LMFDB - Embedded newform 8008.2.a.w.1.10

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} + \cdots + 160 $ x^11 - 3x^10 - 19x^9 + 55x^8 + 128x^7 - 361x^6 - 343x^5 + 1012x^4 + 215x^3 - 1090x^2 + 240x + 160
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.65994$ 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)

$f(q)$	$=$	$q+2.65994 q^{3} +3.55706 q^{5} -1.00000 q^{7} +4.07531 q^{9} +O(q^{10})$	$q+2.65994 q^{3} +3.55706 q^{5} -1.00000 q^{7} +4.07531 q^{9} +1.00000 q^{11} +1.00000 q^{13} +9.46158 q^{15} +4.54294 q^{17} +4.45455 q^{19} -2.65994 q^{21} -4.55552 q^{23} +7.65268 q^{25} +2.86026 q^{27} +6.15552 q^{29} +7.40914 q^{31} +2.65994 q^{33} -3.55706 q^{35} -6.22148 q^{37} +2.65994 q^{39} -8.16590 q^{41} +9.75519 q^{43} +14.4961 q^{45} +0.942550 q^{47} +1.00000 q^{49} +12.0840 q^{51} -3.26085 q^{53} +3.55706 q^{55} +11.8488 q^{57} -9.96964 q^{59} -12.4896 q^{61} -4.07531 q^{63} +3.55706 q^{65} -8.27548 q^{67} -12.1174 q^{69} +4.98120 q^{71} -6.49106 q^{73} +20.3557 q^{75} -1.00000 q^{77} -14.2714 q^{79} -4.61779 q^{81} +16.1053 q^{83} +16.1595 q^{85} +16.3733 q^{87} +0.229569 q^{89} -1.00000 q^{91} +19.7079 q^{93} +15.8451 q^{95} -8.96005 q^{97} +4.07531 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * 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
$3$	2.65994	1.53572	0.767860	−	0.640618i	$-0.221322\pi$
$3$	2.65994	1.53572	0.767860	+	0.640618i	$0.221322\pi$
$4$	0	0
$5$	3.55706	1.59077	0.795383	−	0.606107i	$-0.207270\pi$
$5$	3.55706	1.59077	0.795383	+	0.606107i	$0.207270\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.07531	1.35844
$10$	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
$15$	9.46158	2.44297
$16$	0	0
$17$	4.54294	1.10182	0.550912	−	0.834563i	$-0.314280\pi$
$17$	4.54294	1.10182	0.550912	+	0.834563i	$0.314280\pi$
$18$	0	0
$19$	4.45455	1.02194	0.510971	−	0.859598i	$-0.329286\pi$
$19$	4.45455	1.02194	0.510971	+	0.859598i	$0.329286\pi$
$20$	0	0
$21$	−2.65994	−0.580448
$22$	0	0
$23$	−4.55552	−0.949891	−0.474946	−	0.880015i	$-0.657532\pi$
$23$	−4.55552	−0.949891	−0.474946	+	0.880015i	$0.657532\pi$
$24$	0	0
$25$	7.65268	1.53054
$26$	0	0
$27$	2.86026	0.550457
$28$	0	0
$29$	6.15552	1.14305	0.571526	−	0.820584i	$-0.306352\pi$
$29$	6.15552	1.14305	0.571526	+	0.820584i	$0.306352\pi$
$30$	0	0
$31$	7.40914	1.33072	0.665360	−	0.746522i	$-0.268278\pi$
$31$	7.40914	1.33072	0.665360	+	0.746522i	$0.268278\pi$
$32$	0	0
$33$	2.65994	0.463037
$34$	0	0
$35$	−3.55706	−0.601253
$36$	0	0
$37$	−6.22148	−1.02280	−0.511402	−	0.859342i	$-0.670874\pi$
$37$	−6.22148	−1.02280	−0.511402	+	0.859342i	$0.670874\pi$
$38$	0	0
$39$	2.65994	0.425932
$40$	0	0
$41$	−8.16590	−1.27530	−0.637650	−	0.770326i	$-0.720093\pi$
$41$	−8.16590	−1.27530	−0.637650	+	0.770326i	$0.720093\pi$
$42$	0	0
$43$	9.75519	1.48765	0.743826	−	0.668373i	$-0.233009\pi$
$43$	9.75519	1.48765	0.743826	+	0.668373i	$0.233009\pi$
$44$	0	0
$45$	14.4961	2.16095
$46$	0	0
$47$	0.942550	0.137485	0.0687426	−	0.997634i	$-0.478101\pi$
$47$	0.942550	0.137485	0.0687426	+	0.997634i	$0.478101\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	12.0840	1.69209
$52$	0	0
$53$	−3.26085	−0.447912	−0.223956	−	0.974599i	$-0.571897\pi$
$53$	−3.26085	−0.447912	−0.223956	+	0.974599i	$0.571897\pi$
$54$	0	0
$55$	3.55706	0.479634
$56$	0	0
$57$	11.8488	1.56942
$58$	0	0
$59$	−9.96964	−1.29794	−0.648968	−	0.760815i	$-0.724799\pi$
$59$	−9.96964	−1.29794	−0.648968	+	0.760815i	$0.724799\pi$
$60$	0	0
$61$	−12.4896	−1.59912	−0.799562	−	0.600584i	$-0.794935\pi$
$61$	−12.4896	−1.59912	−0.799562	+	0.600584i	$0.794935\pi$
$62$	0	0
$63$	−4.07531	−0.513440
$64$	0	0
$65$	3.55706	0.441199
$66$	0	0
$67$	−8.27548	−1.01101	−0.505506	−	0.862823i	$-0.668694\pi$
$67$	−8.27548	−1.01101	−0.505506	+	0.862823i	$0.668694\pi$
$68$	0	0
$69$	−12.1174	−1.45877
$70$	0	0
$71$	4.98120	0.591159	0.295580	−	0.955318i	$-0.404487\pi$
$71$	4.98120	0.591159	0.295580	+	0.955318i	$0.404487\pi$
$72$	0	0
$73$	−6.49106	−0.759721	−0.379860	−	0.925044i	$-0.624028\pi$
$73$	−6.49106	−0.759721	−0.379860	+	0.925044i	$0.624028\pi$
$74$	0	0
$75$	20.3557	2.35047
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−14.2714	−1.60566	−0.802830	−	0.596208i	$-0.796673\pi$
$79$	−14.2714	−1.60566	−0.802830	+	0.596208i	$0.796673\pi$
$80$	0	0
$81$	−4.61779	−0.513088
$82$	0	0
$83$	16.1053	1.76779	0.883894	−	0.467688i	$-0.154913\pi$
$83$	16.1053	1.76779	0.883894	+	0.467688i	$0.154913\pi$
$84$	0	0
$85$	16.1595	1.75274
$86$	0	0
$87$	16.3733	1.75541
$88$	0	0
$89$	0.229569	0.0243343	0.0121671	−	0.999926i	$-0.496127\pi$
$89$	0.229569	0.0243343	0.0121671	+	0.999926i	$0.496127\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	19.7079	2.04361
$94$	0	0
$95$	15.8451	1.62567
$96$	0	0
$97$	−8.96005	−0.909755	−0.454878	−	0.890554i	$-0.650317\pi$
$97$	−8.96005	−0.909755	−0.454878	+	0.890554i	$0.650317\pi$
$98$	0	0
$99$	4.07531	0.409584
$100$	0	0
$101$	3.47281	0.345557	0.172779	−	0.984961i	$-0.444725\pi$
$101$	3.47281	0.345557	0.172779	+	0.984961i	$0.444725\pi$
$102$	0	0
$103$	−5.43926	−0.535946	−0.267973	−	0.963426i	$-0.586354\pi$
$103$	−5.43926	−0.535946	−0.267973	+	0.963426i	$0.586354\pi$
$104$	0	0
$105$	−9.46158	−0.923356
$106$	0	0
$107$	1.27979	0.123722	0.0618611	−	0.998085i	$-0.480296\pi$
$107$	1.27979	0.123722	0.0618611	+	0.998085i	$0.480296\pi$
$108$	0	0
$109$	−8.66842	−0.830284	−0.415142	−	0.909757i	$-0.636268\pi$
$109$	−8.66842	−0.830284	−0.415142	+	0.909757i	$0.636268\pi$
$110$	0	0
$111$	−16.5488	−1.57074
$112$	0	0
$113$	−7.70323	−0.724659	−0.362330	−	0.932050i	$-0.618019\pi$
$113$	−7.70323	−0.724659	−0.362330	+	0.932050i	$0.618019\pi$
$114$	0	0
$115$	−16.2043	−1.51105
$116$	0	0
$117$	4.07531	0.376762
$118$	0	0
$119$	−4.54294	−0.416450
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−21.7209	−1.95850
$124$	0	0
$125$	9.43573	0.843958
$126$	0	0
$127$	5.14617	0.456649	0.228324	−	0.973585i	$-0.426675\pi$
$127$	5.14617	0.456649	0.228324	+	0.973585i	$0.426675\pi$
$128$	0	0
$129$	25.9483	2.28462
$130$	0	0
$131$	8.12198	0.709621	0.354810	−	0.934938i	$-0.384545\pi$
$131$	8.12198	0.709621	0.354810	+	0.934938i	$0.384545\pi$
$132$	0	0
$133$	−4.45455	−0.386258
$134$	0	0
$135$	10.1741	0.875648
$136$	0	0
$137$	−13.1177	−1.12072	−0.560361	−	0.828248i	$-0.689338\pi$
$137$	−13.1177	−1.12072	−0.560361	+	0.828248i	$0.689338\pi$
$138$	0	0
$139$	15.6992	1.33159	0.665794	−	0.746135i	$-0.268093\pi$
$139$	15.6992	1.33159	0.665794	+	0.746135i	$0.268093\pi$
$140$	0	0
$141$	2.50713	0.211139
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	21.8956	1.81833
$146$	0	0
$147$	2.65994	0.219389
$148$	0	0
$149$	1.41184	0.115663	0.0578314	−	0.998326i	$-0.481581\pi$
$149$	1.41184	0.115663	0.0578314	+	0.998326i	$0.481581\pi$
$150$	0	0
$151$	−4.99713	−0.406661	−0.203330	−	0.979110i	$-0.565177\pi$
$151$	−4.99713	−0.406661	−0.203330	+	0.979110i	$0.565177\pi$
$152$	0	0
$153$	18.5139	1.49676
$154$	0	0
$155$	26.3548	2.11687
$156$	0	0
$157$	−2.26505	−0.180770	−0.0903852	−	0.995907i	$-0.528810\pi$
$157$	−2.26505	−0.180770	−0.0903852	+	0.995907i	$0.528810\pi$
$158$	0	0
$159$	−8.67369	−0.687868
$160$	0	0
$161$	4.55552	0.359025
$162$	0	0
$163$	9.71101	0.760625	0.380312	−	0.924858i	$-0.375816\pi$
$163$	9.71101	0.760625	0.380312	+	0.924858i	$0.375816\pi$
$164$	0	0
$165$	9.46158	0.736583
$166$	0	0
$167$	11.7839	0.911868	0.455934	−	0.890013i	$-0.349305\pi$
$167$	11.7839	0.911868	0.455934	+	0.890013i	$0.349305\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.1536	1.38824
$172$	0	0
$173$	−23.8510	−1.81335	−0.906677	−	0.421825i	$-0.861390\pi$
$173$	−23.8510	−1.81335	−0.906677	+	0.421825i	$0.861390\pi$
$174$	0	0
$175$	−7.65268	−0.578488
$176$	0	0
$177$	−26.5187	−1.99327
$178$	0	0
$179$	8.54081	0.638370	0.319185	−	0.947692i	$-0.396591\pi$
$179$	8.54081	0.638370	0.319185	+	0.947692i	$0.396591\pi$
$180$	0	0
$181$	5.56281	0.413480	0.206740	−	0.978396i	$-0.433715\pi$
$181$	5.56281	0.413480	0.206740	+	0.978396i	$0.433715\pi$
$182$	0	0
$183$	−33.2215	−2.45581
$184$	0	0
$185$	−22.1302	−1.62704
$186$	0	0
$187$	4.54294	0.332212
$188$	0	0
$189$	−2.86026	−0.208053
$190$	0	0
$191$	18.8453	1.36360	0.681799	−	0.731540i	$-0.261198\pi$
$191$	18.8453	1.36360	0.681799	+	0.731540i	$0.261198\pi$
$192$	0	0
$193$	13.3001	0.957365	0.478682	−	0.877988i	$-0.341115\pi$
$193$	13.3001	0.957365	0.478682	+	0.877988i	$0.341115\pi$
$194$	0	0
$195$	9.46158	0.677558
$196$	0	0
$197$	−24.4113	−1.73923	−0.869616	−	0.493728i	$-0.835634\pi$
$197$	−24.4113	−1.73923	−0.869616	+	0.493728i	$0.835634\pi$
$198$	0	0
$199$	−21.8787	−1.55094	−0.775471	−	0.631383i	$-0.782488\pi$
$199$	−21.8787	−1.55094	−0.775471	+	0.631383i	$0.782488\pi$
$200$	0	0
$201$	−22.0123	−1.55263
$202$	0	0
$203$	−6.15552	−0.432033
$204$	0	0
$205$	−29.0466	−2.02870
$206$	0	0
$207$	−18.5651	−1.29037
$208$	0	0
$209$	4.45455	0.308127
$210$	0	0
$211$	13.6877	0.942299	0.471149	−	0.882053i	$-0.343839\pi$
$211$	13.6877	0.942299	0.471149	+	0.882053i	$0.343839\pi$
$212$	0	0
$213$	13.2497	0.907855
$214$	0	0
$215$	34.6998	2.36651
$216$	0	0
$217$	−7.40914	−0.502965
$218$	0	0
$219$	−17.2659	−1.16672
$220$	0	0
$221$	4.54294	0.305591
$222$	0	0
$223$	−10.5104	−0.703826	−0.351913	−	0.936033i	$-0.614469\pi$
$223$	−10.5104	−0.703826	−0.351913	+	0.936033i	$0.614469\pi$
$224$	0	0
$225$	31.1870	2.07913
$226$	0	0
$227$	21.6478	1.43681	0.718407	−	0.695623i	$-0.244872\pi$
$227$	21.6478	1.43681	0.718407	+	0.695623i	$0.244872\pi$
$228$	0	0
$229$	23.2589	1.53699	0.768496	−	0.639855i	$-0.221006\pi$
$229$	23.2589	1.53699	0.768496	+	0.639855i	$0.221006\pi$
$230$	0	0
$231$	−2.65994	−0.175012
$232$	0	0
$233$	8.66924	0.567941	0.283970	−	0.958833i	$-0.408348\pi$
$233$	8.66924	0.567941	0.283970	+	0.958833i	$0.408348\pi$
$234$	0	0
$235$	3.35271	0.218707
$236$	0	0
$237$	−37.9612	−2.46584
$238$	0	0
$239$	2.85146	0.184446	0.0922230	−	0.995738i	$-0.470603\pi$
$239$	2.85146	0.184446	0.0922230	+	0.995738i	$0.470603\pi$
$240$	0	0
$241$	24.6207	1.58596	0.792978	−	0.609250i	$-0.208529\pi$
$241$	24.6207	1.58596	0.792978	+	0.609250i	$0.208529\pi$
$242$	0	0
$243$	−20.8638	−1.33842
$244$	0	0
$245$	3.55706	0.227252
$246$	0	0
$247$	4.45455	0.283436
$248$	0	0
$249$	42.8392	2.71483
$250$	0	0
$251$	−17.7034	−1.11743	−0.558713	−	0.829361i	$-0.688705\pi$
$251$	−17.7034	−1.11743	−0.558713	+	0.829361i	$0.688705\pi$
$252$	0	0
$253$	−4.55552	−0.286403
$254$	0	0
$255$	42.9834	2.69172
$256$	0	0
$257$	17.7228	1.10552	0.552759	−	0.833341i	$-0.313575\pi$
$257$	17.7228	1.10552	0.552759	+	0.833341i	$0.313575\pi$
$258$	0	0
$259$	6.22148	0.386584
$260$	0	0
$261$	25.0856	1.55276
$262$	0	0
$263$	16.0707	0.990962	0.495481	−	0.868619i	$-0.334992\pi$
$263$	16.0707	0.990962	0.495481	+	0.868619i	$0.334992\pi$
$264$	0	0
$265$	−11.5990	−0.712524
$266$	0	0
$267$	0.610642	0.0373707
$268$	0	0
$269$	−22.7638	−1.38793	−0.693966	−	0.720008i	$-0.744138\pi$
$269$	−22.7638	−1.38793	−0.693966	+	0.720008i	$0.744138\pi$
$270$	0	0
$271$	26.8152	1.62890	0.814452	−	0.580230i	$-0.197038\pi$
$271$	26.8152	1.62890	0.814452	+	0.580230i	$0.197038\pi$
$272$	0	0
$273$	−2.65994	−0.160987
$274$	0	0
$275$	7.65268	0.461474
$276$	0	0
$277$	30.9305	1.85843	0.929217	−	0.369535i	$-0.120483\pi$
$277$	30.9305	1.85843	0.929217	+	0.369535i	$0.120483\pi$
$278$	0	0
$279$	30.1945	1.80770
$280$	0	0
$281$	−5.92662	−0.353552	−0.176776	−	0.984251i	$-0.556567\pi$
$281$	−5.92662	−0.353552	−0.176776	+	0.984251i	$0.556567\pi$
$282$	0	0
$283$	11.7088	0.696015	0.348007	−	0.937492i	$-0.386858\pi$
$283$	11.7088	0.696015	0.348007	+	0.937492i	$0.386858\pi$
$284$	0	0
$285$	42.1471	2.49658
$286$	0	0
$287$	8.16590	0.482018
$288$	0	0
$289$	3.63827	0.214016
$290$	0	0
$291$	−23.8332	−1.39713
$292$	0	0
$293$	20.7754	1.21371	0.606857	−	0.794811i	$-0.292430\pi$
$293$	20.7754	1.21371	0.606857	+	0.794811i	$0.292430\pi$
$294$	0	0
$295$	−35.4626	−2.06471
$296$	0	0
$297$	2.86026	0.165969
$298$	0	0
$299$	−4.55552	−0.263452
$300$	0	0
$301$	−9.75519	−0.562280
$302$	0	0
$303$	9.23748	0.530679
$304$	0	0
$305$	−44.4261	−2.54383
$306$	0	0
$307$	−25.0794	−1.43136	−0.715678	−	0.698430i	$-0.753882\pi$
$307$	−25.0794	−1.43136	−0.715678	+	0.698430i	$0.753882\pi$
$308$	0	0
$309$	−14.4681	−0.823063
$310$	0	0
$311$	−2.14354	−0.121549	−0.0607745	−	0.998152i	$-0.519357\pi$
$311$	−2.14354	−0.121549	−0.0607745	+	0.998152i	$0.519357\pi$
$312$	0	0
$313$	4.51220	0.255044	0.127522	−	0.991836i	$-0.459298\pi$
$313$	4.51220	0.255044	0.127522	+	0.991836i	$0.459298\pi$
$314$	0	0
$315$	−14.4961	−0.816763
$316$	0	0
$317$	12.2283	0.686811	0.343405	−	0.939187i	$-0.388420\pi$
$317$	12.2283	0.686811	0.343405	+	0.939187i	$0.388420\pi$
$318$	0	0
$319$	6.15552	0.344643
$320$	0	0
$321$	3.40418	0.190003
$322$	0	0
$323$	20.2367	1.12600
$324$	0	0
$325$	7.65268	0.424494
$326$	0	0
$327$	−23.0575	−1.27508
$328$	0	0
$329$	−0.942550	−0.0519645
$330$	0	0
$331$	15.7869	0.867724	0.433862	−	0.900979i	$-0.357150\pi$
$331$	15.7869	0.867724	0.433862	+	0.900979i	$0.357150\pi$
$332$	0	0
$333$	−25.3544	−1.38941
$334$	0	0
$335$	−29.4364	−1.60828
$336$	0	0
$337$	1.65027	0.0898959	0.0449480	−	0.998989i	$-0.485688\pi$
$337$	1.65027	0.0898959	0.0449480	+	0.998989i	$0.485688\pi$
$338$	0	0
$339$	−20.4902	−1.11287
$340$	0	0
$341$	7.40914	0.401227
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−43.1024	−2.32056
$346$	0	0
$347$	−27.6363	−1.48360	−0.741798	−	0.670623i	$-0.766027\pi$
$347$	−27.6363	−1.48360	−0.741798	+	0.670623i	$0.766027\pi$
$348$	0	0
$349$	−21.0000	−1.12411	−0.562053	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12411	−0.562053	+	0.827102i	$0.689988\pi$
$350$	0	0
$351$	2.86026	0.152669
$352$	0	0
$353$	−21.1958	−1.12814	−0.564069	−	0.825728i	$-0.690765\pi$
$353$	−21.1958	−1.12814	−0.564069	+	0.825728i	$0.690765\pi$
$354$	0	0
$355$	17.7184	0.940396
$356$	0	0
$357$	−12.0840	−0.639551
$358$	0	0
$359$	−19.5454	−1.03157	−0.515783	−	0.856719i	$-0.672499\pi$
$359$	−19.5454	−1.03157	−0.515783	+	0.856719i	$0.672499\pi$
$360$	0	0
$361$	0.842979	0.0443673
$362$	0	0
$363$	2.65994	0.139611
$364$	0	0
$365$	−23.0891	−1.20854
$366$	0	0
$367$	24.9440	1.30207	0.651033	−	0.759049i	$-0.274336\pi$
$367$	24.9440	1.30207	0.651033	+	0.759049i	$0.274336\pi$
$368$	0	0
$369$	−33.2786	−1.73241
$370$	0	0
$371$	3.26085	0.169295
$372$	0	0
$373$	9.23051	0.477938	0.238969	−	0.971027i	$-0.423191\pi$
$373$	9.23051	0.477938	0.238969	+	0.971027i	$0.423191\pi$
$374$	0	0
$375$	25.0985	1.29608
$376$	0	0
$377$	6.15552	0.317025
$378$	0	0
$379$	12.0728	0.620140	0.310070	−	0.950714i	$-0.399648\pi$
$379$	12.0728	0.620140	0.310070	+	0.950714i	$0.399648\pi$
$380$	0	0
$381$	13.6885	0.701284
$382$	0	0
$383$	5.75954	0.294299	0.147149	−	0.989114i	$-0.452990\pi$
$383$	5.75954	0.294299	0.147149	+	0.989114i	$0.452990\pi$
$384$	0	0
$385$	−3.55706	−0.181285
$386$	0	0
$387$	39.7554	2.02088
$388$	0	0
$389$	0.693589	0.0351664	0.0175832	−	0.999845i	$-0.494403\pi$
$389$	0.693589	0.0351664	0.0175832	+	0.999845i	$0.494403\pi$
$390$	0	0
$391$	−20.6954	−1.04661
$392$	0	0
$393$	21.6040	1.08978
$394$	0	0
$395$	−50.7643	−2.55423
$396$	0	0
$397$	18.0276	0.904778	0.452389	−	0.891821i	$-0.350572\pi$
$397$	18.0276	0.904778	0.452389	+	0.891821i	$0.350572\pi$
$398$	0	0
$399$	−11.8488	−0.593184
$400$	0	0
$401$	−38.3476	−1.91499	−0.957493	−	0.288457i	$-0.906858\pi$
$401$	−38.3476	−1.91499	−0.957493	+	0.288457i	$0.906858\pi$
$402$	0	0
$403$	7.40914	0.369076
$404$	0	0
$405$	−16.4258	−0.816203
$406$	0	0
$407$	−6.22148	−0.308387
$408$	0	0
$409$	8.21133	0.406024	0.203012	−	0.979176i	$-0.434927\pi$
$409$	8.21133	0.406024	0.203012	+	0.979176i	$0.434927\pi$
$410$	0	0
$411$	−34.8924	−1.72112
$412$	0	0
$413$	9.96964	0.490574
$414$	0	0
$415$	57.2876	2.81214
$416$	0	0
$417$	41.7590	2.04495
$418$	0	0
$419$	−20.2322	−0.988408	−0.494204	−	0.869346i	$-0.664540\pi$
$419$	−20.2322	−0.988408	−0.494204	+	0.869346i	$0.664540\pi$
$420$	0	0
$421$	5.51701	0.268883	0.134441	−	0.990922i	$-0.457076\pi$
$421$	5.51701	0.268883	0.134441	+	0.990922i	$0.457076\pi$
$422$	0	0
$423$	3.84118	0.186765
$424$	0	0
$425$	34.7656	1.68638
$426$	0	0
$427$	12.4896	0.604412
$428$	0	0
$429$	2.65994	0.128423
$430$	0	0
$431$	−6.45936	−0.311136	−0.155568	−	0.987825i	$-0.549721\pi$
$431$	−6.45936	−0.311136	−0.155568	+	0.987825i	$0.549721\pi$
$432$	0	0
$433$	22.7168	1.09170	0.545850	−	0.837883i	$-0.316207\pi$
$433$	22.7168	1.09170	0.545850	+	0.837883i	$0.316207\pi$
$434$	0	0
$435$	58.2410	2.79244
$436$	0	0
$437$	−20.2928	−0.970735
$438$	0	0
$439$	−29.0197	−1.38504	−0.692518	−	0.721401i	$-0.743498\pi$
$439$	−29.0197	−1.38504	−0.692518	+	0.721401i	$0.743498\pi$
$440$	0	0
$441$	4.07531	0.194062
$442$	0	0
$443$	−3.25588	−0.154692	−0.0773458	−	0.997004i	$-0.524645\pi$
$443$	−3.25588	−0.154692	−0.0773458	+	0.997004i	$0.524645\pi$
$444$	0	0
$445$	0.816592	0.0387102
$446$	0	0
$447$	3.75543	0.177626
$448$	0	0
$449$	11.2710	0.531909	0.265955	−	0.963986i	$-0.414313\pi$
$449$	11.2710	0.531909	0.265955	+	0.963986i	$0.414313\pi$
$450$	0	0
$451$	−8.16590	−0.384517
$452$	0	0
$453$	−13.2921	−0.624517
$454$	0	0
$455$	−3.55706	−0.166758
$456$	0	0
$457$	−36.3183	−1.69890	−0.849449	−	0.527670i	$-0.823066\pi$
$457$	−36.3183	−1.69890	−0.849449	+	0.527670i	$0.823066\pi$
$458$	0	0
$459$	12.9940	0.606506
$460$	0	0
$461$	−9.22280	−0.429549	−0.214774	−	0.976664i	$-0.568902\pi$
$461$	−9.22280	−0.429549	−0.214774	+	0.976664i	$0.568902\pi$
$462$	0	0
$463$	−40.5840	−1.88610	−0.943049	−	0.332655i	$-0.892056\pi$
$463$	−40.5840	−1.88610	−0.943049	+	0.332655i	$0.892056\pi$
$464$	0	0
$465$	70.1022	3.25091
$466$	0	0
$467$	−2.20699	−0.102127	−0.0510636	−	0.998695i	$-0.516261\pi$
$467$	−2.20699	−0.102127	−0.0510636	+	0.998695i	$0.516261\pi$
$468$	0	0
$469$	8.27548	0.382126
$470$	0	0
$471$	−6.02490	−0.277613
$472$	0	0
$473$	9.75519	0.448544
$474$	0	0
$475$	34.0892	1.56412
$476$	0	0
$477$	−13.2890	−0.608460
$478$	0	0
$479$	−1.47530	−0.0674081	−0.0337040	−	0.999432i	$-0.510730\pi$
$479$	−1.47530	−0.0674081	−0.0337040	+	0.999432i	$0.510730\pi$
$480$	0	0
$481$	−6.22148	−0.283675
$482$	0	0
$483$	12.1174	0.551362
$484$	0	0
$485$	−31.8714	−1.44721
$486$	0	0
$487$	5.41570	0.245409	0.122704	−	0.992443i	$-0.460843\pi$
$487$	5.41570	0.245409	0.122704	+	0.992443i	$0.460843\pi$
$488$	0	0
$489$	25.8307	1.16811
$490$	0	0
$491$	−20.7031	−0.934316	−0.467158	−	0.884174i	$-0.654722\pi$
$491$	−20.7031	−0.934316	−0.467158	+	0.884174i	$0.654722\pi$
$492$	0	0
$493$	27.9641	1.25944
$494$	0	0
$495$	14.4961	0.651552
$496$	0	0
$497$	−4.98120	−0.223437
$498$	0	0
$499$	−22.5281	−1.00850	−0.504248	−	0.863559i	$-0.668230\pi$
$499$	−22.5281	−1.00850	−0.504248	+	0.863559i	$0.668230\pi$
$500$	0	0
$501$	31.3446	1.40037
$502$	0	0
$503$	−34.6753	−1.54609	−0.773047	−	0.634349i	$-0.781268\pi$
$503$	−34.6753	−1.54609	−0.773047	+	0.634349i	$0.781268\pi$
$504$	0	0
$505$	12.3530	0.549701
$506$	0	0
$507$	2.65994	0.118132
$508$	0	0
$509$	−29.7156	−1.31712	−0.658561	−	0.752528i	$-0.728834\pi$
$509$	−29.7156	−1.31712	−0.658561	+	0.752528i	$0.728834\pi$
$510$	0	0
$511$	6.49106	0.287147
$512$	0	0
$513$	12.7411	0.562536
$514$	0	0
$515$	−19.3478	−0.852564
$516$	0	0
$517$	0.942550	0.0414533
$518$	0	0
$519$	−63.4422	−2.78480
$520$	0	0
$521$	9.83718	0.430975	0.215487	−	0.976507i	$-0.430866\pi$
$521$	9.83718	0.430975	0.215487	+	0.976507i	$0.430866\pi$
$522$	0	0
$523$	25.9704	1.13561	0.567804	−	0.823164i	$-0.307793\pi$
$523$	25.9704	1.13561	0.567804	+	0.823164i	$0.307793\pi$
$524$	0	0
$525$	−20.3557	−0.888396
$526$	0	0
$527$	33.6592	1.46622
$528$	0	0
$529$	−2.24725	−0.0977066
$530$	0	0
$531$	−40.6294	−1.76316
$532$	0	0
$533$	−8.16590	−0.353704
$534$	0	0
$535$	4.55230	0.196813
$536$	0	0
$537$	22.7181	0.980358
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−1.63600	−0.0703371	−0.0351685	−	0.999381i	$-0.511197\pi$
$541$	−1.63600	−0.0703371	−0.0351685	+	0.999381i	$0.511197\pi$
$542$	0	0
$543$	14.7968	0.634990
$544$	0	0
$545$	−30.8341	−1.32079
$546$	0	0
$547$	−43.2671	−1.84997	−0.924983	−	0.380009i	$-0.875921\pi$
$547$	−43.2671	−1.84997	−0.924983	+	0.380009i	$0.875921\pi$
$548$	0	0
$549$	−50.8988	−2.17231
$550$	0	0
$551$	27.4200	1.16813
$552$	0	0
$553$	14.2714	0.606882
$554$	0	0
$555$	−58.8650	−2.49868
$556$	0	0
$557$	−30.9617	−1.31189	−0.655945	−	0.754809i	$-0.727730\pi$
$557$	−30.9617	−1.31189	−0.655945	+	0.754809i	$0.727730\pi$
$558$	0	0
$559$	9.75519	0.412601
$560$	0	0
$561$	12.0840	0.510185
$562$	0	0
$563$	8.87774	0.374152	0.187076	−	0.982345i	$-0.440099\pi$
$563$	8.87774	0.374152	0.187076	+	0.982345i	$0.440099\pi$
$564$	0	0
$565$	−27.4009	−1.15276
$566$	0	0
$567$	4.61779	0.193929
$568$	0	0
$569$	−35.1460	−1.47340	−0.736698	−	0.676222i	$-0.763616\pi$
$569$	−35.1460	−1.47340	−0.736698	+	0.676222i	$0.763616\pi$
$570$	0	0
$571$	−16.2907	−0.681745	−0.340873	−	0.940109i	$-0.610722\pi$
$571$	−16.2907	−0.681745	−0.340873	+	0.940109i	$0.610722\pi$
$572$	0	0
$573$	50.1275	2.09410
$574$	0	0
$575$	−34.8619	−1.45384
$576$	0	0
$577$	−33.2316	−1.38345	−0.691724	−	0.722162i	$-0.743149\pi$
$577$	−33.2316	−1.38345	−0.691724	+	0.722162i	$0.743149\pi$
$578$	0	0
$579$	35.3776	1.47024
$580$	0	0
$581$	−16.1053	−0.668161
$582$	0	0
$583$	−3.26085	−0.135051
$584$	0	0
$585$	14.4961	0.599341
$586$	0	0
$587$	−6.89559	−0.284612	−0.142306	−	0.989823i	$-0.545452\pi$
$587$	−6.89559	−0.284612	−0.142306	+	0.989823i	$0.545452\pi$
$588$	0	0
$589$	33.0044	1.35992
$590$	0	0
$591$	−64.9327	−2.67097
$592$	0	0
$593$	46.2156	1.89785	0.948923	−	0.315507i	$-0.102175\pi$
$593$	46.2156	1.89785	0.948923	+	0.315507i	$0.102175\pi$
$594$	0	0
$595$	−16.1595	−0.662475
$596$	0	0
$597$	−58.1962	−2.38181
$598$	0	0
$599$	−34.1242	−1.39428	−0.697138	−	0.716937i	$-0.745544\pi$
$599$	−34.1242	−1.39428	−0.697138	+	0.716937i	$0.745544\pi$
$600$	0	0
$601$	25.9495	1.05850	0.529251	−	0.848465i	$-0.322473\pi$
$601$	25.9495	1.05850	0.529251	+	0.848465i	$0.322473\pi$
$602$	0	0
$603$	−33.7251	−1.37339
$604$	0	0
$605$	3.55706	0.144615
$606$	0	0
$607$	25.7132	1.04367	0.521833	−	0.853047i	$-0.325248\pi$
$607$	25.7132	1.04367	0.521833	+	0.853047i	$0.325248\pi$
$608$	0	0
$609$	−16.3733	−0.663481
$610$	0	0
$611$	0.942550	0.0381315
$612$	0	0
$613$	13.6725	0.552226	0.276113	−	0.961125i	$-0.410954\pi$
$613$	13.6725	0.552226	0.276113	+	0.961125i	$0.410954\pi$
$614$	0	0
$615$	−77.2624	−3.11552
$616$	0	0
$617$	23.3491	0.939998	0.469999	−	0.882667i	$-0.344254\pi$
$617$	23.3491	0.939998	0.469999	+	0.882667i	$0.344254\pi$
$618$	0	0
$619$	−12.1449	−0.488145	−0.244073	−	0.969757i	$-0.578484\pi$
$619$	−12.1449	−0.488145	−0.244073	+	0.969757i	$0.578484\pi$
$620$	0	0
$621$	−13.0300	−0.522874
$622$	0	0
$623$	−0.229569	−0.00919750
$624$	0	0
$625$	−4.69991	−0.187997
$626$	0	0
$627$	11.8488	0.473197
$628$	0	0
$629$	−28.2638	−1.12695
$630$	0	0
$631$	−20.1260	−0.801203	−0.400601	−	0.916252i	$-0.631199\pi$
$631$	−20.1260	−0.801203	−0.400601	+	0.916252i	$0.631199\pi$
$632$	0	0
$633$	36.4085	1.44711
$634$	0	0
$635$	18.3052	0.726421
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	20.2999	0.803052
$640$	0	0
$641$	30.5684	1.20738	0.603690	−	0.797219i	$-0.293696\pi$
$641$	30.5684	1.20738	0.603690	+	0.797219i	$0.293696\pi$
$642$	0	0
$643$	11.8563	0.467568	0.233784	−	0.972289i	$-0.424889\pi$
$643$	11.8563	0.467568	0.233784	+	0.972289i	$0.424889\pi$
$644$	0	0
$645$	92.2996	3.63429
$646$	0	0
$647$	28.6165	1.12503	0.562515	−	0.826787i	$-0.309834\pi$
$647$	28.6165	1.12503	0.562515	+	0.826787i	$0.309834\pi$
$648$	0	0
$649$	−9.96964	−0.391343
$650$	0	0
$651$	−19.7079	−0.772414
$652$	0	0
$653$	8.21489	0.321473	0.160737	−	0.986997i	$-0.448613\pi$
$653$	8.21489	0.321473	0.160737	+	0.986997i	$0.448613\pi$
$654$	0	0
$655$	28.8904	1.12884
$656$	0	0
$657$	−26.4531	−1.03203
$658$	0	0
$659$	7.76495	0.302479	0.151240	−	0.988497i	$-0.451673\pi$
$659$	7.76495	0.302479	0.151240	+	0.988497i	$0.451673\pi$
$660$	0	0
$661$	−29.0396	−1.12951	−0.564754	−	0.825259i	$-0.691029\pi$
$661$	−29.0396	−1.12951	−0.564754	+	0.825259i	$0.691029\pi$
$662$	0	0
$663$	12.0840	0.469302
$664$	0	0
$665$	−15.8451	−0.614446
$666$	0	0
$667$	−28.0416	−1.08577
$668$	0	0
$669$	−27.9570	−1.08088
$670$	0	0
$671$	−12.4896	−0.482154
$672$	0	0
$673$	2.48916	0.0959500	0.0479750	−	0.998849i	$-0.484723\pi$
$673$	2.48916	0.0959500	0.0479750	+	0.998849i	$0.484723\pi$
$674$	0	0
$675$	21.8886	0.842494
$676$	0	0
$677$	−5.28358	−0.203064	−0.101532	−	0.994832i	$-0.532374\pi$
$677$	−5.28358	−0.203064	−0.101532	+	0.994832i	$0.532374\pi$
$678$	0	0
$679$	8.96005	0.343855
$680$	0	0
$681$	57.5819	2.20654
$682$	0	0
$683$	−34.9866	−1.33872	−0.669362	−	0.742937i	$-0.733432\pi$
$683$	−34.9866	−1.33872	−0.669362	+	0.742937i	$0.733432\pi$
$684$	0	0
$685$	−46.6605	−1.78281
$686$	0	0
$687$	61.8674	2.36039
$688$	0	0
$689$	−3.26085	−0.124229
$690$	0	0
$691$	4.74427	0.180481	0.0902403	−	0.995920i	$-0.471236\pi$
$691$	4.74427	0.180481	0.0902403	+	0.995920i	$0.471236\pi$
$692$	0	0
$693$	−4.07531	−0.154808
$694$	0	0
$695$	55.8430	2.11825
$696$	0	0
$697$	−37.0972	−1.40516
$698$	0	0
$699$	23.0597	0.872198
$700$	0	0
$701$	30.4511	1.15012	0.575062	−	0.818110i	$-0.304978\pi$
$701$	30.4511	1.15012	0.575062	+	0.818110i	$0.304978\pi$
$702$	0	0
$703$	−27.7138	−1.04525
$704$	0	0
$705$	8.91802	0.335872
$706$	0	0
$707$	−3.47281	−0.130608
$708$	0	0
$709$	−12.3890	−0.465277	−0.232639	−	0.972563i	$-0.574736\pi$
$709$	−12.3890	−0.465277	−0.232639	+	0.972563i	$0.574736\pi$
$710$	0	0
$711$	−58.1604	−2.18119
$712$	0	0
$713$	−33.7525	−1.26404
$714$	0	0
$715$	3.55706	0.133027
$716$	0	0
$717$	7.58474	0.283257
$718$	0	0
$719$	18.1489	0.676840	0.338420	−	0.940995i	$-0.390108\pi$
$719$	18.1489	0.676840	0.338420	+	0.940995i	$0.390108\pi$
$720$	0	0
$721$	5.43926	0.202568
$722$	0	0
$723$	65.4896	2.43559
$724$	0	0
$725$	47.1062	1.74948
$726$	0	0
$727$	43.4589	1.61180	0.805902	−	0.592050i	$-0.201681\pi$
$727$	43.4589	1.61180	0.805902	+	0.592050i	$0.201681\pi$
$728$	0	0
$729$	−41.6433	−1.54234
$730$	0	0
$731$	44.3172	1.63913
$732$	0	0
$733$	−44.6251	−1.64826	−0.824132	−	0.566397i	$-0.808337\pi$
$733$	−44.6251	−1.64826	−0.824132	+	0.566397i	$0.808337\pi$
$734$	0	0
$735$	9.46158	0.348996
$736$	0	0
$737$	−8.27548	−0.304831
$738$	0	0
$739$	−30.8321	−1.13418	−0.567088	−	0.823657i	$-0.691930\pi$
$739$	−30.8321	−1.13418	−0.567088	+	0.823657i	$0.691930\pi$
$740$	0	0
$741$	11.8488	0.435278
$742$	0	0
$743$	−36.8038	−1.35020	−0.675101	−	0.737725i	$-0.735900\pi$
$743$	−36.8038	−1.35020	−0.675101	+	0.737725i	$0.735900\pi$
$744$	0	0
$745$	5.02202	0.183992
$746$	0	0
$747$	65.6341	2.40143
$748$	0	0
$749$	−1.27979	−0.0467626
$750$	0	0
$751$	−44.6297	−1.62856	−0.814281	−	0.580470i	$-0.802869\pi$
$751$	−44.6297	−1.62856	−0.814281	+	0.580470i	$0.802869\pi$
$752$	0	0
$753$	−47.0900	−1.71605
$754$	0	0
$755$	−17.7751	−0.646902
$756$	0	0
$757$	−24.6990	−0.897700	−0.448850	−	0.893607i	$-0.648166\pi$
$757$	−24.6990	−0.897700	−0.448850	+	0.893607i	$0.648166\pi$
$758$	0	0
$759$	−12.1174	−0.439835
$760$	0	0
$761$	−22.3782	−0.811208	−0.405604	−	0.914049i	$-0.632939\pi$
$761$	−22.3782	−0.811208	−0.405604	+	0.914049i	$0.632939\pi$
$762$	0	0
$763$	8.66842	0.313818
$764$	0	0
$765$	65.8549	2.38099
$766$	0	0
$767$	−9.96964	−0.359983
$768$	0	0
$769$	12.2409	0.441420	0.220710	−	0.975339i	$-0.429163\pi$
$769$	12.2409	0.441420	0.220710	+	0.975339i	$0.429163\pi$
$770$	0	0
$771$	47.1417	1.69777
$772$	0	0
$773$	−16.9495	−0.609630	−0.304815	−	0.952412i	$-0.598595\pi$
$773$	−16.9495	−0.609630	−0.304815	+	0.952412i	$0.598595\pi$
$774$	0	0
$775$	56.6998	2.03672
$776$	0	0
$777$	16.5488	0.593684
$778$	0	0
$779$	−36.3754	−1.30328
$780$	0	0
$781$	4.98120	0.178241
$782$	0	0
$783$	17.6064	0.629200
$784$	0	0
$785$	−8.05691	−0.287564
$786$	0	0
$787$	43.5143	1.55112	0.775559	−	0.631275i	$-0.217468\pi$
$787$	43.5143	1.55112	0.775559	+	0.631275i	$0.217468\pi$
$788$	0	0
$789$	42.7472	1.52184
$790$	0	0
$791$	7.70323	0.273895
$792$	0	0
$793$	−12.4896	−0.443517
$794$	0	0
$795$	−30.8528	−1.09424
$796$	0	0
$797$	21.5664	0.763921	0.381960	−	0.924179i	$-0.375249\pi$
$797$	21.5664	0.763921	0.381960	+	0.924179i	$0.375249\pi$
$798$	0	0
$799$	4.28195	0.151484
$800$	0	0
$801$	0.935565	0.0330566
$802$	0	0
$803$	−6.49106	−0.229064
$804$	0	0
$805$	16.2043	0.571125
$806$	0	0
$807$	−60.5504	−2.13147
$808$	0	0
$809$	−6.47149	−0.227525	−0.113763	−	0.993508i	$-0.536290\pi$
$809$	−6.47149	−0.227525	−0.113763	+	0.993508i	$0.536290\pi$
$810$	0	0
$811$	36.5985	1.28515	0.642574	−	0.766224i	$-0.277867\pi$
$811$	36.5985	1.28515	0.642574	+	0.766224i	$0.277867\pi$
$812$	0	0
$813$	71.3268	2.50154
$814$	0	0
$815$	34.5426	1.20998
$816$	0	0
$817$	43.4549	1.52030
$818$	0	0
$819$	−4.07531	−0.142403
$820$	0	0
$821$	−40.5224	−1.41424	−0.707120	−	0.707093i	$-0.750006\pi$
$821$	−40.5224	−1.41424	−0.707120	+	0.707093i	$0.750006\pi$
$822$	0	0
$823$	28.0693	0.978435	0.489217	−	0.872162i	$-0.337282\pi$
$823$	28.0693	0.978435	0.489217	+	0.872162i	$0.337282\pi$
$824$	0	0
$825$	20.3557	0.708695
$826$	0	0
$827$	−18.9885	−0.660296	−0.330148	−	0.943929i	$-0.607099\pi$
$827$	−18.9885	−0.660296	−0.330148	+	0.943929i	$0.607099\pi$
$828$	0	0
$829$	−37.5215	−1.30317	−0.651587	−	0.758574i	$-0.725897\pi$
$829$	−37.5215	−1.30317	−0.651587	+	0.758574i	$0.725897\pi$
$830$	0	0
$831$	82.2734	2.85403
$832$	0	0
$833$	4.54294	0.157403
$834$	0	0
$835$	41.9162	1.45057
$836$	0	0
$837$	21.1921	0.732504
$838$	0	0
$839$	1.79562	0.0619917	0.0309959	−	0.999520i	$-0.490132\pi$
$839$	1.79562	0.0619917	0.0309959	+	0.999520i	$0.490132\pi$
$840$	0	0
$841$	8.89042	0.306566
$842$	0	0
$843$	−15.7645	−0.542957
$844$	0	0
$845$	3.55706	0.122367
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	31.1447	1.06888
$850$	0	0
$851$	28.3420	0.971553
$852$	0	0
$853$	−27.6979	−0.948358	−0.474179	−	0.880428i	$-0.657255\pi$
$853$	−27.6979	−0.948358	−0.474179	+	0.880428i	$0.657255\pi$
$854$	0	0
$855$	64.5736	2.20837
$856$	0	0
$857$	15.7479	0.537937	0.268969	−	0.963149i	$-0.413317\pi$
$857$	15.7479	0.537937	0.268969	+	0.963149i	$0.413317\pi$
$858$	0	0
$859$	−14.8156	−0.505500	−0.252750	−	0.967532i	$-0.581335\pi$
$859$	−14.8156	−0.505500	−0.252750	+	0.967532i	$0.581335\pi$
$860$	0	0
$861$	21.7209	0.740245
$862$	0	0
$863$	45.9506	1.56418	0.782089	−	0.623167i	$-0.214154\pi$
$863$	45.9506	1.56418	0.782089	+	0.623167i	$0.214154\pi$
$864$	0	0
$865$	−84.8393	−2.88462
$866$	0	0
$867$	9.67759	0.328668
$868$	0	0
$869$	−14.2714	−0.484125
$870$	0	0
$871$	−8.27548	−0.280404
$872$	0	0
$873$	−36.5150	−1.23584
$874$	0	0
$875$	−9.43573	−0.318986
$876$	0	0
$877$	−15.9249	−0.537746	−0.268873	−	0.963176i	$-0.586651\pi$
$877$	−15.9249	−0.537746	−0.268873	+	0.963176i	$0.586651\pi$
$878$	0	0
$879$	55.2615	1.86392
$880$	0	0
$881$	51.8274	1.74611	0.873054	−	0.487623i	$-0.162136\pi$
$881$	51.8274	1.74611	0.873054	+	0.487623i	$0.162136\pi$
$882$	0	0
$883$	0.931861	0.0313596	0.0156798	−	0.999877i	$-0.495009\pi$
$883$	0.931861	0.0313596	0.0156798	+	0.999877i	$0.495009\pi$
$884$	0	0
$885$	−94.3286	−3.17082
$886$	0	0
$887$	1.06741	0.0358402	0.0179201	−	0.999839i	$-0.494296\pi$
$887$	1.06741	0.0358402	0.0179201	+	0.999839i	$0.494296\pi$
$888$	0	0
$889$	−5.14617	−0.172597
$890$	0	0
$891$	−4.61779	−0.154702
$892$	0	0
$893$	4.19863	0.140502
$894$	0	0
$895$	30.3802	1.01550
$896$	0	0
$897$	−12.1174	−0.404589
$898$	0	0
$899$	45.6071	1.52108
$900$	0	0
$901$	−14.8138	−0.493521
$902$	0	0
$903$	−25.9483	−0.863504
$904$	0	0
$905$	19.7872	0.657750
$906$	0	0
$907$	−22.2579	−0.739060	−0.369530	−	0.929219i	$-0.620481\pi$
$907$	−22.2579	−0.739060	−0.369530	+	0.929219i	$0.620481\pi$
$908$	0	0
$909$	14.1528	0.469417
$910$	0	0
$911$	59.7558	1.97980	0.989898	−	0.141779i	$-0.0452823\pi$
$911$	59.7558	1.97980	0.989898	+	0.141779i	$0.0452823\pi$
$912$	0	0
$913$	16.1053	0.533008
$914$	0	0
$915$	−118.171	−3.90661
$916$	0	0
$917$	−8.12198	−0.268212
$918$	0	0
$919$	9.21557	0.303993	0.151997	−	0.988381i	$-0.451430\pi$
$919$	9.21557	0.303993	0.151997	+	0.988381i	$0.451430\pi$
$920$	0	0
$921$	−66.7098	−2.19816
$922$	0	0
$923$	4.98120	0.163958
$924$	0	0
$925$	−47.6109	−1.56544
$926$	0	0
$927$	−22.1666	−0.728048
$928$	0	0
$929$	19.7801	0.648964	0.324482	−	0.945892i	$-0.394810\pi$
$929$	19.7801	0.648964	0.324482	+	0.945892i	$0.394810\pi$
$930$	0	0
$931$	4.45455	0.145992
$932$	0	0
$933$	−5.70170	−0.186665
$934$	0	0
$935$	16.1595	0.528472
$936$	0	0
$937$	11.8061	0.385688	0.192844	−	0.981229i	$-0.438229\pi$
$937$	11.8061	0.385688	0.192844	+	0.981229i	$0.438229\pi$
$938$	0	0
$939$	12.0022	0.391677
$940$	0	0
$941$	−28.4431	−0.927219	−0.463609	−	0.886040i	$-0.653446\pi$
$941$	−28.4431	−0.927219	−0.463609	+	0.886040i	$0.653446\pi$
$942$	0	0
$943$	37.1999	1.21140
$944$	0	0
$945$	−10.1741	−0.330964
$946$	0	0
$947$	2.49546	0.0810915	0.0405458	−	0.999178i	$-0.487090\pi$
$947$	2.49546	0.0810915	0.0405458	+	0.999178i	$0.487090\pi$
$948$	0	0
$949$	−6.49106	−0.210709
$950$	0	0
$951$	32.5267	1.05475
$952$	0	0
$953$	4.46093	0.144504	0.0722518	−	0.997386i	$-0.476981\pi$
$953$	4.46093	0.144504	0.0722518	+	0.997386i	$0.476981\pi$
$954$	0	0
$955$	67.0339	2.16917
$956$	0	0
$957$	16.3733	0.529275
$958$	0	0
$959$	13.1177	0.423593
$960$	0	0
$961$	23.8954	0.770818
$962$	0	0
$963$	5.21555	0.168069
$964$	0	0
$965$	47.3094	1.52294
$966$	0	0
$967$	−15.9932	−0.514306	−0.257153	−	0.966371i	$-0.582784\pi$
$967$	−15.9932	−0.514306	−0.257153	+	0.966371i	$0.582784\pi$
$968$	0	0
$969$	53.8286	1.72922
$970$	0	0
$971$	18.1548	0.582615	0.291307	−	0.956630i	$-0.405910\pi$
$971$	18.1548	0.582615	0.291307	+	0.956630i	$0.405910\pi$
$972$	0	0
$973$	−15.6992	−0.503293
$974$	0	0
$975$	20.3557	0.651904
$976$	0	0
$977$	−17.4416	−0.558007	−0.279003	−	0.960290i	$-0.590004\pi$
$977$	−17.4416	−0.558007	−0.279003	+	0.960290i	$0.590004\pi$
$978$	0	0
$979$	0.229569	0.00733707
$980$	0	0
$981$	−35.3265	−1.12789
$982$	0	0
$983$	47.5604	1.51694	0.758471	−	0.651707i	$-0.225947\pi$
$983$	47.5604	1.51694	0.758471	+	0.651707i	$0.225947\pi$
$984$	0	0
$985$	−86.8324	−2.76671
$986$	0	0
$987$	−2.50713	−0.0798029
$988$	0	0
$989$	−44.4400	−1.41311
$990$	0	0
$991$	−45.1150	−1.43312	−0.716562	−	0.697523i	$-0.754285\pi$
$991$	−45.1150	−1.43312	−0.716562	+	0.697523i	$0.754285\pi$
$992$	0	0
$993$	41.9922	1.33258
$994$	0	0
$995$	−77.8240	−2.46719
$996$	0	0
$997$	−29.7503	−0.942202	−0.471101	−	0.882079i	$-0.656143\pi$
$997$	−29.7503	−0.942202	−0.471101	+	0.882079i	$0.656143\pi$
$998$	0	0
$999$	−17.7950	−0.563010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.w.1.10	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.w.1.10	✓	11	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)

\(f(q)\)	\(=\)	\(q+2.65994 q^{3} +3.55706 q^{5} -1.00000 q^{7} +4.07531 q^{9} +O(q^{10})\)	\(q+2.65994 q^{3} +3.55706 q^{5} -1.00000 q^{7} +4.07531 q^{9} +1.00000 q^{11} +1.00000 q^{13} +9.46158 q^{15} +4.54294 q^{17} +4.45455 q^{19} -2.65994 q^{21} -4.55552 q^{23} +7.65268 q^{25} +2.86026 q^{27} +6.15552 q^{29} +7.40914 q^{31} +2.65994 q^{33} -3.55706 q^{35} -6.22148 q^{37} +2.65994 q^{39} -8.16590 q^{41} +9.75519 q^{43} +14.4961 q^{45} +0.942550 q^{47} +1.00000 q^{49} +12.0840 q^{51} -3.26085 q^{53} +3.55706 q^{55} +11.8488 q^{57} -9.96964 q^{59} -12.4896 q^{61} -4.07531 q^{63} +3.55706 q^{65} -8.27548 q^{67} -12.1174 q^{69} +4.98120 q^{71} -6.49106 q^{73} +20.3557 q^{75} -1.00000 q^{77} -14.2714 q^{79} -4.61779 q^{81} +16.1053 q^{83} +16.1595 q^{85} +16.3733 q^{87} +0.229569 q^{89} -1.00000 q^{91} +19.7079 q^{93} +15.8451 q^{95} -8.96005 q^{97} +4.07531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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