LMFDB - Embedded newform 2320.4.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,4,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2320.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2320.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:	$136.884431213$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 91x^{4} + 90x^{3} + 1784x^{2} - 1456x - 3720 $ x^6 - x^5 - 91x^4 + 90x^3 + 1784x^2 - 1456x - 3720
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.06881$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.49913 q^{3} -5.00000 q^{5} +2.97283 q^{7} +15.2386 q^{9} -26.0077 q^{11} -14.3733 q^{13} -32.4956 q^{15} +100.181 q^{17} +99.3664 q^{19} +19.3208 q^{21} -102.055 q^{23} +25.0000 q^{25} -76.4386 q^{27} -29.0000 q^{29} -253.319 q^{31} -169.027 q^{33} -14.8641 q^{35} -348.399 q^{37} -93.4137 q^{39} +106.824 q^{41} -68.0944 q^{43} -76.1932 q^{45} +285.052 q^{47} -334.162 q^{49} +651.088 q^{51} +409.075 q^{53} +130.038 q^{55} +645.795 q^{57} +395.912 q^{59} -835.967 q^{61} +45.3018 q^{63} +71.8663 q^{65} -4.99986 q^{67} -663.268 q^{69} +67.3198 q^{71} +124.328 q^{73} +162.478 q^{75} -77.3163 q^{77} +923.032 q^{79} -908.227 q^{81} -455.719 q^{83} -500.905 q^{85} -188.475 q^{87} -1017.98 q^{89} -42.7292 q^{91} -1646.35 q^{93} -496.832 q^{95} -5.39427 q^{97} -396.322 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 30 q^{5} - 9 q^{7} + 45 q^{9} + 10 q^{11} + 13 q^{13} - 25 q^{15} - 23 q^{17} - 10 q^{19} - 52 q^{21} - 75 q^{23} + 150 q^{25} + 116 q^{27} - 174 q^{29} + 127 q^{31} - 72 q^{33} + 45 q^{35}+ \cdots + 2976 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 - 30 * q^5 - 9 * q^7 + 45 * q^9 + 10 * q^11 + 13 * q^13 - 25 * q^15 - 23 * q^17 - 10 * q^19 - 52 * q^21 - 75 * q^23 + 150 * q^25 + 116 * q^27 - 174 * q^29 + 127 * q^31 - 72 * q^33 + 45 * q^35 - 158 * q^37 + 407 * q^39 - 622 * q^41 + 311 * q^43 - 225 * q^45 + 628 * q^47 - 879 * q^49 + 886 * q^51 - 813 * q^53 - 50 * q^55 - 1118 * q^57 + 1615 * q^59 - 655 * q^61 + 764 * q^63 - 65 * q^65 + 1000 * q^67 - 1885 * q^69 + 760 * q^71 - 471 * q^73 + 125 * q^75 - 1078 * q^77 + 577 * q^79 - 1402 * q^81 + 1446 * q^83 + 115 * q^85 - 145 * q^87 - 1970 * q^89 + 876 * q^91 - 2274 * q^93 + 50 * q^95 + 1683 * q^97 + 2976 * q^99

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
$3$	6.49913	1.25076	0.625379	−	0.780321i	$-0.284945\pi$
$3$	6.49913	1.25076	0.625379	+	0.780321i	$0.284945\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	2.97283	0.160518	0.0802588	−	0.996774i	$-0.474425\pi$
$7$	2.97283	0.160518	0.0802588	+	0.996774i	$0.474425\pi$
$8$	0	0
$9$	15.2386	0.564394
$10$	0	0
$11$	−26.0077	−0.712874	−0.356437	−	0.934319i	$-0.616009\pi$
$11$	−26.0077	−0.712874	−0.356437	+	0.934319i	$0.616009\pi$
$12$	0	0
$13$	−14.3733	−0.306648	−0.153324	−	0.988176i	$-0.548998\pi$
$13$	−14.3733	−0.306648	−0.153324	+	0.988176i	$0.548998\pi$
$14$	0	0
$15$	−32.4956	−0.559356
$16$	0	0
$17$	100.181	1.42926	0.714631	−	0.699502i	$-0.246595\pi$
$17$	100.181	1.42926	0.714631	+	0.699502i	$0.246595\pi$
$18$	0	0
$19$	99.3664	1.19980	0.599900	−	0.800075i	$-0.295207\pi$
$19$	99.3664	1.19980	0.599900	+	0.800075i	$0.295207\pi$
$20$	0	0
$21$	19.3208	0.200768
$22$	0	0
$23$	−102.055	−0.925215	−0.462607	−	0.886563i	$-0.653086\pi$
$23$	−102.055	−0.925215	−0.462607	+	0.886563i	$0.653086\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−76.4386	−0.544837
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−253.319	−1.46766	−0.733829	−	0.679334i	$-0.762269\pi$
$31$	−253.319	−1.46766	−0.733829	+	0.679334i	$0.762269\pi$
$32$	0	0
$33$	−169.027	−0.891632
$34$	0	0
$35$	−14.8641	−0.0717856
$36$	0	0
$37$	−348.399	−1.54801	−0.774007	−	0.633177i	$-0.781750\pi$
$37$	−348.399	−1.54801	−0.774007	+	0.633177i	$0.781750\pi$
$38$	0	0
$39$	−93.4137	−0.383543
$40$	0	0
$41$	106.824	0.406905	0.203453	−	0.979085i	$-0.434784\pi$
$41$	106.824	0.406905	0.203453	+	0.979085i	$0.434784\pi$
$42$	0	0
$43$	−68.0944	−0.241495	−0.120748	−	0.992683i	$-0.538529\pi$
$43$	−68.0944	−0.241495	−0.120748	+	0.992683i	$0.538529\pi$
$44$	0	0
$45$	−76.1932	−0.252405
$46$	0	0
$47$	285.052	0.884662	0.442331	−	0.896852i	$-0.354152\pi$
$47$	285.052	0.884662	0.442331	+	0.896852i	$0.354152\pi$
$48$	0	0
$49$	−334.162	−0.974234
$50$	0	0
$51$	651.088	1.78766
$52$	0	0
$53$	409.075	1.06020	0.530102	−	0.847934i	$-0.322154\pi$
$53$	409.075	1.06020	0.530102	+	0.847934i	$0.322154\pi$
$54$	0	0
$55$	130.038	0.318807
$56$	0	0
$57$	645.795	1.50066
$58$	0	0
$59$	395.912	0.873615	0.436808	−	0.899555i	$-0.356109\pi$
$59$	395.912	0.873615	0.436808	+	0.899555i	$0.356109\pi$
$60$	0	0
$61$	−835.967	−1.75467	−0.877333	−	0.479882i	$-0.840680\pi$
$61$	−835.967	−1.75467	−0.877333	+	0.479882i	$0.840680\pi$
$62$	0	0
$63$	45.3018	0.0905951
$64$	0	0
$65$	71.8663	0.137137
$66$	0	0
$67$	−4.99986	−0.00911686	−0.00455843	−	0.999990i	$-0.501451\pi$
$67$	−4.99986	−0.00911686	−0.00455843	+	0.999990i	$0.501451\pi$
$68$	0	0
$69$	−663.268	−1.15722
$70$	0	0
$71$	67.3198	0.112527	0.0562633	−	0.998416i	$-0.482081\pi$
$71$	67.3198	0.112527	0.0562633	+	0.998416i	$0.482081\pi$
$72$	0	0
$73$	124.328	0.199336	0.0996679	−	0.995021i	$-0.468222\pi$
$73$	124.328	0.199336	0.0996679	+	0.995021i	$0.468222\pi$
$74$	0	0
$75$	162.478	0.250151
$76$	0	0
$77$	−77.3163	−0.114429
$78$	0	0
$79$	923.032	1.31455	0.657273	−	0.753652i	$-0.271710\pi$
$79$	923.032	1.31455	0.657273	+	0.753652i	$0.271710\pi$
$80$	0	0
$81$	−908.227	−1.24585
$82$	0	0
$83$	−455.719	−0.602671	−0.301335	−	0.953518i	$-0.597432\pi$
$83$	−455.719	−0.602671	−0.301335	+	0.953518i	$0.597432\pi$
$84$	0	0
$85$	−500.905	−0.639185
$86$	0	0
$87$	−188.475	−0.232260
$88$	0	0
$89$	−1017.98	−1.21243	−0.606214	−	0.795301i	$-0.707313\pi$
$89$	−1017.98	−1.21243	−0.606214	+	0.795301i	$0.707313\pi$
$90$	0	0
$91$	−42.7292	−0.0492224
$92$	0	0
$93$	−1646.35	−1.83568
$94$	0	0
$95$	−496.832	−0.536567
$96$	0	0
$97$	−5.39427	−0.00564645	−0.00282322	−	0.999996i	$-0.500899\pi$
$97$	−5.39427	−0.00564645	−0.00282322	+	0.999996i	$0.500899\pi$
$98$	0	0
$99$	−396.322	−0.402342
$100$	0	0
$101$	132.305	0.130345	0.0651727	−	0.997874i	$-0.479240\pi$
$101$	132.305	0.130345	0.0651727	+	0.997874i	$0.479240\pi$
$102$	0	0
$103$	−1966.82	−1.88152	−0.940762	−	0.339068i	$-0.889888\pi$
$103$	−1966.82	−1.88152	−0.940762	+	0.339068i	$0.889888\pi$
$104$	0	0
$105$	−96.6039	−0.0897864
$106$	0	0
$107$	1734.39	1.56701	0.783503	−	0.621388i	$-0.213431\pi$
$107$	1734.39	1.56701	0.783503	+	0.621388i	$0.213431\pi$
$108$	0	0
$109$	−302.909	−0.266178	−0.133089	−	0.991104i	$-0.542490\pi$
$109$	−302.909	−0.266178	−0.133089	+	0.991104i	$0.542490\pi$
$110$	0	0
$111$	−2264.29	−1.93619
$112$	0	0
$113$	−2240.47	−1.86518	−0.932591	−	0.360935i	$-0.882458\pi$
$113$	−2240.47	−1.86518	−0.932591	+	0.360935i	$0.882458\pi$
$114$	0	0
$115$	510.275	0.413769
$116$	0	0
$117$	−219.029	−0.173070
$118$	0	0
$119$	297.820	0.229421
$120$	0	0
$121$	−654.600	−0.491811
$122$	0	0
$123$	694.263	0.508940
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	665.245	0.464811	0.232405	−	0.972619i	$-0.425340\pi$
$127$	665.245	0.464811	0.232405	+	0.972619i	$0.425340\pi$
$128$	0	0
$129$	−442.554	−0.302052
$130$	0	0
$131$	692.115	0.461606	0.230803	−	0.973000i	$-0.425865\pi$
$131$	692.115	0.461606	0.230803	+	0.973000i	$0.425865\pi$
$132$	0	0
$133$	295.399	0.192589
$134$	0	0
$135$	382.193	0.243659
$136$	0	0
$137$	−98.9702	−0.0617197	−0.0308598	−	0.999524i	$-0.509825\pi$
$137$	−98.9702	−0.0617197	−0.0308598	+	0.999524i	$0.509825\pi$
$138$	0	0
$139$	−402.504	−0.245611	−0.122805	−	0.992431i	$-0.539189\pi$
$139$	−402.504	−0.245611	−0.122805	+	0.992431i	$0.539189\pi$
$140$	0	0
$141$	1852.59	1.10650
$142$	0	0
$143$	373.815	0.218602
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	−2171.76	−1.21853
$148$	0	0
$149$	−1711.74	−0.941150	−0.470575	−	0.882360i	$-0.655954\pi$
$149$	−1711.74	−0.941150	−0.470575	+	0.882360i	$0.655954\pi$
$150$	0	0
$151$	−1203.82	−0.648777	−0.324388	−	0.945924i	$-0.605158\pi$
$151$	−1203.82	−0.648777	−0.324388	+	0.945924i	$0.605158\pi$
$152$	0	0
$153$	1526.62	0.806667
$154$	0	0
$155$	1266.59	0.656356
$156$	0	0
$157$	361.941	0.183987	0.0919937	−	0.995760i	$-0.470676\pi$
$157$	361.941	0.183987	0.0919937	+	0.995760i	$0.470676\pi$
$158$	0	0
$159$	2658.63	1.32606
$160$	0	0
$161$	−303.392	−0.148513
$162$	0	0
$163$	−2651.37	−1.27406	−0.637029	−	0.770840i	$-0.719837\pi$
$163$	−2651.37	−1.27406	−0.637029	+	0.770840i	$0.719837\pi$
$164$	0	0
$165$	845.136	0.398750
$166$	0	0
$167$	−3195.76	−1.48081	−0.740405	−	0.672161i	$-0.765366\pi$
$167$	−3195.76	−1.48081	−0.740405	+	0.672161i	$0.765366\pi$
$168$	0	0
$169$	−1990.41	−0.905967
$170$	0	0
$171$	1514.21	0.677160
$172$	0	0
$173$	−2763.62	−1.21453	−0.607266	−	0.794498i	$-0.707734\pi$
$173$	−2763.62	−1.21453	−0.607266	+	0.794498i	$0.707734\pi$
$174$	0	0
$175$	74.3207	0.0321035
$176$	0	0
$177$	2573.08	1.09268
$178$	0	0
$179$	−504.337	−0.210592	−0.105296	−	0.994441i	$-0.533579\pi$
$179$	−504.337	−0.210592	−0.105296	+	0.994441i	$0.533579\pi$
$180$	0	0
$181$	4790.96	1.96746	0.983728	−	0.179664i	$-0.0575010\pi$
$181$	4790.96	1.96746	0.983728	+	0.179664i	$0.0575010\pi$
$182$	0	0
$183$	−5433.06	−2.19466
$184$	0	0
$185$	1742.00	0.692293
$186$	0	0
$187$	−2605.47	−1.01888
$188$	0	0
$189$	−227.239	−0.0874560
$190$	0	0
$191$	−3285.91	−1.24482	−0.622408	−	0.782693i	$-0.713846\pi$
$191$	−3285.91	−1.24482	−0.622408	+	0.782693i	$0.713846\pi$
$192$	0	0
$193$	3294.61	1.22876	0.614382	−	0.789008i	$-0.289405\pi$
$193$	3294.61	1.22876	0.614382	+	0.789008i	$0.289405\pi$
$194$	0	0
$195$	467.068	0.171525
$196$	0	0
$197$	−4449.47	−1.60920	−0.804598	−	0.593819i	$-0.797619\pi$
$197$	−4449.47	−1.60920	−0.804598	+	0.593819i	$0.797619\pi$
$198$	0	0
$199$	3672.29	1.30815	0.654075	−	0.756430i	$-0.273058\pi$
$199$	3672.29	1.30815	0.654075	+	0.756430i	$0.273058\pi$
$200$	0	0
$201$	−32.4947	−0.0114030
$202$	0	0
$203$	−86.2120	−0.0298074
$204$	0	0
$205$	−534.120	−0.181973
$206$	0	0
$207$	−1555.18	−0.522186
$208$	0	0
$209$	−2584.29	−0.855307
$210$	0	0
$211$	668.691	0.218173	0.109087	−	0.994032i	$-0.465207\pi$
$211$	668.691	0.218173	0.109087	+	0.994032i	$0.465207\pi$
$212$	0	0
$213$	437.520	0.140744
$214$	0	0
$215$	340.472	0.108000
$216$	0	0
$217$	−753.072	−0.235585
$218$	0	0
$219$	808.024	0.249321
$220$	0	0
$221$	−1439.93	−0.438281
$222$	0	0
$223$	5643.52	1.69470	0.847350	−	0.531035i	$-0.178197\pi$
$223$	5643.52	1.69470	0.847350	+	0.531035i	$0.178197\pi$
$224$	0	0
$225$	380.966	0.112879
$226$	0	0
$227$	−4175.69	−1.22093	−0.610463	−	0.792045i	$-0.709017\pi$
$227$	−4175.69	−1.22093	−0.610463	+	0.792045i	$0.709017\pi$
$228$	0	0
$229$	867.046	0.250201	0.125100	−	0.992144i	$-0.460075\pi$
$229$	867.046	0.250201	0.125100	+	0.992144i	$0.460075\pi$
$230$	0	0
$231$	−502.489	−0.143123
$232$	0	0
$233$	−2728.36	−0.767128	−0.383564	−	0.923514i	$-0.625303\pi$
$233$	−2728.36	−0.767128	−0.383564	+	0.923514i	$0.625303\pi$
$234$	0	0
$235$	−1425.26	−0.395633
$236$	0	0
$237$	5998.90	1.64418
$238$	0	0
$239$	−3895.60	−1.05433	−0.527166	−	0.849762i	$-0.676746\pi$
$239$	−3895.60	−1.05433	−0.527166	+	0.849762i	$0.676746\pi$
$240$	0	0
$241$	−5911.50	−1.58005	−0.790027	−	0.613072i	$-0.789934\pi$
$241$	−5911.50	−1.58005	−0.790027	+	0.613072i	$0.789934\pi$
$242$	0	0
$243$	−3838.84	−1.01342
$244$	0	0
$245$	1670.81	0.435691
$246$	0	0
$247$	−1428.22	−0.367917
$248$	0	0
$249$	−2961.78	−0.753795
$250$	0	0
$251$	2067.87	0.520012	0.260006	−	0.965607i	$-0.416275\pi$
$251$	2067.87	0.520012	0.260006	+	0.965607i	$0.416275\pi$
$252$	0	0
$253$	2654.21	0.659561
$254$	0	0
$255$	−3255.44	−0.799466
$256$	0	0
$257$	1838.83	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$257$	1838.83	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$258$	0	0
$259$	−1035.73	−0.248483
$260$	0	0
$261$	−441.920	−0.104805
$262$	0	0
$263$	−3993.15	−0.936230	−0.468115	−	0.883668i	$-0.655067\pi$
$263$	−3993.15	−0.936230	−0.468115	+	0.883668i	$0.655067\pi$
$264$	0	0
$265$	−2045.38	−0.474138
$266$	0	0
$267$	−6616.01	−1.51645
$268$	0	0
$269$	−3614.51	−0.819259	−0.409629	−	0.912252i	$-0.634342\pi$
$269$	−3614.51	−0.819259	−0.409629	+	0.912252i	$0.634342\pi$
$270$	0	0
$271$	−3085.36	−0.691596	−0.345798	−	0.938309i	$-0.612392\pi$
$271$	−3085.36	−0.691596	−0.345798	+	0.938309i	$0.612392\pi$
$272$	0	0
$273$	−277.703	−0.0615653
$274$	0	0
$275$	−650.192	−0.142575
$276$	0	0
$277$	−4446.23	−0.964434	−0.482217	−	0.876052i	$-0.660168\pi$
$277$	−4446.23	−0.964434	−0.482217	+	0.876052i	$0.660168\pi$
$278$	0	0
$279$	−3860.23	−0.828337
$280$	0	0
$281$	−1257.07	−0.266870	−0.133435	−	0.991058i	$-0.542601\pi$
$281$	−1257.07	−0.266870	−0.133435	+	0.991058i	$0.542601\pi$
$282$	0	0
$283$	7200.93	1.51255	0.756274	−	0.654255i	$-0.227018\pi$
$283$	7200.93	1.51255	0.756274	+	0.654255i	$0.227018\pi$
$284$	0	0
$285$	−3228.97	−0.671115
$286$	0	0
$287$	317.569	0.0653154
$288$	0	0
$289$	5123.22	1.04279
$290$	0	0
$291$	−35.0581	−0.00706234
$292$	0	0
$293$	198.916	0.0396615	0.0198307	−	0.999803i	$-0.493687\pi$
$293$	198.916	0.0396615	0.0198307	+	0.999803i	$0.493687\pi$
$294$	0	0
$295$	−1979.56	−0.390693
$296$	0	0
$297$	1987.99	0.388400
$298$	0	0
$299$	1466.86	0.283716
$300$	0	0
$301$	−202.433	−0.0387642
$302$	0	0
$303$	859.870	0.163030
$304$	0	0
$305$	4179.84	0.784711
$306$	0	0
$307$	−3888.12	−0.722824	−0.361412	−	0.932406i	$-0.617705\pi$
$307$	−3888.12	−0.722824	−0.361412	+	0.932406i	$0.617705\pi$
$308$	0	0
$309$	−12782.6	−2.35333
$310$	0	0
$311$	10812.0	1.97135	0.985675	−	0.168654i	$-0.0539419\pi$
$311$	10812.0	1.97135	0.985675	+	0.168654i	$0.0539419\pi$
$312$	0	0
$313$	1572.54	0.283979	0.141989	−	0.989868i	$-0.454650\pi$
$313$	1572.54	0.283979	0.141989	+	0.989868i	$0.454650\pi$
$314$	0	0
$315$	−226.509	−0.0405154
$316$	0	0
$317$	−2195.35	−0.388969	−0.194485	−	0.980906i	$-0.562303\pi$
$317$	−2195.35	−0.388969	−0.194485	+	0.980906i	$0.562303\pi$
$318$	0	0
$319$	754.223	0.132377
$320$	0	0
$321$	11272.0	1.95994
$322$	0	0
$323$	9954.62	1.71483
$324$	0	0
$325$	−359.332	−0.0613297
$326$	0	0
$327$	−1968.64	−0.332924
$328$	0	0
$329$	847.410	0.142004
$330$	0	0
$331$	5686.75	0.944326	0.472163	−	0.881511i	$-0.343473\pi$
$331$	5686.75	0.944326	0.472163	+	0.881511i	$0.343473\pi$
$332$	0	0
$333$	−5309.13	−0.873690
$334$	0	0
$335$	24.9993	0.00407718
$336$	0	0
$337$	3628.49	0.586517	0.293259	−	0.956033i	$-0.405260\pi$
$337$	3628.49	0.586517	0.293259	+	0.956033i	$0.405260\pi$
$338$	0	0
$339$	−14561.1	−2.33289
$340$	0	0
$341$	6588.23	1.04625
$342$	0	0
$343$	−2013.09	−0.316899
$344$	0	0
$345$	3316.34	0.517524
$346$	0	0
$347$	−191.273	−0.0295910	−0.0147955	−	0.999891i	$-0.504710\pi$
$347$	−191.273	−0.0295910	−0.0147955	+	0.999891i	$0.504710\pi$
$348$	0	0
$349$	−2249.21	−0.344979	−0.172489	−	0.985011i	$-0.555181\pi$
$349$	−2249.21	−0.344979	−0.172489	+	0.985011i	$0.555181\pi$
$350$	0	0
$351$	1098.67	0.167073
$352$	0	0
$353$	869.155	0.131049	0.0655247	−	0.997851i	$-0.479128\pi$
$353$	869.155	0.131049	0.0655247	+	0.997851i	$0.479128\pi$
$354$	0	0
$355$	−336.599	−0.0503235
$356$	0	0
$357$	1935.57	0.286951
$358$	0	0
$359$	435.740	0.0640598	0.0320299	−	0.999487i	$-0.489803\pi$
$359$	435.740	0.0640598	0.0320299	+	0.999487i	$0.489803\pi$
$360$	0	0
$361$	3014.68	0.439522
$362$	0	0
$363$	−4254.33	−0.615136
$364$	0	0
$365$	−621.641	−0.0891457
$366$	0	0
$367$	−5161.88	−0.734190	−0.367095	−	0.930183i	$-0.619648\pi$
$367$	−5161.88	−0.734190	−0.367095	+	0.930183i	$0.619648\pi$
$368$	0	0
$369$	1627.85	0.229655
$370$	0	0
$371$	1216.11	0.170181
$372$	0	0
$373$	−8548.53	−1.18667	−0.593333	−	0.804957i	$-0.702188\pi$
$373$	−8548.53	−1.18667	−0.593333	+	0.804957i	$0.702188\pi$
$374$	0	0
$375$	−812.391	−0.111871
$376$	0	0
$377$	416.825	0.0569432
$378$	0	0
$379$	12546.5	1.70045	0.850227	−	0.526417i	$-0.176465\pi$
$379$	12546.5	1.70045	0.850227	+	0.526417i	$0.176465\pi$
$380$	0	0
$381$	4323.51	0.581365
$382$	0	0
$383$	3567.15	0.475909	0.237954	−	0.971276i	$-0.423523\pi$
$383$	3567.15	0.475909	0.237954	+	0.971276i	$0.423523\pi$
$384$	0	0
$385$	386.582	0.0511741
$386$	0	0
$387$	−1037.67	−0.136299
$388$	0	0
$389$	−6004.87	−0.782671	−0.391335	−	0.920248i	$-0.627987\pi$
$389$	−6004.87	−0.782671	−0.391335	+	0.920248i	$0.627987\pi$
$390$	0	0
$391$	−10224.0	−1.32237
$392$	0	0
$393$	4498.15	0.577357
$394$	0	0
$395$	−4615.16	−0.587883
$396$	0	0
$397$	13820.9	1.74723	0.873615	−	0.486617i	$-0.161769\pi$
$397$	13820.9	1.74723	0.873615	+	0.486617i	$0.161769\pi$
$398$	0	0
$399$	1919.84	0.240882
$400$	0	0
$401$	−2014.54	−0.250876	−0.125438	−	0.992101i	$-0.540034\pi$
$401$	−2014.54	−0.250876	−0.125438	+	0.992101i	$0.540034\pi$
$402$	0	0
$403$	3641.02	0.450055
$404$	0	0
$405$	4541.14	0.557163
$406$	0	0
$407$	9061.06	1.10354
$408$	0	0
$409$	14205.1	1.71735	0.858673	−	0.512523i	$-0.171289\pi$
$409$	14205.1	1.71735	0.858673	+	0.512523i	$0.171289\pi$
$410$	0	0
$411$	−643.220	−0.0771963
$412$	0	0
$413$	1176.98	0.140231
$414$	0	0
$415$	2278.60	0.269523
$416$	0	0
$417$	−2615.92	−0.307200
$418$	0	0
$419$	10085.5	1.17592	0.587959	−	0.808891i	$-0.299932\pi$
$419$	10085.5	1.17592	0.587959	+	0.808891i	$0.299932\pi$
$420$	0	0
$421$	−6518.42	−0.754604	−0.377302	−	0.926090i	$-0.623148\pi$
$421$	−6518.42	−0.754604	−0.377302	+	0.926090i	$0.623148\pi$
$422$	0	0
$423$	4343.80	0.499298
$424$	0	0
$425$	2504.52	0.285852
$426$	0	0
$427$	−2485.19	−0.281655
$428$	0	0
$429$	2429.47	0.273417
$430$	0	0
$431$	13862.6	1.54928	0.774640	−	0.632402i	$-0.217931\pi$
$431$	13862.6	1.54928	0.774640	+	0.632402i	$0.217931\pi$
$432$	0	0
$433$	−6101.22	−0.677150	−0.338575	−	0.940939i	$-0.609945\pi$
$433$	−6101.22	−0.677150	−0.338575	+	0.940939i	$0.609945\pi$
$434$	0	0
$435$	942.373	0.103870
$436$	0	0
$437$	−10140.8	−1.11007
$438$	0	0
$439$	−10724.4	−1.16594	−0.582969	−	0.812494i	$-0.698109\pi$
$439$	−10724.4	−1.16594	−0.582969	+	0.812494i	$0.698109\pi$
$440$	0	0
$441$	−5092.18	−0.549852
$442$	0	0
$443$	4143.85	0.444425	0.222212	−	0.974998i	$-0.428672\pi$
$443$	4143.85	0.444425	0.222212	+	0.974998i	$0.428672\pi$
$444$	0	0
$445$	5089.92	0.542215
$446$	0	0
$447$	−11124.8	−1.17715
$448$	0	0
$449$	9826.95	1.03288	0.516439	−	0.856324i	$-0.327257\pi$
$449$	9826.95	1.03288	0.516439	+	0.856324i	$0.327257\pi$
$450$	0	0
$451$	−2778.25	−0.290072
$452$	0	0
$453$	−7823.76	−0.811462
$454$	0	0
$455$	213.646	0.0220129
$456$	0	0
$457$	−10652.1	−1.09034	−0.545171	−	0.838325i	$-0.683535\pi$
$457$	−10652.1	−1.09034	−0.545171	+	0.838325i	$0.683535\pi$
$458$	0	0
$459$	−7657.69	−0.778715
$460$	0	0
$461$	−12007.3	−1.21310	−0.606548	−	0.795047i	$-0.707446\pi$
$461$	−12007.3	−1.21310	−0.606548	+	0.795047i	$0.707446\pi$
$462$	0	0
$463$	5616.06	0.563715	0.281858	−	0.959456i	$-0.409049\pi$
$463$	5616.06	0.563715	0.281858	+	0.959456i	$0.409049\pi$
$464$	0	0
$465$	8231.75	0.820943
$466$	0	0
$467$	−12086.0	−1.19759	−0.598796	−	0.800902i	$-0.704354\pi$
$467$	−12086.0	−1.19759	−0.598796	+	0.800902i	$0.704354\pi$
$468$	0	0
$469$	−14.8637	−0.00146342
$470$	0	0
$471$	2352.30	0.230123
$472$	0	0
$473$	1770.98	0.172156
$474$	0	0
$475$	2484.16	0.239960
$476$	0	0
$477$	6233.75	0.598373
$478$	0	0
$479$	−4775.46	−0.455525	−0.227762	−	0.973717i	$-0.573141\pi$
$479$	−4775.46	−0.455525	−0.227762	+	0.973717i	$0.573141\pi$
$480$	0	0
$481$	5007.64	0.474696
$482$	0	0
$483$	−1971.78	−0.185754
$484$	0	0
$485$	26.9714	0.00252517
$486$	0	0
$487$	−3447.42	−0.320776	−0.160388	−	0.987054i	$-0.551274\pi$
$487$	−3447.42	−0.320776	−0.160388	+	0.987054i	$0.551274\pi$
$488$	0	0
$489$	−17231.6	−1.59354
$490$	0	0
$491$	2066.67	0.189954	0.0949770	−	0.995479i	$-0.469722\pi$
$491$	2066.67	0.189954	0.0949770	+	0.995479i	$0.469722\pi$
$492$	0	0
$493$	−2905.25	−0.265407
$494$	0	0
$495$	1981.61	0.179933
$496$	0	0
$497$	200.130	0.0180625
$498$	0	0
$499$	14328.9	1.28547	0.642735	−	0.766089i	$-0.277800\pi$
$499$	14328.9	1.28547	0.642735	+	0.766089i	$0.277800\pi$
$500$	0	0
$501$	−20769.6	−1.85213
$502$	0	0
$503$	−3941.99	−0.349433	−0.174717	−	0.984619i	$-0.555901\pi$
$503$	−3941.99	−0.349433	−0.174717	+	0.984619i	$0.555901\pi$
$504$	0	0
$505$	−661.527	−0.0582922
$506$	0	0
$507$	−12935.9	−1.13314
$508$	0	0
$509$	−20252.4	−1.76360	−0.881798	−	0.471627i	$-0.843667\pi$
$509$	−20252.4	−1.76360	−0.881798	+	0.471627i	$0.843667\pi$
$510$	0	0
$511$	369.606	0.0319969
$512$	0	0
$513$	−7595.43	−0.653696
$514$	0	0
$515$	9834.12	0.841443
$516$	0	0
$517$	−7413.54	−0.630652
$518$	0	0
$519$	−17961.1	−1.51909
$520$	0	0
$521$	1176.42	0.0989252	0.0494626	−	0.998776i	$-0.484249\pi$
$521$	1176.42	0.0989252	0.0494626	+	0.998776i	$0.484249\pi$
$522$	0	0
$523$	21238.2	1.77568	0.887840	−	0.460152i	$-0.152205\pi$
$523$	21238.2	1.77568	0.887840	+	0.460152i	$0.152205\pi$
$524$	0	0
$525$	483.019	0.0401537
$526$	0	0
$527$	−25377.7	−2.09767
$528$	0	0
$529$	−1751.77	−0.143977
$530$	0	0
$531$	6033.15	0.493063
$532$	0	0
$533$	−1535.41	−0.124777
$534$	0	0
$535$	−8671.94	−0.700786
$536$	0	0
$537$	−3277.75	−0.263399
$538$	0	0
$539$	8690.79	0.694506
$540$	0	0
$541$	−12027.2	−0.955806	−0.477903	−	0.878413i	$-0.658603\pi$
$541$	−12027.2	−0.955806	−0.477903	+	0.878413i	$0.658603\pi$
$542$	0	0
$543$	31137.1	2.46081
$544$	0	0
$545$	1514.54	0.119038
$546$	0	0
$547$	−7446.70	−0.582080	−0.291040	−	0.956711i	$-0.594001\pi$
$547$	−7446.70	−0.582080	−0.291040	+	0.956711i	$0.594001\pi$
$548$	0	0
$549$	−12739.0	−0.990323
$550$	0	0
$551$	−2881.63	−0.222797
$552$	0	0
$553$	2744.01	0.211008
$554$	0	0
$555$	11321.5	0.865890
$556$	0	0
$557$	−12073.1	−0.918411	−0.459205	−	0.888330i	$-0.651866\pi$
$557$	−12073.1	−0.918411	−0.459205	+	0.888330i	$0.651866\pi$
$558$	0	0
$559$	978.740	0.0740542
$560$	0	0
$561$	−16933.3	−1.27438
$562$	0	0
$563$	1468.95	0.109962	0.0549811	−	0.998487i	$-0.482490\pi$
$563$	1468.95	0.109962	0.0549811	+	0.998487i	$0.482490\pi$
$564$	0	0
$565$	11202.3	0.834135
$566$	0	0
$567$	−2700.00	−0.199981
$568$	0	0
$569$	18719.0	1.37916	0.689579	−	0.724211i	$-0.257796\pi$
$569$	18719.0	1.37916	0.689579	+	0.724211i	$0.257796\pi$
$570$	0	0
$571$	5871.02	0.430288	0.215144	−	0.976582i	$-0.430978\pi$
$571$	5871.02	0.430288	0.215144	+	0.976582i	$0.430978\pi$
$572$	0	0
$573$	−21355.5	−1.55696
$574$	0	0
$575$	−2551.38	−0.185043
$576$	0	0
$577$	−9824.70	−0.708852	−0.354426	−	0.935084i	$-0.615324\pi$
$577$	−9824.70	−0.708852	−0.354426	+	0.935084i	$0.615324\pi$
$578$	0	0
$579$	21412.1	1.53689
$580$	0	0
$581$	−1354.77	−0.0967392
$582$	0	0
$583$	−10639.1	−0.755792
$584$	0	0
$585$	1095.15	0.0773995
$586$	0	0
$587$	11739.1	0.825428	0.412714	−	0.910861i	$-0.364581\pi$
$587$	11739.1	0.825428	0.412714	+	0.910861i	$0.364581\pi$
$588$	0	0
$589$	−25171.4	−1.76090
$590$	0	0
$591$	−28917.7	−2.01271
$592$	0	0
$593$	−12880.0	−0.891938	−0.445969	−	0.895048i	$-0.647141\pi$
$593$	−12880.0	−0.891938	−0.445969	+	0.895048i	$0.647141\pi$
$594$	0	0
$595$	−1489.10	−0.102600
$596$	0	0
$597$	23866.7	1.63618
$598$	0	0
$599$	15371.2	1.04850	0.524249	−	0.851565i	$-0.324346\pi$
$599$	15371.2	1.04850	0.524249	+	0.851565i	$0.324346\pi$
$600$	0	0
$601$	23982.6	1.62774	0.813869	−	0.581049i	$-0.197357\pi$
$601$	23982.6	1.62774	0.813869	+	0.581049i	$0.197357\pi$
$602$	0	0
$603$	−76.1910	−0.00514550
$604$	0	0
$605$	3273.00	0.219945
$606$	0	0
$607$	13988.2	0.935363	0.467682	−	0.883897i	$-0.345089\pi$
$607$	13988.2	0.935363	0.467682	+	0.883897i	$0.345089\pi$
$608$	0	0
$609$	−560.302	−0.0372818
$610$	0	0
$611$	−4097.13	−0.271280
$612$	0	0
$613$	16612.1	1.09454	0.547271	−	0.836955i	$-0.315667\pi$
$613$	16612.1	1.09454	0.547271	+	0.836955i	$0.315667\pi$
$614$	0	0
$615$	−3471.31	−0.227605
$616$	0	0
$617$	−17478.6	−1.14046	−0.570230	−	0.821485i	$-0.693146\pi$
$617$	−17478.6	−1.14046	−0.570230	+	0.821485i	$0.693146\pi$
$618$	0	0
$619$	4812.20	0.312470	0.156235	−	0.987720i	$-0.450064\pi$
$619$	4812.20	0.312470	0.156235	+	0.987720i	$0.450064\pi$
$620$	0	0
$621$	7800.94	0.504092
$622$	0	0
$623$	−3026.29	−0.194616
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−16795.6	−1.06978
$628$	0	0
$629$	−34903.0	−2.21252
$630$	0	0
$631$	22189.4	1.39992	0.699958	−	0.714184i	$-0.253202\pi$
$631$	22189.4	1.39992	0.699958	+	0.714184i	$0.253202\pi$
$632$	0	0
$633$	4345.91	0.272882
$634$	0	0
$635$	−3326.23	−0.207870
$636$	0	0
$637$	4803.00	0.298747
$638$	0	0
$639$	1025.86	0.0635094
$640$	0	0
$641$	3065.54	0.188895	0.0944475	−	0.995530i	$-0.469892\pi$
$641$	3065.54	0.188895	0.0944475	+	0.995530i	$0.469892\pi$
$642$	0	0
$643$	885.503	0.0543092	0.0271546	−	0.999631i	$-0.491355\pi$
$643$	885.503	0.0543092	0.0271546	+	0.999631i	$0.491355\pi$
$644$	0	0
$645$	2212.77	0.135082
$646$	0	0
$647$	−8412.98	−0.511203	−0.255602	−	0.966782i	$-0.582274\pi$
$647$	−8412.98	−0.511203	−0.255602	+	0.966782i	$0.582274\pi$
$648$	0	0
$649$	−10296.7	−0.622778
$650$	0	0
$651$	−4894.31	−0.294659
$652$	0	0
$653$	−23947.0	−1.43510	−0.717549	−	0.696509i	$-0.754736\pi$
$653$	−23947.0	−1.43510	−0.717549	+	0.696509i	$0.754736\pi$
$654$	0	0
$655$	−3460.58	−0.206437
$656$	0	0
$657$	1894.59	0.112504
$658$	0	0
$659$	−28707.2	−1.69692	−0.848462	−	0.529256i	$-0.822471\pi$
$659$	−28707.2	−1.69692	−0.848462	+	0.529256i	$0.822471\pi$
$660$	0	0
$661$	17136.5	1.00837	0.504186	−	0.863595i	$-0.331792\pi$
$661$	17136.5	1.00837	0.504186	+	0.863595i	$0.331792\pi$
$662$	0	0
$663$	−9358.27	−0.548183
$664$	0	0
$665$	−1477.00	−0.0861284
$666$	0	0
$667$	2959.60	0.171808
$668$	0	0
$669$	36677.9	2.11966
$670$	0	0
$671$	21741.6	1.25086
$672$	0	0
$673$	19734.2	1.13031	0.565153	−	0.824986i	$-0.308817\pi$
$673$	19734.2	1.13031	0.565153	+	0.824986i	$0.308817\pi$
$674$	0	0
$675$	−1910.96	−0.108967
$676$	0	0
$677$	−19825.1	−1.12547	−0.562734	−	0.826638i	$-0.690250\pi$
$677$	−19825.1	−1.12547	−0.562734	+	0.826638i	$0.690250\pi$
$678$	0	0
$679$	−16.0362	−0.000906354 0
$680$	0	0
$681$	−27138.3	−1.52708
$682$	0	0
$683$	15653.0	0.876932	0.438466	−	0.898748i	$-0.355522\pi$
$683$	15653.0	0.876932	0.438466	+	0.898748i	$0.355522\pi$
$684$	0	0
$685$	494.851	0.0276019
$686$	0	0
$687$	5635.04	0.312941
$688$	0	0
$689$	−5879.75	−0.325110
$690$	0	0
$691$	−23665.2	−1.30285	−0.651424	−	0.758714i	$-0.725828\pi$
$691$	−23665.2	−1.30285	−0.651424	+	0.758714i	$0.725828\pi$
$692$	0	0
$693$	−1178.20	−0.0645829
$694$	0	0
$695$	2012.52	0.109840
$696$	0	0
$697$	10701.7	0.581574
$698$	0	0
$699$	−17732.0	−0.959491
$700$	0	0
$701$	30202.1	1.62727	0.813635	−	0.581376i	$-0.197485\pi$
$701$	30202.1	1.62727	0.813635	+	0.581376i	$0.197485\pi$
$702$	0	0
$703$	−34619.2	−1.85731
$704$	0	0
$705$	−9262.94	−0.494841
$706$	0	0
$707$	393.321	0.0209227
$708$	0	0
$709$	23356.3	1.23719	0.618594	−	0.785711i	$-0.287703\pi$
$709$	23356.3	1.23719	0.618594	+	0.785711i	$0.287703\pi$
$710$	0	0
$711$	14065.7	0.741922
$712$	0	0
$713$	25852.4	1.35790
$714$	0	0
$715$	−1869.08	−0.0977616
$716$	0	0
$717$	−25318.0	−1.31871
$718$	0	0
$719$	−558.362	−0.0289616	−0.0144808	−	0.999895i	$-0.504610\pi$
$719$	−558.362	−0.0289616	−0.0144808	+	0.999895i	$0.504610\pi$
$720$	0	0
$721$	−5847.03	−0.302018
$722$	0	0
$723$	−38419.6	−1.97626
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−32033.8	−1.63421	−0.817104	−	0.576490i	$-0.804422\pi$
$727$	−32033.8	−1.63421	−0.817104	+	0.576490i	$0.804422\pi$
$728$	0	0
$729$	−426.979	−0.0216928
$730$	0	0
$731$	−6821.76	−0.345160
$732$	0	0
$733$	16773.2	0.845203	0.422602	−	0.906316i	$-0.361117\pi$
$733$	16773.2	0.845203	0.422602	+	0.906316i	$0.361117\pi$
$734$	0	0
$735$	10858.8	0.544943
$736$	0	0
$737$	130.035	0.00649917
$738$	0	0
$739$	−15616.0	−0.777324	−0.388662	−	0.921380i	$-0.627063\pi$
$739$	−15616.0	−0.777324	−0.388662	+	0.921380i	$0.627063\pi$
$740$	0	0
$741$	−9282.18	−0.460175
$742$	0	0
$743$	−3856.76	−0.190432	−0.0952159	−	0.995457i	$-0.530354\pi$
$743$	−3856.76	−0.190432	−0.0952159	+	0.995457i	$0.530354\pi$
$744$	0	0
$745$	8558.71	0.420895
$746$	0	0
$747$	−6944.54	−0.340144
$748$	0	0
$749$	5156.03	0.251532
$750$	0	0
$751$	−21787.6	−1.05864	−0.529322	−	0.848421i	$-0.677554\pi$
$751$	−21787.6	−1.05864	−0.529322	+	0.848421i	$0.677554\pi$
$752$	0	0
$753$	13439.4	0.650409
$754$	0	0
$755$	6019.09	0.290142
$756$	0	0
$757$	29481.7	1.41550	0.707749	−	0.706464i	$-0.249711\pi$
$757$	29481.7	1.41550	0.707749	+	0.706464i	$0.249711\pi$
$758$	0	0
$759$	17250.1	0.824951
$760$	0	0
$761$	32922.7	1.56826	0.784130	−	0.620596i	$-0.213109\pi$
$761$	32922.7	1.56826	0.784130	+	0.620596i	$0.213109\pi$
$762$	0	0
$763$	−900.495	−0.0427262
$764$	0	0
$765$	−7633.10	−0.360752
$766$	0	0
$767$	−5690.54	−0.267893
$768$	0	0
$769$	−6327.30	−0.296708	−0.148354	−	0.988934i	$-0.547397\pi$
$769$	−6327.30	−0.296708	−0.148354	+	0.988934i	$0.547397\pi$
$770$	0	0
$771$	11950.8	0.558232
$772$	0	0
$773$	23743.5	1.10478	0.552390	−	0.833585i	$-0.313716\pi$
$773$	23743.5	1.10478	0.552390	+	0.833585i	$0.313716\pi$
$774$	0	0
$775$	−6332.97	−0.293532
$776$	0	0
$777$	−6731.34	−0.310792
$778$	0	0
$779$	10614.7	0.488205
$780$	0	0
$781$	−1750.83	−0.0802173
$782$	0	0
$783$	2216.72	0.101174
$784$	0	0
$785$	−1809.70	−0.0822816
$786$	0	0
$787$	−7159.21	−0.324267	−0.162134	−	0.986769i	$-0.551838\pi$
$787$	−7159.21	−0.324267	−0.162134	+	0.986769i	$0.551838\pi$
$788$	0	0
$789$	−25952.0	−1.17100
$790$	0	0
$791$	−6660.52	−0.299394
$792$	0	0
$793$	12015.6	0.538065
$794$	0	0
$795$	−13293.2	−0.593031
$796$	0	0
$797$	4921.88	0.218748	0.109374	−	0.994001i	$-0.465115\pi$
$797$	4921.88	0.218748	0.109374	+	0.994001i	$0.465115\pi$
$798$	0	0
$799$	28556.8	1.26441
$800$	0	0
$801$	−15512.7	−0.684287
$802$	0	0
$803$	−3233.49	−0.142101
$804$	0	0
$805$	1516.96	0.0664171
$806$	0	0
$807$	−23491.2	−1.02469
$808$	0	0
$809$	−30499.1	−1.32545	−0.662727	−	0.748861i	$-0.730601\pi$
$809$	−30499.1	−1.32545	−0.662727	+	0.748861i	$0.730601\pi$
$810$	0	0
$811$	16300.7	0.705791	0.352895	−	0.935663i	$-0.385197\pi$
$811$	16300.7	0.705791	0.352895	+	0.935663i	$0.385197\pi$
$812$	0	0
$813$	−20052.2	−0.865018
$814$	0	0
$815$	13256.9	0.569776
$816$	0	0
$817$	−6766.30	−0.289746
$818$	0	0
$819$	−651.135	−0.0277808
$820$	0	0
$821$	16211.0	0.689121	0.344560	−	0.938764i	$-0.388028\pi$
$821$	16211.0	0.689121	0.344560	+	0.938764i	$0.388028\pi$
$822$	0	0
$823$	294.632	0.0124790	0.00623950	−	0.999981i	$-0.498014\pi$
$823$	294.632	0.0124790	0.00623950	+	0.999981i	$0.498014\pi$
$824$	0	0
$825$	−4225.68	−0.178326
$826$	0	0
$827$	2458.78	0.103386	0.0516929	−	0.998663i	$-0.483538\pi$
$827$	2458.78	0.103386	0.0516929	+	0.998663i	$0.483538\pi$
$828$	0	0
$829$	13131.0	0.550132	0.275066	−	0.961425i	$-0.411300\pi$
$829$	13131.0	0.550132	0.275066	+	0.961425i	$0.411300\pi$
$830$	0	0
$831$	−28896.6	−1.20627
$832$	0	0
$833$	−33476.7	−1.39244
$834$	0	0
$835$	15978.8	0.662238
$836$	0	0
$837$	19363.3	0.799635
$838$	0	0
$839$	−15681.7	−0.645281	−0.322641	−	0.946522i	$-0.604571\pi$
$839$	−15681.7	−0.645281	−0.322641	+	0.946522i	$0.604571\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−8169.86	−0.333790
$844$	0	0
$845$	9952.05	0.405161
$846$	0	0
$847$	−1946.01	−0.0789443
$848$	0	0
$849$	46799.8	1.89183
$850$	0	0
$851$	35555.9	1.43225
$852$	0	0
$853$	35024.3	1.40587	0.702935	−	0.711254i	$-0.251872\pi$
$853$	35024.3	1.40587	0.702935	+	0.711254i	$0.251872\pi$
$854$	0	0
$855$	−7571.04	−0.302835
$856$	0	0
$857$	3236.70	0.129012	0.0645062	−	0.997917i	$-0.479453\pi$
$857$	3236.70	0.129012	0.0645062	+	0.997917i	$0.479453\pi$
$858$	0	0
$859$	−7616.20	−0.302516	−0.151258	−	0.988494i	$-0.548332\pi$
$859$	−7616.20	−0.302516	−0.151258	+	0.988494i	$0.548332\pi$
$860$	0	0
$861$	2063.92	0.0816937
$862$	0	0
$863$	−6154.46	−0.242758	−0.121379	−	0.992606i	$-0.538732\pi$
$863$	−6154.46	−0.242758	−0.121379	+	0.992606i	$0.538732\pi$
$864$	0	0
$865$	13818.1	0.543156
$866$	0	0
$867$	33296.4	1.30427
$868$	0	0
$869$	−24005.9	−0.937106
$870$	0	0
$871$	71.8643	0.00279567
$872$	0	0
$873$	−82.2014	−0.00318682
$874$	0	0
$875$	−371.603	−0.0143571
$876$	0	0
$877$	−19519.1	−0.751555	−0.375777	−	0.926710i	$-0.622624\pi$
$877$	−19519.1	−0.751555	−0.375777	+	0.926710i	$0.622624\pi$
$878$	0	0
$879$	1292.78	0.0496069
$880$	0	0
$881$	35413.4	1.35427	0.677133	−	0.735860i	$-0.263222\pi$
$881$	35413.4	1.35427	0.677133	+	0.735860i	$0.263222\pi$
$882$	0	0
$883$	38576.4	1.47021	0.735107	−	0.677951i	$-0.237132\pi$
$883$	38576.4	1.47021	0.735107	+	0.677951i	$0.237132\pi$
$884$	0	0
$885$	−12865.4	−0.488662
$886$	0	0
$887$	−23091.9	−0.874127	−0.437063	−	0.899431i	$-0.643981\pi$
$887$	−23091.9	−0.874127	−0.437063	+	0.899431i	$0.643981\pi$
$888$	0	0
$889$	1977.66	0.0746103
$890$	0	0
$891$	23620.9	0.888136
$892$	0	0
$893$	28324.6	1.06142
$894$	0	0
$895$	2521.69	0.0941796
$896$	0	0
$897$	9533.33	0.354859
$898$	0	0
$899$	7346.24	0.272537
$900$	0	0
$901$	40981.5	1.51531
$902$	0	0
$903$	−1315.64	−0.0484847
$904$	0	0
$905$	−23954.8	−0.879873
$906$	0	0
$907$	8145.16	0.298187	0.149094	−	0.988823i	$-0.452364\pi$
$907$	8145.16	0.298187	0.149094	+	0.988823i	$0.452364\pi$
$908$	0	0
$909$	2016.15	0.0735662
$910$	0	0
$911$	15778.0	0.573819	0.286909	−	0.957958i	$-0.407372\pi$
$911$	15778.0	0.573819	0.286909	+	0.957958i	$0.407372\pi$
$912$	0	0
$913$	11852.2	0.429628
$914$	0	0
$915$	27165.3	0.981483
$916$	0	0
$917$	2057.54	0.0740959
$918$	0	0
$919$	345.621	0.0124058	0.00620292	−	0.999981i	$-0.498026\pi$
$919$	345.621	0.0124058	0.00620292	+	0.999981i	$0.498026\pi$
$920$	0	0
$921$	−25269.4	−0.904077
$922$	0	0
$923$	−967.606	−0.0345061
$924$	0	0
$925$	−8709.98	−0.309603
$926$	0	0
$927$	−29971.7	−1.06192
$928$	0	0
$929$	−28488.0	−1.00609	−0.503047	−	0.864259i	$-0.667788\pi$
$929$	−28488.0	−1.00609	−0.503047	+	0.864259i	$0.667788\pi$
$930$	0	0
$931$	−33204.5	−1.16889
$932$	0	0
$933$	70268.3	2.46568
$934$	0	0
$935$	13027.4	0.455658
$936$	0	0
$937$	4470.09	0.155850	0.0779250	−	0.996959i	$-0.475171\pi$
$937$	4470.09	0.155850	0.0779250	+	0.996959i	$0.475171\pi$
$938$	0	0
$939$	10220.1	0.355188
$940$	0	0
$941$	713.170	0.0247063	0.0123532	−	0.999924i	$-0.496068\pi$
$941$	713.170	0.0247063	0.0123532	+	0.999924i	$0.496068\pi$
$942$	0	0
$943$	−10901.9	−0.376475
$944$	0	0
$945$	1136.19	0.0391115
$946$	0	0
$947$	16449.1	0.564440	0.282220	−	0.959350i	$-0.408929\pi$
$947$	16449.1	0.564440	0.282220	+	0.959350i	$0.408929\pi$
$948$	0	0
$949$	−1787.00	−0.0611260
$950$	0	0
$951$	−14267.9	−0.486506
$952$	0	0
$953$	−29150.9	−0.990860	−0.495430	−	0.868648i	$-0.664990\pi$
$953$	−29150.9	−0.990860	−0.495430	+	0.868648i	$0.664990\pi$
$954$	0	0
$955$	16429.5	0.556699
$956$	0	0
$957$	4901.79	0.165572
$958$	0	0
$959$	−294.221	−0.00990709
$960$	0	0
$961$	34379.4	1.15402
$962$	0	0
$963$	26429.7	0.884409
$964$	0	0
$965$	−16473.1	−0.549520
$966$	0	0
$967$	42765.7	1.42219	0.711093	−	0.703098i	$-0.248201\pi$
$967$	42765.7	1.42219	0.711093	+	0.703098i	$0.248201\pi$
$968$	0	0
$969$	64696.3	2.14484
$970$	0	0
$971$	−8642.06	−0.285620	−0.142810	−	0.989750i	$-0.545614\pi$
$971$	−8642.06	−0.285620	−0.142810	+	0.989750i	$0.545614\pi$
$972$	0	0
$973$	−1196.57	−0.0394248
$974$	0	0
$975$	−2335.34	−0.0767085
$976$	0	0
$977$	1244.09	0.0407391	0.0203695	−	0.999793i	$-0.493516\pi$
$977$	1244.09	0.0407391	0.0203695	+	0.999793i	$0.493516\pi$
$978$	0	0
$979$	26475.4	0.864309
$980$	0	0
$981$	−4615.92	−0.150229
$982$	0	0
$983$	30642.6	0.994248	0.497124	−	0.867679i	$-0.334389\pi$
$983$	30642.6	0.994248	0.497124	+	0.867679i	$0.334389\pi$
$984$	0	0
$985$	22247.4	0.719655
$986$	0	0
$987$	5507.42	0.177612
$988$	0	0
$989$	6949.38	0.223435
$990$	0	0
$991$	−15647.6	−0.501577	−0.250788	−	0.968042i	$-0.580690\pi$
$991$	−15647.6	−0.501577	−0.250788	+	0.968042i	$0.580690\pi$
$992$	0	0
$993$	36958.9	1.18112
$994$	0	0
$995$	−18361.5	−0.585022
$996$	0	0
$997$	20097.5	0.638411	0.319205	−	0.947686i	$-0.396584\pi$
$997$	20097.5	0.638411	0.319205	+	0.947686i	$0.396584\pi$
$998$	0	0
$999$	26631.1	0.843416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.4.a.o.1.5		6
4.3	odd	2		580.4.a.a.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.4.a.a.1.2	✓	6	4.3	odd	2
2320.4.a.o.1.5		6	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q+6.49913 q^{3} -5.00000 q^{5} +2.97283 q^{7} +15.2386 q^{9} -26.0077 q^{11} -14.3733 q^{13} -32.4956 q^{15} +100.181 q^{17} +99.3664 q^{19} +19.3208 q^{21} -102.055 q^{23} +25.0000 q^{25} -76.4386 q^{27} -29.0000 q^{29} -253.319 q^{31} -169.027 q^{33} -14.8641 q^{35} -348.399 q^{37} -93.4137 q^{39} +106.824 q^{41} -68.0944 q^{43} -76.1932 q^{45} +285.052 q^{47} -334.162 q^{49} +651.088 q^{51} +409.075 q^{53} +130.038 q^{55} +645.795 q^{57} +395.912 q^{59} -835.967 q^{61} +45.3018 q^{63} +71.8663 q^{65} -4.99986 q^{67} -663.268 q^{69} +67.3198 q^{71} +124.328 q^{73} +162.478 q^{75} -77.3163 q^{77} +923.032 q^{79} -908.227 q^{81} -455.719 q^{83} -500.905 q^{85} -188.475 q^{87} -1017.98 q^{89} -42.7292 q^{91} -1646.35 q^{93} -496.832 q^{95} -5.39427 q^{97} -396.322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 30 q^{5} - 9 q^{7} + 45 q^{9} + 10 q^{11} + 13 q^{13} - 25 q^{15} - 23 q^{17} - 10 q^{19} - 52 q^{21} - 75 q^{23} + 150 q^{25} + 116 q^{27} - 174 q^{29} + 127 q^{31} - 72 q^{33} + 45 q^{35}+ \cdots + 2976 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 - 30 * q^5 - 9 * q^7 + 45 * q^9 + 10 * q^11 + 13 * q^13 - 25 * q^15 - 23 * q^17 - 10 * q^19 - 52 * q^21 - 75 * q^23 + 150 * q^25 + 116 * q^27 - 174 * q^29 + 127 * q^31 - 72 * q^33 + 45 * q^35 - 158 * q^37 + 407 * q^39 - 622 * q^41 + 311 * q^43 - 225 * q^45 + 628 * q^47 - 879 * q^49 + 886 * q^51 - 813 * q^53 - 50 * q^55 - 1118 * q^57 + 1615 * q^59 - 655 * q^61 + 764 * q^63 - 65 * q^65 + 1000 * q^67 - 1885 * q^69 + 760 * q^71 - 471 * q^73 + 125 * q^75 - 1078 * q^77 + 577 * q^79 - 1402 * q^81 + 1446 * q^83 + 115 * q^85 - 145 * q^87 - 1970 * q^89 + 876 * q^91 - 2274 * q^93 + 50 * q^95 + 1683 * q^97 + 2976 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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