701685
$716$	0	0
$717$	−22.8164	−0.852096
$718$	0	0
$719$	−17.9430	−0.669162	−0.334581	−	0.942367i	$-0.608595\pi$
$719$	−17.9430	−0.669162	−0.334581	+	0.942367i	$0.608595\pi$
$720$	0	0
$721$	2.65991	0.0990601
$722$	0	0
$723$	5.07344	0.188683
$724$	0	0
$725$	11.5315	0.428268
$726$	0	0
$727$	−35.4511	−1.31481	−0.657405	−	0.753537i	$-0.728346\pi$
$727$	−35.4511	−1.31481	−0.657405	+	0.753537i	$0.728346\pi$
$728$	0	0
$729$	−35.5092	−1.31516
$730$	0	0
$731$	4.37734	0.161902
$732$	0	0
$733$	−33.2910	−1.22963	−0.614815	−	0.788672i	$-0.710769\pi$
$733$	−33.2910	−1.22963	−0.614815	+	0.788672i	$0.710769\pi$
$734$	0	0
$735$	−4.79356	−0.176813
$736$	0	0
$737$	10.2644	0.378096
$738$	0	0
$739$	−19.3588	−0.712123	−0.356062	−	0.934462i	$-0.615881\pi$
$739$	−19.3588	−0.712123	−0.356062	+	0.934462i	$0.615881\pi$
$740$	0	0
$741$	21.1566	0.777207
$742$	0	0
$743$	14.6092	0.535959	0.267979	−	0.963425i	$-0.413644\pi$
$743$	14.6092	0.535959	0.267979	+	0.963425i	$0.413644\pi$
$744$	0	0
$745$	−17.3992	−0.637458
$746$	0	0
$747$	12.3716	0.452654
$748$	0	0
$749$	−3.04339	−0.111203
$750$	0	0
$751$	−11.1499	−0.406864	−0.203432	−	0.979089i	$-0.565210\pi$
$751$	−11.1499	−0.406864	−0.203432	+	0.979089i	$0.565210\pi$
$752$	0	0
$753$	−3.58953	−0.130810
$754$	0	0
$755$	−30.0045	−1.09198
$756$	0	0
$757$	−35.0876	−1.27528	−0.637641	−	0.770334i	$-0.720090\pi$
$757$	−35.0876	−1.27528	−0.637641	+	0.770334i	$0.720090\pi$
$758$	0	0
$759$	0.926017	0.0336123
$760$	0	0
$761$	−3.32937	−0.120690	−0.0603448	−	0.998178i	$-0.519220\pi$
$761$	−3.32937	−0.120690	−0.0603448	+	0.998178i	$0.519220\pi$
$762$	0	0
$763$	−4.44540	−0.160934
$764$	0	0
$765$	13.0178	0.470661
$766$	0	0
$767$	−3.83433	−0.138450
$768$	0	0
$769$	25.8805	0.933275	0.466637	−	0.884449i	$-0.345465\pi$
$769$	25.8805	0.933275	0.466637	+	0.884449i	$0.345465\pi$
$770$	0	0
$771$	−13.6779	−0.492597
$772$	0	0
$773$	36.1954	1.30186	0.650929	−	0.759139i	$-0.274380\pi$
$773$	36.1954	1.30186	0.650929	+	0.759139i	$0.274380\pi$
$774$	0	0
$775$	−10.6976	−0.384269
$776$	0	0
$777$	−2.97256	−0.106640
$778$	0	0
$779$	−50.5838	−1.81235
$780$	0	0
$781$	−13.6978	−0.490147
$782$	0	0
$783$	−10.4972	−0.375139
$784$	0	0
$785$	26.2457	0.936749
$786$	0	0
$787$	17.8963	0.637933	0.318967	−	0.947766i	$-0.396664\pi$
$787$	17.8963	0.637933	0.318967	+	0.947766i	$0.396664\pi$
$788$	0	0
$789$	−28.0228	−0.997640
$790$	0	0
$791$	12.4382	0.442253
$792$	0	0
$793$	13.6910	0.486183
$794$	0	0
$795$	14.6148	0.518333
$796$	0	0
$797$	53.7115	1.90256	0.951279	−	0.308331i	$-0.0997705\pi$
$797$	53.7115	1.90256	0.951279	+	0.308331i	$0.0997705\pi$
$798$	0	0
$799$	15.1283	0.535202
$800$	0	0
$801$	−13.7512	−0.485877
$802$	0	0
$803$	−5.85012	−0.206446
$804$	0	0
$805$	0.680065	0.0239691
$806$	0	0
$807$	33.6267	1.18372
$808$	0	0
$809$	−20.7687	−0.730188	−0.365094	−	0.930971i	$-0.618963\pi$
$809$	−20.7687	−0.730188	−0.365094	+	0.930971i	$0.618963\pi$
$810$	0	0
$811$	−37.1225	−1.30355	−0.651773	−	0.758414i	$-0.725975\pi$
$811$	−37.1225	−1.30355	−0.651773	+	0.758414i	$0.725975\pi$
$812$	0	0
$813$	23.2882	0.816752
$814$	0	0
$815$	15.2702	0.534893
$816$	0	0
$817$	18.4280	0.644715
$818$	0	0
$819$	−3.52720	−0.123250
$820$	0	0
$821$	15.6548	0.546356	0.273178	−	0.961963i	$-0.411925\pi$
$821$	15.6548	0.546356	0.273178	+	0.961963i	$0.411925\pi$
$822$	0	0
$823$	17.1553	0.597998	0.298999	−	0.954253i	$-0.403347\pi$
$823$	17.1553	0.597998	0.298999	+	0.954253i	$0.403347\pi$
$824$	0	0
$825$	3.78018	0.131609
$826$	0	0
$827$	−18.0680	−0.628287	−0.314143	−	0.949376i	$-0.601717\pi$
$827$	−18.0680	−0.628287	−0.314143	+	0.949376i	$0.601717\pi$
$828$	0	0
$829$	39.6919	1.37856	0.689279	−	0.724496i	$-0.257928\pi$
$829$	39.6919	1.37856	0.689279	+	0.724496i	$0.257928\pi$
$830$	0	0
$831$	−22.0814	−0.765997
$832$	0	0
$833$	−1.96704	−0.0681540
$834$	0	0
$835$	−0.702960	−0.0243269
$836$	0	0
$837$	9.73811	0.336598
$838$	0	0
$839$	−35.7673	−1.23482	−0.617412	−	0.786640i	$-0.711819\pi$
$839$	−35.7673	−1.23482	−0.617412	+	0.786640i	$0.711819\pi$
$840$	0	0
$841$	31.7396	1.09447
$842$	0	0
$843$	−37.4884	−1.29117
$844$	0	0
$845$	−1.87627	−0.0645456
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−55.0771	−1.89024
$850$	0	0
$851$	0.421719	0.0144563
$852$	0	0
$853$	−25.4446	−0.871206	−0.435603	−	0.900139i	$-0.643465\pi$
$853$	−25.4446	−0.871206	−0.435603	+	0.900139i	$0.643465\pi$
$854$	0	0
$855$	54.8034	1.87424
$856$	0	0
$857$	−17.7352	−0.605823	−0.302911	−	0.953019i	$-0.597959\pi$
$857$	−17.7352	−0.605823	−0.302911	+	0.953019i	$0.597959\pi$
$858$	0	0
$859$	33.5997	1.14641	0.573204	−	0.819413i	$-0.305700\pi$
$859$	33.5997	1.14641	0.573204	+	0.819413i	$0.305700\pi$
$860$	0	0
$861$	15.6060	0.531852
$862$	0	0
$863$	8.32958	0.283542	0.141771	−	0.989899i	$-0.454720\pi$
$863$	8.32958	0.283542	0.141771	+	0.989899i	$0.454720\pi$
$864$	0	0
$865$	−24.6745	−0.838958
$866$	0	0
$867$	−33.5469	−1.13931
$868$	0	0
$869$	11.0204	0.373841
$870$	0	0
$871$	10.2644	0.347797
$872$	0	0
$873$	−7.24496	−0.245205
$874$	0	0
$875$	12.1575	0.410999
$876$	0	0
$877$	46.5616	1.57227	0.786137	−	0.618052i	$-0.212078\pi$
$877$	46.5616	1.57227	0.786137	+	0.618052i	$0.212078\pi$
$878$	0	0
$879$	−15.1298	−0.510315
$880$	0	0
$881$	−38.4335	−1.29486	−0.647428	−	0.762127i	$-0.724155\pi$
$881$	−38.4335	−1.29486	−0.647428	+	0.762127i	$0.724155\pi$
$882$	0	0
$883$	20.6659	0.695462	0.347731	−	0.937594i	$-0.386952\pi$
$883$	20.6659	0.695462	0.347731	+	0.937594i	$0.386952\pi$
$884$	0	0
$885$	−18.3801	−0.617841
$886$	0	0
$887$	−13.9405	−0.468076	−0.234038	−	0.972227i	$-0.575194\pi$
$887$	−13.9405	−0.468076	−0.234038	+	0.972227i	$0.575194\pi$
$888$	0	0
$889$	17.4283	0.584525
$890$	0	0
$891$	7.14047	0.239215
$892$	0	0
$893$	63.6883	2.13125
$894$	0	0
$895$	2.85450	0.0954155
$896$	0	0
$897$	0.926017	0.0309188
$898$	0	0
$899$	−56.3474	−1.87929
$900$	0	0
$901$	5.99719	0.199795
$902$	0	0
$903$	−5.68538	−0.189198
$904$	0	0
$905$	−21.5777	−0.717268
$906$	0	0
$907$	27.4028	0.909895	0.454947	−	0.890518i	$-0.349658\pi$
$907$	27.4028	0.909895	0.454947	+	0.890518i	$0.349658\pi$
$908$	0	0
$909$	9.06743	0.300748
$910$	0	0
$911$	45.0554	1.49275	0.746376	−	0.665524i	$-0.231792\pi$
$911$	45.0554	1.49275	0.746376	+	0.665524i	$0.231792\pi$
$912$	0	0
$913$	−3.50750	−0.116081
$914$	0	0
$915$	65.6288	2.16962
$916$	0	0
$917$	−19.7907	−0.653547
$918$	0	0
$919$	46.7429	1.54191	0.770953	−	0.636892i	$-0.219780\pi$
$919$	46.7429	1.54191	0.770953	+	0.636892i	$0.219780\pi$
$920$	0	0
$921$	−7.01459	−0.231139
$922$	0	0
$923$	−13.6978	−0.450870
$924$	0	0
$925$	1.72154	0.0566037
$926$	0	0
$927$	9.38202	0.308146
$928$	0	0
$929$	−9.54432	−0.313139	−0.156569	−	0.987667i	$-0.550044\pi$
$929$	−9.54432	−0.313139	−0.156569	+	0.987667i	$0.550044\pi$
$930$	0	0
$931$	−8.28099	−0.271398
$932$	0	0
$933$	−31.4875	−1.03085
$934$	0	0
$935$	−3.69070	−0.120699
$936$	0	0
$937$	−17.6942	−0.578043	−0.289021	−	0.957323i	$-0.593330\pi$
$937$	−17.6942	−0.578043	−0.289021	+	0.957323i	$0.593330\pi$
$938$	0	0
$939$	−1.26893	−0.0414101
$940$	0	0
$941$	−52.1182	−1.69901	−0.849503	−	0.527584i	$-0.823098\pi$
$941$	−52.1182	−1.69901	−0.849503	+	0.527584i	$0.823098\pi$
$942$	0	0
$943$	−2.21404	−0.0720989
$944$	0	0
$945$	−2.52716	−0.0822084
$946$	0	0
$947$	41.5483	1.35014	0.675069	−	0.737754i	$-0.264114\pi$
$947$	41.5483	1.35014	0.675069	+	0.737754i	$0.264114\pi$
$948$	0	0
$949$	−5.85012	−0.189903
$950$	0	0
$951$	16.2229	0.526064
$952$	0	0
$953$	−14.4889	−0.469340	−0.234670	−	0.972075i	$-0.575401\pi$
$953$	−14.4889	−0.469340	−0.234670	+	0.972075i	$0.575401\pi$
$954$	0	0
$955$	8.78363	0.284232
$956$	0	0
$957$	19.9113	0.643641
$958$	0	0
$959$	8.55608	0.276290
$960$	0	0
$961$	21.2727	0.686217
$962$	0	0
$963$	−10.7346	−0.345919
$964$	0	0
$965$	23.9944	0.772407
$966$	0	0
$967$	40.4595	1.30109	0.650544	−	0.759468i	$-0.274541\pi$
$967$	40.4595	1.30109	0.650544	+	0.759468i	$0.274541\pi$
$968$	0	0
$969$	41.6159	1.33690
$970$	0	0
$971$	45.7359	1.46774	0.733868	−	0.679293i	$-0.237713\pi$
$971$	45.7359	1.46774	0.733868	+	0.679293i	$0.237713\pi$
$972$	0	0
$973$	−13.6123	−0.436389
$974$	0	0
$975$	3.78018	0.121063
$976$	0	0
$977$	−52.2940	−1.67303	−0.836517	−	0.547942i	$-0.815412\pi$
$977$	−52.2940	−1.67303	−0.836517	+	0.547942i	$0.815412\pi$
$978$	0	0
$979$	3.89863	0.124601
$980$	0	0
$981$	−15.6798	−0.500618
$982$	0	0
$983$	−28.6798	−0.914744	−0.457372	−	0.889275i	$-0.651209\pi$
$983$	−28.6798	−0.914744	−0.457372	+	0.889275i	$0.651209\pi$
$984$	0	0
$985$	−6.45455	−0.205659
$986$	0	0
$987$	−19.6490	−0.625435
$988$	0	0
$989$	0.806588	0.0256480
$990$	0	0
$991$	32.3257	1.02686	0.513429	−	0.858132i	$-0.328375\pi$
$991$	32.3257	1.02686	0.513429	+	0.858132i	$0.328375\pi$
$992$	0	0
$993$	−49.9229	−1.58425
$994$	0	0
$995$	30.5079	0.967164
$996$	0	0
$997$	27.0031	0.855198	0.427599	−	0.903969i	$-0.359360\pi$
$997$	27.0031	0.855198	0.427599	+	0.903969i	$0.359360\pi$
$998$	0	0
$999$	−1.56713	−0.0495817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.g.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.g.1.6	✓	6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55484 q^{3} -1.87627 q^{5} +1.00000 q^{7} +3.52720 q^{9} +O(q^{10})\)	\(q+2.55484 q^{3} -1.87627 q^{5} +1.00000 q^{7} +3.52720 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.79356 q^{15} -1.96704 q^{17} -8.28099 q^{19} +2.55484 q^{21} -0.362456 q^{23} -1.47962 q^{25} +1.34691 q^{27} -7.79356 q^{29} +7.22999 q^{31} -2.55484 q^{33} -1.87627 q^{35} -1.16350 q^{37} -2.55484 q^{39} +6.10842 q^{41} -2.22534 q^{43} -6.61797 q^{45} -7.69090 q^{47} +1.00000 q^{49} -5.02547 q^{51} -3.04884 q^{53} +1.87627 q^{55} -21.1566 q^{57} +3.83433 q^{59} -13.6910 q^{61} +3.52720 q^{63} +1.87627 q^{65} -10.2644 q^{67} -0.926017 q^{69} +13.6978 q^{71} +5.85012 q^{73} -3.78018 q^{75} -1.00000 q^{77} -11.0204 q^{79} -7.14047 q^{81} +3.50750 q^{83} +3.69070 q^{85} -19.9113 q^{87} -3.89863 q^{89} -1.00000 q^{91} +18.4714 q^{93} +15.5374 q^{95} -2.05403 q^{97} -3.52720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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