LMFDB - Embedded newform 1840.3.c.b.1151.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.12
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.45

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.51586i q^{3} +2.23607 q^{5} +8.23373i q^{7} -3.36124 q^{9} +O(q^{10})$	$q-3.51586i q^{3} +2.23607 q^{5} +8.23373i q^{7} -3.36124 q^{9} -11.4885i q^{11} +11.8367 q^{13} -7.86169i q^{15} +5.43502 q^{17} +19.4129i q^{19} +28.9486 q^{21} +4.79583i q^{23} +5.00000 q^{25} -19.8251i q^{27} -7.48191 q^{29} +30.6301i q^{31} -40.3919 q^{33} +18.4112i q^{35} -32.8263 q^{37} -41.6160i q^{39} +39.0590 q^{41} -5.33166i q^{43} -7.51596 q^{45} +82.4619i q^{47} -18.7943 q^{49} -19.1087i q^{51} +93.9322 q^{53} -25.6890i q^{55} +68.2531 q^{57} -25.8135i q^{59} +97.6558 q^{61} -27.6755i q^{63} +26.4676 q^{65} -101.209i q^{67} +16.8615 q^{69} +105.394i q^{71} -1.04250 q^{73} -17.5793i q^{75} +94.5931 q^{77} +5.12485i q^{79} -99.9532 q^{81} +87.4869i q^{83} +12.1531 q^{85} +26.3053i q^{87} -82.0332 q^{89} +97.4599i q^{91} +107.691 q^{93} +43.4086i q^{95} +117.047 q^{97} +38.6156i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.51586i	− 1.17195i	−0.810328	−	0.585976i	$-0.800711\pi$
$3$	− 3.51586i	− 1.17195i	0.810328	−	0.585976i	$-0.199289\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	8.23373i	1.17625i	0.808771	+	0.588124i	$0.200133\pi$
$7$	8.23373i	1.17625i	−0.808771	+	0.588124i	$0.799867\pi$
$8$	0	0
$9$	−3.36124	−0.373471
$10$	0	0
$11$	− 11.4885i	− 1.04441i	−0.852821	−	0.522204i	$-0.825110\pi$
$11$	− 11.4885i	− 1.04441i	0.852821	−	0.522204i	$-0.174890\pi$
$12$	0	0
$13$	11.8367	0.910513	0.455256	−	0.890360i	$-0.349548\pi$
$13$	11.8367	0.910513	0.455256	+	0.890360i	$0.349548\pi$
$14$	0	0
$15$	− 7.86169i	− 0.524113i
$16$	0	0
$17$	5.43502	0.319707	0.159854	−	0.987141i	$-0.448898\pi$
$17$	5.43502	0.319707	0.159854	+	0.987141i	$0.448898\pi$
$18$	0	0
$19$	19.4129i	1.02173i	0.859660	+	0.510867i	$0.170675\pi$
$19$	19.4129i	1.02173i	−0.859660	+	0.510867i	$0.829325\pi$
$20$	0	0
$21$	28.9486	1.37850
$22$	0	0
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 19.8251i	− 0.734262i
$28$	0	0
$29$	−7.48191	−0.257997	−0.128999	−	0.991645i	$-0.541176\pi$
$29$	−7.48191	−0.257997	−0.128999	+	0.991645i	$0.541176\pi$
$30$	0	0
$31$	30.6301i	0.988068i	0.869443	+	0.494034i	$0.164478\pi$
$31$	30.6301i	0.988068i	−0.869443	+	0.494034i	$0.835522\pi$
$32$	0	0
$33$	−40.3919	−1.22400
$34$	0	0
$35$	18.4112i	0.526034i
$36$	0	0
$37$	−32.8263	−0.887197	−0.443598	−	0.896226i	$-0.646298\pi$
$37$	−32.8263	−0.887197	−0.443598	+	0.896226i	$0.646298\pi$
$38$	0	0
$39$	− 41.6160i	− 1.06708i
$40$	0	0
$41$	39.0590	0.952658	0.476329	−	0.879267i	$-0.341967\pi$
$41$	39.0590	0.952658	0.476329	+	0.879267i	$0.341967\pi$
$42$	0	0
$43$	− 5.33166i	− 0.123992i	−0.998076	−	0.0619960i	$-0.980253\pi$
$43$	− 5.33166i	− 0.123992i	0.998076	−	0.0619960i	$-0.0197466\pi$
$44$	0	0
$45$	−7.51596	−0.167021
$46$	0	0
$47$	82.4619i	1.75451i	0.480026	+	0.877254i	$0.340627\pi$
$47$	82.4619i	1.75451i	−0.480026	+	0.877254i	$0.659373\pi$
$48$	0	0
$49$	−18.7943	−0.383557
$50$	0	0
$51$	− 19.1087i	− 0.374681i
$52$	0	0
$53$	93.9322	1.77230	0.886152	−	0.463394i	$-0.153368\pi$
$53$	93.9322	1.77230	0.886152	+	0.463394i	$0.153368\pi$
$54$	0	0
$55$	− 25.6890i	− 0.467074i
$56$	0	0
$57$	68.2531	1.19742
$58$	0	0
$59$	− 25.8135i	− 0.437517i	−0.975779	−	0.218758i	$-0.929799\pi$
$59$	− 25.8135i	− 0.437517i	0.975779	−	0.218758i	$-0.0702007\pi$
$60$	0	0
$61$	97.6558	1.60092	0.800458	−	0.599389i	$-0.204590\pi$
$61$	97.6558	1.60092	0.800458	+	0.599389i	$0.204590\pi$
$62$	0	0
$63$	− 27.6755i	− 0.439294i
$64$	0	0
$65$	26.4676	0.407194
$66$	0	0
$67$	− 101.209i	− 1.51058i	−0.655390	−	0.755291i	$-0.727496\pi$
$67$	− 101.209i	− 1.51058i	0.655390	−	0.755291i	$-0.272504\pi$
$68$	0	0
$69$	16.8615	0.244369
$70$	0	0
$71$	105.394i	1.48443i	0.670163	+	0.742214i	$0.266224\pi$
$71$	105.394i	1.48443i	−0.670163	+	0.742214i	$0.733776\pi$
$72$	0	0
$73$	−1.04250	−0.0142808	−0.00714040	−	0.999975i	$-0.502273\pi$
$73$	−1.04250	−0.0142808	−0.00714040	+	0.999975i	$0.502273\pi$
$74$	0	0
$75$	− 17.5793i	− 0.234390i
$76$	0	0
$77$	94.5931	1.22848
$78$	0	0
$79$	5.12485i	0.0648715i	0.999474	+	0.0324357i	$0.0103264\pi$
$79$	5.12485i	0.0648715i	−0.999474	+	0.0324357i	$0.989674\pi$
$80$	0	0
$81$	−99.9532	−1.23399
$82$	0	0
$83$	87.4869i	1.05406i	0.849847	+	0.527029i	$0.176694\pi$
$83$	87.4869i	1.05406i	−0.849847	+	0.527029i	$0.823306\pi$
$84$	0	0
$85$	12.1531	0.142977
$86$	0	0
$87$	26.3053i	0.302360i
$88$	0	0
$89$	−82.0332	−0.921722	−0.460861	−	0.887472i	$-0.652459\pi$
$89$	−82.0332	−0.921722	−0.460861	+	0.887472i	$0.652459\pi$
$90$	0	0
$91$	97.4599i	1.07099i
$92$	0	0
$93$	107.691	1.15797
$94$	0	0
$95$	43.4086i	0.456933i
$96$	0	0
$97$	117.047	1.20667	0.603337	−	0.797486i	$-0.293837\pi$
$97$	117.047	1.20667	0.603337	+	0.797486i	$0.293837\pi$
$98$	0	0
$99$	38.6156i	0.390056i
$100$	0	0
$101$	189.820	1.87940	0.939701	−	0.341996i	$-0.111103\pi$
$101$	189.820	1.87940	0.939701	+	0.341996i	$0.111103\pi$
$102$	0	0
$103$	− 200.862i	− 1.95011i	−0.221953	−	0.975057i	$-0.571243\pi$
$103$	− 200.862i	− 1.95011i	0.221953	−	0.975057i	$-0.428757\pi$
$104$	0	0
$105$	64.7310	0.616486
$106$	0	0
$107$	127.275i	1.18948i	0.803916	+	0.594742i	$0.202746\pi$
$107$	127.275i	1.18948i	−0.803916	+	0.594742i	$0.797254\pi$
$108$	0	0
$109$	37.3700	0.342844	0.171422	−	0.985198i	$-0.445164\pi$
$109$	37.3700	0.342844	0.171422	+	0.985198i	$0.445164\pi$
$110$	0	0
$111$	115.412i	1.03975i
$112$	0	0
$113$	−135.488	−1.19901	−0.599506	−	0.800370i	$-0.704636\pi$
$113$	−135.488	−1.19901	−0.599506	+	0.800370i	$0.704636\pi$
$114$	0	0
$115$	10.7238i	0.0932505i
$116$	0	0
$117$	−39.7859	−0.340050
$118$	0	0
$119$	44.7505i	0.376055i
$120$	0	0
$121$	−10.9854	−0.0907887
$122$	0	0
$123$	− 137.326i	− 1.11647i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	− 67.6014i	− 0.532295i	−0.963932	−	0.266147i	$-0.914249\pi$
$127$	− 67.6014i	− 0.532295i	0.963932	−	0.266147i	$-0.0857507\pi$
$128$	0	0
$129$	−18.7453	−0.145313
$130$	0	0
$131$	− 28.9895i	− 0.221294i	−0.993860	−	0.110647i	$-0.964708\pi$
$131$	− 28.9895i	− 0.221294i	0.993860	−	0.110647i	$-0.0352923\pi$
$132$	0	0
$133$	−159.841	−1.20181
$134$	0	0
$135$	− 44.3302i	− 0.328372i
$136$	0	0
$137$	69.0169	0.503773	0.251887	−	0.967757i	$-0.418949\pi$
$137$	69.0169	0.503773	0.251887	+	0.967757i	$0.418949\pi$
$138$	0	0
$139$	− 66.2460i	− 0.476590i	−0.971193	−	0.238295i	$-0.923412\pi$
$139$	− 66.2460i	− 0.476590i	0.971193	−	0.238295i	$-0.0765884\pi$
$140$	0	0
$141$	289.924	2.05620
$142$	0	0
$143$	− 135.985i	− 0.950947i
$144$	0	0
$145$	−16.7301	−0.115380
$146$	0	0
$147$	66.0780i	0.449510i
$148$	0	0
$149$	1.61624	0.0108473	0.00542363	−	0.999985i	$-0.498274\pi$
$149$	1.61624	0.0108473	0.00542363	+	0.999985i	$0.498274\pi$
$150$	0	0
$151$	− 77.7065i	− 0.514613i	−0.966330	−	0.257306i	$-0.917165\pi$
$151$	− 77.7065i	− 0.514613i	0.966330	−	0.257306i	$-0.0828349\pi$
$152$	0	0
$153$	−18.2684	−0.119401
$154$	0	0
$155$	68.4910i	0.441877i
$156$	0	0
$157$	179.151	1.14109	0.570543	−	0.821268i	$-0.306733\pi$
$157$	179.151	1.14109	0.570543	+	0.821268i	$0.306733\pi$
$158$	0	0
$159$	− 330.252i	− 2.07706i
$160$	0	0
$161$	−39.4876	−0.245264
$162$	0	0
$163$	− 257.495i	− 1.57972i	−0.613286	−	0.789861i	$-0.710153\pi$
$163$	− 257.495i	− 1.57972i	0.613286	−	0.789861i	$-0.289847\pi$
$164$	0	0
$165$	−90.3190	−0.547388
$166$	0	0
$167$	164.262i	0.983603i	0.870707	+	0.491802i	$0.163661\pi$
$167$	164.262i	0.983603i	−0.870707	+	0.491802i	$0.836339\pi$
$168$	0	0
$169$	−28.8934	−0.170967
$170$	0	0
$171$	− 65.2515i	− 0.381588i
$172$	0	0
$173$	−155.216	−0.897202	−0.448601	−	0.893732i	$-0.648078\pi$
$173$	−155.216	−0.897202	−0.448601	+	0.893732i	$0.648078\pi$
$174$	0	0
$175$	41.1686i	0.235249i
$176$	0	0
$177$	−90.7565	−0.512749
$178$	0	0
$179$	− 83.8963i	− 0.468694i	−0.972153	−	0.234347i	$-0.924705\pi$
$179$	− 83.8963i	− 0.468694i	0.972153	−	0.234347i	$-0.0752952\pi$
$180$	0	0
$181$	−14.4057	−0.0795896	−0.0397948	−	0.999208i	$-0.512670\pi$
$181$	−14.4057	−0.0795896	−0.0397948	+	0.999208i	$0.512670\pi$
$182$	0	0
$183$	− 343.344i	− 1.87620i
$184$	0	0
$185$	−73.4018	−0.396766
$186$	0	0
$187$	− 62.4402i	− 0.333905i
$188$	0	0
$189$	163.234	0.863673
$190$	0	0
$191$	− 368.606i	− 1.92988i	−0.262477	−	0.964938i	$-0.584539\pi$
$191$	− 368.606i	− 1.92988i	0.262477	−	0.964938i	$-0.415461\pi$
$192$	0	0
$193$	−126.059	−0.653154	−0.326577	−	0.945171i	$-0.605895\pi$
$193$	−126.059	−0.653154	−0.326577	+	0.945171i	$0.605895\pi$
$194$	0	0
$195$	− 93.0562i	− 0.477211i
$196$	0	0
$197$	−48.5215	−0.246302	−0.123151	−	0.992388i	$-0.539300\pi$
$197$	−48.5215	−0.246302	−0.123151	+	0.992388i	$0.539300\pi$
$198$	0	0
$199$	59.8641i	0.300825i	0.988623	+	0.150412i	$0.0480601\pi$
$199$	59.8641i	0.300825i	−0.988623	+	0.150412i	$0.951940\pi$
$200$	0	0
$201$	−355.836	−1.77033
$202$	0	0
$203$	− 61.6041i	− 0.303468i
$204$	0	0
$205$	87.3386	0.426042
$206$	0	0
$207$	− 16.1199i	− 0.0778741i
$208$	0	0
$209$	223.025	1.06711
$210$	0	0
$211$	− 387.063i	− 1.83442i	−0.398401	−	0.917211i	$-0.630435\pi$
$211$	− 387.063i	− 1.83442i	0.398401	−	0.917211i	$-0.369565\pi$
$212$	0	0
$213$	370.552	1.73968
$214$	0	0
$215$	− 11.9219i	− 0.0554509i
$216$	0	0
$217$	−252.200	−1.16221
$218$	0	0
$219$	3.66527i	0.0167364i
$220$	0	0
$221$	64.3325	0.291097
$222$	0	0
$223$	235.925i	1.05796i	0.848634	+	0.528981i	$0.177426\pi$
$223$	235.925i	1.05796i	−0.848634	+	0.528981i	$0.822574\pi$
$224$	0	0
$225$	−16.8062	−0.0746942
$226$	0	0
$227$	176.895i	0.779275i	0.920968	+	0.389637i	$0.127400\pi$
$227$	176.895i	0.779275i	−0.920968	+	0.389637i	$0.872600\pi$
$228$	0	0
$229$	−263.108	−1.14894	−0.574471	−	0.818525i	$-0.694792\pi$
$229$	−263.108	−1.14894	−0.574471	+	0.818525i	$0.694792\pi$
$230$	0	0
$231$	− 332.576i	− 1.43972i
$232$	0	0
$233$	189.786	0.814532	0.407266	−	0.913310i	$-0.366482\pi$
$233$	189.786	0.814532	0.407266	+	0.913310i	$0.366482\pi$
$234$	0	0
$235$	184.390i	0.784640i
$236$	0	0
$237$	18.0182	0.0760262
$238$	0	0
$239$	− 445.980i	− 1.86602i	−0.359845	−	0.933012i	$-0.617170\pi$
$239$	− 445.980i	− 1.86602i	0.359845	−	0.933012i	$-0.382830\pi$
$240$	0	0
$241$	402.583	1.67047	0.835234	−	0.549894i	$-0.185332\pi$
$241$	402.583	1.67047	0.835234	+	0.549894i	$0.185332\pi$
$242$	0	0
$243$	172.995i	0.711915i
$244$	0	0
$245$	−42.0253	−0.171532
$246$	0	0
$247$	229.784i	0.930301i
$248$	0	0
$249$	307.591	1.23531
$250$	0	0
$251$	32.1151i	0.127949i	0.997952	+	0.0639744i	$0.0203776\pi$
$251$	32.1151i	0.127949i	−0.997952	+	0.0639744i	$0.979622\pi$
$252$	0	0
$253$	55.0969	0.217774
$254$	0	0
$255$	− 42.7285i	− 0.167563i
$256$	0	0
$257$	333.878	1.29914	0.649569	−	0.760303i	$-0.274949\pi$
$257$	333.878	1.29914	0.649569	+	0.760303i	$0.274949\pi$
$258$	0	0
$259$	− 270.283i	− 1.04356i
$260$	0	0
$261$	25.1485	0.0963544
$262$	0	0
$263$	369.801i	1.40609i	0.711146	+	0.703044i	$0.248177\pi$
$263$	369.801i	1.40609i	−0.711146	+	0.703044i	$0.751823\pi$
$264$	0	0
$265$	210.039	0.792599
$266$	0	0
$267$	288.417i	1.08021i
$268$	0	0
$269$	244.800	0.910038	0.455019	−	0.890482i	$-0.349632\pi$
$269$	244.800	0.910038	0.455019	+	0.890482i	$0.349632\pi$
$270$	0	0
$271$	− 74.7876i	− 0.275969i	−0.990434	−	0.137984i	$-0.955938\pi$
$271$	− 74.7876i	− 0.275969i	0.990434	−	0.137984i	$-0.0440624\pi$
$272$	0	0
$273$	342.655	1.25515
$274$	0	0
$275$	− 57.4425i	− 0.208882i
$276$	0	0
$277$	48.1151	0.173701	0.0868504	−	0.996221i	$-0.472320\pi$
$277$	48.1151	0.173701	0.0868504	+	0.996221i	$0.472320\pi$
$278$	0	0
$279$	− 102.955i	− 0.369015i
$280$	0	0
$281$	53.2956	0.189664	0.0948321	−	0.995493i	$-0.469769\pi$
$281$	53.2956	0.189664	0.0948321	+	0.995493i	$0.469769\pi$
$282$	0	0
$283$	94.8046i	0.334999i	0.985872	+	0.167499i	$0.0535692\pi$
$283$	94.8046i	0.334999i	−0.985872	+	0.167499i	$0.946431\pi$
$284$	0	0
$285$	152.618	0.535503
$286$	0	0
$287$	321.601i	1.12056i
$288$	0	0
$289$	−259.461	−0.897787
$290$	0	0
$291$	− 411.522i	− 1.41416i
$292$	0	0
$293$	−124.851	−0.426114	−0.213057	−	0.977040i	$-0.568342\pi$
$293$	−124.851	−0.426114	−0.213057	+	0.977040i	$0.568342\pi$
$294$	0	0
$295$	− 57.7207i	− 0.195664i
$296$	0	0
$297$	−227.760	−0.766869
$298$	0	0
$299$	56.7666i	0.189855i
$300$	0	0
$301$	43.8994	0.145845
$302$	0	0
$303$	− 667.378i	− 2.20257i
$304$	0	0
$305$	218.365	0.715951
$306$	0	0
$307$	420.701i	1.37036i	0.728373	+	0.685181i	$0.240277\pi$
$307$	420.701i	1.37036i	−0.728373	+	0.685181i	$0.759723\pi$
$308$	0	0
$309$	−706.201	−2.28544
$310$	0	0
$311$	214.053i	0.688274i	0.938919	+	0.344137i	$0.111828\pi$
$311$	214.053i	0.688274i	−0.938919	+	0.344137i	$0.888172\pi$
$312$	0	0
$313$	−388.074	−1.23985	−0.619926	−	0.784660i	$-0.712837\pi$
$313$	−388.074	−1.23985	−0.619926	+	0.784660i	$0.712837\pi$
$314$	0	0
$315$	− 61.8844i	− 0.196458i
$316$	0	0
$317$	−305.629	−0.964129	−0.482065	−	0.876136i	$-0.660113\pi$
$317$	−305.629	−0.964129	−0.482065	+	0.876136i	$0.660113\pi$
$318$	0	0
$319$	85.9559i	0.269454i
$320$	0	0
$321$	447.480	1.39402
$322$	0	0
$323$	105.510i	0.326655i
$324$	0	0
$325$	59.1833	0.182103
$326$	0	0
$327$	− 131.388i	− 0.401797i
$328$	0	0
$329$	−678.969	−2.06374
$330$	0	0
$331$	387.683i	1.17125i	0.810582	+	0.585625i	$0.199151\pi$
$331$	387.683i	1.17125i	−0.810582	+	0.585625i	$0.800849\pi$
$332$	0	0
$333$	110.337	0.331342
$334$	0	0
$335$	− 226.310i	− 0.675552i
$336$	0	0
$337$	503.645	1.49450	0.747248	−	0.664546i	$-0.231375\pi$
$337$	503.645	1.49450	0.747248	+	0.664546i	$0.231375\pi$
$338$	0	0
$339$	476.358i	1.40519i
$340$	0	0
$341$	351.894	1.03195
$342$	0	0
$343$	248.706i	0.725089i
$344$	0	0
$345$	37.7033	0.109285
$346$	0	0
$347$	114.257i	0.329271i	0.986355	+	0.164635i	$0.0526447\pi$
$347$	114.257i	0.329271i	−0.986355	+	0.164635i	$0.947355\pi$
$348$	0	0
$349$	243.594	0.697978	0.348989	−	0.937127i	$-0.386525\pi$
$349$	243.594	0.697978	0.348989	+	0.937127i	$0.386525\pi$
$350$	0	0
$351$	− 234.663i	− 0.668555i
$352$	0	0
$353$	166.580	0.471899	0.235949	−	0.971765i	$-0.424180\pi$
$353$	166.580	0.471899	0.235949	+	0.971765i	$0.424180\pi$
$354$	0	0
$355$	235.669i	0.663857i
$356$	0	0
$357$	157.336	0.440718
$358$	0	0
$359$	− 182.182i	− 0.507470i	−0.967274	−	0.253735i	$-0.918341\pi$
$359$	− 182.182i	− 0.507470i	0.967274	−	0.253735i	$-0.0816592\pi$
$360$	0	0
$361$	−15.8618	−0.0439386
$362$	0	0
$363$	38.6232i	0.106400i
$364$	0	0
$365$	−2.33110	−0.00638656
$366$	0	0
$367$	− 123.765i	− 0.337236i	−0.985682	−	0.168618i	$-0.946070\pi$
$367$	− 123.765i	− 0.337236i	0.985682	−	0.168618i	$-0.0539304\pi$
$368$	0	0
$369$	−131.287	−0.355790
$370$	0	0
$371$	773.412i	2.08467i
$372$	0	0
$373$	−273.898	−0.734311	−0.367156	−	0.930160i	$-0.619668\pi$
$373$	−273.898	−0.734311	−0.367156	+	0.930160i	$0.619668\pi$
$374$	0	0
$375$	− 39.3085i	− 0.104823i
$376$	0	0
$377$	−88.5609	−0.234910
$378$	0	0
$379$	− 269.675i	− 0.711543i	−0.934573	−	0.355771i	$-0.884218\pi$
$379$	− 269.675i	− 0.711543i	0.934573	−	0.355771i	$-0.115782\pi$
$380$	0	0
$381$	−237.677	−0.623824
$382$	0	0
$383$	689.212i	1.79951i	0.436396	+	0.899755i	$0.356255\pi$
$383$	689.212i	1.79951i	−0.436396	+	0.899755i	$0.643745\pi$
$384$	0	0
$385$	211.517	0.549394
$386$	0	0
$387$	17.9210i	0.0463074i
$388$	0	0
$389$	226.772	0.582960	0.291480	−	0.956577i	$-0.405852\pi$
$389$	226.772	0.582960	0.291480	+	0.956577i	$0.405852\pi$
$390$	0	0
$391$	26.0654i	0.0666635i
$392$	0	0
$393$	−101.923	−0.259346
$394$	0	0
$395$	11.4595i	0.0290114i
$396$	0	0
$397$	−583.004	−1.46852	−0.734262	−	0.678866i	$-0.762472\pi$
$397$	−583.004	−1.46852	−0.734262	+	0.678866i	$0.762472\pi$
$398$	0	0
$399$	561.977i	1.40846i
$400$	0	0
$401$	−346.142	−0.863196	−0.431598	−	0.902066i	$-0.642050\pi$
$401$	−346.142	−0.863196	−0.431598	+	0.902066i	$0.642050\pi$
$402$	0	0
$403$	362.558i	0.899648i
$404$	0	0
$405$	−223.502	−0.551857
$406$	0	0
$407$	377.124i	0.926596i
$408$	0	0
$409$	267.630	0.654352	0.327176	−	0.944963i	$-0.393903\pi$
$409$	267.630	0.654352	0.327176	+	0.944963i	$0.393903\pi$
$410$	0	0
$411$	− 242.653i	− 0.590398i
$412$	0	0
$413$	212.541	0.514628
$414$	0	0
$415$	195.627i	0.471389i
$416$	0	0
$417$	−232.911	−0.558540
$418$	0	0
$419$	− 85.7750i	− 0.204714i	−0.994748	−	0.102357i	$-0.967362\pi$
$419$	− 85.7750i	− 0.204714i	0.994748	−	0.102357i	$-0.0326383\pi$
$420$	0	0
$421$	217.396	0.516381	0.258190	−	0.966094i	$-0.416874\pi$
$421$	217.396	0.516381	0.258190	+	0.966094i	$0.416874\pi$
$422$	0	0
$423$	− 277.174i	− 0.655258i
$424$	0	0
$425$	27.1751	0.0639414
$426$	0	0
$427$	804.072i	1.88307i
$428$	0	0
$429$	−478.105	−1.11446
$430$	0	0
$431$	− 645.107i	− 1.49677i	−0.663266	−	0.748384i	$-0.730830\pi$
$431$	− 645.107i	− 1.49677i	0.663266	−	0.748384i	$-0.269170\pi$
$432$	0	0
$433$	−542.531	−1.25296	−0.626479	−	0.779438i	$-0.715505\pi$
$433$	−542.531	−1.25296	−0.626479	+	0.779438i	$0.715505\pi$
$434$	0	0
$435$	58.8205i	0.135220i
$436$	0	0
$437$	−93.1011	−0.213046
$438$	0	0
$439$	− 416.469i	− 0.948678i	−0.880342	−	0.474339i	$-0.842687\pi$
$439$	− 416.469i	− 0.948678i	0.880342	−	0.474339i	$-0.157313\pi$
$440$	0	0
$441$	63.1721	0.143247
$442$	0	0
$443$	279.885i	0.631795i	0.948793	+	0.315898i	$0.102306\pi$
$443$	279.885i	0.631795i	−0.948793	+	0.315898i	$0.897694\pi$
$444$	0	0
$445$	−183.432	−0.412206
$446$	0	0
$447$	− 5.68247i	− 0.0127125i
$448$	0	0
$449$	−189.388	−0.421799	−0.210899	−	0.977508i	$-0.567639\pi$
$449$	−189.388	−0.421799	−0.210899	+	0.977508i	$0.567639\pi$
$450$	0	0
$451$	− 448.729i	− 0.994964i
$452$	0	0
$453$	−273.205	−0.603101
$454$	0	0
$455$	217.927i	0.478960i
$456$	0	0
$457$	107.641	0.235538	0.117769	−	0.993041i	$-0.462426\pi$
$457$	107.641	0.235538	0.117769	+	0.993041i	$0.462426\pi$
$458$	0	0
$459$	− 107.750i	− 0.234749i
$460$	0	0
$461$	54.7309	0.118722	0.0593610	−	0.998237i	$-0.481094\pi$
$461$	54.7309	0.118722	0.0593610	+	0.998237i	$0.481094\pi$
$462$	0	0
$463$	− 250.575i	− 0.541199i	−0.962692	−	0.270599i	$-0.912778\pi$
$463$	− 250.575i	− 0.541199i	0.962692	−	0.270599i	$-0.0872219\pi$
$464$	0	0
$465$	240.804	0.517859
$466$	0	0
$467$	− 410.909i	− 0.879890i	−0.898025	−	0.439945i	$-0.854998\pi$
$467$	− 410.909i	− 0.879890i	0.898025	−	0.439945i	$-0.145002\pi$
$468$	0	0
$469$	833.327	1.77682
$470$	0	0
$471$	− 629.867i	− 1.33730i
$472$	0	0
$473$	−61.2527	−0.129498
$474$	0	0
$475$	97.0646i	0.204347i
$476$	0	0
$477$	−315.728	−0.661905
$478$	0	0
$479$	16.6360i	0.0347307i	0.999849	+	0.0173654i	$0.00552785\pi$
$479$	16.6360i	0.0347307i	−0.999849	+	0.0173654i	$0.994472\pi$
$480$	0	0
$481$	−388.554	−0.807804
$482$	0	0
$483$	138.833i	0.287438i
$484$	0	0
$485$	261.726	0.539641
$486$	0	0
$487$	144.355i	0.296417i	0.988956	+	0.148209i	$0.0473507\pi$
$487$	144.355i	0.296417i	−0.988956	+	0.148209i	$0.952649\pi$
$488$	0	0
$489$	−905.314	−1.85136
$490$	0	0
$491$	28.8643i	0.0587868i	0.999568	+	0.0293934i	$0.00935755\pi$
$491$	28.8643i	0.0587868i	−0.999568	+	0.0293934i	$0.990642\pi$
$492$	0	0
$493$	−40.6644	−0.0824835
$494$	0	0
$495$	86.3470i	0.174438i
$496$	0	0
$497$	−867.789	−1.74605
$498$	0	0
$499$	− 111.490i	− 0.223426i	−0.993741	−	0.111713i	$-0.964366\pi$
$499$	− 111.490i	− 0.223426i	0.993741	−	0.111713i	$-0.0356337\pi$
$500$	0	0
$501$	577.521	1.15274
$502$	0	0
$503$	22.2511i	0.0442367i	0.999755	+	0.0221184i	$0.00704107\pi$
$503$	22.2511i	0.0442367i	−0.999755	+	0.0221184i	$0.992959\pi$
$504$	0	0
$505$	424.450	0.840494
$506$	0	0
$507$	101.585i	0.200365i
$508$	0	0
$509$	−203.568	−0.399937	−0.199968	−	0.979802i	$-0.564084\pi$
$509$	−203.568	−0.399937	−0.199968	+	0.979802i	$0.564084\pi$
$510$	0	0
$511$	− 8.58364i	− 0.0167977i
$512$	0	0
$513$	384.863	0.750220
$514$	0	0
$515$	− 449.141i	− 0.872118i
$516$	0	0
$517$	947.363	1.83242
$518$	0	0
$519$	545.717i	1.05148i
$520$	0	0
$521$	−286.150	−0.549233	−0.274616	−	0.961554i	$-0.588551\pi$
$521$	−286.150	−0.549233	−0.274616	+	0.961554i	$0.588551\pi$
$522$	0	0
$523$	610.139i	1.16661i	0.812252	+	0.583307i	$0.198242\pi$
$523$	610.139i	1.16661i	−0.812252	+	0.583307i	$0.801758\pi$
$524$	0	0
$525$	144.743	0.275701
$526$	0	0
$527$	166.475i	0.315892i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	86.7653i	0.163400i
$532$	0	0
$533$	462.328	0.867407
$534$	0	0
$535$	284.595i	0.531954i
$536$	0	0
$537$	−294.967	−0.549287
$538$	0	0
$539$	215.918i	0.400590i
$540$	0	0
$541$	630.057	1.16461	0.582307	−	0.812969i	$-0.302150\pi$
$541$	630.057	1.16461	0.582307	+	0.812969i	$0.302150\pi$
$542$	0	0
$543$	50.6484i	0.0932752i
$544$	0	0
$545$	83.5619	0.153325
$546$	0	0
$547$	290.904i	0.531817i	0.963998	+	0.265908i	$0.0856719\pi$
$547$	290.904i	0.531817i	−0.963998	+	0.265908i	$0.914328\pi$
$548$	0	0
$549$	−328.245	−0.597896
$550$	0	0
$551$	− 145.246i	− 0.263604i
$552$	0	0
$553$	−42.1966	−0.0763049
$554$	0	0
$555$	258.070i	0.464991i
$556$	0	0
$557$	−913.550	−1.64013	−0.820063	−	0.572274i	$-0.806062\pi$
$557$	−913.550	−1.64013	−0.820063	+	0.572274i	$0.806062\pi$
$558$	0	0
$559$	− 63.1090i	− 0.112896i
$560$	0	0
$561$	−219.531	−0.391320
$562$	0	0
$563$	727.239i	1.29172i	0.763455	+	0.645861i	$0.223501\pi$
$563$	727.239i	1.29172i	−0.763455	+	0.645861i	$0.776499\pi$
$564$	0	0
$565$	−302.961	−0.536215
$566$	0	0
$567$	− 822.988i	− 1.45148i
$568$	0	0
$569$	−587.187	−1.03196	−0.515981	−	0.856600i	$-0.672573\pi$
$569$	−587.187	−1.03196	−0.515981	+	0.856600i	$0.672573\pi$
$570$	0	0
$571$	− 25.4814i	− 0.0446259i	−0.999751	−	0.0223130i	$-0.992897\pi$
$571$	− 25.4814i	− 0.0446259i	0.999751	−	0.0223130i	$-0.00710303\pi$
$572$	0	0
$573$	−1295.97	−2.26172
$574$	0	0
$575$	23.9792i	0.0417029i
$576$	0	0
$577$	−468.515	−0.811985	−0.405993	−	0.913876i	$-0.633074\pi$
$577$	−468.515	−0.811985	−0.405993	+	0.913876i	$0.633074\pi$
$578$	0	0
$579$	443.204i	0.765465i
$580$	0	0
$581$	−720.343	−1.23983
$582$	0	0
$583$	− 1079.14i	− 1.85101i
$584$	0	0
$585$	−88.9639	−0.152075
$586$	0	0
$587$	− 366.365i	− 0.624130i	−0.950061	−	0.312065i	$-0.898979\pi$
$587$	− 366.365i	− 0.624130i	0.950061	−	0.312065i	$-0.101021\pi$
$588$	0	0
$589$	−594.620	−1.00954
$590$	0	0
$591$	170.595i	0.288654i
$592$	0	0
$593$	1086.34	1.83195	0.915973	−	0.401240i	$-0.131421\pi$
$593$	1086.34	1.83195	0.915973	+	0.401240i	$0.131421\pi$
$594$	0	0
$595$	100.065i	0.168177i
$596$	0	0
$597$	210.474	0.352552
$598$	0	0
$599$	744.779i	1.24337i	0.783267	+	0.621685i	$0.213552\pi$
$599$	744.779i	1.24337i	−0.783267	+	0.621685i	$0.786448\pi$
$600$	0	0
$601$	422.126	0.702372	0.351186	−	0.936306i	$-0.385778\pi$
$601$	422.126	0.702372	0.351186	+	0.936306i	$0.385778\pi$
$602$	0	0
$603$	340.187i	0.564158i
$604$	0	0
$605$	−24.5642	−0.0406019
$606$	0	0
$607$	381.017i	0.627705i	0.949472	+	0.313852i	$0.101620\pi$
$607$	381.017i	0.627705i	−0.949472	+	0.313852i	$0.898380\pi$
$608$	0	0
$609$	−216.591	−0.355650
$610$	0	0
$611$	976.074i	1.59750i
$612$	0	0
$613$	140.923	0.229890	0.114945	−	0.993372i	$-0.463331\pi$
$613$	140.923	0.229890	0.114945	+	0.993372i	$0.463331\pi$
$614$	0	0
$615$	− 307.070i	− 0.499300i
$616$	0	0
$617$	503.918	0.816724	0.408362	−	0.912820i	$-0.366100\pi$
$617$	503.918	0.816724	0.408362	+	0.912820i	$0.366100\pi$
$618$	0	0
$619$	567.405i	0.916648i	0.888785	+	0.458324i	$0.151550\pi$
$619$	567.405i	0.916648i	−0.888785	+	0.458324i	$0.848450\pi$
$620$	0	0
$621$	95.0777	0.153104
$622$	0	0
$623$	− 675.439i	− 1.08417i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 784.125i	− 1.25060i
$628$	0	0
$629$	−178.412	−0.283643
$630$	0	0
$631$	798.202i	1.26498i	0.774569	+	0.632490i	$0.217967\pi$
$631$	798.202i	1.26498i	−0.774569	+	0.632490i	$0.782033\pi$
$632$	0	0
$633$	−1360.86	−2.14985
$634$	0	0
$635$	− 151.161i	− 0.238049i
$636$	0	0
$637$	−222.462	−0.349234
$638$	0	0
$639$	− 354.256i	− 0.554391i
$640$	0	0
$641$	−34.9780	−0.0545679	−0.0272839	−	0.999628i	$-0.508686\pi$
$641$	−34.9780	−0.0545679	−0.0272839	+	0.999628i	$0.508686\pi$
$642$	0	0
$643$	1068.24i	1.66134i	0.556767	+	0.830669i	$0.312042\pi$
$643$	1068.24i	1.66134i	−0.556767	+	0.830669i	$0.687958\pi$
$644$	0	0
$645$	−41.9158	−0.0649858
$646$	0	0
$647$	− 742.131i	− 1.14703i	−0.819194	−	0.573517i	$-0.805579\pi$
$647$	− 742.131i	− 1.14703i	0.819194	−	0.573517i	$-0.194421\pi$
$648$	0	0
$649$	−296.558	−0.456946
$650$	0	0
$651$	886.699i	1.36206i
$652$	0	0
$653$	128.454	0.196714	0.0983568	−	0.995151i	$-0.468641\pi$
$653$	128.454	0.196714	0.0983568	+	0.995151i	$0.468641\pi$
$654$	0	0
$655$	− 64.8225i	− 0.0989657i
$656$	0	0
$657$	3.50408	0.00533346
$658$	0	0
$659$	− 104.636i	− 0.158780i	−0.996844	−	0.0793900i	$-0.974703\pi$
$659$	− 104.636i	− 0.158780i	0.996844	−	0.0793900i	$-0.0252972\pi$
$660$	0	0
$661$	−283.641	−0.429109	−0.214555	−	0.976712i	$-0.568830\pi$
$661$	−283.641	−0.429109	−0.214555	+	0.976712i	$0.568830\pi$
$662$	0	0
$663$	− 226.184i	− 0.341152i
$664$	0	0
$665$	−357.415	−0.537466
$666$	0	0
$667$	− 35.8820i	− 0.0537961i
$668$	0	0
$669$	829.479	1.23988
$670$	0	0
$671$	− 1121.92i	− 1.67201i
$672$	0	0
$673$	498.640	0.740922	0.370461	−	0.928848i	$-0.379200\pi$
$673$	498.640	0.740922	0.370461	+	0.928848i	$0.379200\pi$
$674$	0	0
$675$	− 99.1253i	− 0.146852i
$676$	0	0
$677$	−1064.42	−1.57227	−0.786133	−	0.618058i	$-0.787920\pi$
$677$	−1064.42	−1.57227	−0.786133	+	0.618058i	$0.787920\pi$
$678$	0	0
$679$	963.737i	1.41935i
$680$	0	0
$681$	621.938	0.913272
$682$	0	0
$683$	224.001i	0.327967i	0.986463	+	0.163983i	$0.0524344\pi$
$683$	224.001i	0.327967i	−0.986463	+	0.163983i	$0.947566\pi$
$684$	0	0
$685$	154.326	0.225294
$686$	0	0
$687$	925.048i	1.34650i
$688$	0	0
$689$	1111.84	1.61371
$690$	0	0
$691$	− 571.541i	− 0.827121i	−0.910477	−	0.413560i	$-0.864285\pi$
$691$	− 571.541i	− 0.827121i	0.910477	−	0.413560i	$-0.135715\pi$
$692$	0	0
$693$	−317.950	−0.458802
$694$	0	0
$695$	− 148.130i	− 0.213137i
$696$	0	0
$697$	212.286	0.304572
$698$	0	0
$699$	− 667.260i	− 0.954592i
$700$	0	0
$701$	447.286	0.638069	0.319034	−	0.947743i	$-0.396641\pi$
$701$	447.286	0.638069	0.319034	+	0.947743i	$0.396641\pi$
$702$	0	0
$703$	− 637.254i	− 0.906478i
$704$	0	0
$705$	648.290	0.919560
$706$	0	0
$707$	1562.92i	2.21064i
$708$	0	0
$709$	−1337.93	−1.88707	−0.943535	−	0.331271i	$-0.892522\pi$
$709$	−1337.93	−1.88707	−0.943535	+	0.331271i	$0.892522\pi$
$710$	0	0
$711$	− 17.2258i	− 0.0242276i
$712$	0	0
$713$	−146.897	−0.206026
$714$	0	0
$715$	− 304.073i	− 0.425276i
$716$	0	0
$717$	−1568.00	−2.18689
$718$	0	0
$719$	133.459i	0.185617i	0.995684	+	0.0928086i	$0.0295845\pi$
$719$	133.459i	0.185617i	−0.995684	+	0.0928086i	$0.970416\pi$
$720$	0	0
$721$	1653.84	2.29382
$722$	0	0
$723$	− 1415.42i	− 1.95771i
$724$	0	0
$725$	−37.4096	−0.0515994
$726$	0	0
$727$	− 281.758i	− 0.387563i	−0.981045	−	0.193781i	$-0.937925\pi$
$727$	− 281.758i	− 0.387563i	0.981045	−	0.193781i	$-0.0620752\pi$
$728$	0	0
$729$	−291.352	−0.399660
$730$	0	0
$731$	− 28.9777i	− 0.0396411i
$732$	0	0
$733$	−940.619	−1.28325	−0.641623	−	0.767020i	$-0.721738\pi$
$733$	−940.619	−1.28325	−0.641623	+	0.767020i	$0.721738\pi$
$734$	0	0
$735$	147.755i	0.201027i
$736$	0	0
$737$	−1162.74	−1.57766
$738$	0	0
$739$	− 643.688i	− 0.871026i	−0.900182	−	0.435513i	$-0.856567\pi$
$739$	− 643.688i	− 0.871026i	0.900182	−	0.435513i	$-0.143433\pi$
$740$	0	0
$741$	807.888	1.09027
$742$	0	0
$743$	678.740i	0.913513i	0.889592	+	0.456757i	$0.150989\pi$
$743$	678.740i	0.913513i	−0.889592	+	0.456757i	$0.849011\pi$
$744$	0	0
$745$	3.61403	0.00485104
$746$	0	0
$747$	− 294.064i	− 0.393660i
$748$	0	0
$749$	−1047.95	−1.39913
$750$	0	0
$751$	− 937.442i	− 1.24826i	−0.781321	−	0.624129i	$-0.785454\pi$
$751$	− 937.442i	− 1.24826i	0.781321	−	0.624129i	$-0.214546\pi$
$752$	0	0
$753$	112.912	0.149950
$754$	0	0
$755$	− 173.757i	− 0.230142i
$756$	0	0
$757$	−824.406	−1.08904	−0.544522	−	0.838747i	$-0.683289\pi$
$757$	−824.406	−1.08904	−0.544522	+	0.838747i	$0.683289\pi$
$758$	0	0
$759$	− 193.713i	− 0.255221i
$760$	0	0
$761$	238.581	0.313510	0.156755	−	0.987638i	$-0.449897\pi$
$761$	238.581	0.313510	0.156755	+	0.987638i	$0.449897\pi$
$762$	0	0
$763$	307.695i	0.403269i
$764$	0	0
$765$	−40.8494	−0.0533979
$766$	0	0
$767$	− 305.546i	− 0.398365i
$768$	0	0
$769$	−878.081	−1.14185	−0.570924	−	0.821003i	$-0.693415\pi$
$769$	−878.081	−1.14185	−0.570924	+	0.821003i	$0.693415\pi$
$770$	0	0
$771$	− 1173.87i	− 1.52253i
$772$	0	0
$773$	−1241.10	−1.60556	−0.802780	−	0.596276i	$-0.796646\pi$
$773$	−1241.10	−1.60556	−0.802780	+	0.596276i	$0.796646\pi$
$774$	0	0
$775$	153.151i	0.197614i
$776$	0	0
$777$	−950.275	−1.22300
$778$	0	0
$779$	758.249i	0.973363i
$780$	0	0
$781$	1210.82	1.55035
$782$	0	0
$783$	148.329i	0.189437i
$784$	0	0
$785$	400.593	0.510309
$786$	0	0
$787$	785.990i	0.998717i	0.866395	+	0.499359i	$0.166431\pi$
$787$	785.990i	0.998717i	−0.866395	+	0.499359i	$0.833569\pi$
$788$	0	0
$789$	1300.17	1.64787
$790$	0	0
$791$	− 1115.58i	− 1.41034i
$792$	0	0
$793$	1155.92	1.45765
$794$	0	0
$795$	− 738.466i	− 0.928888i
$796$	0	0
$797$	−808.080	−1.01390	−0.506951	−	0.861975i	$-0.669228\pi$
$797$	−808.080	−1.01390	−0.506951	+	0.861975i	$0.669228\pi$
$798$	0	0
$799$	448.182i	0.560929i
$800$	0	0
$801$	275.733	0.344236
$802$	0	0
$803$	11.9767i	0.0149150i
$804$	0	0
$805$	−88.2969	−0.109686
$806$	0	0
$807$	− 860.682i	− 1.06652i
$808$	0	0
$809$	−1237.21	−1.52930	−0.764652	−	0.644443i	$-0.777089\pi$
$809$	−1237.21	−1.52930	−0.764652	+	0.644443i	$0.777089\pi$
$810$	0	0
$811$	769.398i	0.948703i	0.880336	+	0.474351i	$0.157317\pi$
$811$	769.398i	0.948703i	−0.880336	+	0.474351i	$0.842683\pi$
$812$	0	0
$813$	−262.942	−0.323422
$814$	0	0
$815$	− 575.775i	− 0.706473i
$816$	0	0
$817$	103.503	0.126687
$818$	0	0
$819$	− 327.586i	− 0.399983i
$820$	0	0
$821$	267.162	0.325411	0.162706	−	0.986675i	$-0.447978\pi$
$821$	267.162	0.325411	0.162706	+	0.986675i	$0.447978\pi$
$822$	0	0
$823$	220.689i	0.268152i	0.990971	+	0.134076i	$0.0428067\pi$
$823$	220.689i	0.268152i	−0.990971	+	0.134076i	$0.957193\pi$
$824$	0	0
$825$	−201.959	−0.244799
$826$	0	0
$827$	− 253.677i	− 0.306744i	−0.988169	−	0.153372i	$-0.950987\pi$
$827$	− 253.677i	− 0.306744i	0.988169	−	0.153372i	$-0.0490132\pi$
$828$	0	0
$829$	1071.44	1.29244	0.646222	−	0.763150i	$-0.276348\pi$
$829$	1071.44	1.29244	0.646222	+	0.763150i	$0.276348\pi$
$830$	0	0
$831$	− 169.166i	− 0.203569i
$832$	0	0
$833$	−102.147	−0.122626
$834$	0	0
$835$	367.300i	0.439881i
$836$	0	0
$837$	607.244	0.725501
$838$	0	0
$839$	489.077i	0.582929i	0.956582	+	0.291464i	$0.0941425\pi$
$839$	489.077i	0.582929i	−0.956582	+	0.291464i	$0.905858\pi$
$840$	0	0
$841$	−785.021	−0.933438
$842$	0	0
$843$	− 187.380i	− 0.222277i
$844$	0	0
$845$	−64.6076	−0.0764587
$846$	0	0
$847$	− 90.4511i	− 0.106790i
$848$	0	0
$849$	333.319	0.392602
$850$	0	0
$851$	− 157.429i	− 0.184993i
$852$	0	0
$853$	743.793	0.871972	0.435986	−	0.899953i	$-0.356400\pi$
$853$	743.793	0.871972	0.435986	+	0.899953i	$0.356400\pi$
$854$	0	0
$855$	− 145.907i	− 0.170651i
$856$	0	0
$857$	−1177.53	−1.37401	−0.687005	−	0.726652i	$-0.741075\pi$
$857$	−1177.53	−1.37401	−0.687005	+	0.726652i	$0.741075\pi$
$858$	0	0
$859$	511.683i	0.595673i	0.954617	+	0.297837i	$0.0962651\pi$
$859$	511.683i	0.595673i	−0.954617	+	0.297837i	$0.903735\pi$
$860$	0	0
$861$	1130.70	1.31324
$862$	0	0
$863$	− 549.683i	− 0.636945i	−0.947932	−	0.318472i	$-0.896830\pi$
$863$	− 549.683i	− 0.636945i	0.947932	−	0.318472i	$-0.103170\pi$
$864$	0	0
$865$	−347.073	−0.401241
$866$	0	0
$867$	912.226i	1.05216i
$868$	0	0
$869$	58.8768	0.0677523
$870$	0	0
$871$	− 1197.98i	− 1.37540i
$872$	0	0
$873$	−393.424	−0.450658
$874$	0	0
$875$	92.0559i	0.105207i
$876$	0	0
$877$	−1324.94	−1.51076	−0.755379	−	0.655288i	$-0.772547\pi$
$877$	−1324.94	−1.51076	−0.755379	+	0.655288i	$0.772547\pi$
$878$	0	0
$879$	438.959i	0.499385i
$880$	0	0
$881$	−1365.57	−1.55002	−0.775012	−	0.631947i	$-0.782256\pi$
$881$	−1365.57	−1.55002	−0.775012	+	0.631947i	$0.782256\pi$
$882$	0	0
$883$	− 1173.74i	− 1.32926i	−0.747172	−	0.664631i	$-0.768589\pi$
$883$	− 1173.74i	− 1.32926i	0.747172	−	0.664631i	$-0.231411\pi$
$884$	0	0
$885$	−202.938	−0.229308
$886$	0	0
$887$	− 1057.04i	− 1.19170i	−0.803095	−	0.595851i	$-0.796815\pi$
$887$	− 1057.04i	− 1.19170i	0.803095	−	0.595851i	$-0.203185\pi$
$888$	0	0
$889$	556.612	0.626110
$890$	0	0
$891$	1148.31i	1.28879i
$892$	0	0
$893$	−1600.83	−1.79264
$894$	0	0
$895$	− 187.598i	− 0.209606i
$896$	0	0
$897$	199.583	0.222501
$898$	0	0
$899$	− 229.172i	− 0.254919i
$900$	0	0
$901$	510.523	0.566619
$902$	0	0
$903$	− 154.344i	− 0.170924i
$904$	0	0
$905$	−32.2122	−0.0355936
$906$	0	0
$907$	144.122i	0.158900i	0.996839	+	0.0794498i	$0.0253163\pi$
$907$	144.122i	0.158900i	−0.996839	+	0.0794498i	$0.974684\pi$
$908$	0	0
$909$	−638.029	−0.701902
$910$	0	0
$911$	1403.58i	1.54070i	0.637619	+	0.770352i	$0.279919\pi$
$911$	1403.58i	1.54070i	−0.637619	+	0.770352i	$0.720081\pi$
$912$	0	0
$913$	1005.09	1.10087
$914$	0	0
$915$	− 767.740i	− 0.839060i
$916$	0	0
$917$	238.692	0.260296
$918$	0	0
$919$	− 1196.39i	− 1.30184i	−0.759146	−	0.650921i	$-0.774383\pi$
$919$	− 1196.39i	− 1.30184i	0.759146	−	0.650921i	$-0.225617\pi$
$920$	0	0
$921$	1479.12	1.60600
$922$	0	0
$923$	1247.52i	1.35159i
$924$	0	0
$925$	−164.131	−0.177439
$926$	0	0
$927$	675.145i	0.728311i
$928$	0	0
$929$	896.504	0.965020	0.482510	−	0.875890i	$-0.339725\pi$
$929$	896.504	0.965020	0.482510	+	0.875890i	$0.339725\pi$
$930$	0	0
$931$	− 364.852i	− 0.391893i
$932$	0	0
$933$	752.580	0.806624
$934$	0	0
$935$	− 139.621i	− 0.149327i
$936$	0	0
$937$	−765.792	−0.817280	−0.408640	−	0.912696i	$-0.633997\pi$
$937$	−765.792	−0.817280	−0.408640	+	0.912696i	$0.633997\pi$
$938$	0	0
$939$	1364.41i	1.45305i
$940$	0	0
$941$	−218.667	−0.232378	−0.116189	−	0.993227i	$-0.537068\pi$
$941$	−218.667	−0.232378	−0.116189	+	0.993227i	$0.537068\pi$
$942$	0	0
$943$	187.320i	0.198643i
$944$	0	0
$945$	365.003	0.386246
$946$	0	0
$947$	402.534i	0.425062i	0.977154	+	0.212531i	$0.0681707\pi$
$947$	402.534i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$948$	0	0
$949$	−12.3397	−0.0130028
$950$	0	0
$951$	1074.55i	1.12991i
$952$	0	0
$953$	−381.629	−0.400450	−0.200225	−	0.979750i	$-0.564167\pi$
$953$	−381.629	−0.400450	−0.200225	+	0.979750i	$0.564167\pi$
$954$	0	0
$955$	− 824.229i	− 0.863067i
$956$	0	0
$957$	302.209	0.315787
$958$	0	0
$959$	568.267i	0.592562i
$960$	0	0
$961$	22.7966	0.0237218
$962$	0	0
$963$	− 427.801i	− 0.444238i
$964$	0	0
$965$	−281.876	−0.292099
$966$	0	0
$967$	− 1371.23i	− 1.41802i	−0.705197	−	0.709011i	$-0.749142\pi$
$967$	− 1371.23i	− 1.41802i	0.705197	−	0.709011i	$-0.250858\pi$
$968$	0	0
$969$	370.957	0.382824
$970$	0	0
$971$	451.231i	0.464708i	0.972631	+	0.232354i	$0.0746427\pi$
$971$	451.231i	0.464708i	−0.972631	+	0.232354i	$0.925357\pi$
$972$	0	0
$973$	545.451	0.560587
$974$	0	0
$975$	− 208.080i	− 0.213415i
$976$	0	0
$977$	−614.855	−0.629329	−0.314665	−	0.949203i	$-0.601892\pi$
$977$	−614.855	−0.629329	−0.314665	+	0.949203i	$0.601892\pi$
$978$	0	0
$979$	942.438i	0.962654i
$980$	0	0
$981$	−125.610	−0.128042
$982$	0	0
$983$	853.480i	0.868240i	0.900855	+	0.434120i	$0.142941\pi$
$983$	853.480i	0.868240i	−0.900855	+	0.434120i	$0.857059\pi$
$984$	0	0
$985$	−108.497	−0.110150
$986$	0	0
$987$	2387.16i	2.41860i
$988$	0	0
$989$	25.5697	0.0258541
$990$	0	0
$991$	− 821.376i	− 0.828836i	−0.910087	−	0.414418i	$-0.863985\pi$
$991$	− 821.376i	− 0.828836i	0.910087	−	0.414418i	$-0.136015\pi$
$992$	0	0
$993$	1363.04	1.37265
$994$	0	0
$995$	133.860i	0.134533i
$996$	0	0
$997$	1445.76	1.45011	0.725054	−	0.688692i	$-0.241815\pi$
$997$	1445.76	1.45011	0.725054	+	0.688692i	$0.241815\pi$
$998$	0	0
$999$	650.783i	0.651435i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.12	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.45	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.12	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.45	yes	56	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.51586i q^{3} +2.23607 q^{5} +8.23373i q^{7} -3.36124 q^{9} +O(q^{10})\)	\(q-3.51586i q^{3} +2.23607 q^{5} +8.23373i q^{7} -3.36124 q^{9} -11.4885i q^{11} +11.8367 q^{13} -7.86169i q^{15} +5.43502 q^{17} +19.4129i q^{19} +28.9486 q^{21} +4.79583i q^{23} +5.00000 q^{25} -19.8251i q^{27} -7.48191 q^{29} +30.6301i q^{31} -40.3919 q^{33} +18.4112i q^{35} -32.8263 q^{37} -41.6160i q^{39} +39.0590 q^{41} -5.33166i q^{43} -7.51596 q^{45} +82.4619i q^{47} -18.7943 q^{49} -19.1087i q^{51} +93.9322 q^{53} -25.6890i q^{55} +68.2531 q^{57} -25.8135i q^{59} +97.6558 q^{61} -27.6755i q^{63} +26.4676 q^{65} -101.209i q^{67} +16.8615 q^{69} +105.394i q^{71} -1.04250 q^{73} -17.5793i q^{75} +94.5931 q^{77} +5.12485i q^{79} -99.9532 q^{81} +87.4869i q^{83} +12.1531 q^{85} +26.3053i q^{87} -82.0332 q^{89} +97.4599i q^{91} +107.691 q^{93} +43.4086i q^{95} +117.047 q^{97} +38.6156i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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