LMFDB - Embedded newform 8008.2.a.t.1.4

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 $ x^10 - x^9 - 17x^8 + 13x^7 + 96x^6 - 49x^5 - 207x^4 + 58x^3 + 169x^2 - 12x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.02605$ 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)

$f(q)$	$=$	$q-1.02605 q^{3} +3.56808 q^{5} +1.00000 q^{7} -1.94722 q^{9} +O(q^{10})$	$q-1.02605 q^{3} +3.56808 q^{5} +1.00000 q^{7} -1.94722 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.66103 q^{15} -2.59808 q^{17} +1.78387 q^{19} -1.02605 q^{21} -2.86071 q^{23} +7.73119 q^{25} +5.07609 q^{27} +4.51367 q^{29} +6.44410 q^{31} +1.02605 q^{33} +3.56808 q^{35} -2.49236 q^{37} -1.02605 q^{39} -8.70864 q^{41} -1.37924 q^{43} -6.94784 q^{45} +7.82042 q^{47} +1.00000 q^{49} +2.66576 q^{51} -9.61423 q^{53} -3.56808 q^{55} -1.83034 q^{57} -2.27471 q^{59} -3.88750 q^{61} -1.94722 q^{63} +3.56808 q^{65} +15.1386 q^{67} +2.93523 q^{69} +15.3710 q^{71} -5.51474 q^{73} -7.93258 q^{75} -1.00000 q^{77} -2.13198 q^{79} +0.633343 q^{81} +11.8300 q^{83} -9.27017 q^{85} -4.63125 q^{87} +6.46755 q^{89} +1.00000 q^{91} -6.61196 q^{93} +6.36500 q^{95} -7.82650 q^{97} +1.94722 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.02605	−0.592390	−0.296195	−	0.955128i	$-0.595718\pi$
$3$	−1.02605	−0.592390	−0.296195	+	0.955128i	$0.595718\pi$
$4$	0	0
$5$	3.56808	1.59569	0.797847	−	0.602860i	$-0.205972\pi$
$5$	3.56808	1.59569	0.797847	+	0.602860i	$0.205972\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.94722	−0.649074
$10$	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
$15$	−3.66103	−0.945273
$16$	0	0
$17$	−2.59808	−0.630128	−0.315064	−	0.949070i	$-0.602026\pi$
$17$	−2.59808	−0.630128	−0.315064	+	0.949070i	$0.602026\pi$
$18$	0	0
$19$	1.78387	0.409249	0.204624	−	0.978841i	$-0.434403\pi$
$19$	1.78387	0.409249	0.204624	+	0.978841i	$0.434403\pi$
$20$	0	0
$21$	−1.02605	−0.223902
$22$	0	0
$23$	−2.86071	−0.596499	−0.298249	−	0.954488i	$-0.596403\pi$
$23$	−2.86071	−0.596499	−0.298249	+	0.954488i	$0.596403\pi$
$24$	0	0
$25$	7.73119	1.54624
$26$	0	0
$27$	5.07609	0.976895
$28$	0	0
$29$	4.51367	0.838167	0.419084	−	0.907948i	$-0.362351\pi$
$29$	4.51367	0.838167	0.419084	+	0.907948i	$0.362351\pi$
$30$	0	0
$31$	6.44410	1.15739	0.578697	−	0.815543i	$-0.303561\pi$
$31$	6.44410	1.15739	0.578697	+	0.815543i	$0.303561\pi$
$32$	0	0
$33$	1.02605	0.178612
$34$	0	0
$35$	3.56808	0.603115
$36$	0	0
$37$	−2.49236	−0.409741	−0.204871	−	0.978789i	$-0.565677\pi$
$37$	−2.49236	−0.409741	−0.204871	+	0.978789i	$0.565677\pi$
$38$	0	0
$39$	−1.02605	−0.164299
$40$	0	0
$41$	−8.70864	−1.36006	−0.680030	−	0.733184i	$-0.738033\pi$
$41$	−8.70864	−1.36006	−0.680030	+	0.733184i	$0.738033\pi$
$42$	0	0
$43$	−1.37924	−0.210331	−0.105166	−	0.994455i	$-0.533537\pi$
$43$	−1.37924	−0.210331	−0.105166	+	0.994455i	$0.533537\pi$
$44$	0	0
$45$	−6.94784	−1.03572
$46$	0	0
$47$	7.82042	1.14073	0.570363	−	0.821393i	$-0.306803\pi$
$47$	7.82042	1.14073	0.570363	+	0.821393i	$0.306803\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.66576	0.373282
$52$	0	0
$53$	−9.61423	−1.32062	−0.660308	−	0.750995i	$-0.729574\pi$
$53$	−9.61423	−1.32062	−0.660308	+	0.750995i	$0.729574\pi$
$54$	0	0
$55$	−3.56808	−0.481120
$56$	0	0
$57$	−1.83034	−0.242435
$58$	0	0
$59$	−2.27471	−0.296142	−0.148071	−	0.988977i	$-0.547306\pi$
$59$	−2.27471	−0.296142	−0.148071	+	0.988977i	$0.547306\pi$
$60$	0	0
$61$	−3.88750	−0.497743	−0.248872	−	0.968536i	$-0.580060\pi$
$61$	−3.88750	−0.497743	−0.248872	+	0.968536i	$0.580060\pi$
$62$	0	0
$63$	−1.94722	−0.245327
$64$	0	0
$65$	3.56808	0.442566
$66$	0	0
$67$	15.1386	1.84947	0.924736	−	0.380610i	$-0.124286\pi$
$67$	15.1386	1.84947	0.924736	+	0.380610i	$0.124286\pi$
$68$	0	0
$69$	2.93523	0.353360
$70$	0	0
$71$	15.3710	1.82421	0.912103	−	0.409962i	$-0.134458\pi$
$71$	15.3710	1.82421	0.912103	+	0.409962i	$0.134458\pi$
$72$	0	0
$73$	−5.51474	−0.645452	−0.322726	−	0.946493i	$-0.604599\pi$
$73$	−5.51474	−0.645452	−0.322726	+	0.946493i	$0.604599\pi$
$74$	0	0
$75$	−7.93258	−0.915975
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−2.13198	−0.239866	−0.119933	−	0.992782i	$-0.538268\pi$
$79$	−2.13198	−0.239866	−0.119933	+	0.992782i	$0.538268\pi$
$80$	0	0
$81$	0.633343	0.0703715
$82$	0	0
$83$	11.8300	1.29851	0.649254	−	0.760572i	$-0.275081\pi$
$83$	11.8300	1.29851	0.649254	+	0.760572i	$0.275081\pi$
$84$	0	0
$85$	−9.27017	−1.00549
$86$	0	0
$87$	−4.63125	−0.496522
$88$	0	0
$89$	6.46755	0.685559	0.342779	−	0.939416i	$-0.388632\pi$
$89$	6.46755	0.685559	0.342779	+	0.939416i	$0.388632\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−6.61196	−0.685629
$94$	0	0
$95$	6.36500	0.653035
$96$	0	0
$97$	−7.82650	−0.794660	−0.397330	−	0.917676i	$-0.630063\pi$
$97$	−7.82650	−0.794660	−0.397330	+	0.917676i	$0.630063\pi$
$98$	0	0
$99$	1.94722	0.195703
$100$	0	0
$101$	10.3833	1.03318	0.516589	−	0.856234i	$-0.327202\pi$
$101$	10.3833	1.03318	0.516589	+	0.856234i	$0.327202\pi$
$102$	0	0
$103$	7.19357	0.708803	0.354402	−	0.935093i	$-0.384685\pi$
$103$	7.19357	0.708803	0.354402	+	0.935093i	$0.384685\pi$
$104$	0	0
$105$	−3.66103	−0.357279
$106$	0	0
$107$	4.45862	0.431031	0.215515	−	0.976500i	$-0.430857\pi$
$107$	4.45862	0.431031	0.215515	+	0.976500i	$0.430857\pi$
$108$	0	0
$109$	1.48706	0.142434	0.0712172	−	0.997461i	$-0.477312\pi$
$109$	1.48706	0.142434	0.0712172	+	0.997461i	$0.477312\pi$
$110$	0	0
$111$	2.55728	0.242726
$112$	0	0
$113$	16.4723	1.54958	0.774790	−	0.632219i	$-0.217856\pi$
$113$	16.4723	1.54958	0.774790	+	0.632219i	$0.217856\pi$
$114$	0	0
$115$	−10.2072	−0.951830
$116$	0	0
$117$	−1.94722	−0.180021
$118$	0	0
$119$	−2.59808	−0.238166
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.93549	0.805686
$124$	0	0
$125$	9.74509	0.871627
$126$	0	0
$127$	1.33032	0.118046	0.0590232	−	0.998257i	$-0.481201\pi$
$127$	1.33032	0.118046	0.0590232	+	0.998257i	$0.481201\pi$
$128$	0	0
$129$	1.41516	0.124598
$130$	0	0
$131$	−10.2043	−0.891551	−0.445776	−	0.895145i	$-0.647072\pi$
$131$	−10.2043	−0.891551	−0.445776	+	0.895145i	$0.647072\pi$
$132$	0	0
$133$	1.78387	0.154681
$134$	0	0
$135$	18.1119	1.55882
$136$	0	0
$137$	6.90224	0.589698	0.294849	−	0.955544i	$-0.404731\pi$
$137$	6.90224	0.589698	0.294849	+	0.955544i	$0.404731\pi$
$138$	0	0
$139$	−2.33973	−0.198453	−0.0992265	−	0.995065i	$-0.531637\pi$
$139$	−2.33973	−0.198453	−0.0992265	+	0.995065i	$0.531637\pi$
$140$	0	0
$141$	−8.02414	−0.675755
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	16.1051	1.33746
$146$	0	0
$147$	−1.02605	−0.0846271
$148$	0	0
$149$	−9.35471	−0.766367	−0.383184	−	0.923672i	$-0.625172\pi$
$149$	−9.35471	−0.766367	−0.383184	+	0.923672i	$0.625172\pi$
$150$	0	0
$151$	18.5940	1.51316	0.756578	−	0.653903i	$-0.226870\pi$
$151$	18.5940	1.51316	0.756578	+	0.653903i	$0.226870\pi$
$152$	0	0
$153$	5.05905	0.409000
$154$	0	0
$155$	22.9931	1.84685
$156$	0	0
$157$	13.3555	1.06589	0.532944	−	0.846151i	$-0.321086\pi$
$157$	13.3555	1.06589	0.532944	+	0.846151i	$0.321086\pi$
$158$	0	0
$159$	9.86467	0.782320
$160$	0	0
$161$	−2.86071	−0.225455
$162$	0	0
$163$	−7.15780	−0.560642	−0.280321	−	0.959906i	$-0.590441\pi$
$163$	−7.15780	−0.560642	−0.280321	+	0.959906i	$0.590441\pi$
$164$	0	0
$165$	3.66103	0.285010
$166$	0	0
$167$	9.65270	0.746948	0.373474	−	0.927641i	$-0.378167\pi$
$167$	9.65270	0.746948	0.373474	+	0.927641i	$0.378167\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.47360	−0.265633
$172$	0	0
$173$	18.0144	1.36961	0.684806	−	0.728725i	$-0.259887\pi$
$173$	18.0144	1.36961	0.684806	+	0.728725i	$0.259887\pi$
$174$	0	0
$175$	7.73119	0.584423
$176$	0	0
$177$	2.33396	0.175431
$178$	0	0
$179$	−7.36393	−0.550406	−0.275203	−	0.961386i	$-0.588745\pi$
$179$	−7.36393	−0.550406	−0.275203	+	0.961386i	$0.588745\pi$
$180$	0	0
$181$	−3.44780	−0.256273	−0.128136	−	0.991757i	$-0.540900\pi$
$181$	−3.44780	−0.256273	−0.128136	+	0.991757i	$0.540900\pi$
$182$	0	0
$183$	3.98877	0.294858
$184$	0	0
$185$	−8.89293	−0.653821
$186$	0	0
$187$	2.59808	0.189991
$188$	0	0
$189$	5.07609	0.369232
$190$	0	0
$191$	14.6867	1.06269	0.531345	−	0.847156i	$-0.321687\pi$
$191$	14.6867	1.06269	0.531345	+	0.847156i	$0.321687\pi$
$192$	0	0
$193$	−9.93847	−0.715387	−0.357693	−	0.933839i	$-0.616437\pi$
$193$	−9.93847	−0.715387	−0.357693	+	0.933839i	$0.616437\pi$
$194$	0	0
$195$	−3.66103	−0.262171
$196$	0	0
$197$	9.81176	0.699059	0.349529	−	0.936925i	$-0.386341\pi$
$197$	9.81176	0.699059	0.349529	+	0.936925i	$0.386341\pi$
$198$	0	0
$199$	−16.7202	−1.18527	−0.592633	−	0.805473i	$-0.701911\pi$
$199$	−16.7202	−1.18527	−0.592633	+	0.805473i	$0.701911\pi$
$200$	0	0
$201$	−15.5329	−1.09561
$202$	0	0
$203$	4.51367	0.316797
$204$	0	0
$205$	−31.0731	−2.17024
$206$	0	0
$207$	5.57044	0.387172
$208$	0	0
$209$	−1.78387	−0.123393
$210$	0	0
$211$	−9.20605	−0.633771	−0.316885	−	0.948464i	$-0.602637\pi$
$211$	−9.20605	−0.633771	−0.316885	+	0.948464i	$0.602637\pi$
$212$	0	0
$213$	−15.7714	−1.08064
$214$	0	0
$215$	−4.92122	−0.335624
$216$	0	0
$217$	6.44410	0.437454
$218$	0	0
$219$	5.65840	0.382359
$220$	0	0
$221$	−2.59808	−0.174766
$222$	0	0
$223$	−22.0235	−1.47480	−0.737401	−	0.675455i	$-0.763947\pi$
$223$	−22.0235	−1.47480	−0.737401	+	0.675455i	$0.763947\pi$
$224$	0	0
$225$	−15.0543	−1.00362
$226$	0	0
$227$	−16.9990	−1.12826	−0.564132	−	0.825685i	$-0.690789\pi$
$227$	−16.9990	−1.12826	−0.564132	+	0.825685i	$0.690789\pi$
$228$	0	0
$229$	15.3567	1.01480	0.507400	−	0.861711i	$-0.330607\pi$
$229$	15.3567	1.01480	0.507400	+	0.861711i	$0.330607\pi$
$230$	0	0
$231$	1.02605	0.0675091
$232$	0	0
$233$	16.4702	1.07900	0.539501	−	0.841985i	$-0.318613\pi$
$233$	16.4702	1.07900	0.539501	+	0.841985i	$0.318613\pi$
$234$	0	0
$235$	27.9039	1.82025
$236$	0	0
$237$	2.18752	0.142094
$238$	0	0
$239$	21.9250	1.41821	0.709104	−	0.705104i	$-0.249100\pi$
$239$	21.9250	1.41821	0.709104	+	0.705104i	$0.249100\pi$
$240$	0	0
$241$	1.79333	0.115518	0.0577592	−	0.998331i	$-0.481604\pi$
$241$	1.79333	0.115518	0.0577592	+	0.998331i	$0.481604\pi$
$242$	0	0
$243$	−15.8781	−1.01858
$244$	0	0
$245$	3.56808	0.227956
$246$	0	0
$247$	1.78387	0.113505
$248$	0	0
$249$	−12.1381	−0.769223
$250$	0	0
$251$	5.72751	0.361518	0.180759	−	0.983527i	$-0.442145\pi$
$251$	5.72751	0.361518	0.180759	+	0.983527i	$0.442145\pi$
$252$	0	0
$253$	2.86071	0.179851
$254$	0	0
$255$	9.51165	0.595643
$256$	0	0
$257$	−22.3552	−1.39448	−0.697239	−	0.716839i	$-0.745588\pi$
$257$	−22.3552	−1.39448	−0.697239	+	0.716839i	$0.745588\pi$
$258$	0	0
$259$	−2.49236	−0.154868
$260$	0	0
$261$	−8.78912	−0.544033
$262$	0	0
$263$	−11.3065	−0.697186	−0.348593	−	0.937274i	$-0.613340\pi$
$263$	−11.3065	−0.697186	−0.348593	+	0.937274i	$0.613340\pi$
$264$	0	0
$265$	−34.3043	−2.10730
$266$	0	0
$267$	−6.63603	−0.406118
$268$	0	0
$269$	−0.753116	−0.0459183	−0.0229592	−	0.999736i	$-0.507309\pi$
$269$	−0.753116	−0.0459183	−0.0229592	+	0.999736i	$0.507309\pi$
$270$	0	0
$271$	13.5368	0.822302	0.411151	−	0.911567i	$-0.365127\pi$
$271$	13.5368	0.822302	0.411151	+	0.911567i	$0.365127\pi$
$272$	0	0
$273$	−1.02605	−0.0620993
$274$	0	0
$275$	−7.73119	−0.466208
$276$	0	0
$277$	−0.850239	−0.0510859	−0.0255430	−	0.999674i	$-0.508131\pi$
$277$	−0.850239	−0.0510859	−0.0255430	+	0.999674i	$0.508131\pi$
$278$	0	0
$279$	−12.5481	−0.751235
$280$	0	0
$281$	0.539594	0.0321895	0.0160948	−	0.999870i	$-0.494877\pi$
$281$	0.539594	0.0321895	0.0160948	+	0.999870i	$0.494877\pi$
$282$	0	0
$283$	−25.9740	−1.54399	−0.771997	−	0.635626i	$-0.780742\pi$
$283$	−25.9740	−1.54399	−0.771997	+	0.635626i	$0.780742\pi$
$284$	0	0
$285$	−6.53080	−0.386851
$286$	0	0
$287$	−8.70864	−0.514055
$288$	0	0
$289$	−10.2500	−0.602939
$290$	0	0
$291$	8.03037	0.470749
$292$	0	0
$293$	−7.97125	−0.465685	−0.232843	−	0.972514i	$-0.574803\pi$
$293$	−7.97125	−0.465685	−0.232843	+	0.972514i	$0.574803\pi$
$294$	0	0
$295$	−8.11634	−0.472552
$296$	0	0
$297$	−5.07609	−0.294545
$298$	0	0
$299$	−2.86071	−0.165439
$300$	0	0
$301$	−1.37924	−0.0794978
$302$	0	0
$303$	−10.6538	−0.612044
$304$	0	0
$305$	−13.8709	−0.794245
$306$	0	0
$307$	27.1120	1.54737	0.773683	−	0.633573i	$-0.218412\pi$
$307$	27.1120	1.54737	0.773683	+	0.633573i	$0.218412\pi$
$308$	0	0
$309$	−7.38096	−0.419888
$310$	0	0
$311$	2.68060	0.152003	0.0760013	−	0.997108i	$-0.475785\pi$
$311$	2.68060	0.152003	0.0760013	+	0.997108i	$0.475785\pi$
$312$	0	0
$313$	5.84967	0.330643	0.165321	−	0.986240i	$-0.447134\pi$
$313$	5.84967	0.330643	0.165321	+	0.986240i	$0.447134\pi$
$314$	0	0
$315$	−6.94784	−0.391467
$316$	0	0
$317$	−2.89568	−0.162637	−0.0813187	−	0.996688i	$-0.525913\pi$
$317$	−2.89568	−0.162637	−0.0813187	+	0.996688i	$0.525913\pi$
$318$	0	0
$319$	−4.51367	−0.252717
$320$	0	0
$321$	−4.57476	−0.255338
$322$	0	0
$323$	−4.63465	−0.257879
$324$	0	0
$325$	7.73119	0.428849
$326$	0	0
$327$	−1.52580	−0.0843767
$328$	0	0
$329$	7.82042	0.431154
$330$	0	0
$331$	1.88328	0.103514	0.0517572	−	0.998660i	$-0.483518\pi$
$331$	1.88328	0.103514	0.0517572	+	0.998660i	$0.483518\pi$
$332$	0	0
$333$	4.85318	0.265952
$334$	0	0
$335$	54.0156	2.95119
$336$	0	0
$337$	26.7331	1.45624	0.728122	−	0.685448i	$-0.240393\pi$
$337$	26.7331	1.45624	0.728122	+	0.685448i	$0.240393\pi$
$338$	0	0
$339$	−16.9014	−0.917955
$340$	0	0
$341$	−6.44410	−0.348967
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	10.4731	0.563854
$346$	0	0
$347$	6.46441	0.347028	0.173514	−	0.984831i	$-0.444488\pi$
$347$	6.46441	0.347028	0.173514	+	0.984831i	$0.444488\pi$
$348$	0	0
$349$	−21.0993	−1.12942	−0.564709	−	0.825290i	$-0.691012\pi$
$349$	−21.0993	−1.12942	−0.564709	+	0.825290i	$0.691012\pi$
$350$	0	0
$351$	5.07609	0.270942
$352$	0	0
$353$	16.7629	0.892198	0.446099	−	0.894984i	$-0.352813\pi$
$353$	16.7629	0.892198	0.446099	+	0.894984i	$0.352813\pi$
$354$	0	0
$355$	54.8450	2.91087
$356$	0	0
$357$	2.66576	0.141087
$358$	0	0
$359$	11.5110	0.607530	0.303765	−	0.952747i	$-0.401756\pi$
$359$	11.5110	0.607530	0.303765	+	0.952747i	$0.401756\pi$
$360$	0	0
$361$	−15.8178	−0.832516
$362$	0	0
$363$	−1.02605	−0.0538536
$364$	0	0
$365$	−19.6770	−1.02994
$366$	0	0
$367$	21.7191	1.13373	0.566863	−	0.823812i	$-0.308157\pi$
$367$	21.7191	1.13373	0.566863	+	0.823812i	$0.308157\pi$
$368$	0	0
$369$	16.9577	0.882780
$370$	0	0
$371$	−9.61423	−0.499146
$372$	0	0
$373$	−8.21938	−0.425583	−0.212792	−	0.977098i	$-0.568256\pi$
$373$	−8.21938	−0.425583	−0.212792	+	0.977098i	$0.568256\pi$
$374$	0	0
$375$	−9.99895	−0.516343
$376$	0	0
$377$	4.51367	0.232466
$378$	0	0
$379$	0.438278	0.0225128	0.0112564	−	0.999937i	$-0.496417\pi$
$379$	0.438278	0.0225128	0.0112564	+	0.999937i	$0.496417\pi$
$380$	0	0
$381$	−1.36497	−0.0699295
$382$	0	0
$383$	27.1622	1.38792	0.693961	−	0.720012i	$-0.255864\pi$
$383$	27.1622	1.38792	0.693961	+	0.720012i	$0.255864\pi$
$384$	0	0
$385$	−3.56808	−0.181846
$386$	0	0
$387$	2.68568	0.136521
$388$	0	0
$389$	−11.5363	−0.584914	−0.292457	−	0.956279i	$-0.594473\pi$
$389$	−11.5363	−0.584914	−0.292457	+	0.956279i	$0.594473\pi$
$390$	0	0
$391$	7.43236	0.375871
$392$	0	0
$393$	10.4701	0.528146
$394$	0	0
$395$	−7.60707	−0.382753
$396$	0	0
$397$	9.06512	0.454965	0.227483	−	0.973782i	$-0.426951\pi$
$397$	9.06512	0.454965	0.227483	+	0.973782i	$0.426951\pi$
$398$	0	0
$399$	−1.83034	−0.0916317
$400$	0	0
$401$	−26.1859	−1.30766	−0.653832	−	0.756640i	$-0.726840\pi$
$401$	−26.1859	−1.30766	−0.653832	+	0.756640i	$0.726840\pi$
$402$	0	0
$403$	6.44410	0.321003
$404$	0	0
$405$	2.25982	0.112291
$406$	0	0
$407$	2.49236	0.123542
$408$	0	0
$409$	−22.0588	−1.09074	−0.545368	−	0.838197i	$-0.683610\pi$
$409$	−22.0588	−1.09074	−0.545368	+	0.838197i	$0.683610\pi$
$410$	0	0
$411$	−7.08204	−0.349331
$412$	0	0
$413$	−2.27471	−0.111931
$414$	0	0
$415$	42.2103	2.07202
$416$	0	0
$417$	2.40067	0.117562
$418$	0	0
$419$	−2.66757	−0.130319	−0.0651596	−	0.997875i	$-0.520756\pi$
$419$	−2.66757	−0.130319	−0.0651596	+	0.997875i	$0.520756\pi$
$420$	0	0
$421$	37.0853	1.80743	0.903713	−	0.428139i	$-0.140831\pi$
$421$	37.0853	1.80743	0.903713	+	0.428139i	$0.140831\pi$
$422$	0	0
$423$	−15.2281	−0.740416
$424$	0	0
$425$	−20.0863	−0.974328
$426$	0	0
$427$	−3.88750	−0.188129
$428$	0	0
$429$	1.02605	0.0495381
$430$	0	0
$431$	19.7558	0.951605	0.475803	−	0.879552i	$-0.342158\pi$
$431$	19.7558	0.951605	0.475803	+	0.879552i	$0.342158\pi$
$432$	0	0
$433$	15.7556	0.757167	0.378583	−	0.925567i	$-0.376411\pi$
$433$	15.7556	0.757167	0.378583	+	0.925567i	$0.376411\pi$
$434$	0	0
$435$	−16.5247	−0.792297
$436$	0	0
$437$	−5.10314	−0.244116
$438$	0	0
$439$	8.88749	0.424177	0.212088	−	0.977251i	$-0.431974\pi$
$439$	8.88749	0.424177	0.212088	+	0.977251i	$0.431974\pi$
$440$	0	0
$441$	−1.94722	−0.0927249
$442$	0	0
$443$	3.52052	0.167265	0.0836324	−	0.996497i	$-0.473348\pi$
$443$	3.52052	0.167265	0.0836324	+	0.996497i	$0.473348\pi$
$444$	0	0
$445$	23.0767	1.09394
$446$	0	0
$447$	9.59839	0.453988
$448$	0	0
$449$	−14.9228	−0.704249	−0.352125	−	0.935953i	$-0.614541\pi$
$449$	−14.9228	−0.704249	−0.352125	+	0.935953i	$0.614541\pi$
$450$	0	0
$451$	8.70864	0.410074
$452$	0	0
$453$	−19.0783	−0.896379
$454$	0	0
$455$	3.56808	0.167274
$456$	0	0
$457$	7.21648	0.337573	0.168786	−	0.985653i	$-0.446015\pi$
$457$	7.21648	0.337573	0.168786	+	0.985653i	$0.446015\pi$
$458$	0	0
$459$	−13.1881	−0.615569
$460$	0	0
$461$	40.0169	1.86377	0.931886	−	0.362752i	$-0.118163\pi$
$461$	40.0169	1.86377	0.931886	+	0.362752i	$0.118163\pi$
$462$	0	0
$463$	16.8308	0.782196	0.391098	−	0.920349i	$-0.372095\pi$
$463$	16.8308	0.782196	0.391098	+	0.920349i	$0.372095\pi$
$464$	0	0
$465$	−23.5920	−1.09405
$466$	0	0
$467$	−17.7831	−0.822904	−0.411452	−	0.911431i	$-0.634978\pi$
$467$	−17.7831	−0.822904	−0.411452	+	0.911431i	$0.634978\pi$
$468$	0	0
$469$	15.1386	0.699034
$470$	0	0
$471$	−13.7034	−0.631421
$472$	0	0
$473$	1.37924	0.0634173
$474$	0	0
$475$	13.7915	0.632795
$476$	0	0
$477$	18.7210	0.857178
$478$	0	0
$479$	−23.8549	−1.08996	−0.544979	−	0.838450i	$-0.683462\pi$
$479$	−23.8549	−1.08996	−0.544979	+	0.838450i	$0.683462\pi$
$480$	0	0
$481$	−2.49236	−0.113642
$482$	0	0
$483$	2.93523	0.133558
$484$	0	0
$485$	−27.9256	−1.26803
$486$	0	0
$487$	−10.1004	−0.457691	−0.228845	−	0.973463i	$-0.573495\pi$
$487$	−10.1004	−0.457691	−0.228845	+	0.973463i	$0.573495\pi$
$488$	0	0
$489$	7.34426	0.332119
$490$	0	0
$491$	19.0532	0.859857	0.429928	−	0.902863i	$-0.358539\pi$
$491$	19.0532	0.859857	0.429928	+	0.902863i	$0.358539\pi$
$492$	0	0
$493$	−11.7269	−0.528153
$494$	0	0
$495$	6.94784	0.312282
$496$	0	0
$497$	15.3710	0.689485
$498$	0	0
$499$	17.8469	0.798938	0.399469	−	0.916747i	$-0.369194\pi$
$499$	17.8469	0.798938	0.399469	+	0.916747i	$0.369194\pi$
$500$	0	0
$501$	−9.90414	−0.442484
$502$	0	0
$503$	21.8610	0.974733	0.487367	−	0.873197i	$-0.337958\pi$
$503$	21.8610	0.974733	0.487367	+	0.873197i	$0.337958\pi$
$504$	0	0
$505$	37.0485	1.64863
$506$	0	0
$507$	−1.02605	−0.0455685
$508$	0	0
$509$	17.3832	0.770495	0.385248	−	0.922813i	$-0.374116\pi$
$509$	17.3832	0.770495	0.385248	+	0.922813i	$0.374116\pi$
$510$	0	0
$511$	−5.51474	−0.243958
$512$	0	0
$513$	9.05511	0.399793
$514$	0	0
$515$	25.6672	1.13103
$516$	0	0
$517$	−7.82042	−0.343942
$518$	0	0
$519$	−18.4837	−0.811344
$520$	0	0
$521$	−21.7301	−0.952013	−0.476007	−	0.879442i	$-0.657916\pi$
$521$	−21.7301	−0.952013	−0.476007	+	0.879442i	$0.657916\pi$
$522$	0	0
$523$	−21.8223	−0.954223	−0.477111	−	0.878843i	$-0.658316\pi$
$523$	−21.8223	−0.954223	−0.477111	+	0.878843i	$0.658316\pi$
$524$	0	0
$525$	−7.93258	−0.346206
$526$	0	0
$527$	−16.7423	−0.729307
$528$	0	0
$529$	−14.8163	−0.644189
$530$	0	0
$531$	4.42936	0.192218
$532$	0	0
$533$	−8.70864	−0.377213
$534$	0	0
$535$	15.9087	0.687793
$536$	0	0
$537$	7.55575	0.326055
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	5.34384	0.229749	0.114875	−	0.993380i	$-0.463353\pi$
$541$	5.34384	0.229749	0.114875	+	0.993380i	$0.463353\pi$
$542$	0	0
$543$	3.53761	0.151814
$544$	0	0
$545$	5.30594	0.227282
$546$	0	0
$547$	−7.36450	−0.314883	−0.157442	−	0.987528i	$-0.550325\pi$
$547$	−7.36450	−0.314883	−0.157442	+	0.987528i	$0.550325\pi$
$548$	0	0
$549$	7.56982	0.323072
$550$	0	0
$551$	8.05181	0.343019
$552$	0	0
$553$	−2.13198	−0.0906610
$554$	0	0
$555$	9.12458	0.387317
$556$	0	0
$557$	6.11193	0.258971	0.129485	−	0.991581i	$-0.458667\pi$
$557$	6.11193	0.258971	0.129485	+	0.991581i	$0.458667\pi$
$558$	0	0
$559$	−1.37924	−0.0583354
$560$	0	0
$561$	−2.66576	−0.112549
$562$	0	0
$563$	−15.6224	−0.658407	−0.329204	−	0.944259i	$-0.606780\pi$
$563$	−15.6224	−0.658407	−0.329204	+	0.944259i	$0.606780\pi$
$564$	0	0
$565$	58.7743	2.47265
$566$	0	0
$567$	0.633343	0.0265979
$568$	0	0
$569$	21.4601	0.899654	0.449827	−	0.893116i	$-0.351486\pi$
$569$	21.4601	0.899654	0.449827	+	0.893116i	$0.351486\pi$
$570$	0	0
$571$	−10.3204	−0.431897	−0.215948	−	0.976405i	$-0.569284\pi$
$571$	−10.3204	−0.431897	−0.215948	+	0.976405i	$0.569284\pi$
$572$	0	0
$573$	−15.0692	−0.629527
$574$	0	0
$575$	−22.1167	−0.922329
$576$	0	0
$577$	−27.1669	−1.13097	−0.565486	−	0.824758i	$-0.691311\pi$
$577$	−27.1669	−1.13097	−0.565486	+	0.824758i	$0.691311\pi$
$578$	0	0
$579$	10.1974	0.423788
$580$	0	0
$581$	11.8300	0.490790
$582$	0	0
$583$	9.61423	0.398181
$584$	0	0
$585$	−6.94784	−0.287258
$586$	0	0
$587$	28.6054	1.18067	0.590336	−	0.807158i	$-0.298995\pi$
$587$	28.6054	1.18067	0.590336	+	0.807158i	$0.298995\pi$
$588$	0	0
$589$	11.4955	0.473662
$590$	0	0
$591$	−10.0673	−0.414115
$592$	0	0
$593$	5.85563	0.240462	0.120231	−	0.992746i	$-0.461636\pi$
$593$	5.85563	0.240462	0.120231	+	0.992746i	$0.461636\pi$
$594$	0	0
$595$	−9.27017	−0.380040
$596$	0	0
$597$	17.1558	0.702139
$598$	0	0
$599$	6.11404	0.249813	0.124906	−	0.992169i	$-0.460137\pi$
$599$	6.11404	0.249813	0.124906	+	0.992169i	$0.460137\pi$
$600$	0	0
$601$	7.54078	0.307595	0.153797	−	0.988102i	$-0.450850\pi$
$601$	7.54078	0.307595	0.153797	+	0.988102i	$0.450850\pi$
$602$	0	0
$603$	−29.4782	−1.20044
$604$	0	0
$605$	3.56808	0.145063
$606$	0	0
$607$	−42.7857	−1.73662	−0.868310	−	0.496022i	$-0.834794\pi$
$607$	−42.7857	−1.73662	−0.868310	+	0.496022i	$0.834794\pi$
$608$	0	0
$609$	−4.63125	−0.187668
$610$	0	0
$611$	7.82042	0.316380
$612$	0	0
$613$	43.3124	1.74937	0.874686	−	0.484689i	$-0.161067\pi$
$613$	43.3124	1.74937	0.874686	+	0.484689i	$0.161067\pi$
$614$	0	0
$615$	31.8825	1.28563
$616$	0	0
$617$	−16.3419	−0.657900	−0.328950	−	0.944347i	$-0.606695\pi$
$617$	−16.3419	−0.657900	−0.328950	+	0.944347i	$0.606695\pi$
$618$	0	0
$619$	24.1052	0.968870	0.484435	−	0.874827i	$-0.339025\pi$
$619$	24.1052	0.968870	0.484435	+	0.874827i	$0.339025\pi$
$620$	0	0
$621$	−14.5212	−0.582717
$622$	0	0
$623$	6.46755	0.259117
$624$	0	0
$625$	−3.88468	−0.155387
$626$	0	0
$627$	1.83034	0.0730968
$628$	0	0
$629$	6.47536	0.258189
$630$	0	0
$631$	40.5691	1.61503	0.807516	−	0.589846i	$-0.200812\pi$
$631$	40.5691	1.61503	0.807516	+	0.589846i	$0.200812\pi$
$632$	0	0
$633$	9.44586	0.375439
$634$	0	0
$635$	4.74667	0.188366
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−29.9308	−1.18404
$640$	0	0
$641$	−41.4660	−1.63781	−0.818904	−	0.573931i	$-0.805418\pi$
$641$	−41.4660	−1.63781	−0.818904	+	0.573931i	$0.805418\pi$
$642$	0	0
$643$	29.3223	1.15636	0.578180	−	0.815909i	$-0.303763\pi$
$643$	29.3223	1.15636	0.578180	+	0.815909i	$0.303763\pi$
$644$	0	0
$645$	5.04941	0.198821
$646$	0	0
$647$	18.5797	0.730445	0.365222	−	0.930920i	$-0.380993\pi$
$647$	18.5797	0.730445	0.365222	+	0.930920i	$0.380993\pi$
$648$	0	0
$649$	2.27471	0.0892901
$650$	0	0
$651$	−6.61196	−0.259143
$652$	0	0
$653$	−39.4220	−1.54270	−0.771351	−	0.636410i	$-0.780419\pi$
$653$	−39.4220	−1.54270	−0.771351	+	0.636410i	$0.780419\pi$
$654$	0	0
$655$	−36.4096	−1.42264
$656$	0	0
$657$	10.7384	0.418946
$658$	0	0
$659$	−36.7502	−1.43159	−0.715793	−	0.698313i	$-0.753934\pi$
$659$	−36.7502	−1.43159	−0.715793	+	0.698313i	$0.753934\pi$
$660$	0	0
$661$	−25.3183	−0.984768	−0.492384	−	0.870378i	$-0.663874\pi$
$661$	−25.3183	−0.984768	−0.492384	+	0.870378i	$0.663874\pi$
$662$	0	0
$663$	2.66576	0.103530
$664$	0	0
$665$	6.36500	0.246824
$666$	0	0
$667$	−12.9123	−0.499966
$668$	0	0
$669$	22.5972	0.873658
$670$	0	0
$671$	3.88750	0.150075
$672$	0	0
$673$	−28.6402	−1.10400	−0.552000	−	0.833844i	$-0.686135\pi$
$673$	−28.6402	−1.10400	−0.552000	+	0.833844i	$0.686135\pi$
$674$	0	0
$675$	39.2442	1.51051
$676$	0	0
$677$	−24.7175	−0.949970	−0.474985	−	0.879994i	$-0.657547\pi$
$677$	−24.7175	−0.949970	−0.474985	+	0.879994i	$0.657547\pi$
$678$	0	0
$679$	−7.82650	−0.300353
$680$	0	0
$681$	17.4418	0.668372
$682$	0	0
$683$	−10.4027	−0.398047	−0.199024	−	0.979995i	$-0.563777\pi$
$683$	−10.4027	−0.398047	−0.199024	+	0.979995i	$0.563777\pi$
$684$	0	0
$685$	24.6277	0.940977
$686$	0	0
$687$	−15.7567	−0.601157
$688$	0	0
$689$	−9.61423	−0.366273
$690$	0	0
$691$	−4.98275	−0.189553	−0.0947764	−	0.995499i	$-0.530214\pi$
$691$	−4.98275	−0.189553	−0.0947764	+	0.995499i	$0.530214\pi$
$692$	0	0
$693$	1.94722	0.0739689
$694$	0	0
$695$	−8.34833	−0.316670
$696$	0	0
$697$	22.6258	0.857013
$698$	0	0
$699$	−16.8993	−0.639190
$700$	0	0
$701$	20.7432	0.783458	0.391729	−	0.920081i	$-0.371877\pi$
$701$	20.7432	0.783458	0.391729	+	0.920081i	$0.371877\pi$
$702$	0	0
$703$	−4.44605	−0.167686
$704$	0	0
$705$	−28.6308	−1.07830
$706$	0	0
$707$	10.3833	0.390504
$708$	0	0
$709$	27.7469	1.04206	0.521029	−	0.853539i	$-0.325548\pi$
$709$	27.7469	1.04206	0.521029	+	0.853539i	$0.325548\pi$
$710$	0	0
$711$	4.15144	0.155691
$712$	0	0
$713$	−18.4347	−0.690384
$714$	0	0
$715$	−3.56808	−0.133439
$716$	0	0
$717$	−22.4961	−0.840132
$718$	0	0
$719$	43.4997	1.62226	0.811132	−	0.584863i	$-0.198852\pi$
$719$	43.4997	1.62226	0.811132	+	0.584863i	$0.198852\pi$
$720$	0	0
$721$	7.19357	0.267902
$722$	0	0
$723$	−1.84004	−0.0684320
$724$	0	0
$725$	34.8960	1.29601
$726$	0	0
$727$	26.0396	0.965755	0.482877	−	0.875688i	$-0.339592\pi$
$727$	26.0396	0.965755	0.482877	+	0.875688i	$0.339592\pi$
$728$	0	0
$729$	14.3917	0.533026
$730$	0	0
$731$	3.58337	0.132536
$732$	0	0
$733$	10.3361	0.381773	0.190886	−	0.981612i	$-0.438864\pi$
$733$	10.3361	0.381773	0.190886	+	0.981612i	$0.438864\pi$
$734$	0	0
$735$	−3.66103	−0.135039
$736$	0	0
$737$	−15.1386	−0.557637
$738$	0	0
$739$	3.95578	0.145516	0.0727578	−	0.997350i	$-0.476820\pi$
$739$	3.95578	0.145516	0.0727578	+	0.997350i	$0.476820\pi$
$740$	0	0
$741$	−1.83034	−0.0672393
$742$	0	0
$743$	40.6303	1.49058	0.745291	−	0.666739i	$-0.232311\pi$
$743$	40.6303	1.49058	0.745291	+	0.666739i	$0.232311\pi$
$744$	0	0
$745$	−33.3783	−1.22289
$746$	0	0
$747$	−23.0356	−0.842828
$748$	0	0
$749$	4.45862	0.162914
$750$	0	0
$751$	−11.5960	−0.423142	−0.211571	−	0.977363i	$-0.567858\pi$
$751$	−11.5960	−0.423142	−0.211571	+	0.977363i	$0.567858\pi$
$752$	0	0
$753$	−5.87671	−0.214159
$754$	0	0
$755$	66.3448	2.41453
$756$	0	0
$757$	−30.5830	−1.11156	−0.555780	−	0.831330i	$-0.687580\pi$
$757$	−30.5830	−1.11156	−0.555780	+	0.831330i	$0.687580\pi$
$758$	0	0
$759$	−2.93523	−0.106542
$760$	0	0
$761$	−14.0034	−0.507624	−0.253812	−	0.967254i	$-0.581684\pi$
$761$	−14.0034	−0.507624	−0.253812	+	0.967254i	$0.581684\pi$
$762$	0	0
$763$	1.48706	0.0538351
$764$	0	0
$765$	18.0511	0.652638
$766$	0	0
$767$	−2.27471	−0.0821350
$768$	0	0
$769$	44.2324	1.59506	0.797531	−	0.603278i	$-0.206139\pi$
$769$	44.2324	1.59506	0.797531	+	0.603278i	$0.206139\pi$
$770$	0	0
$771$	22.9375	0.826075
$772$	0	0
$773$	39.7492	1.42968	0.714839	−	0.699289i	$-0.246500\pi$
$773$	39.7492	1.42968	0.714839	+	0.699289i	$0.246500\pi$
$774$	0	0
$775$	49.8205	1.78961
$776$	0	0
$777$	2.55728	0.0917420
$778$	0	0
$779$	−15.5351	−0.556603
$780$	0	0
$781$	−15.3710	−0.550019
$782$	0	0
$783$	22.9118	0.818801
$784$	0	0
$785$	47.6536	1.70083
$786$	0	0
$787$	−47.2809	−1.68538	−0.842690	−	0.538399i	$-0.819029\pi$
$787$	−47.2809	−1.68538	−0.842690	+	0.538399i	$0.819029\pi$
$788$	0	0
$789$	11.6010	0.413006
$790$	0	0
$791$	16.4723	0.585686
$792$	0	0
$793$	−3.88750	−0.138049
$794$	0	0
$795$	35.1979	1.24834
$796$	0	0
$797$	19.0289	0.674037	0.337018	−	0.941498i	$-0.390582\pi$
$797$	19.0289	0.674037	0.337018	+	0.941498i	$0.390582\pi$
$798$	0	0
$799$	−20.3181	−0.718804
$800$	0	0
$801$	−12.5938	−0.444979
$802$	0	0
$803$	5.51474	0.194611
$804$	0	0
$805$	−10.2072	−0.359758
$806$	0	0
$807$	0.772735	0.0272016
$808$	0	0
$809$	3.04861	0.107184	0.0535918	−	0.998563i	$-0.482933\pi$
$809$	3.04861	0.107184	0.0535918	+	0.998563i	$0.482933\pi$
$810$	0	0
$811$	−25.6050	−0.899115	−0.449557	−	0.893251i	$-0.648418\pi$
$811$	−25.6050	−0.899115	−0.449557	+	0.893251i	$0.648418\pi$
$812$	0	0
$813$	−13.8894	−0.487123
$814$	0	0
$815$	−25.5396	−0.894613
$816$	0	0
$817$	−2.46038	−0.0860778
$818$	0	0
$819$	−1.94722	−0.0680415
$820$	0	0
$821$	45.4122	1.58490	0.792449	−	0.609938i	$-0.208806\pi$
$821$	45.4122	1.58490	0.792449	+	0.609938i	$0.208806\pi$
$822$	0	0
$823$	−53.9874	−1.88188	−0.940941	−	0.338571i	$-0.890057\pi$
$823$	−53.9874	−1.88188	−0.940941	+	0.338571i	$0.890057\pi$
$824$	0	0
$825$	7.93258	0.276177
$826$	0	0
$827$	−24.9370	−0.867144	−0.433572	−	0.901119i	$-0.642747\pi$
$827$	−24.9370	−0.867144	−0.433572	+	0.901119i	$0.642747\pi$
$828$	0	0
$829$	42.3040	1.46928	0.734639	−	0.678458i	$-0.237352\pi$
$829$	42.3040	1.46928	0.734639	+	0.678458i	$0.237352\pi$
$830$	0	0
$831$	0.872387	0.0302628
$832$	0	0
$833$	−2.59808	−0.0900183
$834$	0	0
$835$	34.4416	1.19190
$836$	0	0
$837$	32.7109	1.13065
$838$	0	0
$839$	36.8738	1.27303	0.636513	−	0.771266i	$-0.280376\pi$
$839$	36.8738	1.27303	0.636513	+	0.771266i	$0.280376\pi$
$840$	0	0
$841$	−8.62679	−0.297476
$842$	0	0
$843$	−0.553651	−0.0190687
$844$	0	0
$845$	3.56808	0.122746
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	26.6506	0.914647
$850$	0	0
$851$	7.12991	0.244410
$852$	0	0
$853$	−45.5216	−1.55863	−0.779316	−	0.626631i	$-0.784433\pi$
$853$	−45.5216	−1.55863	−0.779316	+	0.626631i	$0.784433\pi$
$854$	0	0
$855$	−12.3941	−0.423868
$856$	0	0
$857$	−54.7135	−1.86898	−0.934489	−	0.355993i	$-0.884143\pi$
$857$	−54.7135	−1.86898	−0.934489	+	0.355993i	$0.884143\pi$
$858$	0	0
$859$	14.0041	0.477812	0.238906	−	0.971043i	$-0.423211\pi$
$859$	14.0041	0.477812	0.238906	+	0.971043i	$0.423211\pi$
$860$	0	0
$861$	8.93549	0.304521
$862$	0	0
$863$	8.28995	0.282193	0.141097	−	0.989996i	$-0.454937\pi$
$863$	8.28995	0.282193	0.141097	+	0.989996i	$0.454937\pi$
$864$	0	0
$865$	64.2769	2.18548
$866$	0	0
$867$	10.5170	0.357175
$868$	0	0
$869$	2.13198	0.0723225
$870$	0	0
$871$	15.1386	0.512951
$872$	0	0
$873$	15.2399	0.515794
$874$	0	0
$875$	9.74509	0.329444
$876$	0	0
$877$	25.6372	0.865707	0.432853	−	0.901464i	$-0.357507\pi$
$877$	25.6372	0.865707	0.432853	+	0.901464i	$0.357507\pi$
$878$	0	0
$879$	8.17890	0.275867
$880$	0	0
$881$	−37.9657	−1.27910	−0.639548	−	0.768751i	$-0.720878\pi$
$881$	−37.9657	−1.27910	−0.639548	+	0.768751i	$0.720878\pi$
$882$	0	0
$883$	25.9909	0.874663	0.437332	−	0.899300i	$-0.355924\pi$
$883$	25.9909	0.874663	0.437332	+	0.899300i	$0.355924\pi$
$884$	0	0
$885$	8.32777	0.279935
$886$	0	0
$887$	10.4651	0.351382	0.175691	−	0.984445i	$-0.443784\pi$
$887$	10.4651	0.351382	0.175691	+	0.984445i	$0.443784\pi$
$888$	0	0
$889$	1.33032	0.0446174
$890$	0	0
$891$	−0.633343	−0.0212178
$892$	0	0
$893$	13.9506	0.466841
$894$	0	0
$895$	−26.2751	−0.878279
$896$	0	0
$897$	2.93523	0.0980044
$898$	0	0
$899$	29.0865	0.970090
$900$	0	0
$901$	24.9786	0.832157
$902$	0	0
$903$	1.41516	0.0470937
$904$	0	0
$905$	−12.3020	−0.408933
$906$	0	0
$907$	27.7079	0.920026	0.460013	−	0.887912i	$-0.347845\pi$
$907$	27.7079	0.920026	0.460013	+	0.887912i	$0.347845\pi$
$908$	0	0
$909$	−20.2186	−0.670609
$910$	0	0
$911$	−17.9547	−0.594867	−0.297433	−	0.954743i	$-0.596131\pi$
$911$	−17.9547	−0.594867	−0.297433	+	0.954743i	$0.596131\pi$
$912$	0	0
$913$	−11.8300	−0.391515
$914$	0	0
$915$	14.2322	0.470503
$916$	0	0
$917$	−10.2043	−0.336975
$918$	0	0
$919$	−48.1654	−1.58883	−0.794415	−	0.607375i	$-0.792222\pi$
$919$	−48.1654	−1.58883	−0.794415	+	0.607375i	$0.792222\pi$
$920$	0	0
$921$	−27.8183	−0.916644
$922$	0	0
$923$	15.3710	0.505944
$924$	0	0
$925$	−19.2689	−0.633557
$926$	0	0
$927$	−14.0075	−0.460066
$928$	0	0
$929$	21.9202	0.719179	0.359590	−	0.933111i	$-0.382917\pi$
$929$	21.9202	0.719179	0.359590	+	0.933111i	$0.382917\pi$
$930$	0	0
$931$	1.78387	0.0584641
$932$	0	0
$933$	−2.75042	−0.0900448
$934$	0	0
$935$	9.27017	0.303167
$936$	0	0
$937$	13.1868	0.430795	0.215398	−	0.976526i	$-0.430895\pi$
$937$	13.1868	0.430795	0.215398	+	0.976526i	$0.430895\pi$
$938$	0	0
$939$	−6.00205	−0.195870
$940$	0	0
$941$	−13.9053	−0.453301	−0.226651	−	0.973976i	$-0.572778\pi$
$941$	−13.9053	−0.453301	−0.226651	+	0.973976i	$0.572778\pi$
$942$	0	0
$943$	24.9129	0.811275
$944$	0	0
$945$	18.1119	0.589180
$946$	0	0
$947$	31.6618	1.02887	0.514434	−	0.857530i	$-0.328002\pi$
$947$	31.6618	1.02887	0.514434	+	0.857530i	$0.328002\pi$
$948$	0	0
$949$	−5.51474	−0.179016
$950$	0	0
$951$	2.97111	0.0963448
$952$	0	0
$953$	6.22106	0.201520	0.100760	−	0.994911i	$-0.467873\pi$
$953$	6.22106	0.201520	0.100760	+	0.994911i	$0.467873\pi$
$954$	0	0
$955$	52.4032	1.69573
$956$	0	0
$957$	4.63125	0.149707
$958$	0	0
$959$	6.90224	0.222885
$960$	0	0
$961$	10.5264	0.339561
$962$	0	0
$963$	−8.68192	−0.279771
$964$	0	0
$965$	−35.4612	−1.14154
$966$	0	0
$967$	−23.2578	−0.747922	−0.373961	−	0.927444i	$-0.622001\pi$
$967$	−23.2578	−0.747922	−0.373961	+	0.927444i	$0.622001\pi$
$968$	0	0
$969$	4.75538	0.152765
$970$	0	0
$971$	−25.6493	−0.823124	−0.411562	−	0.911382i	$-0.635017\pi$
$971$	−25.6493	−0.823124	−0.411562	+	0.911382i	$0.635017\pi$
$972$	0	0
$973$	−2.33973	−0.0750082
$974$	0	0
$975$	−7.93258	−0.254046
$976$	0	0
$977$	−10.0777	−0.322415	−0.161208	−	0.986920i	$-0.551539\pi$
$977$	−10.0777	−0.322415	−0.161208	+	0.986920i	$0.551539\pi$
$978$	0	0
$979$	−6.46755	−0.206704
$980$	0	0
$981$	−2.89563	−0.0924505
$982$	0	0
$983$	16.1862	0.516260	0.258130	−	0.966110i	$-0.416894\pi$
$983$	16.1862	0.516260	0.258130	+	0.966110i	$0.416894\pi$
$984$	0	0
$985$	35.0091	1.11548
$986$	0	0
$987$	−8.02414	−0.255411
$988$	0	0
$989$	3.94559	0.125462
$990$	0	0
$991$	−11.1318	−0.353612	−0.176806	−	0.984246i	$-0.556577\pi$
$991$	−11.1318	−0.353612	−0.176806	+	0.984246i	$0.556577\pi$
$992$	0	0
$993$	−1.93234	−0.0613209
$994$	0	0
$995$	−59.6591	−1.89132
$996$	0	0
$997$	−50.1572	−1.58850	−0.794248	−	0.607594i	$-0.792135\pi$
$997$	−50.1572	−1.58850	−0.794248	+	0.607594i	$0.792135\pi$
$998$	0	0
$999$	−12.6514	−0.400274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.t.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.t.1.4	✓	10	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.02605 q^{3} +3.56808 q^{5} +1.00000 q^{7} -1.94722 q^{9} +O(q^{10})\)	\(q-1.02605 q^{3} +3.56808 q^{5} +1.00000 q^{7} -1.94722 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.66103 q^{15} -2.59808 q^{17} +1.78387 q^{19} -1.02605 q^{21} -2.86071 q^{23} +7.73119 q^{25} +5.07609 q^{27} +4.51367 q^{29} +6.44410 q^{31} +1.02605 q^{33} +3.56808 q^{35} -2.49236 q^{37} -1.02605 q^{39} -8.70864 q^{41} -1.37924 q^{43} -6.94784 q^{45} +7.82042 q^{47} +1.00000 q^{49} +2.66576 q^{51} -9.61423 q^{53} -3.56808 q^{55} -1.83034 q^{57} -2.27471 q^{59} -3.88750 q^{61} -1.94722 q^{63} +3.56808 q^{65} +15.1386 q^{67} +2.93523 q^{69} +15.3710 q^{71} -5.51474 q^{73} -7.93258 q^{75} -1.00000 q^{77} -2.13198 q^{79} +0.633343 q^{81} +11.8300 q^{83} -9.27017 q^{85} -4.63125 q^{87} +6.46755 q^{89} +1.00000 q^{91} -6.61196 q^{93} +6.36500 q^{95} -7.82650 q^{97} +1.94722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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