LMFDB - Embedded newform 4004.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 93x^{6} - 119x^{5} - 174x^{4} + 107x^{3} + 51x^{2} - 17x - 2 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 93x^6 - 119x^5 - 174x^4 + 107x^3 + 51x^2 - 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.672934$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.672934 q^{3} +0.538228 q^{5} -1.00000 q^{7} -2.54716 q^{9} +O(q^{10})$	$q-0.672934 q^{3} +0.538228 q^{5} -1.00000 q^{7} -2.54716 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.362192 q^{15} -1.70606 q^{17} +4.21645 q^{19} +0.672934 q^{21} +3.56036 q^{23} -4.71031 q^{25} +3.73287 q^{27} +9.32798 q^{29} -5.69704 q^{31} +0.672934 q^{33} -0.538228 q^{35} -8.10326 q^{37} +0.672934 q^{39} -6.73633 q^{41} -12.5566 q^{43} -1.37095 q^{45} +12.2622 q^{47} +1.00000 q^{49} +1.14806 q^{51} +7.52462 q^{53} -0.538228 q^{55} -2.83739 q^{57} +1.63094 q^{59} +8.27727 q^{61} +2.54716 q^{63} -0.538228 q^{65} -8.26536 q^{67} -2.39589 q^{69} +9.58999 q^{71} +7.50579 q^{73} +3.16973 q^{75} +1.00000 q^{77} +12.6682 q^{79} +5.12950 q^{81} -10.4121 q^{83} -0.918247 q^{85} -6.27712 q^{87} +0.0335338 q^{89} +1.00000 q^{91} +3.83373 q^{93} +2.26941 q^{95} +12.7427 q^{97} +2.54716 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * 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
$3$	−0.672934	−0.388519	−0.194259	−	0.980950i	$-0.562230\pi$
$3$	−0.672934	−0.388519	−0.194259	+	0.980950i	$0.562230\pi$
$4$	0	0
$5$	0.538228	0.240703	0.120351	−	0.992731i	$-0.461598\pi$
$5$	0.538228	0.240703	0.120351	+	0.992731i	$0.461598\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.54716	−0.849053
$10$	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
$15$	−0.362192	−0.0935176
$16$	0	0
$17$	−1.70606	−0.413779	−0.206890	−	0.978364i	$-0.566334\pi$
$17$	−1.70606	−0.413779	−0.206890	+	0.978364i	$0.566334\pi$
$18$	0	0
$19$	4.21645	0.967320	0.483660	−	0.875256i	$-0.339307\pi$
$19$	4.21645	0.967320	0.483660	+	0.875256i	$0.339307\pi$
$20$	0	0
$21$	0.672934	0.146846
$22$	0	0
$23$	3.56036	0.742387	0.371194	−	0.928555i	$-0.378949\pi$
$23$	3.56036	0.742387	0.371194	+	0.928555i	$0.378949\pi$
$24$	0	0
$25$	−4.71031	−0.942062
$26$	0	0
$27$	3.73287	0.718392
$28$	0	0
$29$	9.32798	1.73216	0.866081	−	0.499903i	$-0.166631\pi$
$29$	9.32798	1.73216	0.866081	+	0.499903i	$0.166631\pi$
$30$	0	0
$31$	−5.69704	−1.02322	−0.511609	−	0.859218i	$-0.670951\pi$
$31$	−5.69704	−1.02322	−0.511609	+	0.859218i	$0.670951\pi$
$32$	0	0
$33$	0.672934	0.117143
$34$	0	0
$35$	−0.538228	−0.0909772
$36$	0	0
$37$	−8.10326	−1.33217	−0.666084	−	0.745877i	$-0.732031\pi$
$37$	−8.10326	−1.33217	−0.666084	+	0.745877i	$0.732031\pi$
$38$	0	0
$39$	0.672934	0.107756
$40$	0	0
$41$	−6.73633	−1.05204	−0.526019	−	0.850473i	$-0.676316\pi$
$41$	−6.73633	−1.05204	−0.526019	+	0.850473i	$0.676316\pi$
$42$	0	0
$43$	−12.5566	−1.91486	−0.957432	−	0.288660i	$-0.906790\pi$
$43$	−12.5566	−1.91486	−0.957432	+	0.288660i	$0.906790\pi$
$44$	0	0
$45$	−1.37095	−0.204370
$46$	0	0
$47$	12.2622	1.78863	0.894313	−	0.447441i	$-0.147665\pi$
$47$	12.2622	1.78863	0.894313	+	0.447441i	$0.147665\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.14806	0.160761
$52$	0	0
$53$	7.52462	1.03359	0.516793	−	0.856110i	$-0.327126\pi$
$53$	7.52462	1.03359	0.516793	+	0.856110i	$0.327126\pi$
$54$	0	0
$55$	−0.538228	−0.0725747
$56$	0	0
$57$	−2.83739	−0.375822
$58$	0	0
$59$	1.63094	0.212330	0.106165	−	0.994349i	$-0.466143\pi$
$59$	1.63094	0.212330	0.106165	+	0.994349i	$0.466143\pi$
$60$	0	0
$61$	8.27727	1.05980	0.529898	−	0.848061i	$-0.322230\pi$
$61$	8.27727	1.05980	0.529898	+	0.848061i	$0.322230\pi$
$62$	0	0
$63$	2.54716	0.320912
$64$	0	0
$65$	−0.538228	−0.0667590
$66$	0	0
$67$	−8.26536	−1.00977	−0.504887	−	0.863185i	$-0.668466\pi$
$67$	−8.26536	−1.00977	−0.504887	+	0.863185i	$0.668466\pi$
$68$	0	0
$69$	−2.39589	−0.288431
$70$	0	0
$71$	9.58999	1.13812	0.569061	−	0.822295i	$-0.307307\pi$
$71$	9.58999	1.13812	0.569061	+	0.822295i	$0.307307\pi$
$72$	0	0
$73$	7.50579	0.878486	0.439243	−	0.898368i	$-0.355247\pi$
$73$	7.50579	0.878486	0.439243	+	0.898368i	$0.355247\pi$
$74$	0	0
$75$	3.16973	0.366009
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	12.6682	1.42529	0.712644	−	0.701526i	$-0.247498\pi$
$79$	12.6682	1.42529	0.712644	+	0.701526i	$0.247498\pi$
$80$	0	0
$81$	5.12950	0.569945
$82$	0	0
$83$	−10.4121	−1.14288	−0.571438	−	0.820645i	$-0.693614\pi$
$83$	−10.4121	−1.14288	−0.571438	+	0.820645i	$0.693614\pi$
$84$	0	0
$85$	−0.918247	−0.0995979
$86$	0	0
$87$	−6.27712	−0.672977
$88$	0	0
$89$	0.0335338	0.00355458	0.00177729	−	0.999998i	$-0.499434\pi$
$89$	0.0335338	0.00355458	0.00177729	+	0.999998i	$0.499434\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	3.83373	0.397540
$94$	0	0
$95$	2.26941	0.232837
$96$	0	0
$97$	12.7427	1.29382	0.646910	−	0.762566i	$-0.276061\pi$
$97$	12.7427	1.29382	0.646910	+	0.762566i	$0.276061\pi$
$98$	0	0
$99$	2.54716	0.255999
$100$	0	0
$101$	11.1490	1.10936	0.554681	−	0.832063i	$-0.312840\pi$
$101$	11.1490	1.10936	0.554681	+	0.832063i	$0.312840\pi$
$102$	0	0
$103$	5.61935	0.553691	0.276846	−	0.960914i	$-0.410711\pi$
$103$	5.61935	0.553691	0.276846	+	0.960914i	$0.410711\pi$
$104$	0	0
$105$	0.362192	0.0353463
$106$	0	0
$107$	7.13010	0.689292	0.344646	−	0.938733i	$-0.387999\pi$
$107$	7.13010	0.689292	0.344646	+	0.938733i	$0.387999\pi$
$108$	0	0
$109$	15.3848	1.47360	0.736798	−	0.676112i	$-0.236337\pi$
$109$	15.3848	1.47360	0.736798	+	0.676112i	$0.236337\pi$
$110$	0	0
$111$	5.45296	0.517572
$112$	0	0
$113$	12.5604	1.18158	0.590789	−	0.806826i	$-0.298816\pi$
$113$	12.5604	1.18158	0.590789	+	0.806826i	$0.298816\pi$
$114$	0	0
$115$	1.91629	0.178695
$116$	0	0
$117$	2.54716	0.235485
$118$	0	0
$119$	1.70606	0.156394
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.53311	0.408736
$124$	0	0
$125$	−5.22636	−0.467460
$126$	0	0
$127$	−17.4061	−1.54455	−0.772273	−	0.635291i	$-0.780880\pi$
$127$	−17.4061	−1.54455	−0.772273	+	0.635291i	$0.780880\pi$
$128$	0	0
$129$	8.44976	0.743960
$130$	0	0
$131$	−15.8268	−1.38280	−0.691398	−	0.722474i	$-0.743005\pi$
$131$	−15.8268	−1.38280	−0.691398	+	0.722474i	$0.743005\pi$
$132$	0	0
$133$	−4.21645	−0.365612
$134$	0	0
$135$	2.00914	0.172919
$136$	0	0
$137$	−5.89307	−0.503479	−0.251739	−	0.967795i	$-0.581003\pi$
$137$	−5.89307	−0.503479	−0.251739	+	0.967795i	$0.581003\pi$
$138$	0	0
$139$	−11.8018	−1.00102	−0.500509	−	0.865731i	$-0.666854\pi$
$139$	−11.8018	−1.00102	−0.500509	+	0.865731i	$0.666854\pi$
$140$	0	0
$141$	−8.25165	−0.694915
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	5.02058	0.416937
$146$	0	0
$147$	−0.672934	−0.0555027
$148$	0	0
$149$	−20.5371	−1.68246	−0.841231	−	0.540676i	$-0.818168\pi$
$149$	−20.5371	−1.68246	−0.841231	+	0.540676i	$0.818168\pi$
$150$	0	0
$151$	7.85237	0.639017	0.319508	−	0.947583i	$-0.396482\pi$
$151$	7.85237	0.639017	0.319508	+	0.947583i	$0.396482\pi$
$152$	0	0
$153$	4.34560	0.351321
$154$	0	0
$155$	−3.06631	−0.246292
$156$	0	0
$157$	−12.0690	−0.963215	−0.481607	−	0.876387i	$-0.659947\pi$
$157$	−12.0690	−0.963215	−0.481607	+	0.876387i	$0.659947\pi$
$158$	0	0
$159$	−5.06357	−0.401567
$160$	0	0
$161$	−3.56036	−0.280596
$162$	0	0
$163$	20.3458	1.59361	0.796804	−	0.604237i	$-0.206522\pi$
$163$	20.3458	1.59361	0.796804	+	0.604237i	$0.206522\pi$
$164$	0	0
$165$	0.362192	0.0281966
$166$	0	0
$167$	−0.151712	−0.0117398	−0.00586990	−	0.999983i	$-0.501868\pi$
$167$	−0.151712	−0.0117398	−0.00586990	+	0.999983i	$0.501868\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.7400	−0.821306
$172$	0	0
$173$	9.52069	0.723845	0.361922	−	0.932208i	$-0.382121\pi$
$173$	9.52069	0.723845	0.361922	+	0.932208i	$0.382121\pi$
$174$	0	0
$175$	4.71031	0.356066
$176$	0	0
$177$	−1.09751	−0.0824943
$178$	0	0
$179$	1.92886	0.144170	0.0720850	−	0.997398i	$-0.477035\pi$
$179$	1.92886	0.144170	0.0720850	+	0.997398i	$0.477035\pi$
$180$	0	0
$181$	21.7724	1.61833	0.809166	−	0.587580i	$-0.199919\pi$
$181$	21.7724	1.61833	0.809166	+	0.587580i	$0.199919\pi$
$182$	0	0
$183$	−5.57005	−0.411750
$184$	0	0
$185$	−4.36140	−0.320657
$186$	0	0
$187$	1.70606	0.124759
$188$	0	0
$189$	−3.73287	−0.271526
$190$	0	0
$191$	−2.82575	−0.204464	−0.102232	−	0.994761i	$-0.532598\pi$
$191$	−2.82575	−0.204464	−0.102232	+	0.994761i	$0.532598\pi$
$192$	0	0
$193$	19.2031	1.38227	0.691134	−	0.722727i	$-0.257111\pi$
$193$	19.2031	1.38227	0.691134	+	0.722727i	$0.257111\pi$
$194$	0	0
$195$	0.362192	0.0259371
$196$	0	0
$197$	15.8466	1.12903	0.564513	−	0.825424i	$-0.309064\pi$
$197$	15.8466	1.12903	0.564513	+	0.825424i	$0.309064\pi$
$198$	0	0
$199$	7.57704	0.537122	0.268561	−	0.963263i	$-0.413452\pi$
$199$	7.57704	0.537122	0.268561	+	0.963263i	$0.413452\pi$
$200$	0	0
$201$	5.56204	0.392316
$202$	0	0
$203$	−9.32798	−0.654696
$204$	0	0
$205$	−3.62568	−0.253229
$206$	0	0
$207$	−9.06882	−0.630326
$208$	0	0
$209$	−4.21645	−0.291658
$210$	0	0
$211$	6.62864	0.456334	0.228167	−	0.973622i	$-0.426727\pi$
$211$	6.62864	0.456334	0.228167	+	0.973622i	$0.426727\pi$
$212$	0	0
$213$	−6.45343	−0.442182
$214$	0	0
$215$	−6.75831	−0.460913
$216$	0	0
$217$	5.69704	0.386740
$218$	0	0
$219$	−5.05090	−0.341308
$220$	0	0
$221$	1.70606	0.114762
$222$	0	0
$223$	−11.9808	−0.802291	−0.401146	−	0.916014i	$-0.631388\pi$
$223$	−11.9808	−0.802291	−0.401146	+	0.916014i	$0.631388\pi$
$224$	0	0
$225$	11.9979	0.799861
$226$	0	0
$227$	−16.1516	−1.07202	−0.536011	−	0.844211i	$-0.680069\pi$
$227$	−16.1516	−1.07202	−0.536011	+	0.844211i	$0.680069\pi$
$228$	0	0
$229$	25.4721	1.68324	0.841621	−	0.540068i	$-0.181602\pi$
$229$	25.4721	1.68324	0.841621	+	0.540068i	$0.181602\pi$
$230$	0	0
$231$	−0.672934	−0.0442758
$232$	0	0
$233$	20.2582	1.32716	0.663579	−	0.748106i	$-0.269037\pi$
$233$	20.2582	1.32716	0.663579	+	0.748106i	$0.269037\pi$
$234$	0	0
$235$	6.59986	0.430528
$236$	0	0
$237$	−8.52488	−0.553751
$238$	0	0
$239$	11.9477	0.772831	0.386415	−	0.922325i	$-0.373713\pi$
$239$	11.9477	0.772831	0.386415	+	0.922325i	$0.373713\pi$
$240$	0	0
$241$	26.9032	1.73299	0.866493	−	0.499189i	$-0.166369\pi$
$241$	26.9032	1.73299	0.866493	+	0.499189i	$0.166369\pi$
$242$	0	0
$243$	−14.6504	−0.939826
$244$	0	0
$245$	0.538228	0.0343861
$246$	0	0
$247$	−4.21645	−0.268286
$248$	0	0
$249$	7.00666	0.444029
$250$	0	0
$251$	1.52645	0.0963490	0.0481745	−	0.998839i	$-0.484660\pi$
$251$	1.52645	0.0963490	0.0481745	+	0.998839i	$0.484660\pi$
$252$	0	0
$253$	−3.56036	−0.223838
$254$	0	0
$255$	0.617919	0.0386956
$256$	0	0
$257$	9.33923	0.582565	0.291283	−	0.956637i	$-0.405918\pi$
$257$	9.33923	0.582565	0.291283	+	0.956637i	$0.405918\pi$
$258$	0	0
$259$	8.10326	0.503512
$260$	0	0
$261$	−23.7599	−1.47070
$262$	0	0
$263$	−10.5211	−0.648758	−0.324379	−	0.945927i	$-0.605155\pi$
$263$	−10.5211	−0.648758	−0.324379	+	0.945927i	$0.605155\pi$
$264$	0	0
$265$	4.04996	0.248787
$266$	0	0
$267$	−0.0225661	−0.00138102
$268$	0	0
$269$	12.1026	0.737908	0.368954	−	0.929448i	$-0.379716\pi$
$269$	12.1026	0.737908	0.368954	+	0.929448i	$0.379716\pi$
$270$	0	0
$271$	11.7953	0.716515	0.358258	−	0.933623i	$-0.383371\pi$
$271$	11.7953	0.716515	0.358258	+	0.933623i	$0.383371\pi$
$272$	0	0
$273$	−0.672934	−0.0407278
$274$	0	0
$275$	4.71031	0.284042
$276$	0	0
$277$	−7.83652	−0.470851	−0.235425	−	0.971892i	$-0.575648\pi$
$277$	−7.83652	−0.470851	−0.235425	+	0.971892i	$0.575648\pi$
$278$	0	0
$279$	14.5113	0.868767
$280$	0	0
$281$	−1.38147	−0.0824113	−0.0412057	−	0.999151i	$-0.513120\pi$
$281$	−1.38147	−0.0824113	−0.0412057	+	0.999151i	$0.513120\pi$
$282$	0	0
$283$	21.4298	1.27387	0.636934	−	0.770918i	$-0.280202\pi$
$283$	21.4298	1.27387	0.636934	+	0.770918i	$0.280202\pi$
$284$	0	0
$285$	−1.52716	−0.0904614
$286$	0	0
$287$	6.73633	0.397633
$288$	0	0
$289$	−14.0894	−0.828787
$290$	0	0
$291$	−8.57497	−0.502673
$292$	0	0
$293$	−16.4337	−0.960066	−0.480033	−	0.877250i	$-0.659375\pi$
$293$	−16.4337	−0.960066	−0.480033	+	0.877250i	$0.659375\pi$
$294$	0	0
$295$	0.877818	0.0511085
$296$	0	0
$297$	−3.73287	−0.216603
$298$	0	0
$299$	−3.56036	−0.205901
$300$	0	0
$301$	12.5566	0.723750
$302$	0	0
$303$	−7.50251	−0.431008
$304$	0	0
$305$	4.45506	0.255096
$306$	0	0
$307$	25.4134	1.45042	0.725209	−	0.688529i	$-0.241743\pi$
$307$	25.4134	1.45042	0.725209	+	0.688529i	$0.241743\pi$
$308$	0	0
$309$	−3.78145	−0.215119
$310$	0	0
$311$	−22.2211	−1.26004	−0.630021	−	0.776578i	$-0.716954\pi$
$311$	−22.2211	−1.26004	−0.630021	+	0.776578i	$0.716954\pi$
$312$	0	0
$313$	−17.4758	−0.987793	−0.493896	−	0.869521i	$-0.664428\pi$
$313$	−17.4758	−0.987793	−0.493896	+	0.869521i	$0.664428\pi$
$314$	0	0
$315$	1.37095	0.0772445
$316$	0	0
$317$	5.35194	0.300595	0.150297	−	0.988641i	$-0.451977\pi$
$317$	5.35194	0.300595	0.150297	+	0.988641i	$0.451977\pi$
$318$	0	0
$319$	−9.32798	−0.522267
$320$	0	0
$321$	−4.79808	−0.267803
$322$	0	0
$323$	−7.19349	−0.400257
$324$	0	0
$325$	4.71031	0.261281
$326$	0	0
$327$	−10.3530	−0.572520
$328$	0	0
$329$	−12.2622	−0.676037
$330$	0	0
$331$	−33.5298	−1.84297	−0.921483	−	0.388419i	$-0.873021\pi$
$331$	−33.5298	−1.84297	−0.921483	+	0.388419i	$0.873021\pi$
$332$	0	0
$333$	20.6403	1.13108
$334$	0	0
$335$	−4.44865	−0.243056
$336$	0	0
$337$	25.5363	1.39105	0.695527	−	0.718500i	$-0.255171\pi$
$337$	25.5363	1.39105	0.695527	+	0.718500i	$0.255171\pi$
$338$	0	0
$339$	−8.45229	−0.459065
$340$	0	0
$341$	5.69704	0.308512
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.28954	−0.0694262
$346$	0	0
$347$	19.2720	1.03458	0.517288	−	0.855811i	$-0.326942\pi$
$347$	19.2720	1.03458	0.517288	+	0.855811i	$0.326942\pi$
$348$	0	0
$349$	−13.0534	−0.698731	−0.349365	−	0.936987i	$-0.613603\pi$
$349$	−13.0534	−0.698731	−0.349365	+	0.936987i	$0.613603\pi$
$350$	0	0
$351$	−3.73287	−0.199246
$352$	0	0
$353$	17.3319	0.922485	0.461243	−	0.887274i	$-0.347404\pi$
$353$	17.3319	0.922485	0.461243	+	0.887274i	$0.347404\pi$
$354$	0	0
$355$	5.16160	0.273949
$356$	0	0
$357$	−1.14806	−0.0607619
$358$	0	0
$359$	0.598945	0.0316111	0.0158056	−	0.999875i	$-0.494969\pi$
$359$	0.598945	0.0316111	0.0158056	+	0.999875i	$0.494969\pi$
$360$	0	0
$361$	−1.22156	−0.0642927
$362$	0	0
$363$	−0.672934	−0.0353199
$364$	0	0
$365$	4.03983	0.211454
$366$	0	0
$367$	6.57145	0.343027	0.171513	−	0.985182i	$-0.445134\pi$
$367$	6.57145	0.343027	0.171513	+	0.985182i	$0.445134\pi$
$368$	0	0
$369$	17.1585	0.893237
$370$	0	0
$371$	−7.52462	−0.390659
$372$	0	0
$373$	15.7645	0.816253	0.408126	−	0.912925i	$-0.366182\pi$
$373$	15.7645	0.816253	0.408126	+	0.912925i	$0.366182\pi$
$374$	0	0
$375$	3.51700	0.181617
$376$	0	0
$377$	−9.32798	−0.480416
$378$	0	0
$379$	11.9448	0.613565	0.306783	−	0.951780i	$-0.400747\pi$
$379$	11.9448	0.613565	0.306783	+	0.951780i	$0.400747\pi$
$380$	0	0
$381$	11.7132	0.600084
$382$	0	0
$383$	18.4537	0.942942	0.471471	−	0.881882i	$-0.343723\pi$
$383$	18.4537	0.942942	0.471471	+	0.881882i	$0.343723\pi$
$384$	0	0
$385$	0.538228	0.0274306
$386$	0	0
$387$	31.9837	1.62582
$388$	0	0
$389$	28.4060	1.44024	0.720122	−	0.693848i	$-0.244086\pi$
$389$	28.4060	1.44024	0.720122	+	0.693848i	$0.244086\pi$
$390$	0	0
$391$	−6.07418	−0.307184
$392$	0	0
$393$	10.6504	0.537242
$394$	0	0
$395$	6.81840	0.343071
$396$	0	0
$397$	−2.07267	−0.104024	−0.0520122	−	0.998646i	$-0.516563\pi$
$397$	−2.07267	−0.104024	−0.0520122	+	0.998646i	$0.516563\pi$
$398$	0	0
$399$	2.83739	0.142047
$400$	0	0
$401$	13.8582	0.692045	0.346022	−	0.938226i	$-0.387532\pi$
$401$	13.8582	0.692045	0.346022	+	0.938226i	$0.387532\pi$
$402$	0	0
$403$	5.69704	0.283790
$404$	0	0
$405$	2.76084	0.137187
$406$	0	0
$407$	8.10326	0.401664
$408$	0	0
$409$	−18.5491	−0.917192	−0.458596	−	0.888645i	$-0.651647\pi$
$409$	−18.5491	−0.917192	−0.458596	+	0.888645i	$0.651647\pi$
$410$	0	0
$411$	3.96565	0.195611
$412$	0	0
$413$	−1.63094	−0.0802533
$414$	0	0
$415$	−5.60408	−0.275094
$416$	0	0
$417$	7.94185	0.388914
$418$	0	0
$419$	−20.4582	−0.999450	−0.499725	−	0.866184i	$-0.666566\pi$
$419$	−20.4582	−0.999450	−0.499725	+	0.866184i	$0.666566\pi$
$420$	0	0
$421$	−5.02752	−0.245026	−0.122513	−	0.992467i	$-0.539095\pi$
$421$	−5.02752	−0.245026	−0.122513	+	0.992467i	$0.539095\pi$
$422$	0	0
$423$	−31.2338	−1.51864
$424$	0	0
$425$	8.03605	0.389806
$426$	0	0
$427$	−8.27727	−0.400565
$428$	0	0
$429$	−0.672934	−0.0324896
$430$	0	0
$431$	4.12654	0.198769	0.0993843	−	0.995049i	$-0.468313\pi$
$431$	4.12654	0.198769	0.0993843	+	0.995049i	$0.468313\pi$
$432$	0	0
$433$	−34.7599	−1.67045	−0.835227	−	0.549905i	$-0.814664\pi$
$433$	−34.7599	−1.67045	−0.835227	+	0.549905i	$0.814664\pi$
$434$	0	0
$435$	−3.37852	−0.161988
$436$	0	0
$437$	15.0121	0.718126
$438$	0	0
$439$	−0.778157	−0.0371394	−0.0185697	−	0.999828i	$-0.505911\pi$
$439$	−0.778157	−0.0371394	−0.0185697	+	0.999828i	$0.505911\pi$
$440$	0	0
$441$	−2.54716	−0.121293
$442$	0	0
$443$	34.6935	1.64834	0.824169	−	0.566343i	$-0.191642\pi$
$443$	34.6935	1.64834	0.824169	+	0.566343i	$0.191642\pi$
$444$	0	0
$445$	0.0180489	0.000855598 0
$446$	0	0
$447$	13.8201	0.653668
$448$	0	0
$449$	−19.4109	−0.916056	−0.458028	−	0.888938i	$-0.651444\pi$
$449$	−19.4109	−0.916056	−0.458028	+	0.888938i	$0.651444\pi$
$450$	0	0
$451$	6.73633	0.317201
$452$	0	0
$453$	−5.28413	−0.248270
$454$	0	0
$455$	0.538228	0.0252325
$456$	0	0
$457$	−11.2186	−0.524783	−0.262391	−	0.964962i	$-0.584511\pi$
$457$	−11.2186	−0.524783	−0.262391	+	0.964962i	$0.584511\pi$
$458$	0	0
$459$	−6.36849	−0.297255
$460$	0	0
$461$	−5.58868	−0.260291	−0.130145	−	0.991495i	$-0.541544\pi$
$461$	−5.58868	−0.260291	−0.130145	+	0.991495i	$0.541544\pi$
$462$	0	0
$463$	−22.4767	−1.04458	−0.522290	−	0.852768i	$-0.674922\pi$
$463$	−22.4767	−1.04458	−0.522290	+	0.852768i	$0.674922\pi$
$464$	0	0
$465$	2.06342	0.0956889
$466$	0	0
$467$	−13.2930	−0.615129	−0.307564	−	0.951527i	$-0.599514\pi$
$467$	−13.2930	−0.615129	−0.307564	+	0.951527i	$0.599514\pi$
$468$	0	0
$469$	8.26536	0.381659
$470$	0	0
$471$	8.12167	0.374227
$472$	0	0
$473$	12.5566	0.577353
$474$	0	0
$475$	−19.8608	−0.911275
$476$	0	0
$477$	−19.1664	−0.877569
$478$	0	0
$479$	11.4999	0.525445	0.262723	−	0.964871i	$-0.415380\pi$
$479$	11.4999	0.525445	0.262723	+	0.964871i	$0.415380\pi$
$480$	0	0
$481$	8.10326	0.369477
$482$	0	0
$483$	2.39589	0.109017
$484$	0	0
$485$	6.85846	0.311426
$486$	0	0
$487$	−16.4335	−0.744671	−0.372335	−	0.928098i	$-0.621443\pi$
$487$	−16.4335	−0.744671	−0.372335	+	0.928098i	$0.621443\pi$
$488$	0	0
$489$	−13.6914	−0.619147
$490$	0	0
$491$	−25.4190	−1.14714	−0.573572	−	0.819155i	$-0.694443\pi$
$491$	−25.4190	−1.14714	−0.573572	+	0.819155i	$0.694443\pi$
$492$	0	0
$493$	−15.9141	−0.716733
$494$	0	0
$495$	1.37095	0.0616198
$496$	0	0
$497$	−9.58999	−0.430170
$498$	0	0
$499$	10.0845	0.451446	0.225723	−	0.974191i	$-0.427526\pi$
$499$	10.0845	0.451446	0.225723	+	0.974191i	$0.427526\pi$
$500$	0	0
$501$	0.102092	0.00456113
$502$	0	0
$503$	−24.2866	−1.08289	−0.541443	−	0.840738i	$-0.682122\pi$
$503$	−24.2866	−1.08289	−0.541443	+	0.840738i	$0.682122\pi$
$504$	0	0
$505$	6.00068	0.267027
$506$	0	0
$507$	−0.672934	−0.0298860
$508$	0	0
$509$	10.3881	0.460444	0.230222	−	0.973138i	$-0.426055\pi$
$509$	10.3881	0.460444	0.230222	+	0.973138i	$0.426055\pi$
$510$	0	0
$511$	−7.50579	−0.332036
$512$	0	0
$513$	15.7395	0.694914
$514$	0	0
$515$	3.02449	0.133275
$516$	0	0
$517$	−12.2622	−0.539291
$518$	0	0
$519$	−6.40680	−0.281227
$520$	0	0
$521$	−41.4927	−1.81783	−0.908915	−	0.416981i	$-0.863088\pi$
$521$	−41.4927	−1.81783	−0.908915	+	0.416981i	$0.863088\pi$
$522$	0	0
$523$	2.19012	0.0957674	0.0478837	−	0.998853i	$-0.484752\pi$
$523$	2.19012	0.0957674	0.0478837	+	0.998853i	$0.484752\pi$
$524$	0	0
$525$	−3.16973	−0.138338
$526$	0	0
$527$	9.71947	0.423387
$528$	0	0
$529$	−10.3238	−0.448861
$530$	0	0
$531$	−4.15427	−0.180280
$532$	0	0
$533$	6.73633	0.291783
$534$	0	0
$535$	3.83762	0.165915
$536$	0	0
$537$	−1.29800	−0.0560127
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	40.0027	1.71985	0.859925	−	0.510420i	$-0.170510\pi$
$541$	40.0027	1.71985	0.859925	+	0.510420i	$0.170510\pi$
$542$	0	0
$543$	−14.6514	−0.628752
$544$	0	0
$545$	8.28053	0.354699
$546$	0	0
$547$	7.70909	0.329617	0.164809	−	0.986326i	$-0.447299\pi$
$547$	7.70909	0.329617	0.164809	+	0.986326i	$0.447299\pi$
$548$	0	0
$549$	−21.0835	−0.899823
$550$	0	0
$551$	39.3310	1.67556
$552$	0	0
$553$	−12.6682	−0.538708
$554$	0	0
$555$	2.93493	0.124581
$556$	0	0
$557$	38.8851	1.64761	0.823807	−	0.566871i	$-0.191846\pi$
$557$	38.8851	1.64761	0.823807	+	0.566871i	$0.191846\pi$
$558$	0	0
$559$	12.5566	0.531087
$560$	0	0
$561$	−1.14806	−0.0484712
$562$	0	0
$563$	−40.3929	−1.70236	−0.851179	−	0.524875i	$-0.824112\pi$
$563$	−40.3929	−1.70236	−0.851179	+	0.524875i	$0.824112\pi$
$564$	0	0
$565$	6.76034	0.284409
$566$	0	0
$567$	−5.12950	−0.215419
$568$	0	0
$569$	10.1163	0.424099	0.212049	−	0.977259i	$-0.431986\pi$
$569$	10.1163	0.424099	0.212049	+	0.977259i	$0.431986\pi$
$570$	0	0
$571$	42.2032	1.76615	0.883074	−	0.469234i	$-0.155470\pi$
$571$	42.2032	1.76615	0.883074	+	0.469234i	$0.155470\pi$
$572$	0	0
$573$	1.90154	0.0794382
$574$	0	0
$575$	−16.7704	−0.699375
$576$	0	0
$577$	−34.0021	−1.41553	−0.707764	−	0.706449i	$-0.750296\pi$
$577$	−34.0021	−1.41553	−0.707764	+	0.706449i	$0.750296\pi$
$578$	0	0
$579$	−12.9224	−0.537037
$580$	0	0
$581$	10.4121	0.431967
$582$	0	0
$583$	−7.52462	−0.311638
$584$	0	0
$585$	1.37095	0.0566819
$586$	0	0
$587$	−11.7956	−0.486857	−0.243428	−	0.969919i	$-0.578272\pi$
$587$	−11.7956	−0.486857	−0.243428	+	0.969919i	$0.578272\pi$
$588$	0	0
$589$	−24.0213	−0.989780
$590$	0	0
$591$	−10.6637	−0.438647
$592$	0	0
$593$	−45.8832	−1.88420	−0.942099	−	0.335334i	$-0.891151\pi$
$593$	−45.8832	−1.88420	−0.942099	+	0.335334i	$0.891151\pi$
$594$	0	0
$595$	0.918247	0.0376445
$596$	0	0
$597$	−5.09885	−0.208682
$598$	0	0
$599$	34.7991	1.42185	0.710927	−	0.703266i	$-0.248276\pi$
$599$	34.7991	1.42185	0.710927	+	0.703266i	$0.248276\pi$
$600$	0	0
$601$	35.8665	1.46302	0.731512	−	0.681828i	$-0.238815\pi$
$601$	35.8665	1.46302	0.731512	+	0.681828i	$0.238815\pi$
$602$	0	0
$603$	21.0532	0.857352
$604$	0	0
$605$	0.538228	0.0218821
$606$	0	0
$607$	12.5866	0.510874	0.255437	−	0.966826i	$-0.417781\pi$
$607$	12.5866	0.510874	0.255437	+	0.966826i	$0.417781\pi$
$608$	0	0
$609$	6.27712	0.254362
$610$	0	0
$611$	−12.2622	−0.496076
$612$	0	0
$613$	−28.3997	−1.14705	−0.573527	−	0.819186i	$-0.694425\pi$
$613$	−28.3997	−1.14705	−0.573527	+	0.819186i	$0.694425\pi$
$614$	0	0
$615$	2.43985	0.0983841
$616$	0	0
$617$	−26.3564	−1.06107	−0.530534	−	0.847663i	$-0.678009\pi$
$617$	−26.3564	−1.06107	−0.530534	+	0.847663i	$0.678009\pi$
$618$	0	0
$619$	11.1729	0.449076	0.224538	−	0.974465i	$-0.427913\pi$
$619$	11.1729	0.449076	0.224538	+	0.974465i	$0.427913\pi$
$620$	0	0
$621$	13.2904	0.533325
$622$	0	0
$623$	−0.0335338	−0.00134351
$624$	0	0
$625$	20.7386	0.829543
$626$	0	0
$627$	2.83739	0.113314
$628$	0	0
$629$	13.8246	0.551223
$630$	0	0
$631$	−34.1256	−1.35852	−0.679259	−	0.733898i	$-0.737699\pi$
$631$	−34.1256	−1.35852	−0.679259	+	0.733898i	$0.737699\pi$
$632$	0	0
$633$	−4.46063	−0.177294
$634$	0	0
$635$	−9.36847	−0.371776
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−24.4272	−0.966327
$640$	0	0
$641$	5.35218	0.211398	0.105699	−	0.994398i	$-0.466292\pi$
$641$	5.35218	0.211398	0.105699	+	0.994398i	$0.466292\pi$
$642$	0	0
$643$	−49.1919	−1.93994	−0.969970	−	0.243226i	$-0.921794\pi$
$643$	−49.1919	−1.93994	−0.969970	+	0.243226i	$0.921794\pi$
$644$	0	0
$645$	4.54790	0.179073
$646$	0	0
$647$	−10.2193	−0.401763	−0.200881	−	0.979616i	$-0.564381\pi$
$647$	−10.2193	−0.401763	−0.200881	+	0.979616i	$0.564381\pi$
$648$	0	0
$649$	−1.63094	−0.0640200
$650$	0	0
$651$	−3.83373	−0.150256
$652$	0	0
$653$	−1.76322	−0.0690003	−0.0345001	−	0.999405i	$-0.510984\pi$
$653$	−1.76322	−0.0690003	−0.0345001	+	0.999405i	$0.510984\pi$
$654$	0	0
$655$	−8.51844	−0.332843
$656$	0	0
$657$	−19.1184	−0.745881
$658$	0	0
$659$	−3.27817	−0.127699	−0.0638496	−	0.997960i	$-0.520338\pi$
$659$	−3.27817	−0.127699	−0.0638496	+	0.997960i	$0.520338\pi$
$660$	0	0
$661$	17.2156	0.669608	0.334804	−	0.942288i	$-0.391330\pi$
$661$	17.2156	0.669608	0.334804	+	0.942288i	$0.391330\pi$
$662$	0	0
$663$	−1.14806	−0.0445870
$664$	0	0
$665$	−2.26941	−0.0880040
$666$	0	0
$667$	33.2110	1.28594
$668$	0	0
$669$	8.06226	0.311705
$670$	0	0
$671$	−8.27727	−0.319540
$672$	0	0
$673$	11.5166	0.443931	0.221966	−	0.975054i	$-0.428753\pi$
$673$	11.5166	0.443931	0.221966	+	0.975054i	$0.428753\pi$
$674$	0	0
$675$	−17.5830	−0.676769
$676$	0	0
$677$	7.99010	0.307084	0.153542	−	0.988142i	$-0.450932\pi$
$677$	7.99010	0.307084	0.153542	+	0.988142i	$0.450932\pi$
$678$	0	0
$679$	−12.7427	−0.489018
$680$	0	0
$681$	10.8690	0.416500
$682$	0	0
$683$	2.67293	0.102277	0.0511384	−	0.998692i	$-0.483715\pi$
$683$	2.67293	0.102277	0.0511384	+	0.998692i	$0.483715\pi$
$684$	0	0
$685$	−3.17182	−0.121189
$686$	0	0
$687$	−17.1410	−0.653971
$688$	0	0
$689$	−7.52462	−0.286665
$690$	0	0
$691$	22.7347	0.864867	0.432434	−	0.901666i	$-0.357655\pi$
$691$	22.7347	0.864867	0.432434	+	0.901666i	$0.357655\pi$
$692$	0	0
$693$	−2.54716	−0.0967586
$694$	0	0
$695$	−6.35208	−0.240948
$696$	0	0
$697$	11.4926	0.435312
$698$	0	0
$699$	−13.6324	−0.515625
$700$	0	0
$701$	13.0708	0.493678	0.246839	−	0.969056i	$-0.420608\pi$
$701$	13.0708	0.493678	0.246839	+	0.969056i	$0.420608\pi$
$702$	0	0
$703$	−34.1670	−1.28863
$704$	0	0
$705$	−4.44127	−0.167268
$706$	0	0
$707$	−11.1490	−0.419300
$708$	0	0
$709$	24.2329	0.910086	0.455043	−	0.890470i	$-0.349624\pi$
$709$	24.2329	0.910086	0.455043	+	0.890470i	$0.349624\pi$
$710$	0	0
$711$	−32.2680	−1.21015
$712$	0	0
$713$	−20.2835	−0.759625
$714$	0	0
$715$	0.538228	0.0201286
$716$	0	0
$717$	−8.03999	−0.300259
$718$	0	0
$719$	−40.5402	−1.51189	−0.755947	−	0.654632i	$-0.772823\pi$
$719$	−40.5402	−1.51189	−0.755947	+	0.654632i	$0.772823\pi$
$720$	0	0
$721$	−5.61935	−0.209276
$722$	0	0
$723$	−18.1041	−0.673297
$724$	0	0
$725$	−43.9377	−1.63180
$726$	0	0
$727$	8.38603	0.311021	0.155510	−	0.987834i	$-0.450298\pi$
$727$	8.38603	0.311021	0.155510	+	0.987834i	$0.450298\pi$
$728$	0	0
$729$	−5.52974	−0.204805
$730$	0	0
$731$	21.4222	0.792330
$732$	0	0
$733$	−23.7946	−0.878873	−0.439437	−	0.898274i	$-0.644822\pi$
$733$	−23.7946	−0.878873	−0.439437	+	0.898274i	$0.644822\pi$
$734$	0	0
$735$	−0.362192	−0.0133597
$736$	0	0
$737$	8.26536	0.304458
$738$	0	0
$739$	27.3233	1.00510	0.502552	−	0.864547i	$-0.332395\pi$
$739$	27.3233	1.00510	0.502552	+	0.864547i	$0.332395\pi$
$740$	0	0
$741$	2.83739	0.104234
$742$	0	0
$743$	−32.2512	−1.18318	−0.591592	−	0.806238i	$-0.701500\pi$
$743$	−32.2512	−1.18318	−0.591592	+	0.806238i	$0.701500\pi$
$744$	0	0
$745$	−11.0536	−0.404973
$746$	0	0
$747$	26.5213	0.970363
$748$	0	0
$749$	−7.13010	−0.260528
$750$	0	0
$751$	−3.78681	−0.138183	−0.0690913	−	0.997610i	$-0.522010\pi$
$751$	−3.78681	−0.138183	−0.0690913	+	0.997610i	$0.522010\pi$
$752$	0	0
$753$	−1.02720	−0.0374334
$754$	0	0
$755$	4.22637	0.153813
$756$	0	0
$757$	20.6522	0.750619	0.375309	−	0.926900i	$-0.377536\pi$
$757$	20.6522	0.750619	0.375309	+	0.926900i	$0.377536\pi$
$758$	0	0
$759$	2.39589	0.0869653
$760$	0	0
$761$	21.1253	0.765790	0.382895	−	0.923792i	$-0.374927\pi$
$761$	21.1253	0.765790	0.382895	+	0.923792i	$0.374927\pi$
$762$	0	0
$763$	−15.3848	−0.556967
$764$	0	0
$765$	2.33892	0.0845639
$766$	0	0
$767$	−1.63094	−0.0588898
$768$	0	0
$769$	−7.64664	−0.275745	−0.137872	−	0.990450i	$-0.544026\pi$
$769$	−7.64664	−0.275745	−0.137872	+	0.990450i	$0.544026\pi$
$770$	0	0
$771$	−6.28468	−0.226337
$772$	0	0
$773$	−17.0537	−0.613379	−0.306689	−	0.951810i	$-0.599221\pi$
$773$	−17.0537	−0.613379	−0.306689	+	0.951810i	$0.599221\pi$
$774$	0	0
$775$	26.8348	0.963936
$776$	0	0
$777$	−5.45296	−0.195624
$778$	0	0
$779$	−28.4034	−1.01766
$780$	0	0
$781$	−9.58999	−0.343157
$782$	0	0
$783$	34.8202	1.24437
$784$	0	0
$785$	−6.49590	−0.231849
$786$	0	0
$787$	−16.1001	−0.573905	−0.286953	−	0.957945i	$-0.592642\pi$
$787$	−16.1001	−0.573905	−0.286953	+	0.957945i	$0.592642\pi$
$788$	0	0
$789$	7.08000	0.252055
$790$	0	0
$791$	−12.5604	−0.446595
$792$	0	0
$793$	−8.27727	−0.293934
$794$	0	0
$795$	−2.72536	−0.0966584
$796$	0	0
$797$	40.1516	1.42224	0.711122	−	0.703069i	$-0.248187\pi$
$797$	40.1516	1.42224	0.711122	+	0.703069i	$0.248187\pi$
$798$	0	0
$799$	−20.9200	−0.740096
$800$	0	0
$801$	−0.0854161	−0.00301803
$802$	0	0
$803$	−7.50579	−0.264873
$804$	0	0
$805$	−1.91629	−0.0675403
$806$	0	0
$807$	−8.14424	−0.286691
$808$	0	0
$809$	−36.0345	−1.26690	−0.633452	−	0.773782i	$-0.718363\pi$
$809$	−36.0345	−1.26690	−0.633452	+	0.773782i	$0.718363\pi$
$810$	0	0
$811$	−42.0083	−1.47511	−0.737555	−	0.675287i	$-0.764020\pi$
$811$	−42.0083	−1.47511	−0.737555	+	0.675287i	$0.764020\pi$
$812$	0	0
$813$	−7.93748	−0.278380
$814$	0	0
$815$	10.9507	0.383586
$816$	0	0
$817$	−52.9442	−1.85228
$818$	0	0
$819$	−2.54716	−0.0890050
$820$	0	0
$821$	−16.2213	−0.566126	−0.283063	−	0.959101i	$-0.591350\pi$
$821$	−16.2213	−0.566126	−0.283063	+	0.959101i	$0.591350\pi$
$822$	0	0
$823$	−28.3340	−0.987661	−0.493830	−	0.869558i	$-0.664404\pi$
$823$	−28.3340	−0.987661	−0.493830	+	0.869558i	$0.664404\pi$
$824$	0	0
$825$	−3.16973	−0.110356
$826$	0	0
$827$	31.0942	1.08125	0.540625	−	0.841264i	$-0.318188\pi$
$827$	31.0942	1.08125	0.540625	+	0.841264i	$0.318188\pi$
$828$	0	0
$829$	12.2909	0.426882	0.213441	−	0.976956i	$-0.431533\pi$
$829$	12.2909	0.426882	0.213441	+	0.976956i	$0.431533\pi$
$830$	0	0
$831$	5.27346	0.182934
$832$	0	0
$833$	−1.70606	−0.0591113
$834$	0	0
$835$	−0.0816555	−0.00282581
$836$	0	0
$837$	−21.2663	−0.735072
$838$	0	0
$839$	−49.0450	−1.69322	−0.846611	−	0.532213i	$-0.821361\pi$
$839$	−49.0450	−1.69322	−0.846611	+	0.532213i	$0.821361\pi$
$840$	0	0
$841$	58.0112	2.00039
$842$	0	0
$843$	0.929635	0.0320183
$844$	0	0
$845$	0.538228	0.0185156
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−14.4208	−0.494921
$850$	0	0
$851$	−28.8506	−0.988984
$852$	0	0
$853$	−16.8025	−0.575305	−0.287653	−	0.957735i	$-0.592875\pi$
$853$	−16.8025	−0.575305	−0.287653	+	0.957735i	$0.592875\pi$
$854$	0	0
$855$	−5.78055	−0.197691
$856$	0	0
$857$	21.6045	0.737995	0.368998	−	0.929430i	$-0.379701\pi$
$857$	21.6045	0.737995	0.368998	+	0.929430i	$0.379701\pi$
$858$	0	0
$859$	−7.06380	−0.241014	−0.120507	−	0.992712i	$-0.538452\pi$
$859$	−7.06380	−0.241014	−0.120507	+	0.992712i	$0.538452\pi$
$860$	0	0
$861$	−4.53311	−0.154488
$862$	0	0
$863$	−40.6779	−1.38469	−0.692345	−	0.721567i	$-0.743422\pi$
$863$	−40.6779	−1.38469	−0.692345	+	0.721567i	$0.743422\pi$
$864$	0	0
$865$	5.12430	0.174232
$866$	0	0
$867$	9.48122	0.321999
$868$	0	0
$869$	−12.6682	−0.429740
$870$	0	0
$871$	8.26536	0.280061
$872$	0	0
$873$	−32.4576	−1.09852
$874$	0	0
$875$	5.22636	0.176683
$876$	0	0
$877$	55.7205	1.88155	0.940775	−	0.339033i	$-0.110100\pi$
$877$	55.7205	1.88155	0.940775	+	0.339033i	$0.110100\pi$
$878$	0	0
$879$	11.0588	0.373004
$880$	0	0
$881$	17.3931	0.585989	0.292994	−	0.956114i	$-0.405348\pi$
$881$	17.3931	0.585989	0.292994	+	0.956114i	$0.405348\pi$
$882$	0	0
$883$	18.4938	0.622367	0.311183	−	0.950350i	$-0.399275\pi$
$883$	18.4938	0.622367	0.311183	+	0.950350i	$0.399275\pi$
$884$	0	0
$885$	−0.590713	−0.0198566
$886$	0	0
$887$	5.90249	0.198186	0.0990931	−	0.995078i	$-0.468406\pi$
$887$	5.90249	0.198186	0.0990931	+	0.995078i	$0.468406\pi$
$888$	0	0
$889$	17.4061	0.583783
$890$	0	0
$891$	−5.12950	−0.171845
$892$	0	0
$893$	51.7030	1.73017
$894$	0	0
$895$	1.03817	0.0347021
$896$	0	0
$897$	2.39589	0.0799964
$898$	0	0
$899$	−53.1419	−1.77238
$900$	0	0
$901$	−12.8374	−0.427676
$902$	0	0
$903$	−8.44976	−0.281190
$904$	0	0
$905$	11.7185	0.389537
$906$	0	0
$907$	22.0760	0.733022	0.366511	−	0.930414i	$-0.380552\pi$
$907$	22.0760	0.733022	0.366511	+	0.930414i	$0.380552\pi$
$908$	0	0
$909$	−28.3982	−0.941908
$910$	0	0
$911$	10.2537	0.339720	0.169860	−	0.985468i	$-0.445668\pi$
$911$	10.2537	0.339720	0.169860	+	0.985468i	$0.445668\pi$
$912$	0	0
$913$	10.4121	0.344590
$914$	0	0
$915$	−2.99796	−0.0991095
$916$	0	0
$917$	15.8268	0.522648
$918$	0	0
$919$	−6.10317	−0.201325	−0.100663	−	0.994921i	$-0.532096\pi$
$919$	−6.10317	−0.201325	−0.100663	+	0.994921i	$0.532096\pi$
$920$	0	0
$921$	−17.1015	−0.563514
$922$	0	0
$923$	−9.58999	−0.315658
$924$	0	0
$925$	38.1689	1.25498
$926$	0	0
$927$	−14.3134	−0.470113
$928$	0	0
$929$	49.8438	1.63532	0.817660	−	0.575701i	$-0.195271\pi$
$929$	49.8438	1.63532	0.817660	+	0.575701i	$0.195271\pi$
$930$	0	0
$931$	4.21645	0.138189
$932$	0	0
$933$	14.9533	0.489550
$934$	0	0
$935$	0.918247	0.0300299
$936$	0	0
$937$	2.75458	0.0899881	0.0449941	−	0.998987i	$-0.485673\pi$
$937$	2.75458	0.0899881	0.0449941	+	0.998987i	$0.485673\pi$
$938$	0	0
$939$	11.7601	0.383776
$940$	0	0
$941$	−56.1728	−1.83118	−0.915591	−	0.402111i	$-0.868276\pi$
$941$	−56.1728	−1.83118	−0.915591	+	0.402111i	$0.868276\pi$
$942$	0	0
$943$	−23.9838	−0.781020
$944$	0	0
$945$	−2.00914	−0.0653572
$946$	0	0
$947$	−7.86955	−0.255726	−0.127863	−	0.991792i	$-0.540812\pi$
$947$	−7.86955	−0.255726	−0.127863	+	0.991792i	$0.540812\pi$
$948$	0	0
$949$	−7.50579	−0.243648
$950$	0	0
$951$	−3.60150	−0.116787
$952$	0	0
$953$	−5.62240	−0.182127	−0.0910637	−	0.995845i	$-0.529027\pi$
$953$	−5.62240	−0.182127	−0.0910637	+	0.995845i	$0.529027\pi$
$954$	0	0
$955$	−1.52090	−0.0492152
$956$	0	0
$957$	6.27712	0.202910
$958$	0	0
$959$	5.89307	0.190297
$960$	0	0
$961$	1.45629	0.0469769
$962$	0	0
$963$	−18.1615	−0.585246
$964$	0	0
$965$	10.3356	0.332716
$966$	0	0
$967$	50.0719	1.61020	0.805102	−	0.593137i	$-0.202111\pi$
$967$	50.0719	1.61020	0.805102	+	0.593137i	$0.202111\pi$
$968$	0	0
$969$	4.84075	0.155507
$970$	0	0
$971$	38.6410	1.24005	0.620024	−	0.784582i	$-0.287123\pi$
$971$	38.6410	1.24005	0.620024	+	0.784582i	$0.287123\pi$
$972$	0	0
$973$	11.8018	0.378349
$974$	0	0
$975$	−3.16973	−0.101513
$976$	0	0
$977$	−10.1764	−0.325573	−0.162786	−	0.986661i	$-0.552048\pi$
$977$	−10.1764	−0.325573	−0.162786	+	0.986661i	$0.552048\pi$
$978$	0	0
$979$	−0.0335338	−0.00107175
$980$	0	0
$981$	−39.1876	−1.25116
$982$	0	0
$983$	10.8713	0.346741	0.173371	−	0.984857i	$-0.444534\pi$
$983$	10.8713	0.346741	0.173371	+	0.984857i	$0.444534\pi$
$984$	0	0
$985$	8.52910	0.271760
$986$	0	0
$987$	8.25165	0.262653
$988$	0	0
$989$	−44.7061	−1.42157
$990$	0	0
$991$	24.8489	0.789353	0.394676	−	0.918820i	$-0.370857\pi$
$991$	24.8489	0.789353	0.394676	+	0.918820i	$0.370857\pi$
$992$	0	0
$993$	22.5634	0.716026
$994$	0	0
$995$	4.07818	0.129287
$996$	0	0
$997$	−19.4611	−0.616340	−0.308170	−	0.951331i	$-0.599717\pi$
$997$	−19.4611	−0.616340	−0.308170	+	0.951331i	$0.599717\pi$
$998$	0	0
$999$	−30.2484	−0.957018

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.j.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.j.1.4	✓	10	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.672934 q^{3} +0.538228 q^{5} -1.00000 q^{7} -2.54716 q^{9} +O(q^{10})\)	\(q-0.672934 q^{3} +0.538228 q^{5} -1.00000 q^{7} -2.54716 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.362192 q^{15} -1.70606 q^{17} +4.21645 q^{19} +0.672934 q^{21} +3.56036 q^{23} -4.71031 q^{25} +3.73287 q^{27} +9.32798 q^{29} -5.69704 q^{31} +0.672934 q^{33} -0.538228 q^{35} -8.10326 q^{37} +0.672934 q^{39} -6.73633 q^{41} -12.5566 q^{43} -1.37095 q^{45} +12.2622 q^{47} +1.00000 q^{49} +1.14806 q^{51} +7.52462 q^{53} -0.538228 q^{55} -2.83739 q^{57} +1.63094 q^{59} +8.27727 q^{61} +2.54716 q^{63} -0.538228 q^{65} -8.26536 q^{67} -2.39589 q^{69} +9.58999 q^{71} +7.50579 q^{73} +3.16973 q^{75} +1.00000 q^{77} +12.6682 q^{79} +5.12950 q^{81} -10.4121 q^{83} -0.918247 q^{85} -6.27712 q^{87} +0.0335338 q^{89} +1.00000 q^{91} +3.83373 q^{93} +2.26941 q^{95} +12.7427 q^{97} +2.54716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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