LMFDB - Embedded newform 1840.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 60x^{2} - 45x + 108 $ x^4 - 2x^3 - 60x^2 - 45*x + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.92711$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.92711 q^{3} +5.00000 q^{5} -24.6272 q^{7} +8.13066 q^{9} +O(q^{10})$	$q-5.92711 q^{3} +5.00000 q^{5} -24.6272 q^{7} +8.13066 q^{9} +24.6573 q^{11} +15.2681 q^{13} -29.6356 q^{15} -78.7383 q^{17} +16.3731 q^{19} +145.968 q^{21} +23.0000 q^{23} +25.0000 q^{25} +111.841 q^{27} +36.5449 q^{29} -186.438 q^{31} -146.147 q^{33} -123.136 q^{35} +327.841 q^{37} -90.4956 q^{39} -313.067 q^{41} +394.105 q^{43} +40.6533 q^{45} +252.906 q^{47} +263.499 q^{49} +466.691 q^{51} -8.12126 q^{53} +123.287 q^{55} -97.0452 q^{57} -173.659 q^{59} +420.666 q^{61} -200.236 q^{63} +76.3404 q^{65} -142.945 q^{67} -136.324 q^{69} -658.533 q^{71} -144.794 q^{73} -148.178 q^{75} -607.241 q^{77} +521.592 q^{79} -882.420 q^{81} +987.368 q^{83} -393.691 q^{85} -216.606 q^{87} +176.234 q^{89} -376.010 q^{91} +1105.04 q^{93} +81.8655 q^{95} +769.092 q^{97} +200.480 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9}+O(q^{10}) $ 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9	$ 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9} - 21 q^{11} + 70 q^{13} - 70 q^{15} + 56 q^{17} - 173 q^{19} - 120 q^{21} + 92 q^{23} + 100 q^{25} - 389 q^{27} - 118 q^{29} - 17 q^{31} - 89 q^{33} - 40 q^{35} - 343 q^{37} + 221 q^{39} + 139 q^{41} + 50 q^{43} + 320 q^{45} - 367 q^{47} - 124 q^{49} - 439 q^{51} - 353 q^{53} - 105 q^{55} - 238 q^{57} + 453 q^{59} - 327 q^{61} + 1723 q^{63} + 350 q^{65} + 455 q^{67} - 322 q^{69} - 195 q^{71} - 633 q^{73} - 350 q^{75} - 2 q^{77} + 1140 q^{79} + 1456 q^{81} + 1199 q^{83} + 280 q^{85} + 1775 q^{87} - 2170 q^{89} - 557 q^{91} - 3241 q^{93} - 865 q^{95} - 703 q^{97} + 2745 q^{99}+O(q^{100}) $ 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9 - 21 * q^11 + 70 * q^13 - 70 * q^15 + 56 * q^17 - 173 * q^19 - 120 * q^21 + 92 * q^23 + 100 * q^25 - 389 * q^27 - 118 * q^29 - 17 * q^31 - 89 * q^33 - 40 * q^35 - 343 * q^37 + 221 * q^39 + 139 * q^41 + 50 * q^43 + 320 * q^45 - 367 * q^47 - 124 * q^49 - 439 * q^51 - 353 * q^53 - 105 * q^55 - 238 * q^57 + 453 * q^59 - 327 * q^61 + 1723 * q^63 + 350 * q^65 + 455 * q^67 - 322 * q^69 - 195 * q^71 - 633 * q^73 - 350 * q^75 - 2 * q^77 + 1140 * q^79 + 1456 * q^81 + 1199 * q^83 + 280 * q^85 + 1775 * q^87 - 2170 * q^89 - 557 * q^91 - 3241 * q^93 - 865 * q^95 - 703 * q^97 + 2745 * 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
$3$	−5.92711	−1.14067	−0.570337	−	0.821411i	$-0.693187\pi$
$3$	−5.92711	−1.14067	−0.570337	+	0.821411i	$0.693187\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−24.6272	−1.32974	−0.664872	−	0.746957i	$-0.731514\pi$
$7$	−24.6272	−1.32974	−0.664872	+	0.746957i	$0.731514\pi$
$8$	0	0
$9$	8.13066	0.301136
$10$	0	0
$11$	24.6573	0.675860	0.337930	−	0.941171i	$-0.390273\pi$
$11$	24.6573	0.675860	0.337930	+	0.941171i	$0.390273\pi$
$12$	0	0
$13$	15.2681	0.325739	0.162869	−	0.986648i	$-0.447925\pi$
$13$	15.2681	0.325739	0.162869	+	0.986648i	$0.447925\pi$
$14$	0	0
$15$	−29.6356	−0.510125
$16$	0	0
$17$	−78.7383	−1.12334	−0.561672	−	0.827360i	$-0.689842\pi$
$17$	−78.7383	−1.12334	−0.561672	+	0.827360i	$0.689842\pi$
$18$	0	0
$19$	16.3731	0.197697	0.0988486	−	0.995102i	$-0.468484\pi$
$19$	16.3731	0.197697	0.0988486	+	0.995102i	$0.468484\pi$
$20$	0	0
$21$	145.968	1.51680
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	111.841	0.797176
$28$	0	0
$29$	36.5449	0.234008	0.117004	−	0.993131i	$-0.462671\pi$
$29$	36.5449	0.234008	0.117004	+	0.993131i	$0.462671\pi$
$30$	0	0
$31$	−186.438	−1.08017	−0.540083	−	0.841612i	$-0.681607\pi$
$31$	−186.438	−1.08017	−0.540083	+	0.841612i	$0.681607\pi$
$32$	0	0
$33$	−146.147	−0.770936
$34$	0	0
$35$	−123.136	−0.594680
$36$	0	0
$37$	327.841	1.45667	0.728335	−	0.685221i	$-0.240294\pi$
$37$	327.841	1.45667	0.728335	+	0.685221i	$0.240294\pi$
$38$	0	0
$39$	−90.4956	−0.371561
$40$	0	0
$41$	−313.067	−1.19251	−0.596254	−	0.802796i	$-0.703345\pi$
$41$	−313.067	−1.19251	−0.596254	+	0.802796i	$0.703345\pi$
$42$	0	0
$43$	394.105	1.39768	0.698842	−	0.715276i	$-0.253699\pi$
$43$	394.105	1.39768	0.698842	+	0.715276i	$0.253699\pi$
$44$	0	0
$45$	40.6533	0.134672
$46$	0	0
$47$	252.906	0.784897	0.392449	−	0.919774i	$-0.371628\pi$
$47$	252.906	0.784897	0.392449	+	0.919774i	$0.371628\pi$
$48$	0	0
$49$	263.499	0.768219
$50$	0	0
$51$	466.691	1.28137
$52$	0	0
$53$	−8.12126	−0.0210480	−0.0105240	−	0.999945i	$-0.503350\pi$
$53$	−8.12126	−0.0210480	−0.0105240	+	0.999945i	$0.503350\pi$
$54$	0	0
$55$	123.287	0.302254
$56$	0	0
$57$	−97.0452	−0.225508
$58$	0	0
$59$	−173.659	−0.383193	−0.191597	−	0.981474i	$-0.561367\pi$
$59$	−173.659	−0.383193	−0.191597	+	0.981474i	$0.561367\pi$
$60$	0	0
$61$	420.666	0.882964	0.441482	−	0.897270i	$-0.354453\pi$
$61$	420.666	0.882964	0.441482	+	0.897270i	$0.354453\pi$
$62$	0	0
$63$	−200.236	−0.400433
$64$	0	0
$65$	76.3404	0.145675
$66$	0	0
$67$	−142.945	−0.260649	−0.130324	−	0.991471i	$-0.541602\pi$
$67$	−142.945	−0.260649	−0.130324	+	0.991471i	$0.541602\pi$
$68$	0	0
$69$	−136.324	−0.237847
$70$	0	0
$71$	−658.533	−1.10075	−0.550376	−	0.834917i	$-0.685516\pi$
$71$	−658.533	−1.10075	−0.550376	+	0.834917i	$0.685516\pi$
$72$	0	0
$73$	−144.794	−0.232149	−0.116075	−	0.993241i	$-0.537031\pi$
$73$	−144.794	−0.232149	−0.116075	+	0.993241i	$0.537031\pi$
$74$	0	0
$75$	−148.178	−0.228135
$76$	0	0
$77$	−607.241	−0.898721
$78$	0	0
$79$	521.592	0.742831	0.371415	−	0.928467i	$-0.378873\pi$
$79$	521.592	0.742831	0.371415	+	0.928467i	$0.378873\pi$
$80$	0	0
$81$	−882.420	−1.21045
$82$	0	0
$83$	987.368	1.30576	0.652878	−	0.757463i	$-0.273562\pi$
$83$	987.368	1.30576	0.652878	+	0.757463i	$0.273562\pi$
$84$	0	0
$85$	−393.691	−0.502375
$86$	0	0
$87$	−216.606	−0.266926
$88$	0	0
$89$	176.234	0.209896	0.104948	−	0.994478i	$-0.466532\pi$
$89$	176.234	0.209896	0.104948	+	0.994478i	$0.466532\pi$
$90$	0	0
$91$	−376.010	−0.433149
$92$	0	0
$93$	1105.04	1.23212
$94$	0	0
$95$	81.8655	0.0884129
$96$	0	0
$97$	769.092	0.805046	0.402523	−	0.915410i	$-0.368133\pi$
$97$	769.092	0.805046	0.402523	+	0.915410i	$0.368133\pi$
$98$	0	0
$99$	200.480	0.203526
$100$	0	0
$101$	−33.6036	−0.0331058	−0.0165529	−	0.999863i	$-0.505269\pi$
$101$	−33.6036	−0.0331058	−0.0165529	+	0.999863i	$0.505269\pi$
$102$	0	0
$103$	1868.28	1.78725	0.893626	−	0.448813i	$-0.148153\pi$
$103$	1868.28	1.78725	0.893626	+	0.448813i	$0.148153\pi$
$104$	0	0
$105$	729.841	0.678335
$106$	0	0
$107$	1438.84	1.29998	0.649992	−	0.759941i	$-0.274772\pi$
$107$	1438.84	1.29998	0.649992	+	0.759941i	$0.274772\pi$
$108$	0	0
$109$	−1864.49	−1.63840	−0.819202	−	0.573504i	$-0.805584\pi$
$109$	−1864.49	−1.63840	−0.819202	+	0.573504i	$0.805584\pi$
$110$	0	0
$111$	−1943.15	−1.66158
$112$	0	0
$113$	−397.768	−0.331141	−0.165570	−	0.986198i	$-0.552947\pi$
$113$	−397.768	−0.331141	−0.165570	+	0.986198i	$0.552947\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	124.140	0.0980915
$118$	0	0
$119$	1939.10	1.49376
$120$	0	0
$121$	−723.016	−0.543213
$122$	0	0
$123$	1855.58	1.36026
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1370.48	−0.957564	−0.478782	−	0.877934i	$-0.658922\pi$
$127$	−1370.48	−0.957564	−0.478782	+	0.877934i	$0.658922\pi$
$128$	0	0
$129$	−2335.90	−1.59430
$130$	0	0
$131$	−1891.14	−1.26130	−0.630649	−	0.776068i	$-0.717211\pi$
$131$	−1891.14	−1.26130	−0.630649	+	0.776068i	$0.717211\pi$
$132$	0	0
$133$	−403.224	−0.262887
$134$	0	0
$135$	559.203	0.356508
$136$	0	0
$137$	2226.21	1.38831	0.694155	−	0.719826i	$-0.255778\pi$
$137$	2226.21	1.38831	0.694155	+	0.719826i	$0.255778\pi$
$138$	0	0
$139$	699.554	0.426873	0.213437	−	0.976957i	$-0.431534\pi$
$139$	699.554	0.426873	0.213437	+	0.976957i	$0.431534\pi$
$140$	0	0
$141$	−1499.00	−0.895311
$142$	0	0
$143$	376.470	0.220154
$144$	0	0
$145$	182.725	0.104651
$146$	0	0
$147$	−1561.79	−0.876287
$148$	0	0
$149$	−1174.11	−0.645549	−0.322775	−	0.946476i	$-0.604616\pi$
$149$	−1174.11	−0.645549	−0.322775	+	0.946476i	$0.604616\pi$
$150$	0	0
$151$	−2733.01	−1.47291	−0.736455	−	0.676487i	$-0.763502\pi$
$151$	−2733.01	−1.47291	−0.736455	+	0.676487i	$0.763502\pi$
$152$	0	0
$153$	−640.195	−0.338279
$154$	0	0
$155$	−932.188	−0.483065
$156$	0	0
$157$	−4.77232	−0.00242594	−0.00121297	−	0.999999i	$-0.500386\pi$
$157$	−4.77232	−0.00242594	−0.00121297	+	0.999999i	$0.500386\pi$
$158$	0	0
$159$	48.1357	0.0240088
$160$	0	0
$161$	−566.426	−0.277271
$162$	0	0
$163$	−1523.51	−0.732090	−0.366045	−	0.930597i	$-0.619288\pi$
$163$	−1523.51	−0.732090	−0.366045	+	0.930597i	$0.619288\pi$
$164$	0	0
$165$	−730.734	−0.344773
$166$	0	0
$167$	−3603.81	−1.66989	−0.834944	−	0.550335i	$-0.814500\pi$
$167$	−3603.81	−1.66989	−0.834944	+	0.550335i	$0.814500\pi$
$168$	0	0
$169$	−1963.89	−0.893894
$170$	0	0
$171$	133.124	0.0595337
$172$	0	0
$173$	349.993	0.153812	0.0769061	−	0.997038i	$-0.475496\pi$
$173$	349.993	0.153812	0.0769061	+	0.997038i	$0.475496\pi$
$174$	0	0
$175$	−615.680	−0.265949
$176$	0	0
$177$	1029.29	0.437099
$178$	0	0
$179$	1283.17	0.535804	0.267902	−	0.963446i	$-0.413670\pi$
$179$	1283.17	0.535804	0.267902	+	0.963446i	$0.413670\pi$
$180$	0	0
$181$	3076.40	1.26336	0.631678	−	0.775231i	$-0.282367\pi$
$181$	3076.40	1.26336	0.631678	+	0.775231i	$0.282367\pi$
$182$	0	0
$183$	−2493.34	−1.00717
$184$	0	0
$185$	1639.21	0.651443
$186$	0	0
$187$	−1941.48	−0.759223
$188$	0	0
$189$	−2754.32	−1.06004
$190$	0	0
$191$	1787.86	0.677306	0.338653	−	0.940911i	$-0.390029\pi$
$191$	1787.86	0.677306	0.338653	+	0.940911i	$0.390029\pi$
$192$	0	0
$193$	−881.506	−0.328768	−0.164384	−	0.986396i	$-0.552564\pi$
$193$	−881.506	−0.328768	−0.164384	+	0.986396i	$0.552564\pi$
$194$	0	0
$195$	−452.478	−0.166167
$196$	0	0
$197$	2217.28	0.801904	0.400952	−	0.916099i	$-0.368679\pi$
$197$	2217.28	0.801904	0.400952	+	0.916099i	$0.368679\pi$
$198$	0	0
$199$	211.313	0.0752742	0.0376371	−	0.999291i	$-0.488017\pi$
$199$	211.313	0.0752742	0.0376371	+	0.999291i	$0.488017\pi$
$200$	0	0
$201$	847.249	0.297315
$202$	0	0
$203$	−899.999	−0.311170
$204$	0	0
$205$	−1565.33	−0.533306
$206$	0	0
$207$	187.005	0.0627911
$208$	0	0
$209$	403.717	0.133616
$210$	0	0
$211$	−3939.97	−1.28549	−0.642745	−	0.766080i	$-0.722205\pi$
$211$	−3939.97	−1.28549	−0.642745	+	0.766080i	$0.722205\pi$
$212$	0	0
$213$	3903.20	1.25560
$214$	0	0
$215$	1970.52	0.625063
$216$	0	0
$217$	4591.43	1.43635
$218$	0	0
$219$	858.212	0.264806
$220$	0	0
$221$	−1202.18	−0.365916
$222$	0	0
$223$	−4154.48	−1.24755	−0.623777	−	0.781603i	$-0.714403\pi$
$223$	−4154.48	−1.24755	−0.623777	+	0.781603i	$0.714403\pi$
$224$	0	0
$225$	203.267	0.0602271
$226$	0	0
$227$	−6133.31	−1.79331	−0.896656	−	0.442727i	$-0.854011\pi$
$227$	−6133.31	−1.79331	−0.896656	+	0.442727i	$0.854011\pi$
$228$	0	0
$229$	2244.20	0.647603	0.323802	−	0.946125i	$-0.395039\pi$
$229$	2244.20	0.647603	0.323802	+	0.946125i	$0.395039\pi$
$230$	0	0
$231$	3599.19	1.02515
$232$	0	0
$233$	2161.23	0.607670	0.303835	−	0.952725i	$-0.401733\pi$
$233$	2161.23	0.607670	0.303835	+	0.952725i	$0.401733\pi$
$234$	0	0
$235$	1264.53	0.351017
$236$	0	0
$237$	−3091.53	−0.847327
$238$	0	0
$239$	−3448.18	−0.933240	−0.466620	−	0.884458i	$-0.654528\pi$
$239$	−3448.18	−0.933240	−0.466620	+	0.884458i	$0.654528\pi$
$240$	0	0
$241$	1654.43	0.442204	0.221102	−	0.975251i	$-0.429035\pi$
$241$	1654.43	0.442204	0.221102	+	0.975251i	$0.429035\pi$
$242$	0	0
$243$	2210.51	0.583556
$244$	0	0
$245$	1317.50	0.343558
$246$	0	0
$247$	249.986	0.0643976
$248$	0	0
$249$	−5852.24	−1.48944
$250$	0	0
$251$	6157.96	1.54855	0.774277	−	0.632846i	$-0.218113\pi$
$251$	6157.96	1.54855	0.774277	+	0.632846i	$0.218113\pi$
$252$	0	0
$253$	567.118	0.140927
$254$	0	0
$255$	2333.45	0.573045
$256$	0	0
$257$	−448.443	−0.108845	−0.0544224	−	0.998518i	$-0.517332\pi$
$257$	−448.443	−0.108845	−0.0544224	+	0.998518i	$0.517332\pi$
$258$	0	0
$259$	−8073.82	−1.93700
$260$	0	0
$261$	297.134	0.0704681
$262$	0	0
$263$	−6298.76	−1.47680	−0.738399	−	0.674364i	$-0.764418\pi$
$263$	−6298.76	−1.47680	−0.738399	+	0.674364i	$0.764418\pi$
$264$	0	0
$265$	−40.6063	−0.00941293
$266$	0	0
$267$	−1044.56	−0.239422
$268$	0	0
$269$	−2106.69	−0.477500	−0.238750	−	0.971081i	$-0.576738\pi$
$269$	−2106.69	−0.477500	−0.238750	+	0.971081i	$0.576738\pi$
$270$	0	0
$271$	3116.92	0.698668	0.349334	−	0.936998i	$-0.386408\pi$
$271$	3116.92	0.698668	0.349334	+	0.936998i	$0.386408\pi$
$272$	0	0
$273$	2228.65	0.494082
$274$	0	0
$275$	616.433	0.135172
$276$	0	0
$277$	1177.86	0.255490	0.127745	−	0.991807i	$-0.459226\pi$
$277$	1177.86	0.255490	0.127745	+	0.991807i	$0.459226\pi$
$278$	0	0
$279$	−1515.86	−0.325277
$280$	0	0
$281$	−5240.30	−1.11249	−0.556246	−	0.831018i	$-0.687759\pi$
$281$	−5240.30	−1.11249	−0.556246	+	0.831018i	$0.687759\pi$
$282$	0	0
$283$	−7162.44	−1.50446	−0.752232	−	0.658899i	$-0.771022\pi$
$283$	−7162.44	−1.50446	−0.752232	+	0.658899i	$0.771022\pi$
$284$	0	0
$285$	−485.226	−0.100850
$286$	0	0
$287$	7709.96	1.58573
$288$	0	0
$289$	1286.72	0.261901
$290$	0	0
$291$	−4558.49	−0.918294
$292$	0	0
$293$	−7564.70	−1.50831	−0.754154	−	0.656698i	$-0.771953\pi$
$293$	−7564.70	−1.50831	−0.754154	+	0.656698i	$0.771953\pi$
$294$	0	0
$295$	−868.293	−0.171369
$296$	0	0
$297$	2757.69	0.538780
$298$	0	0
$299$	351.166	0.0679212
$300$	0	0
$301$	−9705.70	−1.85856
$302$	0	0
$303$	199.172	0.0377629
$304$	0	0
$305$	2103.33	0.394873
$306$	0	0
$307$	−5675.81	−1.05516	−0.527582	−	0.849504i	$-0.676901\pi$
$307$	−5675.81	−1.05516	−0.527582	+	0.849504i	$0.676901\pi$
$308$	0	0
$309$	−11073.5	−2.03867
$310$	0	0
$311$	−4934.29	−0.899671	−0.449835	−	0.893111i	$-0.648517\pi$
$311$	−4934.29	−0.899671	−0.449835	+	0.893111i	$0.648517\pi$
$312$	0	0
$313$	−4458.67	−0.805173	−0.402587	−	0.915382i	$-0.631889\pi$
$313$	−4458.67	−0.805173	−0.402587	+	0.915382i	$0.631889\pi$
$314$	0	0
$315$	−1001.18	−0.179079
$316$	0	0
$317$	4132.87	0.732256	0.366128	−	0.930564i	$-0.380683\pi$
$317$	4132.87	0.732256	0.366128	+	0.930564i	$0.380683\pi$
$318$	0	0
$319$	901.100	0.158156
$320$	0	0
$321$	−8528.19	−1.48286
$322$	0	0
$323$	−1289.19	−0.222082
$324$	0	0
$325$	381.702	0.0651477
$326$	0	0
$327$	11051.1	1.86888
$328$	0	0
$329$	−6228.37	−1.04371
$330$	0	0
$331$	−10319.9	−1.71370	−0.856849	−	0.515567i	$-0.827581\pi$
$331$	−10319.9	−1.71370	−0.856849	+	0.515567i	$0.827581\pi$
$332$	0	0
$333$	2665.57	0.438655
$334$	0	0
$335$	−714.723	−0.116566
$336$	0	0
$337$	−9817.59	−1.58694	−0.793469	−	0.608611i	$-0.791727\pi$
$337$	−9817.59	−1.58694	−0.793469	+	0.608611i	$0.791727\pi$
$338$	0	0
$339$	2357.62	0.377723
$340$	0	0
$341$	−4597.05	−0.730042
$342$	0	0
$343$	1957.89	0.308210
$344$	0	0
$345$	−681.618	−0.106368
$346$	0	0
$347$	5827.53	0.901552	0.450776	−	0.892637i	$-0.351147\pi$
$347$	5827.53	0.901552	0.450776	+	0.892637i	$0.351147\pi$
$348$	0	0
$349$	6483.51	0.994426	0.497213	−	0.867629i	$-0.334357\pi$
$349$	6483.51	0.994426	0.497213	+	0.867629i	$0.334357\pi$
$350$	0	0
$351$	1707.59	0.259671
$352$	0	0
$353$	−11535.2	−1.73925	−0.869627	−	0.493709i	$-0.835641\pi$
$353$	−11535.2	−1.73925	−0.869627	+	0.493709i	$0.835641\pi$
$354$	0	0
$355$	−3292.66	−0.492272
$356$	0	0
$357$	−11493.3	−1.70389
$358$	0	0
$359$	−5922.53	−0.870695	−0.435347	−	0.900263i	$-0.643374\pi$
$359$	−5922.53	−0.870695	−0.435347	+	0.900263i	$0.643374\pi$
$360$	0	0
$361$	−6590.92	−0.960916
$362$	0	0
$363$	4285.40	0.619628
$364$	0	0
$365$	−723.971	−0.103820
$366$	0	0
$367$	4042.72	0.575009	0.287504	−	0.957779i	$-0.407174\pi$
$367$	4042.72	0.575009	0.287504	+	0.957779i	$0.407174\pi$
$368$	0	0
$369$	−2545.44	−0.359107
$370$	0	0
$371$	200.004	0.0279884
$372$	0	0
$373$	−9078.68	−1.26026	−0.630129	−	0.776491i	$-0.716998\pi$
$373$	−9078.68	−1.26026	−0.630129	+	0.776491i	$0.716998\pi$
$374$	0	0
$375$	−740.889	−0.102025
$376$	0	0
$377$	557.971	0.0762253
$378$	0	0
$379$	−6561.42	−0.889280	−0.444640	−	0.895709i	$-0.646668\pi$
$379$	−6561.42	−0.889280	−0.444640	+	0.895709i	$0.646668\pi$
$380$	0	0
$381$	8123.00	1.09227
$382$	0	0
$383$	−587.548	−0.0783872	−0.0391936	−	0.999232i	$-0.512479\pi$
$383$	−587.548	−0.0783872	−0.0391936	+	0.999232i	$0.512479\pi$
$384$	0	0
$385$	−3036.20	−0.401920
$386$	0	0
$387$	3204.33	0.420893
$388$	0	0
$389$	−6470.36	−0.843343	−0.421672	−	0.906749i	$-0.638556\pi$
$389$	−6470.36	−0.843343	−0.421672	+	0.906749i	$0.638556\pi$
$390$	0	0
$391$	−1810.98	−0.234233
$392$	0	0
$393$	11209.0	1.43873
$394$	0	0
$395$	2607.96	0.332204
$396$	0	0
$397$	9118.76	1.15279	0.576395	−	0.817171i	$-0.304459\pi$
$397$	9118.76	1.15279	0.576395	+	0.817171i	$0.304459\pi$
$398$	0	0
$399$	2389.95	0.299868
$400$	0	0
$401$	−3199.75	−0.398474	−0.199237	−	0.979951i	$-0.563846\pi$
$401$	−3199.75	−0.398474	−0.199237	+	0.979951i	$0.563846\pi$
$402$	0	0
$403$	−2846.54	−0.351852
$404$	0	0
$405$	−4412.10	−0.541331
$406$	0	0
$407$	8083.69	0.984505
$408$	0	0
$409$	6146.22	0.743059	0.371529	−	0.928421i	$-0.378833\pi$
$409$	6146.22	0.743059	0.371529	+	0.928421i	$0.378833\pi$
$410$	0	0
$411$	−13195.0	−1.58361
$412$	0	0
$413$	4276.72	0.509549
$414$	0	0
$415$	4936.84	0.583951
$416$	0	0
$417$	−4146.33	−0.486923
$418$	0	0
$419$	9933.76	1.15822	0.579112	−	0.815248i	$-0.303399\pi$
$419$	9933.76	1.15822	0.579112	+	0.815248i	$0.303399\pi$
$420$	0	0
$421$	3653.89	0.422993	0.211496	−	0.977379i	$-0.432166\pi$
$421$	3653.89	0.422993	0.211496	+	0.977379i	$0.432166\pi$
$422$	0	0
$423$	2056.30	0.236361
$424$	0	0
$425$	−1968.46	−0.224669
$426$	0	0
$427$	−10359.8	−1.17412
$428$	0	0
$429$	−2231.38	−0.251124
$430$	0	0
$431$	3056.87	0.341634	0.170817	−	0.985303i	$-0.445359\pi$
$431$	3056.87	0.341634	0.170817	+	0.985303i	$0.445359\pi$
$432$	0	0
$433$	1703.05	0.189014	0.0945072	−	0.995524i	$-0.469872\pi$
$433$	1703.05	0.189014	0.0945072	+	0.995524i	$0.469872\pi$
$434$	0	0
$435$	−1083.03	−0.119373
$436$	0	0
$437$	376.581	0.0412227
$438$	0	0
$439$	758.630	0.0824771	0.0412386	−	0.999149i	$-0.486870\pi$
$439$	758.630	0.0824771	0.0412386	+	0.999149i	$0.486870\pi$
$440$	0	0
$441$	2142.42	0.231338
$442$	0	0
$443$	−5971.42	−0.640430	−0.320215	−	0.947345i	$-0.603755\pi$
$443$	−5971.42	−0.640430	−0.320215	+	0.947345i	$0.603755\pi$
$444$	0	0
$445$	881.168	0.0938682
$446$	0	0
$447$	6959.08	0.736361
$448$	0	0
$449$	−3515.62	−0.369515	−0.184757	−	0.982784i	$-0.559150\pi$
$449$	−3515.62	−0.369515	−0.184757	+	0.982784i	$0.559150\pi$
$450$	0	0
$451$	−7719.39	−0.805969
$452$	0	0
$453$	16198.9	1.68011
$454$	0	0
$455$	−1880.05	−0.193710
$456$	0	0
$457$	12155.6	1.24423	0.622117	−	0.782925i	$-0.286273\pi$
$457$	12155.6	1.24423	0.622117	+	0.782925i	$0.286273\pi$
$458$	0	0
$459$	−8806.14	−0.895502
$460$	0	0
$461$	−8854.82	−0.894599	−0.447299	−	0.894384i	$-0.647614\pi$
$461$	−8854.82	−0.894599	−0.447299	+	0.894384i	$0.647614\pi$
$462$	0	0
$463$	−17908.6	−1.79759	−0.898793	−	0.438374i	$-0.855555\pi$
$463$	−17908.6	−1.79759	−0.898793	+	0.438374i	$0.855555\pi$
$464$	0	0
$465$	5525.18	0.551020
$466$	0	0
$467$	−8596.74	−0.851841	−0.425920	−	0.904761i	$-0.640050\pi$
$467$	−8596.74	−0.851841	−0.425920	+	0.904761i	$0.640050\pi$
$468$	0	0
$469$	3520.33	0.346596
$470$	0	0
$471$	28.2861	0.00276720
$472$	0	0
$473$	9717.57	0.944639
$474$	0	0
$475$	409.328	0.0395394
$476$	0	0
$477$	−66.0313	−0.00633829
$478$	0	0
$479$	12385.2	1.18140	0.590702	−	0.806890i	$-0.298851\pi$
$479$	12385.2	1.18140	0.590702	+	0.806890i	$0.298851\pi$
$480$	0	0
$481$	5005.51	0.474494
$482$	0	0
$483$	3357.27	0.316275
$484$	0	0
$485$	3845.46	0.360027
$486$	0	0
$487$	−4331.97	−0.403081	−0.201541	−	0.979480i	$-0.564595\pi$
$487$	−4331.97	−0.403081	−0.201541	+	0.979480i	$0.564595\pi$
$488$	0	0
$489$	9030.02	0.835075
$490$	0	0
$491$	11366.0	1.04468	0.522341	−	0.852737i	$-0.325059\pi$
$491$	11366.0	1.04468	0.522341	+	0.852737i	$0.325059\pi$
$492$	0	0
$493$	−2877.48	−0.262871
$494$	0	0
$495$	1002.40	0.0910194
$496$	0	0
$497$	16217.8	1.46372
$498$	0	0
$499$	−12620.5	−1.13221	−0.566105	−	0.824333i	$-0.691550\pi$
$499$	−12620.5	−1.13221	−0.566105	+	0.824333i	$0.691550\pi$
$500$	0	0
$501$	21360.2	1.90480
$502$	0	0
$503$	11183.5	0.991348	0.495674	−	0.868509i	$-0.334921\pi$
$503$	11183.5	0.991348	0.495674	+	0.868509i	$0.334921\pi$
$504$	0	0
$505$	−168.018	−0.0148054
$506$	0	0
$507$	11640.2	1.01964
$508$	0	0
$509$	−5066.54	−0.441199	−0.220599	−	0.975364i	$-0.570801\pi$
$509$	−5066.54	−0.441199	−0.220599	+	0.975364i	$0.570801\pi$
$510$	0	0
$511$	3565.88	0.308699
$512$	0	0
$513$	1831.18	0.157599
$514$	0	0
$515$	9341.39	0.799283
$516$	0	0
$517$	6235.99	0.530481
$518$	0	0
$519$	−2074.45	−0.175449
$520$	0	0
$521$	20873.3	1.75523	0.877616	−	0.479364i	$-0.159133\pi$
$521$	20873.3	1.75523	0.877616	+	0.479364i	$0.159133\pi$
$522$	0	0
$523$	−8937.71	−0.747264	−0.373632	−	0.927577i	$-0.621888\pi$
$523$	−8937.71	−0.747264	−0.373632	+	0.927577i	$0.621888\pi$
$524$	0	0
$525$	3649.20	0.303361
$526$	0	0
$527$	14679.8	1.21340
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1411.96	−0.115393
$532$	0	0
$533$	−4779.93	−0.388446
$534$	0	0
$535$	7194.22	0.581370
$536$	0	0
$537$	−7605.51	−0.611177
$538$	0	0
$539$	6497.18	0.519209
$540$	0	0
$541$	−14320.8	−1.13808	−0.569038	−	0.822311i	$-0.692684\pi$
$541$	−14320.8	−1.13808	−0.569038	+	0.822311i	$0.692684\pi$
$542$	0	0
$543$	−18234.2	−1.44108
$544$	0	0
$545$	−9322.47	−0.732717
$546$	0	0
$547$	17084.3	1.33541	0.667706	−	0.744425i	$-0.267276\pi$
$547$	17084.3	1.33541	0.667706	+	0.744425i	$0.267276\pi$
$548$	0	0
$549$	3420.30	0.265892
$550$	0	0
$551$	598.354	0.0462627
$552$	0	0
$553$	−12845.3	−0.987775
$554$	0	0
$555$	−9715.76	−0.743083
$556$	0	0
$557$	−21376.9	−1.62616	−0.813079	−	0.582153i	$-0.802211\pi$
$557$	−21376.9	−1.62616	−0.813079	+	0.582153i	$0.802211\pi$
$558$	0	0
$559$	6017.22	0.455280
$560$	0	0
$561$	11507.3	0.866026
$562$	0	0
$563$	−1867.78	−0.139818	−0.0699091	−	0.997553i	$-0.522271\pi$
$563$	−1867.78	−0.139818	−0.0699091	+	0.997553i	$0.522271\pi$
$564$	0	0
$565$	−1988.84	−0.148091
$566$	0	0
$567$	21731.5	1.60959
$568$	0	0
$569$	9340.49	0.688179	0.344089	−	0.938937i	$-0.388188\pi$
$569$	9340.49	0.688179	0.344089	+	0.938937i	$0.388188\pi$
$570$	0	0
$571$	5452.88	0.399643	0.199821	−	0.979832i	$-0.435964\pi$
$571$	5452.88	0.399643	0.199821	+	0.979832i	$0.435964\pi$
$572$	0	0
$573$	−10596.9	−0.772584
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	3343.59	0.241240	0.120620	−	0.992699i	$-0.461512\pi$
$577$	3343.59	0.241240	0.120620	+	0.992699i	$0.461512\pi$
$578$	0	0
$579$	5224.79	0.375017
$580$	0	0
$581$	−24316.1	−1.73632
$582$	0	0
$583$	−200.249	−0.0142255
$584$	0	0
$585$	620.698	0.0438679
$586$	0	0
$587$	19990.7	1.40563	0.702813	−	0.711375i	$-0.251927\pi$
$587$	19990.7	1.40563	0.702813	+	0.711375i	$0.251927\pi$
$588$	0	0
$589$	−3052.56	−0.213546
$590$	0	0
$591$	−13142.1	−0.914710
$592$	0	0
$593$	15181.6	1.05132	0.525662	−	0.850694i	$-0.323818\pi$
$593$	15181.6	1.05132	0.525662	+	0.850694i	$0.323818\pi$
$594$	0	0
$595$	9695.52	0.668030
$596$	0	0
$597$	−1252.48	−0.0858633
$598$	0	0
$599$	5524.97	0.376869	0.188434	−	0.982086i	$-0.439659\pi$
$599$	5524.97	0.376869	0.188434	+	0.982086i	$0.439659\pi$
$600$	0	0
$601$	22916.0	1.55535	0.777673	−	0.628669i	$-0.216400\pi$
$601$	22916.0	1.55535	0.777673	+	0.628669i	$0.216400\pi$
$602$	0	0
$603$	−1162.23	−0.0784907
$604$	0	0
$605$	−3615.08	−0.242932
$606$	0	0
$607$	10630.6	0.710844	0.355422	−	0.934706i	$-0.384337\pi$
$607$	10630.6	0.710844	0.355422	+	0.934706i	$0.384337\pi$
$608$	0	0
$609$	5334.40	0.354944
$610$	0	0
$611$	3861.39	0.255671
$612$	0	0
$613$	−25267.8	−1.66486	−0.832429	−	0.554132i	$-0.813050\pi$
$613$	−25267.8	−1.66486	−0.832429	+	0.554132i	$0.813050\pi$
$614$	0	0
$615$	9277.91	0.608328
$616$	0	0
$617$	21199.3	1.38323	0.691615	−	0.722266i	$-0.256899\pi$
$617$	21199.3	1.38323	0.691615	+	0.722266i	$0.256899\pi$
$618$	0	0
$619$	8059.26	0.523310	0.261655	−	0.965161i	$-0.415732\pi$
$619$	8059.26	0.523310	0.261655	+	0.965161i	$0.415732\pi$
$620$	0	0
$621$	2572.34	0.166223
$622$	0	0
$623$	−4340.14	−0.279107
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−2392.88	−0.152412
$628$	0	0
$629$	−25813.7	−1.63634
$630$	0	0
$631$	−10528.9	−0.664262	−0.332131	−	0.943233i	$-0.607768\pi$
$631$	−10528.9	−0.664262	−0.332131	+	0.943233i	$0.607768\pi$
$632$	0	0
$633$	23352.6	1.46632
$634$	0	0
$635$	−6852.41	−0.428236
$636$	0	0
$637$	4023.12	0.250239
$638$	0	0
$639$	−5354.31	−0.331476
$640$	0	0
$641$	−13680.3	−0.842961	−0.421481	−	0.906837i	$-0.638489\pi$
$641$	−13680.3	−0.842961	−0.421481	+	0.906837i	$0.638489\pi$
$642$	0	0
$643$	8361.27	0.512809	0.256405	−	0.966570i	$-0.417462\pi$
$643$	8361.27	0.512809	0.256405	+	0.966570i	$0.417462\pi$
$644$	0	0
$645$	−11679.5	−0.712993
$646$	0	0
$647$	31252.2	1.89900	0.949499	−	0.313769i	$-0.101592\pi$
$647$	31252.2	1.89900	0.949499	+	0.313769i	$0.101592\pi$
$648$	0	0
$649$	−4281.95	−0.258985
$650$	0	0
$651$	−27214.0	−1.63840
$652$	0	0
$653$	22983.2	1.37734	0.688670	−	0.725075i	$-0.258195\pi$
$653$	22983.2	1.37734	0.688670	+	0.725075i	$0.258195\pi$
$654$	0	0
$655$	−9455.72	−0.564070
$656$	0	0
$657$	−1177.27	−0.0699084
$658$	0	0
$659$	−5781.74	−0.341767	−0.170884	−	0.985291i	$-0.554662\pi$
$659$	−5781.74	−0.341767	−0.170884	+	0.985291i	$0.554662\pi$
$660$	0	0
$661$	18973.9	1.11649	0.558244	−	0.829677i	$-0.311475\pi$
$661$	18973.9	1.11649	0.558244	+	0.829677i	$0.311475\pi$
$662$	0	0
$663$	7125.47	0.417391
$664$	0	0
$665$	−2016.12	−0.117567
$666$	0	0
$667$	840.533	0.0487940
$668$	0	0
$669$	24624.1	1.42305
$670$	0	0
$671$	10372.5	0.596760
$672$	0	0
$673$	13833.4	0.792328	0.396164	−	0.918180i	$-0.370341\pi$
$673$	13833.4	0.792328	0.396164	+	0.918180i	$0.370341\pi$
$674$	0	0
$675$	2796.02	0.159435
$676$	0	0
$677$	−3041.39	−0.172659	−0.0863295	−	0.996267i	$-0.527514\pi$
$677$	−3041.39	−0.172659	−0.0863295	+	0.996267i	$0.527514\pi$
$678$	0	0
$679$	−18940.6	−1.07050
$680$	0	0
$681$	36352.8	2.04558
$682$	0	0
$683$	−3325.43	−0.186302	−0.0931509	−	0.995652i	$-0.529694\pi$
$683$	−3325.43	−0.186302	−0.0931509	+	0.995652i	$0.529694\pi$
$684$	0	0
$685$	11131.1	0.620871
$686$	0	0
$687$	−13301.6	−0.738704
$688$	0	0
$689$	−123.996	−0.00685613
$690$	0	0
$691$	29403.7	1.61877	0.809384	−	0.587280i	$-0.199801\pi$
$691$	29403.7	1.61877	0.809384	+	0.587280i	$0.199801\pi$
$692$	0	0
$693$	−4937.27	−0.270637
$694$	0	0
$695$	3497.77	0.190904
$696$	0	0
$697$	24650.3	1.33960
$698$	0	0
$699$	−12809.9	−0.693153
$700$	0	0
$701$	19653.5	1.05892	0.529461	−	0.848334i	$-0.322394\pi$
$701$	19653.5	1.05892	0.529461	+	0.848334i	$0.322394\pi$
$702$	0	0
$703$	5367.78	0.287980
$704$	0	0
$705$	−7495.02	−0.400395
$706$	0	0
$707$	827.563	0.0440222
$708$	0	0
$709$	36228.5	1.91903	0.959515	−	0.281659i	$-0.0908846\pi$
$709$	36228.5	1.91903	0.959515	+	0.281659i	$0.0908846\pi$
$710$	0	0
$711$	4240.89	0.223693
$712$	0	0
$713$	−4288.06	−0.225230
$714$	0	0
$715$	1882.35	0.0984558
$716$	0	0
$717$	20437.8	1.06452
$718$	0	0
$719$	3129.71	0.162334	0.0811672	−	0.996700i	$-0.474135\pi$
$719$	3129.71	0.162334	0.0811672	+	0.996700i	$0.474135\pi$
$720$	0	0
$721$	−46010.5	−2.37659
$722$	0	0
$723$	−9805.99	−0.504410
$724$	0	0
$725$	913.623	0.0468015
$726$	0	0
$727$	−7184.40	−0.366513	−0.183256	−	0.983065i	$-0.558664\pi$
$727$	−7184.40	−0.366513	−0.183256	+	0.983065i	$0.558664\pi$
$728$	0	0
$729$	10723.4	0.544807
$730$	0	0
$731$	−31031.1	−1.57008
$732$	0	0
$733$	−37118.8	−1.87042	−0.935208	−	0.354099i	$-0.884788\pi$
$733$	−37118.8	−1.87042	−0.935208	+	0.354099i	$0.884788\pi$
$734$	0	0
$735$	−7808.94	−0.391887
$736$	0	0
$737$	−3524.63	−0.176162
$738$	0	0
$739$	6407.47	0.318948	0.159474	−	0.987202i	$-0.449020\pi$
$739$	6407.47	0.318948	0.159474	+	0.987202i	$0.449020\pi$
$740$	0	0
$741$	−1481.69	−0.0734567
$742$	0	0
$743$	−23437.3	−1.15724	−0.578620	−	0.815597i	$-0.696409\pi$
$743$	−23437.3	−1.15724	−0.578620	+	0.815597i	$0.696409\pi$
$744$	0	0
$745$	−5870.55	−0.288698
$746$	0	0
$747$	8027.95	0.393209
$748$	0	0
$749$	−35434.7	−1.72865
$750$	0	0
$751$	10463.7	0.508423	0.254211	−	0.967149i	$-0.418184\pi$
$751$	10463.7	0.508423	0.254211	+	0.967149i	$0.418184\pi$
$752$	0	0
$753$	−36499.0	−1.76640
$754$	0	0
$755$	−13665.1	−0.658705
$756$	0	0
$757$	−27823.5	−1.33588	−0.667941	−	0.744214i	$-0.732824\pi$
$757$	−27823.5	−1.33588	−0.667941	+	0.744214i	$0.732824\pi$
$758$	0	0
$759$	−3361.38	−0.160751
$760$	0	0
$761$	−30228.5	−1.43992	−0.719962	−	0.694013i	$-0.755841\pi$
$761$	−30228.5	−1.43992	−0.719962	+	0.694013i	$0.755841\pi$
$762$	0	0
$763$	45917.3	2.17866
$764$	0	0
$765$	−3200.97	−0.151283
$766$	0	0
$767$	−2651.43	−0.124821
$768$	0	0
$769$	3522.42	0.165178	0.0825889	−	0.996584i	$-0.473681\pi$
$769$	3522.42	0.165178	0.0825889	+	0.996584i	$0.473681\pi$
$770$	0	0
$771$	2657.97	0.124156
$772$	0	0
$773$	−508.724	−0.0236708	−0.0118354	−	0.999930i	$-0.503767\pi$
$773$	−508.724	−0.0236708	−0.0118354	+	0.999930i	$0.503767\pi$
$774$	0	0
$775$	−4660.94	−0.216033
$776$	0	0
$777$	47854.4	2.20948
$778$	0	0
$779$	−5125.87	−0.235755
$780$	0	0
$781$	−16237.7	−0.743955
$782$	0	0
$783$	4087.21	0.186545
$784$	0	0
$785$	−23.8616	−0.00108491
$786$	0	0
$787$	−30797.9	−1.39495	−0.697476	−	0.716608i	$-0.745694\pi$
$787$	−30797.9	−1.39495	−0.697476	+	0.716608i	$0.745694\pi$
$788$	0	0
$789$	37333.4	1.68454
$790$	0	0
$791$	9795.92	0.440332
$792$	0	0
$793$	6422.76	0.287615
$794$	0	0
$795$	240.678	0.0107371
$796$	0	0
$797$	11682.3	0.519209	0.259604	−	0.965715i	$-0.416408\pi$
$797$	11682.3	0.519209	0.259604	+	0.965715i	$0.416408\pi$
$798$	0	0
$799$	−19913.4	−0.881709
$800$	0	0
$801$	1432.90	0.0632071
$802$	0	0
$803$	−3570.24	−0.156900
$804$	0	0
$805$	−2832.13	−0.123999
$806$	0	0
$807$	12486.6	0.544671
$808$	0	0
$809$	−6661.27	−0.289490	−0.144745	−	0.989469i	$-0.546236\pi$
$809$	−6661.27	−0.289490	−0.144745	+	0.989469i	$0.546236\pi$
$810$	0	0
$811$	19875.6	0.860574	0.430287	−	0.902692i	$-0.358412\pi$
$811$	19875.6	0.860574	0.430287	+	0.902692i	$0.358412\pi$
$812$	0	0
$813$	−18474.3	−0.796952
$814$	0	0
$815$	−7617.56	−0.327400
$816$	0	0
$817$	6452.72	0.276318
$818$	0	0
$819$	−3057.21	−0.130437
$820$	0	0
$821$	−17128.2	−0.728111	−0.364055	−	0.931377i	$-0.618608\pi$
$821$	−17128.2	−0.728111	−0.364055	+	0.931377i	$0.618608\pi$
$822$	0	0
$823$	8.41573	0.000356445 0	0.000178222	−	1.00000i	$-0.499943\pi$
$823$	8.41573	0.000356445 0	0.000178222		1.00000i	$0.499943\pi$
$824$	0	0
$825$	−3653.67	−0.154187
$826$	0	0
$827$	−30880.0	−1.29843	−0.649215	−	0.760605i	$-0.724902\pi$
$827$	−30880.0	−1.29843	−0.649215	+	0.760605i	$0.724902\pi$
$828$	0	0
$829$	−10566.0	−0.442668	−0.221334	−	0.975198i	$-0.571041\pi$
$829$	−10566.0	−0.442668	−0.221334	+	0.975198i	$0.571041\pi$
$830$	0	0
$831$	−6981.31	−0.291431
$832$	0	0
$833$	−20747.5	−0.862974
$834$	0	0
$835$	−18019.1	−0.746797
$836$	0	0
$837$	−20851.3	−0.861083
$838$	0	0
$839$	−32747.3	−1.34751	−0.673755	−	0.738955i	$-0.735320\pi$
$839$	−32747.3	−1.34751	−0.673755	+	0.738955i	$0.735320\pi$
$840$	0	0
$841$	−23053.5	−0.945240
$842$	0	0
$843$	31059.9	1.26899
$844$	0	0
$845$	−9819.43	−0.399762
$846$	0	0
$847$	17805.9	0.722334
$848$	0	0
$849$	42452.6	1.71610
$850$	0	0
$851$	7540.35	0.303737
$852$	0	0
$853$	−37669.3	−1.51204	−0.756021	−	0.654547i	$-0.772859\pi$
$853$	−37669.3	−1.51204	−0.756021	+	0.654547i	$0.772859\pi$
$854$	0	0
$855$	665.621	0.0266243
$856$	0	0
$857$	18749.5	0.747340	0.373670	−	0.927562i	$-0.378099\pi$
$857$	18749.5	0.747340	0.373670	+	0.927562i	$0.378099\pi$
$858$	0	0
$859$	41715.6	1.65695	0.828475	−	0.560026i	$-0.189209\pi$
$859$	41715.6	1.65695	0.828475	+	0.560026i	$0.189209\pi$
$860$	0	0
$861$	−45697.8	−1.80880
$862$	0	0
$863$	−46493.1	−1.83389	−0.916943	−	0.399018i	$-0.869351\pi$
$863$	−46493.1	−1.83389	−0.916943	+	0.399018i	$0.869351\pi$
$864$	0	0
$865$	1749.97	0.0687869
$866$	0	0
$867$	−7626.53	−0.298744
$868$	0	0
$869$	12861.1	0.502050
$870$	0	0
$871$	−2182.49	−0.0849034
$872$	0	0
$873$	6253.23	0.242428
$874$	0	0
$875$	−3078.40	−0.118936
$876$	0	0
$877$	48214.7	1.85644	0.928218	−	0.372038i	$-0.121341\pi$
$877$	48214.7	1.85644	0.928218	+	0.372038i	$0.121341\pi$
$878$	0	0
$879$	44836.8	1.72049
$880$	0	0
$881$	8809.87	0.336903	0.168452	−	0.985710i	$-0.446123\pi$
$881$	8809.87	0.336903	0.168452	+	0.985710i	$0.446123\pi$
$882$	0	0
$883$	50886.4	1.93937	0.969684	−	0.244362i	$-0.0785785\pi$
$883$	50886.4	1.93937	0.969684	+	0.244362i	$0.0785785\pi$
$884$	0	0
$885$	5146.47	0.195476
$886$	0	0
$887$	−42296.0	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$887$	−42296.0	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$888$	0	0
$889$	33751.1	1.27332
$890$	0	0
$891$	−21758.1	−0.818097
$892$	0	0
$893$	4140.86	0.155172
$894$	0	0
$895$	6415.87	0.239619
$896$	0	0
$897$	−2081.40	−0.0774759
$898$	0	0
$899$	−6813.34	−0.252767
$900$	0	0
$901$	639.455	0.0236441
$902$	0	0
$903$	57526.8	2.12001
$904$	0	0
$905$	15382.0	0.564990
$906$	0	0
$907$	37554.9	1.37485	0.687426	−	0.726254i	$-0.258740\pi$
$907$	37554.9	1.37485	0.687426	+	0.726254i	$0.258740\pi$
$908$	0	0
$909$	−273.220	−0.00996933
$910$	0	0
$911$	−23951.7	−0.871080	−0.435540	−	0.900169i	$-0.643442\pi$
$911$	−23951.7	−0.871080	−0.435540	+	0.900169i	$0.643442\pi$
$912$	0	0
$913$	24345.8	0.882508
$914$	0	0
$915$	−12466.7	−0.450422
$916$	0	0
$917$	46573.6	1.67720
$918$	0	0
$919$	8909.66	0.319807	0.159904	−	0.987133i	$-0.448882\pi$
$919$	8909.66	0.319807	0.159904	+	0.987133i	$0.448882\pi$
$920$	0	0
$921$	33641.1	1.20360
$922$	0	0
$923$	−10054.5	−0.358558
$924$	0	0
$925$	8196.03	0.291334
$926$	0	0
$927$	15190.3	0.538205
$928$	0	0
$929$	−25002.6	−0.883003	−0.441501	−	0.897261i	$-0.645554\pi$
$929$	−25002.6	−0.883003	−0.441501	+	0.897261i	$0.645554\pi$
$930$	0	0
$931$	4314.30	0.151875
$932$	0	0
$933$	29246.1	1.02623
$934$	0	0
$935$	−9707.38	−0.339535
$936$	0	0
$937$	−18592.9	−0.648244	−0.324122	−	0.946015i	$-0.605069\pi$
$937$	−18592.9	−0.648244	−0.324122	+	0.946015i	$0.605069\pi$
$938$	0	0
$939$	26427.1	0.918440
$940$	0	0
$941$	−21625.8	−0.749183	−0.374591	−	0.927190i	$-0.622217\pi$
$941$	−21625.8	−0.749183	−0.374591	+	0.927190i	$0.622217\pi$
$942$	0	0
$943$	−7200.54	−0.248655
$944$	0	0
$945$	−13771.6	−0.474064
$946$	0	0
$947$	−4477.58	−0.153645	−0.0768224	−	0.997045i	$-0.524477\pi$
$947$	−4477.58	−0.153645	−0.0768224	+	0.997045i	$0.524477\pi$
$948$	0	0
$949$	−2210.73	−0.0756200
$950$	0	0
$951$	−24496.0	−0.835265
$952$	0	0
$953$	54260.8	1.84436	0.922182	−	0.386755i	$-0.126404\pi$
$953$	54260.8	1.84436	0.922182	+	0.386755i	$0.126404\pi$
$954$	0	0
$955$	8939.32	0.302900
$956$	0	0
$957$	−5340.92	−0.180405
$958$	0	0
$959$	−54825.4	−1.84610
$960$	0	0
$961$	4967.95	0.166760
$962$	0	0
$963$	11698.8	0.391472
$964$	0	0
$965$	−4407.53	−0.147029
$966$	0	0
$967$	26295.4	0.874462	0.437231	−	0.899349i	$-0.355959\pi$
$967$	26295.4	0.874462	0.437231	+	0.899349i	$0.355959\pi$
$968$	0	0
$969$	7641.18	0.253323
$970$	0	0
$971$	−21209.6	−0.700978	−0.350489	−	0.936567i	$-0.613985\pi$
$971$	−21209.6	−0.700978	−0.350489	+	0.936567i	$0.613985\pi$
$972$	0	0
$973$	−17228.1	−0.567632
$974$	0	0
$975$	−2262.39	−0.0743123
$976$	0	0
$977$	−607.767	−0.0199019	−0.00995097	−	0.999950i	$-0.503168\pi$
$977$	−607.767	−0.0199019	−0.00995097	+	0.999950i	$0.503168\pi$
$978$	0	0
$979$	4345.45	0.141860
$980$	0	0
$981$	−15159.6	−0.493382
$982$	0	0
$983$	−47859.5	−1.55288	−0.776440	−	0.630192i	$-0.782976\pi$
$983$	−47859.5	−1.55288	−0.776440	+	0.630192i	$0.782976\pi$
$984$	0	0
$985$	11086.4	0.358622
$986$	0	0
$987$	36916.3	1.19053
$988$	0	0
$989$	9064.41	0.291437
$990$	0	0
$991$	−34510.6	−1.10622	−0.553111	−	0.833108i	$-0.686559\pi$
$991$	−34510.6	−1.10622	−0.553111	+	0.833108i	$0.686559\pi$
$992$	0	0
$993$	61167.3	1.95477
$994$	0	0
$995$	1056.56	0.0336637
$996$	0	0
$997$	21519.8	0.683590	0.341795	−	0.939775i	$-0.388965\pi$
$997$	21519.8	0.683590	0.341795	+	0.939775i	$0.388965\pi$
$998$	0	0
$999$	36666.0	1.16122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.k.1.2		4
4.3	odd	2		230.4.a.j.1.3	✓	4
12.11	even	2		2070.4.a.bg.1.4		4
20.3	even	4		1150.4.b.o.599.3		8
20.7	even	4		1150.4.b.o.599.6		8
20.19	odd	2		1150.4.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.3	✓	4	4.3	odd	2
1150.4.a.n.1.2		4	20.19	odd	2
1150.4.b.o.599.3		8	20.3	even	4
1150.4.b.o.599.6		8	20.7	even	4
1840.4.a.k.1.2		4	1.1	even	1	trivial
2070.4.a.bg.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-5.92711 q^{3} +5.00000 q^{5} -24.6272 q^{7} +8.13066 q^{9} +O(q^{10})\)	\(q-5.92711 q^{3} +5.00000 q^{5} -24.6272 q^{7} +8.13066 q^{9} +24.6573 q^{11} +15.2681 q^{13} -29.6356 q^{15} -78.7383 q^{17} +16.3731 q^{19} +145.968 q^{21} +23.0000 q^{23} +25.0000 q^{25} +111.841 q^{27} +36.5449 q^{29} -186.438 q^{31} -146.147 q^{33} -123.136 q^{35} +327.841 q^{37} -90.4956 q^{39} -313.067 q^{41} +394.105 q^{43} +40.6533 q^{45} +252.906 q^{47} +263.499 q^{49} +466.691 q^{51} -8.12126 q^{53} +123.287 q^{55} -97.0452 q^{57} -173.659 q^{59} +420.666 q^{61} -200.236 q^{63} +76.3404 q^{65} -142.945 q^{67} -136.324 q^{69} -658.533 q^{71} -144.794 q^{73} -148.178 q^{75} -607.241 q^{77} +521.592 q^{79} -882.420 q^{81} +987.368 q^{83} -393.691 q^{85} -216.606 q^{87} +176.234 q^{89} -376.010 q^{91} +1105.04 q^{93} +81.8655 q^{95} +769.092 q^{97} +200.480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9}+O(q^{10}) \) 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9	\( 4 q - 14 q^{3} + 20 q^{5} - 8 q^{7} + 64 q^{9} - 21 q^{11} + 70 q^{13} - 70 q^{15} + 56 q^{17} - 173 q^{19} - 120 q^{21} + 92 q^{23} + 100 q^{25} - 389 q^{27} - 118 q^{29} - 17 q^{31} - 89 q^{33} - 40 q^{35} - 343 q^{37} + 221 q^{39} + 139 q^{41} + 50 q^{43} + 320 q^{45} - 367 q^{47} - 124 q^{49} - 439 q^{51} - 353 q^{53} - 105 q^{55} - 238 q^{57} + 453 q^{59} - 327 q^{61} + 1723 q^{63} + 350 q^{65} + 455 q^{67} - 322 q^{69} - 195 q^{71} - 633 q^{73} - 350 q^{75} - 2 q^{77} + 1140 q^{79} + 1456 q^{81} + 1199 q^{83} + 280 q^{85} + 1775 q^{87} - 2170 q^{89} - 557 q^{91} - 3241 q^{93} - 865 q^{95} - 703 q^{97} + 2745 q^{99}+O(q^{100}) \) 4 * q - 14 * q^3 + 20 * q^5 - 8 * q^7 + 64 * q^9 - 21 * q^11 + 70 * q^13 - 70 * q^15 + 56 * q^17 - 173 * q^19 - 120 * q^21 + 92 * q^23 + 100 * q^25 - 389 * q^27 - 118 * q^29 - 17 * q^31 - 89 * q^33 - 40 * q^35 - 343 * q^37 + 221 * q^39 + 139 * q^41 + 50 * q^43 + 320 * q^45 - 367 * q^47 - 124 * q^49 - 439 * q^51 - 353 * q^53 - 105 * q^55 - 238 * q^57 + 453 * q^59 - 327 * q^61 + 1723 * q^63 + 350 * q^65 + 455 * q^67 - 322 * q^69 - 195 * q^71 - 633 * q^73 - 350 * q^75 - 2 * q^77 + 1140 * q^79 + 1456 * q^81 + 1199 * q^83 + 280 * q^85 + 1775 * q^87 - 2170 * q^89 - 557 * q^91 - 3241 * q^93 - 865 * q^95 - 703 * q^97 + 2745 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more