LMFDB - Embedded newform 8008.2.a.y.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.05490$ 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)

$f(q)$	$=$	$q+2.05490 q^{3} +0.929320 q^{5} +1.00000 q^{7} +1.22261 q^{9} +O(q^{10})$	$q+2.05490 q^{3} +0.929320 q^{5} +1.00000 q^{7} +1.22261 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.90966 q^{15} -5.05388 q^{17} +4.52180 q^{19} +2.05490 q^{21} -3.13093 q^{23} -4.13637 q^{25} -3.65235 q^{27} -3.89235 q^{29} -0.650557 q^{31} -2.05490 q^{33} +0.929320 q^{35} -6.58094 q^{37} -2.05490 q^{39} -11.3620 q^{41} -8.86287 q^{43} +1.13620 q^{45} +7.96562 q^{47} +1.00000 q^{49} -10.3852 q^{51} -9.49550 q^{53} -0.929320 q^{55} +9.29185 q^{57} +12.4166 q^{59} -7.70412 q^{61} +1.22261 q^{63} -0.929320 q^{65} -12.8613 q^{67} -6.43374 q^{69} -10.0543 q^{71} +15.4339 q^{73} -8.49981 q^{75} -1.00000 q^{77} -10.2793 q^{79} -11.1731 q^{81} +8.85960 q^{83} -4.69667 q^{85} -7.99838 q^{87} -1.21794 q^{89} -1.00000 q^{91} -1.33683 q^{93} +4.20220 q^{95} +6.19616 q^{97} -1.22261 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	2.05490	1.18640	0.593198	−	0.805056i	$-0.297865\pi$
$3$	2.05490	1.18640	0.593198	+	0.805056i	$0.297865\pi$
$4$	0	0
$5$	0.929320	0.415604	0.207802	−	0.978171i	$-0.433369\pi$
$5$	0.929320	0.415604	0.207802	+	0.978171i	$0.433369\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.22261	0.407537
$10$	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
$15$	1.90966	0.493072
$16$	0	0
$17$	−5.05388	−1.22575	−0.612873	−	0.790181i	$-0.709986\pi$
$17$	−5.05388	−1.22575	−0.612873	+	0.790181i	$0.709986\pi$
$18$	0	0
$19$	4.52180	1.03737	0.518686	−	0.854965i	$-0.326421\pi$
$19$	4.52180	1.03737	0.518686	+	0.854965i	$0.326421\pi$
$20$	0	0
$21$	2.05490	0.448416
$22$	0	0
$23$	−3.13093	−0.652843	−0.326422	−	0.945224i	$-0.605843\pi$
$23$	−3.13093	−0.652843	−0.326422	+	0.945224i	$0.605843\pi$
$24$	0	0
$25$	−4.13637	−0.827273
$26$	0	0
$27$	−3.65235	−0.702896
$28$	0	0
$29$	−3.89235	−0.722791	−0.361395	−	0.932413i	$-0.617700\pi$
$29$	−3.89235	−0.722791	−0.361395	+	0.932413i	$0.617700\pi$
$30$	0	0
$31$	−0.650557	−0.116844	−0.0584218	−	0.998292i	$-0.518607\pi$
$31$	−0.650557	−0.116844	−0.0584218	+	0.998292i	$0.518607\pi$
$32$	0	0
$33$	−2.05490	−0.357712
$34$	0	0
$35$	0.929320	0.157084
$36$	0	0
$37$	−6.58094	−1.08190	−0.540950	−	0.841055i	$-0.681935\pi$
$37$	−6.58094	−1.08190	−0.540950	+	0.841055i	$0.681935\pi$
$38$	0	0
$39$	−2.05490	−0.329047
$40$	0	0
$41$	−11.3620	−1.77444	−0.887220	−	0.461346i	$-0.847367\pi$
$41$	−11.3620	−1.77444	−0.887220	+	0.461346i	$0.847367\pi$
$42$	0	0
$43$	−8.86287	−1.35158	−0.675788	−	0.737096i	$-0.736196\pi$
$43$	−8.86287	−1.35158	−0.675788	+	0.737096i	$0.736196\pi$
$44$	0	0
$45$	1.13620	0.169374
$46$	0	0
$47$	7.96562	1.16191	0.580953	−	0.813937i	$-0.302680\pi$
$47$	7.96562	1.16191	0.580953	+	0.813937i	$0.302680\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.3852	−1.45422
$52$	0	0
$53$	−9.49550	−1.30431	−0.652154	−	0.758087i	$-0.726134\pi$
$53$	−9.49550	−1.30431	−0.652154	+	0.758087i	$0.726134\pi$
$54$	0	0
$55$	−0.929320	−0.125309
$56$	0	0
$57$	9.29185	1.23074
$58$	0	0
$59$	12.4166	1.61650	0.808251	−	0.588838i	$-0.200414\pi$
$59$	12.4166	1.61650	0.808251	+	0.588838i	$0.200414\pi$
$60$	0	0
$61$	−7.70412	−0.986411	−0.493206	−	0.869913i	$-0.664175\pi$
$61$	−7.70412	−0.986411	−0.493206	+	0.869913i	$0.664175\pi$
$62$	0	0
$63$	1.22261	0.154035
$64$	0	0
$65$	−0.929320	−0.115268
$66$	0	0
$67$	−12.8613	−1.57125	−0.785626	−	0.618702i	$-0.787659\pi$
$67$	−12.8613	−1.57125	−0.785626	+	0.618702i	$0.787659\pi$
$68$	0	0
$69$	−6.43374	−0.774531
$70$	0	0
$71$	−10.0543	−1.19322	−0.596612	−	0.802530i	$-0.703487\pi$
$71$	−10.0543	−1.19322	−0.596612	+	0.802530i	$0.703487\pi$
$72$	0	0
$73$	15.4339	1.80640	0.903199	−	0.429222i	$-0.141212\pi$
$73$	15.4339	1.80640	0.903199	+	0.429222i	$0.141212\pi$
$74$	0	0
$75$	−8.49981	−0.981474
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.2793	−1.15652	−0.578258	−	0.815854i	$-0.696267\pi$
$79$	−10.2793	−1.15652	−0.578258	+	0.815854i	$0.696267\pi$
$80$	0	0
$81$	−11.1731	−1.24145
$82$	0	0
$83$	8.85960	0.972468	0.486234	−	0.873829i	$-0.338370\pi$
$83$	8.85960	0.972468	0.486234	+	0.873829i	$0.338370\pi$
$84$	0	0
$85$	−4.69667	−0.509425
$86$	0	0
$87$	−7.99838	−0.857517
$88$	0	0
$89$	−1.21794	−0.129101	−0.0645506	−	0.997914i	$-0.520561\pi$
$89$	−1.21794	−0.129101	−0.0645506	+	0.997914i	$0.520561\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−1.33683	−0.138623
$94$	0	0
$95$	4.20220	0.431136
$96$	0	0
$97$	6.19616	0.629125	0.314563	−	0.949237i	$-0.398142\pi$
$97$	6.19616	0.629125	0.314563	+	0.949237i	$0.398142\pi$
$98$	0	0
$99$	−1.22261	−0.122877
$100$	0	0
$101$	−4.27848	−0.425725	−0.212863	−	0.977082i	$-0.568279\pi$
$101$	−4.27848	−0.425725	−0.212863	+	0.977082i	$0.568279\pi$
$102$	0	0
$103$	17.0901	1.68393	0.841966	−	0.539530i	$-0.181398\pi$
$103$	17.0901	1.68393	0.841966	+	0.539530i	$0.181398\pi$
$104$	0	0
$105$	1.90966	0.186364
$106$	0	0
$107$	16.9650	1.64006	0.820032	−	0.572318i	$-0.193956\pi$
$107$	16.9650	1.64006	0.820032	+	0.572318i	$0.193956\pi$
$108$	0	0
$109$	14.2886	1.36860	0.684298	−	0.729202i	$-0.260109\pi$
$109$	14.2886	1.36860	0.684298	+	0.729202i	$0.260109\pi$
$110$	0	0
$111$	−13.5232	−1.28356
$112$	0	0
$113$	−5.28934	−0.497579	−0.248790	−	0.968558i	$-0.580033\pi$
$113$	−5.28934	−0.497579	−0.248790	+	0.968558i	$0.580033\pi$
$114$	0	0
$115$	−2.90963	−0.271325
$116$	0	0
$117$	−1.22261	−0.113031
$118$	0	0
$119$	−5.05388	−0.463289
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−23.3477	−2.10519
$124$	0	0
$125$	−8.49060	−0.759423
$126$	0	0
$127$	5.76890	0.511907	0.255953	−	0.966689i	$-0.417611\pi$
$127$	5.76890	0.511907	0.255953	+	0.966689i	$0.417611\pi$
$128$	0	0
$129$	−18.2123	−1.60350
$130$	0	0
$131$	−11.3283	−0.989762	−0.494881	−	0.868961i	$-0.664788\pi$
$131$	−11.3283	−0.989762	−0.494881	+	0.868961i	$0.664788\pi$
$132$	0	0
$133$	4.52180	0.392090
$134$	0	0
$135$	−3.39420	−0.292127
$136$	0	0
$137$	−11.6510	−0.995409	−0.497704	−	0.867347i	$-0.665824\pi$
$137$	−11.6510	−0.995409	−0.497704	+	0.867347i	$0.665824\pi$
$138$	0	0
$139$	−8.37197	−0.710101	−0.355051	−	0.934847i	$-0.615536\pi$
$139$	−8.37197	−0.710101	−0.355051	+	0.934847i	$0.615536\pi$
$140$	0	0
$141$	16.3685	1.37848
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−3.61724	−0.300395
$146$	0	0
$147$	2.05490	0.169485
$148$	0	0
$149$	12.9193	1.05839	0.529196	−	0.848499i	$-0.322494\pi$
$149$	12.9193	1.05839	0.529196	+	0.848499i	$0.322494\pi$
$150$	0	0
$151$	13.1077	1.06669	0.533346	−	0.845897i	$-0.320934\pi$
$151$	13.1077	1.06669	0.533346	+	0.845897i	$0.320934\pi$
$152$	0	0
$153$	−6.17894	−0.499537
$154$	0	0
$155$	−0.604575	−0.0485607
$156$	0	0
$157$	8.26949	0.659978	0.329989	−	0.943985i	$-0.392955\pi$
$157$	8.26949	0.659978	0.329989	+	0.943985i	$0.392955\pi$
$158$	0	0
$159$	−19.5123	−1.54743
$160$	0	0
$161$	−3.13093	−0.246752
$162$	0	0
$163$	−4.91977	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$163$	−4.91977	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$164$	0	0
$165$	−1.90966	−0.148667
$166$	0	0
$167$	3.44439	0.266535	0.133267	−	0.991080i	$-0.457453\pi$
$167$	3.44439	0.266535	0.133267	+	0.991080i	$0.457453\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.52841	0.422768
$172$	0	0
$173$	16.2236	1.23346	0.616729	−	0.787176i	$-0.288457\pi$
$173$	16.2236	1.23346	0.616729	+	0.787176i	$0.288457\pi$
$174$	0	0
$175$	−4.13637	−0.312680
$176$	0	0
$177$	25.5148	1.91781
$178$	0	0
$179$	17.4581	1.30488	0.652441	−	0.757839i	$-0.273745\pi$
$179$	17.4581	1.30488	0.652441	+	0.757839i	$0.273745\pi$
$180$	0	0
$181$	−7.20535	−0.535569	−0.267785	−	0.963479i	$-0.586292\pi$
$181$	−7.20535	−0.535569	−0.267785	+	0.963479i	$0.586292\pi$
$182$	0	0
$183$	−15.8312	−1.17028
$184$	0	0
$185$	−6.11580	−0.449642
$186$	0	0
$187$	5.05388	0.369576
$188$	0	0
$189$	−3.65235	−0.265670
$190$	0	0
$191$	−21.7213	−1.57170	−0.785848	−	0.618420i	$-0.787773\pi$
$191$	−21.7213	−1.57170	−0.785848	+	0.618420i	$0.787773\pi$
$192$	0	0
$193$	4.29040	0.308829	0.154415	−	0.988006i	$-0.450651\pi$
$193$	4.29040	0.308829	0.154415	+	0.988006i	$0.450651\pi$
$194$	0	0
$195$	−1.90966	−0.136753
$196$	0	0
$197$	−6.83071	−0.486668	−0.243334	−	0.969943i	$-0.578241\pi$
$197$	−6.83071	−0.486668	−0.243334	+	0.969943i	$0.578241\pi$
$198$	0	0
$199$	8.16560	0.578844	0.289422	−	0.957202i	$-0.406537\pi$
$199$	8.16560	0.578844	0.289422	+	0.957202i	$0.406537\pi$
$200$	0	0
$201$	−26.4286	−1.86413
$202$	0	0
$203$	−3.89235	−0.273189
$204$	0	0
$205$	−10.5589	−0.737465
$206$	0	0
$207$	−3.82791	−0.266058
$208$	0	0
$209$	−4.52180	−0.312779
$210$	0	0
$211$	6.07057	0.417915	0.208958	−	0.977925i	$-0.432993\pi$
$211$	6.07057	0.417915	0.208958	+	0.977925i	$0.432993\pi$
$212$	0	0
$213$	−20.6605	−1.41564
$214$	0	0
$215$	−8.23644	−0.561721
$216$	0	0
$217$	−0.650557	−0.0441627
$218$	0	0
$219$	31.7151	2.14310
$220$	0	0
$221$	5.05388	0.339961
$222$	0	0
$223$	−11.1208	−0.744705	−0.372353	−	0.928091i	$-0.621449\pi$
$223$	−11.1208	−0.744705	−0.372353	+	0.928091i	$0.621449\pi$
$224$	0	0
$225$	−5.05717	−0.337145
$226$	0	0
$227$	8.64172	0.573571	0.286786	−	0.957995i	$-0.407413\pi$
$227$	8.64172	0.573571	0.286786	+	0.957995i	$0.407413\pi$
$228$	0	0
$229$	−23.3645	−1.54397	−0.771984	−	0.635642i	$-0.780735\pi$
$229$	−23.3645	−1.54397	−0.771984	+	0.635642i	$0.780735\pi$
$230$	0	0
$231$	−2.05490	−0.135202
$232$	0	0
$233$	−25.3459	−1.66046	−0.830231	−	0.557419i	$-0.811792\pi$
$233$	−25.3459	−1.66046	−0.830231	+	0.557419i	$0.811792\pi$
$234$	0	0
$235$	7.40261	0.482893
$236$	0	0
$237$	−21.1230	−1.37209
$238$	0	0
$239$	−22.5131	−1.45625	−0.728125	−	0.685444i	$-0.759608\pi$
$239$	−22.5131	−1.45625	−0.728125	+	0.685444i	$0.759608\pi$
$240$	0	0
$241$	16.5802	1.06803	0.534013	−	0.845476i	$-0.320683\pi$
$241$	16.5802	1.06803	0.534013	+	0.845476i	$0.320683\pi$
$242$	0	0
$243$	−12.0024	−0.769957
$244$	0	0
$245$	0.929320	0.0593720
$246$	0	0
$247$	−4.52180	−0.287715
$248$	0	0
$249$	18.2056	1.15373
$250$	0	0
$251$	−10.6471	−0.672037	−0.336019	−	0.941855i	$-0.609080\pi$
$251$	−10.6471	−0.672037	−0.336019	+	0.941855i	$0.609080\pi$
$252$	0	0
$253$	3.13093	0.196840
$254$	0	0
$255$	−9.65119	−0.604381
$256$	0	0
$257$	5.22557	0.325962	0.162981	−	0.986629i	$-0.447889\pi$
$257$	5.22557	0.325962	0.162981	+	0.986629i	$0.447889\pi$
$258$	0	0
$259$	−6.58094	−0.408920
$260$	0	0
$261$	−4.75883	−0.294564
$262$	0	0
$263$	−8.00459	−0.493584	−0.246792	−	0.969068i	$-0.579376\pi$
$263$	−8.00459	−0.493584	−0.246792	+	0.969068i	$0.579376\pi$
$264$	0	0
$265$	−8.82436	−0.542076
$266$	0	0
$267$	−2.50274	−0.153165
$268$	0	0
$269$	11.9168	0.726581	0.363291	−	0.931676i	$-0.381653\pi$
$269$	11.9168	0.726581	0.363291	+	0.931676i	$0.381653\pi$
$270$	0	0
$271$	10.1920	0.619122	0.309561	−	0.950880i	$-0.399818\pi$
$271$	10.1920	0.619122	0.309561	+	0.950880i	$0.399818\pi$
$272$	0	0
$273$	−2.05490	−0.124368
$274$	0	0
$275$	4.13637	0.249432
$276$	0	0
$277$	17.5066	1.05187	0.525934	−	0.850526i	$-0.323716\pi$
$277$	17.5066	1.05187	0.525934	+	0.850526i	$0.323716\pi$
$278$	0	0
$279$	−0.795379	−0.0476181
$280$	0	0
$281$	−6.56131	−0.391415	−0.195707	−	0.980662i	$-0.562700\pi$
$281$	−6.56131	−0.391415	−0.195707	+	0.980662i	$0.562700\pi$
$282$	0	0
$283$	5.00778	0.297681	0.148841	−	0.988861i	$-0.452446\pi$
$283$	5.00778	0.297681	0.148841	+	0.988861i	$0.452446\pi$
$284$	0	0
$285$	8.63509	0.511499
$286$	0	0
$287$	−11.3620	−0.670676
$288$	0	0
$289$	8.54172	0.502454
$290$	0	0
$291$	12.7325	0.746392
$292$	0	0
$293$	16.8927	0.986883	0.493441	−	0.869779i	$-0.335739\pi$
$293$	16.8927	0.986883	0.493441	+	0.869779i	$0.335739\pi$
$294$	0	0
$295$	11.5390	0.671825
$296$	0	0
$297$	3.65235	0.211931
$298$	0	0
$299$	3.13093	0.181066
$300$	0	0
$301$	−8.86287	−0.510847
$302$	0	0
$303$	−8.79186	−0.505079
$304$	0	0
$305$	−7.15959	−0.409957
$306$	0	0
$307$	5.17232	0.295200	0.147600	−	0.989047i	$-0.452845\pi$
$307$	5.17232	0.295200	0.147600	+	0.989047i	$0.452845\pi$
$308$	0	0
$309$	35.1183	1.99781
$310$	0	0
$311$	13.9152	0.789059	0.394530	−	0.918883i	$-0.370908\pi$
$311$	13.9152	0.789059	0.394530	+	0.918883i	$0.370908\pi$
$312$	0	0
$313$	15.9020	0.898831	0.449416	−	0.893323i	$-0.351632\pi$
$313$	15.9020	0.898831	0.449416	+	0.893323i	$0.351632\pi$
$314$	0	0
$315$	1.13620	0.0640175
$316$	0	0
$317$	−14.6516	−0.822918	−0.411459	−	0.911428i	$-0.634981\pi$
$317$	−14.6516	−0.822918	−0.411459	+	0.911428i	$0.634981\pi$
$318$	0	0
$319$	3.89235	0.217930
$320$	0	0
$321$	34.8613	1.94577
$322$	0	0
$323$	−22.8526	−1.27156
$324$	0	0
$325$	4.13637	0.229444
$326$	0	0
$327$	29.3616	1.62370
$328$	0	0
$329$	7.96562	0.439159
$330$	0	0
$331$	24.5219	1.34784	0.673922	−	0.738803i	$-0.264608\pi$
$331$	24.5219	1.34784	0.673922	+	0.738803i	$0.264608\pi$
$332$	0	0
$333$	−8.04594	−0.440915
$334$	0	0
$335$	−11.9522	−0.653019
$336$	0	0
$337$	−0.125969	−0.00686198	−0.00343099	−	0.999994i	$-0.501092\pi$
$337$	−0.125969	−0.00686198	−0.00343099	+	0.999994i	$0.501092\pi$
$338$	0	0
$339$	−10.8691	−0.590327
$340$	0	0
$341$	0.650557	0.0352296
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.97900	−0.321899
$346$	0	0
$347$	−34.1881	−1.83531	−0.917656	−	0.397375i	$-0.869921\pi$
$347$	−34.1881	−1.83531	−0.917656	+	0.397375i	$0.869921\pi$
$348$	0	0
$349$	4.95886	0.265442	0.132721	−	0.991153i	$-0.457629\pi$
$349$	4.95886	0.265442	0.132721	+	0.991153i	$0.457629\pi$
$350$	0	0
$351$	3.65235	0.194948
$352$	0	0
$353$	−29.1296	−1.55041	−0.775207	−	0.631707i	$-0.782355\pi$
$353$	−29.1296	−1.55041	−0.775207	+	0.631707i	$0.782355\pi$
$354$	0	0
$355$	−9.34364	−0.495909
$356$	0	0
$357$	−10.3852	−0.549644
$358$	0	0
$359$	1.21670	0.0642150	0.0321075	−	0.999484i	$-0.489778\pi$
$359$	1.21670	0.0642150	0.0321075	+	0.999484i	$0.489778\pi$
$360$	0	0
$361$	1.44668	0.0761411
$362$	0	0
$363$	2.05490	0.107854
$364$	0	0
$365$	14.3430	0.750747
$366$	0	0
$367$	−14.4543	−0.754508	−0.377254	−	0.926110i	$-0.623132\pi$
$367$	−14.4543	−0.754508	−0.377254	+	0.926110i	$0.623132\pi$
$368$	0	0
$369$	−13.8913	−0.723151
$370$	0	0
$371$	−9.49550	−0.492982
$372$	0	0
$373$	−22.7743	−1.17921	−0.589605	−	0.807692i	$-0.700716\pi$
$373$	−22.7743	−1.17921	−0.589605	+	0.807692i	$0.700716\pi$
$374$	0	0
$375$	−17.4473	−0.900977
$376$	0	0
$377$	3.89235	0.200466
$378$	0	0
$379$	12.4702	0.640551	0.320275	−	0.947324i	$-0.396225\pi$
$379$	12.4702	0.640551	0.320275	+	0.947324i	$0.396225\pi$
$380$	0	0
$381$	11.8545	0.607325
$382$	0	0
$383$	21.2679	1.08674	0.543371	−	0.839493i	$-0.317148\pi$
$383$	21.2679	1.08674	0.543371	+	0.839493i	$0.317148\pi$
$384$	0	0
$385$	−0.929320	−0.0473625
$386$	0	0
$387$	−10.8359	−0.550817
$388$	0	0
$389$	31.7751	1.61106	0.805530	−	0.592555i	$-0.201881\pi$
$389$	31.7751	1.61106	0.805530	+	0.592555i	$0.201881\pi$
$390$	0	0
$391$	15.8233	0.800220
$392$	0	0
$393$	−23.2786	−1.17425
$394$	0	0
$395$	−9.55280	−0.480653
$396$	0	0
$397$	−12.8096	−0.642895	−0.321448	−	0.946927i	$-0.604169\pi$
$397$	−12.8096	−0.642895	−0.321448	+	0.946927i	$0.604169\pi$
$398$	0	0
$399$	9.29185	0.465174
$400$	0	0
$401$	−34.6309	−1.72939	−0.864693	−	0.502300i	$-0.832487\pi$
$401$	−34.6309	−1.72939	−0.864693	+	0.502300i	$0.832487\pi$
$402$	0	0
$403$	0.650557	0.0324066
$404$	0	0
$405$	−10.3833	−0.515952
$406$	0	0
$407$	6.58094	0.326205
$408$	0	0
$409$	−35.9173	−1.77600	−0.887998	−	0.459847i	$-0.847904\pi$
$409$	−35.9173	−1.77600	−0.887998	+	0.459847i	$0.847904\pi$
$410$	0	0
$411$	−23.9416	−1.18095
$412$	0	0
$413$	12.4166	0.610980
$414$	0	0
$415$	8.23340	0.404162
$416$	0	0
$417$	−17.2036	−0.842462
$418$	0	0
$419$	−19.0102	−0.928708	−0.464354	−	0.885650i	$-0.653713\pi$
$419$	−19.0102	−0.928708	−0.464354	+	0.885650i	$0.653713\pi$
$420$	0	0
$421$	31.2677	1.52389	0.761947	−	0.647639i	$-0.224244\pi$
$421$	31.2677	1.52389	0.761947	+	0.647639i	$0.224244\pi$
$422$	0	0
$423$	9.73886	0.473520
$424$	0	0
$425$	20.9047	1.01403
$426$	0	0
$427$	−7.70412	−0.372828
$428$	0	0
$429$	2.05490	0.0992115
$430$	0	0
$431$	−14.0409	−0.676326	−0.338163	−	0.941088i	$-0.609805\pi$
$431$	−14.0409	−0.676326	−0.338163	+	0.941088i	$0.609805\pi$
$432$	0	0
$433$	19.8510	0.953980	0.476990	−	0.878909i	$-0.341728\pi$
$433$	19.8510	0.953980	0.476990	+	0.878909i	$0.341728\pi$
$434$	0	0
$435$	−7.43305	−0.356388
$436$	0	0
$437$	−14.1574	−0.677242
$438$	0	0
$439$	21.7936	1.04015	0.520076	−	0.854120i	$-0.325904\pi$
$439$	21.7936	1.04015	0.520076	+	0.854120i	$0.325904\pi$
$440$	0	0
$441$	1.22261	0.0582196
$442$	0	0
$443$	−21.4482	−1.01903	−0.509516	−	0.860461i	$-0.670176\pi$
$443$	−21.4482	−1.01903	−0.509516	+	0.860461i	$0.670176\pi$
$444$	0	0
$445$	−1.13185	−0.0536550
$446$	0	0
$447$	26.5479	1.25567
$448$	0	0
$449$	29.1215	1.37433	0.687164	−	0.726503i	$-0.258856\pi$
$449$	29.1215	1.37433	0.687164	+	0.726503i	$0.258856\pi$
$450$	0	0
$451$	11.3620	0.535014
$452$	0	0
$453$	26.9351	1.26552
$454$	0	0
$455$	−0.929320	−0.0435672
$456$	0	0
$457$	−30.7601	−1.43890	−0.719449	−	0.694545i	$-0.755606\pi$
$457$	−30.7601	−1.43890	−0.719449	+	0.694545i	$0.755606\pi$
$458$	0	0
$459$	18.4586	0.861572
$460$	0	0
$461$	7.34317	0.342006	0.171003	−	0.985271i	$-0.445299\pi$
$461$	7.34317	0.342006	0.171003	+	0.985271i	$0.445299\pi$
$462$	0	0
$463$	3.40181	0.158095	0.0790477	−	0.996871i	$-0.474812\pi$
$463$	3.40181	0.158095	0.0790477	+	0.996871i	$0.474812\pi$
$464$	0	0
$465$	−1.24234	−0.0576122
$466$	0	0
$467$	−2.03680	−0.0942521	−0.0471260	−	0.998889i	$-0.515006\pi$
$467$	−2.03680	−0.0942521	−0.0471260	+	0.998889i	$0.515006\pi$
$468$	0	0
$469$	−12.8613	−0.593877
$470$	0	0
$471$	16.9930	0.782995
$472$	0	0
$473$	8.86287	0.407515
$474$	0	0
$475$	−18.7038	−0.858190
$476$	0	0
$477$	−11.6093	−0.531554
$478$	0	0
$479$	12.5052	0.571379	0.285690	−	0.958322i	$-0.407777\pi$
$479$	12.5052	0.571379	0.285690	+	0.958322i	$0.407777\pi$
$480$	0	0
$481$	6.58094	0.300065
$482$	0	0
$483$	−6.43374	−0.292745
$484$	0	0
$485$	5.75822	0.261467
$486$	0	0
$487$	−14.7529	−0.668518	−0.334259	−	0.942481i	$-0.608486\pi$
$487$	−14.7529	−0.668518	−0.334259	+	0.942481i	$0.608486\pi$
$488$	0	0
$489$	−10.1096	−0.457173
$490$	0	0
$491$	−0.0293564	−0.00132483	−0.000662417	−	1.00000i	$-0.500211\pi$
$491$	−0.0293564	−0.00132483	−0.000662417		1.00000i	$0.500211\pi$
$492$	0	0
$493$	19.6715	0.885958
$494$	0	0
$495$	−1.13620	−0.0510683
$496$	0	0
$497$	−10.0543	−0.450996
$498$	0	0
$499$	−31.6586	−1.41723	−0.708617	−	0.705593i	$-0.750681\pi$
$499$	−31.6586	−1.41723	−0.708617	+	0.705593i	$0.750681\pi$
$500$	0	0
$501$	7.07787	0.316216
$502$	0	0
$503$	6.47377	0.288651	0.144326	−	0.989530i	$-0.453899\pi$
$503$	6.47377	0.288651	0.144326	+	0.989530i	$0.453899\pi$
$504$	0	0
$505$	−3.97608	−0.176933
$506$	0	0
$507$	2.05490	0.0912613
$508$	0	0
$509$	26.2339	1.16280	0.581400	−	0.813618i	$-0.302505\pi$
$509$	26.2339	1.16280	0.581400	+	0.813618i	$0.302505\pi$
$510$	0	0
$511$	15.4339	0.682754
$512$	0	0
$513$	−16.5152	−0.729165
$514$	0	0
$515$	15.8821	0.699850
$516$	0	0
$517$	−7.96562	−0.350328
$518$	0	0
$519$	33.3379	1.46337
$520$	0	0
$521$	−2.50570	−0.109777	−0.0548883	−	0.998492i	$-0.517480\pi$
$521$	−2.50570	−0.109777	−0.0548883	+	0.998492i	$0.517480\pi$
$522$	0	0
$523$	−0.325724	−0.0142429	−0.00712146	−	0.999975i	$-0.502267\pi$
$523$	−0.325724	−0.0142429	−0.00712146	+	0.999975i	$0.502267\pi$
$524$	0	0
$525$	−8.49981	−0.370962
$526$	0	0
$527$	3.28784	0.143220
$528$	0	0
$529$	−13.1973	−0.573796
$530$	0	0
$531$	15.1807	0.658785
$532$	0	0
$533$	11.3620	0.492141
$534$	0	0
$535$	15.7659	0.681618
$536$	0	0
$537$	35.8747	1.54811
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−32.7800	−1.40932	−0.704662	−	0.709543i	$-0.748901\pi$
$541$	−32.7800	−1.40932	−0.704662	+	0.709543i	$0.748901\pi$
$542$	0	0
$543$	−14.8063	−0.635398
$544$	0	0
$545$	13.2786	0.568795
$546$	0	0
$547$	22.5393	0.963710	0.481855	−	0.876251i	$-0.339963\pi$
$547$	22.5393	0.963710	0.481855	+	0.876251i	$0.339963\pi$
$548$	0	0
$549$	−9.41915	−0.401999
$550$	0	0
$551$	−17.6004	−0.749803
$552$	0	0
$553$	−10.2793	−0.437122
$554$	0	0
$555$	−12.5673	−0.533454
$556$	0	0
$557$	−37.0145	−1.56835	−0.784177	−	0.620538i	$-0.786914\pi$
$557$	−37.0145	−1.56835	−0.784177	+	0.620538i	$0.786914\pi$
$558$	0	0
$559$	8.86287	0.374860
$560$	0	0
$561$	10.3852	0.438464
$562$	0	0
$563$	10.8927	0.459074	0.229537	−	0.973300i	$-0.426279\pi$
$563$	10.8927	0.459074	0.229537	+	0.973300i	$0.426279\pi$
$564$	0	0
$565$	−4.91549	−0.206796
$566$	0	0
$567$	−11.1731	−0.469224
$568$	0	0
$569$	−10.1624	−0.426028	−0.213014	−	0.977049i	$-0.568328\pi$
$569$	−10.1624	−0.426028	−0.213014	+	0.977049i	$0.568328\pi$
$570$	0	0
$571$	−22.7132	−0.950516	−0.475258	−	0.879846i	$-0.657645\pi$
$571$	−22.7132	−0.950516	−0.475258	+	0.879846i	$0.657645\pi$
$572$	0	0
$573$	−44.6350	−1.86466
$574$	0	0
$575$	12.9507	0.540080
$576$	0	0
$577$	−13.2471	−0.551485	−0.275743	−	0.961231i	$-0.588924\pi$
$577$	−13.2471	−0.551485	−0.275743	+	0.961231i	$0.588924\pi$
$578$	0	0
$579$	8.81633	0.366394
$580$	0	0
$581$	8.85960	0.367558
$582$	0	0
$583$	9.49550	0.393264
$584$	0	0
$585$	−1.13620	−0.0469760
$586$	0	0
$587$	−17.7195	−0.731361	−0.365680	−	0.930741i	$-0.619164\pi$
$587$	−17.7195	−0.731361	−0.365680	+	0.930741i	$0.619164\pi$
$588$	0	0
$589$	−2.94169	−0.121210
$590$	0	0
$591$	−14.0364	−0.577382
$592$	0	0
$593$	15.0647	0.618635	0.309317	−	0.950959i	$-0.399899\pi$
$593$	15.0647	0.618635	0.309317	+	0.950959i	$0.399899\pi$
$594$	0	0
$595$	−4.69667	−0.192545
$596$	0	0
$597$	16.7795	0.686739
$598$	0	0
$599$	−7.88585	−0.322207	−0.161103	−	0.986938i	$-0.551505\pi$
$599$	−7.88585	−0.322207	−0.161103	+	0.986938i	$0.551505\pi$
$600$	0	0
$601$	−7.00278	−0.285649	−0.142825	−	0.989748i	$-0.545619\pi$
$601$	−7.00278	−0.285649	−0.142825	+	0.989748i	$0.545619\pi$
$602$	0	0
$603$	−15.7243	−0.640344
$604$	0	0
$605$	0.929320	0.0377822
$606$	0	0
$607$	5.40076	0.219210	0.109605	−	0.993975i	$-0.465041\pi$
$607$	5.40076	0.219210	0.109605	+	0.993975i	$0.465041\pi$
$608$	0	0
$609$	−7.99838	−0.324111
$610$	0	0
$611$	−7.96562	−0.322254
$612$	0	0
$613$	27.8522	1.12494	0.562470	−	0.826817i	$-0.309851\pi$
$613$	27.8522	1.12494	0.562470	+	0.826817i	$0.309851\pi$
$614$	0	0
$615$	−21.6975	−0.874927
$616$	0	0
$617$	33.6574	1.35499	0.677497	−	0.735525i	$-0.263065\pi$
$617$	33.6574	1.35499	0.677497	+	0.735525i	$0.263065\pi$
$618$	0	0
$619$	−39.0524	−1.56965	−0.784824	−	0.619719i	$-0.787246\pi$
$619$	−39.0524	−1.56965	−0.784824	+	0.619719i	$0.787246\pi$
$620$	0	0
$621$	11.4353	0.458881
$622$	0	0
$623$	−1.21794	−0.0487957
$624$	0	0
$625$	12.7913	0.511654
$626$	0	0
$627$	−9.29185	−0.371081
$628$	0	0
$629$	33.2593	1.32613
$630$	0	0
$631$	35.8663	1.42781	0.713907	−	0.700240i	$-0.246924\pi$
$631$	35.8663	1.42781	0.713907	+	0.700240i	$0.246924\pi$
$632$	0	0
$633$	12.4744	0.495814
$634$	0	0
$635$	5.36115	0.212751
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−12.2925	−0.486283
$640$	0	0
$641$	−20.1431	−0.795606	−0.397803	−	0.917471i	$-0.630227\pi$
$641$	−20.1431	−0.795606	−0.397803	+	0.917471i	$0.630227\pi$
$642$	0	0
$643$	−40.8212	−1.60983	−0.804916	−	0.593389i	$-0.797790\pi$
$643$	−40.8212	−1.60983	−0.804916	+	0.593389i	$0.797790\pi$
$644$	0	0
$645$	−16.9251	−0.666423
$646$	0	0
$647$	−4.47095	−0.175771	−0.0878857	−	0.996131i	$-0.528011\pi$
$647$	−4.47095	−0.175771	−0.0878857	+	0.996131i	$0.528011\pi$
$648$	0	0
$649$	−12.4166	−0.487394
$650$	0	0
$651$	−1.33683	−0.0523945
$652$	0	0
$653$	8.15206	0.319015	0.159507	−	0.987197i	$-0.449009\pi$
$653$	8.15206	0.319015	0.159507	+	0.987197i	$0.449009\pi$
$654$	0	0
$655$	−10.5276	−0.411349
$656$	0	0
$657$	18.8696	0.736175
$658$	0	0
$659$	27.7186	1.07976	0.539881	−	0.841741i	$-0.318469\pi$
$659$	27.7186	1.07976	0.539881	+	0.841741i	$0.318469\pi$
$660$	0	0
$661$	14.5438	0.565689	0.282845	−	0.959166i	$-0.408722\pi$
$661$	14.5438	0.565689	0.282845	+	0.959166i	$0.408722\pi$
$662$	0	0
$663$	10.3852	0.403328
$664$	0	0
$665$	4.20220	0.162954
$666$	0	0
$667$	12.1867	0.471869
$668$	0	0
$669$	−22.8522	−0.883516
$670$	0	0
$671$	7.70412	0.297414
$672$	0	0
$673$	−7.94442	−0.306235	−0.153117	−	0.988208i	$-0.548931\pi$
$673$	−7.94442	−0.306235	−0.153117	+	0.988208i	$0.548931\pi$
$674$	0	0
$675$	15.1075	0.581487
$676$	0	0
$677$	−13.9244	−0.535159	−0.267579	−	0.963536i	$-0.586224\pi$
$677$	−13.9244	−0.535159	−0.267579	+	0.963536i	$0.586224\pi$
$678$	0	0
$679$	6.19616	0.237787
$680$	0	0
$681$	17.7579	0.680483
$682$	0	0
$683$	−25.1515	−0.962396	−0.481198	−	0.876612i	$-0.659798\pi$
$683$	−25.1515	−0.962396	−0.481198	+	0.876612i	$0.659798\pi$
$684$	0	0
$685$	−10.8275	−0.413696
$686$	0	0
$687$	−48.0116	−1.83176
$688$	0	0
$689$	9.49550	0.361750
$690$	0	0
$691$	−1.16034	−0.0441416	−0.0220708	−	0.999756i	$-0.507026\pi$
$691$	−1.16034	−0.0441416	−0.0220708	+	0.999756i	$0.507026\pi$
$692$	0	0
$693$	−1.22261	−0.0464432
$694$	0	0
$695$	−7.78023	−0.295121
$696$	0	0
$697$	57.4220	2.17501
$698$	0	0
$699$	−52.0832	−1.96997
$700$	0	0
$701$	20.8732	0.788370	0.394185	−	0.919031i	$-0.371027\pi$
$701$	20.8732	0.788370	0.394185	+	0.919031i	$0.371027\pi$
$702$	0	0
$703$	−29.7577	−1.12233
$704$	0	0
$705$	15.2116	0.572902
$706$	0	0
$707$	−4.27848	−0.160909
$708$	0	0
$709$	12.3585	0.464132	0.232066	−	0.972700i	$-0.425451\pi$
$709$	12.3585	0.464132	0.232066	+	0.972700i	$0.425451\pi$
$710$	0	0
$711$	−12.5677	−0.471324
$712$	0	0
$713$	2.03685	0.0762805
$714$	0	0
$715$	0.929320	0.0347546
$716$	0	0
$717$	−46.2621	−1.72769
$718$	0	0
$719$	−19.2310	−0.717195	−0.358598	−	0.933492i	$-0.616745\pi$
$719$	−19.2310	−0.717195	−0.358598	+	0.933492i	$0.616745\pi$
$720$	0	0
$721$	17.0901	0.636467
$722$	0	0
$723$	34.0707	1.26710
$724$	0	0
$725$	16.1002	0.597945
$726$	0	0
$727$	9.36247	0.347235	0.173617	−	0.984813i	$-0.444454\pi$
$727$	9.36247	0.347235	0.173617	+	0.984813i	$0.444454\pi$
$728$	0	0
$729$	8.85535	0.327976
$730$	0	0
$731$	44.7919	1.65669
$732$	0	0
$733$	−34.4130	−1.27107	−0.635537	−	0.772070i	$-0.719221\pi$
$733$	−34.4130	−1.27107	−0.635537	+	0.772070i	$0.719221\pi$
$734$	0	0
$735$	1.90966	0.0704388
$736$	0	0
$737$	12.8613	0.473750
$738$	0	0
$739$	13.6474	0.502027	0.251013	−	0.967984i	$-0.419236\pi$
$739$	13.6474	0.502027	0.251013	+	0.967984i	$0.419236\pi$
$740$	0	0
$741$	−9.29185	−0.341344
$742$	0	0
$743$	35.7957	1.31322	0.656608	−	0.754232i	$-0.271991\pi$
$743$	35.7957	1.31322	0.656608	+	0.754232i	$0.271991\pi$
$744$	0	0
$745$	12.0062	0.439872
$746$	0	0
$747$	10.8319	0.396317
$748$	0	0
$749$	16.9650	0.619886
$750$	0	0
$751$	−37.5922	−1.37176	−0.685879	−	0.727716i	$-0.740582\pi$
$751$	−37.5922	−1.37176	−0.685879	+	0.727716i	$0.740582\pi$
$752$	0	0
$753$	−21.8787	−0.797303
$754$	0	0
$755$	12.1813	0.443322
$756$	0	0
$757$	−12.4938	−0.454095	−0.227048	−	0.973884i	$-0.572907\pi$
$757$	−12.4938	−0.454095	−0.227048	+	0.973884i	$0.572907\pi$
$758$	0	0
$759$	6.43374	0.233530
$760$	0	0
$761$	32.6874	1.18492	0.592458	−	0.805601i	$-0.298158\pi$
$761$	32.6874	1.18492	0.592458	+	0.805601i	$0.298158\pi$
$762$	0	0
$763$	14.2886	0.517281
$764$	0	0
$765$	−5.74221	−0.207610
$766$	0	0
$767$	−12.4166	−0.448337
$768$	0	0
$769$	−12.1192	−0.437030	−0.218515	−	0.975834i	$-0.570121\pi$
$769$	−12.1192	−0.437030	−0.218515	+	0.975834i	$0.570121\pi$
$770$	0	0
$771$	10.7380	0.386720
$772$	0	0
$773$	−52.8912	−1.90236	−0.951182	−	0.308631i	$-0.900129\pi$
$773$	−52.8912	−1.90236	−0.951182	+	0.308631i	$0.900129\pi$
$774$	0	0
$775$	2.69094	0.0966615
$776$	0	0
$777$	−13.5232	−0.485141
$778$	0	0
$779$	−51.3765	−1.84076
$780$	0	0
$781$	10.0543	0.359770
$782$	0	0
$783$	14.2162	0.508047
$784$	0	0
$785$	7.68500	0.274290
$786$	0	0
$787$	−1.99290	−0.0710392	−0.0355196	−	0.999369i	$-0.511309\pi$
$787$	−1.99290	−0.0710392	−0.0355196	+	0.999369i	$0.511309\pi$
$788$	0	0
$789$	−16.4486	−0.585586
$790$	0	0
$791$	−5.28934	−0.188067
$792$	0	0
$793$	7.70412	0.273581
$794$	0	0
$795$	−18.1332	−0.643117
$796$	0	0
$797$	−23.8825	−0.845962	−0.422981	−	0.906139i	$-0.639016\pi$
$797$	−23.8825	−0.845962	−0.422981	+	0.906139i	$0.639016\pi$
$798$	0	0
$799$	−40.2573	−1.42420
$800$	0	0
$801$	−1.48907	−0.0526136
$802$	0	0
$803$	−15.4339	−0.544650
$804$	0	0
$805$	−2.90963	−0.102551
$806$	0	0
$807$	24.4879	0.862014
$808$	0	0
$809$	−37.0178	−1.30148	−0.650739	−	0.759302i	$-0.725541\pi$
$809$	−37.0178	−1.30148	−0.650739	+	0.759302i	$0.725541\pi$
$810$	0	0
$811$	25.6575	0.900957	0.450478	−	0.892787i	$-0.351253\pi$
$811$	25.6575	0.900957	0.450478	+	0.892787i	$0.351253\pi$
$812$	0	0
$813$	20.9436	0.734524
$814$	0	0
$815$	−4.57203	−0.160151
$816$	0	0
$817$	−40.0761	−1.40209
$818$	0	0
$819$	−1.22261	−0.0427215
$820$	0	0
$821$	23.2467	0.811316	0.405658	−	0.914025i	$-0.367042\pi$
$821$	23.2467	0.811316	0.405658	+	0.914025i	$0.367042\pi$
$822$	0	0
$823$	−37.8583	−1.31966	−0.659829	−	0.751416i	$-0.729371\pi$
$823$	−37.8583	−1.31966	−0.659829	+	0.751416i	$0.729371\pi$
$824$	0	0
$825$	8.49981	0.295926
$826$	0	0
$827$	−27.9460	−0.971778	−0.485889	−	0.874021i	$-0.661504\pi$
$827$	−27.9460	−0.971778	−0.485889	+	0.874021i	$0.661504\pi$
$828$	0	0
$829$	32.6059	1.13245	0.566225	−	0.824251i	$-0.308404\pi$
$829$	32.6059	1.13245	0.566225	+	0.824251i	$0.308404\pi$
$830$	0	0
$831$	35.9742	1.24793
$832$	0	0
$833$	−5.05388	−0.175107
$834$	0	0
$835$	3.20094	0.110773
$836$	0	0
$837$	2.37606	0.0821288
$838$	0	0
$839$	−31.4877	−1.08708	−0.543538	−	0.839385i	$-0.682916\pi$
$839$	−31.4877	−1.08708	−0.543538	+	0.839385i	$0.682916\pi$
$840$	0	0
$841$	−13.8496	−0.477573
$842$	0	0
$843$	−13.4828	−0.464373
$844$	0	0
$845$	0.929320	0.0319696
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	10.2905	0.353168
$850$	0	0
$851$	20.6044	0.706311
$852$	0	0
$853$	51.2557	1.75496	0.877480	−	0.479612i	$-0.159223\pi$
$853$	51.2557	1.75496	0.877480	+	0.479612i	$0.159223\pi$
$854$	0	0
$855$	5.13766	0.175704
$856$	0	0
$857$	29.9404	1.02275	0.511373	−	0.859359i	$-0.329137\pi$
$857$	29.9404	1.02275	0.511373	+	0.859359i	$0.329137\pi$
$858$	0	0
$859$	9.94140	0.339196	0.169598	−	0.985513i	$-0.445753\pi$
$859$	9.94140	0.339196	0.169598	+	0.985513i	$0.445753\pi$
$860$	0	0
$861$	−23.3477	−0.795687
$862$	0	0
$863$	26.3903	0.898335	0.449168	−	0.893448i	$-0.351721\pi$
$863$	26.3903	0.898335	0.449168	+	0.893448i	$0.351721\pi$
$864$	0	0
$865$	15.0769	0.512630
$866$	0	0
$867$	17.5524	0.596110
$868$	0	0
$869$	10.2793	0.348703
$870$	0	0
$871$	12.8613	0.435787
$872$	0	0
$873$	7.57550	0.256392
$874$	0	0
$875$	−8.49060	−0.287035
$876$	0	0
$877$	48.0860	1.62375	0.811875	−	0.583831i	$-0.198447\pi$
$877$	48.0860	1.62375	0.811875	+	0.583831i	$0.198447\pi$
$878$	0	0
$879$	34.7128	1.17083
$880$	0	0
$881$	51.5152	1.73559	0.867795	−	0.496922i	$-0.165536\pi$
$881$	51.5152	1.73559	0.867795	+	0.496922i	$0.165536\pi$
$882$	0	0
$883$	−48.2091	−1.62236	−0.811182	−	0.584793i	$-0.801176\pi$
$883$	−48.2091	−1.62236	−0.811182	+	0.584793i	$0.801176\pi$
$884$	0	0
$885$	23.7114	0.797051
$886$	0	0
$887$	36.4825	1.22496	0.612481	−	0.790485i	$-0.290171\pi$
$887$	36.4825	1.22496	0.612481	+	0.790485i	$0.290171\pi$
$888$	0	0
$889$	5.76890	0.193483
$890$	0	0
$891$	11.1731	0.374311
$892$	0	0
$893$	36.0189	1.20533
$894$	0	0
$895$	16.2242	0.542315
$896$	0	0
$897$	6.43374	0.214816
$898$	0	0
$899$	2.53219	0.0844534
$900$	0	0
$901$	47.9892	1.59875
$902$	0	0
$903$	−18.2123	−0.606068
$904$	0	0
$905$	−6.69607	−0.222585
$906$	0	0
$907$	−38.6480	−1.28329	−0.641643	−	0.767003i	$-0.721747\pi$
$907$	−38.6480	−1.28329	−0.641643	+	0.767003i	$0.721747\pi$
$908$	0	0
$909$	−5.23093	−0.173499
$910$	0	0
$911$	−43.0649	−1.42680	−0.713402	−	0.700755i	$-0.752847\pi$
$911$	−43.0649	−1.42680	−0.713402	+	0.700755i	$0.752847\pi$
$912$	0	0
$913$	−8.85960	−0.293210
$914$	0	0
$915$	−14.7122	−0.486371
$916$	0	0
$917$	−11.3283	−0.374095
$918$	0	0
$919$	14.0454	0.463314	0.231657	−	0.972798i	$-0.425585\pi$
$919$	14.0454	0.463314	0.231657	+	0.972798i	$0.425585\pi$
$920$	0	0
$921$	10.6286	0.350224
$922$	0	0
$923$	10.0543	0.330941
$924$	0	0
$925$	27.2212	0.895027
$926$	0	0
$927$	20.8945	0.686265
$928$	0	0
$929$	−11.3509	−0.372411	−0.186205	−	0.982511i	$-0.559619\pi$
$929$	−11.3509	−0.372411	−0.186205	+	0.982511i	$0.559619\pi$
$930$	0	0
$931$	4.52180	0.148196
$932$	0	0
$933$	28.5944	0.936137
$934$	0	0
$935$	4.69667	0.153598
$936$	0	0
$937$	27.2212	0.889279	0.444640	−	0.895710i	$-0.353332\pi$
$937$	27.2212	0.889279	0.444640	+	0.895710i	$0.353332\pi$
$938$	0	0
$939$	32.6769	1.06637
$940$	0	0
$941$	2.64879	0.0863479	0.0431740	−	0.999068i	$-0.486253\pi$
$941$	2.64879	0.0863479	0.0431740	+	0.999068i	$0.486253\pi$
$942$	0	0
$943$	35.5735	1.15843
$944$	0	0
$945$	−3.39420	−0.110413
$946$	0	0
$947$	−45.8007	−1.48832	−0.744161	−	0.668000i	$-0.767150\pi$
$947$	−45.8007	−1.48832	−0.744161	+	0.668000i	$0.767150\pi$
$948$	0	0
$949$	−15.4339	−0.501005
$950$	0	0
$951$	−30.1076	−0.976307
$952$	0	0
$953$	15.2570	0.494224	0.247112	−	0.968987i	$-0.420518\pi$
$953$	15.2570	0.494224	0.247112	+	0.968987i	$0.420518\pi$
$954$	0	0
$955$	−20.1860	−0.653204
$956$	0	0
$957$	7.99838	0.258551
$958$	0	0
$959$	−11.6510	−0.376229
$960$	0	0
$961$	−30.5768	−0.986348
$962$	0	0
$963$	20.7416	0.668387
$964$	0	0
$965$	3.98715	0.128351
$966$	0	0
$967$	1.77630	0.0571220	0.0285610	−	0.999592i	$-0.490908\pi$
$967$	1.77630	0.0571220	0.0285610	+	0.999592i	$0.490908\pi$
$968$	0	0
$969$	−46.9599	−1.50857
$970$	0	0
$971$	−13.4161	−0.430544	−0.215272	−	0.976554i	$-0.569064\pi$
$971$	−13.4161	−0.430544	−0.215272	+	0.976554i	$0.569064\pi$
$972$	0	0
$973$	−8.37197	−0.268393
$974$	0	0
$975$	8.49981	0.272212
$976$	0	0
$977$	−10.1992	−0.326302	−0.163151	−	0.986601i	$-0.552166\pi$
$977$	−10.1992	−0.326302	−0.163151	+	0.986601i	$0.552166\pi$
$978$	0	0
$979$	1.21794	0.0389255
$980$	0	0
$981$	17.4694	0.557754
$982$	0	0
$983$	−10.9526	−0.349335	−0.174667	−	0.984627i	$-0.555885\pi$
$983$	−10.9526	−0.349335	−0.174667	+	0.984627i	$0.555885\pi$
$984$	0	0
$985$	−6.34792	−0.202261
$986$	0	0
$987$	16.3685	0.521017
$988$	0	0
$989$	27.7490	0.882367
$990$	0	0
$991$	−39.4006	−1.25160	−0.625800	−	0.779983i	$-0.715228\pi$
$991$	−39.4006	−1.25160	−0.625800	+	0.779983i	$0.715228\pi$
$992$	0	0
$993$	50.3900	1.59908
$994$	0	0
$995$	7.58846	0.240570
$996$	0	0
$997$	27.5869	0.873687	0.436843	−	0.899538i	$-0.356096\pi$
$997$	27.5869	0.873687	0.436843	+	0.899538i	$0.356096\pi$
$998$	0	0
$999$	24.0359	0.760463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.12	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.12	✓	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.05490 q^{3} +0.929320 q^{5} +1.00000 q^{7} +1.22261 q^{9} +O(q^{10})\)	\(q+2.05490 q^{3} +0.929320 q^{5} +1.00000 q^{7} +1.22261 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.90966 q^{15} -5.05388 q^{17} +4.52180 q^{19} +2.05490 q^{21} -3.13093 q^{23} -4.13637 q^{25} -3.65235 q^{27} -3.89235 q^{29} -0.650557 q^{31} -2.05490 q^{33} +0.929320 q^{35} -6.58094 q^{37} -2.05490 q^{39} -11.3620 q^{41} -8.86287 q^{43} +1.13620 q^{45} +7.96562 q^{47} +1.00000 q^{49} -10.3852 q^{51} -9.49550 q^{53} -0.929320 q^{55} +9.29185 q^{57} +12.4166 q^{59} -7.70412 q^{61} +1.22261 q^{63} -0.929320 q^{65} -12.8613 q^{67} -6.43374 q^{69} -10.0543 q^{71} +15.4339 q^{73} -8.49981 q^{75} -1.00000 q^{77} -10.2793 q^{79} -11.1731 q^{81} +8.85960 q^{83} -4.69667 q^{85} -7.99838 q^{87} -1.21794 q^{89} -1.00000 q^{91} -1.33683 q^{93} +4.20220 q^{95} +6.19616 q^{97} -1.22261 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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