LMFDB - Embedded newform 1840.4.a.x.1.7

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 123x^{6} + 335x^{5} + 4492x^{4} - 7035x^{3} - 45582x^{2} + 36684x + 124632 $ x^8 - 4x^7 - 123x^6 + 335x^5 + 4492x^4 - 7035x^3 - 45582x^2 + 36684*x + 124632
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-6.18739$ 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)

$f(q)$	$=$	$q+8.18739 q^{3} +5.00000 q^{5} +31.6165 q^{7} +40.0333 q^{9} +O(q^{10})$	$q+8.18739 q^{3} +5.00000 q^{5} +31.6165 q^{7} +40.0333 q^{9} -33.9339 q^{11} -21.5777 q^{13} +40.9369 q^{15} -18.8750 q^{17} +1.52649 q^{19} +258.856 q^{21} +23.0000 q^{23} +25.0000 q^{25} +106.709 q^{27} +272.128 q^{29} -108.099 q^{31} -277.830 q^{33} +158.082 q^{35} +77.0846 q^{37} -176.665 q^{39} -33.1186 q^{41} +273.096 q^{43} +200.167 q^{45} -40.3157 q^{47} +656.601 q^{49} -154.537 q^{51} +329.041 q^{53} -169.669 q^{55} +12.4979 q^{57} +379.434 q^{59} +595.196 q^{61} +1265.71 q^{63} -107.889 q^{65} -458.623 q^{67} +188.310 q^{69} +984.401 q^{71} +663.378 q^{73} +204.685 q^{75} -1072.87 q^{77} +1036.18 q^{79} -207.234 q^{81} -110.647 q^{83} -94.3751 q^{85} +2228.02 q^{87} +943.469 q^{89} -682.211 q^{91} -885.051 q^{93} +7.63243 q^{95} -958.646 q^{97} -1358.48 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) $ 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9	$ 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) $ 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9 + 35 * q^11 - 54 * q^13 + 60 * q^15 - 11 * q^17 + 107 * q^19 - 148 * q^21 + 184 * q^23 + 200 * q^25 + 405 * q^27 + 37 * q^29 + 290 * q^31 + 289 * q^33 + 155 * q^35 - 172 * q^37 + 311 * q^39 + 308 * q^41 + 538 * q^43 + 310 * q^45 + 1035 * q^47 + 585 * q^49 + 387 * q^51 + 46 * q^53 + 175 * q^55 + 286 * q^57 + 1256 * q^59 + 399 * q^61 + 1234 * q^63 - 270 * q^65 + 1598 * q^67 + 276 * q^69 + 750 * q^71 + 177 * q^73 + 300 * q^75 - 1228 * q^77 + 1292 * q^79 + 380 * q^81 + 2094 * q^83 - 55 * q^85 + 2245 * q^87 + 484 * q^89 + 679 * q^91 - 2111 * q^93 + 535 * q^95 - 2225 * q^97 + 2117 * 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
$3$	8.18739	1.57566	0.787832	−	0.615891i	$-0.211204\pi$
$3$	8.18739	1.57566	0.787832	+	0.615891i	$0.211204\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	31.6165	1.70713	0.853564	−	0.520988i	$-0.174436\pi$
$7$	31.6165	1.70713	0.853564	+	0.520988i	$0.174436\pi$
$8$	0	0
$9$	40.0333	1.48271
$10$	0	0
$11$	−33.9339	−0.930131	−0.465066	−	0.885276i	$-0.653969\pi$
$11$	−33.9339	−0.930131	−0.465066	+	0.885276i	$0.653969\pi$
$12$	0	0
$13$	−21.5777	−0.460352	−0.230176	−	0.973149i	$-0.573930\pi$
$13$	−21.5777	−0.460352	−0.230176	+	0.973149i	$0.573930\pi$
$14$	0	0
$15$	40.9369	0.704658
$16$	0	0
$17$	−18.8750	−0.269286	−0.134643	−	0.990894i	$-0.542989\pi$
$17$	−18.8750	−0.269286	−0.134643	+	0.990894i	$0.542989\pi$
$18$	0	0
$19$	1.52649	0.0184316	0.00921579	−	0.999958i	$-0.497066\pi$
$19$	1.52649	0.0184316	0.00921579	+	0.999958i	$0.497066\pi$
$20$	0	0
$21$	258.856	2.68986
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	106.709	0.760596
$28$	0	0
$29$	272.128	1.74251	0.871256	−	0.490828i	$-0.163306\pi$
$29$	272.128	1.74251	0.871256	+	0.490828i	$0.163306\pi$
$30$	0	0
$31$	−108.099	−0.626297	−0.313148	−	0.949704i	$-0.601384\pi$
$31$	−108.099	−0.626297	−0.313148	+	0.949704i	$0.601384\pi$
$32$	0	0
$33$	−277.830	−1.46557
$34$	0	0
$35$	158.082	0.763451
$36$	0	0
$37$	77.0846	0.342504	0.171252	−	0.985227i	$-0.445219\pi$
$37$	77.0846	0.342504	0.171252	+	0.985227i	$0.445219\pi$
$38$	0	0
$39$	−176.665	−0.725360
$40$	0	0
$41$	−33.1186	−0.126152	−0.0630762	−	0.998009i	$-0.520091\pi$
$41$	−33.1186	−0.126152	−0.0630762	+	0.998009i	$0.520091\pi$
$42$	0	0
$43$	273.096	0.968528	0.484264	−	0.874922i	$-0.339088\pi$
$43$	273.096	0.968528	0.484264	+	0.874922i	$0.339088\pi$
$44$	0	0
$45$	200.167	0.663090
$46$	0	0
$47$	−40.3157	−0.125120	−0.0625600	−	0.998041i	$-0.519926\pi$
$47$	−40.3157	−0.125120	−0.0625600	+	0.998041i	$0.519926\pi$
$48$	0	0
$49$	656.601	1.91429
$50$	0	0
$51$	−154.537	−0.424304
$52$	0	0
$53$	329.041	0.852778	0.426389	−	0.904540i	$-0.359785\pi$
$53$	329.041	0.852778	0.426389	+	0.904540i	$0.359785\pi$
$54$	0	0
$55$	−169.669	−0.415967
$56$	0	0
$57$	12.4979	0.0290420
$58$	0	0
$59$	379.434	0.837257	0.418628	−	0.908158i	$-0.362511\pi$
$59$	379.434	0.837257	0.418628	+	0.908158i	$0.362511\pi$
$60$	0	0
$61$	595.196	1.24929	0.624647	−	0.780907i	$-0.285243\pi$
$61$	595.196	1.24929	0.624647	+	0.780907i	$0.285243\pi$
$62$	0	0
$63$	1265.71	2.53118
$64$	0	0
$65$	−107.889	−0.205876
$66$	0	0
$67$	−458.623	−0.836264	−0.418132	−	0.908386i	$-0.637315\pi$
$67$	−458.623	−0.836264	−0.418132	+	0.908386i	$0.637315\pi$
$68$	0	0
$69$	188.310	0.328549
$70$	0	0
$71$	984.401	1.64545	0.822725	−	0.568440i	$-0.192453\pi$
$71$	984.401	1.64545	0.822725	+	0.568440i	$0.192453\pi$
$72$	0	0
$73$	663.378	1.06360	0.531798	−	0.846871i	$-0.321517\pi$
$73$	663.378	1.06360	0.531798	+	0.846871i	$0.321517\pi$
$74$	0	0
$75$	204.685	0.315133
$76$	0	0
$77$	−1072.87	−1.58785
$78$	0	0
$79$	1036.18	1.47568	0.737841	−	0.674974i	$-0.235845\pi$
$79$	1036.18	1.47568	0.737841	+	0.674974i	$0.235845\pi$
$80$	0	0
$81$	−207.234	−0.284272
$82$	0	0
$83$	−110.647	−0.146327	−0.0731635	−	0.997320i	$-0.523309\pi$
$83$	−110.647	−0.146327	−0.0731635	+	0.997320i	$0.523309\pi$
$84$	0	0
$85$	−94.3751	−0.120428
$86$	0	0
$87$	2228.02	2.74561
$88$	0	0
$89$	943.469	1.12368	0.561840	−	0.827246i	$-0.310094\pi$
$89$	943.469	1.12368	0.561840	+	0.827246i	$0.310094\pi$
$90$	0	0
$91$	−682.211	−0.785880
$92$	0	0
$93$	−885.051	−0.986833
$94$	0	0
$95$	7.63243	0.00824285
$96$	0	0
$97$	−958.646	−1.00346	−0.501731	−	0.865024i	$-0.667303\pi$
$97$	−958.646	−1.00346	−0.501731	+	0.865024i	$0.667303\pi$
$98$	0	0
$99$	−1358.48	−1.37912
$100$	0	0
$101$	−725.798	−0.715045	−0.357523	−	0.933904i	$-0.616378\pi$
$101$	−725.798	−0.715045	−0.357523	+	0.933904i	$0.616378\pi$
$102$	0	0
$103$	−65.9650	−0.0631041	−0.0315520	−	0.999502i	$-0.510045\pi$
$103$	−65.9650	−0.0631041	−0.0315520	+	0.999502i	$0.510045\pi$
$104$	0	0
$105$	1294.28	1.20294
$106$	0	0
$107$	1068.74	0.965594	0.482797	−	0.875732i	$-0.339621\pi$
$107$	1068.74	0.965594	0.482797	+	0.875732i	$0.339621\pi$
$108$	0	0
$109$	−1282.23	−1.12674	−0.563372	−	0.826204i	$-0.690496\pi$
$109$	−1282.23	−1.12674	−0.563372	+	0.826204i	$0.690496\pi$
$110$	0	0
$111$	631.122	0.539670
$112$	0	0
$113$	−1165.15	−0.969980	−0.484990	−	0.874520i	$-0.661177\pi$
$113$	−1165.15	−0.969980	−0.484990	+	0.874520i	$0.661177\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−863.827	−0.682571
$118$	0	0
$119$	−596.761	−0.459706
$120$	0	0
$121$	−179.493	−0.134856
$122$	0	0
$123$	−271.154	−0.198774
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1493.76	−1.04370	−0.521850	−	0.853037i	$-0.674758\pi$
$127$	−1493.76	−1.04370	−0.521850	+	0.853037i	$0.674758\pi$
$128$	0	0
$129$	2235.94	1.52607
$130$	0	0
$131$	−1346.23	−0.897866	−0.448933	−	0.893565i	$-0.648196\pi$
$131$	−1346.23	−0.897866	−0.448933	+	0.893565i	$0.648196\pi$
$132$	0	0
$133$	48.2621	0.0314651
$134$	0	0
$135$	533.543	0.340149
$136$	0	0
$137$	−593.680	−0.370230	−0.185115	−	0.982717i	$-0.559266\pi$
$137$	−593.680	−0.370230	−0.185115	+	0.982717i	$0.559266\pi$
$138$	0	0
$139$	318.764	0.194512	0.0972562	−	0.995259i	$-0.468993\pi$
$139$	318.764	0.194512	0.0972562	+	0.995259i	$0.468993\pi$
$140$	0	0
$141$	−330.080	−0.197147
$142$	0	0
$143$	732.215	0.428188
$144$	0	0
$145$	1360.64	0.779275
$146$	0	0
$147$	5375.84	3.01627
$148$	0	0
$149$	−3180.23	−1.74856	−0.874278	−	0.485426i	$-0.838665\pi$
$149$	−3180.23	−1.74856	−0.874278	+	0.485426i	$0.838665\pi$
$150$	0	0
$151$	−1207.77	−0.650906	−0.325453	−	0.945558i	$-0.605517\pi$
$151$	−1207.77	−0.650906	−0.325453	+	0.945558i	$0.605517\pi$
$152$	0	0
$153$	−755.629	−0.399275
$154$	0	0
$155$	−540.496	−0.280089
$156$	0	0
$157$	1521.10	0.773230	0.386615	−	0.922241i	$-0.373644\pi$
$157$	1521.10	0.773230	0.386615	+	0.922241i	$0.373644\pi$
$158$	0	0
$159$	2693.99	1.34369
$160$	0	0
$161$	727.179	0.355961
$162$	0	0
$163$	1894.79	0.910500	0.455250	−	0.890364i	$-0.349550\pi$
$163$	1894.79	0.910500	0.455250	+	0.890364i	$0.349550\pi$
$164$	0	0
$165$	−1389.15	−0.655425
$166$	0	0
$167$	3223.64	1.49373	0.746864	−	0.664977i	$-0.231559\pi$
$167$	3223.64	1.49373	0.746864	+	0.664977i	$0.231559\pi$
$168$	0	0
$169$	−1731.40	−0.788076
$170$	0	0
$171$	61.1103	0.0273288
$172$	0	0
$173$	−4278.09	−1.88010	−0.940049	−	0.341039i	$-0.889221\pi$
$173$	−4278.09	−1.88010	−0.940049	+	0.341039i	$0.889221\pi$
$174$	0	0
$175$	790.412	0.341426
$176$	0	0
$177$	3106.58	1.31923
$178$	0	0
$179$	3881.76	1.62087	0.810437	−	0.585826i	$-0.199230\pi$
$179$	3881.76	1.62087	0.810437	+	0.585826i	$0.199230\pi$
$180$	0	0
$181$	−1626.90	−0.668103	−0.334051	−	0.942555i	$-0.608416\pi$
$181$	−1626.90	−0.668103	−0.334051	+	0.942555i	$0.608416\pi$
$182$	0	0
$183$	4873.10	1.96847
$184$	0	0
$185$	385.423	0.153172
$186$	0	0
$187$	640.502	0.250471
$188$	0	0
$189$	3373.75	1.29844
$190$	0	0
$191$	−1354.77	−0.513232	−0.256616	−	0.966513i	$-0.582608\pi$
$191$	−1354.77	−0.513232	−0.256616	+	0.966513i	$0.582608\pi$
$192$	0	0
$193$	−613.805	−0.228926	−0.114463	−	0.993428i	$-0.536515\pi$
$193$	−613.805	−0.228926	−0.114463	+	0.993428i	$0.536515\pi$
$194$	0	0
$195$	−883.325	−0.324391
$196$	0	0
$197$	−2156.27	−0.779838	−0.389919	−	0.920849i	$-0.627497\pi$
$197$	−2156.27	−0.779838	−0.389919	+	0.920849i	$0.627497\pi$
$198$	0	0
$199$	−1650.01	−0.587769	−0.293885	−	0.955841i	$-0.594948\pi$
$199$	−1650.01	−0.587769	−0.293885	+	0.955841i	$0.594948\pi$
$200$	0	0
$201$	−3754.92	−1.31767
$202$	0	0
$203$	8603.72	2.97469
$204$	0	0
$205$	−165.593	−0.0564171
$206$	0	0
$207$	920.766	0.309167
$208$	0	0
$209$	−51.7996	−0.0171438
$210$	0	0
$211$	−4228.73	−1.37971	−0.689853	−	0.723950i	$-0.742325\pi$
$211$	−4228.73	−1.37971	−0.689853	+	0.723950i	$0.742325\pi$
$212$	0	0
$213$	8059.67	2.59267
$214$	0	0
$215$	1365.48	0.433139
$216$	0	0
$217$	−3417.72	−1.06917
$218$	0	0
$219$	5431.33	1.67587
$220$	0	0
$221$	407.280	0.123966
$222$	0	0
$223$	−1856.24	−0.557413	−0.278707	−	0.960376i	$-0.589906\pi$
$223$	−1856.24	−0.557413	−0.278707	+	0.960376i	$0.589906\pi$
$224$	0	0
$225$	1000.83	0.296543
$226$	0	0
$227$	−5981.60	−1.74895	−0.874477	−	0.485067i	$-0.838795\pi$
$227$	−5981.60	−1.74895	−0.874477	+	0.485067i	$0.838795\pi$
$228$	0	0
$229$	−2513.75	−0.725386	−0.362693	−	0.931909i	$-0.618143\pi$
$229$	−2513.75	−0.725386	−0.362693	+	0.931909i	$0.618143\pi$
$230$	0	0
$231$	−8783.99	−2.50192
$232$	0	0
$233$	3672.03	1.03246	0.516230	−	0.856450i	$-0.327335\pi$
$233$	3672.03	1.03246	0.516230	+	0.856450i	$0.327335\pi$
$234$	0	0
$235$	−201.578	−0.0559554
$236$	0	0
$237$	8483.58	2.32518
$238$	0	0
$239$	3998.60	1.08221	0.541104	−	0.840956i	$-0.318007\pi$
$239$	3998.60	1.08221	0.541104	+	0.840956i	$0.318007\pi$
$240$	0	0
$241$	347.299	0.0928278	0.0464139	−	0.998922i	$-0.485221\pi$
$241$	347.299	0.0928278	0.0464139	+	0.998922i	$0.485221\pi$
$242$	0	0
$243$	−4577.84	−1.20851
$244$	0	0
$245$	3283.00	0.856095
$246$	0	0
$247$	−32.9381	−0.00848502
$248$	0	0
$249$	−905.913	−0.230562
$250$	0	0
$251$	2739.43	0.688890	0.344445	−	0.938806i	$-0.388067\pi$
$251$	2739.43	0.688890	0.344445	+	0.938806i	$0.388067\pi$
$252$	0	0
$253$	−780.479	−0.193946
$254$	0	0
$255$	−772.685	−0.189755
$256$	0	0
$257$	5841.04	1.41772	0.708860	−	0.705349i	$-0.249210\pi$
$257$	5841.04	1.41772	0.708860	+	0.705349i	$0.249210\pi$
$258$	0	0
$259$	2437.14	0.584698
$260$	0	0
$261$	10894.2	2.58365
$262$	0	0
$263$	−3013.06	−0.706438	−0.353219	−	0.935541i	$-0.614913\pi$
$263$	−3013.06	−0.706438	−0.353219	+	0.935541i	$0.614913\pi$
$264$	0	0
$265$	1645.21	0.381374
$266$	0	0
$267$	7724.55	1.77054
$268$	0	0
$269$	−4817.00	−1.09181	−0.545906	−	0.837846i	$-0.683815\pi$
$269$	−4817.00	−1.09181	−0.545906	+	0.837846i	$0.683815\pi$
$270$	0	0
$271$	−5103.25	−1.14391	−0.571957	−	0.820284i	$-0.693816\pi$
$271$	−5103.25	−1.14391	−0.571957	+	0.820284i	$0.693816\pi$
$272$	0	0
$273$	−5585.52	−1.23828
$274$	0	0
$275$	−848.347	−0.186026
$276$	0	0
$277$	−2754.04	−0.597380	−0.298690	−	0.954350i	$-0.596550\pi$
$277$	−2754.04	−0.597380	−0.298690	+	0.954350i	$0.596550\pi$
$278$	0	0
$279$	−4327.57	−0.928620
$280$	0	0
$281$	2720.75	0.577602	0.288801	−	0.957389i	$-0.406743\pi$
$281$	2720.75	0.577602	0.288801	+	0.957389i	$0.406743\pi$
$282$	0	0
$283$	−4452.02	−0.935142	−0.467571	−	0.883955i	$-0.654871\pi$
$283$	−4452.02	−0.935142	−0.467571	+	0.883955i	$0.654871\pi$
$284$	0	0
$285$	62.4897	0.0129880
$286$	0	0
$287$	−1047.09	−0.215358
$288$	0	0
$289$	−4556.73	−0.927485
$290$	0	0
$291$	−7848.80	−1.58112
$292$	0	0
$293$	5725.65	1.14162	0.570812	−	0.821080i	$-0.306628\pi$
$293$	5725.65	1.14162	0.570812	+	0.821080i	$0.306628\pi$
$294$	0	0
$295$	1897.17	0.374433
$296$	0	0
$297$	−3621.04	−0.707454
$298$	0	0
$299$	−496.287	−0.0959901
$300$	0	0
$301$	8634.32	1.65340
$302$	0	0
$303$	−5942.39	−1.12667
$304$	0	0
$305$	2975.98	0.558702
$306$	0	0
$307$	−4086.88	−0.759773	−0.379887	−	0.925033i	$-0.624037\pi$
$307$	−4086.88	−0.759773	−0.379887	+	0.925033i	$0.624037\pi$
$308$	0	0
$309$	−540.081	−0.0994308
$310$	0	0
$311$	−5179.81	−0.944438	−0.472219	−	0.881481i	$-0.656547\pi$
$311$	−5179.81	−0.944438	−0.472219	+	0.881481i	$0.656547\pi$
$312$	0	0
$313$	−3564.80	−0.643753	−0.321876	−	0.946782i	$-0.604314\pi$
$313$	−3564.80	−0.643753	−0.321876	+	0.946782i	$0.604314\pi$
$314$	0	0
$315$	6328.56	1.13198
$316$	0	0
$317$	−3943.43	−0.698691	−0.349345	−	0.936994i	$-0.613596\pi$
$317$	−3943.43	−0.698691	−0.349345	+	0.936994i	$0.613596\pi$
$318$	0	0
$319$	−9234.35	−1.62077
$320$	0	0
$321$	8750.15	1.52145
$322$	0	0
$323$	−28.8125	−0.00496337
$324$	0	0
$325$	−539.443	−0.0920705
$326$	0	0
$327$	−10498.1	−1.77537
$328$	0	0
$329$	−1274.64	−0.213596
$330$	0	0
$331$	7165.30	1.18985	0.594926	−	0.803781i	$-0.297182\pi$
$331$	7165.30	1.18985	0.594926	+	0.803781i	$0.297182\pi$
$332$	0	0
$333$	3085.95	0.507835
$334$	0	0
$335$	−2293.11	−0.373989
$336$	0	0
$337$	7406.74	1.19724	0.598621	−	0.801032i	$-0.295716\pi$
$337$	7406.74	1.19724	0.598621	+	0.801032i	$0.295716\pi$
$338$	0	0
$339$	−9539.51	−1.52836
$340$	0	0
$341$	3668.23	0.582538
$342$	0	0
$343$	9914.94	1.56081
$344$	0	0
$345$	941.549	0.146931
$346$	0	0
$347$	9184.74	1.42093	0.710465	−	0.703733i	$-0.248485\pi$
$347$	9184.74	1.42093	0.710465	+	0.703733i	$0.248485\pi$
$348$	0	0
$349$	−6425.23	−0.985486	−0.492743	−	0.870175i	$-0.664006\pi$
$349$	−6425.23	−0.985486	−0.492743	+	0.870175i	$0.664006\pi$
$350$	0	0
$351$	−2302.53	−0.350142
$352$	0	0
$353$	1402.90	0.211526	0.105763	−	0.994391i	$-0.466271\pi$
$353$	1402.90	0.211526	0.105763	+	0.994391i	$0.466271\pi$
$354$	0	0
$355$	4922.01	0.735867
$356$	0	0
$357$	−4885.91	−0.724342
$358$	0	0
$359$	681.988	0.100262	0.0501309	−	0.998743i	$-0.484036\pi$
$359$	681.988	0.100262	0.0501309	+	0.998743i	$0.484036\pi$
$360$	0	0
$361$	−6856.67	−0.999660
$362$	0	0
$363$	−1469.58	−0.212487
$364$	0	0
$365$	3316.89	0.475654
$366$	0	0
$367$	4058.06	0.577191	0.288595	−	0.957451i	$-0.406812\pi$
$367$	4058.06	0.577191	0.288595	+	0.957451i	$0.406812\pi$
$368$	0	0
$369$	−1325.85	−0.187048
$370$	0	0
$371$	10403.1	1.45580
$372$	0	0
$373$	−7006.28	−0.972577	−0.486289	−	0.873798i	$-0.661650\pi$
$373$	−7006.28	−0.972577	−0.486289	+	0.873798i	$0.661650\pi$
$374$	0	0
$375$	1023.42	0.140932
$376$	0	0
$377$	−5871.90	−0.802170
$378$	0	0
$379$	−4715.30	−0.639074	−0.319537	−	0.947574i	$-0.603527\pi$
$379$	−4715.30	−0.639074	−0.319537	+	0.947574i	$0.603527\pi$
$380$	0	0
$381$	−12230.0	−1.64452
$382$	0	0
$383$	915.726	0.122171	0.0610854	−	0.998133i	$-0.480544\pi$
$383$	915.726	0.122171	0.0610854	+	0.998133i	$0.480544\pi$
$384$	0	0
$385$	−5364.34	−0.710110
$386$	0	0
$387$	10932.9	1.43605
$388$	0	0
$389$	−11750.6	−1.53156	−0.765782	−	0.643101i	$-0.777648\pi$
$389$	−11750.6	−1.53156	−0.765782	+	0.643101i	$0.777648\pi$
$390$	0	0
$391$	−434.125	−0.0561500
$392$	0	0
$393$	−11022.1	−1.41473
$394$	0	0
$395$	5180.88	0.659945
$396$	0	0
$397$	8231.09	1.04057	0.520285	−	0.853993i	$-0.325826\pi$
$397$	8231.09	1.04057	0.520285	+	0.853993i	$0.325826\pi$
$398$	0	0
$399$	395.141	0.0495784
$400$	0	0
$401$	15701.0	1.95529	0.977644	−	0.210266i	$-0.0674330\pi$
$401$	15701.0	1.95529	0.977644	+	0.210266i	$0.0674330\pi$
$402$	0	0
$403$	2332.53	0.288317
$404$	0	0
$405$	−1036.17	−0.127130
$406$	0	0
$407$	−2615.78	−0.318573
$408$	0	0
$409$	−3306.34	−0.399726	−0.199863	−	0.979824i	$-0.564050\pi$
$409$	−3306.34	−0.399726	−0.199863	+	0.979824i	$0.564050\pi$
$410$	0	0
$411$	−4860.69	−0.583358
$412$	0	0
$413$	11996.4	1.42930
$414$	0	0
$415$	−553.237	−0.0654394
$416$	0	0
$417$	2609.84	0.306486
$418$	0	0
$419$	−3779.53	−0.440673	−0.220336	−	0.975424i	$-0.570716\pi$
$419$	−3779.53	−0.440673	−0.220336	+	0.975424i	$0.570716\pi$
$420$	0	0
$421$	12150.5	1.40660	0.703299	−	0.710894i	$-0.251709\pi$
$421$	12150.5	1.40660	0.703299	+	0.710894i	$0.251709\pi$
$422$	0	0
$423$	−1613.97	−0.185517
$424$	0	0
$425$	−471.875	−0.0538572
$426$	0	0
$427$	18818.0	2.13271
$428$	0	0
$429$	5994.93	0.674680
$430$	0	0
$431$	−2300.61	−0.257115	−0.128557	−	0.991702i	$-0.541035\pi$
$431$	−2300.61	−0.257115	−0.128557	+	0.991702i	$0.541035\pi$
$432$	0	0
$433$	−12293.4	−1.36440	−0.682198	−	0.731167i	$-0.738976\pi$
$433$	−12293.4	−1.36440	−0.682198	+	0.731167i	$0.738976\pi$
$434$	0	0
$435$	11140.1	1.22788
$436$	0	0
$437$	35.1092	0.00384325
$438$	0	0
$439$	−9092.93	−0.988569	−0.494285	−	0.869300i	$-0.664570\pi$
$439$	−9092.93	−0.988569	−0.494285	+	0.869300i	$0.664570\pi$
$440$	0	0
$441$	26285.9	2.83834
$442$	0	0
$443$	21.4741	0.00230308	0.00115154	−	0.999999i	$-0.499633\pi$
$443$	21.4741	0.00230308	0.00115154	+	0.999999i	$0.499633\pi$
$444$	0	0
$445$	4717.35	0.502525
$446$	0	0
$447$	−26037.8	−2.75514
$448$	0	0
$449$	7825.38	0.822500	0.411250	−	0.911523i	$-0.365092\pi$
$449$	7825.38	0.822500	0.411250	+	0.911523i	$0.365092\pi$
$450$	0	0
$451$	1123.84	0.117338
$452$	0	0
$453$	−9888.46	−1.02561
$454$	0	0
$455$	−3411.05	−0.351456
$456$	0	0
$457$	−434.494	−0.0444743	−0.0222372	−	0.999753i	$-0.507079\pi$
$457$	−434.494	−0.0444743	−0.0222372	+	0.999753i	$0.507079\pi$
$458$	0	0
$459$	−2014.13	−0.204818
$460$	0	0
$461$	13343.7	1.34811	0.674056	−	0.738680i	$-0.264550\pi$
$461$	13343.7	1.34811	0.674056	+	0.738680i	$0.264550\pi$
$462$	0	0
$463$	12611.6	1.26590	0.632949	−	0.774194i	$-0.281844\pi$
$463$	12611.6	1.26590	0.632949	+	0.774194i	$0.281844\pi$
$464$	0	0
$465$	−4425.25	−0.441325
$466$	0	0
$467$	850.770	0.0843018	0.0421509	−	0.999111i	$-0.486579\pi$
$467$	850.770	0.0843018	0.0421509	+	0.999111i	$0.486579\pi$
$468$	0	0
$469$	−14500.0	−1.42761
$470$	0	0
$471$	12453.8	1.21835
$472$	0	0
$473$	−9267.19	−0.900858
$474$	0	0
$475$	38.1622	0.00368632
$476$	0	0
$477$	13172.6	1.26443
$478$	0	0
$479$	−6212.13	−0.592567	−0.296284	−	0.955100i	$-0.595747\pi$
$479$	−6212.13	−0.592567	−0.296284	+	0.955100i	$0.595747\pi$
$480$	0	0
$481$	−1663.31	−0.157672
$482$	0	0
$483$	5953.69	0.560874
$484$	0	0
$485$	−4793.23	−0.448761
$486$	0	0
$487$	14681.4	1.36607	0.683036	−	0.730385i	$-0.260659\pi$
$487$	14681.4	1.36607	0.683036	+	0.730385i	$0.260659\pi$
$488$	0	0
$489$	15513.4	1.43464
$490$	0	0
$491$	7022.58	0.645467	0.322734	−	0.946490i	$-0.395398\pi$
$491$	7022.58	0.645467	0.322734	+	0.946490i	$0.395398\pi$
$492$	0	0
$493$	−5136.42	−0.469234
$494$	0	0
$495$	−6792.42	−0.616761
$496$	0	0
$497$	31123.3	2.80899
$498$	0	0
$499$	1538.48	0.138019	0.0690097	−	0.997616i	$-0.478016\pi$
$499$	1538.48	0.138019	0.0690097	+	0.997616i	$0.478016\pi$
$500$	0	0
$501$	26393.2	2.35361
$502$	0	0
$503$	10143.3	0.899141	0.449570	−	0.893245i	$-0.351577\pi$
$503$	10143.3	0.899141	0.449570	+	0.893245i	$0.351577\pi$
$504$	0	0
$505$	−3628.99	−0.319778
$506$	0	0
$507$	−14175.7	−1.24174
$508$	0	0
$509$	−17572.1	−1.53020	−0.765099	−	0.643912i	$-0.777310\pi$
$509$	−17572.1	−1.53020	−0.765099	+	0.643912i	$0.777310\pi$
$510$	0	0
$511$	20973.7	1.81569
$512$	0	0
$513$	162.889	0.0140190
$514$	0	0
$515$	−329.825	−0.0282210
$516$	0	0
$517$	1368.07	0.116378
$518$	0	0
$519$	−35026.4	−2.96240
$520$	0	0
$521$	−1904.97	−0.160188	−0.0800941	−	0.996787i	$-0.525522\pi$
$521$	−1904.97	−0.160188	−0.0800941	+	0.996787i	$0.525522\pi$
$522$	0	0
$523$	11132.9	0.930798	0.465399	−	0.885101i	$-0.345911\pi$
$523$	11132.9	0.930798	0.465399	+	0.885101i	$0.345911\pi$
$524$	0	0
$525$	6471.40	0.537972
$526$	0	0
$527$	2040.38	0.168653
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	15190.0	1.24141
$532$	0	0
$533$	714.623	0.0580746
$534$	0	0
$535$	5343.68	0.431827
$536$	0	0
$537$	31781.5	2.55395
$538$	0	0
$539$	−22281.0	−1.78054
$540$	0	0
$541$	−10655.5	−0.846793	−0.423396	−	0.905945i	$-0.639162\pi$
$541$	−10655.5	−0.846793	−0.423396	+	0.905945i	$0.639162\pi$
$542$	0	0
$543$	−13320.1	−1.05270
$544$	0	0
$545$	−6411.13	−0.503895
$546$	0	0
$547$	6297.15	0.492224	0.246112	−	0.969241i	$-0.420847\pi$
$547$	6297.15	0.492224	0.246112	+	0.969241i	$0.420847\pi$
$548$	0	0
$549$	23827.6	1.85235
$550$	0	0
$551$	415.400	0.0321173
$552$	0	0
$553$	32760.2	2.51918
$554$	0	0
$555$	3155.61	0.241348
$556$	0	0
$557$	−23394.7	−1.77965	−0.889825	−	0.456302i	$-0.849174\pi$
$557$	−23394.7	−1.77965	−0.889825	+	0.456302i	$0.849174\pi$
$558$	0	0
$559$	−5892.78	−0.445864
$560$	0	0
$561$	5244.04	0.394659
$562$	0	0
$563$	7126.24	0.533455	0.266728	−	0.963772i	$-0.414058\pi$
$563$	7126.24	0.533455	0.266728	+	0.963772i	$0.414058\pi$
$564$	0	0
$565$	−5825.73	−0.433788
$566$	0	0
$567$	−6552.00	−0.485288
$568$	0	0
$569$	−15010.5	−1.10593	−0.552965	−	0.833205i	$-0.686504\pi$
$569$	−15010.5	−1.10593	−0.552965	+	0.833205i	$0.686504\pi$
$570$	0	0
$571$	5591.08	0.409771	0.204886	−	0.978786i	$-0.434318\pi$
$571$	5591.08	0.409771	0.204886	+	0.978786i	$0.434318\pi$
$572$	0	0
$573$	−11092.0	−0.808681
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	6460.97	0.466159	0.233079	−	0.972458i	$-0.425120\pi$
$577$	6460.97	0.466159	0.233079	+	0.972458i	$0.425120\pi$
$578$	0	0
$579$	−5025.46	−0.360710
$580$	0	0
$581$	−3498.28	−0.249799
$582$	0	0
$583$	−11165.6	−0.793196
$584$	0	0
$585$	−4319.13	−0.305255
$586$	0	0
$587$	26767.2	1.88211	0.941057	−	0.338248i	$-0.109834\pi$
$587$	26767.2	1.88211	0.941057	+	0.338248i	$0.109834\pi$
$588$	0	0
$589$	−165.012	−0.0115436
$590$	0	0
$591$	−17654.2	−1.22876
$592$	0	0
$593$	3397.27	0.235260	0.117630	−	0.993057i	$-0.462470\pi$
$593$	3397.27	0.235260	0.117630	+	0.993057i	$0.462470\pi$
$594$	0	0
$595$	−2983.81	−0.205587
$596$	0	0
$597$	−13509.3	−0.926126
$598$	0	0
$599$	−8598.39	−0.586512	−0.293256	−	0.956034i	$-0.594739\pi$
$599$	−8598.39	−0.586512	−0.293256	+	0.956034i	$0.594739\pi$
$600$	0	0
$601$	−22554.4	−1.53080	−0.765401	−	0.643554i	$-0.777459\pi$
$601$	−22554.4	−1.53080	−0.765401	+	0.643554i	$0.777459\pi$
$602$	0	0
$603$	−18360.2	−1.23994
$604$	0	0
$605$	−897.465	−0.0603093
$606$	0	0
$607$	−78.4176	−0.00524361	−0.00262181	−	0.999997i	$-0.500835\pi$
$607$	−78.4176	−0.00524361	−0.00262181	+	0.999997i	$0.500835\pi$
$608$	0	0
$609$	70442.0	4.68711
$610$	0	0
$611$	869.920	0.0575993
$612$	0	0
$613$	−18312.9	−1.20661	−0.603305	−	0.797511i	$-0.706150\pi$
$613$	−18312.9	−1.20661	−0.603305	+	0.797511i	$0.706150\pi$
$614$	0	0
$615$	−1355.77	−0.0888943
$616$	0	0
$617$	18893.9	1.23281	0.616403	−	0.787431i	$-0.288589\pi$
$617$	18893.9	1.23281	0.616403	+	0.787431i	$0.288589\pi$
$618$	0	0
$619$	26942.7	1.74946	0.874731	−	0.484608i	$-0.161038\pi$
$619$	26942.7	1.74946	0.874731	+	0.484608i	$0.161038\pi$
$620$	0	0
$621$	2454.30	0.158595
$622$	0	0
$623$	29829.2	1.91827
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−424.103	−0.0270128
$628$	0	0
$629$	−1454.97	−0.0922315
$630$	0	0
$631$	12501.1	0.788689	0.394345	−	0.918963i	$-0.370972\pi$
$631$	12501.1	0.788689	0.394345	+	0.918963i	$0.370972\pi$
$632$	0	0
$633$	−34622.2	−2.17395
$634$	0	0
$635$	−7468.81	−0.466757
$636$	0	0
$637$	−14167.9	−0.881247
$638$	0	0
$639$	39408.8	2.43973
$640$	0	0
$641$	−13845.0	−0.853109	−0.426555	−	0.904462i	$-0.640273\pi$
$641$	−13845.0	−0.853109	−0.426555	+	0.904462i	$0.640273\pi$
$642$	0	0
$643$	−19511.6	−1.19668	−0.598338	−	0.801244i	$-0.704172\pi$
$643$	−19511.6	−1.19668	−0.598338	+	0.801244i	$0.704172\pi$
$644$	0	0
$645$	11179.7	0.682481
$646$	0	0
$647$	−27875.6	−1.69382	−0.846910	−	0.531737i	$-0.821540\pi$
$647$	−27875.6	−1.69382	−0.846910	+	0.531737i	$0.821540\pi$
$648$	0	0
$649$	−12875.7	−0.778759
$650$	0	0
$651$	−27982.2	−1.68465
$652$	0	0
$653$	−30695.4	−1.83952	−0.919758	−	0.392486i	$-0.871615\pi$
$653$	−30695.4	−1.83952	−0.919758	+	0.392486i	$0.871615\pi$
$654$	0	0
$655$	−6731.14	−0.401538
$656$	0	0
$657$	26557.2	1.57701
$658$	0	0
$659$	−29129.8	−1.72190	−0.860951	−	0.508687i	$-0.830131\pi$
$659$	−29129.8	−1.72190	−0.860951	+	0.508687i	$0.830131\pi$
$660$	0	0
$661$	−31280.3	−1.84064	−0.920319	−	0.391169i	$-0.872071\pi$
$661$	−31280.3	−1.84064	−0.920319	+	0.391169i	$0.872071\pi$
$662$	0	0
$663$	3334.56	0.195329
$664$	0	0
$665$	241.311	0.0140716
$666$	0	0
$667$	6258.94	0.363339
$668$	0	0
$669$	−15197.8	−0.878296
$670$	0	0
$671$	−20197.3	−1.16201
$672$	0	0
$673$	19991.9	1.14507	0.572534	−	0.819881i	$-0.305961\pi$
$673$	19991.9	1.14507	0.572534	+	0.819881i	$0.305961\pi$
$674$	0	0
$675$	2667.72	0.152119
$676$	0	0
$677$	−23473.0	−1.33256	−0.666279	−	0.745702i	$-0.732114\pi$
$677$	−23473.0	−1.33256	−0.666279	+	0.745702i	$0.732114\pi$
$678$	0	0
$679$	−30309.0	−1.71304
$680$	0	0
$681$	−48973.7	−2.75576
$682$	0	0
$683$	−21654.0	−1.21313	−0.606565	−	0.795034i	$-0.707453\pi$
$683$	−21654.0	−1.21313	−0.606565	+	0.795034i	$0.707453\pi$
$684$	0	0
$685$	−2968.40	−0.165572
$686$	0	0
$687$	−20581.1	−1.14296
$688$	0	0
$689$	−7099.95	−0.392579
$690$	0	0
$691$	8868.02	0.488213	0.244107	−	0.969748i	$-0.421505\pi$
$691$	8868.02	0.488213	0.244107	+	0.969748i	$0.421505\pi$
$692$	0	0
$693$	−42950.5	−2.35433
$694$	0	0
$695$	1593.82	0.0869886
$696$	0	0
$697$	625.113	0.0339711
$698$	0	0
$699$	30064.4	1.62681
$700$	0	0
$701$	28535.3	1.53747	0.768734	−	0.639569i	$-0.220887\pi$
$701$	28535.3	1.53747	0.768734	+	0.639569i	$0.220887\pi$
$702$	0	0
$703$	117.669	0.00631288
$704$	0	0
$705$	−1650.40	−0.0881669
$706$	0	0
$707$	−22947.2	−1.22067
$708$	0	0
$709$	13405.8	0.710105	0.355053	−	0.934846i	$-0.384463\pi$
$709$	13405.8	0.710105	0.355053	+	0.934846i	$0.384463\pi$
$710$	0	0
$711$	41481.6	2.18802
$712$	0	0
$713$	−2486.28	−0.130592
$714$	0	0
$715$	3661.08	0.191492
$716$	0	0
$717$	32738.0	1.70519
$718$	0	0
$719$	29235.7	1.51642	0.758212	−	0.652008i	$-0.226073\pi$
$719$	29235.7	1.51642	0.758212	+	0.652008i	$0.226073\pi$
$720$	0	0
$721$	−2085.58	−0.107727
$722$	0	0
$723$	2843.47	0.146265
$724$	0	0
$725$	6803.20	0.348503
$726$	0	0
$727$	−19493.6	−0.994469	−0.497234	−	0.867616i	$-0.665651\pi$
$727$	−19493.6	−0.994469	−0.497234	+	0.867616i	$0.665651\pi$
$728$	0	0
$729$	−31885.2	−1.61994
$730$	0	0
$731$	−5154.68	−0.260811
$732$	0	0
$733$	−24201.3	−1.21950	−0.609751	−	0.792593i	$-0.708730\pi$
$733$	−24201.3	−1.21950	−0.609751	+	0.792593i	$0.708730\pi$
$734$	0	0
$735$	26879.2	1.34892
$736$	0	0
$737$	15562.8	0.777835
$738$	0	0
$739$	−9689.33	−0.482311	−0.241156	−	0.970486i	$-0.577526\pi$
$739$	−9689.33	−0.482311	−0.241156	+	0.970486i	$0.577526\pi$
$740$	0	0
$741$	−269.677	−0.0133695
$742$	0	0
$743$	35338.4	1.74487	0.872436	−	0.488729i	$-0.162539\pi$
$743$	35338.4	1.74487	0.872436	+	0.488729i	$0.162539\pi$
$744$	0	0
$745$	−15901.2	−0.781978
$746$	0	0
$747$	−4429.58	−0.216961
$748$	0	0
$749$	33789.6	1.64839
$750$	0	0
$751$	−3388.71	−0.164655	−0.0823275	−	0.996605i	$-0.526235\pi$
$751$	−3388.71	−0.164655	−0.0823275	+	0.996605i	$0.526235\pi$
$752$	0	0
$753$	22428.8	1.08546
$754$	0	0
$755$	−6038.84	−0.291094
$756$	0	0
$757$	27094.6	1.30088	0.650442	−	0.759556i	$-0.274584\pi$
$757$	27094.6	1.30088	0.650442	+	0.759556i	$0.274584\pi$
$758$	0	0
$759$	−6390.08	−0.305593
$760$	0	0
$761$	13330.0	0.634969	0.317485	−	0.948263i	$-0.397162\pi$
$761$	13330.0	0.634969	0.317485	+	0.948263i	$0.397162\pi$
$762$	0	0
$763$	−40539.5	−1.92350
$764$	0	0
$765$	−3778.15	−0.178561
$766$	0	0
$767$	−8187.32	−0.385433
$768$	0	0
$769$	−10947.9	−0.513384	−0.256692	−	0.966493i	$-0.582633\pi$
$769$	−10947.9	−0.513384	−0.256692	+	0.966493i	$0.582633\pi$
$770$	0	0
$771$	47822.9	2.23385
$772$	0	0
$773$	2528.04	0.117629	0.0588144	−	0.998269i	$-0.481268\pi$
$773$	2528.04	0.117629	0.0588144	+	0.998269i	$0.481268\pi$
$774$	0	0
$775$	−2702.48	−0.125259
$776$	0	0
$777$	19953.8	0.921287
$778$	0	0
$779$	−50.5550	−0.00232519
$780$	0	0
$781$	−33404.5	−1.53048
$782$	0	0
$783$	29038.4	1.32535
$784$	0	0
$785$	7605.50	0.345799
$786$	0	0
$787$	28973.3	1.31231	0.656155	−	0.754626i	$-0.272182\pi$
$787$	28973.3	1.31231	0.656155	+	0.754626i	$0.272182\pi$
$788$	0	0
$789$	−24669.1	−1.11311
$790$	0	0
$791$	−36837.8	−1.65588
$792$	0	0
$793$	−12843.0	−0.575116
$794$	0	0
$795$	13469.9	0.600917
$796$	0	0
$797$	−14624.6	−0.649975	−0.324988	−	0.945718i	$-0.605360\pi$
$797$	−14624.6	−0.649975	−0.324988	+	0.945718i	$0.605360\pi$
$798$	0	0
$799$	760.959	0.0336931
$800$	0	0
$801$	37770.2	1.66610
$802$	0	0
$803$	−22511.0	−0.989284
$804$	0	0
$805$	3635.89	0.159191
$806$	0	0
$807$	−39438.6	−1.72033
$808$	0	0
$809$	−6920.91	−0.300774	−0.150387	−	0.988627i	$-0.548052\pi$
$809$	−6920.91	−0.300774	−0.150387	+	0.988627i	$0.548052\pi$
$810$	0	0
$811$	−18153.7	−0.786018	−0.393009	−	0.919535i	$-0.628566\pi$
$811$	−18153.7	−0.786018	−0.393009	+	0.919535i	$0.628566\pi$
$812$	0	0
$813$	−41782.3	−1.80242
$814$	0	0
$815$	9473.95	0.407188
$816$	0	0
$817$	416.877	0.0178515
$818$	0	0
$819$	−27311.1	−1.16524
$820$	0	0
$821$	35502.8	1.50921	0.754603	−	0.656182i	$-0.227830\pi$
$821$	35502.8	1.50921	0.754603	+	0.656182i	$0.227830\pi$
$822$	0	0
$823$	−184.408	−0.00781052	−0.00390526	−	0.999992i	$-0.501243\pi$
$823$	−184.408	−0.00781052	−0.00390526	+	0.999992i	$0.501243\pi$
$824$	0	0
$825$	−6945.74	−0.293115
$826$	0	0
$827$	9522.39	0.400394	0.200197	−	0.979756i	$-0.435842\pi$
$827$	9522.39	0.400394	0.200197	+	0.979756i	$0.435842\pi$
$828$	0	0
$829$	−12652.8	−0.530097	−0.265048	−	0.964235i	$-0.585388\pi$
$829$	−12652.8	−0.530097	−0.265048	+	0.964235i	$0.585388\pi$
$830$	0	0
$831$	−22548.4	−0.941270
$832$	0	0
$833$	−12393.3	−0.515491
$834$	0	0
$835$	16118.2	0.668015
$836$	0	0
$837$	−11535.1	−0.476359
$838$	0	0
$839$	19059.0	0.784255	0.392128	−	0.919911i	$-0.371739\pi$
$839$	19059.0	0.784255	0.392128	+	0.919911i	$0.371739\pi$
$840$	0	0
$841$	49664.6	2.03635
$842$	0	0
$843$	22275.8	0.910106
$844$	0	0
$845$	−8657.01	−0.352438
$846$	0	0
$847$	−5674.93	−0.230216
$848$	0	0
$849$	−36450.4	−1.47347
$850$	0	0
$851$	1772.95	0.0714169
$852$	0	0
$853$	−42005.7	−1.68610	−0.843052	−	0.537832i	$-0.819243\pi$
$853$	−42005.7	−1.68610	−0.843052	+	0.537832i	$0.819243\pi$
$854$	0	0
$855$	305.552	0.0122218
$856$	0	0
$857$	−21476.8	−0.856046	−0.428023	−	0.903768i	$-0.640790\pi$
$857$	−21476.8	−0.856046	−0.428023	+	0.903768i	$0.640790\pi$
$858$	0	0
$859$	−34312.2	−1.36288	−0.681442	−	0.731872i	$-0.738647\pi$
$859$	−34312.2	−1.36288	−0.681442	+	0.731872i	$0.738647\pi$
$860$	0	0
$861$	−8572.95	−0.339332
$862$	0	0
$863$	6661.28	0.262749	0.131375	−	0.991333i	$-0.458061\pi$
$863$	6661.28	0.262749	0.131375	+	0.991333i	$0.458061\pi$
$864$	0	0
$865$	−21390.4	−0.840806
$866$	0	0
$867$	−37307.7	−1.46140
$868$	0	0
$869$	−35161.5	−1.37258
$870$	0	0
$871$	9896.03	0.384976
$872$	0	0
$873$	−38377.8	−1.48785
$874$	0	0
$875$	3952.06	0.152690
$876$	0	0
$877$	−44323.1	−1.70660	−0.853299	−	0.521422i	$-0.825402\pi$
$877$	−44323.1	−1.70660	−0.853299	+	0.521422i	$0.825402\pi$
$878$	0	0
$879$	46878.1	1.79882
$880$	0	0
$881$	14732.2	0.563384	0.281692	−	0.959505i	$-0.409104\pi$
$881$	14732.2	0.563384	0.281692	+	0.959505i	$0.409104\pi$
$882$	0	0
$883$	−25007.5	−0.953081	−0.476541	−	0.879152i	$-0.658109\pi$
$883$	−25007.5	−0.953081	−0.476541	+	0.879152i	$0.658109\pi$
$884$	0	0
$885$	15532.9	0.589980
$886$	0	0
$887$	−15189.1	−0.574971	−0.287486	−	0.957785i	$-0.592819\pi$
$887$	−15189.1	−0.574971	−0.287486	+	0.957785i	$0.592819\pi$
$888$	0	0
$889$	−47227.5	−1.78173
$890$	0	0
$891$	7032.25	0.264410
$892$	0	0
$893$	−61.5413	−0.00230616
$894$	0	0
$895$	19408.8	0.724877
$896$	0	0
$897$	−4063.30	−0.151248
$898$	0	0
$899$	−29416.8	−1.09133
$900$	0	0
$901$	−6210.65	−0.229641
$902$	0	0
$903$	70692.5	2.60520
$904$	0	0
$905$	−8134.50	−0.298785
$906$	0	0
$907$	−4850.81	−0.177584	−0.0887918	−	0.996050i	$-0.528301\pi$
$907$	−4850.81	−0.177584	−0.0887918	+	0.996050i	$0.528301\pi$
$908$	0	0
$909$	−29056.1	−1.06021
$910$	0	0
$911$	−21296.3	−0.774508	−0.387254	−	0.921973i	$-0.626576\pi$
$911$	−21296.3	−0.774508	−0.387254	+	0.921973i	$0.626576\pi$
$912$	0	0
$913$	3754.70	0.136103
$914$	0	0
$915$	24365.5	0.880326
$916$	0	0
$917$	−42562.9	−1.53277
$918$	0	0
$919$	31927.7	1.14603	0.573014	−	0.819546i	$-0.305774\pi$
$919$	31927.7	1.14603	0.573014	+	0.819546i	$0.305774\pi$
$920$	0	0
$921$	−33460.8	−1.19715
$922$	0	0
$923$	−21241.1	−0.757486
$924$	0	0
$925$	1927.12	0.0685007
$926$	0	0
$927$	−2640.79	−0.0935653
$928$	0	0
$929$	−8743.58	−0.308792	−0.154396	−	0.988009i	$-0.549343\pi$
$929$	−8743.58	−0.308792	−0.154396	+	0.988009i	$0.549343\pi$
$930$	0	0
$931$	1002.29	0.0352833
$932$	0	0
$933$	−42409.1	−1.48812
$934$	0	0
$935$	3202.51	0.112014
$936$	0	0
$937$	4778.89	0.166617	0.0833083	−	0.996524i	$-0.473451\pi$
$937$	4778.89	0.166617	0.0833083	+	0.996524i	$0.473451\pi$
$938$	0	0
$939$	−29186.4	−1.01434
$940$	0	0
$941$	24770.1	0.858109	0.429054	−	0.903279i	$-0.358847\pi$
$941$	24770.1	0.858109	0.429054	+	0.903279i	$0.358847\pi$
$942$	0	0
$943$	−761.727	−0.0263046
$944$	0	0
$945$	16868.8	0.580678
$946$	0	0
$947$	−38723.4	−1.32877	−0.664384	−	0.747392i	$-0.731306\pi$
$947$	−38723.4	−1.32877	−0.664384	+	0.747392i	$0.731306\pi$
$948$	0	0
$949$	−14314.2	−0.489629
$950$	0	0
$951$	−32286.4	−1.10090
$952$	0	0
$953$	−3904.08	−0.132703	−0.0663513	−	0.997796i	$-0.521136\pi$
$953$	−3904.08	−0.132703	−0.0663513	+	0.997796i	$0.521136\pi$
$954$	0	0
$955$	−6773.83	−0.229525
$956$	0	0
$957$	−75605.2	−2.55378
$958$	0	0
$959$	−18770.1	−0.632030
$960$	0	0
$961$	−18105.5	−0.607752
$962$	0	0
$963$	42785.0	1.43170
$964$	0	0
$965$	−3069.03	−0.102379
$966$	0	0
$967$	−12182.0	−0.405116	−0.202558	−	0.979270i	$-0.564925\pi$
$967$	−12182.0	−0.405116	−0.202558	+	0.979270i	$0.564925\pi$
$968$	0	0
$969$	−235.899	−0.00782060
$970$	0	0
$971$	−10643.7	−0.351773	−0.175887	−	0.984410i	$-0.556279\pi$
$971$	−10643.7	−0.351773	−0.175887	+	0.984410i	$0.556279\pi$
$972$	0	0
$973$	10078.2	0.332058
$974$	0	0
$975$	−4416.63	−0.145072
$976$	0	0
$977$	−35314.1	−1.15639	−0.578197	−	0.815897i	$-0.696244\pi$
$977$	−35314.1	−1.15639	−0.578197	+	0.815897i	$0.696244\pi$
$978$	0	0
$979$	−32015.6	−1.04517
$980$	0	0
$981$	−51331.8	−1.67064
$982$	0	0
$983$	−11470.8	−0.372188	−0.186094	−	0.982532i	$-0.559583\pi$
$983$	−11470.8	−0.372188	−0.186094	+	0.982532i	$0.559583\pi$
$984$	0	0
$985$	−10781.4	−0.348754
$986$	0	0
$987$	−10436.0	−0.336555
$988$	0	0
$989$	6281.20	0.201952
$990$	0	0
$991$	43077.4	1.38083	0.690413	−	0.723416i	$-0.257429\pi$
$991$	43077.4	1.38083	0.690413	+	0.723416i	$0.257429\pi$
$992$	0	0
$993$	58665.1	1.87480
$994$	0	0
$995$	−8250.05	−0.262858
$996$	0	0
$997$	38891.9	1.23542	0.617712	−	0.786404i	$-0.288060\pi$
$997$	38891.9	1.23542	0.617712	+	0.786404i	$0.288060\pi$
$998$	0	0
$999$	8225.60	0.260507

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.x.1.7		8
4.3	odd	2		920.4.a.c.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.c.1.2	✓	8	4.3	odd	2
1840.4.a.x.1.7		8	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+8.18739 q^{3} +5.00000 q^{5} +31.6165 q^{7} +40.0333 q^{9} +O(q^{10})\)	\(q+8.18739 q^{3} +5.00000 q^{5} +31.6165 q^{7} +40.0333 q^{9} -33.9339 q^{11} -21.5777 q^{13} +40.9369 q^{15} -18.8750 q^{17} +1.52649 q^{19} +258.856 q^{21} +23.0000 q^{23} +25.0000 q^{25} +106.709 q^{27} +272.128 q^{29} -108.099 q^{31} -277.830 q^{33} +158.082 q^{35} +77.0846 q^{37} -176.665 q^{39} -33.1186 q^{41} +273.096 q^{43} +200.167 q^{45} -40.3157 q^{47} +656.601 q^{49} -154.537 q^{51} +329.041 q^{53} -169.669 q^{55} +12.4979 q^{57} +379.434 q^{59} +595.196 q^{61} +1265.71 q^{63} -107.889 q^{65} -458.623 q^{67} +188.310 q^{69} +984.401 q^{71} +663.378 q^{73} +204.685 q^{75} -1072.87 q^{77} +1036.18 q^{79} -207.234 q^{81} -110.647 q^{83} -94.3751 q^{85} +2228.02 q^{87} +943.469 q^{89} -682.211 q^{91} -885.051 q^{93} +7.63243 q^{95} -958.646 q^{97} -1358.48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9}+O(q^{10}) \) 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9	\( 8 q + 12 q^{3} + 40 q^{5} + 31 q^{7} + 62 q^{9} + 35 q^{11} - 54 q^{13} + 60 q^{15} - 11 q^{17} + 107 q^{19} - 148 q^{21} + 184 q^{23} + 200 q^{25} + 405 q^{27} + 37 q^{29} + 290 q^{31} + 289 q^{33} + 155 q^{35} - 172 q^{37} + 311 q^{39} + 308 q^{41} + 538 q^{43} + 310 q^{45} + 1035 q^{47} + 585 q^{49} + 387 q^{51} + 46 q^{53} + 175 q^{55} + 286 q^{57} + 1256 q^{59} + 399 q^{61} + 1234 q^{63} - 270 q^{65} + 1598 q^{67} + 276 q^{69} + 750 q^{71} + 177 q^{73} + 300 q^{75} - 1228 q^{77} + 1292 q^{79} + 380 q^{81} + 2094 q^{83} - 55 q^{85} + 2245 q^{87} + 484 q^{89} + 679 q^{91} - 2111 q^{93} + 535 q^{95} - 2225 q^{97} + 2117 q^{99}+O(q^{100}) \) 8 * q + 12 * q^3 + 40 * q^5 + 31 * q^7 + 62 * q^9 + 35 * q^11 - 54 * q^13 + 60 * q^15 - 11 * q^17 + 107 * q^19 - 148 * q^21 + 184 * q^23 + 200 * q^25 + 405 * q^27 + 37 * q^29 + 290 * q^31 + 289 * q^33 + 155 * q^35 - 172 * q^37 + 311 * q^39 + 308 * q^41 + 538 * q^43 + 310 * q^45 + 1035 * q^47 + 585 * q^49 + 387 * q^51 + 46 * q^53 + 175 * q^55 + 286 * q^57 + 1256 * q^59 + 399 * q^61 + 1234 * q^63 - 270 * q^65 + 1598 * q^67 + 276 * q^69 + 750 * q^71 + 177 * q^73 + 300 * q^75 - 1228 * q^77 + 1292 * q^79 + 380 * q^81 + 2094 * q^83 - 55 * q^85 + 2245 * q^87 + 484 * q^89 + 679 * q^91 - 2111 * q^93 + 535 * q^95 - 2225 * q^97 + 2117 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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