LMFDB - Embedded newform 1840.3.c.b.1151.1

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			1151.1
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.56

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.32909i q^{3} -2.23607 q^{5} +11.3168i q^{7} -19.3992 q^{9} +O(q^{10})$	$q-5.32909i q^{3} -2.23607 q^{5} +11.3168i q^{7} -19.3992 q^{9} +19.0388i q^{11} -3.77788 q^{13} +11.9162i q^{15} +21.0150 q^{17} -27.9466i q^{19} +60.3085 q^{21} -4.79583i q^{23} +5.00000 q^{25} +55.4185i q^{27} -29.6462 q^{29} +9.24449i q^{31} +101.460 q^{33} -25.3052i q^{35} -33.6152 q^{37} +20.1327i q^{39} -79.6929 q^{41} -13.9031i q^{43} +43.3780 q^{45} -78.5790i q^{47} -79.0710 q^{49} -111.991i q^{51} +90.3734 q^{53} -42.5721i q^{55} -148.930 q^{57} -70.7711i q^{59} +83.6229 q^{61} -219.538i q^{63} +8.44760 q^{65} -103.782i q^{67} -25.5574 q^{69} -39.4506i q^{71} +47.9234 q^{73} -26.6455i q^{75} -215.459 q^{77} -12.1693i q^{79} +120.737 q^{81} -28.1325i q^{83} -46.9910 q^{85} +157.987i q^{87} -48.6243 q^{89} -42.7537i q^{91} +49.2648 q^{93} +62.4905i q^{95} -45.5353 q^{97} -369.338i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 5.32909i	− 1.77636i	−0.459491	−	0.888182i	$-0.651968\pi$
$3$	− 5.32909i	− 1.77636i	0.459491	−	0.888182i	$-0.348032\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	11.3168i	1.61669i	0.588707	+	0.808346i	$0.299637\pi$
$7$	11.3168i	1.61669i	−0.588707	+	0.808346i	$0.700363\pi$
$8$	0	0
$9$	−19.3992	−2.15547
$10$	0	0
$11$	19.0388i	1.73080i	0.501081	+	0.865400i	$0.332936\pi$
$11$	19.0388i	1.73080i	−0.501081	+	0.865400i	$0.667064\pi$
$12$	0	0
$13$	−3.77788	−0.290606	−0.145303	−	0.989387i	$-0.546416\pi$
$13$	−3.77788	−0.290606	−0.145303	+	0.989387i	$0.546416\pi$
$14$	0	0
$15$	11.9162i	0.794414i
$16$	0	0
$17$	21.0150	1.23618	0.618088	−	0.786109i	$-0.287907\pi$
$17$	21.0150	1.23618	0.618088	+	0.786109i	$0.287907\pi$
$18$	0	0
$19$	− 27.9466i	− 1.47087i	−0.677593	−	0.735437i	$-0.736977\pi$
$19$	− 27.9466i	− 1.47087i	0.677593	−	0.735437i	$-0.263023\pi$
$20$	0	0
$21$	60.3085	2.87183
$22$	0	0
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	55.4185i	2.05254i
$28$	0	0
$29$	−29.6462	−1.02228	−0.511141	−	0.859497i	$-0.670777\pi$
$29$	−29.6462	−1.02228	−0.511141	+	0.859497i	$0.670777\pi$
$30$	0	0
$31$	9.24449i	0.298209i	0.988821	+	0.149105i	$0.0476391\pi$
$31$	9.24449i	0.298209i	−0.988821	+	0.149105i	$0.952361\pi$
$32$	0	0
$33$	101.460	3.07453
$34$	0	0
$35$	− 25.3052i	− 0.723007i
$36$	0	0
$37$	−33.6152	−0.908518	−0.454259	−	0.890870i	$-0.650096\pi$
$37$	−33.6152	−0.908518	−0.454259	+	0.890870i	$0.650096\pi$
$38$	0	0
$39$	20.1327i	0.516222i
$40$	0	0
$41$	−79.6929	−1.94373	−0.971865	−	0.235539i	$-0.924314\pi$
$41$	−79.6929	−1.94373	−0.971865	+	0.235539i	$0.924314\pi$
$42$	0	0
$43$	− 13.9031i	− 0.323328i	−0.986846	−	0.161664i	$-0.948314\pi$
$43$	− 13.9031i	− 0.323328i	0.986846	−	0.161664i	$-0.0516861\pi$
$44$	0	0
$45$	43.3780	0.963956
$46$	0	0
$47$	− 78.5790i	− 1.67189i	−0.548811	−	0.835946i	$-0.684919\pi$
$47$	− 78.5790i	− 1.67189i	0.548811	−	0.835946i	$-0.315081\pi$
$48$	0	0
$49$	−79.0710	−1.61369
$50$	0	0
$51$	− 111.991i	− 2.19590i
$52$	0	0
$53$	90.3734	1.70516	0.852580	−	0.522597i	$-0.175037\pi$
$53$	90.3734	1.70516	0.852580	+	0.522597i	$0.175037\pi$
$54$	0	0
$55$	− 42.5721i	− 0.774037i
$56$	0	0
$57$	−148.930	−2.61281
$58$	0	0
$59$	− 70.7711i	− 1.19951i	−0.800184	−	0.599755i	$-0.795265\pi$
$59$	− 70.7711i	− 1.19951i	0.800184	−	0.599755i	$-0.204735\pi$
$60$	0	0
$61$	83.6229	1.37087	0.685433	−	0.728135i	$-0.259613\pi$
$61$	83.6229	1.37087	0.685433	+	0.728135i	$0.259613\pi$
$62$	0	0
$63$	− 219.538i	− 3.48473i
$64$	0	0
$65$	8.44760	0.129963
$66$	0	0
$67$	− 103.782i	− 1.54899i	−0.632579	−	0.774496i	$-0.718003\pi$
$67$	− 103.782i	− 1.54899i	0.632579	−	0.774496i	$-0.281997\pi$
$68$	0	0
$69$	−25.5574	−0.370398
$70$	0	0
$71$	− 39.4506i	− 0.555642i	−0.960633	−	0.277821i	$-0.910388\pi$
$71$	− 39.4506i	− 0.555642i	0.960633	−	0.277821i	$-0.0896122\pi$
$72$	0	0
$73$	47.9234	0.656485	0.328242	−	0.944594i	$-0.393544\pi$
$73$	47.9234	0.656485	0.328242	+	0.944594i	$0.393544\pi$
$74$	0	0
$75$	− 26.6455i	− 0.355273i
$76$	0	0
$77$	−215.459	−2.79817
$78$	0	0
$79$	− 12.1693i	− 0.154042i	−0.997029	−	0.0770208i	$-0.975459\pi$
$79$	− 12.1693i	− 0.154042i	0.997029	−	0.0770208i	$-0.0245408\pi$
$80$	0	0
$81$	120.737	1.49058
$82$	0	0
$83$	− 28.1325i	− 0.338946i	−0.985535	−	0.169473i	$-0.945794\pi$
$83$	− 28.1325i	− 0.338946i	0.985535	−	0.169473i	$-0.0542065\pi$
$84$	0	0
$85$	−46.9910	−0.552835
$86$	0	0
$87$	157.987i	1.81595i
$88$	0	0
$89$	−48.6243	−0.546341	−0.273170	−	0.961966i	$-0.588072\pi$
$89$	−48.6243	−0.546341	−0.273170	+	0.961966i	$0.588072\pi$
$90$	0	0
$91$	− 42.7537i	− 0.469821i
$92$	0	0
$93$	49.2648	0.529729
$94$	0	0
$95$	62.4905i	0.657795i
$96$	0	0
$97$	−45.5353	−0.469436	−0.234718	−	0.972064i	$-0.575417\pi$
$97$	−45.5353	−0.469436	−0.234718	+	0.972064i	$0.575417\pi$
$98$	0	0
$99$	− 369.338i	− 3.73069i
$100$	0	0
$101$	112.986	1.11867	0.559337	−	0.828941i	$-0.311056\pi$
$101$	112.986	1.11867	0.559337	+	0.828941i	$0.311056\pi$
$102$	0	0
$103$	− 74.3028i	− 0.721387i	−0.932684	−	0.360693i	$-0.882540\pi$
$103$	− 74.3028i	− 0.721387i	0.932684	−	0.360693i	$-0.117460\pi$
$104$	0	0
$105$	−134.854	−1.28432
$106$	0	0
$107$	96.9513i	0.906087i	0.891488	+	0.453043i	$0.149662\pi$
$107$	96.9513i	0.906087i	−0.891488	+	0.453043i	$0.850338\pi$
$108$	0	0
$109$	90.6591	0.831735	0.415868	−	0.909425i	$-0.363478\pi$
$109$	90.6591	0.831735	0.415868	+	0.909425i	$0.363478\pi$
$110$	0	0
$111$	179.138i	1.61386i
$112$	0	0
$113$	−172.202	−1.52391	−0.761956	−	0.647629i	$-0.775761\pi$
$113$	−172.202	−1.52391	−0.761956	+	0.647629i	$0.775761\pi$
$114$	0	0
$115$	10.7238i	0.0932505i
$116$	0	0
$117$	73.2880	0.626393
$118$	0	0
$119$	237.824i	1.99852i
$120$	0	0
$121$	−241.476	−1.99567
$122$	0	0
$123$	424.691i	3.45277i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	52.0099i	0.409527i	0.978811	+	0.204764i	$0.0656425\pi$
$127$	52.0099i	0.409527i	−0.978811	+	0.204764i	$0.934357\pi$
$128$	0	0
$129$	−74.0909	−0.574348
$130$	0	0
$131$	− 100.700i	− 0.768702i	−0.923187	−	0.384351i	$-0.874425\pi$
$131$	− 100.700i	− 0.768702i	0.923187	−	0.384351i	$-0.125575\pi$
$132$	0	0
$133$	316.267	2.37795
$134$	0	0
$135$	− 123.920i	− 0.917923i
$136$	0	0
$137$	230.335	1.68128	0.840640	−	0.541594i	$-0.182179\pi$
$137$	230.335	1.68128	0.840640	+	0.541594i	$0.182179\pi$
$138$	0	0
$139$	− 18.1430i	− 0.130525i	−0.997868	−	0.0652625i	$-0.979212\pi$
$139$	− 18.1430i	− 0.130525i	0.997868	−	0.0652625i	$-0.0207885\pi$
$140$	0	0
$141$	−418.755	−2.96989
$142$	0	0
$143$	− 71.9263i	− 0.502981i
$144$	0	0
$145$	66.2909	0.457178
$146$	0	0
$147$	421.377i	2.86651i
$148$	0	0
$149$	142.351	0.955375	0.477688	−	0.878530i	$-0.341475\pi$
$149$	142.351	0.955375	0.477688	+	0.878530i	$0.341475\pi$
$150$	0	0
$151$	− 209.937i	− 1.39031i	−0.718861	−	0.695154i	$-0.755336\pi$
$151$	− 209.937i	− 1.39031i	0.718861	−	0.695154i	$-0.244664\pi$
$152$	0	0
$153$	−407.675	−2.66454
$154$	0	0
$155$	− 20.6713i	− 0.133363i
$156$	0	0
$157$	9.25620	0.0589567	0.0294783	−	0.999565i	$-0.490615\pi$
$157$	9.25620	0.0589567	0.0294783	+	0.999565i	$0.490615\pi$
$158$	0	0
$159$	− 481.609i	− 3.02898i
$160$	0	0
$161$	54.2737	0.337104
$162$	0	0
$163$	− 71.3876i	− 0.437961i	−0.975729	−	0.218981i	$-0.929727\pi$
$163$	− 71.3876i	− 0.437961i	0.975729	−	0.218981i	$-0.0702731\pi$
$164$	0	0
$165$	−226.870	−1.37497
$166$	0	0
$167$	− 93.3125i	− 0.558757i	−0.960181	−	0.279379i	$-0.909872\pi$
$167$	− 93.3125i	− 0.558757i	0.960181	−	0.279379i	$-0.0901285\pi$
$168$	0	0
$169$	−154.728	−0.915548
$170$	0	0
$171$	542.143i	3.17043i
$172$	0	0
$173$	83.3071	0.481544	0.240772	−	0.970582i	$-0.422599\pi$
$173$	83.3071	0.481544	0.240772	+	0.970582i	$0.422599\pi$
$174$	0	0
$175$	56.5842i	0.323338i
$176$	0	0
$177$	−377.146	−2.13077
$178$	0	0
$179$	118.105i	0.659806i	0.944015	+	0.329903i	$0.107016\pi$
$179$	118.105i	0.659806i	−0.944015	+	0.329903i	$0.892984\pi$
$180$	0	0
$181$	305.226	1.68633	0.843166	−	0.537653i	$-0.180689\pi$
$181$	305.226	1.68633	0.843166	+	0.537653i	$0.180689\pi$
$182$	0	0
$183$	− 445.634i	− 2.43516i
$184$	0	0
$185$	75.1658	0.406301
$186$	0	0
$187$	400.100i	2.13957i
$188$	0	0
$189$	−627.163	−3.31832
$190$	0	0
$191$	− 40.6062i	− 0.212598i	−0.994334	−	0.106299i	$-0.966100\pi$
$191$	− 40.6062i	− 0.212598i	0.994334	−	0.106299i	$-0.0339001\pi$
$192$	0	0
$193$	−38.3183	−0.198540	−0.0992702	−	0.995061i	$-0.531651\pi$
$193$	−38.3183	−0.198540	−0.0992702	+	0.995061i	$0.531651\pi$
$194$	0	0
$195$	− 45.0180i	− 0.230862i
$196$	0	0
$197$	−127.763	−0.648541	−0.324271	−	0.945964i	$-0.605119\pi$
$197$	−127.763	−0.648541	−0.324271	+	0.945964i	$0.605119\pi$
$198$	0	0
$199$	− 35.7280i	− 0.179538i	−0.995963	−	0.0897688i	$-0.971387\pi$
$199$	− 35.7280i	− 0.179538i	0.995963	−	0.0897688i	$-0.0286128\pi$
$200$	0	0
$201$	−553.067	−2.75158
$202$	0	0
$203$	− 335.501i	− 1.65272i
$204$	0	0
$205$	178.199	0.869262
$206$	0	0
$207$	93.0355i	0.449447i
$208$	0	0
$209$	532.070	2.54579
$210$	0	0
$211$	− 132.206i	− 0.626571i	−0.949659	−	0.313285i	$-0.898570\pi$
$211$	− 132.206i	− 0.626571i	0.949659	−	0.313285i	$-0.101430\pi$
$212$	0	0
$213$	−210.236	−0.987022
$214$	0	0
$215$	31.0883i	0.144597i
$216$	0	0
$217$	−104.618	−0.482113
$218$	0	0
$219$	− 255.388i	− 1.16616i
$220$	0	0
$221$	−79.3921	−0.359240
$222$	0	0
$223$	321.856i	1.44330i	0.692259	+	0.721649i	$0.256616\pi$
$223$	321.856i	1.44330i	−0.692259	+	0.721649i	$0.743384\pi$
$224$	0	0
$225$	−96.9962	−0.431094
$226$	0	0
$227$	232.519i	1.02431i	0.858892	+	0.512156i	$0.171153\pi$
$227$	232.519i	1.02431i	−0.858892	+	0.512156i	$0.828847\pi$
$228$	0	0
$229$	72.6726	0.317348	0.158674	−	0.987331i	$-0.449278\pi$
$229$	72.6726	0.317348	0.158674	+	0.987331i	$0.449278\pi$
$230$	0	0
$231$	1148.20i	4.97057i
$232$	0	0
$233$	−236.628	−1.01557	−0.507785	−	0.861484i	$-0.669536\pi$
$233$	−236.628	−1.01557	−0.507785	+	0.861484i	$0.669536\pi$
$234$	0	0
$235$	175.708i	0.747693i
$236$	0	0
$237$	−64.8513	−0.273634
$238$	0	0
$239$	102.966i	0.430822i	0.976524	+	0.215411i	$0.0691091\pi$
$239$	102.966i	0.430822i	−0.976524	+	0.215411i	$0.930891\pi$
$240$	0	0
$241$	−153.735	−0.637905	−0.318953	−	0.947771i	$-0.603331\pi$
$241$	−153.735	−0.637905	−0.318953	+	0.947771i	$0.603331\pi$
$242$	0	0
$243$	− 144.654i	− 0.595283i
$244$	0	0
$245$	176.808	0.721666
$246$	0	0
$247$	105.579i	0.427445i
$248$	0	0
$249$	−149.921	−0.602091
$250$	0	0
$251$	− 215.580i	− 0.858885i	−0.903094	−	0.429442i	$-0.858710\pi$
$251$	− 215.580i	− 0.858885i	0.903094	−	0.429442i	$-0.141290\pi$
$252$	0	0
$253$	91.3069	0.360897
$254$	0	0
$255$	250.419i	0.982036i
$256$	0	0
$257$	−306.728	−1.19349	−0.596746	−	0.802430i	$-0.703540\pi$
$257$	−306.728	−1.19349	−0.596746	+	0.802430i	$0.703540\pi$
$258$	0	0
$259$	− 380.418i	− 1.46879i
$260$	0	0
$261$	575.113	2.20350
$262$	0	0
$263$	− 145.779i	− 0.554291i	−0.960828	−	0.277146i	$-0.910612\pi$
$263$	− 145.779i	− 0.554291i	0.960828	−	0.277146i	$-0.0893885\pi$
$264$	0	0
$265$	−202.081	−0.762570
$266$	0	0
$267$	259.124i	0.970501i
$268$	0	0
$269$	184.748	0.686795	0.343398	−	0.939190i	$-0.388422\pi$
$269$	184.748	0.686795	0.343398	+	0.939190i	$0.388422\pi$
$270$	0	0
$271$	− 45.8735i	− 0.169275i	−0.996412	−	0.0846374i	$-0.973027\pi$
$271$	− 45.8735i	− 0.169275i	0.996412	−	0.0846374i	$-0.0269732\pi$
$272$	0	0
$273$	−227.838	−0.834573
$274$	0	0
$275$	95.1940i	0.346160i
$276$	0	0
$277$	−534.048	−1.92797	−0.963985	−	0.265956i	$-0.914312\pi$
$277$	−534.048	−1.92797	−0.963985	+	0.265956i	$0.914312\pi$
$278$	0	0
$279$	− 179.336i	− 0.642782i
$280$	0	0
$281$	428.264	1.52407	0.762035	−	0.647536i	$-0.224201\pi$
$281$	428.264	1.52407	0.762035	+	0.647536i	$0.224201\pi$
$282$	0	0
$283$	− 106.981i	− 0.378026i	−0.981975	−	0.189013i	$-0.939471\pi$
$283$	− 106.981i	− 0.378026i	0.981975	−	0.189013i	$-0.0605289\pi$
$284$	0	0
$285$	333.018	1.16848
$286$	0	0
$287$	− 901.873i	− 3.14241i
$288$	0	0
$289$	152.630	0.528133
$290$	0	0
$291$	242.662i	0.833889i
$292$	0	0
$293$	92.2473	0.314837	0.157419	−	0.987532i	$-0.449683\pi$
$293$	92.2473	0.314837	0.157419	+	0.987532i	$0.449683\pi$
$294$	0	0
$295$	158.249i	0.536437i
$296$	0	0
$297$	−1055.10	−3.55253
$298$	0	0
$299$	18.1181i	0.0605956i
$300$	0	0
$301$	157.339	0.522722
$302$	0	0
$303$	− 602.113i	− 1.98717i
$304$	0	0
$305$	−186.986	−0.613070
$306$	0	0
$307$	− 142.685i	− 0.464773i	−0.972623	−	0.232387i	$-0.925347\pi$
$307$	− 142.685i	− 0.464773i	0.972623	−	0.232387i	$-0.0746535\pi$
$308$	0	0
$309$	−395.967	−1.28145
$310$	0	0
$311$	− 516.777i	− 1.66166i	−0.556525	−	0.830831i	$-0.687866\pi$
$311$	− 516.777i	− 1.66166i	0.556525	−	0.830831i	$-0.312134\pi$
$312$	0	0
$313$	−153.280	−0.489711	−0.244856	−	0.969560i	$-0.578741\pi$
$313$	−153.280	−0.489711	−0.244856	+	0.969560i	$0.578741\pi$
$314$	0	0
$315$	490.902i	1.55842i
$316$	0	0
$317$	77.0144	0.242948	0.121474	−	0.992595i	$-0.461238\pi$
$317$	77.0144	0.242948	0.121474	+	0.992595i	$0.461238\pi$
$318$	0	0
$319$	− 564.428i	− 1.76937i
$320$	0	0
$321$	516.663	1.60954
$322$	0	0
$323$	− 587.298i	− 1.81826i
$324$	0	0
$325$	−18.8894	−0.0581212
$326$	0	0
$327$	− 483.131i	− 1.47746i
$328$	0	0
$329$	889.266	2.70294
$330$	0	0
$331$	− 285.914i	− 0.863789i	−0.901924	−	0.431895i	$-0.857845\pi$
$331$	− 285.914i	− 0.863789i	0.901924	−	0.431895i	$-0.142155\pi$
$332$	0	0
$333$	652.108	1.95828
$334$	0	0
$335$	232.065i	0.692731i
$336$	0	0
$337$	−545.372	−1.61831	−0.809157	−	0.587593i	$-0.800076\pi$
$337$	−545.372	−1.61831	−0.809157	+	0.587593i	$0.800076\pi$
$338$	0	0
$339$	917.681i	2.70702i
$340$	0	0
$341$	−176.004	−0.516141
$342$	0	0
$343$	− 340.309i	− 0.992154i
$344$	0	0
$345$	57.1482	0.165647
$346$	0	0
$347$	− 614.942i	− 1.77217i	−0.463525	−	0.886084i	$-0.653416\pi$
$347$	− 614.942i	− 1.77217i	0.463525	−	0.886084i	$-0.346584\pi$
$348$	0	0
$349$	−164.923	−0.472558	−0.236279	−	0.971685i	$-0.575928\pi$
$349$	−164.923	−0.472558	−0.236279	+	0.971685i	$0.575928\pi$
$350$	0	0
$351$	− 209.364i	− 0.596480i
$352$	0	0
$353$	483.967	1.37101	0.685506	−	0.728067i	$-0.259581\pi$
$353$	483.967	1.37101	0.685506	+	0.728067i	$0.259581\pi$
$354$	0	0
$355$	88.2141i	0.248491i
$356$	0	0
$357$	1267.38	3.55010
$358$	0	0
$359$	269.528i	0.750774i	0.926868	+	0.375387i	$0.122490\pi$
$359$	269.528i	0.750774i	−0.926868	+	0.375387i	$0.877510\pi$
$360$	0	0
$361$	−420.013	−1.16347
$362$	0	0
$363$	1286.85i	3.54504i
$364$	0	0
$365$	−107.160	−0.293589
$366$	0	0
$367$	250.817i	0.683426i	0.939804	+	0.341713i	$0.111007\pi$
$367$	250.817i	0.683426i	−0.939804	+	0.341713i	$0.888993\pi$
$368$	0	0
$369$	1545.98	4.18965
$370$	0	0
$371$	1022.74i	2.75672i
$372$	0	0
$373$	−565.320	−1.51560	−0.757801	−	0.652485i	$-0.773726\pi$
$373$	−565.320	−1.51560	−0.757801	+	0.652485i	$0.773726\pi$
$374$	0	0
$375$	59.5811i	0.158883i
$376$	0	0
$377$	112.000	0.297081
$378$	0	0
$379$	− 42.3592i	− 0.111766i	−0.998437	−	0.0558829i	$-0.982203\pi$
$379$	− 42.3592i	− 0.111766i	0.998437	−	0.0558829i	$-0.0177973\pi$
$380$	0	0
$381$	277.166	0.727469
$382$	0	0
$383$	448.739i	1.17164i	0.810440	+	0.585821i	$0.199228\pi$
$383$	448.739i	1.17164i	−0.810440	+	0.585821i	$0.800772\pi$
$384$	0	0
$385$	481.781	1.25138
$386$	0	0
$387$	269.710i	0.696924i
$388$	0	0
$389$	−51.8271	−0.133232	−0.0666158	−	0.997779i	$-0.521220\pi$
$389$	−51.8271	−0.133232	−0.0666158	+	0.997779i	$0.521220\pi$
$390$	0	0
$391$	− 100.784i	− 0.257761i
$392$	0	0
$393$	−536.640	−1.36549
$394$	0	0
$395$	27.2114i	0.0688895i
$396$	0	0
$397$	310.747	0.782737	0.391369	−	0.920234i	$-0.372002\pi$
$397$	310.747	0.782737	0.391369	+	0.920234i	$0.372002\pi$
$398$	0	0
$399$	− 1685.42i	− 4.22411i
$400$	0	0
$401$	517.956	1.29166	0.645830	−	0.763481i	$-0.276511\pi$
$401$	517.956	1.29166	0.645830	+	0.763481i	$0.276511\pi$
$402$	0	0
$403$	− 34.9246i	− 0.0866615i
$404$	0	0
$405$	−269.977	−0.666609
$406$	0	0
$407$	− 639.992i	− 1.57246i
$408$	0	0
$409$	15.6222	0.0381962	0.0190981	−	0.999818i	$-0.493921\pi$
$409$	15.6222	0.0381962	0.0190981	+	0.999818i	$0.493921\pi$
$410$	0	0
$411$	− 1227.48i	− 2.98657i
$412$	0	0
$413$	800.905	1.93924
$414$	0	0
$415$	62.9061i	0.151581i
$416$	0	0
$417$	−96.6856	−0.231860
$418$	0	0
$419$	380.156i	0.907295i	0.891181	+	0.453647i	$0.149877\pi$
$419$	380.156i	0.907295i	−0.891181	+	0.453647i	$0.850123\pi$
$420$	0	0
$421$	−541.050	−1.28515	−0.642577	−	0.766221i	$-0.722135\pi$
$421$	−541.050	−1.28515	−0.642577	+	0.766221i	$0.722135\pi$
$422$	0	0
$423$	1524.37i	3.60372i
$424$	0	0
$425$	105.075	0.247235
$426$	0	0
$427$	946.347i	2.21627i
$428$	0	0
$429$	−383.302	−0.893478
$430$	0	0
$431$	− 640.139i	− 1.48524i	−0.669712	−	0.742621i	$-0.733583\pi$
$431$	− 640.139i	− 1.48524i	0.669712	−	0.742621i	$-0.266417\pi$
$432$	0	0
$433$	−819.624	−1.89290	−0.946448	−	0.322857i	$-0.895357\pi$
$433$	−819.624	−1.89290	−0.946448	+	0.322857i	$0.895357\pi$
$434$	0	0
$435$	− 353.270i	− 0.812115i
$436$	0	0
$437$	−134.027	−0.306698
$438$	0	0
$439$	− 551.539i	− 1.25635i	−0.778071	−	0.628176i	$-0.783802\pi$
$439$	− 551.539i	− 1.25635i	0.778071	−	0.628176i	$-0.216198\pi$
$440$	0	0
$441$	1533.92	3.47827
$442$	0	0
$443$	− 772.219i	− 1.74316i	−0.490254	−	0.871579i	$-0.663096\pi$
$443$	− 772.219i	− 1.74316i	0.490254	−	0.871579i	$-0.336904\pi$
$444$	0	0
$445$	108.727	0.244331
$446$	0	0
$447$	− 758.601i	− 1.69710i
$448$	0	0
$449$	466.750	1.03953	0.519766	−	0.854308i	$-0.326019\pi$
$449$	466.750	1.03953	0.519766	+	0.854308i	$0.326019\pi$
$450$	0	0
$451$	− 1517.26i	− 3.36421i
$452$	0	0
$453$	−1118.77	−2.46969
$454$	0	0
$455$	95.6001i	0.210110i
$456$	0	0
$457$	−455.249	−0.996169	−0.498084	−	0.867129i	$-0.665963\pi$
$457$	−455.249	−0.996169	−0.498084	+	0.867129i	$0.665963\pi$
$458$	0	0
$459$	1164.62i	2.53730i
$460$	0	0
$461$	58.1877	0.126221	0.0631103	−	0.998007i	$-0.479898\pi$
$461$	58.1877	0.126221	0.0631103	+	0.998007i	$0.479898\pi$
$462$	0	0
$463$	− 447.712i	− 0.966981i	−0.875350	−	0.483491i	$-0.839369\pi$
$463$	− 447.712i	− 0.966981i	0.875350	−	0.483491i	$-0.160631\pi$
$464$	0	0
$465$	−110.159	−0.236902
$466$	0	0
$467$	294.616i	0.630868i	0.948947	+	0.315434i	$0.102150\pi$
$467$	294.616i	0.630868i	−0.948947	+	0.315434i	$0.897850\pi$
$468$	0	0
$469$	1174.49	2.50424
$470$	0	0
$471$	− 49.3271i	− 0.104729i
$472$	0	0
$473$	264.698	0.559616
$474$	0	0
$475$	− 139.733i	− 0.294175i
$476$	0	0
$477$	−1753.18	−3.67542
$478$	0	0
$479$	− 3.99955i	− 0.00834978i	−0.999991	−	0.00417489i	$-0.998671\pi$
$479$	− 3.99955i	− 0.00834978i	0.999991	−	0.00417489i	$-0.00132891\pi$
$480$	0	0
$481$	126.994	0.264021
$482$	0	0
$483$	− 289.230i	− 0.598819i
$484$	0	0
$485$	101.820	0.209938
$486$	0	0
$487$	752.438i	1.54505i	0.634986	+	0.772523i	$0.281006\pi$
$487$	752.438i	1.54505i	−0.634986	+	0.772523i	$0.718994\pi$
$488$	0	0
$489$	−380.431	−0.777978
$490$	0	0
$491$	− 561.310i	− 1.14320i	−0.820533	−	0.571598i	$-0.806324\pi$
$491$	− 561.310i	− 1.14320i	0.820533	−	0.571598i	$-0.193676\pi$
$492$	0	0
$493$	−623.014	−1.26372
$494$	0	0
$495$	825.865i	1.66842i
$496$	0	0
$497$	446.456	0.898302
$498$	0	0
$499$	267.804i	0.536682i	0.963324	+	0.268341i	$0.0864754\pi$
$499$	267.804i	0.536682i	−0.963324	+	0.268341i	$0.913525\pi$
$500$	0	0
$501$	−497.271	−0.992557
$502$	0	0
$503$	979.820i	1.94795i	0.226652	+	0.973976i	$0.427222\pi$
$503$	979.820i	1.94795i	−0.226652	+	0.973976i	$0.572778\pi$
$504$	0	0
$505$	−252.644	−0.500286
$506$	0	0
$507$	824.558i	1.62635i
$508$	0	0
$509$	−641.466	−1.26025	−0.630124	−	0.776495i	$-0.716996\pi$
$509$	−641.466	−1.26025	−0.630124	+	0.776495i	$0.716996\pi$
$510$	0	0
$511$	542.342i	1.06133i
$512$	0	0
$513$	1548.76	3.01902
$514$	0	0
$515$	166.146i	0.322614i
$516$	0	0
$517$	1496.05	2.89371
$518$	0	0
$519$	− 443.951i	− 0.855397i
$520$	0	0
$521$	−521.717	−1.00138	−0.500688	−	0.865628i	$-0.666920\pi$
$521$	−521.717	−1.00138	−0.500688	+	0.865628i	$0.666920\pi$
$522$	0	0
$523$	− 1028.20i	− 1.96596i	−0.183702	−	0.982982i	$-0.558808\pi$
$523$	− 1028.20i	− 1.96596i	0.183702	−	0.982982i	$-0.441192\pi$
$524$	0	0
$525$	301.543	0.574367
$526$	0	0
$527$	194.273i	0.368639i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	1372.91i	2.58551i
$532$	0	0
$533$	301.070	0.564860
$534$	0	0
$535$	− 216.790i	− 0.405214i
$536$	0	0
$537$	629.394	1.17206
$538$	0	0
$539$	− 1505.42i	− 2.79298i
$540$	0	0
$541$	474.515	0.877107	0.438554	−	0.898705i	$-0.355491\pi$
$541$	474.515	0.877107	0.438554	+	0.898705i	$0.355491\pi$
$542$	0	0
$543$	− 1626.58i	− 2.99554i
$544$	0	0
$545$	−202.720	−0.371963
$546$	0	0
$547$	− 176.027i	− 0.321805i	−0.986970	−	0.160902i	$-0.948560\pi$
$547$	− 176.027i	− 0.321805i	0.986970	−	0.160902i	$-0.0514404\pi$
$548$	0	0
$549$	−1622.22	−2.95486
$550$	0	0
$551$	828.510i	1.50365i
$552$	0	0
$553$	137.718	0.249038
$554$	0	0
$555$	− 400.565i	− 0.721740i
$556$	0	0
$557$	−754.395	−1.35439	−0.677194	−	0.735804i	$-0.736804\pi$
$557$	−754.395	−1.35439	−0.677194	+	0.735804i	$0.736804\pi$
$558$	0	0
$559$	52.5242i	0.0939611i
$560$	0	0
$561$	2132.17	3.80066
$562$	0	0
$563$	637.291i	1.13195i	0.824421	+	0.565977i	$0.191501\pi$
$563$	637.291i	1.13195i	−0.824421	+	0.565977i	$0.808499\pi$
$564$	0	0
$565$	385.056	0.681514
$566$	0	0
$567$	1366.37i	2.40982i
$568$	0	0
$569$	−448.170	−0.787645	−0.393822	−	0.919187i	$-0.628847\pi$
$569$	−448.170	−0.787645	−0.393822	+	0.919187i	$0.628847\pi$
$570$	0	0
$571$	− 477.970i	− 0.837076i	−0.908199	−	0.418538i	$-0.862543\pi$
$571$	− 477.970i	− 0.837076i	0.908199	−	0.418538i	$-0.137457\pi$
$572$	0	0
$573$	−216.394	−0.377652
$574$	0	0
$575$	− 23.9792i	− 0.0417029i
$576$	0	0
$577$	256.397	0.444363	0.222181	−	0.975005i	$-0.428682\pi$
$577$	256.397	0.444363	0.222181	+	0.975005i	$0.428682\pi$
$578$	0	0
$579$	204.202i	0.352680i
$580$	0	0
$581$	318.371	0.547971
$582$	0	0
$583$	1720.60i	2.95129i
$584$	0	0
$585$	−163.877	−0.280131
$586$	0	0
$587$	− 35.9297i	− 0.0612091i	−0.999532	−	0.0306046i	$-0.990257\pi$
$587$	− 35.9297i	− 0.0612091i	0.999532	−	0.0306046i	$-0.00974325\pi$
$588$	0	0
$589$	258.352	0.438628
$590$	0	0
$591$	680.859i	1.15205i
$592$	0	0
$593$	−177.642	−0.299565	−0.149782	−	0.988719i	$-0.547857\pi$
$593$	−177.642	−0.299565	−0.149782	+	0.988719i	$0.547857\pi$
$594$	0	0
$595$	− 531.790i	− 0.893764i
$596$	0	0
$597$	−190.398	−0.318924
$598$	0	0
$599$	213.744i	0.356834i	0.983955	+	0.178417i	$0.0570976\pi$
$599$	213.744i	0.356834i	−0.983955	+	0.178417i	$0.942902\pi$
$600$	0	0
$601$	−21.9561	−0.0365326	−0.0182663	−	0.999833i	$-0.505815\pi$
$601$	−21.9561	−0.0365326	−0.0182663	+	0.999833i	$0.505815\pi$
$602$	0	0
$603$	2013.30i	3.33881i
$604$	0	0
$605$	539.957	0.892491
$606$	0	0
$607$	352.100i	0.580065i	0.957017	+	0.290033i	$0.0936662\pi$
$607$	352.100i	0.580065i	−0.957017	+	0.290033i	$0.906334\pi$
$608$	0	0
$609$	−1787.92	−2.93582
$610$	0	0
$611$	296.862i	0.485862i
$612$	0	0
$613$	115.896	0.189063	0.0945316	−	0.995522i	$-0.469865\pi$
$613$	115.896	0.189063	0.0945316	+	0.995522i	$0.469865\pi$
$614$	0	0
$615$	− 949.638i	− 1.54413i
$616$	0	0
$617$	644.845	1.04513	0.522565	−	0.852600i	$-0.324975\pi$
$617$	644.845	1.04513	0.522565	+	0.852600i	$0.324975\pi$
$618$	0	0
$619$	89.8983i	0.145232i	0.997360	+	0.0726158i	$0.0231347\pi$
$619$	89.8983i	0.145232i	−0.997360	+	0.0726158i	$0.976865\pi$
$620$	0	0
$621$	265.778	0.427984
$622$	0	0
$623$	− 550.274i	− 0.883265i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 2835.45i	− 4.52225i
$628$	0	0
$629$	−706.423	−1.12309
$630$	0	0
$631$	435.036i	0.689439i	0.938706	+	0.344720i	$0.112026\pi$
$631$	435.036i	0.689439i	−0.938706	+	0.344720i	$0.887974\pi$
$632$	0	0
$633$	−704.541	−1.11302
$634$	0	0
$635$	− 116.298i	− 0.183146i
$636$	0	0
$637$	298.721	0.468949
$638$	0	0
$639$	765.311i	1.19767i
$640$	0	0
$641$	821.910	1.28223	0.641115	−	0.767444i	$-0.278472\pi$
$641$	821.910	1.28223	0.641115	+	0.767444i	$0.278472\pi$
$642$	0	0
$643$	− 319.976i	− 0.497630i	−0.968551	−	0.248815i	$-0.919959\pi$
$643$	− 319.976i	− 0.497630i	0.968551	−	0.248815i	$-0.0800412\pi$
$644$	0	0
$645$	165.672	0.256856
$646$	0	0
$647$	− 278.474i	− 0.430407i	−0.976569	−	0.215204i	$-0.930958\pi$
$647$	− 278.474i	− 0.430407i	0.976569	−	0.215204i	$-0.0690415\pi$
$648$	0	0
$649$	1347.40	2.07611
$650$	0	0
$651$	557.522i	0.856408i
$652$	0	0
$653$	321.834	0.492855	0.246427	−	0.969161i	$-0.420743\pi$
$653$	321.834	0.492855	0.246427	+	0.969161i	$0.420743\pi$
$654$	0	0
$655$	225.172i	0.343774i
$656$	0	0
$657$	−929.677	−1.41503
$658$	0	0
$659$	282.236i	0.428279i	0.976803	+	0.214140i	$0.0686947\pi$
$659$	282.236i	0.428279i	−0.976803	+	0.214140i	$0.931305\pi$
$660$	0	0
$661$	−108.571	−0.164252	−0.0821262	−	0.996622i	$-0.526171\pi$
$661$	−108.571	−0.164252	−0.0821262	+	0.996622i	$0.526171\pi$
$662$	0	0
$663$	423.088i	0.638142i
$664$	0	0
$665$	−707.196	−1.06345
$666$	0	0
$667$	142.178i	0.213161i
$668$	0	0
$669$	1715.20	2.56382
$670$	0	0
$671$	1592.08i	2.37270i
$672$	0	0
$673$	−394.255	−0.585817	−0.292909	−	0.956140i	$-0.594623\pi$
$673$	−394.255	−0.585817	−0.292909	+	0.956140i	$0.594623\pi$
$674$	0	0
$675$	277.093i	0.410507i
$676$	0	0
$677$	−806.247	−1.19091	−0.595456	−	0.803388i	$-0.703029\pi$
$677$	−806.247	−1.19091	−0.595456	+	0.803388i	$0.703029\pi$
$678$	0	0
$679$	− 515.316i	− 0.758933i
$680$	0	0
$681$	1239.11	1.81955
$682$	0	0
$683$	− 351.644i	− 0.514852i	−0.966298	−	0.257426i	$-0.917126\pi$
$683$	− 351.644i	− 0.514852i	0.966298	−	0.257426i	$-0.0828744\pi$
$684$	0	0
$685$	−515.046	−0.751892
$686$	0	0
$687$	− 387.279i	− 0.563725i
$688$	0	0
$689$	−341.420	−0.495530
$690$	0	0
$691$	172.967i	0.250314i	0.992137	+	0.125157i	$0.0399435\pi$
$691$	172.967i	0.250314i	−0.992137	+	0.125157i	$0.960056\pi$
$692$	0	0
$693$	4179.74	6.03138
$694$	0	0
$695$	40.5689i	0.0583726i
$696$	0	0
$697$	−1674.75	−2.40279
$698$	0	0
$699$	1261.01i	1.80402i
$700$	0	0
$701$	85.5406	0.122026	0.0610132	−	0.998137i	$-0.480567\pi$
$701$	85.5406	0.122026	0.0610132	+	0.998137i	$0.480567\pi$
$702$	0	0
$703$	939.430i	1.33632i
$704$	0	0
$705$	936.364	1.32818
$706$	0	0
$707$	1278.65i	1.80855i
$708$	0	0
$709$	−186.008	−0.262353	−0.131177	−	0.991359i	$-0.541875\pi$
$709$	−186.008	−0.262353	−0.131177	+	0.991359i	$0.541875\pi$
$710$	0	0
$711$	236.075i	0.332032i
$712$	0	0
$713$	44.3350	0.0621809
$714$	0	0
$715$	160.832i	0.224940i
$716$	0	0
$717$	548.718	0.765296
$718$	0	0
$719$	187.936i	0.261386i	0.991423	+	0.130693i	$0.0417202\pi$
$719$	187.936i	0.261386i	−0.991423	+	0.130693i	$0.958280\pi$
$720$	0	0
$721$	840.874	1.16626
$722$	0	0
$723$	819.269i	1.13315i
$724$	0	0
$725$	−148.231	−0.204456
$726$	0	0
$727$	− 836.798i	− 1.15103i	−0.817791	−	0.575515i	$-0.804802\pi$
$727$	− 836.798i	− 1.15103i	0.817791	−	0.575515i	$-0.195198\pi$
$728$	0	0
$729$	315.762	0.433145
$730$	0	0
$731$	− 292.174i	− 0.399690i
$732$	0	0
$733$	795.611	1.08542	0.542709	−	0.839921i	$-0.317399\pi$
$733$	795.611	1.08542	0.542709	+	0.839921i	$0.317399\pi$
$734$	0	0
$735$	− 942.227i	− 1.28194i
$736$	0	0
$737$	1975.89	2.68100
$738$	0	0
$739$	− 722.352i	− 0.977472i	−0.872432	−	0.488736i	$-0.837458\pi$
$739$	− 722.352i	− 0.977472i	0.872432	−	0.488736i	$-0.162542\pi$
$740$	0	0
$741$	562.640	0.759298
$742$	0	0
$743$	− 310.124i	− 0.417395i	−0.977980	−	0.208698i	$-0.933078\pi$
$743$	− 310.124i	− 0.417395i	0.977980	−	0.208698i	$-0.0669224\pi$
$744$	0	0
$745$	−318.306	−0.427257
$746$	0	0
$747$	545.749i	0.730587i
$748$	0	0
$749$	−1097.18	−1.46486
$750$	0	0
$751$	− 149.482i	− 0.199044i	−0.995035	−	0.0995219i	$-0.968269\pi$
$751$	− 149.482i	− 0.199044i	0.995035	−	0.0995219i	$-0.0317313\pi$
$752$	0	0
$753$	−1148.85	−1.52569
$754$	0	0
$755$	469.432i	0.621765i
$756$	0	0
$757$	−971.334	−1.28314	−0.641568	−	0.767066i	$-0.721716\pi$
$757$	−971.334	−1.28314	−0.641568	+	0.767066i	$0.721716\pi$
$758$	0	0
$759$	− 486.583i	− 0.641084i
$760$	0	0
$761$	785.612	1.03234	0.516171	−	0.856486i	$-0.327357\pi$
$761$	785.612	1.03234	0.516171	+	0.856486i	$0.327357\pi$
$762$	0	0
$763$	1025.98i	1.34466i
$764$	0	0
$765$	911.589	1.19162
$766$	0	0
$767$	267.365i	0.348585i
$768$	0	0
$769$	−1053.36	−1.36978	−0.684891	−	0.728646i	$-0.740150\pi$
$769$	−1053.36	−1.36978	−0.684891	+	0.728646i	$0.740150\pi$
$770$	0	0
$771$	1634.58i	2.12008i
$772$	0	0
$773$	112.568	0.145625	0.0728127	−	0.997346i	$-0.476802\pi$
$773$	112.568	0.145625	0.0728127	+	0.997346i	$0.476802\pi$
$774$	0	0
$775$	46.2224i	0.0596419i
$776$	0	0
$777$	−2027.28	−2.60911
$778$	0	0
$779$	2227.15i	2.85898i
$780$	0	0
$781$	751.091	0.961705
$782$	0	0
$783$	− 1642.95i	− 2.09827i
$784$	0	0
$785$	−20.6975	−0.0263662
$786$	0	0
$787$	− 618.262i	− 0.785594i	−0.919625	−	0.392797i	$-0.871508\pi$
$787$	− 618.262i	− 0.785594i	0.919625	−	0.392797i	$-0.128492\pi$
$788$	0	0
$789$	−776.868	−0.984624
$790$	0	0
$791$	− 1948.78i	− 2.46370i
$792$	0	0
$793$	−315.917	−0.398382
$794$	0	0
$795$	1076.91i	1.35460i
$796$	0	0
$797$	656.023	0.823115	0.411558	−	0.911384i	$-0.364985\pi$
$797$	656.023	0.823115	0.411558	+	0.911384i	$0.364985\pi$
$798$	0	0
$799$	− 1651.34i	− 2.06675i
$800$	0	0
$801$	943.275	1.17762
$802$	0	0
$803$	912.404i	1.13624i
$804$	0	0
$805$	−121.360	−0.150757
$806$	0	0
$807$	− 984.539i	− 1.22000i
$808$	0	0
$809$	−505.323	−0.624626	−0.312313	−	0.949979i	$-0.601104\pi$
$809$	−505.323	−0.624626	−0.312313	+	0.949979i	$0.601104\pi$
$810$	0	0
$811$	− 316.877i	− 0.390724i	−0.980731	−	0.195362i	$-0.937412\pi$
$811$	− 316.877i	− 0.390724i	0.980731	−	0.195362i	$-0.0625881\pi$
$812$	0	0
$813$	−244.464	−0.300694
$814$	0	0
$815$	159.628i	0.195862i
$816$	0	0
$817$	−388.545	−0.475575
$818$	0	0
$819$	829.389i	1.01268i
$820$	0	0
$821$	278.303	0.338980	0.169490	−	0.985532i	$-0.445788\pi$
$821$	278.303	0.338980	0.169490	+	0.985532i	$0.445788\pi$
$822$	0	0
$823$	− 754.199i	− 0.916402i	−0.888849	−	0.458201i	$-0.848494\pi$
$823$	− 754.199i	− 0.916402i	0.888849	−	0.458201i	$-0.151506\pi$
$824$	0	0
$825$	507.298	0.614906
$826$	0	0
$827$	163.841i	0.198115i	0.995082	+	0.0990575i	$0.0315828\pi$
$827$	163.841i	0.198115i	−0.995082	+	0.0990575i	$0.968417\pi$
$828$	0	0
$829$	−590.539	−0.712351	−0.356176	−	0.934419i	$-0.615919\pi$
$829$	−590.539	−0.712351	−0.356176	+	0.934419i	$0.615919\pi$
$830$	0	0
$831$	2845.99i	3.42478i
$832$	0	0
$833$	−1661.68	−1.99481
$834$	0	0
$835$	208.653i	0.249884i
$836$	0	0
$837$	−512.316	−0.612086
$838$	0	0
$839$	1542.44i	1.83843i	0.393761	+	0.919213i	$0.371174\pi$
$839$	1542.44i	1.83843i	−0.393761	+	0.919213i	$0.628826\pi$
$840$	0	0
$841$	37.8958	0.0450604
$842$	0	0
$843$	− 2282.26i	− 2.70730i
$844$	0	0
$845$	345.981	0.409446
$846$	0	0
$847$	− 2732.75i	− 3.22638i
$848$	0	0
$849$	−570.114	−0.671513
$850$	0	0
$851$	161.213i	0.189439i
$852$	0	0
$853$	1174.20	1.37656	0.688278	−	0.725447i	$-0.258367\pi$
$853$	1174.20	1.37656	0.688278	+	0.725447i	$0.258367\pi$
$854$	0	0
$855$	− 1212.27i	− 1.41786i
$856$	0	0
$857$	−598.531	−0.698402	−0.349201	−	0.937048i	$-0.613547\pi$
$857$	−598.531	−0.698402	−0.349201	+	0.937048i	$0.613547\pi$
$858$	0	0
$859$	780.891i	0.909069i	0.890729	+	0.454535i	$0.150194\pi$
$859$	780.891i	0.909069i	−0.890729	+	0.454535i	$0.849806\pi$
$860$	0	0
$861$	−4806.16	−5.58207
$862$	0	0
$863$	776.579i	0.899860i	0.893064	+	0.449930i	$0.148551\pi$
$863$	776.579i	0.899860i	−0.893064	+	0.449930i	$0.851449\pi$
$864$	0	0
$865$	−186.280	−0.215353
$866$	0	0
$867$	− 813.381i	− 0.938156i
$868$	0	0
$869$	231.689	0.266615
$870$	0	0
$871$	392.078i	0.450147i
$872$	0	0
$873$	883.350	1.01186
$874$	0	0
$875$	− 126.526i	− 0.144601i
$876$	0	0
$877$	357.377	0.407499	0.203749	−	0.979023i	$-0.434687\pi$
$877$	357.377	0.407499	0.203749	+	0.979023i	$0.434687\pi$
$878$	0	0
$879$	− 491.595i	− 0.559266i
$880$	0	0
$881$	1742.16	1.97748	0.988739	−	0.149647i	$-0.0478137\pi$
$881$	1742.16	1.97748	0.988739	+	0.149647i	$0.0478137\pi$
$882$	0	0
$883$	899.024i	1.01815i	0.860723	+	0.509074i	$0.170012\pi$
$883$	899.024i	1.01815i	−0.860723	+	0.509074i	$0.829988\pi$
$884$	0	0
$885$	843.323	0.952908
$886$	0	0
$887$	− 411.206i	− 0.463592i	−0.972764	−	0.231796i	$-0.925540\pi$
$887$	− 411.206i	− 0.463592i	0.972764	−	0.231796i	$-0.0744602\pi$
$888$	0	0
$889$	−588.588	−0.662079
$890$	0	0
$891$	2298.69i	2.57990i
$892$	0	0
$893$	−2196.02	−2.45914
$894$	0	0
$895$	− 264.091i	− 0.295074i
$896$	0	0
$897$	96.5529	0.107640
$898$	0	0
$899$	− 274.064i	− 0.304854i
$900$	0	0
$901$	1899.20	2.10788
$902$	0	0
$903$	− 838.476i	− 0.928545i
$904$	0	0
$905$	−682.506	−0.754151
$906$	0	0
$907$	815.984i	0.899652i	0.893116	+	0.449826i	$0.148514\pi$
$907$	815.984i	0.899652i	−0.893116	+	0.449826i	$0.851486\pi$
$908$	0	0
$909$	−2191.84	−2.41127
$910$	0	0
$911$	905.930i	0.994435i	0.867626	+	0.497218i	$0.165645\pi$
$911$	905.930i	0.994435i	−0.867626	+	0.497218i	$0.834355\pi$
$912$	0	0
$913$	535.609	0.586647
$914$	0	0
$915$	996.468i	1.08904i
$916$	0	0
$917$	1139.61	1.24275
$918$	0	0
$919$	− 1040.14i	− 1.13181i	−0.824470	−	0.565906i	$-0.808526\pi$
$919$	− 1040.14i	− 1.13181i	0.824470	−	0.565906i	$-0.191474\pi$
$920$	0	0
$921$	−760.384	−0.825607
$922$	0	0
$923$	149.039i	0.161473i
$924$	0	0
$925$	−168.076	−0.181704
$926$	0	0
$927$	1441.42i	1.55493i
$928$	0	0
$929$	636.689	0.685349	0.342674	−	0.939454i	$-0.388667\pi$
$929$	636.689	0.685349	0.342674	+	0.939454i	$0.388667\pi$
$930$	0	0
$931$	2209.77i	2.37354i
$932$	0	0
$933$	−2753.95	−2.95172
$934$	0	0
$935$	− 894.652i	− 0.956847i
$936$	0	0
$937$	390.978	0.417265	0.208633	−	0.977994i	$-0.433099\pi$
$937$	390.978	0.417265	0.208633	+	0.977994i	$0.433099\pi$
$938$	0	0
$939$	816.842i	0.869906i
$940$	0	0
$941$	−957.295	−1.01732	−0.508658	−	0.860969i	$-0.669858\pi$
$941$	−957.295	−1.01732	−0.508658	+	0.860969i	$0.669858\pi$
$942$	0	0
$943$	382.194i	0.405296i
$944$	0	0
$945$	1402.38	1.48400
$946$	0	0
$947$	− 271.177i	− 0.286353i	−0.989697	−	0.143177i	$-0.954268\pi$
$947$	− 271.177i	− 0.286353i	0.989697	−	0.143177i	$-0.0457317\pi$
$948$	0	0
$949$	−181.049	−0.190779
$950$	0	0
$951$	− 410.417i	− 0.431563i
$952$	0	0
$953$	805.995	0.845745	0.422872	−	0.906189i	$-0.361022\pi$
$953$	805.995	0.845745	0.422872	+	0.906189i	$0.361022\pi$
$954$	0	0
$955$	90.7983i	0.0950768i
$956$	0	0
$957$	−3007.89	−3.14304
$958$	0	0
$959$	2606.67i	2.71811i
$960$	0	0
$961$	875.539	0.911071
$962$	0	0
$963$	− 1880.78i	− 1.95304i
$964$	0	0
$965$	85.6823	0.0887899
$966$	0	0
$967$	305.659i	0.316090i	0.987432	+	0.158045i	$0.0505191\pi$
$967$	305.659i	0.316090i	−0.987432	+	0.158045i	$0.949481\pi$
$968$	0	0
$969$	−3129.77	−3.22989
$970$	0	0
$971$	− 236.179i	− 0.243233i	−0.992577	−	0.121617i	$-0.961192\pi$
$971$	− 236.179i	− 0.243233i	0.992577	−	0.121617i	$-0.0388078\pi$
$972$	0	0
$973$	205.321	0.211019
$974$	0	0
$975$	100.663i	0.103244i
$976$	0	0
$977$	1281.83	1.31201	0.656003	−	0.754758i	$-0.272246\pi$
$977$	1281.83	1.31201	0.656003	+	0.754758i	$0.272246\pi$
$978$	0	0
$979$	− 925.749i	− 0.945607i
$980$	0	0
$981$	−1758.72	−1.79278
$982$	0	0
$983$	− 1188.67i	− 1.20923i	−0.796517	−	0.604616i	$-0.793327\pi$
$983$	− 1188.67i	− 1.20923i	0.796517	−	0.604616i	$-0.206673\pi$
$984$	0	0
$985$	285.686	0.290037
$986$	0	0
$987$	− 4738.98i	− 4.80140i
$988$	0	0
$989$	−66.6769	−0.0674185
$990$	0	0
$991$	− 1616.74i	− 1.63143i	−0.578456	−	0.815714i	$-0.696344\pi$
$991$	− 1616.74i	− 1.63143i	0.578456	−	0.815714i	$-0.303656\pi$
$992$	0	0
$993$	−1523.66	−1.53440
$994$	0	0
$995$	79.8902i	0.0802916i
$996$	0	0
$997$	−671.210	−0.673230	−0.336615	−	0.941642i	$-0.609282\pi$
$997$	−671.210	−0.673230	−0.336615	+	0.941642i	$0.609282\pi$
$998$	0	0
$999$	− 1862.90i	− 1.86477i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.1	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.56	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.1	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.56	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)

\(f(q)\)	\(=\)	\(q-5.32909i q^{3} -2.23607 q^{5} +11.3168i q^{7} -19.3992 q^{9} +O(q^{10})\)	\(q-5.32909i q^{3} -2.23607 q^{5} +11.3168i q^{7} -19.3992 q^{9} +19.0388i q^{11} -3.77788 q^{13} +11.9162i q^{15} +21.0150 q^{17} -27.9466i q^{19} +60.3085 q^{21} -4.79583i q^{23} +5.00000 q^{25} +55.4185i q^{27} -29.6462 q^{29} +9.24449i q^{31} +101.460 q^{33} -25.3052i q^{35} -33.6152 q^{37} +20.1327i q^{39} -79.6929 q^{41} -13.9031i q^{43} +43.3780 q^{45} -78.5790i q^{47} -79.0710 q^{49} -111.991i q^{51} +90.3734 q^{53} -42.5721i q^{55} -148.930 q^{57} -70.7711i q^{59} +83.6229 q^{61} -219.538i q^{63} +8.44760 q^{65} -103.782i q^{67} -25.5574 q^{69} -39.4506i q^{71} +47.9234 q^{73} -26.6455i q^{75} -215.459 q^{77} -12.1693i q^{79} +120.737 q^{81} -28.1325i q^{83} -46.9910 q^{85} +157.987i q^{87} -48.6243 q^{89} -42.7537i q^{91} +49.2648 q^{93} +62.4905i q^{95} -45.5353 q^{97} -369.338i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more