LMFDB - Embedded newform 2008.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28390 q^{3} +3.72143 q^{5} -0.978625 q^{7} -1.35160 q^{9} +O(q^{10})$	$q-1.28390 q^{3} +3.72143 q^{5} -0.978625 q^{7} -1.35160 q^{9} +0.158851 q^{11} +0.124371 q^{13} -4.77795 q^{15} +0.487197 q^{17} +4.09613 q^{19} +1.25646 q^{21} -5.58572 q^{23} +8.84905 q^{25} +5.58702 q^{27} +9.50175 q^{29} -4.53723 q^{31} -0.203949 q^{33} -3.64189 q^{35} -4.70240 q^{37} -0.159680 q^{39} +8.31648 q^{41} +12.3553 q^{43} -5.02988 q^{45} +4.37024 q^{47} -6.04229 q^{49} -0.625513 q^{51} +10.3814 q^{53} +0.591152 q^{55} -5.25902 q^{57} -6.42238 q^{59} +6.11573 q^{61} +1.32271 q^{63} +0.462838 q^{65} -1.25850 q^{67} +7.17151 q^{69} -15.1543 q^{71} +15.8278 q^{73} -11.3613 q^{75} -0.155455 q^{77} -5.29643 q^{79} -3.11838 q^{81} -14.8743 q^{83} +1.81307 q^{85} -12.1993 q^{87} -5.47610 q^{89} -0.121713 q^{91} +5.82535 q^{93} +15.2435 q^{95} +5.60112 q^{97} -0.214703 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	−1.28390	−0.741260	−0.370630	−	0.928781i	$-0.620858\pi$
$3$	−1.28390	−0.741260	−0.370630	+	0.928781i	$0.620858\pi$
$4$	0	0
$5$	3.72143	1.66427	0.832137	−	0.554570i	$-0.187117\pi$
$5$	3.72143	1.66427	0.832137	+	0.554570i	$0.187117\pi$
$6$	0	0
$7$	−0.978625	−0.369886	−0.184943	−	0.982749i	$-0.559210\pi$
$7$	−0.978625	−0.369886	−0.184943	+	0.982749i	$0.559210\pi$
$8$	0	0
$9$	−1.35160	−0.450533
$10$	0	0
$11$	0.158851	0.0478953	0.0239477	−	0.999713i	$-0.492376\pi$
$11$	0.158851	0.0478953	0.0239477	+	0.999713i	$0.492376\pi$
$12$	0	0
$13$	0.124371	0.0344943	0.0172471	−	0.999851i	$-0.494510\pi$
$13$	0.124371	0.0344943	0.0172471	+	0.999851i	$0.494510\pi$
$14$	0	0
$15$	−4.77795	−1.23366
$16$	0	0
$17$	0.487197	0.118163	0.0590813	−	0.998253i	$-0.481183\pi$
$17$	0.487197	0.118163	0.0590813	+	0.998253i	$0.481183\pi$
$18$	0	0
$19$	4.09613	0.939716	0.469858	−	0.882742i	$-0.344305\pi$
$19$	4.09613	0.939716	0.469858	+	0.882742i	$0.344305\pi$
$20$	0	0
$21$	1.25646	0.274182
$22$	0	0
$23$	−5.58572	−1.16470	−0.582351	−	0.812937i	$-0.697867\pi$
$23$	−5.58572	−1.16470	−0.582351	+	0.812937i	$0.697867\pi$
$24$	0	0
$25$	8.84905	1.76981
$26$	0	0
$27$	5.58702	1.07522
$28$	0	0
$29$	9.50175	1.76443	0.882215	−	0.470847i	$-0.156052\pi$
$29$	9.50175	1.76443	0.882215	+	0.470847i	$0.156052\pi$
$30$	0	0
$31$	−4.53723	−0.814911	−0.407455	−	0.913225i	$-0.633584\pi$
$31$	−4.53723	−0.814911	−0.407455	+	0.913225i	$0.633584\pi$
$32$	0	0
$33$	−0.203949	−0.0355029
$34$	0	0
$35$	−3.64189	−0.615591
$36$	0	0
$37$	−4.70240	−0.773069	−0.386535	−	0.922275i	$-0.626328\pi$
$37$	−4.70240	−0.773069	−0.386535	+	0.922275i	$0.626328\pi$
$38$	0	0
$39$	−0.159680	−0.0255692
$40$	0	0
$41$	8.31648	1.29882	0.649408	−	0.760440i	$-0.275017\pi$
$41$	8.31648	1.29882	0.649408	+	0.760440i	$0.275017\pi$
$42$	0	0
$43$	12.3553	1.88417	0.942084	−	0.335376i	$-0.108864\pi$
$43$	12.3553	1.88417	0.942084	+	0.335376i	$0.108864\pi$
$44$	0	0
$45$	−5.02988	−0.749811
$46$	0	0
$47$	4.37024	0.637465	0.318732	−	0.947845i	$-0.396743\pi$
$47$	4.37024	0.637465	0.318732	+	0.947845i	$0.396743\pi$
$48$	0	0
$49$	−6.04229	−0.863185
$50$	0	0
$51$	−0.625513	−0.0875893
$52$	0	0
$53$	10.3814	1.42600	0.713000	−	0.701164i	$-0.247336\pi$
$53$	10.3814	1.42600	0.713000	+	0.701164i	$0.247336\pi$
$54$	0	0
$55$	0.591152	0.0797110
$56$	0	0
$57$	−5.25902	−0.696574
$58$	0	0
$59$	−6.42238	−0.836123	−0.418062	−	0.908419i	$-0.637290\pi$
$59$	−6.42238	−0.836123	−0.418062	+	0.908419i	$0.637290\pi$
$60$	0	0
$61$	6.11573	0.783039	0.391520	−	0.920170i	$-0.371950\pi$
$61$	6.11573	0.783039	0.391520	+	0.920170i	$0.371950\pi$
$62$	0	0
$63$	1.32271	0.166646
$64$	0	0
$65$	0.462838	0.0574080
$66$	0	0
$67$	−1.25850	−0.153750	−0.0768749	−	0.997041i	$-0.524494\pi$
$67$	−1.25850	−0.153750	−0.0768749	+	0.997041i	$0.524494\pi$
$68$	0	0
$69$	7.17151	0.863348
$70$	0	0
$71$	−15.1543	−1.79849	−0.899244	−	0.437447i	$-0.855883\pi$
$71$	−15.1543	−1.79849	−0.899244	+	0.437447i	$0.855883\pi$
$72$	0	0
$73$	15.8278	1.85250	0.926250	−	0.376909i	$-0.123013\pi$
$73$	15.8278	1.85250	0.926250	+	0.376909i	$0.123013\pi$
$74$	0	0
$75$	−11.3613	−1.31189
$76$	0	0
$77$	−0.155455	−0.0177158
$78$	0	0
$79$	−5.29643	−0.595895	−0.297948	−	0.954582i	$-0.596302\pi$
$79$	−5.29643	−0.595895	−0.297948	+	0.954582i	$0.596302\pi$
$80$	0	0
$81$	−3.11838	−0.346487
$82$	0	0
$83$	−14.8743	−1.63267	−0.816334	−	0.577580i	$-0.803997\pi$
$83$	−14.8743	−1.63267	−0.816334	+	0.577580i	$0.803997\pi$
$84$	0	0
$85$	1.81307	0.196655
$86$	0	0
$87$	−12.1993	−1.30790
$88$	0	0
$89$	−5.47610	−0.580465	−0.290233	−	0.956956i	$-0.593733\pi$
$89$	−5.47610	−0.580465	−0.290233	+	0.956956i	$0.593733\pi$
$90$	0	0
$91$	−0.121713	−0.0127589
$92$	0	0
$93$	5.82535	0.604061
$94$	0	0
$95$	15.2435	1.56395
$96$	0	0
$97$	5.60112	0.568708	0.284354	−	0.958719i	$-0.408221\pi$
$97$	5.60112	0.568708	0.284354	+	0.958719i	$0.408221\pi$
$98$	0	0
$99$	−0.214703	−0.0215784
$100$	0	0
$101$	13.3827	1.33163	0.665815	−	0.746117i	$-0.268084\pi$
$101$	13.3827	1.33163	0.665815	+	0.746117i	$0.268084\pi$
$102$	0	0
$103$	12.1928	1.20139	0.600695	−	0.799478i	$-0.294891\pi$
$103$	12.1928	1.20139	0.600695	+	0.799478i	$0.294891\pi$
$104$	0	0
$105$	4.67582	0.456313
$106$	0	0
$107$	18.0707	1.74696	0.873481	−	0.486859i	$-0.161857\pi$
$107$	18.0707	1.74696	0.873481	+	0.486859i	$0.161857\pi$
$108$	0	0
$109$	−0.754731	−0.0722901	−0.0361451	−	0.999347i	$-0.511508\pi$
$109$	−0.754731	−0.0722901	−0.0361451	+	0.999347i	$0.511508\pi$
$110$	0	0
$111$	6.03741	0.573045
$112$	0	0
$113$	−14.2548	−1.34097	−0.670487	−	0.741921i	$-0.733915\pi$
$113$	−14.2548	−1.34097	−0.670487	+	0.741921i	$0.733915\pi$
$114$	0	0
$115$	−20.7869	−1.93838
$116$	0	0
$117$	−0.168100	−0.0155408
$118$	0	0
$119$	−0.476783	−0.0437067
$120$	0	0
$121$	−10.9748	−0.997706
$122$	0	0
$123$	−10.6775	−0.962761
$124$	0	0
$125$	14.3240	1.28118
$126$	0	0
$127$	16.6350	1.47612	0.738061	−	0.674735i	$-0.235742\pi$
$127$	16.6350	1.47612	0.738061	+	0.674735i	$0.235742\pi$
$128$	0	0
$129$	−15.8630	−1.39666
$130$	0	0
$131$	15.1564	1.32422	0.662110	−	0.749407i	$-0.269661\pi$
$131$	15.1564	1.32422	0.662110	+	0.749407i	$0.269661\pi$
$132$	0	0
$133$	−4.00857	−0.347587
$134$	0	0
$135$	20.7917	1.78947
$136$	0	0
$137$	3.64085	0.311058	0.155529	−	0.987831i	$-0.450292\pi$
$137$	3.64085	0.311058	0.155529	+	0.987831i	$0.450292\pi$
$138$	0	0
$139$	8.66321	0.734804	0.367402	−	0.930062i	$-0.380247\pi$
$139$	8.66321	0.734804	0.367402	+	0.930062i	$0.380247\pi$
$140$	0	0
$141$	−5.61095	−0.472527
$142$	0	0
$143$	0.0197564	0.00165211
$144$	0	0
$145$	35.3601	2.93650
$146$	0	0
$147$	7.75770	0.639845
$148$	0	0
$149$	11.5090	0.942855	0.471428	−	0.881905i	$-0.343739\pi$
$149$	11.5090	0.942855	0.471428	+	0.881905i	$0.343739\pi$
$150$	0	0
$151$	−4.44178	−0.361467	−0.180734	−	0.983532i	$-0.557847\pi$
$151$	−4.44178	−0.361467	−0.180734	+	0.983532i	$0.557847\pi$
$152$	0	0
$153$	−0.658495	−0.0532362
$154$	0	0
$155$	−16.8850	−1.35624
$156$	0	0
$157$	−6.60217	−0.526910	−0.263455	−	0.964672i	$-0.584862\pi$
$157$	−6.60217	−0.526910	−0.263455	+	0.964672i	$0.584862\pi$
$158$	0	0
$159$	−13.3287	−1.05704
$160$	0	0
$161$	5.46632	0.430807
$162$	0	0
$163$	12.2503	0.959520	0.479760	−	0.877400i	$-0.340724\pi$
$163$	12.2503	0.959520	0.479760	+	0.877400i	$0.340724\pi$
$164$	0	0
$165$	−0.758981	−0.0590866
$166$	0	0
$167$	20.1260	1.55740	0.778698	−	0.627399i	$-0.215880\pi$
$167$	20.1260	1.55740	0.778698	+	0.627399i	$0.215880\pi$
$168$	0	0
$169$	−12.9845	−0.998810
$170$	0	0
$171$	−5.53632	−0.423373
$172$	0	0
$173$	−20.9139	−1.59006	−0.795029	−	0.606572i	$-0.792544\pi$
$173$	−20.9139	−1.59006	−0.795029	+	0.606572i	$0.792544\pi$
$174$	0	0
$175$	−8.65990	−0.654627
$176$	0	0
$177$	8.24570	0.619785
$178$	0	0
$179$	−9.33184	−0.697495	−0.348747	−	0.937217i	$-0.613393\pi$
$179$	−9.33184	−0.697495	−0.348747	+	0.937217i	$0.613393\pi$
$180$	0	0
$181$	4.39638	0.326780	0.163390	−	0.986562i	$-0.447757\pi$
$181$	4.39638	0.326780	0.163390	+	0.986562i	$0.447757\pi$
$182$	0	0
$183$	−7.85199	−0.580436
$184$	0	0
$185$	−17.4996	−1.28660
$186$	0	0
$187$	0.0773917	0.00565944
$188$	0	0
$189$	−5.46760	−0.397709
$190$	0	0
$191$	−4.38279	−0.317127	−0.158564	−	0.987349i	$-0.550686\pi$
$191$	−4.38279	−0.317127	−0.158564	+	0.987349i	$0.550686\pi$
$192$	0	0
$193$	25.1081	1.80732	0.903660	−	0.428251i	$-0.140870\pi$
$193$	25.1081	1.80732	0.903660	+	0.428251i	$0.140870\pi$
$194$	0	0
$195$	−0.594238	−0.0425543
$196$	0	0
$197$	8.19301	0.583728	0.291864	−	0.956460i	$-0.405725\pi$
$197$	8.19301	0.583728	0.291864	+	0.956460i	$0.405725\pi$
$198$	0	0
$199$	17.3067	1.22684	0.613420	−	0.789757i	$-0.289793\pi$
$199$	17.3067	1.22684	0.613420	+	0.789757i	$0.289793\pi$
$200$	0	0
$201$	1.61578	0.113969
$202$	0	0
$203$	−9.29865	−0.652637
$204$	0	0
$205$	30.9492	2.16159
$206$	0	0
$207$	7.54965	0.524737
$208$	0	0
$209$	0.650673	0.0450080
$210$	0	0
$211$	7.72391	0.531736	0.265868	−	0.964009i	$-0.414341\pi$
$211$	7.72391	0.531736	0.265868	+	0.964009i	$0.414341\pi$
$212$	0	0
$213$	19.4567	1.33315
$214$	0	0
$215$	45.9795	3.13577
$216$	0	0
$217$	4.44025	0.301424
$218$	0	0
$219$	−20.3213	−1.37319
$220$	0	0
$221$	0.0605932	0.00407594
$222$	0	0
$223$	8.79546	0.588987	0.294494	−	0.955653i	$-0.404849\pi$
$223$	8.79546	0.588987	0.294494	+	0.955653i	$0.404849\pi$
$224$	0	0
$225$	−11.9604	−0.797358
$226$	0	0
$227$	−2.29104	−0.152061	−0.0760307	−	0.997105i	$-0.524225\pi$
$227$	−2.29104	−0.152061	−0.0760307	+	0.997105i	$0.524225\pi$
$228$	0	0
$229$	6.59267	0.435656	0.217828	−	0.975987i	$-0.430103\pi$
$229$	6.59267	0.435656	0.217828	+	0.975987i	$0.430103\pi$
$230$	0	0
$231$	0.199589	0.0131320
$232$	0	0
$233$	−28.8484	−1.88992	−0.944962	−	0.327180i	$-0.893902\pi$
$233$	−28.8484	−1.88992	−0.944962	+	0.327180i	$0.893902\pi$
$234$	0	0
$235$	16.2635	1.06092
$236$	0	0
$237$	6.80010	0.441714
$238$	0	0
$239$	−6.03122	−0.390127	−0.195064	−	0.980791i	$-0.562491\pi$
$239$	−6.03122	−0.390127	−0.195064	+	0.980791i	$0.562491\pi$
$240$	0	0
$241$	−18.5017	−1.19180	−0.595899	−	0.803059i	$-0.703204\pi$
$241$	−18.5017	−1.19180	−0.595899	+	0.803059i	$0.703204\pi$
$242$	0	0
$243$	−12.7574	−0.818386
$244$	0	0
$245$	−22.4860	−1.43658
$246$	0	0
$247$	0.509439	0.0324148
$248$	0	0
$249$	19.0971	1.21023
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.887296	−0.0557838
$254$	0	0
$255$	−2.32780	−0.145773
$256$	0	0
$257$	18.5580	1.15761	0.578807	−	0.815465i	$-0.303519\pi$
$257$	18.5580	1.15761	0.578807	+	0.815465i	$0.303519\pi$
$258$	0	0
$259$	4.60188	0.285947
$260$	0	0
$261$	−12.8426	−0.794934
$262$	0	0
$263$	−13.4324	−0.828276	−0.414138	−	0.910214i	$-0.635917\pi$
$263$	−13.4324	−0.828276	−0.414138	+	0.910214i	$0.635917\pi$
$264$	0	0
$265$	38.6338	2.37326
$266$	0	0
$267$	7.03077	0.430276
$268$	0	0
$269$	−19.5121	−1.18968	−0.594838	−	0.803846i	$-0.702784\pi$
$269$	−19.5121	−1.18968	−0.594838	+	0.803846i	$0.702784\pi$
$270$	0	0
$271$	−19.8337	−1.20481	−0.602405	−	0.798191i	$-0.705791\pi$
$271$	−19.8337	−1.20481	−0.602405	+	0.798191i	$0.705791\pi$
$272$	0	0
$273$	0.156267	0.00945770
$274$	0	0
$275$	1.40568	0.0847656
$276$	0	0
$277$	−24.3978	−1.46592	−0.732960	−	0.680271i	$-0.761862\pi$
$277$	−24.3978	−1.46592	−0.732960	+	0.680271i	$0.761862\pi$
$278$	0	0
$279$	6.13252	0.367144
$280$	0	0
$281$	16.6184	0.991370	0.495685	−	0.868502i	$-0.334917\pi$
$281$	16.6184	0.991370	0.495685	+	0.868502i	$0.334917\pi$
$282$	0	0
$283$	1.11551	0.0663101	0.0331550	−	0.999450i	$-0.489444\pi$
$283$	1.11551	0.0663101	0.0331550	+	0.999450i	$0.489444\pi$
$284$	0	0
$285$	−19.5711	−1.15929
$286$	0	0
$287$	−8.13872	−0.480413
$288$	0	0
$289$	−16.7626	−0.986038
$290$	0	0
$291$	−7.19128	−0.421560
$292$	0	0
$293$	−3.13480	−0.183137	−0.0915685	−	0.995799i	$-0.529188\pi$
$293$	−3.13480	−0.183137	−0.0915685	+	0.995799i	$0.529188\pi$
$294$	0	0
$295$	−23.9005	−1.39154
$296$	0	0
$297$	0.887503	0.0514981
$298$	0	0
$299$	−0.694701	−0.0401756
$300$	0	0
$301$	−12.0912	−0.696927
$302$	0	0
$303$	−17.1821	−0.987084
$304$	0	0
$305$	22.7593	1.30319
$306$	0	0
$307$	7.08635	0.404439	0.202220	−	0.979340i	$-0.435184\pi$
$307$	7.08635	0.404439	0.202220	+	0.979340i	$0.435184\pi$
$308$	0	0
$309$	−15.6543	−0.890543
$310$	0	0
$311$	6.79534	0.385328	0.192664	−	0.981265i	$-0.438287\pi$
$311$	6.79534	0.385328	0.192664	+	0.981265i	$0.438287\pi$
$312$	0	0
$313$	−21.1021	−1.19276	−0.596381	−	0.802701i	$-0.703395\pi$
$313$	−21.1021	−1.19276	−0.596381	+	0.802701i	$0.703395\pi$
$314$	0	0
$315$	4.92237	0.277344
$316$	0	0
$317$	−17.3791	−0.976106	−0.488053	−	0.872814i	$-0.662293\pi$
$317$	−17.3791	−0.976106	−0.488053	+	0.872814i	$0.662293\pi$
$318$	0	0
$319$	1.50936	0.0845079
$320$	0	0
$321$	−23.2010	−1.29495
$322$	0	0
$323$	1.99562	0.111039
$324$	0	0
$325$	1.10056	0.0610483
$326$	0	0
$327$	0.968999	0.0535858
$328$	0	0
$329$	−4.27682	−0.235789
$330$	0	0
$331$	−9.36302	−0.514638	−0.257319	−	0.966326i	$-0.582839\pi$
$331$	−9.36302	−0.514638	−0.257319	+	0.966326i	$0.582839\pi$
$332$	0	0
$333$	6.35576	0.348293
$334$	0	0
$335$	−4.68341	−0.255882
$336$	0	0
$337$	−6.97394	−0.379895	−0.189947	−	0.981794i	$-0.560832\pi$
$337$	−6.97394	−0.379895	−0.189947	+	0.981794i	$0.560832\pi$
$338$	0	0
$339$	18.3017	0.994011
$340$	0	0
$341$	−0.720743	−0.0390304
$342$	0	0
$343$	12.7635	0.689165
$344$	0	0
$345$	26.6883	1.43685
$346$	0	0
$347$	−20.8163	−1.11748	−0.558740	−	0.829343i	$-0.688715\pi$
$347$	−20.8163	−1.11748	−0.558740	+	0.829343i	$0.688715\pi$
$348$	0	0
$349$	−6.73006	−0.360252	−0.180126	−	0.983644i	$-0.557651\pi$
$349$	−6.73006	−0.360252	−0.180126	+	0.983644i	$0.557651\pi$
$350$	0	0
$351$	0.694863	0.0370890
$352$	0	0
$353$	22.9404	1.22099	0.610496	−	0.792019i	$-0.290970\pi$
$353$	22.9404	1.22099	0.610496	+	0.792019i	$0.290970\pi$
$354$	0	0
$355$	−56.3958	−2.99318
$356$	0	0
$357$	0.612143	0.0323980
$358$	0	0
$359$	−9.60134	−0.506739	−0.253370	−	0.967370i	$-0.581539\pi$
$359$	−9.60134	−0.506739	−0.253370	+	0.967370i	$0.581539\pi$
$360$	0	0
$361$	−2.22175	−0.116934
$362$	0	0
$363$	14.0905	0.739560
$364$	0	0
$365$	58.9020	3.08307
$366$	0	0
$367$	−20.2307	−1.05604	−0.528018	−	0.849233i	$-0.677065\pi$
$367$	−20.2307	−1.05604	−0.528018	+	0.849233i	$0.677065\pi$
$368$	0	0
$369$	−11.2406	−0.585160
$370$	0	0
$371$	−10.1595	−0.527457
$372$	0	0
$373$	−22.1957	−1.14925	−0.574625	−	0.818417i	$-0.694852\pi$
$373$	−22.1957	−1.14925	−0.574625	+	0.818417i	$0.694852\pi$
$374$	0	0
$375$	−18.3906	−0.949684
$376$	0	0
$377$	1.18174	0.0608628
$378$	0	0
$379$	−6.31846	−0.324557	−0.162279	−	0.986745i	$-0.551884\pi$
$379$	−6.31846	−0.324557	−0.162279	+	0.986745i	$0.551884\pi$
$380$	0	0
$381$	−21.3577	−1.09419
$382$	0	0
$383$	3.99897	0.204338	0.102169	−	0.994767i	$-0.467422\pi$
$383$	3.99897	0.204338	0.102169	+	0.994767i	$0.467422\pi$
$384$	0	0
$385$	−0.578517	−0.0294839
$386$	0	0
$387$	−16.6994	−0.848880
$388$	0	0
$389$	36.6251	1.85697	0.928484	−	0.371372i	$-0.121113\pi$
$389$	36.6251	1.85697	0.928484	+	0.371372i	$0.121113\pi$
$390$	0	0
$391$	−2.72135	−0.137624
$392$	0	0
$393$	−19.4593	−0.981592
$394$	0	0
$395$	−19.7103	−0.991734
$396$	0	0
$397$	−21.1127	−1.05962	−0.529808	−	0.848118i	$-0.677736\pi$
$397$	−21.1127	−1.05962	−0.529808	+	0.848118i	$0.677736\pi$
$398$	0	0
$399$	5.14661	0.257653
$400$	0	0
$401$	−31.0701	−1.55156	−0.775782	−	0.631001i	$-0.782645\pi$
$401$	−31.0701	−1.55156	−0.775782	+	0.631001i	$0.782645\pi$
$402$	0	0
$403$	−0.564300	−0.0281098
$404$	0	0
$405$	−11.6048	−0.576649
$406$	0	0
$407$	−0.746979	−0.0370264
$408$	0	0
$409$	−1.73077	−0.0855809	−0.0427905	−	0.999084i	$-0.513625\pi$
$409$	−1.73077	−0.0855809	−0.0427905	+	0.999084i	$0.513625\pi$
$410$	0	0
$411$	−4.67448	−0.230575
$412$	0	0
$413$	6.28511	0.309270
$414$	0	0
$415$	−55.3538	−2.71721
$416$	0	0
$417$	−11.1227	−0.544681
$418$	0	0
$419$	11.4727	0.560480	0.280240	−	0.959930i	$-0.409586\pi$
$419$	11.4727	0.560480	0.280240	+	0.959930i	$0.409586\pi$
$420$	0	0
$421$	−30.9162	−1.50676	−0.753382	−	0.657583i	$-0.771579\pi$
$421$	−30.9162	−1.50676	−0.753382	+	0.657583i	$0.771579\pi$
$422$	0	0
$423$	−5.90681	−0.287199
$424$	0	0
$425$	4.31123	0.209125
$426$	0	0
$427$	−5.98501	−0.289635
$428$	0	0
$429$	−0.0253653	−0.00122465
$430$	0	0
$431$	−2.84348	−0.136966	−0.0684829	−	0.997652i	$-0.521816\pi$
$431$	−2.84348	−0.136966	−0.0684829	+	0.997652i	$0.521816\pi$
$432$	0	0
$433$	−9.87776	−0.474695	−0.237347	−	0.971425i	$-0.576278\pi$
$433$	−9.87776	−0.474695	−0.237347	+	0.971425i	$0.576278\pi$
$434$	0	0
$435$	−45.3988	−2.17671
$436$	0	0
$437$	−22.8798	−1.09449
$438$	0	0
$439$	21.7174	1.03652	0.518259	−	0.855224i	$-0.326580\pi$
$439$	21.7174	1.03652	0.518259	+	0.855224i	$0.326580\pi$
$440$	0	0
$441$	8.16676	0.388893
$442$	0	0
$443$	−5.12886	−0.243679	−0.121840	−	0.992550i	$-0.538879\pi$
$443$	−5.12886	−0.243679	−0.121840	+	0.992550i	$0.538879\pi$
$444$	0	0
$445$	−20.3789	−0.966054
$446$	0	0
$447$	−14.7764	−0.698901
$448$	0	0
$449$	20.9927	0.990707	0.495354	−	0.868691i	$-0.335039\pi$
$449$	20.9927	0.990707	0.495354	+	0.868691i	$0.335039\pi$
$450$	0	0
$451$	1.32108	0.0622072
$452$	0	0
$453$	5.70281	0.267941
$454$	0	0
$455$	−0.452945	−0.0212344
$456$	0	0
$457$	16.0836	0.752357	0.376178	−	0.926547i	$-0.377238\pi$
$457$	16.0836	0.752357	0.376178	+	0.926547i	$0.377238\pi$
$458$	0	0
$459$	2.72198	0.127051
$460$	0	0
$461$	−15.4988	−0.721850	−0.360925	−	0.932595i	$-0.617539\pi$
$461$	−15.4988	−0.721850	−0.360925	+	0.932595i	$0.617539\pi$
$462$	0	0
$463$	−39.1923	−1.82142	−0.910710	−	0.413046i	$-0.864465\pi$
$463$	−39.1923	−1.82142	−0.910710	+	0.413046i	$0.864465\pi$
$464$	0	0
$465$	21.6787	1.00532
$466$	0	0
$467$	−12.4339	−0.575372	−0.287686	−	0.957725i	$-0.592886\pi$
$467$	−12.4339	−0.575372	−0.287686	+	0.957725i	$0.592886\pi$
$468$	0	0
$469$	1.23160	0.0568698
$470$	0	0
$471$	8.47653	0.390578
$472$	0	0
$473$	1.96265	0.0902428
$474$	0	0
$475$	36.2468	1.66312
$476$	0	0
$477$	−14.0315	−0.642460
$478$	0	0
$479$	1.77090	0.0809143	0.0404571	−	0.999181i	$-0.487119\pi$
$479$	1.77090	0.0809143	0.0404571	+	0.999181i	$0.487119\pi$
$480$	0	0
$481$	−0.584841	−0.0266665
$482$	0	0
$483$	−7.01822	−0.319340
$484$	0	0
$485$	20.8442	0.946486
$486$	0	0
$487$	−6.05297	−0.274286	−0.137143	−	0.990551i	$-0.543792\pi$
$487$	−6.05297	−0.274286	−0.137143	+	0.990551i	$0.543792\pi$
$488$	0	0
$489$	−15.7282	−0.711254
$490$	0	0
$491$	−0.330789	−0.0149283	−0.00746415	−	0.999972i	$-0.502376\pi$
$491$	−0.330789	−0.0149283	−0.00746415	+	0.999972i	$0.502376\pi$
$492$	0	0
$493$	4.62922	0.208490
$494$	0	0
$495$	−0.799001	−0.0359124
$496$	0	0
$497$	14.8304	0.665235
$498$	0	0
$499$	9.18444	0.411152	0.205576	−	0.978641i	$-0.434093\pi$
$499$	9.18444	0.411152	0.205576	+	0.978641i	$0.434093\pi$
$500$	0	0
$501$	−25.8398	−1.15444
$502$	0	0
$503$	13.1410	0.585930	0.292965	−	0.956123i	$-0.405358\pi$
$503$	13.1410	0.585930	0.292965	+	0.956123i	$0.405358\pi$
$504$	0	0
$505$	49.8028	2.21620
$506$	0	0
$507$	16.6708	0.740378
$508$	0	0
$509$	12.2484	0.542902	0.271451	−	0.962452i	$-0.412497\pi$
$509$	12.2484	0.542902	0.271451	+	0.962452i	$0.412497\pi$
$510$	0	0
$511$	−15.4895	−0.685213
$512$	0	0
$513$	22.8851	1.01040
$514$	0	0
$515$	45.3746	1.99944
$516$	0	0
$517$	0.694216	0.0305316
$518$	0	0
$519$	26.8514	1.17865
$520$	0	0
$521$	−2.33003	−0.102080	−0.0510402	−	0.998697i	$-0.516254\pi$
$521$	−2.33003	−0.102080	−0.0510402	+	0.998697i	$0.516254\pi$
$522$	0	0
$523$	−43.3538	−1.89573	−0.947864	−	0.318675i	$-0.896762\pi$
$523$	−43.3538	−1.89573	−0.947864	+	0.318675i	$0.896762\pi$
$524$	0	0
$525$	11.1185	0.485249
$526$	0	0
$527$	−2.21053	−0.0962920
$528$	0	0
$529$	8.20024	0.356532
$530$	0	0
$531$	8.68049	0.376701
$532$	0	0
$533$	1.03433	0.0448017
$534$	0	0
$535$	67.2489	2.90742
$536$	0	0
$537$	11.9812	0.517025
$538$	0	0
$539$	−0.959823	−0.0413425
$540$	0	0
$541$	−11.7559	−0.505427	−0.252713	−	0.967541i	$-0.581323\pi$
$541$	−11.7559	−0.505427	−0.252713	+	0.967541i	$0.581323\pi$
$542$	0	0
$543$	−5.64451	−0.242229
$544$	0	0
$545$	−2.80868	−0.120311
$546$	0	0
$547$	20.7563	0.887477	0.443738	−	0.896156i	$-0.353652\pi$
$547$	20.7563	0.887477	0.443738	+	0.896156i	$0.353652\pi$
$548$	0	0
$549$	−8.26602	−0.352785
$550$	0	0
$551$	38.9203	1.65806
$552$	0	0
$553$	5.18323	0.220413
$554$	0	0
$555$	22.4678	0.953705
$556$	0	0
$557$	25.0496	1.06139	0.530694	−	0.847564i	$-0.321931\pi$
$557$	25.0496	1.06139	0.530694	+	0.847564i	$0.321931\pi$
$558$	0	0
$559$	1.53664	0.0649930
$560$	0	0
$561$	−0.0993632	−0.00419512
$562$	0	0
$563$	25.3564	1.06865	0.534323	−	0.845280i	$-0.320567\pi$
$563$	25.3564	1.06865	0.534323	+	0.845280i	$0.320567\pi$
$564$	0	0
$565$	−53.0481	−2.23175
$566$	0	0
$567$	3.05173	0.128160
$568$	0	0
$569$	−17.7495	−0.744100	−0.372050	−	0.928213i	$-0.621345\pi$
$569$	−17.7495	−0.744100	−0.372050	+	0.928213i	$0.621345\pi$
$570$	0	0
$571$	−42.6572	−1.78515	−0.892574	−	0.450900i	$-0.851103\pi$
$571$	−42.6572	−1.78515	−0.892574	+	0.450900i	$0.851103\pi$
$572$	0	0
$573$	5.62706	0.235074
$574$	0	0
$575$	−49.4283	−2.06130
$576$	0	0
$577$	35.3567	1.47192	0.735960	−	0.677025i	$-0.236731\pi$
$577$	35.3567	1.47192	0.735960	+	0.677025i	$0.236731\pi$
$578$	0	0
$579$	−32.2363	−1.33969
$580$	0	0
$581$	14.5564	0.603901
$582$	0	0
$583$	1.64910	0.0682987
$584$	0	0
$585$	−0.625571	−0.0258642
$586$	0	0
$587$	38.7462	1.59923	0.799614	−	0.600514i	$-0.205037\pi$
$587$	38.7462	1.59923	0.799614	+	0.600514i	$0.205037\pi$
$588$	0	0
$589$	−18.5851	−0.765785
$590$	0	0
$591$	−10.5190	−0.432694
$592$	0	0
$593$	35.0722	1.44024	0.720121	−	0.693849i	$-0.244086\pi$
$593$	35.0722	1.44024	0.720121	+	0.693849i	$0.244086\pi$
$594$	0	0
$595$	−1.77432	−0.0727399
$596$	0	0
$597$	−22.2201	−0.909408
$598$	0	0
$599$	18.9351	0.773667	0.386834	−	0.922149i	$-0.373569\pi$
$599$	18.9351	0.773667	0.386834	+	0.922149i	$0.373569\pi$
$600$	0	0
$601$	20.2368	0.825476	0.412738	−	0.910850i	$-0.364572\pi$
$601$	20.2368	0.825476	0.412738	+	0.910850i	$0.364572\pi$
$602$	0	0
$603$	1.70098	0.0692694
$604$	0	0
$605$	−40.8418	−1.66046
$606$	0	0
$607$	−11.4054	−0.462931	−0.231466	−	0.972843i	$-0.574352\pi$
$607$	−11.4054	−0.462931	−0.231466	+	0.972843i	$0.574352\pi$
$608$	0	0
$609$	11.9385	0.483774
$610$	0	0
$611$	0.543530	0.0219889
$612$	0	0
$613$	10.5342	0.425473	0.212737	−	0.977110i	$-0.431762\pi$
$613$	10.5342	0.425473	0.212737	+	0.977110i	$0.431762\pi$
$614$	0	0
$615$	−39.7357	−1.60230
$616$	0	0
$617$	8.84836	0.356222	0.178111	−	0.984010i	$-0.443001\pi$
$617$	8.84836	0.356222	0.178111	+	0.984010i	$0.443001\pi$
$618$	0	0
$619$	27.6532	1.11148	0.555738	−	0.831358i	$-0.312436\pi$
$619$	27.6532	1.11148	0.555738	+	0.831358i	$0.312436\pi$
$620$	0	0
$621$	−31.2075	−1.25231
$622$	0	0
$623$	5.35905	0.214706
$624$	0	0
$625$	9.06044	0.362417
$626$	0	0
$627$	−0.835399	−0.0333626
$628$	0	0
$629$	−2.29099	−0.0913479
$630$	0	0
$631$	30.2853	1.20564	0.602820	−	0.797877i	$-0.294044\pi$
$631$	30.2853	1.20564	0.602820	+	0.797877i	$0.294044\pi$
$632$	0	0
$633$	−9.91673	−0.394155
$634$	0	0
$635$	61.9062	2.45667
$636$	0	0
$637$	−0.751486	−0.0297749
$638$	0	0
$639$	20.4826	0.810279
$640$	0	0
$641$	6.48028	0.255956	0.127978	−	0.991777i	$-0.459151\pi$
$641$	6.48028	0.255956	0.127978	+	0.991777i	$0.459151\pi$
$642$	0	0
$643$	−31.9557	−1.26021	−0.630105	−	0.776510i	$-0.716988\pi$
$643$	−31.9557	−1.26021	−0.630105	+	0.776510i	$0.716988\pi$
$644$	0	0
$645$	−59.0331	−2.32442
$646$	0	0
$647$	28.7890	1.13181	0.565907	−	0.824469i	$-0.308526\pi$
$647$	28.7890	1.13181	0.565907	+	0.824469i	$0.308526\pi$
$648$	0	0
$649$	−1.02020	−0.0400464
$650$	0	0
$651$	−5.70084	−0.223434
$652$	0	0
$653$	−35.7344	−1.39840	−0.699198	−	0.714928i	$-0.746459\pi$
$653$	−35.7344	−1.39840	−0.699198	+	0.714928i	$0.746459\pi$
$654$	0	0
$655$	56.4035	2.20387
$656$	0	0
$657$	−21.3928	−0.834613
$658$	0	0
$659$	8.06551	0.314188	0.157094	−	0.987584i	$-0.449787\pi$
$659$	8.06551	0.314188	0.157094	+	0.987584i	$0.449787\pi$
$660$	0	0
$661$	−42.6595	−1.65926	−0.829632	−	0.558311i	$-0.811450\pi$
$661$	−42.6595	−1.65926	−0.829632	+	0.558311i	$0.811450\pi$
$662$	0	0
$663$	−0.0777956	−0.00302133
$664$	0	0
$665$	−14.9176	−0.578481
$666$	0	0
$667$	−53.0741	−2.05504
$668$	0	0
$669$	−11.2925	−0.436593
$670$	0	0
$671$	0.971489	0.0375039
$672$	0	0
$673$	−21.0702	−0.812196	−0.406098	−	0.913830i	$-0.633111\pi$
$673$	−21.0702	−0.812196	−0.406098	+	0.913830i	$0.633111\pi$
$674$	0	0
$675$	49.4398	1.90294
$676$	0	0
$677$	40.8903	1.57154	0.785770	−	0.618519i	$-0.212267\pi$
$677$	40.8903	1.57154	0.785770	+	0.618519i	$0.212267\pi$
$678$	0	0
$679$	−5.48140	−0.210357
$680$	0	0
$681$	2.94146	0.112717
$682$	0	0
$683$	2.36369	0.0904442	0.0452221	−	0.998977i	$-0.485600\pi$
$683$	2.36369	0.0904442	0.0452221	+	0.998977i	$0.485600\pi$
$684$	0	0
$685$	13.5492	0.517687
$686$	0	0
$687$	−8.46433	−0.322934
$688$	0	0
$689$	1.29115	0.0491889
$690$	0	0
$691$	9.94822	0.378448	0.189224	−	0.981934i	$-0.439403\pi$
$691$	9.94822	0.378448	0.189224	+	0.981934i	$0.439403\pi$
$692$	0	0
$693$	0.210113	0.00798155
$694$	0	0
$695$	32.2395	1.22292
$696$	0	0
$697$	4.05177	0.153472
$698$	0	0
$699$	37.0385	1.40093
$700$	0	0
$701$	−20.3292	−0.767825	−0.383912	−	0.923370i	$-0.625424\pi$
$701$	−20.3292	−0.767825	−0.383912	+	0.923370i	$0.625424\pi$
$702$	0	0
$703$	−19.2616	−0.726465
$704$	0	0
$705$	−20.8808	−0.786415
$706$	0	0
$707$	−13.0967	−0.492551
$708$	0	0
$709$	−15.5399	−0.583612	−0.291806	−	0.956478i	$-0.594256\pi$
$709$	−15.5399	−0.583612	−0.291806	+	0.956478i	$0.594256\pi$
$710$	0	0
$711$	7.15866	0.268471
$712$	0	0
$713$	25.3437	0.949129
$714$	0	0
$715$	0.0735222	0.00274957
$716$	0	0
$717$	7.74349	0.289186
$718$	0	0
$719$	−40.0743	−1.49452	−0.747259	−	0.664532i	$-0.768631\pi$
$719$	−40.0743	−1.49452	−0.747259	+	0.664532i	$0.768631\pi$
$720$	0	0
$721$	−11.9322	−0.444377
$722$	0	0
$723$	23.7543	0.883433
$724$	0	0
$725$	84.0814	3.12271
$726$	0	0
$727$	27.3048	1.01268	0.506340	−	0.862334i	$-0.330998\pi$
$727$	27.3048	1.01268	0.506340	+	0.862334i	$0.330998\pi$
$728$	0	0
$729$	25.7343	0.953124
$730$	0	0
$731$	6.01948	0.222638
$732$	0	0
$733$	−11.4085	−0.421382	−0.210691	−	0.977553i	$-0.567571\pi$
$733$	−11.4085	−0.421382	−0.210691	+	0.977553i	$0.567571\pi$
$734$	0	0
$735$	28.8698	1.06488
$736$	0	0
$737$	−0.199913	−0.00736389
$738$	0	0
$739$	−44.5501	−1.63880	−0.819402	−	0.573220i	$-0.805694\pi$
$739$	−44.5501	−1.63880	−0.819402	+	0.573220i	$0.805694\pi$
$740$	0	0
$741$	−0.654069	−0.0240278
$742$	0	0
$743$	−42.3103	−1.55221	−0.776106	−	0.630602i	$-0.782808\pi$
$743$	−42.3103	−1.55221	−0.776106	+	0.630602i	$0.782808\pi$
$744$	0	0
$745$	42.8300	1.56917
$746$	0	0
$747$	20.1041	0.735571
$748$	0	0
$749$	−17.6845	−0.646176
$750$	0	0
$751$	−25.2451	−0.921206	−0.460603	−	0.887606i	$-0.652367\pi$
$751$	−25.2451	−0.921206	−0.460603	+	0.887606i	$0.652367\pi$
$752$	0	0
$753$	1.28390	0.0467879
$754$	0	0
$755$	−16.5298	−0.601581
$756$	0	0
$757$	8.42631	0.306260	0.153130	−	0.988206i	$-0.451065\pi$
$757$	8.42631	0.306260	0.153130	+	0.988206i	$0.451065\pi$
$758$	0	0
$759$	1.13920	0.0413503
$760$	0	0
$761$	−18.7686	−0.680361	−0.340180	−	0.940360i	$-0.610488\pi$
$761$	−18.7686	−0.680361	−0.340180	+	0.940360i	$0.610488\pi$
$762$	0	0
$763$	0.738599	0.0267391
$764$	0	0
$765$	−2.45055	−0.0885997
$766$	0	0
$767$	−0.798758	−0.0288415
$768$	0	0
$769$	−38.5958	−1.39180	−0.695900	−	0.718138i	$-0.744995\pi$
$769$	−38.5958	−1.39180	−0.695900	+	0.718138i	$0.744995\pi$
$770$	0	0
$771$	−23.8266	−0.858093
$772$	0	0
$773$	−7.12987	−0.256444	−0.128222	−	0.991746i	$-0.540927\pi$
$773$	−7.12987	−0.256444	−0.128222	+	0.991746i	$0.540927\pi$
$774$	0	0
$775$	−40.1502	−1.44224
$776$	0	0
$777$	−5.90836	−0.211961
$778$	0	0
$779$	34.0654	1.22052
$780$	0	0
$781$	−2.40728	−0.0861392
$782$	0	0
$783$	53.0865	1.89715
$784$	0	0
$785$	−24.5695	−0.876923
$786$	0	0
$787$	−7.16095	−0.255260	−0.127630	−	0.991822i	$-0.540737\pi$
$787$	−7.16095	−0.255260	−0.127630	+	0.991822i	$0.540737\pi$
$788$	0	0
$789$	17.2458	0.613968
$790$	0	0
$791$	13.9501	0.496007
$792$	0	0
$793$	0.760619	0.0270104
$794$	0	0
$795$	−49.6020	−1.75920
$796$	0	0
$797$	−7.57184	−0.268208	−0.134104	−	0.990967i	$-0.542816\pi$
$797$	−7.57184	−0.268208	−0.134104	+	0.990967i	$0.542816\pi$
$798$	0	0
$799$	2.12917	0.0753245
$800$	0	0
$801$	7.40149	0.261519
$802$	0	0
$803$	2.51425	0.0887261
$804$	0	0
$805$	20.3426	0.716981
$806$	0	0
$807$	25.0516	0.881859
$808$	0	0
$809$	41.3021	1.45211	0.726053	−	0.687639i	$-0.241353\pi$
$809$	41.3021	1.45211	0.726053	+	0.687639i	$0.241353\pi$
$810$	0	0
$811$	−51.5573	−1.81042	−0.905211	−	0.424962i	$-0.860287\pi$
$811$	−51.5573	−1.81042	−0.905211	+	0.424962i	$0.860287\pi$
$812$	0	0
$813$	25.4645	0.893077
$814$	0	0
$815$	45.5888	1.59690
$816$	0	0
$817$	50.6089	1.77058
$818$	0	0
$819$	0.164507	0.00574833
$820$	0	0
$821$	−37.2073	−1.29854	−0.649271	−	0.760557i	$-0.724926\pi$
$821$	−37.2073	−1.29854	−0.649271	+	0.760557i	$0.724926\pi$
$822$	0	0
$823$	11.7945	0.411130	0.205565	−	0.978643i	$-0.434097\pi$
$823$	11.7945	0.411130	0.205565	+	0.978643i	$0.434097\pi$
$824$	0	0
$825$	−1.80475	−0.0628334
$826$	0	0
$827$	54.3940	1.89147	0.945733	−	0.324946i	$-0.105346\pi$
$827$	54.3940	1.89147	0.945733	+	0.324946i	$0.105346\pi$
$828$	0	0
$829$	−19.4462	−0.675394	−0.337697	−	0.941255i	$-0.609648\pi$
$829$	−19.4462	−0.675394	−0.337697	+	0.941255i	$0.609648\pi$
$830$	0	0
$831$	31.3243	1.08663
$832$	0	0
$833$	−2.94379	−0.101996
$834$	0	0
$835$	74.8975	2.59193
$836$	0	0
$837$	−25.3496	−0.876211
$838$	0	0
$839$	−40.1624	−1.38656	−0.693280	−	0.720669i	$-0.743835\pi$
$839$	−40.1624	−1.38656	−0.693280	+	0.720669i	$0.743835\pi$
$840$	0	0
$841$	61.2832	2.11321
$842$	0	0
$843$	−21.3364	−0.734863
$844$	0	0
$845$	−48.3210	−1.66229
$846$	0	0
$847$	10.7402	0.369037
$848$	0	0
$849$	−1.43220	−0.0491530
$850$	0	0
$851$	26.2663	0.900396
$852$	0	0
$853$	28.9463	0.991102	0.495551	−	0.868579i	$-0.334966\pi$
$853$	28.9463	0.991102	0.495551	+	0.868579i	$0.334966\pi$
$854$	0	0
$855$	−20.6030	−0.704609
$856$	0	0
$857$	−0.535180	−0.0182814	−0.00914071	−	0.999958i	$-0.502910\pi$
$857$	−0.535180	−0.0182814	−0.00914071	+	0.999958i	$0.502910\pi$
$858$	0	0
$859$	−15.3975	−0.525357	−0.262678	−	0.964883i	$-0.584606\pi$
$859$	−15.3975	−0.525357	−0.262678	+	0.964883i	$0.584606\pi$
$860$	0	0
$861$	10.4493	0.356111
$862$	0	0
$863$	−29.1111	−0.990955	−0.495477	−	0.868621i	$-0.665007\pi$
$863$	−29.1111	−0.990955	−0.495477	+	0.868621i	$0.665007\pi$
$864$	0	0
$865$	−77.8298	−2.64629
$866$	0	0
$867$	21.5216	0.730911
$868$	0	0
$869$	−0.841343	−0.0285406
$870$	0	0
$871$	−0.156520	−0.00530349
$872$	0	0
$873$	−7.57047	−0.256222
$874$	0	0
$875$	−14.0178	−0.473888
$876$	0	0
$877$	−4.25776	−0.143774	−0.0718871	−	0.997413i	$-0.522902\pi$
$877$	−4.25776	−0.143774	−0.0718871	+	0.997413i	$0.522902\pi$
$878$	0	0
$879$	4.02477	0.135752
$880$	0	0
$881$	−9.60023	−0.323440	−0.161720	−	0.986837i	$-0.551704\pi$
$881$	−9.60023	−0.323440	−0.161720	+	0.986837i	$0.551704\pi$
$882$	0	0
$883$	20.1278	0.677355	0.338678	−	0.940902i	$-0.390020\pi$
$883$	20.1278	0.677355	0.338678	+	0.940902i	$0.390020\pi$
$884$	0	0
$885$	30.6858	1.03149
$886$	0	0
$887$	−30.4028	−1.02082	−0.510412	−	0.859930i	$-0.670507\pi$
$887$	−30.4028	−1.02082	−0.510412	+	0.859930i	$0.670507\pi$
$888$	0	0
$889$	−16.2795	−0.545996
$890$	0	0
$891$	−0.495357	−0.0165951
$892$	0	0
$893$	17.9010	0.599036
$894$	0	0
$895$	−34.7278	−1.16082
$896$	0	0
$897$	0.891927	0.0297806
$898$	0	0
$899$	−43.1116	−1.43785
$900$	0	0
$901$	5.05781	0.168500
$902$	0	0
$903$	15.5239	0.516604
$904$	0	0
$905$	16.3608	0.543852
$906$	0	0
$907$	24.9774	0.829360	0.414680	−	0.909967i	$-0.363893\pi$
$907$	24.9774	0.829360	0.414680	+	0.909967i	$0.363893\pi$
$908$	0	0
$909$	−18.0881	−0.599943
$910$	0	0
$911$	31.4566	1.04221	0.521103	−	0.853494i	$-0.325521\pi$
$911$	31.4566	1.04221	0.521103	+	0.853494i	$0.325521\pi$
$912$	0	0
$913$	−2.36280	−0.0781972
$914$	0	0
$915$	−29.2206	−0.966005
$916$	0	0
$917$	−14.8324	−0.489810
$918$	0	0
$919$	−23.1207	−0.762681	−0.381341	−	0.924435i	$-0.624537\pi$
$919$	−23.1207	−0.762681	−0.381341	+	0.924435i	$0.624537\pi$
$920$	0	0
$921$	−9.09817	−0.299795
$922$	0	0
$923$	−1.88476	−0.0620376
$924$	0	0
$925$	−41.6117	−1.36819
$926$	0	0
$927$	−16.4797	−0.541266
$928$	0	0
$929$	−25.6481	−0.841486	−0.420743	−	0.907180i	$-0.638231\pi$
$929$	−25.6481	−0.841486	−0.420743	+	0.907180i	$0.638231\pi$
$930$	0	0
$931$	−24.7500	−0.811148
$932$	0	0
$933$	−8.72454	−0.285629
$934$	0	0
$935$	0.288008	0.00941886
$936$	0	0
$937$	−19.4551	−0.635572	−0.317786	−	0.948163i	$-0.602939\pi$
$937$	−19.4551	−0.635572	−0.317786	+	0.948163i	$0.602939\pi$
$938$	0	0
$939$	27.0930	0.884148
$940$	0	0
$941$	−10.6946	−0.348634	−0.174317	−	0.984690i	$-0.555772\pi$
$941$	−10.6946	−0.348634	−0.174317	+	0.984690i	$0.555772\pi$
$942$	0	0
$943$	−46.4535	−1.51273
$944$	0	0
$945$	−20.3473	−0.661898
$946$	0	0
$947$	27.6893	0.899781	0.449890	−	0.893084i	$-0.351463\pi$
$947$	27.6893	0.899781	0.449890	+	0.893084i	$0.351463\pi$
$948$	0	0
$949$	1.96852	0.0639007
$950$	0	0
$951$	22.3130	0.723549
$952$	0	0
$953$	−40.6787	−1.31771	−0.658856	−	0.752269i	$-0.728959\pi$
$953$	−40.6787	−1.31771	−0.658856	+	0.752269i	$0.728959\pi$
$954$	0	0
$955$	−16.3102	−0.527787
$956$	0	0
$957$	−1.93787	−0.0626424
$958$	0	0
$959$	−3.56302	−0.115056
$960$	0	0
$961$	−10.4135	−0.335920
$962$	0	0
$963$	−24.4244	−0.787064
$964$	0	0
$965$	93.4380	3.00788
$966$	0	0
$967$	36.7809	1.18279	0.591397	−	0.806381i	$-0.298577\pi$
$967$	36.7809	1.18279	0.591397	+	0.806381i	$0.298577\pi$
$968$	0	0
$969$	−2.56218	−0.0823090
$970$	0	0
$971$	−6.66929	−0.214028	−0.107014	−	0.994258i	$-0.534129\pi$
$971$	−6.66929	−0.214028	−0.107014	+	0.994258i	$0.534129\pi$
$972$	0	0
$973$	−8.47804	−0.271793
$974$	0	0
$975$	−1.41302	−0.0452527
$976$	0	0
$977$	−41.6789	−1.33343	−0.666713	−	0.745315i	$-0.732299\pi$
$977$	−41.6789	−1.33343	−0.666713	+	0.745315i	$0.732299\pi$
$978$	0	0
$979$	−0.869883	−0.0278016
$980$	0	0
$981$	1.02009	0.0325691
$982$	0	0
$983$	57.4121	1.83116	0.915580	−	0.402135i	$-0.131732\pi$
$983$	57.4121	1.83116	0.915580	+	0.402135i	$0.131732\pi$
$984$	0	0
$985$	30.4897	0.971484
$986$	0	0
$987$	5.49102	0.174781
$988$	0	0
$989$	−69.0133	−2.19450
$990$	0	0
$991$	−32.2071	−1.02309	−0.511546	−	0.859256i	$-0.670927\pi$
$991$	−32.2071	−1.02309	−0.511546	+	0.859256i	$0.670927\pi$
$992$	0	0
$993$	12.0212	0.381481
$994$	0	0
$995$	64.4058	2.04180
$996$	0	0
$997$	−31.6186	−1.00137	−0.500686	−	0.865629i	$-0.666919\pi$
$997$	−31.6186	−1.00137	−0.500686	+	0.865629i	$0.666919\pi$
$998$	0	0
$999$	−26.2724	−0.831221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.7	✓	23
4.3	odd	2		4016.2.a.m.1.17		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.7	✓	23	1.1	even	1	trivial
4016.2.a.m.1.17		23	4.3	odd	2

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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28390 q^{3} +3.72143 q^{5} -0.978625 q^{7} -1.35160 q^{9} +O(q^{10})\)	\(q-1.28390 q^{3} +3.72143 q^{5} -0.978625 q^{7} -1.35160 q^{9} +0.158851 q^{11} +0.124371 q^{13} -4.77795 q^{15} +0.487197 q^{17} +4.09613 q^{19} +1.25646 q^{21} -5.58572 q^{23} +8.84905 q^{25} +5.58702 q^{27} +9.50175 q^{29} -4.53723 q^{31} -0.203949 q^{33} -3.64189 q^{35} -4.70240 q^{37} -0.159680 q^{39} +8.31648 q^{41} +12.3553 q^{43} -5.02988 q^{45} +4.37024 q^{47} -6.04229 q^{49} -0.625513 q^{51} +10.3814 q^{53} +0.591152 q^{55} -5.25902 q^{57} -6.42238 q^{59} +6.11573 q^{61} +1.32271 q^{63} +0.462838 q^{65} -1.25850 q^{67} +7.17151 q^{69} -15.1543 q^{71} +15.8278 q^{73} -11.3613 q^{75} -0.155455 q^{77} -5.29643 q^{79} -3.11838 q^{81} -14.8743 q^{83} +1.81307 q^{85} -12.1993 q^{87} -5.47610 q^{89} -0.121713 q^{91} +5.82535 q^{93} +15.2435 q^{95} +5.60112 q^{97} -0.214703 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

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