LMFDB - Embedded newform 6004.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67388 q^{3} +3.55353 q^{5} +1.65788 q^{7} +4.14966 q^{9} +O(q^{10})$	$q-2.67388 q^{3} +3.55353 q^{5} +1.65788 q^{7} +4.14966 q^{9} +1.12044 q^{11} -2.27561 q^{13} -9.50172 q^{15} -3.07611 q^{17} +1.00000 q^{19} -4.43298 q^{21} +4.46777 q^{23} +7.62755 q^{25} -3.07406 q^{27} -3.82490 q^{29} -4.45375 q^{31} -2.99592 q^{33} +5.89132 q^{35} -11.5926 q^{37} +6.08472 q^{39} -9.46040 q^{41} +4.63326 q^{43} +14.7459 q^{45} -2.15431 q^{47} -4.25144 q^{49} +8.22516 q^{51} -7.35612 q^{53} +3.98150 q^{55} -2.67388 q^{57} -3.21366 q^{59} +11.6415 q^{61} +6.87964 q^{63} -8.08644 q^{65} -13.1658 q^{67} -11.9463 q^{69} -12.2547 q^{71} -13.5211 q^{73} -20.3952 q^{75} +1.85755 q^{77} -1.00000 q^{79} -4.22930 q^{81} -10.9635 q^{83} -10.9310 q^{85} +10.2273 q^{87} -4.05620 q^{89} -3.77269 q^{91} +11.9088 q^{93} +3.55353 q^{95} +16.8881 q^{97} +4.64943 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	−2.67388	−1.54377	−0.771884	−	0.635763i	$-0.780685\pi$
$3$	−2.67388	−1.54377	−0.771884	+	0.635763i	$0.780685\pi$
$4$	0	0
$5$	3.55353	1.58919	0.794593	−	0.607143i	$-0.207684\pi$
$5$	3.55353	1.58919	0.794593	+	0.607143i	$0.207684\pi$
$6$	0	0
$7$	1.65788	0.626619	0.313310	−	0.949651i	$-0.398562\pi$
$7$	1.65788	0.626619	0.313310	+	0.949651i	$0.398562\pi$
$8$	0	0
$9$	4.14966	1.38322
$10$	0	0
$11$	1.12044	0.337824	0.168912	−	0.985631i	$-0.445975\pi$
$11$	1.12044	0.337824	0.168912	+	0.985631i	$0.445975\pi$
$12$	0	0
$13$	−2.27561	−0.631141	−0.315570	−	0.948902i	$-0.602196\pi$
$13$	−2.27561	−0.631141	−0.315570	+	0.948902i	$0.602196\pi$
$14$	0	0
$15$	−9.50172	−2.45333
$16$	0	0
$17$	−3.07611	−0.746066	−0.373033	−	0.927818i	$-0.621682\pi$
$17$	−3.07611	−0.746066	−0.373033	+	0.927818i	$0.621682\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.43298	−0.967355
$22$	0	0
$23$	4.46777	0.931593	0.465797	−	0.884892i	$-0.345768\pi$
$23$	4.46777	0.931593	0.465797	+	0.884892i	$0.345768\pi$
$24$	0	0
$25$	7.62755	1.52551
$26$	0	0
$27$	−3.07406	−0.591603
$28$	0	0
$29$	−3.82490	−0.710266	−0.355133	−	0.934816i	$-0.615564\pi$
$29$	−3.82490	−0.710266	−0.355133	+	0.934816i	$0.615564\pi$
$30$	0	0
$31$	−4.45375	−0.799917	−0.399958	−	0.916533i	$-0.630975\pi$
$31$	−4.45375	−0.799917	−0.399958	+	0.916533i	$0.630975\pi$
$32$	0	0
$33$	−2.99592	−0.521523
$34$	0	0
$35$	5.89132	0.995815
$36$	0	0
$37$	−11.5926	−1.90581	−0.952907	−	0.303264i	$-0.901924\pi$
$37$	−11.5926	−1.90581	−0.952907	+	0.303264i	$0.901924\pi$
$38$	0	0
$39$	6.08472	0.974335
$40$	0	0
$41$	−9.46040	−1.47747	−0.738733	−	0.673998i	$-0.764576\pi$
$41$	−9.46040	−1.47747	−0.738733	+	0.673998i	$0.764576\pi$
$42$	0	0
$43$	4.63326	0.706566	0.353283	−	0.935516i	$-0.385065\pi$
$43$	4.63326	0.706566	0.353283	+	0.935516i	$0.385065\pi$
$44$	0	0
$45$	14.7459	2.19819
$46$	0	0
$47$	−2.15431	−0.314238	−0.157119	−	0.987580i	$-0.550221\pi$
$47$	−2.15431	−0.314238	−0.157119	+	0.987580i	$0.550221\pi$
$48$	0	0
$49$	−4.25144	−0.607348
$50$	0	0
$51$	8.22516	1.15175
$52$	0	0
$53$	−7.35612	−1.01044	−0.505220	−	0.862990i	$-0.668589\pi$
$53$	−7.35612	−1.01044	−0.505220	+	0.862990i	$0.668589\pi$
$54$	0	0
$55$	3.98150	0.536866
$56$	0	0
$57$	−2.67388	−0.354165
$58$	0	0
$59$	−3.21366	−0.418383	−0.209192	−	0.977875i	$-0.567083\pi$
$59$	−3.21366	−0.418383	−0.209192	+	0.977875i	$0.567083\pi$
$60$	0	0
$61$	11.6415	1.49054	0.745269	−	0.666764i	$-0.232321\pi$
$61$	11.6415	1.49054	0.745269	+	0.666764i	$0.232321\pi$
$62$	0	0
$63$	6.87964	0.866753
$64$	0	0
$65$	−8.08644	−1.00300
$66$	0	0
$67$	−13.1658	−1.60846	−0.804228	−	0.594321i	$-0.797421\pi$
$67$	−13.1658	−1.60846	−0.804228	+	0.594321i	$0.797421\pi$
$68$	0	0
$69$	−11.9463	−1.43816
$70$	0	0
$71$	−12.2547	−1.45437	−0.727184	−	0.686443i	$-0.759171\pi$
$71$	−12.2547	−1.45437	−0.727184	+	0.686443i	$0.759171\pi$
$72$	0	0
$73$	−13.5211	−1.58253	−0.791263	−	0.611476i	$-0.790576\pi$
$73$	−13.5211	−1.58253	−0.791263	+	0.611476i	$0.790576\pi$
$74$	0	0
$75$	−20.3952	−2.35503
$76$	0	0
$77$	1.85755	0.211687
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−4.22930	−0.469922
$82$	0	0
$83$	−10.9635	−1.20340	−0.601701	−	0.798721i	$-0.705510\pi$
$83$	−10.9635	−1.20340	−0.601701	+	0.798721i	$0.705510\pi$
$84$	0	0
$85$	−10.9310	−1.18564
$86$	0	0
$87$	10.2273	1.09649
$88$	0	0
$89$	−4.05620	−0.429956	−0.214978	−	0.976619i	$-0.568968\pi$
$89$	−4.05620	−0.429956	−0.214978	+	0.976619i	$0.568968\pi$
$90$	0	0
$91$	−3.77269	−0.395485
$92$	0	0
$93$	11.9088	1.23489
$94$	0	0
$95$	3.55353	0.364584
$96$	0	0
$97$	16.8881	1.71473	0.857365	−	0.514709i	$-0.172100\pi$
$97$	16.8881	1.71473	0.857365	+	0.514709i	$0.172100\pi$
$98$	0	0
$99$	4.64943	0.467285
$100$	0	0
$101$	−13.0169	−1.29523	−0.647616	−	0.761967i	$-0.724234\pi$
$101$	−13.0169	−1.29523	−0.647616	+	0.761967i	$0.724234\pi$
$102$	0	0
$103$	6.86859	0.676783	0.338391	−	0.941005i	$-0.390117\pi$
$103$	6.86859	0.676783	0.338391	+	0.941005i	$0.390117\pi$
$104$	0	0
$105$	−15.7527	−1.53731
$106$	0	0
$107$	15.7660	1.52416	0.762079	−	0.647484i	$-0.224179\pi$
$107$	15.7660	1.52416	0.762079	+	0.647484i	$0.224179\pi$
$108$	0	0
$109$	0.491186	0.0470471	0.0235236	−	0.999723i	$-0.492512\pi$
$109$	0.491186	0.0470471	0.0235236	+	0.999723i	$0.492512\pi$
$110$	0	0
$111$	30.9973	2.94213
$112$	0	0
$113$	16.2508	1.52874	0.764371	−	0.644776i	$-0.223050\pi$
$113$	16.2508	1.52874	0.764371	+	0.644776i	$0.223050\pi$
$114$	0	0
$115$	15.8763	1.48047
$116$	0	0
$117$	−9.44301	−0.873007
$118$	0	0
$119$	−5.09981	−0.467499
$120$	0	0
$121$	−9.74462	−0.885875
$122$	0	0
$123$	25.2960	2.28087
$124$	0	0
$125$	9.33707	0.835133
$126$	0	0
$127$	−11.2558	−0.998795	−0.499397	−	0.866373i	$-0.666445\pi$
$127$	−11.2558	−0.998795	−0.499397	+	0.866373i	$0.666445\pi$
$128$	0	0
$129$	−12.3888	−1.09077
$130$	0	0
$131$	15.9990	1.39784	0.698920	−	0.715200i	$-0.253665\pi$
$131$	15.9990	1.39784	0.698920	+	0.715200i	$0.253665\pi$
$132$	0	0
$133$	1.65788	0.143756
$134$	0	0
$135$	−10.9238	−0.940167
$136$	0	0
$137$	14.2950	1.22131	0.610654	−	0.791898i	$-0.290907\pi$
$137$	14.2950	1.22131	0.610654	+	0.791898i	$0.290907\pi$
$138$	0	0
$139$	2.04181	0.173184	0.0865919	−	0.996244i	$-0.472402\pi$
$139$	2.04181	0.173184	0.0865919	+	0.996244i	$0.472402\pi$
$140$	0	0
$141$	5.76037	0.485110
$142$	0	0
$143$	−2.54968	−0.213215
$144$	0	0
$145$	−13.5919	−1.12874
$146$	0	0
$147$	11.3679	0.937605
$148$	0	0
$149$	−11.0853	−0.908143	−0.454072	−	0.890965i	$-0.650029\pi$
$149$	−11.0853	−0.908143	−0.454072	+	0.890965i	$0.650029\pi$
$150$	0	0
$151$	−2.83057	−0.230349	−0.115174	−	0.993345i	$-0.536743\pi$
$151$	−2.83057	−0.230349	−0.115174	+	0.993345i	$0.536743\pi$
$152$	0	0
$153$	−12.7648	−1.03197
$154$	0	0
$155$	−15.8265	−1.27122
$156$	0	0
$157$	−8.00562	−0.638918	−0.319459	−	0.947600i	$-0.603501\pi$
$157$	−8.00562	−0.638918	−0.319459	+	0.947600i	$0.603501\pi$
$158$	0	0
$159$	19.6694	1.55989
$160$	0	0
$161$	7.40702	0.583755
$162$	0	0
$163$	2.75990	0.216172	0.108086	−	0.994142i	$-0.465528\pi$
$163$	2.75990	0.216172	0.108086	+	0.994142i	$0.465528\pi$
$164$	0	0
$165$	−10.6461	−0.828796
$166$	0	0
$167$	24.6556	1.90791	0.953954	−	0.299952i	$-0.0969706\pi$
$167$	24.6556	1.90791	0.953954	+	0.299952i	$0.0969706\pi$
$168$	0	0
$169$	−7.82160	−0.601661
$170$	0	0
$171$	4.14966	0.317332
$172$	0	0
$173$	−5.44441	−0.413931	−0.206965	−	0.978348i	$-0.566359\pi$
$173$	−5.44441	−0.413931	−0.206965	+	0.978348i	$0.566359\pi$
$174$	0	0
$175$	12.6456	0.955914
$176$	0	0
$177$	8.59297	0.645887
$178$	0	0
$179$	17.6753	1.32111	0.660555	−	0.750777i	$-0.270321\pi$
$179$	17.6753	1.32111	0.660555	+	0.750777i	$0.270321\pi$
$180$	0	0
$181$	−4.06274	−0.301981	−0.150991	−	0.988535i	$-0.548246\pi$
$181$	−4.06274	−0.301981	−0.150991	+	0.988535i	$0.548246\pi$
$182$	0	0
$183$	−31.1279	−2.30104
$184$	0	0
$185$	−41.1946	−3.02869
$186$	0	0
$187$	−3.44658	−0.252039
$188$	0	0
$189$	−5.09642	−0.370710
$190$	0	0
$191$	−5.03164	−0.364077	−0.182038	−	0.983291i	$-0.558270\pi$
$191$	−5.03164	−0.364077	−0.182038	+	0.983291i	$0.558270\pi$
$192$	0	0
$193$	−1.75661	−0.126444	−0.0632219	−	0.997999i	$-0.520138\pi$
$193$	−1.75661	−0.126444	−0.0632219	+	0.997999i	$0.520138\pi$
$194$	0	0
$195$	21.6222	1.54840
$196$	0	0
$197$	1.07380	0.0765053	0.0382527	−	0.999268i	$-0.487821\pi$
$197$	1.07380	0.0765053	0.0382527	+	0.999268i	$0.487821\pi$
$198$	0	0
$199$	−17.7158	−1.25584	−0.627918	−	0.778279i	$-0.716093\pi$
$199$	−17.7158	−1.25584	−0.627918	+	0.778279i	$0.716093\pi$
$200$	0	0
$201$	35.2038	2.48308
$202$	0	0
$203$	−6.34122	−0.445066
$204$	0	0
$205$	−33.6178	−2.34797
$206$	0	0
$207$	18.5397	1.28860
$208$	0	0
$209$	1.12044	0.0775022
$210$	0	0
$211$	21.1414	1.45543	0.727716	−	0.685878i	$-0.240582\pi$
$211$	21.1414	1.45543	0.727716	+	0.685878i	$0.240582\pi$
$212$	0	0
$213$	32.7677	2.24521
$214$	0	0
$215$	16.4644	1.12286
$216$	0	0
$217$	−7.38378	−0.501243
$218$	0	0
$219$	36.1539	2.44305
$220$	0	0
$221$	7.00002	0.470872
$222$	0	0
$223$	−4.82163	−0.322881	−0.161440	−	0.986882i	$-0.551614\pi$
$223$	−4.82163	−0.322881	−0.161440	+	0.986882i	$0.551614\pi$
$224$	0	0
$225$	31.6517	2.11012
$226$	0	0
$227$	6.60893	0.438650	0.219325	−	0.975652i	$-0.429615\pi$
$227$	6.60893	0.438650	0.219325	+	0.975652i	$0.429615\pi$
$228$	0	0
$229$	2.87595	0.190048	0.0950240	−	0.995475i	$-0.469707\pi$
$229$	2.87595	0.190048	0.0950240	+	0.995475i	$0.469707\pi$
$230$	0	0
$231$	−4.96687	−0.326796
$232$	0	0
$233$	−17.4858	−1.14553	−0.572765	−	0.819719i	$-0.694129\pi$
$233$	−17.4858	−1.14553	−0.572765	+	0.819719i	$0.694129\pi$
$234$	0	0
$235$	−7.65538	−0.499382
$236$	0	0
$237$	2.67388	0.173687
$238$	0	0
$239$	−14.8527	−0.960739	−0.480370	−	0.877066i	$-0.659497\pi$
$239$	−14.8527	−0.960739	−0.480370	+	0.877066i	$0.659497\pi$
$240$	0	0
$241$	24.4398	1.57431	0.787153	−	0.616758i	$-0.211554\pi$
$241$	24.4398	1.57431	0.787153	+	0.616758i	$0.211554\pi$
$242$	0	0
$243$	20.5308	1.31705
$244$	0	0
$245$	−15.1076	−0.965189
$246$	0	0
$247$	−2.27561	−0.144794
$248$	0	0
$249$	29.3152	1.85777
$250$	0	0
$251$	8.99518	0.567771	0.283885	−	0.958858i	$-0.408376\pi$
$251$	8.99518	0.567771	0.283885	+	0.958858i	$0.408376\pi$
$252$	0	0
$253$	5.00585	0.314715
$254$	0	0
$255$	29.2283	1.83035
$256$	0	0
$257$	−11.2594	−0.702339	−0.351170	−	0.936312i	$-0.614216\pi$
$257$	−11.2594	−0.702339	−0.351170	+	0.936312i	$0.614216\pi$
$258$	0	0
$259$	−19.2191	−1.19422
$260$	0	0
$261$	−15.8720	−0.982454
$262$	0	0
$263$	11.9807	0.738759	0.369380	−	0.929279i	$-0.379570\pi$
$263$	11.9807	0.738759	0.369380	+	0.929279i	$0.379570\pi$
$264$	0	0
$265$	−26.1402	−1.60578
$266$	0	0
$267$	10.8458	0.663752
$268$	0	0
$269$	6.31432	0.384991	0.192495	−	0.981298i	$-0.438342\pi$
$269$	6.31432	0.384991	0.192495	+	0.981298i	$0.438342\pi$
$270$	0	0
$271$	−9.81634	−0.596300	−0.298150	−	0.954519i	$-0.596370\pi$
$271$	−9.81634	−0.596300	−0.298150	+	0.954519i	$0.596370\pi$
$272$	0	0
$273$	10.0877	0.610537
$274$	0	0
$275$	8.54619	0.515355
$276$	0	0
$277$	−9.94650	−0.597628	−0.298814	−	0.954311i	$-0.596591\pi$
$277$	−9.94650	−0.597628	−0.298814	+	0.954311i	$0.596591\pi$
$278$	0	0
$279$	−18.4815	−1.10646
$280$	0	0
$281$	−10.3042	−0.614695	−0.307348	−	0.951597i	$-0.599441\pi$
$281$	−10.3042	−0.614695	−0.307348	+	0.951597i	$0.599441\pi$
$282$	0	0
$283$	−9.10870	−0.541456	−0.270728	−	0.962656i	$-0.587264\pi$
$283$	−9.10870	−0.541456	−0.270728	+	0.962656i	$0.587264\pi$
$284$	0	0
$285$	−9.50172	−0.562833
$286$	0	0
$287$	−15.6842	−0.925809
$288$	0	0
$289$	−7.53756	−0.443386
$290$	0	0
$291$	−45.1569	−2.64714
$292$	0	0
$293$	−34.2179	−1.99903	−0.999514	−	0.0311620i	$-0.990079\pi$
$293$	−34.2179	−1.99903	−0.999514	+	0.0311620i	$0.990079\pi$
$294$	0	0
$295$	−11.4198	−0.664889
$296$	0	0
$297$	−3.44429	−0.199858
$298$	0	0
$299$	−10.1669	−0.587967
$300$	0	0
$301$	7.68139	0.442748
$302$	0	0
$303$	34.8057	1.99954
$304$	0	0
$305$	41.3683	2.36874
$306$	0	0
$307$	−24.6365	−1.40608	−0.703041	−	0.711149i	$-0.748175\pi$
$307$	−24.6365	−1.40608	−0.703041	+	0.711149i	$0.748175\pi$
$308$	0	0
$309$	−18.3658	−1.04480
$310$	0	0
$311$	−24.1337	−1.36850	−0.684249	−	0.729248i	$-0.739870\pi$
$311$	−24.1337	−1.36850	−0.684249	+	0.729248i	$0.739870\pi$
$312$	0	0
$313$	30.2145	1.70783	0.853913	−	0.520416i	$-0.174223\pi$
$313$	30.2145	1.70783	0.853913	+	0.520416i	$0.174223\pi$
$314$	0	0
$315$	24.4470	1.37743
$316$	0	0
$317$	27.8375	1.56351	0.781756	−	0.623585i	$-0.214324\pi$
$317$	27.8375	1.56351	0.781756	+	0.623585i	$0.214324\pi$
$318$	0	0
$319$	−4.28556	−0.239945
$320$	0	0
$321$	−42.1565	−2.35295
$322$	0	0
$323$	−3.07611	−0.171159
$324$	0	0
$325$	−17.3573	−0.962812
$326$	0	0
$327$	−1.31338	−0.0726298
$328$	0	0
$329$	−3.57158	−0.196908
$330$	0	0
$331$	22.5351	1.23864	0.619320	−	0.785139i	$-0.287408\pi$
$331$	22.5351	1.23864	0.619320	+	0.785139i	$0.287408\pi$
$332$	0	0
$333$	−48.1054	−2.63616
$334$	0	0
$335$	−46.7849	−2.55613
$336$	0	0
$337$	16.3766	0.892092	0.446046	−	0.895010i	$-0.352832\pi$
$337$	16.3766	0.892092	0.446046	+	0.895010i	$0.352832\pi$
$338$	0	0
$339$	−43.4526	−2.36002
$340$	0	0
$341$	−4.99014	−0.270231
$342$	0	0
$343$	−18.6535	−1.00720
$344$	0	0
$345$	−42.4515	−2.28551
$346$	0	0
$347$	16.9785	0.911453	0.455726	−	0.890120i	$-0.349380\pi$
$347$	16.9785	0.911453	0.455726	+	0.890120i	$0.349380\pi$
$348$	0	0
$349$	−0.913351	−0.0488906	−0.0244453	−	0.999701i	$-0.507782\pi$
$349$	−0.913351	−0.0488906	−0.0244453	+	0.999701i	$0.507782\pi$
$350$	0	0
$351$	6.99536	0.373385
$352$	0	0
$353$	20.6531	1.09926	0.549628	−	0.835409i	$-0.314769\pi$
$353$	20.6531	1.09926	0.549628	+	0.835409i	$0.314769\pi$
$354$	0	0
$355$	−43.5475	−2.31126
$356$	0	0
$357$	13.6363	0.721710
$358$	0	0
$359$	−30.7753	−1.62426	−0.812129	−	0.583477i	$-0.801692\pi$
$359$	−30.7753	−1.62426	−0.812129	+	0.583477i	$0.801692\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	26.0560	1.36759
$364$	0	0
$365$	−48.0476	−2.51493
$366$	0	0
$367$	21.3499	1.11445	0.557227	−	0.830360i	$-0.311865\pi$
$367$	21.3499	1.11445	0.557227	+	0.830360i	$0.311865\pi$
$368$	0	0
$369$	−39.2575	−2.04366
$370$	0	0
$371$	−12.1956	−0.633162
$372$	0	0
$373$	−31.6340	−1.63795	−0.818973	−	0.573832i	$-0.805456\pi$
$373$	−31.6340	−1.63795	−0.818973	+	0.573832i	$0.805456\pi$
$374$	0	0
$375$	−24.9663	−1.28925
$376$	0	0
$377$	8.70398	0.448278
$378$	0	0
$379$	−12.6379	−0.649164	−0.324582	−	0.945858i	$-0.605224\pi$
$379$	−12.6379	−0.649164	−0.324582	+	0.945858i	$0.605224\pi$
$380$	0	0
$381$	30.0968	1.54191
$382$	0	0
$383$	35.1877	1.79801	0.899004	−	0.437941i	$-0.144292\pi$
$383$	35.1877	1.79801	0.899004	+	0.437941i	$0.144292\pi$
$384$	0	0
$385$	6.60085	0.336410
$386$	0	0
$387$	19.2265	0.977337
$388$	0	0
$389$	−21.1628	−1.07300	−0.536499	−	0.843901i	$-0.680253\pi$
$389$	−21.1628	−1.07300	−0.536499	+	0.843901i	$0.680253\pi$
$390$	0	0
$391$	−13.7433	−0.695030
$392$	0	0
$393$	−42.7795	−2.15794
$394$	0	0
$395$	−3.55353	−0.178797
$396$	0	0
$397$	23.3841	1.17362	0.586808	−	0.809726i	$-0.300385\pi$
$397$	23.3841	1.17362	0.586808	+	0.809726i	$0.300385\pi$
$398$	0	0
$399$	−4.43298	−0.221926
$400$	0	0
$401$	−36.4905	−1.82225	−0.911125	−	0.412131i	$-0.864785\pi$
$401$	−36.4905	−1.82225	−0.911125	+	0.412131i	$0.864785\pi$
$402$	0	0
$403$	10.1350	0.504860
$404$	0	0
$405$	−15.0289	−0.746793
$406$	0	0
$407$	−12.9888	−0.643830
$408$	0	0
$409$	28.2626	1.39749	0.698747	−	0.715368i	$-0.253741\pi$
$409$	28.2626	1.39749	0.698747	+	0.715368i	$0.253741\pi$
$410$	0	0
$411$	−38.2233	−1.88542
$412$	0	0
$413$	−5.32787	−0.262167
$414$	0	0
$415$	−38.9591	−1.91243
$416$	0	0
$417$	−5.45956	−0.267356
$418$	0	0
$419$	−6.34208	−0.309831	−0.154915	−	0.987928i	$-0.549510\pi$
$419$	−6.34208	−0.309831	−0.154915	+	0.987928i	$0.549510\pi$
$420$	0	0
$421$	21.7007	1.05763	0.528813	−	0.848739i	$-0.322637\pi$
$421$	21.7007	1.05763	0.528813	+	0.848739i	$0.322637\pi$
$422$	0	0
$423$	−8.93964	−0.434660
$424$	0	0
$425$	−23.4632	−1.13813
$426$	0	0
$427$	19.3002	0.934000
$428$	0	0
$429$	6.81754	0.329154
$430$	0	0
$431$	−4.81459	−0.231911	−0.115955	−	0.993254i	$-0.536993\pi$
$431$	−4.81459	−0.231911	−0.115955	+	0.993254i	$0.536993\pi$
$432$	0	0
$433$	−32.0527	−1.54035	−0.770177	−	0.637830i	$-0.779832\pi$
$433$	−32.0527	−1.54035	−0.770177	+	0.637830i	$0.779832\pi$
$434$	0	0
$435$	36.3431	1.74252
$436$	0	0
$437$	4.46777	0.213722
$438$	0	0
$439$	33.3217	1.59036	0.795178	−	0.606376i	$-0.207377\pi$
$439$	33.3217	1.59036	0.795178	+	0.606376i	$0.207377\pi$
$440$	0	0
$441$	−17.6420	−0.840096
$442$	0	0
$443$	6.26155	0.297495	0.148748	−	0.988875i	$-0.452476\pi$
$443$	6.26155	0.297495	0.148748	+	0.988875i	$0.452476\pi$
$444$	0	0
$445$	−14.4138	−0.683280
$446$	0	0
$447$	29.6408	1.40196
$448$	0	0
$449$	−28.0340	−1.32301	−0.661504	−	0.749942i	$-0.730082\pi$
$449$	−28.0340	−1.32301	−0.661504	+	0.749942i	$0.730082\pi$
$450$	0	0
$451$	−10.5998	−0.499124
$452$	0	0
$453$	7.56862	0.355605
$454$	0	0
$455$	−13.4063	−0.628499
$456$	0	0
$457$	−30.5800	−1.43047	−0.715236	−	0.698883i	$-0.753681\pi$
$457$	−30.5800	−1.43047	−0.715236	+	0.698883i	$0.753681\pi$
$458$	0	0
$459$	9.45614	0.441375
$460$	0	0
$461$	−2.71325	−0.126369	−0.0631844	−	0.998002i	$-0.520126\pi$
$461$	−2.71325	−0.126369	−0.0631844	+	0.998002i	$0.520126\pi$
$462$	0	0
$463$	−26.9595	−1.25291	−0.626456	−	0.779457i	$-0.715495\pi$
$463$	−26.9595	−1.25291	−0.626456	+	0.779457i	$0.715495\pi$
$464$	0	0
$465$	42.3183	1.96246
$466$	0	0
$467$	6.51255	0.301365	0.150682	−	0.988582i	$-0.451853\pi$
$467$	6.51255	0.301365	0.150682	+	0.988582i	$0.451853\pi$
$468$	0	0
$469$	−21.8273	−1.00789
$470$	0	0
$471$	21.4061	0.986342
$472$	0	0
$473$	5.19128	0.238695
$474$	0	0
$475$	7.62755	0.349976
$476$	0	0
$477$	−30.5254	−1.39766
$478$	0	0
$479$	−0.765359	−0.0349701	−0.0174851	−	0.999847i	$-0.505566\pi$
$479$	−0.765359	−0.0349701	−0.0174851	+	0.999847i	$0.505566\pi$
$480$	0	0
$481$	26.3803	1.20284
$482$	0	0
$483$	−19.8055	−0.901182
$484$	0	0
$485$	60.0124	2.72502
$486$	0	0
$487$	−14.7915	−0.670269	−0.335134	−	0.942170i	$-0.608782\pi$
$487$	−14.7915	−0.670269	−0.335134	+	0.942170i	$0.608782\pi$
$488$	0	0
$489$	−7.37966	−0.333720
$490$	0	0
$491$	−12.8798	−0.581257	−0.290629	−	0.956836i	$-0.593864\pi$
$491$	−12.8798	−0.581257	−0.290629	+	0.956836i	$0.593864\pi$
$492$	0	0
$493$	11.7658	0.529905
$494$	0	0
$495$	16.5219	0.742603
$496$	0	0
$497$	−20.3168	−0.911335
$498$	0	0
$499$	−7.29526	−0.326581	−0.163290	−	0.986578i	$-0.552211\pi$
$499$	−7.29526	−0.326581	−0.163290	+	0.986578i	$0.552211\pi$
$500$	0	0
$501$	−65.9263	−2.94537
$502$	0	0
$503$	−36.6667	−1.63489	−0.817443	−	0.576010i	$-0.804609\pi$
$503$	−36.6667	−1.63489	−0.817443	+	0.576010i	$0.804609\pi$
$504$	0	0
$505$	−46.2560	−2.05836
$506$	0	0
$507$	20.9141	0.928826
$508$	0	0
$509$	−14.7564	−0.654064	−0.327032	−	0.945013i	$-0.606048\pi$
$509$	−14.7564	−0.654064	−0.327032	+	0.945013i	$0.606048\pi$
$510$	0	0
$511$	−22.4164	−0.991642
$512$	0	0
$513$	−3.07406	−0.135723
$514$	0	0
$515$	24.4077	1.07553
$516$	0	0
$517$	−2.41376	−0.106157
$518$	0	0
$519$	14.5577	0.639013
$520$	0	0
$521$	−19.0319	−0.833804	−0.416902	−	0.908951i	$-0.636884\pi$
$521$	−19.0319	−0.833804	−0.416902	+	0.908951i	$0.636884\pi$
$522$	0	0
$523$	27.2712	1.19249	0.596244	−	0.802804i	$-0.296659\pi$
$523$	27.2712	1.19249	0.596244	+	0.802804i	$0.296659\pi$
$524$	0	0
$525$	−33.8128	−1.47571
$526$	0	0
$527$	13.7002	0.596790
$528$	0	0
$529$	−3.03907	−0.132134
$530$	0	0
$531$	−13.3356	−0.578716
$532$	0	0
$533$	21.5282	0.932489
$534$	0	0
$535$	56.0250	2.42217
$536$	0	0
$537$	−47.2616	−2.03949
$538$	0	0
$539$	−4.76347	−0.205177
$540$	0	0
$541$	−6.74519	−0.289998	−0.144999	−	0.989432i	$-0.546318\pi$
$541$	−6.74519	−0.289998	−0.144999	+	0.989432i	$0.546318\pi$
$542$	0	0
$543$	10.8633	0.466189
$544$	0	0
$545$	1.74544	0.0747666
$546$	0	0
$547$	−29.9655	−1.28123	−0.640615	−	0.767862i	$-0.721321\pi$
$547$	−29.9655	−1.28123	−0.640615	+	0.767862i	$0.721321\pi$
$548$	0	0
$549$	48.3081	2.06174
$550$	0	0
$551$	−3.82490	−0.162946
$552$	0	0
$553$	−1.65788	−0.0705002
$554$	0	0
$555$	110.150	4.67560
$556$	0	0
$557$	−5.29491	−0.224352	−0.112176	−	0.993688i	$-0.535782\pi$
$557$	−5.29491	−0.224352	−0.112176	+	0.993688i	$0.535782\pi$
$558$	0	0
$559$	−10.5435	−0.445943
$560$	0	0
$561$	9.21577	0.389090
$562$	0	0
$563$	7.16828	0.302107	0.151054	−	0.988526i	$-0.451733\pi$
$563$	7.16828	0.302107	0.151054	+	0.988526i	$0.451733\pi$
$564$	0	0
$565$	57.7475	2.42946
$566$	0	0
$567$	−7.01167	−0.294462
$568$	0	0
$569$	44.8609	1.88067	0.940334	−	0.340252i	$-0.110512\pi$
$569$	44.8609	1.88067	0.940334	+	0.340252i	$0.110512\pi$
$570$	0	0
$571$	2.96675	0.124154	0.0620772	−	0.998071i	$-0.480227\pi$
$571$	2.96675	0.124154	0.0620772	+	0.998071i	$0.480227\pi$
$572$	0	0
$573$	13.4540	0.562050
$574$	0	0
$575$	34.0781	1.42116
$576$	0	0
$577$	6.29065	0.261883	0.130942	−	0.991390i	$-0.458200\pi$
$577$	6.29065	0.261883	0.130942	+	0.991390i	$0.458200\pi$
$578$	0	0
$579$	4.69698	0.195200
$580$	0	0
$581$	−18.1762	−0.754075
$582$	0	0
$583$	−8.24207	−0.341352
$584$	0	0
$585$	−33.5560	−1.38737
$586$	0	0
$587$	26.4920	1.09344	0.546722	−	0.837314i	$-0.315876\pi$
$587$	26.4920	1.09344	0.546722	+	0.837314i	$0.315876\pi$
$588$	0	0
$589$	−4.45375	−0.183513
$590$	0	0
$591$	−2.87123	−0.118106
$592$	0	0
$593$	11.2771	0.463095	0.231548	−	0.972824i	$-0.425621\pi$
$593$	11.2771	0.463095	0.231548	+	0.972824i	$0.425621\pi$
$594$	0	0
$595$	−18.1223	−0.742943
$596$	0	0
$597$	47.3699	1.93872
$598$	0	0
$599$	20.2223	0.826262	0.413131	−	0.910672i	$-0.364435\pi$
$599$	20.2223	0.826262	0.413131	+	0.910672i	$0.364435\pi$
$600$	0	0
$601$	−11.7713	−0.480162	−0.240081	−	0.970753i	$-0.577174\pi$
$601$	−11.7713	−0.480162	−0.240081	+	0.970753i	$0.577174\pi$
$602$	0	0
$603$	−54.6335	−2.22485
$604$	0	0
$605$	−34.6278	−1.40782
$606$	0	0
$607$	−7.12655	−0.289258	−0.144629	−	0.989486i	$-0.546199\pi$
$607$	−7.12655	−0.289258	−0.144629	+	0.989486i	$0.546199\pi$
$608$	0	0
$609$	16.9557	0.687079
$610$	0	0
$611$	4.90236	0.198328
$612$	0	0
$613$	30.9321	1.24934	0.624668	−	0.780890i	$-0.285234\pi$
$613$	30.9321	1.24934	0.624668	+	0.780890i	$0.285234\pi$
$614$	0	0
$615$	89.8901	3.62472
$616$	0	0
$617$	−5.22160	−0.210214	−0.105107	−	0.994461i	$-0.533518\pi$
$617$	−5.22160	−0.210214	−0.105107	+	0.994461i	$0.533518\pi$
$618$	0	0
$619$	20.5205	0.824788	0.412394	−	0.911006i	$-0.364693\pi$
$619$	20.5205	0.824788	0.412394	+	0.911006i	$0.364693\pi$
$620$	0	0
$621$	−13.7342	−0.551134
$622$	0	0
$623$	−6.72468	−0.269419
$624$	0	0
$625$	−4.95821	−0.198329
$626$	0	0
$627$	−2.99592	−0.119645
$628$	0	0
$629$	35.6601	1.42186
$630$	0	0
$631$	−44.9891	−1.79099	−0.895493	−	0.445075i	$-0.853177\pi$
$631$	−44.9891	−1.79099	−0.895493	+	0.445075i	$0.853177\pi$
$632$	0	0
$633$	−56.5296	−2.24685
$634$	0	0
$635$	−39.9980	−1.58727
$636$	0	0
$637$	9.67461	0.383322
$638$	0	0
$639$	−50.8529	−2.01171
$640$	0	0
$641$	34.6930	1.37029	0.685145	−	0.728407i	$-0.259739\pi$
$641$	34.6930	1.37029	0.685145	+	0.728407i	$0.259739\pi$
$642$	0	0
$643$	−30.7380	−1.21219	−0.606094	−	0.795393i	$-0.707264\pi$
$643$	−30.7380	−1.21219	−0.606094	+	0.795393i	$0.707264\pi$
$644$	0	0
$645$	−44.0240	−1.73344
$646$	0	0
$647$	−25.5916	−1.00611	−0.503055	−	0.864255i	$-0.667791\pi$
$647$	−25.5916	−1.00611	−0.503055	+	0.864255i	$0.667791\pi$
$648$	0	0
$649$	−3.60071	−0.141340
$650$	0	0
$651$	19.7434	0.773804
$652$	0	0
$653$	13.5682	0.530966	0.265483	−	0.964116i	$-0.414469\pi$
$653$	13.5682	0.530966	0.265483	+	0.964116i	$0.414469\pi$
$654$	0	0
$655$	56.8529	2.22143
$656$	0	0
$657$	−56.1080	−2.18898
$658$	0	0
$659$	10.7621	0.419234	0.209617	−	0.977784i	$-0.432778\pi$
$659$	10.7621	0.419234	0.209617	+	0.977784i	$0.432778\pi$
$660$	0	0
$661$	13.2305	0.514606	0.257303	−	0.966331i	$-0.417166\pi$
$661$	13.2305	0.514606	0.257303	+	0.966331i	$0.417166\pi$
$662$	0	0
$663$	−18.7173	−0.726918
$664$	0	0
$665$	5.89132	0.228456
$666$	0	0
$667$	−17.0887	−0.661679
$668$	0	0
$669$	12.8925	0.498453
$670$	0	0
$671$	13.0435	0.503540
$672$	0	0
$673$	37.4043	1.44183	0.720914	−	0.693024i	$-0.243722\pi$
$673$	37.4043	1.44183	0.720914	+	0.693024i	$0.243722\pi$
$674$	0	0
$675$	−23.4476	−0.902497
$676$	0	0
$677$	31.3234	1.20386	0.601928	−	0.798550i	$-0.294399\pi$
$677$	31.3234	1.20386	0.601928	+	0.798550i	$0.294399\pi$
$678$	0	0
$679$	27.9985	1.07448
$680$	0	0
$681$	−17.6715	−0.677174
$682$	0	0
$683$	−6.56990	−0.251390	−0.125695	−	0.992069i	$-0.540116\pi$
$683$	−6.56990	−0.251390	−0.125695	+	0.992069i	$0.540116\pi$
$684$	0	0
$685$	50.7978	1.94088
$686$	0	0
$687$	−7.68995	−0.293390
$688$	0	0
$689$	16.7397	0.637730
$690$	0	0
$691$	−11.3562	−0.432009	−0.216005	−	0.976392i	$-0.569303\pi$
$691$	−11.3562	−0.432009	−0.216005	+	0.976392i	$0.569303\pi$
$692$	0	0
$693$	7.70820	0.292810
$694$	0	0
$695$	7.25562	0.275221
$696$	0	0
$697$	29.1012	1.10229
$698$	0	0
$699$	46.7549	1.76843
$700$	0	0
$701$	−13.9697	−0.527628	−0.263814	−	0.964574i	$-0.584981\pi$
$701$	−13.9697	−0.527628	−0.263814	+	0.964574i	$0.584981\pi$
$702$	0	0
$703$	−11.5926	−0.437224
$704$	0	0
$705$	20.4696	0.770930
$706$	0	0
$707$	−21.5805	−0.811617
$708$	0	0
$709$	−23.3876	−0.878341	−0.439170	−	0.898404i	$-0.644728\pi$
$709$	−23.3876	−0.878341	−0.439170	+	0.898404i	$0.644728\pi$
$710$	0	0
$711$	−4.14966	−0.155624
$712$	0	0
$713$	−19.8983	−0.745197
$714$	0	0
$715$	−9.06035	−0.338838
$716$	0	0
$717$	39.7143	1.48316
$718$	0	0
$719$	18.2219	0.679564	0.339782	−	0.940504i	$-0.389647\pi$
$719$	18.2219	0.679564	0.339782	+	0.940504i	$0.389647\pi$
$720$	0	0
$721$	11.3873	0.424085
$722$	0	0
$723$	−65.3492	−2.43036
$724$	0	0
$725$	−29.1746	−1.08352
$726$	0	0
$727$	12.7594	0.473221	0.236611	−	0.971605i	$-0.423963\pi$
$727$	12.7594	0.473221	0.236611	+	0.971605i	$0.423963\pi$
$728$	0	0
$729$	−42.2092	−1.56330
$730$	0	0
$731$	−14.2524	−0.527145
$732$	0	0
$733$	−14.5936	−0.539029	−0.269514	−	0.962996i	$-0.586863\pi$
$733$	−14.5936	−0.539029	−0.269514	+	0.962996i	$0.586863\pi$
$734$	0	0
$735$	40.3960	1.49003
$736$	0	0
$737$	−14.7514	−0.543375
$738$	0	0
$739$	−7.44091	−0.273718	−0.136859	−	0.990591i	$-0.543701\pi$
$739$	−7.44091	−0.273718	−0.136859	+	0.990591i	$0.543701\pi$
$740$	0	0
$741$	6.08472	0.223528
$742$	0	0
$743$	−34.1645	−1.25337	−0.626687	−	0.779271i	$-0.715590\pi$
$743$	−34.1645	−1.25337	−0.626687	+	0.779271i	$0.715590\pi$
$744$	0	0
$745$	−39.3919	−1.44321
$746$	0	0
$747$	−45.4949	−1.66457
$748$	0	0
$749$	26.1382	0.955068
$750$	0	0
$751$	13.7963	0.503435	0.251718	−	0.967801i	$-0.419005\pi$
$751$	13.7963	0.503435	0.251718	+	0.967801i	$0.419005\pi$
$752$	0	0
$753$	−24.0521	−0.876507
$754$	0	0
$755$	−10.0585	−0.366067
$756$	0	0
$757$	−15.7598	−0.572798	−0.286399	−	0.958110i	$-0.592458\pi$
$757$	−15.7598	−0.572798	−0.286399	+	0.958110i	$0.592458\pi$
$758$	0	0
$759$	−13.3851	−0.485847
$760$	0	0
$761$	13.7578	0.498720	0.249360	−	0.968411i	$-0.419780\pi$
$761$	13.7578	0.498720	0.249360	+	0.968411i	$0.419780\pi$
$762$	0	0
$763$	0.814328	0.0294806
$764$	0	0
$765$	−45.3601	−1.64000
$766$	0	0
$767$	7.31305	0.264059
$768$	0	0
$769$	−53.0357	−1.91252	−0.956259	−	0.292522i	$-0.905506\pi$
$769$	−53.0357	−1.91252	−0.956259	+	0.292522i	$0.905506\pi$
$770$	0	0
$771$	30.1062	1.08425
$772$	0	0
$773$	42.8442	1.54100	0.770500	−	0.637440i	$-0.220007\pi$
$773$	42.8442	1.54100	0.770500	+	0.637440i	$0.220007\pi$
$774$	0	0
$775$	−33.9712	−1.22028
$776$	0	0
$777$	51.3898	1.84360
$778$	0	0
$779$	−9.46040	−0.338954
$780$	0	0
$781$	−13.7306	−0.491321
$782$	0	0
$783$	11.7580	0.420195
$784$	0	0
$785$	−28.4482	−1.01536
$786$	0	0
$787$	−11.7335	−0.418256	−0.209128	−	0.977888i	$-0.567062\pi$
$787$	−11.7335	−0.418256	−0.209128	+	0.977888i	$0.567062\pi$
$788$	0	0
$789$	−32.0349	−1.14047
$790$	0	0
$791$	26.9418	0.957940
$792$	0	0
$793$	−26.4914	−0.940739
$794$	0	0
$795$	69.8958	2.47895
$796$	0	0
$797$	44.3184	1.56984	0.784920	−	0.619598i	$-0.212704\pi$
$797$	44.3184	1.56984	0.784920	+	0.619598i	$0.212704\pi$
$798$	0	0
$799$	6.62688	0.234442
$800$	0	0
$801$	−16.8318	−0.594724
$802$	0	0
$803$	−15.1495	−0.534616
$804$	0	0
$805$	26.3210	0.927694
$806$	0	0
$807$	−16.8838	−0.594336
$808$	0	0
$809$	30.8365	1.08415	0.542077	−	0.840329i	$-0.317638\pi$
$809$	30.8365	1.08415	0.542077	+	0.840329i	$0.317638\pi$
$810$	0	0
$811$	−9.70574	−0.340815	−0.170407	−	0.985374i	$-0.554508\pi$
$811$	−9.70574	−0.340815	−0.170407	+	0.985374i	$0.554508\pi$
$812$	0	0
$813$	26.2478	0.920549
$814$	0	0
$815$	9.80739	0.343538
$816$	0	0
$817$	4.63326	0.162097
$818$	0	0
$819$	−15.6554	−0.547043
$820$	0	0
$821$	3.55255	0.123985	0.0619924	−	0.998077i	$-0.480255\pi$
$821$	3.55255	0.123985	0.0619924	+	0.998077i	$0.480255\pi$
$822$	0	0
$823$	39.8774	1.39004	0.695020	−	0.718990i	$-0.255395\pi$
$823$	39.8774	1.39004	0.695020	+	0.718990i	$0.255395\pi$
$824$	0	0
$825$	−22.8515	−0.795588
$826$	0	0
$827$	−14.0469	−0.488460	−0.244230	−	0.969717i	$-0.578535\pi$
$827$	−14.0469	−0.488460	−0.244230	+	0.969717i	$0.578535\pi$
$828$	0	0
$829$	−15.9389	−0.553581	−0.276791	−	0.960930i	$-0.589271\pi$
$829$	−15.9389	−0.553581	−0.276791	+	0.960930i	$0.589271\pi$
$830$	0	0
$831$	26.5958	0.922598
$832$	0	0
$833$	13.0779	0.453121
$834$	0	0
$835$	87.6144	3.03202
$836$	0	0
$837$	13.6911	0.473233
$838$	0	0
$839$	−12.5192	−0.432210	−0.216105	−	0.976370i	$-0.569335\pi$
$839$	−12.5192	−0.432210	−0.216105	+	0.976370i	$0.569335\pi$
$840$	0	0
$841$	−14.3702	−0.495522
$842$	0	0
$843$	27.5522	0.948947
$844$	0	0
$845$	−27.7943	−0.956151
$846$	0	0
$847$	−16.1554	−0.555106
$848$	0	0
$849$	24.3556	0.835883
$850$	0	0
$851$	−51.7931	−1.77544
$852$	0	0
$853$	46.4699	1.59110	0.795549	−	0.605890i	$-0.207183\pi$
$853$	46.4699	1.59110	0.795549	+	0.605890i	$0.207183\pi$
$854$	0	0
$855$	14.7459	0.504300
$856$	0	0
$857$	−15.0247	−0.513235	−0.256617	−	0.966513i	$-0.582608\pi$
$857$	−15.0247	−0.513235	−0.256617	+	0.966513i	$0.582608\pi$
$858$	0	0
$859$	9.36733	0.319609	0.159805	−	0.987149i	$-0.448914\pi$
$859$	9.36733	0.319609	0.159805	+	0.987149i	$0.448914\pi$
$860$	0	0
$861$	41.9378	1.42924
$862$	0	0
$863$	−42.2506	−1.43823	−0.719114	−	0.694893i	$-0.755452\pi$
$863$	−42.2506	−1.43823	−0.719114	+	0.694893i	$0.755452\pi$
$864$	0	0
$865$	−19.3468	−0.657812
$866$	0	0
$867$	20.1546	0.684485
$868$	0	0
$869$	−1.12044	−0.0380082
$870$	0	0
$871$	29.9602	1.01516
$872$	0	0
$873$	70.0800	2.37185
$874$	0	0
$875$	15.4797	0.523311
$876$	0	0
$877$	−44.0007	−1.48580	−0.742899	−	0.669403i	$-0.766550\pi$
$877$	−44.0007	−1.48580	−0.742899	+	0.669403i	$0.766550\pi$
$878$	0	0
$879$	91.4946	3.08604
$880$	0	0
$881$	−44.6224	−1.50337	−0.751683	−	0.659525i	$-0.770758\pi$
$881$	−44.6224	−1.50337	−0.751683	+	0.659525i	$0.770758\pi$
$882$	0	0
$883$	23.5680	0.793126	0.396563	−	0.918008i	$-0.370203\pi$
$883$	23.5680	0.793126	0.396563	+	0.918008i	$0.370203\pi$
$884$	0	0
$885$	30.5353	1.02643
$886$	0	0
$887$	23.3974	0.785609	0.392804	−	0.919622i	$-0.371505\pi$
$887$	23.3974	0.785609	0.392804	+	0.919622i	$0.371505\pi$
$888$	0	0
$889$	−18.6608	−0.625864
$890$	0	0
$891$	−4.73866	−0.158751
$892$	0	0
$893$	−2.15431	−0.0720911
$894$	0	0
$895$	62.8095	2.09949
$896$	0	0
$897$	27.1851	0.907684
$898$	0	0
$899$	17.0351	0.568153
$900$	0	0
$901$	22.6282	0.753855
$902$	0	0
$903$	−20.5392	−0.683501
$904$	0	0
$905$	−14.4371	−0.479904
$906$	0	0
$907$	−38.9279	−1.29258	−0.646290	−	0.763092i	$-0.723680\pi$
$907$	−38.9279	−1.29258	−0.646290	+	0.763092i	$0.723680\pi$
$908$	0	0
$909$	−54.0158	−1.79159
$910$	0	0
$911$	−3.05729	−0.101293	−0.0506463	−	0.998717i	$-0.516128\pi$
$911$	−3.05729	−0.101293	−0.0506463	+	0.998717i	$0.516128\pi$
$912$	0	0
$913$	−12.2839	−0.406539
$914$	0	0
$915$	−110.614	−3.65679
$916$	0	0
$917$	26.5244	0.875913
$918$	0	0
$919$	33.2902	1.09814	0.549071	−	0.835776i	$-0.314982\pi$
$919$	33.2902	1.09814	0.549071	+	0.835776i	$0.314982\pi$
$920$	0	0
$921$	65.8753	2.17066
$922$	0	0
$923$	27.8870	0.917910
$924$	0	0
$925$	−88.4232	−2.90734
$926$	0	0
$927$	28.5023	0.936139
$928$	0	0
$929$	−22.0090	−0.722093	−0.361046	−	0.932548i	$-0.617580\pi$
$929$	−22.0090	−0.722093	−0.361046	+	0.932548i	$0.617580\pi$
$930$	0	0
$931$	−4.25144	−0.139335
$932$	0	0
$933$	64.5308	2.11264
$934$	0	0
$935$	−12.2475	−0.400537
$936$	0	0
$937$	52.5179	1.71569	0.857843	−	0.513911i	$-0.171804\pi$
$937$	52.5179	1.71569	0.857843	+	0.513911i	$0.171804\pi$
$938$	0	0
$939$	−80.7901	−2.63649
$940$	0	0
$941$	28.9536	0.943862	0.471931	−	0.881636i	$-0.343557\pi$
$941$	28.9536	0.943862	0.471931	+	0.881636i	$0.343557\pi$
$942$	0	0
$943$	−42.2669	−1.37640
$944$	0	0
$945$	−18.1103	−0.589127
$946$	0	0
$947$	54.9697	1.78628	0.893138	−	0.449782i	$-0.148498\pi$
$947$	54.9697	1.78628	0.893138	+	0.449782i	$0.148498\pi$
$948$	0	0
$949$	30.7688	0.998797
$950$	0	0
$951$	−74.4343	−2.41370
$952$	0	0
$953$	−55.1084	−1.78514	−0.892568	−	0.450913i	$-0.851098\pi$
$953$	−55.1084	−1.78514	−0.892568	+	0.450913i	$0.851098\pi$
$954$	0	0
$955$	−17.8801	−0.578586
$956$	0	0
$957$	11.4591	0.370420
$958$	0	0
$959$	23.6995	0.765295
$960$	0	0
$961$	−11.1641	−0.360133
$962$	0	0
$963$	65.4236	2.10825
$964$	0	0
$965$	−6.24217	−0.200943
$966$	0	0
$967$	10.1313	0.325801	0.162900	−	0.986643i	$-0.447915\pi$
$967$	10.1313	0.325801	0.162900	+	0.986643i	$0.447915\pi$
$968$	0	0
$969$	8.22516	0.264230
$970$	0	0
$971$	56.7893	1.82246	0.911228	−	0.411901i	$-0.135135\pi$
$971$	56.7893	1.82246	0.911228	+	0.411901i	$0.135135\pi$
$972$	0	0
$973$	3.38507	0.108520
$974$	0	0
$975$	46.4115	1.48636
$976$	0	0
$977$	24.3146	0.777894	0.388947	−	0.921260i	$-0.372839\pi$
$977$	24.3146	0.777894	0.388947	+	0.921260i	$0.372839\pi$
$978$	0	0
$979$	−4.54471	−0.145250
$980$	0	0
$981$	2.03826	0.0650765
$982$	0	0
$983$	−36.6882	−1.17017	−0.585087	−	0.810971i	$-0.698940\pi$
$983$	−36.6882	−1.17017	−0.585087	+	0.810971i	$0.698940\pi$
$984$	0	0
$985$	3.81579	0.121581
$986$	0	0
$987$	9.54999	0.303980
$988$	0	0
$989$	20.7003	0.658233
$990$	0	0
$991$	−38.6198	−1.22680	−0.613399	−	0.789773i	$-0.710198\pi$
$991$	−38.6198	−1.22680	−0.613399	+	0.789773i	$0.710198\pi$
$992$	0	0
$993$	−60.2562	−1.91217
$994$	0	0
$995$	−62.9534	−1.99576
$996$	0	0
$997$	−58.0512	−1.83850	−0.919251	−	0.393673i	$-0.871204\pi$
$997$	−58.0512	−1.83850	−0.919251	+	0.393673i	$0.871204\pi$
$998$	0	0
$999$	35.6364	1.12748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.4	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.4	✓	27	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67388 q^{3} +3.55353 q^{5} +1.65788 q^{7} +4.14966 q^{9} +O(q^{10})\)	\(q-2.67388 q^{3} +3.55353 q^{5} +1.65788 q^{7} +4.14966 q^{9} +1.12044 q^{11} -2.27561 q^{13} -9.50172 q^{15} -3.07611 q^{17} +1.00000 q^{19} -4.43298 q^{21} +4.46777 q^{23} +7.62755 q^{25} -3.07406 q^{27} -3.82490 q^{29} -4.45375 q^{31} -2.99592 q^{33} +5.89132 q^{35} -11.5926 q^{37} +6.08472 q^{39} -9.46040 q^{41} +4.63326 q^{43} +14.7459 q^{45} -2.15431 q^{47} -4.25144 q^{49} +8.22516 q^{51} -7.35612 q^{53} +3.98150 q^{55} -2.67388 q^{57} -3.21366 q^{59} +11.6415 q^{61} +6.87964 q^{63} -8.08644 q^{65} -13.1658 q^{67} -11.9463 q^{69} -12.2547 q^{71} -13.5211 q^{73} -20.3952 q^{75} +1.85755 q^{77} -1.00000 q^{79} -4.22930 q^{81} -10.9635 q^{83} -10.9310 q^{85} +10.2273 q^{87} -4.05620 q^{89} -3.77269 q^{91} +11.9088 q^{93} +3.55353 q^{95} +16.8881 q^{97} +4.64943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

Varieties

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