LMFDB - Embedded newform 8008.2.a.z.1.5

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.74947$ 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)

$f(q)$	$=$	$q-1.74947 q^{3} -0.876486 q^{5} -1.00000 q^{7} +0.0606454 q^{9} +O(q^{10})$	$q-1.74947 q^{3} -0.876486 q^{5} -1.00000 q^{7} +0.0606454 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.53339 q^{15} +2.59006 q^{17} +4.40706 q^{19} +1.74947 q^{21} -7.88985 q^{23} -4.23177 q^{25} +5.14231 q^{27} -5.50783 q^{29} -0.941850 q^{31} +1.74947 q^{33} +0.876486 q^{35} -1.26431 q^{37} -1.74947 q^{39} +10.7848 q^{41} -5.13415 q^{43} -0.0531548 q^{45} -5.09461 q^{47} +1.00000 q^{49} -4.53123 q^{51} +13.0391 q^{53} +0.876486 q^{55} -7.71003 q^{57} -1.26091 q^{59} -9.35575 q^{61} -0.0606454 q^{63} -0.876486 q^{65} +1.11274 q^{67} +13.8031 q^{69} -8.01306 q^{71} -8.99149 q^{73} +7.40336 q^{75} +1.00000 q^{77} +6.51357 q^{79} -9.17826 q^{81} -9.40002 q^{83} -2.27015 q^{85} +9.63579 q^{87} -12.7404 q^{89} -1.00000 q^{91} +1.64774 q^{93} -3.86273 q^{95} -2.80790 q^{97} -0.0606454 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	−1.74947	−1.01006	−0.505028	−	0.863103i	$-0.668518\pi$
$3$	−1.74947	−1.01006	−0.505028	+	0.863103i	$0.668518\pi$
$4$	0	0
$5$	−0.876486	−0.391976	−0.195988	−	0.980606i	$-0.562791\pi$
$5$	−0.876486	−0.391976	−0.195988	+	0.980606i	$0.562791\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.0606454	0.0202151
$10$	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
$15$	1.53339	0.395918
$16$	0	0
$17$	2.59006	0.628182	0.314091	−	0.949393i	$-0.398300\pi$
$17$	2.59006	0.628182	0.314091	+	0.949393i	$0.398300\pi$
$18$	0	0
$19$	4.40706	1.01105	0.505525	−	0.862812i	$-0.331299\pi$
$19$	4.40706	1.01105	0.505525	+	0.862812i	$0.331299\pi$
$20$	0	0
$21$	1.74947	0.381766
$22$	0	0
$23$	−7.88985	−1.64515	−0.822573	−	0.568659i	$-0.807462\pi$
$23$	−7.88985	−1.64515	−0.822573	+	0.568659i	$0.807462\pi$
$24$	0	0
$25$	−4.23177	−0.846355
$26$	0	0
$27$	5.14231	0.989639
$28$	0	0
$29$	−5.50783	−1.02278	−0.511390	−	0.859349i	$-0.670869\pi$
$29$	−5.50783	−1.02278	−0.511390	+	0.859349i	$0.670869\pi$
$30$	0	0
$31$	−0.941850	−0.169161	−0.0845806	−	0.996417i	$-0.526955\pi$
$31$	−0.941850	−0.169161	−0.0845806	+	0.996417i	$0.526955\pi$
$32$	0	0
$33$	1.74947	0.304544
$34$	0	0
$35$	0.876486	0.148153
$36$	0	0
$37$	−1.26431	−0.207852	−0.103926	−	0.994585i	$-0.533141\pi$
$37$	−1.26431	−0.207852	−0.103926	+	0.994585i	$0.533141\pi$
$38$	0	0
$39$	−1.74947	−0.280139
$40$	0	0
$41$	10.7848	1.68430	0.842148	−	0.539247i	$-0.181291\pi$
$41$	10.7848	1.68430	0.842148	+	0.539247i	$0.181291\pi$
$42$	0	0
$43$	−5.13415	−0.782951	−0.391475	−	0.920189i	$-0.628035\pi$
$43$	−5.13415	−0.782951	−0.391475	+	0.920189i	$0.628035\pi$
$44$	0	0
$45$	−0.0531548	−0.00792385
$46$	0	0
$47$	−5.09461	−0.743125	−0.371563	−	0.928408i	$-0.621178\pi$
$47$	−5.09461	−0.743125	−0.371563	+	0.928408i	$0.621178\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.53123	−0.634499
$52$	0	0
$53$	13.0391	1.79106	0.895531	−	0.444999i	$-0.146796\pi$
$53$	13.0391	1.79106	0.895531	+	0.444999i	$0.146796\pi$
$54$	0	0
$55$	0.876486	0.118185
$56$	0	0
$57$	−7.71003	−1.02122
$58$	0	0
$59$	−1.26091	−0.164157	−0.0820786	−	0.996626i	$-0.526156\pi$
$59$	−1.26091	−0.164157	−0.0820786	+	0.996626i	$0.526156\pi$
$60$	0	0
$61$	−9.35575	−1.19788	−0.598940	−	0.800794i	$-0.704411\pi$
$61$	−9.35575	−1.19788	−0.598940	+	0.800794i	$0.704411\pi$
$62$	0	0
$63$	−0.0606454	−0.00764060
$64$	0	0
$65$	−0.876486	−0.108715
$66$	0	0
$67$	1.11274	0.135942	0.0679712	−	0.997687i	$-0.478347\pi$
$67$	1.11274	0.135942	0.0679712	+	0.997687i	$0.478347\pi$
$68$	0	0
$69$	13.8031	1.66169
$70$	0	0
$71$	−8.01306	−0.950975	−0.475488	−	0.879722i	$-0.657728\pi$
$71$	−8.01306	−0.950975	−0.475488	+	0.879722i	$0.657728\pi$
$72$	0	0
$73$	−8.99149	−1.05237	−0.526187	−	0.850369i	$-0.676379\pi$
$73$	−8.99149	−1.05237	−0.526187	+	0.850369i	$0.676379\pi$
$74$	0	0
$75$	7.40336	0.854866
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	6.51357	0.732834	0.366417	−	0.930451i	$-0.380584\pi$
$79$	6.51357	0.732834	0.366417	+	0.930451i	$0.380584\pi$
$80$	0	0
$81$	−9.17826	−1.01981
$82$	0	0
$83$	−9.40002	−1.03179	−0.515893	−	0.856653i	$-0.672540\pi$
$83$	−9.40002	−1.03179	−0.515893	+	0.856653i	$0.672540\pi$
$84$	0	0
$85$	−2.27015	−0.246232
$86$	0	0
$87$	9.63579	1.03307
$88$	0	0
$89$	−12.7404	−1.35048	−0.675239	−	0.737599i	$-0.735960\pi$
$89$	−12.7404	−1.35048	−0.675239	+	0.737599i	$0.735960\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.64774	0.170862
$94$	0	0
$95$	−3.86273	−0.396308
$96$	0	0
$97$	−2.80790	−0.285099	−0.142549	−	0.989788i	$-0.545530\pi$
$97$	−2.80790	−0.285099	−0.142549	+	0.989788i	$0.545530\pi$
$98$	0	0
$99$	−0.0606454	−0.00609509
$100$	0	0
$101$	16.9436	1.68595	0.842976	−	0.537951i	$-0.180802\pi$
$101$	16.9436	1.68595	0.842976	+	0.537951i	$0.180802\pi$
$102$	0	0
$103$	3.02159	0.297726	0.148863	−	0.988858i	$-0.452439\pi$
$103$	3.02159	0.297726	0.148863	+	0.988858i	$0.452439\pi$
$104$	0	0
$105$	−1.53339	−0.149643
$106$	0	0
$107$	0.609172	0.0588908	0.0294454	−	0.999566i	$-0.490626\pi$
$107$	0.609172	0.0588908	0.0294454	+	0.999566i	$0.490626\pi$
$108$	0	0
$109$	−4.09440	−0.392172	−0.196086	−	0.980587i	$-0.562823\pi$
$109$	−4.09440	−0.392172	−0.196086	+	0.980587i	$0.562823\pi$
$110$	0	0
$111$	2.21188	0.209942
$112$	0	0
$113$	−8.70866	−0.819242	−0.409621	−	0.912256i	$-0.634339\pi$
$113$	−8.70866	−0.819242	−0.409621	+	0.912256i	$0.634339\pi$
$114$	0	0
$115$	6.91534	0.644859
$116$	0	0
$117$	0.0606454	0.00560667
$118$	0	0
$119$	−2.59006	−0.237430
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−18.8676	−1.70123
$124$	0	0
$125$	8.09152	0.723727
$126$	0	0
$127$	12.0248	1.06703	0.533514	−	0.845791i	$-0.320871\pi$
$127$	12.0248	1.06703	0.533514	+	0.845791i	$0.320871\pi$
$128$	0	0
$129$	8.98205	0.790825
$130$	0	0
$131$	−10.5228	−0.919378	−0.459689	−	0.888080i	$-0.652039\pi$
$131$	−10.5228	−0.919378	−0.459689	+	0.888080i	$0.652039\pi$
$132$	0	0
$133$	−4.40706	−0.382141
$134$	0	0
$135$	−4.50716	−0.387915
$136$	0	0
$137$	8.59469	0.734294	0.367147	−	0.930163i	$-0.380335\pi$
$137$	8.59469	0.734294	0.367147	+	0.930163i	$0.380335\pi$
$138$	0	0
$139$	−21.5643	−1.82906	−0.914531	−	0.404517i	$-0.867440\pi$
$139$	−21.5643	−1.82906	−0.914531	+	0.404517i	$0.867440\pi$
$140$	0	0
$141$	8.91287	0.750599
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	4.82754	0.400905
$146$	0	0
$147$	−1.74947	−0.144294
$148$	0	0
$149$	−1.65203	−0.135340	−0.0676700	−	0.997708i	$-0.521556\pi$
$149$	−1.65203	−0.135340	−0.0676700	+	0.997708i	$0.521556\pi$
$150$	0	0
$151$	7.61316	0.619551	0.309775	−	0.950810i	$-0.399746\pi$
$151$	7.61316	0.619551	0.309775	+	0.950810i	$0.399746\pi$
$152$	0	0
$153$	0.157075	0.0126988
$154$	0	0
$155$	0.825518	0.0663072
$156$	0	0
$157$	−1.51561	−0.120958	−0.0604792	−	0.998169i	$-0.519263\pi$
$157$	−1.51561	−0.120958	−0.0604792	+	0.998169i	$0.519263\pi$
$158$	0	0
$159$	−22.8116	−1.80907
$160$	0	0
$161$	7.88985	0.621807
$162$	0	0
$163$	−17.8164	−1.39549	−0.697744	−	0.716348i	$-0.745812\pi$
$163$	−17.8164	−1.39549	−0.697744	+	0.716348i	$0.745812\pi$
$164$	0	0
$165$	−1.53339	−0.119374
$166$	0	0
$167$	4.62838	0.358155	0.179077	−	0.983835i	$-0.442689\pi$
$167$	4.62838	0.358155	0.179077	+	0.983835i	$0.442689\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.267268	0.0204385
$172$	0	0
$173$	10.1962	0.775206	0.387603	−	0.921826i	$-0.373303\pi$
$173$	10.1962	0.775206	0.387603	+	0.921826i	$0.373303\pi$
$174$	0	0
$175$	4.23177	0.319892
$176$	0	0
$177$	2.20593	0.165808
$178$	0	0
$179$	−11.5155	−0.860707	−0.430354	−	0.902660i	$-0.641611\pi$
$179$	−11.5155	−0.860707	−0.430354	+	0.902660i	$0.641611\pi$
$180$	0	0
$181$	6.59438	0.490157	0.245078	−	0.969503i	$-0.421186\pi$
$181$	6.59438	0.490157	0.245078	+	0.969503i	$0.421186\pi$
$182$	0	0
$183$	16.3676	1.20993
$184$	0	0
$185$	1.10815	0.0814731
$186$	0	0
$187$	−2.59006	−0.189404
$188$	0	0
$189$	−5.14231	−0.374048
$190$	0	0
$191$	−5.23668	−0.378913	−0.189456	−	0.981889i	$-0.560673\pi$
$191$	−5.23668	−0.378913	−0.189456	+	0.981889i	$0.560673\pi$
$192$	0	0
$193$	18.5432	1.33477	0.667386	−	0.744712i	$-0.267413\pi$
$193$	18.5432	1.33477	0.667386	+	0.744712i	$0.267413\pi$
$194$	0	0
$195$	1.53339	0.109808
$196$	0	0
$197$	22.5372	1.60571	0.802854	−	0.596175i	$-0.203314\pi$
$197$	22.5372	1.60571	0.802854	+	0.596175i	$0.203314\pi$
$198$	0	0
$199$	0.527499	0.0373934	0.0186967	−	0.999825i	$-0.494048\pi$
$199$	0.527499	0.0373934	0.0186967	+	0.999825i	$0.494048\pi$
$200$	0	0
$201$	−1.94670	−0.137310
$202$	0	0
$203$	5.50783	0.386574
$204$	0	0
$205$	−9.45268	−0.660204
$206$	0	0
$207$	−0.478483	−0.0332569
$208$	0	0
$209$	−4.40706	−0.304843
$210$	0	0
$211$	−24.3234	−1.67449	−0.837246	−	0.546826i	$-0.815836\pi$
$211$	−24.3234	−1.67449	−0.837246	+	0.546826i	$0.815836\pi$
$212$	0	0
$213$	14.0186	0.960539
$214$	0	0
$215$	4.50001	0.306898
$216$	0	0
$217$	0.941850	0.0639369
$218$	0	0
$219$	15.7303	1.06296
$220$	0	0
$221$	2.59006	0.174226
$222$	0	0
$223$	−5.09259	−0.341025	−0.170513	−	0.985355i	$-0.554542\pi$
$223$	−5.09259	−0.341025	−0.170513	+	0.985355i	$0.554542\pi$
$224$	0	0
$225$	−0.256637	−0.0171092
$226$	0	0
$227$	13.8950	0.922241	0.461121	−	0.887337i	$-0.347448\pi$
$227$	13.8950	0.922241	0.461121	+	0.887337i	$0.347448\pi$
$228$	0	0
$229$	7.68185	0.507631	0.253815	−	0.967253i	$-0.418314\pi$
$229$	7.68185	0.507631	0.253815	+	0.967253i	$0.418314\pi$
$230$	0	0
$231$	−1.74947	−0.115107
$232$	0	0
$233$	5.52041	0.361654	0.180827	−	0.983515i	$-0.442123\pi$
$233$	5.52041	0.361654	0.180827	+	0.983515i	$0.442123\pi$
$234$	0	0
$235$	4.46535	0.291288
$236$	0	0
$237$	−11.3953	−0.740204
$238$	0	0
$239$	13.9294	0.901018	0.450509	−	0.892772i	$-0.351242\pi$
$239$	13.9294	0.901018	0.450509	+	0.892772i	$0.351242\pi$
$240$	0	0
$241$	18.6390	1.20064	0.600321	−	0.799759i	$-0.295040\pi$
$241$	18.6390	1.20064	0.600321	+	0.799759i	$0.295040\pi$
$242$	0	0
$243$	0.630149	0.0404241
$244$	0	0
$245$	−0.876486	−0.0559966
$246$	0	0
$247$	4.40706	0.280415
$248$	0	0
$249$	16.4450	1.04216
$250$	0	0
$251$	−19.3036	−1.21843	−0.609217	−	0.793004i	$-0.708516\pi$
$251$	−19.3036	−1.21843	−0.609217	+	0.793004i	$0.708516\pi$
$252$	0	0
$253$	7.88985	0.496030
$254$	0	0
$255$	3.97156	0.248709
$256$	0	0
$257$	12.9089	0.805234	0.402617	−	0.915369i	$-0.368101\pi$
$257$	12.9089	0.805234	0.402617	+	0.915369i	$0.368101\pi$
$258$	0	0
$259$	1.26431	0.0785607
$260$	0	0
$261$	−0.334025	−0.0206756
$262$	0	0
$263$	18.0068	1.11034	0.555172	−	0.831735i	$-0.312652\pi$
$263$	18.0068	1.11034	0.555172	+	0.831735i	$0.312652\pi$
$264$	0	0
$265$	−11.4286	−0.702054
$266$	0	0
$267$	22.2889	1.36406
$268$	0	0
$269$	−29.0271	−1.76982	−0.884908	−	0.465766i	$-0.845779\pi$
$269$	−29.0271	−1.76982	−0.884908	+	0.465766i	$0.845779\pi$
$270$	0	0
$271$	15.7839	0.958802	0.479401	−	0.877596i	$-0.340854\pi$
$271$	15.7839	0.958802	0.479401	+	0.877596i	$0.340854\pi$
$272$	0	0
$273$	1.74947	0.105883
$274$	0	0
$275$	4.23177	0.255185
$276$	0	0
$277$	−18.8898	−1.13498	−0.567489	−	0.823381i	$-0.692085\pi$
$277$	−18.8898	−1.13498	−0.567489	+	0.823381i	$0.692085\pi$
$278$	0	0
$279$	−0.0571188	−0.00341961
$280$	0	0
$281$	19.1103	1.14003	0.570014	−	0.821635i	$-0.306938\pi$
$281$	19.1103	1.14003	0.570014	+	0.821635i	$0.306938\pi$
$282$	0	0
$283$	29.5656	1.75749	0.878746	−	0.477290i	$-0.158381\pi$
$283$	29.5656	1.75749	0.878746	+	0.477290i	$0.158381\pi$
$284$	0	0
$285$	6.75773	0.400293
$286$	0	0
$287$	−10.7848	−0.636604
$288$	0	0
$289$	−10.2916	−0.605388
$290$	0	0
$291$	4.91233	0.287966
$292$	0	0
$293$	17.7971	1.03972	0.519859	−	0.854252i	$-0.325985\pi$
$293$	17.7971	1.03972	0.519859	+	0.854252i	$0.325985\pi$
$294$	0	0
$295$	1.10517	0.0643457
$296$	0	0
$297$	−5.14231	−0.298387
$298$	0	0
$299$	−7.88985	−0.456282
$300$	0	0
$301$	5.13415	0.295928
$302$	0	0
$303$	−29.6423	−1.70291
$304$	0	0
$305$	8.20018	0.469541
$306$	0	0
$307$	−18.4425	−1.05257	−0.526284	−	0.850309i	$-0.676415\pi$
$307$	−18.4425	−1.05257	−0.526284	+	0.850309i	$0.676415\pi$
$308$	0	0
$309$	−5.28617	−0.300720
$310$	0	0
$311$	16.0928	0.912537	0.456268	−	0.889842i	$-0.349186\pi$
$311$	16.0928	0.912537	0.456268	+	0.889842i	$0.349186\pi$
$312$	0	0
$313$	23.8262	1.34674	0.673368	−	0.739307i	$-0.264847\pi$
$313$	23.8262	1.34674	0.673368	+	0.739307i	$0.264847\pi$
$314$	0	0
$315$	0.0531548	0.00299493
$316$	0	0
$317$	30.2347	1.69815	0.849074	−	0.528275i	$-0.177161\pi$
$317$	30.2347	1.69815	0.849074	+	0.528275i	$0.177161\pi$
$318$	0	0
$319$	5.50783	0.308380
$320$	0	0
$321$	−1.06573	−0.0594831
$322$	0	0
$323$	11.4146	0.635123
$324$	0	0
$325$	−4.23177	−0.234737
$326$	0	0
$327$	7.16302	0.396116
$328$	0	0
$329$	5.09461	0.280875
$330$	0	0
$331$	20.5063	1.12713	0.563565	−	0.826072i	$-0.309429\pi$
$331$	20.5063	1.12713	0.563565	+	0.826072i	$0.309429\pi$
$332$	0	0
$333$	−0.0766748	−0.00420175
$334$	0	0
$335$	−0.975298	−0.0532862
$336$	0	0
$337$	5.24022	0.285453	0.142726	−	0.989762i	$-0.454413\pi$
$337$	5.24022	0.285453	0.142726	+	0.989762i	$0.454413\pi$
$338$	0	0
$339$	15.2355	0.827481
$340$	0	0
$341$	0.941850	0.0510040
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−12.0982	−0.651344
$346$	0	0
$347$	33.4946	1.79809	0.899043	−	0.437861i	$-0.144264\pi$
$347$	33.4946	1.79809	0.899043	+	0.437861i	$0.144264\pi$
$348$	0	0
$349$	−7.70354	−0.412361	−0.206181	−	0.978514i	$-0.566103\pi$
$349$	−7.70354	−0.412361	−0.206181	+	0.978514i	$0.566103\pi$
$350$	0	0
$351$	5.14231	0.274476
$352$	0	0
$353$	0.107535	0.00572349	0.00286175	−	0.999996i	$-0.499089\pi$
$353$	0.107535	0.00572349	0.00286175	+	0.999996i	$0.499089\pi$
$354$	0	0
$355$	7.02333	0.372760
$356$	0	0
$357$	4.53123	0.239818
$358$	0	0
$359$	33.1694	1.75061	0.875306	−	0.483569i	$-0.160660\pi$
$359$	33.1694	1.75061	0.875306	+	0.483569i	$0.160660\pi$
$360$	0	0
$361$	0.422208	0.0222215
$362$	0	0
$363$	−1.74947	−0.0918234
$364$	0	0
$365$	7.88091	0.412506
$366$	0	0
$367$	−31.9665	−1.66864	−0.834318	−	0.551283i	$-0.814138\pi$
$367$	−31.9665	−1.66864	−0.834318	+	0.551283i	$0.814138\pi$
$368$	0	0
$369$	0.654046	0.0340482
$370$	0	0
$371$	−13.0391	−0.676958
$372$	0	0
$373$	15.4744	0.801236	0.400618	−	0.916245i	$-0.368795\pi$
$373$	15.4744	0.801236	0.400618	+	0.916245i	$0.368795\pi$
$374$	0	0
$375$	−14.1559	−0.731006
$376$	0	0
$377$	−5.50783	−0.283668
$378$	0	0
$379$	−36.8925	−1.89504	−0.947520	−	0.319696i	$-0.896419\pi$
$379$	−36.8925	−1.89504	−0.947520	+	0.319696i	$0.896419\pi$
$380$	0	0
$381$	−21.0370	−1.07776
$382$	0	0
$383$	−14.2453	−0.727900	−0.363950	−	0.931418i	$-0.618572\pi$
$383$	−14.2453	−0.727900	−0.363950	+	0.931418i	$0.618572\pi$
$384$	0	0
$385$	−0.876486	−0.0446699
$386$	0	0
$387$	−0.311363	−0.0158274
$388$	0	0
$389$	15.6537	0.793672	0.396836	−	0.917890i	$-0.370108\pi$
$389$	15.6537	0.793672	0.396836	+	0.917890i	$0.370108\pi$
$390$	0	0
$391$	−20.4352	−1.03345
$392$	0	0
$393$	18.4093	0.928624
$394$	0	0
$395$	−5.70905	−0.287253
$396$	0	0
$397$	−17.2955	−0.868039	−0.434019	−	0.900904i	$-0.642905\pi$
$397$	−17.2955	−0.868039	−0.434019	+	0.900904i	$0.642905\pi$
$398$	0	0
$399$	7.71003	0.385984
$400$	0	0
$401$	−2.97724	−0.148677	−0.0743383	−	0.997233i	$-0.523684\pi$
$401$	−2.97724	−0.148677	−0.0743383	+	0.997233i	$0.523684\pi$
$402$	0	0
$403$	−0.941850	−0.0469169
$404$	0	0
$405$	8.04461	0.399740
$406$	0	0
$407$	1.26431	0.0626697
$408$	0	0
$409$	23.1019	1.14232	0.571158	−	0.820840i	$-0.306494\pi$
$409$	23.1019	1.14232	0.571158	+	0.820840i	$0.306494\pi$
$410$	0	0
$411$	−15.0362	−0.741679
$412$	0	0
$413$	1.26091	0.0620456
$414$	0	0
$415$	8.23898	0.404436
$416$	0	0
$417$	37.7261	1.84746
$418$	0	0
$419$	39.9601	1.95218	0.976090	−	0.217369i	$-0.0697475\pi$
$419$	39.9601	1.95218	0.976090	+	0.217369i	$0.0697475\pi$
$420$	0	0
$421$	13.5399	0.659895	0.329948	−	0.943999i	$-0.392969\pi$
$421$	13.5399	0.659895	0.329948	+	0.943999i	$0.392969\pi$
$422$	0	0
$423$	−0.308965	−0.0150224
$424$	0	0
$425$	−10.9605	−0.531664
$426$	0	0
$427$	9.35575	0.452756
$428$	0	0
$429$	1.74947	0.0844652
$430$	0	0
$431$	7.37757	0.355365	0.177682	−	0.984088i	$-0.443140\pi$
$431$	7.37757	0.355365	0.177682	+	0.984088i	$0.443140\pi$
$432$	0	0
$433$	0.409308	0.0196701	0.00983505	−	0.999952i	$-0.496869\pi$
$433$	0.409308	0.0196701	0.00983505	+	0.999952i	$0.496869\pi$
$434$	0	0
$435$	−8.44563	−0.404937
$436$	0	0
$437$	−34.7711	−1.66333
$438$	0	0
$439$	−18.1468	−0.866098	−0.433049	−	0.901370i	$-0.642562\pi$
$439$	−18.1468	−0.866098	−0.433049	+	0.901370i	$0.642562\pi$
$440$	0	0
$441$	0.0606454	0.00288787
$442$	0	0
$443$	−4.67583	−0.222155	−0.111078	−	0.993812i	$-0.535430\pi$
$443$	−4.67583	−0.222155	−0.111078	+	0.993812i	$0.535430\pi$
$444$	0	0
$445$	11.1668	0.529356
$446$	0	0
$447$	2.89018	0.136701
$448$	0	0
$449$	2.24075	0.105748	0.0528738	−	0.998601i	$-0.483162\pi$
$449$	2.24075	0.105748	0.0528738	+	0.998601i	$0.483162\pi$
$450$	0	0
$451$	−10.7848	−0.507834
$452$	0	0
$453$	−13.3190	−0.625781
$454$	0	0
$455$	0.876486	0.0410903
$456$	0	0
$457$	−7.12919	−0.333490	−0.166745	−	0.986000i	$-0.553326\pi$
$457$	−7.12919	−0.333490	−0.166745	+	0.986000i	$0.553326\pi$
$458$	0	0
$459$	13.3189	0.621673
$460$	0	0
$461$	6.80839	0.317098	0.158549	−	0.987351i	$-0.449318\pi$
$461$	6.80839	0.317098	0.158549	+	0.987351i	$0.449318\pi$
$462$	0	0
$463$	−13.4501	−0.625079	−0.312539	−	0.949905i	$-0.601180\pi$
$463$	−13.4501	−0.625079	−0.312539	+	0.949905i	$0.601180\pi$
$464$	0	0
$465$	−1.44422	−0.0669740
$466$	0	0
$467$	−30.4302	−1.40814	−0.704070	−	0.710131i	$-0.748636\pi$
$467$	−30.4302	−1.40814	−0.704070	+	0.710131i	$0.748636\pi$
$468$	0	0
$469$	−1.11274	−0.0513814
$470$	0	0
$471$	2.65151	0.122175
$472$	0	0
$473$	5.13415	0.236069
$474$	0	0
$475$	−18.6497	−0.855706
$476$	0	0
$477$	0.790763	0.0362065
$478$	0	0
$479$	15.0671	0.688431	0.344216	−	0.938891i	$-0.388145\pi$
$479$	15.0671	0.688431	0.344216	+	0.938891i	$0.388145\pi$
$480$	0	0
$481$	−1.26431	−0.0576478
$482$	0	0
$483$	−13.8031	−0.628061
$484$	0	0
$485$	2.46108	0.111752
$486$	0	0
$487$	−13.9417	−0.631758	−0.315879	−	0.948800i	$-0.602299\pi$
$487$	−13.9417	−0.631758	−0.315879	+	0.948800i	$0.602299\pi$
$488$	0	0
$489$	31.1692	1.40952
$490$	0	0
$491$	32.4334	1.46370	0.731848	−	0.681468i	$-0.238658\pi$
$491$	32.4334	1.46370	0.731848	+	0.681468i	$0.238658\pi$
$492$	0	0
$493$	−14.2656	−0.642491
$494$	0	0
$495$	0.0531548	0.00238913
$496$	0	0
$497$	8.01306	0.359435
$498$	0	0
$499$	24.3409	1.08965	0.544824	−	0.838550i	$-0.316596\pi$
$499$	24.3409	1.08965	0.544824	+	0.838550i	$0.316596\pi$
$500$	0	0
$501$	−8.09721	−0.361757
$502$	0	0
$503$	−35.7918	−1.59588	−0.797939	−	0.602738i	$-0.794076\pi$
$503$	−35.7918	−1.59588	−0.797939	+	0.602738i	$0.794076\pi$
$504$	0	0
$505$	−14.8508	−0.660853
$506$	0	0
$507$	−1.74947	−0.0776967
$508$	0	0
$509$	−28.1481	−1.24764	−0.623821	−	0.781567i	$-0.714421\pi$
$509$	−28.1481	−1.24764	−0.623821	+	0.781567i	$0.714421\pi$
$510$	0	0
$511$	8.99149	0.397760
$512$	0	0
$513$	22.6625	1.00057
$514$	0	0
$515$	−2.64838	−0.116701
$516$	0	0
$517$	5.09461	0.224061
$518$	0	0
$519$	−17.8380	−0.783002
$520$	0	0
$521$	15.6725	0.686624	0.343312	−	0.939221i	$-0.388451\pi$
$521$	15.6725	0.686624	0.343312	+	0.939221i	$0.388451\pi$
$522$	0	0
$523$	−14.3075	−0.625622	−0.312811	−	0.949815i	$-0.601271\pi$
$523$	−14.3075	−0.625622	−0.312811	+	0.949815i	$0.601271\pi$
$524$	0	0
$525$	−7.40336	−0.323109
$526$	0	0
$527$	−2.43945	−0.106264
$528$	0	0
$529$	39.2497	1.70651
$530$	0	0
$531$	−0.0764687	−0.00331846
$532$	0	0
$533$	10.7848	0.467140
$534$	0	0
$535$	−0.533930	−0.0230838
$536$	0	0
$537$	20.1460	0.869363
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	41.0735	1.76589	0.882944	−	0.469479i	$-0.155558\pi$
$541$	41.0735	1.76589	0.882944	+	0.469479i	$0.155558\pi$
$542$	0	0
$543$	−11.5367	−0.495086
$544$	0	0
$545$	3.58868	0.153722
$546$	0	0
$547$	6.28903	0.268900	0.134450	−	0.990920i	$-0.457073\pi$
$547$	6.28903	0.268900	0.134450	+	0.990920i	$0.457073\pi$
$548$	0	0
$549$	−0.567383	−0.0242153
$550$	0	0
$551$	−24.2734	−1.03408
$552$	0	0
$553$	−6.51357	−0.276985
$554$	0	0
$555$	−1.93868	−0.0822924
$556$	0	0
$557$	36.2172	1.53457	0.767285	−	0.641306i	$-0.221607\pi$
$557$	36.2172	1.53457	0.767285	+	0.641306i	$0.221607\pi$
$558$	0	0
$559$	−5.13415	−0.217151
$560$	0	0
$561$	4.53123	0.191309
$562$	0	0
$563$	3.90254	0.164473	0.0822363	−	0.996613i	$-0.473794\pi$
$563$	3.90254	0.164473	0.0822363	+	0.996613i	$0.473794\pi$
$564$	0	0
$565$	7.63302	0.321123
$566$	0	0
$567$	9.17826	0.385451
$568$	0	0
$569$	−7.70923	−0.323188	−0.161594	−	0.986857i	$-0.551663\pi$
$569$	−7.70923	−0.323188	−0.161594	+	0.986857i	$0.551663\pi$
$570$	0	0
$571$	0.126362	0.00528807	0.00264404	−	0.999997i	$-0.499158\pi$
$571$	0.126362	0.00528807	0.00264404	+	0.999997i	$0.499158\pi$
$572$	0	0
$573$	9.16141	0.382724
$574$	0	0
$575$	33.3880	1.39238
$576$	0	0
$577$	11.1604	0.464613	0.232307	−	0.972643i	$-0.425373\pi$
$577$	11.1604	0.464613	0.232307	+	0.972643i	$0.425373\pi$
$578$	0	0
$579$	−32.4408	−1.34820
$580$	0	0
$581$	9.40002	0.389978
$582$	0	0
$583$	−13.0391	−0.540026
$584$	0	0
$585$	−0.0531548	−0.00219768
$586$	0	0
$587$	19.3436	0.798395	0.399197	−	0.916865i	$-0.369289\pi$
$587$	19.3436	0.798395	0.399197	+	0.916865i	$0.369289\pi$
$588$	0	0
$589$	−4.15079	−0.171030
$590$	0	0
$591$	−39.4281	−1.62186
$592$	0	0
$593$	−31.1454	−1.27899	−0.639495	−	0.768795i	$-0.720856\pi$
$593$	−31.1454	−1.27899	−0.639495	+	0.768795i	$0.720856\pi$
$594$	0	0
$595$	2.27015	0.0930671
$596$	0	0
$597$	−0.922843	−0.0377695
$598$	0	0
$599$	13.1132	0.535789	0.267895	−	0.963448i	$-0.413672\pi$
$599$	13.1132	0.535789	0.267895	+	0.963448i	$0.413672\pi$
$600$	0	0
$601$	25.7011	1.04837	0.524185	−	0.851605i	$-0.324370\pi$
$601$	25.7011	1.04837	0.524185	+	0.851605i	$0.324370\pi$
$602$	0	0
$603$	0.0674823	0.00274809
$604$	0	0
$605$	−0.876486	−0.0356342
$606$	0	0
$607$	42.5625	1.72756	0.863780	−	0.503869i	$-0.168091\pi$
$607$	42.5625	1.72756	0.863780	+	0.503869i	$0.168091\pi$
$608$	0	0
$609$	−9.63579	−0.390462
$610$	0	0
$611$	−5.09461	−0.206106
$612$	0	0
$613$	26.4523	1.06840	0.534199	−	0.845359i	$-0.320613\pi$
$613$	26.4523	1.06840	0.534199	+	0.845359i	$0.320613\pi$
$614$	0	0
$615$	16.5372	0.666844
$616$	0	0
$617$	33.7369	1.35820	0.679098	−	0.734048i	$-0.262371\pi$
$617$	33.7369	1.35820	0.679098	+	0.734048i	$0.262371\pi$
$618$	0	0
$619$	−0.403925	−0.0162351	−0.00811757	−	0.999967i	$-0.502584\pi$
$619$	−0.403925	−0.0162351	−0.00811757	+	0.999967i	$0.502584\pi$
$620$	0	0
$621$	−40.5721	−1.62810
$622$	0	0
$623$	12.7404	0.510433
$624$	0	0
$625$	14.0668	0.562671
$626$	0	0
$627$	7.71003	0.307909
$628$	0	0
$629$	−3.27465	−0.130569
$630$	0	0
$631$	43.7476	1.74156	0.870782	−	0.491670i	$-0.163613\pi$
$631$	43.7476	1.74156	0.870782	+	0.491670i	$0.163613\pi$
$632$	0	0
$633$	42.5531	1.69133
$634$	0	0
$635$	−10.5396	−0.418250
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−0.485955	−0.0192241
$640$	0	0
$641$	−23.5317	−0.929447	−0.464723	−	0.885456i	$-0.653846\pi$
$641$	−23.5317	−0.929447	−0.464723	+	0.885456i	$0.653846\pi$
$642$	0	0
$643$	−16.5075	−0.650994	−0.325497	−	0.945543i	$-0.605532\pi$
$643$	−16.5075	−0.650994	−0.325497	+	0.945543i	$0.605532\pi$
$644$	0	0
$645$	−7.87263	−0.309985
$646$	0	0
$647$	31.8670	1.25282	0.626410	−	0.779493i	$-0.284523\pi$
$647$	31.8670	1.25282	0.626410	+	0.779493i	$0.284523\pi$
$648$	0	0
$649$	1.26091	0.0494952
$650$	0	0
$651$	−1.64774	−0.0645799
$652$	0	0
$653$	3.30348	0.129275	0.0646376	−	0.997909i	$-0.479411\pi$
$653$	3.30348	0.129275	0.0646376	+	0.997909i	$0.479411\pi$
$654$	0	0
$655$	9.22305	0.360375
$656$	0	0
$657$	−0.545292	−0.0212739
$658$	0	0
$659$	−32.5088	−1.26636	−0.633182	−	0.774003i	$-0.718252\pi$
$659$	−32.5088	−1.26636	−0.633182	+	0.774003i	$0.718252\pi$
$660$	0	0
$661$	−20.5454	−0.799122	−0.399561	−	0.916707i	$-0.630837\pi$
$661$	−20.5454	−0.799122	−0.399561	+	0.916707i	$0.630837\pi$
$662$	0	0
$663$	−4.53123	−0.175978
$664$	0	0
$665$	3.86273	0.149790
$666$	0	0
$667$	43.4560	1.68262
$668$	0	0
$669$	8.90934	0.344455
$670$	0	0
$671$	9.35575	0.361175
$672$	0	0
$673$	18.6786	0.720007	0.360003	−	0.932951i	$-0.382776\pi$
$673$	18.6786	0.720007	0.360003	+	0.932951i	$0.382776\pi$
$674$	0	0
$675$	−21.7611	−0.837585
$676$	0	0
$677$	1.16023	0.0445912	0.0222956	−	0.999751i	$-0.492903\pi$
$677$	1.16023	0.0445912	0.0222956	+	0.999751i	$0.492903\pi$
$678$	0	0
$679$	2.80790	0.107757
$680$	0	0
$681$	−24.3088	−0.931516
$682$	0	0
$683$	−49.1206	−1.87955	−0.939775	−	0.341795i	$-0.888965\pi$
$683$	−49.1206	−1.87955	−0.939775	+	0.341795i	$0.888965\pi$
$684$	0	0
$685$	−7.53312	−0.287826
$686$	0	0
$687$	−13.4392	−0.512736
$688$	0	0
$689$	13.0391	0.496751
$690$	0	0
$691$	12.6527	0.481332	0.240666	−	0.970608i	$-0.422634\pi$
$691$	12.6527	0.481332	0.240666	+	0.970608i	$0.422634\pi$
$692$	0	0
$693$	0.0606454	0.00230373
$694$	0	0
$695$	18.9008	0.716949
$696$	0	0
$697$	27.9332	1.05804
$698$	0	0
$699$	−9.65779	−0.365291
$700$	0	0
$701$	−25.6716	−0.969601	−0.484801	−	0.874625i	$-0.661108\pi$
$701$	−25.6716	−0.969601	−0.484801	+	0.874625i	$0.661108\pi$
$702$	0	0
$703$	−5.57191	−0.210149
$704$	0	0
$705$	−7.81200	−0.294217
$706$	0	0
$707$	−16.9436	−0.637230
$708$	0	0
$709$	−31.9272	−1.19905	−0.599525	−	0.800356i	$-0.704644\pi$
$709$	−31.9272	−1.19905	−0.599525	+	0.800356i	$0.704644\pi$
$710$	0	0
$711$	0.395018	0.0148143
$712$	0	0
$713$	7.43105	0.278295
$714$	0	0
$715$	0.876486	0.0327787
$716$	0	0
$717$	−24.3691	−0.910080
$718$	0	0
$719$	16.2469	0.605905	0.302953	−	0.953006i	$-0.402028\pi$
$719$	16.2469	0.605905	0.302953	+	0.953006i	$0.402028\pi$
$720$	0	0
$721$	−3.02159	−0.112530
$722$	0	0
$723$	−32.6083	−1.21272
$724$	0	0
$725$	23.3079	0.865634
$726$	0	0
$727$	21.3498	0.791819	0.395909	−	0.918290i	$-0.370429\pi$
$727$	21.3498	0.791819	0.395909	+	0.918290i	$0.370429\pi$
$728$	0	0
$729$	26.4323	0.978976
$730$	0	0
$731$	−13.2978	−0.491835
$732$	0	0
$733$	16.8270	0.621521	0.310760	−	0.950488i	$-0.399416\pi$
$733$	16.8270	0.621521	0.310760	+	0.950488i	$0.399416\pi$
$734$	0	0
$735$	1.53339	0.0565598
$736$	0	0
$737$	−1.11274	−0.0409882
$738$	0	0
$739$	20.4104	0.750807	0.375404	−	0.926861i	$-0.377504\pi$
$739$	20.4104	0.750807	0.375404	+	0.926861i	$0.377504\pi$
$740$	0	0
$741$	−7.71003	−0.283235
$742$	0	0
$743$	43.8458	1.60855	0.804274	−	0.594259i	$-0.202555\pi$
$743$	43.8458	1.60855	0.804274	+	0.594259i	$0.202555\pi$
$744$	0	0
$745$	1.44798	0.0530501
$746$	0	0
$747$	−0.570068	−0.0208577
$748$	0	0
$749$	−0.609172	−0.0222586
$750$	0	0
$751$	36.3796	1.32751	0.663755	−	0.747950i	$-0.268962\pi$
$751$	36.3796	1.32751	0.663755	+	0.747950i	$0.268962\pi$
$752$	0	0
$753$	33.7711	1.23069
$754$	0	0
$755$	−6.67283	−0.242849
$756$	0	0
$757$	10.3575	0.376452	0.188226	−	0.982126i	$-0.439726\pi$
$757$	10.3575	0.376452	0.188226	+	0.982126i	$0.439726\pi$
$758$	0	0
$759$	−13.8031	−0.501019
$760$	0	0
$761$	37.5518	1.36125	0.680626	−	0.732631i	$-0.261708\pi$
$761$	37.5518	1.36125	0.680626	+	0.732631i	$0.261708\pi$
$762$	0	0
$763$	4.09440	0.148227
$764$	0	0
$765$	−0.137674	−0.00497762
$766$	0	0
$767$	−1.26091	−0.0455290
$768$	0	0
$769$	4.09530	0.147680	0.0738401	−	0.997270i	$-0.476475\pi$
$769$	4.09530	0.147680	0.0738401	+	0.997270i	$0.476475\pi$
$770$	0	0
$771$	−22.5837	−0.813332
$772$	0	0
$773$	17.3389	0.623638	0.311819	−	0.950142i	$-0.399062\pi$
$773$	17.3389	0.623638	0.311819	+	0.950142i	$0.399062\pi$
$774$	0	0
$775$	3.98569	0.143170
$776$	0	0
$777$	−2.21188	−0.0793507
$778$	0	0
$779$	47.5291	1.70291
$780$	0	0
$781$	8.01306	0.286730
$782$	0	0
$783$	−28.3230	−1.01218
$784$	0	0
$785$	1.32841	0.0474129
$786$	0	0
$787$	−1.79407	−0.0639518	−0.0319759	−	0.999489i	$-0.510180\pi$
$787$	−1.79407	−0.0639518	−0.0319759	+	0.999489i	$0.510180\pi$
$788$	0	0
$789$	−31.5023	−1.12151
$790$	0	0
$791$	8.70866	0.309644
$792$	0	0
$793$	−9.35575	−0.332232
$794$	0	0
$795$	19.9940	0.709115
$796$	0	0
$797$	6.53445	0.231462	0.115731	−	0.993281i	$-0.463079\pi$
$797$	6.53445	0.231462	0.115731	+	0.993281i	$0.463079\pi$
$798$	0	0
$799$	−13.1953	−0.466818
$800$	0	0
$801$	−0.772646	−0.0273001
$802$	0	0
$803$	8.99149	0.317303
$804$	0	0
$805$	−6.91534	−0.243734
$806$	0	0
$807$	50.7821	1.78762
$808$	0	0
$809$	−18.7177	−0.658078	−0.329039	−	0.944316i	$-0.606725\pi$
$809$	−18.7177	−0.658078	−0.329039	+	0.944316i	$0.606725\pi$
$810$	0	0
$811$	−36.8924	−1.29547	−0.647734	−	0.761867i	$-0.724283\pi$
$811$	−36.8924	−1.29547	−0.647734	+	0.761867i	$0.724283\pi$
$812$	0	0
$813$	−27.6134	−0.968445
$814$	0	0
$815$	15.6158	0.546998
$816$	0	0
$817$	−22.6265	−0.791602
$818$	0	0
$819$	−0.0606454	−0.00211912
$820$	0	0
$821$	−45.0891	−1.57362	−0.786810	−	0.617195i	$-0.788269\pi$
$821$	−45.0891	−1.57362	−0.786810	+	0.617195i	$0.788269\pi$
$822$	0	0
$823$	9.67829	0.337364	0.168682	−	0.985671i	$-0.446049\pi$
$823$	9.67829	0.337364	0.168682	+	0.985671i	$0.446049\pi$
$824$	0	0
$825$	−7.40336	−0.257752
$826$	0	0
$827$	−25.4997	−0.886712	−0.443356	−	0.896346i	$-0.646212\pi$
$827$	−25.4997	−0.886712	−0.443356	+	0.896346i	$0.646212\pi$
$828$	0	0
$829$	41.0134	1.42445	0.712227	−	0.701950i	$-0.247687\pi$
$829$	41.0134	1.42445	0.712227	+	0.701950i	$0.247687\pi$
$830$	0	0
$831$	33.0471	1.14639
$832$	0	0
$833$	2.59006	0.0897402
$834$	0	0
$835$	−4.05671	−0.140388
$836$	0	0
$837$	−4.84329	−0.167408
$838$	0	0
$839$	26.6262	0.919237	0.459618	−	0.888116i	$-0.347986\pi$
$839$	26.6262	0.919237	0.459618	+	0.888116i	$0.347986\pi$
$840$	0	0
$841$	1.33624	0.0460771
$842$	0	0
$843$	−33.4330	−1.15149
$844$	0	0
$845$	−0.876486	−0.0301520
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−51.7241	−1.77517
$850$	0	0
$851$	9.97525	0.341947
$852$	0	0
$853$	1.79213	0.0613613	0.0306807	−	0.999529i	$-0.490233\pi$
$853$	1.79213	0.0613613	0.0306807	+	0.999529i	$0.490233\pi$
$854$	0	0
$855$	−0.234257	−0.00801141
$856$	0	0
$857$	53.6709	1.83336	0.916682	−	0.399618i	$-0.130857\pi$
$857$	53.6709	1.83336	0.916682	+	0.399618i	$0.130857\pi$
$858$	0	0
$859$	30.5479	1.04228	0.521141	−	0.853471i	$-0.325507\pi$
$859$	30.5479	1.04228	0.521141	+	0.853471i	$0.325507\pi$
$860$	0	0
$861$	18.8676	0.643006
$862$	0	0
$863$	−33.5222	−1.14111	−0.570554	−	0.821260i	$-0.693272\pi$
$863$	−33.5222	−1.14111	−0.570554	+	0.821260i	$0.693272\pi$
$864$	0	0
$865$	−8.93686	−0.303862
$866$	0	0
$867$	18.0048	0.611476
$868$	0	0
$869$	−6.51357	−0.220958
$870$	0	0
$871$	1.11274	0.0377036
$872$	0	0
$873$	−0.170286	−0.00576330
$874$	0	0
$875$	−8.09152	−0.273543
$876$	0	0
$877$	−15.9375	−0.538172	−0.269086	−	0.963116i	$-0.586722\pi$
$877$	−15.9375	−0.538172	−0.269086	+	0.963116i	$0.586722\pi$
$878$	0	0
$879$	−31.1355	−1.05017
$880$	0	0
$881$	−30.2883	−1.02044	−0.510219	−	0.860044i	$-0.670436\pi$
$881$	−30.2883	−1.02044	−0.510219	+	0.860044i	$0.670436\pi$
$882$	0	0
$883$	−6.51688	−0.219310	−0.109655	−	0.993970i	$-0.534975\pi$
$883$	−6.51688	−0.219310	−0.109655	+	0.993970i	$0.534975\pi$
$884$	0	0
$885$	−1.93347	−0.0649928
$886$	0	0
$887$	−33.2276	−1.11567	−0.557836	−	0.829951i	$-0.688368\pi$
$887$	−33.2276	−1.11567	−0.557836	+	0.829951i	$0.688368\pi$
$888$	0	0
$889$	−12.0248	−0.403299
$890$	0	0
$891$	9.17826	0.307483
$892$	0	0
$893$	−22.4523	−0.751337
$894$	0	0
$895$	10.0932	0.337377
$896$	0	0
$897$	13.8031	0.460870
$898$	0	0
$899$	5.18755	0.173015
$900$	0	0
$901$	33.7721	1.12511
$902$	0	0
$903$	−8.98205	−0.298904
$904$	0	0
$905$	−5.77988	−0.192130
$906$	0	0
$907$	−13.0208	−0.432350	−0.216175	−	0.976355i	$-0.569358\pi$
$907$	−13.0208	−0.432350	−0.216175	+	0.976355i	$0.569358\pi$
$908$	0	0
$909$	1.02755	0.0340817
$910$	0	0
$911$	13.2685	0.439604	0.219802	−	0.975544i	$-0.429459\pi$
$911$	13.2685	0.439604	0.219802	+	0.975544i	$0.429459\pi$
$912$	0	0
$913$	9.40002	0.311095
$914$	0	0
$915$	−14.3460	−0.474263
$916$	0	0
$917$	10.5228	0.347492
$918$	0	0
$919$	−14.7223	−0.485644	−0.242822	−	0.970071i	$-0.578073\pi$
$919$	−14.7223	−0.485644	−0.242822	+	0.970071i	$0.578073\pi$
$920$	0	0
$921$	32.2646	1.06315
$922$	0	0
$923$	−8.01306	−0.263753
$924$	0	0
$925$	5.35029	0.175916
$926$	0	0
$927$	0.183245	0.00601856
$928$	0	0
$929$	−3.09314	−0.101483	−0.0507414	−	0.998712i	$-0.516158\pi$
$929$	−3.09314	−0.101483	−0.0507414	+	0.998712i	$0.516158\pi$
$930$	0	0
$931$	4.40706	0.144436
$932$	0	0
$933$	−28.1538	−0.921714
$934$	0	0
$935$	2.27015	0.0742419
$936$	0	0
$937$	14.1390	0.461902	0.230951	−	0.972965i	$-0.425816\pi$
$937$	14.1390	0.461902	0.230951	+	0.972965i	$0.425816\pi$
$938$	0	0
$939$	−41.6832	−1.36028
$940$	0	0
$941$	−12.9562	−0.422361	−0.211180	−	0.977447i	$-0.567731\pi$
$941$	−12.9562	−0.422361	−0.211180	+	0.977447i	$0.567731\pi$
$942$	0	0
$943$	−85.0901	−2.77091
$944$	0	0
$945$	4.50716	0.146618
$946$	0	0
$947$	−39.3097	−1.27740	−0.638698	−	0.769458i	$-0.720526\pi$
$947$	−39.3097	−1.27740	−0.638698	+	0.769458i	$0.720526\pi$
$948$	0	0
$949$	−8.99149	−0.291876
$950$	0	0
$951$	−52.8946	−1.71523
$952$	0	0
$953$	7.02560	0.227581	0.113791	−	0.993505i	$-0.463701\pi$
$953$	7.02560	0.227581	0.113791	+	0.993505i	$0.463701\pi$
$954$	0	0
$955$	4.58987	0.148525
$956$	0	0
$957$	−9.63579	−0.311481
$958$	0	0
$959$	−8.59469	−0.277537
$960$	0	0
$961$	−30.1129	−0.971384
$962$	0	0
$963$	0.0369434	0.00119049
$964$	0	0
$965$	−16.2529	−0.523199
$966$	0	0
$967$	26.0822	0.838747	0.419373	−	0.907814i	$-0.362250\pi$
$967$	26.0822	0.838747	0.419373	+	0.907814i	$0.362250\pi$
$968$	0	0
$969$	−19.9694	−0.641510
$970$	0	0
$971$	−16.8090	−0.539427	−0.269714	−	0.962941i	$-0.586929\pi$
$971$	−16.8090	−0.539427	−0.269714	+	0.962941i	$0.586929\pi$
$972$	0	0
$973$	21.5643	0.691320
$974$	0	0
$975$	7.40336	0.237097
$976$	0	0
$977$	19.7761	0.632694	0.316347	−	0.948644i	$-0.397544\pi$
$977$	19.7761	0.632694	0.316347	+	0.948644i	$0.397544\pi$
$978$	0	0
$979$	12.7404	0.407185
$980$	0	0
$981$	−0.248306	−0.00792781
$982$	0	0
$983$	60.8583	1.94108	0.970538	−	0.240946i	$-0.0774578\pi$
$983$	60.8583	1.94108	0.970538	+	0.240946i	$0.0774578\pi$
$984$	0	0
$985$	−19.7535	−0.629400
$986$	0	0
$987$	−8.91287	−0.283700
$988$	0	0
$989$	40.5077	1.28807
$990$	0	0
$991$	−8.84924	−0.281105	−0.140553	−	0.990073i	$-0.544888\pi$
$991$	−8.84924	−0.281105	−0.140553	+	0.990073i	$0.544888\pi$
$992$	0	0
$993$	−35.8752	−1.13847
$994$	0	0
$995$	−0.462345	−0.0146573
$996$	0	0
$997$	32.6952	1.03547	0.517733	−	0.855542i	$-0.326776\pi$
$997$	32.6952	1.03547	0.517733	+	0.855542i	$0.326776\pi$
$998$	0	0
$999$	−6.50150	−0.205698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.5	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.5	✓	15	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)

\(f(q)\)	\(=\)	\(q-1.74947 q^{3} -0.876486 q^{5} -1.00000 q^{7} +0.0606454 q^{9} +O(q^{10})\)	\(q-1.74947 q^{3} -0.876486 q^{5} -1.00000 q^{7} +0.0606454 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.53339 q^{15} +2.59006 q^{17} +4.40706 q^{19} +1.74947 q^{21} -7.88985 q^{23} -4.23177 q^{25} +5.14231 q^{27} -5.50783 q^{29} -0.941850 q^{31} +1.74947 q^{33} +0.876486 q^{35} -1.26431 q^{37} -1.74947 q^{39} +10.7848 q^{41} -5.13415 q^{43} -0.0531548 q^{45} -5.09461 q^{47} +1.00000 q^{49} -4.53123 q^{51} +13.0391 q^{53} +0.876486 q^{55} -7.71003 q^{57} -1.26091 q^{59} -9.35575 q^{61} -0.0606454 q^{63} -0.876486 q^{65} +1.11274 q^{67} +13.8031 q^{69} -8.01306 q^{71} -8.99149 q^{73} +7.40336 q^{75} +1.00000 q^{77} +6.51357 q^{79} -9.17826 q^{81} -9.40002 q^{83} -2.27015 q^{85} +9.63579 q^{87} -12.7404 q^{89} -1.00000 q^{91} +1.64774 q^{93} -3.86273 q^{95} -2.80790 q^{97} -0.0606454 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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