LMFDB - Embedded newform 4368.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.23607 q^{15} -2.47214 q^{17} +6.47214 q^{19} -1.00000 q^{21} -4.47214 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -3.23607 q^{33} +1.23607 q^{35} +4.47214 q^{37} +1.00000 q^{39} +5.23607 q^{41} -4.00000 q^{43} -1.23607 q^{45} -7.70820 q^{47} +1.00000 q^{49} -2.47214 q^{51} +10.0000 q^{53} +4.00000 q^{55} +6.47214 q^{57} -9.23607 q^{59} +14.9443 q^{61} -1.00000 q^{63} -1.23607 q^{65} +2.47214 q^{67} -4.47214 q^{69} +5.70820 q^{71} +4.47214 q^{73} -3.47214 q^{75} +3.23607 q^{77} +10.4721 q^{79} +1.00000 q^{81} -2.76393 q^{83} +3.05573 q^{85} +8.47214 q^{87} -10.1803 q^{89} -1.00000 q^{91} -8.00000 q^{95} +6.94427 q^{97} -3.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 2 q^{35} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} - 2 q^{47} + 2 q^{49} + 4 q^{51} + 20 q^{53} + 8 q^{55} + 4 q^{57} - 14 q^{59} + 12 q^{61} - 2 q^{63} + 2 q^{65} - 4 q^{67} - 2 q^{71} + 2 q^{75} + 2 q^{77} + 12 q^{79} + 2 q^{81} - 10 q^{83} + 24 q^{85} + 8 q^{87} + 2 q^{89} - 2 q^{91} - 16 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 2 * q^35 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^55 + 4 * q^57 - 14 * q^59 + 12 * q^61 - 2 * q^63 + 2 * q^65 - 4 * q^67 - 2 * q^71 + 2 * q^75 + 2 * q^77 + 12 * q^79 + 2 * q^81 - 10 * q^83 + 24 * q^85 + 8 * q^87 + 2 * q^89 - 2 * q^91 - 16 * q^95 - 4 * q^97 - 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	−2.47214	−0.599581	−0.299791	−	0.954005i	$-0.596917\pi$
$17$	−2.47214	−0.599581	−0.299791	+	0.954005i	$0.596917\pi$
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−3.23607	−0.563327
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.23607	0.817736	0.408868	−	0.912593i	$-0.365924\pi$
$41$	5.23607	0.817736	0.408868	+	0.912593i	$0.365924\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.23607	−0.184262
$46$	0	0
$47$	−7.70820	−1.12436	−0.562179	−	0.827016i	$-0.690037\pi$
$47$	−7.70820	−1.12436	−0.562179	+	0.827016i	$0.690037\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.47214	−0.346168
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	6.47214	0.857255
$58$	0	0
$59$	−9.23607	−1.20243	−0.601217	−	0.799086i	$-0.705317\pi$
$59$	−9.23607	−1.20243	−0.601217	+	0.799086i	$0.705317\pi$
$60$	0	0
$61$	14.9443	1.91342	0.956709	−	0.291046i	$-0.0940034\pi$
$61$	14.9443	1.91342	0.956709	+	0.291046i	$0.0940034\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−1.23607	−0.153315
$66$	0	0
$67$	2.47214	0.302019	0.151010	−	0.988532i	$-0.451748\pi$
$67$	2.47214	0.302019	0.151010	+	0.988532i	$0.451748\pi$
$68$	0	0
$69$	−4.47214	−0.538382
$70$	0	0
$71$	5.70820	0.677439	0.338720	−	0.940887i	$-0.390006\pi$
$71$	5.70820	0.677439	0.338720	+	0.940887i	$0.390006\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	−3.47214	−0.400928
$76$	0	0
$77$	3.23607	0.368784
$78$	0	0
$79$	10.4721	1.17821	0.589104	−	0.808057i	$-0.299481\pi$
$79$	10.4721	1.17821	0.589104	+	0.808057i	$0.299481\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.76393	−0.303381	−0.151690	−	0.988428i	$-0.548472\pi$
$83$	−2.76393	−0.303381	−0.151690	+	0.988428i	$0.548472\pi$
$84$	0	0
$85$	3.05573	0.331440
$86$	0	0
$87$	8.47214	0.908308
$88$	0	0
$89$	−10.1803	−1.07911	−0.539557	−	0.841949i	$-0.681408\pi$
$89$	−10.1803	−1.07911	−0.539557	+	0.841949i	$0.681408\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	6.94427	0.705084	0.352542	−	0.935796i	$-0.385317\pi$
$97$	6.94427	0.705084	0.352542	+	0.935796i	$0.385317\pi$
$98$	0	0
$99$	−3.23607	−0.325237
$100$	0	0
$101$	−8.94427	−0.889988	−0.444994	−	0.895533i	$-0.646794\pi$
$101$	−8.94427	−0.889988	−0.444994	+	0.895533i	$0.646794\pi$
$102$	0	0
$103$	16.9443	1.66957	0.834784	−	0.550577i	$-0.185592\pi$
$103$	16.9443	1.66957	0.834784	+	0.550577i	$0.185592\pi$
$104$	0	0
$105$	1.23607	0.120628
$106$	0	0
$107$	3.52786	0.341051	0.170526	−	0.985353i	$-0.445453\pi$
$107$	3.52786	0.341051	0.170526	+	0.985353i	$0.445453\pi$
$108$	0	0
$109$	3.52786	0.337908	0.168954	−	0.985624i	$-0.445961\pi$
$109$	3.52786	0.337908	0.168954	+	0.985624i	$0.445961\pi$
$110$	0	0
$111$	4.47214	0.424476
$112$	0	0
$113$	20.4721	1.92586	0.962928	−	0.269758i	$-0.0869436\pi$
$113$	20.4721	1.92586	0.962928	+	0.269758i	$0.0869436\pi$
$114$	0	0
$115$	5.52786	0.515476
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.47214	0.226620
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	5.23607	0.472120
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	2.47214	0.219367	0.109683	−	0.993967i	$-0.465016\pi$
$127$	2.47214	0.219367	0.109683	+	0.993967i	$0.465016\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	1.52786	0.133490	0.0667451	−	0.997770i	$-0.478739\pi$
$131$	1.52786	0.133490	0.0667451	+	0.997770i	$0.478739\pi$
$132$	0	0
$133$	−6.47214	−0.561205
$134$	0	0
$135$	−1.23607	−0.106384
$136$	0	0
$137$	15.2361	1.30171	0.650853	−	0.759204i	$-0.274412\pi$
$137$	15.2361	1.30171	0.650853	+	0.759204i	$0.274412\pi$
$138$	0	0
$139$	−17.8885	−1.51729	−0.758643	−	0.651506i	$-0.774137\pi$
$139$	−17.8885	−1.51729	−0.758643	+	0.651506i	$0.774137\pi$
$140$	0	0
$141$	−7.70820	−0.649148
$142$	0	0
$143$	−3.23607	−0.270614
$144$	0	0
$145$	−10.4721	−0.869664
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−12.7639	−1.04566	−0.522831	−	0.852436i	$-0.675124\pi$
$149$	−12.7639	−1.04566	−0.522831	+	0.852436i	$0.675124\pi$
$150$	0	0
$151$	3.05573	0.248672	0.124336	−	0.992240i	$-0.460320\pi$
$151$	3.05573	0.248672	0.124336	+	0.992240i	$0.460320\pi$
$152$	0	0
$153$	−2.47214	−0.199860
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	4.47214	0.352454
$162$	0	0
$163$	2.47214	0.193633	0.0968163	−	0.995302i	$-0.469134\pi$
$163$	2.47214	0.193633	0.0968163	+	0.995302i	$0.469134\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	18.1803	1.40684	0.703418	−	0.710776i	$-0.251656\pi$
$167$	18.1803	1.40684	0.703418	+	0.710776i	$0.251656\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.47214	0.494937
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	0	0
$177$	−9.23607	−0.694225
$178$	0	0
$179$	6.94427	0.519039	0.259520	−	0.965738i	$-0.416436\pi$
$179$	6.94427	0.519039	0.259520	+	0.965738i	$0.416436\pi$
$180$	0	0
$181$	19.8885	1.47830	0.739152	−	0.673539i	$-0.235227\pi$
$181$	19.8885	1.47830	0.739152	+	0.673539i	$0.235227\pi$
$182$	0	0
$183$	14.9443	1.10471
$184$	0	0
$185$	−5.52786	−0.406417
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	6.94427	0.502470	0.251235	−	0.967926i	$-0.419163\pi$
$191$	6.94427	0.502470	0.251235	+	0.967926i	$0.419163\pi$
$192$	0	0
$193$	18.3607	1.32163	0.660815	−	0.750549i	$-0.270211\pi$
$193$	18.3607	1.32163	0.660815	+	0.750549i	$0.270211\pi$
$194$	0	0
$195$	−1.23607	−0.0885167
$196$	0	0
$197$	0.763932	0.0544279	0.0272140	−	0.999630i	$-0.491336\pi$
$197$	0.763932	0.0544279	0.0272140	+	0.999630i	$0.491336\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	2.47214	0.174371
$202$	0	0
$203$	−8.47214	−0.594627
$204$	0	0
$205$	−6.47214	−0.452034
$206$	0	0
$207$	−4.47214	−0.310835
$208$	0	0
$209$	−20.9443	−1.44875
$210$	0	0
$211$	−20.3607	−1.40169	−0.700844	−	0.713315i	$-0.747193\pi$
$211$	−20.3607	−1.40169	−0.700844	+	0.713315i	$0.747193\pi$
$212$	0	0
$213$	5.70820	0.391120
$214$	0	0
$215$	4.94427	0.337197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.47214	0.302199
$220$	0	0
$221$	−2.47214	−0.166294
$222$	0	0
$223$	10.4721	0.701266	0.350633	−	0.936513i	$-0.385966\pi$
$223$	10.4721	0.701266	0.350633	+	0.936513i	$0.385966\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	0	0
$227$	6.76393	0.448938	0.224469	−	0.974481i	$-0.427935\pi$
$227$	6.76393	0.448938	0.224469	+	0.974481i	$0.427935\pi$
$228$	0	0
$229$	−25.4164	−1.67956	−0.839782	−	0.542924i	$-0.817317\pi$
$229$	−25.4164	−1.67956	−0.839782	+	0.542924i	$0.817317\pi$
$230$	0	0
$231$	3.23607	0.212918
$232$	0	0
$233$	5.41641	0.354841	0.177420	−	0.984135i	$-0.443225\pi$
$233$	5.41641	0.354841	0.177420	+	0.984135i	$0.443225\pi$
$234$	0	0
$235$	9.52786	0.621529
$236$	0	0
$237$	10.4721	0.680238
$238$	0	0
$239$	−21.1246	−1.36644	−0.683219	−	0.730214i	$-0.739420\pi$
$239$	−21.1246	−1.36644	−0.683219	+	0.730214i	$0.739420\pi$
$240$	0	0
$241$	−16.4721	−1.06106	−0.530532	−	0.847665i	$-0.678008\pi$
$241$	−16.4721	−1.06106	−0.530532	+	0.847665i	$0.678008\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.23607	−0.0789695
$246$	0	0
$247$	6.47214	0.411812
$248$	0	0
$249$	−2.76393	−0.175157
$250$	0	0
$251$	−5.52786	−0.348916	−0.174458	−	0.984665i	$-0.555817\pi$
$251$	−5.52786	−0.348916	−0.174458	+	0.984665i	$0.555817\pi$
$252$	0	0
$253$	14.4721	0.909855
$254$	0	0
$255$	3.05573	0.191357
$256$	0	0
$257$	26.8328	1.67379	0.836893	−	0.547367i	$-0.184370\pi$
$257$	26.8328	1.67379	0.836893	+	0.547367i	$0.184370\pi$
$258$	0	0
$259$	−4.47214	−0.277885
$260$	0	0
$261$	8.47214	0.524412
$262$	0	0
$263$	0.472136	0.0291132	0.0145566	−	0.999894i	$-0.495366\pi$
$263$	0.472136	0.0291132	0.0145566	+	0.999894i	$0.495366\pi$
$264$	0	0
$265$	−12.3607	−0.759311
$266$	0	0
$267$	−10.1803	−0.623027
$268$	0	0
$269$	4.94427	0.301458	0.150729	−	0.988575i	$-0.451838\pi$
$269$	4.94427	0.301458	0.150729	+	0.988575i	$0.451838\pi$
$270$	0	0
$271$	23.4164	1.42245	0.711223	−	0.702967i	$-0.248142\pi$
$271$	23.4164	1.42245	0.711223	+	0.702967i	$0.248142\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	11.2361	0.677560
$276$	0	0
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.6525	−1.11271	−0.556357	−	0.830944i	$-0.687801\pi$
$281$	−18.6525	−1.11271	−0.556357	+	0.830944i	$0.687801\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−5.23607	−0.309075
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	6.94427	0.407080
$292$	0	0
$293$	5.23607	0.305894	0.152947	−	0.988234i	$-0.451124\pi$
$293$	5.23607	0.305894	0.152947	+	0.988234i	$0.451124\pi$
$294$	0	0
$295$	11.4164	0.664689
$296$	0	0
$297$	−3.23607	−0.187776
$298$	0	0
$299$	−4.47214	−0.258630
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	−8.94427	−0.513835
$304$	0	0
$305$	−18.4721	−1.05771
$306$	0	0
$307$	5.88854	0.336077	0.168038	−	0.985780i	$-0.446257\pi$
$307$	5.88854	0.336077	0.168038	+	0.985780i	$0.446257\pi$
$308$	0	0
$309$	16.9443	0.963926
$310$	0	0
$311$	1.52786	0.0866372	0.0433186	−	0.999061i	$-0.486207\pi$
$311$	1.52786	0.0866372	0.0433186	+	0.999061i	$0.486207\pi$
$312$	0	0
$313$	20.8328	1.17754	0.588770	−	0.808300i	$-0.299612\pi$
$313$	20.8328	1.17754	0.588770	+	0.808300i	$0.299612\pi$
$314$	0	0
$315$	1.23607	0.0696445
$316$	0	0
$317$	−28.1803	−1.58277	−0.791383	−	0.611321i	$-0.790638\pi$
$317$	−28.1803	−1.58277	−0.791383	+	0.611321i	$0.790638\pi$
$318$	0	0
$319$	−27.4164	−1.53502
$320$	0	0
$321$	3.52786	0.196906
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	−3.47214	−0.192599
$326$	0	0
$327$	3.52786	0.195091
$328$	0	0
$329$	7.70820	0.424967
$330$	0	0
$331$	34.8328	1.91458	0.957292	−	0.289122i	$-0.0933632\pi$
$331$	34.8328	1.91458	0.957292	+	0.289122i	$0.0933632\pi$
$332$	0	0
$333$	4.47214	0.245072
$334$	0	0
$335$	−3.05573	−0.166952
$336$	0	0
$337$	−22.3607	−1.21806	−0.609032	−	0.793146i	$-0.708442\pi$
$337$	−22.3607	−1.21806	−0.609032	+	0.793146i	$0.708442\pi$
$338$	0	0
$339$	20.4721	1.11189
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.52786	0.297610
$346$	0	0
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	0	0
$349$	1.41641	0.0758186	0.0379093	−	0.999281i	$-0.487930\pi$
$349$	1.41641	0.0758186	0.0379093	+	0.999281i	$0.487930\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−16.6525	−0.886322	−0.443161	−	0.896442i	$-0.646143\pi$
$353$	−16.6525	−0.886322	−0.443161	+	0.896442i	$0.646143\pi$
$354$	0	0
$355$	−7.05573	−0.374479
$356$	0	0
$357$	2.47214	0.130839
$358$	0	0
$359$	−4.18034	−0.220630	−0.110315	−	0.993897i	$-0.535186\pi$
$359$	−4.18034	−0.220630	−0.110315	+	0.993897i	$0.535186\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	−0.527864	−0.0277057
$364$	0	0
$365$	−5.52786	−0.289342
$366$	0	0
$367$	24.9443	1.30208	0.651040	−	0.759043i	$-0.274333\pi$
$367$	24.9443	1.30208	0.651040	+	0.759043i	$0.274333\pi$
$368$	0	0
$369$	5.23607	0.272579
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	36.4721	1.88846	0.944228	−	0.329293i	$-0.106810\pi$
$373$	36.4721	1.88846	0.944228	+	0.329293i	$0.106810\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	0	0
$377$	8.47214	0.436337
$378$	0	0
$379$	−13.8885	−0.713407	−0.356703	−	0.934218i	$-0.616099\pi$
$379$	−13.8885	−0.713407	−0.356703	+	0.934218i	$0.616099\pi$
$380$	0	0
$381$	2.47214	0.126651
$382$	0	0
$383$	1.81966	0.0929803	0.0464901	−	0.998919i	$-0.485196\pi$
$383$	1.81966	0.0929803	0.0464901	+	0.998919i	$0.485196\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	11.0557	0.559112
$392$	0	0
$393$	1.52786	0.0770705
$394$	0	0
$395$	−12.9443	−0.651297
$396$	0	0
$397$	35.8885	1.80119	0.900597	−	0.434655i	$-0.143130\pi$
$397$	35.8885	1.80119	0.900597	+	0.434655i	$0.143130\pi$
$398$	0	0
$399$	−6.47214	−0.324012
$400$	0	0
$401$	−19.2361	−0.960603	−0.480302	−	0.877103i	$-0.659473\pi$
$401$	−19.2361	−0.960603	−0.480302	+	0.877103i	$0.659473\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.23607	−0.0614207
$406$	0	0
$407$	−14.4721	−0.717357
$408$	0	0
$409$	−17.0557	−0.843351	−0.421676	−	0.906747i	$-0.638558\pi$
$409$	−17.0557	−0.843351	−0.421676	+	0.906747i	$0.638558\pi$
$410$	0	0
$411$	15.2361	0.751540
$412$	0	0
$413$	9.23607	0.454477
$414$	0	0
$415$	3.41641	0.167705
$416$	0	0
$417$	−17.8885	−0.876006
$418$	0	0
$419$	23.4164	1.14397	0.571983	−	0.820265i	$-0.306174\pi$
$419$	23.4164	1.14397	0.571983	+	0.820265i	$0.306174\pi$
$420$	0	0
$421$	0.111456	0.00543204	0.00271602	−	0.999996i	$-0.499135\pi$
$421$	0.111456	0.00543204	0.00271602	+	0.999996i	$0.499135\pi$
$422$	0	0
$423$	−7.70820	−0.374786
$424$	0	0
$425$	8.58359	0.416365
$426$	0	0
$427$	−14.9443	−0.723204
$428$	0	0
$429$	−3.23607	−0.156239
$430$	0	0
$431$	−1.70820	−0.0822813	−0.0411406	−	0.999153i	$-0.513099\pi$
$431$	−1.70820	−0.0822813	−0.0411406	+	0.999153i	$0.513099\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	−10.4721	−0.502100
$436$	0	0
$437$	−28.9443	−1.38459
$438$	0	0
$439$	15.0557	0.718571	0.359285	−	0.933228i	$-0.383020\pi$
$439$	15.0557	0.718571	0.359285	+	0.933228i	$0.383020\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	17.4164	0.827479	0.413739	−	0.910395i	$-0.364223\pi$
$443$	17.4164	0.827479	0.413739	+	0.910395i	$0.364223\pi$
$444$	0	0
$445$	12.5836	0.596519
$446$	0	0
$447$	−12.7639	−0.603713
$448$	0	0
$449$	−33.1246	−1.56325	−0.781624	−	0.623750i	$-0.785608\pi$
$449$	−33.1246	−1.56325	−0.781624	+	0.623750i	$0.785608\pi$
$450$	0	0
$451$	−16.9443	−0.797875
$452$	0	0
$453$	3.05573	0.143571
$454$	0	0
$455$	1.23607	0.0579478
$456$	0	0
$457$	−3.52786	−0.165027	−0.0825133	−	0.996590i	$-0.526295\pi$
$457$	−3.52786	−0.165027	−0.0825133	+	0.996590i	$0.526295\pi$
$458$	0	0
$459$	−2.47214	−0.115389
$460$	0	0
$461$	−16.2918	−0.758785	−0.379392	−	0.925236i	$-0.623867\pi$
$461$	−16.2918	−0.758785	−0.379392	+	0.925236i	$0.623867\pi$
$462$	0	0
$463$	11.4164	0.530565	0.265283	−	0.964171i	$-0.414535\pi$
$463$	11.4164	0.530565	0.265283	+	0.964171i	$0.414535\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.47214	0.114397	0.0571984	−	0.998363i	$-0.481783\pi$
$467$	2.47214	0.114397	0.0571984	+	0.998363i	$0.481783\pi$
$468$	0	0
$469$	−2.47214	−0.114153
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	12.9443	0.595178
$474$	0	0
$475$	−22.4721	−1.03109
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−28.0689	−1.28250	−0.641250	−	0.767332i	$-0.721584\pi$
$479$	−28.0689	−1.28250	−0.641250	+	0.767332i	$0.721584\pi$
$480$	0	0
$481$	4.47214	0.203912
$482$	0	0
$483$	4.47214	0.203489
$484$	0	0
$485$	−8.58359	−0.389761
$486$	0	0
$487$	−30.8328	−1.39717	−0.698584	−	0.715528i	$-0.746186\pi$
$487$	−30.8328	−1.39717	−0.698584	+	0.715528i	$0.746186\pi$
$488$	0	0
$489$	2.47214	0.111794
$490$	0	0
$491$	−16.4721	−0.743377	−0.371689	−	0.928357i	$-0.621221\pi$
$491$	−16.4721	−0.743377	−0.371689	+	0.928357i	$0.621221\pi$
$492$	0	0
$493$	−20.9443	−0.943283
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−5.70820	−0.256048
$498$	0	0
$499$	−21.8885	−0.979866	−0.489933	−	0.871760i	$-0.662979\pi$
$499$	−21.8885	−0.979866	−0.489933	+	0.871760i	$0.662979\pi$
$500$	0	0
$501$	18.1803	0.812238
$502$	0	0
$503$	−10.4721	−0.466929	−0.233465	−	0.972365i	$-0.575006\pi$
$503$	−10.4721	−0.466929	−0.233465	+	0.972365i	$0.575006\pi$
$504$	0	0
$505$	11.0557	0.491973
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	21.5967	0.957259	0.478630	−	0.878017i	$-0.341134\pi$
$509$	21.5967	0.957259	0.478630	+	0.878017i	$0.341134\pi$
$510$	0	0
$511$	−4.47214	−0.197836
$512$	0	0
$513$	6.47214	0.285752
$514$	0	0
$515$	−20.9443	−0.922915
$516$	0	0
$517$	24.9443	1.09705
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−20.9443	−0.917585	−0.458793	−	0.888543i	$-0.651718\pi$
$521$	−20.9443	−0.917585	−0.458793	+	0.888543i	$0.651718\pi$
$522$	0	0
$523$	−33.8885	−1.48184	−0.740921	−	0.671592i	$-0.765611\pi$
$523$	−33.8885	−1.48184	−0.740921	+	0.671592i	$0.765611\pi$
$524$	0	0
$525$	3.47214	0.151536
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−9.23607	−0.400811
$532$	0	0
$533$	5.23607	0.226799
$534$	0	0
$535$	−4.36068	−0.188529
$536$	0	0
$537$	6.94427	0.299667
$538$	0	0
$539$	−3.23607	−0.139387
$540$	0	0
$541$	6.94427	0.298558	0.149279	−	0.988795i	$-0.452305\pi$
$541$	6.94427	0.298558	0.149279	+	0.988795i	$0.452305\pi$
$542$	0	0
$543$	19.8885	0.853499
$544$	0	0
$545$	−4.36068	−0.186791
$546$	0	0
$547$	19.0557	0.814764	0.407382	−	0.913258i	$-0.366442\pi$
$547$	19.0557	0.814764	0.407382	+	0.913258i	$0.366442\pi$
$548$	0	0
$549$	14.9443	0.637806
$550$	0	0
$551$	54.8328	2.33596
$552$	0	0
$553$	−10.4721	−0.445321
$554$	0	0
$555$	−5.52786	−0.234645
$556$	0	0
$557$	−28.7639	−1.21877	−0.609383	−	0.792876i	$-0.708583\pi$
$557$	−28.7639	−1.21877	−0.609383	+	0.792876i	$0.708583\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−19.0557	−0.803103	−0.401552	−	0.915836i	$-0.631529\pi$
$563$	−19.0557	−0.803103	−0.401552	+	0.915836i	$0.631529\pi$
$564$	0	0
$565$	−25.3050	−1.06459
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−17.4164	−0.730134	−0.365067	−	0.930981i	$-0.618954\pi$
$569$	−17.4164	−0.730134	−0.365067	+	0.930981i	$0.618954\pi$
$570$	0	0
$571$	0.360680	0.0150940	0.00754699	−	0.999972i	$-0.497598\pi$
$571$	0.360680	0.0150940	0.00754699	+	0.999972i	$0.497598\pi$
$572$	0	0
$573$	6.94427	0.290101
$574$	0	0
$575$	15.5279	0.647557
$576$	0	0
$577$	18.9443	0.788660	0.394330	−	0.918969i	$-0.370977\pi$
$577$	18.9443	0.788660	0.394330	+	0.918969i	$0.370977\pi$
$578$	0	0
$579$	18.3607	0.763044
$580$	0	0
$581$	2.76393	0.114667
$582$	0	0
$583$	−32.3607	−1.34024
$584$	0	0
$585$	−1.23607	−0.0511051
$586$	0	0
$587$	−21.5967	−0.891393	−0.445697	−	0.895184i	$-0.647044\pi$
$587$	−21.5967	−0.891393	−0.445697	+	0.895184i	$0.647044\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0.763932	0.0314240
$592$	0	0
$593$	17.8197	0.731766	0.365883	−	0.930661i	$-0.380767\pi$
$593$	17.8197	0.731766	0.365883	+	0.930661i	$0.380767\pi$
$594$	0	0
$595$	−3.05573	−0.125273
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	27.3050	1.11565	0.557825	−	0.829959i	$-0.311636\pi$
$599$	27.3050	1.11565	0.557825	+	0.829959i	$0.311636\pi$
$600$	0	0
$601$	9.05573	0.369391	0.184695	−	0.982796i	$-0.440870\pi$
$601$	9.05573	0.369391	0.184695	+	0.982796i	$0.440870\pi$
$602$	0	0
$603$	2.47214	0.100673
$604$	0	0
$605$	0.652476	0.0265269
$606$	0	0
$607$	21.8885	0.888429	0.444214	−	0.895921i	$-0.353483\pi$
$607$	21.8885	0.888429	0.444214	+	0.895921i	$0.353483\pi$
$608$	0	0
$609$	−8.47214	−0.343308
$610$	0	0
$611$	−7.70820	−0.311841
$612$	0	0
$613$	−29.7771	−1.20269	−0.601343	−	0.798991i	$-0.705367\pi$
$613$	−29.7771	−1.20269	−0.601343	+	0.798991i	$0.705367\pi$
$614$	0	0
$615$	−6.47214	−0.260982
$616$	0	0
$617$	−40.7639	−1.64109	−0.820547	−	0.571579i	$-0.806331\pi$
$617$	−40.7639	−1.64109	−0.820547	+	0.571579i	$0.806331\pi$
$618$	0	0
$619$	37.3050	1.49941	0.749706	−	0.661771i	$-0.230195\pi$
$619$	37.3050	1.49941	0.749706	+	0.661771i	$0.230195\pi$
$620$	0	0
$621$	−4.47214	−0.179461
$622$	0	0
$623$	10.1803	0.407867
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−20.9443	−0.836434
$628$	0	0
$629$	−11.0557	−0.440821
$630$	0	0
$631$	−14.4721	−0.576127	−0.288063	−	0.957611i	$-0.593011\pi$
$631$	−14.4721	−0.576127	−0.288063	+	0.957611i	$0.593011\pi$
$632$	0	0
$633$	−20.3607	−0.809264
$634$	0	0
$635$	−3.05573	−0.121263
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	5.70820	0.225813
$640$	0	0
$641$	41.7771	1.65010	0.825048	−	0.565063i	$-0.191148\pi$
$641$	41.7771	1.65010	0.825048	+	0.565063i	$0.191148\pi$
$642$	0	0
$643$	10.8328	0.427205	0.213602	−	0.976921i	$-0.431480\pi$
$643$	10.8328	0.427205	0.213602	+	0.976921i	$0.431480\pi$
$644$	0	0
$645$	4.94427	0.194681
$646$	0	0
$647$	10.8328	0.425882	0.212941	−	0.977065i	$-0.431696\pi$
$647$	10.8328	0.425882	0.212941	+	0.977065i	$0.431696\pi$
$648$	0	0
$649$	29.8885	1.17323
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−38.3607	−1.50117	−0.750585	−	0.660774i	$-0.770228\pi$
$653$	−38.3607	−1.50117	−0.750585	+	0.660774i	$0.770228\pi$
$654$	0	0
$655$	−1.88854	−0.0737915
$656$	0	0
$657$	4.47214	0.174475
$658$	0	0
$659$	−0.111456	−0.00434172	−0.00217086	−	0.999998i	$-0.500691\pi$
$659$	−0.111456	−0.00434172	−0.00217086	+	0.999998i	$0.500691\pi$
$660$	0	0
$661$	−13.4164	−0.521838	−0.260919	−	0.965361i	$-0.584026\pi$
$661$	−13.4164	−0.521838	−0.260919	+	0.965361i	$0.584026\pi$
$662$	0	0
$663$	−2.47214	−0.0960098
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−37.8885	−1.46705
$668$	0	0
$669$	10.4721	0.404876
$670$	0	0
$671$	−48.3607	−1.86694
$672$	0	0
$673$	39.8885	1.53759	0.768795	−	0.639495i	$-0.220857\pi$
$673$	39.8885	1.53759	0.768795	+	0.639495i	$0.220857\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	4.58359	0.176162	0.0880809	−	0.996113i	$-0.471927\pi$
$677$	4.58359	0.176162	0.0880809	+	0.996113i	$0.471927\pi$
$678$	0	0
$679$	−6.94427	−0.266497
$680$	0	0
$681$	6.76393	0.259194
$682$	0	0
$683$	−47.2361	−1.80744	−0.903719	−	0.428126i	$-0.859174\pi$
$683$	−47.2361	−1.80744	−0.903719	+	0.428126i	$0.859174\pi$
$684$	0	0
$685$	−18.8328	−0.719565
$686$	0	0
$687$	−25.4164	−0.969696
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	7.63932	0.290613	0.145307	−	0.989387i	$-0.453583\pi$
$691$	7.63932	0.290613	0.145307	+	0.989387i	$0.453583\pi$
$692$	0	0
$693$	3.23607	0.122928
$694$	0	0
$695$	22.1115	0.838735
$696$	0	0
$697$	−12.9443	−0.490299
$698$	0	0
$699$	5.41641	0.204867
$700$	0	0
$701$	33.7771	1.27574	0.637871	−	0.770143i	$-0.279815\pi$
$701$	33.7771	1.27574	0.637871	+	0.770143i	$0.279815\pi$
$702$	0	0
$703$	28.9443	1.09165
$704$	0	0
$705$	9.52786	0.358840
$706$	0	0
$707$	8.94427	0.336384
$708$	0	0
$709$	18.3607	0.689550	0.344775	−	0.938685i	$-0.387955\pi$
$709$	18.3607	0.689550	0.344775	+	0.938685i	$0.387955\pi$
$710$	0	0
$711$	10.4721	0.392736
$712$	0	0
$713$	0	0
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	−21.1246	−0.788913
$718$	0	0
$719$	9.30495	0.347016	0.173508	−	0.984832i	$-0.444490\pi$
$719$	9.30495	0.347016	0.173508	+	0.984832i	$0.444490\pi$
$720$	0	0
$721$	−16.9443	−0.631038
$722$	0	0
$723$	−16.4721	−0.612605
$724$	0	0
$725$	−29.4164	−1.09250
$726$	0	0
$727$	−30.8328	−1.14353	−0.571763	−	0.820419i	$-0.693740\pi$
$727$	−30.8328	−1.14353	−0.571763	+	0.820419i	$0.693740\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.88854	0.365741
$732$	0	0
$733$	16.4721	0.608412	0.304206	−	0.952606i	$-0.401609\pi$
$733$	16.4721	0.608412	0.304206	+	0.952606i	$0.401609\pi$
$734$	0	0
$735$	−1.23607	−0.0455931
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	31.4164	1.15567	0.577836	−	0.816153i	$-0.303897\pi$
$739$	31.4164	1.15567	0.577836	+	0.816153i	$0.303897\pi$
$740$	0	0
$741$	6.47214	0.237760
$742$	0	0
$743$	42.6525	1.56477	0.782384	−	0.622797i	$-0.214004\pi$
$743$	42.6525	1.56477	0.782384	+	0.622797i	$0.214004\pi$
$744$	0	0
$745$	15.7771	0.578028
$746$	0	0
$747$	−2.76393	−0.101127
$748$	0	0
$749$	−3.52786	−0.128905
$750$	0	0
$751$	−19.0557	−0.695353	−0.347677	−	0.937614i	$-0.613029\pi$
$751$	−19.0557	−0.695353	−0.347677	+	0.937614i	$0.613029\pi$
$752$	0	0
$753$	−5.52786	−0.201447
$754$	0	0
$755$	−3.77709	−0.137462
$756$	0	0
$757$	41.4164	1.50530	0.752652	−	0.658418i	$-0.228774\pi$
$757$	41.4164	1.50530	0.752652	+	0.658418i	$0.228774\pi$
$758$	0	0
$759$	14.4721	0.525305
$760$	0	0
$761$	26.1803	0.949037	0.474518	−	0.880246i	$-0.342622\pi$
$761$	26.1803	0.949037	0.474518	+	0.880246i	$0.342622\pi$
$762$	0	0
$763$	−3.52786	−0.127717
$764$	0	0
$765$	3.05573	0.110480
$766$	0	0
$767$	−9.23607	−0.333495
$768$	0	0
$769$	−7.52786	−0.271462	−0.135731	−	0.990746i	$-0.543338\pi$
$769$	−7.52786	−0.271462	−0.135731	+	0.990746i	$0.543338\pi$
$770$	0	0
$771$	26.8328	0.966360
$772$	0	0
$773$	7.12461	0.256254	0.128127	−	0.991758i	$-0.459103\pi$
$773$	7.12461	0.256254	0.128127	+	0.991758i	$0.459103\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.47214	−0.160437
$778$	0	0
$779$	33.8885	1.21418
$780$	0	0
$781$	−18.4721	−0.660985
$782$	0	0
$783$	8.47214	0.302769
$784$	0	0
$785$	22.2492	0.794109
$786$	0	0
$787$	−7.63932	−0.272312	−0.136156	−	0.990687i	$-0.543475\pi$
$787$	−7.63932	−0.272312	−0.136156	+	0.990687i	$0.543475\pi$
$788$	0	0
$789$	0.472136	0.0168085
$790$	0	0
$791$	−20.4721	−0.727905
$792$	0	0
$793$	14.9443	0.530687
$794$	0	0
$795$	−12.3607	−0.438388
$796$	0	0
$797$	−30.4721	−1.07938	−0.539689	−	0.841864i	$-0.681458\pi$
$797$	−30.4721	−1.07938	−0.539689	+	0.841864i	$0.681458\pi$
$798$	0	0
$799$	19.0557	0.674143
$800$	0	0
$801$	−10.1803	−0.359705
$802$	0	0
$803$	−14.4721	−0.510711
$804$	0	0
$805$	−5.52786	−0.194832
$806$	0	0
$807$	4.94427	0.174047
$808$	0	0
$809$	12.4721	0.438497	0.219248	−	0.975669i	$-0.429639\pi$
$809$	12.4721	0.438497	0.219248	+	0.975669i	$0.429639\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	0	0
$813$	23.4164	0.821249
$814$	0	0
$815$	−3.05573	−0.107037
$816$	0	0
$817$	−25.8885	−0.905725
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−46.6525	−1.62818	−0.814091	−	0.580737i	$-0.802765\pi$
$821$	−46.6525	−1.62818	−0.814091	+	0.580737i	$0.802765\pi$
$822$	0	0
$823$	−47.7771	−1.66540	−0.832702	−	0.553721i	$-0.813207\pi$
$823$	−47.7771	−1.66540	−0.832702	+	0.553721i	$0.813207\pi$
$824$	0	0
$825$	11.2361	0.391190
$826$	0	0
$827$	37.4853	1.30349	0.651746	−	0.758438i	$-0.274037\pi$
$827$	37.4853	1.30349	0.651746	+	0.758438i	$0.274037\pi$
$828$	0	0
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	−15.8885	−0.551167
$832$	0	0
$833$	−2.47214	−0.0856544
$834$	0	0
$835$	−22.4721	−0.777680
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.8197	1.02949	0.514744	−	0.857344i	$-0.327887\pi$
$839$	29.8197	1.02949	0.514744	+	0.857344i	$0.327887\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	−18.6525	−0.642425
$844$	0	0
$845$	−1.23607	−0.0425220
$846$	0	0
$847$	0.527864	0.0181376
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−20.0000	−0.685591
$852$	0	0
$853$	−20.8328	−0.713302	−0.356651	−	0.934238i	$-0.616081\pi$
$853$	−20.8328	−0.713302	−0.356651	+	0.934238i	$0.616081\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	40.9443	1.39863	0.699315	−	0.714814i	$-0.253489\pi$
$857$	40.9443	1.39863	0.699315	+	0.714814i	$0.253489\pi$
$858$	0	0
$859$	10.1115	0.344998	0.172499	−	0.985010i	$-0.444816\pi$
$859$	10.1115	0.344998	0.172499	+	0.985010i	$0.444816\pi$
$860$	0	0
$861$	−5.23607	−0.178445
$862$	0	0
$863$	52.1803	1.77624	0.888120	−	0.459612i	$-0.152012\pi$
$863$	52.1803	1.77624	0.888120	+	0.459612i	$0.152012\pi$
$864$	0	0
$865$	−14.8328	−0.504331
$866$	0	0
$867$	−10.8885	−0.369794
$868$	0	0
$869$	−33.8885	−1.14959
$870$	0	0
$871$	2.47214	0.0837651
$872$	0	0
$873$	6.94427	0.235028
$874$	0	0
$875$	−10.4721	−0.354023
$876$	0	0
$877$	7.88854	0.266377	0.133189	−	0.991091i	$-0.457478\pi$
$877$	7.88854	0.266377	0.133189	+	0.991091i	$0.457478\pi$
$878$	0	0
$879$	5.23607	0.176608
$880$	0	0
$881$	30.8328	1.03878	0.519392	−	0.854536i	$-0.326158\pi$
$881$	30.8328	1.03878	0.519392	+	0.854536i	$0.326158\pi$
$882$	0	0
$883$	15.4164	0.518803	0.259402	−	0.965770i	$-0.416475\pi$
$883$	15.4164	0.518803	0.259402	+	0.965770i	$0.416475\pi$
$884$	0	0
$885$	11.4164	0.383758
$886$	0	0
$887$	5.16718	0.173497	0.0867485	−	0.996230i	$-0.472352\pi$
$887$	5.16718	0.173497	0.0867485	+	0.996230i	$0.472352\pi$
$888$	0	0
$889$	−2.47214	−0.0829128
$890$	0	0
$891$	−3.23607	−0.108412
$892$	0	0
$893$	−49.8885	−1.66946
$894$	0	0
$895$	−8.58359	−0.286918
$896$	0	0
$897$	−4.47214	−0.149320
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−24.7214	−0.823588
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−24.5836	−0.817186
$906$	0	0
$907$	28.3607	0.941701	0.470850	−	0.882213i	$-0.343947\pi$
$907$	28.3607	0.941701	0.470850	+	0.882213i	$0.343947\pi$
$908$	0	0
$909$	−8.94427	−0.296663
$910$	0	0
$911$	−31.8885	−1.05651	−0.528257	−	0.849084i	$-0.677154\pi$
$911$	−31.8885	−1.05651	−0.528257	+	0.849084i	$0.677154\pi$
$912$	0	0
$913$	8.94427	0.296012
$914$	0	0
$915$	−18.4721	−0.610670
$916$	0	0
$917$	−1.52786	−0.0504545
$918$	0	0
$919$	−1.52786	−0.0503996	−0.0251998	−	0.999682i	$-0.508022\pi$
$919$	−1.52786	−0.0503996	−0.0251998	+	0.999682i	$0.508022\pi$
$920$	0	0
$921$	5.88854	0.194034
$922$	0	0
$923$	5.70820	0.187888
$924$	0	0
$925$	−15.5279	−0.510553
$926$	0	0
$927$	16.9443	0.556523
$928$	0	0
$929$	−44.6525	−1.46500	−0.732500	−	0.680767i	$-0.761647\pi$
$929$	−44.6525	−1.46500	−0.732500	+	0.680767i	$0.761647\pi$
$930$	0	0
$931$	6.47214	0.212116
$932$	0	0
$933$	1.52786	0.0500200
$934$	0	0
$935$	−9.88854	−0.323390
$936$	0	0
$937$	−22.9443	−0.749557	−0.374778	−	0.927114i	$-0.622281\pi$
$937$	−22.9443	−0.749557	−0.374778	+	0.927114i	$0.622281\pi$
$938$	0	0
$939$	20.8328	0.679853
$940$	0	0
$941$	−50.5410	−1.64759	−0.823795	−	0.566888i	$-0.808147\pi$
$941$	−50.5410	−1.64759	−0.823795	+	0.566888i	$0.808147\pi$
$942$	0	0
$943$	−23.4164	−0.762543
$944$	0	0
$945$	1.23607	0.0402093
$946$	0	0
$947$	−7.59675	−0.246861	−0.123431	−	0.992353i	$-0.539390\pi$
$947$	−7.59675	−0.246861	−0.123431	+	0.992353i	$0.539390\pi$
$948$	0	0
$949$	4.47214	0.145172
$950$	0	0
$951$	−28.1803	−0.913810
$952$	0	0
$953$	−41.1935	−1.33439	−0.667194	−	0.744884i	$-0.732505\pi$
$953$	−41.1935	−1.33439	−0.667194	+	0.744884i	$0.732505\pi$
$954$	0	0
$955$	−8.58359	−0.277759
$956$	0	0
$957$	−27.4164	−0.886247
$958$	0	0
$959$	−15.2361	−0.491998
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	3.52786	0.113684
$964$	0	0
$965$	−22.6950	−0.730579
$966$	0	0
$967$	40.3607	1.29791	0.648956	−	0.760826i	$-0.275206\pi$
$967$	40.3607	1.29791	0.648956	+	0.760826i	$0.275206\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	−29.8885	−0.959169	−0.479585	−	0.877496i	$-0.659213\pi$
$971$	−29.8885	−0.959169	−0.479585	+	0.877496i	$0.659213\pi$
$972$	0	0
$973$	17.8885	0.573480
$974$	0	0
$975$	−3.47214	−0.111197
$976$	0	0
$977$	−19.2361	−0.615416	−0.307708	−	0.951481i	$-0.599562\pi$
$977$	−19.2361	−0.615416	−0.307708	+	0.951481i	$0.599562\pi$
$978$	0	0
$979$	32.9443	1.05290
$980$	0	0
$981$	3.52786	0.112636
$982$	0	0
$983$	−59.1246	−1.88578	−0.942891	−	0.333101i	$-0.891905\pi$
$983$	−59.1246	−1.88578	−0.942891	+	0.333101i	$0.891905\pi$
$984$	0	0
$985$	−0.944272	−0.0300870
$986$	0	0
$987$	7.70820	0.245355
$988$	0	0
$989$	17.8885	0.568823
$990$	0	0
$991$	−6.11146	−0.194137	−0.0970684	−	0.995278i	$-0.530947\pi$
$991$	−6.11146	−0.194137	−0.0970684	+	0.995278i	$0.530947\pi$
$992$	0	0
$993$	34.8328	1.10539
$994$	0	0
$995$	−19.7771	−0.626976
$996$	0	0
$997$	17.7771	0.563006	0.281503	−	0.959560i	$-0.409167\pi$
$997$	17.7771	0.563006	0.281503	+	0.959560i	$0.409167\pi$
$998$	0	0
$999$	4.47214	0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bl.1.1		2
4.3	odd	2		1092.2.a.f.1.1	✓	2
12.11	even	2		3276.2.a.l.1.2		2
28.27	even	2		7644.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.f.1.1	✓	2	4.3	odd	2
3276.2.a.l.1.2		2	12.11	even	2
4368.2.a.bl.1.1		2	1.1	even	1	trivial
7644.2.a.p.1.2		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.23607 q^{15} -2.47214 q^{17} +6.47214 q^{19} -1.00000 q^{21} -4.47214 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -3.23607 q^{33} +1.23607 q^{35} +4.47214 q^{37} +1.00000 q^{39} +5.23607 q^{41} -4.00000 q^{43} -1.23607 q^{45} -7.70820 q^{47} +1.00000 q^{49} -2.47214 q^{51} +10.0000 q^{53} +4.00000 q^{55} +6.47214 q^{57} -9.23607 q^{59} +14.9443 q^{61} -1.00000 q^{63} -1.23607 q^{65} +2.47214 q^{67} -4.47214 q^{69} +5.70820 q^{71} +4.47214 q^{73} -3.47214 q^{75} +3.23607 q^{77} +10.4721 q^{79} +1.00000 q^{81} -2.76393 q^{83} +3.05573 q^{85} +8.47214 q^{87} -10.1803 q^{89} -1.00000 q^{91} -8.00000 q^{95} +6.94427 q^{97} -3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 2 q^{35} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} - 2 q^{47} + 2 q^{49} + 4 q^{51} + 20 q^{53} + 8 q^{55} + 4 q^{57} - 14 q^{59} + 12 q^{61} - 2 q^{63} + 2 q^{65} - 4 q^{67} - 2 q^{71} + 2 q^{75} + 2 q^{77} + 12 q^{79} + 2 q^{81} - 10 q^{83} + 24 q^{85} + 8 q^{87} + 2 q^{89} - 2 q^{91} - 16 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 2 * q^35 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^55 + 4 * q^57 - 14 * q^59 + 12 * q^61 - 2 * q^63 + 2 * q^65 - 4 * q^67 - 2 * q^71 + 2 * q^75 + 2 * q^77 + 12 * q^79 + 2 * q^81 - 10 * q^83 + 24 * q^85 + 8 * q^87 + 2 * q^89 - 2 * q^91 - 16 * q^95 - 4 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more